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+{"text":"\\section{Introduction}\n\nThe analysis of quantum information processing protocols is a challenging task. Let it be a quantum tomography process, transmission of quantum information over a noisy channel or a cryptographic protocol -- all need to be analysed under general conditions. Since one usually has limited information about the actual quantum state given as input, the analysis should be valid for any given quantum state. For example, a cryptographic protocol should be proven secure independently of the input state, which can be chosen by a malicious adversary. As the space of all possible states can be very large and the structure of the states therein might be complicated due to entanglement, this task can be tedious in the good case, and infeasible in the worst. \n\nThe quantum de Finetti theorems \\cite{hudson1976locally, raggio1989quantum, caves2002unknown, renner2007symmetry} and the post selection theorem \\cite{christandl2009postselection} address the above problem, by exploiting the symmetry of the considered states, namely permutation invariance. These mathematical tools allow us to simplify the analysis of quantum information processing tasks by reducing permutation invariant quantum states to a more structured state, called the quantum de Finetti state.\nIn general, we say that a state is of de Finetti-type if it is a convex combination of independent identically distributed (i.i.d.\\@) states. \n\nde Finetti states are usually much easier to handle than general states due to their simple structure. Moreover, most established information-theoretic techniques can be applied only to i.i.d\\ states, and therefore while not applicable when considering a general state, they can be used when considering de Finetti states. Therefore, a reduction to such states can simplify calculations and proofs of various quantum information processing tasks. Indeed, one of the famous applications of reductions to de Finetti states is a proof which states that in order to establish security of quantum key distribution against general attacks it is sufficient to consider attacks on individual signals~\\cite{christandl2009postselection}. Other applications include quantum tomography \\cite{christandl2012reliable} or quantum reverse Shannon coding \\cite{berta2011reverse}. \n\nUnfortunately, the known variants of the quantum de Finetti theorems are not always applicable. A big class of protocols, commonly used in the past several years, to which those theorems are not applicable is the class of protocols in which the dimension of the states is unknown or cannot be bounded, and in particular, the class of device independent protocols (for a review on the topic, see for example \\cite{scarani2012device, brunner2013bell}). The above mentioned theorems cannot be used in such cases for they depend on the dimension of the quantum state. \n\nIn device independent cryptography \\cite{mayers1998quantum,pironio2009device}, for example, one considers the devices as black boxes, about which we know nothing. The security of such protocols can therefore rely only on the observed statistics and not on the specific quantum states and measurements used in the protocol (in some protocols one does not even assume that the underlying physical system is restricted to be quantum! \\cite{barrett2005no,hanggi2009quantum}). In these cases, one possible framework to work with is the framework of conditional probability distributions.  \n\nConditional probability distributions describe the operational behaviour of physical systems under measurements. That is, if we are only interested in modelling the measurement-outcome behaviour of our physical system, then the system can be described by a conditional probability distribution $\\mathrm{P}_{A|X}$ where $X$ is the input, or the measurement performed on the system, and $A$ is the output. $\\mathrm{P}_{A|X}(a|x)$ is the probability for outcome $a$ given that a measurement $x$ was made. We then say that $\\mathrm{P}_{A|X}$ is the state of the system. Note that the state may have as many inputs and outputs as required and therefore we do not restrict the structure of the underlying system by describing it as a conditional probability distribution.\n\nIn quantum physics, for example, $\\mathrm{P}_{A|X}$ is given by Born's rule. However, conditional probability distributions can also be used to describe states that might not conform with the theory of quantum physics, such as non-signalling states. Consider for example a state $\\mathrm{P}_{AB|XY}$ shared by two space-like separated parties, Alice and Bob, each holding a subsystem of the state. $X$ and $A$ are then, respectively, the input and output of Alice, and $Y$ and $B$ of Bob. We then say that the state is non-signalling if it cannot be used to communicate, i.e., the output of one party is independent of the input of the other. The PR-box \\cite{PR-box} is an example for a non-quantum bipartite state which can be written as a (non-signalling) conditional probability distribution. \n\nGiven all the above, it is thus necessary to see whether de Finetti theorems are unique for quantum states or can be also proven on the level of the correlations in the framework of conditional probability distributions. More specifically, we are interested in a theorem that will allow us to reduce permutation invariant conditional probability distributions to a simple de Finetti-type conditional probability distribution, in a way that will be applicable in device independent protocols and, more generally, when the dimension of the underlying quantum states is unknown. Several different non-signalling de Finetti theorems have been established recently \\cite{barrett2009finetti,christandl2009finite,brandao2012quantum}, but it is yet unknown how these can be applied to device independent cryptography\\footnote{In most of these variants of de Finetti theorems, for example, it is assumed that the subsystems cannot signal each other. For current applications this is a too restrictive condition, since it is equivallent to assuming there is no memory in the devices.}. \n\nIn this letter we prove a general de Finetti reduction theorem, from which we can derive several more specialised statements that are of interest for applications. The different reductions differ from one another in two main aspects -- the set of states to which they can be applied and the specific structure of the de Finetti state. Different de Finetti reductions can therefore be useful in different scenarios and under different assumptions. \n\nThe simplest and most straightforward variant is a de Finetti reduction which can be applied to any permutation invariant conditional probability distribution. The second variant is a reduction which can be applied to a family of states which is relevant for cryptographic protocols based on the CHSH inequality \\cite{CHSH} or the chained Bell inequalities \\cite{braunstein1990wringing,Barrett2006Maximally}. There we connect any state $\\mathrm{P}_{AB|XY}$ out of this family of states to a special \\emph{non-signalling} de Finetti state $\\tau^\\mathcal{CHSH}_{AB|XY}$. We do not assume any non-signalling conditions between the subsystems of $\\mathrm{P}_{AB|XY}$ and therefore the use of the de Finetti reduction is not restricted only to scenarios where each of the subsystems cannot signal each other. \n\nUp to date, almost all known device independent cryptographic protocols are based on the CHSH inequality or the more general chained Bell inequalities. For this reason we pay specific attention to states which are relevant for such protocols. However, our theorem can be applied also to other families of states which might be useful in future protocols. As an example of an application of our theorem we prove that for protocols which are based on the violation of the CHSH and chained Bell inequalities it is sufficient to consider the case where Alice and Bob share the de Finetti state $\\tau^\\mathcal{CHSH}_{AB|XY}$. We do this by bounding the distance between two channels which act on conditional probability distributions. \n\nIn the following we start by describing and explaining the different de Finetti reductions. We then illustrate how the reductions can be used in applications. All the proofs are given in the Appendix.\n\n\\section{Results}\n\nFor stating the different de Finetti reductions we will need some basic definitions. $A$ and $X$ denote discrete random variables over $a \\in \\{0,1, ... , l-1\\}^n$ and $x \\in \\{0,1, ... , m-1\\}^n$ respectively. We use $[n]$ to denote the set $\\{1,\\dotsc,n\\}$. An $n$-partite state $\\mathrm{P}_{A|X}$ is a conditional probability distribution if for every $x$, $\\sum_a \\mathrm{P}_{A|X}(a|x)=1$ and for every $a, x$, $\\mathrm{P}_{A|X}(a|x)\\geq 0$. When we consider two different states $\\mathrm{P}_{A|X}$ and $\\mathrm{Q}_{A|X}$ it is understood that both states are over the same random variables $X$ and $A$. The de Finetti reductions deal with permutation invariant states and de Finetti states. Formally we define these as follows.\n\\begin{defn}\\label{def:permutation}\n\tGiven a state $\\mathrm{P}_{A|X}$ and a permutation $\\pi$ of its subsystems\\footnote{Since we permute $a$ and $x$ together this is exactly as permuting the subsystems.} we denote by $\\mathrm{P}_{A|X}\\circ\\pi$ the state which is defined by \n\t\\[\n\t\t\\forall a,x \\quad \\left(\\mathrm{P}_{A|X}\\circ\\pi \\right) (a|x)=\\mathrm{P}_{A|X}(\\pi(a)|\\pi(x)) \\;.\n\t\\]\n\tAn $n$-partite state $\\mathrm{P}_{A|X}$ is permutation invariant if for any permutation $\\pi$, $\\mathrm{P}_{A|X} = \\mathrm{P}_{A|X}\\circ\\pi$.  \n\\end{defn}\nAs mentioned above, we say that a state is a de Finetti state if it is a convex combination of i.i.d.\\ states. Formally, \n\\begin{defn}\n\tA de Finetti state is a state of the form\n\t\\[\n\t\t\\tau_{A|X} = \\int Q_{A_1|X_1}^{\\otimes n} \\mathrm{d}Q_{A_1|X_1}\n\t\\]\n\twhere $x_1\\in \\{0,1, ... , m-1\\}$, $a_1 \\in \\{0,1, ... , l-1\\}$, $\\mathrm{d}Q_{A_1|X_1}$ is some measure on the space of 1-party states and $Q_{A_1|X_1}^{\\otimes n}$ is a product of $n$ identical 1-party states $Q_{A_1|X_1}$, i.e., it is defined according to \n\t\\[\n\t\tQ_{A_1|X_1}^{\\otimes n}(a|x) = \\prod_{i\\in[n]} Q_{A_1|X_1}(a_i|x_i) \\;.\n\t\\]\n\\end{defn}\nAs seen from the above definition, by choosing different measures $\\mathrm{d}Q_{A_1|X_1}$ we define different de Finetti states. \n\nWe are now ready to state the de Finetti reductions. For simplicity we start by giving the first corollary of the more general theorem (Theorem \\ref{thm:post-selection}). This corollary is a reduction for conditional probability distributions, which connects general permutation invariant states to a specific de Finetti state. \n\n\\begin{cor}[de Finetti reduction for conditional probability distributions]\\label{cor:conditional}\n\tThere exists a de Finetti state $\\tau_{A|X}$ where $x \\in \\{ 0,1, ... ,m-1 \\}^n$ and $a \\in \\{ 0,1, ... ,l-1 \\} ^n$ such that for every permutation invariant state $\\mathrm{P}_{A|X}$  \n\t\\[\n\t\t\\forall a,x \\quad \\mathrm{P}_{A|X} (a|x) \\leq (n+1)^{m(l-1)} \\; \\tau_{A|X} (a|x)  \\;.\n\t\\]\n\\end{cor}\n\nThe de Finetti state $\\tau_{A|X}$ is an \\emph{explicit} state that we construct in the proof of the general theorem in Appendix~\\ref{sec:general-proof}. The proof uses mainly combinatoric arguments; we choose $\\tau_{A|X}$ in a specific way, such that a lower bound on $\\tau_{A|X}(a|x)$ for all $a,x$ can be proven. We then use the permutation invariance of $\\mathrm{P}_{A|X}$  to prove an upper bound on $\\mathrm{P}_{A|X}(a|x)$. The result is then derived by combining the two bounds.\n\nCorollary \\ref{cor:conditional} holds for every permutation invariant state $\\mathrm{P}_{A|X}$, not necessarily quantum or non-signalling. At first sight, the generality of the above mathematical statement might seem as a drawback in applications where only a restricted set of correlations is considered (e.g., only non-signalling correlations). Nevertheless, in a following work \\cite{arnon2014nonsignalling} we show that this is not the case and apply this general theorem to prove parallel repetition theorems for non-signalling games. \nNote that according to Definition~\\ref{def:permutation} we consider permutations which permute the 1-party subsystems of $\\mathrm{P}_{A|X}$\\footnote{This is in contrast to states $\\mathrm{P}_{AB|XY}$ which can also be permuted as $\\left(\\mathrm{P}_{AB|XY}\\circ\\pi \\right) (ab|xy)=\\mathrm{P}_{AB|XY}\\left(\\pi(a)\\pi(b)|\\pi(x)\\pi(y)\\right)$, as is usually the case in cryptographic tasks. For dealing with such states we will consider a different reduction, stated as Corollary~\\ref{cor:chsh-post-selection}.}. \n\nThe multiplicative pre-factor of the de Finetti reduction, $(n+1)^{m(l-1)}$ in Corollary \\ref{cor:conditional} for example, is relevant for applications. Intuitively, this is the ``cost'' for using $\\tau_{A|X}$ instead of $\\mathrm{P}_{A|X}$ in the analysis of the considered protocol. We therefore want it to be as small as possible. Nevertheless, as will be explained later, in many cases a pre-factor polynomial in $n$ suffices.  \n\nCorollary \\ref{cor:conditional} is relevant for scenarios in which one considers permutation invariant conditional probability distributions $\\mathrm{P}_{A|X}$. However, if the states one considers have additional symmetries $\\mathcal{S}$ then we can prove a better de Finetti reduction --- a reduction with a smaller pre-factor and a special de Finetti state with the same symmetries~$\\mathcal{S}$.\n\nIn the following we consider a specific family of symmetries --- symmetries between different inputs and outputs of the subsystems of $\\mathrm{P}_{A|X}$. Formally, the types of symmetries that we consider are described, among other things, by a number $d\\leq m(l-1)$ which we call the degrees of freedom of the symmetry (see Appendix~\\ref{sec:general-proof} for details and formal definition of the symmetries). More symmetry implies less degrees of freedom, i.e., smaller $d$, and as shown in the following theorem, this leads to a smaller pre-factor in the reduction. The general theorem then reads:\n\n\\begin{thm}[de Finetti reduction for conditional probability distributions with symmetries]\\label{thm:post-selection}\n\tThere exists a de Finetti state $\\tau^\\mathcal{S}_{A|X}$ where $x \\in \\{ 0,1, ... ,m-1 \\}^n$ and $a \\in \\{ 0,1, ... ,l-1 \\} ^n$ such that for every permutation invariant state $\\mathrm{P}_{A|X}$ with symmetry $\\mathcal{S}$ (with $d$ degrees of freedom) \n\t\\[\n\t\t\\forall a,x \\quad \\mathrm{P}_{A|X} (a|x) \\leq (n+1)^d \\; \\tau^\\mathcal{S}_{A|X} (a|x) \\;. \n\t\\]\n\\end{thm}\nFor the case of no symmetry we have $d=m(l-1)$ from which Corollary  \\ref{cor:conditional} stated before follows. \n\nThe symmetries $\\mathcal{S}$ that we consider are of particular interest when considering cryptographic protocols which are based on non-signalling states. For example, the states which are relevant for protocols which are based on the violation of the CHSH inequality (such as \\cite{masanes2009universally,hanggi2009quantum}) have a great amount of symmetry. The additional symmetry allows us to prove a corollary of Theorem \\ref{thm:post-selection} which can be used to simplify such protocols. \n\nBefore we state the corollary for the CHSH case, let us define what we mean when we say that a state has a CHSH-type symmetry. In cryptographic protocols based on the CHSH inequality the basic states that we consider are bipartite states $\\mathrm{P}_{AB|XY}$ held by Alice and Bob where $a,b,x,y\\in \\{0,1\\}^n$. \n\\begin{defn}[CHSH-type symmetry]\\label{def:chsh-symmetry}\n\tA state $\\mathrm{P}_{AB|XY}$ where $a,b,x,y\\in \\{0,1\\}^n$ has a CHSH-type symmetry if there exist $p_1,\\dotsc,p_n\\in [0,\\frac{1}{2}]$ such that $\\forall i\\in\\{1,\\dotsc,n\\}$,\n\t\\begin{equation*}\n\t\\begin{split}\n\t\t & \\forall a_i, b_i, x_i, y_i \\\\\n\t\t & a_i\\oplus\\ b_i=x_i\\cdot y_i \\rightarrow \\mathrm{P}_{AB|XY}(a_{\\overline{i}}a_ib_{\\overline{i}}b_i|x_{\\overline{i}}x_iy_{\\overline{i}}y_i) = \\frac{1}{2}-p_i \\\\\n\t\t& a_i\\oplus\\ b_i \\neq x_i\\cdot y_i \\rightarrow \\mathrm{P}_{AB|XY}(a_{\\overline{i}}a_ib_{\\overline{i}}b_i|x_{\\overline{i}}x_iy_{\\overline{i}}y_i) = p_i \\;.\n\t\\end{split}\n\t\\end{equation*}\n\twhere $a_{\\overline{i}}=a_1a_2\\dotsc a_{i-1}a_{i+1} \\dotsc a_n$ and $b_{\\overline{i}}, x_{\\overline{i}}, y_{\\overline{i}}$ are defined in a similar way.\n\\end{defn}\nA simple state $\\mathrm{P}_{AB|XY}$ which has this symmetry for example is a product state of 2-partite states as in Figure~\\ref{fig:CHSH_symmetry} with different values of $p$.\n\n\\begin{figure}\n\\begin{centering}\n\t\\begin{tikzpicture}[scale=0.5]\n\n\t\t\\draw[step=2] (-5,-4) grid (4,5);\n\t\t\\draw[ultra thick] (-6,4)--(4,4);\n\t\t\\draw[ultra thick] (-6,-4)--(4,-4);\n\t\t\\draw[ultra thick] (-4,-4)--(-4,6);\n\t\t\\draw[ultra thick] (4,-4)--(4,6);\n\t\t\\draw[ultra thick] (-6,0)--(4,0);\n\t\t\\draw[ultra thick] (0,-4)--(0,6);\n\t\t\\draw (-4,4)--(-6,6);\n\n\t\t\\draw (-3,3) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (-1,3) node {$p$};\n\t\t\\draw (1,3) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (3,3) node {$p$};\n\n\t\t\\draw[red] (-3,1) node {$p$};\n\t\t\\draw (-1,1) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (1,1) node {$p$};\n\t\t\\draw (3,1) node {$\\frac{1}{2}-p$};\n\n\t\t\\draw (-3,-1) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (-1,-1) node {$p$};\n\t\t\\draw[red] (1,-1) node {$p$};\n\t\t\\draw (3,-1) node {$\\frac{1}{2}-p$};\n\n\t\t\\draw[red] (-3,-3) node {$p$};\n\t\t\\draw (-1,-3) node {$\\frac{1}{2}-p$};\n\t\t\\draw (1,-3) node {$\\frac{1}{2}-p$};\n\t\t\\draw[red] (3,-3) node {$p$};\n\n\t\t\\draw (-5,3) node {0};\n\t\t\\draw (-5,1) node {1};\n\t\t\\draw (-5,-1) node {0};\n\t\t\\draw (-5,-3) node {1};\n\t\t\\draw (-6,2) node {0};\n\t\t\\draw (-6,-2) node {1};\n\n\t\t\\draw (-3,5) node {0};\n\t\t\\draw (-1,5) node {1};\n\t\t\\draw (1,5) node {0};\n\t\t\\draw (3,5) node {1};\n\t\t\\draw (-2,6) node {0};\n\t\t\\draw (2,6) node {1};\n\n\t\t\\draw (-5,4.4) node {$B_1$};\n\t\t\\draw (-4.4,5) node {$A_1$};\n\t\t\\draw (-6,5) node {$Y_1$};\n\t\t\\draw (-5,6) node {$X_1$};\n\t\t\t\n\t\\end{tikzpicture}\n\\end{centering}\n\\caption{A simple 2-partite state $P_{A_1B_1|X_1Y_1}$ with the CHSH symmetry.} \\label{fig:CHSH_symmetry}\n\\end{figure} \n\n\\begin{cor}[de Finetti reduction for states with the CHSH symmetry] \\label{cor:chsh-post-selection}\n\tThere exists a non-signalling de Finetti state $\\tau^\\mathcal{CHSH}_{AB|XY}$ where $a,b,x,y \\in \\{ 0,1 \\}^n$ such that for every permutation invariant\\footnote{Here a permutation acts on the bipartite state as $\\left(\\mathrm{P}_{AB|XY}\\circ\\pi \\right) (ab|xy)=\\mathrm{P}_{AB|XY}\\left(\\pi(a)\\pi(b)|\\pi(x)\\pi(y)\\right)$.} state $\\mathrm{P}_{AB|XY}$ with the CHSH symmetry, for all $a,b,x,y$,\n\t\\[\n\t\t\\mathrm{P}_{AB|XY} (a,b|x,y) \\leq (n+1) \\; \\tau^\\mathcal{CHSH}_{AB|XY} (a,b|x,y)\\;. \n\t\\]\n\\end{cor}\nNote that we do not assume that the state $\\mathrm{P}_{AB|XY}$ satisfies any non-signalling conditions. Our theorem holds even when there is signalling between the subsystems, and therefore can be used in a broad set of applications.\n\nCorollary \\ref{cor:chsh-post-selection} is derived from Theorem \\ref{thm:post-selection} by showing that $d=1$ for the CHSH symmetry\\footnote{Intuitivly, in the CHSH symmetry there is only one degree of freedom, i.e. $d=1$, since we are only free to choose one value $p$ when defining the basic CHSH state given in Figure \\ref{fig:CHSH_symmetry}. Less symmetry implies more degrees of freedom. }. For pedagogical reasons, we also present a self-contained proof including an explicit construction of the state $\\tau^\\mathcal{CHSH}_{AB|XY}$ in Appendix~\\ref{sec:chsh-proof}. \n\nAlthough the assumption about the symmetry of the states in Corollary \\ref{cor:chsh-post-selection} appears to be rather restrictive, the statement turns out to be useful for applications. \n\n\\section{Applications}\n\nTo illustrate the use of the de Finetti reductions, we start by considering the following abstract application. Let $\\mathcal{T}$ be a test which interacts with a state $\\mathrm{P}_{A|X}$ and outputs ``success'' or ``fail'' with some probabilities. One can think about this test, which can be chosen according to the application being considered, as a way to quantify the success probability of the protocol when the state $\\mathrm{P}_{A|X}$ is given as input. For example, if one considers an estimation, or a tomography, protocol a test can be chosen to output ``success'' when the estimated state is close to the actual state \\cite{christandl2009postselection}. \n\nWe denote by $\\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X})$ the probability that $\\mathcal{T}$ outputs ``fail'' after interacting with $\\mathrm{P}_{A|X}$. We consider permutation invariant tests, defined as follows. \n\\begin{defn}\\label{def:permutation-invariant-test}\n\tA test $\\mathcal{T}$ is permutation invariant if for all states $\\mathrm{P}_{A|X}$ and all permutations $\\pi$ we have\n\t\\[\n\t \t\\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X}) = \\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X}\\circ\\pi) \\;.\n\t\\]\n\\end{defn}\n\nUsing the de Finetti reduction in Corollary \\ref{cor:conditional} we can prove upper bounds of the following type: \n\n\\begin{lem}\\label{lem:test-bound}\n\tLet $\\mathcal{T}$ be a permutation invariant test. Then for every state $\\mathrm{P}_{A|X}$ \n\t\\[\n\t\t\t\\mathrm{Pr}_{\\mathrm{fail}}(\\mathrm{P}_{A|X}) \\leq (n+1)^{m(l-1)} \\mathrm{Pr}_{\\mathrm{fail}}(\\tau_{A|X}) \\;.\n\t\\]\n\\end{lem}\n\nThe importance of the de Finetti reductions is obvious from this abstract example --- if one wishes to prove an upper bound on the failure probability of the test $\\mathcal{T}$, instead of proving it for all states $\\mathrm{P}_{A|X}$ it is sufficient to prove it for the de Finetti state $\\tau_{A|X}$ and ``pay'' for it with the additional polynomial pre-factor of $(n+1)^{m(l-1)}$. Since the de Finetti state has an i.i.d.\\ structure this can highly simplify the calculations of the bound. \n\nMoreover, in many cases one finds that the bound on $\\mathrm{Pr}_{\\text{fail}}(\\tau_{A|X})$ is exponentially small in $n$. For an estimation protocol, the failure probability of the test, when interacting with an i.i.d.\\ state, can be shown to be exponentially small in the number of subsystems used for the estimation, using Chernoff bounds. This is also the case when dealing with security proofs -- the failure probability of a protocol, when a de Finetti state is given as input, is usually exponentially small in the number of subsystems used in the protocol. If this is indeed the case then the polynomial pre-factor of  $(n+1)^{m(l-1)}$ will not affect the bound in the asymptotic limit of large $n$. That is, an exponentially small bound on $\\mathrm{Pr}_{\\text{fail}}(\\tau_{A|X})$ implies an exponentially small bound on $\\mathrm{Pr}_{\\text{fail}}(\\mathrm{P}_{A|X})$.\n\nFor an estimation protocol as mentioned above the notion of the test, combined with the de Finetti reductions, can be used to prove that an estimation procedure of permutation invariant states succeeds with high probability. \n\nFor readers who are interested in cryptography, we show in Appendix~\\ref{sec:diamond-proofs} how to derive a similar result when considering the diamond norm~\\cite{kitaev1997quantum}, i.e., the distance between channels acting on conditional probability distributions, instead of the abstract test $\\mathcal{T}$. The diamond norm is the relevant distance measure when considering cryptographic protocols, and therefore using de Finetti reductions to upper bound the diamond norm can simplify the analysis of device independent  protocols. \n\n\\section{Concluding remarks}\nIn this letter we introduced a general de Finetti-type theorem from which various more specialised variants can be derived. Crucially, such theorems can be formulated even without relying on assumptions regarding the non-signalling conditions between the subsystems or the underlying dimension. In the general theorem, Theorem~\\ref{thm:post-selection}, we can also see how additional symmetries of the states can affect the pre-factor in the de Finetti reduction. This suggests that the same relationship might also exist in the quantum post selection theorem \\cite{christandl2009postselection}, which is the quantum variant of the de Finetti reductions presented here.\n\nAs an example for an application we showed how our theorems can be used to bound the failure probability of a test. In a following work \\cite{arnon2014nonsignalling} we show how to use the concept of the test, together with the de Finetti reduction given in Corollary \\ref{cor:conditional} to prove parallel repetition results for non-local games. Previous de Finetti theorems could have not been used in the setting of non-local games due to their dependency on the dimension of the systems or the strict non-signalling conditions they assume. The new de Finetti theorem presented here therefore opens new possibilities and therefore strictly extends the range of applications to which de Finetti reductions can be applied.\n\nAs an additional example, we explain how our theorem can be used in device independent protocols in which the parties are not assumed to be restricted by quantum theory in Appendix~\\ref{sec:diamond-proofs}. We hope that this approach will also be useful for quantum device independent information processing protocols in the future. One possible direction can be to use a similar de Finetti reduction as in Corollary \\ref{cor:chsh-post-selection}, but for a Bell inequality in which the maximal violation is achieved within quantum theory. This way, the resulting de Finetti state will be not only non-signalling but also quantum.\nDue to the general structure of the de Finetti reductions and the increasing use of conditional probability distributions in quantum information theory, we also hope that the presented reductions will be useful in other applications apart from cryptography, such as quantum tomography, as was the case for the quantum post selection theorem~\\cite{christandl2009postselection}.\n\nThe techniques used to prove our theorems (mainly combinatoric arguments) differ from the techniques used in previous papers to establish general de Finetti theorems. We therefore hope that they will shed new light on de Finetti reductions in general. For example, it might be possible to apply some ideas from the proof in (device dependent) quantum de Finetti reductions. \n\n\\begin{acknowledgments}\nThe authors thank Roger Colbeck and Michael Walter for discussing a preliminary version of this work. This work was supported by the Swiss National Science Foundation (via the National Centre of Competence in Research ``QSIT'' and SNF project No. 200020-135048), by the European Research Council (via project No. 258932), by the CHIST-ERA project ``DIQIP'' and by the EC STREP project ``RAQUEL''. \n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\begin{figure}[t!]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/overview-apinet.pdf}\n    \\caption{Overview of proposed Multi Channel Xception Attentive Pairwise Interaction (MCX-API) network. Two inputs are first represented in $n$ kinds of color spaces, $CS_1$ to $CS_n$ to obtain a two N-channel input and subsequently feature vectors. We thereafter obtain $x_{i}^{self}$ and $x_{i}^{other}$ by comparison through MCX-API, where $x_{i}^{self}$ is enhanced by its own images and $x_{i}^{other}$ is activated by the other image. $x_{i}$ is therefore improved with discriminative clues that come from both images. By comparison, we can finally distinguish if an image is pristine or fake.}\n    \\label{fig:overview}\n\\end{figure}\nDeepfakes are synthetic media that are generated by deep learning methods to manipulate the content in images and videos. The manipulations include altering people's identities, faces, expressions, speech or bodies to both entertainment and malicious intent (for example pornographic uses). Benefiting from the remarkable advancement in generation models, amateurish individuals are capable of creating Deepfakes using off-the-shelf models~\\cite{DeepFaceLab, fffs, FaceApp} without tedious efforts. \nIn the meantime, channelized efforts have been dedicated to devising Deepfakes detection algorithms using multiple approaches such as by determining unique artifacts~\\cite{matern2019exploiting,ciftci2020fakecatcher,fernandes2019predicting,haliassos2021lips,agarwal2019protecting,li2020face}, utilizing Convolutional Neural Networks (CNNs) based networks~\\cite{marra2019gans,rossler2019faceforensics++,nguyen2019use}, employing frequency domain information~\\cite{durall2019unmasking,chen2021attentive,qian2020thinking} and other clues~\\cite{cozzolino2021id, cozzolino2022audio}.\n\\par With an atomic effort, these methods could perform well with an average of more than 99\\%~\\cite{rossler2019faceforensics++} accuracy in a closed-set problem where the training and testing data are pulled from the same label and feature spaces. For example, the network is trained on attacks A, B and C and tested on images\/videos drawn from attack A or B or C. However, newer DeepFakes generation mechanisms make the detection algorithms untrustworthy and non-generalizable by degrading the performance of the detector~\\cite{zhao2021learning,zhou2021joint} as no exception to those classifiers trained with machine learning methods. In the context of DeepFakes detection, this can be parallel to detecting attack D when the detector is trained on A, B, and C, making it an open-set problem. The reasons behind the collapse of detection models towards unseen contents can, to some degree, be attributed to various generation algorithms, which often result in different data distributions, feature spaces, and appearance properties of images or videos. While one can see the imperative need for a generalizable detection technique to make reliable decisions on unknown\/unseen generation types in addition to known\/seen generation data, we note low performances of networks in this direction \\cite{zhao2021learning,zhou2021joint,xu2022supervised, aneja2020generalized}. \n\\par We thus motivate our work, focusing on both closed-set and open-set detection in this article. We draw our inspiration from how humans tend to detect altered media in a fine-grained manner by comparing one kind of visual content to another. Human decision making relies on detecting an unseen kind of manipulated images\/videos as fake by comparing the unknown generation type to the known generation types, especially the artifacts and clues~\\cite{zhuang2020learning}. Initial work using on pairwise interaction has shown promising directions to capture subtle differences in a pairwise manner with not only principal parts of the image but also distinct details from the other image \\cite{zhuang2020learning}. Using such a paradigm, we propose to learn the known type of generations in a fine-grained pairwise manner explicitly to improve the performance of a Deepfake detector for unknown types. Further, we also note the complementary information an image\/video can exhibit in different color spaces along the same lines. We therefore incorporate information from four color spaces, including RGB, CIELab, HSV, and YCbCr integrating to boost the attentive pairwise learning to guide the detector to classify the non-manipulated images efficiently. Our proposed approach exploits the information from color channels in a pairwise manner using the strengths of the Xception network and we refer to this as the Multi-Channel Xception Attentive Pairwise Interaction (MCX-API) network between non-manipulated images against a set of manipulated images and to try to generalize the detector towards unknown manipulation types or unseen data. \\Cref{fig:overview} shows an overview of the idea presented in this work. \n\\par To validate our idea in this work, we conduct various experiments using FaceForensics++ dataset~\\cite{rossler2019faceforensics++} which consists of four different manipulation classes including DeepFakes (DF)~\\cite{ffdf}, FaceSwap (FS)~\\cite{fffs}, Face2Face (F2F)~\\cite{thies2016face2face} and NeuralTextures (NT)\\cite{thies2019deferred} where we obtain better state-of-the-art (SOTA) performance or at par detection performance to best performing SOTA approaches in closed-set experiments~\\cite{chollet2017xception,afchar2018mesonet,zhao2021learning,li2020face}. Furthermore, we demonstrate the effectiveness of variants of the proposed approach in detecting Deepfakes in open-set scenarios where our approach achieves better results than SOTA models on three other public datasets such as FakeAV~\\cite{khalid2021fakeavceleb}, KoDF~\\cite{kwon2021kodf}, and Celeb-DF~\\cite{li2020celeb}.\n\\par A detailed ablation study is presented on MCX-API to illustrate the variability of performance of the detector to various design choices in the network. Thus, the main contributions of our paper are \\textbf{(1)} We propose a new framework - Multi-Channel Xception Attentive Pairwise Interaction (MCX-API)  for Deepfakes detection by exploiting color space and pairwise interaction simultaneously, bringing a novel fine-grained idea for the Deepfakes detection field. \\textbf{(2)}We report all results by balanced-open-set-classification (BOSC) accuracy to exemplify the generalizability of our proposed approach. \n   \n   \n\\textbf{(3)}We conduct cross-datasets validations with three SOTA Deepfake datasets, Celeb-DF~\\cite{li2020celeb}, KoDF~\\cite{kwon2021kodf} and FakeAVCelebDF~\\cite{khalid2021fakeavceleb}. Furthermore, we compared the results with SOTA Deepfake detection methods. Our MCX-API obtains 98.48\\% BOSC accuracy on the FF++ dataset and 90.87\\% BOSC accuracy on the Celeb-DF dataset, indicating an optimistic direction for the generalization of DeepFake detection.\n\nIn the rest of the paper, we list a set of directly related works in \\cref{sec:related-works} and then present our proposed approach in \\cref{sec:proposed-approach}. \nWe provide an analysis of explainability in \\cref{sec:ExplainableAnalysisofMCX-API} with the set of experiments and results on generalizability detailed in \\cref{sec:Resutls}. We finally conclude the work in \\cref{sec:conclusion}.\n\n\\begin{figure*}[htp]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/API-Net.pdf}\n    \\caption{The architecture of MCX-API network.}\n    \\label{fig:arr_bar}\n\\end{figure*}\n\n\\section{Related Work}\n\\label{sec:related-works}\n\\textbf{Deepfakes detection methods.} A track of Deepfakes detection focuses on the unique artifacts on human faces, such as eye blinking~\\cite{li2018ictu}, different eye colors~\\cite{matern2019exploiting}, abnormal heartbeat rhythms shown on the face~\\cite{ciftci2020fakecatcher,fernandes2019predicting}. LipForensics~\\cite{haliassos2021lips} targets high-level semantic abnormalities in mouth movements, which the authors observe as a common indicator in many generated videos. Some articles are dedicated to finding inconsistencies in images and videos. These inconsistencies arise out of generation process where landmarks, head pose are inconsistent~\\cite{yang2019exposing,agarwal2019protecting} or observable in image blending~\\cite{li2018exposing,li2020face}. Cozzolino \\textit{et. al.}~\\cite{cozzolino2021id} have introduced ID-Reveal, an identity-aware detection approach leveraging a set of reference videos of a target person and trained in an adversarial manner. \nMany papers have utilized CNNs-based methods for detecting features existing in forged images\\cite{marra2019gans,rossler2019faceforensics++,nguyen2019use}. Using high-frequency features~\\cite{durall2019unmasking,chen2021attentive,qian2020thinking} to distinguish Deepfakes are also gaining more popularity on this topic.\nAlthough pairwise learning have been used for Deepfake detection~\\cite{hsu2020deep, xu2022supervised}, they lack the pairwise interactions by using contrastive learning.\n\n\\par \\textbf{Generalization to unseen manipulations.} While many works are proposed for detecting DeepFakes, they have focused on closed-set experiments where the training and testing set distributions are similar. The open-set experiments indicate that they underperform on unseen manipulations. In the meantime, an increasing number of works have tried to address the problem of generalization of DeepFakes detection. These works have focused on domain adaptation and transfer learning to minimize the task of learning parameters in an end-to-end manner~\\cite{aneja2020generalized,kim2021fretal,lee2021tar}. \\textit{Cozzolino et. al.}~\\cite{cozzolino2018forensictransfer} proposed an autoencoder-like structure ForensicTransfer and the generalization aspect was studied using a single detection method for multiple target domains. The follow-up works like Locality-aware AutoEncoder (LAE)~\\cite{du2019towards} and Multi-task Learning were proposed for detecting and segmenting manipulated facial images and videos~\\cite{nguyen2019multi}. Transfer learning-based Autoencoder with Residuals (TAR)~\\cite{lee2021tar} recently proposed uses the residuals from autoencoders to handle generalizability. \\textit{Kim et. al.}~\\cite{kim2021fretal} employed the Representation Learning (ReL) and Knowledge Distillation (KD) paradigms to introduce a transfer learning-based Feature Representation Transfer Adaptation Learning (FReTAL) method. While these transfer learning and zero-shot\/few-shot learning methods could not wholly deal with the Deepfakes detection generalization problem, because the networks have already seen the manipulated image\/videos. Therefore, strictly speaking, it is not an open-set situation.\n\\par In the meantime, some other novel networks have been proposed dealing with the generalization problem of Deepfakes detection. A new method to detect deepfake images using the cue of the source feature inconsistency within the forged images~\\cite{zhao2021learning} is proposed based on the hypothesis that distinct source features can be preserved and extracted through SOTA deepfake generation processes. Joint Audio-Visual Deepfake Detection~\\cite{zhou2021joint} is proposed by jointly modeling video and audio modalities. This novel visual\/auditory deepfake combined detection task shows that exploiting the intrinsic synchronization between the visual and auditory modalities could benefit deepfake detection. \\textit{Xu et. al.}~\\cite{xu2022supervised} proposed a novel method using supervised contrastive learning to deal with the generalization problem in detecting forged visual media.\n\n\\section{Proposed Method}\n\\label{sec:proposed-approach}\nFine-grained method has been widely used for classification problems where the categories are visually very similar~\\cite{zhuang2020learning, xiao2015application, akata2015evaluation}. We draw similar inspiration to our problem of Deepfake detection following the architecture proposed by earlier~\\cite{zhuang2020learning} and build upon with number of improvements. We assert that architecture for fine-grained classification can help in detecting Deepfakes. Unlike the orginal architecture, we introduce Xception~\\cite{chollet2017xception} to extract the embeddings motivated by earlier works in Deepfake detection~\\cite{rossler2019faceforensics++, zhao2021multi, wang2022m2tr, kim2021fretal}. \n\nSecond, to benefit from information from different color spaces, we make the base network to a multi-channel network. Then, we enforce pairwise learning by following the architecture of Attentive Pairwise Learning \\cite{zhuang2020learning}. We propose using the Multi Channel Xception Attentive Pairwise Interaction Network (MCX-API) to deal with the Deepfakes classification problem as detailed further.\n\n\\subsection{Architecture}\n\\par We first utilize MTCNN\\cite{zhang2016joint} to crop and align the face region of a single frame. Two selected face images are further sent to a Multi-Channel Xception backbone, and this backbone network extracts two corresponding $\\mathrm{D}$-dimension feature vectors $x_{1}$ and $x_{2}$ using the face image represented in $N$ different channels that include RGB, CIELab, HSV, and YCbCr. A mutual vector $x_{m}\\in \\mathbb{R}^{D}$ is further generated by concatenating $x_{1}$ and $x_{2}$ and using a Multi-Layer Perceptron (MLP) function for mapping $x_{m}$ to get a $\\mathrm{D}$ dimension. $x_{m}$ is a joint feature that includes high-level contrastive clues of both input images across multiple color channels.\n\n\\par In order to compare $x_{m}$ with $x_{1}$ and $x_{2}$, we need to activate $x_{m}$ using sigmoid function to increase the positive relation with $x_{i}$ and decrease the negative relation against $x_{i}$~\\cite{zhuang2020learning}. Therefore, two gate vectors $g_{1}$ and $g_{2}$ will be generated. $g_{i}$ is calculated by $x_{m}$ and $x_{i}$, thus containing contrastive clues and acting as discriminative attention spots semantic contrasts with a distinct view of each $x_{i}$. The gate vector $g_{i}$ is the sigmoid of the output of channel-wise product between $x_{m}$ and $x_{i}$, whose formula is provided in \\Cref{eqn:gate-vector}. \n\\vspace{-1mm}\n\\begin{equation}\n    g_{i} = sigmoid(x_{m} \\odot x_{i}), \\;\\; i \\in{\\{1,2\\}}\n    \\label{eqn:gate-vector}\n\\end{equation}\n\n\\par A pairwise interaction between input features $x_{i}$ and gate vectors $g_{i}$ is performed to induce residual attention by comparing one image to the other to distinguish the final class. The sequence of interaction can be shown in \\Cref{eqn:pairwise-interaction}.\n\\begin{equation}\n\\centering\n\\begin{split}\n    x_{1}^{pristine}=x_{1}+x_{1}\\odot g_{1} \\\\\n    x_{1}^{fake}=x_{1}+x_{1}\\odot g_{2} \\\\\n    x_{2}^{pristine}=x_{2}+x_{2}\\odot g_{2} \\\\\n    x_{2}^{fake}=x_{2}+x_{2}\\odot g_{1}\n\\end{split}\n\\label{eqn:pairwise-interaction}\n\\end{equation}\nThrough the pairwise interaction of each feature $x_{i}$, two attentive feature vectors $x_{i}^{pristine}\\in \\mathbb{R}^{D}$ and $x_{i}^{fake}\\in \\mathbb{R}^{D}$ are further produced. The former one is highlighted by its gate vector, and the latter is triggered by the gate vector of the compared image. $x_{i}$ is thus enhanced with discriminative clues from both input features through pairwise interaction.\n\n\n\n\\subsection{Loss calculation}\nThe four attentive features $x_{i}^{j}$ where $i \\in {\\{1,2\\}}$ and $j \\in {\\{pristine,fake\\}}$, the pairwise interaction outputs, are fed into a $softmax$ classifier for the loss calculation~\\cite{zhuang2020learning}. The output of $softmax$ denoted by $p_{i}^{j}$ is the probability of a feature belonging to a specific class (i.e., non-manipulated or Deepfake). The main loss in our case is the cross-entropy loss \n\\begin{equation}\n\\mathcal{L}_{ce} = -\\sum_{i \\in \\{ 1,2 \\}} \\sum_{j \\in \\{ pristine,fake \\}} y_{i}^{\\intercal} log(p_{i}^{j})\n\\label{eqn:lce}\n\\end{equation}\nwhere $y_{i}$ is the one-hot label for image $i$ in the pair and $\\intercal$ represents the transpose. MCX-API can be trained to determine all the attentive features $x_{i}^{j}$ under the supervision of the label $y_{i}$ through this loss.\n\n\\par Furthermore, a hinge loss of score ranking regularization\n\\begin{equation}\n    \\mathcal{L}_{rk} = \\sum_{i\\in {\\{1, 2\\}}} max(0, p_{i}^{fake}(c_{i})-p_{i}^{pristine}(c_{i})+\\epsilon )\n\\label{eqn:lrk}\n\\end{equation}\nis also introduced when computing the complete loss~\\cite{zhuang2020learning}. $c_{i}$ is the corresponding index associated with the ground truth label of image $i$. So $p_{i}^{j}(c_{i})$ is a softmax score of $p_{i}^{j}$. Since $x_{i}^{pristine}$ is activated by its gate vector $g_{i}$, it should contain more discriminative features to identify the corresponding image, compared to $x_{i}^{fake}$. $\\mathcal{L}_{rk}$ is utilized to promote the priority of $x_{i}^{pristine}$ where the score difference between $p_{i}^{fake}(c_{i})$ and $p_{i}^{pristine}(c_{i})$ should be greater than a margin.\nThe whole loss for a pair is composed of two losses, cross-entropy loss $\\mathcal{L}_{ce}$ and score ranking regularization $\\mathcal{L}_{rk}$ with coefficient $\\lambda$. \n\\begin{equation}\n    \\mathcal{L} = \\mathcal{L}_{ce} + \\lambda \\mathcal{L}_{rk}\n\\label{eqn:loss}\n\\end{equation}\nIn this way, MCX-API is able to take feature priorities into account adaptively and learns to recognize each image in the pair.\n\n\\section{Experiments and Results}\n\\label{sec:Resutls}\n\n\\subsection{Datasets}\n\\textbf{Training data: }We select FaceForensics++ \\cite{rossler2019faceforensics++} to train the proposed approach. This forensics dataset consists of 1000 original videos and corresponding number of manipulated videos consisting of 1000 videos for each of the subsets - DeepFakes (denoted as DF)~\\cite{ffdf}, Face2Face (denoted as F2F)~\\cite{thies2016face2face}, FaceSwap (denoted as FS)~\\cite{fffs}, and NeuralTextures (denoted as NT)~\\cite{thies2019deferred}.\n\n\\textbf{Cross-dataset Validation: }We also select three other SOTA datasets for generalization test and comparison. \\textbf{Celeb-DF~\\cite{li2020celeb}}: For Celeb-DF, we choose id51-id61 from Celeb-real, Celeb-synthesis and id240-id299 from YouTube-real for the test set.\n   \n\\textbf{KoDF~\\cite{kwon2021kodf}} We randomly selected 265 real videos and 734 fake ones as our test set.\n\\textbf{FakeAV~\\cite{khalid2021fakeavceleb}} We randomly selected 500 videos as our test set.\n\n\n\\begin{table*}[htp]\n\\caption{\\textbf{Frame-level BOSC Accuracy and AUC for our proposed MCX-API networks and SOTA methods on seen data.} We compare the results with the SOTA methods on DF\/F2F\/FS\/NT respectively. All networks are trained on the whole FF++ c23 dataset. The data of the first three methods are adopted from Table 5 in Appendix of FF++~\\cite{chollet2017xception}.}\n\\label{tab:bosc-ffpp}\n\\begin{threeparttable}\n\\centering\n\\begin{tabular}{llllllll}\n\\toprule\nFF++ c23 &  & \\multicolumn{6}{c}{Frame-level (BOSC(\\%)\/AUC)} \\\\ \\cline{1-1} \\cline{3-8} \nMethod &  & \\multicolumn{1}{c}{DF} & \\multicolumn{1}{c}{F2F} & \\multicolumn{1}{c}{FS} & \\multicolumn{1}{c}{NT} &  & Average \\\\ \\cline{1-6} \\cline{8-8} \nCozzolino \\textit{et al.}~\\cite{cozzolino2017recasting} &  & 75.51\/ - & 86.34\/ - & 76.81\/ - & 75.34\/ - &  & 78.50\/ - \\\\\nBayar and Stamm~\\cite{bayar2016deep} &  & 90.25\/ - & 93.96\/ - & 87.74\/ - & 83.69\/ - &  & 88.91\/ - \\\\\nMesoNet~\\cite{afchar2018mesonet} &  & 89.55\/ - & 88.60\/ - & 81.24\/ - & 92.19\/ - &  & 87.90\/ - \\\\\nXception\\tnote{*} \\cite{chollet2017xception} &  & 96.35\/0.9941 & 96.26\/0.9937 & 96.29\/0.9952 & 92.43\/0.9736 &  & 95.33\/0.9892 \\\\\nSupCon\\tnote{*} \\cite{xu2022supervised} &  & 97.18\/0.9984 & 96.88\/0.9978 & 97.05\/0.9980 & 92.92\/0.9846 &  & 96.01\/0.9947 \\\\ \nAPI-Net(ResNet101)\\tnote{*} \\cite{zhuang2020learning} &  & 88.71\/0.9820 & 90.13\/0.9860 & 87.79\/0.9728 & 82.96\/0.9248 &  & 87.40\/0.9664 \\\\ \\hline\nOurs &  &  &  &  &  &  &  \\\\\n\\textbf{MCX-API(RGB)} &  & \\textbf{98.75}\/0.9996 & \\textbf{99.90}\/0.9986 & \\textbf{98.5}\/\\textbf{0.9993} & \\textbf{96.75}\/0.9896 &  & \\textbf{98.48}\/0.9968 \\\\\n\\textbf{MCX-API(RGB+HSV)} &  & \\textbf{98.75}\/0.9988 & 98.50\/0.9979 & 97.75\/0.9978 & 95.75\/0.9829 &  & 97.69\/0.9943 \\\\\n\\textbf{MCX-API(RGB+CIELab)} &  & 97.00\/0.9996 & 96.50\/0.9985 & 96.25\/0.9989 & 95.25\/0.9909 &  & 96.25\/0.9970 \\\\\n\\textbf{MCX-API(RGB+YCbCr)} &  & 98.00\/\\textbf{0.9998} & 98.25\/\\textbf{0.9991} & 97.75\/\\textbf{0.9993} & \\textbf{96.75}\/0.9920 &  & 97.69\/\\textbf{0.9976} \\\\\n\\textbf{MCX-API(RGB+HSV+CIELab)} &  & 96.50\/0.9990 & 95.50\/0.9888 & 96.00\/0.9835 & 95.50\/\\textbf{0.9933} &  & 95.88\/0.9912 \\\\\n\\textbf{MCX-API(RGB+LAB+YCbCr)} &  & 92.00\/0.9963 & 92.25\/0.9972 & 91.50\/0.9960 & 91.00\/0.9870 &  & 91.69\/0.9941 \\\\\n\\bottomrule\n\\end{tabular}\n    \\begin{tablenotes}\n         \\footnotesize\n         \\item [*] Our implementation of the method.\n    \\end{tablenotes}\n\\end{threeparttable}\n\\end{table*}\n\n\\textbf{Implementation details.} \nWe choose uncompressed videos for our experiments in this work using the Pytorch framework~\\cite{pytorch} to develop the models and the experiments are conducted on Python 3.6 environment on NVIDIA Tesla V100 32Gb in IDUN system owned by NTNU~\\cite{sjalander+:2019epic}. \n\nMulti-task Cascade Convolutional Neural Networks (MTCNN)~\\cite{zhang2016joint} is employed for face detection and face alignment since our experiments are focused on detecting the manipulated face region alone. We allow loose cropping of the face region to capture the entire silhouette against tight cropping. The first 30 frames from each video are extracted, resulting in 150000 total images. We use random cropping in the training phase and center cropping during the testing phase ($512^{2}\\to448^{2}$). In all our experiments, we employ Xception as the backbone where we derive the feature vector $x_{i}\\in \\mathbb{R}^{2048}$ after the global average pooling. We use a batch sampler during the training by randomly sampling three categories in each batch. For each category, we randomly choose nine images due to the limitations of the GPU and memory constraints. We further exercise care to have no sample overlap among all batches, as we exclude the selected sample from the dataset. We locate its most similar image from both its own class and the rest classes for each image by calculating the distance between features by utilizing both Euclidean distance and cosine distance. Each image would get one image as its intra- and inter-pair in the batch, respectively. Each pair is used as input $x_{1}$ and $x_{2}$ as well as generating a mutual vector $x_{m}\\in \\mathbb{R}^{2048}$ through the concatenation and the multilayer perceptron (MLP).\n\\par Based on empirical evaluations, we adopt the coefficient $\\lambda$ in \\Cref{eqn:loss} as 1.0, and 0.05 as the margin value in the score-ranking regularization. We use cosine annealing strategy to alter the learning rate starting from 0.01 \\cite{zhao2021learning}. We train the network with 100 epochs and freeze the parameters in the CNN backbone, and further on train only the classifier in the first eight epochs.\n\n\\textbf{Evaluation Metrics.}\nWe adopt Balanced-Open-Set-Classification (BOSC) accuracy and AUC as evaluation metrics.\n$BOSC = \\frac{Sensitivity+Specificity}{2}$, where $Sensitivity=\\frac{TP}{TP + FN}$ and $Specificity = \\frac{TN}{TN + FP}$.\n\n\\begin{table}[htp]\n\\caption{Comparison of the test results on the FF++ dataset with c23 (high-quality compression) settings. Training for all networks is carried out on FF++ c23. The accuracy and AUC score are at frame-level. The best performances are marked in bold. Data for Xception, $F^3$-Net, and EfficientNet-B4 are adopted from Table 2 in MaDD~\\cite{zhao2021multi}.}\n\\label{tab:ff-sota}\n\\centering\n\\begin{tabular}{llcc}\n\\toprule\nMethod          && ACC            & AUC            \\\\ \\cline{1-1} \\cline{3-4} \nXception        && 95.73          & 0.9909          \\\\ \n$F^3$-Net~\\cite{qian2020thinking}          && 97.52          & 0.9810          \\\\ \nEfficientNet-B4~\\cite{tan2019efficientnet} && 96.63          & 0.9918          \\\\ \nDCL~\\cite{sun2022dual}             && 96.74          & 0.9930          \\\\ \nMaDD~\\cite{zhao2021multi}            && 97.60          & 0.9929          \\\\ \nM2TR~\\cite{wang2022m2tr}            && 97.93          & 0.9951          \\\\ \nAPI-Net            && 87.40          & 0.9664          \\\\ \\hline\nOurs            && \\textbf{98.48} & \\textbf{0.9968} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table*}[htp]\n\\caption{\\textbf{Video-level BOSC Accuracy and AUC for our proposed MCX-API networks and SOTA methods on unseen data.} We compare the results with the SOTA methods on FakeAV\/KoDF\/Celeb-DF respectively. All the networks are trained on the whole FF++ c23 dataset. The data of the SOTA methods are adopted from Table 2 from \\cite{cozzolino2022audio}.}\n\\label{tab:cross-all}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{lllll}\n\\toprule\nFF++ c23 &  & \\multicolumn{3}{c}{Video-level (BOSC(\\%)\/AUC)} \\\\ \\cline{1-1} \\cline{3-5} \nMethod &  & \\multicolumn{1}{c}{FakeAV} & \\multicolumn{1}{c}{KoDF} & \\multicolumn{1}{c}{Celeb-DF} \\\\ \\hline\nXception\\tnote{*} &  & 23.99\/0.450 & 25.97\/0.482 & 31.34\/0.505 \\\\\nSeferbekov~\\cite{seferbekov} &  & \\textbf{95.0\/0.986} & 79.2\/0.884 & 75.3\/0.860 \\\\\nFTCN~\\cite{zheng2021exploring} &  & 64.9\/0.840 & 63.0\/0.765 & - \\\\\nLipForensics~\\cite{haliassos2021lips} &  & 83.3\/0.976 & 56.1\/0.929 & -\/0.820 \\\\\nID-reveal~\\cite{cozzolino2021id} &  & 63.7\/0.876 & 60.3\/0.702 & 71.6\/0.840 \\\\\nPOI~\\cite{cozzolino2022audio} &  & 86.6\/0.941 & 81.1\/0.899 & - \\\\ \nAPI-Net(ResNet101)\\tnote{*} &  & 59.99\/0.72 & 66.92\/0.76 & 58.00\/0.76 \\\\\\hline\nOurs &  &  &  &  \\\\\n\\textbf{MCX-API(RGB)} &  & 74.94\/0.95    & 78.09\/\\underline{0.87}  & 77.88\/0.87 \\\\\n\\textbf{MCX-API(HSV)} &  & 74.63\/0.75 & \\underline{80.64}\/0.85  & 75.67\/0.88 \\\\\n\\textbf{MCX-API(CIELab)} &  & 84.28\/0.90 & \\textbf{81.16\/0.90}  & 64.28\/0.81 \\\\\n\\textbf{MCX-API(RGB+HSV)} &  & 71.58\/0.93    & 78.11\/\\underline{0.87}  & \\underline{80.18}\/0.88 \\\\\n\\textbf{MCX-API(RGB+CIELab)} &  & 83.89\/0.93 & 77.93\/0.83  & 68.34\/\\textbf{0.91} \\\\\n\\textbf{MCX-API(RGB+YCbCr)} &  & 70.41\/0.92   & 78.39\/0.85  & \\textbf{90.87}\/\\underline{0.90} \\\\\n\\textbf{MCX-API(RGB+HSV+CIELab)} &  & \\underline{92.38\/0.98} & 78.91\/0.83 & 59.04\/0.89 \\\\\n\\textbf{MCX-API(RGB+LAB+YCbCr)} &  & 82.93\/0.96  & 76.20\/0.80  &  54.92\/0.85 \\\\\n\\bottomrule\n\\end{tabular}\n    \\begin{tablenotes}\n         \\footnotesize\n         \\item [*] Our implementation of the method.\n    \\end{tablenotes}\n\\end{threeparttable}\n\\end{table*}\n\n\\subsection{Experimental Results}\n\\label{ExperimentalResults}\nWe evaluate the effectiveness of the proposed MCX-API network with both seen and unseen data in this section.  \n\n\\subsubsection{Intra-dataset Evaluation (Closed Set Protocol)}\nWe conduct experiments on six networks with different color spaces on MCX-API whose results are presented in \\cref{tab:bosc-ffpp}. All networks are trained with all four manipulated methods along with pristine in FF++ c23 dataset. We test the frame-level detection performance on the test data of FF++ c23 in a non-overlapping manner regarding the ID. \n\nIn \\cref{tab:bosc-ffpp}, the frame-level test results are listed. We observe that our proposed MCX-API network with RGB inputs reaches the highest average accuracy, 98.48\\%. In addition, this setting also gains the highest accuracy on DF, F2F, and FS with 98.87\\%, 99.90\\% and 98.50\\%, respectively. MCX-API with YCbCr achieves the highest accuracy for NT with 97.00\\%. As RGB provides best performance under  3-channel setting, we combine RGB with HSV, CIELab, and YCbCr, respectively, to create three 6-channel MCX-API networks. From the second block in \\cref{tab:bosc-ffpp}, we can see that RGB+YCbCr obtains the highest average AUC score of 0.9976 and the best performance on DF, F2F, and FS regarding AUC score. This indicates better prediction output scores using MCX-API with the combination of RBG and YCbCr color spaces. The 9-channel MCX-API network with RGB, HSV, and CIELab further gains the highest 0.9933 AUC score for NT.\n\nThe results of the comparison with the SOTA methods are reported in \\cref{tab:ff-sota}. All networks are trained on FF++ c23 (high-quality compression). The accuracy and AUC scores are measured at frame level. The results are averaged on all the test sets from FF++ c23, including pristine and all four kinds of manipulated videos. Our proposed method MCX-API with RGB color space obtains the best performance compared to SOTA methods. The best accuracy of the BOSC is 98.48\\%, and the highest AUC score is 0.9968. The result shows that our idea of pairwise learning in a fine-grained manner could work well in inter-class (closed-set) setting of Deepfake detection problem.\n\n\\subsubsection{Cross-dataset Evaluation}\nWe conduct a comparison on cross-dataset validation with SOTA methods to validate the proposed approach.  We employ FakeAV, KoDF, and Celeb-DF to test the generalizability of our MCX-API network. Training for all networks are carried out on the FF++ c23 dataset and tested on FakeAV, KoDF, and Celeb-DF. We note that MCX-API with CIELab color space gets the best scores for KoDF with an accuracy of 81.86\\% and an AUC score of 0.90 as presented in \\cref{tab:cross-all}. MCX-API with RGB+YCbCr wins in the cross-dataset validation for Celeb-DF with an accuracy of 90.87\\% and the second best AUC score 0.90. MCX-API with color space RGB+HSV+CIELab achieves the second best place for FakeAV with 92.38\\% accuracy and 0.98 AUC score. In general, our proposed network gets a relatively better performance than the SOTA methods which indicates the better generalizability of the proposed MCX-API network.\n\n\\section{Explainable Analysis of MCX-API}\n\\label{sec:ExplainableAnalysisofMCX-API}\nWe further analyze the network to understand the performance gain by analyzing embeddings using t-SNE plots~\\cite{van2008visualizing} and class activation maps~\\cite{selvaraju2017grad, chattopadhay2018grad, draelos2020use, jiang2021layercam, fu2020axiom}. While the t-SNE provides topology explanations of the learned features, the activation maps allow for a better visualization of what has been learned by our network. \n\n\\begin{figure*}[ht]\\centering\n\t\\resizebox{2.\\columnwidth}{!}{\n\t\t\\begin{tabular}{cc}\n\t\t\t\\begin{tikzpicture}[spy using outlines={rectangle,yellow,magnification=3,size=4.0cm, connect spies, every spy on node\/.append style={very thick}}]\n\t\t\t\t\\node {\\includegraphics[width=9cm, height=9cm]{figures\/tsne_MCXapi.png}};\n\t\t\t\t\\spy[red] on (-0.3,0.2) in node [right] at (-5.5,6.3);\n\t\t\t\t\\spy[green,size=2.7cm] on (0.1,-1.2) in node [right] at (-6.9,-2.2);\n\t\t\t\t\\spy[blue,size=3.cm] on (0.5,2.1) in node [right] at (1.,6.);\n                    \\spy[orange,size=2.7cm] on (-2.5,1.0) in node [left] at (-4.2,1);\n\t\t\t\\end{tikzpicture} & \n\n\t\t\t\\begin{tikzpicture}[spy using outlines={rectangle,yellow,magnification=3,size=4cm, connect spies, every spy on node\/.append style={very thick}}]\n\t\t\t\t\\node {\\includegraphics[width=9cm, height=9cm]{figures\/tsne_api.png}};\n\t\t\t\t\\spy[red] on (-0.3,0.2) in node [right] at (-5.5,6.3);\n\t\t\t\t\\spy[green,size=2.7cm] on (0.1,-1.2) in node [right] at (-6.9,-2.2);\n\t\t\t\t\\spy[blue,size=3.cm] on (0.5,2.1) in node [right] at (1.,6.);\n                    \\spy[orange,size=2.7cm] on (-2.,1.0) in node [left] at (-4.2,1);\n\t\t\t\\end{tikzpicture}\\\\\n   \n\t\t\t\\large{MCX-API (RGB)} & \\large{API} \n\t\t\\end{tabular}\n\t}\n\t\\caption{Data visualizations in 2D by t-SNE for MCX-API(RGB) and API. The left plot is t-SNE for our proposed MCX-API. The right plot is t-SNE for base architecture API-Net. We blow up the intersection parts and outliers for a clear view.}\n\\label{fig:tsne}\n\\end{figure*}\n\n\\begin{figure*}[ht]\\centering\n\t\\resizebox{1.7\\columnwidth}{!}{\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/df_frame181.png}};\n\t\t\t\t\\spy[red] on (-0.1,-0.4) in node [right] at (2.,-4);\n\t\t\t\t\\spy[yellow] on (-1.4,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n\t\t\t\n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/df_mcxapi_layercam.png}};\n\t\t\t\t\\spy[red] on (-0.1,-0.4) in node [right] at (2.,-4);\n\t\t\t\t\\spy[yellow] on (-1.4,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n   \n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/df_api_layercam.png}};\n\t\t\t\t\\spy[red] on (-0.1,-0.4) in node [right] at (2.,-4);\n\t\t\t\t\\spy[yellow] on (-1.4,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture}\\\\\n   \n\t\t\t\\huge{DF} &\\huge{MCX-API (RGB)} & \\huge{API} \\\\\n\n   \t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/f2f_frame181.png}};\n\t\t\t\t\\spy[red] on (-1.,1.) in node [left] at (-2.7,2.);\n\t\t\t\t\\spy[yellow] on (0.,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n\t\t\t\n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/f2f_mcxapi_layercam.png}};\n\t\t\t\t\\spy[red] on (-1.,1.) in node [left] at (-2.7,2.);\n\t\t\t\t\\spy[yellow] on (0.,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture} & \n   \n\t\t\t\\begin{tikzpicture}[spy using outlines={circle,yellow,magnification=3,size=4.5cm, connect spies, every spy on node\/.append style={ultra thick}}]\n\t\t\t\t\\node {\\includegraphics[width=6cm, height=6cm]{figures\/blowup\/f2f_api_layercam.png}};\n\t\t\t\t\\spy[red] on (-1.,1.) in node [left] at (-2.7,2.);\n\t\t\t\t\\spy[yellow] on (0.,-1.3) in node [right] at (-5,-5);\n\t\t\t\t\\spy[blue] on (1.6,0.8) in node [right] at (2,4);\n\t\t\t\\end{tikzpicture}\\\\\n   \n\t\t\t\\huge{F2F} &\\huge{MCX-API (RGB)} & \\huge{API} \n   \n\t\t\\end{tabular}\n\t}\n\t\\caption{Blow up in activation maps from LayerCAM analysis of MCX-API(RGB) and base architecture API-Net on DF and F2F faces.}\n\t\\label{fig:blowup-activation}\n\\end{figure*}\n\n\\begin{figure}[htp]\n    \\centering\n    \\subfigure[Visualization of the last block of the exit flow of MCX-API (RGB).]{\\label{fig:visualization-MCXapi}\\includegraphics[width=0.95\\linewidth]{figures\/visualization_MCXapi.pdf}}\n    \\subfigure[Visualization of the last block of the API-Net.]{\\label{fig:visualization-api}\\includegraphics[width=0.95\\linewidth]{figures\/visualization_api.pdf}}\n    \n    \\caption{Visualization of the last layer of MCX-API (RGB) and API networks. We utilize Grad-CAM~\\cite{selvaraju2017grad}, Grad-CAM++~\\cite{chattopadhay2018grad}, HiResCAM~\\cite{draelos2020use}, LayberCAM~\\cite{jiang2021layercam} and XGradCAM~\\cite{fu2020axiom} as our visualization tool. For larger figure, please refer to \\cref{fig:activation-map-large}.}\n    \\label{fig:activation-map}\n\\end{figure}\n\n\\subsection{Data Visualizations With t-SNE}\nThe results of a t-SNE 2D map for the feature vectors are illustrated in \\cref{fig:tsne}. We compare the t-SNE of our MCX-API and base architecture API-Net.\nWe notice that the five classes of Real, DF, F2F, FS, and NT for MCX-API are well separated with five different clusters as against the base architecture of API-Net. There is an unclear boundary between Real and NT, shown in the blue box for MCX-API. This overlapping can be the reason for the relatively lower accuracy obtained on NT. There are small areas overlapping between DF\/NT(yellow\/purple) and Real\/F2F(red\/blue). We further notice a few samples of Real (red dots) distributed in each fake class, leading to the errors of our proposed network.\n\n\\subsection{Visualizing Decisions With Attention Maps}\nWe apply different class activation visualization methods on the last layer of proposed network to analyze MCX-API shown in \\cref{fig:activation-map}. For comparison, we also show the visualization of the base API-Net. Precisely, we adopt Grad-CAM~\\cite{selvaraju2017grad}, Grad-CAM++~\\cite{chattopadhay2018grad}, HiResCAM~\\cite{draelos2020use}, LayberCAM~\\cite{jiang2021layercam} and XGradCAM~\\cite{fu2020axiom}. The visualization results are provided in \\cref{fig:visualization-MCXapi} for our proposed MCX-API and in \\cref{fig:visualization-api} for API-Net.\n\n\nThe activation map for Output Real is on the left part with a green background, and the activation map for Output Fake is on the right part with a pink background. The rows from top to bottom are the visualization for five classes of Real, DF, F2F, FS, and NT, respectively. We can observe that real images gains more attention within Output Real(left part) than Output Fake(right part). In contrast, fake images obtain more attention within Output Fake than Output Real. This explains the ability of our network to detect Deepfakes.\n\nWe further blow up the activation maps from LayerCAM for DF and F2F images in \\cref{fig:blowup-activation}. From visual analysis, it is evident that the MCX-API focuses more on the facial region, such as the eyes and the mouth. For instance, double eyebrows are found in the DF image (blue circle). MCX-API pays more attention than API around this region. \n\n\\section{Limitations of our work}\nWe notice in \\cref{tab:bosc-ffpp} that with the increase in color spaces, there are no apparent improvements in BOSC accuracy. We assume that there is redundant information among channels, and further work would be focused on finding the most helpful color information to extend our proposed approach. We also observe that no single configuration could perform reasonably well for all the unseen data, which is the biggest issue for Deepfake detection field. Introducing other information, such as temporal data and audio, would be a good idea as more inconsistency could be found by extending our work to video based approach.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThere is an imperative need for a generalized DeepFakes detection method to deal with the newer manipulation methods in visual media. In this paper, we proposed to apply the Multi-Channel Xception Attentive Pairwise Interaction (MCX-API) network to the Deepfakes detection field in a fine-grained manner. We conducted experiments on the publicly available FaceForensics++ dataset, and our approach obtained better performance than the SOTA approaches on both seen and unseen manipulation types. We obtain 98.48\\% BOSC accuracy on the FF++ dataset and 90.87\\% BOSC accuracy on the CelebDF dataset suggesting a promising direction for the generalization of DeepFake detection. Comprehensive ablation studies have been conducted to understand our algorithm better. We further explain the performance of our network by using t-SNE and attention maps. The results showed that Deepfake had been well separated from real videos. While our approach has indicated a promising solution to obtain a generalized detection mechanism, we have listed certain limitations that can pave the way for future work.  \n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\n\n\n\\subsection{The aim of this article}\n\n\\subsubsection{}\\label{history}\n\nLet $A$ be an abelian variety defined over a number field $k$.\n\nThen, by a celebrated theorem of Mordell and Weil, the abelian group that is formed by the set $A(k)$ of points of $A$ with coefficients in $k$ is finitely generated.\n\nIt is also conjectured that the Hasse-Weil $L$-series $L(A,z)$ for $A$ over $k$ has a meromorphic continuation to the entire complex plane and satisfies a functional equation with central point $z=1$ and, in addition, that the Tate-Shafarevich group $\\sha(A_k)$ of $A$ over $k$ is finite.\n\nAssuming these conjectures to be true, the Birch and Swinnerton-Dyer Conjecture concerns the leading coefficient $L^\\ast(A,1)$ in the Taylor expansion of $L(A,z)$ at $z=1$. This conjecture was originally formulated for elliptic curves in a series of articles of Birch and Swinnerton-Dyer in the 1960's (see, for example, \\cite{birch}) and then reinterpreted and extended to the setting of abelian varieties by Tate in \\cite{tate} to give the following prediction.\n\\medskip\n\n\\noindent{}{\\bf Conjecture} (Birch and Swinnerton-Dyer)\n\\begin{itemize}\n\\item[(i)] The order of vanishing of $L(A,z)$ at $z=1$ is equal to the rank of $A(k)$.\n\\item[(ii)] One has\n\\begin{equation}\\label{bsd equality} L^\\ast(A,1) = \\frac{\\Omega_A\\cdot R_A\\cdot {\\rm Tam}(A)}{\\sqrt{D_k}^{{\\rm dim}(A)}\\cdot |A(k)_{\\rm tor}|\\cdot |A^t(k)_{\\rm tor}|}\\cdot |\\sha(A_k)|.\\end{equation}\n\\end{itemize}\n\\medskip\n\nHere $\\Omega_A$ is the canonical period of $A$, $R_A$ the discriminant of the canonical N\\'eron-Tate height pairing on $A$, ${\\rm Tam}(A)$ the product over the (finitely many) places of $k$ at which $A$ has bad reduction of local `Tamagawa numbers', $D_k$ the absolute value of the discriminant of $k$,  $A(k)_{\\rm tor}$ the torsion subgroup of $A(k)$ and $A^t$ the dual abelian variety of $A$.\n\nIt should be noted at the outset that, even if one assumes $L(A,z)$ can be meromorphically continued to $z=1$ and $\\sha(A_k)$ is finite, so that both sides of (\\ref{bsd equality}) make sense, the predicted equality is itself quite remarkable.\n\nFor instance, the $L$-series is defined via an Euler product over places of $k$ so that its leading coefficient at $z=1$ is intrinsically local and analytic in nature whilst the most important terms on the right hand side of (\\ref{bsd equality}) are both global and algebraic in nature.\n\nIn addition, and more concretely, whilst isogenous abelian varieties give rise to the same $L$-series, the individual terms that occur in the `Euler characteristic' on the right hand side of  (\\ref{bsd equality}) are not themselves isogeny invariant and it  requires a difficult theorem of Tate to show that the validity of (\\ref{bsd equality}) is invariant under isogeny.\n\nFor these, and many other, reasons, the above conjecture, which in the remainder of the introduction we abbreviate to ${\\rm BSD}(A_k)$, has come to be regarded as one of the most important problems in arithmetic geometry today.\n\nNevertheless, there are various natural contexts in which it seems likely that ${\\rm BSD}(A_k)$ does not encompass the full extent of the interplay between the analytic and algebraic invariants of $A$. Moreover, a good understanding of the finer connections that can arise could lead to much greater insight into concrete questions such as, for example, the growth of ranks of Mordell-Weil groups in extensions of number fields.\n\nFor instance, if $A$ has a large endomorphism ring, then it seems reasonable to expect there to be a version of ${\\rm BSD}(A_k)$ that reflects the existence of such endomorphisms.\n\nThe earliest example of such a refinement appears to be Gross's formulation in \\cite{G-BSD} of an equivariant\nBirch and Swinnerton-Dyer conjecture for elliptic curves $A$ with complex multiplication by the maximal order $\\mathcal{O}$ of an imaginary quadratic field.\n\nThis conjecture incorporates natural refinements of both ${\\rm BSD}(A_k)$(i) and ${\\rm BSD}(A_k)$(ii) and is supported by numerical evidence obtained by Gross and Buhler in \\cite{GrossBuhler} and by theoretical evidence obtained by Rubin in \\cite{rubin2}.\n\nIn a different direction one can study the leading coefficients of the Hasse-Weil-Artin $L$-series $L(A,\\psi,z)$ that are obtained from $A$ and finite dimensional complex characters $\\psi$ of the absolute Galois group $G_k$ of $k$.\n\nIn this setting, general considerations led Deligne and Gross to the expectation that for any finite dimensional character $\\chi$ of $G_k$  over a number field $E$ the order of vanishing ${\\rm ord}_{z=1}L(A,\\sigma\\circ\\chi,z)$ at $z=1$ of $L(A,\\sigma\\circ\\chi,z)$ should be independent of the choice of an embedding $\\sigma: E \\to \\CC$. This prediction in turn led them naturally to the conjecture that for each complex character $\\psi$ one should have\n\\begin{equation}\\label{dg equality} {\\rm ord}_{z=1}L(A,\\psi,z) = {\\rm dim}_\\CC\\bigl( \\Hom_{\\CC[\\Gal(F\/k)]}(V_\\psi,\\CC\\otimes_\\ZZ A^t(F))\\bigr)\\end{equation}\n(cf. \\cite[p.127]{rohrlich}). Here $F$ is any finite Galois extension of $k$ such that $\\psi$ factors through the projection $G_k \\to \\Gal(F\/k)$ and $V_\\psi$ is any  $\\CC[\\Gal(F\/k)]$-module of character $\\psi$ (see also Conjecture \\ref{conj:ebsd}(ii) below).\n\nThis prediction is a natural generalization of ${\\rm BSD}(A_k)$(i) and is known to have important, and explicit, consequences for ${\\rm ord}_{z=1}L(A,\\psi,z)$ (see, for example, the recent article \\cite{bisatt} of Bisatt and Dokchitser).\n\nIn addition, for rational elliptic curves $A$ and characters $\\psi$ for which $L(A,\\psi,1)$ does not vanish, there is by now strong evidence for the conjecture of Deligne and Gross.\n\nSuch evidence has been obtained by Bertolini and Darmon \\cite{BD} in the setting of ring-class characters of imaginary quadratic fields, by Kato \\cite{kato} in the setting of linear characters of $\\QQ$ (in this regard see also Rubin \\cite[\\S8]{rubin}), by Bertolini, Darmon and Rotger \\cite{bdr} for odd, irreducible two-dimensional\nArtin representations of $\\QQ$ and by Darmon and Rotger \\cite{dr} for certain self-dual Artin representations of $\\QQ$ of dimension at most four.\n (We also recall in this context that, in the setting of \\cite{bdr}, recent work of Kings, Loeffler and Zerbes \\cite{klz} proves the finiteness of components of the $p$-primary part of the Tate-Shafarevich group of $A$ over $F$ for a large set of primes $p$.)\n\nWrite $\\mathcal{O}_\\psi$ for the ring of integers of the number field generated by the values of $\\psi$. Then, as a refinement of the conjectural equality (\\ref{dg equality}), and a natural analogue of ${\\rm BSD}(A_k)$(ii) relative to $\\psi$, it would be of interest to understand a precise conjectural formula in terms of suitable `$\\psi$-components' of the standard algebraic invariants of $A$ for the fractional $\\mathcal{O}_\\psi$-ideal that is generated by the product of the leading coefficient of $L(A,\\psi,z)$ at $z=1$ and an appropriate combination of `$\\psi$-isotypic' periods and regulators.\n\nSuch a formula might also reasonably be expected to lead to concrete predictions concerning the behaviour of natural arithmetic invariants attached to the abelian variety.\n\nFor example, in the recent article of Dokchitser, Evans and Wiersema \\cite{vdrehw}, inexplicit versions of such a formula have been shown to lead, under suitable hypotheses, to predictions concerning the non-triviality of Tate-Shafarevich groups and the existence of points of infinite order on $A$ over extension fields of $k$.\n\nHowever, the formulation of an explicit such conjecture has hitherto been straightforward only if one avoids the $p$-primary support of such fractional ideals for primes $p$ that divide the degree of the extension of $k$ that corresponds to the kernel of $\\psi$.\n\nIn addition, such a conjectural formula would not itself take account of any connections that might exist between the leading coefficients of $L(A,\\psi,z)$ for characters $\\psi$ that are not in the same orbit under the action of $G_\\QQ$.\n\nIn this direction, Mazur and Tate \\cite{mt} have in the special case that $k =\\QQ$, $A$ is an elliptic curve and $\\psi$ is linear predicted an explicit family of such congruence relations that refine ${\\rm BSD}(A_k)$(ii).\n\nThese congruences rely heavily on an explicit formula in terms of modular symbols for the values $L(A,\\psi,1)$ for certain classes of tamely ramified Dirichlet characters $\\psi$ that Mazur had obtained in \\cite{mazur79}. They are expressed in terms of the discriminants of integral group-ring valued pairings constructed by using the geometrical theory of bi-extensions and are closely linked to earlier work of Mazur, Tate and Teitelbaum in \\cite{mtt} regarding the formulation of $p$-adic analogues of ${\\rm BSD}(A_k)$(ii).\n\nThe conjecture of Mazur and Tate has in turn motivated much subsequent work and the formulation of several new conjectures involving the values $L(A,\\psi,1)$.\n\nSuch conjectures include the congruence relations that are formulated by Bertolini and Darmon in \\cite{bert} and \\cite{bert2} and involve a natural notion of `derived height pairings' and links to the Galois structure of Selmer modules that are predicted by Kurihara in \\cite{kuri}.\n\nIt has, however, proved to be much more difficult to formulate explicit refinements of ${\\rm BSD}(A_k)$(ii) that involve congruence relations between the values of derivatives of Hasse-Weil-Artin $L$-series.\n\nIn this direction, Darmon \\cite{darmon} has used the theory of Heegner points to formulate an analogue of the Mazur-Tate congruence conjecture for the first derivatives of Hasse-Weil-Artin $L$-series that arise from rational elliptic curves and ring class characters of imaginary quadratic fields.\n\nHowever, aside from this example, the only other such explicit study that we are aware of is due to Kisilevsky and Fearnley who in \\cite{kisilevsky} and \\cite{kisilevsky2} formulated, and studied numerically, conjectures for the `algebraic parts' of the leading coefficients of Hasse-Weil-Artin $L$-series that arise from rational elliptic curves and certain restricted families of Dirichlet characters.\n\n\\subsubsection{}In a more general setting, the formulation by Bloch and Kato \\cite{bk} of the `Tamagawa number conjecture' for the motive $h^1(A_k)(1)$ offers a different approach to the formulation of ${\\rm BSD}(A_k)$.\n\nIn particular, the subsequent re-working of this conjecture by Fontaine and Perrin-Riou in \\cite{fpr}, and its `equivariant' extension to motives with coefficients, as described by Flach and the first author in \\cite{bufl99}, in principle provides a systematic means of studying refined versions of ${\\rm BSD}(A_k)$.\n\nIn this setting it is known, for example, that the conjectures formulated by Gross in \\cite{G-BSD} are equivalent to the equivariant Tamagawa number conjecture for the motive $h^1(A_k)(1)$ with respect to the coefficient ring $\\mathcal{O}$ (cf. \\cite[\\S4.3, Rem. 10]{bufl99}).\n\nTo study Hasse-Weil-Artin $L$-series it is convenient to fix a finite Galois extension $F$ of $k$ of group $G$.\n\nThen the equivariant Tamagawa number conjecture for $h^1(A_{F})(1)$ with respect to the integral group ring $\\ZZ[G]$ is formulated as an equality of the form\n\\begin{equation}\\label{etnc eq} \\delta(L^\\ast(A_{F\/k},1)) = \\chi(h^1(A_{F})(1),\\ZZ[G]).\\end{equation}\n\\noindent{}Here $\\delta$ is a canonical homomorphism from the unit group $\\zeta(\\br [G])^\\times$ of the centre of $\\RR[G]$ to the relative algebraic $K$-group of the ring extension $\\bz [G] \\subseteq \\br\n[G]$ and $L^\\ast(A_{F\/k},1)$ is an element of $\\zeta(\\RR[G])^\\times$ that is defined using the leading coefficients $L^\\ast(A,\\psi,1)$ for each irreducible complex character $\\psi$ of $G$. Also, $\\chi(h^1(A_{F})(1),\\ZZ[G])$ is an adelic Euler characteristic that is constructed by\ncombining virtual objects (in the sense of Deligne) for each prime $p$ of the compactly supported \\'etale cohomology of the $p$-adic Tate modules of $A$ together with the Neron-Tate height pairing and period isomorphisms for $A$ over $F$, as well as an analysis of the finite support cohomology groups introduced by Bloch and Kato.\n\nThe equality (\\ref{etnc eq}) is known to constitute a strong and simultaneous refinement of the conjectures ${\\rm BSD}(A_L)$ as $L$ ranges over the  intermediate fields of $F\/k$.\n\nHowever, the rather technical, and inexplicit, nature of this equality means that it has proved to be very difficult to interpret in a concrete way, let alone to verify either theoretically or numerically.\n\nFor example, in the technically most demanding case in which $A$ has strictly positive rank over $F$ it has still only been verified numerically in a small number of cases by Navilarekallu in \\cite{tejaswi}, by Bley in \\cite{Bley1} and \\cite{Bley2}, by Bley and the second author in \\cite{bleymc} and by Wuthrich and the present authors in \\cite{bmw}.\n\nIn addition, the only theoretical evidence for the conjecture in this setting is its verification for certain restricted dihedral families of the form $F\/\\QQ$ where $F$ is an unramified abelian extension of an imaginary quadratic field (this is the main result of \\cite{bmw} and relies on the theorem of Gross and Zagier). In particular, the restriction on ramification that the latter result imposes on $F\/k$ means that many of the more subtle aspects of the conjecture are avoided.\n\nTo proceed we note that the conjectural equality (\\ref{etnc eq}) naturally decomposes into `components', one for each rational prime $p$, in a way that will be made precise in Appendix \\ref{consistency section}, and that each such $p$-component (which for convenience we refer to as `(\\ref{etnc eq})$_p$' in the remainder of this introduction) is itself of some interest.\n\nFor example, if $A$ has good ordinary reduction at $p$, then the compatibility result proved by Venjakob and the first named author in~\\cite[Th. 8.4]{BV2} shows, modulo the assumed non-degeneracy of classical $p$-adic height pairings, that the equality (\\ref{etnc eq})$_p$ is a consequence of the main conjecture of\nnon-commutative Iwasawa theory for $A$, as formulated by Coates et al in \\cite{cfksv} with respect to any compact $p$-adic Lie extension of $k$ that contains the cyclotomic $\\ZZ_p$-extension of $F$.\n\nThis means that the study of (\\ref{etnc eq}), and of its more explicit consequences, is directly relevant to attempts to properly understand the content of the main conjecture of non-commutative Iwasawa theory. It also shows that the $p$-adic congruence relations that are proved numerically by Dokchitser and Dokchitser in \\cite{dokchitsers} are related to the equality (\\ref{etnc eq}).\n\nTo study congruences in a more general setting we fix an embedding of $\\RR$ into the completion $\\CC_p$ of the algebraic closure of $\\QQ_p$. Then the long exact sequence of relative $K$-theory implies that the equality (\\ref{etnc eq})$_p$ determines the image of $L^\\ast(A_{F\/k},1)$ in $\\zeta(\\CC_p[G])^\\times$ modulo the image under the natural reduced norm map of the Whitehead group $K_1(\\ZZ_p[G])$ of $\\ZZ_p[G]$.\n\nIn view of the explicit description of the latter image that is obtained by Kakde in \\cite{kakde} or, equivalently, by the methods of Ritter and Weiss in \\cite{rw}, this means that (\\ref{etnc eq}) is essentially equivalent to an adelic family of (albeit inexplicit) congruence relations between the leading coefficients $L^\\ast(A,\\psi, 1)$, suitably normalised by a product of explicit equivariant regulators and periods, as $\\psi$ varies over the set of irreducible complex characters of $G$.\n\nThis is also the reason why the study of congruence relations between suitably normalised derivatives of Hasse-Weil-Artin $L$-series should be related to the construction of `$p$-adic $L$-functions' in the setting of non-commutative Iwasawa theory.\n\n\\subsubsection{}The main aim of the present article is then to develop general techniques that will allow one to understand the above congruence relations in a more explicit way, and in a much wider setting, than has previously been possible.\n\nIn this way we are led to the formulation (in Conjecture \\ref{conj:ebsd}) of a seemingly definitive refinement of the Birch and Swinnerton-Dyer formula (\\ref{bsd equality}) in the setting of Hasse-Weil-Artin $L$-series. We then derive a range of concrete consequences of this conjecture that are amenable to explicit investigation, either theoretically or numerically, in cases that (for the first time) involve a thoroughgoing mixture of difficult archimedean considerations that are related to refinements of the conjectural equality (\\ref{dg equality}) of Deligne and Gross, and of delicate $p$-adic congruence relations that are related to aspects of non-commutative Iwasawa theory.\n\nIn particular, we shall show that this family of predictions both refines and extends the explicit refinements of (\\ref{dg equality}) that were recalled in \\S\\ref{history}. It also gives insight into the more subtle aspects of the conjectural equality (\\ref{etnc eq}), and hence (via the results of \\cite{BV2}) of the main conjecture of non-commutative Iwasawa theory, that go well beyond the sort of concrete congruence conjectures that have been considered previously in connection to the central conjectures of either \\cite{bufl99} or \\cite{cfksv}.\n\nWe also believe that some of the general results obtained here can help contribute towards establishing a proper framework for the subsequent investigation of these important questions.\n\n\n\n\\subsection{The main contents}\n\n\\subsubsection{}\n\n\n\n\n\n\n\nAs a key part of our approach, we shall first associate two natural notions of Selmer complex to the $p$-adic Tate module of an abelian variety.\n\n The `classical Selmer complex' that we define in \\S\\ref{selmer section} is closely related to the `finite support cohomology' that was introduced by Bloch and Kato in \\cite{bk} and, as a result, its cohomology can be explicitly described in terms of Mordell-Weil groups and Selmer groups.\n\n Nevertheless, this complex is not well-suited to certain $K$-theoretical calculations since it is not always `perfect' over the relevant $p$-adic group ring.\n\n For this reason we shall in \\S\\ref{selmer section} also associate a notion of `Nekov\\'a\\v r-Selmer complex' to certain choices of $p$-adic submodules of the groups of semi-local points.\n\n This construction is motivated by the general approach of Nekov\\'a\\v r in \\cite{nek} and gives a complex that is always perfect and has cohomology that can be described in terms of the Selmer modules studied by Mazur and Rubin in \\cite{MRkoly}.\n\n Such Nekov\\'a\\v r-Selmer complexes will then play an important role in several subsequent $K$-theoretical computations.\n\nIn \\S\\ref{selmer section} we also explain how a suitably compatible family over all primes $p$ of $p$-adic modules of semi-local points, or a `perfect Selmer structure' for $A$ and $F\/k$ as we shall refer to it, gives rise to a canonical perfect complex of $G$-modules.\n\nWe shall then show that such structures naturally arise from a choice of global differentials and compute the cohomology groups of the associated Selmer complexes.\n\nIn \\S\\ref{ref bsd section} we formulate the Birch and Swinnerton-Dyer Conjecture for the variety $A$ and Galois extension $F$ of $k$, or `${\\rm BSD}(A_{F\/k})$' as we shall abbreviate it.\n\nUnder the assumed validity of an appropriate case of the Generalized Riemann Hypothesis, we shall first associate a $K_1$-valued leading coefficient element to the data $A$ and $F\/k$.\n\nAfter fixing a suitable choice of global differentials we can also associate a $K_1$-valued `period' element to $A$ and $F\/k$.\n\nThe central conjecture of this article then asserts that the image under the natural connecting homomorphism of the quotient of these $K_1$-valued invariants is equal to the Euler characteristic in a relative $K$-group of a pair comprising a Nekov\\'a\\v r-Selmer complex constructed from the given set of differentials and the classical N\\'eron-Tate height pairing for $A$ over $F$.\n\nThis conjectural equality also involves a small number of `Fontaine-Messing' correction terms that we use to\ncompensate for the choice of a finite set of places of $k$ that is necessary to state ${\\rm BSD}(A_{F\/k})$.\n\n\nIt can be directly shown that this conjecture recovers ${\\rm BSD}(A_{k})$ in the case that $F=k$, is consistent in several key respects and has good functorial properties under change of Galois extension $F\/k$.\n\nThe conjecture can also be interpreted as a natural analogue of the `refined Birch and Swinnerton-Dyer Conjecture' for abelian varieties over global function fields that was recently formulated, and in some important cases proved, by Kakde, Kim and the first author in \\cite{bkk}.\n\nIn \\S\\ref{k theory period sect} and \\S\\ref{local points section} we shall then concentrate on proving several technical results that will subsequently help us to derive explicit consequences from the assumed validity of ${\\rm BSD}(A_{F\/k})$.\n\nIn \\S\\ref{k theory period sect} these results include establishing the precise link between $K_1$-valued periods, classical periods, Galois resolvents and suitably modified Galois-Gauss sums.\n\n\nIn \\S\\ref{local points section} we shall prepare for subsequent $K$-theoretical computations with classical Selmer complexes by studying the cohomological-triviality of local points on ordinary abelian varieties.\n\nIn particular, we shall use these results to define a natural $K$-theoretical invariant of the twist matrix of such a variety.\n\n We shall then give a partial computation of these invariants and also explain how, in the case of elliptic curves, the (assumed) compatibility under unramified twist of suitable cases of the local epsilon constant conjecture (as formulated in its most general form by Fukaya and Kato in \\cite{fukaya-kato}) leads to an explicit description of the invariants.\n\nIn \\S\\ref{tmc} we shall impose several mild hypotheses on both the reduction types of $A$ and the ramification invariants of $F\/k$ that together ensure that the classical Selmer complex defined in \\S\\ref{selmer section} is perfect over the relevant $p$-adic group ring.\n\nWorking under these hypotheses, we shall then combine the results of \\S\\ref{k theory period sect} and \\S\\ref{local points section} together with a significant strengthening of the main computations that are made by Wuthrich and the present authors in \\cite{bmw} to derive a more explicit interpretation of ${\\rm BSD}(A_{F\/k})$.\n\nThese results are in many respects the technical heart of this article and rely heavily on the subtle, and still for the most part conjectural, arithmetic properties of wildly ramified Galois-Gauss sums.\n\nThe $K$-theoretical computations in \\S\\ref{tmc} also constitute a natural equivariant refinement and generalisation of several earlier computations in this area including those that are made by Venjakob in~\\cite[\\S3.1]{venjakob}, by the first author in \\cite{ltav}, by Bley in~\\cite{Bley1}, by Kings in~\\cite[Lecture 3]{kings} and by the second author in \\cite{dmc}.\n\nIn \\S\\ref{ecgs} we shall discuss concrete consequences of ${\\rm BSD}(A_{F\/k})$ concerning both the explicit Galois structure of Selmer complexes and modules and the formulation of precise refinements of the Deligne-Gross Conjecture.\n\nIn particular, in this section we shall address a problem explicitly raised by Dokchitser, Evans and Wiersema in \\cite{vdrehw} (see, in particular Remark \\ref{evans}).\n\nIn \\S\\ref{congruence sec} and \\S\\ref{mrsconjecturesection} we shall then specialise to consider abelian extensions $F\/k$ and combine our approach with general techniques recently developed by Sano, Tsoi and the first author in \\cite{bst} in order to derive from ${\\rm BSD}(A_{F\/k})$ several explicit congruence relations between the suitably normalized derivatives of Hasse-Weil-Artin $L$-series.\n\nIn \\S\\ref{comparison section} we then prove that the pairing constructed by Mazur and Tate in \\cite{mt} using the theory of bi-extensions coincides with the inverse of a canonical `Nekov\\'a\\v r height pairing' that we define by using Bockstein homomorphisms arising naturally from Galois descent considerations.\n\nThis comparison result relies, in part, on earlier results of Bertolini and Darmon \\cite{bert2} and of Tan \\cite{kst} and is, we believe, of some independent interest.\n\nThe relations that are discussed in \\S\\ref{congruence sec} and \\S\\ref{mrsconjecturesection} often take a very explicit form (see, for example, the discussion in \\S\\ref{explicit examples intro} below) and, when combined with the results of \\S\\ref{comparison section}, can be seen to extend and refine the earlier conjectures of Mazur and Tate \\cite{mt} and Darmon \\cite{darmon0} amongst others.\n\nThis approach also shows that for certain cyclic and dihedral extensions $F\/k$ the key formula that is predicted by ${\\rm BSD}(A_{F\/k})$  is equivalent to the validity of a family of explicit congruence relations that simultaneously involve both the N\\'eron-Tate and Mazur-Tate height pairings.\n\nIn this way we shall for the first time render refined versions of the Birch and Swinnerton-Dyer Conjecture accessible to numerical verification in cases in which they involve an intricate mixture of both archimedean phenomenon and delicate $p$-adic congruences.\n\nIn particular, in \\S\\ref{mrsconjecturesection} we give details of several such numerical verifications of the `$p$-component' of ${\\rm BSD}(A_{F\/k})$ for primes $p$ that divide the degree of $F\/k$ that Werner Bley has been able to perform by using this approach (see, in particular, Remark \\ref{bleyexamples rem} and Examples \\ref{bleyexamples}).\n\nIn \\S\\ref{mod sect} and \\S\\ref{HHP} we shall then specialise to consider applications of our general approach in two classical settings.\n\nFirstly, in \\S\\ref{mod sect} we consider rational elliptic curves over fields that are both abelian and tamely ramified over $\\QQ$. In this case\nwe can use the theory of modular symbols to give an explicit reinterpretation of ${\\rm BSD}(A_{F\/\\QQ})$ and thereby describe precise conditions under which the conjecture is valid.\n\nAs a concrete application of this result we then use it to deduce from Kato's theorem \\cite{kato} that for every natural number $n$ there are infinitely many primes $p$ and, for each such $p$, infinitely many abelian extensions $F\/\\QQ$ for which the $p$-component of ${\\rm BSD}(A_{F\/\\QQ})$ is valid whilst the degree and discriminant of $F\/\\QQ$ are each divisible by at least $n$ distinct primes and the Sylow $p$-subgroup of $\\Gal(F\/\\QQ)$ has exponent at least $p^n$ and rank at least $n$. This result is a considerable strengthening of the main result of Bley in \\cite{Bley3}.\n\nThen in \\S\\ref{HHP} we consider abelian extensions of imaginary quadratic fields and elliptic curves that satisfy the Heegner hypothesis.\n\nThe main result of this section is a significant extension of the main result of Wuthrich and the present authors in \\cite{bmw} and relies on Zhang's generalization of the theorem of Gross and Zagier relating first derivatives of Hasse-Weil-Artin $L$-series to the heights of Heegner points.\n\nIn this section we shall also point out an inconsistency in the formulation of a conjecture of Bradshaw and Stein in \\cite{BS} regarding Zhang's formula and offer a possible correction.\n\n\nThe article also contains two appendices. In Appendix \\ref{consistency section} we use techniques developed by Wuthrich and the present authors in \\cite{bmw} to explain the precise link between our central conjecture ${\\rm BSD}(A_{F\/k})$ and the conjectural equality (\\ref{etnc eq}).\n\nThis technical result may perhaps be of interest in its own right but also allows us to deduce from the general theory of equivariant Tamagawa numbers that our formulation of ${\\rm BSD}(A_{F\/k})$ is consistent in several key respects.\n\nFinally, in Appendix \\ref{exp rep section} we make explicit certain standard constructions in homological algebra relating to Poitou-Tate duality and also describe a general construction of algebraic height pairings from Bockstein homomorphisms.\n\nThe results of Appendix \\ref{exp rep section} are for the most part routine but nevertheless play an important role in the arguments that we use to compare height pairings in \\S\\ref{comparison section}.\n\n\\subsubsection{}\\label{explicit examples intro}To end the introduction we shall give some concrete examples of the sort of congruence predictions that result from our approach (all taken from the more general material given in \\S\\ref{congruence sec} and \\S\\ref{mrsconjecturesection}).\n\nTo do this we fix a finite abelian extension of number fields $F\/k$ of group $G$ and an elliptic curve $A$ over $k$.\n\nWe also fix an odd prime $p$ that does not divide the order of the torsion subgroup of $A(F)$ and an isomorphism of fields $\\CC\\cong \\CC_p$ (that we do not explicitly indicate in the sequel). We write $\\widehat{G}$ for the set of irreducible complex characters of $G$.\n\nThe first prediction concerns the values at $z=1$ of Hasse-Weil-Artin $L$-series. To state it we set $F_p := \\QQ_p\\otimes_\\QQ F$ and write ${\\rm log}_{A,p}$ for the formal group logarithm of $A$ over $\\QQ_p\\otimes_\\QQ k$ and $\\Sigma(k)$ for the set of embeddings $k \\to \\CC$.\n\nFor each subset $x_\\bullet = \\{x_{\\sigma}: \\sigma \\in \\Sigma(k)\\}$ of $A(F_p)$ and each character $\\psi$ in $\\widehat{G}$ we then define a `$p$-adic logarithmic resolvent' by setting\n\n\\begin{equation}\\label{log resol abelian} \\mathcal{LR}_\\psi(x_\\bullet) := {\\rm det}\\left(\\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}({\\rm log}_{A,p}(x_{\\sigma'})))\\cdot \\psi(g)\\bigr)_{\\sigma,\\sigma' \\in \\Sigma(k)}\\right),\\end{equation}\nwhere we fix an ordering of $\\Sigma(k)$ and an extension $\\hat \\sigma$ to $F$ of each $\\sigma$ in $\\Sigma(k)$.\n\nNow if $S$ is any finite set of places of $k$ that contains all archimedean places, all that ramify in $F$, all at which $A$ has bad reduction and all above $p$, and $L_{S}(A,\\check\\psi,z)$ is the $S$-truncated $L$-series attached to $A$ and the contragredient $\\check\\psi$ of $\\psi$, then our methods predict that for any $x_\\bullet$ the sum\n\n\\begin{equation}\\label{first predict} \\sum_{\\psi \\in \\widehat{G}}\\frac{L_{S}(A,\\check\\psi,1)\\cdot \\mathcal{LR}_\\psi(x_\\bullet)}{\\Omega^\\psi_A\\cdot w_\\psi}\\cdot e_\\psi \\end{equation}\nbelongs to $\\ZZ_p[G]$ and annihilates the $p$-primary part $\\sha(A_{F})[p^\\infty]$ of the Tate-Shafarevich group of $A$ over $F$. Here $e_\\psi$ denotes the idempotent $|G|^{-1}\\sum_{g \\in G}\\check\\psi(g)g$ of $\\CC[G]$ and the periods $\\Omega^\\psi_A$ and Artin root numbers $w_\\psi$ are as explicitly defined in \\S\\ref{k theory period sect2}.\n\nIn particular, if one finds  for some choice of $x_\\bullet$ that the sum in (\\ref{first predict}) is a unit of $\\ZZ_p[G]$, then this implies that $\\sha(A_F)[p^\\infty]$ should be trivial.\n\nMore generally, the fact that each sum in (\\ref{first predict}) should belong to $\\ZZ_p[G]$ implies that for every $g$ in $G$, and every set of local points $x_\\bullet$, there should be a congruence\n\\[ \\sum_{\\psi \\in \\widehat{G}}\\psi(g)\\frac{L_{S}(A,\\check\\psi,1)\\cdot \\mathcal{LR}_\\psi(x_\\bullet)}{\\Omega_A^\\psi\\cdot w_\\psi}\n \\equiv 0 \\,\\,\\,({\\rm mod}\\,\\, |G|\\cdot \\ZZ_p).\\]\n\nIn concrete examples these congruences are strong restrictions on the values $L_{S}(A,\\check\\psi,1)$ that can be investigated  numerically but cannot be deduced by solely considering Birch and Swinnerton-Dyer type formulas for individual Hasse-Weil-Artin $L$-series\n\n\nIn general, our analysis leads to a range of congruence predictions that are both finer than the above and also involve the values at $z=1$ of higher derivatives of Hasse-Weil-Artin $L$-series, suitably normalised by a product of explicit regulators and periods.\n\nTo give an example of this sort of prediction, we shall focus on the simple case that $F\/k$ is cyclic of degree $p$ (although entirely similar predictions can be made in the setting of cyclic extensions of arbitrary $p$-power order and also for certain dihedral families of extensions).\n\nIn this case, under certain natural, and very mild, hypotheses on $A$ and $F\/k$ relative to $p$ there exist non-negative integers $m_0$ and $m_1$ with the property that the pro-$p$ completion $A(F)_p$ of $A(F)$ is isomorphic as a $\\ZZ_p[G]$-module to a direct sum of $m_0$ copies of $\\ZZ_p$ and $m_1$ copies of $\\ZZ_p[G]$.\n\nIn particular, if we further assume $m_0=2$ and $m_1= 1$, then we may fix points $P_{0}^1$ and $P_{0}^2$ in $A(k)$ and $P_1$ in $A(F)$ such that $A(F)_p$ is the direct sum of the $\\ZZ_p[G]$-modules generated by $P_{0}^1,$ $P_{0}^2$ and $P_1.$ (In Example \\ref{wuthrich example} the reader will find explicit examples of such pairs $A$ and $F\/k$ for the prime $p=3$.)\n\n\nThen, writing $L^{(1)}_{S}(A,\\check\\psi,1)$ for the value at $z=1$ of the first derivative of $L_{S}(A,\\check\\psi,z)$, our methods predict that, under mild additional hypotheses, there should exist an element $x$ of $\\ZZ_p[G]$ that annihilates $\\sha(A_{F})[p^\\infty]$ and is such that\n\\begin{equation}\\label{second predict} \\sum_{\\psi \\in \\widehat{G}}\\frac{L^{(1)}_{S}(A,\\check\\psi,1)\\cdot\\tau^*(\\QQ,\\psi)}{\\Omega^\\psi_A\\cdot i^{r_2}}\\cdot e_\\psi  = x\\cdot \\sum_{g\\in G}\\langle g(P_1),P_1\\rangle_{A_F}\\cdot g^{-1},\\end{equation}\nwhere $\\langle -,-\\rangle_{A_F}$ denotes the N\\'eron-Tate height pairing of $A$ relative to $F$, $\\tau^*(\\QQ,\\psi)$ is the (modified, global) Galois-Gauss sum of the character of $G_\\QQ$ that is obtained by inducing $\\psi$ and $r_2$ is the number of complex places of $k$.\n\nTo be more explicit, we write $R_A$ for the determinant of the N\\'eron-Tate regulator matrix of $A$ over $k$ with respect to the ordered $\\QQ$-basis $\\{P_{0}^1,P_{0}^2,\\sum_{g\\in G}g(P_1)\\}$ of $\\QQ\\cdot A(k)$ and for each non-trivial $\\psi$ in $\\widehat{G}$ we define a non-zero complex number by setting\n\\[ h^\\psi(P_1):=\\sum_{g\\in G}\\langle g(P_1),P_1\\rangle_{A_F}\\cdot\\psi(g)^{-1}.\\]\n\nWe finally write $S_{\\rm r}$ for the set of places of $k$ that ramify in $F$, $d_k$ for the discriminant of $k$ and $I_p(G)$ for the augmentation ideal of $\\ZZ_p[G]$. Then, under mild hypotheses, our methods predict that there should be containments\n\\[\n\\sum_{\\psi\\neq {\\bf 1}_G}\\frac{L_{S_{\\rm r}}^{(1)}(A,\\check\\psi,1)\\cdot\\tau^*(\\QQ,\\psi)}{\\Omega^\\psi_{A}\\cdot i^{r_2} \\cdot h^\\psi(P_1)}\\cdot e_\\psi  \\in I_p(G)^2\\,\\,\\text{ and }\\,\\, \\frac{L_{S_{\\rm r}}^{(3)}(A,1)\\sqrt{|d_k|}}{\\Omega_{A}\\cdot R_{A}}\\in \\ZZ_p,\\]\nand a congruence modulo $I_p(G)^3$ of the form\n\\begin{equation}\\label{examplecongruent}\n\\sum_{\\psi\\neq {\\bf 1}_G}\\frac{L_{S_{\\rm r}}^{(1)}(A,\\check\\psi,1)\\cdot\\tau^*(\\QQ,\\psi)}{ \\Omega^\\psi_{A}\\cdot i^{r_2} \\cdot h^\\psi(P_1)}\\cdot\\! e_\\psi\n \\equiv \\!\\frac{L_{S_{\\rm r}}^{(3)}(A,1)\\sqrt{|d_k|}}{\\Omega_{A}\\cdot R_{A}}\\cdot\\det\\left(\\begin{array}{cc}\n\\langle P_{0}^1,P_{0}^1\\rangle^{\\rm MT} & \\langle P_{0}^1,P_{0}^2\\rangle^{\\rm MT}\n\\\\\n\\langle P_{0}^2,P_{0}^1\\rangle^{\\rm MT} & \\langle P_{0}^2,P_{0}^2\\rangle^{\\rm MT}\n\\end{array}\\right)\\!.\n\\end{equation}\nHere $L_{S_{\\rm r}}^{(3)}(A,1)$ denotes the value at $z=1$ of the third derivative of $L_{S_{\\rm r}}(A,z)$ and\n\\[ \\langle\\,,\\rangle^{\\rm MT}:A(k)\\times A(k)\\to I_p(G)\/I_p(G)^2\\]\nis the canonical pairing that Mazur and Tate define in \\cite{mt0} by using the geometrical theory of bi-extensions.\n\nFurther, if $\\sha(A_F)[p^\\infty]$ is trivial, then the $p$-component of ${\\rm BSD}(A_{F\/k})$ is valid if and only if (\\ref{examplecongruent}) holds and, in addition, the $p$-component of the Birch and Swinnerton-Dyer Conjecture is true for $A$ over both $k$ and $F$.\n\n\n\nWe remark that even in the simplest possible case that $k = \\QQ$ and $p=3$, these predictions strongly refine those made by Kisilevsky and Fearnley in \\cite{kisilevsky} and cannot be deduced by simply considering leading term formulas for individual Hasse-Weil-Artin $L$-series.\n\n\n\n\n\n\n\\subsection{General notation} For the reader's convenience we give details here of some of the general notation and terminology that will be used throughout the article.\n\n\\subsubsection{}We write $|X|$ for the cardinality of a finite set $X$.\n\nFor an abelian group $M$ we write $M_{\\rm tor}$ for its torsion subgroup and $M_{\\rm tf}$ for the quotient of $M$ by $M_{\\rm tor}$.\n\nFor a prime $p$ and natural number $n$ we write $M[p^n]$ for the subgroup $\\{m \\in M: p^nm =0\\}$ of the Sylow $p$-subgroup $M[p^{\\infty}]$ of $M_{\\rm tor}$.\n\nWe set $M_p := \\ZZ_p\\otimes_\\ZZ M$, write $M^\\wedge_p$ for the  pro-$p$-completion $\\varprojlim_n M\/p^n M$ of $M$ and denote the Pontryagin dual $\\Hom(M,\\QQ\/\\ZZ)$ of $M$ by $M^\\vee$.\n\nIf $M$ is finite of exponent dividing $p^m$, then $M^\\vee$ identifies with $\\Hom_{\\ZZ_p}(M,\\QQ_p\/\\ZZ_p)$ and we shall (without explicit comment) use the canonical identification $\\QQ_p\/\\ZZ_p=\\varinjlim_n \\ZZ\/p^n\\ZZ$ to identify elements of $M^\\vee$ with their canonical image in the linear dual $\\Hom_{\\ZZ\/p^m\\ZZ}(M,\\ZZ\/p^m\\ZZ)$.\n\nIf $M$ is finitely generated, then for a field extension $E$ of $\\QQ$ we shall often abbreviate $E\\otimes_\\ZZ M$ to $E\\cdot M$.\n\n\nIf $M$ is a $\\Gamma$-module for some group $\\Gamma$, then we always endow $M^\\vee$ with the natural contragredient action of $\\Gamma$.\n\n\nWe recall that if $\\Gamma$ is finite, then a $\\Gamma$-module $M$ is said to be `cohomologically-trivial' if for all subgroups $\\Delta$ of $\\Gamma$ and all integers $i$ the Tate cohomology group $\\hat H^i(\\Delta,M)$ vanishes.\n\n\\subsubsection{}For any ring $R$ we write $R^\\times$ for its multiplicative group and $\\zeta(R)$ for its centre.\n\nUnless otherwise specified we regard all $R$-modules as left $R$-modules. We write ${\\rm Mod}(R)$ for the abelian category of $R$-modules and ${\\rm Mod}^{\\rm fin}(R)$ for the abelian subcategory of ${\\rm Mod}(R)$ comprising all $R$-modules that are finite.\n\nWe write $D(R)$ for the derived category of complexes of $R$-modules. If $R$ is noetherian, then we write $D^{\\rm perf}(R)$ for the full triangulated subcategory of $D(R)$ comprising complexes that are `perfect' (that is, isomorphic in $D(R)$ to a bounded complex of finitely generated projective $R$-modules).\n\nFor a natural number $n$ we write $\\tau_{\\le n}$ for the exact truncation functor on $D(R)$ with the property that for each object $C$ in $D(R)$ and each integer $i$ one has\n\\[ H^i(\\tau_{\\le n}(C)) = \\begin{cases} H^i(C), &\\text{if $i \\le n$}\\\\\n                                          0, &\\text{otherwise.}\\end{cases}\\]\n\n\\subsubsection{}For a Galois extension of number fields $L\/K$ we set $G_{L\/K} := \\Gal(L\/K)$. We also fix an algebraic closure $K^c$ of $K$ and set $G_K := G_{K^c\/K}$.\n\nFor each non-archimedean place $v$ of a number field we write $\\kappa_v$ for its residue field  and denote its  absolute norm $|\\kappa_v|$ by ${\\rm N}v$.\n\nWe write the dual of an abelian variety $A$ as $A^t$ and usually abbreviate its dimension ${\\rm dim}(A)$ to $d$.\n\nIf $A$ is defined over a number field $k$, then for each extension $F$ of $k$ we write the Tate-Shafarevich group of $A$ over $F$ as $\\sha(A_F)$.\n\nWe shall also use the following notation regarding sets of places of $k$.\n\n\\begin{itemize}\n\\item[-] $S_k^\\RR$ is the set of real archimedean places of $k$;\n\\item[-] $S_k^\\CC$ is the set of complex archimedean places of $k$;\n\\item[-] $S_k^\\infty (= S_k^\\RR \\cup S_k^\\CC$) is the set of archimedean places of $k$;\n\\item[-] $S_k^f$ is the set of non-archimedean places of $k$;\n\\item[-] $S_k^v$ is the set of places of $k$ that extend a place $v$ of a given subfield of $k$. In particular,\n\\item[-] $S_k^p$ is the set of $p$-adic places of $k$ for a given prime number $p$;\n\\item[-] $S_k^F$ is the set of places of $k$ that ramify in a given extension $F$ of $k$;\n\\item[-] $S_k^A$ is the set of places of $k$ at which a given abelian variety $A$ has bad reduction.\n\\end{itemize}\n\n For a fixed set of places $S$ of $k$ we also write $S(F)$ for the set of places of $F$ which lie above a place in $S$.\n\n\nFor a place $v$ of $k$ we set $G_{v} := G_{k_v^c\/k_v}$. We also write $k_v^{\\rm un}$ for the maximal unramified extension of $k$ in $k_v^c$ and set $I_{v} := G_{k_v^c\/k_v^{\\rm un}}$.\n\nWe fix a place $w$ of $F$ above $v$ and a corresponding embedding $F\\to k_v^c$. We write $G_w$ and $I_w$ for the images of $G_{v}$ and $I_{v}$ under the induced homomorphism $G_{v} \\to G$. We also fix a lift $\\Phi_v$ to $G$ of the Frobenius automorphism in $G_w\/I_w$.\n\nWe write $\\Sigma(k)$ for the set of embeddings $k \\to \\CC$.\n\n\\subsection{Acknowledgements} We are very grateful to St\\'ephane Vigui\\'e for his help with aspects of the argument presented in \\S\\ref{ptduality} and to Werner Bley for providing us with the material in \\S\\ref{ell curve sect} and for pointing out a sign-error in an earlier version of the argument in \\S\\ref{comparison section}.\n\nWe are also grateful to both Werner Bley and Christian Wuthrich for their interest, helpful correspondence and tremendous generosity regarding numerical computations.\n\nIn addition, we would like to thank  Rob Evans, Masato Kurihara, Jan Nekov\\'a\\v r, Takamichi Sano, Kwok-Wing Tsoi and Stefano Vigni for helpful discussions and correspondence.\n\nFinally, it is a great pleasure to thank Dick Gross for his strong encouragement regarding this project and for several insightful remarks.\n\nThe second author acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the `Severo Ochoa Programme for Centres of Excellence in R\\&D' (SEV-2015-0554) as well as through project MTM2016-79400-P.\n\n\n\\section{Selmer complexes}\\label{selmer section}\n\nIn this section we define the Selmer complexes that play a key role in the conjecture we shall later formulate and establish some of their key properties.\n\nAt the outset we fix a finite set of places $S$ of $k$ with\n\\[ S_k^\\infty\\cup S_k^F \\cup S_k^A\\subseteq S.\\]\n\nWe also write $\\Sigma(k)$ for the set of embeddings $k \\to \\CC$ and $\\Sigma_\\sigma(F)$ for each $\\sigma$ in $\\Sigma(k)$ for the set of embeddings $F \\to \\CC$ that extend $\\sigma$.\n\nFor each $v$ in $S_k^\\infty$ we fix a corresponding embedding $\\sigma_v$ in $\\Sigma(k)$ and an embedding $\\sigma'_v$ in $\\Sigma_{\\sigma_v}(F)$.\n\nWe then write $Y_{v,F}$ for the module $\\prod_{\\Sigma_{\\sigma_v}(F)}\\ZZ$ endowed with its natural action of $G\\times G_v$ (via which $G$ and $G_v$ respectively act via pre-composition and post-composition with the embeddings in $\\Sigma_{\\sigma_v}(F)$).\n\nFor each $v$ in $S_k^\\infty$ we set\n\\[ H_v(A_{F\/k}) := H^0(k_v,Y_{v,F}\\otimes_{\\ZZ}H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)),\\]\nregarded as a $G$-module via the given action on $Y_{v,F}$. We note that this $G$-module is free of rank $2d$ if $v$ is in $S_k^\\CC$ and spans a free $\\ZZ[1\/2][G]$-module of rank $d$ if $v$ is in $S_k^\\RR$.\n\nWe then define a $G$-module by setting\n\\[ H_\\infty(A_{F\/k}) := \\bigoplus_{v \\in S_k^\\infty}H_v(A_{F\/k}).\\]\n\nFinally we write $\\Sigma_k(F)$ for the set of $k$-embeddings $F \\to k^c$ and $Y_{F\/k}$ for the module $\\prod_{\\Sigma_k(F)}\\ZZ$, endowed with its natural action of $G\\times G_k$.\n\n\n\n\n\n\\subsection{Classical Selmer complexes}\\label{p-adiccomplexes} In this section we fix a prime number $p$.\n\n\n\\subsubsection{}We first record a straightforward (and well-known) result regarding pro-$p$ completions that will be useful in the sequel.\n\nWe let $B$ denote either $A$ or its dual variety $A^t$ and write $T_p(B)$ for the $p$-adic Tate module of $B$.\n\n\\begin{lemma}\\label{v not p} For each non-archimedean place $w'$ of $F$ the following claims are valid.\n\\end{lemma}\n\\begin{itemize}\n\\item[(i)] If $w'$ is not $p$-adic then the natural Kummer map $B(F_{w'})^\\wedge_p \\to H^1(F_{w'},T_p(B))$ is bijective.\n\\item[(ii)] There exists a canonical short exact sequence\n\\[ 0 \\to H^1(\\kappa_{w'}, T_p(B)^{I_{w'}}) \\to B(F_{w'})^\\wedge_p \\to H^0(F_{w'}, H^1(I_{w'}, T_{p}(B))_{\\rm tor})\\to 0.\\]\n\\end{itemize}\n\n\\begin{proof} We note first that if $w'$ does not divide $p$, then the module $H^1(F_{w'},T_p(B))$ is finite. This follows, for example, from Tate's local Euler characteristic formula, the vanishing of $H^0(F_{w'},T_p(B))$ and the fact that local duality identifies $H^2(F_{w'},T_p(B))$ with the finite module $H^0(F_{w'},T_p(B^t)\\otimes_{\\ZZ_p}\\QQ_p\/\\ZZ_p)^\\vee$.\n\nGiven this observation, claim (i) is obtained directly upon passing to the inverse limit over $m$ in the natural Kummer theory exact sequence\n\\begin{equation}\\label{kummerseq} 0 \\to B(F_{w'})\/p^m \\to H^1(F_{w'},T_p(B)\/p^m) \\to H^1(F_{w'},B)[p^m]\\to 0.\\end{equation}\n\n\nClaim (ii) is established by Flach and the first author in \\cite[(1.38)]{bufl95} (after recalling the fact that the group denoted $H^1_f(F_{w'},T_p(B))_{\\rm BK}$ in loc. cit. is equal to the image of $B(F_{w'})^\\wedge_p$ in $H^1(F_{w'},T_p(B))$ under the injective Kummer map).\n\\end{proof}\n\n\\begin{remark}\\label{Tamagawa remark}{\\em The cardinality of each module $H^0(F_{w'}, H^1(I_{w'}, T_{p}(B))_{\\rm tor})$ that occurs in Lemma \\ref{v not p}(ii) is the maximal power of $p$ that divides the Tamagawa number of $B$ at $w'$. }\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{}If $B$ denotes either $A$ or $A^t$, then for any subfield $E$ of $k$ and any place $v$ in $S_E^f$ we obtain a $G$-module by setting\n\\[ B(F_v) := \\prod_{w' \\in S_F^v}B(F_{w'}).\\]\n\nFor later purposes we note that if $E = k$, then this module is isomorphic to the module of $G_w$-coinvariants\n$Y_{F\/k}\\otimes_{\\ZZ[G_w]} B(F_w)$ of the tensor product $Y_{F\/k}\\otimes_{\\ZZ} B(F_w)$, upon which $G$ acts only the first factor but $G_k$ acts diagonally on both.\n\nIn a similar way,  the $p$-adic Tate module of the base change of $B$ through $F\/k$ is equal to\n\\[ T_{p,F}(B) := Y_{F\/k,p}\\otimes_{\\ZZ_p}T_p(B)\\]\n(where, again, $G$ acts only on the first factor of the tensor product whilst $G_k$ acts on both). We set $V_{p,F}(B) := \\QQ_p\\cdot T_{p,F}(B)$.\n\n\nWe can now introduce a notion of Selmer complex that will play an important role in the sequel.\n\n\\begin{definition}\\label{bkdefinition}{\\em For any finite subset $\\Sigma$ of $S_k^f$ that contains each of $S_k^p, S_k^F\\cap S_k^f$ and $S_k^A$, the `classical $p$-adic Selmer complex' ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ for the data $A, F\/k$ and $\\Sigma$ is the mapping fibre of the morphism\n\\begin{equation}\\label{bkfibre}\n\\tau_{\\le 3}(R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t))) \\oplus \\left(\\bigoplus_{v\\in\\Sigma} A^t(F_v)^\\wedge_p\\right)[-1]  \\xrightarrow{(\\lambda,\\kappa)} \\bigoplus_{v \\in \\Sigma} R\\Gamma (k_v, T_{p,F}(A^t))\n\\end{equation}\nin $D(\\ZZ_p[G])$. Here $\\lambda$ is the natural diagonal localisation morphism and $\\kappa$ is induced by the Kummer theory maps\n$A^t(F_v)^\\wedge_p\\to H^1(k_v,T_{p,F}(A^t))$ (and the fact that $H^0(k_v, T_{p,F}(A^t))$ vanishes for all $v$ in $\\Sigma$).\n\n}\n\\end{definition}\n\n\\begin{remark}{\\em If $p$ is odd, then $R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t)))$ is acyclic in degrees greater than two and so the natural morphism $\\tau_{\\le 3}(R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t))) \\to R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup \\Sigma},T_{p,F}(A^t)))$ in $D(\\ZZ_p[G])$ is an isomorphism. In this case the truncation functor $\\tau_{\\le 3}$ can therefore be omitted from the above definition.}\\end{remark}\n\nThe following result shows that the Selmer complex ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ is independent, in a natural sense, of the choice of the set of places $\\Sigma$.\n\nFor this reason, in the sequel we shall usually write ${\\rm SC}_{p}(A_{F\/k})$ in place of ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$.\n\n\\begin{lemma}\\label{independenceofsigma} Let $\\Sigma$ and $\\Sigma'$ be any finite subsets of $S_k^f$ as in Definition \\ref{bkdefinition} with $\\Sigma\\subseteq\\Sigma'$. Then there is a canonical isomorphism ${\\rm SC}_{\\Sigma',p}(A_{F\/k})\\to {\\rm SC}_{\\Sigma,p}(A_{F\/k})$ in $D(\\ZZ_p[G])$.\n\\end{lemma}\n\n\\begin{proof} We recall that the compactly supported cohomology complex\n\\[ R\\Gamma_{c,\\Sigma}:=R\\Gamma_c(\\mathcal{O}_{k,S_k^\\infty\\cup\\Sigma},T_{p,F}(A^t))\\]\nis defined to be the mapping fibre of the diagonal localisation morphism\n\\begin{equation}\\label{compactloc} R\\Gamma(\\mathcal{O}_{k,S_k^\\infty\\cup\\Sigma},T_{p,F}(A^t)) \\to \\bigoplus_{v \\in S_k^\\infty\\cup\\Sigma}R\\Gamma(k_v,T_{p,F}(A^t))\\end{equation}\nin $D(\\ZZ_p[G])$.\n\nWe further recall that $R\\Gamma_{c,\\Sigma}$ is acyclic outside degrees $1, 2$ and $3$ (see, for example, \\cite[Prop. 1.6.5]{fukaya-kato}) and that\n $R\\Gamma(k_v,T_{p,F}(A^t))$ for each $v$ in $\\Sigma$ is acyclic outside degrees $1$ and $2$ and hence that the natural morphisms\n\\[ \\tau_{\\le 3}(R\\Gamma_{c,\\Sigma}) \\to R\\Gamma_{c,\\Sigma}\\,\\,\\text{ and } \\,\\,\\tau_{\\le 3}(R\\Gamma(k_v,T_{p,F}(A^t))) \\to R\\Gamma(k_v,T_{p,F}(A^t))\\]\nin $D(\\ZZ_p[G])$ are  isomorphisms.\n\nUpon comparing these facts with the definition of ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ one deduces the existence of a canonical exact triangle in $D(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{comparingtriangles}R\\Gamma_{c,\\Sigma}\\to {\\rm SC}_{\\Sigma,p}(A_{F\/k})\\to \\left(\\bigoplus_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge\\right)[-1]\\oplus \\tau_{\\le 3}(R\\Gamma_\\infty) \\to R\\Gamma_{c,\\Sigma}[1],\n\\end{equation}\nwhere we abbreviate $\\bigoplus_{ v\\in S_k^\\infty} R\\Gamma(k_v,T_{p,F}(A^t))$ to $R\\Gamma_\\infty$.\n\n\nIn addition, by the construction of \\cite[(30)]{bufl99}, there is a canonical exact triangle in $D(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{independencetriangle}\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})[-1]\\to R\\Gamma_{c,\\Sigma'}\\to R\\Gamma_{c,\\Sigma}\\to\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v}).\\end{equation}\n\nFinally we note that, since the choice of $\\Sigma$ implies that $A$ has good reduction at each place $v$ in $\\Sigma'\\setminus\\Sigma$, the module $H^0(F_{w'}, H^1(I_{w'}, T_{p}(A^t))_{\\rm tor})$ vanishes for every $w'$ in $S_F^v$.\n\nThus, since for each $v$ in $\\Sigma'\\setminus\\Sigma$ the complex $R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})$ is acyclic outside degree one, the exact sequences in Lemma \\ref{v not p}(ii) induce a canonical isomorphism\n\\begin{equation}\\label{firstrow}A^t(F_v)_p^\\wedge[-1]\\to R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})\\end{equation}\nin $D(\\ZZ_p[G])$.\n\nThese three facts combine to give a canonical commutative diagram in $D(\\ZZ_p[G])$ of the form\n\n\\begin{equation*}\\label{complexesdiag}\\xymatrix{\n\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}A^t(F_v)_p^\\wedge[-2] \\ar@{^{(}->}[d] \\ar[r]^{\\hskip-0.4truein\\sim}  &\n\\bigoplus\\limits_{ v\\in \\Sigma'\\setminus\\Sigma}R\\Gamma(\\kappa_v,T_{p,F}(A^t)^{I_v})[-1] \\ar[d] &\n\\\\\n\\bigoplus\\limits_{ v\\in \\Sigma'}A^t(F_v)_p^\\wedge[-2]\\oplus \\tau_{\\le 3}(R\\Gamma_\\infty)[-1] \\ar@{->>}[d] \\ar[r] &\nR\\Gamma_{c,\\Sigma'} \\ar[d] \\ar[r]  &\n{\\rm SC}_{\\Sigma',p}(A_{F\/k})\n\\\\\n\\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge[-2]\\oplus \\tau_{\\le 3}(R\\Gamma_\\infty)[-1]  \\ar[r] &\nR\\Gamma_{c,\\Sigma}  \\ar[r]  &\n{\\rm SC}_{\\Sigma,p}(A_{F\/k}).\n}\\end{equation*}\nHere the first row is induced by the isomorphisms (\\ref{firstrow}), the second and third rows by the triangles (\\ref{comparingtriangles}) for $\\Sigma'$ and $\\Sigma$ respectively, the first column is the obvious short exact sequence and the second column is given by the triangle (\\ref{independencetriangle}).\n\nIn particular, since all rows and columns in this diagram are exact triangles, its commutativity implies the existence of a canonical isomorphism in $D(\\ZZ_p[G])$ from\n${\\rm SC}_{\\Sigma',p}(A_{F\/k})$ to ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$, as required.\n\\end{proof}\n\nTaken together, Lemma \\ref{v not p}(ii) and Remark \\ref{Tamagawa remark} imply that if $p$ is odd and the Tamagawa numbers of $A$ over $F$ are divisible by $p$, then the complex ${\\rm SC}_{p}(A_{F\/k})$ differs slightly from the `finite support cohomology' complex $R\\Gamma_f(k,T_{p,F}(A))$ that was defined (for odd $p$) and played a key role in the article \\cite{bmw} of Wuthrich and the present authors.\n\nFor such $p$ we have preferred to use ${\\rm SC}_{p}(A_{F\/k})$ rather than $R\\Gamma_f(k,T_{p,F}(A))$ in this article since it is more amenable to certain explicit constructions that we have to make in later sections.\n\nFor the moment, we record only the following facts about ${\\rm SC}_{p}(A_{F\/k})$ that will be established in Propositions \\ref{explicitbkprop} and \\ref{explicitbkprop2} below. We write $\\Sel_p(A_{F})$ for the classical $p$-primary Selmer group of $A$ over $F$. Then ${\\rm SC}_{p}(A_{F\/k})$ is acyclic outside degrees one, two and three and, assuming the Tate-Shafarevich group $\\sha(A_F)$ of $A$ over $F$ to be finite, there are canonical identifications for each odd $p$ of the form\n\\begin{equation}\\label{bksc cohom} H^i({\\rm SC}_{p}(A_{F\/k})) = \\begin{cases} A^t(F)_p, &\\text{if $i=1$,}\\\\\n\\Sel_p(A_F)^\\vee, &\\txt{if $i=2$,}\\\\\nA(F)[p^{\\infty}]^\\vee, &\\text{if $i=3$,}\\end{cases}\\end{equation}\nwhilst for $p=2$ there is a canonical identification $H^1({\\rm SC}_{2}(A_{F\/k})) =  A^t(F)_2$ and a canonical homomorphism $\\Sel_2(A_F)^\\vee \\to H^2({\\rm SC}_{2}(A_{F\/k}))$ with finite kernel and cokernel, and the module $H^3({\\rm SC}_{2}(A_{F\/k}))$ is finite.\n\n\\begin{remark}\\label{indeptremark}{\\em A closer analysis of the argument in Lemma \\ref{independenceofsigma} shows that, with respect to the identifications (\\ref{bksc cohom}) that are established (under the hypothesis that $\\sha(A_F)$ is finite)  in Proposition \\ref{explicitbkprop} below, the isomorphism ${\\rm SC}_{\\Sigma',p}(A_{F\/k})\\to {\\rm SC}_{\\Sigma,p}(A_{F\/k})$ constructed in Lemma \\ref{independenceofsigma} induces the identity map on all degrees of cohomology.}\\end{remark}\n\n\n\n\\subsection{Nekov\\'a\\v r-Selmer complexes} In this section we again fix a prime number $p$.\n\n\\subsubsection{}Whilst the modules that occur in (\\ref{bksc cohom}) are the primary objects of interest in the theory of abelian varieties, the complex ${\\rm SC}_{p}(A_{F\/k})$ is not always well-suited to our purposes since, except in certain special cases (that will be discussed in detail in \\S\\ref{tmc}), it does not belong to $D^{\\rm perf}(\\ZZ_p[G])$.\n\nFor this reason, we find it convenient to introduce the following alternative notion of Selmer complexes.\n\nThis construction is motivated by the general approach developed by Nekov\\'a\\v r in \\cite{nek}\n\n\\begin{definition}\\label{selmerdefinition}{\\em  Fix $\\ZZ_p[G]$-submodules $X$ of $A^t(F_p)^\\wedge_p$ and $X'$ of $H_{\\infty}(A_{F\/k})_p$. Then the `Nekov\\'a\\v r-Selmer complex' ${\\rm SC}_{S}(A_{F\/k};X,X')$ of the data $(A,F,S,X,X')$ is the mapping fibre of the  morphism\n\\begin{equation}\\label{fibre morphism}\nR\\Gamma(\\mathcal{O}_{k,S\\cup S_k^p},T_{p,F}(A^t)) \\oplus X[-1] \\oplus X'[0] \\xrightarrow{(\\lambda, \\kappa_1,\\kappa_2)} \\bigoplus_{v \\in S \\cup S_k^p} R\\Gamma (k_v, T_{p,F}(A^t))\n\\end{equation}\n in $D(\\ZZ_p[G])$. Here $\\lambda$ is again the natural diagonal localisation morphism, $\\kappa_1$ is the morphism\n\n \\[ X[-1]\\rightarrow \\bigoplus_{v \\in S_k^p}R\\Gamma (k_v, T_{p,F}(A^t))\\]\n\n induced by the sum over $v$ of the local Kummer maps (and the fact each group $H^0(k_v,T_{p,F}(A^t))$ vanishes) and $\\kappa_2$ is the morphism\n\n\\[ X'[0] \\to \\bigoplus_{v \\in S_k^\\infty}R\\Gamma (k_v, T_{p,F}(A^t))\\]\nthat is induced by the canonical comparison isomorphisms\n\\begin{equation}\\label{cancompisom} Y_{v,F,p}\\otimes_{\\ZZ} H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ) \\cong Y_{F\/k,p}\\otimes_{\\ZZ_p}T_{p}(A^t)= T_{p,F}(A^t) \\end{equation}\nfor each $v$ in $S_k^\\infty$\n}\\end{definition}\n\n\n\nIn the next result we establish the basic properties of these Nekov\\'a\\v r-Selmer complexes. In this result we shall write ${\\rm Mod}^\\ast(\\ZZ_p[G])$ for the category ${\\rm Mod}(\\ZZ_p[G])$ in the case that $p$ is odd and for the quotient of ${\\rm Mod}(\\ZZ_2[G])$ by its subcategory ${\\rm Mod}^{\\rm fin}(\\ZZ_2[G])$ in the case that $p = 2$.\n\n\n\n\\begin{proposition}\\label{prop:perfect} Let $X$ be a finite index $\\ZZ_p[G]$-submodule of $A^t(F_p)^\\wedge_p$ that is cohomologically-trivial as a $G$-module.\n\nLet $X'$ be a finite index projective $\\ZZ_p[G]$-submodule of $H_\\infty(A_{F\/k})_p$, with $X' = H_\\infty(A_{F\/k})_p$ if $p$ is odd.\n\nThen the following claims are valid.\n\\begin{itemize}\n\\item[(i)] ${\\rm SC}_{S}(A_{F\/k};X,X')$ is an object of $D^{\\rm perf}(\\ZZ_p[G])$ that is acyclic outside degrees one, two and three.\n\\item[(ii)] $H^3({\\rm SC}_{S}(A_{F\/k};X,X'))$ identifies with $A(F)[p^{\\infty}]^\\vee$.\n\\item[(iii)] If $\\sha(A_F)$ is finite, then in ${\\rm Mod}^\\ast(\\ZZ_p[G])$ there exists a canonical injective homomorphism\n\\[ H^1({\\rm SC}_{S}(A_{F\/k};X,X')) \\to A^t(F)_p \\]\nthat has finite cokernel and a canonical surjective homomorphism\n\\[  H^2({\\rm SC}_{S}(A_{F\/k};X,X')) \\to {\\rm Sel}_p(A_{F})^\\vee\\]\nthat has finite kernel.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} Set $C_{S} := {\\rm SC}_{S}(A_{F\/k};X,X')$.\n\nThen, by comparing the definition of $C_{S}$ as the mapping fibre of (\\ref{fibre morphism}) with the definition of the compactly supported cohomology complex $R\\Gamma_c(A_{F\/k}) := R\\Gamma_c(\\mathcal{O}_{k,S\\cup S_k^p},T_{p,F}(A^t))$ as the mapping fibre of the morphism (\\ref{compactloc})\none finds that there is an exact triangle in $D(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{can tri}  R\\Gamma_c(A_{F\/k})\\to C_{S} \\to X[-1] \\oplus X'[0]\\to R\\Gamma_c(A_{F\/k})[1].\\end{equation}\n\nTo derive claim (i) from this triangle it is then enough to recall (from, for example, \\cite[Prop. 1.6.5]{fukaya-kato}) that $R\\Gamma_c(A_{F\/k})$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ and is acyclic outside degrees one, two and three and note that both of the $\\ZZ_p[G]$-modules $X$ and $X'$ are finitely generated and cohomologically-trivial.\n\nThe above triangle also gives a canonical identification\n\\begin{equation}\\label{artinverdier} H^3(C_{S}) \\cong H^3(R\\Gamma_c(A_{F\/k})) \\cong H^0(k,T_{p,F}(A)\\otimes_{\\ZZ_p}\\QQ_p\/\\ZZ_p)^\\vee = A(F)[p^{\\infty}]^\\vee\\end{equation}\nwhere the second isomorphism is induced by the Artin-Verdier Duality Theorem.\n\nIn a similar way, if we set $\\Sigma:=(S\\cap S_k^f)\\cup S_k^p$ and abbreviate the classical $p$-adic Selmer complex ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ to $C'_\\Sigma$, then a direct comparison of the definitions of $C_{S}$ and $C_\\Sigma'$ shows that $C_{S}$ is isomorphic in $D(\\ZZ_p[G])$ to the mapping fibre of the morphism\n\\begin{equation}\\label{selmer-finite tri} C'_\\Sigma \\oplus X[-1] \\oplus X'[0]\n\\xrightarrow{(\\lambda', \\kappa'_1,\\kappa_2)}\n \\bigoplus_{v \\in \\Sigma} A^t(F_v)^\\wedge_p[-1] \\oplus \\bigoplus_{v \\in S_k^\\infty}\\tau_{\\le 3}(R\\Gamma(k_v,T_{p,F}(A^t)))\\end{equation}\nwhere $\\lambda'$ is the canonical morphism\n\\[ C'_\\Sigma \\to \\bigoplus_{v \\in \\Sigma} A^t(F_v)^\\wedge_p[-1]\\]\ndetermined by the definition of of $C'_\\Sigma$ as the mapping fibre of (\\ref{bkfibre}),\nand the morphism $$\\kappa'_1: X[-1]\\rightarrow \\bigoplus_{v \\in S_k^p} A^t(F_v)^\\wedge_p[-1]$$ is induced by the given inclusion\n $X \\subseteq A^t(F_p)^\\wedge_p$.\n\n\nThis description of $C_{S}$ gives rise to a canonical long exact sequence of $\\ZZ_p[G]$-modules\n\n\\begin{multline}\\label{useful1} 0 \\to {\\rm cok}(H^0(\\kappa_2)) \\to H^1(C_{S}) \\to H^1(C_\\Sigma') \\\\\n \\to (A^t(F_p)^\\wedge_p\/X) \\oplus\\bigoplus_{v \\in (S\\cap S_k^f)\\setminus S_k^p} A^t(F_v)^\\wedge_p \\oplus  \\bigoplus_{v \\in S_k^\\infty}H^1(k_v,T_{p,F}(A^t))\\\\ \\to H^2(C_{S})\\to H^2(C_\\Sigma') \\to \\bigoplus_{v \\in S_k^\\infty}H^2(k_v,T_{p,F}(A^t)). \\end{multline}\n\nIn addition, for each $v \\in S_k^\\infty$ the groups $H^1(k_v,T_{p,F}(A^t))$ and $H^2(k_v,T_{p,F}(A^t))$ vanish if $p$ is odd and are finite if $p=2$, whilst our choice of $X'$ ensures that ${\\rm cok}(H^0(\\kappa_2))$ is also a finite group of $2$-power order.\n\nClaim (iii) therefore follows upon combining the above sequence with the identifications of $H^1(C_\\Sigma')$ and $H^2(C_\\Sigma')$ given in (\\ref{bksc cohom}) for odd $p$, and in the subsequent remarks for $p=2$, that are valid whenever $\\sha(A_F)$ is finite.\n\n\\end{proof}\n\n\n\\begin{remark}\\label{mrselmer}{\\em If $p$ is odd, then the proof of Proposition \\ref{prop:perfect} shows that the cohomology group\n$H^1({\\rm SC}_{S}(A_{F\/k};X,H_\\infty(A_{F\/k})_p))$ coincides with the Selmer group $H^1_{\\mathcal{F}_X}(k,T_{p,F}(A^t))$ in the sense of Mazur and Rubin \\cite{MRkoly}, where $\\mathcal{F}_X$ is the Selmer structure with $\\mathcal{F}_{X,v}$ equal to the image of $X$ in $H^1(k_v,T_{p,F}(A^t))$ for $v\\in S_k^p$ and equal to $0$ for $v \\in S\\setminus S_k^p$.} \\end{remark}\n\n\n\\subsection{Perfect Selmer structures and integral complexes}\\label{perfect selmer integral} We write $\\ell(v)$ for the residue characteristic of a non-archimedean place $v$ of $k$.\n\n\n\\begin{definition}\\label{pgss def}{\\em A `perfect Selmer structure' for the pair $A$ and $F\/k$ is a collection\n\\[ \\mathcal{X} := \\{\\mathcal{X}(v): v \\}\\]\nover all places $v$ of $k$ of $G$-modules that satisfy the following conditions.\n\n\\begin{itemize}\n\\item[(i)] For each $v$ in $S_k^\\infty$ the module $\\mathcal{X}(v)$ is projective and a submodule of $H_v(A_{F\/k})$ of finite $2$-power index.\n\\item[(ii)] For each $v$ in $S_k^f$ the module $\\mathcal{X}(v)$ is cohomologically-trivial and a finite index $\\ZZ_{\\ell(v)}[G]$-submodule of $A^t(F_v)^\\wedge_{\\ell(v)}$.\n\\item[(iii)] For almost all (non-archimedean) places $v$ one has $\\mathcal{X}(v) = A^t(F_v)^\\wedge_{\\ell(v)}.$\n\\end{itemize}\nWe thereby obtain a  projective $G$-submodule\n\\[ \\mathcal{X}(\\infty) := \\bigoplus_{v \\in S_k^\\infty}\\mathcal{X}(v)\\]\nof $H_\\infty(A_{F\/k})$ of finite $2$-power index and, for each rational prime $\\ell$, a finite index cohomologically-trivial $\\ZZ_\\ell[G]$-submodule\n\\[  \\mathcal{X}(\\ell) := \\bigoplus_{v\\in S_k^\\ell}\\mathcal{X}(v)\\]\nof $A^t(F_\\ell)^\\wedge_{\\ell}$.}\n\\end{definition}\n\n\n\\begin{remark}{\\em The conditions (ii) and (iii) in Definition \\ref{pgss def} are consistent since if $\\ell$ does not divide $|G|$, then any $\\ZZ_{\\ell}[G]$-module is automatically cohomologically-trivial for $G$.} \\end{remark}\n\nIn the following result we write $X_\\ZZ(A_F)$ for the `integral Selmer group' of $A$ over $F$ defined by Mazur and Tate in \\cite{mt}.\n\nWe recall that, if the Tate-Shafarevich group $\\sha(A_F)$ is finite, then $X_\\ZZ(A_F)$ is a finitely generated $G$-module and there exists an isomorphism of $\\hat \\ZZ[G]$-modules\n\\[ \\hat\\ZZ\\otimes_\\ZZ X_\\ZZ(A_F) \\cong {\\rm Sel}(A_F)^\\vee\\]\nthat is unique up to automorphisms that induce the identity map on both the submodule $X_\\ZZ(A_F)_{\\rm tor} = \\sha(A_F)^\\vee$ and quotient module $X_\\ZZ(A_F)_{\\rm tf} = \\Hom_\\ZZ(A(F), \\ZZ)$. (Here $\\hat\\ZZ$ denotes the profinite completion of $\\ZZ$).\n\nWe identify ${\\rm Mod}^{\\rm fin}(\\ZZ_2[G])$ as an abelian subcategory of ${\\rm Mod}(\\ZZ[G])$ in the obvious way and write ${\\rm Mod}^\\ast(\\ZZ[G])$ for the associated quotient category.\n\n\\begin{proposition}\\label{prop:perfect2} Assume that $\\sha(A_F)$ is finite. Then for any perfect Selmer structure $\\mathcal{X}$ for $A$ and $F\/k$\n there exists a complex $C_S(\\mathcal{X}) = {\\rm SC}_{S}(A_{F\/k};\\mathcal{X})$ in $D^{\\rm perf}(\\ZZ[G])$ that is unique up to isomorphisms in $D^{\\rm perf}(\\Z[G])$ that induce the identity map in all degrees of cohomology and has all of the following properties.\n\\begin{itemize}\n\\item[(i)] For each prime $\\ell$ there is a canonical isomorphism in $D^{\\rm perf}(\\ZZ_\\ell[G])$\n\\[ \\ZZ_\\ell\\otimes_\\ZZ C_S(\\mathcal{X}) \\cong {\\rm SC}_{S}(A_{F\/k};\\mathcal{X}(\\ell),\\mathcal{X}(\\infty)_\\ell).\\]\n\\item[(ii)] $C_S(\\mathcal{X})$ is acyclic outside degrees one, two and three.\n\\item[(iii)] There is a canonical identification $H^3(C_S(\\mathcal{X})) = (A(F)_{\\rm tor})^\\vee$.\n\n\\item[(iv)] In ${\\rm Mod}^\\ast(\\ZZ[G])$ there exists a canonical injective homomorphism\n\\[ H^1(C_S(\\mathcal{X})) \\to A^t(F) \\]\nthat has finite cokernel and a canonical surjective homomorphism\n\\[ H^2(C_S(\\mathcal{X})) \\to X_\\ZZ(A_F)\\]\nthat has finite kernel.\n\\item[(v)] If $\\mathcal{X}(v)\\subseteq A^t(F_v)$ for all $v$ in $S\\cap S_k^f$ and $\\mathcal{X}(v) = A^t(F_v)^\\wedge_{\\ell(v)}$ for all $v\\notin S$, then there exists an exact sequence in ${\\rm Mod}^\\ast(\\ZZ[G])$ of the form\n\\[ 0 \\to H^1(C_S(\\mathcal{X})) \\to A^t(F) \\xrightarrow{\\Delta_{S,\\mathcal{X}}} \\bigoplus_{v \\in S\\cap S_k^f}\\frac{A^t(F_v)}{\\mathcal{X}(v)} \\to\nH^2(C_S(\\mathcal{X})) \\to X_\\ZZ(A_F)\\to 0\\]\nin which $\\Delta_{S,\\mathcal{X}}$ is the natural diagonal map.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} We write $\\hat \\ZZ$ for the profinite completion of $\\ZZ$ and for each prime $\\ell$ set $C_S(\\ell) := {\\rm SC}_{S}(A_{F\/k};\\mathcal{X}(\\ell),\\mathcal{X}(\\infty)_\\ell)$.\n\nTo construct a suitable complex $C_S(\\mathcal{X})$ we shall use the general result of \\cite[Lem. 3.8]{bkk} with the complex $\\widehat C$ in loc. cit. taken to be the object $\\prod_\\ell C_S(\\ell)$ of $D(\\hat \\ZZ[G])$.\n\nIn fact, since $\\mathcal{X}$ satisfies the conditions (i) and (ii) in Definition \\ref{pgss def}, Proposition \\ref{prop:perfect}(i) implies that each complex $C_S(\\ell)$ belongs to $D^{\\rm perf}(\\ZZ_\\ell[G])$ and is acyclic outside degrees one, two and three and so to apply \\cite[Lem. 3.8]{bkk} it is enough to specify for each $j \\in \\{1,2,3\\}$ a finitely generated $G$-module $M^j$ together with an isomorphism of $\\hat \\Z[G]$-modules of the form $\\iota_j: \\hat \\Z\\otimes_\\Z M^j \\cong \\prod_\\ell H^j(C_S(\\ell))$.\n\nBy Proposition \\ref{prop:perfect}(ii) it is clear that one can take $M^3 = A(F)_{\\rm tor}^\\vee$ and $\\iota_3$ the canonical identification induced by the decomposition $A(F)_{\\rm tor}^\\vee = \\prod_\\ell A(F)[\\ell^\\infty]^\\vee$.\n\nTo construct suitable modules $M^1$ and $M^2$,\n\n\nwe note first that the proof of Proposition \\ref{prop:perfect}(iii) combines with the fact that $\\mathcal{X}$ satisfies condition (iii) in Definition \\ref{pgss def} to give rise to a homomorphism of $\\hat\\ZZ[G]$-modules\n\\[ \\prod_\\ell H^1(C_S(\\ell)) \\xrightarrow{\\theta_1} \\hat\\ZZ\\otimes_\\ZZ A^t(F) \\]\nwith the property that $\\ker(\\theta_1)$ is finite of $2$-power order and ${\\rm cok}(\\theta_1)$ is finite, and to a diagram of homomorphisms of $\\hat\\ZZ[G]$-modules\n\\begin{equation}\\label{derived diag}\n\\prod_\\ell H^2(C_S(\\ell)) \\xrightarrow{\\theta_2} \\prod_\\ell H^2({\\rm SC}_{\\Sigma_\\ell,\\ell}(A_{F\/k})) \\xleftarrow{\\theta_3} \\hat\\ZZ\\otimes_\\ZZ X_{\\ZZ}(A_F)\\end{equation}\nin which $\\ker(\\theta_2)$ is finite whilst ${\\rm cok}(\\theta_2), \\ker(\\theta_3)$ and ${\\rm cok}(\\theta_3)$ are all finite of $2$-power order. Here for each prime number $\\ell$ we have also set $\\Sigma_\\ell:=(S\\cap S_k^f)\\cup S_k^\\ell$.\n\nIt is then straightforward to construct a commutative (pull-back) diagram of $G$-modules\n\\begin{equation}\\label{useful2 diagrams} \\begin{CD}\n M^1 @> >> A^t(F)\\\\\n @V \\iota_{11} VV @VV\\iota_{12} V\\\\\n  \\prod_\\ell H^1(C_S(\\ell)) @> \\theta_1>> \\hat\\ZZ\\otimes_\\ZZ A^t(F)\\end{CD}\\end{equation}\nin which $M^1$ is finitely generated, the upper horizontal arrow has finite kernel of $2$-power order and finite cokernel, the morphism $\\iota_{12}$ is the natural inclusion and the morphism $\\iota_{11}$ induces an isomorphism of $\\hat\\ZZ[G]$-modules $\\iota_1$ of the required sort.\n\nIn a similar way, there is a pull-back diagram of $G$-modules\n\\begin{equation*} \\begin{CD}\n M_2 @> \\theta_2' >> \\theta_3(X_{\\ZZ}(A_F))\\\\\n @V \\iota_{21} VV @VV\\iota_{22} V\\\\\n \\prod_\\ell H^2(C_S(\\ell)) @> \\theta_2 >> \\prod_\\ell H^2({\\rm SC}_{\\Sigma_\\ell,\\ell}(A_{F\/k}))\\end{CD}\\end{equation*}\nin which $M_2$ is finitely generated, $\\iota_{22}$ is the natural inclusion, $\\ker(\\theta_2')$ is finite, ${\\rm cok}(\\theta_2')$ is finite of $2$-power order and the morphism $\\iota_{21}$ induces a short exact sequence\n\\[0\\to \\hat \\ZZ\\otimes_\\ZZ M_2 \\to \\prod_\\ell H^2(C_S(\\ell)) \\to M_2' \\to 0 \\]\nin which $M_2'$ is finite of $2$-power order. Then, since $\\hat \\ZZ$ is a flat $\\ZZ$-module, one has\n\\[ {\\rm Ext}^1_{G}(M_2',M_2) = \\hat\\ZZ\\otimes_\\ZZ{\\rm Ext}^1_{G}(M_2',M_2) = {\\rm Ext}^1_{\\hat \\ZZ[G]}(M_2',\\hat\\ZZ\\otimes_\\ZZ M_2)\\]\nand so there exists an exact commutative diagram of $G$-modules\n\\[ \\begin{CD}\n0 @> >> \\hat \\ZZ\\otimes_\\ZZ M_2 @> \\iota_{21} >> \\prod_\\ell H^2(C_S(\\ell)) @> >>  M_2' @> >> 0\\\\\n& & @A AA @A\\iota_2 AA @\\vert\\\\\n0 @> >> M_2 @> >> M^2 @> >>  M_2' @> >> 0\\end{CD}\\]\nin which the left hand vertical arrow is the natural inclusion and $M^2$ is finitely generated.\n\nIt is then clear that $\\iota_2$ induces an isomorphism $\\hat\\ZZ\\otimes_\\ZZ M^2 \\cong \\prod_\\ell H^2(C_S(\\ell))$ and that the diagram\n\\[ M^2 \\xleftarrow{\\iota_{21}} M_2 \\xrightarrow{\\theta_2'} \\theta_3(X_{\\ZZ}(A_F)) \\xleftarrow{\\theta_3} X_{\\ZZ}(A_F)\\]\nconstitutes a morphism in ${\\rm Mod}^\\ast(\\ZZ[G])$. This morphism is surjective, has finite kernel and lies in a commutative diagram in ${\\rm Mod}^\\ast(\\ZZ[G])$\n\\begin{equation}\\label{derived diag2} \\begin{CD} M^2 @> >> X_\\ZZ(A_F)\\\\\n              @V \\iota_2 VV @VV V\\\\\n              \\prod_{\\ell}H^2(C_S(\\ell)) @> >> \\hat\\ZZ\\otimes_\\ZZ X_\\ZZ(A_F)\\end{CD}\\end{equation}\nin which the right hand vertical arrow is the inclusion map and the lower horizontal arrow corresponds to the diagram (\\ref{derived diag}).\n\nThese observations show that we can apply \\cite[Lem. 3.8]{bkk} in the desired way in order to obtain a complex $C_S(\\mathcal{X})$ in  $D^{\\rm perf}(\\ZZ[G])$ that has $H^j({\\rm SC}_{S}(A_{F\/k};\\mathcal{X})) = M^j$ for each $j$ in $\\{1,2,3\\}$ and satisfies all of the stated properties in claims (i)-(iv).\n\nTurning to claim (v) we note that the given conditions on the modules $\\mathcal{X}(v)$ imply that for each $v$ in $S\\cap S_k^f$ there is a direct sum decomposition of finite modules\n\\[ \\frac{A^t(F_v)}{\\mathcal{X}(v)} = \\frac{A^t(F_v)^\\wedge_{\\ell(v)}}{\\mathcal{X}(v)} \\oplus \\bigoplus_{\\ell \\not= \\ell(v)}A^t(F_v)^\\wedge_\\ell\\]\nand hence also a direct sum decomposition over all primes $\\ell$ of the form\n\\[ \\bigoplus_{v \\in S\\cap S_k^f} \\frac{A^t(F_v)}{\\mathcal{X}(v)} = \\bigoplus_\\ell \\left( \\bigoplus_{v \\in S_k^\\ell}\\frac{A^t(F_v)^\\wedge_{\\ell(v)}}{\\mathcal{X}(v)} \\oplus \\bigoplus_{v \\in (S\\cap S_k^f)\\setminus S_k^\\ell}A^t(F_v)^\\wedge_\\ell\\right).\\]\n\nThis shows that the kernel and cokernel of the map $\\Delta_{S,\\mathcal{X}}$ in claim (v) respectively coincide with the intersection over all primes $\\ell$ of the kernel and the direct sum over all primes $\\ell$ of the cokernel of the diagonal map\n\\[ A^t(F) \\to \\bigoplus_{v \\in S_k^\\ell}\\frac{A^t(F_v)^\\wedge_\\ell}{\\mathcal{X}(v)} \\oplus \\bigoplus_{v \\in (S\\cap S_k^f)\\setminus S_k^\\ell}A^t(F_v)^\\wedge_\\ell\\]\nthat occurs in the sequence (\\ref{useful1}) (with $p$ replaced by $\\ell$ and $X$ by $\\mathcal{X}(\\ell)$).\n\nGiven this fact, the exact sequence follows from the commutativity of the diagrams (\\ref{useful2 diagrams}) and (\\ref{derived diag2}) and the exactness of the sequence (\\ref{useful1}).\n\\end{proof}\n\n\\subsection{Global differentials and perfect Selmer structures}\\label{perf sel sect}\n\nWith a view to the subsequent formulation (in \\S\\ref{statement of conj section}) of our central conjecture we explain how a choice of global differentials gives rise to a natural perfect Selmer structure for $A$ and $F\/k$.\n\nIn the sequel we shall for a natural number $m$ write $[m]$ for the (ordered) set of integers $i$ that satisfy $1 \\le i\\le m$.\n\n\n\\subsubsection{}\\label{gamma section}For each $v$ in $S_k^\\RR$ we fix ordered $\\ZZ$-bases\n\\[ \\{\\gamma_{v,a}^+: a\\in [d]\\}\\]\nof $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)^{c=1}$ and\n\\[ \\{\\gamma_{v,a}^-: a\\in [d]\\}\\]\nof $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)^{c=-1}$, where $c$ denotes complex conjugation.\n\nFor each $v$ in $S_k^\\CC$ we fix an ordered $\\ZZ$-basis\n\\[ \\{\\gamma_{v,a}: a\\in [2d]\\}\\]\nof $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)$.\n\n\nFor each $v$ in $S_k^\\infty$ we then fix $\\tau_v\\in G$ with $\\tau_v(\\sigma_v')=c\\circ\\sigma_v'$ and write $H_v(\\gamma_\\bullet)$ for the free $G$-module with basis\n\\begin{equation}\\label{gamma basis}\\begin{cases} \\{ (1+\\tau_v)\\sigma_v'\\otimes \\gamma^+_{v,a} + (1-\\tau_v)\\sigma_v'\\otimes \\gamma^-_{v,a}: a \\in [d]\\}, &\\text{ if $v$ is real}\\\\\n                 \\{\\sigma_v'\\otimes\\gamma_{v,a}:a \\in [2d]\\}, &\\text{ if $v$ is complex.}\\end{cases}\\end{equation}\n\nThe direct sum\n\\[ H_\\infty(\\gamma_\\bullet) := \\bigoplus_{v \\in S_k^\\infty}H_v(\\gamma_\\bullet)\\]\nis then a free $G$-submodule of $H_\\infty(A_{F\/k})$ of finite $2$-power index.\n\nTo specify an ordered $\\ZZ[G]$-basis of $H_\\infty(\\gamma_\\bullet)$ we fix an ordering of $S_k^\\infty$ and then order the union of the sets\n (\\ref{gamma basis}) lexicographically.\n\n\\subsubsection{}\\label{perf sel construct}We next fix a N\\'eron model $\\mathcal{A}^t$ for $A^t$ over $\\mathcal{O}_k$ and, for each non-archimedean place $v$ of $k$, a N\\'eron model $\\mathcal{A}_v^t$ for $A^t_{\/k_v}$ over $\\mathcal{O}_{k_v}$.\n\nFor any subfield $E$ of $k$ and any non-archimedean place $v$ of $E$ we set $\\mathcal{O}_{F,v}:=\\prod_{w'\\in S_F^v}\\mathcal{O}_{F_{w'}}$.\n\nFor each non-archimedean place $v$ of $k$ we then set\n\\begin{equation}\\label{mathcalD} \\mathcal{D}_F(\\mathcal{A}^t_v) :=  \\mathcal{O}_{F,v}\\otimes_{\\mathcal{O}_{k_v}}\\Hom_{\\mathcal{O}_{k_v}}(H^0(\\mathcal{A}_v^t,\\Omega^1_{\\mathcal{A}_v^t}), \\mathcal{O}_{k_v}).\\end{equation}\n\n\nWe finally fix an ordered $\\QQ[G]$-basis $\\omega_\\bullet$ of the space of invariant differentials\n\\[ H^0(A^t_F,\\Omega^1_{A^t_F}) \\cong F\\otimes_k H^0(A^t,\\Omega^1_{A^t})\\]\nand write $\\mathcal{F}(\\omega_\\bullet)$ for the $G$-module generated by the elements of $\\omega_\\bullet$. In the sequel we often identify $\\omega_\\bullet$ with its dual ordered $\\QQ[G]$-basis in $\\Hom_{F}(H^0(A^t_F,\\Omega^1_{A^t_F}),F)$ and $\\mathcal{F}(\\omega_\\bullet)$ with the $G$-module generated by this dual basis.\n\nIn the sequel, for any subfield $E$ of $k$ and any place $v$ in $S_E^f$ we set $F_v:=\\prod_{w'\\in S_F^v}F_{w'}$.\n\nFor each non-archimedean place $v$ of $k$ we write $\\mathcal{F}(\\omega_\\bullet)_v$ for the $\\ZZ_{\\ell(v)}$-closure of the image of $\\mathcal{F}(\\omega_\\bullet)$ in $F_{v}\\otimes_k\\Hom_{k}(H^0(A^t,\\Omega^1_{A^t}), k)$ and\n\\begin{equation*}\\label{classical exp} {\\rm exp}_{A^t,F_v}: F_v\\otimes_k \\Hom_k(H^0(A^t,\\Omega^1_{A^t}),k) \\cong \\Hom_{F_v}(H^0(A^t_{F_v},\\Omega^1_{A^t_{F_v}}),F_v) \\cong \\QQ_{\\ell(v)}\\cdot A^t(F_v)^\\wedge_{\\ell(v)}\\end{equation*}\nfor the exponential map of $A^t_{F_v}$ relative to some fixed $\\mathcal{O}_{k_v}$-basis of $H^0(\\mathcal{A}_v^t,\\Omega^1_{\\mathcal{A}_v^t})$.\n\nThen, if necessary after multiplying each element of $\\omega_\\bullet$ by a suitable natural number, we may, and will, assume that the following conditions are satisfied:\n\n\\begin{itemize}\n\\item[(i$_{\\omega_\\bullet}$)] for each $v$ in $S_k^f$ one has $\\mathcal{F}(\\omega_\\bullet)_{v}\\subseteq \\mathcal{D}_F(\\mathcal{A}_v^t)$;\n\\item[(ii$_{\\omega_\\bullet}$)] for each $v$ in $S\\cap S_k^f$, the map ${\\rm exp}_{A^t,F_v}$ induces an isomorphism of $\\mathcal{F}(\\omega_\\bullet)_{v}$ with a submodule of $A^t(F_v)$.\n\\end{itemize}\n\n\n\nWe then define $\\mathcal{X}=\\mathcal{X}_S(\\omega_\\bullet) = \\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ to be the perfect Selmer structure for $A$, $F\/k$ and $S$ that has the following properties:\n\\begin{itemize}\n\\item[(i$_\\mathcal{X}$)] If $v\\in S_k^\\infty$, then $\\mathcal{X}(v) = H_v(\\gamma_\\bullet)$.\n\\item[(ii$_\\mathcal{X}$)] If $v \\in S\\cap S_k^f$, then $\\mathcal{X}(v) = {\\rm exp}_{A^t,F_v}(\\mathcal{F}(\\omega_\\bullet)_v)$.\n\\item[(iii$_\\mathcal{X}$)] If $v \\notin S$, then $\\mathcal{X}(v) = A^t(F_v)^\\wedge_{\\ell(v)}$.\n\\end{itemize}\n\n\\begin{remark}{\\em This specification does define a perfect Selmer structure for $A$ and $F\/k$ since if $v$ does not belong to $S$, then the $G$-module $A^t(F_v)^\\wedge_{\\ell(v)}$ is cohomologically-trivial (by Lemma \\ref{useful prel}(ii) below).} \\end{remark}\n\n\\begin{remark}\\label{can structure groups}{\\em The perfect Selmer structure $\\mathcal{X}_S(\\omega_\\bullet) = \\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ defined above satisfies the conditions of Proposition \\ref{prop:perfect2}(v). As a consequence, if one ignores finite modules of $2$-power order, then the cohomology modules of the Selmer complex\n$C_S(\\mathcal{X}(\\omega_\\bullet)) = {\\rm SC}_{S}(A_{F\/k};\\mathcal{X}_S(\\omega_\\bullet))$ can be described as follows: \\\n\n\\noindent{} - $H^1(C_S(\\mathcal{X}(\\omega_\\bullet)))$ is the submodule of $A^t(F)$ comprising all elements $x$ with the property that, for each $v$ in $S$, the image of $x$ in $A^t(F_v)$ belongs to the subgroup ${\\rm exp}_{A^t,F_v}(\\mathcal{F}(\\omega_\\bullet)_v)$.\n\n\\noindent{} - $H^2(C_S(\\mathcal{X}(\\omega_\\bullet)))$ is an extension of the integral Selmer group $X_\\ZZ(A_F)$ by the (finite) cokernel of the diagonal map $A^t(F) \\to \\bigoplus_{v \\in S}\\bigl(A^t(F_v)\/{\\rm exp}_{A^t,F_v}(\\mathcal{F}(\\omega_\\bullet)_v)\\bigr)$.\n\n\\noindent{} - $H^3(C_S(\\mathcal{X}(\\omega_\\bullet)))$ is equal to $(A(F)_{\\rm tor})^\\vee$.}\\end{remark}\n\n\n\n\n\n\n\\section{The refined Birch and Swinnerton-Dyer Conjecture}\\label{ref bsd section}\n\nIn this section we formulate (as Conjecture \\ref{conj:ebsd}) a precise refinement of the Birch and Swinnerton-Dyer Conjecture.\n\n\\subsection{Relative $K$-theory} For the reader's convenience, we first quickly review some relevant facts of algebraic $K$-theory.\n\n\\subsubsection{}\\label{Relative $K$-theory}\n\nFor a Dedekind domain $R$ with field of fractions $F$, an $R$-order $\\mathfrak{A}$ in a finite dimensional separable $F$-algebra $A$ and a field extension $E$ of $F$ we set $A_E := E\\otimes_F A$.\n\nThe relative algebraic $K_0$-group $K_0(\\mathfrak{A},A_E)$ of the ring inclusion $\\mathfrak{A}\\subset A_E$ is described explicitly in terms of generators and relations by Swan in \\cite[p. 215]{swan}.\n\nFor any extension field $E'$ of $E$ there exists a canonical commutative diagram\n\\begin{equation} \\label{E:kcomm}\n\\begin{CD} K_1(\\mathfrak{A}) @> >> K_1(A_{E'}) @> \\partial_{\\mathfrak{A},A_{E'}} >> K_0(\\mathfrak{A},A_{E'}) @> \\partial'_{\\mathfrak{A},A_{E'}} >> K_0(\\mathfrak{A})\\\\\n@\\vert @A\\iota AA @A\\iota' AA @\\vert\\\\\nK_1(\\mathfrak{A}) @> >> K_1(A_E) @> \\partial_{\\mathfrak{A},A_E}  >> K_0(\\mathfrak{A},A_E) @> \\partial'_{\\mathfrak{A},A_E}  >> K_0(\\mathfrak{A})\n\\end{CD}\n\\end{equation}\nin which the upper and lower rows are the respective long exact sequences in relative $K$-theory of the inclusions $\\mathfrak{A}\\subset A_E$ and $\\mathfrak{A}\\subset A_{E'}$ and both of the vertical arrows are injective and induced by the inclusion $A_E \\subseteq A_{E'}$. (For more details see \\cite[Th. 15.5]{swan}.)\n\n\nIn particular, if $R = \\ZZ$ and for each prime $\\ell$ we set $\\mathfrak{A}_\\ell := \\ZZ_\\ell\\otimes_\\ZZ \\mathfrak{A}$ and $A_\\ell:=\n\\QQ_\\ell\\otimes _\\QQ A$, then we can regard each group $K_0(\\mathfrak{A}_\\ell,A_\\ell)$ as a subgroup of $K_0(\\mathfrak{A},A)$ by means of the canonical composite homomorphism\n\\begin{equation}\\label{decomp}\n\\bigoplus_\\ell K_0(\\mathfrak{A}_\\ell,A_\\ell) \\cong K_0(\\mathfrak{A},A)\\subset K_0(\\mathfrak{A},A_\\RR),\n\\end{equation}\nwhere $\\ell$ runs over all primes, the isomorphism is as described in the discussion following \\cite[(49.12)]{curtisr} and the inclusion is induced by the relevant case of $\\iota'$.\n\nFor an element $x$ of $K_0(\\mathfrak{A},A)$ we write $(x_\\ell)_\\ell$ for its image in $\\bigoplus_\\ell K_0(\\mathfrak{A}_\\ell,A_\\ell)$ under the isomorphism in (\\ref{decomp}).\n\nThen, if $G$ is a finite group and $E$ is a field of characteristic zero, taking reduced norms over the semisimple algebra $E[G]$ induces (as per the discussion in \\cite[\\S 45A]{curtisr}) an injective homomorphism\n\\[ {\\rm Nrd}_{E[G]}: K_1(E[G]) \\to \\zeta(E[G])^\\times. \\]\nThis homomorphism is bijective if $E$ is either algebraically closed or complete.\n\n\n\n\\subsubsection{}\\label{nad sec} We shall also use a description of $K_0(\\mathfrak{A},A_E)$ in terms of the formalism of `non-abelian determinants' that is given by Fukaya and Kato in \\cite[\\S1]{fukaya-kato}.\n\nWe recall, in particular, that any pair comprising an object $C$ of $D^{\\rm perf}(\\mathfrak{A})$ and a morphism of non-abelian determinants $\\theta: {\\rm Det}_{A_E}(E\\otimes_R C) \\to {\\rm Det}_{A_E}(0)$ gives rise to a canonical element of $K_0(\\mathfrak{A},A_E)$ that we shall denote by $\\chi_\\mathfrak{A}(C,\\theta)$.\n\nIf $E\\otimes_RC$ is acyclic, then one obtains in this way a canonical element $\\chi_\\mathfrak{A}(C,0)$ of $K_0(\\mathfrak{A},A_E)$.\n\nMore generally, if $E\\otimes_RC$ is acyclic outside of degrees $a$ and $a+1$ for any integer $a$, then a choice of isomorphism of $A_E$-modules $h: E\\otimes_RH^a(C) \\cong E\\otimes_RH^{a+1}(C)$ gives rise to a morphism $h^{\\rm det}: {\\rm Det}_{A_E}(E\\otimes_R C) \\to {\\rm Det}_{A_E}(0)$ of non-abelian determinants and we set\n\\[ \\chi_\\mathfrak{A}(C,h) := \\chi_\\mathfrak{A}(C,h^{\\rm det}).\\]\n\nWe recall the following general result concerning these elements (which follows directly from \\cite[Lem. 1.3.4]{fukaya-kato}) since it will be used often in the sequel.\n\n\\begin{lemma}\\label{fk lemma} Let $C_1 \\to C_2 \\to C_3 \\to C_1[1]$ be an exact triangle in $D^{\\rm perf}(\\mathfrak{A})$ that satisfies the following two conditions:\n\\begin{itemize}\n\\item[(i)] there exists an integer $a$ such that each $C_i$ is acyclic outside degrees $a$ and $a+1$;\n\\item[(ii)] there exists an exact commutative diagram of  $A_E$-modules\n\\[\\begin{CD}\n0 @> >> E\\otimes_R H^a(C_1) @> >> E\\otimes_R H^a(C_2) @> >> E\\otimes_R H^a(C_3) @> >> 0\\\\\n@. @V h_1VV @V h_2VV @V h_3VV \\\\\n0 @> >> E\\otimes_R H^{a+1}(C_1) @> >> E\\otimes_R H^{a+1}(C_2) @> >> E\\otimes_R H^{a+1}(C_3) @> >> 0\\end{CD}\\]\nin which each row is induced by the long exact cohomology sequence of the given exact triangle and each map $h_i$ is bijective.\n\\end{itemize}\n\nThen in $K_0(\\mathfrak{A},A_E)$ one has $\\chi_\\mathfrak{A}(C_2,h_2) = \\chi_\\mathfrak{A}(C_1,h_1) + \\chi_\\mathfrak{A}(C_3,h_3)$.\n\\end{lemma}\n\n\n\n\\begin{remark}\\label{comparingdets}{\\em If $\\mathfrak{A}$ is commutative, then $K_0(\\mathfrak{A},A_E)$ identifies with the multiplicative group of invertible $\\mathfrak{A}$-submodules of $A_E$. If, in this case, $C$ is acyclic outside degrees one and two, then for any isomorphism of $A_E$-modules $h: E\\otimes_RH^1(C)\\to E\\otimes_RH^2(C)$ one finds that the element $\\chi_{\\mathfrak{A}}(C,h)$ defined above corresponds under this identification to the inverse of the ideal $\\vartheta_{h}({\\rm Det}_{\\mathfrak{A}}(C))$ that is defined in \\cite[Def. 3.1]{bst}.}\\end{remark}\n\nFor convenience, we shall often abbreviate the notations $\\chi_{\\ZZ[G]}(C,h)$ and $\\chi_{\\ZZ_p[G]}(C,h)$ to $\\chi_G(C,h)$ and $\\chi_{G,p}(C,h)$ respectively.\n\nWhen the field $E$ is clear from context, we also write $\\partial_{G}$, $\\partial'_{G}$, $\\partial_{G,p}$ and $\\partial'_{G,p}$ in place of $\\partial_{\\ZZ[G],E[G]}$, $\\partial'_{\\ZZ[G],E[G]}$, $\\partial_{\\ZZ_p[G],E[G]}$ and $\\partial'_{\\ZZ_p[G],E[G]}$ respectively.\n\n\n\n\n\n\n\n\\subsection{Statement of the conjecture}\\label{statement of conj section}\n\nIn the sequel we fix a finite set of places $S$ of $k$ as in \\S\\ref{selmer section}. We also fix cycles $\\gamma_\\bullet$ and differentials $\\omega_\\bullet$ as in \\S\\ref{perf sel sect}.\n\n\n\\subsubsection{\n\nWe write $\\Omega_{\\omega_\\bullet}(A_{F\/k})$ for the element of $K_1(\\RR[G])$ that is represented by the matrix of the canonical `period' isomorphism of $\\RR[G]$-modules\n\\begin{multline*} \\RR\\otimes_\\ZZ H_\\infty(\\gamma_\\bullet) = \\RR\\otimes_\\ZZ \\bigoplus_{v \\in S_k^\\infty}H^0(k_v,Y_{v,F}\\otimes_{\\ZZ}H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ))\\\\\n \\cong \\RR\\otimes_{\\QQ} \\Hom_F(H^0(A_F^t,\\Omega^1_{A_F^t}),F),\\end{multline*}\nwith respect to the ordered $\\ZZ[G]$-basis of $H_\\infty(\\gamma_\\bullet)$ specified in \\S\\ref{gamma section} and the ordered $\\QQ[G]$-basis $\\omega_\\bullet$ of $ \\Hom_F(H^0(A_F^t,\\Omega^1_{A_F^t}),F)$.\n\nThis element $\\Omega_{\\omega_\\bullet}(A_{F\/k})$ constitutes a natural `$K$-theoretical period' and can be explicitly computed in terms of the classical periods that are associated to $A$ (see Lemma \\ref{k-theory period} below).\n\n\nTo take account of the local behaviour of the differentials $\\omega_\\bullet$ we define a $G$-module\n\\[ \\mathcal{Q}(\\omega_\\bullet)_S := \\bigoplus_{v \\notin S} \\mathcal{D}_F(\\mathcal{A}_v^t)\/\\mathcal{F}(\\omega_\\bullet)_v,\\]\nwhere $v$ runs over all places of $k$ that do not belong to $S$.\n\nIt is easily seen that almost all terms in this direct sum vanish and hence that $\\mathcal{Q}(\\omega_\\bullet)_S$ is finite. This $G$-module is also cohomologically-trivial since $\\mathcal{D}_F(\\mathcal{A}_v^t)$ and $\\mathcal{F}(\\omega_\\bullet)_v$ are both free $\\ZZ_{\\ell(v)}[G]$-modules for each $v$ outside $S$.\n\nWe can therefore define an object of $D^{\\rm perf}(\\ZZ[G])$ by setting\n\\[ {\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}) := {\\rm SC}_S(A_{F\/k},\\mathcal{X}_S(\\omega_\\bullet)) \\oplus \\mathcal{Q}(\\omega_\\bullet)_S[0],\\]\nwhere we abbreviate the perfect Selmer structure $\\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ defined by the conditions (i$_\\mathcal{X}$), (ii$_\\mathcal{X}$) and (iii$_\\mathcal{X}$) in \\S\\ref{perf sel sect} to $\\mathcal{X}_S(\\omega_\\bullet)$.\n\nWe next write\n \\[\nh_{A,F}: A(F)\\times A^t(F) \\to \\RR\n\\]\nfor the classical N\\'eron-Tate height-pairing for $A$ over $F$.\n\nThis pairing is non-degenerate and hence, assuming $\\sha(A_{F})$ to be finite, it combines with the properties of the Selmer complex\n${\\rm SC}_{S}(A_{F\/k},\\mathcal{X}(\\omega_\\bullet))$ established in Proposition \\ref{prop:perfect}(ii) to induce a canonical isomorphism of $\\RR[G]$-modules\n\\begin{multline*} \\label{height triv}\nh_{A,F}': \\RR\\otimes_\\ZZ H^1({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k})) = \\RR\\otimes_\\ZZ A^t(F)\\\\ \\cong \\RR\\otimes_\\ZZ\\Hom_\\ZZ(A(F),\\ZZ) =   \\RR\\otimes_\\ZZ H^2({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k})).\\end{multline*}\nThis isomorphism then gives rise via the formalism recalled in \\S\\ref{nad sec} to a canonical element\n\\[ \\chi_{G}({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F}) := \\chi_{G}({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h'_{A,F})\\]\nof the relative algebraic $K$-group $K_0(\\ZZ[G],\\RR[G])$.\n\nOur conjecture will predict an explicit formula for this element in terms of Hasse-Weil-Artin $L$-series.\n\n\n\n\\subsubsection{}For every prime $\\ell$ the reduced norm maps ${\\rm Nrd}_{\\QQ_\\ell[G]}$ and ${\\rm Nrd}_{\\CC_\\ell[G]}$ discussed in \\S\\ref{Relative $K$-theory} are bijective and so there exists a composite homomorphism\n\\begin{equation}\\label{G,O hom} \\delta_{G,\\ell}: \\zeta(\\CC_\\ell[G])^\\times \\to K_1(\\CC_\\ell[G]) \\xrightarrow{\\partial_{\\ZZ_\\ell[G],\\CC_\\ell[G]}}\nK_0(\\ZZ_\\ell[G],\\CC_\\ell[G]) \\end{equation}\nin which the first map is the inverse of ${\\rm Nrd}_{\\CC_\\ell[G]}$. This homomorphism maps $\\zeta(\\QQ_\\ell[G])^\\times$ to the subgroup $K_0(\\ZZ_\\ell[G],\\QQ_\\ell[G])$ of $K_0(\\ZZ[G],\\QQ[G])$.\n\nIf now $v$ is any place of $k$ that does not belong to $S$, then $v$ is unramified in $F\/k$ and so the finite $G$-modules $$\\kappa_{F_v}:=\\prod_{w'\\in S_F^v}\\kappa_{F_{w'}}\\,\\,\\,\\, \\text{ and } \\,\\,\\,\\,\\tilde A^t_v(\\kappa_{F_v}):=\\prod_{w'\\in S_F^v}\\tilde A^t(\\kappa_{F_{w'}})$$ are both cohomologically-trivial by  Lemma \\ref{useful prel}(i) below. Here for any place $w'$ in $S_F^v$, $\\tilde A^t$ denotes the reduction of $A^t_{\/F_{w'}}$ to $\\kappa_{F_{w'}}$.\n\nFor any such $v$ we may therefore define an element of the subgroup $K_0(\\ZZ_{\\ell(v)}[G],\\QQ_{\\ell(v)}[G])$ of $K_0(\\ZZ[G],\\QQ[G])$ by setting\n\\begin{equation}\\label{localFM} \\mu_{v}(A_{F\/k}) := \\chi_{G,\\ell(v)}\\bigl(\\kappa_{F_v}^d[0]\\oplus\\tilde A^t_v(\\kappa_{F_v})_{\\ell(v)}[-1],0\\bigr)-\\delta_{G,\\ell(v)}(L_v(A,F\/k))\\end{equation}\nwhere $L_v(A,F\/k)$ is the element of $\\zeta(\\QQ[G])^\\times$ that is equal to the value at $z=1$ of the $\\zeta(\\CC[G])$-valued $L$-factor at $v$ of the motive $h^1(A_{F})(1)$, regarded as defined over $k$ and with coefficients $\\QQ[G]$, as discussed in \\cite[\\S4.1]{bufl99}.\n\nThe sum\n\\[ \\mu_{S}(A_{F\/k}) := \\sum_{v\\notin S}\\mu_{v}(A_{F\/k})\\]\nwill play an important role in our conjecture.\n\nWe shall refer to this sum as the `Fontaine-Messing correction term' for the data $A, F\/k$ and $S$ since, independently of any conjecture, the theory developed by Fontaine and Messing in \\cite{fm} implies that $\\mu_{v}(A_{F\/k})$ vanishes for all but finitely many $v$ and hence that $\\mu_S(A_{F\/k})$ is a well-defined element of $K_0(\\ZZ[G],\\QQ[G])$. (For details see Lemma \\ref{fm} below).\n\n\\subsubsection{}We write $\\widehat{G}$ for the set of irreducible complex characters of $G$. In the sequel, for each $\\psi$ in $\\widehat{G}$ we fix a $\\CC[G]$-module $V_\\psi$ of character $\\psi$.\n\nWe recall that a character $\\psi$ in $\\widehat{G}$ is said to be `symplectic' if the subfield of\n$\\bc$ that is generated by the values of $\\psi$ is totally real\nand $\\End_{\\br [G]}(V_\\psi)$ is isomorphic to the division ring\nof real Quaternions. We write $\\widehat{G}^{\\rm s}$ for the subset of\n$\\widehat{G}$ comprising such characters.\n\nFor each $\\psi$ in $\\widehat{G}$ we write $\\check\\psi$ for its contragredient character and\n\\[ e_\\psi:=\\frac{\\psi(1)}{|G|}\\sum_{g\\in G}\\psi(g^{-1})g\\]\nfor the associated central primitive idempotents of $\\CC[G]$.\n\nThese idempotents induce an identification of $\\zeta(\\CC[G])$ with $\\prod_{\\widehat{G}}\\CC$ and we write $x = (x_\\psi)_\\psi$ for the corresponding decomposition of each element $x$ of $\\zeta(\\CC[G])$.\n\nFor each $\\psi$ in $\\widehat{G}$ we write $L_{S}(A,\\psi,z)$ for the Hasse-Weil-Artin $L$-series of $A$ and $\\psi$, truncated by removing the Euler factors corresponding to places in $S$.\n\nWe can now state the central conjecture of this article.\n\n\\begin{conjecture}\\label{conj:ebsd}\nThe following claims are valid.\n\\begin{itemize}\n\\item[(i)] The group $\\sha(A_F)$ is finite.\n\\item[(ii)] For all $\\psi$ in $\\widehat{G}$ the function $L(A,\\psi,z)$ has an analytic continuation to $z=1$ where it has a zero of order\n $\\psi(1)^{-1}\\cdot {\\rm dim}_{\\CC}(e_\\psi(\\CC\\otimes_\\ZZ A^t(F)))$.\n\\item[(iii)] For all $\\psi$ in $\\widehat{G}^{\\rm s}$ the leading coefficient $L^*_S(A,\\psi,1)$ at $z=1$ of the function $L_S(A,\\psi,z)$ is a strictly positive real number. In particular, there exists a unique element $L^*_{S}(A_{F\/k},1)$ of $K_1(\\RR[G])$ with\n\\[ {\\rm Nrd}_{\\RR[G]}(L_S^*(A_{F\/k},1))_\\psi = L_S^*(A,\\check\\psi,1)\\]\nfor all $\\psi$ in $\\widehat{G}$.\n\\item[(iv)] In $K_0(\\ZZ[G],\\RR[G])$ one has\n\\[ \\partial_G\\left(\\frac{L_S^*(A_{F\/k},1)}{\\Omega_{\\omega_\\bullet}(A_{F\/k})}\\right) = \\chi_G({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F}) + \\mu_{S}(A_{F\/k}).\\]\n\\end{itemize}\n\\end{conjecture}\n\nIn the sequel we shall refer to this conjecture as the `Birch and Swinnerton-Dyer Conjecture for the pair $(A,F\/k)$' and abbreviate it to ${\\rm BSD}(A_{F\/k})$.\n\n\n\n\\begin{remark}{\\em The assertion of ${\\rm BSD}(A_{F\/k})$(i) is the celebrated Shafarevich-Tate conjecture. The quantity $\\psi(1)^{-1}\\cdot {\\rm dim}_{\\CC}(e_\\psi(\\CC\\otimes_\\ZZ A^t(F)))$ is equal to the multiplicity with which the character $\\psi$ occurs in the rational representation $\\QQ\\otimes_\\ZZ A^t(F)$ of $G$ (and hence to the right hand side of the equality (\\ref{dg equality})) and so the assertion of ${\\rm BSD}(A_{F\/k})$(ii) coincides with a conjecture of Deligne and Gross (cf. \\cite[p. 127]{rohrlich}).}\\end{remark}\n\n\\begin{remark}{\\em Write $\\tau$ for complex conjugation. Then, by the Hasse-Schilling-Maass Norm Theorem (cf. \\cite[(7.48)]{curtisr}), the image of ${\\rm Nrd}_{\\RR[G]}$ is the subset of $\\prod_{\\widehat{G}}\\CC^\\times$ comprising $x$ with the property that $x_{\\psi^\\tau} = \\tau(x_\\psi)$ for all $\\psi$ in $\\widehat{G}$ and also that $x_\\psi$ is a strictly positive real number for all $\\psi$ in $\\widehat{G}^{\\rm s}$. This means that the second assertion of ${\\rm BSD}(A_{F\/k})$(iii) follows immediately from the first assertion, the injectivity of ${\\rm Nrd}_{\\RR[G]}$ and the fact that $L_S^*(A,\\psi^\\tau,1) = \\tau(L_S^*(A,\\psi,1))$ for all $\\psi$ in $\\widehat{G}$.\n\nThe first assertion of ${\\rm BSD}(A_{F\/k})$(iii) is itself motivated by the fact that if $\\psi$ belongs to $\\widehat{G}^{\\rm s}$, and $[\\psi]$ denotes the associated Artin motive over $k$, then one can show that $L^*_S(A,\\psi,1)$ is a strictly positive real number whenever the motive $h^1(A)\\otimes [\\psi]$ validates the `Generalized Riemann Hypothesis' discussed by Deninger in \\cite[(7.5)]{den}. However, since this fact does not itself provide any more evidence for ${\\rm BSD}(A_{F\/k})$(iii) we omit the details.} \\end{remark}\n\n\\begin{remark}\\label{weaker BSD}{\\em It is possible to formulate a version of ${\\rm BSD}(A_{F\/k})$ that omits claim (iii) and hence avoids any possible reliance on the validity of the Generalized Riemann Hypothesis. To do this we recall that the argument of \\cite[\\S4.2, Lem. 9]{bufl99} constructs a canonical `extended boundary homomorphism' of relative $K$-theory $\\delta_G: \\zeta(\\RR[G])^\\times \\to K_0(\\ZZ[G],\\RR[G])$ that lies in a commutative diagram\n\n\\[ \\xymatrix{\nK_1(\\RR[G]) \\ar@{^{(}->}[d]^{{\\rm Nrd}_{\\RR[G]}} \\ar[rr]^{\\hskip -0.2truein\\partial_G} & & K_0(\\ZZ[G],\\RR[G])\\\\\n\\zeta(\\RR[G])^\\times . \\ar[urr]^{\\delta_G}}\\]\n\nHence, to obtain a version of the conjecture that omits claim (iii) one need only replace the term on the left hand side of the equality in claim (iv) by the difference\n\\[ \\delta_G\\bigl(\\calL_S^*(A_{F\/k},1)\\bigr) - \\partial_G\\bigl(\\Omega_{\\omega_\\bullet}(A_{F\/k})\\bigr)\\]\nwhere $\\calL_S^*(A_{F\/k},1)$ denotes the element of $\\zeta(\\RR[G])^\\times$ with $\\calL_S^*(A_{F\/k},1)_\\psi = L_S^*(A,\\check\\psi,1)$ for all $\\psi$ in $\\widehat{G}$.}\\end{remark}\n\n\n\\begin{remark}\\label{rbsd etnc rem}{\\em The approach developed by Wuthrich and the present authors in \\cite[\\S4]{bmw} can be extended to show that\n the weaker version of ${\\rm BSD}(A_{F\/k})$ discussed in the last remark is equivalent to the validity of the equivariant Tamagawa number conjecture for the pair $(h^1(A_F)(1),\\ZZ[G])$, as formulated in \\cite[Conj. 4]{bufl99} (for details see Appendix A). Taken in conjunction with the results of Venjakob and the first author in \\cite{BV2}, this observation implies that the study of ${\\rm BSD}(A_{F\/k})$ and its consequences is relevant if one wishes to properly understand the content of the main conjecture of non-commutative Iwasawa theory, as formulated by Coates et al in \\cite{cfksv}.}\\end{remark}\n\n\\begin{remark}\\label{cons1}{\\em If, for each prime $\\ell$, we fix an isomorphism of fields $\\CC\\cong \\CC_\\ell$, then the exactness of the lower row in (\\ref{E:kcomm}) with $\\mathfrak{A} = \\ZZ_\\ell[G]$ and $A_E = \\CC_\\ell[G]$ implies that the equality in ${\\rm BSD}(A_{F\/k})$(iv) determines the image of $(L^*_S(A,\\psi, 1))_{\\psi\\in \\widehat{G}}$ in $\\zeta(\\CC_\\ell[G])^\\times$ modulo the image under the reduced norm map ${\\rm Nrd}_{\\QQ_\\ell[G]}$ of $K_1(\\ZZ_\\ell[G])$. In view of the explicit description of the latter image that is obtained by Kakde in \\cite{kakde} (or, equivalently, by the methods of Ritter and Weiss in \\cite{rw}), this means ${\\rm BSD}(A_{F\/k})$(iv) implicitly incorporates families of congruence relations between the leading coefficients $L^*_S(A,\\psi, 1)$ for varying $\\psi$ in $\\widehat{G}$.}\\end{remark}\n\n\n\n\n\\begin{remark}\\label{consistency remark}{\\em The formulation of ${\\rm BSD}(A_{F\/k})$ is consistent in the following respects.\n\\begin{itemize}\n\\item[(i)] Its validity is independent of the choices of set $S$ and ordered $\\QQ[G]$-basis $\\omega_\\bullet$.\n\n\\item[(ii)] Its validity for the pair $(A,F\/k)$ implies its validity for $(A_E,F\/E)$ for any intermediate field $E$ of $F\/k$ and for $(A,E\/k)$ for any such $E$ that is Galois over $k$.\n\n\\item[(iii)] Its validity for the pair $(A,k\/k)$ is equivalent, up to sign, to the Birch and Swinnerton-Dyer Conjecture for $A$ over $k$.\n\\end{itemize}\nEach of these statements can be proven directly but also follows from the observation in Remark \\ref{rbsd etnc rem} (see Remark \\ref{consistency} for more details).}\\end{remark}\n\n\n\\begin{remark}{\\em A natural analogue of ${\\rm BSD}(A_{F\/k})$ has been formulated, and in some important cases proved, in the setting of abelian varieties over global function fields by Kakde, Kim and the first author in \\cite{bkk}.} \\end{remark}\n\n\n\n\n\n\n\nMotivated at least in part by Remark \\ref{rbsd etnc rem}, our main aim in the rest of this article will be to describe, and in important special cases provide evidence for, a range of explicit consequences that would follow from the validity of ${\\rm BSD}(A_{F\/k})$.\n\n\\subsection{$p$-components}\\label{pro-p sect} To end this section we show that the equality in ${\\rm BSD}(A_{F\/k})$(iv) can be checked by considering separately `$p$-primary components' for each prime $p$.\n\nFor each prime $p$ and each isomorphism of fields $j: \\CC\\cong \\CC_p$, the inclusion $\\RR \\subset \\CC$ combines with the functoriality of $K$-theory to induce a homomorphism\n\\[ K_1(\\RR[G]) \\to K_1(\\CC_p[G])\\]\nand also pairs with the inclusion $\\ZZ \\to \\ZZ_p$ to induce a homomorphism\n\\[ K_0(\\ZZ[G],\\RR[G]) \\to K_0(\\ZZ_p[G],\\CC_p[G]).\\]\nIn the sequel we shall, for convenience, use $j_\\ast$ to denote both of these homomorphisms as well as the inclusion $\\zeta(\\RR[G])^\\times \\to \\zeta(\\CC_p[G])^\\times$ and isomorphism $\\zeta(\\CC[G])^\\times\\cong \\zeta(\\CC_p[G])^\\times$ that are induced by the action of $j$ on coefficients.\n\n\n\\begin{lemma}\\label{pro-p lemma} Fix $\\omega_\\bullet$ and $S$ as in ${\\rm BSD}(A_{F\/k})$. Then, to prove the equality in ${\\rm BSD}(A_{F\/k})$(iv) it suffices to prove, for every prime $p$ and every isomorphism of fields $j:\\CC\\cong \\CC_p$, that\n\\begin{multline}\\label{displayed pj} \\partial_{G,p}\\left(\\frac{j_*(L_S^*(A_{F\/k},1))}{j_*(\\Omega_{\\omega_\\bullet}(A_{F\/k}))}\\right) = \\chi_{G,p}({\\rm SC}_S(A_{F\/k},\\mathcal{X}(p),\\mathcal{X}(\\infty)_p),h^j_{A,F})\\\\ +\\chi_{G,p}( \\mathcal{Q}(\\omega_\\bullet)_{S,p} [0],0) + \\mu_{S}(A_{F\/k})_p,\\end{multline}\nwhere we write $\\mathcal{X}$ for $\\mathcal{X}_S(\\omega_\\bullet)$ and $h^{j}_{A,F}$ for $\\CC_p\\otimes_{\\RR,j}h^{{\\rm det}}_{A,F}$\n\\end{lemma}\n\n\\begin{proof} We consider the diagonal homomorphism of abelian groups\n\\begin{equation}\\label{local iso} K_0(\\ZZ[G],\\RR[G]) \\xrightarrow{(\\prod j_*)_p}  \\prod_p\\left(\\prod_{j: \\CC\\cong \\CC_p} K_0(\\ZZ_p[G],\\CC_p[G])\\right),\\end{equation}\nwhere the products run over all primes $p$ and all choices of isomorphism $j$.\n\nThe key fact that we shall use is that this map is injective. This fact is certainly well-known but, given its importance, we shall, for completeness, prove it.\n\nWe consider the exact sequences that are given by the lower row of (\\ref{E:kcomm}) with $\\mathfrak{A}= R[G]$ and $A_E = E[G]$ for the\npairs $(R,E)=(\\ZZ,\\QQ)$, $(\\ZZ,\\RR)$, $(\\ZZ_p,\\QQ_p)$ and\n$(\\ZZ_p,\\CC_p)$ and the maps between these sequences which are\ninduced by the obvious inclusions and by an embedding\n$j:\\RR\\to\\CC_p$.\n\nBy an easy diagram chase one obtains a\ncommutative diagram of short exact sequences\n\\begin{equation*}\n\\xymatrix{\n0 \\ar[r] & K_0(\\ZZ[G],\\QQ[G]) \\ar[r] \\ar[d] & K_0(\\ZZ[G],\\RR[G]) \\ar[r] \\ar[d] &\nK_1(\\RR[G])\/K_1(\\QQ[G]) \\ar[r] \\ar[d] & 0 \\\\\n0 \\ar[r] & K_0(\\ZZ_p[G],\\QQ_p[G]) \\ar[r] & K_0(\\ZZ_p[G],\\CC_p[G]) \\ar[r] &\nK_1(\\CC_p[G])\/K_1(\\QQ_p[G]) \\ar[r] & 0.\n}\n\\end{equation*}\nTherefore it suffices to show that the maps\n\\begin{equation}\n\\label{equation_K_injectivity_left}\nK_0(\\ZZ[G],\\QQ[G])\\to\\prod_{p,j} K_0(\\ZZ_p[G],\\QQ_p[G])\n\\end{equation}\nand\n\\begin{equation}\n\\label{equation_K_injectivity_right}\nK_1(\\RR[G])\/K_1(\\QQ[G])\\to\\prod_{p,j}\nK_1(\\CC_p[G])\/K_1(\\QQ_p[G])\n\\end{equation}\nare injective. The injectivity of (\\ref{equation_K_injectivity_left})\nfollows immediately from the relevant case of the isomorphism in (\\ref{decomp}).\n\nLet $x\\in K_1(\\RR[G])$ be such that for all $p$ and all $j$ one has\n\\[ j_*(x)\\in K_1(\\QQ_p[G])\\subseteq K_1(\\CC_p[G]).\\]\n\nWe now use the (injective) maps ${\\rm Nrd}_{\\RR[G]}$ and ${\\rm Nrd}_{\\QQ[G]}$ to identify $K_1(\\RR[G])$ and $K_1(\\QQ[G])$ with $\\im({\\rm Nrd}_{\\RR[G]})$ and $\\im({\\rm Nrd}_{\\QQ[G]})$ respectively.\n\nThen, $x=\\sum_{g\\in G} c_gg$ is an element of $\\im({\\rm Nrd}_{\\RR[G]})$ such that\n\\begin{equation}\n\\label{equation_lemma_K_injectivity}\nj_*(x)=\\sum_{g\\in G} j(c_g)g\\in\\zeta(\\QQ_p[G])^\\times.\n\\end{equation}\n\nWe claim that $\\sum_{g\\in G} c_gg\\in\\QQ[G]$. Let $g\\in G$ and consider the\ncoefficient $c_g$.\n\nIf, firstly, $c_g$ was transcendental over $\\QQ$, then\nthere would be an embedding $j:\\RR\\to\\CC_p$ such that\n$j(c_g)\\not\\in\\QQ_p$, thereby contradicting\n(\\ref{equation_lemma_K_injectivity}).\n\nTherefore $c_g$ is algebraic\nover $\\QQ$. Now $j(c_g)\\in\\QQ_p$ for all $p$ and embeddings\n$j$ implies that all primes are completely split in the\nnumber field $\\QQ(c_g)$ and therefore $\\QQ(c_g)=\\QQ$.\n\nHence $x$ belongs to $\\im({\\rm Nrd}_{\\RR[G]})\\cap\\QQ[G]$ which, by the Hasse-Schilling-Maass Norm Theorem, is equal to $\\im({\\rm Nrd}_{\\QQ[G]})$.\n\nThis shows the injectivity of (\\ref{equation_K_injectivity_right}) and hence also of the map (\\ref{local iso}).\nThe injectivity of (\\ref{local iso}) in turn implies that the equality of ${\\rm BSD}(A_{F\/k})$(iv) is valid if and only if its image under each maps $j_*$ is valid.\n\nSet $\\mathcal{X} := \\mathcal{X}_S(\\omega_\\bullet)$. Then\n\\begin{multline*} j_*(\\chi_{G}({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F})) = \\chi_{G,p}(\\ZZ_p\\otimes_\\ZZ {\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}), \\CC_p\\otimes_{\\RR,j}h_{A,F}))\\\\\n=\\chi_{G,p}({\\rm SC}_S(A_{F\/k},\\mathcal{X}(p),\\mathcal{X}(\\infty)_p),h^j_{A,F}) +\\chi_{G,p}( \\mathcal{Q}(\\omega_\\bullet)_{S,p}[0],0),\\end{multline*}\nwhere the first equality is by definition of the map $j_*$ and the second by Proposition \\ref{prop:perfect2}(i).\n\nGiven this, the claim follows from the obvious equality $j_*(\\mu_{S}(A_{F\/k})) = \\mu_{S}(A_{F\/k})_p$ and the commutativity of the diagram\n\\begin{equation*}\\label{commute K thry} \\begin{CD} K_1(\\RR[G]) @> \\partial_G >> K_0(\\ZZ[G],\\RR[G])\\\\\n@VV j_{*} V @VV j_{*}V\\\\\nK_1(\\CC_p[G]) @> \\partial_{G,p} >> K_0(\\ZZ_p[G],\\CC_p[G]).\\end{CD}\\end{equation*}\n\\end{proof}\n\n\n\\begin{remark}{\\em In the sequel we shall say, for any given prime $p$, that the `$p$-primary component' {\\rm BSD}$_p(A_{F\/k})$(iv) of the equality in {\\rm BSD}$(A_{F\/k})$(iv) is valid if for every choice of isomorphism of fields $j:\\CC\\cong \\CC_p$ the equality (\\ref{displayed pj}) is valid.} \\end{remark}\n\n\\section{Periods and Galois-Gauss sums}\\label{k theory period sect}\n\nTo prepare for arguments in subsequent sections, we shall now explain the precise link between the $K$-theoretical period $\\Omega_{\\omega_\\bullet}(A_{F\/k})$ that occurs in ${\\rm BSD}(A_{F\/k})$ and the classical periods that are associated to $A$ over $k$.\n\n\\subsection{Periods and Galois resolvents}\\label{k theory period sect2} At the outset we fix an ordered $k$-basis $\\{\\omega'_j: j \\in [d]\\}$ of $H^0(A^t,\\Omega^1_{A^t})$.\n\nFor each $v$ in $S_\\RR^k$ we then set\n\\[ \\Omega_{A,v}^+ := {\\rm det}\\left(\\left(\\int_{\\gamma_{v,a}^{+}}\\sigma_{v,*}(\\omega'_b)\\right)_{a,b}\\right)\\]\nand\n\n\\[ \\Omega_{A,v}^- := {\\rm det}\\left(\\left(\\int_{\\gamma_{v,a}^{-}}\\sigma_{v,*}(\\omega'_b)\\right)_{a,b}\\right),\\]\nwhere the elements $\\gamma_{v,a}^+$ and $\\gamma_{v,a}^-$ of $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)$ are as specified in \\S\\ref{gamma section} and in both matrices $(a,b)$ runs over $[d]\\times [d]$.\n\nFor each $v$ in $S_k^\\CC$ we also set\n\\[ \\Omega_{A,v} := {\\rm det}\\left(\\left(\\int_{\\gamma_{v,a}}\\sigma_{v,*}(\\omega'_b),c\\!\\left(\\int_{\\gamma_{v,a}}\\sigma_{v,*}(\\omega'_b)\\right)\\right)_{a,b}\\right)\\]\nwhere the elements $\\gamma_{v,a}$ of $H_1((A^t)^{\\sigma_v}(\\CC),\\ZZ)$ are again as specified in \\S\\ref{gamma section} and $(a,b)$ runs over $[2d]\\times [d]$.\n\nWe note that, by explicitly computing integrals, the absolute values of these determinants can be explicitly related to the periods that occur in the classical formulation of the Birch and Swinnerton-Dyer conjecture (see, for example, Gross \\cite[p. 224]{G-BSD}).\n\nFor each archimedean place $v$ of $k$ and character $\\psi$ we then set\n\\[\\Omega^\\psi_{A,v} := \\begin{cases} \\Omega_{A,v}^{\\psi(1)}, &\\text{ if $v \\in S_k^\\CC$,}\\\\\n                                            (\\Omega^+_{A,v})^{1-\\psi_v^-(1)}(\\Omega^-_{A,v})^{\\psi_v^-(1)}, &\\text{ if $v \\in S_k^\\RR$}\\end{cases} \\]\nwith\n\\[\\psi_v^-(1) := \\psi(1) - {\\rm dim}_\\CC(H^0(G_w,V_\\psi)),\\]\nwhere again $V_\\psi$ is a fixed choice of $\\CC[G]$-module of character $\\psi$.\n\nFor each $\\psi$ we set\n\\[ \\Omega_A^\\psi := \\prod_{v \\in S_k^\\infty}\\Omega^\\psi_{A,v}\\]\nand we then finally define an element of $\\zeta(\\CC[G])^\\times$ by setting\n\\begin{equation}\\label{period def} \\Omega_A^{F\/k} := \\sum_{\\psi \\in \\widehat{G}}\\Omega^\\psi_A\\cdot e_\\psi.\\end{equation}\n\nFor each $v$ in $S_k^\\RR$, resp. in $S_k^\\CC$, we also set\n\\[ w_{v,\\psi} := \\begin{cases} i^{\\psi^-_v(1)}, &\\text{if $v\\in S_k^\\RR$,}\\\\\n                              i, &\\text{if $v\\in S_k^\\CC$.}\\end{cases}\\]\n\nFor each character $\\psi$ we then set\n\\[ w_{\\psi} :=  \\prod_{v \\in S_k^\\infty}w_{v,\\psi}\\]\nand then define an element of $\\zeta(\\CC[G])^\\times$ by setting\n\\begin{equation}\\label{root number def} w_{F\/k} := \\sum_{\\psi\\in \\widehat{G}} w_\\psi\\cdot e_\\psi .\\end{equation}\n\n\n\n\\begin{lemma}\\label{k-theory period} Set $n := [k:\\QQ]$. Fix an ordered $\\QQ[G]$-basis $\\{z_i: i \\in [n]\\}$ of $F$ and write $\\omega_\\bullet$ for the (lexicographically ordered) $\\QQ[G]$-basis  $\\{ z_i\\otimes \\omega'_j: i \\in [n], j \\in [d]\\}$ of $H^0(A_F^t,\\Omega^1_{A_F^t})$. Then in $\\zeta(\\RR[G])^\\times$ one has\n\\[ {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k})) = \\Omega_A^{F\/k}\\cdot w_{F\/k}^d\\cdot {\\rm Nrd}_{\\QQ[G]}\n\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^{-d} \\]\nwhere we have fixed an extension $\\hat\\sigma$ to $\\Sigma(F)$ of each embedding $\\sigma$ in $\\Sigma(k)$.\\end{lemma}\n\n\\begin{proof} This follows from the argument of \\cite[Lem. 4.5]{bmw}.\\end{proof}\n\n\n\\subsection{Galois resolvents and Galois-Gauss sums}\n\nUnder suitable conditions, one can also choose the $\\QQ[G]$-basis $\\{z_i: i \\in [n]\\}$ of $F$ so that the reduced norm of the Galois resolvent matrix that occurs in Lemma \\ref{k-theory period} can be explicitly described in terms of Galois-Gauss sums.\n\nBefore explaining this we first recall the relevant notions of Galois-Gauss sums.\n\n\\subsubsection{}\\label{mod GGS section}\n\nThe `global Galois-Gauss sum of $F\/k$' is the element\n\\[ \\tau(F\/k) :=\\sum_{\\psi \\in \\widehat{G}}\\tau(\\QQ,\\psi)\\cdot e_\\psi\\]\nof $\\zeta(\\QQ^c[G])^\\times$.\n\nHere we regard each character $\\psi$ of $G$ as a character of $G_k$ via the projection $G_k \\to G$ and then write\n$\\tau(\\QQ,\\psi)$ for the global Galois-Gauss sum (as defined by Martinet in \\cite{martinet})\nof the induction of $\\psi$ to $G_\\QQ$.\n\n\nTo define suitable modifications of these sums\nwe then define the `unramified characteristic' of $v$ at each character $\\psi$ in $\\widehat{G}$ by setting\n\\[ u_{v,\\psi} := {\\rm det}(-\\Phi_v^{-1}\\mid V_\\psi^{I_w})\\in \\QQ^{c,\\times}.\\]\n\nFor each character $\\psi$ in $\\widehat{G}$ we set\n\\[ u_\\psi :=  \\prod_{v\\in S_k^F}u_{v,\\psi}.\\]\n\nWe then define elements of $\\zeta(\\QQ[G])^\\times$ by setting\n\\begin{equation}\\label{u def} u_v(F\/k) := \\sum_{\\psi\\in \\widehat{G}}u_\\psi\\cdot e_\\psi\\end{equation}\nand\n\\[ u_{F\/k} := \\prod_{v\\in S_k^F}u_v(F\/k) = \\sum_{\\psi\\in \\widehat{G}}u_\\psi\\cdot e_\\psi.\\]\n\nWe finally define the `modified global Galois-Gauss sum of $\\psi$' to be the element\n\\[ \\tau^\\ast(\\QQ,\\psi) := u_\\psi\\cdot \\tau(\\QQ,\\psi)\\]\nof $\\QQ^c$, and the `modified global Galois-Gauss sum of $F\/k$' to be the element\n\\[ \\tau^\\ast(F\/k) := u_{F\/k}\\cdot \\tau(F\/k) = \\sum_{\\psi\\in \\widehat{G}}\\tau^\\ast(\\QQ,\\psi)\\cdot e_\\psi \\]\nof $\\zeta(\\QQ^c[G])^\\times$.\n\n\n\n\\begin{remark}{\\em The modified Galois-Gauss sums $\\tau^\\ast(\\QQ,\\psi)$ defined above play a key role in the proof of the main results of classical Galois module theory, as discussed by Fr\\\"ohlich in  \\cite{frohlich}. In Lemma \\ref{imprimitive GGS} below, one can also find a more concrete reason as to why such terms should arise naturally in the setting of leading term conjectures.}\n\\end{remark}\n\n\\subsubsection{} The next result shows that under mild hypotheses the Galois-resolvent matrix that occurs in Lemma \\ref{k-theory period} can be explicitly interpreted in terms of the elements $\\tau^\\ast(F\/k)$ introduced above.\n\n\\begin{proposition}\\label{lms}The following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] For any ordered $\\QQ[G]$-basis $\\omega_\\bullet$ of $H^0(A_F^t,\\Omega^1_{A_F^t})$ there exists an element $u(\\omega_\\bullet)$ of $\\zeta(\\QQ[G])^\\times$ such that\n\\[ {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k})) = u(\\omega_\\bullet)\\cdot \\Omega_A^{F\/k}\\cdot w_{F\/k}^d\\cdot \\tau^\\ast(F\/k)^{-d}.\\]\n\n\\item[(ii)] Fix a prime $p$ and set $\\mathcal{O}_{F,p} := \\ZZ_p\\otimes_\\ZZ\\mathcal{O}_F.$ Then if no $p$-adic place of $k$ is wildly ramified in $F$, there is an ordered $\\ZZ_p[G]$-basis $\\{z^p_{i}\\}_{i \\in [n]}$ of $\\mathcal{O}_{F,p}$ for which one has\n\\[ {\\rm Nrd}_{\\QQ_p[G]}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z^p_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right) = \\tau^\\ast(F\/k).\\]\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} It is enough to prove claim (i) for any choice of $\\QQ[G]$-basis $\\omega_\\bullet$. Then, choosing $\\omega_\\bullet$ as in Lemma\n\\ref{k-theory period}, the latter result implies that it is enough to prove that the product\n\\[ {\\rm Nrd}_{\\QQ_p[G]}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)\\cdot \\tau^\\ast(F\/k)^{-1}\\]\nbelongs to $\\zeta(\\QQ[G])^\\times$ and this follows from the argument used by Bley and the first author to prove \\cite[Prop. 3.4]{bleyburns}.\n\nTurning to claim (ii) we note that if no $p$-adic place of $k$ is wildly ramified in $F$, then the $\\ZZ_p[G]$-module $\\mathcal{O}_{F,p}$ is free of rank $n$ (by Noether's Theorem) and so we may fix an ordered $\\ZZ_p[G]$-basis $z^p_\\bullet := \\{z^p_{i}: i \\in [n]\\}$.\n\nThe matrix\n\n\\[ M(z^p_\\bullet) := (  (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z^p_b)))\\cdot g)_{\\sigma\\in \\Sigma(k),b\\in [n]})\\]\nin ${\\rm GL}_{n}(\\CC_p[G])$ then represents, with respect to the bases $z^p_\\bullet$ of $F_p$ and $\\{\\hat \\sigma: \\sigma \\in \\Sigma(k)\\}$ of $Y_{F\/k,p}$, the isomorphism of $\\CC_p[G]$-modules\n\\[ \\mu_{F,p}: \\CC_p\\otimes_{\\QQ_p} F_p \\cong \\CC_p\\otimes_{\\ZZ_p}Y_{F\/k,p}\\]\nthat sends each $z\\otimes f$ to $(z\\hat\\sigma(f))_{\\sigma\\in\\Sigma(k)}$.\n\nHence one has\n\\begin{align*} \\delta_{G,p}\\bigl({\\rm Nrd}_{\\CC_p[G]}\\bigl(M(z^p_\\bullet)\\bigr)\\bigr) = \\, &\\partial_{G,p}\\bigl([M(z^p_\\bullet)]\\bigr)\\\\\n                                                                     = \\, &[\\mathcal{O}_{F,p}, Y_{F\/k,p}; \\mu_{F,p}]\\\\\n                                                                     = \\, &\\delta_{G,p}(\\tau^\\ast(F\/k)),\\end{align*}\n                                                                    \nwhere $[M(z^p_\\bullet)]$ denotes the class of $M(z^p_\\bullet)$ in $K_1(\\CC_p[G])$ and the last equality follows from the proof of \\cite[Th. 7.5]{bleyburns}.\n\nNow the exact sequence of relative $K$-theory implies that kernel of $\\delta_{G,p}$ is equal to the image of $K_1(\\ZZ_p[G])$ under the map ${\\rm Nrd}_{\\QQ_p[G]}$.\n\nIn addition, the ring  $\\ZZ_p[G]$ is semi-local and so the natural map ${\\rm GL}_{n}(\\ZZ_p[G]) \\to K_1(\\ZZ_p[G])$ is surjective.\n\nIt follows that there exists a matrix $U$ in ${\\rm GL}_n(\\ZZ_p[G])$ with\n\\[ {\\rm Nrd}_{\\CC_p[G]}(M(z^p_\\bullet))\\cdot {\\rm Nrd}_{\\CC_p[G]}(U) = \\tau^\\ast(F\/k)\\]\nand so it suffices to replace the basis $z^p_\\bullet$ by its image under the automorphism of $\\mathcal{O}_{F,p}$ that corresponds to the matrix $U$. \\end{proof}\n\nTaken together, Lemma \\ref{k-theory period} and Proposition \\ref{lms}(ii) give an explicit interpretation of the $K$-theoretical periods that occur in the formulation of {\\rm BSD}$(A_{F\/k})$.\n\nHowever, the existence of $p$-adic places that ramify wildly in $F$ makes the situation more complicated and this leads to technical difficulties in later sections.\n\n\n\n\n\n\\section{Local points on ordinary varieties}\\label{local points section}\n\nIn \\S\\ref{tmc} we will impose several mild hypotheses on the reduction types of $A$ and the ramification invariants of $F\/k$ which together ensure that the classical Selmer complex is perfect over $\\ZZ_p[G]$. Under these hypotheses, we will then give a more explicit interpretation of the equality in ${\\rm BSD}(A_{F\/k})$(iv).\n\nAs a necessary preparation for these results, in this section we shall establish several preliminary results concerning the properties of local points on varieties with good ordinary reduction.\n\n\n\\subsection{Cohomological-triviality}For this purpose we assume to be given a finite Galois extension $N\/M$ of $p$-adic fields and set $\\Gamma := G_{N\/M}$. We fix a Sylow $p$-subgroup $\\Delta$ of $\\Gamma$. We write $\\Gamma_0$ for the inertia subgroup of $\\Gamma$ and set $N_0 := N^{\\Gamma_0}$.\n\nWe also assume to be given an abelian variety $B$, of dimension $d$, over $M$ that has good reduction and write $\\tilde B$ for the corresponding reduced variety.\n\n\\begin{lemma}\\label{useful prel} The following claims are valid.\n\\begin{itemize}\n\\item[(i)] If $N\/M$ is unramified, then the $\\Gamma$-modules $B(N)$, $\\tilde B(\\kappa_{N})$ and $\\kappa_N$ are cohomologically-trivial.\n\\item[(ii)] If $N\/M$ is at most tamely ramified, then the $\\ZZ_p[\\Gamma]$-modules $B(N)^\\wedge_p$ and $\\tilde B(\\kappa_{N})[p^\\infty]$ are cohomologically-trivial.\n\\item[(iii)] If the variety $B$ is ordinary and $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then the $\\ZZ_p[\\Gamma]$-module $B(N)^\\wedge_p$ is cohomologically-trivial.\n\\item[(iv)] Assume that $B$ is ordinary and write $u$ for the twist matrix (in ${\\rm GL}_{d}(\\ZZ_p)$) of its formal group over the completion of $M^{\\rm un}$. If $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then $B(N)^\\wedge_p$ is torsion-free, and hence projective over $\\ZZ_p[\\Gamma]$, if and only if for any non-trivial $d$-fold vector $\\underline{\\zeta}$ of $p$-th roots of unity in $N^{\\rm un}$ one has\n\\[ \\Phi_N(\\underline{\\zeta}) \\not= \\underline{\\zeta}^u,\\]\nwhere $\\Phi_N$ is the Frobenius automorphism in $G_{N^{\\rm un}\/N}$. In particular, this is the case if any $p$-power root of unity in $N^{\\rm un}$ belongs to $N$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} A standard Hochschild-Serre spectral sequence argument combines with the criterion of \\cite[Thm. 9]{cf} to show that  claim (i) is valid provided that each of the modules $B(N)$, $\\tilde B(\\kappa_{N})$ and $\\kappa_N$ is cohomologically-trivial with respect to every  subgroup $C$ of $\\Gamma$ of prime order (see the proof of \\cite[Lem. 4.1]{bmw} for a similar argument).\n\nWe therefore fix a subgroup $C$ of $\\Gamma$ that has prime order. Now cohomology over $C$ is periodic of order 2 and each of the modules $B(N)$, $\\tilde B(\\kappa_{N})$ and $\\kappa_N$ span free $\\QQ[\\Gamma]$-modules.\nIt thus follows from \\cite[Cor. to Prop. 11]{cf} that the Herbrand Quotient with respect to $C$ of each of these modules is equal to 1.\nTo prove claim (i) it is enough to show that the natural norm maps $B(N)\\to B(N^C)$, $\\tilde B(\\kappa_{N}) \\to \\tilde B(\\kappa_{N^C})$ and $\\kappa_{N}\\to \\kappa_{N^C}$ are surjective.\n\nSince the extension $N\/N^C$ is unramified, this surjectivity is well-known for the module $\\kappa_N$ and for the modules $B(N)$ and $\\tilde B(\\kappa_{N})$ it follows directly from the result of Mazur in \\cite[Cor. 4.4]{m}.\n\nTo prove claim (ii) we assume that $N\/M$ is tamely ramified. In this case the order of $\\Gamma_0$ is prime to $p$ and so the same standard Hochschild-Serre spectral sequence argument as in claim (i) implies claim (ii) is true if the modules $B(N_0)^\\wedge_p = (B(N)^\\wedge_p)^{\\Gamma_0}$ and $\\tilde B(\\kappa_{N})[p^\\infty] = \\tilde B(\\kappa_{N})[p^\\infty]^{\\Gamma_0}$ are cohomologically-trivial with respect to every subgroup $C$ of $\\Gamma\/\\Gamma_0$ of order $p$. Since $N_0\/N_0^C$ is unramified, this follows from the argument in claim (i).\n\nIn a similar way, to prove claim (iii) one is reduced to showing that if $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then for each subgroup $C$ of $\\Gamma_0$ of order $p$, the norm map ${\\rm N}_C: B(N)^\\wedge_p \\to B(N^C)^\\wedge_p$ is surjective.\n\nNow the main result of Lubin and Rosen in \\cite{LR} implies that the cokernel of ${\\rm N}_C$ is isomorphic to the cokernel of the natural action of ${\\rm I}_d-u$ on the direct sum of $d$-copies of $C$ and from the proof of \\cite[Th. 2]{LR} one knows that ${\\rm det}({\\rm I}_d-u)$ is a $p$-adic divisor of $|\\tilde B(\\kappa_N)|$.\n But if $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes, then $\\tilde B(\\kappa_{N})[p^\\infty]$ also vanishes (as $\\Delta$ is a $p$-group) and so\n ${\\rm det}({\\rm I}_d-u)$ is a $p$-adic unit. It follows that ${\\rm cok}(N_C)$ vanishes, as required to prove claim (iii).\n\nTo prove claim (iv) we assume $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes. Then claim (ii) implies $B(N)^\\wedge_p$ is a projective  $\\ZZ_p[\\Gamma]$-module if and only if $B(N)^\\wedge_p[p^\\infty]$ vanishes. In addition, from the lemma in \\cite[\\S1]{LR} (with $L = K = N$), we know that the group $B(N)^\\wedge_p[p^\\infty]$ is isomorphic to the subgroup of $(N^{{\\rm un},\\times})^d$ comprising $p$-torsion elements $\\underline{\\eta}$ which satisfy\n$\\Phi_N(\\underline{\\eta}) = \\underline{\\eta}^u$.\n\nThis directly implies the first assertion of claim (iv) and the second assertion then follows  because ${\\rm det}({\\rm I}_d-u)$ is a $p$-adic unit and so $u\\not\\equiv 1$ (mod $p$). \\end{proof}\n\n\\begin{remark}{\\em A more general analysis of the cohomological properties of formal groups was recently given by Ellerbrock and Nickel in \\cite{ellerbrocknickel}.}\\end{remark}\n\n\\subsection{Twist matrices and $K$-theory}\\label{twist inv prelim}\n\nIn this section we fix an abelian variety $B$ over $M$ of dimension $d$.\nWe assume that $B$ has good ordinary reduction and is such that $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ vanishes.\n\nWe shall then use Lemma \\ref{useful prel} to define a natural invariant in $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ of the twist matrix of $B$ that will play an important role in the explicit interpretation of the equality in ${\\rm BSD}(A_{F\/k})$(iv) that will be given in \\S \\ref{tmc} below.\n\nAt the outset we recall that the complex $R\\Gamma(N,\\ZZ_p(1))$ belongs to $D^{\\rm perf}(\\ZZ_p[\\Gamma])$. Hence, following Lemma \\ref{useful prel}(iii), we obtain a complex in $D^{\\rm perf}(\\ZZ_p[\\Gamma])$ by setting\n\\[ C_{B,N}^{\\bullet} := R\\Gamma(N,\\ZZ_p(1))^d[1] \\oplus B(N)^\\wedge_p[-1].\\]\n\nThis complex is acyclic outside degrees zero and one. In addition, Kummer theory gives an identification $H^1(N,\\ZZ_p(1)) = (N^\\times)^\\wedge_p$ and the invariant map ${\\rm inv}_N$ of $N$ an isomorphism $ H^2(N,\\ZZ_p(1)) \\cong \\ZZ_p$.\n\nWe next fix a choice of isomorphism of $\\QQ_p[\\Gamma]$-modules $\\lambda_{B,N}$  which lies in a commutative diagram\n\\begin{equation}\\label{lambda diag}\\begin{CD}\n0 @> >> \\QQ_p\\cdot (U^{(1)}_{N})^d @> \\subset >> \\QQ_p\\cdot H^0(C_{B,N}^{\\bullet}) @> ({\\rm val}_N)^d >> \\QQ_p^d @> >> 0\\\\\n@. @V {\\rm exp}_{B,N}VV @V \\lambda_{B,N} VV @V \\times f_{N\/M}VV\\\\\n0 @> >> \\QQ_p \\cdot B(N)^\\wedge_p @> \\subset >> \\QQ_p\\cdot H^1(C_{B,N}^{\\bullet}) @> {\\rm can} >> \\QQ_p^d @> >> 0.\\end{CD}\\end{equation}\nHere $U_N^{(1)}$ is the group of 1-units of $N$, ${\\rm val}_N: \\QQ_p\\cdot (N^\\times)^\\wedge_p\\to \\QQ_p$ is the canonical valuation map on $N$, $f_{N\/M}$ is the residue degree of $N\/M$, `{\\rm can}' is induced by ${\\rm inv}_N$ and ${\\rm exp}_{B,N}$ is the composite isomorphism\n\\[ \\QQ_p\\cdot (U^{(1)}_{N})^d \\cong N^d \\cong \\QQ_p\\cdot B(N)^\\wedge_p\\]\nwhere the first isomorphism is induced by the $p$-adic logarithm on $N$ and the second by the exponential map of the formal group of $B$ over $N$.\n\nWe now introduce a useful general convention: for each element $x$ of $\\zeta(\\CC_p[\\Gamma])$ we write $^\\dagger x$ for the unique element of $\\zeta(\\CC_p[\\Gamma])^\\times$ with the property that for each $\\mu$ in $\\widehat{\\Gamma}$ one has\n\\begin{equation}\\label{dagger eq} e_\\mu (^\\dagger x) = \\begin{cases} e_\\mu x, &\\text{ if $e_\\mu x\\not= 0$,}\\\\\n  e_\\mu, &\\text{ otherwise.}\\end{cases}\\end{equation}\n(This construction is written as $x \\mapsto ^*\\!\\! x$ in \\cite{bleyburns,breuning2}).\n\nWe then define an element\n\\[ c_{N\/M} := \\frac{^\\dagger((|\\kappa_M|-\\Phi_{N\/M})e_{\\Gamma_0})}{^\\dagger((1-\\Phi_{N\/M})e_{\\Gamma_0})}\\]\nof $\\zeta(\\QQ[\\Gamma])^\\times$. Here and in the sequel, $\\Phi_{N\/M}$ is a fixed lift to $\\Gamma$ of the Frobenius automorphism in $\\Gamma\/\\Gamma_0$ and, for any subgroup $J$ of $\\Gamma$, $e_{J}$ denotes the idempotent $(1\/|J|)\\sum_{\\gamma\\in J}\\gamma$.\n\nWe finally obtain our desired element of $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ by setting\n\\[ R_{N\/M}(\\tilde B) := \\chi_{\\Gamma,p}(C_{B,N}^{\\bullet},\\lambda_{B,N}) +d\\cdot\\delta_{\\Gamma,p}(c_{N\/M}). \\]\n\n\n\\begin{proposition}\\label{basic props} Assume $B$ is ordinary and $\\tilde B(\\kappa_{N^\\Delta})[p^{\\infty}]$ vanishes.\n\\begin{itemize}\n\\item[(i)] $R_{N\/M}(\\tilde B)$ depends only upon $N\/M$ and the reduced variety $\\tilde B$.\n\\item[(ii)] $R_{N\/M}(\\tilde B)$ has finite order.\n\\item[(iii)] If $N\/M$ is tamely ramified, then\n $R_{N\/M}(\\tilde B)$ vanishes.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}  We set $C^\\bullet := C_{B,N}^\\bullet$, $\\lambda := \\lambda_{B,N}$ and $f := f_{N\/M}$, and also write $\\wp = \\wp_N$ for the maximal ideal in the valuation ring of $N$.\n\nThen, whilst $\\lambda$ can be chosen in many different ways to ensure that (\\ref{lambda diag}) commutes, it is straightforward to check that $\\chi_{\\Gamma,p}(C^{\\bullet},\\lambda)$ is independent of this choice. The fact that this element depends only on the (twist matrix of the) reduced variety $\\tilde B$ follows from Lemma \\ref{twist dependence} below. This proves claim (i).\n\nIt is convenient to prove claim (iii) before claim (ii) and so we assume $N\/M$ is tamely ramified.\n\nWe write $\\hat B$ for the formal group of $B$. In this case Lemma \\ref{ullom}(i) below implies that for each natural number $n$ the $\\ZZ_p[\\Gamma]$-modules $U^{(n)}:= \\mathbb{G}_m(\\wp^n)$ and $V^{(n)}:= \\hat B(\\wp^n)$ are cohomologically-trivial and there exist exact triangles in $D^{\\rm perf}(\\ZZ_p[\\Gamma])$ of the form\n\\begin{equation*}\\label{use tri}\n\\begin{cases} (U^{(n)})^d[0]\\oplus V^{(n)}[-1]\\xrightarrow{\\alpha} C^{\\bullet}\\to C_{\\alpha}^\\bullet \\to (U^{(n)})^d[1]\\oplus V^{(n)}[0]\\\\\nU^{(1)}[0] \\xrightarrow{\\beta} R\\Gamma(N,\\ZZ_p(1))[1] \\to C^\\bullet_{N,1} \\to U_N^1[1]\\\\\n(U^{(1)}\/U^{(n)})^d[0] \\xrightarrow{\\gamma} C_{\\alpha}^\\bullet \\to (C^\\bullet_{N,1})^d\\oplus (V^{(1)}\/V^{(n)})[-1] \\to (U^{(1)}\/U^{(n)})^d[1].\\end{cases}\\end{equation*}\nHere $\\alpha$ is the unique morphism such that $H^0(\\alpha)$ and $H^1(\\alpha)$ are respectively induced by the inclusions $U^{(n)}\\subset (N^\\times)^\\wedge_p$ and $V^{(n)} \\subseteq B(N)^\\wedge_p$ and so that the cohomology sequence of the first triangle induces identifications of $H^0(C_{\\alpha}^\\bullet)$ and $H^1(C_{\\alpha}^\\bullet)$ with $((N^\\times)^\\wedge_p\/U^{(n)})^d$ and $\\ZZ_p^d \\oplus V^{(1)}\/V^{(n)}$; $\\beta$ is the unique morphism so that $H^0(\\beta)$ is induced by the inclusion $U^{(1)} \\subset (N^\\times)^\\wedge_p$ and so the cohomology sequence of the second  triangle induces identifications of $H^0(C_{N,1}^\\bullet)$ and $H^1(C_{N,1}^\\bullet)$ with $(N^\\times)^\\wedge_p\/U^{(1)}$ and $\\ZZ_p$ respectively; $\\gamma$ is the unique morphism so that $H^{0}(\\gamma)$ is the inclusion $(U^{(1)}\/U^{(n)})^d \\subset H^0(C_{\\alpha}^\\bullet)$.\n\nIn particular, if $n$ is sufficiently large, then we may apply Lemma \\ref{fk lemma} to the first and third of the above triangles to deduce that\n\n\\begin{align}\\label{interm} &\\chi_{\\Gamma,p}(C^{\\bullet},\\lambda)\\\\ = \\,&\\chi_{\\Gamma,p}((U^{(n)})^d[0]\\oplus V^{(n)}[-1],{\\rm exp}_{B,N}) +  \\chi_{\\Gamma,p}(C_{\\alpha}^{\\bullet},\\lambda_{\\alpha})\\notag\\\\\n= \\,&\\chi_{\\Gamma,p}(C_{\\alpha}^{\\bullet},\\lambda_{\\alpha})\\notag\\\\\n= \\,& \\chi_{\\Gamma,p}((U^{(1)}\/U^{(n)})^d[0],0) + d\\cdot \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot{\\rm val}_N) + \\chi_{\\Gamma,p}((V^{(1)}\/V^{(n)})[-1],0)\\notag\\\\\n= \\,& \\chi_{\\Gamma,p}((U^{(1)}\/U^{(n)})^d[0],0) + d\\cdot \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot{\\rm val}_N) - \\chi_{\\Gamma,p}((V^{(1)}\/V^{(n)})[0],0)\\notag\\\\\n= \\, &d\\cdot \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot{\\rm val}_N),\\notag\n\n \\end{align}\nwhere we write $\\lambda_{\\alpha}$ for the isomorphism of $\\QQ_p[\\Gamma]$-modules\n\\[ \\QQ_p\\cdot H^0(C_{\\alpha}^\\bullet) = \\QQ_p\\cdot ((N^\\times)^\\wedge_p\/U^{(n)})^d \\cong \\QQ_p^d = \\QQ_p\\cdot H^1(C_{\\alpha}^\\bullet)\\]\nthat is induced by the map $f\\cdot {\\rm val}_N$ and the second and last equalities in (\\ref{interm}) follow from Lemma \\ref{ullom}.\n\nBut $$\\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},f\\cdot {\\rm val}_N) = \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},{\\rm val}_N) + \\delta_{\\Gamma,p}(^\\dagger(f\\cdot e_\\Gamma))$$ whilst from \\cite[Th. 4.3]{bleyburns} one has\n\n\\[ \\chi_{\\Gamma,p}(C_{N,1}^{\\bullet},{\\rm val}_N) = -\\delta_{\\Gamma,p}(c_{N\/M}\\cdot ^\\dagger\\!(f\\cdot e_\\Gamma))= - \\delta_{\\Gamma,p}(c_{N\/M}) - \\delta_{\\Gamma,p}(^\\dagger\\!(f\\cdot e_\\Gamma)).\\]\nClaim (iii) is thus obtained by substituting these facts into the equality (\\ref{interm}).\n\n\nTo deduce claim (ii) from claim (iii) we recall that an element $\\xi$ of $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ has finite order if and only if for cyclic subgroup $\\Upsilon$ of $\\Gamma$ and every quotient $\\Omega = \\Upsilon\/\\Upsilon'$ of\norder prime to $p$ one has $(q^\\Upsilon_{\\Omega}\\circ\\rho^{\\Gamma}_\\Upsilon)(\\xi) = 0$. Here\n$$\\rho^{\\Gamma}_\\Upsilon:K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])\\to K_0(\\ZZ_p[\\Upsilon],\\QQ_p[\\Upsilon])$$ is the natural restriction map,\n$$q^\\Upsilon_{\\Omega}:K_0(\\ZZ_p[\\Upsilon],\\QQ_p[\\Upsilon])\\to K_0(\\ZZ_p[\\Omega],\\QQ_p[\\Omega])$$\nmaps the class of a triple $(P,\\phi,Q)$ to the class of $(P^{\\Upsilon'},\\phi^{\\Upsilon'},Q^{\\Upsilon'})$, and the stated general fact is proved in \\cite[Thm. 4.1]{ewt}.\n\nSince the extension $N^{\\Upsilon'}\/N^\\Upsilon$ is tamely ramified, it is thus enough to show that\n\\[ (q^\\Upsilon_{\\Omega}\\circ\\rho^{\\Gamma}_\\Upsilon)(R_{N\/M}(\\tilde B))=R_{N^{\\Upsilon'}\/N^\\Upsilon}(\\tilde B).\\]\n\nThis is proved by a routine computation in relative $K$-theory that uses the same ideas as in \\cite[Rem. 2.9]{breuning2}.\n In fact, the only point worth mentioning explicitly in this regard is that if $\\Gamma'$ is normal in $\\Gamma$, and we set $N' := N^{\\Gamma'}$, then the natural projection isomorphism $\\iota:\\ZZ_p[\\Gamma\/\\Gamma']\\otimes^{\\mathbb{L}}_{\\ZZ_p[\\Gamma]}R\\Gamma(N,\\ZZ_p(1)) \\cong R\\Gamma(N',\\ZZ_p(1))$ in $D^{\\rm perf}(\\ZZ_p[\\Gamma\/\\Gamma'])$ gives a commutative diagram of (trivial) $\\QQ_p[\\Gamma\/\\Gamma']$-modules\n\n\\[ \\begin{CD} \\QQ_p\\cdot H^2(N,\\ZZ_p(1))^{\\Gamma'} @> {\\rm inv}_N >> \\QQ_p\\\\\n@V H^2(\\iota)VV @VV \\times f_{N\/N'}V\\\\\n\\QQ_p\\cdot H^2(N',\\ZZ_p(1)) @> {\\rm inv}_{N'}>> \\QQ_p.\\end{CD}\\]\n\\end{proof}\n\n\\begin{lemma}\\label{ullom} If $N\/M$ is tamely ramified, the following claims are valid for all natural numbers $a$.\n\\begin{itemize}\n\\item[(i)] The $\\ZZ_p[\\Gamma]$-modules $U^{(a)}$ and $V^{(a)}$ are cohomologically-trivial.\n\\item[(ii)] One has $d\\cdot \\chi_{\\Gamma,p}((U^{(1)}\/U^{(a)})[0],0) = \\chi_{\\Gamma,p}((V^{(1)}\/V^{(a)})[0],0)$.\n\\item[(iii)] For all sufficiently large $a$ one has $\\chi_{\\Gamma,p}((U^{(a)})^d[0]\\oplus V^{(a)}[-1],{\\rm exp}_{B,N})=0$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} The key fact in this case is that for every integer $i$ the $\\ZZ_p[\\Gamma]$-module $\\wp^i$ is cohomologically-trivial (by Ullom \\cite{Ullom}).\n\nIn particular, if we write $i_0$ for the least integer with $i_0 \\ge e\/(p-1)$, where $e$ is the ramification degree of $N\/\\QQ_p$, then for any integer $a \\ge i_0$ the formal logarithm ${\\rm log}_{B}$ and $p$-adic exponential map restrict to give isomorphisms of $\\ZZ_p[\\Gamma]$-modules\n\\begin{equation}\\label{iso1} V^{(a)}\\cong (\\wp^{a})^d,\\,\\,\\,\\,\\,\\,\\,\\,\\wp^a \\cong U^{(a)}\\end{equation}\nand so the $\\ZZ_p[\\Gamma]$-modules $V^{(a)}$ and $U^{(a)}$ are cohomologically-trivial.\n\nIn addition, for all $a$ the natural isomorphisms\n\\begin{equation}\\label{iso2} U^{(a)}\/U^{(a+1)} \\cong\\wp^a\/\\wp^{a+1},\\,\\,\\,\\,\\,\\,\\,\\, \\bigl(\\wp^a\/\\wp^{a+1}\\bigr)^d \\cong V^{(a)}\/V^{(a+1)}\\end{equation}\nimply that these quotient modules are also cohomologically-trivial. By using the tautological exact sequences for each $a < i_0$\n\\begin{equation}\\label{filter1} \\begin{cases} &0 \\to U^{(a+1)}\/U^{(i_0)} \\to U^{(a)}\/U^{(i_0)} \\to U^{(a)}\/U^{(a+1)} \\to 0,\\\\\n &0 \\to V^{(a+1)}\/V^{(i_0)} \\to V^{(a)}\/V^{(i_0)} \\to V^{(a)}\/V^{(i+1)} \\to 0\\end{cases}\\end{equation}\none can therefore deduce (by a downward induction on $a$, starting at $i_0$) that all modules $U^{(a)}$ and $V^{(a)}$ are cohomologically-trivial. This proves claim (i).\n\nIn addition, by repeatedly using the exact sequences (\\ref{filter1}) and isomorphisms (\\ref{iso2}) one computes that $d\\cdot \\chi_{\\Gamma,p}((U^{(1)}\/U^{(a)})[0],0)$ is equal to\n\n\\begin{align*} d\\cdot\\sum_{b=1}^{b=a-1}\\chi_{\\Gamma,p}((U^{(b)}\/U^{(b+1)})[0],0)                                                            = \\, &\\sum_{b=1}^{b=a-1}\\chi_{\\Gamma,p}(\\bigl((U^{(b)}\/U^{(b+1)})\\bigr)^d[0],0)\\\\\n                                                            = \\, &\\sum_{b=1}^{b=a-1}\\chi_{\\Gamma,p}((V^{(b)}\/V^{(b+1)})[0],0)\\\\\n                                                            = \\, &\\chi_{\\Gamma,p}((V^{(1)}\/V^{(b)})[0],0),\\end{align*}\nas required go prove claim (ii).\n\nFinally, claim (iii) is a direct consequence of the isomorphisms (\\ref{iso1}).\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{twist dependence} Let $B$ and $B'$ be abelian varieties over $M$, of the same dimension $d$, that have good ordinary reduction and are such that $\\tilde B(\\kappa_{N^\\Delta})[p^\\infty]$ and $\\tilde B'(\\kappa_{N^\\Delta})[p^{\\infty}]$ both vanish. Then the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] The $\\ZZ_p[\\Gamma]$-modules $B(N)^\\wedge_p$ and $B'(N)^\\wedge_p$ are cohomologically-trivial and the formal group logarithms induce an isomorphism of $\\QQ_p[\\Gamma]$-modules\n\\[ \\QQ_p\\cdot B(N)^\\wedge_p \\xrightarrow{{\\rm log}_{B,N}} N^d \\xrightarrow{{\\rm exp}_{B',N}} \\QQ_p\\cdot B'(N)^\\wedge_p.\\]\n\n\\item[(ii)] If the reduced varieties $\\tilde B$ and $\\tilde B'$ are isomorphic, then in $K_0(\\ZZ_p[\\Gamma],\\QQ_p[\\Gamma])$ one has\n\\[ \\chi_{\\Gamma,p}(B(N)^\\wedge_p[0] \\oplus B'(N)^\\wedge_p[-1], {\\rm exp}_{B',N}\\circ {\\rm log}_{B,N}) = 0.\\]\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} Claim (i) follows directly from Lemma \\ref{useful prel}(iii).\n\nTo prove claim (ii) we write $M^{\\rm un}$ for the  maximal unramified extension of $M$, $\\hat M^{\\rm un}$ for its completion and $\\mathcal{O}$ for the valuation ring of $\\hat M^{\\rm un}$. We write $\\varphi_M$ for the Frobenius automorphism in $G_{M^{\\rm un}\/M}$.\n\nThen the formal group $\\hat B$ of $B$ is toroidal and so there exists an isomorphism of formal groups $f_1: \\hat B \\cong \\mathbb{G}_m^d$ over $\\mathcal{O}$. If we let $\\varphi_M$ act on the coefficients of $f_1$, then one has $f_1^{\\varphi_M} = u\\circ f_1$, where $u$ is the `twist matrix' of $B$. Thus $u$ belongs to ${\\rm GL}_d(\\ZZ_p)$ and depends only on $\\tilde B$ (by the argument of Mazur in \\cite[p. 216]{m}).\n\nIn particular, if $\\tilde B'$ is isomorphic to $\\tilde B$, then there exists an isomorphism of formal groups $f_2: \\hat B' \\cong \\mathbb{G}_m^d$ over $\\mathcal{O}$ for which one also has $f_2^{\\varphi_M} = u\\circ f_2$.\n\nWe now consider the isomorphism $\\phi:= f_2^{-1}\\circ f_1$ from $\\hat B$ to $\\hat B'$ over $\\mathcal{O}$. We fix an element $x$ of $\\mathcal{O}_N\\mathcal{O}^{\\rm un}$ and an element $g$  of $G_{N^{\\rm un}\/M}$ whose image in $G_{M^{\\rm un}\/M}$ is an integral power $\\varphi_M^a$ of $\\varphi_M$.\n\n\nThen one has \n\\begin{multline*} g(\\phi(x)) = \\phi^g(g(x)) = ((f_2^{\\varphi_M^a})^{-1}\\circ f_1^{\\varphi_M^a})(g(x))\\\\ = ((u^a\\circ f_2)^{-1}\\circ (u^a\\circ f_1))(g(x)) =\n(f_2^{-1}\\circ f_1)(g(x)) = \\phi(g(x)).\\end{multline*}\n\nThis means that $\\phi$ is an isomorphism of $\\ZZ_p[[G_{N^{\\rm un}\/M}]]$-modules and so restricts to give an isomorphism of $\\Gamma$-modules\n\\[ \\hat B(N) = \\hat B(N^{\\rm un})^{G_{N^{\\rm un}\/N}} \\cong \\hat B'(N^{\\rm un})^{G_{N^{\\rm un}\/N}} = \\hat B'(N).\\]\n\nUpon passing to pro-$p$-completions, and noting that the groups $\\tilde B(\\kappa_{N})[p^\\infty]$ and $\\tilde B'(\\kappa_{N})[p^\\infty]$ vanish, we deduce that $\\phi$ induces an isomorphism of $\\ZZ_p[\\Gamma]$-modules\n\\[ \\phi_p: B(N)^\\wedge_p = \\hat B(N)^\\wedge_p\\cong \\hat B'(N)^\\wedge_p = B'(N)^\\wedge_p.\\]\n\nThere is also a commutative diagram of formal group isomorphisms\n\\[\n\\begin{CD} \\hat B @> f_1  >> \\mathbb{G}_m^d @> (f_2)^{-1} >> \\hat B' \\\\\n@V {\\rm log}_B VV @V {\\rm log}_{\\mathbb{G}_m} VV @VV {\\rm log}_{B'}V\\\\\n\\mathbb{G}_a^d @> \\times f_1'(0)  >> \\mathbb{G}_a^d @> \\times f_2'(0)^{-1} >> \\mathbb{G}_a^d\\end{CD}\\]\n\nTaken together with the isomorphism $\\phi_p$, this diagram implies that the element\n\\[ \\chi_{\\Gamma,p}(B(N)^\\wedge_p[0] \\oplus B'(N)^\\wedge_p[-1], {\\rm exp}_{B',N}\\circ {\\rm log}_{B,N}) \\]\nis equal to the image under $\\partial_{\\Gamma,p}$ of the automorphism of the $\\QQ_p[\\Gamma]$-module $N^d$ that corresponds to the matrix\n $f_2'(0)^{-1}f_1'(0)$.\n\nIt is thus enough to note that, since the latter matrix belongs to ${\\rm GL}_d(\\mathcal{O}_M)$ it is represented by a matrix in ${\\rm GL}_{d[M:\\QQ_p]}(\\ZZ_p[\\Gamma])$ and so belongs to the kernel of $\\partial_{\\Gamma,p}$, as required. \\end{proof}\n\n\n\n\n\n\\subsection{Elliptic curves}\\label{ell curve sect} If $B$ is an elliptic curve over $\\QQ_p$, then it is possible in certain cases to formulate a precise conjectural formula for the elements $R_{N\/M}(\\tilde B)$ defined above.\n\nThis aspect of the theory will be considered in detail elsewhere. However, to give a brief idea of the general approach we fix an isomorphism of formal groups $f: \\hat{B} \\lra \\mathbb{G}_m$ as in the proof of Lemma \\ref{twist dependence}.\n\nThen, with $\\varphi$ denoting the Frobenius automorphism in $G_{\\Qu_p^{\\rm un} \/ \\Qp}$, the twist matrix of $\\tilde B$ is the unique element $u$ of $\\ZZ_p^\\times$ for which the composite $f^\\varphi\\circ f^{-1}$ is equal to the endomorphism $[u]_{\\mathbb{G}_m}$ of $\\mathbb{G}_m$.\n\n\\begin{lemma} $\\hat{B}$ is a Lubin-Tate formal group with respect to the parameter $u^{-1}p$.\n\\end{lemma}\n\\begin{proof} By using the equalities\n\\[\nf^\\varphi \\circ [u^{-1}p]_{\\hat{B}} \\circ f^{-1} = [u^{-1}p]_{\\mathbb{G}_m} \\circ f^\\varphi \\circ f^{-1} = [p]_{\\mathbb{G}_m}\n\\]\none computes that\n\n\\begin{eqnarray*}\n  [u^{-1}p]_{\\hat{B}} &=& \\left( f^\\varphi \\right)^{-1} \\circ [p]_{\\mathbb{G}_m} \\circ f \\\\\n &\\equiv& \\left( f^\\varphi \\right)^{-1} \\circ X^p \\circ f \\pmod{p} \\\\\n&=& \\left( f^{-1} \\right)^{\\varphi} \\left( f(X)^p \\right)  \\\\\n&=& \\left( f^{-1}(f(X)) \\right)^p \\\\\n&=& X^p.\n\\end{eqnarray*}\n\nThus, since it is well known that $ [u^{-1}p]_{\\hat{B}}  \\equiv u^{-1}p X \\pmod{\\deg 2}$, it follows that $[u^{-1}p]_{\\hat{B}} $ is a Lubin-Tate power series with respect to $u^{-1}p$, as claimed.\n\\end{proof}\n\n\n\nWe write $\\chi^{\\rm ur}$ for the restriction to $G_M$ of the character\n\\[\n\\chi_\\Qp^{\\rm ur} \\colon G_\\Qp \\lra \\Ze_p^\\times, \\quad \\varphi \\mapsto u^{-1}.\n\\]\n\nWe assume that the restriction of $\\chi^{\\rm ur}$ to $G_N$ is non-trivial and write $T$ for the (unramified) twist $\\Zp(\\chi^{\\rm ur})(1)$ of the representation $\\Zp(1)$.\n\nThen, by \\cite[Prop.~2.5]{IV} or \\cite[Lem. 3.2.1]{BC2}, the complex $R\\Gamma(N, T)$ is acyclic outside\ndegrees one and two and there are canonical identifications\n\\[\n H^i(N, T) = \\begin{cases} \\hat{B}(\\frp_N), &\\text{ if $i=1$,}\\\\\n  \\bigl( \\Zp \/ p^{\\omega_N} \\Zp \\bigr) (\\chi^{ur}),&\\text{ if $i=2$,}\\end{cases}\\]\nwhere $\\omega_N$ denotes the $p$-adic valuation of the element $1 - \\chi^{ur}(\\varphi^{f_{N\/\\Qp}})$.\n\nThese explicit descriptions allow one to interpret $R_{N\/M}(\\tilde B)$ in terms of differences between elements that occur in the formulations of the local epsilon constant conjecture for the representations $\\ZZ_p(1)$ and $T$, as studied by Benois and Berger \\cite{benoisberger}, Bley and Cobbe \\cite{BC2} and Izychev and Venjakob \\cite{IV}.\n\nIn this way one finds that the (assumed) compatibility of these conjectures for the representations $\\ZZ_p(1)$  and $T$ implies the following equality\n\n\\begin{equation}\\label{curve local eps conj} R_{N\/M}(\\tilde B) = \\delta_{\\Gamma,p}\\bigl(\\bigl(\\sum_{\\chi}u^{f_{M\/\\Qp}(s_M\\chi(1) + m_\\chi)}e_\\chi\\bigr)\n\\frac{^\\dagger((1 - (u\\cdot\\varphi^{-1})^{f_{M\/\\Qp}})e_{\\Gamma_0})}\n{^\\dagger((|\\kappa_M| - (u\\cdot\\varphi^{-1})^{-f_{M\/\\Qp}})e_{\\Gamma_0})}\\bigr)\n\\end{equation}\nwhere the conductor of each character $\\chi$ is $\\pi_M^{m_\\chi}\\calO_M$ and the different of $M\/\\QQ_p$ is $\\pi_M^{s_M}\\calO_M$.\n\nIn particular, the results of \\cite{BC2} imply that the equality (\\ref{curve local eps conj}) is unconditionally valid for certain natural families of wildly ramified extensions $N\/M$.\n\n\\section{Classical Selmer complexes and refined BSD}\\label{tmc}\n\nIn this section we study ${\\rm BSD}(A_{F\/k})$ under the assumption that $A$ and $F\/k$ satisfy the following list of hypotheses.\n\nIn this list we fix an {\\em odd} prime number $p$ and an intermediate field $K$ of $F\/k$ such that $\\Gal(F\/K)$ is a Sylow $p$-subgroup of $G$\n\\begin{itemize}\n\\item[(H$_1$)] The Tamagawa number of $A_{K}$ at each place in $S_K^A$ is not divisible by $p$;\n\\item[(H$_2$)] $S_K^A \\cap S_K^p = \\emptyset$ (that is, no place of bad reduction for $A_{K}$ is $p$-adic);\n\\item[(H$_3$)] For all $v$ in $S_K^p$ above a place in $S_k^F$ the reduction is ordinary and $A(\\kappa_v)[p^\\infty]$ vanishes;\n\\item[(H$_4$)] For all $v$ in $S_K^f\\setminus S_K^p$ above a place in $S_k^F$ the group $A(\\kappa_v)[p^\\infty]$ vanishes;\n\\item[(H$_5$)] $S_k^A\\cap S_k^F = \\emptyset$ (that is, no place of bad reduction for $A$ is ramified in $F$);\n\\item[(H$_6$)] $\\sha(A_F)$ is finite.\n\\end{itemize}\n\n\\begin{remark}\\label{satisfying H} {\\em For a fixed abelian variety $A$ over $k$ and extension $F\/k$ the hypotheses (H$_1$) and (H$_2$) are clearly satisfied by all but finitely many odd primes $p\n, (H$_4$) and (H$_5$) constitute a mild restriction on the ramification of $F\/k$ and (H$_6$) coincides with the claim of ${\\rm BSD}(A_{F\/k})$(i). However, the hypothesis~ (H$_3$) excludes the case that is called `anomalous' by Mazur in~\\cite{m} and, for a given $A$, there may be infinitely many primes $p$ for which there are $p$-adic places $v$ at which $A$ has good ordinary reduction but $A(\\kappa_v)[p]$ does not vanish.\nNevertheless, it is straightforward to describe examples of abelian varieties $A$ for which there are only finitely many such anomalous places -- see, for example, the result of Mazur and Rubin in~\\cite[Lem. A.5]{mr}.}\n\\end{remark}\n\n\\begin{remark} {\\em The validity of each of the hypotheses listed above is equivalent\nto the validity of the corresponding hypothesis with $A$ replaced by $A^t$ and we will often use this fact without explicit comment.}\n\\end{remark}\n\nIn this section we first verify and render fully explicit the computation (\\ref{bksc cohom}) of the cohomology of the Selmer complex introduced in Definition \\ref{bkdefinition}, thereby extending the computations given by Wuthrich and the present authors in \\cite[Lem. 4.1]{bmw}.\n\nSuch an explicit computation will be useful in the proof of the main result of \\S\\ref{comparison section} below. We shall also use Lemma \\ref{useful prel} to ensure that, under the hypotheses listed above, this complex belongs to the category $D^{\\rm perf}(\\ZZ_p[G])$.\n\nIn the main result of this section we shall then re-interpret ${\\rm BSD}(A_{F\/k})$ in terms of invariants that can be associated to the classical Selmer complex under the above listed hypotheses.\n\n\\subsection{The classical Selmer complex}\\label{explicitbk} We fix an odd prime number $p$ and a finite set of non-archimedean places $\\Sigma$ of $k$ with\n\\[ S_k^p\\cup (S_k^F\\cap S_k^f) \\cup S_k^A \\subseteq \\Sigma.\\]\n\nFor any such set $\\Sigma$ the classical Selmer complex ${\\rm SC}_{\\Sigma,p}(A_{F\/k})$ is defined as the mapping fibre of the morphism (\\ref{bkfibre}) in $D(\\ZZ_p[G])$.\n\nWe further recall from Lemma \\ref{independenceofsigma} that this complex is, in a natural sense, independent of the choice of $\\Sigma$ and so will be abbreviated to ${\\rm SC}_{p}(A_{F\/k})$.\n\nIn the next result we describe consequences of Lemma \\ref{useful prel} for this complex and also give a description of its cohomology that will be useful in the computations that are carried out in \\S \\ref{comparison section} below.\n\n\\begin{proposition}\\label{explicitbkprop}\n\nSet $C:= {\\rm SC}_{p}(A_{F\/k})$.\nThen the following claims are valid.\n\\begin{itemize}\n\\item[(i)] The complex $C$ is acyclic outside degrees one, two and three and there is a canonical identification $H^3(C)=A(F)[p^{\\infty}]^\\vee$ and a canonical inclusion of $H^1(C)$ into $H^1\\bigl(\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)},T_{p}(A^t)\\bigr)$.\n\\item[(ii)] Assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$). Then for every non-archimedean place $v$ of $k$ the $G$-modules $A^t(F_v)^\\wedge_p$ and $\\ZZ_p\\otimes_\\ZZ A^t(F_v)$ are cohomologically-trivial. In addition, the module $A^t(F_v)^\\wedge_p$ vanishes for every place $v$ in $S_k^F\\setminus S_k^p$.\n\nIn particular, the complex $C$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$.\n\\item[(iii)] Assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$). Then for each normal subgroup $J$ of $G$ there is a natural isomorphism in $D^{\\rm perf}(\\ZZ_p[G\/J])$ of the form\n\\[ \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}{\\rm SC}_{p}(A_{F\/k}) \\cong {\\rm SC}_{p}(A_{F^J\/k}).\\]\n\\item[(iv)] If $\\sha(A_F)$ is finite, then $H^1(C)$ identifies with the image of the injective Kummer map $A^t(F)_p\\to H^1\\bigl(\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)},T_{p}(A^t)\\bigr)$ and there is a canonical isomorphism of $H^2(C)$ with $\\Sel_p(A_F)^\\vee$ (that is described in detail in the course of the proof below).\n    \\end{itemize}\\end{proposition}\n\n\n\n\\begin{proof}\nThroughout this argument we abbreviate the rings $\\mathcal{O}_{k,S_k^\\infty\\cup\\Sigma}$ and $\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)}$ to $U_{k}$ and $U_{F}$ respectively.\n\nSince the complexes $R\\Gamma (k_v, T_{p,F}(A^t))$ for $v$ in $\\Sigma$ are acyclic in degrees greater than two, the first assertion of claim (i) follows directly from the definition of $C$ as the mapping fibre of the morphism (\\ref{bkfibre}).\n\nIn addition, the description of the complex ${\\rm SC}_{S_k^\\infty\\cup\\Sigma}(A_{F\/k},X)$ (for any module $X$ as in Proposition \\ref{prop:perfect}) as the mapping fibre of the morphism (\\ref{selmer-finite tri}) in $D(\\ZZ_p[G])$ also implies that $H^3(C)$ is canonically isomorphic to $H^3({\\rm SC}_{S_k^\\infty\\cup\\Sigma}(A_{F\/k},X))$. Proposition \\ref{prop:perfect}(ii) thus implies that $H^3(C)$ identifies with $A(F)[p^\\infty]^\\vee$.\n\nFinally, the explicit definition of $C$ as a mapping fibre (combined with Lemma \\ref{v not p}(i)) also gives an associated canonical long exact sequence\n\\begin{multline}\\label{longexact}0 \\to H^1(C) \\to H^1\\bigl(U_F,T_{p}(A^t)\\bigr) \\to\n\\bigoplus\\limits_{w'\\in S_F^p}T_p\\bigl(H^1(F_{w'},A^t)\\bigr) \\stackrel{\\delta}{\\to} H^2(C)\\\\\n \\to H^2\\bigl(U_{F},T_{p}(A^t)\\bigr) \\to \\bigoplus\\limits_{ w'\\in \\Sigma(F)}H^2\\bigl(F_{w'},T_{p}(A^t)\\bigr) \\to\n H^3(C) \\to 0\n\\end{multline}\nin which the third and sixth arrows are the canonical maps induced by localisation. In this sequence each term $T_p\\bigl(H^1(F_{w'},A^t)\\bigr)$ denotes the $p$-adic Tate module of $H^1(F_{w'},A^t)$, which we have identified with the quotient of $H^1(F_{w'},T_{p}(A^t))$ by the image of $A^t(F_{w'})_p^\\wedge$ under the canonical Kummer map.\n\n\nIn particular, the sequence (\\ref{longexact}) gives a canonical inclusion $H^1(C) \\subseteq H^1\\bigl(U_{F},T_{p}(A^t)\\bigr)$ and this completes the proof of claim (i).\n\n\n\n\n\nTurning to claim (ii) we note first that if $v$ does not belong to $S_k^A\\cup S_k^F$ then the cohomological-triviality of $A^t(F_v)^\\wedge_p$ follows directly from Lemma \\ref{useful prel}(ii).\n\nIn addition, if $v$ is $p$-adic, then $A^t(F_v)^\\wedge_p$ is cohomologically-trivial as a consequence of Lemma \\ref{useful prel}(ii) and (iii) and the given hypotheses (H$_2$) and (H$_3$).\n\nIt suffices therefore to consider the $G$-modules $A^t(F_v)^\\wedge_p$ for places in $(S_k^A\\cup S_k^F)\\setminus S_k^p$. For each such $v$ we write $C_v(A_F)$ for the direct sum over $w'$ in $S_k^v$ of the modules $H^0(F_{w'}, H^1(I_{w'}, T_{p}(A^t))_{\\rm tor})$ that occur in the exact sequence of Lemma \\ref{v not p}(ii).\n\nNow if $v$ belongs to $S_k^A$, then (H$_5$) implies $v$ is unramified in $F\/k$ and so the $\\ZZ_p[G]$-module\n$$T_{p,F}(A^t)^{I_v}\\cong\\ZZ_p[G]\\otimes_{\\ZZ_p}T_p(A^t)^{I_v}$$\nis free. In this case therefore, the natural exact sequence\n$$0\\to T_{p,F}(A^t)^{I_v}\\stackrel{1-\\Phi_v^{-1}}{\\longrightarrow}T_{p,F}(A^t)^{I_v}\\to H^1(\\kappa_v,T_{p,F}(A^t)^{I_v})\\to 0$$\nimplies that the $G$-module $H^1(\\kappa_v,T_{p,F}(A^t)^{I_v})$ is cohomologically-trivial.\n\nSince the conditions (H$_1$) and (H$_5$) combine in this case to imply that $C_v(A_F)$ vanishes (as in the proof of \\cite[Lem. 4.1(ii)]{bmw}) the cohomological-triviality of $A^t(F_v)^\\wedge_p$ therefore follows from the exact sequence in Lemma \\ref{v not p}(ii).\n\nFinally, we claim that $A^t(F_v)^\\wedge_p$ vanishes for each $v$ that belongs to $S_k^F\\setminus S_k^p$. To see this we note that, in this case, (H$_5$) implies $v$ does not belong to $S_k^A$ so that $C_v(A_F)$ vanishes whilst the conditions (H$_4$) and (H$_5$) also combine (again as in the proof of \\cite[Lem. 4.1(i)]{bmw}) to imply $H^1(\\kappa_v,T_{p,F}(A^t)^{I_v})$ vanishes. From the exact sequence of Lemma \\ref{v not p}(ii) we can therefore deduce that $A^t(F_v)^\\wedge_p$ vanishes, as claimed.\n\nAt this stage we have proved that for every non-archimedean place $v$ of $k$, the $G$-module $A^t(F_v)^\\wedge_p$ is cohomologically-trivial. Since each $\\ZZ_p[G]$-module $A^t(F_v)^\\wedge_p$ is finitely generated this implies that each complex $A^t(F_v)^\\wedge_p[-1]$ is an object of $D^{\\rm perf}(\\ZZ_p[G])$.\n\nGiven this fact, the final assertion of claim (ii) is a consequence of the definition of $C$ as the mapping fibre of (\\ref{bkfibre}) and the fact that, since $p$ is odd, the complexes $R\\Gamma(U_k,T_{p,F}(A^t))$ and $R\\Gamma (k_v, T_{p,F}(A^t))$ for each $v$ in $\\Sigma$ each belong to $D^{\\rm perf}(\\ZZ_p[G])$ (as a consequence, for example, of \\cite[Prop. 1.6.5(2)]{fukaya-kato}).\n\nTo complete the proof of claim (ii) we fix a non-archimedean place $v$ of $k$ and consider instead the $G$-module $\\ZZ_p\\otimes_\\ZZ A(F_v)$. We recall that  there exists a short exact sequence of $G$-modules of the form\n\\begin{equation*}\\label{finalassertion}0\\to\\mathcal{O}_{F,v}^d\\to A(F_v)\\to C\\to 0\\end{equation*}\nin which the group $C$ is finite. From this exact sequence one may in turn derive short exact sequences\n\\begin{equation}\\label{completions}0\\to((\\mathcal{O}_{F,v})^\\wedge_p)^d\\to A(F_v)^\\wedge_p\\to C^\\wedge_p\\to 0\\end{equation} and\n\\begin{equation}\\label{tensorproducts}0\\to(\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v})^d\\to\\ZZ_p\\otimes_\\ZZ A(F_v)\\to \\ZZ_p\\otimes_\\ZZ C\\to 0.\\end{equation}\n\nWe assume first that $v$ is $p$-adic. In this case the canonical maps $\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v}\\to(\\mathcal{O}_{F,v})^\\wedge_p$ and $\\ZZ_p\\otimes_\\ZZ C\\to C^\\wedge_p$ are bijective and hence the exactness of the above sequences implies that the canonical map $\\ZZ_p\\otimes_\\ZZ A(F_v)\\to A(F_v)^\\wedge_p$ is also an isomorphism. The $G$-module $\\ZZ_p\\otimes_\\ZZ A(F_v)$ is thus cohomologically-trivial, as required.\n\nWe finally assume that $v$ is not $p$-adic. In this case, the exact sequence (\\ref{completions}) gives an isomorphism $$A(F_v)^\\wedge_p\\cong C^\\wedge_p=\\ZZ_p\\otimes_\\ZZ C$$ and thus from the exact sequence (\\ref{tensorproducts}) we derive a short exact sequence\n$$0\\to(\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v})^d\\to\\ZZ_p\\otimes_\\ZZ A(F_v)\\to A(F_v)^\\wedge_p\\to 0.$$\nSince we have already established the cohomological-triviality of $A(F_v)^\\wedge_p$, we know that the $G$-module $\\ZZ_p\\otimes_\\ZZ A(F_v)$ is cohomologically-trivial if and only if the $G$-module $\\ZZ_p\\otimes_\\ZZ\\mathcal{O}_{F,v}$ is cohomologically-trivial. But the latter module is naturally a $\\QQ$-vector-space, and therefore is indeed cohomologically-trivial. This completes the proof of claim (ii).\n\nTurning to claim (iii) we note that the cohomological-triviality of the $\\ZZ_p[G]$-module $A^t(F_v)^\\wedge_p$ for each $v$ in $\\Sigma$ (as is proved by claim (ii) under the given hypotheses) implies that there are natural isomorphisms in $D(\\ZZ_p[G\/J])$ of the form\n\\begin{align*} \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}A^t(F_v)^\\wedge_p[-1] \\cong\\, &(\\ZZ_p[G\/J]\\otimes_{\\ZZ_p[G]}A^t(F_v)^\\wedge_p)[-1]\\\\\n\\cong\\, &H_0(J,A^t(F_v)^\\wedge_p)[-1]\\\\\n\\cong\\, &H^0(J,A^t(F_v)^\\wedge_p)[-1]\\\\\n= \\, & A^t(F^J_v)^\\wedge_p[-1],\\end{align*}\nwhere the third isomorphism is induced by the map sending each element $x$ of $A^t(F_v)^\\wedge_p$ to its image under the action of $\\sum_{g \\in J}g$.\n\nThe existence of the isomorphism in claim (iii) is then deduced by combining these isomorphisms together with the explicit definitions of the complexes ${\\rm SC}_{p}(A_{F\/k})$ and ${\\rm SC}_{p}(A_{F^J\/k})$ as mapping fibres and the fact (recalled, for example, from \\cite[Prop. 1.6.5(3)]{fukaya-kato}) that there are standard Galois descent isomorphisms in $D(\\ZZ_p[G\/J])$ of the form\n\\begin{equation}\\label{global descent} \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}R\\Gamma(U_k,T_{p,F}(A^t)) \\cong R\\Gamma(U_k,T_{p,F^J}(A^t))\\end{equation}\nand\n\\begin{equation}\\label{local descent} \\ZZ_p[G\/J]\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}R\\Gamma (k_v, T_{p,F}(A^t))\\cong R\\Gamma (k_v, T_{p,F^J}(A^t))\\end{equation}\nfor each $v$ in $\\Sigma$.\n\nTo prove claim (iv) we assume $\\sha(A_F)$ is finite and first prove that\nthe image of $H^1(C)$ in $H^1\\bigl(U_{F},T_{p}(A^t)\\bigr)$ coincides with the image of the injective Kummer map $$\\kappa:A^t(F)_p\\to H^1\\bigl(U_{F},T_{p}(A^t)\\bigr).$$\nIn order to do so,\nwe identify ${\\rm cok}(\\kappa)$ with the $p$-adic Tate module $T_p\\bigl(H^1\\bigl(U_F,A^t\\bigr)\\bigr)$ of $H^1\\bigl(U_F,A^t\\bigr)$.\n\nIt is clear that any element of ${\\rm im}(\\kappa)$ is mapped to the image of $A^t(F_{w'})_p^\\wedge$ in $H^1\\bigl(F_{w'},T_{p}(A^t)\\bigr)$ by localising at any place $w'$ in $S_F^p$.\n\nThe exactness of (\\ref{longexact}) therefore implies that ${\\rm im}(\\kappa)$ is contained in $H^1(C)$ and furthermore that we have a commutative diagram\nwith exact rows\n\\begin{equation}\\label{sha diag} \\xymatrix{\n0 \\ar[r] & A^t(F)_p \\ar[r] \\ar[d]^{\\kappa} & H^1\\bigl(U_F,T_{p}(A^t)\\bigr) \\ar[r] \\ar@{=}[d] & T_p\\bigl(H^1\\bigl(U_F,A^t\\bigr)\\bigr)  \\ar[r] \\ar[d] & 0\\\\\n0 \\ar[r] & H^1(C) \\ar[r] & H^1\\bigl(U_F,T_{p}(A^t)\\bigr) \\ar[r] & \\bigoplus\\limits_{w'\\in S_F^p}T_p\\bigl(H^1(F_{w'},A^t)\\bigr),\n}\\end{equation}\nwhere the right-most vertical arrow is induced by the localisation maps.\nBut the assumed finiteness of $\\sha(A_F)$ (combined with \\cite[Ch. I, Cor 6.6]{milne}) then implies that this arrow is injective, and therefore the Snake Lemma implies that $\\im(\\kappa)=H^1(C)$, as required.\n\n\n\nTo conclude the proof of claim (iv) we use the canonical exact triangle\n\\begin{multline}\\label{compacttriangle}R\\Gamma_c\\bigl(U_F,T_p(A^t)\\bigr)\\to R\\Gamma\\bigl(U_k,T_{p,F}(A^t)\\bigr)\\to \\bigoplus\\limits_{ v\\in S_k^\\infty\\cup\\Sigma} R\\Gamma(k_v,T_{p,F}(A^t))\\\\ \\to R\\Gamma_c\\bigl(U_F,T_p(A^t)\\bigr)[1]\\end{multline}\nin $D(\\ZZ_p[G])$. We also write $\\Delta$ for the canonical composite homomorphism\n$$\\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge\\to\\bigoplus\\limits_{ w'\\in \\Sigma(F)}H^1(F_{w'},T_{p}(A^t))\\to H^2_c(U_F,T_p(A^t)),$$\nwith the first arrow given by the local Kummer maps and the second arrow given by the long exact cohomology sequence associated to the triangle (\\ref{compacttriangle}). We then claim that there is a canonical commutative diagram\n\n\\begin{equation}\\label{Selmerdiagram}\\xymatrix\nH^2_c\\bigl(U_F,T_p(A^t)\\bigr) \\ar[d] \\ar@{=}[r] &\n H^2_c\\bigl(U_F,T_{p}(A^t)\\bigr) \\ar[d] \\ar[r]^-{w\\circ s^{-1}} &\nH^1\\bigl(U_F,A[p^\\infty])^\\vee\n \\ar[d] \\\\\nH^2(C) \\ar^-{\\sim}[r] &\n\\cok(\\Delta) \\ar[r]^-{\\sim} &\n\\Sel_p(A_F)^\\vee\n.\n}\\end{equation}\nHere the second and third vertical arrows are the canonical projection maps and $w$ and $s$ are the isomorphisms defined in (\\ref{themapw}) and (\\ref{themaps}) in Appendix \\ref{exp rep section} below.\n\nThe composition of the horizontal arrows in the bottom row of the diagram (\\ref{Selmerdiagram}) will then define the desired canonical isomorphism of $H^2(C)$ with $\\Sel_p(A_F)^\\vee$.\n\nTo verify the existence of the diagram (\\ref{Selmerdiagram}) we use the canonical exact sequence\n\\begin{multline}\\label{comparingsequence}0\\to\\bigoplus\\limits_{ v\\in S_k^\\infty} H^0(k_v,T_{p,F}(A^t)) \\to H^1_c\\bigl(U_F,T_p(A^t)\\bigr)\\to H^1(C)\\\\ \\to\\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_p^\\wedge\\stackrel{\\Delta}{\\to}H^2_c\\bigl(U_F,T_p(A^t)\\bigr)\\to H^2(C)\\to 0\\end{multline} associated to the exact triangle (\\ref{comparingtriangles}).\n(Here we have used the fact that, as $p$ is odd, the group $H^i(k_v,T_{p,F}(A^t))$ vanishes for every $v$ in $S_k^\\infty$ and every $i > 0$.)\n\nThis exact sequence induces the desired canonical isomorphism of $H^2(C)$ with $\\cok(\\Delta)$ and, by construction, the last map occurring in the sequence gives a vertical map making the first square of the diagram (\\ref{Selmerdiagram}) commute.\n\nIt is finally straightforward, using the commutativity of the diagram in Corollary \\ref{Tatepoitouexplicit} below, to deduce that the isomorphism $$w\\circ s^{-1}:H^2_c\\bigl(U_{F},T_{p}(A^t)\\bigr) \\to\nH^1\\bigl(U_{F},A[p^\\infty])^\\vee$$ induces an isomorphism $$\\cok(\\Delta)\\stackrel{\\sim}{\\to}\\Sel_p(A_F)^\\vee.$$ This induced isomorphism completes the construction of the diagram (\\ref{Selmerdiagram}) and thus also the proof of claim (iv).\\end{proof}\n\n\n\n\n\n\nIn the next result we shall (exceptionally for \\S\\ref{tmc}) consider the prime $2$ and describe an analogue of Proposition \\ref{explicitbkprop} in this case.\n\n\\begin{proposition}\\label{explicitbkprop2} The following claims are valid for the complex $C:= {\\rm SC}_{2}(A_{F\/k})$.\n\\begin{itemize}\n\\item[(i)] $C$ is acyclic outside degrees one, two and three.\n\\item[(ii)] If $\\sha(A_F)$ is finite, then $H^1(C)$ identifies with the image of the injective Kummer map $A^t(F)_2\\to H^1\\bigl(\\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)},T_{2}(A^t)\\bigr)$ and there exists a\n    canonical homomorphism $\\Sel_2(A_F)^\\vee \\to H^2(C)$, the kernel and cokernel of which are both finite.\n\\item[(iii)]  The module $H^3(C)$ is finite.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} Claim (i) is established by the same argument that is used to prove the first assertion of Proposition \\ref{explicitbkprop}(i).\n\nIn a similar way, the analysis concerning the diagram (\\ref{sha diag}) is also valid in the case $p=2$ and proves the first assertion of claim (ii).\n\nTo prove the remaining claims we set $U_F := \\mathcal{O}_{F,S_k^\\infty(F)\\cup \\Sigma(F)}$ and note that the long exact cohomology sequence of the exact triangle (\\ref{comparingtriangles}) gives rise in this case to an exact sequence\n\\begin{multline*} \\bigoplus\\limits_{ v\\in \\Sigma}A^t(F_v)_2^\\wedge \\oplus\\bigoplus\\limits_{ v\\in S_k^\\infty} H^1(k_v,T_{2,F}(A^t)) \\to H^2_c\\bigl(U_F,T_2(A^t)\\bigr) \\to H^2(C)\\\\\n\\to \\bigoplus\\limits_{ v\\in S_k^\\infty} H^2(k_v,T_{2,F}(A^t)) \\to H^3_c(U_F,T_{2}(A^t)) \\to H^3(C) \\to \\bigoplus_{ v\\in S_k^\\infty} H^3(k_v,T_{2,F}(A^t)).\\end{multline*}\nIn addition, for each $v$ in $S_k^\\infty$ and each $j \\in \\{1,2,3\\}$ the group $H^j(k_v,T_{2,F}(A^t))$ is finite.\n\nGiven these facts, the second assertion of claim (ii) is a consequence of Artin-Verdier Duality (just as with the analogous assertion in Proposition \\ref{explicitbkprop}(iv)) and claim (iii) follows directly from the isomorphism (\\ref{artinverdier}).\\end{proof}\n\n\n\n\\subsection{Statement of the main result}\\label{somr tmc sec} We continue to assume that the  hypotheses (H$_1$)-(H$_6$) are satisfied.\n\nIn this case, for any isomorphism of fields $j:\\CC\\cong\\CC_p$, the isomorphism\n\\[ h^{j}_{A,F}:=\\CC_p\\otimes_{\\RR,j}h_{A,F}^{{\\rm det}}\\]\ncombines with the explicit descriptions given in Proposition \\ref{explicitbkprop} to give a canonical element\n\\[ \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})\\]\nof $K_0(\\ZZ_p[G],\\CC_p[G])$. By Lemma \\ref{independenceofsigma} (and Remark \\ref{indeptremark}) this element is in particular independent of the choice of set $\\Sigma$ with respect to which ${\\rm SC}_p(A_{F\/k})={\\rm SC}_{\\Sigma,p}(A_{F\/k})$ is defined. In the rest of \\S\\ref{tmc} we may and will thus set $$\\Sigma:=S_k^p\\cup (S_k^F\\cap S_k^f) \\cup S_k^A.$$\n\nOur aim in the rest of \\S\\ref{tmc} is to interpret ${\\rm BSD}_p(A_{F\/k})$(iv) in terms of an explicit description of\n this element.\n\n\n\n\\subsubsection{} At the outset we note that Hypotheses (H$_2$) and (H$_3$) imply that for each $v$ in $S_k^p$ the restriction $A^t_v$ of $A^t$ to $k_v$ satisfies the conditions that are imposed on $B$ in Proposition \\ref{basic props} and hence that the element $R_{F_w\/k_v}(\\tilde A^t_v)$ of $K_0(\\ZZ_p[G_w],\\QQ_p[G_w])$ is well-defined.\n\nWe write $d$ for ${\\rm dim}(A)$ and then define an element of $K_0(\\ZZ_p[G],\\QQ_p[G])$ by setting\n\n\\[ R_{F\/k}(\\tilde A^t_v) := {\\rm ind}^G_{G_w}(d\\cdot R_{F_w\/k_v}+ R_{F_w\/k_v}(\\tilde A^t_v)).\\]\nHere ${\\rm ind}^G_{G_w}$ is the induction homomorphism $K_0(\\ZZ_p[G_w],\\QQ_p[G_w])\\to K_0(\\ZZ_p[G],\\QQ_p[G])$ and $R_{F_w\/k_v}$ is the canonical element of $K_0(\\ZZ_p[G_w],\\QQ_p[G_w])$ that is defined by Breuning in \\cite{breuning2} (and will be explicitly recalled in the course of the proof of Proposition \\ref{heavy part} below).\n\nIn the sequel we will fix a finite set of places $S$ of $k$ as in \\S\\ref{selmer section} (and hence as in the statement of Conjecture \\ref{conj:ebsd}).\n\nWe abbreviate $S_k^F\\cap S_k^f$ to $S_{\\rm r}$ and set $S_{p,{\\rm r}} := S_k^p\\cap S_k^F$.\nWe shall also write $S_{p,{\\rm w}}$ and $S_{p,{\\rm t}}$ for the (disjoint) subsets of $S_{p,{\\rm r}}$ comprising places that are respectively wildly and tamely ramified in $F$ and $S_{p,{\\rm u}}$ for the set $S_k^p\\setminus S_{p,{\\rm r}}$ of $p$-adic places in $k$ that do not ramify in $F$.\n\nFor each place $v$ in $S_k^p$ and each character $\\psi$ in $\\widehat{G}$ we define a non-zero element\n\\[ \\varrho_{v,\\psi} := {\\rm det}({\\rm N}v\\mid V_\\psi^{I_w})\\]\nof $\\QQ^c$.\n\nWe then define an invertible element of $\\zeta(\\CC[G])$ by setting\n\\begin{equation}\\label{bkcharelement}\n \\mathcal{L}^*_{A,F\/k} := \\sum_{\\psi \\in \\widehat{G}} \\frac{L^{*}_{S_{\\rm r}}(A,\\check{\\psi},1)\\cdot \\tau^{\\ast}(\\QQ,\\psi)^d\\cdot \\prod_{v\\in S_{p,{\\rm r}}}\\varrho_{v,\\psi}^d}{\\Omega_A^\\psi\\cdot w_\\psi^d}\\cdot e_\\psi\\end{equation}\nwhere, for each $\\psi$ in $\\widehat{G}$, the period $\\Omega_A^\\psi$ and root number $w_\\psi$ are as defined in \\S\\ref{k theory period sect2}, the modified global Galois-Gauss sum $\\tau^{\\ast}(\\QQ,\\psi)$ is as defined in \\S \\ref{mod GGS section} and the Hasse-Weil-Artin $L$-series $L_{S_{\\rm r}}(A,\\check{\\psi},z)$ is truncated by removing only the Euler factors corresponding to places in $S_{\\rm r}$.\n\n\n\n\\subsubsection{}\n\nWe can now state the main result of this section. In order to do so we use the homomorphism\n$$\\delta_{G,p}:\\zeta(\\CC_p[G])^\\times\\to K_0(\\ZZ_p[G],\\CC_p[G])$$ defined in (\\ref{G,O hom}) and also the local Fontaine-Messing correction $\\mu_v(A_{F\/k})$ terms defined in (\\ref{localFM}).\n\n\\begin{theorem}\\label{bk explicit} Assume $A$, $F\/k$ and $p$ satisfy all of the hypotheses (H$_1$)-(H$_6$).\n\n\nThen the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if for every isomorphism of fields $j:\\CC\\cong \\CC_p$ one has\n\\[ \\delta_{G,p}(j_\\ast(\\mathcal{L}^*_{A,F\/k}))=\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})  + \\sum_{v \\in S_{p,{\\rm w}}}R_{F\/k}(\\tilde A^t_v)+\n\\sum_{v\\in S_{p,{\\rm u}}^*} \\mu_v(A_{F\/k}). \\\nHere $S^*_{p,{\\rm u}}$ is the subset of $S_{p,{\\rm u}}$ comprising places that divide the different of $k\/\\QQ$.\n\\end{theorem}\n\n\\begin{remark}\\label{emptysets}{\\em If the sets $S_{p,{\\rm w}}$ and $S^*_{p,{\\rm u}}$ are empty, then the above result implies that the equality in ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if one has\n\\[ \\delta_{G,p}(\\mathcal{L}^*_{A,F\/k})=\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}).\\]\nThis is, in particular, the case if $p$ is unramified in $F\/\\QQ$ and, in this way, Theorem \\ref{bk explicit} recovers the results of Wuthrich and the present authors in \\cite[Prop. 4.2 and Th. 4.3]{bmw}. More generally, the above equality is predicted whenever no $p$-adic place of $k$ is wildly ramified in $F$ and, in addition, $p$ is unramified in $k\/\\QQ$ (as is obviously the case if $k = \\QQ$) and this case will play an important role in the special settings considered in \\S\\ref{mod sect} and \\S\\ref{HHP}. }\\end{remark}\n\n\\begin{remark}\\label{breuning remark}{\\em In \\cite[Conj. 3.2]{breuning2} Breuning has conjectured that the terms $R_{F_w\/k_v}$ should always vanish. In \\cite{breuning} and \\cite{breuning2} he has proved this conjecture for all tamely ramified extensions, for all abelian\nextensions of $\\QQ_p$ with $p$ odd, for all $S_3$-extensions of $\\QQ_p$ and for certain families of\ndihedral and quaternion extensions. If $p$ is odd, then Bley and Debeerst \\cite{bleydebeerst} have also given an algorithmic proof of the conjecture for all Galois extensions of $\\QQ_p$ of degree at most $15$. More recently, Bley and Cobbe \\cite{BC} have proved the conjecture for certain natural families of wildly ramified extensions. }\\end{remark}\n\n\n\\begin{remark}{\\em If $A$ is an elliptic curve, then Remark \\ref{breuning remark} combines with the equality in (\\ref{curve local eps conj}) to give a completely explicit description of the elements $R_{F\/k}(\\tilde A_v)$. However, whilst the results of \\cite{BC2} imply that this description is unconditionally valid for certain families of wildly ramified extensions, it is, in general, conjectural.}\\end{remark}\n\n\n\\begin{remark}\\label{bsdinvariants}{\\em If $F=k$ then it can be shown that the element (\\ref{bkcharelement}) is equal to the product $(-1)^d\\cdot(L^\\ast(A,1)\/\\Omega_A)\\cdot (\\sqrt{|d_k|)}^{d}$ with $$\\Omega_A=\\prod_{v\\in S_k^\\CC}\\Omega_{A,v}\\cdot\\prod_{v\\in S_k^\\RR}\\Omega_{A,v}^+,$$ where the classical periods $\\Omega_{A,v}$ and $\\Omega_{A,v}^+$ are as defined in \\S\\ref{k theory period sect2}.}\n\\end{remark}\n\n\n\\subsection{The proof of Theorem \\ref{bk explicit}}\n\n\n\\subsubsection{}\\label{clever peiods}\n\nIn view of Lemma \\ref{pro-p lemma} it is enough for us to fix a field isomorphism $j:\\CC\\cong \\CC_p$ and show that the displayed equality in Theorem \\ref{bk explicit} is equivalent to (\\ref{displayed pj}).\n\n\nTaking advantage of Remark \\ref{consistency remark}(i), we first specify the set $S$ to be equal to $S^\\infty_k\\cup S_k^F\\cup S_k^A$. We shall next use the approach of \\S\\ref{k theory period sect} to make a convenient choice of differentials $\\omega_\\bullet$.\n\nFor each $v$ in $S_k^p$ we set\n\\[ \\mathcal{D}_v := \\Hom_{\\mathcal{O}_{k_v}}(H^0(\\mathcal{A}_v^t,\\Omega^1_{\\mathcal{A}_v^t}), \\mathcal{O}_{k_v}),\\]\nwhere the N\\'eron models $\\mathcal{A}^t_v$ are as fixed at the beginning of \\S\\ref{perf sel sect}.\n\nFor each such $v$ we also fix a free (rank one) $\\mathcal{O}_{k_v}[G]$-submodule $\\mathcal{F}_v$ of $F_v = k_v\\otimes_k F$ and we assume that for each $v \\in S_{p,{\\rm u}}$ one has\n\\[ \\mathcal{F}_v = \\mathcal{O}_{F,v} = \\mathcal{O}_{k_v}\\otimes_{\\mathcal{O}_{k}}\\mathcal{O}_{F}.\\]\n\nWe then set\n\\[ \\Delta(\\mathcal{F}_v) := \\mathcal{F}_v\\otimes_{\\mathcal{O}_{k_v}}\\mathcal{D}_v.\\]\n\nFor each place $w'$ in $S_F^p$ we write $\\Sigma(F_{w'})$ for the set of $\\QQ_p$-linear embeddings $F_{w'} \\to \\QQ_p^c$, we define a $\\ZZ_p[G_{w'}]$-module $Y_{F_{w'}} := \\prod_{\\sigma \\in \\Sigma(F_{w'})}\\ZZ_p$ (upon which $G_{w'}$ acts via precomposition with the embeddings) and write\n\\[ \\pi_{F_{w'}}: \\QQ_p^c\\otimes_{\\ZZ_p}F_{w'} \\to \\QQ_p^c\\otimes_{\\ZZ_p}Y_{F_{w'}}\\]\nfor the isomorphism of $\\QQ_p^c[G_{w'}]$-modules that sends each element $\\ell\\otimes f$ to $(\\ell\\otimes \\sigma(f))_\\sigma$.\n\nFor each $v$ in $S_k^p$ we then consider the isomorphism of $\\QQ_p[G]$-modules\n\\[ \\pi_{F_v}: \\QQ_p^c\\otimes_{\\ZZ_p}F_v = \\prod_{w'\\in S_F^v}(\\QQ_p^c\\otimes_{\\ZZ_p}F_{w'}) \\xrightarrow{(\\pi_{F_{w'}})_{w'}} \\QQ_p^c\\otimes_{\\ZZ_p}\\bigoplus_{w' \\in S_F^v}Y_{F_{w'}} = \\QQ_p^c\\otimes_{\\ZZ_p}Y_{F_v},  \\]\nwhere we set $Y_{F_v} := \\bigoplus_{w'}Y_{F_{w'}}$.\n\nAfter fixing an embedding of $\\QQ^c$ into $\\QQ_p^c$ we obtain an induced identification of $\\bigoplus_{v \\in S_k^p}Y_{F_v}$ with the module $Y_{F,p} := \\bigoplus_{\\Sigma(F)}\\ZZ_p$, upon which $G$ acts via pre-composition on the embeddings.\n\nWe next fix\n\\begin{itemize}\n\\item[$\\bullet$] an ordered $k$-basis $\\{\\omega'_j:j \\in [d]\\}$ of $H^0(A^t,\\Omega^1_{A^t})$ that generates over $\\mathcal{O}_{k,p}$ the module\n $\\mathcal{D}_p := \\prod_{v \\in S_k^p}\\mathcal{D}_v$, and\n\\item[$\\bullet$] an ordered $\\ZZ_p[G]$-basis $\\{z_b:b \\in [n]\\}$ of $\\mathcal{F}_p := \\prod_{v\\in S_k^p}\\mathcal{F}_v$.\n\\end{itemize}\n\nThen the (lexicographically ordered) set\n\\[ \\omega_\\bullet:= \\{ z_b\\otimes \\omega'_j: b \\in [n], j \\in [d]\\}\\]\nis a $\\QQ_p[G]$-basis of $H^0(A_F^t,\\Omega^1_{A_F^t}) = F\\otimes_kH^0(A^t,\\Omega^1_{A^t})$ and the\n\narguments of Lemma \\ref{k-theory period} and Proposition \\ref{lms} combine to show that\n\\begin{equation}\\label{norm resolvents} \\partial_{G,p}\\left(j_*(\\Omega_{\\omega_\\bullet}(A_{F\/k}))\\right)=\\delta_{G,p}\\left(j_*(\\Omega_A^{F\/k}\\cdot w_{F\/k}^d)\\right) + \\sum_{v\\in S_k^p}d\\cdot[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\end{equation}\nin $K_0(\\ZZ_p[G],\\CC_p[G])$.\n\n\\subsubsection{}Now, if necessary, we can multiply $\\mathcal{F}$ by a sufficiently large power of $p$ in order to ensure that for every $v$ in $S_{p,{\\rm r}}$ the following two conditions are satisfied.\n\n\\begin{itemize}\n\\item[$\\bullet$] the $p$-adic exponential map induces a (well-defined) injective homomorphism from $\\mathcal{F}_v$ to $(F_v^\\times)^\\wedge_p$;\n\\item[$\\bullet$] the formal group exponential ${\\rm exp}_{A^t,F_v}$ that arises from the differentials $\\{\\omega'_j:j \\in [d]\\}$ induces an isomorphism of $\\Delta(\\mathcal{F}_v)$ with a submodule of $A^t(F_v)^\\wedge_p$.\n\\end{itemize}\n\nFor each $v$ in $S_k^p$ we now set\n\\[ X(v) := \\begin{cases}{\\rm exp}_{A^t,F_v}(\\Delta(\\mathcal{F}_v)), &\\text{ if $v \\in S_{p,{\\rm r}}$}\\\\\nA^t(F_v)^\\wedge_p, &\\text{ if $v \\in S_{p,{\\rm u}}$.}\n\\end{cases}\\]\nThen it is clear that, for any choice of $\\gamma_\\bullet$ as in \\S \\ref{perf sel sect} and our specific choices of $S$ and $\\omega_\\bullet$, the module $X(v)$ coincides with $\\mathcal{X}(v)$ for $\\mathcal{X}=\\mathcal{X}_S(\\{\\mathcal{A}_v^t\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$.\n\nThe description of the complex\n\\[ C_{X(p)} := {\\rm SC}_{S}(A_{F\/k};X(p),H_\\infty(A_{F\/k})_p)\\]\nas the mapping fibre of the morphism (\\ref{selmer-finite tri}) gives rise to an exact triangle\n\\[ C_{X(p)} \\to {\\rm SC}_p(A_{F\/k}) \\oplus X(p)[-1] \\xrightarrow{(\\lambda', \\kappa'_1)}\n\\bigoplus_{v \\in S_k^p\\cup S_k^A} A^t(F_v)_p^\\wedge[-1]\\to C_{X(p)}[1].\\]\nHere we have used the fact that, for $v\\in S_{\\rm r}\\setminus S_k^p$, the module $A^t(F_v)_p^\\wedge$ vanishes by Proposition \\ref{explicitbkprop}(ii).\n\nFurther, since Proposition \\ref{explicitbkprop}(ii) implies that the $\\ZZ_p[G]$-modules\n\\[ X(p):= \\prod_{v\\in S_k^p}X(v) \\,\\,\\text{ and }\\,\\, \\bigoplus_{v \\in S_k^p\\cup S_k^A} A^t(F_v)_p^\\wedge\\]\nare cohomologically-trivial and that the complex ${\\rm SC}_p(A_{F\/k})$ is perfect, Proposition \\ref{prop:perfect}(i) implies that this is a triangle in $D^{\\rm perf}(\\ZZ_p[G])$.\n\nBy applying Lemma \\ref{fk lemma} to this exact triangle, we can therefore deduce that there is in $K_0(\\ZZ_p[G],\\CC_p[G])$ an equality\n\n\\begin{align}\\label{first comp} &\\chi_{G,p}(C_{X(p)},h^{j}_{A,F})\\\\\n =\\, &\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}) - \\sum_{v \\in S_k^A}\\chi_{G,p}(A^t(F_v)_p^\\wedge[-1],0)\\notag\\\\\n &\\hskip 2truein - \\sum_{v\\in S_k^p}\\chi_{G,p}(X(v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm id})\\notag\\\\\n= \\, & \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}) - \\sum_{v \\in S_k^A}\\chi_{G,p}(A^t(F_v)_p^\\wedge[-1],0)\\notag\\\\\n&\\hskip 2truein - \\sum_{v\\in S_{p,{\\rm r}}}\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v})\\notag\\\\\n= \\, & \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F}) - \\delta_{G,p}\\Bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\Bigr)\\notag\\\\\n&\\hskip 2truein - \\sum_{v\\in S_{p,{\\rm r}}}\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v})\\notag\\end{align}\nwhere the last equality holds because, for each $v\\in S_k^A$, one has\n\\[ \\chi_{G,p}(A^t(F_v)_p^\\wedge[-1],0)=  \\delta_{G,p}\\Bigl( L_v(A,F\/k)\\Bigr).\\]\nThis equality in turn follows upon combining the argument that gives \\cite[(13)]{bmw} with the exactness of the sequence of Lemma \\ref{v not p}(ii) for each place $w'$ of $F$ above a place in $S_k^A$ and the fact that the third term occurring in each of these sequences vanishes, as verified in the course of the proof of Proposition \\ref{explicitbkprop}(ii).\n\n\n\\subsubsection{}The equalities (\\ref{norm resolvents}) and (\\ref{first comp}) lead us to consider for each place $v$ in $S_k^p$ the element\n\\[ c(F\/k,\\tilde A^t_v) := \\begin{cases} d\\cdot[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]-\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v}), &\\text{if $v \\in S_{p,{\\rm r}}$,}\\\\\nd\\cdot[\\mathcal{O}_{F,v},Y_{F_v};\\pi_{F_v}], &\\text{if $v \\in S_{p,{\\rm u}}$}.\\end{cases}\\]\n\nIt is straightforward to check that for each $v \\in S_{p,{\\rm r}}$ this element is independent of the choice of $\\mathcal{F}_v$ and that Lemma \\ref{twist dependence} implies its dependence on $A$ is restricted to (the twist matrix of) the reduction of $A^t$ at $v$.\n\nThe key step, however, in the proof of Theorem \\ref{bk explicit} is the computation of this element in term of local Galois-Gauss sums that is described in the next result.\n\nFor each place $v$ in $S_k^f$ we define an\n`equivariant local Galois-Gauss sum' by setting\n\\[ \\tau_v(F\/k) := \\sum_{\\psi \\in \\widehat{G}}\\tau(\\QQ_{\\ell(v)},\\psi_v)\\cdot e_\\psi\\in \\zeta(\\QQ^c[G])^\\times.\\]\nHere $\\psi_v$ denotes the restriction of $\\psi$ to $G_w$ and $\\tau(\\QQ_{\\ell(v)},\\psi_v)$ is the Galois-Gauss sum (as defined in \\cite{martinet}) of the induction to $G_{\\QQ_{\\ell(v)}}$ of the character of $G_{k_v}$ that is obtained by composing $\\psi_v$ with the natural projection $G_{k_v} \\to G_w$.\n\nWe also define a modified local Galois-Gauss sum by setting\n\\[ \\tau_v^p(F\/k) := \\begin{cases} \\varrho_v(F\/k)\\cdot u_v(F\/k)\\cdot \\tau_v(F\/k), &\\text{ if $v \\in S_{p,{\\rm r}}$,}\\\\\n                                              u_v(F\/k)\\cdot \\tau_v(F\/k), &\\text{ otherwise,}\\end{cases}\\]\nwhere we set\n\\begin{equation}\\label{varrho def} \\varrho_{v}(F\/k) := \\sum_{\\psi\\in \\widehat{G}}\\varrho_{v,\\psi}\\cdot e_\\psi\\end{equation}\nand the element $u_v(F\/k)$ of $\\zeta(\\QQ[G])^\\times$ is as defined in (\\ref{u def}).\n\nFinally, for each $p$-adic place $v$ of $k$ we set\n\n\\[ U_v(F\/k) := {\\rm ind}^{G}_{G_w}(U_{F_w\/k_v}),\\]\nwhere $U_{F_w\/k_v}$ is the `unramified term' in $K_0(\\ZZ_p[G_w],\\QQ_p^c[G_w])$ that is defined by Breuning in \\cite[\\S2.5]{breuning2}.\n\n\\begin{proposition}\\label{heavy part} For each place $v$ in $S_k^p$ and any choice of $j$ one has\n\\[c(F\/k,\\tilde A^t_v) =  d\\cdot \\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + d\\cdot U_v(F\/k) - R_{F\/k}(\\tilde A^t_v).\\]\nin $K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\\end{proposition}\n\n\\begin{proof} The term $\\delta_{G,p}(j_*(\\tau^p_v(F\/k)))$ is independent of the choice of $j$ by \\cite[Lem. 2.2]{breuning2}. To prove the claimed equality we consider separately the cases $v \\in S_{p,{\\rm r}}$ and $v \\in S_{p,{\\rm u}}$.\n\nWe assume first that $v$ belongs to $S_{p,{\\rm r}}$. Then for every place $w'$ in $S_F^v$ we consider the corresponding perfect complex of $\\ZZ_p[G_{w'}]$-modules $R\\Gamma(F_{w'},\\ZZ_p(1))$ as described in \\S \\ref{twist inv prelim} and obtain a perfect complex of $\\ZZ_p[G]$-modules\n$$R\\Gamma(F_v,\\ZZ_p(1)):=\\prod_{w'\\in S_F^v}R\\Gamma(F_{w'},\\ZZ_p(1)).$$\n\nSince Kummer theory canonically identifies the cohomology in degree one of this complex with $(F_v^\\times)_p^\\wedge$ we may define an additional perfect complex of $\\ZZ_p[G]$-modules $C^\\bullet_{\\mathcal{F}_v}$ through the exact triangle\n\\[ \\mathcal{F}_v[0] \\xrightarrow{\\alpha_v} R\\Gamma(F_v,\\ZZ_p(1))[1] \\to C^\\bullet_{\\mathcal{F}_v} \\to \\mathcal{F}_v[1] \\]\nin $D^{\\rm perf}(\\ZZ_p[G])$, with $H^0(\\alpha_v)$ induced by the $p$-adic exponential map ${\\rm exp}_p$.\n\nWe write $f$ for the residue degree of our fixed place $w$ in $S_F^v$. Then the long exact cohomology sequence of the above triangle implies that the normalised valuation map ${\\rm val}_{F\/k,v} := (f\\cdot({\\rm val}_{F_{w'}}))_{w'\\in S_F^v}$\ninduces an isomorphism of $\\QQ_p[G]$-modules\n\\begin{multline*} \\QQ_p \\cdot H^0(C^\\bullet_{\\mathcal{F}_v})  \\cong \\QQ_p\\cdot ((F_v^\\times)^\\wedge_p\/{\\rm exp}_p(\\mathcal{F}_v)) \\xrightarrow{ {\\rm val}_{F\/k,v}}\\prod_{w'\\in S_F^v}\\QQ_p \\\\\n\\xleftarrow{ ({\\rm inv}_{F_{w'}})_{w'}} \\QQ_p\\cdot \\prod_{w'\\in S_F^v}H^2(F_{w'},\\ZZ_p(1)) \\cong \\QQ_p\\cdot H^1(C^\\bullet_{\\mathcal{F}_v})\\end{multline*}\nwhich, by abuse of notation, we also denote by ${\\rm val}_{F\/k,v}$.\n\nIn addition, the chosen differentials $\\{\\omega'_a: a \\in [d]\\}$ induce an isomorphism of\n $\\mathcal{O}_{k_v}$-modules $\\mathcal{D}_{v}\\cong \\mathcal{O}^d_{k_v}$\nand hence an isomorphism of $\\mathcal{O}_{k_v}[G]$-modules $\\omega_{v,*}: \\Delta(\\mathcal{F}_v) \\cong \\mathcal{F}_v^d$.\n\nIn particular, if we write $C_{A^t,F}^{v,\\bullet}$ for the complex $\\prod_{w'\\in S_F^v} C_{A^t_v,F_{w'}}^\\bullet$, where each complex $C_{A^t_v,F_{w'}}^\\bullet$ is as defined at the beginning of \\S\\ref{twist inv prelim}, then there exists a canonical exact triangle\n\\[ \\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1] \\xrightarrow{\\iota_v} C_{A^t,F}^{v,\\bullet} \\xrightarrow{\\iota'_v} C^{\\bullet,d}_{\\mathcal{F}_v} \\to \\Delta(\\mathcal{F}_v)[1]\\oplus A^t(F_v)^\\wedge_p[0]\\]\nin $D^{\\rm perf}(\\ZZ_p[G])$.\nHere $C^{\\bullet,d}_{\\mathcal{F}_v}$ denotes the product of $d$ copies of $C^{\\bullet}_{\\mathcal{F}_v}$, $H^0(\\iota_v)$ is the composite map $({\\rm exp}_p)^d\\circ \\omega_{v,*}$ and $H^1(\\iota_v)$ is the identity map between\n$A^t(F_v)^\\wedge_p = H^1(A^t(F_v)^\\wedge_p[-1])$ and the direct summand $A^t(F_v)^\\wedge_p$ of $H^1(C_{A^t,F}^{v,\\bullet})$.\n\nThe long exact cohomology sequence of this triangle also gives an exact commutative diagram of $\\QQ_p[G]$-modules\n\\[\\minCDarrowwidth1em\\begin{CD} 0 @> >>\\! \\QQ_p\\cdot \\Delta(\\mathcal{F}_v)\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^0(\\iota_v) >>\\! \\QQ_p\\cdot H^0(C_{A^t,F}^{v,\\bullet})\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^0(\\iota'_v) >> \\!\\QQ_p\\cdot H^0(C^{\\bullet,d}_{\\mathcal{F}_v})\\! @> >>\\! 0\\\\\n@. @V{\\rm exp}_{A^t,F_v} VV @V  \\lambda^v_{A^t,F} VV @V ({\\rm val}_{F\/k,v})^d VV \\\\\n0 @> >>\\! \\QQ_p\\cdot A^t(F_v)_p^\\wedge\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^1(\\iota_v) >>\\! \\QQ_p\\cdot H^1(C_{A^t,F}^{v,\\bullet})\\! @> \\QQ_p\\otimes_{\\ZZ_p}H^1(\\iota'_v)>> \\!\\QQ_p\\cdot H^1(C^{\\bullet, d}_{\\mathcal{F}_v}) @> >>\\! 0,\\end{CD}\\]\nin which $\\lambda^v_{A^t,F} = (\\lambda_{A^t_v,F_{w'}})_{w'\\in S_F^v}$, where each map $\\lambda_{A^t_v,F_{w'}}$ is fixed as in diagram (\\ref{lambda diag}).\n\nAfter recalling the definition of $R_{F_w\/k_v}(\\tilde A^t_v)$ and applying Lemma \\ref{fk lemma} to this commutative diagram one can therefore derive an equality\n\\begin{align*}\\label{third}\n&\\chi_{G,p}(\\Delta(\\mathcal{F}_v)[0]\\oplus A^t(F_v)^\\wedge_p[-1],{\\rm exp}_{A^t,F_v})\\\\\n= \\, &\\chi_{G,p}(C_{A^t,F}^{v,\\bullet},\\lambda_{A^t,F}^v) - d\\cdot\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})\\notag\\\\\n= \\, &{\\rm ind}^G_{G_w}(\\chi_{G_w,p}(C_{A^t_v,F_{w}}^\\bullet,\\lambda_{A^t_v,F_{w}})) - d\\cdot\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})\\notag\\\\\n= \\, &{\\rm ind}^G_{G_w}(R_{F_w\/k_v}(\\tilde A^t_v))-{\\rm ind}^G_{G_w}(d\\cdot\\delta_{G_w,p}(c_{F_w\/k_v})) - d\\cdot\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})\\notag\\\\\n=\\, & {\\rm ind}^G_{G_w}(R_{F_w\/k_v}(\\tilde A^t_v)) - d(\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})+\\delta_{G,p}(c_{F_w\/k_v})).\\notag\\end{align*}\n\nIt follows that\n\\[ c(F\/k,\\tilde A^t_v) = d\\cdot[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]-{\\rm ind}^G_{G_w}(R_{F_w\/k_v}(\\tilde A^t_v)) + d(\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})+\\delta_{G,p}(c_{F_w\/k_v}))\\]\nand hence that the claimed result is true in this case if one has\n\\begin{multline}\\label{wanted at last}  [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}] + \\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,v})+\\delta_{G,p}(c_{F_w\/k_v})\n\\\\ = \\delta_{G,p}(j_*(\\tau^p_v(F\/k))) + U_v(F\/k) - {\\rm ind}^G_{G_w}(R_{F_w\/k_v}).\\end{multline}\n\nTo prove this we note that the very definition of $R_{F_w\/K_v}$ in \\cite[\\S3.1]{breuning2} implies that\n\\begin{multline*} {\\rm ind}^G_{G_w}(R_{F_w\/k_v})\\\\ = \\delta_{G,p}(j_*(\\tau_v(F\/k))) - [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}] - \\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}'_{F\/k,v}) + U_v(F\/k) - \\delta_{G,p}(m_{F_w\/k_v}),\\end{multline*}\nwhere ${\\rm val}'_{F\/k,v}$ denotes the isomorphism of $\\QQ_p[G]$-modules\n\\[ \\QQ_p\\cdot H^1(C^\\bullet_{\\mathcal{F}_v})\\cong \\QQ_p\\cdot H^2(C^\\bullet_{\\mathcal{F}_v})\\]\nthat is induced by the maps ($({\\rm val}_{F_{w'}}))_{w'\\in S_F^v}$ and we use the element\n\\[ m_{F_w\/k_v}:= \\frac{^\\dagger(f\\cdot e_{G_w})\\cdot \\, ^\\dagger((1- \\Phi_v\\cdot {\\rm N}v^{-1})e_{I_w})}{^\\dagger((1-\\Phi_v^{-1})e_{I_w})}\\]\nof $\\zeta(\\QQ[G])^\\times$. (Here we use the notational convention introduced in (\\ref{dagger eq}). In addition, to derive the above formula for ${\\rm ind}^G_{G_w}(R_{F_w\/k_v})$ we have relied on \\cite[Prop. 2.6]{breuning2} and the fact that in loc. cit. Breuning uses the `opposite' normalization of Euler characteristics to that used here, so that the term $\\chi_{G,p}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}'_{F\/k,v})$ appears in the corresponding formulas in loc. cit. with a negative sign.)\n\nTo deduce the required equality (\\ref{wanted at last}) from this formula it is then enough to note that\n\\[ \\chi_{G}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}'_{F\/k,p})\n= \\chi_{G}(C^\\bullet_{\\mathcal{F}_v},{\\rm val}_{F\/k,p}) - \\delta_{G,p}(^\\dagger(f\\cdot e_{G_w})),\\]\nand that an explicit comparison of definitions shows that\n\\[  j_*(\\tau^p_v(F\/k))\\cdot m_{F_w\/k_v} = j_*(\\tau_v(F\/k))\\cdot ^\\dagger(f\\cdot e_{G_w})\\cdot  c_{F_w\/k_v}.\\]\n\nTurning now to the case $v\\in S_{p,{\\rm u}}$ we only need to prove that for each such place $v$ one has\n\\[ d\\cdot[\\mathcal{O}_{F,v},Y_{F_v};\\pi_{F_v}] = d\\cdot \\delta_{G,p}(j_*(u_v(F\/k)\\cdot\\tau_v(F\/k))) + d\\cdot U_v(F\/k) - R_{F\/k}(\\tilde A^t_v).\\]\n\nNow, since each such $v$ is unramified in $F\/k$ the term $R_{F\/k}(\\tilde A^t_v)$ vanishes (as a consequence of Proposition \\ref{basic props}(iii) and Remark \\ref{breuning remark}) and so it is enough to note that\n\\[ [\\mathcal{O}_{F,v},Y_{F_v};\\pi_{F_v}] = \\delta_{G,p}((u_v(F\/k)\\cdot \\tau_v(F\/k))) + U_v(F\/k),\\]\nas is shown in the course of the proof of \\cite[Th. 3.6]{breuning2}.\n\\end{proof}\n\n\n\n\\subsubsection{}We next record a result concerning the decomposition of global Galois-Gauss sums as a product of local Galois-Gauss sums.\n\n\\begin{lemma}\\label{gauss} In $K_0(\\ZZ_p[G],\\QQ^c_p[G])$ one has\n\\[ \\delta_{G,p}(j_*(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_{p,{\\rm r}}}\\varrho_v(F\/k))) = \\sum_{v \\in S_{k}^p}(\\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + U_v(F\/k)).\\]\n\\end{lemma}\n\n\\begin{proof} We observe first that the difference $\\xi$ of the left and right hand sides of this claimed equality belongs to $K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\nThis follows from the fact that both the term\n\\[ \\delta_{G,p}(j_*(\\tau^\\ast(F\/k))) - \\sum_{v \\in S_k^p}[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}],\\]\nand for each $v \\in S_k^p$ the term\n\\[ \\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + U_v(F\/k) - [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}],\\]\nbelong to $K_0(\\ZZ_p[G],\\QQ_p[G])$ (the former as a consequence of \\cite[Prop. 3.4 and (34)]{bleyburns} and the latter as a consequence of  \\cite[Prop. 3.4]{breuning2}).\n\nThus, by Taylor's Fixed Point Theorem for group determinants (as discussed in \\cite[Chap. 8]{Taylor}) it is enough for us to show that $\\xi$ belongs to the kernel of the natural homomorphism $\\iota: K_0(\\ZZ_p[G],\\QQ_p^c[G]) \\to K_0(\\mathcal{O}^t_p[G],\\QQ_p^c[G])$ where $\\mathcal{O}_p^t$ is the valuation ring of the maximal tamely ramified extension of $\\QQ_p$ in $\\QQ_p^c$.\n\nNow from \\cite[Prop. 2.12(i)]{breuning2} one has $\\iota(U_v(F\/k)) = 0$ for all $v$ in $S_k^p$. In addition, for each non-archimedean place $v$ of $k$ that is not $p$-adic the vanishing of $\\iota(\\delta_{G,p}(j_*(u_v(F\/k)\\cdot \\tau_v(F\/k))))$ is equivalent to the result proved by  Holland and Wilson in\n\\cite[Th. 3.3(b)]{HW3} (which itself relies crucially on the work of Deligne and Henniart in \\cite{deligne-henniart}).\n\nThe vanishing of $\\iota(\\xi)$ is thus a consequence of the fact that the classical decomposition of global Galois-Gauss sums as a product of local Galois-Gauss sums implies that\n\\[ \\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_{p,{\\rm r}}}\\varrho_v(F\/k) = \\prod_{v}\\tau^p_v(F\/k)\\]\nwhere $v$ runs over all places of $k$ that divide the discriminant of $F$, since for any place $v$ that does not ramify in $F$ one has $\\tau(\\QQ_{\\ell(v)},\\psi_v) = 1$ for all  $\\psi$ in $\\widehat{G}$.\n\\end{proof}\n\n\\subsubsection{}We can now complete the proof of Theorem \\ref{bk explicit}.\n\nTo this end we note first that the definition (\\ref{bkcharelement}) of $\\mathcal{L}^*_{A,F\/k}$ implies that\n\n\\begin{align*} &\\delta_{G,p}(j_\\ast(\\mathcal{L}^*_{A,F\/k})) - \\partial_{G,p}\\left(\\frac{j_*(L_S^*(A_{F\/k},1))}{j_*(\\Omega_{\\omega_\\bullet}(A_{F\/k}))}\\right)\\\\\n=\\, &\\delta_{G,p}\\bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\bigr) + d\\cdot\\bigl(\\delta_{G,p}(j_*(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_{p,{\\rm r}}}\\varrho_v(F\/k))) - \\sum_{v\\in S_k^p}[\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\bigr)\\\\\n= \\, &\\delta_{G,p}\\bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\bigr) + d\\cdot\\sum_{v \\in S_{k}^p}\\bigl(\\delta_{G,p}(j_*(\\tau_v^p(F\/k))) + U_v(F\/k) - [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\bigr)\n\\\\\n= \\, &\\delta_{G,p}\\bigl(\\prod_{v\\in S_k^A} L_v(A,F\/k)\\bigr) +\\sum_{v \\in S_{k}^p} \\bigl(c(F\/k,\\tilde A^t_v) - d\\cdot [\\mathcal{F}_v,Y_{F_v};\\pi_{F_v}]\\bigr) +\\sum_{v \\in S_{k}^p}R_{F\/k}(\\tilde A^t_v)\\\\\n= \\, &\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})+\\sum_{v \\in S_{k}^p}R_{F\/k}(\\tilde A^t_v) - \\chi_{G,p}(C_{X(p)},h^{j}_{A,F})\\\\\n= \\, &\\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h^{j}_{A,F})+\\sum_{v \\in S_{p,{\\rm w}}}R_{F\/k}(\\tilde A^t_v) - \\chi_{G,p}(C_{X(p)},h^{j}_{A,F}).\\end{align*}\nHere the first equality follows from (\\ref{norm resolvents}), the second from Lemma \\ref{gauss}, the third from Proposition \\ref{heavy part}, the fourth from the definition of the terms $c(F\/k,\\tilde A^t_v)$ and the equality (\\ref{first comp}) and the last from the fact that $R_{F\/k}(\\tilde A^t_v)$ vanishes for each $v \\in S_k^p\\setminus S_{p,{\\rm w}}$ as  consequence of Proposition \\ref{basic props}(iii) and Remark \\ref{breuning remark}.\n\nIt is thus sufficient to show that the above displayed equality implies that the equality (\\ref{displayed pj}), with set of places $S$ and differentials $\\omega_\\bullet$ chosen as in \\S\\ref{clever peiods}, is equivalent to the equality stated in Theorem \\ref{bk explicit}.\n\nBut this is true since our choice of periods $\\omega_\\bullet$ and lattices $\\mathcal{F}_v$ as in \\S\\ref{clever peiods} implies that the module $Q(\\omega_\\bullet)_{S,p}$ vanishes and because\n\n\\begin{align*} \\mu_S(A_{F\/k})_p =\\, &\\sum_{v \\in S_k^p\\setminus S}\\mu_v(A_{F\/k})\\\\\n =\\, &\\sum_{v \\in S_{p,{\\rm u}}}\\mu_v(A_{F\/k})\\\\\n  =\\, &\\sum_{v \\in S^\\ast_{p,{\\rm u}}} \\mu_v(A_{F\/k})\\end{align*}\nwhere the last equality follows from Lemma \\ref{fm} below.\n\nThis completes the proof of Theorem \\ref{bk explicit}.\n\n\\begin{lemma}\\label{fm} For any place $v\\in S_k^A$ for which the residue characteristic $\\ell(v)$ is unramified in $F$, the term\n $\\mu_{v}(A_{F\/k})$ vanishes. \\end{lemma}\n\n\\begin{proof} We fix a place $v$ as in the statement of the lemma and set $p:= \\ell(v)$. We write $\\mathcal{O}_{F_v}$ for the integral closure of $\\ZZ_p$ in $F_v$ and set $\\wp_{F_v} := p\\cdot\\mathcal{O}_{F_v}$.\n\nThen, since $p$ does not ramify in $F\/\\QQ$, the $\\ZZ_p[G]$-modules $\\mathcal{O}_{F_v}$ and $\\wp_{F_v}$ are projective and $\\wp_{F_v}$ is the direct sum of the maximal ideals of the valuation rings in each field component of $F_v=\\prod_{w'\\in S_F^v}F_{w'}$.\n\nWe use the canonical comparison isomorphism of $\\QQ_p[G]$-modules\n\\[ \\nu_v: \\Hom_{F_v}(H^0(A^t_{F_v},\\Omega^1_{A^t_{F_v}}),F_v) \\cong {\\rm DR}_v(V_{p,F}(A^t))\/F^0\\]\nand the exponential map ${\\rm exp}_{\\rm BK}: {\\rm DR}_v(V_{p,F}(A^t))\/F^0\\to H^1_f(k,V_{p,F}(A^t))$ of Bloch and Kato.\n\nWe recall, in particular, that in this case there is a natural identification of spaces $H^1_f(k,V_{p,F}(A^t)) = \\QQ_p\\cdot A^t(F_v)^\\wedge_p$ under which the composite ${\\rm exp}_{\\rm BK}\\circ\\nu_v$ sends the free $\\ZZ_p[G]$-lattice $\\mathcal{D}_F(\\mathcal{A}_v^t)$ defined in (\\ref{mathcalD}) to ${\\rm exp}_{A^t,F_v}((\\mathcal{O}_{F_v})^d)$, where ${\\rm exp}_{A^t,F_v}$ is the classical exponential map of $A^t$ over $F_{v}$ (cf. the result of Bloch and Kato in \\cite[Exam. 3.11]{bk}).\n\nIn particular, since $A^t$ has good reduction over the absolutely unramified algebra $F_{v}$, the theory of Fontaine and Messing \\cite{fm} implies (via the proof of \\cite[Lem. 3.4]{bmw}) that there exists an exact sequence of $\\ZZ_p[G]$-modules\n\\[ 0 \\to {\\rm exp}_{A^t,F_v}((\\mathcal{O}_{F_v})^d) \\xrightarrow{\\subseteq} A^t(F_v)^\\wedge_p \\to N \\to 0\\]\nwhere $N$ is a finite module that has finite projective dimension and is such that\n\\[ \\chi_{G,p}(N[-1],0) = \\delta_{G,p}(L_v(A,F\/k))\\]\nin $K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\nOn the other hand, since $F_{v}$ is absolutely unramified, the map ${\\rm exp}_{A^t,F_v}$ restricts to give a short exact sequence of $\\ZZ_p[G]$-modules\n\\[ 0 \\to \\wp_{F_v}^d \\xrightarrow{{\\rm exp}_{A^t,F_v}} A^t(F_v)^\\wedge_p \\to \\tilde A^t_v(\\kappa_{F_v})_p \\to 0.\\]\n\nBy comparing these exact sequences, and noting that $\\mathcal{O}_{F_v}\/\\wp_{F_v}$ identifies with the ring $\\kappa_{F_v}$, one obtains a short exact sequence of $\\ZZ_p[G]$-modules\n\\[ 0 \\to \\kappa_{F_v}^d \\to \\tilde A^t_v(\\kappa_{F_v})_p \\to N \\to 0\\]\nin which each term is both finite and of finite projective dimension.\n\nThus, upon taking Euler characteristics of this exact sequence, one finds that\n\\begin{align*} \\delta_{G,p}(L_v(A,F\/k)) =\\,&\\chi_{G,p}(N[-1],0)\\\\\n = \\, &\\chi_{G,p}(\\tilde A^t_v(\\kappa_{F_v})_p[-1],0) - \\chi_{G,p}(\\kappa_{F_v}^d[-1],0)\\\\\n = \\, &\\chi_{G,p}(\\tilde A^t_v(\\kappa_{F_v})_p[-1],0) + \\chi_{G,p}(\\kappa_{F_v}^d[0],0)\\\\\n  =\\, &\\chi_{G,p}(\\kappa_{F_v}^d[0]\\oplus\\tilde A^t_v(\\kappa_{F_v})_p[-1],0),\\end{align*}\n\n\\noindent{}and hence that the element $\\mu_{v}(A_{F\/k})$ vanishes, as required.\n\\end{proof}\n\n\n\\section{Euler characteristics and Galois structures}\\label{ecgs}\n\nIn this section we consider consequences of ${\\rm BSD}(A_{F\/k})$ concerning both the Galois structure of Selmer complexes and modules and the formulation of refinements of the Deligne-Gross Conjecture.\n\n\n\n\\subsection{Galois structures of Selmer complexes}\\label{Galoiscomplexes} In this section we fix a finite set $S$ of places of $k$ as described at the beginning of \\S\\ref{selmer section} as well as data $\\{\\mathcal{A}^t_v\\}_v$ and $\\omega_\\bullet$ as in \\S\\ref{perf sel construct}. We then write\n\\[ \\Upsilon = \\Upsilon(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,S)\\]\nfor the finite set of non-archimedean places $v$ of $k$ that are such that either $v$ belongs to $S$ or divides the different of $k\/\\QQ$ or the lattice $\\mathcal{F}(\\omega_\\bullet)_{v}$ differs from $\\mathcal{D}_F(\\mathcal{A}_v^t)$.\n\nWe then consider the perfect Selmer structure $\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet)$ that is defined in \\S\\ref{perf sel construct}.\n\n\n\\begin{proposition}\\label{gec} If ${\\rm BSD}(A_{F\/k})$ is valid, then for any data $S$, $\\{\\mathcal{A}^t_v\\}_v$ and $\\omega_\\bullet$ as above the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] The Selmer complex ${\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet))$ is represented by a bounded complex of finitely generated free $G$-modules.\n\\item[(ii)] Set $\\ZZ' := \\ZZ[1\/2]$. If neither of the groups $A(F)$ and $A^t(F)$ has an element of odd order, then the $\\ZZ'[G]$-module $\\ZZ'\\otimes_\\ZZ H^2({\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet)))$ has a presentation with the same number of generators and relations.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} We set $C_{\\omega_\\bullet} := {\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet))$ and write $\\chi_G(C_{\\omega_\\bullet})$ for its Euler characteristic in $K_0(\\ZZ[G])$.\n\nThen, the definition of $\\Upsilon$ implies immediately that the module $\\mathcal{Q}(\\omega_\\bullet)_\\Upsilon$ vanishes and also combines with Lemma \\ref{fm} to imply that $\\mu_{\\Upsilon}(A_{F\/k})$ vanishes.\n\nThe vanishing of $\\mathcal{Q}(\\omega_\\bullet)_\\Upsilon[0]$ implies ${\\rm SC}_{\\Upsilon,\\omega_\\bullet}(A_{F\/k}) = C_{\\omega_\\bullet}$ and hence that\n\\[ \\partial'_G(\\chi_G({\\rm SC}_{\\Upsilon,\\omega_\\bullet}(A_{F\/k}),h_{A,F})) = \\chi_G(C_{\\omega_\\bullet}),\\]\nwhere $\\partial'_{G}$ denotes the canonical connecting homomorphism $K_0(\\ZZ[G],\\RR[G]) \\to K_0(\\ZZ[G])$ of relative $K$-theory.\n\nThen, given the vanishing of $\\mu_{\\Upsilon}(A_{F\/k})$, the equality in ${\\rm BSD}(A,F\/k)$(iv) implies that\n\\[ \\chi_G(C_{\\omega_\\bullet}) = \\partial'_G\\bigl(\\partial_G(L_\\Upsilon^*(A_{F\/k},1)\/\\Omega_{\\omega_\\bullet}(A_{F\/k}))).\\]\n\nHowever, the exactness of the lower row of diagram (\\ref{E:kcomm}), with $\\mathfrak{A} = \\ZZ[G]$ and $A_E = \\RR[G]$, implies that the composite homomorphism $\\partial'_G\\circ \\partial_G$ is zero and so it follows that the Euler characteristic $\\chi_G(C_{\\omega_\\bullet})$ must vanish.\n\nNow, by a standard resolution argument, we may fix a bounded complex of finitely generated $G$-modules $C^\\bullet$ that is isomorphic in $D(\\ZZ[G])$ to $C_{\\omega_\\bullet}$ and is such that, for some integer $a$, all of the following properties are satisfied: $C^i = 0$ for all $i < a$; $C^a$ is projective of rank (over $\\ZZ[G]$) at least two; $C^{i}$ is free for all $i \\not= a$.\n\nFrom the vanishing of $\\chi_G(C_{\\omega_\\bullet}) = \\chi_G(C^\\bullet)$ it then follows that the class of $C^a$ in $K_0(\\ZZ[G])$ coincides with that of a free $G$-module.\n\nThus, since the rank over $\\ZZ[G]$ of $C^a$ is at least two, we may conclude from the Bass Cancellation Theorem (cf. \\cite[(41.20)]{curtisr}) that $C^a$ is a free $G$-module, as required to prove claim (i).\n\nTurning to claim (ii), we note that if $\\ZZ'\\otimes_\\ZZ A(F)$ and $\\ZZ'\\otimes_\\ZZ A^t(F)$ are torsion-free, then Proposition \\ref{prop:perfect2} implies that the complex $C'_{\\omega_\\bullet} := \\ZZ'\\otimes_\\ZZ C_{\\omega_\\bullet}$ is acyclic outside degrees one and two and that $H^1(C'_{\\omega_\\bullet})$ is torsion-free.\n\nThis implies that $C'_{\\omega_\\bullet}$ is isomorphic in $D^{\\rm perf}(\\ZZ'[G])$ to a complex of finitely generated $\\ZZ'[G]$-modules of the form $(C')^1 \\xrightarrow{d'} (C')^2$ where $(C')^1$ is projective and $(C')^2$ is free.\n\nThe vanishing of the Euler characteristic of $C'_{\\omega_\\bullet}$ then implies, by the same argument as in claim (i), that the module $(C')^1$ is free.\n\nIn addition, the fact that the $\\RR[G]$-modules generated by $H^1(C'_{\\omega_\\bullet})$ and $H^2(C'_{\\omega_\\bullet})$ are isomorphic implies that the free modules $(C')^1$ and $(C')^2$ must have the same rank.\n\nGiven this, claim (ii) follows from the tautological exact sequence\n\\[ (C')^1 \\xrightarrow{d'} (C')^2 \\to \\ZZ'\\otimes_\\ZZ H^2({\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet))) \\to 0.\\]\n\\end{proof}\n\n\\begin{remark}{\\em An explicit description of the module $\\ZZ'\\otimes_\\ZZ H^2({\\rm SC}_{\\Upsilon}(A_{F\/k};\\mathcal{X}_{\\Upsilon}(\\omega_\\bullet)))$ that occurs in Proposition \\ref{gec}(ii) can be found in Remark \\ref{can structure groups}.}\\end{remark}\n\n\n\n\n\\subsection{Refined Deligne-Gross-type conjectures}\n\nIn this section we address a problem raised by Dokchitser, Evans and Wiersema in \\cite[Rem. 14]{vdrehw} by explaining how ${\\rm BSD}(A_{F\/k})$ leads to an explicit formula for the fractional ideal that is generated by the product of the leading coefficients of Hasse-Weil-Artin $L$-series\n by a suitable combination of `isotypic' periods and regulators. (See, in particular, Remark \\ref{evans} below.)\n\n\\subsubsection{}We fix a character $\\psi$ in $\\widehat {G}$. We also then fix a subfield $E$ of $\\bc$ that is both Galois and of finite degree over\n$\\QQ$ and also large enough to ensure that, with\n$\\mathcal{O}$ denoting the ring of algebraic integers of $E$, there exists a finitely generated ${\\mathcal\nO}[G]$-lattice $T_\\psi$ that is free over $\\mathcal{O}$ and such that the $\\bc[G]$-module\n$V_\\psi:= \\bc \\otimes_{\\mathcal{O}}T_\\psi$ has character $\\psi$.\n\n\nWe then obtain a left, respectively right,\nexact functor from the category of $G$-modules to the category of $\\mathcal{O}$-modules by setting\n\n\\begin{align*} X^\\psi &:= \\Hom_{{\\mathcal O}}(T_\\psi,{\\mathcal O}\n\\otimes_{\\ZZ} X)^G,\n\\\\ X_\\psi &:= \\Hom_{{\\mathcal O}}(T_\\psi,{\\mathcal O}\\otimes_{\\ZZ} X)_G,\n\\end{align*}\nwhere the $\\Hom$-sets are endowed with the natural diagonal\n$G$-action.\n\nIt is easily seen that for any $G$-module $X$ there is a natural isomorphism of $\\mathcal{O}$-modules\n\\begin{equation}\\label{func iso} \\Hom_\\ZZ(X,\\ZZ)_\\psi \\cong \\Hom_\\mathcal{O}(X^{\\check{\\psi}},\\mathcal{O}).\\end{equation}\n\nFor a given odd prime number $p$, each maximal ideal $\\mathfrak p$ of $\\mathcal{O}$ that divides $p$ and each $\\mathcal{O}$-module $X$ we set $X_\\mathfrak{p} := \\mathcal{O}_\\mathfrak{p}\\otimes_{\\mathcal{O}}X$.\n\nWe also write $I(\\mathcal{O}_\\mathfrak{p})$ for the multiplicative group of invertible $\\mathcal{O}_\\mathfrak{p}$-submodules of $\\CC_p$ and we use the composite homomorphism of abelian groups\n\\[ \\rho_{\\mathfrak{p}}^{\\psi}: K_0(\\ZZ_p [G],\\CC_p[G]) \\to K_0(\\mathcal{O}_\\mathfrak{p} ,\\CC_p) \\xrightarrow{\\iota_\\mathfrak{p}} I(\\mathcal{O}_\\mathfrak{p}).\\]\nHere the first map is induced by the composite functor $X \\mapsto X^\\psi\\to (X^\\psi)_\\mathfrak{p}$ and $\\iota_\\mathfrak{p}$ is the canonical isomorphism induced by\n the upper row of (\\ref{E:kcomm}) with $\\A = {\\mathcal O}_\\mathfrak{p}$ and $E' = \\bc_p$ and the canonical\nisomorphisms $K_1(\\bc_p) \\xrightarrow{\\sim} \\bc_p^\\times$ and\n$K_1({\\mathcal O}_\\mathfrak{p}) \\xrightarrow{\\sim} {\\mathcal O}_\\mathfrak{p}^\\times$.\n\nFor any finite ${\\mathcal O}$-module $X$ we also set\n\\[ {\\rm char}_{\\mathfrak{p}}(X) := \\mathfrak{p}^{{\\rm length}_{{\\mathcal O}_{{\\mathfrak\np}}}(X_{\\mathfrak p})}.\\]\n\n\\subsubsection{}\\label{explicit ec section} Using the isomorphism (\\ref{func iso}), we define $R^\\psi_A$ to be the determinant, with respect to a choice of $\\mathcal{O}$-bases of $A^t(F)^{\\psi}$ and  $A(F)^{\\check\\psi}$ of the isomorphism of $\\CC$-spaces\n\\[ h^\\psi_{A,F}: \\CC\\cdot A^t(F)^{\\psi} \\cong \\CC\\cdot \\Hom_\\ZZ(A(F),\\ZZ)^\\psi \\cong \\CC\\cdot \\Hom_\\mathcal{O}(A(F)^{\\check\\psi},\\mathcal{O})\\]\nthat is induced by the N\\'eron-Tate height pairing of $A$ relative to $F$.\n\nMotivated by \\cite[Def. 12]{vdrehw}, we then define a non-zero complex number by setting\n\\[ \\mathcal{L}^\\ast(A,\\psi) := \\frac{L^\\ast(A,\\check\\psi,1)\\cdot \\tau^\\ast(\\QQ,\\psi)^d}{\\Omega_A^\\psi\\cdot w^d_\\psi\\cdot R_A^\\psi}.\\]\n\nFinally, after recalling the integral Selmer group $X_\\ZZ(A_F)$ of $A$ over $F$ that is defined by Mazur and Tate \\cite{mt} (and discussed in \\S\\ref{perfect selmer integral}), we note that if $\\sha(A_F)$ is finite then the kernel $\\sha_\\psi(A_F)$ of the natural surjective homomorphism of $\\mathcal{O}$-modules\n\\[ X_\\ZZ(A_F)_\\psi \\to \\Hom_\\ZZ(A(F),\\ZZ)_\\psi \\cong \\Hom_\\mathcal{O}(A(F)^{\\check\\psi},\\mathcal{O})\\]\nis finite.\n\nWe can now state the main result of this section.\n\n\\begin{proposition}\\label{ref deligne-gross} If ${\\rm BSD}(A_{F\/k})$ is valid, then so are the following claims.\n\\begin{itemize}\n\\item[(i)] For every $\\omega$ in $G_\\QQ$ one has $\\mathcal{L}^\\ast(A,\\omega\\circ \\psi) = \\omega(\\mathcal{L}^\\ast(A,\\psi))$. In particular, the complex number $\\mathcal{L}^\\ast(A,\\psi)$ belongs to $E$.\n\n\\item[(ii)] Assume that no place of $k$ at which $A$ has bad reduction is ramified in $F$. Then for every odd prime $p$ that satisfies the conditions (H$_1$)-(H$_4$) listed in \\S\\ref{tmc} and for which neither $A(F)$ nor $A^t(F)$ has a point of order $p$, and every maximal ideal $\\mathfrak{p}$ of $\\mathcal{O}$ that divides $p$, there is an equality of fractional $\\mathcal{O}_\\mathfrak{p}$-ideals\n\n\\[ \\mathcal{L}^\\ast(A,\\psi)\\cdot \\mathcal{O}_{\\mathfrak{p}} = \\frac{{\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F))\\cdot \\prod_{v\\in S^*_{p,{\\rm u}}}\\rho_\\mathfrak{p}^\\psi(\\mu_v(A_{F\/k}))}{|G|^{r_{A,\\psi}}\\cdot \\prod_{v \\in S_k^p\\cap S_k^F}\\varrho_\\psi^d\\cdot\\prod_{v\\in S_k^F\\cap S_k^f}P_v(A,\\check\\psi,1)}.\\]\nHere $$r_{A,\\psi}:={\\rm dim}_\\CC(\\CC\\cdot A^t(F)^\\psi)$$\nwhile $S^*_{p,{\\rm u}}$ is the set of $p$-adic places of $k$ that are unramified in $F$ but divide the different of $k\/\\QQ$ and, for every $v\\in S_k^F\\cap S_k^f$, $P_v(A,\\check\\psi,1)$ is the value at $z=1$ of the Euler factor at $v$ of the $\\check\\psi$-twist of $A$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} The first assertion of claim (i) is equivalent to asserting that the element\n\\[ \\mathcal{L}^\\ast := \\sum_{\\psi \\in \\widehat{G}}\\mathcal{L}^\\ast(A,\\psi)\\cdot e_\\psi\\]\nbelongs to the subgroup $\\zeta(\\QQ[G])^\\times$ of $\\zeta(\\RR[G])^\\times$.\n\nRecalling that $\\zeta(\\QQ[G])^\\times$ is the full pre-image under $\\delta_G$ of the subgroup $K_0(\\ZZ[G],\\QQ[G])$ of $K_0(\\ZZ[G],\\RR[G])$, it is therefore enough to prove that $\\delta_G(\\mathcal{L}^\\ast)$ belongs to $K_0(\\ZZ[G],\\QQ[G])$.\n\nTo do this we fix any basis of differentials $\\omega_\\bullet$ as in the statement of ${\\rm BSD}(A_{F\/k})$ and write $\\mathcal{L}^\\ast$ as a product $(\\mathcal{L}^\\ast_1)^{-1}\\cdot \\mathcal{L}^\\ast_2\\cdot (\\mathcal{L}^\\ast_3)^{-1}$ with\n\\[ \\mathcal{L}^\\ast_1 := {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k}))^{-1} \\cdot \\sum_{\\psi\\in \\widehat{G}} \\Omega_A^\\psi\\cdot w^d_\\psi\\cdot\\tau^\\ast(\\QQ,\\psi)^{-d}\\cdot e_\\psi,\\]\n\\[ \\mathcal{L}^\\ast_2 := {\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k}))^{-1}\\cdot \\sum_{\\psi \\in \\widehat{G}}L^\\ast(A,\\check\\psi,1)\\cdot e_\\psi,\\]\nand\n\\[ \\mathcal{L}^\\ast_3 := \\sum_{\\psi\\in \\widehat{G}}R_A^\\psi\\cdot e_\\psi.\\]\n\nProposition \\ref{lms}(i) implies $\\mathcal{L}^\\ast_1$ belongs to $\\zeta(\\QQ[G])^\\times$. In addition, for any set of places $S$ as in the statement of ${\\rm BSD}(A_{F\/k})$, the element $\\mu_S(A_{F\/k})$ belongs to $K_0(\\ZZ[G],\\QQ[G])$. We next note that $\\delta_G(\\mathcal{L}^\\ast_2)$ differs from the left hand side of the equality in ${\\rm BSD}(A_{F\/k})$(iv) by\n$$\\sum_{v\\in S\\cap S_k^f}\\delta_G(L_v(A,F\/k)),$$ where $L_v(A,F\/k)\\in\\zeta(\\QQ[G])^\\times$ is the equivariant Euler factor of $(A,F\/k)$ at $v$ (see Appendix \\ref{consistency section} below).\n\nThis difference belongs to $K_0(\\ZZ[G],\\QQ[G])$ and therefore the validity of  ${\\rm BSD}(A_{F\/k})$ implies that $$\\delta_G(\\mathcal{L}^\\ast_2)- \\chi_G({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F})$$ also belongs to $K_0(\\ZZ[G],\\QQ[G])$.\n\nTo prove $\\mathcal{L}^\\ast \\in \\zeta(\\QQ[G])^\\times$ it is therefore enough to note that $\\delta_G(\\mathcal{L}^\\ast_3)$ also differs from $\\chi_G({\\rm SC}_{S,\\omega_\\bullet}(A_{F\/k}),h_{A,F})$ by an element of $K_0(\\ZZ[G],\\QQ[G])$ (as one verifies by straightforward computation).\n\nThe second assertion of claim (i) is true since if $\\omega$ is any element of $G_{\\QQ^c\/E}$, then $\\omega\\circ \\psi = \\psi$ and so $\\omega(\\mathcal{L}^\\ast(A,\\psi)) = \\mathcal{L}^\\ast(A,\\omega\\circ\\psi) = \\mathcal{L}^\\ast(A,\\psi)$.\n\nTurning to claim (ii) we note that the given hypotheses imply that the data $A, F\/k$ and $p$ satisfy the conditions of Theorem \\ref{bk explicit}. To prove claim (ii) it is therefore enough to show that, if the difference between the left and right hand sides of the equality in Theorem \\ref{bk explicit} belongs to $K_0(\\ZZ_p[G],\\QQ_p[G])$, then its image under $\\rho_{\\mathfrak{p}}^{\\psi}$ is the equality in claim (ii). Since this image is independent of the choice of isomorphism $j:\\CC\\cong\\CC_p$ we will omit it from all notations.\n\nThe group $I(\\mathcal{O}_\\mathfrak{p})$ is torsion-free and so Proposition \\ref{basic props}(ii) implies that each term $R_{F\/k}(\\tilde A^t_v)$ that occurs in Theorem \\ref{bk explicit} belongs to $\\ker(\\rho_{\\mathfrak{p}}^{\\psi})$.\n\nThus, since $\\mathcal{L}^\\ast(A,\\psi)$ differs from the element $\\mathcal{L}^\\ast_{A,\\psi}$ defined in (\\ref{bkcharelement}) by the equality\n\\[ \\mathcal{L}^\\ast(A,\\psi) = \\mathcal{L}^\\ast_{A,\\psi}\\cdot (R_A^\\psi)^{-1}\\prod_{v \\in S_k^p\\cap S_k^F}\\varrho_{\\psi}^{-d}\\cdot\\prod_{v\\in S_k^F\\cap S_k^f}P_v(A,\\check\\psi,1)^{-1}\\]\nthe claimed result will follow if we can show that $\\rho_{\\mathfrak{p}}^{\\psi}$ sends the element\n\\[ \\chi_{G,p}( {\\rm SC}_p(A_{F\/k}),h_{A,F})-\\delta_{G,p}(\\mathcal{L}^\\ast_3)\\]\nof $K_0(\\ZZ_p[G],\\QQ_p[G])$ to the ideal $|G|^{-r_{A,\\psi}}\\cdot {\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F))$.\n\nNow, under the given hypotheses, Proposition \\ref{explicitbkprop} implies that the complex $C := {\\rm SC}_p(A_{F\/k})$ is acyclic outside degrees one and two and has cohomology $A^t(F)_p$ and $X_\\ZZ(A_F)_p=\\Sel_p(A_F)^\\vee$ in these respective degrees.\n\nIn particular, since $A^t(F)_p$ is torsion-free we can fix a representative of $C$ of the form $C^1 \\xrightarrow{d} C^2$, where $C^1$ and $C^2$ are free $\\ZZ_p[G]$-modules of the same rank.\n\nThen the tautological exact sequence\n\\begin{equation} 0 \\rightarrow H^1(C) \\xrightarrow{\\iota} C^1 \\xrightarrow{d}\nC^2 \\xrightarrow{\\pi} H^2(C) \\rightarrow\n0\\label{tatseq}\\end{equation}\ninduces a commutative diagram of ${\\mathcal O}_p$-modules with exact rows\n\\[\\begin{CD}\n@. @. C^1_{\\psi} @> d_\\psi >> C^2_{\\psi} @> \\pi_\\psi\n>> H^2(C)_\\psi @>  >> 0\\\\ @. @. @V {t^1_\\psi} VV @V\n{t^2_\\psi} VV  \\\\ 0 @> >> H^{1}(C)^\\psi @> \\iota^\\psi >>\n C^{1,\\psi} @> d^\\psi >> C^{2,\\psi}.\\end{CD}\\]\nEach vertical morphism $t^i_\\psi$ here is induced by sending each $x$ in $\\Hom_{{\\mathcal O}_p}(T_{\\psi,p}, {\\mathcal O}_p\\otimes_{\\ZZ_p}C^i)$ to $\\sum_{g \\in G}g(x)$ and is bijective since the $\\ZZ_p[G]$-module $C^i$ is free.\n\nThis diagram gives rise to an exact sequence of ${\\mathcal O}_\\mathfrak{p}$-modules\n\\begin{equation}\\label{scal}\n0\\rightarrow\nH^{1}(C)^\\psi_\\mathfrak{p}\n\\xrightarrow{\\iota^\\psi}C_\\mathfrak{p}^{1,\\psi}\\xrightarrow{d^\\psi}\nC_\\mathfrak{p}^{2,\\psi} \\xrightarrow{\\pi_\\psi\\circ (t^2_\\psi)^{-1}} H^2(C)_{\\psi,\\mathfrak{p}}\n\\rightarrow 0\\end{equation}\nwhich in turn implies that\n\\begin{equation}\\label{firstform} \\rho_{\\mathfrak{p}}^{\\psi}(\\chi_{G,p}( {\\rm SC}_p(A_{F\/k}),h_{A,F})) = \\iota_\\mathfrak{p}\\bigl(\\chi_{\\mathcal{O}_\\mathfrak{p}}(C^{\\bullet,\\psi}_{\\mathfrak{p}}, \\tilde h^\\psi)\\bigr).\\end{equation}\nHere $C^{\\bullet,\\psi}_{\\mathfrak{p}}$ denotes the complex $C^{1,\\psi}_\\mathfrak{p} \\xrightarrow{d^\\psi} C^{2,\\psi}_\\mathfrak{p}$ and $\\tilde h^\\psi$ the composite isomorphism of $\\CC_p$-modules\n\\[ \\CC_p\\cdot H^1(C^{\\bullet,\\psi}_{\\mathfrak{p}}) \\cong \\CC_p\\cdot (A^t(F)^\\psi)_\\mathfrak{p} \\xrightarrow{h^\\psi} \\Hom_{\\CC_p}(\\CC_p\\cdot (A(F)^{\\check\\psi})_\\mathfrak{p},\\CC_p) \\cong \\CC_p\\cdot H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\\]\nin which the first and third isomorphisms are induced by the maps in (\\ref{scal}) and $h^\\psi$ is induced by the isomorphism $h^\\psi_{A,F}$.\n\nGiven the definition of each term $R_A^\\psi$ it is, on the other hand, clear that\n\\begin{equation}\\label{secondform} \\rho^\\psi_\\mathfrak{p}(\\delta_{G,p}(\\mathcal{L}^\\ast_3)) = \\iota_\\mathfrak{p}\\bigl(\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^1(C^{\\bullet,\\psi}_{\\mathfrak{p}})[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], h^\\psi)\\bigr).\\end{equation}\n\nNow, since $\\mathcal{O}_\\mathfrak{p}$ is a discrete valuation ring it is straightforward to construct an exact triangle in $D(\\mathcal{O}_\\mathfrak{p})$ of the form\n\\[ H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-2] \\to C^{\\bullet,\\psi}_{\\mathfrak{p}} \\to H^1(C^{\\bullet,\\psi}_{\\mathfrak{p}})[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2] \\to H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-1]\\]\nand Lemma \\ref{fk lemma} applies to this triangle to imply that\n\\begin{align}\\label{thirdform}\n  &\\chi_{\\mathcal{O}_\\mathfrak{p}}(C^{\\bullet,\\psi}_{\\mathfrak{p}}, \\tilde h^\\psi) - \\chi_{\\mathcal{O}_\\mathfrak{p}}(H^1(C^{\\bullet,\\psi}_{p})[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], h^\\psi)\\notag\\\\\n=\\,&\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], \\tilde h^\\psi\\circ ( h^\\psi)^{-1}) +\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-2], 0).\\end{align}\n\nNext we note the definition of $\\sha_\\psi(A_F)$ ensures $\\sha_\\psi(A_F)_\\mathfrak{p} = H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}$ and hence that\n\\begin{equation}\\label{fourthform}\\iota_\\mathfrak{p}\\bigl(\\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tor}[-2], 0)\\bigr) =  {\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F)).\\end{equation}\n\nIn addition, after identifying both $\\CC_p\\cdot H^2(C)_{\\psi,\\mathfrak{p}}$ and $\\CC_p\\cdot H^2(C)^\\psi_{\\mathfrak{p}}$ with $e_\\psi(\\CC_p\\cdot H^2(C)_{\\mathfrak{p}})$ in the natural way, the map $t^2_\\psi$ that occurs in (\\ref{scal}) induces upon the latter space the map given by multiplication by $|G|$.\n\nTo derive claim (ii) from the displayed formulas (\\ref{firstform}), (\\ref{secondform}), (\\ref{thirdform}) and (\\ref{fourthform}) it is thus enough to note that\n\\[ \\dim_{\\CC_p}(\\CC_p\\cdot H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}}))=\\dim_{\\CC_p}(\\CC_p\\cdot (A^t(F)^\\psi)_\\mathfrak{p})=r_{A,\\psi},\\]\nand hence that\n\\begin{multline*} \\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], \\tilde h^\\psi\\circ ( h^\\psi)^{-1})\\\\ = \\chi_{\\mathcal{O}_\\mathfrak{p}}(H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-1]\\oplus H^2(C^{\\bullet,\\psi}_{\\mathfrak{p}})\n_{\\rm tf}[-2], \\cdot |G|^{-1})\\end{multline*}\nis sent by $\\iota_\\mathfrak{p}$ to the ideal generated by $|G|^{-r_{A,\\psi}}$.\n\\end{proof}\n\n\\begin{remark}\\label{evans}{\\em Fix a Galois extension $F$ of $k = \\QQ$ and an elliptic curve $A$ whose conductor $N_A$ is prime to the discriminant $d_F$ of $F$ and is such that $A(F)$ is finite. Then for each $\\psi$ in $\\widehat{G}$ one has $r_{A,\\psi}=0$, and hence $R_A^\\psi = 1$, so that the complex number $\\mathcal{L}^\\ast(A,\\psi)$ agrees up to a unit of $\\mathcal{O}$ with the element $\\mathcal{L}(A,\\psi)$ that is defined in \\cite[Def. 12]{vdrehw}. Now fix an odd prime $p$ that is prime to $d_F$, to $N_A$, to the order of $A(F)$, to the order of the group of points of the reduction of $A$ at any prime divisor of $d_F$ and to the Tamagawa number of $A_F$ at each place of bad reduction. Then the data $A, F\/k$ and $p$ satisfy the hypotheses of Proposition \\ref{ref deligne-gross}(ii) and the sets  $S^*_{p,{\\rm u}}$ and $S_k^p\\cap S_k^F$ are empty (the former since $k = \\QQ$). The explicit formula in the latter result therefore simplifies to give\n\\[ \\mathcal{L}(A,\\psi)\\cdot \\mathcal{O}_{\\mathfrak{p}} = {\\rm char}_\\mathfrak{p}(\\sha_\\psi(A_F)) \\cdot\\prod_{\\ell\\mid d_F}P_\\ell(A,\\check\\psi,1)^{-1}\\]\nwhere in the product $\\ell$ runs over all prime divisors of $d_F$. This formula shows that, in any such case, the fractional $\\mathcal{O}$-ideal generated by $\\mathcal{L}(A,\\psi)$ should depend crucially on the structure of $\\sha(A_F)$ as a $G$-module, as already suggested in this context by Dokchitser, Evans and Wiersema in \\cite[Rem. 40]{vdrehw}. In particular, this observation is both consistent with, and helps to clarify, the result of [loc. cit., Th. 37(2)]. }\n\\end{remark}\n\n\n\\section{Abelian congruence relations and module structures}\\label{congruence sec}\n\nIn both this and the next section we apply the general results of Sano, Tsoi and the first author in \\cite{bst} to derive from the assumed validity of ${\\rm BSD}_p(A_{F\/k})$(iv) families of $p$-adic congruences that can be much more explicit than those discussed in Remark \\ref{cons1}.\n\nIn particular, in this section we will focus on congruence relations that express links to the Galois structures of Selmer and Tate-Shafarevich groups.\n\n\n\nIn contrast to \\S\\ref{tmc}, in this section we are not required to assume any of the hypotheses (H$_1$)-(H$_5$).\n\nHowever, we will now, unless explicitly stated otherwise, restrict to the case that $G$ is abelian and hence will not distinguish between the leading coefficient element $L_S^*(A_{F\/k},1)$ in $K_1(\\RR[G])$ and its reduced norm $\\sum_{\\psi \\in \\widehat G}L_S^*(A,\\check\\chi,1)\\cdot e_\\psi$ in $\\RR[G]^\\times$.\n\nAn extension of the results of this section to the general (non-abelian) setting will be given in the upcoming article \\cite{dmckwt}.\n\n\\subsection{Statement of the main result}\\label{8.1}\n\nThroughout this section we give ourselves a fixed odd prime $p$ and an isomorphism of fields $\\CC\\cong\\CC_p$ (that we will usually not mention).\n\nWe also fix a finite set $S$ of places of $k$ with\n\\[ S_k^\\infty\\cup S_k^p\\cup  S_k^F \\cup S_k^A\\subseteq S.\\]\n\nWe write $x\\mapsto x^\\#$ for the $\\CC$-linear involution of $\\CC[G]$ that inverts elements of $G$. We also set $n := [k:\\QQ]$ and write $d$ for the dimension of $A$.\n\n\\subsubsection{}In order to state the main result of this section we must first extend the definition of logarithmic resolvents given in (\\ref{log resol abelian}) to the setting of abelian varieties.\n\nTo do this we do not require $F\/k$ to be abelian but we do assume to be given an ordered $k$-basis $\\{\\omega'_j: j \\in [d]\\}$ of $H^0(A^t,\\Omega^1_{A^t})$ and we use this basis to define a classical period $\\Omega_A^{F\/k}$ in $\\CC[G]^{\\times}$ as in (\\ref{period def}).\n\n\nFor each index $j$ we then write ${\\rm log}_{\\omega'_j}$ for the formal logarithm of $A^t$ over $F_p$ that is defined with respect to $\\omega_j'$.\n\nWe also fix an ordering of $\\Sigma(k)$. We write $\\CC_p[G]^{nd}$ for the direct sum of $nd$ copies of $\\CC_p[G]$ and fix a bijection between the standard basis of this module and the lexicographically-ordered direct product $[d]\\times \\Sigma(k)$.\n\nThen for any ordered subset\n\\begin{equation}\\label{setx} x_\\bullet:= \\{x_{(i,\\sigma)}: (i,\\sigma) \\in [d]\\times\\Sigma(k)\\}\\end{equation}\nof $A^t(F_p)^\\wedge_p$ we define a logarithmic resolvent element of $\\zeta(\\CC_p[G])$ by setting\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) := {\\rm Nrd}_{\\QQ^c_p[G]}\\left(\\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}({\\rm log}_{\\omega_j'}(x_{(j',\\sigma')})))\\cdot g \\bigr)_{(j,\\sigma),(j',\\sigma')}\\right) \\]\nwhere the indices $(j,\\sigma)$ and $(j',\\sigma')$ run over $[d]\\times \\Sigma(k)$ and ${\\rm Nrd}_{\\QQ^c_p[G]}(-)$ denotes the reduced norm of the given matrix in ${\\rm M}_{dn}(\\QQ_p^c[G])$.\n\nIt is clear that if $A$ is an elliptic curve (so $d=1$) and $F\/k$ is abelian, then the `$\\psi$-component' of this definition agrees with (\\ref{log resol abelian}).\n\n\\subsubsection{}\\label{statementstructure} For each non-archimedean place $v$ of $k$ that does not ramify in $F\/k$ and at which $A$ has good reduction we define an element of $\\QQ [G]$ by setting\n\\[ P_v(A_{F\/k},1) := 1-\\Phi_v\\cdot a_v +  \\Phi_v^2\\cdot {\\rm N}v^{-2}.\\]\nHere $a_v$ is the integer $1 + {\\rm N}v - | A(\\kappa_v)|$\n\nFor a non-negative integer $a$ we write $\\widehat{G}_{A,(a)}$ for the subset of $\\widehat{G}$ comprising characters $\\psi$ for which the  $L$-series $L(A,\\psi,z)$ vanishes at $z=1$ to order at least $a$. This definition ensures that the $\\CC[G]$-valued function\n\\[ L^{(a)}_{S}(A_{F\/k},z) := \\sum_{\\psi \\in \\widehat{G}_{A,(a)}}z^{-a}L_S(A,\\check\\psi,z)\\cdot e_\\psi\\]\nis holomorphic at $z=1$.\n\nFor each $a$ we also define idempotents of $\\QQ[G]$ by setting\n\\[ e_{(a)} = e_{F,(a)} := \\sum_{\\psi \\in \\widehat{G}_{A,(a)}}e_\\psi\\]\nand\n\\[ e_{a} = e_{F,a}:= \\sum_{\\psi \\in \\widehat{G}_{A,(a)}\\setminus \\widehat{G}_{A,(a+1)}}e_\\psi\\]\n(so that $e_{(a)} = \\sum_{b \\ge a}e_b$).\n\nThe N\\'eron-Tate height pairing for $A$ over $F$ induces a canonical isomorphism of $\\RR[G]$-modules\n\\[ h_{A_{F\/k}}:\\RR\\cdot A^t(F) \\cong \\Hom_\\RR(\\RR\\cdot A(F),\\RR) = \\RR\\cdot \\Hom_{\\ZZ[G]}(A(F),\\ZZ[G]).\\]\nFor each non-negative integer $a$ this pairing combines with our fixed isomorphism of fields $\\CC\\cong \\CC_p$ to induce an isomorphism of $\\CC_p[G]$-module\n\n\\begin{equation}\\label{athpower} {\\rm ht}^{a}_{A_{F\/k}}: \\CC_p\\cdot {\\bigwedge}^a_{\\ZZ_p[G]}\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G]) \\cong \\CC_p\\cdot{\\bigwedge}^a_{\\ZZ_p[G]} A^t(F)_p.\\end{equation}\n\n\nIn the following result we shall also use (the scalar extension of) the canonical `evaluation' pairing $${\\bigwedge}^a_{\\ZZ_p[G]} A^t(F)_p\\times{\\bigwedge}^a_{\\ZZ_p[G]} \\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G])\\to \\ZZ_p[G]$$\nand write ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$ for the $a$-th Fitting ideal of the $\\ZZ_p[G]$-module $\\Sel_p(A_F)^\\vee$.\n\n\nThe proof of this result will be given in \\S\\ref{proof of big conj}.\n\n\n\\begin{theorem}\\label{big conj} Fix an ordered maximal subset $x_\\bullet:= \\{x_{(i,\\sigma)}: (i,\\sigma) \\in [d]\\times\\Sigma(k)\\}$ of $A^t(F_p)^\\wedge_p$ that is linearly independent over $\\ZZ_p[G]$ and a finite non-empty set $T$ of places of $k$ that is disjoint from $S$\n\nIf ${\\rm BSD}(A_{F\/k})$ is valid, then for any non-negative integer $a$, any subsets $\\{\\theta_j: j \\in [a]\\}$ and $\\{\\phi_i: i\\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G])$ and $\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G])$ respectively, and any element $\\alpha$ of $\\ZZ_p[G]\\cap \\ZZ_p[G]e_{(a)}$ the product\n\\begin{equation}\\label{key product} \\alpha^{1+2a}\\cdot (\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#) \\cdot  \\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot (\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\end{equation}\nbelongs to ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$  and annihilates $\\sha(A^t_{F})[p^\\infty]$.\n\\end{theorem}\n\n\nWe remark on several ways in which this result either simplifies or becomes more explicit.\n\n\\begin{remarks}\\label{more explicit rem}{\\em \\\n\n\\noindent{}(i) If $A(F)$ does not contain an element of order $p$, then our methods will show that the prediction in Theorem \\ref{big conj} should remain true if the term $\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#$ is omitted. For more details see Remark \\ref{omit T} below.\n\n\\noindent{}(ii) If one fixes a subset $\\{\\phi_i: i \\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G])$ of cardinality $a$ that generates a free direct summand of rank $a$, then our approach combines with \\cite[Th. 3.10(ii)]{bst} to suggest that, as the subset $\\{\\theta_j: j \\in [a]\\}$ varies, elements of the form (\\ref{key product}) can be used to give an explicit description of the $a$-th Fitting ideal ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$.}\\end{remarks}\n\n\\begin{remark}\\label{e=1 case}{\\em In special cases one can either show, or is led to predict, that the idempotent $e_{(a)}$ is equal to $1$ and hence that the element $\\alpha$ in (\\ref{key product}) can be taken to be $1$.\n This is, for example, the case if $a = 0$, since each function $L(A, \\psi,z)$ is holomorphic at $z=1$ and, in the setting of abelian extensions of $\\QQ$, this case will be considered in detail in \\S\\ref{mod sect}. This situation can also arises naturally in cases with $a > 0$, such as the following.\n\n\\noindent{}(i) If $F$ is a ring class field of an imaginary quadratic field $k$ and suitable hypotheses are satisfied by an elliptic curve $A\/\\QQ$ and the extension $F\/\\QQ$, then the existence of a Heegner point in $A(F)$ with non-zero trace to $A(k)$ combines with the theorem of Gross and Zagier to imply that $e_{(1)} = 1$. This case will be considered in detail in \\S\\ref{HHP}.\n\n\\noindent{}(ii) As a generalization of (i), if $F$ is a generalized dihedral extension of a field $F'$, $k$ is the unique quadratic extension of $F'$ in $F$, all $p$-adic places split completely in $k\/F'$ and the rank of $A(k)$ is odd, then the result of Mazur and Rubin in \\cite[Th. B]{mr2} combines with the prediction of ${\\rm BSD}(A_{F\/k})$(ii) to imply that $e_{(1)} = 1$.\n\n\\noindent{}(iii) Let $F$ be a finite extension of $k$ inside a $\\ZZ_p$-extension $k_\\infty$, set $\\Gamma:= G_{k_\\infty\/k}$ and write $r_\\infty$ for the corank of ${\\rm Sel}_p(A_{k_\\infty})$ as a module over the Iwasawa algebra $\\ZZ_p[[\\Gamma]]$. Then, if $A$, $F\/k$  and $p$ satisfy all of the hypotheses listed at the beginning of \\S\\ref{tmc}, one can show that the inverse limit $\\varprojlim_{F'}A^t(F')_p$, where $F'$ runs over all finite extensions of $F$ in $k_\\infty$, is a free $\\ZZ_p[[\\Gamma]]$-module of rank $r_\\infty$ and this in turn combines with the prediction of ${\\rm BSD}(A_{F\/k})$(ii) to imply that $e_{(r_\\infty)}=1$.\n}\n\\end{remark}\n\n\\begin{remark}\\label{new remark SC}{\\em Under suitable additional hypotheses it is also possible to obtain more explicit versions of the containments predicted by Theorem \\ref{big conj}. To describe an example, assume that neither $A(F)$ nor $A^t(F)$ has a point of order $p$, that $p$ is unramified in $k$, that all $p$-adic places of $k$ are at most tamely ramified in $F$ and that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1)$-(H$_5$) that are listed in \\S\\ref{tmc}. Then, after taking account of the equality in Remark \\ref{emptysets}, the argument that is used to prove Theorem \\ref{big conj} can be directly applied to the Selmer complex ${\\rm SC}_p(A_{F\/k})$ rather than to the complex $C_{S,X}$ that occurs in \\S\\ref{proof of big conj}. In this way one finds that ${\\rm BSD}(A_{F\/k})$ predicts under the above hypotheses that for any given non-negative integer $a$ and any data\n $\\alpha$, $\\{\\theta_j:j\\in[a]\\}$ and $\\{\\phi_i:i\\in[a]\\}$ as in Theorem \\ref{big conj}, the product\n\\begin{equation}\\label{key product2} \\alpha^{1+2a}\\cdot(e_{F,a}\\cdot\\mathcal{L}_{A,F\/k}^*)\\cdot (\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\end{equation}\nshould belong to ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$ and annihilate $\\sha(A_F^t)[p^\\infty]$. Here $\\mathcal{L}_{A,F\/k}^*$ is the element defined in (\\ref{bkcharelement}) and hence is related to the values\nof $L$-functions that are truncated only at the places in $S_k^f\\cap S_k^F$ rather than at all places in $S$ (as in the expression (\\ref{key product})).\n}\\end{remark}\n\n\\begin{remark}\\label{integrality rk}{\\em To obtain concrete congruences from the result of Theorem \\ref{big conj} or its variant in Remark \\ref{new remark SC} one can, for example, proceed as follows. The stated results imply that the product expression $\\mathcal{L}$ in either (\\ref{key product}) or (\\ref{key product2}) belongs to $\\ZZ_p[G]$ and hence that for every $g$ in $G$ one has\n\\[ \\sum_{\\psi \\in \\widehat{G}_{A,a}}\\psi(g)\\mathcal{L}_\\psi \\equiv 0 \\,\\,\\,({\\rm mod}\\,\\, |G|\\cdot \\ZZ_p).\\]\nHere each element $\\mathcal{L}_\\psi$ of $\\CC_p$ is defined by the equality $\\mathcal{L} = \\sum_{\\psi\\in \\widehat{G}_{A,a}}\\mathcal{L}_\\psi\\cdot e_\\psi$ and so can be explicitly related to the value at $z=1$ of the function $z^{-a}L(A,\\check\\psi,z)$.}\\end{remark}\n\n\\subsection{Explicit regulator matrices}Motivated by Remark \\ref{e=1 case}, we consider in more detail the case that $e_{(a)} = 1$.\n\nIn this case, there exist $a$-tuples in $A^t(F)$ and $A(F)$ that are each linearly independent over $\\QQ[G]$ and this fact implies the expressions $(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))$ in Theorem \\ref{big conj} can be interpreted in terms of classical Neron-Tate heights.\n\nTo state the result we use the following notation: for ordered $a$-tuples $P_\\bullet = \\{P_i: i \\in [a]\\}$ of $A^t(F)$ and $Q_\\bullet = \\{Q_j: j \\in [a]\\}$ of $A(F)$  we define a matrix in ${\\rm M}_a(\\RR[G])$ by setting\n\\begin{equation}\\label{regulatormatrix} h_{F\/k}(P_\\bullet, Q_\\bullet) := (\\sum_{g \\in G}\\langle g(P_i),Q_j\\rangle_{A_F}\\cdot g^{-1})_{1\\le i,j\\le a},\\end{equation}\nwhere $\\langle -,-\\rangle_{A_F}$ denotes the Neron-Tate height pairing for $A$ over $F$\n\n\\begin{lemma}\\label{height pairing interp} Fix a natural number $a$ such that $e_{(a)} =1$ and then choose ordered $a$-tuples $P_\\bullet$ of $A^t(F)$ and $Q_\\bullet$ of $A(F)$ that are each linearly independent over $\\QQ[G]$. Then the matrix $e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet)$ belongs to ${\\rm GL}_a(\\RR[G]e_a)$ and one has\n\\begin{multline*}  e_a\\cdot\\left\\{(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\mid \\theta_j\\in \\Hom_{\\ZZ[G]}(A^t(F),\\ZZ[G]),\\phi_i\\in \\Hom_{\\ZZ[G]}(A(F),\\ZZ[G])\\right\\}\\\\\n= {\\rm det}(e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet))^{-1}\\cdot{\\rm Fit}^0_{\\ZZ[G]}((A^t(F)_{\\rm tf}\/\\langle P_\\bullet\\rangle)^\\vee_{\\rm tor})\\cdot{\\rm Fit}^0_{\\ZZ[G]}((A(F)_{\\rm tf}\/\\langle Q_\\bullet\\rangle)^\\vee_{\\rm tor}) \\end{multline*}\nwhere $\\langle P_\\bullet\\rangle$ and $\\langle Q_\\bullet\\rangle$ denote the $G$-modules that are generated by $P_\\bullet$ and $Q_\\bullet$.\n  \\end{lemma}\n\n\\begin{proof} We write $N(P_\\bullet)$ and $N(Q_\\bullet)$ for the quotients of $A^t(F)_{\\rm tf}$ and $A(F)_{\\rm tf}$ by $\\langle P_\\bullet\\rangle$ and $\\langle Q_\\bullet\\rangle$. Then, by taking $\\ZZ$-linear duals of the tautological short exact sequence\n\\[ 0 \\to \\langle P_\\bullet\\rangle \\xrightarrow{\\iota_{P_\\bullet}} A^t(F)_{\\rm tf} \\to N(P_\\bullet) \\to 0\\]\none obtains an exact sequence $$A^t(F)^\\ast \\xrightarrow{\\iota_{P_\\bullet}^\\ast} \\langle P_\\bullet\\rangle^\\ast \\to N(P_\\bullet)_{\\rm tor}^\\vee \\to 0$$ and hence an equality\n\\[ \\im({\\bigwedge}^a_{\\ZZ[G]}\\iota_{P_\\bullet}^\\ast) = {\\rm Fit}^0_{\\ZZ[G]}(N(P_\\bullet)_{\\rm tor}^\\vee).\\]\n\nIn the same way one derives an equality\n\\[ \\im({\\bigwedge}^a_{\\ZZ[G]}\\iota_{Q_\\bullet}^\\ast) = {\\rm Fit}^0_{\\ZZ[G]}(N(Q_\\bullet)_{\\rm tor}^\\vee).\\]\n\nSince the maps $e_{a}(\\QQ\\otimes_\\ZZ\\iota_{P_\\bullet}^\\ast)$ and $e_{a}(\\QQ\\otimes_\\ZZ\\iota_{Q_\\bullet}^\\ast)$ are bijective, these equalities imply that the lattice\n\\[ e_a\\cdot\\left\\{(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\mid \\theta_j\\in A^t(F)^\\ast,\\phi_i\\in A(F)^\\ast\\right\\}\\]\nis equal to the product\n\\begin{multline*} \\left\\{e_a(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(e_a(\\wedge_{i=1}^{i=a}\\phi_i)))\\mid \\theta_j\\in \\langle P_\\bullet\\rangle^\\ast,\\phi_i\\in \\langle Q_\\bullet\\rangle^\\ast\\right\\}\\\\\n\\times {\\rm Fit}^0_{\\ZZ[G]}(N(P_\\bullet)_{\\rm tor}^\\vee)\\cdot{\\rm Fit}^0_{\\ZZ[G]}(N(Q_\\bullet)_{\\rm tor}^\\vee). \\end{multline*}\n\nThis implies the claimed result since, writing $P_i^\\ast$ and $Q_j^\\ast$ for the elements of $\\langle P_\\bullet\\rangle^\\ast$ and $\\langle Q_\\bullet\\rangle^\\ast$ that are respectively dual to $P_i$ and $Q_j$, one has\n\\begin{multline*} \\left\\{e_a(\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^a_{A_{F\/k}}(e_a(\\wedge_{i=1}^{i=a}\\phi_i)))\\mid \\theta_j\\in \\langle P_\\bullet\\rangle^\\ast,\\phi_i\\in \\langle Q_\\bullet\\rangle^\\ast\\right\\}\\\\\n=  \\ZZ[G]\\cdot (e_a\\wedge_{i=1}^{i=a}P_i^\\ast)({\\rm ht}^a_{A_{F\/k}}(e_a\\wedge_{i=1}^{i=a}Q^\\ast_i))\\end{multline*}\nand\n\\begin{align*} e_a(\\wedge_{i=1}^{i=a}P_i^\\ast)({\\rm ht}^a_{A_{F\/k}}(e_a\\wedge_{i=1}^{i=a}Q^\\ast_i)) =\\, &{\\rm det}\\bigl( (e_aP_i^\\ast(h^{-1}_{A_{F\/k}}(e_a Q^\\ast_j)))_{1\\le i, j\\le a}\\bigr)\\\\\n =\\, &{\\rm det}(e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet))^{-1}.\\end{align*}\nWe note that the last equality is true because the definition of ${\\rm ht}^a_{A_{F\/k}}$ implies that for every $j$ one has\n\\[ h^{-1}_{A_{F\/k}}(e_a Q^\\ast_j) = \\sum_{b=1}^{b=a} ((e_a\\cdot h_{F\/k}(P_\\bullet, Q_\\bullet))^{-1})_{bj}P_b.\\]\n\\end{proof}\n\n\\subsection{Bounds on logarithmic resolvents}\\label{p-adic sec} The result of Lemma \\ref{height pairing interp} means that in many cases the only term in the product expression (\\ref{key product}) that is not explicitly understood is the logarithmic resolvent of the chosen semi-local points.\n\nWe shall now partly address this issue. For subsequent purposes (related to the upcoming article \\cite{dmckwt}) we do not here require $G$ to be abelian.\n\n\n\n\\subsubsection{} We start by deriving an easy consequence of the arguments in Proposition \\ref{lms} and Lemma \\ref{ullom}. For each natural number $i$ we set $\\wp_{F_p}^i:=\\prod_{w'\\in S_F^p}\\wp_{F_{w'}}^i$, where $\\wp_{F_{w'}}$ denotes the maximal ideal in the valuation ring of $F_{w'}$. We also set $\\hat A^t(\\wp_{F_p}^i):=\\prod_{w'\\in S_F^p}\\hat A^t_{w'}(\\wp_{F_{w'}}^i)$, where $\\hat A^t_{w'}$ denotes the formal group of $A^t_{\/F_{w'}}$.\n\n\\begin{proposition}\\label{explicit log resolve} If all $p$-adic places are tamely ramified in $F\/k$ and $\\hat A^t(\\wp_{F_p})$ is torsion-free, then there exists an ordered $\\ZZ_p[G]$-basis $x_\\bullet$ of $\\hat A^t(\\wp_{F_p})$ for which one has\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) = \\bigl(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr)^d,\\]\nwhere the elements $\\varrho_v(F\/k)$ of $\\zeta(\\QQ[G])^\\times$ are as defined in (\\ref{varrho def}).\n\\end{proposition}\n\n\\begin{proof} Since all $p$-adic places of $k$ are tamely ramified in $F$ Lemma \\ref{ullom} implies that the $\\ZZ_p[G]$-module $\\hat A^t(\\wp^{i}_{F_p})$ is cohomologically-trivial for all $i$. Hence, if $\\hat A^t(\\wp_{F_p})$ is torsion-free, then it is a projective $\\ZZ_p[G]$-module (by \\cite[Th. 8]{cf}) and therefore free of rank $nd$ (since $\\QQ_p\\otimes_{\\ZZ_p}\\hat A^t(\\wp_{F_p})$ is isomorphic to $F^d_p$).\n\nIn this case we fix an ordered basis $x_\\bullet$ of $\\hat A^t(\\wp_{F_p})$ and regard it as a $\\QQ_p^c[G]$-basis of $\\QQ_p^c\\otimes_{\\ZZ_p}\\hat A^t(\\wp_{F_p})$.\n\nWe also regard $\\{(i,\\hat\\sigma): (i,\\sigma) \\in [d]\\times\\Sigma(k)\\}$ as a $\\QQ_p^c[G]$-basis of the direct sum $(\\QQ_p^c\\cdot Y_{F\/k,p})^d$ of $d$ copies of $\\QQ_p^c\\cdot Y_{F\/k,p}$.\n\nThen, with respect to these bases, the matrix\n\\begin{equation}\\label{log resolve matrix} \\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}({\\rm log}_{\\omega_j'}(x_{(j',\\sigma')})))\\cdot g \\bigr)_{(j,\\sigma),(j',\\sigma')}\\end{equation}\nrepresents the composite isomorphism of $\\QQ^c_p[G]$-modules\n\\[ \\mu': \\QQ_p^c\\otimes_{\\ZZ_p}\\hat A^t(\\wp_{F_p}) \\xrightarrow{({\\rm log}_{\\omega_j'})_{j\\in [d]}}\n(\\QQ_p^c\\otimes_\\QQ F)^d \\xrightarrow{(\\mu)_{i \\in [d]}} (\\QQ_p^c\\cdot Y_{F\/k,p})^d,\\]\nwhere $\\mu$ is the isomorphism $\\QQ_p^c\\otimes_\\QQ F \\cong \\QQ_p^c\\cdot Y_{F\/k,p}$ that sends each element $\\lambda\\otimes f$ to $(\\lambda\\cdot \\hat\\sigma(f))_{\\sigma\\in \\Sigma(k)}$. This fact implies that\n\\begin{equation}\\label{eq 1}\\delta_{G,p}(\\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)) = [\\hat A^t(\\wp_{F_p}) , Y^d_{F\/k,p}; \\mu'] \\end{equation}\nin $K_0(\\ZZ_p[G],\\QQ^c_p[G])$.\n\nIn addition, since for any large enough integer $i$ the image of $\\hat A^t(\\wp^i_{F_p})$ under each map ${\\rm log}_{\\omega_j'}$ is equal to $\\wp^i_{F,p}$, the `telescoping' argument of Lemma \\ref{ullom} implies that\n\\begin{align}\\label{eq 2} [\\hat A^t(\\wp_{F_p}) , Y^d_{F\/k,p}; \\mu'] = \\, &d\\cdot [\\wp_{F_p} , Y_{F\/k,p}; \\mu]\\\\\n                                                                 = \\, &d\\cdot [\\mathcal{O}_{F,p}, Y_{F\/k,p}; \\mu] + d\\cdot \\chi_{G,p}\n                                                                 \\bigl((\\mathcal{O}_{F,p}\/\\wp_{F_p})[0],0\\bigr).\\notag\n                                                                 \\end{align}\n\nNext we note that if $f_v$ is the absolute residue degree of a $p$-adic place $v$, then the normal basis theorem for\n$\\mathcal{O}_{F_w}\/\\wp_{F_w}$ over the field with $p$ elements implies that there exists a short\nexact sequence of $G_w\/I_w$-modules\n\n\\[ 0 \\to \\ZZ [G_w\/I_w] ^{f_{v}} \\xrightarrow{\\times p} \\ZZ[G_w\/I_w]^{f_{v}} \\to \\mathcal{O}_{F_w}\/\\wp_{F_w} \\to 0.\\]\nBy using these sequences (for each such $v$) one computes that\n\n\\begin{equation*}\\label{eq 3} \\chi_{G,p}\\bigl((\\mathcal{O}_{F,p}\/\\wp_{F_p})[0],0\\bigr) = \\delta_{G,p}\\bigl(\\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr).\\end{equation*}\n\n\nUpon combining this equality with (\\ref{eq 1}) and (\\ref{eq 2}) we deduce that\n\\[ \\delta_{G,p}(\\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot (\\prod_{v \\in S_k^p}\\varrho_v(F\/k))^{-d}) = d\\cdot [\\mathcal{O}_{F,p}, Y_{F\/k,p}; \\mu]\\]\nand from here one can deduce the claimed result by using the argument of Proposition \\ref{lms}.\\end{proof}\n\n\n\\begin{remark}\\label{new add}{\\em Assume that all $p$-adic places are at most tamely ramified in $F\/k$, that neither $\\widehat{A^t}(\\wp_{F_p})$ nor $A(F)$ has an element of order $p$ and that $e_{(a)} = 1$ so there exist ordered $a$-tuples $P_\\bullet$ and $Q_\\bullet$  as in Lemma \\ref{height pairing interp}. Then Proposition \\ref{explicit log resolve}, Lemma \\ref{height pairing interp} and Remark \\ref{more explicit rem}(i) combine with Theorem \\ref{big conj} to imply ${\\rm BSD}_p(A_{F\/k})$(iv) predicts that any element in the set\n\\begin{equation*}\\label{explicit ann}\n\\frac{L^{(a)}_{S}(A_{F\/k},1)\\cdot \\bigl(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr)^d}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d\\cdot {\\rm det}(h_{F\/k}(P_\\bullet, Q_\\bullet))} \\cdot {\\rm Fit}^0_{\\ZZ[G]}((A^t(F)\/\\langle P_\\bullet\\rangle)^\\vee_{\\rm tor})\\cdot{\\rm Fit}^0_{\\ZZ[G]}((A(F)\/\\langle Q_\\bullet\\rangle)^\\vee_{\\rm tor})\\end{equation*}\nbelongs to ${\\rm Fit}^a_{\\ZZ_{p}[G]}({\\rm Sel}_{p}(A_{F})^\\vee)$ and annihilates $\\sha(A^t_{F})[p^\\infty]$}.\\end{remark}\n\n\\begin{remark}\\label{new add2}{\\em To obtain a variant of the prediction in Remark \\ref{new add} that may in some cases be more amenable to numerical investigation assume that $p$ is unramified in $k$, that all $p$-adic places of $k$ are at most tamely ramified in $F$, that neither $A(F)$ nor $A^t(F)$ has an element of order $p$, that $e_{(a)} = 1$ and that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$). Then by using Remark \\ref{new remark SC} in place of Theorem \\ref{big conj} the same approach as in Remark \\ref{new add} allows one to show that under these hypotheses the prediction in Remark \\ref{new add} should be true after replacing the terms $L^{(a)}_{S}(A_{F\/k},1)$ and $S_k^p$ that occur in the given displayed expression by $L^{(a)}_{S_{\\rm r}}(A_{F\/k},1)$ and $S_{p,{\\rm r}}$ respectively, where, as in (\\ref{bkcharelement}), we set $S_{\\rm r} := S_{k}^f\\cap S_k^F$ and $S_{p,{\\rm r}}:= S_k^p\\cap S_k^F$.\n}\\end{remark}\n\n\\begin{example}\\label{wuthrich example}{\\em Christian Wuthrich kindly supplied us with the following concrete applications of Remark \\ref{new add}.  Set $k = \\QQ$ and $K = \\QQ(\\sqrt{229})$ and write $F$ for the Galois closure of the field $L = \\QQ(\\alpha)$ with $\\alpha^3-4\\alpha+1 = 0$. Then $K \\subset F$ and the group $G := G_{F\/k}$ is dihedral of order six. Let $A$ denote either of the curves 3928b1 (with equation $y^2 = x^3-x^2 + x + 4$) or 5864a1 (with equation $y^2 = x^3-x^2 -24x + 28$). Then ${\\rm rk}(A_\\QQ)= 2$, ${\\rm rk}(A_K) = {\\rm rk}(A_L) = 3$ and ${\\rm rk}(A_F) = 5$ and, since $\\sha_3(A_K)$ vanishes (as can be shown via a computation with Heegner points on the quadratic twist of $A$), these facts combine with \\cite[Cor. 2.10(i)]{bmw0} to imply the $\\ZZ_{3}[G]$-module $A(F)_{3}$ is isomorphic to $\\ZZ_{3}[G](1+\\tau) \\oplus \\ZZ_{3}\\oplus \\ZZ_{3}$, with $\\tau$ the unique non-trivial element in $G_{F\/L}$. In particular, if we set $\\Gamma := G_{F\/K}$, then we can choose a point $P$ that generates over $\\ZZ_{3}[\\Gamma]$ a free direct summand of $A(F)_{3}$. In addition, ${\\rm Fit}^1_{\\ZZ_{3}[\\Gamma]}({\\rm Sel}_{3}(A_{F})^\\vee)$ is contained in\n\\[ {\\rm Fit}^1_{\\ZZ_{3}[\\Gamma]}(\\ZZ_{3}[G](1+\\tau) \\oplus \\ZZ_{3}\\oplus \\ZZ_{3}) = {\\rm Fit}^0_{\\ZZ_{3}[\\Gamma]}(\\ZZ_{3}\\oplus \\ZZ_{3}) = I_{3}(\\Gamma)^2,\\]\nwhere $I_{3}(\\Gamma)$ is the augmentation ideal of $\\ZZ_{3}[\\Gamma]$. Finally, we note that $3$ splits in $K$ and is unramified in $F$ so that for each $3$-adic place $v$ of $K$ the element $\\varrho_v(F\/K)$ is equal to $3$. After taking account of these facts, Remark \\ref{new add} (with $F\/k$ taken to be $F\/K$, $a$ to be $1$ and $P_1 = Q_1$ to be $P$) shows ${\\rm BSD}_3(A_{F\/k})$(iv) predicts that\n\\[ 9\\cdot \\tau^\\ast(F\/K)\\cdot\\frac{L^{(1)}_{S}(A_{F\/K},1)}{\\Omega_A^{F\/k}} = x\\cdot \\sum_{g \\in G}\\langle g(P),P\\rangle_{A_F}\\cdot g^{-1}\\]\nfor an element $x$ of $I_{3}(\\Gamma)^2$ that annihilates the $3$-primary component of $\\sha(A_{F})$. Here we have also used the fact that $w_{F\/K}=1$ because each of the real places $v$ of $K$ has trivial decomposition group in $\\Gamma$ so $\\psi^-_v(1)=1-1=0$ and thus $w_\\psi=1$ for each $\\psi\\in\\widehat\\Gamma$.}\n\\end{example}\n\n\\subsubsection{}It is possible to prove less precise versions of Proposition \\ref{explicit log resolve} without making any hypotheses on ramification and to thereby obtain more explicit versions of the prediction that is made in Theorem \\ref{big conj}.\n\nFor example, if $F_p^\\times$ has no element of order $p$, $\\mathfrak{M}$ is any choice of maximal $\\ZZ_p$-order in $\\QQ_p[G]$ that contains $\\ZZ_p[G]$ and $x$ is any element of $\\zeta(\\ZZ_p[G])$ such that $x\\cdot \\mathfrak{M} \\subseteq \\ZZ_p[G]$, then one can deduce from the result of \\cite[Cor. 7.8]{bleyburns} that for every element $y$ of the ideal\n\\[ \\zeta(\\ZZ_p[G])\\cdot p^d\\cdot x^{1+2d}((1-e_G)+ |G|\\cdot e_G)\\]\nthere exists an ordered $nd$-tuple $x(y)_\\bullet$ of points in $A^t(F_p)$ for which  one has\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x(y)_\\bullet) = y\\cdot \\bigl(\\tau^\\ast(F\/k)\\cdot \\prod_{v \\in S_k^p}\\varrho_v(F\/k)\\bigr)^d\\]\nin $\\zeta(\\QQ^c[G])$.\n\nHowever, we shall not prove this result here both because it is unlikely to be the best possible `bound' that one can give on logarithmic resolvents in terms of Galois-Gauss sums and also because, from the perspective of numerical investigations, the following much easier interpretation of logarithmic resolvents in terms of Galois resolvents is likely to be more helpful.\n\n\\begin{lemma} Let $i_0$ be the least integer with $i_0 > e_{F,p}\/(p-1)$, where $e_{F,p}$ is the maximal absolute ramification degree of any $p$-adic place of $F$. Let $z$ be an element of $F$ that belongs to the $i_0$-th power of every prime ideal of $\\mathcal{O}_F$ above $p$.\n\nThen, for any integral basis $\\{a_i:  i \\in [n]\\}$ of $\\mathcal{O}_k$, there exists an ordered $nd$-tuple $x_\\bullet$ of points in $A^t(F_p)$ for which  one has\n\\[ \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) = {\\rm Nrd}_{\\QQ_p^c[G]}\\left( \\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z\\cdot a_i))\\cdot g\\bigr)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^d. \\]\n\\end{lemma}\n\n\\begin{proof} The definition of $i_0$ ensures that the formal group logarithm of $A^t$ over $F_p$ gives an isomorphism of $\\hat A^t(\\wp_{F_p}^{i_0})$ with a direct sum $(\\wp_{F_p}^{i_0})^d$ of $d$ copies of $\\wp_{F_p}^{i_0}$ (cf. \\cite[Th. 6.4(b)]{silverman}). Here the individual copies of $\\wp_{F_p}^{i_0}$ in the sum are parametrised by the differentials $\\{\\omega_j': j \\in [n]\\}$ that are used to define ${\\rm log}_{A^t}$.\n\nThe choice of $z$ also implies that $Z := \\{z\\cdot a_i:  i \\in [n]\\}$ is a subset of $\\wp_{F_p}^{i_0}$. We may therefore choose a pre-image $x_\\bullet$ in $\\hat A^t(\\wp_{F_p}^{i_0})$ of the ordered $nd$-tuple in $(\\wp_{F_p}^{i_0})^d$ that is obtained by placing a copy of $Z$ in each of the $d$ direct summands.\n\nFor these points $x_\\bullet$ the interpretation of the matrix (\\ref{log resolve matrix}) that is given in the proof of  Proposition \\ref{explicit log resolve} shows that it is equal to a $d\\times d$ diagonal block matrix with each diagonal block equal to the matrix\n\\[  \\bigl(\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z\\cdot a_i))\\cdot g\\bigr)_{\\sigma\\in \\Sigma(k),i\\in [n]}.\\]\n\nSince $\\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)$ is defined to be the reduced norm of the matrix (\\ref{log resolve matrix}) the claimed equality is therefore clear.\\end{proof}\n\n\\subsection{The proof of Theorem \\ref{big conj}}\\label{proof of big conj}\n\n\\subsubsection{}We start by proving a technical result that is both necessary for the proof of Theorem \\ref{big conj} and will also be of further use in the sequel.\n\nIn this result we use the terminology of `characteristic elements' from \\cite[Def. 3.1]{bst}.\n\n\\begin{lemma}\\label{modifiedlemma} Fix an ordered subset $x_\\bullet$ of $A^t(F_p)^\\wedge_p$ as in Theorem \\ref{big conj}. Write $X$ for the $\\ZZ_p[G]$-module generated by $x_\\bullet$ and $C_{S,X}$ for the Selmer complex ${\\rm SC}_S(A_{F\/k};X,H_\\infty(A_{F\/k})_p)$ from Definition \\ref{selmerdefinition}. Then the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)] The module $H^1(C_{S,X})$ is torsion-free.\n\\item[(ii)] For any finite non-empty set of places $T$ of $k$ that is disjoint from $S$, there exists an exact triangle in $D^{\\rm perf}(\\ZZ_p[G])$ of the form\n\\begin{equation}\\label{modifiedtriangle}\\bigoplus_{v\\in T}R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))[-2]\\to C_{S,X}\\stackrel{\\theta}{\\to} C_{S,X,T}\\to\\bigoplus_{v\\in T}R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))[-1]\\end{equation}\nin which $C_{S,X,T}$ is acyclic outside degrees one and two and there are canonical identifications of $H^1(C_{S,X,T})$ with $H^1(C_{S,X})$ and of  $\\Sel_p(A_F)^\\vee$ with a subquotient of $H^2(C_{S,X,T})$ in such a way that $\\QQ_p\\cdot\\Sel_p(A_F)^\\vee=\\QQ_p\\cdot H^2(C_{S,X,T})$.\n \\item[(iii)] Following claim (ii) we write\n\\[ h^{T}_{A,F}: \\CC_p\\cdot H^1(C_{S,X,T}) \\to \\CC_p\\cdot H^2(C_{S,X,T})\\]\nfor the isomorphism $(\\CC_p\\otimes_{\\ZZ_p}H^2(\\theta))\\circ(\\CC_p\\otimes_{\\RR,j}h_{A,F})$ of $\\CC_p[G]$-modules. Then, if ${\\rm BSD}_p(A_{F\/k})$(iv) is valid, there exists a characteristic element $\\mathcal{L}$ for  $(C_{S,X},h_{A,F})$ in $\\CC_p[G]^\\times$ with the property that for any non-negative integer $a$ one has\n\\[ e_a\\cdot\\mathcal{L}^{-1}=\\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet).\\]\n\nIn addition, in this case the element\n\\[ \\mathcal{L}_T := (\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)^{-1}\\cdot\\mathcal{L}\\]\nis a characteristic element for  $(C_{S,X,T},h^{T}_{A,F})$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n\nSince $p$ is odd there exists a homomorphism of $\\ZZ_p[G]$-modules $\\phi$ from the module $H_\\infty(A_{F\/k})_p = \\bigoplus_{v\\in S_k^\\infty}H^0(k_v,T_{p,F}(A^t))$ to $A^t(F_p)^\\wedge_p$ that sends the ordered $\\ZZ_p[G]$-basis of $H_\\infty(A_{F\/k})_p$ specified at the end of \\S\\ref{gamma section} to $x_\\bullet$.\n\nThen, comparing the triangle (\\ref{can tri}) with the construction of \\cite[Prop. 2.8 (ii)]{bst} immediately implies that $C_{S,X}$ is isomorphic in $D^{\\rm perf}(\\ZZ_p[G])$ to the complex that is denoted in loc. cit. by $C_\\phi(T_{p,F}(A^t))$.\n\nGiven this, claim (i) follows directly from \\cite[Prop. 2.8 (ii)]{bst} and claim (ii) from \\cite[Prop. 2.8 (iii)]{bst} with $C_{S,X,T}:=C_{\\phi,T}(T_{p,F}(A^t))$.\n\nTurning to claim (iii) we note that each place $v$ in $T$ is not $p$-adic, does not ramify in $F\/k$ and is of good reduction for $A$.\n\nEach complex $R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))$ is therefore well-defined. Since these complexes are acyclic outside degree one, where they have finite cohomology, we may therefore apply Lemma \\ref{fk lemma} to the triangle (\\ref{modifiedtriangle}) to deduce that\n\\begin{align*}\\chi_{G,p}(C_{S,X},h_{A,F})-\\chi_{G,p}(C_{S,X,T},h^{T}_{A,F})= \\, &\\sum_{v\\in T}\\chi_{G,p}(R\\Gamma(\\kappa_v,T_{p,F}(A^t)(-1))[-2],0)\\\\\n=\\, &\\delta_{G,p}({\\det}_{\\QQ_p[G]}(1-\\Phi_v|\\QQ_p\\cdot T_{p,F}(A^t)(-1)))\\\\\n=\\, &\\delta_{G,p}(P_v(A_{F\/k},1)^\\#)\n\\end{align*}\nin $K_0(\\ZZ_p[G],\\CC_p[G])$.\n\nThus if $\\mathcal{L}$ is a characteristic element of $(C_{S,X},h_{A,F})$, then $(\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)^{-1}\\cdot\\mathcal{L}$ is a characteristic element for  $(C_{S,X,T},h^{T}_{A,F})$, as claimed in the final assertion of claim (iii).\n\nIt is thus enough to deduce from ${\\rm BSD}_p(A_{F\/k})$(iv) the existence of a characteristic element $\\mathcal{L}$ of $(C_{S,X},\\CC_p\\otimes_{\\RR}h_{A,F})$ with the required interpolation property.\n\nNow, since $S$ contains all $p$-adic places of $k$, the module $\\mathcal{Q}(\\omega_\\bullet)_{S,p}$ vanishes and the $p$-primary component of the term $\\mu_{S}(A_{F\/k})$ is also trivial.\n\nIn addition, as the validity of {\\rm BSD}$_p(A_{F\/k})$(iv) is independent of the choice of global periods and we can assume firstly that $\\omega_\\bullet$ is the set $\\{ z_i\\otimes \\omega'_j: i \\in [n], j \\in [d]\\}$ fixed in Lemma \\ref{k-theory period} and secondly that the image of $\\mathcal{F}(\\omega_\\bullet)_p$ under the formal group exponential ${\\rm exp}_{A^t,F_p}$ (defined with respect to the differentials $\\{\\omega_j': j \\in [d]\\})$ is contained in $X$.\n\nThen the assumed validity of the equality (\\ref{displayed pj}) in this case combines with the equality in Lemma \\ref{k-theory period} to imply that the element\n\n\\begin{equation}\\label{char el 1} \\frac{L_S^*(A_{F\/k},1)}{{\\rm Nrd}_{\\RR[G]}(\\Omega_{\\omega_\\bullet}(A_{F\/k}))}= \\frac{L_S^*(A_{F\/k},1)}{ \\Omega_A^{F\/k}\\cdot w_{F\/k}^{d}}\\cdot {\\rm det}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^{d}\\end{equation}\nof $\\zeta(\\CC_p[G])^\\times$ is the inverse of a characteristic element of $(C_{S,\\omega_\\bullet},h_{A,F})$.\n\nHere we write $C_{S,\\omega_\\bullet}$ for the Selmer complex ${\\rm SC}_S(A_{F\/k};\\mathcal{X}(p),H_\\infty(A_{F\/k})_p)$ where $\\mathcal{X}$ is the perfect Selmer structure $\\mathcal{X}_S(\\{\\mathcal{A}^t_v\\}_v,\\omega_\\bullet,\\gamma_\\bullet)$ defined in \\S\\ref{perf sel sect}. (In addition, the fact that it is the inverse of a characteristic element results from a comparison of our chosen normalisation of non-abelian determinants compared with that of \\cite[(10)]{bst}, as described in Remark \\ref{comparingdets}).\n\nIn particular, since $\\mathcal{X}(p)$ is by definition equal to ${\\rm exp}_{A^t,F_p}(\\mathcal{F}(\\omega_\\bullet)_p)$ and hence, by assumption, contained in $X$, a comparison of the definitions of $C_{S,\\omega_\\bullet}$ and $C_{S,X}$ shows that there is an exact triangle\n\\[  C_{S,\\omega_\\bullet} \\to C_{S,X} \\to \\bigl(X\/\\mathcal{X}(p))[-1] \\to C_{S,\\omega_\\bullet}[1]\\]\nin $D^{\\rm perf}(\\ZZ_p[G])$.\n\nSince the product\n\n\\[ \\xi := \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot {\\rm det}\\left( \\left( (\\sum_{g \\in G} \\hat \\sigma(g^{-1}(z_i)))\\cdot g\\right)_{\\sigma\\in \\Sigma(k),i\\in [n]}\\right)^{-d}\\]\nis equal to the determinant of a matrix that expresses a basis of the free $\\ZZ_p[G]$-module $\\mathcal{X}(p)$ in terms of the basis $x_\\bullet$ of $X$, the above triangle implies that the product\n\\[ \\frac{L_S^*(A_{F\/k},1)}{ \\Omega_A^{F\/k}\\cdot w_{F\/k}^{d}}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\]\nof $\\xi$ and the element (\\ref{char el 1}) is the inverse of a characteristic element for  $(C_{S,X},h_{A,F})$.\n\nThe claimed interpolation formula is thus a consequence of the fact that $L_S^{(a)}(A_{F\/k},1)$ is equal to $e_a\\cdot L_S^*(A_{F\/k},1)$. \\end{proof}\n\n\n\\subsubsection{}We are now ready to prove Theorem \\ref{big conj}.\n\nTo do this we will apply the general result of \\cite[Th. 3.10(i)]{bst} to the complex $C_{S,X,T}$, isomorphism $h^{T}_{A,F}$ and characteristic element $\\mathcal{L}_T$ constructed in Lemma \\ref{modifiedlemma}.\n\n In order to do so, we fix an ordered subset $\\Phi:=\\{\\phi_i: i \\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G])$ of cardinality $a$. We fix a pre-image $\\phi_i'$ of each $\\phi_i$ under the surjective composite homomorphism\n\\[ H^2(C_{S,X})\\to\\Sel_p(A_F)^\\vee\\to \\Hom_{\\ZZ_p[G]}(A(F)_p,\\ZZ_p[G]),\\]\nwhere the first arrow is the canonical map from Proposition \\ref{prop:perfect}(iii) and the second is induced by the canonical short exact sequence\n\\begin{equation}\\label{sha-selmer}\n \\xymatrix{0 \\ar[r] & \\sha(A_F)[p^\\infty]^\\vee \\ar[r] & \\Sel_p(A_F)^\\vee \\ar[r]&  \\Hom_{\\ZZ_p}(A(F)_p,\\ZZ_p)\\ar[r] & 0.}\n\\end{equation}\n\nWe set $\\Phi':=\\{\\phi'_i:i \\in [a]\\}$ and consider the image $H^2(\\theta)(\\Phi')$ of $\\Phi'$ in $H^2(C_{S,X,T})$, where $\\theta$ is the morphism that occurs in the triangle (\\ref{modifiedtriangle}) (so that $H^2(\\theta)$ is injective).\n\n\nWe next write $\\iota:H^1(C_{S,X,T})=H^1(C_{S,X})\\to A^t(F)_p$ for the canonical homomorphism in Proposition \\ref{prop:perfect}(iii).\n\nThen, with $\\mathcal{L}_T$ the element specified in Lemma \\ref{modifiedlemma}(iii), a direct comparison of the definitions of $h_{A,F}^{T}$ and ${\\rm ht}_{A_{F\/k}}^{a}$ shows that the `higher special element' that is associated via \\cite[Def. 3.3]{bst} to the data $(C_{S,X,T},h^{T}_{A,F},\\mathcal{L}_T,H^2(\\theta)(\\Phi'))$ coincides with the pre-image under the bijective map $\\CC_p\\cdot\\bigwedge_{\\ZZ_p[G]}^a\\iota$ of the element\n\\begin{equation}\\label{hse interpret}(\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)\\cdot \\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot {\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i).\\end{equation}\n(Here we have also used the fact that, since ${\\rm BSD}(A_{F\/k})$(ii) is assumed to be valid, the idempotent $e_a$ defined here coincides with the idempotent denoted $e_a$ in \\cite[\\S3.1]{bst} for the complex $C_{S,X,T}$.)\n\nTo proceed we fix an ordered subset $\\{\\theta_j: j \\in [a]\\}$ of $\\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G])$ and identify it with its image under the injective map\n\\[ \\Hom_{\\ZZ_p[G]}(A^t(F)_p,\\ZZ_p[G]) \\to \\Hom_{\\ZZ_p[G]}(H^1(C_{S,X,T}),\\ZZ_p[G])\\]\ninduced by $\\iota$. We also set $\\mathfrak{A} := \\ZZ_p[G]e_{(a)}$ and $M := \\mathfrak{A}\\otimes_{\\ZZ_p[G]}H^2(C_{S,X,T})$.\n\nThen the above interpretation of the higher special element in terms of the product (\\ref{hse interpret}) combines with the general result of \\cite[Th. 3.10(i)]{bst} to imply that for any element $\\alpha$ of $\\ZZ_p[G]\\cap\\mathfrak{A}$ and any element $y$ of $\\ZZ_p[G]$ that annihilates ${\\rm Ext}^2_{\\mathfrak{A}}(M,\\mathfrak{A})$\nthe product\n\\[\\alpha \\cdot y^a \\cdot(\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#)\\cdot \\frac{L^{(a)}_{S_F}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet)\\cdot (\\wedge_{j=1}^{j=a}\\theta_j)({\\rm ht}^{a}_{A_{F\/k}}(\\wedge_{i=1}^{i=a}\\phi_i))\\]\nboth belongs to ${\\rm Fit}^a_{\\ZZ_p[G]}(\\Sel_p(A_F)^\\vee)$ and annihilates $(\\Sel_p(A_F)^\\vee)_{\\rm tor}$.\n\nIn addition, the exact sequence (\\ref{sha-selmer}) identifies $(\\Sel_p(A_F)^\\vee)_{\\rm tor}$ with $\\sha(A_F)[p^\\infty]^\\vee$ and the Cassels-Tate pairing identifies $\\sha(A_F)[p^\\infty]^\\vee$ with $\\sha(A^t_F)[p^\\infty]$.\n\nTo deduce the result of Theorem \\ref{big conj} from here, it is therefore enough to show that $\\alpha^2$ annihilates ${\\rm Ext}^2_{\\mathfrak{A}}(M,\\mathfrak{A})$.\n\nTo do this we use the existence of a convergent first quadrant cohomological spectral sequence\n\\[ E_2^{pq} = {\\rm Ext}_{\\mathfrak{A}}^p(M,{\\rm Ext}^q_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A})) \\Rightarrow {\\rm Ext}^{p+q}_{\\ZZ_p[G]}(M,\\mathfrak{A})\\]\n(cf. \\cite[Exer. 5.6.3]{weibel}).\n\nIn particular, since the long exact sequence of low degree terms of this spectral sequence gives an exact sequence of $\\ZZ_p[G]$-modules\n\\[ \\Hom_{\\ZZ_p[G]}(M,{\\rm Ext}^1_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A})) \\to {\\rm Ext}_{\\mathfrak{A}}^2(M,\\mathfrak{A}) \\to {\\rm Ext}^{2}_{\\ZZ_p[G]}(M,\\mathfrak{A}),\\]\nwe find that it is enough to show that the element $\\alpha$ annihilates both ${\\rm Ext}^1_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A})$ and ${\\rm Ext}^{2}_{\\ZZ_p[G]}(M,\\mathfrak{A})$.\n\nTo verify this we write $\\mathfrak{A}^\\dagger$ for the ideal $\\{x \\in \\ZZ_p[G]: x\\cdot e_{(a)} = 0\\}$ so that there is a natural short exact sequence of $\\ZZ_p[G]$-modules $0 \\to \\mathfrak{A}^\\dagger \\to \\ZZ_p[G] \\to \\mathfrak{A} \\to 0$.\n\nThen by applying the exact functor ${\\rm Ext}^\\bullet_{\\ZZ_p[G]}(-,\\mathfrak{A})$ to this sequence one obtains a surjective homomorphism\n\\[ \\Hom_{\\ZZ_p[G]}(\\mathfrak{A}^\\dagger,\\mathfrak{A}) \\twoheadrightarrow {\\rm Ext}^1_{\\ZZ_p[G]}(\\mathfrak{A},\\mathfrak{A}).\\]\n\nIn addition, since $\\ZZ_p[G]$ is Gorenstein, by applying the exact functor ${\\rm Ext}^{\\bullet}_{\\ZZ_p[G]}(M,-)$ to the above sequence one finds that  there is a natural isomorphism\n\\[ {\\rm Ext}^{3}_{\\ZZ_p[G]}(M,\\mathfrak{A}^\\dagger) \\cong {\\rm Ext}^{2}_{\\ZZ_p[G]}(M,\\mathfrak{A}).\\]\n\nTo complete the proof of Theorem \\ref{big conj} it is thus enough to note that the left hand modules in both of the last two displays are annihilated by $\\alpha$ since the definition of $\\mathfrak{A}^\\dagger$ implies immediately that $\\alpha\\cdot \\mathfrak{A}^\\dagger = 0$.\n\n\\begin{remark}\\label{omit T}{\\em If $A(F)$ does not contain an element of order $p$, then \\cite[Th. 3.10(i)]{bst} can be directly applied to the complex $C_{S,X}$ rather than to the auxiliary complex $C_{S,X,T}$. This shows that, in any such case, the prediction in Theorem \\ref{big conj} should remain true if the term $\\prod_{v \\in T}P_v(A_{F\/k},1)^\\#$ is omitted.}\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Abelian congruence relations and height pairings}\\label{mrsconjecturesection}\n\n\n\n\n\n\n\nIn this section we continue to investigate the $p$-adic congruence relations between the leading coefficients of Hasse-Weil-Artin $L$-series that are encoded by ${\\rm BSD}_p(A_{F\/k})$(iv) in the case that $F\/k$ is abelian.\n\nMore concretely in \\S\\ref{mtchd} we will show that, beyond the integrality properties that are discussed in Theorem \\ref{big conj} and Remark \\ref{integrality rk}, elements of the form (\\ref{key product}) can be expected to satisfy additional congruence relations in the integral augmentation filtration that  involve Mazur-Tate regulators.\n\n\nThen in \\S \\ref{cycliccongssection} we specialise to the case of cyclic extensions in order to make these additional congruence relations fully explicit.\n\nIn particular, in this way we render the equality ${\\rm BSD}_p(A_{F\/k})$(iv) amenable to (numerical) verification even in cases in which it incorporates a thorough-going mixture of both archimedean phenomena and delicate $p$-adic congruences.\n\nFinally, in \\S\\ref{dihedral} we explain how these results extend to certain families of non-abelian extensions.\n\nThroughout this section, just as in \\S\\ref{8.1}, we give ourselves a fixed odd prime $p$ and isomorphism of fields $j:\\CC\\cong\\CC_p$ (explicit mention of which we usually omit), a finite set $S$ of places of $k$ with\n\\[ S_k^\\infty\\cup S_k^p\\cup  S_k^F \\cup S_k^A\\subseteq S\\]\nand a fixed ordered $k$-basis $\\{\\omega_j'\\}_{j\\in[d]}$ of $H^0(A^t,\\Omega^1_{A^t})$ with associated classical period $\\Omega_A^{F\/k}$.\n\nExcept in \\S\\ref{dihedral} we shall always assume in this section that $F\/k$ is abelian. In addition, we shall always assume that $p$ is chosen so that neither $A(F)$ nor $A^t(F)$ has a point of order $p$.\n\n\n\n\\subsection{A Mazur-Tate conjecture for higher derivatives}\\label{mtchd}\n\nIn this section we formulate a Mazur-Tate type conjecture for higher derivatives of Hasse-Weil-Artin $L$-series. We then show that, under the hypotheses listed in \\S \\ref{tmc}, this conjecture would follow from the validity of ${\\rm BSD}(A_{F\/k})$.\n\n\\subsubsection{\n\nWe first quickly recall the construction of canonical height pairings of Mazur and Tate \\cite{mt0}.\n\nTo do this we fix a subgroup $J$ of $G$ and set $E := F^J$.\nWe recall that the subgroups of `locally-normed elements' of $A(E)$ and $A^t(E)$ respectively are defined by setting \\begin{equation}\\label{localnorms}U_{F\/E}:=\\bigcap_v \\bigl(A(E)\\cap N_{F_w\/E_v}(A(F_w))\\bigr),\\,\\,\\,\\,U^t_{F\/E}:=\\bigcap_v \\bigl(A^t(E)\\cap N_{F_w\/E_v}(A^t(F_w))\\bigr).\\end{equation}\nHere each intersection runs over all (finite) primes $v$ of $E$ and $w$ is a fixed prime of $F$ above $v$. In addition, $N_{F_w\/E_v}$ denotes the norm map of $F_w\/E_v$ and each intersection of the form $A(E)\\cap N_{F_w\/E_v}(A(F_w))$, resp. $A^t(E)\\cap N_{F_w\/E_v}(A^t(F_w))$, takes place inside $A(E_v)$, resp. $A^t(E_v)$.\n\nWe recall from Lemma \\ref{useful prel}(i) that each of the expressions displayed in (\\ref{localnorms}) is in general a finite intersection of subgroups of $A(E)$, resp. of $A^t(E)$, and that the subgroups $U_{F\/E}$ and $U^t_{F\/E}$ have finite index in $A(E)$ and $A^t(E)$ respectively.\n\nWe note for later use that, whenever $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$) listed in \\S \\ref{tmc}, then Proposition \\ref{explicitbkprop}(ii) (together with the duality of these hypotheses) implies that $$U_{F\/E,p}:=\\ZZ_p\\otimes_{\\ZZ}U_{F\/E}\\,\\,\\text{ and }\\,\\,U^t_{F\/E,p}:=\\ZZ_p\\otimes_{\\ZZ}U^t_{F\/E}$$ are equal to $A(E)_p$ and to $A^t(E)_p$ respectively (for every given subgroup $J$ of $G$).\n\nIn general, Mazur and Tate \\cite{mt0} construct, by using the theory of biextensions, a canonical height pairing \\begin{equation}\\label{tanpairing}\\langle\\,,\\rangle^{\\rm MT}_{F\/E}:U^t_{F\/E}\\otimes_\\ZZ U_{F\/E}\\to J.\\end{equation}\nThis pairing will be a key ingredient of our conjectural congruence relations. To formulate our conjecture we must first describe how to make reasonable choices of points on which to evaluate the Mazur-Tate pairing.\n\n\\begin{definition}\\label{separablepair}{\\em Fix a subgroup $J$ of $G$ and set $E:=F^J$. We define a `$p$-separable choice of points of $A$ for $F\/E$ of rank $(a,a')$' (with $a'\\geq a\\geq 0$) to be a pair $(\\mathcal{Y},\\mathcal{Y}')$ chosen as follows.\n\nLet $\\mathcal{Y}= \\{y_i: i \\in [a]\\}$ be any ordered finite subset of $A(F)_p$ that generates a free $\\ZZ_p[G]$-direct-summand $Y$ of $A(F)_p$ of rank equal to $|\\mathcal{Y}|=a$. Then $\\Tr_J(Y)=Y^J$ is a $\\ZZ_p[G\/J]$-direct-summand of $A(E)_p$\n\nWe then let $$\\mathcal{Y}'=\\Tr_J(\\mathcal{Y})\\cup\\{w_i:a<i\\leq a'\\}$$ be any ordered finite subset of $U_{F\/E,p}$, containing $\\Tr_J(\\mathcal{Y}):=\\{\\Tr_J(y_i):i \\in [a]\\}$, that generates a free $\\ZZ_p[G\/J]$-direct-summand $Y'$ of $A(E)_p$ of rank equal to $|\\mathcal{Y}'|=a'\\geq a$.}\n\\end{definition}\n\n\n\n\n\n\n\n\\begin{remarks}\\label{p-separable}{\\em\nTo identify $p$-separable choices of points as above, one can proceed as follows. Write $G=P\\times H$ where $P$ is the Sylow $p$-subgroup of $G$. Then the argument of \\cite[Prop. 2.1]{burns nagoya} shows that a finite subset $\\mathcal{Y}$ of $A(F)_p$ generates a free $\\ZZ_p[G]$-direct-summand $Y$ of $A(F)_p$ of rank equal to $|\\mathcal{Y}|=a$ if and only if the images in $A(F^P)_p\/p\\cdot A(F^P)_p$ of the elements $$h\\cdot\\sum_{g\\in P}g(y),$$ for each $h\\in H$ and each $y\\in\\mathcal{Y}$, are linearly independent over $\\mathbb{F}_p$. An analogous observation applies to the choice of the set $\\mathcal{Y}'$ in a $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$.}\n\\end{remarks}\n\nIn the rest of \\S \\ref{mrsconjecturesection}, for any $\\ZZ_p[G]$-module $M$ we write $M^*$ for the linear dual $\\Hom_{\\ZZ_p}(M,\\ZZ_p)$, endowed with the natural contragredient action of $G$.\n\n\nFor given integers $a'\\geq a\\geq 0$ and a given $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$, we always use the following notation.\nWe fix a decomposition $A(F)_p=Y\\oplus Z$. Since $A(F)_p^*=Y^*\\oplus Z^*$ and\n$Y^*$ is $\\ZZ_p[G]$-free of rank $a$ we may and will denote by $y_i^*$ the element of $Y^*\\subseteq A(F)_p^*$ which maps $y_i$ to 1 and $y_j$ to 0 for $j\\neq i$.\n\nIn a similar way, to each element $y'\\in\\mathcal{Y}'$ we associate a canonical dual element $(y')^*$ of $A(E)_p^*$.\n\nFix now any maximal subset $x_\\bullet$ of $A^t(F_p)^\\wedge_p$ that is linearly independent over $\\ZZ_p[G]$ and also set $x_\\bullet^J:= \\{\\Tr_J(x):x\\in x_\\bullet\\}\\subseteq A^t(E_p)^\\wedge_p$.\nBy using the isomorphism ${\\rm ht}_{A_{F\/k}}^{a}$ considered in (\\ref{athpower}) as well as the analogous isomorphism ${\\rm ht}_{A_{E\/k}}^{a'}$, one then obtains elements\n\\[ \\eta_{\\mathcal{Y},x_\\bullet} := \\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet) \\cdot {\\rm ht}_{A_{F\/k}}^{a}(\\wedge_{y\\in\\mathcal{Y}}y^*)\\in\\CC_p\\cdot\\bigwedge_{\\ZZ_p[G]}^aA^t(F)_p\\]\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J} := \\frac{L^{(a')}_{S}(A_{E\/k},1)}{\\Omega_A^{E\/k}\\cdot w_{E\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{E\/k}}(x_\\bullet^J)\\cdot {\\rm ht}_{A_{E\/k}}^{a'}(\\wedge_{y'\\in\\mathcal{Y}'}(y')^*)\\in\\CC_p\\cdot\\bigwedge_{\\ZZ_p[G\/J]}^{a'}A^t(E)_p.\\]\n\n\nFor each $a<i\\leq a'$ we may also define composite homomorphisms of $\\ZZ_p[G\/J]$-modules\n\\begin{equation}\\label{htan}{\\rm ht}^{{\\rm MT}}_{w_i}:U^t_{F\/E,p}\\to \\ZZ_p\\otimes_\\ZZ J\\cong I_p(J)\/I_p(J)^2\\to\\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2\\end{equation} by setting\n\\begin{equation}\\label{minussignMT}{\\rm ht}^{{\\rm MT}}_{w_i}(P):=-\\langle P,w_i\\rangle^{\\rm MT}_{F\/E}.\\end{equation} Here $\\mathcal{I}_p(J)$ denotes the ideal of $\\ZZ_p[G]$ generated by the augmentation ideal $I_p(J)$ of $\\ZZ_p[J]$, the isomorphism maps each $g$ in $J$ to the class of $g-1$ and the last arrow is induced by the inclusion $I_p(J)\\subseteq\\mathcal{I}_p(J)$. (We also note that the occurrence of the minus sign in (\\ref{minussignMT}) will be clarified by the result of Theorem \\ref{thecomptheorem} below).\n\n\n\n\\subsubsection{\n\nFor any finitely generated $\\ZZ_p[G]$-module $M$ and any non-negative integer $r$ we write $\\bigcap_{\\ZZ_p[G]}^rM$ for the $r$-th `Rubin lattice' of $M$ inside $\\QQ_p\\cdot\\bigwedge_{\\ZZ_p[G]}^rM$, as defined originally by Rubin \\cite{rub} (see also \\cite[\\S 4.2]{bks}).\n\nThen the homomorphisms ${\\rm ht}^{{\\rm MT}}_{w_i}$ can be combined as in \\cite[\\S3.4.1]{bst} to give a canonical homomorphism\n\\begin{equation}\\label{canbecombined}\n{\\rm ht}_{\\mathcal{Y}'}^{\\rm MT}: {\\bigcap}_{\\ZZ_p[G\/J]}^{a'} U^t_{F\/E,p} \\longrightarrow ({\\bigcap}_{\\ZZ_p[G\/J]}^{a} U^t_{F\/E,p})\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}.\n\\end{equation}\n\nFollowing an original idea of Darmon, we define the `norm operator'\n$$\\mathcal{N}_J : {\\bigcap}_{\\ZZ_p[G]}^a A^t(F)_p \\longrightarrow ({\\bigcap}_{\\ZZ_p[G]}^a A^t(F)_p) \\otimes_{\\ZZ_p} \\ZZ_p[J]\/I_p(J)^{1+a'-a}$$\nto be the homomorphism induced by sending each $x$ to $\\sum_{\\sigma \\in J} \\sigma (x) \\otimes \\sigma^{-1}$.\n\nWe also use \\cite[Prop. 4.12]{bks} to obtain a canonical injective homomorphism of $\\ZZ_p[G]$-modules\n\\begin{multline*} \\nu_J:({\\bigcap}_{\\ZZ_p[G\/J]}^{a} A^t(E)_p)\\otimes_{\\ZZ_p}I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}\\\\ \\rightarrow ({\\bigcap}_{\\ZZ_p[G]}^{a} A^t(F)_p)\\otimes_{\\ZZ_p}I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}.\\end{multline*}\n\n\n\n\nWe can now state the central conjecture of this section.\n\n\\begin{conjecture}\\label{mrsconjecture} Assume that neither $A(F)$ nor $A^t(F)$ has a point of order $p$.\nFix any subgroup $J$ of $G$ (with $E := F^J$) and any $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$. Then one has\n\\[ \\eta_{\\mathcal{Y},x_\\bullet}\\in {\\bigcap}_{\\ZZ_p[G]}^{a}A^t(F)_p\\]\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J}\\in {\\bigcap}_{\\ZZ_p[G\/J]}^{a'}U^t_{F\/E,p}\\]\nand in $({\\bigcap}_{\\ZZ_p[G]}^{a} A^t(F)_p)\\otimes_{\\ZZ_p}I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}$ there is an equality\n\\[ \\mathcal{N}_J(\\eta_{\\mathcal{Y},x_\\bullet})=(-1)^{a(a'-a)}\\cdot \\nu_J({\\rm ht}_{\\mathcal{Y}'}^{\\rm MT}(\\eta_{\\mathcal{Y}',x_\\bullet^J})).\\]\n\\end{conjecture}\n\n\\begin{remark}\\label{explicit pairings}{\\em\nLet $k = \\QQ$, $J = G$, $a=0$, $a' = {\\rm dim}_\\QQ(A(k))$ and $\\mathcal{Y}=\\{y_i: i \\in [a']\\}$ be any $\\ZZ$-basis of $\\Hom_\\ZZ(A(k),\\ZZ)$. \nThen, after taking account of Remark \\ref{new add}, one finds that the congruence in Conjecture \\ref{mrsconjecture} specialises in this case to refine and extend the conjectures formulated (for certain elliptic curves $A$ and abelian fields $F$) by Mazur and Tate in \\cite{mt} and also incorporates the main conjecture of Darmon \\cite{darmon0}. }\\end{remark}\n\n\\begin{remark}\\label{indeptpoints\n{\\em Fix any set $x_\\bullet$, subgroup $J$ of $G$ and integers $a'\\geq a\\geq 0$ as in Conjecture \\ref{mrsconjecture}.\nThen the general result of \\cite[Prop. 4.17]{bks} can be used to show that the validity of Conjecture \\ref{mrsconjecture} is independent of the $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$.}\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{\nThe following result establishes conditions under which Conjecture \\ref{mrsconjecture} is a consequence of ${\\rm BSD}(A_{F\/k})$.\n\nThe proof of this result will be given in \\S \\ref{mrsproof} below.\n\n\\begin{theorem}\\label{rbsdimpliesmt} Fix an odd prime number $p$ and assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$) listed in \\S \\ref{tmc}. Assume also that neither $A(F)$ nor $A^t(F)$ has a point of order $p$ and that Hypothesis \\ref{hypbd} below is satisfied.\n\nThen if ${\\rm BSD}(A_{F\/k})$ is valid, so is Conjecture \\ref{mrsconjecture}.\n\\end{theorem}\n\n\n\n\\begin{remarks}\\label{mrstheorem}{\\em Each of the hypotheses (H$_1$)-(H$_5$) as well as Hypothesis \\ref{hypbd} is satisfied widely  (see Remarks \\ref{satisfying H} and \\ref{rmkbd} respectively). However, without assuming any of these hypotheses it is also possible to deduce from ${\\rm BSD}(A_{F\/k})$ less explicit versions of the congruence relations that are predicted by Conjecture \\ref{mrsconjecture}.\n\nTo be more precise we fix an odd prime number $p$ for which neither $A(F)$ nor $A^t(F)$ has a point of order $p$ and set $$C_{S,x_\\bullet}:={\\rm SC}_S(A_{F\/k};\\langle x_\\bullet\\rangle_{\\ZZ_p[G]},H_\\infty(A_{F\/k})_p).$$ Then, without any additional hypotheses on $A$, $F\/k$ or $p$, the same argument as used to prove Theorem \\ref{rbsdimpliesmt} shows ${\\rm BSD}(A_{F\/k})$ implies that\n\\[ \\eta_{\\mathcal{Y},x_\\bullet}\\in {\\bigcap}_{\\ZZ_p[G]}^{a}H^1(C_{S,x_\\bullet})\\subseteq{\\bigcap}_{\\ZZ_p[G]}^{a}A^t(F)_p\\]\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J}\\in {\\bigcap}_{\\ZZ_p[G\/J]}^{a'}H^1(C_{S,x_\\bullet})^J\\subseteq{\\bigcap}_{\\ZZ_p[G\/J]}^{a'}A^t(E)_p\\]\nand that in $\\left({\\bigcap}_{\\ZZ_p[G\/J]}^{a} A^t(F)_p\\right)\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}$ one has an equality\n\\[ \\mathcal{N}_J(\\eta_{\\mathcal{Y},x_\\bullet})=(-1)^{a(a'-a)}\\cdot \\nu_J({\\rm ht}_{\\mathcal{Y}'}^{C_{S,x_\\bullet}}(\\eta_{\\mathcal{Y}',x_\\bullet^J})).\\]\nHere ${\\rm ht}_{\\mathcal{Y}'}^{C_{S,x_\\bullet}}$ is the `Bockstein homomorphism' (\\ref{htcy}) associated to the Nekov\\'a\\v r-Selmer complex $C_{S,x_\\bullet}$  in the course of the proof of Theorem \\ref{rbsdimpliesmt} below. In this context we recall also from Remark \\ref{mrselmer} that the cohomology group $H^1(C_{S,x_\\bullet})$ may be interpreted as a Selmer group in the sense of Mazur and Rubin \\cite{MRkoly}. }\n\\end{remarks}\n\n\n\n\n\\subsection{Cyclic congruence relations}\\label{cycliccongssection}\n\nIn this section we make the congruences in Conjecture \\ref{mrsconjecture} more explicit in the case of certain cyclic extensions.\n\nWe always assume the hypotheses (H$_1$)-(H$_6$) listed in \\S\\ref{tmc} and, motivated by\nRemarks \\ref{emptysets} and \\ref{new remark SC}, we further strengthen the hypothesis (H$_3$) by assuming that $p$ is unramified in $F$.\n\nIn this situation, the argument that is used in the proof of Theorem \\ref{rbsdimpliesmt} below may be applied to the classical Selmer complex.\n\nIn this way, the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) implies (via Theorem \\ref{bk explicit}) congruence relations in the spirit of Conjecture \\ref{mrsconjecture} that circumvent the choice of semi-local points and apply directly to the elements $\\calL^*_{A,F\/k}$ and $\\calL^*_{A,E\/k}$ as defined in (\\ref{bkcharelement}).\n\nWe shall also now assume that the following additional conditions on $A$, $F\/k$ and  $p$  are satisfied:\n\n\\begin{itemize}\n\\item[(H$_7$)] $F\/k$ is cyclic of order $p^m$ (for some natural number $m$);\n\\item[(H$_8$)] neither $A(k)$ nor $A^t(k)$ has a point of order $p$;\n\\item[(H$_{9}$)] for every field $L$ with $k\\subseteq L\\subset F$ and $L \\not=F$ the group $\\sha(A_L)[p^\\infty]$ vanishes;\n\\item[(H$_{10}$)] the group $H^1({\\rm Gal}(k(A[p^{m}])\/k),A[p^{m}])$ vanishes.\n\\end{itemize}\n\nWe impose the hypothesis (H$_{10}$) since it recovers Hypothesis \\ref{hypbd} below in this specific setting and hence will allow us to apply the result of Theorem \\ref{rbsdimpliesmt}. We also note that this hypothesis is satisfied widely (see Remark \\ref{rmkbd}).\n\n\n\n\nOn the other hand, the hypotheses (H$_7$), (H$_8$) and (H$_9$) are introduced since they allow us to apply the main result of Yakovlev \\cite{yakovlev} in order to deduce that $A(F)_p$ and $A^t(F)_p$ are (isomorphic) permutation $\\ZZ_p[G]$-modules and hence that it is possible to fix an optimal collection of points as in Definition \\ref{separablepair}.\n\n\nThen, by combining such a choice of points with the methods of \\cite{bleymc,dmc}, one can make fully explicit the congruence relations that are implied by ${\\rm BSD}_p(A_{F\/k})$(iv).\n\nIn order to stress the key ideas we shall focus on the simple cases $m=1$ and $m=2$. In particular, in the case $m=1$ we will show that the validity of  ${\\rm BSD}_p(A_{F\/k})$(iv) is under certain natural hypotheses equivalent to the validity of a single explicit congruence relation.\n\n\\subsubsection{\nWe first assume that $m=2$. In this case we denote by $J$ the subgroup of $G$ of order $p$ and set $E := F^J$ and $\\Gamma := G\/J$. For each $i\\in \\{0,1,2\\}$ we also write $F_i$ for the intermediate field of $F\/k$ with $[F_i:k] = p^i$ and set $\\Gamma_i:= G_{F_i\/k}$ (so that $\\Gamma_2 = G$ and $\\Gamma_1 = \\Gamma$).\n\nThen, under the given hypotheses, Yakovlev's result implies (via the general approach of \\cite[\\S7.2.5]{ltav}) that for each $i \\in \\{0,1,2\\}$ there exists a non-negative integers $m_i$ and subsets $\\calP_{(i)}=\\{P_{i,j}\\mid j \\in [m_i]\\}$ of $A(F_i)_p$ and $\\calP^t_{(i)}=\\{P^t_{i,j}:j\\in[m_i]\\}$ of $A^t(F_i)_p$ such that the $\\ZZ_p[\\Gamma_i]$-modules generated by each of the points $P_{i,j}$ and $P^t_{i,j}$ are free of rank one and there are direct sum decompositions of $\\ZZ_p[G]$-modules\n\\begin{equation}\\label{global points}\n A(F)_p = \\bigoplus_{i=0}^{i=2}\\bigoplus_{j=1}^{j=m_i}\\ZZ_p[\\Gamma_i]\\cdot P_{i,j}\\end{equation}\n\n and\n\\begin{equation}\\label{global points2} A^t(F)_p = \\bigoplus_{i=0}^{i=2}\\bigoplus_{j=1}^{j=m_i}\\ZZ_p[\\Gamma_i]\\cdot P^t_{i,j},\n\\end{equation}\n\nso that the integers $m_i$ are uniquely determined by the equalities\n$${\\rm rk}(A_k)=m_0+m_1+m_2,\\,\\,\\,\\,\\,\\,{\\rm rk}(A_E)=m_0+p(m_1+m_2),\\,\\,\\,\\,\\,\\,{\\rm rk}(A_F)=m_0+pm_1+p^2m_2.$$\nFor a full proof of these facts see \\cite[Prop. 2.2]{bleymc}.\n\nWe then obtain `regulator matrices' \\begin{align*}R_{F}(\\calP_{\\bullet}):=\\, &h_{F\/k}(\\calP^t_{(2)};\\,\\calP_{(2)})\\in {\\rm M}_{m_2}(\\RR[G]),\\\\\\ R_{E}(\\calP_{\\bullet}):=\\, &h_{E\/k}(\\calP^t_{(1)}\\cup\\calP_{(2)}^{t,J};\\,\\calP_{(1)}\\cup \\calP_{(2)}^J)\\in {\\rm M}_{m_1+m_2}(\\RR[\\Gamma])\\\\\nR_{k}(\\calP_{\\bullet}) := \\, &h_{k\/k}(\\calP^t_{(0)}\\cup \\calP^{t,\\Gamma}_{(1)}\\cup \\calP_{(2)}^{t,G};\\, \\calP_{(0)}\\cup \\calP^\\Gamma_{(1)}\\cup \\calP^G_{(2)}) \\in {\\rm M}_{{\\rm rk}(A_k)}(\\RR)\\end{align*}\nas in (\\ref{regulatormatrix}), where we write $\\calP_{(2)}^{t,J}$ for the set $\\{\\sum_{g \\in J}g(P_{2,j}^t)\\mid j \\in [m_i]\\}$ and define $\\calP_{(2)}^J$, $\\calP^{t,\\Gamma}_{(1)}$, $\\calP_{(2)}^{t,G}$, $\\calP^\\Gamma_{(1)}$ and $\\calP^G_{(2)}$ similarly, and in each union we order the points  lexicographically\n\nWe define an idempotent\n\\[\ne_{\\langle 2\\rangle} := \\begin{cases}1,& \\text{ if } m_0=m_1=0,\\\\ 1-e_G,& \\text{ if } m_0\\neq 0,m_1=0,\\\\ 1-e_J,& \\text{ if } m_1\\neq 0, \\end{cases}\\]\nin $\\QQ[G]$ and\n\\[ e_{\\langle 1\\rangle} := \\begin{cases}1,& \\text{ if } m_0=0,\\\\ 1-e_\\Gamma,& \\text{ if } m_0\\neq 0, \\end{cases}\n\\]\nin $\\QQ[\\Gamma]$.\n\nThen the matrix $e_{\\langle i\\rangle}\\cdot R_{F_i}(\\calP_{\\bullet})$ belongs to ${\\rm GL}_{m_2}(\\CC[G]e_{\\langle 2\\rangle})$ if $i=2$ and to  ${\\rm GL}_{m_1+ m_2}(\\CC[\\Gamma]e_{\\langle 1\\rangle})$ if $i = 1$ and so we may set\n\\begin{align}\\label{lmin}\n\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}} := \\, &({\\rm det}(e_{\\langle 2\\rangle}\\cdot R_{F}(\\calP_{\\bullet})))^{-1}\\cdot e_{\\langle 2\\rangle}\\calL^*_{A,F\/k}\\in e_{\\langle 2\\rangle}\\cdot\\CC[G]^\\times\\\\ \\notag\n\\mathcal{L}^{(m_1+m_2)}_{E,\\calP_{\\bullet}} := \\, &({\\rm det}(e_{\\langle 1\\rangle}\\cdot R_{E}(\\calP_{\\bullet})))^{-1}\\cdot e_{\\langle 1\\rangle}\\calL^*_{A,E\/k}\\in e_{\\langle 1\\rangle}\\cdot\\CC[\\Gamma_1]^\\times\\\\ \\notag\n\\calL^\\ast_{k,\\calP_{\\bullet}} :=\\, &{\\rm det}(R_k(\\calP_{\\bullet}))^{-1}\\cdot\\calL^*_{A,k}\\in\\RR^\\times.\\end{align\n\n\\begin{remark} {\\em This notation is motivated by fact that the decompositions in (\\ref{global points}) and (\\ref{global points2}) imply $e_{\\langle 2\\rangle}$ and $e_{\\langle 1\\rangle}$ are equal to the idempotents $e_{F,m_2}$ and $e_{E,m_1+m_2}$ that occur in \\S \\ref{statementstructure} with respect to $A$ and the extensions $F\/k$ and $E\/k$ respectively and so the terms $e_{\\langle 2\\rangle}\\calL^*_{A,F\/k}$ and $e_{\\langle 1\\rangle}\\calL^*_{A,E\/k}$ involve the values at $z=1$ of the $m_1$-th and $(m_1+m_2)$-th derivatives of Hasse-Weil-Artin $L$-series. We further recall from Remark \\ref{bsdinvariants} that the term $\\calL^*_{A,k}$ can be explicitly described in terms of invariants occurring in the classical Birch and Swinnerton-Dyer conjecture for $A$ over $k$.}\\end{remark}\n\n\nBy using the approach of the second author in \\cite{dmc}, one finds that Theorem \\ref{rbsdimpliesmt} implies the explicit congruences in the following result. We do not give the proof of this result but rather refer the reader to loc. cit.\n\nIn this result we use the canonical identification\n$$\\mathcal{I}_p(J)^{m_1}\/\\mathcal{I}_p(J)^{m_1+1}\\cong\\ZZ_p[\\Gamma]\\otimes_{\\ZZ_p}I_p(J)^{m_1}\/I_p(J)^{m_1+1}.$$\n\n\n\n\\begin{theorem}\\label{cyclicthm}\nFix an odd prime number $p$ that is unramified in $F$ and such that the data $A, F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_{10}$) with $m=2$. Then if ${\\rm BSD}_p(A_{F\/k})$(iv) is valid one has\n\\begin{equation*}\\label{totallyexplicit}\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}}\\equiv  (-1)^{m_1}\\cdot\\bigl(\\mathcal{L}^{(m_1+m_2)}_{E,\\calP_{\\bullet}}\\otimes {\\rm det}\\bigl(\\langle \\calP_{(1)}^t,\\calP_{(1)}\\rangle^{\\rm MT}_{F\/E}\\bigr)\\bigr)\\,\\,\\,\\,\\,({\\rm mod}\\,\\,\\mathcal{I}_p(J)^{m_1+1})\\end{equation*}\nand\n\\begin{equation*}\\label{totallyexplicit}\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}}\\equiv (-1)^{m_0+m_1}\\cdot\\calL^\\ast_{k,\\calP_{\\bullet}}\\cdot {\\rm det}\\bigl(\\langle\\calP_{(0)}^t\\cup \\calP_{(1)}^{t,\\Gamma},\\calP_{(0)}\\cup \\calP^\\Gamma_{(1)}\\rangle^{\\rm MT}_{F\/k}\\bigr)\\,\\,\\,\\,\\, ({\\rm mod}\\,\\,I_p(G)^{m_0+m_1+1}).\\end{equation*}\n\n\\end{theorem}\n\n\n\\begin{remark}\\label{changeofptsremark}{\\em\nThe Wedderburn components of the elements $\\mathcal{L}^{(m_2)}_{F,\\calP_{\\bullet}}$ and $\\mathcal{L}^{(m_1+m_2)}_{E,\\calP_{\\bullet}}$  recover the products that are studied by Fearnley and Kisilevsky \\cite{kisilevsky,kisilevsky2}. In particular, the numerical computations in loc. cit., as well as the numerical computations performed by Bley and the second author in \\cite{bleymc}, can be interpreted as supporting evidence for ${\\rm BSD}(A_{F\/k})$.\n}\n\\end{remark}\n\n\\subsubsection{\n\n\nWe now assume that $m=1$. In this case Yakovlev's result implies, under the given hypotheses, that for each $i \\in \\{0,1\\}$ there exists a non-negative $m_i$ and subsets $\\calP_{(i)}$ of $A(F_i)_p$ and $\\calP^t_{(i)}$ of $A^t(F_i)_p$ for which there are direct sum decompositions of $\\ZZ_p[G]$-modules\n\\[ A(F)_p = \\bigoplus_{j=1}^{j=m_0}\\ZZ_p\\cdot P_{0,j}\\oplus\\bigoplus_{j=1}^{j=m_1}\\ZZ_p[G]\\cdot P_{1,j}\\]\nand\n\\[ A^t(F)_p = \\bigoplus_{j=1}^{j=m_0}\\ZZ_p\\cdot P^t_{0,j}\\oplus\\bigoplus_{j=1}^{j=m_1}\\ZZ_p[G]\\cdot P^t_{1,j}.\\]\n\nBy using such a choice of points as well as the idempotent $$e_{F,m_1}:= \\begin{cases}1,& \\text{ if } m_0=0,\\\\ 1-e_G,& \\text{ if } m_0\\neq 0, \\end{cases}$$ one obtains elements $\\calL^{(m_1)}_{F,\\calP_\\bullet}$ of $e_{F,m_1}\\cdot\\CC[G]^\\times$ and $\\calL^*_{k,\\calP_\\bullet}$ of $\\RR^\\times$ exactly as in (\\ref{lmin}).\n\n\nIn this special case we obtain the following refinement of Theorem \\ref{cyclicthm}. This result will be proved in \\S\\ref{proofmequals1} below.\n\n\\begin{theorem}\\label{mequals1} Fix an odd prime number $p$ that is unramified in $F$ and such that the data $A, F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_{10}$) with $m=1$. Then the following claims are valid.\n\n\\begin{itemize}\\item[(i)] If ${\\rm BSD}_p(A_{F\/k})$(iv) is valid one has\n\\begin{equation*}\\calL^{(m_1)}_{F,\\calP_\\bullet}\\equiv (-1)^{m_0}\\cdot\\calL^\\ast_{k,\\calP_{\\bullet}}\\cdot {\\rm det}\\bigl(\\langle\\calP_{(0)}^t,\\calP_{(0)}\\rangle^{\\rm MT}_{F\/k}\\bigr)\\,\\,\\,\\,\\, ({\\rm mod}\\,\\,I_p(G)^{m_0+1}).\\end{equation*}\n\n\\item[(ii)] Assume that the $p$-part of the Birch and Swinnerton-Dyer Conjecture is valid for $A$ over both $k$ and $F$ and that the group $\\sha(A_F)[p^\\infty]$ vanishes.\n\nThen ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if the element $\\calL^{(m_1)}_{F,\\calP_\\bullet}$ belongs to $I_p(G)^{m_0}$ and satisfies the displayed congruence in claim (i).\\end{itemize}\n\\end{theorem}\n\n\\begin{remark}\\label{equivalenceeasy}{\\em The argument that we use to prove Theorem \\ref{mequals1} can be extended, under suitable hypotheses, to the setting of cyclic extensions of arbitrary $p$-power degree. In this way one obtains an explicit family of congruence relations (of the same general form as in Theorem \\ref{mequals1}(i)) whose validity is equivalent to that of ${\\rm BSD}_p(A_{F\/k})$(iv) whenever the $p$-part of the Birch and Swinnerton-Dyer Conjecture holds for $A$ over each intermediate field of $F\/k$. This will be explained in forthcoming work of Bley and the second author.\n}\\end{remark}\n\n\\begin{remark}\\label{bleyexamples rem}{\\em Theorems \\ref{cyclicthm} and \\ref{mequals1} both illustrate the fact that, unless the relevant Mordell-Weil groups are projective as Galois modules, the equality ${\\rm BSD}_p(A_{F\/k})$(iv) encodes a family of congruence relations between the values at $z=1$ of higher derivatives of Hasse-Weil-Artin $L$-series that also involve (non-trivial) Mazur-Tate regulators. As far as we are aware, no such congruence relations have previously been identified, let alone verified, in the literature.}\\end{remark}\n\n\\begin{examples}\\label{bleyexamples} {\\em In view of the last remark, we are very grateful to Werner Bley for providing us with the following list of examples for which he has been able to numerically verify the conjectural equality ${\\rm BSD}_p(A_{F\/k})$(iv) by means of the congruence relation in Theorem \\ref{mequals1}. The full details of these computations will appear in future work.\n\nThroughout this list, $A$ is the elliptic curve with indicated Cremona reference, $k=\\QQ$, $p=3$ and $F$ is the unique degree three Galois extension of $\\QQ$ that is contained in $\\QQ(\\zeta_\\ell)$ for any of the indicated primes $\\ell$. In addition, one always has ${\\rm rk}(A_F)={\\rm rk}(A_\\QQ)=1$.\n\n\\begin{itemize}\\item[-] $A$ is the curve $37a1$ (with Weierstrass equation $y^2+y=x^3-x$) and $\\ell\\in\\{13,19\\}.$\n\\item[-] $A$ is  $43a1$ ($y^2+y=x^3+x^2)$ and $\\ell\\in\\{7,13,37\\}.$\n\\item[-] $A$ is $53a1$ ($y^2+xy+y=x^3-x^2$) and $\\ell\\in\\{13,19,31,43\\}.$\n\\item[-] $A$ is $58a1$ ($y^2+xy=x^3-x^2-x+1$) and $\\ell\\in\\{7,13,19,31,43\\}.$\n\\item[-] $A$ is $61a1$ ($y^2+xy=x^3-2x+1$) and $\\ell\\in\\{7,13,43\\}.$\n\\item[-] $A$ is $65a1$ ($y^2+xy=x^3-x$) and $\\ell\\in\\{19,37,43\\}.$\n\\item[-] $A$ is $65a2$ ($y^2+xy=x^3+4x+1$) and $\\ell\\in\\{19,37,43\\}.$\n\\item[-] $A$ is $77a1$ ($y^2+y=x^3+2x$) and $\\ell\\in\\{19,37\\}.$\n\\end{itemize}\n}\n\\end{examples}\n\n\\subsubsection{}\\label{proofmequals1}\nIn this section we prove Theorem \\ref{mequals1}.\n\nAt the outset we set $C:={\\rm SC}_p(A_{F\/k})$ and note that, under the hypotheses of Theorem \\ref{mequals1}, Theorem \\ref{bk explicit} (and Remark \\ref{emptysets}) implies that ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if and only if the element\n$$\\xi:=\\delta_{G,p}(\\calL^*_{A,F\/k})-\\chi_{G,p}(C,h_{A,F})$$\nof $K_0(\\ZZ_p[G],\\CC_p[G])$ vanishes.\n\nIt is a routine exercise to deduce claim (i) of Theorem \\ref{mequals1} from this fact by combining Theorem \\ref{thecomptheorem} below with the approach used by the second author to prove \\cite[Cor. 3.5]{dmc} (and so we leave the details to the reader).\n\nTo prove claim (ii) we assume that the $p$-part of the Birch and Swinnerton-Dyer Conjecture holds for $A$ over both $k$ and $F$ and that $\\sha(A_F)[p^\\infty]$ vanishes. We also assume that $\\calL^{(m_1)}_{F,\\calP_\\bullet}$ belongs to $I_p(G)^{m_0}$ and satisfies the congruence displayed in claim (i), and then proceed to prove that $\\xi$ vanishes.\n\nThe first key step is to explicitly compute the second term that occurs in the definition of $\\xi$ and to do this we  use the canonical identifications\n\\[ r_1:A^t(F)_p\\to H^1(C)\\,\\,\\text{ and }\\,\\, r_2:H^2(C)\\to \\Sel_p(A_F)^\\vee\\stackrel{\\sim}{\\to}A(F)_p^*\\]\nthat are described in Proposition \\ref{explicitbkprop}(iv). Here the second arrow in the definition of $r_2$ is the canonical map and so is bijective since $\\sha(A_F)[p^\\infty]$ is assumed to vanish.\n\nThese identifications imply that the complex $C$ corresponds to a unique element $\\varepsilon = \\varepsilon_{C,r_1,r_2}$ of the Yoneda Ext-group ${\\rm Ext}^2_{\\ZZ_p[G]}(A(F)_p^*,A^t(F)_p)$.\n\nFor each pair $(i,j)$ with $i\\in\\{0,1\\}$ and $j\\in[m_i]$ we define a dual point $P_{i,j}^*$ of $A(F)_p^*$ in the natural way and obtain a direct sum decomposition of $\\ZZ_p[G]$-modules\n$$A(F)_p^*=\\bigoplus_{j=1}^{j=m_0}\\ZZ_p\\cdot P_{0,j}^*\\oplus\\bigoplus_{j=1}^{j=m_1}\\ZZ_p[G]\\cdot P_{1,j}^*$$\nas in \\cite[Lem. 4.1]{bleymc}.\n\nWe also fix a generator $\\sigma$ of $G$ and a free $\\ZZ_p[G]$-module $P:=\\bigoplus_{i,j}\\ZZ_p[G]b_{i,j}$ of rank $m_0+m_1$ and consider the exact sequence\n\\begin{equation}\\label{syzygy}0\\to A^t(F)_p\\stackrel{\\iota}{\\to}P\\stackrel{\\Theta}{\\to}P\\stackrel{\\pi}{\\to}A(F)_p^*\\to 0,\\end{equation} where we set $\\iota(P^t_{0,j})=\\Tr_G(b_{0,j})$, $\\iota(P^t_{1,j})=b_{1,j}$, $\\Theta(b_{i,j})=(\\sigma^{p^i}-1)b_{i,j}$ and $\\pi(b_{i,j})=P^*_{i,j}$.\n\nThis sequence defines a canonical isomorphism\n\\begin{equation}\\label{computeext}{\\rm Ext}^2_{\\ZZ_p[G]}(A(F)_p^*,A^t(F)_p)\\cong{\\rm End}_{\\ZZ_p[G]}(A^t(F)_p)\/\\iota_*\\bigl(\\Hom_{\\ZZ_p[G]}(P,A^t(F)_p)\\bigl),\\end{equation}\nwhere $\\iota_*$ denotes composition with $\\iota$.\n\nThen, since Proposition \\ref{explicitbkprop}(ii) implies $C$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ the results of \\cite[Lem. 4.2 and Lem. 4.3]{bleymc} imply that there exists an automorphism $\\phi$ of the $\\ZZ_p[G]$-module $A^t(F)_p$ that represents the image of $\\varepsilon$ under (\\ref{computeext}) and also fixes the element $P^t_{1,j}$ for every $j$ in $[m_1]$.\n\nIt follows that the exact sequence\n\\begin{equation}\\label{yonedarep}0\\to A^t(F)_p\\stackrel{\\iota\\circ\\phi^{-1}}{\\to}P\\stackrel{\\Theta}{\\to}P\\stackrel{\\pi}{\\to}A(F)_p^*\\to 0\\end{equation}\nis a representative of the extension class $\\varepsilon$.\n\nIn particular, for any choice of $\\CC_p[G]$-equivariant splittings\n$$s_1:\\CC_p\\cdot P\\to \\CC_p\\cdot A^t(F)_p\\oplus\\CC_p\\cdot\\im(\\Theta)$$\nand\n$$s_2:\\CC_p\\cdot P\\to \\CC_p\\cdot A(F)^*_p\\oplus\\CC_p\\cdot\\im(\\Theta)$$\nof the scalar extensions of the canonical exact sequences\n$$0\\to A^t(F)_p\\stackrel{\\iota}{\\to}P\\stackrel{\\Theta}{\\to}\\im(\\Theta)\\to 0$$\nand\n$$0\\to\\im(\\Theta)\\to P\\stackrel{\\pi}{\\to}A(F)_p^*,$$\nthe definition of non-abelian determinants implies that\n\\begin{align}\\label{separatingstuff}&-\\chi_{G,p}(C,h_{A,F})\\\\\n=&-\\delta_{G,p}({\\det}_{\\CC_p[G]}(s_2^{-1}\\circ(h_{A,F}\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ((\\CC_p\\cdot\\phi)\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ s_1))\\notag\\\\\n=&\\,\\delta_{G,p}({\\det}_{\\CC_p[G]}(s_1^{-1}\\circ(h^{-1}_{A,F}\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ s_2))\\notag\\\\\n&\\hskip 1.5truein +\\delta_{G,p}({\\det}_{\\QQ_p[G]}((\\QQ_p\\cdot\\phi^{-1})\\oplus{\\rm id}_{\\QQ_p\\cdot\\im(\\Theta)}))\\notag.\\end{align}\n\nIn addition, by an explicit computation as in \\cite[Prop. 4.4]{bleymc} one shows that\n\\begin{align}\\label{theregulatorterm}&{\\det}_{\\CC_p[G]}(s_1^{-1}\\circ(h^{-1}_{A,F}\\oplus{\\rm id}_{\\CC_p\\cdot\\im(\\Theta)})\\circ s_2)\\\\\n=\\, &\\det(e_{F,m_1}\\cdot h_{F\/k}(\\calP^t_{(1)};\\calP_{(1)}))^{-1}\\cdot(\\sigma-1)^{-m_0}\\notag\\\\\n&\\hskip 1truein +(1-e_{F,m_1})\\cdot\\det(h_{k\/k}(\\calP^t_{(0)}\\cup\\calP^{t,G}_{(1)};\\calP_{(0)}\\cup\\calP^{G}_{(1)}))^{-1}.\\notag\\end{align}\n\nTo compute the second term that occurs in the final equality of (\\ref{separatingstuff}) we fix, as we may, any elements $\\Psi_{(0,j),(0,k)}$ of $\\ZZ_p[G]$ with the property that\n\\begin{equation}\\label{matrixPsi}\\phi^{-1}(P^t_{0,k})=\\sum_{j\\in[m_0]}\\Psi_{(0,j),(0,k)}(P^t_{0,j}).\\end{equation}\nThen the chosen properties of the representative $\\phi$ of $\\varepsilon$ fixed above imply that\n\\begin{equation}\\label{leftovertermMT}{\\det}_{\\QQ_p[G]}((\\QQ_p\\cdot\\phi^{-1})\\oplus{\\rm id}_{\\QQ_p\\cdot\\im(\\Theta)})=\ne_{F,m_1}+(1-e_{F,m_1})\\cdot\\det\\bigl((\\Psi_{(0,j),(0,k)})_{j,k\\in[m_0]}\\bigr).\\end{equation}\n\nThe equalities (\\ref{separatingstuff}), (\\ref{theregulatorterm}) and (\\ref{leftovertermMT}) now combine with the definition of $\\xi$ to imply that\n $\\xi=\\delta_{G,p}(\\mathcal{L})$ with\n\n\\[ \\mathcal{L} := \\calL_{F,\\calP_\\bullet}^{(m_1)}\\cdot(\\sigma-1)^{-m_0}+(1-e_{F,m_1})\\cdot\\calL^*_{k,\\calP_\\bullet}\n\\cdot\\det\\bigl((\\Psi_{(0,j),(0,k)})_{j,k\\in[m_0]}\\bigr).\\\n\nWe shall use this description to show $\\xi$ belongs to the finite group $K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$, or equivalently that $\\mathcal{L}$ belongs to the unit group of the maximal $\\ZZ_p$-order $\\mathcal{M}_p(G)$ in $\\QQ_p[G]$.\n\nTo do this we note first that the assumed containment $\\calL_{F,\\calP_\\bullet}^{(m_1)}\\in I_{G,p}^{m_0}$ and validity of the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $k$ combine to imply $\\mathcal{L}$ belongs to $\\QQ_p[G]^\\times$ and hence that $\\xi$ belongs to\n$K_0(\\ZZ_p[G],\\QQ_p[G])$.\n\nNext we recall from the general result of \\cite[Thm. 4.1]{ewt} that, since $G$ has order $p$, the group $K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$ is the kernel of the homomorphism\n\\[ K_0(\\ZZ_p[G],\\QQ_p[G]) \\xrightarrow{(\\rho,q)}  K_0(\\ZZ_p,\\QQ_p)\\oplus K_0(\\ZZ_p,\\QQ_p)\\]\nwhere $\\rho$ is the canonical restriction of scalars map and $q$ is the homomorphism induced by mapping the class of a triple $(P,\\phi,Q)$ to the class of $(P^G,\\phi^G,Q^G)$.\n\nGiven this, the claimed containment $\\xi \\in K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$ follows from the fact that Remark \\ref{consistency remark}(ii) and (iii) combine to imply that the validities of the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $F$ and $k$ imply that  $\\rho(\\xi)$ and $q(\\xi)$ both vanish.\n\nAt this stage we know that $\\mathcal{L}$ belongs to $\\mathcal{M}_p(G)^\\times$ and hence that it belongs to $\\ZZ_p[G]^\\times$ (so that $\\xi$ vanishes) if and only if it belongs to $\\ZZ_p[G]$.\n\nNow if $m_0=0$ then $\\mathcal{L} =\\calL_{F,\\calP_\\bullet}^{(m_1)}$ and $\\ZZ_p[G]=I_p(G)^{m_0}$ and so our hypotheses imply directly that  $\\mathcal{L} \\in \\ZZ_p[G]$. In the sequel we will therefore assume that $m_0\\neq 0$, and hence that $e_{F,m_1}=1-e_G$.\n\nIn this case $\\mathcal{L}$ belongs to $\\ZZ_p[G]$ if and only if one has\n\\begin{equation*}\\label{totallyexplicit1}\\calL^{(m_1)}_{F,\\calP_\\bullet}\\cdot(\\sigma-1)^{-m_0}\\equiv \\calL^\\ast_{k,\\calP_{\\bullet}}\\cdot \\det\\bigl((\\Psi_{(0,j),(0,k)})_{j,k\\in[m_0]}\\bigr)\\,\\,\\,\\,\\, ({\\rm mod}\\,\\,I_p(G))\\end{equation*}\nand so the assumed validity of the congruence in Theorem \\ref{mequals1}(i) reduces us to verifying that, for any $j$ and $k$ in $[m_0]$, one has\n\\begin{equation}\\label{PsicomputesMT}\\Psi_{(0,j),(0,k)}\\cdot(\\sigma-1)+I_p(G)^2=-\\langle P^t_{0,k},P_{0,j}\\rangle^{\\rm MT}_{F\/k}\\end{equation} in $I_p(G)\/I_p(G)^2$.\n\nTo verify these equalities we use the fact, proved in Theorem \\ref{thecomptheorem} below, that $\\langle \\,,\\rangle^{\\rm MT}_{F\/k}$\ncoincides with the inverse of the pairing induced by\n\\begin{align*}\\beta:A^t(k)_p\\to I_p(G)\/I_p(G)^2\\otimes_{\\ZZ_p}(A(F)_p^*)_{G} \\to  I_p(G)\/I_p(G)^2\\otimes_{\\ZZ_p}A(k)_p^*,\\end{align*}\n\nwhere the first arrow is the Bockstein homomorphism associated to the complex $C$ and the maps $r_1$ and $r_2$ (see \\S\\ref{bocksection} below), while the second arrow is induced by restriction to $A(k)_p$.\n\nNow $\\beta$ may be computed, through the representative (\\ref{yonedarep}) of the extension class $\\varepsilon$, as the connecting homomorphism that  arises when applying the snake lemma to the following commutative diagram (in which both rows and the third column are\nexact and the first column is a complex)\n\\[\\begin{CD}\n@. @. @. A^t(k)_p\\\\ @. @. @. @VV (\\iota\\circ\\phi^{-1})^{G} V  \\\\ 0 @>\n>> I_p(G)\\otimes_{\\ZZ_p[G]}P @> \\subseteq >> P @> \\Tr_{G}  >>\n P^G  @> >> 0\\\\\n@. @VV {\\rm id}\\otimes_{\\ZZ_p[G]}\\Theta V @VV \\Theta V @VV\\Theta^{G} V\\\\\n0 @>\n>> I_p(G)\\otimes_{\\ZZ_p[G]}P @> \\subseteq >> P @> \\Tr_{G} >>\n P^G  @> >> 0\\\\\n@. @VV ({\\rm id}\\otimes_{\\ZZ_p[G]}\\pi)_{G}  V \\\\ @.\nI_p(G)\/I_p(G)^2\\otimes_{\\ZZ_p} (A(F)_p^*)_{G}.\n\\end{CD}\\]\n\nIn addition, the equality (\\ref{matrixPsi}) implies that\n$$\\iota(\\phi^{-1}(P^t_{0,k}))=\\iota(\\sum_{l\\in[m_0]} \\Psi_{(0,l),(0,k)} P^t_{0,l})=\\sum_{l\\in[m_0]} \\Psi_{(0,l),(0,k)}\\cdot\\Tr_G(b_{0,l})$$\nand so, since $$P^*_{0,l}(P_{0,j})=\\begin{cases}1,\\,\\,\\,\\,\\,\\,\\,\\,l=j,\\\\\n0,\\,\\,\\,\\,\\,\\,\\,\\,l\\neq j,\\end{cases}$$\nwe can finally use the above diagram to compute that\n\\begin{align*}-\\langle P^t_{0,k},P_{0,j}\\rangle_{F\/k}^{\\rm MT}=&\\left(({\\rm id}\\otimes_{\\ZZ_p[G]}\\pi)_{G}\\left(\\Theta(\\Psi_{(0,j),(0,k)}\\cdot b_{0,j})\\right)\\right)(P_{0,j})\\\\\n=&\\left(({\\rm id}\\otimes_{\\ZZ_p[G]}\\pi)_{G}\\left((\\sigma-1)\\Psi_{(0,j),(0,k)}\\cdot b_{0,j}\\right)\\right)(P_{0,j})\\\\\n=&\\left(\\left((\\sigma-1)+I_p(G)^2\\right)\\otimes\\Psi_{(0,j),(0,k)}\\cdot P^*_{0,j}\\right)(P_{0,j})\\\\\n=&\\Psi_{(0,j),(0,k)}\\cdot(\\sigma-1)+I_p(G)^2.\n\\end{align*}\nHere the first equality is given by Theorem \\ref{thecomptheorem}. This verifies the equalities (\\ref{PsicomputesMT}) and thus completes the proof of Theorem \\ref{mequals1}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Dihedral congruence relations}\\label{dihedral} With a view to extending the classes of extensions $F\/k$ for which the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) can be made fully explicit we consider the case that $F\/k$ is generalized dihedral of order $2p^n$.\n\nWe recall that this means the Sylow $p$-subgroup $P$ of $G$ is abelian and of index two and that the conjugation action of any lift to $G$ of the generator of $G\/P$ inverts elements of $P$. We write $K$ for the unique quadratic extension of $k$ in $F$.\n\nIn this setting we shall show that, in certain situations, the validity of ${\\rm BSD}_p(A_{F\/k})$(iv) can be checked by verifying congruences relative to the abelian extension $F\/K$.\n\nIn order to state the precise result we fix a finite Galois extension $E$ of $\\QQ$ in $\\bc$ that is large enough to ensure that, with $\\mathcal{O}$ denoting the ring of algebraic integers of $E$, there exists for each character $\\psi$ of $\\widehat{G}$ a finitely generated $\\mathcal{O}[G]$-lattice that is free over $\\mathcal{O}$ and spans a $\\bc[G]$-module of character $\\psi$.\n\nFor each $\\psi$ in $\\widehat{G}$ we recall the non-zero complex number $\\mathcal{L}^\\ast(A,\\psi)$ defined in \\S\\ref{explicit ec section}.\n\n\\begin{proposition}\\label{dihedral prop} Let $F\/k$ be generalized dihedral of degree $2p^n$ as above. Assume that $\\sha(A_F)$ is finite and that no place of $k$ at which $A$ has bad reduction is ramified in $F$. Assume also that $p$ satisfies the conditions (H$_1$)-(H$_4$) listed in \\S\\ref{tmc} and that neither $A(K)$ nor $A^t(K)$ has a point of order $p$.\n\nThen the equality of ${\\rm BSD}_p(A_{F\/k})$(iv) is valid if the following three conditions are satisfied.\n\n\\begin{itemize}\n\\item[(i)] For every $\\psi$ in $\\widehat{G}$ and $\\omega$ in $G_\\QQ$, one has $\\mathcal{L}^\\ast(A,\\omega\\circ \\psi) = \\omega(\\mathcal{L}^\\ast(A,\\psi))$.\n\\item[(ii)] For every $\\psi$ in $\\widehat{G}$ and every prime ideal $\\mathfrak{p}$ of $\\mathcal{O}$ that divides $p$, the explicit formula for $\\mathcal{L}^\\ast(A, \\psi)\\cdot \\mathcal{O}_\\mathfrak{p}$ that is given in Proposition \\ref{ref deligne-gross}(ii) is valid.\n\\item[(iii)] The equality of ${\\rm BSD}_p(A_{F\/K})$(iv) is valid.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof} Since $F\/K$ is an extension of $p$-power degree, the assumption that neither $A(K)$ nor $A^t(K)$ has a point of order $p$ implies that neither $A(F)$ nor $A^t(F)$ has a point of order $p$.\n\nHence, in this case, the given assumptions imply that the data $A$, $F\/k$ and $p$ satisfy all of the hypotheses of Proposition \\ref{ref deligne-gross}.\n\nIn particular, if we write $\\xi$ for the difference between the left and right hand sides of the equality in Theorem \\ref{bk explicit}, then the argument of Proposition \\ref{ref deligne-gross} shows that the assumed validity of the given conditions (i) and (ii) implies that $\\xi$ belongs to\n $K_0(\\ZZ_p[G],\\QQ_p[G])$ and also to the kernel of the homomorphism $\\rho_\\mathfrak{p}^\\psi$ for every $\\psi$ in $\\widehat{G}$ and every prime ideal $\\mathfrak{p}$ of $\\mathcal{O}$ that divides $p$.\n\nThese facts combine with the general result of \\cite[Th. 4.1]{ewt} to imply that $\\xi$ belongs to the finite group $K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor}$.\n\nWe next recall from \\cite[Lem. 5.12(ii)]{bmw} that, since $G$ is assumed to be dihedral, the natural restriction map ${\\rm res}^G_P:K_0(\\ZZ_p[G],\\QQ_p[G])_{\\rm tor} \\to K_0(\\ZZ_p[P],\\QQ_p[P])$ is injective.\n\nIt follows that $\\xi$ vanishes, and hence by Theorem \\ref{bk explicit} that ${\\rm BSD}_p(A_{F\/k})$(iv) is valid, if the element ${\\rm res}^G_P(\\xi)$ vanishes.\n\nTo complete the proof, it is therefore enough to note that the functorial behaviour of the conjecture ${\\rm BSD}(A_{F\/k})$ under change of extension, as described in Remark \\ref{consistency remark}(ii) (and justified via Remark \\ref{consistency}(ii)), implies that ${\\rm res}^G_P(\\xi)$ vanishes if and only if the equality of ${\\rm BSD}_p(A_{F\/K})$(iv) is valid. \\end{proof}\n\n\\begin{remark}{\\em If $P$ is cyclic, then Proposition \\ref{dihedral prop} shows that in certain situations the validity of ${\\rm BSD}_p(A_{F\/k})$(iv) for the non-abelian extension $F\/k$ can be checked by verifying the relevant cases of the refined Deligne-Gross Conjecture formula in Proposition \\ref{ref deligne-gross}(ii) together with explicit congruences for the cyclic extension $F\/K$ of the form that are discussed in \\S\\ref{cycliccongssection}. In addition, if $P$ is cyclic, then the main result of Yakovlev in \\cite{yakovlev2} can be used to show that if the groups $\\sha(A_{F'})[p^\\infty]$ vanish for all proper subfields of $F$ that contain $K$, then the $\\ZZ_p[G]$-module $A(F)_p$ is a `trivial source module' and so has a very explicit structure. }\\end{remark}\n\n\\begin{example}{\\em For the examples described in Example \\ref{wuthrich example} the field $F$ is a dihedral extension of $k = \\QQ$ of degree $6$ and both of the given elliptic curves $A$ satisfy all of the hypotheses that are necessary to apply Proposition \\ref{dihedral prop} (in the case $p=3$ and $n=1$). In this way one finds that the validity of ${\\rm BSD}_3(A_{F\/\\QQ})$(iv) implies, and if $\\sha(A_F)[3^\\infty]$ vanishes is equivalent to, the validity of the relevant cases of the refined Deligne-Gross Conjecture together with the validity of an explicit congruence of the form described in Theorem \\ref{mequals1} for the cyclic extension $F\/K$ (and with $m_1= 1$ and $m_0 = 2$). Unfortunately, however, since $F\/\\QQ$ is of degree $6$ it seems that for the given curves $A$ the latter congruences are at present beyond the range of numerical investigation via the methods that have been used to verify the cases discussed in Example \\ref{bleyexamples}. }\\end{example}\n\n\n\\subsection{The proof of Theorem \\ref{rbsdimpliesmt}}\\label{mrsproof}\n\nWe fix a subgroup $J$ of $G$ and set $E := F^J$ and a maximal subset $x_\\bullet$ of $A^t(F_p)^\\wedge_p$ that is linearly independent over $\\ZZ_p[G]$.\n\nTo study the relationship between ${\\rm BSD}(A_{F\/k})$ and Conjecture \\ref{mrsconjecture} we set $X:=\\langle x_\\bullet\\rangle_{\\ZZ_p[G]}$ and consider both of the complexes $C_{S,X}:={\\rm SC}_S(A_{F\/k};X,H_\\infty(A_{F\/k})_p)$ and $C_{S,X,J}:={\\rm SC}_S(A_{E\/k};X^J,H_\\infty(A_{E\/k})_p)$.\n\nThen the definition of $C_{S,X}$ as the mapping fibre of the morphism (\\ref{fibre morphism}), the well-known properties of \\'etale cohomology under Galois descent and the fact that $X$ is a free $\\ZZ_p[G]$-module (see also (\\ref{global descent}), (\\ref{local descent}) and the argument that precedes them) imply that the object $$(C_{S,X})_J:=\\ZZ_p[G\/J]\\otimes_{\\ZZ_p[G]}^{\\mathbb{L}}C_{S,X}$$ of $D^{\\rm perf}(\\ZZ_p[G\/J])$ is isomorphic to $C_{S,X,J}$. This fact combines with \\cite[Lem. 3.33]{bst} to give canonical identifications\n$$H^1(C_{S,X})^J=H^1((C_{S.X})_J)=H^1(C_{S,X,J})$$ and\n$$H^2(C_{S,X})_J=H^2((C_{S,X})_J)=H^2(C_{S,X,J}).$$\n\nIn addition, our assumption that neither $A(F)$ nor $A^t(F)$ has a point of order $p$ combines with Proposition \\ref{prop:perfect} to imply that,\nin the terminology of \\cite{bst}, the complex $C_{S,X}$ is a `strictly admissible' complex of $\\ZZ_p[G]$-modules. The approach of \\S 3.4.1 in loc. cit. therefore gives a `Bockstein homomorphism' of $\\ZZ_p[G\/J]$-modules \\begin{equation}\\label{htc}{\\rm ht}^{C_{S,X}}:H^1(C_{S,X})^J\\to \\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2\\otimes_{\\ZZ_p[G\/J]}H^2(C_{S,X,J}).\\end{equation}\n\nFor any $p$-separable choice of points $(\\mathcal{Y},\\mathcal{Y}')$ of $A$ for $F\/E$ of rank $(a,a')$\nand each index $i$ with $a< i \\le a'$ we then use the dual point $w_i^*$ to construct a composite homomorphism of $\\ZZ_p[G\/J]$-modules \\begin{equation}\\label{projection}H^2(C_{S,X,J})\\to\\Sel_p(A_E)^\\vee\\to A(E)_p^*\\to (Y')^*\\to\\ZZ_p[G\/J].\\end{equation} Here the first arrow is the canonical homomorphism of Proposition \\ref{prop:perfect}(iii), the second arrow is the canonical homomorpism occurring in (\\ref{sha-selmer}), the third arrow is the natural restriction maps and the fourth arrow maps an element of $(Y')^*$ to its coefficient at the basis element $w_i^*$.\n\nUpon composing ${\\rm ht}^{C_{S,X}}$ with each map (\\ref{projection}) we thereby obtain, for each index $i$ with $a<i\\leq a'$, a homomorphism of $\\ZZ_p[G\/J]$-modules\n\\begin{equation}\\label{htcwi} {\\rm ht}^{C_{S,X}}_{w_i}: H^1(C_{S,X})^J \\to \\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2 \\end{equation}\nthat corresponds to the map denoted by ${\\rm Boc}^C_x$ in \\cite[\\S 3.4.1]{bst}.\nJust as in (\\ref{canbecombined}) these homomorphisms can then be combined to give a canonical homomorphism \\begin{equation}\\label{htcy}{\\rm ht}_{\\mathcal{Y}'}^{C_{S,X}}: {\\bigcap}_{\\ZZ_p[G\/J]}^{a'} H^1(C_{S,X})^J \\longrightarrow ({\\bigcap}_{\\ZZ_p[G\/J]}^{a} H^1(C_{S,X})^J)\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}.\\end{equation}\n\n\n\n\n\n\n\n\nWe will apply the general result of \\cite[Th. 3.34(ii)]{bst} to the complex $C_{S,X}$, the isomorphism $h_{A,F}$ and the characteristic element $\\mathcal{L}_{F\/k}$ for the pair $(C_{S,X},h_{A,F})$ constructed in Lemma \\ref{modifiedlemma}(iii) under the assumption that ${\\rm BSD}(A_{F\/k})$ is valid. We recall that \\begin{equation}\\label{charelementFk}e_{F,a}\\cdot\\mathcal{L}_{F\/k}^{-1}=\\frac{L^{(a)}_{S}(A_{F\/k},1)}{\\Omega_A^{F\/k}\\cdot w_{F\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{F\/k}}(x_\\bullet).\\end{equation}\n\nSimilarly, if ${\\rm BSD}(A_{F\/k})$ is valid then the observations made in Remark \\ref{consistency remark}(ii) (see also Remark \\ref{consistency} below for more details) imply that ${\\rm BSD}(A_{E\/k})$ is also valid. Applying Lemma \\ref{modifiedlemma}(iii) to the complex $C_{S,X,J}$ and the isomorphim $h_{A,E}$ therefore implies the existence of a characteristic element $\\mathcal{L}_{E\/k}$ for this pair with the property that \\begin{equation}\\label{charelementEk}e_{E,a'}\\cdot\\mathcal{L}_{E\/k}^{-1}=\\frac{L^{(a')}_{S}(A_{E\/k},1)}{\\Omega_A^{E\/k}\\cdot w_{E\/k}^d}\\cdot \\mathcal{LR}^p_{A^t_{E\/k}}(x_\\bullet^J).\\end{equation}\n\nWe next fix any (injective) section $s$ to the canonical surjective composite homomorphism $$H^2(C_{S,X})\\to\\Sel_p(A_F)^\\vee\\to A(F)_p^*\\to Y^*,$$ where the last arrow is the natural restriction map. The set $s(\\mathcal{Y}^*):=\\{s(y^*):y\\in\\mathcal{Y}\\}$ is then (in the sense of \\cite[Def. 3.6]{bst}) a `separable subset' of $H^2(C_{S,X})$ (of cardinality $a$).\nThe `higher special element' associated via \\cite[Def. 3.3]{bst} to the data $$(C_{S,X},h_{A,F},\\mathcal{L}_{F\/k},s(\\mathcal{Y}^*))$$ is then, by (\\ref{charelementFk}) and a direct comparison of the definitions of $h_{A,F}$ and ${\\rm ht}_{A_{F\/k}}^{a}$, equal to $\\eta_{\\mathcal{Y},x_\\bullet}$.\n\nWe finally fix any (injective) section $s'$ to the canonical surjective composite homomorphism $$H^2(C_{S,X,J})\\to\\Sel_p(A_E)^\\vee\\to A(E)_p^*\\to (Y')^*,$$ where the last arrow is the natural restriction map. The set $s'((\\mathcal{Y}')^*):=\\{s'(y^*):y\\in\\mathcal{Y}'\\}$ is then a separable subset of $H^2(C_{S,X,J})$ (of cardinality $a'$). The higher special element associated via \\cite[Def. 3.3]{bst} to the data $$(C_{S,X,J},h_{A,E},\\mathcal{L}_{E\/k},s'((\\mathcal{Y}')^*))$$ is then, by (\\ref{charelementEk}) and a direct comparison of the definitions of $h_{A,E}$ and ${\\rm ht}_{A_{E\/k}}^{a'}$, equal to $\\eta_{\\mathcal{Y}',x_\\bullet^J}$.\n\nThe general result \\cite[Th. 3.34(ii)]{bst} therefore implies that\n\\[ \\eta_{\\mathcal{Y},x_\\bullet} \\in {\\bigcap}_{\\ZZ_p[G]}^{a} H^1(C_{S,X})\\subseteq{\\bigcap}_{\\ZZ_p[G]}^{a}A^t(F)_p,\\]\n\nand\n\\[ \\eta_{\\mathcal{Y}',x_\\bullet^J}\\in {\\bigcap}_{\\ZZ_p[G\/J]}^{a'} H^1(C_{S,X})^J\\subseteq{\\bigcap}_{\\ZZ_p[G\/J]}^{a'}A^t(E)_p,\\]\nand that\n\\begin{equation}\\label{forreferenceinremark} \\mathcal{N}_J(\\eta_{\\mathcal{Y},x_\\bullet})=(-1)^{a(a'-a)}\\cdot \\nu_J({\\rm ht}_{\\mathcal{Y}'}^{C_{S,X}}(\\eta_{\\mathcal{Y}',x_\\bullet^J}))\\end{equation}\nin $\\left({\\bigcap}_{\\ZZ_p[G\/J]}^{a} A^t(F)_p\\right)\\otimes_{\\ZZ_p} I_p(J)^{a'-a}\/I_p(J)^{1+a'-a}$.\n\n\nTo complete the proof of Theorem \\ref{rbsdimpliesmt} we will combine the equality (\\ref{forreferenceinremark}) with the comparison of height pairings proved in Theorem \\ref{thecomptheorem} below.\n\n\n\n\nWe first note that $A$, $F\/E$ and $p$ satisfy the hypotheses (H$_1$)-(H$_5$) listed in \\S \\ref{tmc} (see \\cite[Lem. 3.4]{bmw0}). We also note that the validity of these hypotheses (and their duality) combines with Proposition \\ref{explicitbkprop}(ii) to imply that $U_{F\/E,p}=A(E)_p$ and $U^t_{F\/E,p}=A^t(E)_p$.\n\nIt will thus be enough to prove that, for any index $a<i\\leq a'$, the restriction to $H^1(C_{S,X})^J$ of the homomorphism\n$${\\rm ht}_{w_i}^{{\\rm MT}}:A^t(E)_p\\to \\mathcal{I}_p(J)\/\\mathcal{I}_p(J)^2$$ is equal to the homomorphism ${\\rm ht}_{w_i}^{C_{S,X}}$ of (\\ref{htcwi}).\n\nBy Theorem \\ref{thecomptheorem} below applied to $F\/E$, for any $P$ in $A(E)_p$ one has $${\\rm ht}_{w_i}^{{\\rm MT}}(P):=-\\langle P,w_i\\rangle^{\\rm MT}_{F\/E}=\\langle P,w_i\\rangle_{p}^{\\rm N}$$ in $\\ZZ_p\\otimes_\\ZZ J$, where $\\langle \\,,\\rangle_{p}^{\\rm N}$ is the Bockstein pairing associated to the classical Selmer complex ${\\rm SC}_p(A_{F\/E})$ as in (\\ref{thepairing}) below.\n\n\nIt is therefore be enough to prove that, for any index $a<i\\leq a'$ and for any $P\\in H^1(C_{S,X})^J$ one has \\begin{equation*}{\\rm ht}_{w_i}^{C_{S,X}}(P)=\\langle P,w_i\\rangle_{p}^{\\rm N}.\\end{equation*} But this equality follows easily from the compatibility of the maps $$H^1(C_{S,X})^J=H^1(C_{S,X,J})\\to H^1({\\rm SC}_p(A_{F\/E}))$$ and $$H^2(C_{S,X,J})\\to H^2({\\rm SC}_p(A_{F\/E}))$$ occurring in the exact sequence (\\ref{useful1}) with all the other maps involved in the construction of both Bockstein homomorphisms (and the naturality of such homomorphisms). This completes the proof of Theorem \\ref{rbsdimpliesmt}.\n\n\n\n\n\n\\section{Height pairing comparisons}\\label{comparison section}\n\n\n\n\n\n\n\n\n\n\nIn this section we continue to assume that $F\/k$ is abelian.\n\nWe fix an odd prime number $p$ and assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_6$) listed in \\S \\ref{tmc}. We also assume that neither $A(F)$ not $A^t(F)$ has a point of order $p$.\n\nIn this situation, Proposition \\ref{explicitbkprop} implies that the classical Selmer complex ${\\rm SC}_p(A_{F\/k})$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ and is acyclic outside degrees one and two, where its cohomology modules canonically identify with $A^t(F)_p$ and $\\Sel_p(A_F)^\\vee$ respectively.\n\nVia the general formalism outlined in \\S \\ref{bocksection} one therefore obtains an associated canonical algebraic height pairing\n$$\\langle\\,,\\rangle^{\\rm N}_{p}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to \\ZZ_p\\otimes_{\\ZZ}G$$\nthat we refer to as the `Nekov\\'a\\v r height pairing'.\n\nProposition \\ref{explicitbkprop}(ii) also implies that the $p$-completion of the height pairing (\\ref{tanpairing}) (for $J=G$) that is defined using the theory of bi-extensions by Mazur and Tate in \\cite{mt0} gives a well-defined pairing\n$$\\langle\\,,\\rangle_p^{\\rm MT}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to \\ZZ_p\\otimes_{\\ZZ}G.$$\n\nIn this section we shall prove that the Nekov\\'a\\v r height pairing coincides with the inverse of the Mazur-Tate height pairing. This comparison result was already used in the proof of Theorem \\ref{rbsdimpliesmt} and is also, we believe, of some independent interest.\n\n\\subsection{Statement of the result}\n\nTo make the construction of the Nekov\\'a\\v r height pairing more precise we fix a finite subset $\\Sigma$ of $S_k^f$ that contains $S_k^p$, $S_k^F\\cap S_k^f$ and $S_k^A$.\n\nProposition \\ref{explicitbkprop} implies that the Selmer complex $C_{\\Sigma}={\\rm SC}_{\\Sigma,p}(A_{F\/k})$ belongs to $D^{\\rm perf}(\\ZZ_p[G])$ and is acyclic outside degrees one and two.\n\nIn addition,\nwe can use Proposition \\ref{explicitbkprop}(iv) to interpret the global Kummer map as an isomorphism \\begin{equation}\\label{r1}r_1:A^t(F)_p\\to H^1(C_{\\Sigma}),\\end{equation} and also to define a canonical composite homomorphism \\begin{equation}\\label{r2}r_2:H^2(C_{\\Sigma})\\stackrel{\\sim}{\\to}\\Sel_p(A_F)^\\vee{\\to} A(F)_p^*\\to A(k)_p^*.\\end{equation} Here the first arrow is the canonical isomorphism occurring in the diagram (\\ref{Selmerdiagram}) and the last arrow is simply the restriction to $A(k)_p$ of any element of the linear dual $A(F)_p^*$.\n\nBy using the general approach (and notation) of \\S \\ref{bocksection}, we can therefore define the Nekov\\'a\\v r height pairing by setting\n\n\\begin{equation}\\label{thepairing}\\langle\\,,\\rangle^{\\rm N}_{p}:=\\langle\\,,\\rangle_{C_\\Sigma,r_1,r_2}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to I_{p}(G)\/I_{p}(G)^2\\cong \\ZZ_p\\otimes_{\\ZZ}G,\\end{equation}\nwhere $I_p(G)$ denotes the augmentation ideal in $\\ZZ_p[G]$.\n\nThen Lemma \\ref{independenceofsigma} and Remark \\ref{indeptremark} combine with the naturality of Bockstein pairings to imply that this pairing is  independent of the choice of set $\\Sigma$.\n\nWe write $m_p(G)$ for the non-negative integer for which $p^{m_p(G)}$ is the exponent of the group $\\ZZ_p\\otimes_{\\ZZ}G$.\n\nThen the proof of our comparison of height pairings is greatly simplified by imposing the following additional technical hypothesis:\n\n\\begin{hyp}\\label{hypbd\nThe group $H^1({\\rm Gal}(F(A[p^{m_p(G)}])\/F),A[p^{m_p(G)}])$ vanishes.\n\\end{hyp}\n\n\\begin{remarks}\\label{rmkbd}{\\em\nHypothesis \\ref{hypbd} is motivated by the result \\cite[Lem. 2.13]{bert} of Bertolini and Darmon and \n is satisfied whenever multiplication-by-`$-1$' belongs to the image of the canonical Galois representation $G_F\\to\\Aut_{\\mathbb{F}_p}(A[p])$. In particular, if $A$ is an elliptic curve, then the hypothesis only excludes finitely many primes $p$ by a famous result of Serre \\cite{serre}.}\n\\end{remarks}\n\n\n\n\n\n\n\\begin{theorem}\\label{thecomptheorem} Fix an odd prime number $p$ and assume that $A$, $F\/k$ and $p$ satisfy the hypotheses (H$_1$)-(H$_6$) listed in \\S \\ref{tmc}. Assume also that neither $A(F)$ nor $A^t(F)$ has a point of order $p$ and that Hypothesis \\ref{hypbd} is satisfied.\n\nThen the Nekov\\'a\\v r pairing $\\langle\\,,\\rangle^{\\rm N}_{p}$ is equal to the inverse of the Mazur-Tate pairing $\\langle\\,,\\rangle^{\\rm MT}_p$\n\\end{theorem}\n\nThe proof of Theorem \\ref{thecomptheorem} will occupy the rest of this section. We thus assume throughout that all the hypotheses listed in Theorem \\ref{thecomptheorem} are valid.\n\n\\subsection{An auxiliary height pairing}\n\nIn this section we define a $p$-adic height pairing $\\langle\\,,\\rangle^{\\rm BD}_p$  whose construction is motivated by the work of Bertolini and Darmon \\cite{bert,bert2}.\n\nIn \\S\\ref{compproof} below we will then show that $\\langle\\,,\\rangle_p^{\\rm BD}$ coincides with the inverse of $\\langle\\,,\\rangle_p^{\\rm N}$. By then appealing to comparison results due to Bertolini and Darmon and to Tan, we will finally obtain the desired comparison of pairings.\n\nWe will now follow closely some of the constructions from \\cite{bert,bert2} and for this reason we will be rather brief in some of our arguments.\n\n\\subsubsection{} We again write $m:=m_p(G)$ for the non-negative integer such that $p^m$ is the exponent of the finite abelian $p$-group $\\ZZ_p\\otimes_{\\ZZ}G$.\n\nWe set $Z:=\\ZZ\/p^m\\ZZ$ and $R:=Z[G]$ and write $I_{R}$ for the augmentation ideal in $R$. We will also require all of the following notation.\n\nFor any finite set $S$ of non-archimedean places of $k$, we define $\\Sel^S_{p^m}(A^t\/F)$ to be the kernel of the localisation map $$H^1\\bigl(F,A^t[p^m]\\bigr)\\to\\bigoplus_{w'\\notin S(F)}H^1(F_{w'},A^t)$$ (with the sum running over all non-archimedean places of $F$ that do not belong to $S(F)$). When $S$ is taken to be empty one recovers the usual Selmer group associated to multiplication by $p^m$, which we denote by $\\Sel^{(p^m)}(A^t_F)$ instead.\n\nFor such a set $S$ and $B\\in\\{A,A^t\\}$ we also define $R$-modules\n$$B_S(\\mathbb{A}_F)\/p^m:=\\prod_{w'\\in S(F)}B(F_{w'})\/p^mB(F_{w'}),$$ $$H^1_{S}\\bigl(\\mathbb{A}_F,B[p^m]\\bigr):=\\prod_{w'\\in S(F)}H^1(F_{w'},B[p^m])$$ and $$H^1_{S}\\bigl(\\mathbb{A}_F,B\\bigr)[p^m]:=\\prod_{w'\\in S(F)}H^1(F_{w'},B)[p^m].$$\n\nBy abuse of notation, we denote by $\\lambda_{S}$ both the localisation map $$\\Sel^S_{p^m}(A^t\/F){\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)$$ and the induced map $$\\Sel^S_{p^m}(A^t\/F){\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m].$$\n\nWe also write $$\\kappa:A^t_S(\\mathbb{A}_F)\/p^m\\to H^1_{S}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)$$ and $$\\kappa':H^1_{S}\\bigl(\\mathbb{A}_F,A\\bigr)[p^m]^\\vee\\to H^1_{S}\\bigl(\\mathbb{A}_F,A[p^m]\\bigr)^\\vee$$ for the canonical maps induced by the local Kummer sequences.\n\nWe recall (see Corollary \\ref{Tatepoitouexplicit} below) that the local Tate duality isomorphism $$u_S:H^1_{S}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)\\to H^1_{S}\\bigl(\\mathbb{A}_F,A[p^m]\\bigr)^\\vee$$ induces an isomorphism\n\\begin{equation}\\label{inducedbyum}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\cong\\cok(\\kappa)\\stackrel{\\sim}{\\to}\\cok(\\kappa')\\cong\\bigl(A_S(\\mathbb{A}_F)\/p^m\\bigr)^\\vee.\\end{equation}\nBy composing this isomorphism with the dual of the restriction map $$\\Sel^{(p^m)}(A_F)\\to A_S(\\mathbb{A}_F)\/p^m$$ one finally obtains a map\n$$v_S:H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]{\\to}\n{\\rm Sel}^{(p^m)}(A_F)^\\vee.$$\n\n\n\n\n\\subsubsection{\nIn the sequel we denote by $P$ the Sylow $p$-subgroup of $G$ and set $K:=F^P$.\n\n\\begin{lemma}\\label{214} The restriction map $H^1(K,A[p^m])\\to H^1(F(A[p^m]),A[p^m])$ is injective.\\end{lemma}\n\\begin{proof} As in \\cite[Cor. 2.14]{bert}, the result follows immediately upon combining the validity of Hypothesis \\ref{hypbd} with the fact that $A(F)[p^m]$ vanishes (because $A(F)$ does not have a point of order $p$).\n\\end{proof}\n\nLemma \\ref{214} now implies the existence of `admissible' auxiliary sets of primes, as in \\cite[Def. 1.5]{bert2} or \\cite[Def. 2.20, Def. 2.22]{bert}:\n\n\\begin{lemma}\\label{tm} There is a finite set $T$ of non-archimedean places of $K$ such that:\n\\begin{itemize\n\\item[(i)] Every place in $T$ splits completely in $F\/K$.\n\\item[(ii)] If $v\\in T$ then $A(K_v)\/p^mA(K_v)$ is a free $Z$-module of rank $2\\dim(A)$.\n\\item[(iii)] The restriction map \\begin{equation}\\label{Kres}\\Sel^{(p^m)}(A_K)\\to\\bigoplus_{v\\in T}A(K_v)\/p^mA(K_v)\\end{equation} is injective.\n\\end{itemize}\n\\end{lemma}\n\nThe existence of such a set $T$ (of infinitely many such sets $T$, in fact) is proved in \\cite[Lem. 2.21]{bert} using a Tcheboratev density argument (and Lemma \\ref{214}).\n\nIn the sequel, we fix such an admissible set $T$ and, by abuse of notation, also denote by $T$ the set of places of $k$ below places in $T$.\n\nThe existence of the admissible set $T$ now leads us to consider the canonical complex of $R$-modules $$C_{\\rm BD}:=\\bigl[\\Sel^T_{p^m}(A^t\/F)\\stackrel{\\lambda_T}{\\to}H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\bigr],$$ with first term placed in degree 1. The desired auxiliary pairing $\\langle\\,,\\rangle_p^{\\rm BD}$ will then be obtained as a Bockstein pairing associated to $C_{\\rm BD}$.\n\n\n\n\n\n\n\n\n\\begin{lemma}\\label{bdlemma} The following claims are valid.\n\\begin{itemize}\\item[(i)] Let $S$ be any finite set of non-archimedean places of $k$ which contains $T$. Then the sequence\n\\begin{equation}\\label{notexact}0\\to{\\rm Sel}^{(p^m)}(A^t_F)\\to \\Sel^S_{p^m}(A^t\/F)\\stackrel{\\lambda_S}{\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\stackrel{v_S}{\\to}\n{\\rm Sel}^{(p^m)}(A_F)^\\vee\\to 0\\end{equation} is exact.\n\\item[(ii)] The $R$-modules $\\Sel^T_{p^m}(A^t\/F)$, $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ and $A_T(\\mathbb{A}_F)\/p^m$ are finitely generated and $G$-cohomologically-trivial.\n\\item[(iii)] Let $S\\supseteq T$ be any set as in claim (i). Then there is a canonical quasi-isomorphism from the complex $C_{\\rm BD}$ to the complex $$C_{{\\rm BD},S}:=\\bigl[\\Sel^S_{p^m}(A^t\/F)\\stackrel{\\lambda_S}{\\to}H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\bigr],$$ with first term placed in degree 1, which induces the identity map both on ${\\rm Sel}^{(p^m)}(A^t_F)$ and on ${\\rm Sel}^{(p^m)}(A_F)^\\vee$ (viewed as the cohomology modules of $C_{\\rm BD}$ and of $C_{{\\rm BD},S}$ via (\\ref{notexact})).\n\\end{itemize}\n\\end{lemma}\n\\begin{proof} The exactness of (\\ref{notexact}) at the first three modules is proved in \\cite[Lem. 6.15]{milne}, and is independent of the fact that the set $S$ contains $T$. See also \\cite[Prop. 1.2]{bert2} and the discussion that follows it.\n\nIn order to prove claim (i), it is hence enough to show that $v_{T}$ is surjective (since then so must be $v_S$ for any set $S$ containing $T$), or equivalently that the restriction map \\begin{equation}\\label{tmrestriction}\\Sel^{(p^m)}(A_F)\\to A_{T}(\\mathbb{A}_F)\/p^m\\end{equation} is injective. This may be achieved by following the approach of \\cite[Lem. 2.25, Lem. 2.26, Lem 3.1]{bert} or \\cite[Prop. 1.6]{bert2}. We limit ourselves to giving a sketch of this argument.\n\nProposition \\ref{explicitbkprop}(ii) implies that, for any non-archimedean place $v$ of $k$, the module $A(F_v)$ is $P$-cohomologically-trivial.\n\n\n\n\n\nThis fact then combines with our assumption that $A(F)$ has no point of order $p$ to imply the restriction map $\\Sel^{(p^m)}(A_K)\\to\\Sel^{(p^m)}(A_F)^P$ is bijective, by the same argument used to prove \\cite[Lem 2.19]{bert} or \\cite[Lem. 1.4]{bert2}.\n\nSince the restriction map (\\ref{Kres}) is injective, one finally deduces that the restriction map (\\ref{tmrestriction}) is injective, just as in \\cite[Lem 2.25]{bert} or \\cite[Prop. 1.6]{bert2}.\n\nTo next consider claim (ii), it is enough to show that each of the modules $\\Sel^T_{p^m}(A^t\/F)$, $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ and $A_T(\\mathbb{A}_F)\/p^m$ is finitely generated and free over $Z[P]$. These facts in turn follow from the arguments of \\cite[\\S 3.1]{bert}.\n\nIndeed, one first uses properties (i) and (ii) of the set $T$, as given in Lemma \\ref{tm}, to see that $A_T(\\mathbb{A}_F)\/p^m$ is (finitely generated and) $Z[P]$-free.\nThe duality (\\ref{inducedbyum}) between $A_{T}(\\mathbb{A}_F)\/p^m$ and $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ then implies that the same property is true for the latter module.\n\nSince neither $A(F)$ nor $A^t(F)$ has a point of order $p$, the canonical exact sequences\n$$0\\to A(F)\/p^mA(F)\\to\\Sel^{(p^m)}(A_F)\\to\\sha(A_F)[p^m]\\to 0$$ and\n$$0\\to A^t(F)\/p^mA^t(F)\\to\\Sel^{(p^m)}(A^t_F)\\to\\sha(A^t_F)[p^m]\\to 0$$ combine with the Cassels-Tate pairing to imply that the finite groups ${\\rm Sel}^{(p^m)}(A^t_F)$ and ${\\rm Sel}^{(p^m)}(A_F)^\\vee$ have the same order. One may therefore combine the fact that $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ is $Z[P]$-free with the exactness of (\\ref{notexact}) (for $S=T$) to deduce, as in \\cite[Thm. 3.2]{bert}, that $\\Sel^{T}_{p^m}(A^t\/F)$ is also $Z[P]$-free.\n\n\nWe finally consider claim (iii). The canonical inclusions give vertical arrows in the commutative diagram\n\\begin{equation*}\\xymatrix{\n\\Sel^{T}_{p^m}(A^t\/F) \\ar[d] \\ar[r]^(0.45){\\lambda_T} & H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m] \\ar[d] \\\\\n\\Sel^{S}_{p^m}(A^t\/F)  \\ar[r]^(0.45){\\lambda_S} & H^1_{S}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\n}\\end{equation*}\nand hence define a morphism of complexes $C_{\\rm BD}\\to C_{{\\rm BD},S}$.\n\nIt is clear that this morphism  induces the identity map both on ${\\rm Sel}^{(p^m)}(A^t_F)$ and on ${\\rm Sel}^{(p^m)}(A_F)^\\vee$, so by the exactness of the respective sequences of the form (\\ref{notexact}) it is in particular a quasi-isomorphism, as required.\n\\end{proof}\n\n\\subsubsection{\nTo define the useful auxiliary pairing we let $S$ be any finite set of non-archimedean places of $k$ containing $T$.\n\nWe may then combine Lemma \\ref{bdlemma} with the general formalism outlined in \\S \\ref{bocksection} below to define a pairing\n$\\langle\\,,\\rangle_p^{\\rm BD}$ as follows.\n\nWe let $r_{1,m}$ denote the canonical composite map\n$$A^t(k)_p\/p^m\\to A^t(F)_p\/p^m\\to \\Sel^{(p^m)}(A^t_F)\\to H^1(C_{{\\rm BD},S}),$$  and $r_{2,m}$ denote the canonical composite map $$H^2(C_{{\\rm BD},S})\\stackrel{v_{S}}{\\to}{\\rm Sel}^{(p^m)}(A_F)^\\vee\\to(A(F)_p\/p^m)^\\vee\\to(A(k)_p\/p^m)^\\vee.$$\n\nAs in \\S \\ref{bocksection}, these canonical choices induce a canonical Bockstein homomorphism \\begin{equation*}\\label{thebdbock}\\beta_{C_{{\\rm BD},S}}:A^t(k)_p\/p^m{\\to}I_{R}\/I_{R}^2\\otimes_{Z} (A(k)_p\/p^m)^\\vee\\end{equation*} and hence a canonical pairing\n$$\\langle\\,,\\rangle_{C_{{\\rm BD},S}}:A^t(k)_p\/p^m\\otimes_{Z} A(k)_p\/p^m\\to I_{R}\/I_{R}^2\n\\cong Z\\otimes_{\\ZZ}G= \\ZZ_p\\otimes_{\\ZZ}G,$$\nor equivalently a canonical pairing\n$$\\langle\\,,\\rangle_p^{{\\rm BD}}:A^t(k)_p\\otimes_{\\ZZ_p} A(k)_p\\to  \\ZZ_p\\otimes_{\\ZZ}G.$$\n\nIn particular, Lemma \\ref{bdlemma}(iii) combines with the naturality of Bockstein homomorphisms to ensure that this pairing is independent of the choice of set $S$ containing $T$.\n\n\\subsubsection{\n In order to compare $\\langle\\,,\\rangle_p^{\\rm BD}$ to the Nekov\\'a\\v r height pairing we shall need to clarify the relationship between complexes of the form $C_{{\\rm BD},S}$ and classical Selmer complexes.\n\nTo do this we fix a finite subset $\\Sigma$ of $S_k^f$ with\n$$S_k^p\\cup (S_k^F \\cap S_k^f)\\cup S_k^A\\cup T\\subseteq\\Sigma.$$\n\nWe then use the classical Selmer complex $C_\\Sigma$ to define a complex of $R$-modules $$C_{\\Sigma}\/p^m:=Z\\otimes_{\\ZZ_p}^{\\mathbb{L}}C_{\\Sigma}.$$ We refer the reader to Lemma \\ref{reducedbock} in Appendix \\ref{exp rep section} below for the definition of the maps $q_{1}'$ and $q_{2}'$ that are associated, by using the fixed homomorphisms $r_1$ and $r_2$ displayed in (\\ref{r1}) and (\\ref{r2}) respectively, to the construction of this complex.\n\n\\begin{lemma}\\label{morphismphi} There exists a canonical morphism of complexes\n\\[ \\phi:C_{{\\rm BD},\\Sigma}\\to C_{\\Sigma}\/p^m\\]\nwith the following properties:\n\\begin{itemize}\\item[(i)] The restriction of $q'_{1}$ to $A^t(k)_p\/p^m$ is equal to the composition $$A^t(k)_p\/p^m\\stackrel{r_{1,m}}{\\to}H^1(C_{{\\rm BD},\\Sigma})\\stackrel{H^1(\\phi)}{\\to}H^1(C_{\\Sigma}\/p^m).$$\n\\item[(ii)] $r_{2,m}$ is equal to $-1$ times the composition $$H^2(C_{{\\rm BD},\\Sigma})\\stackrel{H^2(\\phi)}{\\to}H^2(C_{\\Sigma}\/p^m)\\stackrel{q'_{2}}{\\to}A(k)_p^*\/p^m\\to(A(k)_p\/p^m)^\\vee,$$ where the last arrow is the canonical map.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof} In order to explicitly define the morphism $\\phi$ we first describe an explicit representative of the classical Selmer complex $C_\\Sigma$. The claimed result is clearly independent of such a choice.\n\nTo do so we use the notation given in \\S \\ref{expcohcomp} below. In particular, for any $w'\\in\\Sigma(F)$ and any $i\\geq 0$ we let $R\\Gamma^i(F_{w'},T_p(A^t))$ denote the group of inhomogeneous $i$-cochains of $G_{w'}$ with coefficients in $T_p(A^t)$, and also write $d_{w'}^i$ for the usual differential.\n\nFor any $v\\in\\Sigma$ we then set\n$$C_v^2:=\\frac{\\prod_{w'\\in S_F^v}\\bigl(R\\Gamma^{1}(F_{w'},T_p(A^t))\/\\im(d^0_{w'})\\bigr)}{\\kappa\\bigl(A^t(F_v)_p^\\wedge\\bigr)}$$\nand also, for any $i\\geq 3$,\n$$C_v^i:=\\prod_{w'\\in S_F^v}R\\Gamma^{i-1}(F_{w'},T_p(A^t)).$$ We note that it is straightforward, by using the relevant exact sequences of the form (\\ref{kummerseq}), to verify that the module $C_v^2$ is $\\ZZ_p$-torsion-free.\n\nIf, for each $v\\in\\Sigma$, we set $D_v^i=C_v^{i-1}$, we let $\\partial_v^{i}$ be induced by $\\prod_{w'\\in S_F^v}d_{w'}^{i-1}$ and consider the complex $D^\\bullet_v=(D_v^{i},\\partial_v^{i})$, then there is a canonical exact triangle\n$$A^t(F_v)_p^\\wedge[-1]\\stackrel{\\kappa}{\\to}R\\Gamma(k_v,T_{p,F}(A^t))\\to D^\\bullet_v\\to A^t(F_v)_p^\\wedge[0]$$ in $D(\\ZZ_p[G])$.\n\nSimilarly, we write $G_{F,\\Sigma}$ for the Galois group of the maximal Galois extension of $F$ wich is unramified outside $\\Sigma(F)\\cup S_F^\\infty$, set $U_{F,\\Sigma}:=\\mathcal{O}_{F,\\Sigma\\cup S_F^\\infty}$, write $R\\Gamma^i(U_{F,\\Sigma},T_p(A^t))$ for the group of inhomogeneous $i$-cochains of $G_{F,\\Sigma}$ with coefficients in $T_p(A^t)$, and also write $d^i$ for the usual differential.\n\n\nThen, since $R\\Gamma^i(U_{F,\\Sigma},T_p(A^t))$ is acyclic in degrees less that one, we may obtain an explicit representative of $C_\\Sigma$ as the $-1$-shift of the explicit mapping cone of the diagonal localisation morphism\n$$\\tau_{\\geq 1}\\bigl(R\\Gamma^i(U_{F,\\Sigma},T_p(A^t))\\bigr)\\stackrel{\\lambda}{\\to}\\bigoplus_{v\\in\\Sigma}D_v^\\bullet.$$\n(Here $\\tau_{\\geq 1}$ denotes the truncation in degree greater than or equal to 1 preserving cohomology).\n\nTo be explicit about this construction of a representative of $C_\\Sigma$, we set\n\\[\nC_{\\Sigma}^i :=\n\\begin{cases}\n  0,    &\\text{ if $i\\leq 0$}\\\\\n  R\\Gamma^1\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr)\/{\\rm im}(d^0),    &\\text{ if $i=1$}\\\\\n\\bigl(\\bigoplus_{v\\in \\Sigma}C^i_v\\bigr)\\oplus R\\Gamma^i\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr),    &\\text{ if $i\\geq 2$.}\n \n\n\\end{cases}\n\\]\nWe let the differential $d_{\\Sigma}^1$ in degree $1$ be given by\n\\[d_{\\Sigma}^1(\\overline{c}):=\\bigl(-(\\overline{c_{w'}})_{w'\\in \\Sigma(F)},d^1(c)\\bigr)\\]\nand the differential $d_{\\Sigma}^i$ in degree $i\\geq 2$ be given by\n\\[d_{\\Sigma}^i(\\bigl((a_{w'})_{w'\\in \\Sigma(F)},b\\bigr)):=\\bigl(-(d^{i-1}_{w'}(a_{w'})+b_{w'})_{w'\\in \\Sigma(F)},d^i(b)\\bigr).\\]\nHere, in any degree $i$ and for any place $w'\\in \\Sigma(F)$, we have written $x_{w'}$ for the localisation at $w'$\nof an inhomogeneous $i$-cochain $x$ in $R\\Gamma^i\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr)$.\n\nThe exactness of the sequence (\\ref{longexact}) combines with our assumption that $A^t(F)$ contains no point of order $p$ to imply that the module $H^1\\bigl(U_{F,\\Sigma},T_{p}(A^t)\\bigr)$ is $\\ZZ_p$-torsion-free. One may deduce from this fact that the module $C_\\Sigma^1$ is also $\\ZZ_p$-torsion-free.\n\nNow, since each module $C_{\\Sigma}^i$ is $\\ZZ_p$-torsion-free, the complex $C_{\\Sigma}\/p^m$ may be represented by the sequence of modules $Z\\otimes_{\\ZZ_p}C_{\\Sigma}^i$ together with the differentials $Z\\otimes_{\\ZZ_p}d_{\\Sigma}^i$.\n\nIn order to define the required morphism $\\phi$, we first define $$\\phi^1:\\Sel^{\\Sigma}_{p^m}(A^t\/F)\\to Z\\otimes_{\\ZZ_p}C_{\\Sigma}^1$$ as follows.\n\nWe first note that, since the set $\\Sigma$ is assumed to contain $S_k^p\\cup S_k^A$, the argument of \\cite[Prop. 6.5]{milne} proves that the canonical inflation map defines an isomorphism $$\\iota:H^1(U_{F,\\Sigma},A^t[p^m])\\stackrel{\\sim}{\\to}\\Sel^{\\Sigma}_{p^m}(A^t\/F).$$\n\nFor any $x\\in\\Sel^{\\Sigma}_{p^m}(A^t\/F)$ we may hence fix a 1-cocycle $y\\in R\\Gamma^1(U_{F,\\Sigma},A^t[p^m])$ representing $\\iota^{-1}(x)$ and use the map $l_\\Sigma$ defined in Lemma \\ref{lifts}(i) below to define $\\phi^1(x)$ as the class in $Z\\otimes_{\\ZZ_p}C_{\\Sigma}^1$ of the lift $$l_\\Sigma(y)\\in Z\\otimes_{\\ZZ_p}R\\Gamma^1(U_{F,\\Sigma},T_p(A^t)).$$\n\nBefore proceeding we note that Lemma \\ref{lifts}(ii) ensures both that $\\phi^1$ is well-defined and also that\n\\begin{equation}\\label{itisacocycle}(Z\\otimes_{\\ZZ_p}d^1)(l_\\Sigma(y))=0.\n\\end{equation}\n\nWe next define $$\\phi^2:H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]\\to Z\\otimes_{\\ZZ_p}C_{\\Sigma}^2$$ as follows.\n\nFor any $(x_{w'})_{w'\\in\\Sigma(F)}$ in\n\\[ H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]=\\frac{H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t[p^m]\\bigr)}\n{\\kappa\\bigl(A^t_\\Sigma(\\mathbb{A}_F)\/p^m\\bigr)}\\]\nwe fix a family of 1-cocycles\n$$(y_{w'})_{w'\\in\\Sigma(F)}\\in\\prod_{w'\\in\\Sigma(F)}R\\Gamma^1(F_{w'},A^t[p^m])$$ representing $(x_{w'})_{w'\\in\\Sigma(F)}$.\nWe then use the maps $l_{w'}$, as $w'$ ranges through $\\Sigma(F)$, defined in Lemma \\ref{lifts}(i) below to obtain a lift\n$$l_{w'}(y_{w'})\\in Z\\otimes_{\\ZZ_p}R\\Gamma^1(F_{w'},T_p(A^t)).$$\n\nWe finally define $\\phi^2((x_{w'})_{w'\\in\\Sigma(F)})$ as the class in $Z\\otimes_{\\ZZ_p}C_{\\Sigma}^2$ of $(-(l_{w'}(y_{w'}))_{w'\\in\\Sigma(F)},0)$. Again one can use Lemma \\ref{lifts}(ii) to verify that $\\phi^2$ is well-defined.\n\n\nOur explicit description of the differential $d_{\\Sigma}^1$ then implies that, for any $x\\in\\Sel^{\\Sigma}_{p^m}(A^t\/F)$, one has\n\\begin{align*} \\bigl((Z\\otimes_{\\ZZ_p}d_{\\Sigma}^1)\\circ\\phi^1\\bigr)(x)=&\\bigl(-(\\overline{l_\\Sigma(y)_{w'}})_{w'},(Z\\otimes_{\\ZZ_p}d^1)(l_\\Sigma(y))\\bigr)\\\\\n=&\\bigl(-(\\overline{l_\\Sigma(y)_{w'}})_{w'},0\\bigr)\\\\\n=&\\bigl(-(\\overline{l_{w'}(y_{w'})})_{w'},0\\bigr)\\\\\n=&\\phi^2((\\overline{y_{w'}})_{w'})\\\\\n=&\\phi^2(\\lambda_\\Sigma(\\iota(\\overline{y})))\\\\\n=&(\\phi^2\\circ\\lambda_\\Sigma)(x).\\end{align*}\nHere again $y$ is a 1-cocycle representing $\\iota^{-1}(x)$, the second equality follows from (\\ref{itisacocycle}) and the third equality holds since, by Lemma \\ref{lifts}(ii), the maps $l_\\Sigma$ and $l_{w'}$ commute with localisation at each $w'$.\n\nPutting then $\\phi^i=0$ for $i\\neq 1,2$ therefore defines a morphism of complexes of $R$-modules $$\\phi:C_{{\\rm BD},\\Sigma}\\to C_{\\Sigma}\/p^m.$$ Indeed, using Lemma \\ref{lifts}(ii) one verifies that\n\\[ \\bigl((Z\\otimes_{\\ZZ_p}d_{\\Sigma}^2)\\circ\\phi^2\\bigr)((x_{w'})_{w'})=0\\] for any $(x_{w'})_{w'}$ in $H^1_{\\Sigma}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$.\n\nWe next prove that $\\phi$ satisfies the stated properties. Now, $q'_{1}$ is by definition equal to the composition\n$$A^t(F)_p\/p^m\\stackrel{r_1\/p^m}{\\to}H^1(C_\\Sigma)\/p^m\\stackrel{q_1}{\\to}H^1(C_\\Sigma\/p^m)$$\nwith $q_{1}$ induced by the canonical projection $$R\\Gamma^1\\bigl(U_{F,\\Sigma},T_p(A^t)\\bigr)\\to Z\\otimes_{\\ZZ_p}R\\Gamma^1\\bigl(U_{F,\\Sigma},T_p(A^t)\\bigr).$$ The defining property of the lifting map $l_\\Sigma$ given in Lemma \\ref{lifts}, combined with our definition of $\\phi^1$, therefore implies that $q'_1$ coincides with the canonical composition\n$$A^t(F)_p\/p^m\\to\\Sel^{(p^m)}(A^t_F)\\to H^1(C_{{\\rm BD},\\Sigma})\\stackrel{H^1(\\phi)}{\\to}H^1(C_\\Sigma\/p^m).$$\nThis fact directly gives property (i) of the morphism $\\phi$.\n\n\n\nTurning now to property (ii), we write $q_{2}$ for the canonical isomorphism $$H^2(C_\\Sigma)\/p^m\\to H^2(C_\\Sigma\/p^m)$$ induced by the canonical projection morphism $C_\\Sigma\\to C_\\Sigma\/p^m$ as in Lemma \\ref{reducedbock} below.\n\nThen an explicit computation using our definition of $\\phi^2$ shows that the composition\n\\begin{multline*}H^1_\\Sigma(\\mathbb{A}_F,A^t[p^m])\\to H^1_\\Sigma(\\mathbb{A}_F,A^t)[p^m]\\to H^2(C_{{\\rm BD},\\Sigma}) \\stackrel{H^2(\\phi)}{\\to}H^2(C_\\Sigma\/p^m)\\stackrel{q_{2}^{-1}}{\\to}H^2(C_{\\Sigma})\/p^m\\\\ \\stackrel{\\sim}{\\to}\\cok(\\Delta)\/p^m\\to\\frac{H^2_c\\bigl(U_{F,\\Sigma},A^t[p^m]\\bigr)}{\\im\\bigl(\\kappa(A^t_\\Sigma(\\mathbb{A}_F)\/p^m)\\bigr)}\\end{multline*}\ncoincides with $-1$ times the canonical map. In this display, the fifth arrow is the canonical isomorphism occurring in the diagram (\\ref{Selmerdiagram}) and the sixth arrow is the canonical projection.\n\n\n\n\nThe commutativity of the diagram (\\ref{Selmerdiagram}), as well as of the diagram in Corollary \\ref{Tatepoitouexplicit} below, can now be combined with the above given fact to show that the composition\n\\begin{equation*} H^2(C_{{\\rm BD},\\Sigma}) \\stackrel{H^2(\\phi)}{\\to}H^2(C_{\\Sigma}\/p^m)\\stackrel{q_{2}^{-1}}{\\to}H^2(C_{\\Sigma})\/p^m\\stackrel{\\sim}{\\to}\\Sel_p(A_F)^\\vee\/p^m\\to\\Sel^{(p^m)}(A_F)^\\vee\\end{equation*} is equal to $-v_{\\Sigma}$.\n\nGiven this fact, and since $q'_{2}$ is by definition equal to $r_2\/p^m\\circ q_{2}^{-1}$, the required equality of composite maps is now clear from their definitions.\n\\end{proof}\n\n\n\n\\subsection{The proof of Theorem \\ref{thecomptheorem}}\\label{compproof}\n\nAs a first step we use Lemma \\ref{morphismphi} to show that the Nekov\\'a\\v r height pairing $\\langle\\,,\\rangle^{\\rm N}_{p}$ is equal to the inverse of the pairing $\\langle\\,,\\rangle_p^{\\rm BD}$.\n\nTo do this we recall that we have set $m:=m_p(G)$ and also fixed a finite set $T$ of places of $k$ as in Lemma \\ref{tm}.\n\nThen for any finite subset $\\Sigma$ of $S_k^f$ that contains $$S_k^p\\cup (S_k^F\\cap S_k^f)\\cup S_k^A\\cup T,$$ any $x\\in A^t(k)_p$ and $y\\in A(k)_p$ one computes that\n\\begin{align}\\label{explicit comp} \\langle x,y\\rangle^{\\rm N}_{p}=\\, &\\langle x,y\\rangle_{C_{\\Sigma},r_1,r_2}\\\\ =\\, &\\langle x+p^m,y+p^m\\rangle_{C_{\\Sigma}\/p^m,q'_{1},q'_{2}}\\notag\\\\\n= \\, &\\langle x+p^m,y+p^m\\rangle_{C_{{\\rm BD},\\Sigma},r_{1,m},-r_{2,m}}\\notag\\\\\n=  &-\\langle x+p^m,y+p^m\\rangle_{C_{{\\rm BD},\\Sigma},r_{1,m},r_{2,m}}\\notag\\\\\n=  &-\\langle x,y\\rangle_p^{\\rm BD}.\\notag\\end{align}\n\nHere the second equality is given by Lemma \\ref{reducedbock} below and the third equality, after fixing a morphism $\\phi:C_{{\\rm BD},\\Sigma}\\to C_{\\Sigma}\/p^m$ as in Lemma \\ref{morphismphi}, is immediate from the naturality of Bockstein homomorphisms and the description of $H^1(\\phi)$ and $H^2(\\phi)$ given in that result.\n\nBertolini and Darmon have defined in \\cite[\\S 3.4.1]{bert} and \\cite[\\S 2.2]{bert2}\na pairing \\[\\langle\\,,\\rangle_1:A^t(k)\\times A(k)\\to  G.\\] (Although the definition of this pairing is only given in the case that $A$ is an elliptic curve and $F\/k$ is an abelian $p$-extension, it extends naturally to this more general setting).\n\nIn addition, the results of Bertolini and Darmon in \\cite[Thm. 2.8, Rem. 2.10]{bert2} and of Tan in \\cite[Prop. 3.1]{kst} combine to imply directly that the pairing obtained by tensoring $\\langle\\,,\\rangle_1$ with $\\ZZ_p$ coincides with the Mazur-Tate pairing $\\langle\\,,\\rangle_p^{\\rm MT}$.\n\nGiven the latter fact, and the equality of (\\ref{explicit comp}), the proof of Theorem \\ref{thecomptheorem} is therefore completed by means of the following result.\n\n\\begin{lemma}\\label{bdcomp}The pairing $\\langle\\,,\\rangle^{\\rm BD}_p$ coincides with the pairing obtained by tensoring $\\langle\\,,\\rangle_{1}$ with $\\ZZ_p$.\n\\end{lemma}\n\\begin{proof}\nAs necessary preparation for the proof of this result, we fix a minimal set of generators $\\gamma_1,\\ldots,\\gamma_r,\\gamma_{r+1},\\ldots,\\gamma_s$ of $G$, ordered so that for $1\\leq i\\leq r$ the order $o(\\gamma_i)$ of $\\gamma_i$ is a power of $p$ and for $r<i\\leq s$ the order $o(\\gamma_i)$ of $\\gamma_i$ is prime to $p$.\n\nWe also denote by $J$ the cartesian product of the sets $\\{0,1,\\ldots,o(\\gamma_i)-1\\}$ as $i$ ranges through all indices between 1 and $s$.\n\nFor each index $1\\leq i\\leq s$ we write $F_i$ for the fixed field in $F$ of the subgroup of $G$ generated by $\\{\\gamma_j:j\\neq i\\}$.\n\nFor each index $1\\leq i\\leq r$ we let $$D_i^{(1)}:=-\\sum_{j=0}^{o(\\gamma_i)-1}j\\gamma_i^j\\in R$$ be the `derivative operator' of \\cite[Lem. 2.1]{bert2}.\n\n\nTo proceed with the proof, it is enough to verify that, for any points $P\\in A^t(k)$ and $Q\\in A(k)$, the image of $\\langle P,Q\\rangle_1$ in $\\ZZ_p\\otimes_{\\ZZ}G\\cong I_R\/I_R^2$ is equal to $\\langle P+p^m,Q+p^m\\rangle_{C_{{\\rm BD},T}}$, since the latter element is by definition equal to $\\langle P,Q\\rangle_p^{{\\rm BD}}$.\n\nTo compute the term $\\langle P,Q\\rangle_1$, we use the fact that the $G$-modules $\\Sel^T_{p^m}(A^t\/F)$ and $A_T(\\mathbb{A}_F)\/p^m$ are cohomologically-trivial established in Lemma \\ref{bdlemma}(ii).\n\nLet $a\\in \\Sel^{T}_{p^m}(A^t\/F)$ be chosen so that $\\Tr_G(a)$ is equal to the image of $P+p^m$ under the composite map\n$$A^t(k)_p\/p^m\\to\\bigl(A^t(F)_p\/p^m\\bigr)^G\\to\\Sel^{T}_{p^m}(A^t\/F)^G.$$\n\nLet $b\\in A_{T}(\\mathbb{A}_F)\/p^m$ be chosen so that $\\Tr_G(b)$ is equal to the image $Q_{\\rm loc}$ of $Q+p^m$ under the composite map\n$$A(k)_p\/p^m\\to\\bigl(A(F)_p\/p^m\\bigr)^G\\to\\bigl(A_{T}(\\mathbb{A}_F)\/p^m\\bigr)^G.$$\n\n\n\nAfter interpreting the local duality pairing $\\langle\\,,\\rangle_{T}$ of (\\ref{localpairing}) below as taking values in $Z$ in the canonical way as is done throughout \\cite{bert2}, the image of $\\langle P,Q\\rangle_1$ in $I_{R}\/I_{R}^2$ is by definition equal to\n\\begin{align}\\label{bdcomput}\\notag &\\bigl(\\sum_{g\\in G}\\langle g^{-1}(\\lambda_{T}(a)),b\\rangle_{T}\\cdot g\\bigr)+I_{R}^2\\\\\n\\notag =&\\bigl(\\sum_{j_\\bullet\\in J}\\langle(\\gamma_1^{-j_1}\\cdot\\ldots\\cdot\\gamma_s^{-j_s})(\\lambda_T(a)),b\\rangle_T\\cdot(\\gamma_1^{j_1}\\cdot\\ldots\\cdot\\gamma_s^{j_s})\\bigr)+I_{R}^2\\\\\n\\notag =&\\bigl(\\sum_{i=1}^{i=r}\\sum_{j_\\bullet\\in J}\\langle(\\gamma_1^{-j_1}\\cdot\\ldots\\cdot\\gamma_s^{-j_s})(\\lambda_T(a)),b\\rangle_T\\cdot j_i(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle(D_i^{(1)}\\cdot\\Tr_{G_{F\/F_i}})(\\lambda_T(a)),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2.\n\\end{align}\n\n\nWe now let, as we may, $a'_1,\\ldots,a'_s\\in H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ be chosen so that\n\\begin{equation}\\label{augmentationdivision}\\sum_{l=1}^{l=s}(\\gamma_l-1)\\cdot a_l'=\\lambda_{T}(a).\\end{equation}\n\nThen (\\ref{bdcomput}) implies that the image of $\\langle P,Q\\rangle_1$ in $I_{R}\/I_{R}^2$ is equal to\n\n\\begin{align*}&\\bigl(\\sum_{i=1}^{i=r}\\langle(D_i^{(1)}\\cdot\\Tr_{G_{F\/F_i}})\\bigl(\\sum_{l=1}^{l=s}(\\gamma_l-1)\\cdot a_l'\\bigr),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle(D_i^{(1)}\\cdot\\Tr_{G_{F\/F_i}}\\cdot(\\gamma_i-1)) (a_i'),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle \\Tr_G(a_i'),b\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle a_i',\\Tr_G(b)\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}\\langle a_i',Q_{\\rm loc}\\rangle_T\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2\\\\\n=&\\bigl(\\sum_{i=1}^{i=r}Q_{\\rm loc}(u_T(a_i'))\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2.\n\\end{align*}\nIn this last expression we have identified $Q_{\\rm loc}$ with its double-dual.\n\nThe claimed result is thus equivalent to the equality $$\\langle P+p^m,Q+p^m\\rangle_{C_{{\\rm BD},T}}=\\bigl(\\sum_{i=1}^{i=r}Q_{\\rm loc}(u_T(a_i'))\\cdot(\\gamma_i-1)\\bigr)+I_{R}^2$$ in $I_{R}\/I_{R}^2$.\n\nBut, taking into account the equality (\\ref{augmentationdivision}), and given that the $R$-modules $\\Sel^T_{p^m}(A^t\/F)$ and $H^1_{T}\\bigl(\\mathbb{A}_F,A^t\\bigr)[p^m]$ are finitely generated and $G$-cohomologically-trivial by Lemma \\ref{bdlemma}(ii),\nthis equality follows directly from a diagram chase to compute the Bockstein homomorphism $\\beta_{C_{{\\rm BD},T}}$.\n\\end{proof}\n\n\n\n\n\n\\section{Modular symbols}\\label{mod sect} In this section we shall use the theory of modular symbols to study ${\\rm BSD}(A_{F\/k})$ in the case $A$ is an elliptic curve and $F$ is a totally real tamely ramified abelian extension of $k = \\QQ$. The results in this section will, in particular, extend the main results of Bley in \\cite{Bley3}.\n\nFor each natural number $d$ we set $\\zeta_d := {\\rm exp}(2\\pi i\/d)$. We write $\\QQ(d)$ for $\\QQ(\\zeta_d)$ and $\\QQ(d)^+$ for the\nmaximal real subfield of $\\QQ(d)$ and set $G_d := G_{\\QQ(d)\/\\QQ}$ and $G^+_d := G_{\\QQ(d)^+\/\\QQ}$.\n\nFor any element $a$ of $(\\ZZ\/d\\ZZ)^\\times$ we write $\\sigma_a$ for the element of $G_d$ that sends $\\zeta_d$ to $\\zeta_d^a$.\n\n\\subsection{Regularized Mazur-Tate elements} We first recall the definition and relevant properties of modular symbols as introduced by Mazur and Tate in \\cite{mt} and normalized by Darmon in \\cite{darmon} and explain their connection to the leading term element that occurs in Theorem \\ref{bk explicit}.\n\n\n\\subsubsection{}\\label{ell cirve section}We fix an elliptic curve $A$ that is defined over $\\QQ$ and has conductor $N$. We also fix a modular parametrisation $\\varphi_A: X_0(N) \\to A$ and write $c(\\varphi_A)$ for the associated Manin constant.\n\nWe write $\\Lambda$ for a N\\'eron lattice for $A$ in $\\CC$ and then identify $A(\\CC)$ with $\\CC\/\\Lambda$ and the N\\'eron differential $\\omega_0$ of $A$ with ${\\rm d}z$.\n\nWe define strictly positive real numbers $\\Omega^+$ and $\\Omega^-$ by the condition that $\\Lambda$ is a sublattice of $\\ZZ\\cdot\\Omega^+ + \\ZZ\\cdot i\\Omega^-$ with minimal index.\n\nThen the pullback under $\\varphi_A$ of $\\omega_0$ defines a cusp form of weight $2$ on $X_0(N)$ so\n\\[ \\varphi_A^*(\\omega_0) = c(\\varphi_A)f(q){\\rm d}q\/q = c(\\varphi_A)2\\pi if (z){\\rm d}z\\]\nwhere $f(q) = \\sum_{n \\ge 1}a_nq^n$, with $q = {\\rm exp}(2\\pi iz)$, is normalized such that $a_1 = 1$.\n\nWe now fix a natural number $c$ that is square-free and prime to $N$. For each integer $a$ we follow Mazur and Tate \\cite{mt} in defining the modular symbols $[ \\frac{a}{c}]^+$ and $[ \\frac{a}{c}]^{-}$ via the equality\n\\[ 2\\pi i\\int_{a\/c}^{a\/c + i\\infty}f(z){\\rm d}z = \\left[ \\frac{a}{c}\\right]^+\\Omega^+ + i\\left[ \\frac{a}{c}\\right]^{-}\\Omega^-.\\]\nRecalling that $[\\frac{a}{c}]^+$ depends only on the residue class of $a$ modulo $t$, we then follow Darmon \\cite{darmon} in defining the regularized modular\nsymbols\n\n\\[ \\left[ \\frac{a}{c}\\right]^* := \\sum_{t\\mid c}\\mu(c\/t)\\left[\\frac{a(c\/t)^{-1}}{t}\\right]^+.\\]\nHere $t$ runs over all natural numbers that divide $c$, $\\mu(-)$ is the M\\\"obius function and $(c\/t)^{-1}$ denotes the inverse of $c\/t$ modulo $t$ (which exists since $c$ is assumed to be squarefree).\n\nWe finally define the `regularized Mazur-Tate element of $A$ at level $c$' to be the element\n\\[ \\Theta^{\\rm MT}_{A,c} := \\frac{1}{2}\\sum_{a\\in (\\ZZ\/c\\ZZ)^\\times}\\left[\\frac{a}{c}\\right]^*\\sigma_a\\]\nof $\\QQ[G_c]$. In the sequel we identify $\\Theta^{\\rm MT}_{A,c}$ with its image under the canonical projection $\\QQ[G_c]\\to\\QQ[G_c^+]$.\n\n\\subsubsection{} In the next result we describe the precise link between the Mazur-Tate element $\\Theta^{\\rm MT}_{A,c}$ and the leading term element $\\calL^*_{A,\\QQ(c)^+\/\\QQ}$ that occurs in Theorem \\ref{bk explicit} (as defined in (\\ref{bkcharelement}) with respect to a fixed odd prime $p$).\n\nTo state the result we write $\\Upsilon_{c,0}$ for the subset of $\\widehat{G_c^+}$ comprising all characters $\\psi$\nfor which $L(A,\\check\\psi,1)$ does not vanish and define an idempotent of $\\CC[G_c^+]$ by setting\n\\[ e_{c,0} := \\sum_{\\psi\\in \\Upsilon_{c,0}} e_\\psi.\\]\n(It is then in fact straightforward to check that $e_{c,0}$ belongs to $\\QQ[G_c^+]$.)\n\n\n\nIn the sequel, we denote by $c_\\psi$ the conductor of a Dirichlet character $\\psi$.\n\n\\begin{lemma}\\label{gammatheta} For any odd prime number $p$, in $\\QQ[G_c^+]$ one has\n\\[ \\Theta^{\\rm MT}_{A,c}=  (c(\\varphi_A)\/c_\\infty)\\cdot \\bigl(\\sum_{\\psi\\in\\widehat{G_c^+}}(c\/c_\\psi)m_p^\\psi \\cdot e_\\psi\\big)\\cdot e_{c,0}\\calL^*_{A,\\QQ(c)^+\/\\QQ}\\]\nwhere $c_\\infty$ is the number of connected components of $A(\\RR)$ and we set\n\\[ m_p^\\psi := \\begin{cases} p^{-1}, &\\text{ if $p$ divides $c$ but not $c_\\psi$},\\\\\n                             1, &\\text{otherwise.}\\end{cases}\\]\n\\end{lemma}\n\n\\begin{proof} The key point is that, by Darmon \\cite[Prop. 2.3]{darmon}, for every $\\psi\\in\\widehat{G_c^+}$ one has \\begin{equation}\\label{chiMT}(\\Theta^{\\rm MT}_{A,c})_\\psi=c(\\varphi_A)\\cdot\\frac{c}{c_\\psi}\\cdot\\frac{L_c(A,\\check{\\psi},1)\\tau_c(\\psi)}{2\\Omega^+}.\\end{equation}\nHere the $L$-function $L_c(A,\\check{\\psi},z)$ is truncated at all rational primes dividing $c$ and we use the (in general, imprimitive) Gauss sum\n\\[ \\tau_c(\\psi):= \\sum_{a\\in (\\ZZ\/c\\ZZ)^\\times}\\psi(a)\\cdot \\zeta^a_c.\\]\n\nSince the definition of $e_{c,0}$ implies that $L_c(A,\\check{\\psi},1)\\not= 0$ if and only if $e_\\psi e_{c,0}\\not= 0$, the equalities (\\ref{chiMT}) combine to imply that $\\Theta^{\\rm MT}_{A,c} = e_{c,0}\\cdot\\Theta^{\\rm MT}_{A,c}$.\n\nIn addition, since each character $\\psi$ in $\\widehat{G_c^+}$ is even one has both $w_{\\infty,\\psi}=1$ and\n\\[ 2\\Omega^+ = c_\\infty\\cdot\\Omega_{A,\\infty}^+ = c_\\infty\\cdot \\Omega_{A,\\infty}^\\psi\\]\nwhilst, if $p$ divides $c$, an explicit computation of each factor $\\rho_{p,\\psi}$ occurring in (\\ref{bkcharelement}) shows that it is equal to $(m_p^\\psi)^{-1}$.\n\nThe claimed equality now follows directly upon substituting these facts and the result of Lemma \\ref{imprimitive GGS} below into the equality of (\\ref{chiMT}), recalling the explicit definition (\\ref{bkcharelement}) of the element $\\mathcal{L}^*_{A,\\QQ(c)^+\/\\QQ}$ and noting that, in terms of the notation of \\S\\ref{somr tmc sec}, one has $L_c(A,\\check{\\psi},1) = L_{S_{\\rm r}}(A,\\check{\\psi},1)$.\n\\end{proof}\n\n\\begin{lemma}\\label{imprimitive GGS} For each $\\psi$ in $\\widehat{G_c^+}$ one has $\\tau_c(\\psi) = \\tau^\\ast(\\QQ,\\psi)$.\\end{lemma}\n\n\\begin{proof} Since $\\psi$ is linear, the definition of the Galois-Gauss sum $\\tau(\\QQ,\\psi)$ implies that it is equal to the primitive Gauss sum $\\tau_{c_\\psi}(\\psi)$.\n\nUpon recalling the explicit definition of $\\tau^\\ast(\\QQ,\\psi)$ from \\S\\ref{mod GGS section}, it is thus enough to show that\n\\[ \\tau_c(\\psi) = \\tau_{c_\\psi}(\\psi)\\cdot \\prod_{\\ell \\mid (c\/c_\\psi)}(-\\psi(\\sigma_\\ell)^{-1})\\]\nwhere $\\ell$ runs over all prime divisors of $c\/c_\\psi$.\n\nTo prove this it is in turn enough to show that for each prime divisor $\\ell$ of $c\/c_\\psi$ one has an equality $\\tau_c(\\psi) = -\\psi(\\sigma_\\ell)^{-1}\\cdot \\tau_{c'}(\\psi)$ with $c' := c\/\\ell$.\n\nTo show this we  fix integers $s$ and $t$ such that $s\\ell + tc' = 1$. Then\n$\\zeta_c = \\zeta_\\ell^t\\cdot \\zeta_{c'}^s$ and hence\n\n\\begin{align*} \\tau_c(\\psi) = \\, &\\sum_{a\\in (\\ZZ\/c\\ZZ)^\\times}\\psi(a)\\cdot \\zeta_c^a\\\\\n= \\, &\\sum_{a_1\\in (\\ZZ\/\\ell\\ZZ)^\\times}\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\psi(a_1a_2)\\cdot \\zeta_\\ell^{ta_1}\\zeta_{c'}^{sa_2}\\\\\n= \\, &\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\left(\\sum_{a_1\\in (\\ZZ\/\\ell\\ZZ)^\\times}\\zeta_\\ell^{ta_1}\\right)\\psi(a_2)\\cdot\\zeta_{c'}^{sa_2}\\\\\n= \\, &-\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\psi(a_2)\\cdot\\zeta_{c'}^{sa_2}\\\\\n= \\, &-\\psi(s)^{-1}\\sum_{a_2\\in (\\ZZ\/c'\\ZZ)^\\times}\\psi(a_2)\\cdot\\zeta_{c'}^{a_2}\\\\\n= \\, &-\\psi(\\sigma_\\ell)^{-1}\\cdot\\tau_{c'}(\\psi).\n\\end{align*}\nThe third equality here follows from the fact that $\\psi(a_1) = 1$ for all $a_1$ in $(\\ZZ\/\\ell\\ZZ)^\\times$, the fourth from the equality $\\sum_{a_1\\in (\\ZZ\/\\ell\\ZZ)^\\times}\\zeta_\\ell^{ta_1} = -1$ and the last from the fact that $\\sigma_\\ell$ acts as multiplication by $s$ on $(\\ZZ\/c'\\ZZ)^\\times$.\n\\end{proof}\n\n\n\\begin{remark}\\label{euler factor rem}{\\em For any abelian extension $F$ of $\\QQ$ and any subfield $F'$ of $F$ we write $\\pi_{F\/F'}$ for the natural projection map $\\QQ[G_{F\/\\QQ}] \\to \\QQ[G_{F'\/\\QQ}]$.\n\nThen, if $p$ is any prime that does not divide $c$, the interpolation formula (\\ref{chiMT}) implies that\n\\[\\pi_{\\QQ(pc)\/\\QQ(c)}(\\Theta^{\\rm MT}_{A,pc}) =- \\sigma_p(p -\\sigma_p^{-1}a_p + \\sigma_p^{-2})\\cdot \\Theta_{A,c}^{\\rm MT}\\]\nwith $a_p=1+p-|A(\\mathbb{F}_p)|$.\n}\\end{remark}\n\n\n\n\\subsection{Mazur-Tate elements and refined BSD}\n\n\\subsubsection{}We now fix a tamely ramified real abelian extension $F$ of $\\QQ$ and set $G := G_{F\/\\QQ}$.\n\nWe also fix an odd prime $p$, write $F'$ for the maximal subfield of $F$ whose degree over $\\QQ$ is prime to $p$ and set $G' := G_{F'\/\\QQ}$.\n\nWe fix an elliptic curve $A$ that is defined over $\\QQ$ and whose conductor $N$ is prime to the conductor $c$ of $F$. We always assume that the Tate-Shafarevich group $\\sha(A_{F'})$ of $A$ over $F'$ is finite.\n\nThen, since the conductor $c$ of $F$ is square-free, we can define an element of $\\QQ[G]$ by setting\n\\[ \\Theta^{\\rm MT}_{A,F} := \\pi_{\\QQ(c)\/F}(\\Theta^{\\rm MT}_{A,c}).\\]\nWe also write $e_{F,0}$ for the idempotent $\\pi_{\\QQ(c)^+\/F}(e_{c,0})$ of $\\QQ[G]$.\n\n\nIn this section we shall prove the following results.\n\n\\begin{theorem}\\label{MT result} We assume that $A$, $F$ and $p$ are such that $(c,N)=1$ and also satisfy all of the following conditions.\n\n\\begin{itemize}\n\\item[(a)]   $p$ is prime to $|A(F')_{\\rm tor}|$;\n\\item[(b)]  $p$ is prime to $N$ and to the Tamagawa number of $A$ at any prime divisor of $N$;\n\\item[(c)] If $p$ divides $c$, then $p$ is prime to $|A(\\kappa_v)|$ for any $p$-adic place $v$ of $F'$ and, in addition, $A$ has ordinary reduction at $p$;\n\\item[(d)] $p$ is prime to $|A(\\kappa_v)|$ for any place $v$ of $F'$ of residue characteristic dividing $c$.\n \\item[(e)] $p$ is prime to the Manin constant $c(\\varphi_A)$.\n\n\\end{itemize}\n\nThen the following assertions are valid.\n\\begin{itemize}\n\\item[(i)] The projection to $K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$ of the equality in ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if one has\n\\begin{equation*}\\label{refMT}\\ZZ_p[G]\\cdot\\Theta^{\\rm MT}_{A,F} = e_{F,0}\\cdot{\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee).\\end{equation*}\n\n\n\\item[(ii)] If $L(A_{F},1)$ is non-zero, then ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if $\\Theta^{\\rm MT}_{A,F}$ belongs to ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ and, in addition, the $p$-part of the Birch and Swinnerton-Dyer conjecture is valid for $A$ over $F'$.\n\\end{itemize}\n\\end{theorem}\n\nThe following explicit corollary of Theorem \\ref{MT result} extends the main results of Bley in \\cite{Bley3} (for more details see Remark \\ref{mod bley rem} below).\n\n\\begin{corollary}\\label{bleycor} Fix an elliptic curve $A$ over $\\QQ$ with $L(A_{\\QQ},1)\\not=0$. Then for each natural number $n$ there are infinitely many primes $p$, and for each such prime $p$ infinitely many abelian extensions $F\/\\QQ$ that satisfy all of the following properties.\n\n\\begin{itemize}\n\\item[$\\bullet$] ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid.\n\\item[$\\bullet$] The exponent of $G_{F\/\\QQ}$ is divisible by $p^n$.\n\\item[$\\bullet$] The $p$-rank of $G_{F\/\\QQ}$ is at least $n$.\n\\item[$\\bullet$] The order of $G_{F\/\\QQ}$ is divisible by at least $n$ distinct primes.\n\\item[$\\bullet$] The discriminant of $F$ is divisible by at least $n$ distinct primes.\n\\end{itemize}\n\\end{corollary}\n\n\n\n\\subsubsection{} We shall now prove Theorem \\ref{MT result}.\n\nAt the outset we note that the given assumptions (a)-(e) imply that the hypotheses (H$_1$)-(H$_5$) listed in \\S\\ref{tmc} are satisfied and so Proposition \\ref{explicitbkprop} implies that the complex\n\\[ C_{F,0}:=\\ZZ_p[G]e_{F,0}\\otimes_{\\ZZ_p[G]}^{\\mathbb{L}}{\\rm SC}_p(A_{F\/\\QQ})\\]\nbelongs to $D^{\\rm perf}(\\ZZ_p[G]e_{F,0})$.\n\nIn addition, by the main result of Kato in \\cite{kato}, one knows that $e_\\psi(\\CC_p\\otimes_{\\ZZ_p} A(F)_p)$ vanishes for every character $\\psi$\n in $\\Upsilon_{c,0}$ and hence that the complex $\\QQ_p\\otimes_{\\ZZ_p}C_{F,0}$ is acyclic.\n\nIt follows that for every field isomorphism $j:\\CC\\cong\\CC_p$, one has\n\\begin{equation*}\\label{zeroEuler}\\pi_{0,*}(\\chi_{G,p}({\\rm SC}_p(A_{F\/\\QQ}),h^j_{A,F}))=\\chi_{\\ZZ_p[G]e_{F,0}}(C_{F,0},0)\\end{equation*}\nin $K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$, where $$\\pi_{0,\\ast}:K_0(\\ZZ_p[G],\\CC_p[G])\\to K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$$ denotes the canonical projection homomorphism.\n\nNow, in Theorem \\ref{bk explicit}, the sets $S_{p,{\\rm w}}$ and $S_{p,{\\rm u}}^\\ast$ are empty (since $F$ is a tamely ramified extension of $k = \\QQ$) and so the above equality implies that the image under $\\pi_{0,\\ast}$ of the equality in ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if one has\n\\begin{equation}\\label{first approx} \\pi_{0,\\ast}(\\delta_{G,p}(\\calL^*_{A,F\/\\QQ})) = \\chi_{\\ZZ_p[G]e_{F,0}}(C_{F,0},0).\\end{equation}\n\nTo interpret this equality we identify $K_0(\\ZZ_p[G]e_{F,0},\\CC_p[G]e_{F,0})$ with the multiplicative group of invertible $\\ZZ_p[G]e_{F,0}$-lattices in $\\CC_p[G]e_{F,0}$ (as in Remark \\ref{comparingdets}).\n\nIn particular, since the description of the cohomology of ${\\rm SC}_p(A_{F\/\\QQ})$ given in (\\ref{bksc cohom}) combines with hypothesis (a) and the acyclicity of $\\QQ_p\\otimes_{\\ZZ_p}C_{F,0}$ to imply\n\\[ H^i(C_{F,0}) = \\begin{cases} \\ZZ_p[G]e_{F,0}\\otimes_{\\ZZ_p[G]}\\Sel_p(A_{F})^\\vee, &\\text{if $i=2$,}\\\\\n0, &\\text{otherwise,}\\end{cases}\\]\nit follows that the right hand side of (\\ref{first approx}) identifies with the lattice\n\\[{\\rm Fit}^0_{\\ZZ_p[G]e_{F,0}}(\\ZZ_p[G]e_{F,0}\\otimes_{\\ZZ_p[G]}\\Sel_p(A_{F})^\\vee) = e_{F,0}\\cdot {\\rm Fit}^0_{\\ZZ_p[G]}(\\Sel_p(A_{F})^\\vee).\\]\n\nIn addition, the left hand side of (\\ref{first approx}) identifies with the $\\ZZ_p[G]$-module generated by $e_{F,0}\\calL^*_{A,F\/\\QQ}$.\n Thus, since both of the terms $c_\\infty$ and $c(\\varphi_A)$ belong to $\\ZZ_p^\\times$ (the former because $p$ is odd and the latter by hypothesis (e)), to derive claim (i) from the formula of Lemma \\ref{gammatheta} it is enough to show that $$\\pi_{\\QQ(c)^+\/F}\\bigl(\\sum_{\\psi\\in\\widehat{G_c^+}}(c\/c_\\psi)m_p^\\psi \\cdot e_\\psi\\bigr)$$ belongs to $\\ZZ_p[G]^\\times$.\n\nTo verify this assertion it is enough to show that $\\sum_{\\psi\\in\\widehat{G_c^+}}(c\/c_\\psi)m_p^\\psi \\cdot e_\\psi$ belongs to $\\ZZ_p[G_c^+]^\\times$ and this fact follows directly from Lemma \\ref{bley lemma} below with $A=G_c^+$, $i=1$ and, for each positive divisor $d$ of $c$, the subgroup $H_d$ of $G_c^+$ taken to be $G_{\\QQ(c)^+\/\\QQ(d)^+}$.\n\nIn order to prove claim (ii), we assume that $L(A_{F},1)\\not= 0$. This implies, by Kato \\cite{kato}, that the groups $\\sha(A_F)$ and $A(F)$ are both finite and also that the idempotent $e_{F,0}$ is equal to $1$.\n\nIn particular,\nthe module $\\Sel_p(A_F)^\\vee$ is canonically isomorphic to $\\sha(A_F)[p^\\infty]^\\vee$ (see the exact sequence (\\ref{sha-selmer})).\n\nThe module $\\Sel_p(A_F)^\\vee$  is therefore also canonically isomorphic to $\\sha(A_F)[p^\\infty]$ via the Cassels-Tate pairing.\n\nClaim (i) therefore implies that ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if ${\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee) = {\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ is generated over $\\ZZ_p[G]$ by $\\Theta^{\\rm MT}_{A,F}$.\n\nNow, by Nakayama's Lemma, the latter assertion is valid if and only if ${\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)$ contains $\\Theta^{\\rm MT}_{A,F}$\n and, in addition, the module of coinvariants\n\n \\[ \\ZZ_p[G']\\otimes_{\\ZZ_p[G]}\\bigl({\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)\/(\\ZZ_p[G]\\cdot \\Theta^{\\rm MT}_{A,F})\\bigr)\\]\n\nvanishes, or equivalently\n\\begin{equation}\\label{conv reinter} \\pi_{F\/F'}\\bigl({\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)\\bigr) = \\ZZ_p[G']\\cdot \\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F}).\\end{equation}\n\nWe next note that, since ${\\rm SC}_p(A_{F\/\\QQ})$ is acyclic outside degree two, the isomorphism\n\n\\[ \\ZZ_p[G']\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}{\\rm SC}_p(A_{F\/\\QQ})\\cong {\\rm SC}_p(A_{F'\/\\QQ})\\]\nin $D(\\ZZ_p[G'])$ coming from Proposition \\ref{explicitbkprop}(iii) induces an isomorphism of $\\ZZ_p[G']$-modules\n\\begin{multline}\\label{comp iso} \\ZZ_p[G']\\otimes_{\\ZZ_p[G]}{\\rm Sel}_p(A_{F})^\\vee \\cong H^2(\\ZZ_p[G']\\otimes^{\\mathbb{L}}_{\\ZZ_p[G]}{\\rm SC}_p(A_{F\/\\QQ})) \\\\ \\cong H^2({\\rm SC}_p(A_{F'\/\\QQ})) \\cong {\\rm Sel}_p(A_{F'})^\\vee , \\end{multline}\nand hence an equality\n\\begin{equation}\\label{proj1} \\pi_{F\/F'}\\bigl({\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)\\bigr) = {\\rm Fit}^0_{\\ZZ_p[G']}({\\rm Sel}_p(A_{F'})^\\vee).\\end{equation}\n\nIn addition, if we write $c'$ for the conductor of $F'$, then the distribution relation in Remark \\ref{euler factor rem} implies that\n\\begin{equation}\\label{dist rel 2} \\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F}) = \\Theta^{\\rm MT}_{A,F'}\\cdot \\prod_{\\ell \\mid (c\/c')}-\\sigma_\\ell(\\ell -\\sigma_\\ell^{-1}a_\\ell + \\sigma_\\ell^{-2}).\\end{equation}\n\nWe claim next that for each prime factor $\\ell$ of $c\/c'$ the element $\\ell -\\sigma_\\ell^{-1}a_\\ell + \\sigma_\\ell^{-2}$ is a unit in $\\ZZ_p[G']$. This is because any such $\\ell$ cannot be equal to $p$ (since $F\/F'$ is a $p$-extension and $F\/\\QQ$ is tamely ramified) so that $\\ell -\\sigma_\\ell^{-1}a_\\ell + \\sigma_\\ell^{-2}$ is a generator of the Fitting ideal of the $\\ZZ_p[G']$-module $\\bigoplus_{v\\in S_{F'}^\\ell} A(\\kappa_v)$, whilst the given condition (d) implies that the latter direct sum contains no element of order $p$.\n\nGiven this, the equality (\\ref{dist rel 2}) implies that $\\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F})$ differs from $\\Theta^{\\rm MT}_{A,F'}$ by a unit in $\\ZZ_p[G']$ and thus combines with (\\ref{proj1}) to imply that (\\ref{conv reinter}) is valid if and only if the invertible $\\ZZ_p[G']$-sublattice\n\\begin{equation}\\label{last ideal} {\\rm Fit}^0_{\\ZZ_p[G']}({\\rm Sel}_p(A_{F'})^\\vee)\\cdot (\\ZZ_p[G']\\cdot (\\Theta^{\\rm MT}_{A,F'}))^{-1}\\end{equation}\nof $\\QQ_p[G']$ is trivial.\n\nIn addition, if ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_F)[p^\\infty])={\\rm Fit}^0_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)$ contains $\\Theta^{\\rm MT}_{A,F}$ then the above arguments would also imply that ${\\rm Fit}^0_{\\ZZ_p[G']}({\\rm Sel}_p(A_{F'})^\\vee)$ contains $\\Theta^{\\rm MT}_{A,F'}$. Since $p\\nmid |G|'$, the lattice (\\ref{last ideal}) would then be trivial if and only if its image under the natural restriction map $\\varrho_{G'}$ to the group of invertible $\\ZZ_p$-sublattices of $\\QQ_p$ is trivial.\n\nIn particular, since the argument of claim (i) shows that the triviality of (\\ref{last ideal}) is equivalent to the validity of ${\\rm BSD}_p(A_{F'\/\\QQ})$(iv), the observations in Remark \\ref{consistency remark}(ii) and (iii) imply that the triviality of its image under $\\varrho_{G'}$ is equivalent to the validity of the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $F'$, as required to complete the proof of Theorem \\ref{MT result}.\n\n\n\n\n\nThe following algebraic result was used above and will also be useful in \\S\\ref{HHP}.\n\n\\begin{lemma}\\label{bley lemma}\nLet $A$ be a finite abelian group, $c$ a square-free positive integer and $(c_\\psi)_{\\psi\\in\\widehat{A}}$ a family of positive divisors of $c$. We fix an odd prime number $p$, we let $n_\\psi$ be equal to $p$ if $p$ divides $c$ but not $c_{\\psi}$, and we let $n_{\\psi}$ be equal to 1 otherwise.\n\nFix $i\\in\\{1,2\\}$ and assume that, for each positive divisor $d$ of $c$, there exists a subgroup $H_d$ which satisfies both of the following properties:\n\\begin{itemize}\\item[(i)] $d$ is divisible by an integer $c_{\\psi}$ if and only if $H_d\\subseteq\\ker(\\psi)$;\n\\item[(ii)] the order of $H_d$ divides $\\prod_{\\ell\\mid\\frac{c}{d}}(l+(-1)^i)$ (where $\\ell$ runs over primes dividing $\\frac{c}{d}$).\n\\end{itemize}\n\nThen the element $$\\sum_{\\psi \\in \\widehat{A}}((c_{\\psi}\/c) n_{\\psi})^i\\cdot e_{\\psi}$$ belongs to $\\ZZ_p[A]^\\times$\n\\end{lemma}\n\n\\begin{proof} Since $((c_{\\psi}\/c)n_{\\psi})^i$ is a $p$-adic unit for all $\\psi$ it suffices to prove that the inverse $x_i := \\sum_{\\psi \\in \\widehat{A}}(c_{\\psi}\/c)^{-i} n^{-i}_{\\psi}\\cdot e_{\\psi}$ of the given sum belongs to $\\ZZ_p[A]$.\n\nFor each divisor $\\delta$ of $c$ we set\n\\[ f_{p,i}(\\delta) := \\begin{cases} \\bigl(c\/\\delta\\bigr)^ip^{-i}, &\\text{ if $p$ divides $c\/\\delta$,}\\\\\n                          \\bigl(c\/\\delta\\bigr)^i, &\\text{ otherwise.}\\end{cases}\\]\n\nThen $x_i$ is equal to $\\sum_{\\psi \\in \\widehat{A}}f_{p,i}(c_{\\psi})\\cdot e_{\\psi}$ and so, when combined with assumption (i), the approach of Bley in \\cite[Prop. 3.1(a)]{Bley3} implies the required containment is true if for each positive divisor $d$ of $c$, the sum\n\\begin{equation}\\label{tricky sum} {\\sum}_{t \\mid \\frac{c}{d}} f_{p,i}\\left(dt\\right)\\mu(t)=\\sum_{\\{t:p\\nmid\\frac{c\/d}{t}\\}}\\left(\\frac{c\/d}{t}\\right)^i\\mu(t)+\\sum_{\\{t:p\\mid\\frac{c\/d}{t}\\}}\\left(\\frac{c\/d}{t}\\right)^ip^{-i}\\mu(t)\\end{equation}\nis divisible by $|H_d|$, where $\\mu(-)$ denotes the M\\\"obius function on square-free natural numbers.\n\n\nIf $d$ is such that $p$ does not divide $c\/d$, then the sum (\\ref{tricky sum}) is equal to\n\\[\\sum_{\\{t:p\\nmid\\frac{c\/d}{t}\\}}\\left(\\frac{c\/d}{t}\\right)^i\\mu(t)=(\\frac{c}{d})^i\\sum_{t\\mid\\frac{c}{d}}t^{-i}\\mu(t)=(\\frac{c}{d})^i\\prod_{\\ell\\mid\\frac{c}{d}}(1-\\ell^{-i})=\\prod_{\\ell\\mid\\frac{c}{d}}(\\ell^{i}-1). \\]\n\n\nThe claimed result is thus true in this case since, using assumption (ii), $|H_d|$ divides (\\ref{tricky sum}) both if $i=1$ and also if $i=2$ because $\\ell+1$ divides $\\ell^2-1$ for each prime divisor $\\ell$ of $c\/d$.\n\nWe now assume that $p$ divides $c\/d$. In this case $p$ divides $c$ but not $d$ so the sum (\\ref{tricky sum}) is equal to\n\\begin{align*} &{\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(ds)\\mu(s)\n+ {\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(dsp)\\mu(sp)\\\\\n= \\, &{\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(ds)\\mu(s)\n- {\\sum}_{s \\mid \\frac{p^{-1}c}{d}} f_p(dsp)\\mu(s)\\\\\n=\\, &{\\sum}_{s \\mid \\frac{p^{-1}c}{d}} \\left(\\frac{c}{dsp}\\right)^i\\mu(s)\n- {\\sum}_{s \\mid \\frac{p^{-1}c}{d}} \\left(\\frac{c}{dsp}\\right)^i\\mu(s)\\\\\n= \\, &0.\n\\end{align*}\nand the claimed result is clear.\n\n\\end{proof}\n\n\\subsubsection{} In this section we prove Corollary \\ref{bleycor}\n\n\nIf $L(A_\\QQ,1)$ is non-zero, then Kolyvagin has proved that $\\sha(A_\\QQ)$ and $A(\\QQ)$ are both finite and one knows that the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A_\\QQ$ is valid for all but finitely many primes $p$.\n\n\nTo prove Corollary \\ref{bleycor} it is therefore enough to fix a natural number $n$, assume to be given an odd prime $p'$ and a tamely ramified real abelian extension $F'\/\\QQ$ that has discriminant $d_{F'}$ prime to $N$ and is such that $L(A_{F'},1)$ is non-zero and all of the stated conditions are satisfied after replacing $p, F$ and $n$ by $p', F'$ and $n-1$, and then deduce from this assumption the assertion of Corollary \\ref{bleycor} for the fixed $n$. Indeed, the validity of Corollary \\ref{bleycor} would then follow from the previous paragraph by induction since ${\\rm BSD}_p(A_{\\QQ\/\\QQ})$(iv) is equivalent to $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A_\\QQ$ (see Remark \\ref{consistency remark}(iii)).\n\nWe hence fix a natural number $n$ as well as $p'$ and $F'$ with the assumed given properties.\nThen the results of Kato \\cite{kato} imply that $\\sha(A_{F'})$ and $A(F')$ are both finite and so we may fix a prime $p$ that is large enough to ensure that all of the following conditions are satisfied: the $p$-torsion subgroups of $A(F')$ and ${\\rm Sel}(A_{F'})^\\vee$ are trivial, $p$ is prime to the degree of $F'\/\\QQ$, to $d_{F'}$, to $N$, to $c(\\varphi_A)$ and to the Tamagawa numbers of $A$ at any place of $F'$ that divides $N$ and the reduction of $A$ at any place $v$ of $F'$ that divides $d_{F'}$ has no element of order $p$.\n\nIn addition, if we set $m := |A(\\QQ)_{\\rm tor}|$ and $G':=G_{F'\/\\QQ}$, then $\\Theta^{\\rm MT}_{A,F'}$ belongs to $\\ZZ[1\/2m][G']$ and so the interpolation property (\\ref{chiMT}) combines with the non-vanishing of $L(A_{F'},1)$ to imply that $\\Theta^{\\rm MT}_{A,F'}$ generates over $\\ZZ[1\/2m][G']$ a finite index submodule of\n $\\ZZ[1\/2m][G']$.\n\nThis means that we can also assume the prime chosen above to be such that both $\\Theta^{\\rm MT}_{A,F'}$ belongs to $\\ZZ_p[G']^\\times$ and the $p$-part of the Birch and Swinnerton-Dyer Conjecture for $A$ over $F'$ is valid.\n\n\nNext we note that if $A$ does not have complex multiplication, then a result of Serre \\cite{serre} implies that for all sufficiently large primes $q$ the group\n$G_{\\QQ(A[q])\/\\QQ}$ is isomorphic to ${\\rm GL}_2(\\mathbb{F}_q)$ and so has order $(q^2-1)(q^2-q)\/2$, whilst if $A$ has complex multiplication by an order in an imaginary quadratic field $K$, then the theory of complex multiplication implies that for all sufficiently large $q$ the degree of $\\QQ(A[q])\/\\QQ$ is either $(q-1)^2\/w_K$ (if $q$ splits in $K$) or $(q^2-1)\/w_K$ (if $q$ is inert in $K$), where $w_K$ is the number of roots of unity in $K$.\n\nIt follows that for all sufficiently large primes $q$ the degree of $\\QQ(A[q])\/\\QQ(\\mu_q)$ is either $(q^2-1)q\/2$, $(q-1)\/w_K$ or $(q+1)\/w_K$.\n In particular, for sufficiently large $q$ the degree of $F'(\\mu_{q^n})\/\\QQ(\\mu_q)$ is not divisible by the degree of $\\QQ(A[q])\/\\QQ(\\mu_q)$ and so $\\QQ(A[q])$ is not contained in $F'(\\mu_{q^n})$.\n\nIn particular, we may assume that the prime $p$ fixed earlier has this property and then, by the Tchebotarev density theorem, it follows that there is an infinite set $\\mathfrak{L}$ of primes that each split completely in $F'(\\mu_{p^n})$ but not in $\\QQ(A[p])$.\n\nThen for any such prime $\\ell\\in\\mathfrak{L}$ one has $\\ell \\equiv 1$ modulo $p^n$ and, by the argument used prove \\cite[Th. 1.3]{Bley3}, also $a_\\ell \\not\\equiv 2$ modulo $p$ so that the order of $A(\\kappa_v) = A(\\mathbb{F}_\\ell)$ is prime to $p$ for any $\\ell$-adic place $v$ of $F'$.\n\nFor any given ordered subset $\\{\\ell_i: i \\in [n]\\}$ of $\\mathfrak{L}$ of cardinality $n$ and for each index $i$ we write $F_i$ for the unique subextension of $\\QQ(\\ell_i)$ of degree $p^n$.\n\nWe claim that the compositum $F$ of $F'$ with each of the fields $F_i$ has all of the  properties stated in Corollary \\ref{bleycor} with respect to the fixed $n$ and $p$.\n\nIt is certainly clear that this field satisfies all of the claimed properties except possibly for the assertion that ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid.\n\nIn addition, our choice of $p$ implies the hypotheses of Theorem \\ref{MT result} are satisfied by the data $A, F\/\\QQ$ and $p$.\n\nIn particular, since the degree of $F\/F'$ is a power of $p$, and our choice of $p$ implies the $p$-part of the Birch and Swinnerton-Dyer Conjecture is valid for $A$ over $F'$, the criterion of Theorem \\ref{MT result}(ii) reduces us to showing that $\\Theta^{\\rm MT}_{A,F}$ belongs to ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ and $L(A_{F},1)$ is non-zero.\n\nNow the isomorphism (\\ref{comp iso}) is valid in this case and combines with our choice of $p$ to imply that $\\sha(A_{F})[p^\\infty]$ is trivial and hence that ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty])$ is equal to $\\ZZ_p[G]$ and so contains $\\Theta^{\\rm MT}_{A,F}$.\n\nIn addition, the interpolation formula (\\ref{chiMT}) implies that $L(A_{F},1)$ is non-zero if and only if $\\Theta^{\\rm MT}_{A,F}$ is an invertible element of $\\QQ[G]$.\n\nTo complete the proof of Corollary \\ref{bleycor} it is therefore enough to show that $\\Theta^{\\rm MT}_{A,F}$ belongs to $\\ZZ_p[G]^\\times$.\n\nBut, since $\\Theta^{\\rm MT}_{A,F}$ belongs to $\\ZZ_p[G]$ and the order of $G_{F\/F'}$ is a power of $p$, it belongs to $\\ZZ_p[G]^\\times$ if and only if its image under $\\pi_{F\/F'}$ belongs to $\\ZZ_p[G']^\\times$.\n\nIn addition, since each prime $\\ell_i$ splits completely in $F'$, the distribution relation (\\ref{dist rel 2}) implies that\n\\[ \\pi_{F\/F'}(\\Theta^{\\rm MT}_{A,F}) = \\Theta^{\\rm MT}_{A,F'}\\cdot \\prod_{i=1}^{i=n}-(\\ell_i- a_{\\ell_i} + 1).\\]\nIn particular, since our choice of $p$ guarantees that both $\\Theta^{\\rm MT}_{A,F'}$ belongs to $\\ZZ_p[G']^\\times$ and for each $i$ one has\n\\[ \\ell_i - a_{\\ell_i} + 1 \\equiv 2-a_{\\ell_i} \\not\\equiv 0 \\,\\, \\text{modulo} \\,\\, p,\\]\nit follows that $\\Theta^{\\rm MT}_{A,F}$ belongs to $\\ZZ_p[G]^\\times$, as required.\n\n\n\\begin{remark}\\label{mod bley rem}{\\em The results of Bley in \\cite[Cor. 1.4 and Rem. 4.1]{Bley3} show that ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid for an infinite family of abelian extensions $F\/\\QQ$ of exponent $p$ and rely on the explicit computations of Fearnley, Kisilevsky and Kuwata in \\cite{fkk} concerning the twists of modular elements by Dirichlet characters of prime order. The above argument avoids these computations by directly using the theorem of Kato. In addition, whilst the argument above only constructs fields $F$ in which $p$ is unramified, it is possible to modify the approach to construct families of abelian fields $F$ in which $p$ can in principle be tamely ramified and ${\\rm BSD}_p(A_{F\/\\QQ})$(iv) is valid if and only if $L(A_F,1)$ does not vanish. }\\end{remark}\n\n\n\\section{Heegner points}\\label{HHP}\n\nIn this section we again fix an elliptic curve $A$ that is defined over $\\QQ$ and has conductor $N$ and also assume that the modular parametrisation $\\varphi_A$ that was fixed in \\S\\ref{mod sect} is of minimal degree.\n\nWe also fix a quadratic imaginary field $K$ in which all prime divisors of $N$ split and a square-free product $c$ of primes that are both inert in $K$ and coprime to $N$. We write $K_c$ for the ring class field of $K$ of conductor $c$: this is an abelian extension of $K$ that is ramified exactly at the primes dividing $c$; moreover $K_c$ contains the Hilbert class field $K_1$ of $K$ and class field theory gives a natural isomorphism\n\\begin{equation}\\label{explicit iso} G_{K_c\/K_1} \\cong (\\mathcal{O}_K\/c\\mathcal{O}_K)^\\times\/(\\ZZ\/c\\ZZ)^\\times. \\end{equation}\n(cf. \\cite[\\S3]{gross_koly}).\n\nWe set $G_c := G_{K_c\/K}$ and $h_c := |G_c|$.\n\n\\subsection{Zhang's Theorem and a conjecture of Bradshaw and Stein}\\label{thm gzz} The above hypothesis on $K$ implies that the ideal $N\\mathcal{O}_K$ factorises as $\\mathcal{N}\\overline{\\mathcal{N}}$ where the natural map $\\ZZ\/N\\ZZ\\to \\mathcal{O}_K\/\\mathcal{N}$ is bijective. We write $\\mathcal{O}_c$ for the order $\\ZZ + c\\mathcal{O}_K$ in $K$ and set ${\\mathcal{N}}_c := \\mathcal{N}\\cap \\mathcal{O}_c$.\n\nFixing an embedding of $K$ into $\\CC$ the pair $(\\CC\/\\mathcal{O}_c,{\\mathcal{N}}_c^{-1}\/\\mathcal{O}_c)$ defines a CM elliptic curve equipped with a cyclic subgroup of order $N$ and the isomorphism class of this pair defines a point $x_c$ on $X_0(N)(K_c)$.\n\nWe define the `higher Heegner point' $y_c$ to be the element $\\varphi_A(x_c)$ of $A(K_c)$.\n\n\\subsubsection{\nAssuming $L(A_K,z)$ vanishes to order one at $z=1$, Zhang's generalisation \\cite{zhang01, zhang} of the seminal results of Gross and Zagier in \\cite{GZ} implies that for every $\\psi$ in $\\widehat{G_c}$ the function $L(A_K,\\psi,z)$ vanishes to order one at $z=1$, that the element $e_\\psi (y_c)$ of $\\CC\\cdot A(K_c)$ is non-zero and that the complex number $\\epsilon_{A,c,\\psi}$ defined by the equality\n\\begin{equation}\\label{e-def} L'(A_{K},\\check{\\psi},1) = \\epsilon_{A,c,\\psi}\\cdot\n\\frac{\\Omega^+\\Omega^-C}{c_\\psi\\sqrt{|d_K|}}\\cdot h_c\\cdot\\langle e_{\\psi}(y_c),e_{\\check\\psi}(y_c)\\rangle _{K_c}\\end{equation}\nsatisfies both\n\\begin{equation}\\label{stark ec} \\epsilon_{A,c,\\psi} \\in \\QQ(\\psi)^\\times \\,\\,\\,\\,\\text{ and }\\,\\,\\,\\, (\\epsilon_{A,c,\\psi})^\\omega = \\epsilon_{A,c,\\psi^\\omega} \\,\\,\\,\\,\\text{ for all $\\omega\\in G_{\\QQ(\\psi)\/\\QQ}$.}\\end{equation}\nHere we have set\n\\[ C := 4 \\cdot c_\\infty \\cdot c(\\varphi_A)^{-2}\\cdot \\bigl\\vert\\mathcal{O}_{K}^\\times\\bigr\\vert^{-2},\\]\nand written $c_\\psi$ for the least divisor of $c$ such that $\\psi$ factors through the restriction map $G_c\\to G_{c_\\psi}$ and $\\langle -,-\\rangle_{K_c}$ for the $\\CC$-linear extension of the N\\'eron-Tate height pairing for $A$ relative to the field $K_c$.\n\n\nTo investigate the conjecture ${\\rm BSD}(A_{K_c\/K})$ it is convenient to consider a version of the equality (\\ref{e-def}) that is adapted to truncated Hasse-Weil-Artin $L$-series.\n\nTo do this we follow Darmon \\cite{darmon} in defining the `regularised higher Heegner points of $A$ at level $c$' by setting\n\n\\[ z_c := \\sum_{m \\mid c}\\mu(m)y_m, \\,\\,\\text{ and }\\,\\,\\,\\,z'_c := \\sum_{m \\mid c}y_m.\\]\nFor each $\\psi$ in $\\widehat{G_c}$ we then define a non-zero complex number $\\epsilon_{A,\\psi}$ by means of the equality\n\\begin{equation}\\label{u-def} L'_{c}(A_{K},\\check{\\psi},1) = \\epsilon_{A,\\psi}\\cdot\n\\frac{\\Omega^+\\Omega^-C}{c_\\psi\\sqrt{|d_K|}}\\cdot \\frac{h_c}{(c\/c_\\psi)^2}\\cdot\\langle e_{\\psi}(z_c),e_{\\check\\psi}(z'_c)\\rangle _{K_c}\n\\end{equation}\nwhere the $L$-series $L_{c}(A_{K},\\check{\\psi},z)$ is truncated at all places of $K$ corresponding to rational prime divisors of $c$.\n\nThe next result describes the precise connection between the algebraic numbers $\\epsilon_{A,c,\\psi}$ and $\\epsilon_{A,\\psi}$ that are defined via the equalities (\\ref{e-def}) and (\\ref{u-def}). This result shows, in particular, that $\\epsilon_{A,\\psi}$ depends only on the character $\\psi$.\n\nIn the sequel, for each prime divisor $\\ell$ of $c$ we write $\\kappa_{(\\ell)}$ for the residue field of the unique prime of $K$ above $\\ell$.\n\n\n\\begin{lemma}\\label{independence} Fix $\\psi$ in $\\widehat{G_c}$. Set $c':= c_\\psi$ and write $\\phi$ for $\\psi$ regarded as an element of $\\widehat{G_{c'}}$. Set $a_{c,c'} := \\prod_{\\ell}a_\\ell$, where $\\ell$ runs over all prime divisors of $c\/c'$ and the terms $a_\\ell$ are as defined at the beginning of \\S\\ref{mod sect}.\n\nThen the following claims are valid.\n\\begin{itemize}\n\\item[(i)] $h_c\\cdot\\langle e_{\\psi}(y_c),e_{\\check\\psi}(y_c)\\rangle _{K_c} = a_{c,c'}^2\\cdot h_{c'}\\cdot\\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_{c'}}$.\n\\item[(ii)] $\\epsilon_{A,c,\\psi} = a_{c,c'}^{-2}\\cdot\\epsilon_{A,c',\\phi}.$\n\\item[(iii)] $\\epsilon_{A,\\psi} = \\epsilon_{A,c',\\phi}$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} We can clearly assume that $c \\not= c'$ and hence that $c > 1$.\n\nIn this case, if we set ${\\rm T}_{c,c'} := \\sum_{g \\in G_{K_c\/K_{c'}}}g$, then the norm-compatibility of Heegner points implies that\n\\begin{equation}\\label{nc heegner} {\\rm T}_{c,c'}(y_c) = a_{c,c'}\\cdot y_{c'}.\\end{equation}\n\nThis implies $e_{\\psi}(y_c) = (h_{c'}\/h_{c})a_{c,c'}\\cdot e_\\phi(y_{c'})$ and $e_{\\check\\psi}(y_c) = (h_{c'}\/h_{c})a_{c,c'}\\cdot e_{\\check\\phi}(y_{c'})$\nand hence\n\\begin{align*} h_c\\langle e_{\\psi}(y_c),e_{\\check\\psi}(y_c)\\rangle _{K_c} =\\, &h_c(h_{c'}\/h_{c})^2(a_{c,c'})^2\\cdot  \\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_c}\\\\\n=\\, &h_c(h_{c'}\/h_{c})(a_{c,c'})^2\\cdot  \\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_{c'}}\\\\\n=\\, &(a_{c,c'})^2\\cdot h_{c'}\\langle e_{\\phi}(y_{c'}),e_{\\check\\phi}(y_{c'})\\rangle _{K_{c'}}.\\end{align*}\nwhere the second equality follows from the general result of \\cite[Chap. VIII, Lem. 5.10]{silverman}. This proves claim (i).\n\nClaim (ii) follows directly from claim (i) and the fact that the terms $L'(A_{K},\\check{\\psi},1)$ and $(\\Omega^+\\Omega^-C)\/(c_\\psi\\sqrt{|d_K|})$ that occur in (\\ref{e-def}) do not change if one replaces $\\psi$ by $\\phi$.\n\nTo prove claim (iii) we set $\\epsilon_{A,\\phi}:= \\epsilon_{A,\\psi}$. Then, since both $e_\\phi(z_{c'}) = e_\\phi(y_{c'})$ and $e_{\\check\\phi}(z'_{c'}) = e_{\\check\\phi}(y_{c'})$, an explicit comparison of the equalities (\\ref{e-def}) and (\\ref{u-def}) shows that it suffices to show (\\ref{u-def}) remains valid if one replaces $c$ by $c'$ and $\\psi$ by $\\phi$.\n\nWe now set $d:=c\/c'$. By a routine computation using (\\ref{nc heegner}) one then finds that\n\\begin{equation}\\label{trace heegner} {\\rm T}_{c,c'}(z_c) = \\prod_{\\ell \\mid d} (\\ell + 1 - a_\\ell)\\cdot z_{c'} \\,\\,\\,\\text{ and }\\,\\,\\,\n{\\rm T}_{c,c'}(z'_c) = \\prod_{\\ell \\mid d} (\\ell + 1 + a_\\ell)\\cdot z'_{c'}.\\end{equation}\n\nIn addition, for each prime divisor $\\ell$ of $c$ one has\n\\begin{equation}\\label{trace heegner2} |A(\\kappa_{(\\ell)})| = (\\ell + 1 - a_\\ell)(\\ell + 1 + a_\\ell)\\end{equation}\nand so\n\n\\begin{align*} \\frac{h_c}{c^2}\\langle e_{\\psi}(z_c),e_{\\check\\psi}(z'_c)\\rangle _{K_c} &=  \\frac{h_c}{c^2}(\\prod_{\\ell \\mid d}(\\ell + 1 - a_\\ell)(\\ell + 1 + a_\\ell))\\langle e_{\\phi}(z_{c'}),e_{\\check\\phi}(z'_{c'})\\rangle _{K_c}\\\\\n&= \\bigl(\\prod_{\\ell \\mid d}\\frac{|A(\\kappa_{(\\ell)})|}{\\ell^2}\\bigr)\\frac{h_{c'}}{(c')^2}\\langle e_{\\phi}(z_{c'}),e_{\\check\\phi}(z'_{c'})\\rangle _{K_{c'}},\\end{align*}\nwhere the second equality uses \\cite[Chap. VIII, Lem. 5.10]{silverman} and the fact $c = c'\\cdot\\prod_{\\ell\\mid d}\\ell$.\n\n\nThe right hand side of (\\ref{u-def}) therefore changes by a factor of $(\\prod_{\\ell \\mid d}|A(\\kappa_{(\\ell)})|\/\\ell^{2})^{-1}$ if one replaces $c$ by $c'$ and $\\psi$ by $\\phi$.\n\nTo show that the left hand side of (\\ref{u-def}) changes by the same factor we note that\n\\begin{align*} L'_{c'}(A_{K},\\check{\\phi},1)L'_{c}(A_{K},\\check{\\psi},1)^{-1} =\\, &L'_{c'}(A_{K},\\check{\\psi},1)L'_{c}(A_{K},\\check{\\psi},1)^{-1}\\\\ =\\, &\\prod_{\\ell\\mid d}P_\\ell(A_{K},\\check{\\psi},1)^{-1}\\end{align*}\nwhere $P_\\ell(A_{K},\\check{\\psi},t)$ denotes the Euler factor at (the unique prime of $K$ above) $\\ell$ of the $\\psi$-twist of $A$.\n\nNow $K_c$ is a dihedral extension of $\\QQ$ and so any prime $\\ell$ that is inert in $K$ must split completely in the maximal subextension of $K_c$ in which it is unramified. In particular, for each prime divisor $\\ell$ of $d$ this implies that $P_\\ell(A_{K},\\check{\\psi},t)$ coincides with the Euler factor $P_\\ell(A_{K},t)$ at $\\ell$ of $A_{K}$ and hence that\n\\[ P_\\ell(A_{K},\\check{\\psi},1) = P_\\ell(A_{K},1) = \\frac{|A(\\kappa_{(\\ell)})|}{{\\rm N}_{K\/\\QQ}(\\ell)} = \\frac{|A(\\kappa_{(\\ell)})|}{\\ell^2},\\]\nas required.\n\\end{proof}\n\n\\subsubsection{}\\label{bs conj section\n\nIf $c=1$, then for each character $\\psi$ in $\\widehat{G_c}$ one has $c_\\psi = 1$ and the results of Gross and Zagier in \\cite[see, in particular, \\S I, (6.5) and the discussion on p. 310]{GZ} imply directly that $\\epsilon_{A,c,\\psi}=1$.\n\nIn addition, for $c > 1$ the work of Zhang in \\cite{zhang01, zhang} implies for each $\\psi$ in $\\widehat{G_c}$ a formula for the algebraic number $\\epsilon_{A,c,\\psi}$.\n\nHowever, as observed by Bradshaw and Stein in \\cite[\\S2]{BS}, this formula is difficult to make explicit and is discussed in the literature in several mutually inconsistent ways.\n\nIn particular, it is explained in loc. cit. that the earlier articles of Hayashi \\cite{hayashi} and Jetchev, Lauter and Stein \\cite{JLS} together contain three distinct formulas for the elements $\\epsilon_{A,c,\\psi}$ that are mutually inconsistent and all apparently incorrect.\n\nIn an attempt to clarify this issue, in \\cite[Conj. 6]{BS} Bradshaw and Stein conjecture that for every non-trivial character $\\psi$ in $\\widehat{G_c^+}$ one should have\n\\begin{equation}\\label{bs conj} \\epsilon_{A,c,\\psi} = 1,\\end{equation}\nand Zhang has asserted that the validity of this conjecture can indeed be deduced from his results in \\cite{zhang} (see, in particular, \\cite[Rem. 7]{BS}).\n\nHowever, if $c > 1$, then Lemma \\ref{independence}(ii) implies $\\epsilon_{A,c,\\psi}$ is not always equal to $\\epsilon_{A,c_\\psi,\\psi}$ and hence that the conjectural equalities (\\ref{bs conj}) are in general mutually compatible only if one restricts to characters $\\psi$ with $c_\\psi = c$.\n\nFor further comments in this regard see Remark \\ref{bs conj rem} below.\n\n\\subsection{Heegner points and refined BSD} In this section we interpret the complex numbers $\\epsilon_{A,\\psi}$ defined above in terms of our refined Birch and Swinnerton-Dyer Conjecture.\n\n\n\nWe define an element of $\\CC[G_c]$ by setting\n\\[ \\epsilon_{A,c} := \\sum_{\\psi\\in \\widehat{G_c}}\\epsilon_{A,\\psi}\\cdot e_\\psi.\\]\nLemma \\ref{independence}(iii) combines with the properties (\\ref{stark ec}) to imply $\\epsilon_{A,c}$ belongs to $\\QQ[G_c]$.\n\nWe also define an element of $\\QQ[G]^\\times$ by setting\n\\[ u_{K,c}:= (-1)^{n(c)}\\sum_{\\psi\\in \\widehat{G_c}}(-1)^{n(c_\\psi)}\\cdot e_\\psi,\\]\nwhere $n(d)$ denotes the number of rational prime divisors of a natural number $d$.\n\n\\begin{theorem}\\label{h-rbsd} Let $F$ be an abelian extension of $K$ of conductor $c$ and set $G := G_{F\/K}$. Fix an odd prime $p$ and assume that all of the following conditions are satisfied:\n\\begin{itemize}\n\\item[$\\bullet$] the data $A$, $F\/K$ and $p$ satisfy the hypotheses (H$_1$)-(H$_6$) listed in \\S\\ref{tmc}.\n\\item[$\\bullet$] $A(F)$ has no point of order $p$.\n\\item[$\\bullet$] The trace to $K$ of $y_1$ is non-zero.\n\\item[$\\bullet$] $p$ is unramified in $K$.\n\\end{itemize}\nSet $z_{F} := {\\rm Tr}_{K_c\/F}(z_c)$ and $z'_{F} := {\\rm Tr}_{K_c\/F}(z'_c)$. Then the following claims are valid.\n\n\\begin{itemize}\n\\item[(i)]\nIf ${\\rm BSD}_p(A_{F\/K})$(iv) is valid then every element of\n\\[ {\\rm Fit}^0_{\\ZZ_p[G]}\\left( \\bigl(A(F)_p\/\\langle z_F\\rangle\\bigr)^\\vee\\right)\\cdot  {\\rm Fit}^0_{\\ZZ_p[G]}\\left(\\bigl( A(F)_p\/\\langle z'_F\\rangle\\bigr)^\\vee\\right)\\cdot C\\cdot u_{K,c}\\cdot\\epsilon_{A,c} \\]\nbelongs to ${\\rm Fit}^1_{\\ZZ_p[G]}({\\rm Sel}_p(A_{F})^\\vee)$ and annihilates $\\sha(A_{F})[p^\\infty]$.\n\n\\item[(ii)] Assume that $F\/K$ is of $p$-power degree and that $p$ does not divide the trace to $K$ of $y_1$. Then ${\\rm BSD}_p(A_{F\/K})$(iv) is valid if and only if one has $${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty]) = \\ZZ_p[G]\\cdot C\\cdot u_{K,c}\\cdot \\epsilon_{A,c}.$$\n\\end{itemize}\n\\end{theorem}\n\n\n\\begin{proof} Since the extension $F\/K$ is tamely ramified we shall derive claim (i) as a consequence of the observation in Remark \\ref{new add2}.\n\n\n\nWe first note that the assumed non-vanishing of the trace to $K$ of $y_1=z_1$ combines with the trace compatibilities in (\\ref{trace heegner}) to imply that the elements $e_\\psi(z_c)$ and $e_\\psi(z'_c)$ are non-zero for all $\\psi$ in $\\widehat{G}$.\n\nTaken in conjunction with (\\ref{u-def}), this fact implies directly that each function $L_{S_{\\rm r}}(A,\\check{\\psi},z)$ vanishes to order one at $z=1$, where as in Remark \\ref{new add2} we have set $S_{\\rm r}=S_k^F\\cap S_k^F$. It also combines with the main result of Bertolini and Darmon in \\cite{BD} to imply that the $\\ZZ_p[G]$-modules generated by $z_c$ and $z_c'$ each have finite index in $A(K_c)$. This implies, in particular, that the idempotent $e_{(1)}$ is equal to $1$.\n\n\nSince every prime divisor of $c$ is inert in $K$ and then splits completely in the maximal subextension of $K_c$ in which it is unramified, the conductor of each character $\\psi$ in $\\widehat{G_c}$ is a divisor $c_\\psi$ of $c$ and the unramified characteristic $u_\\psi$ defined in \\S\\ref{mod GGS section} is equal to $(-1)^{n(c) + n(c_\\psi)}$.\n\nBy using \\cite[(21)]{bmw} one can then compute that for every $\\psi$ in $\\widehat{G_c}$ one has\n\\begin{align}\\label{gauss sums_eq}\n\\tau^\\ast \\bigl(\\QQ,\\psi\\bigr)\\cdot w_\\psi^{-1}=\\, &u_\\psi\\cdot \\tau \\bigl(\\QQ,\\psi\\bigr)\\cdot w_\\psi^{-1}\n\\\\ =\\, &u_\\psi\\sqrt{\\vert d_{K}\\vert} \\sqrt{{\\rm N}c_\\psi}\\notag\\\\\n =\\, &(-1)^{n(c) + n(c_\\psi)}c_\\psi\\sqrt{\\vert d_{K}\\vert}.\\notag\\end{align}\n\nIn addition, for each $\\psi$ in $\\widehat{G}$ one has\n\\[ \\Omega_A^\\psi= \\Omega^+\\Omega^-\\]\nand\n\\begin{align*} e_\\psi\\cdot h_{F\/K}(z_F,z_F') =\\, &\\sum_{g \\in G}\\langle g(z_F),z_F'\\rangle_{F}\\cdot \\psi(g)^{-1}e_\\psi\\\\\n=\\, & |G|\\langle e_\\psi(z_F),z_F'\\rangle_{F}\\cdot e_\\psi\\\\\n=\\, & |G|\\langle e_\\psi(z_F),e_{\\check\\psi}(z_F')\\rangle_{F} \\cdot e_\\psi\\\\\n=\\, & |G|(|G|\/h_c)\\langle e_\\psi(z_F),e_{\\check\\psi}(z_F')\\rangle_{K_c} \\cdot e_\\psi\\\\\n=\\, & h_c\\cdot\\langle e_\\psi(z_c),e_{\\check\\psi}(z_c')\\rangle_{K_c} \\cdot e_\\psi,\\end{align*}\nwhere in the last equality $\\psi$ and $\\check\\psi$ are regarded as characters of $G_c$.\n\nSetting $S_{p,{\\rm r}}=S_{\\rm r}\\cap S_k^p$ one may also explicitly compute, for $\\psi\\in\\widehat{G}$, that $m_\\psi:=\\prod_{v\\in S_{p,{\\rm r}}}\\varrho_{v,\\psi}$ is equal to $p^2$ if $p$ divides $c$ but not $c_\\psi$ and is equal to $1$ otherwise. We use this explicit description to extend the definition of $m_\\psi$ to all characters $\\psi\\in\\widehat{G_c}$.\n\nThese facts combine with (\\ref{u-def}) to imply that for any $\\psi\\in\\widehat{G}$ one has\n\n\\begin{align}\\label{explicit lt}&\\left(\\frac{L^{(1)}_{S_{\\rm r}}(A_{F\/K},1)\\cdot\\tau^*(F\/K)\\cdot\\prod_{v\\in S_{p,{\\rm r}}}\\varrho_{v}(F\/k)}{\\Omega_A^{F\/K}\\cdot w_{F\/k}\\cdot h_{F\/K}(z_F,z_F')}\\right)_\\psi\\\\= \\, &\\frac{L'_{c}(A_K,\\check{\\psi},1)\\cdot\\tau^*(\\QQ,\\psi)\\cdot m_\\psi}{\\Omega_A^\\psi\\cdot w_\\psi\\cdot h_c\\cdot\\langle e_\\psi(z_c),e_{\\check\\psi}(z_c')\\rangle_{K_c}}\\notag\\\\\n = \\, & \\frac{L'_{c}(A_{K},\\check{\\psi},1)(-1)^{n(c)+ n(c_\\psi)}c_\\psi\\sqrt{|d_K|}}{\\Omega^+\\Omega^-\\cdot h_c\\cdot\\langle e_\\psi(z_c),e_{\\check\\psi}(z_c')\\rangle_{K_c}} \\cdot m_\\psi\\notag\\\\\n = \\, & (-1)^{n(c)+ n(c_\\psi)}\\epsilon_{A,\\psi}\\cdot C \\cdot (c_\\psi\/c)^2m_\\psi \\notag\\\\% \\cdot h_c \\langle e_{\\chi}(z_c),e_{\\check\\chi}(z'_c)\\rangle_{K_c}\\notag\\\\\n = \\, & (-1)^{n(c)}(-1)^{n(c_\\psi)}\\epsilon_{A,\\psi}\\cdot C \\cdot (c_\\psi\/c)^2 m_\\psi. \\nota\n  \\end{align}\n\nTo deduce claim (i) from Remark \\ref{new add2} it is thus sufficient to show that the sum\n\\begin{equation}\\label{unit sum} \\sum_{\\psi\\in\\widehat{G_c}}(c_\\psi\/c)^2m_\\psi\\cdot e_\\psi\\end{equation}\n\nbelongs to $\\ZZ_p[G_c]^\\times$. This fact follows from the result of Lemma \\ref{bley lemma} with $A=G_c$ and $i=2$ (so that $n_\\psi^i=m_\\psi$) and, for each positive divisor $d$ of $c$, with the subgroup $H_d$ of $G_c$ specified to be $G_{K_c\/K_d}$. Indeed, this choice of subgroups satisfies the assumption (ii) of Lemma \\ref{bley lemma} because (\\ref{explicit iso}) implies that $|G_{K_c\/K_d}|$ is equal to $\\prod_{\\ell\\mid (c\/d)}(\\ell+1)$.\n\n\nTo prove claim (ii) it suffices to show that, under the given hypotheses, the equality in Theorem \\ref{bk explicit} is valid if and only if one has ${\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_{F})[p^\\infty]) = \\ZZ_p[G]\\cdot C\\cdot u_{K,c}\\cdot\\epsilon_{A,c}.$\n\nNow, in Theorem \\ref{bk explicit}, the set $S_{p,{\\rm w}}$ is empty since $F$ is a tamely ramified extension of $k =K$ and the set $S_{p,{\\rm u}}^\\ast$ is empty since we are assuming that $p$ is unramified in $K$.\n\nWe next note that $z_1 = z'_1 = y_1$ and hence that (\\ref{trace heegner}) implies\n\\[ {\\rm Tr}_{F\/K}(z_F) = {\\rm Tr}_{K_c\/K}(z_c) = \\mu_c\\cdot{\\rm Tr}_{K_1\/K}(z_1) = \\mu_c\\cdot {\\rm Tr}_{K_1\/K}(y_1)\\]\nwith $\\mu_c = \\prod_{\\ell \\mid c} (\\ell + 1 - a_\\ell)$ and, similarly, that\n\\[ {\\rm Tr}_{F\/K}(z'_F) = \\mu_c'\\cdot {\\rm Tr}_{K_1\/K}(y_1)\\]\nwith $\\mu'_c = \\prod_{\\ell \\mid c} (\\ell + 1 +a_\\ell)$.\n\nIn particular, since (\\ref{trace heegner2}) implies that $\\mu_c\\cdot \\mu'_c = \\prod_{\\ell\\mid c}|A(\\kappa_{(\\ell)})|$, our assumption that the hypotheses (H$_3$) and (H$_4$) hold for $F\/K$ means that $\\mu_c\\cdot \\mu'_c$ is not divisible by $p$. This fact in turn combines with our assumption that ${\\rm Tr}_{K_1\/K}(y_1)$ is not divisible by $p$ in $A(K)$ to imply that neither ${\\rm Tr}_{F\/K}(z_F)$ nor ${\\rm Tr}_{F\/K}(z'_F)$ is divisible by $p$ in $A(K)_p$.\n\n Since our hypotheses imply that $A(K)_p = A(F)_p^{G}$ is a free $\\ZZ_p$-module of rank one, it follows that ${\\rm Tr}_{F\/K}(z_F)$ and ${\\rm Tr}_{F\/K}(z'_F)$ are both $\\ZZ_p$-generators of $A(F)_p^G$, and hence, by Nakayama's Lemma, that $A(F)_p$ is itself a free rank one $\\ZZ_p[G]$-module that is generated by both $z_F$ and $z'_F$.\n\nTaken in conjunction with the explicit descriptions of cohomology given in (\\ref{bksc cohom}) (that are valid under the present hypotheses), these facts imply that the Euler characteristic that occurs in Theorem \\ref{bk explicit} can be computed as\n\\[ \\chi_{G,p}({\\rm SC}_p(A_{F\/k}),h_{A,F}) = {\\rm Fit}^0_{\\ZZ_p[G]}(\\sha(A_F)[p^\\infty])\\cdot h_{F\/k}(z_F,z_F'),\\]\nwhere we have identified $K_0(\\ZZ_p[G],\\CC_p[G])$ with the multiplicative group of invertible $\\ZZ_p[G]$-lattices in $\\CC_p[G]$ (as in Remark \\ref{comparingdets}).\n\nWhen combined with the equality (\\ref{explicit lt}) these facts imply that the product $$\\mathcal{L}^*_{A,F\/K}\\cdot h_{F\/k}(z_F,z_F')^{-1},$$ where $\\mathcal{L}^*_{A,F\/K}$ is the leading term element defined in (\\ref{bkcharelement}), is equal to the projection to $\\ZZ_p[G]$ of $C\\cdot u_{K,c}\\cdot\\epsilon_{A,c}$ multiplied by the sum (\\ref{unit sum}).\n\nClaim (ii) is therefore a consequence of Theorem \\ref{bk explicit} and the fact that, as already observed above, the sum (\\ref{unit sum}) belongs to $\\ZZ_p[G_c]^\\times$.\n\\end{proof}\n\n\n\n                         \n\n\n\n\n\n\n\n\\begin{remark}\\label{bs conj rem}{\\em If $p$ is prime to all factors in $C$, then the hypotheses of Theorem \\ref{h-rbsd}(ii) combine with an argument of Kolyvagin to imply $\\sha(A\/K)[p^\\infty]$ vanishes (cf. \\cite[Prop. 2.1]{gross_koly}). This fact combines with the projectivity of $A(F)^\\ast$ to imply $\\sha(A\/F)[p^\\infty]$ vanishes and hence, via Theorem \\ref{h-rbsd}(ii), that ${\\rm BSD}_p(A_{F\/K})$(iv) is valid if and only if the product $u_{K,c}\\cdot\\epsilon_{A,c}$ projects to a unit of $\\ZZ_p[G]$. In the case that $F\/K$ is unramified, this observation was used by Wuthrich and the present authors to prove the main result of \\cite{bmw}. In the general case, it is consistent with an affirmative answer to the question of whether for every $\\psi$ in $\\widehat{G}$ one should always have\n\\[ \\epsilon_{A,\\psi} = (-1)^{n(c_\\psi)}?\\]\nWe observe that such an equality would, if valid, constitute a functorially well-behaved, and consistent, version of the conjecture of Bradshaw and Stein that was discussed in \\S\\ref{bs conj section}.}\\end{remark}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\textit{Chandra} X-ray observatory has found strong X-ray emission from large scale jets of \nmany radio loud quasars. Some of them, most notorious, PKS 0637-752 \n\\citep{schwartz2000,chartes2000} and 3C 273 \\citep{sambruna2001,marshall2001}, \nare hard to explain \nby synchrotron radiation from a high energy extension of radio-optical \nemitting electrons, \nsince the observed X-ray flux is far above the extension from radio-optical \nspectra and since the X-ray spectrum is harder than the optical one.\nThe most conventional synchrotron self-Compton (SSC) model requires \na very small magnetic field strength,\nwhich means a large departure from equi-partition between \nmagnetic field and relativistic electrons with an enormous jet power.\nThe latter has been regarded unlikely. \nIt is then considered that inverse Compton (IC) scattering of cosmic microwave \nbackground (CMB) photons may explain the X-ray emission provided that the jet is \nrelativistic and makes a small angle with the line of \nsight \\citep{tavecchio2000,cgc2001}.\n\\deleted{In this model, however, the intrinsic jet length becomes as large as Mpcs, \nrarely seen in the parent population of radio galaxies. }\nBoth SSC and IC\/CMB models predict that high energy extension of the \nX-ray spectrum reaches GeV energy range which can be detected with \\textit{Fermi} LAT \n\\citep{georganopoulos2006}. \nRecently,  \\textit{Fermi} LAT observations have been reported to put upper limits of \nthe $\\gamma$-ray flux for the jets of 3C 273 \\citep{mg2014}\nand PKS 0637-752 \\citep{meyer2015}. \nThey are an order of magnitude lower than the SSC and IC\/CMB predictions  \nso that these models are incompatible with observations. \n\nIn this situation, several alternative models should be considered. \nWithin the leptonic scenario, a separate population of electrons from \nthose emitting radio-optical photons, extending up to around 100 TeV is considered. \nThose electrons emit X-ray photons by synchrotron radiation. \nThe existence of such an electron \npopulation is rather ad-hoc although not impossible. \nIts origin is usually considered a separate particle acceleration process \nfrom that for the radio-optical emitting population.\nHowever, it is hard to imagine such an efficient acceleration mechanism, \nsince the high energy end of radio-optical \nemitting electrons is determined by the balance of acceleration and radiative cooling. \nActually, for PKS 0637-752, radio, optical, and X-ray emissions are spatially coincident and \nconcentrated in a few bright knots, the size of which is an order of kpc, \nand the broadband spectra imply a break around $10^{12}$-$10^{13}$ Hz which \nis naturally due to radiative cooling across the source.\n\nAlternatively, relativistic hadrons may be responsible for the X-ray emission.\n\\cite{bg2016} and earlier \\cite{aharonian2002} considered proton synchrotron emission \nassuming magnetic field stronger than 10 mG and a large energy density of \naccelerated protons up to more than 10 PeV. \nAlthough they argued that the proton and Poynting powers are modest, it is due to \ntheir adopted time scale. They assumed that the confinement time is as long as \n$10^7$-$10^8$ years,\nwhich is a few orders of magnitude longer than the light \ncrossing time of $10^3$-$10^4$ years. \nSince the jet does not involve a large inertia nor a large confining pressure, \nwe regard that the relevant time scale should be similar to the latter. \nThen, the required proton and Poynting powers become \nan order of $L_p \\approx 10^{50}$ erg s$^{-1}$ \nand $L_\\mathrm{Poy} \\approx 10^{49}$ erg s$^{-1}$, respectively.\nAt the same time, the energy density of relativistic electrons becomes\nabout six orders of magnitude smaller than that of the Poynting power. \n\nRelativistic protons also contribute to X-ray and $\\gamma$-ray emission \nthrough the production of high energy electrons and positrons by\nBethe-Heitler and photo-pion processes. \nThe latter processes were considered before the \\textit{Chandra} era by \\cite{mkb91}\nfor a possible mechanism of the X-ray emission from hot spots of radio galaxies.\n\\cite{aharonian2002} also examined these processes for the knot A1 of 3C 273 and \nconcluded that they are too inefficient.\nBut, the efficiency of these processes depends on the size of the knot as well as \nthe infrared photon spectrum.\n\\added{In fact, the energy density of mm and sub-mm radiation \nin \\cite{aharonian2002} is smaller than our model presented below.}\nThe proton models have been discussed for more compact emission regions of blazars and \nBethe-Heitler process can contribute to the same order, depending on the \nsoft photon spectral shape \\citep{pm2015}.\nThus, in this paper, we consider both Bethe-Heitler and photo-pion processes  \nand examine if these processes can explain the X-ray emission from PKS 0762-752.\n\nIn Section \\ref{sec:estimate} we make a rough estimate of physical quantities \nof the X-ray emitting large scale jet of PKS 0637-752, whose redshift is 0.651. \nIn Section \\ref{sec:model} we formulate the problem, \nand in Section \\ref{sec:results} numerical results are presented. \nIn Section \\ref{sec:conclusion} we draw conclusions.\n  \n\n\\section{Rough Estimate}  \\label{sec:estimate}\n\nWe first make a rough estimate of the physical quantities in order to \ncapture the essence of the problem. For simplicity, \nwe assume a single  uniform sphere of radius $R=R_\\mathrm{kpc}$ kpc for the emission region\nand ignore effects of relativistic beaming and redshift for the time being. \nAlthough the emission region is divided into a few knots in reality, \nwe here treat a combined emission region.\n\n\nObserved spectra at radio through optical frequencies suggest that \nthe synchrotron emission from primary electrons has a peak at infrared band \naround $10^{12}$ Hz with a luminosity $L_\\mathrm{syn}$ about \n$3\\times 10^{44}$ erg s$^{-1}$. \nThus,  energy density of synchrotron photons $u_\\mathrm{syn}$ is about \n\\begin{equation}\n  u_\\mathrm{syn}=\\frac{3L_\\mathrm{syn}}{4\\pi R^2c}\n  \\approx 3 \\times 10^{-10}R_\\mathrm{kpc}^{-2} \\,\\,  \\text{erg} \\,\\, \\text{cm}^{-3} .\n\\end{equation}\nThe number density of photons at radio frequencies may be approximated by \n\\begin{equation}\n  \\nu n_\\nu\\approx 10^6 \\left(\\frac{\\nu}{10^{10} \\, \\text{Hz}}\\right)^{-0.75}\n  R_\\mathrm{kpc}^{-2} \\,\\, \\text{cm}^{-3} \n\\end{equation}\nand that at optical frequencies \n\\begin{equation}\n  \\nu n_\\nu\\approx 10^{2}\\left(\\frac{\\nu}{10^{14} \\, \\text{Hz}}\\right)^{-1.25}\n  R_\\mathrm{kpc}^{-2} \\,\\, \\text{cm}^{-3} .\n\\end{equation}\nFor the magnetic field strength of $B=B_\\mathrm{mG}$ mG, \nthe energy density of magnetic field is \n\\begin{equation}\n  u_\\mathrm{mag}=4 \\times 10^{-8} B_\\mathrm{mG}^2 \\,\\, \\text{erg} \\,\\, \\text{cm}^{-3} ,\n\\end{equation}\nand the Poynting power is estimated as  \n\\begin{equation}\n  L_\\mathrm{Poy}=\\pi R^2 u_\\mathrm{mag}c \n  \\approx 4\\times 10^{46}B_\\mathrm{mG}^{2}R_\\mathrm{kpc}^{2} \n  \\,\\, \\text{erg} \\,\\, \\text{s}^{-1} .\n\\end{equation}\n\nThe Lorentz factor of electrons ranges from \n\\replaced{$\\gamma_{e , \\mathrm{min}}\\approx 3 \\times 10^3B_\\mathrm{mG}^{-0.5}$}\n{$\\gamma_{e , \\mathrm{min}}\\approx 2 \\times 10^3B_\\mathrm{mG}^{-0.5}\n (\\nu_\\mathrm{min}\/10^{10} \\, \\mathrm{Hz})^{0.5}$}\nto $\\gamma_{e, \\mathrm{max}}\\approx 10^6B_\\mathrm{mG}^{-0.5}$\nwith a broken power law spectrum. \nThe power law index of electrons is tentatively taken as 2.5 at low energies \nand 3.5 at high energies in accordance with the above photon spectra. \n\\deleted{The energy density of electrons is governed by the low energy end}\n\\added{The peak luminosity of $3 \\times 10^{44}$ erg s$^{-1}$  at $10^{12}$ Hz\nis emitted by primary electrons with \n$\\gamma_\\mathrm{br} \\sim 2 \\times 10^{4} B_\\mathrm{mG}^{-0.5}$ below which \nthe number spectrum is given by $n_e(\\gamma_e) = K_e \\gamma_e^{-2.5}$.\nOn the other hand, $L_\\mathrm{syn}$ is given by \n$\\sim (4 \\pi R^3\/3) c \\sigma_\\mathrm{T} u_\\mathrm{mag} \\gamma_\\mathrm{br}^2 \nn_e(\\gamma_\\mathrm{br}) \\gamma_\\mathrm{br}$,\nwhere $\\sigma_\\mathrm{T}$ is the Thomson cross section.\nFrom these relations we write $K_e$ in terms of $\\gamma_\\mathrm{br}$, $R$, and $B$.\nThe electron energy density is now given by\n$u_e \\sim 2 m_e c^2  K_e \\gamma_\\mathrm{min}^{-0.5}$ \n}\nand estimated as  \n\\begin{equation}\n  u_e \\approx  8  \\times 10^{-10} B_\\mathrm{mG}^{-1.5} R_\\mathrm{kpc}^{-3}\n  \\left(\\frac{\\nu_\\mathrm{min}}{10^{10} \\mathrm{Hz} }\\right )^{-0.25} \n  \\,\\, \\text{erg} \\,\\, \\text{cm}^{-3} .  \n  \\label{eq:ue}\n\\end{equation}\nIt may seem that \\replaced{inverse Compton}{SSC} luminosity can be large,\nif the magnetic field strength is smaller than 0.1 mG for a typical source size of 1 kpc. \nHowever, in this case, most of the inverse Compton \nemission is produced in the  MeV-GeV range. \nTo reproduce the observed X-ray flux, magnetic field needs to be as small as 0.01 mG.\nIn this case, the kinetic power of electrons given by\n\\begin{equation}\n  L_e= \\frac{4\\pi R^3u_e}{3}\\frac{c}{3R}\n  \\approx  3 \\times  10^{44} B_\\mathrm{mG}^{-1.5} R_\\mathrm{kpc}^{-1}\n  \\left(\\frac{\\nu_\\mathrm{min}}{10^{10} \\text{Hz} }\\right)^{-0.25}\n  \\,\\, \\text{erg} \\,\\, \\text{s}^{-1} \n\\end{equation}\nwould become very large, \n\\added{i.e., $L_e \\sim 3 \\times 10^{47}$ erg s$^{-1}$ for $B = 0.01$ mG\nand $\\nu_\\mathrm{min} = 10$ GHz.}\nHere, we take the escape time of electrons as $3R\/c$. \nHistorically, for this reason, the beamed IC\/CMB model was proposed, \nbut as noted in Section \\ref{sec:intro}, this model is now regarded unlikely. \nThe minimum power for explaining the radio-optical flux \nis realized at $B_\\mathrm{mG}\\approx 0.2$ for $R_\\mathrm{kpc}=1$\nwith $L_\\mathrm{Poy} \\approx L_e\\approx 2 \\times 10^{45} \\, \\text{erg} \\,\\, \\text{s}^{-1}$.\n\n\n\n\\added{Theoretically,} \nthe break energy of the electron energy distribution, \\added{$\\gamma_b$,} \nis determined by the \nbalance between synchrotron cooling and escape; if we equate the cooling time \nwith escape time, we obtain\n\\begin{equation}\n  \\gamma_b \\approx 3 \\times 10^3B_\\mathrm{mG}^{-2}R_\\mathrm{kpc}^{-1} .\n\\end{equation}\nIf the break corresponds to the break of radio-optical spectrum at $10^{12}$ Hz, \nwe obtain the field strength of around 0.2 mG for $R=1 \\, \\text{kpc}$.\nThese considerations lead to $B_\\mathrm{mG}\\approx 0.1-0.3$ as an appropriate choice.\n \nThe maximum possible Lorentz factor of electrons is estimated by equating the \ncooling time with the  \ngyrotime and given by \n\\begin{equation}\n  \\gamma_{e, \\mathrm{lim}} \\approx 10^9 B_\\mathrm{mG}^{-1\/2}  .\n\\end{equation}\nThus, in principle it is possible to obtain $\\gamma_e$ \nas large as $10^8$ with which electrons emit synchrotron X-rays. \nHowever, since the observed X-ray spectrum is much flatter than the optical spectrum, \nsuch a high energy population should be separate from radio-optical emitting one and\nthe acceleration mechanism should be very efficient and distinct. \nAlternatively, such electrons may be supplied from photo-hadronic processes.\nIt should be noted that AGN jets are composed of protons and electrons\/positrons and \nthe inertia is likely to be dominated by protons \\added{\\citep{uchiyama2005}}, \nwhile the existence of electron-positron pairs is also suggested\nby various analysis of observations. \nProton acceleration \\added{may} also naturally \\deleted{takes} \\added{take} place \nand in principle the maximum energy of \nprotons can be as large as  $10^{20}$ eV for 1 mG field and 1 kpc size. \n\nTwo photo-hadronic processes can provide secondary high energy electrons\/positrons, i.e., \nphoto-pion production process and Bethe-Heitler process. \nThe former is through strong interaction with the cross section of about \n$3\\times 10^{-28} \\,\\, \\mathrm{cm}^2$ \nand the threshold energy of\n\\begin{equation}\n  \\gamma_{p, \\mathrm{th}} \\approx m_\\pi c^2 \\epsilon_\\mathrm{soft}^{-1}\n  = 3\\times 10^{12} \\left(\\frac{\\nu_\\mathrm{soft}}{10^{10} \\mathrm{Hz}}\\right)^{-1},\n\\end{equation}\nwhere $\\epsilon_\\mathrm{soft}=h \\nu_\\mathrm{soft}$ is the energy of a target photon. \nThus, for $\\gamma_p=10^{10}$, the energy of target photons should be larger than \n$3\\times 10^{12}$ Hz\nwith the number density about  $10^5 \\,\\, \\text{cm}^{-3}$ for $R_\\mathrm{kpc}=1$. \nThus, a proton interaction probability is around 0.1 for rectilinear propagation. \nCharged pions decay to produce electrons and positrons, while neutral pions decay into \ntwo $\\gamma$-rays, which interact with soft photons to produce electron-positron pairs. \n\\added{About 5 \\% of the inelastic energy goes into electron\/positrons.\nThis is because pion mass is about 15 \\% of proton mass, so that about 15 \\% of\nproton energy goes to pions near the threshold.\nThis energy is further distributed to 4 leptons almost equally,\n\\cite[e.g.,][]{ka2008,dm2009}.}\n\\deleted{Considering that about 5 \\% of the inelastic energy goes into electron\/positrons,  }\n\\added{Considering this,}\nwe estimate 0.5 \\% of the proton power can be used to produce electrons \nand positrons with the Lorentz factor of around $2.5 \\times 10^{11}$, \nwhich subsequently radiate synchrotron radiation below the TeV energy \nregion for $B=1 \\,\\text{mG}$. \nWhile photons with energy higher than TeV is optically thick to \nphoton-photon pair production, most synchrotron photons are emitted below TeV and \npair cascade process does not much develop. \nSince these electrons\/positrons rapidly cool to make \nthe energy distribution of a power law with an index of $-2$, \nthe resultant photon energy spectrum is a power law with an index of $-0.5$\nand the X-ray  luminosity is four orders of magnitudes smaller than the TeV luminosity. \nThus, if we would explain the X-ray observations with this mechanism, \nthe predicted GeV luminosity becomes around \n$3 \\times 10^{47} \\, \\text{erg} \\,\\, \\text{s}^{-1}$, \nwhich exceeds the \\textit{Fermi} LAT upper limit. \nRequired proton power is uncomfortably large \namounting to $3 \\times 10^{51} \\, \\, \\text{erg} \\,\\, \\text{s}^{-1}$.\nHigher energy protons exacerbate the problem and lower energy protons \nhave a negligibly small interaction probability. \n\n\nBethe-Heitler process has a larger cross section of \n$3\\times 10^{-27} \\, \\text{cm}^2$ and the lower threshold energy of \n\\begin{equation}\n  \\gamma_{p, \\mathrm{th}}= m_e c^2\\epsilon_\\mathrm{soft}^{-1}\n  = 3\\times 10^{10} \\left(\\frac{\\nu_\\mathrm{soft}}{10^{10} \\, \\mathrm{Hz}}\\right)^{-1} ,\n\\end{equation}\nbut with a lower efficiency of energy transfer to electrons and positrons.\nFor $\\gamma_p =10^{10}$, target photon energy is larger than $3\\times 10^{10}$ Hz \nwith the number density $5\\times 10^5 \\,\\, \\text{cm}^{-3}$ for $R_\\mathrm{kpc}=1$, \nand on average a proton produces 5 electron-positron pairs before it escapes from the region.\nTaking the efficiency of 0.001, 0.5\\% of the proton power can be used to \npair production, roughly the same order as the photo-pion production. \nIn this case, however, the injection Lorentz factor of electrons and positrons is around \n\\deleted{$10^{10}$} \\added{$\\gamma_e \\sim \\gamma_p = 10^{10}$} \nand the synchrotron frequency is peaked at 100MeV for $B=1$ mG. \nSince the resultant synchrotron spectrum becomes a power law with an index of $-0.5$,\nthe X-ray luminosity  is about two orders of magnitudes lower than the 100 MeV \nluminosity. \nThe required proton power is around $10^{49} \\,\\, \\text{erg} \\, \\, \\text{s}^{-1}$, \nwhich is very large but not inconceivable considering the Poynting power is \nan order of $4 \\times 10^{46} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$. \nWhen the magnetic field strength is 0.1 mG, \nthe required proton power is $3\\times 10^{48} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$,\n\\added{because the beak frequency of synchrotron radiation by secondary leptons\ndecreases as $B$ decreases.}\nThe power of primary electrons becomes \n $10^{46} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$ while the Poynting power \nis $4 \\times 10^{44} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$. \nThus we regard about $B = 0.1 \\,\\text{mG}$ is a best guess. \n\\added{The proton power $\\sim 10^{49}$ erg s$^{-1}$ estimated above\n  is $\\sim 120 L_\\mathrm{Edd}$,\n  where $L_\\mathrm{Edd}$ is the Eddington luminosity with the black hole mass\n  $M_\\mathrm{BH} = 6.5 \\times 10^8 M_\\sun$ \\citep{ljg2006}.\n  It is to be noted that $M_\\mathrm{BH} = 7.8 \\times 10^9 M_\\sun$ has been \n  reported by \\cite{gcj2001}.  For this value of $M_\\mathrm{BH}$, \n  the proton power is $\\sim 10 L_\\mathrm{Edd}$.}\n\n\nSomewhat lower energy protons also contribute to Bethe-Heitler process; although the \ninteraction probability becomes lower, the injection Lorentz factor also becomes smaller, \nwhich makes the synchrotron peak lower.\nFor example, for $\\gamma_p=10^9$, the target photon energy is above $3\\times 10^{11}$ Hz\nwith the number density of $10^5 \\,\\, \\text{cm}^{-3}$, \nand the interaction probability of a proton becomes 1. \nThus 0.1\\% of the proton power is available. \nThe injection Lorentz factor of electrons\/positrons is also $10^9$, which emit \nsynchrotron radiation at 1 MeV for $B=1 \\, \\text{mG}$ or 0.1 MeV for $B=0.1 \\, \\text{mG}$.\nSo the proton power of $3 \\times 10^{48} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$ \ngives rise to the observed X-ray luminosity. \n\nIt is to be noted that for a larger $R_\\mathrm{kpc}$ the interaction probability \nof protons becomes small and required proton power accordingly increases; \na larger $R_\\mathrm{kpc}$ is unfavorable for the photo-hadronic model. \nFor these parameters, proton synchrotron luminosity is a few orders of magnitude \nsmaller than the synchrotron luminosity of secondary pairs.  \nIts peak is at $10^{19}$ Hz for $B=0.1\\, \\text{mG}$.\n\nSince the resultant emission spectra depend on details of the secondary \nelectron\/positron spectra and resultant radiative cooling, \nwe numerically investigate such details in the next section. \nBased on the present investigation, in this paper, \nwe concentrate on Bethe-Heitler process.\n\n\n\\section{The Model} \\label{sec:model}\n\nWe adopt a single zone model in which the size of the emission region of $R$\naround 1 kpc, magnetic field of $B$ around 0.1 mG, and the proton spectrum is \n\\begin{equation}\n  n_p \\added{(\\gamma_p)} = K_p \\gamma^{-p} ,\n  \\label{eq:proton-spec}\n\\end{equation}\nfor $\\gamma_{p, \\mathrm{min}}\\le \\gamma_p \\le \\gamma_{p, \\mathrm{max}}$. \nCanonically we take $p=2$, $\\gamma_{p, \\mathrm{min}}=1$,\nand $\\gamma_{p, \\mathrm{max}}=10^{10}$. \nSince photo-hadronic processes work only for large values of $\\gamma_p$, \nthe value of $\\gamma_{p, \\mathrm{min}}$ does not affect the resultant \nspectrum but only affects the energy density of protons \\added{logarithmically}.\nIf  proton energy distribution is concentrated in the range near $\\gamma_{p, \\mathrm{max}}$,\nthe power requirement below  will be relieved by an order of magnitude.  \nNote that the estimates of the proton power and proton energy density given in the \nprevious section are those for such a high energy population \nand an order of magnitude lower than those for $\\gamma_{p, \\mathrm{min}}=1$.\nIn contrast $\\gamma_{p, \\mathrm{max}}$ critically affects the results.\nThe energy density of protons is \n\\begin{equation}\n  u_p =10^{-3} K_p \\ln(\\gamma_{p, \\mathrm{max}}\/\\gamma_{p, \\mathrm{min}}) \\,\\,\n  \\text{erg} \\,\\, \\text{cm}^{-3} .\n\\end{equation}\nThe proton power is \n\\begin{equation}\n  L_p = \\frac{4\\pi R^3 u_p}{3}\\frac{c}{3R}\n  \\approx 4 \\times 10^{50} K_p\n  \\ln(\\gamma_{p, \\mathrm{max}}\/\\gamma_{p, \\mathrm{min}}) R_\\mathrm{kpc}^2 \n  \\,\\, \\text{erg} \\,\\, \\text{s}^{-1},\n\\end{equation}\nwhere we take the escape time of $3R\/c$.\n\n\nThe target photon spectra of photo-pion and Bethe-Heitler processes \nare based on the observed radio to optical photons,\ntaking into account of the cosmological redshift of 0.651.\nPrimary electrons responsible for the  radio-optical emission are \ndetermined so as to reproduce the radio-optical spectra. \nWe also calculate the inverse Compton scattering of primary electrons \noff radio-optical photons, \nwhose flux is generally orders of magnitude short of X-ray observations.\nIn numerical calculations we solve the kinetic equation of primary electrons\nwith the injection spectrum given by \n$q_\\mathrm{inj}(\\gamma_e) = K_e \\gamma_e^{-\\alpha_e} \\exp(-\\gamma_e\/\\gamma_{e, 0})$,\nwhere $\\gamma_e$ is the Lorentz factor of electrons and $K_e$, $\\alpha_e$, \nand $\\gamma_{e, 0}$ are parameters to fit the observed radio-optical spectrum.\nSince we consider only mildly relativistic beaming if any, we ignore IC\/CMB.  \nAlthough the number density of CMB photons is larger than  \nthat of radio-optical synchrotron photons when $R_\\mathrm{kpc}$ is larger than 10, \nthe efficiency of those processes becomes small for large $R$, \nso that our treatment is justified.\n  \n\nThe emission by high energy leptons produced by the hadronic processes is \ncalculated for  the lepton spectrum\nobtained by solving the kinetic equation given by\n\\begin{equation}\n  \\frac{d n_e (\\gamma_e)}{d t}=q_\\mathrm{BH}(\\gamma_e) + q_\\mathrm{pp}(\\gamma_e)\n  -\\frac{c n_e (\\gamma_e)}{3R}\n  -\\frac{d~}{d\\gamma_e} [\\dot{\\gamma}_e n_e (\\gamma_e)] ,\n  \\label{eq:e-kinetic-hadronic}\n\\end{equation}\nwhere $n_e(\\gamma_e)$ is the lepton density per unit interval of $\\gamma_e$.\nThe lepton injection rate is denoted by $q_\\mathrm{BH}(\\gamma_e)$ for Bethe-Heitler process, \nwhich is calculated according to the formulation  given by \\cite{ka2008},\n\\added{using a given proton spectrum with equation (\\ref{eq:proton-spec}).}\nLepton production via photo-pion processes is denoted by $q_\\mathrm{pp}(\\gamma_e)$.\nHowever, we do not include this term in numerical calculations \nbecause the leptons produced by photo-pion processes do not contribute much \nto the X-ray emission.\nHere $\\dot{\\gamma}_e$ denotes radiative cooling through synchrotron radiation \nand inverse Compton scattering.\nWe also set the lepton escape time to be $3 R\/c$.\nUsing the steady solution of equation (\\ref{eq:e-kinetic-hadronic}), \nwe calculate the emission spectra through synchrotron radiation \nand inverse Compton scattering. \n\n\n\n\\section{Results}  \\label{sec:results}\n\n\\begin{figure}[ht!]  \n  \\centering\\includegraphics[scale=0.5,clip]{figure1.eps}\n  \\caption{The production spectrum of electrons and positrons \n    for \\added{$p = 2$ and} $K_p = 1$ \\added{cm$^{-3}$.}\n    The synchrotron radiation spectrum by primary electrons \n    for $R = 1$ kpc and $B = 0.1$ mG (Fig. \\ref{fig:1kpc-no-beaming-exp})\n    is used as a target photon spectrum in lepton production.\n    The black line shows the pair production rate by Bethe-Heitler process,\n    the red and blue lines show the electron and positron production rates, \n    respectively, by photo-pion process.\n    \\label{fig:pair-production-spec}}\n\\end{figure}\n\n\\begin{figure}[ht!]  \n  \\centering\\includegraphics[scale=0.5,clip]{figure2.eps}\n  \\caption{The lepton energy spectrum for $R = 1$ kpc, $B = 0.1$ mG,\n    \\added{and $p=2$}.\n    The spectrum is calculated by equation (\\ref{eq:e-kinetic-hadronic})\n    for a steady state.\n    The injection of leptons is by Bethe-Heitler process.\n    \\label{fig:lepton-energy-spec}}\n\\end{figure}\n\n\n\n\nIn Figure \\ref{fig:pair-production-spec} we show the production spectrum of electrons \nand positrons through Bethe-Heitler and photo-pion processes\nfor a photon field relevant to PKS 0637-752 jet for the fiducial case $R_\\mathrm{kpc} = 1$\nand $B_\\mathrm{mG}  = 0.1$ with \\added{$p =2$} and $K_p=1$ \\added{cm$^{-3}$}.\n\\added{The pair production rate is calculated based on \\cite{ka2008}.\nThe target photons are synchrotron radiation by primary electrons,\nthe spectra of which are shown in Figure \\ref{fig:1kpc-no-beaming-exp} \nfor various parameters.}\n\\added{Figure  \\ref{fig:pair-production-spec} and Table \\ref{table:prod-rate-2-18} below\ndo not include pair production via the decay of neutral pions.\nThe gamma-rays produced by the decay of neutral pions have energy comparable to\nthe energy of electrons\/positrons produced by charged pions.\nIn collisions with soft photons the gamma-rays produce electron-positron pairs, \nand these pairs contribute mainly to TeV emission.}\nAs we described in Section \\ref{sec:estimate}, the production spectrum\nby Bethe-Heitler process has a rather broad number spectrum \ncentered on $10^6<\\gamma_e<10^{10}$,\nwhile those through photo-pion production have a \\deleted{peaked spectrum}\n\\added{peak} at $\\gamma_e \\approx 10^{11}$-$10^{12}$.\nThe energy injection rate for both processes concentrates on the high energy ends,\nwith the Bethe-Heitler process being an order of magnitude larger than that for the\nphoto-pion production.\n\\added{For $\\gamma_e \\gtrsim 10^{11.5}$, \nphoto-pion processes dominate the Bethe-Heitler process.\nLeptons with $\\gamma_e \\gtrsim 10^{11.5}$ emit synchrotron radiation at \n$\\nu \\gtrsim 10^{25}$ Hz and do not contribute to X-ray emission that is our\ninterest in this paper.}\n\n\nThe resultant steady state electron\/positron energy spectrum is shown \nin Figure  \\ref{fig:lepton-energy-spec}\nfor $R_\\mathrm{kpc}=1$, $B_\\mathrm{mG}=0.1$, \\added{and $p=2$}.\nThe value of $K_p$ is taken as $8.8 \\times 10^{-3}$ \\added{cm$^{-3}$} to reproduce \nthe observed X-ray flux.\nThe proton power amounts to $5 \\times 10^{49} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$.\nAlthough this seems to be too large, it can be reduced by an order of magnitude \nif $\\gamma_{p, \\mathrm{min}}$ is large enough or if the spectral index of protons \nis less than 2, i.e., when the energy of relativistic protons is \nconcentrated in the high energy end.  \nWe tabulate in Table \\ref{table:prod-rate-2-18}\nthe pair production rates for $p = 2$ and 1.8.\n\n\nThe resultant photon spectra are shown in Figure  \\ref{fig:1kpc-no-beaming-exp}.\nAs is seen, while an overall spectral shape is well reproduced, \noptical flux at $5 \\times 10^{14}$ Hz tends to be overproduced.\nAt this frequency primary electrons and secondary pairs equally contribute and \nthe resultant combined flux is a factor of two higher than the observed one.\nThis is due to radiative cooling of secondary pairs and rather a general feature.\nFor a range of the magnetic field strength this feature persists in the present model.\nPossible way out from this problem is discussed later.\nWhen the magnetic field becomes smaller, inverse Compton scattering \nof radio-optical synchrotron photons by primary electrons (SSC) becomes larger.\nIf the magnetic field is as small as 20$\\mu$G, SSC can work as an X-ray emission \nmechanism. \nIn this case, however, the electron power is as large \nas $3 \\times 10^{47} \\, \\text{erg} \\,\\, \\text{s}^{-1}$ with a large deviation \nfrom equi-partition.\nFurthermore, excessive production of optical photons by IC is inevitable.\nThus, SSC model does not work.\n\nThe secondary pairs scatter the synchrotron photons emitted by the primary electrons.  \nThis IC component does not affect the emission below\n$\\sim 10^{24} \\, \\text{Hz}$ but the peak in $\\nu F_\\nu$ appears in the TeV band.\nThe IC component is not much affected by the magnetic field strength\nbecause the spectra of the  (radio-optical) soft photons and the secondary pairs are fixed.\nIt is \\deleted{, however,} to be noted that \nour model does not include the absorption of $\\gamma$-rays\nby extragalactic background light.\n\\added{The TeV bump will be hard to be observed by CTA,\nbecause the expected flux is lower than the lower limit of CTA.}\n\n\n\\floattable\n\\begin{deluxetable}{c|rrrrrr}\n  \\tablecaption{Pair Production Rate for $R = 1$ kpc and  $B = 0.1$ mG\n  \\label{table:prod-rate-2-18}}\n  \\tablewidth{0pt}\n  \\tablehead{\n    \\colhead{} & \\multicolumn{2}{c}{$p=2$} \n    & \\colhead{} &  \\multicolumn{3}{c}{$p=1.8$}\n    \\\\\n    \\cline{2-3}\n    \\cline{5-7}\n    \\colhead{} &  \\colhead{$\\dot{n}_\\pm$} &  \\colhead{$q_\\pm$}\n    & \n    \\colhead{}  &   \\colhead{} &  \\colhead{$\\dot{n}_\\pm$} &  \\colhead{$q_\\pm$}\n    \\\\\n    \\colhead{} & \\colhead{(cm$^{-3}$ s$^{-1}$)} & \\colhead{(erg cm$^{-3}$ s$^{-1}$)} \n    &\n    \\colhead{}  &\\colhead{}   & \\colhead{(cm$^{-3}$ s$^{-1}$)} & \\colhead{(erg cm$^{-3}$ s$^{-1}$)} \n  }\n  \\startdata\n  B-H &  $9.4 \\times 10^{-21}$ \\phn & $4.0 \\times 10^{-18}$  \\phn \n & &  \n  &  $4.8 \\times 10^{-19}$ \\phn & $3.2 \\times 10^{-16}$  \\phn \n  \\\\ \n  $e^-$ &  $2.6 \\times 10^{-25}$ \\phn & $5.4 \\times 10^{-20}$ \\phn \n  & & \n  &  $1.9 \\times 10^{-23}$ \\phn & $3.9 \\times 10^{-18}$ \\phn \n  \\\\ \n  $e^+$ & $2.4 \\times 10^{-24}$ \\phn & $5.5 \\times 10^{-19}$ \\phn \n  & &\n  &   $1.5 \\times 10^{-22}$ \\phn & $4.0 \\times 10^{-17}$ \\phn \n  \\\\ \n  \\enddata\n  \\tablecomments{\n    $\\dot{n}_\\pm$: the electron\/positron production rate per unit volume,\n    $q_\\pm$: the energy production rate per unit volume in electron\/positron production.\n    B-H: Bethe-Heitler pair production, $e^-$ and $e^+$: photo-pion processes.\n    $K_p = 1$ \\added{cm$^{-3}$}, $\\gamma_{p, \\mathrm{min}}=1$,\n    and $\\gamma_{p, \\mathrm{max}} = 10^{10}$ are assumed.\n    All the values in the table are proportional to $K_p$.\n    }\n\\end{deluxetable}\n\n\n\\begin{figure}[t!]\n  \\centering\\includegraphics[scale=0.5,clip]{figure3.eps}\n  \\caption{The emission spectrum of PKS 0637-0752.\n    The black filled circles and crosses are data taken from \\cite{meyer2015} \n    and the crosses show the \\textit{Fermi} upper limit.\n    Models are calculated for $R =1$ kpc.\n    \\added{Solid lines are synchrotron radiation and SSC by primary electrons.\n      The bumps above $\\sim 10^{15}$ - $10^{16}$ Hz of the solid lines are SSC components.}\n    The blue solid line is for $B=0.05$ mG, the black solid line is \n    for $B=0.1$ mG, and the magenta solid line for $B=0.2$ mG.\n    \\added{Dashed and dot-dashed lines are emission by secondary pairs.}\n    The \\deleted{black} dashed line is for $p=2$ and $K_p = 8.8\\times 10^{-3}$ \n    \\added{cm$^{-3}$}\n    and the \\deleted{black} dot-dashed line is for $p=1.8$ and $K_p = 1.3 \\times 10^{-4}$\n    \\added{cm$^{-3}$}.\n    These spectra \\added{(dashed and dot-dashed)} are calculated with $B=0.1 \\,\\text{mG}$.\n    The bumps above $\\sim 10^{26} \\, \\text{Hz}$ are produced by inverse Compton scattering \n    of radio-optical photons off the secondary pairs produced by the Bethe-Heitler process.\n    \\label{fig:1kpc-no-beaming-exp}}\n\\end{figure}\n\n\nWe also examined a smaller size of $R_\\mathrm{kpc}=0.1$ and applied to \nthe knot WK8.9 as shown in Figure \\ref{fig:01kpc-spec}.\nWhen the magnetic field is $B_\\mathrm{mG}=0.3$, $K_p=0.18$ \\added{cm$^{-3}$} can \nreproduce the radio-optical and X-ray spectra, although \noverproduction of optical photons  still persists. \nThe proton power is around $10^{49} \\, \\, \\text{erg} \\,\\, \\text{s}^{-1}$.\n\nFor a larger value of $R$, the required proton power increases; for example \nfor $R_\\mathrm{kpc} = 5$, $K_p =10^{-3}$ \\added{cm$^{-3}$} is needed amounting to \n$L_p =10^{50} \\,\\, \\text{erg}\\,\\, \\text{s}^{-1}$.\nThus, as for the emission region size, a small radius is favored in the energetics of protons. \nThese numerical results are consistent with a rather simple and optimistic estimate made \nin the previous section.\nThe predicted photon spectra show a roll-over at 10 MeV-GeV range \nand are compatible with the reported \\textit{Fermi} upper limits.\nSince the spectra are rather flat, the real problem is in the low energy \nend of the synchrotron emission. \nWe mostly skipped the contribution from electrons\/positrons of photo-pion origin.\nThey contribute mostly in the TeV range\nwith roughly similar luminosity to X-rays so that they do not affect the X-ray spectrum \nand the \\textit{Fermi} upper limit.\nWhen the maximum energy of protons is not so large, this component \ncan be totally ignored. \n\n\nSince the overproduction of optical flux is rather general, \nwe consider the reduction of the emission from the primary electrons.\nIn the above models we assumed the injection spectrum with the exponential cutoff.\nWhen we assume the power-law  injection spectrum without the exponential cutoff,\nthe optical emission is mainly from the secondary pairs.\nOur numerical result is shown in Figure \\ref{fig:1kpc-no-exp-cutoff}.\nSuch an abrupt super-exponential cutoff of primary electron \nenergy distribution may not be unlikely,\nwhen the acceleration is limited by cooling \\citep{krm98}.\n\nAn alternative idea to reduce the overproduction of the optical flux is \ntaking into account a mild relativistic beaming.\nIn this case, the photo-hadronic rates become less frequent\nby a factor of $\\delta^{3.75}$ due to the beaming effects for the same value \nof $\\gamma_p$,\n\\added{where $\\delta$ is the beaming factor.}\n\\added{(The scaling of $\\delta^{3.75}$ is obtained as follows.\nThe observed frequency $\\nu_\\mathrm{obs}$ and the frequency in the source \n$\\nu_s$ are related by $\\nu_\\mathrm{obs} = \\delta \\nu_s$. \nThe photon density at $\\nu_\\mathrm{obs}$ is given by\n$\\nu_\\mathrm{obs} n_{\\nu_\\mathrm{obs}} = \\delta^3 \\nu_s n_{\\nu_s}\n= A \\nu_\\mathrm{obs}^{-0.75} = A \\delta^{-0.75} \\nu_s^{-0.75}$, where $A$ is a constant.\nThen $\\nu_s n_{\\nu_s} = A \\delta^{-3.75} \\nu_s^{-0.75}$ follows.)\n}\nHowever, the source frame X-ray luminosity also decreases by a factor of $\\delta^4$, \nso that  the required proton power in the source frame does not much change,\n\\added{i.e., proportional to $\\delta^{0.25}$.}\nThe required kinetic power of protons increases by a factor of $\\delta^2$,\nif we set the bulk Lorentz factor $\\Gamma$ of the knot equal to $\\delta$.\nThe results for  $R_\\mathrm{kpc}=1$ and $B_\\mathrm{mG}=0.1$ are shown \nin Figure \\ref{fig:beaming-spec} for $\\delta = 3$.\nAs is seen, the overproduction of the optical flux can be \navoided in this case as well.\nThe appropriate value of $K_p=10^{-2}$ \\added{cm$^{-3}$} is similar to the non-beamed case,\nso that the required proton power becomes as high as \n$10^{51} \\,\\, \\text{erg} \\,\\, \\text{s}^{-1}$, which seems to be unlikely.\nHowever, we note that milder value of $\\delta$ may reproduce the observed spectra , \nif the uncertainty of the ultraviolet flux exists by a factor of 1.5 or so.\n\n\n\n\\begin{figure}[ht!]     \n  \\centering\\includegraphics[scale=0.5,clip]{figure4.eps}\n  \\caption{Emission spectrum of the knot WK8.9 (the data are shown by filled green\n    circles taken from \\cite{meyer2015} ).\n    The size of the emission region is assumed to be $0.1$ kpc.\n    The red lines \\added{(solid and dashed)} are for $B = 0.3$ mG, \n    \\added{and} the black lines \\added{(solid and dashed)} are for $B=0.4$ mG.\n    \\deleted{and the green dot-dashed line is for $B=1$ mG.}\n    \\added{For $B = 1$ mG, only synchrotron radiation and SSC by primary electrons is shown\n      by the green dot-dashed line.}\n    To calculate the emission from pairs produced by Bethe-Heitler process, \n    $K_p = 0.18$ and 0.2 \\added{cm$^{-3}$} are assumed for $B=0.3$ and 0.4 mG, respectively.\n    \\label{fig:01kpc-spec}}\n\\end{figure}\n\n\n\\begin{figure}[ht!]     \n  \\centering\\includegraphics[scale=0.5,clip]{figure5.eps}\n  \\caption{\n    The injection of primary electrons without exponential cutoff\n    is assumed for $R = 1$ kpc.\n    The black line is for $B=0.1$ mG, the blue line is for $B=0.2$ mG, and the magenta line \n    is for  $B = 0.3$ mG.\n    The red line is the emission from pairs produced by Bethe-Heitler pair production for\n    $B= 0.1$ mG and $K_p=8 \\times 10^{-3}$ \\added{cm$^{-3}$}.\n    \\label{fig:1kpc-no-exp-cutoff}}\n\\end{figure}\n\n\n\\begin{figure}  [ht!]         \n  \\centering\\includegraphics[scale=0.5,clip]{figure6.eps}\n  \\caption{Beaming factor $\\delta = 3$ is assumed for $R =1$ kpc and $B = 0.1$ mG.\n    To calculate X-ray emission, $K_p = 1.1 \\times 10^{-2}$ \\added{cm$^{-3}$} is assumed.\n    \\label{fig:beaming-spec}}\n\\end{figure}\n\n\n \n\\section{Conclusion}  \\label{sec:conclusion}\n\nWe examined the  photo-hadronic  model of the X-ray emission from large scale jets \nof radio loud quasars specifically PKS 0637-752 and have found that Bethe-Heitler process is \neffective for the high energy electron\/positron injection. \nElectrons and positrons from photo-pion production  mainly radiate\nmulti-TeV photons by synchrotron radiation and do not much contribute to X-ray emission.\nFor an appropriate choice of parameters such as $R=1$ kpc and $B=0.1$ mG, \nthe required proton power is an order of $10^{49} \\, \\text{erg} \\,\\,  \\text{s}^{-1}$,\n\\added{which is $\\sim 120 L_\\mathrm{Edd}$ for $M_\\mathrm{BH} = 6.5 \\times 10^8 M_\\sun$},\nwhen the energy density of protons is concentrated in the region \nof $\\gamma_p =10^{9}$-$10^{10}$. \nCooling tail of these electrons and positrons radiate optical synchrotron emission,\nseparate from the primary electrons. \nTo avoid the overproduction of optical-ultraviolet flux,\neither the energy distribution of the primary electrons has a super-exponential  cutoff or \na mild degree of the relativistic beaming effect ($\\delta \\sim 3$) appears.\nFor the latter case, the required proton power tends to be large.\nThe Poynting and primary electron powers remain moderate. \nBecause the value of the beaming factor is not strongly constrained\n\\citep[e.g.,][]{meyer2015}, further work is needed to determine\nwhich mechanism is applicable to reduce the optical synchrotron emission.\n\nProton synchrotron radiation is a few orders of magnitude smaller than \nthe photo-hadronic model prediction.\nThus, photo-proton model is an alternative option to \nexplain the strong X-ray emission from large scale jets. \n\n\n\\added{The black hole mass of PKS0627-752 given by \\cite{ljg2006} is \n$6.5 \\times 10^8 M_\\odot$.  \nOn the other hand,  \\cite{gcj2001} gives $7.8 \\times 10^9 M_\\odot$.\nThe Eddington luminosity is \n$\\sim 8.2 \\times 10^{46}$ erg s$^{-1}$ and $\\sim 9.8 \\times 10^{47}$ erg s$^{-1}$ \nfor $M_\\mathrm{BH} = 6.5 \\times 10^8 M_\\odot$ and $7.8 \\times 10^9 M_\\odot$,\nrespectively.\nThus $10^{49}$ erg s$^{-1}$ is $\\sim 10$ - $100$ times larger than the Eddington \nluminosity.\nAs for other AGNs, very luminous AGNs have been observed.\nFor example, \\cite{gfv2009} showed that S5 0014+813 has\n$\\nu L_\\nu \\sim 10^{48}$ erg s$^{-1}$ in the optical and this corresponds to \n$0.17 L_\\mathrm{Edd}$ for $M_\\mathrm{BH} = 4 \\times 10^{10} M_\\sun$.\nSome authors, e.g., \\cite{sn2016}, on the other hand,  \nperformed radiation magnetohydrodynamical simulation of super-Eddington mass accretion.\nIn view of uncertainties of mass estimation and theoretical possibility \nof super-Eddington jet power, we believe our model is still viable, \nalthough the required proton power is very large.\n}\n\nFinally, our model predicts TeV emission by inverse Compton scattering of radio-optical\nphotons off pairs produced by the Bethe-Heitler process.\nEmission by electrons\/positrons produced by photo-pion processes will also \ncontribute to TeV emission.\n\n\n\n\\acknowledgments\n\\added{We are grateful to the referee for useful comments that improved the manuscript \nconsiderably. }\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nEntanglement shared among multiple parties is acknowledged as one of the fundamental resources driving the second quantum revolution~\\cite{DowlingMilburn2003}, for instance, as a basis of quantum network proposals~\\cite{EppingKampermannMacchiavelloBruss2017, PivoluskaHuberMalik2018, RibeiroMurtaWehner2018, BaeumlAzuma2017}, as a key resource for improved quantum sensing~\\cite{Toth2012} and quantum error correction~\\cite{Scott2004} or as generic ingredient in quantum algorithms~\\cite{BrussMacchiavello2011} and measurement-based quantum computation~\\cite{RaussendorfBriegel2001, BriegelRaussendorf2001}. Yet, its detection and characterisation are complicated by several factors: among them, the computational hardness of deciding whether any given system even exhibits any entanglement at all~\\cite{Gurvits2004} as well as the fact that the usual paradigm of local operations and classical communication (LOCC) lead to infinitely many types of entanglement~\\cite{VerstraeteDehaeneDeMoorVerschelde2002, OsterlohSiewert2005, DeVicenteSpeeKraus2013, SchwaigerSauerweinCuquetDeVicenteKraus2015, DeVicenteSpeeSauerweinKraus2017, SpeeDeVicenteSauerweinKraus2017, SauerweinWallachGourKraus2018} already for single copies of multipartite states. Significant effort has thus been devoted to devising practical means of entanglement certification from limited experimental data~\\cite{TothGuehne2005b, FriisVitaglianoMalikHuber2019}.\n\n\nOne of the principal challenges for the characterisation of multipartite entanglement lies in distinguishing between \\emph{partial separability} and its counterpart, \\emph{genuine multipartite entanglement} (GME)\\footnote{Note that the term was also coined for multipartite pure states with exclusively non-vanishing $n$-tangle in Ref.~\\cite{OsterlohSiewert2005}.}.\nHere, a multipartite state is called \\emph{partially separable} if it can be decomposed as a mixture of \\emph{partition-separable} states, i.e., of states separable with respect to some (potentially different) partitions of the parties into two or more groups, whereas any state that cannot be decomposed in this way has GME (see Fig.~\\ref{fig:GME structure} and Table~\\ref{tab:term}). One may further classify partially separable states as $k$-separable states according to the maximal number $k$ of tensor factors that all terms in the partially separable decomposition can be factorised into. If a state admits a decomposition where each term is composed of at least two tensor factors ($k=2$), the state is called \\emph{biseparable}. Thus, every partially separable state is $k$-separable for some $k\\geq2$, and hence (at least) biseparable.\nThis distinction arises naturally when considering the resources required to create a specific state:\nany biseparable state can be produced via LOCC in setups where all parties share classical randomness and subsets of parties share entangled states.\nOne of the counter-intuitive features of partially separable states is the possibility for bipartite entanglement across every possible bipartition\\footnote{An explicit example of a $k\/2$-separable (and thus biseparable) $k$-qubit state (for even $k$) with the bipartite entanglement between all neighbouring qubits in a linear arrangement can be found in~\\cite[footnote 30]{FriisMartyEtal2018}.}.\nConsequently, the notion of bipartite entanglement across partitions is insufficient to capture the notion of partial separability, and conventional methods, such as positive maps~\\cite{HorodeckiMPR1996, Peres1996}, cannot be straightforwardly applied to reveal GME (with new concepts for positive maps derived for that purpose in~\\cite{HuberSengupta2014, ClivazHuberLamiMurta2017}), which results in additional challenges compared to the \\textemdash~relatively \\textemdash~simpler scenario of detecting bipartite or partition entanglement (e.g., as in~\\cite{RodriguezBlancoBermudezMuellerShahandeh2021}).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{flower2.png}\n\\vspace*{-8mm}\n\\caption{\\textbf{GME and (partial) separability for three qubits}. All three-qubit states separable with respect to~one of the three bipartitions, $\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}$ (yellow), $\\mathcal{A}_{2}|\\mathcal{A}_{1}\\mathcal{A}_{3}$ (darker green), and $\\mathcal{A}_{3}|\\mathcal{A}_{1}\\mathcal{A}_{2}$ (background), form convex sets, whose intersection (turquoise) contains (but is not limited to) all fully separable states $\\mathcal{A}_{1}|\\mathcal{A}_{2}|\\mathcal{A}_{3}$ (dark blue). The convex hull of these partition-separable states contains all partially separable (the same as biseparable for tripartite systems) states. All states that are not biseparable are GME\\@. States with $k$-copy activatable GME are contained in the set of biseparable but not partition-separable states and are conjectured to form the lighter green areas, with those states for which GME is activatable for higher values of $k$ farther away from the border between GME and biseparability.\nThe horizontal line represents the family of isotropic GHZ states $\\rho(p)$, containing the maximally mixed state ($p=0$) and the GHZ state ($p=1$). The values $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}$ indicate $k$-copy GME activation thresholds, which we discuss in the following.}\n\\label{fig:GME structure}\n\\end{figure}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{GME_activation_3b.pdf}\n\\vspace*{-4mm}\n\\caption{\\textbf{Activation of GME from biseparable states}. (a) Separable bipartite states remain separable, no matter how many copies are shared, e.g., if $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1}\\mathcal{A}_{2}}$ and $\\rho\\subtiny{0}{0}{\\mathcal{B}_{1}\\mathcal{B}_{2}}$ are separable with respect to~the bipartitions $\\mathcal{A}_{1}|\\mathcal{A}_{2}$ and $\\mathcal{B}_{1}|\\mathcal{B}_{2}$, then so is  $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1}\\mathcal{A}_{2}}\\otimes\\rho\\subtiny{0}{0}{\\mathcal{B}_{1}\\mathcal{B}_{2}}$. (b) In contrast, the joint state of multiple copies of biseparable states, e.g., $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{N}}$, $\\rho\\subtiny{0}{0}{\\mathcal{B}_{1},\\mathcal{B}_{2},\\ldots,\\mathcal{B}_{N}}$, and $\\rho\\subtiny{0}{0}{\\mathcal{C}_{1},\\mathcal{C}_{2},\\ldots,\\mathcal{C}_{N}}$, can be GME with respect to~the partition $\\mathcal{A}_{1}\\mathcal{B}_{1}\\mathcal{C}_{1}|\\mathcal{A}_{2}\\mathcal{B}_{2}\\mathcal{C}_{2}|\\ldots|\\mathcal{A}_{\\!N}\\mathcal{B}_{\\!N}\\mathcal{C}_{\\!N}$.\n\\label{fig:GME activation}\n}\n\\end{figure}\n\n\\begin{table*}\n\\centering\n\\caption{\\label{tab:term}Summary of terminology on GME in this paper.}\n\\begin{tabular}{@{}ll@{}}\n\\toprule\nTerm & Meaning\\\\\n\\midrule\n$k$-separable &\\parbox{12cm}{convex combination of pure states, each of which is a product of at least $k$ projectors}\\\\\nbiseparable & synonymous with $2$-separable\\\\\npartially separable& $k$-separable for some $k>1$\\\\\npartition-separable & \\parbox{12cm}{separable for a specific partition of the multipartite Hilbert space, i.e., a convex combination of projectors, each of which is a product with respect to the same partition into subsystems}\\\\\nmultipartite entangled & entangled across all bipartitions\\\\\ngenuine multipartite entangled& non-biseparable\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table*}\n\nAn assumption inherent in the definitions above is that all parties locally act only on a single copy of the distributed state. \nHowever, in many experiments where quantum states are distributed among (potentially distant) parties, multiple independent but identically prepared copies of states are (or at least, can be) shared. For instance, exceptionally high visibilities of photonic states can only be achieved if each detection event stems from almost identical quantum states~\\cite{JoshieEtAl2020,WengerowskyJoshiSteinlechnerHuebelUrsin2018}. Adding noise to the channel then produces the situation we focus on in this article: multiple copies of noisy quantum states produced in a laboratory~\\cite{Ecker-Huber2019,HuEtAl2020}.\nEven limited access to quantum memories or signal delays then allows one to act on multiple copies of the distributed states, which is a recurring theme also in research on quantum networks~\\cite{YamasakiPirkerMuraoDuerKraus2018,NavascuesWolfeRossetPozasKerstjens2020, KraftDesignolleRitzBrunnerGuehneHuber2021}.\nCharacterising properties of GME in multi-copy scenarios is thus not only of fundamental theoretical interest but also crucial for practical applications that require GME to be distributed, such as conference key agreement~\\cite{MurtaGrasselliKampermannBruss2020}.\n\nHowever, we demonstrate here that, unlike the distinction between separable and entangled states, the distinction between biseparability and GME is not maintained in the transition from one to many copies; i.e., partial separability is not a tensor-stable concept.\nAs we show, for $N$ parties $1,\\ldots,N$, there exist multipartite quantum states $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{N}}$ that are biseparable, but which can be \\emph{activated} in the sense that sharing two copies results in a GME state, i.e., such that the joint state $\\rho\\subtiny{0}{0}{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{N}}\\otimes \\rho\\subtiny{0}{0}{\\mathcal{B}_{1},\\mathcal{B}_{2},\\ldots,\\mathcal{B}_{N}}$ of two identical copies (labelled $\\mathcal{A}$ and $\\mathcal{B}$, respectively) is not biseparable with respect to the partition $\\mathcal{A}_{1}\\mathcal{B}_{1}|\\mathcal{A}_{2}\\mathcal{B}_{2}|\\ldots|\\mathcal{A}_{N}\\mathcal{B}_{N}$. (See Fig.~\\ref{fig:GME activation}.)\nThat such activation of GME is in principle possible had previously only been noted in~\\cite{HuberPlesch2011}, where it was observed that two copies of a particular four-qubit state that is itself almost fully separable can become GME\\@. \nHere, we systematically investigate this phenomenon of \\emph{multi-copy GME activation}. As the first main result, we show that the property of biseparability is not tensor stable in general by identifying a family of $N$-qubit isotropic Greenberger-Horne-Zeilinger (GHZ) states with two-copy activatable GME for all $N$.\nWe further demonstrate the existence of biseparable states within this family for which two copies are not enough to activate GME, but three copies are.\nMoreover, we show that the bound for partition-separability coincides with the asymptotic (in terms of the number of copies) GME-activation bound for isotropic GHZ states.\n\nMulti-copy GME activation is particularly remarkable \\textemdash~and may appear surprising at first \\textemdash~because it is in stark contrast to bipartite entanglement:\nTwo copies of states separable with respect to a fixed partition always remain partition-separable and can never become GME\\@.\nHowever, from the perspective of entanglement distillation \\textemdash~the concentration of entanglement from many weakly entangled (copies of) states to few strongly entangled ones \\textemdash~such an activation seems more natural.\nAfter all, if one party shares bipartite maximally entangled states with each other party, these could be used to establish any GME state among all $N$ parties via standard teleportation, thus distributing GME by sharing only two-party entangled states.\nNevertheless, such a procedure would require at least $N-1$ copies of these bipartite entangled states (in addition to a local copy of the GME state to be distributed), and already the example from~\\cite{HuberPlesch2011} suggests that one does not have to go through first distilling bipartite entangled pairs, followed by teleportation, but two copies can naturally feature GME already.\nWhile we have seen that the phenomenon of GME activation is more than just distillation, one may still be tempted to think that distillable entanglement is required for GME activation.\nIt is known that there exist bound entangled states \\textemdash~entangled states that do not admit distillation of entanglement no matter how many copies are provided.\nIn particular, all entangled states with positive partial transpose (PPT) across a given cut are undistillable since any number of copies is also PPT\\@.\nOne might thus suspect that GME activation should not be possible for biseparable states that are PPT across every cut and hence have no distillable entanglement (even if multiple parties are allowed to collaborate).\nAs another main result, we show that this is not the case by constructing a biseparable state that is PPT with respect to~every cut, yet two copies of the state are indeed GME\\@. \nTogether, our results thus support the following conjectures:\n\\begin{enumerate}[(i)]\n\\item{\\label{conjecture i}\nThere exists a hierarchy of states with $k$-copy activatable GME, i.e., for all $k\\geq2$ there exists a biseparable but not partition-separable state $\\rho$ such that $\\rho^{\\otimes k-1}$ is biseparable, but $\\rho^{\\otimes k}$ is GME\\@.\n}\n\\item{\\label{conjecture ii}\nGME may be activated for any biseparable but not partition-separable state (light green areas in Fig.~\\ref{fig:GME structure}) of any number of parties as $k\\rightarrow\\infty$.}\n\\end{enumerate}\n\nIn the following, we first provide the formal definitions for biseparability and GME in Sec.~\\ref{sec:sep and gme} before turning to the family of $N$-qubit isotropic GHZ states in Sec.~\\ref{sec:GME of isotropic GHZ states}. For all biseparable states in this family, we provide upper bounds on the minimal number of copies required to activate GME in Sec.~\\ref{sec:Multi-copy GME criterion}. In Sec.~\\ref{sec:Hierarchy of k-copy activatable states}, we then consider the case of three qubits ($N=3$), for which we can show that the bound on three-copy GME activation is tight in the sense that we identify all states in the family for which one requires at least three copies to activate GME, while two copies remain biseparable, and can also show that GME can indeed be activated for any biseparable but not partition-separable state in this family. Moreover, in Sec.~\\ref{sec:GME activation of PPT entangled states}, we construct an explicit example for two-copy GME activation from biseparable states with no distillable bipartite entanglement. Finally, we discuss the implications of our results and open questions in Sec.~\\ref{sec:Conclusion and Outlook}.\n\n\n\\section{Definitions of biseparability \\& GME}\\label{sec:sep and gme}\nWe summarise the formal definitions of biseparability and GME in this paper.\n(See also Table~\\ref{tab:term} for the summary of the definitions here.)\nFormally, a pure quantum state of an $N$-partite system with Hilbert space $\\mathcal{H}\\suptiny{0}{0}{(N)}=\\bigotimes_{i=1}^{N}\\mathcal{H}_{i}$ is separable with respect to~a $k$-partition $\\{\\mathcal{A}_{1},\\mathcal{A}_{2},\\ldots,\\mathcal{A}_{k}\\}$, with $\\mathcal{A}_{i}\\subset\n\\{1,2,3,\\ldots,N\\}$ and $\\bigcup_{i=1}^{k} \\mathcal{A}_{i}= \\{1,2,3,\\ldots,N\\}$ such that $\\bigotimes_{i=1}^{k}\\mathcal{H}_{\\mathcal{A}_{i}}=\\mathcal{H}\\suptiny{0}{0}{(N)}$, if it can be written as\n\\begin{align}\n    \\ket{\\Phi\\suptiny{0}{0}{(k)}}  &=\\,\\bigotimes\\limits_{i=1}^{k}\\,\\ket{\\phi_{\\mathcal{A}_{i}}},\\quad\\ket{\\phi_{\\mathcal{A}_{i}}}\\in\\mathcal{H}_{\\mathcal{A}_{i}}\n    \\,.\\label{pure}\n\\end{align}\nWhen generalising to density matrices, it is common not to specify all possible partitions, but to use the notion of \\emph{$k$-separability} instead: \nA density operator is called \\emph{$k$-separable} if it can be decomposed as a convex sum of pure states that are all separable with respect to~\\emph{some} $k$-partition, i.e., if it is of the form (see, e.g., the review~\\cite{FriisVitaglianoMalikHuber2019})\n\\begin{align}\n    \\rho\\suptiny{0}{0}{(k)} &=\\,\n    \\sum\\limits_{i} p_{i} \n    \\ket{\\Phi_i\\suptiny{0}{0}{(k)}}\\!\\!\\bra{\\Phi_i\\suptiny{0}{0}{(k)}}\n    \\,.\\label{ksep}\n\\end{align}\nNote that the lack of tensor stability of partial separability shown in the following also implies that the related concept of $k$-producibility~\\cite{GuehneToth2009,Szalay2019} is not tensor stable. Crucially, each $\\ket{\\Phi_i\\suptiny{0}{0}{(k)}}$ may be $k$-separable with respect to~a different $k$-partition. Consequently, $k$-separability does not imply separability of $\\rho\\suptiny{0}{0}{(k)}$ with respect to~a specific partition, except when $\\rho\\suptiny{0}{0}{(k)}$ is a pure state or when $k=N$. In the latter case the state is called \\emph{fully separable}.\nTo make this distinction more explicit, we refer to all (at least) biseparable states that are actually separable with respect to~some bipartition as \\emph{partition-separable}.\nAt the other end of this separability spectrum one encounters \\emph{biseparable states} ($k=2$), while all states that are not at least biseparable (formally, $k=1$) are called \\emph{genuinely $N$-partite entangled}. We will here use the term GME for the case $k=1$.\nThe operational reason for this definition of GME is easily explained: any biseparable state of the form of Eq.~(\\ref{ksep}) can be created by $N$ parties purely by sharing partition-separable states of the form of Eq.~(\\ref{pure}) and some classical randomness. \nIn addition, this conveniently results in a convex notion of biseparability (as illustrated for the example in Fig.~\\ref{fig:GME structure}) amenable to entanglement witness techniques, which inherently rely on convexity.\n\n\n\\section{GME of isotropic GHZ states}\\label{sec:GME of isotropic GHZ states}\nTo overcome the difficulty in analysing GME, the crucial technique here is to use states in $X$-form, i.e., those with nonzero entries of density operators only on the main diagonal and main anti-diagonal with respect to~the computational basis.\nLet us now consider a family of mixed $N$-qubit states, \\emph{isotropic GHZ states}, given by\n\\begin{align}\n    \\rho(p) &=\\,p\\,\\ket{\\mathrm{GHZ}_{N}\\!}\\!\\!\\bra{\\mathrm{GHZ}_{N}\\!}\\,+\\,(1-p)\\,\\tfrac{1}{2^{N}}\\mathds{1}_{2^{N}}\\,,\n    \\label{eq:GHZ with white noise}\n\\end{align}\nobtained as convex combination of the $N$-qubit maximally mixed state $\\tfrac{1}{2^{N}}\\mathds{1}_{2^{N}}$ and a pure \n$N$-qubit GHZ state\n\\begin{align}\n    \\ket{\\mathrm{GHZ}_{N}\\!}  &=\\,\\tfrac{1}{\\sqrt{2}}\\bigl(\\ket{0}^{\\otimes N}+\\ket{1}^{\\otimes N}\\bigr).\n\\end{align}\nwith real mixing parameter $p\\in[-1\/(2^{N}-1),1]$.\nSince states in this family are in $X$-form with respect to~the $N$-qubit computational basis, we can straightforwardly calculate the \\emph{genuine multipartite} (GM) \\emph{concurrence}, an entanglement measure for a multipartite state defined in terms of a polynomial of elements of its density matrix~\\cite{HashemiRafsanjaniHuberBroadbentEberly2012,MaChenChenSpenglerGabrielHuber2011}. For any $N$-qubit density operator $\\rho_{X}$ in $X$-form, i.e., \n\\begin{align}\n    \\rho_{X}=\\begin{pmatrix} \\tilde{a} & \\tilde{z}\\,\\tilde{d} \\\\ \\tilde{d}\\,\\tilde{z}^{\\dagger} & \\tilde{d}\\,\\tilde{b}\\,\\tilde{d} \\end{pmatrix},\n\\end{align}\nwhere $\\tilde{a}=\\diag\\{a_{1},\\ldots,a_{n}\\}$,  $\\tilde{b}=\\diag\\{b_{1},\\ldots,b_{n}\\}$, and $\\tilde{z}=\\diag\\{z_{1},\\ldots,z_{n}\\}$ are diagonal $n\\times n$ matrices with $n=2^{N-1}$, $a_{i},b_{i}\\in\\mathbb{R}$ and $z_{i}\\in\\mathbb{C}$ for all $i=1,2,\\ldots,n$, and $\\tilde{d}=\\operatorname{antidiag}\\{1,1,\\ldots,1\\}$ is antidiagonal,\nthe GM concurrence is given by\n\\begin{align}\n    C_{\\mathrm{GM}}(\\rho_{X}) &=\\,2\\max\\bigl\\{0,\\max_{i}\\{|z_{i}|-\\sum\\limits_{j\\neq i}^{n}\\sqrt{a_{j}b_{j}}\\}\\bigr\\},\n    \\label{eq:GM concurrence}\n\\end{align}\nand provides a necessary and sufficient condition for GME whenever $C_{\\mathrm{GM}}>0$. \nIn the case of the state $\\rho(p)$ from Eq.~(\\ref{eq:GHZ with white noise}), we have $a_{i}=b_{i}=\\tfrac{1-p}{2^{N}}+\\delta_{i1}\\tfrac{p}{2}$ and $z_{i}=\\delta_{i1}\\tfrac{p}{2}$, such that\n\\begin{align}\n    C_{\\mathrm{GM}}\\bigl[\\rho(p)\\bigr] &=\\,\\max\\{0,|p|-(1-p)(1-2^{1-N})\\}.\n\\end{align}\nThus, $\\rho(p)$ is GME if and only if\n\\begin{align}\n    p   &>\\,p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}(N)\\,\\coloneqq\\,\\frac{2^{N-1}-1}{2^{N}-1}\\,,\n\\end{align}\ni.e., if and only if $p$ surpasses the single-copy threshold $p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}$.\nConversely, we can be certain that $\\rho(p)$ is not GME for $p\\leq (2^{N-1}-1)\/(2^{N}-1)$, and hence at least biseparable.\n\n\n\\section{Multi-copy GME criterion}\\label{sec:Multi-copy GME criterion}\nOur first goal is then to check if two copies of $\\rho(p)$ are GME\\@.\nSince the GM concurrence is an entanglement monotone, $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes k}\\bigr]$ is monotonically non-decreasing as $k$ increases~\\cite{MaChenChenSpenglerGabrielHuber2011};\nthat is, if we have $C_\\mathrm{GM}\\bigl[\\rho(p)\\bigr]=0$ for $\\rho(p)$ in $X$-form, it holds that $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes 2}\\bigr]\\geq 0$ in general.\nHowever, \nusing $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes 2}\\bigr]> 0$\nas a necessary and sufficient criterion for GME \nis not an option in this case, \nsince ${\\rho(p)}^{\\otimes 2}$ may not be of $X$-form even if a single copy is, and we therefore generally cannot directly calculate $C_\\mathrm{GM}\\bigl[\\rho(p)^{\\otimes 2}\\bigr]$. \nThe crucial idea here is to make use of the fact that stochastic LOCC (SLOCC) can never create GME from a biseparable state.\n\nTo construct a sufficient GME criterion,\nwe therefore use a map $\\mathcal{E}_{\\circ}$ implementable via SLOCC~\\cite{LamiHuber2016}, which, for any two density operators $\\rho$ and $\\sigma$ acting on $\\mathcal{H}$,\nmaps the state $\\rho\\otimes\\sigma$  acting on $\\mathcal{H}^{\\otimes 2}$ to\n\\begin{align}\n    \\mathcal{E}_{\\circ}[\\rho\\otimes\\sigma] &=\\,\\frac{\\rho\\circ\\sigma}{\\textnormal{Tr}(\\rho\\circ\\sigma)}\\quad\\text{on }\\mathcal{H},\n\\end{align}\nwhere the right-hand side is a density operator acting on $\\mathcal{H}$, and ``$\\circ$'' denotes the Hadamard product (or Schur product), i.e., the component-wise multiplication of the two matrices.\nWhat is useful for us here is that the Hadamard product of two $X$-form matrices results in an $X$-form matrix. Consequently, we can directly calculate the GM concurrence for the state resulting from applying the `\\emph{Hadamard-product map}' $\\mathcal{E}_{\\circ}$ to two copies of an originally biseparable state.\nIf the GM concurrence of $\\mathcal{E}_{\\circ}[\\rho(p)^{\\otimes 2}]$ is nonzero, we can conclude that two copies of $\\rho(p)$ are GME, even if a single copy is not.\nTo decide whether $\\mathcal{E}_{\\circ}[\\rho(p)^{\\otimes 2}]$ is GME or not, i.e., whether the GM concurrence is nonzero or not, we can ignore the normalization and just consider $\\rho(p)\\circ\\rho(p)=\\rho(p)^{\\circ 2}$. Moreover, in the maximization over the index $i$ in Eq.~(\\ref{eq:GM concurrence}), the maximum is obtained for $i=1$. We can thus conclude that $\\rho(p)^{\\otimes 2}$ is GME if\n\\begin{align}\n    |z_{1}^{2}|-\\sum\\limits_{j\\neq 1}^{n}\\sqrt{a_{j}^{2}b_{j}^{2}}\\,=\\,\\tfrac{p^{2}}{4}-(2^{N-1}-1)\\bigl(\\tfrac{1-p}{2^{N}}\\bigr)^{2}\\,>\\,0,\n    \\label{eq:nonzero GM concurrence 2 copies}\n\\end{align}\nwhich translates to the condition $p\/(1-p)>\\sqrt{2^{N-1}-1}\/2^{N-1}$, and in turn can be reformulated to the condition\n\\vspace*{-2mm}\n\\begin{align}\n    p   &>\\,p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}(N)\\,\\coloneqq\\,\\frac{\\sqrt{2^{N-1}-1}}{2^{N-1}+\\sqrt{2^{N-1}-1}}.\n    \\label{eq:GME treshhold 2 copies}\n\\end{align}\nAs we see, we have $p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}>p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}$ for all $N\\geq3$, confirming that \\emph{there exist biseparable states} with values $p<p\\suptiny{0}{0}{(1)}_{\\mathrm{GME}}$ \\emph{for which two copies are GME}, i.e., such that $p>p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}$. \n\nMoreover, we can now concatenate multiple uses of the SLOCC map $\\mathcal{E}_{\\circ}$. For instance, we can identify the threshold value $p\\suptiny{0}{0}{(3)}_{\\mathrm{GME}}$ of $p$ at which the state $\\mathcal{E}_{\\circ}\\bigl[\\rho(p)\\otimes\\mathcal{E}_{\\circ}[\\rho(p)^{\\otimes 2}]\\bigr]$ resulting from $2$ applications of $\\mathcal{E}_{\\circ}$ to a total of $3$ copies of $\\rho(p)$ is GME, or, more generally, the corresponding threshold value $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}$ for which $k$ copies result in a GME state after applying the map $\\mathcal{E}_{\\circ}$ a total of $k-1$ times. From Eq.~(\\ref{eq:nonzero GM concurrence 2 copies}) it is easy to see that these threshold values are obtained as\n\\begin{align}\n    p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}(N)\\,\\coloneqq\\,\\frac{\\sqrt[k]{2^{N-1}-1}}{2^{N-1}+\\sqrt[k]{2^{N-1}-1}}.\n\\end{align}\n\n\n\\section{Hierarchy of $k$-copy activatable states}\\label{sec:Hierarchy of k-copy activatable states}\nThe threshold values $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}$ provide upper bounds on the minimal number of copies required to activate GME\\@: a value $p$ satisfying $p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}<p<p\\suptiny{0}{0}{(k-1)}_{\\mathrm{GME}}$ for $k\\geq2$ implies that $k$ copies are enough to activate GME\\@.\nBut since the map $\\mathcal{E}_{\\circ}$ (does not create and) may reduce GME, it does not imply that $k$ copies are actually needed; up to this point, there is a possibility that two copies are all it takes.\n\nHowever, at least for the case of three qubits ($N=3$) and three copies ($k=3$), we find that this is not the case. That is, for all isotropic three-qubit GHZ states with $p\\le p\\suptiny{0}{0}{(2)}_{\\mathrm{GME}}(N=3)=\\sqrt{3}\/(4+\\sqrt{3})$, we find that two copies are still biseparable, and thus at least three copies are required to activate GME\\@. The explicit biseparable decomposition of two copies of the states in this range is presented in Appendix~\\ref{sec:appendix two copy bisep}. Although it does not constitute conclusive proof, this result nevertheless supports our first conjecture, repeated here for convenience:\n\n\\noindent\n\\emph{Conjecture~(\\ref{conjecture i}):\\ There exists a hierarchy of states with $k$-copy activatable GME, i.e., for all $k\\geq2$ there exists a biseparable but not partition-separable state $\\rho$ such that $\\rho^{\\otimes k-1}$ is biseparable, but $\\rho^{\\otimes k}$ is GME\\@.}\n\nThe conjectured existence of a hierarchy of biseparable states with $k$-copy activatable GME means that states become less and less `valuable' as the number of copies $k$ required to obtain GME increases. At the same time, it is also clear that all partition-separable states cannot be used to activate GME because separability with respect to~any fixed partition is tensor stable. But it is not clear where exactly the boundary between activatable and non-activatable states really lies (see Fig.~\\ref{fig:GME structure}).\n\nTo shed light on this question, let us again examine the isotropic GHZ states from Eq.~(\\ref{eq:GHZ with white noise}) with regards to partition-separability with respect to~the bipartition separating the first qubit $\\mathcal{A}_{1}$ from the remaining $N-1$ qubits (collected in $\\tilde{\\mathcal{A}}_{2}$), i.e., $\\mathcal{A}_{1}|\\tilde{\\mathcal{A}}_{2}$. Using this partition, we can write\n\\begin{align}\n    \\rho(p) &=p\\ket{\\Phi^{+}}\\!\\!\\bra{\\Phi^{+}}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}\n    \\!+\\!\\tfrac{1-p}{2^{N}}\\,\\mathds{1}_{\\!\\mathcal{A}_{1}}\\hspace*{-3pt}\\otimes\\!\n    \\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}}\n    \\!+\\!\\tfrac{1-p}{2^{N}}\\,\\mathds{1}_{\\!\\mathcal{A}_{1}}\\hspace*{-3pt}\\otimes\\!\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}^{\\perp}}\\nonumber\\\\\n    &=\\,\n    \\tfrac{1+p}{2}\\,\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}\\,\n    +\\tfrac{1-p}{2}\\,\\tfrac{1}{2^{N-1}}\\,\n    \\mathds{1}_{\\mathcal{A}_{1}}\\hspace*{-3pt}\\otimes\\!\n    \\mathds{1}_{\\tilde{\\mathcal{A}}_{2}^{\\perp}},\n    \\label{eq:GHZ with white noise rewritten}\n\\end{align}\nwhere $\\ket{\\Phi^{+}}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}=\\tfrac{1}{\\sqrt{2}}\\bigl(\\ket{0}_{\\mathcal{A}_{1}}\\ket{\\tilde{0}}_{\\tilde{\\mathcal{A}}_{2}}+\\ket{1}_{\\mathcal{A}_{1}}\\ket{\\tilde{1}}_{\\tilde{\\mathcal{A}}_{2}}\\bigr)$ with $\\ket{\\tilde{i}}_{\\tilde{\\mathcal{A}}_{2}}=\\bigotimes_{j=2}^{N}\\ket{i}_{\\mathcal{A}_{j}}$ for $i=0,1$, $\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}}=\\sum_{i=0,1}\\ket{\\tilde{i}}\\!\\!\\bra{\\tilde{i}}$ and $\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}^{\\perp}}=\\mathds{1}_{2^{N-1}}-\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}}$. From this decomposition, it becomes clear that the state can be written as a convex sum of a two-qubit state $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$ (where the second qubit lives on the two-dimensional subspace of $\\tilde{\\mathcal{A}}_{2}$, spanned by the states $\\ket{\\tilde{i}}$ for $i=0,1$) and diagonal terms proportional to $\\mathds{1}_{\\mathcal{A}_{1}}\\otimes\\mathds{1}_{\\!\\tilde{\\mathcal{A}}_{2}^{\\perp}}$ with support in a subspace $\\tilde{\\mathcal{A}}_{2}^{\\perp}$ orthogonal to $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$. The latter diagonal terms trivially have a separable decomposition with respect to~the bipartition $\\mathcal{A}_{1}|\\tilde{\\mathcal{A}}_{2}$. For the two-qubit state $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$, the PPT criterion offers a necessary and sufficient separability criterion, and one easily finds that the partial transpose of $\\tilde{\\rho}_{\\mathcal{A}_{1}\\tilde{\\mathcal{A}}_{2}}$ is non-negative if $p\\leq p_{\\mathrm{crit}}\\coloneqq 1\/(1+2^{N-1})$ (see Appendix~\\ref{appendix:PPT criterion for isotropic GHZ states}). Further taking into account its qubit exchange symmetry, we thus find that $\\rho(p)$ is partition-separable with respect to~any bipartition for $p\\leq p_{\\mathrm{crit}}$. At the same time, we find that $\\lim_{k\\rightarrow\\infty}p\\suptiny{0}{0}{(k)}_{\\mathrm{GME}}(N)=p_{\\mathrm{crit}}$, which implies that any isotropic GHZ state with $p>p_{\\mathrm{crit}}$ features $k$-copy activatable GME, at least asymptotically as $k\\rightarrow\\infty$, and is thus also not partition-separable. This leads us to our second conjecture, also repeated here for convenience:\n\n\\noindent\n\\emph{Conjecture~(\\ref{conjecture ii}):\\ \nGME may be activated for any biseparable but not partition-separable state of any number of parties as $k\\rightarrow\\infty$.}\n\nConjecture~(\\ref{conjecture ii}) holds for isotropic GHZ states. But does it hold in general?\n\n\n\\section{GME activation from PPT entangled states}\\label{sec:GME activation of PPT entangled states}\nA situation where one might imagine Conjecture~(\\ref{conjecture ii}) to fail is the situation of biseparable (but not partition-separable) states with PPT entanglement across every bipartition, as discussed in Sec.~\\ref{sec:introduction}.\nFor isotropic GHZ states, however, the PPT criterion across every cut coincides exactly with the threshold $p_{\\mathrm{crit}}$ for biseparability (and GME activation), as one can confirm by calculating the eigenvalues of the partial transpose of $\\rho(p)$ (see Appendix~\\ref{appendix:PPT criterion for isotropic GHZ states}).\nWe thus turn to a different family of states, for which this is not the case.\n\nSpecifically, as we show in detail in Appendix~\\ref{appendix:PPT entangled GME activation}, we construct a family of biseparable three-party states \n\\begin{align}\n    \\rho_{\\mathcal{A}_{1}\\mathcal{A}_{2}\\mathcal{A}_{3}\n    } &=\\,\n    \\sum\\limits_{\\substack{i,j,k=1\\\\ i\\neq j\\neq k\\neq i}}^{3}p_{i}\\ \\rho_{\\mathcal{A}_{i}}\\otimes\\rho_{\\mathcal{A}_{j}\\mathcal{A}_{k}}\\suptiny{0}{0}{\\mathrm{PPT}}\n\\end{align}\nwhere the $\\rho_{\\mathcal{A}_{j}\\mathcal{A}_{k}}\\suptiny{0}{0}{\\mathrm{PPT}}$ are (different) two-qutrit states with PPT entanglement across the respective cuts $\\mathcal{A}_{j}|\\mathcal{A}_{k}$ for $j\\neq k\\in\\{1,2,3\\}$ and $\\sum_{i}p_{i}=1$. Via LOCC, three copies (labelled $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$, respectively) of this state $\\rho_{\\mathcal{A}_{1}\\mathcal{A}_{2}\\mathcal{A}_{3}}$ can be converted to what we call \\emph{PPT-triangle states} of the form\n\\begin{equation}\n    \\rho_{\\mathcal{A}_{2}\\mathcal{A}_{3}}\\suptiny{0}{0}{\\mathrm{PPT}}\\otimes \\rho_{\\mathcal{B}_{1}\\mathcal{B}_{3}}\\suptiny{0}{0}{\\mathrm{PPT}}\\otimes \\rho_{\\mathcal{C}_{1}\\mathcal{C}_{2}}\\suptiny{0}{0}{\\mathrm{PPT}}.\n\\end{equation}\nUsing a GME witness based on the lifted Choi map (cf.~\\cite{HuberSengupta2014, ClivazHuberLamiMurta2017}), we show that there exists a parameter range where these PPT-triangle states are GME.\nTherefore, it is proved that GME activation is possible even from biseparable states only with PPT entanglement across every bipartition.\n\n\n\\section{GME activation and shared randomness}\n\nProvided that our conjectures are true, incoherent mixing (access to shared randomness) can lead to situations where the number of copies needed for GME activation is reduced. In the extreme case, and this is true even based only on the results already proven here (and thus independently of whether or not the conjectures turn out to be true or not), the probabilistic combination of partition-separable states (without activatable GME) can results in a state \\textemdash\\ a biseparable isotropic GHZ state \\textemdash\\ which has activatable GME. Although this may at first glance appear to be at odds with the usual understanding of bipartite entanglement, which cannot arise from forming convex combinations of separable states, we believe this can be understood rather intuitively if we view incoherent mixing as a special case of a more general scenario in which one may have any amount of information on the states that are shared between different observers. As an example, consider the following situation:\n\nThree parties, labelled, $1$, $2$ and $3$, share two identical (as in, the system and its subsystems have the same Hilbert space dimensions and are represented by the same physical degrees of freedom) tripartite quantum systems, labelled $\\mathcal{A}$ and $\\mathcal{B}$, in the states $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}$ and $\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$, respectively, where we assume that $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}$ is separable with respect to the bipartition $\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}$ and $\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$ is separable with respect to the bipartition $\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}$. Clearly, both of these systems and states individually are biseparable, but if the parties have full information about which system is which, e.g., the first system is $A$ and the second system is $B$, then the joint state $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}\\otimes\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$ can be GME with respect to the partition $\\mathcal{A}_{1}\\mathcal{B}_{1}|\\mathcal{A}_{2}\\mathcal{B}_{2}|\\mathcal{A}_{3}\\mathcal{B}_{3}$. In this sense, two biseparable systems can yield one GME system. Now, let us suppose that the parties do not have full information which system is in which state. For simplicity, let us assume that either system may be in either state with the same probability $\\tfrac{1}{2}$. Then the state of either of the systems is described by the convex mixture $\\rho_{\\mathrm{mix}}=\\tfrac{1}{2}\\rho_{A_{1}|A_{2}A_{3}}+\\tfrac{1}{2}\\rho_{B_{1}B_{2}|B_{3}}$, where we have kept the labels $A$ and $B$, but they now refer to the same subsystems, i.e., $A_{i}=B_{i}$ for all~$i$. The state $\\rho_{\\mathrm{mix}}$ may not be partition separable anymore, but is certainly still biseparable. In particular, it may have activatable GME, even though neither $\\rho_{A_{1}|A_{2}A_{3}}$ nor $\\rho_{B_{1}B_{2}|B_{3}}$ do. For the sake of the argument let us assume that the latter is indeed the case and that GME is activated for $2$ copies in this case, such that $\\rho_{\\mathrm{mix}}^{\\otimes 2}$ is GME. That means, if one has access to both systems, $A$ and $B$, even without knowing which system is in which state, one would end up with GME. However, the additional randomness with respect to the case where one knows exactly which state which system is in results in an increased entropy of $\\rho_{\\mathrm{mix}}^{\\otimes 2}$ with respect to $\\rho_{\\mathcal{A}_{1}|\\mathcal{A}_{2}\\mathcal{A}_{3}}\\otimes\\rho_{\\mathcal{B}_{1}\\mathcal{B}_{2}|\\mathcal{B}_{3}}$, and thus represents a disadvantage with respect to the latter case.\n\nIn general, it is therefore not problematic that the conjectures, if true, would imply that incoherent mixtures of $k$-activatable states may result in $k'$-activatable states with $k'<k$. Instead, this can be considered as a sign that scenarios with multiple copies of multipartite quantum states give rise to features that are not captured by convex structures on the level of the single-copy state space.\n\n\\section{Conclusion and outlook}\\label{sec:Conclusion and Outlook}\nOur results show that a modern theory of entanglement in multipartite systems, which includes the potential to locally process multiple copies of distributed quantum states, exhibits a rich structure that goes beyond the convex structure of partially separable states on single copies. While we conjecture that asymptotically, an even simpler description might be possible, i.e., separability in multipartite systems collapses to a simple bipartite concept of separability, we show that two copies are certainly not sufficient for reaching this simple limit, thus leaving the practical certification with finite copies a problem to be studied.\n\nIndeed, our results show that GME is a resource with a complex relationship to bipartite entanglement in the context of local operations and shared randomness (cf.~\\cite{SchmidRossetBuscemi2020}). An array of important open questions arises from our results, which can thus be considered to establish an entirely new direction of research: first and foremost, this includes the quest for conclusive evidence for or against our conjectures. Besides determining whether these conjectures are ultimately correct or not, it will be of high interest to determine which properties (of the biseparable decompositions) of given states permit or prevent GME activation with a certain number of copies. Another open question is the minimal local dimension necessary for GME activation from biseparable states with PPT entanglement across every cut. Furthermore, from a practical point of view, it will be desirable to develop a theory of $k$-copy multipartite entanglement witnesses that are non-linear expressions of density matrices and allow for a more fine-grained characterisation of multipartite entanglement in networks with local memories.\nFinally, although separable states and shared classical randomness are free under LOCC, i.e., under a conventional choice of free operations in the resource theory of bipartite entanglement, our results suggest that convex combinations of different partition-separable states with shared classical randomness can be used as a resource for GME activation in multi-copy scenarios; that is, it may not be straightforward to study GME activation within the usual resource-theoretical framework under LOCC\\@. \nIn view of this situation, it would be interesting for future research to establish a new framework for understanding such a complicated aspect of multipartite entanglement as GME activation by, e.g., considering non-convex quantum resource theories where classical randomness can be used as a resource~\\cite{Kuroiwa2020generalquantum,PhysRevA.104.L020401}.\n\n\\begin{acknowledgments}\nWe acknowledge support from the Austrian Science Fund (FWF) through the START project Y879-N27, the project P 31339-N27, and the Zukunftskolleg ZK03. H.Y.\\ was supported by JSPS Overseas Research Fellowships and JST, PRESTO Grant Number JPMJPR201A, Japan.\n\\end{acknowledgments}\n\n\\bibliographystyle{apsrev4-1fixed_with_article_titles_full_names_new}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe relationship between phase transitions away from thermal equilibrium and open systems in quantum optics was first addressed in early laser days \\cite{degiorgio&scully_1970,graham&haken_1970,grossmann&richter_1971}. The theme was then carried forward by work on optical bistability \\cite{bonifacio&lugiato_1976,bonifacio_etal_1978,drummond&walls_1980,rosenberger_etal_1991}, the degenerate parametric oscillator \\cite{carmichael_2008}---with loss added to the quantum theory of parametric amplification \\cite{mollow&glauber_1967}---and collective radiative phenomena like cooperative fluorescence \\cite{drummond&carmichael_1978,carmichael_1980}, to name just a few of the examples. As counterpoint to these phase transitions of light away from equilibrium, and contemporary with the early laser work, Hepp and Lieb  introduced the celebrated Dicke-model phase transition \\cite{hepp&lieb_1973a,dicke_1954}---a phase transition for photons in thermal equilibrium.\n\nWhile the dissipative platforms provided by the laser, optical bistability, and parametric oscillator encouraged wide experimental activity, Hepp and Lieb's proposal lay\ndormant on the experimental front. Its call for a dipole coupling strength between light and matter in excess of atomic transition frequencies posed an extreme technical challenge, and also undermined approximations adopted in the Dicke model \\cite{rzazewski_etal_1975,bialynicki-birula&rzazewski_1979,gawedzki&rzazewski_1981,rzazewski&wodkiewicz_1991}.  The long wait ended in 2010, however, with the experimental work of Baumann \\emph{et al.} \\cite{baumann_etal_2010,baumann_etal_2011}, who realized the $T=0$ phase transition of Hepp and Lieb with a superfluid gas in an optical cavity.  The key to success was their engineering of the Dicke-model Hamiltonian as an effective Hamiltonian, by employing an external Raman drive to realize the phase transition in a dissipative setting \\cite{dimer_etal_2007,baden_etal_2014}.\n\nIn a separate development rooted in research on open systems in quantum optics, cavity and circuit QED have shown that where many material particles and photons might traditionally be required for the strong  interaction of matter and light, it is now possible to achieve strong interactions, sufficient to access nonlinearities, with one particle (e.g.\\ two-state system) and photon numbers that range from just one to the relatively low tens, hundreds, or thousands.  Thus, with regard to the laser\nand optical bistability, there are cavity QED versions of both \\cite{rice&carmichael_1994,savage&carmichael_1988}, and even realizations of the parametric oscillator where single photons are enough to access the nonlinearity \\cite{leghtas_etal_2015}. Although the thermodynamic limit of equilibrium phase transitions does not apply under these conditions, it still remains that a mean-field treatment and phase transition perspective can guide much of the phenomenology; albeit with the caveat that fluctuations might add more than just minor corrections.\n\nIn this paper we unify the dissipative extension of the Dicke-model quantum phase transition \\cite{baumann_etal_2010,baumann_etal_2011,dimer_etal_2007,baden_etal_2014}\nwith the recently reported  breakdown of photon blockade \\cite{carmichael_2015,fink_etal_2017}. The former is addressed by Hepp and Lieb through the thermodynamic limit ($N\\to\\infty$ two-state systems), while the latter has been approached in a cavity or circuit QED setting \\cite{carmichael_2015,fink_etal_2017,alsing&carmichael_1991,kilin&krinitskaya_1991,armen_etal_2009} (one two-state system). Both phenomena may be engineered, however, in either of the two ways, and we therefore first consider mean-field results for both (Sec.~\\ref{sec:mean-field}) before turning to results specific to one two-state system (Sec.~\\ref{sec:quantum_fluctuations}).\n\nWe achieve the proposed unification within the framework of a generalized Dicke-model Hamiltonian, where two extensions of the analysis in Ref.~\\cite{dimer_etal_2007} are made: first, we allow for rotating and counter-rotating interactions of independently adjustable coupling strength (see Ref.~\\cite{dimer_etal_2007}, Eq.~(12)); and, second, we add external coherent driving of the field mode. The first extension was made by Hepp and Lieb \\cite{hepp&lieb_1973b}, in a quick followup to their original paper; the generalized interaction Hamiltonian is also featured in a number of recent publications \\cite{stepanov_etal_2008,schiro_etal_2012,tomka_etal_2014,xie_etal_2014,tomka_etal_2015,wang_etal_2016,moroz_2016,kirton_etal_2018}. A key link in our unification is a phase that went unreported by Hepp and Lieb. Beyond this, though, the added coherent drive is also key, since the breakdown of photon blockade is organized around a critical drive strength, identified, to date, in the driven Jaynes-Cummings model (no counter-rotating interaction) alone \\cite{alsing&carmichael_1991,alsing_etal_1992,carmichael_2015}. We show that the critical drive is a feature of the generalized Hamiltonian, rotating and counter-rotating interactions included, and thus links the Dicke model quantum phase transition to the breakdown of photon blockade.\n\nWe begin in Sec.~\\ref{sec:background} with a detailed review, building up our generalized Jaynes-Cummings-Rabi model while making connections to prior work. We then survey the mean-field steady states of the model in Sec.~\\ref{sec:mean-field} and show how a common critical drive strength links the dissipative extension of the Dicke-model quantum phase transition to the breakdown of photon blockade.  Finally, in Sec.~\\ref{sec:quantum_fluctuations}, we turn from the mean-field treatment to full quantum mechanical calculations for the special case of one two-state system. We recover the critical drive strength from the quasi-energy spectrum of the model Hamiltonian and show how mean-field predictions can  still provide a guide to the physics in the presence of quantum fluctualtions. Conclusions are presented in Sec.~\\ref{sec:conclusions}\n\n\\section{Background}\n\\label{sec:background}\n\\subsection{The Dicke quantum phase transition in the rotating-wave approximation}\n\\label{sec:Dicke_rotating_wave}\nIn their original paper \\cite{hepp&lieb_1973a} ``On the Superradiant Phase Transition for Molecules in a Quantized Radiation Field: the Dicke Maser Model,'' Hepp and Lieb first introduce an ``interesting caricature$\\ldots$invented by Dicke'' \\cite{dicke_1954} of the interaction between quantized radiation in a box and a system of $N$ molecules. The caricature assumes single-mode radiation, two-state molecules, and the rotating-wave approximation; it generalizes the Tavis-Cummings model \\cite{tavis&cummings_1968} to non-zero detuning, and, adopting natural units with $\\hbar=1$, is defined by the Hamiltonian\n\\begin{equation}\nH_0=\\omega a^\\dagger a+\\omega_0J_z+\\frac{\\lambda}{\\sqrt N}(aJ_++a^\\dagger J_-),\n\\label{eqn:hamiltonian_rotating_wave}\n\\end{equation}\nwhere $\\omega$ is the frequency of the field, $\\omega_0$ the resonance frequency of the two-state molecules, and $\\lambda$ is a coupling strength; annihilation and creation operators for the field mode obey the boson commutation relation, $[a,a^\\dagger]=1$, and the collective operators for $N$ two-state systems obey angular momentum commutation relations, $[J_-,J_+]=-2J_z$, $[J_\\mp,J_z]=\\pm J_\\mp$. Hepp and Lieb exactly compute thermodynamic functions in the limit $N\\to\\infty$ and find a critical temperature, $T_c>0$, for any coupling strength above\n\\begin{equation}\n\\lambda_0=\\sqrt{\\omega\\omega_0}.\n\\label{critical_point_eta=0}\n\\end{equation}\nConsidering zero temperature, as we do in this paper, $\\lambda_0$ has the significance of a critical coupling strength, where for $\\lambda\\le\\lambda_0$ the photon number is zero in the ground state, while it follows the formula\n\\begin{equation}\n\\frac{\\langle a^\\dagger a\\rangle_0}N=\\frac{\\omega_0}{4\\omega}\\frac{\\lambda^4-\\lambda_0^4}{\\lambda^2\\lambda_0^2}\n\\label{eqn:photon_number_eta=0}\n\\end{equation}\nwhen $\\lambda>\\lambda_0$. Soon after the rigorous calculation of Hepp and Lieb, the same result was derived by Wang and Hioe \\cite{wang&hioe_1973} using a simpler method [see their Eq.~(40)].\n\\subsection{Counter-rotating terms}\n\\label{sec:Dicke_counter_rotating}\nThe method of Wang and Hioe readily generalizes to an interaction without the rotating-wave approximation: $aJ_++a^\\dagger J_-\\to (a+a^\\dagger)(J_-+J_+)$. The calculation, made by Hepp and Lieb \\cite{hepp&lieb_1973b} and Carmichael \\emph{et al.} \\cite{carmichael_etal_1973}, retains the phase transition and the form of Eq.~(\\ref{eqn:photon_number_eta=0}), but unlike in the rotating-wave approximation, the state of nonzero photon number now assigns a definite phase to the field, and the critical coupling is changed to $\\sqrt{\\omega\\omega_0}\/2$. In fact Hepp and Lieb \\cite{hepp&lieb_1973b} consider a Hamiltonian generalized in the form\n\\begin{eqnarray}\nH_\\eta&=&\\omega a^\\dagger a+\\omega_0J_z\\notag\\\\\n&&+\\frac{\\lambda}{\\sqrt N}(aJ_++a^\\dagger J_-)+\\eta\\frac{\\lambda}{\\sqrt N}(a^\\dagger J_++aJ_-),\n\\label{eqn:hamiltonian_counter_rotating}\n\\end{eqnarray}\nwith $\\eta$ a parameter. We let $\\eta$ vary from 0 to 1 and show (Sec.~\\ref{sec:epsilon=0}) that there are actually two critical coupling strengths marking transitions to states of definite phase:\n\\begin{equation}\n\\lambda_\\eta^\\pm=\\frac1{1\\pm\\eta}\\sqrt{\\omega\\omega_0}.\n\\label{eqn:critical_points_kappa=0}\n\\end{equation}\nMoreover, photon numbers for solutions bifurcating from both critical points, $\\lambda_\\eta^+$  and $\\lambda_\\eta^-$, follow the same form, that of Eq.~(\\ref{eqn:photon_number_eta=0}):\n\\begin{equation}\n\\frac{\\langle a^\\dagger a\\rangle_\\eta^\\pm}N=\\frac{\\omega_0}{4\\omega}\\frac{\\lambda^4-(\\lambda_\\eta^\\pm)^4}{\\lambda^2(\\lambda_\\eta^\\pm)^2}.\n\\label{eqn:photon_number_kappa=0}\n\\end{equation}\nThe transition at $\\lambda_\\eta^+$ corresponds to the extension of the Dicke phase transition of Ref.~\\cite{hepp&lieb_1973a} discussed in Refs.~\\cite{hepp&lieb_1973b} and \\cite{carmichael_etal_1973}: the zero photon state becomes unstable and is replaced by a stable state of nonzero photon number. The transition at $\\lambda_\\eta^-$, not identified before to our knowledge, marks a restabilization of the zero photon state and the birth of an unstable state of nonzero photon number. It provides the fulcrum upon which the unification of the coherently driven extension of the Dicke phase transition and the breakdown of photon blockade turns.\n\\subsection{Dissipative realization}\n\\label{sec:dissipative_realization}\nWhile Dicke's paper \\cite{dicke_1954} generated enormous interest in superradiance as a transient, away-from-equilibrium process \\cite{gross&haroche_1982}, the Dicke quantum phase transition of Hepp and Lieb was, for many years, largely seen as academic---beyond the reach of experiments due to a needed coupling strength on the order of the transition frequency, and, on the theory side, suspect because of approximations used in the Dicke model \\cite{rzazewski_etal_1975,bialynicki-birula&rzazewski_1979,gawedzki&rzazewski_1981,rzazewski&wodkiewicz_1991}.  Dissipative realizations of the Dicke Hamiltonian as an effective Hamiltonian overcome these obstacles by replacing a transition from a ground to an excited state by one between a pair of ground states. Specifically, we have the scheme introduced by Dimer \\emph{et al.} \\cite{dimer_etal_2007,baden_etal_2014} in mind; although there are essentially parallel setups, where internal states are replaced by momentum states of a Bose-Einstein condensate \\cite{baumann_etal_2010,baumann_etal_2011}.\n\nWe consider a pair of Raman transitions between states $|1\\rangle$ and $|2\\rangle$---the two-state system---as sketched in Fig.~\\ref{fig:fig1}, where one leg of each transition is driven by a laser field, with amplitudes and frequencies $\\Omega_{1,2}$ and $\\omega_{1,2}$, and the other creates and annihilates cavity photons of frequency $\\omega$, with coupling strength to the cavity mode $g$. Adopting this configuration, with the excited states (not shown) adiabatically eliminated, and in an interaction picture---free Hamiltonian $\\omega_+a^\\dagger a+\\omega_-J_z$, $\\omega_{\\pm}=(\\omega_1\\pm\\omega_2)\/2$---an effective Hamiltonian is realized in the form of Eq.~(\\ref{eqn:hamiltonian_counter_rotating}):\n\\begin{eqnarray}\nH_\\eta^\\prime&=&\\Delta a^\\dagger a+\\Delta_0J_z\\notag\\\\\n&&+\\frac{\\lambda}{\\sqrt N}(aJ_++a^\\dagger J_-)+\\eta\\frac{\\lambda}{\\sqrt N}(a^\\dagger J_++aJ_-),\n\\label{eqn:hamiltonian_raman_model}\n\\end{eqnarray}\nwith effective frequencies\n\\begin{eqnarray}\n\\Delta&=&\\omega-\\omega_+=\\frac{\\delta_1+\\delta_2}2,\\\\\n\\label{eqn:delta}\n\\Delta_0&=&\\omega_0-\\omega_-=\\frac{\\delta_1-\\delta_2}2,\n\\label{eqn:delta_zero}\n\\end{eqnarray}\nwhere $\\delta_1$ and $\\delta_2$ are Raman detunings (Fig.~\\ref{fig:fig1}), and the coupling constants $\\lambda$ and $\\eta\\lambda$ follow from the strength of the Raman coupling (see Ref.~\\cite{dimer_etal_2007}). We consider an initial state $|0\\rangle|1\\rangle$, with $|0\\rangle$ the cavity mode vacuum, in which case the Raman driving is a source of photons through the counter-rotating interaction, an external drive that is off-set by the cavity loss; thus, the dissipative realization of the generalized Dicke Hamiltonian, Eq.~(\\ref{eqn:hamiltonian_counter_rotating}), is modeled by the master equation\n\\begin{equation}\n\\frac{d\\rho}{dt}=-i[H_\\eta^\\prime,\\rho]+\\kappa{\\mathcal L}[a]\\rho,\n\\label{eqn:master_equation}\n\\end{equation}\nwhere $\\kappa$ is the loss rate and ${\\mathcal L}[\\xi]\\,\\cdot=2\\xi \\cdot \\xi^\\dagger-\\xi^\\dagger \\xi\\cdot-\\cdot \\xi^\\dagger\\xi$. We show (Sec.~\\ref{sec:epsilon=0}) that in the presence of dissipation, for $\\eta<\\eta_\\kappa$,\n\\begin{equation}\n\\eta_\\kappa\\equiv\\frac{\\kappa}{|\\Delta|}\\left[1+\\sqrt{1+\\frac{\\kappa^2}{\\Delta^2}}\\mkern3mu\\right]^{-1},\n\\label{eqn:eta_critical}\n\\end{equation}\nthere is no critical coupling strength, while for $\\eta\\geq\\eta_\\kappa$, there are two that for $\\kappa\\to0$ reduce to Eq.~(\\ref{eqn:critical_points_kappa=0}):\n\\begin{equation}\n\\lambda_\\eta^\\pm\\equiv\\frac{\\sqrt{|\\Delta\\Delta_0|}}{1-\\eta^2}\\left[1+\\eta^2\\mp2\\eta\\sqrt{1-\\frac{(1-\\eta^2)^2}{4\\eta^2}\\frac{\\kappa^2}{\\Delta^2}}\\mkern3mu\\right]^{1\/2}.\n\\label{eqn:lambda_critical}\n\\end{equation}\nPhoton numbers generalizing Eq.~(\\ref{eqn:photon_number_kappa=0}) are recovered from the mean-field steady state in Sec.~\\ref{sec:epsilon=0} [Eq.~(\\ref{eqn:photon_number_epsilon=0})].\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=2.75in]{figure1.pdf}\n\\end{center}\n\\caption{Schematic of the open system realization of the model Hamiltonian, Eq.~(\\ref{eqn:hamiltonian_raman_model}). A pair of ground states, denoted $|1\\rangle$ and $|2\\rangle$, are coupled to an optical cavity mode, frequency $\\omega$, via far-from-resonance Raman transitions, where bold dashed arrows represent external laser drives while the transfer of photons to and from the cavity mode, coupling strength $g$, is represented by bold solid arrows; $\\Omega_{1,2}$ and $\\omega_{1,2}$ are drive amplitudes and frequencies, and $\\delta_{1,2}$ are detunings; excited states are assumed far from resonance and not shown.}\n\\label{fig:fig1}\n\\end{figure}\n\n\\subsection{Extended model with coherent drive}\nEquations~(\\ref{eqn:hamiltonian_raman_model}) and (\\ref{eqn:master_equation}) set out a driven and dissipative model where the driving of the field mode is mediated by externally driven Raman transitions; the dissipative realization of the effective rotating and counter-rotating interactions amounts to a \\emph{nonlinear} driving of the field mode. In studies of the so-called breakdown of photon blockade  \\cite{alsing&carmichael_1991,carmichael_2015,armen_etal_2009,fink_etal_2017}, the mode is subject to a coherent drive, i.e., \\emph{linear} driving by an external field. We now extend our model by adding a coherent drive of amplitude $\\sqrt N\\epsilon$ and frequency $\\omega_d$---a detuning $\\omega_d-\\omega_+$ in the interaction picture of Eq.~(\\ref{eqn:hamiltonian_raman_model}). Choosing $\\omega_1$ and $\\omega_2$ so that $\\omega_+=\\omega_d$, the master equation then becomes\n\\begin{equation}\n\\frac{d\\rho}{dt}=-i[H_\\eta^\\prime,\\rho]-i\\sqrt N\\epsilon[a^\\dagger+a,\\rho]+\\kappa{\\mathcal L}[a]\\rho,\n\\label{eqn:master_equation_drive}\n\\end{equation}\nwhere, from Eq.~(\\ref{eqn:delta}), $\\Delta=\\omega-\\omega_d$ is now the detuning of the field mode from the drive.\n\nThe next section explores the parameter dependence of the mean-field steady states of Eq.~(\\ref{eqn:master_equation_drive}). In particular, we  connect the breakdown of photon blockade, realized for $\\eta=0$, to the coherently driven extension of the Dicke quantum phase transition. We show that an $\\eta$-dependent critical point organizes behavior as a function of drive strength; we then establish a link through the previously unreported phase of the generalized model presented in Ref.~\\cite{hepp&lieb_1973b}, i.e., the second critical coupling strength $\\lambda_\\eta^-$.\n\n\\section{Mean-Field Steady States}\n\\label{sec:mean-field}\nThe mean-field Maxwell-Bloch equations derived from the master equation, Eq.~(\\ref{eqn:master_equation_drive}), are:\n\\begin{eqnarray}\n\\frac{d\\alpha}{dt}&=&-(\\kappa+i\\Delta)\\alpha-i\\frac{\\lambda}{\\sqrt N}\\frac12(\\beta+\\eta\\beta^*)-i\\sqrt N\\epsilon,\n\\label{eqn:mean-field_alpha}\\\\\n\\frac{d\\beta}{dt}&=&-i\\Delta_0\\beta+2i\\frac{\\lambda}{\\sqrt N}(\\alpha+\\eta\\alpha^*)\\zeta,\n\\label{eqn:mean-field_beta}\\\\\n\\frac{d\\zeta}{dt}&=&-i\\frac{\\lambda}{\\sqrt N}\\left[(\\alpha\\beta^*-\\alpha^*\\beta)-\\eta(\\alpha\\beta-\\alpha^*\\beta^*)\\right],\n\\label{eqn:mean-field_zeta}\n\\end{eqnarray}\nwith $\\alpha\\equiv\\langle a\\rangle$, $\\beta\\equiv2\\langle J_-\\rangle$, and $\\zeta\\equiv2\\langle J_z\\rangle$. We first outline a general approach to their steady state solution, where, introducing intensive variables\n\\begin{equation}\n\\bar\\alpha\\equiv\\alpha\/\\sqrt N,\\qquad\\bar\\beta\\equiv\\beta\/N,\\qquad\\bar\\zeta\\equiv\\zeta\/N,\n\\label{eqn:scaling}\n\\end{equation}\nEqs.~(\\ref{eqn:mean-field_alpha}) and (\\ref{eqn:mean-field_beta}) require\n\\begin{eqnarray}\n\\bar\\beta_x&=&2\\lambda\\frac{1+\\eta}{\\Delta_0}\\bar\\alpha_x\\bar\\zeta,\n\\label{eqn:steady-state_beta1}\\\\\n\\bar\\beta_y&=&2\\lambda\\frac{1-\\eta}{\\Delta_0}\\bar\\alpha_y\\bar\\zeta,\n\\label{eqn:steady-state_beta2}\n\\end{eqnarray}\nwith $\\bar\\alpha_x$ and $\\bar\\alpha_y$ satisfying the simultaneous equations:\n\\begin{eqnarray}\n\\kappa\\bar\\alpha_x-\\left[\\Delta+\\lambda^2\\frac{(1-\\eta)^2}{\\Delta_0}\\bar\\zeta\\right]\\bar\\alpha_y&=&0,\n\\label{eqn:steady-state_alpha1}\\\\\n\\kappa\\bar\\alpha_y+\\left[\\Delta+\\lambda^2\\frac{(1+\\eta)^2}{\\Delta_0}\\bar\\zeta\\right]\\bar\\alpha_x&=&-\\epsilon.\n\\label{eqn:steady-state_alpha2}\n\\end{eqnarray}\nWe may then solve Eqs.~(\\ref{eqn:steady-state_beta1})--(\\ref{eqn:steady-state_alpha2}) for $|\\bar\\beta|^2$ in terms of $\\bar\\zeta$ and impose the conservation law $\\bar\\zeta^2+|\\bar\\beta|^2=1$; hence we find an autonomous equation satisfied by $\\bar\\zeta$,\n\\begin{equation}\n(1-\\bar\\zeta^2)[P(\\bar\\zeta)]^2=\\frac{4\\epsilon^2}{\\lambda^2(1+\\eta)^2}\\bar\\zeta^2Q(\\bar\\zeta),\n\\label{eqn:6th-order_polynomial}\n\\end{equation}\nwith $P(\\bar\\zeta)$ and $Q(\\bar\\zeta)$ both quadratic:\n\\begin{equation}\nP(\\bar\\zeta)=\\bar\\zeta^2+2\\frac{\\Delta\\Delta_0(1+\\eta^2)}{\\lambda^2(1-\\eta^2)^2}\\bar\\zeta+\\frac{\\Delta_0^2(\\kappa^2+\\Delta^2)}{\\lambda^4(1-\\eta^2)^2},\n\\label{eqn:p_quadratic}\n\\end{equation}\nand\n\\begin{equation}\nQ(\\bar\\zeta)=\\bar\\zeta^2+2\\frac{\\Delta\\Delta_0}{\\lambda^2(1-\\eta)^2}\\bar\\zeta+\\frac{\\Delta_0^2\\kappa^2}{\\lambda^4(1-\\eta^2)^2}+\\frac{\\Delta^2\\Delta_0^2}{\\lambda^4(1-\\eta)^4}.\n\\label{eqn:q_quadratic}\n\\end{equation}\nSteady-state solutions for $\\bar\\zeta$ are seen to be roots of a 6th-order polynomial, with a possible six distinct solutions for any setting of the parameters: $\\eta$, $\\Delta$, $\\Delta_0$, $\\lambda$, $\\epsilon$, and $\\kappa$. In the following, for the most part, we set $\\Delta_0=\\Delta$ and keep $\\kappa\/\\lambda$ fixed; we then explore the parameter dependence in the $(\\Delta\/\\lambda,\\epsilon\/\\lambda)$-plane for different choices of $\\eta$. To start, we recover the results summarized in Secs.~\\ref{sec:Dicke_rotating_wave} and \\ref{sec:Dicke_counter_rotating} from our general solution scheme.\n\\subsection{Zero drive: $\\epsilon=0$}\n\\label{sec:epsilon=0}\nIn the absence of a coherent drive, the right-hand side of Eq.~(\\ref{eqn:6th-order_polynomial}) is zero, and the 6th-order polynomial satisfied by $\\bar\\zeta$ reduces to\n\\begin{equation}\n(1-\\bar\\zeta^2)[P(\\bar\\zeta)]^2=0.\n\\label{eqn:6th-order_polynomial_epsilon=0}\n\\end{equation}\nEquations~(\\ref{eqn:steady-state_alpha1}) and (\\ref{eqn:steady-state_alpha2}) are replaced by the homogeneous system\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\Delta_0\\kappa&-\\Delta\\Delta_0-\\lambda^2(1-\\eta)^2\\bar\\zeta\\\\\n\\noalign{\\vskip4pt}\n\\Delta\\Delta_0+\\lambda^2(1+\\eta)^2\\bar\\zeta&\\Delta_0\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip4pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=0.\n\\label{eqn:steady-state_alpha_epsilon=0}\n\\end{equation}\nNoting then that the determinant of this homogeneous system is $\\lambda^4(1-\\lambda^2)^2P(\\bar\\zeta)$, the condition for nontrivial solutions for $\\bar\\alpha$ is $P(\\bar\\zeta)=0$. Thus, the roots $\\bar\\zeta=\\pm1$ of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}) correspond to the trivial solution, $\\bar\\alpha=0$, while the roots of $P(\\bar\\zeta)=0$,\n\\begin{equation}\n\\bar\\zeta_\\pm=-\\frac{\\Delta\\Delta_0}{\\lambda^2(1-\\eta^2)^2}\\mkern-2mu\\left[1+\\eta^2\\mp2\\eta\\sqrt{1-\\frac{(1-\\eta^2)^2}{4\\eta^2}\n\\frac{\\kappa^2}{\\Delta^2}}\\mkern3mu\\right],\n\\label{eqn:nontrivial_zeta_epsilon=0}\n\\end{equation}\nyield nontrivial solutions for $\\bar\\alpha$. The latter are physically acceptable if $\\bar\\zeta_\\pm$ are real and $|\\bar\\zeta_\\pm|\\leq1$; the first condition is satisfied if $\\eta\\geq\\eta_\\kappa$, $\\eta_\\kappa$ defined in Eq.~(\\ref{eqn:eta_critical}), and the second gives the critical coupling strengths, $\\lambda_\\eta^\\pm$, defined in Eq.~(\\ref{eqn:lambda_critical}); for $\\eta\\geq\\eta_\\kappa$ and $\\lambda_\\eta^+\\leq\\lambda\\leq\\lambda_\\eta^-$, $\\bar\\zeta_+$ is the only acceptable root, while $\\bar\\zeta_+$ and $\\bar\\zeta_-$ are both acceptable if $\\lambda\\geq\\lambda_\\eta^-$.\n\nNote that $\\Delta$ and $\\Delta_0$ are detunings and therefore two cases arise, one with $\\Delta\\Delta_0$ positive and $\\bar\\zeta_\\pm<0$, and the other with $\\Delta\\Delta_0$ negative and $\\bar\\zeta_\\pm>0$. Considering steady states only, there is no physical difference between the cases as a quick inspection of Eqs.~(\\ref{eqn:mean-field_alpha})-(\\ref{eqn:mean-field_zeta}) shows---simply reverse the signs of $\\Delta_0$ and $\\bar\\zeta$ in Eq.~(\\ref{eqn:mean-field_beta}); steady state stability can change, though. We always illustrate results with $\\Delta_0=\\Delta$, whence $\\Delta\\Delta_0$ is positive.\n\nBy eliminating $\\Delta_0\\kappa$ from the homogeneous system, Eq.~(\\ref{eqn:steady-state_alpha_epsilon=0}), we may solve for\n\\begin{eqnarray}\n(\\bar\\alpha_x^\\pm)^2&=&-|\\bar\\alpha_\\pm|^2\\frac{\\Delta\\Delta_0+\\lambda^2(1-\\eta)^2\\bar\\zeta_\\pm}{4\\lambda^2\\eta\\bar\\zeta_\\pm},\n\\label{eqn:nontrivial_alphax_epsilon=0}\\\\\n(\\bar\\alpha_y^\\pm)^2&=&+|\\bar\\alpha_\\pm|^2\\frac{\\Delta\\Delta_0+\\lambda^2(1+\\eta)^2\\bar\\zeta_\\pm}{4\\lambda^2\\eta\\bar\\zeta_\\pm},\n\\label{eqn:nontrivial_alphay_epsilon=0}\n\\end{eqnarray}\nand hence, using Eqs.~(\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}), and the conservation law $\\bar\\zeta^2+|\\bar\\beta|^2=1$, find\n\\begin{equation}\n|\\bar\\alpha_\\pm|^2=-\\frac{\\Delta_0}{4\\Delta}\\frac{1-\\bar\\zeta_\\pm^2}{\\bar\\zeta_\\pm}.\n\\label{eqn:photon_number_epsilon=0}\n\\end{equation}\nThis result gives back Eq.~(\\ref{eqn:photon_number_kappa=0}), with $\\omega\\to\\Delta$ and $\\omega_0\\to\\Delta_0$, when $\\kappa=0$.\n\nFigure \\ref{fig:fig2} displays four cross-sections of the parameter space for $\\epsilon=0$ and $\\Delta_0=\\Delta$, each subdivided according to the number of distinct steady-state solutions. Frames (a) and (c) apply to the non-dissipative model ($\\kappa=0$), while frames (b) and (d) include cavity mode loss. Two complementary perspectives are provided: first, in frames (a) and (b), where the cut is the ($\\lambda\/\\Delta$,$\\eta$)-plane, and then, in frames (c) and (d), where the ($\\Delta\/\\lambda$,$\\eta$)-plane is shown. The first view envisages the coupling strength $\\lambda$, at fixed detuning $\\Delta$, as the control parameter, the historical view suggested by Refs.~\\cite{hepp&lieb_1973a,wang&hioe_1973,hepp&lieb_1973b,carmichael_etal_1973}; the second envisages $\\Delta$ as the control parameter, with $\\lambda$ fixed, which is more natural for experiments in optics and the perspective carried through the remainder of the paper. To connect with Secs.~\\ref{sec:Dicke_rotating_wave} and \\ref{sec:Dicke_counter_rotating}, we note the following points:\n\\begin{enumerate}[(i)]\n\\item\nThe Dicke quantum phase transition in the rotating-wave approximation, originally proposed by Hepp and Lieb \\cite{hepp&lieb_1973a}, maps to the line $\\eta=0$ in frames (a) and (c). The critical point $\\lambda\/\\Delta=\\Delta\/\\lambda=1$ marks a transition from the trivial solution to one with photon number $|\\alpha_\\pm|^2=(\\Delta^4-\\lambda^4)\/4\\lambda^2\\Delta^2$ [Eqs.~(\\ref{eqn:photon_number_eta=0}) and (\\ref{eqn:photon_number_epsilon=0})], where $\\bar\\zeta_\\pm=-\\Delta^2\/\\lambda^2$ is a double root of $P(\\bar\\zeta)=0$;  $\\bar\\beta\/\\bar\\alpha=-2\\Delta\/\\lambda$, but there is no preferred phase for $\\bar\\beta$, since Eqs.~(\\ref{eqn:nontrivial_alphax_epsilon=0}) and (\\ref{eqn:nontrivial_alphay_epsilon=0}) reduce to the tautology $0=0$.\n\\item\nThe $\\eta=0$ transition does not occur in the presence of dissipation, as in frames (b) and (d) the $\\eta=0$ axis bounds only the $R_2$ region.\n\\item\nThe critical point on the line $\\eta=0$ [frames (a) and (c)] splits into a pair of critical points when $\\eta>0$, subdividing the plane into regions of two, three, and four distinct solutions (two, four, and six solutions when double roots of $[P(\\bar\\zeta)]^2=0$ are considered). The transition at $\\lambda_{\\eta=1}^+=\\Delta\/2$ from region $R_2$ to $R_3$ recovers the renormalized critical point \\cite{carmichael_etal_1973} when the rotating-wave approximation is lifted---the $R_2\/R_3$ boundary carries that renormalization through as a function of $\\eta$. To our knowledge, the critical point defining the $R_3\/R_4$ boundary has not been reported before, although Hepp and Lieb do discuss a model that embraces our inclusion of the parameter $\\eta$ \\cite{hepp&lieb_1973b}. The transition between regions $R_3$ and $R_4$ is central to the unification we present with a coherent drive included (Sec.~\\ref{sec:coherent_drive_intermediate_eta}).\n\\item\nContrasting the situation in (i), nontrivial solutions in regions $R_3$ and $R_4$ assign $\\bar\\beta$ and $\\bar\\alpha$ a definite phase, through Eqs.~(\\ref{eqn:steady-state_beta1}), (\\ref{eqn:steady-state_beta2}), (\\ref{eqn:nontrivial_alphax_epsilon=0}), and (\\ref{eqn:nontrivial_alphay_epsilon=0}).\n\\item\nWhile the map from frame (b) to frame (d) appears straightforward, the map from frame (c) to frame (d) is not: a diagram with two boundaries at fixed $\\eta$ now acquires three, as the $R_2\/R_4$ boundary bends up to meet $\\eta=1$. This follows from the term $\\kappa^2\/\\Delta^2$ under the square root in Eq.~(\\ref{eqn:nontrivial_zeta_epsilon=0}): when $\\kappa\\neq0$, $\\bar\\zeta_\\pm$ are complex for $\\eta>\\eta_\\kappa$, a $\\Delta$-dependent condition at fixed $\\kappa$ [Eq.~(\\ref{eqn:eta_critical})].\n\\end{enumerate}\n\nFigure \\ref{fig:fig3} further illustrates the parameter dependence of the mean-field steady states in the absence of a drive. The symmetrical presentation of the phase diagram in frame (a) is modelled after Ref.~\\cite{carmichael_2015} (Figs.~1 and 2) and carried through in  Figs.~\\ref{fig:fig4}, \\ref{fig:fig5}, and \\ref{fig:fig7}. Frames (b)-(e) show steady states and their stability as a function of detuning for $\\eta=0.2$ and $\\eta=0.6$; they illustrate how the regions in frame (a) interconnect as solutions track smoothly with the changing detuning and bifurcate at boundaries:\n\\begin{description}\n\\item[Region $R_2$]\nSolutions $\\bar\\zeta=\\pm1$ only; the solution $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\item[Region $R_3$]\nSolutions $\\bar\\zeta=\\pm1$ and the root $\\bar\\zeta_+$ of $P(\\bar\\zeta)=0$; the solutions $\\bar\\zeta=\\pm1$ are both unstable and $\\bar\\zeta^+$ is stable. Three solutions in total.\n\\item[Region $R_4$]\nSolutions $\\bar\\zeta=\\pm1$ and the  roots $\\bar\\zeta_+$ and $\\bar\\zeta_-$ of $P(\\bar\\zeta)=0$; the solutions $\\bar\\zeta=-1$ (+1) and $\\bar\\zeta_+$ ($\\bar\\zeta_-$) are stable (unstable). Four solutions in total.\n\\end{description}\n\\noindent\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.7in]{figure2a.pdf}\\includegraphics[width=1.7in]{figure2b.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.7in]{figure2c.pdf}\\includegraphics[width=1.7in]{figure2d.pdf}\n\\end{center}\n\\caption{Mean-field phase diagram for zero drive and $\\Delta_0=\\Delta$: (a) $\\kappa\/\\Delta=0$, (b) $\\kappa\/\\Delta=0.7$, (c) $\\kappa\/\\lambda=0$, and (d) $\\kappa\/\\lambda=0.1$. The cut through parameter space is the $(\\eta,\\lambda\/\\Delta)$-plane in (a) and (b), and the $(\\eta,\\Delta\/\\lambda)$-plane in (c) and (d).}\n\\label{fig:fig2}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure3a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure3b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure3c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure3d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure3e.pdf}\n\\end{center}\n\\caption{Mean-field steady states for zero drive and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.1$ and $\\eta=0.2$ [(b),(c)] and $\\eta=0.6$ [(d),(e)]. The two sweeps through the phase diagram are indicated by dashed lines in (a); solid red (dashed blue) lines indicate locally stable (unstable) steady states in (b)-(e).}\n\\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Critical drive strength: $\\Delta_0=0$}\n\\label{sec:critical_drive}\nWe turn now to the dependence on the coherent drive strength, where we begin by identifying the critical point that organizes behavior as function of $\\epsilon$. To this end, we must first give special consideration to $\\Delta_0=0$, a limit not readily recovered from our general solution scheme, due to the $\\Delta_0$ in the denominator of Eqs.~(\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}); we essentially review an analysis presented by Alsing and Carmichael \\cite{alsing&carmichael_1991}, but extended here to arbitrary $\\eta$.\n\nFrom Eqs.~(\\ref{eqn:p_quadratic}) and (\\ref{eqn:q_quadratic}), when $\\Delta_0=0$, $P(\\bar\\zeta)=Q(\\bar\\zeta)=\\bar\\zeta^2$, and the 6th-order polynomial satisfied by $\\bar\\zeta$ becomes\n\\begin{equation}\n(1-\\bar\\zeta^2)\\bar\\zeta^4=\\left(\\epsilon\/\\epsilon_{\\rm crit}\\right)^2\\bar\\zeta^4,\n\\label{eqn:6th-order_polynomial_Delta_0=0}\n\\end{equation}\nwith\n\\begin{equation}\n\\epsilon_{\\rm crit}\\equiv\\frac12\\lambda(1+\\eta),\n\\label{eqn:critical_drive}\n\\end{equation}\nwhere the significance of $\\epsilon_{\\rm crit}$ as a critical drive strength is elaborated below.\nEquations (\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}) carry over in the form\n\\begin{equation}\n\\bar\\alpha_x\\bar\\zeta=\\bar\\alpha_y\\bar\\zeta=0,\n\\label{eqn:alpha_zeta_Delta_0=0}\n\\end{equation}\nand Eqs.~(\\ref{eqn:steady-state_alpha1}) and (\\ref{eqn:steady-state_alpha2}) as\n\\begin{eqnarray}\n\\kappa\\bar\\alpha_x-\\Delta\\bar\\alpha_y-\\lambda\\frac12(1-\\eta)\\bar\\beta_y&=&0,\n\\label{eqn:steady-state_alpha1_Delta_0=0}\\\\\n\\kappa\\bar\\alpha_y+\\Delta\\bar\\alpha_x+\\lambda\\frac12(1+\\eta)\\bar\\beta_x&=&-\\epsilon.\n\\label{eqn:steady-state_alpha2_Delta_0=0}\n\\end{eqnarray}\nWorking then from Eq.~(\\ref{eqn:alpha_zeta_Delta_0=0}), we can identify two distinct classes of solutions, one holding below $\\epsilon_{\\rm crit}$ and the other above.\n\n\\subsubsection{Solutions with $\\bar\\alpha_x=\\bar\\alpha_y=0$ ($\\epsilon\\leq\\epsilon_{\\rm crit}$)}\nEquation (\\ref{eqn:alpha_zeta_Delta_0=0}) may be satisfied with $\\bar\\alpha_x=\\bar\\alpha_y=0$, which, from Eqs.~(\\ref{eqn:steady-state_alpha1_Delta_0=0}) and (\\ref{eqn:steady-state_alpha2_Delta_0=0}), requires\n\\begin{equation}\n\\bar\\beta_x=-\\epsilon\/\\epsilon_{\\rm crit},\\qquad\\bar\\beta_y=0,\n\\end{equation}\nand hence, from the conservation law $\\bar\\zeta^2+|\\bar\\beta|^2=1$,\n\\begin{equation}\n\\bar\\zeta=\\pm\\sqrt{1-\\left(\\epsilon\/\\epsilon_{\\rm crit}\\right)^2}.\n\\label{eqn:zeta_below_Delta_0=0}\n\\end{equation}\nThe same result follows directly from Eq.~(\\ref{eqn:6th-order_polynomial_Delta_0=0}) under the assumption $\\bar\\zeta\\neq0$. This solution is physically acceptable for $\\epsilon\\leq\\epsilon_{\\rm crit}$, though larger drives require Eq.~(\\ref{eqn:alpha_zeta_Delta_0=0}) to be satisfied in another way.\n\n\\subsubsection{Solutions with $\\bar\\zeta=0$ ($\\epsilon\\geq\\epsilon_{\\rm crit}$)}\nEquation (\\ref{eqn:alpha_zeta_Delta_0=0}) may also be satisfied with $\\bar\\zeta=0$, which leaves only the phase of $\\bar\\beta$ to be determined:\n\\begin{equation}\n\\bar\\beta=e^{i\\phi}.\n\\end{equation}\nFrom Eq.~(\\ref{eqn:mean-field_zeta}), the phase of $\\bar\\alpha$ must satisfy\n\\begin{equation}\n{\\rm Im}\\big[\\bar\\alpha(e^{-i\\phi}-\\eta e^{i\\phi})\\big]=0,\n\\end{equation}\nand also, from Eq.~(\\ref{eqn:mean-field_alpha}),\n\\begin{equation}\n\\bar\\alpha=-i\\frac{\\epsilon+\\epsilon_{\\rm crit}(e^{i\\phi}+\\eta e^{-i\\phi})\/(1+\\eta)}{\\kappa+i\\Delta}.\n\\end{equation}\nThe phase $\\phi$ is therefore a solution of the transcendental equation\n\\begin{equation}\n\\epsilon\\cos\\phi+\\epsilon_{\\rm crit}=\\frac{\\Delta\\sin\\phi}{\\kappa(1-\\eta^2)}[\\epsilon(1+\\eta)^2+\\epsilon_{\\rm crit}4\\eta\\cos\\phi].\n\\end{equation}\nIf we then take $\\Delta=0$ as well as $\\Delta_0=0$ (and $\\eta\\neq1$), we arrive at the much simpler equation\n\\begin{equation}\n\\phi=\\cos^{-1}(-\\epsilon_{\\rm crit}\/\\epsilon),\n\\end{equation}\nwith solution $\\phi=\\pi$ for $\\epsilon=\\epsilon_{\\rm crit}$ and two solutions for the phase of $\\bar\\beta$ above $\\epsilon_{\\rm crit}$. This prediction of a bistability in phase above $\\epsilon_{\\rm crit}$ recovers the so-called Spontaneous Dressed-State Polarization of Alsing and Carmichael \\cite{alsing&carmichael_1991} (see also \\cite{kilin&krinitskaya_1991}) but generalized to $\\eta\\neq0$.\n\n\\subsection{Rotating-wave approximation with coherent drive: $\\eta=0$}\n\\label{sec:rotating_wave_coherent_drive_eta=0}\nWe now begin to lay out the connection between the breakdown of photon blockade and the coherently driven extension of the Dicke quantum phase transition. In this section, we introduce the breakdown of photon blockade  as the coherently driven extension of Sec.~\\ref{sec:epsilon=0} in the limit $\\eta=0$. In so doing, we introduce a completely new region of nontrivial steady states, one disconnected and distinct from regions $R_3$ and $R_4$ of Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}. What follows recovers results from Ref.~\\cite{carmichael_2015}.\n\nReturning to the 6th-order polynomial satisfied by $\\bar\\zeta$, Eq.~(\\ref{eqn:6th-order_polynomial}), with $\\eta$ zero, $Q(\\bar\\zeta)=P(\\bar\\zeta)$, and the polynomial takes the simpler form\n\\begin{equation}\n(1-\\bar\\zeta^2)[P(\\bar\\zeta)]^2=\\bar\\epsilon^2\\bar\\zeta^2P(\\bar\\zeta),\n\\label{eqn:6th-order_polynomial_eta=0}\n\\end{equation}\nwith\n\\begin{equation}\nP(\\bar\\zeta)=(\\bar\\Delta_0\\bar\\kappa)^2+(\\bar\\Delta_0\\bar\\Delta+\\bar\\zeta)^2,\n\\label{eqn:p_quadratic_eta=0}\n\\end{equation}\nwhere we have introduced parameters scaled by $\\epsilon_{\\rm crit}$:\n\\begin{equation}\n\\bar\\epsilon\\equiv\\epsilon\/\\epsilon_{\\rm crit},\\qquad (\\bar\\kappa,\\bar\\Delta,\\bar\\Delta_0)\\equiv(\\kappa,\\Delta,\\Delta_0)\/2\\epsilon_{\\rm crit}.\n\\label{eqn:scaled_parameters}\n\\end{equation}\nThe roots of $P(\\bar\\zeta)=0$ are nonphysical (complex) when $\\eta=0$ [Eq.~(\\ref{eqn:nontrivial_zeta_epsilon=0})] and therefore $P(\\bar\\zeta)$ may be cancelled on both sides of Eq.~(\\ref{eqn:6th-order_polynomial_eta=0}), which means there are at most four distinct solutions.\n\nTurning then to the field, the homogeneous system, Eq.~(\\ref{eqn:steady-state_alpha_epsilon=0}), is replaced by\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\bar\\kappa&-\\bar\\Delta-\\bar\\Delta_0^{-1}\\bar\\zeta\\\\\n\\noalign{\\vskip2pt}\n\\bar\\Delta+\\bar\\Delta_0^{-1}\\bar\\zeta&\\bar\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip2pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=\\left(\n\\begin{matrix}\n0\\\\\n\\noalign{\\vskip2pt}\n-\\bar\\epsilon\/2\n\\end{matrix}\n\\right)\n\\label{eqn:inhomogeneous_system_eta=0}\n\\end{equation}\nwith solution for the field amplitude ($\\bar\\Delta_0\\neq0$)\n\\begin{equation}\n\\bar\\alpha=-i\\frac{\\bar\\epsilon\/2}{\\bar\\kappa+i\\left(\\bar\\Delta+\\bar\\Delta_0^{-1}\\bar\\zeta\\right)}.\n\\label{eqn:steady-state_alpha3_eta=0}\n\\end{equation}\nThus, the field mode responds to coherent driving as a resonator in the presence of a nonlinear dispersion, where the dispersion is defined by solutions to Eq.~(\\ref{eqn:6th-order_polynomial_eta=0}). If we then note that $P(\\bar\\zeta)=\\bar\\Delta_0^2\\bar\\epsilon^2\/4|\\bar\\alpha|^2$ [Eqs.~(\\ref{eqn:p_quadratic_eta=0}) and (\\ref{eqn:steady-state_alpha3_eta=0})], whence, from Eq.~(\\ref{eqn:6th-order_polynomial_eta=0}),\n\\begin{equation}\n\\bar\\zeta=\\pm\\frac{|\\bar\\Delta_0|}{\\left(\\bar\\Delta_0^2+4|\\bar\\alpha|^2\\right)^{1\/2}},\n\\end{equation}\nwe recover the autonomous equation of state for the field mode \\cite{carmichael_2015}:\n\\begin{equation}\n\\bar\\alpha=-i\\frac{\\bar\\epsilon\/2}{\\bar\\kappa+i\\left[\\bar\\Delta\\pm\\hbox{sgn}\\left(\\bar\\Delta_0)(\\bar\\Delta_0^2+4|\\bar\\alpha|^2\\right)^{-1\/2}\\right]}.\n\\label{eqn:state_equation_eta=0}\n\\end{equation}\n\nFigure \\ref{fig:fig4} illustrates the results for mean-field steady states obtained from Eqs.~(\\ref{eqn:6th-order_polynomial_eta=0}) and (\\ref{eqn:state_equation_eta=0}) when $\\Delta_0=\\Delta$. The phenomenology follows that mapped out in Fig.~4 of Ref.~\\cite{carmichael_2015}, where regions of two and four distinct solutions [frame (a)] interconnect through the frequency pulling of vacuuum Rabi resonances located  at $\\Delta\/2\\epsilon_{\\rm crit}=\\pm1$ for $\\epsilon\/\\epsilon_{\\rm crit}\\to0$:\n\\begin{description}\n\\item[Region $R_2^a$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\item[Region $R_4$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two additional solutions that arise from the bistable folding of the solution that approaches $\\bar\\zeta=-1$; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable), and the two additional solutions are stable and unstable. Four solutions in total.\n\\item[Region $R_2^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of large detuning; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\end{description}\nWe emphasize that regions $R_2^a$ and $R_2^b$ comprise a single connected region of two distinct solutions in frame (a) of Fig.~\\ref{fig:fig4}; region $R_4$ does not touch the $\\Delta\/2\\epsilon_{\\rm crit}$ axis, although it comes close when $\\kappa\/\\lambda$ is small. We note also that regions $R_4$ of Fig.~\\ref{fig:fig3} and $R_4$ of Fig.~\\ref{fig:fig4} are distinct and do not share a common boundary; their interface occurs away from $\\eta=0$ and is discussed in Sec.~\\ref{sec:coherent_drive_intermediate_eta}.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure4a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure4b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure4c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure4d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure4e.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure4f.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure4g.pdf}\n\\end{center}\n\\caption{Mean-field steady states for $\\eta=0$ and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.02$ and $\\epsilon\/\\epsilon_{\\rm crit}=0.6$ [(b),(c)], $\\epsilon\/\\epsilon_{\\rm crit}=1.0$ [(d),(e)], and $\\epsilon\/\\epsilon_{\\rm crit}=1.2$ [(f),(g)]. The three sweeps through the phase diagram are indicated by dashed lines in (a); solid red (dashed blue) lines indicate stable (unstable) steady states in (b)-(g); dashed black lines demark the range of bistability in (c).}\n\\label{fig:fig4}\n\\end{figure}\n\n\\subsection{The quantum Rabi Hamiltonian with coherent drive: $\\eta=1$}\n\\label{sec:Dicke_coherent_drive_eta=1}\nTaking now the opposite limit, $\\eta=1$, we meet with a region of nontrivial steady states that is contiguous with $R_3$ of Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}. The new region supports four distinct solutions, while $R_3$ supports only three. Nonetheless, the boundary forms a continuous interface since one solution in $R_3$ corresponds to a double root of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0})---a root of $[P(\\bar\\zeta)]^2=0$; the coherent drive lifts this degeneracy and splits one distinct solution into two.\n\nIn order to avoid the divergence of $P(\\bar\\zeta)$ and $Q(\\bar\\zeta)$ as $\\eta\\to1$, we take Eqs.~(\\ref{eqn:p_quadratic}) and (\\ref{eqn:q_quadratic}) over in the form\n\\begin{equation}\n(1-\\eta)^2P(\\bar\\zeta)=4\\bar\\Delta\\bar\\Delta_0\\bar\\zeta+4\\bar\\Delta_0^2(\\bar\\kappa^2+\\bar\\Delta^2),\n\\end{equation}\nand\n\\begin{equation}\n(1-\\eta)^4Q(\\bar\\zeta)=16\\bar\\Delta^2\\Delta_0^2,\n\\end{equation}\nin which case the 6th-order polynomial in $\\bar\\zeta$, Eq.~(\\ref{eqn:6th-order_polynomial}), simplifies as\n\\begin{equation}\n(1-\\bar\\zeta^2)\\mkern-2mu\\left[\\bar\\zeta+\\frac{\\bar\\Delta_0}{\\bar\\Delta}(\\bar\\kappa^2+\\bar\\Delta^2)\\right]^2=\\bar\\epsilon^2\\bar\\zeta^2,\n\\label{eqn:6th-order_polynomial_eta=1}\n\\end{equation}\nagain a 4th-order polynomial with two or four physically acceptable solutions. In the $\\bar\\epsilon\\to0$ limit, the range of four solutions is confined by the inequality\n\\begin{equation}\n\\frac{|\\bar\\Delta_0|}{|\\bar\\Delta|}(\\bar\\kappa^2+\\bar\\Delta^2)\\leq1,\n\\end{equation}\nwhich recovers the $\\lambda_{\\eta\\to1}^+$ threshold of Eq.~(\\ref{eqn:lambda_critical}). Note also that, as advertised, the root  $\\bar\\zeta=-(\\bar\\Delta_0\/\\bar\\Delta)(\\bar\\kappa^2+\\bar\\Delta^2)$ on the $\\bar\\epsilon=0$ boundary is a double root; thus the region $R_4$ of Fig.~\\ref{fig:fig5}---four distinct roots in the interior---interfaces continuously with the three distinct roots of region $R_3$ in Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}.\n\nTurning to the field, from Eqs.~(\\ref{eqn:steady-state_alpha1}) and (\\ref{eqn:steady-state_alpha2}), Eq.~(\\ref{eqn:inhomogeneous_system_eta=0}) ($\\eta=0$) is replaced by\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\bar\\kappa&-\\bar\\Delta\\\\\n\\noalign{\\vskip2pt}\n\\bar\\Delta+\\bar\\Delta_0^{-1}\\bar\\zeta&\\bar\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip2pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=\\left(\n\\begin{matrix}\n0\\\\\n\\noalign{\\vskip2pt}\n-\\bar\\epsilon\/2\n\\end{matrix}\n\\right),\n\\label{eqn:inhomogeneous_system_eta=1}\n\\end{equation}\nwhere the coupling through $\\bar\\zeta$ is no longer symmetrical in the off-diagonals of the matrix on the left-hand side, and is therefore not serving the function of a nonlinear dispersion. Indeed, the physical interpretation for $\\eta=1$ says the coupling through $\\bar\\zeta$ belongs on the right-hand side of Eq.~(\\ref{eqn:inhomogeneous_system_eta=1}) where it acts as a nonlinear drive. The interpretation is made particularly clear if we write\n\\begin{equation}\n\\bar\\beta=\\bar\\Delta_0^{-1}2\\bar\\alpha_x\\bar\\zeta,\n\\end{equation}\nEqs.~(\\ref{eqn:steady-state_beta1}) and (\\ref{eqn:steady-state_beta2}), and then, from $\\bar\\zeta^2+|\\bar\\beta|^2=1$,\n\\begin{equation}\n\\bar\\zeta=\\pm|\\bar\\Delta_0|(\\bar\\Delta_0^2+4\\bar\\alpha_x^2)^{-1\/2}.\n\\end{equation}\nNow, moving the term  $\\Delta_0^{-1}\\bar\\alpha_x\\bar\\zeta$ to the right-hand side of Eq.~(\\ref{eqn:inhomogeneous_system_eta=1}), the equation is rewritten as\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\bar\\kappa&-\\bar\\Delta\\\\\n\\noalign{\\vskip2pt}\n\\bar\\Delta&\\bar\\kappa\n\\end{matrix}\n\\mkern3mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n\\bar\\alpha_x\\\\\n\\noalign{\\vskip2pt}\n\\bar\\alpha_y\n\\end{matrix}\n\\right)=\\left(\n\\begin{matrix}\n0\\\\\n\\noalign{\\vskip2pt}\n-\\bar\\epsilon\/2\\mp\\bar\\alpha_x(\\bar\\Delta_0^2+4\\bar\\alpha_x^2)^{-1\/2}\n\\end{matrix}\n\\right),\n\\label{eqn:alpha_eta=1}\n\\end{equation}\nwhere, if we can assume $4\\bar\\alpha_x^2\\gg\\bar\\Delta_0^2$, we find two solutions with the amplitude of the coherent drive simply changed from $\\bar\\epsilon$ to $\\bar\\epsilon\\pm1$:\n\\begin{equation}\n\\bar\\alpha=-i\\frac{(\\bar\\epsilon\\pm1)\/2}{\\bar\\kappa+i\\bar\\Delta},\n\\label{eqn:alpha_eta=1_approx}\n\\end{equation}\nand $\\bar\\zeta=\\pm|\\bar\\Delta_0|\/|\\bar\\alpha_x|$, $\\bar\\beta=\\pm{\\rm sgn}(\\bar\\Delta_0){\\rm sgn}(\\bar\\alpha_x)$.\n\nMore generally, Fig.~\\ref{fig:fig5} shows the dependence of mean-field steady states on drive amplitude and detuning for $\\eta=1$ and $\\bar\\Delta_0=\\bar\\Delta$; frames (b)-(g) illustrate results for three sweeps through a parameter space that divides into just two separate regions [frame (a)]:\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure5a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure5b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure5c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure5d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure5e.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure5f.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure5g.pdf}\n\\end{center}\n\\caption{Mean-field steady states for $\\eta=1$ and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.02$  and $\\epsilon\/\\epsilon_{\\rm crit}=0.6$ [(b),(c)], $\\epsilon\/\\epsilon_{\\rm crit}=1.0$ [(d),(e)], and $\\epsilon\/\\epsilon_{\\rm crit}=1.2$ [(f),(g)]. The three sweeps through the phase diagram are indicated by dashed lines in (a); solid red (dashed blue) lines indicate stable (unstable) steady states in (b)-(g); dashed black lines demark the range of bistability in (c).}\n\\label{fig:fig5}\n\\end{figure}\n\\begin{description}\n\\item[Region $R_4$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two that approach the root $\\bar\\zeta_+=-\\bar\\kappa^2-\\bar\\Delta^2$ of $[P(\\bar\\zeta)]^2=0$; the solutions that approach $\\bar\\zeta=\\bar\\zeta_+$ ($\\pm1$) are stable (unstable); the solution that approaches $\\bar\\zeta=-1$ links in a closed loop to one of the solutions approaching $\\bar\\zeta_+$. Four solutions in total.\n\\item[Region $R_2^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of large detuning; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\end{description}\nWe note the following additional points:\n\\begin{enumerate}[(i)]\n\\item\nTwo of the four solutions in region $R_4$ are consistent with the assumption adopted above Eq.~(\\ref{eqn:alpha_eta=1_approx}) [frame (c) of Fig.~\\ref{fig:fig5}] so long as $\\bar\\kappa\\ll1$; the remaining two solutions satisfy Eq.~(\\ref{eqn:alpha_eta=1}) but do not admit the approximation leading to Eq.~(\\ref{eqn:alpha_eta=1_approx}).\n\\item\nThe boundary between regions $R_4$ and $R_2^b$ in frame (a) of Fig.~\\ref{fig:fig5} follows the curve\n\\begin{equation}\n\\bar\\epsilon=\\left\\{1-\\left[\\frac{|\\bar\\Delta_0|}{|\\bar\\Delta|}(\\bar\\kappa^2+\\bar\\Delta^2)\\right]^{2\/3}\\right\\}^{3\/2}.\n\\end{equation}\nThe boundary is a line of double roots of Eq.~(\\ref{eqn:6th-order_polynomial_eta=1}), and the curve may be found by equating derivatives on the left- and right-hand sides of this equation.\n\\item\nThe critical point $\\epsilon_{\\rm crit}$ [Eq.~(\\ref{eqn:critical_drive})] organizes behavior as a function of drive strength and detuning in much the same way as it does for  $\\eta=0$.\n\\item\nThe closed loop in frame (b) of Fig.~\\ref{fig:fig5} is similar to the loop in frame (b) of Fig.~\\ref{fig:fig4}; both shrink with increasing drive strength to eventually vanish at the critical point---frames (d) of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5}. Note, though, that the stabilities are interchanged; this change is clearly reflected in the accompanying plots of $|\\bar\\alpha|^2$ [frames (c) of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5}].\n\\item\nThe stable solutions displayed in frames (c), (e), and (g) of Fig.~\\ref{fig:fig5} are all single nearly Lorentzian peaks; the splitting in the corresponding frames of Fig.~\\ref{fig:fig4} does not occur.\n\\end{enumerate}\n\n\\subsection{Intermediate regime: $0<\\eta<1$}\n\\label{sec:coherent_drive_intermediate_eta}\nSummarizing what we have learned: with no counter-rotating interaction, the dissipative Dicke system shows no phase transition as a function of coupling strength [$\\eta=0$ in frames (b) and (d) of Fig.~\\ref{fig:fig1}], although the breakdown of photon blockade takes place in the presence of a coherent drive (Fig.~\\ref{fig:fig4}); the dissipative system does, however, show the standard phase transition when $\\eta=1$, where it is deformed by a coherent drive and vanishes with increasing drive strength at a renormalized photon-blockade-breakdown critical point (Fig.~\\ref{fig:fig5}).\n\nIn this section we unify these limiting cases by letting $\\eta$ vary continuously between 0 and 1. We show how the previously unreported phase of the Dicke system, i.e., region $R_4$ of Figs.~\\ref{fig:fig2} and \\ref{fig:fig3}, underlies this unification.\n\nWe begin with the interface between frame (a) of Fig.~\\ref{fig:fig3} and frame (a) of Fig.~\\ref{fig:fig5}, where regions of three and four distinct solutions connect on the boundary $\\bar\\epsilon=0$, $\\eta=1$: moving off the boundary with a perturbation $\\bar\\epsilon\\to\\delta\\bar\\epsilon$ lifts the degeneracy of a double root of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}), and thus provides the link between regions. Something similar is encountered on the $\\bar\\epsilon=0$ boundary with $\\eta_\\kappa<\\eta<1$ (e.g., along the lines $\\eta=0.6$ and $\\eta=0.2$ in Fig.~\\ref{fig:fig3}); however, now two regions, $R_3$ and $R_4$, link to contiguous regions under the perturbation $\\bar\\epsilon\\to\\delta\\bar\\epsilon$. Since $R_4$ accommodates \\emph{two} double roots of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}), we predict its linkage to a contiguous region of six distinct solutions in the presence of a coherent drive.\n\nWe illustrate this situation in Fig.~\\ref{fig:fig6} where we plot the function $\\sqrt{1-\\bar\\zeta^2}P(\\bar\\zeta)$---the square root of the left-hand side of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0})---for four detunings along the $\\eta=0.2$ sweep of Fig.~\\ref{fig:fig3}: frames (a), (b), (c), (d) refer, in sequence, to points in regions $R_2$, $R_3$, $R_4$, $R_2$ along the sweep---moving inwards from either end; they show examples of two, three, four, and again two distinct roots. The trivial roots, $\\bar\\zeta=\\pm1$, appear in every plot, while the additional roots [frames (b) and (c)] are double roots of $[P(\\bar\\zeta)]^2=0$. The perturbation $\\bar\\epsilon\\to\\delta\\bar\\epsilon$ replaces each dashed line in the figure by a pair of curves $\\pm\\delta\\bar\\epsilon|\\bar\\zeta|\\sqrt{Q(\\bar\\zeta)}$, and thus lifts the degeneracy of each double root. [It is readily shown that $Q(\\bar\\zeta)>0$.]\n\nFigure~\\ref{fig:fig7} shows how the results displayed in Figs.~\\ref{fig:fig3}-\\ref{fig:fig5} are generalized for $\\eta=0.2$ and $\\bar\\Delta_0=\\bar\\Delta$. The parameter space is now divided into a larger number of regions, integrating those already met in the three limiting cases:\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=1.7in]{figure6a.pdf}\\includegraphics[width=1.7in]{figure6b.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.7in]{figure6c.pdf}\\includegraphics[width=1.7in]{figure6d.pdf}\n\\end{center}\n\\caption{The square root of the left-hand side of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}) as a function of $\\bar\\zeta$ for $\\eta=0.2$ and $\\Delta_0=\\Delta$: $\\kappa\/2\\epsilon_{\\rm crit}=1\/12$ and $|\\Delta|\/2\\epsilon_{\\rm crit}=1.0$, $0.7$, $0.5$, $0.15$ [(a)-(d)]. Zeros of this function (crossings of the black dashed lines) are roots of Eq.~(\\ref{eqn:6th-order_polynomial_epsilon=0}).}\n\\label{fig:fig6}\n\\end{figure}\n\n\\begin{description}\n\\item[Region $R_2^a$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\item[Region $R_6$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and four additional solutions---two that approach each of the double roots, $\\bar\\zeta_\\pm$, of $[P(\\bar\\zeta)]^2=0$. The solutions approaching $\\bar\\zeta=-1$ and $\\bar\\zeta_+$ ($+1$ and $\\bar\\zeta_-$) are stable (unstable). Six solutions in total.\n\\item[Region $R_4^a$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two that approach the double root $\\bar\\zeta_+$ of $[P(\\bar\\zeta)]^2=0$; the solutions approaching $\\bar\\zeta_+$ ($\\pm1$) are stable (unstable). Four solutions in total.\n\\item[Region $R_4^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of zero drive and two additional solutions that arise from the bistable folding of the solution that approaches $\\bar\\zeta=-1$; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable), and the two additional solutions are one stable\/unstable. Four solutions in total.\n\\item[Region $R_2^b$]\nTwo solutions that approach $\\bar\\zeta=\\pm1$ in the limit of large detuning; the solution approaching $\\bar\\zeta=-1$ ($+1$) is stable (unstable). Two solutions in total.\n\\end{description}\n\nFrames (b)-(e) of Fig.~\\ref{fig:fig7} show how the corresponding plots in Fig.~\\ref{fig:fig3} change when the degeneracy of the double roots ($\\bar\\epsilon=0$) is lifted ($\\bar\\epsilon\\neq0$) to link regions $R_3$ and $R_4$ of Fig.~\\ref{fig:fig3} to regions $R_4^a$ and $R_6$, respectively, of Fig.~\\ref{fig:fig7}. (Note, however, that $\\kappa\/\\lambda$ takes different values in the figures, so region boundaries do not line up.) The change is clearly seen, for example, comparing frame (b) of Fig.~\\ref{fig:fig3} with frames (b) and (d) of Fig.~\\ref{fig:fig7}: a single stable upper branch---Fig.~\\ref{fig:fig3}---is split into two stable upper branches---Fig.~\\ref{fig:fig7}; and a single unstable branch connecting upper and lower stable branches in Fig.~\\ref{fig:fig3}  splits into two unstable branches in Fig.~\\ref{fig:fig7} [near overlapping dashed lines in frame (d)]. In this way features met separately in the limiting cases of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5} are linked together.\n\nFinally, for larger amplitudes of the drive---e.g., adding sweeps at $\\bar\\epsilon=0.6$, 1.0, and 1.2 in frame (a) of Fig.~\\ref{fig:fig7}---mean-field steady states follow the breakdown of photon blockade, as in frames (b)-(g) of Fig.~\\ref{fig:fig4}.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure7a.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure7b.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure7c.pdf}\n\\vskip0.1in\n\\includegraphics[width=1.65in]{figure7d.pdf}\\hskip0.075in\\includegraphics[width=1.65in]{figure7e.pdf}\n\\end{center}\n\\caption{Mean-field steady states for $\\eta=0.2$ and $\\Delta_0=\\Delta$: $\\kappa\/\\lambda=0.02$ and $\\epsilon\/\\epsilon_{\\rm crit}=0.2$; [(d),(e)] expands the view in [(b),(c)]. The sweep through the phase diagram is indicated by the dashed line in (a); solid red (dashed blue and magenta) lines indicate stable (unstable) steady states in (b)-(e); dashed black lines demark the range of bistability or tristability in (c).}\n\\label{fig:fig7}\n\\end{figure}\n\n\\section{Quantum Fluctuations: One Two-State System}\n\\label{sec:quantum_fluctuations}\nWhile the mean field analysis may be highly suggestive of what to expect from an experimental realization of our generalized Jaynes-Cummings-Rabi model, an account in these terms is incomplete---fluctuations are neglected. We encounter coexisting steady states, for example, and although both are stable under small perturbations when Maxwell-Bloch equations are solved, what of the stability once quantum fluctuations are introduced?\n\nIt is beyond the scope of this work to address questions like this in any detail. We limit ourselves here to a few observations about the full quantum treatment for the case $N=1$, where a number of calculations are feasible, some analytical and some numerical, to parallel results for the breakdown of photon blockade \\cite{carmichael_2015}. While it may seem that $N=1$ takes us very far from a many-particle limit where contact with mean-field results may be made, this is not generally the case: it is shown in Ref.~\\cite{carmichael_2015} that the many-photon limit is a strong-coupling limit, and many of the figures from Sec.~\\ref{sec:mean-field} have photon numbers ranging in the hundreds for $N=1$---after the scaling of Eq.~(\\ref{eqn:scaling}) is undone.\n\nIn this section, we show that the $\\eta$-dependence of the critical drive strength (Sec.~\\ref{sec:critical_drive}) follows from the quasi-energy spectrum, extending the previous calculation of the spectrum for $\\eta=0$~\\cite{alsing_etal_1992} to the general case. We then address the role of multi-photon resonances in the limit of small $\\eta$, where we uncover behavior similar to multi-photon blockade \\cite{shamailov_etal_2010} under weak coherent driving, but only for even numbers of photons absorbed. Finally, we use quantum trajectories to explore the accessibility of co-existing mean-field steady states in the presence of fluctuations.\n\n\\subsection{Quasienergies for $\\Delta_0=\\Delta=0$}\n\nEver since the seminal work of Jaynes and Cummings \\cite{jaynes&cummings_1963} (see also \\cite{paul_1963})), the energy spectrum of a single two-state system interacting with one mode of the radiation field has been a fundamental element of quantum optics models and physical understanding. The level scheme is remarkably simple when compared with extensions to the quantum Rabi model \\cite{braak_2011} and generalizations to include a counter-rotating interaction after the manner of Sec.~\\ref{sec:Dicke_counter_rotating} \\cite{tomka_etal_2014}. Alsing \\emph{et al}. \\cite{alsing_etal_1992} showed that the simplicity carries over to the driven Jaynes-Cummings Hamiltonian when the two-state system and radiation mode are resonant with the drive. In this case, a Bogoliubov transformation diagonalizes the interaction picture Hamiltonian, so that quasienergies are recovered. The critical drive $\\epsilon_{\\rm crit}$ is then the point at which all quasienergies collapse to zero. In this section we show that the method employed by Alsing \\emph{et al}. carries through for arbitrary $\\eta$, and the collapse to zero reproduces Eq.~(\\ref{eqn:critical_drive}).\n\nWe consider the Hamiltonian $H_\\eta^{\\prime\\prime}=H_\\eta^\\prime+\\sqrt N\\epsilon(a^\\dagger+a)$, where $H_\\eta^\\prime$ is given by Eq.~(\\ref{eqn:hamiltonian_raman_model}). Taking the coherent drive on resonance and considering just one two-state system, the Hamiltonian is\n\\begin{equation}\nH_\\eta^{\\prime\\prime}=\\lambda(a\\sigma_++a^\\dagger\\sigma_-)+\\eta\\lambda(a^\\dagger\\sigma_++a\\sigma_-)+\\epsilon(a^\\dagger+a).\n\\label{eqn:hamiltonian_zero_detuning_one_atom}\n\\end{equation}\nWe seek solutions to the eigenvalue problem $H_\\eta^{\\prime\\prime}|\\psi_E\\rangle=E|\\psi_E\\rangle$, where $E$ is a quasienergy and\n\\begin{equation}\n|\\psi_E\\rangle=|\\psi^{(1)}_E\\rangle|1\\rangle+|\\psi^{(2)}_E\\rangle|2\\rangle,\n\\label{eqn:eigenket}\n\\end{equation}\nwith the kets $|\\psi^{(1,2)}_E\\rangle$ expanded over the Fock states, $|n\\rangle$, $n=1,2,\\ldots$, of the field mode; we must find allowed values of $E$ and the corresponding field kets.\n\nIt is straightforward to show that the field kets satisfy the homogeneous system of equations\n\\begin{equation}\n\\left(\n\\begin{matrix}\n\\epsilon(a^\\dagger+a)-E&\\lambda(a^\\dagger+\\eta a)\\\\\n\\noalign{\\vskip4pt}\n\\lambda(\\eta a^\\dagger+a)&\\epsilon(a^\\dagger+a)-E\n\\end{matrix}\n\\mkern2mu\\right)\\mkern-4mu\\left(\n\\begin{matrix}\n|\\psi^{(1)}_E\\rangle\\\\\n\\noalign{\\vskip4pt}\n|\\psi^{(2)}_E\\rangle\n\\end{matrix}\n\\right)=0,\n\\label{eqn:quasienergy_homogeneous_system}\n\\end{equation}\nwhence multiplication on the left by ${\\rm diag}(\\eta a^\\dagger+a,a^\\dagger+\\eta a)$ takes us to the coupled equations:\n\\begin{eqnarray}\n-\\epsilon(1-\\eta)|\\psi^{(1)}_E\\rangle&=&[\\epsilon(a^\\dagger+a)-E](\\eta a^\\dagger+a)|\\psi^{(1)}_E\\rangle\\notag\\\\\n\\noalign{\\vskip2pt}\n&&+\\lambda[aa^\\dagger+\\eta(a^{\\dagger 2}+a^2)+\\eta^2a^\\dagger a]|\\psi^{(2)}_E\\rangle,\\notag\\\\\n&&\\label{eqn:eigenket_again1}\\\\\n\\epsilon(1-\\eta)|\\psi^{(2)}_E\\rangle&=&[\\epsilon(a^\\dagger+a)-E](a^\\dagger+\\eta a)|\\psi^{(2)}_E\\rangle\\notag\\\\\n\\noalign{\\vskip2pt}\n&&+\\lambda[a^\\dagger a+\\eta(a^{\\dagger 2}+a^2)+\\eta^2aa^\\dagger]|\\psi^{(1)}_E\\rangle.\\notag\\\\\n&&\\label{eqn:eigenket_again2}\n\\end{eqnarray}\nWe then use Eq.~(\\ref{eqn:quasienergy_homogeneous_system}) to substitute for $(\\eta a^\\dagger+a)|\\psi^{(1)}_E\\rangle$ and $(a^\\dagger+\\eta a)|\\psi^{(2)}_E\\rangle$, respectively, on the right-hand sides of Eqs.~(\\ref{eqn:eigenket_again1}) and (\\ref{eqn:eigenket_again2}), and thus obtain the more compact form:\n\\begin{eqnarray}\n\\left(O(E)-\\lambda^2\\frac{1-\\eta^2}2\\right)|\\psi^{(1)}_E\\rangle-\\epsilon\\lambda(1-\\eta)|\\psi^{(2)}_E\\rangle&=&0,\\mkern20mu\n\\label{eqn:eigenket_yet_again1}\\\\\n\\noalign{\\vskip2pt}\n\\left(O(E)+\\lambda^2\\frac{1-\\eta^2}2\\right)|\\psi^{(2)}_E\\rangle+\\epsilon\\lambda(1-\\eta)|\\psi^{(1)}_E\\rangle&=&0,\n\\label{eqn:eigenket_yet_again2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nO(E)&=&\\lambda^2(1+\\eta^2)\\frac{a^\\dagger a+aa^\\dagger}2+\\lambda^2\\eta\\left(a^{\\dagger 2}+a^2\\right)\\notag\\\\\n&&-\\left[\\epsilon\\left(a^\\dagger+a\\right)-E\\right]^2.\n\\label{eqn:o_operator}\n\\end{eqnarray}\nSince the coefficients of the second terms on the left-hand side are constants, Eqs.~(\\ref{eqn:eigenket_yet_again1}) and (\\ref{eqn:eigenket_yet_again2}) can now be readily uncoupled, and yield the autonomous equation\n\\begin{equation}\nO_+(E)O_-(E)|\\psi^{(1,2)}_E\\rangle=0,\n\\label{eqn:oplus_by_ominus_equation}\n\\end{equation}\nwhere\n\\begin{equation}\nO_{\\pm}(E)=O(E)\\pm\\lambda^2\\frac{1-\\eta^2}2\\Lambda^{1\/2},\n\\label{eqn:oplusminus_operator}\n\\end{equation}\nwith\n\\begin{equation}\n\\Lambda=1-\\frac1{(1+\\eta)^2}\\frac{4\\epsilon^2}{\\lambda^2}.\n\\label{eqn:Lambda_definition}\n\\end{equation}\n\nNote now that $O_+(E)$ and $O_-(E)$ commute, and so the general solution to Eq.~(\\ref{eqn:oplus_by_ominus_equation}) expands as\n\\begin{equation}\n|\\psi_{E}^{(1,2)}\\rangle=c_+^{(1,2)}|\\psi_E^{(+)}\\rangle+c_-^{(1,2)}|\\psi_E^{(-)}\\rangle,\n\\label{eqn:oplus_by_ominus_solution}\n\\end{equation}\nwhere $|\\psi_E^{(+)}\\rangle$ and $|\\psi_E^{(-)}\\rangle$ satisfy\n\\begin{equation}\nO_{\\pm}(E)|\\psi_E^{(\\pm)}\\rangle=0.\n\\label{eqn:oplusminus_equation}\n\\end{equation}\nMoreover, the operators $O_{\\pm}(E)$ are quadratic in creation and annihilation operators, so the diagonalization may be completed by a Bogoliubov transformation: introduce parameters $\\nu$, $\\xi$, $\\alpha(E)$, and $\\mu_\\pm(E)$, such that\n\\begin{equation}\nO_\\pm(E)=\\nu U^\\dagger[\\xi,\\alpha(E)]\\frac{a^\\dagger a+aa^\\dagger}2U[\\xi,\\alpha(E)]+\\mu_\\pm(E),\n\\label{eqn:oplusminus_unitary}\n\\end{equation}\nwhere the unitary $U[\\xi,\\alpha(E)]\\equiv D[\\alpha(E)]S(\\xi)$ executes a displacement and then a squeeze,\n\\begin{equation}\na\\buildrel U\\over\\to[a+\\alpha(E)]\\cosh\\xi+[a^\\dagger+\\alpha(E)]\\sinh\\xi,\n\\label{eqn:a_transform}\n\\end{equation}\nwhence, from Eq.~(\\ref{eqn:oplusminus_equation}),\n\\begin{equation}\n\\left(\\frac{a^\\dagger a+aa^\\dagger}2+\\frac{\\mu_\\pm(E)}\\nu\\right)\\!\\left\\{U[\\xi,\\alpha(E)]|\\psi_E^{(\\pm)}\\rangle\\right\\}=0.\n\\end{equation}\nThe number operator now acts on the left-hand side, and $|\\psi_E^{(+)}\\rangle$ and $|\\psi_E^{(-)}\\rangle$ are displaced and squeezed Fock states:\n\\begin{equation}\n|\\psi_{E_{n_\\pm}}^{(\\pm)}\\rangle=U^\\dagger[\\xi,\\alpha(E_{n_\\pm})]|n_\\pm\\rangle,\n\\end{equation}\n$n_\\pm=0,1,2,\\ldots$, where allowed quasienergies are indexed by the integers $n_\\pm$ and must satisfy\n\\begin{equation}\nn_\\pm+\\frac12+\\frac{\\mu_\\pm(E_{n_\\pm})}\\nu=0.\n\\label{eqn:quasienergy_constraint}\n\\end{equation}\nIt remains to equate terms on both sides of Eq.~(\\ref{eqn:oplusminus_unitary}) to fix the parameters of the Bogoliubov transformation, which yields\n\\begin{equation}\n\\nu=\\lambda^2(1-\\eta^2)\\Lambda^{1\/2},\\qquad\n\\xi=\\frac12\\ln\\left(\\frac{1+\\eta}{1-\\eta}\\Lambda^{1\/2}\\right),\n\\label{eqn:paramters1}\n\\end{equation}\nand\n\\begin{eqnarray}\n\\alpha(E)&=&\\frac{2\\epsilon E}{\\lambda^2(1+\\eta)^2}\\Lambda^{-1},\\\\\n\\noalign{\\vskip4pt}\n\\mu_\\pm(E)&=&\\pm\\frac\\nu2-E^2\\Lambda^{-1},\n\\label{eqn:parameters2}\n\\end{eqnarray}\nand thus the allowed quasienergies follow from\n\\begin{equation}\nn_\\pm+\\frac12\\pm\\frac12-E_{n_{\\pm}}^2\\frac1{\\lambda^2(1-\\eta^2)}\\Lambda^{-3\/2}=0.\n\\label{eqn:energies1}\n\\end{equation}\n\nEquation (\\ref{eqn:energies1}) is the targeted result, which reveals the generalized critical drive strength. It is helpful, however, for clarity, to recognize that $n_+$ and $n_-$ provide a double coverage of the nonnegative integers---traced to the \\emph{two} components on the right-hand side of Eq.~(\\ref{eqn:oplus_by_ominus_solution})---and to replace $n_\\pm$ by a single index $n$: first, associate $n=0$ with $n_-=0$, from which Eq.~(\\ref{eqn:energies1}) yields the quasienergy\n\\begin{equation}\nE_0=0,\n\\label{eqn:energy_zero}\n\\end{equation}\nwith corresponding ket\n\\begin{equation}\n|\\psi_{E_0}^{(-)}\\rangle=U^\\dagger[\\xi,\\alpha(E_0)]|0\\rangle;\n\\end{equation}\nand second, associate $n=1,2,\\dots$ with both $n_+=n-1$ and $n_-=n$, both of which, when substituted in Eq.~(\\ref{eqn:energies1}), yield the quasienergy doublet\n\\begin{equation}\nE_{n,\\pm}=\\pm\\lambda\\sqrt n\\sqrt{1-\\eta^2}\\Lambda^{3\/4},\n\\label{eqn:quasienergy_doublet}\n\\end{equation}\nalthough with distinct corresponding kets:\n\\begin{eqnarray}\n|\\psi_{E_{n,\\pm}}^{(+)}\\rangle&=&U^\\dagger[\\xi,\\alpha(E_{n,\\pm})]|n-1\\rangle,\\\\\n\\noalign{\\vskip2pt}\n|\\psi_{E_{n,\\pm}}^{(-)}\\rangle&=&U^\\dagger[\\xi,\\alpha(E_{n,\\pm})]|n\\rangle.\n\\end{eqnarray}\n\nIt is clear from Eq.~(\\ref{eqn:quasienergy_doublet}) that all quasienergies collapse to zero for $n$ finite and $\\Lambda=0$, a condition that returns, from Eq.~(\\ref{eqn:Lambda_definition}), the critical drive strength $\\epsilon_{\\rm crit}$ [Eq.~(\\ref{eqn:critical_drive})]. From this fully quantum mechanical point of view, $\\epsilon_{\\rm crit}$ marks a transition from a discrete quasienergy spectrum to a continuous one; the continuous side is recovered from the limit $\\Lambda\\to0$, $n\\to\\infty$, $\\sqrt n\\Lambda^{3\/4}$ constant. Note that a continuous spectrum is also recovered in the limit $\\eta\\to1$, $n\\to\\infty$, $\\sqrt n\\sqrt{1-\\eta^2}$ constant. A continuous spectrum is expected for $\\eta=1$, since if we set $\\eta=1$ in Eq.~(\\ref{eqn:quasienergy_homogeneous_system}), $E$ is an eigenvalue of the quadrature operator $a^\\dagger+a$.\n\nThe coefficients $c_\\pm^{(1,2)}$ [Eq.~(\\ref{eqn:oplus_by_ominus_solution})] follow from Eqs.~(\\ref{eqn:quasienergy_homogeneous_system}) and (\\ref{eqn:eigenket_yet_again1}), and normalization (see Ref.~\\cite{alsing_etal_1992}).\n\n\\subsection{Multi-photon resonance}\nWith the focus on just one two-state system, Figs.~\\ref{fig:fig4}, \\ref{fig:fig5}, and \\ref{fig:fig7} show photon numbers ranging from zero to a few thousand, and although numbers are smaller in Fig.~\\ref{fig:fig3}, the range is similar when $\\kappa\/\\lambda$ is set to $0.02$ instead of $0.1$. While we might expect mean-field theory to be broadly reliable for thousands, even hundreds of photons, it will surely miss important features when photon numbers are small. Indeed, photon blockade is a photon by photon effect, underpinned, not by a mean-field nonlinearity, but by a strongly anharmonic ladder of few-photon excited states; it breaks down through multi-photon absorption, where, in Fig.~4 of Ref.~\\cite{carmichael_2015}, for example,  multi-photon resonances dominate the response to weak driving and the mean-field story of dispersive bistability is not picked up until $\\epsilon\/\\epsilon_{\\rm crit}\\sim0.4$.\n\nRecall now that in its dissipate realization (Sec.~\\ref{sec:dissipative_realization}) our generalized model involves not one, but two external\ndrives---a linear drive of strength $\\epsilon$, and a second, nonlinear drive of strength $\\eta$. We show now that the multi-photon response to weak driving carries over, with minor modification, from linear to nonlinear driving.\n\nReinstating detuning and setting $\\Delta_0=\\Delta$, we consider the Hamiltonian $H^{\\prime\\prime\\prime}_\\eta=\\Delta a^\\dagger a+\\Delta\\sigma_z+H^{\\prime\\prime}_\\eta$, where $H^{\\prime\\prime}_\\eta$ is given by Eq.~(\\ref{eqn:hamiltonian_zero_detuning_one_atom}). It is convenient for clarity, however, to adopt an interaction picture, where we define\n\\begin{eqnarray}\nH_\\eta^{\\prime\\prime}(t)&\\equiv&U_0^\\dagger(t)H^{\\prime\\prime}_\\eta U_0(t)\\notag\\\\\n\\noalign{\\vskip2pt}\n&=&H_{\\rm JC}+H_{\\epsilon}(t)+H_\\eta(t),\n\\end{eqnarray}\n$U_0(t)\\equiv\\exp[-i\\Delta(a^\\dagger a+\\sigma_z)t]$, and thus isolate the Jaynes-Cummings interaction,\n\\begin{equation}\nH_{\\rm JC}=\\lambda(a\\sigma_++a^\\dagger\\sigma_-),\n\\end{equation}\nwhich is perturbed by the linear drive\n\\begin{equation}\nH_\\epsilon(t)=\\epsilon(a^\\dagger e^{i\\Delta t}+ae^{-i\\Delta t}),\n\\end{equation}\nand the nonlinear drive\n\\begin{equation}\nH_\\eta(t)=\\eta\\lambda(a^\\dagger\\sigma_+e^{2i\\Delta t}+a\\sigma_-e^{-2i\\Delta t}).\n\\end{equation}\nWe also recall the eigenvalues and eigenkets of $H_{\\rm JC}$:\n\\begin{equation}\nE_0^{\\rm JC}=0,\\qquad E_{n,\\pm}^{\\rm JC}=\\pm\\lambda\\sqrt n,\n\\end{equation}\n$n=1,2,\\ldots$, and\n\\begin{eqnarray}\n|E_0^{\\rm JC}\\rangle&=&|0\\rangle|1\\rangle,\\\\\n|E_{n,\\pm}^{\\rm JC}\\rangle&=&\\frac1{\\sqrt2}\\left(|n\\rangle|1\\rangle\\pm|n-1\\rangle|2\\rangle\\right),\n\\end{eqnarray}\nwhere the first (second) ket refers to the field mode (two-state system) in each product on the right-hand side.\n\nNote now that the perturbation $H_\\epsilon(t)$ has non-zero matrix elements between neighboring pairs of kets in the $n$-step sequence\n\\begin{equation}\n|E_0^{\\rm JC}\\rangle\\rightarrow|E_{1,\\pm}^{\\rm JC}\\rangle\\rightarrow\\cdots\\rightarrow|E_{n-1,\\pm}^{\\rm JC}\\rangle\\rightarrow|E_{n,\\pm}^{\\rm JC}\\rangle,\n\\end{equation}\n$n=1,2,\\ldots$, while $H_\\eta(t)$ has non-zero matrix elements between pairs of kets in the $n\/2$-step sequence\n\\begin{equation}\n|E_0^{\\rm JC}\\rangle\\rightarrow|E_{2,\\pm}^{\\rm JC}\\rangle\\rightarrow\\cdots\\rightarrow|E_{n-2,\\pm}^{\\rm JC}\\rangle\\rightarrow|E_{n,\\pm}^{\\rm JC}\\rangle,\n\\end{equation}\n$n=2,4,\\ldots$. There are thus matrix elements to mediate multi-photon transitions from $|E_0^{\\rm JC}\\rangle$ to $|E_{n,\\pm}^{\\rm JC}\\rangle$ driven by either perturbation, but with the qualification that $H_\\eta(t)$ can only drive those with even $n$; resonance is achieved under the condition\n\\begin{equation}\n\\Delta=\\pm\\lambda\/\\sqrt n,\n\\end{equation}\nwhich is met either by $n$ steps of $\\Delta$ off-setting $\\pm\\lambda\\sqrt n$, or $n\/2$ steps of $2\\Delta$.\n\nFrame (a) of Fig.~\\ref{fig:fig8} illustrates the breakdown of photon blockade from a fully quantum mechanical point of view; we identify up to six multi-photon resonances before they begin to merge and wash out due to power broadening at higher drives. This figure displays quantum corrections, for $N=1$, to the mean-field results of Fig.~\\ref{fig:fig4}, where at high drives---$\\epsilon\/\\epsilon_{\\rm crit}=0.40$ and $0.48$---the layout of frame (a) of Fig.~\\ref{fig:fig4} begins to appear with the photon number averaged over fluctuation-driven switching between the pair of coexisting mean-field steady states.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure8a.pdf}\n\\vskip0.1in\n\\includegraphics[width=3.4in]{figure8b.pdf}\n\\end{center}\n\\caption{Steady-state photon number expectation computed from the master equation, Eq.~(\\ref{eqn:master_equation_drive}), for $N=1$, $\\Delta_0=\\Delta$, and $\\kappa\/\\lambda=0.02$: (a) $\\eta=0$ and $\\epsilon\/\\epsilon_{\\rm crit}=0.08$, $0.16$, $0.24$, $0.32$, $0.40$, $0.48$ (lower to upper) and (b) $\\epsilon\/\\epsilon_{\\rm crit}=0$ and $\\eta=0.04$, $0.08$, $0.12$, $0.16$, $0.20$, $0.24$ (lower to upper); successive curves  are displace upwards by $0.2$ and $0.3$ in (a) and (b), respectively.}\n\\label{fig:fig8}\n\\end{figure}\n\nFrame (b) of Fig.~\\ref{fig:fig8} shows the similar figure for driving through the nonlinear perturbation $H_\\eta(t)$. Once again multi-photon resonances are seen, but only three of the previous six---those corresponding to the absorption of two, four, and six photons. The figure in this case adds quantum corrections to the mean-field results of Fig.~\\ref{fig:fig3} (but note that $\\kappa\/\\lambda$ is $0.02$ in Fig~\\ref{fig:fig8} and $0.1$ in Fig.~\\ref{fig:fig3}).\n\n\\subsection{Quantum induced switching between mean-field steady states}\nWhile multi-photon resonances are completely beyond the scope of mean-field results, Fig.~\\ref{fig:fig8} does provide a hint of  mean-field predictions once photon numbers rise above two or three, where, in the vicinity of zero detuning, we see clear evidence of regions $R_2^a$ in Fig.~\\ref{fig:fig4} and $R_2$ in Fig.~\\ref{fig:fig3}. In this section, we use quantum trajectory simulations to further trace connections between the mean-field theory and a full quantum treatment.\n\nNote, first, that unlike the common situation for phase transitions of light, where the many-photon limit is a weak-coupling limit (Secs.~IVA and IVC of Ref.~\\cite{carmichael_2015}), the photon number for our generalized Jaynes-Cummings Rabi model scales with $N(\\lambda\/\\kappa)^2$---i.e., the many-photon limit is a strong-coupling limit; this is seen, for example, from Eq.~(\\ref{eqn:alpha_eta=1_approx}), which, undoing the scaling of Eqs.~(\\ref{eqn:scaling}) and (\\ref{eqn:scaled_parameters}), reads\n\\begin{equation}\n|\\alpha|^2=N\\left(\\frac{\\lambda}{\\kappa}\\right)^2\\frac{(1+\\eta)^2}4\\frac{(\\bar\\epsilon\\pm 1)^2}{1+(\\Delta\/\\kappa)^2}.\n\\label{eqn:photon_number_eta=1_unscaled}\n\\end{equation}\nThe scaling is also apparent from a comparison between frames (c) and (e) of Fig.~3, and frames (c), (e), and (f) of Figs.~\\ref{fig:fig4} and \\ref{fig:fig5}, and frame (c) of Fig.~\\ref{fig:fig7}: with $\\lambda\/\\kappa=10$ in Fig.~\\ref{fig:fig3}, photon numbers range from 4 to 40, while with five times larger coupling in Figs.~\\ref{fig:fig4}, \\ref{fig:fig5}, and \\ref{fig:fig7} they range in the hundreds and thousands; indeed,  frames (c), (e), and (f) of Fig.~\\ref{fig:fig5} rise to reach photon numbers of $6.4\\times10^3$, $10^4$, and $1.21\\times10^4$, respectively, at zero detuning [Eq.~({\\ref{eqn:photon_number_eta=1_unscaled}}]. Such high numbers can be reached with just one two-state system, since, when the coupling is strong, there is no need for a large value of $N$ to offset a weak nonlinearity per photon.\n\nAmongst the many effects of quantum fluctuations, in the following we target just two: first, mean-field steady states that are stable under Maxwell-Bloch equations are expected to be metastable in the presence of quantum fluctuations; and, second, isolated stable steady states---e.g., the lower state in frames (b) and (c) of Fig.~\\ref{fig:fig5} [the minus sign in Eq.~(\\ref{eqn:photon_number_eta=1_unscaled})]---might be accessed via quantum fluctuations. These effects are illustrated in Figs.~\\ref{fig:fig9} and \\ref{fig:fig10}, where we plot quantum trajectories of the photon number expectation while the detuning is slowly swept, from negative to positive. The coupling $\\lambda\/\\kappa=10$ is used in Fig.~\\ref{fig:fig9} in order to keep the maximum photon number relatively low, while the larger value in Fig.~\\ref{fig:fig10} maps to the mean-field results of Fig.~\\ref{fig:fig7}.\n\n\\begin{figure}[htpb!]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure9a.pdf}\n\\vskip0.2in\n\\includegraphics[width=3.4in]{figure9b.pdf}\n\\vskip0.2in\n\\includegraphics[width=3.4in]{figure9c.pdf}\n\\end{center}\n\\caption{Sample quantum trajectories as a function of scanned detuning and steady-state $Q$ functions for $N=1$, $\\Delta_0=\\Delta$, $\\epsilon\/\\epsilon_{\\rm crit}=0.2$, $\\kappa\/\\lambda=0.1$, and $\\eta=1$, $0.8$, $0.6$ (top to bottom); in all frames the detuning is scanned from $\\Delta\/\\lambda=-1$ to $+1$ in a time $\\kappa T=6\\times10^4$. Two sample scans are plotted in each frame (solid yellow and cyan lines) against the background of mean-field steady states (solid red and dashed blue curves). The inset $Q$ functions are for detunings $\\Delta\/2\\epsilon_{\\rm crit}=0$ (left) and $\\Delta\/2\\epsilon_{\\rm crit}=0.04$, $0.015$, $0.04$ (top to bottom) (right).}\n\\label{fig:fig9}\n\\end{figure}\n\nFigure \\ref{fig:fig9} presents a sequence of plots illustrating the role of quantum fluctuations as we move away from the limit of the coherently driven extension of the Dicke phase transition of Sec.~\\ref{sec:Dicke_coherent_drive_eta=1} into the intermediate regime of Sec.~\\ref{sec:coherent_drive_intermediate_eta}. Beginning with $\\eta=1$, the upper frame shows quantum trajectories tracking the two mean-field curves plotted from Eq.~(\\ref{eqn:photon_number_eta=1_unscaled}). Both trajectories (yellow and cyan lines) start on the left by following the higher mean-field branch, but quantum fluctuations allow the isolated [see frames (b) and (c) of Fig.~\\ref{fig:fig5}] lower branch to be accessed too. The two branches correspond to fields that are $\\pi$ out of phase in the imaginary direction at zero detuning---inset $Q$ function to the left---and rotate to eventually align with the real axis as the detuning is changed---inset $Q$ function to the right.\n\nSimilar results are plotted for $\\eta=0.8$ and $\\eta=0.6$ in the middle and bottom frames, respectively. Once again, mean-field curves are faithfully followed over segments of the path, but the switching between branches is more common. The most prominent feature, however, is the dramatic loss of stability around zero detuning: although the mean-field analysis finds a stable steady state at zero photon number [region $R_2^a$ in frame (a) of Fig.~\\ref{fig:fig5}], the full quantum treatment yields fluctuations spanning the two previously stable coherent states; the fluctuations are particularly apparent from the inset $Q$ functions in the middle frame of Fig~{\\ref{fig:fig9}}. The spikes that accompany switches between branches are not numerical artifacts; they are decaying oscillations---evidence of a spiraling trajectory for the field amplitude in the approach to the new locally stable state.\n\nFigure \\ref{fig:fig10} presents the results of two detuning scans for $\\lambda\/\\kappa=50$ and $\\eta=0.2$, corresponding to the parameters of Fig.~\\ref{fig:fig7}. In one scan the quantum trajectory follows the highest branch of stable mean-field solutions all the way up to its maximum. Much more commonly, though, the trajectory switches between this branch and the vacuum state in the region of $\\Delta\/2\\epsilon_{\\rm crit}=\\pm 0.1$, as illustrated by the second scan. In this region the quantum fluctuations show clear evidence of the three coexisting stable mean-field steady states illustrated in frame (e) of Fig.~\\ref{fig:fig7} (region $R_6$)---inset $Q$ function to the right.\n\\begin{figure}[htpb!]\n\\begin{center}\n\\includegraphics[width=3.4in]{figure10.pdf}\n\\end{center}\n\\caption{As in Fig.~{\\ref{fig:fig9}} but for $\\kappa\/\\lambda=0.02$ and $\\eta=0.2$, and with the detuning scanned from $\\Delta\/\\lambda=-0.6$ to $+0.6$ in a time $\\kappa T=6\\times10^4$. The inset $Q$ functions are for detunings $\\Delta\/2\\epsilon_{\\rm crit}=0$ (left) and $\\Delta\/2\\epsilon_{\\rm crit}=0.12$ (right).}\n\\label{fig:fig10}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\\noindent\nWe have generalized the dissipative extension \\cite{dimer_etal_2007} of the Dicke model \\cite{dicke_1954} of light interacting with matter in two directions, thus linking the superradiant phase transition of Hepp and Lieb \\cite{hepp&lieb_1973a,hepp&lieb_1973b} to the breakdown of blockade \\cite{carmichael_2015,fink_etal_2017}. Although the former was originally approached through exact calculations in the thermodynamic limit for $N$ two-state systems in thermal equilibrium, and the latter as a phenomenon of single systems, both might be engineered in many- and one-two-state-system versions, with the same underlying mean-field phenomenology and where the central issue of photon number in the presence of dissipation is governed not by the number of two-state systems only, but also the ratio of coupling strength to photon loss \\cite{carmichael_2015}---even one two-state system can control many photons in cavity and circuit QED \\cite{armen_etal_2009,fink_etal_2017}.\n\nWe adopted a generalization introduced by Hepp and Lieb \\cite{hepp&lieb_1973b}, and taken up in a number of recent publications \\cite{stepanov_etal_2008,schiro_etal_2012,tomka_etal_2014,xie_etal_2014,tomka_etal_2015,wang_etal_2016,moroz_2016,kirton_etal_2018}, where the interaction Hamiltonian is made from a sum of rotating and counter-rotating terms of variable relative strength; in this way we span the continuum from the Jaynes-Cummings to the quantum Rabi interaction. We also added direct driving of the field mode, since that, not the counter-rotating interaction, creates photons in the breakdown of photon blockade. We analyzed mean-field steady states as a function of adjustable parameters for this extended model and found that a common critical drive strength, $\\epsilon_{\\rm crit}=\\lambda(1+\\eta)\/2$, links the superradiant phase transition  to the breakdown of photon blockade---$\\lambda$ is the coupling strength and $\\eta$ the relative strength of counter-rotating to rotating interactions.  More generally, we found that the extended phase diagram moves from a region of pure superradiant character into the region of broken blockade, passing through a phase that although present in the generalized model of Hepp and Lieb \\cite{hepp&lieb_1973b} is not identified in that work.\n\nWe then carried our analysis beyond mean-field steady states to a fully quantum treatment for the limiting case of one two-state system: we extended a prior calculation of quasi-energies \\cite{alsing_etal_1992} to the generalized Hamiltonian---resonant driving of the field mode and no dissipation---and obtained numerical results with both detuning and photon loss  included. The quasi-energy spectrum for one two-state system was shown to be singular at $\\epsilon_{\\rm crit}$, where it undergoes a transition from discrete to continuous, and numerical simulations broadly support mean-field results, though expanding the view from earlier work \\cite{carmichael_2015,shamailov_etal_2010} of multi-photon resonances at weak drive and exhibiting quantum-fluctuation-induced switching amongst locally stable mean-field steady states.\n\nThe aim of this study has been to uncover connections between different dissipative quantum phase transition for light and we have left many directions untouched; for example, a broader investigation of a very rich parameter space and the fully quantum treatment. We expect future work on the theoretical side will fill the gaps and hope that experiments in the spirit of Refs.~\\cite{baumann_etal_2010,baumann_etal_2011,baden_etal_2014,fink_etal_2017,armen_etal_2009} will prove feasible.\n\n\\section*{Acknowledgments}\nThis work was supported by the Marsden fund of the RSNZ. Quantum trajectory simulations were carried out on the NeSI Pan Cluster at the University of Auckland, supported by the Center for eResearch, University of Auckland.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec-Intro}\nA number of different high-energy, $\\geq$ 1GeV, neutrino sources have been proposed in literature, that include active galactic nuclei (AGNs)\\citep{ste91,sza94,nell93, ato01,alv04}, gamma-ray bursts (GRBs)\\citep{wax97,der03,raz04b,mur06,gup07}, supernova remnants \\citep{alv02,cos05} and core collapse supernovae \\citep{wax01, wan07}, although   long duration GRBs have been found to be tightly connected with core-collapse supernovae \\citep{hjo03,sta03}.  Properties of neutrino fluxes, energy range, shape of the energy spectra and flavor content depend on physical conditions in the sources.  \nNeutrinos are useful for studying sources, especially when photons cannot escape directly. They could be the only prompt signatures of the \"hidden\" sources.  These have been associated to core collapse of massive stars leading to supernovae (SNe) of type Ib,c and II with mildly relativistic jets emitted by a central engine, a black hole or a highly magnetized neutron star.  Depending on the initial density and metallicity, the pre-supernova star could have different radii. Type Ic supernovae are believed to be He stars with radius $R_\\star\\approx $ 10$^{11}$ cm and Supernovae of type II and Ib are thought to have a radius of $R_\\star\\approx$ 3$\\times$10$^{12}$ cm. \\\\\nRecently, IceCube reported the detection of two neutrino-induced events with energies between 1- 10 PeV \\citep{aar13}.  These events have been discussed as having an extragalactic origin, for instance; GRBs\\citep{cho12}  and low-luminosity GRBs \\citep{liu13}. On the other hand, high-energy neutrinos are produced in the decay of charged pions and muons when energetic protons in the jet interact with synchrotron thermalized photons or nucleons\/mesons (pp, pn)\/($\\pi$, K) in the shocks.  For internal shocks, synchrotron radiation and the number density of particles could be calculated with enough accuracy if we know the distribution of the magnetic field and the particle momentum in the shocked region.  These quantities are calculated using the energy equipartition hypothesis through the equipartition parameters; electron equipartition ($\\epsilon_e=U_e\/U$) and magnetic equipartition $\\epsilon_B=U_B\/U$\\citep{mes98}. Many authors \\citep{barn12,fra12,sac12,kum10,she10} have estimated these parameters to be  $\\epsilon_e\\simeq$ 0.1, and  0.1$\\leq \\epsilon_B\\leq 10^{-4}$, to obtain a good description of more than a dozen of GRBs.\\\\\nOn the other hand, the neutrino flavor ratio is expected to be, at the source,  $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$=1 : 2 : 0 and on Earth (due to neutrino oscillations between the source and Earth) $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$=1 : 1 : 1 and  $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$=1 : 1.8 : 1.8  for neutrino energies less and greater than 100 TeV, respectively, for gamma ray bursts ($\\phi^0_{\\nu_l}$ is the sum of $\\nu_l$ and $\\bar{\\nu}_l$) \\citep{kas05}.   Also it has been pointed out that measurements of the deviation of this standard flavor ratio of astrophysical high-energy neutrinos may probe new physics \\citep{lea95,ath00, kas05}. \nAs it is known, neutrino properties get modified when it propagates in a medium.  Depending on their flavor, neutrinos interact via neutral and\/or charged currents, for instance,   $\\nu_e$ interacts with electrons via both neutral and charged currents, whereas $\\nu_\\nu(\\nu_\\tau)$ interacts only via the neutral current. This induces a coherent effect in which maximal conversion of $\\nu_e$ into $\\nu_\\mu (\\nu_\\tau)$ takes place. The resonant conversion of neutrino from one flavor to another due to the medium effect is well known as the Mikheyev-Smirnov-Wolfenstein effect  \\citep{wol78}.  \nResonance condition of high-energy neutrinos in  hidden jets has been studied in the literature \\citep{men07,raz10, sah10}.   Recently,  \\citet{2013arXiv1304.4906O} studied the three-flavor neutrino oscillations on the surface of the star  for neutrino energy  in the range  (0.1 - 100) TeV.  They found that  those neutrinos generated on the surface  with  energies of less than 10 TeV could oscillate.    Unlike previous studies, we show that these sources  are capable of generating  PeV neutrinos pointing them out as possible progenitors of the first observation of PeV-energy neutrinos with IceCube \\citep{aar13}. Besides,  we  do a full analysis of resonance conditions (two- and three-flavors) for neutrinos produced at different places in the star,  estimating the flavor ratios on Earth.\\\\\nIn this paper we both show that PeV neutrinos can be produced in hidden jets and  estimate the flavor ratio of high-energy neutrinos expected on Earth.  Firstly, we compute the energy range of neutrinos produced by cooling down of hadrons and mesons accelerated in a mildly relativistic jet. After that we take different matter density profiles to show that neutrinos may oscillate resonantly depending on the neutrino energy and  mixing neutrino parameters.    Finally, we discuss our results in the fail jet framework.\n\\section{Jet dynamics}\nFor the internal shocks, we consider a mildly relativistic shock propagating with bulk Lorentz factor $\\Gamma_b=10^{0.5}\\Gamma_{b,0.5}$.  Behind the shock, the comoving number density of particles   and density of energy are  $n'_e=n'_p=1\/(8\\,\\pi\\,m_p\\,c^5)\\,\\Gamma_b^{-4}\\,E_j\\,t^{-2}_{\\nu,s}\\,t^{-1}_j=3.1\\times10^{18}$ cm$^{-3}\\,\\,t^{-2}_{\\nu,s}$ and $n'_p m_p c^2$, respectively, where we have taken the set of typical values for which the jet drills but hardly breaks through the stellar envelope:  the jet kinetic energy $E_j=10^{51.5} E_{j,51.5}$ erg, the variability time scale of the central object $t_\\nu=t_{\\nu, {\\rm s}}\\,{\\rm s}$  with $t_{\\nu,{\\rm s}}$= 0.1 and 0.01, and  the jet duration $t_j=10\\,t_{j,1}$ s \\citep{raz05,and05,2013MNRAS.432..857M}.   We assume that electrons and protons  are accelerated in the internal shocks to a power-law distribution $N(\\gamma_j) d\\gamma_j\\propto \\gamma_j^{-p} d\\gamma_j$.   The internal shocks due to shell collisions take place at a radium $r_j=2\\Gamma_b^2\\,c\\,t_\\nu= 6 \\times   10^{11}\\,\\rm{cm}\\,\\Gamma^2_{0.5}\\,t_{v,s}$.  Electrons, with minimum energy $E_{e,m}=\\frac{p-2}{p-1} \\epsilon_e\\,m_p c^2 \\Gamma_b$   and maximum energy limited by the dynamic time scale $t'_{dyn}\\simeq t_\\nu\\Gamma_b$, cool down rapidly by synchrotron radiation in the presence of the magnetic field given by\n\\begin{eqnarray}\nB'&=&\\biggl(\\frac{\\epsilon_B}{c^3}\\,\\Gamma_b^{-4}\\,E_j\\,t^{-2}_\\nu\\,t^{-1}_j  \\biggr)^{1\/2}\\cr\n&=& 3.43\\times10^8\\,{\\rm G}\\, \\Gamma_{b,0.5}^{-2}\\,E^{1\/2}_{j,51.5}\\,t^{-1\/2}_{j,1}\\,\\epsilon_B^{1\/2}\\,t^{-1}_{\\nu,s}\\,,\n\\label{mfield}\n\\end{eqnarray}\nwhere here and further on  the magnetic equipartition parameter and $t_{\\nu,{\\rm s}}$  lie in the range  0.1$\\leq \\epsilon_B\\leq 10^{-4}$ and 0.1 $\\leq t_{\\nu,{\\rm s}} \\leq $  0.01, respectively. The radiated  photon energies by electron synchrotron emission  with energy $E_e$ is $E_{syn,\\gamma}=eB'\/(\\hbar  m_e^3c^5) E^2_e$, and  also  the opacity to Thomson scattering by these photons is\n\\begin{eqnarray}\n\\tau_{th}'&=&\\frac{\\sigma_T}{4\\pi\\,m_p\\,c^4} \\Gamma_b^{-3}\\,E_j\\,t^{-1}_\\nu\\,t^{-1}_j\\cr\n&=&3.9\\times 10^5\\, \\Gamma_{b,0.5}^{-3}\\,E_{j,51.5}\\,t^{-1}_{j,1}\\,t^{-1}_{\\nu,s}\\,.\n\\label{opde}\n\\end{eqnarray}\nDue to the large Thomson optical depth, synchrotron photons will thermalize to a black body temperature, therefore the peak energy is given by\n\\begin{eqnarray}\nE'_{\\gamma}\\sim k_B\\,T_{\\gamma}&=&\\biggl(\\frac{15(\\hbar\\,c)^3}{8\\pi^4\\,c^3}\\biggr)^{1\/4}\\,\\epsilon_e^{1\/4}\\, E_j^{1\/4}\\,\\Gamma^{-1}_b\\,t_v^{-1\/2}\\,t^{-1\/4}_j\\cr\n&=&1.36\\,{\\rm keV}\\, E_{j,51.5}^{1\/4}\\,\\Gamma^{-1}_{b,0.5}\\,t^{-1\/4}_{j,1}\\,\\epsilon^{1\/4}_{e,-1}\\,t_{\\nu,s}^{-1\/2}\\,,\n\\label{enph}\n\\end{eqnarray}\nand the number density of thermalized photons is  \n\\begin{eqnarray}\n\\eta'_\\gamma&=&\\frac{2\\,\\zeta(3)}{\\pi^2\\,(c\\,\\hbar)^3}\\,\\biggl(\\frac{15\\,\\hbar\\,\\epsilon_e\\, E_j}{8\\pi^4\\,\\Gamma^{4}_b\\,t_v^{2}\\,t_j}  \\biggr)^{3\/4}\\cr\n&=&2.86 \\times 10^{23} {\\rm cm^{-3}}\\, E^{3\/4}_{j,51.5}\\,\\Gamma^{-3}_{b,0.5}\\,t^{-3\/4}_{j,1}\\,\\epsilon^{3\/4}_{e,-1}\\,t_{\\nu,s}^{-3\/2}\\,.\n\\label{denph}\n\\end{eqnarray}\nAlthough keV photons can hardly escape due to the high optical depth,  they are  able to interact with relativistic protons accelerated in the jet, producing high-energy neutrinos via  charged pion decay. The pion energies  depend on the proton energy and  characteristics of the jet. \n\\section{Hadronic model}\nProtons accelerated in internal shocks, on the one hand,  radiate photons by  synchrotron radiation and  also scatter  the internal photons by  inverse Compton (IC) scattering, and on the other hand, interact with  thermal keV  photons and  hadrons by  p$\\gamma$ and  p-hadron interactions.  The  optical depths for p$\\gamma$  and  p-hadron  interactions are\n\\begin{eqnarray}\n\\tau'_{p\\gamma}&=&\\frac{4\\,\\zeta(3)\\sigma_{p\\gamma}}{\\pi^2\\,(c\\,\\hbar)^3}\\,\\biggl(\\frac{15\\,\\hbar\\,\\epsilon_e\\, E_j}{8\\pi^4\\,\\Gamma^{8\/3}_b\\,t_v^{2\/3}\\,t_j}  \\biggr)^{3\/4}\\cr\n&=&3.19\\times 10^6\\, E^{3\/4}_{j,51.5}\\,\\Gamma^{-2}_{b,0.5}\\,t^{-3\/4}_{j,-1}\\,\\epsilon^{3\/4}_{e,-1}\\,t_{\\nu,s}^{-1\/2}\\,,\n\\label{optpg}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\tau'_{pp}&=&\\frac{\\sigma_{pp}}{4\\,\\pi\\,m_p\\,c^5}\\,  E_j\\, \\Gamma^{-3}_b\\,t_v^{-1}\\,t_j^{-1}  \\cr\n&=&1.77\\times 10^4\\, E_{j,51.5}\\, \\Gamma^{-3}_{b,0.5}\\,t_{j,1}^{-1}\\,t_{\\nu,s}^{-1}\\,,\n\\label{optpp}\n\\end{eqnarray}\nrespectively. Due to the optical depths  for p$\\gamma$ and p-hadron interactions are very high,  p$\\gamma$ and p-hadron are effective, although p-hadron interactions are more effective at lower energy than p$\\gamma$ interactions \\citep{raz04b}.\n\\subsection{Cooling time scales}\nThe shock acceleration time for an energy  proton, $E'_p$, is\n\\begin{eqnarray}\nt'_{acc}&=&\\frac{2\\pi\\xi}{c} r_L =\\frac{2\\pi\\xi\\,c^{1\/2}\\,B'_{c,p}}{m_p^2}\\,E'_p\\,\\epsilon^{-1\/2}_B\\, E^{-1\/2}_j\\,\\Gamma^{2}_b\\,t_v\\,t^{1\/2}_j\\cr\n&=&2.04\\times 10^{-12}{\\rm s}\\,E'_p\\,\\xi\\, E^{-1\/2}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t^{1\/2}_{j,1}\\,\\epsilon^{-1\/2}_B\\,t_{\\nu,s}\\,,\n\\label{tacc}\n\\end{eqnarray}\nwhere $r_L$ is the Larmor's radius and $\\xi$ is a factor of equality. The acceleration time, $t'_{acc}$, gives an account of the maximum  proton energy achieved, when it is  compared with the maximum cooling time scales.  In the following subsections we are going to calculate the cooling time scales for protons and mesons.\n\n\n\\subsubsection{Proton cooling time scales}\n\n\nThe cooling time scale for proton synchrotron radiation is\n\\begin{eqnarray}\nt'_{p,syn}&=&\\frac{E'_p}{(dE'_p\/dt)_{syn}}=\\frac{6\\pi\\,m_p^4\\,c^6}{\\sigma_T\\,\\beta^2\\,m_e^2\\,E'_p}\\,\\epsilon^{-1}_B\\, E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=& 38.3\\, {\\rm s}\\,E'^{-1}_{p,9}\\, E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_B\\,t^2_{\\nu,s}\\,.\n\\label{tsyn}\n\\end{eqnarray}\nProtons  in the shock region  can upscatter the thermal keV photons $E'_{IC,\\gamma}\\sim\\gamma^2_p\\,E'_\\gamma$  with peak energy and density given in eqs.(\\ref{enph}) and (\\ref{denph}).  The IC cooling time scale in the Thomson regimen is\n\\begin{eqnarray}\nt'^{th}_{p,ic}&=&\\frac{E'_p}{(dE'_p\/dt)^{th}_{ic}}=\\frac{m_p^4\\,c^4\\,\\pi^6(c\\,\\hbar)^2}{5\\,\\sigma_T\\,\\beta^2\\,m_e^2\\,\\zeta(3)\\,E'_p}\\,\\epsilon^{-1}_e\\, E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=& 383.1\\, {\\rm s}\\,E'^{-1}_{p,9}\\, E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_{e,-1}\\,t^2_{\\nu,s}\\, .\n\\label{tic}\n\\end{eqnarray}\nAlso, the IC cooling time scale in the Klein-Nishina (KN) regimen,   $E'_pE'_\\gamma\/m_p^2c^4=\\Gamma_{KN}$ with ($\\Gamma_{KN}=1$),  is\n\\begin{eqnarray}\nt'^{KN}_{p,ic}&=&\\frac{E'_p}{(dE'_p\/dt)^{KN}_{ic}}=\\frac{3\\pi^4(c\\,\\hbar)^3 \\,E'_p\\,\\epsilon^{-1\/2}_e\\, E^{-1\/2}_j\\,\\Gamma^{2}_b\\,t_v\\,t^{1\/2}_j }{2\\sqrt{30\\hbar}\\,\\sigma_T\\,\\beta^2\\,m_e^2\\,c^5\\,\\zeta(3)}\\cr\n&=&5.15 \\times 10^{-10} \\, {\\rm s}\\,E'_{p,9}\\,E^{-1\/2}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t^{1\/2}_{j,1}\\,\\epsilon^{-1\/2}_{e,-1}\\,t_{\\nu,s}\\,.\n\\end{eqnarray}\nOn the other hand,  protons could upscatter thermal photons according  to Bethe-Heitler (BH)  process.  The proton energy loss is taken away by the pairs produced in this process.   The cooling time scale for the BH scattering is \n\\begin{eqnarray}\nt'_{BH}&=&\\frac{E'_p}{(dE'_p\/dt)_{BH}}=\\frac{E'_p}{n'\\,c\\sigma_{BH}\\Delta E'_p}\\cr\n&=&\\frac{E'_p\\,(m^2_p\\,c^4+2E'_pE'_\\gamma)^{1\/2}}{2n'_\\gamma\\,m_e\\,c^3\\sigma_{BH}(E'_p+E'_\\gamma)}\\,,\n\\end{eqnarray}\nwhere $\\sigma_{BH}=\\alpha r^2_e\\, ((28\/9) \\,ln[2E'_p\\,E'_\\gamma\/(m_pm_ec^4)]-106\/9)$.\nThe energy loss rate due to pion production for p$\\gamma$ interactions is \\citep{ste68,bec09}\n\\begin{eqnarray}\nt'_{p\\gamma}&=&\\frac{\\pi^2\\,(c\\,\\hbar)^3}{0.3\\,c\\,\\sigma_{p\\gamma}\\,\\zeta(3)}\\,\\biggl(\\frac{8\\pi^4\\,\\Gamma^{4}_b\\,t_v^{2}\\,t_j}{15\\,\\hbar\\,\\epsilon_e\\, E_j}  \\biggr)^{3\/4}\\cr\n&=&1.32\\times10^{-5}\\,\\,{\\rm s}\\, E^{-3\/4}_{j,51.5}\\Gamma^{3}_{b,0.5}\\,t^{3\/4}_{j,1} \\,\\epsilon^{-3\/4}_{e,-1}\\,t_{\\nu,s}^{3\/2}\\,,\n\\end{eqnarray}\nand for p-hadron interactions  is  \\citep{der03,der09}\n\\begin{eqnarray}\nt'_{pp}&=&\\frac{10\\,\\pi\\,m_p\\,c^4}{\\sigma_{pp}}\\,  E_j^{-1}\\, \\Gamma^{4}_b\\,t_v^{2}\\,t_j \\cr \n&=&4.47\\times 10^{-4}\\,{\\rm s}\\,E_{j,51.5}^{-1}\\, \\Gamma^{4}_{b,0.5}\\,t_{j,1} \\,t_{\\nu,s}^{2}\\,.\n\\end{eqnarray}\nIn figs \\ref{ptime_r1}  and \\ref{ptime_r2}  we have plotted the proton cooling time scales  when the magnetic field is distributed in order to 0.1$\\leq \\epsilon_B\\leq 10^{-4}$ and internal shocks take place at  r=$6\\times 10^{9}$ cm and   r=$6\\times 10^{10}$ cm, respectively.\n\\subsubsection{Meson cooling time scales}\nHigh-energy charged pions and kaons produced by p-hadron and p$\\gamma$  interactions    ($p+\\gamma\/p \\to X+\\pi^{\\pm}\/K^{\\pm}$)  radiate  in the presence of the magnetic field  (eq. \\ref{mfield}). Therefore,  their cooling time scales  are  \n\\begin{eqnarray}\nt'_{\\pi^+, syn}&=&  \\frac{E'_{\\pi^+}}{(dE'_{\\pi^+}\/dt)} \\simeq  \\frac{6\\pi c^6 m^4_{\\pi^+}}{\\sigma_T\\,\\beta^2\\,m_e^2}\\,\\epsilon^{-1}_B\\, E^{-1}_j\\,\\Gamma^{2}_b\\,t^2_v\\,t_j \\,E'^{-1}_{\\pi^+}\\cr\n&=&1.9\\times 10^{-2} \\,{\\rm s}\\, E'^{-1}_{\\pi^+,9}\\,E^{-1}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_B\\,t^2_{\\nu,s}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nt'_{K^+, syn}&=&  \\frac{E'_{k^+}}{(dE'_{k^+}\/dt)} \\simeq  \\frac{6\\pi c^6 m^4_{k^+}}{\\sigma_T\\,\\beta^2\\,m_e^2}\\,\\epsilon^{-1}_B\\, E^{-1}_j\\,\\Gamma^{2}_b\\,t^2_v\\,t_j  \\,E'^{-1}_{k^+}\\cr\n&=&2.94 \\,{\\rm s}\\,E'^{-1}_{k^+,9}\\, E^{-1}_{j,51.5}\\,\\Gamma^{2}_{b,0.5}\\,t_{j,1}\\,\\epsilon^{-1}_B\\,t^2_{\\nu,s}\\,.\n\\end{eqnarray}\nAs protons can collide with secondary pions and kaons ($\\pi^+ p$ and $K^+p$), then its respective cooling time scale is given by\n\\begin{eqnarray}\nt'_{had}&=&\\frac{10\\,\\pi\\,m_p\\,c^4}{\\sigma_{(pK\/p\\pi^+)}}\\,  E_j^{-1}\\, \\Gamma^{4}_b\\,t_v^{2}\\,t_j \\cr \n&=& 4.47\\times 10^{-9} \\,{\\rm s}\\,   E_{j,51.5}^{-1}\\, \\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,t_{\\nu,s}^{2}\\,.\n\\end{eqnarray}\nHere we have used the cross-section   $\\sigma_{(pK^+\/p\\pi^+)} \\approx 3\\times 10^{-26}$ cm$^2$.  Because the mean lifetime of these mesons may be  comparable with the synchrotron  and hadron time scales in some energy range, it is necessary to consider  the cooling time  scales related to their mean lifetime which are given by\n\\begin{eqnarray}\nt'_{\\pi^+,dec}&=&\\frac{E'_{\\pi^+}}{m_{\\pi^+}c^2}\\,\\tau_{\\pi^+}\\cr\n&=&=1.87\\times 10^{-7} \\,{\\rm s}\\,E'_{\\pi^+,9}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nt'_{K^+,dec}&=&\\frac{E'_{K^+}}{m_{K^+}c^2}\\tau_{K^+}\\cr\n&=&2.51\\times 10^{-8} \\,{\\rm s}\\,E'_{K^+,9}\\,,\n\\end{eqnarray}\nwhere $\\tau_{\\pi^+\/K^+}$is the mean lifetime for $\\pi^+\/K^+$ and E$_{\\pi^+\/K^+,9}$= 10$^9$ E$_{\\pi^+\/K^+}$ eV.\\\\\nIn figs \\ref{mtime_r1}  and \\ref{mtime_r2}  we have plotted the meson cooling time scales  when internal shocks happen at  r=$6\\times 10^{9}$ cm and   r=$6\\times 10^{10}$ cm and the magnetic equipartition parameter is in the range  0.1$\\leq \\epsilon_B\\leq 10^{-4}$.\n\\subsection{Neutrino production}\nThe single-pion production channels are $p+\\gamma\\to n+\\pi^+$ and $p+\\gamma\\to p+ \\pi^0$, where the relevant pion decay chains are $\\pi^0\\to 2\\gamma$, $\\pi^+\\to \\mu^++\\nu_\\mu\\to e^++\\nu_e+\\bar{\\nu}_\\mu+\\nu_\\mu$ and $\\pi^-\\to \\mu^-+\\bar{\\nu}_\\mu\\to e^-+\\bar{\\nu}_e+\\nu_\\mu+\\bar{\\nu}_\\mu$ \\citep{der03}, then the threshold neutrino energy from p$\\gamma$ interaction is  \n\\begin{eqnarray}\nE'_{\\nu,\\pi}&=&2.5\\times 10^{-2}\\biggl(\\frac{8\\pi^4}{15\\hbar}\\biggr)^{1\/4}\\,\\,\\frac{(m^2_\\Delta-m_p^2)}{(1-cos\\theta)}\\cr\n&&\\hspace{3.4cm}\\times\\, \\epsilon^{-1\/4}_e\\, E^{-1\/4}_j\\,\\Gamma_b\\,t^{1\/2}_v\\,t^{1\/4}_j\\cr\n&=& 9.72 \\,{\\rm TeV}\\, E^{-1\/4}_{j,51.5}\\,\\Gamma_{b,0.5}\\,t^{1\/4}_{j,1}\\,   \\epsilon^{-1\/4}_{e,-1}\\,t^{1\/2}_{\\nu,s}\\,.\n\\end{eqnarray}\nComparing the time cooling scales  we can estimate the neutrino break energy for each process. Equaling  $t_{acc}\\simeq t'_{p,syn}$, we can  approximately estimate the maximum proton energy\n\\begin{eqnarray}\nE'_{p,max}&=&\\biggl(\\frac{3\\,e\\,m_p^4\\,c^{11\/2}}{\\sigma_T\\,\\xi\\,\\beta^2\\,m_e^2}\\biggr)^{1\/2}\\,\\epsilon^{-1\/4}_B\\, E^{-1\/4}_j\\,\\Gamma_b\\,t^{1\/2}_v\\,t^{1\/4}_j\\cr\n&=&4.3\\times 10^{3} \\,{\\rm GeV}\\, E^{-1\/4}_{j,51.5}\\,\\Gamma_{b,0.5}\\,t^{1\/4}_{j,1}\\,\\epsilon^{-1\/4}_B\\,t^{1\/2}_{\\nu,s}\\,.\n\\end{eqnarray}\nFrom the condition of the synchrotron cooling time scales  for mesons ($t'_{\\pi^+,syn}=t'_{had}$ and $t'_{K^+,syn}=t'_{had}$), one may roughly define the neutrino break energies  as\n\\begin{eqnarray}\nE'_{\\nu,\\pi^+syn}&=&0.15\\times \\frac{m^4_{\\pi^+}\\,c^2\\,\\sigma_{pp}}{m_p\\,\\sigma_T\\,\\beta^2\\,m_e^2}\\epsilon^{-1}_B\\cr\n&=&10.5 \\,{\\rm GeV}\\,  \\epsilon^{-1}_{B}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nE'_{\\nu,k^+syn}&=&0.3\\times \\frac{m^4_{k^+}\\,c^2\\,\\sigma_{pp}}{m_p\\,\\sigma_T\\,\\beta^2\\,m_e^2}\\epsilon^{-1}_B\\cr\n&=&3.28 \\,{\\rm TeV}\\,\\epsilon^{-1}_{B}\\,.\n\\end{eqnarray}\nFrom the lifetime condition of  cooling time scale  ($t'_{\\pi^+,dec}=t'_{had}$ and  $t'_{K^+,dec}=t'_{had}$),  one again we can obtain  the neutrino break energies, which for these cases are \n\\begin{eqnarray}\nE'_{\\nu, \\pi^+lt}&=&2.5\\frac{\\pi\\,m_p\\,m_{\\pi^+}\\,c^6}{\\sigma_{pp}}\\,\\tau^{-1}_{\\pi^+}  \\,E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=&0.6 \\,{\\rm TeV}\\,E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,t^2_{\\nu,s}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nE'_{\\nu, k^+lt}&=&5\\frac{\\pi\\,m_p\\,m_{k^+}\\,c^6}{\\sigma_{pp}}\\,\\tau^{-1}_{k^+}  \\,E^{-1}_j\\,\\Gamma^{4}_b\\,t^2_v\\,t_j\\cr\n&=&8.92 \\,{\\rm TeV}\\,E^{-1}_{j,51.5}\\,\\Gamma^{4}_{b,0.5}\\,t_{j,1}\\,t^2_{\\nu,s}\\,.\n\\end{eqnarray}\nIt is important to say that muons may be suppressed by electromagnetic energy losses and in that case would not contribute  much to high-energy neutrino production. The ratio $\\frac{t'_{\\pi^+\/K^+,cool}}{t'_{\\pi^+\/K^+,dec}}$, where $t'_{\\pi^+\/K^+,cool}=\\frac{t'_{\\pi^+\/K^+,em}\\cdot \\,\\,\\,t'_{\\pi^+\/K^+,had}} {t'_{\\pi^+\/K^+,em}\\,\\,+\\, \\,\\,t'_{\\pi^+\/K^+,had} }$, determines the suppression of mesons before they decay to neutrinos \\citep{raz05}. \\\\\nIn fig. \\ref{prod_neu} we have plotted the neutrino energy created by distinct interaction processes at  different distances,   $6\\times 10^{9}$ cm (above) and $6\\times 10^{10}$ cm (below),  as a function of the magnetic equipartition parameter.\n\\section{Density profile of the source}\nAnalytical and numerical models of density distribution in a  pre-supernova have shown a decreasing dependence  on radius  $\\rho\\propto r^{-n}$, with n=3\/2 - 3 above the core, being 3\/2 and 3  convective and radiative envelopes respectively \\citep{woo93, shi90, arn91}.  In particular,  distributions with $\\rho \\propto r^{-3}$ and $\\rho \\propto r^{-17\/7}$ have been proposed  to describe simple blast wave distributions \\citep{bet90, che89}.   Following \\cite{men07}, we use three models of  density profile; Model [A], Model [B] and Model [C].\\\\ Model [A] ,\n\\begin{equation}\n\\hspace{0.7cm}{\\rm [A]} ~~\\rho(r) = 4.0\\times 10^{-6} \\left( \\frac{R_\\star}{r} -1\\right)^3 ~{\\rm g~cm}^{-3}\\,,\n\\label{dens-pro-A} \\\\\n\\end{equation}\ncorresponds to a polytropic hydrogen envelope with $\\rho(r)\\propto r^{-3}$, scaling valid in the range $r_{jet}\\geq r \\geq R_\\star$.    Model [B], \n\\begin{eqnarray}\n&&{\\rm [B]}~~\\rho(r) = 3.4\\times 10^{-5} ~{\\rm g~cm}^{-3}\\cr\n&&\\times\n\\cases{ \n(R_\\star\/r)^{17\/7}\\,; \\hspace{1cm}10^{10.8} ~{\\rm cm}<r<r_b = 10^{12} ~{\\rm cm}   &\\nonumber \\cr\n(R_\\star\/r_b)^{17\/7} (r-R_\\star)^5\/(r_b - R_\\star)^5\\,;\\hspace{0.3cm} r>r_b\\,,& \\cr\n} \n\\label{dens-pro-B} \\\\\n\\end{eqnarray}\nis a power-law fit with an effective polytropic index $n_{eff}=17\/7$ as done for SN 1987A \\citep{che89}.  Here $r_j \\sim  10^{10.8} ~{\\rm cm} $ is the radius of inner border of the envelope, where the density $\\rho=0.4$ g cm$^{-3}$. Associating the number of electron per nucleon $Y_e=$0.5, we obtained the number density of electrons as $N_e=N_a\\,\\rho(r)\\, Y_e=$1.2$\\times$10$^{23}$ cm$^{-3}$ and    Model [C], \n \\begin{eqnarray}\n{\\rm [C]} ~~\\rho(r) = 6.3\\times 10^{-6} {\\it A}\n\\left( \\frac{R_\\star}{r} -1 \\right)^{n_{\\rm eff}}  \n~{\\rm g~cm}^{-3} \\cr\n(n_{\\rm eff}, {\\it A})=\n\\cases{ \n(2.1,20) ~;~ &$10^{10.8} ~{\\rm cm}< r < 10^{11} ~{\\rm cm}$ \\cr\n(2.5,1) ~;~ &$r > 10^{11} ~{\\rm cm}$\\,, \\cr\n}\n\\label{dens-pro-C}\n\\end{eqnarray}\nincludes a sharp drop in density at the edge of the helium core \\citep{mat99}. \n\\section{Neutrino Mixing}\nIn the following subsections we are going to describe the neutrino oscillations in the matter (along the jet for three density profiles given in section 4 ) and  in vacuum (its path up to  Earth). We will be using  the best fit parameters for two-neutrino mixing (solar, atmospheric and accelerator neutrino experiments) and three-neutrino mixing.\nThe best fit value of   solar, atmospheric and accelerator neutrino experiments  are given as follow.\\\\\n\\textbf{Solar Neutrinos} are electron neutrinos produced in the thermonuclear reactions which generate the solar energy.  The Sudbury Neutrino Observatory (SNO) was designed to measure the flux of neutrinos produced by $^8$B decays in the sun, so-called $^8$B neutrinos, and to study neutrino oscillations, as proposed by \\cite{che85}.  A two-flavor neutrino oscillation analysis gave the following parameters:  $\\delta m^2=(5.6^{+1.9}_{-1.4})\\times 10^{-5}\\,{\\rm eV^2}$ and $\\tan^2\\theta=0.427^{+0.033}_{-0.029}$\\citep{aha11}.\\\\\n\\textbf{Atmospheric Neutrinos} are electron neutrinos  $\\nu_e$  produced mainly from the decay chain $\\pi\\to \\mu+\\nu_\\mu$ followed by $\\mu\\to e+\\nu_\\mu+\\nu_e$.  Super-Kamiokande (SK) observatory  observes interactions between neutrinos with electrons or with nuclei or water   via the water Cherenkov method. Under a two-flavor disappearance model with separate mixing parameters between neutrinos and antineutrinos there were found the following parameters for the SK-I + II + III data: $\\delta m^2=(2.1^{+0.9}_{-0.4})\\times 10^{-3}\\,{\\rm eV^2}$ and $\\sin^22\\theta=1.0^{+0.00}_{-0.07}$ \\citep{abe11a}.\\\\\n \\textbf{Reactor Neutrinos} are  produced in Nuclear reactors.  Kamioka Liquid scintillator Anti-Neutrino Detector (KamLAND) was initially designed to detect reactor neutrinos and so later it was prepared to measure $^7$Be solar neutrinos. A two neutrino oscillation analysis gives  $\\delta m^2=(7.9^{+0.6}_{-0.5})\\times 10^{-5}\\,{\\rm eV^2}$ and $\\tan^2\\theta=0.4^{+0.10}_{-0.07}$\\citep{ara05,shi07,mit11}.\\footnote{this value was obtained using a global analysis of data from KamLAND and solar-neutrino experiments}.\\\\\n\\textbf{Accelerator Neutrinos} are mostly produced by $\\pi$ decays (and some K decays), with the pions produced by the scattering of the accelerated protons on a fixed target. The beam can contain both $\\mu$- and e-neutrinos and antineutrinos.  There are two categories: Long and short baselines.\\\\\nLong-baseline experiments with accelerator beams run  with a baseline of about a hundred of kilometers.  K2K experiment was designed to measure neutrino oscillations using a man-made beam with well controlled systematics, complementing and confirming the measurement made with atmospheric neutrinos.  $\\delta m^2=(2.8^{+0.7}_{-0.9})\\times 10^{-3}\\,{\\rm eV^2}$ and $\\sin^22\\theta=1.0$\\citep{ahn06}.\\\\\nShort-baseline experiments with accelerator beams run  with a baseline of about hundreds of meters. Liquid Scintillator Neutrino Detector (LSND) was designed to search for $\\nu_\\mu\\to\\nu_e$ oscillations using $\\nu_\\mu$ from $\\pi^+$ decay in flight \\citep{ath96, ath98}.  The region of parameter space has been partly tested by Karlsruhe Rutherford medium energy neutrino KARMEN \\citep{arm02} and MiniBooNe experiments. \\cite{chu02} found two well-defined regions of oscillation parameters with either $\\delta m^2  \\approx  7\\, {\\rm eV^2}$ or $\\delta m^2 < 1\\, {\\rm eV^2} $ compatible with both LAND and KARMEN experiments, for the complementary confidence. The MiniBooNE experiment was specially designed to verify the LSND's neutrino data. It  is currently running at Fermilab and is searching for  $\\nu_e (\\bar{\\nu}_e)$ appearance in a $\\nu_\\mu(\\bar{\\nu}_\\mu)$ beam. Although MiniBooNE found no evidence for an excess of $\\nu_e$ candidate events above 475 MeV in the $\\nu_\\mu\\to\\nu_e$ study, there was observed a 3.0$\\sigma$ excess of electron-like events below 475 MeV\\citep{agu09,agu10,agu07}.  In addition, in the $\\bar{\\nu}_\\mu\\to\\bar{\\nu}_e$ study, MiniBooNE found evidence of oscillations in the 0.1 to 1.0 eV$^2$, which are consistent with LSND results \\citep{ath96, ath98}.\\\\\nCombining solar, atmospheric, reactor and accelerator parameters, the best fit values of the three neutrino mixing  are\n\n for $\\sin^2_{13} < 0.053$: \\citep{aha11}\n \\begin{equation}\n \\Delta m_{21}^2= (7.41^{+0.21}_{-0.19})\\times 10^{-5}\\,{\\rm eV^2};   \\hspace{0.1cm} \\tan^2\\theta_{12}=0.446^{+0.030}_{-0.029}\\,,\n \\end{equation}\nand  for $\\sin^2_{13} < 0.04$: \\citep{wen10}\n\\begin{equation}\n\\Delta m_{23}^2=(2.1^{+0.5}_{-0.2})\\times 10^{-3}\\,{\\rm eV^2};    \\hspace{0.1cm} \\sin^2\\theta_{23}=0.50^{+0.083}_{-0.093}\\,\n\\label{3parosc}\n\\end{equation}\n\\subsection{Neutrino oscillation inside the jet}\nWhen  neutrino oscillations take place in the matter, a resonance could occur that dramatically enhances the flavor mixing and can lead to maximal conversion from one neutrino flavor to another.  This resonance depends on the effective potential,  density profile of the medium,  and oscillation parameters. As  $\\nu_e$ is the one that can  interact  via CC,   the effective potential can be obtained  calculating the difference between the potential due to CC and  NC contributions \\citep{kuo89}.\n\\subsubsection{Two-Neutrino Mixing}\nIn this subsection, we will consider the neutrino oscillation process $\\nu_e\\leftrightarrow \\nu_{\\mu, \\tau}$. The evolution equation for the propagation of neutrinos in the above medium is given by\n\\begin{equation}\ni\n{\\pmatrix {\\dot{\\nu}_{e} \\cr \\dot{\\nu}_{\\mu}\\cr}}\n={\\pmatrix\n{V_{eff}-\\Delta \\cos 2\\theta & \\frac{\\Delta}{2}\\sin 2\\theta \\cr\n\\frac{\\Delta}{2}\\sin 2\\theta  & 0\\cr}}\n{\\pmatrix\n{\\nu_{e} \\cr \\nu_{\\mu}\\cr}},\n\\end{equation}\nwhere $\\Delta=\\delta m^2\/2E_{\\nu}$, $V_{eff}=\\sqrt 2G_F\\, N_e$ is the effective potential,     $E_{\\nu}$ is the neutrino energy,   and $\\theta$ is the neutrino mixing angle. For anti-neutrinos one has to replace $N_e$ by $-N_e$. The conversion probability for a given time $t$ is\n\\begin{equation}\nP_{\\nu_e\\rightarrow {\\nu_{\\mu}{(\\nu_\\tau)}}}(t) = \n\\frac{\\Delta^2 \\sin^2 2\\theta}{\\omega^2}\\sin^2\\left (\\frac{\\omega t}{2}\\right\n),\n\\label{prob}\n\\end{equation}\nwith\n\\begin{equation}\n\\omega=\\sqrt{(V_{eff}-\\Delta \\cos 2\\theta)^2+\\Delta^2 \\sin^2\n    2\\theta}.\n\\end{equation}\nThe oscillation length for the neutrino is given by\n\\begin{equation}\nL_{osc}=\\frac{L_v}{\\sqrt{\\cos^2 2\\theta (1-\\frac{V_{eff}}{\\Delta \\cos 2\\theta})^2+\\sin^2 2\\theta}},\n\\label{osclength}\n\\end{equation}\nwhere $L_v=2\\pi\/\\Delta$ is the vacuum oscillation length. If the density of the medium is such that the condition $\\sqrt2 G_F\\,N_e=\\Delta\\,\\cos2\\theta$ is satisfied,   the resonant condition, \n\n\\begin{equation}\nV_{eff,R}=\\Delta \\cos 2\\theta\\,,\n\\label{reso2d}\n\\end{equation}\ncan come about, therefore the resonance length can be written as\n\\begin{equation}\nL_{res}=\\frac{L_v}{\\sin 2\\theta}.\n\\label{oscres}\n\\end{equation}\nCombining eqs (\\ref{oscres}) and (\\ref{reso2d}) we can obtain the resonance density as a function of resonance length\n{\\scriptsize \n\\begin{equation}\\label{p1}\n\\textbf{$\\rho_R$}=\n\\cases{\n\\frac{3.69\\times 10^{-4}}{E_{\\nu,TeV}}\t\\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{4.4 \\times 10^{12} \\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2}{\\rm gr\/cm^3}       &   {\\rm sol.} \\,, \\nonumber\\cr\n\\frac{1.39\\times 10^{-2}}{E_{\\nu,TeV}}\t\\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{1.18 \\times 10^{11}\\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2} {\\rm gr\/cm^3}     &   {\\rm atmosp.}\\,,\\nonumber\\cr\n\\frac{3.29}{E_{\\nu,TeV}}\t                             \\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{4.9 \\times 10^8\\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2} {\\rm gr\/cm^3}           &   {\\rm accel.}\\,, \\cr\n}\n\\end{equation}\n}\n\\noindent where sol, atmosp. and accel. correspond to solar, atmospheric and accelerator  parameters.\\\\\nIn addition of the resonance condition, the dynamics of this transition must be  determined by adiabatic conversion through the adiabaticity  parameter \n \\begin{equation}\n\\gamma\\equiv \\frac{\\delta m^2}{2E}\\sin2 \\theta\\,\\tan2 \\theta\\frac{1}{\\mid \\frac1\\rho\\, \\frac {d\\rho}{dr}\\mid}_R\\,,\n\\end{equation}\nwith  $\\gamma\\gg$ 1  or the flip probability  given by\n\\begin{equation}\n P_f= e^{-\\pi\/2\\,\\gamma}\\,,\n \\label{flip}\n \\end{equation}\n where $\\rho$ is given by eqs. (\\ref{dens-pro-A}), (\\ref{dens-pro-B})  and (\\ref{dens-pro-C}).\n\n\n\\subsubsection{Three-neutrino Mixing}\nTo determine the neutrino oscillation probabilities we have to solve the evolution equation of the neutrino system in the matter. In a three-flavor framework, this equation is given by\n\\begin{equation}\ni\\frac{d\\vec{\\nu}}{dt}=H\\vec{\\nu},\n\\end{equation}\nand the state vector in the flavor basis is defined as\n\\begin{equation}\n\\vec{\\nu}\\equiv(\\nu_e,\\nu_\\mu,\\nu_\\tau)^T.\n\\end{equation}\nThe effective Hamiltonian is\n\\begin{equation}\nH=U\\cdot H^d_0\\cdot U^\\dagger+diag(V_{eff},0,0),\n\\end{equation}\nwith\n\\begin{equation}\nH^d_0=\\frac{1}{2E_\\nu}diag(-\\Delta m^2_{21},0,\\Delta^2_{32}).\n\\end{equation}\nwith the same potential $V_{eff}$ given for two-neutrino mixing subsection and $U$  the three\nneutrino  mixing matrix given by \\cite{gon03,akh04,gon08,gon11}\n\\begin{equation}\nU =\n{\\pmatrix\n{\nc_{13}c_{12}                    & s_{12}c_{13}                    & s_{13}\\cr\n-s_{12}c_{23}-s_{23}s_{13}c_{12} & c_{23}c_{12}-s_{23}s_{13}s_{12}   & s_{23}c_{13}\\cr\ns_{23}s_{12}-s_{13}c_{23}c_{12}  &-s_{23}c_{12}-s_{13}s_{12}c_{23}   &  c_{23}c_{13}\\cr\n}},\n\\end{equation}\nwhere $s_{ij}=\\sin\\theta_{ij}$ and  $c_{ij}=\\cos\\theta_{ij}$ and we have taken the Dirac phase $\\delta=0$. For anti-neutrinos one has to replace $U$ by $U^*$.  The different neutrino probabilities are given as\n{\\scriptsize \n\\begin{eqnarray}\nP_{ee}&=&1-4s^2_{13,m}c^2_{13,m}S_{31}\\,,\\nonumber\\\\\nP_{\\mu\\mu}&=&1-4s^2_{13,m}c^2_{13,m}s^4_{23}S_{31}-4s^2_{13,m}s^2_{23}c^2_{23}S_{21}-4\nc^2_{13,m}s^2_{23}c^2_{23}S_{32}\\,,\\nonumber\\\\\nP_{\\tau\\tau}&=&1-4s^2_{13,m}c^2_{13,m}c^4_{23}S_{31}-4s^2_{13,m}s^2_{23}c^2_{23}S_{21}-4\nc^2_{13,m}s^2_{23}c^2_{23}S_{32}\\,,\\nonumber\\\\\nP_{e\\mu}&=&4s^2_{13,m}c^2_{13,m}s^2_{23}S_{31}\\,,\\nonumber\\\\\nP_{e\\tau}&=&4s^2_{13,m}c^2_{13,m}c^2_{23}S_{31}\\,,\\nonumber\\\\\nP_{\\mu\\tau}&=&-4s^2_{13,m}c^2_{13,m}s^2_{23}c^2_{23}S_{31}+4s^2_{13,m}s^2_{23}c^2_{23}S_{21}+4\nc^2_{13,m}s^2_{23}c^2_{23}S_{32}\\,,\\nonumber\\\\\n\\end{eqnarray}\n}\nwhere\n\\begin{equation}\n\\sin\n2\\theta_{13,m}=\\frac{\\sin2\\theta_{13}}{\\sqrt{(\\cos2\\theta_{13}-2E_{\\nu}V_e\/\\delta\n    m^2_{32})^2+(\\sin2\\theta_{13})^2}},\n\\end{equation}\nand\n\\begin{equation}\nS_{ij}=\\sin^2\\biggl(\\frac{\\Delta\\mu^2_{ij}}{4E_{\\nu}}L\\biggr).\n\\end{equation}\nHere $\\Delta\\mu^2_{ij}$ are given by \n\\begin{eqnarray}\n\\Delta\\mu^2_{21}&=&\\frac{\\Delta\n  m^2_{32}}{2}\\biggl(\\frac{\\sin2\\theta_{13}}{\\sin2\\theta_{13,m}}-1\\biggr)-E_{\\nu}V_e\\,,\\nonumber\\\\\n\\Delta\\mu^2_{32}&=&\\frac{\\Delta\n  m^2_{32}}{2}\\biggl(\\frac{\\sin2\\theta_{13}}{\\sin2\\theta_{13,m}}+1\\biggr)+E_{\\nu}V_e\\,,\\nonumber\\\\\n\\Delta\\mu^2_{31}&=&\\Delta m^2_{32} \\frac{\\sin2\\theta_{13}}{\\sin2\\theta_{13,m}}\\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\sin^2\\theta_{13,m}&=&\\frac12\\biggl(1-\\sqrt{1-\\sin^22\\theta_{13,m}}\\biggr)\\,,\\nonumber\\\\\n\\cos^2\\theta_{13,m}&=&\\frac12\\biggl(1+\\sqrt{1-\\sin^22\\theta_{13,m}}\\biggr)\\,.\n\\end{eqnarray}\nThe oscillation length for the neutrino is given by\n\\begin{equation}\nL_{osc}=\\frac{L_v}{\\sqrt{\\cos^2 2\\theta_{13} (1-\\frac{2 E_{\\nu} V_e}{\\delta m^2_{32} \\cos 2\\theta_{13}} )^2+\\sin^2 2\\theta_{13}}},\n\\label{osclength}\n\\end{equation}\nwhere $L_v=4\\pi E_{\\nu}\/\\delta m^2_{32}$ is the vacuum oscillation length. From the resonance condition,  $\\sqrt2 G_F\\,N_e=\\Delta \\cos2\\theta_{13}$, the resonance length and density  are related as\n\\begin{equation} \n\\rho_R=\\frac{1.9\\times 10^{-2}}{E_{\\nu,TeV}}\t\\,     \\biggl[ 1- E_{\\nu,TeV}^2\\biggl( \\frac{8.2 \\times 10^{10} \\,cm}{l_r}\\biggr)^2  \\biggr]^{1\/2}{\\rm gr\/cm^3}\\,.\n\\label{p2}\n\\end{equation}\nOn the other hand, generalizing the adiabaticity  parameter, $\\gamma$, to three-mixing neutrinos, it can be written as\n \\begin{equation}\n\\gamma\\equiv \\frac{\\delta m_{32}^2}{2E}\\sin2 \\theta_{13}\\,\\tan2 \\theta_{13}\\frac{1}{\\mid \\frac1\\rho\\, \\frac {d\\rho}{dr}\\mid}_R\\,,\n\\end{equation}\nwith the flip probability given by eq.  (\\ref{flip}).\n\\subsection{Neutrino Oscillation from  Source to Earth}\nBetween the surface of the star and the Earth the flavor ratio $\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$   is affected by the full three description  flavor mixing, which is calculated as follow. The probability for a neutrino to oscillate from a flavor estate $\\alpha$ to a flavor state $\\beta$ in a time starting from the emission of neutrino at star t=0, is given as\n\\begin{eqnarray}\nP_{\\nu_\\alpha\\to\\nu_\\beta} &=&\\mid <  \\nu_\\beta(t) | \\nu_\\alpha(t=0) >  \\mid\\cr\n&=&\\delta_{\\alpha\\beta}-4 \\sum_{j>i}\\,U_{\\alpha i}U_{\\beta i}U_{\\alpha j}U_{\\beta i}\\,\\sin^2\\biggl(\\frac{\\delta m^2_{ij} L}{4\\, E_\\nu}   \\biggr)\\,.\n\\end{eqnarray}\nUsing the set of parameters give in eq. (\\ref{3parosc}), we can write the mixing matrix\n\\begin{equation}\nU =\n{\\pmatrix\n{\n0.816669\t   &  0.544650     &     0.190809\\cr\n -0.504583  & 0.513419      &\t 0.694115\\cr\n 0.280085   &  -0.663141    &     0.694115\\cr\n}}\\,.\n\\end{equation}\nAveraging the sin term in the probability to $\\sim 0.5$ for larger distances L \\citep{lea95}, the probability matrix for a neutrino flavor vector of ($\\nu_e$, $\\nu_\\mu$, $\\nu_\\tau$)$_{source}$ changing to a flavor vector  ($\\nu_e$, $\\nu_\\mu$, $\\nu_\\tau$)$_{Earth}$ is given as\n\\begin{equation}\n{\\pmatrix\n{\n\\nu_e   \\cr\n\\nu_\\mu   \\cr\n\\nu_\\tau   \\cr\n}_{Earth}}\n=\n{\\pmatrix\n{\n0.534143\t  & 0.265544\t  & 0.200313\\cr\n 0.265544\t  & 0.366436\t  &  0.368020\\cr\n 0.200313\t  & 0.368020\t  & 0.431667\\cr\n}}\n{\\pmatrix\n{\n\\nu_e   \\cr\n\\nu_\\mu   \\cr\n\\nu_\\tau   \\cr\n}_{source}}\n\\label{matrixosc}\n\\end{equation}\nfor distances longer than the solar system. \n\\section{Results and Discussions}\nWe have considered a core collapse of massive stars leading to supernovae (SNe) of type Ib,c and II with mildly relativistic jets.  Although this mildly relativistic jet may not be able to break through the stellar envelope,  electrons and protons are expected to be accelerated in the internal shocks,   and then to be cooled down by synchrotron radiation, inverse Compton and hadronic processes (p$\\gamma$ and p-hadron\/meson).   Photons from  electron synchrotron radiation thermalized  to a some keV-peak  energy serve  as  cooling mechanism for accelerated protons by means of  p$\\gamma$ interactions. Another cooling mechanism of protons considered here are the  p-p interactions, due to  the high number density of protons  (3.1 $\\times 10^{20}$ cm$^{-3} \\leq n'_p \\leq $ 3.1 $\\times 10^{22}$ cm$^{-3}$ ) \\citep{raz05}.   In p$\\gamma$ and p-p interactions, high-energy pions and kaons are created  which  in turn interact with protons by $\\pi$-p and $K$-p interactions, producing  another hadronic\/meson  cooling mechanism.   To illustrate the degree and  energy region of efficiency of each cooling process, we have plotted the proton (figures \\ref{ptime_r1} and \\ref{ptime_r2}) and meson (figures \\ref{mtime_r1} and \\ref{mtime_r2}) time scales when   internal shocks take place at   $6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm  and, the magnetic field lies in the range  3.4$\\times 10^7$ G  $\\leq B' \\leq$ 1.1$\\times 10^{10}$ G.  Comparing  the time scales in figures \\ref{ptime_r1} and \\ref{ptime_r2}, one can observe that the maximum proton energy is when the acceleration and synchrotron time scales are equal; it happens  when proton energy is in the range $10^{15} eV \\leq E'_p \\leq 10^{16}$  eV which corresponds to  internal shocks at  $6\\times 10^9$ cm  with $B' = 1.1\\times 10^{10}$ G   and  $6\\times 10^{10}$ cm   with $B'=3.4\\times 10^7$ G, respectively.  In figs. \\ref{mtime_r1} and \\ref{mtime_r2},   one can see that hadronic time scales are equal to  other  time scales at different energies.  For instance,  internal shocks at $6\\times 10^{10}$ cm and $B'=1.1\\times 10^{9}$ G,  the time scales of pion synchrotron emission and  hadronic are equal  for  pion energy  $\\sim 5\\times 10^{11}$ eV. Computing the break meson energies for which time scales are equal to each other, we  can estimate the break neutrino energies.   From the equality  of  kaon\/pion lifetime and  synchrotron  cooling time scales we obtain the break neutrino energies  $\\sim$(24\/179) GeV and  $\\sim$428 GeV\/69 TeV, respectively.  Also, considering   p$\\gamma$ interactions  the  threshold neutrino energy $\\sim$ 3 TeV is obtained.  Taking into account the distances of  internal shocks  ($6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm)\nwe have plotted  the neutrino energy  as a function of the magnetic equipartition parameter in the range  0.1$\\leq \\epsilon_B\\leq 10^{-4}$  (3.4$\\times 10^7$ G  $\\leq B' \\leq$ 1.1$\\times 10^{10}$ G).  As shown in the fig.  \\ref{prod_neu},  neutrino energy between 1 - 10 PeV can be generated for  $\\epsilon_B$  between 3.5$\\times 10^{-3}$  and 4.1$\\times 10^{-4}$, that corresponds to a magnetic field in the range  2.02$\\times 10^8$ (2.02$\\times 10^9$) G - 6.9$\\times 10^7$ (6.9$\\times 10^8$) G at $6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm from the central engine, respectively. Under this scenario, chocked jets are bright in high-energy neutrinos and dark in gamma rays.\\\\\nOn the other hand, taking into account the range of neutrino energy  (24 GeV$\\leq E_\\nu\\leq$ 69 TeV),  internal shocks  at a distance of  6$\\times 10^{10}$ cm,  strength of  magnetic field of 1.1$\\times 10^{10}$ G   and considering  three models of density profile  (see section III. eqs. \\ref{dens-pro-A}, \\ref{dens-pro-B} and \\ref{dens-pro-C}) of a pre-supernova star,  we  present a full description of two- and three-flavor neutrino oscillations.   Based on these models of density profiles we calculate the effective potential, the resonance condition and,   the resonance length and density.    From the resonance  condition, we obtain the  resonance density ($\\rho_R$)  as a function of resonance length ($l_R$) for two (eq. \\ref{p1}) and three flavors (eq. \\ref{p2}). We overlap the plots of the density profiles as a function of distance with  the resonance conditions (resonance density as a function of resonance length). They are shown  in Fig \\ref{twoflavor} (two flavors) and in Fig.  \\ref{threeflavor} (three flavors).    For two flavors,   we have taken into account solar (top), atmospheric (middle) and accelerator (bottom) parameters of neutrino experiments.   Using solar parameters, the resonance length is in the range  $\\sim (10^{11} -  10^{14.2})$ cm  and resonance density in $\\sim  (10^{-2} - 10^{-4})$g\/cm$^3$. As can be seen,  neutrinos with energy 24 GeV  are the only ones that meet the resonance condition for all models of density profiles while neutrinos of energy  178 GeV meet marginally the resonance condition just for the model [B].   Neutrinos with other energy cannot meet the resonance condition. Using atmospheric parameters, the resonance length lies in the range $\\sim (10^{9.1} -  10^{13.3})$ cm and the resonance density in $\\sim  (10^{1} - 10^{-4})$g\/cm$^3$.   As shown,  neutrinos in the energy range of 178 GeV - 3 TeV can oscillate many times before leaving the source. Although the resonance length of neutrino with energy 24 GeV is smaller than star radius, the resonance density is greater than other  models.   Using accelerator parameters, the resonance length is less than $\\sim 10^{10.2}$ cm and  the resonance density lies in the range  $\\sim  (10^{2} - 10^{2})$g\/cm$^3$. Although the resonance length is smaller than the star radius for two flavors, the one that meets the resonance density  is the neutrino energy  69 TeV.   For three flavors,  the range of resonance length  is $\\sim (10^{9} -  10^{12.5})$ cm  and resonance density is  $\\sim  (0.9 - 10^{-4})$g\/cm$^3$, presenting a similar behavior to that described by means of atmospheric parameters.   \nAs the dynamics of resonant transitions  is not only determined by the resonance condition, but also by adiabatic conversion,  we plot the flip probability as a  function of neutrino energy for two  (fig \\ref{twoflip}) and three flavors (fig \\ref{threeflip}). Dividing the plots of flip probabilities in three regions of less than 0.2,  between 0.2 and 0.8 and greater than 0.8, we have that  in the first case (P$_\\gamma \\leq$ 0.2), a pure adiabatic conversion  occurs, the last case (P$_\\gamma \\geq$ 0.8) is a strong violation of adiabaticity  and the intermediate region 0.2 $<$ P$_\\gamma$ $<$ 0.8 represents the transition region \\citep{dig00}. In Fig. \\ref{twoflip},  the top, middle and bottom  plots are obtained  using solar, atmospheric and  accelerator parameters of neutrino oscillations, respectively.    As shown in top figure,  the pure adiabatic conversion occurs when neutrino energy is  less than $5\\times 10^{11}$ eV for model [A] and [C] and,   $\\sim 10^{12}$ eV for model [B] and, the strong violation of adiabaticity is given for neutrino energy greater than $6\\times 10^{12}$ eV  in the three profiles.    In the middle figure, one can see that independently of the profile, neutrinos with energy of less than E$_\\nu$=10$^{14}$ eV can have pure adiabatic conversions.    In the bottom figure,  the three models of  density profiles have the same behavior for the whole energy range.   Neutrinos with   energy less than $\\sim 10^{11.3}$ eV and  greater than  $\\sim 10^{13.2}$ eV present conversion adiabatically pure and strong violation, respectively.   In fig. \\ref{threeflip},   the flip probability for three flavors are plotted.   The energy range for each region of P$_\\gamma$ changes marginally   according to the model of  density profile.  Neutrinos with E$\\sim 10^{12}$ eV are capable of having  pure adiabatic conversion in [B] but not in [A] or [C].  The strong violation of adiabaticity begins when the neutrino energies are E$\\sim 10^{13}$ eV and E$\\sim 10^{13.8}$ eV, for [A] and [C], respectively. \\\\\nOn the other hand,   we have also plotted (fig. \\ref{proen}) the oscillation probabilities for three flavors  as a function of energy when neutrinos keep moving at a distance of   r=$10^{11}$ cm (above) and r=$10^{12}$ cm (below) from the core.   In the top figure, the survival probability of electron neutrino, P$_{ee}$, is close to one regardless of  neutrino energy, therefore  the conversion probabilities  P$_{\\mu e}$ and P$_{\\tau e}$ are close to zero, as shown.  Depending on the neutrino energy, the survival probability of muon and tau neutrino, P$_{\\mu \\mu}$ and P$_{\\tau \\tau}$, oscillates between zero and one. For example,   for E$\\sim 430$ GeV,  the conversion probability of muon  P$_{\\mu \\tau}$ is close to zero while the survival probability of muon and tau neutrino, P$_{\\mu \\mu}$ and P$_{\\tau \\tau}$,  are close to one, and for E$\\sim 1$ TeV  probabilities change dramatically, being P$_{\\mu \\tau}\\sim 1$ and P$_{\\tau \\tau}$=P$_{\\mu \\mu}\\sim$ 0. In the bottom figure, neutrinos are moving along the jet  at r=10$^{12}$ cm and  although the survival and conversion probabilities  have  similar behaviors to those moving to  r=10$^{11}$ cm,  they  are changing faster.  To have a better understanding, we have separated all probabilities and plotted them in fig.  \\ref{prosep}.\nFrom up to down,  the probabilities of  electron neutrino and   survival probability of muon neutrino are shown in the first and second graph, respectively,   and  the conversion and survival probability of tau  neutrino are plotted in the third and four graph, respectively. Moreover, we have plotted in figs. \\ref{prob_dist} and \\ref{prob_dist2} the oscillation probabilities  as a function of distance, when  neutrinos are produced at a radius  $6\\times 10^{9}$ cm and $6\\times 10^{10}$ cm, respectively,  and continue to propagate along the jet. We take into account  four neutrino energies $E_\\nu$=178 GeV, $E_\\nu$=428 GeV, $E_\\nu$=3 TeV and $E_\\nu$=69 TeV.  \nAs shown, as neutrino energy  increases, the probabilities oscillate less. For instance,  when an electron neutrino with energy $E_\\nu$=178 GeV  propagates along the jet,  the survival probability of electron changes from one at  $\\sim 8\\times 10^{10}$ cm to zero at $\\sim 9.5\\times 10^{10}$ cm.   For  $E_\\nu$=428 GeV(3 TeV), the survival probabilities  change from one at  $9.1\\times 10^{10}$ ($6.0\\times 10^{10}$) cm  to zero at $1.8\\times 10^{11}$($3.5\\times 10^{11}$) cm  and for $E_\\nu$=69 TeV, the probability  is constant in this range (greater than $\\sim 10^{12}$ cm).  In the last case, neutrino does not oscillate to another flavor during its propagation.\nFinally, considering a flux ratio for $\\pi$, K and $\\mu$ decay  of 1:  2:  0, the density profile [A] and  oscillation probabilities at three distances (10$^{11}$ cm,  10$^{11.5}$ cm and 10$^{12}$ cm), we show in table 1 the flavor ratio on the surface of star. Also, computing the vacuum oscillation effects between the source and Earth (Eq. \\ref{matrixosc}),  we estimate and show in table \\ref{flaratio}  the flavor ratio expected on Earth when neutrinos emerge  from the star at L=(10$^{11}$,  10$^{11.5}$  and 10$^{12}$) cm . \n\n\n\\begin{table}\n\\begin{center}\\renewcommand{\\tabcolsep}{0.2cm}\n\\renewcommand{\\arraystretch}{0.89}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n$E_{\\nu}$  &$\\phi_{\\nu_e}:\\phi_{\\nu_\\mu}:\\phi_{\\nu_\\tau}$ &$\\phi_{\\nu_e}:\\phi_{\\nu_\\mu}:\\phi_{\\nu_\\tau}$&$\\phi_{\\nu_e}:\\phi_{\\nu_\\mu}:\\phi_{\\nu_\\tau}$  \\\\\n(TeV)&(L=10$^{11}$ cm)&(L=10$^{11.5}$ cm)&(L=10$^{12}$ cm)\\\\ \\hline\n\n0.024     &  0.946:1.949:0.115  &  0.697:1.405:0.899   &  0.881:1.578:0.541  \\\\\\hline\n\n0.178     & 0.510:1.814:0.676  &  0.987:1.386:0.627  &  0.507:1.807:0.686 \\\\\\hline\n\n0.428     & 0.983:1.589:0.428  &  0.659:1.871:0.524   &  0.538:1.721:0.741\\\\\\hline\n\n3            & 0.896:1.212:0.892   &  0.502:1.753:0.744   &  0.501:1.762:0.737 \\\\\\hline\n\n68.5     &  0.999:1.997:0.003   &  0.998:1.972:0.030   &  0.979:1.746:0.275 \\\\\\hline\n\n\\end{tabular}\n\\label{tatm}\n\\end{center}\n\\caption{\\small\\sf The flavor ratio on the surface of source for five neutrino energies  (E$_{\\nu}$=24 GeV, 178 GeV, 428 GeV, 3 TeV and 68.5 TeV), leaving the star to three distances L=10$^{11}$ cm, 10$^{11.5}$ cm, and 10$^{12}$ cm.  }\n\\label{flaratio}\n\\end{table}\n\n\n\\begin{table}\n\\begin{center}\\renewcommand{\\tabcolsep}{0.2cm}\n\\renewcommand{\\arraystretch}{0.89}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n$E_{\\nu}$  &$\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$ &$\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$&$\\phi^0_{\\nu_e}:\\phi^0_{\\nu_\\mu}:\\phi^0_{\\nu_\\tau}$  \\\\\n(TeV)&(L=10$^{11}$ cm)&(L=10$^{11.5}$ cm)&(L=10$^{12}$ cm)\\\\ \\hline\n\n0.024     &  1.046:1.008:0.956   &   0.925:1.031:1.045 &  0.998:1.011:0.991 \\\\\\hline\n\n0.178     &   0.889:1.049:1.062  &   1.021:1.000:0.978 &  0.888:1.049:1.063 \\\\\\hline\n\n0.428     &   1.033:1.000:0.966  &  0.954:1.053:1.047  &  0.893:1.046:1.061 \\\\\\hline\n\n3             &   0.979:1.010:1.011  &   0.883:1.049:1.067  &  0.883:1.050:1.067 \\\\\\hline\n\n68.5       &   1.065:0.998:0.936  &   1.063:0.999:0.939 &   1.042:1.001:0.957\\\\\\hline\n\n\\end{tabular}\n\\label{tatm}\n\\end{center}\n\\caption{\\small\\sf The flavor ratio expected on Earth for five neutrino energies  (E$_{\\nu}$=24 GeV, 178 GeV, 428 GeV, 3 TeV and 68.5 TeV), leaving the star to three distances L=10$^{11}$ cm, 10$^{11.5}$ cm, and 10$^{12}$ cm.}\n\\label{flaratio}\n\\end{table}\n\\section{Summary and conclusions}\nWe have done a wide description of  production channels  of high-energy neutrinos in a middle relativistic hidden jet and also shown that neutrinos with energies between  1 - 10 PeV can be generated. Taking  into account a particular range of neutrino energies  generated  in the internal shocks at a distance of  6$\\times 10^{10}$ cm and with a distribution of  magnetic field  1.1$\\times 10^{10}$ G,  we have shown  their oscillations between flavors along the jet for  three models of density profiles.  For two neutrinos mixing,  we have used the fit values of neutrino oscillation parameters from solar, atmospheric, and accelerator experiments and  analyzing the resonance condition we found that  the resonance lengths are the largest  and  resonance densities are the smallest for solar parameters and  using  accelerator parameters we have obtained the opposite situation,  the resonance lengths are the smallest  and  resonance densities are the largest. The most favorable condition for high-energy neutrinos  to oscillate resonantly before going out of the source is given through atmospheric parameters and   these conversions would be pure adiabatic.\nFor three neutrino mixing,  we have calculated the ratio flavor on the surface of the source as well as that expected on Earth. Our analysis shows that deviations  from 1:1:1 are obtained  at different energies and places along the jet, which is given in table 2.     From analysis of flip probability we also show that neutrinos may oscillate depending on their energy and the parameters of neutrino experiments.    As a particular case, when the three-flavor  parameters are considered (fig. \\ref{threeflip}), we obtain that neutrino energies above  $\\geq$ 10 TeV can hardly oscillate, obtaining the same result given by \\citet{2013arXiv1304.4906O}.\\\\  As shown,  depending on the flavor ratio obtained on Earth we could differentiate  the progenitor,  its density profile at different depths in the source,  as well as understand similar features between lGRBs and core collapse supernovae.  Distinct times of arrival of neutrino flavor ratio will provide constraints on density profiles at different places in the star  \\citep{bar12}. \nThese observations in detectors such as IceCube, Antares and KM3Net would be a compelling evidence that chocked jets  are bright in neutrinos \\citep{abb12, abb13, pra10,lei12}.\nThe number of sources with hidden jets  may be much larger than the exhibited one, limited only by the ratio of type Ib\/c and type II SNe to GRB rates. Within 10 Mpc, the rate of core-collapse supernovae is $\\sim$1 - 3 yr$^{-1}$, with a large contribution of galaxies around  3 - 4 Mpc.   At larger distances,  the expected number of neutrino events in IceCube is still several, and the supernova rate is $\\geq$ 10 yr$^{-1}$ at 20 Mpc \\citep{and05}. Recently,  \\citet{tab10} calculated the events expected in DeepCore and neutrino-induced cascades in km$^3$ detectors for neutrinos energies $\\leq$ 10 GeV and $\\leq$  a few TeV respectively and forecast that $\\sim$ 4 events in DeepCore and $\\sim$ 6 neutrino-induced cascades in IceCube\/KM3Net  would be expected. An extension up to higher energies of this  calculation  should be done to correlate the expected events  in these sources with  the number of PeV-neutrinos  observed with IceCube \\citep{aar13}. \\\\  \nInterference effects in the detector by atmospheric neutrino oscillation are very  small (less than 10 \\%) due to short   path traveled by neutrinos in comparison with cosmological distances \\citep{men07}.\n\n\\section*{Acknowledgements}\n\nWe thank the referee  for a critical reading of the paper and valuable suggestions. We also thank B. Zhang, K. Murase, William H. Lee, Fabio de Colle, Enrique Moreno and Antonio Marinelli for useful discussions.   NF gratefully acknowledges a Luc Binette-Fundaci\\'on UNAM Posdoctoral Fellowship.\n \n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{intro}\n\nWe study transition fronts for one-dimensional reaction-diffusion equations\nwith \\emph{compactly-perturbed ignition-monostable reactions}.\nConsider the evolution PDE\n\\begin{equation}  \\label{eq:main}\nu_t = u_{xx} + f(x,u),\\quad (t,x)\\in \\mathbb{R}\\times \\mathbb{R},\n\\end{equation}\nwhere the nonlinearity $f$ satisfies the following on $\\mathbb{R}\\times [0,1]$:\n\\begin{enumerate}[label=(F\\arabic*)]\n  \\setlength\\itemindent{10pt}\n\\item $f\\ge 0$ is Lipschitz continuous with $\\gamma:=\\mbox{Lip}(f)$, and  $f(x,0)=f(x,1) = 0$ for all $x\\in \\mathbb{R}$;\n  \\label{item:general}\n  \n\\item there exists $L>0$ such that $f(x,u)\\equiv f_0(u)$ for all $|x|\\ge L$, where $f_0$ is an ignition reaction with $f_0\\equiv 0$ on $[0,\\theta_0]\\cup \\{1\\}$, $f_0>0$ on $(\\theta_0 ,1)$, and $f_0$ is non-increasing on $[1-\\theta_1,1]$ for some $\\theta_0,\\theta_1\\in (0,1)$;\n  \\label{item:ignition}\n  \n\\item \\label{item:regularity} the (right hand) derivative $a(x):=f_u(x,0)\\ge 0$ exists, and for all $\\varepsilon>0$, there exists $\\zeta=\\zeta(\\varepsilon)\\in(0,\\theta_0)$ such that\n  \\begin{equation*}\n   \n    (1-\\varepsilon)a(x)u\\leq f(x,u)\\leq (a(x)+\\varepsilon)u\\quad \\mbox{for }(x,u)\\in \\mathbb{R}\\times [0,\\zeta].\n  \\end{equation*}\n\\end{enumerate}\nAs described above, $f$ is obtained by perturbing a homogeneous ignition reaction $f_0$ locally on the interval $[-L,L]$ with an inhomogeneous monostable reaction.\nIn the present work, we are interested in how such perturbation affects the existence of transition fronts.\n\nThe PDE \\eqref{eq:main} and its variations are widely used to model a host of natural processes, including thermal, chemical, and ecological dynamics.\nBy \\ref{item:general}, $u\\equiv 0, 1$ are two equilibrium solutions of \\eqref{eq:main}.\nTherefore, one is usually interested in the transition from the (unstable) state $u\\equiv 0$ to the (stable) state $u\\equiv 1$.\n\\emph{Transition fronts} are a class of solutions that model this phenomenon.\nThey are global-in-time solutions $u:\\mathbb{R}^2 \\to (0,1)$ of \\eqref{eq:main} satisfying\n\\begin{equation}  \\label{item:trans_lims}\n\\lim_{x\\to-\\infty} u(t,x)=1, \\quad\\lim_{x\\to+\\infty} u(t,x)=0\\quad \\mbox{for all }t\\in \\mathbb{R}\n\\end{equation}\nand the bounded front width condition, that is, for all $\\mu \\in (0,\\frac 12 )$,\n\\begin{equation}  \\label{item:trans_width}\n\\sup _{t\\in \\mathbb{R}} L_\\mu(t):=\\sup_{t\\in\\mathbb{R}}\\, \\mbox{diam} \\{x\\in\\mathbb{R}|\\; \\mu\\le u(t,x)\\le 1-\\mu\\}<\\infty.\n\\end{equation}\nThis definition was introduced in \\cite{BH12, Matano, Sen}.\n\nThe study of transition fronts has seen much activity since the seminal works by Fisher \\cite{F37} and Kolmogorov, Petrovskii, and Piskunov \\cite{KPP37}, who first studied \\emph{traveling fronts} for \\eqref{eq:main} with \\emph{homogeneous Fisher-KPP reactions}.\nHere, traveling fronts are transition fronts of the form $u(t,x)=U(x-ct)$ for some speed $c\\in \\mathbb{R}$ and profile $U$ with $\\lim_{y\\to-\\infty}U(y)=1$, $\\lim_{y\\to\\infty}U(y)=0$,\nand Fisher-KPP reactions are those $f$ satisfying \\ref{item:general}, $f'(0)>0$, and $0<f(u)\\le f'(0)u$ for $u\\in(0,1)$.\nIn their work, they found that for each $c\\ge c_*:=2\\sqrt{f'(0)}$, \\eqref{eq:main} admits a unique (modulo translation) traveling front, while no front exists for $c<c_*$.\nBy means of phase plane analysis, it can be shown that the same existence result holds for general homogeneous monostable reactions ($f(u)>0$ on $(0,1)$), although $c_*\\ge 2\\sqrt{f'(0)}$ in general (e.g., see \\cite{AW78}).\nIn contrast, for \\emph{homogeneous ignition} (defined as in \\ref{item:ignition}) and \\emph{bistable reactions} (the same as ignition except $f<0$ on $(0,\\theta_0)$ and $\\int _0^1 f(u)du>0$), there is only one speed $c_*>0$ which gives rise to a unique (up to translation) traveling front. The unique speed $c_*$ will be called \\emph{the spreading speed of }$f$.\n\nOver decades, the study of transition fronts extended to spatially periodic reactions (in which case fronts have time-periodic profiles, and are known as \\emph{pulsating fronts}).\nInstead of surveying the vast literature, let us refer to the review articles by Berestycki \\cite{B03} and Xin \\cite{X00}, and the references therein.\nThe development in general inhomogeneous media is considerably more recent.\nThe first existence result was obtained by Vakulenko and Volpert \\cite{VV11} for small perturbations of homogeneous bistable reactions.\nLater, Mellet, Roquejoffre, and Sire \\cite{MARS10} proved the existence of fronts for ignition reactions of the form $f(x,u)=a(x)f_0(u)$, where $f_0$ is ignition, and $a(x)$ is bounded with $\\inf _{\\mathbb{R}}a(x)>0$, which need not be close to being constant (see also \\cite{NR09} for the case of random media, relying on the notion of generalized random traveling waves developed in \\cite{Sen}). Zlato\\v{s} then extended these results (along with uniqueness and stability) to general inhomogeneous ignition and mixed ignition-bistable media \\cite{Z13, ZPreprint}.\n\n\nTransition fronts has also been investigated in inhomogeneous Fisher-KPP media by several authors. As far as Fisher-KPP reactions are concerned, a strong inhomogeneity in the reaction may prevent existence of transition fronts, while a weak inhomogeneity gives rise to them.\nThis is translated into the following result proved by Nolen, Roquejoffre, Ryzhik and Zlato\\v{s} \\cite{Zlatos} for reactions satisfying $0< f(x,u)\\le a(x)u$ for all $(x,u)\\in \\mathbb{R}\\times (0,1)$, with $a(x):=f_u(x,0)$, $a_-:=\\inf_{x\\in \\mathbb{R}}a(x)>0$, and $a(x)-a_-\\in C_c(\\mathbb{R})$.\nThey found that when the inhomogeneity of $f$ is strong, in the sense that the principal eigenvalue $\\lambda$ of the operator $\\partial_{xx}+a(x)$ satisfies $\\lambda>2a_-$, any non-constant global-in-time solution $u$ of \\eqref{eq:main} is \\emph{bump-like} (i.e. $u(t,x)\\le C_t e^{-c|x|}$), preventing the existence of transition fronts.\nThis in fact is the first known example of a reaction function $f$ such that \\eqref{eq:main} does not admit any transition front.\n\nMoreover, in the same work, they also show that the existence criterion is (almost) sharp.\nIn the case of a weak localized inhomogeneity $\\lambda<2a_-$, for each $c\\in (2\\sqrt{a_-},\\lambda\/\\sqrt{\\lambda-a_-})$ the PDE \\eqref{eq:main} admits a transition front with \\emph{global mean speed} $c$, in the sense that if $X(t):=\\sup \\{x\\in\\mathbb{R}:u(t,x)=\\frac 12\\}$, then\n\\begin{equation}\\label{gms}\n  \\lim_{t-s\\to\\infty}\\frac{X(t)-X(s)}{t-s}=c.\n\\end{equation}\nTo construct a front, they find an appropriate pair of ordered global-in-time super- and sub-solutions $w\\ge v$ that propagate with speed $c$, and recover a front $u$ between them as a locally uniform limit along a subsequence of solutions $(u_n)_{n\\in {\\mathbb{N}}}$ of the Cauchy problem \\eqref{eq:main} with initial data $u_n(-n,\\cdot)=w(-n,\\cdot)$.\nThe same method was deployed and extended by Zlato\\v{s} \\cite{Z12} and by Tao, Zhu, and Zlato\\v{s} \\cite{TZZ13} to prove the existence of fronts for general inhomogeneous KPP and monostable reactions when $a(x)-a_-$ is not compactly supported.\n\nIn the present paper, we modify the approach from \\cite{Zlatos} to establish a similar sharp existence criterion for reactions satisfying Hypothesis (F).\nAs mentioned, such $f$ is obtained by locally perturbing the ignition reaction $f_0$ with a monostable reaction.\nWe therefore show that a strong perturbation in the reaction prevents the existence of fronts, while a weak perturbation admits them.\nThe existence criterion in our case is determined by the spreading speed of the reaction $f_0$ and the supremum of the spectrum of the operator $\\partial_{xx}+a(x)$.\nThe spreading speed of $f_0$ is the unique number $c_0>0$ such that the following ODE admits a unique (up to translation) solution:\n\\begin{equation}\n  \\label{tfeq}\n  U''+c_0U' +f_0(U)=0,\\quad \\lim_{x\\to -\\infty} U(x)=1,\\quad \\lim_{x\\to\\infty} U(x)=0.\n\\end{equation}\nOn the other hand, the supremum of the spectrum of $\\partial_{xx}+a(x)$ is given by\n\\begin{equation*}\n \n  \\lambda := \\sup \\sigma (\\partial_{xx}+a(x))= \\sup_{{\\psi \\in H^1(\\mathbb{R})}:\\,||\\psi||_{L^2}=1} {\\int_\\mathbb{R} (-[\\psi'(x)]^2 + a(x) [\\psi (x)]^2)dx}.\n\\end{equation*}\nSince $a(x)\\ge 0$ is compactly supported by \\ref{item:ignition}, the essential spectrum of $\\partial_{xx}+a(x)$ is $(-\\infty,0]$, which implies $\\lambda\\ge 0$.\nIf $\\lambda>0$ (i.e. $a\\not\\equiv 0$), it is in fact the principal eigenvalue.\nThen a corresponding $L^\\infty$-normalized principal eigenfunction $\\psi$ exists, is unique, and satisfies\n\\begin{equation}\n  \\label{eq:eigen}\n  \\psi '' + a(x)\\psi = \\lambda \\psi,\\quad \\psi>0,\\quad ||\\psi||_{L^\\infty}=1.\n\\end{equation}\n\nThe main results of the present work are stated as follows.\n\n\\begin{theorem}\n\t\\label{thm:nonex}\n\tLet $f$ satisfy \\ref{item:general}--\\ref{item:regularity} for some $f_0$ and $a$, $c_0$ be the spreading speed of $f_0$, $\\lambda$ be the supremum of the spectrum of $\\partial_{xx}+a(x)$,\n\tand assume $\\lambda > c_0^2$.\n\t\n\t\\begin{enumerate}\n\t\t\\item All entire solutions $u$ of \\eqref{eq:main} with $0<u<1$, $\\inf_{(t,x)\\in \\mathbb{R}^2} u=0$ are bump-like. That is, there are $c,C_t>0$ such that $u(t,x)\\le C_te^{-c|x|}$ for all $(t,x)\\in \\mathbb{R}^2$.\n\t\tIn particular, \\eqref{eq:main} does not admit a transition front solution.\n\t\t\n\t\t\\item Assume \\ref{item:regularity} is replaced by the following: there exists $\\zeta  \\in (0,\\theta_0)$ such that $f(x,u) = a(x)u$ for $u\\in [0,\\zeta]$.\n\t\tThen a nonzero bump-like solution of \\eqref{eq:main} exists, is unique (up to a time-shift) among all solutions with $0<u<1$, $\\inf _{(t,x)\\in\\mathbb{R}^2}u=0$,\n\t\tand satisfies $u(t,x)=\\zeta e^{\\lambda t}\\psi(x)$ for all $t\\le 0$ after a time translation, where $\\psi$ is the principal eigenfunction given in \\eqref{eq:eigen}.\n\t\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\n\t\\label{thm:ex}\n\tLet $f,$ $c_0,$ and $\\lambda$ be as in Theorem \\ref{thm:nonex}.\n\tIf $\\lambda < c_0^2$, then \\eqref{eq:main} admits a transition front, which is increasing-in-time and has a global mean speed $c_0$.\n\\end{theorem}\n\n\\begin{remark}\nIf \\eqref{eq:main} does not admit a stationary solution other than $u\\equiv 0, 1$ (i.e. $f(\\cdot,\\theta)\\not\\equiv 0$ for all $\\theta \\in (0,1)$), then any entire solution $u\\not\\equiv 1$ to \\eqref{eq:main} must satisfy $\\inf _{(t,x)\\in \\mathbb{R}^2} u=0$ (see the proof of Lemma \\ref{lem:limit}).\nHence in this case, Theorem \\ref{thm:nonex} holds without the infimum assumption.\n\\end{remark}\n\n\\begin{remark}\nIt can been seen in the proof that Theorem \\ref{thm:nonex} still holds if we replace $f(x,u)=f_0(u)$ in \\ref{item:ignition} by $f(x,u)\\le f_0(u)$ for $|x|\\ge L$.\n\\end{remark}\n\n\\begin{remark}\nThe transition front $u$ constructed in Theorem \\ref{thm:ex} satisfies $\\sup _{t\\in \\mathbb{R}} |X\n(t) - c_0t|<\\infty$, which is a stronger condition than \\eqref{gms}. \nIn fact, $u$ can be viewed as a perturbation of a traveling front $U$ of $f_0$.\nBy the asymptotic stability of ignition fronts, one can further show that $u$ converges to $U$ as $t\\to\\pm\\infty$ in the sense that\n\\[ \\lim_{t\\to\\pm \\infty} \\sup_{x\\in\\mathbb{R}} | u(t,x) - U(x-c_0t -x_\\pm) | = 0\\quad \\mbox{for some }x_\\pm\\in \\mathbb{R}. \\]\n\\end{remark}\n\n\\begin{remark}\t\nIn \\cite{Zlatos}, infinitely many transition fronts with different global mean speeds were constructed in the case of Fisher-KPP reactions with weak inhomogeneity.\nIn our case, we only obtain a single transition front with a single global mean speed $c_0$ in Theorem \\ref{thm:ex}. \nIndeed, by a comparison principle argument and stability of front speeds with respect to reactions, one can easily show that any transition front must have a global mean speed $c_0$. \nThough transition fronts in ignition media are expected to be unique \\cite{Z12}, we do not know if the uniqueness still holds in our case of mixed ignition-monostable reactions. \n\n\\end{remark}\n\nOur method is structurally similar to that of \\cite{Zlatos}.\nThe primary difference and technicality in this work is the lack of KPP structure in our reaction.\nIndeed, KPP structure allows one to exploit the intimate connection between solutions of \\eqref{eq:main} and those of the linearized equation $u_t=u_{xx}+a(x)u$ (which are super-solutions to \\eqref{eq:main}).\nTherefore, super- and sub-solutions found in \\cite{Zlatos} are based on (generalized) eigenfunctions of the operator $\\partial_{xx}+a(x)$ and exponential functions.\nIn our case, we clearly lack this convenience.\nHence, to overcome this, we base our super- and sub-solutions on the traveling front of the ignition reaction $f_0$ instead.\nThis modification introduces some difficulties in calculation, but we are still able to obtain a sharp result.\n\nThe body of the paper is organized as follows.\nWe prove the non-existence of transition fronts (Theorem \\ref{thm:nonex}) in the coming section, and present the complimentary existence result (Theorem \\ref{thm:ex}) in Section 3.\n\n\\section*{Acknowledgment}\nAll authors were supported in part by the NSF grant DMS-1056327, and thank Andrej Zlato\\v{s} for directing this research.\nCG gratefuly acknowledges the support of the Fannie and John Hertz Foundation.\nCG, AM, and DW thank the Department of Mathematics at the University of Wisconsin--Madison for its hospitality during the 2013 REU ``Analysis and Differential Equations,'' where this research was mainly conducted.\n\n\\section{Non-existence for $\\lambda >c_0^2$ (proof of Theorem \\ref{thm:nonex})}\n\\label{sec:non}\n\nAs mentioned above, the methods of this section are based on those found in \\cite{Zlatos}.\nIn particular, Theorem \\ref{thm:nonex}, Lemmas \\ref{lem:refined}, \\ref{lem:new}, and their proofs are similar to Theorem 1.2, Lemmas 3.1, 3.2 \\cite{Zlatos}.\nThe primary difference can be found in the proof of Lemma \\ref{lem:new}.\n\nThroughout this section, we assume $f,\\gamma,f_0, \\theta_0,\\theta_1, L, a$ are all from (F), and $\\lambda > c_0^2$.\nFor $\\varepsilon\\in (0,1)$, let $\\lambda_\\varepsilon$ be the principal eigenvalue of the differential operator $\\partial_{xx}+(1-\\varepsilon)a(x)$.\nSince $\\lim_{\\varepsilon\\to 0^+} \\lambda_\\varepsilon=\\lambda>c_0^2$, we may fix $\\varepsilon>0$ such that $\\lambda_\\varepsilon>c_0^2$, and\nlet $\\zeta = \\zeta(\\varepsilon)$ be given in \\ref{item:regularity}.\nFor $M>0$, we let $\\lambda_M = \\lambda_{\\varepsilon,M}$ be the Dirichlet principal eigenvalue of $\\partial_{xx}+(1-\\varepsilon)a(x)$ on $[-M,M]$, and $\\psi_M\\in C^2([-M,M])$ be the corresponding $L^\\infty$-normalized eigenfunction:\n\\begin{gather}\n\t\\label{TS:psi}\n\t\\psi_M'' + (1-\\varepsilon)a(x) \\psi_M = \\lambda_M \\psi_M \\mbox{ on }(-M,M),\\\\\n\t\\psi_M(\\pm M)=0,\\quad ||\\psi_M||_{L^\\infty}=1. \\nonumber\n\\end{gather}\nNote that $\\psi_M>0$ on $(-M,M)$ and $\\lim_{M\\to\\infty} \\lambda_M = \\lambda_\\varepsilon>c_0^2$.\nSo we may again fix $M\\ge L$ large so that $\\lambda_M>c_0^2$. Finally, all constants involved depend on $c_0, M, \\psi_M, \\lambda_M, \\zeta, \\gamma, \\theta_0$.\n\nIn the following, let $u\\in (0,1)$ be an entire solution of \\eqref{eq:main} with $\\inf _{(t,x)\\in \\mathbb{R}}u=0$.\nIn the proofs, we will frequently use the parabolic Harnack inequality for $u$.\nTherefore, for $R,\\sigma>0$, we let $k=k(R,\\sigma)>0$ denote the Harnack constant such that\n\\begin{equation}\n  \\label{h-const}\n\t\\min_{|x-x_0|\\le R} u(t+\\sigma,x) \\ge k \\max _{|x-x_0|\\le R} u(t,x) \\ge k u(t,x_0)\n\\end{equation}\nholds for any $x_0\\in \\mathbb{R}$.\nWe begin with the following simple fact.\n\n\\begin{lemma}\n\t\\label{lem:limit}\n\tThe solution $u$ satisfies $\\lim_{t\\to-\\infty}u(t,x)=0$ locally uniformly.\n\\end{lemma}\n\n\\begin{proof}\n\tBy the Harnack inequality, it suffices to show the limit for $x=0$.\n\tAssume the contrary, so that there exists $\\alpha \\in (0,1)$ and a sequence of times $\\{t_n\\}$ with $t_n\\searrow -\\infty$ such that $u(t_n,0)\\geq \\alpha$.\n\tLet $k=k(M,1)$ be the Harnack constant from \\eqref{h-const},\n\t$\\theta:=\\min\\{k\\alpha, \\zeta\\}$, and extend the eigenfunction $\\psi_M$ from \\eqref{TS:psi} continuously to $\\mathbb{R}$ by setting $\\psi_M\\equiv 0 $ on $[-M,M]^c$.\n\tSince $||\\psi_M||_{L^\\infty}=1$, \\eqref{h-const} (with $(R,\\sigma,x_0,t)=(M,1,0,t_n)$) implies\n\t\\begin{equation}\n\t\t\\label{eq:Harnack_bd}\n\t\tu(t_n+1,x)\\geq \\theta\\psi_M(x)\\quad \\mbox{for all }x\\in {\\mathbb{R}}.\n\t\\end{equation}\n\t\n\tNow let $v:\\mathbb{R}^+\\times \\mathbb{R} \\to [0,1]$ be the solution to the Cauchy problem of \\eqref{eq:main} with initial data $v(0,x) = \\theta\\psi_M(x)$.\n\tWe claim that $v_t\\ge 0$.\n\tBy the comparison principle, it suffices to show $v(s,\\cdot)\\ge \\theta\\psi_M$ for all $s\\ge 0$.\n\tClearly this holds for all $x\\in [-M,M]^c$ because $\\psi_M\\equiv 0$ in this region.\n\tIf $x\\in [-M,M]$ instead, observe that $w(t,x):=\\theta\\psi_M(x)$ is a (stationary) sub-solution of \\eqref{eq:main} by (F3) and \\eqref{TS:psi}. So the comparison principle shows that $v(s,x)\\ge \\theta\\psi_M(x)$ for $(s,x)\\in \\mathbb{R}^+\\times [-M,M]$.\n\tThis implies $v_t\\ge 0$.\n\tLet $v_\\infty(x):= \\lim_{t\\to\\infty}v(t,x)$, which satisfies $v_\\infty '' +f(x,v_\\infty)=0$ on $\\mathbb{R}$ by parabolic regularity.\n\tSince $f\\ge 0$, this forces $v_\\infty \\equiv \\beta$ for some constant $\\beta \\in [\\theta ,1]$.\n\tNow fix $s\\in \\mathbb{R}$.\n\tBy the comparison principle and \\eqref{eq:Harnack_bd}, for all large $n$\n\t\\begin{equation*}\n\t\tu(s,x)\\ge v(s-t_n-1,x)\\quad \\mbox{for all }x\\in \\mathbb{R}.\n\t\\end{equation*}\n\tLetting $n\\to\\infty$, we find that $u(s,\\cdot)\\ge \\beta >0$ for all $s\\in \\mathbb{R}$, contradicting $\\inf_{(t,x)\\in \\mathbb{R}^2} u=0$. Therefore, we must have $\\lim_{t\\to-\\infty}u(t,0)=0$.\n\\end{proof}\n\nWith Lemma \\ref{lem:limit}, after an appropriate time translation we may now assume\n\\begin{gather}\n\t\\label{eq:zetabound}\n\tu(0,0)\\leq \\frac \\zeta 2 \\psi_M(0).\n\\end{gather}\nIn the coming two lemmas, we establish some important bounds on $u$, which play a crucial role in the proof of Theorem \\ref{thm:nonex}.\n\n\\begin{lemma}\n\t\\label{lem:refined}\n\tFor any $c\\in (c_0, \\sqrt{ \\lambda_M} )$, there exists $C_0>0$ (independent of $u$) such that\n\t\\begin{equation}\n\t\t\\label{eq:refined_bound}\n\t\tu(t, x) \\leq C_0 u(0,0) e^{c_0(x+ct)}, \\quad \\text{for }t\\le -1,\\; x\\in [M,\\sqrt{\\lambda_M}(-t-1)-M-1].\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tDenote $u_0 := u(0,0)>0$, $\\psi_0 := \\psi_M(0)>0$, and $D\\subset \\mathbb{R}^2$ the region described in \\eqref{eq:refined_bound}.\n\tTo show \\eqref{eq:refined_bound}, we will prove the following estimate for some $C_0'>0$ (independent of $u)$:\n\t\\begin{equation}\n\t\t\\label{TS:bound2.2}\n\t\tu(t, x) \\leq C'_0 u_0 \\sqrt{|t|}  e^{\\sqrt{\\lambda_M}(x+\\sqrt{\\lambda_M}t)},\\quad \\text{for }(t,x)\\in D.\n\t\\end{equation}\n\tOne can easily show that \\eqref{eq:refined_bound} follows from this with $C_0 := C'_0 \\sup_{t \\leq 0}\\sqrt{|t|}e^{c_0(\\sqrt{\\lambda_M} - c)t}$ (which is finite because $\\sqrt{\\lambda_M}>c$).\n\t\n\tWe prove \\eqref{TS:bound2.2} by contradiction.\n\tLet $k=k(1,1)$ be the Harnack constant from \\eqref{h-const}, and\n\tsuppose there is $(t',x_0)\\in D$ so that \\eqref{TS:bound2.2} does not hold with  $C_0'$ given by\n\t\\[ C_0':=\\frac{\\sqrt {4 \\pi}}{k\\psi_0}e^{\\lambda_M+\\sqrt{\\lambda_M}(M+1)}. \\]\n\tLet $t_0:=t'+1\\le 0$ and \n\t\\begin{gather*}\n\t\t\\beta:=\\frac {x_0+M+1}{2|t_0|\\sqrt{\\lambda_M}},\\quad \n\t\t\\eta:= C'_0ku_0\\sqrt{|t'|}e^{\\sqrt{\\lambda_M}(x_0+\\sqrt \\lambda_Mt') }.\n\t\\end{gather*}\n\tObserve that $\\beta \\in (0,\\frac 12 ]$ as $(t',x_0)\\in D$.\n\tAlso, by \\eqref{h-const} (with $(R,\\sigma,t)=(1,1,t')$) and the opposite of \\eqref{TS:bound2.2} we have $u(t_0,\\cdot)\\ge \\eta \\chi_{[x_0,x_0+1]}$.\n\tApplying the comparison principle ($u$ is a super-solution to the standard heat equation as $f\\ge 0$), for all $x\\in [-M,M]$ we have\n\t\\begin{align}\n\t\tu(t_0+\\beta |t_0|,x) &\\ge \\frac {\\eta}{\\sqrt{4\\pi\\beta |t_0|}} \\int _{x_0}^{x_0+1} e^{-\\frac{(x-y)^2}{4\\beta |t_0|}} dy\\ge \\frac{\\eta}{\\sqrt{4\\pi \\beta |t_0|}} e^{-\\frac{(x_0+M+1)^2}{4\\beta|t_0|}} \\nonumber \\\\\n\t\t&\\ge {2u_0}{\\psi_0}^{-1} e^{\\sqrt{\\lambda_M}(x_0+M+1+\\sqrt{\\lambda_M}t_0)-\\frac{(x_0+M+1)^2}{4\\beta|t_0|}} \\nonumber \\\\\n\t\t& = {2u_0}{\\psi_0}^{-1} e^{\\lambda_M(t_0+\\beta|t_0|)}.\n\t\t\\label{TS:2.2.3}\n\t\\end{align}\n\tHere, the second inequality is due to $-M\\le x \\le y\\le x_0+1$, for then $0\\le y-x\\le M+x_0+1$.\n\tNow let $v(t,x):=2u_0\\psi_0^{-1}e^{\\lambda _M t} \\psi_M(x)$, which by \\eqref{TS:psi} satisfies\n\t\\begin{equation}\\label{pdineq-23}\n\t\tv_t = v_{xx} + (1-\\varepsilon )a(x) v\\quad \\mbox{for all }(t,x)\\in\\mathbb{R}\\times (-M,M),\\quad v(t,\\pm M)=0.\n\t\\end{equation}\n\tFrom \\eqref{TS:2.2.3}, $||\\psi_M||_{L^\\infty}=1$, and \\eqref{eq:zetabound}, we also have\n\t\\begin{equation}\\label{ineq-27}\n\t\t\\begin{split}\n\t\t\tu(t_0+\\beta|t_0|,x)&\\ge v(t_0+\\beta|t_0|,x), \\\\\n\t\t\tv(t,x)\\le v(0,x)&=2u_0\\psi_0^{-1}\\psi_M(x)\\le \\zeta,\n\t\t\\end{split}\n\t\\end{equation}\n\twhere the latter holds for all $t\\le 0$ and $x\\in \\mathbb{R}$.\n\tThe latter with \\eqref{pdineq-23} and \\ref{item:regularity} shows that $v$ is a sub-solution of \\eqref{eq:main} on $\\mathbb{R}^- \\times (-M,M)$.\n\tHence, \\eqref{ineq-27} and the comparison principle (note that $t_0+\\beta |t_0|\\le 0$ as $\\beta\\in (0,\\frac 1 2]$ and $t_0\\le 0$) yield\n\t\\begin{equation*}\n\t\tu(0,x) \\ge v(0 , x)=2u_0\\psi_0^{-1}\\psi_M(x) \\quad  \\mbox{for } x\\in[-M,M].\n\t\\end{equation*}\n\tLetting $x=0$ yields  the contradiction $u_0\\ge 2u_0$ (as $u_0>0$).\n\tTherefore \\eqref{TS:bound2.2} holds.\n\\end{proof}\n\nWith the estimate \\eqref{eq:refined_bound}, we now further refine the bound for $u$ to show that it is bump-like for all large negative time.\n\n\\begin{lemma}\n\t\\label{lem:new}\n\tUnder the same assumptions as Lemma \\ref{lem:refined}, there exist $C>0$ and $\\tau <0$ (both independent of $u$) such that\n\t\\begin{equation}\n\t\t\\label{TS:new}\n\t\tu(t,x)\\le C u(0,0) e^{-c_0|x|+c_0ct},\\quad \\mbox{for } t\\le \\tau,\\, |x|\\ge M.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{remark}\nWe follow the argument in the proof of Theorem 1.2 in \\cite{Zlatos}.\nThe fundamental difference lies in the definition of the super-solution $w=w_1+w_2^{s}$.\nIn \\cite{Zlatos}, $w_2^{s}$ is chosen to be an exponential function (see the definition of $v_{t_0}$ in Section 3 \\cite{Zlatos}).\nIn our case, $w_2^{s}$ is a (leftward-moving) traveling front of a perturbed reaction $f_\\delta$ (defined in the proof).\n\\end{remark}\n\n\\begin{proof}\n\tFirst of all, it suffices to prove \\eqref{TS:new} for the case $x\\ge M$.\n\tThis is because $\\tilde u(t,x):=u(t,-x)$ is still a global solution to \\eqref{eq:main} with $\\tilde f(x,u):=f(-x,u)$ in place of $f(x,u)$.\n\tClearly, $\\tilde f$ and $\\tilde u$ satisfy (F) and \\eqref{eq:zetabound} respectively.\n\tApplying \\eqref{TS:new} to $\\tilde u$ and $x\\ge M$, we find \\eqref{TS:new} for $u$ and $x\\le -M$.\n\t\n\tFix $c\\in (c_0,\\sqrt{\\lambda_M})$.\n\tFor $\\delta>0$ small, consider the perturbation of $f_0$ given by\n\t\\begin{equation*}\n\t\tf_\\delta(u) := \\max_{v\\in [u-\\delta,u+\\delta ]}f_0(v),\n\t\\end{equation*}\n\twhich is a Lipschitz ignition reaction with $\\text{\\rm{supp}}(f_\\delta)= [\\theta_0 -\\delta,1+\\delta]$.\n\tLet $(U_\\delta,c_\\delta)$ be the traveling front solution to the following problem:\n\t\\begin{gather}\n\t\t\\label{eq:tfeq2}\n\t\tU''_\\delta - c_\\delta U' + f_\\delta (U_\\delta ) = 0,\\quad\n\t\t\\lim_{y\\to -\\infty} U_{\\delta}(y) = 0,\\quad  \\lim_{y\\to \\infty} U_{\\delta}(y) = 1+\\delta.\n\t\\end{gather}\n\tNote that $U_\\delta$ is leftward moving, so $U_\\delta'>0$.\n\tBy a simple argument using phase plane analysis and the uniqueness\/stability of solutions to ODEs, one can easily show that $c_\\delta>c_0$ and $\\lim_{\\delta \\searrow 0}c_{\\delta}= c_0$.\n\tThus we may fix $\\delta \\in (0,\\theta_0)$ so that $c_\\delta \\in ( c_0, \\sqrt {c_0c})$.\n\tLet $C_0$ be the constant from Lemma \\ref{lem:refined} and $u_0:=u(0,0)$.\n\tSince $f_\\delta\\equiv 0$ on $[0,\\theta_0-\\delta]$ and \\eqref{eq:tfeq2}, we can specify the translation of $U_\\delta$ so that\n\t\\begin{equation}\\label{Udel-trans}\n\t\tU_\\delta(x)=Ae^{c_\\delta x}\\mbox{ whenever }U_\\delta(x)\\le \\theta_0-\\delta,\\mbox{ where }A:=C_0u_0 e^{(c_0-c_\\delta )M}.\n\t\\end{equation}\n\tFor $s\\in \\mathbb{R}$, define\n\t\\begin{align*}\n\t\tw_1(t,x) := C_0u_0e^{-c_0(x-2M-ct)},\\quad w_2^{s}(t,x) := U_\\delta\\rb {x + c_\\delta t + \\rb{\\frac {c c_0}{c_\\delta} - c_\\delta}s}.\n\t\\end{align*}\n\tLet $\\tau:=\\min\\{T_0,T_1\\}$, where $T_0$, $T_1$ are given by\n\t\\begin{equation*}\n\t\tT_0:= -\\frac {2M+1}{\\sqrt {\\lambda_M}}-1 ,\\qquad C_0 e^{c_0(M+cT_1)}=\\delta.\n\t\\end{equation*}\n\tHere, $T_0$ is defined so that the interval from \\eqref{eq:refined_bound} is non-empty for all $t\\le T_0$.\n\tBy Lemma \\ref{lem:refined},\n\t\\begin{equation}\n\t\t\\label{TS:comp1}\n\t\tu(t,M)\\leq w_1(t,M)\\quad \\mbox{for }t\\le \\tau.\n\t\\end{equation}\n\tWe also claim that for all sufficiently negative $s\\le 0$,\n\t\\begin{equation}\n\t\t\\label{TS:comp2}\n\t\tu(s,x)\\le w_2^{s}(s,x) \\quad \\mbox{for }x\\in [M,\\infty).\n\t\\end{equation}\n\tWe postpone the proof of this claim to first show \\eqref{TS:new}.\n\t\n\tLet $w:=w_1+w_2^{s}$ and $D_{s} := [s,\\tau]\\times [M,\\infty)$.\n\tThen $w$ is a super-solution to \\eqref{eq:main} on $D_{s}$.\n\tAfter all, $w_1(t,x)\\leq w_1(T_1,M)= u_0\\delta\\le \\delta$ for all $(t,x) \\in D_{s}$, so\n\t\\begin{align*}\n\t\tw_t-w_{xx}&=\\partial_t w_1-\\partial_{xx}w_1+\\partial_tw_2^{s}-\\partial_{xx}w_2^{s}\\nonumber \\\\\n\t\t&=c_0(c-c_0)w_1+f_\\delta(w_2^{s})\\geq f_\\delta(w_2^{s})\\nonumber \\\\\n\t\t&\\ge f_0(w)=f(x,w).\n\t\n\t\\end{align*}\n\tThe last inequality follows from $0< w_1\\leq \\delta$ on $D_{s}$ and the definition of $f_\\delta$.\n\tBy \\eqref{TS:comp1}, \\eqref{TS:comp2}, and  the comparison principle, we have $u\\leq w_1+w^{s}_2$ on $D_{s}$, which holds for all large negative $s$.\n\tObserve that the argument of $U_\\delta$ in the definition of $w^{s}_2$ tends to $-\\infty$ as $s\\searrow -\\infty$, since $c_\\delta <\\sqrt {cc_0}$.\n\tHence, $w^{s}_2\\searrow 0$, and $u(t,x)\\le w_1(t,x)$ for all $(t,x)\\in (-\\infty,\\tau]\\times [M,\\infty)$.\n\tThis is \\eqref{TS:new} for $x\\ge M$ if we set $C:=C_0e^{2c_0M}$.\n\t\n\tIt remains to prove \\eqref{TS:comp2}.\n\tLet $\\xi_0 \\in \\mathbb{R}$ satisfy $C_0u_0e^{c_0\\xi_0}=\\theta_0-\\delta$, and define\n\t\\begin{equation*}\n\t\tW(s):= w_2^{s}(s,\\xi_0-cs) = U_\\delta (\\xi_0+c(c_0c_\\delta^{-1}-1)s),\n\t\\end{equation*}\n\twhich is continuous and satisfies $W(-\\infty)=U_\\delta(\\infty)=1+\\delta$ (as $c_0<c_\\delta$).\n\tNow, by this and $c<\\sqrt{\\lambda_M}$, we let $s_0\\le -1$ so that for all $s\\le s_0$ we have\n\t\\begin{equation}\\label{s0-def}\n\t\tW(s)\\ge 1,\\qquad \\xi_0-cs\\le -\\sqrt {\\lambda_M}-M-1-\\sqrt{\\lambda_M}s.\n\t\\end{equation}\n\tNow we show that \\eqref{TS:comp2} holds for all $s\\le s_0$.\n\tIndeed, if $x\\ge \\xi_0-cs$, we have $w_2^{s}(s,x) \\ge w_2^s(s,\\xi_0-cs) =W(s)\\ge 1 > u(s,x)$.\n\tConsider $x\\in [M,\\xi_0-cs)$. By \\eqref{s0-def} and \\eqref{eq:refined_bound}, it suffices to show that $w_2^s(s,x)\\ge C_0u_0e^{c_0(x+cs)}$.\n\tAssume the contrary that it does not hold for some $x_0\\in [M,\\xi_0-cs)$.\n\tIt then follows from the definition of $\\xi_0$ that\n\t\\begin{equation}\\label{cont-05}\n\t\tw_2^s(s,x_0)< C_0u_0e^{c_0(x_0+cs)}\\le C_0u_0e^{c_0\\xi_0}=\\theta_0-\\delta.\n\t\\end{equation}\n\tOn the other hand, by our translation for $U_\\delta$ from \\eqref{Udel-trans}, $c_0<c_\\delta$ and $x_0\\ge M$, it follows\n\t\\begin{align*}\n\t\tw^{s}_2(s,x_0) &= U_\\delta(x_0+cc_0c^{-1}_\\delta s) =  Ae^{c_\\delta x_0+cc_0s}  \\\\\n\t\t&= C_0u_0 e^{c_0M+c_\\delta(x_0-M)+cc_0s}\n\t\t\\ge C_0u_0e^{c_0(x_0+cs)}.\n\t\\end{align*}\n\tThis is a contradiction to the first inequality of \\eqref{cont-05}. Hence, \\eqref{TS:comp2} holds for all $s\\le s_0$.\n\\end{proof}\n\nWe now prove our first main theorem.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:nonex}]\n\t(i) We have already shown in Lemma \\ref{lem:new} that $u$ is bump-like for all $t\\le \\tau$.\n\tIt follows from a standard argument using the comparison principle that $u$ remains bump-like for all time. This clearly implies that transition front does not exist.\n\t\n\t\n\t(ii) Denote $\\varphi(t,x):=\\zeta e^{\\lambda t }\\psi(x)$, which solves the following PDE globally in time:\n\t\\begin{equation}\n\t\t\\label{eq:linearized}\n\t\t\\varphi_t = \\varphi_{xx} + a(x)\\varphi.\n\t\\end{equation}\n\tTo construct a bump-like solution, we let $u(t,x):= \\varphi(t,x)$ for $t\\le 0$, and propagate forward in time by solving the Cauchy problem for \\eqref{eq:main} with initial data $u(0,x) = \\zeta \\psi(x)$ for $t >0$.\n\tNext we prove that this solution is unique up to time translation. Let $\\tilde u\\in (0,1)$ be another solution of \\eqref{eq:main} with $\\inf _{(t,x)\\in \\mathbb{R}^2}\\tilde u=0$.\n\tBy Lemmas \\ref{lem:limit} and  \\ref{lem:new}, $\\tilde u(t,\\cdot)\\to 0$ uniformly as $t\\to-\\infty$.\n\tTherefore, after a time-shift we may assume\n\t\\begin{equation}\\label{sup-trans}\n\t\t\\sup _{(t,x)\\in \\mathbb{R}^-\\times \\mathbb{R}}\\tilde u(t,x)\\le \\frac \\zeta 2 \\psi_M(0).\n\t\\end{equation}\n\tLet $\\tilde \\varphi(t,x):=\\tilde u(t,x)$ for all $t\\le 0$, and propagate forward in time as the solution of \\eqref{eq:linearized} with $\\tilde \\varphi(0,x)=\\tilde u(0,x)$.\n\tSince $f(x,u)=a(x)u$ for $u\\in [0,\\zeta]$ by the assumption, $\\tilde \\varphi$ is an entire solution of \\eqref{eq:linearized}.\n\tBy \\cite[Proposition 2.5]{HP07}, we have $\\tilde \\varphi= q \\varphi$ for some $q>0$, provided that Conditions (A), (H1), and (2.12) from \\cite{HP07} are met (which will be shown shortly).\n\tTherefore, we have $\\tilde \\varphi(t,\\cdot) = \\varphi(t-T,\\cdot)$ with $T:=-\\lambda^{-1}\\log({q}{\\zeta}^{-1})$.\n\tSince $u\\equiv \\varphi$, $\\tilde u \\equiv \\tilde \\varphi$ for all $t\\le 0$, it clearly follows $\\tilde u(t,\\cdot)=u(t-T,\\cdot)$, which shows the uniqueness of solution.\n\t\n\tIt remains to check all the conditions from \\cite{HP07}.\n\tNote that (A) follows from $0\\le a\\le \\gamma\\chi_{[-L,L]}$, and (H1) holds for the PDE \\eqref{eq:linearized} because $\\lambda >0$.\n\tTo show \\cite[(2.12)]{HP07}, we will prove that\n\t\\begin{equation}\n\t\t\\label{TS:b1}\n\t\t\\sup_{x\\in\\mathbb{R}} \\tilde \\varphi(s,x)\\le K \\tilde \\varphi(s,0) \\quad \\mbox{for all }s\\in \\mathbb{R},\n\t\\end{equation}\n\tfor some $K>0$ independent in time.\n\tConsider the above for $s\\le 0$\n\tLet $\\tilde \\varphi^s(t,x):= \\tilde \\varphi(t+s,x)$, which again satisfies \\eqref{eq:zetabound} (by \\eqref{sup-trans}).\n\tFollow from Lemma \\ref{lem:new}, $\\tilde \\varphi^s$ satisfies the estimate \\eqref{TS:new}. With $t=\\tau\\le 0$, we find that\n\t\\begin{equation*}\n\t\t\\sup_{|x|\\ge M}\\tilde \\varphi^s(\\tau,x)\\le C \\tilde\\varphi^s(0,0).\n\t\\end{equation*}\n\tOn the other hand, by the Harnack inequality \\eqref{h-const}, we have $\\max_{|x|\\le M}\\tilde \\varphi^s(\\tau,x)\\le k^{-1}\\tilde \\varphi^s(0,0)$, where $k=k(M,-\\tau)$.\n\tHence, $\\sup_{x\\in \\mathbb{R}}\\tilde \\varphi^s(\\tau,x)\\le A\\varphi^s(0,0)$ with $A:=\\max\\{C,k^{-1}\\}$.\n\tApplying the comparison principle (noting that $w(t,x)=A\\varphi^s(0,0)e^{\\gamma (t-\\tau)}$ is a super-solution to \\eqref{eq:linearized}), we find $\\sup_{x\\in \\mathbb{R}}\\tilde\\varphi^s (0,x)\\le A e^{-\\gamma \\tau} \\tilde\\varphi^s(0,0)$, which is \\eqref{TS:b1} with $K:=Ae^{-\\gamma\\tau}$.\n\t\n\tNow consider \\eqref{TS:b1} with $s>0$.\n\tDecompose $\\tilde \\varphi(0,x)=\\alpha\\psi(x)+\\psi^\\perp(x)$, where $\\psi$, $\\psi^\\perp$ are orthogonal in $L^2(\\mathbb{R})$ (recalling that $\\psi$ is the eigenfunction from \\eqref{eq:eigen}).\n\tLet $\\phi(t,x) := e^{-\\lambda t}\\tilde\\varphi(t,x)$, which by \\eqref{eq:linearized} satisfies\n\t\\begin{equation*}\n\t\n\t\t\\phi_t = (\\partial_{xx}+a(x)-\\lambda)\\phi.\n\t\\end{equation*}\n\tSince the principal eigenvalue $0$ of $\\partial_{xx}+a(x)-\\lambda$ is isolated, it is well-known that $\\phi (t,\\cdot)\\to \\alpha \\psi $ uniformly as time progresses. This clearly implies \\eqref{TS:b1} for $s>0$, as desired.\n\\end{proof}\n\n\\section{Existence for $\\lambda<c_0^2$ (proof of Theorem \\ref{thm:ex})}\n\\label{sec:ex}\n\nIn this section we will prove Theorem \\ref{thm:ex} by finding a transition front solution of \\eqref{eq:main} under the assumption $\\lambda<c_0^2$.\nThe construction is based on the standard solution method mentioned in Section \\ref{intro}---we first construct appropriate pairs of super- and sub-solutions, then obtain an entire solution $u$ sandwiched between them, and use their asymptotic behavior to show that $u$ is a transition front.\n\nThroughout the section, $f,\\gamma,f_0, \\theta_0,\\theta_1, L$, and $a$ are all from Hypothesis (F),\nand we extend $f(x,\\cdot),f_0(\\cdot)$ continuously to $\\mathbb{R}$ by setting $f_0(u)=f(x,u)\\equiv 0$ for $u\\notin [0,1]$.\nFor $\\varepsilon > 0$, let $\\lambda^\\varepsilon>0,\\psi^\\varepsilon> 0$ be the principal eigenvalue and (normalized) eigenfunction of the differential operator $\\partial_{xx} + a(x) + 2\\varepsilon\\chi_{[-L,L]}(x)$, which satisfy\n\\begin{gather*}\n\\partial_{xx}\\psi^\\varepsilon+[a(x)+2\\varepsilon\\chi_{[-L,L]}(x)] \\psi ^\\varepsilon = \\lambda^\\varepsilon \\psi^\\varepsilon \\mbox{ on }\\mathbb{R}\\\\\n\\lim_{|x|\\to\\infty}\\psi^\\eps(x)=0,\\quad ||\\psi^\\eps||_{L^\\infty}=1.\n\\end{gather*}\nSince $a(x)=0$ for $|x|\\ge L$ and $||\\psi^\\varepsilon||_{L^\\infty}=1$, $\\psi^\\varepsilon$ satisfies the exponential bound\n\\begin{equation}\n\\label{TS:psi-exp}\n\\psi^\\eps (x) \\le \\min\\{1,e^{-\\sqrt{\\lambda^\\varepsilon}(|x|-L)}\\}\\quad \\mbox{for all }x\\in \\mathbb{R}.\n\\end{equation}\nNote also that $\\varepsilon\\mapsto \\lambda^\\varepsilon$ is increasing and continuous, with $\\lambda^0 = \\lambda$ and $\\lim_{\\varepsilon \\to \\infty}\\lambda^\\varepsilon \\to \\infty$, so we may fix $\\varepsilon>0$ such that $\\lambda^{\\varepsilon}\\in (\\frac{c_0^2}{16},c_0^2)$, and let $\\zeta = \\zeta(\\varepsilon)\\in (0,\\theta_0)$ be from \\ref{item:regularity}.\nLet $U$ be the unique traveling front of $f_0$ in the sense of \\eqref{tfeq} with $U(0)=\\frac {\\theta_0} 2$.\nGiven $y\\in \\mathbb{R}$, we define\n\\begin{equation}\n\\label{TS:v1-def}\nv^{y}(t,x) := U(x-c_0t + y),\n\\end{equation}\nwhich satisfies \\eqref{eq:main} with $f(x,u)$ replaced by $f_0(u)$.\nFinally, let \\begin{equation}\n\\label{TS:om-eta-def}\n\\omega := \\inf_{x\\in[-L,L]} \\psi^\\varepsilon(x),\\quad \\eta: = c_0\\inf \\left\\{-U'(s);\\;s\\in \\mathbb{R},\\,U(s)\\in \\left[\\frac{\\theta_0}{2},1-\\theta_1\\right]\\right\\}.\n\\end{equation}\nAll the constants involved in this section will depend on $\\gamma, \\theta_0,\\theta_1, L, \\varepsilon, \\zeta, U, c_0, \\omega,$ and $\\eta$.\n\nWe begin with the construction of sub- and super-solutions for $t\\le 0$.\n\\begin{lemma}\n\t\\label{lem:supersub}\n\t\\begin{enumerate}\n\t\t\\item For all $y\\ge L$, $v^y$ given in \\eqref{TS:v1-def} is a sub-solution to \\eqref{eq:main} on $(-\\infty,0)\\times \\mathbb{R}$.\n\t\t\\item  There exists $y_0\\ge L+c_0\\beta(0)$, such that $w$ given as follows is a super-solution to \\eqref{eq:main} on $(-\\infty,0)\\times \\mathbb{R}$\\emph{:}\n\t\t\\begin{gather*}\n\t\tw(t,x) := v^{y_0}(t+\\beta(t),x) + \\phi(t,x),\\\\\n\t\t\\beta(t):= \\frac{\\gamma\\zeta}{4\\sqrt{\\lambda^\\eps }c_0\\eta}e^{2\\sqrt{\\lambda^\\eps }c_0 t},\\quad\n\t\t\\phi(t,x):= \\frac \\zeta 2 e^{\\sqrt{\\lambda^\\eps }c_0 t}\\psi^\\varepsilon(x).\n\t\t\\end{gather*}\n\t\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\t(i) Abbreviate $v=v^y$, recalling that $v$ satisfies \\eqref{eq:main} with $f(x,u)\\equiv f_0(u)$.\n\tThen we must show that\n\t\\begin{equation}\n\t\\label{eq:sub1}\n\tv_t-v_{xx}-f(x,v)=f_0(v)-f(x,v)\\leq 0\\quad \\mbox{for }(t,x)\\in (-\\infty,0)\\times \\mathbb{R}.\n\t\\end{equation}\n\tFor $x\\in [-L,L]^c$, we have $f(x,v)\\equiv f_0(v)$ by \\ref{item:ignition}, and the above holds trivially.\n\tIf $x\\in [-L,L]$ instead, by $U'<0$ we have $v(t,x)=U(x-c_0t+y)\\le U(y-L)\\le U(0)= \\frac {\\theta_0}2$.\n\tTherefore $f_0(v)=0$, and \\eqref{eq:sub1} holds.\n\t\n\t(ii) Let $y_0\\in \\mathbb{R}$ be the unique number such that\n\t\\begin{equation}\\label{y0-def}\n\tU(y_0- c_0\\beta(0)-L) = \\min \\cb{\\frac\\zeta 2, \\frac{\\varepsilon \\omega \\zeta }{2(\\gamma +\\varepsilon )}}.\n\t\\end{equation}\n\tClearly, $y_0\\ge L+c_0\\beta(0)$ because $U(0)=\\frac {\\theta_0}{2} >\\frac{\\zeta}{2}$ and $U'<0$.\n\tAbbreviate $v^{y_0} = v$. Then we must show for all $(t,x)\\in (-\\infty,0)\\times \\mathbb{R}$ that\n\t\\begin{align}\n\tw_t-w_{xx}-f(x,w)\n\t=&f_0(v) + v_t\\,\\beta' + \\sqrt{\\lambda^\\eps }\\left(c_0-\\sqrt{\\lambda^\\eps }\\right)\\phi\\nonumber \\\\\n\t& +[a(x)+2\\varepsilon\\chi_{[-L,L]}(x)] \\phi - f(x,w)\\geq 0,\n\t\\label{eq:super1}\n\t\\end{align}\n\twhere $v$ and $v_t$ are evaluated at $(t+\\beta(t),x)$ in what follows.\n\tConsider first $x\\in [-L,L]$.\n\tSince the first four terms in \\eqref{eq:super1} are nonnegative, it suffices to show that $[a(x)+2\\varepsilon]\\phi\\geq f(x,w)$.\n\tNote that $\\phi(t,x)\\le \\frac{\\zeta}{2}$ as the eigenfunction $\\psi^\\varepsilon$ satisfies $||\\psi^\\varepsilon||_{L^\\infty}=1$.\n\tMoreover, by $U'<0$, $\\beta '>0$ and \\eqref{y0-def}, we have\n\t\\begin{equation*}\n\tv(t+\\beta(t),x)\\le U(y_0-c_0\\beta(0)-L)\\le \\frac{\\zeta}{2}.\n\t\\end{equation*}\n\tTherefore $w=v + \\phi \\le \\zeta$, and $f(x,w)\\leq\\left[a(x)+\\varepsilon\\right]w$ by \\ref{item:regularity}.\n\tOn the other hand, by \\eqref{tfeq} and $f_0\\equiv 0$ on $[0,\\theta_0]$, we have $U(y)=\\frac{\\theta_0}{2}e^{-c_0y}$ for $y\\ge 0$.\n\t\\eqref{y0-def} then implies\n\t\\begin{align*}\n\tv(t+\\beta(t),x)&\\le U(y_0-c_0\\beta(0)-L-c_0t)\\\\\n\t&=U(y_0-c_0\\beta(0)-L)e^{c_0^2 t} \\le \\frac {\\varepsilon \\omega \\zeta}{2(\\gamma+\\varepsilon )} e^{c_0^2t}.\n\t\\end{align*}\n\tCombining this, $\\sqrt{\\lambda^\\varepsilon}<c_0$, $a(x)\\le \\gamma$ and $\\psi^\\varepsilon(x)\\ge \\omega $ on $[-L,L]$ (by \\eqref{TS:om-eta-def}), we compute\n\t\\begin{align*}\n\t[a(x)+2\\varepsilon]\\phi-f(x,w)\n\t&\\geq [a(x)+2\\varepsilon]\\phi-\\left[a(x)+\\varepsilon\\right]w\\\\\n\t&= \\varepsilon\\phi-\\left[a(x)+\\varepsilon\\right]v \n\t\\geq \\frac \\zeta 2 \\left[\\varepsilon \\phi-\\left[a(x)+\\varepsilon\\right]\\left(\\frac{\\varepsilon\\omega}{\\gamma+\\varepsilon }e^{c_0^2t}\\right)\\right]\\\\\n\t&\\geq \\frac {\\varepsilon\\zeta }2 e^{\\sqrt{\\lambda^\\varepsilon}c_0 t} [\\psi^\\varepsilon(x)-\\omega]\\geq 0.\n\t\\end{align*}\n\tHence \\eqref{eq:super1} holds for $(t,x)\\in (-\\infty,0)\\times [-L,L]$.\n\t\n\tNow consider \\eqref{eq:super1} for $x\\in [-L,L]^c$.\n\tSince $f(x,w)=f_0(w)$ and the first four terms of \\eqref{eq:super1} are non-negative, it suffices to show that\n\t\\begin{align}\n\tf_0(v) + v_t\\,\\beta' - f_0(w)\\geq 0.\n\t\\label{eq:super2}\n\t\\end{align}\n\tWe consider three cases for the value of $v$.\n\tWhen $v\\leq \\frac{\\theta_0}{2}$, we have $w=v+\\phi\\leq \\theta_0$ (recalling that $\\phi\\le \\frac{\\zeta}{2}$), so $f_0(w)=0$ and \\eqref{eq:super2} holds.\n\tIf $v\\geq 1-\\theta_1$, $f_0(v)\\geq f_0(w)$ because $f_0$ is non-increasing on $[1-\\theta_1,\\infty)$ by \\ref{item:ignition}; \\eqref{eq:super2} follows.\n\tFinally, suppose $\\frac{\\theta_0}{2} \\leq v\\leq 1-\\theta_1$.\n\tIn this case, by $y_0\\ge c_0\\beta(0)+L$, $U'<0$, and $U(0)=\\frac {\\theta_0}2$, we have $x\\le c_0t-L$.\n\tUsing $\\mbox{Lip}(f)=\\gamma$, \\eqref{TS:psi-exp}, and $v_t\\ge \\eta$ from \\eqref{TS:om-eta-def}, we deduce\n\t\\begin{align*}\n\t|f_0(v+\\phi)-f_0(v)|&\\leq \\gamma \\phi(t,x) \\le \\frac {\\gamma\\zeta } 2 e^{\\sqrt{\\lambda^\\eps }(x+L)+\\sqrt{\\lambda^\\eps }c_0 t}\\le \\frac {\\gamma\\zeta } {2} e^{2\\sqrt{\\lambda^\\eps }c_0t} \\\\\n\t&\\leq v_t\\frac{\\gamma \\zeta }{2\\eta}e^{2\\sqrt{\\lambda^\\eps }c_0 t} =  v_t\\,\\beta'.\n\t\\end{align*}\n\tHence \\eqref{eq:super2} always holds, and $w$ is a super-solution to \\eqref{eq:main} on $(-\\infty,0)\\times\\mathbb{R}$.\n\\end{proof}\n\nNext, we construct super- and sub-solutions for $t\\ge 0$.\n\n\\begin{lemma}\\label{lem:sup-t+}\n\t\\begin{enumerate}\n\t\t\\item There exist $y_1\\le -L$ and $B_1>0$ such that\n\t\t$\\tilde w$ given as follows is a super-solution to \\eqref{eq:main}\n\t\ton $(0,\\infty)\\times \\mathbb{R}$\\emph{:}\n\t\t\\begin{gather*}\n\t\t\\tilde w(t,x):= v^{y_1} (t+\\beta_1(t),x)+\\phi_1 (t,x),\\\\\n\t\t\\beta_1 (t):=B_1(1-e^{-c_0^2t\/8}),\\quad \\phi_1(t,x) := e^{-\\frac{c_0}{4} (x-L -\\frac{c_0}{2} t)}.\n\t\t\\end{gather*}\n\t\t\\item For all $y\\in \\mathbb{R}$, there exists $B_2=B_2(y)>0$ such that\n\t\t$\\tilde v^y$ given as follows is a sub-solution to \\eqref{eq:main}\n\t\ton $(0,\\infty)\\times \\mathbb{R}$\\emph{:}\n\t\t\\begin{gather*}\n\t\t\\tilde v^y(t,x):=  v^{y} (t+\\beta_2(t),x) -\\phi_2 (t,x),\\\\\n\t\t\\beta_2 (t):=B_2 e^{-c_0^2t\/8},\\quad \\phi_2(t,x):= \\frac{16\\gamma }{c_0^2} e^{-\\frac{c_0}{4} (x-L-\\frac{c_0}{2}t)}.\n\t\t\\end{gather*}\n\t\\end{enumerate}\n\n\\end{lemma}\n\n\\begin{proof}\n\t(i) Let $y_1,\\ell,B_1\\in \\mathbb{R}$ be defined as follows:\n\t\\begin{equation}\n\t\\label{TS:3.2-def}\n\t\\phi_1(0,-y_1)=\\frac{\\theta_0}{2},\\quad U(\\ell)=1-\\theta_1,\\quad B_1 := \\frac{8\\gamma }{c_0^2 \\eta }e^{-\\frac{c_0}{4}(\\ell-y_1 -L)}.\n\t\\end{equation}\n\tNote that $y_1\\le -L$ because $\\phi_1(0,L)=1$. As before, we abbreviate $v=v^{y_1}$, and we need to show for all $(t,x)\\in(-\\infty,0)\\times \\mathbb{R}$ that\n\t\\begin{equation}\n\t\\label{eq:tgreater}\n\t\\tilde w_t - \\tilde w_{xx} - f(x,\\tilde w) =  v_t\\,\\beta_1' + \\frac{c_0^2}{16}\\phi_1+f_0(v)- f(x,\\tilde w)\\geq 0,\n\t\\end{equation}\n\twhere $v$ and $v_t$ are evaluated at $(t+\\beta_1(t),x)$ for the remainder of this part.\n\tNote that the first three terms are nonnegative.\n\tIf $x\\le L$, then \\eqref{eq:tgreater} holds by $f(x,\\tilde w) =0$ as $\\tilde w(t,x)\\geq \\phi_1(t,x)\\ge 1$.\n\tNow consider $x>L$, noting that $f(x,\\tilde w) = f_0(\\tilde w)$.\n\tWe again consider three cases for the value of $v$.\n\tWhen $v\\geq 1-\\theta_1$, we have $f_0(v)-f_0(\\tilde w)\\geq 0$ because $f_0$ is non-increasing on $\\left[1-\\theta_1,\\infty\\right)$.\n\t\\eqref{eq:tgreater} follows.\n\tIf $v\\leq \\frac{\\theta_0}{2}$, then \\eqref{TS:v1-def} and $U(0)=\\frac {\\theta_0}{2}$ imply $x-c_0t \\ge -y_1$.\n\tTherefore,\n\t\\begin{equation*}\n\t\\phi_1(t,x)\\le \\phi_1(2t,x)=\\phi_1(0,x-c_0t)\\le \\phi_1(0,-y_1)=\\frac{\\theta_0}{2}.\n\t\\end{equation*}\n\tIt follows that $\\tilde w= v+\\phi_1 \\le \\theta_0$, so $f_0(\\tilde w)=0$, and \\eqref{eq:tgreater} holds again.\n\tFinally, suppose $v\\in [\\frac{\\theta_0}{2}, 1-\\theta_1] $.\n\tFrom \\eqref{TS:3.2-def}, we again have $x-c_0t \\geq \\ell-y_1 $.\n\tUsing the definition of $\\eta$ from \\eqref{TS:om-eta-def},\n\t\\begin{align*}\n\t|f_0(v)-f_0(\\tilde w)|&\\leq \\gamma \\phi_1(t,x) = \\gamma e^{-\\frac{c_0}{4} (x-L -\\frac{c_0}{2} t)}\\\\\n\t&\\leq \\gamma e^{-\\frac{c_0 ^2}{8}t-\\frac{c_0}{4}(\\ell-y_1 -L)} \\le v_t \\frac{\\gamma}{\\eta} e^{-\\frac{c^2_0}{8}t-\\frac{c_0}{4}(\\ell-y_1 -L)} = v_t\\,\\beta_1'.\n\t\\end{align*}\n\t\\eqref{eq:tgreater} again follows, so $\\tilde w$ is a super-solution to \\eqref{eq:main} on $(0,\\infty)\\times \\mathbb{R}$.\n\t\n\t(ii) Define $\\tilde \\theta,\\tilde \\eta,\\tilde \\ell, B_2=B_2(y)$ as follows:\n\t\\begin{align*}\n\t\\tilde \\theta := \\min\\cb{\\frac{\\theta_0}{2},\\frac{\\theta_1}{2},\\frac{c_0^2 \\theta_1}{32\\gamma}},&\\quad \\tilde \\eta:= c_0\\inf\\left\\{-U'(s):s\\in \\mathbb{R},\\,U(s)\\in \\left[\\til \\theta,1-\\til \\theta\\right]\\right\\},\\\\\n\tU({\\til \\ell})= 1-\\til \\theta,&\\quad B_2:= \\frac{2^7 \\gamma^2}{c_0^4\\tilde \\eta }e^{-\\frac{c_0}{4}\\left({\\til \\ell}-y -L\\right)}.\n\t\\end{align*}\n\tAgain abbreviate $v=v^{y}$, $\\tilde v=\\tilde v^y$.\n\tWe will show for all $(t,x)\\in (0,\\infty)\\times \\mathbb{R}$ that\n\t\\begin{align}\n\t\\tilde v_t- \\tilde v_{xx} - f(x, \\tilde v ) = v_t\\,\\be_2' - \\frac{c_0^2}{16}\\phi_2  + f_0(v) - f(x, \\tilde v)\\leq 0,\n\t\\label{eq:subsol}\n\t\\end{align}\n\twhere $v$ and $v_t$ are evaluated at $(t+\\be_2(t),x)$ for the rest of the proof.\n\tNote that $f_0(v)$ is the only term in \\eqref{eq:subsol} which is potentially positive.\n\tConsider $x\\in [-L,L]$, where we have $\\phi_2(t,x)\\geq \\phi_2(0,L) = {16\\gamma}{c_0}^{-2}$.\n\tThen \\eqref{eq:subsol} follows from\n\t\\begin{equation*}\n\t- \\frac{c_0^2}{16}\\phi_2 + f_0(v) \\le -\\gamma+f_0(v) \\leq 0.\n\t\\end{equation*}\n\tNow consider $x\\in[-L,L]^c$, where $f(x,\\tilde v) = f_0(\\tilde v)$.\n\t\\eqref{eq:subsol} holds whenever $v \\leq \\til \\theta(\\le \\theta_0)$, as $f_0(v)=0$.\n\tIf $v\\in[\\til \\theta, 1 - \\til \\theta]$, then $x\\ge c_0 t+{\\til \\ell}-y$ because $U(\\tilde \\ell)=1-\\tilde \\theta$. It then follows from $v_t \\ge \\tilde \\eta>0 $ that\n\t\\begin{align*}\n\t\\abs{f_0(v) - f_0(\\tilde v)} &\\leq \\gamma \\phi_2(t,x) =  \\frac{16\\gamma^2}{c_0^2}e^{-\\frac{c_0}{4}(x-L-\\frac{c_0}{2}t)}\\le \\frac{16\\gamma^2}{c_0^2} e^{-\\frac{c_0^2}{8}t-\\frac{c_0}{4}({\\til \\ell}-y-L)} \\\\\n\t&\\le v_t\\frac{16\\gamma^2}{c_0^2\\tilde \\eta}e^{-\\frac{c_0^2}{8}t-\\frac{c_0}{4}({\\til \\ell}-y-L)} = -v_t\\,\\be_2',\n\t\\end{align*}\n\twhich implies \\eqref{eq:subsol}.\n\tFinally, consider $v \\geq 1 - \\til \\theta(\\ge 1-\\frac{\\theta_1}{2})$.\n\tIf $\\phi_2 \\leq \\frac{\\theta_1}{2}$ then $\\tilde v \\geq  1 - \\theta_1$.\n\tSince $f_0$ is non-increasing on $[1 - \\theta_1,\\infty)$, $f_0( v) \\leq f_0( \\tilde v)$, and \\eqref{eq:subsol} holds again.\n\tIf $\\phi_2\\geq \\frac{\\theta_1}{2}$, then\n\t\\begin{equation*}\n\t\\frac{c_0^2}{16}\\phi_2\\geq \\frac{c_0^2\\theta_1}{32}\\geq \\gamma \\til \\theta \\geq \\gamma \\abs{v - 1} \\geq f_0(v).\n\t\\end{equation*}\n\tHence \\eqref{eq:subsol} holds in all cases, as claimed.\n\\end{proof}\n\nWith the requisite super- and sub-solutions in place, we may now establish Theorem \\ref{thm:ex}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:ex}]\n\tIn what follows, $y_0,y_1, B_1, \\beta, \\phi, w,\\beta_1, \\phi_1,\\tilde w,$ and $\\phi_2$ are constants and functions defined in Lemmas \\ref{lem:supersub}(ii) and \\ref{lem:sup-t+}, and let $v=v^{y_0}$ be given as in \\eqref{TS:v1-def}.\n\t\n\tThe construction of an entire solution $u$ follows the standard procedure.\n\tFor $n\\in \\mathbb N$, let $u_n$ be the solution to \\eqref{eq:main} on $(-n,\\infty)\\times \\mathbb{R}$ with initial data $u_n(-n,x) = v(-n,x)$.\n\tFirst, observe that $u_n$ is increasing in time.\n\tIndeed, since $v$ is a sub-solution of \\eqref{eq:main} on $(-\\infty,0)\\times \\mathbb{R}$ with $v_t>0$ by Lemma \\ref{lem:supersub}, the comparison principle implies $u_n(t,\\cdot)> u_n(0,\\cdot)$ for all $t> 0$.\n\tIt follows thast $\\partial_t u_n> 0$ by the maximum principle.\n\tMoreover, since $v(-n,\\cdot)\\le w(-n,\\cdot)$ by their definition, Lemma \\ref{lem:supersub} and the comparison principle ensure that\n\t\\begin{equation*}\n\tv(t,\\cdot)\\le u_n(t,\\cdot)\\le w(t,\\cdot)\\quad\\mbox{for all }t\\in [-n,0].\n\t\\end{equation*}\n\tBy parabolic regularity, we obtain an increasing in time entire solution $u$ to \\eqref{eq:main} as a locally uniform limit along a subsequence of $(u_n)$ satisfying\n\t\\begin{equation}\n\t\\label{TS:ex-comp}\n\tv(t,\\cdot)\\le u(t,\\cdot)\\le w(t,\\cdot)\\quad \\mbox{for all }t\\le 0.\n\t\\end{equation}\n\t\n\tNext, we check that $u$ fulfills \\eqref{item:trans_lims}.\n\tIt obviously holds for $t\\le 0$ by \\eqref{TS:ex-comp} and the limit behavior of $v,w$ at $\\pm \\infty$.\n\tFor $t>0$, the first limit of \\eqref{item:trans_lims} still holds because $u_t>0$ and $u<1$.\n\tTo prove the second limit condition, we first claim that\n\t\\begin{equation}\n\t\\label{TS:ex-comp1}\n\tu(0,x)\\le \\min\\{ 1, w(0,x) \\} \\le \\tilde w(0,x).\n\t\\end{equation}\n\tFrom this, Lemma \\ref{lem:sup-t+}(i), and the comparison principle,\n\t\\begin{equation}\n\t\\label{TS:ex-comp3}\n\tu(t,\\cdot)\\le \\tilde w(t,\\cdot)\\quad \\mbox{for all }t\\ge 0.\n\t\\end{equation}\n\tThe second limit of \\eqref{item:trans_lims} then follows from $\\lim_{x\\to\\infty}\\tilde w(t,x) =0$ and $u>0$.\n\tNow consider \\eqref{TS:ex-comp1}.\n\tThe first inequality is simply \\eqref{TS:ex-comp} with $t=0$.\n\tFor the second, when $x\\le L$, $\\tilde w(0,x)\\ge \\phi_1(0,x) \\ge 1$.\n\tFor $x>L$, \\eqref{TS:psi-exp}, $\\lambda^\\varepsilon >\\frac{c_0^2}{16}$, $U'<0$, and $y_1\\le y_0-c_0\\beta(0)$ (by Lemmas \\ref{lem:supersub}(ii) and \\ref{lem:sup-t+}(i)) imply\n\t\\begin{align*}\n\tw(0,x) &= U(x+y_0-c_0\\beta(0)) + \\frac{\\zeta}{2}\\psi^\\varepsilon(x)\\le U(x+y_1) + e^{-\\frac{c_0}{4}(x-L)} = \\tilde w(0,x).\n\t\\end{align*}\n\tTherefore \\eqref{TS:ex-comp1} holds.\n\tThis completes the proof of \\eqref{item:trans_lims}.\n\t\n\tIt remains to show the bounded front width condition \\eqref{item:trans_width}.\n\tFix $\\mu \\in (0,\\frac 12 )$ and let\n\t\\begin{equation*}\n\tX^-(t):=\\inf \\{x\\in \\mathbb{R}:u(t,x)\\le 1-\\mu\\},\\quad X^+(t):=\\sup \\{x\\in \\mathbb{R}:u(t,x)\\ge \\mu\\},\n\t\\end{equation*}\n\tfor then we have $L_\\mu(t) = X^+(t)-X^-(t)$.\n\tWe will show that $L_\\mu(t)$ is uniformly bounded in $t\\in \\mathbb{R}$ by considering these three cases: $t\\le 0$, $t> t_\\mu,$ and $t\\in (0,t_\\mu]$, where $t_\\mu$ will be defined shortly.\n\tFor the first case $t<0$, let\n\t\\begin{gather*}\n\t\\rho_-:= U^{-1}(1-\\mu),\\quad \n\t \\rho_+:= \\max\\left\\{ U^{-1}\\left (\\frac \\mu 2\\right), \\frac 1{\\sqrt{\\lambda^\\eps }}\\left |\\log \\frac \\mu 2\\right|+L+y_0-c_0\\beta(0) \\right\\}.\n\t\\end{gather*}\n\tFor all $x< c_0 t + \\rho_--y_0$, by \\eqref{TS:ex-comp} we have\n\t\\begin{equation*}\n\tu(t,x)\\ge v(t,x) = U(x-c_0t +y_0)> U(\\rho_-)= 1-\\mu.\n\t\\end{equation*}\n\tTherefore $X^-(t) \\ge c_0t +\\rho_--y_0$.\n\tNow, suppose $x> c_0t+\\rho_+ -y_0 +c_0\\beta(0)$.\n\tCombining \\eqref{TS:psi-exp}, $||\\psi^\\varepsilon||_{L^\\infty}=1$, \\eqref{TS:ex-comp} and the definition of $\\rho_+$, we compute\n\t\\begin{align*}\n\tu(t,x) &\\le w(t,x) = U(x-c_0(t+\\beta(t))+y_0)+\\frac \\zeta 2 e^{c_0\\sqrt{\\lambda^\\eps }t}\\psi^\\eps(x) \\\\\n\t&\\le  U(x-c_0(t+\\beta(0))+y_0)+ \\frac \\zeta 2e^{-\\sqrt{\\lambda^\\eps }(x-L-c_0t)} \\\\\n\t&< U(\\rho_+)+ e^{-\\sqrt{\\lambda^\\eps }(\\rho_+-L-y_0+c_0\\beta(0))}\\le \\mu .\n\t\\end{align*}\n\tThis implies $X^+(t) \\le c_0t +\\rho_+ - y_0+c_0\\beta(0)$, so\n\t\\begin{equation}\n\t\\label{TS:L-1}\n\tL_\\mu(t) \\le c_0\\beta(0)+\\rho_+ - \\rho_-\\quad \\mbox{for }t\\le 0.\n\t\\end{equation}\n\t\n\tWe now define $t_\\mu$.\n\tGiven $y\\in \\mathbb{R}$, let $\\bar v^y(t,x):= v^{y}(t,x)- \\phi_2(t,x)$, where $v^y,\\phi_2$ are from \\eqref{TS:v1-def} and Lemma \\ref{lem:sup-t+}(ii).\n\tOne can easily verify that $M(y):=\\sup_{x\\in \\mathbb{R}} \\bar v^{y}(0,x)$ is continuous, non-increasing in $y\\in \\mathbb{R}$, $\\lim_{y\\to-\\infty} M(y)=1$, $\\lim_{y\\to\\infty}M(y)=0$, and the supremum is achieved somewhere if $M(y)>0$.\n\tSo we may fix $y_2=y_2(\\mu)$ such that $M(y_2)=1-\\mu$.\n\tFor the remainder of the proof, we abbreviate $\\bar v = \\bar v ^{y_2}$ and let $B_2=B_2(y_2)$, $\\be_2$, and $\\tilde v=\\tilde v^{y_2}$ be from Lemma \\ref{lem:sup-t+}(ii).\n\tLet $x_\\mu\\in \\mathbb{R}$ be a maximizer so that $\\bar v (0,x_\\mu)=1-\\mu$, and define\n\t\\begin{equation}\\label{t-mu-def}\n\tt_\\mu := \\inf \\cb{t>0: u(t,\\cdot)\\ge \\max \\{\\tilde v(0,\\cdot), (1-\\mu)\\chi_{(-\\infty,x_\\mu]}\\}  }\\ge 0.\n\t\\end{equation}\n\tWe claim that $t_\\mu$ is finite.\n\tAfter all, we have $\\tilde v(0,\\cdot)\\le (1-\\mu')\\chi_{I}$ for some bounded interval $I\\subset \\mathbb{R}$ and $\\mu'\\in (0,1)$.\n\tRecall that $u_t>0$ and the limit condition \\eqref{item:trans_lims} holds.\n\tTherefore, we have $u(t,\\cdot)\\nearrow 1$ uniformly on each $(-\\infty,R)$, $R\\in \\mathbb{R}$, which implies $t_\\mu<\\infty$.\n\t\n\tNow consider the front width for $t>t_\\mu$.\n\tCombining \\eqref{t-mu-def}, Lemma \\ref{lem:sup-t+}(ii), the comparison principle, and $\\be_2>0$, we have\n\t\\begin{equation}\n\t\\label{TS:ex-comp2}\n\tu(t_\\mu+t,\\cdot)\\ge \\tilde v(t,\\cdot)\\ge \\bar v(t,\\cdot)\\quad \\mbox{for all }t\\ge 0.\n\t\\end{equation}\n\tRecall $B_1$ from Lemma \\ref{lem:sup-t+}(i) and set\n\t\\begin{equation*}\n\t\\tilde\\rho_+ := \\max \\cb{U^{-1}\\rb{\\frac \\mu 2}, \\frac{4}{c_0} \\abs{\\log\\frac{\\mu}{2}} +y_1+L-c_0B_1}.\n\t\\end{equation*}\n\tThen $X^+(t) \\le c_0(t+B_1) +\\tilde \\rho_+-y_1$.\n\tIndeed, for $x> c_0(t+B_1)+\\tilde\\rho_+-y_1$, \\eqref{TS:ex-comp3} and $\\beta_1< B_1$ imply\n\t\\begin{align*}\n\tu(t,x)&\\le \\tilde w(t,x) = U(x-c_0(t+\\beta_1(t))+y_1) + e^{-\\frac{c_0}{4} (x-L-\\frac{c_0}{2}t)}\\\\\n\t&< U(\\tilde\\rho_+)+ e^{-\\frac{c_0}{4}(c_0B_1+\\tilde \\rho_+-y_1-L)} \\le \\mu.\n\t\\end{align*}\n\tOn the other hand, we claim that $X^-(t)\\ge c_0(t-t_\\mu)+x_\\mu$, implying:\n\t\\begin{equation}\n\t\\label{TS:L-2}\n\tL_\\mu(t) \\le c_0(t_\\mu+ B_1) +\\tilde \\rho_+ -y_1 - x_\\mu,\\quad \\text{for }t\\in (t_\\mu,\\infty).\n\t\\end{equation}\n\tTo prove the claimed bound, it suffices to check that $u(t,x)\\ge 1-\\mu$ for all $x< c_0(t-t_\\mu)+x_\\mu$.\n\tIf $x\\le x_\\mu$, then $u_t>0$ and \\eqref{t-mu-def} show that $u(t,x)> u(t_\\mu ,x)\\ge 1-\\mu$.\n\tNow consider $x\\in (x_\\mu, c_0(t-t_\\mu)+x_\\mu)$.\n\tNote that $v^{y_2}(t,x_\\mu+c_0t)=U(x_\\mu+y_2)$, while $t\\mapsto \\phi_2(t,x_\\mu+c_0t)$ is decreasing.\n\tHence, their difference $\\bar v (t,x_\\mu+c_0t) $ is increasing in $t$, and\n\t\\begin{equation*}\n\t\\bar v(t,x_\\mu+c_0t)> \\bar v(0,x_\\mu)=1-\\mu\\quad  \\mbox{ for all }t>0.\n\t\\end{equation*}\n\tLet $t_*:=c_0^{-1}(x-x_\\mu)\\in (0,t-t_\\mu)$.\n\tThen by $u_t> 0$ and \\eqref{TS:ex-comp2}, it follows that\n\t\\begin{equation*}\n\tu(t,x) > u(t_\\mu+t_*,x)=u(t_\\mu+t_*,x_\\mu+c_0t_*)\\ge \\bar v(t_*,x_\\mu+c_0t_*) > 1-\\mu.\n\t\\end{equation*}\n\tThis proves the claim.\n\t\n\tFinally, consider $t\\in (0,t_\\mu]$.\n\tSince $u_t>0$, the width is bounded by\n\t\\begin{equation*}\n\tL_\\mu(t)\\le X^+(t_\\mu)-X^-(0) \\le c_0 (t_\\mu+B_1) +\\tilde\\rho_+-\\rho_- +y_0-y_1.\n\t\\end{equation*}\n\tWith this, \\eqref{TS:L-1}, and \\eqref{TS:L-2}, $L_\\mu(t)$ is uniformly bounded for all $t\\in \\mathbb{R}$.\n\tThis concludes the proof of \\eqref{item:trans_width}.\n\tTherefore $u$ is an increasing-in-time transition front solution of \\eqref{eq:main}. \n\tIt also obviously holds from the comparisons \\eqref{TS:ex-comp}, \\eqref{TS:ex-comp3} and \\eqref{TS:ex-comp2} that $u$ has a global mean speed $c_0$. \n\tThis completes the proof of Theorem \\ref{thm:ex}. \n\\end{proof}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGeneralized Nash equilibrium problems (GNEPs) have been widely studied in the literature \\cite{Ros65, facchinei2007, facchinei2010} and such a strong interest is motivated by numerous applications ranging from economics to engineering \\cite{pavel2007,kulkarni2012}. In a GNEP, each agent seeks to minimize his own cost function, under local and coupled feasibility constraints. In fact, both the cost function and the constraints depend on the strategies chosen by the other agents. Due to the presence of these shared constraints, the search for generalized Nash equilibria is usually a quite challenging task. \n\nFor the computation of a GNE, various algorithms have been proposed, both distributed \\cite{belgioioso2018,yi2019}, and semi-decentralized \\cite{facchinei2010,belgioioso2017}. When dealing with coupling constraints, a common principle is the focus on a special class of equilibria, which reflect some notion of fairness among the agents. This class is known as \\emph{variational equilibria} (v-GNE) \\cite{Deb52,facchinei2010}. Besides fairness considerations, v-GNE is computationally attractive since it can be formulated in terms of variational inequality, which makes it possible to solve them via operator splitting techniques \\cite{BauCom16,facchinei2007}. A recent breakthrough along these lines is the distributed, preconditioned, forward-backward (FB) algorithm conceived in \\cite{yi2019} for strongly-monotone games. The key lesson from \\cite{yi2019} is that the FB method cannot be directly applied to GNEPs, thus a suitable preconditioning is necessary. From a technical perspective, the FB operator splitting requires that the pseudo-gradient mapping of the game is strongly monotone, an assumption which is not always satisfied\n\nIn this paper we investigate two distributed algorithmic schemes for computing a v-GNE. Motivated by the need to relax the strong monotonicity assumption on the pseudo-gradient of the game, we first investigate a distributed \\textit{forward-backward-forward} (FBF) algorithm \\cite{tseng2000}. We show that a suitably constructed FBF operator splitting guarantees not only fully distributed computation, but also convergence to a v-GNE under the mere assumption of monotonicity of the involved operators. This enables us to drop the strong monotonicity assumption, which is the main advantage with respect to the FB-based splitting methods \\cite{yi2019}.\nAs a second condition, in order to exploit the structure of the monotone inclusion defining the v-GNE problem, we also investigate the \\textit{forward-backward-half-forward} (FBHF) algorithm \\cite{briceno2018}. We would like to point out that both our algorithms are distributed in the sense that each agent needs to know only his local cost function and its local feasible set, and there is no central coordinator that updates and broadcasts the dual variables. The latter is the main difference with semi-decentralized schemes for aggregative games \\cite{Gram17,belgioioso2017}.\n\n\nCompared with the FB and the FBHF algorithms, the FBF requires less restrictive assumptions to guarantee convergence, i.e., plain monotonicity of the pseudo-gradient mapping. On the other hand, the FBF algorithm requires two evaluations of the pseudo-gradient mapping, which means that the agents must communicate at least twice at each iterative step. Confronted with the FBF algorithm, our second proposal, the FBHF algorithm requires only one evaluation of the pseudo-gradient mapping, but needs strong monotonicity to provide theoretical convergence guarantees. Effectively, the FBHF algorithm is guaranteed to converge under the same assumptions as the preconditioned FB \\cite{yi2019}.\n\n\n\n\n\n\n\\paragraph*{Notation}\n${\\mathbb{R}}$ indicates the set of real numbers and $\\bar{\\mathbb{R}}={\\mathbb{R}}\\cup\\{+\\infty\\}$. $\\mathbf{0}_N$ ($\\mathbf{1}_N$) is the vector of N zeros (ones). The Euclidean inner product and norm are indicated with $\\inner{\\cdot,\\cdot}$ and $\\norm{\\cdot}$, respectively.\nLet $\\Phi$ be a symmetric, positive definite matrix, $\\Phi \\succ 0$. The induced inner product is $\\inner{\\cdot,\\cdot}_{\\Phi}:=\\inner{\\Phi\\cdot,\\cdot}$, and the associated norm is $\\norm{\\cdot}_{\\Phi}:=\\inner{\\cdot,\\cdot}_{\\Phi}^{1\/2}$. We call $\\mathcal H_\\Phi$ the Hilbert space with norm $\\norm{\\cdot}_\\Phi$. \nGiven a set $\\mathcal X\\subseteq{\\mathbb{R}}^n$, the normal cone mapping is denoted with $\\NC_{\\mathcal{X}}(x)$. $\\Id$ is the identity mapping. Given a set-valued operator $A$, the graph of $A$ is the set $\\gr(A)=\\{(x,y)\\vert y\\in Ax\\}$ The set of zeros is $\\Zer A=\\{x\\in{\\mathbb{R}}^n \\mid 0\\in Ax\\}$.  \nThe resolvent of a maximally monotone operator $A$ is the map $\\mathrm{J}_{A}=(\\Id+A)^{-1}$. If $g:{\\mathbb{R}}^{n}\\to(-\\infty,\\infty]$ is a proper, lower semi-continuous, convex function, its subdifferential is the maximal monotone operator $\\partial g(x)$. The proximal operator is defined as $\\operatorname{prox}^{\\Phi}_{g}(v)=\\operatorname{J}_{\\Phi\\partial g}(v)$ \\cite{BauCom16}.\nGiven $x_{1}, \\ldots, x_{N} \\in {\\mathbb{R}}^{n}, \\boldsymbol{x} :=\\operatorname{col}\\left(x_{1}, \\dots, x_{N}\\right)=\\left[x_{1}^{\\top}, \\dots, x_{N}^{\\top}\\right]^{\\top}$.\n\n\\section{Mathematical Setup: The Monotone Game and Variational Generalized Nash Equilibria}\n\\label{sec:problem}\nWe consider a game with $N$ agents where each agent chooses an action $x_{i}\\in{\\mathbb{R}}^{n_i}$, $i\\in\\mathcal I=\\{1,\\dots,N\\}$.  \n\nEach agent $i$ has an extended-valued local cost function $J_{i}: {\\mathbb{R}}^n \\to (-\\infty,\\infty]$ of the form \n\\vspace{-.15cm}\\begin{equation}\\label{eq:f}\nJ_{i}(x_{i}, \\boldsymbol x_{-i}):=f_{i}(x_{i},\\boldsymbol x_{-i}) + g_{i}(x_{i}).\n\\vspace{-.15cm}\\end{equation}\nwhere $\\boldsymbol x_{-i}=\\operatorname{col}(\\{x_j\\}_{j\\neq i})$ is the vector of all decision variables except for $x_i$, and $g_{i}:{\\mathbb{R}}^{n_{i}}\\to(-\\infty,\\infty]$ is a local idiosyncratic costs function which is possibly non-smooth. Thus, the function $J_{i}$ in \\eqref{eq:f} has the typical splitting into smooth and non-smooth parts.\n\\begin{standassumption}[Local cost]\nFor each $i\\in\\mathcal I$, the function $g_i$ in \\eqref{eq:f} is lower semicontinuous and convex.\nFor each $i\\in\\mathcal I$, $\\dom(g_{i})=\\Omega_i$ is a closed convex set.\n\\hfill\\small$\\blacksquare$\n\\end{standassumption}\n\n\nExamples for the local cost function are indicator functions to enforce set constraints, or penalty functions that promote sparsity, or other desirable structure. \n\nFor the function $f_i$ in \\eqref{eq:f}, we assume convexity and differentiability, as usual in the GNEP literature \\cite{facchinei2010}.\n\\begin{standassumption}[Local convexity]\n\\label{ass:IC}\nFor each $i \\in \\mathcal{I}$ and for all $\\boldsymbol{x}_{-i} \\in {\\mathbb{R}}^{n-n_i}$, the function $f_{i}(\\cdot, \\boldsymbol{x}_{-i})$ in \\eqref{eq:f} is convex and continuously differentiable. \n\\hfill$\\blacksquare$\n\\end{standassumption}\n\n\n\nWe assume that the game displays joint convexity with affine coupling constraints defining the collective feasible set \n\\vspace{-.15cm}\\begin{equation}\\label{eq:coupling}\n\\boldsymbol{\\mathcal{X}}:=\\left\\{\\boldsymbol x \\in\\boldsymbol\\Omega \\mid A \\boldsymbol{x}-b \\leq {\\boldsymbol{0}}_{m}\\right\\}\n\\vspace{-.15cm}\\end{equation}\nwhere $A:=[A_1,\\dots, A_N]\\in{\\mathbb{R}}^{m\\times n}$ and $b:=\\sum_{i=1}^{N}b_{i}\\in{\\mathbb{R}}^m$. \nEffectively, each matrix $A_i\\in{\\mathbb{R}}^{m\\times n_i}$ defines how agent $i$ is involved in the coupling constraints, thus we consider it to be private information of agent $i$. \nThen, for each $i$, given the strategies of all other agents $\\boldsymbol x_{-i}$, the feasible decision set is\n\\vspace{-.15cm}\\begin{equation}\n\\mathcal{X}_{i}(\\boldsymbol{x}_{-i}) := \\left\\{y_i \\in \\Omega_i \\mid \\, A_i y_i \\leq b-\\textstyle\\sum_{j \\neq i}^{N} A_j x_j\\right\\}.\n\\vspace{-.15cm}\\end{equation}\n\nNext, we assume a constraint qualification condition.\n\n\\begin{standassumption}\\label{ass_X}\n(\\textit{Constraint qualification})\nThe set $\\boldsymbol{\\mathcal{X}}$ in \\eqref{eq:coupling} satisfies Slater's constraint qualification. \n\\hfill\\small$\\blacksquare$\n\\end{standassumption}\nThe aim of each agent is to solve its local optimization problem\n\\vspace{-.15cm}\\begin{equation}\\label{game}\n\\forall i\\in\\mathcal I: \\quad\\left\\{\\begin{array}{cl}\n\\min_{x_i \\in  \\Omega_i} & J_{i}(x_i, \\boldsymbol x_{-i}) \\\\ \n\\text { s.t. } & A_i x_i \\leq b-\\sum_{j \\neq i}^{N} A_j x_j.\n\\end{array}\\right.\n\\vspace{-.15cm}\\end{equation}\n\n\n\n\n\nThus, the solution concept for such a competitive scenario is the generalized Nash equilibrium \\cite{Deb52,facchinei2010}. \n\n\\begin{definition} (\\textit{Generalized Nash equilibrium})\nA collective strategy $\\boldsymbol x^{\\ast}=\\operatorname{col}(x_{1}^{\\ast},\\ldots,x_{N}^{\\ast})\\in \\boldsymbol{\\mathcal{X}}$\nis a generalized Nash equilibrium of the game in \\eqref{game} if, for all $i\\in\\mathcal I$,\n\\vspace{-.15cm}\\begin{equation*}\nJ_i(x^*_i,\\boldsymbol x^*_{-i}) \\leq \\inf\\{ J_i(y,\\boldsymbol x^*_{-i}) \\, \\mid  \\, y\\in\\mathcal X_i(\\boldsymbol x_{-i})\\}.\\vspace{-.4cm}\n\\vspace{-.15cm}\\end{equation*}\n\\hfill\\small$\\blacksquare$\n\\end{definition}\nTo derive optimality conditions characterizing GNE, we define agent $i$'s Lagrangian function as\n$\\mathcal L_{i}(x_{i},\\lambda_i,  \\boldsymbol x_{-i}):=J_{i}(x_i, \\boldsymbol x_{-i})+\\lambda_i^{\\top}(A\\boldsymbol{x}-b)$\nwhere $\\lambda_i\\in{\\mathbb{R}}^m_{\\geq 0}$ is the Lagrange multiplier associated with the coupling constraint $A \\boldsymbol{x} \\leq b$. Thanks to the sum rule of the subgradient for Lipschitz continuous functions \\cite[\\S 1.8]{Cla98}, we can write the subgradient of agent $i$ as \n$ \\partial_{x_{i}}J_{i}(x_{i},\\boldsymbol x_{-i})=\\nabla_{x_{i}} f_{i}(x_{i},\\boldsymbol x_{-i})+\\partial g_{i}(x_{i})$. Therefore, \nUnder Assumption \\ref{ass_X}, the Karush--Kuhn--Tucker (KKT) theorem ensures the existence of a pair $(x^*_{i},\\lambda^*_i)\\in\\Omega_{i}\\times\\mathbb{R}^{m}_{\\geq 0}$, such that \n\\begin{equation}\\label{KKT_game}\n\\forall i\\in\\mathcal I:\\begin{cases}\n\\mathbf{0}_{n_i}\\in \\nabla_{x_{i}} f_{i}(x^*_{i};\\boldsymbol x^*_{-i})+\\partial g_{i}(x^*_{i})+A^{\\top}_{i}\\lambda_i^*\\\\\n\\mathbf{0}_{m}\\in \\NC_{\\mathbb{R}^{m}_{\\geq0}}(\\lambda_i^*)-(A\\boldsymbol{x}^*-b).\n\\end{cases}\n\\vspace{-.15cm}\\end{equation}\n\nWe conclude the section by postulating a standard assumption for GNEP's \\cite{facchinei2010}, and inclusion problems in general \\cite{BauCom16}, concerning the monotonicity and Lipschitz continuity of the mapping that collects the partial gradients $\\nabla_{i} f_{i}$.\n\n\\begin{standassumption}[Monotonicity]\n\\label{ass:GM}\nThe mapping \n\\vspace{-.15cm}\\begin{equation}\\label{eq:F}\nF(\\boldsymbol x):= \\mathrm{col}\\left( \\nabla_{x_1}f_{1}(\\boldsymbol x),\\ldots,\\nabla_{x_N}f_{N}(\\boldsymbol x)\\right)\n\\vspace{-.15cm}\\end{equation}\nis monotone on $\\boldsymbol\\Omega$, i.e., for all $\\boldsymbol x,\\boldsymbol y\\in\\boldsymbol\\Omega$\n$\\langle F(\\boldsymbol x)-F(\\boldsymbol y),\\boldsymbol x-\\boldsymbol y\\rangle\\geq 0.$\nand $\\frac{1}{\\beta}$-Lipschitz continuous, $\\beta > 0$, i.e., for all $\\boldsymbol x,\\boldsymbol y\\in\\boldsymbol\\Omega$,\\vspace{-.15cm}\n$\\norm{F(\\boldsymbol x)-F(\\boldsymbol y)} \\leq \\tfrac{1}{\\beta}\\norm{\\boldsymbol x-\\boldsymbol y}.\n\\hfill\\small$\\blacksquare$\n\\end{standassumption}\n\\vspace{.15cm}\nAmong all possible GNEs of the game, this work focuses on the computation of a \\emph{variational GNE} (v-GNE) \\cite[Def. 3.10]{facchinei2010}, i.e. a GNE in which all players share consensus on the dual variables:\n\\vspace{-.15cm}\\begin{equation}\\label{KKT_VI}\n\\forall i\\in\\mathcal I:\\begin{cases}\n\\mathbf{0}_{n_i}\\in \\nabla_{x_{i}}f_{i}(x^{*}_{i};\\boldsymbol{x}^{*}_{-i})+\\partial g_{i}(x^{*}_{i})+A_{i}^{\\top}\\lambda^{\\ast}\\\\\n\\mathbf{0}_{m}\\in \\NC_{\\mathbb{R}^{m}_{\\geq0}}(\\lambda^*)-(A\\boldsymbol{x}^*-b).\\\\\n\\end{cases}\n\\vspace{-.15cm}\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Distributed Generalized Nash equilibrium seeking via Operator Splitting}\nIn this section, we present the proposed distributed algorithms. We allow each agent to have information on his own local problem data only, i.e., $J_{i},\\Omega_{i}, A_{i}$ and $b_{i}$. We let each agent $i$ control its local decision $x_{i}$, and a local copy $\\lambda_{i}\\in\\mathbb{R}^{m}_{\\geq0}$ of dual variables, as well as a local auxiliary variable $z_{i}\\in\\mathbb{R}^{m}$ used to enforce consensus of the dual variables.\nTo actually reach consensus on the dual variables, we let the agents exchange information via an undirected weighted \\emph{communication graph}, represented by its weighted adjacency matrix $\\boldsymbol W = [w_{i,j}]_{i,j}\\in{\\mathbb{R}}^{N\\times N}$. We assume $w_{ij}>0$ iff $(i,j)$ is an edge in the communication graph. The set of neighbours of agent $i$ in the graph is $\\mathcal{N}_{i}^{\\lambda}=\\{j |w_{i,j}>0\\}$.\n\\vspace{.15cm}\n\\begin{standassumption}[Graph connectivity]\\label{ass:graph}\nThe matrix $\\mathbf{W}$ is symmetric and irreducible.\\hfill\\small$\\blacksquare$\n\\end{standassumption}\nDefine the weighted Laplacian as $\\mathbf{L}:=\\diag\\left\\{(\\mathbf{W}\\mathbf{1}_{N})_{1}, \\dots, (\\mathbf{W}\\mathbf{1}_{N})_{N}\\right\\}-\\mathbf{W}$. It holds that $\\mathbf{L}^{\\top}=\\mathbf{L}$, $\\ker(\\mathbf{L})=\\Span(\\mathbf{1}_{N})$ and that, given Standing Assumption \\ref{ass:graph}, $\\mathbf{L}$ is positive semi-definite with real and distinct eigenvalues $0=s_{1}<s_{2}\\leq \\ldots \\leq s_{N}$.\nMoreover, given the maximum degree of the graph $\\mathcal G^\\lambda$, $\\Delta:=\\max_{i\\in\\mathcal I}(\\mathbf{W}\\mathbf{1}_{N})_{i}$, it holds that $\\Delta \\leq s_{N}\\leq 2\\Delta$. Denoting by $\\kappa:=\\abs{\\mathbf{L}}$, it holds that $\\kappa\\leq 2\\Delta$.  We define the tensorized Laplacian as the matrix $\\bar{\\mathbf{L}}=\\mathbf{L}\\otimes I_{m}$. We set $\\bar{b}=(b_1,\\ldots,b_{N})^{\\top}$, $\\boldsymbol x=\\operatorname{col}(x_{1},\\ldots,x_{N})$ and similarly $\\boldsymbol z$ and $\\boldsymbol \\lambda$.\n\nLet ${\\bf{A}}=\\operatorname{diag}(A_1,\\dots,A_N)$ and define \n\n\\vspace{-.15cm}\\begin{equation}\\label{eq:AB}\n\\begin{aligned}\n\\mathcal{A}(\\boldsymbol{x},\\boldsymbol z,\\boldsymbol \\lambda):=\\operatorname{col}(F({\\bf x}),{\\bf{0}}_{mN},\\bar{\\bf{L}}\\boldsymbol\\lambda+\\bar b),\\\\\n\\mathcal{B}(\\boldsymbol{x},\\boldsymbol z,\\boldsymbol \\lambda):=\\operatorname{col}({\\bf{A}}^{\\top}\\boldsymbol \\lambda, \\bar{\\bf{L}}\\boldsymbol \\lambda,-{\\bf{A}}\\boldsymbol{x}-\\bar{\\bf{L}}\\boldsymbol z)\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nLet us also define the operator $\\mathcal D :=\\mathcal A+\\mathcal B$, and the set-valued operator\n\\vspace{-.15cm}\\begin{equation}\\label{eq:C}\n\\mathcal C(\\boldsymbol{x},\\boldsymbol z,\\boldsymbol \\lambda)=G(\\boldsymbol{x})\\times\\{{\\bf{0}}_{mN}\\}\\times \\NC_{{\\mathbb{R}}_{ \\geq 0}^{mN}}(\\boldsymbol \\lambda)\n\\vspace{-.15cm}\\end{equation}\nwhere $G(\\boldsymbol x)=\\partial g_{1}(x_{1})\\times\\cdots\\times\\partial g_{N}(x_{N})$.\n\nLet us summarize the properties of the operators above.\n\\begin{lemma}\\label{lemma_op}\nThe following statements hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal A$ is maximally monotone and $L_{\\mathcal A}=( \\frac{1}{\\beta}+\\kappa)$-Lipschitz continuous.\n\\item[(ii)] $\\mathcal B$ is maximally monotone and $L_{\\mathcal B}=(2|{\\bf {A}}|+2\\kappa)$-Lipschitz continuous.\n\\item[(iii)] $\\mathcal D$is maximally monotone and $L_{\\mathcal D}=L_{\\mathcal A}+L_{\\mathcal B}$-Lipschitz continuous.\n\\item[(iv)] $\\mathcal C+\\mathcal D$ is maximally monotone.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n(i) The operator $\\mathcal A$ is maximally monotone by \\cite[Prop. 20.23]{BauCom16}.\nFurthermore, given $\\kappa=\\abs{\\mathbf{L}}$ and $\\boldsymbol u:=\\operatorname{col}(\\boldsymbol x,\\boldsymbol z,\\boldsymbol \\lambda)$, it holds that \n\\vspace{-.15cm}\\begin{equation}\\label{A_lip}\n\\begin{aligned}\n\\norm{ \\mathcal A\\boldsymbol u-\\mathcal A\\boldsymbol u'}\\leq& \\norm{F(\\boldsymbol x)-F(\\boldsymbol x')}+\\norm{\\bar{\\mathbf{L}}(\\boldsymbol z-\\boldsymbol z')}\\\\\n\\leq&( \\frac{1}{\\beta}+\\kappa)\\left(\\norm{\\boldsymbol x-\\boldsymbol x'}+\\norm{\\boldsymbol z-\\boldsymbol z'}\\right),\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nshowing that $\\mathcal A$ is $L_{\\mathcal A}:=(\\frac{1}{\\beta}+\\kappa)$-Lipschitz continuous.\\\\\n(ii) The operator $\\mathcal B$ is maximally monotone by \\cite[Cor. 20.28]{BauCom16}.\nBy a computation similar to \\eqref{A_lip}, it can be shown that $\\mathcal B$ is $L_{\\mathcal B}=(2|{\\bf {A}}|+2\\kappa)$-Lipschitz continuous. \\\\\n(iii) The operator $\\mathcal D$ is maximally monotone since $\\dom(\\mathcal B)=\\boldsymbol\\Omega$ \\cite[Prop. 20.23]{BauCom16}. It is $L_{\\mathcal D}=L_{\\mathcal A}+L_{\\mathcal B}$-Lipschitz continuous because sum of Lipschitz operators.\\\\\n(iv) The operator $\\mathcal C$ is maximally monotone by \\cite[Lem. 5]{yi2019} and $\\mathcal C+\\mathcal D$ is maximally monotone by \\cite[Prop. 20.23]{BauCom16}.\n\\end{proof}\n\nThe following result, which immediately follows from the definition of the inner product $\\langle\\cdot,\\cdot\\rangle_\\Phi$, holds for monotone operators and it will be recalled later on.\n\\begin{lemma}\\label{lemma_mono}\nLet $\\Phi\\succ0$ and $\\mathcal T$ be a monotone operator, then $\\Phi^{-1}\\mathcal T$ is monotone in the Hilbert space $\\mathcal H_\\Phi$.\n\\hfill\\small$\\blacksquare$\\end{lemma}\n\n\n\nNow, given the operators $\\mathcal A$, $\\mathcal B$ and $\\mathcal C$ as in \\eqref{eq:AB} and \\eqref{eq:C}, a simple proof shows that the zeros of the sum $\\mathcal A+\\mathcal B+\\mathcal C$ are v-GNE of the game in \\eqref{game}. In fact, this follows verbatim from \\cite[Thm. 2]{yi2019}, so we omit the detailed proof here.\n\\begin{lemma}\nThe set $\\Zer(\\mathcal A+\\mathcal B+\\mathcal C)$ is the set of v-GNE of the game in \\eqref{game} and it is non-empty.\n\\hfill\\small$\\blacksquare$\n\\end{lemma}\n\n\n\n\n\n\\subsection{Forward-backward operator splitting} \n\\label{sec:FB}\nThe aim of this section is to revisit a distributed forward-backward (FB) splitting algorithm for the  distributed computation of a v-GNE, see Algorithm \\ref{FB_algo} \\cite{yi2019}. \n\\begin{algorithm}[h]\n\\caption{Preconditioned Forward Backward}\\label{FB_algo}\nInitialization: $x_i^0 \\in \\Omega_i, \\lambda_i^0 \\in {\\mathbb{R}}_{\\geq0}^{m},$ and $z_i^0 \\in {\\mathbb{R}}^{m} .$\\\\\nIteration $k$: Agent $i$\\\\\n($1$) Receives $x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}, \\lambda_j^k$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$, then updates\n$$\\begin{aligned}\n&x_i^{k+1}=\\operatorname{prox}^{\\rho_i}_{g_{i}}[x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-A_{i}^{T} \\lambda_i^k)]\\\\\n&z_i^{k+1}=z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)\n\\end{aligned}$$\n($2$) Receives $z_{j}^{k+1}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$, then updates\n$$\\begin{aligned}\n \\lambda_i^{k+1}=&\\operatorname{proj}_{{\\mathbb{R}}^m_{\\geq 0}}\\{\\lambda_i^k-\\tau_{i}[A_{i}(2 x_i^{k+1}-x_i^k)-b_{i}\\\\\n&+\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[2(z_i^{k+1}-z_{j, k+1})-(z_i^k-z_j^k)]\\\\\n&+\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)]\\}\\\\\n\\end{aligned}$$\n\\end{algorithm}\nFrom now on, the triplet $\\boldsymbol u:=\\operatorname{col}(\\boldsymbol x,\\boldsymbol z,\\boldsymbol \\lambda)$ defines the state variable of a distributed algorithm. Given the state at iteration $k$, $\\boldsymbol u^k=(\\boldsymbol x^k,\\boldsymbol z^k,\\boldsymbol \\lambda^k)$, the FB algorithm can be written as fixed-point iteration of the form $\\boldsymbol u^{k+1}=T_{\\text{FB}}\\boldsymbol u^k,$ where\n\\vspace{-.15cm}\\begin{equation}\\label{eq:FB}\nT_{\\text{FB}}:= \\mathrm{J}_{\\Phi_{\\text{FB}}^{-1}(\\mathcal C+\\mathcal B)}(\\operatorname{Id}-\\Phi_{\\text{FB}}^{-1}\\mathcal A).\n\\vspace{-.15cm}\\end{equation}\nand $\\Phi_{\\text{FB}}$ is the preconditioning matrix defined as \n\\vspace{-.15cm}\\begin{equation}\\label{eq:phi}\n\\Phi_{\\text{FB}}:=\\left[\\begin{array}{ccc} \n\\rho^{-1} & 0 & -{\\mathbf A}^{\\top}\\\\\n0 & \\sigma^{-1} & -\\bar{\\mathbf{L}}\\\\\n-{\\mathbf A} & -\\bar{\\mathbf{L}} &  \\tau^{-1}\n\\end{array}\\right].\n\\vspace{-.15cm}\\end{equation}\nThe matrices $\\rho=\\diag\\{\\rho_{1}I_{n_{1}},\\ldots,\\rho_{N} I_{n_{N}}\\}$, $\\sigma$ and $\\tau$ (defined analoguosly)\ncollect the step sizes of the primal, the auxiliary and the dual updates, respectively. By choosing the step sizes appropriately, the preconditioning matrix $\\Phi$ can be made positive definite \\cite{belgioioso2018}. The FB algorithm is known to converge to a zero of a monotone inclusion $0 \\in \\mathcal A+\\mathcal B+\\mathcal C$ when the operators are maximally monotone and the single-valued operator $\\Phi_{\\text{FB}}^{-1}\\mathcal A$ is cocoercive \\cite[Thm. 26.14]{BauCom16}. Thus, the pseudo-gradient mapping $F$ in \\eqref{eq:F} should satisfy the following assumption.\n\\begin{assumption}[Strong monotonicity]\n\\label{ass:Hstrong}\nFor all $\\boldsymbol x,\\boldsymbol x'\\in \\boldsymbol{\\Omega}$,\n$\\inner{F(\\boldsymbol x)-F(\\boldsymbol{x}'),\\boldsymbol x-\\boldsymbol x'}\\geq\\eta\\norm{\\boldsymbol{x}-\\boldsymbol{x}'}^{2}$\nfor some $\\eta>0$.\n\\hfill\\small$\\blacksquare$\n\\end{assumption}\n\nTo ensure the cocoercivity condition, we refer to the following result.\n\n\\begin{lemma}\\cite[Lem. 5 and Lem. 7]{yi2019}\\label{lemma_coco}\nLet $\\Phi\\succ0$ and $F$ as in \\eqref{eq:F} satisfy Assumption \\ref{ass:Hstrong}. Then, the following hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal A$ is $\\theta$-cocoercive with $\\theta\\leq\\min\\{1\/2\\Delta,\\eta\\beta^2\\}$.\n\\item[(ii)] $\\Phi^{-1}\\mathcal A$ is $\\alpha\\theta$-cocoercive with $\\alpha=1\/\\abs{\\Phi^{-1}}$. \n\\hfill\\small$\\blacksquare$\n\\end{itemize}\n\\end{lemma}\n\nWe recall that convergence to a v-GNE has been demonstrated in \\cite[Th. 3]{yi2019}, if the step sizes in \\eqref{eq:phi} are chosen small enough \\cite[Lem. 6]{yi2019}. \n\n\n\\subsection{Forward-backward-forward splitting}\n\\label{sec:FBF}\nIn this section, we propose our distributed forward-backward-forward (FBF) scheme, Algorithm \\ref{FBF_algo}.\n\n\\begin{algorithm}\n\\caption{Distributed Forward Backward Forward}\\label{FBF_algo}\nInitialization: $x_i^0 \\in \\Omega_i, \\lambda_i^0 \\in {\\mathbb{R}}_{\\geq0}^{m},$ and $z_i^0 \\in {\\mathbb{R}}^{m} .$\\\\\nIteration $k$: Agent $i$\\\\\n($1$) Receives $x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}$, $ \\lambda_j^k$ and $z_{j,k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&\\tilde x_i^k=\\operatorname{prox}^{\\rho_i}_{g_{i}}[x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-A_{i}^{T} \\lambda_i^k)]\\\\\n&\\tilde z_i^k=z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)\\\\\n&\\tilde\\lambda_i^k=\\operatorname{proj}_{{\\mathbb{R}}^m_{\\geq 0}}\\{\\lambda_i^k-\\tau_{i}(A_{i}x_i^k-b_{i})\\\\\n&\\quad\\quad+\\tau\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_{i}^{k}-z_j^k)-(\\lambda_i^k-\\lambda_j^k)]\\}\n\\end{aligned}$$\n($2$) Receives $\\tilde x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}$, $ \\tilde \\lambda_j^k$and $\\tilde z_{j}^{k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&x_i^{k+1}=\\tilde x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-\\nabla_{\\tilde x_{i}} f_{i}(\\tilde x_i^k,\\tilde {\\boldsymbol{x}}_{-i}^k))\\\\\n&\\quad\\quad\\quad-\\rho_iA_{i}^{T} (\\lambda_i^k-\\tilde \\lambda_{i,k})\\\\\n&z_i^{k+1}=\\tilde z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(\\lambda_i^k-\\lambda_j^k)-(\\tilde\\lambda_i^k-\\tilde\\lambda_j^k)]\\\\\n&\\lambda_i^{k+1}=\\tilde{\\lambda}_i^{k}+\\tau_iA_i(\\tilde{x}_{i}^{k}-x_{i}^{k})\\\\\n&\\quad\\quad\\quad-\\tau_i\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_i^k-z_j^k)-(\\tilde z_i^k-\\tilde z_j^k)]\\\\\n&\\quad\\quad\\quad+\\tau_i\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(\\lambda_{i,k}-\\lambda_j^k)-(\\tilde\\lambda_i^k-\\tilde\\lambda_j^k)]\\\\\n\\end{aligned}$$\n\\end{algorithm}\n\nIn compact form, the FBF algorithm generates two sequences $(\\boldsymbol u^{k},\\boldsymbol v^{k})_{k\\geq 0}$ as follows: \n\\vspace{-.15cm}\\begin{equation}\\label{FBF}\n\\begin{aligned}\n\\boldsymbol u^{k}&=J_{\\Psi^{-1} \\mathcal C}(\\boldsymbol v^{k}-\\Psi^{-1} \\mathcal D \\boldsymbol v^{k})\\\\\n\\boldsymbol v^{k+1}&=\\boldsymbol u^{k}+\\Psi^{-1} (\\mathcal D\\boldsymbol v^{k}-\\mathcal D\\boldsymbol u^{k}).\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\n\nIn \\eqref{FBF}, $\\Psi$ is the block-diagonal matrix of the step sizes: \n\\vspace{-.15cm}\\begin{equation}\\label{Psi}\n\\Psi=\\operatorname{diag}(\\rho^{-1},\\sigma^{-1}, \\tau^{-1}),\n\\vspace{-.15cm}\\end{equation}\n\nWe recall that $\\mathcal D=\\mathcal A+\\mathcal B$ is single-valued, maximally monotone and Lipschitz continuous by Lemma \\ref{lemma_op}. Each iteration differs from the scheme in \\eqref{eq:FB} by one additional forward step and the fact that the resolvent is now defined in terms of the operator $\\mathcal C$ only. Writing the coordinates as $\\boldsymbol u^{k}=(\\tilde{\\boldsymbol{x}}^{k},\\tilde{\\boldsymbol z}^{k},\\tilde{\\boldsymbol \\lambda}^{k})$ and $\\boldsymbol v^{k}=(\\boldsymbol{x}^{k},\\boldsymbol z^{k},\\boldsymbol \\lambda^{k})$, the updates are explicitly given in Algorithm \\ref{FBF_algo}.\n\nFBF operates on the splitting $\\mathcal C+\\mathcal D$ and it can be compactly written as the fixed-point iteration\n$\\boldsymbol v^{k+1}=T_{\\text{FBF}} \\, \\boldsymbol v^{k},$\nwhere the mapping $T_{\\text{FBF}}$ is defined as  \n\\vspace{-.15cm}\\begin{equation}\\label{eq:T_FBF}\nT_{\\text{FBF}}:=\\Psi^{-1}\\mathcal D+(\\Id-\\Psi^{-1}\\mathcal D)\\circ \\operatorname{J}_{\\Psi^{-1}\\mathcal C}\\circ(\\Id-\\Psi^{-1}\\mathcal D). \n\\vspace{-.15cm}\\end{equation}\n\n\nTo ensure convergence of Algorithm \\ref{FBF_algo} to a v-GNE of the game in \\eqref{game}, we need the next assumption.\n\n \n\\begin{assumption}\\label{step_FBF}\n$\\abs{\\Psi^{-1}} < 1\/L_{\\mathcal D}$, with $\\Psi$ as in \\eqref{Psi} and $L_{\\mathcal D}$ being the Lipschitz constant of $\\mathcal D$ as in Lemma \\ref{lemma_op}.\n\\hfill\\small$\\blacksquare$\n\\end{assumption}\n\n\n\n\n\n\n\\begin{theorem}\\label{theo_FBF}\nLet Assumption \\ref{step_FBF} hold. The sequence $(\\boldsymbol x^k,\\boldsymbol \\lambda^k)$ generated by Algorithm \\ref{FBF_algo} converges to \n$\\operatorname{zer}(\\mathcal A+\\mathcal B+\\mathcal C)$, thus the primal variable converges to a v-GNE of the game in \\eqref{game}.\n\\end{theorem}\n\\begin{proof}\nThe fixed-point iteration with $T_{\\text{FBF}}$ as in \\eqref{eq:T_FBF} can be derived from \\eqref{FBF} by substituting $\\boldsymbol u_k$.\nTherefore, the sequence $(\\boldsymbol x^k,\\boldsymbol\\lambda^k)$ generated by Algorithm \\ref{FBF_algo} converges to a v-GNE by \\cite[Th.26.17]{BauCom16} and \\cite[Th.3.4]{tseng2000} since $\\Psi^{-1}\\mathcal A$ is monotone by Lemma \\ref{lemma_mono} and $\\mathcal A+\\mathcal B+\\mathcal C$ is maximally monotone by Lemma \\ref{lemma_op}. See Appendix \\ref{sec:FBF} for details.\n\\end{proof}\n \n\nWe emphasize that Algorithm \\ref{FBF_algo} does not require strong monotonicity (Assumption \\ref{ass:Hstrong}) of the pseudo-gradient mapping $F$ in \\eqref{eq:F}.\nMoreover, we note that the FBF algorithm requires two evaluations of the individual gradients, which requires computing the operator $\\mathcal D$ twice per iteration. At the level of the individual agents, this means that we need two communication rounds per iteration in order to exchange the necessary information. Compared with the FB algorithm, the non-strong monotonicity assumption comes at the price of increased communications at each iteration.\n\n\n\\subsection{Forward-backward-half forward splitting}\n\\label{sec:FBHF}\n\nShould the strong monotonicity condition (Assumption \\ref{ass:Hstrong}) be satisfied, an alternative to the FB is the \\emph{forward-backward-half-forward} (FBHF) operator splitting, developed in \\cite{briceno2018}. Thus, our second GNE seeking algorithm is a distributed FBHF, described in Algorithm \\ref{FBHF_algo}.\n\n\\begin{algorithm}\n\\caption{Distributed Forward Backward Half Forward}\\label{FBHF_algo}\nInitialization: $x_i^0 \\in \\Omega_i, \\lambda_i^0 \\in {\\mathbb{R}}_{\\geq0}^{m},$ and $z_i^0 \\in {\\mathbb{R}}^{m} .$\\\\\nIteration $k$: Agent $i$\\\\\n($1$) Receives $x_j^k$ for $j \\in \\mathcal{N}_{i}^{J}$, $ \\lambda_j^k$ and $z_{j,k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&\\tilde x_i^k=\\operatorname{prox}^{\\rho_i}_{g_{i}}[x_i^k-\\rho_{i}(\\nabla_{x_{i}} f_{i}(x_i^k,\\boldsymbol x_{-i}^k)-A_{i}^{T} \\lambda_i^k)]\\\\\n&\\tilde z_i^k=z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}(\\lambda_i^k-\\lambda_j^k)\\\\\n&\\tilde\\lambda_i^k=\\operatorname{proj}_{{\\mathbb{R}}^m_{\\geq 0}}\\{\\lambda_i^k-\\tau_{i}(A_{i}x_i^k-b_{i})\\\\\n&\\quad\\quad+\\tau\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_{i}^{k}-z_j^k)-(\\lambda_i^k-\\lambda_j^k)]\\}\n\\end{aligned}$$\n($2$) Receives $ \\tilde \\lambda_j^k$and $\\tilde z_{j,k}$ for $j \\in \\mathcal{N}_{i}^{\\lambda}$ then updates\n$$\\begin{aligned}\n&x_i^{k+1}=\\tilde x_i^k+\\rho_iA_{i}^{T} (\\lambda_i^k-\\tilde \\lambda_{i,k})]\\\\\n&z_i^{k+1}=\\tilde z_i^k+\\sigma_{i} \\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(\\lambda_i^k-\\lambda_j^k)-(\\tilde\\lambda_i^k-\\tilde\\lambda_j^k)]\\\\\n&\\lambda_i^{k+1}=\\tilde{\\lambda}_i^{k}+\\tau_iA_i(\\tilde{x}_{i}^{k}-x_{i}^{k})\\\\\n&\\quad\\quad\\quad-\\tau_i\\sum\\nolimits_{j \\in \\mathcal{N}_{i}^{\\lambda}} w_{i,j}[(z_i^k-z_j^k)-(\\tilde z_i^k-\\tilde z_j^k)]\\\\\n\\end{aligned}$$\n\\end{algorithm}\nIn compact form, the FBHF algorithm reads as\n\\vspace{-.15cm}\\begin{equation}\\label{FBHF}\n\\begin{aligned}\n\\boldsymbol u^{k} & = \\mathrm{J}_{\\Psi^{-1}\\mathcal C}(\\boldsymbol v^{k}-\\Psi^{-1} (\\mathcal A+\\mathcal B) \\boldsymbol v^{k}) \\\\\n\\boldsymbol v^{k+1} & =\\boldsymbol u^{k}+\\Psi^{-1}(\\mathcal B\\boldsymbol v^k- \\mathcal B\\boldsymbol u^k).\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nWe note that the iterates of FBHF are similar to those of the FBF, but the second forward step requires the operator $\\mathcal B$ only. \nMore simply, we can write the FBHF as the fixed-point iteration\n$\\boldsymbol v^{k+1}=T_{\\text{FBHF}}\\boldsymbol v^{k},$\nwhere \n\\vspace{-.15cm}\\begin{equation}\\label{eq:T_FBHF}\nT_{\\text{FBHF}}=(\\Id-\\Psi^{-1}\\mathcal B)\\circ \\operatorname{J}_{\\Psi^{-1}\\mathcal C}\\circ (\\Id-\\Psi^{-1}\\mathcal D)+\\Psi^{-1}\\mathcal B. \n\\vspace{-.15cm}\\end{equation}\n\nAlso in this case, we have a bound on the step sizes.\n\n \n\\begin{assumption}\\label{step_FBHF}\n$|\\Psi^{-1}| \\leq \\min\\{2\\theta_{\\mathcal A},1\/L_{\\mathcal B}\\}$,\nwith $\\theta_{\\mathcal A}$ as in Lemma \\ref{lemma_coco} and $L_{\\mathcal B}$ as in Lemma \\ref{lemma_op}.\n\\hfill\\small$\\blacksquare$\n\\end{assumption}\n \n\nWe note that in Assumption \\ref{step_FBHF}, the step sizes in $\\Psi$ can be chosen larger compared to those in Assumption \\ref{step_FBF}, since the upper bound is related to the Lipschitz constant of the operator $\\mathcal B$, not of $L_{\\mathcal D}=L_{\\mathcal A}+L_{\\mathcal B}$ as for the FBF (Assumption \\ref{step_FBF}). A similar comparison can be done with respect to the FB algorithm. Intuitively, larger step sizes should be beneficial in term of convergence speed.\n\nWe can now establish our convergence result for the FBHF algorithm.\n\n \n\\begin{theorem}\nLet Assumptions \\ref{ass:Hstrong} and \\ref{step_FBHF} hold. The sequence $(\\boldsymbol x^k,\\boldsymbol \\lambda^k)$ generated by Algorithm \\ref{FBHF_algo} converges to $\\operatorname{zer}(\\mathcal A+\\mathcal B+\\mathcal C)$, thus the primal variable converges to \na v-GNE of the game in \\eqref{game}. \\hfill\\small$\\blacksquare$\n\\end{theorem}\n\n\\begin{proof}\nAlgorithm \\ref{FBHF_algo} is the fixed point iteration in \\eqref{eq:T_FBHF} whose convergence is guaranteed by \\cite[Th. 2.3]{briceno2018} under Assumption \\ref{step_FBHF} because $\\Psi^{-1}\\mathcal A$ is cocoercive by Lemma \\ref{lemma_coco}. See Appendix \\ref{sec:FBHF} for details.\n\\end{proof}\n\n\n\n\n\n\n\\section{Case study and numerical simulations}\nWe consider a networked Cournot game with market capacity constraints \\cite{yi2019}.\nAs a numerical setting, we use a set of 20 companies and 7 markets, similarly to \\cite{yi2019}. Each company $i$ has a local constraint $x_i\\in(0,\\delta_i)$ where each component of $\\delta_i$ is randomly drawn from $[1, 1.5]$. The maximal capacity of each market $j$ is $b_j$, randomly drawn from $[0.5, 1]$. The local cost function of company $i$ is $c_i(x_i) = \\pi_i\\sum_{j=1}^{n_i} ([x_i]_j)^2 + r^\\top x_i$, where $[x_i]_j$ indicates the $j$ component of $x_i$.\nFor all $i\\in\\mathcal I$, $\\pi_i$ is randomly drawn from $[1, 8]$, and the components of $r_i$ are randomly drawn from $[0.1, 0.6]$. Notice that $c_i(x_i)$ is strongly convex with Lipschitz continuous gradient. The price is taken as a linear function $P= \\bar P-DA\\boldsymbol x$ where each component of $\\bar P =\\operatorname{col}(\\bar P_1,\\dots,\\bar P_7)$ is randomly drawn from $[2,4]$ while the entries of $D=\\operatorname{diag}(d_1,\\dots,d_7)$ are randomly drawn from $[0.5,1]$. Recall that the cost function of company $i$ is influenced by the variables of the agents selling in the same market. Such informations can be retrieved from \\cite[Fig. 1]{yi2019}. Since $c_i(x_i)$ is strongly convex with Lipschitz continuous gradient and the prices are linear, the pseudo gradient of $f_i$ is strongly monotone. The communication graph $\\mathcal G^\\lambda$ for the dual variables is a cycle graph with the addition of the edges $(2,15)$ and $(6,13)$. As local cost functions $g_i$ we use the indicator functions. In this way, the proximal step is a projection on the local constraints sets.\n\nThe aim of these simulations is to compare the proposed schemes.\nThe step sizes are taken differently for every algorithm. In particular, we take $\\rho_{\\text{FB}}$, $\\sigma_{\\text{FB}}$ and $\\tau_{\\text{FB}}$ as in \\cite[Lem. 6]{yi2019}, $\\rho_{\\text{FBF}}$, $\\sigma_{\\text{FBF}}$ and $\\tau_{\\text{FBF}}$ such that Assumption \\ref{step_FBF} is satisfied and $\\rho_{\\text{FBHF}}$, $\\sigma_{\\text{FBHF}}$ and $\\tau_{\\text{FBHF}}$ such that Assumption \\ref{step_FBHF} holds. We select them to be the maximum possible.\n\nThe initial points $\\lambda_i^0$ and $z_i^0$ are set to 0 while the local decision variable $x_i^0$ is randomly taken in the feasible sets.\n\nThe plots in Fig. \\ref{distance_sol} show the performance parameter $\\frac{\\norm{\\boldsymbol{x}_{k+1}-\\boldsymbol{x}^{*}}}{\\norm{\\boldsymbol{x}^{*}}}$, that is, the convergence to a solution $\\boldsymbol x^*$, and the CPU time (in seconds) used by each algorithm. We run 10 simulations, changing the parameters of the cost function to show that the result are replicable. The darker line represent the average path towards the solution.\n\n\nThe plot in Fig \\ref{distance_sol} shows that with suitable parameters convergence to a solution is faster with the FBF algorithm which, however, is computationally more expansive than the FB and FBHF algorithms\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=.22]{ave_sol.eps}\\hspace{-.2cm}\n\\includegraphics[scale=.22]{ave_sec.eps}\n\\caption{Relative distance from v-GNE (left) and cumulative CPU time (right).}\\label{distance_sol}\n\\end{figure}\n\\vspace{-.2cm}\n\n\n\n\\section{Conclusion}\n\nThe FBF and the FBHF splitting methods generate distributed equilibrium seeking algorithms for solving generalized Nash equilibrium problems. Compared to the FB, the FBF has the advantage to converge under the non-strong monotonicity assumption. This comes at the price of increased communications between the agents. If strong monotonicity holds, an alternative to the FBF is the FBHF that, in our numerical experience is less computationally expensive than the FBF.\n\n\\section{Appendix}\n\\subsection{Convergence of the forward-backward-forward}\\label{app:FBF}\nWe show the convergence proof for the FBF. From now on, $\\mathsf{H}={\\mathbb{R}}^n\\times{\\mathbb{R}}^{mN}\\times{\\mathbb{R}}^{mN}$ and $\\operatorname{fix}(T)=\\{x\\in\\mathsf{H}:Tx=x\\}$.\n\\begin{proposition}\nIf Assumption \\ref{step_FBF} holds, $\\operatorname{fix}(T_{\\text{FBF}})=\\mathcal Z$. \n\\end{proposition}\n\\begin{proof}\nWe first show that $\\mathcal Z\\subseteq \\operatorname{fix}(T_{\\text{FBF}})$. Let $u^{\\ast}\\in\\mathcal Z$: \n\\vspace{-.15cm}\\begin{equation*}\n\\begin{aligned}\n0\\in \\mathcal Cu^{\\ast} +\\mathcal Du^{\\ast} & \\Leftrightarrow -\\mathcal Du^{\\ast} \\in \\mathcal Cu^{\\ast} \\\\\n&\\Leftrightarrow u^{\\ast}=J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})\\\\\n&\\Leftrightarrow \\Psi^{-1}\\mathcal Du^{\\ast}=\\Psi^{-1}\\mathcal DJ_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})\\\\\n&\\Leftrightarrow u^{\\ast}=T_{\\text{FBF}}u^{\\ast}. \n\\end{aligned}\n\\vspace{-.15cm}\\end{equation*}\nConversely, let $u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}})$. Then \n$u^{\\ast}-J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})=\n\\Psi^{-1}\\mathcal Du^{\\ast}-\\Psi^{-1}\\mathcal DJ_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})$\nans\n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\norm{u^{\\ast}-J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})}\\leq\\\\\n&\\leq \\alpha^{-1}\\norm{\\mathcal Du^{\\ast} -\\mathcal DJ_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})}\\\\\n&\\leq \\tfrac{L}{\\alpha}\\norm{u^{\\ast}-J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nHence,\n$u^{\\ast}=J_{\\Psi^{-1}\\mathcal C}(u^{\\ast}-\\Psi^{-1}\\mathcal Du^{\\ast})$. \n\\end{proof}\n\n\\begin{proposition}\nFor all $u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}})$ and $v\\in\\mathsf{H}$, there exists $\\varepsilon\\geq 0$ such that \n\\vspace{-.15cm}\\begin{equation}\\label{eq:Fejer}\n\\norm{T_{\\text{FBF}}v-u^{\\ast}}^{2}_{\\Psi}= \\norm{v-u^{\\ast}}^{2}_{\\Psi}-\\left(1-(L\/\\alpha)^{2}\\right)\\norm{u-v}^{2}_{\\Psi}-2\\varepsilon.\n\\vspace{-.15cm}\\end{equation}\n\\end{proposition}\n\\begin{proof}\nLet $u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}})$ and $u=J_{\\Psi^{-1}\\mathcal C}(v-\\Psi^{-1}\\mathcal D v),v^{+}=T_{\\text{FBF}}v$, for $v\\in\\mathsf{H}$ arbitrary. Then, \n\\vspace{-.18cm}\\begin{equation*}\\begin{aligned}\n\\norm{v-u^{\\ast}}_{\\Psi}^{2}&=\\norm{v-u+u-v^{+}+v^{+}-u^{\\ast}}^{2}_{\\Psi}\\\\\n&=\\norm{v-u}^{2}_{\\Psi}+\\norm{u-v^{+}}^{2}_{\\Psi}+\\norm{v^{+}-u^{\\ast}}^{2}_{\\Psi}\\\\\n&+2\\inner{v-u,u-u^{\\ast}}_{\\Psi}+2\\inner{u-v^{+},v^{+}-u^{\\ast}}_{\\Psi}.\n\\end{aligned}\\vspace{-.18cm}\\end{equation*}\nSince, \n$2\\inner{u-v^{+},v^{+}-u^{\\ast}}_{\\Psi}=2\\inner{u-v^{+},v^{+}-u}_{\\Psi}\n+2\\inner{u-v^{+},u-u^{\\ast}}_{\\Psi}=-2\\norm{u-v^{+}}_{\\Psi}^{2}+2\\inner{u-v^{+},u-u^{\\ast}}_{\\Psi}.$\nThis gives \n$\\norm{v-u^{\\ast}}_{\\Psi}^{2}=\\norm{v-u}^{2}_{\\Psi}-\\norm{u-v^{+}}_{\\Psi}^{2}+\\norm{v^{+}-u^{\\ast}}_{\\Psi}^{2}+2\\inner{u-u^{\\ast},v-v^{+}}_{\\Psi}. $\nBy definition of the updates, we have for $\\bar{v}\\equiv Bv,\\bar{u}\\equiv Bu,\\hat{v}\\in Cu$, the identities\n$u+\\Psi^{-1}\\hat{v}=v-\\Psi^{-1}\\bar{v}$ and $v^{+}=u+\\Psi^{-1}(\\bar{v}-\\bar{u}).$\nFurthermore, since $0\\in \\mathcal Du^{\\ast} +\\mathcal Cu^{\\ast} $, there exists $\\hat{v}^{\\ast}\\in \\mathcal Cu^{\\ast} $ and $\\bar{u}^{\\ast}\\equiv \\mathcal Du^{\\ast} $ such that\n$0=\\bar{u}^{\\ast}+\\hat{v}^{\\ast}.$\nIt follows that \n$v-v^{+}=v-u-\\Psi^{-1}(\\bar{v}-\\bar{u})=\\Psi^{-1}(\\hat{v}+\\bar{u}).$\nHence, \n\\vspace{-.18cm}\\begin{equation*}\n\\begin{aligned}\n\\norm{v-u^{\\ast}}_{\\Psi}^{2}=&\\norm{v-u}^{2}_{\\Psi}-\\norm{u-v^{+}}_{\\Psi}^{2}+\\\\\n&+\\norm{v^{+}-u^{\\ast}}_{\\Psi}^{2}+2\\inner{u-u^{\\ast},\\hat{v}+\\bar{u}}\\\\\n=&\\norm{v-u}^{2}_{\\Psi}-\\norm{u-v^{+}}_{\\Psi}^{2}+\\norm{v^{+}-u^{\\ast}}_{\\Psi}^{2}+\\\\\n&+2\\inner{\\hat{v}-\\hat{v}^{\\ast}-\\bar{u}^{\\ast}+\\bar{u},u-u^{\\ast},u-u^{\\ast}}.\n\\end{aligned}\n\\vspace{-.18cm}\\end{equation*}\nSince $(u,\\hat{v}),(u^{\\ast},\\hat{v}^{\\ast})\\in\\gr(C),(u^{\\ast},\\bar{u}^{\\ast}),(u,\\bar{u})\\in\\gr(B)$, it follows from the monotonicity that \n$\\varepsilon:=\\inner{\\hat{v}-\\hat{v}^{\\ast}-\\bar{u}^{\\ast}+\\bar{u},u-u^{\\ast},u-u^{\\ast}}\\geq 0.$\nFinally, observe that $u-v^{+}=\\Psi^{-1}(\\mathcal Du-\\mathcal Dv)$, and that \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\norm{\\Psi^{-1}(\\mathcal Du-\\mathcal Dv)}_{\\Psi}^{2}=\\inner{\\Psi^{-1}(\\mathcal Du-\\mathcal Dv),\\mathcal Du-\\mathcal Dv}\\\\\n&\\leq \\lambda_{\\max}(\\Psi^{-1})\\norm{\\mathcal Du-\\mathcal Dv}^{2}\\leq L^{2}\\lambda_{\\max}(\\Psi^{-1})\\norm{u-v}^{2}\\\\\n&\\leq L^{2}\\tfrac{\\lambda_{\\max}(\\Psi^{-1})}{\\lambda_{\\min}(\\Psi)}\\norm{u-v}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nSince $\\alpha=1\/\\lambda_{\\max}(\\Psi^{-1})=\\lambda_{\\min}(\\Psi)$, it follows from the Lipschitz continuity of the operator $B$ that\n$\\norm{u-v^{+}}_{\\Psi}^{2}\\leq (L\/\\alpha)^{2}\\norm{u-v}^{2}_{\\Psi} $\nand the statement is proven.\n\\end{proof}\n\\begin{corollary}\nIf $L\/\\alpha<1$, the map $T_{\\text{FBF}}:\\mathsf{H}\\to\\mathsf{H}$ is quasinonexpansive in the Hilbert space $(\\mathsf{H},\\inner{\\cdot,\\cdot}_{\\Psi})$, i.e. \n\\vspace{-.15cm}\\begin{equation*}\n\\forall v\\in\\mathsf{H}\\; \\forall u^{\\ast}\\in\\operatorname{fix}(T_{\\text{FBF}}) \\;\\norm{T_{\\text{FBF}}v-u^{\\ast}}_{\\Psi}\\leq\\norm{v-u^{\\ast}}_{\\Psi}.\n\\vspace{-.15cm}\\end{equation*}\n\\end{corollary}\n\\begin{proposition}\nIf Assumption \\ref{step_FBF} holds, the sequence generated by the FBF algorithm, $(v^{k})_{k\\geq 0}$, is bounded in norm, and all its accumulation points are elements in $\\mathcal Z$.\n\\end{proposition}\n\\begin{proof}\nForm \\eqref{eq:Fejer} we deduce that $(v^{k})_{k\\geq 0}$ is Fej\\'{e}r monotone with respect to $\\operatorname{fix}(T_{\\text{FBF}})=\\mathcal Z$. Therefore, it is bounded norm. It remains to show that all accumulation points are in $\\mathcal Z$. By an obvious abuse of notation, let $(v^{k})_{k\\geq 0}$ denote a converging subsequence with limit $u^{\\ast}$. From \\eqref{eq:Fejer} it follows $\\norm{u^{k}-v^{k}}_{\\Psi}\\to 0$, and hence $\\norm{u^{k}-v^{k}}\\to 0$ as $k\\to\\infty$. By continuity, it therefore follows as well $\\norm{\\mathcal Du^k-\\mathcal Dv^k}\\to 0$ as $k\\to\\infty$. Since $u^{k}=J_{\\Psi^{-1}\\mathcal C}(v^{k}-\\Psi^{-1}\\mathcal Dv^{k})$, it follows that $w^{k}:=\\Psi(v^{k}-u^{k})+\\mathcal Du^k-\\mathcal Dv^k\\in \\mathcal Du^k+\\mathcal Cu^{k}.$\nSince $w^{k}\\to 0$ and the operator $\\mathcal C+\\mathcal D$ is maximally monotone by Lemma \\ref{lemma_op} and has a closed graph \\cite[Lem. 3.2]{tseng2000}, we conclude $0\\in \\mathcal Du^{\\ast} +\\mathcal Cu^{\\ast} $. Hence, $u^{\\ast}\\in\\mathcal Z$.\n\\end{proof}\n\n\\subsection{Convergence of the forward-backward-half-forward}\n\\label{sec:FBHF}\nWe here provide the convergence proof for the FBHF.\n\\begin{proposition}\nIf Assumption \\ref{step_FBHF} holds, the sequence generated by the FBHF algorithm converges to $\\mathcal Z$.\n\\end{proposition}\nSince, $w-u\\in\\Psi^{-1}\\mathcal C u$, it follows that $(u,w-u)\\in\\gr(\\Psi^{-1}\\mathcal C)$. Additionally, $0\\in \\mathcal Du^{\\ast} +\\mathcal Cu^{\\ast} $, implying that $(u^{\\ast},-\\Psi^{-1}\\mathcal Du^{\\ast})\\in \\gr(\\Psi^{-1}\\mathcal C)$. Monotonicity of the involved operators, implies that \n$\\inner{u-u^{\\ast},w-u-\\Psi^{-1}\\mathcal Du^{\\ast}}_{\\Psi}\\leq 0,\\text{ and }\n\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bu^{\\ast} -\\mathcal Bu)}_{\\Psi}\\leq 0, $\nUsing these two inequalities, we see \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\inner{u-u^{\\ast},u-w-\\Psi^{-1}\\mathcal Bu}_{\\Psi}=\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Au^{\\ast}}_{\\Psi}\\\\\n&+\\inner{u-u^{\\ast},u-w-\\Psi^{-1}\\mathcal Du^{\\ast}}_{\\Psi}\\\\\n&+\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Du^{\\ast} -\\mathcal Bu)}_{\\Psi}\\leq \\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Au^{\\ast}}_{\\Psi}\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nTherefore, \n\\vspace{-.15cm}\\begin{equation}\\label{step}\n\\begin{aligned}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}=\\\\\n&2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Bv+w-u}_{\\Psi}+2\\inner{u-u^{\\ast},u-w-\\Psi^{-1}\\mathcal Bu}_{\\Psi}\\\\\n&\\leq 2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Bv+w-u}_{\\Psi}+2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Au^{\\ast}}_{\\Psi}\\\\\n&=2\\inner{u-u^{\\ast},\\Psi^{-1}\\mathcal Dv+w-u}_{\\Psi}\\\\\n&+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}\\\\\n&=2\\inner{u-u^{\\ast},v-u}_{\\Psi}+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi},\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nwhere in the last equality we have used the identity $w=v-\\Psi^{-1}\\mathcal Dv$. Using the cosine formula, \n\\eqref{step} becomes \n\\vspace{-.15cm}\\begin{equation}\n\\begin{aligned}\\label{eq:step}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}\\leq \\norm{v-u^{\\ast}}_{\\Psi}^{2}-\\norm{u-u^{\\ast}}_{\\Psi}^{2}\\\\\n&-\\norm{v-u}_{\\Psi}^{2}+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}.\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation}\nThe cocoercivity of $\\Psi^{-1}\\mathcal A$ in $(\\mathsf{H},\\inner{\\cdot,\\cdot}_{\\Psi})$ gives for all $\\varepsilon>0$\n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}=\\\\\n&2\\inner{v-u^{\\ast},\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}+2\\inner{u-v,\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}\\\\\n&\\leq -2\\alpha\\theta\\norm{\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}^{2}+2\\inner{u-v,\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}\\\\\n&=-2\\alpha\\theta\\norm{\\Psi^{-1}(\\mathcal Au^{\\ast}-\\mathcal Av)}_{\\Psi}^{2}+\\tfrac{1}{\\varepsilon}\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&+\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}\\\\\n&=\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\left(2\\alpha\\theta-\\tfrac{1}{\\varepsilon}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nCombining this estimate with (\\ref{eq:step}), we see \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}\\leq \\norm{v-u^{\\ast}}_{\\Psi}^{2}-\\norm{u-u^{\\ast}}_{\\Psi}^{2}\\\\\n&-\\norm{v-u}_{\\Psi}^{2}+\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\left(2\\alpha\\theta-\\tfrac{1}{\\varepsilon}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nTherefore, \n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n&\\norm{v^{+}-u^{\\ast}}^{2}_{\\Psi\n=\\norm{u+\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)-u^{\\ast}}^{2}_{\\Psi}\\\\\n&=\\norm{u-u^{\\ast}}^{2}_{\\Psi}+2\\inner{u-u^{\\ast},\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}\\\\\n&+\\norm{\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}^{2}\\\\\n&\\leq \\norm{u-u^{\\ast}}^{2}_{\\Psi}+\\norm{\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}_{\\Psi}^{2}-\\norm{u-u^{\\ast}}_{\\Psi}^{2}\\\\\n&-\\hspace{-.1cm}\\left(2\\alpha\\theta-\\tfrac{1}{\\varepsilon}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}+\\norm{v-u^{\\ast}}_{\\Psi}^{2}\\hspace{-.1cm}-\\hspace{-.1cm}\\norm{v-u}_{\\Psi}^{2}\\\\\n&+\\varepsilon\\norm{v-u}^{2}_{\\Psi}-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nSince, \n$\\norm{\\Psi^{-1}(\\mathcal Bv-\\mathcal Bu)}^{2}_{\\Psi}\\leq (L\/\\alpha)^{2}\\norm{v-u}^{2}_{\\Psi},$\nthe above reads as\n\\vspace{-.15cm}\\begin{equation*}\n\\begin{aligned}\n\\norm{T_{\\text{FBHF}}v-u^{\\ast}}_{\\Psi}^{2}\\leq& \\norm{v-u^{\\ast}}^{2}_{\\Psi}-L^{2}\\left(\\tfrac{1-\\varepsilon}{L^{2}}-\\tfrac{1}{\\alpha^{2}}\\right)\\norm{v-u}^{2}_{\\Psi}\\\\\n&-\\tfrac{1}{\\alpha\\varepsilon}\\left(2\\theta\\varepsilon-\\tfrac{1}{\\alpha}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\varepsilon\\norm{v-u-\\tfrac{1}{\\varepsilon}\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}^{2}_{\\Psi}.\n\\end{aligned}\n\\vspace{-.15cm}\\end{equation*}\nIn order to choose the largest interval for $1\/\\alpha$ ensuring that the second and third terms are negative, we set\n$\\chi\\leq\\min\\{2\\theta,1\/L\\}$.\nThen,\n\\vspace{-.15cm}\\begin{equation*}\\begin{aligned}\n\\norm{T_{\\text{FBHF}}v-u^{\\ast}}_{\\Psi}^{2}\\leq& \\norm{v-u^{\\ast}}^{2}_{\\Psi}-L^{2}\\left(\\chi^{2}-\\tfrac{1}{\\alpha^{2}}\\right)\\norm{v-u}^{2}_{\\Psi}\\\\\n&-\\tfrac{2\\theta}{\\alpha\\chi}\\left(\\chi-\\tfrac{1}{\\alpha}\\right)\\norm{\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast})}_{\\Psi}^{2}\\\\\n&-\\tfrac{\\chi}{2\\theta}\\norm{v-u-\\tfrac{2\\theta}{\\chi}(\\Psi^{-1}(\\mathcal Av-\\mathcal Au^{\\ast}))}^{2}_{\\Psi}.\n\\end{aligned}\\vspace{-.15cm}\\end{equation*}\nFrom here, we obtain convergence of the sequence $(v^{k})_{k\\geq 0}$ as a consequence of \\cite[Thm. 2.3]{briceno2018} for $1\/\\alpha\\in(0,\\chi)$.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe 2014 Nobel Prize in Chemistry was awarded for development of methods of super-resolution optical imaging, which, in particular, relied on single optical emitters imaging and localization \\cite{Betzig2006science,Dickson1997nature}. Despite a tremendous progress of nanoscopic fluorescence-based imaging, which has made possible through those pioneering work, identification of single emitters remains to be a challenge \\cite{Mortensen2010natmeth} and often relies on ultra-bright emission, which is not always affordable for biological systems, and some prior knowledge of the system, which is often not available. Thus, it would be highly desirable to be able to characterize individual emitters and quantify their presence in any given imaging volume. This can be generalized to a broader fundamental problem of counting the number of emitters in a sample from optically collected data, which has significant implications beyond the commonly used fluorescence imaging. For example,\nis it possible to determine the number of emitters contributing to a fluorescence or Raman signal, on the basis of imaging data alone?  Furthermore, if it is possible, what is the limit to which we may determine the number of emitters?  In particular we want to be able to determine the number of molecules, which might be in the range of 10, 100, or 1,000.  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{schematic.pdf}\n    \\caption{Schematic showing the concept to determine the  number of emitters $M$ in one sample with unknown detection probability $P$. A pulsed laser excites the sample and emitted photons are measured using a the photon-number-resolving detector, PNRD.  a histogram of the number of photons in each collected pulse in analysed using a Maximum Likelihood Estimator, which enables the determination of both $M$ and $P$.  Such discrimination is not possible using only classical intensity measurements.}\n    \\label{fig:schematic}\n\\end{figure}\n\nThe problem with determination of the number of emitters from intensity is fundamental.  For a classical fluorescence measurement, we observe an intensity, which we may express as $I = M I_0$, where $I$ is the total intensity, $M$ the number of emitters and $I_0$ is the intensity of each emitter, where we assume each emitter has the same emission intensity, for simplicity.  However, without \\emph{a priori} knowledge of $I_0$ or $M$, it is not possible to discriminate between cases where there are more dimmer emitters, or fewer brighter emitters.  Additional practical confouding issues include uncertainty in the probability of collecting light emitted from the emitters, which may vary due to experimental conditions.\n\nQuantum mechanically, however, there is the prospect for distinguishing different emitter configurations.  It is known that photon anti-bunching signals (Hanbury Brown and Twiss experiment) can be used to distinguish the number of emitters \\cite{Monticone2014prl}.  This technique is typically used when the number of emitters is few (e.g. to determine the difference between 1, 2 or 3 emitters) \\cite{Worboys2020pra, Davin2021arxiv, Chen2017nphoton, Schwartz2013nanoletter}. The Hanbury Brown and Twiss (HBT) experiment, in its simplest form, uses two SPDs that sample the same optical field of view via a beamsplitter \\cite{Stevens2013book}.  By observing the signal in coincidence, some information about the number of emitters can be obtained, which leads to the well-known result for the background-free, equal brightness HBT signal at coincidence:\n\\begin{align}\n    g^{(2)}(0) = 1 - \\frac{1}{M}.\n\\end{align}\nOne way to improve determination of the number of emitters is to increase the number of SPDs. This approach leads to considerable complexity due to the increased number of beamsplitters and coincidence electronics that is required \\cite{Steven2014oe}.  \n\nMeasurement of photon number has traditionally been a difficult task.  Originally, this task was performed using single photon detectors (SPD) such as photomultiplier tubes, and later avalanche photodiodes. Such devices permit a binary measurement of the number of photons: they measure either 0 photons, or more than 0 photons, but a single device typically does not allow for a  more sophisticated determination of the number of photons. Alternatively, new generations of photon number resolving (PNR) detectors are becoming available. PNR detectors have the ability to perform a direct projective measurement of the number of photons in a pulse of light. Compared with non-PNR detection, PNR detection can provide more information about noise and receiver imperfections \\cite{Becerra2015natphotonics}. Several techniques have been applied in realising photon number resolving \\cite{Provaznik2020oe, Thekkadath2020thesis}, including multiplexed APD \\cite{Kardynal2008nphoton}, CMOS image sensors \\cite{Ma2017optica}, superconductor nanowire \\cite{Cahall2017optica}, and superconducting transition-edge sensor (TES) \\cite{Schmidt2018LowTemPhys}. Additionally, there are  multipixel photon counters (MPPC) that have the ability to distinguish from one up to 10 photons \\cite{Kalashnikov2011oe}. Superconducting transition-edge sensors have recently been reported to resolve photon numbers up to 16 with the efficiency of over 90\\% \\cite{Morais2020arXiv}. A study has reported a 24-pixel PNR detector based on superconducting nanowires that achieves the detection of $n=0-24$ photons \\cite{Mattioli2016oe}. Given the increase in technology it is is expected that this upper limit will soon be exceeded, and the availability of such detectors will become more widespread.  It is therefore timely to see the effects that such detectors will have on the determination of the number of emitters in an unknown sample.  \n\nHere we show that photon number resolving measurements enable the determination of emitter number more generally. The schematic is shown in Fig.\\ref{fig:schematic}.  We theoretically determine the photon number probability distribution for $M$ emitters, with photon detection probability $p$.  On the basis of this, we show maximum likelihood estimation and the Cramer-Rao lower bound for the simultaneous determination of both the number of emitters and the probability of detection.  This analysis enables us to provide scaling laws for the number of experiments required to distinguish between different configurations.  \n\nThis paper is organised as follows: We first discuss the photon statistics from an ensemble of $M$ classically identical emitters (ie emitters with the same emission probability in the same field of view with the same emission properties such as polarisation and wavelength, although we stress that the emitters are assumed to be not quantum indistinguishable).  We then show the maximum likelihood determination of the number of emitters and photon detection probability for particular cases, as a function of the number of experiments.  Lastly we present the Cramer-Rao lower bound for the scaling.\n\n\n\\section{Photon number resolving detection probabilities}\n\nWe are concerned with the problem of simultaneously determining the number of emitters, and the collection probability for a number of emitters.  We consider an experimental configuration where $M$ (unknown) emitters are excited by a short pulse laser, and the fluorescence signal collected confocally.  Each emitter is assumed to emit no more than one photon per excitation pulse, and we assume that the probability of detecting a photon from each emitter in that pulse is $p$.  The photon resolving detector performs a projective measurement in the photon number basis, and we may write down the binomially distributed probability of detecting $N$ photons from the $M$ emitters as \n\n\\begin{align}\n\\mathcal{P}(N|M,p) = \\frac{M!}{\\left(M-N\\right)! N!} p^N \\left(1 - p\\right)^{M-N}. \\label{eq:P(N)}\n\\end{align}\n\n\nWe can explore Eq.~\\ref{eq:P(N)} in various limits, however to address the original problem, the clearest case to consider is where we have a known (measured) brightness, but where the actual number of emitters and their probability of emission is unknown.  Therefore,  we set $\\lambda = M p$, so that $\\lambda$ is the expected number of photons emitted per experiment, where each experiment is a Bernouilli trial.  Note that although $N$ is quantised, $\\lambda$ is not.  \n\nEq.~\\ref{eq:P(N)} in terms of $\\lambda$ becomes\n\\begin{align}\n    \\mathcal{P}(N|\\lambda,M) = \\frac{M!}{\\left(M-N\\right)! N!} \\left(\\frac{\\lambda}{M}\\right)^N \\left(1 - \\frac{\\lambda}{M}\\right)^{M-N}.\n\\end{align}\nThis result should be compared with the standard Poisson distribution, which is expected in the limit $M\\rightarrow \\infty$\n\\begin{align}\n    \\lim_{M\\rightarrow\\infty}\\mathcal{P}(N|M,p) = \\frac{\\lambda^N e^{-\\lambda}}{N!}.\n\\end{align}\nAnalytical results for this are shown in Fig.~\\ref{fig:poissonDistri}. Fig.~\\ref{fig:poissonDistri}(a) shows the probability of obtaining $N$ photons for different $M$.  As shown in Fig.~\\ref{fig:poissonDistri}(a), the greatest change in $\\mathcal{P}(N)$ occurs at $N\\approx \\lambda$, although it is clear that the \\emph{entire} distribution provides information about $M$.  Hence it is important that any photon resolving detector should be at least able to detect $\\lambda$ photons for maximum ability to determine $M$.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[trim=0cm 5cm 0cm 5cm, width=\\textwidth]{Poisson.pdf}\n\\caption{(a) Series of curves showing the distribution of probability distribution function for the number of photons, $N$, for different $M$ and $p$ such that $\\lambda = Mp = 20$. The dotted line shows the limit for a Poisson distribution.  As $M$ increases the peak broadens, and approaches the Poisson limit. By measuring the distribution, the number of emitters should be distinguishable, however it is important to stress that the differences are small, and noise will make sure determination difficult. (b) The peak of the probability distribution, at $N = \n\\lambda$, is the point that shows the largest dependence on number of emitters however as this curve shows, even variation of $M$ over three orders of magnitude only leads to a change in the probability of $N=20$ photon events from  $\\mathcal{P}(N = 20| M = 30, \\lambda = 20) = 15.3\\%$ to $\\mathcal{P}(N = 20| M = 10^5, \\lambda = 20) = 8.88\\%$}\n\\label{fig:poissonDistri}\n\\end{figure}\n\n\nTo explore the determination of both $M$ and $p$, we begin by generating synthetic data obtained by sampling Eq.~\\ref{eq:P(N)} for a finite number of numerical experiments.  This yields a histogram of events, such as that shown in Fig.~\\ref{fig:BarFig}.  This data was generated on the basis of $\\nu = 100$ experiments, with $M=40$ atoms and probability of detection $p=0.2$.  Also shown is the probability distribution function, $\\mathcal{P}(N)$ under the same circumstances. As the number of experiments increases, the synthetic data and probability distribution function should converge.  \n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{BarFig.pdf}\n    \\caption{Plot showing relative frequency data (bars) and expected (ideal) probability of detecting a number of photons given $M=40$ emitters, each with a detection probability of $p=0.2$.  The synthetic data was generated on the basis of 100 experiments and assumes no noise other than random fluctuations provided by Eq.~\\ref{eq:P(N)}.\n    }\n    \\label{fig:BarFig} \n\\end{figure}\n\nTo address the issue of how well we can determine the model parameters using the data, we turn to maximum likelihood estimation.  For a given set of data $\\bm{N} = \\left(N_1, N_2, ...N_{\\nu}\\right)$, where the $N_i$ correspond to the number of photons resolved in experiment $i$, which are independent and identically distributed with probability mass function $\\mathcal{P}(N_i|\\bm{\\theta})$, where $\\bm{\\theta} = \\left(M,p\\right)\\in\\mathbb{Z}^+\\bigtimes[0,1]$ is the vector of parameters to be estimated. Furthermore, let $\\bm{\\theta}_0$ be the ground truth and $L(\\bm{\\theta}|N_i)$ the associated likelihood function of $\\bm{\\theta}$ given data $N_i$. \n\nThen we can write the joint log likelihood function as follows:\n\\begin{align}\n\\ell\\left(\\bm{\\theta}\\big|\\mathbf{N}\\right) = \\log L\\left(\\bm{\\theta}\\big|\\mathbf{N}\\right)=\\sum_{i = 1}^{\\nu}\\log L\\left(\\bm{\\theta}\\big|{N}_i\\right),\n\\end{align}\nwhere $\\nu$ is the number of experiments.  \n\nAccordingly, the MLE of $\\bm{\\theta}$ is given by\n\\begin{align}\n\\hat{\\bm{\\theta}}=\\mathop{\\arg \\max}\\limits_{\\bm{\\theta}\\in\\mathbb{Z}^+\\bigtimes [0,1]} \\ell\\left(\\bm{\\theta}\\big|\\mathbf{N}\\right).\n\\end{align}\nAs $\\nu$ increases, from the consistency of MLE \\cite{rao1973linear}, we expect that the $\\hat{\\bm{\\theta}}$ approaches $\\bm{\\theta}_0$. \n\nIt is easier to determine sample brightness than the number of emitters.  This accords with our classical intuition, namely that on the basis of intensity-only measurements it should be \\emph{only} possible to determine the mean brightness, and \\emph{impossible} to determine the number of emitters (ie few bright emitters should be indistinguishable from many dim emitters).  It is therefore useful to transform our parameters from $\\bm{\\theta}=(M,p)$ to $\\bm{\\beta}=(\\lambda, \\xi)$ where $\\xi = M\/p$.  With this parameterisation, the probability distribution function becomes\n\\begin{align}\n\\mathcal{P}(N|\\bm{\\beta})=\\mathcal{P}(N|\\lambda,\\xi)=\\frac{(\\sqrt{\\lambda\\xi})!}{(\\sqrt{\\lambda\\xi}-N)!N!} \\left(\\frac{\\lambda}{\\xi}\\right)^{N}\\left[1-\\left(\\frac{\\lambda}{\\xi}\\right)\\right]^{\\sqrt{\\lambda\\xi}-N}\\label{pdf2}\n\\end{align}\n\n\n\n\\section{Uncertainty of the estimates: Cramer Rao lower bound}\n\nTo obtain the scaling laws for estimating $M$, and $p$, we now proceed to calculate the Cramer-Rao lower bound (CRLB). The Cramer-Rao Lower Bound (CRLB) gives a lower estimate for the variance of an unbiased estimator. The Fisher Information Matrix (FIM)\\cite{Nishiyama2019arxiv,Ly2017jmp} is required to calculate the CRLB.  To do this, we need to find the derivative of (\\ref{pdf2}) w.r.t $\\lambda$ and $\\xi$. However the likelihood function $L(\\bm{\\beta}|N)$ is not differentiable since $\\sqrt{\\lambda\\xi}\\in\\mathbb{Z}^+$. To implement the derivative we use the $x!=x\\Gamma(x)$ to transfer $(\\sqrt{\\lambda\\xi})!$ into a continuous function with respect to $\\lambda$ and $\\xi$. Additionally, we have $\\left[x\\Gamma(x)\\right]'=\\Gamma(x)+x\\Gamma(x)\\psi(x),$\nwhere $\\psi(\\cdot)$ is digamma function.\n\nLet $\\bar{L}(\\bm{\\beta}|N)$ be the approximated likelihood function (after replacing the factorial term associated to $\\lambda$ and $\\xi$ by the interpolation function), then \n\\begin{align}\n &\\frac{\\partial \\bar{L}(\\bm{\\beta}|N)}{\\partial\\lambda}\\notag\\\\ \n=&\\sqrt{\\frac{\\xi}{\\lambda  }}\\alpha_1\\left\\{\\xi  \\lambda ^2+N^2 \\sqrt{\\xi  \\lambda }-N \\left[\\left(\\sqrt{\\xi  \\lambda }-1+\\lambda \\right) \\sqrt{\\xi  \\lambda }+\\lambda  \\right]+\\left(\\lambda -\\sqrt{\\xi  \\lambda }\\right)\\alpha_2\\right\\}\\\\\n&\\frac{\\partial \\bar{L}(\\bm{\\beta}|N)}{\\partial\\xi}\\notag\\\\ \n=&\\lambda\\alpha_1\\left[-\\lambda  \\sqrt{\\xi  \\lambda }-N^2+N \\left(-\\sqrt{\\frac{\\lambda }{\\xi }}+\\sqrt{\\xi  \\lambda }+\\lambda +1\\right)+\\left(\\sqrt{\\frac{\\lambda }{\\xi }}-1\\right)\\alpha_2\\right]\n\\end{align}\nwhere\n\\begin{align}\n    \\alpha_1&=\\frac{\\Gamma \\left(\\sqrt{\\xi  \\lambda }\\right) \\left(\\frac{\\lambda }{\\xi }\\right)^{N\/2} \\left(1-\\sqrt{\\frac{\\lambda }{\\xi }}\\right)^{\\sqrt{\\xi  \\lambda }-N-1}}{2 N! \\sqrt{\\xi  \\lambda } \\left(\\sqrt{\\xi  \\lambda }-N\\right)^2 \\Gamma \\left(\\sqrt{\\xi  \\lambda }-N\\right)}\\\\\n    \\alpha_2&= \\left(\\xi  \\lambda -N \\sqrt{\\xi  \\lambda }\\right) \\left[\\log \\left(1-\\sqrt{\\frac{\\lambda }{\\xi }}\\right)+\\psi\\left(\\sqrt{\\xi  \\lambda }\\right)-\\psi \\left(\\sqrt{\\xi  \\lambda }-N\\right)\\right]\n\\end{align}\nThen the $(i,j)$-th element, $\\forall i,j=1,2$, in FIM, i.e. $\\mathbf{I}_N(\\bm{\\beta})_{i,j}$, is\n\\begin{align}\n\\mathbf{I}_N(\\bm{\\beta})_{1,1}&=\\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\lambda}\\right]^2\\frac{1}{f(\\lambda,\\xi|N)}\\right\\}\\label{I11_2}\\\\\n\\mathbf{I}_N(\\bm{\\beta})_{2,2}&=\\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\xi}\\right]^2\\frac{1}{f(\\lambda,\\xi|N)}\\right\\}\\label{I22_2}\\\\\n\\mathbf{I}_N(\\bm{\\beta})_{1,2}=\\mathbf{I}_N(\\bm{\\beta})_{2,1}&=\\sum_{N=0}^n\\left[\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\lambda}\\frac{ \\partial \\bar{f}(\\lambda,\\xi|N)}{ \\partial \\xi}\\frac{1}{f(\\lambda,\\xi|N)}\\right]\\label{I21_2}\n\\end{align}\n\n\nEquivalently, the FIM for $\\bm{\\theta}=(M,p)$ is\n\\begin{align}\n    \\mathbf{I}_N(\\bm{\\theta})_{1,1}\n    \\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial L(\\bm{\\theta}|N)}{ \\partial M}\\right]^2\\frac{1}{L(\\bm{\\theta}|N)}\\right\\},\\notag\n   \n    \\label{eq:I11}\n\\end{align}\nwhere $L(\\bm{\\theta}|N)$ is not differentiable again since it is discrete in $M$. We can find a approximated $f(M,p|N)$, i.e.  $\\bar{f}(M,p|N)$, using the similar method in obtaining $\\bar{f}(\\lambda,\\xi|N)$. Then we have\n\\begin{align}\n&\\frac{ \\partial \\bar{f}(M,p|N)}{ \\partial M}\\notag\\\\\n=&\\frac{\\Gamma (M) p^N (1-p)^{M-N} }{N! (M-N)^2 \\Gamma (M-N)}\\left\\{M (N-M) \\left[\\psi(M-N)-\\psi(M)-\\log (1-p)\\right]-N\\right\\}\n\\end{align}\n\n\nSimilarly, we have \n\\begin{align}\n\\mathbf{I}_N(\\bm{\\theta})_{2,2}=\\sum_{N=0}^n\\left\\{\\left[\\frac{ \\partial f(M,p|N)}{ \\partial p}\\right]^2\\frac{1}{f(M,p|N)}\\right\\},\\label{I22}\n\\end{align}\nand\n\\begin{align}\n\\mathbf{I}_N(\\bm{\\theta})_{2,1}&=\\mathbf{I}_N(\\bm{\\theta})_{1,2}\\notag\\\\\n&\\approx\\sum_{N=0}^n\\left\\{\\frac{ \\partial \\bar{f}(M,p|N)}{ \\partial M}\\frac{ \\partial f(M,p|N)}{ \\partial p}\\frac{1}{f(M,p|N)}\\right).\\label{I12}\n\\end{align}\nwhere \n\\begin{align}\n\\frac{ \\partial f(M,p|N)}{ \\partial p}=-\\frac{M! p^{N-1} (1-p)^{M-N-1} (M p-N)}{N! (M-N)!}\n\\end{align}\nThe CRLB is given by the inverse of the FIM,\n\\begin{align}\n\\mathbf{C} = \\mathbf{I}_N(\\bm{\\theta})^{-1}\\big|_{M=M_0,p=p_0}.\n\\end{align}\nGiven that there are $\\nu$ i.i.d. experiments,  so the underlying Cramer Rao lower bound  is\n\\begin{align}\n\\mathbf{C}_{\\nu} = \\frac{\\mathbf{C}}{\\nu} =  \\frac{1}{\\nu}\\mathbf{I}_N(\\bm{\\theta})^{-1}\\big|_{M=M_0,p=p_0}\\label{eq:CRLB}\n\\end{align}\n\nWe proceeded to compare our maximum likelihood simulations with the CRLB. The parameters $(M,p)$ are estimated using increasing number of experiments, i.e. $\\nu$. For each $\\nu$, we performed $500$ independent Monte Carlo simulations.  By performing an ensemble of numerical experiments, we could compare the estimated values in the $(M,p)$ space with a 2D confidence region, i.e. the CRLB 95\\% error ellipse (within which the probability that the random estimated value $\\bm{\\theta}=(M,p)$ will fall inside the ellipse is 95\\%).\nThe simulation results are shown in Fig.~\\ref{fig:CRLB} with ground truth $p_0=0.2$ and $M_0=40$. \n\nFig.~\\ref{fig:CRLB} shows a series of maximum likelihood determinations of the number of emitters and probability of detection per emitter, for ground truth $\\bm{\\theta_0} = (M,p) = (40,0.2)$.  We show the results in $\\left(\\lambda,\\xi\\right)$ and the data converted back into $\\left(M,p\\right)$ space, for increasing experiments $\\nu$.  Each point represents the maximum likelihood determination and the solid curve is the 95\\% confidence interval.  Observe that in $\\left(\\lambda,\\xi\\right)$ space we obtain a standard error ellipse in Fig.\\ref{fig:CRLB} (b), whereas in $\\left(M,p\\right)$ space in Fig.\\ref{fig:CRLB} (a), the ellipse is converted according to reciprocal functions: $p = \\sqrt{\\lambda\/\\xi}$ and $M = \\sqrt{\\lambda \\xi}$. The shape of the error region in $\\left(M,p\\right)$ space is a consequence of the classical ambiguity between more dim emitters and fewer brighter emitters.  Nevertheless, as can be seen, by applying quantum measurements, some bounding on the number of emitters can be obtained, with increasing certainty as the number of experiments increases.  For simplicity, we have not enforced the requirement that CRLBs of $\\left(M,p\\right)$ and $\\left(\\lambda,\\xi\\right)$ are positive, although these values are strictly positive in the simulation data, hence the maximum likelihood and CRLB values do not agree for small $\\nu$. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.8\\textwidth]{CRLB_MP_LX.pdf}\n    \\caption{ The Monte-Carlo simulation results (blue dots) using maximum likelihood and the Cramer-Rao lower bound (red) showing as 95\\% confidence interval ellipse in both (a) ($M,p$) space and (b) ($\\lambda,\\xi$) space. $p_0=0.2$ and $M_0=40$.}\n    \\label{fig:CRLB}\n\\end{figure}\n\n\nTo justify the performance of the maximum likelihood model, we compare the variances of predicted $M$ and $p$ results with CRLB. Fig.\\ref{fig:scalinglaw} presents two configurations, $\\bm{\\theta_0} = \\left(40,0.2\\right)$ and $\\bm{\\theta_0} = \\left(100,0.1\\right)$ with CRLB in $(\\lambda,\\xi)$ space and $(M,P)$ space. Both showing asymptotic trend to CRLB. A log-log scaling law is observed here, i.e the $\\log(\\text{Variance})$ scales with $-\\log(\\nu)$. Ideally the variance of estimated data cannot be lower than CRLB, but for small $\\nu$ the estimator is biased when there are too few measurement data, and CRLB only holds when the estimator is unbiased.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.8\\textwidth]{ScalingLaw.pdf}\n    \\caption{The scaling of Cramer-Rao lower bound (red) and the Monte-Carlo simulation (blue) using maximum likelihood. Ground truth (a) $p_0=0.2$, $M_0=40$ (b) $p_0=0.1$, $M_0=100$. (a)i-ii, (b)i-ii present variances in (M,P) space and (a)iii-iv, (b)iii-iv are variances in ($\\lambda,\\xi$) space. It can be observed that the sampled variance of $p$ is slightly smaller than the CRLB when the number of experiments is small. This is because the MLE is biased on those points due to the small size of data \\cite{WANG201579}.}\n    \\label{fig:scalinglaw}\n\\end{figure}\n\n\nFig.\\ref{fig:NexpMap} (a) shows the number of experiments that required to meet the CRLB criterion, here we define the CRLB criterion as the relative variance of $M$: $\\mathrm{Var}[M]\/M=1\\%$. Here the lower bound of variance of $M$ is the corresponding CRLB element in the matrix of Eq.~\\ref{eq:CRLB}. The contours show the resolved photon number with max likelihood $\\lambda=MP$. In the top half of the map where the probability $p$ roughly larger than 0.5, the number of experiments to achieve CRLB is relatively small ($<10^6$), even for large emitter numbers $10^3$. In the bottom half of the map where $p<0.5$, with the decrease of the probability $p$ the number of experiments to achieve CRLB increases dramatically. With such low brightness or detected probability when it reaches to large emitter numbers $M$, measurements that required to determine $M$ is several orders higher than high brightness scenario.\n \n Along one contour with a fixed $\\lambda$, $\\nu$ increases with the increase of $M$ and decrease of $p$, which means given a detected photon number distribution with the peak occurrence locating at $\\lambda$ (similar to Fig.\\ref{fig:BarFig}), more measurements are required to resolve many low brightness emitters than few high brighter emitters. Fig.\\ref{fig:NexpMap} (b) shows the relationships of $\\nu$ and $M$ along one contour with fixed $\\lambda$, from $\\lambda=5$ to 50. The curves show an elbow shape when approaching large amount of emitters. To the left of the elbow shape, measurements $\\nu$ increases dramatically with emitter numbers. To the right of elbow shape, e.g.$M=200$, the small $\\lambda$ curves stay on top of the large $\\lambda$ ones. This is because the small $\\lambda$ indicates a small probability of detecting photons from each emitter, and results in more measurements being required to determine the number of emitters.\n \nWe now consider the example of quantitative fluoresence. If we consider a sample of 1,000 fluorophores, in a field of view with probability of photon collection of $1\\%$ from each emitter, then the number of measurements required to achieve a determination of the number of emitters with a relative variance of $1\\%$ is around $1.96 \\times 10^9$.  Photon number resolving measurements can be performed at of order microsecond timescales \\cite{Morais2020arXiv}.  This means that the length of time required to achieve to determine the number of identical (but unknown) fluorescent emitters is if order $\\sim$30min.  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[trim=0cm 5cm 0cm 5cm, width=\\textwidth]{NuMap.pdf}\n    \\caption{(a) Number of experiments to achieve CRLB with the relative variance of M: {$\\mathrm{Var}[M]\/M=1\\%$}. Here the lower bound of variance of M is the CRLB(M) element in the matrix of Eq.\\ref{eq:CRLB}. The contours show the $\\lambda=MP$. (b) The relationships of $\\nu$ and $M$ along contours with fixed $\\lambda$ value, from $\\lambda=5$ to 50, generated by extracting the $\\nu$ values along contours in (a). }\n    \\label{fig:NexpMap}\n\\end{figure}\n\n\n\\section{Conclusions}\nWe have shown that photon number resolving measurements can help to identify the number of emitters in a field of view, even without \\emph{a priori} knowledge of the brightness of the emitters.    Our results enable the prediction of the number of experiments required for a particular variance to be achieved.  As the number of emitters increases, the photon distribution approaches Poissonian, and in this limit, resolution of the number of emitters becomes increasingly difficult.\n\nOur results show the idealised case, of equal brightness emitters.  Naturally, variations in the emission probability (for example by particles locate in different parts of the optical point spread function, or with different local environments) will lead to increased number of experiments.  Nevertheless, our analysis is likely to guide future experiments in quantitative tests of biological pathways.  Practical systems will also have to contend with variations in photo-bleaching of emitters that may limit the practically achievable number of experiments.  As such, our results  provide an opportunity to bound the expected sample variance, and hence to give limits on the number of emitters that might be contributing to a signal - bounds that are not possible to impose given the current limits of classical fluorescence based imaging.  \n\n\n\\section*{Acknowledgements}\nThis work was funded by the Air Force Office of Scientific Research (FA9550-20-1-0276).  ADG also acknowledges funding from the Australian Research Council (CE140100003 and FT160100357).\nVVY acknowledges partial support from the National Science Foundation (NSF) (DBI-1455671, ECCS-1509268, CMMI-1826078), the Air Force Office of Scientific Research (AFOSR) (FA9550-15-1-0517, FA9550-20-1-0366, FA9550-20-1-0367), Army Medical Research Grant (W81XWH2010777), the National Institutes of Health (NIH) (1R01GM127696-01, 1R21GM142107-01), the Cancer Prevention and Research Institute of Texas (CPRIT) (RP180588).\n\n\n\\section*{Reference}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nMotivated by  the desire for greater efficiency in drug development and the low success rates in confirmatory (Phase 3) studies, methodological research on adaptive designs and interest in their application has grown tremendously over the last 30 years. In an adaptive design, accumulating data can be used to modify the course of the trial. Several possible adaptations can be considered in interim analyses, for example, adaptive randomization for dose finding, dropping and\/or adding treatment arms, sample size re-estimation, and early stopping for safety, futility or efficacy, to name a few.\n\nValidity and integrity are two major considerations in adaptive designs (Dragalin, 2006). Because data from one stage of the trial can inform the design of future stages of the trial, careful steps need to be taken to maintain the validity of the trial, i.e., control of the Type I error probability and minimization of bias. To maintain trial integrity, it is important that all adaptations be pre-planned, prior to the unblinded examination of data, and that all trial personnel other than those responsible for making the adaptations are blind to the results of any interim analysis (Food and Drug Administration, 2019). It is also important to ensure consistency in trial conduct among the different stages.\n\nA general method for hypothesis testing in experiments with adaptive interim analyses based on combining stage-wise $p$-values was proposed by Bauer and K\\\"{o}hne (1994). The basic idea behind the construction of a combination test in a two-stage adaptive design is to transform the stage-wise test statistics to $p$-values, with independence of the $p$-values following from the conditional invariance principle (Brannath {\\it{et al}}., 2007, 2012; Wassmer and Brannath, 2016), regardless of the adaptation performed after the first stage. The principle holds as long as the null distribution of the first-stage $p$-value ($p_1$) as well as the conditional distribution of the second-stage $p$-value ($p_2$) given $p_1$ are stochastically larger than the $U(0,1)$ distribution (the so-called ``p-clud\" property). A specified combination function is used to combine the $p$-values obtained before and after the preplanned adaptation of the design into a single global test statistic. An extension of combination tests to allow more flexibility regarding the number of stages and the choice of decision boundaries was provided by Brannath {\\it{et al}}. (2002).\n\nIn dose-response studies, a component of the MCP-Mod procedure (Bretz {\\it{et al}}., 2005) has gained popularity for the purpose of detecting a proof-of-concept (PoC) signal in learning-phase trials. The procedure consists of specifying a set of candidate dose-response models, determining the optimal contrast statistic for each candidate model, and using the maximum contrast as the overall test statistic. Other authors have considered extensions of this procedure to adaptive dose-response designs. Miller (2010) investigated a two-stage adaptive dose-response design for PoC testing incorporating adaptation of the dosages, and possibly the contrast vectors. He developed an adaptive multiple contrast test (AMCT) that combines the multiple contrast test statistics across two stages under the assumption that the variance is known. Franchetti {\\it{et al}}. (2013) extended the MCP-Mod procedure to a two-stage dose-response design with a pre-specified rule of adding and\/or dropping dosage groups in Stage 2 based on the Stage 1 results. The PoC test uses Fisher's (1932) combination method to combine the two stage-wise $p$-values, each obtained by applying the MCP-Mod procedure to the data from each stage. This method includes a restrictive requirement of equal total sample sizes for each stage. Also, the authors claimed that the independence of the two stage-wise $p$-values is potentially compromised if the number of dosages used in Stage 2 is not the same as that used in Stage 1 and proposed a method for assigning weights to the different dosage groups to deal with this problem. We do not believe that such weighting is necessary as long as the statistic used to combine the stage-wise $p$-values (Fisher's, in this case) does not include weights that depend on the Stage 1 data.\n\nEarly work related to adaptive designs for dose-response testing includes a general procedure with multi-stage designs proposed by Bauer and R\\\"{o}hmel (1995), in which dosage adaptations were performed at interim analyses. Other goals of adaptive dose-response studies include determining if any dosage yields a clinically relevant benefit, estimating the dose-response curve, and selecting a target dosage for further study (Dragalin {\\it{et al}}., 2010). Several model-based adaptive dose-ranging designs that utilize principles of optimal experimental design to address these objectives were studied by Dragalin {\\it{et al}}. (2010). Bornkamp {\\it{et al}}. (2011) proposed a response-adaptive dose-finding design under model uncertainty, which uses a Bayesian approach to update the parameters of the candidate dose-response models and model probabilities at each interim analysis.\n\nIn this article, we propose new methods to address the specific objective of detecting a PoC signal in adaptive dose-response studies with normally-distributed outcomes. We extend the MCP-Mod procedure to include generalized multiple contrast tests (GMCTs; Ma and McDermott, 2020) and apply them to adaptive designs; we refer to these as adaptive generalized multiple contrast tests (AGMCTs). These tests are introduced in Section \\ref{Adaptive Generalized Multiple Contrast Tests}. In Section \\ref{Adaptive Multiple Contrast Test} we extend the AMCT of Miller (2010) to accommodate more flexible adaptations and to the important case where the variance is unknown using the conditional rejection probability (CRP) principle (M\\\"{u}ller and Sch\\\"{a}fer, 2001, 2004). Numerical examples are provided in Section \\ref{Numerical Example} to illustrate the application of the AGMCTs and AMCT. In Section \\ref{Simulation studies}, we conduct simulation studies to evaluate the operating characteristics of the various methods as well as the corresponding tests for non-adaptive designs. The conclusions are given in Section \\ref{Conclusion}.\n\n\n\\section{Adaptive Generalized Multiple Contrast Tests}\\label{Adaptive Generalized Multiple Contrast Tests}\n\nIn this section, we propose a two-stage adaptive design in which we use data from Stage 1 to get a better sense of the true dose-response model and make adaptations to the design for Stage 2. We then use data from both Stage 1 and Stage 2 to perform an overall test to detect the PoC signal. The rationale is to overcome the problem of potential model misspecification at the design stage.\n\n\\subsection{General Procedure}\nWe consider the case of a normally distributed outcome variable. Suppose that there are $n_{i1}$ subjects in dosage group $i$ in Stage 1, $i=1,\\ldots,k_1$. Denote the first stage data as $\\pmb{Y}_1=(Y_{111},\\ldots,Y_{1 n_{11} 1},\\ldots, $ $Y_{k_1 1 1},\\ldots,Y_{k_1 n_{k_1 1}1})^\\prime.$\nThe statistical model is\n$$Y_{ij1}=\\mu_i +\\epsilon_{ij1},\\quad\\epsilon_{ij1}\\stackrel{iid}{\\sim} N(0, \\sigma^2),\\quad i=1,\\ldots, k_1,\\ j=1,\\ldots, n_{i1}.$$\nThe true mean configuration is postulated to follow some dose-response model $\\mu_i=f(d_i,\\pmb{\\theta})$, where $d_i$ is the dosage in the $i^{\\text{th}}$ group, $i=1,\\ldots,k_1$. The dose-response model is restricted to be of the form $f(\\cdot;\\pmb{\\theta})=\\theta_0+\\theta_1 f^0(\\cdot;\\pmb{\\theta}^0)$, where $f^0(\\cdot;\\pmb{\\theta}^0)$ is a standardized dose-response model indexed by a parameter vector $\\pmb{\\theta}^0$ (Thomas, 2017). A candidate set of $M$ dose-response models $f_m(\\cdot,\\pmb{\\theta})$, $m=1,\\ldots,M$, including values for $\\pmb{\\theta}$, is pre-specified. For each candidate model, an optimal contrast is determined to maximize the power to detect differences among the mean responses; the contrast coefficients are chosen to be perfectly correlated with the mean responses if that model is correct (Bretz {\\it{et al}}., 2005; Pinheiro {\\it{et al}}., 2014).\n\nFor each candidate model, the following hypothesis is tested:\n$$H_{0m 1}: \\sum_{i=1}^{k_1} c_{mi1} \\mu_i=0,\\quad\\text{vs.}\\quad H_{1m 1}: \\sum_{i=1}^{k_1} c_{mi1} \\mu_i>0,\\quad m=1,\\ldots, M,$$\nwhere $c_{m11},\\ldots, c_{mk_11}$ are the optimal contrast coefficients associated with the $m^{\\text{th}}$ candidate model in Stage 1. The multiple contrast test statistics are\n$$T_{m1}=\\sum_{i=1}^{k_1} c_{mi1} \\bar{Y}_{i1}\\Bigg\/\\left(S_1\\sqrt{\\sum_{i=1}^{k_1}\\frac{c_{mi1}^2}{n_{i1}}}\\right),\\quad m=1,\\ldots, M,$$\nwhere $\\bar{Y}_{i1}=\\sum_{j=1}^{n_{i1}} Y_{ij1}\/n_{i1}$ and the pooled variance estimator is $S_1^2=\\sum_{i=1}^{k_1}\\sum_{j=1}^{n_{i1}} (Y_{ij1}-\\bar{Y}_{i1})^2\/\\nu_1$, where $\\nu_1=\\sum_{i=1}^{k_1} n_{i1}-k_1$. The joint null distribution of $(T_{11},\\ldots,T_{M1})^\\prime$ is multivariate $t$ (with $\\nu_1$ degrees of freedom) with common denominator and correlation matrix having elements\n$$\\rho_{m m^{\\prime}1}=\\sum_{i=1}^{k_1}\\frac{c_{mi1} c_{m^{\\prime} i1}}{n_{i1}}\\Bigg\/\\sqrt{\\sum_{i=1}^{k_1}\\frac{c_{m i1}^2}{n_{i1}}\\sum_{i=1}^{k_1}\\frac{c_{m^{\\prime} i1}^2}{n_{i1}}}, \\quad m, m^{\\prime}=1,\\ldots, M.$$\n\nLet $p_{m1}=1-\\mathcal{T}_{\\nu_1} (T_{m1})$ be the $p$-values derived from $T_{m1}$, $m=1, \\ldots, M$, where $\\mathcal{T}_{\\nu_1} (\\cdot)$ is the cumulative distribution function of the $t$ distribution with $\\nu_1$ degrees of freedom. We consider three combination statistics to combine the $M$ dependent one-sided $p$-values in Stage 1 (Ma and McDermott, 2020):\n\\begin{enumerate}[(i)]\n\\item Tippett's (1931) combination statistic,\n$$\\Psi_{T1}=\\min_{1 \\leq m \\leq M} \\ p_{m1};$$\n\\item Fisher's (1932) combination statistic,\n$$\\Psi_{F1}=-2 \\sum_{m=1}^M \\log (p_{m1});$$\n\\item Inverse normal combination statistic (Stouffer, 1949),\n$$\\Psi_{N1}=\\sum_{m=1}^M \\Phi^{-1} (1-p_{m1}).$$\n\\end{enumerate}\n\n\nNote that the use of Tippett's combination statistic is equivalent to the original MCP-Mod procedure; the use of different combination statistics results in a generalization of the MCP-Mod procedure, yielding GMCTs (Ma and McDermott, 2020). When the $p$-values are independent, these statistics have simple null distributions. In our case the $p$-values are dependent, but the correlations among $T_{11},\\ldots,T_{M1}$ are known. For Tippett's combination method, one can obtain multiplicity-adjusted $p$-values from $T_{m1}$, $m=1, \\ldots, M$, given the correlation structure using the {\\tt{mvtnorm}} package in {\\tt{R}}. A PoC signal is established in Stage 1 if the minimum adjusted $p$-value $p_{\\text{min, adj}1}< \\alpha$ (Bretz {\\it{et al}}., 2005). For Fisher's and the inverse normal combination methods, excellent approximations to the null distributions of $\\Psi_{F1}$ and $\\Psi_{N1}$ have been developed (Kost and McDermott, 2002), enabling computation of the overall $p$-value $p_1$ for Stage 1 using a GMCT (Ma and McDermott, 2020).\n\nAfter obtaining the Stage 1 data, we make design adaptations and determine the optimal contrasts for the updated models in Stage 2 (see Sections \\ref{Adapting the Candidate Dose-Response Models} and \\ref{Adapting the Dosage Groups} below). We then conduct a GMCT in Stage 2 and obtain the second-stage $p$-value $p_2$. Under the overall null hypothesis $H_0: \\mu_1= \\cdots =\\mu_{k^*}$, where $k^*$ is the total number of unique dosage groups in Stages 1 and 2 combined, the independence of the stage-wise $p$-values $p_1$ and $p_2$ can be established using the conditional invariance principle (Brannath {\\it{et al}}., 2007). To perform the overall PoC test in the two-stage adaptive design, we combine $p_1$ and $p_2$ using one of the above combination statistics.\n\nA procedure that ignores the adaptation, i.e., that simply pools the data from Stage 1 and Stage 2 and applies a GMCT to the pooled data as if no adaptation had been performed, would substantially increase the Type I error probability.\n\n\n\\subsection{Adapting the Candidate Dose-Response Models}\\label{Adapting the Candidate Dose-Response Models}\nHere and in Section \\ref{Adapting the Dosage Groups} below, we consider adaptations for the second stage that are arguably most relevant for PoC testing, namely those of the candidate dose-response models and the dosages to be studied. The choice of the candidate dose-response models and dosages for Stage 1 would depend on prior knowledge from pre-clinical or early-stage clinical experience with the investigative agent. If there is great uncertainty concerning the nature of the dose-response relationship, it would seem sensible to select a more diverse set of candidate dose-response models with pre-specified parameters when the trial begins.\n\nAfter collecting the Stage 1 data, these data can be used to estimate $\\pmb{\\theta}$ for each of the $M$ candidate dose-response models and adapt each of the models by substituting $\\pmb{\\hat{\\theta}}$ for the original specification (guess) of $\\pmb{\\theta}$. The optimal contrast vectors can be constructed for each of the updated models $f_m(\\cdot,\\pmb{\\hat{\\theta}})$, $m=1,\\ldots,M$, for use in Stage 2.\n\n\n\nA potential problem occurs when the true dose-response model differs markedly from some of the specified candidate models and if those candidate models are nonlinear models with several unknown parameters. In such cases there can be a failure to fit the models using the Stage 1 data. To handle this problem, one can consider fall-back approaches to determine the corresponding contrasts to be used in Stage 2. These include using isotonic regression (Robertson {\\it{et al}}., 1988), imposing reasonable bounds on the nonlinear parameters during model-fitting (as is done in the {\\tt{R}}-package {\\tt{DoseFinding}} to ensure the existence of the maximum likelihood estimates), and retaining the Stage 1 contrast for use in Stage 2. Different strategies can be used for different models in cases where more than one model cannot be fit using the Stage 1 data.\n\nSpecifically, consider the following 5 candidate dose-response models:\n\\begin{itemize}\n  \\item[] $E_{\\max}$ model: $f_1(d,\\pmb{\\theta})=E_0+E_{\\max} d\/ (ED_{50}+d)$\n  \\item[] Linear-log model: $f_2 (d,\\pmb{\\theta})=\\theta_0+\\theta_1\\log(5d+1)$\n  \\item[] Linear model: $f_3 (d,\\pmb{\\theta})=\\theta_0+\\theta_1 d$\n  \\item[] Quadratic model: $f_4 (d,\\pmb{\\theta})=\\theta_0+\\theta_1 d + \\theta_2 d^2$\n  \\item[] Logistic model: $f_5 (d,\\pmb{\\theta})=E_0+E_{\\max} \/[1 + \\exp\\{(ED_{50}-d)\/ \\delta\\}]$\n\\end{itemize}\nAmong these 5 candidate models, the $E_{\\max}$ and Logistic models are the ones that may fail to converge since the others can be expressed as linear models in $d$ (or a simple function of $d$). A possible fall-back strategy could be as follows: if only one of the $E_{\\max}$ and Logistic models fails to converge in Stage 1, isotonic regression is used to generate the corresponding contrast for use in Stage 2; if both the $E_{\\max}$ and Logistic models fail to converge in Stage 1, then isotonic regression is used to generate the corresponding contrast for the Logistic model and the same contrast that was used in Stage 1 is used in Stage 2 for the $E_{\\max}$ model (see Section \\ref{Numerical Example, AGMCT} for a numerical example).\n\nAnother potential concern arises if the data from Stage 1 suggest that there is a negative dose-response relationship, i.e., that higher dosages are associated with worse outcomes. In this case, the adapted contrast associated with the linear model, say, in Stage 2 would be the negative of that used in Stage 1. If a similar dose-response pattern is observed in Stage 2, then the contrast associated with the linear model would incorrectly indicate (possibly strong) evidence against the null hypothesis. One way to avoid this problem would be to not adapt the dose-response models in such a case, but instead to consider adapting the dosage groups by retaining only dosages, if any, that appear to be associated with increasing sample means (see Section \\ref{Adapting the Dosage Groups} below).\n\nIdeally, of course, it would be required to pre-specify the measures that would be taken to deal with the problems noted above (non-convergence of non-linear models, negative dose-response relationship) prior to examination of the data.\n\nOne could also consider different numbers of candidate models (or contrast vectors) in Stage 1 and Stage 2. One non-model-based option, for example, would be to use a single contrast in Stage 2 based on the sample means of the dosage groups from Stage 1. We found that this strategy, while intuitively appealing, yielded tests with reduced power, likely due to the reliance on a single contrast combined with the uncertainty associated with estimation of the means of each dosage group in Stage 1. One could also consider a small number of other contrasts based on values that are within the bounds of uncertainty reflected in the sample means, though how to choose these contrasts is somewhat arbitrary.\n\n\n\\subsection{Adapting the Dosage Groups}\\label{Adapting the Dosage Groups}\n\nAdaptation of the dosage groups in Stage 2, including the number of dosage groups, could also be considered. One would have to establish principles for adding and\/or dropping dosages; for example, dropping active dosages that appear to be less efficacious than placebo or that appear to be less efficacious than other active dosages, or adding a dosage (within a safe range) when there appears to be no indication of a dose-response relationship in Stage 1. Relevant discussion of these issues can be found in Bauer and R\\\"{o}hmel (1995), Miller (2010), and Franchetti {\\it{et al}}. (2013).\n\nTo illustrate this type of adaptation, we create an example dosage adaptation rule to drop the active dosage groups that appear to be less efficacious than placebo and the adjacent group. Suppose that there are $k_1$ dosage groups in Stage 1 and denote the dosage vector in Stage 1 as $\\pmb{d}_{\\text{Stage1}}=(d_{11},\\ldots, d_{k_11})^\\prime$, where $d_{11}=0$ (placebo group). We will select $k_2$ dosage groups from the $k_1$ Stage 1 dosage groups, $k_2\\leq k_1$. Denote the dosage vector in Stage 2 as $\\pmb{d}_{\\text{Stage2}}=(d_{12}, \\ldots, d_{k_22})^\\prime$, where $d_{12}=0$ (placebo group). The example dosage adaptation rule is as follows:\n\\begin{itemize}\n  \\item[] \\textbf{Step 1}: Always select the placebo group to be included in Stage 2, i.e., $d_{12}=d_{11}=0$.\n\n  \\item[] \\textbf{Step 2}: Consider the difference in the means between each active dosage group and the placebo group in Stage 1.\n\nDenote $\\hat{\\Delta}_{21}=\\bar{Y}_{21}-\\bar{Y}_{11},\\ldots,\\hat{\\Delta}_{k_11}=\\bar{Y}_{k_11}-\\bar{Y}_{11}$. If there exists dosage group(s) $i$, $i=2,\\ldots, k_1$, such that $\\hat{\\Delta}_{i 1}<-\\delta$, where $\\delta\\ge 0$, then we remove dosage(s) $d_{i 1}$ from consideration; however, if $\\hat{\\Delta}_{i 1}<-\\delta$ for all $i=2,\\ldots, k_1$, then we stop the trial at the interim analysis and fail to reject $H_0$.\n\n  \\item[] \\textbf{Step 3}: Consider the differences in the means between two adjacent dosage groups among the remaining dosage groups, ordered from smallest to largest.\n\nAfter Steps 1 and 2, we have selected $d_{11}$ (placebo) into Stage 2 and have several remaining dosage groups $d_{\\tilde{2}1},\\ldots,d_{\\tilde{k}1}$, where $\\tilde{k}\\leq k_1$.\n\nWe first examine the difference in the means between dosages $d_{11}$ and $d_{\\tilde{2}1}$. If $\\hat{\\Delta}_{\\tilde{2}1}=\\bar{Y}_{\\tilde{2}1}-\\bar{Y}_{11} > -\\delta$, then $d_{\\tilde{2}1}$ is selected to be included in Stage 2, i.e., $d_{22}=d_{\\tilde{2}1}$; otherwise, $d_{\\tilde{2}1}$ is discarded and we proceed to the next possible dosage $d_{\\tilde{3}1}$.\n\nIf $d_{\\tilde{2}1}$ is selected to be included in Stage 2, then we proceed to compare the means between dosages $d_{\\tilde{2}1}$ and $d_{\\tilde{3}1}$. If $\\hat{\\Delta}_{\\tilde{3} \\tilde{2}}=\\bar{Y}_{\\tilde{3}1}-\\bar{Y}_{\\tilde{2}1}> -\\delta$, then $d_{\\tilde{3}1}$ is selected to be included in Stage 2, i.e., $d_{32}=d_{\\tilde{3}1}$; otherwise, $d_{\\tilde{3}1}$ is discarded. However, if $d_{\\tilde{2}1}$ is discarded, then the means should be compared between dosages $d_{11}$ and $d_{\\tilde{3}1}$, since these are now adjacent dosages among those remaining.\n\nThis procedure is repeated until the last possible dosage $d_{\\tilde{k}1}$ is reached and its associated mean is compared with that of the remaining adjacent dosage. This results in a final number $k_2 \\leq \\tilde{k}$ of dosage groups selected to be included in Stage 2, i.e., $\\pmb{d}_{\\text{Stage2}}=(d_{12}, \\ldots, d_{k_22})^\\prime$.\n\\end{itemize}\n\nHere we consider the threshold of adaptive dosing $\\delta=0$, which simply considers the difference between two sample means and retains the dosage with the larger sample mean. This threshold might be strict since it does not consider the variability of the difference between two sample means. An alternative threshold could be $\\delta=\\sqrt{\\text{var}(\\bar{Y}_{i1}-\\bar{Y}_{i^\\prime 1})}$, $i,i^\\prime=1, \\ldots, k_1$, which retains a dosage with a mean that is no more than one standard error lower than the mean of the adjacent dosage (or placebo). Users are free to choose their own threshold $\\delta$ based on considerations specific to their problem.\n\n\n\n\n\nWe emphasize that this is just one possible rule to adapt the dosage groups for Stage 2, and this rule only considers dropping dosages at the end of Stage 1. One could consider different adaptation rules that allow adding and\/or dropping dosages at the end of Stage 1, i.e., $k_2$ does not need to be less than or equal to $k_1$, and some of the dosage groups selected in Stage 2 may differ from those included in Stage 1. Also, as in Miller (2010), such a rule is based on heuristic considerations and is relatively easy to communicate to non-statisticians. Mercier {\\it{et al}}. (2015) provide an approach to selecting dosages for Stage 2 based on the hypothetical dose-response shape (out of several pre-specified) that correlates highest with the data observed in Stage 1.\n\nOne can adapt both the candidate dose-response models and the dosage groups in Stage 2. The optimal contrast vectors for Stage 2 would then be determined by the updated candidate dose-response models with parameters $\\pmb{\\hat{\\theta}}$ and the adapted dosages $\\pmb{d}_{\\text{Stage2}}$. The overall p-value for Stage 2, $p_2$, would be obtained from a GMCT that uses the updated optimal contrast vectors. We incorporate this strategy in our simulation studies below. It should be noted that if one adapts only the candidate dose-response models and not the dosage groups, the contrasts for the Linear and Linear-log models would not change based on the Stage 1 data.  This would not be the case if one also adapted the dosage groups.\n\n\\section{Adaptive Multiple Contrast Test}\\label{Adaptive Multiple Contrast Test}\n\n\\subsection{Known Variance Case}\\label{AMCT, Known Variance Case}\n\nInstead of combining the stage-wise $p$-values $p_1$ and $p_2$, each based on a GMCT, Miller (2010) suggested combining the test statistics for each candidate dose-response model across the two stages, and then derving an overall $p$-value from a multiple contrast test applied to those statistics, assuming a known variance $\\sigma^2$. For each candidate model, we have\n$$Z_m=\\left(\\sum_{i=1}^{k_1} c_{mi1} \\bar{Y}_{i1}+\\sum_{i=1}^{k_2} c_{mi2} \\bar{Y}_{i2}\\right)\\Bigg\/\\sigma\\sqrt{\\sum_{i=1}^{k_1} \\frac{c^2_{mi1}}{n_{i1}}+\\sum_{i=1}^{k_2}\\frac{c^2_{mi2}}{n_{i2}}},\\quad m= 1,\\ldots,M.$$\nSince $k_2$, $c_{mi2}$, and $n_{i2}$ can depend on the interim data (adaptation), the null distribution of $Z_m$ is not standard normal in general.\n\nIn order to control the Type I error probability of the overall test, Miller (2010) applies a conditional error approach based on the conditional rejection probability (CRP) principle (M\\\"{u}ller and Sch\\\"{a}fer, 2001, 2004). Computation of the conditional Type I error probability requires pre-specification of what Miller (2010) calls a ``base test\", i.e., pre-specified values for the contrast coefficients ($c^*_{mi2}$), number of dosage groups ($k_2^*$), and group sample sizes ($n^*_{i2}$) in Stage 2, $i=1, \\ldots, k^*_2$, $m=1,\\ldots, M$. There is not a clear best strategy for choosing these pre-specified values. Miller (2010) considers an example where all possible Stage 2 designs can be enumerated and have $k_1=k_2$ and $n_{i1}=n_{i2}$, $i=1,\\ldots,k_1$, and the pre-specified values involving $c_{mi2}^*$, $i=1,\\ldots,k_2$, $m=1,\\ldots,M$, are averaged over the possible Stage 2 designs. More generally one cannot enumerate all possible Stage 2 designs, so in the development below we pre-specify $c^*_{mi2}=c_{mi1}$, $k^*_2=k_1$, and $n^*_{i2}=n_{i1}$, $i=1,\\ldots,k_2$, $m=1,\\ldots,M$. Since the dosages can also be adapted, we suggest pre-specifying $\\pmb{d}^*_{\\text{Stage2}}=\\pmb{d}_{\\text{Stage1}}=(d_{11},\\ldots,d_{k_1 1})^\\prime$. One can think of this ``base test\" as one that is based on a study that uses the same design in Stage 2 as was used in Stage 1.\n\nThe $Z$-statistics for the base test are\n$$Z^*_m=\\sum_{i=1}^{k_1} c_{mi1}\\left(\\bar{Y}_{i1}+\\bar{Y}_{i2}\\right)\\Bigg\/\\sigma\\sqrt{2\\sum_{i=1}^{k_1}\\frac{c_{mi1}^2 }{n_{i1}}},\\quad m=1,\\ldots,M.$$\nUnder $H_0$, the joint distribution of $\\pmb{Z}^*=(Z^*_1,\\ldots,Z^*_M)^\\prime$ is multivariate normal with mean $\\pmb{0}$ and covariance matrix $\\pmb{R}^*=(\\rho_{m m^{\\prime}1})$, $m, m^{\\prime}=1,\\ldots, M$. One can then obtain the non-adaptive $\\alpha$-level critical value $u^*_{1-\\alpha}$ based on the null distribution of $Z_{\\max}^*=\\max \\{\\pmb{Z}^*\\}$ using the {\\tt{R}}-package {\\tt{mvtnorm}}.\n\nIn order to obtain the conditional Type I error probability $A=P_{H_0}(Z^*_{\\max}\\ge u^*_{1-\\alpha}\\,|\\,\\pmb{Y}_1)$, where $\\pmb{Y}_1$ are the Stage 1 data, it can be seen that the conditional distribution of $\\pmb{Z}^*$ given $\\pmb{Y}_1=\\pmb{y}_1$ is multivariate normal with mean vector\n$$\\left(\\sum_{i=1}^{k_1} c_{1i1}\\bar{y}_{i1}\\Bigg\/\\sigma\\sqrt{2\\sum_{i=1}^{k_1}\\frac{c_{1i1}^2}{n_{i1}}},\\ldots,\\sum_{i=1}^{k_1} c_{Mi1}\\bar{y}_{i1}\\Bigg\/\\sigma\\sqrt{2\\sum_{i=1}^{k_1}\\frac{c_{Mi1}^2 }{n_{i1}}}\\right)^\\prime$$\nand covariance matrix $\\pmb{R_2}^*=\\pmb{R}^*\/2$, where $\\bar{y}_{i1}=\\sum_{j=1}^{n_{i1}} y_{ij1}\/n_{i1}$, $i=1,\\ldots,k_1$. Hence, the conditional Type I error probability is\n$$A=P_{H_0} (Z^*_{\\max}\\geq u^*_{1-\\alpha}\\,|\\,\\pmb{Y}_1)=1- P_{H_0} (\\pmb{Z}^* \\leq (u^*_{1-\\alpha},\\ldots,u^*_{1-\\alpha})^\\prime\\,|\\,\\pmb{Y}_1),$$\nwhich can be obtained using the {\\tt{pmvnorm}} function in the {\\tt{R}}-package {\\tt{mvtnorm}}.\n\nIn general, the interim analysis at the end of Stage 1 could yield adapted values of $c_{mi2}$, $k_2$, and $n_{i2}$ for Stage 2 and, hence, the adapted $Z$-statistics $Z_m$, $m=1, \\ldots, M$. Denote $\\pmb{Z}=(Z_1,\\ldots,Z_M)^\\prime$ and $Z_{\\max}=\\max \\{\\pmb{Z}\\}$. The adaptive critical value $\\tilde{u}_{1-\\alpha}$ can be obtained by solving the equation\n$$\\tilde{A}=P_{H_0}(Z_{\\max}\\geq\\tilde{u}_{1-\\alpha}\\,|\\,\\pmb{Y}_1)=1- P_{H_0}(\\pmb{Z} \\leq (\\tilde{u}_{1-\\alpha},\\ldots,\\tilde{u}_{1-\\alpha})^\\prime\\,|\\,\\pmb{Y}_1)= A,$$\nwhere the conditional distribution of $\\pmb{Z}$ given $\\pmb{Y}_1$ is multivariate normal with mean vector\n$$\\left(\\sum_{i=1}^{k_1} c_{1i1}\\bar{y}_{i1}\\Bigg\/\\sigma\\sqrt{\\sum_{i=1}^{k_1}\\frac{c_{1i1}^2}{n_{i1}}+\\sum_{i=1}^{k_2}\\frac{c_{1i2}^2}{n_{i2}}},\\ldots,\\sum_{i=1}^{k_1} c_{Mi1}\\bar{y}_{i1}\\Bigg\/\\sigma\\sqrt{\\sum_{i=1}^{k_1}\\frac{c_{Mi1}^2}{n_{i1}}+\\sum_{i=1}^{k_2}\\frac{c_{Mi2}^2}{n_{i2}}}\\right)^{\\prime}$$\nand covariance matrix $\\pmb{\\tilde{R}}=(\\text{cov}(Z_m,Z_{m^\\prime}\\,|\\,\\pmb{Y}_1))$, $m,m^\\prime=1,\\ldots,M$, where\n$$\\text{cov}(Z_m,Z_{m^\\prime}\\,|\\,\\pmb{Y}_1 )={\\sum_{i=1}^{k_2}}\\frac{c_{mi2} c_{m^\\prime i2}}{n_{i2}} \\Bigg\/\n\\sqrt{\\left( {\\sum_{i=1}^{k_1}}\\frac{c_{mi1}^2}{n_{i1}}+ {\\sum_{i=1}^{k_2}}\\frac{c_{mi2}^2}{n_{i2}}\\right)\\left( {\\sum_{i=1}^{k_1}}\\frac{c_{m^\\prime i1}^2}{n_{i1}}+ {\\sum_{i=1}^{k_2}} \\frac{c_{m^\\prime i2}^2}{n_{i2}}\\right)}.$$\nUse of $\\tilde{u}_{1-\\alpha}$ as the critical value for the AMCT controls the Type I error probability at level $\\alpha$ (M\\\"{u}ller and Sch\\\"{a}fer, 2001, 2004; Miller, 2010).\n\n\\subsection{Unknown Variance Case}\\label{AMCT, Unknown Variance Case}\n\nMiller (2010) briefly discusses the possibility of extending the AMCT to accommodate estimation of the variance $\\sigma^2$, the complication being that the conditional Type I error probability depends on the unknown variance. Posch {\\it{et al}}. (2004) developed methods to calculate the conditional Type I error probability for the one sample $t$-test given the interim data, but the authors only consider the univariate case and the approach does not directly apply to either the single contrast test or the multiple contrast test.\n\nIn this subsection, we extend the AMCT to the unknown variance case by considering the combined $T$-statistics\n$$T_m=\\frac{\\displaystyle{\\sum_{i=1}^{k_1}} c_{mi1}\\bar{Y}_{i1}+\\displaystyle{\\sum_{i=1}^{k_2}} c_{mi2} \\bar{Y}_{i2}}{S\\sqrt{\\displaystyle{\\sum_{i=1}^{k_1}}  \\frac{c^2_{mi1}}{n_{i1}}+\\displaystyle{\\sum_{i=1}^{k_2}}\\frac{c^2_{mi2}}{n_{i2}}}}=\\frac{\\sigma Z_m}{S},\\quad m= 1, \\ldots ,M,$$\nwhere the pooled variance estimator is\n$$S^2=\\left(\\sum_{i=1}^{k_1}\\sum_{j=1}^{n_{i1}} (Y_{ij1}-\\bar{Y}_{i1})^2+\\sum_{i=1}^{k_2}\\sum_{j=1}^{n_{i2}} (Y_{ij2}-\\bar{Y}_{i2})^2\\right)\\Bigg\/\\left(\\sum_{i=1}^{k_1} n_{i1}-k_1+\\sum_{i=1}^{k_2} n_{i2}-k_2\\right).$$\nAs in Section \\ref{AMCT, Known Variance Case}, we pre-specify $c^*_{mi2}=c_{mi1}$, $k^*_2=k_1$, $n^*_{i2}=n_{i1}$, and $\\pmb{d}^*_{\\text{Stage2}}=\\pmb{d}_{\\text{Stage1}}$, $i=1,\\ldots,k^*_2$, $m=1,\\ldots,M$. The $T$-statistics for the base test are\n$$T^*_m=\\sum_{i=1}^{k_1} c_{mi1} (\\bar{Y}_{i1}+\\bar{Y}_{i2})\\Bigg\/S^*\\sqrt{2\\sum_{i=1}^{k_1}\\frac{c^2_{mi1}}{n_{i1}}}=\\frac{\\sigma Z^*_m }{S^*},\\quad m=1,\\ldots,M,$$\nwhere\n$$S^{*2}=\\sum_{i=1}^{k_1}\\sum_{j=1}^{n_{i1}}\\left[(Y_{ij1}-\\bar{Y}_{i1})^2+(Y_{ij2}-\\bar{Y}_{i2})^2\\right]\\Bigg\/(2\\nu_1),\\text{ where }\\nu_1=\\sum_{i=1}^{k_1} n_{i1}-k_1.$$\nSince $S^{*2}$ is independent of $Z^*_m$ and $2\\nu_1 S^{*2}\/\\sigma^2 \\sim \\chi^2_{2 \\nu_1}$, the null joint distribution of $\\pmb{T}^*=(T_1^*, \\ldots, T_M^*)^\\prime$ is multivariate $t$ with $2\\nu_1$ degrees of freedom and correlation matrix $\\pmb{R}^*$. The non-adaptive $\\alpha$-level critical value $c^*_{1-\\alpha}$ can then be obtained using the {\\tt{qmvt}} function in the {\\tt{R}}-package {\\tt{mvtnorm}}.\n\nThe main difficulty in the unknown variance case is that the approach outlined in Section \\ref{AMCT, Known Variance Case} cannot be employed because the conditional distribution of $T_m^*$ given $\\pmb{Y}_1$ is not central $t$ under $H_0$. We develop the conditional Type I error probability as follows. Denote\n$$T_m^*\\,|\\,\\pmb{Y}_1=\n\\frac{\\displaystyle{\\sum_{i=1}^{k_1}} c_{mi1} (\\bar{y}_{i1}+\\bar{Y}_{i2})}{\\sqrt{\\displaystyle{\\sum_{i=1}^{k_1}}\\frac{c^2_{mi1}}{n_{i1}}}\\sqrt{ \\displaystyle{\\sum_{i=1}^{k_1}\\sum_{j=1}^{n_{i1}}}\\left\\{ (y_{ij1}-\\bar{y}_{i1})^2+ (Y_{ij2}-\\bar{Y}_{i2})^2\\right\\} \\Bigg\/\\nu_1}}=\\frac{U^*_m}{\\displaystyle{\\sqrt{\\frac{V^*}{\\nu_1}+q^*}}},\\quad m=1,\\ldots, M,$$\nwhere\n$$U_m^*=\\frac{\\displaystyle{\\sum_{i=1}^{k_1}} c_{mi1} (\\bar{y}_{i1}+\\bar{Y}_{i2})}{\\sigma\\sqrt{\\displaystyle{\\sum_{i=1}^{k_1}}\\frac{c^2_{mi1}}{n_{i1}}}},\\quad V^*=\\sum_{i=1}^{k_1}\\sum_{j=1}^{n_{i1}} (Y_{ij2}-\\bar{Y}_{i2})^2 \\Big\/\\sigma^2,$$\nand the constant\n$$q^*= {\\sum_{i=1}^{k_1}\\sum_{j=1}^{n_{i1}}} (y_{ij1}-\\bar{y}_{i1})^2\\Big\/(\\nu_1\\sigma^2).$$\n\nUnder $H_0$, the joint distribution of $(U_1^*,\\ldots,U_M^*)^\\prime$ is multivariate normal with mean vector $(b^*_1,\\ldots, b^*_M)^\\prime$ and variance-covariance matrix $\\pmb{R}^*$, where\n$$b^*_m=\\sum_{i=1}^{k_1} c_{mi1}\\bar{y}_{i1}\\Bigg\/\\sigma\\sqrt{\\sum_{i=1}^{k_1}\\frac{c^2_{mi1}}{n_{i1}}},\\quad m=1,\\ldots,M.$$\nSince $V^*\\sim\\chi^2_{\\nu_1}$ and is independent of $(U_1^*, \\ldots,U_M^*)^\\prime$, the joint density function of $(U_1^*,\\ldots,U_M^*,V^*)^\\prime$ is\n\\begin{eqnarray*}\n& &f_{(U_1^*,\\ldots,U_M^*,V^*)}(u_1^*,\\ldots,u_M^*,v^*)=\\frac{1}{(2 \\pi)^{M\/2} |\\pmb{R}^*|^{1\/2}}\\frac{1}{\\Gamma(\\nu_1\/2) 2^{\\nu_1\/2}}\\times\n\\\\[0.5\\baselineskip]\n& &(v^*)^{\\nu_1\/2-1}e^{-v^*\/2}\\exp\\left \\{-\\frac{1}{2}(u^*_1-b^*_1,\\ldots,u^*_M-b^*_M)(\\pmb{R}^*)^{-1}(u^*_1-b^*_1,\\ldots,u^*_M-b^*_M)^\\prime\\right \\},\n\\end{eqnarray*}\nwhere $\\Gamma (\\cdot)$ is the Gamma function. Now make the transformation\n$$T_m^*\\,|\\,\\pmb{Y}_1=\\frac{U_m^*}{\\displaystyle{\\sqrt{\\frac{V^*}{\\nu_1}+q^*}}},\\quad m=1,\\ldots,M,\\quad\\text{and}\\quad\nW^*=V^*$$\nwith Jacobian $(W^*\/\\nu_1+q^*)^{M\/2}$. The joint density function of $\\pmb{T}^*\\,|\\,\\pmb{Y}_1$ is\n\\begin{eqnarray*}\n& &\\displaystyle{f_{\\pmb{T}^*\\,|\\,\\pmb{Y}_1}\\left((t_1^*,\\ldots,t_M^*)\\,|\\,\\pmb{y}_1 \\right)} \\\\\n&=&\\frac{1}{(2\\pi)^{M\/2} |\\pmb{R}^*|^{1\/2}}\\frac{1}{\\Gamma (\\nu_1\/2) 2^{\\nu_1\/2}}\n\\int_0^{+ \\infty}\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{M\/2}(w^*)^{\\nu_1\/2-1} e^{-w^*\/2}\\times \\\\\n& &\\exp\\Bigg[-\\frac{1}{2}\\left\\{t_1^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_1,\\ldots,t_M^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_M\\right \\}(\\pmb{R}^*)^{-1} \\\\\n& &\\left \\{t_1^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_1,\\ldots,t_M^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_M\\right\\}^\\prime\\, \\Bigg ] dw^*\n\\end{eqnarray*}\nWe then obtain the conditional Type I error probability\n\\begin{eqnarray*}\nA&=&1- P_{H_0}\\left(\\pmb{T}^*\\leq (c^*_{1-\\alpha},\\ldots,c^*_{1-\\alpha})^\\prime\\,|\\,\\pmb{Y}_1\\right) \\\\\n&=&1- \\int\\cdots\\int_{(t_1^*,\\ldots, t_M^*) \\leq (c^*_{1-\\alpha},\\ldots,c^*_{1-\\alpha})} f_{\\pmb{T}^*\\,|\\,\\pmb{Y}_1}\\left((t_1^*,\\ldots,t_M^*)\\,|\\,\\pmb{y}_1\\right)\\ d t_1^* \\cdots d t_M^*.\n\\end{eqnarray*}\n\nAfter making the adaptations at the interim analysis, from the conditional distribution of $\\pmb{T}=(T_1, \\ldots, T_M)^\\prime$ given $\\pmb{Y}_1$, the adaptive critical value $\\tilde{c}_{1-\\alpha}$ can be determined as a solution to the following equation:\n\\begin{eqnarray*}\n\\tilde{A}&=&1-P_{H_0}\\left(\\pmb{T} \\leq (\\tilde{c}_{1-\\alpha},\\ldots,\\tilde{c}_{1-\\alpha})^\\prime\\,|\\,\\pmb{Y}_1\\right) \\\\\n&=&1- \\int\\cdots\\int_{(t_1,\\ldots,t_M) \\leq (\\tilde{c}_{1-\\alpha},\\ldots,\\tilde{c}_{1-\\alpha})} f_{\\pmb{T}\\,|\\,\\pmb{Y}_1}\\left((t_1,\\ldots,t_M)\\,|\\,\\pmb{y}_1\\right)\\ d t_1 \\cdots d t_M=A,\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\n&\\displaystyle f_{\\pmb{T}\\,|\\,\\pmb{Y}_1}\\left((t_1, \\ldots, t_M)\\,|\\,\\pmb{y}_1\\right)=\\displaystyle\\frac{1}{(2\\pi)^{M\/2} |\\pmb{\\tilde{R}}|^{1\/2}}\\displaystyle \\frac{1}{\\Gamma (\\nu_2\/2) 2^{\\nu_2\/2}}\\int_0^{+ \\infty}\\left(\\frac{w}{\\nu}+q\\right)^{M\/2} w^{\\nu_2\/2-1} e^{-w\/2}\\times \\\\\n&\\exp\\Bigg[-\\displaystyle\\frac{1}{2}\\left\\{t_1\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_1,\\ldots,t_M\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_M\\right\\}\\pmb{\\tilde{R}}^{-1} \\\\\n&\\displaystyle\\left\\{t_1\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_1,\\ldots,t_M\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_M\\right\\}^\\prime\\, \\Bigg] dw,\\\\\n&\\nu_2=\\displaystyle{\\sum_{i=1}^{k_2}} n_{i2}-k_2, \\quad \\nu=\\nu_1+\\nu_2, \\quad q=\\displaystyle{\\sum_{i=1}^{k_1}\\sum_{j=1}^{n_{i1}}} (y_{ij1}-\\bar{y}_{i1})^2\\Big\/(\\nu\\sigma^2),\n\\end{eqnarray*}\nand\n\\[b_m=\\displaystyle{\\sum_{i=1}^{k_1}} c_{mi1}\\bar{y}_{i1}\\Bigg\/\\sigma\\sqrt{\\displaystyle{\\sum_{i=1}^{k_1}}\\frac{c^2_{mi1}}{n_{i1}}+\\displaystyle{\\sum_{i=1}^{k_2}}\\frac{c^2_{mi2}}{n_{i2}}},\\quad m=1,\\ldots,M.\\]\n$H_0$ is rejected if $T_{\\max}=\\max \\{\\pmb{T}\\}\\geq\\tilde{c}_{1-\\alpha}$. Use of the critical value $\\tilde{c}_{1-\\alpha}$ provides control of the Type I error probability at level $\\alpha$ according to the CRP principle (M\\\"{u}ller and Sch\\\"{a}fer, 2001, 2004).\n\n\n\\section{Numerical Example}\\label{Numerical Example}\n\n\n\\subsection{Adaptive Generalized Multiple Contrast Tests}\\label{Numerical Example, AGMCT}\n\nTo illustrate the adaptive generalized multiple contrast tests (AGMCTs), we generated a numerical example. The example data set is available as Supporting Information. Suppose that there are $k_1=5$ dosage groups in Stage 1, with $\\pmb{d}_{\\text{Stage1}}=(0,0.05,0.20,0.60,1.00)^\\prime$. The total sample sizes in two stages are the same ($N_1=N_2=120$) and the group sample sizes are equal in Stage 1 ($n_{11}=\\cdots =n_{51}=N_1\/5=24$). The $M=5$ candidate dose-response models with the original specifications of $\\pmb{\\theta}$ are shown in Table \\ref{Table.original candidate models}.\n\nWe assume that the true dose-response model is the $E_{\\max}$ 2 model:\n$$f_{E_{\\max}2}(d,\\pmb{\\theta})=E_0+E_{\\text{max}} d\/ (ED_{50}+d)=0.2+0.6 d\/(0.1+d).$$\nWe generate the Stage 1 data from a multivariate normal distribution with mean $f_{E_{\\max}2}(\\pmb{d}_{\\text{Stage1}},\\pmb{\\theta})=$ $(0.20, 0.40, 0.60, 0.71, 0.75)^\\prime$ and covariance matrix $\\sigma^2\\pmb{I}=1.478^2\\pmb{I}$. The sample mean and variance estimates from the Stage 1 data are $\\bar{\\pmb{y}}_1=(0.52, 0.47, 1.09, 1.70, 0.45)^\\prime$ and $s^2_1=1.58^2$, respectively.\n\nThe optimal contrast vectors in Stage 1 based on the $M=5$ candidate dose-response models in Table \\ref{Table.original candidate models} are as follows.\n\\begin{align*}\n&E_{\\max}: \\pmb{c}_{11}=(-0.64, -0.36, 0.06, 0.41, 0.53)^\\prime, \\\\\n&\\text{Linear-log}: \\pmb{c}_{21}=(-0.54, -0.39, -0.08, 0.37, 0.64)^\\prime, \\\\\n&\\text{Linear}: \\pmb{c}_{31}=(-0.44, -0.38, -0.20, 0.27, 0.74)^\\prime, \\\\\n&\\text{Quadratic}: \\pmb{c}_{41}=(-0.57, -0.36,  0.16, 0.71, 0.07)^\\prime, \\\\\n&\\text{Logistic}: \\pmb{c}_{51}=(-0.40, -0.39, -0.31, 0.50, 0.59)^\\prime.\\end{align*}\nAfter conducting three different GMCTs using Tippett's, Fisher's, and inverse normal combination statistics, we obtain the following Stage 1 p-values: $p_{T1}=0.005$, $p_{F1}=0.047$, and $p_{N1}=0.06$.\n\nWe then adapt the candidate dose-response models and the dosage groups. We fit the 5 original candidate dose-response models using the Stage 1 data. Unfortunately, the Logistic model failed to converge on a solution so we replaced it with isotonic regression. Also, we use the dosage adaptation rule described in Section \\ref{Adapting the Dosage Groups} with $\\delta=0$ to drop the active dosage groups that appear to be less efficacious than placebo or the adjacent dosage. Finally, we obtain $k_2=3$ dosage groups in Stage 2: $\\pmb{d}_{\\text{Stage2}}=(0,0.20,0.60)^\\prime$ and $n_{12}=n_{22}=n_{32}=N_2\/3=40$.\n\nThe optimal contrast vectors in Stage 2 based on the adapted dose-response models and dosage groups are as follows:\n\\begin{align*}\n&E_{\\max}: \\pmb{c}_{12}=(-0.433, -0.383, 0.816)^\\prime,\\\\\n&\\text{Linear-log}: \\pmb{c}_{22}=(-0.707, 0.000, 0.707)^\\prime,\\\\\n&\\text{Linear}: \\pmb{c}_{32}=(-0.617, -0.154, 0.772)^\\prime, \\\\\n&\\text{Quadratic}: \\pmb{c}_{42}=(-0.766, 0.137, 0.629)^\\prime,\\\\\n&\\text{Isotonic regression}: \\pmb{c}_{52}=(-0.816, 0.408, 0.408)^\\prime.\\end{align*}\n\nThe Stage 2 data are then generated from a multivariate normal distribution with mean $f_{E_{\\max}2}(\\pmb{d}_{\\text{Stage2}},\\pmb{\\theta})=$ $(0.20, 0.60, 0.71)^\\prime$ and covariance matrix $\\sigma^2\\pmb{I}=1.478^2\\pmb{I}$. The sample mean and variance estimates from the Stage 2 data under adaptation are $\\bar{\\pmb{y}}_2=(-0.09, 0.77, 0.73)^\\prime$ and $s^2_2=1.52^2$, respectively. After conducting three different GMCTs using Tippett's, Fisher's, and inverse normal combination statistics, we obtain the following Stage 2 p-values: $p_{T2}=0.005$, $p_{F2}=0.008$, and $p_{N2}=0.008$. The p-values from Stage 1 and Stage 2 are then combined using Fisher's combination statistic and the inverse normal combination statistic. The combination statistics and resulting overall p-values are shown in Table \\ref{num.results AGMCT}.\n\n\\subsection{Adaptive Multiple Contrast Test}\\label{Numerical Example, AMCT}\n\n\\subsubsection{Known Variance Case}\nWe use the same simulated data as in Section \\ref{Numerical Example, AGMCT} to illustrate the adaptive multiple contrast test (AMCT) for the known variance case (for purposes of this illustration, we use $\\sigma^2=1.478^2$). We first obtain the non-adaptive critical value $u^*_{1-\\alpha}$. The joint null distribution of $\\pmb{Z}^*=(Z_1^*,\\ldots,Z_5^*)^\\prime$ is multivariate normal with mean $\\pmb{0}$ and covariance matrix $\\pmb{R}^*$, where\n\\begin{eqnarray*}\n\\pmb{R}^*=\\left(\\begin{array}{ccccc}\n1 &0.977 &0.912& 0.842& 0.896 \\\\\n0.977 &1& 0.977& 0.750& 0.956 \\\\\n0.912 &0.977& 1& 0.602& 0.957 \\\\\n0.842 &0.750& 0.602& 1& 0.715 \\\\\n0.896 &0.956& 0.957& 0.715& 1\n\\end{array}\\right).\n\\end{eqnarray*}\n\nThe value of $u^*_{1-\\alpha}$ is obtained using the {\\tt{qmvnorm}} function in the {\\tt{R}}-package {\\tt{mvtnorm}}, resulting in $u^*_{1-\\alpha}=1.968$. We then calculate the conditional mean of $\\pmb{Z}^*$ given $\\pmb{Y}_1$,\n$$\\left(\\frac{\\displaystyle{\\sum_{i=1}^{k_1}} c_{1i1} \\bar{y}_{i1}}{\\sigma\\sqrt{2\\displaystyle{\\sum_{i=1}^{k_1}}\\frac{c_{1i1}^2}{n_{i1}}}},\\ldots,\n\\frac{\\displaystyle{\\sum_{i=1}^{k_1}} c_{Mi1} \\bar{y}_{i1}}{\\sigma\\sqrt{2\\displaystyle{\\sum_{i=1}^{k_1}}\\frac{c_{Mi1}^2}{n_{i1}}}}\\right)^\\prime=(1.19, 0.87, 0.42, 2.22, 0.92)^\\prime,$$\nand the conditional covariance matrix $\\pmb{R_2}^*=\\pmb{R}^*\/2$. The conditional error is obtained using the {\\tt{pmvnorm}} function in the {\\tt{R}}-package {\\tt{mvtnorm}} as\n$$A=1- P_{H_0}\\left(\\pmb{Z}^*  \\leq (u^*_{1-\\alpha},\\ldots, u^*_{1-\\alpha})^{\\prime}\\,|\\,\\pmb{Y}_1\\right)=0.64.$$\n\nAfter adapting the dose-response models and dosage groups as in Section \\ref{Numerical Example, AGMCT} above, we obtain the conditional distribution of $\\pmb{Z}\\,|\\,\\pmb{Y}_1$, which is multivariate normal with mean\n$$\n\\left(\\frac{\\displaystyle{\\sum_{i=1}^{k_1}} c_{1i1} \\bar{y}_{i1}}{\\sigma\\sqrt{\\displaystyle{\\sum_{i=1}^{k_1}}\\frac{c_{1i1}^2}{n_{i1}}+\\displaystyle{\\sum_{i=1}^{k_2}}\\frac{c_{1i2}^2}{n_{i2}}}},\\ldots,\\frac{\\displaystyle{\\sum_{i=1}^{k_1}} c_{Mi1} \\bar{y}_{i1}}{\\sigma\\sqrt{\\displaystyle{\\sum_{i=1}^{k_1}}\\frac{c_{Mi1}^2}{n_{i1}}+\\displaystyle{\\sum_{i=1}^{k_2}} \\frac{c_{Mi2}^2}{n_{i2}}}},\\right)^{\\prime} \\\\[0.5\\baselineskip]=(1.33, 0.98, 0.47, 2.48, 1.03)^{\\prime}\n$$\nand covariance matrix\n\\begin{eqnarray*}\n\\pmb{\\tilde{R}}=\\left(\\begin{array}{ccccc}\n0.375& 0.331& 0.358& 0.297& 0.199 \\\\\n0.331& 0.375& 0.368& 0.370& 0.325 \\\\\n0.358& 0.368& 0.375& 0.351& 0.283 \\\\\n0.297& 0.370& 0.351& 0.375& 0.352 \\\\\n0.199& 0.325& 0.283& 0.352& 0.375\n\\end{array}\\right).\n\\end{eqnarray*}\n\nFinally, we obtain the adaptive critical value $\\tilde{u}_{1-\\alpha}=2.263$ and the combined test statistics $\\pmb{Z}=(Z_1,\\ldots, Z_M)^\\prime=(2.22, 2.50, 1.78, 4.15, 2.83)^\\prime$. We reject $H_0$ since $Z_{\\max}=4.15 \\geq \\tilde{u}_{1-\\alpha}$.\n\n\\subsubsection{Unknown Variance Case}\n\nTo illustrate the AMCT in the unknown variance case (Section \\ref{AMCT, Unknown Variance Case}), we use the same example data as in Section \\ref{Numerical Example, AGMCT} for $M=2$ candidate dose-response models. Here, we only consider the $E_{\\max}$ and Linear-log candidate dose-response models in Table \\ref{Table.original candidate models}. Other settings are the same as in Section \\ref{Numerical Example, AGMCT}, including the optimal contrasts for\nboth Stage 1 and Stage 2, and the adapted dosage groups for Stage 2.\n\nWe first obtain the non-adaptive critical value $c^*_{1-\\alpha}$. The joint null distribution of $\\pmb{T}^*=(T_1^*,T_2^*)^\\prime$ is bivariate $t$ with degrees of freedom $2 \\nu_1$ and correlation matrix $\\pmb{R}^*$, where $\\nu_1=N_1-5=115$ and\n\\begin{eqnarray*}\n\\pmb{R}^*=\\left(\\begin{array}{cc}\n1 &0.977\\\\\n0.977 &1\n\\end{array}\\right).\n\\end{eqnarray*}\nThe value of $c^*_{1-\\alpha}$ is obtained using the {\\tt{qmvt}} function in the {\\tt{R}}-package {\\tt{mvtnorm}}, resulting in $c^*_{1-\\alpha}=1.732$.\n\nWe then obtain the conditional error by numerically calculating the three-dimensional integral below using the {\\tt{adaptIntegrate}} function in the {\\tt{R}}-package {\\tt{cubature}}.\n\\begin{eqnarray*}\nA&=&1- \\frac{1}{(2\\pi)^{M\/2} |\\pmb{R}^*|^{1\/2}}\\frac{1}{\\Gamma (\\nu_1\/2) 2^{\\nu_1\/2}} \\int_0^{+ \\infty} \\int_{-\\infty}^{c^*_{1-\\alpha}} \\int_{-\\infty}^{c^*_{1-\\alpha}} \\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{M\/2}(w^*)^{\\nu_1\/2-1} e^{-w^*\/2}\\times\\\\\n& &\\exp\\Bigg[-\\frac{1}{2}\\left\\{t_1^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_1, t_2^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_2\\right\\}(\\pmb{R}^*)^{-1} \\\\\n& & \\left\\{t_1^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_1, t_2^*\\left(\\frac{w^*}{\\nu_1}+q^*\\right)^{1\/2}-b^*_2\\right\\}^\\prime\\, \\Bigg] dw^* \\ dt_1^* \\ dt_2^* =  0.198.\n\\end{eqnarray*}\n\nAfter adapting the dose-response models and dosage groups at the end of Stage 1, we consider the conditional distribution of $\\pmb{T}\\,|\\,\\pmb{Y}_1$. The adaptive critical value $\\tilde{c}_{1-\\alpha}$ can be obtained by solving the following equation using a bisection algorithm:\n\n\\begin{eqnarray*}\n\\tilde{A}&=& \\frac{1}{(2\\pi)^{M\/2} |\\pmb{\\tilde{R}}|^{1\/2}}\\frac{1}{\\Gamma (\\nu_2\/2) 2^{\\nu_2\/2}} \\int_0^{+ \\infty} \\int_{-\\infty}^{\\tilde{c}_{1-\\alpha}} \\int_{-\\infty}^{\\tilde{c}_{1-\\alpha}} \\left(\\frac{w}{\\nu}+q\\right)^{M\/2} w^{\\nu_2\/2-1} e^{-w\/2}\\times \\\\\n& &\\exp\\Bigg[-\\frac{1}{2}\\left\\{t_1\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_1, t_2\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_2\\right\\}\\pmb{\\tilde{R}}^{-1} \\\\\n& & \\left\\{t_1\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_1, t_2\\left(\\frac{w}{\\nu}+q\\right)^{1\/2}-b_2\\right\\}^\\prime\\, \\Bigg] dw \\ dt_1\\ dt_2 = A ,\n\\end{eqnarray*}\nwhere the covariance matrix $\\pmb{\\tilde{R}}$ is\n\\begin{eqnarray*}\n\\pmb{\\tilde{R}}=\\left(\\begin{array}{cc}\n0.375 &0.331\\\\\n0.331 &0.375\n\\end{array}\\right).\n\\end{eqnarray*}\n\nFinally, we obtain the adaptive critical value $\\tilde{c}_{1-\\alpha}=1.802$ with tolerance $10^{-7}$. The combined test statistics are $\\pmb{T}=(T_1,T_2)^\\prime=(2.11, 2.38)^\\prime$ and we reject $H_0$ since $T_{\\max}=2.38 \\geq \\tilde{c}_{1-\\alpha}$.\n\n\n\\section{Simulation studies}\\label{Simulation studies}\n\n\nIn this section, we conduct simulation studies to compare the operating characteristics of the AGMCTs with those of the AMCT in the setting of a design that adapts both the candidate dose-response models and the dosage groups based on data from Stage 1. We also compare these with the operating characteristics of the corresponding tests in a non-adaptive design.\n\nAssume $k_1=5$ and $\\pmb{d}_{\\text{Stage1}}=(0,0.05,0.20,0.60,1.00)^\\prime$. The total sample size is the same for each of the two stages ($N_1=N_2$) and the group sample sizes within each stage are equal, with $N_1=N_2=60$, 120, 180, and 240. The $M=5$ candidate dose-response models with the original specifications of $\\pmb{\\theta}$ are shown in Table \\ref{Table.original candidate models}. The outcome for each patient is distributed as $N(\\mu(d),\\sigma^2)$, where the true mean configuration $\\mu(d)$ follows one of the eight different dose-response models in Table \\ref{Sim.eight dose-response models}, and $\\sigma=1.478$. The dose-response curves for the five candidate models and the eight true dose-response models are shown in Figure \\ref{fig: candidate and true model}.\n\nFor the (true) $E_{\\max}$ 2 and Double-logistic models, the optimal contrasts are highly correlated with those of the candidate models. In contrast, for the (true) $E_{\\max}$ 3, Exponential 1, Exponential 2, Quadratic 2, Step and Truncated-logistic models, the optimal contrasts are not highly correlated with those of the candidate models (Figure \\ref{fig: True vs. candidate}).\n\nFor the AGMCTs, we use three GMCTs to combine the $M=5$ dependent $p$-values \\emph{within} each stage: Tippett's ($T$), Fisher's ($F$) and inverse normal ($N$) combination methods (Ma and McDermott, 2020). The same GMCT is used in both Stage 1 and Stage 2. To perform the overall test, only the inverse normal ($\\Psi_N$) combination statistic is used to combine $p_1$ and $p_2$ \\emph{across} stages since our preliminary simulation studies showed that, in general, using $\\Psi_N$ to combine $p_1$ and $p_2$ yielded greater power than using $\\Psi_F$. The reason for this is that under the alternative hypothesis, $p_1$ and $p_2$ both tend to be small and the rejection region of $\\Psi_N$ is larger than that of $\\Psi_F$ when $p_1$ and $p_2$ are both small (Wassmer and Brannath 2016, Section 6.2).\n\n\n\nFor the AGMCTs, we report the results of the operating characteristics for both the known and unknown variance cases. The results for the corresponding GMCTs in a non-adaptive design are also reported. For the AMCT, the simulation studies of the operating characteristics are presented only for the known variance case. The corresponding test in a non-adaptive design is just the MCP-Mod procedure, which is equivalent to the GMCT based on Tippett's combination method in a non-adaptive design.\n\nAll dosage adaptations are made according to the example rule described in Section \\ref{Adapting the Dosage Groups}. To deal with the problems outlined in Section \\ref{Adapting the Candidate Dose-Response Models} above, if only one of the $E_{\\max}$ and Logistic models fails to converge in Stage 1, isotonic regression is used to generate the corresponding contrast for use in Stage 2; if both the $E_{\\max}$ and Logistic models fail to converge in Stage 1, then isotonic regression is used to generate the corresponding contrast for the Logistic model and the same contrast that was used in Stage 1 is used in Stage 2 for the $E_{\\max}$ model.  Also, if there is a negative dose-response relationship suggested by the Stage 1 data (i.e., a negative estimated slope in the Linear model), no adaptation of the dose-response models is performed for Stage 2 and we only adapt the dosage groups.\n\nAll estimated values of Type I error probability and power are based on 10,000 replications of the simulations. The Type I error probabilities for the AGMCTs and the AMCT (Tables \\ref{sim: type I error, known variance} and \\ref{sim: type I error, unknown variance} in the Appendix) agree with theory that the tests being considered all exhibit control of the Type I error probability at $\\alpha=0.05$; all values fall within the 95\\% confidence interval (0.0457, 0.0543).\n\nFor the known variance case, the power curves of the competing tests are shown in Figure \\ref{fig: Power AGMCTs and AMCT, known variance}. When the optimal contrasts associated with the true dose-response models are highly correlated with those of the candidate models ($E_{\\max}$ 2 and Double-logistic models), the AGMCTs and the AMCT are, in general, slightly less powerful than the corresponding tests in a non-adaptive design. When the optimal contrasts associated with the true dose-response models are not highly correlated with those of the candidate models ($E_{\\max}$ 3, Exponential 1, Exponential 2, Quadratic 2, Step and Truncated-logistic models), however, the AGMCTs and AMCT are more powerful than the corresponding tests in a non-adaptive design. Another observation is that the overall performance of the AMCT is the best among all the adaptive designs.\n\nFor the unknown variance case, the power curves of the competing tests are shown in Figure \\ref{fig: Power AGMCTs, unknown variance}. The overall results for these comparisons are very similar to those for the known variance case.\n\n\\section{Conclusion}\\label{Conclusion}\n\nIn this article, we extend the MCP-Mod procedure with GMCTs (Bretz {\\it{et al}}., 2005; Ma and McDermott, 2020) to two-stage adaptive designs. We perform a GMCT within each stage and combine the stage-wise $p$-values using a specified combination method to test the overall null hypothesis of no dose-response relationship. We also consider and extend an alternative AMCT approach proposed by Miller (2010), which uses the maximum standardized stratified contrast across Stage 1 and Stage 2 as the test statistic. One issue that deserves further exploration is how to best determine the ``base test\" for the AMCT. Our development in Sections \\ref{AMCT, Known Variance Case} and \\ref{AMCT, Unknown Variance Case} is based on pre-specification of the contrasts, number of candidate dose-response models, and group sample sizes to be the same in Stage 2 as they were in Stage 1. While this is not necessarily the best choice, in the absence of the ability to enumerate all possible two-stage designs being considered, it might be quite reasonable in practice. An issue that remains unresolved is that of efficiently computing the conditional error and adaptive critical value for the AMCT when the variance is unknown since these involve multidimensional integrals that can take a long time to compute.\n\nSimulation studies demonstrate that the AGMCTs and AMCT are generally more powerful for PoC testing than the corresponding tests in a non-adaptive design if the true dose-response model is, in a sense, not ``close'' to the models included in the initial candidate set. This might occur, for example, if the selection of the candidate set of dose-response models is not well informed by evidence from preclinical and early-phase studies. This is consistent with intuition: if the dose-response models are badly misspecified at the design stage, using data from Stage 1 to get a better sense of the true dose-response model and using data from both Stage 1 and Stage 2 to perform an overall test for $H_0$ should result in increased power. On the other hand, if the true dose-response model is ``close\" to the models specified in the initial candidate set, the non-adaptive design is sufficient to detect the PoC signal. In this case, the adaptive design does not provide any benefit and results in a small loss of efficiency.\n\nComparisons among the different AGMCTs and the AMCT did not reveal major differences in their operating characteristics in general. Differences among the AGMCTs tended to be larger in the setting of a non-adaptive design (Ma and McDermott, 2020). In principle, the AGMCTs proposed here for two-stage adaptive designs could be extended to multiple stages, although the circumstances under which that would be beneficial are not clear.\n\nFinally, we note that baseline covariates can easily be incorporated into the AGMCTs, as outlined in Section 2.3 of Ma and McDermott (2020).\\\\\n\n\\setstretch{1}\n\n\\makeatletter\n\\renewcommand{\\@biblabel}[1]{#1.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAdvances in maritime robotics over the last two decades have fostered an emergence of unmanned surface vehicles (USVs). These autonomous boats range from small vessels used for automated inspection of dangerous areas and automation of repetitive tasks like bathymetry or environmental control, to massive cargo and transport ships. This next stage of maritime automation holds a potential to transform maritime-related tasks and will likely impact the global economy. The safety of autonomous navigation systems hinges on their environment perception capability, in particular obstacle detection, which is responsible for timely reaction and collision avoidance.\n\n\nCameras as low-power and information rich sensors are particularly appealing due to their large success in perception for autonomous cars~\\cite{Cordts2016Cityscapesa,Chen2018Encoder}. However, recent works~\\cite{Bovcon2019Mastr,Bovcon2020MODS} have shown that methods developed for autonomous cars do not translate well to USVs due to the specifics of the maritime domain. As a result, several approaches that exploit the domain specifics for improved detection accuracy have been recently proposed~\\cite{Bovcon2021WaSR,Cane2019Evaluating,Yao2021Shoreline,Zust2021SLR}. Since everything but water can be an obstacle, classical detectors for individual obstacle classes cannot address all obstacle types. State-of-the-art methods~\\cite{Bovcon2021WaSR} instead casts maritime obstacle detection as an anomaly segmentation problem by segmenting the image into the water, sky and obstacle categories.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/motivation.pdf}\n\\caption{Single-frame obstacle detection methods (top right) struggle to distinguish between object reflections and true objects. However, reflections exhibit a distinctive temporal pattern compared to true objects (bottom left). WaSR-T (bottom right) considers the temporal context from recent frames to learn these patterns and increase segmentation robustness.}\n\\label{fig:motivation}\n\\end{figure}\n\nDespite significant advances reported in the recent maritime benchmark~\\cite{Bovcon2020MODS}, the state-of-the-art is still challenged by the reflective properties of the water surface, which cause objects reflections and sun glitter. In fact, given a single image, it is quite difficult to distinguish a reflected object or a spot of sun glitter from a true obstacle (Figure~\\ref{fig:motivation}). This results in a number of false positive detections, which in practice leads to frequent and unnecessary slowdowns of the boat, rendering current camera-based obstacle detection methods impractical.\n\n\nWe note that while correctly classifying reflections from a single image is challenging, the problem might become simpler when considering the temporal context. As illustrated in Figure~\\ref{fig:motivation}, due to water dynamics, the reflection appearance is not locally static, like that of an obstacle, but undergoes warped deformations. Based on this insight, we propose a new maritime obstacle segmentation network WaSR-T, which is our main contribution. WaSR-T introduces a new temporal context module that allows the network to extract the temporal context from a sequence of frames to differentiate objects from reflections. To the best of our knowledge, this is the first deep maritime obstacle detection architecture with a temporal component.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/architecture.pdf}\n\\caption{Overview of WaSR-T (left). Target frame and preceding context frames are fed into a shared encoder producing per-frame feature maps $X_F$ and $M_F$. The Temporal Context Module (right) extracts the temporal information from per-frame embeddings using a 3D convolution. The resulting temporal context embeddings $C_V$ are combined with target frame embeddings $X_V$ and fed into the decoder which predicts the target frame segmentation.}\n\\label{fig:architecture}\n\\end{figure*}\n\n\nWe also observe that the challenging maritime mirroring and glitter scenes are underrepresented in the standard training sets. We therefore extend the existing single-frame maritime segmentation training dataset MaSTr1325~\\cite{Bovcon2019Mastr} with corresponding preceding frames and introduce additional training images representing challenging reflection conditions, which is our secondary contribution. To maintain the notation convention, we name the extended dataset MaSTr1478. Experiments show that the dataset extension delivers significant performance improvement. Results on the recent maritime benchmark MODS~\\cite{Bovcon2020MODS} show that, compared to the single-frame WaSR~\\cite{Bovcon2021WaSR}, the proposed \\mbox{WaSR-T} reduces the number of false positive detections by 30\\% with a low computational overhead and sets a new state-of-the-art in maritime obstacle detection.\n\nIn summary, our main contributions are: (i) WaSR-T, a temporal extension of WaSR~\\cite{Bovcon2021WaSR} that leverages the temporal context for increased robustness and (ii) MaSTr1478, an extension of the existing single-frame training dataset~\\cite{Bovcon2019Mastr} with challenging reflection scenes that facilitates the training of temporal maritime segmentation networks. The new dataset and the WaSR-T source code will be publicly released to facilitate further research of temporal features in maritime obstacle detection.\n\n\n\n\n\n\n\n\\section{Related work}\n\n\nSemantic segmentation has become a common approach for obstacle detection in the marine domain~\\cite{Bovcon2019Mastr,Cane2019Evaluating,Bovcon2020MODS}, as it can address both dynamic (\\eg boats, swimmers, buoys) and static obstacles (\\eg shore, piers, floating fences) in a unified way by posing the problem as anomaly segmentation. Recently, several specialized networks for the marine domain have been proposed for this task~\\cite{Bovcon2021WaSR,Yao2021Shoreline,Qiao2022Automated}. These methods address reflections and increase detection robustness in multiple ways, including regularization techniques~\\cite{Yao2021Shoreline}, specialized loss functions~\\cite{Bovcon2021WaSR} and obstacle-oriented training regimes~\\cite{Zust2021SLR}. \n\nHowever, robustness to reflections is still lacking and causes comparatively low performance within the 15m area near the boat~\\cite{Bovcon2020MODS}, where segmentation errors are most critical. In practice, obstacle detection methods receive frames sequentially, thus the temporal component of the data is also available and could be used to distinguish between reflections and objects. So far, the additional temporal information has not yet been explored in context of maritime obstacle detection.\n\nIn other domains with similar access to sequential image data, effort has been made to harness the temporal information to improve the segmentation performance. Some approaches investigate the use of temporal information only during training to improve the temporal consistency of single-frame networks. \\cite{Varghese2021Unsupervised} and \\cite{Liu2020Efficient} achieve this by propagating the segmentation masks in consecutive frames by optical flow.\n\nIncorporating temporal information into the network for improved prediction has been explored as well,\nwith attention-based approaches being the most prevalent method. In video object segmentation~\\cite{Oh2019Video,Li2020Fast,Duke2021SSTVOS} attention is used to aggregate the information from features and segmentation masks of previous \"memory\" frames based on the attention between the target and memory features. However, these methods are designed mainly for propagating initial segmentation masks of large foreground objects over the video sequence and are not directly suitable for general purpose discriminative semantic segmentation required for obstacle detection.\n\nSimilarly, in video semantic segmentation~\\cite{Wang2021Temporal,Yuan2021CSANet} attention-based approaches are used to aggregate the temporal information from recent frames to improve general purpose semantic segmentation. \\cite{Yuan2021CSANet} additionally introduces auxiliary losses, which guide the learning of attention based on inter-frame consistency. Instead of a global attention which aggregates information from semantically similar regions from past frames, we propose a convolutional approach to facilitate the learning of local temporal texture, which is characteristic for reflections.\n\n\\section{Temporal context for obstacle detection}\n\n\nGiven a target frame $X \\in \\mathbb{R}^{3 \\times H \\times W}$, the task of the segmentation-based obstacle detection method is to predict the segmentation mask, \\ie to classify each location in $X$ as either water, sky or obstacle. We propose using the temporal context to improve the prediction accuracy. Our network (Figure~\\ref{fig:architecture}), denoted as \\mbox{WaSR-T}, is based on the state-of-the-art single-frame network for maritime obstacle detection\nWaSR~\\cite{Bovcon2021WaSR}. \nWe design WaSR-T to encode the discriminative temporal information about local appearance changes of a region over $T$ preceding context frames $M \\in \\mathbb{R}^{T \\times 3 \\times H \\times W}$. The temporal context is added to the high-level features at the deepest level of the network as shown in Figure~\\ref{fig:architecture}.\n\nFollowing \\cite{Oh2019Video} and \\cite{Wang2021Temporal}, the target and context frames are first individually encoded with a shared encoder network, producing per-frame feature maps $X_F \\in \\mathbb{R}^{N \\times H \\times W}$ and $M_F \\in \\mathbb{R}^{T \\times N \\times H \\times W}$, where $N$ is the number of channels.\nThe Temporal Context Module (Section~\\ref{sec:method\/temporal_descriptors}) then extracts dicriminative temporal context embeddings from per-frame features. Finally, the temporal context embeddings are concatenated with target frame embeddings and fed into a decoder network. Following \\cite{Bovcon2021WaSR}, the decoder gradually merges the information with multi-level features of the target frame (\\ie skip connections) and outputs the final target frame segmentation.\n\n\n\\subsection{Temporal Context Module}\\label{sec:method\/temporal_descriptors}\n\nThe Temporal Context Module (TCM) extracts the temporal information from embeddings of the context and target frames and combines it with embeddings of the target frame using concatenation (Figure~\\ref{fig:architecture}). For this reason, the number of input channels to the decoder doubles compared to the single-frame network. Thus, in order to preserve the structure and number of input channels to the decoder, TCM first reduces the dimensionality of per-frame feature maps $X_F$ and $M_F$ accordingly -- a shared $1 \\times 1$ convolutional layer is used to project the per-frame feature maps into $N\/2$ dimensional per-frame embeddings $X_V$ and $M_V$ as shown in Figure~\\ref{fig:architecture}.\n\nTo extract the temporal information from a sequence of frame embeddings, attention-based approaches~\\cite{Oh2019Video,Duke2021SSTVOS,Wang2021Temporal} are often utilized, as they allow aggregation of information from semantically similar regions across multiple frames to account for movement and appearance changes of objects. Reflections, however often feature significant local texture changes as demonstrated in Figure~\\ref{fig:motivation}.\nThus, instead of globally aligning semantically similar regions using attention mechanisms, we utilize a spatio-temporal convolution to extract the local texture changes.\n\nFirst we stack the context and target frame embeddings $M_V$ and $X_V$ into a spatio-temporal context volume $C \\in \\mathbb{R}^{(T+1) \\times N\/2 \\times H \\times W}$. Then a 3D convolution layer is used to extract discriminative spatio-temporal features from $C$. To account for minor inter-frame object and camera movements, a kernel size of $(T+1) \\times 3 \\times 3$ is used to capture temporal information in a local spatial region around locations in the context volume. We apply padding in the spatial dimensions to preserve the spatial size of the output context features $C_V  \\in \\mathbb{R}^{N\/2 \\times H \\times W}$.\n\n\\subsection{Efficient inference}\n\nDuring training, for each input image $X$, WaSR-T needs to extracts all per-frame context embeddings $M_F$ in addition to target frame embeddings $X_V$. However, during inference the frames are passed to the network sequentially, thus recent frame embeddings can be stored in memory and feature extraction only needs to be performed on the newly observed target frame. Specifically, WaSR-T stores a buffer of $T$ most recent frame embeddings $X_V$ in memory and uses them as the context frame embeddings $M_V$ in TCM. The memory buffer is initialized with $T$ copies of the $X_V$ embeddings of the first frame in the sequence. Using sequential inference, the efficiency of WaSR-T is not significantly impacted compared to single-frame methods, differing only due to the temporal context extraction in TCM.\n\n\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/cmp_real_compressed.pdf}\n\\caption{Qualitative results on MODS (top) and web-sourced sequences (bottom) reveal that in WaSR-T the handling of reflections and sun glitter is significantly improved compared to WaSR, resulting in a smaller number of FP detections and increased temporal consistency.}\n\\label{fig:cmp_mods}\n\\end{figure*}\n\n\\subsection{Implementation details}\n\nWaSR-T follows the architecture of WaSR~\\cite{Bovcon2021WaSR} and applies ResNet101 as the feature encoder. In a preliminary study we observed that in contrast to WaSR, the inertial measurements (IMU) do not bring improvements in our temporal extension. Therefore the IMU is not used in the decoder for simplicity. We apply the original WaSR training procedure, i.e., the water separation loss function, hyper-parameters, optimizers, learning rate schedule and image augmentation. We set the number of past frames in the temporal context module to $T=5$. Because of training memory constraints, the backbone gradients are restricted to the current and previous frame. WaSR-T is trained for 100 epochs on 2$\\times$NVIDIA Titan A100S GPUs with a minibatch size of 4 per GPU.\n\nThe networks in our experiments are trained on the training set Mastr1478 (Section~\\ref{sec:mastr1478}) and tested on the most recent maritime obstacle detection benchmark MODS~\\cite{Bovcon2020MODS}, which contains approximately 100 annotated sequences captured under various conditions. The evaluation protocol reflects the detection performance meaningful for practical USV navigation and separately evaluates the detection of obstacle-water edge for static obstacles and the detection of dynamic obstacles. The water-edge detection robustness ($\\mu_R$) is computed from the ground truth edge, while dynamic obstacle detection is evaluated in terms of true-positive (TP), false-positive (FP) and false-negative (FN) detections, and summarized by the F1 measure, precision (Pr) and recall (Re). A dynamic obstacle counts as detected (TP) if the coverage of the segmentation inside the ground truth bounding box is sufficient, otherwise the obstacle counts as undetected (FN). Predicted segmentations outside of the ground truth bounding boxes count as false positive detections. Detection performance is reported over the entire visible navigable area and separately within a 15m \\textit{danger zone} from the USV, where the detection performance is critical for immediate collision prevention.\n\n\\subsection{Temporally extended training dataset MaSTr1478}\\label{sec:mastr1478}\n\nTo facilitate the training of temporal networks, we extended the recent MaSTr1325~\\cite{Bovcon2019Mastr} dataset, which contains 1325 fully segmented images recorded by a USV. First, the dataset was extended by adding $T=5$ preceding frames for each annotated frame, to allow learning of the temporal context. We noticed that while MaSTr1325 is focused on the broader challenges in maritime segmentation, it contains relatively few examples of challenging reflections and glitter. We have thus extended the original dataset with additional 153 images (including their preceding frames) and use the codename \\textit{MaSTr1478} for this new dataset. The additional images were obtained from online videos or were additionally recorded by us to represent difficult scenarios for current single-frame methods, where the temporal information is important for accurate prediction, such as object mirroring, reflections and sun glitter. Examples are shown in Figure~\\ref{fig:dataset}. The frames are labeled with per-pixel ground truth following~\\cite{Bovcon2019Mastr}. To emphasize the challenging conditions, the training samples in the training batches are sampled from the original MaSTr1325 images and the additional images with equal probability. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/ds_examples.jpg}\n\\caption{Examples of the additional training sequences in the MaSTr1478 with object reflections, sun glitter and low-light conditions.}\n\\label{fig:dataset}\n\\end{figure}\n\n\n\n\\subsection{Comparison with state of the art}\n\nWe compare WaSR-T with single-frame state-of-the-art segmentation methods (DeepLabV3+~\\cite{Chen2018Encoder}, BiSeNet~\\cite{Yu2018Bisenet}, RefineNet~\\cite{Lin2017RefineNet}, WaSR~\\cite{Bovcon2021WaSR}), which scored as top performers on the recent maritime obstacle detection benchmark MODS~\\cite{Bovcon2020MODS}, as well as with state-of-the-art segmentation methods that rely on temporal information. For the latter we considered the video object segmentation method STM~\\cite{Oh2019Video} and a recent video semantic segmentation method TMANet~\\cite{Wang2021Temporal}, which use memory attention to encode the temporal information from past frames. Since a relatively simple backbone is used in the original STM, we extended it to the same backbone and decoder architecture as used in WaSR~\\cite{Bovcon2021WaSR}.\n\n\n\nResults in Table~\\ref{tab:sota} show that multi-frame methods outperform the single-frame networks in detection precision (particularly within the danger zone), and except from TMANet, preserve a high recall.\nWaSR-T outperforms the original WaSR by 1.8 points in precision and 0.9 points in the overall F1, while substantially outperforming it within the danger zone resulting in a 6.0 points F1 score improvement. This is primarily due to reduction of false positives (see Figures \\ref{fig:cmp_mods} and \\ref{fig:cmp_davimar}), which is reflected in a 10.5 point improvement of the Pr score within the danger zone. \\mbox{WaSR-T} also outperforms the temporal state-of-the-art networks especially inside the danger zone, resulting in approximately 2 points performance improvement of danger-zone F1 score. \n\nIn terms of speed, the new temporal module does not substantially increase the computation. The original WaSR runs at 15 FPS, while WaSR-T runs at approximately 10 FPS, which matches the sequence acquisition framerate.\n\nDespite the large improvements in robustness to reflections, WaSR-T also shares some limitations (\\eg detection of thin objects) with existing methods as shown in Figure~\\ref{fig:cmp_failure}. For example, the temporal context is still not able to fully address reflections in rare situations where the water is completely still and the temporal texture changes cannot be observed. We aim to tackle these challenges in our future work. \n\n\\begin{table}[]\n    \\centering\n    \\caption{Comparison of SOTA single-frame and multi-frame methods on MODS in terms of water-edge detection robustness ($\\mu_R$), precision, recall and F1 score for obstacle detection. Danger-zone performance is reported in parentheses.}\n    \\label{tab:sota}\n    \\begin{tabular}{lcccc}\n   \n    method    & $\\mu_R$ &   Pr & Re  &      F1 \\\\\n    \\midrule\n    DeepLabV3+~\\cite{Chen2018Encoder}  & 96.8 & 80.1 (18.6) & \\textbf{92.7} (\\textbf{98.4}) & 86.0 (31.3) \\\\\n    BiSeNet~\\cite{Yu2018Bisenet}     & 97.4 & 90.5 (53.7) & 89.9 (97.0) & 90.2 (69.1) \\\\\n    RefineNet~\\cite{Lin2017RefineNet}   & 97.3 & 89.0 (45.1) & 93.0 (98.1) & 91.0 (61.8) \\\\\n    WaSR~\\cite{Bovcon2021WaSR}   &  97.8 & 95.1 (80.3) &  91.9 (96.2) &  93.5 (87.6) \\\\\n    \\midrule\n    TMANet~\\cite{Wang2021Temporal} &  98.3 &  96.4 (90.0) &  85.1 (93.0) &  90.4 (91.5) \\\\\n    STM~\\cite{Oh2019Video}    &  \\textbf{98.4} &  96.3 (86.2) &  92.5 (96.4) &  \\textbf{94.4} (91.0) \\\\\n    WaSR-T &  \\textbf{98.4} &  \\textbf{96.9} (\\textbf{90.8}) &  92.0 (96.5) &  \\textbf{94.4} (\\textbf{93.6}) \\\\\n   \n    \\end{tabular}\n\\end{table}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/cmp_davimar.pdf}\n\\caption{Qualitative results on challenging inland water sequences demonstrates large improvements of WaSR-T in terms of practical robustness to reflections.}\n\\label{fig:cmp_davimar}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/cmp_failure.pdf}\n\\caption{Failure cases of both methods include small objects hiding in reflections (column 1), reflections on very still water (column 2), thin objects (column 3) and challenging water-land boundaries (columns 4 and 5).}\n\\label{fig:cmp_failure}\n\\end{figure*}\n\n\\subsection{Analysis of the alternative temporal aggregation methods}\n\nNext, we analyzed alternatives to the feature fusion in the temporal context module proposed in Section~\\ref{sec:method\/temporal_descriptors}: (i) pixel-wise average pooling of temporal features (window size of $T + 1 \\times 1 \\times 1$) and (ii) local average pooling of temporal features ($T + 1 \\times 3 \\times 3$). Table~\\ref{tab:temp_agg} shows that, compared to singe-frame WaSR, the simple pixel-wise temporal average pooling of context features already improves the performance over single-frame inference by 0.8 points (overall) and 1.9 points (danger zone) in F1. Increasing the pooling window size to a local window does not improve performance. In contrast, the 3D convolution approach described in Section~\\ref{sec:method\/temporal_descriptors} is able to learn discriminative local temporal relations and increases the F1 by an additional 0.2 points overall, and by 3.5 points inside the danger zone. The improvement is primarily on the account of substantial reduction of false positive detections.\n\\begin{table}[htb]\n    \\centering\n    \\caption{WaSR-T performance with different temporal aggregation methods in terms of water-edge detection robustness ($\\mu_R$), number of FP detections and F1 score. Performance inside the danger-zone is reported in parentheses.}\n    \\label{tab:temp_agg}\n    \\begin{tabular}{lccc}\n   \n    aggregation        & $\\mu_R$ &         FP &       F1 \\\\\n    \\midrule\n    Single-frame       &  97.8 &  2492 (629) &  93.5 (87.6) \\\\\n    Avg pool ($T+1 \\times 1 \\times 1$)    &  \\textbf{98.4} &  1771 (474) &  94.2 (90.1) \\\\\n    Avg pool ($T+1 \\times 3 \\times 3$)    &  98.3 &  2152 (537) &  93.5 (89.2) \\\\\n   \n   \n    3D Convolution     &  \\textbf{98.4} &  \\textbf{1540} (\\textbf{261}) &  \\textbf{94.4} (\\textbf{93.6}) \\\\\n   \n    \\end{tabular}\n\\end{table}\n\n\\subsection{Influence of the temporal and spatial context size}\n\nTo gain further insights, we analyzed the influence of temporal context module parameters, i.e., the temporal context length $T$ and spatial kernel size. Table~\\ref{tab:abl} shows that utilizing even a single temporal context frame (i.e., $T=1$) significantly improves the performance over single-frame inference ($T=0$) by decreasing the number of false positive detections by 30\\% overall and 39\\% inside the danger zone.\nIncreasing the temporal context length $T$ further, brings consistent, but smaller improvements in reduction of FP detections and danger-zone F1 scores.\n\n\nThe spatial context size, determined by the kernel size of the 3D convolution of the temporal context module also importantly affects the performance. Using $1 \\times 1$ spatial kernel size encodes only pixel-wise temporal relations, which negatively impacts the performance inside the danger-zone within which the objects are typically large. Increasing the kernel size to $3 \\times 3$ addresses this issue, while the performance does not improve with further increasing the spatial context size.\n\\begin{table}[htb]\n    \\centering\n    \\caption{Influence of parameters in WaSR-T in terms of water-edge detection robustness ($\\mu_R$), number of FP detections and F1 score. Performance inside the danger-zone is reported in parentheses.}\n    \\label{tab:abl}\n    \\begin{tabular}{cccc}\n   \n    $T$    & $\\mu_R$ &         FP &       F1 \\\\\n    \\midrule\n    0      &  97.8 &  2492 (629) &  93.5 (87.6) \\\\\n    1      &  98.4 &  1745 (383) &  94.2 (91.5) \\\\\n    3      &  \\textbf{98.6} &  1606 (323) &  94.0 (92.6) \\\\\n    5      &  98.4 &  \\textbf{1540} (\\textbf{261}) &  \\textbf{94.4} (\\textbf{93.6}) \\\\\n    \\midrule\n    kernel size & & & \\\\\n    \\midrule\n    $1 \\times 1$     &  98.1 &  \\textbf{1456} (357) &  \\textbf{94.6} (92.0) \\\\\n    $3 \\times 3$     &  \\textbf{98.4} &  1540 (\\textbf{261}) &  94.4 (\\textbf{93.6}) \\\\\n    $5 \\times 5$     &  98.3 &  1639 (318) &  94.2 (92.6) \\\\\n   \n    \\end{tabular}\n\\end{table}\n\n\n\\subsection{Influence of the extended MaSTr1478}\n\nFinally, several experiments were performed to evaluate the contribution of the extended training dataset MaSTr1478. In particular, how much performance improvement is brought by the temporal extension and how much by the new scenes with reflections and glitter.  The results in Table~\\ref{tab:wasr_comp} show that the single-frame WaSR does not benefit from the additional sequences in MaSTr1478. While the overall detection performance improves by 0.1 points F1, the performance decreases by 0.6 points inside the danger zone. \nUsing only temporally extended MaSTr1325 does not improve WaSR-T performance. However, considering also the new sequences in MaSTr1478, the performance improves substantially. We observe a 41\\% overall reduction in the number of FP detections and a 53\\% reduction of FPs inside the danger zone. The overall performance is thus increased by 1.0 F1 points overall and by 5.4 F1 points inside the danger zone. \n\nFigure~\\ref{fig:cmp_mods} provides qualitative results. In contrast to \\mbox{WaSR-T}, the single-frame WaSR is unable to correctly segment regions of water containing the reflections and glitter, despite using the reflection-specific training examples of MaSTr1478. We conclude that both the new scenes and the temporal extension allow learning of the temporal appearance in WaSR-T and are responsible for improved segmentation.\n\n\n\n\\begin{table}[htb]\n    \\centering\n    \\caption{Influence of training dataset extensions in terms of water-edge detection robustness ($\\mu_R$), number of FP detections and F1 score. Performance inside the danger-zone is reported in parentheses.}\n    \\label{tab:wasr_comp}\n    \\begin{tabular}{lccc}\n   \n    model        & $\\mu_R$ &         FP &       F1 \\\\\n    \\midrule\n    WaSR (MaSTr1325)       &  97.2 &  2625 (561) &  93.4 (88.2) \\\\\n    WaSR (MaSTr1478)   &  97.8 &  2492 (629) &  93.5 (87.6) \\\\\n    WaSR-T (MaSTr1325)     &  97.5 &  2273 (655) &  93.7 (87.3) \\\\\n    WaSR-T (MaSTr1478) &  \\textbf{98.4} &  \\textbf{1540} (\\textbf{261}) &  \\textbf{94.4} (\\textbf{93.6}) \\\\\n   \n    \\end{tabular}\n\\end{table}\n\n\\section{Conclusion}\n\nWe presented WaSR-T, a novel maritime obstacle detection network that harnesses the temporal context to improve obstacle detection by segmentation on water regions with ambiguous appearance. We also extended the well-known training dataset MaSTr1325~\\cite{Bovcon2019Mastr} by including preceding images for each training image and added new 153 training images with challenging scenes containing object mirroring and glitter -- the new dataset is called MaSTr1478. Experiments show that the new images and temporal extension lead to substantial improvement on maritime obstacle detection. WaSR-T outperforms single-frame maritime obstacle detection networks as well as other networks that use temporal contexts and sets a new state-of-the-art on the maritime obstacle detection benchmark MODS~\\cite{Bovcon2020MODS}. \n\n\n\n\n\\addtolength{\\textheight}{-12cm}  \n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe task of generating textual descriptions of images tests a machine's ability to understand visual data and interpret it in natural language. \nIt is a fundamental research problem lying at the intersection of natural language processing, computer vision, and cognitive science.\nFor example, single-image captioning~\\citep{farhadi2010every, kulkarni2013babytalk, vinyals2015show, xu2015show} has been extensively studied.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=.9\\linewidth]{images\/overview.pdf}\n  \\caption{Overview of the visual comparison task and our motivation. The key is to understand both images and compare them. Explicit semantic structures can be compared between images and used to generate comparative descriptions aligned to the image saliency.}\n  \\label{fig:task}\n\\end{figure}\n\nRecently, a new intriguing task, visual comparison, along with several benchmarks ~\\citep{jhamtani2018learning, tan2019expressing, park2019robust, forbes2019neural} has drawn increasing attention in the community.\nTo complete the task and generate comparative descriptions, a machine should understand the visual differences between a pair of images (see \\cref{fig:task}).\nPrevious methods~\\cite{jhamtani2018learning} often consider the pair of pre-trained visual features such as the ResNet features~\\cite{he2016deep} as a whole, and build end-to-end neural networks to predict the description of visual comparison directly.\nIn contrast, humans can easily reason about the visual components of a single image and describe the visual differences between two images based on their semantic understanding of each one. \nHumans do not need to look at thousands of image pairs to describe the difference of new image pairs, as they can leverage their understanding of single images for visual comparison. \n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{images\/model.pdf}\n\\caption{Our \\textsc{L2C} model. It consists of a segmentation encoder, a graph convolutional module, and an LSTM decoder with an auxiliary loss for single-image captioning. Details are in \\cref{sec:method}.}\n\\label{fig:model}\n\\end{figure*}\n\nTherefore, we believe that visual differences should be learned by understanding and comparing every single image's semantic representation.\nA most recent work~\\cite{zhang2020diagnosing} conceptually supports this argument, where they show that low-level ResNet visual features lead to poor generalization in vision-and-language navigation, and high-level semantic segmentation helps the agent generalize to unseen scenarios. \n\nMotivated by humans, we propose a Learning-to-Compare (\\textsc{L2C}) method that focuses on reasoning about the semantic structures of individual images and then compares the difference of the image pair. \nOur contributions are three-fold: \n\\begin{itemize}\n    \\setlength\\itemsep{-0.2em}\n    \\item We construct a structured image representation by leveraging image segmentation with a novel semantic pooling, and use graph convolutional networks to perform reasoning on these learned representations.\n    \\item We utilize single-image captioning data to boost semantic understanding of each image with its language counterpart.\n    \\item Our \\textsc{L2C} model outperforms the baseline on both automatic evaluation and human evaluation, and generalizes better on the testing image pairs.\n\\end{itemize}\n\n\\section{\\textsc{L2C} Model}\n\\label{sec:method}\nWe present a novel framework in \\cref{fig:model}, which consists of three main components. \nFirst, a \\emph{segmentation encoder} is used to extract structured visual features with strong semantic priors.\nThen, a \\emph{graph convolutional module} performs reasoning on the learned semantic representations. \nTo enhance the understanding of each image, we introduce a \\emph{single-image captioning auxiliary loss} to associate the single-image graph representation with the semantic meaning conveyed by its language counterpart.\nFinally, a decoder generates the visual descriptions comparing two images based on differences in graph representations. \nAll parameters are shared for both images and both tasks.\n\n\\subsection{Semantic Representation Construction}\nTo extract semantic visual features, we utilize pre-trained fully convolutional networks (FCN)~\\citep{long2015fully} with ResNet-101 as the backbone. \nAn image $\\mathcal{I}$ is fed into the ResNet backbone to produce a feature map $\\mathcal{F}$ $\\in \\mathbb{R}^{D\\times H\\times W}$, which is then forwarded into an FCN head that generates a binary segmentation mask $B$ for the bird class. \nHowever, the shapes of these masks are variable for each image, and simple pooling methods such as average pooling and max pooling would lose some information of spatial relations within the mask.\n\nTo address this issue and enable efficient aggregation over the area of interest (the masked area), we add a module after the ResNet to cluster each pixel within the mask into $K$ classes. Feature map $\\mathcal{F}$ is forwarded through this pooling module to obtain a confidence map $\\mathcal{C}$ $\\in \\mathbb{R}^{K\\times H\\times W}$, whose entry at each pixel is a $K$-dimensional vector that represents the probability distribution of $K$ classes.\n\nThen a set of nodes $V = \\{v_1, ..., v_K\\}, v_k \\in \\mathbb{R}^D$ is constructed as following: \n\\begin{equation}\n    v_k= \\sum_{i, j} \\mathcal{F} \\odot \\mathcal{B} \\odot \\mathcal{C}_k\n\\end{equation}\nwhere $i$=$1,... H,$ $j$=$1,...,W ,$, $\\mathcal{C}_k$ is the $k$-th probability map and $\\odot$ denotes element-wise multiplication.\n\nTo enforce local smoothness, i.e., pixels in a neighborhood are more likely belong to one class, we employ total variation norm as a regularization term:\n\\begin{equation}\n    \\mathcal{L}_{TV} = \\sum_{i,j}|C_{i+1,j}-C{i,j}|+|C_{i,j+1}-C{i,j}|\n\\end{equation}\n\n\\subsection{Comparative Relational Reasoning}\nInspired by recent advances in visual reasoning and graph neural networks ~\\citep{chen2018iterative, li2019visual}, we introduce a relational reasoning module to enhance the semantic representation of each image.\nA fully-connected visual semantic graph $G = (V, E)$ is built, where $V$ is the set of nodes, each containing a regional feature, and $E$ is constructed by measuring the pairwise affinity between each two nodes $v_i, v_j$ in a latent space.\n\\begin{equation}\n    A(v_i, v_j) = (W_i v_i)^T (W_j v_j)\n\\end{equation}\nwhere $W_i, W_j$ are learnable matrices, and $A$ is the constructed adjacency matrix. \n\nWe apply Graph Convolutional Networks (GCN) ~\\citep{kipf2016semi} to perform reasoning on the graph.\nAfter the GCN module, the output $V^o = \\{v_1^o, ..., v_K^o\\}, v_k^o \\in \\mathbb{R}^D$ will be a relationship enhanced representation of a bird.\nFor the visual comparison task, we compute the difference of each two visual nodes from two sets, denoted as  $V^g_{diff} = \\{v_{diff,1}^o, ..., v_{diff,K}^o\\}, v_{diff,k}^o = v_{k,1}^o - v_{k, 2}^o \\in \\mathbb{R}^D$.\n\n\\subsection{Learning to Compare while Learning to Describe}\nAfter obtaining relation-enhanced semantic features, we use a Long Short-Term Memory (LSTM) ~\\citep{hochreiter1997long} to generate captions. \nAs discussed in \\cref{sec:intro}, semantic understanding of each image is key to solve the task. However, there is no single dataset that contains both visual comparison and single-image annotations.\nHence, we leverage two datasets from similar domains to facilitate training. One is for visual comparison, and the other is for single-image captioning. Alternate training is utilized such that for each iteration, two mini-batches of images from both datasets are sampled independently and fed into the encoder to obtain visual representations $V^o$ (for single-image captioning) or $V^o_{diff}$ (for visual comparison).\n\nThe LSTM takes $V^o$ or $V^o_{diff}$ with previous output word embedding $y_{t-1}$ as input, updates the hidden state from $h_{t-1}$ to $h_t$, and predicts the word for the next time step.\nThe generation process of bi-image comparison is learned by maximizing the log-likelihood of the predicted output sentence. The loss function is defined as follows:\n\\begin{equation}\n    \\mathcal{L}_{diff}=-\\sum_t {\\log P(y_{t}|y_{1:t-1}, V^o_{diff})}\n\\end{equation}\nSimilar loss is applied for learning single-image captioning:\n\\begin{equation}\n    \\mathcal{L}_{single}=-\\sum_t {\\log P(y_{t}|y_{1:t-1}, V^o)}\n\\end{equation}\n\nOverall, the model is optimized with a mixture of cross-entropy losses and total variation loss:\n\\begin{equation}\n    \\begin{split}\n    \\mathcal{L}_{loss} = \\mathcal{L}_{diff} + \\mathcal{L}_{single} +  \\lambda \\mathcal{L}_{TV}\n    \\end{split}\n\\end{equation}\nwhere $\\lambda$ is an adaptive factor that weighs the total variation loss.\n\n\n\\section{Experiments}\n\\subsection{Experimental Setup}\n\\paragraph{Datasets} \nThe Birds-to-Words (B2W) has 3347 image pairs, and each has around 5 descriptions of visual difference. This leads to 12890\/1556\/1604 captions for train\/val\/test splits. Since B2W contains only visual comparisons, We use the CUB-200-2011 dataset (CUB) ~\\citep{wah2011caltech}, which consists of single-image captions as an auxiliary to facilitate the training of semantic understanding. \nCUB has 8855\/2933 images of birds for train\/val splits, and each image has 10 captions.\n\n\\paragraph{Evaluation Metrics}\nPerformances are first evaluated on three automatic metrics\\footnote{\\url{https:\/\/www.nltk.org}}: BLEU-4~\\citep{papineni2002bleu}, ROUGE-L~\\citep{lin-2004-rouge}, and CIDEr-D~\\citep{vedantam2015cider}. Each generated description is compared to all five reference paragraphs. Note for this particular task, researchers observe that CIDEr-D is susceptible to common patterns in the data (See \\cref{tab:main} for proof), and ROUGE-L is anecdotally correlated with higher-quality descriptions (which is noted in previous work~\\citep{forbes2019neural}). Hence we consider ROUGE-L as the major metric for evaluating performances.\nWe then perform a human evaluation to further verify the performance.\n\n\\begin{table*}[t]\n\\small\n\\centering\n\\setlength{\\tabcolsep}{8pt}\n\\begin{tabular}{l rrrrr rrrrr}\n\\toprule\n & \\multicolumn{3}{c}{\\textbf{Validation}} & \\multicolumn{3}{c}{\\textbf{Test}}\\\\\n\\cmidrule(lr){2-4} \\cmidrule(lr){5-7}\nModel & BLEU-4 $\\uparrow$ & ROUGE-L $\\uparrow$ & CIDEr-D $\\uparrow$ & BLEU-4 $\\uparrow$ & ROUGE-L $\\uparrow$ & CIDEr-D $\\uparrow$ \\\\\n\\toprule\nMost Frequent & 20.0 & 31.0 & \\textbf{42.0} & 20.0 & 30.0 & \\textbf{43.0} \\\\\nText-Only     & 14.0 & 36.0 & 5.0 & 14.0 & 36.0 & 7.0 \\\\\nNeural Naturalist & 24.0 & 46.0 & 28.0 & 22.0 & 43.0 & 25.0 \\\\\nCNN+LSTM & 25.1 & 43.4 & 10.2 & 24.9 & 43.2 & 9.9  \\\\\n\\midrule \n\\textsc{L2C} [B2W] & 31.9 & 45.7 & 15.2 & 31.3 & 45.3 & 15.1 \\\\\n\\textsc{L2C} [CUB+B2W] & \\textbf{32.3} & \\textbf{46.2} & 16.4 & \\textbf{31.8} & \\textbf{45.6} & 16.3 \\\\\n\\midrule\nHuman & 26.0 & 47.0 & 39.0 & 27.0 & 47.0 & 42.0 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Results for visual comparison on the Birds-to-Words dataset~\\citep{forbes2019neural}. \\textit{Most Frequent} produces only the most observed description in the dataset: ``the two animals appear to be exactly the same\". \\textit{Text-Only} samples captions from the training data according to their empirical distribution. \\textit{Neural Naturalist} is a transformer model in ~\\citet{forbes2019neural}. \\textit{CNN+LSTM} is a commonly-used CNN encoder and LSTM decoder model.\n}\n\\label{tab:main}\n\\end{table*}\n\n\n\\paragraph{Implementation Details}\nWe use Adam as the optimizer with an initial learning rate set to 1e-4. The pooling module to generate $K$ classes is composed of two convolutional layers and batch normalization, with kernel sizes 3 and 1 respectively. We set $K$ to 9 and $\\lambda$ to 1. The dimension of graph representations is 512. The hidden size of the decoder is also 512. The batch sizes of B2W and CUB are 16 and 128. Following the advice from ~\\citep{forbes2019neural}, we report the results using models with the highest ROUGE-L on the validation set, since it could correlate better with high-quality outputs for this task.\n\n\n\\subsection{Automatic Evaluation}\nAs shown in \\cref{tab:main}, first, L2C[B2W] (training with visual comparison task only) outperforms baseline methods on BLEU-4 and ROUGE-L. Previous approaches and architectures failed to bring superior results by directly modeling the visual relationship on ResNet features.\nSecond, joint learning with a single-image caption L2C[B2W+CUB] can help improve the ability of semantic understanding, thus, the overall performance of the model.\nFinally, our method also has a smaller gap between validation and test set compared to \\textit{neural naturalist}, indicating its potential capability to generalize for unseen samples.\n\n\\begin{table}\n\\small\n\\centering\n\\begin{tabular}{c c|c|c}\n\\toprule\nChoice (\\%) & L2C & CNN+LSTM & Tie \\\\\n\\midrule\nScore & \\textbf{50.8} & 39.4 & 9.8 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Human evaluation results. We present workers with two generations by L2C and CNN+LSTM for each image pair and let them choose the better one.\n}\n\\label{tab:human}\n\\end{table}\n\n\\subsection{Human Evaluation}\nTo fully evaluate our model, we conduct a pairwise human evaluation on Amazon Mechanical Turk with 100 image pairs randomly sampled from the test set, each sample was assigned to 5 workers to eliminate human variance. Following~\\citet{wang2018arel}, for each image pair, workers are presented with two paragraphs from different models and asked to choose the better one based on text quality\\footnote{We instruct the annotators to consider two perspectives, relevance (the text describes the context of two images) and expressiveness (grammatically and semantically correct).}. As shown in  \\cref{tab:human}, \\textsc{L2C} outperforms \\textsc{CNN+LSTM}, which is consistent with automatic metrics.\n\n\n\\subsection{Ablation Studies} \n\n\\paragraph{Effect of Individual Components}\nWe perform ablation studies to show the effectiveness of semantic pooling, total variance loss, and graph reasoning, as shown in \\cref{tab:ablation}.\nFirst, without semantic pooling, the model degrades to average pooling, and results show that semantic pooling can better preserve the spatial relations for the visual representations. \nMoreover, the total variation loss can further boost the performance by injecting the prior local smoothness.\nFinally, the results without GCN are lower than the full L2C model, indicating graph convolutions can efficiently modeling relations among visual regions.\n\n\\begin{table}[t]\n\\small\n\\centering\n\\setlength{\\tabcolsep}{2pt}\n\\begin{tabular}{l rrr}\n\\toprule\n  & \\multicolumn{3}{c}{\\textbf{Validation}}\\\\\n\\cmidrule(lr){2-4}\nModel  & BLEU-4 $\\uparrow$ & ROUGE-L $\\uparrow$ & CIDEr-D $\\uparrow$ \\\\\n\\toprule\nL2C  & \\textbf{31.9} & \\textbf{45.7}  & \\textbf{15.2} \\\\\n\\midrule \n$-$ Semantic Pooling & 24.5 & 43.2 & 7.2 \\\\\n$-$ TV Loss & 29.3 & 44.8 & 13.6 \\\\\n$-$ GCN  & 30.2 & 43.5 & 10.7 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Ablation study on the B2W dataset. We individually remove Semantic Pooling, total variation (TV) loss, and GCN to test their effects.\n}\n\\label{tab:ablation}\n\\end{table}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=.8\\linewidth]{images\/robust.pdf}\n  \\caption{Sensitivity test on number of K chosen.}\n  \\label{fig:robust}\n\\end{figure}\n\n\\paragraph{Sensitivity Test}\nWe analyze model performance under a varying number of $K$ ($K$ is the number of classes for confidence map $\\mathcal{C}$), as shown in \\cref{fig:robust}. Empirically, we found the results are comparable when $K$ is small. \n\n\n\\section{Conclusion}\nIn this paper, we present a learning-to-compare framework for generating visual comparisons. \nOur segmentation encoder with semantic pooling and graph reasoning could construct structured image representations. \nWe also show that learning to describe visual differences benefits from understanding the semantics of each image.\n\n\\section*{Acknowledgments}\nThe research was partly sponsored by the U.S. Army Research Office and was accomplished under Contract Number W911NF19-D-0001 for the Institute for Collaborative Biotechnologies. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{intro}}\nWR124 ($\\equiv$BAC209) is a Galactic massive star characterized by a very high heliocentric recession velocity of $\\sim$175 km~s$^{-1}$ \\citep{1982A&A...116...54S}, and it  is regarded to be among the fastest moving massive stars in the Galaxy \\citep{1982A&A...114..135M}. It was classified by \\citet{1938PASP...50..350M} as a nitrogen-sequence Wolf-Rayet star (WN) and later as Population I WN8 star \\citep{1969ApJ...157.1245S}. \\\\ \n \nWR stars are thought to be a late stage in the evolution of stars more massive than 25~$M_{\\mathrm{\\sun}}$ and they are characterized by significant stellar winds with high mass-loss rate and terminal velocity. Many WRs are surrounded by nebular emission, some of which are members of a class of object called \\textit{ring nebulae}. The structure of this type of nebula is attributed to a continual process of mass loss from the exciting WR star, which sweeps the surrounding interestellar gas into a shell \\citep{1965ApJ...142.1033J}. The study of nebulae around WRs gives us clues to the mass-loss history of massive stars, as well as to the chemical enrichment of the interstellar medium (ISM).\\\\\n\n\\begin{table*}[t]\n\t\t \\caption{Main physical parameters of WR\\,124 and M1-67.}   \n\t\t\\label{table:parameter}    \n\t\t\\centering                       \n        \t\\begin{tabular}{ l l l l l}\t\\hline\n\t\t\t\t\\\\\n             Object &  Parameter   &  Value   & Reference   \\\\  \\hline \\hline \\\\\n\t\t\tWR\\,124 & ($\\alpha$,$\\delta$) (J2000) & (19:11:30.88, +16:51:38.16) & \\citet{1997ESASP1200.....P} \\\\ \t\n\t\t\t& Spectral type & WN 8 &  \\citet{1969ApJ...157.1245S} \\\\\n\t\t    & $v_{\\mathrm{\\infty}}$  (km~s$^{-1}$)  & 710 &   \\citet{2001NewAR..45..135V} \\\\\n\t\t    & $T_{\\mathrm{eff}}$  (kK)  & 44.7 & Hamann et al. (2006) \\\\\n\t\t\t&  Distance (kpc )& 4-5  & Crawford \\& Barlow (1991)  \\\\\n\t\t\t& $R_{\\mathrm{G}} $ (kpc )& 8-10&  Esteban et al. (1992)   \\\\\n\t\t\t&\t $v_{\\mathrm{hel}}$  (km~s$^{-1}$)  &  175 &  Solf \\& Carsenty (1982)  \\\\\n     \t\t& $M_{\\mathrm{v}} $ (mag) & -7.22 &  Hamann et al. (2006)  \\\\\n            & $E_{\\mathrm{b-v}} $ (mag) & 1.08  &  Hamann et al. (2006) \\\\ \t \t\t\n     \\\\ \n\t\t\tM1-67 & H${\\alpha}$ diameter (arcsec) & 110-120 & \\citet{1998ApJ...506L.127G}   \\\\\n\t\t\t&\t $v_{\\mathrm{hel}}$  (km~s$^{-1}$)  & 150-185  &   \\citet{1981ApJ...249..586C} \\\\\n\t\t\t&\t $v_{\\mathrm{exp}}$  (km~s$^{-1}$)  &  46 &  Sirianni et al. (1998) \\\\\n\t\t\t&$ M_{\\mathrm{ionized}}$ ($M_{\\mathrm{\\sun}}$) &  1.73  &   \\citet{1998ApJ...506L.127G} \\\\\n\t\t\t\t\\hline\t\t\\\\ \n\t\t\t\\end{tabular}\n\t\t\\end{table*}\n\nM1-67 \\citep[$\\equiv$Sh2-80,][]{1959ApJS....4..257S} is a bright nebula surrounding WR124, and it shows a clumpy and irregular distribution of gas that is mostly condensed in bright knots and filaments. It was first reported by \\citet{1946PASP...58..305M} during an H${\\alpha}$ objective prism survey. Classification of the nebula and its distance have been subjects of debate in the past years. Although it was first considered an H{\\sc ii} region \\citep{1959ApJS....4..257S}, the classification of M1-67 has been alternating between a planetary nebula (PN) and a ring nebula. \\citet{1964PASP...76..241B} adopted a distance of 0.9~kpc and suggested that M1-67 might be a PN since both star and nebula have high radial velocity. Studies from optical, infrared, and radio data by \\citet{1975ApL....16..165C} prompted its classification as ring nebula around a WR, with a distance of 4.33~kpc (in agreement with \\citet{1979RMxAA...4..271P} estimations). Nevertheless, \\citet{1985A&A...145L..13V} supported the PN status based on the energy distribution in the far infrared. The issue was settled by  \\citet{1991A&A...244..205E} and \\citet{1991A&A...249..518C}. The detailed abundance analysis of the nebula by \\citet{1991A&A...244..205E} revealed nitrogen enhancement and oxygen deficiency, which is typical of material ejected in a previous evolutionary phase, and pointed to a progenitor more massive than those usually associated with PN central stars. \\citet{1991A&A...249..518C} estimated a distance between 4 kpc and 5 kpc using the interstellar Na{\\sc i}D$_{2}$ absorption spectrum of the star, ruling out the PN nature. Recently, \\citet{2010ApJ...724L..90M} have used a comprehensive model of the nebular expansion to estimate a distance of 3.35~kpc. Currently, M1-67 is classified as an ejected type WR ring-nebula.\n\nAlthough M1-67 shows an apparent spherical symmetry, ground-based coronographic images revealed a bipolar structure \\citep{1995IAUS..163...78N}. The emission lines seems to be caused by condensations of gas in clumps and radial filaments \\citep{1998ApJ...506L.127G}. One of the most striking characteristics of the nebula is the virtual absence of optical oxygen emission lines \\citep{1978ApJ...219..914B,1991A&A...244..205E}. Nevertheless, \\citet{1981ApJ...249..586C} reported a bright spot of [O{\\sc iii}]$\\lambda$5007\\AA{} 15\\arcsec\\, to the NE of the central star. Spectroscopic investigations of the physical conditions and abundances of the nebular shell have shown that the ionized gas is nitrogen-enriched and oxygen-depleted, suggesting that O has been processed into N mainly via the ON cycle \\citep{1991A&A...244..205E}. This implies that M1-67 is almost completely composed of stellar material that is poorly mixed with the surrounding ISM. The long-slit spectroscopy of M1-67 established that the bulk of the nebula is expanding at $v_{\\mathrm{exp}}$=42-46~km~s$^{-1}$ \\citep{1982A&A...116...54S,1998A&A...335.1029S} and \\citet{1981ApJ...249..586C} confirmed the high heliocentric velocity of the nebula $v_{\\mathrm{hel}}$=150-185~km~s$^{-1}$, which is comparable to the velocity of the star. The main parameters of the central star, WR124, and the nebula, M1-67, are summarized in Table \\ref{table:parameter}. \n\nMany studies have tried to disentangle the geometry and dynamics of M1-67 and its interaction with WR124. \\citet{1982A&A...116...54S} proposed a simple expanding \\textquotedblleft empty\\textquotedblright shell with condensation of stellar material that was ejected by the high-velocity parent star; indeed, the leading edge of the shell is considerably brighter than the trailing part. \\citet{1998A&A...335.1029S} found two components in the environment of the central star and interpreted them as the consequence of two different events in the past: a spherical hollow shell of 92\\arcsec\\, in diameter expanding at 46~km~s$^{-1}$ and a bipolar outflow with a semi-dimension of 48\\arcsec\\, and a velocity of 88~km~s$^{-1}$ with respect to the expansion centre. On the other hand, some authors explained the asymmetry as the result of a possible low-mass companion for WR124 \\citep{1981ApJ...249..586C,1982A&A...114..135M}. \\\\\n\nDespite the important findings of the last years, some relevant aspects of the evolution and formation of the ring nebula associated with WR124 remain unknown. In particular, a 2D study of the ionization structure of the nebula covering all the morphologies and\/or the structural components can shed light on the formation process of the nebula from the ejecta of the central star. The late spectral type of the WR ionizing star (WN8) is also very remarkable to study, as well as the degree of homogeneity in the chemical composition of its ejecta. \\\\\n\nTo do this, we included M1-67 in our programme of integral field spectroscopy (IFS) observations to compare the 2D structure with the integrated properties of certain selected areas and with models of WR evolution. The paper is organized as follows. First, we describe the observations and data reduction in Sect. \\ref{obsandred}. Then, we present the 2D results for morphology, ionization structure, and kinematic in Sect. \\ref{2d}. In Sect. \\ref{1d} we show the physical conditions and chemical abundances of eight selected areas. We perform a study of M1-67 in the mid-infrared range by analysing the IRS spectrum and the 24$\\,\\mu$m MIPS image from Spitzer in Sect. \\ref{ir}. In Sect. \\ref{discussion} we discuss the chemical composition of M1-67, the observed structure, and its relation with the evolution of the central WR star. Finally, a summary of the main conclusion is given in Sect. \\ref{conclusions}.\n\n\n\n\\section{Observation and data reduction \\label{obsandred}} \n\\begin{figure*}\n\\centering\n\\includegraphics[width=14cm]{m167_INT.pdf}\n\\caption{Narrow-band image of M1-67 in H${\\alpha}$+continuum taken with the Wide Field Camera at the Isaac Newton Telescope. North is up and east left. Red hexagons show the two zones of our IFS observations: \\emph{Edge} to the NE (left) and \\emph{Centre} to the SW (right).}\n\\label{fig:rgb}\n\\end{figure*}\n\nThe observations were carried out on July 5, 2005 using the Potsdam Multi-Aperture Spectrograph instrument (PMAS) \\citep{2005PASP..117..620R} in PPAK mode (PMAS fibre Package, \\citealt{2006PASP..118..129K}) at the 3.5~m telescope of the Centro Astron\\'omico Hispano Alem\\'an (CAHA) at the observatory of Calar Alto (Almer\\'ia, Spain). \nThe PPAK fibre bundle consists of 382 fibres with a diameter of 2.7 arcsec. The 331 science fibres are concentrated in an hexagonal bundle covering a field of view (FoV) of $74\\arcsec \\times 65\\arcsec$. The surrounding sky is sampled by 36 fibres distributed in six bundles located following a circle at about 90\\arcsec \\,from the centre. There are 15 fibres for calibration purposes too \\citep[see Fig. 5 in][]{2006PASP..118..129K}. We used the V300 grating, covering the spectral range from 3660 to 7040 \\AA{} with a dispersion of 1.67~\\AA{}\/pix, giving a spectral resolution of FWHM$\\sim$8.7 \\AA{} (R = $\\lambda\/\\Delta\\lambda\\sim$ 660) at 5577\\AA{}. The weather was photometric throughout the observations with the typical seeing subarsecond. \\\\\n\nTo choose the regions of M1-67 to be mapped, we resorted to the narrow-band images observed by our group at the Isaac Newton Telescope (INT) with the Wide Field Camera (WFC). The first PPAK pointing (called \\emph{Centre}) was centred on the WR star and it covers almost the whole nebula. The second zone (called \\emph{Edge}) was selected to study the NE edge of the object containing nebular emission and surrounding medium. Both regions can be seen in Fig. \\ref{fig:rgb}. Table \\ref{table:log} shows the observational log for M1-67.\n\nBias frames, continuum, arcs, and one spectrophotometric standard star (Hz\\,44) were also acquired during the observations.\\\\\n\n\\begin{table*}\n\\caption{M1-67 PPAK observational log.} \n\\label{table:log} \n\\centering \n\\begin{tabular}{l c c c c c c }\n\\hline\nZone & Coordinates (J2000) &Grating & Spectral range & Exp. time & Airmass & Date \\\\\n& ($\\alpha$,$\\delta$) & & ( \\AA{} ) & (s) & &\\\\\n\\hline\\hline\n\\\\\nCentre & (19:11:30.9 , +16:51:39.2) & V300 & 3640-7040 & 3 $\\times$ 30 & 1.08 & July, 5, 2005 \\\\\nEdge & (19:12:14.8 , +16:52:12.9) & V300 & 3640-7040 & 3 $\\times$ 450 & 1.07 &July, 5, 2005 \\\\\n\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\nThe data were reduced using the R3D software \\citep{2006AN....327..850S} in combination with IRAF\\footnote{The Image Reduction and Analysis Facility IRAF is distributed by the National Optical Astronomy Observatories, which are operated by Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. Website: http:\/\/iraf.noao.edu\/.} and the Euro3D packages \\citep{2004AN....325..167S}. The reduction consisted of the standard steps for fibre-fed IFS observations.\n\nAt first, a master bias frame was created and subtracted from all the images. The different exposures taken at the same position on the sky were combined to reject cosmic rays using IRAF routines. A trace mask was generated from a continuum-illuminated exposure, identifying the location of each spectrum on the detector along the dispersion axis. Then, each spectrum was extracted from the science and standard star frames, co-adding the flux within an aperture of five pixels at the location of the spectral peak in the raw data using the tracing information, and storing it in a 2D image called row-stacked spectrum (RSS) \\citep{2004AN....325..167S}. We checked that the contamination from flux coming from adjacent fibres using this aperture was negligible \\citep{2004PASP..116..565B,2006AN....327..850S}. For a given aperture and $FWHM\\sim(0.5\\times~aperture)$, we found a level of cross-talk that was always $<$10$\\%$. This seems to be an acceptable compromise between maximizing the recovered flux and minimizing the cross-talk. \n\nDistortion and dispersion solutions were obtained using a He calibration-lamp exposure and applied to the science data to perform the wavelength calibration. The accuracy achieved was better than $\\sim$0.1~\\AA{} (rms) for the arc exposures. Corrections to minimize the differences between fibre-to-fibre transmission throughput were also applied, creating a fibre flat from the exposure of a continuum source. Observations of the spectrophotometric standard star Hz\\,44 were used to perform the flux calibration. \n\nThe sky emission was determined using the science data spectra obtained throughout the additional fibres for sampling the sky. As explained above, the second pointing was made at the edge of M1-67, and some of its sky bundles are located within an area containing signals from the nebula. We inspected the 36 sky-fibres of each pointing, selecting those that did not show nebular emission. The spectra of all the selected fibres were combined with a mean in a single spectrum, and a 2D spectrum was created by copying the combined spectrum in each fibre. These sky spectra were then subtracted from every science spectrum, each pointing with its own sky.\n\nFinally, considering the wavelength range and the airmass of the observations, and using \\citet{1982PASP...94..715F}, we estimated that offsets due to the differential atmospheric refraction (DAR) were always smaller than one third of the fibre diameter. Correction for DAR was not necessary in our data.\n\n\n\n\\section{Two-dimensional analysis \\label{2d}} \nTo perform a detailed analysis of the 2D structure of the nebula, we built easy-to-use datacubes of the two reduced pointings with three dimensions: two spatials and one spectral. We studied all the interpolation routines included in the E3D software and verified which conserves the flux and the apparent morphology observed in the spaxels. Finally, we generated our cubes with the linear Delaunay interpolation and a pixel size of 1.5$\\times$1.5~arcsec$^{2}$.\\\\\n\nIn a first attempt to understand the morphology of the two observed zones, we extracted images by sliding the cubes at different wavelength ranges. They are presented in Fig. \\ref{fig:morphology_all} on logarithmic scale.\n\n\\begin{figure}\n\\includegraphics[width=9cm]{morph_O3.pdf}\n\\includegraphics[width=9cm]{morph_ha.pdf}\n\\includegraphics[width=9cm]{morph_S2.pdf}\n\\caption{Interpolated images of M1-67 of the two observed regions: In the left column the edge pointing and in the right the central one. In each row we represent the flux (including continuum) integrated in a wavelength range. Top: range 5006\\AA{}-5014\\AA{} including [O{\\sc iii}]$\\lambda$5007\\AA{}. Middle: range 6562\\AA{}-6590\\AA{} including H${\\alpha}$ and [N{\\sc ii}]$\\lambda$6584\\AA{}. Bottom: range 6729\\AA{}-6737\\AA{} including [S{\\sc ii}]$\\lambda \\lambda$6717,6731$\\AA{}$. \nAll the maps are represented on logarithmic scales with units of $\\log$(erg~cm$^{-2}$~s$^{-1}$). The size of the hexagon side is 38\\arcsec. In all the maps, north is up and east to the left (see Fig. \\ref{fig:rgb}).}\n\\label{fig:morphology_all}\n\\end{figure}\n\nIn the 5006\\AA{}-5014\\AA{} range, which includes the [O{\\sc iii}]$\\lambda$5007\\AA{} line, no significant extended emission can be observed in both regions, supporting previous studies that revealed no oxygen emission in M1-67. Several spots appear in the FoV (including the central WR124 star) with fluxes lower than\n$\\sim10^{-17}$~erg~cm$^{-2}$~s$^{-1}$, but we checked that their nature was not nebular. They probably are stars in our line of sight. We gave special attention to the spot described by \\citet{1981ApJ...249..586C} at 15\\arcsec\\ to the NE of the star. Although we can observe some emission, we cannot confirm that it comes from the nebula. A more detailed analysis of this spot is performed in Sect. \\ref{1d} by means of the integrated spectrum. \n\nAs for other lines from the central pointing maps (H${\\alpha}$, [N{\\sc ii}], and [S{\\sc ii}]), most of the emission seems to be concentrated in at least five knots distributed in the NE-SW direction without counterpart in the [O{\\sc iii}]$\\lambda$5007\\AA{} image. In addition, two regions with very faint surface brightness (or even no emission) can be seen at the opposite sides (NW and SE). This orientation agrees with the bipolar structure observed by \\citet{1995IAUS..163...78N} and \\citet{1998A&A...335.1029S} with coronographic studies. H${\\alpha}$ and [S{\\sc ii}] emission shows a discontinuity in the edge pointing of the nebula with higher surface brightness in its SW area. When we move to the NE, the emission decreases until it disappears. The purple coloured area reaches non-negligible emission up to H${\\alpha}\\sim10^{-16}$~erg~cm$^{-2}$~s$^{-1}$ per pixel (1pixel=2.25~arcsec$^2$).\\\\\n\n\n\n\n\\subsection{2D study of the emission-line maps \\label{maps}}\nMaps were created from cubes by fitting the emission lines in each spatial element following the methodology presented in \\citet{2012A&A...541A.119F}. Basically, we performed a Gaussian fit to the emission lines using our own routine, which returns maps of the flux, centre, and FWHM among other properties. To prevent contamination by low signal-to-noise (S\/N) data, we masked out all pixels with S\/N lower than 5. During the creation of the S\/N masks, we visually inspected all the pixels, rejecting interactively those with non-nebular spectra or with contamination from the central WR and other field stars.\n\nFor both regions (centre and edge pointings), maps of parameters from the Gaussian fitting were generated for seven emission lines: H${\\gamma}$, H${\\beta}$, H${\\alpha}$, [N{\\sc ii}]$\\lambda \\lambda$6548,6584\\AA{}, and [S{\\sc ii}]$\\lambda \\lambda$6717,6731$\\AA{}$. Although our spectral range includes the [O{\\sc ii}]$\\lambda \\lambda$3726,3728\\AA{} lines, their automatic fit was not considered because these lines are faint and placed at the edge of the CCD, where the distortion correction bended and deformed them.\\\\\n\nAll the emission line maps were reddening-corrected using the reddening coefficient, c(H${\\beta}$), map (each pointing was corrected with its own c(H${\\beta}$) map). To determine this coefficient we resorted to the H${\\alpha}$\/H${\\beta}$ line ratio. We analysed the maps of the three Balmer lines detected in our wavelength spectral range (H${\\gamma}$, H${\\beta}$, and H${\\alpha}$) and decided to discard the H${\\gamma}$ flux because the S\/N was lower than 5 in $\\sim$20$\\%$ of the spaxels. However, we checked that both derivations were consistent in the spaxels with good S\/N. We used an intrinsic Balmer emission line ratio of  H${\\alpha}$\/H${\\beta}$ =3.03 obtained from the public software of \\citet{1995MNRAS.272...41S}, assuming Case B recombination with an electron density of $n_{\\mathrm{e}}\\sim1000$ cm$^{-3}$ \\citep{1991A&A...244..205E} and an electron temperature of $T_{\\mathrm{e}}\\sim 7000$ K (the mean value between the estimations of \\citealt{1978ApJ...219..914B} ($\\sim$7500 K) and \\citealt{1991A&A...244..205E} ($\\sim$6000 K)). Statistical frequency distributions of the reddening coefficient were also created for the two maps taking the mean error ($\\sim$0.1) as binning. Figure \\ref{fig:reddening} shows the spatial distribution of the two derived c(H${\\beta}$) maps and their corresponding histograms.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{chbeta_paper.pdf}\n\\caption{Spatial structure of the derived c(H${\\beta}$) maps and their corresponding statistical frequency distributions with a binning of 0.1. On the left the edge pointing and on the right the central one. Orientation and sizes of maps are as in Fig. \\ref{fig:morphology_all}.}\n\\label{fig:reddening}\n\\end{figure}\n\nThe structure of the reddening map of the central region is mostly uniform with values ranging from 1.3 to 2.5 and a mean value of $\\sim$1.85$\\pm$0.10; the histogram reveals that the most probable value for c(H${\\beta}$) in this zone is 1.90. Single pixels with very high or very low values have a large error in the coefficient estimations. Big holes in the map correspond to the masked pixels.\n\nThe derived reddening coefficient map of the edge pointing has a less homogeneous structure, with a c(H${\\beta}$) mean value of $\\sim$2.11$\\pm$0.08 over the 1.7-2.8 range. It is interesting to notice that all the pixels with c(H${\\beta}$)$>$2.5 are placed in the NW area, where the discontinuity of the H${\\alpha}$ image is observed (Fig. \\ref{fig:morphology_all}). In this region, pixels with c(H${\\beta}$)$>$2.5 were inspected individually; after checking the S\/N of the Balmer lines and the c(H${\\beta}$) errors we decided not to mask them and to pay special attention to the rest of properties derived there. We study this region in detail in the 1D analysis (Sect. \\ref {1d}). The statistical frequency distribution of the reddening coefficient for the edge pointing gives 2.0 as the most probable value. If we exclude values higher than 2.5, the distribution can be fitted by a Gaussian function. \n\nTo compare our results with the literature, we estimated the extinction as $A_{\\mathrm{v}}=2.145 \\times c(H{\\beta})$, using the \\citet{1989ApJ...345..245C} extinction law with $R_{\\mathrm{v}}=3.1$ and the colour excess as $E(B-V)=0.692 \\times c(H{\\beta})$. The mean values derived were $A_{\\mathrm{v}}$=3.9 and E(B-V)=1.3 for the central pointing, and $A_{\\mathrm{v}}$=4.5 and E(B-V)=1.5 for the edge. The reddening coefficients derived from our data are higher than those estimated by \\citet{1991A&A...244..205E} ($\\sim$1.35), whereas the $A_{\\mathrm{v}}\\sim3.8$ obtained by \\citet{1981ApJ...249..586C} and $E(B-V)\\sim1.35$ from \\citet{1975ApL....16..165C} agree with our values.\\\\\n\nThe electron density (n$_{\\mathrm{e}}$) maps were produced from the [S{\\sc ii}]$\\lambda\\lambda$6717\/6731 ratios using the IRAF package TEMDEN based on a five-level statistical equilibrium model \\citep{1987JRASC..81..195D,1995PASP..107..896S}. Using these maps, we created statistical frequency distribution of the electron density with a binning of 100~cm$^{-3}$ (low density limit). They are shown with the derived n$_{\\mathrm{e}}$ maps in Fig. \\ref{fig:density}. In general terms, the values of n$_{\\mathrm{e}}$ presented in these maps are in good agreement with values reported in the literature. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{density_paper.pdf}\n\\caption{Electron density, n$_{\\mathrm{e}}$, maps derived from the [S{\\sc ii}]$\\lambda\\lambda$6717\/6731 line ratios in units of cm$^{-3}$. Orientations and sizes as in Fig. \\ref{fig:morphology_all}. On the bottom the statistical frequency distributions with a binning of 100~cm$^{-3}$. The edge pointing is on the left, and the central on the right. The white lines across the maps (from NE to SW) represent the direction along which the cuts were extracted to study the radial variation of n$_{\\mathrm{e}}$. See text for details.}\n\\label{fig:density}\n\\end{figure}\n\nThe histogram for the central pointing shows elements distributed in a wide range of density from $\\sim$200 to $\\sim$3000~cm$^{-3}$ with 1000~cm$^{-3}$ as the most probable n$_{\\mathrm{e}}$. The mean value of the distribution is 1008~cm$^{-3}$. Some isolated pixels appear very intense in the image with densities as high as 3000~cm$^{-3}$, but with large errors. The density distribution follows the low-ionization emission elements supporting the idea of a bipolar structure. It is interesting to notice that the knots with the higher surface brightness correspond to the denser zones.\n\nThe histogram for the edge pointing shows a distribution close to a Gaussian centred on 500~cm$^{-3}$. The map ranges from 100~cm$^{-3}$ to 1000~cm$^{-3}$ with a mean value of 507~cm$^{-3}$. As happens in the other region, some pixels (7) show higher densities (up to 1000~cm$^{-3}$), and we removed them from our estimations. The majority of the pixels with high reddening coefficient (c(H${\\beta}$)$>$2.5) were rejected by the S\/N mask for the sulphur line, but the unmasked pixels present a mean density of 613~cm$^{-3}$.\n\nThe morphological analyses showed that the bright knots are aligned in a preferred axis along the NE-SW direction with a bipolar structure (see Fig. \\ref{fig:morphology_all}); to check that the electron density is related with the bipolarity, we performed a cut in the density maps along this direction (see cuts in Fig. \\ref{fig:density}), the density profiles obtained are presented in Fig. \\ref{fig:dens_rad}. For them, we performed four fits using the least-squares method: the first from the star towards the SW, the second from the star towards the NE including pixels from the two pointings (centre and edge), and the last two fits from the star towards the NE, but differentiating the two pointings (see Fig. \\ref{fig:dens_rad}). It can be seen that the density decreases when we move away from the WR star. In addition, the fits show a symmetric gradient in the central points with a tendency to flatten out towards the ends.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{neVSr.pdf}\n\\caption{Radial variation in electronic density (in cm$^{-3}$) with distance (in pc) along the direction of bipolarity (from NE to SW). We consider negative radius from the star towards the NE and positive from the star towards the SW. Lines indicate least-squares fits: solid lines correspond to pixels differentiating the two pointings, and the dashed line represents the fit along the direction star-NE including pixels from both pointings.}\n\\label{fig:dens_rad}\n\\end{figure}\n\n\n\n\\subsection{Emission line relations \\label{diagdiag}}\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{ratio_SH.pdf}\n\\includegraphics[width=9cm]{ratio_NS.pdf}\n\\includegraphics[width=9cm]{ratio_NH.pdf}\n\\caption{Derived maps of the emission line ratios of the two pointings: edge (left) and centre (right). Top: [S{\\sc ii}]$\\lambda \\lambda$6717,6731\/H${\\alpha}$. Middle: [N{\\sc ii}]$\\lambda $6584\/[S{\\sc ii}]$\\lambda \\lambda$6717,6731. Bottom: ([N{\\sc ii}]$\\lambda\\lambda$6584\/H${\\alpha}$. Orientations and sizes as in Fig. \\ref{fig:morphology_all}.}\n\\label{fig:ratio_maps}\n\\end{figure}\n\nFigure \\ref{fig:ratio_maps} shows maps of the emission line ratios for the two pointings. Their mean values are summarized in Table \\ref{table:line_ratios}. All the intensities presented in the table and figure are reddening-corrected. \n\nIn both regions the [S{\\sc ii}]$\\lambda\\lambda$6717,6731\/H${\\alpha}$ map presents an inhomogeneus and patchy structure. [S{\\sc ii}] lines are fainter than H${\\alpha}$ in all the spaxels, with a maximum logarithmic ratio of -1.3 to the north of the edge pointing. Some isolated pixels show higher values, but they are over the limits of the region masked, so unreliable. \n\nThe distribution of the [N{\\sc ii}]$\\lambda $6584\/[S{\\sc ii}]$\\lambda \\lambda$6717,6731 map presents a structure opposite to [S{\\sc ii}]\/H${\\alpha}$ in both regions. The [N{\\sc ii}]  emission is stronger than [S{\\sc ii}], reaching $\\log$([N{\\sc ii}]\/[S{\\sc ii}])=1.4 in areas close to the ISM.\n\nStudying the [N{\\sc ii}]$\\lambda\\lambda$6584\/H${\\alpha}$ maps led to more interesting results. The central pointing shows positive values, except in some regions in the direction of the bipolarity, where [N{\\sc ii}] and H${\\alpha}$ fluxes are equal. In the edge pointing, regions with different ratios are clearly separated. In most of the pixels, [N{\\sc ii}]$\\ge$H${\\alpha}$ with an increasing ratio towards the side. To the north, an area can be seen where H${\\alpha}$ $>$[N{\\sc ii}]; this region possesses the higher derived c(H${\\beta})$, and it was masked in the sulphur maps because of its low S\/N ($<$5). The NW area has the highest ratio, possibly produced by the contamination of a nearby field star.\\\\\n\n\\begin{table}[!h]\n\\caption{Mean values of the emission line ratio maps}\n\\label{table:line_ratios} \n\\centering \n\\begin{tabular}{l c c}\n\\hline\nLine ratios & Edge & Centre \\\\\n\\hline\n\\hline \\\\\n$\\log$([S{\\sc ii}]$\\lambda\\lambda$6717,6731\/H${\\alpha}$)& -1.03 &  -1.01 \\\\\n$\\log$([N{\\sc ii}]$\\lambda$6584\/[S{\\sc ii}]$\\lambda \\lambda$6717,6731)& 1.15 &  1.07 \\\\\n$\\log$([N{\\sc ii}]$\\lambda\\lambda$6584\/H${\\alpha}$)& 0.07 &  0.06 \\\\\n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nTo understand the differences of the [N{\\sc ii}]$\\lambda\\lambda$6584\/H${\\alpha}$ ratio in the edge pointing, we complement the study by generating statistical frequency distributions of the ratio map and plotting all the spaxels from emission line maps in the diagram [N{\\sc ii}]$\\lambda$6584 vs. H${\\alpha}$ (Fig. \\ref{fig:NHa}).The diagram [N{\\sc ii}]$\\lambda$6584 vs. H${\\alpha}$ showed in Fig. \\ref{fig:NHa} a, presents double behaviour. We considered two lines with unity slope as upper and lower limits. Points above the upper line are pixels where the [N{\\sc ii}] emission is stronger than H${\\alpha}$, while below the lower line they have the opposite behaviour. Points between these two lines are pixels with $\\log$(H${\\alpha}$)=$\\log$([N{\\sc ii}]) $\\pm$ 0.05. Then, we located the points of the diagram in the FoV of PPAK, taking these limits into account, to identify their spatial locations (see Fig. \\ref{fig:NHa} b); they appear grouped. The statistical frequency distribution of the [N{\\sc ii}]\/H${\\alpha}$ map shows a bimodal distribution as we can see in Fig.\\ref{fig:NHa} c. When we identified the spaxels of the three regions defined above, we found that the left peak (centred in $\\sim$-0.3) includes all the points behind the lower limit, and the right peak (centred in $\\sim$0.1) includes the points of the two other zones. We can conclude that at least two spatial regions exist in this pointing: one with [N{\\sc ii}]$\\geq$H${\\alpha}$ to the SW and another one to the north with [N{\\sc ii}]$<$H${\\alpha}$. All the pixels with c(H${\\beta}$)$>$2.5 are included in the second region, along with the spectra with very low S\/N of the sulphur lines.\\\\\n\nFor the central pointing, the same analysis shows that [N{\\sc ii}] follows H${\\alpha}$ for all the points in a one-to-one relation line of unity slope. Relations between the other lines were also studied by means of two diagrams ([N{\\sc ii}]$\\lambda$6584 vs. [S{\\sc ii}]$\\lambda\\lambda$6717,6731 and [S{\\sc ii}]$\\lambda\\lambda$6717,6731 vs. H${\\alpha}$), showing strong correlations in both the pointings. The statistical frequency distribution of all the emission line ratios showed single peaks with distributions close to Gaussian functions,  except [N{\\sc ii}]$\\lambda\\lambda$6584\/H${\\alpha}$ on the edge.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{NHa_paper.pdf}\n\\caption{Relations between [N{\\sc ii}] and H${\\alpha}$ for the edge pointing. Colours help us locate points spatially: red corresponds to points with $\\log$(H${\\alpha}$)=$\\log$([N{\\sc ii}]) $\\pm$ 0.05, blue to points with $\\log$(H${\\alpha}$)$>\\log$([N{\\sc ii}]), and green to points where $\\log$(H${\\alpha}$)$<\\log$([N{\\sc ii}]). From the top to the bottom: (a) $\\log$([N{\\sc ii}]$\\lambda$6584) vs. $\\log$(H${\\alpha}$). All the spaxels of the intensity maps (in units of $\\log$(erg~cm$^{-2}$~s$^{-1}$) are represented in the diagram with crosses. Black lines with unitary slope represent the limits. (b) PPAK FoV of the edge pointing with the zones defined in plot \\emph{a}. (c) Statistical frequency distributions of the $\\log$([N{\\sc ii}]\/H${\\alpha}$) map. Black solid line represents the distributions of all the spaxels, and coloured dashed lines represent the regions defined above. See text for details.}\n\\label{fig:NHa}\n\\end{figure}\n\n\n\n\n\\subsection{The radial velocity field \\label{kinematics}}\nLimitations on the instrument resolution prevented us from carrying out an exhaustive analysis of the kinematics of M1-67. Nevertheless, the resolution was sufficient for studying the distribution of the radial velocity field and relating it to the morphology and ionization structure.\n\nUsing the central wavelength of the Gaussian fit performed in the cubes, we created radial velocity maps for the two observed regions. Two corrections were carried out over the measured radial velocities. First, we estimated the error in the wavelength calibration by comparing the wavelength of a sky emission line with its theoretical value, we obtained a difference of -0.303~\\AA{} ($\\sim$-16~km~s$^{-1}$ for [OI]$\\lambda$5577\\AA{}), and this zero point was added to the measured velocities. Then, we translated maps into the local standard of rest (LSR) and corrected for the Earth's motions, taking coordinates and universal time of the observations into account.\n\nWith the corrected radial velocity fields of H${\\alpha}$, we scaled the measured velocities using the overlapping region of the two pointings to avoid deviations. Then, we calculated the total mean velocity, obtaining a value of 139~km~s$^{-1}$, and we established it as the heliocentric velocity of the nebula. This velocity is in very good agreement with the 137~km~s$^{-1}$ obtained by \\citet{1998A&A...335.1029S}. We present the relative radial velocity field of H${\\alpha}$ for the two regions mosaicked in Fig. \\ref{fig:ha_velocity}.\\\\\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{vradial.pdf}\n\\caption{Relative radial velocity field derived for H${\\alpha}$ in units of km~s$^{-1}$. Zero is the global mean velocity (139~km~s$^{-1}$), see text for details. The two pointings are mosaicked. The red cross marks the position of the central star. North is up and east left (see Fig. \\ref{fig:rgb}).}\n\\label{fig:ha_velocity}\n\\end{figure}\n\nPrevious kinematic studies \\citep{1982A&A...116...54S,1998A&A...335.1029S} have found two components (one redshifted and another blueshifted) supporting the idea of a shell in expansion. With the low resolution of our data we cannot resolve both components, and the velocity field shown is dominated by the radial velocity of the brightest knots, a kind of intensity- weighted radial velocity distribution. Despite the low resolution, a study of the overall structure of both regions can be carried out. The gas of the nebula seems to move faster near to the WR star, decreasing its relative velocity when moving away from the centre. The velocity field changes its tendency (increasing) in the \\textquotedblleft peculiar\\textquotedblright ~zone towards the north of the edge pointing, where other properties were also found to differ from the rest of the nebula.\n\n\nFigure \\ref{fig:hist_vr} shows the statistical frequency distributions of the radial velocity maps with a binning of 5~km~s$^{-1}$. To consolidate the differences found in the diagram [N{\\sc ii}]$\\lambda$6584 vs. H${\\alpha}$ for the edge pointing (Fig. \\ref{fig:NHa}), we represented pixels from the two regions separately  The region where [N{\\sc ii}]$\\geq$H${\\alpha}$ presents a Gaussian distribution that covers a wide range in velocity, suggesting that some regions are moving away from us and others towards us. The distribution of the regions where H${\\alpha}$ emission is higher than [N{\\sc ii}] is narrower and it is centred near to the zero velocity.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{hist_vr.pdf}\n\\caption{Statistical frequency distributions of the radial velocity of H${\\alpha}$ relative to the heliocentric velocity, with a binning of 5~km~s$^{-1}$. The central region is on the top with all the spaxeles represented. On the bottom the edge pointing: black solid line represents all the pixels, the short-dashed red line is the region where [N{\\sc ii}]$\\geq$H${\\alpha}$, and the long-dashed blue line represents the regions with [N{\\sc ii}]$<$H${\\alpha}$.}\n\\label{fig:hist_vr}\n\\end{figure}\n\n\n\n\n\\section{Properties of the integrated spectra \\label{1d}} \n\nWe created 1D spectra by combining fibres to describe the integrated properties of several interesting zones. Eight integrated spectra were generated over the two pointings (see Fig. \\ref{fig:integratedregions}).\\\\\n\nThe regions selected over the central pointing are:\n\n-\\textit{Region 1} (R1): examining the emission of the low ionization elements showed in Fig. \\ref{fig:morphology_all}, three bright knots appear to the south of the nebula. Eight fibres over these knots were selected and combined to create a single spectrum. The offset from central star is $\\Delta\\alpha\\sim$4.05$\\arcsec$, $\\Delta\\delta\\sim$13.5$\\arcsec$.\n\n-\\textit{Region 2} (R2): we combined three spaxels to the north of the star coinciding with another isolated knot. The offset from central star is $\\Delta\\alpha\\sim$1.35$\\arcsec$, $\\Delta\\delta\\sim$14.85$\\arcsec$.\n\n-\\textit{Region 3} (R3): we chose those fibres placed to the east of the star in a zone where an extended emission is seen in H${\\alpha}$, paying attention to not include light from any star. The offset from central star is $\\Delta\\alpha\\sim$12.15$\\arcsec$, $\\Delta\\delta\\sim$4.05$\\arcsec$.\n\n-\\textit{Region 4} (R4): we were interested in analysing a large region in the NW masked in the 2D analysis where all the emission line maps showed S\/N lower than 5. The fourth integrated spectrum was created there to check that there is emission in this area. The offset from central star is $\\Delta\\alpha\\sim$14.85$\\arcsec$, $\\Delta\\delta\\sim$14.85$\\arcsec$.\\\\\n\nThe regions selected over the edge pointing are:\n\n-\\textit{Region 5} (R5): nine fibres were selected on the south of the edge pointing, close to the discontinuity. This spectrum belongs to the region showed in Fig. \\ref{fig:NHa} b where [N{\\sc ii}] is stronger than H${\\alpha}$. The offset from central star is $\\Delta\\alpha\\sim$31.05$\\arcsec$, $\\Delta\\delta\\sim$16.2$\\arcsec$.\n\n-\\textit{Region 6} (R6): we combined several spaxels at the SW limit of the FoV to check that [N{\\sc ii}]$\\sim$H${\\alpha}$ in this area, as we found in Sect. \\ref{diagdiag}. The offset from central star is $\\Delta\\alpha\\sim$14.85$\\arcsec$, $\\Delta\\delta\\sim$20.25$\\arcsec$.\n\n-\\textit{Region 7} (R7): in a faint region to the north of the edge pointing where interesting properties were obtained in the 2D analysis: some pixels show c(H${\\beta}$)$>$2.5, the S\/N of the sulphur lines is very low so they were masked in several maps, the [N{\\sc ii}]\/H${\\alpha}$ ratio has its minimum values, and the kinematic study revealed that here the radial velocity increases opposite to the general trend. Seven spaxels were combined in this region to analyse the properties in detail. The offset from central star is $\\Delta\\alpha\\sim$27$\\arcsec$, $\\Delta\\delta\\sim$40.5$\\arcsec$.\n\n-\\textit{Region 8} (R8): six fibres were selected on the left of the discontinuity to checked whether this region has nebular emission. The offset from central star is $\\Delta\\alpha\\sim$47.25$\\arcsec$, $\\Delta\\delta\\sim$52.65$\\arcsec$.\\\\\n\n \n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{integrated_regions.pdf}\n\\caption{H${\\alpha}$ images of the two areas of M1-67 observed with PPAK. Boxes represent the eight regions where the integrated spectra were generated. For the offsets of each region from the central star (green cross), see the text. Orientations and sizes are as in Fig. \\ref{fig:morphology_all}. Edge on the left and centre on the right.}\n\\label{fig:integratedregions}\n\\end{figure}\n\nIn addition, another three integrated spectra were extracted to perform several tests. From the central spaxel of the central pointing FoV, we obtained the spectrum of WR124 (\\textit{Region WR}). The other two were extracted at $\\sim$15\\arcsec\\ to the NE of the star (common region in both pointings) to study the zone where \\citet{1981ApJ...249..586C} found emission in [O{\\sc iii}]$\\lambda$5007\\AA{} (\\textit{Regions S1 and S2} in the central and edge pointings, respectively). Figure \\ref{fig:integratedspectra} shows six representative 1D spectra from the 11 created. \\\\\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{integrated_all.pdf}\n\\caption{Examples of integrated spectra. From left to right and top to bottom: (a) Whole spectrum of \\textit{Region 3}. (b) Spectrum of \\textit{Region 5} in the range of H${\\alpha}$. (c) \\textit{Region 7}, in same range as \\textit{b}, where the absence of sulphur lines can be seen. (d) Spectrum of \\textit{Region 5} centred on the [N{\\sc ii}]$\\lambda$5755\\AA{} emission line, used to calculate the electron temperature. (e) Zoom over the spectrum in \\textit{Region S2} without any emission in the [O{\\sc iii}]$\\lambda$5007\\AA{} emission line.(f) Whole spectrum of WR124 obtained from the central spaxel. }\n\\label{fig:integratedspectra}\n\\end{figure*}\n\n\nFluxes of the main emission lines were measured by fitting Gaussian functions using SPLOT within IRAF. All the measured fluxes are in units of erg~cm$^{-2}$~s$^{-1}$ per fibre (area of fibre $\\sim$5.7~arcsec$^2$). Statistical errors were estimated using the formula presented  in \\citet{2003MNRAS.346..105P}:\n\\begin{equation}\n\\sigma_{\\mathrm{1}}=\\sigma_{\\mathrm{c}} N^{1\/2} [1+EW\/(N\\Delta)]^{1\/2}\n\\end{equation}\nwhere $\\sigma_{\\mathrm{1}}$ represents the error on the observed line flux, N is the number of pixels used to measure the line, EW the line equivalent width, $\\sigma_{\\mathrm{c}} $ the standard deviation in the continuum close to the line of interest, and $\\Delta$ represents the dispersion in \\AA{}\/pix.\\\\\n\nWe derived the reddening coefficient, c(H${\\beta}$), from the H${\\alpha}$\/H${\\beta}$ and H${\\gamma}$\/H${\\beta}$ line ratios using the procedure described in Sect. \\ref{maps}. In \\textit{Region 7}, H${\\gamma}$  was measured with low S\/N, and only the other two Balmer lines were used to estimate c(H${\\beta}$). The reddening coefficients agree with values obtained in the 2D study. Table \\ref{table:all_lines} lists the reddening-corrected fluxes of the emission lines measured in every zone labelled with their standard identification. The third column reports the adopted reddening curve using extinction law by \\citet{1989ApJ...345..245C} with $R_{\\mathrm{V}}=3.1$.  Errors in the emission line intensities were derived by propagating the observational errors in the fluxes and the reddening constant uncertainties. The estimated fluxes and errors were normalized to $F(H{\\beta})=100$. The values obtained for c(H${\\beta}$) are also presented in the last row of Table \\ref{table:all_lines}. \\\\\n\nFive integrated spectra deserve special attention. First, the R4 created in the NW dark area of the central pointing only showed three emission lines: H${\\alpha}$, [N{\\sc ii}]$\\lambda$6548\\AA{}, and [N{\\sc ii}]$\\lambda$6584\\AA{}. We deduce that, in areas out of the bipolar structure a faint, but not negligible, emission exists coming from the nebular gas rather than the ISM. We estimated the H${\\beta}$ flux by means of the reddening coefficient: assuming c(H${\\beta}$)=1.87 (the mean value of the other integrated spectra of this pointing), we performed the inverse process of the extinction correction and obtained F($H{\\beta}$)=3.21~10$^{-16}$~erg~cm$^{-2}$~s$^{-1}$. \n\nThe spectrum of R8 does not show any emission, thus physical and chemical properties could not be estimated here. We did not include this region in tables. We extracted a spectrum of the WR star (\\textit{Region WR}) shown in Fig.\\ref{fig:integratedspectra}f, and not perform a detailed analysis here. Finally, the study performed over \\textit{Regions S1 and S2} revealed typical nebular spectra (very similar to R4), but we did not find any emission of the [O{\\sc iii}]$\\lambda$5007\\AA{} line as seen in Fig. \\ref{fig:integratedspectra}e.\n\n\n\n\n\\subsection{Physical properties and chemical abundances \\label{prop_and_ab}}\nElectron density (n$_{\\mathrm{e}}$) was calculated from the [S{\\sc ii}]$\\lambda\\lambda$6717\/6731 line ratio using the IRAF package TEMDEN. The derived density ranges from $\\sim$1500~cm$^{-3}$ near the star, to $\\sim$650~cm$^{-3}$ towards the edge. These values are consistent with our 2D maps and with previous studies \\citep{1991A&A...244..205E,1998A&A...335.1029S}. \\\\\n\nElectron temperature, T$_{\\mathrm{e}}$, can be derived using the line ratio R$_{\\mathrm{N2}}$: \n\\begin{equation}\nR_{\\mathrm{N2}}={I([\\mathrm{N \\textsc{ii}}]\\lambda 6548)+I([\\mathrm{N \\textsc{ii}}]\\lambda 6584) \\over I([\\mathrm{N \\textsc{ii}}]\\lambda 5755)}.\n\\end{equation}\nThe [N{\\sc ii}]$\\lambda$5755\\AA{} auroral line that appears close to the \\textquotedblleft sky\\textquotedblright\\, line Hg{\\sc i} 5770\\AA{}, was detected in two zones (R5 and R6). We measured this line again in the spectra before sky subtraction and conclude that the flux of [N{\\sc ii}]$\\lambda$5755\\AA{} line in R6 is contaminated by the Hg{\\sc i} emission line, thus not reliable. We obtained a direct estimate of T$_{\\mathrm{e}}$([N{\\sc ii}]) from R$_{\\mathrm{N2}}$ only for R5.\\\\\n\nTo reinforce the validity of the chemical abundances estimations and to provide ionization correction factors (ICFs) for those species whose ionizations stages were not all observed in the optical spectrum, we performed photoionization models of R5. To do so, we used the code CLOUDY v.10 \\citep{1998PASP..110..761F}, assuming a central ionizing source from a WR star atmosphere \\citep{2002MNRAS.337.1309S} with $Z = 0.008$ and an effective temperature of the star of 45~000 K which are, respectively, the closest values to the measured total metallicity of the gas and the estimated temperature of WR124 \\citep{2006A&A...457.1015H}. \n\nWe considered a spherical geometry putting the gas at a distance of 1~pc from the star and assumed a constant density of 700~cm$^{-3}$, a value similar to the one derived from [S{\\sc ii}] emission lines. The model that fits the emission line intensities of [O{\\sc ii}], [O{\\sc iii}], He{\\sc i}, [N{\\sc ii}], and [S{\\sc ii}]  better in R5 was obtained by varying the ionization parameter (U) and the relative chemical abundances of He, O, N, and S. The emission lines from this model are listed in Table \\ref{table:all_lines}, while the derived physical properties and the ionic and total chemical abundances are listed in Table \\ref{table:paramyabun}. The ICFs obtained were ICF(N$^{+}$)=1.21 and ICF(S$^{+}$)=1.58. Regarding the resulting geometry, the final radius is 1.22~pc, which is of the same order of magnitude as the apparent size of the nebula in the images. \\\\\n\nTo estimate chemical abundances, electron density and electron temperature are required. We used T$_{\\mathrm{e}}$([N{\\sc ii}]) as temperature representative of the low ionization ions, S$^{+}$, N$^{+}$, and O$^{+}$, and T$_{e}$([O{\\sc iii}]) for  deriving the O$^{2+}$ and He$^{+}$ abundances. In those zones where the electron temperature was not calculated, we adopted the value of R5. In previous studies, T$_{\\mathrm{e}}$([N{\\sc ii}]) ranges from 5900~K \\citep{1998A&A...335.1029S} to 8000~K \\citep{1981ApJ...249..586C}; maybe, the supposition of T$_{\\mathrm{e}}$=8200~K leads to our abundances being underestimated.  Since the photionization model predicts T$_{\\mathrm{e}}$([N{\\sc ii}])$\\sim$8550~K and T$_{e}$([O{\\sc iii}])$\\sim$8330~K in R5, we considered T$_{\\mathrm{e}}$([N{\\sc ii}])$\\sim$T$_{e}$([O{\\sc iii}]) in the estimations. To infer abundances in R4 and R7, where sulphur lines were not measured, we adopted the electron density of R5, n$_{e}$=631~cm$^{-3}$. We checked that variations in density do not affect this estimation. \\\\\n\nIonic abundances were derived from the forbidden-to-hydrogen emission line ratios using the functional forms given by \\citet{2008MNRAS.383..209H}, which are based on the IRAF package IONIC. We used equations from \\citet{2004ApJ...617...29O} to obtain the singly ionized helium abundance. To determine the total abundance of O\/H we added the two ionic abundances (O\/H~$\\sim$ O$^{+}$\/H$^{+}$ + O$^{2+}$\/H$^{+}$). The total N\/H and S\/H abundances were inferred thanks to the ICFs obtained in the photoionization model, X\/H$\\sim$(X$^{+}$\/H) $\\times$ ICF(X$^{+}$). In the case of helium abundances we used the relation between X(O$^{2+}$)=O$^{2+}$\/(O$^{2+}$+O$^{+}$) and ICF(He$^{+}$+He$^{++}$) from \\citet[Fig. 7]{2007ApJ...662...15I} and we deduced that ICF(y$^{+}$)$\\gg$1. Since our helium measurements are uncertain, we do not venture to estimate the total helium abundances.\\\\\n\nIn R5 all the useful emission lines were measured and the abundances determined as explained above. In the rest of the regions, we did not measure all the necessary lines to calculate abundances, and we resorted to the empirical parameter N2S2 \\citep{2009MNRAS.398..949P} to estimate N\/O from the nitrogen and sulphur emission lines:\n\n\\begin{equation}\n\\log(N\/O)=1.26\\times N2S2 - 0.86\n\\end{equation}\nwhere\n\n\\begin{equation}\nN2S2=\\log \\left ({I([\\mathrm{N \\textsc{ii}}]\\lambda 6584) \\over I([\\mathrm{S\\textsc{ii}}] \\lambda\\lambda 6717,6731)} \\right).\n\\end{equation}\n\nBefore, we estimated the N\/O in R5 with the N2S2 parameter and checked that the result was in good agreement with the value obtained with the direct method. In Table \\ref{table:paramyabun} we present all the ionic and total abundances, with their corresponding errors, derived for the integrated spectra. We discuss the results in Sect. \\ref{chemical}.\n\n\n\n\n\\section{Infrared study \\label{ir}}\nTo enhance the morphological and chemical analysis, a study in the mid-infrared was performed. We obtained IRS (Infrared Spectrograph, \\citealt{2004ApJS..154...18H}) data in mapping mode and the MIPS (Multiband Imaging Photometer, \\citealt{2004ApJS..154...25R}) 24$\\,\\mu$m image from the Spitzer Heritage Archive (SHA)\\footnote{Website: sha.ipac.caltech.edu\/applications\/Spitzer\/SHA}. M1-67 has already been studied in the past in the infrared range by \\citet{1985A&A...145L..13V}. They presented the energy distribution of the central star WR\\,124 and flux densities, finding thermal emission of dust at T$_{\\mathrm{c}}\\sim$100~K.\\\\\n\nFigure \\ref{fig:spitz_24micr} shows the MIPS 24$\\mu$m image of M1-67. This image has already been presented by \\cite{2010MNRAS.405.1047G}. In a nebula, the origin of the 24$\\mu$m emission can be mainly due to two factors: presence of the [O{\\sc iv}]25.90$\\mu$m line from highly ionized gas or warm dust. Since M1-67 presents a low degree of ionization, we deduce that the observed emission shown in Fig. \\ref{fig:spitz_24micr} traces the warm dust distribution of the nebula. The emission has an elliptical shape along the NE-SW direction in very good agreement with the bipolar axis observed in Fig. \\ref{fig:morphology_all}, thus suggesting that the structure is composed of a mixture of ionized gas and warm dust. Furthermore, an external and spherical structure can be seen extending around the ellipsoidal shell. This faint bubble is not seen in our optical images.\\\\\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{spitzer24micr.pdf}\n\\caption{MIPS 24$\\mu$m image of M1-67. North is up and east left. Boxes indicate the two regions where IR spectra were obtained. Contourns represent the H${\\alpha}$ emission derived from Fig. \\ref{fig:rgb}.}\n\\label{fig:spitz_24micr}\n\\end{figure}\n\nFor the low-resolution short-low (SL) and long-low (LL) modules spectroscopic observations, basic calibrated data (BCD, pipeline version 18.18) were processed and analysed with the CUBISM software \\citep{2007ApJ...656..770S}. Data were background-subtracted using averaged off-source observations and flux-calibrated. Bad pixels were removed with the automatic correction routine within CUBISM and a datacube assembled for each module. CUBISM allows extracting of 1D spectra over polygonal apertures: given the different spatial coverage of the SL and LL module, we chose two apertures (with an area of $\\sim$60~arcsec$^2$) on the outskirts of the nebula, observed by both modules. The spectra from the different modules were stitched together, ignoring the noisy region at the red end of each order. We called them Regions A and B (see Fig. \\ref{fig:spitz_24micr}). In Fig. \\ref{fig:spitz_spec} we present the spectrum obtained in Region B.\\\\\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{spectrum_ir.pdf}\n\\caption{Infrared spectrum obtained in Region B. The most relevant lines are indicated.}\n\\label{fig:spitz_spec}\n\\end{figure}\n\nWe measured the most important lines by fitting Gaussian functions with IRAF. Errors were calculated as we explained before (Sect. \\ref{1d}). Fluxes and their corresponding errors are presented for the two regions in Table \\ref{table:infrared}.\\\\\n\nAssuming the electron temperature of \\textit{Region 5} (T$_{e}$=8200~K) and an electron density n$_{e}$=600~cm$^{-3}$, we inferred the chemical abundances. To obtain the fluxes relatives to H${\\beta}$ we used the theoretical ratio of H(7-6)\/H${\\beta}$=0.0109 from \\citet{1995MNRAS.272...41S}. The ionic abundances, Ne$^{+}$\/H$^{+}$, Ne$^{2+}$\/H$^{+}$, S$^{2+}$\/H$^{+}$, and S$^{3+}$\/H$^{+}$ were inferred by using the IRAF package IONIC. We estimated the total neon abundance by adding the two ionic abundances, Ne\/H$\\sim$Ne$^{+}$\/H$^{+}$+Ne$^{2+}$\/H$^{+}$. For deriving the total S\/H abundance we need to add the S$^{+}$\/H$^{+}$ from the optical spectra. To do so we compared the regions from which the IR and the optical 1D spectra were taken. Noticing the proximity between Region A and R3, we approximated the total sulphur abundance in the spectrum A as S\/H$\\sim$(S$^{+}$\/H$^{+}$)$_{R3}$ + (S$^{++}$\/H$^{+}$)$_{A}$. The 1D spectrum nearest to B is R4, but in R4 we did not measure sulphur lines. Since the S$^{+}$\/H$^{+}$ is similar in all the integrated spectra, we considered the mean value, (S$^{+}$\/H$^{+}$)$_{mean}$=6.17, so that the total sulphur abundance in B can be written as S\/H$\\sim$(S$^{+}$\/H$^{+}$)$_{mean}$ + (S$^{++}$\/H$^{+}$)$_{B}$. We assumed that S$^{3+}$\/H$^{+}$ is negligible. Results are presented in Table \\ref{table:abinfrared} and discussed in Sect. \\ref{chemical}.\n\n\n\\begin{table}[h!]\n\\caption{Lines measured over the two spectra studied in the infrared range. Integrated fluxes are in units of 10$^{-5}$~erg~cm$^{-2}$~s$^{-1}$. }\n\\label{table:infrared} \n\\centering \n\\begin{tabular}{l c c c}\n\\hline\n&&\\multicolumn{2}{c}{F($\\lambda$)} \\\\\n\\cline{3-4}\nLine & $\\lambda$~($\\mu$m) & Region A & Region B \\\\\n\\hline \\hline \\\\\n{[}S{\\sc iv}] & 10.51 & ...& 4.1 $\\pm$ 0.5 \\\\\nH(7-6) & 12.37 & 4.9 $\\pm$ 0.5 & 5.1 $\\pm$ 1.0 \\\\\n{[}Ne{\\sc ii}] & 12.81 & 121.7 $\\pm$ 3.2 &105.3 $\\pm$ 3.5 \\\\\n{[}Ne{\\sc iii}] & 15.56 & 5.0 $\\pm$ 0.4 & 1.1 $\\pm$ 0.3 \\\\\n{[}S{\\sc iii}] & 18.71 & 133.9 $\\pm$ 5.5 & 99.2 $\\pm$ 1.9 \\\\\n{[}S{\\sc iii}] & 33.48 & 156.6 $\\pm$ 5.3 & 135.2 $\\pm$ 4.7 \\\\\n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[h!]\n\\caption{Ionic and total chemical abundances estimated in Regions A and B with infrared spectroscopy.}\n\\label{table:abinfrared} \n\\centering \n\\begin{tabular}{l c c}\n\\hline\n& Region A & Region B \\\\\n\\hline \\hline \\\\\n12+log(Ne$^{+}$\/H$^{+})$ & 7.56 $\\pm$ 0.04 & 7.47 $\\pm$ 0.08 \\\\\n12+log(Ne$^{2+}$\/H$^{+})$ & 5.85 $\\pm$ 0.05 & 5.18 $\\pm$ 0.16 \\\\\n12+log(S$^{2+}$(18.71$\\mu$m)\/H$^{+}$) & 6.59 $\\pm$ 0.09 & 6.44 $\\pm$ 0.17\\\\\n12+log(S$^{2+}$(33.48$\\mu$m)\/H$^{+}$) & 6.64 $\\pm$ 0.16 & 6.56 $\\pm$ 0.21\\\\\n12+log(S$^{3+}$\/H$^{+}$) & ... & 4.41 $\\pm$ 0.19 \\\\\n\\\\\n12+log(Ne\/H) & 7.57 $\\pm$ 0.04 & 7.48 $\\pm$ 0.08\\\\\n12+log(S\/H)$\\dagger$ & 6.72 $\\pm$ 0.07 & 6.63 $\\pm$ 0.11  \\\\\n\\hline\n\\end{tabular}\n        \\begin{list}{}{}\n\t\t\t\\item {$\\dagger$} Assuming S$^{+}$\/H$^{+}$ from the optical spectroscopy.\\\\\n\t\t\\end{list}\n\n\\end{table}\n\n\n\n\n\\section{Dicussion \\label{discussion}}\nWe included M1-67 in our IFS observational programme to provide answers to some questions that still surround this object: degree of gas homogeneity (both kinematic and chemical), stellar evolutionary phase origin of the gas, interaction with the ISM, influence of the star spectral type at WR stage, etcetera. To do this, we put together our results for the optical (1D + 2D) and infrared analysis, and we complemented them with theoretical models of stellar evolution and previous kinematic studies of this nebula.\n\n\n\\subsection{Chemical content of M1-67 \\label{chemical}}\nThe chemical abundances derived from the 1D optical and infrared studies, presented in Tables \\ref{table:abinfrared} and \\ref{table:paramyabun}, give us relevant information on the chemical content across the nebula. To compare the derived abundances with the expected ISM values at the location of the nebula, we use the solar values from \\citet{2009ARA&A..47..481A} as our primary reference. For the sake of consistency as reference for M1-67, we consider here gas abundances derived following the same methodology, i.e. H{\\sc ii} region collisional emission lines. We adopted the chemical abundances of the prototypical H{\\sc ii} region M\\,42 as a reference (\\citealt{2007A&A...465..207S,2011MNRAS.412.1367T} with t$^{2}$=0, and references therein). Then, we corrected the t$^{2}$=0 abundances from the effect of the radial abundance gradient of the Milky Way \\citep{2006ApJS..162..346R} to the galactocentric radius of M1-67\\footnote{Assumed R$_{G}\\sim$10 kpc as the the galactocentric distance of the representative ISM at the location of M1-67 \\citep{1992A&A...259..629E} and taking the distance from the Sun to M\\,42 into account (d$\\sim$0.414\\,kpc, \\citealt{2007A&A...474..515M}).}. We considered the constant ratio $\\log \\mathrm{(Ne\/O)}$=-0.73$\\pm$0.08 since they are products of the same nucleosynthesis. After these corrections the expected ISM abundances to be compared with M1-67 are 12+$\\log \\mathrm{(O\/H)}\\sim$8.42$\\pm$0.03, 12+$\\log \\mathrm{(N\/H)}\\sim$7.54$\\pm$0.09, 12+$\\log \\mathrm{(S\/H)}\\sim$6.99$\\pm$0.12, and 12+$\\log \\mathrm{(Ne\/H)}\\sim$7.69$\\pm$0.09.\\\\\n\n\nFirst of all, it can be observed that our derived oxygen abundances in R5 and R6 (12+$\\log \\mathrm{(O\/H)}\\sim$7.73$\\pm$0.06, 7.67$\\pm$0.07, respectively) are substantially lower than the expected value by factors $\\sim$10 with respect to the solar reference, and $\\sim$7 with respect to the ISM. This result implies that in the M1-67 nebula oxygen is strongly under-abundant. Comparing the derived N\/H abundance with the expected ISM value, we find that nitrogen is strongly enriched in M1-67 (factor $\\ge$ 6). \n\n\nOverall, this chemical composition can be seen in all the nebular regions observed; the N\/O ratio appears extremely high due to the effect of both nitrogen enhancement and oxygen deficiency. This fact can be understood when assuming we are seeing regions composed of material processed in the CNO cycle. This result for N\/O abundance is consistent with previous 1D studies \\citep{1991A&A...244..205E}, but here it has been extended across the whole (2D) nebular geometry and physical conditions. The only region where the N\/H abundance is close to the ISM expected value is R7 (the region with different properties in the 2D analysis, see Sect. \\ref{2d}).\n\nWe did not estimate the total helium abundances since our helium lines are very faint and the measures uncertain. Nonetheless, given the low limit of the value of He{\\sc i} ($<$0.03), the absence of He{\\sc ii} and the ICF inferred from \\citet{2007ApJ...662...15I} (ICF(y$^{+}$)$\\gg$1), we deduced that in M1-67 the largest part of helium is unseen and in neutral form. \\\\\n\nThe analysis of the chemical abundances obtained here is reinforced by the information derived from the infrared study. The infrared spectrum allowed us to derive the sulphur and neon abundances for the main ionic species, Ne$^{+}$, Ne$^{++}$, S$^{++}$, and S$^{+3}$. The total neon abundance derived in M1-67 is consistent within the errors with the expected ISM abundance for the two apertures (Table \\ref{table:abinfrared}). The noble gas neon is not expected to suffer nucleosynthetic transformation in the stellar interior, and its abundance should be preserved.\n\nIn the case of sulphur, the derivation of the total abundance requires the contribution of the optical S$^{+}$ to be added to the ionic fractions derived from the infrared. Once this approximation has been assumed, the total abundance of S\/H obtained is close to, though still slightly lower than, the expected ISM value at the galactocentric distance of M1-67. Thus we cannot rule out the possibility that the nebular material could be slightly sulphur-poor: either a certain degree of depletion on dust or maybe a nucleosynthetic origin (or both) could be at work as reported for some planetary nebulae \\citep{2012ApJ...749...61H}.\\\\\n\nTaking the abundance ratios into account, we can obtain clear indications of the excitation degree of the nebula. The values N$^{+}$\/N~$\\sim$1 and O$^{+}$\/O$^{++}$~$>$1 from the optical and the derived ratios of Ne$^{+}$\/Ne$^{++}$ and S$^{++}$\/S$^{3+}$ from the IR study point to the very low ionization degree of the gas in M1-67. The ionization parameter obtained from the photoionization model of R5, $\\log\\mathrm{(U)}=-3.84$, is fully consistent with this very low excitation observed.\\\\\n\n\nTo provide a summary of the chemical abundances obtained across the nebula in the optical and infrared ranges, we have grouped regions with similar physical and chemical properties (whenever possible). In Table \\ref{table:summary} we show the results: $<$1,2,3$>$ represents the average of R1, R2, and R3, $<$5,6$>$ the average of R5 and R6, and $<$A,B$>$ the average of zones A and B from the IR study. In these cases the corresponding parameters were estimated as the mean weighted by the error in each zone. The two last columns represent the expected ISM values and solar abundances from \\citet{2009ARA&A..47..481A}, respectively.\n\n\\begin{table*}\n\t\t \\caption{Summary of inferred properties in M1-67.}   \n\t\t\\label{table:summary}    \n\t\t\\centering                      \n\t\t\\begin{tabular}{l c c c c c c c}\n\t\t\\hline\n\t\t& $<$1,2,3$>$ & 4  & $<$5,6$>$ & 7 & $<$A,B$>$ & ISM$^{a}$  & Solar$^{b}$ \\\\\n\t\t\\hline\n\t\t\\hline\n   \t\t\\\\\n\t\tc(H${\\beta}$) &  1.87 $\\pm$ 0.01  & 1.87 $\\pm$  0.01 & 1.90 $\\pm$ 0.02 & 2.15 $\\pm$ 0.04 & ...  & ... & ... \\\\\n\t\tn$_{\\mathrm{e}}$([S{\\sc ii}]) (cm$^{-3}$)& 1581 $\\pm$ 49 & ... & 677 $\\pm$ 62 & ...  & ...  & ...  & ...\\\\\n\t\t12+log(O\/H)  & ...  &... & 7.70 $\\pm$  0.03  & ... & 8.28 $\\pm$ 0.09 $^{c}$  & 8.42$\\pm$ 0.03 & 8.69 $\\pm$ 0.05\\\\\n\t\t12+log(S\/H)  & 6.35 $\\pm$ 0.02 & ... & 6.40 $\\pm$ 0.02   & ...  & 6.69 $\\pm$ 0.04 & 6.99$\\pm$ 0.12  & 7.12 $\\pm$ 0.03\\\\\n\t\t12+log(N\/H) & 8.13 $\\pm$ 0.01 & 8.36 $\\pm$ 0.03  & 8.21 $\\pm$ 0.03 & 7.92 $\\pm$ 0.03 & ... & 7.54 $\\pm$ 0.09 & 7.83 $\\pm$ 0.05  \\\\\n\t\t12+log(Ne\/H) & ...  & ... & ...  & ...  & 7.55 $\\pm$ 0.04 & 7.69 $\\pm$ 0.09  & 7.93 $\\pm$ 0.10\\\\\n\t\t$\\Delta$(log(N\/H))$^{d}$ & 0.59 $\\pm$ 0.09 & 0.82 $\\pm$ 0.09 & 0.67 $\\pm$ 0.10 & 0.38 $\\pm$ 0.09 & ...  & ...& ...  \\\\\n\t\t$\\Delta$(log(O\/H))$^{d}$& ...  & ...  & -0.72 $\\pm$ 0.04 & ...  & -0.14 $\\pm$ 0.09 $^{c}$  & ... & ... \\\\\n\t\t\\hline\n\t\t\\end{tabular}\n        \\begin{list}{}{}\n\t\t\t\\item {$^{a}$} Expected ISM abundances at R$_{G}\\sim$10 kpc. \\\\\n\t\t\t\\item {$^{b}$} Solar abundances from \\citet{2009ARA&A..47..481A}.\\\\\n\t\t\t\\item {$^{c}$} Estimated assuming $\\log \\mathrm{(Ne\/O)}$=-0.73$\\pm$0.08.\\\\\t\n\t\t\t\\item {$^{d}$} Variations with respect to the expected ISM abundance. \\\\ \n\t\t\\end{list}\n\t\t\\end{table*}\n\n\n\n\n\n\\subsection{M1-67 structure \\label{structure}}\nAlthough the first observations of M1-67 showed a nearly spherical shape, the high contrast achieved by coronographic studies in the inner regions made a bipolar symmetry clearly visible \\citep{1995IAUS..163...78N}. Owing to the field of view of our PPAK observations, we cannot detect this bipolarity; however, the narrow-band images from the INT and the interpolated maps from PPAK (see Figs. \\ref{fig:rgb} and \\ref{fig:morphology_all}) show that the bright knots are aligned along a preferred axis with \\textquotedblleft holes\\textquotedblright ~in the perpendicular direction. The integrated spectrum of R4, confirms that the emission in the holes is very faint (i.e. H${\\beta}$ was not detected). Furthermore, the MIPS image from Spitzer (Fig. \\ref{fig:spitz_24micr}) also reveals the bipolar appearance at 24$\\mu$m, suggesting that the ionized gas is mixed with warm dust. We emphasize that the knots are not only regions with high surface brightness but also very dense areas where the [N{\\sc ii}]\/H${\\alpha}$ and [N{\\sc ii}]\/[S{\\sc ii}] ratios show the maximum values.\\\\\n\nWe support the idea of a preferred axis, but either way, is the bipolarity the footprint of an ejection from the star? Looking at the radial velocity map of Fig. \\ref{fig:ha_velocity}, we can see that velocity decreases when we move away from the centre (except in the far NE where the gas has peculiar properties, see below). This agrees with the studies from \\citet{1981ApJ...249..586C}, who predict a faster movement near the star, and \\citet{1998A&A...335.1029S} who discuss the idea of a bipolar outflow. The spatial distribution of the electron density shows similar behaviour: the mean values of maps and integrated spectra of the central pointing are higher than at the edge ($\\sim$1500~cm$^{-3}$ and $\\sim$650~cm$^{-3}$, respectively). Also the electron density decreases along the radial cut seen in Fig. \\ref{fig:dens_rad} with a symmetric gradient in two directions (from the centre to NE and to SW) and flattening towards the edges. Both analyses lead us to think that the preferred axis is not only morphological, but is also the footprint of a mechanism that could have expelled material in the past and, later, interacted with the ISM diluted and decelerated the gas.\\\\\n\nLeaving aside for a moment the discussion of bipolarity, there is another striking morphological feature in this object. The IR study at 24$\\mu$m reveals a spherical bubble surrounding the bipolar structure. Kinematic studies from \\citet{1998A&A...335.1029S} show two different motions in the enviroment of WR124: a bipolar outflow and an external spherical hollow shell expanding into the ISM. In our narrow-band images from the INT, this bubble is not detected possibly because the material is diluted in the ISM and very weak in the optical range. A simple sketch of the proposed structure of M1-67 is presented in Figure \\ref{fig:sketch}: an inner region with bipolar or elliptical shape along the direction NE-SW surrounded by an external spherical bubble.\\\\\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{sketch.pdf}\n\\caption{Sketch showing the structure of M1-67 around the central star WR124: the bipolar axis along the direction NE-SW and the spherical bubble.}\n\\label{fig:sketch}\n\\end{figure}\n\nThe study of the edge pointing region suggests that the gas in the NE possesses different properties from the gas of the bipolar outflow. In summary, the properties found for this region are a) the largest reddening coefficient of the nebula with c(H${\\beta})>$2.5; b) the only area where we measure [N{\\sc ii}]$<$H${\\alpha}$ and with the smallest N\/H abundance estimated, close to the solar neighbourhood value; c) increase rather than decrease in the relative radial velocity; d) absence of [S{\\sc ii}]$\\lambda\\lambda$6717\/6731emission lines; e) the minimum H${\\beta}$ flux measured (R7) and lack of [O{\\sc iii}]$\\lambda$5007 or helium lines. The presence of these properties puzzles us, but here we propose a possible scenario to explain the origin of this region. The nitrogen abundance of our \\textquotedblleft peculiar\\textquotedblright ~region points towards material not processed in the CNO cycle (e.g. ISM or MS bubble) while the morphology (Fig. \\ref{fig:rgb}) and kinematics (Fig. \\ref{fig:ha_velocity}) suggest that it does not belong to the bipolar ejection. When looking at the bow-shock simulations of \\citet{2003A&A...398..181V} and taking the external IR bubble into account, it is possible that the high velocity of the runaway WR124 causes a paraboloid-like bow shock around the star sweeping up the surrounding medium, so that we are seeing the remaining of this bow-shock placed in our line of sight. We should bear in mind that the peculiar region is spatially close to the small reversed bow-shock like structures found by \\citet{2001ApJ...562..753G} at the NE periphery of M1-67.\n\n\n\n\\subsection{M1-67: a consequence of the evolution of the central star WR124 \\label{evolution}}\nThe theory of evolution of massive stars can help us explain the observed structure. We compare the stellar parameters from the central star in M1-67 (effective temperature and luminosity from \\citealt{2006A&A...457.1015H}) with the stellar evolution models from STARS \\citep{1971MNRAS.151..351E,1995MNRAS.274..964P,2004MNRAS.353...87E}, \\citet{2003A&A...404..975M}, and the most recent models from \\citet{2012A&A...537A.146E} to estimate the initial mass of the WR star. Regardless of small discrepancies, all the models predict initial mass for WR124 of 60-80~M$\\sun$ (J. Toal\\'a, \\textit{private communication}). The evolutionary scenario for a single massive star with 60~M$\\sun<$M$_{\\mathrm{i}}<$90~M$\\sun$ follows the sequence O-Of\/WNL$\\longleftrightarrow$LBV-WNL-WCL-SN \\citep{2011BSRSL..80..266M}. After spending a normal life as O stars on the main sequence (MS), they evolve towards cooler temperatures becoming luminous blue variables (LBVs) \\citep{1994PASP..106.1025H}. These stars undergo extremely strong mass loss (up to 10$^{-3\\ldots -4}~$M$\\sun$~yr$^{-1}$) through winds and occasionally giant eruptions, and thus peel off parts of their stellar envelope to form a small LBV nebulae (LBVN) \\citep{1995ApJ...448..788N}. LBV stars lose their mass so fast that they rapidly evolve away from LBV stage to become WR stars. With an initial mass range of 60-80~M$\\sun$, we can derive that the central star in M1-67 has experienced an LBV phase instead of a red or a yellow supergiant phase before becoming a WR star. This idea is in good agreement with previous studies of the nature of M1-67 based on different observational approaches: M1-67 is very likely the imprint of a previous LBV wind instead of a classical red super-giant (RSG) wind-blown nebula \\citep{1998ApJ...506L.127G, 2003A&A...398..181V}.\\\\\n\nThe spectral type of the central star (WN8), tells us that it is a \\textquotedblleft young\\textquotedblright  ~Wolf-Rayet and that it has most likely entered the WR phase recently. Under this hypothesis, we propose that the WR winds have not had enough time to substantially interact with the previous nebular material and, therefore, the layers and observed features originate in stellar material ejected during the MS and\/or LBV phases. Considering a representative expansion velocity and the linear size of the nebula, we estimate that the ejection happened $\\sim$5$\\times$10$^{4}$~yr ago. This value is slightly higher than the LBV phase duration ($\\sim$1.3$\\times$10$^{4}$~yr, \\citealt{1996A&A...305..229G}) thus supporting the hypothesis that the star has recently entered the WR phase.\\\\\n\nTaking the physical sizes and morphologies from an hydrodynamical simulations of a 60 M$\\sun$ star as reference \\citep{1996A&A...305..229G}, it is possible that the external bubble of M1-67 contains material expelled during the MS phase, which is very tenuous in the optical because of the dilution with the ISM. \\citet{Castor1975} and \\citet{Weaver1977} both built models the derive analytical solutions for the dynamic evolution of shock bubbles created by interaction between the ISM and the stellar wind in the MS phase.\\\\ \n\nSeveral observational reasons have led us to think that the bipolar ejection (or axis of preference) is composed by material ejected during the LBV stage. First, the abundances in the knots along this axis show enrichment in nitrogen and deficiency in oxygen, a behaviour typical of CNO-processed material in phases after the MS stage\\footnote{We should bear in mind that models that include rotation \\citep{Meynet2005} and recent observations of O stars in the LMC \\citep{2012A&A...537A..79R} have revealed that some stars of the MS stage can also show CNO-processed material.}. It is common in observations of LBVN to find very intense [N{\\sc ii}] emission and absence of [O{\\sc iii}] \\citep{1995ApJ...448..788N,1998ApJ...503..278S,2002A&A...393..503W}, this being also indicative of a low effective temperature and low degree of excitation. Furthermore, many of these nebulae show clumpy radial structures (not multiple shells) and morphologies with preferred axes \\citep{1993ApJ...410L..35C}. The presence of a bipolar ejection in M1-67 enhances the similarity of the nebula to other LBVN, which almost all display some degree of bipolarity \\citep{1995ApJ...448..788N, 2001RvMA...14..261W}. In short, M1-67 shows the general properties of LBV nebulae: linear size, total ionized gas, velocity field, IR emission, chemical abundances, line intensities, and dynamical characteristics; this clearly points to an LBV progenitor \\citep[among others]{1995ApJ...448..788N, 2001ApJ...551..764L,2011BSRSL..80..440W}.\n\nThe idea of M1-67 being made up of material ejected during the LBV stage was suggested in the past by \\citet{1998A&A...335.1029S} based on the total mass of ionized gas, the expansion velocity, and the linear size of the nebula. Also \\citet{1998ApJ...506L.127G} explain the clumpy appearance of M1-67 by assuming the interaction of winds in a previous LBV phase.\\\\\n\nOur study depicts M1-67 as a nebula with two regions: an external spherical bubble with material likely produced during the MS and an inner nearly elliptical region along the NE-SW direction produced due to an ejection in the LBV phase. We are observing a WR nebula with LBVN appearance.\\\\\n\n\n\n\n\\section{Summary and conclusions \\label{conclusions}}\n\\renewcommand {\\labelenumi} {\\arabic {enumi}$)$}\n\\renewcommand {\\labelenumii} {$\\bullet$}\nIn this work, we have presented the first integral field spectroscopy study of the ring-nebula M1-67 around the Wolf-Rayet star WR124 in the optical range with PPAK. Two regions of the nebula were observed and analysed by means of 2D and 1D studies. We also obtained and analysed IR spectroscopic data and the MIPS 24$\\mu$m image of M1-67 from Spitzer. In the following, we present the main results derived from this work.\n\n\\begin{enumerate}\n\\item We obtain maps from the emission lines that allow us to perform a detailed study of the 2D structure of the nebula:\n\\begin{enumerate}\n\\item Interpolated maps from the main emission lines show a clumpy structure with bright knots aligned along a preferred axis in the NE-SW direction. The [O{\\sc iii}]$\\lambda$5007\\AA{} emission is absent over the whole nebula.\n\\item The spatial distribution of the reddening coefficient maps, c(H${\\beta}$), presents slight variations between the two pointings. In the central region c(H${\\beta}$) ranges from 1.3 to 2.5 with a mean value of $\\sim$1.85, while in the edge pointing the mean is 2.11, ranging from 1.7 to 2.8.\n\\item Electron density maps, n$_{e}$, derived from the [S{\\sc ii}]$\\lambda\\lambda$6717\/6731 ratios, show a non-uniform structure. Knots with higher surface brightness in H${\\alpha}$ possess the highest densities. We also find that density decreases with increasing the distance from the star showing a symmetric gradient.\n\\item We analysed the ionization structure by means of line ratios maps. In particular, the [N{\\sc ii}]\/H${\\alpha}$ map of the edge pointing field reveals two behaviours, thus defining two spatially well delimited regions: one in the NE with [N{\\sc ii}]$<$H${\\alpha}$ and the second one with [N{\\sc ii}]$\\ge$H${\\alpha}$.\n\\item With radial velocity maps we studied the kinematics of the nebula. The derived heliocentric velocity for M1-67 is $\\sim$139~km~s$^{-1}$, in agreement with previous results. The relative radial velocity seems to decrease as it moves away from the central star along the preferred axis. \\\\\n\\end{enumerate}\n\n\\item We derived the physical parameters and chemical abundances of M1-67 using the integrated spectra of eight regions:\n\\begin{enumerate}\n\\item The electron densities inferred on the central region present higher values than on the edge ($\\sim$1500~cm$^{-3}$ and  $\\sim$650~cm$^{-3}$, respectively). This result agrees with the radial variations of the 2D study.\n\\item We derived an electron temperature of $\\sim$8200~K  in R5 by using our measurement of the [N{\\sc ii}]$\\lambda$5755\\AA{} emission line.\n\\item The chemical abundances show, in all the studied areas, an enrichment in nitrogen and a deficiency in oxygen. The nitrogen enhancement in each region is different, suggesting an inhomogeneous chemical enrichment.\\\\\n\\end{enumerate}\n\n\\item The 24$\\mu$m image reveals an inner bipolar-like structure in the NE-SW direction and an outer faint spherical bubble interacting with the surrounding ISM. From the low-resolution mid-IR spectroscopic data, we measured the main emission lines and estimate ionic and total chemical abundances, verifying the low ionization degree of the gas. \\\\\n\n\\item Overall, this study revealed the clumpy structure of M1-67 with knots aligned along a preferred axis and with \\textquotedblleft holes\\textquotedblright~ along the perpendicular direction. The gas along this bipolar axis possesses a low ionization degree, and it is well mixed with warm dust. The optical analysis of these knots revealed chemical abundances typical  of material processed in the CNO cycle, suggesting that the material comes from an evolved stage of the star. The radial variations in electron density and velocity indicate that the gas of the bipolar feature was ejected by the star. \\\\\n\n\\item A region placed to the NE of the nebula shows different kinematic, chemical, and morphological properties. We propose that this region comprises the remaining of a bow-shock caused by the runaway WR124 with ISM material mixed up with the MS bubble.\\\\\n\\end{enumerate}\n\nBased on our observational results and taking theoretical models from the literature into account (e.g. \\citealt{1996A&A...305..229G}), we propose a scenario where the central star has recently entered the WR phase. This implies that the interaction of WR winds with previous surrounding material is not visible yet. After comparing our results with stellar evolution models and taking the inferred initial mass of the star (60~M$\\sun < $ M$_{i} <$ 80~M$\\sun$)  into account, we deduced that the central star experienced an LBV stage before becoming a WR. The bipolar material observed belongs to an ejection during the LBV stage since the morphology, kinematics, and chemistry are in good agreement with previous studies of LBV nebulae.\\\\\n\n\n\n\n\n\\begin{acknowledgements}\nThis work is supported by the Spanish Ministry of Science and Innovation (MICINN) under the grant BES-2008-008120. This work has been partially funded by the projects: AYA2010-21887-C04-01 of the Spanish PNAYA and CSD2006 - 00070 \"1st Science with GTC from the CONSOLIDER 2010 programme of the Spanish MICINN and TIC114 of the Junta de Andaluc\\'ia. We thank J. Toal\\'a for providing estimations for the initial mass of the WR star and for useful suggestions. We are also very grateful to M. Fern\\'andez-Lorenzo, A. Monreal-Ibero, K. Weis, and the ESTALLIDOS collaboration for their useful comments and scientific support.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nOpen source projects have an increasing importance in modern software development.\nFor example, several open source projects are daily used by millions of users.\nHowever, it is very important to continually attract more participants and contributors to these projects, in order to increase the chances of long-term success~\\cite{comino2007}.\nParticularly, several channels can be used to promote open source software, helping to keep the interest of the community and also to attract new members.\n\nIn this article, we investigate the most common channels used by developers to promote open source projects.\nWe manually inspected a large set of popular projects on GitHub, which is the world's largest collection of open source software, with around 27 million users and 77 million repositories~\\cite{githubsearch}.\nOur contributions include: (i) data about the promotion channels frequently used by popular open source projects; (ii) a comparison on the use of promotion channels by popular projects and by random ones; and (iii) an analysis of the impact of promotion on Hacker News, a popular news aggregation site, in the popularity of the studied projects.\nOur findings help practitioners to understand the importance of using promotion channels in the open source development context.\n\n\\section{Study Design}\n\\label{sec:design}\n\nTo reveal the most common promotion channels used by developers, we manually inspected the documentation of the top-100 projects with most stars on GitHub (stars is a popular feature to manifest interest or satisfaction with GitHub projects~\\cite{icsme2016}).\nWe restricted our analysis to popular projects because they have a large number of users and therefore need better and efficient ways to communicate with users and also to attract new contributors.\n\nFigure~\\ref{fig:repos-overview} shows the distribution of the number of stars of the projects considered in this study.\nThis number ranges from 291,138 stars ({\\sc \\mbox{freeCodeCamp\/freeCodeCamp}}) to 23,322 stars ({\\sc \\mbox{tiimgreen\/github-cheat-sheet}}).\nThe considered projects are primarily developed on 17 programming languages; JavaScript is the most common one (40 projects), followed by Python (9 projects) and Go (5 projects).\nFurthermore, 14 projects only include markdown files with documentation purposes (e.g., projects with tutorials, books, awesome lists, etc).\nFinally, regarding the project owners, 69 are organizational accounts and 31 are user accounts.\\medskip\n\n\\begin{figure}[!ht]\n\t\\center\n\t\\includegraphics[width=0.65\\textwidth,keepaspectratio,trim={0 2em 0 2em},clip]{images\/repos_overview.pdf}\n\t\\caption{Number of GitHub stars of the analyzed projects}\n\t\\label{fig:repos-overview}\n\\end{figure}\n\nFor each of these 100 projects, the first author of this paper initially inspected their READMEs on GitHub to identify the channels used to promote the projects and to keep the users up-to-date with important information about them.\nFor example, the following sentence is available on the README of {\\sc adobe\/brackets}: \\aspas{\\it You can see some screenshots of Brackets on the \\underline{wiki}, intro videos on \\underline{YouTube}, and news on the Brackets \\underline{blog}}.\nIn this case, wiki and YouTube are used to support users whereas blog is a channel used to disseminate news about {\\sc Brackets}.\nThus, only blog is considered a promotion channel in our study.\nNext, we inspected the projects' website, for those projects having one.\nWe navigated through the site pages, searching for more channels used to promote the projects.\n\nAfter this manual inspection, the following promotion channels emerged:\n\n\\begin{itemize}\n\t\\item {\\bf Blogs}, which are used, for example, to publish announcements of new software versions, upcoming events, and improvements.\n\n\t\\item {\\bf Events and Users Meetings:} Organizing events and supporting users meetings are other strategies commonly followed to promote projects. On events the initiative usually comes from the development team or from the organization that supports the project, whereas on user meeting the initiative comes from the users, usually from a specific region or country. We rely on Meetup (\\url{https:\/\/meetup.com}) to discover users meetings.\n\n\t\\item {\\bf Twitter, Facebook, and Google+}, which are also used to connect the projects to users. We considered only official accounts, which are explicitly advertised on the project documentation or are verified by the social network (e.g., \\url{https:\/\/support.twitter.com\/articles\/20174631}).\n\n\t\\item {\\bf Newsletter and RSS feeds}, which refer to e-mails with the most relevant news about the projects and RSS feeds.\n\n\\end{itemize}\n\nIn addition, we found that developers use Q\\&A forums (e.g., StackOverflow), discussion groups (e.g., Google Groups), and messaging tools (e.g., IRC and Slack) to promote their projects.\nHowever, these channels are mostly used to discuss the projects and to provide answers to common questions raised by users.\nFor example, from the 155 topics opened in 2017 in the {\\sc adobe\/brackets} discussion group at Google Groups, only eight (5.1\\%) are related to announcements of new versions, mostly pre-releases for community testing.\nMoreover, from almost 500 topics on {\\sc facebook\/react} official forum, we could not identify any announcement related to the project development.\nThus, in this study, we do not consider forums, discussion groups, and messaging tools as promotion channels.\n\n\\section{Results}\n\\label{results}\n\n\n\\subsection{What are the most common promotion channels?}\n\\label{sec:results:rq1}\n\nFigure~\\ref{fig:rq1} presents the most common promotion channels used by the top-100 projects on GitHub.\nThe most common channel is Twitter, which is used by 56 projects.\nThe second one is Users Meetings (41 projects), followed by Blogs (38 projects), Events (33 projects), and RSS feeds (33 projects).\nThe least common channels are Facebook and Google+, which are used by 18 and 7 projects, respectively.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth,keepaspectratio,trim={0 1em 0 3em},clip]{images\/channels_binary.pdf}\n\t\\caption{Most common promotion channels}\n\t\\label{fig:rq1}\n\\end{figure}\n\nFigure~\\ref{fig:rq1_2} shows the distribution of the number of promotion channels per project.\nAlmost one third of the projects (32 projects) do not use any channel.\nBy contrast, more than half of the projects (55 projects) use at least two promotion channels.\nThe highest number of promotion channels is seven, which is the case of {\\sc \\mbox{facebook\/react}}, {\\sc \\mbox{facebook\/react-native}}, {\\sc \\mbox{meteor\/meteor}}, {\\sc \\mbox{golang\/go}}, {\\sc \\mbox{ionic-team\/ionic}}, {\\sc \\mbox{angular\/angular}}, and {\\sc adobe\/\\\\brackets}.\nWe also found that Blog and Twitter is the most frequent combination of channels (35 projects).\nOther frequent combinations include, for example, Blog and RSS (31 projects), Events and Users Meetings (31 projects), and Twitter, Events and User Meetings (31 projects).\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.725\\textwidth,keepaspectratio,trim={0 1em 0 3em},clip]{images\/channels_binary_histogram.pdf}\n\t\\caption{Number of promotion channels per project}\n\t\\label{fig:rq1_2}\n\\end{figure}\n\n\n\\subsection{How often do developers promote their projects?}\n\\label{sec:results:rq2}\n\nIn this second question, we investigate how often developers promote their projects on blogs and social networks.\nFor blogs, we calculate the promotion frequency as the number of posts on the last 12 months.\nFor social networks, we could not retrieve all posts for all projects because their  APIs restrict the search to a recent period (e.g., last seven days for Twitter and last 100 posts for Facebook).\nThus, in this case, we only classified each social network account in two distinct groups: active and inactive.\nAn {\\em active} account has at least three posts on the last three months; otherwise, it is considered an {\\em inactive} account.\nThis classification was performed by manually counting the number of posts on the social network pages.\n\nFigure~\\ref{fig:rq2} presents the distribution of the number of blog posts on the last 12 months.\nThe number ranges from 1 ({\\sc nylas\/nylas-mail}) to 1,300 ({\\sc freeCodeCamp\/freeCodeCamp}); the first, second, and third quartile values are 7, 19, and 54 posts, respectively.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=.325\\linewidth,keepaspectratio,trim={0 2em 0 2em},clip,page=2]{images\/activity_blog.pdf}\n\t\\caption{Distribution of the number of posts on the last 12 months (outliers are omitted)}\n\t\\label{fig:rq2}\n\\end{figure}\n\nTable~\\ref{tab:rq2:social} lists the activity status of the Twitter, Facebook, and Google+ accounts.\nWe found that 83.9\\% of the projects that use Twitter have an active account; 55.6\\% of the projects have an active Facebook account and only 28.6\\% have an active Google+ account.\n\n\\begin{table}[!ht]\n  \\caption{Active Twitter, Facebook, and Google+ accounts}\n  \\label{tab:rq2:social}\n  \\centering\n  \\begin{tabular}{@{}ccrr@{}}\n    \\toprule\n    \\multicolumn{1}{c}{\\bf Channel} && \\multicolumn{1}{c}{\\bf Active (\\%)} & \\multicolumn{1}{c}{\\bf Inactive (\\%)} \\\\\n    \\midrule\n     Twitter  && 47 (83.9\\%) & 9 (16.1\\%) \\\\\n     Facebook && 10 (55.6\\%) & 8 (44.4\\%) \\\\\n     Google+  && 2  (28.6\\%) & 5 (71.4\\%) \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\nFinally, we investigate the characteristics of the user meeting groups promoted on Meetup (such meetings are the 3rd most common promotion channel studied in this article).\nA Meetup group is a local community of people that is responsible for organizing meeting events~\\cite{meetupgroup}. \nThese groups are identified by topics to help members find them. \nHere, we rely on these topics to collect meetups about the studied open source projects, along with their locations (i.e., city and country).\nFor example, the topic for {\\sc jquery\/jquery}  is {\\em jquery} and a summary of the meeting groups about this topic can be found at \\url{https:\/\/www.meetup.com\/topics\/jquery\/all}.\nFigure~\\ref{fig:rq3_meetups} presents the distribution of the number of groups, cities, and countries of the projects with meetings registered at Meetup. For groups, the values ranges from 2 to 2,261 groups; considering the cities, the values range from 2 to 725; finally, for countries, the values range from 2 to 96. The maximum values always refer to {\\sc torvalds\/linux}. In other words, {\\sc torvalds\/linux} has 2,261 meetup groups, which are spread over 725 cities from 96 countries.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\begin{subfigure}{.325\\textwidth}\n\t  \\centering\n\t  \\includegraphics[width=.95\\linewidth,keepaspectratio,trim={1.5em 2em 1em 2em},clip,page=2]{images\/meetups.pdf}\n\t\t\\caption{Groups}\n\t  \\label{fig:rq3_meetups_sub1}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.325\\textwidth}\n\t  \\centering\n\t  \\includegraphics[width=.95\\linewidth,keepaspectratio,trim={1.5em 2em 1em 2em},clip,page=3]{images\/meetups.pdf}\n\t\t\\caption{Cities}\n\t  \\label{fig:rq3_meetups_sub2}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.325\\textwidth}\n\t  \\centering\n\t  \\includegraphics[width=.95\\linewidth,keepaspectratio,trim={1.5em 2em 1em 2em},clip,page=4]{images\/meetups.pdf}\n\t\t\\caption{Countries}\n\t  \\label{fig:rq3_meetups_sub3}\n\t\\end{subfigure}%\n\t\\caption{Number of groups, cities, and countries of the user meetings}\n\t\\label{fig:rq3_meetups}\n\\end{figure}\n\n\n\\subsection{How popular and random projects differ on the usage of promotion channels?}\n\\label{sec:results:rq3}\n\nIn Section~\\ref{sec:results:rq1}, we investigated the most common promotion channels used by popular GitHub projects.\nIn this section, we contrast the usage of promotion channels by these projects and by a random sample of GitHub projects.\nFor this purpose, we randomly selected 100 projects from the top-5,000 repositories by number of stars and manually inspected their documentation using the same methodology reported in Section~\\ref{sec:design}.\nThe number of stars of this random sample ranges from 2,297 stars ({\\sc uber-archive\/image-diff}) to 22,558 ({\\sc vsouza\/awesome-ios}).\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth,keepaspectratio,trim={0 1em 0 2em},clip]{images\/channels_random.pdf}\n\t\\caption{Most common promotion channels used by random projects}\n\t\\label{fig:rq3}\n\\end{figure}\n\nFigure~\\ref{fig:rq3} compares the usage of promotion channels by the random projects and by the most popular ones.\nIn the random sample, the number of projects using the investigated promotion channels is significantly lower compared to the most popular ones.\nHowever, by applying the Spearman's rank correlation test, we found a strong correlation between the number of projects using the promotion channels on each group ($rho =$ 0.904 and \\emph{p-value} $<$ 0.01).\nFor example, Twitter is also the most used promotion channel among the random projects (31 projects), followed by Blogs (17 projects) and RSS (13 projects).\nCompared to the most popular projects, Users meetings and Newsletter are less common (13 and 6 projects, respectively).\nFinally, Facebook and Google+ also have a very limited usage (7 and 4 projects, respectively).\n\n\n\\subsection{What is the impact of promotion on Hacker News?}\n\\label{results:rq4}\n\nAfter publishing content on blogs, Twitter, etc., open source developers can also promote this content on social news aggregator sites. These sites aggregate contents from distinct sources for easing viewing by a large public.\nThe most popular and important example is Hacker News (\\url{https:\/\/news.ycombinator.com}), which is dedicated to Computer Science and related technologies content. Hacker News posts just include a title and the URL of the promoted content (e.g.,~a blog post about a new version of an open source project). Any user registered in the site can post a link on Hacker News, i.e., not necessarily the links are posted by the contributors of an open source project, for example. Other Hacker News users can discuss the posts and upvote\nthem. An upvote is similar to a {\\em like} in social networks; posts are listed on Hacker News according to the number of upvotes.\nIn this research question, we use Hacker News due to its popularity;  posts that reach the front page of the site receive for example 10-100K page views, in one or two days (\\url{https:\/\/goo.gl\/evyP4w}). Furthermore, Hacker News\nprovides a public API, which allows search and metadata collection.\n\nFor each popular project considered in our study (100 projects), we searched for Hacker News posts with a URL referencing the project sites or pages, including GitHub pages (READMEs, issues, etc). As result, we found 3,019 posts on Hacker News referencing content from 96 studied projects (i.e., only four projects are never referenced on Hacker News).\nFigure~\\ref{fig:rq4_overview} presents the distributions of the number of posts per project, upvotes, and comments.\nThe number of posts ranges from 1 to 298 posts per project ({\\sc rails\/rails}); the first, second, and third quartile values are 4, 10, and 43 posts, respectively.\nRegarding their upvotes, the most popular post is about {\\sc appple\/swift} (\\aspas{\\em Swift is Open Source}), with 1,824 upvotes; the quartile values are 2, 3, and 12 upvotes, respectively.\nFinally, the highest number of comments is 760, about a GitHub issue opened for Microsoft Visual Studio (\\aspas{\\em VS Code uses 13\\% CPU when idle due to blinking cursor rendering}); the quartile values are 0, 0, and 2 comments, respectively. \nOn the one hand, these results show that most Hacker News posts do not attract attention. \nBy contrast, a small number of posts attract a lot of attention. For example, the top-10\\% posts have at least 132 upvotes. These posts are called {\\em successful posts} in this investigation.\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\begin{subfigure}{.325\\textwidth}\n\t  \\centering\n\t  \\includegraphics[width=.95\\linewidth,keepaspectratio,trim={0 2em 1em 2em},clip,page=2]{images\/hn_posts_overview.pdf}\n\t\t\\caption{Posts}\n\t  \\label{fig:rq4_overview_sub1}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.325\\textwidth}\n\t  \\centering\n\t  \\includegraphics[width=.95\\linewidth,keepaspectratio,trim={0 2em 1em 2em},clip,page=4]{images\/hn_posts_overview.pdf}\n\t\t\\caption{Upvotes}\n\t  \\label{fig:rq4_overview_sub2}\n\t\\end{subfigure}%\n\t\\begin{subfigure}{.325\\textwidth}\n\t  \\centering\n\t  \\includegraphics[width=.95\\linewidth,keepaspectratio,trim={0 2em 1em 2em},clip,page=6]{images\/hn_posts_overview.pdf}\n\t\t\\caption{Comments}\n\t  \\label{fig:rq4_overview_sub3}\n\t\\end{subfigure}%\n\t\\caption{Number of posts, upvotes, and comments (outliers are omitted)}\n\t\\label{fig:rq4_overview}\n\\end{figure}\n\nFigure~\\ref{fig:rq4_stars_before_after} shows boxplots with the number of GitHub stars gained by projects covered by successful posts, in the first three days before and  after the publication date on Hacker News. The  intention is to investigate the impact of a successful promotion on Hacker News, by comparing the number of stars gained before and after each successful post publication. On the median, the projects covered by successful posts gained 74 stars in the first three days before their appearance on Hacker News; in the first three days after the publication, the projects gained 138 stars. Therefore, Hacker News has a positive impact on the project's popularity, measured by GitHub stars. \nIndeed, the distributions are statistically different, according to the one-tailed variant of the Mann-Whitney U test (p-value $\\leq 0.05$). By computing Cliff's delta, we found a {\\em medium} effect size ($d = -0.372$).\n\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth,keepaspectratio,trim={0 0 0 2em},clip,page=2]{images\/hn_stars_before_after.pdf}\n\t\\caption{Number of GitHub stars received by projects covered by successful Hacker News posts in the first three days before and after the post publication}\n\t\\label{fig:rq4_stars_before_after}\n\\end{figure}\n\nFinally, we inspected the titles of each successful post, aiming to categorize the post purpose. The most common category includes posts announcing new releases of open source projects (44.9\\%; e.g., \\aspas{\\em Angular 2 Final Released}). Other popular categories include posts promoting articles or reports about the projects (25.4\\%; e.g., \\aspas{\\em Vue.js vs.~React}), announcing the first release of a project (16.5\\%; e.g., \\aspas{\\em YouTube-dl: Open-source YouTube downloader}), highlighting new project features (10.6\\%; e.g., \\aspas{\\em Git and GitHub Integration Comes to Atom}) and open sourcing products (1.6\\%; e.g., \\aspas{\\em Visual Studio Code is now open source}).\n\n\n\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{Related Work}\n\\label{sec:related}\n\nAlthough open source software has been exhaustively explored recently, little is known about how developers promote these projects.\nThe main exception is a work conducted by Bianco et al. where the authors analyze marketing and communication strategies of three companies that develop open source software~\\cite{Bianco12}.\nBy means of interviews, they found that websites and product launch events are adopted by the three organizations; however, the organizations differ considerably on the use of other communication channels, mainly when promoting the projects in open source communities and among industrial users.\n\nOn the one hand, most communication channels investigated in this paper are explored in other studies, but with different intentions.\nSinger et al. report a qualitative study focused on discovering the benefits that Twitter brings to developers~\\cite{singer2014}.\nThey found that Twitter adopters use it to stay aware of industry changes, for learning, and for building relationships.\nBy correlating the blogging and committing behavior of developers, Pagano and Maleej observed an intensive use of blogs, frequently detailing activities described shortly before in commit messages~\\cite{pagano2011}.\nBajic and Lyons analyze how software companies use social media techniques to gather feedback from users collectively~\\cite{Bajic2011}.\nTheir results suggest that startups use social media mainly for competitive advantage and established organizations use it to monitor the buzz among their users.\nBy studying a successful software development company, Hansson et al. identified that user meetings and newsletter are adopted to include and increase the participation of users in the development process~\\cite{Hansson2006}.\nFinally, Aniche et al. conduct a study to understand how developers use modern news aggregator sites (Reddit and Hacker News)~\\cite{aniche2018}. According to their results, the two main reasons for posting links on these sites is to promote own work and to share relevant content.\n\n\n\\section{Conclusion and Practical Implications}\n\\label{sec:conclusion}\n\nIn this paper, we investigated the most common promotion channels used by popular GitHub projects. This investigation supports the following practical recommendations to open source project managers and leaders:\n\n\\begin{enumerate}\n\\item Promotion is an important aspect of open source project management, which should be emphasized by project leaders. For example, most popular GitHub projects (two thirds) use at least one promotion channel; half of the projects invest on two channels. By contrast, the use of promotion channels is less common among projects with lower popularity. \n\n\\item Open source project managers should  consider  the use of Twitter (47 projects among the top-100 most popular GitHub projects have active Twitter accounts), Users meetings (which are organized or supported by 41 projects), and blogs (which are used by 38 projects).\n\n\\item Open source project managers should also consider promotion on social news aggregator sites. Successful posts on Hacker News may have an important impact on the popularity of GitHub projects. However, only 10\\% of the Hacker News posts about the studied projects have had some success\n\\end{enumerate}\n\n\n\n\\section*{Acknowledgments}\n\n\\noindent Our research is supported by CAPES, FAPEMIG, and CNPq.\n\n\\small\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\par With the fast development of deep learning techniques, face recognition systems (FRSs) have become a popular technique for identifying and verifying people due to the ease of capturing biometrics from the face. In our daily lives, one of the most relevant applications of FRS is the Automatic Border Control system, which can quickly verify the identity of a person with his electronic machine-readable travel document (eMRTD) \\cite{icao20159303} by comparing the face image of the traveler with a reference in the database. Although high-accuracy FRS can effectively distinguish an individual from others, it is vulnerable to adversarial attacks that conceal the real identity. Recent research found that attacks based on morphed faces \\cite{ferrara2014magic,scherhag2017vulnerability} pose a serious security risk in various applications. \n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=.75\\linewidth]{image\/fingerprint.eps}\n\\caption{Few-shot learning for morphing attack fingerprinting (MAF), a multiclass extension of MAD. Each class (morphing attack model) of the training set contains a few examples. After training, the model can classify unseen test samples for each class.}\n\\label{fig:MAF}\n\\end{figure*}\n\n\\par Morphing attacks were first introduced in 2014 \\cite{ferrara2014magic}. The morphed face is combined by two or more bona fide faces, and it was shown that commercial face recognition software tools are highly vulnerable to such attacks. In a further study \\cite{ferrara2016effects}, the authors showed that the images of morphed faces are realistic enough to fool human examiners. With the emergence of face morphing generation techniques \\cite{gimp,damer2018morgan,zhang2020mipgan,karras2019style} and numerous easy-to-use face morphing softwares (e.g., MorphThing \\cite{morphthing}, 3Dthis Face Morph \\cite{3dmorph}, Face Swap Online \\cite{faceswap}, Abrosoft FantaMorph \\cite{fanta}, FaceMorpher \\cite{morpher}), there is an imminent need to protect FRS security by detecting morphing attacks \\cite{scherhag2019face}. \n\nSome morphing attack detection (MAD) approaches have been developed since 2018 (for a recent review, see \\cite{venkatesh2021face}). They can be categorized into two types: single image-based (S-MAD) and differential image-based (D-MAD) \\cite{raja2020morphing}. The deep face representation for D-MAD has been studied in \\cite{scherhag2020deep}; existing S-MAD methods can be further classified into two subtypes \\cite{venkatesh2020face}: model-based (using handcraft characteristics) and deep learning-based. Noise-based photo-response non-uniformity (PRNU) methods \\cite{debiasi2018prnu,debiasi2018prnu2,scherhag2019detection,zhang2018face} represent the former subtype due to its popularity and outstanding performance. Originally proposed for camera identification, PRNU turns out to be useful for detecting the liveness of face photos. For the latter subtype, Noiseprint \\cite{cozzolino2018noiseprint} used a CNN to learn the important features, with the objective of improving detection performance and supporting fingerprinting applications. \n\n\\par Despite rapid progress, existing MAD methods are often constructed on a small training dataset and a single modality, which makes them lacking good generalization properties \\cite{raja2020morphing,venkatesh2020face}. The performance of existing MAD methods might be satisfactory for predefined morphing attack models, but degrades rapidly when deployed in the real world facing newly evolved attacks. Although it is possible to alleviate this problem by fine-tuning the existing MAD model, the cost of collecting labeled data for every new morphing attack is often formidable. Furthermore, we argue that MAD alone is not sufficient to meet the demand for increased security risk facing FRS. A more aggressive countermeasure than MAD to formulate the problem of morphing attack fingerprinting (MAF), that is, we aim at a multiclass classification of morphing attack models, as shown in Fig. \\ref{fig:MAF}.\n\n\\par Based on the above observations, we propose to formulate MAF as a few-shot learning problem in this paper. Conventional few-shot learning (FSL) \\cite{snell2017prototypical} learns the knowledge from a few examples of each class and predicts the class label of the new test samples. Similarly, we train the detector using data from both predefined models and new attack models (only a few samples are required) to predict unknown new test samples. This task is named the few-shot MAD (FS-MAD) problem. Unlike existing MAD research, few-shot MAF (FS-MAF) aims at learning general discriminative features, which can be generalized from predefined to new attack models. The problem of few-shot MAF is closely related to camera identification (ID) \\cite{lukas2006digital}, camera model fingerprinting \\cite{cozzolino2018noiseprint}, and GAN fingerprinting (a.k.a. model attribution \\cite{yu2019attributing}) in the literature. The main contributions of this paper are summarized below.\n\n$\\bullet$ Problem formulation of few-shot learning for MAD\/MAF. We challenge the widely accepted assumptions of the MAD community, including the NIST's FRVT MORPH competition. The generalization property of MAD\/MAF methods will be as important as the optimization of recognition accuracy. \n\n$\\bullet$ Feature-level fusion for MAD applications. Although both PRNU and Noiseprint have shown promising performance in camera identification applications, no one has demonstrated their complementary nature in the open literature. We believe that this work is the first to combine them through feature-level fusion and to study the optimal fusion strategy.\n\n$\\bullet$ Design a fusion-based FSL method with adaptive posterior learning (APL) for MAD\/MAF. By adaptively combining the most surprising observations encountered by PRNU and Noiseprint, we can achieve a good generalization property by optimizing the performance of FS-MAD\/FS-MAF at the system level. \n\n$\\bullet$ Construction of a large-scale benchmark dataset to support MAD\/MAF research. More than 20,000 images with varying spatial resolution have been collected from various sources. Extensive experimental results have justified the superior generalization performance of FS-MAD and FS-MAF over all other competing methods. \n\n\n\\section{Related Work}\n\\label{related}\n\n\\subsection{Morphing Attack Detection (MAD)}\n\\noindent \\textbf{Model-based S-MAD}. Residual noise feature-based methods are designed to analyze pixel discontinuity, which may be greatly affected by the morphing process. Generally, noise patterns are extracted by subtracting the given image from a denoised version of the same image using different models, such as the deep multiscale context aggregate network (MS-CAN) \\cite{venkatesh2020detecting}. The most popular should be sensor noise patterns, such as PRNU. Recently, both PRNU-based \\cite{zhang2018face,debiasi2018prnu,debiasi2018prnu2,scherhag2019detection} and scale-space ensemble approaches \\cite{raja2020morphing,raja2017transferable} have been studied. \n\n\n\\noindent \\textbf{Learning-based S-MAD}. Along with rapid advances in deep learning, many methods have considered the extraction of deep learning features for detection. The use of a convolutional neural network (CNN) has reported promising results \\cite{8897214}. Most works are based on pre-trained networks and transfer learning. Commonly adopted deep models contain AlexNet \\cite{krizhevsky2012imagenet}, VGG16 \\cite{simonyan2014very}, VGG19 \\cite{simonyan2014very,raja2017transferable}, GoogleNet \\cite{szegedy2015going}, ResNet \\cite{he2016deep}, etc. In addition, several self-design models were also proposed. More recently, a deep residual color noise pattern was proposed for MAD in \\cite{venkatesh2019morphed}; and an attention-based deep neural network (DNN) \\cite{aghdaie2021attention} was studied, focusing on the salient regions of interest (ROI) that have the most spatial support for the morph detector decision function.\n\n\n\\noindent\\textbf{Learning-based D-MAD}. The presented D-MAD methods mainly focus on feature differences and demorphing. For feature difference-based methods, features of the suspected image and the live image are subtracted and further classified. Texture information, 3D information, gradient information, landmark points, and deep feature information (ArcFace \\cite{scherhag2020deep}, VGG19 \\cite{seibold2020accurate}) are the most popular features used. The authors in \\cite{scherhag2018detecting} computed distance-based and angle-based features of landmark points for analysis. In \\cite{singh2019robust}, a robust method using diffuse reflectance in a deep decomposed 3D shape was proposed. Fusion methods were commonly adopted by concatenating hand-crafted Local Binary Pattern Histogram (LBPH) and transferable deep CNN features \\cite{damer2019multi}, or concatenating feature vectors extracted from texture descriptors, keypoint extractors, gradient estimators and deep neural networks \\cite{scherhag2018towards}. More recently, a discriminative DMAD method in the wavelet subband domain was developed to discern the disparity between a real and a morphed image.\n\n\n\\subsection{Few-Shot Learning (FSL)}\nFew-shot learning addresses the challenge with the generalization property of deep neural networks, i.e., how can a model quickly generalize after only seeing a few examples from each class? Early approaches include meta-learning models \\cite{ravi2016optimization} and deep metric learning techniques \\cite{snell2017prototypical}. More recent advances have explored new directions such as the relation network \\cite{sung2018learning}, meta-transfer learning \\cite{sun2019meta}, adaptive posterior learning (APL) \\cite{ramalho2019adaptive}, and cluster-based object seeker with shared object concentrator (COSOC) \\cite{luo2021rectifying}.\n\n\n\\subsection{Camera and Deepfake Fingerprinting}\n\\par PRNU, as a model-based device fingerprint, has been used to perform multiple digital forensic tasks, such as device identification \\cite{cozzolino2020combining}, device linking \\cite{salazar2021evaluation}, forgery localization \\cite{lin2020prnu}, detection of digital forgeries \\cite{lugstein2021prnu}. It can find any type of forgery, irrespective of its nature, since the lack of PRNU is seen as a possible clue of manipulation. Furthermore, PRNU-based MAD methods \\cite{debiasi2018prnu,debiasi2018prnu2,scherhag2019detection,zhang2018face} also confirm the usefulness of the sensor fingerprint in MAD.\nIn recent years, PRNU has been applied successfully in MAD \\cite{debiasi2018prnu2,debiasi2018prnu,scherhag2019detection}. The method in \\cite{debiasi2018prnu2} shows that region-based PRNU spectral analysis reliably detects morphed face images, while it fails if image post-processing is applied to generated morphs. Based on previous work, a PRNU variance analysis was performed in \\cite{debiasi2018prnu}. It focused on local variations of face images, which can be useful as a reliable indicator for image morphing. The work in \\cite{scherhag2019detection} proposed an improved version of the scheme based on the previous PRNU variance analysis in image blocks. Another work \\cite{marra2019gans} showed that each GAN model leaves a specific fingerprint in the generated images, just as the PRNU traces left by different cameras in real-world photos.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{image\/pipeline.eps}\n\\vspace{-0.75cm}\n\\caption{An overview of the proposed system (FBC-APL) for few-shot MAF (FS-MAF). It consists of factorized bilinear coding (FBC) and adaptive posterior learning (APL) modules. The output contains the probability that the input image will be classified into one of the known morphing models.}\n\\label{fig:pipeline}\n\\end{figure*}\n\n\n\\section{Methodology}\n\\label{funda}\n\nMorphing attack fingerprinting (MAF) refers to the multiclass generalization of the existing binary MAD problem. In addition to detecting the presence of morphing attacks, we aim at finer-granularity classification about the specific model generating the face morph. It is hypothesized that different attack models inevitably leave fingerprints in morphed images (conceptually similar to the sensor noise fingerprint left by different camera models \\cite{lukas2006digital}).\nFig. \\ref{fig:pipeline} shows the overall system consisting of two stages: feature fusion through factorized bilinear coding (FBC) and few-shot learning (FSL) for MAF. We will first elaborate on fusion-based MAD in detail and then discuss the extension to few-shot MAF.\n\n\n\n\\subsection{Fusion-based Single-Image MAD}\n\\par Noise is often embedded in the image data during acquisition or manipulation. The uniqueness of the noise pattern is determined by the physical source or an artificial algorithm, which can be characterized as a statistical property to reveal the source of the noise \\cite{popescu2004statistical}. The noise of the sensor pattern was first used for the MAD task by performing a facial quantification statistics analysis, which confirmed its effectiveness \\cite{zhang2018face}. Here, we consider two types of sensor noise patterns: Photo Response Non-Uniformity (PRNU) \\cite{fridrich2009digital} and Noiseprint \\cite{cozzolino2018noiseprint}.\n\n\\noindent \\textbf{Photo Response Non-Uniformity (PRNU)}. PRNU originates from slight variations between individual pixels during photoelectric conversion in digital image sensors \\cite{lukas2006digital}. Different image sensors embed this weak signal into acquired images as a unique signature. Although the weak signal itself is mostly imperceptible to the human eye, its uniqueness can be characterized by statistical techniques and exploited by sophisticated fingerprinting methods such as PRNU \\cite{fridrich2009digital}. \nThis systemic and individual pattern, which plays the role of a sensor fingerprint, has proven robust to various innocent image processing operations such as JPEG compression. Although PRNU is stochastic in nature, it is a relatively stable component of the sensor over its lifetime. \n\nPRNU has been widely studied in camera identification because it is not related to image content and is present in every image acquired by the same camera. Most recently, PRNU has been proposed as a promising tool for detecting morphed face images \\cite{debiasi2018prnu,debiasi2018prnu2}.\nThe spatial feature of PRNU can be extracted using the approach presented by Fridrich \\cite{fridrich2009digital}. For each image $I$, the residual noise ${W}_{I}$ is estimated as described in Equation \\eqref{eq1}:\n\\begin{equation}\n\\label{eq1}\n\\vspace{-0.1in}\n{W}_{I} = I - F(I) \n\\end{equation}\nwhere $F$ is a denoising function that filters the noise from the sensor pattern. The clever design of the mapping function $F$ (e.g., wavelet-based filter \\cite{lukas2006digital}) makes PRNU an effective tool for various forensic applications.\n\n\n\\noindent \\textbf{Noiseprint}. Unlike model-based PRNU, data-driven or learning-based methods tackle the problem of camera identification by assuming the availability of training data. Instead of mathematically constructing unique signatures, Noiseprint \\cite{cozzolino2018noiseprint} attempts to learn the embedded noise pattern from the training data. A popular learning methodology adopted by Noiseprint is to construct a Siamese network \\cite{bertinetto2016fully}. The Siamese network is trained with pairs of image patches that come from the same or different cameras in an unsupervised manner. Similarly to PRNU, Noiseprint has shown clear traces of camera fingerprints. It should be noted that Noiseprint has performed better than PRNU when cropped image patches become smaller, implying the benefit of exploiting spatial diversity \\cite{cozzolino2018noiseprint}.\n\n\\par To the best of our knowledge, Noiseprint has not been proposed for MAD in the open literature. Existing deep learning-based S-MADs often use pre-trained networks such as VGG-face \\cite{raja2020morphing}. Our empirical study shows that morphing-related image manipulation leaves evident traces in Noiseprint, suggesting the feasibility of Noiseprint-based MAD. Moreover, morphed faces are often manipulated across the face, whose spatial diversity can be exploited by cropping image patches using Noiseprint. To justify this claim, Fig. \\ref{fig:featurefig} (d) presents the Noiseprint comparison between bona fide and morphed faces averaged over 1,000 examples. Visual inspection clearly shows that the areas around the eyes and nose have more significant (bright) traces than the bona fide faces. In contrast, Fig. \\ref{fig:featurefig} (c) shows the comparison of the extracted PRNU patterns with the same experimental setting. Similar visual differences between bona fide and morphed faces can be observed; more importantly, PRNU and Noiseprint demonstrate complementary patterns (low vs. high frequency) begging for fusion.\n\n\\noindent \\textbf{Feature Fusion Strategy}. Fusion methods are usually based on multiple feature representations or classification models. Taking advantage of diversity, the strategy of combining classifiers \\cite{kittler1998combining} has shown improved recognition performance compared to single-mode approaches. Recent work has shown that fusion methods based on Dempster-Shafer theory can improve the performance of face morphing detectors \\cite{makrushin2019dempster}. However, previous work \\cite{makrushin2019dempster} only considered ensemble models of the scale space and pre-trained CNN models. For the first time, we propose to combine PRNU and Noiseprint using a recently developed similarity-based fusion method, called factorized bilinear coding (FBC) \\cite{gao2020revisiting}.\n\nFBC is a sparse coding formulation that generates a compact and discriminative representation with substantially fewer parameters by learning a dictionary $\\boldsymbol{B}$ to capture the structure of the entire data space. It can preserve as much information as possible and activate as few dictionary atoms as possible. Let $\\boldsymbol{x}_i$, $\\boldsymbol{y}_j$ be the two features extracted from PRNU and Noiseprint, respectively. The key idea behind FBC is to encode the extracted features based on sparse coding and to learn a dictionary $\\boldsymbol{B}$ with $k$ atoms by matrix factorization. Specifically, the sparsity FBC opts to encode the two input features $(\\boldsymbol{x}_i, \\boldsymbol{y}_j)$ in the FBC code $\\boldsymbol{c}_v$ by solving the following optimization problem:\n\n\\begin{equation}\n\\underset{{{\\boldsymbol{c}}_{v}}}{\\mathop{\\min }}\\,\\bigg|\\bigg|{{\\boldsymbol{x}}_{i}}\\boldsymbol{y}_{j}^{\\top}-\\sum\\limits_{l=1}^{k}{c_{v}^{l}}{{\\boldsymbol{U}}_{l}}\\boldsymbol{V}_{l}^{\\top}\\bigg|{{\\bigg|}^{2}}+\\lambda||{{\\boldsymbol{c}}_{v}}|{{|}_{1}}\n\\end{equation}\nwhere $\\lambda$ is a trade-off parameter between the reconstruction error and the sparsity. The dictionary atom $b_l$ of $\\boldsymbol{B}$ is factorized into $\\boldsymbol{U}_{l}\\boldsymbol{V}_{l}^{\\top}$ where \n    $\\boldsymbol{U}_{l}$ and $\\boldsymbol{V}_{l}^{\\top}$ are low-rank matrices. The $l_1$ norm $|| \\cdot ||_1$ is used to impose the sparsity constraint on $\\boldsymbol{c}_{v}$. In essence, the bilinear feature $\\boldsymbol{x}_{i}\\boldsymbol{y}_{j}^{\\top}$ is reconstructed by $\\sum\\limits_{l=1}^{k}{c_{v}^{l}} \\boldsymbol{U}_{l}\\boldsymbol{V}_{l}^{\\top}$\nwith $\\boldsymbol{c}_{v}$ being the FBC code and $c_v^l$ representing the $l$-th element of $\\boldsymbol{c}_{v}$.\n\nThis optimization can be solved using well-studied methods such as LASSO \\cite{tibshirani1996regression}. With two groups of features $\\{\\boldsymbol{x}_i\\}_{i=1}^m$ and $\\{\\boldsymbol{y}_j\\}_{j=1}^n$ at our disposal, we first calculate all FBC codes $\\{\\boldsymbol{c}_v\\}_{v=1}^N$ and then fuse them by the operation $max$ to achieve global representation $\\boldsymbol{z}$: \n\\begin{equation}\n    \\boldsymbol{z}=max\\left\\{\\boldsymbol{c}_{v}\\right\\}_{i=1}^{N}.\n    \\label{eq:3}\n\\end{equation}\nThe entire FBC module is shown in Fig. \\ref{fig:fbc}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{image\/fbc.eps}\n\\vspace{-0.65cm}\n\\caption{The architecture of the FBC module to combine PRNU and Noiseprint. $\\tilde{\\boldsymbol{U}}$ and $\\tilde{\\boldsymbol{V}}$ replace $\\boldsymbol{U}$ and $\\boldsymbol{V}$ to avoid numerically unstable matrix inversion operations; $\\boldsymbol{P}$ is a fixed binary matrix.}\n\\label{fig:fbc}\n\\end{figure}\n\n\n\n\\subsection{Few-shot learning for Morphing Attack Fingerprinting}\n\\par Based on the FBC-fused feature $\\boldsymbol{z}$, we construct a few-shot learning module as follows. Inspired by recent work on adaptive posterior learning (APL) \\cite{ramalho2019adaptive}, we have redesigned the FSL module to adaptively select feature vectors of any size (e.g., FBC-fused feature) as input. This newly designed module consists of three parts: an encoder, a decoder, and an external memory store. The encoder is used to generate a compact representation for the incoming query data; the memory saves the previously seen representation by the encoder; the decoder aims at generating a probability distribution over targets by analyzing the query representation and pairwise data returned from the memory block. Next, we will elaborate on the design of these three components.\n\n\\noindent\\textbf{Encoder}. The encoder can convert input data of any size to a compact embedding with low dimensionality. It is implemented by a convolutional network, which is composed of a single first convolution to map the input to 64 feature channels, followed by 15 convolutional blocks. Each block is made up of a batch normalization step, followed by a ReLU activation and a convolutional layer with kernel size 3. For every three blocks (one combo), the convolution contains a stride 2 to down-sample the image. All layers have 64 features. Finally, the feature is flattened to a 1D vector and passed through Layer Normalization, generating a 64-dimensional embedding as an encoded representation.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{image\/MAD_APL_module2.eps}\n\\vspace{-0.4cm}\n\\caption{(a) APL training procedure for iterations. We train the APL module on a sequence of episodes ($x_t$, $y_t$), where $x_t$ is the FBC feature and $y_t$ is the true label. At first, the memory is empty; at each iteration, a batch of samples is fed to the module, and a prediction is made. Cross-entropy loss L($\\hat{y}_t$, $y_t$) is calculated and a gradient update step is performed to minimize the loss in that batch alone. The loss is also fed to the memory controller so that the network can decide whether to write to memory. (b) and (c) show the behavior of the accuracy and memory size in a 9-class training scenario. APL stops writing to memory after having about 7 examples per class for classification.}\n\\label{fig:fscnn}\n\\end{figure*}\n\n\n\\noindent\\textbf{Memory}. The external memory store is a database to store experiences. It is key-value data. Each row represents the information for one data point. Each column is decomposed into an embedding (encoded representation) and a true label. The memory store is managed by a controller that decides which embeddings can be written into the memory while at the same time tries to minimize the amount of written embeddings. During the writing process, a quantity metric surprise is defined. The higher the probability that the model assigns to the true class correctly, the less surprised it will be. If the confidence in the prediction in the correct class is smaller than the probability assigned by a uniform prediction, the embedding should be written into memory. During the querying process, the memory is queried for the k-nearest-neighbors of the embeddings of queries from the encoder. The distance metric used to calculate the proximity between points is an open choice, and here we use two types (euclidean distance and cosine distance). Both the full-row data for each of the neighbors and query embeddings are concatenated and fed to the decoder. \n\n\\begin{figure\n\\centering\n\\includegraphics[width=0.8\\linewidth]{image\/db_sample.eps}\n\\caption{Face samples in five merged datasets. (a) FERET-Morphs (bona fide faces come from FERET \\cite{feret}), (b) FRGC-Morphs (bona fide faces come from FRGC V2.0 \\cite{frgc}), (c) FRLL-Morphs (bona fide faces come from Face Research Lab London Set (FRLL) \\cite{amslraw}), (d) CelebA-Morphs (bona fide faces come from CelebA \\cite{liu2015deep}), and (e) Doppelg\u00e4nger Morphs (bona fide faces come from the Web collection).}\n\\label{fig:sample}\n\\end{figure}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{image\/self_sample.eps}\n\\caption{Some sample pairs of bona-fide face images from the Doppelg\u00e4nger dataset (note that these look-alike pairs do not have biological connections).}.\n\\label{fig:self}\n\\end{figure*}\n\n\n\\noindent\\textbf{Decoder}. The decoder takes the concatenation of query embedding, recalled neighbor embeddings from memory, labels, and distances as input. The architecture is a self-attention-based relational feedforward module. It processes each of the neighbors individually by comparing them with the query and then does a cross-element comparison with a self-attention module before reducing the activations with an attention vector calculated from neighbor distances. The self-attention blocks are repeated five times in a residual manner. The resulting tensors are called activation tensors. In addition, the distances between neighbors and the query are passed through a softmax layer to generate an attention vector, which is summed with the activation tensor on the first axis to obtain the final logit result for classification. The self-attention block comprises a multihead attention layer, a multihead dot product attention (MHDPA) layer \\cite{santoro2018relational} for cross-element comparison, and a nonlinear multilayer perceptron (MLP) layer to process each element individually. \n\n\n\n\\noindent\\textbf{Training}. During APL training, as shown in Fig. \\ref{fig:fscnn} (a), the query data (that is, the FBC-fused feature vector $\\boldsymbol{z}$) are passed through the encoder to generate an embedding, and this representation is used to query an external memory store. At first, the memory is empty; at each training episode, a batch of examples is fed to the model, and a prediction is made. Cross-entropy loss is used to be fed to the memory controller to decide whether to write to memory. After the query is searched in memory, the returned memory contents, as well as the query, are fed to the decoder for classification. Figs. \\ref{fig:fscnn} (b) and (c) show the behavior (accuracy and memory size) of APL during a single episode. The accuracy of APL increases as it sees more samples and saturates at some point, indicating that the additional inputs do not surprise the module anymore. In the case of the 9-class classification scenario, we have observed that about 7 examples per class are sufficient to reach performance saturation.\n\n\\noindent\\textbf{Morphing Attack Fingerprinting}. Both PRNU \\cite{lukas2006digital} and Noiseprint \\cite{cozzolino2018noiseprint} were originally proposed for the identification of camera models, which is known to be a fingerprint in image forensics. The duality between image generation in the cyber and physical worlds inspires us to extend the existing problem formulation of binary MAD \\cite{debiasi2018prnu,debiasi2018prnu2,scherhag2019detection,zhang2018face} into multiclass fingerprinting. Different camera models (e.g., Sony vs. Nikon) are analogous to varying face morphing methods (e.g., LMA \\cite{damer2018morgan} vs. StyleGAN2 \\cite{karras2020analyzing}); therefore, it is desirable to go beyond MAD by exploring the feasibility of distinguishing one morphing attack from another. Fortunately, the system shown in Fig. \\ref{fig:pipeline} easily lends itself to generalization from binary to multiclass classification by resetting the hyperparameters, like the number of classes, the data path for each class, etc. To learn a discriminative FBC feature for fingerprinting, multiclass labeled data for training and testing should be prepared to be fed to the FBC module for retraining. When the FBC feature is available, it will be fed to the APL module for multiclass classification.\n\n\\section{Experiments}\n\\label{result}\n\n\n\\begin{table}[t]\n\\begin{center}\n\\small\n\\caption{The hybrid face morphing benchmark database consists of five image sources and 3-6 different morphing methods for each.}\n\\label{tab:dbinfo}\n\\vspace{-0.3cm}\n\\begin{threeparttable}\n\\begin{tabular}{ l |l c c}\n\\hline\\noalign{\\smallskip}\nDatabase & Subset & \\#Images & Resolution \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\multirow{4}{*}{FERET-Morphs} &\tbona fide \\cite{feret} &\t576 & 512x768\\\\\n\t& FaceMorpher \\cite{sarkar2020vulnerability} &\t529 & 512x768\\\\\n\t& OpenCV \\cite{sarkar2020vulnerability} &529 & 512x768\\\\\n\t& StyleGAN2 \\cite{sarkar2020vulnerability} & 529 & 1024x1024 \\\\\n\\hline\n\\multirow{4}{*}{FRGC-Morphs} & bona fide \\cite{frgc} & \t964 & 1704x2272\\\\\n\t& FaceMorpher \\cite{sarkar2020vulnerability} &\t964 & 512x768\\\\\n\t& OpenCV \\cite{sarkar2020vulnerability} & 964 & 512x768\\\\\n\t& StyleGAN2 \\cite{sarkar2020vulnerability} & 964 & 1024x1024\\\\\n\\hline\n\\multirow{7}{*}{FRLL-Morphs} & bona fide \\cite{amslraw} & 102+1932 & 413x531\\\\\n\t& AMSL \\cite{neubert2018extended} & 2175 & 413x531 \\\\\n\t& FaceMorpher \\cite{sarkar2020vulnerability} &\t1222 & 431x513 \\\\\n\t& OpenCV \\cite{sarkar2020vulnerability}& 1221 & 431x513 \\\\\n\t& LMA  &768 & 413x531\\\\\n\t& WebMorph \\cite{sarkar2020vulnerability} & 1221 & 413x531\\\\\n\t& StyleGAN2 \\cite{sarkar2020vulnerability} & 1222 & 1024x1024 \\\\\n\\hline\n\\multirow{4}{*}{CelebA-Morphs*} & bona fide \\cite{liu2015deep} & 2989 & 128x128 \\\\\n\t& MorGAN \\cite{damer2018morgan}& 1000 & 64x64\\\\\n\t& CIEMorGAN \\cite{damer2019realistic} & 1000 & 128x128 \\\\\n\t& LMA \\cite{damer2018morgan} & 1000 & 128x128 \\\\\n\\hline\n\\multirow{4}{*}{Doppelg\u00e4nger} & bona fide & 306 & 1024x1024 \\\\\n\t& FaceMorpher &\t150 & 1024x1024 \\\\\n\t& OpenCV &\t153 & 1024x1024\\\\\n\t& StyleGAN2\t& 153 & 1024x1024 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\begin{tablenotes}\n\\small\n\\item * means only the cropped faces from raw images are used; no facial cropping is used for other datasets. The raw number of bona fide images in FRLL-Morphs is 102. Based on the raw faces, data augmentation is implemented to obtain extra 1932 images. \n\\end{tablenotes}\n\\end{threeparttable}\n\\vspace{-0.2in}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Large-scale Morphing Benchmark Dataset}\n\\noindent \\textbf{Benchmark Dataset Description.} To simulate the amount and distribution of data in real-world applications, we have combined five datasets to build a large-scale evaluation benchmark for detecting and fingerprinting few-shot morphing attacks. It contains four publicly available datasets, namely, FERET-Morphs \\cite{feret,sarkar2020vulnerability}, FRGC-Morphs \\cite{frgc,sarkar2020vulnerability}, FRLL-Morphs \\cite{amslraw,neubert2018extended,sarkar2020vulnerability}, and CelebA-Morphs \\cite{liu2015deep,damer2018morgan,damer2019realistic}. We also generated a new dataset with high-resolution faces collected from the Web, named Doppelg\u00e4nger Morphs, which contains morphing attacks from three algorithms and satisfies the so-called Doppelg\u00e4nger constraint \\cite{rottcher2020finding} (that is, look-alike faces without biological connections, refer to Fig. \\ref{fig:self}). A total of more than 20,000 images (6,869 bona fide faces and 15,764 morphed faces) have been collected, as shown in Table \\ref{tab:dbinfo}. Eight morphing algorithms are involved, including five landmark-based methods, OpenCV \\cite{opencv}, FaceMorpher \\cite{facemorpher}, LMA \\cite{damer2018morgan}, WebMorph \\cite{webmorph}, and AMSL \\cite{neubert2018extended}, and three adversarial generative networks based, including MorGAN \\cite{damer2018morgan}, CIEMorGAN \\cite{damer2019realistic}, and StyleGAN2 \\cite{karras2020analyzing}. Fig. \\ref{fig:sample} provides some cropped face samples with real faces and morphed faces from different morphing algorithms in these five datasets. To the best of our knowledge, this is one of the largest and most diverse face morphing benchmarks that can be used for MAD and MAF evaluations. \n\n\n\\noindent \\textbf{Evaluation Protocols.}\nBased on the large-scale dataset collected for few-shot MAD and MAF benchmarks, we have designed the evaluation protocols for each task as follows:\n\n$\\bullet$ Protocol FS-MAD (few-shot MAD). This protocol is designed for the few-shot binary classification (bona fide\/morphed). Training data comes from predefined types and a few (1 or 5) samples per new type. The test data come from new types. Here, the predefined types in our experiment contain five types of morphing results generated by FaceMorpher \\cite{facemorpher}, OpenCV \\cite{opencv}, WebMorph \\cite{webmorph}, StyleGAN2 \\cite{karras2020analyzing}, and AMSL \\cite{neubert2018extended}, and their corresponding bona fide faces. Faces of these types are from the FERET-Morphs, FRGC-Morphs, FRLL-Morphs, and Doppelg\u00e4nger-Morphs datasets. The morphing faces generated by LMA \\cite{damer2018morgan}, MorGAN \\cite{damer2018morgan}, and CIEMorGAN \\cite{damer2019realistic}, and their corresponding bona fide faces, are treated as new types. Faces of these types are from the CelebA-Morphs dataset.\n \n\n$\\bullet$ Protocol FS-MAF (few-shot MAF). This protocol is designed for multiclass fingerprint classification on the hybrid large-scale benchmark and for five separate morph datasets. Each morphing type and bona fide type are treated as different categories, namely FERET-Morphs, FRGC-Morphs, CelebA-Morphs, and Doppelg\u00e4nger datasets all with 4 classes, FRLL-Morphs with 7 classes, and the hybrid with 9 classes. For each data set, the data are split according to the rule of 8: 2. Training data consist of 1 and 5 images per class for 1 shot and 5-shot learning, respectively. The testing data contains non-overlapping data with the training in each dataset. To reduce the bias of the imbalanced distribution of the data, a similar number of faces is maintained for each class in each test set. \n\n\\begin{table}[!t]\n\\begin{center}\n\\caption{Traditional MAD performance (Accuracy-\\%) comparison of different feature-level fusion methods. NP - Noiseprint; CN - Concatenation; CC - Convex Compression; $\\bot$ - spatial; $\\square$ - spectral.}\n\\vspace{-0.3cm}\n\\label{tab:toyexp}\n\\begin{tabular}{ l c c c c c c }\n\\hline\\noalign{\\smallskip}\n{Feature} & CN & Sum & Max & CC & FBC (ours) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nPRNU $\\bot$+PRNU $\\square$ & 83.78 & 84.23 & 83.78 & 84.23 & 84.42 \\\\\nNP $\\bot$ + NP $\\square$ & 89.19 & 89.64 & 89.64 & 89.64 & 96.40\\\\\nPRNU $\\bot$ + NP $\\square$ & 89.19 & 89.19 & 89.64 & 89.19 & 89.59\\\\\nPRNU $\\square$ + NP $\\bot$ & 83.78 & 84.23 & 83.78 & 85.59 & 86.04\\\\\nPRNU $\\square$. + NP $\\square$ & 86.94 & 85.59 & 85.59 & 86.94 & 84.68 \\\\\nPRNU $\\bot$ + NP $\\bot$ & \\textbf{91.44} & \\textbf{91.89} & \\textbf{91.89} & \\textbf{94.59} & \\textbf{96.85}\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[!t]\n\\begin{center}\n\\caption{Performance (\\%) comparison of few-shot MAD. Accu. - Accuracy.}\n\\vspace{-0.3cm}\n\\label{tab:madfs}\n\\resizebox{.95\\linewidth}{!}{\n\\begin{tabular}{ l |c c c |c c c}\n\\hline\\noalign{\\smallskip}\n & \\multicolumn{3}{c}{1-shot} & \\multicolumn{3}{|c}{5-shot} \\\\\nMethod & Accu. & D-EER & ACER & Accu. &\tD-EER &\tACER \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nXception \\cite{chollet2017xception} & 66.50 & 32.50 & 33.50 & 73.25 & 27.00 & 26.75 \\\\\nMobileNetV2 \\cite{sandler2018mobilenetv2} & 67.00 & 36.50 & 33.00 & 71.25 & 29.00 & 28.75 \\\\\nNasNetMobile \\cite{zoph2018learning} & 59.00 & 40.50 & 41.00 & 66.25 & 35.00 & 33.75 \\\\\nDenseNet121 \\cite{huang2017densely} &68.25 & 31.50 & 31.75 & 73.50 & 24.50 & 26.50 \\\\\nArcFace \\cite{deng2019arcface} & 58.00 & 41.00 & 42.00 & 62.25 &\t37.50 & 37.75 \\\\\n\\hline\nRaghavendra. et al. \\cite{raghavendra2017face} & 49.25 & 48.00 & 50.75 & 46.75 & 47.50 & 53.25 \\\\\nMB-LBP \\cite{scherhag2020face} & 61.00 & 38.50 & 39.00 & 69.25 & 31.00 & 30.75 \\\\\nFS-SPN \\cite{zhang2018face} & 51.50 & 45.00 & 48.50 & 58.25 & 43.50 & 41.75 \\\\\t\nPipeline Footprint \\cite{neubert2018reducing} & 54.25 & 44.50 &45.75 &\t60.25 &\t38.50 &\t39.75 \\\\\nPRNU Analysis \\cite{debiasi2018prnu} & 56.50 & 57.00 & 43.50 & 64.25 & 66.70 & 35.75 \\\\\nInception-MAD \\cite{damer2022privacy} & 62.00 & 34.50 & 38.00 & 67.75 & 32.50 & 32.25 \\\\\nMixFaceNet-MAD \\cite{damer2022privacy} & 76.10 & 27.50 & 28.00 & 82.16 & 24.50 & 24.25 \\\\\nNoiseprint-SVM \\cite{cozzolino2018noiseprint} & 53.75 & 50.50 & 46.25 & 61.25 & 38.50 & 38.75 \\\\\n\\hline\nMeta-Baseline \\cite{chen2021meta} & 60.45 & - & - & 71.38 & - & -  \\\\\nCOSOC \\cite{luo2021rectifying} & 66.89 & -&-& 74.54 &-&-  \\\\\n\\hline\n\\textbf{FBC-APL} & \\textbf{99.25} & \\textbf{1.50} & \\textbf{0.75} & \\textbf{99.75} & \\textbf{0.50} & \\textbf{0.25} \\\\\n\\noalign{\\smallskip}  \\hline\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Experimental Settings}\n\\noindent \\textbf{Data Preprocessing}. Dlib face detector \\cite{king2009dlib} is used to detect and crop the face region. The cropped face is normalized according to the coordinates of the eye and resized to a fixed size of $270\\times270$ pixels. The feature extraction of PRNU and Noiseprint is performed on the processed faces, respectively. The resulting vector dimension for each type of feature is 72,900 ($270\\times270$).\n\n\\noindent \\textbf{Performance Metrics}. Following previous MAD studies \\cite{raja2020morphing,scherhag2020deep}, we report performance using four metrics, including: (1) Accuracy; (2) D-EER; (3) ACER; (4) Confusion Matrix. Detection Equal-Error-Rate(D-EER) is the error rate for which both BPCER and APCER are identical. Average Classification Error Rate (ACER) is calculated by the mean of the APCER and BPCER values. Attack Presentation Classification Error Rate (APCER) reports the proportion of morph attack samples incorrectly classified as bona fide presentation, and the Bona Fide Presentation Classification Error Rate (BPCER) refers to the proportion of bona fide samples incorrectly classified as morphed samples. Both APCER and BPCER are commonly used in previous studies of MAD \\cite{raja2020morphing,scherhag2020deep}.\n\n\n\\subsection{Comparison of Feature Extraction and Fusion Strategies}\nFirst, we show the visual comparison of extracted features by different methods.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\columnwidth]{image\/avg_feature.eps}\n\\caption{Average of (a) MB-LBP, (b) FS-SPN, (c) PRNU and (b) Noiseprint features over 1000 randomly selected face images. Left: bona fide; right: morphed faces.}\n\\label{fig:featurefig}\n\\end{figure}\n\n\\par We first compare different feature-level fusion strategies to combine PRNU and Noiseprint patterns, including element-wise operation (sum\/max), convex compression (CC) \\cite{norouzi2013zero}, vector concatenation, and our factorized bilinear coding (FBC) method \\cite{gao2020revisiting}. We consider the features in both the spatial and the spectral domains. The PRNU and Noiseprint features extracted from the images are treated as spatial features. The spectral features are obtained by applying the discrete Fourier transform to the spatial features. Any two types of feature are fused to perform traditional MAD tasks on a subset of the test data. Therefore, six different fusion features are generated. For concatenation, the final dimension of the feature is 145,800. For sum, max, and CC, it is 72,900. The fusion feature of FBC is as compact as 2,048 dimensions. All generated features are fed into the SVM with a linear kernel for binary classification. As shown in Table \\ref{tab:toyexp}, the fusion of spatial features of PRNU and Noiseprint performs best for the six features, which can be attributed to the fact that the two patterns in the spatial domain contain more discriminative features (as shown in Fig.~\\ref{fig:featurefig}). Furthermore, our FBC-based fusion achieves the highest accuracy among the five fusion strategies.\n\n\n\\begin{table*}[!t]\n\\small\n\\caption{Accuracy(\\%) of 1-shot MAF classification on single and hybrid datasets.}\n\\vspace{-0.3cm}\n\\label{tab:maf1}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ l |c c c c c c}\n\\hline\n\\multirow{2}{*}{Method} & FERET-Morphs & FRGC-Morphs & FRLL-Morphs & CelebA-Morphs & Doppelg\u00e4nger & Hybrid \\\\\n& 4-class & 4-class & 7-class & 4-class & 4-class & 9-class \\\\\n\\hline\nXception \\cite{chollet2017xception} & 29.47&\t25.26&\t17.68&\t16.67&\t21.05&\t15.11\\\\\nMobileNetV2 \\cite{sandler2018mobilenetv2} & 31.58&\t33.68&\t31.30&\t55.19&\t25.26&\t17.33\\\\\nNasNetMobile \\cite{zoph2018learning} & 32.63&\t27.37&\t22.61&\t19.26&\t23.16&\t12.88\\\\\nDenseNet121 \\cite{huang2017densely} & 46.32&\t26.32&\t22.03&\t47.04&\t23.16&\t19.33\\\\\nArcFace \\cite{deng2019arcface} & 29.33&\t39.64&\t26.12&\t28.33&\t18.03&\t15.22\\\\\n\\hline\nRaghavendra. et al. \\cite{raghavendra2017face} & 38.95&\t43.16&\t29.28&\t89.63&\t31.58&\t11.11 \\\\ \nMB-LBP \\cite{scherhag2020face} & 33.95 &\t33.42&\t34.59&\t34.50 &\t21.31&\t14.89 \\\\\nFS-SPN \\cite{zhang2018face} & 25.41&\t31.22&\t23.71&\t61.50 &\t32.79&\t29.44 \\\\\t\nPipeline Footprint \\cite{neubert2018reducing} & 26.32&\t29.47&\t29.28&\t25.93&\t25.26&\t21.89 \\\\\nPRNU Analysis \\cite{debiasi2018prnu} & 34.74 & 26.32 & 11.01 & 37.04 &\t25.26 &\t18.56 \\\\\nInception-MAD \\cite{damer2022privacy} & 23.16 &\t30.53 &\t20.00\t& 44.81 & 29.47 & 21.78 \\\\\nMixFaceNet-MAD \\cite{damer2022privacy} & 36.84 & 37.89 & 35.94 & 57.04 & 49.47 & 33.56 \\\\\nNoiseprint-SVM \\cite{cozzolino2018noiseprint} & 50.53 & 43.16 & 22.61 & 84.44 & 31.58 & 22.00 \\\\\n\\hline\nMeta-Baseline \\cite{chen2021meta} & 51.05 & 51.44 & 34.77 & 61.43 & 33.43 & 53.46 \\\\\nCOSOC \\cite{luo2021rectifying} & 54.58 & 64.37 & 35.22 & 63.19 & 34.30 & 59.55 \\\\\n\\hline\nFBC & 96.93 & 98.83 & 94.06 & 99.50 & 56.67 & 96.11 \\\\\nFBC-all & 98.11 & 99.48 & 98.42 & 100 & 84.17 & 96.78 \\\\\n\\textbf{FBC-APL} & \\textbf{98.82} & \\textbf{99.61} & \\textbf{98.24} & \\textbf{99.67} & \\textbf{91.67} & \\textbf{98.11} \\\\\n\\hline\n\\end{tabular}}\n\\end{table*}\n\n\n\\begin{table*}[!t]\n\\small\n\\caption{Accuracy(\\%) of 5-shot MAF classification on single and hybrid datasets.}\n\\vspace{-0.3cm}\n\\label{tab:maf5}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{ l |c c c c c c}\n\\hline\n\\multirow{2}{*}{Method} & FERET-Morphs & FRGC-Morphs & FRLL-Morphs & CelebA-Morphs & Doppelg\u00e4nger & Hybrid \\\\\n& 4-class & 4-class & 7-class & 4-class & 4-class & 9-class \\\\\n\\hline\nXception \\cite{chollet2017xception} & 46.32& 43.16&\t31.01&\t73.70&\t28.42&\t43.67\\\\\nMobileNetV2 \\cite{sandler2018mobilenetv2} & 55.79 & 53.68 &\t40.00 & 89.26 & 26.32 & 54.56 \\\\\nNasNetMobile \\cite{zoph2018learning} & 48.42 & 40.00 & 24.35 &\t67.41&\t27.37&\t37.33\\\\\nDenseNet121 \\cite{huang2017densely} & 54.74 & 55.79 &\t36.23&\t89.26&\t25.26\t&53.33\\\\\nArcFace \\cite{deng2019arcface} & 44.34 & 50.91 & 33.81 & 39.67 & 20.49 & 29.11 \\\\\n\\hline\nRaghavendra. et al. \\cite{raghavendra2017face} & 45.26 & 61.05 & 31.59 & 42.96 &\t28.42 &\t11.11 \\\\\nMB-LBP \\cite{scherhag2020face} & 69.28 & 74.87 & 42.67 & 63.00\t& 26.23 & 42.11 \\\\\nFS-SPN \\cite{zhang2018face} & 41.34 & 41.97 & 26.91 & 82.67 & 27.04 & 43.89 \\\\\t\nPipeline Footprint \\cite{neubert2018reducing} & 45.26 & 61.05 & 31.59 &\t42.96 & 28.42 & 37.78 \\\\\nPRNU Analysis \\cite{debiasi2018prnu} & 53.68 & 32.63 & 29.86 & 78.15 & 26.32 & 39.22 \\\\\nInception-MAD \\cite{damer2022privacy} & 50.53 &\t51.58 &\t37.39 &\t82.59 &\t29.47 & 44.00 \\\\\nMixFaceNet-MAD \\cite{damer2022privacy} & 63.16\t& 63.68 & 53.48 & 82.59 & 33.68 & 51.00 \\\\\nNoiseprint-SVM \\cite{cozzolino2018noiseprint} & 69.47 & 69.47 & 57.39  & 87.41 & 37.89 & 51.89 \\\\\n\\hline\nMeta-Baseline \\cite{chen2021meta} & 60.60 & 64.72 & 50.74 & 81.42 & 36.80 & 61.98 \\\\\nCOSOC \\cite{luo2021rectifying} & 65.98 & 75.04 & 54.90 & 89.60 & 41.81 & 72.62 \\\\\n\\hline\nFBC & 97.64 & 99.09 & 96.94 & 99.50 & 65.83 & 96.22 \\\\\nFBC-all & 98.11 & 99.48 & 98.42 & 100 & 84.17 & 96.78 \\\\\n\\textbf{FBC-APL} & \\textbf{98.82} & \\textbf{99.61} & \\textbf{98.24} & \\textbf{99.67} & \\textbf{96.67} & \\textbf{98.22} \\\\\n\\hline\n\\end{tabular}\n}\n\\end{table*}\n\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{image\/confusion_matrix.eps}\n\\vspace{-0.75cm}\n\\caption{Confusion matrix of few-shot MAF classification on hybrid dataset.}\n\\label{fig:confus}\n\\end{figure*}\n\n\n\\subsection{Few-shot Learning for MAD}\n\n\n\nWe extend the traditional MAD problem to a few-shot learning problem. First, the PRNU and Noiseprint features are extracted, respectively. Then an FBC module (VGG-16 \\cite{simonyan2014very} as the backbone) is trained as a binary classifier for feature fusion, taking PRNU and Noiseprint features from the entire training set (all images of predefined types) as input. Based on the pre-trained FBC module, 2,048-dimensional fusion representations are generated and then fed to the APL module for binary few-shot learning using the cross-entropy loss. Here, the Euclidean distance is used to query the top five nearest neighbors of the memory component. The APL output is a tuple of the probability distribution for each class. The results in terms of accuracy, D-EER, and ACER are shown in Table \\ref{tab:madfs}. Two methods based on FSL \\cite{luo2021rectifying,chen2021meta}, two methods based on face recognition (FR) \\cite{schroff2015facenet,deng2019arcface}, several popular deep models pre-trained \\cite{chollet2017xception,sandler2018mobilenetv2,zoph2018learning,huang2017densely} on ImageNet \\cite{deng2009imagenet}, and eight current MAD methods \\cite{raghavendra2017face,scherhag2020face,zhang2018face,neubert2018reducing,debiasi2018prnu,damer2022privacy,cozzolino2018noiseprint}, are adopted for comparison. Due to the effective fusion of two complementary patterns (i.e., PRNU and Noiseprint) and the APL module, our proposed FBC-APL clearly outperforms other competing methods by a large margin.\n\n\n\n\n\n\\subsection{Few-shot Learning for MAF}\nUnlike the few-shot MAD problem, in MAF, the FBC module uses ResNet50 \\cite{he2016deep} as the backbone and is pre-trained as a nine-class classifier using all the training data (about 80\\%) of the collected database. The FBC fusion feature obtained from the training samples is then fed to the APL module for multiclass few-shot learning. A cosine similarity score is adopted to compute the similarity between queries and the data stored in memory to find the three nearest neighbors. From Tables \\ref{tab:maf1} and \\ref{tab:maf5}, one can see that our FBC-APL has achieved outstanding performance, and some results are even better than the FBC-all method, which uses FBC features from all training data to fit SVM for classification. To better illustrate the effectiveness of the proposed FBC-FSL method, we have compared the confusion matrix for nine different classes (including bona fide and eight different morphing models), as shown in Fig. \\ref{fig:confus}.\n\n\n\n\\subsection{Discussions and Limitations}\nWhy did the proposed method outperform other competing methods by a large margin? We believe there are three contributing reasons. First, PRNU and Noiseprint feature maps as shown in Fig. \\ref{fig:featurefig} have shown better discriminative capability than others; meanwhile, their complementary property makes fusion an efficient strategy for improving the accuracy. Second, we have specifically taken the few-shot constraints into the design (i.e., the adoption of APL module) while other competing approaches often assume numerous training samples. Third, from binary MAD to multi-class MAF, our FBC fusion strategy is more effective on distinguishing different classes as shown in Fig. \\ref{fig:confus}. Note that we have achieved unanimously better results than other methods across six different datasets, as shown in Table \\ref{tab:maf5}, which justifies the good generalization property of our approach.\n\nThe overall pipeline in Fig. \\ref{fig:pipeline} can be further optimized by end-to-end training. In our current implementation, the three steps are separated, that is, the extraction of PRNU and Noiseprint features, FBC-based fusion, and APL-based FSL. From the perspective of network design, end-to-end training could further improve the performance of the FBC-APL model. Moreover, there are still smaller and more challenging datasets for morphing attacks in the public domain. Validation of the generalization property for the FBC-APL model remains to be completed, especially when novel face morphing attacks (e.g., adversarial morphing attack \\cite{wang2020amora}, transformer-based, and 3D reconstruction-based face morphing) are invented. Finally, we have not considered the so-called post-morphing process \\cite{damer2021pw} where the print and scan operations are performed when issuing a passport or identity document.\n\n\n\n\\section{Conclusion and Future Work}\n\\label{con}\n\\par Face morphing attacks pose a serious security threat to FRS. In this work, we proposed a few-shot learning framework for the detection of non-reference morphing attacks and fingerprinting problems based on factorized bilinear coding of two types of camera fingerprint feature, PRNU and Noiseprint. Additionally, a large-scale database is collected that contains five types of face dataset and eight different morphing methods to evaluate the proposed few-shot MAD and fingerprinting problem. The results show outstanding performance of the proposed fusion-based few-shot MAF framework on our newly collected large-scale morphing dataset. \nWe note that face-morphing attack and defense research is likely to coevolve in the future. Future work on the attack side will include the invention of more powerful morphing attacks, such as GANformer-based \\cite{hudson2021generative} and diffusion model-based \\cite{dhariwal2021diffusion}. Consequently, defense models that include MAD and MAF could focus on the study of the feasibility of detecting novel attacks and morphed face images from printed and scanned image data. In practical applications, optimizing differential morphing attack detection with live trusted capture is also an interesting new research direction.\n\n\n\\section*{Acknowledgments}\nThis work was partially supported by the NSF Center for Identification (CITeR) awards 20s14l and 21s3li.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Acknowledgements}\nThe IceCube collaboration gratefully acknowledges the support from the following agencies and institutions: USA {\\textendash} U.S. National Science Foundation-Office of Polar Programs,\nU.S. National Science Foundation-Physics Division,\nU.S. National Science Foundation-EPSCoR,\nWisconsin Alumni Research Foundation,\nCenter for High Throughput Computing (CHTC) at the University of Wisconsin{\\textendash}Madison,\nOpen Science Grid (OSG),\nExtreme Science and Engineering Discovery Environment (XSEDE),\nFrontera computing project at the Texas Advanced Computing Center,\nU.S. Department of Energy-National Energy Research Scientific Computing Center,\nParticle astrophysics research computing center at the University of Maryland,\nInstitute for Cyber-Enabled Research at Michigan State University,\nand Astroparticle physics computational facility at Marquette University;\nBelgium {\\textendash} Funds for Scientific Research (FRS-FNRS and FWO),\nFWO Odysseus and Big Science programmes,\nand Belgian Federal Science Policy Office (Belspo);\nGermany {\\textendash} Bundesministerium f{\\\"u}r Bildung und Forschung (BMBF),\nDeutsche Forschungsgemeinschaft (DFG),\nHelmholtz Alliance for Astroparticle Physics (HAP),\nInitiative and Networking Fund of the Helmholtz Association,\nDeutsches Elektronen Synchrotron (DESY),\nand High Performance Computing cluster of the RWTH Aachen;\nSweden {\\textendash} Swedish Research Council,\nSwedish Polar Research Secretariat,\nSwedish National Infrastructure for Computing (SNIC),\nand Knut and Alice Wallenberg Foundation;\nAustralia {\\textendash} Australian Research Council;\nCanada {\\textendash} Natural Sciences and Engineering Research Council of Canada,\nCalcul Qu{\\'e}bec, Compute Ontario, Canada Foundation for Innovation, WestGrid, and Compute Canada;\nDenmark {\\textendash} Villum Fonden and Carlsberg Foundation;\nNew Zealand {\\textendash} Marsden Fund;\nJapan {\\textendash} Japan Society for Promotion of Science (JSPS)\nand Institute for Global Prominent Research (IGPR) of Chiba University;\nKorea {\\textendash} National Research Foundation of Korea (NRF);\nSwitzerland {\\textendash} Swiss National Science Foundation (SNSF);\nUnited Kingdom {\\textendash} Department of Physics, University of Oxford.\n\\section{Investigation of the significance of TXS 0506+056}\n\\label{sec:TXS_significance_investigation}\nThe significance of TXS 0506+056 found by this multi-flare algorithm is smaller than the (single-flare) time-dependent significance that was determined in \\cite{IceCube:2018cha}. The goal of this Appendix is to show that the decrease of significance is only due to the different event selection of the sample used in this analysis, and not due to the different likelihood algorithms. It is mainly related to 2 cascade events that are rejected in the new event selection, presented in~\\citep{Aartsen:2019fau}. This was discussed also in IceCube~\\citep{Abbasi:2021bvk}. As a matter of fact, the new selection was focused on muon tracks for achieving best angular resolutions for the point-source search.\n\nThe differences between this analysis and the one described in \\cite{IceCube:2018cha} are mainly of three types. These are investigated using the analysis described in this letter and the one presented in \\cite{IceCube:2018cha} to find out how much each of them contributes to the change in significance of TXS 0506+056. The results are summarized in Table~\\ref{tab:TXS_comparisons}.\n\n\\paragraph{\\textbf{Different datasets:}}\n    As mentioned also in Section \\ref{sec:detector}, the event selections used to produce the dataset analyzed in \\cite{IceCube:2018cha} and the one analyzed in this work (from~\\cite{Aartsen:2019fau}) are different. According to the internal IceCube nomenclature, the two datasets are referred to as \\MA{{\\tt PSTracks v2}} and \\MA{{\\tt PSTracks v3}}, respectively. In some cases the different event selection results in the reconstruction of slightly different energy and local angles. An extensive and detailed description of the two datasets can be found in~\\cite{Abbasi:2021bvk}.\n    \n     The significance of TXS 0506+056 is estimated on \\MA{{\\tt PSTracks v2}} and \\MA{{\\tt PSTracks v3}} by applying the multi-flare algorithm to the years 2012-2015 (containing only one of the two flares detected by this analysis). We observe the same drop in significance (from $4.0~\\sigma$ in \\MA{{\\tt PSTracks v2}} to $2.6~\\sigma$ in \\MA{{\\tt PSTracks v3}}) described in~\\cite{Abbasi:2021bvk}. The significance observed for \\MA{{\\tt PSTracks v3}} increases to $3.4~\\sigma$ if the two high-energy events, present in \\MA{{\\tt PSTracks v2}} but absent in \\MA{{\\tt PSTracks v3}}, are added by hand to the dataset. It is worth noticing also that the pre-trial significance observed for \\MA{{\\tt PSTracks v2}} with the multi-flare algorithm is not different to the pre-trial significance reported in \\citep{IceCube:2018cha}, which was obtained with a single-flare algorithm.\n     \n   \n   \n   \n   \n   \n\n   \n    \n   \n   \n   \n   \n   \n   \n   \n    \n    \\paragraph{\\textbf{Different algorithms:}}\n    The multi-flare algorithm has been developed for this analysis and applied for the first time in this work. \n    This is a crucial difference between this work and the one presented in \\cite{IceCube:2018cha}, since a multi-flare likelihood could in principle consist of more fit parameters than a single-flare likelihood. The increased parameter space of the fit may thus degrade the sensitivity. This degradation was avoided by requiring a pre-selection of candidate flares with $\\mathrm{TS}\\ge2$ (see Section \\ref{sec:analysis} and Appendix~\\ref{sec:multi-flare_algorithm}).\n   \n    Other minor improvements between the two analyses concern:\n   \n    a Gaussian integral factor, included in the marginalization term to correct for boundary effects;\n    the time PDF normalization, set to 1 across each IceCube sample by considering only up times of the detector (in \\cite{IceCube:2018cha} it was set to 1 in an infinite range, regardless the up times). The results, shown in Table~\\ref{tab:TXS_comparisons} for the single- and multi-flare algorithm applied to the 2012-2015 data, suggest that the multi-flare algorithm is not responsible for the drop of the significance, when applied to the same dataset. \n\n    \n    \\paragraph{\\textbf{Different strategies for combining independent samples:}} \n    The third and last potential source of change in significance is due to the different strategies adopted to combine the IceCube samples.\n    Since the  10-year data sample of IceCube concerns different IceCube detector configurations, triggers and event cuts, this analysis is based on the maximization of the joint likelihood defined as the product of the likelihoods of each IceCube sample (see Section \\ref{sec:analysis}). The strategy adopted in~\\cite{IceCube:2018cha}, instead, consists in maximizing the likelihood of each IceCube sample, picking up the most significant p-value and reporting it as post-trial after correcting for the look-elsewhere effect. Such a correction is made by penalizing the most significant $p$-value by the ratio of the livetime of the sample with the most significant $p$-value to the total time. To investigate this difference, the single-flare algorithm is applied to \\MA{{\\tt PSTracks v3}}. To reproduce the analysis in~\\citep{IceCube:2018cha}, the TS is maximized only across the 3 years between 2012-2015 (containing the most significant flare) and the $p$-value is penalized by the ratio of 10 years to 3 years, adopting the same logic described in \\cite{IceCube:2018cha}. In the analysis presented in this letter, the whole 10-year data are analyzed with a single joint likelihood (as described in Section \\ref{sec:analysis} but without the multiple flare feature), and the same penalization of the $p$-value is not needed in this case. As seen in Table~\\ref{tab:TXS_comparisons}, it can be stated that the results obtained in the two cases are comparable and that the strategy adopted to combine the different samples is not responsible for a substantial change in significance.\n\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{>{\\centering\\arraybackslash}m{5cm} >{\\centering\\arraybackslash}m{3.5cm} >{\\centering\\arraybackslash}m{3.5cm}}\n    \\multicolumn{3}{c}{TXS 0506+056 change in significance}\\\\\n    \\hline\n    \\hline\n    \\multirow{3}{*}{\\parbox{4.2cm}{\\centering Different datasets (multi-flare, 2012-2015 only)}} & \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\MA{{\\tt PSTracks v2}}\\\\(\\cite{IceCube:2018cha})}} & \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\MA{{\\tt PSTracks v3}}\\\\(This work)}}\\\\\n    & & \\\\\n    & $4.0~\\sigma$ & $2.6~\\sigma$ \\\\[3pt]\n    \\hline\n    \\multirow{6}{*}{\\parbox{3cm}{\\centering Different algorithms (2012-2015 only)}} &\\multirow{2}{*}{\\parbox{3.5cm}{\\centering Single-flare\\\\(\\cite{IceCube:2018cha})}} & \\multirow{2}{*}{\\parbox{3.5cm}{\\centering Multi-flare\\\\(This work)}}\\\\\n    & & \\\\\n    & \\multicolumn{2}{c}{\\MA{{\\tt PSTracks v2}}}\\\\\n    & $4.0~\\sigma$ & $4.0~\\sigma$ \\\\\n    & \\multicolumn{2}{c}{\\MA{{\\tt PSTracks v3}}}\\\\\n    & $2.7~\\sigma$ & $2.6~\\sigma$ \\\\\n    \\hline\n    \\multirow{3}{*}{\\parbox{5cm}{\\centering Strategy of sample combination (single-flare, \\MA{{\\tt PSTracks v3}})}} & \\multirow{2}{*}{\\parbox{3.5cm}{Separate likelihoods\\\\(\\centering\\cite{IceCube:2018cha})}} & \\multirow{2}{*}{\\parbox{3.5cm}{\\centering Joint likelihood\\\\(This work)}}\\\\\n    & & \\\\\n    & $2.2~\\sigma$ (post-trial) & $2.3~\\sigma$ \\\\\n    \\hline\n    \\hline\n\\end{tabular}\n\\caption{Results of the comparison between the significance obtained for TXS 0506+056 when using an analysis with features similar to the one in \\cite{IceCube:2018cha} and the one presented in this paper. When testing the impact of different datasets, the years 2012-2015 are analyzed with the multi-flare algorithm. \nWhen testing the impact of a different strategy in the combination of the samples, the single-flare algorithm is used on the dataset \\MA{{\\tt PSTracks v3}}: in one case only the IceCube sample containing the known flare is analyzed and the p-value penalized, adopting the same logic as in~\\cite{IceCube:2018cha}; in the other case all the 10-year samples are combined in a joint likelihood, as described in Section~\\ref{sec:analysis}, and no penalization is needed.}\n\\label{tab:TXS_comparisons}\n\\end{table}\n\n\n\n\n\\section{\\textit{A posteriori} comparisons with the time-integrated analysis}\n\\label{app:variab}\n\nThe results of these time-dependent analyses, despite unveiling new features of the source catalog, partly overlap with the results of the time-integrated search~\\citep{Aartsen:2019fau}. In fact, the time-dependent and time-integrated analyses are based on similar likelihood functions, sharing the same space and energy PDFs, but the time-dependent analysis distinguishes itself by adding a time PDF. This time-dependent analysis was planned together with the time-integrated analysis, and it was not developed based on the time-integrated unblinded results. Nonetheless, one might wonder how the results of the time-dependent analysis can be interpreted in the light of the prior knowledge of the time-integrated results. To address such a question, two tests are proposed in this Appendix. A first test estimates the time variability of the four most significant sources of the time-integrated analysis. A second test estimates the probability of obtaining the observed pre-trial significance of $3.8~\\sigma$ from a time-dependent binomial test (see Section~\\ref{sec:results}) on the source catalog, in the assumption that the neutrino excess observed by the time-integrated analysis~\\citep{Aartsen:2019fau} does not have any time structure. Both tests exploit the same approach, based on producing pseudo-realizations of the data by randomizing the time of the events and, unlike for the standard time-dependent analysis, keeping fixed the associated equatorial coordinates.\n\n\\paragraph{\\bf Time-variability test:}\nThis test aims at quantifying the time variability of the highly-significant events detected from the directions of NGC 1068, TXS 0506+056, PKS 1424+240 and GB6 J1542+6129 and at testing the compatibility of their arrival time with a flat distribution.\nThis test is sensitive only to the time information of the events and is unavoidably less sensitive than the time-dependent search described in Section~\\ref{sec:analysis} (referred to as standard time-dependent analysis), which is sensitive to energy, space and time information. Moreover, the significance of the likelihood method using the three variables at the same time is not equivalent to the product of likelihood methods that use one variable at a time. \n\nThe null (or background) hypothesis of the time-variability test assumes that the time-integrated signal-like events (i.e. the events with the highest time-integrated signal-over-background ratio, that mostly contribute to the significance around each source direction) are not clustered in time. Pseudo-realizations of the data for this null hypothesis (also called background samples which allow to count trials) are obtained similarly to the standard time-dependent analysis: events in a declination band around the location of the tested sources are selected and assigned a new time taken from a real up time of the detector. This procedure destroys any time correlation among events. However, while the standard analysis keeps the local coordinates of an event (azimuth and zenith) fixed and recalculates the right ascension using the new randomized time, the time-variability test freezes the equatorial coordinates of the events at the measured values, and randomizes the azimuth (notably the zenith angle, corresponding to an equatorial coordinate, at the South Pole does not depend on the time). This method guarantees that the same time-integrated signal-like events from the direction of a given source are present in the background sample with randomized times. On the other hand, this method flattens out the sub-daily modulation of the event rate in local coordinates due to the increased reconstruction efficiency along azimuth directions where more strings are aligned. As described in Section~\\ref{sec:analysis}, in the standard analysis this sub-daily modulation of the event rate is taken into account by using a correction in local coordinates to the background PDF. The azimuth dependency of the reconstruction efficiency is averaged out for flares longer than $\\sigma_T=0.2$ days as a consequence of the Earth rotation, while it might induce a change up to 5\\% in the TS for flares as short as $\\sigma_T = 10^{-3}$ days. Given that the variability observed for the four most significant time-integrated sources was beyond a flare duration of $\\sigma_T\\gg 1$ day, a lower limit $\\sigma_T^{min} = 0.2$ days is used for this time-variability test.\n\nWhereas for the standard analysis signal samples are produced by injecting Monte Carlo events on top of the background events, for the time-variability test $n_s$ events among real signal-like events are selected in the data and their times are sampled from a Gaussian distribution. The real signal-like events, potentially usable for signal injection in the time-variability test, are randomly chosen among the $2\\hat{N}_s^{t-int}$ events with the highest time-integrated signal-over-background ratio, where $\\hat{N}_s^{t-int}$ is the best-fit number of signal-like events reported by the time-integrated analysis~\\citep{Aartsen:2019fau}.\n\nThe likelihood in Eq.~\\ref{eq:10-year-likelihood} is maximized on the background and signal samples of the time-variability test and the corresponding TS distributions (for illustration at the location of NGC 1068) are shown in Fig.~\\ref{fig:ts_comparison}, for comparison with the same distributions for the standard analysis. For both analyses the separation of the signal and background TS is better for shorter flares (left plots) than longer ones (right plots). A notable feature concerns the background TS distributions in blue. For the standard analysis the TS distribution has a characteristic spike in the first bin populated by under-fluctuations set to zero. On the other hand, the TS distribution for the time-variability test is on average shifted towards larger values of TS, showing a more signal-like behavior. This is a consequence of preserving the same time-integrated space and energy variables of signal-like events in the background sample with the method described above. \nIt is to be noted that the time-integrated analysis in~\\cite{Aartsen:2019fau} fits a spectral index of NGC 1068 of 3.16, while the best-fit spectral index for the time-dependent analysis is harder, namely 2.8 (see Tab.~\\ref{tab:PS_results1}). As a consequence of preserving the spatial and energy information of the events, the background and signal samples of the time-variability test (used to make the distributions in the last row in Fig.~\\ref{fig:time-variability_comparison}) have a varying spectral index centered around 2.8. Notably, about 89\\% of the spectral indices of the 100,000 generated background samples are contained between $\\gamma^f=2$ and of $\\gamma^f=3$. Hence, these values of the spectral indices are used for the signal injection in the standard analysis when comparing the TS distributions of the standard analysis with the same distributions of the time-variability test in Fig.~\\ref{fig:time-variability_comparison}. In general, for harder spectral indices and the same flare duration $\\sigma_T^f$, the time-variability test characterizes the difference between background and signal less powerfully than the standard analysis. In fact, in the time-variability test the coordinates of the events are frozen to the true values, hence the differences between the spatial and energy PDFs of signal and background are not exploited, unlike for the standard analysis. \n\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=.95\\linewidth]{figures\/TSdistributions.png}\n\t\\caption{Comparison of the TS distributions for signals of different intensity $n_s$ and for the background between the standard analysis (first and second row) and the time-variability test (third row) at the location of NGC 1068. The left plots are made for a flare duration of $\\sigma_T=1$~d, the right plots for 100~d. Spectral indices of $\\gamma^f=2$ (first row) and $\\gamma^f=3$ (second row) are used for the injected signal in the standard analysis.}\n\t\\label{fig:ts_comparison}\n\\end{figure}\n\n \nTo complete the comparison between the standard time-dependent analysis and the time-variability test, the sensitivity and $5~\\sigma$ DP at the location of NGC 1068 are shown for the two analyses in Fig.~\\ref{fig:time-variability_comparison}. The times of signal events are sampled from a Gaussian distribution with fixed mean $t_0=58000$ MJD and variable width $\\sigma_T$. This plot can be understood by observing the TS distributions in Figure~\\ref{fig:ts_comparison}.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=.6\\linewidth]{figures\/sens_DP_time-variability_VS_std-ana.png}\n\t\\caption{Comparison of the sensitivity (dashed lines) and $5~\\sigma$ DP (solid lines) of the standard analysis (blue and orange, respectively for $\\gamma=2$ and $\\gamma=3$) and the time-variability test (green lines) in terms of the time-integrated flux per flare $F_0^f$ described in equation~\\ref{eq:time-integrated_flux}. These curves are produced at the location of NGC 1068 under the hypothesis of a single signal flare. Notice that the reconstructed value of the spectral index for NGC 1068 in \\cite{Aartsen:2019fau} is 3.16.}\n\t\\label{fig:time-variability_comparison}\n\\end{figure}\n\nFor each of the four aforementioned sources, the likelihood in Eq.~\\ref{eq:10-year-likelihood} is maximized on the real data and an observed TS is reported. A pre-trial p-value for the time-variability test is then evaluated by counting the fraction of generated background samples with TS larger than the observed TS. The post-trial p-value of each source is obtained by applying a Sidak correction (\\cite{Abdi2007}) to the pre-trial p-values with penalty factor 4 (the number of sources). The results of this test are shown in Table~\\ref{tab:time_var_analysis}. None of the four sources shows a significant time variability for the signal-like neutrino events. \n\n\\begin{table}\n\t\\centering\n\t\\begin{tabular}{>{\\centering\\arraybackslash}m{2.8cm} >{\\centering\\arraybackslash}m{2.8cm} >{\\centering\\arraybackslash}m{2.8cm} }\n\t\t\\multicolumn{3}{c}{Time-variability results}\\\\\n\t\t\\hline\n\t\t\\hline\n\t\tSource & Pre-trial p-value & post-trial p-value\\\\[3pt] \\hline\n\t\t\\vspace{3pt}\n\t\tNGC 1068 & 0.13 & 0.43 \\\\[3pt]\n\t\tTXS 0506+056 & 0.24 & 0.67\\\\  [3pt]\n\t\tPKS 1424+240 & 0.33 & 0.80 \\\\[3pt]\n\t\tGB6 J1542+6129 & 0.029 & 0.11 \\\\[3pt]\n\t\t\\hline\n\t\t\\hline\n\t\\end{tabular}\n\t\\caption{Results of the time-variability test applied to the four most significant sources of the time-integrated analysis of Ref.~\\cite{Aartsen:2019fau}. The table shows the p-values before (pre-trial) and after (post-trial) the Sidak correction with penalty factor 4. As described in this Appendix, the time-variability test only assesses the time distribution of the recorded events, by comparing with simulated samples in which the event directions and energies remain fixed as recorded, but times are randomized according to a uniform distribution.}\n\t\\label{tab:time_var_analysis}\n\\end{table}\n\nIt is worth noticing the case of M87: this source was an under-fluctuation for the time-integrated analysis, with no signal-like events identified in \\citep{Aartsen:2019fau}, but it shows up as the most significant source of the catalog when a time-dependent analysis is performed. Although a time-variability test is not made in this case, with $\\hat{n}_s=3$ signal-like neutrino events in a time scale of $\\hat{\\sigma}_T=2.0$ minutes, almost the entire significance of this source is expected to come from the time variability of the detected events.\n\n\\paragraph{\\bf Posterior time-dependent binomial test:} The second test determines the \\textit{a posteriori} probability that the time-dependent binomial test (see Section~\\ref{sec:analysis} referred to as \"standard\" in this Appendix) produces a pre-trial significance as high or higher than the observed value of $3.8~\\sigma$, in the assumption that the time-integrated neutrino excess is steady in time (background hypothesis). To do so, the same binomial test described in Section~\\ref{sec:analysis} is repeated on the list of p-values of the Northern sources. The per-source p-values are computed in the same way, by comparing the TS of each source with a distribution of TS from fully-scrambled (randomized times and recalculated right ascensions) background samples at the respective declination. As a matter of fact, the binomial p-value of the data for this test (referred to as \"posterior binomial test\") is the same as reported in Section~\\ref{sec:results} ($3.8\\sigma$). Nevertheless, the difference between the standard and the posterior binomial test is in the realization of the background samples used to translate the binomial p-value into a post-trial p-value.\n\nIn the posterior binomial test, background pseudo-realizations of the data for all the Northern sources of the catalog are obtained in the same way as described for the time-variability test: the times of the events are randomized while the equatorial coordinates are fixed at the recorded values, such that the spatial correlations among the events are preserved and the time-integrated information is effectively incorporated in the background hypothesis. For each pseudo-realization of the Northern catalog, the likelihood in Eq.~\\ref{eq:10-year-likelihood} is maximized at the location of each source and the corresponding TS is converted into a pre-trial p-value as described in Section~\\ref{sec:analysis}, by comparison with a distribution of TS from fully-scrambled background samples at the same declination. The lower limit on the flare duration $\\sigma_T^{min}$ is removed in this test to allow a proper comparison with the standard time-dependent binomial test. As a consequence, the azimuth-dependent correction to the background spatial PDF is neglected. However, this is a minor correction that has an impact at most of 5\\% only for time scales of the flares as short as $\\sim10^{-3}$ days.\n\nOnce a pre-trial p-value is computed for all the sources in a particular pseudo-realization of the Northern catalog, the binomial test is performed on this set of p-values, resulting in a background binomial p-value $P_{bin}$. This method is then repeated on many pseudo-realizations of the Northern catalog to produce the distribution of background binomial p-values for the posterior binomial test shown in blue in Fig.~\\ref{fig:binomial_p-value_distr}. These p-values are the typical binomial p-values that the binomial test produces if the neutrino events of the data (including the time-integrated excess) have no time structure. For comparison, the orange histogram in Fig.~\\ref{fig:binomial_p-value_distr} is the distribution of background binomial p-values for the standard binomial test, used in Section~\\ref{sec:results} to calculate the post-trial binomial p-value in the assumption that the time-integrated information is also randomized. Note that when the time-integrated information is preserved, the overall distribution is shifted towards higher values of $-\\log_{10}(P_{bin})$ as a consequence of including the additional information about the time-integrated excess in the background samples.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=.8\\linewidth]{figures\/binomial_test_comparison.png}\n\t\\caption{Background distribution of the binomial p-value $P_{bin}$ for the posterior (blue) and standard (orange) binomial test. For the posterior binomial test, the background sample is produced by randomizing the time of the events while keeping fixed the equatorial coordinates; for the standard analysis, the equatorial coordinates are recalculated (assuming fixed local coordinates) after the time is randomized.}\n\t\\label{fig:binomial_p-value_distr}\n\\end{figure}\n\nFinally, the probability that the time-dependent binomial test produces a more significant result than the one observed in the real data ($3.8~\\sigma$ pre-trial), given the prior knowledge about the time-integrated excess and under the assumption that the neutrino events do not have any time correlation, is estimated by counting the fraction of background binomial p-values of the posterior binomial test that are more significant than the observed result. Such estimation leads to a probability of $0.9\\%$.\n\n\\section{Multi-flare algorithm}\n\\label{sec:multi-flare_algorithm}\n\n\nThe multi-flare algorithm aims at determining the number of flares to fit in the data. This is done by evaluating the TS of clusters of events with the highest signal-over-background ratio of the spatial and energy PDFs and selecting as candidate flares those that pass a given TS threshold. On the one hand, a high value of TS threshold is required to suppress background fluctuations (fake flares), on the other hand a low value is desired to avoid the rejection of signal flares of low intensity.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.80\\linewidth]{figures\/background_reco_flares_VS_declinations.png} \n\t\\caption{Fraction of trials in which, under the null hypothesis, 1 single flare (blue line) or more than 1 flare (orange line) are reconstructed as a function of the declination if a TS threshold of 2 is applied to select the candidate flares.}\n\t\\label{fig:bkg_reco_flares}\n\\end{figure}\n\nThis multi-flare algorithm selects as candidate flares the cluster of events with the highest TS and all additional clusters of events passing a TS threshold of 2. The choice of this threshold ensures a high efficiency in rejecting fake flares, with a frequency of multiple flare reconstruction under the null hypothesis of $\\lesssim 0.1\\%$ as shown in Fig.~\\ref{fig:bkg_reco_flares}. Such a high rejection efficiency allows to preserve a sensitivity and a DP comparable to the single-flare algorithm, as shown in Fig.~\\ref{fig:single_VS_multi_sensDP} at the declination of TXS 0506+056. Note additionally that if only one candidate flare is selected by the multi-flare algorithm, the multi-flare algorithm is completely equivalent to the single-flare algorithm. \n\n\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.49\\linewidth]{figures\/sensitivity_singleflare_VS_multiflare.png}\n\t\\includegraphics[width=.49\\linewidth]{figures\/5sigmaDP_singleflare_VS_multiflare.png}\n\t\\caption{Sensitivity (left) and discovery potential (right) of the single-flare (orange lines) and multi-flare (blue lines) algorithm as a function of the flare duration $\\sigma_T$. Sensitivity and discovery potential are evaluate at the declination of TXS 0506+056 under the hypothesis of a single signal flare with a spectrum $E^{-2}$ (solid lines) and $E^{-3}$ (dashed lines). The bottom plots show the ratio of the multi-flare to single-flare curves above.}\n\t\\label{fig:single_VS_multi_sensDP}\n\\end{figure}\n\nTo quantify the goodness of the multi-flare reconstruction, two quantities are introduced: the multi-flare efficiency, defined as the fraction of trials in which all the signal flares are identified (no matter if additional fake flares are also reconstructed), and the multi-flare purity, defined as the fraction of trials in which no fake flares are reconstructed (no matter if all the signal flares are identified). The former is an indicator of how frequent the algorithm is able to identify \\textit{all} the signal flares injected in the data; the latter is an indicator of how well the algorithm is able to reject fake flares. Note that a partially reconstructed flare is considered as a fake flare in the estimation of efficiency and purity. These two quantities are shown in Fig.~\\ref{fig:efficiency_and_purity} under the hypothesis of two flares of equal intensity as a function of the time-integrated flux of each flare, for spectra $E^{-2}$ and $E^{-3}$ and for some values of $\\sigma_T$. The efficiency asymptotically reaches the value of 1: if the signal is strong enough the algorithm is always able to identify it. However, the flux required to reach such asymptotic \\textit{plateau} depends on the parameters of the flare (spectral index $\\gamma$ and flare duration $\\sigma_T$), and notably in extreme cases (soft spectra, long flare duration) the convergence is very slow, as a consequence of the high TS threshold. Nevertheless, note that in such extreme cases the flare intensity is mostly below the sensitivity level. The purity also tends to an asymptotic \\textit{plateau} at $\\gtrsim 95\\%$ with a rapidity that depends on the flare parameters.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.49\\linewidth]{figures\/working_point_efficiency_gamma2.png} \n\t\\includegraphics[width=.49\\linewidth]{figures\/working_point_purity_gamma2.png}\n\t\\includegraphics[width=.49\\linewidth]{figures\/working_point_efficiency_gamma3.png} \n\t\\includegraphics[width=.49\\linewidth]{figures\/working_point_purity_gamma3.png}\n\t\\caption{Efficiency (left plots) and purity (right plots) under the hypothesis of a two flares of equal intensity as a function of the time-integrated flux of each flare. Efficiency and purity are calculated for a spectrum $E^{-2}$ (top plots, declination of TXS 0506+056) and $E^{-3}$ (bottom plots, declination of NGC 1068) and for some values of $\\sigma_T$ (see legend). Efficiency is defined as the fraction of trials in which \\textit{all} the injected flares are correctly reconstructed (no matter if additional fake flares are also reconstructed); purity is defined as the fraction of trials in which no fake flares are reconstructed. Note that a partially reconstructed flare is considered as a fake flare when calculating the efficiency and purity.}\n\t\\label{fig:efficiency_and_purity}\n\\end{figure}\n\n\n\\section{Sensitivity, discovery potential and upper limits}\n\\label{sec:sens_DP_upLims}\n\nThe sensitivity and discovery potential (DP) are evaluated by injecting a fake signal in the dataset and looking at the signal-like TS distributions. The sensitivity is defined as the signal flux required such that the resulting TS is greater than the background median in 90\\% of the trials; the $5~\\sigma$ DP is defined as the signal flux required such that the resulting TS is greater than the $5~\\sigma$ threshold of the background TS distribution in 50\\% of the trials. The sensitivity and $5~\\sigma$ discovery potential (DP) of the multi-flare analysis as a function of the declination are shown in Fig.~\\ref{fig:sensDP} for a single (left) and a double (right) signal flare hypothesis. In the latter case, the intensity and spectral shape of the two flares are the same.\n\nThe sensitivity and $5~\\sigma$ DP are expressed in terms of a time-integrated flux:\n\\begin{equation}\n    \\label{eq:time-integrated_flux}\n    F = \\int_{T_{live}}E^2\\Phi(E,t)dt=\\sum_{f=\\mathrm{flares}}F_0^f\\left(\\frac{E}{\\mathrm{TeV}}\\right)^{2-\\gamma^f},\n\\end{equation}\nwhere $F_0^f$ is the time-integrated flux normalization factor of the $f$-th flare, independent of the flare duration $\\sigma_T^f$ and carrying the units of an energy divided by an area, and $\\Phi(E,t)$ is the overall flux, defined as the sum of the flux of all the flares from a single direction:\n\\begin{equation}\n    \\label{eq:flux_definition}\n    \\Phi(E,t)=\\sum_{f=\\mathrm{flares}}\\frac{F_0^f}{\\sqrt{2\\pi}\\sigma_T^f}\\left(\\frac{E}{\\mathrm{TeV}}\\right)^{-\\gamma^f}G^f(t|t_0^f,\\sigma_T^f).\n\\end{equation}\nIn Eq. \\ref{eq:flux_definition}, $G^f(t|t_0^f,\\sigma_T^f)=\\exp\\left[-\\frac{1}{2}\\left(\\frac{t-t_0^f}{\\sigma_T^f}\\right)^2\\right]$ is the Gaussian time profile of the $f$-th flare.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=.49\\linewidth]{figures\/timeInt_sensitivity_singleflare.png} \n\t\\includegraphics[width=.49\\linewidth]{figures\/timeInt_sensitivity_doubleflare.png} \n\t\\caption{Sensitivity (dashed lines) and $5~\\sigma$ DP (solid lines) of the multi-flare analysis vs declination, expressed in terms of the flux normalization factor per flare $F_0^f$ defined in Eq.~\\ref{eq:time-integrated_flux}, under the hypothesis of a single (left plot) and a double (right plot) signal flare. \n\n\tThe assumed energy dependence of the flares has a spectral index of $\\gamma^f = 2$ and $\\gamma^f = 3$ (see labels), and a flare duration of $\\sigma_T^f = 1$~day (blue lines) and $\\sigma_T^f = 100$~days (orange lines). The double-flare\n\tassumes two identical and well separated flares.}\n\t\\label{fig:sensDP}\n\\end{figure}\n\nSensitivities and DPs are shown in Fig.~\\ref{fig:sensDP} for two different hypotheses of the spectral index of the flares ($\\gamma^f=2$ and $\\gamma^f=3$) and two different flare durations ($\\sigma_T^f=1$ day and $\\sigma_T^f=100$ days). In the double-flare case, two identical and well separated flares are assumed, with the same spectral index $\\gamma^f$, flare duration $\\sigma_T^f$ and time-integrated flux normalization per flare $F_0^f$.\n\nThe 90\\% confidence level (CL) upper limits on the flux of each source of the catalog are defined as the flux required to produce a TS distribution that exceeds the unblinded TS of the respective source in 90\\% of the trials. These upper limits are expressed in terms of a time-integrated flux by mean of the factor $F_{90\\%}$, defined as:\n\n\\begin{equation}\n    \\label{eq:time-int_flux_upLims}\n    F = F_{90\\%}\\sum_{f=\\mathrm{flares}}\\left(\\frac{E}{\\mathrm{TeV}}\\right)^{2-\\gamma^f}.\n\\end{equation}\n\nIn the case of TXS 0506+056, the only observed multi-flare source of the catalog, the upper limits are evaluated assuming the same flare intensity for the two flares. As a matter of fact, only one global factor $F_{90\\%}$ appears in Eq.~\\ref{eq:time-int_flux_upLims}.\n\nThe upper limits of the not under-fluctuating sources of the catalog, together with the coordinates, maximum-likelihood parameters and pre-trial p-values, are reported in Table~\\ref{tab:PS_results1}. To calculate these upper limits, a spectral index $\\gamma^f=2$ in Eq. \\ref{eq:time-int_flux_upLims} is assumed for all the flares, whereas the flare time $t_0^f$ and duration $\\sigma_T^f$ are taken as the maximum-likelihood parameters. Only one flare is injected for each source, except for TXS 0506+056 for which two flares are injected, according to the maximum-likelihood results.\n\n\\setlength\\LTleft{-3cm}\n\\begin{center}\n\t\\begin{longtable}{>{\\centering\\arraybackslash}m{3.1cm} >{\\centering\\arraybackslash}m{1.0cm} >{\\centering\\arraybackslash}m{1.0cm} >{\\centering\\arraybackslash}m{1.1cm} >{\\centering\\arraybackslash}m{1.0cm} >{\\centering\\arraybackslash}m{2.3cm} >{\\centering\\arraybackslash}m{2.2cm} >{\\centering\\arraybackslash}m{1.5cm} >{\\centering\\arraybackslash}m{1.9cm}}\n\t\\hline\n\t\t\\multicolumn{9}{|c|}{catalog results}\\\\ \\hline\n\t\tSource & R.A. & $\\delta$ & $\\hat{n}_s$ & $\\hat{\\gamma}$ & $\\hat{t}_0$ & $\\hat{\\sigma}_T$ & $-\\log_{10}(p_{loc})$ & $F_{90\\%}\\times10^{4}$\\\\\n\t\t & [ deg ] & [ deg ] & & & [ MJD ] & [ days ] & & [ TeV cm$^{-2}$ ] \\\\ [3pt]\n\t\t \\midrule\n\t\t\\endfirsthead\n\t\t\\midrule\n\t\tSource & R.A. & $\\delta$ & $\\hat{n}_s$ & $\\hat{\\gamma}$ & $\\hat{t}_0$ & $\\hat{\\sigma}_T$ & $-\\log_{10}(p_{loc})$  & $F_{90\\%}\\times10^{4}$\\\\\n\t\t & [ deg ] & [ deg ] & & & [ MJD ] & [ days ] & & [ TeV cm$^{-2}$ ] \\\\[3pt]\n\t\t\\midrule\n\t\t\\endhead\n\t\tS5 0716+71 & 110.49 & 71.34 & -- & -- & -- & -- & -- & --\\\\\n\t\tS4 1749+70 & 267.15 & 70.10 & -- & -- & -- & -- & -- & --\\\\\n\t\tM82 & 148.95 & 69.67 & 27.8 &4.0 & 57395.8 & 200.0 & 1.51 & 5.7\\\\\n\t\t1ES 1959+650 & 300.01 & 65.15 & 3.9 &3.3 & 55028.4 & $1.8\\times10^{-1}$ & 2.21 & 3.8\\\\\n\t\t\\textbf{GB6 J1542+6129} & \\textbf{235.75} & \\textbf{61.50} & $\\mathbf{23.7^{+9.7}_{-7.9}}$ & $\\mathbf{2.7^{+0.5}_{-0.3}}$ & $\\mathbf{57740^{+80}_{-60}}$ & $\\mathbf{147^{+110}_{-25}}$ & \\textbf{2.67} & \\textbf{5.3}\\\\\n\t\tPG 1246+586 & 192.08 & 58.34 & -- & -- & -- & -- & -- & --\\\\\n\t\tTXS 1902+556 & 285.80 & 55.68 & 3.2 &4.0 & 54862.5 & 6.0 & 0.46 & 3.6\\\\ \n\t\t4C +55.17 & 149.42 & 55.38 & 11.2 &3.6 & 58303.7 & 59.7 & 1.00 & 2.5\\\\ \n\t\tS4 1250+53 & 193.31 & 53.02 & 6.1 &2.2 & 55062.9 & 35.9 & 0.74 & 3.7\\\\ \n\t\t1ES 0806+524 & 122.46 & 52.31 & 6.5 &3.1 & 55248.3 & 43.3 & 0.39 & 2.8\\\\ \n\t\t1H 1013+498 & 153.77 & 49.43 & 3.1 &2.2 & 58053.6 & $2.7\\times10^{-1}$ & 0.41 & 1.2\\\\ \n\t\tB3 1343+451 & 206.40 & 44.88 & 4.0 &2.7 & 57856.5 & $2.8\\times10^{-1}$ & 0.49 & 1.2\\\\ \n\t\tMG4 J200112+4352 & 300.30 & 43.89 & 11.6 &2.0 & 56776.2 & 105.9 & 1.00 & 2.6\\\\\n\t\t3C 66A & 35.67 & 43.04 & -- & -- & -- & -- & -- & --\\\\\n\t\tS4 0814+42 & 124.56 & 42.38 & 3.4 &2.6 & 56301.3 & 3.1 & 0.47 & 1.3\\\\ \n\t\tBL Lac & 330.69 & 42.28 & 3.8 &4.0 & 54637.6 & 5.6 & 0.48 & 2.5\\\\\n\t\t2HWC J2031+415 & 307.93 & 41.51 & 18.8 & 3.4 & 58056.8 & 114.0 & 0.93 & 2.4\\\\\n\t\tNGC 1275 & 49.96 & 41.51 & -- & -- & -- & -- & -- & --\\\\\n\t\tB3 0609+413 & 93.22 & 41.37 & 8.7 &1.7 & 56736.2 & 163.7 & 0.90 & 2.5\\\\ \n\t\tM31 & 10.82 & 41.24 & 8.6 &2.3 & 57900.7 & 16.4 & 1.29 & 2.1\\\\\n\t\tTXS 2241+406 & 341.06 & 40.96 & 3.8 &2.9 & 55334.5 & $2.5\\times10^{-1}$ & 0.55 & 1.7\\\\\n\t\tGamma Cygni & 305.56 & 40.26 & 5.8 &1.5 & 57336.8 & 13.0 & 0.95 & 1.8\\\\\n\t\tMkn 501 & 253.47 & 39.76 & -- & -- & -- & -- & -- & --\\\\\n\t\tB3 0133+388 & 24.14 & 39.10 & -- & -- & -- & -- & -- & --\\\\\n\t\tMkn 421 & 166.12 & 38.21 & 2.9 &2.1 & 54875.0 & $7.6\\times10^{-1}$ & 1.23 & 2.8\\\\\n\t\t4C +38.41 & 248.82 & 38.14 & 6.2 &2.1 & 56751.6 & 9.0 & 0.60 & 1.5\\\\ \n\t\tMG2 J201534+3710 & 303.92 & 37.19 & 3.9 &1.3 & 57326.7 & 129.4 & 0.45 & 1.8\\\\ \n\t\tMGRO J2019+37 & 304.85 & 36.80 & 4.2 &1.4 & 57330.9 & 135.0 & 0.40 & 1.7\\\\\n\t\tB2 0218+357 & 35.28 & 35.94 & -- & -- & -- & -- & -- & --\\\\\n\t\tB2 2114+33 & 319.06 & 33.66 & -- & -- & -- & -- & -- & --\\\\\n\t\tB2 1520+31 & 230.55 & 31.74 & 5.0 &2.4 & 55999.0 & 2.7 & 0.66 & 1.2\\\\\n\t\tNGC 598 & 23.52 & 30.62 & 4.9 &1.8 & 56520.7 & 33.0 & 0.75 & 1.7\\\\\n\t\tPG 1218+304 & 185.34 & 30.17 & 2.0 &2.4 & 54647.8 & $2.1\\times10^{-2}$ & 1.12 & 2.1\\\\ \n\t\tB2 1215+30 & 184.48 & 30.12 & 2.0 &2.4 & 54647.8 & $2.1\\times10^{-2}$ & 1.21 & 2.2\\\\\n\t\tTon 599 & 179.88 & 29.24 & 2.0 &1.7 & 55024.2 & $3.0\\times10^{-3}$ & 0.45 & 1.2\\\\\n\t\tMG2 J043337+2905 & 68.41 & 29.10 & -- & -- & -- & -- & -- & --\\\\\n\t\t4C +28.07 & 39.48 & 28.80 & -- & -- & -- & -- & -- & --\\\\\n\t\tW Comae & 185.38 & 28.24 & 3.1 &3.4 & 55682.4 & 1.5 & 0.49 & 1.2\\\\\n\t\tTXS 0141+268 & 26.15 & 27.09 & -- & -- & -- & -- & -- & --\\\\\n\t\tON 246 & 187.56 & 25.30 & -- & -- & -- & -- & -- & --\\\\\n\t\t1ES 0647+250 & 102.70 & 25.06 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 1441+25 & 220.99 & 25.03 & 4.1 &1.7 & 56994.7 & 105.6 & 0.69 & 1.8\\\\ \n\t\tPKS 1424+240 & 216.76 & 23.80 & 17.9 &2.8 & 57182.6 & 200.0 & 1.00 & 2.2\\\\\n\t\tS2 0109+22 & 18.03 & 22.75 & 4.6 &4.0 & 55153.2 & $9.2\\times10^{-1}$ & 0.93 & 1.6\\\\\n\t\tCrab nebula & 83.63 & 22.01 & -- & -- & -- & -- & -- & --\\\\\n\t\t4C +21.35 & 186.23 & 21.38 & 2.0 &2.3 & 55690.3 & $1.2\\times10^{-3}$ & 0.64 & 0.9\\\\\n\t\tTXS 0518+211 & 80.44 & 21.21 & -- & -- & -- & -- & -- & --\\\\\n\t\tRGB J2243+203 & 340.99 & 20.36 & 11.2 &3.6 & 57300.1 & 33.0 & 0.81 & 1.5\\\\ \n\t\tOJ 287 & 133.71 & 20.12 & 3.6 &2.6 & 56416.8 & $8.4\\times10^{-1}$ & 0.72 & 1.0\\\\ \n\t\tPKS 1717+177 & 259.81 & 17.75 & 2.0 &3.3 & 54587.2 & $2.0\\times10^{-1}$ & 0.45 & 1.3\\\\ \n\t\tOX 169 & 325.89 & 17.73 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0735+17 & 114.54 & 17.71 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0235+164 & 39.67 & 16.62 & -- & -- & -- & -- & -- & --\\\\\n\t\t3C 454.3 & 343.50 & 16.15 & 5.1 &2.0 & 56119.1 & 200.0 & 0.46 & 1.3\\\\\n\t\t4C +14.23 & 111.33 & 14.42 & 3.1 &2.0 & 58076.6 & 1.2 & 0.81 & 1.0\\\\\n\t\tPSR B0656+14 & 104.95 & 14.24 & -- & -- & -- & -- & -- & --\\\\\n\t\t\\textbf{M87} & \\textbf{187.71} & \\textbf{12.39} & $\\mathbf{3.0^{+2.0}_{-1.4}}$ &$\\mathbf{4.0^{+0.9}_{-0.9}}$ & $\\mathbf{57730.031^{+0.001}_{-0.001}}$ & $\\mathbf{1.4^{+1.3}_{-0.4}\\times10^{-3}}$ & \\textbf{3.35} & \\textbf{0.9}\\\\\n\t\t1H 1720+117 & 261.27 & 11.88 & -- & -- & -- & -- & -- & --\\\\\n\t\tCTA 102 & 338.15 & 11.73 & -- & -- & -- & -- & -- & --\\\\\n\t\tPG 1553+113 & 238.93 & 11.19 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 2032+107 & 308.85 & 10.94 & -- & -- & -- & -- & -- & --\\\\\n\t\tMG1 J021114+1051 & 32.81 & 10.86 & 2.8 &2.1 & 56179.2 & $8.9\\times10^{-1}$ & 0.52 & 0.9\\\\ \n\t\t1RXS J194246.3+1 & 295.70 & 10.56 & 4.2 &3.4 & 54904.8 & 24.3 & 0.51 & 1.4\\\\\n\t\tPKS 1502+106 & 226.10 & 10.50 & 9.8 &2.5 & 55509.5 & 21.6 & 1.97 & 1.8\\\\ \n\t\tOT 081 & 267.87 & 9.65 & 9.7 &2.9 & 57751.6 & 45.7 & 0.79 & 1.3\\\\\n\t\tRX J1931.1+0937 & 292.78 & 9.63 & -- & -- & -- & -- & -- & --\\\\\n\t\tOG +050 & 83.18 & 7.55 & -- & -- & -- & -- & -- & --\\\\\n\t\tMGRO J1908+06 & 287.17 & 6.18 & 2.9 &2.1 & 57045.2 & 4.6 & 0.63 & 0.9\\\\\n\t\tPKS 0019+058 & 5.64 & 6.14 & -- & -- & -- & -- & -- & --\\\\\n\t\t\\multirow{2}{*}{\\textbf{TXS 0506+056}} & \\multirow{2}{*}{\\textbf{77.35}} & \\multirow{2}{*}{\\textbf{5.70}} &$\\mathbf{10.0^{+5.2}_{-4.2}}$ & $\\mathbf{2.2^{+0.3}_{-0.3}}$ & $\\mathbf{57000^{+30}_{-30}}$ & $\\mathbf{62^{+27}_{-27}}$ & \\multirow{2}{*}{\\textbf{2.77}} & \\multirow{2}{*}{\\textbf{1.7}}\\\\ & & & $\\mathbf{7.6^{+6.1}_{-5.8}}$ & $\\mathbf{2.6^{+0.5}_{-0.6}}$ & $\\mathbf{58020^{+40}_{-40}}$ & $\\mathbf{42^{+42}_{-28}}$ & & \\\\\n\t\tPKS 0502+049 & 76.34 & 5.00 & 2.7 &2.0 & 57072.1 & 1.2 & 0.81 & 0.9\\\\\n\t\tMG1 J123931+0443 & 189.89 & 4.73 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0829+046 & 127.97 & 4.49 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 1502+036 & 226.26 & 3.44 & 2.0 &2.9 & 54606.9 & $3.4\\times10^{-1}$ & 0.53 & 1.2\\\\\n\t\tHESS J1857+026 & 284.30 & 2.67 & 3.6 &2.3 & 54984.4 & $2.0\\times10^{-1}$ & 0.71 & 0.9\\\\\n\t\t3C 273 & 187.27 & 2.04 & -- & -- & -- & -- & -- & --\\\\\n\t\tOJ 014 & 122.87 & 1.78 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0215+015 & 34.46 & 1.74 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0736+01 & 114.82 & 1.62 & -- & -- & -- & -- & -- & --\\\\\n\t\t4C +01.02 & 17.16 & 1.59 & -- & -- & -- & -- & -- & --\\\\\n\t\t4C +01.28 & 164.61 & 1.56 & -- & -- & -- & -- & -- & --\\\\\n\t\tGRS 1285.0 & 283.15 & 0.69 & 6.5 &2.8 & 54808.6 & 87.3 & 0.39 & 1.9\\\\\n\t\tPKS 0422+00 & 66.19 & 0.60 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS B1130+008 & 173.20 & 0.58 & -- & -- & -- & -- & -- & --\\\\\n\t\tPMN J0948+0022 & 147.24 & 0.37 & 2.0 &2.4 & 55610.7 & $4.3\\times10^{-4}$ & 0.90 & 0.6\\\\ \n\t\tHESS J1852-000 & 283.00 & 0.00 & 5.4 &2.8 & 54751.9 & 100.3 & 0.38 & 1.9\\\\\n\t\t\\textbf{NGC 1068} & \\textbf{40.67} & \\textbf{-0.01} & $\\mathbf{23.0^{+8.7}_{-7.9}}$ &$\\mathbf{2.8^{+0.3}_{-0.3}}$ & $\\mathbf{56290^{+90}_{-80}}$ & $\\mathbf{198^{+64}_{-64}}$ & \\textbf{2.65} & \\textbf{1.9}\\\\\n\t\tHESS J1849-000 & 282.26 & -0.02 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0440-00 & 70.66 & -0.29 & 6.2 &2.6 & 57896.8 & 66.8 & 0.51 & 0.9\\\\\n\t\tPKS 1216-010 & 184.64 & -1.33 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0420-01 & 65.83 & -1.33 & -- & -- & -- & -- & -- & --\\\\\n\t\tNVSS J190836-012 & 287.20 & -1.53 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0336-01 & 54.88 & -1.77 & -- & -- & -- & -- & -- & --\\\\\n\t\tS3 0458-02 & 75.30 & -1.97 & 4.6 &2.5 & 56974.6 & $7.0\\times10^{-1}$ & 0.65 & 0.7\\\\ \n\t\tNVSS J141826-023 & 214.61 & -2.56 & 3.7 &2.9 & 57733.0 & $3.4\\times10^{-1}$ & 0.44 & 0.6\\\\\n\t\tPKS 2320-035 & 350.88 & -3.29 & 10.8 &3.2 & 56176.8 & 160.2 & 0.57 & 1.1\\\\\n\t\tHESS J1843-033 & 280.75 & -3.30 & -- & -- & -- & -- & -- & --\\\\[3pt]\n\t\t\\midrule\n\t\tPKS 1329-049 & 203.02 & -5.16 & -- & -- & -- & -- & -- & --\\\\\n\t\tHESS J1841-055 & 280.23 & -5.55 & -- & -- & -- & -- & -- & --\\\\\n\t\t3C 279 & 194.04 & -5.79 & -- & -- & -- & -- & -- & --\\\\\n\t\tHESS J1837-069 & 279.43 & -6.93 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0805-07 & 122.07 & -7.86 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 1510-089 & 228.21 & -9.10 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0048-09 & 12.68 & -9.49 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 0727-11 & 112.58 & -11.69 & -- & -- & -- & -- & -- & --\\\\\n\t\tPKS 2233-148 & 339.14 & -14.56 & 2.0 &2.8 & 54877.5 & $2.6\\times10^{-3}$ & 1.04 & 12.0\\\\ \n\t\tNGC 253 & 11.90 & -25.29 & 4.1 &2.5 & 56511.7 & 22.7 & 0.52 & 8.7\\\\\n\t\tNGC 4945 & 196.36 & -49.47 & 2.0 &1.9 & 54739.8 & $2.4\\times10^{-1}$ & 0.63 & 55.3\\\\\n\t\tLMC & 80.00 & -68.75 & -- & -- & -- & -- & -- & --\\\\\n\t\tSMC & 14.50 & -72.75 & -- & -- & -- & -- & -- & --\\\\\n\t\t\\hline\\hline\n\t\n\t\t\\caption{Coordinates (Right Ascension R.A. and declination $\\delta$), maximum-likelihood flare parameters, logarithm of the local pre-trial p-values $p_{loc}$ of the sources of the catalog and the 90\\% CL upper limits on the time-integrated flux $F_{90\\%}$ (in units of TeV cm$^{-2}$) defined in equation~\\ref{eq:time-int_flux_upLims} for an $E^{-2}$ spectrum. Under-fluctuating results are shown with hyphens. For the four sources that give rise to the $3.0~\\sigma$ excess of the binomial test in the Northern hemisphere (highlighted in bold), the fit parameters are shown with the confidence interval at $68\\%$ CL. A line is used to separate the Northern from Southern sources. The parameters of the flare from TXS 0506+056 at 58020 MJD and related to the neutrino alert ($n_s=7.6$, $\\gamma=2.6$, $\\sigma_T=42$ days) are different from those reported in \\cite{IceCube:2018cha}, when the data available for analysis extended up to 40 days after the central time of the flare. This analysis includes 7 additional months and reconstructs a longer, more significant flare associated with the same alert.}\n\t\t\\label{tab:PS_results1}\n\t\\end{longtable}\n\\end{center}\n\n\\section{Estimation of the single-flare significance of TXS 0506+056}\n\\label{sec:singleflare_significance}\n\nThis Appendix is intended to describe how the single-flare significances of the two flares of TXS 0506+056, that are shown in Fig.~\\ref{fig:best_fit_flares}, are estimated.\n\nBy factorizing the background PDF, the multi-flare likelihood ratio $\\Lambda_{mf}^{-1}$ in Eq.~\\ref{eq:teststatistic} can be written as follows:\n\\begin{equation}\n    \\label{eq:likelihood_ratio_simple}\n    \\Lambda^{-1}_{mf}=\\frac{\\mathcal{L}(\\vec{\\hat{n}}_s, \\vec{\\hat{\\gamma}}, \\vec{\\hat{t}}_0,\\vec{\\hat{\\sigma}}_T)}{\\mathcal{L}(n_s=0)}=\\prod_{j=\\mathrm{sample}}\\prod_{i=\\mathrm{1}}^{N_j}\\left(1+\\sum_{f=\\mathrm{flares}}\\mathcal{F}^f_{i,j}\\right), \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathcal{F}_{i,j}^f \\coloneqq\\frac{n_s^f(\\mathcal{S}^f_{i,j}\/\\mathcal{B}_{i,j}-1)}{N}.\n\\end{equation}\nThe single-flare signal and background PDFs in Eq.~\\ref{eq:likelihood_ratio_simple} are the same as in Eq.~\\ref{eq:multi-likelihood}, but for the sake of clarity here they explicitly show the flare ($f$), event ($i$) and sample ($j$) indices. In addition, the dependency on the parameters, being the same as in Eq.~\\ref{eq:multi-likelihood}, is omitted to simplify the notation.\n\n\nFor TXS 0506+056 there are two identified flares, thus $\\sum_f \\mathcal{F}^f_{i,j}=\\mathcal{F}^1_{i,j}+\\mathcal{F}^2_{i,j}$. In addition, when an event $i$ does not contribute significantly to $\\mathcal{F}^f_{i,j}$, then $\\mathcal{F}^f_{i,j}\\sim10^{-6}\\text{--}10^{-4}$. Since an event can contribute significantly only to one flare, the crossed terms $\\mathcal{F}^1_{i,j}\\mathcal{F}^2_{i,j}$ can be neglected and it is meaningful to retain only terms at first order in $\\mathcal{F}^f_{i,j}$. Based on these observations, the likelihood ratio in Eq. \\ref{eq:likelihood_ratio_simple} can be well approximated as:\n\\begin{equation}\n    \\Lambda^{-1}_{mf}=\\prod_{j=\\mathrm{sample}}\\prod_{i=\\mathrm{1}}^{N_j}\\left(1+\\mathcal{F}^1_{i,j}+\\mathcal{F}^2_{i,j}\\right)\\simeq\n   \\prod_{j=\\mathrm{sample}}\\prod_{i=\\mathrm{1}}^{N_j}\\left(1+\\mathcal{F}^1_{i,j}\\right)\\left(1+\\mathcal{F}^2_{i,j}\\right)=\\left(\\Lambda_{sf}^{f=1}\\right)^{-1}\\left(\\Lambda_{sf}^{f=2}\\right)^{-1}.\n\\end{equation}\nThus, it can be factorized into single-flare components, that are equivalent to the multi-flare likelihood ratio when only one flare is considered. This result can be easily generalised to $N_f>2$ flares.\n\nSuch a factorization can be exploited to disentangle the contribution of each flare to the multi-flare TS in Eq. \\ref{eq:teststatistic}:\n\\begin{equation}\n    \\mathrm{TS}\\simeq-2\\log\\left[\\frac{1}{2}\\prod_{f=\\mathrm{flares}}\\left(\\frac{T_{live}}{\\hat{\\sigma}_T^fI\\left[\\hat{t}_0^f,\\hat{\\sigma}_T^f\\right]}(\\Lambda_{sf}^f)^{-1}\\right)\\right]=-2\\sum_{f=\\mathrm{flares}}\\log\\left[\\left(\\frac{1}{2}\\right)^{1\/N_f}\\frac{T_{live}}{\\hat{\\sigma}_T^fI\\left[\\hat{t}_0^f,\\hat{\\sigma}_T^f\\right]}(\\Lambda_{sf}^f)^{-1}\\right]=\\sum_{f=\\mathrm{flares}}\\mathrm{TS}_{sf}^{f},\n\\end{equation}\nwhere $\\mathrm{TS}_{sf}^f$ is the contribution of the $f$-th flare to the multi-flare TS and can be interpreted as a single-flare TS.\n\nThe single-flare significance $\\sigma_{sf}^f$ can be obtained in the same way as the multi-flare significance, but using the single-flare TS instead of the multi-flare TS. Assuming that the two flares of TXS 0506+056 are independent, one might expect to retrieve the multi-flare TS by summing linearly the single-flare TS and to retrieve the multi-flare significance $\\sigma_{mf}$ by summing in quadrature the single-flare significance. Although this is effectively observed for the TS, the summation in quadrature of the single-fare significance results in a mismatch of nearly 2.5\\% with respect to the multi-flare significance. To correct for this mismatch, the single-flare significance is redefined as $\\sigma^{\\prime f}_{sf}$ through the following relation:\n\\begin{equation}\n    \\frac{\\sigma^{\\prime 1}_{sf}}{\\sigma^{\\prime 2}_{sf}}=\\frac{\\sigma^1_{sf}}{\\sigma_{sf}^2}, \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\sqrt{\\left(\\sigma^{\\prime 1}_{sf}\\right)^2+\\left(\\sigma^{\\prime 2}_{sf}\\right)^2}=\\sigma_{mf}.\n\\end{equation}\n\nFor TXS 0506+056 this method is used to disentangle the single-flare significance $\\sigma^{\\prime f}_{sf}$ of the 2 flares used in Fig.~\\ref{fig:best_fit_flares}.\n\n\\section{Discussion on the multi-messenger context}\n\\label{sec:MM}\n\nAs shown in Section~\\ref{sec:results}, M87 is the most significant source of the catalog, exhibiting 3 events over a time lag of minute scale with post-trail significance of $1.7~\\sigma$. It is one of the closest ($z=0.00436$) potential cosmic ray accelerators, hosting a supermassive black hole of $6.5\\times10^9M_\\odot$. Its jet was already observed more than a century ago ~\\citep{blanford_agn} in a large elliptical radio galaxy of Fanaroff-Riley type I in the Virgo cluster.\nIt has been observed in $>100$~GeV energy region: VERITAS detected a flare extending beyond 350~GeV with a spectral index at the peak of $2.19 \\pm 0.07$ \\citep{Aliu_2012} in Apr. 2010. In a 2008 flare, a clear correlation between the X-ray emission and the TeV one \\cite{Acciari_2008,Albert:2008kb}. Previous positive detection was reported by HEGRA in 1998\/99 above 700 GeV~\\citep{2003A&A...403L...1A} , and up to $\\sim 10$~TeV by H.E.S.S. in 89 hours of observation between 2003-6, showing a variability at the time scale of a few days in the 2005 high state associated to the Schwarzschild radius of M87 \\cite{Aharonian_2006}. Recently, MAGIC reported the results on the monitoring of M87 for 156 hours in 2012-15 \\cite{MAGIC2020}. It is worth noting that HAWC set an upper limit above 2 TeV for 760 d of data. The non-observation of gamma-rays at $>$~TeV energies, may indicate a cut-off in the spectrum. Such cut-off may differ for neutrinos, being less affected by the absorption in the source and by the extra-galactic background light. \n\nThe gamma-ray observations from M87 are summarized in Fig.~\\ref{fig:MM}, together with the 10-year time-integrated upper limits on the neutrino flux estimated in~\\cite{Aartsen:2019fau} for a spectrum of the form $dN\/dE\\sim E^{-2}$. \n\nPrecise radio observations \\cite{Sikora_2016} indicate a persistent central ridge structure, namely a spine flow in the interior of M87 jet, in addition to the well-known limb-brightening profile, which needs further measurements. A composite structure of the jet has been speculated also for TXS 0506+056 based on observations months after the detection of the IceCube high-energy event that triggered its multi-wavelength observations. With the millimeter-VLBI it was observed that the core jet expands in size with apparent super-luminal velocity \\cite{Ros:2019bgo}. This can be interpreted as deceleration due to proton loading from jet-star interactions in the inner host galaxy and\/or spine-sheath structure of the jet \\cite{2005A&A...432..401G,Tavecchio:2008be}.  This sort of spine-sheat structure has been advocated as a possible explanation for the higher flux of neutrinos than gamma-rays and also suggested for TXS 0506+056 by MAGIC \\citep{2018ApJ...863L..10A}, while models with a single zone struggle to explain the 2014-2015 flare of TXS 0506+056 (see e.g. \\cite{Murase_2018,Zhang:2019htg,2018ApJ...864...84K}).\n\nOther models, e.g \\cite{Inoue:2019yfs,Murase_2020}, have been revised to explain the more recent observations of IceCube on NGC 1068 \\citep{Aartsen:2019fau}. These models focus on the higher observed flux of IceCube neutrino events in the $\\sim 1-50$~TeV region with respect to the level of gamma-ray fluxes observed at lower energy by Fermi and the limits of MAGIC. The corona super-hot plasma around the super-massive black hole accelerates protons, carrying few percent of the thermal energy, through plasma turbulence \\cite{Murase_2020} or shock acceleration \\cite{Inoue:2019yfs} leading to the creation of neutrinos and gamma rays. The environment is dense enough to prevent the escape of $\\gg$ 100 MeV gamma rays while $\\sim \\mathrm{MeV}$ gamma-rays would be their result from cascading down.\nFurther insights will be needed in both messengers and all wavelengths to better constrain the structure of jets and acceleration mechanisms in one or multiple zones.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{figures\/MM_plot.png} \n\t\\caption{The gamma-ray flux in the steady state of the source observed between 2012-2015 \\cite{MAGIC2020} is shown with black (Fermi-LAT) and blue dots (MAGIC). The higher level dashed lines are levels of flux observed during flares (see references in the text). The purple line with downing arrows corresponds to the 10-year time-integrated upper limits taken from ~\\cite{Aartsen:2019fau}, with an assumed spectrum $dN\/dE\\sim E^{-2}$.}\n\\label{fig:MM}\n\\end{figure}\n\n\n\n\\section{Results} \\label{sec:results}\n\nThe point-source search identifies M87 as the most significant source in the Northern hemisphere, with a pre-trial p-value of $p_{loc}=4.6\\times10^{-4}$, which becomes $4.3\\times10^{-2}$ ($1.7~\\sigma$) post-trial. In the Southern hemisphere, the most significant source is PKS 2233-148 with a pre-trial p-value of $p_{loc}=0.092$ and post-trial p-value of $0.72$. TXS 0506+056 is the only source of the catalog for which 2 flares are found. The time profiles of the neutrino flares reconstructed by this analysis at the location of each source, together with their pre-trial significance $\\sigma_{loc}^f$, are visualized in Fig.~\\ref{fig:best_fit_flares}. For the sake of clarity, the flare significance is denoted as $\\sigma_{loc}^f$ while the overall multi-flare significance is referred to as $\\sigma_{loc}=\\sqrt{\\sum_f\\sigma_{loc}^{f2}}$. For single-flare sources (all but TXS 0506+056) the flare and multi-flare significances coincide. \n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{Gaussians_TXS_2weights.png} \n  \\caption{Pre-trial flare significance $\\sigma_{loc}^f$ for the sources of the catalog. For all sources a single flare has been found, except for TXS 0506+056 for which 2 flares are found. In this case, the pre-trial significance of each individual flare is calculated as described in Appendix \\ref{sec:singleflare_significance}. The sources of the catalog with multi-flare pre-trial significance $\\sigma_{loc}\\ge2$ are labeled with their names.}\n  \n  \\label{fig:best_fit_flares}\n\\end{figure}\n\nThe cumulative distributions of pre-trial p-values at the location of the sources of the catalog, used as inputs to the population study, are shown in Fig.~\\ref{fig:pvalues_distribution}.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=.49\\linewidth]{figures\/cumulative_pvals_north_distribution.png}  \n  \\includegraphics[width=.49\\linewidth]{figures\/cumulative_pvals_south_distribution.png}  \n  \\caption{Cumulative distributions of the pre-trial p-values of the sources of the catalog in the Northern (left) and Southern (right) hemispheres. The cumulative p-values of the unblinded data are shown in red and compared to the background expectations in blue.}\n  \\label{fig:pvalues_distribution}\n\\end{figure}\n\nThe pre-trial binomial p-value is shown in Fig.~\\ref{fig:binomial_test} as a function of the source index $k$. The smallest binomial p-value is selected in each hemisphere and converted into a post-trial binomial p-value as described in Section~\\ref{sec:analysis}. In the Northern hemisphere the smallest pre-trial binomial p-value is $7.3\\times10^{-5}$ ($3.8~\\sigma$) when $k=4$ sources are considered (M87, TXS 0506+056, GB6 J1542+6129, NGC 1068), corresponding to a post-trial p-value of $1.6\\times 10^{-3}$ ($3.0~\\sigma$). In the Southern hemisphere the smallest pre-trial binomial p-value is 0.71, obtained by $k=1$ source (PKS 2233-148) and corresponding to a post-trial p-value of $0.89$.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics[width=.47\\linewidth]{figures\/northern_binomial_pval_noBkg.png}  \n  \\includegraphics[width=.49\\linewidth]{figures\/southern_binomial_pval_noBkg.png}   \n  \\caption{Pre-trial binomial p-value $P_{bin}(k)$ as a function of the source index $k$ in the Northern (left) and Southern (right) hemispheres. The edge with the under-fluctuating sources, with binomial p-value set to 1, is shown in blue.}\n  \\label{fig:binomial_test}\n\\end{figure}\n\nThe results of the two searches are summarized in Table~\\ref{tab:summary_results}. Having not found any significant time-dependent excess, upper limits on the neutrino emission from the sources of the catalog are estimated as discussed in Appendix~\\ref{sec:sens_DP_upLims}, using Eq.~\\ref{eq:time-integrated_flux} and~\\ref{eq:flux_definition}.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{>{\\centering\\arraybackslash}m{2.8cm} >{\\centering\\arraybackslash}m{2.5cm} >{\\centering\\arraybackslash}m{1.7cm} >{\\centering\\arraybackslash}m{2.5cm}}\n    \\multicolumn{4}{c}{Summary of the results}\\\\\n    \\hline\n    \\hline\n    \\multirow{2}{*}{Analysis} & \\multirow{2}{*}{Hemisphere} & \\multicolumn{2}{c}{p-value}\\\\ & & Pre-trial & Post-trial \\\\[3pt] \\hline\n    \\multirow{2}{*}{Point-source} & North & $4.6\\times10^{-4}$ & $4.3\\times10^{-2}$ ($1.7~\\sigma$)\\\\ & South & $9.2\\times 10^{-2}$ & 0.72\\\\  [3pt] \\hline\n    \\multirow{2}{*}{Binomial test} & North & $7.3\\times10^{-5}$ & $1.6\\times10^{-3}$ ($3.0~\\sigma$) \\\\ & South & $0.71$ & $0.89$\\\\\n    \\hline\n    \\hline\n\\end{tabular}\n\\caption{Summary of the results of the two analyses: for the point-source search the results of the best sources in the Northern (M87) and Southern (PKS 2233-148) hemisphere are reported.}\n\\label{tab:summary_results}\n\\end{table}\n\\section{Introduction}\n\\label{sec:intro}\n\nAfter more than 100 years since their discovery, the origin and acceleration processes of cosmic rays (CRs) remain unsolved. Relevant hints exist, one being provided by a neutrino event detected by IceCube with most probable energy of 290 TeV which triggered follow-up gamma-ray observations ~\\citep{IceCube:2018dnn}. These observations identified in the 50\\% containment region for the arrival direction of the IceCube event a classified BL Lac object, though possibly a Flat-Spectrum Radio Quasar (FSRQ) \\citep{Padovani:2019xcv}, at redshift $z = 0.34$, known as TXS 0506+056. It was in a flaring state \\citep{IceCube:2018dnn} with a chance correlation between the neutrino event and the photon counterpart rejected at the $3~\\sigma$ level. The intriguing aspect of the possible coincidence between the neutrino event and the gamma-ray flare hints at TXS 0506+056 being a potential CR source. Additionally, in the analysis of the data prior to the event alert IceCube found a neutrino flare of 110 day duration between 2014\/2015~\\citep{IceCube:2018cha} at a significance of $3.7~\\sigma$, if a Gaussian time window is assumed. In this case, no clear flare has been identified in available gamma-ray data from TXS 0506+056~\\citep{Aartsen:2019gxs,Glauch:2019emd}. \nThe total contribution of the observed TXS 0506+056 neutrino flares to the diffuse astrophysical flux observed by IceCube~\\citep{Aartsen:2013jdh,Aartsen:2014gkd,Aartsen:2016xlq,Aartsen:2017mau} is at most a few percent~\\citep{IceCube:2018cha}.\nIn addition, time-integrated upper limits on stacked catalogs of classes of sources (e.g. tidal disruption events \\citep{Stein:2019ivm}, blazars \\citep{Aartsen:2016lir}, gamma ray bursts \\citep{Aartsen:2017wea}, compact binary mergers \\citep{Aartsen:2020mla} and pulsar wind nebulae \\citep{Aartsen:2020eof}), \nconstrain their contribution to the measured diffuse flux. While these limits depend on assumptions on the emission of such classes of sources, such as their spectral shapes and their uniformity within the class, they indicate that there might be a mixture of contributing classes and still unidentified contributors.\n\nRecently, IceCube performed another analysis on neutrino sources: a time-integrated search for point-like neutrino source signals using ten years of data~\\citep{Aartsen:2019fau}. This search uses a maximum-likelihood (ML) method to test the locations of a catalog of 110 selected sources and the full sky. As an intriguing coincidence, the two searches find the hottest spot to be a region including the Seyfert II galaxy NGC 1068, with a significance reported from the catalog search of $2.9~\\sigma$. Additionally, a population study of the catalog revealed a $3.3~\\sigma$-level incompatibility of the neutrino events from the directions of four Northern sources with respect to the estimated background: NGC 1068, TXS 0506+056, PKS 1424+240 and GB6 J1542+6129.\n\n\nTo fully investigate this catalog of sources, this letter shows the results of a complementary time-dependent study. Time-dependent searches are particularly interesting not only because of their better sensitivity to time-integrated searches for flares of duration $\\lesssim 200$ d, due to the suppression of the time-constant background of atmospheric neutrinos, but also because flare events are particularly suitable periods for neutrino production in blazars. In fact, the injection rate of accelerated protons and the density of target photon fields for photo-meson interactions can be noticeably increased during flaring periods of blazars, leading to an enhanced neutrino luminosity $L_\\nu \\propto L^{1.5\\text{--}2}_\\gamma$ (see \\cite{Zhang:2019htg} and references therein), where $L_\\gamma$ is the photon luminosity. \nApart from the aforementioned evidence of the 2014\/2015 flare from the direction of TXS 0506+056, other IceCube time-dependent searches did not find any significant excess. Nevertheless, they constrained specific emission models \\citep{Abbasi:2020dfi} or set upper limits on the neutrino emission from selected sources \\citep{Aartsen:2015wto}. Triggered searches adopt lightcurves or flare directions from gamma-ray experiments, while sky scans search for largest flares anywhere in the sky.\nIn this paper, we extend these searches to a multiple flare scan based on a ML method. \n\\section{Conclusions} \\label{sec:conclusions}\n\nThe time-dependent point-source search presented in this letter identified M87 as the most significant source in the Northern hemisphere, with $\\hat{n}_s=3$ signal-like neutrino events in a time window of $\\hat{\\sigma}_T=2.0$ minutes and with a soft spectrum ($\\hat{\\gamma}=3.95$). The post-trial significance of M87 is found to be $1.7~\\sigma$. Because of the quite short time lag between the events, the time-dependent search is more sensitive than the time-integrated one, which explains the absence of significant signals in previous IceCube time-integrated analyses that had included M87. For the case of~\\cite{OSullivan:2019rpq}, a smaller data sample from Apr. 26, 2012 to May 11, 2017 was used. The difference in significance is due to small changes in the event reconstruction and angular uncertainty estimation between the two samples.\n\nThis analysis also identifies the two known flares at the location of TXS 0506+056, one corresponding to the most significant flare at $\\sim 57000$ MJD \\citep{IceCube:2018cha} and the other related to the high-energy event alert IceCube-170922A detected on 22 Sep. 2017 \\citep{IceCube:2018dnn}. Although these two flares are consistently identified, the significance of the result at the location of TXS 0506+056 is lower than the one reported in~\\citep{IceCube:2018cha}. This is due to the new data selection \\citep{Abbasi:2021bvk} described in Section~\\ref{sec:detector}, which introduces a different energy reconstruction from the past one~\\citep{Abbasi:2021bvk}. Further information about the reduced significance of TXS 0506+056 resulting from this analysis are provided in Appendix~\\ref{sec:TXS_significance_investigation}.\n\n\n\nThe time-dependent binomial test of the Northern hemisphere suggests an incompatibility at $3.0~\\sigma$ significance of the neutrino events from four sources with respect to the overall Northern background expectation. Of the four most significant sources in the Northern hemisphere, three are common with the time-integrated analysis~\\citep{Aartsen:2019fau}, namely NGC 1068, TXS 0506+056, GB6 J1542+6129, whereas a fourth source (M87) is different  and shows a strong time-dependent behavior. However, the results of the time-dependent and time-integrated binomial test partly overlap, as both share the same space and energy PDFs in the likelihood definition in Eq.~\\ref{eq:multi-likelihood} and both select the same three out of four sources. For this reason, although a time-dependent structure of the data is suggested by the binomial test, a time-independent scenario cannot be excluded by this analysis (see Appendix~\\ref{app:variab} for a further discussion).\n\n\nNo significant result is found in the Southern hemisphere. This is consistent with the lower sensitivity due to the substantially larger background of atmospheric muons in the Southern hemisphere.\n\\section{Data analysis methods}\n\\label{sec:analysis}\n\nThe presented analyses are based on an unbinned ML method similar to previous IceCube analyses, extended to allow the detection of multiple flares and to handle different IceCube samples (IC40, IC59, IC79, IC86-I, IC86-II-VII) with different detector configurations. Since each IceCube sample is independent, the total 10-year likelihood $\\mathcal{L}$ is defined as the product of the likelihoods of each single IceCube sample $\\mathcal{L}_j$:\n\\begin{equation}\n    \\label{eq:10-year-likelihood}\n    \\mathcal{L}(\\vec{n}_s, \\vec{\\gamma}, \\vec{t}_0, \\vec{\\sigma}_T)=\\prod_{j=\\mathrm{sample}}\\mathcal{L}_j(\\vec{n}_{s,j}, \\vec{\\gamma}, \\vec{t}_0, \\vec{\\sigma}_T),\n\\end{equation}\nwhere $\\mathcal{L}_j$ is defined as\n\n\\begin{equation}\n\\label{eq:multi-likelihood}\n\\mathcal{L}_j(\\vec{n}_{s,j}, \\vec{\\gamma}, \\vec{t}_0, \\vec{\\sigma}_T) = \\prod_{i=1}^{N_j}\\left[\\frac{\\sum_{f=\\mathrm{flares}}n_{s,j}^f\\mathcal{S}_j(|\\vect{x_s}-\\vect{x_i}|,\\sigma_i, E_i,t_i; \\gamma^f, t_0^f, \\sigma_T^f)}{N_j}+\\left(1-\\frac{\\sum_fn_{s,j}^f}{N_j}\\right)\\mathcal{B}_j(\\sin\\delta_i, E_i)\\right] .\n\\end{equation}\n\nFor each flare $f$, the likelihood in Eq.~\\ref{eq:10-year-likelihood} is a function of four parameters described below: the total number of signal-like events in the flare $n_s^f$, the flare spectral index $\\gamma^f$, the flaring time $t_0^f$ and the flare duration $\\sigma_T^f$. They are denoted with an arrow in the likelihood arguments to indicate that there are as many sets of these four parameters as the number of flares. For each flare $f$, $n_{s,j}^f$ in Eq.~\\ref{eq:multi-likelihood} denotes the partial contribution of the $j$-th sample to the total number of signal-like events in that flare, such that $n_s^f=\\sum_j n_{s,j}^f$. Such partial contribution $n_{s,j}^f$ is estimated from the relative effective area of the IceCube configuration of the $j$-th sample (determined by Monte Carlo simulations of the detector and varying with spectral index and declination) and the fraction of time that the $f$-th flare stretches on the data-taking period of the $j$-th sample.\n\nFor each IceCube sample $j$, with $N_j$ total events, the likelihood in Eq.~\\ref{eq:multi-likelihood} is constructed from a single-flare signal probability density function (PDF) $\\mathcal{S}_j$, weighted by $n_{s,j}^f$ and summed over all flares from a source (multi-flare signal PDF), and a background PDF $\\mathcal{B}_j$. The single-flare signal PDF and the background PDF are the product of a space, energy and time PDFs, as also described in \\cite{Aartsen:2015wto}. The spatial signal PDF assumes a cluster of events distributed according to a 2D Gaussian around the source position $\\vect{x_s}$, with $\\sigma_i$ being the estimated angular uncertainty on the $\\vect{x_i}$ position of the $i$-th event. For the signal energy PDF, that depends on the declination $\\delta_i$ and the energy proxy $E_i$ of the events (the energy as measured by IceCube from visible light released in the detector by muon tracks), an unbroken power law $\\propto E^{-\\gamma^f}$ is used. The spectral index $\\gamma^f$ is bound within $1\\le\\gamma^f\\le4$ and can be different for each flare $f$. The signal time PDF of each flare $f$ is provided by a one-dimensional Gaussian $\\propto \\exp{[-(t_i-t_0^f)^2\/(2\\sigma_T^{f2})]}$, where $t_i$ is the time of the $i$-th event. Its normalization is such that the integral of the time PDF across the up times of each IceCube sample is 1. The central time of each Gaussian flare $t_0^f$ is constrained within the 10-year period of the analyzed data and the flare duration $\\sigma_T^f$ cannot exceed an upper limit of 200 days, above which time-integrated searches are more sensitive than time-dependent ones. For computational efficiency, the signal time PDF of each flare is truncated at $\\pm 4\\sigma_T^f$, where the flare can be considered concluded.\n\nThe spatial background PDF is obtained through a data-driven method by scrambling the time of the events and correcting the right ascension accordingly, assuming fixed local coordinates (azimuth, zenith). It depends only on the declination $\\delta_i$ of the events and it is uniform in right ascension. Due to the natural tendency of the reconstruction to be more efficient if the direction of the source is aligned with the strings of the detector, an azimuth-dependent correction is applied to the spatial background PDF. Such correction is relevant for time scales shorter than one day, whereas it is negligible for longer time scales, since any azimuth dependency is averaged out by the Earth rotation. The background energy PDF is taken from scrambled data as well, and it is fully described in~\\cite{Aartsen:2013uuv}. It depends on the declination $\\delta_i$ and the energy proxy $E_i$ of the events. The background time PDF is uniform, as expected for atmospheric muons and neutrinos if seasonal sinusoidal variations are neglected. The maximal amplitude for these variations is 10\\% for the downgoing muons produced in the polar atmosphere  and smaller for atmospheric neutrinos coming from all latitudes \\citep{Gaisser:2013lrk}.\n\nThe test statistic (TS) is defined as:\n\\begin{equation}\n\\label{eq:teststatistic}\n\\mathrm{TS}=-2\\ln\\left[\\frac{1}{2}\\left(\\prod_{f=\\mathrm{flares}}\\frac{T_{live}}{\\hat{\\sigma}_T^f I\\left[\\hat{t}_0^f, \\hat{\\sigma}_T^f\\right]}\\right)\\times\\frac{\\mathcal{L}(\\vec{n}_s=\\vec{0})}{\\mathcal{L}(\\vec{\\hat{n}}_s, \\vec{\\hat{\\gamma}}, \\vec{\\hat{t}}_0, \\vec{\\hat{\\sigma}}_T)}\\right] ,\n\\end{equation} \nwhere the parameters that maximize the likelihood function in Eq.~\\ref{eq:10-year-likelihood} are denoted with a hat and $\\mathcal{L}(\\vec{n}_s=\\vec{0})$ is the background likelihood, obtained from Eq.~\\ref{eq:10-year-likelihood} by setting $n_s^f=0$ for all the flares.\nThe likelihood ratio is multiplied by a marginalization term intended to penalize short flares, similarly used in previous time-dependent single-flare IceCube analyses to correct a natural bias of the likelihood towards selecting short flares. This was discussed in~\\cite{Braun:2009wp} for the single-flare analysis. For the multi-flare analysis, the numerical factor $1\/2$ in the equation above is chosen such that the marginalization term has the same form as the single-flare one when the true hypothesis is a single flare. The factor $0<I\\left[\\hat{t}_0^f, \\hat{\\sigma}_T^f\\right]<1$ is defined as\n\\begin{equation}\nI\\left[\\hat{t}_0^f, \\hat{\\sigma}_T^f\\right]=\\int_{T_{live}}\\frac{1}{\\sqrt{2\\pi}\\sigma_T^f}\\exp{\\left[-\\frac{(t-t_0^f)^2}{2\\sigma_T^{f2}}\\right]}dt ,\n\\end{equation}\nwhere $T_{live}$ is the full period of the analysis. It is introduced in this analysis to correct for boundary effects of Gaussian-modeled time-dependent searches.\n\nA pre-selection of interesting time frames is made by evaluating the TS of the events, looking for clusters of high-energy events in time and space, and imposing that $\\mathrm{TS} \\ge 2$.\nThis value is chosen to reduce the reconstruction probability of `fake' multiple flares to $\\lesssim0.1\\%$ under the null hypothesis (see Appendix~\\ref{sec:multi-flare_algorithm}). Clusters of events fulfilling the above requirement are called candidate flares. For the candidate flares with central times $t_0^f$ overlapping within $\\pm 4\\sigma^f_T$, only the one with the highest TS is retained. If no candidate flares are found, the pre-selection reports as candidate flare the one cluster with the highest TS. The parameters of these candidate flares are used as seed values for the ML fit.\n\nThe first analysis presented in this letter, the point source search, looks for the most significant flaring source in each hemisphere. To do so, the TS is maximized at the location of each source and converted into a local pre-trial p-value $p_{loc}$ by comparing the observed result with a distribution of TS produced under the null hypothesis ($\\vec{n}_s=\\vec{0}$). The source corresponding to the lowest pre-trial p-value in the respective hemisphere is reported, together with its post-trial p-value after accounting for the number of trials, i.e. often called look-elsewhere effect. The post-trial p-value is estimated in each hemisphere by repeating the analysis on background scrambles, picking up the lowest p-value in each scramble and counting the fraction of such background p-values that are smaller than the lowest pre-trial p-value observed in the data.\n\nThe second analysis presented in this paper, a population study, is a binomial test that looks for an excess of neutrino emission from a subset of sources that are not strong enough to emerge individually. It aims at determining whether the pre-trial p-values observed in a particular hemisphere are compatible as a whole with a background scenario. The p-values of the sources are ranked from the lowest to the highest, and in each hemisphere the pre-trial binomial p-value $P_{bin}(k)$ is calculated as a function of the source index $k$ \\cite{OSullivan:2019rpq,Aartsen:2019fau}:\n\\begin{equation}\nP_{bin}(k) = \\sum_{m=k}^N \\frac{N!}{(N-m)!\\,m!}p_k^m(1-p_k)^{N-m} .\n\\end{equation}\n\nHere $p_k$ is the pre-trial p-value of the $k$-th source in the sorted list of $N$ sources ($N=97$ in the North, $N=13$ in the South). After scanning all the source indices $k$, the smallest binomial p-value is corrected for trials and reported as post-trial in each hemisphere, together with the corresponding number of sources. The post-trial binomial p-value is estimated in each hemisphere by producing many background realizations of the catalog, picking up the smallest binomial p-value in each background realization and counting the fraction of such background binomial p-values that are smaller than the binomial p-value observed in the data.\n\\section{Apparatus and data sample}\n\\label{sec:detector}\nThe IceCube Neutrino Observatory~\\citep{Achterberg:2006md} is a 1~km$^3$-sized neutrino telescope optimized for detection of high-energy neutrinos above $\\sim 100$~GeV. It is located at the South Pole at a depth from $\\sim 1.5$ to 2.5~km in the Antarctic ice. It consists\nof 86 strings instrumented with 5,160 digital optical modules (DOMs), each equipped with a 10-inch photomultiplier tube~\\citep{Abbasi:2010vc} in a pressure-resistant sphere with associated digitizing electronics~\\citep{Abbasi:2008aa}.\n\nThe dataset used in this analysis comprises 10 years of IceCube data, from \nApril 6, 2008 to July 10, 2018~\\citep{Aartsen:2019fau,Abbasi:2021bvk}. It includes data from partially-built detector configurations, with 40, 59, 79 strings (IC40, IC59, IC79) described in ~\\cite{Abbasi:2010rd,Aartsen:2013uuv,\nAartsen:2016oji}, and from the full detector configuration of 86 strings (IC86-I), described\nin~\\cite{Aartsen:2014cva}. An updated selection of track-like events that reduces the atmospheric background, described in \\cite{Aartsen:2019fau}, is applied to the data recorded in 2012-2018. It leads to an all-sky event rate of $\\sim 4$~mHz, dominated by muons from interactions of atmospheric neutrinos from the Northern hemisphere (up-going region, declination $\\delta \\geqslant -5^{\\circ}$) and by high-energy, well-reconstructed atmospheric muons from the Southern hemisphere (down-going region, $\\delta < -5^{\\circ}$).\nThe resulting median angular resolution (the difference between the reconstructed simulated event direction and the true primary neutrino) is smaller than $0.60^{\\circ}$ above 10~TeV, where it shows an improvement of $\\sim 10\\%$ with respect to the previous event selection~\\citep{Aartsen:2016oji}.\n\nThe selection of the candidate neutrino sources used in these analyses is described in~\\cite{Aartsen:2019fau}. The catalog is composed of extragalactic sources selected from the \\textit{Fermi}-LAT 4FGL catalog~\\citep{Fermi-LAT:2019yla} that include eight starburst galaxies detected by \\textit{Fermi}-LAT, which may host hadronic interactions of cosmic rays with ambient matter, and galactic sources from TeVCat~\\citep{Wakely:2007qpa} and gammaCat~\\citep{gammaCat:2018}. The catalog is aimed at maximizing the detection probability of IceCube, while considering relevant GeV-TeV information from gamma-ray experiments. In total, 110 sources are selected (see Fig.~\\ref{fig:skymap}), 97 located in the Northern hemisphere and 13 in the Southern one. \n\n\\begin{figure}[hbt]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{figures\/Source_skymap.png}\n    \\caption{Distribution of the sources in the catalog in equatorial coordinates. They are classified as BL Lacs (BLLs), Flat-Spectrum Radio Quasars (FSRQs), Active Galactic Nuclei (AGNs), Starburst Galaxies (SBGs), Unidentified Blazars (UNIDBs) and galactic sources (GALs). The red line divides the Northern hemisphere (up-going region) and the Southern hemisphere (downgoing region) at declination $\\delta=-5^\\circ$, where the background is substantially different.}\n    \\label{fig:skymap}\n\\end{figure}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of open quantum systems provides the necessary means to study and characterize the dynamics of quantum systems that are in an inevitable interaction with their surrounding environments~\\cite{openbook}. It is well known that although closed systems evolve in time unitarily, dynamics of open quantum systems is no longer unitary due to the coupling to their environments. Such an interaction between the principal open system and the environment typically results in decoherence of the principal system, resulting in the loss of characteristic quantum properties such as coherent phase relations. From the standpoint of dynamical memory effects, time evolution of open quantum systems can be classified as exhibiting Markovian (memoryless) or non-Markovian behaviour. As a consequence of the increasing experimental control over quantum systems and the development of reservoir engineering techniques in recent years, the study of quantum non-Markovianity has become a significant line of research~\\cite{rivas14,breuer16}. Various different methods have been introduced for quantifying and characterizing the non-Markovian behaviour in the dynamics of open quantum systems. Among others, the approaches based on the information dynamics between the open system and its environment have become prominent~\\cite{fanchini14,lu10,luo12,rivas10,breuer09} since when information flows from the environment back to the system throughout the dynamics, the future states of the open system might depend on its earlier states.\n\nQuantum walks have been proposed as the quantum counterpart of the classical random walks~\\cite{aharonov1993} and they attracted considerable attention in the first place due to their quadratically faster spreading rates compared to their classical analogues~\\cite{kempe2003}. In nearly two decades, the subject has gained its place in quantum computation as a substantial field of research on both the theoretical and the experimental sides~\\cite{venegas-andraca2012}. It turned out, for example, that quantum walks are promising resources for developing new quantum algorithms~\\cite{ambainis2003} and are universal for quantum computation~\\cite{lovett2010}. They also provide a powerful framework for simulating physical systems~\\cite{buerschaper2004,oka2005}, quantum state transfer \\cite{kurzynski2011, zhan2014, yalcinkaya2015, stefanak2016, stefanak2017} and for examining the topological quantum matter~\\cite{kitagawa2010,kitagawa2012,Cedzich2016,Cedzich2018a,Cedzich2018}. The open system dynamics of quantum walks on the other hand has also been extensively studied in existence of various noise channels~\\cite{brun2003, romanelli2005, kendon2007, annabestani2010}. One of the main consequences of these studies is that decoherence may supress the spreading rate and give rise to a quantum to classical transition in the probability distribution of the walker. Some specific types of noise may even lead to Anderson localisation~\\cite{schreiber2011}. Many of these theoretical works are supported by experiments performed over various physical setups as well, including ultracold atoms in optical lattices~\\cite{karski2009}, trapped ions~\\cite{schmitz2009, zahringer2010}, photons in a fibre loop~\\cite{schreiber2010} and waveguide lattices~\\cite{perets2008}.\n\n\\ALS{}{The connection between localisation and non-Markovianity has been studied in different settings, such as atomic impurities embedded in a disordered coupled cavity array ~\\cite{lorenzo2017quantum}, in a quasi periodic Fermi lattice\\cite{cosco2018memory} and in a Bose Hubbard lattice\\cite{cosco2018bose}. These studies show that non-Markovian memory effects emerge  as excitations localise in the vicinity of the impurity.} \nIn quantum walks, non-Markovian effects can be analysed without considering an extra external environment and focusing on the coin space reduced dynamics where the spatial degree of freedom is traced out \\cite{hinarejos2014}. In that case, the position space itself is treated as the environment of the coin space. Even though this reduced dynamics is known to be non-Markovian for the standard quantum walk, the presence of decoherence in the form of broken links wipes out the non-Markovian behaviour and gives rise to a Markovian process. On the other hand, it has also been shown that the non-Markovian behaviour can be enhanced when the walker is subjected to some specific static or dynamic disorder \\cite{kumar2018}. In this paper, we analyse non-Markovianity for the quantum walk on the line with a single impurity at the origin. In particular, we examine the relationship between non-Markovianity and  emerging bound states depending on the impurity's phase angle. After solving for the bound states by a transfer matrix approach in section~\\ref{sec:qwwspi}, we focus on the effects of these bound states on the system's evolution and investigate the reduced coin dynamics from the point of view of quantum non-Markovianity. We characterize the localisation in position space using the properties of bound states and participation ratio of the probability distribution of quantum walks. We compare the trends in these quantities with those from two non-Markovianity measures of the reduced coin dynamics to reveal the connection between the phenomenon of bound state localisation and non-Markovianity. We point out the differences between the outcomes of the two non-Markovianity measures, which are based on state distinguishability and system-ancilla entanglement, in relation with the emergence of bound states. Our results for the localisation and non-Markovianity in comparison with each other follow in section~\\ref{sec:locAndNM}. We present a discussion of our results and conclude in section~\\ref{sec:conc}.\n\n\\section{\\label{sec:qwwspi}Quantum walk with a phase impurity}\n\nIn analogy to the classical random walk, a single step $\\hat{W}$ for the quantum walk on the line is defined to include two consecutive unitary operations, a coin-toss $\\hat{C}$ and a conditional translation $\\hat{T}$, which is given in the form $\\hat{W}=\\hat{T}\\hat{C}$ \\cite{aharonov1993,kempe2003,venegas-andraca2012}. The time evolution therefore takes place in the discrete bipartite coin-position Hilbert space $H_c \\otimes H_p$ spanned by the states $\\ket{c,n}$, where $c\\in \\set{\\uparrow,\\downarrow}$ describes the two coin states being the eigenstates of the Pauli operator in z-direction, $\\hat{\\sigma}_z$, and $n$ is an integer labelling the discrete sites that the walker can be found. Thus, $t$ steps of the walk are realized by applying $\\hat{W}$ repeatedly to an arbitrary initial state $\\ket{\\Psi_0}$ and the total state of the system at step $t$ can be written as\n\\begin{equation}\n\\ket{\\Psi_t} = W^t \\ket{\\Psi_0} = \\sum_{c,n} a_{c,n}(t) \\ket{c,n},\n\\end{equation}\nwhere $a_{c,n}(t)$ are complex coefficients belonging to the appropriate coin and position states at time $t$. The probability to find the walker at any site $n$ after $t$ steps is accordingly given as $P_n(t)\\!=\\!\\sum_c|a_{c,n}(t)|^2$. The coin operator can preferably be chosen as any unitary operator in $SU(2)$. Here, we will restrict it to the rotations about x-axis by an angle of $2\\theta \\in [0,4\\pi]$, which are written as\n\\begin{equation}\n\\hat{C}_\\theta = \\rme^{-i \\theta \\hat{\\sigma}_x } \\otimes \\hat{I}_N\n\\end{equation}\nwhere $\\hat{\\sigma}_x$ is the Pauli-X operator and $\\hat{I}_N=\\sum_n |n\\rangle\\langle n|$ is the identity operator acting on the position space with dimension $N$. As it is seen, the coin operator only affects the coin state and keeps the position state intact. The conditional translation operator on the other hand moves the walker to the left or to the right depending on the coin state shown here by \n\\begin{equation}\n\\hat{T} =\n\\sum_n \n\\bigg[\n\\outerp{\\uparrow}{\\uparrow} \\otimes \\outerp{n+1}{n}\n+ \n\\outerp{\\downarrow}{\\downarrow} \\otimes \\outerp{n-1}{n}\n\\bigg],\n\\end{equation}\nand thus entangling the coin and the position degrees of freedom in general~\\cite{carneiro2005}.\n\nThe walk can also be considered as a stroboscopic simulation of a quantum evolution generated by an effective Hamiltonian $\\hat{H}_\\theta$ such that $\\hat{W}_\\theta\\equiv\\exp{(-i \\hat{H}_\\theta)}$, where we assume that the time required for taking one step and $\\hbar$ are both set to unity \\cite{kitagawa2010,kitagawa2012}. It is well-known that the spectrum of this Hamiltonian has a band structure with a period of $2\\pi$, which arises from the discrete time-translation symmetry of the walk, i.e., the Hamiltonian is in fact a representative of a recurring single-step evolution (see figure~\\ref{fig:bandStr} (a)). The energy eigenvalues $E$ here are called quasi-energies, similar to the quasi-momentum $k$ showing up due to discrete spatial translation symmetry of $\\hat{W}_\\theta$ in the standard quantum walk by a unit lattice spacing. The standard walk Hamiltonian becomes diagonal in this quasi-momentum basis via the transformation $\\ket{c,k}=\\frac{1}{\\sqrt{2\\pi}}\\sum_n \\rme^{-ikn}\\ket{c,n}$ with $k\\in [-\\pi,\\pi]$ and one obtains the dispersion relation $\\cos E(k) = \\pm \\cos \\theta \\cos k$, where $k$ is in units of $\\hbar$ over the lattice spacing. Two quasi-energy bands associated with the coin states are symmetric about $E=0$ and the band gap closes for $\\theta=0$ or $\\pi$. Any pair of eigenstates with a quasi-energy difference of $\\pi$ can be associated with one another via the sublattice operator \n\\begin{equation}\n\\hat{S} = \\hat{I}_2 \\otimes \\sum_n (-1)^n \\outerp{n}{n},\n\\label{eq:sublatOpr}\n\\end{equation}\nwhich is both unitary and Hermitian, i.e., $\\hat{S}^2=\\hat{I}_{2N}$ \\cite{asboth2012}. Thus, for each eigenstate $|E\\rangle $ of $\\hat{H}_\\theta $ with quasi-energy $E$ in a given band, there exists another eigenstate $\\hat{S}|E\\rangle $ with quasi-energy $E+\\pi$ in the other band, which can actually be deduced directly using  (\\ref{eq:sublatOpr}),\n\\begin{equation}\n\\hat{S} \\hat{W}_\\theta \\hat{S} = -\\hat{W}_\\theta~~\n\\Rightarrow~~\n\\hat{S} \\hat{H}_\\theta \\hat{S} = \\hat{H}_\\theta + \\pi.\n\\label{eq:enAlt}\n\\end{equation}\nThe sublattice operator has two degenerate eigenvalues $\\pm 1$ such that eigenstates corresponding $+1$ ($-1$) parity, which we will denote by $\\ket{S_e}$ ($\\ket{S_o}$), occupy only even (odd) labelled sites in the position space. These eigenstates can also be written in terms of the symmetric (for~$+1$) and anti-symmetric (for~$-1$) superpositions of $\\ket{E}$ and $\\hat{S}\\ket{E}$ such that\n\\begin{equation}\n\\ket{S_{e,o}} = \\ket{E} \\pm \\hat{S} \\ket{E}.\n\\end{equation}\nNote that the step operator $\\hat{W}_\\theta$ transforms $\\ket{S_{e}}$ to $\\ket{S_{o}}$, or vice versa, up to an overall phase of $\\rme^{-iE}$. Therefore, if one initializes the walk with either of $\\ket{\\Psi_0}=\\ket{S_{e,o}}$, the total state will oscillate from one to the other forever, i.e., step operator moves the state back and forth between two sublattices.  It is worth mentioning here that this periodic oscillation with a period of $2$ will play an important role in our discussion of non-Markovianity. \n\nWe will study the non-Markovian behaviour from the point of view of the coin sub-system, which will be considered as an open system with the position space interacting with it as the environment \\cite{hinarejos2014}. Therefore, the time evolution of the coin density matrix will be investigated. Since the stationary states of the standard quantum walk are product states of the form $\\ket{E_k} = \\ket{\\chi_k}\\otimes\\ket{k}$, the reduced pure coin density matrix can be written as $\\rho^\\mathrm{coin}_k=\\outerp{\\chi_k}{\\chi_k} = (I_2 + \\vec{r}_k \\cdot \\sigma)\/2$, where\n\\begin{equation}\n\\vec{r}_k = \\frac{1}{\\sin E(k)} (\\cos k \\sin \\theta, - \\sin k \\sin \\theta, -\\sin k \\cos \\theta).\n\\label{eq:ideal_rxyz}\n\\end{equation}\n\nWe consider a modified version of the standard quantum walk such that whenever the walker passes through the origin $n\\!=\\!0$, it acquires a phase of $\\rme^{i\\phi}$ as illustrated in figure~\\ref{fig:transfer}. This effect can be introduced to the step operator by rewriting it as $\\hat{W}_\\theta'=\\hat{T} \\hat{C}_\\theta \\hat{P}\\equiv \\rme^{-i\\hat{H}_\\theta'}$ to include the phase operator \n\\begin{equation}\n\\hat{P} = \\hat{I}_2 \\otimes \\sum_n \\rme^{i \\phi_n } \\outerp{n}{n},\n\\label{eq:phase_op}\n\\end{equation}\nwhere $\\phi_n=\\phi\\delta_{n,0}$ and $\\delta_{n,m}$ denotes the Kronecker delta function. This model was studied by Wojcik et al. and the bound eigenstates of the double step operator $\\hat{W}_\\theta'^2$ are obtained \\cite{Wojcik2012}. We will provide an alternative solution for the stationary bound states in this model for the single step operator $\\hat{W}_\\theta'$ using a transfer matrix approach \\cite{zhao2015disordered}. We note that a double step operator would be useless in our case since it restricts the evolution to one of the sublattices, and hence, no oscillation takes place between $\\ket{S_{e,o}}$ and $\\ket{S_{o,e}}$ contrary to the discussion we will have, where these oscillations lie at the center of non-Markovian behaviour.\n\nThe phase operator $\\hat{P}$ breaks the translation invariance of the step operator $\\hat{W}_\\theta'$ in the considered model. However, we can still employ the reflection symmetry which is introduced by the reflection operator\n\\begin{equation}\n\\hat{R} = \\sigma_x \\otimes \\sum_n |-n\\rangle \\langle n |,\n\\label{eq:reflOpr}\n\\end{equation}\nin finding the stationary states of $\\hat{H}_\\theta'$ as follows. Similar to the sublattice operator, $\\hat{R}$ also has the property $\\hat{R}^2=\\hat{I}_{2N}$ and it possesses two degenerate eigenvalues $\\pm 1$. Also, by considering the commutation relations that $[\\hat{R},\\hat{H}_\\theta']=0=[\\hat{R},\\hat{S}]$, eigenstates of $\\hat{H}_\\theta'$ can be labelled by a definite $\\pm$ parity, i.e. $\\hat{R} | E^{\\pm} \\rangle =\\pm | E^{\\pm} \\rangle$, and application of $\\hat{S}$ does not change this parity and it yields the eigenstates with quasi-energies $E^\\pm+\\pi$ since the condition (\\ref{eq:enAlt}) is still valid for $\\hat{W}_\\theta'$. These energy eigenstates can be written in the component form\n\\begin{equation}\n\\ket{E^\\pm} =\\sum_{n} \\alpha^\\pm_n \\ket{\\uparrow,n}+\\beta^\\pm_n\\ket{\\downarrow,n},\n\\label{eq:stStates}\n\\end{equation}\nwhere the coefficients obey the constraint $\\alpha^{\\pm}_{-n} = \\pm \\beta^{\\pm}_{n}$ \nas a direct consequence of the reflection symmetry. Thus, $\\ket{E^\\pm}$ can be constructed by knowing only half of their components. Also, by using $\\hat{W}_\\theta'\\ket{E^\\pm}~=~\\exp(-i E^\\pm)\\ket{E^\\pm}$, one can obtain the recursion relations between these coefficients and rearrange them in the following way \n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1.55]{fig1.pdf}\n\\caption{Schematic representation of a bound state and its probability distribution over the position space. The walker acquires a phase $\\rme^{i\\phi}$ due to the impurity at the origin.  As given in (\\ref{eq:trRec}), the coefficients $(\\alpha_n,\\beta_{n-1})$ are related to each other by transfer matrices $T$ and $T(\\phi)$ for $n\\neq 0$ and $n=0$, respectively. The localisation length $l$ is determined from eigenvalue $\\lambda$ of the transfer matrix $T$. Blue arrows represents the coin state in the phase space and their lengths are drawn proportional to $P_n$.\n\\label{fig:transfer}}\n\\end{figure}\n\\begin{equation}\n\\begin{pmatrix}\n\t\\alpha_{n+1}^\\pm \\\\\n  \\beta_n^\\pm\n\\end{pmatrix}\n=\nT(\\phi_n)\n\\begin{pmatrix}\n    \\alpha_{n}^{\\pm} \\\\\n    \\beta_{n-1}^{\\pm} \n  \\end{pmatrix}.\n\\label{eq:trRec}\n\\end{equation}\nHere, the matrix $T(\\phi_n)$ is called the transfer matrix for site $n$ and it connects the adjacent coefficients in (\\ref{eq:stStates}) (See figure~\\ref{fig:transfer}), which is given in general as\n\\begin{equation}\nT(\\phi_n)=\n\\begin{pmatrix}\n\\rme^{i(E^\\pm+\\phi_n)}\\sec\\theta & -i\\tan\\theta \\\\\ni\\tan\\theta & \\rme^{-i(E^\\pm+\\phi_n)}\\sec\\theta \n\\end{pmatrix},\n\\end{equation}\nwith the inverse $T^{-1}(\\phi_n) = \\sigma_x T(\\phi_n) \\sigma_x$.\nReflection symmetry implies $T(\\phi_{-n}) = \\sigma_x T^{-1}(\\phi_n) \\sigma_x=T(\\phi_{n})$ and thus $\\phi_n = \\phi_{-n} $. \nSince $\\phi_n = \\phi \\delta_{n,0}$ in our model, we set $T(\\phi_n=0) = T$ for $n \\neq 0$. Now, the problem of finding the stationary states is boiled down to determine a suitable pair $(\\alpha_1^{\\pm},\\beta_0^{\\pm})^T$ in (\\ref{eq:stStates}) satisfying the reflection property. We can look for stationary states whose coefficients as simultaneous eigenvectors of the transfer matrices must satisfy\n\\begin{equation}\nT(\\phi) \\sigma_x \\!\n\\begin{pmatrix}\n    \\alpha_{1}^{\\pm} \\\\\n    \\beta_{0}^{\\pm} \n  \\end{pmatrix} \\!\\!\n  = \\!\n\\pm \\!\n\\begin{pmatrix}\n    \\alpha_{1}^{\\pm} \\\\\n    \\beta_{0}^{\\pm} \n  \\end{pmatrix}\n\\mathrm{and}~\nT \\!\n\\begin{pmatrix}\n    \\alpha_{n}^{\\pm} \\\\\n    \\beta_{n-1}^{\\pm} \n  \\end{pmatrix} \\!\\!\n= \\!\\!\n\\lambda \\!\n\\begin{pmatrix}\n    \\alpha_{n}^{\\pm} \\\\\n    \\beta_{n-1}^{\\pm} \n\\end{pmatrix}\n\\label{eq:eigvalue_eq_tmatrix}\n\\end{equation}\nsuch that\n\\begin{equation}\n\\begin{pmatrix}\n    \\alpha_{n+1}^{\\pm} \\\\\n    \\beta_{n}^{\\pm} \n  \\end{pmatrix}\n=\n\\lambda^n\n\\begin{pmatrix}\n    \\alpha_{1}^{\\pm} \\\\\n    \\beta_{0}^{\\pm} \n  \\end{pmatrix}, \\qquad n\\ge 1.\n\\label{eq:eigvalue_eq_tmatrix2}\n\\end{equation}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.75]{fig2.pdf}\n\t\\caption{(a) Energy band diagram of the walk ($\\theta=\\pi\/4$) as a function of the phase parameter $\\phi$, where $\\phi=0$ corresponds to the standard quantum walk. The analytic result given in (\\ref{eq:sol1}) is drawn for reflection symmetric (blue dashed) and anti-symmetric (red dot-dashed) bound state quasi-energy values, where the numerical values (black dots) are omitted since they exactly coincide with the analytic result. (b) Inverse localisation length of reflection symmetric (blue dashed) and anti-symmetric (red dot-dashed) energy eigenstates and their sum as the effective inverse localisation length (solid black) as a function of $\\phi$, which is used as a measure of localisation. (c) Reduced coin density matrix parameter $r_x$ for reflection symmetric (blue dashed) and anti-symmetric (red dot-dashed) bound states as a function of $\\phi$ gives average value about which oscillations in the reduced density matrix take place.\n\\label{fig:bandStr}}\n\\end{figure}\nIn the infinite chain limit, vanishing boundary conditions impose that $|\\lambda|<1$ for bound states and matching the two forms \n\\begin{eqnarray}\n\\begin{pmatrix}\n    \\alpha_{1}^{\\pm}\\\\\n    \\beta_{0}^{\\pm}\n\\end{pmatrix}\n&=& \nC(E^\\pm)\n\\begin{pmatrix}\n    \\sin \\theta \\\\\n    \\sin (E^\\pm) - i \\sqrt{\\sin^2 \\theta - \\sin^2 (E^\\pm) }  \n\\end{pmatrix}, \\nonumber \\\\\n\\begin{pmatrix}\n    \\alpha_{1}^{\\pm} \\\\\n    \\beta_{0}^{\\pm} \n\\end{pmatrix}\n&=&\nC(E^\\pm) \\sin \\theta\n\\begin{pmatrix}\n    \\rme^{i(E^\\pm+\\phi)} \\\\\n    \\pm \\rme^{\\pm i \\theta} \n\\end{pmatrix}\n\\label{eq:transfer_eigvecs}\n\\end{eqnarray}\ngives the quasi-energy\n\\begin{equation}\nE^{\\pm}  = \\cot^{-1} \\left( \\pm \\frac\n{1- \\sin (\\theta \\mp \\phi) \\sin \\theta }\n{\\sin \\theta \\cos (\\theta \\mp \\phi) }  \\right),\n\\label{eq:sol1}\n\\end{equation}\nprovided that\n\\begin{equation}\n\\sin {(E^{\\pm}+\\phi \\mp \\theta)} = \\pm \\frac{ \\sqrt{\\sin^2 \\theta  - \\sin^2 E^{\\pm} }}{\\sin \\theta} .\n\\label{eq:sol1_cond}\n\\end{equation}\nThe normalization constant is given by $|C(E^{\\pm})|^2=\\left(1-\\lambda (E^{\\pm})^2\\right)\/2$ with the transfer matrix eigenvalue \n\\begin{equation}\n\\lambda_\\pm \\!\\!=\\!\\!\\lambda (E^{\\pm})\n\\!=\\! \\frac{1}{\\cos \\theta} \\! \\left(\\! \\cos (E^{\\pm}) \\!-\\! \\sqrt{\\sin^2 \\theta \\!-\\! \\sin^2 (E^{\\pm})} \\!  \\right)\\!\\!.\n\\label{eq:eigenvalues}\n\\end{equation}\nIt is seen that $|\\lambda_{\\pm}|\\le 1$ provided that $|\\sin E^\\pm| \\le |\\sin \\theta|$.\nWhen a solution with definite parity and an eigenvalue $-\\theta < E^\\pm < \\theta$ exists, then there is a sublattice symmetric solution, which  has $\\lambda_\\pm \\rightarrow -\\lambda_\\pm $ and \n$\\begin{pmatrix}\n    \\alpha_{1}^{\\pm} \\\\\n    \\beta_{0}^{\\pm} \n\\end{pmatrix} \\rightarrow\n\\hat{S}\n\\begin{pmatrix}\n    \\alpha_{1}^{\\pm} \\\\\n    \\beta_{0}^{\\pm} \n\\end{pmatrix}$, with the same parity and shifted eigenvalue $-\\theta+\\pi < E^\\pm < \\theta+\\pi$.\nThe combinations of the sublattice symmetric pairs of reflection symmetric states, for example, give $\\ket{S^+_{e,o}}\\!=\\!\\ket{E^+} \\pm \\ket{E^++\\pi}$ which are supported on even\/odd sites. Similar combinations $\\ket{S^-_{e,o}}$ exist for reflection anti-symmetric bound states.\n\nThe reflection symmetric bound states exist only in the interval $\\phi \\in (0 , 2\\pi- 2\\theta )$ and the reflection anti-symmetric bound states exist for $\\phi \\in (2\\theta , 2\\pi)$. The quasi-energies are shown with blue dashed and red dot-dashed lines in the quasi-energy diagram as a function of $\\phi$ in figure~\\ref{fig:bandStr}(a) for $\\theta=\\pi\/4$, which corresponds to balanced walks.\nThe numerical solution is performed for a large lattice with periodic boundary conditions and the results are indistinguishable on this scale from the quasi-energies of the bound states determined from the eigenvalues of the transfer matrix.\n\nWe would like to quantify the localisation in our model as a function of the impurity phase $\\phi$. For this purpose one of the quantities we calculate is an effective inverse localisation length $\\ell_\\mathrm{eff}^{-1}$ which is the sum of the inverse localisation lengths over all energy eigenstates. Since the localisation length diverges for an extended state in the infinite chain limit, extended states' contribution to $\\ell_\\mathrm{eff}^{-1}$ also vanishes in that limit and we are left with only the inverse localisation lengths of the bound states given by the eigenvalues of the transfer matrix, i.e. $\\ell^{-1} = \\ln \\lambda$. Therefore, the sum of $\\ell^{-1}$ over all stationary states is determined by the number of  bound states and their localisation lengths. The total $\\ell_\\mathrm{eff}^{-1}$ (solid black curve) is plotted in figure~\\ref{fig:bandStr}(b) as a function of $\\phi$, where the individual contributions of symmetric (blue dashed) and anti-symmetric (red dot-dashed) bound states are also shown. The localisation length of a bound state is minimum when its quasienergy is at the center of the band gap (see (\\ref{eq:eigenvalues})), so that the reflection symmetric and anti-symmetric bound states become maximally localised when $\\phi=3 \\pi\/4$ and $\\phi=5 \\pi\/4$, respectively. The total $\\ell_\\mathrm{eff}^{-1}$ is symmetric about $\\phi=\\pi$ where it attains its maximum value. Although it monotonically decreases towards both sides of $\\phi=\\pi$, kinks in $\\ell_\\mathrm{eff}^{-1}$ occur at $\\phi=\\pi\/2$ ($\\phi=3 \\pi\/2$) at the (dis)appearance of new bound states as $\\phi$ increases from $0$ to $2\\pi$.\n\nUnlike the homogeneous quantum walk, in the presence of phase impurity the energy eigenstates become entangled in the composite coin-position space in this model and $\\rho^\\mathrm{coin}$ becomes a mixed state. Furthermore, the $y$- and $z$-components of the density matrix vector $\\vec{r}$ become zero for all energy eigenstates (see (\\ref{eq:ideal_rxyz}) for the standard walk). Here, we note the $x$-components of the reflection symmetric and anti-symmetric bound states\n\\begin{equation} \nr_{x,\\pm} = \\pm \\frac{1}{2} \n\\left[2\\lambda_{\\pm} \\cos \\left(E^{\\pm}+\\phi \\mp \\frac{\\pi}{4}\\right)\n+\\left(1-\\lambda_{\\pm}^2\\right)\\right].\n\\end{equation}\nAs $\\phi$ changes, $r_x$ changes from $+1$ to $-1$ for both the symmetric and anti-symmetric bound states over the regions of their existence as shown in figure~\\ref{fig:bandStr}(c). (Since the action of the unitary sublattice operator on the coin space is trivial, sublattice symmetric bound state give the same  $\\rho^\\mathrm{coin}$.) At $\\phi \\sim 0.6 \\pi $ and $ \\phi \\sim 1.4 \\pi $, the difference in $r_{x,\\pm}$ becomes zero, which means the two coin states become indistinguishable for bound states of either parity. \n\n\\section{\\label{sec:locAndNM}Localisation \\& Non-Markovianity}\n\n\\subsection{Bound state localisation}\n\nWe commence this section by presenting our results for time evolving states in balanced quantum walks ($\\theta=\\pi\/4$) making connection to the stationary bound states obtained in the previous section. In general, the dynamical properties resulting from the evolution of the quantum walk depend on the initial state. We consider initial states such as $\\ket{\\Psi_0}=\\ket{\\psi_c} \\otimes \\ket{0}$ which are localised at the origin, for examining the effects due to the existence of bound states around the origin. Here, $\\ket{\\psi_c}$ is the initial coin state and the reflection symmetry of $\\ket{\\Psi_0}$  is defined by the symmetry of $\\ket{\\psi_c}$ under rotation by $\\pi$ about the $x$-direction. For our purposes, we choose to concentrate on the reflection symmetric and anti-symmetric initial states whose coin subsystems are the eigenstates of $\\sigma_x$, i.e., the initial states $\\ket{\\Psi_\\textrm{S}}= \\frac{1}{\\sqrt{2}} [\\ket{\\uparrow}+\\ket{\\downarrow}]\\otimes\\ket{0}$ and $\\ket{\\Psi_\\textrm{A}}~=~\\frac{1}{\\sqrt{2}} [\\ket{\\uparrow}-\\ket{\\downarrow}]\\otimes\\ket{0}$, respectively. Time evolution starting from these states yields symmetric probability distributions about the origin independent of the presence of a phase impurity. For example, in case of $\\ket{\\Psi_0}\\equiv \\ket{\\Psi_\\textrm{A}}$, figure~\\ref{fig:single_state_distributions}(a) shows the probability distribution in position $P_n(t)$ at $t=300$ for all $\\phi$ values. The significance of the reflection anti-symmetric bound state for a given $\\phi$ becomes clear when we compare the degree of localisation and the overlap of the initial state with the corresponding bound states $\\ket{E^-}$. For an arbitrary initial coin state $\\ket{\\psi_c}=\\cos\\frac{\\gamma}{2}\\ket{\\uparrow}+\\rme^{i\\eta}\\sin \\frac{\\gamma}{2}\\ket{\\downarrow}$, this overlap is given by \n\\begin{eqnarray}\nF^\\mathrm{bound}_{\\gamma,\\eta} &=&\n\\sum_{E \\in \\{ E_\\mathrm{bound} \\}}  | \\braket{E}{\\Psi_t} |^2\n\\nonumber \\\\\n&=&\n\\frac{(2-\\lambda_+^2-\\lambda_-^2)-(\\lambda_+^2-\\lambda_-^2)\\sin\\gamma\\cos\\eta}{2}\n\\label{eq:pBound}\n\\end{eqnarray}\nFor our selection of initial coins, (\\ref{eq:pBound}) simplifies to $F^\\mathrm{bound}_{\\pm\\pi\/2,0} = 1-\\lambda_\\pm^2$. The localisation region apparent in figure~\\ref{fig:single_state_distributions}(a), when $\\phi \\in (\\pi\/2,2\\pi)$, is directly related with $F^\\mathrm{bound}_{-\\pi\/2,0}$ shown in figure~\\ref{fig:single_state_distributions}(b). The walker is most localised when the overlap becomes maximum at $\\phi=5 \\pi\/4$. Similarly, in the interval $\\phi \\in (0 , \\pi\/2 )$ the overlap with the bound subspace is zero and the probability distribution has a dip around the initial site. Four representative probability distributions for different $\\phi$ values are shown in detail in figure~\\ref{fig:single_state_distributions}(c). For the standard quantum walk $\\phi=0$, the probability distribution is uniform in the middle region and there are two peaks near the edges which move in opposite directions with constant speed. For any $\\phi$, the resulting probability distribution spreads balistically in the position space, i.e., it's variance is proportional to $t^2$. However, the proportionality constants may vary depending on the amount of localisation, and hence, on $\\phi$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.95]{fig3.pdf}\n\\caption{(a) The probability distribution $P_n(t)$ of the walk after $300$ steps for the reflection anti-symmetric initial state $\\ket{\\Psi_\\textrm{A}}$ as a function of the impurity phase angle $\\phi$. (b-c) Depending on the overlap $F_{-\\pi\/2,0}^\\textrm{bound}$ between $\\ket{\\Psi_\\textrm{A}}$ and bound states, the probability distribution can have a peak or a dip at the starting site of the walker.\n\\label{fig:single_state_distributions}}\n\\end{figure}\n\nThe average probability distribution of the walk $\\left<P_n(t)\\right>$ is obtained by averaging $P_n(t)$ over all possible initial coin states. However, we observe that we get exactly the same result by only taking into account any pair of orthogonal coin states. This is due to the fact that the average probability distribution resulting from two walks starting with any two orthogonal coin states at the origin is equal to the one resulting from the evolution of a completely mixed coin state. (The resulting distribution is symmetric since the completely mixed coin state at the origin is reflection invariant.)  Also, for the long-time limit, the bound states stay in the vicinity of the origin, whereas the extended states get spread over the infinite position space yielding probabilities going to zero. Based on these facts, we can obtain an analytic expression to estimate the long-time behaviour of $\\left<P_n(t)\\right>$ by projecting the evolved state onto the bound subspace and averaging the corresponding probabilities over two orthogonal initial states, such that\n\\begin{eqnarray}\n\\left< P_0 \\right > &=& \\frac{1}{2}\\left[(1-\\lambda_+^2)^2 + (1-\\lambda_-^2)^2\\right]~~\\textrm{and} \\\\\n\\left< P_n \\right >\n&=&\n\\frac{1}{4}\n[\\lambda_+^{2|n|-2} (1+\\lambda_+^2) (1-\\lambda_+^2)^2\n\\nonumber \\\\\n&+&\n\\lambda_-^{2|n|-2} (1+\\lambda_-^2)  (1-\\lambda_-^2)^2],\n\\label{eq:Pn}\n\\end{eqnarray}\nwhere $n\\neq 0$ and non-zero probabilities appear for even (odd) sites only after even (odd) number of steps. To quantify the localisation, we utilize the participation ratio of the averaged probability distribution, which is given by\n\\begin{equation}\n\\mathrm{PR} = \\sum_n \\left<P_n(t)\\right>^2.\n\t\\label{eq:PR}\n\\end{equation}\nFor a uniform probability distribution over $N$ sites, PR yields its minimum value $\\sim N^{-1}$. At the other extreme of localisation at one site, PR takes its maximum value of one. In figure~\\ref{fig:average_loc}, the numeric results for the PR (green solid curve) and $\\left<P_0\\right>$ (orange dashed curve) for $150$ steps are represented. Both of them is calculated by using the average probability distribution $\\left<P_n(t)\\right>$ which is averaged over a pair of orthogonal initial coin states as we mentioned before. We also provide the analytic prediction of PR (black dots) for the long-time behaviour using (\\ref{eq:Pn}) and (\\ref{eq:PR}) which slightly differs from its numerical simulation, whereas we omitted that of $\\left<P_0\\right>$ for clarity since it exactly fits to the numerical data. First of all, both curves exhibit similar behaviour with respect to $\\phi$ and $\\left<P_0\\right>$ pointing out that localisation occurs around the impurity site. They get maximized at $\\phi=\\pi$ and vanish at the standard quantum walk limit $\\phi=0,2\\pi$. The kinks at $\\phi=\\pi\/2,3\\pi\/2$ are due to bound states appearing or disappearing in this model as discussed previously. This behaviour matches exactly that of the effective localisation length determined by the bound states in figure~\\ref{fig:bandStr}(b), which consequently shows that the localisation properties of the walk in the long-time limit is determined by the number and character of the stationary bound states. The slight difference between the numerical and analytical results of PR stems from the finite number of time steps in the numerical simulation and the fact that contribution from the extended states is completely excluded in the analytical expression. As a consequence of this, the numerical data stays above the analytical prediction. For example, as we approach the standard walk case, the wavefunction for a finite-step walk stays relatively ``localised'' in comparison to that of the long-time case which spreads infinitely over the position space without any localisation. Hence, the numerical prediction  will become zero in the standard walk in this limit as well. The very good agreement between the numerical and analytical results in figure~\\ref{fig:average_loc} implies that the effect of the extended states on the PR is negligible even after $150$ steps.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.85]{fig4.pdf}\n\\caption{The numerical results for the participation ratio (PR) and the average probability at the origin $\\left<P_0\\right>$ with respect to $\\phi$ after $150$ steps. The analytical prediction for PR (black dots) is also provided.\n\\label{fig:average_loc}}\n\\end{figure}\n\n\\subsection{Non-Markovianity}\n\nWe now turn our attention to the non-Markovian behaviour of the dynamics of the coin for the quantum walk with a phase impurity. As mentioned before, we are interested in the effects of localised bound states and their symmetry on the degree of non-Markovianity of the reduced coin evolution. In order to quantify the amount of memory effects in the open system dynamics from different perspectives, we will comparatively study two well-established measures of quantum non-Markovianity that are based on the information flow dynamics between the coin and the spatial degrees of freedom.\n\nLet us first briefly discuss how to characterize the non-Markovian nature of an open system evolution and identify the existence of possible memory effects in the dynamics. Assume that we have a quantum map \\ALS{$\\Lambda_{t,0}$}{$\\Lambda(t,0)$}, i.e., a completely positive trace preserving (CPTP) map describing the evolution of the open quantum system. The property of divisibility implies that divisible maps satisfy the decomposition rule $\\Lambda(t,0) = \\Lambda(t,s) \\Lambda(s,0)$, where $\\Lambda(t,s)$ is a CPTP map for all $s\\leq t$. Markovian or so-called memoryless dynamical maps are recognized as the ones that satisfy this decomposition rule. On the other hand, when the divisibility rule is violated, i.e., when \\ALS{$\\Lambda_{t,s}$}{$\\Lambda(t,s)$} is not a CPTP map or when it does not even exist, then the dynamical map $\\Lambda$ is said to be non-divisible and the evolution it describes non-Markovian. The concept of divisibility can also be discussed in the context of discrete dynamics, such as quantum walk, where $t,s \\in \\mathbb{N}$~\\cite{luoma15}.\n\nThe first non-Markovianity measure that we utilize in our work is known as \\ALS{the BLP}{Breuer-Laine-Piilo (BLP)} measure~\\cite{breuer09} which is based on the idea of distinguishability of two open system states under a given dynamical evolution. In this approach, the changes in the distinguishability between two arbitrary initial states of the open system during the dynamics are interpreted as the information flow between the open system and its environment. In particular, if distinguishability between the initial states decreases monotonically in time throughout the evolution, the dynamics is said to be Markovian, since in this case information flows from the open system to its environment in a monotonic fashion. However, if distinguishability temporarily increases during the dynamics, then this is understood as a back-flow of information from the environment to the open system giving rise to non-Markovian memory effects. The distinguishability of two systems can be quantified through trace distance between their density matrices $\\rho_1$  and $\\rho_2$ as\n\\begin{equation}\nD(\\rho_1, \\rho_2)\\!=\\!\n\\frac{1}{2}\n||\\rho_1\\!-\\!\\rho_2||_1\n\\!=\\!\n\\frac{1}{2}\n\\Tr \\left[(\\rho_1\\!-\\!\\rho_2)^{\\dagger} (\\rho_1\\!-\\!\\rho_2)\\right]^{1\/2}\n\\label{eq:trace_dist}\n\\end{equation}\nwhich acquires its maximum value of one, when the states $\\rho_1$  and $\\rho_2$ are orthogonal. At this point, we should stress that since CPTP maps are contractions for the trace distance, BLP measure vanishes for divisible maps, resulting in a memoryless evolution. However, we also emphasize that it is possible for trace distance to monotonically decrease for certain non-divisible maps as well. Therefore, as is well known in the recent literature, even though widely used as a measure for non-Markovianity on its own, BLP measure is actually a witness for the non-divisibility of quantum dynamical maps. The BLP measure can be expressed in discrete time as \\cite{luoma15}\n\\begin{equation}\n{\\cal{N}}\n=\n\\max_{\\rho_{1,2}}\n\\sum_{t, \\Delta D>0} \\Delta D_t\n=\n\\sum_{t} \\Delta D_t \\Theta(\\Delta D_t),\n\\label{eq:nonmarkov}\n\\end{equation}\nwhere $\\Theta(x)$ denotes the Heaviside step function,\n\\begin{equation}\n\\Delta D_t\n=\nD(\\rho_{1,t}, \\rho_{2,t})-D(\\rho_{1,t-1}, \\rho_{2,t-1}).\n\\end{equation}\nand the maximization is carried out over all possible initial state pairs. It has been shown that the pair which maximizes the sum in  (\\ref{eq:nonmarkov}) is a pair of orthogonal of states~\\cite{wissmann12}. In our analysis, we study the reduced system dynamics of a pair of such initial states, namely, $\\ket{\\psi_{S,A}}$ introduced before, with opposite reflection symmetry, which will be later on revealed as the optimal initial state pair optimizing the BLP measure.\n\nThe time evolution of $\\rho^\\mathrm{coin}_{S,A}$ is particularly easy to visualize because the parametrization $\\rho^\\mathrm{coin}_t = (I + \\vec{r}_t\\cdot \\vec{\\sigma})\/2$ has only one non-zero component, i.e. $r_{x,t}$, throughout the time evolution which is shown in figure~\\ref{fig:spinxoscillations} for representative values of the phase $\\phi$. For $\\phi=0$, which gives the standart quantum walk, both $r^S_{x,t}$ (black dotted line in figure~\\ref{fig:spinxoscillations}(a)) and $r^A_{x,t}=-r^S_{x,t}$ (black dotted line in figure~\\ref{fig:spinxoscillations}(b)) undergo damped oscillations with a period of four steps as the  steady-state is reached. Since the oscillations are out of phase for these orthogonal initial states, the trace distance between such states also oscillates in time with decreasing amplitude (black dotted line in figure~\\ref{fig:spinxoscillations}(c)). Therefore, even though there is a back-flow of information from the environment to the open system in the standard walk, the damping in oscillations shows that information flow between the two subsystems reduces and eventually vanishes in time~\\cite{hinarejos2014}. For non-zero values of $\\phi$, oscillations in the initial state component $r^{A(S)}_{x,t}$ arise depending on the overlap with the bound states. When $\\phi=\\pi\/4$, the oscillations in $r^A_{x,t}$ die out very quickly, whereas oscillations with period two between sublattice symmetric pair of localised states survive for $r^S_{x,t}$ as shown by the blue dot-dashed line in figure~\\ref{fig:spinxoscillations}(a)-(b). For $\\phi=\\pi\/2$, similar oscillations exist, except they die out more slowly for $r^A_{x,t}$ which has a finite overlap with the emerging reflection anti-symmetric bound state whereas oscillations continue with higher amplitudes for $r^S_{x,t}$ since the reflection symmetric bound-state becomes more localised for this value of $\\phi$. At $\\phi=\\pi$ where bound states of both parities exist, oscillations in $r_{x,t}$ occur with higher amplitudes for both of the initial states in comparison with the other shown phase values.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.90]{fig5.pdf}\n\\caption{Oscillations in the reduced coin density matrices starting from $\\ket{\\Psi_\\text{S}}$ in (a) and from $\\ket{\\Psi_\\text{A}}$ in (b) as a function of time for representative values of the phase parameter $\\phi$. The trace distance of these coin states $D(\\rho_S,\\rho_A)=|r_{x,A}-r_{x,S}|$ is shown in (c) and the oscillating behaviour gives rise to non-zero BLP measure.} \n\\label{fig:spinxoscillations}\n\\end{figure}\n\nHaving obtained the time dependence of $\\rho^\\mathrm{coin}_{S,A}$, we calculate the trace distance $D(\\rho_S, \\rho_A) = |r_{S,x}-r_{A,x}|$, and display our findings in figure~\\ref{fig:spinxoscillations}(c), as a function of $\\phi$. In contrast to the standard quantum walk where the trace distance oscillations die out in time, we find that they survive for non-zero $\\phi$, as at least one of $r^{S,A}_{x,t}$ keeps oscillating in time. However, we should keep in mind that the value of the trace distance also depends on the mean values $\\overline{r^{S,A}_{x,t}}$ about which oscillations take place. For example, when $\\phi=\\pi\/2$ we get oscillations in $D(\\rho_1,\\rho_2)$ with smaller amplitudes than in $r^S_{x,t}$, which will be of importance in our later discussions.\n\nAs the persistent oscillations in trace distance play a crucial role for the evaluation of the BLP measure in our model, the oscillation means $\\overline{r^{S,A}_{x,t}}$ and the oscillation amplitudes are plotted in figure~\\ref{fig:oscillations}(a). Comparison with figure~\\ref{fig:bandStr}(c) reveals that, as the overlap between one of the the initial states and the bound states increases, $\\overline{r^{S,A}_{x,t}}$ converges to the $r_x$ of the corresponding bound state and oscillations appear. For the interval $\\phi \\in (\\pi \/2, 3 \\pi \/2)$, $\\overline{r^{S,A}_{x,t}}$ becomes the same as $r_x$ in the long time limit. The difference in $\\overline{r^{S,A}_{x,t}}$ approaches to zero at $\\phi \\sim 0.6 \\pi$ and $\\phi \\sim 1.4 \\pi$, yielding very small values for the trace distance together with the fact that essentially one of $r^{S,A}_{x,t}$ oscillates about their common mean. For other values of $\\phi$, the trace distance is mainly determined by the oscillations in $r^{S,A}_{x,t}$. Since the period of the oscillations is two time steps due to the sublattice symmetry, the changes in trace distance can be obtained by subtracting the value at even time step from the neighbouring odd time step which is plotted in figure~\\ref{fig:oscillations} (b) at three different times. These plots clearly demonstrate that the trace distance oscillations quickly converge to their long time limit. As the bound states get more localised for certain $\\phi$ values and also the overlap of the initial states with them increases, so do the amplitude of the oscillations in the trace distance.\n\nTo evaluate the BLP measure, we maximize the sum of the positive increases in trace distance over all possible orthogonal pairs of initial states starting at the impurity site which is shown in figure~\\ref{fig:oscillations}(c) as a function of $\\phi$ for three increasing values of time. The result reveals that the pair $\\ket{\\psi_{S,A}}$ that we used for the preceeding analysis actually maximizes the sum in the BLP measure in the long-time limit. In contrast to the standard walk, the initial states maximizing BLP measure are equal superposition of symmetric and anti-symmetric states and these states do not change under other decoherence mechanisms~\\cite{hinarejos2014}. \nNear $\\phi=0,\\pi\/2, 3\\pi\/2, 2\\pi$, where bound states are weakly localised, we find that other orthogonal pairs actually maximize the BLP measure. However these regions get smaller as we consider longer time evolutions. The sudden drop in BLP at $\\phi=\\pi\/2,3\\pi\/2$ is related to the fact that oscillations take place about similar mean values. More importantly, we establish that the BLP measure of non-Markovianity increases with the emergence of bound states and reaches its maximum value at $\\phi=\\pi$ when the number and localisation of bound states assumes their maximum, as demonstrated by the effective localisation length in figure~\\ref{fig:bandStr}(b). The relation of non-Markovianity and localisation is also apparent comparing the BLP curve with the average PR shown in figure~\\ref{fig:average_loc}, which is maximum at $\\phi=\\pi$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.80]{fig6.pdf}\n\\caption{(a) Long-time limit time average of the reduced coin density matrix parameter $r_x$ for reflection symmetric ($\\ket{\\Psi_\\text{S}}$) and anti-symmetric ($\\ket{\\Psi_\\text{A}}$) initial states as a function of $\\phi$. (Time average is taken over 100 steps between $t=400$ and  $t=500$.) Instantaneous values at even and odd time steps are shown by square and triangle markers, respectively. (b) Trace distance oscillation amplitudes between initial states $\\ket{\\Psi_\\text{S}}$ and $\\ket{\\Psi_\\text{A}}$ at different times show that they quickly converge to their long-time limit values for all $\\phi$. (c) BLP measure $\\mathcal{N}$ (\\ref{eq:nonmarkov}) at three different times. The maximization is performed over all the initial coin states for quantum walks starting at the impurity site. The linear increase in time reflects trace distance oscillations with constant amplitude. (See (b).) \n\\label{fig:oscillations}}\n\\end{figure}\n\nNext, we consider \\ALS{the RHP}{Rivas-Huelga-Plenio (RHP)}~\\cite{rivas10} measure of non-Markovianity, which is based on the dynamics of entanglement between the system of interest and an ancillary system. The ancillary system $A$ is assumed to have no dynamics of its own and is completely isolated so that any initial entanglement between the system and the ancilla can be affected by the open system dynamics only. In fact, similar to the BLP measure, this measure is also a witness for the violation of the divisibility. Considering the fact that no entanglement measure $E$ can increase under local CPTP maps, it is rather straightforward to observe that \n\\begin{equation}\n\tE[(\\Lambda(t,0) \\otimes I) \\rho_{\\mathrm{coin},A}] \\leq E[(\\Lambda(s,0) \\otimes I) \\rho_{\\mathrm{coin},A}]\n\\end{equation}\nfor all times $0\\leq s \\leq t$. Hence, any increase in the entanglement between the open system and its ancillary can be understood as a signature of non-Markovian memory effects in the time evolution. In other words, while the entanglement contained in $\\rho_{\\mathrm{coin},A}$ decreases monotonically for all Markovian processes, non-Markovian behaviour in the dynamics can be captured through the temporary increase of entanglement. In the same spirit of the BLP measure, one can then measure the degree of non-Markovianity using the following quantity:\n\\begin{equation}\n{\\cal{I}}^{(E)}\n=\n\\max_{\\rho_{CA}}\n\\sum_{t,\\Delta E_\\mathrm{CA}>0}\n\\Delta E_{\\mathrm{CA},t}\n\\end{equation}\nwhere $E_\\mathrm{CA}$ denotes the entanglement between the coin and a two level ancillary system. For any entanglement measure $E_{CA}$, the RHP measure is found by maximizing ${\\cal{I}}^{(E)}$ over all initial reduced density matrices $\\rho_{CA}$ of the composite coin-ancilla system. In order to calculate this measure, we start the evolution from composite initial state  $|\\Phi^+ \\rangle \\vert 0 \\rangle = \\frac{1}{\\sqrt{2}}(|\\leftarrow \\rangle_C|\\downarrow\\rangle_A+|\\rightarrow \\rangle_C |\\uparrow \\rangle_A)\\vert 0\\rangle$ and use concurrence~\\cite{wooters97} as the entanglement measure. It has been shown that when concurrence is used as entanglement measure, the optimum initial state maximizing the RHP measure is a Bell state, for a single qubit interacting with an environment~\\cite{neto16}. \n\nFigure~\\ref{fig:non_makovianity}(a) shows the variation of the concurrence in time which is calculated from the reduced coin-ancilla state after tracing out the spatial degrees of the walker during the evolution. For the standard quantum walk, the entanglement oscillations with period of four steps are damped and slowly die out with time. Therefore, the RHP measure accumulates a finite amount of non-Markovianity in the long time limit which is similar to the behaviour of the BLP measure for the standard walk. On the other hand, in contrast to the BLP measure, the nature of bound states emerging with non-zero phase $\\phi$ plays a key role for the coin-ancilla entanglement. In the presence of reflection symmetric or anti-symmetric bound states only, the concurrence dies out very quickly. This is due to the fact that the symmetric and anti-symmetric states couple to different environmental degrees of freedom. For example, with only symmetric bound states present, the symmetric part of the coin-position state remains mostly localised in the vicinity of the impurity site whereas the anti-symmetric part moves away from the origin. Hence, the coin-ancilla entanglement is quickly destroyed upon tracing out the environmental degrees of position, as the coin-ancilla state becomes an incoherent mixture. An example of this situation is displayed in figure~\\ref{fig:non_makovianity}(a) for $\\phi=\\pi\/3$. It is only when both reflection symmetric and anti-symmetric stationary states exist that some entanglement can survive which shows non-decaying oscillations. These oscillations are due to the finite dimension of the bound state subspace and the frequencies of concurrence oscillations can easily be obtained from the quasi-energy differences. Such a case is displayed in figure~\\ref{fig:non_makovianity}(a) for $\\phi=\\pi$ with two dominant periods. One period is of two steps due to the sublattice symmetric bound states with quasi-energy difference $\\pi$ and another one is approximately ten steps due to the quasi-energy difference of $\\Delta E \\approx 0.205\\pi$ between reflection symmetric and anti-symmetric states. The latter dependence again shows the importance of bound states of both parities for the RHP measure. The energy difference $\\Delta E$ does not change much as $\\phi$ changes in the domain of four bound states unless one group of bound states is very weakly bound. (See figure~\\ref{fig:bandStr})\n\nUsing the time evolution of the coin-ancilla entanglement as shown in figure~\\ref{fig:non_makovianity}(a), we evaluate the RHP measure for all values of the impurity phase $\\phi$. The results are plotted in figure~\\ref{fig:non_makovianity}(b) for three increasing values of the final time. The amount of non-Markovianity measured by the RHP measure drastically depends on whether the reflection symmetric and anti-symmetric bound states are both supported for a given $\\phi$ or not. In the interval $\\phi \\in (0, \\pi\/2)$ where only the symmetric bound states exist, the concurrence vanishes quickly in time since the coin-ancilla Bell state can only be supported if both symmetric and anti-symmetric bound states exist. Therefore, the coupling of the symmetric and anti-symmetric coin states to different environmental degrees of freedom completely destroys the Bell state of the coin-ancilla system and results in a vanishing value for the RHP measure. A similar situation occurs in the interval $\\phi \\in (3\\pi\/2, 2\\pi)$ where only reflection anti-symmetric bound states exist and coin-ancilla entanglement is destroyed. In the interval $\\phi \\in (\\pi\/2, 3\\pi\/2)$ where bound states of both symmetries exist, the coin-ancilla entanglement is more robust  and the RHP measure captures the non-Markovianity increasing linearly with $t$ in the long time limit due to non-decaying oscillations in the coin-ancilla entanglement. In this $\\phi$ interval, the RHP displays the same behaviour as seen for the BLP measure in figure~\\ref{fig:oscillations}(c). \n\n\\section{\\label{sec:conc}Conclusion}\n\nWe have provided a comprehensive and systematic analysis of non-Markovianity in a quantum walk model with a phase impurity in relation with the phenomenon of localisation. At the heart of analysis lies the manifestation of bound states emerging due to the existence of the phase impurity at the starting site of the walker. We have first presented a technique to analytically obtain the bound states of the model making use of the transfer matrix method. These bound states emerge in one or two sublattice symmetric pairs possessing definite reflection symmetry. With this knowledge at hand, we have explored the localisation properties of the walker in the position space. To this end, we have adopted two initial state independent quantities to measure the degree of localisation, namely, the effective localisation length for all eigenstates and an average participation ratio after time evolution over all initial states starting at the impurity site. Our analysis clearly demonstrates that the degree of localisation of the walker is directly determined by the properties of the bound states.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.8]{fig7.pdf}\n\\caption{(a) Concurrence between the coin and the ancilla qubit as a function of time for representative values of the phase parameter $\\phi$. When bound states with both positive and negative reflection parity exist, the concurrence shows oscillations. (See text for the involved frequencies.) (b) Concurrence based RHP measure as a function of $\\phi$ at three different time steps showing linear increase with time for $\\phi \\in (\\pi\/2,3\\pi\/2)$. RHP has a vanishing value when well-formed bound states of only positive or negative reflection parity exist.\n\\label{fig:non_makovianity}}\n\\end{figure}\n\nMore importantly, our main contribution in this work is the unveiling of an intrinsic relation between the emergence of bound states and the degree of non-Markovianity of the dynamics of the walker. In order to study non-Markovian behaviour in the time evolution of the walker, after tracing out the spatial degrees of freedom, we have utilized two distinct measures of quantum non-Markovianity, i.e., the BLP and the RHP measures based on the dynamics of trace distance and entanglement, respectively. These measures help us to understand the information flow between the principal coin system and the position system forming the environment from different perspectives. We show that, in the case of the existence of spatial decoherence in the form of a phase impurity, the BLP measure is optimized by the eigenstates of the coin operator for almost all values of the phase $\\phi$. Note that when one has decoherence in terms of broken links instead, the degree of decoherence does not change the optimal state maximizing the BLP measure~\\cite{hinarejos2014}. Our investigation also proves that phase impurity amplifies the degree of non-Markovianity quantified by the BLP measure.\\ALS{, similar to the disorder model studied in~\\cite{kumar2018}}{}\nThe underlying reason behind this behaviour is the oscillations in the state of the coin which essentially takes place between the sublattice symmetric bound state components with a period of two steps. Then, in general, increasing overlap between the initial and the bound states implies a greater degree of non-Markovianity. However, also note that when the time average of the reduced coin states corresponding to two orthogonal initial states are close to each other, the BLP measure drops abruptly.\n\nNext, we employed the RHP measure to analyse the degree of non-Markovianity in the dynamics of the walker. When the coin state is maximally entangled with an ancillary system initially, the amount of entanglement is known to oscillate in time for the standard walk. However, our examination demonstrates that, in case of the existence of a phase impurity, if the bound subspace supports only one type of reflection symmetric state, the coin-ancilla entanglement vanishes after a few time steps and the RHP measure becomes very small compared to the standard walk case. On the other hand, when both reflection symmetric and anti-symmetric bound states are present, the entanglement oscillations are persistent in time, leading to high values of RHP measure. Thus, while the RHP measure is generally in good agreement with the BLP measure when both even and odd parity bound states exist, the RHP measure fails to reliably detect the non-Markovian behaviour when only symmetric or anti-symmetric bound states are present. Most importantly, as can be clearly seen from both measures, maximum non-Markovianity is reached where our localisation measures determined by the bound states become also maximum.\n\\ALS{}{Relationship between non-Markovianity and localisation have been discussed in random static disorder models~\\cite{lorenzo2017quantum,kumar2018} where non-Markovianity increases with disorder.\nWe observe more nuanced behaviour between bound states and non-Markovianity as discussed above.} \n\nWe would like to indicate that the experimental realization of the model we presented here is quite feasible with today's technology. The time-multiplexing quantum walk employs laser light pulses going successively around a fiber loop where the position space is effectively encoded in the time domain from the point of view of the detectors \\cite{schreiber2010}. The main advantage of this setup is it's scalability and it's long coherence times, i.e., it only requires a fixed number of optical elements to realize the quantum walk for relatively large number of steps. The recent developments in the setup allow deterministic out-coupling of the light pulses from any site by utilizing electro-optic modulators \\cite{nitsche2018}. It is also possible to introduce arbitrary phases specific to any site by programming of the electro-optic modulators accordingly, which actually would allow the realization of the model we provided here \\cite{schreiber2011, nitsche2016}. \n\nAs a concluding remark, it would be interesting to study whether the oscillations due to the bound states become robust in the case of many-body interactions with more degrees of freedom in the context of quantum walks as a future work.\n \n\n\\ack{\n\\.{I}.Y. is supported by M\\v{S}MT under Grant No. RVO 14000 and the Czech Science Foundation under Grant No. GA CR 19-15744Y.\nG.K. is supported by the BAGEP Award of the Science\nAcademy and the TUBA-GEBIP Award of the Turkish Academy of Sciences. G.K. is also supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under\nGrant No. 117F317.\nB.D. and A.L.S. are supported by Istanbul Technical University Scientific Research Projects Department (ITU BAP No. 40881). A.L.S. would like to acknowledge useful discussions with {\\c S}.E. Kocaba{\\c s} at earlier stages of this work.\n}\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Background \\& Summary}\n\nMany assessments of future electricity demand in India project large increases in electricity consumption from adoption of air conditioning technologies in the buildings sector over the next two decades \\cite{weo20, ev, iea}. This large growth is likely to make India among the top nations in terms of electricity consumption, implying that technology choices related to energy consumption and production in India are likely to play a significant impact on global climate change mitigation efforts. Additionally, the Indian government has been pushing for the transportation sector's electrification, starting with two- and three-wheel vehicles,which is further likely to increase overall electricity demand. As of 2020 in India, there are 152,000 registered electric vehicles \\cite{ev}. Air conditioning (AC) related electricity demand accounted for 32.7 TWh, contributing to less than 2.5\\% of the total demand in 2019 \\cite{iea}. However, both air conditioning and transport electrification are anticipated to introduce structural changes in the temporal and spatial trends in electricity consumption patterns, that has important ramifications for long-term resource planning for the electricity sector \\cite{teri}. This paper presents an bottom-up approach to estimate electricity consumption in India for various scenarios of technology and policy adoption with a specific focus on providing aggregated consumption estimates as well as spatio-temporally resolved consumption profiles that would be relevant for regional and national electricity system planning studies. The approach enables quantifying the impact of  various growth and technology adoption scenarios on quantity and pattern in electricity consumption. The datasets detailed in this paper include annual energy consumption at India's state, regional, and national levels as visualized in Fig. \\ref{fig:demand}, as well as underlying consumption profiles at an hourly time resolution. The annual energy consumption is forecasted on a five-year increment to 2050. Fig. \\ref{fig:summary_results} shows one scenario of national electricity demand forecast. In addition to the snapshot of annual consumption, hourly load profiles are developed at the same resolution as seen in Fig. \\ref{fig:profile_results}. \n\nThe forecasting is divided into two steps: business-as-usual and technology. Business-as-usual is a statistical model that infers data it can be trained on i.e. historical electricity demand. The technology model is a bottom-up approach that adds new loads to the total demand. Among new loads, we focus on residential and commercial cooling as well as various electric vehicles (EV). Some key insights from cooling\\cite{iea} and EV\\cite{ev} studies highlighting peak demand development motivate the need for demand forecasting at the hourly resolution. Cooling demand due to mainly split unit air conditioning installation in India is expected to increase the peak to mean ratio (also sometimes referred to the \"peakiness\") of electricity demand in India as well as shift the timing of peak demand from evenings to midnight\\cite{iea}. While electric vehicles do not constitute a large portion of the total demand, certain charging schemes can contribute significantly to the peak demand\\cite{ev}. Numerous energy demand forecast for India have recently been published as decadal snapshots \\cite{weo20, teri, brookings}, however granularity of demand at an hourly resolution has not been presented in these studies. Our approach enables quantifying the impact of different technology and structural elements, such as adopting energy efficient vs. baseline cooling technology or work-place charging vs. home charging for EVs, on the hourly electricity consumption profiles. These insights and the accompanying data sets are essential to carry out generation and transmission expansion as well as distribution network planning,and are thus essential for a sustainable energy infrastructure development in the Indian context. \n\n\nSimilar to other forecasting studies, we model Gross domestic product (GDP) growth \\cite{mospi} to be the main econometric driver of the business-as-usual demand forecasting, and thus three scenarios are introduced: slow, stable, and rapid GDP growth. We examine two AC load scenarios: energy-efficient equipment and baseline equipment per the International Energy Agency's Future of Cooling study \\cite{iea}. Finally, we evaluate three EV charging mechanisms: home, work, and public charging. This totals the number of data sets spanning three input dimensions to 18 scenarios. Technology adoption growth has been correlated with economic growth under the assumption that new technologies are adopted faster when the economy is growing faster and vice versa. We present two cooling scenarios to highlight the difference in energy-efficient and regular air conditioning units and bring attention to the need for policy and programs that favor energy-efficient cooling unit sales. Furthermore, we present various EV charging mechanisms to inspect the demand impacts that electric vehicle charging can have on the electric grid at different times. The produced data can be used as input to electricity infrastructure planning both at the distribution and transmission level. \n\n\\section*{Methods}\nFig. \\ref{fig:schematic} illustrates the major steps of our proposed demand forecasting approach. We use two models to estimate future electricity demand in India. In the first model --- business-as-usual --- we use a linear regression model to project daily peak and consumption on a regional basis; this is the business-as-usual scenario. We then add natural variation to the projections by finding the error between the training data and results and scaling it to every region based on seasonality. Then we fit the projected peak and total consumption to an annual hourly load profile for 2015 \\cite{shakti} featuring an evening peak \\cite{ivan}. In the second model --- technology model --- we take AC and EV adoption into account as an additive component on top of the business-as-usual predictions. GDP data, which is an independent variable in the model, is chosen to be the main driver of growth of the business-as-usual scenario as well as technology adoption rates. The input data used are publicly available and are referenced in Table \\ref{table:data}.\n\n\\subsection*{Input data processing}\n\nAlthough GDP is widely used for forecasting energy demand, it is specifically essential in the case of India, where economic growth is expected to ramp up over the next few decades similar to the recent trends in China \\cite{mckinsey}. We based our demand forecast on GDP projections from a PricewaterhouseCoopers (PwC) report \\cite{pwc}, that projected India's GDP to grow from 3.6 trillion in 2020 to reach 28 trillion USD in 2050. Considering the historical national GDP data for India starting in 1990, we fit and project an exponential curve for rapid growth and an Gompertz curve for slow growth \\cite{gompertz} as detailed in Table \\ref{table:gdp}. We use PwC's projections to define the stable GDP growth scenario. Curve fitting and projection results are illustrated in Supplementary Fig. \\ref{fig:sup-gdp}. The rapid growth scenario produces an annual average growth rate of 9.5\\% , PwC's growth rates start at 7.8\\% for the first projected decade and ends at 6.2 \\% in the final projected decade. The slow growth scenario starts at 7.2\\% growth rate in the first projected decade and ends at 3.9\\% in the final projected decade. To break down the regional energy consumption projections to state level we use the ratio of GDP per capita of the corresponding state to the GDP per capita of the region it is in. For each GDP growth scenario, we fit the same functions given state-wise data to produce GDP forecast at the same resolution.  GDP per capita at state-level is computed using the projected GDP data and state level population projections \\cite{ssrn}.\n\n\\subsubsection*{GDP dependence and limitation}\n\nRelating growth in electricity demand to GDP is a strong generalization, however it is not a novel one in the case of India. Strong correlation between economic growth and energy consumption has been established in the Indian context in this study and other studies \\cite{eia} given data from the past two decades \\cite{mospi}. We recognize that GDP as a metric of economic growth has several limitations particularly related to projecting how economic growth is distributed among society within a state or nation. This may be the strongest limitation of the data we are presenting in the manuscript. However, lack of historical record and long-term projections of alternative open-access economic data at the desired spatial and temporal resolution limit the development of a framework to project energy consumption with other metrics. While GDP and energy consumption growths may differ in the long-run, there is an evident correlation between the two that can be used to estimate long-run energy consumption growth. Deviating away from linear regression may yield better results, however, data scarcity is again a limitation to the development of more complex models. Furthermore, this manuscript motivates the need for more bottom-up projections and not just regression models because historical consumption cannot infer consumption trends from new demand sources such as cooling and EVs.\n\nAdditionally, since the Future of Cooling study by the International Energy Agency relies on GDP forecasts developed by the International Monetary Fund\\cite{iea}, we elected to use a similar metric. We intentionally develop a large bandwidth of projection scenarios to mitigate the limitation of an individual snapshot representing a singular assumption. The motivation behind presenting the described results is ability to compare different scenarios and post-analyze the demand growth and the trade-offs. To produce a large bandwidth of growth scenarios we needed to use a straightforward metric that has enough historical data to produce various fitted curves for projections.\n\n\\subsection*{Business-as-usual model}\n\nThe business as usual projections are modeled with a linear regression considering weather and economic growth features. The ground truth historical daily peak and total consumption for each electric grid were obtained from the Power System Operation Corporation (POSOCO) for 2014-2019 \\cite{posoco}. The GDP used in the model was obtained, as explained in the previous section. Weather data was secured from the NASA Merra-2 data set \\cite{nasa}. The choice of features for the regression model is limited to GDP and weather variation due to the limitation in availability of data, both historical and future projections, at the desired spatial and temporal resolution. GDP is identified as a long-term parameter driving growth in year over year demand projections as highlighted in Fig. \\ref{fig:longrun}. Weather data is identified as a short-term parameter driving seasonal variation within a year's demand projections as highlighted in Fig. \\ref{fig:shortrun}. Previous parametric analysis on these features and their coefficient for short and long term demand forecasting in both time and frequency domain \\cite{meia} reinforce their use as features for the business-as-usual regression model. We present detailed outcomes for the Southern region, with further details available in \\cite{meia}.\n\n\\subsubsection*{NASA Merra 2 data acquisition}\n\nFor each of the five electric grid demand regions highlighted in right panel of Fig. \\ref{fig:demand}, the largest cities in each region were identified using population data made available by the United Nations\\cite{pop}. Then, the city's latitude and longitude were used to pull down the corresponding environmental data from the Nasa Merra-2 data set. The cities used for each of the five regions are listed here:\n\\begin{itemize}\n    \\item Northern: Delhi, Jaipur, Lucknow, Kanpur, Ghaziabad, Ludhiana, Agra\n    \\item Western: Mumbai, Ahmadabad, Surat, Pune, Nagpur, Thane, Bhopal, Indore, Pimpri-Chinchwad\n    \\item Eastern: Kolkata, Patna, Ranchi (Howrah was ignored because the environmental factors are the same as Kolkata)\n    \\item Southern: Hyderabad, Bangalore, Chennai, Visakhapatnam, Coimbatore, Vijayawada, Madurai\n    \\item Northeast: Guwahati, Agartala, Imphal\n\\end{itemize}\n\nFrom the NASA set, 11 variables were included for each city: specific humidity, temperature, eastward wind, and northward wind (all 2m above the surface and 10m above the surface - eight total variables), precipitable ice water, precipitable liquid water, and precipitable water vapor. In particular, the instantaneous two-dimensional collection \"inst1\\_2d\\_asm\\_Nx (M2I1NXASM)\" from NASA was used. Detailed descriptions of these variables are available in the Merra-2 file specification provided by NASA\\cite{nasa} . The environmental variables available from the NASA MERRA-2 dataset were given on an hourly basis.  The daily minimum, daily, maximum, and daily average was calculated for each of the 11 variables for each day.\n\n\\subsubsection*{Forecasts}\nThe business-as-usual demand forecasting problem was divided into ten separate problems,corresponding to one problem each peak and total consumption for each of the five regional grids shown in Figure \\ref{fig:demand}.  To ensure the model would not overfit the data, the model was trained with Elastic Net \\cite{scikit-learn} to regularize results, and validated on held out 2019 data. An L1 ratio (Lasso) of .9 was chosen to minimize error in 2019 as the validation set. Then all of the models were trained with .9 L1 ratio on the full dataset.\n\n\\subsubsection*{Addition of natural variation}\nThis step aimed to match the statistical characteristics of an actual load year with the projected year. 2019 was used to derive the differences. Natural variation was estimated by a distribution characterized by the mean and standard deviation of the differences (in absolute value). Then, a natural variation adjustment was added to that day (with a random true\/false bit for positive or negative variation). The noise was calculated for each region and peak demand and daily consumption separately. The natural variation (noise) vectors used are on the Github repository for this paper \\cite{git}. This part of the process is non-deterministic and replication of the results requires using the same natural variation vector used in our projections.\n\n\\subsubsection*{Hourly profiles}\nThe statistical inference model presented above forecasts daily consumption driven by state-level economic parameters and weather data. The produced projections are at a daily resolution. We downscaled the data to hourly load profiles based on the 2015 hourly load profile data \\cite{shakti}. The result of the regression model is at regional level, breaking it down state-wise is pro-rated based on state-wise to region-wise GDP per capita projections ratios for the respective year. To do so, we tag each day of the year by the month it corresponds to and whether it is a weekday or weekend. We cluster demand for each hour by month and day. Each hour of the day then has its own cluster of demand data from 2015 based on the assumption that the same hour of the day for a given month and the same day type will exhibit similar demand behavior. This biases the construction of the profiles to demand patterns from 2015 only. To minimize the impact of this bias, we use the historical weather data\\cite{nasa} of the testing data years (2014-2019) for each day to simulate daily temperatures variations that are reflected in higher or lower demand. We sample weather data for each day and compare it to 2015, and subsequently use normalized the difference to scale the demand on a daily basis. Finally, we sample demand for each hour of the year from the corresponding cluster (defined by month and weekend or weekday) and scale it accordingly. Constructing the hourly load profile and fitting them to match the projected daily consumption and the projected daily peak demand then becomes a trivial exercise of sampling and fitting from the corresponding clusters and weather data space. The 2015 hourly demand data used in this study is documented in detail elsewhere and has been used in projecting demand for supply-side modeling efforts \\cite{ivan}. Limited availability of complete hourly data at state and regional level in India biases the hourly profiles to the 2015 datasets. However, the business-as-usual projections are for existing demands composed mainly of lighting and appliance at the residential level and large daytime loads at the commercial level \\cite{usaid}. Our approach implicitly assumes that energy consumption trends for these loads will follow historical patterns and therefore sampling from a given year with post-processed noise variation can yield reasonable results.\n\n\\subsubsection*{Impact of Climate change on business-as-usual demand}\nAs per the International Energy Agency (IEA) World Energy Outlook (WEO) 2019\\cite{weo19} only 5\\% of households in India currently own air conditioning units and 2.6\\% of commercial building energy use is from space cooling. Historically, electricity consumption in India has been driven by lighting and appliances in the residential sector \\cite{usaid} with commercial and industrial sector contributing via larger daytime loads. Since cooling demand is not historically available in the data that the business-as-usual regression model is learning from, there is no parametric value to projecting increase in temperatures since there is no evident correlation between temperature increase and lighting or appliance use. Moreover, since space cooling is a small percentage of current electricity demand in India, no major trends can be identified given the limited daily training data that is being used for the business-as-usual regression. It is then safe to assume that weather remains constant for the business-as-usual demand.\n\n\\subsection*{Technology model}\nSince a regression model can only produce forecasts of data it can learn from, additional bottom-up processing must be carried out to get a full picture of India's demand in the future. We identify trends and data points at the state level of the country to build a regional profile as well as the national one. \n\n\\subsubsection*{Cooling}\nCooling is divided into two main categories: residential and commercial. The ratio of commercial to residential consumption is computed from state-level data \\cite{stats} and is used as the ratio of commercial to residential cooling demand. Using the IEA's baseline and efficient cooling projections from the Future of Cooling study \\cite{iea}, we use the annual sales and unit types to calculate the energy consumption and growth rate at a national level and pro-rate it down to state level given GDP per capita. Surveyed hourly demand profiles \\cite{usaid} are indicators of behavioral cooling energy consumption patterns as exemplified in Supplementary Fig. \\ref{fig:sup-ac-res} and \\ref{fig:sup-ac-com}. The survey produce various profiles given climate seasons, household income and size. We apply a time-domain convolution of these profiles to generate a representative profile for each state for the various climates and seasons.\n\nWe can generate the air conditioning demand profiles for two weather seasons (winter and summer) by convolution of the sample profiles to generate a smooth aggregated demand profile. Moreover, coincidence factors must be applied to properly estimate the simultaneity of the demand and its peak. Two coincidence factors are identified: weekday and weekend, values are extracted from a Reference Network Model Toolkit \\cite{5504171}. We break down the national cooling demand to residential and commercial at state level by identifying state-level sector size and growth trends. Scaling the profiles to match the projected cooling energy demand produces hourly energy consumption profiles from residential and commercial cooling. Aggregating the appropriate states together will produce the same results at the regional level.\n\nMore importantly, the IEA's future of cooling study \\cite{iea} stresses the usage of Cooling Degree Days (CDD) to project cooling demand dependency on temperature. The unit consumption pattern and projections of capacity for India's share of global cooling demand is based on growth in electrification, urbanization as well as Purchasing Power Parity. The IEA future of cooling study estimates that a 1-degree Celsius increase in decadal average temperature in 2050 will to lead to 25\\% more CDD and a 2-degree Celsius increase will lead to 50\\% more CDD. Climate change impacts are considered in the unit sales and energy consumption data used from the IEA's future of cooling study. In our analysis, we use IEA's 50\\% increase in CDD to model cooling demand in 2050. For prior periods, we interpolate CDD between 2018 and 2050 to model cooling demand. The increase in CDD and the addition of noise variation are introduced for the purpose of modeling the projected increase in peak demand due to climate change. Specifically, this analysis does not consider frequency nor forecast of extreme weather events.\n\n\\subsubsection*{Electric vehicles}\nThe second component of the technology model projects EV demand in India. The data presented here considered electric two, three, and four-wheel vehicles. Two-wheelers, being the dominating vehicle in terms of annual sales in India \\cite{vehicle_sales}, are expected to be electrified first, followed by the three-wheelers and regular cars \\cite{ey}. The Indian government has set a goal of converting 100\\% of two-wheeler sales and 30\\% of all vehicle sales to electric by 2030 \\cite{nitiaayog}, so the starting point is vehicle sales at the state level \\cite{vehicle_sales}. Using the regression equations of the corresponding GDP growth scenarios, we can project car sales with the EV targets by 2030 met in the rapid growth scenario. From vehicle sales and conversion rates, we get an estimate of the number of EV that will require charging. From a market survey on the average commute distance of vehicles in urban areas and rural areas \\cite{ey}, long and short-range battery capacity and EV energy can be estimated. We introduce a mix of EV sales starting with short-range as the dominant market product and shifting to long-range, a market-dominant market in 2050. This trends reflects the current economic competitiveness of short-range EVs vs. existing internal combustion engine vehicles  as well as the long-term competitiveness of long-range EVs with declining battery costs.\n\nSimilar to the construction of the cooling profiles, a coincidence factor must be implemented, so as to not over-predict peak EV charging demand. Since this is a new consumption behavior and given the relatively small batteries of two-wheelers and three-wheelers, it is assumed that every vehicle needs to charge every other day on average for urban drivers and every day for rural ones. This yields an average daily consumption from EV charging. As shown in Supplementary Fig. \\ref{fig:sup-ev-profiles}, three different charging profiles  --- home, work, public -- are identified in an EV pilot project study in Mexico City \\cite{berkeley}. While Mexico and India differ greatly in many socio-economic aspects. The different hourly EV charging profiles collected were for a pilot project to deploy electric two-wheelers and small sedans in the metropolitan area of Mexico City. This presents two synergies enabling the usage of the charging profiles in India. Under the assumptions that EV deployment will be more prevalent in urban areas in India with initial conversion of smaller vehicles (two-wheelers and three-wheelers), the charging data collected \\cite{berkeley} is a suitable fit for potential EV charging schemes in India. Energy consumption is computed from vehicle sales, projections, and electrification conversion. That calculated number is then fitted under the chosen charging profile. Time domain convolution of the profiles is applied to smoothen the peakiness of the total constructed hourly time series.\n\n\\subsubsection*{Data Dependence}\n\nThe technology model relies heavily on surveyed data to produce the representative hourly profiles for cooling and electric vehicle demands at state levels. This is indeed a limitation, and our projections assumes that future technology adopters will behave just like initial adopters. In the absence of a better alternative at a similar spatial and temporal resolution, the bottom-up modeling effort provides a reasonable estimate of temporal patterns expected from these new demand sources. For the hourly sample cooling profiles, the main assumption is that cooling demand consumption is only dependent on weather patterns and econometric patterns. Specifically, we apply a weighted sum convolution of the income level cooling profiles based on the states' GDP per capita ranking. For the total cooling demand at national level, we depend on the air cooling unit sales projection as well as break down of unit energy consumption under baseline and efficient scenarios of the IEA's Future of Cooling report \\cite {iea}. We pro-rate residential cooling at state level using the GDP per capita projections. For commercial cooling we use the state-wise sector growth trends \\cite{energystatistics}. A sanity check for this break down is to sum both residential and commercial state-wise cooling demand and compare to the IEA's all India cooling demand annual electricity consumption projections to 2050, the difference is highlighted in Supplementary Fig. \\ref{fig:sup-ac_compare} and \\ref{fig:sup-cooling}. Regarding the EV profiles, while there are alternative choices of charging schemes, we identified the synergies with the Berkeley study \\cite{berkeley} to be best reflective of the bookend EV charging scenarios across India.\n\n\\section*{Data Records}\nThe data is uploaded on Zenodo \\cite{marc_barbar_2020_4564581} and is available to download at \\hyperlink{https:\/\/doi.org\/10.5281\/zenodo.4564581}{https:\/\/doi.org\/10.5281\/zenodo.4564581}. The path leading to a CSV file indicate the scenario corresponding to the results of that file. Breakdown of the folder hierarchy listed as:\n\n\\begin{enumerate}\n    \\item GDP Growth: slow, stable, rapid\n    \\item EV charging: home, work, public\n    \\item Cooling: baseline, efficient\n    \\item Type: detailed, summary\n\\end{enumerate}\n\nThe \\textit{detailed} results are tables of the itemized hourly demand profile of each considered scenario; all files will produce 8760 rows (number of hours in a year). The \\textit{summary} are tables of the itemized annual energy consumption for the considered years; all files will produce seven rows (number of considered future years). Both file types are itemized the same way as per Table \\ref{table:headers}. The path of each file is the reference to the specific scenario the data in the tables represents. For example \\textit{SR.csv} file under \\textit{slow\/home\/efficient\/summary} is the summary file of the case of slow economic growth, home EV and energy efficient air conditioning consumption.\n\n\\section*{Technical Validation}\nThe Business-as-usual statistical model is validated using standard statistical metrics when backtesting is applied. Further details on the backtesting are available elsewhere \\cite{meia}. For the technology model, we compare our estimates to the IEA's WEO \\cite{weo20,weo19,weo18,weo17} and Brookings India \\cite{brookings}. Furthermore, our projections compare favorably against the EV projections to the IEA's Global Electric Vehicle Outlook 2020 \\cite{ev}.\n\n\\subsection*{Back testing}\n\nDaily consumption and peak are projected for all five regions, we show the daily consumption back tests of the Southern Region in Fig. \\ref{fig:regression}. More results can be found on the GitHub repository. It is important to note that the regression model captures the organic growth of the historical demand as well as the seasonal variation in demand but is not accurate at predicting daily variation. This shortcoming can be attributed to the small training dataset that is available. To compensate for this short-coming, we add additional noise variation as discussed earlier in the Methods section. We compare the R-squared value of the regression only versus the regression and noise time series as shown in Table \\ref{table:r2}. Additionally, selected parameter performance metrics of the model for the Southern Region are presented in Table \\ref{table:params}. The model's independent variables are the 2 meters and 10 meters elevation historic temperature and humidity data for the selected cities and GDP data for the state. Various weather parameters will have a higher coefficient then GDP since the latter is not as granular as a metric but will still be factored in for longer term growth as interpreted by its Fourier component \\cite{meia}.\n\n\\subsection*{Cross-comparison}\nSupplementary Fig. \\ref{fig:sup-stated-weo} and \\ref{fig:sup-sustainable-weo} compare the forecasting results to the WEO 2020 projections of India's Energy Demand to 2040. Our band of projections is notably wider due to the large number of scenarios that are combined to forecast energy demand. We further compare our results to Brookings India's study in Supplementary Fig. \\ref{fig:sup-brookings}. We also compare our electric vehicle projections to those of the Global EV Outlook in Supplementary Fig. \\ref{fig:sup-ev}. Finally, we compare our air conditioning demand contribution to the peak demand to the Future of Cooling study in Supplementary Fig. \\ref{fig:sup-ac_compare}.\n\n\\subsection*{COVID-19 pandemic impact on year 2020}\nThe COVID-19 pandemic has drastically affected the global population in various ways. Energy consumption dropped severely as people were advised to stay at home. While it is not possible to project such \"Black Swan\" events from historical data, their long-term effects can be modeled as delayed growth under various recovery schemes. Fig. \\ref{fig:comparison} shows that our projections for the month of January 2020 align with the realized demand, which is prior to the global outbreak of COVID-19. Evidently, there is a strong mismatch in the following months as the outbreak developed into a global pandemic. However, in the later part of the year, signs of recovery are noticed where the historical daily consumption once again reaches projected levels.\n\nThe impact of extreme events on energy consumption are difficult to predict at a granular level. Our projections are at a five year increment so that such yearly variations are smoothed out and the regression towards the mean phenomenon is observed. Moreover, the recovery from extreme events and their long-term impact can depend on many factors: economic, social, scientific and more. Without modeling those events in detail, projected growth can model the long-term average growth rate. In case of a negative extreme event, a smaller growth rate can model the long-term impact caused by the slow down. Similarly, a positive extreme event can be modeled as larger growth rate to include the long-term impact by the rapid growth. With signals of a fast recovery in total daily consumption for most regions, we elected to disregard projections that model long-term COVID-19 pandemic impact to avoid confirmation bias. Moreover, there is little data to support projections modeling a long-term impact on Indian energy consumption. We believe that the model and data presented in this paper are valid beyond the COVID-19 pandemic.\n\n\\section*{Usage Notes}\n\nThe format of the results is comma-separated values (CSV). All the results are available on the Zenodo Open-Access repository \\cite{marc_barbar_2020_4564581}.\n\n\\section*{Code availability}\n\nThe code used in the generation of the data sets is open-sourced on Github repository \\cite{git}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n    \\input{tex\/11_intro}\n\n\\section{Background}\n\t\\subsection{Generative Design}\\label{sec:sec31}\t\n\t    \\input{tex\/21_gd}\n\t\\subsection{Exploration and Optimization with Non-Objective Criteria}\\label{sec:sec32}\t\n        \\input{tex\/22_mcx}\n\n\\section{Method}\n    \\input{tex\/31_method}\n        \t      \n\\section{Benchmarks}\n    \\subsection{Setup}\n        \\input{tex\/41_exp1_setup}\n    \\subsection{Result}        \n        \\input{tex\/42_exp1_result}\n\\section{Case Study}\n        \\subsection{Setup}\n        \\input{tex\/51_exp2_setup}\n        \\subsection{Result}\n        \\input{tex\/52_exp2_result}\n        \n\\section{Discussion}\n        \\input{tex\/60_discussion}\n        \n\\section{Acknowledgements}\n        \\input{tex\/62_ack}        \n        \n\\section{Supplemental Material and Code}\nSupplemental material and code available at:\\\\ \\href{https:\/\/github.com\/agaier\/tdomino_ppsn}{https:\/\/github.com\/agaier\/tdomino\\_ppsn}\n\n\n\n\n\n\\subsubsection{Benchmark Functions}\\hfill\\\\\n\\indent\\textit{RastriginMOO}. To judge the performance of T-DominO on Multi-Objective QD problems, we test on a version of RastriginMOO as introduced in~\\cite{moo_qd}. The Rastrigin function is a classic optimization benchmark, often used to test QD algorithms because it contains many local minima~\\cite{cmame,cully2021multi}. Here it is converted into a multiobjective benchmark by optimizing a pair of Rastrigin functions with shifted centers. We use a 10-D version with constants added so that every discovered bin has a positive effect on the aggregate QD Score. These objectives can be explicitly defined as:\n\\begin{align}\n    \\begin{cases}\n      f_1(\\mathbf{x}) = 200 - (\\sum\\limits_{i=1}^n [(x_i - \\textcolor{blue}{\\lambda_1})^2 - 10\\cos (2\\pi (x_i - \\textcolor{blue}{\\lambda_1}))]) \\\\\n      f_2(\\mathbf{x}) = 200 - (\\sum\\limits_{i=1}^n [(x_i - \\textcolor{blue}{\\lambda_2})^2 - 10\\cos (2\\pi (x_i - \\textcolor{blue}{\\lambda_2}))])\n    \\end{cases}       \n\\end{align}\nwhere $\\lambda_1 = 0.0$ and $\\lambda_2 = 2.2$ for $f_1$ and $f_2$. All parameters are limited to the range $[-2, 2]$, with the feature space defined by the first two parameters.\n\n\\textit{ZDT3}.\nWhen spread across the objective space is desired, objectives themselves could be used as features. This use case is demonstrated with the ZDT3 benchmark, a 30 variable problem from the ZDT MOO benchmark problem suite ~\\cite{zitzler2000comparison} whose hallmark is a set of disconnected Pareto-optimal fronts, and whose first parameter is value of the first objective. Parameter ranges span 0-1 with the first two parameters used as features, enforcing a spread of solutions across the range of the first objective.\n\n\\textit{DTLZ3}. To illustrate T-DominO's bias toward balanced solutions we analyze its performance on DTLZ3, a many-objective benchmark with a tunable number of objectives and variables\\cite{deb2002scalable}. We test with 10 parameters and 5 objectives, with the 6th and 7th parameters use as features.\\footnote{The first $n$ parameters are explicitly linked to the first $n$ objectives as in ZDT3 -- later parameters are used to avoid explicitly exploring the objective space.}. \n\n\n\\subsubsection{Baseline Approaches}\\hfill\\\\\n\\indent\\textit{ME Single.} MAP-Elites~\\cite{mapelites} optimizing only a single objective is used to establish an upper and lower bound of performance we can expect from MAP-Elites. Blind to the second objective we can expect it to find the top performing solutions for the first. Equally important, the exploration of all bins without regard to the performance on the second objective establishes a floor for performance -- the performance we could expect for having any solution in the bin.\n\n\n\\textit{ME Sum.} \nWe compare using the T-Domino objective with MAP-Elites~\\cite{mapelites} using the most naive way of combining multiple objective -- simply adding them. Our benchmarks all have well-scaled objectives, but this is typically not the case. To simulate this difficulty we use a weighted sum, with each additional objective values increased by an order of magnitude (e.g $\\times$1, $\\times$10, $\\times$100...). \n\n\\textit{NSGA-II.}\nNSGA-II~\\cite{nsga2} is used as a benchmark for conventional multi-objective optimization without feature space exploration, reaching near the Pareto front on these simple benchmarks. Though it is not our goal to compete with MOO algorithms, they provide a useful metric to contextualize the difference between exploratory approaches and pure optimizers.\n\n\\subsubsection{Settings.} In all MAP-Elites approaches the feature space is partitioned a 20x20 grid, with 2 CMA-ME improvement emitters~\\cite{cmame} performing optimization. T-Domino was computed using the neighbors from 4 bins away, using a history of the 10 most recent elites in each bin. Hyperparameters for NSGA-II were kept comparable, a population of 400 matched the 400 bins of the MAP-Elites grids, with the same number of new solutions generated per generation for the same number of generations. A standard implementation of NSGA-II from the PyMoo library~\\cite{pymoo} is used, as well as the library's formulations for the ZDT3 and DTLZ benchmarks whose the exact formulation is included in the Supplemental \\ref{sssec:zdt3}. The PyRibs~\\cite{pyribs} library was used as a basis for all MAP-Elites experiments, with T-DominO implemented as a specialized archive type. All experiments were replicated 30 times, additional plots are provided in the Supplemental.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Supplemental Material}\nUpon publication all supplemental material, along with all source code used to produce the results in this paper will be published online.\n\n\n\\subsection{Building Layout Objectives, Features, and Constraints}\n\\input{tex\/53_exp2_table}\n\n\\newpage\n\\subsection{Wave Function Collapse Tiles Set and Seed Examples}\n\\input{tex\/fig_supp_01_tiles}\n\n\\newpage\n\\subsection{Single Building Example Outputs of Wave Function Collapse}\n\\input{tex\/fig_supp_02_example}\n\n\\newpage\n\\subsection{QD Score}\n\\input{tex\/fig_supp_03_qdscore}\n\n\\newpage\n\\subsection{MOO Benchmark Functions}\n\\input{tex\/supp_moo_obj}\n\n\n\n\n\\subsubsection{ZDT3}\\label{sssec:zdt3}\nThe ZDT3 benchmark objective function is defined as:\n\n$\n\\begin{aligned}\nf_{1}(x) &=x_{1} \\\\\ng(x) &=1+\\frac{9}{n-1} \\sum_{i=2}^{n} x_{i} \\\\\nh\\left(f_{1}, g\\right) &=1-\\sqrt{f_{1} \/ g}-\\left(f_{1} \/ g\\right) \\sin \\left(10 \\pi f_{1}\\right) \\\\\n0 & \\leq x_{i} \\leq 1 \\quad i=1, \\ldots, n\n\\end{aligned}\n$\n\n\n\\subsubsection{DTLZ3}\\label{sssec:dltz3}\nThe DTLZ3 benchmark objective function is defined as:\n\nMin. $f_{1}(\\mathbf{x})=\\left(1+g\\left(\\mathbf{x}_{M}\\right)\\right) \\cos \\left(x_{1} \\pi \/ 2\\right) \\cdots \\cos \\left(x_{M-2} \\pi \/ 2\\right) \\cos \\left(x_{M-1} \\pi \/ 2\\right)$,\n\nMin. $f_{2}(\\mathbf{x})=\\left(1+g\\left(\\mathbf{x}_{M}\\right)\\right) \\cos \\left(x_{1} \\pi \/ 2\\right) \\cdots \\cos \\left(x_{M-2} \\pi \/ 2\\right) \\sin \\left(x_{M-1} \\pi \/ 2\\right)$,\n\nMin. $f_{3}(\\mathbf{x})=\\left(1+g\\left(\\mathbf{x}_{M}\\right)\\right) \\cos \\left(x_{1} \\pi \/ 2\\right) \\cdots \\sin \\left(x_{M-2} \\pi \/ 2\\right)$,\n\n$\\vdots \\quad \\vdots$\n\nMin. $f_{M}(\\mathbf{x})=\\left(1+g\\left(\\mathbf{x}_{M}\\right)\\right) \\sin \\left(x_{1} \\pi \/ 2\\right)$,\n\nwith $g\\left(\\mathbf{x}_{M}\\right)=100\\left[\\left|\\mathbf{x}_{M}\\right|+\\sum_{x_{i} \\in \\mathbf{x}_{M}}\\left(x_{i}-0.5\\right)^{2}-\\cos \\left(20 \\pi\\left(x_{i}-0.5\\right)\\right)\\right]$,\n$0 \\leq x_{i} \\leq 1, \\quad$ for $i=1,2, \\ldots, n$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe Epoch of Reionization is the interval of time during which the cosmic gas evolves from an almost completely neutral state (neglecting the recombination leftovers) to an ionized state.\nThis ionization process is believed to happen due to the onset of star formation at redshifts $z\\simeq 12$, and it is believed to last until $z\\simeq 6$.\nSeveral astrophysical observables (quasars~\\cite{Fan:2005es,Becker:2014oga}, Lyman $\\alpha$ emitters~\\cite{Stark:2010qj,Treu:2013ida,Pentericci:2014nia,Schenker:2014tda,Tilvi:2014oia}, $\\gamma$ ray bursts~\\cite{Wang:2015ira,Gallerani:2009aw}) seem to agree with this hypothesis.\nHowever, the precise details of the overall reionization process still remain obscure.\nThe main reason is that the currently available most precise information on the reionization period comes from Cosmic Microwave Background (CMB) measurements through a redshift-integrated quantity.\nDuring reionization, the number density of free electrons which can scatter the CMB, $n_e$, increases. As a consequence, the reionization optical depth $\\tau$ increases according to a line of-sight integral of $n_e$, generating a suppression of the CMB peaks at any scale within the horizon at the reionization period.\nThis suppression, however, can be easily compensated with an enhancement of the primordial power spectrum amplitude, $A_{\\rm s}$.\nA much better and cleaner measurement of $\\tau$ can be obtained via measurements of the CMB polarization, which is linearly affected by reionization (see e.g. Refs.~\\cite{Kaplinghat:2002vt,Haiman:2003ea,Holder:2003eb,Hu:2003gh} for seminal works and \\cite{Reichardt:2015cos} for a recent review). The latest measurements of the Planck collaboration provide a value of $\\tau = 0.055 \\pm 0.009$~\\cite{Aghanim:2016yuo, Adam:2016hgk} based exclusively on the CMB polarization spectrum.\nThis value of $\\tau$ is in a much better agreement than previous WMAP~\\cite{Hinshaw:2012aka} and Planck~\\cite{Ade:2015xua} estimates with observations of Lyman-$\\alpha$ (Ly-$\\alpha$) emitters at $z\\simeq 7$~\\cite{Stark:2010qj,Treu:2013ida,Pentericci:2014nia,Schenker:2014tda,Tilvi:2014oia}, which require that reionization is complete by $z\\simeq 6$.\nEven if now cosmological and astrophysical tests of the reionization process seem to agree, the measurement of $\\tau$ provides only integrated information on the free electron fraction $x_e$, and not on its precise redshift evolution.\nConsequently, the same measured value of $\\tau$ may correspond to very different reionization histories.\n\nTraditionally, the most commonly exploited model for the time evolution of the free electron fraction, $x_e(z)$, uses a step-like transition, implemented via a hyperbolic tangent~\\cite{Lewis:2008wr}.\nModel independent attempts have been carried out in several works in the past~\\cite{Hu:2003gh,Mortonson:2007hq,Mortonson:2007tb,Mortonson:2008rx,Mortonson:2009qv,Mortonson:2009xk,Mitra:2010sr,Lewis:2006ym,Pandolfi:2010dz,Pandolfi:2010mv} and also more recently~\\cite{Heinrich:2016ojb,Hazra:2017gtx,Mitra:2017oxx}, based either on a redshift-node decomposition of $x_e(z)$ or on a Principal Component Analysis (PCA) of the CMB polarization angular power spectrum.\nMore concretely, using the latter approach, the authors of \\cite{Heinrich:2016ojb} claimed that Planck 2015 data favors a high-redshift ($z>15$) component to the reionization optical depth.\nThe quoted $2\\sigma$ evidence would come from the excess in power in the low multipole range of the Planck 2015 CMB polarization spectrum. \nAccordingly to their results, the functional form of the usual step-like model prevents a priori for such an early component in the reionization history of our universe.\nHowever, the authors of \\cite{Hazra:2017gtx}, using a different method, which implements reionization through a non-parametric reconstruction that uses a Piecewise Cubic Hermite Interpolating Polynomial (\\texttt{PCHIP}), find only marginal evidence for extended reionization histories.\nSince an early component in the reionization history $x_e(z)$ (or, in other words, a high redshift contribution to the reionization optical depth $\\tau$) may either imply the need for a high-redshift population of ionizing sources (hypothesis that will be tested by the future James Webb Space Telescope~\\cite{Gardner:2006ky}),\nor give hints about a possible energy injection from dark matter annihilations or decays~\\cite{Pierpaoli:2003rz,Mapelli:2006ej,Natarajan:2008pk,Natarajan:2009bm,Belikov:2009qx,Huetsi:2009ex,Cirelli:2009bb,Kanzaki:2009hf,Natarajan:2010dc,Giesen:2012rp,Diamanti:2013bia,Lopez-Honorez:2013lcm,Lopez-Honorez:2016sur,Poulin:2016nat,Poulin:2015pna},\nor accreting massive primordial black holes~\\cite{Ricotti:2007au,Horowitz:2016lib,Ali-Haimoud:2016mbv,Blum:2016cjs,Poulin:2017bwe},\nit is mandatory to robustly establish what current data prefer, regardless of the model used to describe the redshift evolution of the free electron fraction. \n\nHere we first analyze several possible parameterizations for reionization (PCA with several fiducial cosmologies and the \\texttt{PCHIP}\\ method)\nand explore the corresponding constraints on the reionization history of the universe.\nWe then shall exploit tools related to model selection among competing models, using both the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), which will allow us to quantitatively decide which model is currently preferred and whether it exists or not an indication for an early reionization component in our universe.\n\nThe structure of the paper is as follows.\nWe start by discussing the different reionization approaches that we shall test against current data in Sec.~\\ref{sec:histories}.\nIn Sec.~\\ref{sec:data} we describe the cosmological observations exploited in our numerical analyses, whose results are shown in Sec.~\\ref{sec:results}.\nOur conclusions are summarized in Sec.~\\ref{sec:conclusions}.\n\n\n\\section{Reionization histories}\n\\label{sec:histories}\n\nIn the following, we will derive the constraints on the reionization history of our universe from cosmological observations exploring several possible scenarios, focusing on a possible early reionization component in our universe.\nFor that, we shall exploit the reionization optical depth:\n\\begin{equation}\n\\tau(z) = \\int_z^{\\infty} dz' \\frac{c ~dt'}{dz'}  (n_{\\rm e}(z')- n_{\\rm e, 0}(z'))\\sigma_{\\rm T}\\,~,\n\\label{eq:cumtau}\n\\end{equation}\nwhere $n_{\\rm e}(z)=n_{\\rm H}(0)(1+z)^3x_{\\rm e}(z)$ and $n_{\\rm e,0}(z)=n_{\\rm H}(0)(1+z)^3x_{\\rm e, 0}(z)$, being $n_{\\rm H}(0)$ the number density of hydrogen at present, $x_{\\rm e}(z)$ the free electron fraction and $x_{e,0}(z)$ the free electron fraction leftover from the recombination epoch (see e.g.\\ \\cite{Kolb:1990vq,2009fflr,2010gfe}). Therefore, Eq.~\\eqref{eq:cumtau} just accounts for the cumulative Compton optical depth after recombination, subtracting the pre-reionization contribution.\n\n\n\\subsection{Canonical scenarios}\n\\label{subsec:canonical}\nWe start describing the free electron fraction by means of the most simple and commonly exploited parameterizations in the literature, i.e.\\ the so-called \\emph{redshift-symmetric} and \\emph{redshift-asymmetric} parameterizations (see e.g.~\\cite{Adam:2016hgk}).   \n\n\\begin{itemize}\n\n\\item \\emph{Redshift-symmetric} parameterization.\n\nThe most economical and widely employed approach to describe the reionization process in our universe assumes that the free electron fraction follows a step-like function, taking the recombination leftover value at high redshifts and becoming close to one at low redshifts, and being described by the hyperbolic tangent function~\\cite{Lewis:2008wr}\n\\begin{equation}\nx_e^{\\rm tanh}(z) = \\frac{1+f_{\\rm He}}{2} \\left(1+ \\tanh \\left[ \\frac{y(z_{\\rm{re}})-y(z)}{\\Delta y} \\right] \\right),\n\\label{eqn:tanh}\n\\end{equation}\nwhere $y(z)=(1+z)^{3\/2}$, $\\Delta y=3\/2(1+z_{\\rm{re}})^{1\/2}\\Delta z$, and $\\Delta z$ is the width of the transition, fixed in the following to $\\Delta z=0.5$.\nThis parametrization is named ``redshift symmetric'' because the redshift interval between the beginning of reionization and its half completion equals the corresponding one between half completion and the reionization offset, and it is the default one implemented in Boltzmann solver codes such as \\texttt{CAMB}~\\footnote{\\href{http:\/\/camb.info}{http:\/\/camb.info}}~\\cite{Lewis:1999bs}.\nThis parameterization, as well as the following ones, also accounts for the first ionization of helium $f_{\\rm He}=n_{\\rm{He}}\/n_{\\rm{H}}$, assumed to happen at the same time than that of hydrogen.\nThe full helium reionization is modeled via another hyperbolic tangent function with $z_{\\rm{re,He}}=3.5$ and $\\Delta z=0.5$.\nTherefore, the only free parameter in this simple approach is the reionization redshift $z_{\\rm{re}}$.\nWhen this redshift-symmetric parameterization is used as the fiducial model in our PCA analyses (see next subsection), we fix $z_{\\rm{re}}=8.8$, following the results quoted in Ref.~\\cite{Adam:2016hgk}.\n\n\\item \\emph{Redshift-asymmetric} reionization.\n\nBesides the previous case, alternative reionization parameterizations with a non redshift-symmetric transition have been proposed in the literature.\nOne of the most flexible choices, which shows good agreement with current measurements from quasars, Ly$\\alpha$ emitters and star-forming galaxies, is represented by a power law, described via three parameters~\\cite{Adam:2016hgk,Douspis:2015nca}:\n\\begin{equation}\n  x_e^{asym}(z) =\n  \\begin{cases}\n\t1+f_{\\rm He} & \\mbox{for } z<z_{\\rm end}, \\\\\n    (1+f_{\\rm He}) \\left(\\frac{z_{\\rm early}-z}{z_{\\rm early}-z_{\\rm end}}\\right)^\\alpha\n      & \\mbox{for } z_{\\rm end} < z < z_{early}, \\\\\n      0 & \\mbox{for } z > z_{\\rm early}.\n  \\end{cases}\n  \\label{eqn:asym}\n\\end{equation}\nFollowing Planck 2016 reionization analyses~\\cite{Adam:2016hgk}, when using this redshift-asymmetric model as a fiducial model in our PCA analyses, we shall fix the redshift at which the first sources in our universe switch on, $z_{\\rm early} = 20$, the redshift at which reionization is fully complete, $z_{\\rm end} = 6$, and the exponent $\\alpha = 6.10$. \n\\end{itemize}\n\n\\subsection{Principal Component Analysis (PCA)}\nThe second method we follow here to model the reionization process is the Principal Component Analysis (PCA) approach of Refs.~\\cite{Hu:2003gh,Mortonson:2007hq,Mortonson:2007tb,Mortonson:2008rx,Mortonson:2009qv,Mortonson:2009xk,Mitra:2010sr}, exploited more recently in Refs.~\\cite{Heinrich:2016ojb,Mitra:2017oxx}.\nFollowing these previous works, we discretize the redshift range from $z_{\\rm{min}}=6$ to $z_{\\rm{max}}=30$ in $N_z$ bins of width of $\\delta z = 0.25$.\nWe set the ionization fraction to $x_e=0$ for $z \\geq z_{\\rm{max}}$, when the reionization processes have not started yet, while for $z \\leq 6$ we assume fully ionized hydrogen and singly ionized helium, i.e.\\ $x_e=1+f_{\\rm He}$.\nThe full helium reionization is modeled as aforementioned.\nThis approach makes use of the Fisher information matrix~\\cite{Tegmark:1996bz}, that we compute as:\n\\begin{equation}\nF_{ij} = \\sum_{\\ell=2}^{\\ell_{\\rm max}}\\frac{1}{\\sigma_{ C_{\\ell}}^2}\n \\frac{\\partial  C_{\\ell}}{\\partial x_e(z_i)}\n \\frac{\\partial  C_{\\ell}}{\\partial x_e(z_j)} =\\sum_{\\ell=2}^{\\ell_{\\rm max}}\\left(\\ell+\\frac{1}{2}\\right)\n \\frac{\\partial \\ln C_{\\ell}}{\\partial x_e(z_i)}\n \\frac{\\partial \\ln C_{\\ell}}{\\partial x_e(z_j)} \\,,\n\\label{eq:fisher}\n\\end{equation}\nwhere the $C_{\\ell}$ are the components of the large angle $EE$ polarization spectrum.\nThe sum above is truncated at $\\ell_{\\rm max}=100$, because the reionization imprint is mostly located in the lowest modes of the CMB polarization spectrum.\nIn Eq.~\\eqref{eq:fisher} we have used the well-known result for the cosmic variance: $\\sigma_{ C_{\\ell}}^2 = C_{\\ell}^2\\, 2\/(2\\ell+1)$.\nHaving the Fisher matrix, we can diagonalize it and find that the eigenfunctions are the principal components $S_{\\mu}(z)$ and the eigenvalues are proportional to the inverse of the estimated variance of each eigenmode, $\\sigma^2_{\\mu}$.\nUsing the normalization of Ref.~\\cite{Mortonson:2007hq}, we can write the Fisher matrix as\n\\begin{equation}\nF_{ij}=\\frac{1}{(N_z+1)^2}\\sum_{\\mu=1}^{N_z}\n \\frac{1}{\\sigma^2_{\\mu}}S_{\\mu}(z_i) S_{\\mu}(z_j)~.\n\\label{eq:fisher2}\n\\end{equation}\nWe sort the different eigenfunctions in order to have the smallest uncertainties at the lowest modes, being therefore the $\\mu=1$ case the best constrained mode.\nDue to completeness and orthogonality of the principal components, the following properties are fulfilled:\n\\begin{align}\n\\int_{z_{\\rm min}}^{z_{\\rm max}} dz \\, S_{\\mu}(z)S_{\\nu}(z)&=(z_{\\rm max}-z_{\\rm min})\\delta_{\\mu\\nu} \\, , \\\\\n\\sum_{\\mu=1}^{N_z} S_{\\mu}(z_i)S_{\\mu}(z_j)&= (N_z+1) \\delta_{ij}\\,.\n\\end{align}\nSince the width of the bins is chosen to be sufficiently small, in practice we can replace the integrals over redshift by discrete sums.\nOne of the ideas behind the PCA approach is that one can write redshift-dependent quantities such as the ionization fraction as a linear combination of the principal components.\nSince the lowest modes have the smallest uncertainties, we truncate the sum, using only the first 5 principal components, following Ref.~\\cite{Mortonson:2007hq}.\nWe apply the PCA analysis to the ionization history in two different ways, which are explained below.\n\n\\begin{itemize}\n\n\\item \\textbf{Case A}\n\nIn the first PCA approach, named \\textbf{PCA-A} in what follows, the reionization history reads as\n\\begin{equation}\nx_e^A (z) = x^{\\rm{fid}}_{\\rm e} (z)+ \\sum_\\mu m_{\\mu}^{A} S_\\mu (z)~.\n\\label{eq:pca_a}\n\\end{equation}\nGiven a fiducial model $x^{\\rm{fid}}_{\\rm e} (z)$, and knowing the amplitudes derived from the Fisher matrix (see Eq.~\\eqref{eq:fisher}), one can recover an arbitrary ionization history using a PCA analysis.\nThis is the standard approach adopted in Refs.~\\cite{Mortonson:2007hq,Heinrich:2016ojb} in order to constrain the ionization history with CMB data.\nFollowing \\cite{Mortonson:2007hq}, we can derive upper and lower bounds for each amplitude $m_{\\mu}$:\n\\begin{equation}\nm_{\\mu}^{\\pm} = \\int_{z_{\\rm min}}^{z_{\\rm max} } dz \\frac{S_{\\mu}(z)[x_e^{\\rm max} -2 x_e^{\\rm fid}(z)]\n\\pm x_e^{\\rm max} | S_{\\mu}(z)|}{2(z_{\\rm max}-z_{\\rm min})}~.\n\\label{eq:mbounds}\n\\end{equation}\nAdditionally, in order to guarantee physical ionization histories, the choice of our amplitudes $m_{\\mu}$ has to fulfill the condition $0 \\leq x_e(z) \\leq 1+f_{\\rm He}$ at any redshift $z$~\\footnote{Notice that this constraint for physicality is stronger than that followed in Ref.~\\cite{Heinrich:2016ojb}, as any unphysical model will be retained for the Monte Carlo analyses.}.\n\n\\item \\textbf{Case B}\n\nIn the second of our PCA analyses, named \\textbf{PCA-B}, we choose a different approach to the standard PCA analysis described above, in which the free electron fraction is proportional to the fiducial model plus the PCA decomposition.\nHere, we exploit the functional form of the fiducial model in order to test other possible reionization parameterizations.\nFollowing this idea, for the redshift-symmetric, \\textit{tanh} description, we insert the PCA decomposition inside the argument of the hyperbolic tangent:\n\\begin{equation}\nx_e^{B,tanh}(z) = \\frac{1+f_{\\rm He}}{2} \\left(1+ \\tanh \\left[ \\frac{y(z_{\\rm{re}})-y(z)}{\\Delta y} + \\sum_\\mu m_\\mu^B S_\\mu (z)  \\right] \\right)~.\n\\label{eqn:tanh_b}\n\\end{equation}\nNotice that we recover the fiducial \\textit{tanh} model by setting the amplitudes $m_{\\mu}$ to $0$.\nWe perform an analogous replacement for the redshift-asymmetric parameterization:\n\\begin{equation}\n  x_e^{B,asym}(z) =\n  \\begin{cases}\n\t1+f_{\\rm He} & \\mbox{for } z<z_{\\rm end}, \\\\\n    (1+f_{\\rm He}) \\left(\\left(\\frac{z_{\\rm early}-z}{z_{\\rm early}-z_{\\rm end}} \\right)+ \\sum_\\mu m_\\mu^B S_\\mu (z)\\right)^\\alpha\n      & \\mbox{for }z_{\\rm end} < z < z_{\\rm early}, \\\\\n      0 & \\mbox{for } z > z_{\\rm early}.\n  \\end{cases}\n  \\label{eqn:asym_b}\n\\end{equation}\nWe take for the specific parameters of the \\textit{tanh} and \\textit{asym} cases the fiducial values given in Sec.~\\ref{subsec:canonical}.\n\n\\end{itemize}\n\n\\subsection{\\texttt{PCHIP}}\nThe third and last method we adopt in order to describe the reionization history is based on a non-parametric form for the free electron fraction $x_e(z)$, which is described using the function values $x_e(z_i)$ in a number $n$ of fixed redshift points $z_1,\\ \\ldots,\\ z_n$.\nFollowing the procedure adopted for the PCA analyses, we fix the function to be a constant\nboth at low\nredshifts ($z\\leq6$) and at high redshifts ($z\\geq30$).\nThe first and the last redshift nodes we use to parameterize the function at intermediate redshifts are therefore $z_1=6$ and $z_n=30$, where we also want the function to be continuous:\nas a consequence, the values $x_e(z_1)=1+f_{\\rm He}$ and $x_e(z_n)=0$ are fixed\nand the number of varying parameters that describe $x_e(z)$ is always $n-2$.\nWe consider a case with a total of $n=7$ nodes (5 free parameters),\nlocated at redshifts\n\\begin{equation}\\label{eq:nodes7}\n z_i \\in \\{6,\\,7,\\,8.5,\\,10,\\,13,\\,20,\\,30\\}\\,,\n\\end{equation}\nin order to have the same number of free parameter than in the PCA cases.\n\nThe function $x_e(z)$ at $z\\neq z_i$ is computed through an interpolation among its values in the nodes.\nWe employ the\n``piecewise cubic Hermite interpolating\npolynomial'' (\\texttt{PCHIP})~\\cite{Fritsch:1980,Fritsch:1984}\nin a very similar way to Refs.~\\cite{Gariazzo:2014dla,DiValentino:2015zta,Gariazzo:2015qea,DiValentino:2016ikp},\nwhere the \\texttt{PCHIP}\\ function was adopted to describe the power spectrum\nof initial curvature perturbations, or the more recent work of \\cite{Hazra:2017gtx}, where the \\texttt{PCHIP}\\ method has also been used to model the evolution of $x_e(z)$.\nThe idea behind the \\texttt{PCHIP}\\ function is similar to that of the natural cubic spline,\nwith the difference that the monotonicity of the series of interpolating points\nmust be preserved.\nSpurious oscillations that may be introduced by the standard spline\nare avoided by imposing a condition on the first derivative of the function in the nodes,\nwhich must be zero if there is a change in the monotonicity\nof the point series.\nA more detailed discussion on the \\texttt{PCHIP}\\ function can be found in the appendix of Ref.~\\cite{Gariazzo:2014dla}.\n\nSummarizing, the free electron fraction in the \\texttt{PCHIP}\\ case is described by:\n\\begin{equation}\\label{eq:xe_pchip}\n  x_e(z) =\n  \\begin{cases}\n    1+f_{\\rm He}\n    & \\mbox{for } z \\leq z_1, \\\\\n    \\texttt{PCHIP}(z;\\ x_e(z_1),\\ \\ldots,\\ x_e(z_n))\n    & \\mbox{for } z_1 < z < z_n, \\\\\n    0\n    & \\mbox{for } z \\geq z_n,\n  \\end{cases}\n\\end{equation}\nwhere $n$ will be 7 and the redshifts $z_i$ are reported in Eq.~\\eqref{eq:nodes7}.\n\nFor the values of the function in the varying nodes,\nwhich are the free reionization parameters in our Markov Chain Monte Carlo analyses,\nwe impose a linear prior $0 \\leq x_e(z_i) \\leq 1+f_{\\rm He}$,\nwith $i=2,\\ \\ldots,\\ n-1$.\nThis ensures that the free electron fraction is always positive and smaller than its value today.\nThe value of the reionization optical depth $\\tau$  \nthat we report in our results is derived from Eq.~\\eqref{eq:cumtau}.\n\n\\section{Cosmological data}\n\\label{sec:data}\nWe use Planck satellite 2015 measurements of the CMB temperature,\npolarization, and cross-correlation spectra~\\cite{Adam:2015rua,Ade:2015xua}\nto derive the constraints on the possible reionization histories~\\footnote{%\nWe make use of the publicly available Planck likelihoods~\\cite{Aghanim:2015xee}, see \\href{http:\/\/www.cosmos.esa.int\/web\/planck\/pla}{www.cosmos.esa.int\/web\/planck\/pla}.\n}.\nMore precisely, we exploit both\nthe high-$\\ell$ ($30 \\leq \\ell \\leq 2508$) and\nthe low-$\\ell$ ($2 \\leq \\ell \\leq 29$) $TT$\nlikelihoods\nbased on the reconstructed CMB maps\nand\nwe include the Planck\npolarization likelihoods in the low-multipole regime\n($2 \\leq \\ell \\leq 29$), plus the high-multipole ($30 \\leq \\ell \\leq 1996$) $EE$ and $TE$ likelihoods~\\footnote{The latest reionization constraints from the Planck collaboration do not consider the TE data in the analyses, due to its larger cosmic variance and its weaker dependence on the reionization optical depth, when compared to EE measurements, see \\cite{Adam:2016hgk}.}. \nAll these CMB likelihood functions depend on several nuisance parameters\n(e.g.\\ residual foreground contamination, calibration, and\nbeam-leakage~\\cite{Ade:2015xua,Aghanim:2015xee}),\nwhich have been properly considered and marginalized over.  \nTo derive constraints on the reionization history and related parameters, we have modified the Boltzmann equations solver \\texttt{CAMB} code \\cite{Lewis:1999bs} and apply\nMarkov Chain Monte Carlo (MCMC) methods by means of an adapted version of the \\texttt{CosmoMC} package~\\cite{Lewis:2002ah}.\nAs for current constraints, we consider a minimal version of the $\\Lambda$CDM model, described by the following set of parameters: \n\\begin{equation}\\label{parameterPPS}\n\\{\\omega_{\\rm{b}},\\,\\omega_{\\rm{c}},\\, \\Theta_{\\rm{s}},\\,\\ln{(10^{10} A_{\\rm{s}})},\\,n_{\\rm{s}}\\}~,\n\\end{equation}\nwhere $\\omega_{\\rm{b}}\\equiv\\Omega_{\\rm{b}}h^2$ and $\\omega_{\\rm{c}}\\equiv\\Omega_{\\rm{c}}h^2$\nrepresent the physical baryon and cold dark matter energy densities, $\\Theta_{\\rm{s}}$\nis the angular scale of recombination, $A_{\\rm{s}}$ is the primordial power spectrum amplitude and $n_{\\rm s}$ the spectral index.\nNotice that we do not have $\\tau$ among the parameters included in our analyses, as $\\tau$ is a derived parameter.\nInstead, we will add the additional parameters describing the PCA and \\texttt{PCHIP}\\ reionization models, that will lead to the constraints presented in what follows.    \n\n\\section{Results}\n\\label{sec:results}\nFigure~\\ref{fig:tau} shows the most relevant results from our analyses of Planck 2015 temperature and polarization data assuming different reionization histories.\nAs aforementioned, we shall focus on the cumulative redshift distribution function of the reionization optical depth, Eq.~\\eqref{eq:cumtau}.\nA large departure from $0$ at redshifts $z>10$ would indicate evidence for an early reionization contribution, and therefore for non-standard reionization sources as, for instance, energy injection from dark matter annihilations or from matter accretion on massive primordial black holes.\nNotice that the PCA-A method of Ref.~\\cite{Heinrich:2016ojb}, in which the PCA decomposition is added linearly to a fiducial $x^{\\rm{fid}}_{\\rm{e}}(z)$, leads \\emph{always} to an early contribution to the optical depth $\\tau$, i.e.\\ $\\tau$ is significantly different from 0 at $z>10$, in contrast to standard reionization scenarios.\nFurthermore, the presence of this early contribution is independent of the fiducial model,\nas we can see from\nthe four PCA-A cases depicted in Fig.~\\ref{fig:tau}, which provide the same predictions at $z>10$, differing only mildly at small redshifts, regardless whether the fiducial model is a constant function or it depends on the redshift instead. \n\n\\begin{figure}[t]\n\\centering \n\\includegraphics[width=0.85\\textwidth]{reioPCHIP_pol_sm_bands_taue_new.pdf}\n\\caption{\\label{fig:tau} Cumulative redshift evolution of the reionization optical depth $\\tau(z)$ for several possible reionization scenarios.\nThe black thin solid and dot-dashed lines illustrate the PCA-A scenario for the case of two fiducial models constant in redshift.\nThe two upper dot-dashed lines refer also to the PCA-A parameterization but with redshift-dependent fiducial models.\nThe two lower colored solid lines depict the PCA-B scenarios, while the thick solid black line and the blue contours show the mean value and the $1$, $2$ and $3\\sigma$ allowed regions within the \\texttt{PCHIP}\\ prescription.}\n\\end{figure}\n\nIn order to unravel the origin of this early reionization component present when using the PCA-A description, several tests have been carried out.\nFirstly, we have eliminated the physical limits in the PCA amplitudes, finding very similar results.\nSecondly, we have simulated mock Planck data with the hyperbolic tangent description and then fitted these data to a PCA-A modeling, using different fiducial models.\nWe always find two bumps in the recovered $x_e$, see Fig.~\\ref{fig:xe}, one located between $z=10$ and $z=15$ and a second one located between $z=20$ and $z=25$.\nUpcoming measurements from the Planck satellite could disentangle if this early reionization component is truly indicated by the data or instead it is due to the adopted modeling or to other effects (i.e.\\ systematics).\n\nFurthermore, this early reionization component is definitely absent when other possible reionization histories are used in the analyses. \nFor instance, in the case of PCA-B parameterizations (see Eqs.~\\eqref{eqn:tanh_b} and \\eqref{eqn:asym}), there is no early reionization contribution, as $\\tau(z)$ is negligibly small for $z>10$.\nThe same happens for the \\texttt{PCHIP}\\ method, in which the mean reconstructed value of $\\tau(z)$ is also very small at high redshifts, showing little evidence for an early reionization component (see also Ref.~\\cite{Hazra:2017gtx}).\nNotice that the value of $\\tau$ today is smaller in the PCA-B approaches than in the PCA-A and \\texttt{PCHIP}\\ descriptions.\nHowever, this behavior is the expected one, as the PCA-B scenarios are very close to those explored by the Planck collaboration in Ref.~\\cite{Adam:2016hgk}, where it was found that the current value of $\\tau$ is $0.058\\pm 0.012$ for the hyperbolic tangent case, in perfect agreement with our findings here, even if we make use of the 2015 Planck likelihood only (the mean value is $\\tau=0.068$ for the very same model).\nThe differences between the PCA-A and PCA-B cases can be understood from the fact that the case B imposes a more restrictive functional form on the ionization history.\n \n\\begin{figure}\n\\centering \n\\includegraphics[width=0.85\\textwidth]{reioPCHIP_pol_sm_bands_xe_new_xe.pdf}\n\\caption{\\label{fig:xe}\nFree electron fraction as a function of the redshift for several possible reionization scenarios.\nLine styles and colors are the same as in Fig.\\ref{fig:tau}.}\n\\end{figure} \n\nThe findings above are fully consistent with our limits on the free electron fraction $x_{e}(z)$ at a given redshift.\nFigure~\\ref{fig:xe} shows the free electron fraction for the \\texttt{PCHIP}\\ parameterization together with the other PCA-A and PCA-B models explored here.\nThe color coding is identical to that used in Fig.~\\ref{fig:tau}. \nNotice that for the PCA-A models the free electron fraction is almost constant in the redshift interval $z=10-30$, as a consequence of the choice of the fiducial model, and therefore there will always be an early reionization component \\emph{within this approach}.\nHowever, when considering either the \\texttt{PCHIP}\\ or the PCA-B models, the free electron fraction is significantly smaller than $0.2$ for redshifts above $z=15$ and it is almost negligible above $z=20$.\nTherefore, the fact that current CMB observations need an early contribution to reionization is highly questionable, as it strongly depends on the framework used to analyze the data.\nUsing Planck CMB temperature and polarization data within the \\texttt{PCHIP}\\ analysis,\nwe find $x_e<0.90$, $<0.49$ and $<0.13$ at $2\\sigma$ in the nodes at $z=10$, $13$ and $20$, respectively.\nFluctuations in the lower $1\\sigma$ limits shown in Fig.~\\ref{fig:xe}\nare numerical artifacts that appear when computing the error bands at intermediate positions between the fixed \\texttt{PCHIP}\\ nodes and cannot be considered as significant.\nFigure~\\ref{fig:pchipamplitudes} shows the $68\\%$ and $95\\%$~CL allowed regions for the amplitudes of the \\texttt{PCHIP}\\ nodes, i.e.\\ the $x_{\\rm e} (z)$ at the redshifts listed in Eq.~\\eqref{eq:nodes7},\nfrom the Planck CMB measurements considered here.\nA quick inspection of Fig.~\\ref{fig:pchipamplitudes} tells us that all the amplitudes are perfectly compatible with a vanishing value.\nOnly one of them, $m_5$, the node corresponding to $z=13$, shows a very mild departure from $0$. However, this mild departure is far from being a significant effect, as it barely appears at $1\\sigma$.\nWe can therefore conclude that there is no evidence for a high redshift component in $x_{\\rm e}(z)$.\nNotice also from Fig.~\\ref{fig:pchipamplitudes} that, in general, the \\texttt{PCHIP}\\ amplitudes are anti-correlated among themselves.\nWe also illustrate the derived distribution for the value of the reionization optical depth, $\\tau_{\\rm PC}$, which is significantly correlated with the nodes at the higher redshifts.\nEven a modest increase of $x_{\\rm e}$ at $z=13$ or at $z=20$ would imply a significant shift towards larger values of the current reionization optical depth. \n\n\\begin{figure}\n\\centering \n\\includegraphics[width=0.85\\textwidth]{pchipNodes}\n\\caption{\\label{fig:pchipamplitudes} $68\\%$ and $95\\%$~CL allowed regions from the Planck CMB measurements considered here on the amplitudes in the \\texttt{PCHIP}\\  approach, together with the one-dimensional posterior probability distributions.}\n\\end{figure} \n\nIn order to further assess our findings above, we adopt here two information criteria \nwhich have been widely exploited\nin astrophysical and cosmological contexts (see Refs.~\\cite{Liddle:2007fy,Trotta:2008qt} for details), namely the frequentist Akaike Information Criterion (AIC)\n\\begin{equation}\n\\textrm{AIC}\\equiv -2 \\ln \\mathcal{L }_{\\rm{max}} +2k~,\n\\end{equation}\nwhich establishes that the penalty term between competing models is twice the number of free parameters in the model, $k$; and the Bayesian Information Criterion (BIC)\n\\begin{equation}\n\\textrm{BIC}\\equiv -2 \\ln \\mathcal{L }_{\\rm{max}} +k\\ln N~,\n\\end{equation}\nin which the penalty is proportional to the number of free parameters in the model times the logarithm of the number of data points $N$.\nThe best model is the one minimizing either the AIC or the BIC criteria. \nFollowing Ref.~\\cite{Liddle:2007fy}, the significance against a given model will be judged based on the Jeffreys' scale, which will characterize a difference $\\Delta$AIC (BIC)$>5$ ($>10$) as a strong (decisive) evidence against the cosmological model with higher AIC (BIC) value.\n\nAdopting first the AIC prescription, we shall compare the different models explored here to the standard scenario, in which reionization is described via just only one parameter, $\\tau$. \nThis \\emph{tau-only} cosmological model gives $-2\\ln \\mathcal{L }_{\\rm{max}}=12956.2$~\\cite{Ade:2015xua}.\nAs a comparison, the PCA-A case with constant fiducial model $x_{e}=0.15$ ($0.05$) provides $-2\\ln \\mathcal{L }_{\\rm{max}}=12954.0$ ($12953.2$). \nNotice that both the PCA-A cases have a higher AIC value than the \\emph{tau-only} cosmology because of the larger number of parameters.\nThe values for $\\Delta$AIC are $\\Delta$AIC $=5.8$ and $5$, respectively, and therefore there is strong evidence against these possible reionization histories.\nAlso within the PCA-A description, we get $-2\\ln \\mathcal{L }_{\\rm{max}}=12956.5$ ($12958.3$) in the PCA-A \\emph{tanh} (\\emph{asym}) fiducial approach.\nThese two models also provide a larger AIC than the \\emph{tau-only} scenario, and again, there will be strong (decisive) evidence against the PCA-A \\emph{tanh} (\\emph{asym}), in favor of the simplest and most economical \\emph{tau-only}  reionization paradigm.\nIn the case of the \\texttt{PCHIP}\\ approach, our results lead to $-2\\ln \\mathcal{L}_{\\rm{max}}=12954.5$, which \nalso indicates strong preference for the \\emph{tau-only} scheme. We point out that all the reported values of $-2\\ln \\mathcal{L }_{\\rm{max}}$\nare taken from the corresponding MCMC chains, and not from a specific minimization algorithm.\nFor this reason, they may not be extremely precise and they must be considered only as fair estimates of the true values of each $-2\\ln \\mathcal{L }_{\\rm{max}}$, with possible errors of order unity, as estimated from the different parallel MCMC chains.\nIn the case of the PCA-B parameterizations,\nthe difference in the minimum $-2\\ln \\mathcal{L }_{\\rm{max}}$ from the different MCMC parallel chains is too large to give even a fair estimate of the true minimum,\nand we decide not to claim any evidence against these two descriptions, for the reasons listed above.\nHowever, we expect that these two models are equally good in fitting the CMB data, at a comparable level with respect to the \\emph{tau-only} scenario, as their reionization histories are extremely close to the standard cosmological framework, see Fig.~\\ref{fig:xe}.\nNevertheless, given the fact that the number of parameters in the PCA-B scheme is larger, the \\emph{tau-only} reionization description, with current data, will always be favored over the PCA-B parameterization.\n\nWe can also compare the different reionization descriptions among themselves using the BIC approach, as all of them have the same number of free parameters (five in total) and also the same number of data points.\nThe result of comparing the PCA-A and \\texttt{PCHIP}\\ scenarios among themselves will always give very weak or inconclusive answers, as none of them in particular is preferred over the other possible formulations.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nUnraveling the reionization period, which is still a poorly known period in the evolution of our universe, is one of the most important goals of current and future cosmological probes.\nThis is a mandatory step, not only towards a complete understanding of star formation and evolution, but also to answer questions such as the nature of the dark matter component~\\cite{Barkana:2001gr,Yoshida:2003rm,Somerville:2003sh,Yue:2012na,Pacucci:2013jfa,Mesinger:2013nua,Schultz:2014eia,Dayal:2015vca,Lapi:2015zea,Bose:2016hlz,Bose:2016irl,Corasaniti:2016epp,Menci:2016eui,Lopez-Honorez:2017csg,Villanueva-Domingo:2017lae}, constraining dark matter properties or the abundance of accreting massive primordial black holes~\\cite{Pierpaoli:2003rz,Mapelli:2006ej,Natarajan:2008pk,Natarajan:2009bm,Belikov:2009qx,Huetsi:2009ex,Cirelli:2009bb,Kanzaki:2009hf,Natarajan:2010dc,Giesen:2012rp,Diamanti:2013bia,Lopez-Honorez:2013lcm,Lopez-Honorez:2016sur,Poulin:2016nat,Poulin:2015pna,Ricotti:2007au,Horowitz:2016lib,Ali-Haimoud:2016mbv,Blum:2016cjs,Poulin:2017bwe}.\nCurrently, the most accurate measurement of the reionization period comes from Cosmic Microwave Background data through a redshift-integrated quantity: the reionization optical depth $\\tau$.\nThe latest measurements of the Planck collaboration provide a value of $\\tau = 0.055 \\pm 0.009$~\\cite{Aghanim:2016yuo, Adam:2016hgk}, which shows a very good agreement with observations of Lyman-$\\alpha$ emitters at $z\\simeq 7$~\\cite{Stark:2010qj,Treu:2013ida,Pentericci:2014nia,Schenker:2014tda,Tilvi:2014oia}.\nHowever, this measured value of $\\tau$ may correspond to very different reionization histories.\n\nThe most commonly exploited model for the time evolution of the free electron fraction, $x_e(z)$, uses a step-like transition, implemented via a hyperbolic tangent~\\cite{Lewis:2008wr}.\nRecently, there have been several studies in the literature claiming that Planck 2015 data may prefer a high-redshift ($z>15$) component to the reionization optical depth, implying a clear departure from the hyperbolic tangent picture.\nHere we consider a number of possible reionization scenarios, some of them previously explored in the literature, such as the Principal Component Analysis (PCA) approach of  Refs.~\\cite{Hu:2003gh,Mortonson:2007hq,Mortonson:2007tb,Mortonson:2008rx,Mortonson:2009qv,Mortonson:2009xk,Mitra:2010sr,Heinrich:2016ojb}, or the \\texttt{PCHIP}\\ framework~\\cite{Hazra:2017gtx}. \nWe find that the claimed need for an early reionization component from present data is highly debatable, as it is only motivated by a particular set of reionization descriptions.\nIn other possible reionization prescriptions, equally allowed by data, we do not find such a preference.\nTo assess this, we have applied the frequentist Akaike Information Criterion (AIC), which provides an unbiased model comparison method.\nThe AIC results show that there is strong evidence from current data against more complicated reionization scenarios, always favoring the minimal  scenario with the symmetric hyperbolic tangent function and described by one single parameter, the reionization optical depth $\\tau$. In other words, current Planck CMB analyses are unable to provide more information beyond that based on a single value of the $\\tau$.  Upcoming data from the Planck mission will help in further disentangling the reionization history of our universe. \n\n\n\\acknowledgments\nThis work makes use of the publicly available \\texttt{CosmoMC}~\\cite{Lewis:2002ah} and \\texttt{CAMB}~\\cite{Lewis:1999bs} codes and of the Planck data release 2015 Likelihood Code~\\cite{Aghanim:2015xee}. OM and PVD would like to thank the Fermilab Theoretical Physics Department for hospitality.\nOM and PVD are supported by PROMETEO II\/2014\/050, by the Spanish Grants SEV-2014-0398 and FPA2014--57816-P of MINECO and by the European Union's Horizon 2020 research and innovation program under the Marie Sk\\l odowska-Curie grant agreements No.\\ 690575 and 674896.   \nThe work of SG was supported by the Spanish grants\nFPA2014-58183-P,\nMultidark CSD2009-00064 and\nSEV-2014-0398 (MINECO),\nand PROMETEOII\/2014\/084 (Generalitat Valenciana).\n\nThis manuscript has been authored in part by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. This work made extensive use of the NASA Astrophysics Data System and {\\tt arXiv.org} preprint server.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe ALICE experiment at the LHC is dedicated to the study of strongly interacting matter under extreme conditions, i.e. high temperature, which can be reached in heavy-ion collisions. In such collisions, the formation of a Quark-Gluon Plasma (QGP) is expected. Dielectrons are produced at all stages of the collision and therefore carry information about the whole evolution of the system. Since they do not interact strongly with the medium, they are a prime probe to study the properties of the QGP. Dielectrons stem from decays of pseudoscalar and vector mesons, from semi-leptonic decays of correlated open-charm and open-beauty hadrons and from internal conversions of direct photons. In heavy-ion collisions, additional sources are expected, i.e. thermal radiation from the QGP and hadron gas. The medium introduces modifications of the vector meson properties, in particular the short-lived $\\rho$, related to chiral symmetry restoration. In addition, the initial conditions of the collisions are expected to change compared to elementary collisions due to modifications of the parton distribution functions in nuclei. The latter can be studied in proton-lead (p--Pb) collisions, whereas pp collisions provide an important vacuum baseline. It is crucial to first understand the dielectron production in vacuum to single out the signal characteristics of the QGP. Moreover, proton-proton (pp) collisions can also be used to study the heavy-flavour (HF) and direct photon production.\n\nIn the following, the steps of the data analysis are explained and the first measurements of the dielectron production in pp collisions at $\\sqrt{s} = 7$\\,TeV are presented and discussed~\\cite{ref-ee}.\n\n\n\\section{Data analysis and results}\n\n\nThe analysis is performed with pp data taken during the first data-taking period of the LHC in 2010 with the ALICE detector. The integrated luminosity of the data sample is $L_{\\rm int} = 6.0\\pm0.2$\\,nb$^{-1}$.\nAfter identifying electrons in the ALICE detector it is not a priori clear which electrons belong to the same pair. We follow a statistical approach to obtain the final spectrum. The electrons and positrons are combined to an opposite-sign spectrum (OS), which includes not only the signal but also background, that can be purely combinatorial or have some residual correlation from jets or cross pairs from double Dalitz decays. This background is estimated by the same-sign spectrum (SS). Residual acceptance differences for OS and SS pairs are estimated with mixed events and taken into account during the subtraction of the background. Finally, the background-subtracted spectrum is corrected for tracking and particle identification inefficiencies within the ALICE acceptance ($p_{\\rm T,e} > 0.2$\\,GeV\/$c$, $ \\eta_{\\rm e}<0.8 $).\n\nIn Fig. 1 the measured dielectron cross section as a function of $m_{\\rm ee}$ is compared to a so-called hadronic cocktail, which includes all known sources of dielectron production from hadron decays and uses parameterisations of measured spectra as input when available. Where no measurements are available $m_{\\rm T}$-scaling~\\cite{ref-mt-scaling} is applied. The HF contributions are simulated using the Perugia2011 tune of PYTHIA~6~\\cite{ref-pythia,ref-pythia2011}. The resulting dielectron pairs from the hadron decays are then filtered through the ALICE acceptance.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.355]{.\/2018-09-03-2018-09-03-invmassintegrated.pdf}\n\\caption{Dielectron cross section as a function of $m_{\\rm ee}$ compared to a cocktail of known hadronic sources.}\n\\end{figure}\n\nA good agreement is observed between the cocktail and the data. The charm contribution already dominates the spectrum for $m_{\\rm ee} \\geq 0.5$\\,GeV\/$c^{2}$. The very large heavy-flavour contribution makes the measurement of thermal radiation from the medium in heavy-ion collisions very challenging at LHC energies. To separate the heavy-flavour background from thermal radiation from the QGP in a future heavy-ion run in the intermediate-mass range (IMR, $\\phi < m_{\\rm ee} < J\/\\psi$), an additional variable, the pair-distance-of-closest-approach ($\\rm DCA_{ee}$), is added to the traditional analysis as a function of $m_{\\rm ee}$ and $p_{\\rm T,ee}$. $\\rm DCA_{ee}$~is defined as:\n\\begin{equation}\n{\\rm DCA_{ee}} = \\sqrt{\\frac{({\\rm DCA_{{\\it xy},1}}\/\\sigma_{xy{ \\rm ,1}})^{2}+({\\rm DCA_{{\\it xy},2}}\/\\sigma_{xy,2})^{2}}{2}}\n\\end{equation}\n\nHere ${\\rm DCA}_{xy,i}$ is the closest distance between the reconstructed electron track and the primary collision vertex in the transverse plane. Its resolution is estimated from the covariance matrix of the track reconstruction and denoted as $\\sigma_{xy,i}$. In the case of weak decays, the decay electron candidates do not point to the vertex which leads to a broader DCA distribution than for tracks from prompt decays.\nThis can be seen in Fig. 2 and Fig. 3, where the $\\rm DCA_{ee}$~spectra are shown for two invariant mass regions. Fig. 2 shows the mass region between the $\\pi^{0}$ and the $\\phi$ mass. The light flavour template is taken from the $\\pi^{0}$ shape, normalised to the expected contribution from all light flavour sources. Fig. 3 shows the mass region around the $J\/\\psi$ mass peak. In both mass regions we can see a clear peak which can be described by the expected prompt contributions, whereas the tail of the spectrum is described by the broader contributions from charm and beauty.\n\\begin{figure}\n\\centering\n  \\begin{minipage}{0.47\\textwidth}\n    \\includegraphics[scale=0.35]{.\/2018-May-09-resonancedca.pdf}\n    \\caption{Dielectron spectrum as a function of $\\rm DCA_{ee}$~for $0.14 < m_{\\rm ee} < 1.1$\\,GeV\/$c^2$~\\cite{ref-ee}.}\n  \\end{minipage}\n  \\begin{minipage}{0.47\\textwidth}\n    \\includegraphics[scale=0.35]{.\/2018-May-09-jpsidca.pdf}\n    \\caption{Dielectron spectrum as a function of $\\rm DCA_{ee}$~for $2.7 < m_{\\rm ee} < 3.3$\\,GeV\/$c^2$~\\cite{ref-ee}.}\n  \\end{minipage}\n\\end{figure}\nIn Fig. 3 the $J\/\\psi$ from $B$-mesons  can be seen in addition to the open HF contributions.\nIn the so-called intermediate mass region, located between the $\\phi$ and $J\/\\psi$ in the mass spectrum, the dominant contribution is from open HF.\nThe dielectron cross section as function of $p_{\\rm T,ee}$ and $\\rm DCA_{ee}$~is compared to a hadronic cocktail using PYTHIA 6 Perugia0~\\cite{ref-pythia2011} to estimate the $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ contributions in the left and right panels of Fig. 4, respectively.\n\\begin{figure}\n\\centering\n  \\begin{minipage}{0.47\\textwidth}\n    \\includegraphics[scale=0.35]{.\/2018-May-09-heavyflavourptee.pdf}\n  \\end{minipage}\n  \\begin{minipage}{0.47\\textwidth}\n    \\includegraphics[scale=0.35]{.\/2018-May-09-heavyflavourdca.pdf}\n  \\end{minipage}\n\\caption{Dielectron cross section as a function of $p_{\\rm T,ee}$ (left) and $\\rm DCA_{ee}$~(right) in the IMR compared to a cocktail calculated with PYTHIA~6~\\cite{ref-ee}.}\n\\end{figure}\nThe data are well described by the hadronic cocktail within the statistical and systematic uncertainties. The contribution from $\\rm c\\bar{c}$ dominates the dielectron yield at low $p_{\\rm T,ee}$ and relatively small $\\rm DCA_{ee}$, whereas the $\\rm b\\bar{b}$ becomes relevant at high $p_{\\rm T,ee}$ and large $\\rm DCA_{ee}$.\nTo investigate the processes of heavy-quark production we changed the generator from PYTHIA to POWHEG, switching from leading order in the HF quark generation to next-to-leading order. To quantify the differences the total $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ cross sections are extracted from the data by fitting the results two-dimensionally as a function of $p_{\\rm T,ee}$ and $m_{\\rm ee}$ and one-dimensionally as a function of $\\rm DCA_{ee}$~in the IMR allowing the contributions of the two HF components to vary. The results are shown in the left and right panels of Fig. 5  for PYTHIA and POWHEG\\cite{ref-powheg}, respectively.\nBoth fits give consistent results for a given MC event generator. The uncertainties are fully correlated between the cross sections extracted with PYTHIA and POWHEG. Significant model dependences are observed which reflect the different rapidity distribution of charm quarks and different $p_{\\rm T,ee}$ spectra of the $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ contributions predicted by the two models.\n\\begin{figure}\n\\centering\n  \\begin{minipage}{0.47\\textwidth}\n    \\includegraphics[trim={0, 0, 0, 1.5cm},clip,scale=0.357]{.\/2018-May-09-oneSigmaPythiaDCA0to8.pdf}\n  \\end{minipage}\n  \\begin{minipage}{0.47\\textwidth}\n    \\includegraphics[trim={0, 0, 0, 1.5cm},clip,scale=0.357]{.\/2018-May-09-oneSigmaPowhegDCA0to8.pdf}\n  \\end{minipage}\n\\caption{Total $\\rm c\\bar{c}$ and $\\rm b\\bar{b}$ cross sections with their systematic and statistical uncertainties, extracted from a fit of the measured dielectron yield from heavy-flavour hadron decays in ($m_{\\rm ee}$, $p_{\\rm T,ee}$) and in $\\rm DCA_{ee}$ with PYTHIA (left) and POWHEG (right) are compared to published cross sections (lines)~\\cite{ref-ee}.}\n\\end{figure}\nThe results are compared to independent measurements of $\\sigma_{\\rm c\\bar{c}}$\\cite{ref-ccbar} and $\\sigma_{\\rm b\\bar{b}}$\\cite{ref-bbbar} from single heavy-flavour particle spectra and found to be consistent within the large uncertainties. Once these uncertainties are reduced, the dielectron measurements can give further constraints on the MC event generators aiming to reproduce the heavy-flavour production mechanisms.\n\n\\section{Conclusion}\n\nTo summarise, ALICE measured the dielectron cross sections as a function of $m_{\\rm ee}$, $p_{\\rm T,ee}$, and $\\rm DCA_{ee}$~in pp collisions at $\\sqrt{s} = 7$\\,TeV. The hadronic cocktail is in good agreement with the measured  dielectron cross sections in the three discussed observables, which suggests a good understanding of the dielectron cross section in the ALICE acceptance. We show that $\\rm DCA_{ee}$~makes it possible to separate prompt from non-prompt dielectron pairs, and thus will be a key tool to determine the average temperature of the QGP formed in heavy-ion collisions in the future. In the heavy flavour sector we can report a significant dependence of the total cross sections of charm and beauty when using PYTHIA and POWHEG, which reflects the sensitivity of the dielectron measurement to the underlying heavy-flavour production mechanisms implemented in the models.\n\n\n\n\\bibliographystyle{unsrt}  \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec-1}\n\nThis technical report is an extension of the paper of the same title, which is to appear at MUCOCOS'13. The technical report proves correctness of the ELB-trees operations' semantics and that the operations are lock-free.\n\nThe following is a brief summary of the design of the datastructure, which is detailed in section 3 of the paper.\nAll ELB-trees have a permanent root node $r$ with a single child.\nELB-trees are $k$-ary leaf-oriented search tree, or multiway search trees, so internal nodes have up to $k$ children and $k-1$ keys. An ELB-trees contain a set $E_r$ of integer keys in the range $(0;2^{63})$. The key 0 is reserved. Keys have an additional read-only bit: when the read-only bit is set, the key cannot be written to.  ELB-trees offer 3 main operations:\n\\begin{itemize}\n\\item Search($e_1$, $e_2$) returns a key $e$ from $E_r$ satisfying $e_1 \\le e \\le e_2$, if such a key exists. Otherwise it returns $0$.\n\\item Remove($e_1$, $e_2$) removes and returns a key $e$ from $E_r$ satisfying $e_1 \\le e \\le e_2$, if such a key exists. Otherwise it returns $0$.\n\\item Insert($e$) adds $e$ to $E_r$, if $e$ was not in $E_r$ before.\n\\end{itemize}\nELB-trees can also be used as dictionaries or priority queues by storing values in the least significant bits of the keys.\n\nThe operations of ELB-trees cannot generally be expressed as atomic operations, as they occur over a time interval. As a consequence, series of concurrent operations cannot generally be expressed as ocurring serially, that is the semantics are not linearizable.\nHowever, the set $E_r$ is atomic.\n$E_r$ is the union of the keys in the leaf nodes of the ELB-tree.\nThe keys in internal nodes guide tree search.\n\nSection 2 provides formal definitions for terms used throughout the proof.\nThe proof starts in Section 3 by proving that ELB-trees are leaf-oriented search trees.\nWe prove through induction, that \nELB-trees are leaf-oriented search trees initially, and that all operations maintain that property.\nThe inductive step is assisted by two significant subproofs:\n\\begin{enumerate}\n\\item Rebalancing does not change the keys in $E_r$.\n\\item The keys in leaf nodes are within a permanent range.\n\\end{enumerate}\n\nThese properties hold due to the behavior of rebalancing.\nThe first subproof shows that rebalancing is deterministic, even when concurrent.\nThe second shows that leaf nodes have a range of keys they may contain and it never changes.\n\nGiven these properties, Section 4 derives the operations' semantics.\nSection 5 follows up by proving that the operations are lock-free.\nFirst we prove that some operation has made progress whenever a node is rebalanced.\nNext we prove that some operation has made progress whenever any part of an operation is restarted.\n\nSection 6 concludes the technical report with a summary.\n\\section{Definitions}\n\\label{sec-2}\n\nThis section introduces definitions used in the following proofs of the ELB-trees' properties. The definitions start with the terms used, before moving on to the contents and properties of nodes. Finally the intitial state of ELB-trees is formally defined.\n\nLet $L$ be the set of leaf ndoes, $I$ the set of internal nodes, and $T$ the set of points in time. The sets are disjoint.\n\nNodes contain:\n\\begin{description}\n\\item[$C_i (t)$] list of children of internal node $i$ at time $t$\n\\item[$S_i (t)$] list of keys in internal node $i$ at time $t$\n\\item[$E_n (t)$] keys represeted by the node $n$ where at time $t$:\\begin{center}$E_n (t) =$ \\begin{math} \\left\\{\n     \\begin{array}{lr}\n       $Non-zero keys in $l & n \\in L \\\\\n       \\bigcup _{c \\in C_i (t)} E_c (t) & : n \\in I\n     \\end{array}\n   \\right. \\end{math}\\end{center} \n\\end{description}\n\nThe following node properties can be derived from their content:\n\\begin{description}\n\\item[$D_n (t)$] the descendants of node $n$ at time $t$: \\\\ $D_n (t)$ = \\begin{math} \\left\\{\n     \\begin{array}{lr}\n       \\emptyset & : n \\in L\\\\\n       C_n (t) \\cup \\bigcup _{d \\in C_n(t)} D_d (t) & : n \\in I\n     \\end{array}\n   \\right.\\end{math} \\\\ $n$ is reachable when $reachable_n (t) \\equiv n \\in (\\{r\\} \\cup D_r (t))$\n\\item[$parent_n (t)$] the parents of node $n$: \\\\ $parent_n (t) = \\{i \\in reachable_r (t) | n \\in C_i (t)\\}, t \\in T$\n\\end{description}\n\nInitially $r$ has one child $C_r (0) = \\langle ic \\rangle$, and one grandchild $C_{ic} (0) = \\\\ \\langle ln \\rangle$.\nThe grandchild is an empty leaf node $E_{ln} (0) = \\emptyset \\wedge E_r (0) = \\emptyset$.\n\n\\section{Search tree proof}\n\\label{sec-3}\n\nThis section proves that ELB-trees are $k$-ary leaf-oriented search trees.\nIn such a tree, all nodes except the root have one parent, and all internal nodes have strictly ordered keys.\nSpecifically the $i$'th key in a node provides an upper bound for the $i$'th child of the node, and a lower bound for the $i + 1$'th child.\nThe key ordering is formally expressed as: \n\\begin{center} $W_i (t) \\equiv \\forall j \\in [0;C_i (t)). E_{{C_i(t)} _t} \\subseteq (0; {S_i}_j] \\wedge E_{{C_i (t)} _t} \\subseteq ({S_i} _j; 2^{63})$ \\end{center}\nThe tree property is formally expressed as:\n\\begin{center} $\\forall n \\in reachable_n (t). \\left\\vert parent_n (t) \\right\\vert = 1 \\vee n = r$ \\end{center}\nThe properties are proven inductively, but doing so requires several intermediate steps.\nTo begin with, we will show that the behavior of rebalancing of search trees is deterministic, and does not change $E_r$.\n\n\\begin{lemma}\\label{ro-reb}\nUnbalanced nodes and their parent are read-only while rebalancing. \\end{lemma}\n\\begin{proof} While finding the nodes involved in rebalancing, they are made read-only: internal nodes are made read-only by setting their status field, and\nleaf nodes are made read-only by setting the read-only bit of all their keys, see Figure~16 in the paper.\\end{proof}\n\n\\begin{lemma}\\label{reb-nodes}\nIf $W_r$ holds and the unbalanced nodes' parent is still reachable, all threads can find the nodes involved in a rebalancing from the status field of the unbalanced nodes grandparent, . \\end{lemma}\n\\begin{proof} The status field stores the key of the unbalanced node and its parent.\nSince $W_r$ holds, the nodes can be found by searching for the key in the grandparent and parent of the unbalanced node.\\end{proof}\n\n\\begin{lemma}\\label{inv-detreb}\nRebalancing completes deterministically exactly once, if $W_r$ holds. \\end{lemma}\n\\begin{proof} Rebalancing finds the involved nodes (Lemma \\ref{reb-nodes}) and decides how to rebalance (Lemma \\ref{ro-reb}) determinstically. The parent is replaced, and the grandparent's status field is cleared using ABA safe CAS operations, see Section~3b of the paper. The grandparent has the status field \\{*,*,*,STEP2\\} when replacing the parent, ensuring that the grandparent is reachable when replacing the parent node.\\end{proof}\n\n\\begin{lemma}\\label{Er-reb}\n$E_r (t)$ does not change when rebalancing, if $W_r$ holds. \\end{lemma}\n\\begin{proof} The content of balanced nodes and their new parent is copied from the old nodes, while their content is read-only (Lemma \\ref{ro-reb}).\\end{proof}\n\nThe preceding lemmas show that rebalancing is well-behaved in search trees. The following lemmas will show that all operations maintain the tree property and $W_r$.\n\n\\begin{lemma}\\label{inv-tree}\nAll operations maintain the tree property, if $W_r$ holds. \\end{lemma}\n\\begin{proof} $descendants_n$ only changes when rebalancing. Specifically, $descendants_n$ changes when replacing an internal node $op$ with a new node $np$. \nThe children of $op$ had $op$ as their only parent, so all the children $np$ and $op$ share, will have $np$ as their only parent after rebalancing. The new children have $np$ as their only parent, because they have just been introduced, and the descendants of the new nodes have their parents replaced. Formally: \\begin{center} $(\\forall c \\in C_{op} (t_1). parent_c (t_1) = \\{op\\}) \\Rightarrow \\forall c \\in C_{np} (t_2). parent_c (t_2) = \\{np\\}$ \\end{center}\\end{proof}\n\n\\begin{lemma}\\label{lrange}\nLeaf nodes $l$ have a permanent range $R_l$ of keys they may contain, if $W_r$ holds.\\end{lemma}\n\\begin{proof} The lower bound is given by the keys of its ancestors. The ancestors change deterministically when $W_r$ holds (Lemma \\ref{inv-detreb}). Although the ancestors may change, their replacements use the same keys. \nInternal node keys are only introduced or removed when splitting and merging nodes, which results in two or three new nodes. \nWhen rebalancing results in two new nodes, the new parent has one less key. When rebalancing results in three new nodes, the new parent has one updated or additional key, which the old parent did not have. The updated or new key is copied from its the unbalanced nodes, so it only affects the new nodes. \\end{proof}\n\n\\begin{lemma}\\label{res-si}\nIf $W_r$ holds, the leaf node $l$ reached by $Search(e, e)$ satisfies: $W_r \\Rightarrow e \\in R_l$. \\end{lemma}\n\\begin{proof} Search visiting a node $n$ where $\\neg reachable_n (t)$ eventually restarts, so a terminating search only visits reachable nodes in the tree (Lemma \\ref{inv-tree}). Search of reachable nodes when $W_r$ holds is regular $k$-ary tree search.\\end{proof}\n\n\\begin{lemma}\\label{res-sl}\nIf $W_r$ holds, searching the leaf node $l$ from $t_{l1}$ to $t_{l2}$ must read the keys $O(t_{l1}, t_{l2}) \\cap R_l$. \\end{lemma}\n\\begin{proof} $l$ is read after a memory barrier, ensuring that $O(t_{l1}, t_{l2}) \\cap R_l$ are read.\\end{proof}\n\n\n\\begin{lemma}\\label{inv-Wr}\nAll writes to the tree maintain $W_r$. Formally: \\begin{center} $\\forall t_1, t_2 \\in T . (t_1 \\le t_2 \\wedge W_r (t_1)) \\Rightarrow W_r (t_2)$ \\end{center}\\end{lemma}\n\\begin{proof} Writes to the tree can be classified into: key insertion, key removal, and rebalancing. \nRebalancing maintains $W_r$ (Lemma \\ref{lrange}).\nKey removal and insertion only affects the keys in the tree.\n$remove(e_1, e_2, t_1, t_2)$ removes an key from a leaf node $l$, which maintain $W_r$. \n$insert(e, t_1, t_2)$ inserts into leaf nodes for which $\\forall t \\in T. W_r (t) \\Rightarrow e \\in R_l$ (Lemma \\ref{res-si}), which maintain $W_r$. \\end{proof}\n\n\\begin{theorem}\\label{lost}\nELB-trees are leaf-oriented search trees. \\end{theorem}\n\\begin{proof} ELB-trees are trees and $W_r$ holds initially. All operation on ELB-trees maintains the tree property (Lemma \\ref{inv-tree}) and $W_r$ (Lemma \\ref{inv-Wr}).\\end{proof}\n\nThis section proves that ELB-trees are leaf-oriented search trees. Such proofs are sufficient to derive the semantics of concurrent searches and serial insertions and removals. The next section will derive the semantics of the concurrent operations, which requires a few additional lemmas.\n\n\\section{Correctness}\n\\label{sec-4}\n\nThis section derives the semantics of the operations. But first we will introduce some terms to reason about the results of such operations. Let: \n\\begin{description}\n\\item[$search(e_1, e_2, t_1, t_2)$] be the result of a search operation matching against keys $e \\in [e_1;e_2]$ starting at $t_1$ and ending at $t_2$;\n\\item[$remove(e_1, e_2, t_1, t_2)$] be the result of a remove operation matching against keys $e \\in [e_1;e_2]$ starting at $t_1$ and ending at $t_2$;\n\\item[$insert(e, t_1, t_2)$] be an insert $e$ operation starting at $t_1$ and ending at $t_2$;\n\\item[$O(t_1, t_2)$] be the keys that were in $E_r$ at all times during $[t_1;t_2)$: \\begin{center} $O(t_1, t_2) = \\left\\{ e |  \\forall t \\in [t_1;t_2) . e \\in E_r (t)  \\right\\}$; and \\end{center}\n\\item[$U(t_1, t_2)$] be the keys that were in $E_r$ at some time during $[t_1;t_2)$: \\begin{center} $U(t_1, t_2) = \\left\\{ e |  \\exists t \\in [t_1;t_2). e \\in E_r (t)  \\right\\}$. \\end{center}\n\\end{description}\n\nWe first prove properties of search operations, then derive the operations' semantics:\n\n\\begin{lemma}\\label{sl}\nSearching a set of leaf nodes $RL$ from $t_1$ to $t_2$ reads the keys $\\bigcup _{l \\in RL} R_l \\cap O(t_1, t_2)$. \\end{lemma}\n\\begin{proof}\nThe search reads the keys $\\bigcup _ {l \\in RL} R_l \\cap O(t_{l1}, t_{l2})$ (Lemma \\ref{res-sl}). $\\forall l \\in RL . O(t_{l1}, t_{l2}) \\subseteq O(t_1, t_2)$ holds, as any key in the tree during $t_1$ to $t_2$ must have been in the tree for all fragments of that duration.\n\\end{proof}\n\n\\begin{theorem} $search(e_1, e_2, t_1, t_2)$ can only return $0$ (fail) if there are no matching entries in $E_r$ at all times during $[t_1, t_2)$: \\begin{center} $search(e_1, e_2, t_1, t_2) = 0 \\Rightarrow [e_1; e_2] \\cap O(t_1, t_2) = \\emptyset$ \\end{center} \\end{theorem}\n\\begin{proof}\n$search(e_1, e_2, t_1, t_2) = 0$ implies that a set of leaf nodes $RL$ have been searched, where $[e_1;e_2] \\subseteq \\bigcup _{l \\in RL} R_l$. If there was an key in $[e_1;e_2] \\cap O(t_1, t_2)$ it would have been read (Theorem \\ref{lost}, Lemma \\ref{sl}).\\end{proof}\n\n\\begin{theorem} Successful searches return a matching key that was in $E_r$ at some point in time during $[t_1;t_2)$: \\begin{center}$e = search(e_1, e_2, t_1, t_2) \\Rightarrow (e \\in U(t_1, t_2) \\wedge e \\in [e_1; e_2])$\\end{center} \\end{theorem}\n\\begin{proof} Successful searches return a key $e$ that was read from a leaf. Since $e$ was read it must have been in $E_r$ (Lemma \\ref{sl}).\\end{proof}\n\n\n\\begin{theorem} Remove can only return $0$ (fail) if there are no matching entries in $E_r$ at all times during $[t_1, t_2)$:\n\\begin{center} $remove(e_1, e_2, t_1, t_2) = 0 \\Rightarrow O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset$. \\end{center}\\end{theorem}\n\\begin{proof} Terminating remove operations that return $0$ have searched a set of leafs $RL$ satisfying $[e_1; e_2] \\subseteq \\bigcup _{l \\in RL} R_l$ (Lemma \\ref{sl}), so any keys in $O(t_1, t_2) \\cup [e_1;e_2]$ would have been read.\\end{proof}\n\n\\begin{theorem} Successful remove operations remove matching a key $e$ from $E_r$ that was in $E_r$ at some point in time during $[t_1;t_2)$: \\begin{center} $e = remove(e_1, e_2, t_1, t_2) \\ne 0 \\Rightarrow$ \\ $(e_1 \\le e \\le min(O(t_1, t_2) \\cap [e_1 ; e_2]) \\le e_2 \\wedge e \\in U(t_1, t_2))$ \\end{center} \\end{theorem}\n\\begin{proof} Terminating remove operations have searched a set of leafs $RL$ satisfying $[e_1; e] \\subseteq \\bigcup _{l \\in RL} R_l$ (Lemma \\ref{sl}). Any keys smaller than $e$ in $O(t_1, t_2) \\cup [e_1;e_2]$ would have been read.\\end{proof}\n\n\\begin{theorem} $insert(e, t_1 , t_2)$ adds $e$ to the $E_r$, if $e \\notin U (t_1 , t_2 )$. \\end{theorem}\n\\begin{proof} Insert operations terminate when they use a successful CAS operation to write the key into an empty key of a leaf node $l$ where $e \\in R_l$ (Lemma \\ref{res-si}). The CAS operations success implies the key is not read-only, and hence $reachable_l (t_2)$.\\end{proof}\n\nTheorem 2-6 can be summarized as: \\\\\n$e = search( e_1, e_2, t_1, t_2 ) \\Rightarrow$ \\begin{math} \\left\\{\n     \\begin{array}{lr}\n       O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n       e_1 \\le e \\le e_2 \\wedge e \\in U(t_1, t_2) & : e \\neq 0\n     \\end{array}\n   \\right. \\end{math}\n\\\\$e = remove(e_1, e_2, t_1, t_2) \\Rightarrow$ \\begin{math} \\left\\{\n     \\begin{array}{lr}\n        O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n       e_1 \\le e \\le min([e_1; e_2] \\cap O(t_1, t_2)) \\\\ ~ \\wedge e \\in U(t_1, t_2)  & \\raisebox{11pt}{$: e \\neq 0$}\n     \\end{array}\n   \\right. \\end{math}\n\\ $insert(e, t_1, t_2)$ adds $e$ to $E_r$, if $e \\notin U(t_1, t_2)$.\n\n\n\\section{Lock-freedom}\n\\label{sec-5}\n\nLock-freedom guarantees that as long as some thread is working on an operation $o_1$, some operation $o_2$ is coming closer to terminating. In this case we say $o_1$ is causing progress, and $o_2$ is making progress. The operations $o_1$ and $o_2$ can be different.\nFor ELB-trees, this means that whenever a thread is searching, inserting, or removing, some thread must be making progress. \nThe following is proof that the operations are lock-free:\n\n\\begin{lemma}\\label{ter-op}\nOperations eventually terminate or restart part of their operation. \\end{lemma}\n\\begin{proof} The operations' algorithms have loops in the following for: node search, tree search, rebalancing, and updating keys in leafs. The algorithms are given in the paper~\\cite{bkp13}. Without concurrency, they iterate up to K, tree height, tree height, and 1 times. With concurrency, tree search, rebalancing, and key update loops may restart part of their operation.\\end{proof}\n\n\\begin{lemma}\\label{lf-rebl}\nRebalancing leaf nodes cause progress. \\end{lemma}\n\\begin{proof} If the nodes are written to between deciding to rebalance and rebalancing,  some operation has made progress.\nIf there are no writes, the size of the first node is either D or S, resulting in balanced nodes of $size \\in [min(2 S, 0.5 D); D-1]$. Such nodes can be removed from and inserted into at least once before requiring additional rebalancing. As such, every time a rebalancing completes, one operation has made progress.\\end{proof}\n\n\\begin{lemma}\\label{lf-rebi}\nRebalancing internal nodes cause progress. \\end{lemma}\n\\begin{proof} Rebalancing internal nodes leads to child nodes that can be rebalanced at least one. Each leaf rebalancing cause progress (Lemma \\ref{lf-rebl}), hence each internal rebalancing cause progress. \\end{proof}\n\n\\begin{theorem}\\label{lf-s}\nSearch causes progress. \\end{theorem}\n\\begin{proof} Search eventually terminates, similar to $k$-ary tree search, or rebalances a node (Lemma \\ref{res-si}). In the first case the search operation is making progress. In the second case some operation is making progress (Lemma \\ref{lf-rebl}, Lemma \\ref{lf-rebi}).\\end{proof}\n\n\\begin{theorem}\\label{lf-ri}\nRemove and insert operations cause progress. \\end{theorem}\n\\begin{proof} The operations proceed as searches followed by writes to leaf nodes. The leaf node write takes a bounded number of steps, as each key may be read once, but the steps can be restarted due to rebalancing, or other insertions and removals terminating. In the first case, some operation is nearing termination, and in the second case some operation terminated (Lemma \\ref{lf-rebl}, Lemma \\ref{lf-rebi}).\\end{proof}\n\n\\section{Conclusion}\n\\label{sec-6}\n\nThis technical report has introduced, proved, and derived properties of ELB-trees. ELB-trees have been proven to be leaf-oriented search trees. Their operations' semantics have been derived as:\\\\\n$e = search( e_1, e_2, t_1, t_2 ) \\Rightarrow$ \\begin{math} \\left\\{\n     \\begin{array}{lr}\n       O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n       e_1 \\le e \\le e_2 \\wedge e \\in U(t_1, t_2) & : e \\neq 0\n     \\end{array}\n   \\right. \\end{math}\n\\\\$e = remove(e_1, e_2, t_1, t_2) \\Rightarrow$ \\begin{math} \\left\\{\n     \\begin{array}{lr}\n        O(t_1, t_2) \\cap [e_1 ; e_2] = \\emptyset & : e = 0 \\\\\n       e_1 \\le e \\le min([e_1; e_2] \\cap O(t_1, t_2)) \\\\ ~ \\wedge e \\in U(t_1, t_2)  & \\raisebox{11pt}{$: e \\neq 0$}\n     \\end{array}\n   \\right. \\end{math}\n\\ $insert(e, t_1, t_2)$ adds $e$ to $E_r$, if $e \\notin U(t_1, t_2)$.\nFinally the operations have been proven to be lock-free.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOur notation is standard (e.g., see \\cite{Bol98}, \\cite{CDS80}, and\n\\cite{HoJo88}); in particular, all graphs are defined on the vertex set\n$\\left\\{  1,2,...,n\\right\\}  =\\left[  n\\right]  $ and $G\\left(  n,m\\right)  $\nstands for a graph with $n$ vertices and $m$ edges. We write $\\Gamma\\left(\nu\\right)  $ for the set of neighbors of the vertex $u$ and set $d\\left(\nu\\right)  =\\left\\vert \\Gamma\\left(  u\\right)  \\right\\vert .$ Given a graph $G$\nof order $n,$ we assume that the eigenvalues of the adjacency matrix of $G$\nare ordered as $\\mu\\left(  G\\right)  =\\mu_{1}\\left(  G\\right)  \\geq...\\geq\n\\mu_{n}\\left(  G\\right)  $. As usual, $\\overline{G}$ denotes the complement of\na graph $G$ and $\\omega(G)$ stands for the clique number of $G.$\n\nNosal \\cite{Nos70} showed that for every graph $G$ of order $n,$\n\\begin{equation}\nn-1\\leq\\mu\\left(  G\\right)  +\\mu\\left(  \\overline{G}\\right)  <\\sqrt{2}n.\n\\label{Nosin}%\n\\end{equation}\nQuite of attention has been given to second of these inequalities. In\n\\cite{Nik02} it was shown that%\n\\begin{equation}\n\\mu\\left(  G\\right)  +\\mu\\left(  \\overline{G}\\right)  \\leq\\sqrt{\\left(\n2-\\frac{1}{\\omega(G)}-\\frac{1}{\\omega(\\overline{G})}\\right)  n\\left(\nn-1\\right)  }, \\label{Nikin}%\n\\end{equation}\nimproving earlier results in \\cite{Hon95}, \\cite{HoSh00}, \\cite{Li96}, and\n\\cite{Zho97}. Unfortunately inequality (\\ref{Nikin}) is not much better then\n(\\ref{Nosin}) when both $\\omega(G)$ and $\\omega(\\overline{G})$ are large\nenough. Thus, it is natural to ask whether $\\sqrt{2}$ in (\\ref{Nosin}) can be\nreplaced by a smaller absolute constant for $n$ sufficiently large. In this\nnote we answer this question in the positive but first we state a more general problem.\n\n\\begin{problem}\nFor every $1\\leq k\\leq n$ find%\n\\[\nf_{k}\\left(  n\\right)  =\\max_{v\\left(  G\\right)  =n}\\left\\vert \\mu_{k}\\left(\nG\\right)  \\right\\vert +\\left\\vert \\mu_{k}\\left(  \\overline{G}\\right)\n\\right\\vert .\n\\]\n\n\\end{problem}\n\nIt is difficult to determine precisely $f_{k}\\left(  n\\right)  $ for every $n$\nand $k,$ so at this stage it seems more practical to estimate it\nasymptotically. In this note we show that\n\\begin{equation}\n\\frac{4}{3}n-2\\leq f_{1}\\left(  n\\right)  <\\left(  \\sqrt{2}-c\\right)  n\n\\label{mainin1}%\n\\end{equation}\nfor some $c>8\\times10^{-7}$ independent of $n.$ For $f_{2}\\left(  n\\right)  $\nwe give the following tight bounds%\n\\begin{equation}\n\\frac{\\sqrt{2}}{2}n-3<f_{2}\\left(  n\\right)  <\\frac{\\sqrt{2}}{2}n.\n\\label{mainin2}%\n\\end{equation}\n\n\nWe also show that\n\\begin{equation}\n\\frac{\\sqrt{2}}{2}n-3<f_{n}\\left(  n\\right)  \\leq\\frac{\\sqrt{3}}{2}n\n\\label{mainin3}%\n\\end{equation}\n\n\nFinally for fixed $k,$ $2<k<n,$ and $n$ large, we prove that%\n\\begin{align*}\n\\left\\lfloor \\frac{n}{k}\\right\\rfloor -1  &  \\leq f_{k}\\left(  n\\right)\n\\leq\\sqrt{\\frac{2}{k}}n,\\\\\n\\left\\lfloor \\frac{n}{k}\\right\\rfloor +1  &  \\leq f_{n-k}\\left(  n\\right)\n\\leq\\sqrt{\\frac{2}{k}}n.\n\\end{align*}\n\n\n\\section{Bounds on $f_{1}\\left(  n\\right)  $}\n\nBefore stating the main result of this section, we shall recall two auxiliary\nresults whose proofs can be found in \\cite{Nik05}. Given a graph $G=G\\left(\nn,m\\right)  ,$ let%\n\\[\ns\\left(  G\\right)  =\\sum_{u\\in V\\left(  G\\right)  }\\left\\vert d\\left(\nu\\right)  -\\frac{2m}{n}\\right\\vert .\n\\]\n\n\n\\begin{proposition}\nFor every graph $G=G\\left(  n,m\\right)  ,$%\n\\begin{equation}\n\\frac{s^{2}\\left(  G\\right)  }{2n^{2}\\sqrt{2m}}\\leq\\mu_{1}\\left(  G\\right)\n-\\frac{2m}{n}\\leq\\sqrt{s\\left(  G\\right)  }, \\label{prpin1}%\n\\end{equation}\nand%\n\\begin{equation}\n\\mu_{n}\\left(  G\\right)  +\\mu_{n}\\left(  \\overline{G}\\right)  \\leq\n-1-\\frac{s^{2}\\left(  G\\right)  }{n^{3}}. \\label{prpin2}%\n\\end{equation}\n\n\\end{proposition}\n\nDecreasing the constant $\\sqrt{2}$ in (\\ref{Nosin}) happened to be a\nsurprisingly challenging task for the author. The little progress that has\nbeen made is given in the following theorem.\n\n\\begin{theorem}\n\\label{thf1} There exists $c\\geq8\\times10^{-7}$ such that%\n\\[\n\\mu_{1}\\left(  G\\right)  +\\mu_{1}\\left(  \\overline{G}\\right)  \\leq\\left(\n\\sqrt{2}-c\\right)  n.\n\\]\nfor every graph $G$ of order $n$.\n\\end{theorem}\n\n\\begin{proof}\nAssume the opposite: let $\\varepsilon=8\\times10^{-7}$ and let there exist a\ngraph $G$ of order $n$ such that\n\\[\n\\mu_{1}\\left(  G\\right)  +\\mu_{1}\\left(  \\overline{G}\\right)  >\\left(\n\\sqrt{2}-\\varepsilon\\right)  n.\n\\]\nWriting $A\\left(  G\\right)  $ for the adjacency matrix of $G,$ we have\n\\begin{equation}\n\\sum_{i=1}^{n}\\mu_{i}^{2}\\left(  G\\right)  =tr\\left(  A^{2}\\left(  G\\right)\n\\right)  =2e\\left(  G\\right)  , \\label{basin}%\n\\end{equation}\nimplying that%\n\\[\n\\mu_{1}^{2}\\left(  G\\right)  +\\mu_{n}^{2}\\left(  G\\right)  +\\mu_{1}^{2}\\left(\n\\overline{G}\\right)  +\\mu_{n}^{2}\\left(  \\overline{G}\\right)  \\leq2e\\left(\nG\\right)  +2e\\left(  \\overline{G}\\right)  <n^{2}.\n\\]\nFrom%\n\\[\n\\mu_{1}^{2}\\left(  G\\right)  +\\mu_{1}^{2}\\left(  \\overline{G}\\right)\n\\geq\\frac{1}{2}\\left(  \\mu_{1}\\left(  G\\right)  +\\mu_{1}\\left(  \\overline\n{G}\\right)  \\right)  ^{2}>\\left(  1-\\frac{\\varepsilon}{\\sqrt{2}}\\right)\n^{2}n^{2}>\\left(  1-2\\varepsilon\\right)  n^{2}%\n\\]\nwe find that\n\\begin{equation}\n\\left\\vert \\mu_{n}\\left(  G\\right)  \\right\\vert +\\left\\vert \\mu_{n}\\left(\n\\overline{G}\\right)  \\right\\vert \\leq\\sqrt{2\\left(  \\mu_{n}^{2}\\left(\nG\\right)  +\\mu_{n}^{2}\\left(  \\overline{G}\\right)  \\right)  }<\\sqrt\n{4\\varepsilon}n^{2}, \\label{in1}%\n\\end{equation}\nand so, $\\mu_{n}\\left(  G\\right)  +\\mu_{n}\\left(  \\overline{G}\\right)\n>-\\sqrt{4\\varepsilon}n.$ We thus have $\\sqrt{4\\varepsilon}n^{4}\\geq\ns^{2}\\left(  G\\right)  .$ On the other hand, by (\\ref{prpin1}) and in view of\n$s\\left(  G\\right)  =s\\left(  \\overline{G}\\right)  ,$ we see that\n\\[\n\\mu_{1}\\left(  G\\right)  +\\mu_{1}\\left(  \\overline{G}\\right)  \\leq\nn-1+2\\sqrt{s\\left(  G\\right)  }<n+2\\sqrt{s\\left(  G\\right)  },\n\\]\nand, by (\\ref{in1}), it follows that\n\\[\n\\left(  \\sqrt{2}-\\varepsilon\\right)  n<n+2\\left(  4\\varepsilon\\right)\n^{1\/8}n.\n\\]\nDividing by $n$, we obtain $\\left(  \\sqrt{2}-1\\right)  <\\varepsilon\n+2^{5\/4}\\varepsilon^{1\/8}$, a contradiction for $\\varepsilon=8\\times10^{-7}.$\n\\end{proof}\n\nIt is certain that the upper bound given by Theorem \\ref{thf1} is far from the\nbest one. We shall give below a lower bound on $f_{1}\\left(  n\\right)  $ which\nseems to tight.\n\nFor every $1\\leq r<n$ the graph $G=K_{r}+\\overline{K_{n-r}}$ (see, e.g.\n\\cite{HSF01}) satisfies\n\\[\n\\mu_{1}\\left(  G\\right)  +\\mu_{1}\\left(  \\overline{G}\\right)  =\\frac{r-1}%\n{2}+\\sqrt{nr-\\frac{3r^{2}+2r-1}{4}}+n-r-1=n-\\frac{r+3}{2}+\\sqrt{nr-\\frac\n{3r^{2}+2r-1}{4}}.\n\\]\nThe right-hand side of this inequality is increasing in $r$ for $0\\leq\nr\\leq\\left(  n-1\\right)  \/3$ and we find that\n\\[\nf_{1}\\left(  n\\right)  >\\frac{4n}{3}-2.\n\\]\nThis gives some evidence for the following conjecture.\n\n\\begin{conjecture}%\n\\[\nf_{1}\\left(  n\\right)  =\\frac{4n}{3}+O\\left(  1\\right)  .\n\\]\n\n\\end{conjecture}\n\nWe conclude this section with an improvement of the lower bound in\n(\\ref{Nosin}). Using the first of inequalities (\\ref{prpin1}) we obtain\n\\begin{align*}\n\\mu_{1}\\left(  G\\right)  +\\mu_{1}\\left(  \\overline{G}\\right)   &  \\geq\nn-1+\\frac{s^{2}\\left(  G\\right)  }{2n^{2}}\\left(  \\frac{1}{\\sqrt{2e\\left(\nG\\right)  }}+\\frac{1}{\\sqrt{2e\\left(  \\overline{G}\\right)  }}\\right)  \\geq\\\\\n&  \\geq n-1+\\sqrt{2}\\frac{s^{2}\\left(  G\\right)  }{n^{3}}.\n\\end{align*}\n\n\n\\section{A class of graphs}\n\nIn this section we shall describe a class of graphs that give the right order\nof $f_{2}\\left(  G\\right)  $ and, we believe, also of $f_{n}\\left(  G\\right)\n.$\n\nLet $n\\geq4$. Partition $\\left[  n\\right]  $ in $4$ classes $A,B,C,D$ so that\n$\\left\\vert A\\right\\vert \\geq\\left\\vert B\\right\\vert \\geq\\left\\vert\nC\\right\\vert \\geq\\left\\vert D\\right\\vert \\geq\\left\\vert A\\right\\vert -1.$ Join\nevery two vertices inside $A$ and $D,$ join each vertex in $B$ to each vertex\nin $A\\cup C,$ join each vertex in $D$ to each vertex in $C.$ Write $G\\left(\nn\\right)  $ for the resulting graph.\n\nNote that if $n$ is divisible by $4,$ the sets $A,B,C,D$ have equal\ncardinality and we see that $G\\left(  n\\right)  $ is isomorphic to its complement.\n\nOur main goal to the end of this section is to estimate the eigenvalues of\n$G\\left(  n\\right)  .$ Write $ch\\left(  A\\right)  $ for the characteristic\npolynomial of a matrix $A.$ The following general theorem holds.\n\n\\begin{theorem}\n\\label{thch}Suppose $G$ is a graph and $V\\left(  G\\right)  =\\cup_{i=1}%\n^{k}V_{i}$ is a partition in sets of size $n$ such that\n\n(i) for all $1\\leq i\\leq k,$ either $e\\left(  V_{i}\\right)  =\\binom{n}{2}$ or\n$e\\left(  V_{i}\\right)  =0$;\n\n(ii) for all $1\\leq i<j\\leq k,$ either $e\\left(  V_{i},V_{j}\\right)  =n^{2}$\nor $e\\left(  V_{i},V_{j}\\right)  =0.$\n\nLet the sets $V_{1},...,V_{p}$ be independent and $V_{p+1},...,V_{k}$ induce a\ncomplete graph. Then for the characteristic polynomial of the adjacency matrix\nof $G$ we have%\n\\[\nch\\left(  A\\left(  G\\right)  \\right)  =x^{pn-p}\\left(  1-x\\right)  ^{\\left(\nk-p\\right)  n-\\left(  k-p\\right)  }ch\\left(  R\\right)  ,\n\\]\nwhere $R=\\left(  r_{ij}\\right)  $ is a $k\\times k$ matrix such that\n\\[\nr_{ij}=\\left\\{\n\\begin{array}\n[c]{lll}%\n0 & \\text{if }i\\neq j & \\text{and }e\\left(  V_{i},V_{j}\\right)  =0\\\\\nn & \\text{if }i\\neq j & \\text{and }e\\left(  V_{i},V_{j}\\right)  =n^{2}\\text{\n}\\\\\n0 & \\text{if }i=j & \\text{and }e\\left(  V_{i}\\right)  =0\\\\\nn-1 & \\text{if }i=j & \\text{and }e\\left(  V_{i}\\right)  =\\binom{n}{2}%\n\\end{array}\n\\right.  .\n\\]\n\n\\end{theorem}\n\nThe proof of this theorem is a straight exercise in determinants, so we shall\nomit it.\n\nIf $n$ is divisible by $4,$ say $n=4k,$ by Theorem \\ref{thch}, for the\ncharacteristic polynomial of $A\\left(  G\\left(  n\\right)  \\right)  $ we have%\n\\[\nch\\left(  A\\left(  G\\left(  n\\right)  \\right)  \\right)  =x^{2k-2}\\left(\n1-x\\right)  ^{2k-2}\\left[\n\\begin{array}\n[c]{cccc}%\nk-1-x & k & 0 & 0\\\\\nk & -x & k & 0\\\\\n0 & k & -x & k\\\\\n0 & 0 & k & k-1-x\n\\end{array}\n\\right]  .\n\\]\nBy straightforward calculations, setting $a=1-1\/k$ and $y=x\/k$, we see that\n\\begin{align*}\nch\\left(  A\\left(  G\\left(  n\\right)  \\right)  \\right)   &  =x^{2k-2}\\left(\n1-x\\right)  ^{2k-2}\\left[  \\left(  a-y\\right)  \\left(  y^{2}\\left(\na-y\\right)  +2y-a\\right)  -\\left(  y^{2}-ay-1\\right)  \\right] \\\\\n&  =x^{2k-2}\\left(  1-x\\right)  ^{2k-2}\\left(  y^{2}-\\left(  1+a\\right)\ny-\\left(  1-a\\right)  \\right)  \\left(  y^{2}+\\left(  1-a\\right)  y-\\left(\na+1\\right)  \\right)  .\n\\end{align*}\nHence, we find that%\n\\begin{align*}\n\\mu_{2}\\left(  G\\right)   &  =-\\frac{1}{2}+\\sqrt{\\frac{1}{4}+2\\left\\lfloor\n\\frac{n}{4}\\right\\rfloor ^{2}-\\left\\lfloor \\frac{n}{4}\\right\\rfloor }\\\\\n\\mu_{n}\\left(  G\\right)   &  =-\\frac{1}{2}-\\sqrt{\\frac{1}{4}+2\\left\\lfloor\n\\frac{n}{4}\\right\\rfloor ^{2}-\\left\\lfloor \\frac{n}{4}\\right\\rfloor .}%\n\\end{align*}\n\n\nIf $n$ is not divisible by $4,$ we will give some tight estimates of $\\mu\n_{2}\\left(  G\\right)  $ and $\\mu_{n}\\left(  G\\right)  .$ Notice first that\n$G\\left(  4\\left\\lfloor n\/4\\right\\rfloor \\right)  $ is an induced graph of\n$G\\left(  n\\right)  $ which in turn is an induced graph of $G\\left(\n4\\left\\lceil n\/4\\right\\rceil \\right)  .$ Thus the adjacency matrix of\n$G\\left(  4\\left\\lfloor n\/4\\right\\rfloor \\right)  $ is a principal submatrix\nof the adjacency matrix of $G\\left(  n\\right)  $ which in turn is a principal\nsubmatrix of the adjacency matrix of $G\\left(  4\\left\\lceil n\/4\\right\\rceil\n\\right)  .$ Since the eigenvalues of a matrix and its principal matrices are\ninterlaced (\\cite{HoJo88}, Theorem 4.3.15), we obtain%\n\\begin{align}\n-\\frac{1}{2}+\\sqrt{\\frac{1}{4}+2\\left\\lfloor \\frac{n}{4}\\right\\rfloor\n^{2}-\\left\\lfloor \\frac{n}{4}\\right\\rfloor }  &  \\leq\\mu_{2}\\left(  G\\right)\n\\leq-\\frac{1}{2}+\\sqrt{\\frac{1}{4}+2\\left\\lceil \\frac{n}{4}\\right\\rceil\n^{2}-\\left\\lceil \\frac{n}{4}\\right\\rceil },\\label{in3}\\\\\n-\\frac{1}{2}-\\sqrt{\\frac{1}{4}+2\\left\\lfloor \\frac{n}{4}\\right\\rfloor\n^{2}-\\left\\lfloor \\frac{n}{4}\\right\\rfloor }  &  \\leq\\mu_{n}\\left(  G\\right)\n\\leq-\\frac{1}{2}-\\sqrt{\\frac{1}{4}+2\\left\\lceil \\frac{n}{4}\\right\\rceil\n^{2}-\\left\\lceil \\frac{n}{4}\\right\\rceil }. \\label{in4}%\n\\end{align}\n\n\n\\section{The asymptotics of $f_{2}\\left(  n\\right)  $}\n\nIn this section we shall prove inequalities (\\ref{mainin2}). From (\\ref{in3})\nwe readily have%\n\\[\nf_{2}\\left(  n\\right)  \\geq-\\frac{1}{2}+\\sqrt{\\frac{1}{4}+2\\left\\lfloor\n\\frac{n}{4}\\right\\rfloor ^{2}-\\left\\lfloor \\frac{n}{4}\\right\\rfloor }%\n>\\frac{\\sqrt{2}}{2}n-3,\n\\]\nso all we need to prove is that $f_{2}\\left(  n\\right)  \\leq n\/\\sqrt{2}$.\n\nBy (\\ref{basin}) we have\n\\begin{equation}\n\\mu_{1}^{2}\\left(  G\\right)  +\\mu_{2}^{2}\\left(  G\\right)  +\\mu_{n}^{2}\\left(\nG\\right)  +\\mu_{1}^{2}\\left(  \\overline{G}\\right)  +\\mu_{2}^{2}\\left(\n\\overline{G}\\right)  +\\mu_{n}^{2}\\left(  \\overline{G}\\right)  \\leq n\\left(\nn-1\\right)  . \\label{in2}%\n\\end{equation}\nBy Weyl's inequalities (\\cite{HoJo88}, p. 181), for every graph $G$ of order\n$n,$ we have\n\\[\n\\mu_{2}\\left(  G\\right)  +\\mu_{n}\\left(  \\overline{G}\\right)  \\leq\\mu\n_{2}\\left(  K_{n}\\right)  =-1.\n\\]\nHence, using $\\mu_{2}\\geq0$ and $\\mu_{n}\\leq-1$ we obtain\n\\[\n\\mu_{2}^{2}\\left(  G\\right)  \\leq\\mu_{n}^{2}\\left(  \\overline{G}\\right)\n+2\\mu_{n}\\left(  \\overline{G}\\right)  +1<\\mu_{n}^{2}\\left(  \\overline\n{G}\\right)  .\n\\]\nHence, from (\\ref{in2}) and $\\mu_{1}\\left(  G\\right)  +\\mu_{1}\\left(\n\\overline{G}\\right)  \\geq n-1,$ we find that\n\\[\n\\frac{\\left(  n-1\\right)  ^{2}}{2}+2\\mu_{2}^{2}\\left(  G\\right)  +2\\mu_{2}%\n^{2}\\left(  \\overline{G}\\right)  \\leq\\mu_{1}^{2}\\left(  G\\right)  +\\mu_{2}%\n^{2}\\left(  G\\right)  +\\mu_{n}^{2}\\left(  G\\right)  +\\mu_{1}^{2}\\left(\n\\overline{G}\\right)  +\\mu_{2}^{2}\\left(  \\overline{G}\\right)  +\\mu_{n}%\n^{2}\\left(  \\overline{G}\\right)  \\leq n\\left(  n-1\\right)  .\n\\]\nAfter some algebra, we deduce that\n\\[\n\\mu_{2}\\left(  G\\right)  +\\mu_{2}\\left(  \\overline{G}\\right)  \\leq\\frac\n{\\sqrt{2}}{2}n,\n\\]\ncompleting the proof of inequalities (\\ref{mainin2}).\n\n\\section{Bounds on $f_{n}\\left(  n\\right)  $}\n\nIn this section we shall prove inequalities (\\ref{mainin3}). From (\\ref{in4}),\nas above, we have%\n\\[\nf_{n}\\left(  n\\right)  >\\frac{\\sqrt{2}}{2}n-3.\n\\]\nWe believe that, in fact, the following conjecture is true.\n\n\\begin{conjecture}%\n\\[\nf_{n}\\left(  G\\right)  =\\frac{\\sqrt{2}n}{2}+O\\left(  1\\right)  .\n\\]\n\n\\end{conjecture}\n\nHowever we can only prove that $f_{n}\\left(  G\\right)  <\\left(  \\sqrt\n{3}\/2\\right)  n$ which is implied by the following theorem.\n\n\\begin{theorem}\nFor every graph $G$ of order $n,$\n\\[\n\\mu_{n}^{2}\\left(  G\\right)  +\\mu_{n}^{2}\\left(  \\overline{G}\\right)\n\\leq\\frac{3}{8}n^{2}.\n\\]\n\n\\end{theorem}\n\n\\begin{proof}\nIndeed, suppose $\\left(  u_{1},...,u_{n}\\right)  $ and $\\left(  w_{1}%\n,...,w_{n}\\right)  $ are eigenvectors to $\\mu_{n}\\left(  G\\right)  $ and\n$\\mu_{n}\\left(  \\overline{G}\\right)  .$ Let\n\\[\nU=\\left\\{  i:u_{i}>0\\right\\}  ,\\text{ \\ \\ \\ }W=\\left\\{  i:w_{i}>0\\right\\}  .\n\\]\nSetting $V=\\left[  n\\right]  ,$ we clearly have $\\mu_{n}^{2}\\left(  G\\right)\n\\leq E_{G}\\left(  U,V\\backslash U\\right)  $ and $\\mu_{n}^{2}\\left(\n\\overline{G}\\right)  \\leq E_{\\overline{G}}\\left(  W,V\\backslash W\\right)  $.\nSince $E_{G}\\left(  U,V\\backslash U\\right)  \\cap E_{\\overline{G}}\\left(\nW,V\\backslash W\\right)  =\\varnothing,$ we see that the graph\n\\[\nG^{\\prime}=\\left(  V,E_{G}\\left(  U,V\\backslash U\\right)  \\cup E_{\\overline\n{G}}\\left(  W,V\\backslash W\\right)  \\right)\n\\]\nis at most $4$-colorable and hence $G^{\\prime}$ contains no $4$-cliques. By\nTur\\'{a}n's theorem (e.g., see \\cite{Bol98}), we obtain $e\\left(  G^{\\prime\n}\\right)  \\leq\\left(  3\/8\\right)  n^{2},$ completing the proof.\n\\end{proof}\n\n\\section{Bounds on $f_{k}\\left(  n\\right)  ,$ $2<k<n$}\n\nIn this section we shall give simple bounds on $f_{k}\\left(  n\\right)  $ for\n$2<k<n$. Write for the Tur\\'{a}n graph of order $n$ with $k$ classes. Recall\nthat $T_{k}\\left(  n\\right)  $ is a complete $k$-partite graph whose vertex\nclasses differ by at most $1$ in size. We assume that $k$ is fixed and $n$ is\nlarge enough. Since $\\mu_{k}\\left(  T_{k}\\left(  n\\right)  \\right)  =0$ and\n$\\mu_{n-k}\\left(  T_{k}\\left(  n\\right)  \\right)  \\leq-\\left\\lfloor\nn\/k\\right\\rfloor $ for $n$ large, we immediately have\n\\begin{align*}\nf_{k}\\left(  n\\right)   &  \\geq\\left\\lfloor n\/k\\right\\rfloor -1,\\\\\nf_{n-k}\\left(  n\\right)   &  \\geq\\left\\lfloor n\/k\\right\\rfloor +1.\n\\end{align*}\n\n\nWe next turn to upper bounds on $f_{k}\\left(  n\\right)  .$\n\n\\begin{theorem}\nFor any fixed $k$ and any graph $G$ of sufficiently large order $n,$\n\\begin{equation}\n\\left\\vert \\mu_{k}\\left(  G\\right)  \\right\\vert +\\left\\vert \\mu_{k}\\left(\n\\overline{G}\\right)  \\right\\vert <\\sqrt{\\frac{2}{k}}n. \\label{in5}%\n\\end{equation}\nand%\n\\begin{equation}\n\\left\\vert \\mu_{n-k}\\left(  G\\right)  \\right\\vert +\\left\\vert \\mu_{n-k}\\left(\n\\overline{G}\\right)  \\right\\vert <\\sqrt{\\frac{2}{k}}n. \\label{in6}%\n\\end{equation}\n\n\\end{theorem}\n\n\\begin{proof}\nSet $e\\left(  G\\right)  =m.$ Our first goal is to prove that $\\left\\vert\n\\mu_{k}\\left(  G\\right)  \\right\\vert \\leq\\sqrt{2e\\left(  G\\right)  \/k}.$ If\n$\\mu_{k}\\left(  G\\right)  \\geq0,$ we have in view of (\\ref{basin})\n\\[\nk\\mu_{k}^{2}\\left(  G\\right)  \\leq\\sum_{i=1}^{n}\\mu_{i}^{2}\\left(  G\\right)\n=2m.\n\\]\nIf $\\mu_{k}\\left(  G\\right)  <0$ and $\\left\\vert \\mu_{k}\\left(  G\\right)\n\\right\\vert >\\sqrt{2m\/k}$ then\n\\[\n\\sum_{i=1}^{n}\\mu_{i}^{2}\\left(  G\\right)  \\geq\\left(  n-k\\right)  \\mu_{k}%\n^{2}\\left(  G\\right)  >2m\\frac{n-k}{k}>2m,\n\\]\na contradiction. Hence, $\\left\\vert \\mu_{k}\\left(  G\\right)  \\right\\vert\n\\leq\\sqrt{2e\\left(  G\\right)  \/k},$ and, by symmetry, $\\left\\vert \\mu\n_{k}\\left(  \\overline{G}\\right)  \\right\\vert \\leq\\sqrt{2e\\left(  \\overline\n{G}\\right)  \/k}.$ Now\n\\[\n\\left\\vert \\mu_{k}\\left(  G\\right)  \\right\\vert +\\left\\vert \\mu_{k}\\left(\n\\overline{G}\\right)  \\right\\vert \\leq\\sqrt{2e\\left(  G\\right)  \/k}%\n+\\sqrt{2e\\left(  \\overline{G}\\right)  \/k}\\leq\\sqrt{\\frac{2}{k}n\\left(\nn-1\\right)  }<\\sqrt{\\frac{2}{k}}n,\n\\]\nproving inequality (\\ref{in5}). The proof of inequality (\\ref{in6}) goes along\nthe same lines, so we will omit it.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Janus and Hades}\n\n\nJanus solutions in string\/M-theory were originally introduced in the context of type IIB supergravity as a simple deformation of the $\\,\\textrm{AdS}_{5} \\times \\textrm{S}^{5}\\,$ background involving a non-trivial dilaton profile \\cite{Bak:2003jk}. The deformation breaks the $\\,\\textrm{SO}(2,4)\\,$ isometries of $\\,\\textrm{AdS}_{5}\\,$ to the $\\,\\textrm{SO}(2,3)\\,$ isometries of $\\,\\textrm{AdS}_{4}\\,$, but preserves the $\\,\\textrm{SO}(6)\\,$ isometries of the round $\\,\\textrm{S}^{5}\\,$. Soon after, a holographic interpretation of the solutions in \\cite{Bak:2003jk} was proposed in terms of a planar $\\,(1+2)$-dimensional interface in super Yang--Mills (SYM) separating two half-spaces with different coupling constants \\cite{Clark:2004sb}. The supersymmetric Janus was constructed in \\cite{Clark:2005te} using a 5D effective SO(6) gauged supergravity approach. Its ten-dimensional incarnation was put forward in \\cite{DHoker:2006vfr}, which provided the gravity dual of the $\\,\\mathcal{N}=1\\,$ supersymmetric interface first anticipated in \\cite{Clark:2004sb} and then constructed in \\cite{DHoker:2006qeo}. The $\\,\\mathcal{N}=1\\,$ supersymmetric Janus turns out to break the symmetry of the original (non-supersymmetric) Janus down to at least $\\,\\textrm{SU(3)} \\subset \\textrm{SO}(6)\\,$. The $\\,\\mathcal{N}=4\\,$ Janus solution with $\\,\\textrm{SO}(4)\\,$ symmetry was constructed in \\cite{DHoker:2007zhm}. However it was only recently that the $\\,\\mathcal{N}=2\\,$ supersymmetric Janus with $\\,\\textrm{SU}(2) \\times  \\textrm{U}(1)\\,$ symmetry was constructed in five and ten dimensions \\cite{Bobev:2020fon}, thus completing the list of Janus solutions dual to the SYM interfaces scrutinised in \\cite{DHoker:2006qeo}.\n\n\nJanus solutions have been much less investigated in the context of M-theory. The first examples were constructed in \\cite{DHoker:2009lky} (and generalised in \\cite{Bachas:2013vza}) as $\\,\\mathcal{N}=4\\,$ deformations of the $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ background preserving a subgroup $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4) \\subset \\textrm{SO}(8)\\,$ of the isometries of the round $\\,\\textrm{S}^{7}\\,$. This time the deformation breaks the $\\,\\textrm{SO}(2,3)\\,$ isometries of $\\,\\textrm{AdS}_{4}\\,$ to the $\\,\\textrm{SO}(2,2)\\,$ isometries of $\\,\\textrm{AdS}_{3}\\,$. The M-theory Janus solutions can still be holographically understood as $\\,(1+1)$-dimensional interfaces in ABJM theory \\cite{Aharony:2008ug} despite the absence of a dilaton field in the M-theory context \\cite{DHoker:2009lky,Bobev:2013yra,Kim:2018qle}. Interestingly, it was shown in \\cite{Bobev:2013yra} that the $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetric Janus can be alternatively found using a 4D effective SO(8) gauged supergravity description. Using this 4D approach, an $\\,\\mathcal{N}=1\\,$ supersymmetric and $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetric Janus was constructed in \\cite{Bobev:2013yra} using numerical methods, for which 11D uplift formuli were provided in \\cite{Pilch:2015dwa}. More numerical Janus solutions were also presented in \\cite{Bobev:2013yra} by studying the $\\,\\textrm{G}_{2}$-invariant sector of the SO(8) gauged supergravity.\\footnote{See \\cite{Karndumri:2020bkc} for a numerical study of Janus solutions in the one-parameter family of $\\omega$-deformed SO(8) gauged supergravities \\cite{Dall'Agata:2012bb}. See also \\cite{Suh:2018nmp,Karndumri:2021pva} for a similar study in the context of massive IIA compactified on $\\,\\textrm{S}^{6}\\,$ and its effective description in terms of the ISO(7) gauged supergravity \\cite{Guarino:2015jca,Guarino:2015qaa,Guarino:2015vca}.}\n\n\nAmongst the various interesting questions raised in the discussion section of \\cite{DHoker:2009lky} we will provide a positive answer to that of whether exact M-theory Janus solutions exist with no supersymmetry. We will use the four-dimensional SO(8) gauged supergravity that arises upon reduction of eleven-dimensional supergravity on $\\,\\textrm{S}^{7}\\,$ \\cite{deWit:1982ig,deWit:1986oxb} and construct non-supersymmetric, yet analytic and regular, AdS$_{3}$-sliced domain-wall solutions of the form\n\\begin{equation}\n\\label{metric_ansatz_intro}\nds_{4}^{2} = d\\mu^{2} + e^{2 A(\\mu)} \\, ds_{\\textrm{AdS}_{3}}^{2} \\ ,\n\\end{equation}\nfor which the metric function $\\,A(\\mu)\\,$ depends arbitrarily on three real constants $\\,\\alpha_{i} \\in \\mathbb{R}\\,$ with $\\,{i=1,2,3}\\,$. The geometry is supported by three complex scalar fields $\\,z_{i}(\\alpha_{i},\\beta_{i};\\mu)\\,$ which depend on three additional compact parameters, or phases $\\,\\beta_{i} \\in [0,2\\pi]\\,$, and develop non-trivial profiles along the radial coordinate $\\,\\mu\\,$ transverse to the domain-wall. The effective 4D gauge coupling $\\,g\\,$ -- which relates to the inverse radius of $\\,\\textrm{S}^{7}\\,$ -- and the set of real parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ fully determine a particular Janus configuration.\n\n\n\nThe Janus parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ specify the boundary values of the complex scalars at $\\,{\\mu \\rightarrow \\pm \\infty}\\,$. In particular, the parameters $\\,\\beta_{i}\\,$ encode the source\/VEV and bosonic\/fermionic nature of the dual operators turned on on each side of the interface living at the boundary. A generic choice of Janus parameters breaks all the supersymmetries and the $\\,\\textrm{S}^{7}\\,$ isometry group down to its Cartan subgroup $\\,\\textrm{U}(1)^4 \\subset \\textrm{SO}(8)\\,$. On the contrary, the very special choice $\\,\\alpha_{i}=0\\,$ $\\forall i\\,$ trivialises the Janus and the maximally supersymmetric AdS$_{4}$ vacuum of the SO(8) supergravity that uplifts to the $\\,\\textrm{AdS}_4 \\times \\textrm{S}^7\\,$ Freund--Rubin background of eleven-dimensional supergravity with a round $\\,\\textrm{S}^{7}\\,$ metric is recovered \\cite{Freund:1980xh}. Interestingly, (super) symmetry enhancements occur upon suitable identifications between the parameters. For instance, the supersymmetric Janus of \\cite{DHoker:2009lky,Bobev:2013yra} with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetry is recovered upon setting two of the $\\,\\alpha_{i}\\,$ parameters to zero. In this work we will pay special attention to the Janus with $\\,{\\alpha_{1}=\\alpha_{2}=\\alpha_{3}}\\,$ and $\\,{\\beta_{1}=\\beta_{2}=\\beta_{3}}\\,$ which is non-supersymmetric and features an $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetry enhancement. We will present the uplift of this 4D Janus to eleven-dimensions providing, to the best of our knowledge, the first example of an exact M-theory Janus with no supersymmetry. \n\n\n\n\nIn addition to the Janus, we will construct another class of analytic solutions -- we refer to them as flows to Hades following standard terminology in the literature -- which are non-supersymmetric and display a singularity at $\\,\\mu =0\\,$ where the $\\,e^{2 A(\\mu)}\\,$ factor in (\\ref{metric_ansatz_intro}) shrinks to zero size and the complex scalars run to the boundary of moduli space. Some similar curved-sliced \\cite{Bobev:2013yra} and flat-sliced \\cite{Cvetic:1999xx,Pope:2003jp,Pilch:2015vha,Pilch:2015dwa} singular flows have been constructed within the $\\,\\textrm{SO}(8)\\,$ gauged supergravity and argued to holographically describe an interface between a superconformal ABJM phase and a non-conformal phase with potentially interesting physics.\\footnote{The scalar potential of the maximal $\\,\\textrm{SO}(8)\\,$ gauged supergravity is bounded above by its value at the maximally supersymmetric AdS$_{4}$ vacuum thus satisfying the \\textit{good} condition of \\cite{Gubser:2000nd}.} In their simplest realisation, these flat-sliced singular flows in M-theory are the analogue of the type IIB flows to the Coulomb branch of $\\,\\mathcal{N}=4\\,$ SYM investigated in \\cite{Cvetic:1999xx,Freedman:1999gp,Freedman:1999gk,Gubser:2000nd}. \n\n\n\nThere are similarities and differences between the Janus and the Hades. As for the Janus, the Hades solutions depend on a set of six parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. Unlike for the Janus, no supersymmetric limit can be taken on the Hades parameters, and the very special choice $\\,\\alpha_{i}=0\\,$ $\\forall i\\,$ does not trivialise the Hades to recover AdS$_{4}$. Instead, a special class of Hades flows -- we will refer to them as \\textit{ridge flows} adopting the terminology of \\cite{Pilch:2015vha} -- appears in this limit. As before, we will concentrate on the simple case with $\\,{\\alpha_{1}=\\alpha_{2}=\\alpha_{3}} \\equiv \\alpha\\,$ and $\\,{\\beta_{1}=\\beta_{2}=\\beta_{3}}\\equiv \\beta\\,$ for which the flows to Hades feature an $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetry, and present their uplift to eleven-dimensional supergravity. \n\n\n\nSpecial attention will then be paid to the $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetric ridge flows with $\\,\\alpha=0\\,$ for which there is just one free parameter left, \\textit{i.e.} the phase $\\,\\beta \\in [0,2\\pi]\\,$. This phase specifies the boundary values of the complex scalars at $\\,\\mu \\rightarrow \\infty\\,$ and, therefore, the source\/VEV and bosonic\/fermionic nature of the dual operators turned on on the ultraviolet (UV) side of the conformal interface. In the infrared (IR) side $\\,\\mu \\rightarrow 0\\,$ of the interface, the four-dimensional solution becomes singular and the dual field theory is expected to enter the non-conformal phase. Interestingly, the parameter $\\,\\beta\\,$ determining the boundary conditions of the complex scalars is associated with a $\\,\\textrm{U}(1)_{\\xi}\\,$ duality symmetry of the four-dimensional supergravity Lagrangian. However, as originally noticed in \\cite{Pope:2003jp} for a class of conventional flat-sliced RG-flows (see also \\cite{Pilch:2015vha,Pilch:2015dwa}), the $\\,\\textrm{U}(1)_{\\xi}\\,$ changes the physics of the ridge flows once they are uplifted to eleven dimensions: it takes metric modes into three-form gauge field modes. \n\n\nWe will illustrate this phenomenon by analysing in some detail the simple cases of setting $\\,\\beta= \\frac{\\pi}{2}\\,$ and $\\,\\beta=0\\,$. The corresponding ridge flows are triggered from the UV solely by bosonic VEV's or fermionic sources, respectively. The resulting M-theory ridge flows will be shown to be drastically different as far as the persistence of the singularity and the presence of magnetic M5-branes sources in the background are concerned. Setting $\\,\\beta= \\frac{\\pi}{2}\\,$ produces a singular M-theory background without magnetic M5-branes sources akin the (flat-sliced) Coulomb branch flows constructed in \\cite{Cvetic:1999xx}. Modifying the phase $\\,\\beta\\,$ by acting with $\\,\\textrm{U}(1)_{\\xi}\\,$ turns out to induce a transformation on the eleven-dimensional backgrounds that parallels the dielectric rotation of Coulomb branch flows investigated in \\cite{Pope:2003jp,Pilch:2015vha,Pilch:2015dwa}. We will look in detail at the limiting case $\\,\\beta=0\\,$ and conclude that the $\\,\\textrm{U}(1)_{\\xi}\\,$ transformation totally polarises M2-branes into M5-branes when flowing from the UV to the IR, leaving no M2-branes. We will provide some evidence for this phenomenon to occur also at generic values of $\\,\\beta\\,$.\n\n\nThe paper is organised in four sections plus appendices. In Section~\\ref{sec:Janus} we present our multi-parametric $(\\alpha_{i},\\beta_{i})$-families of analytic Janus and Hades solutions and discuss the ridge flow limit of the latter. We investigate the various possibilities of (super) symmetry enhancement depending on the choice of $\\,\\alpha_{i}\\,$ parameters, as well as the various possibilities of boundary conditions for the complex scalars (sources\/VEV's of dual operators) depending on the choice of $\\,\\beta_{i}\\,$ parameters. In Section~\\ref{sec:Uplift_11D} we present the uplift of the Janus and Hades solutions with $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetry to eleven-dimensional supergravity. We then focus on the ridge flows and discuss some eleven-dimensional aspects of the solutions, like the presence of singularities or the characterisation of the M2\/M5-brane sourcing the backgrounds, as a function of the parameter $\\,\\beta\\,$. We summarise the results and conclude in Section~\\ref{sec:conclusions}. Two additional appendices accompany the main text which contain technical results regarding the BPS equations as well as some relevant uplift formuli for the STU model. This is the subsector of the four-dimensional maximal $\\,\\textrm{SO}(8)\\,$ gauged supergravity within which we have constructed all the solutions presented in this work.\n\n\n\n\n\\section{Four-dimensional Janus and Hades}\n\\label{sec:Janus}\n\n\n\\subsection{The model}\n\n\nOur starting point is the $\\mathcal{N}=2$ gauged STU supergravity in four dimensions \\cite{Cvetic:1999xp}. This theory has a gauge group $\\textrm{U}(1)^4$, the maximal Abelian subgroup of $\\textrm{SO}(8)$, and can be embedded into the maximal $\\mathcal{N}=8$ $\\textrm{SO}(8)$-gauged supergravity \\cite{deWit:1982ig} as its $\\textrm{U}(1)^4$ invariant sector \\cite{Cvetic:1999xp}. The field content consists of the $\\mathcal{N}=2$ supergravity multiplet coupled to three vector multiplets. Upon setting vector fields to zero, the bosonic Lagrangian reduces to an Einstein-scalar model given by\n\\begin{equation}\n\\label{Lagrangian_model_U1^4_Einstein-scalars}\n\\begin{array}{lll}\n\\mathcal{L} & = & \\left(  \\dfrac{R}{2} - V \\right)  * 1 - \\dfrac{1}{4} \\displaystyle\\sum_{i=1}^3 \\left[\nd\\varphi_{i} \\wedge* d\\varphi_{i} + e^{2 \\varphi_{i}} \\, d\\chi_{i} \\wedge* d\\chi_{i} \\right] \\\\[4mm]\n& = & \\left(  \\dfrac{R}{2} - V \\right)  * 1 - \\displaystyle\\sum_{i=1}^{3}\n\\dfrac{1}{\\left(  1-|\\tilde{z_{i}}|^{2} \\right)  ^{2}} \\, d\\tilde{z}_{i}\n\\wedge* d\\tilde{z}_{i}^{*} \\ .\n\\end{array}\n\\end{equation}\nIn passing from the first line to the second one in (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) we have changed the parameterisation of the scalar  fields $\\,z_{i}\\,$ ($i=1,2,3$) in the vector multiplets -- which serve as coordinates in the scalar coset geometry $[\\textrm{SL}(2)\/\\textrm{SO}(2)]^3$ -- from the upper-half plane to the unit-disk parameterisation via the field redefinition\n\\begin{equation}\n\\label{ztilde&z}\n\\tilde{z}_{i}=\\frac{z_{i}-i}{z_{i}+i} \n\\hspace{10mm} \\textrm{ with } \\hspace{10mm} \nz_{i} = -\\chi_{i} + i \\, e^{-\\varphi_{i}} \\ .\n\\end{equation}\nThe non-trivial scalar potential in the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) is given by\n\\begin{equation}\n\\label{V_U1^4}\nV = - \\tfrac{1}{2} \\, g^{2} \\sum_{i} \\left(  2 \\, \\cosh\\varphi_{i} + \\chi\n_{i}^{2} \\, e^{\\varphi_{i}} \\right)  =  g^{2} \\, \\left( 3 - \\sum_{i} \\frac{2}{1-|\\tilde{z}_{i}|^{2}}   \\right) \\ ,\n\\end{equation}\nwhere $\\,g\\,$ is the gauge coupling in the gauged four-dimensional supergravity. From (\\ref{V_U1^4}) one immediately sees that only $|\\tilde{z}_{i}|$ enter the\npotential. As a result, the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) is\ninvariant under the three $\\,\\textrm{U}(1)_{\\xi_{i}}\\,$ shifts of $\\,\\arg\\tilde{z}_{i}\\,$, namely, $\\,\\delta_{\\xi_{i}}\\tilde{z}_{i} = i \\, \\xi_{i} \\, \\tilde{z}_{i}\\,$, with constant parameters $\\,\\xi_{i}\\,$. However, as we will see shortly, the phases $\\,\\arg\\tilde{z}_{i}\\,$ will play a central role when discussing boundary conditions for Janus- and Hades-like solutions in this supergravity model.\n\n\n\nIn this work we will investigate Janus-like solutions for which the space-time metric takes the form\n\\begin{equation}\n\\label{metric_ansatz}\nds_{4}^{2} = d\\mu^{2} + e^{2 A(\\mu)} \\, d\\Sigma^{2} \\ ,\n\\end{equation}\nwith $\\,\\mu\\in (-\\infty, \\infty)\\,$ or $\\,\\mu\\in [0, \\infty)\\,$ being the coordinate along which space-time is foliated with $\\Sigma$ slices, and $A(\\mu)$ being a scale function. The line element $\\,d\\Sigma^{2}\\,$ describes a globally AdS$_{3}$ space-time of radius $\\,\\ell= 1\\,$. The second-order Euler-Lagrange equations for the scalar fields that follow from the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) read\n\\begin{equation}\n\\label{EOM_scalars}\n\\begin{array}{rll}\n\\varphi_{i} '' - e^{2 \\varphi_{i}} \\,(\\chi_{i}')^2 + 3 \\, A' \\, \\varphi_{i}' + g^2 \\, \\left( 2 \\, \\sinh\\varphi_{i} +  e^{\\varphi_{i}} \\, \\chi_{i}^2\\right) & = & 0  \\ , \\\\[2mm]\n\\chi_{i}'' + \\left( 3 \\, A' + 2  \\, \\varphi_{i}' \\right) \\chi_{i}'  + 2 \\, g^2 \\, e^{-\\varphi_{i}} \\, \\chi_{i}  & = & 0 \\ ,\n\\end{array}\n\\end{equation}\nwith $i=1,2,3$ and where primes denote derivatives with respect to the coordinate $\\,\\mu\\,$. The Einstein equations impose two additional independent equations given by\n\\begin{equation}\n\\label{EOM_Einstein}\n\\begin{array}{rll}\n1 - e^{2 A} \\Big[ A'' + \\frac{1}{4} \\displaystyle\\sum_{i} \\Big( \\, (\\varphi_{i}')^2 + e^{2 \\varphi_{i}} \\, (\\chi_{i}')^2 \\Big) \\, \\Big] &=& 0 \\ , \\\\[2mm]\n2 + e^{2 A} \\Big[  A'' + 3 \\, (A')^2 - \\tfrac{1}{2} \\, g^{2} \\displaystyle\\sum_{i}  \\left( 2 \\, \\cosh\\varphi_{i} + \\chi_{i}^2  \\,  e^{\\varphi_{i}} \\right)   \\Big] & = & 0 \\ .\n\\end{array}\n\\end{equation}\nWe will now present analytic and multi-parametric families of Janus and Hades solutions to this system of second-order differential equations.\n\n\n\n\\subsection{Multi-parametric Janus solutions}\n\nThe second-order equations of motion in (\\ref{EOM_scalars}) and (\\ref{EOM_Einstein}) have a multi-parametric family of analytic Janus solutions. The scale factor in the space-time metric is given by\n\\begin{equation}\n\\label{A(mu)_func_U1^4}\ne^{2A(\\mu)} = {(g k)}^{-2} \\cosh^2(g\\mu) \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{k_factor}\nk^2= 1 + \\sum_{i}\\sinh^{2}\\alpha_{i}  \\, \\ge \\, 1\n\\hspace{10mm} \\text{ with } \\hspace{10mm} \\alpha_{i} \\in \\mathbb{R} \\ .\n\\end{equation}\nUsing the unit-disk parameterisation in (\\ref{ztilde&z}) to describe the scalar fields in the three vector multiplets, they acquire simple $\\mu$-dependent profiles of the form\n\\begin{equation}\n\\label{Janus_solution_U1^4_ztil}\n\\tilde{z}_{i}(\\mu) = e^{i \\beta_{i}}\\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\sinh(g \\mu) }  \n\\hspace{10mm} \\text{ with } \\hspace{10mm} \\beta_{i} \\in [0,2\\pi] \\ ,\n\\end{equation}\nso that $\\,|\\tilde{z}_{i}(0)|=\\tanh\\alpha_{i}\\,$. Eqs (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) describe a multi-parametric family of Janus solutions parameterised by $3+3$ arbitrary real constants $(\\alpha_{i},\\beta_{i})$. Importantly, the presence of non-trivial axions $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ (spin $0$ pseudo-scalars) turns out to be crucial for the existence of regular Janus solutions, as first noticed in \\cite{Bobev:2013yra}. Parametric plots of the complex scalars $\\,\\tilde{z}_{i}(\\mu)\\,$ in (\\ref{Janus_solution_U1^4_ztil}) are displayed in Figure~\\ref{fig:ztilde_U1^4}. The real $\\,\\textrm{Re}\\tilde{z}_{i}\\,$ and imaginary $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ components of $\\,\\tilde{z}_{i}\\,$ are shown in Figure~\\ref{fig:Rez&Imz}. Note the special limiting case of $\\,\\alpha_{i} \\gg 1\\,$ (\\textit{i.e.} $\\tanh\\alpha_{i} \\approx 1$) for which the flows become singular. In this limit, the complex scalar $\\,\\tilde{z}_{i}\\,$ gets to the boundary of the moduli space, which is located at $\\,|\\tilde{z}_{i}|=1\\,$ in the unit-disk parameterisation of the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}), and the scalar potential in (\\ref{V_U1^4}) diverges.\n\n\n\nOn the other hand, the value $\\,\\alpha_{i}=0\\,$ is certainly special. At this value an AdS$_{4}$ maximally supersymmetric solution with radius $\\,L_{\\text{AdS}_{4}}={g}^{-1}\\,$ is recovered with the scalars being fixed at the constant value $\\,\\tilde{z}_{i}=0\\,$. This AdS$_{4}$ vacuum uplifts to the $\\,\\textrm{AdS}_4 \\times \\textrm{S}^7\\,$ Freund--Rubin background of eleven-dimensional supergravity with a round $\\,\\textrm{S}^{7}\\,$ metric \\cite{Freund:1980xh}. Moreover, it describes the near-horizon geometry of a stack of M2-branes and is holographically dual to the three-dimensional ABJM theory \\cite{Aharony:2008ug}. When evaluated at this AdS$_{4}$ vacuum, the three $\\textrm{U}(1)^{4}$ invariant complex scalars have a normalised mass\n\\begin{equation}\n\\label{m^2L^2_AdS4}\nm_{i}^2 L^2 = -2 \\ ,\n\\end{equation}\nthus lying within the mass range $\\, -9\/4 < m_{i}^2 \\, L^2 <  - 5\/4\\,$ for wich two possible quantisations of scalar fields in AdS$_{4}$ exist \\cite{Klebanov:1999tb}: the mode with conformal dimension $\\,\\Delta_{i}=\\Delta_{-}=1\\,$ and the mode with conformal dimension $\\,\\Delta_{i}=\\Delta_{+}=2\\,$ (where $\\,\\Delta_{\\pm}\\,$ are the two roots of $\\,m_{i}^2 \\, L^2=(\\Delta_{i}-3)\\Delta_{i}\\,$) can be interpreted as the source and the VEV of the corresponding dual operators (standard quantisation) or \\textit{viceversa} (alternative quantisation). However, as shown in \\cite{Breitenlohner:1982bm}, proper scalars $\\,\\textrm{Re}\\tilde{z}_{i}\\,$ and pseudo-scalars $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ must be quantised in exactly opposite ways in order to preserve maximal supersymmetry. And, moreover, only the choice of proper scalars having alternative quantisation yields a perfect matching between the scaling dimensions of the supergravity modes and those of the dual operators in the  M2-brane  theory \\cite{Bobev:2011rv} (see footnote~\\ref{footnote:operators}). \n\n\n\nThe class of Janus solutions in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) depends on the set of parameters $\\,g\\,$ and $\\,(\\alpha_{i},\\beta_{i})\\,$. As discussed in \\cite{Bobev:2013yra}, the four-dimensional gauge coupling $\\,g\\,$ sets the scale of the asymptotic AdS$_{4}$ vacuum and, via the AdS\/CFT correspondence, the number of M2-branes as well as the rank of the Chern--Simons gauge groups in ABJM theory. The parameters $\\,\\alpha_{i}\\,$ set the height of the bump, \\textit{i.e.}  $\\,|\\tilde{z}_{i}(0)|=\\tanh\\alpha_{i}\\,$, and therefore the strength of the coupling between the (1+1)-dimensional defect and the three-dimensional ambient field theory. The parameters $\\,\\beta_{i}\\,$ set the boundary conditions of the bulk scalars at $\\,\\mu \\rightarrow \\pm \\infty\\,$ and, again via the AdS\/CFT correspondence (see footnote~\\ref{footnote:operators}), the specific linear combinations of bosonic and fermionic bilinear operators that are activated in the field theory. We will analyse the possible choices of boundary conditions in detail in Section~\\ref{sec:boundary conditions}.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.50\\textwidth]{Plots\/ztilde_plot.pdf} \n\\put(0,90){$\\textrm{Re}\\tilde{z}_{i}$} \\put(-105,210){$\\textrm{Im}\\tilde{z}_{i}$}\n\\put(-114,101){{\\color{red}{$\\bullet$}}}\n\\end{center}\n\\caption{Parametric plot of $\\,\\tilde{z}_{i}(\\mu)\\,$ in (\\ref{Janus_solution_U1^4_ztil}) for the Janus solutions with $\\,\\alpha_{i}=1\\,$ and $\\,\\beta_{i}=\\frac{n\\pi}{4}\\,$ with $\\,n=0,\\ldots,7\\,$. The central red point at $\\,\\tilde{z}_{i}=0\\,$ $\\,\\forall i\\,$ corresponds to the maximally supersymmetric AdS$_{4}$ vacuum and describes the asymptotic values at $\\,\\mu\\rightarrow \\pm\\infty\\,$.}\n\\label{fig:ztilde_U1^4}\n\\end{figure}\n\n\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=1_b=0.pdf} \n\\hspace{5mm}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=10_b=0.pdf}\n\\put(-307,15){\\small{$\\alpha_{i}=1 \\, , \\, \\beta_{i}=0$}}\\put(-87,15){\\small{$\\alpha_{i} \\gg 1 \\, , \\, \\beta_{i}=0$}}\n\\\\[5mm]\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=1_b=Pi4.pdf} \n\\hspace{5mm}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=10_b=Pi4.pdf} \n\\put(-307,22){\\small{$\\alpha_{i}=1 \\, , \\, \\beta_{i}=\\frac{\\pi}{4}$}}\\put(-87,22){\\small{$\\alpha_{i} \\gg 1 \\, , \\, \\beta_{i}=\\frac{\\pi}{4}$}}\n\\\\[5mm]\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=1_b=Pi2.pdf} \n\\hspace{5mm}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Plot_a=10_b=Pi2.pdf} \n\\put(-307,15){\\small{$\\alpha_{i}=1 \\, , \\, \\beta_{i}=\\frac{\\pi}{2}$}}\\put(-87,15){\\small{$\\alpha_{i} \\gg 1 \\, , \\, \\beta_{i}=\\frac{\\pi}{2}$}}\n\\put(-412,-8){\\scriptsize{$-\\infty$}}\\put(-323,-8){\\scriptsize{$0$}}\\put(-238,-8){\\scriptsize{$\\infty$}}\n\\put(-330,-22){\\small{$g \\mu$}}\n\\put(-190,-8){\\scriptsize{$-\\infty$}}\\put(-101,-8){\\scriptsize{$0$}}\\put(-16,-8){\\scriptsize{$\\infty$}}\n\\put(-109,-22){\\small{$g \\mu$}}\n\\end{center}\n\\caption{Plots of $\\,\\textrm{Re}\\tilde{z}_{i}\\,$ (blue dotted line), $\\,\\textrm{Im}\\tilde{z}_{i}\\,$ (orange dashed line) and $\\,A'(\\mu)\\,$ (green solid line) as a function of the radial coordinate $\\,g \\mu \\in (-\\infty , \\infty)\\,$ for different values of the Janus parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. The limit $\\,\\alpha_{i} \\gg 1\\,$ (\\textit{i.e.} $\\tanh\\alpha_{i} \\approx 1$) renders the Janus solution singular. In this limit, $\\,\\tilde{z}_{i}\\,$ gets to the boundary of the moduli space which is located at $\\,|\\tilde{z}_{i}|=1\\,$ in the unit-disk parameterisation of (\\ref{ztilde&z}).}\n\\label{fig:Rez&Imz}\n\\end{figure}\n\n\n\nLastly, a study of the supersymmetry preserved by this family of solutions is presented in the Appendix~\\ref{app:susy}. The BPS equations (\\ref{BPS_A}) and (\\ref{BPS_scalars}) are not satisfied by the Janus solution in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) for generic values of $(\\alpha_{i} , \\beta_{i})$ thus implying that such a solution is generically non-supersymmetric. However, as we will see in a moment, some supersymmetry can be restored upon suitable choice of $(\\alpha_{i} , \\beta_{i})$, namely, upon suitable adjustment of the Janus boundary conditions.\n\n\n\n\\subsection{Janus with (super) symmetry enhancements}\n\\label{sec:Janus_sym_enhancement}\n\nSpecific choices of the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ translate into various (super) symmetry enhancements of the general Janus solution in (\\ref{A(mu)_func_U1^4}) and (\\ref{Janus_solution_U1^4_ztil}).\n\n\\subsubsection{\\texorpdfstring{$\\text{SO}(4) \\times\\text{SO}(4)$}{SO(4)xSO(4)} symmetry enhancement}\n\\label{sec:Janus_SO4xSO4}\n\nSetting two vector multiplets to zero, \\textit{e.g.} $\\,\\tilde{z}_{2}=\\tilde{z}_{3}=0$, by setting\n\\begin{equation}\n\\label{alpha_2_3=0}\n\\alpha_{2} = \\alpha_{3} = 0 \\ ,\n\\end{equation}\nand renaming $\\tilde{z}_{1} \\equiv\\tilde{z}\\,$, the $\\text{SO}(4) \\times\\text{SO}(4)$ invariant sector of the SO(8) gauged supergravity investigated in Section~$5$ of \\cite{Bobev:2013yra} is recovered upon the identification $\\tilde{z}=z_{\\text{there}}$. The Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) reduces to\n\\begin{equation}\n\\label{Lagrangian_model_SO4xSO4}\n\\begin{array}{lll}\n\\mathcal{L} & = & \\left(  \\frac{R}{2} - V \\right)  * 1 - \\frac{1}{4} \\left[ (d\\varphi)^{2} + e^{2 \\varphi} \\, (d\\chi)^{2} \\right] \\\\[2mm]\n& = & \\left(  \\frac{R}{2} - V \\right)  * 1 - \\dfrac{1}{\\left(  1-|\\tilde{z}|^{2} \\right)  ^{2}} \\, d\\tilde{z} \\wedge* d\\tilde{z}^{*} \\ ,\n\\end{array}\n\\end{equation}\nand the scalar potential in (\\ref{V_U1^4}) simplifies to\n\\begin{equation}\n\\label{V_SO4xSO4}\nV = - \\tfrac{1}{2} \\, g^{2} \\left(  4 + 2 \\cosh\\varphi+ \\chi^{2} \\, e^{\\varphi} \\right)  = -  g^{2} \\, \\dfrac{3 - |\\tilde{z}|^{2}}%\n{1-|\\tilde{z}|^{2}} \\ .\n\\end{equation}\nThe Janus solution then reads\n\\begin{equation}\n\\label{Janus_solution_SO4xSO4^4_alt}\nds_{4}^{2} = d\\mu^{2}+ e^{2 A(\\mu)} \\, d\\Sigma^{2} \n\\hspace{10mm} , \\hspace{10mm} \n\\tilde{z}(\\mu) = e^{i \\beta} \\,\n\\frac{\\sinh\\alpha}{\\cosh\\alpha + i \\, \\sinh(g \\mu) } \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{A(mu)_func_SO4xSO4}\ne^{2A(\\mu)} = (g k)^{-2} \\cosh^2(g\\mu)\n\\hspace{8mm} \\textrm{ and } \\hspace{8mm}\nk= \\cosh \\alpha \\ge 1\\ ,\n\\end{equation}\nwhere $(\\alpha, \\beta) = (\\alpha_{1} , \\beta_{1})$. This solution precisely matches the one presented in Section~$5$ of \\cite{Bobev:2013yra} upon the identification $\\cosh\\alpha=(1-a^{2}_{\\text{there}})^{-\\frac{1}{2}}$. As noticed therein, the Janus solution is half-PBS and preserves $\\,16\\,$ real supercharges. From a holographic perspective, the $(1+1)$-dimensional defect dual to the AdS$_{3}$ factor in the geometry features $(4,4)$ supersymmetry and therefore has an $\\,\\textrm{SO}(4)_{\\textrm{R}} \\times \\textrm{SO}(4)_\\textrm{R}\\,$ R-symmetry group. We have explicitly verified that, when selecting the minus sign in (\\ref{Janus_solution_SO4xSO4^4_alt}), the Janus solution satisfies the 1\/2-BPS equations (\\ref{BPS_A}) and (\\ref{BPS_scalars}) for the eight gravitino mass terms (superpotentials) of the maximal theory (see Footnote~\\ref{Footnote:axions}). Finally, the original M-theory supersymmetric Janus with $\\text{SO}(4) \\times\\text{SO}(4)$ symmetry was presented in \\cite{DHoker:2009lky}.\n\n\n\n\n\\subsubsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2$}{SU(3)xU(1)2} symmetry enhancement}\n\nIdentifying the three vector multiplets, namely $\\,\\tilde{z}_{1}=\\tilde{z}_{2}=\\tilde{z}_{3} \\equiv\\tilde{z}\\,$, so that\n\\begin{equation}\n\\alpha_{1} = \\alpha_{2} = \\alpha_{3} \\equiv\\alpha\n\\hspace{5mm} , \\hspace{5mm}\n\\beta_{1} = \\beta_{2} = \\beta_{3} \\equiv\\beta\\ ,\n\\end{equation}\nthe $\\text{SU}(3) \\times\\text{U}(1)^2$ invariant sector of Section~$6$ of \\cite{Bobev:2013yra} (see also \\cite{Pilch:2015dwa} for the 11D uplift) is recovered upon the identification $\\tilde{z}=z_{\\text{there}}$. The\nLagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) simplifies to\n\\begin{equation}\n\\label{Lagrangian_model_SU3xU1xU1}\n\\begin{array}{lll}\n\\mathcal{L} & = & \\left(  \\frac{R}{2} - V \\right)  * 1 - \\frac{3}{4} \\left[(d\\varphi)^{2} + e^{2 \\varphi} \\, (d\\chi)^{2} \\right] \\\\[2mm]\n& = & \\left(  \\frac{R}{2} - V \\right)  * 1 - \\dfrac{3}{\\left(  1-|\\tilde{z}|^{2} \\right)  ^{2}} \\, d\\tilde{z} \\wedge* d\\tilde{z}^{*} \\ ,\n\\end{array}\n\\end{equation}\nand the scalar potential in (\\ref{V_U1^4}) reduces to\n\\begin{equation}\n\\label{V_SU3xU1^2}\nV = - \\tfrac{3}{2}  \\, g^{2} \\left(  2 \\cosh\\varphi+ \\chi^{2} \\, e^{\\varphi} \\right)  = - 3 \\, g^{2} \\, \\dfrac{1+|\\tilde{z}|^{2}}{1-|\\tilde{z}|^{2}} \\ .\n\\end{equation}\nThe Janus solution takes the form\n\\begin{equation}\n\\label{Janus_solution_SU3xU1xU1}\nds_{4}^{2} = d\\mu^{2}+ e^{2 A(\\mu)} \\, d\\Sigma^{2} \n\\hspace{10mm} , \\hspace{10mm} \n\\tilde{z}(\\mu) = e^{i \\beta} \\, \\frac{\\sinh\\alpha}{\\cosh\\alpha + i \\, \\sinh(g \\mu) } \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{A(mu)_func_SU3xU1xU1}\ne^{2A(\\mu)} = (g k)^{-2} \\cosh^2(g\\mu) \n\\hspace{8mm} \\textrm{ and } \\hspace{8mm}\nk^2= 1 + 3 \\sinh^{2}\\alpha \\ge 1\\ .\n\\end{equation}\nThis provides an analytic solution in the $\\text{SU}(3) \\times\\text{U}(1)^2 $ invariant sector of the SO(8) maximal supergravity investigated in Section~$6$ of \\cite{Bobev:2013yra}. The solution (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}) satisfies the second-order equations of motion in (\\ref{EOM_scalars}) and (\\ref{EOM_Einstein}). However, we have verified that the BPS equations (\\ref{BPS_A}) and (\\ref{BPS_scalars}) are not satisfied for any of the eight gravitino mass terms (superpotentials) in the maximal SO(8) gauged supergravity, so the solution is non-supersymmetric.\n\n\n\n\\subsection{Janus geometry and boundary conditions}\n\nLet us discuss the geometry of the multi-parametric family of Janus solutions presented in the previous sections. Introducing embedding coordiantes in $\\mathbb{R}^{2,3}$, the $k$-family of Janus metrics in (\\ref{metric_ansatz}) and (\\ref{A(mu)_func_U1^4}) corresponds to\n\\begin{equation}\n\\label{embedding_coordinates}\n\\begin{array}{lll}\nX_{0} &=& (g k)^{-1} \\, \\dfrac{\\cos\\tau}{\\cos\\eta} \\, \\cosh(g \\mu) \\ , \\\\[6mm]\nX_{4} &=& (g k)^{-1} \\, \\dfrac{\\sin\\tau}{\\cos\\eta} \\, \\cosh(g \\mu)   \\ , \\\\[6mm]\nX_{1} &=&  (g k)^{-1} \\, \\tan\\eta \\cos\\theta \\, \\cosh(g \\mu)  \\ , \\\\[6mm]\nX_{2} &=&  (g k)^{-1} \\, \\tan\\eta \\sin\\theta \\, \\cosh(g \\mu)   \\ , \\\\[6mm]\nX_{3} &=&  g^{-1} \\,   i \\, \\textrm{E}(i g \\mu \\, ; \\, k^{-2}) \\ ,\n\\end{array}\n\\end{equation}\nwith \n\\begin{equation}\nk^2 = 1 + \\sum_{i}\\sinh^{2}\\alpha_{i} \\, \\ge \\, 1 \\ ,\n\\end{equation}\nand $\\textrm{E}(i g \\mu \\, ; \\, k^{-2})$ being the incomplete elliptic integral of the second kind. The solution describes the hyper-surface\n\\begin{equation}\n\\label{hypersurface_X}\n-X_{0}^2 - X_{4}^2 + X_{1}^2 + X_{2}^2 +(g k)^{-2} \\sinh^2(g \\mu) = -  (g k)^{-2} \\ ,\n\\end{equation}\nwhere the term $\\,(g k)^{-2} \\sinh^2(g \\mu)\\,$ is implicitly given in terms of $X_{3}$ by the last relation in (\\ref{embedding_coordinates}). For $\\,k=1\\,$ one has that $\\,i \\, \\textrm{E}(i g \\mu \\, ; \\,1) = - \\sinh(g \\mu)\\,$ and (\\ref{hypersurface_X}) reduces to the hyperboloid describing AdS$_{4}$.\n\n\\subsubsection{Global coordinates and boundary structure}\n\n\nLet us perform a change of coordinates that will help us to understand the Janus geometry in (\\ref{metric_ansatz}) and (\\ref{A(mu)_func_U1^4})-(\\ref{k_factor}), especially its boundary structure. We start by performing a change of radial coordinate to make its range compact\n\\begin{equation}\n\\tilde{\\mu} = 2 \\, k \\, \\textrm{arctan} \\left[ \\tanh \\left( \\frac{g \\, \\mu}{2}  \\right)\\right] \\ ,\n\\end{equation}\nand then choose global coordinates to describe the AdS$_{3}$ slicing in (\\ref{metric_ansatz}). The Janus metric in (\\ref{metric_ansatz}) and (\\ref{A(mu)_func_U1^4})-(\\ref{k_factor}) then becomes conformal to $\\,\\mathbb{R} \\times \\textrm{S}^{3}\\,$\n\\begin{equation}\n\\label{Janus_metric_original}\nds_{4}^{2} = \\frac{(g k)^{-2} }{\\cos^2\\left( \\frac{\\tilde{\\mu}}{k}\\right) \\cos^{2}\\eta} \\,  \\left( - d\\tau^2  + \\cos^{2}\\eta \\, d\\tilde{\\mu}^{2} + d\\eta^2 + \\sin^2\\eta \\, d\\theta^2\\right) \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{global_coords_ranges}\n\\tau \\in (-\\infty \\, , \\infty)\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\tilde{\\mu} \\in [-\\frac{\\pi k}{2} \\, , \\frac{\\pi k}{2}]\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\eta \\in [0 \\, , \\frac{\\pi}{2}] \n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\theta \\in [0 \\, , 2 \\pi]  \\ .\n\\end{equation}\nThese are the global coordinates used to describe the original type IIB Janus solution in \\cite{Bak:2003jk,Clark:2004sb}. The geometry (\\ref{Janus_metric_original}) has a boundary that consists of two hemi-spheres of $\\,\\textrm{S}^2\\,$ at $\\,\\tilde{\\mu} = \\pm \\tilde{\\mu}_{0}\\,$, with $\\, \\tilde{\\mu}_{0}=\\frac{\\pi k}{2} \\,$, joined at the $\\textrm{S}^1$ equator at $\\,\\eta = \\frac{\\pi}{2}\\,$. Lastly, using the new radial coordinate $\\,\\tilde{\\mu}\\,$, the profiles for the complex scalars in (\\ref{Janus_solution_U1^4_ztil}) become\n\\begin{equation}\n\\label{Janus_solution_U1^4_new}\n\\tilde{z}_{i}(\\tilde{\\mu}) = e^{i \\beta_{i}}\\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\tan\\left(\\frac{\\tilde{\\mu}}{k}\\right) }  \\ ,\n\\end{equation}\nso that $\\,\\tilde{z}_{i}(\\tilde{\\mu}) \\rightarrow 0\\,$ when approaching the two hemi-spheres of $\\,\\textrm{S}^2\\,$ at $\\,\\tilde{\\mu} \\rightarrow \\pm \\tilde{\\mu}_{0} \\,$ in the Janus boundary. Note that $\\,\\textrm{arg}\\left[\\tilde{z}_{i}(\\tilde{\\mu}_{0})\\right] - \\textrm{arg}\\left[\\tilde{z}_{i}(-\\tilde{\\mu}_{0})\\right] = \\pi$, thus creating an interface discontinuity at the $\\,\\textrm{S}^{1}\\,$ equator where the defect lives.\n\n\n\\subsubsection{\\texorpdfstring{AdS$_{3}$}{AdS3} slicing and boundary conditions}\n\\label{sec:boundary conditions}\n\n\nIn order to investigate the boundary conditions of the the family of Janus solution in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) we will perform a regular change of radial coordinate\n\\begin{equation}\n\\label{new_coordinate_Janus}\n\\rho= \\sinh({g} \\mu) \n\\hspace{10mm} , \\hspace{10mm}\nd\\mu = g^{-1} \\, \\dfrac{d\\rho}{\\sqrt{\\rho^2+1}}  \\ ,\n\\end{equation}\nso that the family of Janus solutions in (\\ref{A(mu)_func_U1^4})-(\\ref{Janus_solution_U1^4_ztil}) becomes\\footnote{The Ricci scalar constructed from the metric (\\ref{Janus_U1^4_rho_1}) reads\n\\begin{equation}\n\\label{Janus_Ricci}\nR(\\rho) = - 6 \\, g^{2} \\left( \\,  1 +  \\frac{\\rho^2 + k^2}{\\rho^2 + 1} \\, \\right) \\ ,\n\\end{equation}\nthus ensuring regularity of the Janus geometry within the whole range $\\,\\rho \\in ( -\\infty , \\infty)\\,$.}\n\\begin{equation}\n\\label{Janus_U1^4_rho_1}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left(  \\frac{d\\rho^{2}}{\\rho^{2}+1} + \n\\frac{\\rho^{2}+1  }{ k^2} \\, d\\Sigma^{2}  \\right)\n\\hspace{5mm} , \\hspace{5mm}\nd\\Sigma^{2} = \\frac{1}{\\cos^{2}\\eta}\\left( - d\\tau^2 + d\\eta^2 + \\sin^2\\eta \\, d\\theta^2\\right) \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{global_coords_ranges_2}\n\\tau \\in (-\\infty \\, , \\infty)\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\rho  \\in (-\\infty \\, , \\infty)\n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\eta \\in [0 \\, , \\frac{\\pi}{2}] \n\\hspace{5mm} \\textrm{ , } \\hspace{5mm}\n\\theta \\in [0 \\, , 2 \\pi]  \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Janus_U1^4_rho_2}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}} \\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\rho} \\ .\n\\end{equation}\nThe Janus geometry (\\ref{Janus_U1^4_rho_1}) has a three-dimensional conformal boundary at $\\,\\rho \\rightarrow \\pm \\infty\\,$ that is conformal to $\\,\\mathbb{R} \\times \\textrm{S}^{2}\\,$ with a $k$-dependent prefactor $\\,(g k)^{-2} \\, \\rho^2\\,$. This is the geometry we will use to analyse the asymptotic behaviour of the $\\,\\textrm{U}(1)^4\\,$ invariant complex scalars (\\ref{Janus_U1^4_rho_2}).\n\n\n\nWhen approaching the maximally supersymmetric AdS$_4$ vacuum dual to ABJM theory\\footnote{\\label{footnote:operators}The 35 pseudo-scalars and 35 proper scalars of the maximal supergravity multiplet are dual to single-trace deformations of ABJM theory \\cite{Aharony:2008ug}. More concretely, pseudo-scalars are dual to fermionic bilinears $\\,\\mathcal{O}_{F}=\\textrm{Tr}(\\psi^{\\dot{A}} \\psi^{\\dot{B}}) - \\frac{1}{8} \\delta^{\\dot{A}\\dot{B}} \\textrm{Tr}(\\psi^{\\dot{C}} \\psi^{\\dot{C}} )\\,$ with $\\,\\dot{A}=1,\\ldots,8\\,$ and $\\,\\textrm{dim}(\\mathcal{O}_{F})=2\\,$. Proper scalars are dual to bosonic bilinears $\\,\\mathcal{O}_{B}=\\textrm{Tr}(X^{A} X^{B}) - \\frac{1}{8} \\delta^{AB} \\textrm{Tr}(X^{C}X^{C})\\,$ with $\\,A=1,\\ldots,8\\,$ and $\\,\\textrm{dim}(\\mathcal{O}_{B})=1\\,$.}, the asymptotic behaviour of (\\ref{Janus_U1^4_rho_2}) around the endpoints $\\,\\rho \\rightarrow \\pm \\infty\\,$ of the Janus solution reads\n\\begin{equation}\n\\label{source&vevs_zt}\n\\tilde{z}_{i}(\\rho) = \\dfrac{\\tilde{z}_{i,0}}{\\rho} +   \\dfrac{\\tilde{z}_{i,1}}{\\rho^2}  + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) \n\\hspace{10mm} \\textrm{ with } \\hspace{10mm} \ni=1,2,3 \\ , \n\\end{equation}\nin terms of normalisable modes $\\,\\tilde{z}_{i,0}\\,$ with $\\,\\Delta_{i}=1\\,$ specified by the parameters $\\,(\\alpha_{i} , \\beta_{i})\\,$,\n\\begin{equation}\n\\label{zt_0}\n\\tilde{z}_{i,0} = \\sinh\\alpha_{i} \\, e^{i (\\beta_i - \\frac{\\pi}{2})}  \\ ,\n\\end{equation}\nas well as normalisable modes $\\,\\tilde{z}_{i,1}\\,$ with  $\\,\\Delta_{i}=2\\,$. These modes satisfy a set of $\\alpha_{i}$-dependent algebraic relations\n\\begin{equation}\n\\label{zt_1}\n\\tilde{z}_{i,1} - i \\cosh\\alpha_{i}  \\, \\tilde{z}_{i,0} = 0  \\ .\n\\end{equation}\nThe on-shell relations (\\ref{zt_1}) will help us to characterise the deformations in the field theory dual of the Janus solution upon appropriate manipulation of boundary terms and finite counterterms. \n\n\n\nIn order to discuss the boundary conditions (\\ref{source&vevs_zt})--(\\ref{zt_1}) in more detail, we will resort to an expansion of $\\,\\text{Re}\\tilde{z}_{i}\\,$ (proper scalars) and $\\,\\text{Im}\\tilde{z}_{i}\\,$ (pseudo-scalars) around $\\rho\\rightarrow\\pm\\infty$. This yields\n\\begin{equation}\n\\label{source&vevs_rho_f}\n\\begin{array}{llll}\n\\text{Re}\\tilde{z}_{i}(\\rho) & = & \\dfrac{a^{(v)}_{i,0}}{\\rho} +   \\dfrac{a^{(s)}_{i,1}}{\\rho^2}  + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & , \\\\[4mm]\n\\text{Im}\\tilde{z}_{i}(\\rho) & = & \\dfrac{b^{(s)}_{i,0}}{\\rho} + \\dfrac{b^{(v)}_{i,1}}{\\rho^2}  + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & ,\n\\end{array}\n\\end{equation}\nso that\n\\begin{equation}\n\\label{zToab}\n\\tilde{z}_{i,0} = a^{(v)}_{i,0} + i \\, b^{(s)}_{i,0}\n\\hspace{5mm} , \\hspace{5mm}\n\\tilde{z}_{i,1} = a^{(s)}_{i,1} + i \\, b^{(v)}_{i,1} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{ab_definition}\na^{(v)}_{i,0}=\\sinh\\alpha_{i} \\, \\sin\\beta_i\n\\hspace{5mm} , \\hspace{5mm}\nb^{(s)}_{i,0} = - \\sinh\\alpha_{i} \\, \\cos\\beta_i\n\\ .\n\\end{equation}\nThe algebraic relations in (\\ref{zt_1}) then become\n\\begin{equation}\n\\label{ab_algebraic}\na^{(s)}_{i,1}  + \\cosh\\alpha_{i} \\,\\,  b^{(s)}_{i,0} = 0\n\\hspace{5mm} , \\hspace{5mm}\nb^{(v\t)}_{i,1} - \\cosh\\alpha_{i} \\,\\, a^{(v)}_{i,0} = 0 \\ .\n\\end{equation}\nNote that the independent parameters specifying the boundary conditions in (\\ref{ab_definition}) are $\\,(\\alpha_{i} , \\beta_{i})\\,$. As a consequence, the coefficients in the expansions (\\ref{source&vevs_rho_f}) obey the following two sets of algebraic relations \n\\begin{equation}\n\\dfrac{\\left(a^{(s)}_{i,1}\\right)^2}{\\left(b^{(s)}_{i,0}\\right)^2} = 1+ |\\tilde{z}_{i,0} |^2\n\\hspace{8mm} , \\hspace{8mm}\n\\dfrac{\\left(b^{(v)}_{i,1}\\right)^2}{\\left(a^{(v)}_{i,0}\\right)^2} = 1+  |\\tilde{z}_{i,0} |^2 \\  .\n\\end{equation}\nLastly, following \\cite{Bobev:2011rv} (see also \\cite{Bobev:2013yra}), we have attached the labels ``source\" $\\,^{(s)}\\,$ and ``VEV\" $\\,^{(v)}\\,$ to the modes in (\\ref{source&vevs_rho_f}) to highlight that, in order to preserve maximal supersymmetry, proper scalars should feature the alternative quantisation and pseudo-scalars the standard quantisation. Note that setting $\\,\\beta_{i}=\\pm\\frac{\\pi}{2}\\,$ switches off the sources in (\\ref{source&vevs_rho_f}) leaving only the VEV's. This is in agreement with the standard AdS\/CFT prescription and renders $\\,\\tilde{z}_{i,0}\\,$ in (\\ref{zt_0}) real.\n\n\n\n\n\n\\subsubsection{Janus solutions and boundary conditions}\n\n\n\nLet us compute the on-shell variation of the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}). A standard computation yields the boundary term\n\\begin{equation}\n\\label{deltaS}\n\\delta S = \\displaystyle\\sum_{i} \\delta S_{i}  =  \\displaystyle\\sum_{i} \\int d^{4}x \\,\\, \\partial_{\\mu} \\theta^{\\mu}_{i} = - \\displaystyle\\sum_{i}  \\int_{\\partial M}d^{3}x  \\frac{\\sqrt{-h}}{\\left( 1-\\left\\vert\n\\tilde{z}_{i}\\right\\vert ^{2}\\right)^{2}} \\, N^{\\mu} \\, ( \\partial_{\\mu } \\tilde{z}_{i} \\, \\delta\\tilde{z}_{i}^{\\ast } + \\textrm{c.c.} ) \\ ,\n\\end{equation}\nwhere \n\\begin{equation}\n\\theta^{\\mu}_{i}  \\equiv  - \\frac{\\sqrt{-g}}{\\left( 1-\\left\\vert\n\\tilde{z}_{i}\\right\\vert^{2}\\right)^{2}} \\, g^{\\mu \\nu }  \\left( \\partial_{\\nu} \\tilde{z}_{i} \\,\\,\n\\delta\\tilde{z}_{i}^{\\ast} + \\textrm{c.c.}  \\right) \\ ,    \n\\end{equation}\nand c.c stands for complex conjugation. In (\\ref{deltaS}) we have introduced the standard foliation $\\,g_{\\mu \\nu }=h_{\\mu \\nu }+N_{\\mu} N_{\\nu }\\,$ with $\\,N_{\\mu }=\\sqrt{g_{\\rho \\rho }} \\, \\delta _{\\mu }^{\\rho }\\,$ being the vector normal to the AdS$_{3}$ leaves. \n\nPlugging into (\\ref{deltaS}) the asymptotic expansion of the scalars in (\\ref{source&vevs_zt}) around $\\,\\rho \\rightarrow \\pm \\infty\\,$, and using the asymptotic form of the metric (\\ref{Janus_U1^4_rho_1}), we encounter the well known linearly divergent term. In order to regularise the above boundary action and have a well-defined variational principle we introduce, for each complex field $\\,\\tilde{z}_{i}\\,$, the counter-term\n\\begin{equation}\n\\label{counterterm}\nS_{\\textrm{ct},i} = - \\, g  \\lim_{\\rho \\rightarrow \\pm\\infty } \\, \\int_{\\partial M}   \nd^{3}x  \\sqrt{-h}  \\,\\,\n\\tilde{z}_{i} \\, \\tilde{z}_{i}^{\\ast }  \\ ,\n\\end{equation}\nso that\n\\begin{equation}\n\\label{boundary_contributions}\n\\delta S_{i} + \\delta S_{\\textrm{ct},i}  = g^{-2} \\, k^{-3} \\int_{\\partial M}     \\left(   \\tilde{z}_{i,1} \\, \\delta \\tilde{z}_{i,0}^{\\ast }  +  \\tilde{z}_{i,1}^{\\ast} \\, \\delta \\tilde{z}_{i,0}  \\right) \\, d\\Sigma \\ ,\n\\end{equation}\nin terms of the volume element at the boundary $\\,d\\Sigma=\\sqrt{-\\gamma} \\, d^{3}x \\,$ with $\\, \\sqrt{-\\gamma} =\\sin\\eta \\, \\cos^{-3}\\eta\\,$. Substituting the scalar mode parameterisation of (\\ref{zToab}) into the boundary contributions in (\\ref{boundary_contributions}) one obtains\n\\begin{equation}\n\\label{boundary_contributions_ab}\n\\delta S_i + \\delta S_{\\textrm{ct},i} =  2 \\, g^{-2} \\, k^{-3}  \\int_{\\partial M} \\left(   a^{(s)}_{i,1} \\, \\delta a^{(v)}_{i,0}  +  b^{(v)}_{i,1} \\, \\delta b^{(s)}_{i,0}  \\right) \\, d\\Sigma \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{k_factor_alt}\nk^2= 1 + \\sum_{i}\\sinh^{2}\\alpha_{i}  = 1 + \\sum_{i} |\\tilde{z}_{i,0} |^2 \\, \\ge \\, 1 \\ .\n\\end{equation}\nIn order to remove the $\\,k^{-3}\\,$ factor in (\\ref{boundary_contributions_ab}) we could rescale the radial coordinate as $\\,\\hat{\\rho} = k \\, \\rho\\,$ or, instead, perform the non-linear mode redefinitions\n\\begin{equation}\n\\label{rescaled_ab}\n\\hat{a}^{(v)}_{i,0}  = k^{-1} \\, a^{(v)}_{i,0} \n\\hspace{5mm} , \\hspace{5mm}\n\\hat{a}^{(s)}_{i,1}  = k^{-2} \\, a^{(s)}_{i,1}\n\\hspace{5mm} , \\hspace{5mm}\n\\hat{b}^{(s)}_{i,0}  = k^{-1} \\, b^{(s)}_{i,0} \n\\hspace{5mm} , \\hspace{5mm}\n\\hat{b}^{(v)}_{i,1}  = k^{-2} \\, b^{(v)}_{i,1}  \\ .\n\\end{equation}\nFollowing the latter prescription, the boundary contribution in (\\ref{boundary_contributions_ab}) becomes\n\\begin{equation}\n\\label{boundary_contributions_ab_hat}\n\\delta S_{i} + \\delta S_{\\textrm{ct},i} \\, =  \\, 2 \\, g^{-2}  \\int_{\\partial M} \\left(   \\hat{a}^{(s)}_{i,1} \\, \\delta \\hat{a}^{(v)}_{i,0}  +  \\hat{b}^{(v)}_{i,1} \\, \\delta \\hat{b}^{(s)}_{i,0}  \\right) \\, d\\Sigma \\ ,\n\\end{equation}\nand, due to the alternative quantisation featured by the proper scalars, we must add an extra boundary term such that\n\\begin{equation}\n\\label{boundary_contributions_ab_hat_final}\n\\delta S_{i} + \\delta S_{\\textrm{ct},i} - \\delta \\left( 2 \\, g^{-2}  \\int_{\\partial M}  \\hat{a}^{(s)}_{i,1} \\, \\hat{a}^{(v)}_{i,0}  \\right)  \\, =  \\, 2 \\, g^{-2}  \\int_{\\partial M} \\left( \\hat{b}^{(v)}_{i,1} \\, \\delta \\hat{b}^{(s)}_{i,0}  -   \\hat{a}^{(v)}_{i,0} \\, \\delta \\hat{a}^{(s)}_{i,1} \\right) \\, d\\Sigma \\ .\n\\end{equation}\nTherefore, having a well-defined variational principle therefore requires $\\,\\delta \\hat{b}^{(s)}_{i,0} = \\delta \\hat{a}^{(s)}_{i,1} = 0\\,$. Recalling from (\\ref{ab_definition})-(\\ref{ab_algebraic}) that \n\\begin{equation}\nb^{(s)}_{i,0} = - \\sinh\\alpha_{i} \\, \\cos\\beta_i \n\\hspace{10mm} \\textrm{ and } \\hspace{10mm}\na^{(s)}_{i,1} = - \\cosh\\alpha_{i} \\,\\,  b^{(s)}_{i,0} \\ ,\n\\end{equation}\nwe conclude that sources are generically present at the boundary theory of the Janus ($\\alpha_{i} \\neq 0$) except for the particular choice of boundary conditions $\\,\\beta_{i}= \\pm \\frac{\\pi}{2}\\,$. This implies that every choice of $\\,(\\alpha_{i}, \\beta_{i})\\,$ with $\\,\\beta_{i} \\neq \\pm \\frac{\\pi}{2}\\,$ corresponds to a different theory with a different value of the sources in the variational principle. On the contrary, when $\\,\\beta_{i}= \\pm \\frac{\\pi}{2}\\,$, the sources are zero on-shell and the boundary theory is unique.\n\n\n\n\n\n\n\n\\subsection{Multi-parametric Hades solutions}\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.50\\textwidth]{Plots\/boundary_Hades_straight.pdf} \n\\put(10,95){$\\textrm{Re}\\tilde{z}_{i}$} \\put(-105,228){$\\textrm{Im}\\tilde{z}_{i}$}\n\\put(-112.5,107.5){{\\color{red}{$\\bullet$}}}\n\\end{center}\n\\caption{Parametric plot of $\\,\\tilde{z}_{i}(\\rho)\\,$ in (\\ref{Hades_U1^4_rho_2}) for the Hades solutions with $\\,\\alpha_{i}=1\\,$ (blue-solid lines) and the ridge flows with $\\,\\alpha_{i}=0\\,$ (brown-dashed lines) upon setting $\\,\\beta_{i}=\\frac{n\\pi}{4}\\,$ with $\\,n=0,\\ldots,7\\,$. The central red point at $\\,\\tilde{z}_{i}=0\\,$ $\\,\\forall i\\,$ corresponds to the maximally supersymmetric AdS$_{4}$ vacuum and describes the asymptotic values at $\\,\\rho\\rightarrow \\infty\\,$. The boundary circle at $\\,|\\tilde{z}_{i}|=1\\,$ corresponds to the singularity at $\\,\\rho = 1\\,$.}\n\\label{fig:Hades_ztilde_U1^4}\n\\end{figure}\n\n\n\n\nStarting from the field equations in (\\ref{EOM_scalars})-(\\ref{EOM_Einstein}) and performing a change of radial coordinate\n\\begin{equation}\n\\label{new_coordinate_Hades}\n\\rho= \\cosh({g} \\mu) \n\\hspace{10mm} , \\hspace{10mm}\nd\\mu = g^{-1} \\, \\dfrac{d\\rho}{\\sqrt{\\rho^2-1}}  \\ ,\n\\end{equation}\nwe find a new class of singular solutions of the form\n\\begin{equation}\n\\label{Hades_U1^4_rho_1}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left(  \\frac{d\\rho^{2}}{\\rho^{2}-1} + \n\\frac{\\rho^{2}-1  }{ k^2} \\, d\\Sigma^{2}   \\right)\n\\hspace{10mm} \\textrm{ with } \\hspace{10mm}\nk^2= - 1 + \\sum_{i} \\cosh^2\\alpha_{i} \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Hades_U1^4_rho_2}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}}\\, \\frac{\\cosh\\alpha_{i}}{\\sinh\\alpha_{i} + i \\rho} \\ .\n\\end{equation}\nThese solutions are defined in the domain $\\,\\rho \\in [1,\\infty)\\,$ and feature a singularity at $\\,\\rho=1\\,$ where the change of radial coordinate in (\\ref{new_coordinate_Hades}) is ill-defined, the warping factor in front of the AdS$_{3}$ piece in the geometry collapses to zero size and $\\,|\\tilde{z}_{i}(1)|=1\\,$ (see Figure~\\ref{fig:Hades_ztilde_U1^4}). More concretely, the Ricci scalar constructed from the metric (\\ref{Hades_U1^4_rho_1}) reads\n\\begin{equation}\n\\label{Hades_Ricci}\nR(\\rho) = - 6 \\, g^{2} \\left( \\,  1 +  \\frac{\\rho^2 + k^2}{\\rho^2 - 1} \\, \\right) \\ ,\n\\end{equation}\nand becomes singular at $\\,\\rho= 1\\,$. An analysis of the BPS equations (\\ref{BPS_scalars}) shows that the flows in (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}) turn out to be non-supersymmetric. We will refer to these singular solutions as flows to Hades. This term was coined for singular (flat-sliced) domain-walls dual to conventional RG-flows in \\cite{Freedman:1999gp,Gubser:2000nd}.\n\n\nAs previously done for the Janus solution, let us expand $\\,\\text{Re}\\tilde{z}_{i}\\,$ (proper scalars) and $\\,\\text{Im}\\tilde{z}_{i}\\,$ (pseudo-scalars) around $\\,\\rho\\rightarrow \\infty\\,$. One finds\n\\begin{equation}\n\\label{Hades_source&vevs_rho}\n\\begin{array}{llll}\n\\text{Re}\\tilde{z}_{i}(\\rho) & = & \\dfrac{a^{(v)}_{i,0}}{\\rho} +   \\dfrac{a^{(s)}_{i,1}}{\\rho^2}  + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & , \\\\[4mm]\n\\text{Im}\\tilde{z}_{i}(\\rho) & = & \\dfrac{b^{(s)}_{i,0}}{\\rho} + \\dfrac{b^{(v)}_{i,1}}{\\rho^2}  + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & ,\n\\end{array}\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Hades_ab_vevs}\na^{(v)}_{i,0}=\\cosh\\alpha_{i} \\, \\sin\\beta_i\n\\hspace{5mm} , \\hspace{5mm}\nb^{(v\t)}_{i,1} = \\sinh\\alpha_{i} \\,\\, a^{(v)}_{i,0} \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Hades_ab_sources}\nb^{(s)}_{i,0} = - \\cosh\\alpha_{i} \\, \\cos\\beta_i\n\\hspace{5mm} , \\hspace{5mm}\na^{(s)}_{i,1}  = - \\sinh\\alpha_{i} \\,\\,  b^{(s)}_{i,0} \\ .\n\\end{equation}\nDifferent choices of the Hades parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ translate into different boundary conditions in the expansions (\\ref{Hades_source&vevs_rho}). Note that the boundary theory has sources in (\\ref{Hades_ab_sources}) generically activated except if setting $\\,\\beta_{i}=\\pm\\frac{\\pi}{2}\\,$.\n\n\n\n\\subsubsection{Ridge flows with \\texorpdfstring{$\\,\\alpha_{i}=0\\,$}{alpha=0}}\n\\label{sec:ridge_4D}\n\n\nUnlike for the Janus solutions, setting $\\,\\alpha_{i}=0\\,$ does not recover a regular AdS$_{4}$ vacuum. Instead, the complex scalars in (\\ref{Hades_U1^4_rho_2}) reduce to\n\\begin{equation}\n\\label{Ridge_U1^4_rho_2}\n\\tilde{z}_{i}(\\rho) = \\rho^{-1}  \\, e^{i \\left( \\beta_{i} - \\frac{\\pi}{2} \\right)} \\ ,\n\\end{equation}\nand \\textit{ridge flows} of the type investigated in \\cite{Pilch:2015dwa,Pilch:2015vha} appear with constant $\\,\\textrm{arg}\\tilde{z}_{i} = \\beta_{i} - \\frac{\\pi}{2}\\,$ and $\\,k^2=2\\,$ in the singular geometry (\\ref{Hades_U1^4_rho_1}). The $\\,\\rho \\rightarrow \\infty\\,$ expansion in (\\ref{Hades_source&vevs_rho}) and the boundary conditions in (\\ref{Hades_ab_vevs})-(\\ref{Hades_ab_sources}) also simplify drastically\n\n\\begin{equation}\n\\label{Ridge_source&vevs_rho}\n\\begin{array}{llll}\n\\text{Re}\\tilde{z}_{i}(\\rho) & = & \\dfrac{a^{(v)}_{i,0}}{\\rho} +   \\dfrac{a^{(s)}_{i,1}}{\\rho^2}  + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & , \\\\[4mm]\n\\text{Im}\\tilde{z}_{i}(\\rho) & = & \\dfrac{b^{(s)}_{i,0}}{\\rho} + \\dfrac{b^{(v)}_{i,1}}{\\rho^2}  + \\mathcal{O}\\left( \\dfrac{1}{\\rho^3}\\right) & ,\n\\end{array}\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Ridge_ab_definition}\na^{(v)}_{i,0}= \\sin\\beta_i\n\\hspace{8mm} , \\hspace{8mm}\nb^{(s)}_{i,0} = - \\cos\\beta_i\n\\hspace{8mm} , \\hspace{8mm}\na^{(s)}_{i,1}  = 0\n\\hspace{8mm} , \\hspace{8mm}\nb^{(v\t)}_{i,1}  = 0 \n\\ .\n\\end{equation}\nTwo special cases are immediately identified. Setting $\\,\\beta_{i}=0,\\pi\\,$ renders $\\,\\tilde{z}_{i}(\\rho)\\,$ purely imaginary and the ridge flow from the maximally supersymmetric AdS$_{4}$ vacuum at $\\,\\rho \\rightarrow \\infty\\,$ is triggered by the source modes $\\,b^{(s)}_{i,0}\\,$ of the pseudo-scalars dual to fermion bilinears. On the contrary, setting $\\,\\beta_{i}=\\pm\\frac{\\pi}{2}\\,$ renders $\\,\\tilde{z}_{i}(\\rho)\\,$ purely real and the ridge flow is triggered by the VEV modes $\\,a^{(v)}_{i,0}\\,$ of the proper scalars dual to boson bilinears. As we will see in Section~\\ref{sec:Uplift_11D}, the uplift of these special ridge flows to eleven dimensions will be very different. This is to be contrasted with the situation in four dimensions where the Lagrangian (\\ref{Lagrangian_model_U1^4_Einstein-scalars}) is invariant under constant shifts of $\\,\\beta_{i}\\,$. Note also that a shift of the form $\\,\\beta_{i} \\rightarrow \\beta_{i} + \\pi\\,$ amounts to a reflection $\\,\\rho \\rightarrow -\\rho\\,$ in the respective field $\\,\\tilde{z}_{i}\\,$ in (\\ref{Ridge_U1^4_rho_2}) while leaving the Hades metric in (\\ref{Hades_U1^4_rho_1}) invariant. Since the domain of the radial coordinate is fixed to $\\,\\rho \\in [1,\\infty)\\,$, the shift $\\,\\beta_{i} \\rightarrow \\beta_{i} + \\pi\\,$ generically generates a new solution. \n\n\n\n\n\nA fundamental difference between our ridge flows in (\\ref{Hades_U1^4_rho_1}) and (\\ref{Ridge_U1^4_rho_2}) and the ones investigated in \\cite{Pope:2003jp,Pilch:2015dwa,Pilch:2015vha} is that the ones there have a flat-sliced geometry. Therefore they correspond to conventional holographic RG-flows. Our solutions have an AdS$_{3}$-slicing of the geometry, instead. It was further shown in \\cite{Pilch:2015dwa,Pilch:2015vha} that, for the flat-sliced solutions, only a set of discrete values of $\\,\\textrm{arg}\\tilde{z}_{i}\\,$ was compatible with supersymmetry. However, if relaxing supersymmetry, any value of $\\,\\textrm{arg}\\tilde{z}_{i}\\,$ was permitted. In our non-supersymmetric ridge flows, any possible value of $\\,\\beta_{i}\\,$ is permitted too. Generic flows to Hades with $\\,\\alpha_{i} \\neq 0\\,$ and ridge flows with $\\,\\alpha_{i} = 0\\,$ are depicted in Figure~\\ref{fig:Hades_ztilde_U1^4}.\n\n\n\n\n\\subsubsection{Hades with (super) symmetry enhancements}\n\n\n\nAs already discussed for the Janus solutions in Section~\\ref{sec:Janus_sym_enhancement}, imposing identifications between the complex fields $\\,\\tilde{z}_{i}(\\rho)\\,$ translates into different patterns of (super) symmetry enhancements. For example, non-supersymmetric Hades solutions with $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetry are obtained upon identifying the three complex scalars, namely, upon setting $\\,{\\alpha_{1}=\\alpha_{2}=\\alpha_{3}}\\,$ and $\\,{\\beta_{1}=\\beta_{2}=\\beta_{3}}\\,$ in the general Hades solution (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}).\n\n\n\n\nSupersymmetric Hades solutions with an AdS$_{3}$ slicing have previously been constructed in \\cite{Bobev:2013yra} within the $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ invariant sector of the $\\,\\textrm{SO}(8)\\,$ gauged supergravity. As discussed in Section~\\ref{sec:Janus_SO4xSO4}, this sector of the theory is recovered upon setting two of the three complex fields $\\,\\tilde{z}_{i}\\,$ to zero, \\textit{i.e.}, $\\,\\tilde{z}_{2}(\\rho)=\\tilde{z}_{3}(\\rho)=0\\,$. However, it is easy to see that this cannot be achieved by tuning the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ in (\\ref{Hades_U1^4_rho_2}) to any real value. Instead, one must set two complex fields to zero from the start and search for solutions of the field equations. In this manner, one finds Hades solutions of the form \n\\begin{equation}\n\\label{Hades_no_ridge_SO(4)xSO(4)_rho_1}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left(  \\frac{d\\rho^{2}}{\\rho^{2}-1} + \n\\frac{\\rho^{2}-1  }{ k^2} \\, d\\Sigma^{2}   \\right) \n\\hspace{10mm} \\textrm{ with } \\hspace{10mm}\nk^2= \\sinh^2\\alpha_{1}\n\\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{Hades_no_ridge_SO(4)xSO(4)_rho_2}\n\\tilde{z}_{1}(\\rho) = e^{i \\beta_{1}}\\, \\frac{\\cosh\\alpha_{1}}{\\sinh\\alpha_{1} + i \\rho} \n\\hspace{8mm} , \\hspace{8mm} \\tilde{z}_{2}(\\rho)=\\tilde{z}_{3}(\\rho)=0 \\ ,\n\\end{equation}\nwhich turn out to solve the BPS equations in (\\ref{BPS_A})-(\\ref{BPS_scalars}). It is worth emphasising that these supersymmetric Hades with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)$ symmetry do not belong to the same class of solutions as the non-supersymmetric Hades in (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}). Also, they do not admit a ridge flow limit since setting $\\,\\alpha_{1}=0\\,$ implies having a pathological ($k^2=0$) warping of AdS$_{3}$ in the geometry (\\ref{Hades_no_ridge_SO(4)xSO(4)_rho_1}).\n\n\n\n\\section{Uplift to eleven-dimensional supergravity}\n\\label{sec:Uplift_11D}\n\nIn this section we present the uplift to eleven-dimensional supergravity of the Janus and Hades solutions constructed within the four-dimensional SO(8) gauged supergravity. We use the conventions of \\cite{Gauntlett:2002fz} according to which the Lagrangian of eleven-dimensional supergravity \\cite{Cremmer:1978km} takes the form\n\\begin{equation}\n\\mathcal{L}_{11} = \\hat{R} \\, \\text{vol}_{11} - \\tfrac{1}{2} \\, \\hat{F}_{(4)} \\wedge*_{11} \\hat{F}_{(4)} - \\tfrac{1}{6} \\, \\hat{A}_{(3)} \\wedge\\hat{F}_{(4)} \\wedge\\hat{F}_{(4)} \\ .\n\\end{equation}\nA consistent background is then subject to the source-less Bianchi identity\n\\begin{equation}\n\\label{BI_F4}\nd\\hat{F}_{(4)} = 0 \\ ,\n\\end{equation}\nas well as the equations of motion\n\\begin{equation}\n\\label{EOM_11D}\n\\begin{array}\n[c]{rll}%\nd(*_{11} \\hat{F}_{(4)}) + \\frac{1}{2} \\, \\hat{F}_{(4)} \\wedge\\hat{F}_{(4)} & = & 0 \\ ,\\\\[2mm]%\n\\hat{R}_{MN} - \\frac{1}{12} \\left(  \\hat{F}_{MPQR} \\, \\hat{F}_{N}{}^{PQR} - \\frac{1}{12} \\, \\hat{F}_{PQRS} \\, \\hat{F}^{PQRS} \\, \\hat{G}_{MN} \\right)  & = & 0 \\ .\n\\end{array}\n\\end{equation}\nThe equation of motion for $\\,\\hat{F}_{(4)}\\,$ in (\\ref{EOM_11D}) can be used to introduce the dual flux \n\\begin{equation}\n\\label{F7_definition}\n\\hat{F}_{(7)} \\equiv *_{11} \\hat{F}_{(4)} + \\tfrac{1}{2} \\hat{A}_{(3)} \\wedge \\hat{F}_{(4)} \\ ,\n\\end{equation}\nwhich therefore obeys the Bianchi identity $\\,d\\hat{F}_{(7)}=0\\,$. The flux in (\\ref{F7_definition}) determines the conserved Page charge of M2-branes in the background\\footnote{We have set the string length to unity, \\textit{i.e.}, $\\,\\ell_{s} = 1\\,$.}\n\\begin{equation}\n\\label{M2_brane_charge}\nN_{2} = \\frac{1}{(2 \\pi)^{6}} \\int_{M_{7}} \\hat{F}_{(7)}  = \\frac{1}{(2 \\pi)^{6}} \\int_{M_{7}}  *_{11} \\hat{F}_{(4)} + \\tfrac{1}{2} \\hat{A}_{(3)} \\wedge \\hat{F}_{(4)}   \\ ,\n\\end{equation}\nwhere $\\,M_{7}\\,$ is the internal space. The contribution $\\,*_{11} \\hat{F}_{(4)}\\,$ comes from electric M2-branes and the contribution $\\,{\\tfrac{1}{2} \\hat{A}_{(3)} \\wedge \\hat{F}_{(4)}}\\,$ originates from magnetic M5-branes. \n\n\n\n\\subsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2\\,$}{SU(3) x U(1)2} invariant sector}\n\\label{sec:11D_Janus}\n\nThe eleven-dimensional uplift of the $\\,\\text{SU}(3) \\times\\text{U}(1)^2\\,$ invariant sector of the maximal SO(8) supergravity has been worked out in \\cite{Pilch:2015dwa,Azizi:2016noi} (see also \\cite{Larios:2019kbw}). To describe the internal geometry, we will closely follow the Appendix~B.2 of \\cite{Larios:2019kbw} and use intrinsic coordinates on $\\,\\textrm{S}^{7}\\,$ adapted to its seven-dimensional Sasaki--Einstein structure. In these coordinates, the round metric on $\\,\\textrm{S}^{7}\\,$ takes the form\n\\begin{equation}\n\\label{metric_round_S7}\nds_{7}^{2} = ds_{\\mathbb{CP}_{3}}^{2} + \\left( d\\psi_{-}+ \\sigma_{-} \\right)^2 \\ ,\n\\end{equation}\nwhere $\\,ds_{\\mathbb{CP}_{3}}^{2}\\,$ is the Fubini-Study line element (normalised as in \\cite{Larios:2019kbw})\n\\begin{equation}\n\\label{metric_round_CP3}\nds_{\\mathbb{CP}_{3}}^{2} = d\\tilde{\\alpha}^{2} + \\cos^{2} \\tilde{\\alpha} \\, \\big( \\, ds^{2}_{\\mathbb{CP}_{2}} + \\sin^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma)^{2} \\, \\big)\n\\hspace{6mm} \\textrm{ with } \\hspace{6mm}\n\\sigma_{-} = \\cos^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma) \\ .\n\\end{equation}\nThe ranges of the angles in (\\ref{metric_round_S7})-(\\ref{metric_round_CP3}) are $\\,\\tilde{\\alpha} \\in[0,\\frac\n{\\pi}{2}]\\,$, $\\,\\tau_{-} \\in[0,2 \\pi]\\,$ and $\\,\\psi_{-} \\in[0,2 \\pi]\\,$. Moreover, $\\,\\sigma\\,$ in (\\ref{metric_round_CP3}) is the one-form on $\\,\\mathbb{CP}_{2}\\,$ such that $\\,d\\sigma=2\\boldsymbol{J}\\,$ with $\\,\\boldsymbol{J}\\,$ being the K\\\"ahler form on $\\,\\mathbb{CP}_{2}\\,$. The round metric in (\\ref{metric_round_S7}) occurs when the scalar field in the four-dimensional Lagrangian (\\ref{Lagrangian_model_SU3xU1xU1}) vanishes, \\textit{i.e.}, $\\,\\tilde{z}=0\\,$, and the $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ Freund--Rubin vacuum of eleven-dimensional supergravity is recovered \\cite{Freund:1980xh}. However, whenever non-vanishing, the scalar $\\,\\tilde{z}\\,$ in (\\ref{Janus_solution_SU3xU1xU1}) inflicts a deformation on the Freund--Rubin vacuum so that a new background is generated which displays a smaller $\\,\\text{SU}(3) \\times \\text{U}(1)^{2} \\subset \\textrm{SO}(8)\\,$ isometry group.\n\n\n\nWe are encoding the breaking of isometries caused by $\\,\\tilde{z}\\,$ into a set of metric functions $\\,f\\,$'s and flux functions $\\,h\\,$'s. The eleven-dimensional metric takes the form\n\\begin{equation}\n\\label{11D_metric}\n\\begin{array}{lll}\nd\\hat{s}_{11}^{2} & = & \\frac{1}{2} \\, f_{1} \\, ds_{4}^{2} + 2 \\,  g^{-2} \\Big[ f_{2} \\, d\\tilde{\\alpha}^{2} + \\cos^{2} \\tilde{\\alpha} \\, \\big( \\, f_{3} \\, \\, ds^{2}_{\\mathbb{CP}_{2}} + \\sin^{2} \\tilde{\\alpha} \\, f_{4} \\, (d\\tau_{-} + \\sigma)^{2} \\, \\big)\\\\[2mm]\n& + & f_{5} \\, \\big( d\\psi_{-} + \\cos^{2} \\tilde{\\alpha} \\, f_{6} \\, (d\\tau_{-} + \\sigma) \\big)^{2} \\Big] \\ ,\n\\end{array}\n\\end{equation}\nwith $\\,ds_{4}^{2}\\,$ given in (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}). Note that the eleven-dimensional metric  (\\ref{11D_metric}) displays an $\\,\\text{SU}(3) \\times \\text{U}(1)_{\\tau_{-}} \\times\\text{U}(1)_{\\psi_{-}}\\,$ symmetry. The $\\,\\textrm{SU}(3)\\,$ factor accounts for the $\\,\\mathbb{CP}_{2}\\,$ isometries and the two $\\,\\textrm{U}(1)\\,$ factors correspond with shifts along the angles $\\,\\tau_{-}\\,$ and $\\,\\psi_{-}\\,$, hence the attached labels. The various metric functions in (\\ref{11D_metric}) depend on the complex scalar $\\tilde{z}$ in\n(\\ref{Janus_solution_SU3xU1xU1}) and on the angle $\\,\\tilde{\\alpha} \\,$ on S$^{7}$. They are given by\n\\begin{equation}\n\\label{f_functions}\n\\begin{array}{c}\nf_{1}^{3} = \\dfrac{ (1+\\tilde{z}) (1+\\tilde{z}^{*})}{(1-|\\tilde{z}|^{2})^{3}} \\, H^{2}\n\\hspace{5mm} , \\hspace{5mm}\nf_{2}^{3\/2} = \\dfrac{H}{(1+\\tilde{z}) (1+\\tilde{z}^{*})}\n\\hspace{5mm} , \\hspace{5mm}\nf_{3}^{3} = \\dfrac{(1+\\tilde{z}) (1+\\tilde{z}^{*})}{H} \\ , \\\\[6mm]\nf_{4}^{3\/2} = \\dfrac{(1-|\\tilde{z}|^{2})^{3}}{ (1+\\tilde{z}) (1+\\tilde{z}^{*})} \\, H \\, K^{-\\frac{3}{2}} \n\\hspace{5mm} , \\hspace{5mm}\nf_{5}^{3\/2} = \\dfrac{1}{ (1+\\tilde{z}) (1+\\tilde{z}^{*})} \\, H^{-2} \\, K^{\\frac{3}{2}} \\ , \\\\[8mm]\nf_{6} = \\Big[ (1+\\tilde{z}) (1+\\tilde{z}^{*}) \\, H + ( \\tilde{z} - \\tilde{z}^{*})^{2} \\cos(2\\tilde{\\alpha}) \\Big] \\, K^{-1} \\ ,\n\\end{array}\n\\end{equation}\nwith\n\\begin{equation}\nH = 1+|\\tilde{z}|^{2} - ( \\tilde{z} +\\tilde{z}^{*}) \\cos(2\\tilde{\\alpha}) \\hspace{8mm} \\text{ and } \\hspace{8mm} K = 1+|\\tilde{z}|^{4} - 2 \\, |\\tilde{z}|^{2} \\, \\cos(4\\tilde{\\alpha}) \\ .\n\\end{equation}\nThe round metric on $\\,\\textrm{S}^{7}\\,$ is recovered from (\\ref{11D_metric}) upon setting $\\,\\tilde{z}=0\\,$, what implies that all the metric functions $\\,H=K=f_{1,\\ldots,6}=1\\,$. The part of the internal geometry in the upper line of (\\ref{11D_metric}) then reconstructs the $\\,\\mathbb{CP}_{3}\\,$ metric in (\\ref{metric_round_CP3}). \n\n\n\nThe eleven-dimensional four-form flux takes a more lengthy expression given in terms of three-, one- and zero-form deformations in four dimensions which we collectively denote $\\,h$'s. Adopting the terminology of \\cite{Pilch:2015dwa}, the four-form flux naturally splits as\n\\begin{equation}\n\\label{11D_F4}\n\\hat{F}_{(4)} = \\hat{F}_{(4)}^{\\textrm{st}} + \\hat{F}_{(4)}^{\\textrm{tr}} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{11D_F4_st}\n\\hat{F}_{(4)}^{\\textrm{st}} =\n-\\frac{1}{2\\sqrt{2}} \\, g \\, h_{1} \\, \\text{vol}_{4} + \\frac{1}{\\sqrt{2}} \\, g^{-1} \\, \\sin(2 \\tilde{\\alpha}) \\,\\, h^{(3)}_{2} \\wedge d\\tilde{\\alpha} \\ ,\n\\end{equation}\nand\n\\begin{equation}\n\\label{11D_F4_tr}\n\\begin{array}{lll}\n\\hat{F}_{(4)}^{\\textrm{tr}} &=& - 2 \\sqrt{2} \\,  g^{-3} \\Big[  \\, \\sin(2 \\tilde{\\alpha}) \\, h^{(1)}_{3} \\wedge d\\tilde{\\alpha} \\wedge d\\psi_{-} \\wedge(d\\tau_{-}+\\sigma)\\\\[2mm]\n& + &   \\cos^{4} \\tilde{\\alpha} \\,\\, h^{(1)}_{4} \\wedge (d\\tau_{-} + \\sigma) \\wedge\\boldsymbol{J} +   \\cos^{2} \\tilde{\\alpha} \\, \\cos(2 \\tilde{\\alpha}) \\,\\, h_{5}^{(1)} \\wedge d\\psi_{-} \\wedge\\boldsymbol{J}\\\\[2mm]\n& + &   \\sin(2 \\tilde{\\alpha}) \\, h_{6} \\, d\\tilde{\\alpha} \\wedge d\\psi_{-} \\wedge\\boldsymbol{J} +   \\cos^{4} \\tilde{\\alpha} \\,\\, h_{7} \\, \\boldsymbol{J} \\wedge\\boldsymbol{J}\\\\[2mm]\n& + &   \\cos^{2}\\tilde{\\alpha} \\, \\sin(2 \\tilde{\\alpha}) \\,\\, h_{8} \\, d\\tilde{\\alpha} \\wedge(d\\tau_{-} + \\sigma) \\wedge\\boldsymbol{J} \\, \\Big] \\ .\n\\end{array}\n\\end{equation}\nFor the space-time part in (\\ref{11D_F4_st}) we have introduced a zero-form\n\\begin{equation}\n\\label{h1_func}\nh_{1} = \\dfrac{1}{(1-|\\tilde{z}|^{2})} \\Big( \\, 3 \\, (1+|\\tilde{z}|^{2}) + ( \\tilde{z} +\\tilde{z}^{*}) \\, (1 - 2 \\cos(2 \\tilde{\\alpha}) ) \\, \\Big) \\ ,    \n\\end{equation}\nand a three-form\n\\begin{equation}\n\\label{h2_func}\n\\begin{array}{lll}\nh^{(3)}_{2} & = & \\dfrac{1}{(1-|\\tilde{z}|^{2})^{2}} \\Big( (\\tilde{z}^{2}-1) *_{4}  d\\tilde{z}^{*} + ((\\tilde{z}^{*})^{2}-1) *_{4} d\\tilde{z} \\Big) \\ ,\n\\end{array}\n\\end{equation}\nwhich has legs along the AdS$_{3}$ factor in the external geometry\\footnote{The Hodge dual $\\,*_{4}\\,$ is defined in four-dimensions using the metric $\\,ds^{2}_{4}\\,$ in (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}).}. For the transverse part in (\\ref{11D_F4_tr}) we have introduced a set of one-forms\n\\begin{equation}\n\\label{11D_F4_one-forms}\n\\begin{array}{lll}\nh^{(1)}_{3} & = & \\dfrac{i}{2} \\left(  \\dfrac{d\\tilde{z}^{*} }{(1+\\tilde{z}^{*})^{2}} - \\dfrac{d\\tilde{z} }{(1+\\tilde{z})^{2}} \\right)  \\ ,\\\\[4mm]\nh^{(1)}_{4} & = & h^{(1)}_{5} \\,\\, = \\,\\, i \\, H^{-2} \\, \\Big( \\left(  1 - 2 \\cos(2 \\tilde{\\alpha}) \\, \\tilde{z}^{*} + (\\tilde{z}^{*})^{2} \\right)  \\, d\\tilde{z} - \\left(  1 - 2 \\cos(2 \\tilde{\\alpha}) \\, \\tilde{z} + \\tilde{z}^{2} \\right)  \\, d\\tilde{z}^{*}\n\\Big) \\ ,\n\\end{array}\n\\end{equation}\ntogether with zero-forms\n\\begin{equation}\n\\begin{array}{lll}\nh_{6} & = & i \\, 4 \\, H^{-2} \\, (\\tilde{z}^{*}-\\tilde{z}) \\dfrac{(1+|\\tilde{z}|^{2})}{(1+\\tilde{z})(1+\\tilde{z}^{*})} \\Big( 1+ |\\tilde{z}|^{2} + (\\tilde{z}+\\tilde{z}^{*}) \\sin^{2}\\tilde{\\alpha} \\Big) \\ ,\\\\[4mm]\nh_{7} & = & -i \\, 2 \\, H^{-1} \\, (\\tilde{z}^{*}-\\tilde{z}) \\ ,\\\\[4mm]\nh_{8} & = & i \\, 2 \\, H^{-2} \\, (\\tilde{z}^{*}-\\tilde{z}) \\, \\Big( 1+ |\\tilde{z}|^{2} + (\\tilde{z}+\\tilde{z}^{*}) \\sin^{2}\\tilde{\\alpha} \\Big) \\ .\n\\end{array}\n\\end{equation}\nThe zero-forms $\\,h_{6}\\,$, $\\,h_{7}\\,$ and $\\,h_{8}\\,$ determine the purely internal components in (\\ref{11D_F4_tr}) and vanish if $\\,\\tilde{z}^{*}=\\tilde{z}\\,$. Also the one-form deformations in (\\ref{11D_F4_one-forms}) vanish in this case so that $\\,\\hat{F}_{(4)}^{\\textrm{tr}}=0\\,$. Lastly, the entire eleven-dimensional flux in (\\ref{11D_F4}) preserves an $\\,\\text{SU}(3)\\times\\text{U}(1)_{\\tau_{-}} \\times\\text{U}(1)_{\\psi_{-}}\\,$ symmetry since there is no explicit dependence on the angle $\\,\\psi_{-}\\,$ and, moreover, the two-form $\\,\\boldsymbol{J}\\,$ on $\\,\\mathbb{CP}_{2}\\,$ is not charged under $\\,\\text{U}(1)_{\\tau_{-}}\\,$.\n\n\nTo complete the uplift, the above quantities must be evaluated at the value of the complex scalar $\\,\\tilde{z} \\equiv \\tilde{z}_{1}=\\tilde{z}_{2}=\\tilde{z}_{3}\\,$ both for the Janus (\\ref{Janus_U1^4_rho_2}) and Hades (\\ref{Hades_U1^4_rho_2}) solutions. We have explicitly verified that the resulting eleven-dimensional backgrounds in (\\ref{11D_metric}) and (\\ref{11D_F4}) satisfy the source-less Bianchi identity and equations of motion in (\\ref{BI_F4}) and (\\ref{EOM_11D}), respectively.\n\n\n\\subsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2\\,$}{SU(3) x U(1)2} symmetric Janus}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f1_beta0.pdf} \n\\put(-100,100){$f_{1}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f2_beta0.pdf} \n\\put(-100,100){$f_{2}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f3_beta0.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{3}$}\n\\vspace{5mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f4_beta0.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, \\sin^2\\tilde{\\alpha}  \\,f_{4}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f5_beta0.pdf} \n\\put(-100,100){$f_{5}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f6_beta0.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{5}\\, f_{6}$}\n\\end{center}\n\\caption{Regular metric functions in (\\ref{11D_metric}) for the Janus solution with $\\alpha=1$ and $\\beta=0$.}\n\\label{fig:f_functions_beta0}\n\\end{figure}\n\nWe have performed the explicit uplift of the analytic and non-supersymmetric Janus solution in (\\ref{Janus_solution_SU3xU1xU1})-(\\ref{A(mu)_func_SU3xU1xU1}). The resulting eleven-dimensional backgrounds are everywhere regular and depend on the choice of parameters $\\,(\\alpha, \\beta)\\,$ specifying the boundary conditions (\\ref{ab_definition})-(\\ref{ab_algebraic}) of the four-dimensional Janus solution. Plots of the functions entering the metric (\\ref{11D_metric}) for $\\,\\beta=0\\,$ and $\\,\\beta=\\frac{\\pi}{2}\\,$ are depicted in Figure~\\ref{fig:f_functions_beta0} and Figure~\\ref{fig:f_functions_betaPi}. These two choices respectively activate only sources or VEV's in the Janus boundary conditions (\\ref{ab_definition})-(\\ref{ab_algebraic}). In addition, the scalar $\\,\\tilde{z}\\,$ in the $\\text{SU}(3) \\times\\text{U}(1)^2\\,$ symmetric Janus solution of (\\ref{Janus_solution_SU3xU1xU1}) is necessarily complex so that no limit to a real Janus solution exists even in the general case of (\\ref{Janus_solution_U1^4_ztil}). This further implies that all the $\\,h\\,$ functions (and also three- and one-forms) entering $\\,\\hat{F}_{(4)}^{\\textrm{st}}\\,$ in (\\ref{11D_F4_st}) and $\\,\\hat{F}_{(4)}^{\\textrm{tr}}\\,$ in (\\ref{11D_F4_tr}) are generically activated.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f1_betaPi2.pdf} \n\\put(-100,100){$f_{1}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f2_betaPi2.pdf} \n\\put(-100,100){$f_{2}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f3_betaPi2.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{3}$}\n\\vspace{5mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f4_betaPi2.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, \\sin^2\\tilde{\\alpha}  \\,f_{4}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f5_betaPi2.pdf} \n\\put(-100,100){$f_{5}$} \\hspace{10mm}\n\\includegraphics[width=0.25\\textwidth]{Plots\/f6_betaPi2.pdf} \n\\put(-100,100){$\\cos^2\\tilde{\\alpha} \\, f_{5}\\, f_{6}$}\n\\end{center}\n\\caption{Regular metric functions in (\\ref{11D_metric}) for the Janus solution with $\\alpha=1$ and $\\beta=\\frac{\\pi}{2}$.}\n\\label{fig:f_functions_betaPi}%\n\\end{figure}\n\n\n\nIn order to compute the M$2$-brane charge for the $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ symmetric Janus, we first note that the dual seven-form flux can be expressed as\n\\begin{equation}\n\\label{F7_general}\n\\hat{F}_{(7)} = d\\hat{\\alpha} \\wedge h^{(6)}  + \\ldots \\ , \n\\end{equation}\nwith $\\,h^{(6)}=\\frac{1}{2}  \\boldsymbol{J} \\wedge \\boldsymbol{J} \\wedge d\\tau_{-} \\wedge d\\psi_{-}\\,$ being the volume form of $\\,M_{6}\\,$ spanned by $\\,(\\mathbb{CP}_{2}, \\tau_{-}, \\psi_{-})$, and $\\,\\hat{\\alpha}\\,$ playing the role of an ``adapted'' angular coordinate threaded by the flux. This adapted coordiante is in general a complicated function\n\\begin{equation}\n\\label{hat_alpha}\n\\hat{\\alpha}=\\hat{\\alpha}(\\rho,\\tilde{\\alpha} \\, ; \\,\\alpha,\\beta) \\ ,\n\\end{equation}\nthat depends on the original coordinates $\\,(\\rho,\\tilde{\\alpha})\\,$ as well as on the Janus parameters $\\,(\\alpha,\\beta)\\,$. Lastly, the ellipsis in (\\ref{F7_general}) stand for additional terms with legs on the AdS$_3$ piece of the geometry which do not play a relevant role when computing M$2$-brane charges. Therefore, all the relevant information regarding M$2$-brane charges gets codified into the one-form $\\,d\\hat{\\alpha}\\,$ as it defines an adapted angular direction. It is important to highlight that, when taking the limit $\\,\\rho \\rightarrow \\pm \\infty\\,$, one finds that $\\,d\\hat{\\alpha} \\propto \\sin(2\\tilde{\\alpha}) \\cos^4\\tilde{\\alpha} \\, d\\tilde{\\alpha}\\,$ no longer depends on the Janus parameters $\\,(\\alpha,\\beta)\\,$. In this limit, the dual seven-form flux threads the $\\,\\textrm{S}^7\\,$ as required by the asymptotic $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ geometry of the flow. \n\n\n\nThe computation of the M$2$-brane charge in the Janus solution gives\n\\begin{equation}\n\\label{N2_Janus}\nN_2 = \\frac{1}{(2\\pi)^6}\\int_{\\Gamma \\times \\textrm{M}_{6}} \\hat{F}_{(7)} = \\frac{1}{32 \\pi^2}\\int_{\\partial\\Gamma} \\hat{\\alpha}  = \\frac{1}{4\\pi^2 g^6} \\ ,\n\\end{equation}\nwhere the relevant curves $\\,\\Gamma$'s threaded by the purely internal part of the seven-form flux in (\\ref{F7_general}) are specified by their tangent vector field $\\,\\boldsymbol{v} = (\\boldsymbol{v}_{\\mu},\\boldsymbol{v}_{\\tilde{\\alpha}}) = (g \\sqrt{\\rho^2+1} \\, \\partial_{\\rho}\\hat{\\alpha} \\,,\\, \\partial_{\\tilde{\\alpha}}\\hat{\\alpha})\\,$\\footnote{Note that $\\,\\boldsymbol{v}_{\\mu}=\\partial_{\\mu}\\hat{\\alpha}=g \\sqrt{\\rho^2+1} \\, \\partial_{\\rho}\\hat{\\alpha}\\,$ as a consequence of the change of radial coordinate in (\\ref{new_coordinate_Janus}).}. For the Janus, all the curves $\\,\\Gamma\\,$ start at $\\,\\tilde{\\alpha}=0\\,$ and end at $\\,\\tilde{\\alpha}=\\frac{\\pi}{2}\\,$ pointing at the $\\,\\tilde{\\alpha}\\,$ direction on $\\,\\textrm{S}^{7}\\,$ -- see Figure~\\ref{fig:vec_field} for an illustration of such curves in various examples --. Since the $\\,N_{2}\\,$ charge in (\\ref{N2_Janus}) is independent of $\\,\\Gamma\\,$ and also of the Janus parameters $\\,(\\alpha,\\beta)$, it matches the one of the $\\textrm{AdS}_{4} \\times \\textrm{S}^7\\,$ background controlling the asymptotic behaviour of the (regular) Janus solutions at $\\,\\rho \\rightarrow \\pm \\infty\\,$.\n\n\nLastly, it is also interesting to compute the volume of the internal manifold $\\,\\textrm{vol}_7\\,$ along the Janus flow as a function of the radial coordinate $\\,\\rho\\,$ and the Janus parameters $\\,(\\alpha,\\beta)\\,$. The result is a lengthy expression not very illuminating that we have evaluated and plotted in Figure~\\ref{Fig:Janus_S7} for various choices of the Janus parameters. The behaviour is akin a wormhole: the $\\,\\textrm{S}^7\\,$ is a non-contractible seven-manifold  whose volume does not vanish anywhere in the flow along the radial direction $\\,\\rho\\,$. Moreover, for a given value of $\\,\\alpha\\,$, there is a range of the parameter $\\,\\beta\\,$ for which the eleven-dimensional Janus features two throats (see right plot in Figure~\\ref{Fig:Janus_S7}).\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{Plots\/Shell_Janus.pdf} \n\\put(-35,30){$\\textrm{Re}\\tilde{z}$}\n\\put(-150,10){$\\textrm{Im}\\tilde{z}$}\n\\hspace{8mm}\n\\includegraphics[width=0.42\\textwidth]{Plots\/Throats_Janus.pdf} \n\\put(0,-5){$\\rho$}\n\\put(-100,135){$\\frac{3 \\, g^7}{ \\, 2^{7\/2} \\pi^{4}} \\, \\textrm{vol}_7$}\n\\end{center}\n\\caption{Left: Volume of the internal seven-sphere as a function of the complex scalar $\\,\\tilde{z}\\,$ (orange dome). Examples of regular Janus flows (loops) are superimposed. Right: Volume of the internal seven-sphere as a function of the radial coordinate $\\,\\rho\\,$ for the regular Janus solutions. The parameters of the curves are: $\\,\\alpha=1\\,$ and $\\,\\beta=-\\frac{\\pi}{2}\\,$ (blue line), $\\,\\beta=\\pi\\,$ (black line) and $\\,\\beta=\\frac{17}{16}\\pi\\,$ (green line). Note the presence of two minima (throats) in the black and green lines.}\n\\label{Fig:Janus_S7}\n\\end{figure}\n\n\n\n\n\n\\subsection{\\texorpdfstring{$\\text{SU}(3) \\times\\text{U}(1)^2\\,$}{SU3 x U(1)2} symmetric Hades and ridge flows}\n\nSetting $\\,\\alpha \\equiv \\alpha_{1}=\\alpha_{2}=\\alpha_{3}\\,$ and $\\,\\beta \\equiv \\beta_{1}=\\beta_{2}=\\beta_{3}\\,$ enhances the symmetry of the general Hades solution in (\\ref{Hades_U1^4_rho_1})-(\\ref{Hades_U1^4_rho_2}) from $\\,\\textrm{U}(1)^4\\,$ to $\\,\\text{SU}(3) \\times\\text{U}(1)^2\\,$. Setting $\\,\\alpha \\neq 0\\,$ renders the running of the scalar field (\\ref{Hades_U1^4_rho_2}) along the flow intrinsically complex, as it happened for the Janus case. This again implies that all the $\\,h\\,$ functions (and also three- and one-forms) entering $\\,\\hat{F}_{(4)}^{\\textrm{st}}\\,$ in (\\ref{11D_F4_st}) and $\\,\\hat{F}_{(4)}^{\\textrm{tr}}\\,$ in (\\ref{11D_F4_tr}) are generically activated.\n\n\n\n\n\n\n\n\nThe decomposition of the seven-form flux $\\,\\hat{F}_{(7)} \\,$ in  (\\ref{F7_general}) is still at work for the Hades solutions. The computation of the M$2$-brane charge gives\n\\begin{equation}\n\\label{N2_Hades}\nN_2 = \\frac{1}{(2\\pi)^6}\\int_{\\Gamma \\times \\textrm{M}_{6}} \\hat{F}_{(7)} = \\frac{1}{32 \\pi^2}\\int_{\\partial\\Gamma} \\hat{\\alpha}  = \\frac{1}{4\\pi^2 g^6} \\ ,\n\\end{equation}\nso that it matches the one of the $\\textrm{AdS}_{4} \\times \\textrm{S}^7\\,$ background controlling the asymptotic behaviour of the Hades solutions at $\\,\\rho \\rightarrow \\infty\\,$. Some examples of Hades flows on the $\\,\\tilde{z}\\,$ complex plane are displayed in Figure~\\ref{fig:Shell_Hades} and superimposed on the volume of the internal seven-sphere.\n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.50\\textwidth]{Plots\/Shell_Hades.pdf} \n\\put(-80,20){$\\textrm{Re}\\tilde{z}$}\n\\put(-210,60){$\\textrm{Im}\\tilde{z}$}\n\\end{center}\n\\caption{Volume of the internal seven-sphere (orange dome) as a function of the complex scalar $\\,\\tilde{z}\\,$. Examples of singular Hades flows are superimposed with $\\,(\\alpha,\\beta)=(0,0)\\,$ (green straight line), $\\,{(\\alpha,\\beta)=(0,\\frac{\\pi}{2})}\\,$ (blue straight line), $\\,{(\\alpha,\\beta)=(0,-\\frac{\\pi}{2})}\\,$ (red straight line), $\\,{(\\alpha,\\lambda)=(2,\\frac{\\pi}{2}})\\,$ (blue curved line) and $\\,{(\\alpha,\\lambda)=(2,-\\frac{\\pi}{2}})\\,$ (red curved line).}\n\\label{fig:Shell_Hades}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_janus_beta_0.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\hspace{12mm}\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_janus_beta_pi.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\\\[5mm]\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_ridge_beta_minus_pihalf.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\hspace{12mm}\n\\includegraphics[width=0.40\\textwidth]{Plots\/1-form_ridge_beta_pihalf.pdf} \n\\put(-190,88){$\\tilde{\\alpha}$} \n\\put(-88,-10){$\\rho$}\n\\end{center}\n\\caption{Plots of the vector field $\\,\\boldsymbol{v} = (\\,g \\sqrt{\\rho^2 \\pm 1} \\,  \\partial_{\\rho}\\hat{\\alpha} \\,,\\, \\partial_{\\tilde{\\alpha}}\\hat{\\alpha}\\,)\\,$ on the strip spanned by $\\,(\\rho,\\tilde{\\alpha})\\,$. The $\\,+\\,$ sign must be chosen for the Janus solutions whereas the $\\,-\\,$ sign corresponds to the Hades solutions as a consequence of the change of radial coordinate $\\,d\\rho = g \\, \\sqrt{\\rho^2 \\pm 1} \\, d\\mu \\,$. Top-Left: Janus flow with $\\,\\alpha=1\\,$ and $\\,\\beta=0\\,$. Top-Right: Janus flow with $\\,\\alpha=1\\,$ and $\\,\\beta=\\pi\\,$. Bottom-Left: Ridge flow with $\\,\\beta = -\\frac{\\pi}{2}\\,$. Bottom-Right: Ridge flow with $\\,\\beta = \\frac{\\pi}{2}\\,$.}\n\\label{fig:vec_field}\n\\end{figure}\n\n\n\\subsubsection*{Ridge flows and singularities}\n\n\nIn order to investigate the possible eleven-dimensional resolution of the four-dimensional Hades singularity at $\\,\\rho=1\\,$, we will look at the metric (\\ref{11D_metric}) and analyse the relevant function\n\\begin{equation}\n\\label{Omega_func}\n\\Omega \\equiv f_{1}^{\\frac{1}{2}} \\, e^{A} \\ ,     \n\\end{equation}\nlying in front of the $\\,\\textrm{AdS}_{3}\\,$ factor of the eleven-dimensional metric describing the world-volume of the (curved) M2-branes in the UV. For simplicity, we will take the limiting case of $\\,\\alpha=0\\,$ and focus on the ridge flows with\n\\begin{equation}\n\\label{Ridge_SU3xU1xU1_rho}\nds_{4}^{2}=\\frac{1}{{g}^{2}} \\left(  \\frac{d\\rho^{2}}{\\rho^{2}-1} + \n\\frac{\\left(  \\rho^{2}-1\\right)  }{2} \\, d\\Sigma^{2}  \\right)\n\\hspace{8mm} \\textrm{ and } \\hspace{8mm}\n\\tilde{z}(\\rho) =  \\rho^{-1} \\, e^{i \\left( \\beta - \\frac{\\pi}{2} \\right)} \\ .\n\\end{equation}\nRemarkably, for these flows, the four-dimensional singularity at $\\,\\rho = 1\\,$ gets resolved when uplifting the solutions to eleven dimensions provided $\\,\\beta \\neq \\pm \\frac{\\pi}{2}\\,$. \n\n\n\n\nThe explicit computation of the $\\,\\Omega\\,$ factor in (\\ref{Omega_func}) for the ridge flows yields\n\\begin{equation}\n\\label{Omega_func_ridge}\n\\Omega = (2 g)^{-1} \\left(  1 + \\rho^2 + 2 \\, \\rho \\, \\sin\\beta \\right)^{\\frac{1}{6}} \\left( 1 + \\rho^2 - 2 \\, \\rho  \\, \\sin\\beta \\, \\cos(2 \\tilde{\\alpha})\\right)^{\\frac{1}{3}} \\ .\n\\end{equation}\nEvaluating (\\ref{Omega_func_ridge}) at $\\,\\rho=1\\,$ where the four-dimensional singularity is located, one concludes that $\\,\\Omega\\,$ vanishes at $\\,(\\beta,\\tilde{\\alpha})=(\\frac{\\pi}{2},0)\\,$ as well as at $\\,(\\beta,\\tilde{\\alpha})=(-\\frac{\\pi}{2},\\tilde{\\alpha})\\,$ $\\,\\forall \\tilde{\\alpha}\\,$. In other words, the pathology at $\\,\\rho=1\\,$ persists for $\\,\\beta=\\pm \\frac{\\pi}{2}\\,$ and it either localises at $\\,\\tilde{\\alpha}=0\\,$ or gets delocalised along the interval $\\,\\tilde{\\alpha} \\in [0, \\frac{\\pi}{2}]\\,$.\\footnote{A similar class of conventional (flat-sliced) RG-flows with $\\,{\\text{SU}(3) \\times\\text{U}(1)^2}\\,$ symmetry was constructed in \\cite{Pilch:2015vha}. For the sake of comparison, there is a redefinition of the relevant parameter given by $\\,\\zeta_{\\tiny{\\cite{Pilch:2015vha}}}=\\beta-\\frac{\\pi}{2}\\,$. The singularity of the ridge flows we study here would be similar to that of a (yet to be constructed) non-supersymmetric generalisation of the flows in \\cite{Pilch:2015vha} with $\\,\\cos(3\\zeta)=+1\\,$.} We will look at some limiting examples of ridge flows in order to illustrate their main physical implications.\n\n\n\\subsubsection*{$\\circ\\,$ Singular $\\,\\boldsymbol{\\beta=\\pm\\frac{\\pi}{2}\\,}$ ridge flows:}\n\n\nThe scalar in (\\ref{Ridge_SU3xU1xU1_rho}) becomes real when setting $\\,\\beta=\\frac{\\pi}{2}\\,$. The eleven-dimensional geometry gets simplified to\n\\begin{equation}\n\\label{11D_metric_ridge_beta_pi\/2}\n\\begin{array}{lll}\nds_{11}^2 &=& \\dfrac{f_{-}^{\\frac{2}{3}}}{g^2} \\dfrac{(\\rho+1)^\\frac{2}{3}}{4} \\left[ \\,  ds_{\\textrm{AdS}_{3}}^2 +\n\\dfrac{2 \\, d\\rho^2}{(\\rho^2-1)^2}  + 8 \\dfrac{d\\tilde{\\alpha}^2}{(\\rho+1)^{2}}  \\right. \\\\[6mm]\n& + & \\left.  \\dfrac{8}{f_{-}}   \\cos^2\\tilde{\\alpha} \\, \\left(  ds^2_{\\mathbb{CP}^2} +  \\dfrac{(\\rho-1)^2}{f_{+}} \\,  \\sin^{2} \\tilde{\\alpha} \\,  (d\\tau_{-} + \\sigma)^{2} \\right)  \\right. \\\\[6mm]\n& + & \\left. \n\\dfrac{8}{f_{-}}  \\,  \\left( \\dfrac{f_{+}^{\\frac{1}{2}}}{\\rho+1} d\\psi_{-} + \\, \\dfrac{\\rho+1}{f_{+}^{\\frac{1}{2}}}  \\, \\cos^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma) \\right)^{2} \\, \\right]  \\ ,\n\\end{array}\n\\end{equation}\nin terms of the functions\n\\begin{equation}\n\\label{f+-_functions}\nf_{\\pm}= (\\rho \\pm 1)^2 \\mp 4 \\, \\rho \\, \\sin^2 \\tilde{\\alpha} \\ .\n\\end{equation}\nMoreover, since the scalar in (\\ref{Ridge_SU3xU1xU1_rho}) becomes real, one has that\n\\begin{equation}\n\\label{F4tr_ridge_beta_pi\/2}\n\\hat{F}_{(4)}^{\\textrm{tr}}=0 \\ ,   \n\\end{equation}\nin (\\ref{11D_F4_tr}). The non-vanishing contribution to the three-form gauge potential in this case is given by \n\\begin{equation}\n\\label{Ast_ridge_beta_pi\/2}\n\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{\\rho \\, (3+\\rho+\\rho^2) - 2 \\, (\\rho^2-1) \\cos(2\\tilde{\\alpha})}{8 \\, g^3} \\, \\textrm{vol}_{\\textrm{AdS}_{3}} \\ ,\n\\end{equation}\nproducing a space-time four-form flux in (\\ref{11D_F4_st}) of the form\n\\begin{equation}\n\\label{F4st_ridge_beta_pi\/2}\n\\hat{F}_{(4)}^{\\textrm{st}}=d\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{1}{2 g^3} \\left(   \n\\frac{3 + \\rho \\, (2 + 3 \\, \\rho) - 4 \\, \\rho \\,  \\cos(2\\tilde{\\alpha})}{4} \\, d\\rho   \n+ (\\rho^2-1) \\, \\sin(2\\tilde{\\alpha}) \\, d\\tilde{\\alpha} \\right) \\wedge  \\textrm{vol}_{\\textrm{AdS}_{3}}\\ . \n\\end{equation}\nTwo facts suggest an interpretation of this flow as a Coulomb branch type flow very much along the line of \\cite{Cvetic:1999xx}. Firstly, this singular ridge flow lies in the purely proper scalar sector of maximal supergravity as a consequence of $\\,\\beta=\\frac{\\pi}{2}\\,$. Namely, it is triggered from the UV solely by the VEV of the proper scalar dual to the boson bilinears. Secondly, the internal flux in (\\ref{F4tr_ridge_beta_pi\/2}) vanishes all along the flow so there are no magnetic M5-branes sourcing $\\,\\hat{F}_{(7)}\\,$.\n\n\nLet us now investigate the four-dimensional singularity at $\\,\\rho=1\\,$ from a higher-dimensional perspective. To study the eleven-dimensional geometry around $\\,\\rho=1\\,$ it is convenient to look at the Ricci scalar which, in this case, takes the form\n\\begin{equation}\n\\label{Ricci_beta_pi\/2}\n\\hat{R}(\\rho) = g^2 \\, \\frac{ (\\rho -1)^2}{3 \\, (\\rho +1)^{\\frac{2}{3}} f_{-}^{\\frac{8}{3}}} \\,\\, r(\\rho,\\tilde{\\alpha}) \\ ,\n\\end{equation}\nin terms of the negative-definite function\n\\begin{equation}\n\\begin{array}{lll}\nr(\\rho,\\tilde{\\alpha}) &=&  \n-(9 \\rho^4 + 12 \\rho^3 + 32 \\rho^2 + 16 \\rho + 11)\n+ 8 \\, \\rho \\, (3 \\rho^2 + 2 \\rho + 3)   \\cos (2 \\tilde{\\alpha}) \\\\[2mm]\n& & -2 \\, (\\rho -1) (3 \\rho +1) \\, \\cos(4 \\tilde{\\alpha}) \\ .\n\\end{array}\n\\end{equation}\nThe Ricci scalar in (\\ref{Ricci_beta_pi\/2}) becomes singular at $\\,(\\rho,\\tilde{\\alpha})=(1,0)\\,$. On the other hand, the evaluation of the four-form flux in (\\ref{F4st_ridge_beta_pi\/2}) around the singular value $\\,\\rho=1\\,$ is more subtle. The change of radial coordinate in (\\ref{new_coordinate_Hades}) becomes ill-defined and one must resort to the original coordinate $\\,\\mu\\,$ in (\\ref{metric_ansatz}) using $\\,d\\rho = g \\, \\sqrt{\\rho^2-1} \\, d\\mu \\,$. Then, it becomes clear from (\\ref{F4st_ridge_beta_pi\/2}) that\n\\begin{equation}\n\\left. \\hat{F}_{(4)}^{\\textrm{st}} \\right|_{\\rho = 1} = 0 \\ .\n\\end{equation}\n\n\n\n\nIt is also instructive to look at the flux $\\,\\hat{F}_{(7)} = d\\hat{\\alpha} \\wedge h^{(6)}  + \\ldots\\,$ by analysing the expression of the adapted angular variable $\\,\\hat{\\alpha}\\,$. In this case it takes the form\n\\begin{equation}\n\\hat{\\alpha}(\\rho,\\tilde{\\alpha}) = - 8 \\, g^{-6} \\,  f_{-}^{-1} (\\rho-1)^2 \\cos^6\\tilde{\\alpha} \\ ,  \n\\end{equation}\nwith $\\,f_{-}\\,$ given in (\\ref{f+-_functions}). A plot of the curves $\\,\\Gamma\\,$ is presented in Figure~\\ref{fig:vec_field} (bottom-right plot). Note that not all of them start at $\\,\\tilde{\\alpha}=0\\,$ and end at $\\,\\tilde{\\alpha}=\\frac{\\pi}{2}\\,$. There are curves that start at $\\,\\tilde{\\alpha}=0\\,$ but end at some value $\\,0 < \\tilde{\\alpha} < \\frac{\\pi}{2}\\,$ when reaching the singularity at $\\,\\rho = 1\\,$. These curves display a strong singularity bending: the one-form $\\,d\\hat{\\alpha}\\,$ interpolates between being aligned with the $\\,\\textrm{S}^{7}\\,$ direction $\\,d\\tilde{\\alpha}\\,$ at $\\,\\rho \\rightarrow \\infty\\,$ and being aligned with the non-compact direction $\\,d\\rho\\,$ when reaching the singularity at $\\,\\rho =1\\,$.\n\nFinally, recalling the result in Section~\\ref{sec:ridge_4D}, setting $\\,\\beta=-\\frac{\\pi}{2}\\,$ amounts to a reflection of the radial coordinate $\\,\\rho \\rightarrow -\\rho\\,$ (which implies an exchange $\\,f_{+} \\leftrightarrow f_{-}\\,$) while keeping the domain $\\,\\rho \\in [1,\\infty)\\,$. This reflection drastically modifies the eleven-dimensional geometry in (\\ref{11D_metric_ridge_beta_pi\/2}) and (\\ref{f+-_functions}) which becomes singular at $\\,\\rho=1\\,$ for any value of the angular coordinate within the interval $\\,\\tilde{\\alpha} \\in [0,\\frac{\\pi}{2}]\\,$. This can also be viewed in the eleven-dimensional Ricci scalar which reads\n\\begin{equation}\n\\label{Ricci_beta_-pi\/2}\n\\hat{R}(\\rho) = g^2 \\, \\frac{ (\\rho+1)^2}{3 \\, (\\rho -1)^{\\frac{2}{3}} f_{+}^{\\frac{8}{3}}} \\,\\, r(-\\rho,\\tilde{\\alpha}) \\ .\n\\end{equation}\nSince there is no special value of $\\,\\tilde{\\alpha}\\,$ as far as singularities are concerned, the $\\,\\Gamma\\,$ curves constructed from the adapted angular variable \n\\begin{equation}\n\\hat{\\alpha}(\\rho,\\tilde{\\alpha}) = - 8 \\, g^{-6} \\,  f_{+}^{-1} (\\rho+1)^2 \\cos^6\\tilde{\\alpha} \\ , \n\\end{equation}\ndo not display any bending when approaching $\\,\\rho=1\\,$. These curves are presented in Figure~\\ref{fig:vec_field} (bottom-left plot). Lastly, the three-form gauge potential at $\\,\\beta=-\\frac{\\pi}{2}\\,$ is given by\n\\begin{equation}\n\\label{Ast_ridge_beta_minus_pi\/2}\n\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{\\rho \\, (3-\\rho+\\rho^2) + 2 \\, (\\rho^2-1) \\cos(2\\tilde{\\alpha})}{8 \\, g^3} \\, \\textrm{vol}_{\\textrm{AdS}_{3}} \\ .\n\\end{equation}\n\n\n\n \n \n \n\n\\subsubsection*{$\\circ\\,$ Regular $\\,\\boldsymbol{\\beta=0,\\pi\\,}$ ridge flows:}\n\nThe scalar in (\\ref{Ridge_SU3xU1xU1_rho}) becomes purely imaginary when setting $\\,\\beta=0\\,$. As a result, this ridge flow is triggered from the UV solely by the source mode of the pseudo-scalar dual to the fermion bilinears. \n\n\n\nThe eleven-dimensional metric reduces in this case to\n\\begin{equation}\n\\label{11D_metric_ridge_beta_0}\n\\begin{array}{lll}\nds_{11}^2 &=& \\dfrac{\\rho^2+1}{4 \\, g^2} \\left[ \\,  ds_{\\textrm{AdS}_{3}}^2 +\n\\dfrac{2 \\, d\\rho^2}{(\\rho^2-1)^2}  + 8 \\dfrac{d\\tilde{\\alpha}^2}{\\rho^2+1}  \\right. \\\\[6mm]\n& + & \\left. 8 \\, \\cos^2\\tilde{\\alpha} \\, \\left( \\dfrac{1}{\\rho^2+1}  ds^2_{\\mathbb{CP}^2} + \\dfrac{1}{j_{2}} \\dfrac{(\\rho^2-1)^2}{\\rho^2+1} \\,  \\sin^{2} \\tilde{\\alpha} \\,  (d\\tau_{-} + \\sigma)^{2} \\right)  \\right. \\\\[6mm]\n& + & \\left. \n\\dfrac{8 \\, j_{1}}{(\\rho^2+1)^3}  \\,  \\left( \\sqrt{\\dfrac{j_{2}}{j_{1}}} \\, d\\psi_{-} + \\sqrt{\\dfrac{j_{1}}{j_{2}}}  \\, \\cos^{2} \\tilde{\\alpha} \\, (d\\tau_{-} + \\sigma) \\right)^{2} \\, \\right]  \\ ,\n\\end{array}\n\\end{equation}\nin terms of the two functions\n\\begin{equation}\n\\label{j1_j2_functions}\nj_{1} = (\\rho^2 + 1)^2 - 4 \\, \\rho^2 \\, \\cos(2 \\tilde{\\alpha}) \n\\hspace{8mm} , \\hspace{8mm}\nj_{2} = (\\rho^2 + 1)^2 - 4 \\, \\rho^2 \\, \\cos^2(2 \\tilde{\\alpha}) \\ .\n\\end{equation}\nThe four-form flux in (\\ref{11D_F4}) comes with both space-time and transverse contributions. The former is given by\n\\begin{equation}\n\\label{Ast_ridge_beta_0}\n\\hat{F}_{(4)}^{\\textrm{st}}=d\\hat{A}_{(3)}^{\\textrm{st}}\n\\hspace{10mm} \\textrm{ with } \\hspace{10mm}\n\\hat{A}_{(3)}^{\\textrm{st}} = \\frac{\\rho \\, (3+\\rho^2)}{8 \\, g^3}  \\, \\textrm{vol}_{\\textrm{AdS}_{3}} \\ ,\n\\end{equation}\nwhereas the latter reads\n\\begin{equation}\n\\hat{F}_{(4)}^{\\textrm{tr}}=d\\hat{A}_{(3)}^{\\textrm{tr}} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{Atr_ridge_beta_0}\n\\begin{array}{lll}\n\\hat{A}_{(3)}^{\\textrm{tr}} &=& - \\dfrac{4 \\sqrt{2}}{g^{3}} \\dfrac{\\rho}{\\rho^2 +1} \\Big[ \\frac{1}{2} \\sin(2\\tilde{\\alpha}) \\, d\\tilde{\\alpha}  \\wedge (d\\tau_{-} + \\sigma) \\wedge d\\psi_{-}  \\\\[2mm]\n&& \\qquad\\qquad\\qquad + \\cos^4\\tilde{\\alpha} \\, \\boldsymbol{J}  \\wedge (d\\tau_{-} + \\sigma) + \\cos^2\\tilde{\\alpha} \\, \\cos(2\\tilde{\\alpha}) \\, \\boldsymbol{J} \\wedge d\\psi_{-} \\Big] \\ .\n\\end{array}\n\\end{equation}\nThis signals the presence of both electric M2-branes and magnetic M5-branes at a generic point along the flow.\n\n\n\n\nIn order to investigate the four-dimensional singularity at $\\,\\rho=1\\,$ from a higher-dimensional perspective we will look again at the eleven-dimensional Ricci scalar. It reads\n\\begin{equation}\n\\label{Ricci_scalar_Hades_beta=0}\n\\hat{R}(\\rho) = g^2 \\, \\left(1 + \\rho^2\\right)^{-3} \\, \\left(1 + 3 \\, \\rho^2\\right) \\, \\left[ 1 + \\rho^2 \\left( 8 - \\rho^2 \\right) \\right] \\ ,\n\\end{equation}\nand becomes this time independent of the angular variable $\\,\\tilde{\\alpha}\\,$. The Ricci scalar in  (\\ref{Ricci_scalar_Hades_beta=0}) features no singularity within the domain $\\,\\rho \\in [1 , \\infty )\\,$. It has a boundary value $\\,\\hat{R}(\\infty) = -3 \\, g^2\\,$ and changes smoothly until reaching the finite value $\\,\\hat{R}(1)=4 \\, g^2\\,$, thus making the eleven-dimensional solution regular. The space-time (\\ref{Ast_ridge_beta_0}) and transverse  (\\ref{Atr_ridge_beta_0}) components of the three-form gauge potential are both non-zero when approaching the IR region at $\\,\\rho =1\\,$. However, recalling again the change of radial coordinate $\\,d\\rho = g \\, \\sqrt{\\rho^2-1} \\, d\\mu \\,$, it follows from (\\ref{Ast_ridge_beta_0}) that\n\\begin{equation}\n\\left. \\hat{F}_{(4)}^{\\textrm{st}} \\right|_{\\rho = 1} = 0 \\ .\n\\end{equation}\nTherefore, only magnetic M5-branes source the geometry in the deep IR. The same behaviour was observed for the similar, but flat-sliced, $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2\\,$ invariant flows constructed in \\cite{Pilch:2015vha}. Such flows were argued to describe how M2-branes in the UV totally dissolve along the flow into magnetic M5-branes, leaving no M2-branes at the core of the regular flows.\\footnote{The same type of behaviour was also observed in the flat-sliced dielectric flows with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetry of \\cite{Pope:2003jp}, although the M2-branes do not totally polarise into M5-branes at the core of these flows.} Moving back to the original radial coordinate\n\\begin{equation}\n\\rho = \\cosh(g \\, \\mu) \n\\hspace{15mm}  \\textrm{ with }  \\hspace{15mm}\\mu \\in [0,\\infty) \\ ,\n\\end{equation}\nand expanding around $\\,\\mu = 0\\,$ one arrives at\n\\begin{equation}\n\\label{11D_metric_ridge_beta_0_IR}\n\\begin{array}{lll}\n\\left. ds_{11}^2 \\, \\right|_{\\textrm{IR}} & \\approx & \\dfrac{1}{4 \\, g^2} \\left[ \\, \\left(\\dfrac{4}{(g\\mu)^2} + \\dfrac{2}{3} + \\dfrac{4}{15}(g\\mu)^2 + \\ldots \\right) d(g\\mu)^2  +  \\left( 2 + (g \\mu)^2 + \\ldots \\right)\\, ds_{\\textrm{AdS}_{3}}^2 \\right. \\\\[6mm]\n& + & \\left. \n\\left( \\dfrac{(g\\mu)^4}{2} - \\dfrac{(g\\mu)^6}{6} + \\ldots \\right)   \\,   \\Big( (d\\tau_{-} + \\sigma) + 2  \\cos(2\\tilde{\\alpha})  \\, \\left( d\\psi_{-} + \\frac{1}{2} (d\\tau_{-} + \\sigma) \\right) \\Big)^{2} \\,  \\right. \\\\[6mm]\n& + &\n\\left.  8 \\, \\left( d\\tilde{\\alpha}^2 + \\cos^2\\tilde{\\alpha} \\, ds^2_{\\mathbb{CP}^2} +  \\sin^2(2 \\tilde{\\alpha}) \\, \\left(d\\psi_{-} + \\frac{1}{2}(d\\tau_{-} + \\sigma) \\right)^2 \\right)  \\right] \\ .\n\\end{array}\n\\end{equation}\nNote that the $\\,\\mu$-dependent part of the metric only involves the first two lines in (\\ref{11D_metric_ridge_beta_0_IR}). This $\\,\\mu$-dependent part describes a five-dimensional section of the eleven-dimensional geometry that involves the original four coordinates of the ridge flow and an additional $\\,\\textrm{S}^1\\,$ that is non-trivially fibered over a six-dimensional manifold. The latter is described by the last line in (\\ref{11D_metric_ridge_beta_0_IR}). Ignoring this fibration, the five-dimensional section of the geometry verifies $\\,R^{\\textrm{(\\tiny{5D})}}_{\\mu\\nu}= \\frac{1}{5} \\, R^{\\textrm{(\\tiny{5D})}} \\, g^{\\textrm{(\\tiny{5D})}}_{\\mu\\nu}\\,$ with $\\,R^{\\textrm{(\\tiny{5D})}} = -20 \\, g^2 < 0\\,$ at leading order in the radial coordinate $\\,\\mu\\,$. Therefore, up to \nthe non-trivial fibration over the six-dimensional manifold, this ridge flow develops a five-dimensional Einstein geometry in the deep IR.\n\n\n\nThe regularity of the ridge flow at $\\,\\beta=0\\,$ is also reflected in the flux $\\,\\hat{F}_{(7)} = d\\hat{\\alpha} \\wedge h^{(6)}  + \\ldots\\,$. The adapted angular variable $\\,\\hat{\\alpha}\\,$ simplifies in this case to\n\\begin{equation}\n\\hat{\\alpha}(\\tilde{\\alpha}) = - 8 \\, g^{-6} \\,  \\cos^6\\tilde{\\alpha} \\ ,  \n\\end{equation}\nso it is independent of $\\,\\rho\\,$. Therefore, all the $\\,\\Gamma\\,$ curves start at $\\,\\tilde{\\alpha}=0\\,$, end at $\\,\\tilde{\\alpha}=\\frac{\\pi}{2}\\,$ and flow parallel to the $\\,\\textrm{S}^7\\,$ angular direction $\\,\\tilde{\\alpha}\\,$ without displaying any bending or pathological behaviour.\n\n\n\n\n\n\n\nFinally, as discussed in Section~\\ref{sec:ridge_4D}, setting $\\,\\beta=\\pi\\,$ amounts to a shift $\\,\\rho \\rightarrow -\\rho\\,$ in the four-dimensional ridge flow solution while keeping the domain $\\,\\rho \\in [1,\\infty)\\,$. This reflection of the radial coordinate leaves the eleven-dimensional metric in (\\ref{11D_metric_ridge_beta_0}) and (\\ref{j1_j2_functions}) invariant.  The three-form gauge potential in (\\ref{Ast_ridge_beta_0}) and (\\ref{Atr_ridge_beta_0}) simply flips its sign.\n\n\n\n\n\n\n\n\n\n\\section{Summary and discussion}\n\\label{sec:conclusions}\n\n\nIn this paper we have presented new analytic families of $\\,\\textrm{AdS}_{3} \\times \\mathbb{R}\\,$ Janus and $\\,\\textrm{AdS}_{3} \\times \\mathbb{R}^{+}\\,$ Hades solutions in the $\\,\\mathcal{N}=2\\,$ gauged STU-model in four dimensions \\cite{Cvetic:1999xp}. This supergravity model corresponds to the $\\textrm{U}(1)^{4}$ invariant sector of the maximal SO(8) gauged supergravity that arises upon reduction of eleven-dimensional supergravity on a seven sphere. \n\n\nThe Janus solutions turn out to be surprisingly simple. Using a radial coordinate $\\,\\rho  \\in (-\\infty \\, , \\infty)\\,$, the geometry is given by \n\\begin{equation}\n\\label{Janus_metric_conclus}\ng^{2} \\, ds_{4}^{2} =  \\frac{d\\rho^{2}}{\\rho^{2}+1} + \n\\frac{ \\rho^{2} + 1  }{ k^2} \\,ds_{\\textrm{AdS}_{3}}^{2} \\ ,\n\\end{equation}\nin terms of the supergravity gauge coupling $\\,g\\,$ and three constant parameters $\\, \\alpha_{i} \\in \\mathbb{R} \\,$. The latter enter (\\ref{Janus_metric_conclus}) through the specific combination\n\\begin{equation}\n\\label{k_factor_Janus_conclus}\nk^2= 1 + \\sum_{i=1}^{3} \\sinh^{2}\\alpha_{i} \\, \\ge \\, 1 \\ .\n\\end{equation}\nThe Janus geometry (\\ref{Janus_metric_conclus}) is supported by $\\rho$-dependent profiles for the three complex scalars in the STU-model. Using the unit-disk parameterisation of the SL(2)\/SO(2) scalar coset, they adopt the form\n\\begin{equation}\n\\label{Janus_scalar_conclus}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}} \\, \\frac{\\sinh\\alpha_{i}}{\\cosh\\alpha_{i} + i \\, \\rho}\n\\hspace{8mm} \\textrm{ with } \\hspace{8mm} i=1,2,3  \\ ,\n\\end{equation}\nand depend on three additional phases $\\, \\beta_{i} \\in [0,2 \\pi] \\,$. The result is then a six-parameter family $\\,(\\alpha_{i},\\beta_{i})\\,$ of Janus solutions in the STU-model which are everywhere regular for arbitrary choices of the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. These are generically non-supersymmetric solutions (they solve second-order equations of motion) but there is a supersymmetry enhancement when two $\\,\\alpha_{i}\\,$ parameters are set to zero. In this limit the supersymmetric Janus with $\\,\\textrm{SO}(4) \\times \\textrm{SO}(4)\\,$ symmetry of \\cite{Bobev:2013yra} is recovered. The very special choice $\\,\\alpha_{i}=0\\,$ $\\forall i \\,$ sets the three scalars to zero. In this limit the maximally supersymmetric AdS$_{4}$ vacuum of the $\\,\\textrm{SO}(8)\\,$ supergravity is recovered which uplifts to the Freund--Rubin $\\,\\textrm{AdS}_{4} \\times \\textrm{S}^{7}\\,$ vacuum of eleven-dimensional supergravity \\cite{Freund:1980xh}. Note that this vacuum controls the asymptotic behaviour of the Janus solutions at $\\,\\rho \\rightarrow \\pm \\infty\\,$.\\footnote{The Janus solutions in (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) might resemble the ``boomerang RG flows\" studied in \\cite{Donos:2017ljs} within the STU-model. These are flows in supergravity both starting and ending at the maximally supersymmetric AdS$_{4}$ vacuum of the SO(8) gauged supergravity, thus being also relevant for ABJM theory. However the Ansatz for the scalar fields in \\cite{Donos:2017ljs} explicitly breaks translation invariance in the spatial directions of the dual field theory. This is not the case for the Janus solutions (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) which have no dependence on the spatial directions of AdS$_3$.} It is also worth emphasising that the Janus solutions in (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) are everywhere regular and genuinely \\textit{axionic} in nature: $\\,\\textrm{Im}\\tilde{z}_{i}(\\rho) \\neq 0\\,$ for the solution to exist. This fact makes the study of similar solutions in the Euclidean theory (where pseudo-scalars pick up an extra factor of $\\,i\\,$ with respect to proper scalars) interesting in the AdS\/CFT spirit of \\cite{Arkani-Hamed:2007cpn,Bobev:2020pjk}. This could help to understand instanton-like solutions in the context of M-theory, as it has been done for the type IIB non-extremal D-instantons \\cite{Bergshoeff:2004fq,Bergshoeff:2004pg,Bergshoeff:2005zf} (see also \\cite{Hertog:2017owm}), and perhaps to shed new light on axionic wormholes in M-theory. This issue certainly deserves further investigation.\n\n\nThe Hades solutions are closely related to the Janus solutions and turn out to be very simple too. Using this time a radial coordinate $\\,\\rho  \\in [1 \\, , \\infty)$, the geometry is given by \n\\begin{equation}\n\\label{Hades_metric_conclus}\ng^{2} \\, ds_{4}^{2} =  \\frac{d\\rho^{2}}{\\rho^{2} - 1} + \n\\frac{  \\rho^{2} - 1  }{ k^2} \\,ds_{\\textrm{AdS}_{3}}^{2} \\ ,\n\\end{equation}\nwith\n\\begin{equation}\n\\label{k_factor_Hades_conclus}\nk^2= -1 + \\sum_{i=1}^3 \\cosh^2\\alpha_{i} \\ ,\n\\end{equation}\nand the scalar profiles read\n\\begin{equation}\n\\label{Hades_scalar_conclus}\n\\tilde{z}_{i}(\\rho) = e^{i \\beta_{i}}\\, \\frac{\\cosh\\alpha_{i}}{\\sinh\\alpha_{i} + i \\rho} \n\\hspace{8mm} \\textrm{ with } \\hspace{8mm} i=1,2,3  \\ .\n\\end{equation}\nUnlike the Janus, the Hades solutions are singular at $\\,\\rho=1\\,$ and do not possess a supersymmetric limit upon tuning of the parameters $\\,(\\alpha_{i},\\beta_{i})\\,$. Still the maximally supersymmetric AdS$_{4}$ vacuum controls the asymptotic behaviour of the Hades at $\\,\\rho \\rightarrow \\infty\\,$. The special limit $\\,\\alpha_{i}=0\\,$ $\\forall i \\,$ drastically simplifies the Hades solutions giving rise to the so-called ridge flows (see Figure~\\ref{fig:Hades_ztilde_U1^4}).\n\n\nBeing obtained within the $\\textrm{U}(1)^4$ invariant sector of the massless $\\,\\mathcal{N}=8\\,$ supergravity multiplet in four dimensions, the analytic Janus solutions in (\\ref{Janus_metric_conclus})-(\\ref{Janus_scalar_conclus}) generalise the supersymmetric ones with $\\,\\text{SO}(4) \\times\\text{SO}(4)\\,$ symmetry constructed in \\cite{Bobev:2013yra}. The non-supersymmetric Hades solutions in (\\ref{Hades_metric_conclus})-(\\ref{Hades_scalar_conclus}) are genuinely new an cannot be continuously connected with the supersymmetric Hades with $\\,\\text{SO}(4) \\times\\text{SO}(4)\\,$ symmetry of \\cite{Bobev:2013yra} upon tuning of $\\,\\alpha_{i}\\,$. In addition, the Janus and Hades solutions presented in this work can be readily uplifted to eleven-dimensional supergravity using the general results for the oxidation of the STU-model worked out in \\cite{Cvetic:1999xp,Azizi:2016noi} and the uplift building blocks collected in the Appendix~\\ref{app:general_uplift}. Instead of uplifting the general $\\textrm{U}(1)^4$ symmetric Janus and Hades solutions, and for the sake of simplicity, we have restricted to the case \n\\begin{equation}\n\\alpha_{1}=\\alpha_{2}=\\alpha_{3}=\\alpha\n\\hspace{10mm} \\textrm{ and } \\hspace{10mm}\n\\beta_{1}=\\beta_{2}=\\beta_{3}=\\beta\n\\end{equation}\nfor which a larger symmetry group $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^2 \\subset \\textrm{SO}(8)\\,$ is preserved by the solutions. The Janus solutions are non-supersymmetric and fully regular, both in four and eleven dimensions, for arbitrary values of the parameters $\\,(\\alpha,\\beta)\\,$. The four-dimensional singularity of the Hades may or may not be cured when the solutions are uplifted to eleven-dimensions depending on the choice of parameters $\\,(\\alpha,\\beta)\\,$. For example, in the ridge flow limit $\\,\\alpha=0\\,$, the choice $\\,\\beta=0,\\pi\\,$ eliminates the singularity whereas, if setting $\\,\\beta=\\pm\\frac{\\pi}{2}\\,$, the singularity remains either localised or delocalised in the internal space. It would be interesting to understand the ultimate fate of the singularity in the general Hades solution with $\\,\\textrm{U}(1)^4\\,$ symmetry, as well as to investigate the process of taking the ridge flow limit sequentially on the three scalars $\\,\\tilde{z}_{i}\\,$. Also to further investigate a possible holographic interpretation of these more general flows as interfaces connecting an $\\,\\mathcal{N}=8\\,$ Chern--Simons matter theory to new (non-)conformal phases.\n\n\n\n\n\nSome open questions and follow-up directions regarding the Janus and Hades presented in this work are immediately envisaged. The first one is the issue of the stability, both perturbative and non-perturbative, of the general class of non-supersymmetric Janus and Hades with $\\textrm{U}(1)^4$ symmetry. These solutions can be viewed as AdS$_{3}$ vacua in M-theory, so it would be interesting to investigate their stability in light of the Weak Gravity and Swampland conjectures \\cite{ArkaniHamed:2006dz,Ooguri:2016pdq}. In this respect, and unlike for the Hades, the Janus solutions presented here are continuously connected (in parameter space) to the supersymmetric, and thus stable, Janus solutions with $\\,{\\textrm{SO}(4) \\times \\textrm{SO}(4)}\\,$ symmetry of \\cite{Bobev:2013yra}. This could help in improving the stability properties of the generic non-supersymmetric Janus solution at least within some region in the parameter space $\\,(\\alpha_{i},\\beta_{i})\\,$. Along this line, it would also be interesting to perform a probe brane analysis as a first step towards assessing the non-perturbative stability of the solutions. \n\n\nThe second issue is to understand the higher-dimensional brane picture of the various flows constructed in this work. For a related class of flat-sliced ridge flows, it was shown in \\cite{Pilch:2015vha} (motivated by \\cite{Pope:2003jp}) that the M2-branes in the UV totally polarise into a $\\,(1+3)$-dimensional intersection of M5-branes in the IR generating an AdS$_{5}$ metric at the core of the flow that is non-trivially fibered over a six-dimensional manifold.\\footnote{The appearance of a new strongly-coupled IR phase on the M2-brane involving an extra dimension was argued in \\cite{Pilch:2015vha} to originate from charged solitons that become massless, very much in the spirit of (massless) type IIA string theory and 11D supergravity.} This phenomenon was signaled by the vanishing of the space-time flux component at the IR endpoint of the flow. In our ridge flows with $\\,\\textrm{SU}(3) \\times \\textrm{U}(1)^{2}\\,$ symmetry, the expression of the space-time four-form flux (\\ref{11D_F4_st}) at generic $\\,\\beta\\,$ is given by\n\\begin{equation}\n\\label{F4st_ridge_beta_general}\n\\hat{F}_{(4)}^{\\textrm{st}} = \\frac{1}{2 g^3} \\left(   \n\\frac{3 \\, (1+\\rho^2) + 2 \\, \\rho \\, \\sin\\beta \\, (1- 2 \\cos(2\\tilde{\\alpha}))}{4} \\, d\\rho   \n+ \\sin\\beta \\, (\\rho^2-1) \\, \\sin(2\\tilde{\\alpha}) \\, d\\tilde{\\alpha} \\right) \\wedge  \\textrm{vol}_{\\textrm{AdS}_{3}} \\ ,\n\\end{equation}\nso that\n\\begin{equation}\n\\left. \\hat{F}_{(4)}^{\\textrm{st}} \\right|_{\\rho = 1} = 0 \\ ,\n\\end{equation}\nin the deep IR by virtue of the change of radial coordinate $\\,d\\rho = g \\, \\sqrt{\\rho^2-1} \\, d\\mu \\,$. This suggests a possible interpretation in terms of non-supersymmetric dielectric flows with M2-branes being polarised into intersecting M5-branes. Also, in the case of $\\,\\beta=0\\,$, we have shown the appearence of a five-dimensional geometry in the IR non-trivially fibered over a six-dimensional manifold along the lines of \\cite{Pilch:2015vha}. The generalisation to ridge and Hades flows with $\\,\\textrm{U}(1)^4\\,$ symmetry also deserves further investigation.\n\n\nThe third issue has to do with the holographic interpretation of the general Janus and Hades solutions in terms of non-supersymmetric interfaces in the field theory living at the boundary. We have made manifest the strong correlation between the choice of Janus\/Hades parameters $\\,(\\alpha_{i},\\beta_{i})\\,$ (\\textit{i.e.} boundary conditions for the complex scalars $\\,\\tilde{z}_{i}\\,$), the possible emergence of supersymmetry, the source\/VEV and bosonic\/fermionic nature of the dual operators that are turned on in the interface and the (dis)appearance of gravitational singularities. But much work remains to be done to better understand and characterise the physics of the non-supersymmetric interfaces we have presented. \n\n\nFinally, it would also be very interesting to construct charged solutions generalising the Janus and Hades constructed in this work, as well as to investigate the effect of including hypermultiplets in the setup thus going beyond the STU-model. We plan to come back to these and related issues in the future.\n\n\n\n\n\n\\section*{Acknowledgements}\n\nWe are grateful to Ant\\'on Faedo, Carlos Hoyos and Anayeli Ram\\'irez for conversations. The research of AA is supported in part by the Fondecyt Grants 1210635, 1221504 and 1181047 and by the FAPESP\/ANID project 13231-7. The work of AG and MCh-B is partially supported by the AEI through the Spanish grant PGC2018-096894-B-100 and by FICYT through the Asturian grant SV-PA-21-AYUD\/2021\/52177.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThis paper is devoted to a spectral analysis of the biharmonic operator subject to   Neumann boundary conditions  on a domain which undergoes a singular perturbation.\nThe focus is on    planar dumbbell-shaped domains $\\Omega_{\\epsilon}$, with $\\epsilon >0$, described in Figure~\\ref{fig: dumbbell}. Namely,\ngiven two   bounded  smooth domains  $\\Omega_L, \\Omega_R$ in $\\numberset{R}^2$ with $\\Omega_L\\cap \\Omega_R=\\emptyset $  such that $\\partial \\Omega_L \\supset \\{(0,y)\\in \\numberset{R}^2 : -1<y<1\\}$, $\\partial \\Omega_R  \\supset \\{(1,y)\\in \\numberset{R}^2 : -1<y<1\\}$, and $(\\Omega_R\\cup \\Omega_L) \\cap \\left([0,1]\\times[-1,1]\\right) = \\emptyset$, we\nset\n\\[\\Omega = \\Omega _{L}\\cup \\Omega_R, \\ \\ {\\rm and}\\ \\  \\Omega_\\epsilon = \\Omega \\cup R_\\epsilon \\cup L_\\epsilon\\, , \\]\nfor all $\\epsilon >0$ small enough. Here\n$ R_\\epsilon \\cup L_\\epsilon$ is a thin channel connecting $\\Omega_L$ and $\\Omega_R$  defined by\n\\begin{equation}\n\\label{def: R_eps}\nR_\\epsilon = \\{(x,y)\\in\\numberset{R}^2 : x\\in(0,1), 0< y< \\epsilon g(x) \\},\n\\end{equation}\n\\[ L_\\epsilon =( \\{0\\} \\times (0, \\epsilon g(0)) \\cup (\\{1\\}\\times (0, \\epsilon g(1))) ), \\]\nwhere  $g \\in C^2[0,1]$ is a positive real-valued function. Note that $\\Omega_\\epsilon$ collapses to the limit set $\\Omega_0 = \\Omega \\cup ([0,1] \\times \\{0\\})$ as $\\epsilon \\to 0$.\n\nWe consider  the eigenvalue problem\n\\begin{equation} \\label{PDE: main problem_eigenvalues}\n\\begin{cases}\n\\Delta^2u - \\tau \\Delta u + u = \\lambda \\, u,   &\\textup{in $\\Omega_\\epsilon$,}\\\\\n(1-\\sigma) \\frac{\\partial^2 u}{\\partial n^2} + \\sigma \\Delta u = 0, &\\textup{on $\\partial \\Omega_\\epsilon$,}\\\\\n\\tau \\frac{\\partial u}{\\partial n} - (1-\\sigma)\\, \\Div_{\\partial \\Omega_\\epsilon}(D^2u \\cdot n)_{\\partial \\Omega_\\epsilon} - \\frac{\\partial(\\Delta u)}{\\partial n} = 0, &\\textup{on $\\partial \\Omega_\\epsilon$,}\n\\end{cases}\n\\end{equation}\nwhere  $\\tau \\geq 0$, $\\sigma \\in (-1,1)$ are fixed parameters,  and we analyse the behaviour of the eigenvalues  and of the corresponding eigenfunctions as $\\epsilon \\to 0$.  Here  $\\Div_{\\partial \\Omega_\\epsilon}$ is the tangential divergence operator, and $(\\cdot)_{\\partial \\Omega_\\epsilon}$ is the projection on the tangent line to $\\partial \\Omega_\\epsilon$.\n The corresponding Poisson problem reads\n\\begin{equation} \\label{PDE: main problem}\n\\begin{cases}\n\\Delta^2u - \\tau \\Delta u +u= f, &\\textup{in $\\Omega_\\epsilon$},\\\\\n(1-\\sigma) \\frac{\\partial^2 u}{\\partial n^2} + \\sigma \\Delta u = 0, &\\textup{on $\\partial \\Omega_\\epsilon$},\\\\\n\\tau \\frac{\\partial u}{\\partial n} - (1-\\sigma) \\Div_{\\partial \\Omega_\\epsilon}(D^2u \\cdot n)_{\\partial \\Omega_\\epsilon} - \\frac{\\partial(\\Delta u)}{\\partial n} = 0, &\\textup{on $\\partial \\Omega_\\epsilon$},\n\\end{cases}\n\\end{equation}\nwith  datum\n$f \\in L^2(\\Omega_\\epsilon)$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{omega_eps_crop}\n\\caption{The dumbbell domain $\\Omega_\\epsilon$.}\n\\label{fig: dumbbell}\n\\end{figure}\n\n Since $\\partial\\Omega_{\\epsilon}$ has corner singularities at the junctions $(0,0)$,  $(0,\\epsilon g(0))$, $(1,0)$, $(1,\\epsilon g(1))$ and $H^{4}$\nregularity does not hold around those points, we shall  always understand problems  \\eqref{PDE: main problem_eigenvalues}, \\eqref{PDE: main problem},\n(as well as analogous problems) in a weak (variational) sense, in which case only $H^2$ regularity is required.\n\nNamely, the variational formulation of problem \\eqref{PDE: main problem} is the following: find $u\\in H^2(\\Omega_\\epsilon)$ such that\n\\begin{equation} \\label{PDE: main problem weak}\n\\int_{\\Omega_\\epsilon} (1-\\sigma) D^2u : D^2\\varphi + \\sigma \\Delta u \\Delta \\varphi + \\tau \\nabla u \\cdot \\nabla \\varphi +u\\varphi \\,dx = \\int_{\\Omega_\\epsilon} f \\varphi\\,dx\\, ,\n\\end{equation}\nfor all $\\varphi \\in H^2(\\Omega_\\epsilon)$. The quadratic form associated with the left-hand side    of (\\ref{PDE: main problem weak})  -  call it $B_{\\Omega_{\\epsilon}}(u, \\varphi )$ -    is coercive for all $\\tau \\geq 0$  and  $\\sigma \\in (-1,1)$, see e.g. \\cite{ChAppl}, \\cite{Ch}.\nIn particular, by standard spectral theory  this quadratic form allows to define a non-negative self-adjoint operator $T=(\\Delta^2 - \\tau \\Delta +I)_{N(\\sigma )}$  in $L^2(\\Omega_{\\epsilon})$ which plays the role of   the classical operator $\\Delta^2 - \\tau \\Delta +I$   subject to the  boundary conditions  above.\nMore precisely, $T$ is uniquely defined by the relation\n$$ B_{\\Omega_{\\epsilon}}(u, \\varphi )=<T^{1\/2}u,T^{1\/2}\\varphi >_{L^2(\\Omega_{\\epsilon})} \\, , $$\nfor all\n$ u,\\varphi \\in H^2(\\Omega_{\\epsilon})$. In particular the domain of the square root $T^{1\/2}$ of $T$ is $H^2(\\Omega_{\\epsilon})$ and\n a function $u$ belongs to the domain of $T$ if and only if\n$u\\in H^2(\\Omega_{\\epsilon})$\nand there exists $f\\in L^2(\\Omega_{\\epsilon})$ such that\n$B_{\\Omega_{\\epsilon}}(u, \\varphi )=  <f,\\varphi >_{L^2(\\Omega_{\\epsilon})} $ for all $\\varphi \\in H^2(\\Omega_{\\epsilon})$, in which case\n$Tu=f$. We refer to \\cite[Chp.~4]{Daviesbook} for a general introduction to the variational approach  to the spectral analysis of partial  differential operators on non-smooth domains.\n\n The operator $T$  is densely defined  and its eigenvalues and eigenfunctions are exactly those of  problem (\\ref{PDE: main problem_eigenvalues}).\nMoreover,  since the embedding $H^2(\\Omega_{\\epsilon} )\\subset L^2(\\Omega_{\\epsilon}  )$ is compact,   $(\\Delta^2 - \\tau \\Delta +I)_{N(\\sigma )}$ has compact resolvent, hence the spectrum is discrete and consists of a divergent increasing sequence of positive eigenvalues\n$\\lambda_{n}(\\Omega_{\\epsilon}),\\  n\\in \\numberset{N}$, with finite multiplicity (here each eigenvalue is repeated as many times as its multiplicity).\n\nProblem (\\ref{PDE: main problem_eigenvalues}) arises in linear elasticity in connection with the Kirchhoff-Love model for  the study of vibrations and deformations of free plates, in which case  $\\sigma $ represents the\n Poisson ratio of the material and  $\\tau$ the lateral tension. In this sense, the dumbbell domain $\\Omega_{\\epsilon}$  could  represent a plate and $R_{\\epsilon }$\n  a part of it which  degenerates to the   segment   $[0,1] \\times \\{0\\}$.\n\nWe note that problem (\\ref{PDE: main problem_eigenvalues}) can be considered as a natural fourth order version of the corresponding  eigenvalue problem for the\nNeumann Laplacian $-\\Delta_N$, namely\n\\begin{equation} \\label{PDE: second problem_eigenvalues}\n\\begin{cases}\n-\\Delta  u  + u = \\lambda \\, u,   &\\textup{in $\\Omega_\\epsilon$,}\\\\\n\\frac{\\partial u}{\\partial n} = 0, &\\textup{on $\\partial \\Omega_\\epsilon$,}\\\\\n\\end{cases}\n\\end{equation}\nthe variational formulation of which reads\n\n\\begin{equation} \\label{PDE: second problem weak}\n\\int_{\\Omega_\\epsilon}  Du \\cdot D \\varphi +  u  \\varphi \\,dx =\\lambda   \\int_{\\Omega_\\epsilon} u   \\varphi\\,dx ,\n\\end{equation}\nwhere the test functions $\\varphi$ and the unknown $u$ are considered in $H^1(\\Omega_{\\epsilon})$.\n\nAlthough the terminology used in the literature  to refer to boundary value problems for fourth order operators is sometimes a bit misleading, we emphasise\nthat the formulation  of  problems  \\eqref{PDE: main problem_eigenvalues}, \\eqref{PDE: main problem} is rather classical, see e.g.   \\cite[Example~2.15]{necas}\nwhere problem  \\eqref{PDE: main problem} with $\\tau =0$ is referred to as the Neumann problem for the biharmonic operator. Moreover, we point out that a number of  recent papers devoted to the  analysis of \\eqref{PDE: main problem_eigenvalues} have confirmed that problem  (\\ref{PDE: main problem_eigenvalues})\ncan be considered as   the natural Neumann problem for the biharmonic operator, see  \\cite{arlacras}, \\cite{ArrLamb}, \\cite{BuosoProv}, \\cite{BuosoProvCH}, \\cite{bulacompl}, \\cite{ChAppl}, \\cite{Ch}, \\cite{Prov}.\nWe also refer to  \\cite{GazzGS}  for an extensive discussion on boundary value problems for higher order elliptic operators.\n\nIt is  known that  the eigenelements of the Neumann Laplacian   on a typical dumbbell domain as above have a singular behaviour, see \\cite{ArrPhD}, \\cite{Arr1}, \\cite{Arr2}, \\cite{ACJdE}, \\cite{ACL}, \\cite{AHH}, and the references therein.  For example,   it is  known that  not all the eigenvalues of  $-\\Delta_N$  on $\\Omega_{\\epsilon}$ converge to  the eigenvalues of $-\\Delta_N$ in $\\Omega$; indeed, some of the eigenvalues of the dumbbell domain are asymptotically close to the eigenvalues of a boundary value problem  defined  in the channel $R_\\epsilon$. This allows the appearance in the limit of extra eigenvalues associated with an ordinary differential equation in the segment $(0,1)$, which are  generally  different from the eigenvalues of $-\\Delta_N$ in $\\Omega$.\nSuch singular behaviour reflects a general characteristic of boundary value problems with Neumann boundary conditions, the stability of which requires rather strong assumptions on the admissible  domain perturbations, see e.g., \\cite{ACJdE}, \\cite{ArrLamb}, \\cite{lalaneu}. We refer to \\cite[p.~420]{C-H} for a classical counterexample.\n\n\nThe aim of the present paper is to clarify how Neumann boundary conditions affect the spectral behaviour of the operator $\\Delta^2-\\tau \\Delta $ on dumbbell domains, by extending the validity of some results   known for the second order operator $-\\Delta_N$ to the fourth-order operator $(\\Delta^2 - \\tau \\Delta)_{N(\\sigma )}$.\n\nFirst of all, we  prove that the eigenvalues of problem (\\ref{PDE: main problem_eigenvalues})\ncan be asymptotically decomposed into two families of eigenvalues as\n\\begin{equation}\\label{dec}\n(\\lambda_n(\\Omega_\\epsilon))_{n\\geq 1} \\approx (\\omega_k)_{k\\geq 1} \\cup (\\theta^\\epsilon_l)_{l\\geq 1}, \\ \\ {\\rm as }\\  \\epsilon \\to 0,\n\\end{equation}\n where $(\\omega_k)_{k\\geq 1}$ are the eigenvalues  of problem\n\\begin{equation} \\label{PDE: Omega}\n\\begin{cases}\n\\Delta^2 w - \\tau \\Delta w + w = \\omega_k\\, w, &\\text{in $\\Omega$},\\\\\n(1-\\sigma) \\frac{\\partial^2 w}{\\partial n^2} + \\sigma \\Delta w = 0, &\\textup{on $\\partial \\Omega$},\\\\\n\\tau \\frac{\\partial w}{\\partial n} - (1-\\sigma) \\Div_{\\partial \\Omega}(D^2w \\cdot n)_{\\partial \\Omega} - \\frac{\\partial(\\Delta w)}{\\partial n} = 0, &\\textup{on $\\partial \\Omega$,}\n\\end{cases}\n\\end{equation}\nand  $(\\theta^\\epsilon_l)_{l\\geq 1}$ are the eigenvalues of problem\n\\begin{equation} \\label{PDE: R_eps}\n\\begin{cases}\n\\Delta^2 v - \\tau \\Delta v + v = \\theta^\\epsilon_l\\, v, &\\text{in $R_\\epsilon$},\\\\\n(1-\\sigma) \\frac{\\partial^2 v}{\\partial n^2} + \\sigma \\Delta v = 0, &\\textup{on $\\Gamma_\\epsilon$},\\\\\n\\tau \\frac{\\partial v}{\\partial n} - (1-\\sigma) \\Div_{\\Gamma_\\epsilon}(D^2v \\cdot n)_{\\Gamma_\\epsilon} - \\frac{\\partial(\\Delta v)}{\\partial n} = 0, &\\textup{on $\\Gamma_\\epsilon$,}\\\\\nv = 0 = \\frac{\\partial v}{\\partial n}, &\\text{on $L_\\epsilon$.}\n\\end{cases}\n\\end{equation}\nThe decomposition \\eqref{dec} is proved under the assumption that a certain condition on $R_\\epsilon$, called H-Condition, is satisfied. We provide in particular a simple condition on the profile function $g$ which guarantees the validity of the H-Condition.\n\nThus, in order to analyse the behaviour of $\\lambda_n(\\Omega_\\epsilon)$ as $\\epsilon \\to 0$, it suffices to study $\\theta^\\epsilon_l$ as $\\epsilon \\to 0$. To do so, we need to  pass to the limit in the variational formulation of problem \\eqref{PDE: R_eps}. Since the domain $R_\\epsilon$ collapses to a segment as $\\epsilon \\to 0$, we use thin domain techniques in order to find the appropriate limiting problem. As in the case of the Laplace operator, the limiting problem depends on the shape of the channel $R_\\epsilon$ via the profile function $g(x)$. More precisely it can be written as follows\n\\begin{equation}\\label{ODE: limit problem}\n\\begin{cases}\n\\frac{1 - \\sigma^2}{g} (gh'')'' - \\frac{\\tau}{g}(gh')' + h = \\theta h, &\\text{in $(0,1)$}\\\\\nh(0)=h(1)=0,&\\\\\nh'(0)=h'(1)=0.&\n\\end{cases}\n\\end{equation}\nThis allows to  prove convergence results for the eigenvalues and eigenfunctions of problem \\eqref{PDE: main problem}. The precise statement can be found in Theorem~\\ref{lastthm}.\nRoughly speaking, Theorem~\\ref{lastthm} establishes the following alternative:\n\\begin{itemize} \\item[(A)] either $\\lambda_n(\\Omega_\\epsilon) \\to \\omega_k$, for some $k\\geq 1$  in which case  the corresponding eigenfunctions  converge in $\\Omega$ to the eigenfunctions associated with   $\\omega_k$.\n\\item[(B)] or $\\lambda_n(\\Omega_\\epsilon) \\to \\theta_l$ as $\\epsilon \\to 0$ for some $l\\in \\numberset{N}$  in which case the corresponding eigenfunctions  behave in $R_\\epsilon$ like the eigenfunctions\nassociated with $ \\theta_l$.\n\\end{itemize}\nMoreover, all eigenvalues $\\omega_k$ and $\\theta_l$ are reached in the limit by the eigenvalues $\\lambda_n(\\Omega_{\\epsilon})$.\n\nWe find it  remarkable that for $\\sigma\\ne 0$ the limiting equation in (\\ref{ODE: limit problem}) is distorted by the coefficient $1-\\sigma^2\\ne 1$. This phenomenon\nshows that  the dumbbell problem for our fourth order problem  \\eqref{PDE: main problem_eigenvalues} with $\\sigma \\ne 0$ is significantly different from the second order problem \\eqref{PDE: second problem_eigenvalues} considered in the literature.\n\nWe also note that the Dirichlet problem for the operator $\\Delta^2u - \\tau \\Delta u + u$, namely\n\\begin{equation} \\label{PDE: dir}\n\\begin{cases}\n\\Delta^2u - \\tau \\Delta u + u = \\lambda \\, u,   &\\textup{in $\\Omega_\\epsilon$,}\\\\\n u = 0, &\\textup{on $\\partial \\Omega_\\epsilon$,}\\\\\n \\frac{\\partial u}{\\partial n} = 0, &\\textup{on $\\partial \\Omega_\\epsilon$}\n\\end{cases}\n\\end{equation}\nis stable in the sense that its  eigenelements converge to those of the operator $\\Delta^2- \\tau \\Delta + I$ in $\\Omega$ as $\\epsilon\\to 0$. In other words,  as for the Laplace operator, in the case of Dirichlet boundary conditions, no eigenvalues from the channel $R_{\\epsilon}$ appear in the limit as $\\epsilon \\to 0$. In fact, it is well known that Dirichlet eigenvalues on thin domains diverge to $+\\infty$ as $\\epsilon \\to 0$, because of the Poincar\\'e inequality.\n\nIn order to prove our results, we study  the convergence of the resolvent operators  $(\\Delta^2 - \\tau \\Delta +I)_{N(\\sigma , \\tau)}^{-1}$ and this is done by using the notion of  $\\mathcal{E}$-convergence, which is  a useful tool  in the analysis  of boundary value problems defined on variable domains, see e.g., \\cite{ACL}, \\cite{arlacras}, \\cite{ArrLamb}.\n\n\nWe point out that, although many papers in the literature have been devoted to the spectral analysis of second order operators with either Neumann or Dirichlet boundary conditions  on dumbbell domains,  see \\cite{Arr1}, \\cite{Arr2}, \\cite{Jimbo1}, \\cite{Jimbo2} and references therein, very little seems to be known about these problems for higher order operators. We refer to \\cite{taylor} for a recent analysis of\nthe dumbbell problem in the case of elliptic systems subject to Dirichlet boundary conditions.\n\nFinally, we observe that it would be interesting to provide precise rates of convergence  for the eigenvalues $\\lambda_n(\\Omega_{\\epsilon})$ and the corresponding eigenfunctions as $\\epsilon \\to 0$ in the spirit of the asymptotic analysis performed e.g., in \\cite{Arr2}, \\cite{Gady1}, \\cite{Gady2}, \\cite{Gady3}, \\cite{Gady4},  \\cite{Jimbo1}, \\cite{Jimbo2}  for second order operators. However, in case of higher order operators, this seems a challenging problem and is not addressed here.\n\n\nThe paper is organized as follows. In Section \\ref{sec: decomposition} we prove the asymptotic decomposition \\eqref{dec} of the eigenvalues $\\lambda_n(\\Omega_\\epsilon)$. This is achieved in several steps. In Theorem \\ref{thm: upper bound} we provide a suitable  upper bound for the eigenvalue $\\lambda_n(\\Omega_\\epsilon)$.  Then,  in Definition~\\ref{def: H condition} we introduce an assumption on the shape of the channel $R_\\epsilon$, called H-Condition,  which is needed to prove a lower bound for $\\lambda_n(\\Omega_\\epsilon)$ as $\\epsilon \\to 0$,  see Theorem~\\ref{thm: lower bound}. Finally,   we collect the results of the section in Theorem \\ref{thm: eigenvalues decomposition} to deduce a convergence result for the eigenvalues and the eigenfunctions of problem \\eqref{PDE: main problem_eigenvalues} under the assumption that the H-Condition holds. In Section \\ref{sec: proof H condition regular dumbbells} we  show that a wide class of  regular dumbbell domains satisfy the H-Condition. In Section \\ref{sec: thin plates} we study the convergence of the solutions of problem \\eqref{PDE: R_eps} as $\\epsilon \\to 0$, we  identify the limiting problem in $(0,1)$,  and we prove the spectral convergence of problem \\eqref{PDE: R_eps} to problem \\eqref{ODE: limit problem}.\nFinally, in Section \\ref{conclusionsec} we combine the results of the previous sections and prove Theorem~\\ref{lastthm}.\n\n\n\n\n\n\n\n\n\n\\section{Decomposition of the eigenvalues} \\label{sec: decomposition}\nThe main goal of this section is to prove the decomposition of the eigenvalues of problem \\eqref{PDE: main problem_eigenvalues} into the two families of eigenvalues coming from \\eqref{PDE: Omega} and \\eqref{PDE: R_eps}. First of all we note that, since $\\Omega_{\\epsilon} $, $\\Omega $ and $R_{\\epsilon }$ are sufficiently regular, by standard spectral theory for differential operators it follows that  the operators associated with the quadratic forms appearing in the weak formulation of problems \\eqref{PDE: main problem_eigenvalues}, \\eqref{PDE: Omega}, \\eqref{PDE: R_eps} have compact resolvents.  Thus, the spectra of such problems are discrete and consist of positive eigenvalues of finite multiplicity. The eigenpairs of problems \\eqref{PDE: main problem_eigenvalues}, \\eqref{PDE: Omega}, \\eqref{PDE: R_eps} will be denoted by $(\\lambda_n(\\Omega_\\epsilon), \\varphi_n^\\epsilon)_{n \\geq 1}$, $(\\omega_n, \\varphi_n^\\Omega)_{n \\geq 1}$,  $(\\theta_n^\\epsilon, \\gamma_n^\\epsilon)_{n\\geq 1}$ respectively, where    the three families of eigenfunctions $\\varphi_n^\\epsilon$, $\\varphi_n^\\Omega$,  $\\gamma_n^\\epsilon$ are complete orthonormal bases of the spaces $L^2(\\Omega_{\\epsilon})$, $L^2(\\Omega )$, $L^2(R_{\\epsilon})$ respectively.\nMoreover we set $(\\lambda_n^\\epsilon)_{n\\geq 1} = (\\omega_k)_{k \\geq 1} \\cup (\\theta_l^\\epsilon)_{l\\geq 1}$, where it is understood that the eigenvalues are arranged in increasing order and repeated according to their multiplicity. In particular if $\\omega_k = \\theta_l^\\epsilon$ for some $k,l \\in \\numberset{N}$, then such an eigenvalue is repeated in the sequence $(\\lambda_n^\\epsilon)_{n \\geq 1}$ as many times as the sum of the multiplicities of $\\omega_k$ and $\\theta_l^\\epsilon$. Let us note explicitly that the order in the sequence $(\\lambda_n^\\epsilon)_{n\\geq 1}$ depends on $\\epsilon$. For each $\\lambda_n^\\epsilon$ we define the function  $\\phi^\\epsilon_n \\in H^2(\\Omega) \\oplus H^2(R_\\epsilon)$ in the following way:\n\\begin{equation}\n\\label{def: phi_n 1}\n\\phi^\\epsilon_n = \\begin{cases}\n        \\varphi_k^\\Omega, &\\text{in $\\Omega$},\\\\\n        0, &\\text{in $R_\\epsilon$},\n        \\end{cases}\n\\end{equation}\nif $\\lambda_n^\\epsilon = \\omega_k$, for some $k \\in \\numberset{N}$; otherwise\n\\begin{equation}\n\\label{def: phi_n 2}\n\\phi^\\epsilon_n=\\begin{cases}\n        0, &\\text{in $\\Omega$},\\\\\n        \\gamma_l^\\epsilon, &\\text{in $R_\\epsilon$},\n        \\end{cases}\n\\end{equation}\nif $\\lambda_n^\\epsilon = \\theta_l^\\epsilon$, for some $l \\in \\numberset{N}$. We observe that in the case $\\lambda_n^\\epsilon= \\omega_k = \\theta_l^\\epsilon$ for some $k,l \\in \\numberset{N}$, with $\\omega_k$ of multiplicity $m_1$ and $\\theta_l^\\epsilon$ of multiplicity $m_2$  we agree  to order the eigenvalues (and the corresponding functions $\\phi^\\epsilon_n$) by listing first the $m_1$ eigenvalues $\\omega_k$, then the remaining $m_2$ eigenvalues $\\theta_l^\\epsilon$.\n\nNote that $(\\phi^\\epsilon_i, \\phi^\\epsilon_j)_{L^2(\\Omega_\\epsilon)} = \\delta_{ij}$ where  $\\delta_{ij}$ is the Kronecker symbol, that is $\\delta_{ij}=0$ for $i\\ne j$ and $\\delta_{ij}=1$ for $i=j$. Note also that  although $\\phi_n^\\epsilon$ defined by \\eqref{def: phi_n 2} are in $H^2(\\Omega_\\epsilon)$ (due to the Dirichlet boundary condition imposed in $L_\\epsilon$), the function $\\phi_n^\\epsilon$ defined by  \\eqref{def: phi_n 1} do not lie in $H^2(\\Omega_\\epsilon)$.\nTo bypass this problem we define a sequence of functions in $H^2(\\Omega_\\epsilon)$  by setting\n\\[\n\\xi_n^\\epsilon =\n\\begin{cases}\nE\\varphi_k^\\Omega, &\\text{if $\\lambda_n^\\epsilon = \\omega_k$,}\\\\\n\\phi^\\epsilon_n, &\\text{if $\\lambda_n^\\epsilon = \\theta_l^\\epsilon$},\n\\end{cases}\n\\]\nwhere $E$ is a linear continuous extension operator mapping $H^2(\\Omega)$ to $H^2(\\numberset{R}^N)$. Then it is easy to verify that for fixed $i,j$, we have  $(\\xi^\\epsilon_i, \\xi^\\epsilon_j)_{L^2(\\Omega_\\epsilon)}=\\delta_{ij}+o(1)$ as $\\epsilon \\to 0$. Then for fixed $n$ and for  $\\epsilon$ small enough, $\\xi_1^\\epsilon,\\ldots,\\xi_n^\\epsilon$  are linearly independent.\n\nNow we  prove an upper bound for the eigenvalues $\\lambda_n(\\Omega_\\epsilon)$.\n\\begin{theorem}[Upper bound] \\label{thm: upper bound}\nLet $n\\geq 1$ be fixed.  The eigenvalues $\\lambda_n^\\epsilon$ are uniformly bounded in  $\\epsilon$ and\n\\begin{equation} \\label{eq: upper bound}\n\\lambda_n(\\Omega_\\epsilon) \\leq \\lambda_n^\\epsilon + o(1), \\quad \\text{as $\\epsilon \\to 0$.}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\n\nThe fact that $\\lambda_n^\\epsilon$ remains bounded as $\\epsilon \\to 0$  is an easy consequence of the inequality \\begin{equation} \\label{eq: boundedness lambda_n^eps}\n\\lambda_n^\\epsilon \\leq \\omega_n < \\infty,\n\\end{equation}\nwhich holds by definition of $\\lambda_n^\\epsilon$.\nIn the sequel we write $\\perp$ to denote the orthogonality in $L^2$, and $[f_1, \\dots, f_m]$ for the linear span of the functions $f_1, \\dots, f_m$.\n\nBy the variational characterization of the eigenvalues $\\lambda_n(\\Omega_\\epsilon)$ we have\n\\begin{multline} \\label{eq: lambda_n(Omega_eps)var}\n\\lambda_n(\\Omega_\\epsilon) = \\min \\left\\{ \\frac{\\displaystyle \\int_{\\Omega_\\epsilon} (1-\\sigma) |D^2\\psi|^2 + \\sigma |\\Delta \\psi|^2 + \\tau |\\nabla \\psi|^2 + |\\psi|^2 }{\\displaystyle\\int_{\\Omega_\\epsilon} |\\psi|^2}  \\right.\\\\\n\\left. : \\text{$\\psi \\in H^2(\\Omega_\\epsilon)$, $\\psi \\not\\equiv 0$ and $\\psi \\perp \\varphi_1^\\epsilon, \\dots, \\varphi_{n-1}^\\epsilon$}   \\right\\}.\n\\end{multline}\nSince the functions $\\xi^\\epsilon_1,\\dots,\\xi^\\epsilon_n$ are linearly independent, by a dimension argument there exists $\\xi^\\epsilon \\in [\\xi^\\epsilon_1,\\dots,\\xi^\\epsilon_n]$ such that $\\norma{\\xi^\\epsilon}_{L^2(\\Omega_\\epsilon)}=1$, and $\\xi^\\epsilon \\perp \\varphi_1^\\epsilon, \\dots, \\varphi_{n-1}^\\epsilon$.\n\nWe can write\n$ \\xi^\\epsilon = \\sum_{i=1}^n \\alpha_i \\xi_i^\\epsilon$,\nfor some $\\alpha_1,\\dots, \\alpha_n \\in \\numberset{R}$ depending on $\\epsilon$ such that $\\sum_{i=1}^n \\alpha_i^2 = 1 + o(1)$ as $\\epsilon \\to 0$. By using $\\xi^\\epsilon$ as a test function in \\eqref{eq: lambda_n(Omega_eps)var} we get\n\n\\begin{equation} \\label{proof: computationsRQ}\n\\begin{split}\n&\\lambda_n(\\Omega_\\epsilon) \\leq \\int_{\\Omega_\\epsilon} (1-\\sigma) |D^2\\xi^\\epsilon|^2 + \\sigma |\\Delta \\xi^\\epsilon|^2 + \\tau |\\nabla \\xi^\\epsilon|^2 +|\\xi^\\epsilon|^2\\\\\n&= \\sum_{i=1}^n \\alpha_i^2 \\biggl( \\int_{\\Omega_\\epsilon} (1-\\sigma) |D^2\\xi_i^\\epsilon|^2 + \\sigma  |\\Delta\\xi_i^\\epsilon|^2 + \\tau  |\\nabla \\xi_i^\\epsilon|^2 + |\\xi_i^\\epsilon|^2 \\biggr) \\\\\n&+ \\sum_{i\\neq j}\\alpha_i\\alpha_j \\biggl( \\int_{\\Omega_\\epsilon} (1-\\sigma) (D^2\\xi^\\epsilon_i : D^2\\xi_j^\\epsilon) + \\sigma \\Delta \\xi_i^\\epsilon \\Delta\\xi_j^\\epsilon + \\tau \\nabla\\xi_i^\\epsilon \\cdot \\nabla \\xi_j^\\epsilon +  \\xi_i^\\epsilon \\xi_j^\\epsilon  \\biggr).\n\\end{split}\n\\end{equation}\n\nBy definition of $\\xi_i^\\epsilon$ and the absolute continuity of the Lebesgue integral, we have\n{\\small\\[\\int_{\\Omega_\\epsilon} (1-\\sigma) |D^2\\xi_i^\\epsilon|^2 + \\sigma  |\\Delta\\xi_i^\\epsilon|^2 + \\tau  |\\nabla \\xi_i^\\epsilon|^2 + |\\xi_i^\\epsilon|^2=\n\\begin{cases}\n\\omega_k+o(1), & \\hbox{if }\\textup{$\\exists\\, k$ s.t. $\\lambda_i^\\epsilon=\\omega_k$} ,\\\\\n\\theta_\\epsilon^l, &\\hbox{if }\\textup{$\\exists\\, l$ s.t. $\\lambda_i^\\epsilon=\\theta_\\epsilon^l$},\n\\end{cases}\n\\]}\nwhich implies that\n$\\int_{\\Omega_\\epsilon} (1-\\sigma) |D^2\\xi_i^\\epsilon|^2 + \\sigma  |\\Delta\\xi_i^\\epsilon|^2 + \\tau  |\\nabla \\xi_i^\\epsilon|^2 + |\\xi_i^\\epsilon|^2\\leq \\lambda_n^\\epsilon+o(1).$\n\nNote that\n\\[\n\\begin{split}\n&\\sum_{i\\neq j}\\alpha_i\\alpha_j \\biggl( \\int_{\\Omega_\\epsilon} (1-\\sigma) (D^2\\xi^\\epsilon_i : D^2\\xi_j^\\epsilon) + \\sigma \\Delta \\xi_i^\\epsilon \\Delta\\xi_j^\\epsilon + \\tau \\nabla\\xi_i^\\epsilon \\cdot \\nabla \\xi_j^\\epsilon  +  \\xi_i^\\epsilon \\xi_j^\\epsilon   \\biggr)=o(1)\n\\end{split}\n\\]\n\nHence,\n$\\lambda_n(\\Omega_\\epsilon)\\leq \\sum_{i=1}^n \\alpha_i^2 ( \\lambda_n^\\epsilon+o(1))+o(1)\\leq \\lambda_n^\\epsilon+o(1)$\nwhich concludes the proof of \\eqref{eq: upper bound}.\n\\end{proof}\n\n\\begin{remark}\nNote  that the shape of the channel $R_\\epsilon$ does not play any role in establishing the upper bound. The only fact needed is that the measure of $R_\\epsilon$ tends to $0$ as $\\epsilon \\to 0$.\n\\end{remark}\n\nIn the sequel we shall provide a lower bound for the eigenvalues $\\lambda_n(\\Omega_\\epsilon)$. Before doing so, let us introduce some notation.\n\n\\begin{definition}\\label{definitionNorm}\nLet $\\sigma \\in (-1,1)$, $\\tau \\geq 0$. We denote by $H^2_{L_\\epsilon}(R_\\epsilon)$ the space obtained as the closure in $H^2 (R_\\epsilon)$ of $C^{\\infty}(\\overline{R_{\\epsilon}})$ functions which vanish in a neighbourhood of  $L_\\epsilon$.\nFurthermore, for any Lipschitz bounded open set $U$  we define\n\\[\n[f]_{H^2_{\\sigma,\\tau}(U)} = \\bigl|(1-\\sigma) \\norma{D^2 f}_{L^2(U)}^2 + \\sigma \\norma{\\Delta f}_{L^2(U)}^2 + \\tau \\norma{\\nabla f}_{L^2(U)}^2 + \\norma{f}_{L^2(U)}^2   \\bigr|^{1\/2}\\, ,\n\\]\nfor all $f \\in H^2(U)$.\n\\end{definition}\n\nNote the functions $u$ in $ H^2_{L_\\epsilon}(R_\\epsilon) $ satisfy the conditions $u=0$ and $\\nabla u =0$ on $L_{\\epsilon}$ in the sense of traces.\n\n\n\\begin{proposition} \\label{prop: convergence eigenprojections}\nLet $n \\in \\numberset{N}$ be such that the following two conditions are satisfied:\n\\begin{enumerate}[label=(\\roman*)]\n\\item  For all $i=1,\\dots,n$,\n\\begin{equation} \\label{prop: lambda_i}\n\\abs{\\lambda_i^\\epsilon - \\lambda_i(\\Omega_\\epsilon)}\\to 0 \\quad \\quad \\text{as $\\epsilon \\to 0$,}\n\\end{equation}\n\\item There exists $\\delta>0$ such that\n\\begin{equation} \\label{prop: lambda_n+1}\n\\lambda_n^\\epsilon \\leq \\lambda_{n+1}(\\Omega_\\epsilon) - \\delta\n\\end{equation}\nfor any $\\epsilon >0$ small enough.\n\\end{enumerate}\nLet  $P_n$  be the projector from $L^2(\\Omega_\\epsilon)$ onto the linear span $[\\phi_1^\\epsilon,\\dots,\\phi_n^\\epsilon]$ defined by\n\\begin{equation}\nP_n g = \\sum_{i=1}^n (g, \\phi_i^\\epsilon)_{L^2(\\Omega_\\epsilon)} \\phi_i^\\epsilon\\, ,\n\\end{equation}\nfor all $g\\in L^2(\\Omega_\\epsilon)$, where $\\phi_i^\\epsilon$ is defined in \\eqref{def: phi_n 1}, \\eqref{def: phi_n 2}. Then\n\\begin{equation} \\label{prop: thesis}\n\\norma{\\varphi_i^\\epsilon - P_n \\varphi_i^\\epsilon}_{H^2(\\Omega) \\oplus H^2(R_\\epsilon)} \\to 0,\n\\end{equation}\nas $\\epsilon \\to 0$, for all $i=1,\\dots,n$.\n\\end{proposition}\n\\begin{proof}\nBy \\eqref{eq: upper bound} and \\eqref{eq: boundedness lambda_n^eps} we can extract a subsequence from both the sequences $(\\lambda_i^\\epsilon)_{\\epsilon>0}$ and $(\\lambda_i(\\Omega_\\epsilon))_{\\epsilon>0}$ such that\n\\[\n\\lambda_i^{\\epsilon_k} \\to \\lambda_i,\\ \\ {\\rm and}\\ \\\n\\lambda_i(\\Omega_{\\epsilon_k}) \\to \\widehat{\\lambda}_i,\n\\]\nas $k\\to \\infty$, for all $i=1,\\dots,n+1$.\\\\\nBy assumption we have $\\lambda_i = \\widehat{\\lambda}_i$ for all $i=1,\\dots,n$. Thus, by passing to the limit as $\\epsilon \\to 0$ in \\eqref{eq: upper bound} (with $n$ replaced by $n+1$) and in \\eqref{prop: lambda_n+1}, we get\n\\[ \\lambda_n \\leq \\widehat{\\lambda}_{n+1} - \\delta \\leq \\lambda_{n+1} - \\delta. \\]\n\nWe rewrite $\\lambda_1,\\dots,\\lambda_n$ without repetitions due to multiplicity in order to get a new sequence\n\\begin{equation} \\label{proof: nonoverlappeigenvalues}\n\\widetilde{\\lambda}_1< \\widetilde{\\lambda}_2<\\dots< \\widetilde{\\lambda}_s = \\lambda_n\n\\end{equation}\nand set  $\\widetilde{\\lambda}_{s+1}:= \\widehat{\\lambda}_{n+1} \\leq \\lambda_{n+1}$. Thus, by assumption \\eqref{prop: lambda_n+1} we have that\n\\begin{equation} \\label{proof: nonoverlappeigenvalues2}\n\\widetilde{\\lambda}_s < \\widetilde{\\lambda}_{s+1}.\n\\end{equation}\nFor each $r=1,\\dots,s$, let $\\widetilde{\\lambda}_r = \\lambda_{i_r} = \\dots = \\lambda_{j_r}$, for some $i_r \\leq j_r$, $i_r, j_r \\in \\{1,\\dots,n \\}$, where it is understood that $j_r - i_r + 1$ is the multiplicity of $\\widetilde{\\lambda}_r$. Furthermore, we define the eigenprojector $Q_r$ from $L^2(\\Omega_\\epsilon)$ onto the linear span $[\\varphi_{i_r}^\\epsilon, \\dots, \\varphi_{j_r}^\\epsilon]$ by\n\\begin{equation} \\label{proof: def Q_r}\nQ_r g = \\sum_{i=i_r}^{j_r} (g, \\varphi_{i_r}^\\epsilon)_{L^2(\\Omega_\\epsilon)} \\varphi_{i_r}^\\epsilon.\n\\end{equation}\nWe now proceed to prove the following\\\\ \\smallskip\n\n\\noindent\\emph{Claim:} $\\norma{\\xi_i^{\\epsilon_k} - Q_r \\xi_i^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})} \\to 0$ as $\\epsilon \\to 0$, for all $i_r \\leq i \\leq j_r$ and $r \\leq s$.\n\n\\noindent Let us prove it by induction on $1 \\leq r \\leq s$.\\\\\nIf $r=1$, we define the function\n\\[\n\\chi_{\\epsilon_k} = \\xi_i^{\\epsilon_k} - Q_1 \\xi_i^{\\epsilon_k} = \\xi_i^{\\epsilon_k} - \\sum_{l=1}^{j_1} (\\xi_i^{\\epsilon_k}, \\varphi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\varphi_l^{\\epsilon_k}.\n\\]\nThen $\\chi_{\\epsilon_k} \\in H^2(\\Omega_{\\epsilon_k})$, $(\\chi_{\\epsilon_k} , \\varphi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})}= 0$ for all $l=1,\\dots, j_1$ and by the min-max representation of $\\lambda_2(\\Omega_{\\epsilon_k})$ we have that\n\\begin{equation} \\label{proof: bigger lambda_2}\n[\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} \\geq \\lambda_2(\\Omega_{\\epsilon_k)} \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} \\geq \\widetilde{\\lambda}_2 \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} - o(1).\n\\end{equation}\nOn the other hand, it is easy to prove by definition of $\\chi_{\\epsilon_k}$ that\n\\begin{multline}\n\\int_{\\Omega_{\\epsilon_k}} (1-\\sigma) \\big(D^2\\chi_{\\epsilon_k} : D^2 \\psi\\big) + \\sigma \\Delta \\chi_{\\epsilon_k} \\Delta\\psi + \\tau \\nabla\\chi_{\\epsilon_k}\\cdot \\nabla \\psi + \\chi_{\\epsilon_k}\\psi \\, dx\\\\\n= \\lambda_1(\\Omega_{\\epsilon_k})\\int_{\\Omega_{\\epsilon_k}} \\chi_{\\epsilon_k} \\psi \\, dx + o(1)\n\\end{multline}\nfor all $\\psi\\in H^2(\\Omega_{\\epsilon_k})$. This in particular implies that\n\\begin{equation} \\label{proof: chi_eps equality}\n[\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} = \\lambda_1(\\Omega_{\\epsilon_k})  \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} +o(1)\n\\end{equation}\nand consequently,\n\\begin{equation} \\label{proof: less lambda_1}\n[\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} \\leq \\widetilde{\\lambda}_1 \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} + o(1).\n\\end{equation}\nHence, inequalities \\eqref{proof: bigger lambda_2}, \\eqref{proof: less lambda_1} imply that\n\\[\n\\widetilde{\\lambda}_2 \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} - o(1) \\leq \\widetilde{\\lambda}_1 \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} + o(1),\n\\]\nwhich implies that $\\norma{\\chi_{\\epsilon_k}}_{L^2(\\Omega_{\\epsilon_k})} = o(1)$ (otherwise we would have $\\widetilde{\\lambda}_2 \\leq \\widetilde{\\lambda}_1 + o(1)$, against \\eqref{proof: nonoverlappeigenvalues}). Finally, equation \\eqref{proof: chi_eps equality} implies that $[\\chi_{\\epsilon_k}]_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} = o(1)$, so that also $\\norma{\\chi_{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})}= o(1)$.\n\nLet $r>1$ and assume by induction hypothesis that\n\\begin{equation} \\label{proof: ind_hyp}\n\\norma{\\xi^{\\epsilon_k}_i - Q_t \\xi^{\\epsilon_k}_i}_{H^2(\\Omega_{\\epsilon_k})} \\to 0\n\\end{equation}\nas $k \\to \\infty$, for all $i_t \\leq i \\leq j_t$ and for all $t=1,\\dots,r-1$. We have to prove that \\eqref{proof: ind_hyp} holds also for $t=r$. Let $i_r \\leq i \\leq j_r$ and let $\\chi_{\\epsilon_k} = \\xi_i^{\\epsilon_k} - Q_r \\xi_i^{\\epsilon_k}$. Then\n\\begin{equation} \\label{proof: almost orthog}\n(\\chi_{\\epsilon_k}, \\varphi_h^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\to 0 \\quad \\text{as $k \\to \\infty$, for all $h=1,\\dots,j_r$ }.\n\\end{equation}\nIndeed, if $h \\in \\{i_r, \\dots, j_r\\}$ then by definition of $\\chi_{\\epsilon_k}$, $(\\chi_{\\epsilon_k}, \\varphi_h^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} = 0$. Otherwise, if $h < i_r$, note that the function $\\varphi_h^{\\epsilon_k}$ satisfies\n\\begin{multline*}\n\\int_{\\Omega_{\\epsilon_k}} (1-\\sigma) \\left( D^2\\varphi_h^{\\epsilon_k} : D^2\\psi \\right) + \\sigma \\Delta \\varphi_h^{\\epsilon_k} \\Delta \\psi + \\tau \\nabla \\varphi_h^{\\epsilon_k} \\nabla\\psi + \\varphi_h^{\\epsilon_k} \\psi\\, dx \\\\\n= \\lambda_h(\\Omega_{\\epsilon_k}) \\int_{\\Omega_{\\epsilon_k}} \\varphi_h^{\\epsilon_k} \\psi\\,dx\\, ,\n\\end{multline*}\nfor all $\\psi \\in H^2(\\Omega_{\\epsilon_k})$, briefly\n$\nB_{\\Omega_{\\epsilon_k}}(\\varphi_h^{\\epsilon_k}, \\psi) = \\lambda_h(\\Omega_{\\epsilon_k})(\\varphi_h^{\\epsilon_k}, \\psi)_{L^2(\\Omega_{\\epsilon_k})}\\, ,\n$\nfor all $\\psi \\in H^2(\\Omega_{\\epsilon_k})$, where $B_U$ denotes the quadratic form associated with the operator\n$\\Delta^2-\\tau\\Delta +I$\non an open set $U$.  Similarly,\n$\nB_{\\Omega_{\\epsilon_k}}(\\xi_i^{\\epsilon_k}, \\psi) = \\lambda_i^{\\epsilon_k}(\\xi_i^{\\epsilon_k}, \\psi)_{L^2(\\Omega_{\\epsilon_k})} + o(1)\n$\nfor all $\\psi \\in H^2(\\Omega_{\\epsilon_k})$. Thus,\n$\\lambda_h(\\Omega_{\\epsilon_k})(\\varphi_h^{\\epsilon_k}, \\xi_i^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} = \\lambda_i^{\\epsilon_k}(\\xi_i^{\\epsilon_k}, \\varphi_h^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} + o(1)\n$\nwhich implies\n\\begin{equation}\n\\label{proof: difference eigenvalues}\n( \\lambda_h(\\Omega_{\\epsilon_k}) - \\lambda_i^{\\epsilon_k}) (\\varphi_h^{\\epsilon_k}, \\xi_i^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} = o(1)\n\\end{equation}\nand since $( \\lambda_h(\\Omega_{\\epsilon_k}) - \\lambda_i^{\\epsilon_k}) \\to (\\widetilde{\\lambda}_h - \\widetilde{\\lambda}_i) \\neq 0$ by assumption, by \\eqref{proof: difference eigenvalues} we deduce that $(\\varphi_h^{\\epsilon_k}, \\xi_i^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} = o(1)$ as $\\epsilon_k \\to 0$, for all $h=1,\\dots,j_r$, which implies \\eqref{proof: almost orthog}.\n\nAs in the case $r=1$ we may deduce that\n\\begin{equation} \\label{proof: bigger lambda_r+1}\n[\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} \\geq \\widetilde{\\lambda}_{r+1} \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} - o(1).\n\\end{equation}\nOn the other hand, by definition of $\\chi_{\\epsilon_k}$ we have\n\\begin{equation} \\label{proof: less lambda_r}\n[\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} \\leq \\widetilde{\\lambda}_r \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} + o(1).\n\\end{equation}\nBy \\eqref{proof: bigger lambda_r+1}, \\eqref{proof: less lambda_r} and \\eqref{proof: nonoverlappeigenvalues} it must be $\\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} = o(1)$ and by \\eqref{proof: less lambda_r} we deduce that $[\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} = o(1)$, hence  $\\norma{\\chi_{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})} \\to 0$,\nas $k\\to \\infty$. This concludes the proof of the Claim.\\\\\n\nNow define the projector $\\widetilde{Q}_n$ from $L^2(\\Omega_\\epsilon)$ into the linear span $[\\varphi_1^{\\epsilon}, \\dots, \\varphi_n^{\\epsilon}]$ by\n\\[\n\\widetilde{Q}_n g = \\sum_{i=1}^n (g,\\varphi_i^\\epsilon)_{L^2(\\Omega_\\epsilon)} \\varphi_i^\\epsilon.\n\\]\nThen, as a consequence of the Claim we have that\n\\begin{equation}\n\\label{convergence xi}\n\\norma{\\xi_i^{\\epsilon_k} - \\widetilde{Q}_n \\xi_i^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})} \\to 0\n\\end{equation}\nas $k \\to \\infty$, for all $i=1,\\dots,n$. Indeed for all indexes $i=1,\\dots,n$ there exists $1 \\leq r\\leq s$ such that $i_r \\leq i \\leq j_r$; let assume for simplicity that $r=1$. Then we have $\\norma{\\xi_i^{\\epsilon_k} - Q_1 \\xi_i^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})} \\to 0$ as $k\\to \\infty$; and also\n\\[\n\\norma{\\xi_i^{\\epsilon_k} - \\widetilde{Q}_n \\xi_i^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})} \\leq \\norma{\\xi_i^{\\epsilon_k} - Q_1 \\xi_i^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})} + \\sum_{l > j_1}^n \\big\\lvert(\\xi_i^{\\epsilon_k}, \\varphi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})}\\big\\rvert \\norma{\\varphi_l^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})}\n\\]\nand the right-hand side tends to 0 as $k \\to \\infty$ because $\\norma{\\varphi_l^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})}$ is uniformly bounded in $k$ and $(\\xi_i^{\\epsilon_k}, \\varphi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\to 0$ as $k\\to \\infty$ (to see this it is sufficient to argue as in the proof of \\eqref{proof: difference eigenvalues}). Moreover, since $\\norma{\\xi_i^{\\epsilon_k} - \\phi_i^{\\epsilon_k}}_{H^2(\\Omega) \\oplus H^2(R_{\\epsilon_k})} \\to 0$ as $k \\to \\infty$ for all $i=1,\\dots,n$, we also have $\\norma{\\phi_i^{\\epsilon_k} - \\widetilde{Q}_n \\phi_i^{\\epsilon_k}}_{H^2(\\Omega) \\oplus H^2(R_{\\epsilon_k})} \\to 0$\nas $k \\to \\infty$, for all $i=1,\\dots,n$. Thus $(\\widetilde{Q}_n \\phi_1^{\\epsilon_k}, \\dots,$ $ \\widetilde{Q}_n \\phi_n^{\\epsilon_k} )$ is a basis in $(L^2(\\Omega_{\\epsilon_k})^n)$ for $[\\varphi_1^{\\epsilon_k}, \\dots, \\varphi_n^{\\epsilon_k}]$. Hence,\n$\n\\varphi_i^{\\epsilon_k} = \\sum_{l=1}^n a_{li}^{\\epsilon_k} \\widetilde{Q}_n \\phi_l^{\\epsilon_k}\n$\nfor some coefficients $a_{li}^{\\epsilon_k} = (\\varphi_i^{\\epsilon_k}, \\phi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} + o(1)$ as $k\\to\\infty$. Then for all $i =1,\\dots,n$ we have\n\\begin{multline*}\n\\norma{\\varphi_i^{\\epsilon_k} - P_n \\varphi_i^{\\epsilon_k}}_{H^2(\\Omega) \\oplus H^2(R_{\\epsilon_k}\\!)}\\\\\n = \\bigg\\lVert \\sum_{l=1}^n (\\varphi_i^{\\epsilon_k}, \\phi_l^{\\epsilon_k})_{L^2} [\\phi_l^{\\epsilon_k} - \\widetilde{Q}_n \\phi_l^{\\epsilon_k}] + o(1) \\sum_{l=1}^n \\widetilde{Q}_n \\phi_l^{\\epsilon_k} \\bigg \\rVert_{H^2(\\Omega) \\oplus H^2(R_{\\epsilon_k}\\!)}\n\\end{multline*}\nand the right-hand side tends to 0 as $k \\to \\infty$.\n\\end{proof}\n\n\\begin{remark}\n\\label{rmk: orthogonal matrix A}\nIn the previous proof one could prove that the matrix\n$A = (a_{li} ^{\\epsilon_k}    )_{l,i=1,\\dots,n}  $\nis almost orthogonal, in the sense that $A A^t = A^t A = \\mathbb{I} + o(1)$ as $k \\to \\infty$. To prove this it is sufficient to show that the matrix\n $\\tilde A= \\bigl((\\phi^{\\epsilon_k}_l, \\varphi^{\\epsilon_k}_m)_{L^2(\\Omega_{\\epsilon_k})}\\bigr)_{l,m=1,\\dots,n}$\nis almost orthogonal.   Let $l$ be fixed and note that\n$\n\\phi^{\\epsilon_k}_l = \\sum_{m=1}^n (\\phi^{\\epsilon_k}_l, \\varphi_m^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\varphi^{\\epsilon_k}_m + (\\mathbb{I}-\\widetilde{Q}_m) \\phi^{\\epsilon_k}_l,\n$\nhence, by \\eqref{convergence xi} we deduce that\n\\begin{equation}\\label{almost orthogonal matrix}\n\\delta_{li} = (\\phi^{\\epsilon_k}_l, \\phi^{\\epsilon_k}_i)_{L^2(\\Omega_{\\epsilon_k})} = \\sum_{m=1}^n (\\phi^{\\epsilon_k}_l, \\varphi^{\\epsilon_k}_m)_{L^2(\\Omega_{\\epsilon_k})} (\\varphi^{\\epsilon_k}_m, \\phi^{\\epsilon_k}_i)_{L^2(\\Omega_{\\epsilon_k})} + o(1)\\, ,\n\\end{equation}\nas $k \\to \\infty$.\nNote that we can rewrite \\eqref{almost orthogonal matrix} as  $\\tilde A \\tilde A^t = \\mathbb{I} + o(1)$, and in a similar way we also get  that $\\tilde A^t \\tilde A = \\mathbb{I} + o(1)$, concluding the proof.\n\\end{remark}\n\nIn the sequel we shall need the following lemma.\n\n\\begin{lemma} \\label{lemma: equation for chi} Let  $1 \\leq i \\leq j \\leq n$.     Assume that $\\widehat \\lambda \\in \\numberset{R}$ is such that, possibly passing to a subsequence,  $\\lambda_m(\\Omega_\\epsilon )\\to  \\widehat{\\lambda}$ as $\\epsilon \\to 0$ for all $m \\in \\{i, \\dots, j \\}$.\nIf  $\\chi_{\\epsilon} \\in [\\varphi_i^{\\epsilon}, \\dots, \\varphi_j^{\\epsilon}]$, $\\norma{\\chi_{\\epsilon}}_{L^2(\\Omega_{\\epsilon})}=1$ and $\\chi_{\\epsilon}|_{\\Omega} \\rightharpoonup \\chi$ in $H^2(\\Omega)$\nthen\n\\begin{equation}\n\\label{eq: equation for chi}\n\\int_{\\Omega} (1-\\sigma) (D^2 \\chi : D^2 \\psi) + \\sigma \\Delta \\chi \\Delta \\psi + \\tau \\nabla \\chi \\cdot\\nabla \\psi + \\chi \\psi\\, dx = \\widehat{\\lambda} \\int_{\\Omega} \\chi \\psi\\, dx\\, ,\n\\end{equation}\nfor all $\\psi \\in H^2(\\Omega)$.\n\\end{lemma}\n\\begin{proof}\nSince $\\chi_{\\epsilon} \\in [\\varphi_i^{\\epsilon}, \\dots, \\varphi_j^{\\epsilon}]$ and $\\norma{\\chi_{\\epsilon}}_{L^2(\\Omega_{\\epsilon})}=1$ there exist coefficients $(a_l(\\epsilon))_{l=i}^j$ such that\n$\n\\chi_{\\epsilon} = \\sum_{l=i}^j a_l(\\epsilon) \\varphi_l^{\\epsilon}$ and $\\sum_{l=i}^j a_l^2(\\epsilon) = 1.$\nNote that for all $m \\in \\{i, \\dots, j \\}$, possibly passing to a subsequence, there exists $\\widehat{\\varphi}_m \\in H^2(\\Omega)$ such that $\\varphi_m^{\\epsilon}|_{\\Omega} \\rightharpoonup \\widehat{\\varphi}_m$ in $H^2(\\Omega)$. Since $\\chi_{\\epsilon}|_{\\Omega} \\rightharpoonup \\chi$ in $H^2(\\Omega)$ by assumption, we get that $\\chi = \\sum_{l=i}^j a_l \\widehat{\\varphi}_l$ in $\\Omega$ for some coefficients $(a_l)_{l=i}^j$. Let $\\psi \\in H^2(\\Omega)$ be fixed and consider an extension $\\widetilde{\\psi} = E \\psi \\in H^2(\\numberset{R}^N)$. Then\n\\begin{equation} \\label{proof: lemma chi}\n\\begin{split}\n&\\int_{\\Omega_{\\epsilon}} (1-\\sigma) \\bigl(D^2\\chi_{\\epsilon} : D^2 \\widetilde{\\psi}\\bigr) + \\sigma \\Delta \\chi_{\\epsilon} \\Delta \\widetilde{\\psi} + \\tau \\nabla \\chi_{\\epsilon} \\nabla \\widetilde{\\psi} + \\chi_{\\epsilon}\\widetilde{\\psi}\\\\\n&= \\sum_{l=i}^j a_l(\\epsilon) \\biggl[ \\int_{\\Omega_{\\epsilon}} (1-\\sigma) \\bigl(D^2 \\varphi_l^{\\epsilon} : D^2 \\widetilde{\\psi}\\bigr) + \\sigma \\Delta \\varphi_l^{\\epsilon} \\Delta \\widetilde{\\psi} + \\tau \\nabla \\varphi_l^{\\epsilon} \\nabla \\widetilde{\\psi} + \\varphi_l^{\\epsilon}\\widetilde{\\psi}  \\biggr]\\\\\n&= \\sum_{l=i}^j a_l(\\epsilon) \\lambda_l(\\Omega_{\\epsilon}) \\int_{\\Omega_{\\epsilon}} \\varphi_l^{\\epsilon} \\widetilde{\\psi}.\n\\end{split}\n\\end{equation}\nThen it is possible to pass to the limit in both sides of \\eqref{proof: lemma chi} by splitting the integrals over $\\Omega_{\\epsilon}$ into an integral over $R_{\\epsilon}$ (that tends to $0$ as $\\epsilon \\to 0$) and an integral over $\\Omega$. Moreover, the integrals over $\\Omega$ will converge to the corresponding integrals in \\eqref{eq: equation for chi} as $\\epsilon \\to 0$, because of the weak convergence of $\\chi_{\\epsilon}$ in $H^2(\\Omega)$ and the strong convergence of $E\\psi$ to $\\psi$ in $H^2(\\Omega)$.\n\\end{proof}\n\n\nWe proceed to prove the lower bound for $\\lambda_n(\\Omega_\\epsilon)$.\nTo do so,  we need to add an extra assumption on the shape of $\\Omega_{\\epsilon}$. Hence,\n  we introduce the following condition in the spirit of what is known for the Neumann Laplacian (see e.g., \\cite{ArrPhD}, \\cite{Arr1}, \\cite{AHH}).\n\n\\begin{definition}[H-Condition]\n\\label{def: H condition}\nWe say that the family of dumbbell domains   $\\Omega_\\epsilon$, $\\epsilon>0$,  satisfies the H-Condition if, given functions $u_\\epsilon \\in H^2(\\Omega_\\epsilon)$ such that $\\norma{u_\\epsilon}_{H^2(\\Omega_\\epsilon)} \\leq R$ for all $\\epsilon>0$, there exist functions  $\\bar{u}_\\epsilon \\in H^2_{L_\\epsilon}(R_\\epsilon)$ such that\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\norma{u_\\epsilon - \\bar{u}_\\epsilon }_{L^2(R_\\epsilon)} \\to 0$ as $\\epsilon \\to 0$,\n\\item $[\\bar{u}_\\epsilon]^2_{H^2_{\\sigma, \\tau}(R_\\epsilon)} \\leq [u_\\epsilon]^2_{H^2_{\\sigma, \\tau}(\\Omega_\\epsilon)} + o(1)$ as $\\epsilon \\to 0$.\n\\end{enumerate}\n\\end{definition}\n\nRecall that $[\\cdot ]_{H^2_{\\sigma,\\tau}}$ is defined above in Definition \\ref{definitionNorm}. We will show in Section \\ref{sec: proof H condition regular dumbbells} that a wide class of channels $R_\\epsilon$ satisfies the H-Condition.\n\n\n\\begin{theorem}[Lower bound] \\label{thm: lower bound}\nAssume that the family of dumbbell domains $\\Omega_\\epsilon$, $\\epsilon>0$, satisfies the H-Condition.  Then for every $n\\in \\numberset{N}$ we have $\\lambda_n(\\Omega_\\epsilon) \\geq \\lambda_n^\\epsilon - o(1)$ as $\\epsilon \\to 0$.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{thm: upper bound} and its proof we know that both  $\\lambda_i(\\Omega_\\epsilon)$ and $\\lambda_i^\\epsilon$ are uniformly bounded in $\\epsilon$. Then, for each subsequence $\\epsilon_k$ we can find a subsequence (which we still call $\\epsilon_k$), sequences of real numbers $(\\lambda_i)_{i\\in \\numberset{N}}$, $(\\widehat{\\lambda}_i)_{i\\in \\numberset{N}}$, and sequences of $H^2(\\Omega)$ functions $(\\phi_i)_{i \\in \\numberset{N}}$, $(\\widehat{\\varphi}_i)_{i \\in \\numberset{N}}$, such that the following conditions are satisfied:\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\lambda_i^{\\epsilon_k} \\longrightarrow \\lambda_i$, for all $i \\geq 1$;\n\\item $\\lambda_i(\\Omega_{\\epsilon_k}) \\longrightarrow \\widehat{\\lambda}_i$, for all $i \\geq 1$;\n\\item $\\xi^{\\epsilon_k}_i|_{\\Omega} \\longrightarrow \\phi_i $ strongly in $H^2(\\Omega)$, for all $i \\geq 1$;\n\\item $\\varphi_i^{\\epsilon_k}|_{\\Omega} \\longrightarrow \\widehat{\\varphi}_i$ weakly in $H^2(\\Omega)$, for all $i \\geq 1$;\n\\end{enumerate}\nNote that $(iii)$ immediately follows by recalling that $\\xi^{\\epsilon_k}_i|_{\\Omega}$ either it is zero or it coincides with $\\varphi_i^\\Omega$. Then $(iv)$ is deduced by the estimate\n$\\norma{\\varphi_i^{\\epsilon_k}}_{H^2(\\Omega_{\\epsilon_k})} \\leq c\\, \\lambda_i(\\Omega_{\\epsilon_k})$ and by the boundedness of the sequence $\\lambda_i(\\Omega_{\\epsilon_k})$,  $k \\in \\numberset{N}$.\n\nWe plan to prove that $\\widehat{\\lambda}_i = \\lambda_i$ for all $i\\geq 1$. We do it by induction.\nFor $i=1$ we clearly have $\\lambda_1 = \\lambda_1(\\Omega) = 1 = \\lambda(\\Omega_{\\epsilon_k})$ for all $k$; hence, passing to the limit as $k\\to\\infty$ in the right-hand side of the former equality we get $\\lambda_1 = \\widehat{\\lambda_1}$.\nThen, we assume by induction hypothesis that $\\widehat{\\lambda}_i = \\lambda_i$ for all $i=1,\\dots,n$ and we prove that $\\widehat{\\lambda}_{n+1} = \\lambda_{n+1}$. There are two possibilities: either $\\lambda_n = \\lambda_{n+1}$ or $\\lambda_n < \\lambda_{n+1}$. In the first case we deduce by \\eqref{eq: upper bound} that\n\\[\n\\lambda_n = \\widehat{\\lambda}_n \\leq \\widehat{\\lambda}_{n+1} \\leq \\lambda_{n+1} = \\lambda_n,\n\\]\nhence all the inequalities are equalities and in particular $\\widehat{\\lambda}_{n+1} = \\lambda_{n+1}$.\nConsequently we can assume without loss of generality that $\\lambda_n < \\lambda_{n+1}$. In this case we must have $\\widehat{\\lambda}_{n+1} \\in [\\lambda_n, \\lambda_{n+1}]$ because $\\lambda_n=\\widehat{\\lambda_n}$ and $\\lambda_n(\\Omega_{\\epsilon_k}) \\leq \\lambda_{n+1}(\\Omega_{\\epsilon_k}) \\leq \\lambda_{n+1}^{\\epsilon_k} + o(1)$ as $k\\to \\infty$. Let $r= \\max\\{\\lambda_i : i<n, \\lambda_i < \\lambda_n \\}$. Then $\\lambda_r < \\lambda_{r+1}= \\dots = \\lambda_n < \\lambda_{n+1}$. In particular we can apply Proposition \\ref{prop: convergence eigenprojections} with $n$ replaced by $r$ in order to get\n\\begin{equation} \\label{proof: E5}\n\\norma{\\varphi_i^{\\epsilon_k} - P_r \\varphi_i^{\\epsilon_k}}_{H^2(\\Omega) \\oplus H^2(R_{\\epsilon_k})} \\to 0\n\\end{equation}\nas $k \\to \\infty$, for all $i=1,\\dots,r$. We now divide the proof in two steps.\n\\smallskip\n\n\\noindent\\emph{Step 1}: we prove that $\\lambda_n < \\widehat{\\lambda}_{n+1}$.\\\\\nLet us assume by contradiction that $\\lambda_n = \\widehat{\\lambda}_{n+1}$; then $\\widehat{\\lambda}_{r+1}=\\dots=\\widehat{\\lambda}_n=\\widehat{\\lambda}_{n+1}$. Define the subspace $S$ by $S = [\\varphi^{\\epsilon_k}_{r+1}, \\dots, \\varphi^{\\epsilon_k}_{n+1}]$. Hence, $S$ is $(n-r+1)$-dimensional. We then choose $\\chi_{\\epsilon_k} \\in S$ with the following properties:\n\\begin{enumerate}[label=(\\Roman*)]\n\\item $\\norma{\\chi_{\\epsilon_k}}_{L^2(\\Omega_{\\epsilon_k})} = 1$.\n\\item $\\chi_{\\epsilon_k} \\perp \\phi^{\\epsilon_k}_{r+1}, \\dots, \\phi^{\\epsilon_k}_{n}$ in $L^2(\\Omega_{\\epsilon_k})$.\n\\end{enumerate}\nThis choice is possible because $[\\phi^{\\epsilon_k}_{r+1}, \\dots, \\phi^{\\epsilon_k}_{n}]$ is $(n-r)$-dimensional. Moreover, we have that\n\\begin{equation} \\label{proof: E6}\n(\\,\\chi_{\\epsilon_k}, \\phi^{\\epsilon_k}_i)_{L^2(\\Omega_{\\epsilon_k})} \\longrightarrow 0\n\\end{equation}\nas $k\\to \\infty$, for all $i=1, \\dots, r$. To see this, recall that $\\chi_{\\epsilon_k} \\in S$, hence\n\\begin{equation}\n\\label{proof: orthogonality chi_eps}\n(\\chi_{\\epsilon_k}, \\varphi^{\\epsilon_k}_j)_{L^2(\\Omega_{\\epsilon_k})} = 0, \\quad \\forall j \\leq r.\n\\end{equation}\nBy \\eqref{proof: E5} and \\eqref{proof: orthogonality chi_eps}, we have\n\\[\n(\\chi_{\\epsilon_k}, P_r \\varphi^{\\epsilon_k}_j)_{L^2(\\Omega_{\\epsilon_k})} \\longrightarrow 0, \\quad \\forall j \\leq r\n\\]\nas $k\\to \\infty$. Thus,\n\\begin{equation}\n\\label{proof: matrix times vector}\n\\sum_{l=1}^r (\\varphi_j^{\\epsilon_k}, \\phi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} (\\chi_{\\epsilon_k}, \\phi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\longrightarrow 0, \\quad \\forall j \\leq r,\n\\end{equation}\nas $k\\to \\infty$. We can rewrite \\eqref{proof: matrix times vector} as $A^t b \\to 0$ as $k \\to \\infty$, where $A$ is the matrix defined in Remark \\ref{rmk: orthogonal matrix A} and $b \\in \\numberset{R}^r$ is the vector defined by $b_l = ((\\chi_{\\epsilon_k}, \\phi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})})_l$ for all $l \\in \\{1,\\dots,r\\}$. Hence, also $A A^t b \\to 0$ as $k \\to \\infty$ and by Remark \\ref{rmk: orthogonal matrix A} we deduce that $A A^t b =\\bigl(\\mathbb{I} + o(1)\\bigr) b = b + o(1) \\to 0$ as $k \\to \\infty$, since $b$ is bounded in $k$. This implies that each component of $b$, which is $(\\chi_{\\epsilon_k}, \\phi_l^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})}$ tends to zero as $k \\to \\infty$, which is \\eqref{proof: E6}.\n\n\\noindent It is now clear that \\eqref{proof: E6} and property (II) of $ \\chi_{\\epsilon_k}$ yield\n\\begin{equation} \\label{proof: E7}\n(\\chi_{\\epsilon_k}, \\phi^{\\epsilon_k}_i)_{L^2(\\Omega_{\\epsilon_k})} \\longrightarrow 0, \\quad \\text{for all $i=1,\\dots,n$}\n\\end{equation}\nas $k\\to \\infty$. Since $\\norma{\\chi_{\\epsilon_k}}_{H^2(\\Omega)} \\leq C \\max_{r+1 \\leq j \\leq n+1} \\norma{\\varphi_j^{\\epsilon_k}}_{H^2(\\Omega)} < \\infty$ there exists a function $\\chi \\in H^2(\\Omega)$     such that possibly passing to a subsequence\n\\begin{equation} \\label{proof: E8}\n\\chi_{\\epsilon_k} |_{\\Omega} \\rightharpoonup \\chi \\quad \\text{in $H^2(\\Omega)$},\n\\end{equation}\nas $k\\to \\infty$. By \\eqref{proof: E7} and \\eqref{proof: E8} we deduce that $(\\chi, \\phi_i)_{L^2(\\Omega)} = 0$, for all $i=1,\\dots,n$. By Lemma \\ref{lemma: equation for chi} $\\chi$ is a $n$-th eigenfunction of $(\\Delta^2 - \\tau \\Delta + \\mathbb{I})_{N(\\sigma)}$ in $\\Omega$ associated with $\\widetilde{\\lambda}_n$ which is orthogonal to $\\phi_1, \\dots, \\phi_n$, among which there are all the possible $n$-th eigenfunctions. Since $\\lambda_n < \\lambda_{n+1}$, the only way to avoid a contradiction is that $\\chi \\equiv 0$ in $\\Omega$, that is\n\\begin{equation} \\label{proof: E9}\n\\norma{\\chi_{\\epsilon_k}}_{L^2(\\Omega)} \\to 0, \\quad\\quad \\norma{\\chi_{\\epsilon_k}}_{L^2(R_{\\epsilon_k})} \\to 1\n\\end{equation}\nas $k\\to \\infty$. We use now the H-Condition; let us choose a sequence of functions $\\overline{\\chi}_{\\epsilon_k} \\in H^2_{L_{\\epsilon_k}}(R_{\\epsilon_k})$ such that $\\norma{\\chi_{\\epsilon_k} - \\overline{\\chi}_{\\epsilon_k}}_{L^2(R_{\\epsilon_k})} \\to 0$ as $k\\to \\infty$ and\n\\begin{equation}\n\\label{proof: ineq chibar}\n[\\overline{\\chi}_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(R_{\\epsilon_k})} \\leq [\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} + o(1)\n\\end{equation}\nas $k\\to \\infty$. Then we can extend by zero $\\overline{\\chi}_{\\epsilon_k}$ to get a function (that we still call $\\overline{\\chi}_{\\epsilon_k}$) in $H^2(\\Omega_{\\epsilon_k})$. Hence,\n\\begin{equation} \\label{proof: eq}\n\\begin{split}\n(\\overline{\\chi}_{\\epsilon_k}, &\\phi^{\\epsilon_k}_i)_{L^2(\\Omega_{\\epsilon_k})}= (\\overline{\\chi}_{\\epsilon_k}, \\phi^{\\epsilon_k}_i)_{L^2(R_{\\epsilon_k})}\\\\\n&= (\\overline{\\chi}_{\\epsilon_k} - \\chi_{\\epsilon_k} , \\phi^{\\epsilon_k}_i)_{L^2(R_{\\epsilon_k})} + (\\chi_{\\epsilon_k}, \\phi^{\\epsilon_k}_i)_{L^2(\\Omega_{\\epsilon_k})} - (\\chi_{\\epsilon_k}, \\phi^{\\epsilon_k}_i)_{L^2(\\Omega)}\n\\end{split}\n\\end{equation}\nfor all $i=1,\\dots,n$. By \\eqref{proof: E7}, \\eqref{proof: E9}, and the definition of $\\overline{\\chi}_{\\epsilon_k}$ the right hand side of \\eqref{proof: eq} tends to $0$ as $k\\to \\infty$, for all $i=1,\\dots,n$. Thus, $\\overline{\\chi}_{\\epsilon_k}$ is  asymptotically orthogonal\n to $\\phi^{\\epsilon_k}_1, \\dots, \\phi^{\\epsilon_k}_n$. In particular, by the variational characterization of the eigenvalues $\\lambda_i^{\\epsilon_k}$ we get that\n\\begin{equation} \\label{proof: E10}\n[\\overline{\\chi}_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(R_{\\epsilon_k})} \\geq \\lambda^{\\epsilon_k}_{n+1} \\norma{\\overline{\\chi}_{\\epsilon_k}}_{L^2(R_{\\epsilon_k})} - o(1) \\geq \\lambda_{n+1} \\norma{\\overline{\\chi}_{\\epsilon_k}}_{L^2(R_{\\epsilon_k})} - o(1)\n\\end{equation}\nOn the other hand, by \\eqref{proof: ineq chibar} we deduce that\n\\[\n\\begin{split}\n[\\overline{\\chi}_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(R_{\\epsilon_k})}&\\leq [\\chi_{\\epsilon_k}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} + o(1)\\\\\n&=\\lambda_n \\norma{\\chi_{\\epsilon_k}}^2_{L^2(\\Omega_{\\epsilon_k})} + o(1) = \\lambda_n \\norma{\\overline{\\chi}_{\\epsilon_k}}^2_{L^2(R_{\\epsilon_k})} + o(1).\n\\end{split}\n\\]\nThis is a contradiction to \\eqref{proof: E10} because $\\lambda_n < \\lambda_{n+1}$. Step 1 is complete.\n\\smallskip\n\n\\noindent \\emph{Step 2}: we prove that $\\widehat{\\lambda}_{n+1} = \\lambda_{n+1}$.\\\\\nAssume by contradiction that $\\widehat{\\lambda}_{n+1} < \\lambda_{n+1}$. Let us note that as a consequence of Step 1 we can use Proposition \\ref{prop: convergence eigenprojections} for the $n$-th eigenvalues in order to obtain\n\\begin{equation} \\label{proof: E11}\n\\norma{\\varphi_i^{\\epsilon_k} - P_n \\varphi_i^{\\epsilon_k}}_{H^2(\\Omega) \\oplus H^2(R_{\\epsilon_k})} \\to 0,\n\\end{equation}\nfor all $i =1,\\dots,n$. Then we can use the same argument we used in Step 1 for $\\chi_{\\epsilon_k}$ to show that\n\\begin{equation} \\label{proof: E12}\n\\norma{\\varphi_{n+1}^{\\epsilon_k}}_{L^2(\\Omega)} \\longrightarrow 0,\n\\end{equation}\nas $k\\to \\infty$. To see this, just note that $\\varphi_{n+1}^{\\epsilon_k}$ is orthogonal to $\\varphi_1^{\\epsilon_k},\\dots, \\varphi_n^{\\epsilon_k}$, and by \\eqref{proof: E11} we deduce that $(\\varphi_{n+1}^{\\epsilon_k}, \\phi_i^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\to 0$, as $k \\to \\infty$, for all $i=1,\\dots,n$. Moreover,\n\\begin{equation}\n\\label{proof: limit varphi_n+1}\n(\\varphi_{n+1}^{\\epsilon_k}, \\phi_{n+1}^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\longrightarrow 0,\n\\end{equation}\nas $k \\to \\infty$.  Indeed, looking  at the weak formulation of problem \\eqref{PDE: main problem_eigenvalues} and denoting  by $B_U$ denotes the quadratic problem associated with the operator\n$\\Delta^2-\\tau\\Delta +I$\non an open set $U$,  we deduce both\n\\[\nB_{\\Omega_{\\epsilon_k}}(\\varphi_{n+1}^{\\epsilon_k}, \\phi_{n+1}^{\\epsilon_k}) = \\lambda_{n+1}(\\Omega_{\\epsilon_k})(\\varphi_{n+1}^{\\epsilon_k}, \\phi_{n+1}^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} +o(1)\n\\]\nand\n\\[\nB_{\\Omega_{\\epsilon_k}}( \\phi_{n+1}^{\\epsilon_k},\\varphi_{n+1}^{\\epsilon_k}) = \\lambda^{\\epsilon_k}_{n+1}(\\phi_{n+1}^{\\epsilon_k}, \\varphi_{n+1}^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} + o(1),\n\\]\nand subtracting the above equalities and passing to the limit as $k \\to \\infty$ we obtain\n$(\\widehat{\\lambda}_{n+1} - \\lambda_{n+1}) \\lim_{k \\to \\infty}(\\phi_{n+1}^{\\epsilon_k}, \\varphi_{n+1}^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} = 0$, which implies \\eqref{proof: limit varphi_n+1}. Then\n\\begin{equation}\n\\label{proof: varphi_n+1 almost orth}\n(\\varphi_{n+1}^{\\epsilon_k}, \\phi_i^{\\epsilon_k})_{L^2(\\Omega_{\\epsilon_k})} \\longrightarrow 0,\n\\end{equation}\nas $k \\to \\infty$, for all $i=1,\\dots,n +1$. Passing to the limit in $k$ we have $(\\widehat{\\varphi}_{n+1}, \\phi_i)_{L^2(\\Omega)} = 0$ for all $i=1,\\dots,n+1$. However, as in Step 1 we would have $[\\widehat{\\varphi}_{n+1}]^2_{H^2_{\\sigma,\\tau}(\\Omega)} = \\widehat{\\lambda}_{n+1} \\norma{\\widehat{\\varphi}_{n+1}}_{L^2(\\Omega)}$, which contradicts the assumption $\\widehat{\\lambda}_{n+1} < \\lambda_{n+1}$ unless $\\widehat{\\varphi}_{n+1} \\equiv 0$, which gives \\eqref{proof: E12}.\\\\\nNow we use the H-Condition and \\eqref{proof: E12} in order to find a function $\\overline{\\varphi}^{\\epsilon_k}_{n+1} \\in H^2_{L_{\\epsilon_k}}(R_{\\epsilon_k})$ such that\n$ \\norma{\\overline{\\varphi}^{\\epsilon_k}_{n+1}}_{L^2(R_{\\epsilon_k})} = 1 + o(1)$ and\n\\[\n[\\overline{\\varphi}^{\\epsilon_k}_{n+1}]^2_{H^2_{\\sigma,\\tau}(R_{\\epsilon_k})} \\leq [\\varphi^{\\epsilon_k}_{n+1}]^2_{H^2_{\\sigma,\\tau}(\\Omega_{\\epsilon_k})} + o(1) = \\lambda_{n+1}(\\Omega_{\\epsilon_k}) + o(1) \\leq \\widehat{\\lambda}_{n+1} + o(1),\n\\]\nas $k\\to\\infty$. On the other hand, by the variational characterization of $\\lambda_{n+1}^{\\epsilon_k}$ and by \\eqref{proof: E12}, \\eqref{proof: varphi_n+1 almost orth} we deduce that\n$\n[\\overline{\\varphi}^{\\epsilon_k}_{n+1}]^2_{H^2_{\\sigma,\\tau}(R_{\\epsilon_k})} \\geq \\lambda_{n+1}^{\\epsilon_k}\\norma{\\overline{\\varphi}^{\\epsilon_k}_{n+1}}_{L^2(R_{\\epsilon_k})} - o(1) \\geq \\lambda_{n+1} -o(1)\n$\nas $k \\to \\infty$, hence $\\lambda_{n+1} \\leq \\widehat{\\lambda}_{n+1}$, a contradiction. Thus it must be $\\lambda_{n+1} = \\widehat{\\lambda}_{n+1}$.\n\\end{proof}\n\nWe will say that $x_\\epsilon \\in (0,\\infty )$ \\textit{divides the spectrum} of a family of  nonnegative self-adjoint operators $A_\\epsilon$, $\\epsilon >0$, with compact resolvents in $L^2(\\Omega_\\epsilon)$ if there exist $\\delta, M, N, \\epsilon_0 > 0$ such that\n\\begin{align}\n[x_\\epsilon - \\delta, x_\\epsilon + \\delta] \\cap \\{ \\lambda_n^\\epsilon \\}_{n=1}^\\infty = \\emptyset,& \\quad\\forall \\epsilon < \\epsilon_0\\\\\nx_\\epsilon \\leq M,& \\quad\\forall \\epsilon < \\epsilon_0\\\\\nN(x_{\\epsilon}) := \\#\\{ \\lambda_i^{\\epsilon} : \\lambda_i^{\\epsilon} \\leq x_\\epsilon\\}\\leq N < \\infty.\n\\end{align}\nIf $x_\\epsilon$ divides the spectrum we  define the projector $P_{x_\\epsilon}$ from $L^2(\\Omega_\\epsilon)$ onto the linear span $[\\phi_1^{\\epsilon}, \\dots, \\phi_{N(x_\\epsilon)}^{\\epsilon}]$ of the first $N(x_\\epsilon)$ eigenfunctions  by\n\\[\nP_{x_\\epsilon} g = \\sum_{i=1}^{N(x_\\epsilon)} (g,\\phi_i^\\epsilon)_{L^2(\\Omega_\\epsilon)} \\phi_i^\\epsilon\\, ,\n\\]\nfor all $g\\in L^2(\\Omega_\\epsilon)$. Then, recalling Theorem \\ref{thm: upper bound} and Theorem \\ref{thm: lower bound} we deduce the following.\n\n\\begin{theorem}[(Decomposition of the eigenvalues)] \\label{thm: eigenvalues decomposition}\nLet  $\\Omega_\\epsilon$, $\\epsilon>0$, be a family of  dumbbell domains satisfying the H-Condition. Then the following statements hold:\n\\begin{enumerate}[label =(\\roman*)]\n\\item   $\\lim_{\\epsilon \\to 0}\\, \\abs{\\lambda_n(\\Omega_\\epsilon) - \\lambda_n^\\epsilon} = 0$,  for all  $n\\in \\numberset{N} $.\n\n\\item   For any $x_\\epsilon$ dividing the spectrum,\n $\\lim_{\\epsilon \\to 0}\\, \\norma{\\varphi^\\epsilon_{r_\\epsilon} - P_{x_\\epsilon} \\varphi^\\epsilon_{r_\\epsilon}}_{H^2(\\Omega) \\oplus H^2(R_\\epsilon)} = 0$, for all $r_\\epsilon = 1,\\dots, N(x_\\epsilon)$.\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\section{Proof of the H-Condition for regular dumbbells}\n\\label{sec: proof H condition regular dumbbells}\nThe goal of this section is to prove that the H-Condition holds for regular dumbbell domains. More precisely, we will consider channels $R_\\epsilon$ such that the profile function $g$ has the following monotonicity property:\\vspace{8pt}\\\\\n (MP): \\textit{ there exists $\\delta \\in ]0, 1\/2[$ such that $g$ is decreasing on $[0,\\delta)$ and increasing on $(1-\\delta, 1]$. }  \\vspace{8pt}\\\\\n\\noindent If (MP) is satisfied then the set $A_{\\epsilon} = \\{ (x,y) \\in \\numberset{R}^2 : x \\in (0,\\delta) \\cup (1-\\delta, 1), 0<y< \\epsilon g(x)      \\}$ is contained in the union of the two rectangles $[0, \\delta] \\times [0, \\epsilon g(0)]$ and $[1-\\delta, 1] \\times [0, \\epsilon g(1)]$. This fact will be used in the proof of the following theorem in order to control the $H^2$ norm of the candidate function $\\overline{u}_\\epsilon$ appearing in the H-Condition.\n\n\\begin{theorem} \\label{thm: (MP) implies (H)}\nThe validity of condition (MP) implies the validity of the H-Condition.\n\\end{theorem}\n\nBefore writing the proof of this theorem we need to introduce some notation. First, for the sake of clarity we will consider a ``one-sided'' dumbbell $\\Omega_\\epsilon = \\Omega \\cup \\overline{R_\\epsilon}$ where $\\Omega$ is a smooth bounded domain in $\\numberset{R}^2$ such that the segment $\\{0\\} \\times [-1, 1]$ is contained in the boundary of $\\Omega$, $\\Omega \\cap \\{x \\geq 0\\} = \\emptyset$ and $R_\\epsilon$ is defined as in \\eqref{def: R_eps}. We will assume that $R_\\epsilon$ satisfies the (MP) condition on $0<x<\\delta$ only. Let $L_\\epsilon$ be the segment $\\{0\\} \\times (0, \\epsilon g(0))$.\n\nFor any  $\\gamma \\in (0,1)$, we define a function $\\chi_\\epsilon^\\gamma \\in C^{1,1}[-\\epsilon^\\gamma, 1]$,  such that   $\\chi_\\epsilon^\\gamma(-\\epsilon^\\gamma) = (\\chi_\\epsilon^\\gamma)'(-\\epsilon^\\gamma) = 0$,  $\\chi_\\epsilon^\\gamma(x) \\equiv 1$ for all $0 \\leq x \\leq 1$ and such that the following bounds on the derivatives\n\\[\n\\norma{(\\chi_\\epsilon^\\gamma)'}_{L^\\infty(-\\epsilon^\\gamma, 0)} \\leq \\frac{c_1}{\\epsilon^\\gamma}, \\quad \\norma{(\\chi_\\epsilon^\\gamma)''}_{L^\\infty(-\\epsilon^\\gamma, 0)} \\leq \\frac{c_2}{\\epsilon^{2\\gamma}},\n\\]\nare satisfied for some positive real numbers $c_1, c_2$. A possible choice for $\\chi_\\epsilon^\\gamma$ is\n\\[\n\\chi_\\epsilon^\\gamma(x) =\n\\begin{cases} -2 \\bigg( \\displaystyle \\frac{x + \\epsilon^\\gamma}{\\epsilon^\\gamma} \\bigg)^3 + 3 \\bigg( \\frac{x + \\epsilon^\\gamma}{\\epsilon^\\gamma} \\bigg)^2, & x\\in (-\\epsilon^{\\gamma},0), \\\\  \\\\\n\\qquad 1, & x\\in (0,1),\n\\end{cases}\n\\]\nwhich gives the (non-optimal) bounds $c_1 = 3\/2$, $c_2 = 6$. For any  $\\gamma, \\beta>0$ we define the function $f_{\\gamma, \\beta} \\in C^{1,1}(0,1)$ by setting\n\\begin{equation}\nf=f_{\\gamma,\\beta}(x) =\n\\begin{cases} -\\epsilon^\\gamma \\Big(\\frac{x}{\\epsilon^\\beta}\\Big)^2 + (\\epsilon^\\beta+2\\epsilon^\\gamma) \\Big( \\frac{x}{\\epsilon^\\beta} \\Big) - \\epsilon^\\gamma, & x\\in (0,\\epsilon^\\beta), \\\\\n\\qquad x, & x\\in (\\epsilon^\\beta, 1).\n\\end{cases}\n\\end{equation}\nNote that $f$ is a $C^{1,1}$-diffeomorphism from  $(0, \\epsilon^\\beta)$ onto $(-\\epsilon^\\gamma, \\epsilon^\\beta)$. Then,\n\\[\nf'(x) =\n\\begin{cases} 1+2 \\epsilon^{\\gamma-\\beta} \\, (1-\\frac{x}{\\epsilon^\\beta}), & x\\in (0,\\epsilon^\\beta), \\\\\n\\qquad 1, & x\\in (\\epsilon^\\beta, 1),\n\\end{cases}\n\\]\nand\n\\[\nf''(x) =\n\\begin{cases} - 2 \\epsilon^{\\gamma-2\\beta}, & x\\in (0,\\epsilon^\\beta), \\\\\n\\qquad 0, & x\\in (\\epsilon^\\beta, 1),\n\\end{cases}\n\\]\nwhich implies  that $|f'(x)-1|\\leq 2 \\epsilon^{\\gamma-\\beta}$, for all $x\\in (0,1)$, and $|f''(x)|\\leq 2  \\epsilon^{\\gamma-2\\beta}$, for all $x\\in (0,1)$. Thus,\n if  $\\gamma>\\beta$ then\n\\begin{equation}\n\\label{eq: asymptotics f'}\nf'(x) = 1 + o(1) \\quad \\hbox{ as } \\epsilon\\to 0.\n\\end{equation}\n\nFor any  $\\theta \\in (0,1)$, we define the following sets:\n\\begin{align*}\n&K_\\epsilon^\\theta = \\{ (x,y) \\in \\Omega: - \\epsilon^\\theta < x < 0,\\, 0 < y < \\epsilon g(0) \\}\\, ,   \\\\\n&\\Gamma_\\epsilon^\\theta = \\{ (-\\epsilon^\\theta, y) : 0<y<\\epsilon g(0) \\} \\, ,         \\\\\n&J_\\epsilon^\\theta = \\{ (x,y) \\in R_\\epsilon : 0< x < \\epsilon^\\theta    \\} \\, ,   \\\\\n&Q_\\epsilon^\\theta = \\{(x,y) \\in \\numberset{R}^2: 0<x<\\epsilon^\\theta, 0<y<\\epsilon g(0)  \\}\\,  .\n\\end{align*}\n\n\nFinally,  if  $\\gamma \/3 < \\beta < \\gamma \/2$, for every  $u_\\epsilon \\in H^2(\\Omega_\\epsilon)$ we define  the function $\\overline{u}_\\epsilon \\in H^2(R_\\epsilon)$\nby setting\n\\begin{equation} \\label{def: u bar}\n\\overline{u}_\\epsilon (x,y) = u_\\epsilon(f(x),y) \\,\\chi^\\gamma_\\epsilon(f(x)),\n\\end{equation}\nfor all $(x,y) \\in R_\\epsilon$.   Function $\\overline{u}_\\epsilon$ will be used to prove the validity of\nthe   H-Condition. Before doing so, we need to prove the following proposition.\n\n\\bigskip\n\n\n\\begin{proposition}\\label{prop: sym arg}\nLet $\\Omega_\\epsilon = \\Omega \\cup R_\\epsilon$ with $R_\\epsilon$ satisfying the (MP) condition. Let $u_\\epsilon \\in H^2(\\Omega_\\epsilon)$ be such that $\\norma{u_\\epsilon}_ {H^2(\\Omega_\\epsilon)} \\leq R$ for all $\\epsilon > 0$. Then, with the notation above and for $0<\\theta<\\frac{1}{3}$, we have\n\\begin{equation}\n \\norma{u_\\epsilon}_{L^2(J_\\epsilon^\\theta)} = O(\\epsilon^{2\\theta}), \\quad \\norma{\\nabla u_\\epsilon}_{L^2(J_\\epsilon^\\theta)}=O(\\epsilon^\\theta), \\hbox{  as  } \\epsilon\\to 0\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWe define the function $u_\\epsilon^s \\in H^2(J_\\epsilon^\\theta)$ by setting\n\\[\nu_\\epsilon^s (x,y) = -3 u_\\epsilon(-x,y) + 4 u_\\epsilon \\Bigl(-\\frac{x}{2}, y \\Bigr)\n\\]\nfor all $(x,y) \\in J_\\epsilon^\\theta$. The function $u_\\epsilon^s$ can be viewed as a higher order reflection of $u_\\epsilon$ with respect to the $y$-axis. Let us note that we can estimate the $L^2$ norm of $u^s_\\epsilon$, of its gradient and of its derivatives of order 2, in the following way:\n\\begin{align}\n&\\norma{u_\\epsilon^s}_{L^2(J_\\epsilon^\\theta)} \\leq C \\norma{u_\\epsilon}_{L^2(K_\\epsilon^\\theta)}, \\label{proof: ineq 1}  \\\\\n&\\norma{\\nabla u_\\epsilon^s}_{L^2(J_\\epsilon^\\theta)} \\leq C \\norma{\\nabla u_\\epsilon}_{L^2(K_\\epsilon^\\theta)}, \\label{proof: ineq 2} \\\\\n&\\norma{D^\\alpha u_\\epsilon^s}_{L^2(J_\\epsilon^\\theta)} \\leq C\\norma{D^\\alpha u_\\epsilon}_{L^2(K_\\epsilon^\\theta)}, \\label{proof: ineq 3}\n\\end{align}\nfor any multiindex $\\alpha$ of length $2$ and for some constant $C$ independent of $\\epsilon$.  To obtain the three inequalities above, we are using that the image of $K_\\epsilon^\\theta$ under the reflexion about the $y$-axis contains $J_\\epsilon^\\theta$. This is a consequence of (MP).\nSince the $L^2$ norms on the right-hand sides of the inequalities above are taken on a subset of $\\Omega$, we can improve the estimate of  \\eqref{proof: ineq 1}  and  \\eqref{proof: ineq 2}  using H\\\"older's inequality and Sobolev embeddings to obtain\n\\begin{equation}\\label{sobolev-1}\n\\norma{u_\\epsilon}_{L^2(K_\\epsilon^\\theta)} \\leq  |K_\\epsilon^\\theta|^{1\/2} \\norma{u_\\epsilon}_{L^\\infty(\\Omega)} \\leq c \\bigl( \\epsilon^{\\theta + 1}\\bigr)^{1\/2} \\norma{u_\\epsilon}_{H^2(\\Omega)}\n\\end{equation}\nand in a similar way\n\\begin{equation}\\label{sobolev-2}\n\\norma{\\nabla u_\\epsilon}_{L^2(K_\\epsilon^\\theta)} \\leq  |K_\\epsilon^\\theta|^{\\frac{1}{2} - \\frac{1}{p}} \\norma{\\nabla u_\\epsilon}_{L^p(\\Omega)} \\leq c \\bigl(\\epsilon^{\\theta + 1}\\bigr)^{\\frac{1}{2} - \\frac{1}{p}} \\norma{u_\\epsilon}_{H^2(\\Omega)}\n\\end{equation}\nfor any $2<p<\\infty$. Thus\n\\begin{equation}\n\\norma{u_\\epsilon^s}_{L^2(J_\\epsilon^\\theta)} \\leq C \\epsilon^{\\frac{\\theta + 1}{2}} \\norma{u_\\epsilon}_{H^2(\\Omega)},\\ \\ {\\rm and}\\ \\\n\\norma{\\nabla u_\\epsilon^s}_{L^2(J_\\epsilon^\\theta)} \\leq C \\bigl(\\epsilon^{\\theta + 1}\\bigr)^{\\frac{1}{2} - \\frac{1}{p}} \\norma{u_\\epsilon}_{H^2(\\Omega)} \\label{proof: decay ineq nabla u eps^s}.\n\\end{equation}\n\nWe also get\n\\begin{equation}\\label{bound-second-derivatives}\n\\norma{D^\\alpha u_\\epsilon^s}_{L^2(J_\\epsilon^\\theta)} \\leq C\\norma{u_\\epsilon}_{H^2(\\Omega)}\\, .\n\\end{equation}\n\n\\noindent We define now the function\n\\[\n\\psi_\\epsilon = (u_\\epsilon - u_\\epsilon^s) |_{J_\\epsilon^\\theta}  \\in H^2(J_\\epsilon^\\theta).\n\\]\nThen $\\psi_\\epsilon = 0 = \\nabla \\psi_\\epsilon$ on $L_\\epsilon$. Let us first estimate $\\norma{\\nabla u_\\epsilon}_{L^2(J_\\epsilon^\\theta)}$. Since we have\n\\[\n\\norma{\\nabla u_\\epsilon}^2_{L^2(J_\\epsilon^\\theta)} = \\sum_{i=1}^2 \\int_{J_\\epsilon^\\theta} \\Big\\lvert \\frac{\\partial u_\\epsilon}{\\partial x_i} \\Big\\rvert^2 dx,\n\\]\nwe can directly estimate the $L^2$-norm of the partial derivatives.\nSince $ \\partial_{x_i} \\psi_\\epsilon = 0 $ on $L_\\epsilon$ for all $i=1,2$ we apply a one-dimensional Poincar\u00e9 inequality in the $x$-direction. We proceed as follows. For each $x_2\\in (\\epsilon g(\\epsilon^\\theta),\\epsilon g(0))$ we denote by $h_\\epsilon(x_2)$ the unique number such that $\\epsilon g(h_\\epsilon(x_2))=x_2$  (that is, the inverse function of $\\epsilon g(\\cdot)$,  which exists because of hypothesis (MP)). For $x_2\\in (0,\\epsilon g(\\epsilon^\\theta))$ we define $h_\\epsilon(x_2)=\\epsilon^{\\theta}$.  Observe that $0\\leq h_\\epsilon(x_2)\\leq \\epsilon^\\theta $  and that $J_\\epsilon^\\theta$ can be expressed as $J_\\epsilon^\\theta=\\{(x_1,x_2):  0<x_2<\\epsilon g(0); 0<x_1<h_\\epsilon(x_2)\\}$.  Hence, for $i=1,2$ we have\n\\begin{equation} \\label{proof: Poincare ineq-1D}\n\\bigg \\lVert \\frac{\\partial \\psi_\\epsilon}{\\partial x_i}(\\cdot,x_2) \\bigg \\rVert^2_{L^2(0,h_\\epsilon(x_2))} \\leq \\frac{1}{\\lambda_1(h_\\epsilon(x_2))} \\bigg \\lVert   \\frac{\\partial}{\\partial x_1} \\bigg(\\frac{\\partial \\psi_\\epsilon}{\\partial x_i}(\\cdot, x_2) \\bigg)\\bigg \\rVert^2_{L^2(0,h_\\epsilon(x_2))}\n\\end{equation}\nwhere $\\lambda_1(\\rho)=\\bigl(\\frac{\\pi}{2}\\bigr)^2 \\rho^{-2}$ is the first eigenvalue of the problem\n\\[\n\\begin{cases}\n&-v''=\\lambda v, \\quad \\text{in $(0,\\rho)$,}\\\\\n&v(0)= 0,\\\\\n&v'(\\rho)= 0.\n\\end{cases}\n\\]\nSince $0\\leq h_\\epsilon(x_2)\\leq \\epsilon^\\theta$, we get the bound $\\lambda_1(h_\\epsilon(x_2))\\geq \\bigl(\\frac{\\pi}{2\\epsilon^\\theta }\\bigr)^2$ and integrating in \\eqref{proof: Poincare ineq-1D} with respect to  $x_2\\in(0,\\epsilon g(0))$, we get\n\\begin{equation} \\label{proof: Poincare ineq}\n\\bigg \\lVert \\frac{\\partial \\psi_\\epsilon}{\\partial x_i} \\bigg \\rVert^2_{L^2(J_\\epsilon^\\theta)} \\leq \\biggl(\\frac{2\\epsilon^\\theta }{\\pi}\\biggr)^2  \\bigg \\lVert \\frac{\\partial^2 \\psi_\\epsilon}{\\partial x \\partial x_i} \\bigg \\rVert^2_{L^2(J_\\epsilon^\\theta)}.\n\\end{equation}\n\n\\noindent Now note that\n$\n\\Big\\lVert \\frac{\\partial^2 \\psi_\\epsilon}{\\partial x \\partial x_i} \\Big\\rVert_{L^2(J_\\epsilon^\\theta)} \\leq C \\norma{u_\\epsilon}_{H^2(\\Omega_\\epsilon)} \\leq C R\n$\nfor all $\\epsilon > 0$, where we have used \\eqref{bound-second-derivatives}. Hence we rewrite inequality \\eqref{proof: Poincare ineq} in the following way:\n\\begin{equation} \\label{proof: decay ineq psi_eps}\n\\Big \\lVert \\frac{\\partial \\psi_\\epsilon}{\\partial x_i} \\Big \\rVert_{L^2(J_\\epsilon^\\theta)} \\leq \\frac{2}{\\pi} \\epsilon^\\theta  (C R + o(1)) = O(\\epsilon^\\theta)\n\\end{equation}\nas $\\epsilon \\to 0$, for $i=1,2$.\n\nFinally, by the inequalities  \\eqref{proof: decay ineq nabla u eps^s}, \\eqref{proof: decay ineq psi_eps} we deduce that\n\\begin{equation}\n\\begin{split}\n\\norma{\\nabla u_\\epsilon}_{L^2(J_\\epsilon^\\theta)} &\\leq \\norma{\\nabla \\psi_\\epsilon}_{L^2(J_\\epsilon^\\theta)} + \\norma{\\nabla u^s_\\epsilon}_{L^2(J_\\epsilon^\\theta)}\\\\\n&\\leq O(\\epsilon^\\theta) + C \\bigl(\\epsilon^{\\theta + 1}\\bigr)^{\\frac{1}{2} - \\frac{1}{p}} \\norma{u_\\epsilon}_{H^2(\\Omega)}\\leq O(\\epsilon^\\theta),\n\\end{split}\n\\end{equation}\nwhere we  have used that $(\\theta + 1) (1\/2 - 1\/p) > \\theta$ for large enough $p$.\n\nIt remains to prove that $\\norma{u_\\epsilon}_{L^2(J_\\epsilon^\\theta)}= O(\\epsilon^{2\\theta})$ as $\\epsilon \\to 0$. We can repeat the argument for $u_\\epsilon$ instead of $\\partial_{x_i} u_\\epsilon$, with the difference that now we can improve the decay of $\\norma{\\psi_\\epsilon}_{L^2(J_\\epsilon^\\theta)}$ by using  the one-dimensional Poincar\\'{e} inequality twice. More precisely we have that\n\\[\n\\norma{\\psi_\\epsilon}_{L^2(J_\\epsilon^\\theta)} \\leq \\Big(\\frac{2}{\\pi}\\Big)^2 \\epsilon^{2\\theta} \\bigg \\lVert \\frac{\\partial^2 \\psi_\\epsilon}{\\partial x^2} \\bigg \\rVert_{L^2(J_\\epsilon^\\theta)}\n\\]\nfrom which we deduce\n$\n\\norma{\\psi_\\epsilon}_{L^2(J_\\epsilon^\\theta)} = O(\\epsilon^{2\\theta})\n$\nas $\\epsilon \\to 0$. Hence,\n\\begin{equation}\\label{eq:estimate ueps}\n\\begin{split}\n\\norma{u_\\epsilon}_{L^2(J_\\epsilon^\\theta)} &\\leq \\norma{\\psi_\\epsilon}_{L^2(J_\\epsilon^\\theta)} + \\norma{u^s_\\epsilon}_{L^2(J_\\epsilon^\\theta)}\n\\leq O(\\epsilon^{2\\theta}) + C \\epsilon^{\\frac{\\theta + 1}{2}} \\norma{u_\\epsilon}_{H^2(\\Omega)} = O(\\epsilon^{2\\theta})\n\\end{split}\n\\end{equation}\nas $\\epsilon \\to 0$, concluding the proof.\n\\end{proof}\n\nWe can now give a proof of Theorem \\ref{thm: (MP) implies (H)}.\n\\begin{proof}[Proof of Theorem \\ref{thm: (MP) implies (H)}]\nLet $u_\\epsilon\\in H^2(\\Omega_\\epsilon)$ be such that $\\norma{u_\\epsilon}_{H^2(\\Omega_\\epsilon)}\\leq R$ for any $\\epsilon > 0$. We prove that the H-Condition holds if we choose $\\overline{u}_\\epsilon$ as in \\eqref{def: u bar} with $\\gamma <1\/3$. Note that $u_\\epsilon \\equiv \\overline{u}_\\epsilon$ on $R_\\epsilon \\setminus J_\\epsilon^\\beta$. Let us first estimate $\\norma{\\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)}$. By a change of variable and by \\eqref{eq: asymptotics f'} we deduce that\n\\begin{equation}\n\\begin{split}\n\\norma{\\overline{u}_\\epsilon}^2_{L^2(J_\\epsilon^\\beta)} &= \\int_0^{\\epsilon^\\beta} \\int_0^{\\epsilon g(x)} |(u_\\epsilon \\chi_\\epsilon^\\gamma)(f(x),y)|^2\\, dy dx\\\\\n&= \\int_{-\\epsilon^\\gamma}^{\\epsilon^\\beta} \\int_0^{\\epsilon g(f^{-1}(z))} |(u_\\epsilon \\chi_\\epsilon^\\gamma)(z,y)|^2 |f'(f^{-1}(z))|^{-1}\\, dy dz\\\\\n&\\leq (1+o(1)) \\int_{-\\epsilon^\\gamma}^{\\epsilon^\\beta} \\int_0^{\\epsilon g(f^{-1}(z))} |(u_\\epsilon \\chi_\\epsilon^\\gamma)(z,y)|^2 dy dz\\\\\n&\\leq (1+o(1)) \\norma{u_\\epsilon}^2_{L^2(Z_\\epsilon^\\gamma)},\n\\end{split}\n\\end{equation}\nwhere $Z_\\epsilon^\\gamma = \\{ (x,y) \\in \\Omega_\\epsilon : -\\epsilon^\\gamma < x < \\epsilon^\\beta, 0 < y < \\epsilon g(f^{-1}(x)) \\}$. Note that since the function $g$ is non increasing, then $Z_\\epsilon^\\gamma\\subset K_\\epsilon^{\\gamma}\\cup J_\\epsilon^\\beta$. Hence,\n\\begin{equation} \\label{proof: estimate overline(u)}\n\\norma{\\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)}^2 \\leq (1+o(1))( \\norma{u_\\epsilon}_{L^2(K^\\gamma_\\epsilon)}^2+ \\norma{u_\\epsilon}_{L^2(J_\\epsilon^\\beta)}^2).\n\\end{equation}\nNote that the last summand in the right-hand side of \\eqref{proof: estimate overline(u)} behaves as $O(\\epsilon^{4\\beta})$ as $\\epsilon \\to 0$ because of Proposition \\ref{prop: sym arg}. Also by \\eqref{sobolev-1} with $\\theta$ replaced by $\\gamma$, we get\n\\[\n\\norma{u_\\epsilon}_{L^2(K^\\gamma_\\epsilon)} \\leq c \\epsilon^{\\frac{\\gamma+1}{2}} \\norma{u_\\epsilon}_{H^2(\\Omega)},\n\\]\n\n\\noindent Thus,\n\\[\n\\norma{\\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)}^2 \\leq (1+o(1)) (O(\\epsilon^{4\\beta}) + O(\\epsilon^{\\gamma+1}) = O(\\epsilon^{4\\beta})\n\\]\nas $\\epsilon \\to 0$. We then have by Proposition \\ref{prop: sym arg} that\n\\[\n\\norma{u_\\epsilon - \\overline{u}_\\epsilon}_{L^2(R_\\epsilon)} = \\norma{u_\\epsilon - \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)} \\leq \\norma{u_\\epsilon}_{L^2(J_\\epsilon^\\beta)} + \\norma{\\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)} = O(\\epsilon^{2\\beta})\n\\]\nas $\\epsilon \\to 0$. This concludes the proof of $(i)$ in the H-Condition.\n\nIn order to prove $(ii)$  from Definition \\ref{def: H condition}, we first need to compute $\\norma{\\nabla \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)}$ and $\\norma{D^2 \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)}$. We have\n\\[\n\\begin{split}\n&\\frac{\\partial  \\overline{u}_\\epsilon}{\\partial x} (x,y)= \\Bigg[\\bigg(\\frac{\\partial u_\\epsilon }{\\partial x} \\chi_\\epsilon^\\gamma\\bigg) (f(x),y) + (u_\\epsilon (\\chi_\\epsilon^\\gamma)')(f(x),y) \\Bigg] f'(x)\\\\\n&\\frac{\\partial  \\overline{u}_\\epsilon}{\\partial y} (x,y)= \\bigg(\\frac{\\partial u_\\epsilon}{\\partial y} \\chi_\\epsilon^\\gamma\\bigg)(f(x),y).\n\\end{split}\n\\]\nHence,\n\\begin{equation}\n\\label{proof: gradient estimate}\n\\begin{split}\n\\norma{\\nabla \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)} &\\leq \\norma{f'}_{L^\\infty} \\bigl( \\norma{\\nabla u_\\epsilon (f(\\cdot),\\cdot)}_{L^2(J_\\epsilon^\\beta)} + \\norma{(u_\\epsilon (\\chi_\\epsilon^\\gamma)')(f(\\cdot),\\cdot)}_{L^2(J_\\epsilon^\\beta)}\\bigr)\\\\\n&\\leq  \\norma{f'}_{L^\\infty} \\norma{f'}_{L^\\infty}^{-1\/2}\\bigl(\\norma{\\nabla u_\\epsilon}_{L^2(K_\\epsilon^\\gamma \\cup J_\\epsilon^\\beta)} + c_1 \\norma{\\epsilon^{-\\gamma} u_\\epsilon}_{L^2(K_\\epsilon^\\gamma)}\\bigr)\\\\\n&\\leq (1 + o(1)) \\bigl(\\norma{\\nabla u_\\epsilon}_{L^2(K_\\epsilon^\\gamma)} + \\norma{\\nabla u_\\epsilon}_{L^2(J_\\epsilon^\\beta)} + c_1 \\epsilon^{-\\gamma} \\norma{u_\\epsilon}_{L^2(K_\\epsilon^\\gamma)}\\bigr)\n\\end{split}\n\\end{equation}\nwhere we have used the definition of $\\chi_\\epsilon^\\gamma$ and the change of variables $(f(x), y) \\mapsto (x, y)$. By Proposition \\ref{prop: sym arg} we know that $\\norma{\\nabla u_\\epsilon}_{L^2(J_\\epsilon^\\beta)} = O(\\epsilon^{\\beta})$ as $\\epsilon \\to 0$. Moreover, by \\eqref{sobolev-1}, \\eqref{sobolev-2} with $\\theta$ replaced by $\\gamma$, we deduce that\n\\[\n\\norma{u_\\epsilon}_{L^2(K_\\epsilon^\\gamma)} = O(\\epsilon^{  \\frac{\\gamma+1}{2}} ), \\quad\\quad \\norma{\\nabla u_\\epsilon}_{L^2(K_\\epsilon^\\gamma)} = O(\\epsilon^{\\gamma_p}),\n\\]\nfor any $p < \\infty$, where we have set\n\\[\n\\gamma_p = \\biggl(\\frac{1}{2} - \\frac{1}{p}\\biggr)(\\gamma + 1).\n\\]\nFinally, we deduce by \\eqref{proof: gradient estimate} that\n\\begin{equation} \\label{proof: gradient estimate final}\n\\norma{\\nabla \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)} \\leq (1+o(1)) (O(\\epsilon^{\\gamma_p}) + O(\\epsilon^{\\beta}) + \\epsilon^{-\\gamma} O(\\epsilon^{\\gamma_p}) ) = O(\\epsilon^{\\beta})\n\\end{equation}\nbecause $\\gamma_p-\\gamma>\\beta$,  for sufficiently large $p$ (note that  $\\beta < (1-\\gamma )\/2$ for $\\gamma < 1\/3$).\n\n\nWe now estimate the $L^2$ norm of $D^2 \\overline{u}_\\epsilon$. In order to simplify our  notation we write $F(x,y) = (f(x),y)$, $\\chi_\\epsilon^\\gamma = \\chi$, $\\bar u_\\epsilon=\\bar u$, $u_\\epsilon=u$ and we use the subindex notation for the partial derivatives, that is, $u_x=\\frac{\\partial u}{\\partial x}$ and so on. First, note that\n\\begin{equation}\\label{2D-eq1}\n\\begin{split}\n&\\bar u_{xx} = \\Big[\\Big(u_{xx} \\chi + 2 u_x \\chi' + u \\chi''\\Big) \\circ F \\Big] \\cdot |f'|^2 + \\Big[\\Big(u_x \\chi + u \\chi' \\Big) \\circ F\\Big] \\cdot f'', \\\\\n&\\bar u_{xy} = \\Big[\\Big( u_{xy} \\chi + u_y \\chi' \\Big) \\circ F \\Big] \\cdot f', \\\\\n&\\bar u_{yy} =\\Big( u_{yy} \\chi \\Big) \\circ F,\n\\end{split}\n\\end{equation}\nand we may write\n\\begin{equation*}\n\\bar u_{xx} = [u_{xx} \\chi \\circ F]\\cdot |f'|^2+ R_1,  \\quad  \\bar u_{xy} = [u_{xy} \\chi \\circ F] \\cdot f'+ R_2, \\quad \\bar u_{yy} = u_{yy} \\chi  \\circ F.\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\begin{split}\n&R_1= \\Big[\\Big( 2 u_x \\chi' + u \\chi''\\Big) \\circ F \\Big] \\cdot |f'|^2 + \\Big[\\Big(u_x \\chi + u \\chi' \\Big) \\circ F\\Big] \\cdot f'', \\\\\n&R_2 =  u_y \\chi' \\circ F \\cdot f'.\n\\end{split}\n\\end{equation*}\n\nWe now show that $\\|R_1\\|_{L^2(J_\\epsilon^\\beta)}=o(1)$, $\\|R_2\\|_{L^2(J_\\epsilon^\\beta)}=o(1)$ as $\\epsilon\\to 0$. For this, we will prove that each single term in $R_1$ and $R_2$ is $o(1)$ as $\\epsilon\\to 0$.  Recall that $f'(x)=1+o(1)$ and $f''(x)=o(1)$, $\\chi'=O(\\epsilon^{-\\gamma})$ and $\\chi''=O(\\epsilon^{-2\\gamma})$  for $x\\in (0,\\epsilon^\\beta)$. By a change of variables, by the Sobolev Embedding Theorem and the definition of $\\chi$ it is easy to deduce that\n\\begin{align*}\n&\\norma{(u_x \\chi')\\circ F}_{L^2(J_\\epsilon^\\beta)} \\leq (1+o(1)) \\norma{u_x \\chi'}_{L^2(K^\\gamma_\\epsilon)} \\leq C R \\epsilon^{\\gamma_p - \\gamma} = O(\\epsilon^\\beta)\\\\\n&\\norma{(u \\chi'')\\circ F}_{L^2(J_\\epsilon^\\beta)} \\leq c_2 (1+o(1)) \\norma{u \\epsilon^{-2\\gamma}}_{L^2(K_\\epsilon^\\gamma)} \\leq C R \\epsilon^{\\frac{1-3\\gamma}{2}}\\\\\n&\\norma{(u_y \\chi')\\circ F}_{L^2(J_\\epsilon^\\beta)} \\leq c_1 (1+o(1))  \\norma{\\epsilon^{-\\gamma} u_y}_{L^2(K^\\gamma_\\epsilon)} \\leq C R \\epsilon^{\\gamma_p - \\gamma}= O(\\epsilon^\\beta)\\, .\n\\end{align*}\nBy \\eqref{proof: gradient estimate final} we also have\n\\begin{equation}\n\\norma{(u_x \\chi + u \\chi' ) \\circ F}_{L^2(J_\\epsilon^\\beta)} \\leq (1+o(1)) \\norma{\\nabla \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)} = O(\\epsilon^\\beta).\n\\end{equation}\nHence the $L^2$ norms of  $R_1$, $R_2$ vanish as $\\epsilon \\to 0$. In particular,\n\\begin{equation*}\n\\norma{D^2 \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)} =  (1 + o(1))\\norma{D^2 u_\\epsilon}_{L^2(K_\\epsilon^\\gamma \\cup J_\\epsilon^\\beta)} + O(\\epsilon^{\\frac{1-3\\gamma}{2}}) + O(\\epsilon^\\beta),\n\\end{equation*}\nas $\\epsilon\\to 0$. In a similar way we can also prove that\n\\begin{equation*}\n\\norma{\\Delta \\overline{u}_\\epsilon}_{L^2(J_\\epsilon^\\beta)}  =   (1 + o(1))\\norma{\\Delta u_\\epsilon}_{L^2(K_\\epsilon^\\gamma \\cup J_\\epsilon^\\beta)} + O(\\epsilon^{\\frac{1-3\\gamma}{2}}) + O(\\epsilon^\\beta),\n\\end{equation*}\nas $\\epsilon\\to 0$. Hence,\n\\begin{multline}\n\\label{proof: channel energy}\n(1-\\sigma) \\norma{D^2 \\overline{u}_\\epsilon}^2_{L^2(J_\\epsilon^\\beta)} + \\sigma \\norma{\\Delta \\overline{u}_\\epsilon}^2_{L^2(J_\\epsilon^\\beta)} + \\tau \\norma{\\nabla \\overline{u}_\\epsilon }^2_{L^2(J_\\epsilon^\\beta)}\\\\\n  =(1-\\sigma)\\norma{D^2 u_\\epsilon}^2_{L^2(K_\\epsilon^\\gamma \\cup J_\\epsilon^\\beta)} + \\sigma \\norma{\\Delta u_\\epsilon}^2_{L^2(K_\\epsilon^\\gamma \\cup J_\\epsilon^\\beta)} + o(1).\n\\end{multline}\nBy adding to both handsides of \\eqref{proof: channel energy} {\\small$(1-\\sigma)\\norma{D^2 \\overline{u}_\\epsilon}^2_{L^2(R_\\epsilon \\setminus J_\\epsilon^\\beta)}$, $\\sigma \\norma{\\Delta \\overline{u}_\\epsilon}^2_{L^2(R_\\epsilon \\setminus J_\\epsilon^\\beta)}$} and the lower order term $\\tau \\norma{\\nabla \\overline{u}_\\epsilon }^2_{L^2(R_\\epsilon \\setminus J_\\epsilon^\\beta)}$, and keeping in account that $\\overline{u}_\\epsilon \\equiv u_\\epsilon$ on $R_\\epsilon \\setminus J_\\epsilon^\\beta$ we deduce that\n\\begin{multline}\\label{mon}\n(1-\\sigma) \\norma{D^2 \\overline{u}_\\epsilon}^2_{L^2(R_\\epsilon)} + \\sigma \\norma{\\Delta \\overline{u}_\\epsilon}^2_{L^2(R_\\epsilon)} + \\tau \\norma{\\nabla \\overline{u}_\\epsilon }^2_{L^2(R_\\epsilon)}\\\\\n  = (1-\\sigma)\\norma{D^2 u_\\epsilon}^2_{L^2(K_\\epsilon^\\gamma \\cup R_\\epsilon)} + \\sigma\\norma{\\Delta u_\\epsilon}^2_{L^2(K_\\epsilon^\\gamma \\cup R_\\epsilon)} + \\tau \\norma{\\nabla u_\\epsilon }^2_{L^2(R_\\epsilon \\setminus J_\\epsilon^\\beta)} + o(1)\\\\\n\\leq  (1-\\sigma)\\norma{D^2 u_\\epsilon}^2_{L^2(\\Omega_\\epsilon)} + \\sigma \\norma{\\Delta u_\\epsilon}^2_{L^2(\\Omega_\\epsilon)} + \\tau \\norma{\\nabla u_\\epsilon }^2_{L^2(\\Omega_\\epsilon)} + o(1),\n\\end{multline}\nas $\\epsilon \\to 0$, concluding the proof of $(ii)$ in the H-Condition.\n Note that  in \\eqref{mon}, we have used the monotonicity of the quadratic form with respect to inclusion of sets. Such property is straightforward\nfor $\\sigma \\in [0,1)$. In the case $\\sigma \\in (-1,0)$ it follows by observing that\n\\begin{multline*}\n(1-\\sigma) \\bigl[ u^2_{xx} + 2 u^2_{xy} + u^2_{yy} \\bigr] + \\sigma \\bigl[ u^2_{xx} + 2 u_{xx} u_{yy} + u^2_{yy} \\bigr]\\\\\n \\geq u^2_{xx} + u^2_{yy} + \\sigma (u^2_{xx} + u^2_{yy} ) = (1+\\sigma) (u^2_{xx} + u^2_{yy} ) > 0,\n\\end{multline*}\nfor all $u \\in H^2(\\Omega_\\epsilon)$.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Asymptotic analysis on the thin domain}\n\\label{sec: thin plates}\nThe purpose of this section is to study the convergence of the eigenvalue  problem \\eqref{PDE: R_eps} as $\\epsilon \\to 0$. Since the thin domain $R_\\epsilon$ is shrinking to the segment $(0,1)$ as $\\epsilon \\to 0$, we plan to identify the  limiting problem in $(0,1)$ and to prove that the resolvent operator of problem \\eqref{PDE: R_eps} converges as $\\epsilon \\to 0$  to the resolvent operator of the limiting problem in a suitable sense which guarantees the spectral convergence.\n\nMore precisely, we shall prove that the  the limiting eigenvalue problem in $[0,1]$ is\n\\begin{equation}\\label{classiceigenode}\n\\begin{cases}\n\\frac{1-\\sigma^2}{g} (gh'')''- \\frac{\\tau}{g}(gh')' + h = \\theta h, &\\text{in $(0,1)$,}\\\\\nh(0)=h(1)=0,&\\\\\nh'(0)=h'(1)=0.&\n\\end{cases}\n\\end{equation}\nNote that the weak formulation of (\\ref{classiceigenode}) is\n\\[\n(1-\\sigma^2)\\int_0^1  h''\\psi''gdx+\\tau \\int_0^1h'\\psi'gdx+\\int_0^1h\\psi g dx=\\theta \\int_0^1h\\psi g dx,\n\\]\nfor all $\\psi\\in H^2_0(0,1)$, where $h$ is to be found in the Sobolev space $H^2_0(0,1)$. In the sequel, we shall denote by $L^2_g(0,1)$ the Hilbert space $L^2((0,1); g(x)dx)$.\n\n\\subsection{Finding the limiting problem}\n\\label{subsection: finding limit prb}\n\nIn order to use  thin domain techniques in the spirit of \\cite{HR}, we need to fix a reference domain $R_1$ and pull-back the eigenvalue problem defined on $R_{\\epsilon}$ onto $R_1$ by means of a suitable diffeomorphism.\n\nLet $R_1$ be the rescaled domain obtained by setting $\\epsilon = 1$ in the definition of $R_\\epsilon$ (see \\eqref{def: R_eps}). For any fixed $\\epsilon >0$, let  $\\Phi_\\epsilon$ be the map from $R_1$ to $R_\\epsilon$ defined by $\\Phi_\\epsilon(x',y') = (x', \\epsilon y')= (x,y)$ for all $(x',y') \\in R_1$. We consider the composition operator $T_\\epsilon$ from $L^2(R_\\epsilon; \\epsilon^{-1}dxdy)$ to $L^2(R_1)$ defined by\n\\[\nT_\\epsilon u(x',y') = u \\circ \\Phi_\\epsilon (x', y') = u(x', \\epsilon y')\\, ,\n\\]\nfor all $u \\in L^2(R_\\epsilon)$, $(x',y') \\in R_1$.  We also endow the spaces $H^2(R_1)$ and $H^2(R_\\epsilon)$    with the norms  defined by\n\\begin{multline}\n\\|\\varphi\\|_{H^2_{\\epsilon, \\sigma, \\tau}(R_1)}^2 =\\int_{R_1} \\Bigg((1-\\sigma) \\Bigg[ \\abs*{\\frac{\\partial^2 \\varphi}{\\partial x^2}}^2 + \\frac{2}{\\epsilon^2}\\abs*{\\frac{\\partial^2 \\varphi}{\\partial x \\partial y}}^2 + \\frac{1}{\\epsilon^4} \\abs*{\\frac{\\partial^2 \\varphi}{\\partial y^2}}^2 \\Bigg]\\\\\n+ \\sigma \\abs*{\\frac{\\partial^2 \\varphi}{\\partial x^2} + \\frac{1}{\\epsilon^2}\\frac{\\partial^2 \\varphi}{\\partial y^2}}^2  + \\tau \\Bigg[ \\abs*{\\frac{\\partial \\varphi}{\\partial x}}^2 + \\frac{1}{\\epsilon}\\abs*{\\frac{\\partial \\varphi}{\\partial y}}^2 \\Bigg] + \\abs{\\varphi}^2\\, \\Bigg)dxdy\\, ,\n\\end{multline}\n\n\\begin{multline}\n\\|\\varphi\\|_{H^2_{\\sigma, \\tau}(R_\\epsilon)}^2 =\\int_{R_\\epsilon} \\Bigg((1-\\sigma) \\Bigg[ \\abs*{\\frac{\\partial^2 \\varphi}{\\partial x^2}}^2 + 2\\abs*{\\frac{\\partial^2 \\varphi}{\\partial x \\partial y}}^2 +  \\abs*{\\frac{\\partial^2 \\varphi}{\\partial y^2}}^2 \\Bigg]\\\\\n+ \\sigma \\abs*{\\frac{\\partial^2 \\varphi}{\\partial x^2}+ \\frac{\\partial^2 \\varphi}{\\partial y^2}}^2 + \\tau \\Bigg[ \\abs*{\\frac{\\partial \\varphi}{\\partial x}}^2 + \\abs*{\\frac{\\partial \\varphi}{\\partial y}}^2 \\Bigg] + \\abs{\\varphi}^2\\,\\Bigg) dxdy\\, .\n\\end{multline}\nIt is not difficult to see that if $\\varphi\\in H^2(R_\\epsilon)$ then\n\\[\\|T_\\epsilon \\varphi\\|_{H^2_{\\epsilon,\\sigma,\\tau}(R_1)}^2= \\epsilon^{-1} \\|\\varphi\\|_{H^2_{\\sigma,\\tau}(R_\\epsilon)}^2.\\]\n\nWe consider  the following Poisson problem with datum $f_\\epsilon \\in L^2(R_\\epsilon)$:\n\\begin{equation} \\label{PDE: R_eps f_eps}\n\\begin{cases}\n\\Delta^2 v_\\epsilon - \\tau \\Delta v_\\epsilon + v_\\epsilon = f_\\epsilon, &\\text{in $R_\\epsilon$},\\\\\n(1-\\sigma) \\frac{\\partial^2 v_\\epsilon}{\\partial n_\\epsilon^2} + \\sigma \\Delta v_\\epsilon = 0, &\\textup{on $\\Gamma_\\epsilon$},\\\\\n\\tau \\frac{\\partial v_\\epsilon}{\\partial n_\\epsilon} - (1-\\sigma) \\Div_{\\partial \\Omega_\\epsilon}(D^2v_\\epsilon \\cdot n_\\epsilon)_{\\partial \\Omega_\\epsilon} - \\frac{\\partial(\\Delta v_\\epsilon)}{\\partial n_\\epsilon} = 0, &\\textup{on $\\Gamma_\\epsilon$,}\\\\\nv = 0 = \\frac{\\partial v_\\epsilon}{\\partial n_\\epsilon}, &\\text{on $L_\\epsilon$.}\n\\end{cases}\n\\end{equation}\nNote that the energy space associated with Problem \\eqref{PDE: R_eps f_eps} is exactly $H^2_{L_{\\epsilon}}(R_\\epsilon)$.\nBy setting  $\\tilde{v}_\\epsilon = v_\\epsilon (x', \\epsilon y')$, $\\tilde{f}_\\epsilon = f(x', \\epsilon y')$ and  pulling-back problem   (\\ref{PDE: R_eps f_eps}) to $R_1$ by means of  $\\Phi_\\epsilon$, we get the following equivalent problem in $R_1$ in the unknown  $\\tilde{v}_\\epsilon $ (we use again the variables $(x,y)$ instead of $(x',y')$ to simplify the notation):\n{\\small \\begin{equation} \\label{PDE: R_1}\n\\begin{cases}\n\\frac{\\partial^4 \\tilde{v}_\\epsilon}{\\partial x^4} + \\frac{2}{\\epsilon^2} \\frac{\\partial^4 \\tilde{v}_\\epsilon}{\\partial x^2 \\partial y^2} + \\frac{1}{\\epsilon^4} \\frac{\\partial^4 \\tilde{v}_\\epsilon}{\\partial y^4} - \\tau \\Big( \\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial x^2} + \\frac{1}{\\epsilon^2} \\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial y^2} \\Big) + \\tilde{v}_\\epsilon = \\tilde{f}_\\epsilon, &\\text{in $R_1$},\\\\\n(1-\\sigma) \\Big( \\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial x^2}\\tilde{n}_x^2 + \\frac{2}{\\epsilon} \\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial x \\partial y}\\tilde{n}_x \\tilde{n}_y + \\frac{1}{\\epsilon^2}\\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial y^2}\\tilde{n}_y^2 \\Big)  + \\sigma \\Big( \\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial x^2} + \\frac{1}{\\epsilon^2} \\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial y^2} \\Big)= 0, &\\textup{on $\\Gamma_1$},\\\\\n\\tau \\Big(\\frac{\\partial \\tilde{v}_\\epsilon}{\\partial x} \\tilde{n}_x + \\frac{1}{\\epsilon} \\frac{\\partial \\tilde{v}_\\epsilon}{\\partial y} \\tilde{n}_y \\Big)  - (1-\\sigma) \\Div_{\\Gamma_{1,\\epsilon}}(D_\\epsilon^2 \\tilde{v}_\\epsilon \\cdot \\tilde{n}){_{\\Gamma_{1,\\epsilon}} }- \\nabla_\\epsilon(\\Delta_\\epsilon \\tilde{v}_\\epsilon) \\cdot \\tilde{n} = 0, &\\textup{on $\\Gamma_{1}$,}\\\\\n\\tilde{v}_\\epsilon = 0 = \\frac{\\partial \\tilde{v}_\\epsilon}{\\partial x} n_x + \\frac{1}{\\epsilon}\\frac{\\partial \\tilde{v}_\\epsilon}{\\partial y} \\tilde{n}_y , &\\text{on $L_1$.}\n\\end{cases}\n\\end{equation}}\nHere   $\\tilde{n} = (\\tilde{n}_x, \\tilde{n}_y) = (n_x, \\epsilon^{-1}n_y)$ and  the operators $\\Delta_\\epsilon, \\nabla_\\epsilon$ are the standard differential operators associated with $(\\partial_x, \\epsilon^{-1} \\partial_y)$. Moreover,\n\\[\\Div_{\\Gamma_{1,\\epsilon}} F = \\frac{\\partial F_1}{\\partial x} + \\frac{1}{\\epsilon}\\frac{\\partial F_2}{ \\partial y} - \\tilde{n}_\\epsilon \\nabla_{\\!\\epsilon} F\\, \\tilde{n}_\\epsilon\\]\nand $(F)_{\\Gamma_{1,\\epsilon}} = F - (F, \\tilde{n})\\, \\tilde{n}$ for any vector field $F =(F_1,F_2)$.\n\nAssume now that   the data  $f_{\\epsilon }$, $\\epsilon>0$ are such that    $(\\tilde{f}_\\epsilon)_{\\epsilon>0}$ is an equibounded family in $L^2(R_1)$, i.e.,\n\\begin{equation} \\label{hypotesis on f eps}\n\\int_{R_1} \\abs{\\tilde{f}_\\epsilon}^2\\,dxdy' \\leq c, \\quad \\textup{or equivalently} \\quad \\int_{R_\\epsilon} \\abs*{f_\\epsilon}^2 dxdy \\leq c \\epsilon\\, ,\n\\end{equation}\nfor all $\\epsilon>0$, where  $c$  is a positive  constant not depending on $\\epsilon$.\n\nWe plan to pass to the limit in  \\eqref{PDE: R_1} as $\\epsilon \\to 0$ by arguing as follows. If $\\tilde{v}_\\epsilon \\in H^2_{L_1}(R_1)$ is the solution to problem \\eqref{PDE: R_1}, then we have the following integral equality\n\\small\n\\begin{multline}\n\\label{eq: weak formulation R1}\n(1-\\sigma) \\int_{R_1} \\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial x^2} \\frac{\\partial^2 \\varphi }{\\partial x^2} + \\frac{2}{\\epsilon^2} \\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial x \\partial y} \\frac{\\partial^2 \\varphi }{\\partial x \\partial y} + \\frac{1}{\\epsilon^4} \\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial y^2} \\frac{\\partial^2 \\varphi }{\\partial y^2} dx\\\\\n+ \\sigma \\int_{R_1} \\Big(\\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial x^2} + \\frac{1}{\\epsilon^2} \\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial y^2}  \\Big) \\Big( \\frac{\\partial^2 \\varphi }{\\partial x^2} + \\frac{1}{\\epsilon^2} \\frac{\\partial^2 \\varphi }{\\partial y^2}     \\Big) dx\\\\\n + \\tau \\int_{R_1}  \\frac{\\partial \\tilde{v}_\\epsilon }{\\partial x} \\frac{\\partial \\varphi }{\\partial x} + \\frac{1}{\\epsilon^2} \\frac{\\partial \\tilde{v}_\\epsilon }{\\partial y} \\frac{\\partial \\varphi }{\\partial y} dx + \\int_{R_1} \\tilde{v}_\\epsilon \\varphi dx = \\int_{R_1} \\tilde{f}_\\epsilon \\varphi dx\n\\end{multline}\n\\normalsize\nfor all $\\varphi \\in H^2_{L_1}(R_1)$. By choosing $\\varphi = \\tilde{v}_\\epsilon$ we deduce the following apriori estimate:\n\\small\n\\begin{multline}\n\\label{ineq: apriori ineq tilde v eps}\n(1-\\sigma) \\int_{R_1} \\abs*{\\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial x^2}}^2 + \\frac{2}{\\epsilon^2} \\abs*{\\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial x \\partial y}}^2 + \\frac{1}{\\epsilon^4} \\abs*{\\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial y^2}}^2 dx + \\sigma \\int_{R_1} \\abs*{\\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial x^2} + \\frac{1}{\\epsilon^2} \\frac{\\partial^2 \\tilde{v}_\\epsilon }{\\partial y^2}}^2  dx\\\\\n+ \\tau \\int_{R_1} \\abs*{\\frac{\\partial \\tilde{v}_\\epsilon }{\\partial x}}^2 + \\frac{1}{\\epsilon^2} \\abs*{\\frac{\\partial \\tilde{v}_\\epsilon }{\\partial y}}^2 dx + \\int_{R_1} \\abs{\\tilde{v}_\\epsilon}^2 dx \\leq \\frac{1}{2} \\int_{R_1} \\abs{\\tilde{f}_\\epsilon}^2\\,dx + \\frac{1}{2} \\int_{R_1} \\abs{\\tilde{v}_\\epsilon}^2\\,dx\n\\end{multline}\n\\normalsize\nfor all $\\epsilon>0$. This implies that $\\norma{\\tilde{v}_\\epsilon}_{H^2_{\\epsilon,\\sigma, \\tau}(R_1)} \\leq C$ for all $\\epsilon> 0$, in particular $\\norma{\\tilde{v}_\\epsilon}_{H^2(R_1)} \\leq C(\\sigma, \\tau)$ for all $\\epsilon>0$; hence, there exists $v \\in H^2(R_1)$ such that, up to a subsequence\n$\\tilde{v}_\\epsilon \\to v$, weakly in $H^2(R_1)$, strongly in $H^1(R_1)$. Moreover from \\eqref{ineq: apriori ineq tilde v eps} we deduce that\n\\begin{align}\n\\label{ineq: decay y derivatives 1} &\\norma*{\\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial x \\partial y}}_{L^2(R_1)} \\leq C \\epsilon,    \\quad\\quad  \\norma*{\\frac{\\partial \\tilde{v}_\\epsilon}{\\partial y}}_{L^2(R_1)} \\leq C \\epsilon , \\\\\n\\label{ineq: decay y derivatives 2}&\\norma*{\\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial y^2}}_{L^2(R_1)} \\leq C \\epsilon^2,   \\end{align}\nfor all $\\epsilon > 0$, hence there exists $u \\in L^2(R_1)$ such that, up to a subsequence\n\\begin{equation}\\label{weakly-to-u}\n\\frac{1}{\\epsilon^2} \\frac{\\partial^2 \\tilde{v}_\\epsilon}{\\partial y^2} \\rightharpoonup u, \\hbox{ weakly in }L^2(R_1)\n\\end{equation}\n as $\\epsilon \\to 0$. By \\eqref{ineq: decay y derivatives 1} we deduce that the limit function $v$ is constant in $y$. Indeed, if we choose any function $\\phi \\in C^\\infty_c(R_1)$, then\n\\[\n\\int_{R_1} v \\frac{\\partial \\phi}{ \\partial y} = \\lim_{\\epsilon \\to 0} \\int_{R_1} \\tilde{v}_\\epsilon \\frac{\\partial \\phi}{ \\partial y} = - \\lim_{\\epsilon \\to 0} \\int_{R_1} \\frac{\\partial \\tilde{v}_\\epsilon}{\\partial y} \\phi = 0,\n\\]\nhence $\\frac{\\partial v}{\\partial y} = 0$ and then $v(x,y) \\equiv v(x)$ for almost all $(x,y) \\in R_1$. This suggests to choose test functions $\\psi$ depending only on $x$ in the weak formulation \\eqref{eq: weak formulation R1}. Possibly passing to a subsequence, there exists $f \\in L^2(R_1)$ such that\n\\[\n\\tilde{f}_\\epsilon \\rightharpoonup f \\quad \\quad \\text{in $L^2(R_1)$, as $\\epsilon \\to 0$}.\n\\]\nLet $\\psi \\in H^2_0(0,1)$. Then $\\psi \\in H^2(R_1)$ (here it is understood that the function is extended to the whole of $R_1$ by setting $\\psi(x,y) = \\psi(x)$ for all $(x,y) \\in R_1$) and clearly $\\psi \\equiv 0$ on $L_1$. Use $\\psi$ as a test function in \\eqref{eq: weak formulation R1},  pass to the limit as $\\epsilon \\to 0$  and consider \\eqref{weakly-to-u} to get\n\\begin{equation} \\label{eq: limit probl weak x}\n\\int_0^1 \\Big( \\frac{\\partial^2 v}{\\partial x^2} \\frac{\\partial^2 \\psi}{\\partial x^2} + \\sigma \\mathcal{M}(u) \\frac{\\partial^2 \\psi}{\\partial x^2} + \\tau \\frac{\\partial v}{\\partial x} \\frac{\\partial \\psi}{\\partial x} + v \\psi   \\Big) g(x)\\, dx = \\int_0^1 \\mathcal{M}(f) \\psi g(x)\\, dx\n\\end{equation}\nfor all $\\psi \\in H^2_0(0,1)$.\nHere,  the averaging operator $\\mathcal{M} $ is defined from $L^2(R_1)$ to $L^2_g(0,1)$ by\n\\[\n\\mathcal{M}  h (x) = \\frac{1}{ g(x)} \\int_0^{ g(x)} h(x,y)\\, dy\\, ,\n\\]\nfor all $h\\in  L^2(R_1)$ and for almost all $x \\in (0,1)$.\n\nFrom \\eqref{eq: limit probl weak x} we deduce that\n\\[\n\\frac{1}{g} (v'' g)'' + \\frac{\\sigma}{g} (\\mathcal{M}(u) g)'' - \\frac{\\tau}{g} (v' g)' + v = \\mathcal{M}(f), \\quad\\quad \\text{in (0,1)},\n\\]\nwhere the equality is understood in the sense of distributions.\\\\\nComing back to \\eqref{eq: weak formulation R1} we may also choose test functions $\\varphi(x,y) = \\epsilon^2 \\zeta(x,y)$, where $\\zeta \\in H_{L_1}^2(R_1)$.  Using \\eqref{ineq: decay y derivatives 1}, \\eqref{ineq: decay y derivatives 2} and letting $\\epsilon \\to 0$ we deduce\n\\begin{equation*}\n(1-\\sigma) \\int_{R_1} u \\frac{\\partial^2 \\zeta}{ \\partial y^2} + \\sigma \\int_{R_1}\\Big( \\frac{\\partial^2 v}{\\partial x^2} \\frac{\\partial^2 \\zeta}{\\partial y^2} + u \\frac{\\partial^2 \\zeta}{\\partial y^2}    \\Big) = 0\n\\end{equation*}\nwhich can be rewritten as\n\\begin{equation}\n\\label{eq: identity 2 order deriv}\n\\int_{R_1} \\Big( u + \\sigma \\frac{\\partial^2 v}{\\partial x^2} \\Big) \\frac{\\partial^2 \\zeta}{\\partial y^2} = 0\n\\end{equation}\nfor all $\\zeta \\in H_{L_1}^2(R_1)$. In particular this holds for all $\\zeta \\in C^\\infty_c(R_1)$, hence there exists the second order derivative\n\\begin{equation} \\label{eq: second derivative yy = 0}\n\\frac{\\partial^2}{\\partial y^2}\\Big( u + \\sigma \\frac{\\partial^2 v}{\\partial x^2}  \\Big) = 0.\n\\end{equation}\nHence, $u(x,y) + \\sigma \\frac{\\partial^2 v}{\\partial x^2} = \\psi_1(x) + \\psi_2(x) y$ for almost all $(x,y) \\in R_1$ and for some functions $\\psi_1, \\psi_2 \\in L^2(R_1)$, and then   \\eqref{eq: identity 2 order deriv} can be written as\n\\begin{equation} \\label{eq: limit probl weak y}\n  \\int_{R_1} (\\psi_1(x)+y\\psi_2(x)) \\frac{\\partial^2 \\zeta}{\\partial y^2} = 0\n\\end{equation}\nIntegrating twice by parts in $y$ in equation \\eqref{eq: limit probl weak y} we deduce that\n\\begin{equation}\n\\label{eq: boundary identity}\n- \\int_{\\partial R_1} \\psi_2(x) \\zeta n_y dS + \\int_{\\partial R_1} (\\psi_1(x)+y\\psi_2(x)) \\frac{\\partial \\zeta}{\\partial y} n_y dS = 0\n\\end{equation}\nfor all $\\zeta \\in H_{L_1}^2(R_1)$. We are going to choose now particular functions $\\zeta$ in \\eqref{eq: boundary identity}.  Consider first  $b=\\frac{1}{2}\\min_{x\\in [0,1]} g(x)>0$ so that the rectangle $(0,1)\\times (0,b)\\subset R_1$ and consider a function $\\eta=\\eta(y)$ with $\\eta\\in C^\\infty(0,b)$ such that $\\eta(y)=1+\\alpha y$ for $0<y<b\/4$,  where $\\alpha\\in \\numberset{R}$ is a parameter, and $\\eta(y)\\equiv 0$ for $y\\in(\\frac{3}{4}b,b)$. If we define $\\zeta(x,y)=\\theta(x)\\eta(y)$ for $(x,y)\\in (0,1)\\times (0,b)$  where $\\theta\\in C_c^\\infty(0,1)$ and we extend this function $\\zeta$  by 0 to all of $R_1$, then we can use $\\zeta$ in \\eqref{eq: boundary identity} in order to obtain\n\\[\n\\alpha \\int_0^1 \\psi_1(x)\\theta(x) dx - \\int_0^1\\psi_2(x)\\theta(x)dx=0\n\\]\nfor all $\\alpha\\in \\numberset{R}$ and all $\\theta\\in C_0^\\infty(0,1)$. But this easily implies that $\\psi_1\\equiv \\psi_2\\equiv 0$. Thus, we obtain\n\\[\nu(x,y) = u(x) = - \\sigma \\frac{\\partial^2 v(x)}{\\partial x^2}\n\\]\nfor almost all $(x,y) \\in R_1$, i.e., $\\frac{1}{\\epsilon^2} \\frac{\\partial^2 \\tilde{v}_\\epsilon}{ \\partial y^2} \\rightharpoonup - \\sigma  \\frac{\\partial^2 v(x)}{\\partial x^2}$ in $L^2(R_1)$. Hence $v$ solves the following limit problem\n\\begin{equation}\\label{ODE: auxiliary problem sigma2}\n\\begin{cases}\n\\frac{1 - \\sigma^2}{g} (gv'')'' - \\frac{\\tau}{g}(gv')' + v = \\mathcal{M}(f) , &\\text{in $(0,1)$}\\\\\nv(0)=v(1)=0,&\\\\\nv'(0)=v'(1)=0,&\n\\end{cases}\n\\end{equation}\nand then by regularity theory we deduce that $v \\in H^4(0,1)$.\n\n\n\n\n\n\\subsection{Spectral convergence }\n\\label{sec: spectral convergence}\nWe aim at proving the spectral convergence of the eigenvalues and eigenfunctions of problem \\eqref{PDE: R_eps} to the corresponding eigenvalues and eigenfunctions of the one dimensional problem \\eqref{ODE: limit problem}. To do so we shall prove the compact convergence of the  associated  resolvent operators combined with the computations carried out in the previous section.  Note that  the domain $R_\\epsilon$ varies with $\\epsilon$, hence the corresponding Hilbert spaces vary as well.  To bypass this problem we will use the notion of $\\mathcal{E}$-convergence of the resolvent operators in $L^2$.  We  recall the basic definitions and results.\\\\\nLet $\\mathcal{H}_\\epsilon$, $\\epsilon >0$, be a family of Hilbert spaces. We assume the existence of a family of linear operators $\\mathcal{E}_\\epsilon \\in \\mathcal{L}(\\mathcal{H}_0, \\mathcal{H}_\\epsilon)$, $\\epsilon >0$,  such that\n\\begin{equation}\n\\label{def: basic property E_eps}\n\\norma{\\mathcal{E}_\\epsilon u_0}_{\\mathcal{H}_\\epsilon} \\to \\norma{u_0}_{\\mathcal{H}_0},\\ \\ {\\rm as}\\ \\epsilon\\to 0,\n\\end{equation}\nfor all $u_0 \\in \\mathcal{H}_0$.\n\n\n\n\\begin{definition} Let $\\mathcal{H}_\\epsilon$ and $\\mathcal{E}_\\epsilon$ be as above.\n\\begin{enumerate}[label =(\\roman*)]\n\\item Let $u_\\epsilon\\in   \\mathcal{H}_\\epsilon$, $\\epsilon >0$.  We say that $u_\\epsilon$ $\\mathcal{E}$-converges to $u$ as $\\epsilon \\to 0$ if $\\norma{u_\\epsilon - \\mathcal{E}_\\epsilon u}_{\\mathcal{H}_\\epsilon} \\to 0$ as $\\epsilon \\to 0$. We write $u_\\epsilon \\overset{E}{\\longrightarrow} u$.\n\\item Let $ B_\\epsilon \\in \\mathcal{L}(\\mathcal{H}_\\epsilon)$,  $\\epsilon >0$. We say that $B_\\epsilon$ $\\mathcal{E}\\E$-converges to a linear operator $B_0 \\in \\mathcal{L}(\\mathcal{H}_0)$ if $B_\\epsilon u_\\epsilon \\overset{E}{\\longrightarrow} B_0 u$ whenever $u_\\epsilon \\overset{E}{\\longrightarrow} u\\in \\mathcal{H}_0$. We write $B_\\epsilon \\overset{EE}{\\longrightarrow} B_0$.\n\\item Let $ B_\\epsilon \\in \\mathcal{L}(\\mathcal{H}_\\epsilon)$, $  \\epsilon >0$. We say that $B_\\epsilon$ compactly converges to $B_0 \\in \\mathcal{L}(\\mathcal{H}_0)$ (and we write $B_\\epsilon \\overset{C}{\\longrightarrow} B_0$) if the following two conditions are satisfied:\n    \\begin{enumerate}[label=(\\alph*)]\n    \\item $B_\\epsilon \\overset{EE}{\\longrightarrow} B_0$ as $\\epsilon \\to 0$;\n    \\item for any family $u_\\epsilon \\in \\mathcal{H}_{\\epsilon}$, $\\epsilon>0$, such that $\\norma{u_\\epsilon}_{\\mathcal{H}_\\epsilon}=1$ for all $\\epsilon \\in (0,1)$, there exists a subsequence $B_{\\epsilon_k}u_{\\epsilon_k}$ of $B_\\epsilon u_\\epsilon$ and $\\bar{u} \\in \\mathcal{H}_0$ such that $B_{\\epsilon_k}u_{\\epsilon_k} \\overset{E}{\\longrightarrow} \\bar{u}$ as $k \\to \\infty$.\n    \\end{enumerate}\n\\end{enumerate}\n\\end{definition}\n\nFor any $\\epsilon \\geq 0$, let $A_\\epsilon$ be a (densely defined) closed, nonnegative differential operator on $\\mathcal{H}_\\epsilon$ with domain $\\mathscr{D}(A_\\epsilon) \\subset \\mathcal{H}_\\epsilon$.  We assume for simplicity that $0$ does not belong to the spectrum of $A_{\\epsilon}$ and that\n\\[\n\\textup{(H1): $A_\\epsilon$ has compact resolvent $B_\\epsilon := A_\\epsilon^{-1}$ for any $\\epsilon \\in [0,1)$,}\n\\]\nand\n\\[\n\\textup{(H2): $B_\\epsilon \\overset{C}{\\longrightarrow} B_0 $, as $\\epsilon \\to 0$.}\n\\]\nGiven an eigenvalue $\\lambda$ of $A_0$ we consider the generalized eigenspace $S(\\lambda, A_0) := Q(\\lambda, A_0)\\mathcal{H}_0$, where\n\\[\nQ(\\lambda, A_0) = \\frac{1}{2\\pi i} \\int_{|\\xi - \\lambda| = \\delta} (\\xi \\mathbb{I} - A_0)^{-1}\\, d\\xi\n\\]\nand $\\delta > 0$ is such that the disk $\\{\\xi \\in \\mathbb{C} : |\\xi - \\lambda| \\leq \\delta \\}$ does not contain any eigenvalue except for $\\lambda$. In a similar way, if (H1),(H2) hold, then we can define   $S(\\lambda, A_{\\epsilon}) := Q(\\lambda, A_{\\epsilon})\\mathcal{H}_{\\epsilon}$, where\n\\[\nQ(\\lambda, A_\\epsilon) = \\frac{1}{2\\pi i} \\int_{|\\xi - \\lambda| = \\delta} (\\xi \\mathbb{I} - A_\\epsilon)^{-1}\\, d\\xi.\n\\]\nThis definition makes sense because for $\\epsilon$ small enough $(\\xi \\mathbb{I} - A_\\epsilon)$ is invertible for all $\\xi$ such that $|\\xi - \\lambda| = \\delta$, see \\cite[Lemma 4.9]{ACJdE}. Then the following theorem holds.\n\n\\begin{theorem}\n\\label{thm: E conv -> spectral conv}\nLet $A_\\epsilon$, $A_0$ be operators as above satisfying  conditions (H1), (H2). Then the operators $A_{\\epsilon}$ {\\rm are spectrally convergent} to $A_0$ as $\\epsilon\\to 0$, i.e., the following statements hold:\n\\begin{enumerate}[label=(\\roman*)]\n\\item If $\\lambda_0$ is an eigenvalue of $A_0$, then there exists a sequence of eigenvalues $\\lambda_\\epsilon$ of $A_\\epsilon$ such that $\\lambda_\\epsilon \\to \\lambda_0$ as $\\epsilon \\to 0$. Conversely, if $\\lambda_\\epsilon$ is an eigenvalue of $A_\\epsilon$ for all $\\epsilon >0$, and $\\lambda_\\epsilon \\to \\lambda_0$, then $\\lambda_0$ is an eigenvalue of $A_0$.\n\\item There exists $\\epsilon_0 > 0$ such that the dimension of the generalized eigenspace $S(\\lambda_0, A_\\epsilon)$ equals the dimension of $S(\\lambda_0, A_0)$, for any eigenvalue $\\lambda_0$ of $A_0$, for any $\\epsilon \\in [0,\\epsilon_0)$.\n\\item If $\\varphi_0 \\in S(\\lambda_0, A_0)$ then for any $\\epsilon >0$ there exists  $\\varphi_\\epsilon \\in S(\\lambda_0, A_\\epsilon)$ such that  $\\varphi_\\epsilon \\overset{E}{\\longrightarrow} \\varphi_0$ as $\\epsilon \\to 0$.\n\\item If $\\varphi_\\epsilon \\in S(\\lambda_0, A_\\epsilon)$ satisfies $\\norma{\\varphi_\\epsilon}_{\\mathcal{H}_\\epsilon} = 1$ for all $\\epsilon >0$, then  $\\varphi_\\epsilon $, $\\epsilon >0$,  has an $\\mathcal{E}$-convergent subsequence whose limit is in $S(\\lambda_0, A_0)$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nSee \\cite[Theorem 4.10]{ACL}.\n\\end{proof}\n\nWe now apply Theorem \\ref{thm: E conv -> spectral conv} to problem \\eqref{PDE: R_eps}.  To do so, we consider the following Hilbert spaces\n\\[\n\\mathcal{H}_\\epsilon = L^2(R_\\epsilon; \\epsilon^{-1}dxdy),\\ \\ {\\rm and}\\ \\  \\mathcal{H}_0 = L^2_g(0,1),\n\\]\nand we denote by   $\\mathcal{E}_\\epsilon$   the extension operator from $L^2_g(0,1)$ to  $L^2(R_\\epsilon; \\epsilon^{-1}dxdy)$, defined by\n\\begin{equation}\n\\label{def: extension}\n(\\mathcal{E}_\\epsilon v)(x,y) = v(x),\n\\end{equation}\nfor all $v \\in L^2_g(0,1)$, for almost all $(x,y) \\in R_\\epsilon$. Clearly $\\norma{E_\\epsilon u_0}_{(R_\\epsilon; \\epsilon^{-1}dxdy)} = \\norma{u_0}_{L^2_g(0,1)}$, hence $\\mathcal{E}_\\epsilon$ trivially satisfies property \\eqref{def: basic property E_eps}.\n\nWe consider the operators $A_{\\epsilon}=(\\Delta^2 - \\tau \\Delta +I)_{ L_{\\epsilon }}$,  $A_0=(\\Delta^2 - \\tau \\Delta +I)_{D }$  on $\\mathcal{H}_{\\epsilon}$ and $\\mathcal{H}_0$ respectively, associated with the eigenvalue problems \\eqref{PDE: R_eps} and  \\eqref{ODE: limit problem}, respectively. Namely, $(\\Delta^2 - \\tau \\Delta +I)_{ L_{\\epsilon }}$ is the operator $\\Delta^2 - \\tau \\Delta +I$ on $R_{\\epsilon}$ subject to Dirichlet boundary\nconditions on $L_{\\epsilon}$ and Neumann  boundary conditions on $\\partial R_{\\epsilon}\\setminus L_{\\epsilon }$ as described in  \\eqref{PDE: R_eps}. Similarly, $(\\Delta^2 - \\tau \\Delta +I)_{D }$\nis the operator $\\Delta^2 - \\tau \\Delta +I$ on $(0,1)$ subject to Dirichlet boundary conditions as described in \\eqref{ODE: limit problem}.\n\nThen we can prove the following\n\\begin{theorem}\\label{spectfin} The operators  $(\\Delta^2 - \\tau \\Delta +I)_{ L_{\\epsilon }}$  spectrally converge to\\\\  $(\\Delta^2 - \\tau \\Delta +I)_{D }$ as $\\epsilon \\to 0$,  in the sense of Theorem~\\ref{thm: E conv -> spectral conv}.\n\\end{theorem}\n\n\\begin{proof}\nIn view of Theorem \\ref{thm: E conv -> spectral conv},  it is sufficient to prove the following two facts:\n\\begin{enumerate}[label=(\\arabic*)]\n\\item if  $f_\\epsilon \\in L^2(R_\\epsilon; \\epsilon^{-1}dxdy)$ is  such that $\\epsilon^{-1\/2}\\norma{f_\\epsilon}_{L^2(R_\\epsilon)} = 1$ for any $\\epsilon >0$, and  $v_\\epsilon$ is the corresponding solutions of Problem \\eqref{PDE: R_eps f_eps}, then there exists a subsequence $\\epsilon_k \\to 0$ as $k \\to \\infty$ and $\\bar{v} \\in L^2_g(0,1)$ such that $v_{\\epsilon_k}$ $\\mathcal{E}$-converge to $\\bar{v}$ as $k \\to \\infty$.\n\\item if  $f_\\epsilon \\in L^2(R_\\epsilon; \\epsilon^{-1}dxdy)$ and $f_\\epsilon  \\overset{E}{\\longrightarrow} f$ as $\\epsilon \\to 0$, then the corresponding solutions $v_\\epsilon$ of Problem \\eqref{PDE: R_eps f_eps} $\\mathcal{E}$-converge to the solution of Problem \\eqref{ODE: auxiliary problem sigma2} with datum $f$.\n\\end{enumerate}\nNote that (1) follows immediately from the computations in Section \\ref{subsection: finding limit prb}. Indeed, if $f_\\epsilon \\in L^2(R_\\epsilon; \\epsilon^{-1}dxdy)$ is as in (1), up to a subsequence, $\\tilde{f}_\\epsilon \\rightharpoonup f$ in $L^2(R_1)$, which implies that $\\tilde{v}_\\epsilon \\rightharpoonup v_0\\in H_0^2(0,1)$ in $H^2(R_1)$, where $v_0$ is the solution of Problem \\eqref{ODE: auxiliary problem sigma2}. This implies that $\\norma{v_\\epsilon - \\mathcal{E} v_0}_{L^2(R_\\epsilon; \\epsilon^{-1}dxdy)} \\to 0$, hence (1) is proved.\\\\\nIn order to show (2) we take a sequence of functions $f_\\epsilon \\in L^2(R_\\epsilon; \\epsilon^{-1}dxdy)$ and $f\\in L^2_g(0,1)$ such that $\\epsilon^{-1\/2}\\norma{f_\\epsilon - \\mathcal{E}_\\epsilon f}_{L^2(R_\\epsilon)} \\to 0$ as $\\epsilon \\to 0$. After a change of variable, this is equivalent to $\\norma{\\tilde{f}_\\epsilon - \\mathcal{E} f}_{L^2(R_1)} \\to 0$ as $\\epsilon \\to 0$. Arguing as in Section \\ref{subsection: finding limit prb}, one show that the $\\tilde{v}_\\epsilon \\rightharpoonup v \\in L^2_g(0,1)$ in $H^2(R_1)$ and that $v$ solves problem \\eqref{ODE: auxiliary problem sigma2}. Hence $\\norma{\\tilde{v}_\\epsilon - \\mathcal{E} v}_{L^2(R_1)} \\to 0$ as $\\epsilon \\to 0$, or equivalently, $\\norma{v_\\epsilon - \\mathcal{E}_\\epsilon v}_{L^2(R_\\epsilon; \\epsilon^{-1}dxdy)} \\to 0$ as $\\epsilon \\to 0$, proving (2).\n\n\n\\end{proof}\n\n\n\n\n\n\n\\section{Conclusion}\\label{conclusionsec}\n\nRecall that the eigenpairs of problems \\eqref{PDE: main problem_eigenvalues}, \\eqref{PDE: Omega} are denoted by $(\\lambda_n(\\Omega_\\epsilon), \\varphi_n^\\epsilon)$, $(\\omega_n, \\varphi_n^\\Omega)_{n\\geq1}$ respectively, where the two families of eigenfunctions $\\varphi_n^\\epsilon$, $\\varphi_n^\\Omega$  are complete orthonormal bases of the spaces $L^2(\\Omega_{\\epsilon})$, $L^2(\\Omega )$, respectively.    Denote now by $(h_n, \\theta_n)_{n\\geq 1}$ the eigenpairs of problem $\\eqref{ODE: limit problem}$\nwhere  the eigenfunctions $h_n$ define an orthonormal basis of the space $L^2_g(0,1)$.\nIn the spirit of the definition of $\\lambda_n^{\\epsilon} $ given in Section 2, we  set now  $(\\lambda_n^0)_{n\\geq 1} = (\\omega_k)_{k \\geq 1} \\cup (\\theta_l )_{l\\geq 1}$, where it is understood that the eigenvalues are arranged in increasing order and repeated according to their multiplicity.  For each $\\lambda_n^0$ we define the function  $\\phi_n^0 \\in H^2(\\Omega) \\oplus H^2(R_\\epsilon)$ in the following way:\n\\begin{equation*}\n\\phi^0_n =\n\\begin{cases}\n    \\varphi_k^\\Omega, &\\text{in $\\Omega$}\\\\\n    0, &\\text{in $R_\\epsilon$},\n\\end{cases}\n\\end{equation*}\nif $\\lambda_n^0 = \\omega_k$, for some $k \\in \\numberset{N}$; otherwise\n\\begin{equation*}\n\\phi^0_n=\n\\begin{cases}\n    0, &\\text{in $\\Omega$}, \\\\\n    \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_l, &\\text{in $R_\\epsilon$}\n\\end{cases}\n\\end{equation*}\nif $\\lambda_n^\\epsilon = \\theta_l$, for some $l \\in \\numberset{N}$ (here we agree to order the eigenvalues and the  eigenfunctions following the same rule used in the definition of $\\lambda_n^{\\epsilon}$ and $\\phi_n^{\\epsilon }$ in Section 2).\n\nFinally,  if $x>0$ divides the spectrum $\\lambda_n(\\Omega_{\\epsilon})$ for all $\\epsilon >0 $ sufficiently small  (see the end of Section 2)   and $N(x)$ is the number of eigenvalues with $\\lambda_n(\\Omega_{\\epsilon})\\le x$ (counting their multiplicity),  we  define the projector $P_{x}^0$ from $L^2(\\Omega_\\epsilon)$ onto the linear span $[\\phi_1^{0}, \\dots, \\phi_{N(x)}^{0}]$  by setting\n\\[\nP_{x}^0 u = \\sum_{i=1}^{N(x)} (u,\\phi_i^0)_{L^2(\\Omega_\\epsilon)} \\phi_i^0\n\\]\nfor all $u\\in L^2(\\Omega_\\epsilon)$. (Note that choosing $x$ independent of $\\epsilon$ is possible by the limiting behaviour of the eigenvalues.) Then, using Theorems \\ref{thm: eigenvalues decomposition} and  \\ref{spectfin}  we deduce the following.\n\n\\begin{theorem} \\label{lastthm}\nLet $\\Omega_\\epsilon$, $\\epsilon>0$, be a family of  dumbbell domains satisfying the H-Condition. Then the following statements hold:\n\\begin{enumerate}[label =(\\roman*)]\n\\item   $\\lim_{\\epsilon \\to 0}\\, \\abs{\\lambda_n(\\Omega_\\epsilon) - \\lambda_n^0} = 0$,  for all  $n\\in \\numberset{N} $.\n\\item   For any $x$ dividing the spectrum,\n $\\lim_{\\epsilon \\to 0}\\, \\norma{\\varphi^\\epsilon_{n} - P^0_{x} \\varphi^\\epsilon_{n}}_{H^2(\\Omega) \\oplus L^2(R_\\epsilon)} = 0$, for all $n = 1,\\dots, N(x)$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nThe convergence of the eigenvalues follows directly by  Theorems \\ref{thm: eigenvalues decomposition} and  \\ref{spectfin}.  Indeed, by Theorem \\ref{thm: eigenvalues decomposition} we know that $|\\lambda_n(\\Omega_\\epsilon) - \\lambda_n^\\epsilon| \\to 0$ as $\\epsilon \\to 0$. If $\\lambda_n^\\epsilon = \\omega_k$ for some $k\\in \\numberset{N}$, for all sufficiently small $\\epsilon$, then we are done; otherwise, if $\\lambda_n^\\epsilon = \\theta_l^\\epsilon$ for some $l\\in \\numberset{N}$, definitely in $\\epsilon$, by Theorem \\ref{spectfin} we deduce that $\\theta_l^\\epsilon \\to \\theta_l$ as $\\epsilon \\to 0$, hence $|\\lambda_n(\\Omega_\\epsilon) - \\theta_l| \\leq |\\lambda_n(\\Omega_\\epsilon) - \\theta_l^\\epsilon| + |\\theta_l^\\epsilon - \\theta_l| \\to 0 $ as $\\epsilon \\to 0$.\\\\\nConsider now the convergence of the eigenfunctions. By Theorems~\\ref{thm: E conv -> spectral conv},~\\ref{spectfin} it follows that for any $\\epsilon >0$ there exists an orthonormal  sequence of generalized eigenfunctions $\\delta_j^{\\epsilon }$ in $L^2(R_{\\epsilon}, \\epsilon^{-1}dxdy)$ associated with the eigenvalues\n$\\theta_j^{\\epsilon}$ of problem \\eqref{PDE: R_eps} such that for every $j\\in \\numberset{N}$\n\\begin{equation}\\label{lastthm1}\n\\norma{\\delta^\\epsilon_j - \\mathcal{E}_{\\epsilon} h_j}_{L^2(R_\\epsilon,  \\epsilon^{-1}dxdy )} \\to 0,\n\\end{equation}\nas $\\epsilon \\to 0$.  Recall that  a generalized eigenfunction is an element of a generalized eigenspace, see Section~\\ref{sec: spectral convergence}.  We set\n$\n\\gamma_j^\\epsilon =\\epsilon^{-1\/2}\\delta^\\epsilon_j\n$\nand we note that $\\gamma_j^{\\epsilon}$ is a sequence of generalized eigenfunctions of Problem \\eqref{PDE: R_eps} which is orthonormal in $L^2(R_{\\epsilon})$, as required in Theorem~\\ref{thm: eigenvalues decomposition}. Thus by Theorem~\\ref{thm: eigenvalues decomposition} $(ii)$,  we deduce that\n\\small{\n\\begin{equation*}\n\\begin{split}\n&\\norma*{\\varphi_n^{\\epsilon} - \\sum^{N(x) }_{i=1} (\\varphi_n^\\epsilon, \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i)_{L^2(R_\\epsilon)} \\epsilon^{-1\/2} \\mathcal{E}_\\epsilon h_i}_{L^2(R_\\epsilon)} \\leq \\norma*{\\varphi_n^{\\epsilon} - \\sum^{N(x)}_{i=1} (\\varphi_n^\\epsilon, \\gamma_i^\\epsilon)_{L^2(R_\\epsilon)} \\gamma_i^\\epsilon }_{L^2(R_\\epsilon)}\\\\\n&+ \\norma*{\\sum^{N(x)}_{i=1} (\\varphi_n^\\epsilon, \\gamma_i^\\epsilon)_{L^2(R_\\epsilon)} \\gamma_i^\\epsilon - \\sum^{N(x)}_{i=1} (\\varphi_n^\\epsilon, \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i)_{L^2(R_\\epsilon)} \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i  }_{L^2(R_\\epsilon)}\\\\\n&\\leq o(1) + \\norma*{\\sum^{N(x)}_{i=1} (\\varphi_n^\\epsilon, \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i)_{L^2(R_\\epsilon)} ( \\gamma_i^\\epsilon -\\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i)  }_{L^2(R_\\epsilon)} + \\norma*{\\sum^{N(x)}_{i=1} (\\varphi_n^\\epsilon, \\gamma_i^\\epsilon - \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i)_{L^2(R_\\epsilon)} \\gamma_i^\\epsilon}_{L^2(R_\\epsilon)}\\\\\n\\end{split}\n\\end{equation*}}\n\\normalsize\nHence, \n\\begin{equation}\\label{lastthm2}\n\\begin{split}\n&\\norma*{\\varphi_n^{\\epsilon} - \\sum^{N(x) }_{i=1} (\\varphi_n^\\epsilon, \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i)_{L^2(R_\\epsilon)} \\epsilon^{-1\/2} \\mathcal{E}_\\epsilon h_i}_{L^2(R_\\epsilon)}\\\\ \n&\\leq o(1) + C \\sum^{N(x)}_{i=1} \\norma{\\gamma_i^\\epsilon - \\epsilon^{-1\/2}\\mathcal{E}_\\epsilon h_i}_{L^2(R_\\epsilon)} =o(1) + C \\sum^{N(x)}_{i=1} \\norma{\\delta _i^\\epsilon - \\mathcal{E}_\\epsilon h_i}_{L^2(R_\\epsilon ,\\epsilon^{-1}dxdy   )}.\n\\end{split}\n\\end{equation}\nSince the right-hand side of the last inequality in  \\eqref{lastthm2} goes to zero as $\\epsilon \\to 0$ by (\\ref{lastthm1}), we conclude that $\\lim_{\\epsilon \\to 0}\\, \\norma{\\varphi^\\epsilon_{n} - P^0_{x} \\varphi^\\epsilon_{n}}_{L^2(R_\\epsilon)} = 0$. Finally, the fact that $\\lim_{\\epsilon \\to 0}\\, \\norma{\\varphi^\\epsilon_{n} - P^0_{x} \\varphi^\\epsilon_{n}}_{H^2(\\Omega)} = 0$ follows directly from Theorem \\ref{thm: eigenvalues decomposition}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection*{Acknowledgment}\nThe first author is partially supported by grants MTM2012-31298, MTM2016-75465, ICMAT Severo Ochoa project SEV-2015-0554, MINECO, Spain and Grupo de Investigaci\\'on CADEDIF, UCM. The third author acknowledges financial support from the INDAM - GNAMPA project 2016 ``Equazioni differenziali con applicazioni alla meccanica\". The second and third authors are also members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\\`{a} e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).\\\\\nThe second and the third authors are very thankful to the Departamento de Matem\\'atica Aplicada of the Universidad Complutense de Madrid for the warm hospitality received on the occasion of their visits. The authors are thankful to an anonymous referee for pointing out a number of items in the reference list.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn semiconductors or insulators with broken inversion symmetry, an intrinsic electromechanical\ncoupling between stresses and electric polarizations can be observed, which is called piezoelectric effect.\nTwo-dimensional (2D) materials can show  unique  properties compared to their bulk counterparts, and the reduction in dimensionality of 2D\nmaterials can  often  eliminate inversion symmetry,  which allows these materials to become piezoelectric\\cite{q4}.\nIt has been theoretically reported that many  2D\nmaterials break inversion symmetry\nand hence can exhibit piezoelectricity, such as  group IIA and IIB metal oxides, group-V binary semiconductors, transition metal dichalchogenides (TMD), Janus TMD and group III-V semiconductors\\cite{q7,q7-2,q7-3,q7-3-1,q7-3-2,q9,q10,q11,q12,qr,nr,nr1}.  A majority of  structures have piezoelectric coefficients greater than  a typical value of  bulk piezoelectric materials (5 pm\/V). Significantly,  the  monolayer SnSe,\nSnS, GeSe and GeS  with puckered structure  possess   giant piezoelectricity,  as high as  75-251 pm\/V\\cite{q10}, which  may have huge potential application in the field of sensors, actuators and energy harvesters.\n The different crystal symmetry can induce  a only in-plane piezoelectricity like TMD monolayers\\cite{q9}, both in-plane and out-of-plane piezoelectricity for example 2D  Janus monolayers\\cite{q7,q7-3},  or a pure out-of-plane piezoelectricity such as penta-graphene\\cite{q7-4}. It has been proved that strain may be a effective strategy to tune piezoelectric properties of 2D materials\\cite{r1,r3}.\nExperimentally discovered piezoelectricity of $\\mathrm{MoS_2}$\\cite{q5,q6}, MoSSe\\cite{q8}  and $\\mathrm{In_2Se_3}$\\cite{q8-1}  has  triggered an intense interest in piezoelectric properties of  2D materials.\n\n\\begin{figure*}\n \n  \\includegraphics[width=16.0cm]{Fig1.eps}\n  \\caption{(Color online)The crystal structure of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MA_2Z_4}$  including  top  and  side views.   The purple balls represent M atoms, and the  blue balls for A atoms, and the yellow balls for Z atoms. These crystal structure  can be divided into three categories: A ($\\alpha_1$, $\\alpha_2$), B ($\\alpha_3$, $\\alpha_4$) and C ($\\alpha_5$, $\\alpha_6$) according to the relative positions of M and A atoms. The different categories can be connected by translation operation, and the different structures in the same category can be related by mirror or rotation operations. The green lines represent mirror face,  translation direction or rotation axis.}\\label{t0}\n\\end{figure*}\n\n\nIt is  meaningful to  explore piezoelectricity of new 2D family. Recently, the layered\n2D $\\mathrm{MoSi_2N_4}$ has been  experimentally achieved by chemical vapor deposition (CVD)\\cite{msn}, which possesses  semiconducting behavior, high strength and excellent ambient stability.  In rapid sequence, 2D\n$\\mathrm{WSi_2N_4}$ has also been synthesized by CVD. In the wake of $\\mathrm{MSi_2N_4}$ (M=Mo and W), $\\mathrm{MA_2Z_4}$ family are constructed with twelve different structures ($\\alpha_i$ and  $\\beta_i$ ($i$=1 to 6))  by intercalating $\\mathrm{MoS_2}$-type  $\\mathrm{MZ_2}$ monolayer into InSe-type  $\\mathrm{A_2Z_2}$ monolayer\\cite{m20}.  The  $\\mathrm{MA_2Z_4}$ family spans a wide range of properties  from semiconductor to topological insulator to Ising superconductor upon the number of valence electrons (VEC).   Intrinsic piezoelectricity in monolayer  $\\mathrm{XSi_2N_4}$ (X=Ti, Zr, Hf, Cr, Mo and W) with $\\alpha_1$ phase  are studied  by the first principle calculations\\cite{m21}, and the independent  in-plane piezoelectric constants $d_{11}$ is predicted to be 0.78 pm\/V-1.24 pm\/V. The valley-dependent properties of monolayer $\\mathrm{MoSi_2N_4}$, $\\mathrm{WSi_2N_4}$ and $\\mathrm{MoSi_2As_4}$ have been investigated by the first-principle calculations\\cite{g1}. The structural, mechanical, thermal, electronic, optical and photocatalytic properties of $\\mathrm{MoSi_2N_4}$ are studied by using hybrid density functional theory (HSE06-DFT)\\cite{g2}.\n\nIn this work, the role of crystal structure  on intrinsic piezoelectricity in monolayer  $\\mathrm{MSi_2N_4}$ (M=Mo and W) are studied by using density functional perturbation theory (DFPT)\\cite{pv6}. It is interesting to note that  the same structural dependence on $d_{11}$ and $e_{11}$  between monolayer $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ is observed. Calculated results show  that the atomic arrangement of $\\mathrm{A_2Z_2}$ double layers has important effect on the in-plane piezoelectric polarization of  $\\mathrm{MSi_2N_4}$ (M=Mo and W) monolayers.\nFinally, we investigate the intrinsic piezoelectricity of  monolayer $\\alpha_1$- and $\\alpha_2$-  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$.  It is found that the  $\\mathrm{MA_2P_4}$ have more stronger  piezoelectricity than  $\\mathrm{MA_2N_4}$. So,\n experimentally synthesizing monolayer $\\mathrm{MA_2Z_4}$ containing P atoms  is  very promising for energy harvesting and piezoelectric sensing.\n\n\n\n\n\nThe rest of the paper is organized as follows. In the next\nsection, we shall give our computational details and methods  about piezoelectric coefficients.\n In the third section, we perform symmetry analysis for elastic and piezoelectric coefficients of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MA_2Z_4}$. In the fourth  sections, we shall present main results and analysis. Finally, we shall give our  conclusions in the fifth section.\n\\begin{table}\n\\centering \\caption{The optimized lattice constants of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MSi_2N_4}$ (M=Mo and W)  using GGA ($\\mathrm{{\\AA}}$). }\\label{tab0}\n  \\begin{tabular*}{0.48\\textwidth}{@{\\extracolsep{\\fill}}ccccccc}\n  \\hline\\hline\nName & $\\alpha_1$ &  $\\alpha_2$ & $\\alpha_3$ & $\\alpha_4$ & $\\alpha_5$ & $\\alpha_6$ \\\\\\hline\\hline\n $\\mathrm{MoSi_2N_4}$   &2.91&  2.90 &  2.84 & 2.84 & 2.86 & 2.85\\\\\n $\\mathrm{WSi_2N_4}$   &2.91 & 2.90  & 2.84  & 2.84 & 2.87 & 2.85\\\\ \\hline\\hline\n\\end{tabular*}\n\\end{table}\n\n\n\\section{Computational detail}\nBased on  the density functional theory (DFT)\\cite{1}, our  simulations are carried out   as implemented\nin the plane-wave code VASP\\cite{pv1,pv2,pv3}. The exchange-correlation functional is treated within  popular generalized gradient\napproximation of Perdew, Burke and  Ernzerhof  (GGA-PBE)\\cite{pbe}  to perform the structural relaxation and the calculations of the elastic and\npiezoelectric tensors.  For energy band calculations, the spin orbital coupling (SOC)\nis also taken into account due to containing early transition metal.\nProjector-augmented wave pseudopotentials are used with a cutoff energy of 500 eV for plane-wave expansions.\nA vacuum spacing of more than 20 $\\mathrm{{\\AA}}$ is adopted to prevent any interactions\nbetween the adjacent periodic images of the 2D monolayers. The total energy  convergence criterion is set\nto $10^{-8}$ eV, and  the\natomic positions  are optimized until all components of\nthe forces on each atom are reduced to values  less than 0.0001 $\\mathrm{eV.{\\AA}^{-1}}$.\nWe calculate the coefficients of  elastic stiffness tensor  $C_{ij}$  by using strain-stress relationship (SSR)  and   the piezoelectric stress coefficients $e_{ij}$  by DFPT method\\cite{pv6}.\nA Monkhorst-Pack mesh of 15$\\times$15$\\times$1  in the first Brillouin zone\nis sampled   for $C_{ij}$, and  9$\\times$16$\\times$1 for $e_{ij}$.\nThe 2D elastic coefficients $C^{2D}_{ij}$\n and   piezoelectric stress coefficients $e^{2D}_{ij}$\nhave been renormalized by the the length of unit cell along z direction ($Lz$):  $C^{2D}_{ij}$=$Lz$$C^{3D}_{ij}$ and $e^{2D}_{ij}$=$Lz$$e^{3D}_{ij}$.\nHowever, the $d_{ij}$ is independent of $Lz$.\n\n\\begin{figure*}\n \n  \\includegraphics[width=12cm]{Fig2.eps}\n  \\caption{(Color online) The energy band structures of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ using GGA+SOC, and the VBM and CBM are connected by red arrow.  }\\label{band}\n\\end{figure*}\n\n\\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig3.eps}\n  \\caption{(Color online)The energy band gaps of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$  using GGA+SOC.}\\label{band1}\n\\end{figure}\n\n\n\n\\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig4.eps}\n  \\caption{(Color online) The elastic constants  $C_{ij}$ of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$.}\\label{c}\n\\end{figure}\n\\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig5.eps}\n  \\caption{(Color online) The  piezoelectric stress coefficients  $e_{11}$,  the ionic contribution and electronic contribution to $e_{11}$ of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$.  }\\label{e}\n\\end{figure}\n\n\\begin{table}\n\\centering \\caption{The $d_{11}$ of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MSi_2N_4}$ (M=Mo and W)  using GGA (pm\/V). }\\label{tab0-1}\n  \\begin{tabular*}{0.48\\textwidth}{@{\\extracolsep{\\fill}}ccccccc}\n  \\hline\\hline\nName & $\\alpha_1$ &  $\\alpha_2$ & $\\alpha_3$ & $\\alpha_4$ & $\\alpha_5$ & $\\alpha_6$ \\\\\\hline\\hline\n $\\mathrm{MoSi_2N_4}$   &1.15 &  0.65&  0.34 & -1.98 & 3.53 & 1.32\\\\\n $\\mathrm{WSi_2N_4}$   &0.78 &0.25 & -0.07  & -2.05 & 2.91 & 0.88\\\\ \\hline\\hline\n\\end{tabular*}\n\\end{table}\n\n\\section{Symmetry Analysis}\n The piezoelectric  stress tensors  $e_{ijk}$ and strain tensor $d_{ijk}$  is defined as:\n \\begin{equation}\\label{pe0}\n      e_{ijk}=\\frac{\\partial P_i}{\\partial \\varepsilon_{jk}}=e_{ijk}^{elc}+e_{ijk}^{ion}\n \\end{equation}\nand\n \\begin{equation}\\label{pe0-1}\n   d_{ijk}=\\frac{\\partial P_i}{\\partial \\sigma_{jk}}=d_{ijk}^{elc}+d_{ijk}^{ion}\n \\end{equation}\nwhere  $P_i$, $\\varepsilon_{jk}$ and $\\sigma_{jk}$ are polarization vector, strain and stress, respectively.\nThe $e_{ijk}^{elc}$  ($d_{ijk}^{elc}$) is the clamped-ion\npiezoelectric tensors resulting from the pure electronic contribution.  The relaxed-ion\npiezoelectric tensors $e_{ijk}$ ($d_{ijk}$) is obtained from the sum of ionic\nand electronic contributions. The $d_{ijk}$ can be connected with  $e_{ijk}$ by the elastic stiffness tensor  $C_{ijkl}$.\nBy employing the frequently used Voigt notation (11$\\rightarrow$1,\n22$\\rightarrow$2, 33$\\rightarrow$3, 23$\\rightarrow$4, 31$\\rightarrow$5 and 12$\\rightarrow$6),\nthe elastic tensor  $C_{ijkl}$, piezoelectric tensors $e_{ijk}$ and $d_{ijk}$ become into $C_{ij}$ (6$\\times$6  matrix), $e_{ij}$  (3$\\times$6  matrix) and $d_{ij}$ (3$\\times$6  matrix). The symmetry of crystal\nstructure will further reduce the number of independent  $C_{ij}$, $e_{ij}$ and $d_{ij}$ tensors.\n\nBy intercalating  $\\mathrm{MoS_2}$-type  $\\mathrm{MZ_2}$ monolayer into InSe-type  $\\mathrm{A_2Z_2}$ monolayer,\nsix $\\alpha_i$ and six $\\beta_i$ ($i$=1 to 6)  $\\mathrm{MA_2Z_4}$ monolayers can be constructed\\cite{m20}.\nThe six $\\alpha_i$ have the same $P\\bar{6}m2$ space group due to inserting  2H-$\\mathrm{MoS_2}$-type  $\\mathrm{MZ_2}$ monolayer   into $\\alpha$-InSe-type  $\\mathrm{A_2Z_2}$ double layers, which break inversion symmetry. The six $\\beta_i$ are built by intercalating 1T-$\\mathrm{MoS_2}$-type  $\\mathrm{MZ_2}$ monolayer into $\\beta$-InSe-type  $\\mathrm{A_2Z_2}$ double layers with the same $P\\bar{3}m1$ space group, which keep inversion symmetry.\nTherefore, $\\alpha_i$-  ($i$=1 to 6)  $\\mathrm{MA_2Z_4}$ monolayers are piezoelectric.\n\n The six  $\\alpha_i$  geometric structures of the   $\\mathrm{MA_2Z_4}$ monolayer are\nplotted in \\autoref{t0}. All considered six $\\alpha_i$ crystal structures  have  the same $\\bar{6}m2$ point group. Only  the in-plane  piezoelectric effect is  allowed in\nmonolayer $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MA_2Z_4}$, when a uniaxial in-plane  strain is applied. For 2D semiconductors, in general,\nin-plane  stresses and strains are only allowed,\nwhile the out-of-plane is strain\/stress free\\cite{q9,q10,q11}. And then the  $e_{ij}$,  $d_{ij}$ and $C_{ij}$  can be written as:\n \\begin{equation}\\label{pe1}\n  \\left(\n    \\begin{array}{ccc}\n      e_{11} &-e_{11} & 0 \\\\\n    0 &0 & -e_{11}\\\\\n      0 & 0 & 0 \\\\\n    \\end{array}\n  \\right)\n  \\end{equation}\n  \\begin{equation}\\label{pe1}\n  \\left(\n    \\begin{array}{ccc}\n        d_{11} & -d_{11} & 0 \\\\\n    0 &0 & -2d_{11} \\\\\n      0 & 0 & 0 \\\\\n    \\end{array}\n  \\right)\n  \\end{equation}\n  \\begin{equation}\\label{pe1}\n    \\left(\n    \\begin{array}{ccc}\n      C_{11} & C_{12} &0 \\\\\n     C_{12} & C_{11} &0 \\\\\n     0 & 0 & \\frac{C_{11}-C_{12}}{2} \\\\\n    \\end{array}\n  \\right)\n   \\end{equation}\n The forms of these piezoelectric and stiffness constants  are the same as those for TMD monolayers\\cite{q9,q11} due to the same  point group.\nBy  $e_{ik}$=$d_{ij}C_{jk}$, the only in-plane $d_{11}$ is found to be:\n\\begin{equation}\\label{pe2-7}\n    d_{11}=\\frac{e_{11}}{C_{11}-C_{12}}\n\\end{equation}\n\n\n\n\\begin{table*}\n\\centering \\caption{The optimized lattice constants of $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$  using GGA ($\\mathrm{{\\AA}}$). }\\label{tab1}\n  \\begin{tabular*}{0.96\\textwidth}{@{\\extracolsep{\\fill}}cccccccccccc}\n  \\hline\\hline\nName & $\\mathrm{CrSi_2N_4}$ &  $\\mathrm{MoSi_2N_4}$ & $\\mathrm{WSi_2N_4}$ & $\\mathrm{MoGe_2N_4}$ & $\\mathrm{WGe_2N_4}$ & $\\mathrm{CrSi_2P_4}$&$\\mathrm{MoSi_2P_4}$&$\\mathrm{WSi_2P_4}$&$\\mathrm{CrGe_2P_4}$&$\\mathrm{MoGe_2P_4}$&$\\mathrm{WGe_2P_4}$ \\\\\\hline\\hline\n $\\alpha_1$ &2.84&  2.91 & 2.91 &  3.02 & 3.02&  3.42 & 3.47 & 3.47 &3.50 & 3.54 &3.54\\\\\n  $\\alpha_2$  &2.84 & 2.90 & 2.90  & 3.01  &3.01 & 3.41 & 3.45 & 3.46 &3.49 & 3.53 &  3.53\\\\ \\hline\\hline\n\\end{tabular*}\n\\end{table*}\n\n\n\\section{Main calculated results}\n Firstly,  we discuss the  structural relation among six $\\alpha_i$  crystal structures of  $\\mathrm{MA_2Z_4}$.\n According to the relative positions of M and A atoms, the six  $\\alpha_i$  crystal structure  can be divided into three categories: A ($\\alpha_1$, $\\alpha_2$), B ($\\alpha_3$, $\\alpha_4$) and C ($\\alpha_5$, $\\alpha_6$).\nThe different categories can be connected by translation operation.  The $\\alpha_3$  can be attained by  translating $\\mathrm{A_2Z_2}$ double layers of   $\\alpha_2$ along the green line of top view of $\\alpha_2$ in \\autoref{t0} with  the transfixion of  $\\mathrm{MZ_2}$ monolayer.\n The $\\alpha_6$  can be attained from  $\\alpha_4$ by similar translation operation. The different structures in the same category can be related by mirror or rotation operations.  The  $\\alpha_2$ can be built by mirroring $\\mathrm{A_2Z_2}$ double layers of $\\alpha_1$ with respect to the vertical  surface defined by two green lines of top and side views of $\\alpha_1$. The $\\alpha_4$ ($\\alpha_6$) can be constructed by rotating the $\\mathrm{A_2Z_2}$ double layers of $\\alpha_3$ ($\\alpha_5$) with $\\pi$\/3 along the vertical axis defined by linking two A atoms.\n\n  \\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig6.eps}\n  \\caption{(Color online) The  piezoelectric strain coefficients  $d_{11}$ of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$.  }\\label{d}\n\\end{figure}\n\n\n\n\n\n\n\\begin{table*}\n\\centering \\caption{The $d_{11}$ of $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$  using GGA (pm\/V). }\\label{tab1-1}\n  \\begin{tabular*}{0.96\\textwidth}{@{\\extracolsep{\\fill}}cccccccccccc}\n  \\hline\\hline\nName & $\\mathrm{CrSi_2N_4}$ &  $\\mathrm{MoSi_2N_4}$ & $\\mathrm{WSi_2N_4}$ & $\\mathrm{MoGe_2N_4}$ & $\\mathrm{WGe_2N_4}$ & $\\mathrm{CrSi_2P_4}$&$\\mathrm{MoSi_2P_4}$&$\\mathrm{WSi_2P_4}$&$\\mathrm{CrGe_2P_4}$&$\\mathrm{MoGe_2P_4}$&$\\mathrm{WGe_2P_4}$ \\\\\\hline\\hline\n $\\alpha_1$ &1.24&\t1.15&\t0.78&\t1.85&\t1.31&\t6.03&\t4.91&\t4.16&\t6.12&\t5.27&\t4.36\\\\\n  $\\alpha_2$  &1.42&\t0.65&\t0.25&\t0.75&\t0.26&\t3.96&\t2.64&\t1.65&\t5.06&\t3.87&\t2.77\\\\ \\hline\\hline\n\\end{tabular*}\n\\end{table*}\n\n\n It has been proved that monolayer  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$  with 34 VEC are non-magnetic with $\\alpha_1$ or $\\alpha_2$ crystal structure, and are both dynamically and thermodynamically\\cite{m20}.  A piezoelectric material  should  be a semiconductor for\nprohibiting current leakage. Only $\\mathrm{MSi_2N_4}$ (M=Mo and W) monolayers are  semiconductors for all six $\\alpha_i$ crystal structures.\nSo, we mainly study the structure  effect on intrinsic piezoelectricity of $\\mathrm{MSi_2N_4}$ (M=Mo and W). The\nstructural parameters  of $\\alpha_i$- ($i$=1 to 6)  $\\mathrm{MSi_2N_4}$ (M=Mo and W) are  optimized, and the lattice constants are listed in \\autoref{tab0}.\nIt is found that the lattice constants between $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$  with the same phase are almost the same.\nThe size of these  lattice constants can also be  classified into A, B and C, which declare that the relative positions of M and A atoms determine lattice constants.\n\nNext, we use optimized crystal structures  to investigate their electronic structures.\n Although the SOC has little effects on the energy band gaps of  $\\mathrm{MSi_2N_4}$ (M=Mo and W) monolayers,  the SOC  can produce  observed  spin-orbit splitting  in the valence bands   at K point\\cite{m21}. Because the energy band outlines between $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ are very similar, only the energy bands of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ are shown in \\autoref{band} using GGA+SOC. It is clearly seen that they all are indirect gap semiconductors, and the gap range is 0.18 eV to 1.99 eV.  The position  of  conduction band minimum (CBM) for all six $\\alpha_i$  is  at K point, except for $\\alpha_6$ at M point.  The valence band maximum (VBM) of  $\\alpha_1$, $\\alpha_2$  and $\\alpha_5$ is  at $\\Gamma$ point, while the one of  $\\alpha_3$, $\\alpha_4$  and $\\alpha_6$ is slightly off $\\Gamma$ point, and  at one point along the $\\Gamma$-K line.\nThe energy band gaps of $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$  using GGA+SOC are plotted \\autoref{band1}. It is clearly seen that the structural dependence of band gap of  $\\mathrm{WSi_2N_4}$ is the same with one of  $\\mathrm{MoSi_2N_4}$, and the gap ranges from 0.08 eV to 2.37 eV. Therefore, it is very effective to tune the electronic structures  of  $\\mathrm{MSi_2N_4}$ (M=Mo and W) monolayers by  translating or rotating  $\\mathrm{Si_2N_2}$ bilayer.\n\\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig7.eps}\n  \\caption{(Color online)The energy band gaps of  $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$  using GGA+SOC. The direct band gap is marked by \"D\", and the unmarked one is indirect band gap.}\\label{band2}\n\\end{figure}\n\n\n\n\n\n\nTo calculate the $d_{11}$,  two independent elastic stiffness coefficients  ($C_{11}$ and $C_{12}$) of  $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ are attained by SSR, which are plotted in \\autoref{c}, together with $C_{11}$-$C_{12}$.  For six structures,  all calculated elastic coefficients of $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ satisfy the Born stability criteria\\cite{ela}, which means that they all are mechanically stable. Similar structural dependence of $C_{11}$, $C_{12}$ and $C_{11}$-$C_{12}$ can be observed between $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$.\nIt is found that the  $C_{12}$  of two structures in the same category are very close, if the two structures are connected by mirror operation  ($\\alpha_1$ and  $\\alpha_2$).  However, the  $C_{12}$  has obvious difference for two structures related by rotation operation  ($\\alpha_3$  and  $\\alpha_4$  or $\\alpha_5$  and  $\\alpha_6$). The  $\\alpha_4$ ($\\alpha_5$)  has the larger\n $C_{12}$ than $\\alpha_3$ ($\\alpha_6$). Between  $\\alpha_4$ ($\\alpha_3$) and $\\alpha_5$ ($\\alpha_6$), the difference is only the position of Si atom.\nIt is found that the $C_{11}$-$C_{12}$  of $\\alpha_1$, $\\alpha_2$,   $\\alpha_4$ and  $\\alpha_5$ are close, and  $\\alpha_3$ and  $\\alpha_6$ have the larger $C_{11}$-$C_{12}$, which is against the $d_{11}$ according to \\autoref{pe2-7}. These elastic stiffness coefficients are very larger than ones of\n TMD monolayers\\cite{q9,q11},  which  indicates that\n$\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ are not easy to be deformed.\n\\begin{figure}\n \n  \\includegraphics[width=8cm]{Fig8.eps}\n  \\caption{The energy band structures of  $\\alpha_i$- ($i$=1 to 2)   $\\mathrm{CrGe_2P_4}$  using GGA+SOC.}\\label{band-c}\n\\end{figure}\n\n\n\n\n\n\n\n Another key physical quantity $e_{11}$ of   $\\alpha_i$- ($i$=1 to 6) $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ are calculated to attain $d_{11}$. Their   piezoelectric coefficients  $e_{11}$ along with the ionic contribution and electronic contribution to $e_{11}$  are shown \\autoref{e}.\n It is clearly seen that the similar structural dependence between  $\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ can be observed.\n It is found that the ionic contribution of two structures  connected by mirror or rotation operations  in the same category  has opposite sign.  In the different category, the ionic contribution of two structures  connected by translation operation   has the same sign.\n Calculated results show that the ionic contribution and electronic contribution  have opposite sign for all $\\alpha_i$ except $\\alpha_5$.\n  For A and B categories, the electronic contribution has similar structural dependence  with  ionic contribution.\n  In the C category, the rotation operation gives rise to the identical signs for electronic contribution from $\\alpha_6$ to $\\alpha_5$.\nIn considered six structures, the $e_{11}$ of $\\alpha_5$ has the largest value, which is due to superposed  ionic contribution and electronic contribution. The  $e_{11}$ with  $\\alpha_5$ phase is 13.95$\\times$$10^{-10}$ C\/m for $\\mathrm{MoSi_2N_4}$,  and\t12.17$\\times$$10^{-10}$ C\/m for $\\mathrm{WSi_2N_4}$.  These $e_{11}$ are very larger than ones of 2D TMD, metal oxides,  III-V\nsemiconductor and Janus TMD materials\\cite{q7,q9,q11}.\nThe  $e_{11}$  of  experimentally synthesized  $\\alpha_1$-$\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$ is 4.40$\\times$$10^{-10}$ C\/m  and 3.14$\\times$$10^{-10}$ C\/m, which\nare  comparable to that of most  2D materials, such as  TMD and Janus TMD materials\\cite{q7,q9,q11}.\nUsing the calculated $C_{11}$-$C_{12}$ and $e_{11}$, the $d_{11}$ can be attained according to \\autoref{pe2-7}, which are shown in \\autoref{d}. From  $\\alpha_1$ to  $\\alpha_6$, the $d_{11}$ and $e_{11}$ show very analogical structural dependence. For $\\alpha_5$ phase,  the  $d_{11}$ has the largest value of  3.53 pm\/V for $\\mathrm{MoSi_2N_4}$,  and\t2.91 pm\/V for $\\mathrm{WSi_2N_4}$.\n For experimentally synthesized  $\\alpha_1$-$\\mathrm{MoSi_2N_4}$ and $\\mathrm{WSi_2N_4}$,  the  $d_{11}$  is 1.14 pm\/V  and 0.78 pm\/V, which are  smaller than that of 2D TMD\\cite{q9,q11}  due to very large $C_{11}$-$C_{12}$. The related $d_{11}$ are listed in \\autoref{tab0-1}.\n\\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig9.eps}\n  \\caption{(Color online) The elastic constants  $C_{ij}$ of $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$.}\\label{c1}\n\\end{figure}\n\n\n\n\nFor  $\\alpha_1$ and  $\\alpha_2$ phases, the  monolayer  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$ are all semiconductors using GGA+SOC. The energy band gaps of  $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$  using GGA+SOC are plotted in \\autoref{band2}. It is found that the gap of  $\\mathrm{CrGe_2P_4}$ is very small, and  0.008 eV for  $\\alpha_1$ phase and 0.061 eV for $\\alpha_2$ phase. To unambiguously indicate them to be semiconductors,  the energy band structures of  $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{CrGe_2P_4}$  using GGA+SOC are shown in \\autoref{band-c}.\n For the same material, the gap with $\\alpha_2$ phase is larger than one of $\\alpha_1$ phase.  For $\\mathrm{MA_2N_4}$, the gap increases with M from Cr to Mo to W, while  the gap of  $\\mathrm{MA_2P_4}$ firstly increases, and then decreases.  Another reason is that their  enthalpies of formation between  $\\alpha_1$ and  $\\alpha_2$ phases are very close\\cite{m20}.\nSo, we investigate the intrinsic piezoelectricity of the 11 kinds of materials with $\\alpha_1$ and  $\\alpha_2$ phases.\nThe optimize lattice constants are listed in \\autoref{tab1}, and the lattice constants between $\\alpha_1$ and  $\\alpha_2$ phases for the same material\nalmost the same, which is because the $\\alpha_1$ and  $\\alpha_2$ phases are in the same A class.\nWith element changing from Cr to Mo to W, from Si to Ge, and from N to P, the lattice constants of both $\\alpha_1$ and $\\alpha_2$ phases increase, which is due to increasing atomic radius.\n\n\n\n\\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig10.eps}\n  \\caption{(Color online) The  piezoelectric stress coefficients  $e_{11}$,  the ionic contribution and electronic contribution to $e_{11}$ of $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$.  }\\label{e1}\n\\end{figure}\n\n\nThe elastic constants  $C_{ij}$ of $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$ are plotted in \\autoref{c1}.  For all studied materials, the $C_{11}$-$C_{12}$  with  $\\alpha_2$ phase are larger than ones of  $\\alpha_1$ phase, which is due to larger $C_{11}$ and smaller $C_{12}$. The  $C_{ij}$ of $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ containing P atom are very smaller than ones including N atom, which is favor of $d_{11}$. The  piezoelectric stress coefficients  $e_{11}$  of $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$ together with  the ionic contribution and electronic contribution to $e_{11}$ are shown in \\autoref{e1}. When M changes from Cr to Mo to W with the same A and Z atoms,   the electronic contribution of  $\\mathrm{MA_2Z_4}$ with  $\\alpha_1$ phase decreases, while the one of $\\alpha_2$ phase changes toward more negative value. It is found that the electronic contribution (absolute value) of all  $\\mathrm{MA_2Z_4}$ with  $\\alpha_2$ phase is smaller than one of  $\\alpha_1$.  The ionic contribution of all materials with $\\alpha_2$ phase is positive, which is the same with the electronic contribution of $\\alpha_1$.\n With M from Cr to Mo to W,   the ionic contribution of  $\\mathrm{MA_2Z_4}$ of   $\\alpha_2$ phase with the same A and Z atoms  decreases, while the one of $\\alpha_1$ phase changes toward more negative value except for  $\\mathrm{CrSi_2N_4}$. It is clearly seen that the $e_{11}$ of all materials are positive values. The $e_{11}$ of  $\\mathrm{MA_2Z_4}$ containing P atom with the same M and A atoms is larger than one including N atom for both  $\\alpha_1$ and  $\\alpha_2$ phases.  For  $\\alpha_1$ phase, the $e_{11}$ ranges from 3.14$\\times$$10^{-10}$ C\/m to 9.31$\\times$$10^{-10}$ C\/m, and the whole range for $\\alpha_2$ phase is 0.85$\\times$$10^{-10}$ C\/m  to 7.39$\\times$$10^{-10}$ C\/m.\n  \\begin{figure}\n \n  \\includegraphics[width=7cm]{Fig11.eps}\n  \\caption{(Color online) The  piezoelectric strain coefficients  $d_{11}$ of  $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$.  }\\label{d1}\n\\end{figure}\n\n\n Finally, the  piezoelectric strain coefficients  $d_{11}$ of  $\\alpha_i$- ($i$=1 to 2)  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) expect  $\\mathrm{CrGe_2N_4}$ are plotted in \\autoref{d1}.  The related $d_{11}$ are also summarized in \\autoref{tab1-1}. For  $\\alpha_1$ phase, the range of  $d_{11}$ is 0.78 pm\/V to 6.12 pm\/V, and the  range changes from 0.25 pm\/V to 5.06 pm\/V  for $\\alpha_2$ phase. The change trend of  $d_{11}$  as a function of material  is very similar with one of $e_{11}$. It is clearly seen that monolayer $\\mathrm{MA_2Z_4}$ containing P atom have more excellent piezoelectric response due to high $d_{11}$.\n The most  $d_{11}$ of them  are larger than $d_{33}$ = 3.1 pm\/V of  familiar bulk piezoelectric  wurtzite GaN\\cite{zh1}.\nSo, it is highly recommended to synthesize monolayer $\\mathrm{MA_2Z_4}$ containing P atom, such as  $\\alpha_1$-$\\mathrm{CrSi_2P_4}$, $\\alpha_1$-$\\mathrm{MoSi_2P_4}$, $\\alpha_1$-$\\mathrm{CrGe_2P_4}$, $\\alpha_1$-$\\mathrm{MoGe_2P_4}$ and $\\alpha_2$-$\\mathrm{CrGe_2P_4}$.\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nWe have demonstrated strong structure  effect on intrinsic piezoelectricity in septuple-atomic-layer  $\\mathrm{MSi_2N_4}$ (M=Mo and W)\nthrough first-principles simulations.   The same structural dependence on $d_{11}$ and $e_{11}$, together with the  ionic and  electronic contributions to  $e_{11}$ between $\\mathrm{MoSi_2N_4}$ and  $\\mathrm{WSi_2N_4}$ monolayers is found, and the  $\\alpha_5$  phase  has  large piezoelectric coefficients. The intrinsic piezoelectricity of  monolayer  $\\mathrm{MA_2Z_4}$ (M=Cr, Mo and W; A=Si and Ge; Z=N and P) with $\\alpha_1$ and $\\alpha_2$ phases expect  $\\mathrm{CrGe_2N_4}$ are explored, and the  monolayer $\\mathrm{MA_2P_4}$  have more stronger  piezoelectric polarization than  monolayer $\\mathrm{MA_2Z_4}$  including N atom.\nThe largest $d_{11}$  among $\\mathrm{MA_2N_4}$ materials  only is 1.85 pm\/V, and  the largest $d_{11}$ of $\\mathrm{MA_2P_4}$ is up to 6.12 pm\/V.\n Among the studied 22 materials, the $d_{11}$ of monolayer  $\\alpha_1$-$\\mathrm{CrSi_2P_4}$, $\\alpha_1$-$\\mathrm{MoSi_2P_4}$, $\\alpha_1$-$\\mathrm{CrGe_2P_4}$, $\\alpha_1$-$\\mathrm{MoGe_2P_4}$ and $\\alpha_2$-$\\mathrm{CrGe_2P_4}$ are  greater than or close to 5 pm\/V.\n These $d_{11}$ of  $\\mathrm{MA_2P_4}$ compare favorably with piezoelectric coefficients of\nfamiliar bulk piezoelectrics such as $\\alpha$-quartz ($d_{11}$ = 2.3\npm\/V), wurtzite GaN ($d_{33}$ = 3.1 pm\/V) and wurtzite AlN ($d_{33}$ = 5.1 pm\/V)\\cite{zh1,zh2}.\nOur works provide valuable guidance for\nexperimental synthesis efforts, and hope our study will stimulate more\nresearch interest into $\\mathrm{MA_2Z_4}$ family, especially for\nits applications in piezoelectric field.\n\n\n\n\n\\section{Data availability}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\n\\begin{acknowledgments}\nThis work is supported by the Natural Science Foundation of Shaanxi Provincial Department of Education (19JK0809). We are grateful to the Advanced Analysis and Computation Center of China University of Mining and Technology (CUMT) for the award of CPU hours and WIEN2k\/VASP software to accomplish this work.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\bigskip\nThere have been a number of recent investigations into the \n{\\em chiral odd} structure functions of the nucleon. As in the case \nof the polarized structure functions there are two quantities of \ninterest at leading twist: The transverse spin chiral odd structure\nfunction $h_T(x,Q^2)$ and the longitudinal spin chiral odd structure \nfunction $h_L(x,Q^2)$. Within the context of the operator product \nexpansion (OPE) the analysis in terms of twist reveals that the \ntransverse chiral odd structure function $h_T(x,Q^2)$ is purely twist--2, \nwhile  the longitudinal structure function $h_L(x,Q^2)$ contains \nboth twist--2 and twist--3 contributions.\nAccordingly, the  decomposition of $h_L(x,Q^2)$ into twist--2 \nand twist--3 ($\\overline{h}_L(x,Q^2)$) pieces is given by\n\\begin{eqnarray}\nh_L(x,Q^2)=2x\\, \\int_{x}^1\\ dy\\frac{h_T(y,Q^2)}{y^2}+ \n\\overline{h}_L(x,Q^2)\\ .\n\\label{hL}\n\\end{eqnarray}\nAs a reminder we note that the kinematics are defined such that $q$ denotes \nthe momentum transferred to a nucleon of momentum $p$. In the Bjorken\nlimit, {\\it i.e.} $Q^2=-q^2\\to\\infty$ with $x=Q^2\/2p\\cdot\\! q$ fixed,\nthe leading twist contributions to the nucleon structure\nfunctions dominate the $1\/Q^2$ expansion. The additional and \nimportant logarithmic dependence on $Q^2$, which is associated \nwith soft gluon emission, is included via the evolution program \nof perturbative quantum--chromo--dynamics (QCD).\n\nWhile the chiral odd structure functions are not directly accessible \nin deep inelastic lepton nucleon scattering (DIS) there is the well known \nproposal at {\\em RHIC} to extract the quark transversality distributions\n$h_T^{(a)}(x,Q^2)$ ($a$ being the flavor index) from Drell--Yan \ndilepton--production resulting from transversely polarized proton \nbeams \\cite{Ra79}. Unfortunately dilepton production processes \nare difficult to extract from proton--proton collisions as the purely\nhadronic processes dominate. Furthermore this experiment will provide\nonly the product of the chiral odd distributions for quarks and \nantiquarks. As the latter are presumably small these flavor distributions \nare not easily measurable in the Drell--Yan process. In the light of these \ndisadvantages it has recently been pointed out that the transversality \ndistributions may also be measured in the fragmentation region of \nDIS \\cite{Ja98}. The key observation is that these distribution \nfunctions can be extracted from an asymmetry in the two meson production \nin the special case that this two meson state (like $\\pi^+\\pi^-$) is a \nsuperposition of different $C$--parity states, as {\\it e.g.} $\\sigma$ \nand $\\rho$. Then the phases in the final state interactions do not \nvanish on the average and the differential cross section is proportional \nto the product of chiral odd distributions and the interference \nfragmentation functions. The latter describe the emission and \nsubsequent absorption of a two pion intermediate state from quarks \nof different helicity. In case these fragmentation functions are not \nanomalously small the chiral odd distribution functions can then be \nobtained from DIS processes\\footnote{The relevant fragmentation and \ndistribution functions depend on different kinematical variables: the \ntwo meson state momentum fraction and the Bjorken variable, respectively.} \nlike $eN\\to e^\\prime \\pi^+\\pi^-X$ with the nucleon $N$ being transversely \npolarized. Assuming isospin covariance for the fragmentation functions \nthese DIS processes will provide access to the charge squared weighted \nchiral odd distribution functions \\cite{Ja98}. Such processes \nshould be measurable in the transversely polarized target experiments at \n{\\it HERMES}. Knowledge of the chiral odd structure \nfunctions will serve to complete our picture of the spin structure\nof the nucleon as they correspond to the distribution of the quark \ntransverse spin in a nucleon which is transversely polarized \n\\cite{Ja96}. With these data being expected in the near future \nit is, of course, interesting to understand the structure of \nthe nucleon from the theoretical point of view. As we are still \nlacking a bound state wave function for nucleon in terms of quarks \nand gluons, {\\it i.e.} computed from first principles in QCD,\nit is both mandatory and fruitful to investigate these chiral odd  \nflavor distributions and their charge weighted average\nnucleon structure functions within hadronic models of the\nnucleon \\cite{Ja92,St93,Ba97,Sc97,Sc97a,Ka97,Po96}. \n\nIn the context of the spin structure of the nucleon chiral soliton\nmodels are particularly interesting as they provide an  explanation\nfor the small magnitude of the quark spin contribution to the proton \nspin, {\\it i.e.} the vanishingly small matrix element of the singlet \naxial current \\cite{We96}. In these models the nucleon is described as a \nnon--perturbative field configuration in some non--linear effective \nmeson theory \\cite{Sk61,Ad83,Al96}. Unfortunately in many of these soliton \nmodels the evaluation of structure functions is infeasible due to the \nhighly non--linear structure of the current operators and the inclusion \nof higher derivative operators which complicates the current commutation \nrelations. However, it has recently been recognized that the soliton \nsolution \\cite{Al96} which emerges after bosonization \\cite{Eb86} of \nthe Nambu--Jona--Lasinio (NJL) \\cite{Na61} chiral quark model can be \nemployed to compute nucleon structure functions \\cite{We96a,We97}. In \norder to project this soliton configuration onto nucleon states with good \nspin and flavor a cranking procedure must be employed \\cite{Ad83,Re89} \nwhich implements significant $1\/N_C$ contributions ($N_C$ is the number \nof color degrees of freedom.). When extracting the structure functions \nfrom the NJL chiral soliton model the full calculation which also\nincludes effects of the vacuum polarized by the background soliton is \nquite laborious. In addition we are still lacking a regularization \nprescription of the vacuum contribution to the structure functions\nwhich is derived from the action functional and which yields algebraic \nexpressions for their moments which are {\\em consistent} with those for \nthe static nucleon properties. Fortunately it is known that the \ndominant contribution to \nstatic nucleon properties stems from the single quark level which has the \nlowest energy eigenvalue (in magnitude) and is strongly bound by the \nsoliton \\cite{Al96}. This is particularly the case for spin related \nquantities. Hence it is a reasonable approximation to consider\nonly the contribution of this level to the structure functions. In \nthe proceeding section the NJL chiral soliton model together with the \nabove mentioned approximation, which we will call {\\it valence quark \napproximation}\\footnote{This notation refers to the valence quark in the \nNJL chiral soliton model and should not be confused with the valence quark\nin the parton model.} will be described in more detail. \n\nThe NJL model for the quark flavor dynamics incorporates spontaneous\nbreaking of chiral symmetry in a dynamic fashion. Hence the quark fields \nwhich built up the soliton self--consistently \\cite{Re88} are {\\em \nconstituent quarks} with a constituent quark mass of several hundred \n{\\rm MeV}. Keeping this in mind we calculate both the {\\em effective}\nconstituent quark distributions and in turn the corresponding leading twist\ncontributions to nucleon structure functions ({\\it cf.} eq (\\ref{chgw}))\nat a low scale $Q_0^2$. In the language of Feynman diagrams \nthe DIS processes are described by a constituent quark of the nucleon \nabsorbing a quanta of the external source. In the Bjorken limit the quark \nthen propagates highly off--shell before emitting a quanta of the external \nsource. The intermediate quark may propagate forward and backward.\nHence the complete structure functions acquire contributions from \nboth distributions where the intermediate constituent quark moves \nforward and backward. We will focus on nucleon structure functions which\nare defined as the sum over the charge--weighted flavor distributions \n\\cite{Ja92}\n\\begin{eqnarray}\nh_{T\/L}^{(\\pm)}(x,Q_0^2)=\\frac{1}{2} \\sum_a  \ne_{a}^2 h^{(a,\\pm)}_{T\/L}(x,Q_0^2) \\ ,\n\\label{chgw}\n\\end{eqnarray}\nin analogy to those of the chiral even spin polarized and unpolarized \nnucleon structure functions \\cite{Ja98,Ja96}. Here $a$ represents a \nquark label, while $(\\pm)$ refers to the forward $(+)$ and backward\n$(-)$ propagating intermediate constituent quarks. Furthermore $e_{a}$ \ndenotes the charge fraction of the considered quark flavor $a$. The \ncomplete chiral odd structure functions are finally obtained as the sum\n\\begin{eqnarray}\nh_{T\/L}(x,Q_0^2)=h_{T\/L}^{(+)}(x,Q_0^2)+h_{T\/L}^{(-)}(x,Q_0^2)\\ .\n\\label{defintro}\n\\end{eqnarray}\nThe calculation of the flavor distributions $h^{(a)}_{T\/L}$ in the \nvalence approximation to the NJL chiral soliton model \\cite{We96a,We97} \nis summarized in section 3.\n\nFurther it is important to note that when considering model structure \nfunctions the OPE implies that the initial conditions,\n$\\mu^2=Q_0^2$, for the evolution is, \n{\\it a priori}, a free parameter in any baryon model \\cite{Sc91}. \nFor the model under consideration it has previously been determined to \n$Q_0^2\\approx0.4{\\rm GeV}^2$ by studying the evolution dependence of \nthe model prediction for the unpolarized structure functions \\cite{We96a}. \nIn a subsequent step to compute the chiral odd structure functions we \nemploy a leading order evolution program \\cite{Ba97,Ka97} to obtain the \nchiral odd structure functions at a larger scale, {\\it e.g.} \n$Q^2\\approx 4{\\rm GeV}^2$ relevant to the experimental conditions. This \nevolution program incorporates the leading logarithmic corrections to \nthe leading twist pieces. The evolution procedure as applied to our \nmodel structure functions will be explained in section 4.\n\nThe numerical results for the chiral odd structure functions are\npresented in section 5 while concluding remarks are contained in \nsection 6. Technical details on the model calculations and the QCD \nevolution procedure are relegated to appendices. Let us also mention\nthat there has been a previous calculation of $h_T(x,Q_0^2)$ \\cite{Po96} \nwhich, however, ignored both the projection onto good nucleon states \nand the QCD evolution. Furthermore in that calculation an (arbitrary) \nmeson profile was employed rather than a self--consistent soliton \nsolution to the static equations of motion. \n\n\\bigskip\n\\section{The NJL--Model Chiral Soliton}\n\\bigskip\n \nBefore continuing with the discussion of the chiral odd structure\nfunctions, we will review the issue of the \nchiral soliton in the NJL model.\n\nThe Lagrangian of the NJL model in terms of quark degrees of freedom \nreads \\cite{Na61,Eb86}\n\\begin{eqnarray}\n{\\cal L} = \\bar q (i\\partial \\hskip -0.5em \/ -  m^0 ) q +\n      2G_{\\rm NJL} \\sum _{i=0}^{3}\n\\left( (\\bar q \\frac {\\tau^i}{2} q )^2\n      +(\\bar q \\frac {\\tau^i}{2} i\\gamma _5 q )^2 \\right) .\n\\label{NJL}\n\\end{eqnarray}\nHere $q$, $\\hat m^0$ and $G_{\\rm NJL}$ denote the quark field, the \ncurrent quark mass and a dimensionful coupling constant, respectively.\nThis model is motivated as follows:\nIntegrating out the gluon fields from QCD yields a current--current \ninteraction mediated by one gluon exchange to leading order\nin powers of the quark current.  Replacing the gluon mediating \npropagator with a local contact interaction and \nperforming the appropriate Fierz--transformations yields the \nLagrangian (\\ref{NJL}) in leading order of $1\/N_C$ \\cite{Ca87,Re90}, \nwhere $N_C$ refers to the number of color degrees of freedom. Although\nonly a subset of possible non--perturbative gluonic modes are \ncontained in the contact interaction term in eq (\\ref{NJL}) \nit is important to stress that gluonic effects are contained in the \nmodel (\\ref{NJL}). Furthermore the NJL model embodies the approximate \nchiral symmetry of QCD and has to be understood as an effective \n(non--renormalizable) theory of the low--energy quark flavor dynamics.\n\nApplication of functional bosonization techniques \\cite{Eb86} to the \nLagrangian (\\ref{NJL}) yields the mesonic action\n\\begin{eqnarray}\n{\\cal A}&=&{\\rm Tr}_\\Lambda\\log(D)+\\frac{1}{4G_{\\rm NJL}}\n\\int d^4x\\ {\\rm tr}\n\\left(m^0\\left(M+M^{\\dag}\\right)-MM^{\\dag}\\right)\\ , \n\\label{bosact} \\\\\nD&=&i\\partial \\hskip -0.5em \/-\\left(M+M^{\\dag}\\right)\n-\\gamma_5\\left(M-M^{\\dag}\\right)\\ ,\n\\label{dirac}\n\\end{eqnarray}\nwhere $M=S+iP$ comprises composite scalar ($S$) and pseudoscalar ($P$) \nmeson fields which appear as quark--antiquark bound states.  \nFor regularization, which is indicated by the cut--off $\\Lambda$, we \nwill adopt the proper--time scheme \\cite{Sch51}. The free parameters \nof the model are the current quark mass $m^0$, the coupling constant \n$G_{\\rm NJL}$ and the cut--off $\\Lambda$. The equation of motion for \nthe scalar field $S$ may be considered as the gap--equation for the \norder parameter $\\langle {\\bar q} q\\rangle$ of chiral symmetry breaking. \nThis equation relates the vacuum expectation value \n$\\langle M\\rangle=m{\\mbox{{\\sf 1}\\zr{-0.16}\\rule{0.04em}{1.55ex}\\zr{0.1}}}$ to the model parameters $m^0$, $G_{\\rm NJL}$ \nand $\\Lambda$. For apparent reasons $m$ is called the {\\em constituent} \nquark mass. The occurrence of this vacuum expectation value reflects the \nspontaneous breaking of chiral symmetry and causes the pseudoscalar fields \nto emerge as (would--be) Goldstone bosons. Expanding ${\\cal A}$ to quadratic \norder in $P$ (around $\\langle M\\rangle$) these parameters are related to \nphysical quantities; that is, the pion mass, $m_\\pi=135{\\rm MeV}$ and the \npion decay constant, $f_\\pi=93{\\rm MeV}$. This leaves one undetermined \nparameter which we choose to be the constituent quark mass \\cite{Eb86}.\n\nThe NJL model chiral soliton \\cite{Al96,Re88} is given \nby a non--perturbative meson configuration which is assumed of the \nhedgehog type\n\\begin{eqnarray}\nM_{\\rm H}(\\mbox{\\boldmath $x$})=m\\ {\\rm exp}\n\\left(i\\mbox{\\boldmath $\\tau$}\\cdot{\\hat{\\mbox{\\boldmath $x$}}}\n\\Theta(r)\\right)\\ .\n\\label{hedgehog}\n\\end{eqnarray}\nIn order to compute the functional trace in eq (\\ref{bosact}) for this \nstatic configuration we express the \nDirac operator (\\ref{dirac}) in terms of a Hamiltonian\noperator $h$, {\\it i.e.}  $D=i\\beta(\\partial_t-h)$ with\n\\begin{eqnarray}\nh=\\mbox{\\boldmath $\\alpha$}\\cdot\\mbox{\\boldmath $p$}+m\\ \\beta\\\n{\\rm exp}\\left(i\\gamma_5\\mbox{\\boldmath $\\tau$}\n\\cdot{\\hat{\\mbox{\\boldmath $x$}}}\\Theta(r)\\right)\\ .\n\\label{hamil}\n\\end{eqnarray}\nWe denote the eigenvalues and eigenfunctions of $h$ by \n$\\epsilon_\\mu$ and $\\Psi_\\mu$, respectively. Explicit expressions for \nthese wave--functions are displayed in appendix B of ref \\cite{Al96}. \nIn the proper--time regularization scheme the energy functional of \nthe NJL model is found to be \\cite{Re89,Al96}, \n\\begin{eqnarray}\nE[\\Theta]=\n\\frac{N_C}{2}\\epsilon_{\\rm v}\n\\left(1+{\\rm sgn}(\\epsilon_{\\rm v})\\right)\n&+&\\frac{N_C}{2}\\int^\\infty_{1\/\\Lambda^2}\n\\frac{ds}{\\sqrt{4\\pi s^3}}\\sum_\\nu{\\rm exp}\n\\left(-s\\epsilon_\\nu^2\\right)\n\\nonumber \\\\* && \\hspace{1.5cm}\n+\\ m_\\pi^2 f_\\pi^2\\int d^3r  \\left(1-{\\rm cos}\\Theta(r)\\right) .\n\\label{efunct}\n\\end{eqnarray}\nThe subscript ``${\\rm v}$\" denotes the valence quark level. This state \nis the distinct level bound in the soliton background, {\\it i.e.}\n$-m<\\epsilon_{\\rm v}<m$. The chiral angle, $\\Theta(r)$, is\nobtained by self--consistently extremizing $E[\\Theta]$ \\cite{Re88}.\n\nStates possessing nucleon quantum numbers of spin and isospin are \ngenerated by elevating the rotational zero modes to time dependent \nlarge amplitude rotational fluctuations about the hedgehog field\n\\cite{Ad83}\n\\begin{eqnarray}\nM(\\mbox{\\boldmath $x$},t)=\nA(t)M_{\\rm H}(\\mbox{\\boldmath $x$})A^{\\dag}(t)\\ ,\n\\label{collrot}\n\\end{eqnarray}\nwhich introduces the collective coordinates $A(t)\\in SU(2)$. \nSubstituting the {\\it ansatz} (\\ref{collrot})\ninto the action functional (\\ref{bosact}) and expanding\n\\cite{Re89} in the angular velocities \n\\begin{eqnarray} \n2A^{\\dag}(t)\\dot A(t)=\ni\\mbox{\\boldmath $\\tau$}\\cdot\\mbox{\\boldmath $\\Omega$}\n\\label{angvel}\n\\end{eqnarray}\nto quadratic order yields the Lagrange function for the collective \ncoordinates.  Upon canonical quantization the angular velocity \n$\\mbox{\\boldmath $\\Omega$}$ is substituted by the nucleon spin \noperator $\\mbox{\\boldmath $J$}=\\alpha^2\\mbox{\\boldmath $\\Omega$}$,\nwith $\\alpha^2$ being the moment of inertia \\cite{Re89,Al96}.\nThe eigenfunctions of the resulting Hamiltonian are the \nWigner $D$--functions\n\\begin{eqnarray}\n\\langle A|N\\rangle=\\frac{1}{2\\pi}\nD^{1\/2}_{I_3,-J_3}(A)\\ ,\n\\label{nwfct}\n\\end{eqnarray}\nwith $I_3$ and $J_3$ being respectively the isospin and spin projection \nquantum numbers of the nucleon. The nucleon matrix elements of the \ncollective rotations are obtained via\n$\\langle N | {\\rm tr}(\\tau_i A \\tau_j A^\\dagger) |N\\rangle=\n-(8\/3)\\langle N | I_i J_j |N\\rangle$ \\cite{Ad83}.\nThis approach to \ngenerate nucleon states from the hedgehog corresponds to the\ncranking technique in nuclear physics \\cite{In54}.\n\nExpectation values of bilocal quark--bilinears appearing in the \nevaluation of nucleon structure functions are expressed as (regularized) \nsums over bilocal and bilinear combinations of all eigenfunctions \n$\\Psi_\\mu$ including the Dirac sea states. In practice this is quite \na painful task, in particular when cranking corrections (\\ref{collrot}) \nare included. Also the problem of regularization is not consistently \nsolved. Fortunately it \nturns out that the dominant contributions ($\\ge80\\%$) to static nucleon \nproperties (which are moments of the structure functions) stems from the \ndistinct valence level $\\Psi_{\\rm v}$ \\cite{Al96}. It is therefore \nreasonable to approximate the relevant bilinears by their valence quark \ncontribution. In order to obtain quark distributions of the {\\em nucleon}\nand the corresponding nucleon structure functions ({\\it c.f.} \\ref{chgw}),\nrather than {\\em soliton} structure functions the cranking contribution\nto the wave--function, which is induced by the collective rotation $A(t)$, \nmust be included. That is, the valence quark wave--function employed to \napproximate the bilinears in the structure functions reads\n\\begin{eqnarray}\n\\Psi_{\\rm v}(\\mbox{\\boldmath $x$},t)=\n{\\rm e}^{-i\\epsilon_{\\rm v}t}A(t)\n\\left\\{\\Psi_{\\rm v}(\\mbox{\\boldmath $x$})\n+\\frac{1}{2}\\sum_{\\mu\\ne{\\rm v}}\n\\Psi_\\mu(\\mbox{\\boldmath $x$})\n\\frac{\\langle \\mu |\\mbox{\\boldmath $\\tau$}\\cdot\n\\mbox{\\boldmath $\\Omega$}|{\\rm v}\\rangle}\n{\\epsilon_{\\rm v}-\\epsilon_\\mu}\\right\\}=:\n{\\rm e}^{-i\\epsilon_{\\rm v}t}A(t)\n\\psi_{\\rm v}(\\mbox{\\boldmath $x$}).\n\\label{valrot}\n\\end{eqnarray}\nHere $\\psi_{\\rm v}(\\mbox{\\boldmath $x$})$ refers to the spatial part \nof the body--fixed valence quark wave--function with the rotational \ncorrections included and \n$\\Psi_\\mu=\\langle\\mu|\\mbox{\\boldmath $x$}\\rangle$ are eigenfunctions \nof the Dirac Hamiltonian (\\ref{hamil}). This replacement of the bilocal \nand bilinear quark fields when computing nucleon structure functions defines \nthe valence quark approximation.\n\n\n\\bigskip\n\\section{Chiral Odd Structure Functions, \n${\\lowercase{h}}_T({\\lowercase{x}})$ \nand ${\\lowercase{h}}_L({\\lowercase{x}})$\nin the NJL model}\n\\bigskip\n\nHere we present the major topic of this paper, namely the calculation \nof the twist--2 and twist--3 chiral odd structure functions in the \nNJL chiral soliton model. Like their deep inelastic chiral even \n(un)polarized counterparts, the chiral odd structure functions are\ncomputed as Fourier transformations of nucleon matrix elements of  \nbilocal quark operators on the light--cone \\cite{Ja92}. The key \nfeatures of the relevant light--cone kinematics are given in Appendix A.  \n\nWe begin by listing the forward propagating intermediate quark $(+)$ \ncontribution to the chiral odd nucleon structure functions.\nBefore, however, straightforwardly transcribing the expressions from \nappendix A we must recall that the soliton represents a localized \nfield configuration. Therefore a collective coordinate \n$\\mbox{\\boldmath $x$}_0$ is introduced which parameterizes the position \nof the soliton. This collective coordinate is employed to generate \nstates with good linear momentum \\cite{Ge75}. When computing matrix \nelements between states of identical momenta one is essentially left \nwith an integration over $\\mbox{\\boldmath $x$}_0$. In the nucleon \nrest--frame (RF) the contribution of the forward moving intermediate \nquark to the chiral odd structure functions may therefore be \nexpressed as\\footnote{The following expressions constitute \na generalization of Jaffe's original definition \\cite{Ja75} for nucleon \nstructure functions.}\n\\begin{eqnarray}\nh^{(+)}_T(x)&=&N_C\\frac{2M\\sqrt{2}}{8\\pi}\\ \\int d\\xi^-\n{\\rm exp}(-i\\xi^-\\frac{Mx}{\\sqrt{2}})\n\\nonumber \\\\ && \\hspace{1.5cm}\n\\times\\int d^3\\bbox{x}_0\\ \n\\langle \\bbox{S}_{\\perp}|\\Psi_+^\\dagger(\\xi - \\bbox{x}_0) \n\\gamma_\\perp\\gamma_5{\\cal Q}^2\\Psi_+(-\\bbox{x}_0)\n|\\bbox{S}_\\perp\\Big\\rangle_{\\xi^+=\\mbox{\\scriptsize\\boldmath $\\xi$}_\\perp=0}\\ .\n\\label{cht}\n\\end{eqnarray}\nFor convenience we have omitted the subscript ``v'' for the valence \nquark wave function. Note that $\\xi$ refers to a four vector which \nin light--cone coordinates reads $(\\xi^+,\\xi^-,\\bbox{\\xi}_\\perp)$.\nThis coordinate \nenters the light--cone variables via  $\\xi^{\\pm}=(t\\pm z)\/\\sqrt{2}$. \nAlso, the notation $\\bbox{S}_{\\perp}$ is synonymous for \nthe spin being perpendicular to the coordinate $z$. \nOn the other hand for the longitudinal counterpart\n\\begin{eqnarray}\nh^{(+)}_L(x)&=&N_C\\frac{2M\\sqrt{2}}{16\\pi}\\ \\int  d\\xi^-\n{\\rm exp}(-i\\xi^-\\frac{Mx}{\\sqrt{2}})\n\\nonumber \\\\ && \\hspace{1.5cm}\n\\times\\int d^3\\bbox{x}_0\\ \n\\langle \\bbox{S}_z|\\Psi_+^\\dagger(\\xi - \\bbox{x}_0) \n\\gamma_0\\gamma_5{\\cal Q}^2\\Psi_-(-\\bbox{x}_0)\n\\nonumber \\\\ &&\\hspace{4cm}\n-\\Psi_-^\\dagger(\\xi - \\bbox{x}_0) \n\\gamma_0\\gamma_5{\\cal Q}^2\\Psi_+(-\\bbox{x}_0)| \n\\bbox{S}_z\\rangle_{\\xi^+=\\mbox{\\scriptsize\\boldmath $\\xi$}_\\perp=0}\n\\label{chl}\n\\end{eqnarray}\nthe spin is aligned with the $z$--axis.\nThe ``good'' and ``bad'' light--cone components of the quark wave \nfunctions are the projections $\\Psi_{\\pm}=P_{\\pm}\\Psi$, with\n$P_{\\pm}=\\frac{1}{2}\\gamma^{\\mp}\\gamma^{\\pm}$ being the corresponding \nprojections operators. Above, ${\\cal Q}={\\rm diag}(2\/3,-1\/3)$ refers \nto the matrix containing the quark charge fractions and the zero momentum \nnucleon states are given by, \n$|\\bbox{p}=0,\\bbox{S}\\rangle\n=\\left[(2\\pi)^3 2M\\right]^{\\frac{1}{2}}\n|\\bbox{S}\\rangle$.\nIntroducing Fourier transforms for the spatial part of \nthe valence quark wave functions ({\\it cf.} eq (\\ref{valrot})),\n\\begin{eqnarray}\n\\psi\\left(\\bbox{\\xi}_\\bot,\n\\xi_3=- \\frac{\\xi^-}{\\scriptstyle\\sqrt{2}}\\right)=\n\\int \\frac{d^2p_\\bot dp_3}{2\\pi^2}\\\n{\\rm exp}\\left[i\\left(\\frac{p_3\\xi^-}{\\scriptstyle\\sqrt{2}}-\n\\bbox{p}_\\bot\\cdot\n\\bbox{\\xi}_\\bot\\right)\\right]\n\\tilde{\\psi}\\left(\\bbox{p}_\\bot,p_3\\right) \\ \n\\label{ftrans}\n\\end{eqnarray}\nyields,\n\\begin{eqnarray}\nh^{(+)}_T(x)&=&N_C\\frac{M}{\\sqrt{2}\\pi^2}\\int d\\xi^-\\ \np^2dp\\ d({\\rm cos}\\, \\theta)\\ d\\phi\\ \n{\\rm exp}\\left(\\frac{-i\\xi^-\n\\left(Mx-{\\rm\\epsilon_{\\rm v}}+p{\\rm cos}\\, \\theta\\right)}{\\sqrt 2}\\right)\n\\nonumber \\\\ &&\\hspace{6cm}\n\\times\\langle \\bbox{S}_\\perp |\\tilde{\\psi}_+^\\dagger(\\bbox{p})\n\\gamma_\\perp\\gamma_5{\\cal Q}^2\n\\tilde{\\psi}_+(\\bbox{p})|\\bbox{S}_\\perp\\rangle\\ ,\n\\label{htpx}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nh^{(+)}_L(x)&=&N_C\\frac{M}{\\sqrt{2}\\pi^2}\\int d\\xi^-\\ \np^2dp\\ d({\\rm cos}\\, \\theta)\\ d\\phi\\ {\\rm exp}\\left(\\frac{-i\\xi^-\n\\left(Mx-{\\rm\\epsilon_{\\rm v}}+p{\\rm cos}\\, \\theta\\right)}{\\sqrt 2}\\right)\n\\nonumber \\\\ &&\\hspace{1cm}\n\\times\\langle \\bbox{S}_z |\\tilde{\\psi}_+^\\dagger(\\bbox{p})\n\\gamma_0\\gamma_5{\\cal Q}^2\\tilde{\\psi}_-(\\bbox{p})\n-\\tilde{\\psi}_-^\\dagger(\\bbox{p})\n\\gamma_0\\gamma_5{\\cal Q}^2\\tilde{\\psi}_+(\\bbox{p})\n|\\bbox{S}_z\\rangle\\ .\n\\label{hlpx}\n\\end{eqnarray}\nIn anticipation of the decomposition (\\ref{valrot}) the square of the \ncharge operator is redefined to\n\\begin{eqnarray}\n{\\cal Q}^2=\\frac{5}{18}{\\mbox{{\\sf 1}\\zr{-0.16}\\rule{0.04em}{1.55ex}\\zr{0.1}}} +\\frac{1}{6}D_{3i}\\tau_{i}\\ .\n\\label{qsquare}\n\\end{eqnarray}\nHere $D_{ij}=\\frac{1}{2}\\\n{\\rm tr}\\left(\\tau_{i}A(t)\\tau_{j}A^{\\dagger}\\right)$ denotes the\nadjoint representation of the collective rotation which is defined\nin eq (\\ref{collrot}). The integrals over $\\xi^-$ and $\\theta$ \nenforce both the constraint \n${\\rm cos}\\, \\theta=({\\rm\\epsilon_{\\rm v}}-Mx)\/p$ and the lower bound,  \n$p_{\\rm min}=|Mx-{\\rm\\epsilon_{\\rm v}|}$ on the $p$ integration. This \nresults in the forward moving quark contributions to the transverse \nand longitudinal chiral odd nucleon structure functions\n\\begin{eqnarray}\nh^{(+)}_T(x)=N_C\\frac{2M}{\\pi}\\int_{p_{\\rm min}}^\\infty \npdp\\ d\\phi\\langle \\bbox{S}_\\perp |\\tilde{\\psi}_+^\\dagger(\\bbox{p})\n\\gamma_\\perp\\gamma_5 {\\cal Q}^2\\tilde{\\psi}_+(\\bbox{p})\n|\\bbox{S}_\\perp\\rangle\\ \\Big|\\, _{{\\rm cos}\\, \n\\theta=\\frac{{\\rm\\epsilon}-Mx}{p}}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nh^{(+)}_L(x)&=&N_C\\frac{2M}{\\pi}\\int_{p_{\\rm min}}^\\infty pdp\\ d\\phi\n\\langle \\bbox{S}_z |\\tilde{\\psi}_+^\\dagger(\\bbox{p})\n\\gamma_0\\gamma_5{\\cal Q}^2\\tilde{\\psi}_-(\\bbox{p})\n\\nonumber \\\\ &&\\hspace{3cm}\n-\\tilde{\\psi}_-^\\dagger(\\bbox{p})\n\\gamma_0\\gamma_5{\\cal Q}^2\\tilde{\\psi}_+(\\bbox{p})\n|\\bbox{S}_z\\rangle\\ \n\\Big|\\, _{{\\rm cos}\\, \\theta=\\frac{{\\rm\\epsilon}-Mx}{p}}\\ .\n\\end{eqnarray}\nIn order to obtain the full structure functions the contribution of \nbackward moving quarks $h_{T,L}^{(-)}$ must be considered as well. These \ncontributions are easily obtained from $h_{T,L}^{(+)}(-x)$ by \nreversing the appropriate signs in equations (\\ref{htpx}) and (\\ref{hlpx}). \nThe two contributions may be comprised in\\footnote{We have used that \nthe valence quark level has positive parity, {\\it i.e.} under\n$\\bbox{p}\\rightarrow\\, -\\bbox{p}$ we find\n$\\tilde{\\psi}(-\\bbox{p})=\\gamma^0\\tilde{\\psi}(\\bbox{p})$.}\n\\begin{eqnarray}\nh^{(\\pm)}_T(x)&=&\\pm N_C\\frac{M}{\\pi}\\int_{p_{\\rm min}^{\\mp}}^\\infty \npdp\\ d\\phi\\langle \\bbox{S}_\\perp |\\tilde{\\psi}^\\dagger(\\bbox{p}_{\\mp})\n\\left(1\\mp\\alpha_3\\right)\\gamma_\\perp\\gamma_5\n{\\cal Q}^2\\tilde{\\psi}(\\bbox{p}_{\\mp})\n|\\bbox{S}_\\perp\\rangle\\ \n\\Big|\\, _{{\\rm cos}\\, \\theta^{\\mp}_p}\n\\label{ht11}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nh^{(\\pm)}_L(x)&=&\\pm N_C\\frac{M}{\\pi}\\int_{p_{\\rm min}^{\\mp}}^\\infty pdp\\\nd\\phi \\langle \\bbox{S}_z |\\tilde{\\psi}^\\dagger(\\bbox{p}_{\\mp})\n\\alpha_3\\gamma_0\\gamma_5{\\cal Q}^2\\tilde{\\psi}(\\bbox{p}_{\\mp})\n|\\bbox{S}_z\\rangle\\ \n\\Big|\\, _{{\\rm cos}\\, \\theta^{\\mp}_p}\n\\label{hl11}\n\\end{eqnarray}\nwhere $p_{\\rm min}^{\\pm}=|Mx\\pm{\\rm \\epsilon_{\\rm v}}|$ \nand ${\\rm cos}\\, \\theta^{\\pm}=(Mx \\pm \\epsilon_{\\rm v})\/p$ and \n$\\tilde{\\psi}(\\bbox{p}_{\\pm})\n=\\tilde{\\psi}(p,{\\rm cos}\\,\\theta^{\\pm}_{p},\\phi)$.\nFinally we summarize our results by decomposing the proton structure \nfunctions into their (iso)scalar and vector components,\n\\begin{eqnarray}\nh_T(x)&=&h^{I=0}_{T,+}(x)+h^{I=1}_{T,+}(x)\\, \n+ \\left(h^{I=0}_{T,-}(x)+h^{I=1}_{T,-}(x)\\right)\\ , \\nonumber \\\\\nh_L(x)&=&h^{I=0}_{L,+}(x)+h^{I=1}_{L,+}(x)\\, \n+ \\left(h^{I=0}_{L,-}(x)+h^{I=1}_{L,-}(x)\\right)\\ .\n\\label{hltnjl}\n\\end{eqnarray}\nThe isoscalar piece ($I=0$) originates from the unit matrix in the \ndecomposition (\\ref{qsquare}) while the isovector part ($I=1$) stems from \nthe terms involving the collective coordinates. The explicit expressions \nfor the structure functions (\\ref{hltnjl}) in terms of the static quark\nwave functions are computed in Appendix B.\n\n\n\\bigskip\n\\section{Projection And Evolution}\n\\bigskip\n\nWe consider that our model approximates QCD at a low scale $Q_0^2$. In oder \nto compare the predicted structure functions with data they must be\nevolved to a (larger) $Q^2$ commensurate with experimental conditions.\nA direct comparison with data gathered at a low scale cannot be made as \nthe latter structure functions contain sizable contributions from higher \ntwist.  Thus we evolve the chiral odd model structure functions\nof the preceding section utilizing the results of perturbative\nQCD.\n\nIn the soliton approach the baryon states are built from localized \nfield configurations. In fact, these states do not carry good \nfour--momentum. Therefore the calculated structure functions \n({\\it cf.} Figs. \\ref{fig_htud}, \\ref{fig_htnp} and \\ref{fig_hlnp}) do \nnot vanish exactly for $x>1$ although the contributions for $x>1$ \nare very small. \n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=updown.400.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=updown.450.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_htud}\nThe valence quark approximation of the transverse chiral--odd nucleon \ndistribution function as a function of Bjorken--$x$ for the up and down \nquark flavor content in the rest frame. For comparison also the model \ncalculation \\protect\\cite{We97} for the twist--2 polarized structure \nfunction $g_1(x,Q_0^2)$ is shown for the respective flavor channels.\nTwo values of the constituent quark mass are considered:\n$m=400 {\\rm MeV}$ (left panel) and $m=450 {\\rm MeV}$ (right panel).}\n\\end{figure}\n\nThe calculation of nucleon \nstructure functions in the Bjorken limit, however, singles out \nthe null plane, $\\xi^+=0$. This condition can be satisfied upon \ntransformation to the infinite momentum frame (IMF) even for models \nwhere the nucleon emerges as a (static) localized object \\cite{Hu77}.  \nFor the quark soliton model under consideration this transformation \ncorresponds to performing a boost in the space of the collective \ncoordinate $\\bbox{x}_0$, {\\it cf.} eq (\\ref{cht}). Upon this boost \nto the IMF we have observed \\cite{Ga97} that the common problem of \nimproper support for the structure functions, {\\it i.e.} non--vanishing \nstructure functions for $x>1$, is cured along the line suggested by \nJaffe \\cite{Ja80} some time ago. The reason simply is that the Lorentz \ncontraction associated with the boost to the IMF maps the infinite line \nexactly onto the interval $x\\in [0,1[$. In addition we have observed that \nthis Lorentz contraction effects the structure functions also at small \nand moderate $x$.  Incorporating these results for the general set \nof leading twist structure functions within the NJL--chiral soliton model\nyields the following form for the forward and backward\nmoving  intermediate quark\nstate contributions to the chiral odd transverse\nspin structure function, $h^{(\\pm)}_T\\left(x,Q^2\\right)$,\n\\begin{eqnarray}\n\\hspace{-0.3cm}\nh^{(\\pm)}_T(x)&=&\\pm N_C\\frac{M}{\\pi(1-x)}\n\\int_{p_{\\rm min}}^\\infty \\hspace{-0.2cm} pdp d\\varphi \\\n\\nonumber \\\\ && \\hspace{1.0cm}\n\\times\\langle N |\\tilde{\\psi}^\\dagger (\\bbox{p}_{\\mp})\n\\left(1\\mp\\alpha_3\\right)\\gamma_{\\perp}\\gamma_5{\\cal Q}^2\n\\tilde{\\psi}(\\bbox{p}_{\\mp})|N\\rangle\n\\Big|_{{\\rm cos}\\theta=-\n{\\textstyle \\frac{M\\ {\\rm ln}(1-x)\\pm\\epsilon_{\\rm v}}{p}}} \\ .\n\\label{htp}\n\\end{eqnarray}\nIn general the resulting relation between structure functions \nin the IMF and the rest frame (RF) reads\n\\begin{eqnarray}\nf_{\\rm IMF}(x)=\\frac{\\Theta(1-x)}{1-x} f_{\\rm RF}\n\\Big(-{\\rm ln}(1-x)\\Big)\\ .\n\\label{fboost}\n\\end{eqnarray}\nOf course, in the context of the chiral odd structure functions \n$f_{\\rm RF}$  is to be identified with the expressions in \neqs (\\ref{ht11},\\ref{hl11},\\ref{hltnjl}). As will be recognized \nshortly the solution to\nthe proper support problem is essential in order to \napply the evolution program of perturbative QCD.\n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=updownpr.400.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=updownpr.450.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_htudpr}\nSame as figure \\protect\\ref{fig_htud} \nin the IMF (\\protect\\ref{fboost}).}\n\\end{figure}\nThe chiral odd and polarized structure functions resulting from this\ntransformation are shown in figure \\ref{fig_htudpr}.\n\nIn order to include the logarithmic corrections to the \ntwist--2 pieces of the chiral odd structure functions we \napply the well--established GLAP procedure \\cite{Gr72}.\nFor the transverse component $h_T(x,Q^2)$ this is \nstraightforward as it is pure twist--2. For the longitudinal\npiece $h_L(x,Q^2)$ one first has to extract the twist--2\ncomponent through $h_T(x,Q^2)$ namely,\n$h_L^{(2)}(x,Q^2)=2x\\, \\int_{x}^1\\ dy\\,h_T(y,Q^2)\/y^2$.\n\nWe simultaneously denote by $h^{(2)}$ the twist--2 parts of $h_T$ \nand $h_L$. To leading order (in $\\alpha_{QCD}(Q^2)$) the variations\nof the structure functions from a change $\\delta t$ of the \nmomentum scale is given by\n\\begin{eqnarray}\nh^{(2)}(x,t+\\delta{t})=h^{(2)}(x,t)\\ \n+ \\frac{dh^{(2)}(x,t)}{dt} \\ \\delta t\\ ,\n\\label{h2var}\n\\end{eqnarray}\nwhere $t={\\rm log}\\left(Q^2\/\\Lambda_{QCD}^2\\right)$. The variation\n(\\ref{h2var}) is essentially due to the emission and absorption of \nsoft gluons. The explicit expression for the evolution differential \nequation is given by the convolution integral,\n\\begin{eqnarray}\n\\frac{d\\, h^{(2)}(x,t)}{dt} =\\frac{\\alpha_{QCD}(t)}{2\\pi}\nC_{R}(F)\\int^1_{x}\\ \\frac{dy}{y}P_{qq}^h\\left(y\\right)\nh^{(2)}\\left(\\frac{x}{y},t\\right)\n\\label{convl}\n\\end{eqnarray}\nwhere the leading order splitting function \\cite{Ar90,Ba97} is\ngiven by,\n\\begin{eqnarray} \nP_{qq}^{h}\\left(z\\right)=\n\\frac{4}{3}\\left[\\frac{2}{\\left(1-z\\right)_+}-2\n+\\frac{3}{2}\\ \\delta(z-1)\\right]\n\\end{eqnarray}\nand $C_R(f)=\\left(n_f^2-1\\right)\/2n_f$ for $n_f$ active flavors,\n$\\alpha_{QCD}(t)=4\\pi\/\\left[b_0\\log\\left(Q^2\/ \\Lambda^2\\right)\\right]$\nand $b_0=(11N_C-2n_f)\/3$.\nEmploying the ``+\" prescription yields for three light flavors and\n$N_C=3$\n\\begin{eqnarray}\n\\frac{d h^{(2)}(x,t)}{dt}&=&\\frac{\\alpha_{QCD}(t)}{2\\pi}\n\\left\\{\\ \\left(2 + \\frac{8}{3}\\log(1-x)\\right)h^{(2)}(x,t)\n\\right.\n\\nonumber \\\\*&& \\hspace{-0.7cm} \n\\left. \n+\\, \\frac{8}{3}\\int^{1}_{x}\\ \\frac{dy}{y}\\left[\n\\frac{1}{1-y}\\left(h^{(2)}(\\frac{x}{y},t)-yh^{(2)}(x,t)\\right)\n- h^{(2)}(\\frac{x}{y},t)\\right]\\right\\}\\ .\n\\label{evhtw2}\n\\end{eqnarray}\nAs indicated above, the structure functions must vanish at the boundary \n$x=1$ in order to cancel the divergence of the logarithm in eq \n(\\ref{evhtw2}) and thus for the GLAP procedure to be applicable. This \nmakes the projection of the rest frame structure functions mandatory.\nThe variation of the structure functions for finite intervals \nin $t$ is straightforwardly obtained by iteration of these \nequations, {\\it i.e.} as a solution to the differential \nequation (\\ref{evhtw2}). As discussed previously the initial value \nfor integrating the differential equation is given by the scale \n$Q_0^2$ at which the model is defined. It should be emphasized that \nthis scale essentially is a new parameter of the model. For a given \nconstituent quark mass $m$ we adjust $Q_0^2$ to maximize the \nagreement of the predictions with the experimental data on \npreviously \\cite{We96a} calculated unpolarized structure functions for \nelectron--nucleon DIS: $F_2^{ep}-F_2^{en}$. For the constituent \nquark mass $m=400{\\rm MeV}$ we have obtained $Q_0^2\\approx0.4{\\rm GeV}^2$.\nNote that this value of $Q_0^2$ is indeed (as it should) smaller than \nthe ultraviolet cut--off of the underlying NJL soliton model as \n$\\Lambda^2\\approx 0.56{\\rm GeV}^2$. The latter quantity indicates the range \nof validity of the model. In figure \\ref{fig_ht2p}a we compare the un--evolved, \nprojected, proton structure function $h_T^{p}\\left(x,Q_0^2\\right)$ with \nthe one evolved from $Q_0^2=0.4{\\rm GeV}^2$ to $Q^2=4.0{\\rm GeV}^2$. As \nexpected the evolution pronounces the structure function at low $x$. \n\nThis change towards small $x$ is a generic feature of the projection \nand evolution process and presumably not very sensitive to the \nprescription applied here. In particular, choosing a projection \ntechnique \\cite{Tr97} alternative to (\\ref{fboost}) may easily be \ncompensated by an appropriate variation of the scale $Q_0^2$. In \nfigure \\ref{fig_ht2p}b the same calculation for $h_L^{(2)}(x,Q^2)$ is \npresented.\n\nIn the evolution of the twist--2 pieces we have restricted ourselves\nto the leading order in $\\alpha_s$ because for the twist--3 piece of\n$h_L$, the necessary ingredients are not known in next--to--leading \norder. Even the leading order evolution is only known in the large \n$N_C$ limit. It should be noted that such an approach seems \nparticularly suited for soliton models which equally utilize large \n$N_C$ arguments. As pointed out by Balitskii et al. \\cite{Bal96} the \nadmixture of independent quark and quark--gluon operators contributing \nto the twist--3 portion ${\\overline{h}}_L(x,Q^2)$ grows with $n$ \nwhere $n$ refers to the $n^{\\rm th}$ moment,\n${\\cal M}_n\\left[ \\overline{h}_L(Q^2)\\right]$ of $h_L(x,Q^2)$.\nHowever, much like the case with\nthe spin--polarized structure function, $g_2(x,Q^2)$ \\cite{Ali91}\nin the $N_C\\rightarrow \\infty$ limit the quark operators of \ntwist--3 decouple from the quark--gluon operators of the same twist.\nThen the anomalous dimensions $\\gamma_n$ which govern the \nlogarithmic $Q^2$ dependence of ${\\cal M}_n$ can be computed. Once the \n$\\gamma_n$'s are known an evolution kernel can be constructed that \n``propagates'' the the twist--3 part $\\overline{h}(x,Q^2)$ in momentum\n\\begin{eqnarray}\n\\overline{h}_L(x,Q^2)&=&\\int_x^1 \\frac{dy}{y} b(x,y;Q^2,Q_0^2)\n\\overline{h}_L(y,Q_0^2)\\ .\n\\label{evkern}\n\\end{eqnarray}\nWe relegate the detailed discussion of the kernel $b(x,y;Q^2,Q_0^2)$,\nwhich is obtained by inverting the $Q^2$ dependence of ${\\cal M}_n$,\nto appendix C. In figure \\ref{fig_h2bllp}a we show the evolution of \n$\\overline{h}_L(x)$. Again we used $Q_0^2=0.4{\\rm GeV}^2$ and \n$Q^2=4.0{\\rm GeV}^2$.\n\nAs discussed in ref \\cite{Bal96} the merit of this \napproach is that to leading order in $N_C$ the knowledge of \n$h_L(x,Q^2)$ at one scale is sufficient to predict it at any arbitrary \nscale, which is not the case at finite $N_C$.\\footnote{As noted in \n\\cite{Bal96}, next to leading order corrections are estimated to go \nlike $O\\left(1\/N^2_c\\times{\\rm ln}(n)\/\\, n\\right)$ at large $n$.}\nThus $h_L(x,Q^2)$ obeys a generalized GLAP evolution equation. \nThis finally enables us (in much the same manner as was the case \nfor $g_2(x,Q^2)$ in \\cite{We97}) to compute the longitudinal chiral odd \nstructure function $h_L(x,Q^2)$ by combining the separately evolved \ntwist--2 and twist--3 components together. The result for \n$Q_0^2=0.4{\\rm GeV}^2$ and $Q^2=4.0{\\rm GeV}^2$ is shown in figure\n\\ref{fig_h2bllp}b. We recall that the only ingredients have been the leading \ntwist pieces of the chiral odd structure functions at the model \nscale $Q_0$.\\footnote{A feature of $h_L(x)$ compared with $g_2(x)$ \nis that as $h_L(x)$ does not mix with gluon distributions \nowing to its chiral-odd nature and its $Q^2$ evolution is given by \n(\\ref{mom}), (\\ref{adm}) even for the flavor singlet piece.} \n\n\\bigskip\n\\section{Discussion of the Numerical Results}\n\\bigskip\nIn this section we discuss the results of the chiral-odd structure\nfunctions calculated from eqs (\\ref{hT0})--(\\ref{hL1}) for constituent\nquark masses $m=400 {\\rm MeV}$ and $m=450 {\\rm MeV}$. In figure\n\\ref{fig_htud} we have shown the up and down quark contributions\nto the transverse chiral odd structure function of the proton. Figure\n\\ref{fig_htudpr} displays them boosted to the IMF.  We observe\nthat these structure functions are always smaller (in magnitude) than\nthe twist--2 polarized structure function $g_1$ with the same flavor\ncontent. This relation is also known from the bag model \\cite{Ja92}.\nSimilar to the confinement model calculation of Barone {\\it et al.}\n\\cite{Ba97} we find that $h_T^{(d)}(x)$ is negative at small $x$. In\ncontrast to $g_1^{(d)}(x)$, however, it might change sign although\nthe positive contribution appears to be small and diminishing with\nincreasing constituent quark mass.\n\nAs already indicated in the introduction the DIS processes which are \nsensitive to these distributions will provide access to the charge \nweighted combinations thereof. We will hence concentrate on this flavor \ncontent. In any event, as we will be discussing both, the proton and \nthe neutron chiral odd distributions, other flavor combinations can \nstraightforwardly be extracted by disentangling the isoscalar \nand isovector pieces in eq (\\ref{qsquare}). In \nconnection with the chiral--odd transverse nucleon structure function \nwe also calculate its zeroth moment which is referred to as the isoscalar \nand isovector nucleon tensor charges \\cite{Ja92},\n\\begin{eqnarray}\n\\Gamma^S_{T}(Q^2) &=& \\frac{18}{5} \\int_0^1\\, \n\\left[ dx\\ h_T^p\\left(x,Q^2\\right)\\\n+ h_T^n\\left(x,Q^2\\right)\\right]\n\\label{gtens} \\\\\n\\Gamma^V_{T}(Q^2) &=& 6 \\int_0^1\\, \\left[ dx\\ h_T^p\\left(x,Q^2\\right)\\ \n- h_T^n\\left(x,Q^2\\right)\\right] \n\\label{gtenv}\n\\end{eqnarray}\nat both the low scale, $Q_0^2=0.4 {\\rm GeV}^2$ and a scale commensurate\nwith experiment, $Q^2= 4 {\\rm GeV}^2$. Of course, for the neutron we \nhave to reverse the signs of the isovector pieces in eq (\\ref{hltnjl}).\nIn eqs (\\ref{gtens}) and (\\ref{gtenv}) the normalization factors are \ndue to the separation into isosinglet and isovector contributions, \n{\\it cf.} eq (\\ref{qsquare}). Note that due to \n$\\int_0^1 dz P_{qq}^h(z)\\ne0$ the tensor charge is not protected against \nlogarithmic corrections. Our results for the valence quark approximation \nare summarized in Table 1. For completeness we also add the vacuum \ncontribution to the tensor charges at the model scale $Q_0^2$. Their \nanalytic expressions are given in appendix D. Obviously this \nvacuum contribution is negligibly small. This is a strong\njustification of the valence quark approximation to the chiral \nodd structure functions. \n\\begin{table}[ht]\n\\caption{\\label{tab_1}\nNucleon tensor charges calculated from eqs (\\ref{gtens}) and\n(\\ref{gtenv}) as a function of the constituent quark mass $m$ in the\nNJL chiral--soliton model. The momentum scales are $Q_0^2=0.4{\\rm GeV}^2$\nand $Q^2=4.0{\\rm GeV}^2$. The numbers in parenthesis in the respective\nupper rows include the negligible contribution from the polarized quark\nvacuum. We compare with results from the Lattice \\protect\\cite{Ao97},\nQCD sum rules \\protect\\cite{He95}, the constituent quark model with\nGoldstone boson effects \\protect\\cite{Su97} and a quark soliton model \ncalculation \\protect\\cite{Ki96} including multiplicative $1\/N_C$ corrections \nviolating PCAC in the similar case of the axial vector current \n\\protect\\cite{Al93}. Finally the predictions from the confinement model \nof ref \\protect\\cite{Ba97} with the associated momentum scales \n(in ${\\rm GeV}^2$) are shown.}\n~ \\vskip0.1cm\n\\centerline{\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{c|lll|llll|ll}\n$m$ ({\\rm MeV})  &~~~350 &~~~400  &~~~450 \n& Lat. & ~SR & ~CQ & ~QS & ~$Q^2$ & CM\n\\\\ \\hline\n$\\Gamma^S_T(Q_0^2)$ & 0.80 (0.82)\n& 0.72 (0.76) & 0.67 (0.72)\n& 0.61 & 0.61 & 1.31 & 0.69 & 0.16 & 0.90 \\\\\n$\\Gamma^S_T(Q^2) $ & 0.73  & 0.65 & 0.61 \n&\\multicolumn{4}{c|}{no scale attributed} \n&25.0 & 0.72\\\\\n\\hline\n$\\Gamma^V_T(Q_0^2)$ & 0.88 (0.89) \n& 0.86 (0.87) & 0.86 (0.85) \n& 1.07 & 1.37 & 1.07 & 1.45 & 0.16 & 1.53 \\\\\n$\\Gamma^V_T(Q^2) $ & 0.80  & 0.78 & 0.77 \n&\\multicolumn{4}{c|}{no scale attributed}\n&25.0 & 1.22 \\\\\n\\end{tabular}}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\nA further justification comes from a recent\nstudy of the Gottfried sum rule within the same model \\cite{Wa98}.\nAlso in that case the contribution of the distorted quark vacuum\nto the relevant structure function turned out to be negligibly\nsmall.\n\nBesides justifying the valence quark approximation for the chiral \nodd distributions table \\ref{tab_1} contains the comparison to other \nmodel calculations of the nucleon tensor charges. We note that in obtaining \nthe isovector tensor charge $\\Gamma_T^V$ we have omitted contributions \nwhich are suppressed by $1\/N_C$ ({\\it cf.} appendix D). These contributions\narise when one adopts a non--symmetric ordering of the operators in \nthe space of the collective operators \\cite{Ki96}. The main reason for \ntaking the symmetric ordering is that in the case of the isovector axial \ncharge, $g_A$, any non--symmetric ordering of the collective operators \nleads to a sizable violation of PCAC unless the meson profile is not \nmodified \\cite{Al93}. These multiplicative $1\/N_C$ corrections \\cite{Da94} \nmay be the reason why our predictions for $\\Gamma_T^V$ are somewhat lower \nthan those of other models. In the case of the flavor singlet component, \nwhich does not have such corrections, our results compare nicely with \nother model calculations except for the constituent quark model of \nref \\cite{Su97}.\n\nIn figure \\ref{fig_htnp} we display the transverse chiral odd proton \n$h_T^{p}\\left(x,Q_0^2\\right)$ and neutron \n$h_T^{n}\\left(x,Q_0^2\\right)$ structure functions at the low momentum \nscale $Q_0^2$, while in figure \\ref{fig_hlnp} we do the same for the \ncorresponding chiral odd longitudinal structure functions \n$h_L^{p}\\left(x,Q_0^2\\right)$\nand $h_L^{n}\\left(x,Q_0^2\\right)$.\n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=hTp.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=hTn.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_htnp}\nThe valence quark approximation of the chiral--odd\nnucleon structure functions as a function of Bjorken--$x$.\nLeft panel: $h_{T}^{p}\\left(x ,Q_0^2\\right)$ for constituent\nquark masses $m=400 {\\rm MeV}$ (solid line) and\n$m=450 {\\rm MeV}$ (long--dashed line).\nRight panel: $h_{T}^{n}\\left(x,Q_0^2\\right)$.}\n\\end{figure}\n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=hLp.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=hLn.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_hlnp}\nThe valence quark approximation of the chiral--odd\nnucleon structure functions as a function of Bjorken--$x$.\nLeft panel: $h_{L}^{p}\\left(x ,Q_0^2\\right)$ for constituent\nquark masses $m=400 {\\rm MeV}$ (solid line)\nand $m=450 {\\rm MeV}$ (long--dashed line).\nRight panel: $h_{L}^{n}\\left(x,Q_0^2\\right)$.}\n\\end{figure}\nWe observe that the structure \nfunctions $h_{T}^{N}(x,Q_0^{2})$ and $h_{L}^{N}(x,Q_0^{2})$ are \nreasonably localized in the interval $0\\le x\\le1$. In particular, this\nis the case for the chiral odd structure functions of the neutron. \nNevertheless a projection as in eq (\\ref{fboost}) is required to \nimplement Lorentz covariance. In addition the computed structure functions \nexhibit a pronounced maximum at $x\\approx0.3$ which is smeared out when the \nconstituent quark mass $m$ increases. This can be understood as follows:\nIn our chiral soliton model the constituent mass serves as a coupling\nconstant of the quarks to the chiral field (see eqs (\\ref{bosact})\nand (\\ref{hamil})). The valence quark level becomes more strongly bound \nas the constituent quark mass increases. Hence the lower components of \nthe valence quark wave--function increase with $m$ and relativistic \neffects become more important. This effect results in the above \nmentioned broadening of the maximum.\n\nAs discussed above a sensible comparison with (eventually available)\ndata requires either to evolve the model results upward according to\nthe QCD renormalization group equations or to compare the model \nresults with a low momentum scale parameterization of the leading \ntwist pieces of the structure functions. The latter requires the \nknowledge of the structure functions at some scale in the whole \ninterval $x\\in[0,1[$. At present no such data are available for \nthe chiral odd structure functions $h_T(x)$ and $h_L(x)$. Therefore \nand in anticipation of results from {\\em RHIC} and or {\\em HERMES} we \napply leading order evolution procedures to evolve the structure \nfunction from the model scale, $Q_0^2=0.4 {\\rm GeV}^2$ to \n$Q^2=4{\\rm GeV}^2$. In Figs. \\ref{fig_ht2p}a and \\ref{fig_ht2p}b we \ndisplay the results of the two step process of projection and evolution \nfor the twist--2 transverse structure function, $h_T^{p}(x,Q^2)$ and \n$h_L^{p(2)}(x,Q^2)$, respectively for a constituent quark mass\nof $m=400 {\\rm MeV}$. \n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=hTpe.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=h2Lpe.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_ht2p}\nLeft panel: The evolution of $h_{T}^{p}\\left(x ,Q^{2}\\right)$\nfrom $Q^2_0=0.4 {\\rm GeV}^2$ (solid line) to $Q^2=4 {\\rm GeV}^2$ \n(long--dashed line) for the constituent quark mass $m=400 {\\rm MeV}$.\nRight panel: The evolution of the twist--2 contribution to the \nlongitudinal chiral odd structure function,\n$h_{L}^{p(2)}\\left(x ,Q^{2}\\right)$\nfrom $Q^2_0=0.4 {\\rm GeV}^2$ (solid line) to\n$Q^2=4 {\\rm GeV}^2$ (long--dashed line) for $m=400 {\\rm MeV}$.}\n\\end{figure}\nIn figure \\ref{fig_h2bllp} we present the evolution of \n$h_L^{p}(x)$ along with its decomposition into terms of the leading \ntwist--2 contribution, $2x \\int_{x}^1\\ dy h^p_T(y,Q^2)\/y^2$, and the \nremaining twist--3 piece, $\\overline{h}^p_L(x,Q^2)$.\n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=h2bLpe.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=hLpe.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_h2bllp}\nLeft panel (\\protect\\ref{fig_h2bllp}a): \nThe evolution of the twist--3 contribution to the longitudinal \nchiral odd structure function, $\\overline{h}_L^p(x,Q^2)$\nalong with the corresponding twist--2 piece,\n$h_{L}^{p(2)}\\left(x ,Q^{2}\\right)$.\nRight panel (\\protect\\ref{fig_h2bllp}b): The evolution\nof $h_{L}^{p}\\left(x ,Q^{2}\\right)=h_{L}^{p(2)}\\left(x ,Q^{2}\\right)\n+\\overline{h}_L^p(x,Q^2)$ from $Q^2_0=0.4 {\\rm GeV}^2$ (solid line) to\n$Q^2=4 {\\rm GeV}^2$ (long--dashed line) for the constituent\nquark mass $m=400 {\\rm MeV}$.}\n\\end{figure}\nAs in the case of the polarized structure\nfunction, $g_2(x,Q^2)$, the non--trivial twist--3\npiece arises as a result of the binding of the constituent \nquarks through the pion fields acting as effective non--perturbative \ngluonic modes. The twist--3 contribution is evolved according to the \nlarge $N_C$ scheme \\cite{Bal96,Ali91,Io95} outlined in the preceding\nsection (and in Appendix C).  Similarly in Figs. \\ref{fig_ht2n} and \n\\ref{fig_h2blln} we display the projection and \nevolution procedure to the twist--2 and 3 contribution to the neutron \nstructure functions, $h_L^{n(2)}(x,Q^2)$ and $\\overline{h}_L^n(x,Q^2)$,\nrespectively.\n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=hTne.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=h2Lne.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_ht2n}\nLeft panel:The evolution\nof $h_{T}^{n}\\left(x ,Q^{2}\\right)$\nfrom $Q^2_0=0.4 {\\rm GeV}^2$ (solid line) to\n$Q^2=4 {\\rm GeV}^2$ (long--dashed line) for the constituent\nquark mass $m=400 {\\rm MeV}$.\nRight panel: The evolution\nof the twist--2 contribution to the longitudinal chiral odd\nstructure function,\n$h_{L}^{n(2)}\\left(x ,Q^{2}\\right)$\nfrom $Q^2_0=0.4 {\\rm GeV}^2$ (solid line) to\n$Q^2=4 {\\rm GeV}^2$ (long--dashed line) for $m=400 {\\rm MeV}$.}\n\\end{figure}\n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=h2bLne.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=hLne.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_h2blln}\nLeft panel:  The evolution of the twist--3\ncontribution to the longitudinal chiral odd\nstructure function, $\\overline{h}_L^n(x,Q^2)$\nalong with the corresponding twist--2 piece,\n$h_{L}^{n(2)}\\left(x ,Q^{2}\\right)$.\nRight panel: The evolution\nof $h_{L}^{n}\\left(x ,Q^{2}\\right)=h_{L}^{n(2)}\\left(x ,Q^{2}\\right)\n+\\overline{h}_L^n(x,Q^2)$\nfrom $Q^2_0=0.4 {\\rm GeV}^2$ (solid line) to\n$Q^2=4 {\\rm GeV}^2$ (long--dashed line) for the constituent\nquark mass $m=400 {\\rm MeV}$.}\n\\end{figure}\n\nBesides the absolute magnitudes, the major difference between the chiral \nodd structure functions of the proton and the neutron is that the latter \ndrop to zero at a lower value of $x$. As can be observed from figure \n\\ref{fig_htnp} this is inherited from the model chiral odd structure \nfunction at the low momentum scale and can be linked to the smallness of \nthe down quark component of $h_T$, {\\it cf.} figure \\ref{fig_htud}. \nApparently the projection and evolution program does not alter this \npicture.\n\nWe would also like to compare our results from the NJL chiral soliton\nmodel to those obtained in other approaches. A MIT bag model calculation of \nthe isovector contribution $6(h_T^{p}-h_T^{n})$ has been presented \nin ref \\cite{Ja92}. In shape ({\\it e.g.} position of the maximum) that \nresult is quite similar to ours. However, the absolute value is a bit \nlarger in the MIT bag model. This reflects the fact that in the MIT bag \nmodel the isovector combinations of the axial and tensor charges turn \nout to be bigger than in the present model. Additionally, the QCD evolution \nof the MIT bag model prediction for $h_T$ has been studied in \nref \\cite{St93} utilizing the Peierls--Yoccoz projection as in ref \\cite{Sc91}. \nIn that case the maximum at $x\\approx0.5$ gets shifted to a value as low \nas  $x=0.2$. Also the structure function becomes rather broad at the large \nscale.  The fact that in that calculation the evolution effects are more \npronounced than in the present approach is caused by the significantly \nlower scale ($\\mu_{\\rm bag}=0.08{\\rm GeV}^2$) used in  ref \\cite{St93}.\nOn the other hand our results \nare quite different to those obtained in the QCD sum rule approach of \nref \\cite{Io95}. The sum rule approach essentially predicts $h_T$ to be \nconstant in the interval $0.3<x<0.8$. For small values of $x$ the authors \nof ref \\cite{Io95} assume a Regge behavior. In the (covariant) constituent \nquark model of Suzuki and Shijetamin \\cite{Su97} a result similar to ours \nis obtained when effects attributed to Goldstone bosons are included. \nOtherwise the maximum of their distribution is about 50\\% larger than in our\ncalculation. These authors also observe that in magnitude the down quark \ncomponent is significantly smaller than the up quark piece. The chiral \nchromo--dielectric model of Barone et al. \\cite{Ba97} predicts a similar \nshape for $h_T$ but their distribution $h_T^{(u)}$ is larger than the one \nin NJL chiral soliton model. This is also reflected by the sizable value \nfor the isovector tensor charge $\\Gamma_T^V(Q^2=25{\\rm GeV}^2)=1.22$ and \n$\\Gamma_T^V(Q^2=0.16{\\rm GeV}^2)=1.53$ in that approach. \nIn the Isgur Karl model (which has $h_1=g_1$) the maximal value of $h_1$\nis only about have as big as in our calculation, {\\it cf.} figure 1\nin ref \\cite{Sc97a}.\n\nFor completeness we also demonstrate in figs \\ref{fig_sofu} and \n\\ref{fig_sofd} that at the \nlow model scale $Q_0^2=0.4{\\rm GeV^2}$ Soffer's inequality \\cite{So95} is \nsatisfied. This inequality relates the nucleon chiral odd distribution\nfunctions to both the unpolarized $f_1^{(q)}(x,Q_0^2)$ and polarized \n$g_1^{(q)}(x,Q_0^2)$ structure functions\n\\begin{eqnarray}\nf_1^{(q)}(x,Q^2)+g_1^{(q)}(x,Q^2)\\ge 2\\ | h_T^{(q)}(x,Q^2) | \\ .\n\\label{sofineq}\n\\end{eqnarray}\nHere the superscript refers to the flavor combination which projects \nonto up and down quark quantum numbers ($q=u,d$). \nNote again, that this projection refers\nto the constituent quarks which contain some non--perturbative \ngluonic distributions. \n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=sofu.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=sofupr.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_sofu}\nLeft panel (\\protect\\ref{fig_sofu}a): \nThe Soffer inequality for the chiral even \ncombination $f_1^{(u)}(x,Q_0^2)+g_1^{(u)}(x,Q_0^2)$ (solid line)\nof the effective up--quark distributions\nand the chiral odd structure function $2\\ h_T^{(u)}(x,Q_0^2)$\n(long--dashed line) for a constituent quark mass of $m=400 {\\rm MeV}$,\ncalculated in the nucleon rest frame (RF). \nRight panel: (\\protect\\ref{fig_sofu}b) Same as \nfigure \\ref{fig_sofu}a calculated in the infinite momentum\nframe (IMF). The transformation prescription is given in\neq (\\protect\\ref{fboost}).}\n\\end{figure}\n\\begin{figure}[ht]\n\\centerline{\n\\epsfig{figure=sofd.ps,height=8.5cm,width=8.0cm,angle=270}\n\\hspace{-0.5cm}\n\\epsfig{figure=sofdpr.ps,height=8.5cm,width=8.0cm,angle=270}}\n\\caption{\\label{fig_sofd}\nSame as figure \\protect\\ref{fig_sofu} for the down quark combination.}\n\\end{figure}\nIn figure \\ref{fig_sofd} we display the down quark component \nof the inequality (\\ref{sofineq}). As this component of $h_T$ is almost\nnegligible the inequality is satisfied by the unpolarized structure\nfunction $f_1^{(d)}$ being larger in magnitude than the polarized one \n$g_1^{(d)}$.  For the constituent quark mass $m=450{\\rm MeV}$ we also find \nthat Soffer's inequality is satisfied at the model scale $Q_0$. The \nonly remarkable difference to $m=400{\\rm MeV}$ is that the \ndown quark component of $h_T^{(d)}$ does not possess nodes. This can already\nbe inferred from figures \\ref{fig_htud} and \\ref{fig_htudpr}.\n\nA more thorough model study of Soffer's inequality for scales other \nthan $Q_0^2$ which also contains the next--to--leading order \ncontributions in the evolution program is subject to further \ninvestigations \\cite{Ga98}. The next--to--leading order calculation \nfor $h_T$ \\cite{Sc97a} in the Isgur--Karl model indicates\nthat its scale dependence is slightly mitigated by the inclusion\nof next--to--leading order contributions.\n\n\\bigskip\n\\section{Conclusions}\n\\bigskip\n\nIn this paper we have presented the NJL chiral soliton model calculation \nof the leading twist parts of the transverse and longitudinal chiral odd \nstructure (distribution) functions of the nucleon. Data on these \ndistribution functions should eventually be available from DIS experiments \nin the fragmentation regions (in conjunction with fragmentation functions) \nor be extracted from Drell--Yan experiments. These structure \nfunctions serve to complete our picture of the spin distributions of \nthe nucleon. The most important feature of the present quark based model \nis that it is chirally invariant and that this symmetry is dynamically\nbroken. After bosonization the NJL model becomes an effective meson theory \nin which baryons emerge as self--consistent soliton solutions exactly the \nway as expected from large $N_C$ considerations in QCD. Chiral soliton \nmodels are particularly interesting in the context of the nucleon spin \nstructure as these models nicely explain the small contribution of \nthe quarks the total nucleon spin. \n\nIn the NJL chiral soliton model there are two contributions to nucleon \nproperties.  First, there is the contribution of the distinct valence \nquark level. This is the lowest level in the quark spectrum and bound in \nthe background of the chiral soliton.  Second, there is the part which \nis associated to the polarization (by the soliton) of the vacuum. For \nmany static nucleon properties the latter contribution is quite small,\nin particular for those which are related to the axial (spin) properties \nof the nucleon. This is a strong indication that the vacuum contribution\nto the chiral odd structure functions is negligible as well. Hence \nit seems more important to include substantial $1\/N_C$ corrections\nto the valence quark contribution. These corrections come about when \nprojecting the soliton onto states with good spin and isospin, {\\it i.e.} \nproton and neutron. Inclusion of these $1\/N_C$ corrections together with \na consistently regularized treatment of the vacuum polarization is \ntechnically rather involved and beyond the scope of the present paper.\nThe numerical results for the tensor charge indicate that the \nvacuum contributions are in fact negligibly small. \n\nWhen the model structure functions are computed one immediately \nrecognizes that they have improper support due to the breaking \nof translational invariance by the background soliton, {\\it i.e.} \nthe structure functions do not exactly vanish for $x>1$. This \ncan be cured by Lorentz boosting to the infinite momentum frame which \nis particularly suited for DIS processes. Although the un--boosted\nstructure functions are negligibly small at $x>1$ the transformation \nto this frame is essential and has sizable effects on the structure\nfunctions at moderate $x$. However, the most important \nissue when comparing the model predictions to (not yet available)\nexperimental data is the observation that the model represents \nQCD at a low momentum scale $Q_0^2$. {\\it A priori} this scale \nrepresents an additional parameter to the model calculation \nwhich, for consistency, has to be smaller than the ultraviolet \ncut--off of the model $\\Lambda^2=0.56{\\rm GeV}^2$. For the model \nunder consideration we previously fixed $Q_0^2$ when studying the \nunpolarized structure functions and found $Q_0^2=0.4{\\rm GeV}^2$. \nThe important logarithmic corrections\nto the model structure functions are then obtained within a generalized\nGLAP evolution program. In this context we have restricted ourselves\nto a leading order (in $\\alpha_{\\rm QCD}$) calculation because \nthe anomalous dimensions, which govern the QCD evolution, for the \ntwist--3 piece of the longitudinal part of the chiral odd structure \nare only known to that order. As the full evolution to the longitudinal\nstructure function involves both twist--2 and twist--3 pieces this \nrestriction is consistent. We have seen that the QCD evolution of the \nchiral odd structure function leads to sizable enhancements at low $x$, \n{\\it i.e.} in the region $0.01\\le x\\le 0.10$. In this respect the present \nsituation is similar to that for the polarized structure functions.\nA difference to the polarized structure function is that the lowest moment\nis not protected against logarithmic corrections, even at leading order\nin $\\alpha_{\\rm QCD}$. For the nucleon tensor charge we thus find a \nreduction of about 10\\% upon evolution to $Q^2=4.0{\\rm GeV}^2$.\nWe have also compared the neutron and proton chiral odd structure \nfunctions. This has been achieved by the inclusion of the $1\/N_C$\ncranking corrections. In absolute value the proton structure functions\nare about twice as large as those of the neutron. Furthermore the \nneutron structure functions drop to zero at a lower value of $x$.\nThese two effects can be linked to the down quark component of the \ntransverse nucleon chiral odd distribution functions being significantly \nsmaller than the component with up--quark quantum numbers. We have also \nobserved that neither of these features is effected by the evolution \nprogram.\n\n\\bigskip\n\\section*{Acknowledgements}\n\\bigskip\nThis work has been supported in part by the\nDeutsche Forschungsgemeinschaft (DFG) under contract Re 856\/2-3,\nand by the U.S. Department of Energy (D.O.E.) under\ncontract DE--FE--02--95ER40923.\nOne of us (LG) is grateful to G. R. Goldstein for helpful comments\nand to K. A. Milton for encouragement and support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFinding the spectrum of a transition matrix is a very popular subject in graph theory and Markov chain theory. There are only a few techniques known to describe the exact spectrum of a Markov chain, and they usually work under very specific conditions, such as when the Markov chain is a random walk on a finite group, generated only by a conjugacy class \\cite{DiaSha}. Most well-known examples where a transition matrix has been diagonalized usually rely on combination of advanced representation theory, Fourier analysis, and combinatorial arguments \\cite{DSaliola},  \\cite{Hough}, \\cite{Star}, \\cite{hyp}, \\cite{hyp2}, \\cite{hyp3}, \\cite{Brown}, \\cite{Pike}. But even in most of these cases, there is no description of what an eigenbasis of the transition matrix would look like, which in general is needed as well in order to understand the transition matrix. \n\nIn this work, we present the full spectrum of the simple random walk on complete, finite $d-$ary trees and a corresponding eigenbasis, and we use this information to produce a lower bound for the interchange process on the trees, which we conjecture is sharp. Consider the complete, $d-$ary tree $\\mathcal T_h$ of height $h$, which has $n = 1+d+\\dots+d^{h} = \\frac{d^{h+1}-1}{d-1}$ vertices. We study the simple random walk on $\\mathcal T_h$ ,whose transition matrix is denoted by $Q_h$, according to which when we are at the root we stay fixed with probability $1\/(d+1)$, or we move to a child with probability $1\/(d+1)$ each. When we are at a leaf, we stay fixed with probability $d\/(d+1)$ otherwise we move to the unique parent with probability $1\/(d+1)$. For any other node, we choose one of the neighbors with probability $1\/(d+1)$. \n\nThis is a well studied Markov chain. Aldous \\cite{Aldous} proved that the cover time is asymptotic to $2 h^2 d^{h+1} \\log h\/(h-1)$. The order of the spectral gap and the mixing time of this Markov chain have been widely known for a long time. In fact, the random walk on $\\mathcal T_h$ is one of the most famous examples of a random walk not exhibiting cutoff (see Example 18.6 of \\cite{AMPY4}). However, finding the exact value of the spectral gap has been an open question for years, let alone finding the entire spectrum and an eigenbasis of the transition matrix $Q_h$.\n\n\nWe denote by $\\rho$ the root of $\\mathcal T_h$, by $V(\\mathcal T_h)$ the vertex set of $\\mathcal T_h$, and by $E(\\mathcal T_h)$ the set of edges of $\\mathcal T_h$. Let $\\ell: V(\\mathcal T_h) \\rightarrow [0,\\ldots, h]$ denote the distance from the root. For every node $v$, let $\\mathcal T^v$ be the complete $d-$ary subtree rooted at $v$, namely consisting of $v$ and all vertices of $V( \\mathcal T_h)$ that are descendants of $v$ in $\\mathcal T_h$. Let $\\mathcal T_i^v$ be the complete $d-$ary subtree of $\\mathcal T^v$ rooted at the $i-$th child of $v$.\n\nThe next theorem includes the first result of this paper, presenting the eigenvalues and an eigenbasis of $Q_h$.\n\\begin{theorem}\\label{thm:spectrum} \n\t\\begin{enumerate}  [label = (\\alph*)]\n\t\t\\item $Q_h$ is diagonalizable with $1$ being an eigenvalue with multiplicity 1. Every other eigenvalue $\\lambda\\neq 1$ of $Q_h$ is of the form\n\t\t\\begin{equation}\\label{eq:lambda:x:thm}\n\t\t\\lambda = \\frac{d}{d+1}\\left (x+ \\frac{1}{xd}\\right ),\n\t\t\\end{equation}\n\t\twhere $x\\neq \\pm \\frac{1}{\\sqrt d}$ is a solution of one of the following $h+1$ equations:\n\t\t\\begin{equation} \\label{eq:x:sym:thm}\n\t\td^{h+1}x^{2h+2} = 1\n\t\t\\end{equation}\n\t\tand \n\t\t\\begin{equation}  \\label{eq:x:antisym}\n\t\td^{k+2} x^{2k+4}- d^{k+2} x^{2k+3}+ dx -1 = 0,\\quad \\text{for some } 0 \\leq k \\leq h-1.\n\t\t\\end{equation}\n\t\t\n\t\tReversely, each solution $x\\neq \\pm \\frac{1}{\\sqrt d}$ of these equations corresponds to an eigenvalue $\\lambda$ according to \\eqref{eq:lambda:x:thm}. \n\t\tFor each of these equations, if $x$ is a solution then so is $\\frac{1}{xd}$. Both $x$ and $\\frac{1}{xd}$ correspond to the same $\\lambda$. The correspondence between $x$ and $\\lambda$ is $2$-to-$1$.\n\t\t\n\t\t\\item \\label{thm:spectrum:eigenvector} For each solution $x\\neq \\pm \\frac{1}{\\sqrt d}$ of \\eqref{eq:x:sym:thm}, an eigenvector $f_{\\lambda}$ with respect to $\\lambda$ is given by the formula \n\t\t\\begin{equation}\\label{eq:thm:spectrum:sym}\n\t\tf_{\\lambda}(v) = \\frac{dx^{2}-x}{dx^{2}-1}  x^i +  \\frac{x-1}{dx^{2}-1}\\frac{1}{d^{i} x^{i}} \\quad\\text{for every $v$ with $\\ell(v)=i$, $0\\le i\\le h$}.\n\t\t\\end{equation}\n\t\t\n\t\tFor each $0 \\leq k \\leq h-1$, each solution $x\\neq \\pm \\frac{1}{\\sqrt d}$ of \\eqref{eq:x:antisym}, each $v \\in V(\\mathcal T_h)$ such that $\\ell(v)= h-1-k$, and each $j\\in [1, \\dots, d-1]$, an eigenvector $f_{v, j, j+1}$ with respect to $\\lambda$ is given by the formula \n\t\t\\begin{equation}\\label{eq:thm:spectrum:antisym}\n\t\tf_{ v, j, j+1}(w) = \n\t\t\\begin{cases}\n\t\t& \\frac{dx^{i+2}}{dx^{2}-1} - \\frac{1}{(dx^{2}-1)d^{i} x^{i}} \\quad \\mbox{ for } w\\in \\mathcal T^v_j,\\mbox{ where } i= \\ell(w) -h+k,\\\\\n\t\t& -\\frac{dx^{i+2}}{dx^{2}-1} + \\frac{1}{(dx^{2}-1)d^{i }x^{i}} \\quad \\mbox{ for } w\\in \\mathcal T^v_{j+1}, \\mbox{ where } i= \\ell(w) -h+k,\\\\\n\t\t& 0, \\mbox{ otherwise.} \n\t\t\\end{cases}\n\t\t\\end{equation}\n\t\t\\item The collection of these eigenvectors together with the all-1 vector form an eigenbasis of $Q_h$.\n\t\\end{enumerate}\n\n\\end{theorem}\n\n\n\nIn Lemma \\ref{lm:sym:antisym} and Figure \\ref{fig:sym:antisym}, we describe and illustrate the eigenvectors in more detail.\n\nThe idea behind the proof is to consider appropriate projections of the random walk. For example, let $X_t$ be the state of the random walk at time $t$ and let $Y_t$ be the distance of $X_t$ from the root. Then $Y_t$ is a Markov chain on $[0,h],$ whose eigenvalues are also eigenvalues of $Q_h$. Also, the eigenvectors of $Y_t$ lift to give the eigenvectors presented in \\eqref{eq:thm:spectrum:sym}. This computation is not going to give us the full spectrum, however. \n\nFor example, in the case of the binary tree, another type of projection to consider is as follows. We consider the process $W_t,$ which is equal to $-Y_t$ if $X_t \\in \\mathcal T^{\\rho}_1$ and equal to $Y_t$ otherwise. The second largest eigenvalue can be derived by this new process, while the eigenvectors are of the form presented in \\eqref{eq:thm:spectrum:antisym}. The reason why this is the right process to study is hidden in the mixing time of the random walk on $\\mathcal T_h$. A coupling argument roughly says that we have to wait until $X_t$ reaches the root $\\rho$. The first time that $X_t$ hits $\\rho$ is captured by $W_t$, since $W_t$ is a Markov chain on $[-h,h]$, where the bias is towards the ends and away from zero. The projections that we consider form birth and death processes, whose mixing properties have been thoroughly studied by Ding, Lubetzky, and Peres \\cite{DLP}. To capture the entire spectrum, our method is to find in each eigenspace a well-structured eigenvector, which occurs by considering an appropriate projection.\n\nOur analysis has immediate applications to card shuffling, namely the interchange process on $\\mathcal T_h$, and to the exclusion process. We enumerate the nodes in $V (\\mathcal T _h)$ and we assign cards to the nodes. At time zero, card $i$ is assigned to node $i$. The interchange process on $\\mathcal T_h$ chooses an edge uniformly at random and then flips a fair coin. If heads, interchange the cards on the ends of $e$; if tails, stay fixed. A configuration of the deck corresponds to an element of the symmetric group.\n\nLet $g \\in S_n$. Let $P$ be the transition matrix of the interchange process on the complete, finite $d-$ary tree $\\mathcal T_h$ and let $P^t_{id}(g)$ be the probability that we are at $g$ after $t$ steps, given that we start at the identity. We define the total variation distance between $P^t_{id}$ and the uniform measure $U$ to be \n\\begin{equation}\n\td(t)= \\frac{1}{2} \\sum_{x \\in S_n} \\left \\vert P^t_{id}(x) -\\frac{1}{n!}\\right \\vert. \\nonumber\n\\end{equation}\n\nA celebrated result concerning the interchange process was the proof of Aldous conjecture \\cite[Theorem 1.1]{CLR}, which states that the spectral gap of P is the same as the spectral gap of the Markov chain that the ace of spades performs. Adjusting our computations, we now get the following result.\n\\begin{theorem}\\label{thm:lowerbound}\n\tFor the interchange process on the complete $d$-ary tree of depth $h$, we have that \n\t\\begin{itemize}\n\t\t\\item[(a)]  The spectral gap of the transition matrix is $\\frac{(d-1)^{2}}{2(n-1) d^{h+1}} + O \\left (\\frac{\\log_{d} n}{n^{3}}\\right )$, \n\t\t\\item[(b)] And if $t=\\frac{1}{d-1}n^{2}\\log n- \\frac{1}{d-1}n^2 \\log \\left( \\frac{1}{\\varepsilon} \\right) + O\\left (  n^{2}\\right ) $, then \n\t\t$$d(t) \\geq1-  \\varepsilon,$$\n\t\twhere $\\varepsilon$ is any positive constant. \n\t\\end{itemize}\n\\end{theorem}\nThis is already much faster than the interchange process on the path, another card shuffle that uses $n-1$ transpositions, which Lacoin   \\cite{Lacoin} recently proved exhibits cutoff at $\\frac{1}{2\\pi^2} n^3 \\log n$. We conjecture that the lower bound in part $(b)$ of Theorem \\ref{thm:lowerbound} is sharp and that the interchange process on $\\mathcal T_h$ exhibits cutoff at $\\frac{1}{d-1} n^{2}\\log n$.\n\n\nWe can get lower bounds for the mixing time of another well studied process, the exclusion process on the complete $d$-ary tree. This is a famous interacting particle system process, according to which at time zero, $k \\leq n\/2$ nodes of the tree are occupied by indistinguishable particles. At time $t$, we pick an edge uniformly at random and we flip the two ends. Similar computations to the ones of the proof of Theorem \\ref{thm:lowerbound} give that if $t=\\frac{1}{d-1}n^{2}\\log k- \\frac{1}{d-1}n^2 \\log \\left( \\frac{1}{\\varepsilon} \\right) + o\\left (  n^{2} \\log k\\right ) $, then \n$$d(t) \\geq1-  \\varepsilon,$$\nwhere $\\varepsilon>0$ is a constant. Combining Oliveira's result \\cite{OL} with Theorem \\ref{thm:lowerbound} $(b)$,  we get that the order of the mixing time of the exclusion process on the complete $d-$ary tree is $n^2 \\log k$. \n\nAs potential open questions, we suggest trying to find the spectrum or just the exact value of the spectral gap for the simple random on finite Galton-Watson trees or for the frog model as presented in \\cite{Jon}.\n \n  \n\\section{The spectrum of $Q_h$}\nThis section is devoted to the proof of Theorem \\ref{thm:spectrum}.\n\n\nLet $\\lambda$ be an eigenvalue of $Q_h$ and let $E(\\lambda)$ be the corresponding eigenvalue. We first show that there exists an eigenvector in $E(\\lambda)$ that has the form described in Theorem \\ref{thm:spectrum} \\ref{thm:spectrum:eigenvector}.\n\\begin{lemma}\\label{lm:sym:antisym}\nThe eigenspace $E(\\lambda)$ contains an eigenvector $f$ that has one of the following forms:\n\\begin{enumerate}\n\\item[(a)] [Completely symmetric] $f(v)=f(w)$ for every $v,w \\in V(\\mathcal T_h) $ such that $\\ell(v)=\\ell(w)$. In this case we will call $f$ completely symmetric for $\\mathcal T_h$;\n\\item[(b)] [Pseudo anti-symmetric] There is a node $v$ and $i,j \\in \\{ 1,\\ldots, d\\}$ such that  $f(w)=0$ for every $w \\notin V(\\mathcal T_i^v\\cup \\mathcal T_j^v)$, $f \\vert_{\\mathcal T_i^v}$ and $f \\vert_{\\mathcal T_j^v}$ are completely symmetric, and $f \\vert_{\\mathcal T_i^v}=-f \\vert_{\\mathcal T_j^v}$. We call such $f$ pseudo anti-symmetric.\n\\end{enumerate}\n\\end{lemma}\nThe following illustrations explain what the described eigenvectors look like for binary trees.\n\n\\tikzset{every tree node\/.style={minimum width=2em,draw,circle},\n         blank\/.style={draw=none},\n         edge from parent\/.style=\n         {draw,edge from parent path={(\\tikzparentnode) -- (\\tikzchildnode)}},\n         level distance=1.5cm}\n \n\n\\begin{figure}[H] \n\t\\centering\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\\Tree\n[.$y_0$     \n[.$y_1$ \n[.$y_2$\n[.$y_3$ ]\n[.$y_3$ ]\n]\n[.$y_2$\n[.$y_3$ ]\n[.$y_3$ ]\n]]\n[.$y_1$\n[.$y_2$ \n[.$y_3$ ]\n[.$y_3$ ]]\n[.$y_2$ \n[.$y_3$ ]\n[.$y_3$ ]]]\n]\n\t\\end{minipage}%\n\t\\begin{minipage}{.5\\textwidth}\n\t\t\\centering\n\\Tree\n[.0     \n[.0 \n[.$y_0$ \n[.$y_1$ ]\n[.$y_1$ ]]\n[.-$y_0$ \n[.-$y_1$ ]\n[.-$y_1$ ]]]\n[.0 \n[.0 \n[.0 ]\n[.0 ]]\n[.0 \n[.0 ]\n[.0 ]\n]]\n]\n\t\\end{minipage}\t\n\\caption{Completely symmetric eigenvectors (left) and Pseudo anti-symmetric eigenvectors (right)}\n\\label{fig:sym:antisym}\n\t\\end{figure}\n\n\n\n\\begin{proof}\nAssume that $E(\\lambda)$ does not contain a completely symmetric eigenvector. Let $f$ be a nonzero element of $E(\\lambda)$. Since $f$ is not completely symmetric, there exist vertices of the same level at which $f$ takes different values. Let $v$ be a vertex with the largest $l(v)$ such that there are at least two of its children, say the $i$-th and $j$-th children, at which $f$ has different values. For example, if there are two leaves $u$ and $w$ at which $f(u)\\neq f(w)$ that have the same parent $v'$ then we simply take $v$ to be $v'$. \n\nBy the choice of $v$, $f\\vert _{\\mathcal T_k^v}$ is completely symmetric for all $k\\in [d]$. Indeed, let $u$ be the $k$-th child of  $v$. We have $\\mathcal T_k^v = \\mathcal T^{u}$. By the choice of $v$, $f$ takes the same value at all children of $u$. Let $u_1, u_2$ be two arbitrary children of $u$. Again by the choice of $v$, $f$ takes the same value, denoted by $f_1$, at all children of $u_1$, and the same value, denoted by $f_2$, at all children of $u_2$. Since $f$ is an eigenvector of $Q_h$, \n\\begin{equation}\\label{key}\n\\lambda f(u_1) = \\frac{d}{d+1} f_1+\\frac{1}{d+1} f(u) \\quad \\text{and}\\quad \\lambda f(u_2) = \\frac{d}{d+1} f_2+\\frac{1}{d+1} f(u). \\nonumber\n\\end{equation}\nSince $f(u_1) = f(u_2)$, $f_1 = f_2$. Thus, $f$ takes the same value at all grandchildren of $u$. Repeating this argument shows that  $f\\vert _{\\mathcal T^u}$ is completely symmetric.\n\nConsider the vector $g$ obtained from $f$ by switching its values on $\\mathcal T^{v}_{i}$ and $\\mathcal T^{v}_{j}$. More specifically, $g\\vert _{\\mathcal T^{v}_{i}} = f\\vert _{\\mathcal T^{v}_{j}}$,  $g\\vert _{\\mathcal T^{v}_{j}} = f\\vert _{\\mathcal T^{v}_{i}}$, and $g = f$ elsewhere.\n \n By the symmetry of the tree and the matrix $Q_h$, $g$ also belongs to $E(\\lambda)$. So is $f-g$, which we denote by $h$. Observe that $h$ is an eigenvector that is 0 everywhere except on $\\mathcal T^{v}_{i} \\cup \\mathcal T^{v}_{j}$ and $h \\vert_{\\mathcal T_i^v}=f\\vert_{\\mathcal T_i^v} - f\\vert_{\\mathcal T_j^v}=-h \\vert_{\\mathcal T_j^v}$. Moreover, $h$ is completely symmetric when restricted to $\\mathcal T^{v}_{i}$ and $ \\mathcal T^{v}_{j}$ because both $f$ and $g$ are, as seen above. Thus, $h \\in E(\\lambda)$ and is pseudo anti-symmetric.\n\\end{proof}\n\n\n\n\n\\subsection{Completely symmetric eigenvectors}\\label{subsection:sym}\n\nIn this section, we describe completely symmetric eigenvectors. We shall show that the completely symmetric eigenvectors of $Q_h$ are given by the formula \\eqref{eq:thm:spectrum:sym} and correspond to $\\lambda$ and $x$ satisfying \\eqref{eq:lambda:x:thm} and \\eqref{eq:x:sym:thm} as in Theorem \\ref{thm:spectrum}.\n\n\nSince a completely symmetric eigenvector of $Q_h$ has the same value at every node of the same level (see Figure \\ref{fig:1}), we can project it onto the path $[0, h]$ and obtain an eigenvector of the corresponding random walk on the path.\n \n\\begin{figure}[H]\\label{figure:sym:3}\n\t\\centering\n\\begin{tikzpicture}\n\\Tree\n[.$y_0$     \n    [.$y_{1}$ \n    [.$y_{2}$ \n     [.$\\ldots$ ]\n    [.$\\ldots$ ]]\n    [.$y_{2}$ \n     [.$\\ldots$ ]\n    [.$\\ldots$ ]]]\n    [.$y_{1}$ \n      [.$y_2$ \n      [.$\\ldots$ ]\n      [.$\\ldots$ ]]\n    [.$y_2$ \n    [.$\\ldots$ ]\n      [.$\\ldots$ ]\n    ]]\n] \n\\end{tikzpicture}\n\\caption{Completely symmetric eigenvectors}\n\\label{fig:1}\n\\end{figure}\n\n \n\\begin{lemma} \\label{lm:sym}\n\tThere are exactly $h+1$ linearly independent completely symmetric eigenvectors of $Q_h$.\n\\end{lemma}\n\\begin{proof} Each symmetric eigenvector of $Q_h$ corresponds one-to-one to an eigenvector of the following projection onto the path $[0, h]$ with transition matrix $R_h$:\n\t\\begin{itemize}\n\t\t\\item $R_h(0, 1) = \\frac{d}{d+1},  R_h(0, 0) = \\frac{1}{d+1}$,\n\t\t\\item $R_h(l, l-1) = \\frac{1}{d+1}$, $R_h(l, l+1) = \\frac{d}{d+1}$ for all $1\\le l\\le h-1$,\n\t\t\\item $R_h(h, h-1) = \\frac{1}{d+1}$, $R_h(h, h) = \\frac{d}{d+1}$,\n\t\\end{itemize}\n\n\tSince $R_h$ is a reversible transition matrix with stationary distribution $\\pi := [1, d, d^{2}, \\dots, d^{h}]$, the matrix $A:= D^{1\/2} R_h D^{-1\/2}$ is symmetric where $D$ is the diagonal matrix with $D(x, x)= \\pi(x)$. Therefore, $A$ is diagonalizable and so is $R_h$. In other words, $R_h$ has $h+1$ independent real eigenvectors. This implies that $Q_h$ has $h+1$ linearly independent completely symmetric eigenvectors.\n\\end{proof}\n\n\\begin{lemma}\\label{lm:sym:detail}\n\tThe matrix $R_n$ has 1 as an eigenvalue with multiplicity 1. Each of the remaining $h$ eigenvalues $\\lambda\\neq 1$ of $R_n$ is of the form \n\t$$\\lambda = \\frac{d}{d+1}\\left (x+ \\frac{1}{xd}\\right )$$\n\twhere $x \\neq \\pm \\frac{1}{\\sqrt{d}}$ is a non-real solution of the equation\n\t\\begin{equation} \n\td^{h+1}x^{2h+2} = 1.\\nonumber\n\t\\end{equation}\n\tThis equation has exactly $2h$ such solutions. If $x$ is a solution, so is $\\frac{1}{xd}$. There is a 2-to-1 correspondence between $x$ and $\\lambda$. An eigenvector $y = (y_0, y_1, \\dots, y_h)$ of $R_h$ with respect to $\\lambda$ is given by  \n\t\\begin{equation} \n\ty_i = \\frac{dx^{2}-x}{dx^{2}-1}  x^i +  \\frac{x-1}{dx^{2}-1}\\frac{1}{d^{i} x^{i}} \\quad\\text{for every $0\\le i\\le h$}.\\nonumber\n\t\\end{equation}\n\tThe vector $f:\\mathcal T_h\\to \\mathbb R$ that takes value $y_i$ at all nodes of depth $i$ is an eigenvector of $Q_h$ with respect to $\\lambda$.\n\\end{lemma}\n\\begin{proof}\n\tLet $\\lambda$ be an eigenvalue of $R_h$ and $y = (y_0, y_1, \\dots, y_h)$ be an eigenvector corresponding to $\\lambda$. We have\n\t\\begin{enumerate}[label=(R\\arabic{*}), ref=R\\arabic{*}]\n\t\t\\item\\label{eq:R:0} $\\frac{d}{d+1} y_{1} +\\frac{1}{d+1} y_{0} = \\lambda y_0$,\n\t\t\\item\\label{eq:R:i}  $\\frac{1}{d+1} y_{i-1} +\\frac{d}{d+1} y_{i+1} = \\lambda y_i$ for all $1\\le i\\le h-1$,\n\t\t\\item\\label{eq:R:h} $\\frac{1}{d+1} y_{h-1} +\\frac{d}{d+1} y_{h} = \\lambda y_h$.\n\t\\end{enumerate}\n\nSince $y$ is not the zero vector, the above equations imply that $y_0\\neq 0$. Without loss of generality, we assume $y_0=1$. \n\t\n\tLet $x_1, x_2$ be the solutions to the characteristic equation of \\eqref{eq:R:i}: \n\t$$\\frac{1}{d+1} - \\lambda x + \\frac{d}{d+1}x^{2} = 0$$\n\tor equivalently\n\t\\begin{equation}\\label{eq:x:lambda:1}\n\td x^{2} - (d+1)\\lambda x+1 = 0.\n\t\\end{equation}\n\t\n\tBy \\eqref{eq:x:lambda:1}, we have\n\t$$x_1 x_2 = \\frac{1}{d}$$\n\tand \n\t\\begin{equation} \n\t\\lambda = \\frac{d}{d+1}(x_1+ x_2) = \\frac{d}{d+1}\\left (x_1+ \\frac{1}{x_1 d}\\right ) .\\label{eq:lambda:x}\n\t\\end{equation}\n\t\n\t\n\t\n\tIf $x_1\\neq x_2$ then we can write $y_0 = \\alpha_1 - \\alpha_2$, $y_1 = \\alpha_1 x_1 -\\alpha_2 x_2$ for some $\\alpha_1, \\alpha_2$. We show that for all $0\\le i\\le h$,\n\t\\begin{equation}\\label{eq:recurrent:y:1}\n\ty_i = \\alpha_1 x_1^{i} - \\alpha_2 x_2^{i}.\n\t\\end{equation}\n\tIndeed, assuming that \\eqref{eq:recurrent:y:1} holds for $y_0, \\dots, y_i$ for some $1\\le i\\le h-1$ then by \\eqref{eq:x:lambda:1},\n\t\\begin{eqnarray}\n\t\\lambda y_i - \\frac{1}{d+1} y_{i-1} = \\alpha_1 x_1^{i-1}\\left (\\lambda x_1 - \\frac{1}{d+1}\\right )-\\alpha_2 x_2^{i-1}\\left (\\lambda x_2 - \\frac{1}{d+1}\\right ) = \\frac{d}{d+1} \\left (\\alpha_1 x_1^{i+1}-  \\alpha_2 x_2^{i+1}\\right ).\\nonumber\n\t\\end{eqnarray}\n\tThus, by \\eqref{eq:R:i}, \n\t\\begin{equation}\\label{key}\n\t\\frac{d}{d+1} y_{i+1} = \\frac{d}{d+1} \\alpha_1 x_1^{i+1}- \\frac{d}{d+1} \\alpha_2 x_2^{i+1}\\nonumber\n\t\\end{equation}\n\tand so\n\t\\begin{equation}\\label{key}\n\ty_{i+1} = \\alpha_1 x_1^{i+1}- \\alpha_2 x_2^{i+1}.\\nonumber\n\t\\end{equation}\n\tThus, \\eqref{eq:recurrent:y:1} also holds for $y_{i+1}$ and hence, for all $y_0, \\dots, y_h$. \n\t\n\tSimilarly, by \\eqref{eq:R:h}, we get\n\t\\begin{eqnarray}\n\t\\frac{d}{d+1} y_h &=&\\lambda y_h - \\frac{1}{d+1} y_{h-1} = \\alpha_1 x_1^{h-1}\\left (\\lambda x_1 - \\frac{1}{d+1}\\right )-\\alpha_2 x_2^{h-1}\\left (\\lambda x_2 - \\frac{1}{d+1}\\right )\\nonumber\\\\\n\t& =& \\frac{d}{d+1} \\left (\\alpha_1 x_1^{h+1}-  \\alpha_2 x_2^{h+1}\\right ).\\nonumber\n\t\\end{eqnarray}\n\tThus, \n\t\\begin{equation}\\label{eq:R:h:1}\n\t\\alpha_1 x_1^{h+1}-  \\alpha_2 x_2^{h+1} = \\alpha_1 x_1^{h}-  \\alpha_2 x_2^{h}\n\t\\end{equation}\n\tas they are both equal to $y_h$.\n\t\n\t\n\tBy \\eqref{eq:recurrent:y:1}, \\eqref{eq:R:0} becomes\n\t\\begin{equation}\\label{eq:R:0:1}\n\td(\\alpha_1x_1 - \\alpha_2 x_2) = \\left (xd+\\frac{1}{x} - 1\\right ) (\\alpha_1 - \\alpha_2).\n\t\\end{equation}\n\t\n\t\n\t\n\tFor simplicity, we write $\\alpha = \\alpha_1$ and $x = x_1$. By \\eqref{eq:recurrent:y:1} for $i=0$, we get\n\t$$\\alpha_2 = \\alpha-1.$$\n\t\n\n\tEquations \\eqref{eq:R:0:1} becomes\n\t\\begin{equation} \n\td\\alpha x -  \\frac{\\alpha-1}{x} =  dx+\\frac{1}{x} - 1\\nonumber\n\t\\end{equation}\n\twhich gives\n\t\\begin{equation}\\label{eq:R:0:2}\n\t\\alpha_1 = \\alpha = \\frac{dx^{2}-x}{dx^{2}-1} \\quad\\text{and}\\quad \\alpha_2 = \\alpha-1 = \\frac{1-x}{dx^{2}-1}. \n\t\\end{equation}\n\t\n \n\t\n\tPlugging \\eqref{eq:R:0:2} into \\eqref{eq:R:h:1} and taking into account $x_2 = \\frac{1}{xd}$, we get\n\t\\begin{equation}\\label{key}\n\t(dx-1)(x-1)(d^{h+1}x^{2h+2}-1) = 0.\\nonumber\n\t\\end{equation}\n\t\n\tIf $x = 1$ then $\\alpha_2 = \\alpha-1 = 0$ by \\eqref{eq:R:0:2}. And so, $y = \\alpha(1, \\dots, 1)$ which is an eigenvector of the eigenvalue 1. Since $\\lambda\\neq 1$, $x\\neq 1$. If $x=\\frac{1}{d}$ then $x_2 = \\frac{1}{xd} = 1$. By the symmetry of $x_1$ and $x_2$, this also corresponds to $\\lambda=1$ which is not the case. \n\t\n\tThus, $x$ satisfies\n\t\\begin{equation} \n\td^{h+1}x^{2h+2}-1=0.\\nonumber\n\t\\end{equation}\n\t \n\t\n\tThis equation has $2h$ non-real solutions and 2 real solutions $\\pm \\frac{1}{\\sqrt{d}}$. For each non-real solution $x_1$, observe that $x_2:=\\frac{1}{dx_1}$ is also a non-real solution. Note that $x_1\\neq x_2$ and by setting $\\lambda$ and $y$ as in \\eqref{eq:lambda:x} and \\eqref{eq:recurrent:y:1} with $\\alpha_1$ and $\\alpha_2$ as in \\eqref{eq:R:0:2}, one can check that $y$ is indeed an eigenvector corresponding to $\\lambda$. Thus, these $2h$ non-real solutions correspond to exactly $h$ eigenvalues $\\lambda\\neq 1$ of $R_n$. Since $R_n$ has exactly $h+1$ eigenvalues, these are all.\n\\end{proof}\n\n\n\\subsection{Pseudo anti-symmetric eigenvectors}\\label{subsection:anti}\nIn this section, we describe pseudo anti-symmetric eigenvectors. We shall show that the pseudo anti-symmetric eigenvectors of $Q_h$ are given by the formula \\eqref{eq:thm:spectrum:antisym} and correspond to $\\lambda$ and $x$ satisfying \\eqref{eq:lambda:x:thm} and \\eqref{eq:x:antisym} as in Theorem \\ref{thm:spectrum}.\n\n\nConsider a pseudo anti-symmetric eigenvector $f$  with node $v$ and indices $i, j$ as described in Lemma \\ref{lm:sym:antisym} (see Figure \\ref{fig:sym:antisym}). Let $k = h-\\ell(v)-1\\in [0, h-1]$. As in Figure \\ref{fig:sym:antisym} and Figure \\ref{fig:anti}, let $y=(y_0, y_1, \\dots, y_k)$ where $y_0$ is the value of $f$ at the $i$-th child of $v$, which is denoted by $u$, $y_1$ is the value of $f$ at the children of $u$ and so on. With these notations, we also write $f$ as $f_{y, v, i, j}$. Observe that $y$ is an eigenvector of the following matrix $S_k$:\n\\begin{itemize}\n\t\\item $S_k(0, 1) = \\frac{d}{d+1}$,\n\t\\item $S_k(l, l-1) = \\frac{1}{d+1}$, $S_k(l, l+1) = \\frac{d}{d+1}$ for all $1\\le l\\le k-1$,\n\t\\item $S_k(k, k-1) = \\frac{1}{d+1}$, $S_k(k, k) = \\frac{d}{d+1}$.\n\\end{itemize}\n\nReversely, for any eigenvector $y$ of $S_k$, for any node $v$ at depth $h-k-1$ and for any choice of $i, j\\in [1, d]$ with $i\\neq j$, we can lift it to a pseudo anti-symmetric eigenvector $f_{y, v, i, j}$.\n\n\n\\begin{figure}[H]\t\\centering\n\t\\begin{tikzpicture}\n\t\\Tree\n\t[.0\n\t[.$0$     \n\t[.$y_0$ \n\t[.$y_1$ \n\t[.$y_2$ ]\n\t[.$y_2$ ]]\n\t[.$y_1$ \n\t[.$y_2$ ]\n\t[.$y_2$ ]]]\n\t[.$-y_{0}$ \n\t[.$-y_1$ \n\t[.$-y_2$ ]\n\t[.$-y_2$ ]]\n\t[.$-y_1$ \n\t[.$-y_2$ ]\n\t[.$-y_2$ ]\n\t]]\n\t]\n\t[.$0$     \n\t[.$0$ \n\t[.$0$ \n\t[.$0$ ]\n\t[.$0$ ]]\n\t[.$0$ \n\t[.$0$ ]\n\t[.$0$ ]]]\n\t[.$0$ \n\t[.$0$ \n\t[.$0$ ]\n\t[.$0$ ]]\n\t[.$0$ \n\t[.$0$ ]\n\t[.$0$ ]\n\t]]\n\t]\n\t]\n\t\\end{tikzpicture}\n\t\\caption{Pseudo anti-symmetric eigenvectors}\\label{fig:anti}\n\n\\end{figure}\n\n  \\begin{lemma} \\label{lm:antisym}\n\tFor each $k\\in [0, h-1]$, $S_k$ has $k+1$ eigenvectors. For each eigenvector $y$ of $S_k$ and for each $v$ with $l(v) = h-k-1$, there are $d-1$ linearly independent pseudo anti-symmetric eigenvectors of $Q_h$ of the form $f_{y,v,i,j}$.\n\\end{lemma}\n\\begin{proof}\n\tSince $S_k$ differs from $R_k$ only at the $(0, 0)$ entry, it also satisfies the equation $\\pi(x) S_k(x, y) = \\pi(y)S_k(y, x)$ where $\\pi = [1, d, d^{2}, \\dots, d^{k}]$. Thus, like $R_k$, the matrix $DS_k D^{-1}$ is symmetric where $D$ is the diagonal matrix with $D(x, x) = \\pi(x)^{1\/2}$. By symmetry, $D S_k D^{-1}$ has $k+1$ eigenvalues and so does $S_k$. \n\t\n\tFor each eigenvector $y$ of $S_k$, we create $d-1$ independent vectors $f_{y, v, i, i+1}$ for $1\\le i\\le d-1$. It is clear that any $f_{y, v, i, j}$ can be written as a linear combination of these vectors. This completes the proof.\n\\end{proof}\nWe now describe the eigenvectors of $S_k$.\n\\begin{lemma}\\label{lm:anti:detail}\n\tEach of the $k+1$ eigenvalue $\\lambda$ of $S_k$ is of the form \n\t$$\\lambda = \\frac{d}{d+1}\\left (x+ \\frac{1}{dx}\\right )$$\n\twhere $x\\neq \\pm \\frac{1}{\\sqrt d}$ is a solution of the equation\n\t\\begin{equation} \n\td^{k+2} x^{2k+4}- d^{k+2} x^{2k+3}+ dx -1 = 0.\\nonumber\n\t\\end{equation}\n\tThis equation has $2k+2$ solutions that differ from $\\frac{1}{\\sqrt d}$. If $x$ is a solution, so is $\\frac{1}{dx}$. There is a 2-to-1 correspondence between $x$ and $\\lambda$. An eigenvector $y = (y_0, y_1, \\dots, y_k)$ of $S_k$ with respect to $\\lambda$ is given by  \n\t\\begin{equation} \n\ty_i = \\frac{dx^{i+2}}{dx^{2}-1} - \\frac{1}{(dx^{2}-1)d^{i} x^{i}} \\quad\\text{for every $0\\le i\\le k$}.\\nonumber\n\t\\end{equation}\n\\end{lemma}\n\n\n\\begin{proof}\n\tLet $\\lambda$ be an eigenvalue of $S_k$ and $y = (y_0, y_1, \\dots, y_k)$ be an eigenvector corresponding to $\\lambda$. We have\n\t\\begin{enumerate}[label=(S\\arabic{*}), ref=S\\arabic{*}]\n\t\t\\item\\label{eq:S:0} $\\frac{d}{d+1} y_{1} = \\lambda y_0$,\n\t\t\\item\\label{eq:S:i}  $\\frac{1}{d+1} y_{i-1} +\\frac{d}{d+1} y_{i+1} = \\lambda y_i$ for all $1\\le i\\le k-1$,\n\t\t\\item\\label{eq:S:k} $\\frac{1}{d+1} y_{k-1} +\\frac{d}{d+1} y_{k} = \\lambda y_k$.\n\t\\end{enumerate}\n\t\n\tAs before, we let $x_1, x_2$ be the solutions to the equation \n\t$$\\frac{1}{d+1} - \\lambda x + \\frac{d}{d+1}x^{2} = 0.$$\n\tBy exactly the same argument as in the proof of Lemma \\ref{lm:sym:detail}, we derive by setting $y_0=1$ that \n\t$$y_i = \\alpha_1 x_1^{i} - \\alpha_2 x_2^{i}$$\nwhere\n\t$$\\alpha_1 = \\frac{dx^{2}}{dx^{2}-1}\\quad\\text{and}\\quad \\alpha_2 =\\frac{1}{dx^{2}-1}$$\n\tand $x_1$ and $x_2$ satisfy\n\t\\begin{equation}\\label{eq:x:2}\n\td^{k+2} x^{2k+4}- d^{k+2} x^{2k+3}+ dx -1 = 0\n\t\\end{equation}\n\t\n \tNote that, $x = \\pm \\frac{1}{\\sqrt d}$ are solutions of \\eqref{eq:x:2}. The remaining $2k+2$ solutions split into pairs $(x, \\frac{1}{dx})$ of distinct components. For each of these pairs, let $x_1 := x$ and $x_2:=\\frac{1}{dx}$. We have $x_1\\neq x_2$ and by setting $\\lambda$ and $y$ as in \\eqref{eq:lambda:x} and \\eqref{eq:recurrent:y:1} with $\\alpha_1 = \\frac{dx^{2}}{dx^{2}-1}$ and $\\alpha_2 =\\frac{1}{dx^{2}-1}$, one can check that $y$ is indeed an eigenvector corresponding to $\\lambda$. Thus, these $2k+2$ solutions correspond to exactly $k+1$ eigenvalues $\\lambda$ of $S_k$. Since $S_k$ has exactly $k+1$ eigenvalues, these are all.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm:spectrum}}\\label{subsection:proof:spectrum}\nThe following lemma shows that we can retrieve all eigenvectors of $Q_h$ from completely symmetric and pseudo anti-symmetric eigenvectors. Let $\\mathcal A_{S_k}$ be the eigenbasis of $S_k$ as described in Lemma \\ref{lm:anti:detail} and $\\mathcal B$ be a collection of $h+1$ independent completely symmetric eigenvectors of $Q_h$ as in Lemma \\ref{lm:sym:detail}. Let \n$$\\mathcal A: = \\lbrace f_{y, v, i, i+1}, v \\in V(\\mathcal T_{h-1}), y \\in \\mathcal A_{S_{h-\\ell(v)-1}}, i \\in [d-1] \\rbrace.$$\n\\begin{lemma}\\label{lm:spanning}\n\tThe collection $\\mathcal A \\cup \\mathcal B$ is an eigenbasis for $Q_h$.\n\\end{lemma}\n\nAssuming Lemma \\ref{lm:spanning}, we now put everything together to complete the proof of Theorem \\ref{thm:spectrum}.\n\\begin{proof}[Proof of Theorem \\ref{thm:spectrum}]\n\tThe first part of the theorem follows from Lemmas \\ref{lm:sym:detail} and \\ref{lm:anti:detail}. As seen in Lemma \\ref{lm:sym:detail}, the set $\\mathcal B$ in Lemma \\ref{lm:spanning} consists of eigenvectors as in \\eqref{eq:thm:spectrum:sym} and the all-1 vector. By Lemmas \\ref{lm:antisym} and \\ref{lm:anti:detail}, the set $\\mathcal A$ consists of eigenvectors as in \\eqref{eq:thm:spectrum:antisym}. That gives the second part. Finally, the third part follows from Lemma \\ref{lm:spanning}.\n\\end{proof}\n \n\nBefore proving Lemma \\ref{lm:spanning}, we make the following simple observation. For a rooted-tree $T$ that is not necessarily regular, recall that a vector $f: T\\to \\mathbb R$ is said to be \\textit{completely symmetric} if $f(u) = f(v)$ for all pairs of vertices $u, v$ at the same level. A vector $f$ is said to be \\textit{energy-preserving} if for all level $l$, \n$$\\sum_{v\\in T: l(v)=l} f(v)=0.$$\n\\begin{observation}\\label{obs}\n\tFor any rooted-tree $T$ and any vector $f: T\\to \\mathbb R$, if $f$ is both energy-preserving and completely symmetric then it is the zero vector.\n\\end{observation}\n\n\n \\begin{proof}[Proof of Lemma \\ref{lm:spanning}]\nFirst of all, we check that their number is equal to $n$. By Lemmas \\ref{lm:sym} and \\ref{lm:antisym}, the total number of vectors is\n\\begin{align*}\nh+1+  \\sum_{k=0}^{h-1} (k+1)(d-1) d^{h-k-1} \n\\end{align*}\nwhere $d^{h-k-1}$ is the number of nodes $v$ of depth $h-k-1$. By algebraic manipulation, this number is exactly $\\frac{d^{h+1} -1}{d-1}=n$.\n\n\nWe will now prove that the vectors considered are linearly independent. Assume that there exist coefficients $c_{y,v,i}$ and $c_g$ such that\n\\begin{equation} \n\\sum c_{y,v,i} f_{y,v,i,i+1} + \\sum_{g\\in \\mathcal B} c_g g = 0\\nonumber\n\\end{equation}\nwhere the first sum runs over all $v \\in V(\\mathcal T_{h-1}), y \\in \\mathcal A_{S_{h-\\ell(v)-1}}, i \\in [d-1]$. We need to show that $ c_{y,v,i}$ and $c_g$ are all 0.\n\nSince pseudo anti-symmetric vectors are energy-preserving on $\\mathcal T_{h}$, the sum $\\sum_{g\\in \\mathcal B} c_g g = -\\sum c_{y,v,i} f_{y,v,i,i+1}$ is both completely symmetric and energy-preserving. And so, by Observation \\ref{obs},\n\\begin{equation}\\label{eq:indep:sum}\n\\sum c_{y,v,i} f_{y,v,i,i+1} = \\sum_{g\\in \\mathcal B} c_g g = 0\n\\end{equation}\nBy the independence of vectors in $\\mathcal B$, we conclude that $c_g = 0$ for all $g\\in \\mathcal B$. \n \nWe now prove by induction on the vertices of $v\\in V(\\mathcal T_{h-1})$ and $i\\in [d-1]$ that $c_{y,v,i} = 0$ for all $y\\in \\mathcal A_{S_{h-\\ell(v)-1}}$. For this induction, we shall use the natural ordering of pairs $(v, i)$ as follows.\n$$(v, i)< (v', i')\\quad \\text{if and only if} \\quad l(v)< l(v') \\text{ or } l(v) = l(v') \\text{ and } i< i'.$$ \n \n \n For the base case, which is for $v := \\rho$ and $i:=1$, from \\eqref{eq:indep:sum}, we have\n \\begin{equation}\\label{key}\nF_{\\rho, 1}:= \\sum_{y\\in \\mathcal A_{S_{h-1}}}  c_{y,\\rho,1} f_{y,\\rho,1,2} = -\\sum c_{y,u,j} f_{y,u,j,j+1} \\nonumber\n \\end{equation}\n where the second sum runs over all $u \\in V(\\mathcal T_{h-1})$ and $j \\in [d-1]$ with $(\\rho, 1)< (u, j)$ and all $y \\in \\mathcal A_{S_{h-\\ell(v)-1}}$. Note that when restricting on the subtree $\\mathcal T_{1}^{\\rho}$, $F_{\\rho, 1}$ is a completely symmetric vector because all of the $f_{y,\\rho,1,2}$ are completely symmetric. Likewise, $F_{\\rho, 1}$ is energy-preserving on $\\mathcal T_{1}^{\\rho}$, because of the vectors $ f_{y,u,j,j+1}$. By Observation \\ref{obs}, $F_{\\rho, 1}=0$ on $\\mathcal T_{1}^{\\rho}$. Since the $f_{y,\\rho,1,2}$ are only supported on $\\mathcal T_{1}^{\\rho}\\cup \\mathcal T_{2}^{\\rho}$ and $f_{y,\\rho,1,2}\\vert_{\\mathcal T_{1}^{\\rho}} = -f_{y,\\rho,1,2}\\vert_{\\mathcal T_{2}^{\\rho}} $, so is $F_{\\rho, 1}$. Therefore, $F_{\\rho, 1} = 0$ on $\\mathcal T_{2}^{\\rho}$ and thus on $\\mathcal T_{h}$. So, \n \\begin{equation}\\label{key}\n\\sum_{y\\in \\mathcal A_{S_{h-1}}}  c_{y,\\rho,1} f_{y,\\rho,1,2} = 0 \\nonumber.\n \\end{equation}\n By the independence of vectors in $ \\mathcal A_{S_{h-1}}$, we conclude that $c_{y,\\rho,1} = 0$ for all $y\\in \\mathcal A_{S_{h-1}}$, establishing the base case.\n \n For the induction step, assume that for some $(v, i)$, it is proven that $c_{y,w, k} = 0$ for all $(w, k)< (v, i)$ and  $y\\in \\mathcal A_{S_{h-\\ell(w)-1}}$. We now show that $c_{y,v, i} = 0$ for all $y\\in \\mathcal A_{S_{h-\\ell(v)-1}}$. By this assumption, the left-most side in \\eqref{eq:indep:sum} reduces to\n \\begin{equation} \\label{eq:indep:induction}\n \\sum c_{y,u, j} f_{y,u, j, j+1} = 0 \n \\end{equation}\n where the sum runs over all $(u, j)\\ge (v, i)$. Our argument now is similar to the base case. From \\eqref{eq:indep:induction}, we have\n \\begin{equation} \n F_{v, i}:=\\sum_{y\\in \\mathcal A_{S_{h-\\ell(v)-1}}} c_{y,v,i} f_{y,v,i,i+1} = -\\sum_{y, (v, i)<(u, j)} c_{y,u,j} f_{y,u,j,j+1}.\\nonumber\n \\end{equation}\n Similarly to the base case, when restricting on the subtree $\\mathcal T_{i}^{v}$, $F_{v, i}$ is both completely symmetric and energy-preserving on $\\mathcal T_{j}^{v}$. By Observation \\ref{obs}, $F_{v, i}=0$ on $\\mathcal T_{j}^{v}$. This leads to $F_{v, i} = 0$ on $\\mathcal T_{i+1}^{v}$ and thus $F_{v, i}=0$ on $\\mathcal T_{h}$. So, \n \\begin{equation}\\label{key}\n \\sum_{y\\in \\mathcal A_{S_{h-\\ell(v)-1}}} c_{y,v,i} f_{y,v,i,i+1}  =0 \\nonumber.\n \\end{equation}\n By the independence of vectors in $ \\mathcal A_{S_{h-\\ell(v)-1}}$, we conclude that $c_{y,v, i} = 0$ for all $y\\in \\mathcal A_{S_{h-\\ell(v)-1}}$, establishing the induction step and thus finishing the proof.\n\\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{thm:lowerbound}}\n\\subsection{Proof of Theorem \\ref{thm:lowerbound}(a)}\nConsider the interchange process on $\\mathcal T_h$. Let $Q_h'$ be the transition matrix of the ace of spades. In other words, $Q_h'$ is the transition matrix of any fixed card on the tree. \nBy \\cite[Theorem 1.1]{CLR}, the spectral gap of the interchange process on the complete $d$-ary tree of depth $h$ is the same as the spectral gap of $Q_h'$. We note that\n\\begin{equation}\\label{key}\nQ_h' = \\frac{2n-d-3}{2(n-1)} I_n + \\frac{d+1}{2(n-1)} Q_h.\n\\end{equation}\nAnd therefore, the spectral gap of $Q_h'$ is $\\frac{d+1}{2(n-1)}$ times the spectral gap of $Q_h$.\n\n\n\nThus \\ref{thm:lowerbound} (a) is deduced from the following.\n\\begin{lemma}\\label{lm:Q_h:gap} For sufficiently large $h$, the spectral gap of $Q_h$ is equal $1 - \\lambda_2$ where $\\lambda_2$ is the second largest eigenvalue of $Q_h$. Moreover,\n\\begin{equation}\\label{eq:lm:gap}\n\\lambda_2 = 1- \\frac{(d-1)^{2}}{(d+1)\\cdot d^{h+1}} + O \\left (\\frac{\\log_{d} n}{n^{2}}\\right )\n\\end{equation}\n\\end{lemma} \n\n \n\n\n\nTo prove Lemma \\ref{lm:Q_h:gap}, we shall use Theorem \\ref{thm:spectrum}. Let $\\lambda$ be an eigenvalue of $Q_h$ and $x$ be a solution of\n\\begin{equation} \\label{eq:x:lambda:1:1}\nd x^{2} - (d+1)\\lambda x+1 = 0\n\\end{equation}\nwhich we have encountered in \\eqref{eq:x:lambda:1}.\n\nNote that if $\\lambda^{2}\\ge \\frac{4d}{(d+1)^{2}}$ then this equation has two real solutions both of which have the same sign as $\\lambda$.\n \nSince the equation \\eqref{eq:x:sym:thm} only has nonreal solutions except $x = \\pm \\frac{1}{\\sqrt d}$, combining this observation with Theorem \\ref{thm:spectrum}, each eigenvalue $\\lambda^{2}\\ge \\frac{4d}{(d+1)^{2}}$ is given by Equation \\eqref{eq:lambda:x:thm} for some $x \\neq \\pm \\frac{1}{\\sqrt d}$ satisfying\n \\begin{equation} \\label{eq:x:anti:k}\n d^{k+1} x ^{2k+2}- d^{k+1} x ^{2k+1}+ dx  -1 = 0,\n \\end{equation}\nfor some $k\\in [1, h]$ which is simply Equation \\eqref{eq:x:antisym} (with $k$ being shifted for notational convenience). \n  \n \n We shall show the following\n \\begin{lemma}\\label{lm:bound:lambda}\n \t\\begin{enumerate}  [label = (\\alph*)]\n \t\t\\item For all $k\\in [1, h]$, Equation \\eqref{eq:x:anti:k} has no solutions in $\\left (-\\infty, -\\frac{1}{\\sqrt d}\\right )$. There are no eigenvalues of $Q_h$ less than $-\\sqrt  \\frac{4d}{(d+1)^{2}}$.\n\\item  \t There exists a constant $h_0>0$ such that for all $k\\ge h_0$, the largest solution $x$ of \\eqref{eq:x:anti:k} satisfies\n \t\\begin{equation}\\label{eq:bound:x}\n \t1 - \\frac{a}{d^{k+1}} < x<1 - \\frac{d-1}{d^{k+1}} \\quad\\text{where}\\quad a=d-1 + \\frac{2(d-1)^{2}(k+1)}{d^{k+1}}.\n \t\\end{equation}\n \tFurthermore, for $k=h$, the eigenvalue that corresponds to this $x$ satisfies\n \t\\begin{equation}\\label{eq:bound:lambda}\n \t\\left |\\lambda - \\left (1-\\frac{(d-1)^{2}}{(d+1)\\cdot d^{h+1}}\\right )\\right | = O\\left (\\frac{\\log_{d} n}{n^{2}}\\right ).\n \t\\end{equation}\n \\end{enumerate}\n \\end{lemma}\n \n Assuming Lemma \\ref{lm:bound:lambda}, we conclude that for sufficiently large $h$, the largest $x$ that satisfies one of the equations \\eqref{eq:x:anti:k} for some $k$ in $[1, h]$ satisfies\n \\begin{equation} \n 1 - \\frac{a}{d^{h+1}} < x<1 - \\frac{d-1}{d^{h+1}} \\quad\\text{where}\\quad a=d-1 + \\frac{2(d-1)^{2}(h+1)}{d^{h+1}}.\\nonumber\n \\end{equation}\nSince the right-hand side of \\eqref{eq:lambda:x:thm} is increasing in $x$ for $x\\ge \\frac{1}{\\sqrt d}$, the second largest eigenvalue $\\lambda_2$ of $Q_h$ corresponds to such $x$ and so it satisfies \\eqref{eq:bound:lambda}, proving \\eqref{eq:lm:gap}. By the first part of Lemma \\ref{lm:bound:lambda}, there are no eigenvalues of $Q_h$ whose absolute value is larger than $\\lambda_2$. This proves Lemma \\ref{lm:Q_h:gap}.\n \n \\begin{proof}[Proof of Lemma \\ref{lm:bound:lambda}]\n \tLet $f(x) = d^{k+1} x^{2k+2} - d^{k+1} x^{2k+1} + dx-1$. \n \t\n \tTo prove part (a), for all $x< -\\frac{1}{\\sqrt d}$, we have\n \t$$d^{k+1}x^{2k+2}> 1\\quad\\text{and}\\quad - d^{k+1} x^{2k+1} > - dx$$\n \tand so $f$ has no roots in $\\left (-\\infty, -\\frac{1}{\\sqrt d}\\right )$. Assume that there were an eigenvalue $\\lambda<-\\sqrt  \\frac{4d}{(d+1)^{2}}$. By the argument right before \\eqref{eq:x:anti:k}, Equation \\eqref{eq:x:lambda:1:1} has two negative solutions $x_1<x_2$ with $x_1 x_2=\\frac{1}{d}$. We conclude that $x_1<-\\frac{1}{\\sqrt d}$. This is a contradiction because $x_1$ satisfies \\eqref{eq:x:anti:k} for some $k$ while for all $k$, the function $f$ has no roots less than $-\\frac{1}{\\sqrt d}$.\n \t\n \t\n \tTo prove part (b), for all $x\\ge \\frac{2k+1}{2k+2}$, we have\n \t\\begin{equation}\\label{key}\n \tf'(x) = d^{k+1} x^{2k}\\left ((2k+2)x-(2k+1)\\right ) +d > 0.\\nonumber\n \t\\end{equation}\n \tThus, $f$ is increasing on the interval $[1 - \\frac{1}{2k+2} , \\infty)$ which contains $[1 - \\frac{a}{d^{k+1}}, 1 - \\frac{d-1}{d^{k+1}}]$ for sufficiently large $k$. Thus, to prove \\eqref{eq:bound:x}, it suffices to show that \n \t\\begin{equation}\\label{eq:derivative:test}\n \tf\\left (1 - \\frac{a}{d^{k+1}}\\right )<0<f\\left (1 - \\frac{d-1}{d^{k+1}}\\right )\n \t\\end{equation}\n \tfor sufficiently large $k$. \n \tIndeed, \n \t\\begin{equation}\\label{key}\n \tf\\left (1 - \\frac{a}{d^{k+1}}\\right ) = d-1 - a\\left (1 - \\frac{a}{d^{k+1}}\\right )^{2k+1} -\\frac{a}{d^{k}} < d-1- a \\left (1 - \\frac{a}{d^{k+1}}\\right )^{2k+1}<d-1- a \\left (1 - \\frac{a(2k+1)}{d^{k+1}}\\right ) \\nonumber\n \t\\end{equation}\n \twhich is, by plugging in $a=d-1 +\\frac{2(d-1)^{2}(k+1)}{d^{k+1}}$, at most\n \t\\begin{equation}\\label{key}\n \t-\\frac{2(d-1)^{2}(k+1)}{d^{k+1}} + \\frac{a^{2}(2k+1)}{d^{k+1}}\\le 0.\\nonumber\n \t\\end{equation}\n \tThus the first inequality of \\eqref{eq:derivative:test} holds. \n \tFor the second inequality, we have\n \t\\begin{eqnarray} \n \tf\\left (1 - \\frac{d-1}{d^{k+1}}\\right ) &=& d-1 - (d-1)\\left (1 - \\frac{d-1}{d^{k+1}}\\right )^{2k+1} -\\frac{d-1}{d^{k}}\\nonumber\\\\\n \t& \\ge& d-1 - (d-1)\\left (1 - \\frac{3(d-1)}{d^{k+1}}\\right )  -\\frac{d-1}{d^{k}} >0,\\nonumber\n \t\\end{eqnarray}\n \tproving the \\eqref{eq:derivative:test}. \n \t\n \tWe have shown that there exists a solution $x = 1-\\alpha$ where $\\frac{d-1}{d^{k+1}}\\le \\alpha \\le \\frac{a}{d^{k+1}}$. Let $\\lambda$ be the eigenvalue corresponding to $x$ as in \\eqref{eq:lambda:x}. We have\n \t\\begin{equation}\\label{key}\n \t\\frac{d+1}{d}\\lambda = 1 - \\alpha+\\frac{1}{d(1-\\alpha)} \\in  \\left (1 - \\alpha+\\frac{1}{d} (1 +\\alpha), 1 - \\alpha+\\frac{1}{d} (1 +\\alpha+2\\alpha^{2})\\right ). \\nonumber \n \t\\end{equation}\n \tIn other words, \n \t\\begin{equation}\\label{key}\n \t\\frac{d+1}{d}\\lambda \\in\\left (\\frac{d+1}{d} -\\frac{d-1}{d} \\alpha , \\frac{d+1}{d} -\\frac{d-1}{d} \\alpha +\\frac{2}{d}\\alpha^{2}\\right ). \\nonumber\n \t\\end{equation}\n \tUsing the bounds $\\frac{d-1}{d^{k+1}}\\le \\alpha \\le \\frac{a}{d^{k+1}}$, we obtain\n \t\\begin{equation}\\label{key}\n \t\\lambda - \\left (1-\\frac{(d-1)^{2}}{(d+1)\\cdot d^{k+1}}\\right )  \\le  \\frac{2}{d+1}\\alpha^{2}\\le \\frac{2a^{2}}{(d+1)\\cdot d^{2k+2}}\\le \\frac{2}{(d+1)\\cdot d^{2k+1}}\\nonumber \n \t\\end{equation}\n \tand\n \t\\begin{equation}\\label{key}\n \t\\lambda - \\left (1-\\frac{(d-1)^{2}}{(d+1)\\cdot d^{k+1}}\\right )  \\ge  -\\frac{d-1}{d+1}\\alpha+\\frac{(d-1)^{2}}{(d+1)\\cdot d^{k+1}} \\ge -\\frac{2(d-1)^{3}(k+1)}{(d+1)\\cdot d^{2k+2}}\\ge - \\frac{2(k+1)}{d^{2k}}.\\nonumber \n \t\\end{equation}\n \tThus, for $k=h$,\n \t\\begin{equation} \n \t\\left |\\lambda - \\left (1-\\frac{(d-1)^{2}}{(d+1)\\cdot d^{h+1}}\\right )\\right | \\le \\frac{2(h+1)}{d^{2h}} .\\nonumber\n \t\\end{equation}\n \tThese bounds together with the equation $n = \\frac{d^{h+1}-1}{d-1} \\in (d^{h}, 2d^{h})$ give \\eqref{eq:bound:lambda}.\n \\end{proof}\n \n \n \\subsection{Proof of Theorem \\ref{thm:lowerbound} (b)}\n For the proof of the lower bound, we will use Wilson's lemma.\n \\begin{lemma}[Lemma 5, \\cite{Wilson}]\\label{W}\n \tLet $\\varepsilon, R$ be positive numbers and $0<\\gamma< 2-\\sqrt{2} $. Let $F: X\\to \\mathbb R$ be a function on the state space $X$ of a Markov chain $(C_t)$ such that \n \t$$\\expect{F(C_{t+1})\\vert C_t) }= (1 - \\gamma )F(C_t), \\quad \\expect{\\left [F(C_{t+1})- F(C_{t})\\right ]^2 \\vert C_t} \\leq  R,$$ and \n \t$$t \\leq \\frac{ \\log \\max_{x\\in X}F(x) + \\frac{1}{2} \\log( \\gamma  \\varepsilon\/(4R))}{-\\log (1 - \\gamma )}.\n \t$$ Then the total variation distance from stationarity at time $t$ is at least $1-\\varepsilon$.\n \\end{lemma}\n \n \n \\begin{proof}[Proof of Theorem \\ref{thm:lowerbound} (b)]\n Let $0<x<1$ be a solution of \\eqref{eq:x:antisym} (for $k=h$) satisfying \\eqref{eq:bound:x} and $\\lambda$ be the eigenvalue of $Q_h$ corresponding to $x$. In particular,\n\\begin{equation}\n\\lambda = \\frac{d}{d+1}\\left (x+ \\frac{1}{dx}\\right ).\\nonumber\n\\end{equation}\nLet $f:\\mathcal T_{h}\\to \\mathbb R$ be an eigenvector of $Q_h$ corresponding to $\\lambda$. As in the proof of Lemma \\ref{lm:anti:detail}, we can choose $f$ as follows.\n\\begin{equation}\\label{eq:def:f}\nf(v) = \\begin{cases}\n0\\quad \\text{if } v\\notin T_{1}^{\\rho}\\cup T_{2}^{\\rho},  \\\\\ndx^{l(v)+2} - \\frac{1}{d^{l(v)-1}x^{l(v)-2}}, \\quad \\text{if } v\\in T_{1}^{\\rho},\\\\\n-dx^{l(v)+2} + \\frac{1}{d^{l(v)-1}x^{l(v)-2}}, \\quad \\text{if } v\\in T_{2}^{\\rho}.\n\\end{cases}\n\\end{equation}\n\nWe now consider the interchange process on the $d$-ary tree $\\mathcal T_h$. Fix an arbitrary enumeration of the vertices of $\\mathcal T_{h}$ by $1, 2, \\dots, n$. Let $\\sigma \\in S_n$. We define $F (\\sigma)= \\sum_{v=1}^n f(v) f(\\sigma(v))$.\nThen, we have that \n\\begin{eqnarray}\n\\expect{F(\\sigma_{t+1})\\vert \\sigma_t) }= \\frac{1}{n-1}\\sum_{e} \\E\\left (F(\\sigma_{t+1})\\vert \\sigma_t, e\\right ), \\nonumber\n\\end{eqnarray}\nwhere the sum runs over all $n-1$ edges $e$ of the tree and the conditioning on the right is conditioning on the edge $e$ being chosen. \nSo, \n\\begin{eqnarray}\n\\expect{F(\\sigma_{t+1})\\vert \\sigma_t) }= \\frac{1}{n-1}\\sum_{e} \\sum_{v=1}^{n} f(v)  \\E\\left [f({\\sigma}_{t+1}(v))\\vert \\sigma_t, e\\right ]=\\frac{1}{n-1}  \\sum_{v=1}^{n} f(v) \\sum_{e} \\E\\left [f({\\sigma}_{t+1}(v))\\vert \\sigma_t, e\\right ]\\nonumber.\n\\end{eqnarray}\n\nBy direct computation, we obtain\n\\begin{equation}\\label{key}\n\\sum_{e} \\E\\left (f(\\sigma_{t+1}(v)\\vert \\sigma_t, e\\right )= \\left (n-\\frac{d}{2}-\\frac{3}{2} + \\frac{\\lambda(d+1)}{2}\\right ) f(\\sigma_t(v)).\\nonumber\n\\end{equation}\n\nSo, we have\n\\begin{eqnarray}\n\\expect{F(\\sigma_{t+1})\\vert \\sigma_t) }= \\frac{n-\\frac{d}{2}-\\frac{3}{2} + \\frac{\\lambda(d+1)}{2}}{n-1} F(\\sigma_t)\\nonumber.\n\\end{eqnarray}\n\n\nFor $n $ sufficiently big, we  have that \n$$F(id)= \\sum_{v=1}^n f^{2}(v)  \\geq 2 d^{h-1} \\left (dx^{h+2} - \\frac{1}{d^{h-1}x^{h-2}}\\right )^2 \\geq \\frac{dn}{2},$$\nwhere the first inequality occurs from keeping only the leaves of $T_{1}^{\\rho}\\cup T_{2}^{\\rho}$ and the last inequality follows from \\eqref{eq:bound:x}.\n\n\n\nFinally, we consider what happens if we change a configuration $\\sigma_t$ by transposing an edge $e$, which connects two vertices $u$ and $v$. We have that \n$$| F(\\sigma_{t+1})- F(\\sigma_t)|= |(f(u)-f(v)) (f(\\sigma(u)) - f(\\sigma(v)))|.$$\nNote that for all vertices $w$, we have by definition of $f$,\n$$|f(w)| \\le d$$\nand by \\eqref{eq:bound:x}, assuming wlog that $l(u) = l(v)+1 =: l+1$,\n$$|f(v) - f(u)|\\le d x^{l+2} (1-x) + \\frac{1}{d^{l-1}x^{l-2}}\\left (1 - \\frac{1}{dx}\\right )\\le \\frac{1}{d^{h-1}} + \\frac{1}{d^{l-1}}\\le \\frac{2}{d^{l-1}}.$$\nThus, $|(f(u)-f(v)) (f(\\sigma(u)) - f(\\sigma(v)))|^{2}\\le \\frac{16}{d^{2l-4}}$. Note that by \\eqref{eq:def:f}, the left-hand side is 0 if neither $u$ nor $v$ belongs to $T_{1}^{\\rho}\\cup T_{2}^{\\rho}$. And so,\n$$\\E_{e} \\left (\\left (F(\\sigma_{t+1})- F(\\sigma_t)\\right )^{2}\\vert \\sigma_t\\right ) \\le \\frac{1}{n-1} \\sum_{l=0}^{h} 2d^{l} \\frac{16}{d^{2l-4}} \\le \\frac{64 d^{4}}{n-1},$$\nwhere $2d^{l}$ is the number of edges $e$ that connect levels $l$ and $l+1$ of $T^{1}_{\\rho}\\cup T^{2}_{\\rho}$.\n\nSo we can take $R = \\frac{64d^{4}}{n-1}$.\n\nSet $\\lambda'= \\frac{n-\\frac{d}{2}-\\frac{3}{2} + \\frac{\\lambda(d+1)}{2}}{n-1}$. Using Wilson's lemma, we have that if $t \\le t_0:=\\frac{ \\log (F(id)) + \\frac{1}{2} \\log( (1-\\lambda') \\varepsilon\/(4R))}{-\\log (\\lambda')}$ then the total variation distance is at least $1- \\varepsilon$. By Lemma \\ref{lm:Q_h:gap}, \n$$\\lambda=  1-\\frac{(d-1)^{2}}{(d+1)\\cdot d^{h+1}}  + O\\left (\\frac{\\log_{d} n}{n^{2}}\\right ),$$ \nwhich gives\n$$1 - \\lambda' = \\frac{(d+1)(1-\\lambda)}{2(n-1)} =\\frac{(d-1)^{2}}{2(n-1)d^{h+1}} + O\\left (\\frac{d\\log_{d} n}{n^{3}}\\right ).$$\nwe get\n\\begin{eqnarray}\nt_0 &=& \\frac{\\frac{\\log n}{2}+\\frac{\\log \\varepsilon}{2}+O(\\log d)}{\\frac{(d-1)^{2}}{2(n-1)d^{h+1}} + O\\left (\\frac{d\\log_{d} n}{n^{3}}\\right )} =\\left (\\frac{\\log n}{2}+\\frac{\\log \\varepsilon}{2}+O(\\log d)\\right )\\left (\\frac{2n^{2}}{d-1} + O\\left (n\\log_{d}n\\right )\\right )  \\nonumber\\\\\n&=& \\frac{n^{2}\\left (\\log n+\\log \\varepsilon\\right ) }{d-1}+ O\\left (n^{2}\\right )\\nonumber.\n\\end{eqnarray}\nThis completes the proof.\n\\end{proof}\n\n \n\n \n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Causality-Based Game Solving}\n\\label{sec:gamesolving}\n\n\\Cref{lem:slicing} in the preceding section foreshadows how subgoals can be employed in building a recursive approach for the solution of reachability games. \nBefore turning to our actual algorithm, we describe a way to symbolically represent nondeterministic memoryless strategies. As discussed in \\cite{FarzanK17}, there is no ideal strategy description language for the class of games we consider. Our approach allows us to describe sets of concrete strategies as defined in Section~\\ref{sec:prelims} with linear arithmetic formulas.\nThis framework will prove convenient for \\emph{strategy synthesis}, i.e., the computation of winning strategies instead of simply determining the winner of the game.\n\n\n\\subsection{Symbolically Represented Strategies}\nWe will represent strategies for both players using transition predicates $\\mathfrak{S} \\in \\cal L(\\mathcal{V} \\cup \\mathcal{V'})$, henceforth called \\emph{symbolic strategies}, where we only require that $(\\mathfrak{S} \\implies (\\mathit{Safe} \\lor \\mathit{Reach}))$ is valid.\nA sequence $s_0 \\ldots s_n \\in S^+$ is called a \\emph{play prefix} if it is a prefix of some play in $\\mathcal{G}$, $(\\neg \\mathit{Goal})(s_j)$ holds for all $0 \\leq j \\leq n$, and $s_n$ is not a trap state.\nWe say that a play prefix $\\rho = s_0 \\ldots s_n$ \\emph{conforms to} a symbolic reachability strategy $\\mathfrak{S}$ if for all $j < n$ we have that $\\mathfrak{S}(s_j,s_{j+1})$ holds whenever $s_j \\in S_{\\texttt{{REACH}}}$ (and analogously for safety strategies).\nA play conforms to~$\\mathfrak{S}$ if all its play prefixes conform to $\\mathfrak{S}$.\nWe say that $\\mathfrak{S}$ is winning for \\texttt{{REACH}}{} in $s$ if all plays from $s$ that conform to $\\mathfrak{S}$ are winning for \\texttt{{REACH}}{}  and all play prefixes $s_0 \\ldots s_n \\in S^*S_{\\texttt{{REACH}}{}}$ from $s$ that conform to $\\mathfrak{S}$ are such that $(\\mathfrak{S} \\land \\mathit{Reach})(s_n)$ is satisfiable (and analogously for \\texttt{{SAFE}}{}).\nThe second condition ensures that the player cannot be forced to play a transition outside of $\\mathfrak{S}$ by their opponent while the play has not reached a trap state or $\\mathit{Goal}$, and in particular guarantees the existence of a concrete strategy (as defined in~\\Cref{sec:prelims}) conforming to~$\\mathfrak{S}$.\n\n\\begin{lemma}\n  If \\texttt{{REACH}}{} (\\texttt{{SAFE}}{}) has a winning symbolic strategy in $s$, then \\texttt{{REACH}}{} (\\texttt{{SAFE}}{}) has a concrete winning strategy in $s$.\n\\end{lemma}\n\\begin{proof}\n  Let $\\mathfrak{S}$ by a symbolic winning strategy for \\texttt{{REACH}}{}.\n  Let $\\sigma_R$ be any reachability strategy such that for all play prefixes $\\omega s \\in S^* S_{\\texttt{{REACH}}}$ that conform to $\\mathfrak{S}$ the formula $\\mathfrak{S}(s,\\sigma_R(\\omega s))$ is valid. \n  Such a function is guaranteed to exist, as $(\\mathfrak{S} \\land \\mathit{Reach})(s)$ is satisfiable for all such play prefixes by definition.\n  Furthermore, $\\sigma_R$ is winning as all play prefixes of plays consistent with $\\sigma_R$ conform to $\\mathfrak{S}$, and hence all these plays are winning by assumption. \n  The proof for \\texttt{{SAFE}}{} is analogous.\n  \\qed\n\\end{proof}\n\nThis representation allows us to specify nondeterministic strategies, but classical memoryless strategies on finite arenas (specified as a function $\\sigma\\colon S_{\\texttt{{REACH}}}\\to S$ or $\\S_{\\texttt{{SAFE}}}\\to S$) can also be represented in this form using a disjunction over formulas $\\bigwedge_{v\\in \\mathcal{V}} (v =s(v)\\land v' =\\sigma(s)(v))$ for varying $s\\in S$.\n\nThe following lemma shows that a necessary subgoal directly yields a symbolic strategy for \\texttt{{SAFE}}{} if the subgoal is, in a certain sense, \\emph{locally avoidable} by \\texttt{{SAFE}}{}. It will be our main tool for synthesizing safety player strategies.\n\\begin{restatable}{lemma}{avoidingC}\n\t\\label{lem:avoidingC}\n\tLet $C$ be a necessary subgoal for $\\mathcal{G}$ and suppose that $\\operatorname{Unsat}(\\operatorname{Enf}(C,\\mathcal{G}))$ holds.\n\tThen, $\\mathit{Safe} \\land \\neg C$ is a winning symbolic strategy for $\\texttt{{SAFE}}$ in $\\mathcal{G}$.\n\\end{restatable}\n\n\\subsection{A Recursive Algorithm}\n\n\n\\begin{algorithm}\n\t\\caption{$\\operatorname{Reach}(\\mathcal{G})$}\n\t\\label{alg:algreach}\n\t\\SetKwInOut{Input}{In}\n\t\\SetKwInOut{Output}{Out}\n\t\\Input{ reachability game $\\G = \\langle \\init, \\safe, \\reach, \\goal \\rangle${}}\n\t\\Output{ triple $(R, \\mathfrak{S}_{\\texttt{{REACH}}}, \\mathfrak{S}_{\\texttt{{SAFE}}})$ s.t.\n\t\t\\begin{itemize}\n\t\t\t\\item $\\mathit{R}\\in\\cal L(\\mathcal{V})$ represents the set of initial states winning for \\texttt{{REACH}}{};\n\t\t\t\\item $\\mathfrak{S}_{\\texttt{{REACH}}}$ is a winning symbolic reachability strategy  for states in $R$;\n\t\t\t\\item $\\mathfrak{S}_{\\texttt{{SAFE}}}$ is a winning symbolic safety strategy for states in $\\mathit{Init} \\land \\neg R$.\n\t\t\\end{itemize}}\n\t\\Begin{\n\t\t$\\mathit{R} \\leftarrow \\mathit{Init} \\land \\mathit{Goal}$\\\\\n\t\t$\\mathit{I} \\leftarrow \\mathit{Init} \\land \\lnot\\mathit{Goal}$\\\\\n\t\t\\If{$\\operatorname{Unsat}(\\mathit{I})$}{\\KwRet{$\\mathit{R}, {\\normalfont \\texttt{false}}, \\normalfont \\texttt{false}$ \\label{line:ret1}}}\n\t\t$\\varphi \\leftarrow  \\operatorname{Interpolate}(\\mathit{I},\\mathit{Goal})$\\label{line: phi}\\\\\n\t\t$\\mathit{C} \\leftarrow \\operatorname{Instantiate}(\\varphi,\\mathcal{G})$\\label{line: subgoal}\\\\\n\t\t\\If{$\\operatorname{Unsat}(\\operatorname{Enf}(\\mathit{C},\\mathcal{G}))$\\label{line: cpre}}{\\KwRet{$\\mathit{R}, {\\normalfont \\texttt{false}},\\mathit{Safe} \\land \\neg C$}\\label{line:safety wins}}\n\t\t$\\mathcal{G}_{post} \\leftarrow \\langle \\operatorname{Post}(C)[\\varp\/\\var], \\mathit{Safe} \\land \\varphi, \\mathit{Reach} \\land \\varphi, \\mathit{Goal} \\rangle$\\label{line:gpost}\\\\\n\t\t$\\mathit{R}_{post},\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}, \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post} \\leftarrow \\operatorname{Reach} \\big(\\mathcal{G}_{post}\\big)$\\label{line: recursion1}\\\\\n\t\t$\\mathit{F} \\leftarrow C \\land \\mathit{R}_{post}[\\var\/\\varp]$\\\\\n\t\t\\If{$\\operatorname{Unsat}(\\operatorname{Enf}(\\mathit{F},\\mathcal{G}))$\\label{line:safeavoidsE}}{\\KwRet{$\\mathit{R}, {\\normalfont \\texttt{false}}, \\mathit{Safe} \\land \\neg F \\land (\\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})$}\\label{line:thirdreturn}}\n\t\t\\If{$\\operatorname{Sat}((\\mathit{Reach} \\lor \\mathit{Safe}) \\land \\varphi \\land \\lnot \\varphi' \\land \\lnot \\mathit{Goal})$\\label{line:transback}}{\n\t\t\t$\\mathit{F} \\leftarrow \\mathit{F} \\lor \\mathit{Goal} [\\var\/\\varp]$\\label{line: E2}\\\\\n\t\t\t$\\varphi \\leftarrow \\texttt{false}$\\label{line:tpostfalse}\n\t\t}\n\t\t$\\mathcal{G}_{pre} \\leftarrow \\langle I,\\mathit{Safe} \\land \\lnot \\mathit{F},\\mathit{Reach} \\land \\lnot \\mathit{F}, \\operatorname{Pre}(\\operatorname{Enf}(\\mathit{F},\\mathcal{G}))\\rangle$\\label{line:gpre2}\\\\\n\t\t$\\mathit{R}_{pre}, \\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre}, \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre} \\leftarrow \\operatorname{Reach} \\big(\\mathcal{G}_{pre}\\big)$\\label{line: recursion2}\\\\\n    \\KwRet{$\\mathit{R} \\lor \\mathit{R}_{pre}$, \\label{line: last return}\\\\ $\\qquad \\quad \\;\\; \\operatorname{combine}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre},F, \\mathfrak{S}_{\\texttt{{REACH}}{}}^{post})$\\label{line:reachstrat}, \\\\$\\qquad \\quad \\;\\;(\\neg \\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre}) \\land (\\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})$\\label{line:safetystrat}}\n\t}\n\\end{algorithm}\n\nWe now describe our algorithm which utilizes necessary subgoals to decompose and solve two-player reachability games (Algorithm~\\ref{alg:algreach}).\nIt is incomplete in the sense that it does not return on every input (\\Cref{sec:termination} discusses special cases with guaranteed termination).\nIf the algorithm returns on input $\\mathcal{G}$, it returns a triple $(R,\\mathfrak{S}_{\\texttt{{REACH}}{}},\\mathfrak{S}_{\\texttt{{SAFE}}{}})$, where (1) $R$ is a state predicate characterizing the initial states that are winning for \\texttt{{REACH}}{} in $\\mathcal{G}$, (2) $\\mathfrak{S}_{\\texttt{{REACH}}{}}$ is a symbolic strategy for \\texttt{{REACH}}{} that wins in all initial states satisfying $R$, and (3) $\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ is a symbolic strategy for \\texttt{{SAFE}}{} that wins in all initial states satisfying $\\mathit{Init} \\land \\neg R$.\nThe returned safety strategy $\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ is such that $\\neg \\mathfrak{S}_{\\texttt{{SAFE}}{}}$ is a necessary subgoal that \\texttt{{SAFE}}{} can avoid locally in the game $\\mathcal{G}$ restricted to intial states $\\mathit{Init} \\land \\neg R$ (see~\\Cref{lem:avoidingC}).\n\nAlgorithm~\\ref{alg:algreach} works as follows.\nStates satisfying $\\mathit{Init}$ and $\\mathit{Goal}$ are immediately winning for $\\texttt{{REACH}}$ and thus always part of the returned formula $R$.\nFollowing the discussion at the beginning of Section~\\ref{sec:subgoals}, further analysis considers the game starting in the remaining initial states $I = \\mathit{Init} \\land \\lnot \\mathit{Goal}$.\nIf there is no such state, we may return that all initial states are winning (line~\\ref{line:ret1}).\nHere, \\texttt{{REACH}}{} wins from $R$ without playing any move, and hence $\\mathfrak{S}_{\\texttt{{REACH}}{}} = \\texttt{false}$ is a valid winning symbolic strategy (winning symbolic strategies are only required to provide moves in prefixes that have not seen $\\mathit{Goal}$ so far).\nWe may choose $\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ arbitrarily as there is no initial state winning for $\\texttt{{SAFE}}{}$.\n\nIf the algorithm does not return in line~\\ref{line:ret1}, a necessary subgoal $C$ between $I$ and $\\mathit{Goal}$ is computed by instantiating a Craig interpolant $\\varphi$ for the two predicates (lines \\ref{line: phi} and \\ref{line: subgoal}, see also \\Cref{prop_necessary}).\nWe break up the remaining description of the algorithm into three parts, which correspond to the main cases that occur when splitting the game along the subgoal $C$.\n\n\\subsubsection{Case 1: \\texttt{{SAFE}}{} can avoid the subgoal $C$.}\nIf the necessary subgoal $C$ qualifies for~\\Cref{lem:avoidingC}, we can immediately conclude that \\texttt{{SAFE}}{} is winning for all states statisfying $I$ (lines~\\ref{line: cpre} and~\\ref{line:safety wins}).\nAn instance of this case occurs if the interpolant describes a bottleneck in the game which is fully controlled by \\texttt{{SAFE}}{}.\nThe winning symbolic reachability strategy is $\\mathit{Safe} \\land \\neg C$ in this case (line~\\ref{line:safety wins}), and we will assume that safety strategies returned by recursive calls of the algorithm are essentially negations of necessary subgoals that can be avoided by \\texttt{{SAFE}}{}. \n\nIf Lemma \\ref{lem:avoidingC} is not applicable, we next find those transitions in $C$ that move into a winning state for the safety player.\nThis is achieved by analyzing the \\emph{post-game} (line~\\ref{line:gpost}):\n\\[\\mathcal{G}_{post} = \\langle \\operatorname{Post}(C)[\\varp\/\\var], \\mathit{Safe} \\land \\varphi, \\mathit{Reach} \\land \\varphi, \\mathit{Goal} \\rangle.\\]\n\nIts initial states are exactly the states one sees after bridging the subgoal $C$.\nIn order to make sure that $\\mathcal{G}_{post}$ is, in some sense, easier to solve than $\\mathcal{G}$, we restrict both $\\mathit{Safe}$ and $\\mathit{Reach}$ to $\\varphi$, which is the interpolant used to compute the subgoal $C$.\nThis has the effect of removing all transitions in states \\emph{not} satisfying~$\\varphi$, making them trap states.\nFor the safety player this makes $\\mathcal{G}_{post}$ easier to win than $\\mathcal{G}$ as all plays ending in such a trap state without seeing $\\mathit{Goal}$ before are winning for \\texttt{{SAFE}}{} in $\\mathcal{G}_{post}$.\nHence we formally have:\n\\begin{restatable}{lemma}{gpost}\n  \\label{lem:gpost}\n  If $\\mathfrak{S}$ is a winning symbolic reachability strategy from $s$ in $\\mathcal{G}_{post}$, then $\\mathfrak{S}$ is also winning from $s$ in $\\mathcal{G}$.\n\\end{restatable}\n\n\n\nDue to the restriction to $\\varphi$, intuitively \\texttt{{REACH}}{} wins from a state $s$ in $\\mathcal{G}_{post}$ if they can win from $s$ in $\\mathcal{G}$ \\emph{while staying inside the interpolant} $\\varphi$.\nIn other words, \\texttt{{REACH}}{} must guarantee that the necessary subgoal $C$ is not visited again in the play.\nStill, the set $R_{post}$, as returned in line \\ref{line: recursion1} by the recursive call to Algorithm~\\ref{alg:algreach} on $\\mathcal{G}_{post}$, is a sufficient subgoal in $\\mathcal{G}$, by the above lemma.\nFurthermore, if \\texttt{{SAFE}}{} can avoid all states satisfying $R_{post}$ (see line~\\ref{line:safeavoidsE}), then this also implies a winning strategy from all initial states in $I$. \nThe reason is that \\texttt{{REACH}}{} can only win by eventually visiting a state from which they can win without leaving $\\varphi$ again, as $(\\mathit{Goal} \\implies \\varphi)$ is valid.\nThis is not possible if \\texttt{{SAFE}}{} can avoid all states in $R_{post}$. \n\nIn this case we construct $\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ as follows.\nWe assume that $\\neg \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post}$ is a necessary subgoal that can be locally avoided in $\\mathcal{G}_{post}$ from all states satisfying $\\operatorname{Post}(C)[\\varp\/\\var] \\land \\neg R_{post}$, and furthermore, we know that $F := C \\land R_{post}[\\var\/\\varp]$ can be locally avoided in $\\mathcal{G}$ (line~\\ref{line:safeavoidsE}).\nIntuitively, playing according to $\\mathfrak{S}^{post}_{\\texttt{{SAFE}}{}}$ in $\\mathcal{G}_{post}$ yields a strategy for \\texttt{{SAFE}}{} which avoids $\\mathit{Goal}$ and may move back into a state satisfying $\\neg \\varphi$, which forces \\texttt{{REACH}}{} to bridge the subgoal $C$ again in order to win.\nIt follows that $F \\lor (\\varphi \\land \\neg \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})$ is a necessary subgoal from $I$ that can be locally avoided by \\texttt{{SAFE}}{} in $\\mathcal{G}$, and the corresponding symbolic strategy is $\\mathit{Safe} \\land \\neg F \\land (\\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})$ (we additionally intersect the negated necessary subgoal with $\\mathit{Safe}$ to ensure that the symbolic strategy only includes legal transitions).\n\n\\medskip\nSo far, the subgoal was such that \\texttt{{SAFE}}{} could avoid it entirely, or at least avoid all states from which \\texttt{{REACH}}{} would win when forced to remain inside the post-game.\nIf this is not the case, then we also need to consider the \\emph{pre-game} (line~\\ref{line:gpre2}):\n\\[\\mathcal{G}_{pre} = \\langle I,\\mathit{Safe} \\land \\lnot \\mathit{F},\\mathit{Reach} \\land \\lnot \\mathit{F}, \\operatorname{Pre}(\\operatorname{Enf}(\\mathit{F},\\mathcal{G}))\\rangle.\\]\nwhich intuitively describes the game before bridging the interpolant $C$ for the last time.\nThe exact definition of $F$ will depend on whether $C$ \\emph{perfectly partitions} the game or not.\nIn both cases $F$ will be the largest sufficient subgoal contained in a necessary subgoal, which lets us apply~\\Cref{lem:slicing} to conclude that the initial winning regions of $\\mathcal{G}$ and $\\mathcal{G}_{pre}$ coincide.\n\n\\subsubsection{Case 2: The subgoal perfectly partitions $\\mathcal{G}$.}\nWe say that $\\varphi$ \\emph{perfectly partitions} $\\mathcal{G}$ if $(\\mathit{Reach} \\lor \\mathit{Safe}) \\land \\varphi \\land \\neg \\varphi' \\land \\neg \\mathit{Goal}$ is unsatisfiable (cf. line~\\ref{line:transback}).\nIntuitively, this means that there is no transition that ``undoes'' the effect of the subgoal~$C$.\nIf this holds, then the restriction of $\\mathcal{G}_{post}$ to states satisfying $\\varphi$ is \\emph{de facto} no longer a restriction, as no play can reach such a state anyway after passing through the subgoal.\nThis intuition is formalized by the following lemma.\n\\begin{restatable}{lemma}{perfectpartition}\n  Assume that $\\varphi$ \\emph{perfectly partitions} $\\mathcal{G}$, and let $s$ be a state satisfying $\\operatorname{Post}(C)[\\varp\/\\var]$. Then \\texttt{{REACH}}{} wins from $s$ in $\\mathcal{G}_{post}$ if and only if \\texttt{{REACH}}{} wins from $s$ in~$\\mathcal{G}$.\n\\end{restatable}\nIt follows that $F = C \\land R_{post} [\\var\/\\varp]$ is the largest sufficient subgoal included in~$C$.\nBy~\\Cref{lem:slicing}, the same initial states are winning for \\texttt{{REACH}}{} in $\\mathcal{G}_{pre}$ and in~$\\mathcal{G}$. \nIn this case, we construct the desired safety strategy (line~\\ref{line:safetystrat}) as \n\\[\\mathfrak{S}_{\\texttt{{SAFE}}{}} = (\\neg \\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre}) \\land (\\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post}),\\]\nwhere $\\neg \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre\/post}$ are assumed to be necessary subgoals avoidable by \\texttt{{SAFE}}{} in the corresponding subgames. \nIntuitively, the combined strategy consists of following $\\mathfrak{S}_{\\texttt{{SAFE}}}^{pre}$ as long as one remains in the pre-game, which, by induction hypothesis, allows \\texttt{{SAFE}}{} to avoid all transitions from $F$ if starting in $R_{pre}$. \nIf the play crosses $C \\land \\neg F$, the strategy is to play according to the winning strategy of the post-game.\n\nA symbolic strategy for \\texttt{{REACH}}{} can be given by combining pre- and post-strategies as follows (line~\\ref{line:reachstrat}):\n\\begin{align*}\n  \\operatorname{combine}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre},F, \\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) := \\;\\;&(\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\implies \\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\\\\n  \\land \\;\\; &((\\neg \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\land \\operatorname{Pre}(F)) \\implies F) \\\\\n  \\land \\;\\; &((\\neg \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\land \\neg \\operatorname{Pre}(F)) \\implies \\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre}) \\\\\n  \\land \\;\\; &(\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\lor \\operatorname{Pre}(F) \\lor \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre})).\n\\end{align*}\nThis represents a nested conditional strategy that prefers the strategies of the subgames in the priority order $\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}, F$, and finally $\\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre}$.\nThe reason for this order is that the winning condition in the post-game coincides with the global winning objective (to reach $\\mathit{Goal}$), while in the pre-game \\texttt{{REACH}}{} tries to reach a winning state in the post-game.\nThe set $F$ is exactly the bridge between these two.\nThe last condition makes sure that the strategy only includes transitions of states in which it is winning.\n\n\\subsubsection{Case 3: The subgoal does not perfectly partition $\\mathcal{G}$.}\n\nIf $\\operatorname{Sat}((\\mathit{Reach} \\lor \\mathit{Safe}) \\land \\varphi \\land \\lnot \\varphi' \\land \\neg \\mathit{Goal})$ is true in line~\\ref{line:transback}, we can no longer assume that $F$ is the largest sufficient subgoal in $C$.\nThe reason is that \\texttt{{SAFE}}{} may win in $\\mathcal{G}_{post}$ by moving out of the subgame, but if this move leads to a winning state for \\texttt{{REACH}}{} in $\\mathcal{G}$, then such a strategy is winning in $\\mathcal{G}_{post}$, but not in $\\mathcal{G}$.\nSo we can only assume that $F$ is sufficient (this follows by~\\Cref{lem:gpost}).\nIn order to apply~\\Cref{lem:slicing} we extend~$F$ by all transitions that move directly into $\\mathit{Goal}$ (line~\\ref{line: E2}). \nThis immediately yields a necessary and sufficient subgoal, and so again~\\Cref{lem:slicing} applies to $\\mathcal{G}_{pre}$ (line~\\ref{line:gpre2}).\nWe could have also added $\\mathit{Goal}$-states to $F$ in Case 2, but we have observed that not doing so improves the performance of our procedure considerably. \n\nThe reachability strategy is composed of $\\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre}, F$, and $\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}$ exactly as in Case 2 (line~\\ref{line:reachstrat}).\nAs all transitions in $F$ are losing for \\texttt{{SAFE}}{}, and these are the only ones that are removed in $\\mathcal{G}_{pre}$, essentially \\texttt{{SAFE}}{} can play using the same strategies in $\\mathcal{G}$ and $\\mathcal{G}_{pre}$.\nWe implement this by setting $\\varphi$ to $\\texttt{false}$ (line~\\ref{line:tpostfalse}), in which case $\\mathfrak{S}_{\\texttt{{SAFE}}}$ (line~\\ref{line:safetystrat}) equals \n$(\\texttt{true} \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre}) \\land (\\texttt{false} \\implies \\mathfrak{S}_{\\texttt{{SAFE}}}^{post}) \\equiv \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre}$.\n\n\\medskip\nFinally, we formally state the partial correctness of the algorithm, using the ideas outlined above. \nThe proof can be found in the appendix.\n\n\\begin{restatable}[Partial correctness]{theorem}{partialcorrectness}\n  \\label{thm:partcorr}\n  If $\\operatorname{Reach}(\\mathcal{G})$ returns $(R,\\mathfrak{S}_{\\texttt{{REACH}}{}},\\mathfrak{S}_{\\texttt{{SAFE}}{}})$, then\n  \\begin{itemize}\n  \\item $R$ characterizes the set of initial states that are winning for $\\texttt{{REACH}}{}$ in $\\mathcal{G}$,\n  \\item $\\mathfrak{S}_{\\texttt{{REACH}}{}}$ is a winning symbolic reachability strategy from $R$,\n  \\item $\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ is a winning symbolic safety strategy from $\\mathit{Init} \\land \\neg R$.\n  \\end{itemize}\n\\end{restatable}\n\n\\begin{remark}[Simulating the attractor]\n  \\label{rem:attractor}\n\tNote that Craig interpolants are by no means unique. If we choose the interpolation function so that $\\operatorname{Interpolate}(I,\\mathit{Goal})$ always returns $\\mathit{Goal}$ (this is a valid interpolant), Algorithm~\\ref{alg:algreach} essentially simulates the attractor.\n  In this case the subgoal $C$ (line~\\ref{line: subgoal}) contains exactly the transitions that move directly into $\\mathit{Goal}$.\n  All states in $\\operatorname{Post}(C)[\\varp\/\\var]$ are then winning for \\texttt{{REACH}}{} and hence $R_{post}$ would be equivalent to $\\operatorname{Post}(C)[\\varp\/\\var]$, which implies that $C \\equiv F$ holds in this case.\n  The new goal states in $\\mathcal{G}_{pre}$ are set to $\\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G}))$, which are exactly the states in $\\operatorname{Pre}(C)$ that either are controlled by \\texttt{{REACH}}{}, or such that all their transitions are included in $F$.\n  Hence the set $\\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G}))$ is exactly the classical controlled predecessor.\n\\end{remark}\n\nOne effect of slicing the game along general subgoals is that the initial predicate of the post-game (which describes all states satisfying the post-condition of the subgoal) may be satisfied by many states that do not necessarily need to be considered in order to decide who wins from the initial states of $\\mathcal{G}$ (for example, because they are not reachable from any initial state, or cannot reach $\\mathit{Goal}$).\nThis can be a drawback if the (superfluous) size of the subgames makes them hard to solve.\nNotably, this is in general less of an issue for approaches based on unrolling of the transition relation: \nThe method of solving increasingly large step-bounded games~\\cite{FarzanK17} will only consider states that are reachable from $\\mathit{Init}$, while backwards fixpoint computations will not explore states that do not reach $\\mathit{Goal}$.\n A way of coping with this is to provide additional information on the domains of variables, whenever this is available (we discuss the effect of bounding variable domains in \\Cref{sec:experiments}).\nIndeed, in the case where all variable domains are finite, Algorithm~\\ref{alg:algreach} is guaranteed to terminate, as shown in the next subsection.\n\n\\subsection{Special Cases with Guaranteed Termination}\n\\label{sec:termination}\n\nDeciding the winner in the types of games we consider is generally undecidable (see~\\cite{FarzanK17} for the case that $\\cal L$ is linear real arithmetic).\nSince Algorithm \\ref{alg:algreach} returns a correct result whenever it terminates, this implies that it cannot always terminate.\nIn this section, we give two important cases in which we can prove termination. \nThe proofs can be found in the appendix.\n\n\\begin{restatable}{theorem}{terminationfinite}\n  If the domains of all variables in $\\mathcal{G}$ are finite, then $\\operatorname{Reach}(\\mathcal{G})$ terminates.\n\\end{restatable}\n\n\\begin{remark}[Time complexity]\n  The termination argument given in the appendix yields a single-exponential upper bound on the runtime of the algorithm, where the input size is measured in the number of concrete transitions of the game.\n  This is because in both recursive calls the subgames may be ``almost'' as large as the input -- they are only guaranteed to have at least one concrete transition less.\n\\end{remark}\n\nWe now show that, under certain assumptions, our algorithm also terminates for games that have a finite bisimulation quotient.\nTo this end, we first clarify what bisimilarity means in our setting.\nA relation $R \\subseteq S \\times S$ over the states of $\\mathcal{G}$ is called a \\emph{bisimulation} on $\\mathcal{G}$, if\n\\begin{itemize}\n\t\\item for all $(s_1,s_2) \\in R$ the formulas $\\mathit{Goal}(s_1) \\iff \\mathit{Goal}(s_2)$, $\\mathit{Init}(s_1) \\iff \\mathit{Init}(s_2)$ and $\\mathbf{r}(s_1) \\iff \\mathbf{r}(s_2)$ are valid (recall that $\\mathbf{r}$ holds exactly in states controlled by \\texttt{{REACH}}{}).\n\\item for all $(s_1,s_2) \\in R$ and $s_1' \\in S$ such that $(\\mathit{Safe} \\lor \\mathit{Reach})(s_1,s_1')$ holds, there exists $s_2' \\in S$ such that $(\\mathit{Safe} \\lor \\mathit{Reach})(s_2,s_2')$ holds, and $(s_1',s_2') \\in R$.\n\\item for all $(s_1,s_2) \\in R$ and $s_2' \\in S$ such that $(\\mathit{Safe} \\lor \\mathit{Reach})(s_2,s_2')$ holds, there exists $s_1' \\in S$ such that $(\\mathit{Safe} \\lor \\mathit{Reach})(s_1,s_1')$ holds, and $(s_1',s_2') \\in R$.\n\\end{itemize}\nWe say that $s_1$ and $s_2$ are \\emph{bisimilar} (denoted by $s_1 \\sim s_2$) if there exists a bisimulation $R$ such that $(s_1,s_2) \\in R$.\nBisimilarity is an equivalence relation, and it is the coarsest bisimulation on $\\mathcal{G}$.\nThe equivalence classes are called \\emph{bisimulation classes}.\nAs the winning region of any player can be expressed in the $\\mu$-calculus~\\cite{Zappe01} and the $\\mu$-calculus is invariant under bisimulation~\\cite{BradfieldS07}, it follows that bisimilar states are won by the same player.\n\\begin{lemma}\n  \\label{lem:bisimpreservesreach}\n  Let $R$ be a bisimulation on $\\mathcal{G}$.\n  If $(s_1, s_2) \\in R$, then \\texttt{{REACH}}{} wins from $s_1$ in $\\mathcal{G}$ if and only if \\texttt{{REACH}}{} wins from $s_2$ in $\\mathcal{G}$.\n\\end{lemma}\n\nWe will assume that for each bisimulation class $S_i$ there exists a formula $\\psi_i \\in \\cal L(\\mathcal{V})$ that \\emph{defines} $S_i$, formally: \nFor all $s \\in S$, $\\psi_i(s)$ holds if and only if $s \\in S_i$.\nFurthermore, we will assume that the interpolation procedure \\emph{respects}~$\\sim$, formally: $\\operatorname{Interpolate}(\\varphi,\\psi)$ is equivalent to a disjunction of formulas $\\psi_i$.\nSuch an interpolant exists if $\\psi$ or $\\varphi$ already satisfy this assumption.\n\n\\begin{restatable}{theorem}{terminationbisim}\n  Let $\\mathcal{G}$ be a reachability game with finite bisimulation quotient under $\\sim$ and assume that all bisimulation classes of $\\mathcal{G}$ are definable in $\\cal L$.\n  Furthermore, assume that $\\operatorname{Interpolate}$ respects $\\sim$.\n  Then, $\\operatorname{Reach}(\\mathcal{G})$ terminates.\n\\end{restatable}\n\n\n\\section*{Appendix}\n\n\\avoidingC*\n\\begin{proof}\n\tWe first show that all plays that conform to the symbolic safety strategy $\\mathit{Safe} \\land \\neg C$ are winning for \\texttt{{SAFE}}{}.\n\tLet $\\rho = s_0s_1 \\ldots$ be such a play and assume, for contradiction, that $n \\in \\mathbb{N}$ exist such that $\\mathit{Goal}(s_n)$ is valid, and let $n$ be the least such $n$.\n\tFrom $\\operatorname{Unsat}(\\operatorname{Enf}(C,\\mathcal{G}))$ it follows that all states in $\\operatorname{Pre}(C)$ are under the control of \\texttt{{SAFE}}{}, and hence $C(s_j,s_{j+1})$ can only be true if $s_j \\in S_{\\texttt{{SAFE}}}$.\n\tBut as $\\rho$ conforms to $\\mathit{Safe} \\land \\neg C$, it follows that all play prefixes of $\\rho$ conform to $\\mathit{Safe} \\land \\neg C$, in particular, for all $j < n$ the formula $\\neg C(s_j,s_{j+1})$ must hold.\n  This is in contradiction to the fact that $C$ is a necessary subgoal.\n\t\n\tNow we show that if $\\pi = s_0 \\ldots s_n \\in S^*S_{\\texttt{{SAFE}}}$ is a play prefix that conforms to $\\mathit{Safe} \\land \\neg C$, then $\\operatorname{Pre}(\\mathit{Safe} \\land \\neg C)(s_n)$ holds.\n\tAs $\\pi$ is a play prefix and $s_n \\in S_{\\texttt{{SAFE}}}$, $\\mathit{Safe}(s_n)$ is satisfiable.\n\tFrom $\\operatorname{Unsat}(\\operatorname{Enf}(C,\\mathcal{G}))$ it follows that if $(\\mathit{Safe} \\land C)(s_j)$ is satisfiable, then so is $(\\mathit{Safe} \\land \\neg C)(s_j)$, which concludes the proof.\n\n\t\\qed\n\\end{proof}\n\n\n\\gpost*\n\\begin{proof}\n\tFirst, we observe that $\\mathfrak{S} \\implies ((\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\varphi)$ is valid by assumption, and hence so is $\\mathfrak{S} \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$.\n\tWe show that all plays of $\\mathcal{G}$ that conform to $\\mathfrak{S}$ are winning for \\texttt{{REACH}}{}.\n\tLet $\\rho = s_0 s_1 \\ldots$ be such a play, and assume for contradiction that it is winning for \\texttt{{SAFE}}{}.\n\tAs playing according to $\\mathfrak{S}$ is winning for \\texttt{{REACH}}{} in $\\mathcal{G}_{post}$, the play needs to see a transition that is not included in $\\mathcal{G}_{post}$.\n\tSuch a transition can only be played by \\texttt{{SAFE}}{}, as \\texttt{{REACH}}{} plays according to $\\mathfrak{S}$, and $\\mathfrak{S} \\implies ((\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\varphi)$ is valid.\n\tThis can only happen if there exists $i \\in \\mathbb{N}$ such that $(\\mathit{Safe}\\land \\lnot\\varphi)(s_i, s_{i+1})$ holds, or in other words, $\\mathit{Safe}(s_i, s_{i+1})$ and $\\lnot\\varphi(s_i)$. Let $i$ be the first such index. Then $s_i$ does not have any outgoing transitions in $\\mathcal{G}_{post}$, meaning that $s_0\\ldots s_i$ is a finite play in $\\mathcal{G}_{post}$, and hence there must exist $0\\leq j\\leq i$ such that $\\mathit{Goal}(s_j)$ is satisfied.\n\tOtherwise, the play would be winning for \\texttt{{SAFE}}{} in $\\mathcal{G}_{post}$, contradicting the assumption.\n\tHence, $\\rho$ is winning for \\texttt{{REACH}}{}.\n\t\n\tIt remains to show that any play prefix $s_0 \\ldots s_n \\in S^*S_{\\texttt{{REACH}}}$ that conforms to $\\mathfrak{S}$ is such that $(\\mathfrak{S} \\land \\mathit{Reach})(s_n)$ is satisfiable.\n  If the play prefix remains inside $\\mathcal{G}_{post}$, then this follows from the fact that for all such prefixes $(\\mathfrak{S} \\land \\mathit{Reach}(s_n) \\land \\varphi)(s_n)$ is satisfiable by assumption, as $\\mathfrak{S}$ is winning for \\texttt{{REACH}}{} in $\\mathcal{G}_{post}$.\n  Otherwise, the play prefix corresponds to a play that conforms to $\\mathfrak{S}$ and is loosing for $\\texttt{{REACH}}{}$ in $\\mathcal{G}_{post}$.\n  But such a play does not exist by assumption that $\\mathfrak{S}$ is a winning reachability strategy for $\\texttt{{REACH}}{}$ in $\\mathcal{G}_{post}$.\n\t\\qed\n\\end{proof}\n\n\n\\perfectpartition*\n\\begin{proof}\n  \\Cref{lem:gpost} can be adapted slightly to show that if \\texttt{{REACH}}{} has \\emph{any} (not necessarily expressable as a symbolic strategy) winning strategy from $s$ in $\\mathcal{G}_{post}$, then they have a winning stratey from $s$ in $\\mathcal{G}$. \n\n\tHence, it is enough to prove the backward direction. Let us assume that there is a reachability player strategy $\\sigma_R$ that is winning from $s$ in $\\mathcal{G}$, and, for contradiction, a safety player strategy $\\sigma_S$ that is winning from $s$ in $\\mathcal{G}_{post}$. Consider strategy $\\sigma'_S$ such that $\\sigma_{\\mathit{S}}'(\\omega s') = \\sigma_{\\mathit{S}}(\\omega s')$ if $(\\mathit{Safe} \\land \\varphi)(s')$ is satisfiable, and else $\\sigma_{\\mathit{S}}'(\\omega s') = \\sigma_{\\mathit{S}}''(\\omega s')$, where $\\sigma_{\\mathit{S}}''$ is an arbitrary safety player strategy in $\\mathcal{G}$. There exists a unique play $\\rho = s_0s_1\\ldots$ in $\\mathcal{G}$ consistent with both $\\sigma_R$ and $\\sigma'_S$ with $s_0 = s$.  Since s satisfies $\\operatorname{Post}(C)[\\mathcal{V}'\/\\mathcal{V}]$, we know that $\\varphi(s_0)$ is valid (i.e., $\\rho$ starts in $\\mathcal{G}_{post}$), and since $\\varphi$ perfectly partitions $\\mathcal{G}$, we know that there is no $0 \\leq i$ such that $\\left((\\mathit{Reach} \\lor \\mathit{Safe}) \\land \\varphi \\land \\neg \\varphi'\\right)(s_i, s_{i+1})$ holds. It follows that for all $0 \\leq j$ we have that $\\varphi(s_j)$ holds (i.e., $\\rho$ stays in $\\mathcal{G}_{post}$). Hence, $\\rho$ is a play in $\\mathcal{G}_{post}$ consistent with $\\sigma_S$ and $\\sigma_R$ (considered as a strategy in $\\mathcal{G}_{post}$). Since $\\sigma_R$ is winning in $\\mathcal{G}$, there must exist some $n\\in\\mathbb{N}$ such that $\\mathit{Goal}(s_n)$ is valid. This is a contradiction with $\\sigma_S$ being a winning strategy in $\\mathcal{G}_{post}$.\n\t\\qed\n\n\n\n\\end{proof}\n\n\n\n\\begin{lemma}\n\t\\label{lem:unsat_sum}\n\tLet $T,F$ be transition predicates such that $\\operatorname{Unsat}(\\operatorname{Pre}(T)\\land\\operatorname{Pre}(F))$ holds. Suppose that $\\mathcal{G}_T$ and $\\mathcal{G}_F$ are games such that  $\\operatorname{Unsat}(\\operatorname{Enf}(T,\\mathcal{G}_T))$ and $\\operatorname{Unsat}(\\operatorname{Enf}(F,\\mathcal{G}_F))$ both evaluate to true, and we have \n\t\\begin{gather*}\n\tT \\implies (\\mathit{Safe}_T \\lor \\mathit{Reach}_T) \\implies (\\mathit{Safe} \\lor \\mathit{Reach}),\\\\\n\tF \\implies (\\mathit{Safe}_F \\lor \\mathit{Reach}_F) \\implies (\\mathit{Safe} \\lor \\mathit{Reach}).\n\t\\end{gather*}\n\tThen it follows that $\\operatorname{Unsat}(\\operatorname{Enf}(T \\lor F,\\mathcal{G}))$ is also true.\n\\end{lemma}\n\\begin{proof}\n\tWe proceed by contradiction. Suppose there were $s,v \\in \\S$ such that $\\operatorname{Enf}(T \\lor F,\\mathcal{G})(s,v)$ holds. It follows that $(T \\lor F)(s,v)$ must hold. Since we have $\\operatorname{Unsat}(\\operatorname{Pre}(T)\\land\\operatorname{Pre}(F))$, two cases may occur: either (1) $\\operatorname{Pre}(T)(s)$ holds, or (2) $\\operatorname{Pre}(F)(s)$ holds, but not both. Since they work completely symmetrically, we give the argument only for case (2).\n\t\n\tSo we know that $F(s,v)$ holds and $T(s,v)$ does not. As $F \\implies (\\mathit{Safe}_F \\lor Reach_F)$, we know $(\\mathit{Safe}_F \\lor Reach_F)(s,v)$ holds. The only way we can still have $\\operatorname{Unsat}(\\operatorname{Enf}(F,\\mathcal{G}_F))$ is if there is some $s'$ such that $\\big(\\mathit{Safe}_F(s,s')\\land\\lnot F(s,s')\\big)$ holds. However, as $(\\mathit{Safe}_F \\lor \\mathit{Reach}_F) \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$, then also $\\big(\\mathit{Safe}(s,s')\\land\\lnot F(s,s')\\big)$, which is a contradiction to the fact that $\\operatorname{Enf}(F \\lor T,\\mathcal{G})(s,v)$ holds. \n\t\\qed\n\\end{proof}\n\n\\partialcorrectness*\n\\begin{proof}\n\tWe show by induction on the recursion depth that if $\\operatorname{Reach}(\\mathcal{G})$ returns $(R,\\mathfrak{S}_{\\texttt{{REACH}}{}},\\mathfrak{S}_{\\texttt{{SAFE}}{}})$, where $R \\in \\cal L(\\mathcal{V})$, $\\mathfrak{S}_{\\texttt{{REACH}}}$ and $\\mathfrak{S}_{\\texttt{{SAFE}}}$ are symbolic strategies for \\texttt{{REACH}}{}\/\\texttt{{SAFE}}{} respectively, and\n\t\\begin{enumerate}\n\t\t\\item[(a)] $\\mathfrak{S}_{\\texttt{{REACH}}{}}$  is a winning strategy for \\texttt{{REACH}}{} from all states satisfying $R \\lor \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}})$,\n\t\t\\item[(b)] $\\lnot \\mathfrak{S}_{\\texttt{{SAFE}}{}}$  is a necessary subgoal in $\\mathcal{G}^I = \\langle \\mathit{Init} \\land \\lnot \\mathit{R}, \\mathit{Safe}, \\mathit{Reach}, \\mathit{Goal} \\rangle$, and\n\t\t\\item[(c)] if $\\mathit{Init} \\land \\neg R$ is satisfiable, then $\\operatorname{Unsat}(\\operatorname{Enf}(\\lnot \\mathfrak{S}_{\\texttt{{SAFE}}{}},\\mathcal{G}))$ holds.\n\t\\end{enumerate}\n\tNote that the last condition is equivalent to $\\operatorname{Unsat}(\\operatorname{Enf}(\\lnot \\mathfrak{S}_{\\texttt{{SAFE}}{}},\\mathcal{G}^I))$ as $\\operatorname{Enf}$ does not take initial states into account. Properties (b) and (c) ensure, in view of \\Cref{lem:avoidingC}, that $\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ is indeed a winning symbolic strategy from all initial states of $\\mathcal{G}^I$ (provided that $\\mathfrak{S}_{\\texttt{{SAFE}}} \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$ holds, which we will show is the case). Together, these properties ensure that $R$ characterizes the set of initial states that are winning for $\\texttt{{REACH}}$ in $\\mathcal{G}$. We distinguish the five cases that can occur when the algorithm terminates (there are four \\textbf{return} statements, the last of which depends on the \\textbf{if} statement in line~\\ref{line:transback}).\n\t\n\t\\medskip\n\t{\\it Case 1:  $\\operatorname{Reach}(\\mathcal{G})$ returns in line~\\ref{line:ret1}}. Then the formula $(\\mathit{Init} \\implies \\mathit{Goal})$ is valid, which means that all initial states are goal states.\n\tThe algorithm returns $R = \\mathit{Init}\\land\\mathit{Goal}$, which in this case is equivalent to $\\mathit{Init}$.\n\tTrivially, any strategy is winning for \\texttt{{REACH}}{} for all initial states.\n  Then, $\\mathfrak{S}_{\\texttt{{REACH}}} = \\texttt{false}$ is a winning symbolic strategy for any state satisfying $R$.\n  This is because clearly any play starting in $R$ is winning for \\texttt{{REACH}}{}, and we only need to show that \\texttt{{REACH}}{} has a move satisfying $\\mathfrak{S}_{\\texttt{{REACH}}}$ in a play prefix, which by definition has not seen $\\mathit{Goal}$ already.\n  In this case, there is no such play prefix that we need to cover.\n  Observe also that $\\operatorname{Pre}(\\texttt{false}) \\equiv \\texttt{false}$, which is required to show (a).\n\n As $\\mathcal{G}^I$ has no initial states, one can choose $\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ arbitrarily, as anything qualifies as necessary subgoal if $I \\equiv \\texttt{false}$, in which case (c) is also directly satisfied. So the properties (a)--(c) above are satisfied.\n\t\n\t\\medskip\n\t{\\it Case 2:  $\\operatorname{Reach}(\\mathcal{G})$ returns in line~\\ref{line:safety wins}}. Then $\\operatorname{Enf}(C,\\mathcal{G})$ is unsatisfiable, where $C$ is the instantiation of the interpolant $\\varphi$ (lines~\\ref{line: phi} and \\ref{line: subgoal}). \n \t\\Cref{prop_necessary} states that $C$ is a necessary subgoal in $\\mathcal{G}^I$. Since $\\operatorname{Enf}(C,\\mathcal{G})$ is unsatisfiable, \\Cref{lem:avoidingC} states that $\\mathfrak{S}_{\\texttt{{SAFE}}{}} = \\mathit{Safe} \\land \\lnot C$ is a winning symbolic strategy for $\\texttt{{SAFE}}$ in $\\mathcal{G}^I$. For \\texttt{{REACH}}{}, we have the same argument as in Case 1. We conclude that the properties (a)--(c) above are satisfied.\n\t\n\t\\medskip\n\t{\\it Case 3:  $\\operatorname{Reach}(\\mathcal{G})$ returns in line~\\ref{line:thirdreturn}}. By induction hypothesis we assume that the recursive call in line~\\ref{line: recursion1} returned a tuple $(R_{post}, \\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}, \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})$ satisfying properties (a)--(c) above for $\\mathcal{G}_{post}$. We now show these properties in $\\mathcal{G}$ for $R = \\mathit{Init}\\land\\mathit{Goal}$, $\\mathfrak{S}_{\\texttt{{REACH}}{}} = \\texttt{false}$, and $\\mathfrak{S}_{\\texttt{{SAFE}}{}} = \\mathit{Safe} \\land \\neg F  \\land (\\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})$, where $F= C \\land \\mathit{R}_{post}[\\var\/\\varp]$.\n\t\n\tAs in the previous cases, property (a) is satisfied. In order to prove (b) let $\\rho = s_0 s_1 \\ldots$ be a play such that $(\\mathit{Init}\\land\\lnot R)(s_0)$ and $\\mathit{Goal}(s_n)$ hold for some $n \\in \\mathbb{N}$. As $C$ is a necessary subgoal in $\\mathcal{G}$, there exists $m \\in \\mathbb{N}$ with $m < n$ such that $C(s_m,s_{m+1})$ holds. Take $m$ to be the last index with this property.  If $F(s_m,s_{m+1})$ holds, then also $(F  \\lor (\\varphi \\land \\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post}))(s_m,s_{m+1})$, which implies $\\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}(s_m, s_{m+1})$. If $\\lnot F(s_m,s_{m+1})$ holds, then $(\\operatorname{Post}(C)[\\varp\/\\var] \\land \\lnot R_{post})(s_{k+1})$ holds. By induction hypothesis, all winning plays for $\\texttt{{REACH}}$ in $\\mathcal{G}_{post}$ starting in $\\lnot R_{post}$ have some $n > j > m$ such that $\\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post}(s_j,s_{j+1})$ holds.\n  Furthermore, by construction of $\\mathcal{G}_{post}$, all states $s_j$ with $n > j > m$ satisfy $\\varphi$.\n  Hence $\\varphi \\land \\lnot\\mathfrak{S}^{post}_{\\texttt{{SAFE}}{}}(s_j,s_{j+1})$ holds, which implies that $\\lnot \\mathfrak{S}_{\\texttt{{SAFE}}}(s_j,s_{j+1})$ holds. We conclude that $\\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}$ is a necessary subgoal in $\\mathcal{G}^I$.\n\t\n\tWe now show (c). Since we return in  line~\\ref{line:thirdreturn}, we have $\\operatorname{Unsat}(\\operatorname{Enf}(F,\\mathcal{G}))$ and, by induction hypothesis, $\\operatorname{Unsat}(\\operatorname{Enf}(\\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post},\\mathcal{G}_{post}))$.\n  As the transition relation of $\\mathcal{G}_{post}$ is restricted to $\\varphi$, this implies $\\operatorname{Unsat}(\\operatorname{Enf}(\\varphi \\land \\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post},\\mathcal{G}_{post}))$.\n We also have $F \\implies \\lnot \\varphi$ and $(\\varphi \\land \\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post}) \\implies \\varphi$. As $(\\mathit{Safe}_{post} \\lor \\mathit{Reach}_{post}) \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$ holds, we can apply \\Cref{lem:unsat_sum} to conclude $\\operatorname{Unsat}(\\operatorname{Enf}(F  \\lor (\\varphi\\land\\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})),\\mathcal{G})$, which implies $\\operatorname{Unsat}(\\operatorname{Enf}(\\neg \\mathit{Safe} \\lor F  \\lor (\\varphi\\land\\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})),\\mathcal{G}) = \\operatorname{Unsat}(\\operatorname{Enf}(\\lnot\\mathfrak{S}_{\\texttt{{SAFE}}{}}),\\mathcal{G}))$.\n\t\t\n\t\\medskip\n\t{\\it Case 4:  $\\operatorname{Reach}(\\mathcal{G})$ returns in line~\\ref{line: last return} and the {\\upshape\\textbf{if}} statement in line~\\ref{line:transback} is false}.\n\tBy induction hypothesis we assume that the recursive calls in lines~\\ref{line: recursion1} and \\ref{line: recursion2} returned tuples $(R_{post}, \\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}, \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})$ and $(R_{pre}, \\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre}, \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre})$ satisfying properties (a)--(c) above for $\\mathcal{G}_{post}$ and $\\mathcal{G}_{pre}$. We now show these properties in $\\mathcal{G}$ for $R\\lor R_{pre}$, and \n\t\\begin{align*}\n\t\t\\mathfrak{S}_{\\texttt{{REACH}}{}} &=  \\;\\;(\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\implies \\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\\\\n  & \\land \\;\\; ((\\neg \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\land \\operatorname{Pre}(F)) \\implies F) \\\\\n  & \\land \\;\\; ((\\neg \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\land \\neg \\operatorname{Pre}(F)) \\implies \\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre}) \\\\\n    & \\land \\;\\; (\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{post}) \\lor \\operatorname{Pre}(F) \\lor \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}{}}^{pre})) \\\\\n\t\t\\mathfrak{S}_{\\texttt{{SAFE}}{}} &= (\\neg \\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre}) \\land (\\varphi \\implies \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post}),\n\t\\end{align*} \n\twith $F= C \\land \\mathit{R}_{post}[\\var\/\\varp]$.\n\nWe first show that (a) $\\mathfrak{S}_{\\texttt{{REACH}}{}}$ is winning for \\texttt{{REACH}}{} from states satisfying $R\\lor R_{pre} \\lor \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}})$. For states in $R$ this is trivial, so let  $\\rho = s_0 s_1 \\ldots$ be a play in $\\mathcal{G}$ conforming to $\\mathfrak{S}_{\\texttt{{REACH}}{}}$ such that $R_{pre}(s_0)$ holds.\n  Our first claim is that if there exists $k \\in \\mathbb{N}$ such that $\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{post})(s_k)$ holds, then $\\rho$ must be winning for \\texttt{{REACH}}{}.\n  This is due to the fact that $\\mathfrak{S}_{\\texttt{{REACH}}}^{post}$ is winning in $\\mathcal{G}_{post}$ from all states satisfying $\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{post})$, which allows us to use~\\Cref{lem:gpost}.\n  To argue that $\\mathfrak{S}_{\\texttt{{REACH}}}^{post}$ keeps playing according to $\\mathfrak{S}_{\\texttt{{REACH}}}^{post}$ once such a state is reached, we observe that if a symbolic reachability strategy $\\mathfrak{S}$ wins from $s$, then $\\operatorname{Pre}(\\mathfrak{S})$ holds in any state in $S_{\\texttt{{REACH}}}$ reachable from $s$ via a play prefix conforming to $\\mathfrak{S}$, by definition.\n  \n  Now we show that such a position $k$ must exist.\n  First, for $j \\in \\mathbb{N}$ such that $(\\neg \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{post}) \\land \\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G})))(s_j)$ holds, the transition $(s_j,s_{j+1})$ must satisfy $F$.\n  This is because if $s_j \\in S_{\\texttt{{SAFE}}}$, then all outgoing transitions from $s_j$ satisfy $F$.\n  Otherwise, it follows by the fact that $\\rho$ conforms to $\\mathfrak{S}_{\\texttt{{REACH}}}$.\n  As $\\operatorname{Post}(F) \\equiv R_{post}[\\var\/\\varp]$ and $\\mathfrak{S}_{\\texttt{{REACH}}}^{pos}$ wins from all states satisfying $R_{post}$ by assumption, it follows that $s_{j+1}$ satisfies $\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{post})$.\n\n  As long as $\\rho$ visits only states satisfying $(\\neg \\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G})) \\land \\neg \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{post}))$, the strategy $\\mathfrak{S}_{\\texttt{{REACH}}}$ prescribes to play according to $\\mathfrak{S}_{\\texttt{{REACH}}}^{pre}$.\n  By assumption, this strategy is winning for \\texttt{{REACH}}{} in $\\mathcal{G}_{pre}$, and hence the play $\\rho$ eventually visits a state in $\\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G}))$.\n  As above, the play is guaranteed to stay in $\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{pre})$ until that position.\n\n  The above argument also shows that $\\mathfrak{S}_{\\texttt{{REACH}}}$ is winning for all states satisfying $\\operatorname{Pre}(F) \\lor \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{post}) \\lor \\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}}^{pre})$, which is implied by $\\operatorname{Pre}(\\mathfrak{S}_{\\texttt{{REACH}}})$.\n  Also, $\\mathfrak{S}_{\\texttt{{REACH}}} \\implies (\\mathit{Reach} \\lor \\mathit{Safe})$ is valid, as the corresponding statements hold for the pre- and post-strategies, and $F \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$ is valid.\n\n\n\n\t\n\n\t \n\n\t\n\tNext we show that (b) $\\lnot \\mathfrak{S}_{\\texttt{{SAFE}}{}}$  is a necessary subgoal in $\\mathcal{G}^I$.\n\tNo player can play back from $\\mathcal{G}_{post}$ to $\\mathcal{G}_{pre}$ without $\\texttt{{REACH}}$ having already won in $\\mathcal{G}_{post}$. We first show that under this condition,\n  \\[\\neg \\mathfrak{S}_{\\texttt{{SAFE}}} = (\\neg \\varphi \\land \\neg \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{pre}) \\lor (\\varphi \\land \\neg \\mathfrak{S}_{\\texttt{{SAFE}}{}}^{post})\\]\n  qualifies as a necessary subgoal in $\\mathcal{G}^I$. For this, consider the necessary subgoal $C$.\n  For any play $\\rho = s_0 s_1 \\ldots$ with $n \\in \\mathbb{N}$ such that $\\mathit{Goal}(s_n)$ there is some $k \\in \\mathbb{N}$ with $k < n$ and $C(s_k,s_{k+1})$. As $F$ characterizes a subset of $C$, we check two cases: Either (1) $\\lnot F(s_k,s_{k+1})$ or (2) $F(s_k,s_{k+1})$. In case (1), we have $\\lnot R_{post}(s_{k+1})$ and because of our assumption that no transition of the game satisfies $\\varphi \\land \\neg \\varphi'$, for all $j \\in \\mathbb{N}$ with $k < j < n: \\varphi(s_j)$. It follows by induction hypothesis that there is some $l \\in \\mathbb{N}$ such that $\\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{post}(s_l,s_{l+1})$.\n  In case (2) we use that $\\mathfrak{S}_{\\texttt{{SAFE}}}^{pre}$ plays only moves available in $\\mathcal{G}_{pre}$, and hence $\\mathfrak{S}_{\\texttt{{SAFE}}}^{pre} \\implies \\neg F$ is valid.\n  Furthermore $F \\implies \\neg \\varphi$, as $F$ characterizes a subset of the subgoal $C$.\n  Hence we can conclude that $F \\implies (\\neg \\varphi \\land \\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{pre})$ is valid.\n  It follows that $\\neg \\mathfrak{S}_{\\texttt{{SAFE}}}$ qualifies as necessary subgoal.\n\n  Finally we show (c) that $\\operatorname{Unsat}(\\operatorname{Enf}(\\neg \\mathfrak{S}_{\\texttt{{SAFE}}},\\mathcal{G}))$ holds.\n  We have $(\\mathit{Safe}_{pre} \\lor \\mathit{Reach}_{pre}) \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$ and $(\\mathit{Safe}_{post} \\lor \\mathit{Reach}_{post}) \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$. As $\\operatorname{Pre}(\\neg \\varphi \\land \\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{pre}) \\land \\operatorname{Pre}(\\varphi \\land \\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{post})$ is cleary unsatisfiable, we can again apply Lemma \\ref{lem:unsat_sum} to infer $\\operatorname{Unsat}(\\operatorname{Enf}(\\neg \\mathfrak{S}_{\\texttt{{SAFE}}},\\mathcal{G}))$.\n  This uses that any transitions reachable in $\\mathcal{G}_{post}$ has to satisfy $\\varphi$ in this case.\n\t\n\t\\medskip\n\t{\\it Case 5:  $\\operatorname{Reach}(\\mathcal{G})$ returns in line~\\ref{line: last return} and the {\\upshape\\textbf{if}} statement in line~\\ref{line:transback} is true}. \n\n\tWe  assume that both recursive calls terminated and, by induction, returned triples $(R_{post},\\mathfrak{S}_{\\texttt{{REACH}}}^{post},\\mathfrak{S}_{\\texttt{{SAFE}}}^{post})$ and $(R_{pre},\\mathfrak{S}_{\\texttt{{REACH}}}^{pre},\\mathfrak{S}_{\\texttt{{SAFE}}}^{pre})$ satisfying (a)-(c).\n\n  (a) is shown exactly as in Case 4.\n\n\tFor (b) we first observe that by setting $\\varphi$ to $\\texttt{false}$ (see line~\\ref{line:tpostfalse}) in this case we get $\\mathfrak{S}_{\\texttt{{SAFE}}} = \\mathfrak{S}_{\\texttt{{SAFE}}}^{pre}$.\n  We show that $\\neg \\mathfrak{S}_{\\texttt{{SAFE}}} = \\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{pre}$ is a necessary subgoal in $\\mathcal{G}_I$.\n  The transition predicate $F$ in line~\\ref{line: recursion1} is a sufficient subgoal by induction hypothesis, but due to the restriction on the post-game, we cannot conclude that states in $\\operatorname{Post}(C)$ that are not in $\\operatorname{Post}(F)$ are winning for \\texttt{{SAFE}}{}.\n\tBy adding all transitions to $\\mathit{Goal}$ (line~\\ref{line: E2}) we get that $F$ in line~\\ref{line: recursion2} is a necessary and sufficient subgoal (clearly, any winning play must go through $\\mathit{Goal} [\\var\/\\varp]$).\n\tAs we have ensured that $F$ is necessary, we know for all plays $\\rho = s_0 s_1 \\ldots$ with some $n \\in \\mathbb{N}$ such that $\\mathit{Goal}(s_n)$ there is some $k \\in \\mathbb{N}$ with $k < n$ and $F(s_k,s_{k+1})$. As in Case 4 we may conclude that $F \\implies \\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{pre}$.\nIt follows that $\\neg \\mathfrak{S}_{\\texttt{{SAFE}}}$ is a necessary subgoal in $\\mathcal{G}_I$.\n\nFor (c) we observe that $\\operatorname{Unsat}(\\operatorname{Pre}(\\operatorname{Enf}(\\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{pre},\\mathcal{G}_{pre})))$ holds by induction hypothesis, which directly implies $\\operatorname{Unsat}(\\operatorname{Pre}(\\operatorname{Enf}(\\neg \\mathfrak{S}_{\\texttt{{SAFE}}}^{pre},\\mathcal{G})))$. This concludes the argument for the final case, and the proof is complete.\n\t\\qed\n\n\\end{proof}\n\n\n\\terminationfinite*\n\\begin{proof}\n\tWe denote by $\\operatorname{size}(\\mathcal{G})$ the number of concrete transitions of $\\mathcal{G}$, formally: $\\operatorname{size}(\\mathcal{G}) = |\\{ (s,s') \\in S \\times S \\mid (\\mathit{Safe} \\lor \\mathit{Reach})(s,s') \\text{ is valid}\\}|$.\n\tIf the domains of all variables are finite, then so is $\\operatorname{size}(\\mathcal{G})$.\n\tWe assume that this is the case and show that the subgames on which $\\operatorname{Reach}(\\mathcal{G})$ recurses are strictly smaller in this measure.\n\tThis is enough to guarantee termination.\n\t\n\tThe first subgame is constructed in line~\\ref{line:gpost} and takes the form:\n\t\\[\\mathcal{G}_{post} = \\langle \\operatorname{Post}(\\mathit{C})[\\varp\/\\var],\\mathit{Safe} \\land \\varphi, \\mathit{Reach} \\land \\varphi, \\mathit{Goal} \\rangle.\\]\n\tThe important restriction of this game is that both safety and reachability player transitions have the additional precondition $\\varphi$.\n\tWe may assume that $\\operatorname{Enf}(C,\\mathcal{G})$ is satisfiable, as otherwise the algorithm does not reach line~\\ref{line:gpost}.\n\tThen, in particular, $C$ is satisfiable, by the definition of $\\operatorname{Enf}(C,\\mathcal{G})$.\n\tBut $C = \\operatorname{Instantiate}(\\varphi,\\mathcal{G}) = (\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\neg \\varphi \\land \\varphi'$, which means that there exist states $s,s'$ such that $ (\\mathit{Safe} \\lor \\mathit{Reach})(s,s')$, $\\neg \\varphi(s)$, and $\\varphi(s')$ are all valid. \n\tThis transition from $s$ to $s'$ in $\\mathcal{G}$ is excluded in $\\mathcal{G}_{post}$, and as no new transitions are included, it follows that $\\operatorname{size}(\\mathcal{G}_{post}) < \\operatorname{size}(\\mathcal{G})$.\n\t\n\tThe second subgame is constructed in line~\\ref{line:gpre2} and takes the form:\n\t\\[\\mathcal{G}_{pre} = \\langle I,\\mathit{Safe} \\land \\lnot F,\\mathit{Reach} \\land \\lnot F, \\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G}))\\rangle.\\]\n  We may assume that $F \\land (\\mathit{Safe} \\lor \\mathit{Reach})$ is satisfiable, as otherwise the algorithm would not have moved past line~\\ref{line:safeavoidsE}.\n  Observe that if $F$ is changed in line~\\ref{line: E2} then it is only extended and hence satisfiability is preserved.\n  As no transition satisfying $F$ exists in $\\mathcal{G}_{pre}$ it follows that $\\operatorname{size}(\\mathcal{G}_{pre}) < \\operatorname{size}(\\mathcal{G})$.\n  This concludes the proof.\n\t\\qed\n\\end{proof}\n\n\\terminationbisim*\n\\begin{proof}\n\tLet $S_1,\\ldots,S_n$ be the bisimulation classes of $\\mathcal{G}$, and $\\psi_1, \\ldots, \\psi_n \\in \\cal L(\\mathcal{V})$ be the formulas that define them.\n\tWe define\n\t\\[\\operatorname{size}(\\mathcal{G}) = |\\{(S_i,S_j) \\mid (\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\psi_i \\land \\psi_j' \\text{ is satisfiable} \\}|,\\]\n\twhich equals the number of transitions in the bisimulation quotient of $\\mathcal{G}$ under $\\sim$.\n\tOur aim is to show that $\\operatorname{Reach}(\\cdot)$ terminates for all subgames that are considered in any recursive call of $\\operatorname{Reach}(\\mathcal{G})$.\n\t\n\tTo this end, we show that $\\operatorname{Reach}(\\mathcal{G})$ terminates for all reachability games $\\mathcal{G} = \\langle \\mathit{Init}, \\mathit{Safe}, \\mathit{Reach}, \\mathit{Goal} \\rangle$ such that\n\t\\begin{itemize}\n\t\\item $\\operatorname{size}(\\mathcal{G})$ is finite,\n  \\item the relation $\\sim$ is a bisimulation on $\\mathcal{G}$, and\n\t\\item $\\mathit{Goal}$ is equivalent to a disjunction of formulas $\\psi_i$.\n\t\\end{itemize}\n\tWe show this by induction on $\\operatorname{size}(\\mathcal{G})$.\n\t\n\tLet $\\mathcal{G} = \\langle \\mathit{Init}, \\mathit{Safe}, \\mathit{Reach}, \\mathit{Goal} \\rangle$ satisfy these conditions, and assume that $\\operatorname{size}(\\mathcal{G}) = 0$.\n\tThen it follows that $\\mathit{Safe} \\lor \\mathit{Reach}$ is unsatisfiable.\n\tThis is because if any $(s_1,s_2)$ would satisfy $\\mathit{Safe} \\lor \\mathit{Reach}$, then in particular $(\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\psi_i \\land \\psi_j'$ would be satisfied by $(s_1,s_2)$, where we assume $s_1 \\in \\S_i$ and $s_2 \\in \\S_j$.\n\tIt follows that $\\operatorname{Unsat}(\\operatorname{Enf}(C,\\mathcal{G}))$ in line~\\ref{line: cpre} is true, as $\\operatorname{Enf}(C,\\mathcal{G}) \\implies (\\mathit{Safe} \\lor \\mathit{Reach})$ is valid for any $C$.\n\tBut then Algorithm~\\ref{alg:algreach} terminates on input $\\mathcal{G}$.\n\t\n\tNow suppose that we have $\\mathcal{G}$ with $\\operatorname{size}(\\mathcal{G}) > 0$.\n\tIf the algorithm does not return in lines~\\ref{line:ret1} or~\\ref{line:safety wins}, we have to consider the first subgame\n\t\\[\\mathcal{G}_{post} = \\langle \\operatorname{Post}(C)[\\varp\/\\var], \\mathit{Safe} \\land \\varphi, \\mathit{Reach} \\land \\varphi, \\mathit{Goal} \\rangle,\\]\n\twhich is constructed in line~\\ref{line:gpost}.\n\tWe may assume that for some $I \\subseteq \\{1,\\ldots,n\\}$ we have $\\varphi \\equiv \\bigvee_{i \\in I} \\psi_i$, due to our assumption on the function $\\operatorname{Interpolate}$.\n\tHence the effect of restricting all transitions to $\\varphi$ is to remove all transitions in states not in $\\bigcup \\{S_i \\mid i \\in I\\}$, which are exactly the states in $\\bigcup \\{S_i \\mid i \\in \\{1,\\ldots,n\\} \\setminus I\\}$.\n\tIt is clear that $\\sim$ is still a bisimulation in the resulting game, and that the goal states are preserved.\n\tTo see that $\\operatorname{size}(\\mathcal{G}_{post}) < \\operatorname{size}(\\mathcal{G})$ we may assume that $\\operatorname{Unsat}(\\operatorname{Enf}(C,\\mathcal{G}))$ is false, otherwise we would have returned in line~\\ref{line:safety wins}.\n\tThen, in particular, there is a transition in $\\mathcal{G}$ satisfying $\\neg \\varphi$, which means that there is a pair $S_i,S_j$ such that $(\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\neg \\varphi \\land \\psi_i \\land \\psi_j'$ is satisfiable.\n\tThis is cleary unsatisfiable when replacing $(\\mathit{Safe} \\lor \\mathit{Reach})$ by $(\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\varphi$.\n\tHence, $\\operatorname{size}(\\mathcal{G}_{post}) < \\operatorname{size}(\\mathcal{G})$.\n\tAs a result, we can apply the induction hypothesis to conclude that the recursive call $\\operatorname{Reach}(\\mathcal{G}_{post})$ in line~\\ref{line: recursion1} terminates.\n\t\n\tNow let us consider the second subgame $\\mathcal{G}_{pre}$, as constructed in line~\\ref{line:gpre2}.\n\tFirst, we observe that $F \\equiv (\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\neg \\varphi \\land \\varphi' \\land R_{post}[\\var\/\\varp]$, where $R_{post}$ is a state predicate characterizing the initial winning states of $\\mathcal{G}_{post}$ (this uses~\\Cref{thm:partcorr}).\n\tAs $\\sim$ is a bisimulation on $\\mathcal{G}_{post}$, it follows by~\\Cref{lem:bisimpreservesreach} that $R_{post}$ is equivalent to a disjunction of formulas $\\psi_i$.\n\tAs a consequence, we can equivalently write $F$ as $\\phi_1 \\land (\\phi_2[\\var\/\\varp])$ for two formulas $\\phi_1,\\phi_2 \\in \\cal L(\\mathcal{V})$ that are both equivalent to disjunctions of $\\psi_i$.\n\tBy~\\Cref{lem:preenfbisim} it follows that $\\operatorname{Pre}(\\operatorname{Enf}(E,\\mathcal{G}))$ is also equivalent to a disjunction of $\\psi_i$.\n\t\n\tRestricting transitions to $\\neg F$ in $\\mathcal{G}_{pre}$ has the effect of removing all transitions from states in $\\bigcup\\{S_i \\mid \\psi_i \\implies \\phi_1 \\text{ is valid}\\}$ to states in $\\bigcup\\{S_i \\mid \\psi_i \\implies \\phi_2  \\text{ is valid}\\}$.\n\tIt is clear that $\\sim$ is still a bisimulation in the resulting game.\n\tFurthermore, as $\\operatorname{Enf}(F,\\mathcal{G})$ is satisfiable, there is at least one such transition in $\\mathcal{G}$.\n\tIt follows that $\\operatorname{size}(\\mathcal{G}_{pre}) < \\operatorname{size}(\\mathcal{G})$ and hence the algorithm terminates by induction hypothesis.\n\t\\qed\n\\end{proof}\n\n\\begin{lemma}\n\t\\label{lem:preenfbisim}\n\tLet $\\sim$ be a bisimulation on $\\mathcal{G}$ which is also an equivalence relation, and $S_1,\\ldots, S_n$ be its equivalence classes.\n\tAssume that $S_1,\\ldots, S_n$ are defined by $\\psi_1,\\ldots, \\psi_n \\in \\cal L(\\mathcal{V})$.\n\tLet $\\phi_1 \\land (\\phi_2[\\var\/\\varp]) \\in \\cal L(\\mathcal{V} \\cup \\mathcal{V'})$ be such that both $\\phi_1,\\phi_2$ are equivalent to disjunctions of formulas $\\psi_i$.\n\t\n\tThen, $\\operatorname{Pre}(\\operatorname{Enf}(\\phi_1 \\land (\\phi_2[\\var\/\\varp]),\\mathcal{G}))$ is equivalent to a disjunction of formulas $\\psi_i$.\n\\end{lemma}\n\\begin{proof}\n\tWe show that if there exists a state in $S_i$ that satisfies $\\operatorname{Pre}(\\operatorname{Enf}(\\phi_1 \\land (\\phi_2[\\var\/\\varp]),\\mathcal{G}))$, then so do all states in $S_i$.\n\tLet $s_1 \\in S_i$ be such that $\\operatorname{Pre}(\\operatorname{Enf}(\\phi_1 \\land (\\phi_2[\\var\/\\varp]),\\mathcal{G}))(s_1)$ is valid.\n\t\n\tWe make a case distinction on whether $s_1 \\in S_{\\texttt{{REACH}}}$ holds.\n\tIf so, then there exists a state $q_1$ such that $(\\mathit{Reach} \\land \\phi_1 \\land (\\phi_2[\\var\/\\varp]))(s_1,q_1)$ is valid.\n\tIn particular, $\\phi_1(s_1)$ and $\\phi_2(q_1)$ are both valid.\n\tAssuming that $q_1 \\in S_j$ holds, both $\\psi_i \\implies \\phi_1$ and $\\psi_j \\implies \\phi_2$ are valid, as both $\\phi_1$ and $\\phi_2$ are equivalent to disjunctions of $\\psi$-formulas (which have pairwise disjoint sets of models).\n\tNow take any other state $s_2 \\in S_i$.\n\tAs $s_1 \\sim s_2$ and $\\mathit{Reach}(s_1,q_1)$ is valid, there exists a state $q_2 \\in S_j$ such that $\\mathit{Reach}(s_2,q_2)$ is valid.\n\tFurthermore, as $\\psi_i(s_2)$ and $\\psi_j(q_2)$ are both valid, so is $(\\mathit{Reach} \\land \\phi_1 \\land (\\phi_2[\\var\/\\varp]))(s_2,q_2)$.\n\tHence, $\\operatorname{Pre}(\\operatorname{Enf}(\\phi_1 \\land (\\phi_2[\\var\/\\varp]),\\mathcal{G}))(s_2)$ is valid.\n\t\n\tNow assume that $s_1 \\in S_{\\texttt{{SAFE}}}$.\n\tThen, for all states $q_1$ such that $\\mathit{Safe}(s_1,q_1)$ is valid, $(\\phi_1 \\land (\\phi_2[\\var\/\\varp]))(s_1,q_1)$ holds.\n\tWhenever this is the case, and $q_1 \\in S_j$ holds, it follows that $\\psi_j \\implies \\phi_2$ is valid.\n\n\tNow take any other state $s_2 \\in S_i$ and assume, for contradiction, that there exists a $q_2$ such that $\\mathit{Safe}(s_2,q_2)$ is valid, but not $(\\phi_1 \\land (\\phi_2[\\var\/\\varp]))(s_2,q_2)$.\n\tAssuming $q_2 \\in S_j$, we have that $\\psi_j \\land \\phi_2$ is unsatisfiable.\n\tAs $s_1 \\sim s_2$ holds, we find $q_1$ such that $\\mathit{Safe}(s_1,q_1) \\land \\psi_j(q_1)$ is valid.\n  By the previous reasoning, this would imply that $\\psi_j \\implies \\phi_2$ is valid.\n\tThis is a contradiction as $\\psi_j$ is satisfiable.\n\t\\qed\n\\end{proof}\n\n\\section{Conclusion}\nOur work is a step towards the fully automated synthesis of software. \nIt targets symbolically represented reachability games which are expressive enough to model a variety of problems, from common game benchmarks to program synthesis problems. \nThe presented approach exploits causal information in the form of \\emph{subgoals}, which are parts of the game that the reachability player needs to pass through in order to win.\nHaving computed a subgoal, which can be done using Craig interpolation, the game is split along the subgoal and solved recursively.\nAt the same time, the algorithm infers a structured symbolic strategy for the winning player.\nThe evaluation of our prototype implementation \\textsc{CabPy} shows that our approach is practically applicable and scales much better than previously available tools on several benchmarks.  \nWhile termination is only guaranteed for games with finite bisimulation quotient, the experiments demonstrate that several infinite games can be solved as well.\n\nThis work opens up several interesting questions for further research.\nOne concerns the quality of the returned strategies.\nDue to its compositional nature, at first sight it seems that our approach is not well-suited to handle global optimization criteria, such as reaching the goal in fewest possible steps.\nOn the other hand, the returned strategies often involve only a few key decisions and we believe that therefore the strategies are often very sparse, although this has to be further investigated.\nWe also plan to automatically extract deterministic strategies from the symbolic ones~\\cite{Bloem,Ehlers} we currently consider.\n\nAnother question regards the computation of subgoals. \nThe performance of our algorithm is highly influenced by which interpolant is returned by the solver.\nIn particular this affects the number of subgames that have to be solved, and how complex they are.\nWe believe that template-based interpolation \\cite{template_interpolation} could be a promising candidate to explore with the goal to compute good interpolants. \nThis could be combined with the possibility for the user to provide templates or expressive interpolants directly, thereby benefiting from the user's domain knowledge.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\n\\section{Motivating Example} \\label{sec:motivation}\n\nConsider the scenario that an expensive painting is displayed in a large exhibition room of a museum.\nIt is secured with an alarm system that is controlled via a control panel on the opposite side of the room.\nA security guard is sleeping at the control panel and occasionally wakes up to check whether the alarm is still armed.\nTo steal the painting, a thief first needs to disable the alarm and then reach the painting before the alarm has been reactivated. We model this scenario as a two-player game between a safety player (the guard) and a reachability player (the thief) in the theory of linear arithmetic.\nThe moves of both players, their initial positions, and the goal condition are described by the formulas:\n\n\n\\iffalse\n\\begin{figure}[t]\n\t\\centering\n\t\\scalebox{0.4}{\n\t\t\\begin{tikzpicture}[node distance=2cm]\n\t\t\\node(mona) at (10,5){\\def\\svgwidth{1cm}\\input{monalisa_color.pdf_tex}};\n\t\t\\node(thief) at (0,0){\\def\\svgwidth{1.15cm}\\input{thief_color.pdf_tex}};\n\t\t\\node(guard) at (0,10){\\def\\svgwidth{2cm}\\input{guard2_color.pdf_tex}};\n\t\t\\draw[step=2.0,black,thin] (-0.5,-0.5) grid (10.5,10.5);\n\t\t\\node(guard) at (-1,0.0){$0$};\n\t\t\\node(guard) at (0,-1){$0$};\n\t\t\\node(guard) at (10.0,-1){$10$};\n\t\t\\node(guard) at (-1,10.0){$10$};\n\t\t\\end{tikzpicture}}\n\t\\caption{The Mona Lisa problem. The painting is secured with an alarm the thief has to disable first in order to remain undetected. The sleeping guard will occasionally wake up to check whether the alarm is still on.}\n\t\\label{exp_monalisa_pic}\n\\end{figure}\n\\fi\n\n\n\\begin{align*}\n\\mathit{Init} &\\equiv && \\lnot \\mathbf{r} \\land x = 0 \\land y = 0 \\land p = 0 \\land a = 1 \\land t = 0, &&&\\\\\n\\mathit{Guard} &\\equiv &&\\neg\\mathbf{r} \\land \\mathbf{r}' \\land x' = x \\land y' = y \\land p' = p  &&&\\\\\n& &&\\land ((t' = t - 1 \\land a' = a)\\lor (t \\leq 0 \\land t' = 2)),&&&(\\text{sleep or wake up})\\\\\n\\mathit{Thief} &\\equiv &&  \\mathbf{r} \\land \\neg\\mathbf{r}' \\land t' = t \\\\\n\t&&&\\land x + 1 \\ge x' \\ge x - 1 \\land y + 1 \\ge y' \\ge y - 1&&&(\\text{move})\\\\\n& &&\\land (x' \\neq 0 \\lor y' \\neq 10 \\implies a' = a)&&&(\\text{alarm off})\\\\\n& &&\\land (x' \\neq 10 \\lor y' \\neq 5 \\lor a = 1 \\implies p' = p),&&&(\\text{steal})\\\\\n\\mathit{Goal} &\\equiv && \\lnot \\mathbf{r} \\land p = 1. &&&\n\\end{align*} \n\nThe thief's position in the room is modeled by two coordinates $x,y \\in \\mathbb{R}$ with initial value $(0,0)$, and with every transition the thief can move some bounded distance. \nNote that we use primed variables to represent the value of variables after taking a transition.\nThe control panel is located at $(0,10)$ and the painting at $(10,5)$. \nThe status of the alarm and the painting are described by two boolean variables $a,p \\in \\{0,1\\}$. \nThe guard wakes up every two time units, modeled by the variable $t \\in \\mathbb{R}$. \nThe variables $x,y$ are bounded to the interval $[0,10]$ and $t$ to $[0,2]$. \nThe boolean variable \\textbf{r} encodes who makes the next move. \nIn the presented configuration, the thief needs more time to move from the control panel to the painting than the guard will sleep. \nIt follows that there is a winning strategy for the guard, namely, to always reactivate the alarm upon waking up.\n\nAlthough it is intuitively fairly easy to come up with this strategy for the guard, it is surprisingly hard for game solving tools to find it. The main obstacle is the infinite state space of this game.\nOur approach for solving games represented in this logical way imitates \\emph{causal reasoning}: \nHumans observe that in order for the thief to steal the painting (i.e., the effect $p=1$), a transition must have been taken whose source state does not satisfy the pre-condition of (steal) while the target state does.  \nPart of this cause is the condition $a=0$, i.e., the alarm is off. Recursively, in order for the effect $a=0$ to happen, a transition setting $a$ from $1$ to $0$ must have occurred, and so on. \n\nOur approach captures these cause-effect relationships through the notion of \\emph{necessary subgoals}, which are essential milestones that the reachability player has to transition through in order to achieve their goal.\nThe first necessary subgoal corresponding to the intuitive description above is\n$$C_1 = (\\mathit{Guard} \\lor \\mathit{Thief}) \\land p \\neq 1 \\land p' = 1.$$\nIn this case, it easy to see that $C_1$ is also a \\emph{sufficient subgoal}, meaning that all successor states of $C_1$ are winning for the thief. Therefore, it is enough to solve the game with the modified objective to reach those predecessor states of $C_1$ from which the thief can \\emph{enforce} $C_1$ being the next move (even if it is not their turn). Doing so recursively produces the necessary subgoal\n$$C_2 = (\\mathit{Guard} \\lor \\mathit{Thief}) \\land a \\neq 0 \\land a' = 0,$$\nmeaning that some transition must have caused the effect that the alarm is disabled. However, $C_2$ is \\emph{not} sufficient which can be seen by recursively solving the game spanning from successor states of $C_2$ to $C_1$. This computation has an important caveat: After passing through $C_2$, it may happen that $a$ is reset to $1$ at a later point (in this particular case, this constitutes precisely the winning strategy of the safety player), which means that there is no canonical way to slice the game along this subgoal into smaller parts. Hence the recursive call solves the game from $C_2$ to $C_1$ \\emph{subject to} the bold assumption that any move from $a = 0$ to $a' = 1$ is winning for the guard. This generally underapproximates the winning states of the thief. Remarkably, we show that this approximation is enough to build winning strategies for \\emph{both} players from their respective winning regions. In this case, it allows us to infer that moving through $C_2$ is always a losing move for the thief. However, at the same time, any play reaching $\\mathit{Goal}$ has to move through $C_2$. It follows that the thief loses the global game.\n\nWe evaluated our method on several configurations of this game, which we call \\emph{Mona Lisa}. The results in Section \\ref{sec:experiments} support our conjecture that the room size has little influence on the time our technique needs to solve the game.\n\n\\section{Case Studies}\n\\label{sec:experiments}\n\nIn this section we evaluate our approach on a number of case studies. \nOur prototype \\textsc{CabPy}{}\\footnote[2]{The source code of \\textsc{CabPy}{} and our experimental data are both available at \\url{https:\/\/github.com\/reactive-systems\/cabpy}. We provide a virtual machine image with \\textsc{CabPy}{} already installed for reproducing our evaluation~\\cite{VM}.} is written in Python and implements the game solving part of the presented algorithm. \nExtending it to returning a symbolic strategy using the ideas outlined above is straightforward.\nWe compared our prototype with \\textsc{SimSynth} \\cite{FarzanK17}, the only other readily available tool for solving linear arithmetic games. \nThe evaluation was carried out with Ubuntu 20.04,  a 4-core Intel\\textsuperscript{\\textregistered} Core\\texttrademark~i5 2.30GHz processor, as well as 8GB of memory. \\textsc{CabPy}{} uses the PySMT~\\cite{pysmt2015} library as an interface to the MathSAT5~\\cite{mathsat5} and Z3~\\cite{z3solver} SMT solvers.\nOn all benchmarks, the timeout was set to 10 minutes. In addition to the winner, we report the runtime and the number of subgames our algorithm visits. Both may vary with different SMT solvers or in different environments.\n\\subsection{Game of Nim}\n\nGame of Nim is a classic game from the literature \\cite{Bouton1901} and played on a number of heaps of stones. Both players take turns of choosing a single heap and removing at least one stone from it. We consider the version where the player that removes the last stone wins. Our results are shown in \\Cref{exp_nim}. In instances with three heaps or more we bounded the domains of the variables in the instance description, by specifying that no heap exceeds its initial size and does not go below zero.\n\nFollowing the discussion in \\Cref{sec:termination}, we need to bound the domains to ensure the termination of our tool on these instances.\nRemarkably, bounding the variables was not necessary for instances with only two heaps, where our tool \\textsc{CabPy}{} scales to considerably larger instances than \\textsc{SimSynth}.\nWe did not add the same constraints to the input of \\textsc{SimSynth}{}, as for \\textsc{SimSynth}{} this resulted in longer runtimes rather than shorter.\nIn Game of Nim, there are no natural necessary subgoals that the safety player can locally control.\n\nThe results (see~\\Cref{exp_nim}) demonstrate that our approach is not completely dependent on finding the right interpolants and is in particular also competitive when the reachability player wins the game. We suspect that \\textsc{SimSynth}{} performs worse in these cases because the safety player has a large range of possible moves in most states, and inferring the win of the reachability player requires the tool to backtrack and try our all of them.\n\n\\begin{figure}[tbp]\n\t\\centering\n\t{\\def\\arraystretch{1.1}\\tabcolsep=5pt\n    \\small\n\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{2}{c|}{\\textsc{CabPy}} & \\textsc{SimSynth} & \\\\\n\t\t\t\\hline\n\t\t\tHeaps & Subgames & Time(s) &Time(s) & Winner\\\\\n\t\t\t\\hline\n\t\t\t(4,4) & 19 & 1.50 & 10.44 & $\\texttt{{REACH}}$\\\\\n\t\t\t(4,5) & 23 & 1.92 & 12.74 & $\\texttt{{SAFE}}$\\\\\n\t\t\t(5,5) & 23 & 1.99 & 85.75 & $\\texttt{{REACH}}$\\\\\n\t\t\t(5,6) & 27 & 2.90 & 91.66 & $\\texttt{{SAFE}}$\\\\\n\t\t\t(6,6) & 28 & 3.04 & Timeout & $\\texttt{{REACH}}$\\\\\n\t\t\t(6,7) & 31 & 3.76 & Timeout & $\\texttt{{SAFE}}$\\\\\n\t\t\t(20,20) & 88 & 94.85 & Timeout & $\\texttt{{REACH}}$\\\\\n\t\t\t(20,21) & 94 & 113.04 & Timeout & $\\texttt{{SAFE}}$\\\\\n\t\t\t(30,30) & 128 & 364.13 & Timeout & $\\texttt{{REACH}}$\\\\\n\t\t\t(30,31) & 135 & 404.02 & Timeout & $\\texttt{{SAFE}}$\\\\\\hline\n\t\t\t(3,3,3)b & 23 & 13.63 & 2.85 & $\\texttt{{SAFE}}$\\\\\n\t\t\t(1,4,5)b & 32 & 7.00 & 289.85 & $\\texttt{{REACH}}$\\\\\n\t\t\t(4,4,4)b & 33 & 50.55 & 24.39 & $\\texttt{{SAFE}}$\\\\\n\t\t\t(2,4,6)b & 38 & 19.77 & Timeout & $\\texttt{{REACH}}$\\\\\n\t\t\t(5,5,5)b & 33 & 127.89 & 162.50 & $\\texttt{{SAFE}}$\\\\\n\t\t\t(3,5,6)b & 40 & 86.56 & Timeout & $\\texttt{{REACH}}$\\\\\\hline\n\t\t\t(2,2,2,2)b & 39 & 84.79 & 213.79 & $\\texttt{{REACH}}$\\\\\n\t\t\t(2,2,2,3)b & 41 & 102.01 & Timeout & $\\texttt{{SAFE}}$\\\\\n\t\t\t\\hline\n\t\\end{tabular}}\n\t\\caption{Experimental results for the Game of Nim. The notation $(h_1,\\ldots, h_n)$ denotes the instance played on $n$ heaps, each of which consists of $h_i$ stones. Instances marked with b indicate that the variable domains were explicitly bounded in the input for \\textsc{CabPy}{}.}\n\t\\label{exp_nim}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\t\\centering\n\t{\\def\\arraystretch{1.1}\\tabcolsep=5pt\n    \\small\n\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{2}{c|}{\\textsc{CabPy}} & \\textsc{SimSynth} & \\\\\n\t\t\t\\hline\n\t\t\t$r$ & Subgames & Time(s) &Time(s) & Winner\\\\\n\t\t\t\\hline\n\t\t\t10 & 10 & 0.57 & 3.93 & $\\texttt{{SAFE}}$\\\\\n\t\t\t20 & 20 & 1.23 & 20.48 & $\\texttt{{SAFE}}$\\\\\n\t\t\t40 & 40 & 3.42 & 121.96 & $\\texttt{{SAFE}}$\\\\\n\t\t\t60 & 60 & 7.36 & Timeout & $\\texttt{{SAFE}}$\\\\\n\t\t\t80 & 80 & 17.72 & Timeout & $\\texttt{{SAFE}}$\\\\\n\t\t\t100 & 100 & 26.36 & Timeout & $\\texttt{{SAFE}}$\\\\\n\t\t\t\\hline\n\t\\end{tabular}}\n\t\\caption{Experimental results for the Corridor game. The safety player controls the door between rooms $r-1$ and $r$.}\n\t\\label{exp_corridor}\n\\end{figure}\n\n\\subsection{Corridor}\n\nWe now consider an example that demonstrates the potential of our method in case the game structure contains natural bottlenecks. Consider a corridor of $100$ rooms arranged in sequence, i.e., each room $i$ with $0 \\leq i < 100$ is connected to room $i+1$ with a door. The objective of the reachability player is to reach room 100 and they are free to choose valid values from $\\mathbb{R}^2$ for the position in each room at every other turn. The safety player controls some door to a room $r \\leq 100$. Naturally, a winning strategy is to prevent the reachability player from passing that door, which is a natural bottleneck and necessary subgoal on the way to the last room.\n\n The experimental results are summarized in Figure \\ref{exp_corridor}. We evaluated several versions of this game, increasing the length from the start to the controlled door. The results confirm that our causal synthesis algorithm finds the trivial strategy of closing the door quickly. This is because Craig interpolation focuses the subgoals on the room number variable while ignoring the movement in the rooms in between, as can be seen by the number of considered subgames. \\textsc{SimSynth}, which tries to generalize a strategy obtained from a step-bounded game, struggles because the tool solves the games that happen between each of the doors before reaching the controlled one.\n\n\\subsection{Mona Lisa}\n\nThe game described in Section \\ref{sec:motivation} between a thief and a security guard is very well suited to further assess the strength and limitations of both our approach as well as of \\textsc{SimSynth}{}. We ran several experiments with this scenario, scaling the size of the room and the sleep time of the guard, as well as trying a scenario where the guard does not sleep at all. Scaling the size of the room makes it harder for \\textsc{SimSynth}{} to solve this game with a forward unrolling approach, while our approach extracts the necessary subgoals irrespective of the room size. However, scaling the guard's sleep time makes it harder to solve the subgame between the two necessary subgoals, while it only has a minor effect on the length of the unrolling needed to stabilize the play in a safe region, as done by \\textsc{SimSynth}.\n\n The results in Figure \\ref{exp_monalisa} support this conjecture. The size of the room has \\emph{almost no effect at all} on both the runtime of \\textsc{CabPy}{} and the number of considered subgames. However, as the results for a sleep value of 4 show, the employed combination of quantifier elimination and interpolation introduces some instability in the produced formulas. This means we may get different Craig interpolants and slice the game with more or less subgoals. Therefore, we see a lot of potential in optimizing the interplay between the employed tools for quantifier elimination and interpolation. The phenomenon of the runtime being sensitive to these small changes in values is also seen with \\textsc{SimSynth}, where a longer sleep time sometimes means a faster execution.\n\n\\begin{figure}[tbp]\n\t\\centering\n    {\\def\\arraystretch{1.1}\\tabcolsep=5pt\n      \\small\n\t\t\\begin{tabular}{|c|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t&  & \\multicolumn{2}{c|}{\\textsc{CabPy}} & \\textsc{SimSynth} & \\\\\n\t\t\t\\hline\n\t\t\tSize & Sleep & Subgames & Time(s) & Time(s) & Winner\\\\\n\t\t\t\\hline\n\t\t\t$10\\times10$ & - & 7 & 0.61 & 4.79 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$20\\times20$ & - & 7 & 0.60 & 25.26 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$40\\times40$ & - & 7 & 0.61 & 157.62 & $\\texttt{{SAFE}}$\\\\\n\t\t\t\\hline\n\t\t\t$10\\times10$ & 1 & 10 & 4.22 & 20.31 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$20\\times20$ & 1 & 11 & 4.34 & 36.44 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$40\\times40$ & 1 & 11 & 4.65 & 226.14 & $\\texttt{{SAFE}}$\\\\\n\t\t\t\\hline\n\t\t\t$10\\times10$ & 2 & 13 & 5.88 & 7.40 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$20\\times20$ & 2 & 14 & 5.98 & 60.00 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$40\\times40$ & 2 & 13 & 5.92 & 270.48 & $\\texttt{{SAFE}}$\\\\\n\t\t\t\\hline\n\t\t\t$10\\times10$ & 3 & 18 & 26.58 & 13.94 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$20\\times20$ & 3 & 17 & 26.19 & 115.53 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$40\\times40$ & 3 & 18 & 27.85 & 290.12 & $\\texttt{{SAFE}}$\\\\\n\t\t\t\\hline\n\t\t\t$10\\times10$ & 4 & 30 & 175.27 & 13.96 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$20\\times20$ & 4 & 22 & 204.79 & 60.08 & $\\texttt{{SAFE}}$\\\\\n\t\t\t$40\\times40$ & 4 & 27 & 123.95 & 319.47 & $\\texttt{{SAFE}}$\\\\\n\t\t\t\\hline\n\t\\end{tabular}}\n\t\\caption{Experimental results for the Mona Lisa game.}\n\t\\label{exp_monalisa}\n\\end{figure}\n\n\\subsection{Program Synthesis}\n\nLastly, we study two benchmarks that are directly related to program synthesis. \nThe first problem is to synthesize a controller for a thermostat by filling out an incomplete program, as described in \\cite{BeyeneCPR14}. A range of possible initial values of the room temperature $c$ is given, e.g., $20.8 \\leq c \\leq 23.5$, together with the temperature dynamics which depend on whether the heater is on (variable $o \\in \\mathbb{B}$).\nThe objective for $\\texttt{{SAFE}}$ is to control the value of $o$ in every round such that $c$ stays between $20$ and $25$. This is a common benchmark for program synthesis tools and both \\textsc{CabPy}{} and \\textsc{SimSynth}{} solve it quickly.\nThe other problem relates to Lamport's bakery algorithm\\cite{Lamport1974}. We consider two processes using this protocol to ensure mutually exclusive access to a shared resource. The game describes the task of synthesizing a scheduler that violates the mutual exclusion. This essentially is a model checking problem, and we study it to see how well the tools can infer a safety invariant that is out of control of the safety player. For our approach, this makes no difference, as both players may play through a subgoal and the framework is well suited to find a safety invariant. The forward unrolling approach of \\textsc{SimSynth}, however, seems to explore the whole state space before inferring safety, and fails to find an invariant before a timeout. \n\n\\begin{figure}[tbp]\n\t\\centering\n\t{\\def\\arraystretch{1.1}\\tabcolsep=5pt\n    \\small\n\t\t\\begin{tabular}{|c|c|c|c|c|}\n\t\t\t\\hline\n\t\t\t& \\multicolumn{2}{c|}{\\textsc{CabPy}} & \\textsc{SimSynth} & \\\\\n\t\t\t\\hline\n\t\t\tName & Subgames & Time(s) &Time(s) & Winner\\\\\n\t\t\t\\hline\n\t\t\tThermostat & 6 & 0.44 & 0.39 & $\\texttt{{SAFE}}$ \\\\\n\t\t\tBakery & 46 & 18.25 & Timeout & $\\texttt{{SAFE}}$ \\\\\n\t\t\t\\hline\n\t\\end{tabular}}\n\t\\caption{Experimental results for program synthesis problems.}\n\t\\label{exp_synth}\n\\end{figure}\n\n\n\n\n\\section{Introduction}\nTwo-player games are a fundamental model in logic and verification due to their connection to a wide range of topics such as decision procedures, synthesis and control~\\cite{Harding05,Alur15,Alur16,BLOEM2012911,BLOEM20073,6225075}. \nAlgorithmic techniques for \\emph{finite-state} two-player games have been studied extensively for many acceptance conditions~\\cite{GraedelTW2002}.\nFor \\emph{infinite-state} games most problems are directly undecidable. \nHowever, infinite state spaces occur naturally in domains like software synthesis~\\cite{RyzhykCKLH09} and cyber-physical systems~\\cite{JessenRLD07}, and hence handling such games is of great interest. \nAn elegant classification of infinite-state games that can be algorithmically handled, depending on the acceptance condition of the game, was given in~\\cite{deAlfaroHM2001}.\nThe authors assume a symbolic encoding of the game in a very general form.\nMore recently, incomplete procedures for solving infinite-state two-player games specified using logical constraints were studied~\\cite{BeyeneCPR14,FarzanK17}.\nWhile~\\cite{BeyeneCPR14} is based on automated theorem-proving for Horn formulas and handles a wide class of acceptance conditions, the work in~\\cite{FarzanK17} focusses on reachability games specified in the theory of linear arithmetic, and uses sophisticated decision procedures for that theory.\n\nIn this paper, we present a novel technique for solving logically represented reachability games based on the notion of \\emph{subgoals}.\nA \\emph{necessary} subgoal is a transition predicate that is satisfied at least once on every play that reaches the overall goal.\nIt represents an intermediate target that the reachability player must reach in order to win. Subgoals open up game solving to the study of cause-effect relationships in the form of counterfactual reasoning~\\cite{counterfactual}: If a cause (the subgoal) had not occurred, then the effect (reaching the goal) would not have happened.\nThus for the safety player, a necessary subgoal provides a chance to win the game based on local information:\nIf they control all states satisfying the pre-condition of the subgoal, then any strategy that in these states picks a transition outside of the subgoal is  winning. \nFinding such a necessary subgoal may let us conclude that the safety player wins without ever having to unroll the transition relation.\n\nOn the other hand, passing through a necessary subgoal is in general not enough for the reachability player to win. We call a subgoal \\emph{sufficient} if indeed the reachability player has a winning strategy from every state satisfying the post-condition of the subgoal.\nDual to the description in the preceding paragraph, sufficient subgoals provide a chance for the reachability player to win the global game as they must merely reach this intermediate target. The two properties differ in one key aspect: While necessity of a subgoal only considers the paths of the game arena, for sufficiency the game structure is crucial.  \n\nWe show how Craig interpolants can be used to compute necessary subgoals, making our methods applicable to games represented by any logic that supports interpolation. In contrast, determining whether a subgoal is sufficient requires a partial solution of the given game. This motivates the following recursive approach. We slice the game along a necessary subgoal into two parts, the pre-game and the post-game.\nIn order to guarantee these games to be smaller, we solve the post-game under the assumption that the considered subgoal was bridged \\emph{for the last time}.\nWe conclude that the safety player wins the overall game if they can avoid all initial states of the post-game that are winning for the reachability player.\nOtherwise, the pre-game is solved subject to the winning condition given by the sufficient subgoal consisting of these states. \nThis approach does not only determine which player wins from each initial state, but also computes symbolically represented winning strategies with a causal structure. \nWinning safety player strategies induce necessary subgoals that the reachability player cannot pass, which constitutes a cause for their loss. Winning reachability player strategies represent a sequence of sufficient subgoals that will be passed, providing an explanation for the win.\n\nThe Python-based implementation \\textsc{CabPy}{} of our approach was used to compare its performance to \\textsc{SimSynth} \\cite{FarzanK17}, which is, to the best of our knowledge, the only other available tool for solving linear arithmetic reachability games. Our experiments demonstrate that our algorithm is competitive in many case studies. We can also confirm the expectation that our approach heavily benefits from qualitatively expressive Craig interpolants. It is noteworthy that like \\textsc{SimSynth}{} our approach is fully automated and does not require any input in the form of hints or templates. \nOur contributions are summarized as follows:\n\\begin{itemize}[topsep=0.5ex]\n\t\\item We introduce the concept of \\emph{necessary} and \\emph{sufficient subgoals} and show how Craig interpolation can be used to compute necessary subgoals (\\Cref{sec:subgoals}).\n\t\\item We describe an algorithm for solving logically represented two-player reachability games using these concepts.\n    We also discuss how to compute representations of winning strategies in our approach (\\Cref{sec:gamesolving}).\n\t\\item We evaluate our approach experimentally through our Python-based tool \\textsc{CabPy}, demonstrating a competitive performance compared to the previously available tool \\textsc{SimSynth}{} on various case studies (\\Cref{sec:experiments}).\n\\end{itemize}\n\n\\textbf{Related Work. } \nThe problem of solving linear arithmetic games is addressed in~\\cite{FarzanK17} using an approach that relies on a dedicated decision procedure for quantified linear arithmetic formulas, together with a method to generalize safety strategies from truncated versions of the game  that end after a prescribed number of rounds.\nOther approaches for solving infinite-state games include deductive methods that compute the winning regions of both players using proof rules~\\cite{BeyeneCPR14}, \npredicate abstraction where an abstract controlled predecessor operation is used on the abstract game representation~\\cite{WalkerR14}, \nand symbolic BDD-based exploration of the state space~\\cite{Edelkamp2002}. \nAdditional techniques are available for finite-state games, e.g., generalizing winning runs into a winning strategy for one of the players~\\cite{NarodytskaLBRW14}. \n\nOur notion of subgoal is related to the concept of landmarks as used in planning~\\cite{HoffmannPS11}. Landmarks are milestones that must be true on every successful plan, and they can be used to decompose a planning task into smaller sub-tasks. \nLandmarks have also been used in a game setting to prevent the opponent from reaching their goal using counter-planning~\\cite{PozancoEFB18}. \nWhenever a planning task is unsolvable, one method to find out why is checking hierarchical abstractions for solvability and finding the components causing the problem~\\cite{SreedharanSSK19}. \n\nCausality-based approaches have also been used for model checking of multi-threaded concurrent programs~\\cite{DBLP:conf\/concur\/KupriyanovF13,DBLP:conf\/cav\/KupriyanovF14}. \nIn our approach, we use Craig interpolation to compute the subgoals. \nInterpolation has already been used in similar contexts before, for example to extract winning strategies from game trees~\\cite{EenLNR15} or to compute new predicates to refine the game abstractions~\\cite{SlicingAbstractions}. \nIn ~\\cite{FarzanK17}, interpolation is used to synthesize concrete winning strategies from so called \\emph{winning strategy skeletons}, which describe a set of strategies of which at least one is winning.\n\n\n\n\\section{Preliminaries}\n\\label{sec:prelims}\nWe consider two-player reachability games defined by formulas in a given logic $\\cal L$.\nWe let $\\cal L(\\mathcal{V})$ be the $\\cal L$-formulas over a finite set of variables $\\mathcal{V}$, also called \\emph{state predicates} in the following.\nWe call $\\mathcal{V'} = \\{\\mathit{v'} \\mid \\mathit{v} \\in \\mathcal{V}\\}$ the set of \\emph{primed variables}, which are used to represent the value of variables after taking a transition.\nTransitions are expressed by formulas in the set $\\cal L(\\mathcal{V} \\cup \\mathcal{V'})$, called \\emph{transition predicates}.\nFor some formula $\\varphi \\in \\cal L(\\mathcal{V})$, we denote the substitution of all variables by their primed variant by $\\varphi[\\var\/\\varp]$. Similarly, we define $\\varphi[\\varp\/\\var]$.\n\nFor our algorithm we will require the satisfiability problem of $\\cal L$-formulas to be decidable and \\emph{Craig interpolants} \\cite{Craig1957} to exist for any two mutually unsatisfiable formulas.\nFormally, we assume there is a function $\\operatorname{Sat} : \\cal L(\\mathcal{V}) \\to \\mathbb{B}$ that checks the satisfiability of some formula $\\varphi \\in \\cal L(\\mathcal{V})$ and an unsatisfiability check $\\operatorname{Unsat} : \\cal L(\\mathcal{V}) \\rightarrow \\mathbb{B}$. \nFor interpolation, we assume that there is a function $\\operatorname{Interpolate} : \\cal L(\\mathcal{V}) \\times \\cal L(\\mathcal{V}) \\to \\cal L(\\mathcal{V})$ computing a \\emph{Craig interpolant} for mutually unsatisfiable formulas: If $\\varphi ,\\psi \\in \\cal L(\\mathcal{V})$ are such that $\\operatorname{Unsat}(\\varphi\\land\\psi)$ holds, then $\\psi \\implies \\operatorname{Interpolate}(\\varphi,\\psi)$ is valid, $\\operatorname{Interpolate}(\\varphi,\\psi)\\land \\varphi$ is unsatisfiable, and $\\operatorname{Interpolate}(\\varphi,\\psi)$ only contains variables shared by $\\varphi$ and $\\psi$.\n\nThese functions are provided by many modern \\emph{Satisfiability Modulo Theories} (SMT) solvers, in particular for the theories of linear integer arithmetic and linear real arithmetic, which we will use for all our examples. Note that interpolation is usually only supported for the quantifier-free fragments of these logics, while our algorithm will introduce existential quantifiers.\nTherefore, we resort to quantifier elimination wherever necessary, for which there are known procedures for both linear integer arithmetic and linear real arithmetic formulas~\\cite{presburger1929uber,Monniaux2008}.\n\nIn order to distinguish the two players, we will assume that a Boolean variable called $\\mathbf{r} \\in \\mathcal{V}$ exists, which holds exactly in the states controlled by the reachability player.\nFor all other variables $v \\in \\mathcal{V}$, we let $\\mathcal{D}(v)$ be the domain of $v$, and we define $\\mathcal{D} = \\bigcup \\{\\mathcal{D}(v) \\mid v \\in \\mathcal{V}\\}$.\nIn the remainder of the paper, we consider the variables $\\mathcal{V}$ and their domains to be fixed.\n\n\\begin{definition}[Reachability Game]\n\tA reachability game is defined by a tuple $\\G = \\langle \\init, \\safe, \\reach, \\goal \\rangle${}, where $\\mathit{Init} \\in \\cal L(\\mathcal{V})$ is the \\emph{initial condition}, $\\mathit{Safe} \\in \\cal L(\\mathcal{V} \\cup \\mathcal{V'})$ defines the transitions of player $\\texttt{{SAFE}}$, $\\mathit{Reach} \\in \\cal L(\\mathcal{V} \\cup \\mathcal{V'})$ defines the transitions of player $\\texttt{{REACH}}$ and $\\mathit{Goal} \\in \\cal L(\\mathcal{V})$ is the \\emph{goal condition}.\n\n  We require the formulas $(\\mathit{Safe} \\implies \\neg \\mathbf{r})$ and $(\\mathit{Reach} \\implies \\mathbf{r})$ to be valid.\n\\end{definition}\n\nA \\emph{state} $s$ of $\\mathcal{G}$ is a valuation of the variables $\\mathcal{V}$, i.e., a function $s\\colon\\mathcal{V} \\to \\mathcal{D}$ that satisfies $s(v) \\in \\mathcal{D}(v)$ for all $v \\in \\mathcal{V}$.\nWe denote the set of states by $S$, and we let $S_\\texttt{{SAFE}}$ be the states $s$ such that $s(\\mathbf{r}) = \\texttt{false}$, and $S_\\texttt{{REACH}}$ be the states $s$ such that $s(\\mathbf{r}) = \\texttt{true}$. The variable $\\mathbf{r}$ determines whether \\texttt{{REACH}}{} or \\texttt{{SAFE}}{} makes the move out of the current state, and in particular $\\mathit{Safe} \\land \\mathit{Reach}$ is unsatisfiable. \n\nGiven a state predicate $\\varphi \\in \\cal L(\\mathcal{V})$, we denote by $\\varphi(s)$ the closed formula we get by replacing each occurrence of variable $v \\in \\mathcal{V}$ in $\\varphi$ by $s(v)$.\nSimilarly, given a transition predicate $\\tau \\in \\cal L(\\mathcal{V} \\cup \\mathcal{V'})$ and states $s,s'$, we let $\\tau(s,s')$ be the formula we obtain by replacing all occurrences of $v \\in \\mathcal{V}$ in $\\tau$ by $s(v)$, and all occurrences of $v' \\in \\mathcal{V'}$ in $\\tau$ by $s'(v)$. For replacing only $v \\in \\mathcal{V}$ by $s(v)$, we define $\\tau(s)\\in\\cal L(\\mathcal{V}')$. A \\emph{trap state} of $\\mathcal{G}$ is a state $s$ such that $(\\mathit{Safe} \\lor \\mathit{Reach})(s)\\in\\cal L(\\mathcal{V}')$ is unsatisfiable (i.e., $s$ has no outgoing transitions).\n\nA \\emph{play} of $\\mathcal{G}$ starting in state $s_0$ is a finite or infinite sequence of states $\\rho = s_0 s_1 s_2 \\ldots \\in \\S^+ \\cup \\S^\\omega$ such that for all $i < \\operatorname{len}(\\rho)$ either $\\mathit{Safe}(s_i,s_{i+1})$ or $\\mathit{Reach}(s_i,s_{i+1})$ is valid, and if $\\rho$ is a finite play, then $s_{\\operatorname{len}(\\rho)}$ is required to be a trap state.\nHere, $\\operatorname{len}(s_0\\ldots s_n) = n$ for finite plays, and $\\operatorname{len}(\\rho) = \\infty$ if $\\rho$ is an infinite play. The set of plays of some game $\\G = \\langle \\init, \\safe, \\reach, \\goal \\rangle${} is defined as $\\operatorname{Plays}(\\mathcal{G}) = \\{\\rho = s_0 s_1 s_2 \\ldots \\mid \\rho\\text{ is a play in } \\mathcal{G} \\text{ s.t. } \\mathit{Init}(s_0)\\text{ holds} \\}$.\n$\\texttt{{REACH}}$ \\emph{wins} some play $\\rho = s_0 s_1 \\ldots$ if the play reaches a goal state, i.e., if there exists some integer $0\\leq k \\leq\\operatorname{len}(\\rho)$ such that $\\mathit{Goal}(s_k)$ is valid.\nOtherwise, $\\texttt{{SAFE}}$ wins play $\\rho$.\nA \\emph{reachability strategy} $\\sigma_{\\mathit{R}}$ is a function $\\sigma_{\\mathit{R}} : \\S^*S_\\texttt{{REACH}} \\to \\S$ such that if $\\sigma_{\\mathit{R}}(\\omega s) =s'$ and $s$ is not a trap state, then $\\mathit{Reach}(s,s')$ is valid. \nWe say that a play $\\rho = s_0 s_1 s_2 \\ldots$ is \\emph{consistent} with $\\sigma_{\\mathit{R}}$ if for all $i$ such that $s_i(\\mathbf{r}) = \\texttt{true}$ we have $s_{i+1} = \\sigma_{\\mathit{R}}(s_0 \\ldots s_i)$.\nA reachability strategy $\\sigma_{\\mathit{R}}$ is \\emph{winning} from some state $s$ if $\\texttt{{REACH}}$ wins every play consistent with $\\sigma_{\\mathit{R}}$ starting in $s$. We define \\emph{safety strategies} $\\sigma_{\\mathit{S}}$ for $\\texttt{{SAFE}}$ analogously. We say that a player \\emph{wins in or from a state}~$s$ if they have a winning strategy from $s$. Lastly, $\\texttt{{REACH}}$ \\emph{wins the game} $\\mathcal{G}$ if they win from some initial state.\nOtherwise, $\\texttt{{SAFE}}$ wins.\n\nWe often project a transition predicate $T$ onto the source or target states of transitions satisfying $T$, which is taken care of by the formulas $\\operatorname{Pre}(\\mathit{T})=\\exists \\mathcal{V'}.\\:\\mathit{T}$ and $\\operatorname{Post}(\\mathit{T})=\\exists \\mathcal{V}.\\:\\mathit{T}$.\nThe notation $\\exists \\mathcal{V}$ (resp. $\\exists \\mathcal{V'}$) represents the existential quantification over all variables in the corresponding set.\nGiven $\\varphi \\in \\cal L(\\mathcal{V})$, we call the set of transitions in $\\mathcal{G}$ that move from states not satisfying $\\varphi$, to states satisfying $\\varphi$, the \\emph{instantiation} of $\\varphi$, formally:\n\\[\\operatorname{Instantiate}(\\varphi,\\mathcal{G})=(\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\lnot\\varphi\\land\\varphi'.\\]\n\n\\section{Subgoals}\n\\label{sec:subgoals}\n\nWe formally define the notion of subgoals.\nLet $\\mathcal{G} = \\langle \\mathit{Init}, \\mathit{Safe}, \\mathit{Reach}, \\mathit{Goal} \\rangle$ be a fixed reachability game throughout this section, where we assume that $\\mathit{Init} \\land \\mathit{Goal}$ is unsatisfiable.\nWhenever this assumption is not satisfied in our algorithm, we will instead consider the game $\\mathcal{G}' = \\langle \\mathit{Init} \\land \\neg \\mathit{Goal}, \\mathit{Safe}, \\mathit{Reach}, \\mathit{Goal} \\rangle$ which does satisfy it.\nAs states in $\\mathit{Init} \\land \\mathit{Goal}$ are immediately winning for \\texttt{{REACH}}, this is not a real restriction.\n\n\\begin{definition}[Enforceable transitions]\n\tThe set of \\emph{enforceable transitions} relative to a transition predicate $T \\in \\cal L(\\mathcal{V} \\cup \\mathcal{V'})$ is defined by the formula\n\t\\[\\operatorname{Enf}(\\mathit{T},\\mathcal{G})=\\; (\\mathit{Safe} \\lor \\mathit{Reach}) \\land \\mathit{T} \\land \\lnot \\exists \\mathcal{V'}.\\:\\big(\\mathit{Safe}\\land\\lnot\\mathit{T}\\big).\\]\n\t\n\\end{definition}\n\nThe enforceable transitions operator serves a purpose similar to the \\emph{controlled predecessors} operator commonly known in the literature, which is often used in a backwards fixed point computation, called \\emph{attractor construction} \\cite{Thomas95}. For both operations, the idea is to determine controllability by \\texttt{{REACH}}{}. The main difference is that we do not consider the whole transition relation, but only a predetermined set of transitions and check from which predecessor states the post-condition of the set can be enforced by \\texttt{{REACH}}{}. These include all transitions in $T$ controlled by \\texttt{{REACH}}{} and additionally transitions in $T$ controlled by \\texttt{{SAFE}}{} such that \\emph{all other transitions} in the origin state of the transition also satisfy $T$. The similarity with the controlled predecessor is exemplified by the following lemma:\n\\begin{lemma}\n  \\label{lem:enf}\n  Let $T$ be a transition predicate, and suppose that all states satisfying $\\operatorname{Post}(T)[\\varp\/\\var]$ are winning for \\texttt{{REACH}}{} in $\\mathcal{G}$.\n  Then all states in $\\operatorname{Pre}(\\operatorname{Enf}(T,\\mathcal{G}))$ are winning for \\texttt{{REACH}}{} in $\\mathcal{G}$.\n\\end{lemma}\n\\begin{proof}\n  Clearly, all states in $\\operatorname{Pre}(\\operatorname{Enf}(T,\\mathcal{G}))$ that are under the control of \\texttt{{REACH}}{} are winning for \\texttt{{REACH}}{}, as in any such state they have a transition satisfying $T$ (observe that $\\operatorname{Enf}(T,\\mathcal{G}) \\implies T$ is valid), which leads to a winning state by assumption.\n\n  So let $s$ be a state satisfying $\\operatorname{Pre}(\\operatorname{Enf}(T,\\mathcal{G}))$ that is under the control of \\texttt{{SAFE}}{}.\n  As $\\operatorname{Pre}(\\operatorname{Enf}(T,\\mathcal{G}))(s)$ is valid, $s$ has a transition that satisfies $T$ (in particular, $s$ is not a trap state).\n  Furthermore, we know that there is no $s' \\in \\S$ such that $\\mathit{Safe}(s,s')\\land\\lnot\\mathit{T}(s,s')$ holds, and hence there is no transition satisfying $\\lnot\\mathit{T}$ from $s$. Since $\\operatorname{Post}(T)[\\varp\/\\var]$ is winning for \\texttt{{REACH}}{}, it follows that from $s$ player \\texttt{{SAFE}}{} cannot avoid playing into a winning state of \\texttt{{REACH}}{}.\n  \\qed\n\\end{proof}\n\nWe now turn to a formal definition of \\emph{necessary subgoals}, which intuitively are sets of transitions that appear on every play that is winning for \\texttt{{REACH}}{}. \n\\begin{definition}[Necessary subgoal]\\label{necessary_subgoal}\n\tA \\emph{necessary subgoal} $C \\in \\cal L(\\mathcal{V} \\cup \\mathcal{V'})$  for~$\\mathcal{G}$ is a transition predicate such that for every play $\\rho = s_0 s_1 \\ldots$ of $\\mathcal{G}$ and $n \\in \\mathbb{N}$ such that $\\mathit{Goal}(s_{n})$ is valid, there exists some $k < n$ such that $C(s_k,s_{k+1})$ is valid.\n\\end{definition}\n\nNecessary subgoals provide a means by which winning safety player strategies can be identified, as formalized in the following lemma.\n\n\\begin{lemma}\\label{prop_safestrat}\n\tA safety strategy $\\sigma_{\\mathit{S}}$ is winning in $\\mathcal{G}$ if and only if there exists a necessary subgoal $\\mathit{C}$ for $\\mathcal{G}$ such that for all plays $\\rho = s_0 s_1 \\ldots$ of $\\mathcal{G}$ consistent with~$\\sigma_{\\mathit{S}}$ there is no $n \\in \\mathbb{N}$ such that $C(s_n,s_{n+1})$ holds. \n\\end{lemma}\n\\begin{proof}\n  ``$\\implies$''. The transition predicate $\\mathit{Goal}[\\var\/\\varp]$ (i.e., transitions with endpoints satisfying $\\mathit{Goal}$) is clearly a necessary subgoal. If $\\sigma_{\\mathit{S}}$ is winning for \\texttt{{SAFE}}, then no play consistent with $\\sigma_{\\mathit{S}}$ contains a transition in this necessary subgoal. \\\\\n  \\noindent ``$\\Longleftarrow$''. Let $C$ be a necessary subgoal such that no play consistent with $\\sigma_{\\mathit{S}}$ contains a transition of $C$. Then by  \\Cref{necessary_subgoal} no play consistent with $\\sigma_{\\mathit{S}}$ contains a state satisfying $\\mathit{Goal}$. Hence $\\sigma_{\\mathit{S}}$ is a winning strategy for \\texttt{{SAFE}}.\n\t\\qed\n\\end{proof}\n\nOf course, the question remains how to compute non-trivial subgoals. Indeed, using $\\mathit{Goal}$ as outlined in the proof above provides no further benefit over a simple backwards exploration (see~\\Cref{rem:attractor} in the following section).\n\nIdeally, a subgoal should represent an interesting key decision to focus the strategy search.\nAs we show next, Craig interpolation allows to extract partial causes for the mutual unsatisfiability of $\\mathit{Init}$ and $\\mathit{Goal}$ and can in this way provide necessary subgoals. \nRecall that a Craig interpolant $\\varphi$ between $\\mathit{Init}$ and $\\mathit{Goal}$ is a state predicate that is implied by $\\mathit{Goal}$, and unsatisfiable in conjunction with $\\mathit{Init}$. \nIn this sense, $\\varphi$ describes an observable \\emph{effect} that must occur if $\\texttt{{REACH}}{}$ wins, and the concrete transition that instantiates the interpolant \\emph{causes} this effect.\n\n\\begin{proposition}\\label{prop_necessary}\n\tLet $\\varphi$ be a Craig interpolant for $\\mathit{Init}$ and $\\mathit{Goal}$. Then the transition predicate $\\operatorname{Instantiate}(\\varphi,\\mathcal{G})$ is a necessary subgoal.\n\\end{proposition}\n\\begin{proof}\n  As $\\varphi$ is an interpolant, it holds that $\\mathit{Goal} \\implies \\varphi$ is valid and $\\mathit{Init} \\land \\varphi$ is unsatisfiable.\n  Consider any play $\\rho = s_0 s_1 \\ldots$ of $\\mathcal{G}$ such that $\\mathit{Goal}(s_n)$ is valid for some $n \\in \\mathbb{N}$.\n  It follows that $\\lnot \\varphi(s_0)$ and $\\varphi(s_n)$ are both valid.\n  Consequently, there is some $0 \\leq i < n$ such that $\\lnot \\varphi(s_i)$ and $\\varphi(s_{i+1})$ are both valid.\n  As all pairs $(s_k,s_{k+1})$ satisfy either $\\mathit{Safe}$ or $\\mathit{Reach}$, it follows that $\\big(\\operatorname{Instantiate}(\\varphi,\\mathcal{G})\\big)(s_i,s_{i+1})$ is valid.\n  Hence, $\\operatorname{Instantiate}(\\varphi,\\mathcal{G})$ is a necessary subgoal.\n\t\\qed\n\\end{proof}\n\nWhile avoiding a necessary subgoal is a winning strategy for \\texttt{{SAFE}}{}, reaching a necessary subgoal is in general not sufficient to guarantee a win for \\texttt{{REACH}}{}.\nThis is because there might be some transitions in the necessary subgoal that produce the desired effect described by the Craig interpolant, but that trap \\texttt{{REACH}}{} in a region of the state space where they cannot enforce some other necessary effect to reach goal. \nFor the purpose of describing a set of transitions that is guaranteed to be winning for the reachability player, we introduce \\emph{sufficient subgoals}.\n\n \\begin{definition}[Sufficient subgoal]\n   A transition predicate $\\mathit{F}\\in\\cal L(\\mathcal{V}\\cup\\mathcal{V'})$ is called a \\emph{sufficient subgoal} if $\\texttt{{REACH}}$ wins from every state satisfying $\\operatorname{Post}(\\mathit{F})[\\varp\/\\var]$.\n \\end{definition}\n\n\\begin{example}\n\tConsider the Mona Lisa game $\\mathcal{G}$ described in Section \\ref{sec:motivation}.\n\t\\[C_1 = (\\mathit{Guard} \\lor \\mathit{Thief}) \\land p \\neq 1 \\land p' = 1\\]\n\tqualifies as sufficient subgoal, because $\\texttt{{REACH}}$ wins from every successor state as all those states satisfy $\\mathit{Goal}$. \n\tAlso, every play reaching $\\mathit{Goal}$ eventually passes $C_1$, and hence $C_1$ is also necessary. On the other hand, \n\t\\[C_2 = (\\mathit{Guard} \\lor \\mathit{Thief}) \\land a \\neq 0 \\land a' = 0\\]\n\tis only a necessary subgoal in $\\mathcal{G}$, because $\\texttt{{SAFE}}$ wins from some (in fact all) states satisfying $\\operatorname{Post}(C_2)$.\n\t\n\\end{example}\n \nIf the set of transitions in the necessary subgoal $C$ that lead to winning states of \\texttt{{REACH}}{} is definable in $\\cal L$ then we call the transition predicate $F$ that defines it the \\emph{largest sufficient subgoal} included in $C$. \nIt is characterized by the properties (1) $F \\implies C$ is valid, and (2) if $F'$ is such that $F \\implies F'$ is valid, then either $F \\equiv F'$, or $F'$ is not a sufficient subgoal. Since $C$ is a necessary subgoal and $F$ is maximal with the properties above, \\texttt{{REACH}}{} needs to see a transition in $F$ eventually in order to win. This balance of necessity and sufficiency allows us to partition the game along $F$ into a game that happens after the subgoal and one that happens before.\n\n\\begin{proposition}\n\t\t\\label{lem:slicing}\n\t  Let $C$ be a necessary subgoal, and $F$ be the largest sufficient subgoal included in $C$. Then \\texttt{{REACH}}{} wins from an initial state $s$ in $\\mathcal{G}$ if and only if \\texttt{{REACH}}{} wins from $s$ in the pre-game \n    \\[\\mathcal{G}_{pre} = \\langle \\mathit{Init}, \\mathit{Safe} \\land \\neg F, \\mathit{Reach} \\land \\neg F, \\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G})) \\rangle.\\]\n\\end{proposition}\n\\begin{proof}\n  ``$\\implies$''. Suppose that \\texttt{{REACH}}{} wins in $\\mathcal{G}$ from $s$ using strategy $\\sigma_R$. Assume for a contradiction that \\texttt{{SAFE}}{} wins in $\\mathcal{G}_{pre}$ from $s$ using strategy $\\sigma_S$. Consider strategy $\\sigma'_S$ such that $\\sigma_{\\mathit{S}}'(\\omega s') = \\sigma_{\\mathit{S}}(\\omega s')$ if $(\\mathit{Safe} \\land \\lnot F)(s')$ is satisfiable, and else $\\sigma_{\\mathit{S}}'(\\omega s') = \\sigma_{\\mathit{S}}''(\\omega s')$, where $\\sigma_{\\mathit{S}}''$ is an arbitrary safety player strategy in $\\mathcal{G}$. Let $\\rho = s_0s_1\\ldots$ be the (unique) play of $\\mathcal{G}$ consistent with both $\\sigma_R$ and $\\sigma'_S$, where $s_0 = s$. Since $\\sigma_R$ is winning in $\\mathcal{G}$ and $C$ is a necessary subgoal in $\\mathcal{G}$, there must exist some $m\\in\\mathbb{N}$ such that $C(s_m, s_{m+1})$ is valid. Let $m$ be the smallest such index. Since $F \\implies C$, we know for all $0 \\leq k < m$ that $\\lnot F (s_k,s_{k+1})$ holds. Hence, there is the play $\\rho' = s_0s_1\\ldots s_m \\ldots$ in $\\mathcal{G}_{pre}$ consistent with $\\sigma_S$. The state $s_{m+1}$ is winning for \\texttt{{REACH}}{} in $\\mathcal{G}$, as it is reached on a play consistent with the winning strategy $\\sigma_R$. Hence, we know that $F(s_m, s_{m+1})$ holds, because $F$ is the largest sufficient subgoal included in $C$.\n  If $(\\mathit{Reach} \\land F)(s_m, s_{m+1})$ held, we would have that $\\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G})(s_m)$ holds: a contradiction with $\\rho'$ being consistent with $\\sigma_S$, which we assumed to be winning in $\\mathcal{G}_{pre}$. It follows that $(\\mathit{Safe} \\land F)(s_m, s_{m+1})$ holds. We can conclude that $(\\mathit{Safe} \\land \\lnot F)(s_m)$ is unsatisfiable (i.e., $s_m$ is a trap state in $\\mathcal{G}_{pre}$), because in all other cases $\\texttt{{SAFE}}$ plays according to $\\sigma_{\\mathit{S}}$, which cannot choose a transition satisfying $F$. However, this implies that  $\\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G})(s_m)$ holds, again a contradiction with $\\rho'$ being consistent with winning strategy $\\sigma_S$.\n\n\t\\noindent ``$\\Longleftarrow$''. If $\\texttt{{REACH}}$ wins in $\\mathcal{G}_{pre}$ they have a strategy $\\sigma_{\\mathit{R}}$ such that every play consistent with $\\sigma_{\\mathit{R}}$ reaches the set $\\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G}))$.\n  As $F$ is a sufficient subgoal, the states $\\operatorname{Post}(F)$ are winning for \\texttt{{REACH}}{} by definition.\n  It follows by~\\Cref{lem:enf} that all states satisfying $\\operatorname{Pre}(\\operatorname{Enf}(F,\\mathcal{G}))$ are winning in $\\mathcal{G}$.\n  Combining $\\sigma_{R}$ with a strategy that wins in all these states yields a winning strategy for \\texttt{{REACH}}{} in $\\mathcal{G}$.\n  \\qed\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{}}\n\n\n\\section{Introduction}\n\\label{sec:Introduction}\n\nWe consider the problem of Bayesian inference from cosmological data, in the common scenario where we can generate synthetic data through forward simulations, but where the exact likelihood function is intractable. The generative process can be extremely general: it may be a noisy non-linear dynamical system involving an unrestricted number of latent variables. Likelihood-free inference methods, also known as approximate Bayesian computation \\citep[ABC, see][for reviews]{Marin2011,Lintusaari2017a} replace likelihood calculations with data model evaluations. In recent years, they have emerged as a viable alternative to likelihood-based techniques, when the simulator is sufficiently cheap. Applications in cosmology include measuring cosmological parameters from type Ia supernovae \\citep{Weyant2013} and weak lensing peak counts \\citep{Lin2015}, analysing the galaxy halo connection \\citep{Hahn2017}, inferring the photometric and size evolution of galaxies \\citep{Carassou2017}, measuring cosmological redshift distributions \\citep{Kacprzak2018}, estimating the ionising background from the Lyman-$\\alpha$ and Lyman-$\\beta$ forests \\citep{Davies2018}.\n\nIn its simplest form, ABC takes the form of likelihood-free rejection sampling and involves forward simulating data from parameters drawn from the prior, then accepting parameters when the discrepancy (by some measure) between simulated data and observed data is smaller than a user-specified threshold $\\varepsilon$. Such an approach tends to be extremely expensive since many simulated data sets get rejected, due to the lack of knowledge about the relation between the model parameters and the corresponding discrepancy. Variants of likelihood-free rejection sampling such as Population (or Sequential) Monte Carlo ABC \\citep[\\textsc{pmc}-\\textsc{abc} or \\textsc{smc}-\\textsc{abc}, see][for implementations aimed at astrophysical applications]{Akeret2015,Ishida2015,Jennings2017} improve upon this scheme by making the proposal adaptive; however, they do not use a probabilistic model for the relation between parameters and discrepancies (also known as a surrogate surface), so that their practical use usually necessitates $\\mathcal{O}(10^4-10^6)$ evaluations of the simulator. \n\nIn this paper, we address the challenging problem where the number of simulations is extremely limited, e.g. to a few thousand, rendering the use of sampling-based ABC methods impossible. To this end, we use Bayesian optimisation for likelihood-free inference \\citep[{\\textsc{bolfi}},][]{GutmannCorander2016}, an algorithm which combines probabilistic modelling of the discrepancy with optimisation to facilitate likelihood-free inference. Since it was introduced, {\\textsc{bolfi}} has been applied to various statistical problems in science, including inference of the Ricker model \\citep{GutmannCorander2016}, the Lotka-Volterra predator-prey model and population genetic models \\citep{Jaervenpaeae2018}, pathogen spread models \\citep{Lintusaari2017a}, atomistic structure models in materials \\citep{Todorovic2017}, and cognitive models in human-computer interaction \\citep{Kangasraeaesioe2017}. This work aims at introducing {\\textsc{bolfi}} in cosmological data analysis and at presenting its first cosmological application. We focus on computable parametric approximations to the true likelihood (also known as synthetic likelihoods), rendering the approach completely $\\varepsilon$-free. Recently, \\citet{Jaervenpaeae2017} introduced an acquisition function for Bayesian optimisation (the expected integrated variance), specifically tailored to perform efficient and accurate ABC. We extend their work by deriving the expression of the expected integrated variance in the parametric approach. This acquisition function measures the expected uncertainty in the estimate of the {\\textsc{bolfi}} posterior density, which is due to the limited number of simulations, over the future evaluation of the simulation model. The next simulation location is proposed so that this expected uncertainty is minimised. As a result, high-fidelity posterior inferences can be obtained with orders of magnitude fewer simulations than with likelihood-free rejection sampling. As examples, we demonstrate the use of {\\textsc{bolfi}} on the problems of summarising Gaussian signals and inferring cosmological parameters from the Joint Lightcurve Analysis (JLA) supernovae data set \\citep{Betoule2014}.\n\nThe structure of this paper is as follows. In section \\ref{sec:Inference of simulator-based statistical models}, we provide a review of the formalism for the inference of simulator-based statistical models. In section \\ref{sec:Regression and Optimisation for likelihood-free inference}, we describe {\\textsc{bolfi}} and discuss the regression and optimisation strategies. In particular, we provide the optimal acquisition rule for ABC in the parametric approach to likelihood approximation. Applications are given in section \\ref{sec:Applications}. The developed method is discussed in section \\ref{sec:Discussion} in the context of cosmological data analysis. Section \\ref{sec:Conclusion} concludes the paper. Mathematical details and descriptions of the case studies are presented in the appendices. \n\n\\section{Inference of simulator-based statistical models}\n\\label{sec:Inference of simulator-based statistical models}\n\n\\subsection{Simulator-based statistical models}\n\\label{ssec:Simulator-based statistical models}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}\n\t\\pgfdeclarelayer{background}\n\t\\pgfdeclarelayer{foreground}\n\t\\pgfsetlayers{background,main,foreground}\n\n\t\\tikzstyle{probability}=[draw, thick, text centered, rounded corners, minimum height=1em, minimum width=1em, fill=green!20]\n\t\\tikzstyle{variabl}=[draw, thick, text centered, circle, minimum height=1em, minimum width=1em]\n\n\t\\def0.7{0.7}\n\t\\def2.0{2.0}\n\n\n    \\node (thetaprobaii) [probability]\n    {$\\mathpzc{P}(\\boldsymbol{\\uptheta})$};\n    \\path (thetaprobaii.south)+(0,-0.7) node (thetaii) [variabl]\n    {$\\boldsymbol{\\uptheta}$};\n    \\path (thetaii.south)+(0,-0.7) node (dprobaii) [probability]\n    {$\\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\uptheta})$};\n    \\path (dprobaii.south)+(0,-0.7) node (dii) [variabl]\n    {$\\textbf{d}$};\n    \n   \n    \\path (thetaprobaii.west)+(-2.0,0) node (thetaprobai) [probability]\n    {$\\mathpzc{P}(\\boldsymbol{\\uptheta})$};\n    \\path (thetaprobai.south)+(0,-0.7) node (thetai) [variabl]\n    {$\\boldsymbol{\\uptheta}$};\n    \\path (thetai.south)+(0,-0.7) node (di) [variabl]\n    {$\\textbf{d}$};\n\n\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (thetaprobaii) -- (thetaii);\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (thetaii) -- (dprobaii);\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (dprobaii) -- (dii);\n\n\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (thetaprobai) -- (thetai);\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (thetai) -- (di);\t\n\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Hierarchical representation of the exact Bayesian problem for simulator-based statistical models of different complexities: a deterministic simulator (left), and a stochastic simulator (right).\\label{fig:BHM_exact}}\n\\end{figure}\n\nSimulator-based statistical models (also known as generative models) can be written in a hierarchical form (figure \\ref{fig:BHM_exact}), where $\\boldsymbol{\\uptheta}$ are the parameters of interest, and $\\textbf{d}$ the simulated data. $\\mathpzc{P}(\\boldsymbol{\\uptheta})$ is the prior probability distribution of $\\boldsymbol{\\uptheta}$ and $\\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\uptheta})$ is the sampling distribution of $\\textbf{d}$ given $\\boldsymbol{\\uptheta}$.\n\nThe simplest case (figure \\ref{fig:BHM_exact}, left) is when the simulator is a deterministic function of its input and does not use any random variable, i.e.\n\\begin{equation}\n\\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\uptheta}) = \\updelta_\\mathrm{D}(\\textbf{d} - \\boldsymbol{\\hat{\\mathrm{d}}}(\\boldsymbol{\\uptheta})) ,\\label{eq:Dirac_deterministic_simulator}\n\\end{equation}\nwhere $\\updelta_\\mathrm{D}$ is a Dirac delta distribution and $\\boldsymbol{\\hat{\\mathrm{d}}}$ a deterministic function of $\\boldsymbol{\\uptheta}$.\n\nIn a more generic scenario (figure \\ref{fig:BHM_exact}, right), the simulator is stochastic, in the sense that the data are drawn from an overall (but often unknown analytically) probability distribution function (pdf) $\\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\uptheta})$. Equation \\eqref{eq:Dirac_deterministic_simulator} does not hold in this case. The scatter between different realisations of $\\textbf{d}$ given the same $\\boldsymbol{\\uptheta}$ can have various origins. In the simplest case, it only reflects the intrinsic uncertainty, which is of interest. More generically, additional nuisance parameters can be at play to produce the data $\\textbf{d}$ and will contribute to the uncertainty. This ``latent space'' can often be hundred-to-multi-million dimensional. Simulator-based cosmological models are typically of this kind: although the physical and observational processes simulated are repeatable features about which inferences can be made, the particular realisation of Fourier phases of the data is entirely noise-driven. Ideally, phase-dependent quantities should not contribute to any measure of match or mismatch between model and data.\n\n\\subsection{The exact Bayesian problem}\n\\label{ssec:The exact Bayesian problem}\n\nThe inference problem is to evaluate the probability of $\\boldsymbol{\\uptheta}$ given $\\textbf{d}$,\n\\begin{equation}\n\\mathpzc{P}(\\boldsymbol{\\uptheta}|\\textbf{d}) = \\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\uptheta}) \\, \\frac{\\mathpzc{P}(\\boldsymbol{\\uptheta})}{\\mathpzc{P}(\\textbf{d})},\n\\label{eq:exact_problem_Bayes}\n\\end{equation}\nfor the observed data $\\textbf{d}_\\mathrm{O}$, i.e.\n\\begin{equation}\n\\mathpzc{P}(\\boldsymbol{\\uptheta}|\\textbf{d})_\\mathrm{|\\textbf{d}=\\textbf{d}_O} = \\mathcal{L}(\\boldsymbol\\uptheta) \\, \\frac{\\mathpzc{P}(\\boldsymbol{\\uptheta})}{Z_\\textbf{d}} ,\n\\end{equation}\nwhere the exact likelihood for the problem is defined as\n\\begin{equation}\n\\mathcal{L}(\\boldsymbol\\uptheta) \\equiv \\mathpzc{P}(\\textbf{d}|\\boldsymbol\\uptheta)_\\mathrm{|\\textbf{d}=\\textbf{d}_O} .\n\\end{equation}\nIt is generally of unknown analytical form. The normalisation constant is $Z_\\textbf{d} \\equiv \\mathpzc{P}(\\textbf{d})_\\mathrm{|\\textbf{d}=\\textbf{d}_O}$, where $\\mathpzc{P}(\\textbf{d})$ is the marginal distribution of $\\textbf{d}$.\n\n\\subsection{Approximate Bayesian computation}\n\\label{ssec:Approximate Bayesian computation}\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}\n\t\\pgfdeclarelayer{background}\n\t\\pgfdeclarelayer{foreground}\n\t\\pgfsetlayers{background,main,foreground}\n\n\t\\tikzstyle{probability}=[draw, thick, text centered, rounded corners, minimum height=1em, minimum width=1em, fill=green!20]\n\t\\tikzstyle{variabl}=[draw, thick, text centered, circle, minimum height=1em, minimum width=1em]\n\n\t\\def0.7{0.7}\n\n    \\node (thetaproba) [probability]\n    {$\\mathpzc{P}(\\boldsymbol{\\uptheta})$};\n    \\path (thetaproba.south)+(0,-0.7) node (theta) [variabl]\n    {$\\boldsymbol{\\uptheta}$};\n    \\path (theta.south)+(0,-0.7) node (dproba) [probability]\n    {$\\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\uptheta})$};\n    \\path (dproba.south)+(0,-0.7) node (d) [variabl]\n    {$\\textbf{d}$};\n    \\path (d.south)+(0,-0.7) node (phiproba) [probability]\n    {$\\mathpzc{P}(\\boldsymbol{\\Phi}|\\textbf{d})$};\n    \\path (phiproba.south)+(0,-0.7) node (phi) [variabl]\n    {$\\boldsymbol{\\Phi}$};\n\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (thetaproba) -- (theta);\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (theta) -- (dproba);\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (dproba) -- (d);\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (d) -- (phiproba);\n\t\\path [draw, line width=0.7pt, arrows={-latex}] (phiproba) -- (phi);\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Hierarchical representation of the approximate Bayesian inference problem for simulator-based statistical models, with a compression of the raw data to a set of summary statistics.\\label{fig:BHM_approx}}\n\\end{figure}\n\nInference of simulator-based statistical models is usually based on a finite set of simulated data $\\textbf{d}_{\\boldsymbol{\\uptheta}}$, generated with parameter value $\\boldsymbol{\\uptheta}$, and on a measurement of the discrepancy between simulated data and observed data $\\textbf{d}_\\mathrm{O}$. This discrepancy is used to define an approximation to the exact likelihood $\\mathcal{L}(\\boldsymbol{\\uptheta})$. The approximation happens on multiple levels.\n\nOn a physical and statistical level, the approximation consists of compressing the full data $\\textbf{d}_\\mathrm{O}$ to a set of summary statistics $\\boldsymbol{\\Phi}_\\mathrm{O}$ before performing inference. Similarly, simulated data $\\textbf{d}_{\\boldsymbol{\\uptheta}}$ are compressed to simulated summary statistics $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$. This can be seen as adding a layer to the Bayesian hierarchical model (figure \\ref{fig:BHM_approx}). The purpose of this operation is to filter out the information in $\\textbf{d}$ that is not deemed relevant to the inference of $\\boldsymbol{\\uptheta}$, so as to reduce the dimensionality of the problem. Ideally, $\\boldsymbol{\\Phi}$ should be \\textit{sufficient} for parameters $\\boldsymbol{\\uptheta}$, i.e. formally $\\mathpzc{P}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}) = \\mathpzc{P}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi},\\textbf{d})$ or equivalently $\\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\Phi},\\boldsymbol{\\uptheta}) = \\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\Phi})$, which happens when the compression is lossless. However, sufficient summary statistics are generally unknown or even impossible to design; therefore the compression from $\\textbf{d}$ to $\\boldsymbol{\\Phi}$ will usually be lossy. The approximate inference problem to be solved is now $\\mathpzc{P}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}) = \\mathpzc{P}(\\boldsymbol{\\Phi}|\\boldsymbol{\\uptheta}) \\, \\dfrac{\\mathpzc{P}(\\boldsymbol{\\uptheta})}{\\mathpzc{P}(\\boldsymbol{\\Phi})}$ for the observed summary statistics $\\boldsymbol{\\Phi}_\\mathrm{O}$, i.e.\n\\begin{equation}\n\\mathpzc{P}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi})_\\mathrm{|\\boldsymbol{\\Phi}=\\boldsymbol{\\Phi}_O} = L(\\boldsymbol\\uptheta) \\, \\frac{\\mathpzc{P}(\\boldsymbol{\\uptheta})}{Z_{\\boldsymbol{\\Phi}}} .\n\\label{eq:approx_problem_Bayes}\n\\end{equation}\nIn other words, $\\mathcal{L}(\\boldsymbol{\\uptheta})$ is replaced by\n\\begin{equation}\nL(\\boldsymbol{\\uptheta}) \\equiv \\mathpzc{P}(\\boldsymbol\\Phi|\\boldsymbol\\uptheta)_\\mathrm{|\\boldsymbol{\\Phi}=\\boldsymbol{\\Phi}_O} ,\n\\label{eq:L_theta}\n\\end{equation}\nand $Z_\\textbf{d}$ by $Z_{\\boldsymbol{\\Phi}} \\equiv \\mathpzc{P}(\\boldsymbol{\\Phi})_{|\\boldsymbol{\\Phi}=\\boldsymbol{\\Phi}_\\mathrm{O}}$. Inference of model \\ref{fig:BHM_approx} gives\n\\begin{equation}\n\\mathpzc{P}(\\boldsymbol\\uptheta, \\textbf{d} | \\boldsymbol\\Phi) \\propto \\mathpzc{P}(\\boldsymbol\\Phi|\\textbf{d}) \\, \\mathpzc{P}(\\textbf{d}|\\boldsymbol\\uptheta) \\, \\mathpzc{P}(\\boldsymbol\\uptheta),\n\\label{eq:BHM_approx_expansion}\n\\end{equation}\nwith, after marginalisation over $\\textbf{d}$,\n\\begin{equation}\n\\mathpzc{P}(\\boldsymbol\\uptheta | \\boldsymbol\\Phi) = \\int \\mathpzc{P}(\\boldsymbol\\uptheta, \\textbf{d} | \\boldsymbol\\Phi) \\, \\mathrm{d}\\textbf{d} .\n\\label{eq:BHM_approx_marginalisation}\n\\end{equation}\nTherefore, the approximate likelihood $L(\\boldsymbol{\\uptheta})$ must satisfy\n\\begin{equation}\nL(\\boldsymbol{\\uptheta}) \\propto \\int \\mathpzc{P}(\\boldsymbol\\Phi|\\textbf{d})_\\mathrm{|\\boldsymbol{\\Phi}=\\boldsymbol{\\Phi}_O} \\, \\mathpzc{P}(\\textbf{d}|\\boldsymbol\\uptheta) \\, \\mathrm{d}\\textbf{d} .\n\\label{eq:BHM_approx_likelihood}\n\\end{equation}\nIn many cases, the compression from $\\textbf{d}$ to $\\boldsymbol{\\Phi}$ is deterministic, i.e.\n\\begin{equation}\n\\mathpzc{P}(\\boldsymbol{\\Phi}|\\textbf{d}) = \\updelta_\\mathrm{D}(\\boldsymbol{\\Phi} - \\boldsymbol{\\hat{\\Phi}}(\\textbf{d})) ,\n\\label{eq:Dirac_compression}\n\\end{equation}\nwhich simplifies the integral over $\\textbf{d}$ in equations \\eqref{eq:BHM_approx_marginalisation} and \\eqref{eq:BHM_approx_likelihood}.\n\nOn a practical level, $L(\\boldsymbol{\\uptheta})$ is still of unknown analytical form (which is a property of $\\mathpzc{P}(\\boldsymbol\\Phi|\\boldsymbol\\uptheta)$ inherited from $\\mathpzc{P}(\\textbf{d}|\\boldsymbol{\\uptheta})$ in model \\ref{fig:BHM_approx}). Therefore, it has to be approximated using the simulator. We denote by $\\widehat{L}^N(\\boldsymbol{\\uptheta})$ an estimate of $L(\\boldsymbol{\\uptheta})$ computed using $N$ realisations of the simulator. The limiting approximation, in the case where infinite computer resources were available, is denoted by $\\widetilde{L}(\\boldsymbol{\\uptheta})$, such that\n\\begin{equation}\n\\widehat{L}^N(\\boldsymbol{\\uptheta}) \\xrightarrow[N \\rightarrow \\infty]{} \\widetilde{L}(\\boldsymbol{\\uptheta}) .\n\\end{equation}\nNote that $\\widetilde{L}(\\boldsymbol{\\uptheta})$ can be different from $L(\\boldsymbol{\\uptheta})$, depending on the assumptions made to construct $\\widehat{L}^N(\\boldsymbol{\\uptheta})$. These are discussed in section \\ref{ssec:Computable approximations of the likelihood}.\n\n\\subsection{Computable approximations of the likelihood}\n\\label{ssec:Computable approximations of the likelihood}\n\n\\subsubsection{Deterministic simulators}\n\\label{sssec:Deterministic simulators}\n\nThe simplest possible case is when the simulator does not use any random variable, i.e. $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ is an entirely deterministic function of $\\boldsymbol{\\uptheta}$ (see figure \\ref{fig:BHM_exact}, left). Equivalently, all the conditional probabilities appearing in equation \\eqref{eq:BHM_approx_expansion} reduce to Dirac delta distributions given by equations \\eqref{eq:Dirac_deterministic_simulator} and \\eqref{eq:Dirac_compression}. In this case, one can directly use the approximate likelihood given by equation \\eqref{eq:L_theta}, complemented by an assumption on the functional shape of $\\mathpzc{P}(\\boldsymbol{\\Phi}|\\boldsymbol{\\uptheta})$.\n\n\\subsubsection{Parametric approximations and the synthetic likelihood}\n\\label{sssec:Parametric approximations and the synthetic likelihood}\n\nWhen the simulator is not deterministic, the pdf $\\mathpzc{P}(\\boldsymbol{\\Phi}|\\boldsymbol{\\uptheta})$ is unknown analytically. Nonetheless, in some situations, it may be reasonably assumed to follow specific parametric forms.\n\nFor example, if $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ is obtained through averaging a sufficient number of independent and identically distributed variables contained in $\\textbf{d}$, the central limit theorem suggests that a Gaussian distribution is appropriate, i.e. $\\widetilde{L}(\\boldsymbol{\\uptheta}) = \\exp\\left[\\tilde{\\ell}(\\boldsymbol{\\uptheta})\\right]$ with \n\\begin{equation}\n-2 \\tilde{\\ell}(\\boldsymbol{\\uptheta}) \\equiv \\log \\left| 2\\pi \\boldsymbol{\\Sigma}_{\\boldsymbol{\\uptheta}} \\right| + (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\upmu}_{\\boldsymbol{\\uptheta}})^\\intercal \\boldsymbol{\\Sigma}_{\\boldsymbol{\\uptheta}}^{-1} (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\upmu}_{\\boldsymbol{\\uptheta}}),\n\\end{equation}\nwhere the mean and covariance matrix,\n\\begin{equation}\n\\boldsymbol{\\upmu}_{\\boldsymbol{\\uptheta}} \\equiv \\mathrm{E}\\left[ \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}} \\right] \\enskip \\mathrm{and} \\enskip \\boldsymbol{\\Sigma}_{\\boldsymbol{\\uptheta}} \\equiv \\mathrm{E}\\left[ (\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}-\\boldsymbol{\\upmu}_{\\boldsymbol{\\uptheta}}) (\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}-\\boldsymbol{\\upmu}_{\\boldsymbol{\\uptheta}})^\\intercal \\right],\n\\end{equation}\ncan depend on $\\boldsymbol{\\uptheta}$. This is an approximation of $L(\\boldsymbol{\\uptheta})$, unless the summary statistics $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ are indeed Gaussian-distributed. $\\boldsymbol{\\upmu}_{\\boldsymbol{\\uptheta}}$ and $\\boldsymbol{\\Sigma}_{\\boldsymbol{\\uptheta}}$ are generally unknown, but can be estimated using the simulator: given a set of $N$ simulations $\\lbrace \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}^{(i)} \\rbrace$, drawn independently from $\\mathpzc{P}(\\boldsymbol{\\Phi}|\\boldsymbol{\\uptheta})$, one can define\n\\begin{equation}\n\\boldsymbol{\\hat{\\upmu}}_{\\boldsymbol{\\uptheta}} \\equiv \\mathrm{E}^N\\left[ \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}} \\right] \\enskip \\mathrm{and} \\enskip \\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}} \\equiv \\mathrm{E}^N\\left[ (\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}-\\boldsymbol{\\hat{\\upmu}}_{\\boldsymbol{\\uptheta}}) (\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}-\\boldsymbol{\\hat{\\upmu}}_{\\boldsymbol{\\uptheta}})^\\intercal \\right],\n\\label{eq:mean_covariance_empirical}\n\\end{equation}\nwhere $\\mathrm{E}^N$ stands for the empirical average over the set of simulations. A computable approximation of the likelihood is therefore $\\widehat{L}^N(\\boldsymbol{\\uptheta}) = \\exp\\left[ \\hat{\\ell}^N(\\boldsymbol{\\uptheta}) \\right]$, where\n\\begin{equation}\n-2 \\hat{\\ell}^N(\\boldsymbol{\\uptheta}) \\equiv \\log \\left| 2\\pi \\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}} \\right| + (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\hat{\\upmu}}_{\\boldsymbol{\\uptheta}})^\\intercal \\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}^{-1} (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\hat{\\upmu}}_{\\boldsymbol{\\uptheta}}).\n\\label{eq:synthetic_likelihood}\n\\end{equation}\nDue to the approximation of the expectation $\\mathrm{E}$ with an empirical average $\\mathrm{E}^N$, both $\\boldsymbol{\\hat{\\upmu}}_{\\boldsymbol{\\uptheta}}$ and $\\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}$ become random objects. The approximation of the likelihood $\\widehat{L}^N(\\boldsymbol{\\uptheta})$ is therefore a random function with some intrinsic uncertainty itself, and its computation is a stochastic process. This is further discussed using a simple example in section \\ref{ssec:Summarising Gaussian signals}.\n\nThe approximation given in equation \\eqref{eq:synthetic_likelihood}, known as the synthetic likelihood \\citep{Wood2010,Price2017}, has already been applied successfully to perform approximate inference in several scientific fields. However, as pointed out by \\citet{SellentinHeavens2016}, for inference from Gaussian-distributed summaries $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ with an estimated covariance matrix $\\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}$, a different parametric form, namely a multivariate $t$-distribution, should rather be used. The investigation of a synthetic $t$-likelihood is left to future investigations.\n\nIn section \\ref{ssec:Summarising Gaussian signals} and appendix \\ref{apx:Summarising Gaussian signals}, we extend previous work on the Gaussian synthetic likelihood and introduce a Gamma synthetic likelihood for case where the $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ are (or can be assumed to be) Gamma-distributed. \n\n\\subsubsection{Non-parametric approximations and likelihood-free rejection sampling}\n\\label{sssec:Non-parametric approximations and likelihood-free rejection sampling}\n\nAn alternative to assuming a parametric form for $L(\\boldsymbol{\\uptheta})$ is to replace it by a kernel density estimate of the distribution of a discrepancy between simulated and observed summary statistics, i.e.\n\\begin{equation}\n\\widetilde{L}(\\boldsymbol{\\uptheta}) \\equiv \\mathrm{E}\\left[ \\kappa(\\Delta_{\\boldsymbol{\\uptheta}}) \\right],\n\\end{equation}\nwhere $\\Delta_{\\boldsymbol{\\uptheta}}$ is a non-negative function of $\\boldsymbol{\\Phi}_\\mathrm{O}$ and $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ (usually of $\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$) which can also possibly depend on $\\boldsymbol{\\uptheta}$ and any variable used internally by the simulator, and the kernel $\\kappa$ is a non-negative, univariate function independent of $\\boldsymbol{\\uptheta}$ (usually with a maximum at zero). A computable approximation of the likelihood is then given by\n\\begin{equation}\n\\widehat{L}^N(\\boldsymbol{\\uptheta}) \\equiv \\mathrm{E}^N\\left[ \\kappa(\\Delta_{\\boldsymbol{\\uptheta}}) \\right] .\n\\end{equation}\n\nFor likelihood-free inference, $\\kappa$ is often chosen as the uniform kernel on the interval $\\left[ 0, \\varepsilon \\right)$, i.e. $\\kappa(u) \\propto \\chi_{\\left[ 0, \\varepsilon \\right)}(u)$, where $\\varepsilon$ is called the threshold and the indicator function $\\chi_{\\left[ 0, \\varepsilon \\right)}$ equals one if $u \\in \\left[ 0, \\varepsilon \\right)$ and zero otherwise. This yields\n\\begin{equation}\n\\widetilde{L}(\\boldsymbol{\\uptheta}) \\propto \\mathpzc{P}(\\Delta_{\\boldsymbol{\\uptheta}} \\leq \\varepsilon) \\quad \\mathrm{and} \\quad \\widehat{L}^N(\\boldsymbol{\\uptheta}) \\propto \\mathpzc{P}^N(\\Delta_{\\boldsymbol{\\uptheta}} \\leq \\varepsilon),\n\\label{eq:approximate_likelihood_acceptance}\n\\end{equation}\nwhere $\\mathpzc{P}^N(\\Delta_{\\boldsymbol{\\uptheta}} \\leq \\varepsilon)$ is the empirical probability that the discrepancy is below the threshold. $\\widehat{L}^N(\\boldsymbol{\\uptheta})$ can be straightforwardly evaluated by running simulations, computing $\\Delta_{\\boldsymbol{\\uptheta}}$ and using $\\Delta_{\\boldsymbol{\\uptheta}} \\leq \\varepsilon$ as a criterion for acceptance or rejection of proposed samples. Such an approach is often simply (or mistakenly) referred to as approximate Bayesian computation (ABC) in the astrophysics literature, although the more appropriate and explicit denomination is likelihood-free rejection sampling \\citep[see e.g.][]{Marin2011}.\n\nIt is interesting to note that the parametric approximate likelihood approach of section \\ref{sssec:Parametric approximations and the synthetic likelihood} can be embedded into the non-parametric approach. Indeed, $\\Delta_{\\boldsymbol{\\uptheta}}$ can be defined as\n\\begin{equation}\n\\Delta^{\\textbf{C}_{\\boldsymbol{\\uptheta}}}_{\\boldsymbol{\\uptheta}} \\equiv \\log|2\\pi \\textbf{C}_{\\boldsymbol{\\uptheta}}| + (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}})^\\intercal \\textbf{C}_{\\boldsymbol{\\uptheta}}^{-1} (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}})\n\\end{equation}\nfor some positive semidefinite matrix $\\textbf{C}_{\\boldsymbol{\\uptheta}}$. The second term is the square of the Mahalanobis distance, which includes the Euclidean distance as a special case, when $\\textbf{C}_{\\boldsymbol{\\uptheta}}$ is the identity matrix. Using an exponential kernel $\\kappa(u) = \\exp(-u\/2)$ and $\\textbf{C}_{\\boldsymbol{\\uptheta}} = \\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}$ gives $\\widetilde{L}(\\boldsymbol{\\uptheta}) = \\mathrm{E}\\left[ \\kappa(\\Delta_{\\boldsymbol{\\uptheta}}^{\\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}}) \\right]$ and $\\widehat{L}^N(\\boldsymbol{\\uptheta}) = \\mathrm{E}^N\\left[ \\kappa(\\Delta_{\\boldsymbol{\\uptheta}}^{\\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}}) \\right]$ with\n\\begin{eqnarray}\n-2 \\log\\left[\\kappa(\\Delta_{\\boldsymbol{\\uptheta}}^{\\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}}) \\right] & = & \\log \\left| 2\\pi \\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}} \\right| \\\\\n& & + (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}})^\\intercal \\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}^{-1} (\\boldsymbol{\\Phi}_\\mathrm{O} - \\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}), \\nonumber\n\\end{eqnarray}\nthe form of which is similar to equation \\eqref{eq:synthetic_likelihood}. In fact, \\citet[][proposition 1]{GutmannCorander2016} show that the synthetic likelihood satisfies\n\\begin{eqnarray}\n-2\\tilde{\\ell}(\\boldsymbol{\\uptheta}) & = & J(\\boldsymbol{\\uptheta}) + \\mathrm{constant}, \\quad \\mathrm{and}\\label{eq:l_J_proposition_1}\\\\\n-2\\hat{\\ell}^N(\\boldsymbol{\\uptheta}) & = & \\widehat{J}^N(\\boldsymbol{\\uptheta}) + \\mathrm{constant},\\label{eq:l_J_proposition_2}\n\\end{eqnarray}\nwhere \n\\begin{equation}\nJ(\\boldsymbol{\\uptheta}) \\equiv \\mathrm{E}\\left[ \\Delta_{\\boldsymbol{\\uptheta}}^{\\textbf{C}_{\\boldsymbol{\\uptheta}}} \\right]\n\\label{eq:def_J}\n\\end{equation}\nand\n\\begin{equation}\n\\widehat{J}^N(\\boldsymbol{\\uptheta}) \\equiv \\mathrm{E}^N\\left[ \\Delta_{\\boldsymbol{\\uptheta}}^{\\textbf{C}_{\\boldsymbol{\\uptheta}}} \\right]\n\\label{eq:def_J_N}\n\\end{equation}\nare respectively the expectation and the empirical average of the discrepancy $\\Delta_{\\boldsymbol{\\uptheta}}^{\\textbf{C}_{\\boldsymbol{\\uptheta}}}$, for $\\textbf{C}_{\\boldsymbol{\\uptheta}}= \\boldsymbol{\\hat{\\Sigma}}_{\\boldsymbol{\\uptheta}}$.\n\n\\section{Regression and Optimisation for likelihood-free inference}\n\\label{sec:Regression and Optimisation for likelihood-free inference}\n\n\\subsection{Computational difficulties with likelihood-free rejection sampling}\n\\label{ssec:Computational difficulties with likelihood-free rejection sampling}\n\nWe have seen in section \\ref{ssec:Computable approximations of the likelihood} that computable approximations $\\widehat{L}^N(\\boldsymbol{\\uptheta})$ of the likelihood $L(\\boldsymbol{\\uptheta})$ are stochastic processes, due to the use of simulations to approximate intractable expectations. In the most popular ABC approach, i.e. likelihood-free rejection sampling (see section \\ref{sssec:Non-parametric approximations and likelihood-free rejection sampling}), the expectations are approximated by empirical probabilities that the discrepancy is below the threshold $\\varepsilon$. While this approach allows inference of simulator-based statistical models with minimal assumptions, it suffers from several limitations that can make its use impossible in practice.\n\\begin{enumerate}\n\\item It rejects most of the proposed samples when $\\varepsilon$ is small, leading to a computationally inefficient algorithm.\n\\item It does not make assumptions about the shape or smoothness of the target function $L(\\boldsymbol{\\uptheta})$, hence accepted samples cannot ``share'' information in parameter space.\n\\item It uses a fixed proposal distribution (typically the prior $\\mathpzc{P}(\\boldsymbol{\\uptheta})$) and does not make use of already accepted samples to update the proposal of new points.\n\\item It aims at equal accuracy for all regions in parameter space, regardless of the values of the likelihood.\n\\end{enumerate}\n\nTo overcome these issues, the proposed approach follows closely \\citet{GutmannCorander2016}, who combine regression of the discrepancy (addressing issues 1 and 2) with Bayesian optimisation (addressing issues 3 and 4) in order to improve the computational efficiency of inference of simulator-based models. In this work, we focus on parametric approximations of the likelihood; we refer to \\citet{GutmannCorander2016} for a treatment of the non-parametric approach.\n\n\\subsection{Regression of the discrepancy}\n\\label{ssec:Regression of the discrepancy}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{GP_illustration.pdf} \n\\caption{Illustration of Gaussian process regression in one dimension, for the target test function $f: \\theta \\mapsto 2 - \\exp\\left[-(\\theta - 2)^2\\right] - \\exp\\left[-(\\theta - 6)^2\/10\\right] - 1\/ (\\theta^2 + 1)$ (dashed line). Training data are acquired (red dots); they are subject to a Gaussian observation noise with standard deviation $\\sigma_\\mathrm{n} = 0.03$. The blue line shows the mean prediction $\\mu(\\theta)$ of the Gaussian process regression, and the shaded region the corresponding $2\\sigma(\\theta)$ uncertainty. Gaussian processes allow interpolating and extrapolating predictions in regions of parameter space where training data are absent.\\label{fig:GP_illustration}}\n\\end{center}\n\\end{figure}\n\nThe standard approach to obtain a computable approximate likelihood relies on empirical averages (equations \\eqref{eq:mean_covariance_empirical} and \\eqref{eq:def_J_N}). However, such sample averages are not the only way to approximate intractable expectations. Equations \\eqref{eq:l_J_proposition_1} and \\eqref{eq:def_J} show that, up to constants and the sign, $\\tilde{\\ell}(\\boldsymbol{\\uptheta})$ can be interpreted as a regression function with the model parameters $\\boldsymbol{\\uptheta}$ (the ``predictors'') as the independent input variables and the discrepancy $\\Delta_{\\boldsymbol{\\uptheta}}$ as the response variable. Therefore, in the present approach, we consider an approximation of the intractable expectation defining $J(\\boldsymbol{\\uptheta})$ in equation \\eqref{eq:def_J} based on a regression analysis of $\\Delta_{\\boldsymbol{\\uptheta}}$, instead of sample averages. Explicitly, we consider\n\\begin{equation}\n\\widehat{J}^{(\\mathrm{t})}(\\boldsymbol{\\uptheta}) \\equiv \\mathrm{E}^{(\\mathrm{t})}\\left[ \\Delta_{\\boldsymbol{\\uptheta}}^{\\textbf{C}_{\\boldsymbol{\\uptheta}}} \\right],\n\\label{eq:def_J_t}\n\\end{equation}\nwhere the superscript $(\\mathrm{t})$ stands for ``training'' and the expectation $\\mathrm{E}^{(\\mathrm{t})}$ is taken under the probabilistic model defined in the following.\n\nInferring $J(\\boldsymbol{\\uptheta})$ via regression requires a training data set $\\lbrace (\\boldsymbol{\\uptheta}^{(i)} , \\Delta_{\\boldsymbol{\\uptheta}}^{(i)}) \\rbrace\\vspace*{-2pt}$ where the discrepancies are computed from the simulated summary statistics $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}^{(i)}$. Building this training set requires to run simulations, but does not involve an accept\/reject criterion as does likelihood-free rejection sampling (thus addressing issue 1, see section \\ref{ssec:Computational difficulties with likelihood-free rejection sampling}). A regression-based approach also allows incorporating a smoothness assumption about $J(\\boldsymbol{\\uptheta})$. In this way, samples of the training set can ``share'' the information of the computed $\\Delta_{\\boldsymbol{\\uptheta}}$ in the neighbourhood of $\\boldsymbol{\\uptheta}$ (thus addressing issue 2). This suggests that fewer simulated data are needed to reach a certain level of accuracy when learning the target function $J(\\boldsymbol{\\uptheta})$.\n\nIn this work, we rely on Gaussian process (GP) regression in order to construct a prediction for $J(\\boldsymbol{\\uptheta})$. There are several reasons why this choice is advantageous for likelihood-free inference. First, GPs are a general-purpose regressor, able to deal with a large variety of functional shapes for $J(\\boldsymbol{\\uptheta})$, including potentially complex non-linear, or multi-modal features. Second, GPs provide not only a prediction (the mean of the regressed function), but also the uncertainty of the regression. This is useful for actively constructing the training data via Bayesian optimisation, as we show in section \\ref{ssec:Acquisition rules}. Finally, GPs allow extrapolating the prediction into regions of the parameter space where no training points are available. These three properties are shown in figure \\ref{fig:GP_illustration} for a multi-modal test function subject to observation noise.\n\nWe now briefly review Gaussian process regression. Suppose that we have a set of $t$ training points, $(\\boldsymbol{\\Theta}, \\textbf{f}) \\equiv \\lbrace (\\boldsymbol{\\uptheta}^{(i)}, f^{(i)} = f(\\boldsymbol{\\uptheta}^{(i)}) \\rbrace$, of the function $f$ that we want to regress. We assume that $f$ is a Gaussian process with prior mean function $m(\\boldsymbol{\\uptheta})$ and covariance function $\\kappa(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}')$ also known as the kernel \\citep[see][]{RasmussenWilliams2006}. The joint probability distribution of the training set is therefore $\\mathpzc{P}(\\textbf{f}|\\boldsymbol{\\Theta}) \\propto \\exp\\left[ \\ell(\\textbf{f}|\\boldsymbol{\\Theta}) \\right]$, where the exponent $\\ell(\\textbf{f}|\\boldsymbol{\\Theta})$ is\n\\begin{equation}\n- \\frac{1}{2} \\sum_{i,j=1}^t \\left[f^{(i)}-m(\\boldsymbol{\\uptheta}^{(i)})\\right]^\\intercal \\kappa(\\boldsymbol{\\uptheta}^{(i)},\\boldsymbol{\\uptheta}^{(j)})^{-1} \\left[f^{(j)}-m(\\boldsymbol{\\uptheta}^{(j)})\\right] .\n\\end{equation}\nThe mean function $m(\\boldsymbol{\\uptheta})$ and the kernel $\\kappa(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}')$ define the functional shape and smoothness allowed for the prediction. Standard choices are respectively a constant and a squared exponential (the radial basis function, RBF), subject to additive Gaussian observation noise with variance $\\sigma_\\mathrm{n}^2$. Explicitly, $m(\\boldsymbol{\\uptheta}) \\equiv C$ and \n\\begin{equation}\n\\kappa(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}') \\equiv \\sigma_f^2 \\exp\\left[ -\\frac{1}{2} \\sum_p \\left( \\frac{\\theta_p - \\theta_p'}{\\lambda_p} \\right)^2 \\right] + \\sigma_\\mathrm{n}^2 \\, \\updelta_\\mathrm{K}(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}').\n\\end{equation}\nThe $\\theta_p$ and $\\theta_p'$ are the components of $\\boldsymbol{\\uptheta}$ and $\\boldsymbol{\\uptheta}'$, respectively. In the last term, $\\updelta_\\mathrm{K}(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}')$ is one if and only if $\\boldsymbol{\\uptheta} = \\boldsymbol{\\uptheta}'$ and zero otherwise. The hyperparameters are $C$, the $\\lambda_p$ (the length scales controlling the amount of correlation between points, and hence the allowed wiggliness of $f$), $\\sigma_f^2$ (the signal variance, i.e. the marginal variance of $f$ at a point $\\boldsymbol{\\uptheta}$ if the observation noise was zero), and $\\sigma_\\mathrm{n}^2$ (the observation noise). For the results of this paper, GP hyperparameters were learned from the training set using L-BFGS \\citep{L-BFGS}, a popular optimiser for machine learning, and updated every time the training set was augmented with ten samples.\n\nThe predicted value $f_\\star$ at a new point $\\boldsymbol{\\uptheta}_\\star$ can be obtained from the fact that $(\\lbrace \\boldsymbol{\\Theta}, \\boldsymbol{\\uptheta}_\\star \\rbrace, \\lbrace \\textbf{f} , f_\\star \\rbrace)$ form jointly a random realisation of the Gaussian process $f$. Thus, the target pdf $\\mathpzc{P}(f_\\star|\\textbf{f}, \\boldsymbol{\\Theta}, \\boldsymbol{\\uptheta}_\\star)$ can be obtained from conditioning the joint pdf $\\mathpzc{P}(\\textbf{f},f_\\star | \\boldsymbol{\\Theta}, \\boldsymbol{\\uptheta}_\\star)$ to the values of the training set $\\textbf{f}$. The result is \\citep[see][section 2.7]{RasmussenWilliams2006}\n\\begin{eqnarray}\n\\mathpzc{P}(f_\\star|\\textbf{f}, \\boldsymbol{\\Theta}, \\boldsymbol{\\uptheta}_\\star) & \\propto & \\exp\\left[ -\\frac{1}{2} \\left( \\frac{f_\\star - \\mu(\\boldsymbol{\\uptheta}_\\star)}{\\sigma(\\boldsymbol{\\uptheta}_\\star)} \\right)^2 \\right], \\label{eq:GP_posterior_predictive_distribution}\\\\\n\\mu(\\boldsymbol{\\uptheta}_\\star) & \\equiv & m(\\boldsymbol{\\uptheta}_\\star) + \\uline{\\textbf{K}}_\\star^\\intercal \\uuline{\\textbf{K}}^{-1} (\\textbf{f} - \\textbf{m}), \\label{eq:GP_mean}\\\\\n\\sigma^2(\\boldsymbol{\\uptheta}_\\star) & \\equiv & K_{\\star\\star} - \\uline{\\textbf{K}}_\\star^\\intercal \\uuline{\\textbf{K}}^{-1} \\uline{\\textbf{K}}_\\star, \\label{eq:GP_variance}\n\\end{eqnarray}\nwhere we use the definitions\n\\begin{eqnarray}\nK_{\\star\\star} & \\equiv & \\kappa(\\boldsymbol{\\uptheta}_\\star, \\boldsymbol{\\uptheta}_\\star), \\label{eq:GP_notation_def_1}\\\\\n\\textbf{m} & \\equiv & \\left(m(\\boldsymbol{\\uptheta}^{(i)})\\right)^\\intercal \\quad \\mathrm{for}~\\boldsymbol{\\uptheta}^{(i)} \\in \\boldsymbol{\\Theta}, \\label{eq:GP_notation_def_2}\\\\\n\\uline{\\textbf{K}}_\\star & \\equiv & \\left(\\kappa(\\boldsymbol{\\uptheta}_\\star, \\boldsymbol{\\uptheta}^{(i)})\\right)^\\intercal \\quad \\mathrm{for}~\\boldsymbol{\\uptheta}^{(i)} \\in \\boldsymbol{\\Theta}, \\label{eq:GP_notation_def_3}\\\\\n(\\uuline{\\textbf{K}})_{ij} & \\equiv & \\kappa(\\boldsymbol{\\uptheta}^{(i)}, \\boldsymbol{\\uptheta}^{(j)}) \\quad \\mathrm{for}~\\lbrace \\boldsymbol{\\uptheta}^{(i)}, \\boldsymbol{\\uptheta}^{(j)} \\rbrace \\in \\boldsymbol{\\Theta}^2. \\label{eq:GP_notation_def_4}\n\\end{eqnarray}\n\n\\subsection{Bayesian optimisation}\n\\label{ssec:Bayesian optimisation}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{BO_illustration_11.pdf} \n\\includegraphics[width=0.49\\textwidth]{BO_illustration_12.pdf} \\\\\n\\includegraphics[width=0.49\\textwidth]{BO_illustration_13.pdf} \n\\includegraphics[width=0.49\\textwidth]{BO_illustration_14.pdf} \n\\caption{Illustration of four consecutive steps of Bayesian optimisation to learn the test function of figure \\ref{fig:GP_illustration}. For each step, the top panel shows the training data points (red dots) and the regression (blue line and shaded region). The bottom panel shows the acquisition function (the expected improvement, solid green line) with its maximiser (dashed green line). The next acquisition point, i.e. where to run a simulation to be added to the training set, is shown in orange; it differs from the maximiser of the acquisition function by a small random number. The acquisition function used is the expected improvement, aiming at finding the minimum of $f$. Hyperparameters of the regression kernel are optimised after each acquisition. As can observed, Bayesian optimisation implements a trade-off between exploration (evaluation of the target function where the variance is large, e.g. after 12 points) and exploitation (evaluation of the target function close to the predicted minimum, e.g. after 11, 13, and 14 points).  \\label{fig:BO_illustration}}\n\\end{center}\n\\end{figure*}\n\nThe second major ingredient of the proposed approach is Bayesian optimisation, which allows the inference of the regression function $J(\\boldsymbol{\\uptheta})$ while avoiding unnecessary computations. It allows active construction of the training data set $\\lbrace (\\boldsymbol{\\uptheta}^{(i)} , \\Delta_{\\boldsymbol{\\uptheta}}^{(i)}) \\rbrace$, updating the proposal of new points using the regressed $\\widehat{J}^{(\\mathrm{t})}(\\boldsymbol{\\uptheta})$ (thus addressing issue 3 with likelihood-free rejection sampling, see section \\ref{ssec:Computational difficulties with likelihood-free rejection sampling}). Further, since we are mostly interested in the regions of the parameter space where the variance of the approximate posterior is large (due to its stochasticity), the acquisition rules can prioritise these regions, so as to obtain a better approximation of $J(\\boldsymbol{\\uptheta})$ there (thus addressing issue 4).\n\nBayesian optimisation is a decision-making framework under uncertainty, for the automatic learning of unknown functions. It aims at gathering training data in such a manner as to evaluate the regression model the least number of times while revealing as much information as possible about the target function and, in particular, the location of the optimum or optima. The method proceeds by iteratively picking predictors to be probed (i.e. simulations to be run) in a manner that trades off \\textit{exploration} (parameters for which the outcome is most uncertain) and \\textit{exploitation} (parameters which are expected to have a good outcome for the targeted application). In many contexts, Bayesian optimisation has been shown to obtain better results with fewer simulations than grid search or random search, due to its ability to reason about the interest of simulations before they are run \\citep[see][for a review]{Brochu2010}. Figure \\ref{fig:BO_illustration} illustrates Bayesian optimisation in combination with Gaussian process regression, applied to finding the minimum of the test function of figure \\ref{fig:GP_illustration}.\n\nIn the following, we give a brief overview of the elements of Bayesian optimisation used in this paper. In order to add a new point to the training data set $(\\boldsymbol{\\Theta}, \\textbf{f}) \\equiv \\lbrace (\\boldsymbol{\\uptheta}^{(i)}, f^{(i)} = f(\\boldsymbol{\\uptheta}^{(i)}) \\rbrace$, Bayesian optimisation uses an acquisition function $\\mathcal{A}(\\boldsymbol{\\uptheta})$ that estimates how useful the evaluation of the simulator at $\\boldsymbol{\\uptheta}$ will be in order to learn the target function. The acquisition function is constructed from the posterior predictive distribution of $f$ given the training set $(\\boldsymbol{\\Theta}, \\textbf{f})$, i.e. from the mean prediction $\\mu(\\boldsymbol{\\uptheta})$ and the uncertainty $\\sigma(\\boldsymbol{\\uptheta})$ of the regression analysis (equations \\eqref{eq:GP_mean} and \\eqref{eq:GP_variance}). The optimum of the acquisition function in parameter space determines the next point $\\boldsymbol{\\uptheta}_\\star \\equiv \\mathrm{argopt}_{\\boldsymbol{\\uptheta}} \\mathcal{A}(\\boldsymbol{\\uptheta})$ to be evaluated by the simulator ($\\mathrm{argopt} = \\mathrm{argmax}$ or $\\mathrm{argmin}$ depending on how the acquisition function is defined), so that the training set can be augmented with $(\\boldsymbol{\\uptheta}_\\star, f(\\boldsymbol{\\uptheta}_\\star))$. The acquisition function is a scalar function whose evaluation should be reasonably expensive, so that its optimum can be found by simple search methods such as gradient descent. \n\nThe algorithm needs to be initialised with an initial training set. In numerical experiments, we found that building this initial set by drawing from the prior (as would typically be done in likelihood-free rejection sampling) can result in difficulties with the first iterations of Gaussian process regression. Uniformly-distributed points within the boundaries of the GP are also a poor choice, as they will result in an uneven initial sampling of the parameter space. To circumvent this issue, we build the initial training set using a low-discrepancy quasi-random Sobol sequence \\citep{Sobol1967}, which covers the parameter space more evenly.\n\n\\subsection{Expressions for the approximate posterior}\n\\label{ssec:Expressions for the approximate posterior}\n\nAs discussed in section \\ref{ssec:Regression of the discrepancy}, using $\\Delta_{\\boldsymbol{\\uptheta}}^{\\textbf{C}_{\\boldsymbol{\\uptheta}}}$ as the regressed quantity directly gives an estimate of $J(\\boldsymbol{\\uptheta})$ in equation \\eqref{eq:def_J}. The response variable is thus $f(\\boldsymbol{\\uptheta}) \\equiv \\Delta_{\\boldsymbol{\\uptheta}}^{\\textbf{C}_{\\boldsymbol{\\uptheta}}}$ and the regression then gives\n\\begin{equation}\n\\widehat{J}^{(\\mathrm{t})}(\\boldsymbol{\\uptheta}) = \\mu(\\boldsymbol{\\uptheta}).\n\\label{eq:J_t_equals_mu}\n\\end{equation}\n\nIn the parametric approach to likelihood approximation, this is equivalent to an approximation of $-2\\tilde{\\ell}(\\boldsymbol{\\uptheta}) = -2\\log \\widetilde{L}(\\boldsymbol{\\uptheta})$ (see equation \\eqref{eq:l_J_proposition_1}). The expectation of the (unnormalised) approximate posterior is therefore directly given as (see equation \\eqref{eq:approx_problem_Bayes})\n\\begin{equation}\n\\mathrm{E}^{(\\mathrm{t})} \\left[ \\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}) \\right] \\equiv \\mathpzc{P}(\\boldsymbol{\\uptheta}) \\exp\\left( -\\frac{1}{2} \\mu(\\boldsymbol{\\uptheta}) \\right),\n\\label{eq:approximate_posterior_expectation}\n\\end{equation}\nwhere $\\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}) \\approx Z_{\\boldsymbol{\\Phi}} \\times \\mathpzc{P}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi})_{|\\boldsymbol{\\Phi}=\\boldsymbol{\\Phi}_\\mathrm{O}}$.\n\nThe estimate of the variance of $f(\\boldsymbol{\\uptheta})$ can also be propagated to the approximate posterior, giving\n\\begin{equation}\n\\mathrm{V}^{(\\mathrm{t})} \\left[ \\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}) \\right] \\equiv \\frac{\\mathpzc{P}(\\boldsymbol{\\uptheta})^2}{4} \\exp\\left[ -\\mu(\\boldsymbol{\\uptheta}) \\right] \\sigma^2(\\boldsymbol{\\uptheta}) .\n\\label{eq:approximate_posterior_variance}\n\\end{equation}\nDetails of the computations can be found in appendix \\ref{sapx:Expressions for the approximate posterior}.\n\nExpressions for the {\\textsc{bolfi}} posterior in the non-parametric approach with the uniform kernel can also be derived \\citep[][lemma 3.1]{Jaervenpaeae2017}. As this paper focuses on the parametric approach, we refer to the literature for the former case.\n\n\\subsection{Acquisition rules}\n\\label{ssec:Acquisition rules}\n\n\\subsubsection{Expected improvement}\n\\label{sssec:Expected improvement}\n\nStandard Bayesian optimisation uses acquisition functions that estimate how useful the next evaluation of the simulator will be in order to find the minimum or minima of the target function. While several other choices are possible \\citep[see e.g.][]{Brochu2010}, in this work we discuss the acquisition function known as \\textit{expected improvement} (EI). The \\textit{improvement} is defined by $I(\\boldsymbol{\\uptheta}_\\star) = \\max\\left[\\min(\\textbf{f}) - f(\\boldsymbol{\\uptheta}_\\star), 0\\right]$, and the expected improvement is $\\mathrm{EI}(\\boldsymbol{\\uptheta}_\\star) \\equiv \\mathrm{E}^{(\\mathrm{t})}\\left[ I(\\boldsymbol{\\uptheta}_\\star) \\right]$, where the expectation is taken with respect to the random observation assuming decision $\\boldsymbol{\\uptheta}_\\star$. For a Gaussian process regressor, this evaluates to \\citep[see][section 2.3]{Brochu2010}\n\\begin{equation}\n\\mathrm{EI}(\\boldsymbol{\\uptheta}_\\star) \\equiv \\sigma(\\boldsymbol{\\uptheta}_\\star) \\left[ z\\Phi(z) + \\phi(z) \\right], \\, \\mathrm{with}~z \\equiv \\frac{\\min(\\textbf{f}) - \\mu(\\boldsymbol{\\uptheta}_\\star)}{\\sigma(\\boldsymbol{\\uptheta}_\\star)},\n\\label{eq:EI}\n\\end{equation}\nor $\\mathrm{EI}(\\boldsymbol{\\uptheta}_\\star) \\equiv 0$ if $\\sigma(\\boldsymbol{\\uptheta}_\\star)=0$, where $\\phi$ and $\\Phi$ denote respectively the pdf and the cumulative distribution function (cdf) of the unit-variance zero-mean Gaussian. The decision rule is to select the location $\\boldsymbol{\\uptheta}_\\star$ that maximises $\\mathrm{EI}(\\boldsymbol{\\uptheta}_\\star)$.\n\nThe EI criterion can be interpreted as follows: since the goal is to find the minimum of $f$, a reward equal to the improvement $\\min(\\textbf{f}) - f(\\boldsymbol{\\uptheta}_\\star)$ is received if $f(\\boldsymbol{\\uptheta}_\\star)$ is smaller than all the values observed so far, otherwise no reward is received. The first term appearing in equation \\eqref{eq:EI} is maximised when evaluating at points with high uncertainty (exploration); and, at fixed variance, the second term is maximised by evaluating at points with low mean (exploitation). The expected improvement therefore automatically captures the exploration-exploitation trade-off as a result of the Bayesian decision-theoretic treatment.\n\n\\subsubsection{Expected integrated variance}\n\\label{sssec:Expected integrated variance}\n\nAs pointed out by \\citet{Jaervenpaeae2017}, in Bayesian optimisation for approximate Bayesian computation, the goal should not be to find the minimum of $J(\\boldsymbol{\\uptheta})$, but rather to minimise the expected uncertainty in the estimate of the approximate posterior over the future evaluation of the simulator at $\\boldsymbol{\\uptheta}_\\star$. Consequently, they propose an acquisition function, known as the \\textit{expected integrated variance} (ExpIntVar or EIV in the following) that selects the next evaluation location to minimise the expected variance of the future posterior density $\\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}, \\boldsymbol{\\uptheta}_\\star)$ over the parameter space. The framework used is Bayesian decision theory. Formally, the loss due to our uncertain knowledge of the approximate posterior density can be defined as\n\\begin{equation}\n\\mathpzc{L}\\left[ \\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}) \\right] = \\int \\mathrm{V}^{(\\mathrm{t})}\\left[ \\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}) \\right] \\, \\mathrm{d}\\boldsymbol{\\uptheta},\n\\end{equation}\nand the acquisition rule is to select the location $\\boldsymbol{\\uptheta}_\\star$ that minimises\n\\begin{equation}\n\\begin{split}\n& \\mathrm{EIV}(\\boldsymbol{\\uptheta}_\\star) \\equiv \\mathrm{E}^{(\\mathrm{t})} \\left[ \\mathpzc{L}\\left[ \\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}, f_\\star, \\boldsymbol{\\uptheta}_\\star) \\right] \\right] \\\\\n& = \\int \\mathpzc{L}\\left[ \\mathpzc{P}_{\\textsc{bolfi}}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi}_\\mathrm{O}, \\textbf{f}, \\boldsymbol{\\Theta}, f_\\star, \\boldsymbol{\\uptheta}_\\star) \\right] \\mathpzc{P}(f_\\star|\\textbf{f}, \\boldsymbol{\\Theta}, \\boldsymbol{\\uptheta}_\\star) \\,  \\mathrm{d}f_\\star\n\\end{split}\n\\end{equation}\nwith respect to $\\boldsymbol{\\uptheta}_\\star$, where we have to marginalise over the unknown simulator output $f_\\star$ using the probabilistic model $\\mathpzc{P}(f_\\star|\\textbf{f}, \\boldsymbol{\\Theta}, \\boldsymbol{\\uptheta}_\\star)$ (equations \\eqref{eq:GP_posterior_predictive_distribution}--\\eqref{eq:GP_variance}).\n\n\\citet[][proposition 3.2]{Jaervenpaeae2017} derive the expressions for the expected integrated variance for a GP model in the non-parametric approach. In appendix \\ref{apx:Derivations of the mathematical results}, we extend this work and derive the ExpIntVar acquisition function and its gradient in the parametric approach. The result is the following: under the GP model, the expected integrated variance after running the simulation model with parameter $\\boldsymbol{\\uptheta}_\\star$ is given by\n\\begin{equation}\n\\mathrm{EIV}(\\boldsymbol{\\uptheta}_\\star) = \\int \\frac{\\mathpzc{P}(\\boldsymbol{\\uptheta})^2}{4} \\exp\\left[ -\\mu(\\boldsymbol{\\uptheta}) \\right] \\left[ \\sigma^2(\\boldsymbol{\\uptheta}) - \\tau^2(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}_\\star) \\right] \\, \\mathrm{d}\\boldsymbol{\\uptheta},\n\\label{eq:EIV}\n\\end{equation}\nwith\n\\begin{equation}\n\\tau^2(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}_\\star) \\equiv \\dfrac{\\mathrm{cov}^2(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}_\\star)}{\\sigma^2(\\boldsymbol{\\uptheta}_\\star)},\n\\label{eq:def_tau}\n\\end{equation}\nwhere $\\mathrm{cov}(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}_\\star) \\equiv \\kappa(\\boldsymbol{\\uptheta},\\boldsymbol{\\uptheta}_\\star) - \\uline{\\textbf{K}}^\\intercal \\uuline{\\textbf{K}}^{-1}\\vspace{-4pt} \\uline{\\textbf{K}}_\\star$ is the GP posterior predicted covariance between the evaluation point $\\boldsymbol{\\uptheta}$ in the integral and the candidate location for the next evaluation $\\boldsymbol{\\uptheta}_\\star$. Note that in addition to the notations given by equations \\eqref{eq:GP_notation_def_1}--\\eqref{eq:GP_notation_def_4}, we have introduced the vector\n\\begin{equation}\n\\uline{\\textbf{K}} \\equiv \\left(\\kappa(\\boldsymbol{\\uptheta}, \\boldsymbol{\\uptheta}^{(i)})\\right)^\\intercal \\quad \\mathrm{for}~\\boldsymbol{\\uptheta}^{(i)} \\in \\boldsymbol{\\Theta}.\n\\label{eq:GP_notation_def_5}\n\\end{equation}\n\nIt is of interest to examine when the integrand in equation \\eqref{eq:EIV} is small. As for the EI (equation \\eqref{eq:EI}), optimal values are found when the mean of the discrepancy $\\mu(\\boldsymbol{\\uptheta})$ is small or the variance $\\sigma^2(\\boldsymbol{\\uptheta})$ is large. This effect is what yields the trade-off between exploitation and exploration for the ExpIntVar acquisition rule. However, unlike in standard Bayesian optimisation strategies such as the EI, the trade-off is a non-local process (due to the integration over the parameter space), and also depends on the prior, so as to minimise the uncertainty in the posterior (and not likelihood) approximation.\n\nComputing the expected integrated variance requires integration over the parameter space. In this work, the integration is performed on a regular grid of $50$ points per dimension within the GP boundaries. In high dimension, the integral can become prohibitively expensive to compute on a grid. As discussed by \\citet{Jaervenpaeae2017}, it can then be evaluated with Monte Carlo or quasi-Monte Carlo methods such as importance sampling.\n\nIn numerical experiments, we have found that the ExpIntVar criterion (as any acquisition function for Bayesian optimisation) has some sensitivity to the initial training set. In particular, the initial set (built from a Sobol sequence or otherwise) shall sample sufficiently well the GP domain, which shall encompass the prior. This ensures that the prior volume is never wider than the training data. Under this condition, as \\citet{Jaervenpaeae2017}, we have found that ExpIntVar is stable, in the sense that it produces consistent {\\textsc{bolfi}} posteriors over different realisations of the initial training data set and simulator outputs.\n\n\\subsubsection{Stochastic versus deterministic acquisition rules}\n\\label{sssec:Stochastic versus deterministic acquisition rules}\n\nThe above rules do not guarantee that the selected $\\boldsymbol{\\uptheta}_\\star$ is different from a previously acquired $\\boldsymbol{\\uptheta}^{(i)}$. \\citet[][see in particular appendix C]{GutmannCorander2016} found that this can result in a poor exploration of the parameter space, and propose to add a stochastic element to the decision rule in order to avoid getting stuck at one point. In some experiments, we followed this prescription by adding an ``acquisition noise'' of strength $\\sigma_\\mathrm{a}^p$ to each component of the optimiser of the acquisition function. More precisely, $\\boldsymbol{\\uptheta}_\\star$ is sampled from the Gaussian distribution $\\mathpzc{G}(\\boldsymbol{\\uptheta}_\\mathrm{opt}, \\textbf{D})$, where $\\boldsymbol{\\uptheta}_\\mathrm{opt} \\equiv \\mathrm{argopt}_{\\boldsymbol{\\uptheta}} \\mathcal{A}(\\boldsymbol{\\uptheta})$ and $\\textbf{D}$ is the diagonal covariance matrix of components $(\\sigma_\\mathrm{a}^p)^2$. The $\\sigma_\\mathrm{a}^p$ are chosen to be of order $\\lambda_p\/10$.\n\nFor a more extensive discussion and comparison of various stochastic and deterministic acquisition rules, the reader is referred to \\citet{Jaervenpaeae2017}.\n\n\\section{Applications}\n\\label{sec:Applications}\n\nIn this section, we show the application of {\\textsc{bolfi}} to several application studies. In particular, we discuss the simulator and the computable approximation of the likelihood to be used, and compare {\\textsc{bolfi}} to likelihood-free rejection sampling in terms of computational efficiency. In all cases, we show that {\\textsc{bolfi}} reduces the amount of required simulations by several orders of magnitude.\n\nIn section \\ref{ssec:Summarising Gaussian signals}, we discuss the toy problem of summarising Gaussian signals (i.e. inferring the unknown mean and\/or variance of Gaussian-distributed data). In section \\ref{ssec:Supernova cosmology}, we show the first application of {\\textsc{bolfi}} to a real cosmological problem using actual observational data: the inference of cosmological parameters from supernovae data. For each test case, we refer to the corresponding section in the appendices for the details of the data model and inference assumptions.\n\n\\subsection{Summarising Gaussian signals}\n\\label{ssec:Summarising Gaussian signals}\n\nA simple toy model can be constructed from the general problem of summarising Gaussian signals with unknown mean, or with unknown mean and variance. This example allows for the comparison of {\\textsc{bolfi}} and likelihood-free rejection sampling to the true posterior conditional on the full data, which is known analytically. All the details of this model are given in appendix \\ref{apx:Summarising Gaussian signals}.\n\n\\subsubsection{Unknown mean, known variance}\n\\label{sssec:Unknown mean, known variance}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{Gaussian_mean_illustration.pdf} \n\\caption{Illustration of {\\textsc{bolfi}} for a one-dimensional problem, the inference of the unknown mean $\\mu$ of a Gaussian. \\textit{Lower panel}. The discrepancy $\\Delta_\\mu$ (i.e. twice the negative log-likelihood) is a stochastic process due to the limited computational resources. Its mean and the $2\\sigma$ credible interval are shown in red. The dashed red line shows one realisation of the stochastic process as a function of $\\mu$. Simulations at different $\\mu$ are shown as black dots. {\\textsc{bolfi}} builds a probabilistic model for the discrepancy, the mean and $2\\sigma$ credible interval of which are shown in blue. \\textit{Upper panel}. The expectation of the (rescaled) {\\textsc{bolfi}} posterior and its $2\\sigma$ credible interval are shown in comparison to the exact posterior for the problem. The dashed red line shows the posterior obtained from the corresponding realisation of the stochastic process of the lower panel. \\label{fig:Gaussian_mean_illustration}}\n\\end{center}\n\\end{figure}\n\nWe first consider the problem, already discussed by \\citet{GutmannCorander2016}, where the data $\\textbf{d}$ are a vector of $n$ components drawn from a Gaussian with unknown mean $\\mu$ and known variance $\\sigma^2_\\mathrm{true}$. The empirical mean $\\Phi^1$ is a sufficient summary statistic for the problem of inferring $\\mu$. The distribution of simulated $\\Phi^1_\\mu$ takes a simple form, $\\Phi^1_\\mu \\sim \\mathpzc{G}\\left( \\mu, \\sigma^2_\\mathrm{true}\/n \\right)$. Using here the true variance, the discrepancy and synthetic likelihood are\n\\begin{equation}\n\\Delta^1_\\mu = -2 \\hat{\\ell}^N_1(\\mu) = \\log \\left(\\frac{2\\pi \\sigma^2_\\mathrm{true}}{n} \\right) + n\\frac{(\\Phi^1_\\mathrm{O}-\\hat{\\mu}^1_\\mu)^2}{\\sigma^2_\\mathrm{true}},\n\\end{equation}\nwhere $\\hat{\\mu}^1_\\mu$ is an average of $N$ realisations of $\\Phi^1_\\mu$. In figure \\ref{fig:Gaussian_mean_illustration} (lower panel), the black dots show simulations of $\\Delta^1_\\mu$ for different values of $\\mu$. We have $\\hat{\\mu}^1_\\mu \\sim \\mathpzc{G}\\left( \\mu, \\sigma^2_\\mathrm{true}\/(Nn) \\right)$, therefore the stochastic process defining the discrepancy can be written\n\\begin{equation}\n\\Delta^1_\\mu = \\log \\left(\\frac{2\\pi \\sigma^2_\\mathrm{true}}{n} \\right) + n\\frac{(\\Phi^1_\\mathrm{O}- \\mu -g )^2}{\\sigma^2_\\mathrm{true}}, \\quad g \\sim \\mathpzc{G}\\left(0, \\sigma^2_g \\right),\n\\end{equation}\nwhere $\\sigma^2_g \\equiv \\sigma^2_\\mathrm{true}\/(Nn)$. Each realisation of $g$ gives a different mapping $\\mu \\mapsto \\Delta^1_\\mu$. In figure \\ref{fig:Gaussian_mean_illustration}, we show one such realisation in the lower panel, and the corresponding approximate posterior in the upper panel. Using the percent point function (inverse of the cdf) of the Gaussian $\\mathpzc{G}\\left(0, \\sigma^2_g \\right)$, we also show in red the mean and $2\\sigma$ credible interval of the true stochastic process.\n\nThe GP regression using the simulations shown as the training set is represented in blue in the lower panel of figure \\ref{fig:Gaussian_mean_illustration}. The corresponding {\\textsc{bolfi}} posterior and its variance, defined by equations \\eqref{eq:approximate_posterior_expectation} and \\eqref{eq:approximate_posterior_variance}, are shown in purple in the upper panel. The uncertainty in the estimate of the posterior (shaded purple region) is due to the limited number of available simulations (and not to the noisiness of individual training points). It is the expectation of this uncertainty under the next evaluation of the simulator which is minimised in parameter space by the ExpIntVar acquisition rule.\n\n\\subsubsection{Unknown mean and variance}\n\\label{sssec:Unknown mean and variance}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Gaussian_mean_variance.pdf} \n\\caption{Prior and posterior for the joint inference of the mean and variance of Gaussian signals. The prior and exact posterior (from the analytic solution) are Gaussian-inverse-Gamma distributed and shown in blue and orange, respectively. In the left panel, the approximate rejection-sampling posterior, based on $5,000$ samples accepted out of $\\sim 350,000$ simulations, is shown in green. It loosely encloses the exact posterior. In the right panel, the approximate {\\textsc{bolfi}} posterior, based on $2,500$ simulations only, is shown in red. It is a much finer approximation of the exact posterior. For all distributions, the $1\\sigma$, $2\\sigma$ and $3\\sigma$ contours are shown.\\label{fig:Gaussian_mean_variance}}\n\\end{center}\n\\end{figure*}\n\nWe now consider the problem where the full data set $\\textbf{d}$ is a vector of $n$ components drawn from a Gaussian with unknown mean $\\mu$ and unknown variance $\\sigma^2$. The aim is the two-dimensional inference of $\\boldsymbol{\\uptheta} \\equiv (\\mu, \\sigma^2)$. Evidently, the true likelihood $\\mathcal{L}(\\mu, \\sigma^2)$ for this problem is the Gaussian characterised by $(\\mu, \\sigma^2)$. The Gaussian-inverse-Gamma distribution is the conjugate prior for this likelihood. It is described by four parameters. Adopting a Gaussian-inverse-Gamma prior characterised by $(\\alpha, \\beta, \\eta, \\lambda)$ yields a Gaussian-inverse-Gamma posterior characterised by $(\\alpha', \\beta', \\eta', \\lambda')$ given by equations \\eqref{eq:Gaussian_analytic_solution_alpha}--\\eqref{eq:Gaussian_analytic_solution_lambda}. This is the analytic solution to which we compare our approximate results.\n\nFor the numerical approach, we forward model the problem using a simulator that draws from the prior, simulates $N = 10$ realisations of the Gaussian signal, and compresses them to two summary statistics, the empirical mean and variance, respectively $\\Phi^1$ and $\\Phi^2$. The graphical probabilistic model is given in figure \\ref{fig:BHM_Gaussian_model}. It is a noise-free simulator without latent variables (of the type given by figure \\ref{fig:BHM_exact}, right) completed by a deterministic compression of the full data. Note that the vector $\\boldsymbol{\\Phi} \\equiv (\\Phi^1 , \\Phi^2)$ is a sufficient statistic for the inference of $(\\mu, \\sigma^2)$. To perform likelihood-free inference, we also need a computable approximation $\\widehat{L}^N(\\mu, \\sigma^2)$ of the true likelihood. We derive such an approximation in section \\ref{sapx:Derivation of the Gaussian-Gamma synthetic likelihood for likelihood-free inference} using a parametric approach, under the assumptions (exactly verified in this example) that $\\Phi^1$ is Gaussian-distributed and $\\Phi^2$ is Gamma-distributed. We name it the Gaussian-Gamma synthetic likelihood.\n\nThe posterior obtained from likelihood-free rejection sampling is shown in green in figure \\ref{fig:Gaussian_mean_variance} (left) in comparison to the prior (in blue) and the analytic posterior (in orange). It was obtained from $5,000$ accepted samples using a threshold of $\\varepsilon = 4$ on $-2\\hat{\\ell}^N$. The entire run required $\\sim 350,000$ forward simulations in total, the vast majority of which have been rejected. The rejection-sampling posterior is a fair approximation to the true posterior, unbiased but broader, as expected from a rejection-sampling method. \n\nFor comparison, the posterior obtained via {\\textsc{bolfi}} is shown in red in figure \\ref{fig:Gaussian_mean_variance} (right). {\\textsc{bolfi}} was initialised using a Sobol sequence of $20$ members to compute the original surrogate surface, and Bayesian optimisation with the ExpIntVar acquisition function and acquisition noise was run to acquire $230$ more samples. As can be observed, {\\textsc{bolfi}} allows very precise likelihood-free inference; in particular, the $1\\sigma$, $2\\sigma$ and $3\\sigma$ contours (the latter corresponding to the $0.27\\%$ least likely events) of the analytic posterior are reconstructed almost perfectly. The overall cost to get these results is only $2,500$ simulations with {\\textsc{bolfi}} versus $\\sim 350,000$ with rejection sampling (for a poorer approximation of the analytic posterior), which corresponds to a reduction by $2$ orders of magnitude. \n\n\\subsection{Supernova cosmology}\n\\label{ssec:Supernova cosmology}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{Supernovae.pdf} \n\\caption{Prior and posterior distributions for the joint inference of the matter density of the Universe, $\\Omega_\\mathrm{m}$, and the dark energy equation of state, $w$, from the JLA supernovae data set. The prior and exact posterior distribution (obtained from a long MCMC run requiring $\\sim 6 \\times 10^6$ data model evaluations) are shown in blue and orange, respectively. In the left panel, the approximate rejection-sampling posterior, based on $5,000$ samples accepted out of $\\sim 450,000$ simulations, is shown in green. In the right panel, the approximate {\\textsc{bolfi}} posterior, based on $6,000$ simulations only, is shown in red. For all distributions, the $1\\sigma$, $2\\sigma$ and $3\\sigma$ contours are shown. \\label{fig:Supernovae}}\n\\end{center}\n\\end{figure*}\n\nIn this section, we present the first application of {\\textsc{bolfi}} to a cosmological inference problem. Specifically, we perform an analysis of the Joint Lightcurve Analysis (JLA) data set, consisting of the B-band peak apparent magnitudes $m_\\mathrm{B}$ of $740$ type Ia supernovae (SN Ia) with redshift $z$ between $0.01$ and $1.3$ \\citep{Betoule2014}: $\\textbf{d}_\\mathrm{O} \\equiv \\left( m_{\\mathrm{B},\\mathrm{O}}^k \\right)$ for $k \\in \\llbracket 1,740 \\rrbracket$. The details of the data model and inference assumptions are given in appendix \\ref{apx:Supernova cosmology}. For the purpose of validating {\\textsc{bolfi}}, we assume a Gaussian synthetic likelihood (see section \\ref{sapx:Discrepancy}), allowing us to demonstrate the fidelity of the {\\textsc{bolfi}} posterior against the exact likelihood-based solution obtained via Markov Chain Monte Carlo (MCMC). This analysis can also be compared to the proof of concept for another likelihood-free method, {\\textsc{delfi}} \\citep[Density Estimation for Likelihood-Free Inference,][]{Papamakarios2016,Alsing2018}, as the assumptions are very similar.\n\nAs described in appendix \\ref{apx:Supernova cosmology}, the full problem is six dimensional; however, in this work, we focus on the inference of the two physically relevant quantities, namely $\\Omega_\\mathrm{m}$ (the matter density of the Universe) and $w$ (the equation of state of dark energy, assumed constant), and marginalise over the other four (nuisance) parameters ($\\alpha$, $\\beta$, $M_\\mathrm{B}$, $\\delta\\hspace{-0.1em}M$). We assume a Gaussian prior,\n\\begin{equation}\n\\begin{pmatrix}\n\\Omega_\\mathrm{m} \\\\\nw\n\\end{pmatrix} \\sim\n\\mathpzc{G}\\left[\n\\begin{pmatrix}\n0.3 \\\\\n-0.75\n\\end{pmatrix},\n\\begin{pmatrix}\n0.4^2 & -0.24 \\\\\n-0.24 & 0.75^2\n\\end{pmatrix}\n\\right],\n\\label{eq:SNe_prior_Omegam_w}\n\\end{equation}\nwhich is roughly aligned with the direction of the well-known $\\Omega_\\mathrm{m}-w$ degeneracy. We generated $10^6$ samples (out of $\\sim 6\\times 10^6$ data model evaluations) of the posterior for the exact six-dimensional Bayesian problem via MCMC \\citep[performed using the \\textsc{emcee} code,][]{Foreman-Mackey2013}, ensuring sufficient convergence to characterise the $3\\sigma$ contours of the distribution.\\footnote{The final Gelman-Rubin statistic \\citep{Gelman1992} was $R -1 \\leq 5 \\times 10^{-4}$ for each of the six parameters.} The prior and the exact posterior are shown in blue and orange, respectively, in figure \\ref{fig:Supernovae}.\n\nFor likelihood-free inference, the simulator takes as input $\\Omega_\\mathrm{m}$ and $w$ and simulates $N$ realisations of the magnitudes $m_\\mathrm{B}$ of the 740 supernovae at their redshifts. Consistently with the Gaussian likelihood used in the MCMC analysis, we assume a Gaussian synthetic likelihood with a fixed covariance matrix $\\textbf{C}$. The observed data $\\textbf{d}_\\mathrm{O}$ and the covariance matrix $\\textbf{C}$ are shown in figure \\ref{fig:JLA_Hubble_correlation}. \n\nThe approximate posterior obtained from likelihood-free rejection sampling is shown in green in figure \\ref{fig:Supernovae}. It was obtained from $5,000$ accepted samples using a (conservative) threshold of $\\varepsilon = 650$ on $\\Delta_{(\\Omega_\\mathrm{m},w)}$, chosen so that the acceptance ratio was not below $0.01$. The entire run required $\\sim 450,000$ simulations in total. The approximate posterior obtained via {\\textsc{bolfi}} is shown in red in figure \\ref{fig:Supernovae}. {\\textsc{bolfi}} was initialised with a Sobol sequence of $20$ samples, and $100$ acquisitions were performed according to the ExpIntVar criterion, without acquisition noise. The {\\textsc{bolfi}} posterior is a much finer approximation to the true posterior than the one obtained from likelihood-free rejection sampling. It is remarkable that only $100$ acquisitions are enough to learn the non-trivial banana shape of the posterior. Only the $3\\sigma$ contour \\citep[which is usually not shown in cosmology papers, e.g.][]{Betoule2014} notably deviates from the MCMC posterior. This is due to the fact that we used one realisation of the stochastic process defining $\\Delta_{(\\Omega_\\mathrm{m},w)}$ and only $N=50$ realisations per $(\\Omega_\\mathrm{m},w)$; the marginalisation over the four nuisance parameters is therefore partial, yielding slightly smaller credible contours. However, a better approximation could be obtained straightfowardly, if desired, by investing more computational resources (increasing $N$), without requiring more acquisitions. \n\nAs we used $N=50$, the total cost for {\\textsc{bolfi}} is $6,000$ simulations. This is a reduction by $\\sim 2$ orders of magnitude with respect to likelihood-free rejection sampling ($\\sim 450,000$ simulations) and $3$ orders of magnitude with respect to MCMC sampling of the exact posterior ($6 \\times 10^6$ simulations). It is also interesting to note that our {\\textsc{bolfi}} analysis required a factor of $\\sim 3$ fewer simulations than the recently introduced \\textsc{delfi}\\ procedure \\citep{Alsing2018}, which used $20,000$ simulations drawn from the prior for the analysis of the JLA.\\footnote{A notable difference is that {\\textsc{delfi}} allowed the authors to perform the joint inference of the six parameters of the problem, whereas we only get the distribution of $\\Omega_\\mathrm{m}$ and $w$. However, since these are the only two physically interesting parameters, inference of the nuisance parameters is not deemed crucial for this example.}\n\n\\section{Discussion}\n\\label{sec:Discussion}\n\n\\subsection{Benefits and limitations of the proposed approach for cosmological inferences}\n\\label{ssec:Benefits and limitations of the proposed approach for cosmological inferences}\n\nAs noted in the introduction, likelihood-free rejection sampling, when at all viable, is extremely costly in terms of the number of required simulations. In contrast, the {\\textsc{bolfi}} approach relies on a GP probabilistic model for the discrepancy, and therefore allows the incorporation of a smoothness assumption about the approximate likelihood $L(\\boldsymbol{\\uptheta})$. The smoothness assumption allows simulations in the training set to ``share'' information about their value of $\\Delta_{\\boldsymbol{\\uptheta}}$ in the neighbourhood of $\\boldsymbol{\\uptheta}$, which suggests that fewer simulations are needed to reach a certain level of accuracy. Indeed, the number of simulations required is typically reduced by $2$ to $3$ orders of magnitude, for a better final approximation of the posterior, as demonstrated by our tests in section \\ref{sec:Applications} and in the statistical literature \\citep[see][]{GutmannCorander2016}. \n\nA second benefit of {\\textsc{bolfi}} is that it actively acquires training data through Bayesian optimisation. The trade-off between computational cost and statistical performance is still present, but in a modified form: the trade-off parameter is the size of the training set used in the regression. Within the training set, the user is free to choose which areas of the parameter space should be prioritised, so as to approximate the regression function more accurately there. In contrast, in ABC strategies that rely on drawing from a fixed proposal distribution (often the prior), or variants such as \\textsc{pmc}-\\textsc{abc}, a fixed computational cost needs to be paid per value of $\\boldsymbol{\\uptheta}$ regardless of the value of $\\Delta_{\\boldsymbol{\\uptheta}}$. \n\nFinally, by focusing on parametric approximations to the exact likelihood, the approach proposed in this work is totally ``$\\varepsilon$-free'', meaning that no threshold (which is often regarded as an unappealing \\textit{ad hock} element) is required. As likelihood-based techniques, the parametric version of {\\textsc{bolfi}} has the drawback that assuming a wrong form for the synthetic likelihood or miscalculating values of its parameters (such as the covariance matrix) can potentially bias the approximate posterior and\/or lead to an underestimation of credible regions. Nevertheless, massive data compression procedures can make the assumptions going into the choice of a Gaussian synthetic likelihood (almost) true by construction (see section \\ref{sssec:Data compression}).\n\nOf course, regressing the discrepancy and optimising the acquisition function are not free of computational cost. However, the run-time for realistic cosmological simulation models can be hours or days. In comparison, the computational overhead introduced by {\\textsc{bolfi}} is negligible.\n\nLikelihood-free inference should also be compared to existing likelihood-based techniques for cosmology such as Gibbs sampling or Hamiltonian Monte Carlo (e.g. \\citealp{Wandelt2004,Eriksen2004} for the cosmic microwave background; \\citealp{Jasche2010b,Jasche2015,Jasche2015BORGSDSS} for galaxy clustering; \\citealp{Alsing2016} for weak lensing). The principal difference between these techniques and {\\textsc{bolfi}} lies in its likelihood-free nature. Likelihood-free inference has particular appeal for cosmological data analysis, since encoding complex physical phenomena and realistic observational effects into forward simulations is much easier than designing an approximate likelihood which incorporates these effects and solving the inverse problem. While the numerical complexity of likelihood-based techniques typically requires to approximate complex data models in order to access required products (conditionals or gradients of the pdfs) and to allow for sufficiently fast execution speeds, {\\textsc{bolfi}} performs inference from full-scale black-box data models. In the future, such an approach is expected to allow previously infeasible analyses, relying on a much more precise modelling of cosmological data, including in particular the complicated systematics they experience. However, while the physics and instruments will be more accurately modelled, the statistical approximation introduced with respect to likelihood-based techniques should be kept in mind.\n\nOther key aspects of {\\textsc{bolfi}} for cosmological data analysis are the arbitrary choice of the statistical summaries and the easy joint treatment of different data sets. Indeed, as the data compression from $\\textbf{d}$ to $\\boldsymbol{\\Phi}$ is included in the simulator (see section \\ref{ssec:Approximate Bayesian computation}), summary statistics do not need to be quantities that can be physically modelled (such as the power spectrum) and can be chosen robustly to model misspecification. For example, for the microwave sky, the summaries could be the cross-spectra between different frequency maps; and for imaging surveys, the cross-correlation between different bands. Furthermore, joint analyses of correlated data sets, which is usually challenging in likelihood-based approaches (as they require a good model for the joint likelihood) can be performed straightforwardly in a likelihood-free approach. \n\nImportantly, as a general inference technique, {\\textsc{bolfi}} can be embedded into larger probabilistic schemes such as Gibbs or Hamiltonian-within-Gibbs samplers. Indeed, as posterior predictive distributions for conditionals and gradients of GPs are analytically tractable, it is easy to obtain samples of the {\\textsc{bolfi}} approximate posterior for use in larger models. {\\textsc{bolfi}} can therefore allow parts of a larger Bayesian hierarchical model to be treated as black boxes, without compromising the tractability of the entire model. \n\n\\subsection{Possible extensions}\n\\label{ssec:Possible extensions}\n\n\\subsubsection{High-dimensional inference}\n\\label{sssec:High-dimensional inference}\n\nIn this proof-of-concept paper, we focused on two-dimensional problems. Likelihood-free inference is in general very difficult when the dimensionality of the parameter space is large, due to the curse of dimensionality, which makes the volume exponentially larger with $\\mathrm{dim}~\\boldsymbol{\\uptheta}$. In {\\textsc{bolfi}}, this difficulty manifests itself in the form of a hard regression problem which needs to be solved. The areas in the parameter space where the discrepancy is small tend to be narrow in high dimension, therefore discovering these areas becomes more challenging as the dimension increases. The optimisation of GP kernel parameters, which control the shapes of allowed features, also becomes more difficult. Furthermore, finding the global optimum of the acquisition function becomes more demanding (especially with the ones designed for ABC such as ExpIntVar, which have a high degree of structure -- see figure \\ref{fig:Supernovae_acquisition}, bottom right panel).\n\nNevertheless, \\citet{Jaervenpaeae2017} showed on a toy simulation model (a Gaussian) that up to ten-dimensional inference is possible with {\\textsc{bolfi}}. As usual cosmological models do not include more than ten free physical parameters, we do not expect this limitation to be a hindrance. Any additional nuisance parameter or latent variable used internally by the simulator (such as $\\alpha$, $\\beta$, $M_\\mathrm{B}$, $\\delta\\hspace{-0.1em}M$ in supernova cosmology, see section \\ref{ssec:Supernova cosmology}) can be automatically marginalised over, by using $N$ realisations per $\\boldsymbol{\\uptheta}$. Recent advances in high-dimensional implementation of the synthetic likelihood \\citep{Ong2017} and high-dimensional Bayesian optimisation \\citep[e.g.][]{Wang2013:BOH:2540128.2540383,Kandasamy2015} could also be exploited. In future work, we will address the problem of high-dimensional likelihood-free inference in a cosmological context.\n\n\\subsubsection{Scalability with the number of acquisitions and probabilistic model for the discrepancy}\n\\label{sssec:Scalability with the number of acquisitions and probabilistic model for the discrepancy}\n\nIn addition to the fundamental issues with high-dimensional likelihood-free inference described in the previous section, practical difficulties can be met.\n\nGaussian process regression requires the inversion of a matrix $\\uuline{\\textbf{K}}\\vspace{-4pt}$ of size $t \\times t$, where $t$ is the size of the training set. The complexity is $\\mathcal{O}(t^3)$, which limits the size of the training set to a few thousand. Improving GPs with respect to this inversion is still subject to research \\citep[see][chapter 8]{RasmussenWilliams2006}. For example, ``sparse'' Gaussian process regression reduces the complexity by introducing auxiliary ``inducing variables''. Techniques inspired by the solution to the Wiener filtering problem in cosmology, such as preconditioned conjugate gradient or messenger field algorithms could also be used \\citep{Elsner2013,KodiRamanah2017,Papez2018}. Another strategy would be to divide the regression problem spatially into several patches with a lower number of training points \\citep{Park2017}. Such approaches are possible extensions of the presented method.\n\nIn the GP probabilistic model employed to model the discrepancy, the variance depends only on the training locations, not on the obtained values (see equation \\eqref{eq:GP_variance}). Furthermore, a stationary kernel is assumed. However, depending on the simulator, the discrepancy can show heteroscedasticity (i.e. its variance can depend on $\\boldsymbol{\\uptheta}$ -- see e.g. figure \\ref{fig:Gaussian_mean_illustration}, bottom panel). Such cases could be handled by non-stationary GP kernels or different probabilistic models for the discrepancy, allowing a heteroscedastic regression.\n\n\\subsubsection{Acquisition rules}\n\\label{sssec:Acquisition rules}\n\nAs shown in our examples, attention should be given to the selection of an efficient acquisition rule. Although standard Bayesian optimisation strategies such as the EI are reasonably effective, they are usually too greedy, focusing nearly all the sampling effort near the estimated minimum of the discrepancy and gathering too little information about other regions in the domain (see figure \\ref{fig:Supernovae_acquisition}, bottom left panel). This implies that, unless the acquisition noise is high, the tails of the posterior will not be as well approximated as the modal areas. In contrast, the ExpIntVar acquisition rule, derived in this work for the parametric approach, addresses the inefficient use of resources in likelihood-free rejection sampling by directly targeting the regions of the parameter space where improvement in the estimation accuracy of the approximate posterior is needed most. In our experiments, ExpIntVar seems to correct -- at least partially -- for the well-known effect in Bayesian optimisation of overexploration of the domain boundaries, which becomes more problematic in high dimension.\n\nAcquisition strategies examined so far in the literature \\citep[see][for a comparative study]{Jaervenpaeae2017} have focused on single acquisitions and are all ``myopic'', in the sense that they reason only about the expected utility of the next acquisition, and the number of simulations left in a limited budget is not taken into account. Improvement of acquisition rules enabling batch acquisitions and non-myopic reasoning are left to future extensions of {\\textsc{bolfi}}.\n\n\\subsubsection{Data compression}\n\\label{sssec:Data compression}\n\nIn addition to the problem of the curse of dimensionality in parameter space, discussed in section \\ref{sssec:High-dimensional inference}, likelihood-free inference usually suffers from difficulties in the measuring the (mis)match between simulations and observations if the data space also has high dimension. As discussed in section \\ref{ssec:Approximate Bayesian computation}, simulator-based models include a data compression step. The comparison in data space can be made more easily if $\\mathrm{dim}~\\boldsymbol{\\Phi}$ is reduced. In future work, we will therefore aim at combining {\\textsc{bolfi}} with massive and (close to) optimal data compression strategies. These include \\textsc{moped} \\citep{Heavens2000}, the score function \\citep{AlsingWandelt2018}, or information-maximising neural networks \\citep{Charnock2018}. Using such efficient data compression techniques, the number of simulations required for inference with {\\textsc{bolfi}} will be reduced even more, and the number of parameters treated could be increased.\n\nParametric approximations to the exact likelihood depend on quantities that have to be estimated using the simulator (typically for the Gaussian synthetic likelihood, the inverse covariance matrix of the summaries). Unlike supernova cosmology where the covariance matrix is easily obtained, in many cases it is prohibitively expensive to run enough simulations to estimate the required quantities, especially when they vary with the model parameters. In this context, massive data compression offers a way forward, reducing enormously the number of required simulations and making the analysis feasible when otherwise it might be essentially impossible \\citep{Heavens2017,Gualdi2018}.\n\nAn additional advantage of several data compression strategies is that they support the choice of a Gaussian synthetic likelihood. Indeed, the central limit theorem (for \\textsc{moped}) or the form of the network's reward function (for information-maximising neural networks) assist in giving the compressed data a near-Gaussian distribution. Furthermore, testing the Gaussian assumption for the synthetic likelihood will be far easier in a smaller number of dimensions than in the original high-dimensional data space.\n\n\\subsection{Parallelisation and computational efficiency}\n\\label{ssec:Parallelisation and computational efficiency}\n\nWhile MCMC sampling has to be done sequentially, {\\textsc{bolfi}} lends itself to more parallelisation. In an efficient strategy, a master process performs the regression and decides on acquisition locations, then dispatches simulations to be run by different workers. In this way, many simulations can be run simultaneously in parallel, or even on different machines. This allows fast application of the method and makes it particularly suitable for grid computing. Extensions of the probabilistic model and of the acquisition rules, discussed in section \\ref{sssec:Scalability with the number of acquisitions and probabilistic model for the discrepancy} and \\ref{sssec:Acquisition rules}, would open the possibility of doing asynchronous acquisitions. Different workers would then work completely independently and decide on their acquisitions locally, while just sharing a pool of simulations to update their beliefs given all the evidence available.\n\nWhile the construction of the training set depends on the observed data $\\boldsymbol{\\Phi}_\\mathrm{O}$ (through the acquisition function), simulations can nevertheless be reused as long as summaries $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ are saved. This means that if one acquires new data $\\boldsymbol{\\Phi}_\\mathrm{O}'$, the existing $\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}$ (or a subset of them) can be used to compute the new discrepancy $\\Delta_{\\boldsymbol{\\uptheta}}(\\boldsymbol{\\Phi}_{\\boldsymbol{\\uptheta}}, \\boldsymbol{\\Phi}_\\mathrm{O}')$. Building an initial training set in this fashion can massively speed up the inference of $\\mathpzc{P}(\\boldsymbol{\\uptheta}|\\boldsymbol{\\Phi})_{\\boldsymbol{\\Phi}=\\boldsymbol{\\Phi}_\\mathrm{O}'}$, whereas likelihood-based techniques would require a new MCMC.\n\n\\subsection{Comparison to previous work}\n\\label{ssec:Comparison to previous work}\n\nAs discussed in the introduction, likelihood-free rejection sampling is not a viable strategy for various problems that {\\textsc{bolfi}} can tackle. In recent work, an other algorithm for scalable likelihood-free inference in cosmology \\citep[{\\textsc{delfi}},][]{Papamakarios2016,Alsing2018} was introduced. The approach relies on estimating the joint probability $\\mathpzc{P}(\\boldsymbol{\\uptheta},\\boldsymbol{\\Phi})$ via density estimation. This idea also relates to the work of \\citet{Hahn2018}, who fit the sampling distribution of summaries $\\mathpzc{P}(\\boldsymbol{\\Phi}|\\boldsymbol{\\uptheta})$ using Gaussian mixture density estimation or independent component analysis, before using it for parameter estimation. This section discusses the principal similarities and differences.\n\nThe main difference between {\\textsc{bolfi}} and {\\textsc{delfi}} is the data acquisition. Training data are actively acquired in {\\textsc{bolfi}}, contrary to {\\textsc{delfi}} which, in the simplest scheme, draws from the prior. The reduction in the number of simulations for the inference of cosmological parameters (see section \\ref{ssec:Supernova cosmology}) can be interpreted as the effect of the Bayesian optimisation procedure in combination with the ExpIntVar acquisition function. Using a purposefully constructed surrogate surface instead of a fixed proposal distribution, {\\textsc{bolfi}} focuses the simulation effort to reveal as much information as possible about the target posterior. In particular, its ability to reason about the quality of simulations before they are run is an essential element. Acquisition via Bayesian optimisation almost certainly remains more efficient than even the \\textsc{pmc} version of {\\textsc{delfi}}, which learns a better proposal distribution but still chooses parameters randomly. In future cosmological applications with simulators that are expensive and\/or have a large latent space, an active data acquisition procedure could be crucial in order to provide a good model for the noisy approximate likelihood in the interesting regions of parameter space, and to reduce the computational cost. This comes at the expense of a reduction of the parallelisation potential: with a fixed proposal distribution (like in {\\textsc{delfi}} and unlike in {\\textsc{bolfi}}), the entire set of simulations can be run at the same time.\n\nThe second comment is related to the dimensionality of problems which can be addressed. Like {\\textsc{delfi}}, {\\textsc{bolfi}} relies on a probabilistic model to make ABC more efficient. However, the quantities employed differ, since in {\\textsc{delfi}} the relation between the parameters $\\boldsymbol{\\uptheta}$ and the summary statistics $\\boldsymbol{\\Phi}$ is modelled (via density estimation), while {\\textsc{bolfi}} focuses on the relation between the parameters $\\boldsymbol{\\uptheta}$ and the discrepancy $\\Delta_{\\boldsymbol{\\uptheta}}$ (via regression). Summary statistics are multi-dimensional while the discrepancy is a univariate scalar quantity. Thus, {\\textsc{delfi}} requires to solve a density estimation problem in $\\mathrm{dim}~\\boldsymbol{\\uptheta} + \\mathrm{dim}~\\boldsymbol{\\Phi}$ (which equals $2 \\times \\mathrm{dim}~\\boldsymbol{\\uptheta}$ if the compression from \\citealp{AlsingWandelt2018} is used), while {\\textsc{bolfi}} requires to solve a regression problem in $\\mathrm{dim}~\\boldsymbol{\\uptheta}$. Both tasks are expected to become more difficult as $\\mathrm{dim}~\\boldsymbol{\\uptheta}$ increases (a symptom of the curse of dimensionality, see section \\ref{sssec:High-dimensional inference}), but the upper limits on $\\mathrm{dim}~\\boldsymbol{\\uptheta}$ for practical applications may differ. Further investigations are required to compare the respective maximal dimensions of problems that can be addressed by {\\textsc{bolfi}} and {\\textsc{delfi}}.\n\nFinally, as argued by \\citet{Alsing2018}, {\\textsc{delfi}} readily provides an estimate of the approximate evidence. In contrast, as in likelihood-based techniques, integration over parameter space is required with {\\textsc{bolfi}} to get\n\\begin{equation}\nZ_{\\boldsymbol{\\Phi}} = \\left(\\int \\mathpzc{P}(\\boldsymbol{\\Phi}|\\boldsymbol{\\uptheta}) \\, \\mathrm{d}\\boldsymbol{\\uptheta} \\right)_{\\boldsymbol{\\Phi}=\\boldsymbol{\\Phi}_\\mathrm{O}}.\n\\end{equation}\nHowever, due to the GP model, the integral can be more easily computed, using the same strategies as for the integral appearing in ExpIntVar (see section \\ref{sssec:Expected integrated variance}): only the GP predicted values are required at discrete locations on a grid (in low dimension) or at the positions of importance samples. A potential caveat is that {\\textsc{delfi}} has only been demonstrated to work in combination with the score function \\citep{AlsingWandelt2018}, which is necessary to reduce the dimensionality of $\\boldsymbol{\\Phi}$ before estimating the density.\\footnote{In contrast, section \\ref{ssec:Supernova cosmology} showed, for the same supernovae problem, that {\\textsc{bolfi}} can still operate if the comparison is done in the full $740$-dimensional data space.} The score function produces summaries that are only sufficient up to linear order in the log-likelihood. However, in ABC, care is required to perform model selection if the summary statistics are insufficient. Indeed, \\citet[][equation 1]{Robert2011} show that, in such a case, the approximate Bayes factor can be arbitrarily biased and that the approximation error is unrelated to the computational effort invested in running the ABC algorithm. Moreover, sufficiency for models $\\mathcal{M}_1$ and $\\mathcal{M}_2$ alone, or even for both of them -- even if approximately realised via Alsing \\& Wandelt's procedure -- does not guarantee sufficiency to compare the two different models $\\mathcal{M}_1$ and $\\mathcal{M}_2$ \\citep{Didelot2011}. As the assumptions behind {\\textsc{bolfi}} do not necessarily necessitate to reduce $\\mathrm{dim}~\\boldsymbol{\\Phi}$ ($\\Delta_{\\boldsymbol{\\uptheta}}$ is always a univariate scalar quantity, see above), these difficulties could be alleviated with {\\textsc{bolfi}} by carefully designing sufficient summary statistics for model comparison within the black-box simulator, if they exist.\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nLikelihood-free inference methods allow Bayesian inference of the parameters of simulator-based statistical models with no reference to the likelihood function. This is of particular interest for data analysis in cosmology, where complex physical and observational processes can usually be simulated forward but not handled in the inverse problem. \n\nIn this paper, we considered the demanding problem of performing Bayesian inference when simulating data from the model is extremely costly. We have seen that likelihood-free rejection sampling suffers from a vanishingly small acceptance rate when the threshold $\\varepsilon$ goes to zero, leading to the need for a prohibitively large number of simulations. This high cost is largely due to the lack of knowledge about the functional relation between the model parameters and the discrepancy. As a response, we have described a new approach to likelihood-free inference, {\\textsc{bolfi}}, that uses regression to infer this relation, and optimisation to actively build the training data set. A crucial ingredient is the acquisition function derived in this work, with which training data are acquired such that the expected uncertainty in the final estimate of the posterior is minimised.\n\nIn case studies, we have shown that {\\textsc{bolfi}} is able to precisely recover the true posterior, even far in its tails, with as few as $6,000$ simulations, in contrast to likelihood-free rejection sampling or likelihood-based MCMC techniques which require orders of magnitude more simulations. The reduction in the number of required simulations accelerated the inference massively.\n\nThis study opens up a wide range of possible extensions, discussed in section \\ref{ssec:Possible extensions}. It also allows for novel analyses of cosmological data from fully non-linear simulator-based models, as required e.g. for the cosmic web \\citep[see the discussions in][]{Leclercq2015ST,Leclercq2016CIT,Leclercq2017DMSHEET}. Other applications may include the cosmic microwave background, weak gravitational lensing or intensity mapping experiments. We therefore anticipate that {\\textsc{bolfi}} will be a major ingredient in principled, simulator-based inference for the coming era of massive cosmological data.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStrong infrared behavior is a characteristic feature of the inflationary physics \\cite{w1} (for reviews see e.g. \\cite{r1,r2}). The modes are continuously pushed from subhorizon to superhorizon regimes, which enlarges short distance quantum effects on cosmologically interesting scales. Most of the time, the loop corrections contain infrared infinities that must properly be handled by viable physical reasoning. The so called infrared logarithms\\footnote{In the presence of the entropy perturbations the infrared loop divergences are power law rather than logarithmic, see \\cite{pl}.} show up in loops as a reminiscent of the peculiar infrared behavior  (see e.g. \\cite{il1,il2,il3,il4,il5,il6,il7,il8,il9,il10,il11}).  \n\nIn this paper, we apply  cosmological perturbation theory to the standard scalar slow-roll inflationary model in the minisuperspace approximation. The minisuperspace theory is expected to capture the dynamics of the zero modes of the full theory. Besides, it is free from loop infinities and renormalization issues, which are intricate in the presence of gravity, and it still contains special features related to the gauge invariance and nonlinearities. Therefore the results obtained in the minisuperspace approximation must shed light on some crucial questions in the full theory and our aim here is to examine the appearance of the infrared logarithms.  \n\nWe first consider the single scalar field model and obtain the {\\it complete} gauge fixed action for the curvature perturbation $\\zeta$. The corresponding Hamiltonian can be expanded in powers of the momentum conjugate to $\\zeta$, which becomes an expansion in the inverse powers of the background scale factor of the universe $a_B(t)$. Asymptotically at late times, the quadratic momentum term, which is still nonlinear in $\\zeta$, dominates the dynamics. In this case, $\\zeta$ is conserved and no $\\ln a_B$ behavior appears, which is consistent with \\cite{zc1,zc2}. We then add a self-interacting spectator scalar to the system. This time there arises a specific asymptotically dominant interaction term, which yields an $(\\ln a_B)^n$ correction to $\\zeta$ in the $n$'th order perturbation theory. The emergence of this infrared logarithm is similar to what has been observed in the field theory calculations. For models having large number of e-folds such a correction may invalidate the perturbation theory. On the other hand, in the minisuperspace theory a nonperturbative argument shows that asymptotically $\\zeta$ has actually a slowly evolving $\\ln a_B$ correction to the constant mode, which indicates other infrared logarithms involving higher powers of $\\ln a_B$ might be the artifacts of the perturbation theory.   \n\n\\section{Single Scalar Slow-roll Inflation}\n\nWe start from the following minisuperspace action:\n\\begin{equation}\\label{1}\nS=L^3\\int dt\\, a^3 \\, \\left\\{\\frac{1}{N} \\left[-6\\frac{\\dot{a}^2}{a^2}+\\fr12 \\dot{\\Phi}^2\\right]-NV(\\Phi)\\right\\},\n\\end{equation}\nwhere the dot denotes the time derivative, $N$ is the lapse function, $a(t)$ is the scale factor of the universe, $\\Phi$ is the inflaton and $V(\\Phi)$ is the inflaton potential (we set the reduced Planck mass $M_p=1$. The proper $M_p$ factors can easily be reinstated by dimensional analysis as we will do below). The minisuperspace action \\eq{1} can be obtained from the usual Einstein-Hilbert action by setting $N^i=0$, $h_{ij}=a^2\\delta_{ij}$, where $N$, $N^i$ and $h_{ij}$ refer to the standard ADM decomposition of the metric, and by assuming that the variables $N$, $a$ and $\\Phi$ depend only on time. The parameter $L$ denotes the size of the comoving spatial coordinates and the factor $L^3$ in \\eq{1} arises from their integration reducing the field theory to a quantum mechanical system. \n\nThe action \\eq{1} is invariant under a local time transformation with the parameter $k^0$:\n\\begin{equation}\\label{2}\n\\delta N=k^0\\dot{N}+N\\dot{k}^0,\\hs{10}\\delta\\Phi=k^0\\dot{\\Phi},\\hs{10}\\delta a=k^0\\dot{a}.\n\\end{equation}\nTo fix the gauge invariance one may define the background field variables $a_B$ and $\\Phi_B$ obeying\n\\begin{equation}\\label{3}\n6H_B^2=\\fr12\\dot{\\Phi}_B^2+V(\\Phi_B),\\hs{10}\\dot{H}_B^2=-\\fr14\\dot{\\Phi}_B^2,\n\\end{equation}\nwhere $H_B=\\dot{a}_B\/a_B$. Next, one may introduce the fluctuation fields $\\zeta$ and $\\phi$ as\n\\begin{equation}\\label{4}\na=a_Be^{\\zeta},\\hs{10}\\Phi=\\Phi_B+\\phi,\n\\end{equation}\nand impose the gauge $\\phi=0$. After algebraically solving the lapse $N$  from its own equation of motion one may obtain \n\\begin{equation}\\label{5}\nS=-2L^3\\int dt\\,a_B^3V_B\\,e^{3\\zeta}\\,\\left[1+\\frac{12H_B}{V_B}\\dot{\\zeta}+\\frac{6}{V_B}\\dot{\\zeta}^2\\right]^{1\/2},\n\\end{equation}\nwhere $V_B=V(\\Phi_B)$. We take the background \\eq{3} to be a slow-roll inflationary solution with $\\dot{\\Phi}_B<0$. \n\nAssuming that the physical size of the pre-inflationary patch is determined by the initial Hubble parameter $H_i$, one has\n\\begin{equation}\\label{6}\na_B(t_i)L=\\frac{1}{H_i},\n\\end{equation}\nwhere $t_i$ is the initial time of the inflation. Normalizing the scale factor as \n\\begin{equation}\\label{7}\na_B(t_i)=\\frac{1}{H_i}\n\\end{equation}\ncorresponds to setting $L=1$, which eliminates the unphysical comoving scale from the equations. \n\nIt is instructive to repeat  the gauge fixing procedure in the Hamiltonian formulation where the minisuperspace action \\eq{1} can be written as \n\\begin{eqnarray}\n&&S=\\int dt\\left[\\hat{P}_\\zeta \\dot{\\hat{\\zeta}}+P_\\Phi\\dot{\\Phi}-NH\\right],\\nonumber\\\\\n&&H=-\\frac{1}{24}e^{-3\\hat{\\zeta}}\\hat{P}_\\zeta^2+\\fr12e^{-3\\hat{\\zeta}}P_\\Phi^2+e^{3\\hat{\\zeta}}V(\\Phi)\\label{8},\n\\end{eqnarray}\nand $\\hat{\\zeta}=\\ln a$. Expanding the variables around their background values\n\\begin{eqnarray}\n&&\\hat{\\zeta}=\\ln a_B+\\zeta,\\hs{10}\\Phi=\\Phi_B+\\phi,\\hs{10}N=1+n,\\nonumber\\\\\n&&\\hat{P}_\\zeta=-12a_B^3H_B+P_\\zeta,\\hs{10}P_\\Phi=a_B^3\\dot{\\Phi}_B+P_\\phi,\\label{9}\n\\end{eqnarray} \nthe action becomes\n\\begin{equation}\nS=\\int dt \\left[P_\\zeta\\dot{\\zeta}+P_\\phi\\dot{\\phi}-H_F-nC\\right],\\label{10}\n\\end{equation}\nwhere $H_F$ is the fluctuation Hamiltonian involving all variables but $n$ and $C$ is the constraint given by\n\\begin{equation}\\label{11}\nC=-\\frac{1}{24a_B^3}e^{-3\\zeta}\\left(-12a_B^3H_B+P_\\zeta\\right)^2+\\frac{1}{2a_B^3}e^{-3\\zeta}\\left(a_B^3\\dot{\\Phi}_B+P_\\phi\\right)^2+a_B^3e^{3\\zeta}V(\\Phi_B+\\phi).\n\\end{equation}\nAfter imposing the gauge $\\phi=0$, one may solve\\footnote{In solving $P_\\phi$ one should keep in mind that $a_B^3\\dot{\\Phi}_B+P_\\phi<0$ since we are making an expansion around a background solution with $\\dot{\\Phi}_B<0$.} the constraint $C=0$ for $P_\\phi$, which would give the reduced action for $\\zeta$ and $P_\\zeta$ \n\\begin{equation}\\label{12}\nS=\\int\\, dt\\, P_\\zeta\\dot{\\zeta}-\\dot{\\Phi}_B\\left[\\frac{1}{12}\\left(P_\\zeta-12a_B^3H_B\\right)^2-2a_B^6V_Be^{6\\zeta}\\right]^{1\/2}+H_BP_\\zeta+6a_B^3V_B\\zeta.\n\\end{equation}\nIn the phase space path integral quantization, this procedure corresponds to the Faddeev-Popov gauge fixing\n\\begin{equation}\\label{13}\n\\delta(\\phi)\\delta(C) \\det\\left\\{\\phi,C\\right\\}=\\delta(\\phi)\\delta(C)\\frac{\\partial C}{\\partial\\phi}=\\delta(\\phi)\\delta(P_\\phi-P_\\phi^*),\n\\end{equation}\nwhere $P_\\phi^*$ is the solution of $C=0$. One can check that the two actions \\eq{5} and \\eq{12} are related by the Legendre transformation exchanging the Lagrangian and the Hamiltonian. \n\nOne may see that a constant $\\zeta$ solves the equations of motion that follows from \\eq{5} provided that the background equations \\eq{3} are satisfied. At first, this is not obvious from \\eq{5} since it contains a pure $\\zeta$ term with no time derivatives when the square root in \\eq{5} is expanded. In any case, it is possible to add \\eq{5} a total derivative term so that\n\\begin{eqnarray}\nS&=&-2\\int dt\\,a_B^3V_B\\,e^{3\\zeta}\\,\\left[1+\\frac{12H_B}{V_B}\\dot{\\zeta}+\\frac{6}{V_B}\\dot{\\zeta}^2\\right]^{1\/2}+2\\int dt \\,a_B^3\\,e^{3\\zeta}\\left[V_B+6H_B\\dot{\\zeta}\\right],\\label{14}\\\\\n&=&\\int dt\\, a_B^3\\,e^{3\\zeta}\\left[\\frac{3\\dot{\\Phi}_B^2}{V_B}\\dot{\\zeta}^2-\\frac{18H_B\\dot{\\Phi}_B^2}{V_B^2}\\dot{\\zeta}^3+. . . \\right].\\label{15}\n\\end{eqnarray}\nIt is now clear from \\eq{15} that the equation of motion involves only the time derivatives of $\\zeta$ and a constant mode  is a trivial solution. Note that by normalization \\eq{7}, the scale factor $a_B$ has mass dimension $-1$. In the Hamiltonian language the extra surface term added in \\eq{14} corresponds to a canonical transformation $P_\\zeta\\to P_\\zeta+12a_B^3H_B-12 a_B^3H_B e^{3\\zeta}$ as compared to \\eq{12}. The Hamiltonian of \\eq{14} can be found as \n\\begin{eqnarray}\nH&=&-a_B^3\\dot{\\Phi}_B^2e^{3\\zeta}\\left[1-\\frac{2H_B}{a_B^3\\dot{\\Phi}_B^2}e^{-3\\zeta}P_\\zeta+\\frac{1}{12a_B^6\\dot{\\Phi}_B^2}e^{-6\\zeta}P_\\zeta^2\\right]^{1\/2}-H_B P_\\zeta+a_B^3\\dot{\\Phi}_B^2e^{3\\zeta},\\label{16}\\\\\n&=&\\frac{V_B}{12a_B^3\\dot{\\Phi}^2}e^{-3\\zeta}P_\\zeta^2+\\frac{H_BV_B}{12a_B^6\\dot{\\Phi}_B^4}e^{-6\\zeta}P_\\zeta^3+. . . \\label{17}\n\\end{eqnarray}\nwhere the dotted terms are suppressed with more powers of the background scale factor. Evidently, the first term in \\eq{17} dominates the dynamics at late times in inflation. \n\nThe interaction picture operators are governed by the free Hamiltonian\n\\begin{equation}\\label{18}\nH_0=\\frac{V_B}{12a_B^3\\dot{\\Phi}_B^2}P_\\zeta^2.\n\\end{equation}\nTheir time evolution can be found as\n\\begin{eqnarray}\n&&\\zeta_I(t)=\\zeta_i+\\fr16\\int_{t_i}^t dt'\\frac{V_B(t')}{a_B(t')^3\\dot{\\Phi}_B(t')^2}\\,P_i,\\nonumber\\\\\n&&P_{\\zeta I}=P_i,\\label{19}\n\\end{eqnarray}\nwhere $\\zeta_i$ and $P_i$ are the initial time independent (Schr\\\"{o}dinger) operators obeying $[\\zeta_i,P_i]=i$. \n\nAn operator in the Heisenberg picture $O_H$ can be related to the corresponding interaction picture operator $O_I$ by\n\\begin{equation}\nO_H=U_I^\\dagger O_I U_I,\\label{20}\n\\end{equation}\nwhere $i\\dot{U}_I=H_IU_I$, $U(t_i)=I$ and $H_I$ is the interaction Hamiltonian in the interaction picture. As shown by Weinberg \\cite{il1}, \\eq{20} can be expanded as\n\\begin{equation}\\label{21}\nO_H(t)=O_I(t)-i\\int_{t_i}^t dt'\\,[O_I(t),H_I(t')]-\\int_{t_i}^t dt''\\int_{t''}^t dt'\\,[[O_I(t),H_I(t')],H_I(t'')]+. . . \n\\end{equation}\nwhere the dotted terms contain more nested commutators of $O_I$ with $H_I$. Eq. \\eq{21} can be used as the basis for the in-in perturbation theory. From \\eq{17} the interaction Hamiltonian can be determined as\n\\begin{equation}\\label{22}\nH_I=\\frac{V_B}{24a_B^3\\dot{\\Phi}_B^2}\\left\\{\\left(e^{3\\zeta_I}-1\\right),P_{\\zeta I}^2\\right\\}+. . .\n\\end{equation}\nwhere we apply symmetric ordering to make $H_I$ Hermitian. \n\nOne may approximate the time integrals during slow-roll inflation by taking (note the normalization \\eq{7}) \n\\begin{equation}\na_B\\simeq \\frac{1}{H_B}e^{H_B(t-t_i)}\\label{23}\n\\end{equation}\nand by treating the slowly changing variables $H_B$, $V_B$ and $\\dot{\\Phi}_B$ as constants. Using \\eq{22} in \\eq{21} for $\\zeta$, one finds that at the end of inflation after $N$ e-folds \n\\begin{equation}\\label{24}\n\\zeta_H=\\zeta_i+\\frac{H_B^2}{12M_p^2\\epsilon}P_i+. . .+O\\left(e^{-3N}\\right),\n\\end{equation}\nwhere the slow-roll parameter is defined as\n\\begin{equation}\n\\epsilon=\\frac{3\\dot{\\Phi}_B^2}{V_B}\\simeq-\\frac{\\dot{H}_B}{H_B},\\label{25}\n\\end{equation}\nand dots denote time independent but nonlinear terms in $\\zeta_i$ and $P_i$ coming from the lower limits of the time integrals in \\eq{21} at $t_i$. Consequently, one sees that at late times $\\zeta_H$ exponentially asymptotes to a constant operator and no infrared logarithms appear.\n\nWe observe that neglecting all but the first term in \\eq{17}, which are exponentially suppressed at late times, gives an explicitly integrable system. Namely, the (classical) equations corresponding to the Hamiltonian\n\\begin{equation}\nH=\\frac{V_B}{12a_B^2\\dot{\\Phi}_B^2}e^{-3\\zeta}P_\\zeta^2,\\label{26}\n\\end{equation}\ncan be integrated to get\n\\begin{eqnarray}\n&&\\zeta(t)=\\zeta_i+\\fr23 \\ln\\left[1+P_i e^{-3\\zeta_i}\\int_{t_i}^t dt'\\frac{V_B}{4a_B^3\\dot{\\Phi}_B^2}\\right],\\nonumber\\\\\n&&P_\\zeta(t)=P_i+P_i^2e^{-3\\zeta_i}\\int_{t_i}^t dt'\\frac{V_B}{4a_B^3\\dot{\\Phi}_B^2}.\\label{27}\n\\end{eqnarray}\nIn the quantum theory \\eq{27} should be true for Heisenberg operators provided that operator orderings are solved in a suitable way. Eq. \\eq{27} shows that the asymptotic change of $\\zeta$ compared to its initial value is determined by the dimensionless parameter $H^2\/(M_p^2\\epsilon)$. \n\nTill now in our discussion we have focused on the evolution of the Heisenberg operator $\\zeta_H$. As for the initial state it is natural to take a minimum uncertainty Gaussian wave function $\\psi(\\zeta_i)$, which has zero mean $<\\zeta_i>=0$ and the deviations $<\\zeta_i>=\\sigma^2$, $<P_i^2>=1\/4\\sigma^2$. Although this choice can be motivated from field theory side, the value of the deviation $\\sigma$ cannot be directly deduced from the field theory, which has a continuous spectrum of wave numbers and the zero mode is not isolated (unless the space is not compact). On the other hand, one must also note that the validity of the perturbation theory actually depends on the initial state. Choosing $\\sigma$ to be extremely small would yield a large momentum that may invalidate the perturbative expansion in $P_\\zeta$, at least at early times during inflation when the exponential suppression is not effective yet. \n\nIn showing the constancy of $\\zeta$ in single scalar inflationary models, the consistency condition in the squeezed limit \\cite{c1,c2}, hence the choice of the Bunch-Davies vacuum, plays an important role, see \\cite{c3}. In the minisuperspace model, no such property is needed since the time independence of $\\zeta$ becomes an operator statement, i.e. the Heisenberg picture $\\zeta_H$ exponentially approaches to a constant operator as in \\eq{24}. We anticipate this should also be the case in field theory since at late times semi-classical approximation becomes excellent \\cite{cl} and $\\zeta$ is conserved in the classical theory.\\footnote{Indeed, one naturally expects that some form of minisuperspace description of superhorizon modes, which is similar to the one considered here, must be valid at late times. However, such an approximation, if ever exists, is only possible in a suitable gauge that allows a smooth soft limit, which is not the case for the standard $\\zeta$-gauge because the shift $N^i$ is non-local.} Therefore, the constancy of $\\zeta$ in cosmological perturbation theory must hold not just for the Bunch-Davies vacuum but for a wider range of states. \n\n\\section{Adding a Spectator}\n\nWe have seen in the previous section that the $\\zeta$-self interactions cannot yield infrared logarithms in the minisuperspace perturbation theory. From \\eq{19} and \\eq{23} one sees that $[\\zeta_I(t),\\zeta_I(t')]\\propto 1\/a_B^3$, thus the time integrals in the perturbative series in \\eq{21} can produce an infrared logarithm provided that $H_I\\propto a_B^3$. To produce such an interaction term one may add a self-interacting {\\it massless} spectator scalar $\\varphi$ which has the potential $V(\\varphi)$. It is easy to repeat the gauge fixing in the presence of the spectator to get the following gauge fixed action: \n\\begin{equation}\nS=-2\\int dt\\,a_B^3V_B\\,e^{3\\zeta}\\,\\left[1+\\frac{V(\\varphi)}{V_B}\\right]^{1\/2}\\left[1+\\frac{12H_B}{V_B}\\dot{\\zeta}+\\frac{6}{V_B}\\dot{\\zeta}^2-\\frac{1}{2V_B}\\dot{\\varphi}^2\\right]^{1\/2}.\\label{28}\n\\end{equation}\nBy expanding the square roots one may obtain the free Lagrangian and various interactions, where the quadratic spectator action is given by\n\\begin{equation}\\label{29}\nS=\\fr12\\,\\int \\,dt\\, a_B^3\\,\\dot{\\varphi}^2.\n\\end{equation}\nHence, the interaction picture spectator operators evolve like \n\\begin{eqnarray}\n&&\\varphi_I=\\varphi_i+\\int_{t_i}^t \\frac{dt'}{a_B(t')^3}P_{\\varphi i},\\nonumber\\\\\n&&P_{\\varphi I}=P_{\\varphi i},\\label{30}\n\\end{eqnarray}\nwhere $P_{\\varphi I}$ is the momentum conjugate to $\\varphi_I$, and $\\varphi_i$ and $P_{\\varphi i}$ are time independent initial operators obeying $[\\varphi_i,P_{\\varphi i}]=i$. \n\nAmong the interactions that follow from \\eq{28} we focus on the following one\n\\begin{equation}\nH_I=a_B^3V(\\varphi_I)\\left(e^{3\\zeta_I}-1\\right),\\label{31}\n\\end{equation}\nwhich would potentially yield infrared logarithms in the perturbation theory as noted above. Indeed, from the first order correction in \\eq{21} one may find\n\\begin{equation}\\label{32}\n\\zeta_H=\\zeta_I-\\fr12 \\int_{t_i}^t dt'a_B(t')^3\\,V(\\varphi_I(t'))e^{3\\zeta_I(t')}\\int_{t'}^t\\,dt''\\frac{V_B}{a_B^3\\dot{\\Phi}_B^2}.\n\\end{equation}\nUsing \\eq{23} one can get a late time expansion of the interaction picture operators so that at the end of inflation after $N$ e-folds one has \n\\begin{eqnarray}\n&&\\zeta_I=\\zeta_i+\\frac{H_B^2}{12M_p^2\\epsilon}P_{i}+O\\left(e^{-3N}\\right),\\nonumber\\\\\n&& \\varphi_I=\\varphi_i+\\frac{H_B^2}{3}P_{\\varphi i}+O\\left(e^{-3N}\\right).\\label{33}\n\\end{eqnarray}\nUtilizing this expansion in \\eq{32} gives\n\\begin{equation}\n\\zeta_H(t)=\\zeta_c+\\frac{1}{12H_B^2M_p^2\\epsilon}\\left[1-3\\ln\\left(\\frac{a_B(t)}{a_B(t_i)}\\right)\\right]\\,V(\\varphi_c)\\,e^{3\\zeta_c}+O\\left(e^{-3N}\\right),\\label{34}\n\\end{equation}\nwhere the constant operators $\\zeta_c$ and $\\varphi_c$ are defined from \\eq{33} by\n\\begin{eqnarray}\n&&\\zeta_c=\\zeta_i+\\frac{H_B^2}{12M_p^2\\epsilon}P_{i},\\nonumber\\\\\n&&\\varphi_c=\\varphi_i+\\frac{H_B^2}{3}P_{\\varphi i}.\\label{344}\n\\end{eqnarray}\nAs it is anticipated, the interaction \\eq{31} yields an infrared logarithm in the first order perturbation theory. \n\nThe above calculation hints how one should handle the higher order perturbative corrections. Namely, one should first evaluate the commutators in \\eq{21} using\n\\begin{equation}\n[\\zeta_I(t),\\zeta_I(t')]=\\frac{i}{6}\\int_t^{t'}dt''\\frac{V_B}{a_B^3\\dot{\\Phi}_B^2},\\hs{10}[\\varphi_I(t),\\varphi_I(t')]=i\\int_t^{t'}\\frac{dt''}{a_B^3}.\\label{35}\n\\end{equation}\nOne can then apply the late time expansion of the interaction picture operators given in \\eq{33} and calculate the time integrals of the leading order terms. Using this strategy one may obtain the following second order correction to $\\zeta_H$:\n\\begin{equation}\n\\left[\\frac{V(\\varphi_c)^2}{16M_p^4H_B^2\\epsilon^2}e^{6\\zeta_c}+\\frac{V'(\\varphi_c)^2}{36M_p^2H_B^2\\epsilon}\\left(e^{6\\zeta_c}-e^{3\\zeta_c}\\right)\\right]\\left[1-2\\ln\\left(\\frac{a_B(t)}{a_B(t_i)}\\right)+\\fr32 \\ln^2\\left(\\frac{a_B(t)}{a_B(t_i)}\\right)\\right],\\label{36}\n\\end{equation}\nwhere $V'(\\varphi)=dV\/d\\varphi$. Note that \\eq{36} contains a different type of infrared logarithm, i.e. a log square. \n\nIt is possible to argue that the interaction \\eq{31} yields the factor $\\ln^n(a_B(t)\/a_B(t_i))$ in the $n$'th order perturbation theory. In the $n$'th term of \\eq{21} there are $n$ factors of $a_B^3$ coming from the interaction Hamiltonian and there are $n$ factors of $a_B^{-3}$ coming from $n$-commutators. At late times, these cancel each other and the interaction picture operators asymptote to constant operators as in \\eq{33}. Hence, to leading order one ends up with an $n$-dimensional time integral of a constant operator giving $(t-t_i)^n=H_B^n\\ln^n(a_B(t)\/a_B(t_i))$. \n\nThese findings are consistent with what has been observed in the field theory calculations \\cite{il1,il2,il3}. In the minisuperspace approximation one can further make a nonperturbative estimate as follows: Using the asymptotic form \\eq{33}, the interaction Hamiltonian converges to\n\\begin{equation}\\label{37}\nH_I=a_B^3V(\\varphi_c)\\left(e^{3\\varphi_c}-1\\right)\\left\\{1+O\\left(e^{-3N}\\right)\\right\\}.\n\\end{equation}\nOne can check that at late times the commutator $[H_I(t),H_I(t')]$ is suppressed by a huge factor related to the number of e-folds as compared to the product $H_I(t)H_I(t')$. Therefore, to a very good approximation $H_I(t)$ becomes a self-commuting operator of its argument after some time $t_m$ corresponding to, say, 10 e-folds (the fact that $\\zeta$ has a similar property has been used to argue the classicality of the cosmological perturbations \\cite{cl}). The unitary interaction picture evolution operator can be decomposed like\n\\begin{equation}\nU_I(t,t_i)=U_2(t,t_m)U_1(t_m,t_i).\\label{38}\n\\end{equation}\nSince $H_I(t)$ can be treated as a self-commuting operator when $t>t_m$, one may approximate\n\\begin{equation}\nU_2(t,t_m)=Te^{-i\\int_{t_m}^t dt'H_I(t')}\\simeq e^{-i\\int_{t_m}^t dt'H_I(t')}.\\label{39}\n\\end{equation}\nFurthermore one has\n\\begin{equation}\n\\zeta_H=U_I^\\dagger\\zeta_IU_I=U_1^\\dagger U_2^\\dagger\\zeta_I U_2U_1.\\label{40}\n\\end{equation}\nTo proceed we note \n\\begin{equation}\n[\\zeta_I(t),H_I(t')]=3[\\zeta_I(t),\\zeta_I(t')]H_I(t'),\\label{41}\n\\end{equation}\nthus using \\eq{39} one may find\n\\begin{eqnarray}\n[\\zeta_I(t),U_2]&&\\simeq\\fr12 \\int_{t_m}^tdt'a_B(t')^3V(\\varphi(t'))e^{3\\zeta_I(t')}\\int_t^{t'}dt''\\frac{V_B(t'')}{a_B(t'')^3\\dot{\\Phi}_B(t'')^2}\\,U_2\\nonumber\\\\\n&&\\simeq \\frac{1}{12H_B^2M_p^2\\epsilon}\\left[1-3\\ln\\left(\\frac{a_B(t)}{a_B(t_i)}\\right)\\right]V(\\varphi_c)e^{3\\varphi_c}\\,U_2.\\label{42}\n\\end{eqnarray}\nSo \\eq{40} becomes\n\\begin{equation}\n\\zeta_H\\simeq U_1^\\dagger\\zeta_cU_1+\\frac{1}{12H_B^2M_p^2\\epsilon}\\left[1-3\\ln\\left(\\frac{a_B(t)}{a_B(t_i)}\\right)\\right]U_1^\\dagger V(\\varphi_c)e^{3\\varphi_c}U_1.\\label{43}\n\\end{equation}\nThe unitary operator $U_1$ only mixes the operators up to time $t_m$ and its action merely produces constant operators since these do not depend on the final time. As a result, in \\eq{43} we are able to extract the leading order time dependence of $\\zeta$, which is a single infrared logarithm of the form $\\ln a_B$, and other corrections are exponentially suppressed. On dimensional grounds one may estimate that (in expectation values) $V(\\varphi_c)\\propto H_B^4$, therefore the infrared logarithm correction is suppressed by the factor $H^2\/(M_p^2\\epsilon)$, which is generically small in realistic models. \n\nThe above nonperturbative argument shows that the $(\\ln a_B)^n$ behavior for $n>1$ that arises in the $n$'th order perturbation theory may be an artifact of that approximation. For $t<t_m$ the sum of all these terms must yield the time evolution dictated by the unitary operator $U_1$, which we manage to bypass in our analysis. The minisuperspace approximation clearly shows the emergence of a single infrared logarithm related to the asymptotic form of the interaction picture operators. Specifically, after some time during inflation an interaction picture operator becomes almost self-commutative when evaluated at different time arguments. In that case, in relating the Heisenberg and the interaction picture operators by a unitary transformation, it is enough to carry out a single commutator that cancels $a_B^3$ factor in the interaction Hamiltonian.  The time integral of the leading order constant operators then gives a single infrared logarithm as in \\eq{43}.  \n\nFinally, it is instructive to consider a {\\it massive} spectator to see how the emergence of infrared logarithms depends on the mass parameter. In that case the quadratic spectator action \\eq{29} becomes \n\\begin{equation}\\label{29m}\nS=\\fr12\\,\\int \\,dt\\, a_B^3\\,\\left[\\dot{\\varphi}^2-m^2\\varphi^2\\right]\n\\end{equation}\nand the interaction picture operators can be solved as \n\\begin{eqnarray}\n&&\\varphi_I=f_1(t)\\varphi_i+f_2(t)P_{\\varphi i},\\nonumber\\\\\n&&P_{\\varphi I}=a_B^3\\dot{f}_1(t)\\,\\varphi_i+ a_B^3\\dot{f}_2(t)\\,P_{\\varphi i},\\label{30m}\n\\end{eqnarray}\nwhere $f_1$ and $f_2$ are the two solutions of \n\\begin{equation}\\label{de}\n\\ddot{f}+3H_B\\,\\dot{f}+m^2\\,f=0\n\\end{equation}\ndetermined by the initial conditions\n\\begin{eqnarray}\n&&f_1(t_i)=1,\\hs{5}\\dot{f}_1(t_i)=0,\\nonumber\\\\\n&&f_2(t_i)=0,\\hs{5}\\dot{f}_2(t_i)=a_B(t_i)^{-3}.\n\\end{eqnarray}\nLet us now examine the first order correction which is still given by \\eq{32}. For the massless spectator when $m=0$, one has $f_1=1$ and $f_2=[a_B(t_i)^{-3}-a_B(t)^{-3}]\/(3H_B)$, where we assume \\eq{23}. Here, $f_1$ is already a constant and $f_2$ exponentially approaches to a constant, consequently one has \\eq{33} which then gives the infrared logarithm \\eq{34}. However, when $m\\not=0$ the two solutions of \\eq{de} for constant $H_B$ are given by\n\\begin{equation}\\label{sol}\n\\exp\\left[n_\\pm H_B(t-t_i)\\right],\\hs{5}n_\\pm=-\\fr32\\pm\\sqrt{\\fr94-\\frac{m^2}{H_B^2}},\n\\end{equation}\nwhich are decreasing functions of time. Accordingly, both $f_1$ and $f_2$ turn out to decay either as $\\exp(-m^2(t-t_i)\/3H_B)$ for $m\\ll H_B$ or as $a_B^{-3\/2}$ for $m>3H_B\/2$. Using \\eq{30m} in \\eq{32}, one may then see that the integrand in \\eq{32} does {\\it not} approach to a time independent operator yielding the infrared logarithm as in the case of a massless spectator, but instead it becomes an exponentially decreasing function of $t'$ whose integral gives a smaller correction for larger mass. \n\n\\section{Conclusions}\n\nIn this paper we investigate the appearance of the infrared logarithms in the cosmological perturbation theory by studying the scalar slow-roll inflationary model in the minisuperspace approximation, which simplifies the field theoretical system involving gravity to a quantum mechanical one. The minisuperspace  theory is still highly nontrivial because of the nonlinearities and the local gauge invariance related to the time reparametrizations. We obtain the complete gauge fixed action for the curvature perturbation $\\zeta$, both in the single scalar case and when a self-interacting spectator is added. The full action can be expanded around the inflationary background yielding an infinite number of interaction terms. \n\nIn our analysis we focus on the time evolution of the Heisenberg operators, which can be calculated using in-in perturbation theory. Thus, our findings are state independent provided that the expectation values do not break down the series expansion. We verify that in the single scalar case no infrared logarithms appear and $\\zeta$  exponentially asymptotes to a constant operator. In the presence of a spectator we find that the $n$'th order perturbation theory gives an infrared logarithm of the form $(\\ln a_B)^n$. Note that supposing the existence of a spectator is not unnatural for inflation; in any model where Higgs is not the inflaton, it actually becomes a self-interacting spectator scalar.\n\nIn the minisuperspace approximation it is possible to examine the time evolution of the Heisenberg operators nonperturbatively. Following some time after the beginning of inflation, the interaction picture operators including the interaction Hamiltonian become nearly self-commuting at different times. This allows one to extract the leading order time evolution of $\\zeta$ where all other corrections are exponentially suppressed. In the presence of a spectator, this leading order correction turns out to be a single infrared logarithm $\\ln a_B$. It would be interesting to generalize this argument to field theory to understand the structure of the infrared logarithms in the cosmological perturbation theory. \n\nOne usually attributes the emergence of infrared logarithms to loop effects. This is natural since in field theory calculations they normally appear in loop corrections. However, loops do not exist in the minisuperspace approach yet we still encounter infrared logarithms implied by the Heisenberg picture equations of motion. This indicates that neither the loops nor the modes running in them may not be the primary reason for the existence of infrared logarithms. Indeed, consider as an example the three point function $\\left< \\phi(k_1)\\phi(k_2)\\phi(k_3)\\right>$ of a self-interacting {\\it massless} test scalar field. It is not difficult to see that this correlation function is time dependent {\\it at tree level} because of a cubic interaction term $H_I=g\\, a^3 \\int d^3 x\\,\\phi^3$, even at late times when $k_1$, $k_2$ and $k_3$ become superhorizon. Choosing the vacuum and correspondingly the mode functions determine the precise form of this time dependence, e.g. for the Bunch-Davies vacuum one gets an infrared logarithm. The crucial point is that the cubic interaction induces nontrivial superhorizon evolution which is {\\it not} exponentially suppressed. In single field inflation, one may see that $\\zeta$ self interactions cannot produce such effects mainly because of the shift symmetry (as shown above, pure $\\zeta$ interactions containing no derivatives, which are potentially dangerous, actually disappear after integration by parts). Nevertheless, in the presence of a spectator field there are interactions which yield non-negligible superhorizon evolution both in classical and quantum theories. We expect these conclusions to hold irrespective of the gauge conditions or possible explicit non-localities present in the action. \n\nHence, the emergence of infrared logarithms or some other form of nontrivial superhorizon motion has a dynamical origin related to suitable interactions. Remember that in the minisuperspace model, solving the unitary evolution nonperturbatively gives only a single infrared logarithm as opposed to perturbation theory and this shows the importance of determining dynamics correctly. On the other hand, the initial state chosen is also crucial in fixing the exact form of the superhorizon time dependence. Of course, loops also arise from such interactions sandwiched between states. As discussed in \\cite{proj}, some corrections related to superhorizon modes are projection effects that define the mapping between physical scales of inflation and post-inflationary universe, and these disappear in observable quantities. One way of understanding projection effects is to keep in mind that the decomposition of the metric into  background and  fluctuation parts is introduced for computational convenience. Strictly, the {\\it full} metric must be used in relating the comoving and physical scales. In the minisuperspace theory any constant piece of $\\zeta$ will disappear as a projection artifact but a time dependent part represents a real physical effect, modifying for instance the Hubble expansion rate. \n\n\n\\begin{acknowledgments}\nThis work, which has been done at C.\\.{I}.K. Ferizli, Sakarya, Turkey without any possibility of using references, is dedicated to my friends at rooms C-1 and E-10 who made my stay bearable at hell for 440 days between 7.10.2016 and 20.12.2017. I am also indebted to the colleagues who show support in these difficult times. \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\emph{INTEGRAL} (INTErnational Gamma Ray Astrophysics Laboratory, \\citet{Winetal03}) was launched in 2002 and since then has performed high-quality observations in the energy band from 3 keV up to $\\sim 10$ MeV. The \\emph{INTEGRAL} payload consists of two main soft gamma-ray instruments (the imager IBIS \\citep{Ubeetal03}, and the spectrometer SPI \\citep{Vedetal03}) and two monitors (in X-rays JEM-X \\citep{Lunetal03}, and in optical OMC \\citep{Masetal03}). The wide field-of-view of the imager IBIS provides an ideal opportunity to survey the sky in hard X-rays.\\\\\n\nDuring its first 6 years in orbit, \\emph{INTEGRAL} has covered nearly the whole sky. The observational data have been mainly used to study the soft gamma-ray emission from the Galactic plane (GP) \\citep{Bouetal05, Krietal06} through the Galactic plane scans and the Galactic centre (GC) \\citet{Beletal04, Revetal04, Beletal06, Krietal06} through the Galactic centre deep exposure programme. A number of papers have already presented general surveys \\citep{Bazetal06, Biretal06} of the sky as well as of specific regions \\citep{Gotetal05, Haretal06, Moletal04} and population types \\citep{Baretal06, Basetal06, Sazetal07, Becetal06, Becetal09}.\\\\\n\nThe majority of the classified sources detected by \\emph{INTEGRAL} are either low and high mass X-ray binaries (LMXBs and HMXBs) or AGNs \\citep{Bodetal07}. However, a significant fraction of the detected sources remain unidentified. A special approach to population classification is required for the GC region to resolve the population types because of the high density of sources. Fortunately, the physics of the sources may help us to unveil their type. Indeed, the bulk of the \\emph{INTEGRAL} sources are accreting systems that are expected to be intrinsically variable on multiple timescales depending on the source type and the nature of the variability. For instance, X-ray binaries (XRBs) may exhibit variability on timescales that range from milliseconds (supporting the idea that emission originates close to the compact object in the inner accretion radius) to hours and days, indicating that the variability can originate throughout the accretion flow at multiple radii and propagate inwards to modulate the central X-ray emission \\citep{AreUtt05}. This idea is supported by the known correlation between millisecond\/second and hour\/day scale variability in XRBs \\citep{Utt03}. LMXBs may exhibit flaring behavior with an increase in both emission intensity and hardness over a period of a few hundred to a few thousand seconds. X-ray bursts with rise times of a few seconds and decay times of hunderds of seconds or even several hours \\citep{Baretal02} are also common to these objects. On the other hand, HMXBs are known to exhibit variability on timescales ranging from a fraction of a day up to several days, generated by the clumpiness of the stellar wind acreting onto the compact object \\citep{Ducetal09}. Hour-long outbursts caused by variable accretion rates are observed in supergiant fast X-ray transients, a sub-class of HMXBs discovered by INTEGRAL \\citep{Rometal2009}. Owing to their larger size, AGNs of different types exhibit day-to-month(s) variability depending on the black hole mass \\citep{IshCou09}. Gamma-ray loud blazars have variability timescales in the range from $10^{1.6}$ to $10^{5.6}$~s \\citep{LiaLiu03}. Therefore, a list of \\emph{INTEGRAL} sources with quantitative measurements of their variability would be an important help to classifying the unidentified sources and more detailed studies of their physics.\\\\\n\nThe variability of \\emph{INTEGRAL} sources was addressed in the latest 4th IBIS\/ISGRI survey catalog paper \\citep{Biretal09} when the authors performed the so-called \\textit{bursticity} analysis intended to facilitate the detection of variable sources.\\\\\n\nHere we present a catalog of \\emph{INTEGRAL} variable sources identified in a large fraction of the archival public data. In addition to standard maps produced by the standard data analysis software, we compiled a $\\chi^2$ all-sky map and applied the newly developed method to measure the fractional variability of the sources detected by the IBIS\/ISGRI instrument onboard \\emph{INTEGRAL}. The method is sensitive to variability on timescales longer than those of single ScW exposures ($\\approx 2000$ seconds), i.e., to variability on timescales of hour(s)-day(s)-month(s). The catalog is compiled from the sources detected in the variability map. In addition, we implemented an online service providing the community with all-sky maps in the 20-40, 40-100, and 100-200 keV energy bands generated during the course of this research.\\\\\n\nIn the following, we describe the data selection procedure and the implemented data analysis pipeline (Sect.~\\ref{sec:datana}). In Sect.~\\ref{sec:method}, we outline our systematic approach to the detection of variability in \\emph{INTEGRAL} sources and describe our detection procedure in Sect.~\\ref{sec:detect}. We compile the variability catalog in Sect.~\\ref{sec:catvar}. In Sect.~\\ref{sec:skyview}, we briefly describe the implemented all-sky map online service. We make some concluding remarks in Sect.~\\ref{sec:conclu}.\n\n\\section{Data and analysis}\n\\label{sec:datana}\n\n\\subsection{Data selection and filtering}\n\nSince its launch, \\emph{INTEGRAL} has performed over 800 revolutions each lasting for three days. We utilized the ISDC Data Centre for Astrophysics \\citep{Couetal03} archive\\footnote{http:\/\/isdc.unige.ch} to obtain all public data available up to June 2009 and the Offline Scientific Analysis (OSA) v. 7.0 to process the data. \\\\\n\n\\emph{INTEGRAL} data are organized into science windows (ScWs), each being an individual observation that can be either of pointing or slew type. Each observation (pointing type) lasts 1 -- 3 ksecs. For our analysis, we chose all pointing ScWs with an exposure time of at least 1 ksec. We filtered out revolutions up to and including 0025 belonging to the early performance verification phase, observations taken in staring mode, and ScWs marked as bad time intervals in instrument characteristics data including ScWs taken during solar flares and radiation belt passages. Finally, after the reconstruction of sky images we applied the following statistical filtering. We calculated the standard deviation of the pixel distribution for each ScW and found the mean value of standard deviations for the whole data set. We then rejected all the ScWs in which the standard deviation exceeded the mean for the whole data set by more than 3$\\sigma$. We assumed the distribution of standard deviations of individual and independent ScWs to be normal. While calculating standard deviations in individual ScWs, image pixels were assumed to be independent. Thus, the filtering procedure allowed us to remove all ScWs affected by a high background level.  In the end, 43~724 unique pointing-type ScWs were selected for the analysis, giving us a total exposure time of 80.0~Msec and a more than 95 percent sky coverage.\n\n\\subsection{Instrument and background}\n\nIn the present study we use only the low-energy detector layer of the IBIS coded-mask instrument, called ISGRI (\\emph{INTEGRAL} Soft Gamma Ray Imager, \\citet{Lebetal03}), which consists of 16~384 independent CdTe pixels. It features an angular resolution of $12^{\\prime}$ (FWHM) and a source location accuracy of $\\sim$1 arcmin, depending on the signal significance \\citep{Groetal03}. Its field of view (FOV) is $29^{\\circ} \\times 29^{\\circ}$. The fully-coded part of the FOV (FCFOV), i.e., the area of the sky where the detector is fully illuminated by the hard X-ray sources, is $9^{\\circ} \\times 9^{\\circ}$. It operates in the energy range between 15 keV and 1 MeV.\\\\\n\nOver short timescales, the variability of the background of the instrument is assumed to be smaller than the statistical uncertainties. However, this is not the case for mosaic images constructed from long exposures. In general, it is assumed that the mean ISGRI background in each individual pixel changes very little with time, and therefore the standard OSA software provides only one background map for the entire mission. During the construction of the all-sky map, we noted that the quality of the mosaics of the extragalactic sky region depends on the time period over which the data were taken. We therefore, concluded that the long-term variation in the background of the instrument \\citep{Lebetal05} significantly affects the extragalactic sky mosaic. On the other hand, in the GC and inner GP regions (l$\\left\\lbrace-90;90\\right\\rbrace$, b$\\left\\lbrace -20;20\\right\\rbrace$) the standard background maps provided by OSA provide better results (noise distributions are narrower). This might be because of the large number of bright sources and the Galactic ridge emission \\citep{Krietal06}, although we leave this question open for the future research.\\\\\n\nTo produce time-dependent background maps, we extracted raw detector images for each \\emph{INTEGRAL} revolution (3 days) and calculated the mean count rate in each individual pixel during the corresponding time period. To remove the influence of the bright sources on the neighboring background, we fitted and removed these sources from the raw detector images, i.e., in each ScW we constructed a model of the source pattern on the detector (pixel illumination fraction, PIF) and fitted the raw detector images using the model\n\\begin{equation}\nS_{k,l}=\\sum_{i=1}^M f_i \\times PIF_{k,l}+B,\n\\end{equation}\nwhere $S_{k,l}$ are the detector count rate, $PIF_{k,l}$ are the respective pattern model of source $i$ in the detector pixel with coordinates $(k,l)$, $f_i, i=1..M$ is the flux of source $i$ in the given ScW, and $B$ is the mean background level. This procedure was applied to all the detected sources in the FOV. The stability of the fitting procedure was tested using a large set ($>1~000$) of simulated ScWs with variable source fluxes. The results of the fit were normally distributed around the expected source flux, and therefore we can conclude that our procedure is sufficiently accurate to remove the point sources\nfrom the construction of the background maps. The results of the fitting procedure were then used to create a transformed detector image, $\\hat S_{k,l}$, defined as\n\\begin{equation}\n\\hat S_{k,l}=S_{k,l}-\\sum_{i=1}^M f_i \\times PIF_{k,l}.\n\\end{equation}\nBackground maps were then constructed by averaging the transformed detector images of a given data set.\n\nFrom our time-dependent background maps, we found that the shape of the ISGRI background varies with time, in particular after each solar flare. A long-term change in the background was noticed as well. This result agrees with the findings of \\citet{Lebetal05}. To take these variations into account, we generated background maps for each spacecraft revolution and in the image reconstruction step applied them to the extragalactic sky region.\n\nBesides the real physical background of the sky, there is also artificial component, because IBIS\/ISGRI is a coded-mask instrument with a periodic mask pattern. Therefore, the deconvolution of ISGRI images creates structures of fake sources that usually appear around bright sources. Apart from the periodicity of the mask, insufficient knowledge of the response function leads to residuals in the deconvolved sky images. The orientation of the spacecraft changes from one observation of the real source to another, so fake sources and structures around the real source contribute to the noise level of the local background. To reduce this contribution, we used a method described in Sect.~\\ref{subsec:imarec}.\\\\\n\n\\subsection{Image reconstruction}\n\\label{subsec:imarec}\nAfter producing the background maps as described in the previous subsection, we started the analysis of the data using the standard Offline Scientific Analysis (OSA) package, version 7.0, distributed by ISDC \\citep{Couetal03}. For image reconstruction, we used a modified version of the method described in \\citet{Ecketal08}.  It is known that screws and glue strips attaching the IBIS mask to the supporting structure can introduce systematic effects in the presence of very bright sources \\citep{Nevetal09}. To remove these effects, we identified the mask areas where screws and glue absorb incoming photons, and we disregarded the pixels illuminated by these mask areas for the 11 brightest sources in the hard X-ray band. No more than 1\\% of the detector area was disregarded for each of the brightest sources. For weaker sources, the level of systematic errors produced by the standard OSA software was found to be consistent with the noise, so the modified method was not required. Finally, we summed all the processed images weighting by variances to create the all-sky mosaic. For this work, we produced mosaics in 3 energy bands (20-40, 40-100, and 100-200 keV). Both our all-sky map images and corresponding exposure maps are available online and we direct the reader to our online web service\\footnote{http:\/\/skyview.virgo.org.ua}. As an example, we provide here the image of the inner part (36$^{\\circ}$ by 12$^{\\circ}$) of the Galaxy in the 20-40 keV energy band (see Fig.~\\ref{fig:gc}).\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\includegraphics[width=0.40\\textwidth]{fig1a.eps}\n\\includegraphics[width=0.14\\textwidth]{fig1b.eps}\n\\includegraphics[width=0.40\\textwidth]{fig1c.eps}\n\\caption{Lightcurves and variability map of HMXB 4U~1700-377 and LMXB GX~349+2. The solid line indicates the mean flux of the sources during the observation time, the dotted line shows the mean flux minus $S_{int}$, the dashed line shows the mean flux plus $S_{int}$.}\n\\label{fig:lc}\n\\end{center}\n\\end{figure*}\n\n\\section{Method of variability detection}\n\\label{sec:method}\nThe variability of \\emph{INTEGRAL} sources can be analyzed in a standard way by studying the inconsistency of the detected signal with that expected from a constant source by performing the $\\chi^2$ test. Here we consider introducing a variability measurement for the \\emph{INTEGRAL} sources and show how to apply it to the specific case of the coded-mask instrument. For an alternative approach based on the maximum likelihood function for the determination of intrinsic variability of X-ray sources the reader is referred to \\citet{Almetal00} and \\citet{Becetal07}.\n\nThe \\emph{INTEGRAL} data are naturally organized by pointings (ScW) with average duration of $\\sim 1-3$~ksec. Therefore, the simplest way to detect the variability of a source on ksec and longer timescales is to analyse the evolution of the flux from the source on a ScW-by-ScW basis. We define $F_i$ and $\\sigma_i^2$ to be the flux and the variance of a given source, respectively, in the $i$-th ScW. The weighted mean flux from the source is then given by\n\\begin{equation}\n\\langle F \\rangle=\\frac {\\sum_{i=1}^N \\frac {F_i}{\\sigma_i^2}} {\\sum_{i=1}^N \\frac{1}{\\sigma_i^2}},\n\\end{equation}\nwhere $N$ is the total number of ScWs. The variance of the source's flux, which is the mean squared deviation of the flux from its mean value during the observation time, is given by\n\\begin{equation}\nS_{tot}^2 = \\frac {\\sum_{i=1}^N \\frac{(F_i - \\langle F \\rangle)^2}{\\sigma_i^2}} {\\sum_{i=1}^N \\frac{1}{\\sigma_i^2}} = \\chi^2\\sigma^2,\n\\label{SV}\n\\end{equation}\nwhere $\\chi^2 = \\sum_{i=1}^N \\frac{(F_i - \\langle F \\rangle)^2}{\\sigma_i^2}$ and $\\sigma^2 = \\left(\\sum_{i=1}^N \\frac{1}{\\sigma_i^2}\\right)^{-1}$ is the variance of the weighted mean flux.\n\nHowever, in addition to intrinsic variance of the source, this value includes the uncertainty in the flux measurements during individual ScWs, i.e., the contribution of the noise. If the source variance is caused only by the noise, i.e., $F_i = \\langle F \\rangle \\pm \\sigma_i$, Eq.~(\\ref{SV}) is given by $S_{noise}^2 = N \\sigma^2$. To eliminate the noise contribution, we can subtract the noise term of the variance from the source variance and derive the \\emph{intrinsic variance} of the source\n\n\\begin{equation}\nS_{int}^2 = \\chi^2\\sigma^2 - N\\sigma^2.\n\\label{intvar}\n\\end{equation}\nWhen all measurement errors are equal ($\\sigma_i = \\sigma_0$, $\\sigma^2 = \\sigma_{0}^2\/N$), our case reduces to the method used by \\cite{Nanetal97}\n\n\\begin{equation}\nS_{int}^2 = \\frac {1}{N} \\sum_{i=1}^N (F_i - \\overline{F})^2 - \\sigma_{0}^2,\n\\end{equation}\nwhere $\\overline{F}$ is the unweighted mean flux and $S_{int}^2$ is called the \\emph{excess variance}. In the absence of measurement errors, our case reduces to the standard definition of the variance\n\n\\begin{equation}\nS_{int}^2 = \\frac {1}{N} \\sum_{i=1}^N (F_i - \\overline{F})^2.\n\\end{equation}\n\nGiven that different sources have different fluxes, the variability of sources can be quantified by using the normalized measure of variability, which we call here the \\emph{fractional variability}\n\\begin{equation}\nV = \\frac {S_{int}}{\\langle F \\rangle}.\n\\label{simplefracvar}\n\\end{equation}\nHowever, in reality, if one were to apply the above method to detect the variable sources in a crowded field (i.e., containing many sources) of a coded-mask instrument such as IBIS, one would infer {\\it all} the detected sources to be highly variable. This is because in coded-mask instruments, each source casts a shadow of the mask on the detector plane. If there are several sources in the field of view, each of them produces a shadow that is spread over the whole detector plane. Some detector pixels are illuminated by more than one source. If the signal in a detector pixel is variable, one can tell, only with a certain probability, which of the sources illuminating this pixel is responsible for the variable signal. Thus, in a coded-mask instrument, the presence of bright variable sources in the field of view introduces an ``artificial'' variability for all the other sources illuminating the same pixels. Since the overlap between the PIF of the bright variable source and the sources at different positions on the sky varies with the position on the sky, one is also unable to determine in advance the level of this ``artificial'' variability in a given region of the deconvolved sky image.\n\nTo overcome this difficulty, one has to measure the variability of the flux not only directly in the sky pixels at the position of the source of interest, but also in the background pixels around the source. Obviously, the ``artificial'' variability introduced by the nearby bright sources is similar in the adjacent background pixels to that in the pixel(s) at the source position. Therefore, one can produce the variability map for the whole sky and compare the values of variability at the position of the source of interest to the mean values of variability in the adjacent background pixels. The variable sources should be visible as local excesses in the variability map of the region of interest. If a source can be localized in the variability image, then the true fractional variability of the source is calculated as\n\n\\begin{equation}\nV_r = \\frac {\\sqrt{S_{int,s}^2 - S_{int,b}^2}} {\\langle F_s \\rangle - \\langle F_b \\rangle},\n\\label{fracvar}\n\\end{equation}\nwhere the subscript $b$ represents the values of the background in the area adjacent to the source and the subscript $s$ the values taken from the source position.\\\\\n\nTo illustrate the method, we present the lightcurves (Fig.~\\ref{fig:lc}) of two objects that are typical bright \\emph{INTEGRAL} sources: the HMXB 4U~1700-377, which is a very bright and very variable source ($V_{r} \\simeq 104$~\\%), and the LMXB GX~349+2, which is a moderately bright and variable source ($V_{r} \\simeq 45$~\\%). The solid line indicates the mean flux of the sources, $\\langle F \\rangle$. We can see that the mean flux deviation (dotted lines), calculated as the square root of the intrinsic variance, $S_{int}^2$, measures the average flux variation of the sources during the corresponding time. However, we note that in the present case we consider calculations based solely on a lightcurve. If one wishes to obtain a fractional variability value dividing the mean flux deviation by the mean flux, one will obtain the $V$ value, but not $V_{r}$, i.e., the contribution of bright variable neighbor sources is not treated properly. It is impossible to extract the variability of the background, $S_{int,b}$, and the mean background flux $\\left\\langle F_b \\right\\rangle$ using the source lightcurve only. A number of lightcurves of the neighboring pixels should also be compiled to estimate $S_{int,b}$ and $\\left\\langle F_b \\right\\rangle$. This number should be sufficiently high to obtain good estimates. Therefore, an all-sky approach is justified. In the current example, no sources are much brighter in the vicinity of the ones considered that could strongly affect them, so the difference between $V$ and the catalog $V_r$ value is around 3\\%, but for the weak sources in the vicinity of bright ones the difference would be higher.\n\n\\begin{figure*}[!th]\n\\begin{center}\n\\includegraphics[width=0.95\\textwidth]{fig2a.eps}\n\\includegraphics[width=0.95\\textwidth]{fig2b.eps}\n\\caption{Inner parts (36$^{\\circ}$ by 12$^{\\circ}$) of the INTEGRAL\/ISGRI all-sky maps in Galactic coordinates, Aitoff projection. The significance image (top) in the 20-40 keV energy band, has square root scaling. The bottom image shows the corresponding intrinsic variance map, and also has square root scaling. The circle shows the inner 4$^{\\circ}$ for which the variability background extraction was made from the box region.}\n\\label{fig:gc}\n\\end{center}\n\\end{figure*}\n\nLooking at Eq.~\\ref{fracvar}, one can indeed see that the effect of ``artificial'' fractional variability is strong for moderate and faint sources in the vicinity of the bright variable sources, while for bright sources the effect is small. The ``artificial'' variability introduced by the bright sources in their vicinity ($S_{int,b}^2$ for the surrounding sources) is in range from a fraction of a percent up to a few percent of their own variability (dictated by the PIF accuracy). When we consider the persistent source in the vicinity of the bright variable source, $S_{int,s}^2$ is defined by $S_{int,b}^2$ only (i.e., by variability introduced by the bright variable source). For moderate or faint sources, $S_{int,b}$ may well be comparable to their own flux, and if we apply Eq.~\\ref{simplefracvar} directly we will infer substantial fractional variability, which may well be between ten and fifty percent, or even higher. The bright sources are less sensitive to this effect because $S_{int,b}$ will be only a small fraction of their flux. We checked these conclusions by performing simulations of a moderate persistent source ($F = 1$~ct\/s) in the vicinity of a bright variable source ($\\langle F \\rangle = 20$~ct\/s). By applying Eq.~\\ref{simplefracvar} directly to measure the fractional variability of a moderate source, we found that $V \\simeq 25$\\% while Eq.~\\ref{fracvar} infered that the source was constant, i.e., $V_r \\simeq 0$\\%. \n\n\\begin{figure*}[!th]\n\\begin{center}\n\\includegraphics[width=0.61\\textwidth]{fig3a.eps}\n\\includegraphics[width=0.37\\textwidth]{fig3b.eps}\n\\includegraphics[width=0.61\\textwidth]{fig3c.eps}\n\\includegraphics[width=0.37\\textwidth]{fig3d.eps}\n\\caption{Top: the distributions of the intrinsic variability in pixels of $3.5^{\\circ} \\times 3.5^{\\circ}$ area (shown nearby) centered on the IGR~J18450-0435. In green is $B_1$ distribution in the range $(min, 2m-min)$ here representing the local intrinsic variance background, in red is the sources contribution, f(x) is the gaussian distribution. Bottom: the $3.5^{\\circ} \\times 3.5^{\\circ}$ area (shown nearby) centered on the GC source 1A~1743-288. In green is $B_1$ distribution in the range $(min, 2m-min)$, in blue is $B_2$ distribution from the empty region in GC here representing the local intrinsic variance background, in red and f(x) are same as above.}\n\\label{fig:sintbhist}\n\\end{center}\n\\end{figure*}\n\nIn the course of simulations, we also checked the behavior of the ``artificial'' variability in the case a moderate persistent source situated at the ghost position of the bright variable source. We considered two situations: a) when the mosaic image consisted mostly of ScWs in a configuration being determined almost entirely by the spacecraft orientation, which remained constant (i.e., sky region of the Crab), and b) when the mosaic image contained only a chance fraction of ScWs in that specific configuration because of different spacecraft orientations (i.e., sky region of the Cyg X-1). The simulations showed that the flux and therefore the variability measurement of the mosaic deviated from the input source parameters in case a) only, while in case b), there were no significant deviations. As expected in case a), the moderate source was affected. This was caused by the coincidence of the sources shadowgrams. The deconvolution procedure was unable to distinguish the sources correctly on the ScW level, therefore the mosaic results were affected. We detected an incorrect flux and variability for the moderate source. In reality, this particular simulated situation is very rare (see Sect.~\\ref{sec:catvar} for discussion of this situation in real case). Even if the constant orientation of some observation is kept, different observation patterns applied during the observation will significantly reduce the undesirable effect considered in situation a). \n\\\\\n\n\\section{The detection of variable sources}\n\\label{sec:detect}\n\nFor our study, we used the latest distributed \\emph{INTEGRAL} reference catalog\\footnote{http:\/\/isdc.unige.ch\/?Data+catalogs} \\citep{Ebietal03} version 30 and selected the sources with ISGRI\\_FLAG == 1, i.e., all the sources ever detected by IBIS\/ISGRI.\\\\\n\nBased on the aforementioned method, we compiled the instrinsic variance maps ($S_{int}^2$) of the \\emph{INTEGRAL} sky in three energy bands (20-40, 40-100, and 100-200 keV). This was accomplished by performing pixel operations following Eq.~(\\ref{intvar}) on the constructed all-sky mosaic maps of $\\chi^2$, $\\sigma^2$ (variance), and $N$, which is the map showing the number of ScWs used in a given pixel for the all-sky mosaic. As an example, the instrinsic variance image of the inner region of our Galaxy is given in Fig.~\\ref{fig:gc} (see our online service for all-sky maps).\\\\\n\nAfter compiling an intrinsic variance map in each band, we calculated the local (or background) intrinsic variance, $S_{int,b}^2$, and its scatter, $\\Sigma$, in the region around each catalog source. This was performed by the following scheme. First, the values of the mean $m$, the minimum $min$, and their difference were calculated in a square of $3.5^{\\circ} \\times 3.5^{\\circ}$ centered on the source position. We then assumed that the pixel values in the corresponding area are distributed normally and there is the always-positive contribution from the sources in the field. Since the sources occupy a small fraction of the considered sky region, the initial mean value, $m$, is almost unaffected by their presence because of their small contribution. To reject the source contribution and to obtain the parameters of the normal component, we calculated the mean and the standard deviation in the range $(min, 2m-min)$. The newly found mean value is $S_{int,b}^2$ in Eq.~\\ref{fracvar} and the standard deviation shows its scatter $\\Sigma$. The detectability of the sources in the intrinsic variance map is then defined by the condition that $S_{int}^2 \\geq S_{int,b}^2 + 3 \\Sigma$. For an illustration (see top of Fig.~\\ref{fig:sintbhist}) we present a region around INTEGRAL source IGR~J18450-0435 with respective distributions. The green solid area is the distribution in the range $(min, 2m-min)$ with the mean value representing $S_{int,b}^2$ for the current source, and in red the always-positive contribution from the sources in the field is given. The distribution in the range $(min, 2m-min)$ is well fitted by the Gaussian shown with a dashed line. The top of Fig.~\\ref{fig:sintbhist} justifies well the approximate rejection of the source contribution to the overall pixel value distribution in the field. Applying this rejection procedure allows us to obtain the true scatter in the background rather than the scatter in the overall distribution (including sources), which is obviously higher.\n\nThe detection of variable sources in the innermost region of our Galaxy (circle of 4$^\\circ$ from the GC, see Fig.~\\ref{fig:gc}) was performed differently because it is a crowded field and therefore a large number of sources contribute to the intrinsic variance background of each other. In contrast to the individual source case, the sources in the inner GC region occupy a significant fraction of the region around the source of interest and therefore influence the $m$ value significantly. This results in improper estimation of the background distribution if one applies the rejection procedure based on $(min, 2m-min)$ range ($B_1$ distribution at the bottom of Fig.~\\ref{fig:sintbhist}). In place of calculating the intrinsic variance background and its scatter for each GC source individually, these values were therefore estimated from an empty field near the GC (box at the bottom image of Fig.~\\ref{fig:gc}) and assumed to be equal for all the sources in the GC region. The $B_2$ distribution shown at Fig.~\\ref{fig:sintbhist} (bottom) is accurately determined and well fitted by the Gaussian shown by the dashed line.\\\\\n\n\\section{The catalog of variable sources}\n\\label{sec:catvar}\n\nThe search for variable sources from the reference catalog was performed on the maps generated from ScWs divided into three equal subsequent time periods (maps 1,2, and 3, approximately 2 years each). This was done to detect transient sources that would be difficult (or even impossible) to detect on the map integrated over the whole time period (map T). The search was performed separately on each map (1,2,3) and in each energy band. The results of the search were put into one list from which the sources were selected. Finally, the search was performed on the total map (T) to find sources that were detected only on the map integrated over the whole time period and identify the sources that were active only during specific time periods. The resulting catalog of variable sources detected by \\emph{INTEGRAL} can be found in Table~\\ref{varcat}.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.99\\columnwidth]{fig4.eps}\n\\caption{Number of significantly variable sources detected in different energy bands classified by types.}\n\\label{chart:poptypes}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{chart:poptypes} shows the number of significantly variable sources from our catalog for each source type (HMXB, LMXB, AGN, unidentified, and miscellaneous). The majority of the variable sources ($\\sim$76\\%) in all energy bands are Galactic X-ray binaries. We see that in the 100-200 keV band there are four times more LMXBs than HMXBs. The remaining variable sources are AGNs ($\\sim$10\\%), unidentified ($\\sim$7.5\\%),  and other ($\\sim$6.5\\%) source types (cataclysmic variables, gamma-ray bursts, and pulsars). The number of significantly variable sources decreases with energy for each population type, which only reflects the sensitivity of the instruments.\n\nThe distribution of the variability of sources from Table~\\ref{varcat} is presented in Fig.~\\ref{chart:histo}. The variability distribution is approximately normal with one evident outlier, the gamma-ray burst IGR J00245+6251 \\citep{Vesetal05}. However, this is mainly caused by the upper limit to the mean flux of this source being too low. The gamma-ray burst IGR J00245+6251 is detected in all three energy bands. Figure~\\ref{fig:V1-V2} shows the fractional variability in the 40-100 keV band versus the variability in the 20-40 keV band. The majority of the sources that are found to be variable in both the 20-40 keV and 40-100 keV energy bands exhibit nearly equal variability in both bands. \n\nTo show the detection threshold for the variability of a source, we compiled a diagram (see Fig.~\\ref{fig:logV-logF}) where we plot the fractional variability versus flux for all detected variable sources along with the upper limit to the fractional variability versus flux for all the reference catalog sources detected in our significance map in 20-40 keV energy band. The upper limit was determined by substituting $S_{int,s}^2$ with $S_{int,b}^2 + 3 \\Sigma$ in Eq.~(\\ref{fracvar}). Although we chose all the sources detected in our significance map, we used the mean flux of the source because, unlike significance, it is an exposure-independent value. According to our definition, the fractional variability is also an exposure-independent value, so we plot it versus the exposure-independent mean flux to clearly see the detection threshold. We can see that starting from a limiting flux ($6.2\\times10^{-11}$~ergs\/cm$^2$s or $\\sim10$~mCrab) nearly all catalog sources are found to be variable. The majority of the bright sources are binary systems, which explains why nearly all of them are found to be variable.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.99\\columnwidth]{fig5.eps}\n\\caption{Number of sources versus fractional variability of the sources detected in different energy bands.}\n\\label{chart:histo}\n\\end{center}\n\\end{figure}\n\n\\begin{table*}[!h]\n\\begin{center}\n\\caption{The transient sources detected at the intrinsic variance map and not detected at the significance map. }\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\nName & Class & $V_{r,notnorm} \\pm Err$ (c\/s) & Exposure (ksec) & Map & Band\\\\\n\\hline\n\\input{varnstab.tex}\n\\hline\n\\end{tabular}\n\\label{transou}\n\\end{center}\n\\end{table*}\n\nWe also found a number of variable sources that have no counterparts in the significance map, which means that we were unable to measure the mean flux of these sources as it is compatible with the background mean flux. Hence, we are not able to give their fractional variability value, which is a normalized value and therefore tends to infinity with infinite errors. For these sources, we provide a 3-$\\sigma$ lower limit to their fractional variability by taking a 3-$\\sigma$ upper limit on their mean flux. Most of the sources are transient, and sometimes (in a specific observation period in a given energy band) the source is not detected in the significance map because the sensitivity of the instrument decreases with energy. Therefore, we can see that the variability map provides a tool to detect transient or faint but variable sources that would be missed in mosaics averaged over long timescales. To illustrate the detectability of the sources in the variability map, we provide a list (see Table~\\ref{transou}) of the sources that are smeared out in the significance map because of their high exposures. The values given in the table are not normalized variability values, $V_{r,notnorm} = \\sqrt{S_{int,s}^2 - S_{int,b}^2}$ along with their 3-$\\sigma$ errors, $Err$. To verify that the sources that are absent in the significance maps but detected in the variability maps are not the result of the low detection threshold, we ran the same detection procedures on the mock catalog of 2500 false sources distributed randomly and uniformly over the sky. The test detected 11 of 2500 false sources seen in variability maps and absent from the significance maps, compared to 8 of 576 in the case of the real catalog. This means that our detection criteria are rather strict.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.99\\columnwidth]{fig6.eps}\n\\caption{Fractional variability of sources in the 20-40 keV band versus fractional variability in the 40-100 keV band.}\n\\label{fig:V1-V2}\n\\end{center}\n\\end{figure}\n\nWe comment on the inclusion of Crab in our catalog. It is known to be a constant source that is commonly used as a ``standard candle'' in high-energy astrophysics. There are two reasons why it appears in the catalog. The first is the deterioration of the detector electronics onboard the spacecraft. The loss of detector gain is around 10\\% over the entire mission. Although this loss is partially compensated by the software, it introduces around 3-5\\% variability in our method. The remaining variability can be ascribed to systematic errors in the spacecraft alignment \\citep{Waletal03}, which for OSA 7.0 are of the order of 7 arcsec \\citep{Ecketal09}, hence slightly different positions of the Crab are found in each observation. Since Crab is a very bright source, its Gaussian profile is very steep. When the peak is slightly offset, we measure a sharp decrease in the flux at the catalog position of Crab. The combination of the two effects leads to an artifical variability of Crab in all energy bands. A similar effect occurs in the pixels adjacent to the catalog position of Crab.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.99\\columnwidth]{fig7.eps}\n\\caption{Fractional variability (V) versus flux (F) for all significantly variable sources (red crosses) from Table~\\ref{varcat}. Energy band is 20-40 keV. For comparison, the green crosses show the fractional variability detection threshold (3-$\\sigma$) versus flux for all reference catalog sources detected in the significance map. The sources not detected at the variability map are indicated by blue stars and are coincident with their green cross counterparts. Pink squares indicate 3-$\\sigma$ lower limits to the fractional variability of the sources that are not detected in the significance map.}\n\\label{fig:logV-logF}\n\\end{center}\n\\end{figure}\nWhen the peak of the PSF is found at a slightly displaced position, we find a sharp increase in the flux in this pixel. Our interpretation is confirmed by the image of Crab in the variability map, where the closest pixels to the source are found to be very variable, creating a ring-like structure. Moreover, it has been found that the observed position of Crab does not coincide with the position of the pulsar \\citep{Ecketal09}, which thus explains such an effect at the expected source position. To determine the influence of this effect on other sources, we inspected the 11 brightest sources in the 20-40 keV band and looked for a similar situation. In the case of Cyg~X-1 and Sco~X-1, the same effect, albeit weaker, was also seen. However, the derived value of their variability was not found to be affected by this effect. This effect contributes mainly to the variability of the pixels around the catalog position of these sources. Cyg X-1 and Sco X-1 are intrinsically very variable so the value extracted from the source position is much higher than for surrounding pixels, which is the opposite of the situation found in the Crab. For all the other sources, the influence of the misalignment was found to be negligible.\n\n\\begin{table*}[!h]\n\\begin{center}\n\\caption{Sources with additional error induced by the ``ghosts''.}\n\\begin{scriptsize}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}\n\n\\hline\n\\multicolumn{1}{|l|}{Name} & \\multicolumn{5}{c|}{20-40 keV} &\\multicolumn{5}{c|}{40-100 keV} &\\multicolumn{5}{c|}{100-200 keV} \\\\\n\\cline{2-16}\n\\multicolumn{1}{|l|}{} & $V_r$ & $V_{r,err}$ & $G_{err}$ & Gexp & Exp & $V_r$ & $V_{r,err}$ & $G_{err}$ & Gexp & Exp & $V_r$ & $V_{r,err}$ & $G_{err}$ & Gexp & Exp \\\\\n\n\\hline\n\\input{ghostab.tex}\n\\hline\n\\end{tabular}\n\\label{ghostab}\n\\end{scriptsize}\n\\end{center}\n\\hspace{-6mm} Here, $V_r$ is the true fractional variability of the source, $V_{r,err}$ is the fractional variability error same as in catalog, $G_{err}$ is the error induced by the source, Gexp, in ksec, is the exposure time the source was affected by the ``ghost'', Exp, in ksec, is the total exposure time.\n\\end{table*}\n\nFinally, we performed a test to find cases in which a source is coincident with the ghost of another source within one ScW (a case described in Sect.~\\ref{sec:method}). We considered all the reference catalog sources and searched for ``ghost-source'' pairs in individual ScWs from the list used for our all-sky maps. If one of the sources in a given pair was present in our variability catalog, the pair was selected for further analysis. According to our simulations, if two sources are in the ghosts of each other, the fainter one loses up to half of its flux to the flux of the brighter one. If one of the sources is substantially brighter than the other, the relative distortion of the flux of the bright source is minor. Therefore, the flux of the source is significantly distorted if its ``ghost companion'' is brighter or comparable in brightness to the source itself. In the latter case, the ``ghost companion'' is also affected. Thus, we assume that if the source in the ghost is more than ten times fainter than its ``ghost companion'', its contaminating influence is negligible, whereas its own flux is contaminated significantly. After adopting this condition, we obtained a list of the sources affected by such position coincidences and the exposure times for each of them during which they where affected. For nearly all of the sources from the list, the fraction of exposures with distorted flux is less than 1\\%, which infers a relative uncertainty in the average flux and fractional variability of the same order. This is much smaller than the error set on variability in our catalog and, as can be seen from the Fig.~\\ref{fig:logV-logF} is smaller than the variability detection threshold even for the brighest sources. Nonetheless, a number of sources have larger than 1\\% errors induced by the ghosts, which we indicate with a $^{g}$ superscript in the catalog and provide a Table~\\ref{ghostab} where both errors are given. As can be seen, in all cases the ghost induced error is much smaller than the catalog variability error.\n\n\n\\section{The All-Sky online}\n\\label{sec:skyview}\n\nTo make our results available to the community, we decided to take advantage of the SkyView interface \\citep{Mcgetal98} (i.e., a Virtual Observatory service available online\\footnote{http:\/\/skyview.gsfc.nasa.gov} developed at and hosted by the High Energy Astrophysics Science Archive Research Centre (HEASARC)). SkyView provides a straightforward interface where users can retrieve images of the sky at all wavelengths from radio to gamma-rays \\citep{Mcgetal98}. SkyView uses NED and SIMBAD name resolvers to translate names into positions and is connected to the HEASARC catalog services. The user can retrieve images in various coordinate systems, projections, scalings, and sizes as ordinary FITS as well as standard JPEG and GIF file formats. The output image is centered on the queried object or sky position. SkyView is also available as a standalone Java application \\citep{Mcgetal97}. The ease-of-use of this system allowed us to incorporate INTEGRAL\/ISGRI variability and significance all-sky maps into the SkyView interface. We developed a simple web interface for the SkyView Java application and have made all-sky mosaics available online\n\n\n\\section{Concluding remarks}\n\\label{sec:conclu}\n\nOur study of variability of the \\emph{INTEGRAL} sky has found that 202 sources exhibit significantly variable hard X-ray emission. To compile the catalog of variable sources, we have developed and implemented a method to detect variable sources, and compiled all-sky variability maps. A search for variable sources from the \\emph{INTEGRAL} reference catalog was carried out in three energy bands: 20-40, 40-100, and 100-200 keV. The variable sources were detected in all studied energy bands, although their number sharply decreases with increasing energy. A number of sources were detected only during specific time periods and not detected on the map integrated over the whole time period. These sources are most likely transient. On the other hand, some sources were found to be variable only on the total variability map. This means that they might be persistent and not extremely variable sources.\n\nWe found that around 76\\% of all variable sources of our catalog are binary systems and nearly 24\\% of variable sources are either AGNs, unidentified, or other source types. The variability measurements of our catalog sources have rather normal distributions in all energy bands. We found that in the majority of cases the variability of the source in the first band correlates with its variability in the second band. We derived the limits to the fractional variability value to be detected as a function of the source flux (Fig.~\\ref{fig:logV-logF}). We also found that variability map can be a tool to detecting transient or faint but variable sources that would be missed in mosaics averaged over long timescales.  In a forthcoming paper, we will discuss in more detail the properties of the variable sources detected during this study in order to gain some physical insights into the population of hard X-ray sources.\\\\\n\nFinally, we emphasize that the sky maps generated during this study represent 6 years of \\emph{INTEGRAL} operation in orbit. In addition to the variability maps, we have compiled significance maps in three energy bands (20-40, 40-100, and 100-200 keV). All our maps are available as an online service to the community using the SkyView engine. \n\n\\section*{Acknowledgements}\n\nBased on observations with INTEGRAL, an ESA project with instruments and science data centre funded by ESA member states (especially the PI countries: Denmark, France, Germany, Italy, Switzerland, Spain), Czech Republic and Poland, and with participation of Russia and the USA.\n\nThis work was supported by the Swiss National Science Foundation and the Swiss Agency for Development and Cooperation in the framework of the programme SCOPES - Scientific co-operation between Eastern Europe and Switzerland. The computational part of the work was done at VIRGO.UA\\footnote{http:\/\/virgo.org.ua} and BITP\\footnote{http:\/\/bitp.kiev.ua} computing resources.\n\nWe are grateful to the anonymous referee for the critical remarks which helped us improve the paper.\n\nIT acknowledges the support from the INTAS YSF grant No.~06-1000014-6348. \n\n\\section*{Appendix: Error Calulation on $V_r$}\n\nWe use the standard error propagation formula to find the error, $\\sigma_{V_r}$, of the function $V_r = f(S_{int,s}^2=a, S_{int,b}^2=b, F_{s}=c, F_{b}=d)$ so :\n\\begin{equation}\n\\sigma_{V_r} = \\sigma_{f} = \\sqrt{\\left(\\frac{\\partial f}{\\partial a}\\sigma_a\\right)^2 + \\left(\\frac{\\partial f}{\\partial b}\\sigma_b\\right)^2 + \\left(\\frac{\\partial f}{\\partial c}\\sigma_c\\right)^2 + \\left(\\frac{\\partial f}{\\partial d}\\sigma_d\\right)^2},\n\\label{VERR}\n\\end{equation}\nwhere $\\sigma_a = \\sigma_b = \\Sigma$ and $\\sigma_c = \\sigma_d = \\sigma$ for a given source. By substituting the appropriate values in Eq.~\\ref{VERR} and by taking derivatives we find that:\n\\begin{equation}\n\\sigma_{V_r} = \\sqrt{\\frac{\\Sigma^2}{2\\left(F_s - F_b\\right)^2 \\left(S_{int,s}^2 - S_{int,b}^2 \\right)} + \\frac{2 \\sigma^2 \\left(S_{int,s}^2 - S_{int,b}^2 \\right) }{\\left(F_s - F_b\\right)^4}}.\n\\end{equation}\n\n\n\\bibliographystyle{aa}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Band Structure Calculations}\nTo support these claims, we performed extensive ab-initio numerical simulations of adatom-decorated stanene. For all of our simulations, we used the high-buckled, free-standing structure of stanene shown in Fig.~\\ref{fig:struc_BC}, with buckling height $\\delta$=0.859\\AA~, and in-plane lattice parameter $a=4.676$\\AA,~as found by relaxing the structure of free-standing stanene using DFT. We decorated stanene with Zn adatoms at one of each of the four structural sites for the buckled honeycomb lattice: hollow (H), bridge (B), valley (V), and top (T)~\\cite{naqvi2017exploring}, and relaxed the height of the adatoms. Because the phenomena in which we are interested require sublattice site symmetry breaking, we primarily focus on the V and T adatom sites for the remainder of this work.\n\nTo determine the stability of Zn atoms in the V and T positions, we used density functional theory to calculate the adsorption energy of Zn at each site using the definition  $E_{ads} = E_{Zn+stanene} - E_{stanene} - E_{Zn}$. We found the adsorption energies to be $E_{ads}^{V} = -0.404$ eV and $E_{ads}^{T} = -0.545$ eV. To explore the possibility that the adatoms would migrate from the V and T positions, we also performed both a nudged elastic band calculation~\\cite{henkelman2000climbing} to determine the most favorable pathway for adatom transport away from the V or T sites, and a dimer method calculation~\\cite{henkelman1999dimer} to precisely determine the activation barriers for Zn migration. These calculations indicated that the Zn atoms on V and T sites are most likely to move to the H site, with diffusion barriers $E^{V}_{barrier}$ = 0.008 eV and $E^{T}_{barrier}$ = 0.011 eV. This is similar to other reports of barrier heights for the migration of adatoms on stanene for the V\/T $\\rightarrow$ H processes~\\cite{mortazavi2016staneneAdatomNaLi} and indicates that the adatoms will be slow to diffuse at temperatures below $\\sim$100K. These migration barriers, while limiting stability at higher temperature, would also allow the adatoms to be manipulated by STM techniques more easily. However, for cases where a higher operating temperature is desirable, we identify other candidate adatoms with higher diffusion barriers in the supplemental material.\n    \nNext we calculated the band structure for bare stanene and stanene decorated with Zn adatoms at the T or V sites, shown in Fig~\\ref{fig:struc_BC}a-c.  Bare stanene has massive Dirac cones with negative band gaps of magnitude $E_g^{bare} = 0.073$ eV at the $\\v{K}$ and $\\v{K}'$ points. When decorated with Zn adatoms in either position, the degenerate Dirac cones are spin-split, resulting in a smaller $E_g^{V}=0.095$ eV gap for V decoration and a larger $E_g^{T}=0.398$ eV gap for T decoration. Crucially, we find that the decoration leaves the bands away from the Fermi level qualitatively unchanged, ensuring that the significant physical changes in the material can be captured by the topological indices near the Fermi-level. \n\nWe repeated this analysis for stanene decorated at the T and V sites with each element in rows 2 through 5 of the periodic table. We find that nearly all elements produce bands that differ significantly from bare stanene. Additionally, many elements that do not qualitatively change the band structure of stanene end up doping the system to result in a metallic character. We provide more details of the viability of these adatom species and how they compare to Zn in the supplementary material.\n\n\n\n\\section{QSH and QVH Indicators}\nUsing the results of the ab-initio calculations, we generated tight-binding parameters via the maximally-localized Wannier function procedure. From the resulting Hamiltonians we calculate the Berry curvature for bare and decorated stanene~\\cite{marzari2012maximally}. As shown by the coloration of the band structures in Fig.~\\ref{fig:struc_BC}b and c, with red and blue representing positive and negative Berry curvature, both decorations produce equal and opposite Berry curvature concentrations at the $\\v{K}$ and $\\v{K}'$ points, indicating that one set of bands at each of these points was inverted by the Zn decoration. \n\nThe origin and consequences of these band inversions are best understood via a low-energy effective model for the massive Dirac cones at the $\\v{K}$ and $\\v{K}'$ points in stanene~\\cite{yao_spin-orbit_2007, Liu11, molle_buckled_2017}:\n\\begin{equation}\n    \\begin{split}\n        H=\\hbar v_F(\\eta k_x\\tau^x+ k_y\\tau^y)+\\eta\\tau^z\\sigma^z\\lambda_{SO}+\\Delta\\tau^z,\n    \\end{split}\n    \\label{eqn:ham_tb}\n\\end{equation}\nwhere $\\tau^\\alpha$ and $\\sigma^\\alpha$ are Pauli matrices for the sublattice and spin degrees of freedom respectively, $v_F$ is the Fermi velocity, $\\eta=+1$ for $\\v{K}$ and $\\eta=-1$ for $\\v{K}'$, and $\\lambda_{SO}$ is the spin-orbit coupling strength. The final term describes a staggered potential of strength $\\Delta$ between the sublattice sites generated by the adatom decoration.\n\nIn the absence of the staggered potential $\\Delta$, this model describes the QSH phase realized by bare stanene. Because the Berry curvature distribution is concentrated around the $\\v{K}$ and $\\v{K}'$ points, and the $z$-component of the spin is conserved, we can define spin-valley resolved Chern numbers, which are protected by time-reversal symmetry and spin-conservation, by integrating the Berry curvature of a particular spin around $\\v{K}$ or $\\v{K}'$~\\cite{ezawa_monolayer_2015}. We obtain $C_{K\\uparrow}=C_{K'\\uparrow}=\\pm1\/2$ and $C_{K\\downarrow}=C_{K'\\downarrow}=\\mp1\/2$, with the signs dependent on the sign of $\\lambda_{SO}$. In terms of these spin-valley resolved indices, the total Chern number $C \\in \\mathbb{Z}$ and the spin Chern number $C_s \\in \\mathbb{Z}_2$ are\n\\begin{equation}\n    \\begin{aligned}\n        C &= C_{K\\uparrow} + C_{K\\downarrow} + C_{K'\\uparrow} + C_{K'\\downarrow} = 0 \\\\\n        C_s &= \\frac{1}{2}(C_{K\\uparrow} - C_{K\\downarrow} + C_{K'\\uparrow} - C_{K'\\downarrow}) = 1 \\text{ mod } 2.\n    \\end{aligned}\n\\end{equation}\nAccording to the bulk-boundary correspondence, we expect that interfaces between regions with different spin Chern numbers host gapless helical modes that carry a spin current\\cite{kanemele05, Bernevig06, hasan_colloquium_2010, qi_topological_2011}. Pairs of helical modes are not protected by time-reversal symmetry, so the spin Chern number is defined modulo $2$, $C_s\\in\\mathbbm{Z}_2$, and interfaces either have zero or one pair of stable helical modes.\n\nNow when we consider the adatom decoration we find that a sufficiently large positive (negative) sublattice potential $\\Delta$ induces a spin-valley resolved band inversion and changes the signs of $C_{K,\\uparrow}$ and $C_{K',\\downarrow}$ ($C_{K,\\downarrow}$ and $C_{K',\\uparrow}$)~\\cite{Ni12, ezawa_monolayer_2015}. The resulting Chern and spin Chern numbers both vanish, but we can instead assign a translation symmetry protected \\emph{momentum} vector charge to each valley.\nThis charge is equal to the vector describing the position of the valley in momentum space and defines the valley vector index: $\\vec{C}_{v}=\\hbar\\vec{K}(C_{K}-C_{K'})=2\\hbar\\vec{K}$.\nSystems with a non-vanishing $\\vec{C}_v$ are called quantum valley Hall (QVH) insulators~\\cite{xiao_valley-contrasting_2007, ren_topological_2016}. \n\nInterestingly, a change in the QVH index across an interface is accompanied by a translation symmetry protected current~\\cite{Xiao07} carrying \\emph{lattice momentum} along the interface. \nThe amount of momentum transported along the edge is characterized by a scalar quantity $C_v$, equal to the projection of $\\vec{C}_v$ onto the unit vector $\\vec{\\tau}$ tangential to the interface.\nIn particular, a straight open edge satisfying $\\vec{\\tau} \\cdot \\vec{C}_v = 0$ projects the valleys on top of each other, resulting in a trivial edge as indicated by the vanishing scalar index $C_v=0$. In contrast, when $\\vec{\\tau}$ is parallel to $\\vec{C}_v$, we obtain $C_v=2\\hbar|\\vec{K}|$. It is customary to drop the factor of momentum $\\hbar|\\vec{K}|$ from the definition of $C_v$ altogether and work with the dimensionless valley index $C_{v}=C_{K}-C_{K'}=\\pm2$. We list the values of the Chern numbers and valley index realized by bare and decorated stanene in Table~\\ref{tab:indices}.\n\n\\begin{table}[t]\n    \\centering\n    \\def1.25{1.25}\n    \\begin{tabular}{| c | c c c | c |}\n        \\hline\n        Phase & $C$ & $C_{s}$ & $C_{v}$ & Decoration pattern \\\\\n        \\hline\n        \\hline\n        \n        \\multirow{2}{*}{QVH} & 0 & 0 & 2 & Zn at V  \\\\\n                              \\cline{2-5}\n                              & 0 & 0 & $-2$ & Zn at T \\\\\n        \\hline\n        QSH & 0 & 1 & 0 & No adatom decoration \\\\\n        \\hline\n    \\end{tabular}\n  \\caption{The Chern number, $C$, spin-Chern number, $C_s$, and valley-Chern number, $C_{v}$, for each topologically nontrivial phase realized by Hamiltonian (\\ref{eqn:ham_tb}), along with the corresponding adatom decoration patterns.}\n  \\label{tab:indices}\n\\end{table}\n    \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{images\/figure2.pdf}\n    \\caption{\n    %\n    \\textbf{Interfacial edge modes from the QSH-QVH structure.}\n    %\n    The (a, b, d, e) band structure and (c, f) representative edge state probability distributions for the QSH-QVH ribbon for zigzag (top) and armchair (bottom) orientations. In (a, b, d, e), the thin gray and thick black lines represent the bulk and edge states, respectively. (b, e) zoom in on the near-gap region at the $\\v{K}'$ point. The red dots marks a spin-up edge state propagating down the right interface and the blue dots marks a spin-up edge state propagating up the left interface. The line plots in (c) and (f) represent the probability density of the edge states indicated by the red and blue dots integrated over the plane perpendicular to the width of the ribbon. The shapes of the probability density plots for the zigzag interfaces differ because the adatom decoration makes the two interfaces asymmetric. An in-plane view of the interface structure, with colored atoms corresponding to those in Fig.~\\ref{fig:struc_BC} is shown in the bottom row of (c) and (f). The locking of the spin and valley degrees of freedom at each interface is the hallmark of the QSH-QVH edge state.\n    %\n    }\n    \\label{fig:qsh_qvh}\n\\end{figure}\n\n\\section{First-Principles Interface Calculations}\nWith this understanding of the bulk properties of decorated and bare stanene we can now consider interfaces between different spatial domains. Three interfaces can be constructed from the three phases realized by bare and decorated stanene: two distinct QSH-QVH interfaces where the spin Chern number changes from 1 to 0 and the valley Chern number changes from 0 to $\\pm2$, and a QVH-QVH interface where the spin Chern number is zero on both sides while the valley Chern number changes from $-2$ to $2$. As discussed above, the QVH-QVH interface is sensitive to the orientation of the interface relative to the valley separation, $\\mathbf{K}-\\mathbf{K}'$, so we consider only ``zigzag'' QVH-QVH interfaces for which the edge is perpendicular to the valley separation. The QSH-QVH interfaces are insensitive to the edge orientation because the change in $C_s$ does not depend on the valley degree of freedom, so we consider both zigzag and armchair interfaces for this case. \n\nNow let us consider the possible interface states. At QVH-QVH interfaces, the valley Chern number changes from $\\mp2$ to $\\pm2$, indicating that four gapless edge modes will appear. Each valley hosts two chiral modes, with the chirality determined by the valley such that a net momentum current will flow along the interface. At QSH-QVH interfaces, the spin Chern number changes by one, and the valley Chern number changes by two, producing a pair of oppositely propagating spin-valley polarized modes. The modes in each valley are of opposite spin and opposite chirality, resulting in both spin and momentum currents at the interface.\n\nTo determine the characteristics of these interface modes in decorated stanene, we performed large-scale first principles electronic structure simulations of stanene nanoribbons decorated to produce QVH-QSH and QVH-QVH interfaces. The translation-invariant direction points in the $\\vec{b}$ and $\\frac{1}{2}\\vec{a} + \\vec{b}$ directions to realize zigzag and armchair interfaces, respectively. The unit vectors $\\vec{a}$ and $\\vec{b}$ point along the primitive lattice vectors, as shown in Fig.~\\ref{fig:struc_BC}. To create topological interfaces, we selectively decorated domains in the transverse direction (the x-axis in Fig.~\\ref{fig:qsh_qvh}c, \\ref{fig:qsh_qvh}f, and \\ref{fig:qvh_qvh}c). We used periodic boundary conditions in the transverse direction to eliminate spurious interfaces with the vacuum, forming two topological interfaces per ribbon. All zigzag ribbons were 145.67~\\AA ~wide and the armchair ribbon was 149.64~\\AA  ~wide.\n\n To make the most of finite computational resources, the relative sizes of the decoration domains were chosen to minimize wavefunction overlap between the exponentially decaying interface states. Accordingly, the zigzag QSH-QVH ribbon is T-decorated on 10 unit cells and bare on 26 unit cells, the armchair QSH-QVH ribbon is T-decorated on 10 unit cells and bare on 22 unit cells, and the QVH-QVH ribbon is T-decorated on 10 unit cells and V-decorated on 26 unit cells. We note that the overlap of the interface wavefunctions in the bulk of the ribbon leads to undesired gaps in the interface spectrum produced by finite-size effects. We show in the supplementary material that these interface spectrum gaps vanish for sufficiently wide ribbons, and we find that the interface states decay exponentially into the insulating bulk with a decay length roughly determined by the ratio of the Fermi velocity to the bulk gap. \n\nThe resulting band structure and interface wavefunction plots are shown in Fig.~\\ref{fig:qsh_qvh} for the QSH-QVH zigzag and armchair ribbons. Each interface in the zigzag ribbon hosts a helical pair of spin-valley locked modes that produce both equilibrium spin and momentum currents on the interface. Each interface of the armchair ribbon also hosts a helical pair of modes, but since the valleys in this case are projected to the $\\Gamma$ point of the Brillouin zone the interfaces only carry spin current, not momentum current. The configurations of edge states we find agree with the $\\v{k}\\cdot\\v{p}$ model predictions of the previous section. As mentioned above, both ribbons have a small gap in the interface state spectrum, $E_g \\approx 0.03$ eV, that originates from the overlap and hybridization of the  wavefunctions on the two interfaces and would vanish for a larger system. The decay lengths of the interface states in the zigzag ribbon are $\\lambda_T=5.95$ \\AA{} and $\\lambda_0=32.4$ \\AA{} in the T-decorated and bare regions, respectively. The decay lengths in the armchair ribbon are $\\lambda_T=6.55$ \\AA{} and $\\lambda_0=35.7$ \\AA{}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{images\/figure3.pdf}\n    \\caption{\n    %\n    \\textbf{Interfacial edge modes from the QVH-QVH structure.}\n    %\n    The (a, b) band structure and (c) representative edge state probability densities of the QVH-QVH interface with a zigzag orientation. In (a, b), the thin gray and thick black lines represent the bulk and edge states, respectively. (b) Zoom in on the near-gap region at the $\\v{K}'$ point. The red and purple dots mark states propagating along the right interface and the blue and yellow dots mark edge states propagating along the left interface. The line plots in (c) represent the probability density integrated over the plane perpendicular to the width of the ribbon. The shapes of the probability density plots for each interface differ because the adatom decoration makes the two interfaces asymmetric. An in-plane view of the structure, with colored atoms corresponding to those in Fig.~\\ref{fig:struc_BC}, is shown in the bottom row of (c). Each valley contributes two unpolarized edge modes to each interface, as the indicated by the change of the valley Chern number by four across each interface.\n    %\n    }\n    \\label{fig:qvh_qvh}\n\\end{figure}\n\nThe results for the QVH-QVH ribbon are shown in the same format as the QSH-QVH ribbon in Fig.~\\ref{fig:qvh_qvh}. Each interface hosts two chiral modes from each valley, with the valleys contributing modes of opposite chirality. This leads to the equilibrium edge momentum current predicted above. In this case the decay lengths of the edge states are $\\lambda_T=5.68$ \\AA{} and $\\lambda_V=23.8$ \\AA{} in the T- and V-decorated regions, respectively. The gaps in the interface state spectrum resulting from wavefunction overlap are $E_g \\approx 0.02$ eV and and $0.006$ eV. We report two gaps here because there are two sets of interface states at each valley for this ribbon. For interfaces with a finite projection onto the valley separation direction, the momentum carried by the edge is reduced. In the extreme case of an armchair interface, the valleys exactly overlap, the edge carries no momentum current, and any local perturbation can gap out the interface states. \n    \n\\section{Technological Applications}\nThe above calculations demonstrate that decorating stanene with Zn adatoms presents a uniquely promising platform for technological applications. The topological domains can be patterned with a high degree of control by manipulating the adatom positions with an STM tip, permitting the fabrication of many topological devices, two of which are depicted schematically in Fig.~\\ref{fig:devices}. Furthermore, the interface states residing at domain walls are localized on the scale of tens of nanometers, which permits extremely dense packing of features. One of the first proposals for an application of topological edge modes was designer interconnect networks, which are a possible solution to the ``interconnect bottleneck'', wherein scattering and parasitic capacitance in interconnects leads to signal delays that prohibit further miniaturization of semiconductor devices~\\cite{george_chiral_2012}. The minimum metal pitch, or center to center distance between interconnects, of the current semiconductor manufacturing technology node is 24 nm to 36 nm~\\cite{ITRS}. At this scale, grain boundary and defect scattering leads to large resistances that inhibit the performance of traditional copper interconnects~\\cite{graham_resistivity_2010}. Considering the edge state decay lengths obtained in our simulations, the minimum pitch that could be achieved with Zn-decorated stanene interconnects is also on the order of tens of nanometers. However, the topological protection of the interface modes eliminates the issue of scattering, drastically improving performance with no compromise on feature size.\n\nThe interface modes of Zn-decorated stanene also have many applications beyond the world of conventional electronics. The fields of spin- and valley-tronics attempt to process information by exploiting the spin and valley degrees of freedom, rather than the charge degree of freedom~\\cite{bader_spintronics_2010, vitale_valleytronics_2018}. Topological interface modes are useful for engineering spin- and valleytronic devices such as waveguides, splitters, valves, and filters~\\cite{li_valley_2018, ezawa_topological_2013, xu_manipulating_2017,yang_topological_2020, qiao_spin-polarized_2011, jana_robust_2021}, and STM manipulation of Zn adatoms on stanene provides an ideal platform to fabricate the precise geometries of such devices. The same is true of electron quantum optics devices, such as valley Hall beam splitters, Mach-Zehnder interferometers, and Fabry-Perot resonators~\\cite{jo_quantum_2021, rickhaus_transport_2018, wei_mach-zehnder_2017}. This approach to engineering topological interface modes is also well suited to fabricating quantum computing gates out of helical edge states decorated with magnetic impurities~\\cite{niyazov_coherent_2020, chen_quantum_2014}, as STM manipulation of adatoms can be used to both create the edge states \\emph{and} deposit magnetic impurities.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{images\/figure4}\n    \\caption{\n    %\n    \\textbf{Schematic drawing of example devices.}\n    %\n    A schematic showing two possible devices constructed via adatom decoration of stanene. The blue spheres are Sn tin atoms and the red and yellow spheres are Zn adatoms located at T and V sites, respectively. The blue, red, and yellow shading is a guide to the eye, indicating the regions that are bare, T-decorated, and V-decorated, respectively. The red and yellow arrows indicate quantum spin Hall edge modes, the color determined by the decoration site of the quantum valley Hall region. The left side of the image shows two densely-packed chiral interconnects constructed by decorating a thin region of stanene with Zn adatoms. The right side of the image shows a Mach-Zehnder interferometer built out of the edge modes of two adjacent T- and V-decorated regions.\n    %\n    }\n    \\label{fig:devices}\n\\end{figure}\n\n\\section{Conclusion}\nWe have demonstrated that sublattice-selective decoration of stanene with Zn adatoms is an excellent platform for engineering topological interface modes. Because Zn adatoms bond relatively weakly to stanene, they act as an ideal sublattice potential and induce a QSH to QVH transition in stanene. The weak nature of the bond also does not transfer significant charge to stanene and permits STM manipulation of the adatoms allowing detailed patterning of topological interfaces. Importantly, the Zn-Sn bond is also strong enough for the decoration to remain stable at liquid nitrogen temperatures. The combined result of these effects is a platform suitable for fabricating arbitrary networks of topological interface modes without any of the deleterious effects that plague existing proposals for topological devices. These ideal topological interface-state networks have applications in semiconductor devices, spintronics, valleytronics, quantum electron optics, and even quantum computing. Implementing this technique is possible with existing fabrication and STM technology and can lead to transformative advances in topological device engineering.\n\n\\section*{Methods}\n    The investigation of all possible adatom species in rows two through five of the periodic table was completed using JDFTx~\\cite{sundararaman2017jdftx} to take advantage of GPU functionality. These calculations were carried out with ONCV pseudopotentials~\\cite{hamann2013optimized, van2018pseudodojo} using the Perdew-Burke-Enzerhof (PBE)~\\cite{pbe} exchange-correlation functional, a 1090 eV plane-wave energy cutoff, a 15x15x1 $\\Gamma$-centered k-mesh, and Methfessel-Paxton smearing of $\\sigma = 0.0272$ eV. \n    \n    The electronic structure of the decorated interface structures was determined via the Vienna Ab-initio Software Package (VASP)~\\cite{vasp1,vasp2,vasp3} using the PBE functional with the projector-augmented wave (PAW)~\\cite{paw_pseudopotentials} potentials provided by VASP. The calculations were performed on a 1x15x1 $\\Gamma$-centered k-mesh with a plane-wave energy cutoff of 450 eV and Methfessel-Paxton smearing of $\\sigma = 0.01$ eV.\n    \n    All calculations included spin-orbit coupling, 20 \\AA{} of z-axis vacuum between periodic images, and the many-body dispersion (MBD) van der Waals correction~\\cite{ambrosetti2014long}. Probability density data was visualized using the pawpyseed package~\\cite{pawpyseed}.\n\n\\nolinenumbers\n\n\\bibliographystyle{naturemag}\n\\footnotesize{","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The collisions of heavy ions at ultra relativistic energies are performed to create and\nstudy bulk strongly interacting matter at high temperatures. \nThe data from RHIC and LHC provide strong evidences of formation of a new state\nof matter known as quark gluon\nplasma (QGP) in these collisions \\cite{Proceedings:2019drx}.\n\nThe light charged hadrons and jets \ntransverse momentum ($p_{\\rm{T}}$) spectra give insight into the particle production\nmechanism in pp collisions. The partonic energy energy loss is reflected in these\nparticles when measured in heavy ion collisions \ndue to jet-quenching \\cite{Wang:2003aw} which measures the opacity of the medium.\nA modified power law distribution \\cite{Tsallis:1987eu, Biro:2008hz, Khandai:2013gva} describes\nthe $p_{\\rm{T}}$ spectra of the hadrons in pp collisions in terms of a\npower index $n$ which determines  the initial production in partonic\ncollisions. In Ref.\\cite{Saraswat:2017kpg}, the power law function is applied to heavy ion\ncollisions as well which includes the transverse\nflow in low $p_{\\rm{T}}$ region and the in-medium energy loss (also in terms of power law)\nin high $p_{\\rm{T}}$ region.\n\n\n\n  The spectra of hadrons are measured in both pp and AA collisions and\nnuclear modification factor ($R_{\\rm{AA}}$) is obtained.\nThe energy loss of partons can be connected to horizontal shift in the scaled hadron \nspectra in AA with respect to pp spectra as done by PHENIX measurement \\cite{Adler:2006bw}.\nTheir measurements of neutral pions upto $p_{\\rm T} \\sim 10$ GeV\/$c$ are consistent with the\nscenario where the momentum shift $\\Delta p_{\\rm T}$ is proportional to $p_{\\rm T}$.\n  In similar approach, the authors in Ref.~\\cite{Wang:2008se}, extracted the fractional energy loss\nfrom the measured nuclear modification factor  of hadrons as a function of $p_{\\rm{T}}$ below\n10 GeV\/$c$ in AuAu collisions at $\\sqrt{s_{\\rm{NN}}}$ = 200 GeV. They also considered that \nthe energy loss increases linearly with $p_{\\rm T}$.\n  In recent PHENIX work \\cite{Adare:2015cua},\nfractional energy loss was obtained in the hadron spectrum measured upto $p_{\\rm T}=20$ GeV\/$c$\nin heavy ions collisions at RHIC and LHC energy and is not found to be constant.\nThis means that a constant fractional energy loss (energy loss varying linearly with $p_{\\rm T}$)\ncan be applicable only to low $p_{\\rm T}$ RHIC measurements.\n\n There are many recent studies which use so-called shift formalism to study the energy loss.\nThe work in Ref.\\cite{Spousta:2015fca} is based on \nshift formalism and  describes the transverse momentum ($p_{\\rm{T}}$), rapidity ($y$)\nand centrality dependences of the measured jet nuclear modification factor ($R_{\\rm{AA}}$)\nin PbPb collisions at LHC.\nThey assume that the energy loss is given by a power law in terms of $p_{\\rm T}$, the value of power\nindex is obtained between 0.4 to 0.8 by fitting the $R_{\\rm{AA}}$ as a function of $p_{\\rm{T}}$\nand centralities.\n They also found that the energy loss linearly increases with number of participants. \nUsing the same method they study the magnitude and the colour charge dependence of the\nenergy loss in PbPb collisions at LHC energies using the measured data of the inclusive\njet suppression~\\cite{Spousta:2016agr}.\n The authors of the Ref.\\cite{Ortiz:2017cul} work on inclusive charged particle spectra\nmeasured in the range ($5 < p_{\\rm T} < 20$ GeV\/$c$) in heavy ion collisions at RHIC and LHC.\nThey assume that the energy loss linearly increases with $p_{\\rm T}$ and pathlength. \n\n\n\n\n\n\nThere are detailed calculations of energy loss of partons in the hot medium\n[see e.g. Refs.~\\cite{Wang:1994fx,Baier:1996kr}.\n Phenomenological models tend to define simple dependence of the radiative energy\nloss of the parton on the energy of the parton inside the medium [for a discussions\nsee Ref.~\\cite{Baier:2000mf}]. The energy loss can be characterized in terms of \ncoherence length $l_{\\rm{coh}}$, which is associated with the formation time of\ngluon radiation by a group of scattering centres. If $l_{\\rm{coh}}$ is less then \nthe mean free path $(\\lambda)$ of the parton, the energy loss is proportional to the\nenergy of the parton.\nIf $l_{\\rm{coh}}$ is greater than $\\lambda$ but less than the path length ($L$) of the\nparton ($\\lambda < l_{\\rm{coh}} < L)$, the energy loss is proportional to the square\nroot of the energy of the parton.\nIn the complete coherent regime, $l_{\\rm{coh}} > L$, the energy loss per unit length\nis independent on energy but proportional to the parton path length implying that\n$\\Delta p_T$ is proportional to square of pathlength.\n There is a nice description of charged particle spectra at RHIC and LHC using such a\nprescription by dividing the $p_{\\rm T}$ spectra in three regions \\cite{De:2011fe, De:2011aa}.\n  For low and intermediate energy partons, $\\Delta p_T$ is assumed to be linearly\ndependent on $L$ \\cite{Muller:2002fa}. The work in Ref.~\\cite{Betz:2011tu} studies \nthe energy loss of jets in terms of exponent of the number of participants.\n It should be remembered that the fragmentation changes the momentum between the partonic\nstage (at which energy is lost) and hadron formation.\n There are models which say that softening occurs at fragmentation stage \ndue to color dynamics [See e.g. Ref.~\\cite{Beraudo:2012bq}].\n \n\n In general, one can assume that the energy loss of partons in the hot medium as a function of\nparton energy is in the form of power law where the power index ranges from 0 (constant) to\n1 (linear). Guided by these considerations, in the present work, the $p_{\\rm T}$ loss has been assumed\nas power law with different power indices in three different $p_{\\rm T}$ regions.\nThe energy loss in different collisions centralities are described in terms of fractional power\nof number of participants.\n  The $p_{\\rm T}$ distributions in pp collisions are fitted with a modified power law and\n$R_{\\rm AA}$ in PbPb collisions can be obtained using effective shift ($\\Delta p_{\\rm T}$) in the\n$p_{\\rm T}$ spectrum measured at different centralities.\n The power index and the boundaries of three $p_{\\rm T}$ regions are obtained by fitting the\nmeasured $R_{\\rm{AA}}$ of charged particles and jets in PbPb collisions at\n$\\sqrt{s_{\\rm NN}}$ =  2.76 and 5.02 TeV in large transverse momentum ($p_{\\rm T}$) and\ncentrality range.\nThe shift $\\Delta p_{\\rm{T}}$ includes the medium effect, mainly energy loss of parent\nquark inside the plasma.\n The shift $\\Delta p_T$ can be approximatively understood as the partonic energy loss in the\ncase of jets while in case of hadrons it is not simple due to complicated correlations.\nOften we refer to the shift $\\Delta p_{\\rm{T}}$ as the energy loss.\n\n\n\n\\section{Nuclear Modification Factor and Energy Loss}\n\nThe nuclear modification factor $R_{\\rm{AA}}$ of hadrons is defined as the ratio\nof yield of the hadrons in AA collision and the yield in pp collision with a\nsuitable normalization\n\\begin{equation}\nR_{\\rm{AA}} (p_{\\rm{T}}, b) = \\frac{1}{T_{\\rm{AA}}} {\\frac{d^2N^{AA}(p_{\\rm{T}}, b)}{dp_{\\rm{T}}dy}}\/\n{\\frac{d^2\\sigma^{pp}(p_{\\rm{T}}, b)}{dp_{\\rm{T}}dy}}~.\n\\label{raa_definition}\n\\end{equation}\nHere,  $T_{\\rm{AA}}$ is the nuclear overlap function which can be calculated from the\nnuclear density distribution. High $p_{\\rm T}$ partons traversing the medium loose energy and\ncause the suppression of hadrons at high $p_{\\rm T}$ indicated by value of $R_{\\rm{AA}}$\nwhich is less than one.\n The transverse momentum distribution of hadrons in pp collisions\ncan be described by the Hagedorn function which is a  QCD-inspired summed power\nlaw \\cite{Hagedorn:1983wk} given as\n\\begin{equation}\n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,   \\frac{d^2\\sigma^{\\rm{pp}}}{dp_{\\rm{T}}dy}\n= A_n~2\\pi p_{\\rm{T}} ~\\Bigg(1 + \\frac{p_{\\rm{T}}}{p_{0}}\\Bigg)^{-n}~.\n\\label{Hag}\n\\end{equation}\nwhere $n$ is the power and $A_n$ and $p_{0}$ are other parameters which are obtained\nby fitting the experimental pp spectrum.\n  The yield in the AA collision can be obtained by shifting the spectrum by \n$\\Delta p_{\\rm T}$ as\n\\begin{eqnarray}\n \\,\\,\\,\\,\\,  \\frac{1}{T_{\\rm{AA}}}\\frac{d^{2}N^{\\rm{AA}}}{dp_{\\rm{T}}dy}\n   &  = \\frac{d^{2}\\sigma^{\\rm{pp}}(p'_{\\rm{T}} = p_{\\rm{T}} +  \\Delta p_{\\rm{T}})}{dp'_{\\rm{T}}dy}\n  \\frac{dp'_{\\rm{T}}}{dp_{\\rm{T}}}  \\nonumber \\\\\n  & = \\frac{d^{2}\\sigma^{\\rm{pp}}(p'_{\\rm{T}})}{dp'_{\\rm{T}}dy}\n  \\Bigg(1 + \\frac{d(\\Delta p_{\\rm{T}})}{dp_{\\rm{T}}}\\Bigg)~.\n\\label{shiftRAA}\n\\end{eqnarray}\nThe reasoning behind writing Eq.~\\ref{shiftRAA} lies in the assumption that particle yield at\na given $p_{\\rm{T}}$ in AA collisions would have been equal to the yield in pp collisions\nat $p_{\\rm{T}} + \\Delta p_{\\rm{T}}$. The shift $\\Delta p_{\\rm{T}}$ includes the medium effect,\nmainly energy loss of parent quark inside the plasma.\n\nThe nuclear modification factor $R_{\\rm{AA}}$ can be obtained as \n\\begin{eqnarray}\nR_{\\rm{AA}} = \\left(1 + { \\Delta p_{\\rm{T}} \\over p_{0}+p_{\\rm{T}} } \\right)^{-n} \\,\\,\n\\left({p_{\\rm{T}} + \\Delta p_{\\rm{T}} \\over  p_{\\rm{T}}}\\right) \\, \n\\left(1 + {d(\\Delta p_{\\rm{T}}) \\over dp_{\\rm{T}}}\\right)\n\\label{nmf_raa_fitting_function}\n\\end{eqnarray}\nThe energy loss given by $p_{T}$ loss, $\\Delta p_{T}$ can be extracted by fitting the\nexperimental data on $R_{\\rm AA}$\nwith Eq.~\\ref{nmf_raa_fitting_function}.\nThe $\\Delta p_{\\rm T}$ and its derivative will go as input in the above equation\nand can be assumed to be in the form of the power law\nwith different values of power indices in three different $p_{\\rm T}$ regions as follows\n\n\n\n\\begin{eqnarray}\n\\Delta p_{\\rm T}  =   \\left\\{\n\\begin{array}{l}\na_1~(p_{\\rm T} - C_1)^{\\alpha_1} ~~~ {\\rm for} ~~~ p_{\\rm T} < p_{\\rm T_1}~~~, \\\\\na_2~(p_{\\rm T} - C_2)^{\\alpha_2} ~~~ {\\rm for} ~~~ p_{\\rm T_1} \\leq p_{\\rm T} < p_{\\rm T_2}~~,\\\\\na_3~(p_{\\rm T} - C_3)^{\\alpha_3} ~~~ {\\rm for} ~~~ p_{\\rm T} \\geq p_{\\rm T_2}~.\n\\end{array}\n\\right\\}\n\\label{Equation_Two}\n\\end{eqnarray}\n\n\nThe parameter $a_1$ in our work contains the pathlength dependence. The pathlength $L$\n scales as the square root of number of participants as $\\sqrt{N_{\\rm part}}$.\n For low and intermediate energy partons, $\\Delta p_T$ can be assumed to be\nlinearly dependent on $L$ \\cite{Muller:2002fa}. If the scattering happens\nin complete coherent regime where the whole medium acts as one coherent source of radiation, \nthe $\\Delta p_T$ approaches quadratic dependence on $L$.\n The work in Ref.~\\cite{Betz:2011tu} studies \nthe energy loss of jets in terms of exponent of the number of participants.\nWithout complicating the calculations we can assume that\n$a_1 = M \\, (N_{\\rm{part}}\/(2A))^\\beta$. The exponent $\\beta$ is obtained separately\nfor each dataset. \n The parameter $M$ relies on the energy density of the medium depending on the\ncollision energy but has a same value for all centralities.\nThe boundaries of the $p_{\\rm T}$ regions $p_{{\\rm{T}}_{1}}$, $p_{{\\rm{T}}_{2}}$ and the  power\nindices $\\alpha_{1}$, $\\alpha_{2}$ and $\\alpha_{3}$ in the three different regions are\nused as free parameters while fitting the $R_{\\rm{AA}}$ measured at different centralities\nsimultaneously.\n The parameter $C_{1}$ is fixed to a suitable value to choose a lower $p_{\\rm T}$ cutoff\n and the parameters $C_{2}$, $C_{3}$, $a_2$ and $a_3$ are obtained by assuming the function\n and its derivative to be continuous at boundaries.\n\nDemanding that the function in Eq.~\\ref{Equation_Two} to be continuous\nat $p_{\\rm{T}} = p_{{\\rm{T}}_{1}}$ and at $p_{\\rm{T}} = p_{{\\rm{T}}_{2}}$ we obtain\n\\begin{equation}\n  a_{2} = a_{1}~ \\frac{(p_{{\\rm{T}}_{1}} - C_{1})^{\\alpha_{1}}}{(p_{{\\rm{T}}_{1}} - C_{2})^{\\alpha_{2}}}~~.\n\\label{Equation_Three}\n\\end{equation}\n\n\\begin{equation}\n  a_{3} = a_{2}~ \\frac{(p_{{\\rm{T}}_{2}} - C_{2})^{\\alpha_{2}}}{(p_{{\\rm{T}}_{2}} - C_{3})^{\\alpha_{3}}}~~.\n\\label{Equation_Four}\n\\end{equation}\n Demanding that at $p_{\\rm{T}} = p_{{\\rm{T}}_{1}}$, the derivative of Eq.~\\ref{Equation_Two} is  continuous.\n\\begin{equation}\n a_{1}~\\alpha_{1}~(p_{{\\rm{T}}_{1}} - C_{1})^{(\\alpha_{1}-1)} = \n a_{2}~\\alpha_{2}~(p_{{\\rm{T}}_{1}} - C_{2})^{(\\alpha_{2}-1)}~~,\n\\label{Equation_Seven}\n\\end{equation}\nUsing the value of $a_{2}$ from Eq.~\\ref{Equation_Three}\n\\begin{equation}\n\\frac{\\alpha_{1}}{(p_{{\\rm{T}}_{1}} - C_{1})} = \\frac{\\alpha_{2}}{(p_{{\\rm{T}}_{1}} - C_{2})}~~,\n\\label{Equation_Eight}\n\\end{equation}\n\n\\begin{equation}\nC_{2} = p_{{\\rm{T}}_{1}} - \\frac{\\alpha_{2}}{\\alpha_{1}}(p_{{\\rm{T}}_{1}} -C_{1})~.\n\\label{Equation_Nine}\n\\end{equation}\nSimilarly, demanding the derivative to be continuous at\n$p_{\\rm{T}} = p_{{\\rm{T}}_{2}}$, we get $C_{3}$ \n\\begin{equation}\nC_{3} = p_{{\\rm{T}}_{2}} - \\frac{\\alpha_{3}}{\\alpha_{2}}(p_{{\\rm{T}}_{2}} -C_{2})~.\n\\label{Equation_Ten}\n\\end{equation}\nIn case of jets we consider only one region as the data starts from very high $p_T$\nabove 40 GeV\/$c$.\n\n\n\\section{Results and Discussions}\n\n\nFigure~\\ref{Figure1_charged_particles_pT_ALICE_spectra_Tsallis_fit_pp_276TeV} shows the\ninvariant yields of the charged particles as a function of the transverse momentum $p_{\\rm{T}}$\nfor pp collisions at $\\sqrt{s}$ = 2.76 TeV measured by the ALICE\nexperiment \\cite{Abelev:2013ala}. The solid curve is the Hagedorn distribution fitted\nto the $p_{\\rm{T}}$ spectra with the parameters given in \nTable~\\ref{table0_charged_particles_jet_pT_spectra_tsallis_fitting_parameters_276_502_TeV}. \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure1_charged_particles_pT_ALICE_spectra_Tsallis_fit_pp_276TeV.pdf}\n\\caption{The invariant yields of the charged particles as a function of transverse momentum \n$p_{\\rm{T}}$ for pp collision at $\\sqrt{s}$ = 2.76 TeV measured by the ALICE experiment\n \\cite{Abelev:2013ala}. The solid curve is the fitted Hagedorn function.}\n\\label{Figure1_charged_particles_pT_ALICE_spectra_Tsallis_fit_pp_276TeV}\n\\end{figure}\n\n\\begin{table}[ht]\n  \\caption{Parameters for the Hagedorn function obtained by fitting the\ntransverse momentum spectra of charged particles and jets measured in pp \ncollisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 and 5.02 TeV.}\n\\begin{center}\n\\scalebox{0.8}{\n\\begin{tabular}{| c || c | c | c | c |} \n\\hline\nParameters  & \\multicolumn{2}{c|}{Charged particles}\n                         & \\multicolumn{2}{c|}{Jets } \\\\ \\hline\n            & $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV & $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV \n                        & $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV & $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV  \\\\  \\hline \\hline\n  $n$                & 7.26  $\\pm$ 0.08 & 6.70  $\\pm$ 0.14     & 8.21  $\\pm$ 1.55   & 7.90 $\\pm$ 0.50  \\\\\\hline\n  $p_{0}$ (GeV\/$c$)  & 1.02  $\\pm$ 0.04  & 0.86  $\\pm$ 0.16     & 18.23  $\\pm$ 1.69  & 19.21 $\\pm$ 3.20 \\\\\\hline\n$\\chi^{2}\/\\rm{NDF}$ & 0.15 & 0.06 & 0.23 & 0.95  \\\\\\hline  \n\\end{tabular}}\n\\end{center}\n\\label{table0_charged_particles_jet_pT_spectra_tsallis_fitting_parameters_276_502_TeV}\n\\end{table}\n\n\n\nFigure~\\ref{Figure2_charged_particles_RAA_ALICE_spectra_com_fit_PbPb_276TeV} shows the nuclear\nmodification factor $R_{\\rm{AA}}$ of the charged particles as a function of the transverse\nmomentum $p_{\\rm{T}}$ for different centrality classes in PbPb collisions at $\\sqrt{s_{\\rm{NN}}}$\n= 2.76 TeV measured by the ALICE experiment \\cite{Abelev:2012hxa}. The solid lines are the\nfunction given by Eq.~\\ref{nmf_raa_fitting_function}. The modeling of centrality dependence using\n$N_{\\rm part}^\\beta$ with $\\beta=0.58$ gives a very good description of the data. \n The extracted parameters of the shift $\\Delta p_{\\rm{T}}$ \nobtained by fitting the $R_{\\rm{AA}}$ measured in different centrality classes of PbPb\ncollisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV are given in\nTable~\\ref{table1_charged_particles_raa_fitting_parameter_276_502_TeV}\nalong with value of $\\chi^{2}\/\\rm{NDF}$. It shows that the $\\Delta p_{\\rm{T}}$ increases\nalmost linearly ($p_{\\rm{T}}^{0.97}$) upto $p_{\\rm{T}} \\simeq$  5 GeV\/$c$ in confirmation with\nearlier studies. After that it increases slowly with power $\\alpha=0.224$ upto\na $p_{\\rm{T}}$ value 29 GeV\/$c$ and then\nbecomes constant for higher values of $p_{\\rm{T}}$.\n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Figure2_charged_particles_RAA_ALICE_spectra_com_fit_PbPb_276TeV.pdf}\n\\caption{The nuclear modification factor $R_{\\rm{AA}}$  of the charged particles as a \nfunction of transverse momentum $p_{\\rm{T}}$ for various centrality classes in PbPb \ncollisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV measured by the ALICE experiment \\cite{Abelev:2012hxa}.\n The solid curves are the $R_{\\rm{AA}}$ fitting function\n(Eq.~\\ref{nmf_raa_fitting_function}).} \n\\label{Figure2_charged_particles_RAA_ALICE_spectra_com_fit_PbPb_276TeV}\n\\end{figure}\n\n\n\\begin{table}[ht]\n  \\caption[]{The extracted parameters of the shift $\\Delta p_{\\rm{T}}$  obtained by fitting the charged\n    particles $R_{\\rm{AA}}$\nmeasured in different centrality classes of PbPb collisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 and 5.02 TeV.}\n\\label{table1_charged_particles_raa_fitting_parameter_276_502_TeV}\n\\begin{center}\n\\begin{tabular}{| c || c | c |} \\hline\n  ~ Parameters & $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV   & $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV  \\\\ \\hline\\hline\n~ $M$                        &  0.75  $\\pm$  0.02   &   0.80   $\\pm$ 0.038    \\\\ \\hline      \n~ $p_{{\\rm{T}}_{1}}$ (GeV\/$c$)     &  5.03  $\\pm$  0.15    &   5.10  $\\pm$ 0.22     \\\\ \\hline   \n~ $p_{{\\rm{T}}_{2}}$ (GeV\/$c$)     &  29.0   $\\pm$  0.1    &   22.2 $\\pm$ 4.1      \\\\ \\hline   \n~ $C_{1}$         (GeV\/$c$)     &   1.0 (fixed)            &   1.0                   \\\\ \\hline  \n~ $\\alpha_{1}$                &   0.97  $\\pm$  0.02    &   0.95   $\\pm$ 0.04     \\\\ \\hline  \n~ $\\alpha_{2}$                &   0.22  $\\pm$  0.02    &   0.22  $\\pm$ 0.03     \\\\ \\hline    \n~ $\\alpha_{3}$                &   0.05   $\\pm$  0.13     &   0.05   $\\pm$ 0.10     \\\\ \\hline     \n~ $\\frac{\\chi^{2}}{\\rm{NDF}}$ &   0.35                  &   0.38                  \\\\ \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n Figure~\\ref{Figure3_charged_particles_com_fit_Del_pT_PbPb_276TeV} shows the energy loss\n$\\Delta p_{\\rm{T}}$ of the charged particles as a function of the transverse momentum\n$p_{\\rm{T}}$ for different centrality classes in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$\n= 2.76 TeV. The $\\Delta p_{\\rm{T}}$ is obtained from Eq.~\\ref{Equation_Two}\nwith parameters given in \nTable~\\ref{table1_charged_particles_raa_fitting_parameter_276_502_TeV}.\n The $\\Delta p_{\\rm{T}}$ increases from peripheral to the most\n central collision regions as per $N_{\\rm part}^{0.58}$.\n  The figure shows that the $\\Delta p_{\\rm{T}}$ increases \nalmost linearly upto $p_{\\rm{T}} \\sim 5$ GeV\/$c$. After that it\nincreases slowly upto a $p_{\\rm{T}}$ value 29 GeV\/$c$ and then\nbecomes constant for higher values of $p_{\\rm{T}}$.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure3_charged_particles_com_fit_Del_pT_PbPb_276TeV.pdf}\n\\caption{The energy loss $\\Delta p_{\\rm{T}}$ of the charged particles as a function of \ntransverse momentum $p_{\\rm{T}}$ in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV for \ndifferent centrality classes.}\n\\label{Figure3_charged_particles_com_fit_Del_pT_PbPb_276TeV}\n\\end{figure}\n\n\n Figure~\\ref{Figure4_charged_particles_cms_pT_spectra_pp_502TeV} shows the invariant\nyields of the charged particles as a function of the transverse momentum $p_{\\rm{T}}$\nfor pp collisions at $\\sqrt{s}$ = 5.02 TeV measured by the CMS experiment\n\\cite{Khachatryan:2016odn}. The solid lines are the Hagedorn function fitted to\nthe measured $p_{\\rm{T}}$ spectra the parameters of which are given in \nTable~\\ref{table0_charged_particles_jet_pT_spectra_tsallis_fitting_parameters_276_502_TeV}. \n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure4_charged_particles_cms_pT_spectra_pp_502TeV.pdf}\n\\caption{The invariant yields of the charged particles as a function of transverse momentum \n$p_{\\rm{T}}$ for pp collision at $\\sqrt{s}$ = 5.02 TeV measured by the CMS experiment \n\\cite{Khachatryan:2016odn}. The solid curve is the fitted Hagedorn function.}\n\\label{Figure4_charged_particles_cms_pT_spectra_pp_502TeV}\n\\end{figure}\n\n\nFigure~\\ref{Figure5_charged_particles_cms_RAA_spectra_com_fit_PbPb_502TeV} shows the\nnuclear modification factor $R_{\\rm{AA}}$ of the charged particles as a function of the\ntransverse momentum $p_{\\rm{T}}$ for different centrality classes in PbPb collisions at\n$\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV measured by the CMS experiment \\cite{Khachatryan:2016odn}.\nThe solid curves are the $R_{\\rm{AA}}$ fitting function (Eq.~\\ref{nmf_raa_fitting_function}).\n Here also the modeling of centrality dependence using \n$N_{\\rm part}^\\beta$ with $\\beta=0.58$ gives a good description of the data. \n The extracted parameters of the shift $\\Delta p_{\\rm{T}}$ \nobtained by fitting the $R_{\\rm{AA}}$ measured in different centrality classes of PbPb\ncollisions at $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV are given in\nTable~\\ref{table1_charged_particles_raa_fitting_parameter_276_502_TeV}\nalong with the value of $\\chi^{2}\/\\rm{NDF}$. It shows that the $\\Delta p_{\\rm{T}}$ increases\nalmost linearly ($p_{\\rm{T}}^{0.96}$) similar to the case at 2.76 TeV for $p_{\\rm T}$ upto 5.1 GeV\/$c$.\nAfter that it increases slowly with power $\\alpha=0.22$ upto a $p_{\\rm{T}}$ value 22.2 GeV\/$c$ and then\nbecomes constant for higher values of $p_{\\rm{T}}$ right upto 160 GeV\/$c$.\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{Figure5_charged_particles_cms_RAA_spectra_com_fit_PbPb_502TeV.pdf}\n\\caption{The nuclear modification factor $R_{\\rm{AA}}$  of the charged particles as a function \nof transverse momentum $p_{\\rm{T}}$ for various centrality classes in PbPb collisions at \n$\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV measured by the CMS  experiment \\cite{Khachatryan:2016odn}. The\nsolid lines are the $R_{\\rm{AA}}$ fitting function (Eq.~\\ref{nmf_raa_fitting_function}).}\n\\label{Figure5_charged_particles_cms_RAA_spectra_com_fit_PbPb_502TeV}\n\\end{figure}\n \n\nFigure~\\ref{Figure6_charged_particles_com_fit_Del_pT_PbPb_502TeV} shows the energy loss\n$\\Delta p_{\\rm{T}}$ of the charged particles as a function of the transverse momentum\n$p_{\\rm{T}}$ for different centrality classes in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ =\n5.02 TeV. The $\\Delta p_{\\rm{T}}$ is obtained from Eq.~\\ref{Equation_Two} with the\nparameters given in \nTable~\\ref{table1_charged_particles_raa_fitting_parameter_276_502_TeV}.\nThe $\\Delta p_{\\rm{T}}$ becomes constant for $p_{\\rm{T}}$ in the range 22 GeV\/$c$ to 160 GeV\/c\nand increases as the collisions become more central.\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure6_charged_particles_com_fit_Del_pT_PbPb_502TeV.pdf}\n\\caption{The energy loss $\\Delta p_{\\rm{T}}$ of the charged particles as a function of \ntransverse momentum $p_{\\rm{T}}$ in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV for \ndifferent centrality classes.}\n\\label{Figure6_charged_particles_com_fit_Del_pT_PbPb_502TeV}\n\\end{figure}\n\n\nFigure~\\ref{Figure7_charged_particles_Del_pT_cen_0_5_PbPb_276_502TeV} shows the comparison of energy\nloss $\\Delta p_{\\rm{T}}$ of the charged particles as a function of the transverse momentum\n$p_{\\rm{T}}$ for 0 - 5 $\\%$ centrality class in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ =\n2.76 and at 5.02 TeV. The $\\Delta p_{\\rm{T}}$ at 5.02 TeV is similar but slightly more than\nthat at 2.76 TeV. \n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure7_charged_particles_Del_pT_cen_0_5_PbPb_276_502TeV.pdf}\n\\caption{The energy loss $\\Delta p_{\\rm{T}}$ of the charged particles as a function \nof transverse momentum $p_{\\rm{T}}$ in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 \nand 5.02 TeV for 0 - 5 $\\%$ centrality.}\n\\label{Figure7_charged_particles_Del_pT_cen_0_5_PbPb_276_502TeV}\n\\end{figure}\n\n\nFigure~\\ref{Figure8_jet_atlas_pT_spectra_pp_276TeV} shows the yields of the\njets as a function of the transverse momentum $p_{\\rm{T}}$ for pp collisions at $\\sqrt{s}$\n= 2.76 TeV measured by the ATLAS experiment~\\cite{Aad:2014bxa}. The solid curve is the\nHagedorn distribution fitted to the $p_{\\rm{T}}$ spectra, the parameters of which\nare given in \nTable~\\ref{table0_charged_particles_jet_pT_spectra_tsallis_fitting_parameters_276_502_TeV}. \n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure8_jet_atlas_pT_spectra_pp_276TeV.pdf}\n\\caption{The yields of the jets as a function of transverse momentum \n$p_{\\rm{T}}$ for pp collision at $\\sqrt{s}$ = 2.76 TeV measured by the ATLAS experiment \n  \\cite{Aad:2014bxa}. The solid curve is the fitted Hagedorn distribution.}\n\\label{Figure8_jet_atlas_pT_spectra_pp_276TeV}\n\\end{figure}\n\n\nFigure~\\ref{Figure9_Jet_particles_cms_RAA_spectra_com_fit_PbPb_276TeV} shows the nuclear\nmodification factor $R_{\\rm{AA}}$ of the jets as a function of the transverse\nmomentum $p_{\\rm{T}}$ for different centrality classes in PbPb collisions at $\\sqrt{s_{\\rm{NN}}}$\n= 2.76 TeV measured by the ATLAS experiment \\cite{Aad:2014bxa}. \nThe solid curves are the $R_{\\rm{AA}}$ fitting function (Eq.~\\ref{nmf_raa_fitting_function}).\n Here the modeling of centrality dependence using $N_{\\rm part}^\\beta$ with $\\beta=0.60$\ngives a good description of the data. \n  The extracted parameters of the energy loss\nobtained by fitting the $R_{\\rm{AA}}$ measured in different centrality classes of PbPb\ncollisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV are given in\nTable~\\ref{Table2_Jet_raa_fitting_parameter_PbPb_276_502_TeV}\nalong with the value of $\\chi^{2}\/\\rm{NDF}$.\nIt shows that the $\\Delta p_{\\rm{T}}$ increases as $p_{\\rm{T}}^{0.76}$\nat all the values of $p_{\\rm T}$ measured for jets.\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{Figure9_Jet_particles_cms_RAA_spectra_com_fit_PbPb_276TeV.pdf}\n\\caption{The nuclear modification factor $R_{\\rm{AA}}$  of jets as a function of\ntransverse momentum $p_{\\rm{T}}$ for various centrality classes in PbPb collisions at \n$\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV measured by the ATLAS experiment \\cite{Aad:2014bxa}. The solid\ncurves are the $R_{\\rm{AA}}$ fitting function given by Eq.~\\ref{nmf_raa_fitting_function}.}\n\\label{Figure9_Jet_particles_cms_RAA_spectra_com_fit_PbPb_276TeV}\n\\end{figure}\n\n\n\n\\begin{table}[ht]\n  \\caption[]{The extracted parameters of the shift $\\Delta p_{\\rm{T}}$  obtained by fitting the jet $R_{\\rm{AA}}$\n    measured in different centrality classes of PbPb collisions at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 and\n    5.02 TeV.}\n\\label{Table2_Jet_raa_fitting_parameter_PbPb_276_502_TeV}\n\\begin{center}\n\\begin{tabular}{| c || c | c |} \\hline\n~ Parameters~  & $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV  & $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV \\\\ \\hline\\hline\n~ $M$          &  0.33   $\\pm$  0.1     &  0.40   $\\pm$  0.12     \\\\ \\hline \n~ $C$  (GeV\/$c$) &  -55.1  $\\pm$ 22.7      &  -119   $\\pm$  15        \\\\ \\hline \n~ $\\alpha$     &  0.76   $\\pm$  0.08    &  0.72  $\\pm$  0.01     \\\\ \\hline \n~ $\\frac{\\chi^{2}}{\\rm{NDF}}$  &  0.30    &  0.25                   \\\\ \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nFigure~\\ref{Figure10_Jet_particles_Del_pT_PbPb_276TeV} shows the shift $\\Delta p_{\\rm{T}}$\nof the jets as a function of the transverse momentum $p_{\\rm{T}}$\nfor different centrality classes in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV.\nThe $\\Delta p_{\\rm{T}}$ is obtained from Eq.~\\ref{Equation_Two} using the \nparameters given in Table~\\ref{Table2_Jet_raa_fitting_parameter_PbPb_276_502_TeV}.\nThe $\\Delta p_{\\rm{T}}$ increases from \nperipheral to the most central collision regions.\n The figure shows that the $\\Delta p_{\\rm{T}}$ increases almost linearly \nat all the values of $p_{\\rm T}$ measured for jets.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure10_Jet_particles_Del_pT_PbPb_276TeV.pdf}\n\\caption{The shift  $\\Delta p_{\\rm{T}}$ of the jets as a function of transverse \nmomentum $p_{\\rm{T}}$ in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 TeV for different \ncentrality classes.}\n\\label{Figure10_Jet_particles_Del_pT_PbPb_276TeV}\n\\end{figure}\n\n\n\nFigure~\\ref{Figure11_jet_yield_pp_502tev} shows the yields of the jets\nas a function of the transverse momentum $p_{\\rm{T}}$ for pp collisions at $\\sqrt{s}$\n= 5.02 TeV measured by the ATLAS experiment \\cite{Aaboud:2018twu}. The solid curve\nis the Hagedorn distribution with the parameters given in \nTable~\\ref{table0_charged_particles_jet_pT_spectra_tsallis_fitting_parameters_276_502_TeV}. \n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.60\\linewidth]{Figure11_jet_yield_pp_502tev.pdf}\n\\caption{The yields of the jets as a function of transverse \nmomentum $p_{\\rm{T}}$ for pp collision at $\\sqrt{s}$ = 5.02 TeV measured by the \nATLAS experiment \\cite{Aaboud:2018twu}. The solid curve is the fitted Hagedorn \ndistribution.}\n\\label{Figure11_jet_yield_pp_502tev}\n\\end{figure}\n\n\n\nFigure~\\ref{Figure12_Jet_particles_cms_RAA_spectra_com_fit_PbPb_502TeV} shows the\nnuclear modification factor $R_{\\rm{AA}}$ of the jets as a function of the\ntransverse momentum $p_{\\rm{T}}$ for different centrality classes in PbPb collisions\nat $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment \\cite{Aaboud:2018twu}.\n The solid curves are the $R_{\\rm{AA}}$ fitting function (Eq.~\\ref{nmf_raa_fitting_function}).\n The modeling of centrality dependence is done with $N_{\\rm part}^\\beta$ and the\nvalue of exponent is obtained as $\\beta=0.75$. \n  The extracted parameters of the energy loss\nobtained by fitting the $R_{\\rm{AA}}$ measured in different centrality classes of PbPb\ncollisions at $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV are given in\nTable~\\ref{Table2_Jet_raa_fitting_parameter_PbPb_276_502_TeV},\nalong with the value of $\\chi^{2}\/\\rm{NDF}$.\nIt shows that the $\\Delta p_{\\rm{T}}$ increases as $p_{\\rm{T}}^{0.72}$\nat all the values of $p_{\\rm T}$ measured for jets similar to the case of jets at \n$\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.85\\linewidth]{Figure12_Jet_particles_cms_RAA_spectra_com_fit_PbPb_502TeV.pdf}\n\\caption{The nuclear modification factor $R_{\\rm{AA}}$  of the jets as a function\nof transverse momentum $p_{\\rm{T}}$ for various centrality classes in PbPb collisions at\n$\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment \\cite{Aaboud:2018twu}. \nThe solid curves are the $R_{\\rm{AA}}$ fitting function given by Eq.~\\ref{nmf_raa_fitting_function}.}\n\\label{Figure12_Jet_particles_cms_RAA_spectra_com_fit_PbPb_502TeV}\n\\end{figure}\n\n\n\nFigure~\\ref{Figure13_Jet_particles_Del_pT_PbPb_502TeV} shows the energy loss\n$\\Delta p_{\\rm{T}}$ of the jets as a function of the transverse momentum $p_{\\rm{T}}$\nfor different centrality classes in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV.\nThe $\\Delta p_{\\rm{T}}$ is obtained from  Eq.~\\ref{Equation_Two} with the \nparameters given in Table~\\ref{Table2_Jet_raa_fitting_parameter_PbPb_276_502_TeV}.\nThe $\\Delta p_{\\rm{T}}$ increases from peripheral to the most central collision regions.\nThe figure shows that the $\\Delta p_{\\rm{T}}$ increases almost linearly\nat all the values of $p_{\\rm T}$ measured for jets.\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.61\\linewidth]{Figure13_Jet_particles_Del_pT_PbPb_502TeV.pdf}\n\\caption{The energy loss $\\Delta p_{\\rm{T}}$ of the jets as a function of transverse \nmomentum $p_{\\rm{T}}$ in PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV for different\ncentrality classes.}\n\\label{Figure13_Jet_particles_Del_pT_PbPb_502TeV}\n\\end{figure}\n\n\n\nFigure~\\ref{Figure14_jet_Del_pT_cen_0_10_PbPb_276_502TeV} shows the energy loss\n$\\Delta p_{\\rm{T}}$ of the jets as a function of the transverse momentum $p_{\\rm{T}}$ \nin the most central (0-10\\%) PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 and 5.02 TeV.\nThese are compared with the $\\Delta p_{\\rm{T}}$ obtained for charged particles in\nthe 0-5\\% centrality class of PbPb collision\nat $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV.\nThe energy loss $\\Delta p_{\\rm{T}}$ in case of jets for both the energies increases\nwith $p_{\\rm{T}}$. The values of $\\Delta p_{\\rm{T}}$ for jets at 5.02 TeV is more than\nthat at 2.76 TeV. This behaviour at high $p_{\\rm{T}}$ is very different from the\nenergy loss of charged particles which becomes  constant in these $p_{\\rm{T}}$ regions.\nThe modeling of centrality dependence of energy loss has been done using\n$N_{\\rm part}^\\beta$.\nFor charged particles, the centrality dependence of $p_T$ shift is found to\nbe $N_{\\rm part}^{0.58}$ which corresponds to $L^{1.18}$.\nThe centrality dependence for jets at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV is found to be\n$N_{\\rm part}^{0.60}$. \nIn case of jets at 5 TeV, the centrality dependence of energy loss is found to be\n$N_{\\rm part}^{0.75}$ corresponding to $L^{1.5}$ which means that the jets even at\nvery high energy are still away from complete coherent regime. \n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.61\\linewidth]{Figure14_jet_Del_pT_cen_0_10_PbPb_276_502TeV.pdf}\n\\caption{The energy loss $\\Delta p_{\\rm{T}}$ of the jets as a function of transverse  \nmomentum $p_{\\rm{T}}$ in the most central PbPb collision at $\\sqrt{s_{\\rm{NN}}}$ = 2.76 and 5.02 TeV. \nThe $\\Delta p_{\\rm{T}}$ obtained for charged particles in the most central PbPb collision\nat $\\sqrt{s_{\\rm{NN}}}$ = 5.02 TeV is also shown.}\n\\label{Figure14_jet_Del_pT_cen_0_10_PbPb_276_502TeV}\n\\end{figure}\n\n\n\n\\clearpage\n\n\\section{Conclusions}\n\nWe presented a study of partonic energy loss with $p_T$ shift extracted from the measured\n$R_{\\rm{AA}}$ of charged particles and jets in PbPb collisions at $\\sqrt{s_{\\rm NN}}$ = 2.76\nand 5.02 TeV in wide transverse momentum and centrality range.\n  The functional form of energy loss given by\n$\\Delta p_{\\rm T}$ has been assumed as power law with different power indices\nin three different $p_{\\rm T}$ regions driven by physics considerations.\nThe power indices and the boundaries of three $p_{\\rm T}$ regions are obtained by\nfitting the experimental data of $R_{\\rm{AA}}$ as a function of $p_{\\rm T}$ and centrality.\n The energy loss for light \ncharged particles is found to increase linearly with $p_{\\rm T}$ in low $p_{\\rm T}$ region\nbelow 5-6 GeV\/$c$ and approaches a constant value in high $p_{\\rm T}$ region above 25 GeV\/$c$\nwith an intermediate power law connecting the two regions.\n The $\\Delta p_{\\rm{T}}$ at 5.02 TeV is similar but slightly more than\nthat at 2.76 TeV. \nIn case of jets we consider only one $p_T$ region and it is found that for jets, the\nenergy loss increases almost linearly even at very\nhigh $p_{\\rm T}$.\n The modeling of centrality dependence of energy loss has been done using\n$N_{\\rm part}^\\beta$.\n For charged particles, the centrality dependence of $p_T$ shift is found to\nbe $N_{\\rm part}^{0.58}$ which corresponds to $L^{1.18}$.\n The centrality dependence for jets at $\\sqrt{s_{\\rm NN}}$ = 2.76 TeV goes as\n$N_{\\rm part}^{0.60}$. \nIn case of jets at 5 TeV, the centrality dependence of energy loss is found to be \n$N_{\\rm part}^{0.75}$ corresponding to $L^{1.5}$ which means that the jets even at\nvery high energy are still away from complete coherent regime. \n\n\n\n\\ \\\\\n\n\\noindent\n{\\bf References}\n\n\\noindent\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\\subsection{Motivation and problem description}\n\n\n\\IEEEPARstart{T}o study the coordination and control features of a group\ntask, the multiple groups' performances must be fitted together. \n An enduring postulate in organization science\nis that coordination and control cannot be achieved strictly by the\nauthority structure, but must also entail informal communication and\ninfluence networks that link the members of different task-oriented groups;\nwe focus on formation of such network structures. As the size of a\nconnected social network increases, multigroup formations that are\ndistinguishable clusters of individuals become a characteristic and\nimportant feature of network topology. The connectivity of multigroup\nnetworks may be based on edge bundles connecting multiple individuals in\ntwo disjoint groups, bridges connecting two individuals in two disjoint\ngroups, or co-memberships. A large-scale network may include instances of\nall of these connectivity modalities.  We set up populations of multiple\ngroups and propose a dynamic model for formation of these intergroup\nconnectivity structures.\n\nOur economic dynamical model explains and predicts whether a network\nevolves into different coordination and control structures. Medium and\nlarge scale organizations adopt these multigroup structures to tackle\ncomplex nested tasks. Among the multitude of possible coordination and\ncontrol structures, we study formation of multigroup connectivity\nstructures shown in Fig.~\\ref{fig:schematic}, which are familiar constructs\nin the field of social network science.\n\\begin{figure}[h]\n\t\\begin{center} \n\t\t\\subfloat[Co-memberships]{\\includegraphics[height=.89in]{ControlStructure-3}\\label{fig:control-structure-3}}\\qquad\n\t\t\\subfloat[Edge Bundles]{\\includegraphics[height=.8in]{ControlStructure-2}\\label{fig:control-structure-2}}\\qquad\n\t\t\\subfloat[Bridges]{\\includegraphics[height=.8in]{ControlStructure-1}\\label{fig:control-structure-1}}\\\\\n\t\t\\caption{\\small Schematic illustration of the three possible control\nand coordination structures}\\label{fig:schematic}\n\t\\end{center}\n\\end{figure}\nFor this purpose we apply a game-theoretic framework in which strategic agents take actions based on the rate or importance of coordination problems. In other words, a value is assigned to the coordination problem between any two distinct groups, so that all control and coordination problems among groups are described by a square non-negative matrix, as illustrated in Fig.~\\ref{fig:matrix-F}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.3\\linewidth , height=0.3\\linewidth]{matrixGrayscale.png}\\\\\n\\caption{$F$ = frequency\/importance of intergroup coordination problem}\\label{fig:matrix-F} \n\\end{figure} \nIn our setting, agents are myopic, self-interested, and have thorough knowledge of graph topology and the utility they acquire from any other agent. \n\n\n\\subsection{Related literature}\n\nBridge, edge bundle, and co-membership connectivity models have been studied extensively in~\\cite{SM-PA-FB-NEF:17c}, where implications of these structures are investigated and generative models are proposed for each. These prototypical structures can mitigate coordination and control loss in an organization. Coordination and control importance of bridge connected structure, in which\ncommunication between subgroups are based on single contact edges, is the emphasis of the~\\cite{MSG:73, MT-DK:10}, and~\\cite{WS-TE:08} models. \nCoordination and control importance of the redundant ties structure, in which multiple redundant contact edges connect pairs of groups, is the emphasis\nof~\\cite{NEF:98}, Chapter~8, ~\\cite{NEF:83}, and~\\cite{HCW-SAB-RLB:76}. Co-membership intersection\nstructures, in which subgroups have common members, is the emphasis of the linking-pin\nmodel by Likert~\\cite{RL:67}, as well as~\\cite{BC-JAH:04} and~\\cite{SPB-DSH:11}.~\\cite{XZ-CW-YS-LP-HZ:17} and \\cite{JY-JL:12} propose a community detection algorithm for overlapping networks.\n\nJackson and Wolinsky introduced a strategic network formation model in their seminal paper~\\cite{MOJ-AW:96}. They studied pairwise stability, where bilateral agreement is required for link formation. Homogeneity and common knowledge of current network to all players are two assumptions in this model. Jackson and Watts studied strategic network formation in a dynamic framework in~\\cite{MOJ-AW:02}. The network formation model we present in this work is closely related to~\\cite{MOJ-AW:96} and~\\cite{MOJ-AW:02}. Jackson and Rogers examined an economic model of network formation in \\cite{MOJ-BWR:05} where agents benefit from indirect relationships. They showed that small-world features necessarily emerge for a wide set of parameters.\n\nIn \\cite{VB-SG:00}, Bala and Goyal proposed a dynamic model to study Nash and strict Nash stability. In their model, starting from any initial network, each player with some positive probability plays a best response (or randomizes across them when there is more than one); otherwise the player exhibits inertia. A Markov chain on the state space of all networks is defined whose absorbing states are strict Nash networks. The authors proved that starting from any network, the dynamic process converges to a strict Nash network (i.e., the empty network or a center-sponsored star) with probability 1.\n\nIn \\cite{NO-FV:13}, Olaizola and Valenciano extended the model in~\\cite{VB-SG:00} and studied network formation under linking constraints. An exogenous link-constraining system specifies the admissible links. Players in the same component of the link-constraining network have common knowledge of that component. This model collapses to the unrestricted setting in~\\cite{VB-SG:00} (when the underling constraining network is complete graph). The set of Nash networks is a subset of  Bala and Goyal's unrestricted Nash network sets.\n\nIn the network formation game by Chasparis and Shamma in~\\cite{GCC-JSS:13} and~\\cite{GCC-JSS:08}, agents form and sever unidirectional links with other nodes, and stable networks are characterized through the notion of Nash equilibrium. Pagan and D{\\\"o}rfler~\\cite{NP-FD:19} studied network formation on directed weighted graphs and considered two notions of stability: Nash equilibrium to model purely selfish actors, and pairwise-Nash stability which combines the selfish attitude with the possibility of coordination among agents. McBride dropped the common knowledge assumption and studied the effects of limited perception (each player perceives the current network only up to a certain distance) in \\cite{MMB:06}. Song and van der Schaar \\cite{YS-MVDS:15} studied a dynamic network formation model with incomplete information.\n\nCommunity networks and their growth into potential socially robust\nstructures is studied in~\\cite{LM:19}. Bringmann et al. analyzed the\nevolution of large networks to predict link creation among the nodes\nin~\\cite{BB-MB-FB-AG:10}. \\cite{YJ-YW-XJ-ZZ-XC:17} studied link\ninference problem in heterogeneous information networks by proposing a\nknapsack-constrained inference method.\n\n\\subsection{Statement of contribution} \nWe consider a strategic network formation game described by a cost of maintaining links, a benefit of having connections, and an importance of coordination problems among pre-specified groups. Our setup is a heterogeneous generalization of the famous connection model. For this game, we study the resulting multi-group  structures that are pairwise stable and socially efficient.\n\n\nFor this game, we also introduce a formation dynamics  whereby \nlink formations require mutual consent and link removals can be initiated unilaterally. We study the conditions that give rise to formation of multigroup structures, as well as conditions which cause the multigroup structures be stable and\/or efficient.\nOur contributions are as follows:\n \n\nWe introduce certain threshold functions and provide bounds based on these functions to study pairwise stable and efficient structures. We also investigate the convergence of Formation Dynamics. For our analysis, we first focus on the structure of each group and formation of intra-connections. We particularly study the conditions which result in each group being a clique, and present results on pairwise stability, efficiency, and convergence of these cliques.\n\nWe then focus on the interconnections among those cliques. We present results on the pairwise stability and convergence to disjoint union of cliques for multigroup structures of arbitrary sizes. The rest of the analysis for density of interconnections is divided into two sections: two-group connectivity structures and multigroup connectivity structures. \n\nFor the two group structures, we provide a complete characterization of full ranges of parameters for stability and  efficiency. We present results on the pairwise stability and efficiency of minimally connected, redundantly connected, and maximally connected structures. We identify the ranges of parameter in which the efficient and the pairwise stable structure overlap and those in which they have a conflict.\n\nWe then investigate the multigroup structures. We study the pairwise stability of minimally connected cliques along arbitrary interconnection structures. We show that for the special case of the interconnection being a star graph, it is possible to identify the boundaries of parameters for stability of all interconnections being minimally connected. We also present results on formation of redundancies and for efficiency.\n\n\\subsection{Preliminaries}\n\n\nEach undirected graph is identified with the pair $\\mathcal{(V, E)}$. The set of graph nodes $\\mathcal{V} \\neq \\emptyset$ represents individuals or groups of individuals in a social network. $|\\mathcal{V}|=n$ is the size of the network. The pair $(i,j)$ is called an edge and it indicates the interaction between the two individuals $i$ and $j$. The set of graph edges $\\mathcal{E}$ represents the social interactions or ties among all individuals. Throughout this paper, since the individuals are unchangeable, we refer to the network $\\mathcal{(V, E)}$ simply as $\\mathcal{E}$. \n\nThe density of a graph is given by the ratio of the number of its observed to possible edges, $ \\ddfrac{2|\\mathcal{E}|}{n(n-1)}$. In a complete graph every pair of distinct nodes is connected by an edge. We denote the complete graph of size $n$ by $K_n$. A clique is a subset of vertices of a graph in which every two distinct vertices are adjacent. We say two graphs are adjacent if they differ in precisely one edge. A path of length $k$ is a sequence of nodes $i_{1}i_{2}\\dots i_{k}$ such that $\\{(i_{s},i_{s+1}) \\} \\in \\mathcal{E}$.  A walk of minimum length between two nodes is the shortest path. $d_{ij}(\\mathcal{E})$ denotes the distance between nodes $i$ and $j$, which is defined as the length of the shortest path beginning at $i$ and ending at $j$. \n\n\\section{Multigroup Network Formation Model}\n\\label{sec:model}\nConsider a society of $n$ individuals $\\mathcal{V}$, divided into $m$ groups. The set of $m$ groups is denoted by $\\until{m}, m \\leq n$.\n$P= \\{ P_1, \\dots, P_m\\}$ represents the partitioning of individuals into\nthe groups and is a set partition of size $n$, i.e, $\\mathcal{V}=\n\\bigcup \\limits_{\\gamma=1}^{m} P_{\\gamma}$, and $\\bigcap \\limits_{\\gamma=1}^{m} P_{\\gamma} =\n\\emptyset$. We use the shorthand notation $s_{\\gamma}= |P_{\\gamma}|$\ndenoting the size of group $\\gamma$. Throughout this paper, we assume that\n$s_{\\gamma} \\geq 3$ for all $\\gamma \\in \\until{m}$.\n\n\\textit{Group coordination importance matrix (data)}: is given as $F \\in\n\\mathbb{R}^{m \\times m}$, where $ 0 \\leq F_{\\alpha \\beta} \\leq 1$ for\n$\\alpha, \\beta \\in \\until{m}$ represents importance\/frequency of coordination\nproblem between groups $\\alpha$ and $\\beta$. We assume $F$ is a symmetric\nmatrix with diagonal entries equal to $1$.\n\n\\textit{Individual coordination importance matrix}: $\\hat{F}  \\in \\mathbb{R}^{n \\times n}$, is obtained from $F$ and the partition $P$, i.e., $\\hat{F}= f(F, P)$. We construct $\\hat{F}$ as follows:\n  \n  \\begin{equation*}  \n    \\hat{F}_{ij} =\\begin{cases}\n    F_{\\alpha \\beta}, & i \\in P_{\\alpha},  j \\in P_{\\beta},  i \\neq j \\\\\n    0, & i=j. \n    \\end{cases}\n  \\end{equation*}\n  \n  For the setting where groups are all of equal size $s$, one can write\n  \\begin{equation*}\n     \\hat{F}=F \\otimes  \\vectorones[s] \\vectorones[s]^T -I_n \n  \\end{equation*}\n  \n\nAt edge set $\\mathcal{E}$, the payoff function for individual $i \\in\n\\mathcal{V}$ is \n\\begin{equation}\\label{payoff-function}\n  U_i (\\mathcal{E}) = \\sum_{k=1}^n \\hat{F}_{ik} \\delta^{d_{ik}\n    (\\mathcal{E})} - \\sum\\nolimits_{k \\in N_i(\\mathcal{E}) } c,\n\\end{equation}\nwhere $d_{ik}(\\mathcal{E})$ is the number of steps from individual $i$\nto $k$, $\\delta < 1$ is the one-hop benefit, and $c$ is the cost of each\nlink. The value of network $\\mathcal{E}$ is defined as the sum of all individuals' payoffs, i.e., $v(\\mathcal{E})=\\sum_{i=1}^{n}U_{i}(\\mathcal{E})$, and it indicates the social welfare. For a given society $\\mathcal{V}$ and value function $v$, $\\mathcal{E}^*$ is an \\emph{efficient structure} if its social welfare(value) is maximized over all possible edge sets on $\\mathcal{V}$, i.e., $\\mathcal{E}^*=  \\arg \\max\\limits_{\\mathcal{E}} v(\\mathcal{E})$. Given the pair $(i, j)$ in network $\\mathcal{E}$, we say that individual $i$ \\emph{benefits from edge} $\\{(i,j)\\}$ if\n$\n  U_i \\big (\\mathcal{E}\\cup \\{( i,j) \\} \\big ) > U_i \\big(\\mathcal{E} \\setminus \\{( i,j) \\}\n  \\big).\n$\n\n\\textit{Formation Dynamics}: Time periods are represented with countable\ninfinite set $\\mathbb{N}= \\{ 1, 2, \\dots, t, \\dots \\}$.  In each period, a\npair $(i,j)$ is uniformly randomly selected and is added to, or removed\nfrom, the network $\\mathcal{E}$ according to the following rules:\n\\begin{itemize}\n\n\\item if $\\{(i,j)\\} \\notin \\mathcal{E}$, then it is added when its\n  addition is marginally beneficial to the pair of individuals (i.e.,\n  either both individuals benefit or one individual is indifferent and the\n  other benefits); the edge $(i,j)$ is not added when its addition causes\n  a drop in the payoff of either or both individuals or both individuals\n  are indifferent towards it; and\n\\item if $\\{(i,j)\\} \\in \\mathcal{E}$, then $(i,j)$ is removed when its\n  removal benefits at least one of the two individuals; no action is taken\n  when both sides are either indifferent or benefit from the existence of\n  the edge.\n\\end{itemize} \n  \n\n\n\n\\begin{definition}(Pairwise Stability)\\label{def:pairwise_stable}\nA network $\\mathcal{E}$ is \\emph{pairwise stable} if,\n\\begin{equation*}\n\\begin{aligned}\n& \\text{for all } \\{(i,j)\\}\\in\\mathcal{E},\\\\\n&  \\quad U_i(\\mathcal{E}) \\geq U_i(\\mathcal{E} \\setminus \\{(i,j)\\} ) \n\\text{ and } U_j(\\mathcal{E}) \\geq U_j(\\mathcal{E} \\setminus \\{( i,j) \\}); \\\\\n & \\text{and } \\text{for all } \\{(i,j)\\} \\notin \\mathcal{E}, \\\\\n &\\quad \\text{if } U_i(\\mathcal{E})<U_i(\\mathcal{E}\\cup\\{(i,j)\\}), \\text{ then }\nU_j(\\mathcal{E})>U_j(\\mathcal{E}\\cup\\{(i,j)\\}).\n\\end{aligned}\n\\end{equation*}\n\\end{definition}\n \\begin{remark}\\label{remark:pairwise-stable}\nAccording to Definition~\\ref{def:pairwise_stable}, if the edge $(i,j)$ belongs to the pairwise stable network, removing it results in a loss for $i$ or $j$; and if the edge $(i,j)$ does not belong to the pairwise stable network, adding it makes no difference or causes loss for $i$ or $j$. \n \\end{remark}\n \n \n\\begin{definition}\n$\\mathcal{E}'$ defeats $\\mathcal{E}$ if either $\\mathcal{E}'=\\mathcal{E} \\setminus\n\\{(i,j)\\}$ and $U_i(\\mathcal{E}')>U_i(\\mathcal{E})$, or $\\mathcal{E}'=\\mathcal{E} \\cup \\{(i,j)\\}$\nand $U_i(\\mathcal{E}') \\geq U_i(\\mathcal{E})$ and $U_j(\\mathcal{E}') \\geq U_j(\\mathcal{E})$ with at least one inequality holding strictly. \n\\end{definition}\\label{def:improving-path}\n\n\\begin{lemma}\\label{lem:stable-by-dynamics}\nA network is pairwise stable if and only if it does not change under Formation Dynamics.\n\\end{lemma}\n\\begin{proof}\nTo prove necessity, we refer to Remark~\\ref{remark:pairwise-stable}. According to the definition, if a network is pairwise stable, no network can defeat it, i.e., no links can be added to or severed from it. To show sufficiency, note that a network not being changed by Formation Dynamics, implies that: \n\\begin{enumerate}\n\\item adding a link makes no difference or causes loss for at least one individual;\n\\item removing a link results in loss for at least one individual. \n\\end{enumerate}\nTherefore, the network is pairwise stable. \n\\end{proof}\t\n\n\nAccording to Lemma~\\ref{lem:stable-by-dynamics}, if there exists some time $t^*$\nsuch that from $t^*$ on, no additional links are added to or severed from a network by Formation Dynamics, then the network has reached the pairwise stable structure.\n\nWe define the following terms that we will frequently use throughout this paper indicating the density of the interconnections among the groups. \n\\begin{definition}\nWe say that a society of individuals consists of {\\it the disjoint union of groups} if there exists no interconnection among any pairs of groups.  \nFor a connected pair, we say it is\n\\begin{enumerate}\n\\item {\\it minimally connected} if there exists exactly one interconnection among the pair;\n\\item {\\it redundantly connected} if there exist at least two interconnections among the pair;\n\\item {\\it maximally connected} if all of the possible interconnections among the pair of groups exist.\n\\end{enumerate}\n\\end{definition}\nFig. \\ref{fig:density_interC} represents a schematic illustration of the terms discussed above.\n\\begin{remark}\nA minimally connected pair corresponds to the bridge connection (Fig.~\\ref{fig:control-structure-1}), redundantly connected to the ridge connection (Fig.~\\ref{fig:control-structure-2}), and maximally connected to a full co-membership connection (Fig.~\\ref{fig:control-structure-3}.)\n\\end{remark}\n\\begin{figure}[h]\n\t\\begin{center} \t\n\t\\includegraphics[width=0.99\\linewidth]{interconnection_density}\\label{fig:}\n\t\t\\caption{\\small Schematic illustration of interconnection densities}\\label{fig:density_interC}\n\t\\end{center}\n\\end{figure}\n\nWe next define the Price of Anarchy (PoA) as a measure of how the efficiency of a system degrades due to the selfish behavior of its individuals. It is calculated as follows:\n\\[\nPoA=\\dfrac{\\max_{\\mathcal{E}}v(\\mathcal{E})}{\\min_{p.w. stable\\mathcal{E}}v(\\mathcal{E})}.\n\\]\n\nThroughout this paper we use the following threshold functions\n$y_1(s, \\delta)$, $y_2(s, \\delta)$, and $y_3(\\delta)$ defined by\n\\begin{equation*}\n\\begin{aligned}\n  & y_1(s, \\delta)= {\\delta + \\big(s-1\\big) \\delta^2},\\\\\n  & y_2(s, \\delta) =  {\\delta - \\delta^2 + \\big(s-1\\big) \\delta^2 - \\big(s-1\\big) \\delta^3} = { \\big (1-\\delta \\big )y_1(s)},\\qquad \\\\\n & y_3(\\delta)={\\delta - \\delta^2}.\n\\end{aligned}\n\\end{equation*}\nIn what follows we will often suppress the argument $\\delta$ in the interest of simplicity.\n\nUnder the conditions $0<\\delta<1$ and $s\\geq 3$, we claim that,\n\\begin{equation*}\n  0<y_3<y_2(s)<y_1(s).\n\\end{equation*}\nThe proof is as follows: it is easy to see that $y_2(s) < y_1(s)$. To\nverify $y_3 < y_2(s) $, we rewrite $y_2(s)$ as $ \\big (\\delta - \\delta^2\n\\big) \\big(1+ \\delta (s-1) \\big) = y_3 \\big(1+ \\delta (s-1) \\big) > y_3$.\nPlots of these three threshold functions for $0<\\delta<1$, where $s=3$ are depicted in\nFig.~\\ref{fig:thresholds}.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.6\\linewidth]{y.pdf}\n        \\caption{Plots of $y_1$, $y_2$, and $y_3$ for $s=3$.}\\label{fig:thresholds} \n\\end{figure} \n  In what follows we provide bounds based on these functions to study pairwise stable and efficient structures, and investigate the convergence of the Formation Dynamics when possible.\n\n\n\\section{Results on Formation of Disjoint Cliques}\n\n We first study the inner structure of each group in a pairwise stable network.  Throughout this paper, we assume that the dynamics does not start with an initial state containing any interconnection.  We define the invariant set of all subgraphs of disjoint cliques as $S= \\Big \\{ \\bigcup \\limits_{\\gamma=1}^{m} \\mathcal{E}_{\\gamma} \\ | \\  \\mathcal{E}_{\\gamma} \\subset \\mathcal{E}_{ K_{s_{\\gamma}}}  \\}$ where $\\mathcal{E}_{\\gamma}$ indicates the inner-network of group $P_{\\gamma}$.    \n\n \n\\begin{theorem}[Formation of Cliques: Pairwise Stability, Efficiency, Convergence]\\label{thm: Pairwise Stability of Cliques}\nConsider $n$ individuals partitioned into groups $P_1,\\dots,P_m$. Then, each one of these $m$ groups is a clique in the pairwise stable and in the efficient structure if and only if $c<y_3$.  Moreover, starting from any state in the invariant set $S$, each group $P_{\\alpha}$ will form a $s_{\\alpha}$-size clique along Formation Dynamics (introduced in Section~\\ref{sec:model}).\n\\end{theorem}\n\\begin{proof}\nWe first provide the proof of sufficiency for pairwise stability: for any individual $i\\in P_{\\alpha}$, a direct link with individual $j$, $(j\\neq i)$ from the same group provides a profit of $\\delta-c$. Without a direct link, this profit is equal to $\\delta^{d_{ij}}$ where $d_{ij}>1$ is the distance between $i$ and $j$ in $\\mathcal{E} \\setminus \\{(i,j)\\}$. Since $\\delta-\\delta^2>c$, we have \n  $\\delta-c>\\delta^2 >\\dots >\\delta^n$; meaning that all\n  agents prefer direct links to any indirect link. Thus, if agents $i$ and\n  $j$ in group $P_\\alpha$ are not directly connected, they will form a link\n  and each will gain at least $(\\delta-c)-\\delta^{d_{ij}}>0$,\n  i.e.,\n  \\[\n  \\begin{aligned} \n  &\\text{for all } \\{(i,j)\\} \\notin \\mathcal{E},\\quad i, j\n    \\in P_{\\alpha}, \\ i \\neq j \\\\\n   & \\quad  U_i(\\mathcal{E})<U_i(\\mathcal{E}\\cup\\{(i,j)\\}),\n    \\text{ and } U_j(\\mathcal{E})<U_j(\\mathcal{E}\\cup\\{(i,j)\\}).\n  \\end{aligned}\n  \\]\n  Moreover, no node has an incentive to break any link since its payoff strictly\n  decreases if it do so, i.e., \n  \\[\n  \\begin{aligned}\n &   \\text{for all } \\{(i,j)\\} \\in \\mathcal{E}, \\quad i, j \\in P_{\\alpha},\\ i \\neq j , \\\\\n      &  \\quad U_i(\\mathcal{E}) > U_i(\\mathcal{E} \\setminus \\{(i,j)\\}), \\text{ and }\n    U_j(\\mathcal{E}) > U_j(\\mathcal{E} \\setminus \\{( i,j) \\}).\n  \\end{aligned}\n  \\]\n  Thus, each group forms a clique and no intra-connection is removed after being formed, and according to Lemma \\ref{lem:stable-by-dynamics},  these $m$ groups are cliques in the pairwise stable structure.\nTo prove necessity,  assume we have a pairwise stable clique. \nFor $P_\\alpha$ to remain a clique, all pairs of nodes belonging to the same group should prefer to keep one-hop links rather than having links with larger lengths, and thus $\\delta-c>\\delta^2>\\delta^3> \\dots$. This proves the claim that each group $P_\\alpha$ is a clique if and only if $c<\\delta-\\delta^2$. Convergence of dynamics to cliques can be obtained directly from the same argument.\n\n We now continue by first proving that if $c<\\delta - \\delta^2$, in the efficient structure each group is a clique. From the analysis above, when $c<\\delta-\\delta^2$, we have:\n\\[\n\\begin{aligned}\nv\\big(\\mathcal{E}\\cup\\{(i,j)\\}\\big)&-v\\big(\\mathcal{E}\\setminus\\{(i,j)\\}\\big) \\\\\n  & \\geq U_i\\big(\\mathcal{E}\\cup\\{(i,j)\\}\\big)+U_j\\big(\\mathcal{E}\\cup\\{(i,j)\\}\\big)\n    \\\\\n    &\\quad-U_i\\big(\\mathcal{E}\\setminus\\{(i,j)\\}\\big) -U_j\\big(\\mathcal{E}\\setminus\\{(i,j)\\}\\big)\\\\\n &   \\geq 2(\\delta -c-\\delta^{2})>0\n\\end{aligned}\n\\] \nwhich holds for each pair $(i,j)$ belonging to the same group, meaning that each group is a clique in the efficient structure.  \nWe next prove necessity for efficiency: assume $\\mathcal{E}$ is the efficient structure and each group is a clique, i.e., $\\{(i,j)\\}\\in \\mathcal{E}$ for any two individuals $i, j$, $(i \\neq j)$ from the same group. Then, we have:\n\\[\n\\begin{aligned}\nv(\\mathcal{E})&-v(\\mathcal{E}\\setminus\\{(i,j)\\})\\\\\n & =U_i(\\mathcal{E})+U_j(\\mathcal{E})-U_i(\\mathcal{E}\\setminus\\{(i,j)\\})-U_j(\\mathcal{E}\\setminus\\{(i,j)\\})\\\\\n &=2(\\delta -c-\\delta^{2})>0,\n\\end{aligned}\n\\]\nwhich results in $c<y_3$.\n\\end{proof}\n\n\\textbf{Note}: Theorem~\\ref{thm: Pairwise Stability of Cliques} implies that formation of cliques requires $c < 1\/4$.\n\n\n\\begin{theorem}[Pairwise Stable Structures and Convergence: Disjoint Union of Cliques]\\label{thm:notequal-Disjoint} \nConsider $n$ individuals partitioned into groups $P_1,\\dots,P_m$ of sizes $s_{1},\\dots,s_{m}$ respectively. Assume that $c<y_3$. Then, the unique pairwise stable structures consists of disjoint union of cliques equal to the groups $P_1,\\dots,P_m$ if and only if $F_{\\alpha \\beta} \\leq \\max\\limits_{s \\in\\{s_{\\alpha}, s_{\\beta}\\}} \\dfrac{c}{ y_1(s)} $ for all $\\alpha, \\beta \\in \\until{m}$, $\\alpha \\neq \\beta$. Moreover, starting from any state in the invariant set $S$, Formation Dynamics (introduced in Section~\\ref{sec:model}) converges to this pairwise stable structure.\n\\end{theorem}\n\n\\begin{proof}\nWe first prove sufficiency.  Since no interconnection exists in the invariant set $S$, for any network $\\mathcal{E}\\in S$, suppose two individuals $i\\in P_{\\alpha}$ and $j\\in P_{\\beta}$ are picked to decide whether to add the corresponding interconnection or not. We know that $U_{i}(\\mathcal{E} \\cup \\{(i,j)\\})\\leq F_{\\alpha \\beta}y_{1}(s_{\\beta})-c$ and \n$U_{j}(\\mathcal{E} \\cup \\{(i,j)\\})\\leq F_{\\alpha \\beta}y_{1}(s_{\\alpha})-c$, \nwhere equalities hold when both groups form cliques. Since $F_{\\alpha \\beta} \\leq \\max\\limits_{s \\in\\{s_{\\alpha}, s_{\\beta}\\}} \\dfrac{c}{ y_1(s)} $, at least one of $U_{i}\\big(\\mathcal{E} \\cup \\{(i,j)\\}\\big)\\leq 0$ and $U_{i}\\big(\\mathcal{E} \\cup \\{(i,j)\\}\\big)\\leq 0$ holds. Therefore, the interconnection \n$\\{(i,j)\\}$ does not belong to pairwise stable structure and it does not form. From Theorem \\ref{thm: Pairwise Stability of Cliques}, we know that unique stable state consists of the disjoint union of cliques equal to the groups $\\until{m}$. Suppose that all groups form cliques at a time $t^{*}$. From then on, no link will be added or removed. \nAccording to Lemma \\ref{lem:stable-by-dynamics},  the pairwise stable structure consists of the disjoint union of cliques equal to the groups $\\until{m}$, and the network converges to this unique stable state.\n\nTo prove necessity, take any two individuals $i\\in P_{\\alpha}$ and $j\\in P_{\\beta}$. Since $\\{(i,j)\\}$ does not belong to pairwise stable structure, we have at least one of \n$U_{i}(\\mathcal{E} \\cup \\{(i,j)\\})\\leq 0$ and $U_{i}(\\mathcal{E} \\cup \\{(i,j)\\})\\leq 0$ holds, and therefore, $F_{\\alpha \\beta} \\leq \\max\\limits_{s \\in\\{s_{\\alpha}, s_{\\beta}\\}} \\dfrac{c}{ y_1(s)} $. \n\n\\end{proof}\n\n\\section{Two Group Connectivity Structure}\n\nIn this section we study pairwise stable and efficient structures when individuals are partitioned into two groups. \n\n\\subsection{Pairwise Stability}\nIn what follows we give the sufficient and necessary condition for pairwise stable structures. \n\n\n\\begin{theorem}[Pairwise Stability and Convergence with Two groups]\\label{thm:existence-convergence-two}\n  Consider $n$ individuals partitioned into groups $P_1, P_2$ of sizes  $s_1$ and $s_2$ respectively. Then, under the assumption $c<y_3$, the network has \n  \\begin{enumerate}\n   \\item\\label{fact:bridge-2groups} a unique\n     pairwise stable structure consisting of minimally connected cliques if and\n     only if  \n     \\[ \n     \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}}  \\dfrac{c}{ y_1(s)}  \\leq   \n     F_{12} < \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{  y_2(s)}  ;\n     \\]  \n   \\item\\label{fact:redundant-2groups} a unique pairwise stable structure consisting of exact $k~ (2\\leq k\\leq \\min\\{s_{1},s_{2}\\})$  interconnections if and only if \n     \\begin{multline}\n    \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)-(k-2)\\delta y_{3}}  \\\\\n    < F_{12} \n      < \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)-(k-1)\\delta y_{3}}.\n     \\end{multline}\n   \\item\\label{fact:comember-2groups} a unique\n     pairwise stable structure consisting of maximally connected cliques if and\n     only if \n     \\[\\dfrac{c}{ y_3 } < F_{12} \\leq 1.\\]      \n  \\end{enumerate} \n  Moreover, starting from any state in the invariant set $S$, the Formation Dynamics (introduced in Section~\\ref{sec:model}) converges to the corresponding pairwise stable structure. \n \\end{theorem}\n\\begin{figure}[ht]\n\t\\includegraphics[width=1\\linewidth]{spectrum.pdf}\n        \\caption{An illustration of the four ranges of parameter space in Theorems ~\\ref{thm:notequal-Disjoint} and~\\ref{thm:existence-convergence-two}.} \\label{fig:spectrum}  \n\\end{figure} \n\\begin{proof}\n\n\nFrom Theorem \\ref{thm:notequal-Disjoint}, we know that at least one interconnection will be formed under Formation Dynamics if and only if $ F_{12} \\geq \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)} $. Without loss of generality, suppose that $\\mathcal{E}$ contain two cliques and at least one interconnection between them. Assume that agents $i$ and $j$, respectively from $P_{1}$ and $P_{2}$, are connected in $\\mathcal{E}$. \n\nTake  agents $\\hat{i} $ from $P_{1}$ and $\\hat j$ from $P_{2}$. For $\\hat{i} \\neq i, \\hat j=j$, we have\n$\n  U_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} )=  (s_{1} -1) \\delta+ F_{12} \\delta + (s_{2} -1) F_{12} \\delta^2 - s_{1} c\n$\nand \n$\n U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j) \\} )=(s_{1} -1) (\\delta- c) +  F_{12} \\delta^2 + (s_{2} -1) F_{12} \\delta^3,\n$\nimplying $ U_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} )> U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j)\\} )   \n   \\iff F_{12} > \\dfrac{c}{y_2(s_2)}.$\n   From \n$\n U_{j}(\\mathcal{E} \\cup \\{ (\\hat{i},j)\\} )=  (s_{2} -1) \\delta +2 F_{12} \\delta + (s_{2} -2) F_{12} \\delta^2 +(s_{2} +1) c\n$\nand \n$\n U_{j}(\\mathcal{E} \\setminus \\{ (\\hat{i},j )\\} )=  (s_{2} -1) \\delta + F_{12} \\delta + (s_{2} -1) F_{12} \\delta^2 +s_{2} c\n$,\nwe obtain:  \n$ U_j(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} ) > U_j(\\mathcal{E} \\setminus \\{ (\\hat{i},j)\\} )   \n   \\iff F_{12} > \\dfrac{c}{y_{3}}.\n$\nThen, from $y_2(s)>y_{3}>0$, we conclude that an additional interconnection $\\{(\\hat i, j)\\}$ is added and maintained if and only if  $F_{12} > \\dfrac{c}{y_{3}}$. \nSimilarly, an additional interconnection $\\{( i,\\hat j)\\}$ is added and maintained if and only if  $F_{12}> \\dfrac{c}{y_{3}}$. \nFor $\\hat i \\neq i, \\hat j\\neq j$, using a similar argument, we obtain that $U_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i},\\hat j ) \\} )> U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i},\\hat j )\\} )   \n   \\iff F_{12} > \\dfrac{c}{y_2(s_2)}$ and \n   $U_{\\hat j} (\\mathcal{E} \\cup \\{ (\\hat{i},\\hat j) \\} )> U_{\\hat j} (\\mathcal{E} \\setminus \\{( \\hat{i},\\hat j )\\} )   \n   \\iff F_{12}> \\dfrac{c}{y_2(s_1)}$; which means that an additional interconnection $\\{(\\hat i, \\hat j)\\} $ is added and maintained if and only if $ F_{12} > \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)} $ (strictly holds when $s_{1} = s_{2}$). Thus, we conclude that at least two interconnections are added and maintained if and only if $F_{12} > \\min\\left\\{\\dfrac{c}{y_{3}},\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)}\\right\\}=\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)}$  (strictly holds when $s_{1} = s_{2}$.)  \nTherefore, the network contains precisely one interconnection if and only if $ \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}}  \\dfrac{c}{ y_1(s)}  <  \n     F_{12} < \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{  y_2(s)} $. Moreover, from the moment when two group form cliques and this unique interconnection builds, the network will not change.  According to Lemma \\ref{lem:stable-by-dynamics}, this concludes the proof of statement~\\ref{fact:bridge-2groups}. \n\nTo prove~\\ref{fact:comember-2groups}, assume that $F_{12}>  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)} $. We have shown above that $\\mathcal{E}$ contains at least two interconnections between two cliques. As a result, for any agent $\\hat i$ from $P_{1}$ and $\\hat j$ from $P_{2}$, the distance between $\\hat i$ and $\\hat j$ in $\\mathcal{E} \\setminus \\{(\\hat i, \\hat j)\\}$ is equal to either 2 or 3.  If it is equal to 2, then  $U_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i}, \\hat j ) \\} -U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i}, \\hat j )\\}=U_{\\hat j}(\\mathcal{E} \\cup \\{( \\hat{i}, \\hat j )\\} -U_{\\hat{j}}(\\mathcal{E} \\setminus \\{ (\\hat{i}, \\hat j )\\}=F_{12}(\\delta-\\delta^{2})-c$; and if it is equal to 3,  then $U_{\\hat i}(\\mathcal{E} \\cup \\{( \\hat{i}, \\hat j  )\\} -U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i}, \\hat j) \\}=U_{\\hat j}(\\mathcal{E} \\cup \\{ (\\hat{i}, \\hat j )\\} -U_{\\hat{j}}(\\mathcal{E} \\setminus \\{ (\\hat{i}, \\hat j )\\}=F_{12}(\\delta-\\delta^{3})-c$. Interconnection $\\{(\\hat i, \\hat j)\\}$ is added and maintained if and only if $U_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i},\\hat j ) \\} -U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i}, \\hat j) \\}> 0$ and $U_{\\hat j}(\\mathcal{E} \\cup \\{( \\hat{i}, \\hat j) \\} -U_{\\hat{j}}(\\mathcal{E} \\setminus \\{ (\\hat{i}, \\hat j )\\}>0$. \nThus, we conclude that $\\{(\\hat i, \\hat j)\\}$ is added and maintained if and only if $F_{12}>\\max\\left\\{\\dfrac{c}{\\delta-\\delta^{2}},\\dfrac{c}{\\delta-\\delta^{3}}\\right\\}$. Since $\\max\\left\\{\\dfrac{c}{\\delta-\\delta^{2}},\\dfrac{c}{\\delta-\\delta^{3}}\\right\\}=\\dfrac{c}{\\delta-\\delta^{2}}$ for $0<\\delta<1$,  $\\{(\\hat i, \\hat j)\\}$ is added and maintained maintained if and only if $F_{12}>\\dfrac{c}{\\delta-\\delta^{2}}=\\dfrac{c}{y_{3}}$. Therefore, the network will not be changed when all agents link with each other. By Lemma \\ref{lem:stable-by-dynamics}, this concludes the proof of~\\ref{fact:comember-2groups}.\n\nFrom statements \\ref{fact:bridge-2groups} and \\ref{fact:comember-2groups}, we know that the pairwise stable structure contains at least 2 but not fully numbers of interconnections if $\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)} < F_{12} < \\dfrac{c}{ y_3 }$.  Suppose that $(i_{1},j_{1}), \\dots, (i_{k-1},j_{k-1})$ are $k-1$ interconnections between $P_{1}$ and $P_{2}$. Take  agents $\\hat{i} $ from $P_{1}$ and $\\hat j$ from $P_{2}$. Similar to the analysis in the proof of statement~\\ref{fact:bridge-2groups}, we have the following two cases:\n\\begin{enumerate}[(a)]\n\\item For $\\hat{i} \\notin \\{i_{1},\\dots,i_{k-1}\\}, \\hat j\\notin \\{j_{1},\\dots, j_{k-1}\\}$, we have\n\\[\n\\begin{aligned}\n   U_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} )&- U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j) \\} )\\\\\n &\\quad = F_{12}(y_{2}(s_{2})-(k-2)\\delta y_{3}), \\text{ and}\\\\\n    U_{\\hat j}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} ) &- U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j) \\} )\\\\\n&\\quad  = F_{12}(y_{2}(s_{1})-(k-2)\\delta y_{3}),\n\\end{aligned}\n\\]\nimplying \n\\[\\begin{aligned}\nU_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} )&>U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j)\\} ), \\text {~and~}  \\\\\nU_{\\hat j}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} )&> U_{\\hat{j}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j)\\} ) \\\\\n  \\iff \\qquad F_{12} &> \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}}  \\dfrac{c}{y_2(s)-(k-2)\\delta y_{3}}.\n   \\end{aligned}\\]\n\\item For $\\hat{i} \\in \\{i_{1},\\dots,i_{k-1}\\}, \\hat j\\notin \\{j_{1},\\dots, j_{k-1}\\}$ or  $\\hat{i} \\notin \\{i_{1},\\dots,i_{k-1}\\}, \\hat j\\in \\{j_{1},\\dots, j_{k-1}\\}$, we have \n\\[\\begin{aligned}\n U_{\\hat i}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} )&> U_{\\hat{i}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j)\\} ) \n   \\text {~ and~}\\\\\n U_{\\hat j}(\\mathcal{E} \\cup \\{ (\\hat{i},j) \\} )&>U_{\\hat{j}}(\\mathcal{E} \\setminus \\{ (\\hat{i},j)\\} ) \\\\\n    \\iff \\qquad F_{12} &>   \\dfrac{c}{y_{3}}.   \n   \\end{aligned}\\]\n\\end{enumerate}\nTherefore,  we conclude then $k-th$ interconnection is added and maintained if and only if  $F_{12} >\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}}  \\dfrac{c}{y_2(s)-(k-2)\\delta y_{3}}.$ Likewise, the $(k+1)-th$ interconnection is added and maintained if and only if  $F_{12} > \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}}  \\dfrac{c}{y_2(s)-(k-1)\\delta y_{3}}.$ It follows that the unique pair-wise stable structure has exact $k~ (2\\leq k\\leq \\min\\{s_{1},s_{2}\\})$  interconnections if and only if $\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)-(k-2)\\delta y_{3}}< F_{12} < \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)-(k-1)\\delta y_{3}}$. \nThis concludes the proof of~\\ref{fact:redundant-2groups}.\n\n Finally, we complete the proof of Theorem~\\ref{thm:existence-convergence-two} by proving the convergence statement. Since  $c<y_3$, from Theorem \\ref{thm: Pairwise Stability of Cliques} we know that the network structure, regardless of the density of intra-group connections, consists of cliques of sizes $s_1$ and $s_2$.  From above analysis,  starting from invariable set $S$, intra-connections will increase one by one until one more intra-connection can not bring increasing of benefit for two players. Then, it follows from Lemma \\ref{lem:stable-by-dynamics} that the network can not be changed from then on, i.e., the network will converge to the corresponding pairwise stable structure under the formation dynamics in Section~\\ref{sec:model}. \n\\end{proof}  \n\nTheorem \\ref{thm:existence-convergence-two} implies that if $F_{12}$ equals the boundaries, i.e., \n$\n\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{y_2(s)}, \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{y_2(s)-\\delta y_{3}}, \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)-2\\delta y_{3}}$, $\\dots, \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{y_{3}},\n$\n the pairwise stable structure is not unique. To show this, we provide the following example for a society consisting of 8 individuals.\n\n\\begin{example}{\\label{example:non-unique-pwStable}}\n Suppose that individuals are partitioned into groups $P_{1}=\\{1,2,3\\}$ and $P_{2}=\\{4,5,6,7,8\\}$. Let $c=0.2$, $\\delta=0.5$ and  $F_{12}=\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{y_2(s)}=0.4$. Assume at time $0$, each of the two groups have a line structure, illustrated in Fig~\\ref{fig:stabilityatBoundary}. Now consider the following two processes:\n\\begin{enumerate}[Process A.]\n\\item At first, individuals 2 and 6 are chosen to play the game and interconnection $(2,6)$ is formed. Then, individuals 3 and 7 are chosen to play the game, and interconnection $(3,7)$ is also formed.  After that, all possible intra-connections are considered and formed. Finally, the network contains two interconnections and reaches pairwise stability.\n\\item At first, all possible intra-connections are considered and formed. Then individuals 2 and 6 are chosen to play the game and interconnection $(2,6)$ is formed. As a result, the network contains only 1 interconnection and reaches pairwise stability.\n\\end{enumerate}\n\n\\end{example}\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{stabilityatBoundary.pdf}\n        \\caption{An illustration of two processes in Example~\\ref{example:non-unique-pwStable}. At each step, the darker dots are chosen to play the game.} \\label{fig:stabilityatBoundary}  \n       \\end{figure} \n\n\nFig.~\\ref{fig:spectrum} illustrates the scenarios specified in Theorems ~\\ref{thm:notequal-Disjoint} and~\\ref{thm:existence-convergence-two} for $F_{12}$.\n\n\\begin{remark}{}\n\nAs the difference between group sizes $|s_2- s_1|$ increases, the bound for formation of bridges increases, and thus communication between groups becomes harder.\nTo see why this is true, for a society of $n=s_1+s_2$ individuals, without loss of generality, we assume that $s_1 \\leq s_2.$.\nWe know $\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)}= \\dfrac{c}{ \\min\\limits_{s \\in\\{s_{1}, s_{2}\\}} y_1(s)}= \\dfrac{c}{ y_1(\\min\\{s_{1}, s_{2}\\})}$, and since $\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)}$ is a monotonically decreasing function of $s$, as $s_1$ increases, $|s_2-s_1|$, and therefore, $\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)}$ decreases, and communication is facilitated. As illustrated in Fig.~\\ref{fig:2x2_var}, for a society of fixed size, as the sizes of the two groups becomes closer to each other, the number of interconnections increases.\n\\end{remark}\n\\begin{figure}\n\t\\begin{center} \n\t\n\t\t{\\includegraphics[width=0.9\\linewidth]{interC_sizeVar_20_0}\\label{fig:2x2-interC}}\n\t\n\t\t\\caption{\\small Plots of no. of interconnections with fixed $F_{12}=0.25$ and $n=20$ as a function of groups size $s_1$, $\\delta=0.5$, $c=0.2$. Note that $s_2=n-s_1$, and thus the plot is symmetric about the line $s_1=\\frac{n}{2}$ and maximized at $s_1=\\frac{n}{2}$.}\\label{fig:2x2_var}\n\t\\end{center}\n\\end{figure}\n\n\\subsection{Efficiency}\n\nWe now study the efficiency of two group structure.\n\n\n\\begin{theorem}[Efficiency with Two groups]\\label{thm:efficiency-two}\n   Consider $n$ individuals partitioned into groups $P_1, P_2$ of sizes  $s_1$ and $s_2$ respectively. Under the assumption that $c<y_3$, the efficient structure consists of:\n  \\begin{enumerate}\n  \\item\\label{fact:eff-disjoint-2groups} \n     disjoint union of cliques if and only if  \n             \\[ 0 \\leq F_{12}  \\leq  \\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) } ;\\]\n  \\item\\label{fact:eff-bridge-2groups} \n       minimally connected cliques if and only if  \n         \\[\n          \\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) } \n            \\leq   F_{12} \\leq \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_3};\n            \\]\n  \\item\\label{fact:eff-redundant-2groups} \n       redundantly connected cliques if  and only if \n               \\[ \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_{3}}\n                \\leq F_{12} \\leq \\dfrac{c}{ y_3 };\\]\n   \\item\\label{fact:eff-comember-2groups} \n        maximally connected cliques if and   only if \n        \\[\\dfrac{c}{ y_3 }\\leq F_{12} \\leq 1.\\]      \n  \\end{enumerate}  \n\\end{theorem}\n\\begin{proof}\nFrom Theorem \\ref{thm: Pairwise Stability of Cliques}, we know that addition of an intra-connection results in increasing the payoff of both individuals involved in that link, as well as the social welfare, and removal of any intra-connection causes loss for both individuals and decreases social welfare. \nAs a result, the efficient structure consists of two cliques. Suppose $i\\in P_{1}$ and $j\\in P_{2}$. Let $\\mathcal{E}_{0}=\\mathcal{E}_{K_{s_{1}}}\\bigcup \\mathcal{E}_{K_{s_{2}}}$ be the union of two cliques of sizes $s_1$ and $s_2$. We have \n\\begin{equation}\n\\begin{aligned}\n U_{k}(\\mathcal{E}_{0}\\cup&\\{(i,j)\\}) -U_{k}(\\mathcal{E}_{0}) \n\\\\\n&= \\left\\{\n\\begin{aligned}\n& F_{12}\\big(\\delta+(s_{2}-1)\\delta^{2}\\big)-c, && k=i;\\\\\n& F_{12}\\big(\\delta+(s_{1}-1)\\delta^{2}\\big)-c, && k=j;\\\\\n& F_{12}\\big(\\delta^{2}+(s_{2}-1)\\delta^{3}\\big), && k\\in P_{1}, k\\neq i;\\\\\n& F_{12}\\big(\\delta^{2}+(s_{1}-1)\\delta^{3}\\big), && k\\in P_{2}, k\\neq j.\n\\end{aligned}\n\\right.\n\\end{aligned}\n\\end{equation}\nIt follows that \n\\begin{multline*}\nv\\big((\\mathcal{E}_{0}\\cup \\{(i,j)\\}\\big) -v(\\mathcal{E}_{0})\n=2F_{12}[\\delta+(s_{1}-1)\\delta^{2}+(s_{2}-1)\\delta^{2}\\\\\n+(s_{1}-1)(s_{2}-1)\\delta^{3}]-2c.\n\\end{multline*}\n$\\mathcal{E}_{0}$ is the efficient structure if and only if $v\\big((\\mathcal{E}_{0}\\cup \\{(i,j)\\}\\big) -v(\\mathcal{E}_{0})\\leq 0$, which is equivalent to:  \n\\begin{equation*}\\label{eqn:F12-efficent-bound-i}\n\\begin{aligned}\nF_{12} &\\leq   \\dfrac{c}{ \\delta+(s_{1}-1)\\delta^{2}+(s_{2}-1)\\delta^{2} +(s_{1}-1)(s_{2}-1)\\delta^{3}} \\\\\n&= \\dfrac{c \\delta}{ y_1(s_{1})y_1(s_{2})}.\n\\end{aligned}\\end{equation*}\nThis concludes the proof of~\\ref{fact:eff-disjoint-2groups}.\n\nNow let $\\mathcal{E}_{1}=\\mathcal{E}_{0}\\bigcup \\{(i,j)\\}$.  As elaborated above,  $v(\\mathcal{E}_{1}) -v(\\mathcal{E}_{0})\\geq 0$ if and only if \n\\[\\begin{aligned}\nF_{12} &\\geq   \\dfrac{c}{\\delta+(s_{1}-1)\\delta^{2}+(s_{2}-1)\\delta^{2} +(s_{1}-1)(s_{2}-1)\\delta^{3}}\\\\\n&=\\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) } .\n\\end{aligned}\\]\nSuppose  $\\hat i \\in P_{1}$ and $\\hat j\\in P_{2}$. For the case of $\\hat i \\neq i$ and $\\hat j\\neq j$, we have \n\\begin{equation*}\nU_{k}(\\mathcal{E}_{1}\\cup \\{(\\hat i,\\hat j)\\})-U_{k}(\\mathcal{E}_{1})=\n \\left\\{\n\\begin{aligned}\n& F_{12}y_{2}(s_{2})-c, && k=\\hat i \\\\\n&F_{12}y_{2}(s_{1})-c, && k=\\hat j\\\\\n& 0, && k =  i, j \\\\\n& F_{12}(\\delta^{2}-\\delta^{3}), && \\text{otherwise.}\n\\end{aligned}\n\\right.\n\\end{equation*}\nFor the case of $\\hat i =i$ and $\\hat j\\neq j$, we have \n\\begin{equation*}\n\\begin{aligned}\nU_{k}(\\mathcal{E}_{1}\\cup&\\{( i,\\hat j)\\})-U_{k}(\\mathcal{E}_{1})\\\\\n&= \\left\\{\n\\begin{aligned}\n& F_{12}(\\delta-\\delta^{2})-c, && k=\ti, \\\\\n&F_{12}y_{2}(s_{1})-c, && k=\\hat j,\\\\\n& 0, && k\\in P_{2} ~\\text{and}~ k\\neq \\hat j ,\\\\\n& F_{12}(\\delta^{2}-\\delta^{3}), && k\\in P_{1} ~\\text{and}~ k\\neq i.\n\\end{aligned}\n\\right.\n\\end{aligned}\n\\end{equation*}\nFor the case of $\\hat i \\neq i$ and $\\hat j=j$, we have \n\\begin{equation*}\n\\begin{aligned}\nU_{k}(\\mathcal{E}_{1}\\cup&\\{(\\hat i, j)\\})-U_{k}(\\mathcal{E}_{1})\\\\\n&= \\left\\{\n\\begin{aligned}\n& F_{12}(\\delta-\\delta^{2})-c, && k=\tj, \\\\\n&F_{12}y_{2}(s_{2})-c, && k=\\hat i,\\\\\n& 0, && k\\in P_{1} ~\\text{and}~ k\\neq \\hat i,\\\\\n& F_{12}(\\delta^{2}-\\delta^{3}), && k\\in P_{2} ~\\text{and}~ k\\neq j.\n\\end{aligned}\n\\right.\n\\end{aligned}\n\\end{equation*}\nIt follows that \n\\begin{equation*}\n\\begin{aligned}\n&v(\\mathcal{E}_{1}\\cup \\{(\\hat i,\\hat j)\\})-v(\\mathcal{E}_{1})\\\\\n&=\\left\\{\n\\begin{aligned}\n& F_{12}[y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)(\\delta^{2}-\\delta^{3})]-2c, \\\\\n&\\quad  \\quad \\quad \\quad \\quad\\quad \\quad     \\hat j \\neq j, \\hat i \\neq i,\\\\\n&2F_{12}y_{2}(s_{1})-2c , \\hat i  =  i,\\hat j \\neq j,\\\\\n&2F_{12}y_{2}(s_{2})-2c ,  \\hat i  \\neq i,\\hat j = j.\\\\\n\\end{aligned}\n\\right.\n\\end{aligned}\n\\end{equation*}\n\nSince $y_2(s)$ is a monotonically increasing function of $s$ and  \n\\[ \\begin{aligned}\n&\\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)(\\delta^{2}-\\delta^{3})}\\\\\n&\\qquad=\\dfrac{c}{y_{2}(s_{1})+(s_{2}-2)(\\delta^{2}-\\delta^{3})}\\\\\n&\\qquad=\\dfrac{c}{y_{2}(s_{2})+(s_{1}-2)(\\delta^{2}-\\delta^{3})},\n\\end{aligned}\\]\n we have\n\\begin{multline*}\n\\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)(\\delta^{2}-\\delta^{3})}\\\\\n< \\min\\limits_{s \\in\\{s_{1}, s_{2}\\}}\\dfrac{c}{y_{2}(s)}.\n\\end{multline*}\nWe then conclude that $v(\\mathcal{E}_{2}\\cup \\{(\\hat i,\\hat j)\\})-v(\\mathcal{E}_{2})\\leq 0$ if and only if $F_{12} \\leq \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_3}$.  Statement~\\ref{fact:eff-bridge-2groups} follows accordingly.\n\nAssume $F_{12}\\geq  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_2(s)} $. From statement~\\ref{fact:eff-bridge-2groups}, we know that the efficient structure has at least two interconnections. For any agent $\\hat i$ from $P_{1}$ and $\\hat j$ from $P_{2}$, as elaborated in the proof of Theorem \\ref{thm:existence-convergence-two}, one has\n\\[\nv(\\mathcal{E} \\cup \\{ (\\hat{i},\\hat j ) \\} -v(\\mathcal{E} \\setminus \\{ (\\hat{i},\\hat j) \\}\\geq 2[F_{12}(\\delta-\\delta^{2})-c],\n\\]\nwhich concludes statements \\ref{fact:eff-redundant-2groups} and \\ref{fact:eff-comember-2groups}.\n\n\\end{proof} \n\n\\begin{figure*}\n\t\\begin{center} \n\t\t\\subfloat[Ranges of parameter space for high value benefit ($\\delta \\geq  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{s-3}{n-3}$)]{\\includegraphics[width=0.9\\linewidth]{efficiency-stability-2groups}\\label{fig:}}\\\\\n\t\t\\subfloat[Ranges of parameter space for low value benefit ($\\delta <  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{s-3}{n-3}$)]{\\includegraphics[width=0.9\\linewidth]{efficiency-stability-2groups_2}\\label{fig:}}\n\t\t\\caption{\\small An illustration of the ranges of parameter space in Theorems~\\ref{thm:existence-convergence-two} and~\\ref{thm:efficiency-two}. The intervals indicated above the horizontal line refer to bounds of pairwise stability and those below the horizontal line refer to bounds of efficiency.}\n\t\t\\label{fig:efficiency-stability-2groups}\n\t\\end{center}\n\\end{figure*}\n\n\nFrom the properties of $y_{1}(s)$, $y_{2}(s)$, and $y_{3}$, we have:\n\\begin{enumerate}\n\\item if $\\delta \\geq  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{s-3}{n-3}$, then\n\\[\\begin{aligned}\n \\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) }    \n<&\n \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)} \\\\\n  \\leq &\n   \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_3} \\\\\n <&\n  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{  y_2(s)};\n\\end{aligned}\n\\]\n\\item  if $\\delta <  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{s-3}{n-3}$, then\n\\[\\begin{aligned}\n \\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) }   & < \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_3} \n \\\\&<\n \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)}\\\\\n & <\n  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{  y_2(s)}.\n\\end{aligned}\\]\n\\end{enumerate}\nFrom Theorems \\ref{thm:existence-convergence-two} and \\ref{thm:efficiency-two}, we can directly obtain the conditions for the equivalence between pairwise stability and efficiency. \n\n\n\\begin{corollary}\\label{the: equivalence-stability-efficiency-two}\nConsider $n$ individuals partitioned into groups, $P_1$ and $P_2$  of sizes  $s_1$ and $s_2$ respectively. Under the assumption $c<y_3$, the efficient structure has equal or more interconnections than the pairwise stable structure. Moreover, the efficient structure is the pairwise stable structure if \n\\begin{enumerate}\n\\item for $\\delta \\geq  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{s-3}{n-3},$\n\\[\\begin{aligned}\n&F_{12}\\in \\left[0, \\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) } \\right] \\\\\n&\\bigcup  \\left[ \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)} ,  \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_3}\\right] \\\\\n&\\bigcup \n\\left[ \\dfrac{c}{  y_3},1\\right], \\text{ or}\n\\end{aligned}\\] \n\\item  for $ \\delta <  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{s-3}{n-3}, $\n\n$F_{12}\\in \\left[0,  \\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) }  \\right] \\bigcup \n\\left[\\dfrac{c}{  y_3},1\\right], $\n\\end{enumerate}\n\n\\end{corollary}\nWe illustrate the compatibility (and incompatibility) between the pairwise stable and efficient structures in Fig.~\\ref{fig:efficiency-stability-2groups}. The intersections of efficient and pairwise stable structures are highlighted. \n\n\n\\begin{corollary}\nConsider $n$ individuals partitioned into groups $P_1$ and $P_2$  of sizes  $s_1$ and $s_2$, respectively. Under the assumption $c<y_3$,  if the efficient structure is not pairwise stable, then the efficient structure has more interconnections than the pairwise stable structure.\n\\end{corollary}\n\\begin{proof}\nIf the efficient structure is not pairwise stable, it implies that it can be changed by Formation Dynamics. Suppose that it has less interconnections than pairwise stable structure. Then, adding interconnections can make it pairwise stable, which means that both players involved in the interconnection benefit from this interconnection. Then the social welfare increases which is conflict with the fact that the structure is efficient. Thus, the efficient structure has more interconnections than the pairwise stable structure.\n\\end{proof}\n\n \n\\begin{remark}\\label{rm:PoA}\nFor the two intervals $\\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) }   <F_{12}<\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)}$ and $ \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_3}<F_{12}<  \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{  y_2(s)}$, the pairwise stable structure is not efficient. For $ \\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{  y_2(s)}\\geq F_{12}< \\dfrac{c}{  y_3}$, the pairwise stable structure might be not efficient. This is because individuals are rational and selfish (i.e., they only care about their own payoffs.) Therefore, total social utility might drop when an individual tries to maximize its own payoff. Specifically, \n\\begin{enumerate}\n\\item when $ \\dfrac{c\\delta}{y_{1}(s_{1})y_1(s_{2}) }  <F_{12}<\\max\\limits_{s \\in\\{s_{1}, s_{2}\\}} \\dfrac{c}{ y_1(s)}$,  an individual experiences loss by  having an interconnection, although that interconnection could bring profit for other individuals and, therefore, result in an increase of total utility. The individual refuses to {add} or maintain this interconnection.\n\\item when $ \\dfrac{2c}{ y_{2}(s_{2})+y_{2}(s_{1})+(s_{1}+s_{2}-4)\\delta y_3}<F_{12}< \\dfrac{c}{  y_3}$,  as mentioned in the proof of Theorem \\ref{thm:efficiency-two}, there exists at least one interconnection $\\{(i,j)\\}, i\\in P_{1}, j\\in P_{2}$. For the two individuals $\\hat i\\in P_{1}$ and $\\hat j \\in P_{2}$ ($\\hat i\\neq i, \\hat j \\neq j$), an interconnection causes loss for the player from the larger group, but brings profit for all other individuals, which results in removal of the interconnection or rejection in adding it and, therefore, in total utility loss. \n\\end{enumerate}\n\\end{remark}\n\nFig.~\\ref{fig:2x2} shows the number of interconnections and social welfare for both efficient and stable structures for two cliques of sizes 3 and 5, where the value of $\\delta=0.5$ and $c=0.2$. Note that:\n\\begin{enumerate}\n\\item As shown in Fig.~\\ref{fig:interC_2x2}, when $F_{12}=\\dfrac{c}{y_3}$ which evaluates to 0.8 for the choice of our parameters, there exist non unique pairwise stable and efficient structures, and the number of interconnections for those structures vary between 3 and 15.\n\\item  As shown in Fig.~\\ref{fig:interC_2x2} and Fig.~\\ref{fig:value_2x2}, for $F_{12}\\in (0.67, 0.2) \\cup (0.23,0.53)$, the pairwise stable structure is not efficient and thus the social welfare of efficient structures is always greater than that of pairwise stable structures. Fig. \\ref{fig:PoA} shows plot of PoA as a function of $F_{12}$. We observe that the slope for $F_{12}\\in (0.67, 0.2) $ is the highest. This is because in this interval, the pairwise stable structure dose not have any interconnection, whereas the efficient structure has one interconnection. This link brings large value for the overall network and makes a large difference in social welfare of two kinds of structures. \n\\end{enumerate}\n\n \\begin{figure}\n\t\\begin{center} \n\t\t\\subfloat[Number of Interconnections]{\\includegraphics[width=0.99\\linewidth]{interC_2x2_2}\\label{fig:interC_2x2}}\\\\\n\t\t\\subfloat[Social Welfare]{\\includegraphics[width=0.95\\linewidth]{values2x2}\\label{fig:value_2x2}}\n\t\t\\caption{\\small Plots of no. of interconnections and social welfare as a function of $F_{12}$ for two groups of sizes 3 and 5, $\\delta=0.5$, $c=0.2$.}\\label{fig:2x2}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t{\\includegraphics[width=0.9\\linewidth]{PoA}}\n\t\t\\caption{\\small Plot of price of anarchy as a function of $F_{12}$ for two groups of sizes 3 and 5, $\\delta=0.5$, $c=0.2$.}\\label{fig:PoA}\n\\end{figure}\nWe next discuss the evolution of intra vs. interconnections according to Formation Dynamics, when starting from an empty graph.\n\n\\begin{remark}[Formation of intra- vs. interconnections] Starting from an empty (or sparse) graph, initially the speed of formation of intra-connections is generally higher than that of interconnections. Fig.~\\ref{fig:intra_inter} illustrates the fact that at the beginning of the formation dynamics, a certain amount of intra-connections is required to produce enough incentives for both groups to communicate through an interconnection.\n\\end{remark}\n\\begin{figure}[]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{intra_inter}\n        \\caption{Time evolution of the number of inter- and intra-connections for two cliques of size 7.} \\label{fig:intra_inter}\n\\end{figure} \n\n\\section{Multigroup Connectivity Structure}\n\nIn what follows we analyze the more general case of having more than two groups. Consider $n$ individuals partitioned into groups $P_1,\\dots,P_m$, and the payoff function defined above with 1-benefit $\\delta<1$ and edge cost $c$, and functions $y_1$, $y_2$, and $y_3$. From Theorem \\ref{thm: Pairwise Stability of Cliques} we know that each group forms a clique when $y_3>c$. We introduce the undirected graph $\\mathcal T=(\\mathcal V_{P}, \\mathcal{E}_{\\mathcal T} )$, whose nodes represent groups and $(\\alpha,\\beta)\\in \\mathcal{E}_{\\mathcal T} $ if there exists at least one connection between $P_{\\alpha}$ and $P_{\\beta}$.\n\n\\begin{theorem}[Sufficiency Condition for Minimally Connected Cliques]\\label{thm:multigroup_stability}\nConsider $n$ individuals partitioned into groups $P_1,\\dots,P_m$ of sizes $s_{1},\\dots, s_{m}$ respectively. \nAssume that $c<y_3$ and $\\mathcal T$ is connected. Then, there exists a pairwise stable structure consisting of minimally connected cliques along $\\mathcal T$, if\n \\begin{enumerate}\n   \\item \\label{fact:nonedge-equal} for all $(\\alpha, \\beta)\\in \\mathcal{E}_{\\mathcal T}$, $\\alpha, \\beta \\in \\until{m} $,  \\\\   \n   \\begin{equation}\\label{utility-change-1}\n   \\begin{aligned}\n  & \\sum _{\\lambda\\neq \\alpha, \\lambda=1}^{m} F_{\\alpha \\lambda} \n   ( \\delta^{d'_{\\alpha\\lambda}}-\\delta^{d_{\\alpha\\lambda}})(1+(s_{\\lambda}-1)\\delta)>c,  \\\\\n  & \\sum _{\\lambda\\neq \\beta, \\lambda=1}^{m} F_{\\beta \\lambda} \n   ( \\delta^{d'_{\\beta\\lambda}}-\\delta^{d_{\\beta\\lambda}})(1+(s_{\\lambda}-1)\\delta)>c, \\\\\n & F_{\\alpha \\beta} <\\max_{s\\in \\{s_{\\alpha},s_{\\beta}\\}}\\frac{c}{y_{2}(s)};  \\text{ and}\n   \\end{aligned}\n   \\end{equation}\n  \\item \\label{fact:tree-equal} for all $(\\alpha, \\beta) \\notin \\mathcal{E}_{\\mathcal T}$, $\\alpha, \\beta \\in \\until{m} $,   \\\\\n     \\begin{equation}\\label{utility-change-2}\n     \\begin{aligned}\n      \\sum _{\\lambda\\neq \\alpha, \\lambda=1}^{m} F_{\\alpha \\lambda} \n   ( \\delta^{d'_{\\alpha\\lambda}}-\\delta^{d_{\\alpha\\lambda}})(1+(s_{\\lambda}-1)\\delta)<c,\\\\\n   ~ or  \\sum _{\\lambda\\neq \\beta, \\lambda=1}^{m} F_{\\beta \\lambda} \n   ( \\delta^{d'_{\\beta\\lambda}}-\\delta^{d_{\\beta\\lambda}})(1+(s_{\\lambda}-1)\\delta)<c,\n     \\end{aligned}\n   \\end{equation}\n\\end{enumerate}    \nwhere $d'_{\\mu \\lambda}=d_{\\mu \\lambda}(\\mathcal{E}_{\\mathcal T}\\cup \\{(\\alpha, \\beta)\\})$ and $d_{\\mu \\lambda}=d_{\\mu \\lambda}(\\mathcal{E}_{\\mathcal T}\\setminus \\{(\\alpha, \\beta)\\})$.\n\\end{theorem}\n\\begin{proof}\nSuppose the network $\\mathcal{E}_{0}$ (consisting of disjoint cliques) be connected along $\\mathcal T$ and satisfy\n\\begin{enumerate}\n\\item if $(\\alpha, \\beta)\\in \\mathcal{E}_{\\mathcal T}$, there is only one inter-link between $P_{\\alpha}$ and $P_{\\beta}$;\n\\item for every group $P_{\\alpha}$, only one agent $i_{\\alpha}$ has inter-links. \n\\end{enumerate}\n For any pair of $(\\alpha, \\beta)$,  let $\\mathcal{E}''=\\mathcal{E}_{0}\\cup\\{(i_{\\alpha},i_{\\beta})\\}$ and $\\mathcal{E}'=\\mathcal{E}_{0}\\setminus\\{(i_{\\alpha},i_{\\beta})\\}$. it is easy to find that \n \\begin{align*}\nU_i(\\mathcal{E}'')-U_i(\\mathcal{E}')=\\sum_{k=1,k\\neq \\alpha}^{m} F_{k \\alpha}\\sum_{l\\in P_{k}}(\\delta^{d_{il}(\\mathcal{E}'')}-\\delta^{d_{il}(\\mathcal{E}')}) -c.\n\\end{align*}\n Therefore, we have\n \\[\n U_{i_{\\alpha}}(\\mathcal{E}'')- U_{i_{\\alpha}}(\\mathcal{E}')=\n  \\sum _{\\lambda\\neq \\alpha, \\lambda=1}^{m} F_{\\alpha \\lambda} \n   ( \\delta^{d'_{\\alpha\\lambda}}-\\delta^{d_{\\alpha\\lambda}})(1+(s_{\\lambda}-1)\\delta) -c\n\\]\nand \n\\[\n  U_{i_{\\beta}}(\\mathcal{E}'')- U_{i_{\\alpha}}(\\mathcal{E}')=  \\sum _{\\lambda\\neq \\beta, \\lambda=1}^{m} F_{\\beta \\lambda} \n   ( \\delta^{d'_{\\beta\\lambda}}-\\delta^{d_{\\beta\\lambda}})(1+(s_{\\lambda}-1)\\delta)-c.\n \\]\nTherefore, for $(\\alpha, \\beta)\\in \\mathcal{E}_{\\mathcal T}$,  it follows from (\\ref{utility-change-1}) that $\\mathcal{E}''$ defeats $\\mathcal{E}'$.  \n Similar to the proof of Theorem \\ref{thm:existence-convergence-two}, since $F_{\\alpha \\beta} <\\max_{s\\in \\{s_{\\alpha},s_{\\beta}\\}}\\frac{c}{y_{2}(s)}$, there only exists one inter-link between $P_{\\alpha}$ and $P_{\\beta}$.\n For $(\\alpha, \\beta)\\notin \\mathcal{E}_{\\mathcal T}$, (\\ref{utility-change-2}) implies that there exist no inter-link between $P_{\\alpha}$ and $P_{\\beta}$.\n \n By Theorem \\ref{lem:stable-by-dynamics}, since network $\\mathcal{E}_{0}$ can not be changed under dynamics, we can conclude that network $\\mathcal{E}_{0}$ is stable.\n\\end{proof}\n\n\\begin{remark}\nTheorem~\\ref{thm:multigroup_stability} answers the question: given a certain matrix $F$ and graph structure $\\mathcal{E}$ is $\\mathcal{E}$ pairwise stable or not?\n\\end{remark}\n\n\\begin{corollary}\nFor the special case of interconnection structure being a star, with $P_{\\gamma}$ as the central group, the sufficient condition of Theorem~\\ref{thm:multigroup_stability} can be simplified as follows:\\begin{enumerate}\n   \\item for all $\\alpha \\in \\until{m} $, $(\\alpha \\neq \\gamma)$ \\\\   \n   \\begin{equation*}\n   \\begin{aligned}\n & F_{\\alpha \\gamma} >\\max_{s\\in \\{s_{\\alpha},s_{\\gamma}\\}}\\frac{c}{y_{1}(s)}; \\text{ and}\n   \\end{aligned}\n   \\end{equation*}\n  \\item for all $(\\alpha, \\beta)  \\in \\until{m} $, $(\\alpha, \\beta \\neq \\gamma)$,   \\\\\n  \\begin{equation*}\n   \\begin{aligned}\n & F_{\\alpha \\beta} <\\max_{s\\in \\{s_{\\alpha},s_{\\beta}\\}}\\frac{c}{y_{2}(s)}.\n   \\end{aligned}\n   \\end{equation*}\n\\end{enumerate}    \n\\end{corollary}\n\n\nIn the following example, we illustrate that due to randomness in choosing the pair of players, Formation Dynamic does not always converge to a unique stable structure even for the same initial network structure and matrix $F$.\n\n\n\\begin{example}\\label{ex:multigroup-convergence}\nConsider the case where we have $\\dfrac{c}{y_{1}(s)}<F_{\\alpha \\beta}<\\dfrac{c}{y_2(s)}$ for all $\\alpha, \\beta$, and that $\\delta<\\dfrac{\\sqrt{5}-1}{2}$. We have five equal size groups named $\\{1, 2, \\dots, 5\\}$. Two different processes are shown in Fig.~\\ref{fig:dynamics_ex}. \n\\begin{enumerate}[Process A.]\n\\item  For the process shown in Fig.~\\ref{fig:dynamics_ex2} the order of pair selection is as follows: $(1, 2) \\rightarrow (1, 3) \\rightarrow (1, 4) \\rightarrow (1, 4) \\rightarrow (1,5)$. Note that by $(1,2)$ we mean an individual selected from group 1 paired with an individual from group 2, which results in the star graph being the convergent pairwise stable structure.\n\\item Now, consider the process shown in Fig.~\\ref{fig:dynamics_ex1} for which the order of pairs of groups selected is as follows:  $(2, 3) \\rightarrow (1, 2) \\rightarrow (1, 3) \\rightarrow (1,4)\\rightarrow (3, 5) \\rightarrow (2,4) \\rightarrow (4,5)$. At the very last step, we have:\n\\[\n\\begin{aligned}\n&U_i\\big( \\mathcal{E} \\cup \\{4,5\\} -U_i(\\mathcal{E} \\setminus \\{ 4,5\\}) \\big)\\\\\n=& F_{45} \\big(y_1(s) \\big) - F_{45} \\big( \\delta^3 y_1(s) \\big) \\\\\n&\\quad + F_{35} \\big( \\delta y_1(s) \\big) - F_{35} \\big( \\delta^2 y_1(s)  \\big)-c .\n\\end{aligned}\\]\nSince $F_{35}, F_{45}<\\dfrac{c}{y_1(s)}$ and $\\delta<\\dfrac{\\sqrt{5}-1}{2}$, we conclude that  \n\\[F_{45} (1-\\delta^3) + F_{35} (\\delta -\\delta^2) > \\dfrac{c}{y_1(s)}.\n\\]\n Consequently, we obtain \n \\[U_i\\big( \\mathcal{E} \\cup \\{4,5\\} -U_i(\\mathcal{E} \\setminus \\{ 4,5\\}) \\big)>0\\]\n  which means that the connection (4,5) is formed. Now since we have $F_{\\alpha \\beta}< \\dfrac{c}{y_2(s)}$, no connected triad and thereby, no additional links will be formed. Also no link will be removed. Therefore, the final structure in Fig.~\\ref{fig:dynamics_ex1}, which is a ring, is stable.\n\n\\end{enumerate}\n\\end{example}\n \\begin{figure}[h]\n\t\\begin{center} \n\t\t\\subfloat[Process A]{\\includegraphics[width=0.9\\linewidth]{dynamics_ex2}\\label{fig:dynamics_ex2}}\\quad\n\t\t\\subfloat[Process B]{\\includegraphics[width=0.9\\linewidth]{dynamics_ex1}\\label{fig:dynamics_ex1}}\n\t\t\\caption{The processes of Example \\ref{ex:multigroup-convergence}. At each step, shaded nodes represent the groups which the selected individuals belong to, and the outcome of the game (action taken regarding link addition, link removal, or indifference) is represented in the next.}\\label{fig:dynamics_ex}\n\t\\end{center}\n\\end{figure}\n\n\nExample \\ref{ex:multigroup-convergence} shows that, based on the order of the sequence of selected pairs, we can have two or possibly more convergent stable structures, and therefore, the convergence results cannot be generalized and the convergent structure is not always unique.\n\n\nFrom Theorem \\ref{thm: Pairwise Stability of Cliques} we know that each group forms a clique. We now analyze the interconnections among those cliques. Theorem~\\ref{statements on redundant and comembers} addresses the redundancy of interconnections.\n\n\n\\begin{theorem}[Formation of Redundancies]\\label{statements on redundant and comembers}\n  Consider $n$ individuals partitioned into groups $P_1,\\dots,P_m$ of sizes $s_1, s_2, \\dots s_m$.  Suppose that  $c<y_3$. Then, under Formation Dynamics, \n  \\begin{enumerate}\n   \\item\\label{fact:redundant} redundant interconnections between $P_{\\alpha}$ and $P_{\\beta}$ will be formed and never removed,  if \n  $  F_{\\alpha \\beta} > \\max\\limits_{s \\in\\{s_{\\alpha}, s_{\\beta}\\}} \\dfrac{c}{ y_2(s)}$, and \n   \\item\\label{fact:comember} maximal interconnections between $P_{\\alpha}$ and $P_{\\beta}$ will be formed and never removed, if $\\dfrac{c}{ y_3 }<F_{\\alpha \\beta} \\leq 1$.      \n  \\end{enumerate}  \n\\end{theorem}\n\n\nFig.~\\ref{fig:bounds-samesize} illustrates the four scenarios for $F_{\\alpha \\beta}$'s. The horizontal axis corresponds to the values of $F_{\\alpha \\beta}$ where $\\{\\alpha \\beta\\}$ belongs to the edge-set of the spanning tree, and the vertical axis corresponds to all other $F_{\\alpha \\beta}$'s. \n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=0.99\\linewidth]{bounds-size5-thm2.pdf}\n        \\caption{An illustration of ranges of parameter space in Theorem~\\ref{statements on redundant and comembers} \n        } \\label{fig:bounds-samesize}  \n       \\end{figure} \n\n\n\n\\begin{figure}[ht]\n\t\\begin{center} \n\t\t\\subfloat[Number of Interconnections]{\\includegraphics[width=0.8\\linewidth]{interC_3x3_general_ave_mesh}\\label{}}\\\\\n\t\t\\subfloat[Social Welfare]{\\includegraphics[width=0.8\\linewidth]{value_3x3_general_ave_mesh}\\label{}}\n\t\t\\caption{\\small Three groups of sizes 3, 4, and 5, $\\delta=0.5$, $c=0.2$}\\label{fig:3x3_general}\n\t\\end{center}\n\\end{figure}\n\n\\begin{theorem}\\label{thm:efficiency1} (Efficiency) For $n$ individuals partitioned into groups according to $P$, the efficient structure requires the same node to be chosen from each clique to provide bridges to other cliques, i.e., for a fixed interconnection structure and number of nodes, choosing the same representative from each clique increases the social welfare. \n\\end{theorem}\n\\begin{proof}\nSuppose that the density and the structure of interconnections are fixed. It is easy to see that by choosing the same representative from each clique, the distance between individuals from different cliques would be shortened, resulting in the term $\\delta^d_{ij}$ in equations~\\eqref{payoff-function} being larger. Therefore, the social welfare will increase. \n\\end{proof}\n\n\n\nHowever, this structure is unlikely to be pairwise stable since the\nrepresentative would bear a high cost for maintaining these\ninterconnections. Figure~\\ref{fig:eff-thm} illustrates two networks with\nthe same interconnection structureswhere one network has higher social\nwelfare due to each group having only one representative.\n\n\\begin{figure}[ht]\n\t\\begin{center} \n\t\t\\includegraphics[width=0.95\\linewidth]{Efficiency_thm.pdf}\\label{fig:efficient-line}\t\t\\caption{\\small In network a) each cliques has only one representative, whereas in figure b) some cliques have multiple representatives.}\\label{fig:eff-thm}\n\t\\end{center}\n\\end{figure}\n\n\\section{Conclusion}   \nThis paper proposes the first strategic network formation model that, given a matrix specifying the frequency of a coordination problem among groups, identifies the conditions that result in multigroup formation. The model deviates from the seminal papers on strategic network formation in that it accounts for heterogeneous frequency of control problems arising among the individuals and investigates pairwise stability and efficiency of multigroup connectivity structures, as well as convergence of the formation dynamics. \n\nIn our model link formations occur bilaterally and thus many of the classical game-theoretic concepts do not apply to our framework. In particular, to study equilibrium structures, we utilize the concept of pairwise stability. A key challenge in our problem stems from the fact that not many tools are available for rigorous analysis or that they cannot be applied to the case of heterogeneous coordination problems among groups.\n\nWe identified the ranges of parameters where pairwise stable and efficient structures do and do not coincide and concluded that, for two-group structures, the efficient structures always has the same or a larger number of links than the pairwise stable ones. We also considered the price of anarchy and observed that the highest value occurs for the case when pairwise stable structures consist of disjoint union of cliques and the efficient structure has one link. Similar to the classical models, at the two ends of the spectrum of link values there is an overlap between efficient and stable structures.\n\nWe presented the conditions that result in the formation dynamics starting from an invariant set converge to cliques, and provided rigorous results for the number of interconnections in two-group structures. However, exact identification of the boundaries that result in certain number of interconnections among arbitrary number of groups with arbitrary size and interconnection structure is out of scope of this paper.\n\nWe note that by providing a full characterization of pairwise stability and efficiency for a two-group model, we focus on local topologies versus global topologies, as the individual interconnections can capture valuable information about the whole network and that all interconnections have subsets of two groups. This can be  interpreted into taking the distance only for the people in one's group or in the next immediate group in the utility function.\n\n\n\n\\section*{Acknowledgment}\nThe authors thank professors Ambuj K.\\ Singh and Noah E.\\ Friedkin for\ntheir valuable comments and suggestions. This material is based upon work\nsupported by, or in part by, the U.S.~Army Research Laboratory and the\nU.S.~Army Research Office under grant numbers W911NF-15-1-0577, and the National Natural Science Foundation of China under grant 61963032.\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\\bibliographystyle{plainurl+isbn}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe large variety of unique physical phenomena offered by quasi low-dimensional systems arouse the unfading interest of researchers exploring the field of condensed matter physics. Large anisotropy of the electronic structure is a precursor for the Peierls transition towards charge density wave (CDW) with electronic carrier condensation and Fermi surface nesting\\cite{Monceau2012, Gruner2000, Gruner1988}.\nLow dimensionality has been suggested to play a crucial role in high temperature superconductivity while charge density wave has been found to emerge in all the phase diagrams of the cuprate superconductors\\cite{Caprara2017, Caplan2017, Laliberte2018, Cyr2018}. This fact additionally amplifies the interest in the interactions between various types of electronic, quantum, and magnetic ordering\\cite{Jacques2014, Chang2012, Chang2016, Graf2004, Gruner2017, Cho2018}. \n\nRecently, extensive attention has been devoted to the family of ternary carbides $R$NiC$_2$, where $R$ is rare earth element.  This group of materials offers the rare opportunity to tune the magnetic ground state via replacement of the rare earth atom (the nickel atoms carry no magnetic moment). Most of the lanthanide-based members of this family are antiferromagnets ($R$ = Ce, Nd, Gd, Tb, Dy, Ho, Er and Tm) with N\\'eel temperature varying from 3.4 K for HoNiC$_2$ to 25 K for TbNiC$_2$\\cite{Schafer1997, Onodera1995, Onodera1998, Hanasaki2011, Roman2018_1, Steiner2018}; SmNiC$_2$ is a ferromagnet with $T_C$ = 17.5 K \\cite{Onodera1998}, PrNiC$_2$ shows a weak magnetic anomaly at $T^*$ = 8 K, LaNiC$_2$ is a noncentrosymmetric superconductor with $T_{c}$ = 2.9 K\\cite{Wiendlocha2016, lee_superconductivity_1996, Pecharsky1998, Quintanilla2010, Landaeta2017} while LuNiC$_2$ was recently reported as a plain paramagnet down to 1.9 K\\cite{Steiner2018}. Furthermore, most of the $R$NiC$_2$ compounds (with the exception of $R$ = La and Ce) show charge density wave with Peierls  temperature ($T_{CDW}$) not only higher than $T_N$, $T_C$ and $T^*$, but exceeding 300 K for the late lanthanides ($R$ = Ho - Lu)\\cite{Murase2004, Ahmad2015, Michor2017, Laverock2009, Shimomura2016}. So far, the charge density wave has never been found in $R$NiC$_2$ compounds, with $R$ outside of the lanthanide group. The electronic structure calculations however, have revealed the resemblance between the Fermi surface topology of YNiC$_2$\\cite{Hase2009} and lanthanide-based $R$NiC$_2$\\cite{Laverock2009, Steiner2018} showing CDW. Alas, the fermiology of this compound has not been discussed in the terms of nesting.\n\nRemarkably, magnetic ordering has been found to mutually interact with the CDW; in SmNiC$_2$ the ferromagnetic transition destroys the charge density wave\\cite{Shimomura2009, Hanasaki2012, Kim2012}, while in NdNiC$_2$ and GdNiC$_2$ antiferromagnetism only partially suppresses the CDW, and both entities coexist below $T_N$\\cite{Yamamoto2013, Kolincio20161, Hanasaki2017, Kolincio2017}. On the other hand, this magnetic anomaly has been found to enhance the nesting properties in PrNiC$_2$ \\cite{Yamamoto2013, Kolincio2017} and some signatures of a constructive influence of CDW on AFM were recently observed in GdNiC$_2$ \\cite{Kolincio20161, Hanasaki2017}, Nd$_{1-x}$Gd$_x$NiC$_2$\\cite{Roman2018_2}, and Nd$_{1-x}$La$_x$NiC$_2$ \\cite{Roman_tmp} despite the clear competition between these types of ordering.\nFor the majority of the the $R$NiC$_2$ family members, the CDW state is influenced (mostly suppressed) by magnetism, and therefore the Fermi surface nesting is partially or even completely disturbed. \n\nIn this paper we study the physical properties of paramagnetic YNiC$_2$ and LuNiC$_2$ in order to explore the consequences of a pure and fully developed Peierls transition in $R$NiC$_2$ in the absence of magnetic ordering. We report for the first time the charge density wave in YNiC$_2$ with $T_{CDW}$ = 318 K followed by a putative lock-in transition at $T_1$ = 275 K. We also observe the large positive and remarkably linear magnetoresistance, which at the lowest temperatures reaches 470 \\% for YNiC$_2$ and 50 \\% for LuNiC$_2$ with no signs of saturation up to $B$ = 9 T. By detailed analysis of magnetotransport and galvanomagnetic properties we find that this effect stems from the multiband character of the electrical conductivity and existence of the high mobility pockets remaining in the Fermi surface after uninterrupted, yet imperfect nesting characteristic for quasi-2D metals undergoing a Peierls transition\\cite{Monceau2012, Gruner2000, Gruner1988}.\n\n\n\n\\section{Experimental}\nThe polycrystalline samples of YNiC$_2$ and LuNiC$_2$  were synthesized by arc melting of elemental precursors (no excess of Y or Lu was added) followed by annealing at 900$^o$ C, according to the procedure described in detail in ref. \\cite{Kolincio2017}. Powder x-ray diffraction (pXRD) was conducted with PANalytical X'Pert PRO-MPD diffractometer using K$_\\alpha$ line of Cu spectrum. Experimental data was analyzed with the use of Fullprof software\\cite{FULLPROF}.\n\nAll the physical properties measurements were performed with a commercial Quantum Design Physical Properties Measurement System (PPMS) in the temperature range from 1.9 to 400 K and magnetic field up to 9 T. Thin ($\\phi=37 \\mu m$) Pt wires serving as electrical contacts for four-probe transport and Hall measurements were spark-welded to the polished samples surfaces. The magnetic field for Hall and magnetoresistance measurements was oriented perpendicularly to the current direction.\nThe Hall signal was measured with the reversal of the direction of the magnetic field and corrected for the parasitic longitudinal resistance component via antisymmetrization of the measured data. Specific heat measurements were conducted with a standard relaxation method on flat samples with polished surfaces. Apiezon N grease was used as a heat conducting medium for measurements in temperature range 1.9 K - 300 K. Since Apiezon N reveals a glass transition above 300 K, \\cite{Bunting1969, Kreitman1971}, the data collection at elevated temperatures was performed with Apiezon L.\n\n\\section{Results and discussion}\n\\subsection{X-ray diffraction}\n\\begin{figure} [t]\n  \\includegraphics[angle=0,width=1.0\\columnwidth]{XRD.eps}\n \\caption{\\label{XRD} Powder x-ray diffraction patterns (black open points) with LeBail fit refinement (blue solid line) for YNiC$_2$ (a) and LuNiC$_2$ (b). Bragg peak positions for YNiC$_2$ and LuNiC$_2$ phases are marked by vertical violet lines. Similar, yet green lines in panel (b) apply to the impurity phase Lu$_4$Ni$_2$C$_5$. Difference between observed and calculated pattern is represented by solid red line.}\n  \\end{figure}\n\nThe quality and phase purity of polycrystalline YNiC$_2$ and LuNiC$_2$ samples were confirmed with pXRD. The analysis of the obtained diffraction patterns, shown in Fig. \\ref{XRD} revealed that the observed peaks for both compounds are successfully indexed in the orthorombic CeNiC$_2$-type structure with space group Amm2 (\\#38). Only for LuNiC$_2$, additional reflections corresponding to Lu$_4$Ni$_2$C$_5$ impurity phase were detected. From the comparison of the highest peaks corresponding to main and impurity phases respectively, we have estimated the relative amount of Lu$_4$Ni$_2$C$_5$ to 9\\%. The pXRD results for both $R$NiC$_2$ compounds were analyzed by LeBail refinement which revealed lattice constants of YNiC$_2$: $a$ = 3.5733(1) \\AA, $b$ = 4.5082(1) \\AA, $c$ = 6.0351(1) \\AA~ and of LuNiC$_2$: $a$ = 3.4468(1) \\AA, $b$ = 4.4734(2) \\AA, $c$ = 5.9787(2) \\AA. These structural parameters are in good agreement with previous reports\\cite{Jeitschko1986, Steiner2018}.\n\n\\subsection{Transport and magnetoresistance}\n\\begin{figure*} [ht!]\n  \\includegraphics[angle=0,width=2.1\\columnwidth]{rt.eps}\n \\caption{\\label{RT} (a), (b) Temperature dependence of the normalized electrical resistivity $\\rho_{xx}\/\\rho_{xx}(400 K)$(T) measured in magnetic fields varying from 0 to 9 T for YNiC$_2$ (a) and LuNiC$_2$ (b) respectively. Upper inset of panel (a) depicits the resistivity of YNiC$_2$ in the low temperature limit. Fit to the data above superconducting transition with the equation \\ref{EQres} is shown with red solid line. Lower inset of shows the thermal hysteresis limited from below and above with temperatures $T_{1L}$ and  $T_{1H}$ respectively, in the vicinity of the lock-in transition. Legend for pannels (a) and (b) is displayed on panel (b). (c) the characteristic temperatures $T_{CDW}$ and $T_1$ for YNiC$_2$ and LuNiC$_2$ compared with the analogous values for the other members of the $R$NiC$_2$ family. The solid lines correspond to the linear scaling with unit cell volume ($V$), reported  previously \\cite{Roman2018_1}. Green points correspond to the YNiC$_2$ and LuNiC$_2$ compounds studied in this paper. The Peierls temperature for LuNiC$_2$ has been determined in ref. \\cite{Roman2018_1, Steiner2018}, while to the authors knowledge, there are no previous reports on CDW in YNiC$_2$ }\n  \\end{figure*}\n  \nThe thermal dependence of the resistivity of YNiC$_2$ and LuNiC$_2$ is depicted respectively in panels a and b of Fig. \\ref{RT}. At high temperatures, the transport properties of YNiC$_2$ show a regular metallic behavior with the resistivity lowering as the temperature is decreased. At $T$ = 318 K a local minimum followed by a hump is observed in $\\rho_{xx}$($T$). At lower temperatures, the resistivity returns to the metallic character with $\n\\frac{d\\rho_{xx}}{dT}<0$. Such behavior is typical of a Peierls transition in quasi-2D metals, where the nesting is not complete and a certain number of free carriers remain in the Fermi surface. One unusual feature here, however, is the sharp decrease of the resistivity at temperatures slightly below the maximum of $\\rho_{xx}$($T$) indicated by an arrow in the Fig. \\ref{RT}a. This stands in contrast to a smooth crossover observed for canonical Peierls transitions\\cite{McMillan1977}. To determine the character of this additional anomaly, we have performed slow heating and cooling temperature sweeps in the vicinity of the CDW transition (0.1 K\/min). The results, shown in the lower inset of Fig. \\ref{RT}a reveal the presence of a narrow thermal hysteresis opening approximately at  $T_{1H}\\simeq$  285 K and closing at  $T_{1L}\\simeq$ 265 K. The transition temperature is determined as the average of  $T_{1H}$ and  $T_{1L}$, giving $T_1$ = 275 K. Such a first order transition is expected for the preexisting incommensurate CDW locking in with the lattice and transforming into a commensurate modulation\\cite{McMillan1975} as reported beforehand for the late lanthanide-based $R$NiC$_2$\\cite{Shimomura2016, Roman2018_1}. Interestingly both the Peierls and presumed lock-in transition temperatures stand in agreement with the previously proposed linear scaling with the unit cell volume\\cite{Roman2018_1}, as shown in Fig. \\ref{RT}c. One must, however, remember that despite the analogy with the other $R$NiC$_2$, an x-ray diffuse scattering experiment performed with single crystals is required to deliver an ultimate evidence of the lock-in nature of the transition seen at $T_1$ = 275 K. An additional feature observed in the low temperature resistivity curve is a decrease of $\\rho_{xx}$ at $T$ = 4 K (upper inset of Fig. \\ref{RT}a), reminiscent of the onset of a superconducting transition. The absence of any signature of the bulk superconductivity in specific heat at this temperature allows us to attribute this anomaly to the presence of a trace amount of YC$_2$ impurity\\cite{Gulden_1997}, not detected by x-ray diffraction. To estimate the genuine value of $\\rho_{xx}$ for YNiC$_2$ at lowest $T$, we have fitted the temperature range 5 K $<T<$ 50 K with the power law:\n\\begin{equation}\n\\label{EQres}\n\\rho_{xx} = \\rho_{xx0}+AT^p,\n\\end{equation}\n(where $\\rho_{xx0}$ is the residual resistivity and exponent $p$ depends on the prevailing scattering mechanism) and extended the obtained function to lowest temperature.\n\n For LuNiC$_2$ (see Fig. \\ref{RT}c) the character of the conductivity is metallic up to 400 K, in agreement with previous reports of a similar metal-metal CDW transition at $T_{CDW}\\simeq$ 450 K\\cite{Roman2018_1, Steiner2018}, which is beyond the upper temperature limit of the PPMS, thus not revealed by our current measurement. Interestingly, despite that the CDW modulation wavevector determined previously for LuNiC$_2$ \\cite{Steiner2018} differs from the vectors reported for $R$ = Pr, Nd, Sm, Gd and Tb \\cite{Yamamoto2013, Shimomura2009, Shimomura2016}, the Peierls temperature corresponding to Lu bearing compound (reported in references\\cite{Roman2018_1, Steiner2018}) obeys the scaling proposed for magnetic $R$NiC$_2$ \\cite{Roman2018_1} as well, which is shown in Fig. \\ref{RT}c.\n\n\\begin{figure*} [ht!]\n  \\includegraphics[angle=0,width=2.1\\columnwidth]{Panel1.eps}\n \\caption{\\label{Panel1} (a)-(b) Magnetoresistance in YNiC$_2$ (a) and LuNiC$_2$ (b) as a function of applied magnetic field. (c)-(d) Kohler plots of the magnetoresistance in YNiC$_2$ (c) and LuNiC$_2$ (d) Inset of panel (c) shows the low temperature range for YNiC$_2$ (e)-(f) Thermal dependence of the normalized Hall resistivity $\\frac{\\rho_{yx}}{B}$ in YNiC$_2$ (e) and LuNiC$_2$ (f). The legend for (a), (b), (c) and (d) is displayed on panel (a).}\n\\end{figure*}  \n  \n  \n  An external magnetic field has a negligible impact on the transport properties in the normal state of YNiC$_2$, typical for a casual metal. On the other hand, the application of $B$ significantly increases the electrical resistivity in the CDW state of this material. This is also true, however, to a lesser extent, for the CDW state of LuNiC$_2$.\nThe magnetic field dependence of magnetoresistance defined as:\n\\begin{equation}\n\\label{EQMR}\nMR=\\frac{\\rho_{xx}(B)-\\rho_{xx}(B=0)}{\\rho_{xx}(B=0)}\\cdot 100\\%,\n\\end{equation} \nis presented in Fig. \\ref{Panel1}a and b for YNiC$_2$ and LuNiC$_2$, respectively. For YNiC$_2$, the MR curve for $T$ = 1.9 K where the zero field value of $\\rho_{xx}$ was estimated from equation \\ref{EQres} is close to the one measured for $T$ = 5 K, above the superconducting transition and where the $\\rho_{xx}$ is already close to the residual value. This result confirms that, the MR($B$) curve for $T$ = 1.9 K is not contaminated by any spurious contribution that could stem from the superconducting impurity phase.\nAt low field, the electrical resistance of both compounds follows a MR$\\sim B^q$ dependence on magnetic field. MR is subquadratic ($q \\simeq$  1.4 at $T$ = 1.9 K) for YNiC$_2$ and approximately parabolic ($q \\simeq$ 2.2 K at $T$ = 1.9 K ) for LuNiC$_2$. As the magnetic field is raised, the character of MR$(B)$ curves for both compounds evolves towards a linear manner, without any signs of saturation up to 9 T. The magnitude of the magnetoresistance increases as the temperature is lowered \nand at $T$ = 1.9 K and $B$ = 9 T reaches 470 \\% for the former and 50 \\% for the latter compound, respectively. \nTypically, the linear magnetoresistance term is described by one of two prevailing models: the classical approach of Parish and Littlewood based on mobility fluctuations in inhomogeneous material\\cite{Parish2003, Parish2005} or Abrikosov's quantum model used beyond the quantum critical limit, when only a single Landau level is occupied\\cite{Abrikosov1969, Abrikosov1988, Abrikosov2000}. In a CDW metal however, the linear MR can also originate from the fluctuations of the charge density wave order parameter enhancing the scattering in certain regions of the Fermi surface \\cite{Sinchenko2017, Frolov2018} or from sharp curves on the carrier path due to nesting induced reduction of FS \\cite{Feng_2019}. Another possible scenario is that the linearity in MR originates from the complicated geometry of the Fermi surface (which is additionally modified by nesting), containing both closed and open orbits, which in a polycrystalline sample contribute to MR with saturating and non-saturating signals \\cite{Lifshitz1957} respectively.\n\nIt shall be noted that, such a large positive magnetoresistive effect has never been reported for any members of the $R$NiC$_2$ family. Typically, in $R$NiC$_2$ compounds exhibiting Peierls transition, MR shows a negative sign due to partial or complete suppression of charge density wave, induced either by magnetic field or magnetic ordering\\cite{Yamamoto2013, Hanasaki2012, Kolincio20161, Kolincio2017, Hanasaki2017, Lei2017}. In  YNiC$_2$ and LuNiC$_2$ however, no signatures of magnetism has been found above $T$ = 1.9 K.\n\nThe magnetoresistance can be discussed in the framework of the semiclassical Kohler approach\\cite{Pippard2009}. The prediction of this model is that on the condition of uniform scattering over the whole Fermi surface and a single type of electronic carriers with constant concentration, all the plots:\n\\begin{equation}\n\\label{EQKohler}\nMR=f(\\omega_c\\tau)=f\\left( \\frac{B}{\\rho_{xx}(B=0)}\\right)\n\\end{equation}\n(where $\\tau$ and $\\omega_c$ are relaxation time and cyclotron frequency, respectively) superimpose into a single line. The plot of MR as a function of \n$\\frac{B}{R_0}$ ($R_0$ is the zero field resistance) for YNiC$_2$ is shown in Fig. \\ref{Panel1}c. The deviations from Kohler scaling, shown in Fig. \\ref{Panel1}c, are weak yet visible. Upon decreasing the temperature, the plots are subsequently moved higher on the vertical scale (faster increase of MR($\\frac{B}{R_0}$)) which is observed in a wide temperature range above $T$ = 40 K. For $T<$ 40 K, curves are pressed slightly lower (slower $\\frac{B}{R_0}$ growth), as depicted in the inset of Fig. \\ref{Panel1}c. At lowest temperatures, where the zero field resistance is already close to the residual value, the curves superimpose (within the experimental resolution).\n  As shown in Fig. \\ref{Panel1}d, for LuNiC$_2$ the divergence from Kohler's scaling is even less pronounced but still visible. The strongest effect is seen in the temperature range $T>\\frac{T_{CDW}}{2}$, where the CDW gap is expected not to be completely open yet\\cite{Gruner1988}. This violation of the Kohler rule can be attributed both to the reconstruction of the Fermi surface due to nesting and to presence of more than one type of carriers in the CDW state\\cite{McKenzie1998, Yasuzuka2005}. We suggest that a stronger manifestation of the deviation from the MR scaling could be observed at temperatures in the close vicinity of $T_{CDW}$ as in tungsten bronzes also showing Peierls transition \\cite{Kolincio20162}. This range is, however, beyond the scope of our experimental equipment.\n\n\\subsection{Hall effect}\n\\begin{figure*} [ht!]\n  \\includegraphics[angle=0,width=2.1\\columnwidth]{panel2.eps}\n \\caption{\\label{Panel2} (a)-(b) Magnetic field dependence of Hall resistivity $\\rho_{yx}$ in YNiC$_2$ (a) and LuNiC$_2$(b). The plots have been vertically shifted for clarity and the vertical scale applies to the plot for corresponding to $T$ = 1.9 K. (c)-(d) Hall conductivity $\\sigma_{xy}$ in YNiC$_2$ (c) and LuNiC$_2$ (d). The black solid lines are representative fits to the experimental data with equation \\ref{EQsigmaxyAP}. (e)-(f) The results of the analysis of Hall resistivity and conductivity: mobilities $\\mu_H$, $\\mu_{ext}$ and concentrations $n_H$, $n_{eff}$ plotted as a function of temperature for YNiC$_2$ (e) and LuNiC$_2$ (f). The legend for (a), (b), (c) and (d) is displayed in panel (c).}\n\\end{figure*}\n\nTo explore the evolution of carrier concentrations, we have examined the Hall effect for both compounds. The thermal dependence of the Hall resistivity ($\\rho_{yx}$) is depicted in Fig. \\ref{Panel1}e (YNiC$_2$) and \\ref{Panel1}f (LuNiC$_2$). For YNiC$_2$, $\\frac{\\rho_{yx}}{B}$ is almost temperature independent above $T_{CDW}$. At this characteristic temperature, the Hall resistivity shows an abrupt downturn, indicating the loss of free electrons due to the CDW condensation. The presumed lock-in transition is indicated by a kink in  $\\frac{\\rho_{yx}}{B}(T)$. At lower temperatures, the Hall resistance shows a minimum and then returns to less negative values. Previously, such an effect was observed in magnetic $R$NiC$_2$, and was attributed both to the suppression of charge density wave by the magnetic ordering and to the onset of the anomalous component of the Hall effect\\cite{Kim2012, Kolincio20161, Kolincio2017, Lei2017}. Due to the absence of long range magnetism in YNiC$_2$, these two terms appear to be irrelevant in this case.\nAt temperatures below $T_1$, the  $\\frac{\\rho_{yx}}{B}(T)$ curves do not superimpose into a single line which suggests that in the CDW state,  $\\rho_{yx}$ is not linear with $B$. \n\nFor LuNiC$_2$ the Peierls temperature $T_{CDW}\\simeq$ 450 K\\cite{Roman2018_1, Steiner2018}, thus at 400 K, which is the maximum temperature limit of our experiment, the system is already in the charge density wave state. All the curves reveal a kink at $T\\simeq$ 355 K. Its origin is not clear, however, while this weak anomaly is not detected by other measurements, it might result from the experimental artifact instead of being truly intrinsic to the sample. Another scenario is, that this anomaly originates from the Lu$_4$Ni$_2$C$_5$ impurity phase. Similarly to YNiC$_2$, the sign of $\\rho_{yx}$ is negative in the whole temperature range, indicating the dominance of electrons. This is not the only similarity between the $\\frac{\\rho_{yx}}{B}$ curves for both compounds. Here we also find that for LuNiC$_2$ the Hall resistivity is also driven to more negative values as the free electrons are condensed in the CDW state, which is followed by the return of $\\rho_{yx}$ to close to zero at lower temperatures.  We find that the $\\frac{\\rho_{yx}}{B}$ superimpose at temperatures above approximately 250 K. At lower temperatures, the plots do not coincide with each other, indicating a nonlinearity of $\\rho_{yx}(B)$ also in LuNiC$_2$. Similarly to the case of YNiC$_2$, further temperature decrease leads to the upturn of the Hall resistivity, which also cannot be attributed to magnetic ordering. A plausible scenario to explain these features is the existence of more than one type of electronic carrier, originating from unnested pockets remaining in the Fermi surface after imperfect nesting, a situation characteristic of quasi-2D metals showing charge density wave\\cite{Monceau2012, Gruner2000}. \n\nTo obtain a more detailed picture of the electronic parameters, we have examined the magnetic field dependence of $\\rho_{yx}$. The results of field sweeps at constant temperatures, shown in the Fig. \\ref{Panel2}a for YNiC$_2$ and \\ref{Panel2}b for LuNiC$_2$ reveal a visible deviation of Hall signal from linearity.\nIn the absence of long range magnetic interactions or ordering, this effect is a clear manifestation of the multiband character of electrical conductivity\\cite{Akiba2017, Li2016, Liu2017, Luo2015, Wang2014}. \nIn the two-band model, the Hall resistivity is expressed with equation (\\ref{EQHall})\\cite{Hurd1972}:\n\\begin{equation}\n\\label{EQHall}\n\\frac{\\rho_{yx}}{B}=\\frac{1}{e}\\frac{n_h\\mu_h^2-n_e\\mu_e^2+(n_h-n_e)\\mu_e^2\\mu_h^2B^2}{(n_h\\mu_h+n_e\\mu_e)^2+(n_h-n_e)^2\\mu_h^2\\mu_e^2B^2}\n\\end{equation}\nwhere $n_h$, $n_e$, $\\mu_h$ and $\\mu_e$ are respectively concentrations and mobilities corresponding to two (hole and electron) conduction channels. The direct $\\rho_{yx}$ fit with eq. (\\ref{EQHall}) gives four dependent parameters, which may lead to misguiding conclusions\\cite{Rotella20151}. However, the high field  limit of this equation gives an approximate measure of the effective carrier concentration $n_{eff}$\\cite{Sun2014}, which will be discussed in section D:\n\\begin{equation}\n\\label{EQHallHF}\n\\frac{\\rho_{yx}}{B}=\\frac{1}{e}\\frac{1}{n_h-n_e}=\\frac{1}{e}\\frac{1}{n_{eff}}\n\\end{equation}\n\\subsection{Multiband conductivity}\nMore detailed information can be extracted by transforming components of resistivity tensor  $\\rho_{yx}$ and $\\rho_{xx}$ to obtain  Hall conductivity $\\sigma_{xy}$ via the following equation:\n\n\\begin{equation}\n\\label{EQsigmaxy}\n\\sigma_{xy}(B)=\\frac{\\rho_{yx}}{\\rho_{yx}^2+\\rho_{xx}^2}\n\\end{equation}\n\n\nIn the multiband system, $\\sigma_{xy}$ is a superposition of the terms originating from subsequent contributing bands.  Equation (\\ref{EQsigmaxy}) can be then rewritten as\\cite{Lin2016}:\n\n\\begin{equation}\n\\label{EQsigmaxymulti}\n\\sigma_{xy}(B)=\\sum_{i}\\frac{\\sigma_i\\mu_iB}{1+\\mu_i^2B^2}\n\\end{equation}\n\nHall conductivity is commonly used to determine the electronic parameters, since the extremum of $\\sigma_{xy}(B)$ is a direct measure (or at least a good approximation in a multiband system) of the dominant mobility $\\mu_{ext}$ calculated from the inverse of the magnetic field $B_{ext}$, at which $\\sigma_{xy}$ peaks\\cite{Liang2014}:\n\n\\begin{equation}\n\\label{EQmuinv}\n\\mu_{ext}=\\frac{1}{B_{ext}}\n\\end{equation}\n\nThe Hall conductivity for both compounds is negative in the whole temperature range and at low temperatures shows a minimum, which for YNiC$_2$ is visibly sharper than for LuNiC$_2$. The position of this minimum shifts from high fields to lower values of $B$ as temperature is lowered. For YNiC$_2$, the {$B_{ext}$ is clearly defined, while the broad extremum seen in LuNiC$_2$ precludes the precise determination of the peak position. Since the direct fitting of $\\sigma_{xy}$ with equation \\ref{EQsigmaxymulti} assuming one hole and one electron bands once again requires using four dependent parameters, for further analysis we have used an approach\\cite{Takahashi2011, Ishiwata2013}, in which we have assumed the existence of a single band with high mobility carriers and the remaining band(s) to show significantly lower mobility:\n\n\\begin{equation}\n\\label{EQsigmaxyAP}\n\\sigma_{xy}(B)=n_{xy}e\\mu_{xy}^2B  \\left( \\frac{1}{1+\\mu_{xy}^2B^2} +C_{xy} \\right)\n\\end{equation}\n\n\n\nEquation \\ref{EQsigmaxyAP} allows the estimation of the mobility $\\mu_{xy}$, and concentration $n_{xy}$ of this single 'fast' band (pocket), while other 'slower' bands contribute to $C_{xy}$ parameter. The typical fits are shown by solid lines in panels c and d of Fig. \\ref{Panel2} respectively. We have found that $\\sigma_{xy}$ can be reasonably well-described with equation (\\ref{EQsigmaxyAP}) despite of the fact that the zero field values of $\\rho_{xx}$ can be significantly increased due to the polycrystalline character of the samples. \n\n\\begin{figure} [ht!]\n  \\includegraphics[angle=0,width=1.1\\columnwidth]{Cpar.eps}\n \\caption{\\label{Cpar} $C_{xy}$  parameters resulting from least square fit of $\\sigma_{xx}$ with equation \\ref{EQsigmaxyAP} for YNiC$_2$ (red color) and LuNiC$_2$ (blue color).}\n  \\end{figure}\n\nThe parameters derived from this procedure, as well as the values of $n_{eff}$ and $\\mu_{ext}$, are summarized in Fig.  \\ref{Panel2}e for YNiC$_2$ and \\ref{Panel2}f, for LuNiC$_2$. The mobilities $\\mu_{ext}$ and $\\mu_{xy}$ coincide with each other for YNiC$_2$, and both quantities reach very large values of $7\\cdot 10^3$ cm$^2$ V$^{-1}$ s$^{-1}$ at $T$ = 1.9 K. \nThe electronic mobility $\\mu_{xy}$ of LuNiC$_2$ is twice as  small as in the case YNiC$_2$, yet still considerable.  The coincidence of $\\mu_{xy}$ and $\\mu_{ext}$ is an additional argument for the correctness of the value calculated from $\\sigma_{xy}$. It shall be, however, noted that, next to the increase of the residual resistivity, the polycrystalline samples character is expected also to substantially lower the electronic mobility in comparison with the single crystal.\n\n As seen in Fig. \\ref{Panel2}e and f, for both compounds, the concentration of the carriers originating from the high mobility band increases as temperature is lowered. \n The growth of $n_{xy}$  is concomitant with the decrease of the effective carrier concentration $n_{eff}$ below Peierls temperature. This is consistent with the nesting picture: while the majority of electrons are removed from the conducting band and condenses towards CDW, the parallel opening of unnested pockets results in the increase of the high mobility carriers. Interestingly, while the results of the $\\sigma_{xy}$ analysis suggests the electron origin of the carriers described with concentration $n_{xy}$, the upturn of Hall resistivity and $n_{eff}$ at lowest temperatures can possibly be caused by the existence of holes with not so large mobility, thus contributing only to $C_{xy}$ parameter in equation (\\ref{EQsigmaxyAP}). The temperature interval in which this effect is observed corresponds to the range in which the a turnover of deviations from Kohler scaling is observed in YNiC$_2$ (inset of Fig. \\ref{Panel1}c).\n \nThe $C_{xy}$ parameter serves as an estimate of the ratio of the conductivities stemming from 'slow' to 'fast' bands respectively. Thermal dependence of $C_{xy}$ for both compounds is shown in figure \\ref{Cpar}. For both compounds, the values of this parameter show values close to unity at high $T$ and  decrease as temperature is lowered.  $C_{xy}$ reaches $\\simeq$ 0.001 for YNiC$_2$ and $\\simeq$ 0.1 for LuNiC$_2$. Small upturn is seen at low temperatures at low temperatures, which can be associated with the existence of an additional band as suggested above. The relatively low values of $C_{xy}$, especially in the former compound, underline the major role played by the carriers originating from the 'fast' pocket in the terms of electronic transport and show that the used approximate model can be used to describe the properties of YNiC$_2$ and LuNiC$_2$. \n\nThe presence of both electron and hole pockets in the CDW state of LuNiC$_2$ is also consistent with the results of band structure calculations\\cite{Steiner2018}. Owing to the similarities between the Fermi surfaces of YNiC$_2$ \\cite{Hase2009} and other $R$NiC$_2$ showing CDW, it is reasonable to assume the relevance of the same scenario for Y bearing compound as well. \n\nThe high mobility of carriers contained in these pockets is then likely responsible for the high magnitude of MR in both compounds. Opening of such pockets was reported in a number of quasi-2D CDW materials showing strong, yet imperfect Fermi surface nesting, leading to the enhancement of magnetoresistance\\cite{Rotger1994, Rotger1996, Yasuzuka1999, Chen2017},\n themopower\\cite{Rhyee2009, Rhyee2015} and galvanothermomagnetic properties\\cite{Bel2003, Kolincio20163}.\n\nThis result supports the scenario of strong Fermi surface reconstruction in YNiC$_2$ and LuNiC$_2$, which is possible due to the absence of any competing magnetic ordering which was responsible for the CDW suppression in majority of $R$NiC$_2$ family members \\cite{Yamamoto2013, Hanasaki2012, Kolincio20161, Kolincio2017, Hanasaki2017, Lei2017}.\n\n\n\n\n\\subsection{Specific heat}\n\n\nTo complement the results of transport, magnetotransport, and Hall experiments, and to further characterize the CDW transition in YNiC$_2$, we have measured the specific heat $C_p$. Fig. \\ref{CP}a depicts the temperature dependence of the specific heat capacity $C_p(T)$ in the temperature range 1.9 - 300 K. At 300 K, $C_p$ reaches approximately 80\\% of the value expected by Dulong-Petit law (3nR $\\sim$ 100 J mol$^{-1}$ K$^{-1}$), suggesting that the Debye temperature for YNiC$_2$ exceeds 300 K. \n\nNo anomalies have been detected at low temperatures, which confirms the absence of bulk superconductivity or magnetic ordering. \nThe specific heat data plotted as $\\frac{C_p}{T} $ vs. $T^2$ presented in Fig. \\ref{CP}b has been fitted to the equation (\\ref{CPeq}) with both sides divided by $T$.\n\n\\begin{equation}\n\\label{CPeq}\nC_p  =  \\gamma T + \\beta T^3\n\\end{equation}\n\n\\noindent where the first and second terms represent electronic and lattice contributions, respectively.\nThe fit revealed values of Sommerfeld coefficient $\\gamma$ = 1.65(1) mJ mol$^{-1}$K$^{-2}$ and $\\beta$ = 0.326(4) mJ mol$^{-1}$K$^{-4}$, the latter corresponds to the Debye temperature $\\Theta_D$ = 620 K according to:\n\n\\begin{equation}\n\\label{Debye}\n\\Theta_D = \\left(  \\frac{12\\pi^4 nR}{5\\beta} \\right)^{\\frac{1}{3}} \n\\end{equation}\n\n\\noindent where R =  8.314 J mol$^{-1}$K$^{-1}$  and $n$ is the number of atoms per formula unit ($n$ =  4 for YNiC$_2$). This value is larger than the $\\Theta_D$ = 456 K reported previously for YNiC$_2$ \\cite{long_heat_2001}. The Debye temperature found here is also larger than the value reported for LaNiC$_2$ ($\\Theta_D$ = 445 K) \\cite{Prathiba2016}. Such behavior can be reasonably explained by a mass relationship: for molar mass of Y smaller than La, one expects higher $\\Theta_D$.\n\n  \\begin{figure} [t]\n  \\includegraphics[angle=0,width=1.0\\columnwidth]{Cp.eps}\n \\caption{\\label{CP} (a) Specific heat of YNiC$_2$ as a function of temperature. The inset shows an expanded view on the vicinity of the Peierls transition. The anomalies are marked with arrows. Dashed line corresponds to the background subtracted to evaluate the excess specific heat corresponding to the transitions - highlighted with light violet color. The high temperature measurements were performed with Apiezon L grease - see experimental section for details. (b) $\\frac{C_p}{T} (T^2)$ in the low temperature region. Black solid line corresponds to the fit with equation (\\ref{CPeq}), divided by $T$ on both sides.}\n  \\end{figure}\n  \nResults of the detailed measurements of $C_p(T)$ above room temperature are shown in the inset of Fig. \\ref{CP}a. The Peierls transition is signaled by a small maximum of $C_p(T)$ at $T$ = 310 K, being in rough agreement with the transition temperature $T_{CDW}$ established from resistivity measurements. The relative increase of specific heat at the charge density wave formation temperature denotes $\\frac{\\Delta C_p}{C_p(T_{CDW})} \\simeq 1.1 \\% $, thus is at the same order of magnitude as in canonical CDW systems as NbSe$_3$\\cite{Tomic1981}, K$_{0.9}$Mo$_6$O$_{17}$\\cite{Escribe1984}, or tungsten bronzes\\cite{Chung1993}. \n\nThe mean-field weak coupling description of the Peierls transition predicts the specific heat jump of:\n\n\\begin{equation}\n\\label{BCSeq}\n\\frac{\\Delta C_p}{\\gamma T_{CDW}}=1.43\n\\end{equation}\n\nIn the case of YNiC$_2$, the equation (\\ref{BCSeq}) gives the value of 1.79, slightly larger than the BCS prediction, indicating the relevance of a weak coupling scenario.\n\n\nVisibly stronger and sharper anomaly accompanies the presumed lock-in crossover at $T_1$ = 275 K. Here the specific heat increases by $\\frac{\\Delta C_p}{C_p(T_1)} \\simeq 2.9 \\% $, with $C_p(T_1)$ estimated from the background. The magnitude of this anomaly is noticeably larger than for the features typically observed at the incommensurate-commensurate CDW transformation\\cite{Craven1977, Kuo2004}.\n\n\n\nThe entropy $\\Delta S$ and enthalpy $\\Delta H$ of both anomalies were estimated from the excess specific heat at each transition by integrating the $\\frac{\\Delta C_p}{T}dT$ of and $\\Delta C_pdT$ respectively, after evaluating and subtracting the background values of $C_p$.\nThe integrated regions are highlighted by light violet color in Fig. \\ref{CP}a.\n\\begin{table}[t!]\n\\caption{Thermodynamic parameters: relative increase of specific heat $\\frac{\\Delta C_p}{C_p(T)}$, entropy $\\Delta S$ and enthalpy $\\Delta H$ at transition temperatures $T_{CDW}$ and $T_1$ in YNiC$_2$.}\n\\label{tableHC}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\n& $\\frac{\\Delta C_p}{C_p(T)}$ (\\%) & $\\Delta S$ (mJ mol$^{-1}$K$^{-1}$) & $\\Delta H$ (J mol$^{-1}$)\\\\\n \\hline\n$T_{CDW}$ & 1.1 & 30.6 & 9.4\\\\\n$T_1$ & 2.9 & 77.8 & 21.3\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\n\\end{table}\nThe results of the integration of $C_p$ excess  accompanying the phase transitions are summarized in Tab. \\ref{tableHC}. While the size of $\\Delta C_p$ step at $T_{CDW}$ stands in agreement with the BCS predictions as well as with the values found in other materials exhibiting a weakly coupled charge density wave, we find an unusualy low value of $\\Delta S$ accompanying this transition. This can be imposed by the high Peierls temperature, resulting in a large denominator of $\\frac{\\Delta C_p}{T}$ and thus small result of the integral. The value of enthalpy however, does not diverge from the typically observed values in CDW systems\\cite{Escribe1984, Wang2006}.\nIn agreement with the comparison of $\\Delta C_p$ jump, for the crossover at $T_1$, the values $\\Delta S$ and $\\Delta H$, are significantly larger than for the Peierls transition at $T_{CDW}$. This result is unexpected, since typically the lock-in transition is not associated with the opening of a new electronic gap, next the one already existing in the CDW state.  \nThe sharp peak shape of this anomaly can suggest a large role played by CDW order parameter fluctuations \\cite{McMillan1977, Kuo2001, Kwok1990}. The detailed analysis of crystal structure, as well as of the phonon spectra, performed on a single crystal is required to elucidate this issue.\n\n\n  \n\\section{conclusions}\nWe have examined the physical properties of polycrystalline YNiC$_2$ and LuNiC$_2$. The former compound shows at $T_{CDW}$ = 318 K Peierls transition with signatures of BCS - mean field weak coupling scenario, followed by presumed lock-in crossover at $T_1$ = 275 K. The  temperatures corresponding to these anomalies, revealed by transport, Hall effect and specific heat measurements, are found to obey the linear scaling with the unit cell volume, observed previously with lanthanide-based $R$NiC$_2$ compounds. \nBoth studied materials show large magnetoresistance in the CDW state, reaching 470 \\% for YNiC$_2$ and 50 \\% for LuNiC$_2$ at $T$ = 1.9 K and $B$ = 9 T. To discuss its origin, we have combined the analysis of thermal and magnetic field depencence of Hall effect and magnetoresistance. We have found that the effect standing behind such strong magnetoresistive features in YNiC$_2$ and LuNiC$_2$ is the existence of pockets, including at least one with high mobility carriers, remaining in the Fermi surface after nesting, caused by fully developed CDW transition not interrupted by competing orders such as magnetism or superconductivity.\n\n\\begin{acknowledgments}\n The authors gratefully acknowledge the financial support from National Science Centre (Poland), grant number:  UMO-2015\/19\/B\/ST3\/03127. The authors would also like to thank to M. Hirshberger and C. Zhang (both RIKEN), Alain Pautrat (CRISMAT), Helen Walker (ISIS), Nathan Runyon, and Jesse Sprowes for their helpful advice. \n \\end{acknowledgments}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA standard model of cosmology is emerging (often dubbed the\nConcordance Model), in which the universe consists of 5\\% ordinary\nbaryonic matter, $\\sim 26$\\% dark matter, and $\\sim 69$\\% dark energy.\n\\cite{Komatsu:2008hk,concordo2}  The baryonic content is\nwell-known, both from element abundances produced in primordial\nnucleosynthesis roughly 100 seconds after the Big Bang, and from\nmeasurements of anisotropies in the cosmic microwave background (CMB).\nThe evidence for the existence of dark matter is overwhelming, and\ncomes from a wide variety of astrophysical measurements.\n\n\n\\section{Dark Matter in Galaxies and Clusters}\n\n\\subsection{The Beginnings of the Dark Matter Problem and Rotation Curves}\n\nThe dark matter problem is perhaps the longest outstanding problem in all of modern physics.\nThe puzzle dates back to the 1930's, to the work first of Knut Lundmark in Sweden and shortly after that Fritz Zwicky at Caltech.\nZwicky noticed that galaxies in the Coma Cluster were moving too rapidly to be explained by the stellar\nmaterial in the cluster.  He postulated that additional mass in the form of something dark must\nbe providing the gravitational pull to speed up the orbits.  Subsequent work continued to find\nsimilar evidence, but it wasn't until the work of Ford and Rubin \\cite{FordRubin1970} in the 1970's \nthat the same unexplained rapid orbits were found\nto exist in every single galaxy.  At that point the scientific consensus for dark matter emerged.\nFor a review of dark matter history, see the review of Ref.~\\refcite{Bertone:2016nfn}.\n\nRotation curves of\ngalaxies are flat.  The velocities of objects (stars or gas) orbiting\nthe centers of galaxies, rather than decreasing as a function of the\ndistance from the galactic centers as had been expected, remain\nconstant out to very large radii.  Similar observations of flat\nrotation curves have now been found for all galaxies studied,\nincluding our Milky Way.  The simplest explanation is that galaxies\ncontain far more mass than can be explained by the bright stellar\nobjects residing in galactic disks.  This mass provides the force to\nspeed up the orbits.  To explain the data, galaxies must have enormous\ndark halos made of unknown `dark matter.' Indeed, more than 95\\% of\nthe mass of galaxies consists of dark matter.  This is illustrated in\nFig. 1, where the velocity profile of galaxy NGC 6503 is displayed as\na function of radial distance from the galactic center. The baryonic\nmatter which accounts for the gas and disk cannot alone explain the\ngalactic rotation curve. However, adding a dark matter halo allows a\ngood fit to data.\\footnote{It is interesting to note that alternative scenarios without dark matter\nbegan with modified Newtonian dynamics (MOND). \\cite{milgrom}  While these models\nhave been shown to fail, particularly by cosmic microwave background observations, \nthey may provide an interesting phenomenological fit on small scales. \\cite{mcgaugh}}\n\nThe limitations of rotation curves are that one can only look out as\nfar as there is light or neutral hydrogen (21 cm), namely to distances\nof tens of kpc.  Thus one can see the beginnings of dark matter haloes, but\ncannot trace where most of the dark matter is. The lensing experiments\ndiscussed in the next section go beyond these limitations.\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{fig1}\n\\caption{Galactic rotation curve for NGC 6503 showing disk and gas\n  contribution plus the dark matter halo contribution needed to match\n  the data.}\n\\end{figure}\n\n\\subsection{Gravitational Lensing}\n\nEinstein's theory of general relativity predicts that mass bends, or\nlenses, light.  This effect can be used to gravitationally ascertain\nthe existence of mass even when it emits no light.  Lensing\nmeasurements confirm the existence of enormous quantities of dark\nmatter both in galaxies and in clusters of galaxies.\n\nObservations are made of distant bright objects such as galaxies or\nquasars.  As the result of intervening matter, the light from these\ndistant objects is bent towards the regions of large mass.  Hence\nthere may be multiple images of the distant objects, or, if these\nimages cannot be individually resolved, the background object may\nappear brighter.  Some of these images may be distorted or sheared.\nThe Sloan Digital Sky Survey used weak lensing (statistical studies of\nlensed galaxies) to conclude that galaxies, including the Milky Way,\nare even larger and more massive than previously thought, and require\neven more dark matter out to great distances. \\cite{Sloan2005}  Again, the\npredominance of dark matter in galaxies is observed.\n\nA beautiful example of a strong lens is shown in Fig.~2.  The panel\non the right shows a computer reconstruction of a foreground cluster\ninferred by lensing observations made by Tyson et al.\\ \\cite{tyson} using the Hubble\nSpace Telescope.  This extremely rich cluster contains many galaxies,\nindicated by the peaks in the figure.  In addition to these galaxies,\nthere is clearly a smooth component, which is the dark matter\ncontained in clusters in between the galaxies.\n\nThe key success of the lensing of dark matter to date is the evidence that dark matter\nis seen out to much larger distances than could be probed by rotation\ncurves: the dark matter is seen in galaxies out to 200 kpc from the centers\nof galaxies, in agreement with\n$N$-body simulations.  On even larger Mpc scales, there is\nevidence for dark matter in filaments (the cosmic web).\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{fig2}\n\\caption{Left: The foreground cluster of galaxies gravitationally\n  lenses the blue background galaxy into multiple images. Right: A\n computer reconstruction of the lens shows a\n  smooth background component not accounted for by the mass of the\n  luminous objects.}\n\\end{figure}\n\n\\subsection{Hot Gas in Clusters}\n\nAnother piece of gravitational evidence for dark matter is the hot gas\nin clusters.  Fig.~3 illustrates the Coma Cluster. The left panel is\nin the optical, while the right panel is emission in the X-ray\nobserved by ROSAT. \\cite{Coma1997}\n[Note that these two images are not\non the same scale.]  The X-ray image indicates the presence of hot\ngas.  The existence of this gas in the cluster can only be explained\nby a large dark matter component that provides the potential well to\nhold on to the gas.\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{fig3a}\n\\includegraphics[width=0.49\\textwidth]{fig3b}\n\\caption{COMA Cluster: without dark matter, the hot gas would\n  evaporate. Left panel: optical image. Right panel: X-ray image from\n  ROSAT satellite.}\n\\end{figure}\n\n\n\\subsection{Bullet Cluster}\n\nAn image (shown in Fig.~4) of the Bullet Cluster of galaxies (a cluster formed out\nof a collision of two smaller clusters) taken by the Chandra X-ray\nobservatory shows in pink the baryonic matter; in blue is an image of\nthe dark matter, deduced from gravitational lensing. In the process of\nthe merging of the two smaller clusters, the dark matter has passed\nthrough the collision point, while the baryonic matter slowed due to\nfriction and coalesced to a single region at the center of the new\ncluster.  The Bullet Cluster provides clear evidence of the existence\nof  two different types\nof matter: baryons and dark matter behave differently.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7cm]{fig4}\n\\end{center}\n\\caption{The Bullet Cluster: A collision of galactic clusters shows\n  baryonic matter (pink) as separate from dark matter (blue), whose\n  distribution is deduced from gravitational lensing.}\n\\end{figure}\n\nThus the evidence that most of the mass of galaxies and clusters is\nmade of\nsome unknown component of dark matter is overwhelming.\nAs I've shown, dark matter shows its existence gravitationally in many ways, including rotation\ncurves (out to tens of kpc), gravitational lensing (out to 200\nkpc), hot gas in clusters, and the Bullet Cluster.\n\nAdditionally,\nwithout dark matter, large scale structure could not have formed by the present time\nand we would not exist.  Until recombination at $z=1100$, the universe is ionized, baryons are\ntied to photons, and both photons and baryons stream out of structures as they are forming.  It is the dark matter\nthat clumps together first, before recombination, and provides the potential wells for the ordinary matter to fall\ninto at a later time.  In order for dark matter to initiate the formation of galaxies and clusters,\nit must be cold rather than hot.  Hot dark matter would be moving relativistically and would\nstream out of structures in the same way that photons do; hence it was already known in the\n1980s that neutrinos cannot\nprovide the potential wells for structure formation and cannot constitute the dark matter.\nNonrelativistic cold dark matter has become the standard paradigm for the dark matter in the universe.\\footnote{Alternatives do exist including warm dark matter.}\n\nBelow I turn to the cosmic microwave background which\nprovides  irrefutable evidence for dark matter.\n\n\n\n\\section{Cosmic Abundances}\nThe cosmic abundances tell a consistent story in which the\npreponderance of the mass in the universe consists of an unknown dark matter\ncomponent.  The cosmic microwave background provides the most powerful\nmeasurements of the cosmological parameters; primordial\nnucleosynthesis restricts the abundance of baryonic matter; Type IA\nsupernovae provided the first evidence for the acceleration of the\nuniverse, possibly explained by dark energy as the major constituent\nof the cosmic energy density.\n\n\\subsection{The Cosmic Microwave Background}\n\nFurther evidence for dark matter comes from measurements on\ncosmological scales of anisotropies in the cosmic microwave background. \n\\cite{Komatsu:2008hk,concordo2}  The CMB is the remnant\nradiation from the hot early days of the universe. The photons\nunderwent oscillations that froze in just before decoupling from the\nbaryonic matter at a redshift of 1100.  The angular scale and height\nof the peaks (and troughs) of these oscillations are powerful\nprobes of cosmological parameters, including the total energy density,\nthe baryonic fraction, and the dark matter component, as shown in Fig.~5.  The sound\nhorizon at last scattering provides a ruler stick for the geometry of\nthe universe: if the light travels in a straight line (as would be the\ncase for a flat geometry), then the angular scale of the first Doppler\npeak was expected to be found at 1 degree; indeed this is found to be\ncorrect.  Thus the geometry is flat, corresponding to an energy\ndensity of the universe of $\\sim 10^{-29} {\\rm gm\/cm}^3$.  The height\nof the second peak implies that 5\\% of the total is ordinary atoms,\nwhile matching all the peaks implies that 26\\% of the total is dark matter.\nIndeed the CMB by itself provides irrefutable evidence for dark matter.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7cm]{planck}\n\\end{center}\n\\caption{Planck's power spectrum of temperature fluctuations in the cosmic microwave background.\nThe fluctuations are shown at different angular scales on the sky.  Red dots with error bars are the\nPlanck data.  The green curve represents the standard model of cosmology, $\\Lambda$CDM. The\npeak at 1 degree is consistent with a flat geometry of the universe, the height of the second peak with 5\\%, and the second and third peaks with 26\\% dark matter. }\n\\end{figure}\n\n\n\n\\subsection{Primordial nucleosynthesis}\nWhen the universe was a few hundred seconds old, at a temperature of\nten billion degrees, deuterium became stable: $p + n \\rightarrow D +\n\\gamma$.  Once deuterium forms, helium and lithium form as well. The\nformation of heavier elements such as C, N, and O must wait a billion\nyears until stars form, with densities high enough for triple\ninteractions of three helium atoms into a single carbon atom. The\npredictions from the Big Bang are 25\\% Helium-4, $10^{-5}$\ndeuterium, and $10^{-10}$ Li-7 abundance by mass.  These predictions\nexactly match the data as long as atoms are only 5\\% of the total\nconstituents of the universe.\n\n\\subsection{Dark Energy}\nThe first evidence for the $\\sim$70\\% dark energy in the universe came from\nobservations of distant supernovae (Perlmutter et al.,\n\\cite{sn1999a} Riess et al., \\cite{sn1999b} Riess et\n  al.\\ \\cite{sn2004}). The supernovae are dimmer than expected, as\nis most easily explained by an accelerating universe.  There are two\ndifferent theoretical approaches currently pursued  to explain the dark energy: (i) a vacuum energy such as a\ncosmological constant or time-dependent vacuum \\cite{fafm1987} may be responsible, or (ii) it is possible that\nGeneral Relativity is incomplete and that Einstein's equations need to\nbe modified. \\cite{modgenrel2002a,modgenrel2005,modgenrel2002b,Carroll_etal2004}  Note, however, that\nthis dark energy does not resolve or contribute to the question of\ndark matter in galaxies, which remains as puzzling (if not more) than\ntwenty years ago.  We now have a concordance model of the universe, in\nwhich roughly a quarter of its content consists of dark matter.\n\n\\section{Dark Matter Candidates}\n\n\\subsection{MACHOs}\n\nTwenty years ago, it seemed reasonable that dark matter might consist of faint stars, substellar objects,\nor stellar remnants (white dwarfs or neutron stars),\ni.e., stars that simply were too faint to have yet been discovered. These fall into the category\nof massive compact halo objects, or MACHOs.  Other MACHO candidates would include\n primordial black holes or mirror matter. \\cite{MohapatraTeplitz1999}\n\nA combination of theory and observation have ruled these out\nas solving the dark matter problem of the Milky Way.  First, \nRefs.~\\refcite{faintstars,graff1996} used HST data to show that low mass stars could be at most 3\\% of the Milky Way dark matter.  Next, a combination of theory plus Hipparchos parallax data was \nused to rule out substellar objects, or brown dwarfs, as the primary constituent of the Galaxy's dark matter. \\cite{browndwarfs}  Stellar \nremnants were also potential DM candidates. Bounds on white dwarfs (WD) as dark matter came from many arguments (see Refs.~\\refcite{machoreview,machoreview2} for a review).  Stellar precursors of white dwarfs would have produced too much IR radiation that would have swallowed TeV gamma-rays seen from objects like Markarian 451; a too large fraction of the Universe's baryonic mass budget would have been required to produce the progenitor stars of the white dwarfs; WD would have overproduced carbon and nitrogen.   From\n these constraints we argued that at most 15\\% of the Milky Way Halo could be\nmade of white dwarfs (Freese et al.,\n\\cite{Freese_etal2000} Fields et al., \\cite{ffgwpb},\nGraff et al.\\ \\cite{ffgwpc}); at that time we disagreed with claims made by\n the MACHO microlensing experimental  that 100\\% of the dark matter could be in the\n form of MACHOs (the experiments originally overestimated the MACHO contribution).\n\n  Microlensing experiments (the MACHO (Alcock et al.\\\n\\cite{alcock2000}) and EROS experiments (Ansari et al.\\\n\\cite{eros2004}))  eventually showed that MACHOs less massive than 0.1 $M_\\odot$\nmake an insignificant contribution to the energy density of the Galaxy.  However, there is a possible detection\n(Alcock et al. \\cite{alcock2000}) of a roughly 15\\% halo\nfraction made of $\\sim 0.5 M_\\odot$ objects which might be made of\nstellar remnants such as white dwarfs.  These estimates agree with the numbers we found earlier\nfrom a combination of theory and other data sets. \\cite{machoreview,machoreview2}  \nThe white dwarf contribution to the dark matter halo could be significant, yet\nnot enough to explain all of the dark matter of the Milky Way.\n\n\\subsection{Nonbaryonic Dark Matter}\nFrom primordial nucleosynthesis and microwave background data, it has\nbecome clear that dark matter consists of nonbaryonic material.\nThere is a plethora of dark matter candidates.\nOf the many candidates, the most popular are the weakly interacting massive particles\n(WIMPS) and the axions, as these\nparticles have been proposed for other reasons in particle physics.\nThese are discussed further below.\nOrdinary  neutrinos are too light to be cosmologically\nsignificant, though sterile neutrinos remain a possibility.  Other\ncandidates include primordial black holes (for the latest bounds, see Ref.~\\refcite{Carr:2016drx}),\nself-interacting dark matter, light dark matter,\nasymmetric dark matter, nonthermal WIMPzillas, Q-balls, and many others.\n\n\n \\subsection{Axions}\n\\label{sec:axions}\n\nThe good news is that cosmologists don't need to ``invent'' new\nparticles.  Two candidates already exist in particle physics for other\nreasons: axions and WIMPs.  Axions arise in the Peccei-Quinn solution to the strong-CP\nproblem in the theory of strong interactions, \\cite{peccei} and are suitable dark matter candidates \\cite{weinberg1978,wilczek1978} if the mass lies in the range $m_a \\sim 10^{-(3-6)}\\,$eV. An upper bound on the axion mass $m_a < 15\\,$meV can be derived from astrophysical considerations,  \\cite{raffelt1986, raffelt2008,viaux,miller_bertolami} while a lower bound comes from cosmology \\cite{preskill1983,abbott1983,dine1983} and its value strongly depends on the thermal history of the universe and on the amount of topological defects. \\cite{visinelli2010} An exclusion region $6 \\times 10^{-13}{\\rm\\,eV} < m_a < 2 \\times 10^{-11}{\\rm \\, eV}$ that is independent of the cosmological history and comes from black hole super-radiance has been obtained  \\cite{arvanitaki} using aLIGO measurements.\nMicrowave cavity searches  \\cite{sikivie1983} allow for a direct detection of axions. The Adark matterX cavity experiment  (ADMX)\\cite{rosenberg} has already excluded a portion of the axion mass range and is currently searching for axions with a mass $\\sim 10^{-5}\\,$eV. A different technique consisting of searching for keV photons from axion-photon conversion in the Sun (through the Primakoff effect) has also been used in the KEK, CAST, and IAXO observatories. Such ``axion helioscopes'' are sensitive to the heavier end of the axion mass window.  In addition, new ideas for axion searches include the Cosmic Axion Spin Precession Experiment (CASPEr) \\cite{Budker:2013hfa} and broadband and resonant approaches. \\cite{Kahn:2016aff}\nAxion searches continue to reach into the theoretically best motivated regions of mass and coupling.\n\n\n\\subsection{WIMPs}\n\\label{sec:WIMPs}\n\nWIMPs are \nthought to be good dark matter candidates from particle physics for two reasons.\nThey are defined to be particles that participate in weak interactions (but not strong or electromagnetic)\nand their masses are in the range GeV--10 TeV. \nThese particles, if present in thermal abundance in the early\nuniverse, annihilate with one another so that a predictable number of\nthem remain today.  The relic density of these particles comes out to\nbe the right value:\n\\begin{equation}\n\\Omega_\\chi h^2 = (3 \\times 10^{-27} {\\rm cm}^3\/{\\rm sec})\n\/ \\langle \\sigma v \\rangle_{ann}\\,.\n\\end{equation}\nHere $h$ is the Hubble constant in units of 100 km\/s\/Mpc, and\n the annihilation cross section $\\langle \\sigma v \\rangle_{ann} $\nof weak interaction strength automatically gives the correct abundance of these particles today.\nThis coincidence is known as ``the WIMP miracle\" and is the first reason why\nWIMPs are taken so seriously as dark matter candidates.\n\nSecondly, WIMP candidates automatically exist in models that have been proposed to resolve problems\nin theoretical particle physics. These models contain WIMPs as a byproduct of the theory. \nFor example WIMP candidates exist in supersymmetric models (SUSY), including the lightest\nneutralino in the minimal supersymmetric standard model.\nSupersymmetry in particle theory is designed to keep particle masses\nat the right value.  As a consequence, each particle we know has a\npartner: the photino is the partner of the photon, the squark is the\nquark's partner, and the selectron is the partner of the electron.\nThe lightest superysmmetric partner is a good dark matter candidate.\nAnother type of WIMP exists in models of universal extra dimensions.\nIn these theories all standard model fields propagate in a higher dimensional\nbulk that is compactified on a space that is TeV$^{-1}$ in extent.  Higher\ndimensional momentum conservation in the bulk translates in four dimensions\nto Kaluza-Klein (KK) number (with boundary conditions to KK parity).  The lightest\nKK particle, known as the LKP, does not decay and is a WIMP candidate. \\cite{Servant:2002aq}\nWIMP candidates are well-motivated from the point of view of particle physics and relic density;\nthe key issue now is whether or not nature agrees with our theoretical prejudice.\nThe experimental hunt for WIMPs is ongoing.\n\n\\section{Four Pronged Approach to WIMP Detection}\n\nThere are several ways to search for  WIMPs based on their interactions\nwith standard model particles: production at the Large Hadron Collider, scattering in underground direct detection experiments,\nindirect detection of the products of annihilating dark matter, and discovery of dark stars. I will discuss each of these in turn.\n\n\\subsection{Production at the Large Hadron Collider at CERN}\n\nAt the Large Hadron Collider (LHC), protons are accelerated to 13 TeV.  Two beams travel in opposing directions\naround a 27 kilometer long ring, and then collide in several detectors.  The two general purpose detectors ATLAS and CMS\nwere built with the goal of discovering the Higgs, discovering SUSY and dark matter, and discovering the unknown.\nThe first goal of finding the Higgs boson, the last missing piece of the standard model of particle physics,\n was successful as of July 2012 and immediately led to a Nobel Prize for\nHiggs and Englert.  The other goals have as yet been elusive.\n\nSUSY dark matter particles could manifest at the LHC in a variety of ways.\nA possible signature would be missing transverse energy as the dark matter\nparticle leaves undetected, together with jets of particles created during the decay chain of SUSY particles emerging from the\ncollision.  Such a signature has not yet been seen, leading to ever higher bounds on SUSY particle masses.\nThe minimal supersymmetric standard model  (MSSM) has 105 free parameters.  If one makes some simplifying assumptions that\nunify all fermion masses $m_{1\/2}$ and all scalar masses $m_0$ at a high scale, then in the resulting constrained minimal\nsupersymmetric model (CMSSM, or MSUGRA), only five parameters remain.  The experimental results are often quoted in the context\nof this CMSSM\/MSUGRA.  For example, Fig.~6 illustrates the bounds from ATLAS on the supersymmetric parameter space.\nThe remaining parameter space is being pushed to above the TeV scale. However, it is important to note that\nthese bounds apply only to the MSUGRA\/CMSSM.\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{atlasbounds}\n\\caption{Bounds on MSUGRA\/CMSSM from 8 TeV  ATLAS data.  The remaining allowed parameter space is above the lines.}\n\\end{figure}\n\n\n\nThe LHC will never be able to kill even minimal supersymmetry. \\cite{gates}  Even in the MSSM, a 25 GeV neutralino currently\nsurvives as a possibility. \\cite{Pierce:2013rda}  If the LHC sees nothing, SUSY can survive.  It may be at high scale.\nOr, it may be less simple than the assumption that all scalars and all fermions unify at some high scale; e.g. the non-universal\nHiggs model (NUHM) or the non-universal gaugino model (NUGM).\n\nSUSY particles may be\ndiscovered at the LHC as missing transverse energy plus jets in an event.  In that case one\nknows that the particles live long enough to escape the detector, but\nit will still be unclear whether they\nare long-lived enough to be the dark matter.  Thus\ncomplementary astrophysical experiments are needed.\nProof that the dark matter has been found requires astrophysical particles to be found,\nvia the other prongs of the dark matter search techniques.\n\n\\subsection{Direct Detection Experiments}\n\nDirect detection experiments take advantage of the large number of WIMPs in the Galaxy.\nA WIMP travels through the detector,\n scatters off of a nucleus, and deposits a small amount of energy that may be detected\n The experiments are extraordinarily difficult and the progress has been impressive:\nthe count rates are less than one count\/kg\/day and the energy deposited is O(keV).\n\nThe history of dark matter direct detection began with the ideas and theoretical calculations in the 1980s.\nIn 1984 Drukier and Stodolsky \\cite{stodolsky} proposed neutrino detection via weak scattering off nuclei.\nThen Goodman and Witten \\cite{gw} turned the same approach to dark matter detection.\n Drukier, Freese, and Spergel \\cite{dfs}\nfirst  included a Maxwellian distribution of WIMPs in the Galaxy,\ncomputed cross sections for a variety of candidates, and proposed the idea of annual modulation to identify\na WIMP signal.  In another paper we further studied the idea of using annual modulation, not only for background rejection but also to tease out a\nsignal even in the presence of overwhelming noise; \\cite{wgould} this is the technique used by the DAMA experiment described below.\nFor  reviews, see Refs.~\\refcite{Jungman_etal1996,jkgb,jkgc,Bertone_etal2004,lisanti}.\n\nThe text in the subsequent few paragraphs outlines dark matter direct detection and\n is taken from my review paper with Lisanti and Savage. \\cite{lisanti}\nWhen a WIMP strikes a nucleus, the nucleus recoils with energy $E$.\nThe differential recoil rate per unit detector mass is\n\\begin{equation}\\label{eqn:dRdEnr}\n dR\/dE\n    = \\frac{n_\\chi}{M} \\Big\\langle v \\frac{d\\sigma}{dE} \\; \\Big\\rangle\n    = \\frac{2\\rho_\\chi}{m_\\chi}\n      \\int d^3v \\, v f(v,t) \\frac{d\\sigma}{dq^2}(q^2,v) \\, ,\n\\end{equation}\nwhere $n_\\chi = \\rho_\\chi\/m_\\chi$ is the number density of WIMPs, with\n$\\rho_\\chi$ the local dark matter mass density; $f(v,t)$ is the\ntime-dependent WIMP velocity distribution; and\n$\\frac{d\\sigma}{dq^2}(q^2,v)$ is the velocity-dependent differential\ncross-section, with $q^2 = 2 M E$ the momentum exchange in the\nscatter.  The differential rate is typically given in units of\ncpd\\,kg$^{-1}$\\,keV$^{-1}$, where cpd is counts per day.  Using the\nform of the differential cross-section for the most commonly assumed\ncouplings, to be discussed below,\n\\begin{equation}\\label{eqn:dRdEnr2}\n  dR\/dE\n    = \\frac{1}{2 m_\\chi \\mu^2} \\, \\sigma(q) \\, \\rho_\\chi \\eta(v_{min}(E),t),\n\\end{equation}\nwhere $\\sigma(q)$ is an effective scattering cross-section and\n\\begin{equation} \\label{eqn:eta}\n  \\eta(v_{min},t) = \\int_{v > v_{min}} d^3v \\, \\frac{f(v,t)}{v}\n\\end{equation}\nis the mean inverse speed, with\n\\begin{equation} \\label{eqn:vmin}\n  v_{min} =\n      \\sqrt{\\frac{M E}{2\\mu^2}}\n       \\end{equation}\nThe benefit of writing the recoil spectrum in the\nform of Eqn.(\\ref{eqn:dRdEnr2}) is that the particle physics and astrophysics\nseparate into two factors, $\\sigma(q)$ and $\\rho_\\chi \\eta(v_{min},t)$,\nrespectively.\nIt is traditional to define a form-factor\ncorrected cross-section\n\\begin{equation}\\label{eqn:sigmaq}\n  \\sigma(q) \\equiv \\sigma_0 F^2(q) \\, ,\n\\end{equation}\nHere $\\sigma_0$ is the scattering cross-section in the\nzero-momentum-transfer limit and\n$F^2(q)$ is a form factor to account for the finite size of the\nnucleus.\n\nTwo types of interactions are most commonly studied.\nIn spin independent (SI) interactions,  the\nscattering is coherent and scales as the atomic mass squared, $A^2$.\nThe SI\ncross-section can be written as\n\\begin{equation} \\label{eqn:sigmaSI2}\n  \\sigma_{SI} = \\frac{\\mu^2}{\\mu_p^2} A^2 \\, \\sigma_{p,SI} \\, ,\n\\end{equation}\nwhere $\\mu_p$ is the WIMP-proton reduced mass.\nThe SI cross-section grows rapidly with nuclear mass.  The explicit\n$A^2$ factor arises from the fact that the\ncontributions to the total SI cross-section of a nucleus is a coherent\nsum over the individual protons and neutrons within.\n\nSpin dependent (SD) scattering is due to the interaction of a WIMP with the spin of the\nnucleus and takes place only in those\ndetector isotopes with an unpaired proton and\/or unpaired neutron.\nThe SD WIMP-nucleus cross-section is\n\\begin{equation} \\label{eqn:sigmaSD}\n  \\sigma_{SD} = \\frac{32 \\mu^2}{\\pi} G_{F}^{2} J(J+1) \\Lambda^2 \\, ,\n\\end{equation}\nwhere $G_F$ is the Fermi constant, $J$ is the spin of the nucleus,\n\\begin{equation} \\label{eqn:Lambda}\n  \\Lambda \\equiv \\frac{1}{J} \\Big( a_p\\langle S_p \\rangle + a_n \\langle S_n \\rangle\n \\Big) \\, ,\n\\end{equation}\nwhere $\\langle S_p \\rangle$ and $\\langle S_n \\rangle $\n are the average spin contributions from the\nproton and neutron groups, respectively, and $a_p$ ($a_n$) are the\neffective couplings to the proton (neutron) (these need not be the same).\n\nThe dark matter halo in the local neighborhood is most likely\ndominated by a smooth and well-mixed (virialized) component with an\naverage density $\\rho_\\chi \\approx 0.4$~GeV\/cm$^3$.\nThe simplest model for this smooth component is often taken to be the\nstandard halo model (SHM) \\cite{dfs,wgould} of an\nisothermal sphere with an isotropic, Maxwellian velocity distribution\nand rms velocity dispersion $\\sigma_v$.  The SHM is written as\n \n   \n     \n       \n       \n     \n   \n\\begin{equation} \\label{eqn:TruncMaxwellian}\n  \\widetilde{f}(v) =\n  \n      \\frac{1}{N_{esc}} \\left( \\frac{3}{2 \\pi \\sigma_v^2} \\right)^{3\/2}\n        \\, e^{-3v^2\\!\/2\\sigma_v^2} ,\n         \\textrm{for} \\,\\, |v| < v_{esc}  \\\\\n         \\end{equation}\nand     $ \\widetilde{f}(v) =  0 $ otherwise.  Here,\n\\begin{equation} \\label{eqn:Nesc}\n  N_{esc} = \\textrm{erf}(z) - \\frac{2}{\\sqrt{\\pi}} z e^{-z^2} \\, ,\n\\end{equation}\nwith $z \\equiv v_{esc}\/v_0$, is a normalization factor and\n\\begin{equation} \\label{eqn:vmp}\n  v_0 = \\sqrt{2\/3} \\, \\sigma_v\n\\end{equation}\nis the most probable speed, with an approximate value of 235~km\/s\n(see Refs.~\\refcite{Kerr:1986hz,Reid:2009nj,McMillan:2009yr,Bovy:2009dr}).\n\nOur early work \\cite{dfs,wgould} used this Maxwellian dark matter distribution.  Although there has been concern\nthat the velocity distribution of the dark matter might deviate significantly from Maxwellian, Refs.~\\refcite{withbaryons,bozorgnia,sloane2016} showed that results obtained for dark matter with a Maxwellian profile are consistent with those obtained when baryons are included in dark matter simulations, though there is as yet possible disagreement for the high velocity tail.  We concluded that\nthe Maxwellian approximation\n is a perfectly good approximation when comparing results of dark matter experiments to data.\n\n We also showed \\cite{dfs} that the dark matter signal should experience an annual modulation\n(for a review, see Ref.~\\refcite{lisanti}.)\n As the Sun orbits around the Galactic Center, Earth-based detectors are effectively moving into a ``wind\" of WIMPs.\n The WIMPs are moving in random directions in the Galaxy, and the Sun's motion creates (on the\n average) a relative velocity between us and the WIMPs.\n  On top of that, because the Earth is moving around the Sun, the relative velocity of the Earth with the WIMP wind\n varies with the time of year. Thus the count rate should modulate sinusoidally with the time of year, peaking in June\n and with a minimum in December.\n   We predicted that the annually modulating\nrecoil rate can be approximated by\n\\begin{equation} \\label{eqn:dRdES}\n  dR\/dE(E,t) \\approx S_0(E) + S_m(E) \\cos{\\omega(t-t_0)} ,\n\\end{equation}\nwith $|S_m| \\ll S_0$, where $S_0$ is the time-averaged rate, $S_m$ is\nreferred to as the modulation amplitude, $\\omega = 2\\pi$\/year and $t_0$ is the phase of the\nmodulation.\nSince typical backgrounds do not experience the same annual modulation, this effect\ncan be used to tease the signal out of the background. \\cite{wgould}\n\n These first papers convinced experimentalists that they would be able to build detectors sensitive\n enough to search for WIMPs. The detectors must be placed deep underground in order to filter out cosmic rays,\n in underground mines or underneath mountains.  The first experimental effort to search for and bound WIMP dark\n matter was Ref.~\\refcite{ahlenavignone}.\n Now, 30 years later, direct detection searches are\n ongoing worldwide, in US, Canada, Europe, Asia, and the South Pole, see Fig.~7.\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{undergroundlabs}\n\\caption{Underground dark matter laboratories worldwide (courtesy of M. Tripathi and M. Woods). The CanFranc underground\nlaboratory in Spain is missing from the figure.}\n\\end{figure}\n\n Of all of these experiments, only one, the Italian DAMA experiment, \\cite{Bernabei:2014tea} has positive signal.\n They use NaI crystals in the Gran Sasso tunnel under the Apennine Mountains near Rome.\n The signal they have is the annual modulation we predicted for the WIMP signal. \\cite{dfs,wgould}\n DAMA has observed exactly this annual modulation with the correct phase, see Fig.~8. \n Indeed DAMA has 10 years of cycles corresponding to a 9 $\\sigma$ detection of modulation.\n\n Now the question is, have they detected dark matter? Unfortunately they have not released the data for others to study.\n In addition, no experiment other than DAMA has found any signal at all.  Indeed the null results from other experiments\n place strong bounds on the WIMP elastic scattering cross section.  Naively it might seem that the other experiments\n rule out the DAMA results as being due to WIMPs. Yet, this may not be true, because all the detectors are made of\n different materials.  DAMA is the only experiment to date that uses NaI crystals.\n  For example, LUX \\cite{Akerib:2016vxi}\nand XENON \\cite{Aprile:2016swn}\nare made of xenon while CDMS (and SuperCDMS) \\cite{cdms} is made of germanium, which are far heavier nuclei\n than the components of DAMA's NaI crystals.  To compare the different experiments, theoretical input is required.\n For example, if one assumes the scattering is SI so that the cross section scales as $A^2$, one can then plot\n the different experiments as in Fig.~9 in the cross section\/ WIMP mass plane.  DAMA signals\ncould  be due to roughly 10 GeV WIMPs if the scattering is with Na atoms, while the signal would be due to 80 GeV WIMPs\nif the scattering were off of iodine atoms.  The higher mass region is in severe conflict with bounds from other\nexperiments, while the lower mass region also appears to be ruled out.  However, if one abandons the $A^2$ assumption\nthen this comparison plot is no longer valid.  For all known theoretical assumptions it is hard to reconcile the positive results\nof DAMA with the negative results of other experiments.  Perhaps uncertain nuclear physics may be responsible.\n\\cite{Anand:2014kea,Dent:2015zpa}\n Many alternate explanations to the discovery of DM\nhave been proposed (e.g. radon contamination, muons, etc.) but all have been shown to be wrong.\nThe reason DAMA remains so interesting is that there is no other known explanation of the\nannual modulation they are seeing. \n\n \\begin{figure}\n\\includegraphics[width=\\textwidth]{DAMAdata}\n\\caption{DAMA data (including DAMA\/LIBRA) has a 9 $\\sigma$ detection of annual modulation consistent\nwith WIMPs. \\cite{Bernabei:2014tea}}\n\\end{figure}\n\nWhat is needed are further experimental tests using the same detector material as DAMA (NaI crystals) but in a different\nlocation.  These experiments are now taking place:  SABRE, \\cite{Froborg:2016ova}\nCOSINE-100 \\cite{COSINE100} (KIMS has joined with dark matter-ICE \\cite{deSouza:2016fxg}), and ANAIS. \\cite{Amare:2015rpa}\n Thus in the next five years there should be either\nconfirmation of DAMA or it will be ruled out.\n\n \\begin{figure}\n\\includegraphics[width=\\textwidth]{SIbounds}\n\\caption{Spin independent scattering bounds from direct detection experiments as shown, as well as regions compatible with DAMA data,\nin the SI elastic scattering cross section vs. WIMP mass plane. Plot taken from Particle Data Book 2015 (PANDA-X and LUX bounds need to be updated).}\n\\end{figure}\n\nI also wanted to mention a new idea we have for dark matter direct detection using DNA (see Fig.~10). We proposed \\cite{DNA} to\nuse nanometer thin sheets of gold (or other material) with $\\sim10^{60}$ strands of DNA attached.  When a WIMP hits the\ngold sheet, it knocks a gold atom forward into the DNA.  The gold atom then severs whatever DNA strands it hits.\nThe broken strand of DNA then falls down and is collected.\nThe DNA has been carefully constructed to have a well-known sequence of bases (A,G,C...).\nUsing well known biological techniques (PCR and sequencing), the location of the break can be identified.\nThus the track of the recoiling gold nucleus can be reconstructed.  Since the distance between the bases in the DNA\nstrand is nanometer in size, this technique provides a nanometer tracker.\nOnce the track of the gold nucleus is known, since the WIMP traveled in roughly the same direction, the direction that\nthe WIMP came in from is also known.  This idea thus provides a directional dark matter detector.   The importance of this\nis as follows.  It allows clear proof of dark matter discovery.  We expect ten times as many counts when the detector\nis pointing into the direction of the WIMP wind than when it is pointing in the opposite direction.  This head\/tail asymmetry\nwould be hard to explain with any background.  Additionally, both the annual and daily (due to Earth's rotation)\n variation of the signal would\nbe detectable and would give superb background rejection.  In the long run, a directional detector would allow\nthe discovery of where the WIMPs are in the Galaxy and how they are moving.\n\nA second radically new idea we have proposed for dark matter detection is ``nanobooms\". \\cite{nanobooms}\nThe WIMP sets off a very small explosion when\nit deposits heat in the detector.  For example, the detector might consist of thermites. Then the WIMP's energy deposit would\ncause the exothermic reaction between a metal and a metal oxide to take place, i.e., there is a small explosion, which can then\nbe detected acoustically, optically, or more likely via gas expansion.\n\n \\begin{figure}\n\\includegraphics[width=\\textwidth]{DNA}\n\\caption{DNA based Dark Matter Directional Detector.\nA WIMP hits the nanometer-thin gold plate, knocks a gold atom into the hanging\nstrands of DNA. Whenever the gold atom strikes a DNA strand, the strand breaks and is collected.  Since the base sequence\nof the strands is controlled, sequencing the broken strand allow the location of the break to be identified.\nHence the DNA serves as a tracker\nwith nanometer accuracy.  Since the WIMP travels in roughly the same direction as the gold atom, the detector discovers the\ndirection the WIMP came from.}\n\\end{figure}\n\nThe next five years stand to lead to tests of the DAMA annual modulation signal and a confirmation or refutation of WIMP\ndiscovery as well as progress in directional sensitivity.\n\n \\subsection{Indirect Detection}\nWIMP annihilation in today's universe takes place wherever there is an overdensity of WIMPs.\nThe final products of WIMP annihilation are neutrinos, e$^+$\/e$^-$ pairs, and photons.\nAll three of these are being looked for in detectors.\nPromising places to look are the Galactic Center, dwarf galaxies, clusters of galaxies, \\cite{Adams:2016alz}\nand in the case of neutrinos, the Earth and the Sun. The first papers suggesting\nthe latter neutrino searches were by Silk et al.\\, \\cite{SOS} in the Sun; and\nby Freese \\cite{Freese1986} as well as Krauss, Srednicki and\nWilczek \\cite{Krauss_etal1986} in the Earth.  As yet no signal of neutrinos due to WIMP annihilation\nin the Sun or Earth \\cite{Aartsen:2016fep}\nhas been found\nin the IceCube\/DeepCore detectors at the South Pole.\n\nThe AMS experiment on board the International Space Station has found an excess of positrons. \\cite{AMS}\n However, this excess is not likely to be due to WIMP annihilation.\nA combination of two papers has shown that such an explanation is extremely unlikely.\nFirst, the work of Lopez, Savage, Spolyar, and Adams \\cite{Lopez:2015uma}\npointed out that\nsuch a positron excess would predict also gamma-rays from dwarf galaxies, which are not seen in the Fermi Gamma Ray\nSpace Telescope (Fermi-LAT) data.\nThey used the bounds on gamma-rays from dwarfs in Fermi-LAT data \nto show that all WIMP annihilation channels are excluded as explanations of AMS data except one (via  a mediator to four muons).\nThis latter channel was further examined by Scaffidi et al.\\ \\cite{Scaffidi:2016ind}\nSecond, the Planck satellite\nexamined the effects such an excess would imply for the CMB and ruled out a large swath of parameter space. \\cite{Ade:2015xua}\nThe work of  Ref.~\\refcite{Lopez:2015uma} using Fermi-LAT data to rule out a DM explanation of the AMS positron excess\nwas placed on the arXiv a month prior to the Planck bounds.\nIt is far more likely that the AMS positron excess is due to pulsars or other point sources than due to WIMP\nannihilation.\n\nOf great interest over the last few years has been Fermi-LAT's discovery of a gamma-ray excess towards the Galactic Center.\nHooper and Goodenough \\cite{GCexcess}  pointed out that it could be from the annihilation of a 40 GeV WIMP.\nMore recent studies of cosmic ray backgrounds have widened the possible range of masses \\cite{agrawal}\nand therefore SUSY\nexplanations of this excess.  \\cite{Freese:2015ysa}\nHowever, studies \\cite{pointsources} have shown that  a point source explanation (e.g., pulsars) is at least as likely as\na dark matter explanation.  Though tantalizing, a dark matter explanation of this gamma ray excess will be hard to prove as there\nis much astrophysical competition at the Galactic Center.\n\n\\subsection{Summary of WIMP Searches}\n\nTo summarize the current status of WIMP searches, there is possible evidence for WIMP detection already  now, but\nnone of it is certain.\nThe direct detection experiment DAMA has found annual modulation of its signal that would be compatible\nwith a WIMP origin. However, other experiments have null results in conflict with DAMA's result.  Since the\nexperiments are made of different detector materials, further tests of the same material as DAMA are now\ntaking place around the world and will result in confirmation or refutation in the next five years.\n\nAs far as indirect detection of WIMP annihilation products, the positron excess seen by AMS likely has a different\norigin than WIMPs.  The gamma-ray excess seen from the direction of the Galactic Center by the Fermi Gamma Ray Space Telescope\nis compatible with a WIMP origin but other astrophysical explanations are at least as likely.\n\nTheorists are looking for models in which some of these results are consistent with one another, given a WIMP interpretation.\nWhat will it take for us to believe dark matter has been found? We need a compatible signal in a variety of experiments made\nof different detector materials and all the parties agree.\n\n\n\\section{Dark Stars}\nA fourth prong of the hunt for dark matter is the search to discover dark stars.\nThe first stars to form in the universe, at redshifts $z \\sim 10-50$,\nmay be very unusual; these dark stars are made almost entirely of atomic matter (hydrogen and helium, with only $10^{-3}$ of the mass made of dark matter)\nand yet are powered by dark matter heating rather than by fusion.   Dark stars were first proposed by Spolyar, Freese, and Gondolo \\cite{SpolyarFreeseGondolo08} and are reviewed in Ref.~\\refcite{Freese:2015mta}.\n\n As discussed in the last section, WIMP dark matter annihilation in the\nearly universe provides the right abundance today to explain the dark\nmatter content of our universe. This same annihilation process will\ntake place at later epochs in the universe wherever the dark matter\ndensity is sufficiently high to provide rapid annihilation.  The first\nstars to form in the universe are a natural place to look for\nsignificant amounts of dark matter annihilation, because they form at\nthe right place and the right time. They form at high redshifts, when\nthe universe was still substantially denser than it is today, and at\nthe high density centers of dark matter haloes.\n\nThe first stars form inside dark matter haloes of $\\sim10^6 M_\\odot$\n(for reviews see e.g.,  Ripamonti \\& Abel, \\cite{RipamontiAbel05}\nBarkana \\& Loeb, \\cite{BarkanaLoeb01} and Bromm \\& Larson;\n\\cite{BrommLarson03} see also Yoshida et al.\\ \\cite{Yoshida_etal06}).\nOne star is thought to form inside one such dark matter halo.   It was our idea to ask, what is the\neffect of the dark matter on these first stars?  We studied the behavior of\nWIMPs in the first stars.  As our canonical values, we take $m_\\chi =\n100$GeV for the WIMP mass and $\\langle \\sigma v \\rangle_{ann} = 3\n\\times 10^{-26} {\\rm cm^3\/sec}$ for the annihilation cross section\n(motivated above).  However, the same behavior results for a wide variety\nof WIMP masses and cross sections over many orders of magnitude.\nWe find that the annihilation products of the\ndark matter inside the star can be trapped and\ndeposit enough energy to heat the star and prevent it from further\ncollapse.  A new stellar phase results, a dark star, powered\nby dark matter annihilation as long as there is dark matter fuel.\n\n\\subsection{Three Criteria for Dark Matter Heating}\n\n WIMP annihilation produces energy at a rate per\nunit volume\n\\begin{equation}\n  Q_{\\rm ann} = \\langle \\sigma v \\rangle_{ann} \\rho_\\chi^2\/m_\\chi\n  \\linebreak \\simeq  10^{-29} {{\\rm erg} \\over {\\rm cm^3\/s}} \\,\\,\\, {\\langle\n    \\sigma v \\rangle \\over (3 \\times 10^{-26} {\\rm cm^3\/s})} \\left({n \\over {\\rm\n        cm^{-3}}}\\right)^{1.6} \\left({100 {\\rm GeV}\\over m_\\chi}\\right)\n\\end{equation}\nwhere $\\rho_\\chi$ is the dark matter energy density inside the star and $n$ is\nthe stellar hydrogen density.  Spolyar, Freese and Gondolo\n\\cite{SpolyarFreeseGondolo08} outlined the three key ingredients\nfor dark stars: 1) high dark matter densities, 2) the annihilation\nproducts get stuck inside the star, and 3) dark matter heating wins over other\ncooling or heating mechanisms.  These  ingredients are required\nthroughout the evolution of the dark stars.\n\n{\\bf First criterion: high dark matter density inside the star.}  Dark\nmatter annihilation is a powerful energy source in these first stars\nbecause the dark matter density is high. To find the dark matter density\nprofile, we started with an NFW (Navarro, Frenk \\& White\n\\cite{NavarroFrenkWhite96}) profile for both dark matter and gas in the $10^6\nM_\\odot$ halo.  However, we find the same behavior results for even\na completely flat profile; the dark star is born regardless.\nOriginally we used adiabatic contraction ($M(r)r$ =\nconstant) (Blumenthal et al.\\ \\cite{Blumenthal_etal85}) and\nmatched onto the baryon density profiles given by Abel, Bryan \\&\nNorman \\cite{AbelBryanNorman02} and Gao et\nal.\\ \\cite{Gao_etal07} to obtain dark matter profiles; see also Natarajan,\nTan \\& O'Shea \\cite{NatarajanTanO'Shea08} for a recent\ndiscussion.  Subsequent to our original work, we have done an exact\ncalculation (which includes radial orbits) (Freese, Gondolo, Sellwood\n\\& Spolyar \\cite{FreeseGondoloSellwoodSpolyar08}) and found that\nour original results were remarkably accurate, to within a factor of\ntwo.  At later stages, we also consider possible further enhancements\ndue to capture of dark matter into the star (discussed below).\n\n{\\bf Second Criterion: dark matter annihilation products get stuck\n  inside the star}.  In the early stages of Population III star formation,\nwhen the gas density is low, most of the annihilation energy is\nradiated away (Ripamonti, Mapelli \\& Ferrara\n\\cite{RipamontiMapelliFerrara06}). However, as the gas collapses\nand its density increases, a substantial fraction $f_Q$ of the\nannihilation energy is deposited into the gas, heating it up at a rate\n$f_Q Q_{\\rm ann}$ per unit volume.  While neutrinos escape from the\ncloud without depositing an appreciable amount of energy, electrons\nand photons can transmit energy to the core.  We have computed\nestimates of this fraction $f_Q$ as the core becomes more dense. Once\n$n\\sim 10^{11} {\\rm cm}^{-3}$ (for 100 GeV WIMPs), e$^-$ and photons\nare trapped and we can take $f_Q \\sim 2\/3$.\n\n{\\bf Third Criterion: dark matter heating is the dominant heating\/cooling\n  mechanism in the star}.  We find that, for WIMP mass $m_\\chi =\n100$GeV (1 GeV), a crucial transition takes place when the gas density\nreaches $n> 10^{13} {\\rm cm}^{-3}$ ($n>10^9 {\\rm cm}^{-3}$).  Above\nthis density, dark matter heating dominates over all relevant cooling\nmechanisms, the most important being H$_2$ cooling (Hollenbach\n\\& McKee \\cite{HollenbachMcKee79}).\n\nFig.~11 shows evolutionary tracks of the protostar in the\ntemperature-density phase plane with dark matter heating included\n(Yoshida et al.\\ \\cite{Yoshida_etal08}), for two dark matter particle\nmasses (10 GeV and 100 GeV).  Moving to the right on this plot is\nequivalent to moving forward in time.  Once the black dots are\nreached, dark matter heating dominates over cooling inside the star.\nThe protostar collapses somewhat further until it reaches equilibrium, at which point the\ndark star phase begins.  The protostellar core is prevented from\ncooling and collapsing further.  The size of the core at this point is\n$\\sim 17$ A.U. and its mass is $\\sim 1 M_\\odot$ for 100 GeV mass\nWIMPs.  A new type of object is created, a dark star supported by dark matter\nannihilation rather than fusion.\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{naoki-bob}\n\\caption{ Temperature (in degrees K) as a function of hydrogen density\n  (in cm$^{-3}$) for the first protostars, with dark matter annihilation\n  included, for two different dark matter particle masses (10 GeV and 100 GeV).\n  Moving to the right in the figure corresponds to moving forward in\n  time.  When the ``dots'' are reached, dark matter annihilation wins over H2\n  cooling.  After that the protostar collapses somewhat further until\n  it reaches equilibrium.  At that point a dark star is created.}\n\\end{figure}\n\n\\subsection{Building up the Mass}\n\nWe have found the stellar structure of the dark stars\n(hereafter DS) (Freese, Bodenheimer, Spolyar \\& Gondolo\n\\cite{FreeseBodenheimerSpolyarGondolo08}).  They accrete mass from the\nsurrounding medium.  We built up the DS mass as it grows\nfrom $\\sim 1 M_\\odot$ to possibly become supermassive.\nThe studies were done in two different ways, first assuming polytropic interiors and\nmore recently using the MESA stellar evolution code; the basic results are the same.\n\\cite{Rindler-Daller:2014uja}\nAs the mass increases, the DS radius adjusts\nuntil the dark matter heating matches its radiated luminosity.  We find\nsolutions for dark stars in hydrostatic and thermal\nequilibrium. We build up the DS by accreting $1 M_\\odot$ at a time\nwith a variety of possible accretion rates, always\nfinding equilibrium solutions.  We find that initially the DS are in\nconvective equilibrium; from $(100-400) M_\\odot$ there is a transition\nto radiative; and heavier DS are radiative.  As the DS grows, it pulls\nin more dark matter, which then annihilates.  Fig.~12 shows the hydrogen and dark matter density profiles. One can see ``the power of\ndarkness'': although the dark matter constitutes a tiny fraction ($<10^{-3}$)\nof the mass of the DS, it can power the star. The reason is that WIMP\nannihilation is a very efficient power source: 2\/3 of the initial\nenergy of the WIMPs is converted into useful energy for the star,\nwhereas only 1\\% of baryonic rest mass energy is useful to a star via\nfusion.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{fig6}\n\\caption{Evolution of a dark star as mass is accreted onto the\n  initial protostellar core of 3 M$_\\odot$.  The set of upper (lower)\n  curves correspond to the baryonic (dark matter) density profile at different\n  masses and times. Note that dark matter constitutes $<10^{-3}$ of the mass of\n  the DS.}\n\\end{figure}\n\n\\subsection{Later stages: Capture}\n\nThe dark stars will last as long as the dark matter fuel inside them persists.\nOnce the gravitationally attracted dark matter runs out, the star collapses somewhat,\nat which point the star is dense enough to capture more dark matter.\n\nThe new source of dark matter in the first stars is capture of dark matter particles\nfrom the ambient medium.  Any dark matter particle that passes through the\nDS has some probability of interacting with a nucleus in the star\nand being captured. The new particle physics ingredient required\nhere is a significant scattering cross section between the WIMPs\nand nuclei. Whereas the annihilation cross section is\nfixed by the relic density, the scattering cross section is a\nsomewhat free parameter, set only by bounds from direct detection\nexperiments.\nTwo simultaneous papers (Freese, Spolyar \\& Aguirre,\n\\cite{FreeseSpolyarAguirre08} Iocco \\cite{Iocco08}) found the\nsame basic idea: the dark matter luminosity from captured WIMPs can be larger\nthan fusion for the DS. Two uncertainties exist here: the scattering\ncross section, and the amount of dark matter in the ambient medium to capture\nfrom.  DS studies following the original papers that include\ncapture have assumed (i)  the maximal scattering cross sections allowed by\nexperimental bounds and (ii) ambient dark matter densities that are never depleted.\nWith these assumptions, DS evolution models with dark matter heating after the\nonset of fusion were studied in several papers.  \\cite{Taoso_etal08,Yoon_etal08}\n\n\\subsection{Supermassive Dark Stars}\n\nDark stars are very unusual stars --- they are made of atomic matter (hydrogen and helium)\nbut they are powered by dark matter heating (Freese, Bodenheimer, Spolyar \\& Gondolo\n\\cite{FreeseBodenheimerSpolyarGondolo08}). They are very puffy (10 A.U. in size) and cool\n(surface temperatures $\\sim$ 10,000 K.  Reionization\nduring this period is likely to be slowed down, as these stars can\nheat the surroundings but not ionize them.\nBecause they are so cool, they can keep accreting matter and growing\nas long as there is dark matter fuel.\nStandard Population III stars are hot, give off ionizing photons, and prevent further accretion\nabove $\\sim 140 M_\\odot$.  Dark stars, on the other hand, can keep growing to become supermassive,\neven as massive as $10^7 M_\\odot$ and as bright as $10^{10} L_\\odot$.\nThere should be a variety of dark star masses ranging from a few solar masses all the way up to these\nvery large masses.\n\nFig.~13 shows the Hertzsprung-Russell  diagram for dark stars as they grow from $\\sim 1 M_\\odot$ to become supermassive.\nThe two cases of matter being accreted gravitationally and via capture are shown separately. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{HR}\n\\caption{Hertzsprung-Russell diagram for DSs for a variety of WIMP masses as labeled for the two cases: (i) with gravitationally\nattracted dark matter only (dotted lines), assuming no significant depletion of dark matter due to annihilation, which is equivalent to assuming a replenishment of dark matter due to centrophilic orbits; (ii) with capture (solid lines). Results were obtained assuming polytropic interiors for the DS. The case with capture is for product of scattering cross section times ambient WIMP density\n$\\sigma_c \\rho_\\chi = 10^{-39}$ GeV\/cm$^3$ (the maximum allowed cross section for all WIMP masses and the maximum reasonable ambient density for 100 GeV WIMPs).  Once the gravitational dark matter runs out,  DSs must first become dense enough in order\nfor dark matter capture to happen. This explains the horizontal lines in the evolution of the case with capture. Labeled are also stellar masses reached by the DS on its way to becoming supermassive. The final DS mass was taken to be $10^5 M_\\odot$ (the baryonic mass inside the initial halo), but could vary from halo to halo, depending on the specifics of the halo mergers (figure taken from Ref.~\\protect\\refcite{HR}).\n}\n\\end{figure}\n\n\\subsection{Dark Stars are Detectable in James Webb Space Telescope}\nSupermassive dark stars may be detectable in the JWST as J, H, or K-band dropouts.  Detailed discussion may be found\nin Refs.~\\refcite{HR,ruiz,cosmin}.  Comparison of light output with sensitivity of JWST filters is shown in Fig.~14\nfor a $10^6 M_\\odot$ DS.  Predictions for numbers of these objects, based on cosmological simulations, is also found in Ref.~\\refcite{HR}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{JWST}\n\\caption{Supermassive Dark Stars in JWST.  Spectra for $10^6 M_\\odot$ supermassive DSs formed at  redshift $z_{form}=15$ compared with sensitivity of JWST filters. The formation mechanism in this figure is gravitational attraction of dark matter only. The surface temperature $T_{\\rm eff} = 1.9 \\times 10^4$K. The fluxes are shown at z = 15 (dashed line), 10 (solid line) and 5 (dotted line) and compared to the detection limits of NirCam wide passband filters. The colored horizontal lines represent the sensitivity limits for the filters as labeled in the legend for exposure times $10^4$ sec (upper lines) and $10^6$ sec (lower lines). IGM absorption will decrease the observed fluxes for wavelengths shortward of the vertical red lines, which indicate the Lyman-$\\alpha$ line (1216 Angstroms) redshifted from the rest-frame of the star (figure taken from Ref.~\\protect\\refcite{cosmin}).}\n\\end{figure}\n\n\n\n\\subsection{Supermassive Black Holes}\nOnce these supermassive dark stars (SMDS) run out of dark matter fuel, they collapse to black holes.\nThey may provide large seeds for the supermassive black holes that have been found\nat high redshift ($10^9-10^{10} M_\\odot$ BH at $z=6$) and are, as yet,\nunexplained (Li et al., \\cite{Li_etal07} Pelupessy et\nal., \\cite{Pelupessy_etal07} Wu et al.\\ \\cite{WuNature}).\n\n\\subsection{Pulsations} \n\nAn interesting new research direction is the fact that DS pulsate, like all stars.  As a first step, we used the MESA stellar\nevolution code to calculate  the adiabatic pulsation periods of radial $p$-modes (where the\nrestoring force is pressure and those for which there is no angular dependence, so $l = 0$).\nWe found that our DS models pulsate on timescales which range from less than a day to more than two years in their restframes at about $z = 15$, depending on the WIMP mass and overtone number. The pulsation periods are significantly shorter for higher WIMP mass. Converting to the observer frame, the shortest periods we found are less than about 50 days for modes with overtone number\n$n > 6$ and a WIMP mass of 1 TeV (Ref.~\\refcite{Rindler-Daller:2014uja}).\nWe are currently investigating other pulsation modes: nonadiabatic modes and also dark matter density driven modes.\n\n\nIn short, the first stars to form in the universe may be dark stars\npowered by dark matter heating rather than by fusion.  Our work indicates that\nthey may become very large (up to $10^7 M_\\odot$) and bright (up to $10^{10} L_\\odot$),\nthereby detectable in upcoming JWST observations.\nThey may provide seeds for the many supermassive black holes found in the universe.  \nThe observational possibilities of discovering dark matter by finding these stars with JWST data\nis intriguing.  Further, once DS\nare found, one can use them as a tool to study the properties of WIMPs.\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{sterileneutrinos}\n\\caption{ Sterile Neutrinos:\nObservations consistent with and bounding the sterile neutrino mass and mixing angle to ordinary neutrinos\n(figure courtesy of K. Abazajian ``Cosmology of Sterile Neutrinos,\" in preparation (2016)).}\n\\end{figure}\n\n\n\\section{Sterile Neutrinos}\n\nAnother intriguing dark matter candidate is a  sterile neutrino.\nWhereas the three known neutrino species are far too light\nto constitute dark matter, it is possible that one or more additional neutrino types, that do not\ninteract via the fundamental interactions of the standard model of particle physics\ncould make up the dark matter.\nThese sterile neutrinos could, however, mix with ordinary neutrinos.\n\nIn the past few years, several X-ray astronomy groups \\cite{Bulbul:2014sua,Boyarsky:2014jta}\nhave found evidence for a 3.5 keV line in clusters\nof galaxies and in M31.  This line would be consistent with a dark matter origin, corresponding to a 7 keV\nrest mass sterile neutrino with vacuum mixing with active neutrinos\n${\\rm sin}^2 2 \\theta \\sim (2-20) \\times 10^{-11}$.  Fig.~15 illustrates some of the observations.\nHowever, others argue against this interpretation, e.g. Ref.~\\refcite{Jeltema:2015mee} claims that the\nline is not seen from the dwarf galaxy DRACO and thus the 7 keV sterile neutrino is ruled out.\nThis is a subject of\ndeep controversy.\n\nTheoretical studies of sterile neutrinos are also ongoing.\nThe sterile neutrino is a singlet under the standard model; it is likely a right handed neutrino.\nThe production of these particles is difficult.  If thermal, they tend to overclose the universe.\nOther mechanisms \\cite{Barbieri:1990vx,Dodelson:1993je,Dolgov:2002wy}\n or resonance using a large lepton\nasymmetry \\cite{Canetti:2012vf}\nare difficult but being investigated.\nIn many models the sterile neutrino constitutes warm dark matter, which leads to testable predictions such as the core\/cusp\nof galaxies and the numbers of substructures (objects smaller than our Galaxy), see e.g. Ref.~\\refcite{Bozek:2015bdo}.\n\n\\section{What's Hot in Dark Matter}\n\nAs I've discussed, unexplained signals in a variety of data sets point to four hints of possible dark matter detection.\nFirst,  the DAMA \\cite{Bernabei:2014tea}\nannual modulation \\cite{dfs} signal could be compatible with a $\\sim$10GeV WIMP.\nHowever, since other experiments do not see any signal at all, the DAMA results must be checked.  Currently three\ndifferent experiments are planning to repeat the DAMA setup with NaI crystals:  SABRE, COSINE, and ANAIS.\n\nSecond, the Fermi-LAT $\\gamma$-ray excess from the direction of the Galactic Center could be due to WIMP annihilation.\nHowever, point sources (such as pulsars) constitute another explanation of the excess that is at least as good or better.\n\nThird, the possible 3.5 keV X-ray line from clusters and from M31 could be explained by a 7 keV sterile neutrino, but this \ninterpretation\nis very controversial.\n\nA fourth intriguing signal, not yet mentioned in this article, is the 511 keV $\\gamma$-ray line in INTEGRAL data. \n\\cite{Knodlseder:2003sv,Jean:2003ci}\nThis is seen in the Galactic Bulge out to 6 degrees (3 kpc).  There is no clear astrophysical explanation.\nLow mass X-ray binaries were thought to be a compelling explanation which is now being ruled out.\nThe explanation for the line could be dark matter annihilation to e$^+$e$^-$ pairs.  This would be MeV dark matter.\n\\cite{Boehm:2003bt}\n\nThe future holds interesting studies of these signals as well as the continuing hunt for dark matter.\n\n\\section{Conclusion}\nMost of the mass in the universe is in the form\nof an unknown type of dark matter. The need for dark matter has become more\nand more clear since the 1930s, with evidence from rotation curves,\ngravitational lensing, hot gas in clusters, the Bullet Cluster, structure formation, and\nthe cosmic microwave background.    A consensus picture has emerged, in which the dark matter\ncontributes 26\\% of the overall energy density of the universe.  Its\nnature is still unknown.  At most 15\\% of the dark matter in galaxies can be\nwhite dwarfs (or other MACHO candidates), but most is likely to be an\nexotic particle candidate.  Dark matter searches for the best motivated\ncandidates, axions and WMPs are ongoing and promising over the next\ndecade.\n\nThe interesting unexplained signals that may herald the discovery of dark matter have\nbeen reviewed:  DAMA's annual modulation signal and the\nFermi-LAT gamma-rays from the Galactic Center might be due to WIMPs, a 3.5 keV X-ray line from various\nastrophysical sources is possibly from sterile neutrinos,\nand the 511 keV line in INTEGRAL  might be due to MeV dark matter.\nAll of these would require further confirmation in\nother experiments or data sets to be proven correct.\nIn addition, a new line of research on dark stars was reviewed\nwhich suggests that the first stars to exist in the universe\nwere powered by dark matter heating rather than by fusion:\nthe observational possibilities of discovering dark matter by finding these stars with JWST data\n were discussed.  The goal of the searches over the next decade is to decipher the nature of the unknown dark matter.\n\n\n\\section{Acknowledgments}\nKF would like to thank Luca Visinelli for commenting on the draft.\nKF acknowledges support through a grant from the Swedish Research Council (Contract No. 638-2013-8993). KF  acknowledges support from DoE grant DE-SC007859 at the University of Michigan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nExchange anisotropy, or exchange bias, is an interfacial phenomenon between \nferromagnetic and antiferromagnetic domains which results in the shifting and \nbroadening of magnetic hysteresis loops.  Exchange bias is believed to result \nfrom the interaction of ferromagnetic (FM) spins with uncompensated\nantiferromagnetic (AFM) spins at the FM\/AFM \ninterface.\\cite{blamire_uncomp,kuch_uncomp,nogues_schuller}  \nSince its discovery in partially oxidized Co\/CoO nanoparticles by Meiklejohn \nand Bean,\\cite{mbean} exchange bias has been observed and engineered in\ncore-shell nanoparticles,\\cite{coreshell} thin films,\\cite{thinfilms} and \ngranular composites.\\cite{nanostructure_review} These\narchitectures are utilized because a high proportion of FM spins must be\ninterfacial in order for the AFM switching behavior to appreciably\naffect the FM coercivity.  While they achieve a high interface\/volume\nratio, core-shell nanoparticles and thin film architectures\ndo not result in large quantities of exchange-biased material.\nAs an alternative, novel methods of processing exchange biased systems have\nbeen explored, including coevaporation,\\cite{coevap} mechanical\nmilling,\\cite{sort_milling} and spontaneous phase \nseparation.\\cite{separation_alloy} \n\nInitial reports from Sort \\textit{et al.}\\cite{sort} have demonstrated \nhydrogen reduction of Fe$_{0.2}$Cr$_{1.8}$O$_3$ to produce metal\/oxide \ncomposites. Different transition metals reduce sequentially, resulting in\nnanosized Fe particles within micron sized Cr$_2$O$_3$ grains. \nInteraction between the $\\sim$10\\,nm Fe precipitates and the bulk\nCr$_2$O$_3$ provides exchange bias shifts of 10\\,Oe.  Reduction\nkinetics of the system CoCr$_2$O$_4$--Co$_3$O$_4$ have been reported by\nBracconi and Dufour\\cite{bracconi}, and Kumar and Mandal\\cite{kumar} have\nproduced Co\/Cr$_2$O$_3$ composites directly from nitrate precursors.  \nRecently, Toberer \\textit{et al.}\\cite{toberer_advmat05} have demonstrated \nthat remarkable microstructures with aligned porosity can be\nobserved when the reduction product shares a common oxygen sublattice\nwith the precursor. \n\nHere we report on hydrogen reduction of the system Ni$_x$Mn$_{3-x}$O$_4$ \nto form Ni\/MnO composites with striking microstructures associated\nwith substantial exchange biasing.\nThe Ni particles exhibit bulk saturation magnetization values, and\nexchange bias is observed below the N\\'eel temperature of MnO at $T_N$ =  \n119\\,K.  Surface and interior particle size analysis reveals\nthat this system produces Ni nanoparticles on the order of 15\\,nm to 30\\,nm. \nSize-dependent exchange bias phenomena are manifested in trends between\nthe Ni content of the precursor spinel and the exchange and coercive\nfields of the reduced composite.\n\n\\section{Experimental}\n\nSingle-phase ceramic monoliths were prepared\nby solid-state reactions of oxalates, similar to that of\nWickham.\\cite{wickham} Oxalates are versatile precursors for \nmixed metal oxides, and have found extensive use in recent years\nto produce substituted binary\\cite{risbud_co-zno,lawes_co-mn-zno} and \nternary\\cite{toberer_chemmat05,toberer_advmat05,toberer_chemmat06} compounds. \n\nStoichiometric amounts of\nnickel acetate and manganese acetate [Ni(CH$_2$COOH)$_2$$\\cdot$4H$_2$O and\nMn(CH$_2$COOH)$_2$$\\cdot$4H$_2$O, Aldrich 99\\,\\%] were added to a solution\ncontaining one equivalent of glacial acetic acid.  Excess oxalic acid\nmonohydrate [H$_2$(C$_2$O$_4$)$\\cdot$H$_2$O, Fisher 99.9\\,\\%] was mixed in a\nseparate solution and both were stirred at $90^{\\circ}$C.  Addition of\nthe oxalic acid to the dissolved acetates results in coprecipitation of\nvery fine, single-phase nickel-manganese oxalates in which the metals are \nmixed on the atomic scale. The oxalate powders, \nNi$_x$Mn$_{3-x}$(C$_2$O$_4$)$_3$$\\cdot$2H$_2$O,\nwere washed with deionized water and dried at $90^{\\circ}$C,\ncalcined in alumina boats in\nair at temperatures ranging from 780$^\\circ$ to $1200^\\circ$C for 10\\,h,\nthen quenched into water to prevent conversion to $\\alpha$-Mn$_2$O$_3$ or\nto NiMnO$_3$.  The resulting single-phase Ni$_x$Mn$_{3-x}$O$_4$ spinel\npowder was pressed into pellets at 100\\,MPa and sintered at\n$1325^{\\circ}$C for 24\\,h, then annealed at the previous calcination\ntemperature and water quenched.\n\nReductions were performed in \nalumina boats in a tube furnace under \n5\\,\\% H$_2$\/N$_2$ with a flow rate of approximately 30\\,sccm. Once the gas \nmixture had equilibrated, the specimens, as pellets, were heated at \n$2^{\\circ}$C\/min to 650$^\\circ$C, 700$^\\circ$C, or 725$^\\circ$C, held for \n2\\,h, then cooled at $10^\\circ$C\/min to room temperature. Reduced samples \nwere verified to be Ni\/MnO by x-ray diffraction (XRD, Philips X'Pert with \nCu$K_{\\alpha}$ radiation) and Rietveld refinement using the \\textsc{xnd}\ncode.\\cite{xnd} Composites were characterized by\nthermogravimetic analysis (TGA, Cahn TG-2141), scanning electron\nmicroscopy (SEM, FEI Sirion XL40), focused ion beam milling and\nmicroscopy (FIB, FEI DB235), and SQUID magnetometry (Quantum Design MPMS\n5XL). \n\n\\section{Results and Discussion}\n\n\\begin{figure}\n\\centering\\epsfig{file=fig01.eps,width=7cm}\\\\\n\\caption{X-ray diffraction Rietveld refinements of \n(a) Ni$_{0.3}$Mn$_{2.7}$O$_4$ single-phase tetragonal spinel (hausmannite)\nprecursor, and (b) the \\textit{fcc}-Ni\/rock-salt MnO composite produced by \nreduction of the above spinel in 5\\,\\%H$_2$\/N$_2$.}\n\\label{rietveld10}\n\\end{figure}\n\n\nThe calcining of the single-phase Ni\/Mn oxalates, according to the phase \ndiagram presented by Wickham,\\cite{wickham} results in single phase  \nspinel-related compounds that are not all cubic. Wickham\\cite{wickham}\nhas reported that in their high-temperature state, bulk samples \nof Ni$_x$Mn$_{3-x}$O$_4$ with $x$ between 0.15 and 1.00 are cubic spinels \nbefore decomposing into NiMnO$_3$ and $\\alpha$-Mn$_2$O$_3$ in the temperature\nrange of 705$^\\circ$ to 1000$^\\circ$C.  Upon\nwater quenching, samples prepared with $x < 1$ and fired at\n$\\geq1000^{\\circ}$C are observed to distort from the high-temperature \ncubic spinel reported by Wickham into single-phase hausmannite-type tetragonal\nspinels in space group $I4_1\/amd$. Slow-cooling, air-quenching, or quenching \nin flowing nitrogen are insufficient to prevent decomposition of the solid\nsolution. \n\nRietveld refinement of the room-temperature XRD pattern for the\nwater-quenched compound Ni$_{0.30}$Mn$_{2.7}$O$_4$ is shown in \nFig.\\,\\ref{rietveld10}(a).  Only peaks for the hausmannite-type solid solution\nare evident; this is a requirement for the final reduced composite to be \nhomogeneous in terms of the distribution of Ni precipitates.\nThe refinement assumes a ``normal\" spinel, where Ni$^{2+}$ and Mn$^{2+}$ \noccupy the 4b tetrahedral sites. Mn$^{3+}$ in the 8c octahedral \nsites causes a cooperative Jahn-Teller distortion which leads to a loss of \ncubic symmetry.\\cite{goodenough_jt}  An accurate determination of the cation\ndistribution may be obtained by neutron diffraction and has been investigated \nby Larson \\textit{et al.}\\cite{cation_dist}  When sintered at 1325$^{\\circ}$C, \nsamples with $x$ near 1 partially decompose into mixtures of NiO and\nNi$_{1-\\delta}$Mn$_{2+\\delta}$O$_4$ as described by Wickham,\\cite{wickham} \nbut subsequent annealing at 800$^{\\circ}$C for 72\\,h\nensures the formation of a single-phase tetragonal spinel.  Dense pellets\nand micron-sized powder are both suitable precursors for hydrogen\nreduction because the dimensions of the precipitates and pores are\norders of magnitude smaller than the grain size in either case. \nAdequately high oxygen mobility at the reduction temperature allows the \nreaction to permeate the sample regardless of any lack of preexisting porosity.\n\n\\begin{figure}\n\\centering\\epsfig{file=fig02.eps,width=7cm}\\\\\n\\caption{TGA of a Ni$_{0.3}$Mn$_{2.7}$O$_4$ sample shows that reduction \nproceeds by an initial reaction to rocksalt Ni$_{0.1}$Mn$_{0.9}$O solid \nsolution, followed by a reduction of Ni$^{2+}$ into metallic Ni.} \n\\label{tga}\n\\end{figure}\n\nIn all cases, TGA analysis confirms the total amount\nof nickel precipitated (and thus the stoichiometry of the precursor\nspinel) during hydrogen reduction.  A TGA weight loss curve for\nNi$_{0.3}$Mn$_{2.7}$O$_4$ is shown in Fig.\\,\\ref{tga}.  The\nweight loss curve reveals that the single-phase spinel first reduces to\na rocksalt (Ni$_{0.1}$,Mn$_{0.9}$)O solid solution, followed by\nprecipitation of metallic Ni. This progression is verified by\nthe fact that incompletely reduced samples display an MnO lattice\nparameter that is smaller than the theoretical value, due to Ni substitution.\nX-ray diffraction Rietveld refinement of the final composite\nproduct obtained after reduction in 5\\% H$_2$\/N$_2$ indicate only\nrocksalt MnO and face-centered cubic Ni [Fig.\\,\\ref{rietveld10}(b)].  \n\n\n\\begin{figure}\n\\centering\\epsfig{file=fig03.eps,width=7cm}\\\\\n\\caption{(a) Lattice parameter of MnO obtained by Rietveld refinement \nof samples after hydrogen reduction at varying temperatures. The diagonal \ndotted line is the calculated lattice parameter of a (Ni,Mn)O solid solution, \nwhile the top line represents the desired conversion to pure MnO.\n(b) Magnetic saturation of reduced Ni\/MnO samples increases linearly \nwith the completeness of Ni reduction as determined by $a_{\\rm{MnO}}$\nfrom XRD. All magnetic data concerning coercivity or \nexchange bias was measured from samples with complete Ni reduction and \nthus $M_S \\approx 0.6 \\mu_B$\/Ni.}\n\\label{reducheck}\n\\end{figure}\n\nHigh-spin Mn$^{2+}$ in octahedral coordination has an ionic radius of \n0.83\\,\\AA\\\/ in contrast to octahedral Ni$^{2+}$ which has a radius of only \n0.69\\,\\AA.\\cite{shannon-prewitt} Consequently, when Ni$^{2+}$ enters product\nMnO lattice, there is significant shrinkage of the cell parameter, which can \nbe used to estimate the degree of conversion of the starting phases into pure \nNi\/MnO. The MnO lattice parameter obtained from Rietveld refinement is plotted\nin Fig.\\,\\ref{reducheck}(a) as a function of the Ni content in the single-phase\nhausmannite\/spinel precursor. The cell parameter of pure MnO, 4.444\\,\\AA\\,\\ is \nalso indicated as a horizontal dashed line. It is seen that for \nsmall substitution of Ni ($x$ in the starting phases) the reduction \ntemperature must be increased from 650$^\\circ$C to 725$^{\\circ}$C to ensure \ncomplete reduction and avoid the rock-salt (Ni,Mn)O solid solution.  \nDepression of the required reduction temperature of Ni$_x$Mn$_{3-x}$O$_4$ as \n$x$ deviates from Mn$_3$O$_4$ is a consequence of the higher ionization \nenergy of Ni$^{2+}$.  In other words, more energy is released by reduction \nof Ni$^{2+}$ ions than of Mn$^{2+}$, so the reduction to metal occurs more \nreadily when $x$ is larger. The greater ease of reduction of Ni over Mn\nis suggested by the appropriate Ellingham diagram.\\cite{ellingham} \nThe saturation magnetization $M_S$ of the magnetic\nNi nanoparticle precipitates can be used in tandem with the values of \n$a_{\\rm{MnO}}$ \nobtained from Rietveld refinement to determine the completeness of Ni \nreduction.  This is shown in Fig.\\,\\ref{reducheck}(b), where \nagreement is seen between the convergence of $a_{\\rm{MnO}}$ and $M_S$ to their \nrespective theoretical values of 4.444~\\AA\\, and 0.6\\,$\\mu_B$\/Ni for a\ncompletely reduced $x$Ni\/MnO composite, regardless of $x$.  \n\n\n\\begin{figure}\n\\centering\\epsfig{file=fig04.eps,width=7cm}\\\\\n\\caption{Representative scanning electron microscope images: \n(a) The as-sintered surface of a dense pellet of Ni$_x$Mn$_{3-x}$O$_4$ with \n$x$ = 0.3. (b) The fracture surface of the material obtained from reducing\nthe sample in (a) at 725$^\\circ$C for 2~h. (c) and (d) are the sample\nin (b) shown at higher magnification. The highly porous and crystalline matrix\nof MnO is seen in (c), and at higher magnification, small Ni particles with\nsizes in the 30\\,nm to 40\\,nm range are seen as bright objects on a darker\nbackground.}\n\\label{semprog}\n\\end{figure}\n\nHydrogen reduction of single-phase oxide monoliths can lead to striking \nhierarchically porous microstructures, which have been characterized by \nToberer \\textit{et al.}\\cite{toberer_advmat05,toberer_chemmat06,toberer_chemmat07}\nAt first glance, low-magnification SEM micrographs of Ni$_x$Mn$_{3-x}$O$_4$\nprecursor spinels and Ni\/MnO reduced samples [Fig.\\,\\ref{semprog}(a)\nand Fig.~\\ref{semprog}(b), respectively] appear nearly identical.  However,\nhigher magnification [Fig.\\,\\ref{semprog}(c) and (d)]\nreveals that reduced composites contain aligned pores in rock-salt MnO\ncovered with Ni metal nanoprecipitates.  It has been previously \nsuggested \\cite{toberer_advmat05,toberer_chemmat07} that the shared\noxide sublattice of spinel and rocksalt allows the transformation from\none to the other to take place without reconstruction. \nPorosity is introduced during the spinel to rocksalt transformation \nwhile leaving the oxygen framework largely intact. The associated volume loss\ngives rise to a pore structure that can be regarded as negative crystals --\nvoids in crystals that possess the same facets as the crystals themselves do. \n\nAlthough the pores are as small as 20\\,nm, the pore and surface edges are \naligned at right angles over the entire breadth of the 20\\,$\\mu$m grains.  This\nlong-range alignment implies that the MnO grains are in fact single\ncrystals with the same orientation and extent as the pre-reduction\nspinel grains.\\cite{toberer_advmat05,toberer_chemmat06}  \nIncreasing the reduction temperatures should lead to densification \nand closing of the pores in the MnO monolith.\nHowever, in the interest of maintaining\nsmall Ni nanoparticles (and thus a high interface\/volume ratio), and\nbecause the majority of nanoparticles are completely encased in MnO even\nin porous samples, reduction was performed at the lowest temperature\nthat allowed complete Ni precipitation.\n\n\\begin{figure}\n\\centering\\epsfig{file=fig05.eps,width=7cm}\\\\\n\\caption{The mean diameters of the Ni particles on the surface of the MnO matrix\nas a function of the initial Ni content $x$ in Ni$_x$Mn$_{3-x}$O$_4$.\nError bars indicate one standard deviation in the particle diameter. Typically\nat least 30 distinct particle's were counted in preparing the distributions. \nIt is seen that most in most samples, the sizes are somewhat independent of $x$\nand are clustered around 30\\,nm.}\n\\label{size_nifrac}\n\\end{figure}\n\nIf we assume that for the different values of $x$, the number of nuclei are \nthe same, and that increasing $x$ only affects the growth (\\textit{ie.} the \ndiameter) of the particles, then we would expect only a weak dependence \n(changing as $x{^\\frac13}$) of the particle diameter rate on $x$. \nIf we assume that increasing $x$ also increases the number of Ni nuclei \nupon reduction, then average particle diameter would show an even weaker \ndependence on $x$.\nWe have analyzed the Ni particles in the SEM images of the surfaces of the\nmonoliths by using the program \\textsc{imageJ}\\cite{imagej} to prepare \nhistograms of particle size distributions. These are plotted in \nFig.\\,\\ref{size_nifrac} for the different\nmonoliths. It is seen that mean particle diameters \nrange from $\\sim$15\\,nm to 35\\,nm, but there is no clear trend in size,\nat least until a nickel content of $x = 0.60$ is reached. \n\n\\begin{figure}\n\\centering\\epsfig{file=fig06.eps,width=7cm}\\\\\n\\caption{(a) Fracture surface of a reduced sample (700$^\\circ$C, 2\\,h)\nof Ni$_x$Mn$_{3-x}$O$_4$ with $x$ = 0.45, and (b) the surface of a \nFIB-cut sample of the product formed on reduction (700$^\\circ$C, 2\\,h)\nof Ni$_x$Mn$_{3-x}$O$_4$ with $x$ = 0.60.\nThe two images are displayed at the same magnification. It is seen in (a) \nthat different faces of the underlying MnO\nseem to nucleate different particle sizes of \\textit{fcc}-Ni. In (b),\nit is seen that the Ni particles are found within the MnO matrix as well, and\nnot simply on the surface.}\n\\label{fib}\n\\end{figure}\n\nIndeed, in the different monoliths, a clearer correlation is found \nfor Ni particle size with the specific crystallographic face of MnO\nupon which it grows, rather than the starting $x$ value.\nIt is evident in Fig.\\,\\ref{fib}(a) that for a $x = 0.45$\nspecimen, regions can be found which exhibit a wide variety of surface\nparticle sizes and spacings depending on the nucleation environment. \nThe coherent pore structure introduced by reduction produces square or\ntriangular facets seen in Fig.\\,\\ref{fib}(a) which correspond to\nexposed \\{100\\} or \\{111\\} faces.\n\nCross-sections of reduced grains produced by FIB milling, shown in \nFig.\\,\\ref{fib}, reveal that the bulk\nMnO contains Ni nanoparticles of similar dimensions as those on the surface. \nPorosity is still prevalent in the bulk of the monoliths as it is in the\nimages of the monolith surface. This is necessary to accommodate the volume loss\nof the structure while retaining the size and alignment of the MnO\ngrains.  By a comparison of lattice parameters, and assuming no\nsintering during reduction, the fraction of intragranular porosity\nproduced by the conversion of Ni$_x$Mn$_{3-x}$O$_4$ to $x$Ni\/MnO\nincreases linearly from 16\\% when $x=0$ to 39\\% when $x=0.6$, which is\nin rough agreement with observations of the intragranular pore volume in\nFIB-milled samples.  Most Ni nanoparticles observed in\ncross-section [Fig.\\ \\ref{fib}(b) are completely encased within the MnO \nmatrix.  Based upon the observed surface density of Ni particles and assuming\n100\\,nm diameter pores, it can be determined\nthat the observed surface Ni particles only constitute about 20\\% of the volume of\nNi that must be precipitated.  Therefore, we estimate that\napproximately 80\\% of the Ni grains are encased within the MnO matrix.\n\n\\begin{figure}\n\\centering\\epsfig{file=fig07.eps,width=7cm}\\\\\n\\caption{Magnetization $M$ as a function of magnetic field $H$ \nfor an Ni\/MnO composite with $x = 0.3$ above $T_{N}$ (dashed) and field-cooled \nunder a 50\\,KOe field to 5\\,K (solid). \nExchange bias leads to a broadening  (associated with the coercivity $H_C$)\nand shift (associated with the exchange field $H_E$) of the \nfield-cooled loop at 5\\,K.}\n\\label{hys}\n\\end{figure}\n\n\\begin{figure}\n\\centering\\epsfig{file=fig08.eps,height=7cm}\\\\\n\\caption{(a) Coercive field $H_C$ and (b) Exchange-bias field $H_E$ as a \nfunction of the initial Ni content $x$ in the reduced Ni\/MnO composites. \nThe coercivity $H_C$ of the Ni nanoparticles decreases with Ni content for \neach firing temperature in response to the increased fraction of interfacial \nspins as Ni content is decreased. Exchange bias effects become less pronounced \nas well, with increasing Ni content in the oxide precursor.} \n\\label{mag_nifrac}\n\\end{figure}\n\nMagnetic hysteresis loops for an $x = 0.3$ sample (Fig.\\ \\ref{hys}) display\nat 5\\,K, a loop shift $H_E$ characteristic of exchange\nbiased systems.  Above the N\\'{e}el temperature of MnO, $T_N$ = 119\\,K,\nthe hysteresis loop is centered about $H = 0$.  After field cooling at\n$H$ = 50\\,kOe, the coercive field is broadened\nand shifted $H_E$ = 100\\,Oe in opposition of the cooling field direction.\nThe exchange behavior can be influenced by many\nfactors, including Ni particle size, the amount and orientation of the\nFM-AFM interface, temperature, and the cooling field.\\cite{nogues_schuller}\nWe anticipate that in the size regime studied here (near 20\\,nm) the Ni\nnanoparticles are single-domain magnets and that the coercivity below\nthe blocking temperature should not show a strong size-dependence.\\cite{cullity}\nFig.\\ \\ref{mag_nifrac}(a) shows that as the\nnickel content $x$ increases, $H_C$ decreases for samples reduced at\neither 700$^{\\circ}$C or 725$^{\\circ}$C. \nAt both reduction temperatures, the highest $H_C$ is found for the smallest \n$x$, and the smallest $H_C$ is found for the largest $x$.  \n\nAdditionally, the decrease in $H_C$ for $x$ = 0.3 samples reduced at \n725$^{\\circ}$C as opposed to 700$^{\\circ}$C implies increased coalescence \nof Ni particles as the temperature increases. We therefore anticipate that \nthe increased coercivity as size is decreased arises from the same\ninterfacial coupling that results in the increased exchange bias.\n\nIn exchange biased nanostructures of spherical FM particles in an AFM\nmatrix, the strength of the exchange field $H_E$ has been suggested to\nvary as\n\n\\[ H_E = \\frac{6 E_A}{M_S d_{\\rm{FM}}} \\] \n\n\\noindent where $E_A$ is the interfacial coupling energy per unit area, \n$M_S$ is the saturation magnetization of the FM, and $d_{\\rm{FM}}$ is the \ndiameter of the FM particle.\\cite{nanostructure_review} Assuming this model\nto be correct, we anticipate that the exchange field $H_E$ should decrease \nwith in increasing ferromagnetic particle size. If, with increasing $x$ in our \nsystems, Ni particle particle size indeed increases, then our results are \nbroadly consistent with this model.\n\n\\begin{figure}\n\\centering\\epsfig{file=fig09.eps,width=7cm}\\\\\n\\caption{Coercive field as a function of exchange field at 5\\,K for the\ndifferent Ni\/MnO composites. For most of the composite systems, $H_C$ varies \nnearly linearly with $H_E$.}\n\\label{hc_xb}\n\\end{figure}\n\nIn Fig.\\,\\ref{hc_xb} we plot the 5\\,K coercivity as a function of the exchange\nfield for the different systems measured, data for which are displayed in \nFig.\\,\\ref{mag_nifrac}. We see that the coercivity varies nearly linearly with\nthe exchange field, with the exception of one outlier. G\\\"okemeijer \n\\textit{et al.}\\cite{PhysRevB.63.174422} have recently measured biasing of \nferromagnets on different CoO surfaces and have concluded that on the \nuncompensated CoO surfaces, exchange biasing, and the associated shift of \nhysteresis is found, but on compensated CoO surfaces, the effect of the \ninterface is simply to increase coercivity. The magnetic structure of MnO\nis not simple\\cite{goodwin:047209} and the architectures described here of\nnearly spherical ferromagnetic particles embedded in an antiferromagnetic\nhost cannot be described in terms of simple interfaces. Given this, we suggest \nthat perhaps both effects, of the uncompensated as well as the compensated\nsurfaces are playing a role, and the linear relation between coercivity and \nexchange is simply an indication of increasing interfacial area between the\ntwo magnetic components. \n\n\\section{Conclusions} \n\nWe have demonstrated that hydrogen reduction of Ni$_x$Mn$_{3-x}$O$_4$\nspinels produces Ni\/MnO composites with significant interfacial area\nbetween antiferromagnetic MnO and ferromagnetic Ni, and associated exchange \nbias. With increasing nickel content $x$, these effects decrease, presumably\nbecause of a decrease in the relative proportion of interfacial spins\nin the ferromagnet. Exchange bias effects at the FM--AFM interface lead to \nan increase in $H_C$ with decreasing Ni content, along with a $1\/x$ dependence \nof $H_E$.  A nearly linear relationship is found between $H_C$ and $H_E$ in \nthese systems.\n\n\\section{Acknowledgments}\n\nThis work was supported by the donors of the American Chemical Society \nPetroleum Research Fund, and the National Science Foundation through\na Career Award (DMR 0449354) to RS, and for the use of MRSEC facilities\n(DMR 0520415). MG was supported by a RISE undergraduate fellowship.\n\n\\bibliographystyle{apsrev} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\nSuppose that $A$ is a Banach algebra. We denote by $\\textbf{A-mod}$\nand $\\textbf{mod-A}$ the categories of Banach left $A$-modules and\nBanach right $A$-modules, respectively. In the case where $A$ is\nunital, we also denote by $\\textbf{A-unmod}$ the categories of\nunital Banach left $A$-modules. For each $E,F\\in\\textbf{A-mod},$ let\n$_{A}B(E,F)$ be the closed subspace of $B(E,F)$ consisting of the\nleft $A$-module morphisms. An operator $T\\in B(E,F)$ is called {\\it\nadmissible} if $\\text{ker} T$ and $\\text{Im} T$ are closed\ncomplemented subspaces of $E$ and $F$, respectively. It is easy to\nverify that $T$ is admissible if and only if there exists $S\\in\nB(F,E)$ such that $T\\circ S\\circ T=T.$\n\nA Banach left $A$-module $E$ is called {\\it injective} if for each\n$F, K\\in\\textbf{A-mod}$ and admissible monomorphism\n$T\\in_{A}\\hspace{-0.1cm}B(F,K),$ the induced map\n$_{A}B(K,E)\\longrightarrow\\hspace{-0.1cm}_{A}B(F,E)$ is onto. We\nalso say $E\\in\\textbf{mod-A}$ is {\\it flat} if the dual module of\n$E^{*}\\in\\textbf{A-mod}$ is injective with the following left module\naction:\n$$(a\\cdot f)(x)=f(x\\cdot a)\\qquad (a\\in A, x\\in E).$$\nThe notions of injectivity and flatness of Banach algebras were\nintroduced by A. Ya. Helemskii. These notions have been studied for\nvarious classes of Banach modules; see \\cite{Dales. Polyakov},\n\\cite{Helem}, \\cite{Ramsden Paper} and \\cite{white} for more\ndetails. Recently, Ramsden in \\cite{Ramsden Paper} studied\ninjectivity and flatness of Banach modules over semigroup algebras.\nIt is well known that if $A$ is amenable, then every Banach\n$A$-modules is flat but the converse is a long standing open\nproblem. We recall that the answer is positive for some classes of\nBanach algebras associated with locally compact groups such as, the\nclass of group algebras and measure algebras; see \\cite{Dales.\nPolyakov} and \\cite{Ramsden}.\n\nKaniuth, Lau and Pym introduced and studied in \\cite{klp} and\n\\cite{klp2} the notion of $\\phi$-amenability for Banach algebras,\nwhere $\\phi:A\\longrightarrow\\mathbb{C}$ is a character, i.e., a\nnon-zero homomorphism on $A$. Afterwards, Monfared introduced and\nstudied in \\cite{m} the notion of character amenability for Banach\nalgebras. Let $\\Delta(A)$ be the set of all characters of the Banach\nalgebra $A$, and let $\\phi\\in\\Delta(A)$. The Banach algebra $A$ is\ncalled {\\it left $\\phi$-amenable} if for all Banach $A$-bimodules\n$E$ for which the left module action is given by\n$$a\\cdot x=\\phi(a)x \\qquad (a\\in A, x\\in E),$$ every\nderivation $D:A\\longrightarrow E^{*}$ is inner. It is clear that\namenability of $A$ implies $\\phi$-amenability for all\n$\\phi\\in\\Delta(A).$\n\nRecently, Nasr-Isfahani and Soltani Renani in \\cite{Nasr} introduced\nand studied the notion of $\\phi$-injectivity and $\\phi$-flatness for\nBanach modules (see Definition $2.1$). As an important result, it is\nshown in \\cite[Proposition 3.1]{Nasr} that the Banach algebra $A$ is\nleft $\\phi$-amenable if and only if every Banach left $A$-modules\n$E$ is $\\phi$-flat. Indeed, this result gives a positive answer to\nthe above open problem arises by A. Ya. Helemskii in this homology\nsetting based on character $\\phi$. Furthermore, they obtained some\nnecessary and sufficient conditions for $\\phi$-injectivity and\ncharacterized $\\phi$-injectivity of Banach modules in terms of a\ncoretraction problem; see \\cite[Theorem 2.4]{Nasr}.\n\nThis paper is organized as follows. In Section 2, after recalling\nsome definitions, we investigate some properties of\n$\\phi$-injectivity for Banach modules. Indeed, we obtain a\nsufficient condition for $\\phi$-injectivity of Banach left\n$A$-modules, in the case where $A$ is a commutative Banach algebra.\nMoreover, we give some hereditary properties of $\\phi$-injectivity\nfor Banach $A$-modules related to the closed ideals of Banach\nalgebra $A$. As the main result, we show that if $J$ is a left\ninvariant complemented ideal in $A$, then $\\phi$-injectivity of $J$\nand $A\/J$ in $\\textbf{A-mod}$ is equivalent to the\n$\\phi$-injectivity of $A$ in $\\textbf{A-mod}$ (Theorem \\ref{Cor:\nDirectsum}). In Section 3, by using the results of Section 2, we\nstudy $\\phi$-injectivity of certain $\\ell^{1}$-semilattice algebras\nand show that $\\ell^{1}(\\mathbb{N}_{\\wedge})$ as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$-module is $\\phi$-injective for each\ncharacter $\\phi,$ although is not injective.\n\n\\section{$\\phi$-injectivity and some hereditary properties}\nFirst, we recall some standard notations that we shall use and\ndefine the notions of $\\phi$-injectivity and $\\phi$-flatness of\nBanach modules.\n\nLet $A$ be a Banach algebra and $E\\in\\textbf{A-mod}.$ Throughout the\npaper, we regard $E$ as a Banach left $A^{\\sharp}$-module (the\nunitization of $A$) with the following left module action:\n$$(a,\\lambda)\\cdot x=a\\cdot x+\\lambda x \\qquad(a\\in A, \\lambda\\in\\mathbb{C}, x\\in E).$$\nMoreover, the space $B(A,E)$ is a Banach $A$-bimodule with the\nfollowing module actions:\n$$(a\\cdot T)(b)=T(ba), \\qquad (T\\cdot a)(b)=T(ab)\\quad (T\\in B(A,E), a, b\\in A).$$\nSuppose that $A$ is a Banach algebra and $\\phi\\in\\Delta(A)$. For\neach $E\\in\\textbf{A-mod}$ we define,\n$$I(\\phi,E)=\\textrm{span}\\{a\\cdot x-\\phi(a)x : a\\in A, x\\in E\\}.$$\nFollowing \\cite{Nasr}, we also consider\n$$_{\\phi}B(A^{\\sharp},E)=\\{T\\in B(A^{\\sharp},E) :\nT(ab-\\phi(b)a)=a\\cdot T(b-\\phi(b)e^{\\sharp})\\:\\:\\text{for all}\\:\\:\na, b\\in A\\},$$ where $e^{\\sharp}=(0,1)$ denotes the unite of\n$A^{\\sharp}.$ It is straightforward to check that\n$_{\\phi}B(A^{\\sharp},E)$ is a closed $A$-submodule of\n$B(A^{\\sharp},E).$ Moreover, we define {\\it the canonical morphism}\n$_{\\phi}\\Pi^{\\sharp}:E\\longrightarrow _{\\phi}B(A^{\\sharp},E)$ as\nfollows:\n$$_{\\phi}\\Pi^{\\sharp}(x)(a)=a\\cdot x\\qquad (x\\in E, a\\in A^{\\sharp}).$$\n\\begin{definition}\nLet $A$ be a Banach algebra, $\\phi\\in \\Delta(A)$ and $E\\in\n\\textbf{A-mod}$. We say that $E$ is {\\it $\\phi$-injective} if, for\neach $F, K\\in \\textbf{A-mod}$ and admissible monomorphism\n$T:F\\longrightarrow K$ with $I(\\phi,K)\\subseteq\\text{Im}(T)$, the\ninduced map $T_{E}:\\hspace{-0.1cm}_{A}B(K,E)\\longrightarrow\\\n_{A}B(F,E)$ defined by $T_{E}(R)=R\\circ T$ is onto.\n\\end{definition}\nThe following theorem gives a characterization of $\\phi$-injectivity\nin terms of a coretraction problem.\n\\begin{theorem}$($\\cite[Theorem 2.4]{Nasr}$)$\nLet $A$ be a Banach algebra and $\\phi\\in\\Delta(A)$. For\n$E\\in\\textbf{A-mod}$ the following statements are equivalent.\n\\begin{enumerate}\n\\item[(i)] $E$ is $\\phi$-injective.\n\\item[(ii)] $_{\\phi}\\Pi^{\\sharp}\\in\\hspace{-0.1cm}_{A}B(E,_{\\phi}B(A^{\\sharp},E))$ is a coretraction, $($that is there exists\n$_{\\phi}\\rho^{\\sharp}\\in\\: _{A}B(_{\\phi}B(A^{\\sharp},E), E)$ such\nthat is a left inverse for $_{\\phi}\\Pi^{\\sharp})$.\n\\end{enumerate}\n\\end{theorem}\nA Banach right (left) $A$-module $E$ is $\\phi$-flat if $E^{*}$ is\n$\\phi$-injective as a left (right) $A$-module. It is shown that\nBanach algebra $A$ is left $\\phi$-amenable if and only if each\nBanach left $A$-module $E$ is $\\phi$-flat \\cite[Proposition\n3.1]{Nasr}.\n\nIn this section, we give some hereditary properties of\n$\\phi$-injectivity for certain classes of Banach modules. We also\nconsider some hereditary properties of $\\phi$-injectivity of Banach\nleft $A$-modules with their ideals. We first give a sufficient\ncondition for $\\phi$-injectivity of Banach left $A$-module $E$ in\nthe case where $A$ is a commutative Banach algebra. Following\n\\cite[Definition 1.4.4]{Dales}, {\\it the annihilator} of $E$ is\ndefined by $E^{\\perp}=\\{a\\in A: a\\cdot E=\\{0\\}\\}$.\n\\begin{theorem}\\label{Th: 2}\nLet $A$ be a commutative Banach algebra, $\\phi\\in \\Delta(A)$ and\n$E\\in \\textbf{A-mod}.$ If\n$E^{\\perp}\\cap(\\ker(\\phi))^{c}\\neq\\emptyset$, then $E$ is\n$\\phi$-injective.\n\\end{theorem}\n\\begin{proof}\nLet $a_{0}\\in E^{\\perp}\\cap(\\ker(\\phi))^{c}$. We can assume that\n$\\phi(a_{0})=1$ and define the map\n$_{\\phi}\\rho^{\\sharp}:$$_{\\phi}B(A^{\\sharp},E)\\longrightarrow E$ by\n\\begin{center}\n$_{\\phi}\\rho^{\\sharp}(T)=T(e^{\\sharp}-a_{0})\\hspace{0.5cm}(T\\in$$\n_{\\phi}B(A^{\\sharp},E)).$\n\\end{center}\nHence, for each $x\\in E$ we have\n\\begin{center}\n$_{\\phi}\\rho^{\\sharp}\\circ$$_{\\phi}\\Pi^{\\sharp}(x)=$$_{\\phi}\\Pi^{\\sharp}(x)(e^{\\sharp}-a_{0})=(e^{\\sharp}-a_{0})\\cdot\nx=x.$\n\\end{center}\nTherefore, $_{\\phi}\\rho^{\\sharp}\\circ$$_{\\phi}\\Pi^{\\sharp}=I_{E}$.\nOn the other hand, for each $a\\in A$ and $T\\in$$\n_{\\phi}B(A^{\\sharp},E)$ we have\n\\begin{equation}\n\\begin{split}\n_{\\phi}\\rho^{\\sharp}(a\\cdot T)&=(a\\cdot T)(e^{\\sharp}-a_{0})=T((e^{\\sharp}-a_{0})\\cdot a)\\\\\n&=T(a-a_{0}a)\\\\\n&=T(a\\phi(a_{0})-aa_{0}).\n\\end{split}\n\\end{equation}\nSince $T\\in$$ _{\\phi}B(A^{\\sharp},E)$ we have\n$T(a\\phi(a_{0})-aa_{0})=a\\cdot T(e^{\\sharp}-a_{0}).$ Now, using\n($2.1$) we conclude that\n\\begin{align*}\n_{\\phi}\\rho^{\\sharp}(a\\cdot T)=a\\cdot\nT(e^{\\sharp}-a_{0})=a\\cdot_{\\phi}\\rho^{\\sharp}(T).\n\\end{align*}\nIt follows that $_{\\phi}\\rho^{\\sharp}$ is a left $A$-module\nmorphism. Hence, $E$ is a $\\phi$-injective Banach left $A$-module.\n\\end{proof}\n\\begin{corollary}\\label{Cor: 2} Let $A$ be a commutative Banach algebra, and $J$ be a closed ideal of $A$\nsuch that $\\phi_{|J}\\neq 0$. Then $A\/J$ is $\\phi$-injective as a\nBanach left $A$-module.\n\\end{corollary}\n\\begin{proof}\nSince $\\phi_{|J}\\neq 0$, it is easy to check that\n$(A\/J)^{\\perp}\\cap(\\ker(\\phi))^{c}\\neq \\emptyset$. Now, apply\nTheorem \\ref{Th: 2}.\n\\end{proof}\n\\begin{corollary}\\label{Th: Inj Module}\nLet $A$ be a commutative Banach algebra, $\\phi\\in\\Delta(A)$ and let $E\\in\\textbf{A-mod}$ with $I(\\phi,E)=\\{0\\}$. Then for all $\\psi\\in\\Delta(A)\\setminus\\{\\phi\\}$, $E\\in \\textbf{A-mod}$ is $\\psi$-injective.\n\\end{corollary}\n\\begin{proof} Since $\\phi\\neq\\psi$ there exists $a_{0}\\in A$ such that $\\phi(a_{0})=0$ and $\\psi(a_{0})=1$.\nOn the other hand, since $I(\\phi,E)=\\{0\\}$ we conclude that $a_{0}\\in E^{\\perp}\\cap(\\ker(\\psi))^{c}$ and the proof\nis complete.\n\\end{proof}\nNow, we give some hereditary properties of $\\phi$-injectivity of\nBanach modules that we shall use. Recall that $E\\in\\textbf{A-mod}$\nis {\\it faithful} if $ A\\cdot x=\\{0\\}$ implies that $x=0.$\n\\begin{theorem}\\label{Th: Her,of,Id}\nLet $A$ be a Banach algebra,$E\\in\\textbf{A-mod}$, $\\phi\\in\\Delta(A)$ and $J$ be a closed\nideal of $A$ such that $\\phi_{|J}\\neq 0$.\n\\begin{enumerate}\n\\item[(i)] Suppose that $J$ has an identity and $E\\in\\textbf{J-unmod}$. If $E\\in\\textbf{A-mod}$ is\n$\\phi$-injective, then $E\\in\\textbf{J-unmod}$ is\n$\\phi_{|J}$-injective.\n\\item[(ii)] If $E\\in\\textbf{J-mod}$ is $\\phi_{|J}$-injective and\nfaithful, then $E\\in\\textbf{A-mod}$ is $\\phi$-injective.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}(i) Suppose that $E\\in \\textbf{A-mod}$ is $\\phi$-injective. Let $F$ and $K$ be in $\\textbf{J-mod}$ and $T:F\\longrightarrow K$ be an\nadmissible monomorphism with $I(\\phi_{|J},K)\\subseteq \\textrm{Im}T$.\nWe claim that the induced map, $_{J}B(K,E)\\longrightarrow\\\n_{J}B(F,E)$ defined by, $R\\longrightarrow R\\circ T$ is onto. Suppose\nthat $e_{J}$ is the identity of $J$. We can consider $F$ and $K$ as\nBanach left $A$-modules with the following module actions:\n\\begin{align*}\n&a\\bullet f=(ae_{J})\\cdot f\\hspace{0.5cm}(a\\in A, f\\in F),\\\\\n&a\\bullet k=(ae_{J})\\cdot k\\hspace{0.5cm}(a\\in A, k\\in K).\n\\end{align*}\nWe denote these $A$-modules with $\\widetilde{F}$ and\n$\\widetilde{K},$ respectively. Take $W\\in$$_{J}B(F,E)$ and define\nthe map $\\widetilde{W}:\\widetilde{F}\\longrightarrow E$ by\n$\\widetilde{W}(f)=W(f)$. For each $a\\in A$ and $f\\in F$ we have,\n\\begin{align*}\n\\widetilde{W}(a\\bullet f)&=W((ae_{J})\\cdot f)=(ae_{J})\\cdot\nW(f)\\\\\n&=a\\cdot (e_{J}\\cdot W(f))=a\\cdot W(f)\\\\\n&=a\\cdot\\widetilde{W}(f).\n\\end{align*}\nSo $\\widetilde{W}$ is a left $A$-module morphism. Moreover, the map\n$\\widetilde{T}:\\widetilde{F}\\longrightarrow\\widetilde{K}$ defined by\n$\\widetilde{T}(f)=T(f)$ is an admissible monomorphism such that\n\\begin{align*}\nI(\\phi,\\widetilde{K})&=\\textrm{span}\\{a\\bullet k-\\phi(a)k : a\\in A, k\\in K\\}\\\\\n&=\\textrm{span}\\{(ae_{J})\\cdot k-\\phi(ae_{J})k : a\\in A, k\\in K\\}\\\\\n&\\subseteq \\textrm{Im}T=\\textrm{Im}\\widetilde{T}.\n\\end{align*}\nSince $E\\in \\textbf{A-mod}$ is $\\phi$-injective, there exist\n$S\\in$$_{A}B(\\widetilde{K},E)$ such that $S\\circ\n\\widetilde{T}=\\widetilde{W}$. On the other hand, for each $a\\in J$\nand $k\\in K$ we have\n\\begin{equation*}\na\\cdot S(k)=S(a\\bullet k)=S((ae_{J})\\cdot k)=S(a\\cdot k).\n\\end{equation*}\nIt follows that $S\\in$$_{J}B(K,E)$. Now, we conclude that $E\\in\n\\textbf{J-unmod}$ is $\\phi_{|J}$-injective.\n\n(ii) Let $F$ and $K$ be in $\\textbf{A-mod}$ and $T:F\\longrightarrow\nK$ be an admissible monomorphism and take\n$W\\in\\hspace{-0.1cm}_{A}B(F,E)$. So $W\\in\\hspace{-0.1cm}_{J}B(F,E)$\nand there exists $S\\in\\hspace{-0.1cm}_{J}B(K,E)$ such that $S\\circ\nT=W$. For each $a\\in J$, $b\\in A$ and $k\\in K$, we have\n\\begin{align*}\na\\cdot(S(b\\cdot k)-b\\cdot S(k))&=a\\cdot S(b\\cdot k)-(ab)\\cdot S(k)\\\\\n&=S(ab\\cdot k)-S(ab\\cdot k)=0.\n\\end{align*}\nSince $E\\in \\textbf{J-mod}$ is faithful, we conclude that $S(b\\cdot\nk)=b\\cdot S(k).$ It follows that $S\\in\\hspace{-0.1cm}_{A}B(K,E)$ and\nthe proof is complete.\n\\end{proof}\n\\begin{corollary}\nLet $A$ be a Banach algebra, $\\phi\\in\\Delta(A)$ and $J$ be a closed\nideal of $A$ with an identity such that $\\phi_{|J}\\neq 0$. Then\n$J\\in\\textbf{A-mod}$ is $\\phi$-injective if and only if\n$J\\in\\textbf{J-mod}$ is $\\phi_{|J}$-injective.\n\\end{corollary}\nB. E. Forrest in \\cite{Forrest} introduced the notion of invariantly\ncomplemented submodules in categories of Banach modules. In the\nsequel, we obtain some results for $\\phi$-injectivity of invariantly\ncomplemented ideals.\n\\begin{definition}(\\cite[Definition\n6.3]{Forrest}) Let $X$ be a Banach left $A$-module and $Y$ be a\nBanach $A$-submodule of $X$. We say that $Y$ is {\\it left\n$($right$)$ invariantly complemented} in $X$ if there exists\n$P\\in\\hspace{-0.1cm}_{A}B(X,Y)$ ($P\\in B_{A}(X,Y)$) such that\n$P^{2}=P$ and $P(X)=Y$.\n\\end{definition}\n\\begin{theorem}\\label{Th: 3} Let $\\{E_{\\alpha}\\}_{\\alpha\\in \\Gamma}$ be a collection of Banach left $A$-modules and consider\n$E=\\ell^{1}-\\bigoplus_{\\alpha\\in \\Gamma} E_{\\alpha}$ as a Banach\nleft $A$-module with the natural module action.\n\\begin{enumerate}\n\\item[(i)] If $E$ is $\\phi$-injective, then for each $\\alpha\\in\\Gamma,$ $E_{\\alpha}$ is $\\phi$-injective.\n\\item[(ii)] Conversely, if $\\Gamma$ is finite and each $E_{\\alpha}$ is $\\phi$-injective, then $E$ is $\\phi$-injective.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof} (i) It is obvious that each $E_{\\alpha}$ is left invariantly complemented in $E$. Hence, for each $\\alpha\\in\\Gamma$,\nlet $P_{\\alpha}\\in$$_{A}B(E,E_{\\alpha})$ such that\n$P_{\\alpha}(E)=E_{\\alpha}$ and $P^{2}_{\\alpha}=P_{\\alpha}$. Also,\nlet $i_{\\alpha}:E_{\\alpha}\\longrightarrow E$ be the natural\nembedding of $E_{\\alpha}$ into $E$.\n\nLet $E$ be $\\phi$-injective. Then there exists\n$_{\\phi}\\rho^{E}\\in$$_{A}B(_{\\phi}B(A^{\\sharp},E),E)$ such that is a\nleft inverse for\n$_{\\phi}\\Pi^{E}:E\\longrightarrow\\hspace{-0.1cm}_{\\phi}B(A^{\\sharp},E)$.\nFor each $\\alpha\\in \\Gamma$, we define the map\n$_{\\phi}\\rho^{\\alpha}:$$_{\\phi}B(A^{\\sharp},E_{\\alpha})\\longrightarrow\nE_{\\alpha}$ by\n\\begin{center}\n$_{\\phi}\\rho^{\\alpha}(T)=P_{\\alpha}\\circ$$_{\\phi}\\rho^{E}(i_{\\alpha}\\circ\nT)\\qquad(T\\in$$_{\\phi}B(A^{\\sharp},E_{\\alpha})).$\n\\end{center}\nWe claim that $_{\\phi}\\rho^{\\alpha}$ is a left $A$-module morphism\nsuch that\n$_{\\phi}\\rho^{\\alpha}\\circ$$_{\\phi}\\Pi^{\\alpha}=I_{E_{\\alpha}}$.\nIndeed, since for each $x\\in E_{\\alpha}$,\n$i_{\\alpha}\\circ$$_{\\phi}\\Pi^{\\alpha}(x)=$$_{\\phi}\\Pi^{E}(i_{\\alpha}(x))$,\nso we have\n\\begin{align*}\n_{\\phi}\\rho^{\\alpha}\\circ_{\\phi}\\Pi^{\\alpha}(x)&=P_{\\alpha}\\circ_{\\phi}\\rho^{E}(i_{\\alpha}\\circ\n_{\\phi}\\Pi^{\\alpha}(x))=P_{\\alpha}\\circ_{\\phi}\\rho^{E}(_{\\phi}\\Pi^{E}(i_{\\alpha}(x)))\\\\\n&=P_{\\alpha}(i_{\\alpha}(x))=x.\n\\end{align*}\nTherefore,\n$_{\\phi}\\rho^{\\alpha}\\circ$$_{\\phi}\\Pi^{\\alpha}=I_{E_{\\alpha}}$. On\nthe other hand, since $P_{\\alpha}\\in$$_{A}B(E,E_{\\alpha})$ it is\neasy to see that $_{\\phi}\\rho^{\\alpha}$ is a left $A$-module\nmorphism and the proof is complete.\n\n(ii) Suppose that for each $\\alpha\\in \\Gamma$, $E_{\\alpha}$  is\n$\\phi$-injective. So, for each $\\alpha\\in \\Gamma$ there exists\n$_{\\phi}\\rho^{\\alpha}\\in$$_{A}B(_{\\phi}B(A^{\\sharp},E_{\\alpha}),E_{\\alpha})$\nfor which\n$_{\\phi}\\rho^{\\alpha}\\circ$$_{\\phi}\\Pi^{\\alpha}=I_{E_{\\alpha}}$.\nDefine the map $\\rho:$$_{\\phi}B(A^{\\sharp},E)\\longrightarrow E$ by\n\\begin{center}\n$\\rho(T)=(_{\\phi}\\rho^{\\alpha}(P_{\\alpha}\\circ T))_{\\alpha\\in\n\\Gamma}\\qquad(T\\in$$_{\\phi}B(A^{\\sharp},E)).$\n\\end{center}\nSince $\\Gamma$ is finite, $\\rho$ is well-defined. For each $a\\in A$ and $T\\in$$_{\\phi}B(A^{\\sharp},E)$ we have\n\\begin{align*}\n\\rho(a\\cdot T)&=(_{\\phi}\\rho^{\\alpha}(P_{\\alpha}\\circ (a\\cdot T)))_{\\alpha\\in \\Gamma}=(_{\\phi}\\rho^{\\alpha}(a\\cdot(P_{\\alpha}\\circ T)))_{\\alpha\\in \\Gamma}\\\\\n&=(a\\cdot_{\\phi}\\rho^{\\alpha}(P_{\\alpha}\\circ T))_{\\alpha\\in \\Gamma}=a\\cdot(_{\\phi}\\rho^{\\alpha}(P_{\\alpha}\\circ T))_{\\alpha\\in \\Gamma}\\\\\n&=a\\cdot\\rho(T).\n\\end{align*}\nMoreover, if $x=(x_{\\alpha})_{\\alpha\\in\\Gamma}$ is an arbitrary\nelement of $E$, it is easy to see that\n$P_{\\alpha}\\circ$$_{\\phi}\\Pi^{E}(x)=$$_{\\phi}\\Pi^{\\alpha}(x_{\\alpha})$.\nHence,\n\\begin{align*}\n\\rho\\circ_{\\phi}\\Pi^{E}(x)&=(_{\\phi}\\rho^{\\alpha}(P_{\\alpha}\\circ\n_{\\phi}\\Pi^{E}(x)))_{\\alpha\\in\n\\Gamma}=(_{\\phi}\\rho^{\\alpha}(_{\\phi}\\Pi^{\\alpha}(x_{\\alpha})))_{\\alpha\\in\n\\Gamma}\\\\\n&=(x_{\\alpha})_{\\alpha\\in\\Gamma}=x.\n\\end{align*}\nTherefore, we conclude that $E$ is $\\phi$-injective.\n\\end{proof}\n\\begin{theorem}\\label{Cor: Directsum}\nLet $A$ be a Banach algebra, $\\phi\\in \\Delta(A)$, $B$ be a\nsubalgebra of $A$ and $J$ be a closed left ideal of $A$. Then the\nfollowing assertions holds:\n\\begin{enumerate}\n\\item[(i)] If $B$ is left invariantly complemented in $A$ and $A$  is $\\phi$-injective in\n\\textbf{A-mod},  then $B$ is $\\phi$-injective in \\textbf{A-mod}.\n\\item[(ii)] If $J$ is left invariantly complemented, then $J$ and $A\/J$ are $\\phi$-injective in \\textbf{A-mod}\nif and only if $A$ is $\\phi$-injective in \\textbf{A-mod}.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}(i) Since $B$ is left invariantly complemented in $A$,\nthere exists an onto projection $P\\in$$_{A}B(A,B)$. Hence\n$$A\\cong\\textrm{Im} P\\oplus\\ker P=B\\oplus\\ker P,$$ as a Banach\nleft $A$-module. Therefore, by Theorem \\ref{Th: 3} it follows that\n$B$ is $\\phi$-injective in $\\textbf{A-mod}.$\n\n(ii) Since $J$ is a left invariant complemented ideal in $A,$ there\nexists an onto projection $P\\in$$_{A}B(A,J)$. We claim that $A\\cong\nJ\\oplus \\frac{A}{J}$ as a Banach left $A$-module. To see this,\ndefine the map $T:A\\longrightarrow J\\oplus \\frac{A}{J}$ by\n\\begin{equation*}\nT(a)=(P(a), a+J)\\qquad(a\\in A).\n\\end{equation*}\nFirst, $T$ is a left $A$-module morphism, because for each $a, b\\in\nA$ we have,\n\\begin{align*}\nT(ab)&=(P(ab),ab+J)=(aP(b),a(b+J))\\\\\n&=a\\cdot (P(b),b+J)=a\\cdot T(b).\n\\end{align*}\nOn the other hand, if $a\\in \\textrm{Im}P\\ \\cap\\ \\ker P$, then there\nexists $b\\in A$ such that $P(b)=a$. Hence, $a=P(b)=P(P(b))=P(a)=0$.\nThis follows that $\\textrm{Im}P\\cap \\ker P=\\{0\\}$ and so $T$ is\none-to-one. Moreover, $T$ is onto because for each $(a,b+J)\\in\nJ\\oplus \\frac{A}{J}$ if we put $c=a+b-P(b)$, then $T(c)=(a,b+J)$.\nNow, the result follows from Theorem \\ref{Th: 3}.\n\\end{proof}\nAs an application of Theorem \\ref{Cor: Directsum} and Corollary\n\\ref{Cor: 2}, we have the following result for commutative Banach\nalgebras.\n\\begin{corollary}\\label{Cor: A and I. A is Com}\nLet $A$ be a commutative Banach algebra, $\\phi\\in \\Delta(A)$ and let\n$J$ be a closed invariant complemented ideal of $A$ such that\n$\\phi_{|J}\\neq 0$. Then $A\\in\\textbf{A-mod}$ is $\\phi$-injective if\nand only if $J\\in\\textbf{A-mod}$ is $\\phi$-injective.\n\\end{corollary}\n\\section{Applications to semigroup algebras}\nIn this section, we apply our later results to study\n$\\phi$-injectivity of certain commutative semigroup algebras and\ngive some examples of non-injective modules which are\n$\\phi$-injective for each character $\\phi.$ First, we need some\nbasic facts about semigroup algebras.\n\nLet $S$ be a semigroup and let $E(S)=\\{s\\in S : s^{2}=s\\}.$ We say\nthat $S$ is a {\\it semilattice} if $S$ is commutative and $E(S)=S.$\nA {\\it semi-character} on $S$ is a non-zero homomorphism\n$\\widehat{\\phi}:S\\longrightarrow\\{z\\in\\mathbb{C}: |z|\\leq 1\\}.$ The\nspace of semi-characters on $S$ is denoted by $\\Phi_{S}.$ The\nsemi-character $\\widehat{\\phi}_{S}:S\\longrightarrow\\{z\\in\\mathbb{C}:\n|z|\\leq 1\\}$, defined by\n$$\\widehat{\\phi}_{S}(t)=1 \\qquad (t\\in S),$$ is called the {\\it\naugmentation character} on $S$. For each\n$\\widehat{\\phi}\\in\\Phi_{S},$ we associate the map\n$\\phi:\\ell^{1}(S)\\longrightarrow \\mathbb{C}$ defined by\n$$\\phi(f)=\\sum_{s\\in S}\\widehat{\\phi}(s)f(s) \\qquad (f\\in\\ell^{1}(S)).$$\nIt is easy to verify that $\\phi\\in\\Delta(\\ell^{1}(S))$ and every\ncharacter on $\\ell^{1}(S)$ arises in this way. Indeed, we have\n$$\\Delta(\\ell^{1}(S))=\\{\\phi: \\widehat{\\phi}\\in\\Phi_{S}\\}.$$\nWe also define the convolution of two elements $f,g\\in \\ell^{1}(S)$\nby\n$$(f\\ast g)(s)= \\sum_{uv=s}f(u)g(v) \\qquad (s\\in S),$$\nwhere $\\sum_{uv=s}f(u)g(v)=0,$ when there are no elements $u,v\\in S$\nwith $uv=s.$ Then $(\\ell^{1}(S),\\ast,{\\|\\cdot\\|}_{1})$ becomes a\nBanach algebra that is called the {\\it semigroup algebra} of $S.$\nThe following proposition immediately follows from Corollary\n\\ref{Cor: A and I. A is Com}.\n\\begin{proposition}\\label{pp}\nLet $S$ be a semilattice, $\\phi\\in\\Delta(\\ell^{1}(S))$ and $I$ be a\nclosed invariant complemented ideal in $\\ell^{1}(S)$ such that\n$\\phi_{|_{I}}\\neq 0.$ Then $\\ell^{1}(S)\\in\\ell^{1}(S)\\textbf{-mod}$\nis $\\phi$-injective if and only if $I\\in\\ell^{1}(S)\\textbf{-mod}$ is\n$\\phi$-injective.\n\\end{proposition}\nLet $\\ell^{1}(\\mathbb{N}_{\\wedge})$ be the semigroup algebra on semigroup $S=(\\mathbb{N},{\\wedge})$ with the following\nproduct:\n\\begin{equation*}\n\\mathbb{N}\\times \\mathbb{N}\\longrightarrow \\mathbb{N},\\quad\n(m,n)\\longrightarrow m\\wedge n=\\min\\{m, n\\}.\n\\end{equation*}\nIt is easy to check that $\\Phi_{S}=\\{\\widehat{\\phi}_{n} :\nn\\in\\mathbb{N}\\},$ where for each $n\\in\\mathbb{N},$\n$$\\begin{array}{lll}\n\\widehat{\\phi}_{n}(m) =\\left\\{\\begin{array}{l} 1 \\quad\\text{if}\\quad\nm\\geq n\n\\\\ 0 \\quad \\text{if} \\quad m< n\n\\end{array}\\qquad (m\\in\\mathbb{N}).\\right.\n\\end{array}$$\nFor each $n\\in \\mathbb{N}$, let $I_{n}$ be the ideal of\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$ generated by the set $\\{\\delta_{1},\n\\delta_{2}, \\delta_{3},\\ldots, \\delta_{n}\\}$. It is easy to see that\n$\\ell^{1}(\\mathbb{N}_{\\wedge})\/I_{n}$ does not have an identity. So\n$I_{n}$ does not have a modular identity and using \\cite[Corollary\n2.2.8 (ii)]{Ramsden}, we conclude that\n$\\ell^{1}(\\mathbb{N}_{\\wedge})\/I_{n}$ is not injective as a Banach\nleft $\\ell^{1}(\\mathbb{N}_{\\wedge})$-module. Furthermore, we recall\nthat $\\ell^{1}(\\mathbb{N}_{\\wedge})$ as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$-module is not injective, because it\ndoes not have a right identity \\cite[Example 4.10]{Dales. Lau}. As\nmentioned above, we regard the map\n$\\phi_{n}:\\ell^{1}(\\mathbb{N}_{\\wedge})\\longrightarrow \\mathbb{C}$\nas a character on $\\ell^{1}(\\mathbb{N}_{\\wedge})$ which is defined\nby\n\\begin{equation*}\n\\phi_{n}(f)=\\sum_{i=n}^{\\infty}f(i)\\qquad(f\\in\n\\ell^{1}(\\mathbb{N}_{\\wedge})).\n\\end{equation*}\nThe following theorem shows that $\\ell^{1}(\\mathbb{N}_{\\wedge})$ as\na left $\\ell^{1}(\\mathbb{N}_{\\wedge})$-module is $\\phi$-injective\nfor each $\\phi\\in \\Delta(\\ell^{1}(\\mathbb{N}_{\\wedge}))$, although\nis not injective.\n\\begin{theorem}\\label{Th: L1Min is 1-inj}\nWith the above notations, we have following assertions:\n\\begin{enumerate}\n\\item[(i)] $\\ell^{1}(\\mathbb{N}_{\\wedge})\/I_{n}$ as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$-module is $\\phi_{n}$-injective, for\neach $n\\in\\mathbb{N}$.\n\\item[(ii)] $\\ell^{1}(\\mathbb{N}_{\\wedge})$ as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$-module is $\\phi_{n}$-injective,  for\neach $n\\in\\mathbb{N}$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}(i) Since $\\ell^{1}(\\mathbb{N}_{\\wedge})$ is commutative and\n$(\\phi_{n})_{|I_{n}}\\neq 0$, by Corollary \\ref{Cor: 2}, it follows\nthat $\\ell^{1}(\\mathbb{N}_{\\wedge})\/I_{n}$ as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$-module is $\\phi_{n}$-injective.\n\n(ii) First, we show that for each $n\\in \\mathbb{N}$, $I_{n}$ is an\ninvariant complemented ideal of $\\ell^{1}(\\mathbb{N}_{\\wedge})$. To\nsee this, suppose that the map\n$P_{n}:\\ell^{1}(\\mathbb{N}_{\\wedge})\\longrightarrow I_{n}$ is\ndefined by\n\\begin{equation*}\nP_{n}(f)=\\sum_{i=1}^{n-1}f(i)\\delta_{i}+(\\sum_{i=n}^{\\infty}f(i))\\delta_{n}\\qquad(f\\in\n\\ell^{1}(\\mathbb{N}_{\\wedge})).\n\\end{equation*}\nIt is easy to check that $P_{n}$ is a projection on $I_{n}.$\nMoreover, if $f$ or $g$ belong to $I_{n}$ we have $P_{n}(f\\ast\ng)=f\\ast P_{n}(g)$. Now suppose that $f, g$ are not in $I_{n}$. We\ncan suppose that $f=\\delta_{i}$ and $g=\\delta_{j}$ such that\n$n<i\\leq j$. Hence, we have\n\\begin{align*}\nP_{n}(\\delta_{i}\\ast\\delta_{j})=P_{n}(\\delta_{i})=\\delta_{n}=\\delta_{i}\\ast\\delta_{n}=\\delta_{i}\\ast\nP_{n}(\\delta_{j}).\n\\end{align*}\nThis follows that $I_{n}$ is an invariant complemented closed ideal\nof $\\ell^{1}(\\mathbb{N}_{\\wedge})$.\n\n$\\mathbf{Case\\ 1}$: We prove that $A=\\ell^{1}(\\mathbb{N}_{\\wedge})$\nis $\\phi_{1}$-injective. Since $I_{1}$ is invariantly complemented\nin $A,$ $(\\phi_{1})_{|I_{1}}\\neq 0$ and $A$ is commutative, by\nProposition \\ref{pp}, it suffices to show that\n$I_{1}\\in\\textbf{A-mod}$ is $\\phi_{1}$-injective.\n\nDefine the map $\\rho:\\ _{\\phi_{1}}B(A^{\\sharp},I_{1})\\longrightarrow\nI_{1}$ by\n$$\\rho(T)=T(e^{\\sharp})\\qquad  (T\\in\\\n_{\\phi_{1}}B(A^{\\sharp},I_{1}).$$\nClearly $\\rho$ is a left inverse\nfor $\\ _{\\phi_{1}}\\Pi^{\\sharp}:I_{1}\\longrightarrow \\\n_{\\phi_{1}}B(A^{\\sharp},I_{1})$. We claim that $\\rho$ is a left\n$A$-module morphism.\n\nFor each $f\\in A$ and $g\\in I_{1}$ we note that $f\\ast\ng=\\phi_{1}(f)g$. Hence, if $T\\in\\ _{\\phi_{1}}B(A^{\\sharp},I_{1})$,\nthen for each $f, g\\in A$ we have\n\\begin{equation*}\nT(f\\ast g-\\phi_{1}(g)f)=f\\cdot T(g-\\phi_{1}(g)e^{\\sharp}).\n\\end{equation*}\nIn the case where $g=\\delta_{1}$, we conclude that\n\\begin{align*}\nT(\\phi_{1}(f)\\delta_{1}-\\phi_{1}(g)f)&=T(f\\ast g-\\phi_{1}(g)f)\\\\\n&=f\\cdot T(g-\\phi_{1}(g)e^{\\sharp})\\\\\n&=f\\ast T(g-\\phi_{1}(g)e^{\\sharp})\\\\\n&=\\phi_{1}(f)T(\\delta_{1}-\\phi_{1}(\\delta_{1})e^{\\sharp})\\\\\n&=\\phi_{1}(f)T(\\delta_{1}-e^{\\sharp}).\n\\end{align*}\nTherefore, $T(f)=\\phi_{1}(f)T(e^{\\sharp})$. This follows that for\neach $f\\in A$ and $T\\in\\ _{\\phi_{1}}B(A^{\\sharp},I_{1})$ we have\n\\begin{align*}\n\\rho(f\\cdot T)&=(f\\cdot T)(e^{\\sharp})=T(e^{\\sharp}f)=T(f)\\\\\n&=\\phi_{1}(f)T(e^{\\sharp}) =f\\cdot T(e^{\\sharp})=f\\cdot\\rho(T).\n\\end{align*}\nHence, $\\rho$ is a left $A$-module morphism which is also a left\ninverse for $_{\\phi_{1}}\\Pi^{\\sharp}$. Therefore,\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$ is $\\phi_{1}$-injective as a Banach\nleft $\\ell^{1}(\\mathbb{N}_{\\wedge})$-module.\n\n$\\mathbf{Case\\ 2}$: We prove that  $\\ell^{1}(\\mathbb{N}_{\\wedge})$ is\n$\\phi_{n}$-injective as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\wedge})$-module for each  $n\\geq 2$. Since\n$I_{n}\\in\\textbf{I}_{n}\\textbf{-mod}$ has the identity $\\delta_{n}$,\nso it is faithful. By Theorem \\ref{Th: Her,of,Id}, it suffices to\nshow that $I_{n}\\in\\textbf{I}_{n}\\textbf{-mod}$ is\n$(\\phi_{n})_{|I_{n}}$-injective. Let $\\widehat{\\phi_{n}}$ be the\nsemi-character on $\\mathbb{N}_{\\wedge}$ associated with $\\phi_{n}$.\nSince $S=(\\{1,2,3,...,n\\},\\wedge)$ is a semilattice and\n$\\widehat{\\phi_{n}}(1)=0$, it follows from \\cite[Theorem\n2.1]{Essmaili} that $I_{n}=\\ell^{1}(\\{1,2,3,\\ldots,n\\},\\wedge)$ is\n$(\\phi_{n})_{|I_{n}}$-amenable. Therefore, we conclude that\n$c_{0}(S)$ is $(\\phi_{n})_{|I_{n}}$-flat in\n$\\textbf{mod-}\\textbf{I}_{n}$ by \\cite[Proposition 3.1]{Nasr}.\nHence, $I_{n}=c_{0}(S)^{*}$  is $(\\phi_{n})_{|I_{n}}$-injective in\n$\\textbf{I}_{n}\\textbf{-mod}$ and the proof is complete.\n\\end{proof}\nLet $\\ell^{1}(\\mathbb{N}_{\\vee})$ be the semigroup algebra on\nsemigroup $S=(\\mathbb{N},{\\vee})$ with the following product:\n\\begin{equation*}\n\\mathbb{N}\\times \\mathbb{N}\\longrightarrow \\mathbb{N},\\quad\n(m,n)\\longrightarrow m\\vee n=\\max\\{m, n\\}.\n\\end{equation*}\nIt is easy to check that $\\Phi_{S}=\\{\\widehat{\\psi}_{n} :\nn\\in\\mathbb{N}\\}\\cup\\{\\widehat{\\psi_{S}}\\},$ where\n$\\widehat{\\psi_{S}}$ is the augmentation character and for each\n$n\\in\\mathbb{N},$\n$$\\begin{array}{lll}\n\\widehat{\\psi}_{n}(m) =\\left\\{\\begin{array}{l} 1 \\quad\\text{if}\\quad\nm\\leq n\n\\\\ 0 \\quad \\text{if} \\quad m> n\n\\end{array}\\qquad (m\\in\\mathbb{N}).\\right.\n\\end{array}$$\nIn \\cite[Example 5.6]{Ramsden Paper}, it is proved that\n$\\ell^{1}(\\mathbb{N}_{\\vee})$ is not injective as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\vee})$-module. In the following theorem, we\nshow that $\\ell^{1}(\\mathbb{N}_{\\vee})$ is $\\phi$-injective for each\n$\\phi\\in \\Delta(\\ell^{1}(\\mathbb{N}_{\\vee}))$.\n\\begin{theorem}\\label{aa}\n$\\ell^{1}(\\mathbb{N}_{\\vee})$ as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\vee})$-module is $\\phi$-injective for each\n$\\phi\\in \\Delta(\\ell^{1}(\\mathbb{N}_{\\vee}))$.\n\\end{theorem}\n\\begin{proof} By \\cite[Corollary 2.2]{Essmaili}, it follows that $\\ell^{1}(\\mathbb{N}_{\\vee})$ is character amenable . Hence,\nfor each $\\phi\\in\\Delta(\\ell^{1}(\\mathbb{N}_{\\vee}))$,\n$\\ell^{1}(\\mathbb{N}_{\\vee})$ is $\\phi$-amenable. On the other hand,\nsince $\\mathbb{N}_{\\vee}$ is weakly cancellative by \\cite[Theorem\n4.6]{Dales. Lau}, we conclude that $c_{0}(\\mathbb{N}_{\\vee})$ is a\nBanach $\\ell^{1}(\\mathbb{N}_{\\vee})$-module . Hence,\n$c_{0}(\\mathbb{N}_{\\vee})$ is $\\phi$-flat as a Banach right\n$\\ell^{1}(\\mathbb{N}_{\\vee})$-module \\cite[Proposition 3.1]{Nasr}.\nThis follows that\n$c_{0}(\\mathbb{N}_{\\vee})^{*}=\\ell^{1}(\\mathbb{N}_{\\vee})$ is\n$\\phi$-injective as a Banach left\n$\\ell^{1}(\\mathbb{N}_{\\vee})$-module.\n\\end{proof}\nFor each $n\\in \\mathbb{N}$, let $J_{n}$ be the closed ideal of\n$A=\\ell^{1}(\\mathbb{N}_{\\vee})$ generated by\n$\\{\\delta_{n},\\delta_{n+1},\\ldots\\}$. It is easy to see that $J_{n}$\nis invariantly complemented in $\\ell^{1}(\\mathbb{N}_{\\vee})$.\nIndeed, it is sufficient to consider the map $Q_{n}:A\\longrightarrow\nJ_{n}$ defined by\n$$Q_{n}(f)=(\\sum_{i=1}^{n}f(i))\\delta_{n}+\\sum_{i=n+1}^{\\infty}f(i)\\delta_{i} \\qquad(f\\in A).$$\nIt is straightforward to check that $Q_{n}$ is an onto projection in\n$_{A}B(A,J_{n})$.\n\nAs a consequence of Corollary \\ref{Cor: A and I. A is Com} and\nTheorem \\ref{aa}, we give the following result.\n\\begin{corollary} For each $n\\in \\mathbb{N}$, $J_{n}$ as a Banach left $\\ell^{1}(\\mathbb{N}_{\\vee})$-module is\n$\\phi$-injective for each\n$\\phi\\in\\Delta(\\ell^{1}(\\mathbb{N}_{\\vee}))$ with $\\phi_{|J_{n}}\\neq\n0$.\n\\end{corollary}\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\nThe field of interpretability aims at providing users and practitioners with techniques meant to explain either globally a trained machine learning model or locally a particular prediction made by a model. This can be achieved either by training directly an interpretable model, or in a post hoc approach, using model-agnostic or model-specific interpretability techniques.\n\nThis paper focuses on post hoc surrogate models that globally approximate a machine learning classifier while providing explanations at the local level of each prediction.\nWe are interested in model-agnostic interpretability approaches meant to be applied on standard feature spaces composed of tabular data. Our goal is to explain any type of trained model: the classifier is a black-box left to the discretion of the practitioners.\nWe refer the reader to recent published surveys for a global picture of the interpretability field as for instance~\\citep{Guidotti2018a}.\n\n\\begin{figure}\n    \\centering\n    \\resizebox*{\\columnwidth}{3.5cm}{\n    \\input{illustration_tree_concept.tex}\n    }\n    \\caption{Concept Tree trained on FRED-MD macroeconomic dataset. Variables are grouped by Concepts to constraint the training of an interpretable surrogate decision tree}\n    \\label{fig:my_label}\n\\end{figure}\n\nSurrogate models aiming at providing post hoc interpretability may induce confusion by conveying a false sense of simplicity, especially when subgroups of dependent variables are involved. We refer to dependent variables as variables sharing similar information and possibly generated by a common phenomenon. It may include the various lags of a given time series, various features of a variables, or various measures of a given fact. Surrogate models may arbitrarily select one given variable among a group of dependent variables, thus obscuring the global picture.\nSubsequently, practitioners may better understand a surrogate model that retains the whole set of dependent variables and depicts a bigger picture than a simpler model. \n\nThis paper introduces the idea of \\textit{concept}. A \\textit{concept} is a representation gathering a group of dependent variables. It can be defined using either domain knowledge or statistical properties of dependent variables (such as the Pearson correlation). The use of \\textit{concepts} allows to provide high-level representations that practitioners may find easier to interpret. We contend that \\textit{concept}-based methods may be better suited to human understanding and provide more practitioner-friendly representations of a black-box classifier. \n\nWe substantiate that claim with an application to decision tree surrogates. Decision trees are universally considered interpretable by domain experts~\\cite{freitas2014comprehensible}. We compare standard surrogate tree models to trees whose training is constrained by the grouping of subgroups of variables in \\textit{concepts}. More specifically, we embed the idea of \\textit{concept} in the TREPAN algorithm~\\citep{Craven1996b}, an interpretable decision tree originally instantiating a variant of \\emph{id2-of-3}~\\citep{murphy1991id2} with a mechanism of oracle querying aiming at populating areas of the training set where the fidelity of the surrogate can be improved. In our approach, the \\textit{concepts} are used at each node of the decision tree to constrain the training of the split rule based on \\emph{id2-of-3}. We compare the resulting \\textit{Concept Trees} to the surrogates provided by the original TREPAN algorithm. \n\nThe next section expands on the motivation and formally introduces the idea of \\textit{high-level concepts}. Section 3 introduces \\textit{Concept Trees}, a version of the TREPAN algorithm that builds on \\textit{concepts}, and shows that \\textit{Concept Trees} meet the prerequisites of a global-to-local, post-hoc and model-agnostic surrogate. Section 4 assesses both the qualitative and quantitative relevance of our proposition through experiments led on FRED-MD, a monthly macroeconomic database designed for empirical analysis of the US economy \\citep{mccracken2016fred}.\n\n\n\\section{Concept: Grouping Dependent Variables into High-Level Representation of Variables}\n\nIt is often the case that groupings of variables in a given dataset may naturally appear. Such grouping can derive from similar meaning or a similar origin (\\emph{e.g.} unemployment among men, unemployment among women, unemployment among young people...). A grouping can also be the result of multiple transformations applied to a given source of data (such as multiple lags of a time series, or features engineered from the same variable).\n\nIn this work, we consider two types of \\textit{concepts}: expert-defined grouping of features and automatically-defined grouping based on a statistical criterion such as feature correlation.\nExpert-based \\textit{concepts} may be used when domain knowledge is available. Automatically-defined concepts do not require prior domain knowledge.\n\nExploiting the group structure of variables has already been used in the literature to train more accurate sparse models, for instance with \\emph{group-lasso}~\\citep{yuan2006model} or \\emph{sparse-group-lasso}~\\citep{simon2013sparse}. In the latter, improved accuracy is observed with variable groupings such as gene pathways or factor level indicators in categorical data.\nOther machine learning fields also cover the idea of grouping dimensions, such as subspace clustering \\citep{vidal2011subspace}.\n\nIn the field of interpretability, the idea of exploiting a meaningful grouping of features to generate better explanations has emerged, for instance with topic-modeling-based feature compression~\\citep{kim2015scalable} or on image classification with deep learning models~\\citep{Kim2017,Ghorbani2019}.\n\nCorrelated features is a known challenge when building machine learning models and interpreting feature importances~\\cite{Buhlmann2013,Gregorutti2017,Strobl2008,Tolosi2011}. For instance, \\emph{lasso}-based methods for feature selection tend to select only one representative from a group of correlated features and to discard the others~\\cite{Buhlmann2013}. It has been pointed out that correlated features severely impact variable importance measures of random forests~\\cite{Strobl2008, Gregorutti2017}. Also, many feature selection methods suffer from a \\emph{correlation bias}: features belonging to a group of correlated features receive weights inversely proportional to the size of the group~\\cite{Tolosi2011}. This issue creates instability in the feature selection process. Small changes in the training data can result in significant changes in the selected set of features. This instability prevents a robust interpretation of variable importance.\n\nWe propose to use the idea of \\textit{concept} to address both expert-defined grouping of features and automatically (correlated)-defined grouping. \\textit{Concepts} are embedded into surrogate models in order to constrain their training, which provides two levels of granularity for the explanations: at high-level (concept) and at finer level (raw variables).\nThe next paragraph offers a formal presentation of the idea of \\textit{concepts}. \n\nWe consider a set of training examples $\\mathbb{X}$ where each example is denoted $x^{(i)}$ with $i \\in [1...|\\mathbb{X}|]$ and associated with a label $y^{(i)}$. The set of training examples $\\mathbb{X}$ is composed of a set of features $j \\in \\mathbb{J}$ and each feature vector is noted  $x_j$ with $j \\in \\mathbb{J}=[1...N]$.\n\nA \\emph{concept} is a subset of features denoted $c_k \\subset \\mathbb{J}$. $K$ concepts $c_k$ co-exist to form the set of concepts $c_k \\in \\mathbb{C}, k \\in [1...K]$. The instantiation of a concept $c_k$ is the process of populating it with dependent features. Every feature $j \\in \\mathbb{J}$ belongs to a single concept $c_k$ and one concept only:\n$$c_k \\cap c_l = \\emptyset \\mid \\forall l \\in \\mathbb{J} \\text{ and } l \\neq k$$\n\n\\subsection{Expert knowledge concepts}\nThe instantiation of a concept $c_k$ can be either driven by domain knowledge or performed automatically. The former requires that all variables belong to user-defined groups that be meaningful to domain experts. The variable classifications are sometimes to be found in the documentation of a dataset. That is the case of the FRED-MD data, which is used in the experimentation section of this work. The paper accompanying the dataset \\citep{mccracken2016fred} includes in appendix a table that classifies the 134 monthly macroeconomic indicators into 8 categories: output and income, labour market, housing, consumption orders and inventories, money and credit, interest and exchange rates, prices, and stock markets. Table~\\ref{tab:fred_succint_desc} provides a sample of these categories. \n\n\\begin{table*}[]\n\\centering\n\\caption{Overview of the grouping of variables by concept in FRED-MD database~\\citep{mccracken2016fred}}\n\\label{tab:fred_succint_desc}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{@{}ccc@{}}\n\\toprule\n\\thead{Concept 1: Output and Income}     & \\thead{Concept 2: Labor Market} & \\thead{Concept 5: Money and Credit}        \\\\ \\midrule\nReal Personal Income                      & Civilian Labor Force             & Total Reserves of Depository Institutions   \\\\\nReal personal income ex transfer receipts & Civilian Employment              & Commercial and Industrial Loans             \\\\\nIP: Consumer Goods                        & Civilian Unemployment Rate       & Total Consumer Loans and Leases Outstanding \\\\\n...                                       & ...                              & ...                                         \\\\ \\bottomrule\n\\end{tabular}\n}\n\\end{table*}\n\n\\subsection{Automatic concepts: a simple approach}\nFailing that user may rely on domain knowledge, the set of \\textit{concepts} $\\mathbb{C}$ can be built automatically using a clustering algorithm based on feature correlations. Features (indexed by $j$) can be grouped in a concept $c_k$ using any dependence measure $\\rho$. The most straightforward is the Pearson correlation, that measures linear correlation between variables. Assuming the measure has values between $[-1;1]$ (an absolute value of $1$ meaning two variables perfectly dependent), a user-defined threshold $\\epsilon$ is set on the absolute value of the measure of dependence between two features $x_j$ and $x_{j'}$ in order to decide whether these features belong to the same concept $c_k$:\n\n$$\\left|\\rho(x_j,x_{j'})\\right| \\geq \\epsilon \\mid \\forall (j, j') \\in c_k$$\n\nThe clustering algorithm is greedy: for each iteration a feature is tested against all features and existing groups. A feature $j'$ is affected to a \\textit{concept} $c_k$ if its dependence to each feature in $c_k$ is higher than $\\epsilon$:\n$$\\left|\\rho(x_j,x_{j'})\\right| \\geq \\epsilon \\mid \\forall j \\in c_k \\rightarrow c_k = c_k \\cup j'$$\n\nIf a given feature is independent from all the others, it belongs to a singleton.\nThis formalization is also adequate for the expert's knowledge grouping. In that case, $\\rho$ and $\\epsilon$ would be the criteria of group assignment by the expert.\n\n\nThe next section explains how the notion of  \\textit{concepts} may be used to constrain the training of a decision tree in order to produce more interpretable surrogates. \n\n\n\\section{Concept Tree: Embedding Concepts For More Interpretable Surrogate Decision Tree}\n\nDecision trees are a well-known interpretable machine learning model. A decision tree has a graphical structure, its decisions rely on a sparse subset of features, and features are used in a hierarchical way, thus conveying an intuitive sense of feature importance and providing several levels of explanation granularity~\\citep{freitas2014comprehensible}. Training a decision tree on the training set $\\mathbb{X}$ yields an interpretable classification algorithm, provided that the number of nodes is kept under a certain threshold. The limit on the tree complexity may come at the expense of predictive performance. Decision trees appear as good candidate surrogates to black-box classifiers. \n\nA decision tree surrogate is produced as follows. A black-box $b$ is trained on $\\mathbb{X}$ with the true class labels $y^{(i)}$ ; the surrogate $f$ is then trained on the black-box predictions $\\hat{y}^{(i)} = b(x^{(i)})$. In production, the classification is performed by the black-box while the explanations are provided by the surrogate decision tree. The fidelity of the surrogate is assessed as the proportion of instances where the surrogate makes the same prediction than the black-box classifier. \n\n\nThe TREPAN algorithm is an instance of interpretable surrogate tree model~\\citep{Craven1996}. It is model-agnostic and aims at mimicking the classification behaviour of a black-box $b$. It queries the black-box with instances to get predictions $\\hat{y}^{(i)} = b(x^{(i)})$ and then fits an interpretable decision tree. The outline of TREPAN is shown in Algorithm~\\ref{alg:trepan}. The querying of extra instances allows to populate the critical areas of the feature space and thus significantly curb the tendency of decision trees to overfit. \n\n\nTREPAN uses $m-of-n$ decision rules, that are inspired from $id2-of3$ decision trees~\\citep{murphy1991id2}. To fit an $m-of-n$ decision rule, the set of the $n$ most discriminative tests on the features for the node is discovered using the information gain. Then, in order to validate a node, an instance must validate at least $m$ tests among the $n$. For instance, given a decision rule with 3 tests $x_1$, $x_2$ and $x_3$, such as $2$-of-$\\{x_1, \\neg x_2, x_3\\}$ is equivalent to the logical expression $(x_1 \\vee \\neg x_2) \\wedge (x_1 \\vee x_3) \\wedge (\\neg x_2 \\vee x_3)$.\nThe parameters $m$ and $n$ are user-defined upper-bounds: their final values are learnt by the node. The $m-of-n$ decision rules are learnt in a greedy way for computational efficiency. For the outline of the fitting algorithm of an $m-of-n$ decision rule, we refer the reader to the original paper~\\citep{Craven1996b} for the sake of conciseness and precision.\n\nWhile the original TREPAN paper is two decades old already, researchers have kept reassessing its relevance up until recently~\\citep{Sarkar2016}.\nExperimentations show that TREPAN has a good fidelity to the black-box and a better accuracy on the test set than a decision tree directly trained on the training set $\\mathbb{X}$~\\citep{Craven1996}. This good performance is attributed to the additional-instance-drawing mechanism, which yields a denser support to the fit of a decision rule and thus a better prediction accuracy. \n\nThe $m-of-n$ decision rule structure improves the accuracy and the fidelity of the decision tree as it allows to learn more complex decision boundaries. However, it comes at the price of interpretability of both the node's decision rule and the decision tree overall. A practitioner may find it hard to understand all the possible ${n \\choose m}$ combinations of variables at the same time. Moreover, the contrary of a $m-of-n$ literal may be challenging to conceive as soon as $m>1$ and $1<n<m$. Alternatively, simpler nodes such as the ones in $C4.5$~\\citep{quinlan1993c4} would be easier to interpret, but would yield significantly larger trees for a given fidelity level, thus reducing overall interpretability ~\\citep{Craven1996}.\n\nSimilar to TREPAN, the Concept Tree aims at mimicking the classification behaviour of $b$, using the additional-instance-drawing procedure at the node level to improve fidelity. The chief difference between Concept Tree and TREPAN lies in the learning of the decision rule at each node. The $m-of-n$ decision rule is no longer fitted on the whole feature space. It is constrained to use subsets of features defined by the \\emph{concepts}. At a given node, the Concept Tree fits a $m-of-n$ decision rule using only the variables related to the \\textit{concept} $c_k$ for each \\textit{concept} $c_k$, and selects the one that yields the best information gain. Each node thus splits the sample on a $m-of-n$ decision rule based on a \\emph{concept}, using related variables only. A part from that restriction, the Concept Tree uses the same expanding procedure as TREPAN, described in Algorithm~\\ref{alg:trepan}.\n\nThis paper aims at improving the interpretability of surrogate trees built with $m-of-n$ decision rules by introducing the Concept Tree, a tree-based surrogate methods based on TREPAN and the use of \\textit{concepts}. \nThe use of \\textit{concepts} is expected to help practitioners better understand the surrogate. Each node relies on variables belonging to one \\textit{concept}-grouping only. Nodes thus use complex $m-of-n$ literals but ensure better human-understandability by organizing information at a \\textit{concept} level.\n\nWe argue that concept-based decision rules have a better interpretability than standard $m-of-n$ decision rules while having the exact same informational complexity (the number of bits needed to write down the decision rule). Defying the conventional notion of complexity-interpretablity trade-off, Concept Trees achieve a higher interpretability at the same level of complexity, thus preserving predictive accuracy. \n\n\n\\begin{algorithm}[tb]\n   \\caption{Simplified overview of Trepan}\n   \\label{alg:trepan}\n\\begin{algorithmic}\n    \\STATE {\\bfseries Trepan}($b$, $X$,  $max\\_nodes$, $min\\_sample$, $m$, $n$)\n\\STATE Initialize the tree with root $R$\n\\STATE $S \\gets X \\cup DrawSample(min\\_sample-|X|)$\n\\STATE Get labels from black-box $b$ for train set $S$\n\\STATE Initialize $Queue$ with $<R, S>$\n\\STATE $n\\_nodes = 1$\n\\WHILE{$Queue \\neq \\emptyset$ and $n\\_nodes < max\\_nodes$}\n\\STATE Remove $<node N, S_N>$ from head of $Queue$ \n\\STATE Fit decision rule of node $N$\n\\FOR{each outcome $t$ of the test}\n    \\STATE Initialize a child node $C$\n    \\STATE $S_c \\gets$ instances of $S_N$ with outcome $t$ for the test\n    \\STATE $S_C \\gets S_c \\cup DrawSample(min\\_sample-|S_c|)$\n    \\STATE Get labels from black-box $b$ for $S_C$\n    \\IF{$C$ is not pure enough}\n        \\STATE Add $<node C, S_C>$ to $Queue$\n    \\ENDIF\n    \\STATE $n\\_nodes = n\\_nodes + 1$\n\\ENDFOR\n\\ENDWHILE\n\n\\STATE {\\bfseries Return} $R$\n\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{algorithm}[tb]\n   \\caption{Construction of a Concept Tree decision rule}\n   \\label{alg:concepttreenode}\n\\begin{algorithmic}\n   \\STATE {\\bfseries ConstructConceptDecisionRule}($X$, $y$, $concepts$)\n   \n   \\STATE $best\\_candidate \\leftarrow \\emptyset$\n   \\STATE $best\\_ig \\leftarrow = 0$\n   \\FOR{$c \\in concepts$}\n   \\STATE $X_c \\leftarrow$ Select features from $X$ belonging to $c$\n   \\STATE $candidate \\leftarrow MofNDecisionRule(X_c, y, m, n)$\n   \\STATE $ig \\leftarrow$ Compute information gain for $candidate$\n   \\IF{$ig > best\\_ig$}\n    \\STATE $best\\_ig \\leftarrow ig$\n    \\STATE $best\\_candidate \\leftarrow candidate$\n   \\ENDIF\n   \\ENDFOR\n   \n   \\STATE {\\bfseries Return} $best\\_candidate$\n \n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experimentation: FRED-MD Macroeconomic Database}\n\nThis paper has introduced the ideas of Concept and Concept Tree, whose main objectives are to provide an accurate surrogate $f$ mimicking a black-box classifier $b$ while being as interpretable as possible. The next paragraphs describe experimentations made with the FRED-MD dataset ~\\citep{mccracken2016fred}, a publicly released macroeconomic database of 134 monthly U.S. indicators with more than 700 instances. Interpretability is critical in economics and our experimentations show how Concept Trees may match the requirements of the field. \n\nThe experimentations are conduced as follows. The classification target is computed from the \\emph{civilian unemployment rate}: if the value for an instance is lower than in the previous period, the target value is set to label 0 and the label is set to 1 otherwise. Domain-knowledge-based experts are extracted from the FRED-MD official documentation, which classifies variables into 8 subgroups (see Table~\\ref{tab:fred_succint_desc}). \n\nThe competitors are both flavors of Concept Tree (Concept Tree-Expert and Concept Tree-Correlated for automatically-defined concepts) and the original TREPAN.\nSince the Concept Tree and TREPAN have a similar structure, they share the same parameters for the experimentation. The maximal number of nodes $max\\_nodes$ is set to 10. For the split rules, the values of $m-of-n$ are set to $1-of-1$, $3-of-3$ and $5-of-5$. The minimal value of samples $min\\_samples$ to fit a split rule at a node is 100, thus additional samples are drawn from the fitted distribution if the $\\mathbb{X}$ is not large enough. For Concept Tree-Correlation, the threshold $\\epsilon$ on the correlation $\\rho$ is set to $0.9$ such as $\\left|\\rho(x_j,x_{j'})\\right| \\geq 0.9$.\n\nThe black-box $b$ used is a Random Forest with 200 estimators, with the scikit-learn default values for the other parameters.\nOut-sample-fidelity is computed by 5-fold cross-validation. At each split the black-box is fitted on the train set and makes predictions for the train set and the test set. The Concept Tree and TREPAN instances are then fitted on the train set with black-box predictions as targets, and their fidelities are measured against the black-box predictions made on the test set. Out-of-sample accuracy is assessed using the same procedure. Fidelity measures the proportion of predictions made by the surrogate that match the predictions made by the black-box, while accuracy measures the proportion of predictions made by the surrogate that match the actual value of the target. Interpretability is assessed by economic expert judgement.\n\n\\begin{table}[!htbp]\n\\centering\n\\caption{Experimental results: surrogate accuracy and fidelity as a function of the algorithm, the concept type and the split rule}\n\\label{tab:results}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}ccc||ccc@{}}\n\\toprule\n\\thead{Algorithm}   & \\thead{Concept Type} & \\thead{Split Rule}  & \\thead{Surr. Accuracy} & \\thead{Surr. Fidelity} \\\\ \\midrule \\midrule\nConcept Tree & Expert & \\multirow{ 3}{*}{$1-of-1$}  & $63\\% \\pm 4\\%$ & $65\\% \\pm 9\\%$ \\\\\nConcept Tree & Correlation &  & $68\\% \\pm 6\\%$ & $71\\% \\pm 6\\%$ \\\\\nTREPAN & \/ &  & $\\bm{75\\% \\pm 9\\%}$ & $\\bm{74\\% \\pm 7\\%}$ \\\\ \\hline\nConcept Tree & Expert & \\multirow{ 3}{*}{$3-of-3$}  & $69\\% \\pm 9\\%$ & $\\bm{76\\% \\pm 4\\%}$ \\\\\nConcept Tree & Correlation &  & $\\bm{72\\% \\pm 11\\%}$ & $75\\% \\pm 5\\%$ \\\\\nTREPAN & \/ &  & $68\\% \\pm 8\\%$ & $72\\% \\pm 6\\%$ \\\\ \\hline\nConcept Tree & Expert & \\multirow{ 3}{*}{$5-of-5$} & $\\bm{71\\% \\pm 4\\%}$ & $\\bm{73\\% \\pm 2\\%}$ \\\\\nConcept Tree & Correlation &  & $70\\% \\pm 8\\%$ & $71\\% \\pm 8\\%$ \\\\\nTREPAN & \/ &  & $67\\% \\pm 5\\%$ & $71\\% \\pm 4\\%$ \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\end{table}\n\n\\subsection{Results}\nTable~\\ref{tab:results} presents the cross-validated accuracies and fidelities for TREPAN, the Concept Tree with expert-defined concepts and the Concept Tree with automatically defined clusters. The black-box mean accuracy over the folds is $82\\% \\pm 4\\%$.\n\nThe experimentations show that Concept Tree provides surrogates whose fidelity and accuracy matches the performance of TREPAN trees and whose interpretability may be significantly enhanced. Although, TREPAN leads in terms of accuracy and fidelity for $1-of-1$ nodes; Concept Tree-Expert for $3-of-3$; and Concept Tree-Correlated for $5-of-5$ nodes, the non-negligible standard-deviations and the setup of this preliminary experiment (number of folds and datasets) don't allow for a final conclusion. However, the experiment highlights the relevance of Concept Tree in terms of accuracy and fidelity and as things stand, Concept Tree is at least as relevant as TREPAN.\n\nFigure \\ref{fig:trees} in Appendix displays the trees generated by TREPAN (Figure \\ref{fig:tree_trepan}), the Concept Tree with expert-defined concepts (Figure \\ref{fig:concept_expert}) and the Concept Tree with correlation-based defined concepts (Figure \\ref{fig:concept_correlated}). \n\nFrom a macroeconomic point of view, the Concept Tree yields meaningful high level explanations of the workings of the black-box classifiers. The Concept Tree-Expert highlights that Labor Market related variables are the most important in the prediction of the target, followed by Output and Income related variables and Consumption related variables. The Concept Tree-Correlated also sheds light on the importance of nodes referring to Labour market data. Overall, Concept Tree enhances the interpretability of surrogate trees by \\textbf{structuring} the explanations.\n\nIn Concept Tree-Expert (based on domain-knowledge), explanations are structured by sharing a common \"language\" with users or experts. It provides the big picture with one general \\textit{concept} by node. The detailed analysis of a node is eased because only related, homogeneous, variables are assembled. There is an intuitive relations between high-level explanations (concepts) and low-level explanations (raw variables).\n\nIn Concept Tree-Correlation, computed automatically based on variable correlations, part of the domain-knowledge can be recovered. Concept Tree-Correlation presents also the advantage of gathering dependent variables for each node, avoiding arbitrary choices between correlated variables to build a test.\n\nIn contrast, TREPAN trees use an idiosyncratic language not shared by the practitioner. Associations of tests in a TREPAN node generate confusion by gathering variables that are hardly related from a domain-knowledge point of view. Such nodes obstruct the understanding by preventing the user from getting the big picture.\n    \nTo illustrate these arguments, we focus on the top three nodes of the trees in Figures \\ref{fig:tree_trepan}, \\ref{fig:concept_expert} and \\ref{fig:concept_correlated}. In the TREPAN tree (Figure \\ref{fig:tree_trepan}), the colored variables names highlight that 8 out of the 9 chosen variables are part of the Labor Market concept. This structure is explicitly displayed by the Concept Tree-Expert (Figure \\ref{fig:concept_expert}) as the concept chosen in the root node, facilitating the interpretation of the tree by referring to high level concept. We can also notice in the TREPAN tree that the left child of the root node chose the \\textit{Civilian Employment, gr} feature for its first rule, whereas the right child node chose the \\textit{All Employees: Total nonfarm, gr} feature instead. However, the cluster 3 in the Concept Tree-Correlated (Figure \\ref{fig:concept_correlated}) explicitly shows that these features are highly correlated, suggesting that they are interchangeable.\n\n\n\n\n\\section{Conclusion}\n\nThe present paper introduces \\textit{concepts}, a meaningful manner to group dependent variables, and Concept Trees, an alternative tree-based surrogate model that provides both high-level and detailed explanation to black-box classifiers. The grouping of variables in \\textit{concepts} allows to overcome the false sense of simplicity conveyed by simpler decision tree surrogate that may give an artificially high importance to a given variable picked among a set of correlated variables, thus obscuring the bigger picture. The use of \\textit{concepts} also helps practitioners make sense of otherwise cryptic $m-of-n$ literals, by relying on a higher-level representation of the data. Compared to TREPAN, Concept Trees produce surrogates that have a comparable size and are as accurate, but more easily understandable to a human thanks to a better organization of the information along higher-level representations that significantly enhance the interpretability of the surrogate.\nExperiments were conduced using FRED-MD, a macroeconomic database whose documentation includes a grouping of variables. The Concept Tree was applied to this data using both expert-defined concepts derived from the data documentations and concepts built using a simple correlation-based clustering algorithm. First results show a notable improvement in human-readability while accuracy and fidelity of the surrogate are preserved.\nFurther research could involve a deeper assessment of our propositions, both quantitatively and qualitatively. It could also be relevant to explore alternative clustering algorithms designed to produce more relevant \\textit{concepts}.\nFurther modification to the Concept Tree algorithm may improve performance: currently, following TREPAN's principle, one concept can only be used once in a decision path. Considering a concept encompass several variables, the accuracy and fidelity of the surrogate may suffer from this probably too severe constraint.\n\n\n\\bibliographystyle{icml2019}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter[Pupil Plane Phase]{Pupil Plane Phase Apodization}\\label{sec:ppp}\n\n\\author[M. Kenworthy, J. Codona and F. Snik]{Matthew A. Kenworthy, Johanan\nL. Codona\\footnote{Steward Observatory, 933 N. Cherry Avenue, Tucson, AZ\n85721, USA} and Frans Snik}\n\n\\address{Leiden Observatory, Leiden University, \\\\\nP.O. Box 9513, 2300 RA Leiden, The Netherlands, \\\\\nkenworthy@strw.leidenuniv.nl}\n\n\\begin{abstract}\n\nPhase apodization coronagraphs are implemented in a pupil plane to\ncreate a dark hole in the science camera focal plane.\nThey are successfully created as ``Apodizing Phase Plates'' (APPs) using\nclassical optical manufacturing, and as ``vector-APPs'' using\nliquid-crystal patterning with essentially achromatic performance.\nThis type of coronagraph currently delivers excellent broadband contrast\n($\\sim$10$^{-5}$) at small angular separations (few $\\lambda\/D$) at\nground-based telescopes, owing to their insensitivity to tip\/tilt\nerrors.\n\n\\end{abstract}\n\n\\body\n\n\\section{Introduction}\n\nPupil-plane apodization techniques (amplitude, phase, or complex) differ\nfrom focal plane mask coronagraphs in that they affect all objects in\nthe field in an identical fashion.\nThe main goal of such pupil-plane coronagraphs is to enforce dark holes\nin the ensuing point spread function (PSF) in which faint companions can\nbe directly detected and characterized.\nSince the star and companion have the same PSF, the halo should be\nsuppressed while preserving the starlight in the core as much as\npossible, {\\it i.e.}~a high Strehl ratio PSF.\nIn this situation, the ``noise'' is governed by the PSF diffraction halo\nplus any diffuse background, while the ``signal'' is contained in the\nPSF core.\n\nThe phase-only ``Apodizing Phase Plate''\\cite{Kenworthy07,\nKenworthy13,Kenworthy10a,Kenworthy10b} (APP) coronagraphs have now been\nsuccessfully applied on-sky at ground-based telescopes.\nThe main benefits of APPs include a high contrast inside the dark hole\n($\\sim$10$^{-4}$--10$^{-6}$), at a small inner working angle $\\sim$$1.5\n\\lambda\/D$,  with complete insensitivity to tip\/tilt errors (and\npartially resolved stellar disks) that usually limit focal-plane\ncoronagraphs.\nThis invariance of the PSF additionally enables beam-switching for\nthermal background removal, and observations of multiple star systems.\nWith the introduction of advanced liquid-crystal technology for the\nvector-APP coronagraph\\cite{vAPP,vAPP-prototype,vAPP-MagAO}, it has also\nbecome efficient over spectral bandwidths of more than an octave, at\nwavelengths from 300 to 30,000 nm\\cite{Packham2010}.\nThe extreme phase patterns enabled by liquid-crystal writing techniques\ncan now also produce dark holes with various shapes, including\ncomplementary 180$^\\circ$ D-shaped dark holes and 360$^\\circ$\ndonut-shaped dark holes.\nAs a single pupil-plane optic, the (vector-)APP is easily implemented in\na filter wheel in existing instruments, and is fully compatible with\ncryovacuum (and likely also space-based) operation.\n\n\\section{Theory}\n\nThe 1-D apodization problem has been studied for a long time, including\nslit apodization in spectroscopy and pulse shaping to reduce channel\nbandwidth in telegraphy, by apodizing in amplitude\\cite{Jacquinot64}.\nThe family of functions to describe this are the Slepian functions and\nthe Prolate Spheroidal wavefunction\\cite{slepian1965ast}.\nSince transmission apodization is linear, it can achieve a high degree\nof suppression between the PSF and the halo beyond a selected inner\nworking angle (IWA), and in general the apodizations are complex with\nboth transmission and phase.\nThe accurate manufacture of complex amplitude masks is non-trivial and\ncan result in low transmission efficiencies.\n\nPhase-only apodization theory was initially developed for removing\nspeckles generated by residual optical aberrations in high contrast\nimaging experiments\\cite{Malbet95}, where wavefront sensing in the final\nfocal plane of a coronagraph forms a closed loop with a deformable\nmirror (DM) in the optical system.\nA sinusoidal ripple on the DM forms a diffraction grating in the phase\nof the wavefront, generating a pair of speckles that are copies of the\nAiry core of the central PSF.\nThe appropriate choice of spatial phase and amplitude of the ripple\napplied to the DM destructively interferes with speckles generated by\naberrations in the optical system.\nThe same principle can be generalized to cancel out the diffraction\nrings of the PSF itself, as demonstrated on-sky by the addition of coma\ninto an adaptive optics system to cancel out part of the first Airy\nring\\cite{Serabyn07}.\nApodization in phase over a two-dimensional region does not yet have an\nanalytic solution.\nSuperposing many different phase ripples in the pupil plane to suppress\nthe diffraction pattern over a region of interest (ROI - typically\ndefined as a D-shaped region next to the Airy core of the PSF) is\nchallenging, since the speckles add vectorially  and interfere with each\nother, making it a nonlinear problem.\nRef.~\\refcite{Codona2007} searched for phase-only apodization solutions\nthrough a modal basis approach.  An ROI is defined in a complex\namplitude focal plane, where the diffraction halo is to be minimized.\nA complex amplitude field is defined in the pupil plane, and a Fourier\nimaging operator is defined that maps from the pupil plane into the ROI.\nSingular Value Decomposition of this operator produces a modal basis set\nof complex pupil amplitudes, ordered canonically from the most power\ncontained within the ROI to the least.\nThese modes typically have complex amplitudes in the pupil plane, so\ntheir complex amplitude is normalized to unity to make them phase-only\napodization.\nThese ``antihalo'' modes are subtracted off the complex amplitude of the\npupil plane, and the process is repeated.\nThe antihalo modes extend a short distance beyond the ROI, and if the\nIWA is within the first Airy ring, flux from the core of the PSF is\ndetrimentally removed as well.\nCare is needed to suppress these modes by imposing additional\nconstraints to maximize the PSF core encircled energy.\nIf not properly accounted for, phase wrapping can also occur when the\npeak-to-valley phase apodization is greater than $2\\pi$.\n\nNew algorithms have been developed at Leiden Observatory by Doelman,\nKeller and Por.\nDoelman generates focal plane dark zones using a combination of\nphase-only pupil modes.\\cite{Doelman16}\nA simulated annealing approach is used, where the mode amplitudes are\nrandomly adjusted.\nSolutions that improve the dark region are kept, but worse solutions are\noccasionally accepted as well to escape local minima.\nKeller uses a Gerchberg-Saxton\\cite{Gerchberg72} method, switching\nbetween the pupil plane and focal plane.\nConvergence to a given contrast level is increased by an order of\nmagnitude using Douglas-Rachford operator splitting\\cite{Douglas56}.\nPor\\cite{por2017optimal} generalizes an algorithm by\nCarlotti\\cite{Carlotti2013} for general complex amplitudes in the pupil\nplane.\nStrehl ratio maximisation for this mask is a linear operation solved by\nlarge scale optimizer, and phase-only solutions are naturally found\nthrough this approach.\n\n\\section{First generation APPs using classical phase}\n\nThe manufacture of APP solutions requires the variation of phase across\nthe pupil plane of the camera, and the development of free-form optic\nmanufacture with notable departures from sphericity using\ncomputer-controlled diamond turning\\cite{Davis07} encoded the phase\npatterns as variations in the thickness of a high refractive index\ntransmissive substrate.\nFirst light observations of an APP with diamond turned\noptics\\cite{Kenworthy07} demonstrated the viability of the manufacturing\ntechnique and of the theory.\nThe success of the prototype led to APP coronagraphs on the 6.5m MMTO\ntelescope in Arizona\\cite{Kenworthy13} and on the Very Large Telescope\nin Chile\\cite{Kenworthy10a,Kenworthy10b}.\nThe VLT APP led to the first coronagraphic image of the extrasolar\nplanet $\\beta$ Pictoris b\\cite{Quanz10} and the discovery of the\nextrasolar planet HD~100546b\\cite{Quanz12}.\n\nDiamond turning only allows for low spatial frequencies in the azimuthal\ndirection of the cutting tip, and the classical phase plate\nmanufacturing was inherently chromatic.\nAttempts to achromatize the APP using doublets proved highly\nchallenging\\cite{Kenworthy10c}.\n\n\\section{The Vector-APP}\n\nThe main limitations of the APP coronagraph (chromaticity, limited\ncoverage around the star, limited phase pattern accuracy) were solved by\nthe introduction of the vector-APP (vAPP)\\cite{vAPP}.\nIn a similar way as for the Vector Vortex Coronagraph\\cite{VVC}, the\nvAPP replaces the classical phase pattern ($\\phi_{\\textrm{c}}[u,v] =\nn(\\lambda) \\Delta d[u,v]$) with the so-called\nPancharatnam\\cite{Pancharatnam}-Berry\\cite{Berry} phase or ``geometric\nphase''\\cite{Escuti-geometricphase}.\nThe vAPP phase pattern is imposed by a half-wave retarder with a\npatterned fast axis orientation $\\theta[u,v]$.\nThe geometric phase is imprinted on incident beams decomposed according\nto circular polarization state: $\\phi_{\\textrm{g}}[u,v] =\n\\pm2\\cdot\\theta[u,v]$, with the sign depending on the circular\npolarization handedness.\nAs this fast axis orientation pattern does not vary as a function of\nwavelength (with the possible exception of an inconsequential\noffset\/piston term), the geometric phase is strictly achromatic.\nA simple Fraunhofer propagation from the pupil $[u,v]$ to the focal\nplane $[x,y]$ shows that after splitting circular polarization states\nthe two ensuing coronagraphic PSFs are point-symmetric\n($PSF_{\\textrm{L}}[x,y] = PSF_{\\textrm{R}}[-x,-y]$), and therefore, in\nthe case of D-shaped dark holes, delivers complementary PSFs that\nfurnish instantaneous 3$60^\\circ$ search space around each star.\n\nVector-APP devices are produced by applying two breakthrough\nliquid-crystal techniques: any desired phase pattern is applied onto a\nsubstrate glass through a \\textit{direct-write\nprocedure}\\cite{directwrite} that applies the orientation pattern\n$\\theta[u,v]$ by locally polymerizing the alignment layer material in\nthe direction set by the controllable polarization of a scanning UV\nlaser.\nConsecutively, birefringent liquid-crystal layers are deposited on top\nof this alignment layer.\nSeveral self-aligning layers (``\\textit{Multi-Twist Retarders}'';\nMTR\\cite{MTR}) with predetermined parameters (birefringence dispersion,\nthickness, nematic twist) yield a linear retardance that is close to\nhalf-wave over the specified wavelength range.\nThe vAPP can become efficient over a large wavelength range (up to more\nthan an octave), while any phase pattern can be written with high\naccuracy.\n\n\\subsection{Prototyping and first on-sky results}\n\nThe first broad-band vAPP device was fully characterized in the lab at\nvisible wavelengths (500--900 nm)\\cite{vAPP-prototype}.\nThe main limitation of the contrast performance inside the dark hole was\nthe occurrence of leakage terms that produced a faint copy of the\nregular PSF on top of the coronagraphic PSFs.\nThese leakage terms are caused by small offsets to the half-wave\nretardance of the vAPP device, and offsets from quarter-wave retardance\nof the quarter-wave plate that, together with a Wollaston prism,\naccomplishes the (broad-band) circular polarization splitting.\nThis issue was resolved with the introduction of the\n``grating-vAPP''\\cite{grating-vAPP}, which implements the circular\npolarization splitting by superimposing a tilt (i.e.~a ``polarization\ngrating''\\cite{Packham2010}) pattern on top of the coronagraphic pupil\nphase pattern, which, by virtue of the properties of the geometric\nphase, very efficiently sends the coronagraphic PSFs into grating orders\n$\\pm$1, and leaves all the leakage terms in the zeroth order.\nThe grating-vAPP also greatly simplifies the optical configuration, as\nall the manipulation takes place within one single (flat) optic.\nThe coronagraphic PSFs are now subject to a lateral grating dispersion\nterm and so the grating-vAPP can only be used in combination with\nnarrow-band filters, although the wavelength range throughout which\nthese filters can be applied can still be very large.\n\nThe first grating-vAPP successfully demonstrated on-sky was installed at\nthe MagAO\/Clio instrument attached to the 6.5-m Magallan-Clay telescope\nin Chile\\cite{vAPP-MagAO} (\\fref{MagAO-vAPPs}a--c).\nThe device was designed and built to operate from 2--5 $\\mu$m, covering\nthe infrared atmospheric K, L and M-bands.\nThe first-light observations demonstrated excellent suppression of the\nstellar diffraction halo in the complementary dark holes (see\n\\fref{MagAO-vAPPs}c).\nDetailed analysis of the data demonstrated a 5-$\\sigma$ contrast for\npoint source detection of $\\sim$$10^{-5}$ at 2.5--7\n$\\lambda\/D$\\cite{vAPP-MagAO}.\nThe contrast performance is greatly enhanced by combining the two\ncomplementary dark holes through a simple rotation-subtraction procedure\nto further suppress the wind-driven starlight halo in the dark holes,\nwhich is caused by finite AO loop speed.\n\\Fref{MagAO-vAPPs}c shows the presence of the leakage term PSF in\nbetween the coronagraphic PSFs, which can be used as an astrometric and\nphotometric reference, in the (frequent) case that the coronagraphic PSF\ncores are saturated.\n\n\\begin{figure}[ht] \\includegraphics[width=\\textwidth]{MagAO-vAPPs}\n\n \\caption{Phase patterns, theoretical and on-sky PSFs (logarithmic\nscale) for the two vAPP devices installed at MagAO.\na) Theoretical phase pattern for a 180$^\\circ$ dark hole covering 2--7\n$\\lambda\/D$, b) the ensuing theoretical PSF, c) the on-sky PSFs at MagAO\nfor the star $\\eta$ Crucis at 3.9 $\\mu$m.\nd) Theoretical phase pattern for a 360$^\\circ$ dark hole covering 3--7\n$\\lambda\/D$, e) the ensuing theoretical PSF, f) the on-sky PSFs at MagAO\nfor the binary star $\\beta$ Centauri at 3.9 $\\mu$m.\nPhase pattern designs by Christoph Keller. Data processing by Gilles\nOtten\\cite{vAPP-MagAO}.}\n\n\\label{MagAO-vAPPs}\n\\end{figure}\n\n\\subsection{360 degree APP solutions}\n\nAs part of the algorithm exploration of the APP surface, a family of\nfunctions was found that showed 360 degrees of suppression around the\ncentral star.\nThese solutions have lower Strehl ratios for the star (typically\n20--40\\%) with larger IWA compared to the 180$^\\circ$ dark holes, and\nthese phase pattern solutions are pathological in nature, with rapid\nphase changes over small scales.\nThe advent of liquid-crystal patterning encouraged us to revisit these\n360$^\\circ$ solutions, and test them in the lab and on-sky.\n\\Fref{MagAO-vAPPs}d-f shows the phase pattern and ensuing PSFs for the\nexperimental vAPP device at MagAO.\nThe lower row of figures shows that the liquid-crystal manufacture\nsuccessfully reproduces the complex phase pattern, and this on-sky image\n(\\fref{MagAO-vAPPs}f) shows a fainter binary stellar companion to the\nright of the primary star's PSF.\n\n\\section{Future Directions}\n\nOur team is currently installing different vAPP coronagraphs at several\ninstruments at large telescopes around the world, and working on novel\ndesigns for the future extremely large telescopes.\nForeseeable future developments of the vector-APP as a separate optical\ncomponent, and as integral part of a high-contrast imaging system\ninclude:\n\n\\begin{itemize}\n\n\\item The combination of several grating layers in a\n``\\textit{double-grating-vAPP}'' to recombine the two coronagraphic PSFs\nwith 360$^\\circ$ dark holes to feed an integral-field unit while\nrejecting the leakage terms. \n\n\\item By prescribing a specific retardance profile as a function of\nwavelength, it is possible to build a \\textit{wavelength-selective vAPP}\ndevice, that operates as a regular vAPP coronagraph at the science\nwavelengths, and acts like a regular glass plate at the spectral range\nof a wavefront sensor behind it. \n\n\\item The pupil phase manipulation of the vAPP can be extended by\namplitude manipulation in the pupil to create complex\napodizers\\cite{Carlotti2013}, and by phase\/amplitude masks in the focal\nplane to yield \\textit{hybrid coronagraphy}\\cite{Ruane2015}.\n\n\\item As this technology is likely compatible with operation in space,\nit is opportune to characterize the performance of vAPP-like\ncoronagraphs at the \\textit{extreme contrast levels} ($\\sim$10$^{-9}$)\nof space-based high-contrast imaging.\n\n\\item To adapt the vAPP phase pattern to the observational needs, the\nobserving conditions, and segmented pupils with variable configurations,\nactive liquid-crystal devices will be developed to establish\n``\\textit{adaptive coronagraphy}''. Such a system can then deliver dark\nholes of various geometry and depth, depending on whether the observer\nis interested in detecting exoplanets or characterizing known targets.\n\n\\item As the vAPP relies on polarization splitting, it is possible to\ndesign an optimal system for \\textit{coronagraphic\npolarimetry}\\cite{vAPP-polarimetry}, particularly with the\n360$^\\circ$-designs.\n\n\\item The fact that the vAPP produces several PSFs for the same star at\nthe focal plane makes it an attractive option for implementing\n\\textit{focal-plane wavefront sensing}, for instance through\nphase-diversity techniques. Another promising approach involves the\nincorporation of an additional pupil phase pattern which generates pairs\nof PSF copies around the main PSFs, with each pair encoding a wavefront\nerror mode through an intensity difference\\cite{Wilby2016}.\n\n\\end{itemize}\n\n\\section{Acknowledgments}\n\nThe research of FS leading to these results has received funding from\nthe European Research Council under ERC Starting Grant agreement 678194\n(FALCONER).\n\n\\bibliographystyle{ws-rv-van}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHeavy-flavour hadrons are suitable to probe the conditions of the high-energy-density Quark-Gluon Plasma (QGP) medium formed in ultra-relativistic heavy-ion collisions. Heavy quarks are mainly produced in hard scattering processes in the early stage of Pb--Pb collisions. The time-scale of their production ($\\Delta\\tau< 1\/2m_{c,b}\\sim 0.07~{\\rm fm}\/c~{\\rm for~charm~and}\\sim 0.02~{\\rm fm}\/c~{\\rm for~beauty}$) is, in general, shorter than the formation time of the QGP, $\\tau_{0}\\sim0.1-1$~fm\/$c$. During their propagation through the medium, heavy quarks interact with its constituents and lose energy. QCD energy loss is expected to occur via both inelastic (radiative energy loss via medium-induced gluon radiation)~\\cite{Baier1997265} and elastic (collisional energy loss) processes~\\cite{PhysRevD.44.1298}. The energy loss for quarks is expected to be smaller than for gluons, due to the smaller colour coupling factor of quarks with respect to gluons. In addition, the ``dead-cone effect'' should reduce small-angle gluon radiation for heavy quarks with moderate energy compared to their mass, thus further attenuating the effect of the medium~\\cite{PhysRevD.69.114003}.  \n\nThe nuclear modification factor $R_{\\rm AA}(p_{\\rm T})=({\\rm d}N_{\\rm AA}\/{\\rm d}p_{\\rm T})\/(\\langle T_{\\rm AA}\\rangle\\cdot{\\rm d}\\sigma_{\\rm pp}\/{\\rm d}p_{\\rm T})$, where $\\langle T_{\\rm AA}\\rangle$ is the average nuclear overlap function for a given centrality class, is sensitive to the interaction of hard partons with the medium. At large $p_{\\rm T}$, $R_{\\rm AA}$ is expected to be mostly sensitive to the average energy loss of heavy-quarks in the hot and dense medium. The questions whether low-momentum heavy quarks can reach thermal equilibrium with the medium constituents and participate in the collective expansion of the system~\\cite{Batsouli200326,Greco2004202}, and whether heavy quarks can hadronise also via recombination with other quarks from the medium~\\cite{Greco2004202,Andronic200336} are still open. \nThese questions are addressed by studying $R_{\\rm AA}$ at low and intermediate $p_{\\rm T}$ and measuring the azimuthal anisotropy of heavy-flavour hadron production with respect to the reaction plane, defined by the beam axis and the impact parameter of the collision. The hadronisation mechanisms of the c quark are also investigated through the measurement of  ${\\rm D}_{\\rm s}^+$ production in nucleus--nucleus collisions compared to that in pp collisions~\\cite{Anastasia}.\n\n\\section{D-meson reconstruction}\n\nThe decays ${\\rm D^0}\\rightarrow {\\rm K^-}\\pi^{+}$, ${\\rm D^+}\\rightarrow {\\rm K^-}\\pi^{+}\\pi^{+}$ and ${\\rm D^{*+}}\\rightarrow {\\rm D^0}\\pi^{+}$, and their charge conjugates, were reconstructed at mid-rapidity ($|y|<0.8$) in minimum-bias Pb--Pb collisions using the ALICE central barrel detectors. The D-meson selection was based on the precise reconstruction of the primary and secondary (decay) vertices, which is provided by the Inner Tracking System (ITS).\nCharged pions and kaons were identified using the information provided by the Time Projection Chamber (TPC) and the Time Of Flight (TOF) detectors~\\cite{Abelev:2014ffa}.\nThe reference proton--proton cross section at $\\sqrt{s_{\\rm NN}}=2.76$~TeV, needed to compute $R_{\\rm AA}$, was obtained by a pQCD-based energy scaling of the $p_{\\rm T}$-differential cross section measured at $\\sqrt{s}=7$~TeV~\\cite{ALICE:2011aa}. \n\n\\section{D-meson nuclear modification factor}\n\n\\begin{figure}[!t]\n\\begin{minipage}[t]{0.48\\textwidth}\n\\includegraphics[height=0.92\\textwidth]{2014-May-15-pPbAndPbPb.eps}\n\\caption{\\label{fig:DmesonRAARpA}Average $R_{\\rm pPb}$ of prompt ${\\rm D^0}$, ${\\rm D^+}$ and ${\\rm D^{*+}}$ mesons~\\cite{Abelev:2014hha} compared to the prompt D-meson $R_{\\rm AA}$ in Pb--Pb collisions in the 0--20\\% and 40--80\\% centrality classes~\\cite{ALICE:2012ab}.}\n\\end{minipage}\\hspace{1pc\n\\begin{minipage}[t]{0.48\\textwidth}\n\\includegraphics[height=0.9\\textwidth]{2014-May-16-AverageDMesonsRaa_075_ComparisonWithModels_150514.eps}\n\\caption{\\label{fig:DmesonRAACentral}Average prompt D-meson $R_{\\rm AA}$ in Pb--Pb collisions in the 0--7.5\\% centrality class compared to theoretical models including parton energy loss.}\n\\end{minipage} \n\\end{figure}\n\nA large suppression of the D-meson $R_{\\rm AA}$ (factor 3-5) was observed for $p_{\\rm T}>5$~GeV\/$c$ in central Pb--Pb collisions at $\\sqrt{s_{\\rm NN}}=2.76~{\\rm TeV}$ (Figure~\\ref{fig:DmesonRAARpA})~\\cite{ALICE:2012ab}. The comparison of $R_{\\rm AA}$ with the D-meson nuclear modification factor measured in p--Pb collisions at $\\sqrt{s_{\\rm NN}}=5.02~{\\rm TeV}$~\\cite{Abelev:2014hha} shows that the expected cold nuclear matter effects are smaller than the uncertainties on $R_{\\rm AA}$ for $p_{\\rm T}>3~{\\rm GeV\/}c$. Therefore, the suppression observed in central Pb--Pb collisions is predominantly induced by final-state effects due to the presence of a hot and dense partonic medium. Figure~\\ref{fig:DmesonRAACentral} shows the D-meson $R_{\\rm AA}$ measured in Pb--Pb collisions in the centrality class 0--7.5\\%, compared to theoretical models including charm interactions with the medium constituents.\nThe large suppression observed, e.g. of a factor 6 at $p_{\\rm T}=10~{\\rm GeV\/}c$, is described by the models that include radiative and collisional heavy-quark energy loss.\n\n\\begin{figure}[!t]\n\\begin{minipage}[t]{0.48\\textwidth}\n\\includegraphics[height=0.93\\textwidth]{2015-Jun-25-DmesPions_8to16_CompDjordjevic_200515.eps}\n\\caption{\\label{fig:DmesonPionRAA}$R_{\\rm AA}$ of D mesons~\\cite{Adam:2015nna} and charged pions~\\cite{Abelev2014196} as a function of centrality compared to a pQCD model including mass dependent radiative and collisional energy loss~\\cite{Djordjevic2014298}.}\n\\end{minipage}\\hspace{1pc\n\\begin{minipage}[t]{0.48\\textwidth}\n\\includegraphics[height=0.93\\textwidth]{2015-Jun-26-DmesNonPromptJpsi_8to16_CompDjordjevic_110615.eps}\n\\caption{\\label{fig:DmesonJPsiRAA}D~\\cite{Adam:2015nna} and non-prompt J\/$\\psi$ meson~\\cite{CMSnonprompt} $R_{\\rm AA}$ vs. centrality compared to a pQCD model including mass dependent radiative and collisional energy loss~\\cite{Djordjevic2014298}.}\n\\end{minipage} \n\\end{figure}\n\nFigures~\\ref{fig:DmesonPionRAA} and~\\ref{fig:DmesonJPsiRAA} show the D-meson $R_{\\rm AA}$ as a function of centrality (quantified in terms of the average number of participant nucleons in the Pb--Pb collision)~\\cite{Adam:2015nna} along with the $R_{\\rm AA}$ of charged pions~\\cite{Abelev2014196} and non-prompt J\/$\\psi$ mesons measured by the CMS Collaboration~\\cite{CMSnonprompt}, respectively. The focus here is on the study of the parton energy loss, thus, the results are presented for the high-$p_{\\rm T}$ interval $8-16~{\\rm GeV\/}c$ for the pions and D mesons and for $6.5<p_{\\rm T}<30~{\\rm GeV\/}c$ for J\/$\\psi$ from B-meson decays. The $R_{\\rm AA}$ of charged pions and D mesons are compatible within uncertainties in all centrality classes. The average difference of the $R_{\\rm AA}$ values of D mesons and non-prompt J\/$\\psi$ in the 0--10\\% and 10--20\\% centrality classes is larger than zero with a significance of 3.5~$\\sigma$. A pQCD model including mass-dependent radiative and collisional energy loss~\\cite{Djordjevic2014298} describes well the similarity of the D-meson and charged-pion $R_{\\rm AA}$.\nIn this model, the colour-charge dependence of energy loss is compensated by the softer fragmentation and $p_{\\rm T}$ spectrum of gluons with respect to those of c quarks leading to a very similar $R_{\\rm AA}$ for D mesons and pions. This calculation results in a larger suppression of D mesons with respect to non-prompt J\/$\\psi$, in agreement with the data. The large difference in the two $R_{\\rm AA}$ derives predominantly from the quark-mass dependence of parton energy loss, as demonstrated by the non-prompt J\/$\\psi$ $R_{\\rm AA}$ calculated considering the c-quark mass in the energy loss of b quarks (dotted lines in Figure~\\ref{fig:DmesonJPsiRAA})~\\cite{Djordjevic2014298,PhysRevC.89.014905,He2013409}.\n\n\\section{D-meson azimuthal anisotropy}\n\n\\begin{figure}[!t]\n\\begin{minipage}[t]{0.49\\textwidth}\n\\includegraphics[height=0.81\\textwidth]{2014-May-09-v2AverDandModels-HalfTPCEP.eps}\n\\caption{\\label{fig:Dmesonv2}Model comparison for the average D-meson $v_2$ in the 30--50\\% centrality class~\\cite{Abelev:2013lca,Abelev:2014ipa}.}\n\\end{minipage}\\hspace{1pc\n\\begin{minipage}[t]{0.48\\textwidth}\n\\centering\n\\includegraphics[height=0.85\\textwidth]{2014-May-09-D0RaavsEP_080514_paper.eps}\n\\caption{\\label{fig:DmesonRAAvsEP}${\\rm D^0}$ $R_{\\rm AA}$ in the 30--50\\% centrality class in two $90^{\\circ}$-wide orthogonal azimuthal intervals~\\cite{Abelev:2014ipa}.}\n\\end{minipage} \n\\end{figure}\n\n$v_2$ is defined as the second coefficient of the Fourier expansion of the $p_{\\rm T}$-dependent particle distribution ${\\rm d}^2N\/{\\rm d}p_{\\rm T}{\\rm d}\\varphi$ in azimuthal angle $\\varphi$ relative to the reaction plane. It quantifies the momentum azimuthal anisotropy of final-state D mesons that might result from the spatial asymmetry of the initial parton distribution in non-central collisions, as a consequence of the collective expansion of the QGP. Figure~\\ref{fig:Dmesonv2} shows the D-meson $v_2$ in Pb--Pb collisions in the 30--50\\% centrality class~\\cite{Abelev:2013lca,Abelev:2014ipa} compared to theoretical models including charm-quark interactions with the medium constituents.\nThe value of $v_2$ is larger than zero in the interval $2<p_{\\rm T}<6$~GeV\/$c$ with a significance of about 5~$\\sigma$ and it is best described by models including mechanisms that transfer to the charm quarks the elliptic flow of the medium during the system expansion (namely collisional processes and hadronisation by recombination with light quarks). A smaller suppression is observed in the in-plane direction (Figure~\\ref{fig:DmesonRAAvsEP})~\\cite{Abelev:2014ipa}; the ordering $R_{\\rm AA}^{\\rm out-of-plane}<R_{\\rm AA}^{\\rm in-plane}$ is equivalent to the observation of $v_2>0$ and, thus, consistent with the expectations from collective flow.\n\n\\section{Conclusions}\nThe results obtained by ALICE using the data from the LHC Run-1 (2010--2013) indicate a strong suppression of the D-meson production in central Pb--Pb collisions for $p_{\\rm T}>3~{\\rm GeV\/}c$, which is mainly due the interactions of heavy quarks with the hot and dense medium. \nThe smaller $R_{\\rm AA}$ observed for D mesons with respect to non-prompt J\/$\\psi$ confirms the mass-dependent nature of the energy-loss mechanisms.\nThe non-zero $v_2$ of D mesons and the azimuthal dependence of the ${\\rm D^0}$ $R_{\\rm AA}$ indicate that, during the collective expansion of the medium, the interactions between its constituents and the charm quarks transfer to the latter information on the azimuthal anisotropy of the system. During the LHC Run-2 we expect to collect a data sample larger by a factor 5-10 with respect to Run-1, depending on collision centrality. It will be, thus, possible to measure the D-meson $R_{\\rm AA}$ and $v_2$ with a better precision and in an extended $p_{\\rm T}$ range.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:intro}\n\nMassive MIMO is one of the most relevant technologies in wireless communications \\cite{marzetta,rusek}. Among the key features of this technology are high spectral efficiency and improved link reliability, making it a key enabler for 5G. Massive MIMO exploits spatial diversity far beyond traditional MIMO systems by employing a large scale antenna array in the base-station (BS) with hundreds or possibly even thousands of elements. This large number of elements allows for unprecedented spatial resolution and high spectral efficiency, while providing simultaneous service to several users within the same time-frequency resource.\n\nDespite all the advantages of Massive MIMO, there are still challenges from an implementation point of view. One of the most critical ones is sending data from the BS antennas to the central processing unit (CPU) and vice-versa, and the high interconnection throughput it requires. In current set-ups, uplink detection algorithms based on zero-forcing (ZF) equalizer typically rely on a centralized architecture, shown in Fig. \\ref{fig:BS_centralized}, where baseband samples are collected in the CPU for obtaining channel state information (CSI) and further matrix inversion, which allows data estimation and further detection. The same argument is valid for downlink precoding. In order to avoid dedicated links between antenna modules and CPU, a shared bus is typically used to exchange this data. In case of LuMaMi testbed \\cite{lumami,lumami2}, the shared bus was reported to support an aggregated data-rate of 384Gps, which exceed base-station internal interface standards such as eCPRI \\cite{ecpri}. Additionally, the pin-count of integrated circuits (IC) limits the number of links the IC can handle simultaneously and thus the throughput. Due to this high data-rate, the power appears as another potential limitation. This combination of factors are considered as the main bottleneck in the system and a clear limitation for array scalability. In this paper we will address the inter-connection throughput limitation by decreasing its value per link and consequently reducing the impact of the other two (pin-count and power).\n\nThe inter-connection bottleneck has been noted in several previous studies on different architectures for Massive MIMO BSs \\cite{argos,Bertilsson,puglielli,lumami,cavallaro,li_jeon,jeon_li}. As a solution, most of these studies recommend moving to a decentralized approach where uplink estimation and downlink precoding can be performed locally in processing nodes close to the antennas (final detection can still be done in a CPU). However, to achieve that, CSI still needs to be collected in the CPU, where matrix inversion is performed \\cite{argos,Bertilsson,lumami}, imposing an overhead in data shuffling.\n\nThe CSI problem is addressed in \\cite{cavallaro}, where CSI is obtained and used only locally (not shared) for precoding and estimation, with performance close to MMSE. However, this architecture relies on the CPU for exchanging a certain amount of consensus information between the nodes, and this exchange negatively impacts the processing latency and throughput \\cite{li_jeon}, and therefore limits the scalability of this solution. In order to solve these problems, feedforward architectures for detection \\cite{jeon_li} and precoding \\cite{li_jeon} have been proposed recently, where the authors present a partially decentralized (PD) architecture for detection and precoding, which achieves the same results as linear methods (MRC, ZF, L-MMSE), and therefore becomes optimal when $M$ is large enough. Partial Gramian matrices from antennas are added up before arriving to a processing unit where the Gramian is inverted.\n\nIn \\cite{argos}, a flat-tree structure with daisy-chained nodes was presented. The authors propose conjugate beamforming as a fully decentralized method with the corresponding penalty in system capacity. In the same work it is also pointed out that by following this topology the latency was being severely compromised. The more detailed analysis on latency is thus needed to evaluate the algorithm.\n\nIn this article we propose a fully decentralized architecture and a recursive algorithm for Massive MIMO detection and precoding, which is able to achieve very low inter-connection data-rate without compromising latency.\nThe proposed algorithm is pipelined so that it runs in a distributed way at the antenna processing units, providing local vectors for estimation\/detection that approximate to the zero-forcing solution.\nWe make use of the Coordinate Descent (CD) algorithm, which is detailed in Section \\ref{section:CD}, to compute these vectors.\n\nThere is previous work based on CD, such as \\cite{li_CD}. The main difference is that the coordinate update in \\cite{li_CD} is done per user basis, i.e., a different user index is updated every iteration, while in our proposed method the coordinate update is done per antenna basis, updating all users at once.\n\nWe extend the work presented in \\cite{jesus} and \\cite{muris}, which are also based on decentralized daisy-chain architecture. The novelties of the present work compared to these two is as follows:\n\\begin{itemize}\n\\item A common strategy for downlink precoding and uplink equalization is presented, in contrast to \\cite{jesus} and \\cite{muris}, which only covers uplink and downlink separately.\n\\item The algorithm has been modified that serial processing is only needed when new CSIs are estimated. The corresponding filtering phase can be conducted in parallel to reduce latency, in contrast to \\cite{jesus}, where serial processing is always needed, which increases the latency.\n\\item A recommended step-size is provided, in contrast to \\cite{jesus}.\n\\item An analytical expression for resulting SINR and a complete performance analysis is presented in this paper.\n\\item Complexity analysis from a general point of view (not attached to any specific implementation) is provided, which includes: inter-connection data-rate, memory size and latency. In \\cite{jesus}, only inter-connection data-rates are analyzed.\n\\end{itemize}\n\nDecentralized architectures, as shown in Fig. \\ref{fig:BS_decentralized}, have several advantages compared to the centralized counterpart, as shown in Fig. \\ref{fig:BS_centralized}. For example, they overcome bottlenecks by finding a more equal distribution of the system requirements among the processing nodes of the system. Apart from this, data localization is a key characteristic of decentralized architectures. In uplink, the architecture allows data to be consumed as close as possible to where it is generated, minimizing the amount to transfer, and therefore saving throughput and energy. To achieve data localization, processing nodes need to be located near the antenna, where they perform processing tasks locally such as channel and data estimation. Local CSI is estimated and stored locally in each, without any need to share it with any other nodes in the system. This approach has been suggested previously in \\cite{argos,Bertilsson,jeon_li,cavallaro,li_jeon,puglielli}, and we take advantage of it in the proposed solution.\n\nThe remainder of the paper is organized as follows. In Section \\ref{section:background} the preliminaries are presented, comprising the system model for uplink and downlink, together with an introduction to linear processing and the ZF method. Section \\ref{section:central_vs_decentral} is dedicated to a comparison between the centralized and decentralized architectures and reasoning why the latter one is needed, together with an overview of the daisy-chain topology. The proposed algorithm, based on CD, is presented in Section \\ref{section:CD}. In \\ref{section:analysis} closed-form expressions of the SIR and SINR are provided for this algorithm, together with  interconnection data-rates, latency and memory requirements of the proposed solution. Finally, Section \\ref{section:conclusions} summarizes the conclusions of this publication. \n\nNotation: In this paper, lowercase, bold lowercase and upper bold face\nletters stand for scalar, column vector and matrix, respectively. The\noperations $(.)^T$, $(.)^*$ and $(.)^H$ denote transpose, conjugate and conjugate transpose respectively.\nThe $i$-th element of vector $\\h$ is denoted as $h_{i}$. A vector $\\w$ and a matrix $\\A$ related to the $m$th antenna is denoted by $\\w_m$ and $\\A_{m}$, respectively. $A_{i,j}$ denotes element $(i,j)$ of $\\A$. $\\mathbf{A}_{m}(i,j)$ denotes element $(i,j)$ of the $m$-th matrix in the sequence $\\{\\A_{m}\\}$. The $k$th coordinate vector in $\\mathbb{R}^{K}$ is defined as $\\e_{k}$. Kronecker delta is represented as $\\delta_{ij}$. Probability density function and cumulative density function are denoted respectively as $f_{\\mathbf{X}}(x)$ and $F_{\\mathbf{X}}(x)$. Computational complexity is measured in terms of the number of complex-valued multiplications.\\\\\n\n\\section{Background}\n\\label{section:background}\n\\subsection{System model}\nFor uplink, we consider a scenario with $K$ single-antenna users transmitting to a BS with an antenna array with $M$ elements. Assuming time-frequency-based channel access, a Resource\nElement (RE) represents a unit in the time-frequency grid (also\nnamed subcarrier in OFDM) where the channel is expected to be approximately flat. Under this scenario, the input-output relation is\n\\begin{equation}\n\\yu = \\Hbf\\xu + \\nup,\n\\label{eq:ul_model}\n\\end{equation}\nwhere $\\yu$ is the $M \\times 1$ received vector, $\\xu$ is the transmitted user data vector ($K \\times 1$), $\\Hbf=[\\h_1 \\; \\h_2 \\, \\cdots \\, \\h_M]^{{T}}$ is the channel matrix ($M \\times K$) and $\\nup$ an $M \\times 1$ vector of white, zero-mean complex Gaussian noise. The entries of $\\Hbf$ are i.i.d. zero-mean circularly-symmetric complex-gaussian entries, with rows $\\h_{i} \\sim \\mathcal{CN}(0, \\I)$ for all $i$. The noise covariance at the receiver is  $N_{0}\\I$. The average transmitted power is assumed to be equal across all users and we assume, without any loss of generality, a unit transmit power. SNR is defined as $\\frac{1}{N_{0}}$ and represents the average \"transmit\" signal-to-noise ratio.\n\nFor downlink, if Time Division Duplex (TDD) is assumed, then according to channel reciprocity principle and by employing reciprocity calibration techniques \\cite{joao}, it is assumed that within the same coherence time, the channel matrix is the same as in the uplink case, and the system model follows\n\\begin{equation}\n\\xdd = \\Hbf^{T}\\yd + \\nd,\n\\label{eq:dl_model}\n\\end{equation}\nfor a RE, where $\\yd$ is the $M \\times 1$ transmitted vector, $\\xdd$ is the received data vector by users ($K \\times 1$), and $\\nd$ samples of noise ($K \\times 1$).\n\nOnce the system model is established, we introduce the linear processing fundamentals used for downlink precoding and uplink estimation.\n\n\\begin{figure*}\\centering\n\t\\footnotesize\n\t\\subfloat[Centralized architecture]{\n\t\t\\psfrag{1}{$1$}\n\t\t\\psfrag{M}{$M$}\n\t\t\\psfrag{RPU}[][][0.7]{$\\mathrm{RPU}$}\n\t\t\\psfrag{RF}[][][0.7]{$\\mathrm{RF}$}\n\t\t\\psfrag{OFDM}[][][0.55]{$\\mathrm{OFDM}$}\n\t\t\\psfrag{CPU}{$\\mathrm{CPU}$}\n\t\t\\psfrag{CHEST}[][][0.6]{$\\mathrm{CHEST}$}\n\t\t\\psfrag{EST}[][][0.6]{$\\mathrm{EST}$}\n\t\t\\psfrag{DET}[][][0.6]{$\\mathrm{DET}$}\n\t\t\\psfrag{DEC}[][][0.6]{$\\mathrm{DEC}$}\n\t\t\\psfrag{Bs}[][][1.0]{$\\text{Base Station}$}\n\t\t\\psfrag{Rc}{$R_\\mathrm{c}$}\t\n\t\t\\includegraphics[width=0.35\\textwidth]{BS_centralized.eps}\n\t\t\\label{fig:BS_centralized}\n\t}\n\t\\subfloat[Decentralized architecture]{\n\t\t\\psfrag{1}{$1$}\n\t\t\\psfrag{M}{$M$}\n\t\t\\psfrag{RPU}[][][0.7]{$\\mathrm{RPU}$}\n\t\t\\psfrag{RF}[][][0.7]{$\\mathrm{RF}$}\n\t\t\\psfrag{OFDM}[][][0.55]{$\\mathrm{OFDM}$}\n\t\t\\psfrag{CPU}{$\\mathrm{CPU}$}\n\t\t\\psfrag{CHEST}[][][0.5]{$\\mathrm{CHEST}$}\n\t\t\\psfrag{EST}[][][0.6]{$\\mathrm{EST}$}\n\t\t\\psfrag{DET}[][][0.6]{$\\mathrm{DET}$}\n\t\t\\psfrag{DEC}[][][0.6]{$\\mathrm{DEC}$}\n\t\t\\psfrag{Bs}{$\\text{Base Station}$}\n\t\t\\psfrag{Rd}{$R_\\mathrm{d}$}\n\t\t\\includegraphics[width=0.35\\textwidth]{BS_decentralized.eps}\n\t\t\\label{fig:BS_decentralized}\n\t}\n\t\n\t\\caption{Comparison between base station receiver chain in centralized and fully decentralized architectures for Massive MIMO uplink. Antenna array with $M$ elements is divided into RPUs, each containing a set of antennas. (a): Centralized architecture. Each RPU has one link to transfer baseband samples to the CPU, where the rest of processing tasks are done. (b): Fully decentralized architecture for detection. Each RPU performs RF, ADC, OFDM, channel estimation (CHEST) and data estimation (EST) locally. Detection (DET) and decoding (DEC) is centralized. RPUs are connected to each other by uni-directional links. Only one RPU has a direct connection with the CPU. Proposed algorithms are executed in EST blocks in parallel mode. The points where the interconnection data-rate is estimated are marked by circles and the value is denoted by $\\mathrm{R}_{c}$ and $\\mathrm{R}_{d}$ for centralized and decentralized respectively. The goal is to have  $\\mathrm{R}_{d} \\ll \\mathrm{R}_{c}$ without compromising performance and latency.}\n\t\\label{fig:comparison}\n\\end{figure*}\n\n\\subsection{Linear Processing}\nIn this article we focus on linear estimators and precoders, because they show close to optimal performance in Massive MIMO regime while requiring low complexity.\n\nA linear estimator provides $\\hatx^u$, which is an estimate of $\\xu$, by applying an equalizer filter matrix $\\Wbf$ to the vector of observations, $\\yu$:\n\\begin{equation}\n\\begin{split}\n\\hatxu &= \\Wbf^{H} \\yu\\\\\n&= \\sum_{m=1}^{M} \\w_{m}^{*} \\ymu,\\\\\n\\end{split}\n\\label{eq:linear_det}\n\\end{equation}\nwhere $\\Wbf = [\\w_{1} \\; \\w_{2} \\, \\cdots \\, \\w_{M}]^{T}$ is an $M \\times K$ matrix, $\\w_{m}$ is a $K \\times 1$ filter vector related to antenna $m$ and $\\ymu$ the observation at antenna $m$. As it can be seen the estimate $\\hatxu$ is computed by the sum of $M$ partial products. If $\\w_{m}$ is obtained and stored locally in the m$th$ antenna module, then the partial products can be computed with local data only, reducing the amount of data to exchange between nodes. From implementation point of view, the linear estimator relies on the accumulation of all partial results according to \\eqref{eq:linear_det}, which can be done centrally (fusion node) or distributed.\n\nFor downlink, the data vector intended to the users, $\\xd$, is precoded with matrix $\\Pbf$ as\n\\begin{equation}\n\\yd = \\Pbf\\xd,\\\\\n\\label{eq:linear_prec}\n\\end{equation}\nwhere $\\Pbf = [\\p_{1} \\; \\p_{2} \\, \\cdots \\, \\p_M]^{T}$ is an $M \\times K$ matrix, which fulfills a power constraint $\\|\\Pbf\\|_{F}^{2}\\leq P$, such that $P$ is the maximum transmitted power. Particularly for antenna $m$ we have\n\\begin{equation}\n\\ymd = \\p_{m}^T \\xd.\\\\\n\\label{eq:linear_prec_i}\n\\end{equation}\nSimilarly to uplink, if $\\p_{m}$ is obtained and stored locally at the m$th$ antenna module, then $\\ymd$ can be computed only with local data after $\\xd$ is broadcasted to all antennas.\n\nThe zero-forcing (ZF) equalizer, which is one type of linear estimator, constitutes a reference in our analysis. It is defined for uplink estimation as\n\\begin{equation}\n\\Wbf_\\text{ZF}^{H} = (\\Hbf^H \\Hbf)^{-1}\\Hbf^H,\n\\label{eq:W_ZF}\n\\end{equation}\nand $\\Pbf_\\text{ZF}=\\Wbf_\\text{ZF}^{*}$ for the downlink precoding.\n\nZF is able to completely cancel inter-user interference (IUI) and reach the promised spectral efficiency of Massive MIMO. However, as ZF is performed in a central processor, the Gramian matrix $\\Hbf^{H}\\Hbf$ needs to be collected and inverted, which increases the average inter-connection data-rate. The computational load is also increased due to the matrix inversion and posterior matrix multiplication during estimation phase. Taking this into consideration, we look for methods with IUI-cancellation capabilities but with lower requirements for the system.\n\n\\subsection{Uplink \\& Downlink reciprocity}\nSubstituting \\eqref{eq:ul_model} into \\eqref{eq:linear_det} leads to\n\\begin{equation}\n\\begin{split}\n\\hatxu\n&= \\Eu \\xu + \\zu\\\\\n\\end{split}\n\\label{eq:Eu}\n\\end{equation}\nfor uplink, where $\\Eu = \\Wbf^{H} \\Hbf$ is a $K \\times K$ matrix containing the equivalent uplink channel with IUI information and $\\mathbf{z}^u$ is the $K \\times 1$ post-equalization noise term.\n\nOn the other hand, in the downlink, substituting \\eqref{eq:linear_prec} into \\eqref{eq:dl_model} leads to\n\\begin{equation}\n\\begin{split}\n\\xdd\n&= \\Ed \\xd + \\nd,\\\\\n\\end{split}\n\\label{eq:Ed}\n\\end{equation}\nwhere $\\Ed = \\Hbf^{T} \\Pbf$ is a $K \\times K$ matrix containing the equivalent downlink channel with IUI information. For the particular case that $\\Pbf^{T} = \\Wbf^{H}$, we have $\\Ed = \\Eu^{T}$, meaning that both equivalent channels are transposed, and therefore experiment the same IUI cancellation properties.\nFrom this result it is clear that once an equalization matrix $\\Wbf$ is obtained for uplink detection, it can also be applied for downlink precoding with no extra effort. It is interesting to note that, since $\\Pbf^{T} = \\Wbf^{H}$, it follows that $\\p_i = \\w_i^{*}$, so each antenna node can re-use same vector for detection and precoding, ideally reducing complexity and storage needs by half. Said that, in this article we focus mainly on uplink estimation without limiting the results to downlink. In reality, there is a downlink power constraint as the total transmitted power, which is addressed in \\ref{section:analysis}.\n\n\\section{Centralized vs Decentralized}\n\\label{section:central_vs_decentral}\nIn this section we describe the differences between centralized and decentralized Massive MIMO processing and the justification to study the later one.\n\nUplink estimation based on ZF equalization has two components that should be multiplied: $\\Wbf_\\text{ZF}$  and $\\yu$. The former includes a $K \\times K$ matrix inversion, which typically is done in one place, and for that, CSI from all antennas needs to be collected. Apart from that, the observation data vector, $\\yu$, is also needed for estimation. This vector is $M \\times 1$, increasing considerably the amount of data to transfer and limiting the scalability of the array. Based on those considerations, we can think of two possible architectures for the Massive MIMO base-station: centralized and decentralized.\n\nFig. \\ref{fig:BS_centralized} presents an architecture based on a central baseband processing node, where baseband samples are exchanged between Remote Processing Units (RPUs) and CPU. Each antenna is connected to a receiver and transmitter circuitry, which involves: RF front-end, ADC\/DAC and OFDM processing. For simplicity, only uplink is represented in this figure. We can identify some common tasks among these processing elements across different antennas, such as: time synchronization, automatic gain control, local oscillator generation, carrier frequency and sampling rate offset estimation, phase noise compensation, among others. Therefore, a few antennas (together with corresponding receivers\/transmitters) can be grouped into one RPU for efficient implementation of such common tasks. However, for simplicity, in this work we only analyze the case where each RPU manages one antenna.\n\nDedicated physical links would easily exceed the number of I\/O connections in current standards, in addition to the increment of the cost of adding a new RPUs when needed. To overcome this, we consider that RPUs are connected to the CPU node by a shared bus as shown in Fig. \\ref{fig:BS_centralized}. \n\nEven though, this approach can support ZF detection (and precoding) from a functionality point of view, from the implementation point of view, it requires a very high inter-connection data-rate in the bus and at the input of the CPU ($R_\\mathrm{c}$ in the figure). As an example, consider a 5G NR-based system with 128 antennas and OFDM as an access technology, then the average data-rate can be calculated as\n\\begin{equation}\nR_{\\mathrm{c}} = \\frac{2w M N_{\\mathrm{u}}}{T_{\\mathrm{OFDM}}},\n\\label{eq:R_central}\n\\end{equation}\nwhere $N_{\\mathrm{u}}$ is the number of active subcarriers, $w$ is the bit-width for the baseband samples (real\/imaginary parts) after FFT, and $T_{\\mathrm{OFDM}}$ is the OFDM symbol duration. For $N_{\\mathrm{u}}=3300$, $w=12$ and $T_{\\mathrm{OFDM}}=1\/120\\mathrm{kHz}$ then $R_{\\mathrm{c}}=1.2 \\mathrm{Tbps}$. This result clearly exceed the limit data-rate for common interfaces, such as eCPRI \\cite{ecpri} and PCIe, and furthermore, it is proportional to $M$, which clearly limits the scalability of the system.\n\nAs a solution to this limitation, we propose the fully-decentralized architecture for baseband detection and precoding shown in Figure \\ref{fig:BS_decentralized}. We can observe that channel estimation and estimation\/precoding have been moved from the CPU to the RPUs, with detection and decoding as a remaining task in the CPU from physical layer point of view. The benefit of this move is manifold. Firstly, the inter-connection data-rate scales with $K$ instead of $M$. Secondly, the high complexity requirement in the CPU for channel estimation and data estimation\/precoding is now equally distributed among RPUs, which highly simplifies the implementation and overcomes the computational bottleneck and, additionally, CSI is obtained and consumed locally in each RPU without the need for exchange, with the consequent reduction in the required inter-connection data-rate. In addition to the advantages already mentioned, which are common to other decentralized schemes, the proposed architecture presented in this work achieves an unprecedented low inter-connection data-rate by the direct connection of RPUs forming a daisy-chain, where the CPU is at one of the ends.\n\nIn the daisy-chain, depicted in Fig. \\ref{fig:BS_decentralized}, nodes are connected serially to each other by a dedicated connection. All elements in the chain work simultaneously in pipeline mode, processing and transmitting\/receiving to\/from the respective next\/previous neighbor in the chain. The data is passed through the nodes sequentially, being updated at every RPU. There is an unique connection to the root node where the last estimate is transmitted and therefore been detected by the CPU. An important remark is the average inter-connection data-rate between nodes is the same regardless of the number of elements in the chain. This topology was proposed in \\cite{argos} and further studied in \\cite{jesus} and \\cite{muris} with specific algorithms designed for this topology.\n\nWhen the decentralized architecture in Fig. \\ref{fig:BS_decentralized} needs to be deployed, antennas can be collocated in the same physical place or distributed over a large area. These antennas and therefore their corresponding RPUs can behave as nodes in the chain, whilst the CPU remains as the root node. There may be multiple chains in a network. The selection of the RPUs to form a chain may depend on the users they are serving. RPUs which serve the same set of users should be in the same chain, so they can work jointly to cancel IUI. This concept fits very well with the distributed wireless communication system \\cite{DWCS}, the recent cell-free Massive MIMO concept \\cite{cell-free} and the promising large intelligent surface \\cite{lis}.\n\nDecentralized architectures, such as the one shown in Fig. \\ref{fig:BS_decentralized}, require other type of algorithms compared to Fig. \\ref{fig:BS_centralized}. In the next section we introduce our proposed algorithm, which is a method for obtaining $\\w_{m}$ and $\\p_{m}$ as the equalization and precoding vectors, respectively.\n\n\\section{Coordinate Descent}\n\\label{section:CD}\n\nOur proposed algorithm is an iterative algorithm based on the gradient descent (GD) optimization, in which the gradient information is approximated with a set of observations in every step. From this, each antenna can obtain its own equalization\/precoding vector sequentially in a coordinate descent approach. The main advantage of this method is that it does not require access to all observations at each iteration, becoming an ideal choice for large scale distributed systems.\n\n\\subsection{Preliminaries}\nFrom \\eqref{eq:Eu} we know that in the non-IUI case, $\\Eu$ is a diagonal matrix, which is the case when zero-forcing (ZF) is applied. In the general case, IUI is not zero and as consequence $\\Eu$ contains non-zero entries outside the main diagonal.\n\nThe objective is to find a matrix $\\Wbf$, which cancels IUI to a high extent ($\\Eu \\approx \\I$), while fulfilling the following conditions:\n\\begin{itemize}\n\t\\item Uses daisy-chain as a base topology, so we exploit the advantages seen in Section \\ref{section:central_vs_decentral}.\n\t\\item No exchange of CSI between nodes. Only local CSI. \n\t\\item Limited amount of data to pass between antenna nodes. It should depend on $K$ instead of $M$, to enable scalability.\n\t\\item Limit the dependency on the central processing unit in order to reduce data transfer, processing and memory requirements of that unit. One consequence of this is to avoid matrix inversion in the central unit.\n\\end{itemize}\n\n\\subsection{Algorithm formulation}\nThe algorithm setup is that one intends to solve the unconstrained Least Squares (LS) problem in the uplink\n\\begin{equation} \\label{eq:CD_R} \\hatx = \\argmin_{\\mathbf{x}} \\|\\mathbf{y}-\\mathbf{H}\\mathbf{x}\\|^2\n\\end{equation}\nvia a GD approach. The gradient of \\eqref{eq:CD_R} equals $\\nabla_{\\mathbf{x}}=\\Hbf^{H}\\Hbf\\mathbf{x}-\\Hbf^{H}\\mathbf{y}$.\nEven though $\\Hbf^{H}\\Hbf$ and $\\Hbf^{H}\\mathbf{y}$ can be formulated in a decentralized way, the selection of $\\mathbf{x}$ and the product with $\\Hbf^{H}\\Hbf$ is preferably done in a central processing unit to limit latency and inter-connection data-rates. Following the fully-decentralized approach and the intention to off-load the equalization\/precoding computation from the CPU to the RPUs, we propose a different approach.\n\nThe proposed method can be derived as an approximate version of GD that can be operated in a decentralized architecture with minimum CPU intervention. It does so by computing, at each antenna, as much as possible of $\\nabla_{\\mathbf{x}}$ with the information available at the antenna. Then the estimate $\\hatx$ is updated by using a scaled version of the \"local\" gradient and the antenna passes the updated estimate on to the next antenna.\n\nThe above described procedure can, formally, be stated as\n\\begin{equation}\n\\begin{split}\n\\varepsilon_m &= y_{m} - \\h_{m}^{T} \\hatx_{m-1} \\\\\n\\hatx_{m} &= \\hatx_{m-1} + \\mu_m \\h_{m}^{*} \\varepsilon_m,\n\\end{split}\n\\label{eq:CD_sm}\n\\end{equation}\nfor antenna $m$, where $\\mu_m$ is a scalar step-size. The update rule in \\eqref{eq:CD_sm} corresponds to the Kaczmarz method \\cite{kaczmarz}, whose step-size is according to \\cite{censor}\n\\begin{equation}\n\\mu_{m} = \\frac{\\mu}{\\|\\h_{m}\\|^2},\n\\label{eq:mu_m}\n\\end{equation}\nwhere $\\mu \\in \\mathbb{R}$ is a relaxation parameter. In case of consistent systems, this is $\\mathbf{y}=\\Hbf \\mathbf{x}$ (if SNR is high enough or there is no noise), $\\mu=1$ is optimum and the method converge to the unique solution. Otherwise, when the system is inconsistent, $\\mu$ give us an extra degree of freedom, which allows to outperform the $\\mu=1$ case, as we will see in Section \\ref{section:analysis}.\n\nAfter $M$ iterations of \\eqref{eq:CD_sm} we have\n\\begin{equation}\n\\begin{split}\n\\hatx_{M} &= \\prod_{m=1}^{M} \\left( \\I_K - \\mu_{m} \\h_{m}^{*} \\h_{m}^{T} \\right) \\hatx_0 \\\\\n&+ \\sum_{m=1}^{M} \\prod_{i=m+1}^{M} \\left(\\I_K - \\mu_{i} \\h_{i}^{*} \\h_{i}^{T} \\right) \\mu_{m} \\h_{m}^{*} y_m.\n\\nonumber\n\\end{split}\n\\end{equation}\nIf we assume $\\hatx_0 = \\mathbf{0}_{K\\times1}$ \\footnote[1]{If prior information of $\\mathbf{x}$ is available, it can be used here.}, then it is possible to express $\\hatx_M$ as linear combination of $\\mathbf{y}$, in the same way as \\eqref{eq:linear_det}, and identify\n$\\w_m$ (the equalization vector associated to antenna $m$) as\n\\begin{equation}\n\\w_m = \\left[ \\prod_{i=m+1}^{M} \\left(\\I_K - \\mu_{i} \\h_{i} \\h_{i}^{H} \\right) \\right] \\mu_{m} \\h_{m}.\n\\label{eq:CD_W}\n\\end{equation}\nIf \\eqref{eq:CD_sm} is applied in reverse antenna order ($m=M \\cdots 1$), then we obtain a different estimation. The expression for $\\w_{m}$ when using the alternative approach is\n\\begin{equation}\n\\w_m = \\mu_{m} \\A_{m-1} \\h_{m},\n\\label{eq:CD_W2}\n\\end{equation}\nwhere matrix $\\A_m$ is defined as\n\\begin{equation}\n\\A_m = \\prod_{i=1}^{m} \\left(\\I_K - \\mu_{i} \\h_{i} \\h_{i}^{H} \\right).\n\\label{eq:CD_A_impl}\n\\end{equation}\n\nIt is important to remark that both approaches lead to different $\\w_{m}$ sequences, however the overall performance should be the same if CSI in all antennas shows same statistical properties (stationarity across antennas).\n\n\n\\subsection{Algorithm design and pseudocode}\n\\label{section:alg}\nIn this subsection we derive an equivalent and more attractive form for the calculation of the weights of the algorithm in \\eqref{eq:CD_W2} in an easy and low-complexity way, suitable for hardware implementation.\n\nThe algorithm description is shown in Algorithm \\ref{algo:CD}. The vector $\\w_{m}$ is computed in each antenna, while the matrix $\\A_{m-1}$ gets updated according to the recursive rule: $\\A_{m} = \\A_{m-1} - \\w_{m} \\h_{m}^{H}$. Then, $\\w_{m}$ is stored for the detection and precoding phase, and $\\A_{m}$ is passed to the next antenna node for further processing.\n\\IncMargin{1em}\n\\begin{algorithm}[ht]\n\t\\SetKwInOut{Input}{Input}\n\t\\SetKwInOut{Output}{Output}\n\t\\SetKwInOut{Preprocessing}{Preprocessing}\n\t\\SetKwInOut{Init}{Init}\n\t\\Input{ $\\Hbf = \\left[ \\h_{1}, \\h_{2} \\cdots \\h_{M} \\right]^{T}$}\n\t\\Preprocessing{}\n\t$\\A_0 = \\I_K$\\\\\n\t\\For{$m = 1,2,...,M$}{\n\t\t$\\w_m = \\mu_{m} \\A_{m-1} \\h_m$\\\\\n\t\t$\\A_m = \\A_{m-1} - \\w_{m} \\h_{m}^{H}$\n\t}\n\t\\caption{Proposed algorithm}\n\t\\label{algo:CD}\n\t\\Output{$\\Wbf = \\left[ \\w_{1}, \\w_{2} \\cdots \\w_{M} \\right]^{T}$}\n\t\n\\end{algorithm}\\DecMargin{1em}\n\nFrom Algorithm \\ref{algo:CD} we can observe that after $M$ steps we achieve the following expression: $\\A_M = \\I_{K} - \\Eu^{*}$. Then, if perfect IUI cancellation is achieved, $\\Eu=\\I_{K}$ and therefore $\\A_{M} = \\mathbf{0}$. As a consequence we can take $\\|\\A_{m}\\|^{2}$ as a metric for residual IUI. The interpretation of Algorithm \\ref{algo:CD} is as follows.  $\\|\\A_{m}\\|$ is reduced by subtracting from $\\A_{m}$ a rank-1 approximation to itself. In order to achieve that, $\\A_{m}$ is projected onto $\\h_{m}$ to obtain $\\w_{m}$, therefore $\\w_{m} \\h^{H}_m$ is the best rank-1 approximation to $\\A_{m}$, having $\\h_{m}$ as vector base. Ideally, if the channel is rich enough, vectors $\\h_{m}$ are weakly correlated and assuming $M$ is large (Massive MIMO scenario) then IUI can be canceled out to a high extent \\footnote[2]{The selection of Coordinate Descent as our method's name is because we consider the vectors $\\{\\w_i\\}$ as the outcome of the method, and these can be seen as coordinates of a cost function to minimize. Such optimization problem can be written as: $\\w_{m} = \\argmin_{z} f(\\w_{1},\\cdots,\\w_{m-1},\\mathbf{z},\\w_{m+1},\\cdots,\\w_{M})$, where $f = \\|\\A_{m-1} - \\mathbf{z} \\h_{m}^{H}\\|_{F}^{2}$, and $\\A_{m-1} = \\I_{K}-\\sum_{i \\neq m} \\w_{i} \\h_{i}^{H}$. Each antenna solves this optimization problem in a sequential fashion, obtaining one coordinate as a result, while keeping the rest fixed. This is valid for single and multiple iterations to the array, which is presented in the next subsection.}.\n\nThe role of step-size $\\mu$ is to control how much IUI is removed at every iteration. High values will tend to reduce IUI faster at the beginning when the amount to remove is high, but will lead to oscillating or unstable residual IUI after some iterations because the steps are too big, so the introduced error dominates. Low values for $\\mu$ will ensure convergence of the algorithm and a relatively good IUI cancellation at the expense of a slower convergence.\n\n\\subsection{Multiple-iterations along the array}\n\\label{sub:multiple-iter}\nRecalling from Section \\ref{section:alg}, Algorithm \\ref{algo:CD} reduces the norm of $\\A$ at each step, providing as a result $\\A_{M}$, which contains the residual IUI after the algorithm is run along the array. It is possible to expand the algorithm and apply $\\A_{M}$ as initial value, $\\A_{0}$ for a new iteration through the array, with the intention of decreasing even more the norm of $\\A$. The pseudocode of the expanded version is shown in Algorithm \\ref{algo:CD_multiple}, with $n_{iter}$ iterations, and as it can be seen, an increment of $\\w_{m}$ is computed at each iteration. From topology point of view, it requires an extra connection between last and first RPUs, closing the daisy-chain and becoming a ring. It is expected to improve the performance at the expense of increasing the latency.\n\\IncMargin{1em}\n\\begin{algorithm}[ht]\n\t\\SetKwInOut{Input}{Input}\n\t\\SetKwInOut{Output}{Output}\n\t\\SetKwInOut{Preprocessing}{Preprocessing}\n\t\\SetKwInOut{Init}{Init}\n\t\\Input{ $\\Hbf = \\left[ \\h_{1}, \\h_{2} \\cdots \\h_{M} \\right]^{T}$}\n\t\\Preprocessing{}\n\t$\\A_{0,1} = \\I_K$\\\\\n\t$\\w_{m,1} = \\mathbf{0},m=1,...,M$\\\\\n\t\\For{$n = 1,2,...,n_{iter}$}{\n\t\t\\For{$m = 1,2,...,M$}{\n\t\t\t$\\w_{m,n} = \\w_{m,n-1} + \\mu_{m} \\A_{m-1,n} \\h_m$\\\\\n\t\t\t$\\A_{m,n} = \\A_{m-1,n} - \\w_{m,n} \\h_{m}^{H}$\\\\\n\t\t}\n\t\t$\\A_{0,n+1} = \\A_{M,n}$\n\t}\n\t\\caption{Proposed algorithm multiple iterations}\n\t\\label{algo:CD_multiple}\n\t\\Output{$\\Wbf = \\left[ \\w_{1,n_{iter}}, \\w_{2,n_{iter}} \\cdots \\w_{M,n_{iter}} \\right]^{T}$}\n\t\n\t\\end{algorithm}\\DecMargin{1em}\n\n\n\\section{Analysis}\n\\label{section:analysis}\nIn this section we present an analysis of the proposed solution. The main points are:\n\\begin{itemize}\n\t\\item Performance analysis of the presented solution based on SIR, SINR and BER evaluation, and comparison with other methods. \n\t\\item Complexity and timing analysis, including computational complexity, inter-connection throughput, memory requirement and latency.\n\\end{itemize}\n\nAs was commented in the Introduction, the analysis presented in this section is quite general and not dependent on any specific hardware implementation. The idea is to provide high level guidelines on algorithm-hardware trade-offs, system parameter selections, and hardware architectures. A more specific analysis can be performed when one has decided the dedicated implementation strategy.\n\n\\subsection{Performance}\n\\label{section:performance}\nIn this subsection we obtain and present different metrics to evaluate and compare the performance of the proposed algorithm. The analysis we present is divided as follows: Derivation of SIR and SINR closed form expressions, bit-error-rate (BER) analysis of the proposed algorithm based on ideal and measured channels and comparison with other methods, such as MF and ZF. The performance analysis that follows is focused on uplink, but it can be extended to downlink.\n\n\\subsubsection{SIR \\& SINR}\nSpecifically for user $k$, \\eqref{eq:Eu} is reduced to\n\\begin{equation}\n\\hat{x}^{u}_{k} = E_{k,k} x^{u}_{k} + \\sum_{i=1,i \\neq k}^{K} E_{k,i} x^{u}_{i} + z_{k},\n\\nonumber\n\\end{equation}\nwhere the first term represents the desired value to estimate (scaled version), the second one is the interference from other users and the third one is due to noise. The signal-to-interference ratio (SIR) for user $k$ is defined as\n\\begin{equation}\n\\text{SIR}_{k} = \\frac{\\E|E_{k,k}|^{2}}{ \\E \\left\\lbrace \\sum_{i=1,i \\neq k}^{K} |E_{k,i}|^2 \\right\\rbrace}.\n\\label{eq:SIR}\n\\end{equation}\n\nAnd for the signal-to-interference-and-noise ratio (SINR) we have\n\\begin{equation}\n\\text{SINR}_{k} = \\frac{\\E|E_{k,k}|^{2}}{\\E \\left\\lbrace \\sum_{i=1,i \\neq k}^{K} |E_{k,i}|^2 \\right\\rbrace + \\E|z_{k}|^2 }.\n\\label{eq:SINR}\n\\end{equation}\n\nA list of parameters and their corresponding values are presented in Table \\ref{table:parameters}, which are used in the following propositions.\n\n\\begin{table}[h!]\n\t\\begin{center}\n\t\t\\caption{Parameters}\n\t\t\\label{table:parameters}\n\t\t\\begin{tabular}{llr}\n\t\t\t\\cline{1-2}\n\t\t\tParameter & Description \\\\\n\t\t\t\\hline\n\t\t\t$\\alpha$ &  $1-\\frac{2 \\mu}{K} +\\frac{\\mu^2}{K(K+1)}$\\\\\n\t\t\t\\hline\n\t\t\t$\\beta$ & $\\frac{\\mu^2}{K(K+1)}$ \\\\\n\t\t\t\\hline\n\t\t\t$\\nu$ & $1 - \\frac{\\mu}{K}$  \\\\\n\t\t\t\\hline\n\t\t\t$\\epsilon$ & $1 - \\frac{2\\mu}{K} + \\frac{\\mu^2}{K}$\n\t\t\\end{tabular}\n\t\\end{center}\n\\end{table}\n\nFrom \\eqref{eq:SIR} it is possible to obtain a closed-form expression of the SIR as follows:\n\\begin{theorem}\n\t\\label{prop:SIR}\n\tWith perfect $\\mathrm{CSI}$ and channel model as defined in Section \\ref{section:background}, $\\mathrm{SIR}$ per user in uplink with $\\mathrm{CD}$ algorithm for estimation is\n\n\t\\begin{equation}\n\t\\mathrm{SIR} = \\frac{1 - 2\\nu^{M} +  \\alpha^{M} \\left(1-\\frac{1}{K}\\right) + \\epsilon^M \\frac{1}{K} }{\\left(1-\\frac{1}{K}\\right) \\cdot \\left( \\epsilon^{M} - \\alpha^{M} \\right)},\n\t\\end{equation}\n\n\twhich can be simplified in case of relatively large $M$, $K$, and $\\frac{M}{K}$, which is the case of Massive MIMO, as\n\n\t\\begin{equation}\n\t\\mathrm{SIR} \\approx e^{\\mu(2-\\mu)\\frac{M}{K}}.\n\t\\label{eq:SIR_approx}\n\t\\end{equation}\n\n\\end{theorem}\n\\begin{proof}\n\tSee Appendix-\\ref{proof:SIR}.\n\\end{proof}\n\n\\begin{figure*}\\centering\n\t\\subfloat[SINR vs $\\mu$ under different SNR. M=128 and K=16.]{\n\t\t\\includegraphics[width=0.48\\textwidth]{SINR_vs_mu_M128_K16}\n\t\t\\label{fig:SINR_vs_mu_M128_K16}\n\t}\n\t\\subfloat[SINR vs $\\mu$ under different channels. M=128 and K=5. SNR=0dB.]{\n\t\t\\includegraphics[width=0.48\\textwidth]{SINR_vs_mu_sim}\n\t\t\\label{fig:SINR_vs_mu_sim}\n\t}\\\\[-2ex]\n\t\\caption{ }\n\t\\label{fig:SINR_vs_mu}\n\t\\vspace*{-4mm}\n\\end{figure*}\n\nThe maximum value of \\eqref{eq:SIR_approx} is achieved for $\\mu=1$ and the SIR value only depends on the ratio $\\frac{M}{K}$ in an exponential fashion, showing how fast the IUI is canceled as $M$ grows, and therefore ZF is approached. As an example, for a target value of SIR = 40dB, $\\frac{M}{K}=10$ meets the requirement, which is a typical ratio in Massive MIMO regime.\n\nRegarding SINR, it can be derived based on previous results as\n\\begin{theorem}\n\t\\label{prop:SINR}\n\tWith perfect $\\mathrm{CSI}$ and channel model as defined in Section \\ref{section:background}, $\\mathrm{SINR}$ per user in uplink with $\\mathrm{CD}$ algorithm for estimation is given by\n\n\t\\begin{equation}\n\t\\begin{split}\n\t\\mathrm{SINR} = \\frac{1 - 2\\nu^{M} +  \\alpha^{M} \\left(1-\\frac{1}{K}\\right) + \\epsilon^M \\frac{1}{K} }{\\left(1-\\frac{1}{K}\\right) \\left(\\epsilon^{M} - \\alpha^{M} \\right) + \\frac{N_{0}}{K-1} \\left( \\frac{\\mu}{2-\\mu}\\right) (1-\\epsilon^{M})},\n\t\\end{split}\n\t\\label{eq:SINR_CD}\n\t\\end{equation}\n\n\twhich can be simplified in case of relatively large $M$, $K$, and $\\frac{M}{K}$, which is the case of Massive MIMO, as\n\n\t\\begin{equation}\n\t\\mathrm{SINR} \\approx \\left[ e^{-\\mu(2-\\mu)\\frac{M}{K}} + \\frac{1}{K \\cdot \\mathrm{SNR}} \\left( \\frac{\\mu}{2-\\mu}\\right) \\right]^{-1}.\n\t\\label{eq:SINR_CD_limit}\n\t\\end{equation}\n\n\\end{theorem}\n\n\\begin{proof}\n\tSee Appendix-\\ref{proof:SINR}.\n\\end{proof}\n\nThe first term in \\eqref{eq:SINR_CD_limit} represents SIR containing IUI information, while the second one takes into account the post-equalized noise power. For high SNR, the first term is dominant and $\\mathrm{SINR} \\to e^{\\mu(2-\\mu)\\frac{M}{K}}$, which depends on $\\frac{M}{K}$ and $\\mu$, but not on $\\mathrm{SNR}$. On the other hand, when SNR is low, the second term is dominant and $\\mathrm{SINR} \\to \\mathrm{SNR} \\cdot K \\left(\\frac{2 - \\mu}{\\mu}\\right)$ as $M$ grows, which grows linearly with $\\mathrm{SNR}$ and $K$ (up to certain value). This linear dependency on $K$ is due to the post-equalization noise is equally distributed among the users. While the noise power per antenna remains constant, the portion assigned to each user decays as $K$ grows, so the SINR per user grows linearly. However, as $K$ increases the IUI does so (first term in \\eqref{eq:SINR_CD_limit} grows), and both effects cancel out at some point, being IUI dominant afterwards, with the corresponding decay of SINR.\n\nThe optimal value of $\\mu$, denoted as $\\mu^{*}$, depends on $M$, $K$, and the specific channel. For the i.i.d. case, defined in Section \\ref{section:background}, it is possible to obtain $\\mu^{*}$ by numerical optimization over \\eqref{eq:SINR_CD}. An approximate value, denoted as $\\mu_{0}$, is presented as follows.\n\\begin{theorem}\n\t\\label{prop:mu_init}\n\tA recommended value for $\\mu_{0}$, in the vicinity of $\\mu^{*}$, under CD and i.i.d. channel as defined in Section \\ref{section:background}, is given by\n\n\t\\begin{equation}\n\t\\mu_{0} = \\frac{1}{2} \\frac{K}{M} \\log (4 M \\cdot \\mathrm{SNR} ).\n\t\\label{eq:mu_init}\n\t\\end{equation}\n\n\\end{theorem}\n\n\\begin{proof}\n\tSee Appendix-\\ref{proof:mu_init}.\n\\end{proof}\n\nAs a side result, from the analysis performed in this section, we can extract interesting properties of the matrix $\\Wbf$, such the following one:\n\\begin{theorem}\n\t\\label{prop:W_power}\n\tThe equalization matrix $\\Wbf$ as result of $\\mathrm{CD}$ algorithm satisfies the next properties for $\\mu \\in [0,2)$\n\n\t\\begin{equation}\n\t\\E \\| \\Wbf \\|^{2}_{F} = \\frac{K}{K-1} \\cdot \\frac{\\mu}{2-\\mu} \\cdot \\left( 1-\\epsilon^{M} \\right).\n\t\\label{eq:W_power}\n\t\\end{equation}\n\n\t\\nonumber\n\\end{theorem}\n\\begin{proof}\n\tSee Appendix-\\ref{proof:W_power}.\n\\end{proof}\n\nThis result is relevant in downlink, where a transmission power budget is needed. Expression in \\eqref{eq:W_power} is a monotonically growing function of $\\mu$. It can be shown that total transmitted mean power is bounded by $4\\frac{M}{K}$, reaching this value at $\\mu=2$. However, as we will see in next section, optimal $\\mu$ for i.i.d. Gaussian channel is within the range $(0,1]$, therefore for a large enough $K$, we have $\\E \\| \\Wbf \\|^{2}_{F} \\leq 1$, which does not depend on $M$, therefore ensure the scalability of the proposed solution.\\\\\n\nExpression \\eqref{eq:SINR_CD} is plotted in Figure \\ref{fig:SINR_vs_mu_M128_K16} showing SINR vs $\\mu$ for CD under different SNR values and step-size according to \\eqref{eq:mu_m}. As expected, optimal $\\mu$ approaches 1 as SNR grows. Simulation results shows a good match with \\eqref{eq:SINR_CD}. The curve with $\\mu_{0}$ values obtained from \\eqref{eq:mu_init} is also plotted for a wide range of SNR. It is observed how the $\\mu_{0}$ value is reasonably close to the optimum for the SNR range depicted. Furthermore, the result is much closer to ZF than MRC values, which are $\\{40.5, 30.5, 20.5, 10.5\\}$dB and $\\{9.0, 9.0, 8.8, 6.8\\}$dB respectively for the different SNR values used in the figure.\n\nFigure \\ref{fig:SINR_vs_mu_sim} shows simulation results for the CD algorithm performance under different channels. For some of them we use a model (i.i.d and WINNER II) and others are based on real measurements (Rich A and LOS A). For this comparison we use different $\\frac{M}{K}$ ratio and the step-size according to \\eqref{eq:mu_m}. Rich A is non-line-of-sight (NLOS) channel, rich in scatters, while LOS A is predominantly line-of-sight (LOS) channel. WINNER II is obtained from a NLOS scenario with a uniform linear array at the BS, with M elements separated by $\\lambda$\/2. Users are randomly located in a 300m$\\times$300m area, with the BS at the center. It is noticed how rich channels (i.i.d and WINNER II) provide better performance. SINR levels reached by ZF are \\{20.9, 20.9, 19.8, 17.6\\}dB and for MRC they are \\{14.3, 15.2, 7.8, 4.8\\}dB, in both cases for the i.i.d., WINNER II, Rich A and LOS A channels, respectively. It is also noticed that CD performance lies in between ZF and MRC for these scenarios.\n\nFigure $\\ref{fig:SINR_vs_M_over_K_SNR0}$ shows SINR versus $\\frac{M}{K}$ for $M=128$ and SNR = 0dB. SINR for CD is shown comparing the effect of using  $\\mu^{*}$ and $\\mu_{0}$ according to \\eqref{eq:mu_init}. We observe that $\\frac{M}{K} \\approx 10$ (equivalent to $K\\approx12$) is the preferred working point, where SINR reaches the maximum value and $\\mu_{0}$ gives the same result as $\\mu^{*}$. We also compare the performance with ZF and MRC algorithms.\n\nAs presented in Subsection \\ref{sub:multiple-iter}, the algorithm can be extended to perform multiple iterations through the array, in order to increase the performance. Figure $\\ref{fig:SINR_vs_mu_sim_num_iter}$ shows SINR versus $\\mu$ for a different number of iterations through the array together with ZF for comparison. From the figure we can notice that the maximum SINR increases after each iteration, approaching to ZF. It is also relevant to note that $\\mu^{*}$ changes with the number of iterations. \n\n\\begin{figure}[t]\\centering\n\t\\includegraphics[width=1\\linewidth]{SINR_vs_M_over_K_SNR0}\n\t\\vspace*{-4mm}\n\t\\caption{SINR (dB) versus $\\frac{M}{K}$ for SNR=0dB and M=128. CD SINR is plotted in the case of $\\mu^{*}$ (dashed) and $\\mu_{0}$ (solid) are used. i.i.d. channel. SNR = 0dB.}\n\t\\label{fig:SINR_vs_M_over_K_SNR0}\n\\end{figure}\n\n\\begin{figure}\\centering\n\t\\includegraphics[width=1\\linewidth]{SINR_vs_mu_sim_num_iter}\n\t\\vspace*{-4mm}\n\t\\caption{SINR vs. SNR for $M$=128, $K=16$. 16QAM. i.i.d. channel. SNR=0dB. SINR after a certain number of iterations through the array. ZF added for comparison.}\n\t\\label{fig:SINR_vs_mu_sim_num_iter}\n\\end{figure}\n\n\n\\subsubsection{BER}\nBER versus SNR is shown in Figure \\ref{fig:BER_vs_SNR} under i.i.d. channel for three different methods: CD, ZF and MRC. CD is shown using two different values for $\\mu$: 1 and $\\mu^*$. It is noticeable the great impact of the selected $\\mu$ and therefore the importance of selecting an appropriate value.\n\nThe effect of non-ideal CSI in the BER is shown in Figure \\ref{fig:BER_vs_SNR_non-ideal-CSI} for ZF and CD (for $\\mu^{*}$). The non-ideal CSI is modeled as an ideal-CSI with a noise contribution (complex normal distributed) with a variance equal to $N_{0}$, therefore it depends inversely on SNR. No boosting in pilots is used. As it can be observed, for SNR$<$0dB the SNR gap is very small and increases as long as SNR increases too, in a similar fashion as the ideal CSI case. For SNR$>$0 the SNR gap in both cases is similar.\n\n\\begin{figure}\\centering\n\t\\includegraphics[width=1\\linewidth]{BER_vs_SNR_M128_K16_16QAM_IID}\n\t\\vspace*{-4mm}\n\t\\caption{BER vs. SNR for $M$=128, $K=16$. 16QAM. i.i.d. channel.}\n\t\\label{fig:BER_vs_SNR}\n\\end{figure}\n\n\\begin{figure}\\centering\n\t\\includegraphics[width=1\\linewidth]{BER_vs_SNR_M128_K16_16QAM_IID_non-ideal-CSI}\n\t\\vspace*{-4mm}\n\t\\caption{BER vs. SNR for $M$=128, $K=16$. 16QAM. i.i.d. channel. Comparison between ideal and non-ideal CSI.}\n\t\\label{fig:BER_vs_SNR_non-ideal-CSI}\n\\end{figure}\n\n\\subsection{Complexity \\& Timing}\nIn this subsection we analyze the complexity of the proposed solution from three different domains: computational complexity (data processing), inter-connection throughput (data movement) and memory (data storage). Timing in the form of total system latency is also analyzed.\n\nFor this analysis we assume a frame structure based on OFDM, which contains one dedicated OFDM symbol per frame for channel estimation based on orthogonal pilots, so each one is dedicated to one of the users in a consecutive way. The other symbols convey users' data. Under the TDD assumption, some of them are used for DL and others for UL. We also assume that all RPUs perform IFFT\/FFT in parallel with an output data-rate of $\\frac{N_{\\mathrm{u}}}{T_{\\mathrm{OFDM}}}$.\n\n\\begin{figure*}[ht]\n\t\\footnotesize\n\t\\centering\n\t\\psfrag{P1}{$P_1$}\n\t\\psfrag{P2}{$P_2$}\n\t\\psfrag{P3}{$P_3$}\n\t\\psfrag{PN}{$P_N$}\n\t\\psfrag{D1}{$D_1$}\n\t\\psfrag{D2}{$D_2$}\n\t\\psfrag{D3}{$D_3$}\n\t\\psfrag{DN}{$D_N$}\n\t\\psfrag{M1}{$M_1$}\n\t\\psfrag{M2}{$M_2$}\n\t\\psfrag{M3}{$M_3$}\n\t\\psfrag{MN}{$M_N$}\n\t\\psfrag{C1}{$C_1$}\n\t\\psfrag{C2}{$C_2$}\n\t\\psfrag{C3}{$C_3$}\n\t\\psfrag{CN}{$C_N$}\n\t\\psfrag{W11}{$w_{1}^{(1)}$}\n\t\\psfrag{W12}{$w_{1}^{(2)}$}\n\t\\psfrag{W13}{$w_{1}^{(3)}$}\n\t\\psfrag{W1N}{$w_{1}^{(N)}$}\n\t\\psfrag{W21}{$w_{2}^{(1)}$}\n\t\\psfrag{W22}{$w_{2}^{(2)}$}\n\t\\psfrag{W23}{$w_{2}^{(3)}$}\n\t\\psfrag{W2N}{$w_{2}^{(N)}$}\n\t\\psfrag{WM1}{$w_{M}^{(1)}$}\n\t\\psfrag{WM2}{$w_{M}^{(2)}$}\n\t\\psfrag{WM3}{$w_{M}^{(3)}$}\n\t\\psfrag{WMN}{$w_{M}^{(N)}$}\n\t\\psfrag{M1}{$M_1$}\n\t\\psfrag{M2}{$M_2$}\n\t\\psfrag{M3}{$M_3$}\n\t\\psfrag{MN}{$M_N$}\n\t\\psfrag{ant1}{$1$}\n\t\\psfrag{ant2}{$2$}\n\t\\psfrag{ant3}{$3$}\n\t\\psfrag{antM}{$M$}\n\t\\psfrag{OFDM1}{$\\mathrm{OFDM} 1$}\n\t\\psfrag{OFDM2}{$\\mathrm{OFDM} 2$}\n\t\\psfrag{OFDM3}{$\\mathrm{OFDM} 3$}\n\t\\psfrag{OFDML}{$\\mathrm{OFDM} L$}\n\t\\psfrag{A1}{$\\mathbf{A}_{1}^{(n)}$}\n\t\\psfrag{A2}{$\\mathbf{A}_{2}^{(n)}$}\n\t\\psfrag{A3}{$\\mathbf{A}_{3}^{(n)}$}\n\t\\psfrag{AM1}{$\\mathbf{A}_{M-1}^{(n)}$}\n\t\\psfrag{A11}{$\\mathbf{A}_{1}^{(1)}$}\n\t\\psfrag{A1N}{$\\mathbf{A}_{1}^{(N)}$}\n\t\\psfrag{A21}{$\\mathbf{A}_{2}^{(1)}$}\t\n\t\\psfrag{A2N}{$\\mathbf{A}_{2}^{(N)}$}\n\t\\psfrag{TS}{$\\cdots$}\n\t\\psfrag{TPRB}{$T_{\\mathrm{PRB}}$}\n\t\\includegraphics[width=1.0\\linewidth]{time_diagram.eps}\n\t\\caption{Time diagram representing formulation and filtering\/precoding activities performed in the antenna modules. Each OFDM symbol is split into $N_\\mathrm{PRB}$ blocks ($N$ in the figure) in the same order as data come out of any of the receiver FFT. Those blocks which contains pilots are shown as $P_{i}$, while those carrying data are denoted as $D_{i}$. Channel estimation is performed during $C_{i}$ blocks, while formulation is done in $\\w_{i}$ blocks. Filtering\/precoding data is carried out during the MIMO processing blocks, named $\\mathrm{M}_{i}$. As it can be observed, all antennas perform their tasks simultaneously, while formulation is done sequentially as a matrix $\\A^{(n)}$ passes through the array. In total, $N$ matrices are passed sequentially through antenna $m$, corresponding to $\\A_{m}^{(n)}, n=1 \\cdots N$. $\\w_{i}$ vectors need to be available in the antenna modules before the corresponding data comes out of the receiver FFT so it can be properly processed. Daisy-chain topology exploits the parallelism of the operations by allowing the pipeline of the operations and the fully usage of all dedicated links simultaneously.}\n\t\\label{fig:time_diagram}\n\\end{figure*}\n\nWe can exploit channel correlation based on the Physical Resource Block (PRB) concept in 3GPP. A PRB is a region in frequency-time domain where the channel response is assumed to be approximately constant across all subcarriers within that PRB. Within an OFDM symbol, the number of subcarriers in each PRB and the number of PRB per symbol, defined as $N_{\\mathrm{sc,PRB}}$ and $N_{\\mathrm{PRB}}$ respectively, are related as follows: $N_{\\mathrm{u}} = N_{\\mathrm{PRB}} N_{\\mathrm{sc,PRB}}$. We define $T_{\\mathrm{PRB}}$ as the time needed by $N_{\\mathrm{sc,PRB}}$ consecutive subcarriers to come out the FFT.\n\nFor each PRB we have a different channel matrix and also MIMO model as in \\eqref{eq:ul_model} and \\eqref{eq:dl_model}. Then, it is required to have a unique set of vectors $\\w_m$ and $\\p_m (m=1...M)$ per antenna, as in \\eqref{eq:linear_det} and \\eqref{eq:linear_prec_i}, for uplink detection and downlink precoding respectively. The phase where these vectors are computed is named $\\textit{formulation}$, while the phase where user's data is processed is named $\\textit{filtering}$ and $\\textit{precoding}$ for UL and DL respectively. To minimize data buffering, formulation needs to be completed before filtering\/precoding starts. This imposes the constraint that the formulation phase needs to be finished within one OFDM symbol, or in other words, all antennas need to obtain these vectors and the matrix $\\A$ needs also to pass through the array within one OFDM symbol. A diagram of the main activities involved and their timing relationship is shown in Figure \\ref{fig:time_diagram}. The analysis assumes that the processing and data transmission are pipelined in each RPU so they concurrently operate.\n\n\\subsubsection{Computational complexity}\n\n\\begin{itemize}  \n\\item Formulation phase:\nThe number of complex multiplications needed to formulate one precoding\/filtering vector per antenna are $C_{\\mathrm{form}} \\approx 2K^{2}$, which represents the matrix-vector product to obtain $\\w_{m}$ and the outer product to update $\\A_{m}$ according to algorithm \\ref{algo:CD}. Other possible required operations such as norm, square root or division are assumed to be negligible.\n\n\\item Filtering phase:\nDuring the filtering phase, each RPU performs the required operations  for UL detection. Vectors $\\w_{m}$ are applied to all observations (data subcarriers), $y^{u}_{m}$, under the same PRB. The complexity measured in number of complex multiplications per antenna and per $N_{\\mathrm{sc,PRB}}$ subcarriers is $C_{\\mathrm{filt}} = KN_{\\mathrm{sc,PRB}}$.\n\n\\item Precoding phase:\nDuring the precoding phase, each RPU performs the operations required by \\eqref{eq:linear_prec_i}. Similarly to the filtering case, the same vector $\\p_{m}$ is applied to all data vectors $x^{d}_{m}$ under same PRB. The complexity measured in number of complex multiplications per antenna and PRB is $C_{\\mathrm{prec}} = KN_{\\mathrm{sc,PRB}}$.\n\\end{itemize}\n\n\\subsubsection{Inter-connection data-rate}\n\\label{section:data-rate}\n\\begin{itemize}  \n\t\\item Formulation phase:\n\tThe average inter-connection data-rate during formulation can be calculated assuming that the average time to complete a transfer of a matrix $\\A$ is $T_{\\mathrm{PRB}}$, which leads to an average rate of\n\n\t\\begin{equation}\n\tR_{\\mathrm{d,form}} = \\frac{2w_{\\A} K^{2} N_{\\mathrm{PRB}}}{T_{\\mathrm{OFDM}}},\n\t\\nonumber\n\t\\end{equation}\n\n\twhere the numerator represents the amount of bits to transfer (all matrices $\\A$ in a symbol) and $w_{\\A}$ is the bit-width of $\\A$ entries (real\/imaginary parts).\n\t\n\t\\item Filtering phase:\n\tPartial filtering results from each RPU are added up through the chain. The average inter-connection data-rate per dedicated link can be calculated as\n\n\t\\begin{equation}\n\tR_{\\mathrm{d,filt}} = \\frac{2 w_{\\mathrm{d}} KN_{\\mathrm{u}}}{T_{\\mathrm{OFDM}}},\n\t\\nonumber\n\t\\end{equation}\n\n\twhere $w_{\\mathrm{d}}$ is the bit-width of baseband samples exchanged among RPUs.\n\t\n\t\\item Precoding phase:\n\tIn the precoding phase, the data vectors $\\xd$ are passed through the array for processing. Each node receives a vector which is passed to next node without any required pause (broadcasting). This leads to the same data-rate as in the filtering case.\n\t\n\\end{itemize}\n\n\\subsubsection{Latency}\nThe processing latency in the formulation phase for one antenna is given from next expression\n\\begin{equation}\n\\begin{split}\nT_{\\mathrm{proc,form}} &= \\frac{C_{\\mathrm{form}} T_{\\mathrm{CLK}}}{N_{\\mathrm{mult}}} \\\\\n&\\approx \\frac{2K^{2} T_{\\mathrm{CLK}}}{N_{\\mathrm{mult}}},\n\\end{split}\n\\nonumber\n\\end{equation}\nwhere $N_{\\mathrm{mult}}$ is the number of multipliers available in each RPU that can be used in parallel, $T_{\\mathrm{CLK}}$ is the clock period and we assume that one complex multiplication can be done within one  $T_{\\mathrm{CLK}}$. Total latency is expressed as\n\\begin{equation}\n\\begin{split}\nLat_{form} &= M \\cdot T_{\\mathrm{proc, form}} + (N_{\\mathrm{RPU}}-1) \\cdot T_{\\mathrm{trans}},\n\\end{split}\n\\nonumber\n\\end{equation}\nwhere $N_{\\mathrm{RPU}}$ is the number of RPUs in the system, and $T_{\\mathrm{trans}}$ is the transmission latency between two consecutive RPUs. As said before, formulation needs to be finished within one $T_\\mathrm{OFDM}$, therefore the formulation latency is constrained as $Lat_{form} < T_{\\mathrm{OFDM}}$. This leads to an upper limit for M as\n\\begin{equation}\nM < \\frac{T_\\mathrm{OFDM}+T_{\\mathrm{trans}}}{T_{\\mathrm{proc, form}} + \\frac{T_{\\mathrm{trans}}}{M_{\\mathrm{RPU}}}},\n\\nonumber\n\\end{equation}\nwhere $M_{\\mathrm{RPU}}=\\frac{M}{N_{\\mathrm{RPU}}}$ is the number of antennas per RPU, which is considered as a design parameter. We can consider another limit, slightly lower than previous one but easier to extract conclusions as follows\n\\begin{equation}\nM < \\frac{T_\\mathrm{OFDM}}{T_{\\mathrm{proc, form}} + \\frac{T_{\\mathrm{trans}}}{M_{\\mathrm{RPU}}}}.\n\\nonumber\n\\end{equation}\n\nWe analyze three scenarios:\n\\begin{itemize}  \n\t\\item $T_{\\mathrm{proc, form}} \\rightarrow 0$: When processing time is reduced, by increasing $N_{\\mathrm{mult}}$ or decreasing $T_{\\mathrm{CLK}}$, then transaction time becomes dominant and a reduction in the number of links allow for higher values of $M$. Formally, the upper value for $M$ scales proportionally to $M_{\\mathrm{RPU}}$ as follows\n\n\t\\begin{equation}\n\tM < M_{\\mathrm{RPU}} \\cdot \\frac{T_\\mathrm{OFDM}}{T_{\\mathrm{trans}}}.\n\t\\nonumber\n\t\\end{equation}\n\n\t\\item $T_{\\mathrm{trans}} \\rightarrow 0$: By decreasing the transaction time the upper limit of $M$ converges to a certain value, which is inversely proportional to the processing time as follows\n\n\t\\begin{equation}\n\tM < \\frac{T_\\mathrm{OFDM}}{T_{\\mathrm{proc, form}}}.\n\t\\nonumber\n\t\\end{equation}\n\n\t\\item $M_{\\mathrm{RPU}} \\gg \\frac{T_{\\mathrm{trans}}}{T_{\\mathrm{proc, form}}}$. When $M_{\\mathrm{RPU}}$ increases beyond a certain value, processing time becomes dominant and we obtain the same limit as previous point.\n\\end{itemize}\n\nIn case of filtering, its related processing is done in parallel as soon as data comes out of the FFT. However, partial results needs to be accumulated through the array from RPU 1 to $N_{\\mathrm{RPU}}$. This latency is uniquely due to data transfer through the dedicated links, then\n\\begin{equation}\n\\begin{split}\nLat_{\\mathrm{filt}} &= (N_{\\mathrm{RPU}}-1) \\cdot T_{\\mathrm{trans}}\\\\\n& < Lat_{\\mathrm{form}}\\\\ & < T_{\\mathrm{OFDM}}.\n\\end{split}\n\\label{eq:lat_filt}\n\\end{equation}\n\n\\subsubsection{Memory}\nIn terms of memory requirement, a centralized architecture requires to store the channel matrix $\\mathbf{H}$ fully at the CPU, previous to the inversion. There is a channel matrix per PRB, so CSI storage requires $M_{\\mathrm{H}} = 2 w_{\\mathrm{h}} M K N_{\\mathrm{PRB}}$ bits, where $w_{\\h}$ represents the bit-width of $\\Hbf$ entries (real\/imaginary parts), and in order to store the resulting square matrix, $(\\Hbf^{H}\\Hbf)^{-1}$ requires $M_{\\mathrm{inv}} = 2 w_{\\mathrm{h}} K^{2} N_{\\mathrm{PRB}}$ and therefore the total requirement is: $M_{\\mathrm{central}} = M_{\\mathrm{H}} + M_{\\mathrm{inv}} \\approx M_{\\mathrm{H}}$.\n\nIn the decentralized architecture, each antenna module needs to store the corresponding $\\h$, which gets replaced by $\\mathbf{w}$ after formulation. Both of them requires the same amount of memory if same bit-width is assumed, which is $M_{\\mathrm{w}} = 2 w_{\\mathrm{h}} K N_{\\mathrm{PRB}}$, and the total amount of memory in the system is: $M_{\\mathrm{daisy}} = M \\cdot M_{\\w} \\approx M_{\\mathrm{central}}$. Therefore, the total amount of memory required for $\\Hbf$ and $\\Wbf$ is the same in both systems, however the daisy-chain allows a uniform distribution of the memory requirements across all antenna modules, reducing design complexity, time and cost. As a drawback, we point out the need for data buffering during the filtering phase due to latency in the transfer of partial results, as discussed in the previous subsection (Latency). The buffer size for the RPU closest to the CPU (worst case) can, based on \\eqref{eq:lat_filt}, be obtained as\n\\begin{equation}\nM_{\\mathrm{buffer}} = \\frac{ 2 w_{\\mathrm{d}} K N_{\\mathrm{u}} Lat_{\\mathrm{filt}}}{T_{\\mathrm{OFDM}}},\n\\nonumber\n\\end{equation}\nwhich is shared by all antennas belonging to that RPU.\n\n\\subsection{Comparison}\n\\begin{table}\n\t\\renewcommand{\\arraystretch}{1.3} \n\t\\caption{Inter-connection data-rate comparison for different system parameters [$G\\lowercase{b\/s}$]}\n\t\\label{tab:data-rate}\n\t\\centering\n\t\\begin{tabular}{l*{5}{c}}\n\t\t\\hline\n\t\tScenario    &  &   &  & \\\\\n\t\t$M$\t   & 32     & 64     & 128    & 256 \\\\\n\t\t$K$\t   & 4      & 8      & 12     & 12 \\\\\n\t\t\\hline\n\t\t$R_{\\mathrm{d,form}} $ & 12.67 & 50.69 & 114.05 & 114.05\\\\\n\t\t$R_{\\mathrm{d,filt\/prec}} $ & 38.02 & 76.03 & 114.05 & 114.05\\\\\n\t\t\\hline\n\t\t$R_{\\mathrm{c}} $ & 304.13 & 608.26 & 1216.51 & 2433.02\\\\\n\t\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:data-rate} shows a comparison of interconnection data-rate between daisy-chain and centralized architecture for different scenarios of $M$ and $K$. It is important to remark that $R_{\\mathrm{c}}$ corresponds to the aggregated data\/rate at the shared bus, while $R_{\\mathrm{d}}$ is the average data\/rate in each of the RPU-RPU dedicated links. For the centralized case, \\eqref{eq:R_central} is used, while for the daisy-chain case, data-rates are detailed according to the different tasks (formulation, filtering and precoding) as described in Section \\ref{section:data-rate}. For the numerical results we employ $T_{\\mathrm{CLK}}=1\\mathrm{ns}$ and $w=12$. The rest of system parameters are as follows according to worst case in 5G NR: $N_{\\mathrm{u}}=3300$, $N_{\\mathrm{PRB}}=275$, $N_{\\mathrm{sc,PRB}}=12$ and $T_{\\mathrm{OFDM}}=\\frac{1}{120\\mathrm{KHz}}$. We observe that for $M=128$ case, daisy-chain requires $\\sim 10\\%$ of the inter-connection data-rate needed by the centralized case. This number can even decrease as $\\frac{M}{K}$ grows. As it is observed, daisy-chain requires much lower inter-connection data-rates than the centralized counterpart. We remark that if we take into account the total inter-connection data-rate in the decentralized case, which is $N_{\\mathrm{RPU}} R_{\\mathrm{d,form}}$, may easily exceed the centralized counterpart $R_{\\mathrm{c}}$, however the decentralized architecture is able to distribute this data-rate equally across all links, reducing considerably the requirements for each of them.\n\n\\begin{table}\n\t\\renewcommand{\\arraystretch}{1.3} \n\t\\caption{Computational complexity comparison for different system parameters [$GOPS$]}\n\t\\label{tab:complexity}\n\t\\centering\n\t\\begin{tabular}{l*{5}{c}}\n\t\t\\hline\n\t\tScenario    &  &   &  & \\\\\n\t\t$M$\t   & 32     & 64     & 128    & 256 \\\\\n\t\t$K$\t   & 4      & 8      & 12     & 12 \\\\\n\t\t\\hline\n\t\t$C_{\\mathrm{d,ant}} $ & 1.58 & 3.17 & 4.75 & 4.75\\\\\n\t\t\\hline\n\t\t$C_{\\mathrm{c}} $ & 50.69 & 202.75 & 608.26 & 1216.51\\\\\n\t\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:complexity} shows a computational complexity comparison between centralized and decentralized architectures. $C_{\\mathrm{d,ant}}$ represents complex multiplications per second and per antenna in the decentralized case, while $C_{\\mathrm{c}}$ is the computational complexity required by CPU in centralized system. In both cases, only filtering\/precoding is taken into account because formulation depends on how often channel estimation is available. The result of the comparison is meaningful. Even tough, the total complexity in the decentralized system is approximately equal to the centralized counterpart, this is $M \\cdot C_{\\mathrm{d,ant}} \\approx C_{\\mathrm{c}}$, our decentralized solution is able to divide equally the total computational complexity among all existing RPUs, relaxing considerably the requirements compared to the CPU in centralized case. The relatively low number obtained for the daisy-chain allows the employment of cheap and general processing units in each RPU, in opposite to the centralized architecture where the total complexity requirement is on the CPU.\n\nNumerical results for latency are shown in table \\ref{tab:latency} for $N_{\\mathrm{mult}}=8$, $T_{\\mathrm{trans}}=100ns$ and $N_{\\mathrm{RPU}}=\\frac{M}{4}$. These design parameters meets the constraint $Lat < T_\\mathrm{OFDM}$ up to $M=128$. For larger arrays there are different solutions: allows the latency to increase and buffer the needed input data (need for larger memory), group more antennas in each RPU (which reduces the number of links but increase the complexity of the CPU controlling each RPU), and\/or employ low-latency link connections (reducing $T_{\\mathrm{trans}}$ at the expense of higher cost). It is relevant to note that $T_\\mathrm{OFDM}$ value in the table is the worst case $1\/120KHz$.\n\\begin{table}\n\t\\renewcommand{\\arraystretch}{1.3} \n\t\\caption{Latency comparison for different system parameters}\n\t\\label{tab:latency}\n\t\\centering\n\t\\begin{tabular}{l*{5}{c}}\n\t\t\\hline\n\t\tScenario    &  &   &  & \\\\\n\t\t$M$\t   & 32     & 64     & 128    & 256 \\\\\n\t\t$K$\t   & 4      & 8      & 12     & 12 \\\\\n\t\t\\hline\n\t\t$Lat(\\mu s)$ & 0.83 & 2.52 & 7.71 & 15.52\\\\\n\t\t$Lat\/T_{\\mathrm{OFDM}}$ & 0.10 & 0.30 & 0.92 & 1.86\\\\\n\t\\end{tabular}\n\\end{table}\n\nIn table $\\ref{tab:memory}$ a comparison between both systems from memory perspective is shown. If $w_{\\h}=12$ and $N_{\\mathrm{PRB}}=275$ are assumed, then for the $M=128$ case, each antenna module in the daisy-chain only needs $\\sim 80 \\mathrm{kbits}$ of memory and each RPU needs at maximum $354 \\mathrm{kbits}$ for buffering, while in the centralized architecture, the central processor requires $\\sim 11 \\mathrm{Mbits}$, which is a challenging number for a cache memory. The memory requirement grows proportionally to M in the centralized system, while that does not happen in $M_{\\w}$. In order to reduce the buffer size we can group more antennas in each RPU, so all of them share the same buffer memory.\n\\begin{table}\n\t\\renewcommand{\\arraystretch}{1.3} \n\t\\caption{Memory requirement comparison for different system parameters [$kbits$]}\n\t\\label{tab:memory}\n\t\\centering\n\t\\begin{tabular}{l*{5}{c}}\n\t\t\\hline\n\t\tScenario    &  &   &  & \\\\\n\t\t$M$\t   & 32     & 64     & 128    & 256 \\\\\n\t\t$K$\t   & 4      & 8      & 12     & 12 \\\\\n\t\t\\hline\n\t\t$M_{\\w} (ant) $ & 26.4 & 52.8 & 79.2 & 79.2\\\\\n\t\t$M_{\\mathrm{buffer}} (RPU) $ & 26.6 & 114.1 & 353.6 & 718.5\\\\\t\n\t\t\\hline\n\t\t$M_{\\mathrm{H}}  $ & 844.8 & 3379.2 & 10137.6 & 20275.2\\\\\n\t\t$M_{\\mathrm{inv}}  $ & 105.6 & 422.4 & 950.4 & 950.4\\\\\n\t\\end{tabular}\n\\end{table}\n\n\\section{Conclusions}\n\\label{section:conclusions}\n\nIn this article we proposed an architecture for Massive MIMO base-station for uplink detection and downlink precoding, which is based on the fully distribution of the required baseband processing across all antenna modules in the system. The main goal is to reduce the inter-connection data-rate needed to carry out the processing tasks and enable the scalability needed in Massive MIMO. We continued our previous work in this topic \\cite{jesus} \\cite{muris} by a detailed introduction to the CD algorithm and its application to the Massive MIMO case. We also presented an extensive analysis of the expected performance of the system, the inter-connection data-rate, complexity, latency and memory requirements. The results show that there is a performance loss compared to ZF, but unlike MF, our proposed method does not have an error floor, from which we can not recover, while the inter-connection data-rate is distributed avoiding the aggregation of the centralized approach. At the same time, complexity and memory requirements per antenna module are easy to meet with commercial off-the-self hardware, which proves the scalability of this solution.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nIn high dimensional linear regression\n\\begin{align*}\ny=X\\beta^\\ast+w,\\quad\\mbox{with noise }w \\sim \\mathcal{N}(0,\\sigma^2 I),\n\\end{align*}\nthe goal is to parsimoniously predict the response $y\\in \\mathbb R^n$ as a linear combination of a large number of covariates $X=(X_1,X_2,\\ldots,X_p)\\in\\mathbb R^{n\\times p}$, and conduct statistical inference on the linear combination coefficients $\\beta^\\ast=(\\beta_1^\\ast,\\ldots,\\beta_p^\\ast)^T\\in\\mathbb R^p$ \\citep{tibshirani1996regression,donoho2006compressed}. By leveraging on certain lower dimensional structure in the regression coefficient vector $\\beta^\\ast\\in\\mathbb R^p$ such as a sparsity constraint $s=\\|\\beta^\\ast\\|_0\\ll n$, where $\\|\\beta^\\ast\\|_0$ counts the number of nonzeros in $\\beta^\\ast$, the number $p$ of covariates is allowed to be substantially larger than the sample size $n$. Due to the intrinsic computational hardness in dealing with the $\\ell_0$ metric reflecting sparsity, people instead use different metrics as surrogates, and cast the estimation problem into various convex or nonconvex optimization problems. Many approaches have been proposed for high dimensional regression by solving certain penalized optimization problem, including basis pursuit \\citep{chen2001atomic}, the Lasso \\citep{tibshirani1996regression}, the Dantzig selector \\citep{candes2007dantzig}, SCAD \\citep{fan2001variable}, MCP \\citep{zhang2010nearly} and so on. In this work, we focus on the recovery of $\\beta^\\ast\\in\\mathbb R^p$ without explicitly specifying a penalty.\n\nRecent work~\\citep{hoff2017lasso} shows that through a change-of-variable (over-parametrization) via Hadamard product parametrization, the Lagrangian (dual) form of the non-smooth convex optimization problem for the Lasso~\\eqref{Eqn:CS}:\n\\begin{equation}\\label{Eqn:CS}\n\\min_{\\beta} \\frac{1}{2n}\\|X\\beta-y\\|^2+\\lambda \\|\\beta\\|_{1}, \\quad\\mbox{with }\\|\\beta\\|_1:\\,=\\sum_{j=1}^p|\\beta_j|,\n\\end{equation}\ncan be reformulated as a smoothed optimization problem at a cost of introducing non-convexity. Due to the smoothness feature, simple and low-cost first-order optimization methods such as gradient descent and coordinate descent can be applied to recover $\\beta^\\ast$. Despite the non-convexity and exponentially many stationary points induced by the change-of-variable, these first-order algorithms exhibit encouraging empirical performance~\\citep{hoff2017lasso}.\n\nIn this work, we consider the same Hadamard product over-parametrization $\\beta = g\\circ l$ as in \\cite{hoff2017lasso}, where $g, \\,l\\in\\mathbb R^p$ and $\\circ$ denotes the Hadamard product (element-wise product). Instead of solving the penalized optimization problem~\\eqref{Eqn:CS}, we consider directly applying the gradient descent to the quadratic loss function\n\\begin{equation}\\label{eq_opt}\nf(g,l)=\\frac{1}{2n}\\,\\|X(g\\circ l)-y\\|^2.\n\\end{equation}\nIn the noiseless case where $\\sigma=0$, minimizing $f(g,\\,l)$ jointly over $(g,\\,l)$ is a highly non-convex optimization problem with exponentially many saddle points. To see this, notice that each non-zero pattern of $\\beta$ corresponds to at least one saddle point by choosing $g_j=l_j=0$ for each $j$ such that $\\beta_j=0$. In addition, due to the column rank deficiency of the design matrix $X$ (for example, when $p>n$), there are infinitely many global minimizers of \\eqref{eq_opt} as potential convergent points of the gradient descent. \nInterestingly, we show that despite these seemingly hopeless difficulties, in the noiseless case if we initialize the gradient descent arbitrarily close to $g=l=0$, then under\nthe prominent Restricted Isometry Property (RIP) condition~\\citep{candes2008restricted} on the design matrix $X$, a properly tuned gradient descent converges to least $\\ell_1$-norm solution within error $\\varepsilon$ in $\\mathcal O(\\log\\frac{C}{\\varepsilon})$ iterations, where constant $C$ depends on the RIP constant, step size of the gradient descent, and some other characteristics of the problem. Our proofs borrow ideas from \\cite{li2018algorithmic}, where they prove the algorithmic convergence of matrix factorized gradient descent in the context of noiseless matrix sensing under the RIP.\n\n\nIn high dimensional regression,  the usual regularized least square is known to suffer from the so-called saturation phenomenon \\citep{vito2005learning,yao2007early}, where the overall estimation error is dominated by a bias term due to the penalty. In particular, since regularization is artificially introduced for restricting the ``effective size'' of the parameter space, the resulting estimator may be deteriorated and the estimation error cannot fall below the penalty level to adapt to a possibly faster convergence rate. For example, the estimator by solving the Lasso achieves the minimax rate of $\\sqrt{s}\\lambda\\asymp\\sqrt{s \\log p\/n}$. However, this worse-case rate only happens when there exist weak signals, meaning that some nonzero $\\beta^\\ast_j$'s have a borderline magnitude of order $\\sqrt{s\\log p\/n}$. In fact, if all signals are sufficiently strong, or significantly larger this borderline magnitude,\nthen the faster dimension-independent parametric rate $\\sqrt{s\/n}$ is attainable. For regularized approaches such the Lasso, one possible way to remove the penalty-induced bias term (whose order is $\\lambda$) is to refit the model with the selected variables. However, this two stage procedure requires stringent assumptions on the minimal signal strength to guarantee variable selection consistency for the first stage, and will suffer from weak signals. Interestingly, we show that by combining the Hadamard product over-parametrization with early stopping, a widely used regularization technique in boosting \\citep{zhang2005boosting} and nonparametric regression \\citep{raskutti2014early}, our method can overcome the saturation issue and lead to more accurate estimation. \nMore precisely, in the presence of random noise $w$ in the linear model, the solution path \nof minimizing the quadratic loss function~\\eqref{eq_opt} as we increase the gradient descent iteration still tends to converge to the least $\\ell_1$-norm solution that will overfit the data.\nFortunately, by terminating the gradient descent updating procedure earlier within a proper number of iterations, we may find a solution that optimally balances between the model complexity (reflected by the increasing $\\ell_1$-norm of the iterate) and goodness fit of the model, akin the bias-variance trade-off.\nIn particular, we show that the estimator can adapt to an optimal convergence rate of $\\sqrt{s\/n}$ when all signals are relatively strong. Generally, when both strong signals and weak signals exist, our estimator attains the rate $\\sqrt{s_1\/n}+\\sqrt{s_2 \\log p\/n}$ (with $s_1, s_2$ denoting thenumber of strong signals and weak signals, respectively).\n\n\nOur result also complements the recent surge of literature on over-parametrization for implicit regularization of the first-order iterative method for non-convex optimization in machine learning. \\cite{gunasekar2017implicit} introduce the phenomenon of implicit regularization in matrix factorization, where they empirically observe the convergence of gradient methods in matrix factorization problem to the minimal nuclear norm solution as the the initial value tends to zero. However, they only provide some heuristic illustration under some hard-to-check assumptions such as the continuity of the solution relative to the change in the initialization. Later, \\cite{li2018algorithmic} rigorously prove the implicit regularization in matrix sensing problem under a matrix RIP condition. Some other very recent works such \\cite{pmlr-v80-gunasekar18a} and \\cite{soudry2018implicit} study implicit regularizations in mirror descent and in classification problems. Note that all above implicit regularization literatures only consider data that are either noiseless (regression) or perfectly separable (classification). To our best knowledge, we are the first to rigorously study and utilize implicit regularization in high dimensional linear regression where responses are noisy. \n\n\nIn a nutshell, we show that through a simple change-of-variable, the non-smooth $\\ell_1$- penalized optimization problem~\\eqref{Eqn:CS} can be transformed to an unconstrained smooth quadratic loss minimization one; moreover, a simple gradient descent initialized near zero efficiently solves this non-convex optimization problem with provable guarantees. Furthermore, our method enjoy several advantages over existing procedures for high dimensional linear regression under sparsity constraints. First, our method is computationally efficient --- its time complexity is $O(np)$, which is linear in both $n$ and $p$.  Second, despite the non-convexity nature, our method has a natural initialization that provably leads the optimal solution.\nIn comparison, penalized $M$-estimators based on non-convex penalties such as SCAD and MCP require stringent conditions on their initializations: to obtain good estimators, they require good initial values that are sufficiently closed to the truth (theoretically) or satisfy some restricted strong convexity conditions \\citep{zhao2018pathwise}, otherwise their optimization algorithms will suffer from bad local minima with bad generalization errors. In contrast, our algorithm only requires the initialization to be closed to zero. Moreover, unlike penalized approaches such as SCAD and MCP, where both parameters for the noise level and the concavity of the penalty need to be tuned, our method only need to tune the number of iterations. \n\n\n\n\nTo conclude, our main contributions with respect to the relative literatures are as follows:\n\\begin{enumerate}\n\t\\item We propose an estimator by combining early stopping with implicit regularization to overcome the saturation issues in high dimensional regression with explicit regularizations;\n\t\\item Unlike recent implicit regularization literatures that exclusively focus on noiseless data, we are the first to rigorously study the effect of implicit regularization for noisy data;\n\t\\item From computational perspective, we transform the non-smooth optimization problem to an unconstrained smooth quadratic loss minimization problem for which standard optimization tools can be applied.\n\\end{enumerate}\n\n\n\\section{Background and Our Method}\nTo begin with, we formally introduce the setup and notations used throughout the paper. After that, we introduce the intuition for our new implicit regularized algorithm for high dimensional linear regression via Hadamard product parameterization.\n\n\\subsection{Setup and notations}\nRecall that $\\beta^\\ast$ is the unknown $s$-sparse signal in $\\mathbb{R}^{p}$ to be recovered. Let $S\\subset\\{1,\\ldots,p\\}$ denote the index set that corresponds to the nonzero components of $\\beta^\\ast$, and the size $|S|$ of $S$ is then $s$.\nFor two vectors $g, l\\in \\mathbb{R}^{p}$, we call $\\beta = g\\circ l \\in\\mathbb R^p$ as their Hadamard product, whose components are $\\beta_j = g_jl_j$ for $j=1,\\ldots p$. For two vectors $a,b\\in\\mathbb R^p$, we use the notation $a\\geq b$ to indicate element-wise ``great than or equal to''.\nWhen there is no ambiguity, we use $\\beta^2=\\beta\\circ \\beta$ to denote the self-Hadamard product of $\\beta$.  For a function $f:\\mathbb R^p\\times \\mathbb R^p \\to \\mathbb R$, $(g,\\,l)\\mapsto f(g,\\,l)$, we use $\\nabla_g f$ and $\\nabla_l f$ to denote its partial derivative relative to $g$ and $l$, respectively. \nFor any index set $J\\subset \\{1,\\ldots,p\\}$ and vector $a\\in \\mathbb R^p$, we use $a_J=(a_j:\\, j\\in J)$ to denote the sub-vector of $a$ formed by concatenating the components indexed by $J$. Let $\\mathbf 1\\in\\mathbb R^p$ denote the vector with all entries as $1$, and $I$ as the identity matrix in $\\mathbb R^p$.  Let $I_J$ be the diagonal matrix with one on the $j$th diagonal for $j\\in J$ and $0$ elsewhere. For a vector $a\\in\\mathbb R^p$, we use $\\|a\\|$ to denote its vector-$\\ell_2$-norm, and $\\|a\\|_\\infty=\\max_{j}|a_j|$ its $\\ell_\\infty$-norm. Let $\\mbox{Unif}(a,b)$ to denote the uniform distribution over interval $(a,b)$. For a symmetric matrix $A$, let \n$\\lambda_{\\min}(A)$ denote its smallest eigenvalue. For two sequences $\\{a_n\\}$ and $\\{b_n\\}$, we use the notation $a_n\\lesssim b_n$ or $a_n\\gtrsim b_n$ to mean there exist some constant $c$ and $C$ independent of $n$ such that $a_n \\leq Cb_n$ or $a_n \\geq cb_n$ for all $n<0$, respectively, and $a_n\\asymp b_n$ to mean $a_n\\lesssim b_n$ and $b_n \\lesssim a_n$.\n\n\\subsection{Gradient descent with Hadamard product parametrization}\nAs we mentioned in the introduction, we consider augmenting the $p$-dimensional vector $\\beta$ into two $p$-dimensional vectors $g,\\,l$ through $\\beta=g\\circ l$. Instead of solving the Lasso problem~\\eqref{Eqn:CS} with $\\beta$ replaced with $g\\circ l$, we consider directly applying gradient descent to the quadratic loss function $f(g,l)=(2n)^{-1}\\|X(g\\circ l)-y\\|^2$. \nIn particular, we apply the updating formula $g_{t+1}=g_t-\\eta \\nabla f_g(g_t,\\,l_t)$, $l_{t+1}=l_t-\\eta \\nabla_l f(g_t,l_t)$, with random initial values $g_0$ and $l_0$ chosen close enough to $0$ (notice that $(0,0)$ is a saddle point of the objective function, so we need to apply a small perturbation $\\alpha$ on the initial values).\nThis leads to the following algorithm:\n\\smallskip\n\n\\begin{algorithm}[H]\n\\KwData{Design matrix $X\\in\\mathbb R^{n\\times p}$,\\, measurement vector $y\\in\\mathbb R^n$, initialization magnitude $\\alpha$, step size $\\eta$, and stopping threshold $\\epsilon$;}\n Initialize variables $[g_0]_j\\overset{iid}{\\sim}\\mbox{Unif}(-\\alpha,\\alpha)$, $[l_0]_j\\overset{iid}{\\sim}\\mbox{Unif}(-\\alpha,\\alpha)$ for $j=1,\\ldots,p$, and iteration number $t=0$;\\\\\n \\While{$\\ \\|X(g_t\\circ l_t)-y\\|\/\\sqrt{n}>\\epsilon\\ $}{\n  ${g}_{t+1}=g_t-\\eta \\ l_t \\circ \\big[n^{-1}\\,X^{T}\\big(X(g_t\\circ l_t)-y\\big)\\big]$;\\\\ \n  $\\, {l}_{t+1}=l_t-\\eta \\ g_t \\circ \\big[n^{-1}\\,X^{T}\\big(X(g_t\\circ l_t)-y\\big)\\big]$; \\\\[0.2em]\n  $\\, t=t+1$;\\\\\n}\n \\KwResult{Output the final estimate $\\widehat \\beta=g_t\\circ l_t$;\n }\\label{alg1}\n \\caption{Gradient Descent for linear regression}\n\\end{algorithm}\n\\smallskip\n\nAlgorithm~\\ref{alg1} is the standard gradient descent algorithm, and the iterates $(g_{t+1},l_{t+1})$ tend to converge to a stationary point $(g_\\infty, l_{\\infty})$ of $f(g,l)$ that satisfies the first order optimality condition $\\nabla f_g(g_\\infty,\\,l_\\infty) = 0$ and $\\nabla f_l(g_\\infty,\\,l_\\infty) = 0$. However, stationary points of $f(g,l)$ can be local minimum, local maximum, or saddle points (when the Hessian matrix $\\nabla^2_{g,l} f(g,l)$ contains both positive and negative eigenvalues).\nThe following result provides the optimization landscape of $f(g,l)$, showing that $f(g,l)$ does not have local maximum, all its local minimums are global minimum, and all saddle points are strict. The strict saddle points are saddle points with negative smallest eigenvalues for Hessian matrix.\n\n\\begin{lem}\\label{Lem:global_min}\n$f(g,l)=(2n)^{-1}\\|X(g\\circ l)-y\\|^2$ does not have local maximum, and all its local minimums are global minimum. In particular, $(\\bar g, \\bar l\\,)$ is a global minimum of $f(g,l)$ if and only if\n\\begin{align*}\nX^T\\big(X(\\bar g\\circ \\bar l)-y\\big) = 0.\n\\end{align*}\nIn addition, any saddle point $(g^\\dagger,l^\\dagger)$ of $f(g,l)$ is a strict saddle, that is,\n$\\lambda_{\\min}\\big(\\nabla^2_{g,l} f(g^\\dagger,l^\\dagger)\\big)<0$.\n\\end{lem}\n\n\\noindent According to the first order condition associated with $f(g,l)$\n\\begin{align*}\ng\\circ \\big[X^T\\big(X(g\\circ l)-y\\big)\\big]=l\\circ \\big[X^T\\big(X(g\\circ l)-y\\big)\\big] = 0,\n\\end{align*}\nthere could be exponentially many (at least $2^p-1$) saddle points as a solution to this equation, for example, for those $(g,l)$ satisfying\n\\begin{align*}\ng_A=l_A =0\\in\\mathbb R^{|A|}, \\qquad\\mbox{and}\\qquad \\big[X^T\\big(X(g\\circ l)-y\\big)\\big]_{A^c} = 0\\in \\mathbb R^{p-|A|},\n\\end{align*}\nfor any non-empty subset $A$ of $\\{1,\\ldots,p\\}$. Consequently, the gradient descent algorithm may converge to any of these bad saddle points. To see this, if we initialize $(g,l)$ in a way such that the components in the index set $A$ are zero, or $[g_0]_A=[l_0]_A=0$, then these components will remain zero forever in the gradient iterations, or $[g_t]_A=[l_t]_A=0$ for all $t>0$. Fortunately, the following result implies that as long as we use a random initialization for $(g,l)$ with continuous pdf over $\\mathbb R^{2p}$ as in Algorithm~\\ref{alg1}, then the gradient descent almost surely converges to a global minimum.\n\n\\begin{lem}\\label{Lem:GD_global_min}\nSuppose the step size $\\eta$ is sufficiently small. Then with probability one, Algorithm~\\ref{alg1} converges to a global minimum of $f(g,l)$.\n\\end{lem}\n\nIn the low-dimensional regime where the design matrix $X$ has full column rank, the solution $\\bar \\beta$ to the normal equation $X^T(X\\beta-y) = 0$ is unique, which is also the least squares estimator. Under this scenario, Lemma~\\ref{Lem:global_min} and Lemma~\\ref{Lem:GD_global_min} together certify that Algorithm~\\ref{alg1} will converge to this optimal least squares estimator.  However, in the high-dimensional regime which is the main focus in the paper, the normal equation $X^T(X\\beta-y) = 0$ have infinitely many solutions, and it is not clear which solution the algorithm tends to converge to. For example, if we consider instead applying the gradient descent to the original parameter $\\beta$ in the objective function $(2n)^{-1}\\|X\\beta-y\\|^2$ with initialization $\\beta_0=0$, then the iterates will converge to the minimal $\\ell_2$-norm solution of the normal equation (see below for details). Interestingly, as we will illustrate in the following, under the Hadamard parametrization the gradient descent Algorithm~\\ref{alg1} now tends to converge to the minimal $\\ell_1$-norm solution under certain conditions for initialization, thereby inducing sparsity and naturally facilitating variable selection.\n\n\n\n\\subsection{Gradient descent converges to sparse solution}\\label{Sec:Heuristic}\nIn this subsection, we provide two different perspectives for understanding the following informal statement on the behavior of simple gradient descent for the loss function $f(g,\\,l)$ defined in~\\eqref{eq_opt} under the Hadamard product parameterization $\\beta =g\\circ l$. For simplicity, we assume that the response $y$ in the linear model is perfect, that is, the noise variance $\\sigma^2=0$, throughout this subsection.\n Then in the next subsection, we turn to general noisy observations, and propose methods that lead to optimal estimation when the true regression coefficient $\\beta^\\ast$ is sparse. \n\n\\paragraph{Informal Statement:} \\emph{If we initialize the algorithm to be arbitrarily close to $g=l=0$, then under suitable conditions on design $X$, a simple gradient descent converges to a solution of basis pursuit problem:\n\t\\begin{equation}\\label{bs}\n\t\\min_{\\beta\\in\\mathbb R^p}{\\|\\beta\\|_1}  \\quad \\mbox{subject to}  \\quad X\\beta =y.\n\t\\end{equation}}\nOur first perspective is based on the $\\ell_2$-norm implicit regularization in linear system, and the second is based on analyzing the gradient dynamical system as the limit of the gradient descent algorithm as the step size $\\eta\\to 0_{+}$. However, both perspectives in this section are heuristic, and formal statements and proofs (based on a different strategy) will be provided in Section~\\ref{Sec:Theory}.\n\n\\paragraph{$\\ell_2$-norm implicit regularization perspective:} Consider the under-determined system $X\\beta=y$, where $X\\in\\mathbb R^{n\\times p}$ has full row rank.\nOur first intuition comes from the fact that a zero-initialized gradient descent algorithm over $\\beta\\in\\mathbb R^p$ for solving \n\\begin{align*}\n\\min_{\\beta\\in\\mathbb R^p} \\frac{1}{2n}\\,\\|X\\beta - y\\|^2:\\,= g(\\beta)\n\\end{align*}\nfinds a minimal $\\ell_2$-norm solution to $X\\beta=y$. \n\nIn fact, we know that any solution to the linear system $X\\beta=y$ takes the form of\n\\begin{align*}\n\\beta = X^{+} y + [I - X^+X]w,\\quad \\mbox{over all }w\\in\\mathbb R^p,\n\\end{align*}\nwhere $X^+=\\lim_{\\lambda\\to 0_+}(X^TX+\\lambda\\, I)^{-1}X^T$ is the Moore-Penrose inverse of $X$. Since $X(I-X^+X)=0$, we have \n\\begin{align*}\n\\|\\beta\\|^2 = \\|X^{+} y\\|^2 + \\| [I - X^+X]w\\|^2 \\geq \\|X^{+} y\\|^2,\n\\end{align*}\nimplying that $X^{+}y$ is the unique solution of $X\\beta=y$ in the column space of $X^T$, which is also the minimal $\\ell_2$-norm solution. \nNow since the gradient updating formula for $\\beta$, $\\beta_{t+1}=\\beta_t-\\eta X^T(X\\beta_t-y)\/n$, implies that the difference $\\beta_t-\\beta_0$ always lies in the column span of $X^T$. Let $\\beta_\\infty:\\,=\\lim_{t\\to\\infty}\\beta_t$ be the limiting point of the gradient algorithm. Then $\\beta_\\infty$ must be a solution to $X\\beta=y$. On the other hand , when $\\beta_0$ is initialized at zero, $\\beta_\\infty$ should also belong to the column span of $X^T$. These two properties combined imply that $\\beta_\\infty$ must be the minimal $\\ell_2$-norm solution $X^{+}y$.\n\nIn high dimensional linear regression with perfect observations, a popular class of penalization methods attempts find the minimal $\\ell_1$-norm solution to $X\\beta=y$ .  \nUnder the Hadamard product parameterization $\\beta =g\\circ l$, the fact that gradient descent tends to find the  minimal $\\ell_2$-norm solution suggests (this is not rigorous) that the gradient descent algorithm for jointly minimizing $f(g,\\,l)$ over $(g,\\,l)$ tends to converge to a solution to $X(g\\circ l) =y$ with a minimal $\\ell_2$-norm $\\sqrt{\\|g\\|^2+\\|l\\|^2}$. However, a minimal $\\ell_2$-norm solution to $X(g\\circ l) =y$ must satisfy $|g_j|=|l_j|$ for each $j=1,\\ldots,p$ (otherwise we can always construct another solution with strictly smaller $\\ell_2$-norm), which implies $\\sqrt{\\|g\\|^2+\\|l\\|^2} = \\sqrt{2}\\, \\|g\\circ l\\|_1=\\sqrt{2}\\,\\|\\beta\\|_1$. As a consequence, $\\beta_{\\infty}=g_{\\infty}\\circ l_{\\infty}$ should be the minimal $\\ell_1$-norm solution to $X\\beta = y$.\n\nAnother way to understand the difference in the evolutions of gradient descents for $f(g,l)$ and $h(\\beta)$ is by noticing that the gradient $\\nabla_{g_j} f(g,l) = l_j\\cdot \\nabla_{\\beta_j} h(\\beta)\\big|_{\\beta =g\\circ l}$ in the new parametrization, for each $j=1,\\ldots,p$, has an extra multiplicative factor of $l_j$ than the gradient $\\nabla_{\\beta_j} h(\\beta)$ in the usual least squares of minimizing $g(\\beta)$. It is precisely this extra multiplicative factor $l_j$ that helps select important signals (nonzero regression coefficients) and prevent unimportant signals (zero regression coefficients) to grow too fast at the early stage of the evolution when both $g$ and $l$ are close to zero. Precisely, as we will show in our theory part (Section~\\ref{Sec:Theory}), under suitable conditions on the model, all unimportant signals remain to stay in a $\\mathcal{O}(p^{-1})$ neighbourhood of zero, while important ones tend to grow exponentially fast. \n\n\n\\paragraph{Gradient dynamical system perspective:} Our second perspective comes from considering the limiting gradient dynamical system of the problem (i.e.~gradient descent with an infinitesimally small step size), which is motivated by the interpretation for matrix factorization problems in~\\cite{gunasekar2017implicit} and \\cite{pmlr-v80-gunasekar18a}. In particular, the behavior of this limiting dynamical system is captured by the ordinary differential equations\n\\begin{equation}\\label{de}\n\\begin{cases} \n\\ \\dot{g}(t)=-\\big[ X^{T}r(t)\\big] \\circ l(t),\\\\ \n\\ \\, \\dot{l}(t)=-\\big[X^{T}r(t)\\big] \\circ g(t),\n\\end{cases}\\quad\\mbox{with initialization}\\quad \n\\begin{cases} \n\\ g(0)=\\alpha\\mathbf 1,\\\\ \n\\ \\, l(0)= 0,\n\\end{cases}\n\\end{equation}\nwhere $r(t)=n^{-1}\\big[X(g(t)\\circ l(t))-y\\big]\\in\\mathbb R^p$, and for simplicity we  fixed the initialization. To emphasize the dependence of the solution on $\\alpha$, we instead write $g(t),\\,l(t),\\,r(t)$ as $g(t,\\alpha), \\,l(t,\\alpha),\\,r(t,\\alpha)$.\nFor illustration purposes, we assume that the limiting point of this system is continuous and bounded as the initialization value $\\alpha\\to 0_+$, that is, both limits $g_\\infty=\\lim_{t\\to\\infty, \\alpha\\to 0_+}g(t,\\alpha)$ and $l_\\infty=\\lim_{t\\to\\infty, \\alpha\\to 0_+}l(t,\\alpha)$ exist in $\\mathbb R^p$ and are finite. \n\nLet $s(t,\\alpha)=\\int_0^t r(\\tau,\\alpha) d\\tau\\in\\mathbb R^p$, then simple calculation leads to the relation\n\\begin{equation*}\n\\left[ \\begin{array}{c} g_j(t,\\alpha)+l_j(t,\\alpha) \\\\[0.3em] g_j(t,\\alpha)-l_j(t,\\alpha) \\end{array} \\right] \n= \\alpha\\,\\left[ \\begin{array}{c}\\exp(-X_j^{T}s(t,\\alpha)) \\\\[0.3em]  \\exp(X_j^{T}s(t,\\alpha)) \\end{array} \\right],\\quad\\mbox{for each }j=1,\\ldots,p.\n\\end{equation*}\nUnder the aforementioned assumption on the existence of limits as $t\\to\\infty$ and $\\alpha\\to 0_+$, the preceding display implies one of the following three situations for each $j$:\n\\begin{enumerate}\n\t\\item[Case 1:] $g_{j,\\infty}=l_{j,\\infty}\\neq 0$, and \n\t$\\displaystyle \\lim_{t\\to\\infty,\\alpha\\to 0_+}X_j^{T}s(t,\\alpha)\/\\log(1\/\\alpha)=  1$.\n\t\\item[Case 2:] $g_{j,\\infty}=-l_{j,\\infty}\\neq 0$, and \n\t$\\displaystyle \\lim_{t\\to\\infty,\\alpha\\to 0_+}X_j^{T} s(t,\\alpha)\/\\log(1\/\\alpha)=  -1$.\n\t\\item[Case 3:] $g_{j,\\infty}=l_{j,\\infty}=0$, and \n\t$\\displaystyle \\lim_{t\\to\\infty,\\alpha\\to 0_+}X_j^{T} s(t,\\alpha)\/\\log(1\/\\alpha) =\\gamma_j\\in[-1,1]$.\n\\end{enumerate}\nDenote $s_\\infty$ as the limit $\\lim_{t\\to\\infty,\\alpha\\to 0_+} s(t,\\alpha)\/\\log(1\/\\alpha)$. Recall $\\beta_\\infty = g_\\infty\\circ l_\\infty$, and the previous three cases can be unified into\n\\begin{equation*}\nX_j^{T}s_\\infty=\n\\begin{cases} \n\\mbox{sign}(\\beta_{j,\\infty}), & \\mbox{if}\\  \\beta_{j,\\infty}\\neq 0, \\\\ \n\\gamma_j\\in [-1,1], &  \\mbox{if}\\ \\beta_{j,\\infty}= 0,\n\\end{cases}\\quad\\mbox{for each }j=1,\\ldots,p.\n\\end{equation*}\nThis identity together with the limiting point condition $X\\beta_{\\infty}=y$\ncoincides with the KKT condition for the basis pursuit problem~\\eqref{bs}.\n\n\n\nAgain, this argument is based on the hard-to-check solution continuity assumption. In the next section, we provide a formal proof of the result without making this assumption, albeit under a somewhat strong RIP condition on $X$. We conjecture this gradient descent implicit regularization property to be generally true for a much wider class of design matrices (see our simulation section for numerical evidences).\n\n\n\n\\subsection{Gradient descent with early stopping}\nIn this subsection, we consider the general case where the response $y$ contains noise, or $\\sigma^2\\neq 0$. In particular, we propose the use of early stopping, a widely used implicit regularization technique \\citep{zhang2005boosting,raskutti2014early}, to the gradient descent Algorithm~\\ref{alg1}. As the name suggests, we will use certain criterion to decide whether to terminate the gradient descent updating to prevent overfitting of the data.  Algorithm~\\ref{alg2} below summarizes a particular stopping criterion widely-used in the machine learning community via validation. In principal, we can also treat the number of iterations as a hyperparameter and repeat this procedure multiple times  in same spirits as data splitting and cross validation.\n\n\\smallskip\n\n\\begin{algorithm}[H]\n\t\\KwData{Training design matrix $X\\in\\mathbb R^{n\\times p}$,\\, measurement vector $y\\in\\mathbb R^n$, validation data $X'$, $y'$, initialization magnitude $\\alpha$, step size $\\eta$, and maximal number of iterations $T_{max}$;}\n\tInitialize variables $[g_0]_j\\overset{iid}{\\sim}\\mbox{Unif}(-\\alpha,\\alpha)$, $[l_0]_j\\overset{iid}{\\sim}\\mbox{Unif}(-\\alpha,\\alpha)$ for $j=1,\\ldots,p$, and iteration number $t=0$;\\\\\n\t\\While{$t<T_{max}$}{\n\t\t${g}_{t+1}=g_t-\\eta \\ l_t \\circ \\big[n^{-1}\\,X^{T}\\big(X(g_t\\circ l_t)-y\\big)\\big]$;\\\\ \n\t\t$\\, {l}_{t+1}=l_t-\\eta \\ g_t \\circ \\big[n^{-1}\\,X^{T}\\big(X(g_t\\circ l_t)-y\\big)\\big]$; \\\\[0.2em]\n\t\t$\\, t=t+1$;\\\\\n\t}\n\t\\KwResult{Choose the first ${\\tilde t}$ such that $\\|X'(g_{{\\tilde t}}\\circ l_{{\\tilde t}})-y'\\|> \\|X'(g_{{\\tilde t}+1}\\circ l_{{\\tilde t}+1})-y'\\|$ or $\\|X'(g_{{\\tilde t}}\\circ l_{{\\tilde t}})-y'\\|$ is minimized over all iterations, then output the final estimate $\\widehat \\beta=g_{{\\tilde t}}\\circ l_{{\\tilde t}}$.\n\t}\\label{alg2}\n\t\\caption{Gradient Descent for Linear Regression with Validation Data} \n\\end{algorithm} \n\n\\smallskip\n\nRecall that in the introduction, we discussed about the saturation issue suffered by explicit penalized methods such as the Lasso. Now we turn to our method and illustrate that it is unaffected, or at least less suffered from the saturation issue. In the next section, we will provide rigorous theory showing that our method can achieve a faster $\\sqrt{s\/n}$ rate of convergence then the typical $\\sqrt{s\\log p\/n}$ rate when all nonzero signals are relatively large.\n\nDue to the connection of our method with the basis pursuit problem~\\eqref{bs}, one may naturally think that our method in the noisy case should be equivalent to a basis pursuit denoising problem:\n\\begin{equation}\\label{bsdn}\n\\min \\|\\beta\\|_1  \\quad \\mbox{subject to} \\quad \\|X \\beta -y\\|_2 \\leq \\epsilon,\n\\end{equation}\nwith some error tolerance level $\\varepsilon$ depending on the stopping criterion, and therefore is equivalent to the Lasso. Surprisingly, a simulation example below shows that the iterate path of the gradient descent Algorithm~\\ref{alg1} contains estimates with much smaller error than the Lasso. Precisely, we adopt the simulation setting S2 in section~\\ref{sec:simu} . As comparisons, we also report the Lasso solution path (as a function of the regularization parameter $\\lambda$) solved by ISTA and FISTA \\citep{beck2009fast}. For our gradient descent algorithm, we set $\\alpha = 10^{-5}$ in the random initialization. From figure~\\ref{fig:31}, when the iteration number is around $1000$, even though the prediction error in panel~(c) of our algorithm and the Lasso (with an optimally tuned $\\lambda$, see panel~(b) for the entire Lasso solution path), the estimation error in panel~(a) of our method is significantly lower than that of the Lasso, illustrating the occurrence of the saturation phenomenon of the Lasso. Moreover, the stabilized region (iterations $200$--$1000$) of our method GD in panel~(a) is relatively wide, and therefore the performance tends to be robust to the stopping criterion. \n\n\n\n\\begin{figure}[H]\n\t\\begin{subfigure}{0.32\\textwidth}\n\t\t\\includegraphics[width=\\linewidth]{estimation2.jpg}\n\t\t\\caption{Estimation error vs Iteration}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.32\\textwidth}\n\t\t\\includegraphics[width=\\linewidth]{lassopath.jpg}\n\t\t\\caption{Estimation error vs Regularization for Lasso}\n\t\\end{subfigure} \n\t\\begin{subfigure}{0.32\\textwidth}\n\t\t\\includegraphics[width=\\linewidth]{prediction2.jpg}\n\t\t\\caption{Prediction error vs Iteration}\n\t\t\\label{fig:32}\n\t\\end{subfigure} \n\t\\caption{Panel (a) is a log-log plot of standardized estimation error $\\|\\widehat \\beta-\\beta^\\ast\\|^2_2\/\\|\\beta^\\ast\\|^2_2$ versus iteration number $t$. Panel (b) is a log-log plot of standardized estimation error versus regularization parameter $\\lambda$ for Lasso. Panel (c) is a log-log plot of mean prediction error $\\sqrt{\\|\\widehat y-y\\|^2_2\/n}$ versus iteration number $t$.}\\label{fig:31}\n\t\\vspace{-0.7em}\n\\end{figure}\n\n\nNext, let us briefly illustrate why implicit regularization with early stopping works, while explicit regularized methods may fail.\nWe know that early stopping, serving as algorithmic regularization, is based on the intuition that as the iteration number grows, the bias keeps decreasing while the variance increasing. Consequently, the iteration number $T$, acting as an implicit regularization parameter, aims to optimally balance between the bias and the variance, akin to the bias-variance trade-off. In our parametrization, the iteration number $T$ controls the $\\ell_1$ norm of the solution, reflecting the model space complexity. To see this, we plot the $\\ell_1$ norm versus the iteration number, and also the estimation errors versus the $\\ell_1$ norm, all in logarithmic scales, in figure~\\ref{fig:33}. As we expected, as the number of iterations increases, the $\\ell_1$ norm of the iterate also increases. When the logarithm of the iteration number is within $(2.3,3)$, the $\\ell_1$ norm of the estimated coefficients tends to be stabilized at the $\\ell_1$ norm of the true $\\beta^\\ast$ as $0.9$, corresponding to the most accurate estimation region in panel~(a) of figure~\\ref{fig:31}. In contrast, as we can see from panel~(b) of figure~\\ref{fig:33}, the estimation error is very sensitive in the regularization parameter domain --- the region corresponds to smallest estimation accuracy is very narrow, and a small change in the $\\ell_1$ norm in the solution leads to a drastic deterioration in the estimation accuracy. This numerical example provides evidences of why explicit regularized approaches may suffer from large bias and low accuracy.\n\n\n\n\\begin{figure}[t]\n\t\\begin{subfigure}{0.48\\textwidth}\n\t\t\\includegraphics[width=\\linewidth]{L1IN.jpg}\n\t\t\\caption{$\\ell_1$ norm vs Iteration}\n\t\\end{subfigure}\n\t\\begin{subfigure}{0.48\\textwidth}\n\t\t\\includegraphics[width=\\linewidth]{EEL1.jpg}\n\t\t\\caption{Prediction error vs Iteration Comparison}\n\t\\end{subfigure} \n\t\\caption{Panel (a) is a log-log plot of $\\ell_1$ norm of the estimated coefficients versus iteration number $t$. Panel (b) is a log-log plot of standardized estimation error versus $\\ell_1$ norm of the estimated coefficients.} \\label{fig:33}\n\t\\vspace{-0.7em}\n\\end{figure}\n\nFinally, we discuss several commonly used early stopping rules by treating the iteration number as a tuning parameter. \n\n\n\\paragraph{Hold-out or cross validation:} The simplest method is to use hold-out data as validation: for example, we can split the training data into half-half, and then run gradient descent on the first half $D_1$ of the data while calculate the prediction error $R(t)=\\sum_{i\\in D_2}({X^{(i)}}^T (g_t\\circ l_t)-y_i)^2$ for all $t  \\leq T_{max} $ on the second half $D_2$, then the final iteration number is decided by (cf.~Algorithm~\\ref{alg2}):\n\t\\begin{equation}\\label{cv}\n\t\\tilde t : = \\arg \\min \\{ t \\leq T_{max}  \\,|\\, R ( t+1 ) > R (t) \\} \\quad \\mbox{or}\n\t\\end{equation} \n\t\\begin{equation}\\label{cv2}\n\t\\tilde t : = \\arg \\min \\{ t \\leq T_{max} \\,|\\, R ( t ) = \\min_{\\tau \\leq T_{max}} R (\\tau) \\}. \\quad \n\t\\end{equation} \n\tTo make use of the whole dataset, we can perform cross validation: first split data into $K$ folds, then apply gradient descent on all possible combinations of $K-1$ folds without replacement and evaluate at the corresponding rest $1$ folds. The final risk $R(t)$ can be the sum of all the evaluations on each fold, and the criterion~\\eqref{cv} or~\\eqref{cv2} can be used to obtain the iteration number. Finally we can apply the same iteration number obtained from cross validation to approximate the optimal one for the entire training data.\n\\paragraph{Stein's unbiased risk estimate (SURE):} \\cite{stein1981estimation} suggested the use of degrees of freedom as surrogate to the true prediction error given the standard derivation $\\sigma$. Under our settings, ignoring the second order term of step size $\\eta$, the updating of the prediction error (up to rescaling) $r_t=n^{-1}\\big[X(g_t\\circ l_t) - y\\big]\\in\\mathbb R^n$ through gradient descent can be approximated by (by ignoring second order terms of order $\\eta^2$):\n\t\\begin{equation}\n\tr_{t+1}\\approx [I - 2 \\eta n^{-1} X \\mbox{diag}(|g_t\\circ l_t|) X^T] \\,r_t, \n\t\\end{equation} \n\twhere for a vector $u$, diag$(u)$ denotes the diagonal matrix with components of $u$ in the its diagonals. Define $S_t= \\Pi_{i=1}^{t-1} (I - 2 \\eta n^{-1} X \\mbox{diag}(|g_t\\circ l_t|) X^T)$, then the estimated degrees of freedom at time $t$ can be approximated by $n-\\mbox{trace}(S_t)$. Consequently, the risk based on the $C_p$-type statistic \\citep{efron2004estimation} is\n\t\\begin{equation}\n\tR(t) = \\frac { \\| r_t \\| ^ { 2 } } { n } + \\Big(2-\\frac { 2 \\mbox{trace}(S_t) } { n } \\Big)\\sigma ^ { 2 }.\n\t\\end{equation} \n\tThe total iteration number as a tunign parameter can then be selected by minimizing $R(t)$ in equation~\\eqref{cv} or~\\eqref{cv2} . In practice, we can use the plug-in estimator $\\hat{\\sigma}$ to replace the unknown $\\sigma$ in $R(t)$. According to our simulation studies (for example, see figure~\\ref{fig:es}), early stopping based on SURE generally works not as good as the hold-out or cross validation method.\n\t\n\\paragraph{Measure of model complexity:}  \\citep{raskutti2014early} proposed an early stopping rule based on  local empirical Rademacher complexity of the Reproducing kernel Hilbert space. However, their method can not be directly applied in our case: their stopping rule is based on the eigenvalues of empirical kernel matrix, which is $ n^{-1} X \\mbox{diag}(|g_t\\circ l_t|)  X^T$ in our settings. Since our empirical kernel matrix keeps updated throughout the iterations, their method is not directly applicable.\n\t\n\t\\smallskip \n\tIn the end of this subsection, we adopt the simulation framework S1-S4 (only change the standard derivation to $\\sigma=0.1$) in section~\\ref{sec:simu} to compare different early stopping criteria. We record the mean estimation errors averaging over $50$ trials  and report the errors in figure~\\ref{fig:es}. From the figure, we can see that cross-validation tends to be more robust than SURE. \t\n\tTherefore, we recommend using hold-out or cross validation to select the iteration number, and will stick to this method in the rest of the paper.\n\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\t\\includegraphics[scale=0.7]{ESrule.jpeg}\n\t\t\\caption{Comparison of the estimation errors for different early stopping rules, `Oracle' stands for the optimal early stopping rule with knowldege on the truth. 'CV' stand for early stopping through $5$ fold cross validation.} \\label{fig:es}\n\t\t\\vspace{-0.7em}\n\t\\end{figure}\n\t\n\n\\subsection{Adaptive step size and variable selection} \\label{sec:2.5}\n\nA nature extension of gradient descent algorithm~\\ref{alg1} is to assign different weights (step sizes) to different coordinates of $\\beta$, which is related to the adaptive Lasso \\citep{zou2006adaptive}. It can be seen from the differential equation interpretation: by inserting a constant weighting matrix $D(\\Omega)=\\mbox{diag}(\\omega_1,...,\\omega_p)$ into the equation~\\eqref{de}, we obtain the limiting dynamical system as\n\\begin{equation*}\n\\begin{cases} \n\\ \\dot{g}(t)=-\\big[D(\\Omega) X^{T}r(t)\\big] \\circ l(t),\\\\ \n\\ \\, \\dot{l}(t)=-\\big[D(\\Omega)X^{T}r(t)\\big] \\circ g(t).\n\\end{cases}\n\\end{equation*}\nBased on similar heuristic analysis as in Section~\\ref{Sec:Heuristic} for the noiseless case, the limiting point of the dynamic system satisfies:\n\\begin{equation*}\nX_j^{T}s_\\infty=\n\\begin{cases} \n\\mbox{sign}(\\beta_{j,\\infty}) \/\\omega_j, & \\mbox{if}\\  \\beta_{j,\\infty}\\neq 0, \\\\ \n\\gamma_j\\in [-\\frac{1}{\\omega_j},\\frac{1}{\\omega_j}], &  \\mbox{if}\\ \\beta_{j,\\infty}= 0,\n\\end{cases}\\quad\\mbox{for each }j=1,\\ldots,p.\n\\end{equation*}\nwhich is the KKT condition for the dual form of the adaptive Lasso \n\\begin{align*}\n\\min_{\\beta\\in\\mathbb R^p}\\sum_{j=1}^p\\frac{|\\beta_j|}{w_j} \\qquad\\mbox{subject to } X\\beta =y.\n\\end{align*}\nIn the limiting case when the step size $\\omega_j$ of a particular component $\\beta_j$ tends to $0$, we are equivalently adding an $+\\infty$ when $\\beta_j\\neq 0$. In contrast, if we apply a larger step size $\\omega_j$ to $\\beta_j$, then $\\beta_j=g_j\\circ l_j$ tend to move faster and more freely in the parameter space, which is equivalent to a smaller penalty on $\\beta_j$.\nThe original paper in \\cite{zou2006adaptive} constructed the weights based on the ordinary least square solution, which requires $n\\geq p$. In practice when $p>n$, we can construct weights through a preprocessing step. For example, variable screening can be applied to introduce sparse weights.\n\n\nTo enable variable selection in our method, we can perform a component-wise hard thresholding operation to the final estimator $\\hat{\\beta}=g_{{\\tilde t}}\\circ l_{{\\tilde t}}$. Based on our theoretical analysis, since our method tries to shrink both weak signals and errors into very small order $p^{-2}$, it is more robust to false detections than other explicit regularizations when the same tuning parameter for noise level is applied. Let us consider a simple example to illustrate the basic idea: we set $n=10$, $p=20$, $X_{ij} \\overset{iid}{\\sim} \\mathbb{ N }(0,1)$ for $i=1,2,...,n$ and $j=1,2,...,p$,  $\\beta^*_1=0.5 \\sigma\\sqrt{\\log p\/n}$, $\\beta^*_2=5 \\sigma \\sqrt{\\log p\/n}$, and all other components are zeros in the data generating model $y=X\\beta^* +w$ with $w \\sim \\mathbb{ N }(0,I)$.  Since the strength of the first components of truth is weak, it is hard to be detected by all methods we have tried. However, the effect of the weak signals on $y$ still pertains. In particular, when applying the cross validation, traditional penalized methods tends to over-select the predictors, leading a lot of false discoveries.  In comparison, due to the implicit regularization our method tend to be more robust to the weak signals---our estimate is typically non-sparse, the effect of the non-detected weak signals can be distributed to all components of the estimated vector, and no component is particularly spiked. As a consequence, our method tends to be more robust to false discoveries after applying the hard thresholding. The variable selection results are shown in figure~\\ref{fi:vsl}. As we can see, the Lasso can not detect the weak signal, and two components, indexed by $6,19$, appear to be false detected through cross validation (note that in Lasso, a soft thresholding has already been applied). In contrast, in our method most unimportant signals remains small. Performing a hard thresholding with the same regularization parameter selected by the Lasso can erase all false detections, leading to the selection of strong signal only.  \n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\linewidth]{selection.jpg}\n\t\\caption{The values versus index for truth $\\beta^*$, $\\beta$ estimator through lasso by minimizing the cross validation error in prediction, $\\beta$ estimator through gradient descent by minimizing the cross validation error in prediction and $\\beta$ estimator through `post estimation' selection for gradient descent.} \\label{fi:vsl}\n\t\\vspace{-0.7em}\n\\end{figure}\n\n\n\n\\subsection{Related literatures}\n\\cite{li2018algorithmic} studies the theory for implicit regularization in matrix sensing, which requires the data to be perfect measured and has different geometric structures as linear regression. \\cite{hoff2017lasso} considers the Hadamard product parametrization to optimize the parameters in high-dimensional linear regression. However, his method is computational-wise in order to reformulate the non-smooth Lasso optimization problem into a smooth one.\nIn particular, their objective function involves an $\\ell_2$ penalty on $(g,l)$ (which is equivalent to the $\\ell_1$ penalty on $\\beta$), and the solution is precisely the Lasso solution. \n\n\n\n\\section{Theoretical Analysis}\\label{Sec:Theory}\nIn this section, we provide formal statements for characterizing the behavior of the gradient descent algorithm for minimizing the Hadamard product parametrized quadratic loss $f(g,l)$ as defined in \\eqref{eq_opt}. We start with a description of our assumptions. Then we turn to the case of non-negative parameters, where a simpler parametrization $\\beta=u\\circ u$ can be applied, as a warm-up to convey the main proof ideas. Finally, we consider the general signal case and illustrate when the fast parametric root-$n$ rate independent of the dimension can be achieved. All proofs are deferred to the appendix in the supplementary material of the paper.\n\n\\subsection{Assumptions}\nRecall that the underlying true data generating model is $y=X\\beta^*+w$ with $w \\sim \\mathcal{N}(0,\\sigma^2 I)$, where the true parameter $\\beta^\\ast$ is $s$-sparse.\nWithin the $s$ nonzero signal components of $\\beta^\\ast$, we define the index set of strong signals as $S_1=\\{i\\in S: |\\beta^*_i| \\geq  2\\sigma \\log p \\sqrt{\\frac{\\log p}{n}}\\}$ and weak signals as $S_2=\\{i\\in S: |\\beta^*_i| \\leq  2\\sigma \\sqrt{\\frac{\\log p}{n}}\\}$, where $|S_1|=s_1$, $|S_2|=s_2$. According to the information-theoretic limits from \\cite{wainwright2009information}, weak signals of order $\\sigma \\sqrt{\\log p\/n}$ in sparse linear regression are generally impossible to be jointly recovered or selected (but can be detected in terms of the type I\/II error control in hypothesis testings, e.g.~see \\cite{jin2016rare}). Therefore, our primary focus would be the estimation and selection of strong signals.\nWe use the notation $\\theta_{s_1}(\\beta)$ to denote the $s_1$-th largest absolute component value of $\\beta$, and let $m=\\theta_{s_1}(\\beta^\\ast)$, which reflects the minimal strength for strong signals. We also use $\\kappa$ to denote the strong signal-condition number as the ratio between the largest absolute signal value to the smallest strong signal. \nWe will also make use of the notation of Restricted Isometry Property (RIP, \\cite{candes2008restricted}), which is a commonly used assumption (e.g. \\cite{candes2007dantzig}) in the high dimensional linear regression literatures.\n\n\\begin{defin}[Restricted Isometry Property]\n\tA matrix $X\\in \\mathbb{R}^{n\\times p}$ is said to satisfy the $(s,\\delta)$-Restricted Isometry Property (RIP) if for any $s$-sparse vector $u$ in $\\mathbb{R}^{p} $, we have:\n\t\\begin{equation*}\n\t(1-\\delta)\\|u\\|^2 \\leq \\frac{1}{n}\\,\\|Xu\\|^2 \\leq (1+\\delta)\\|u\\|^2\n\t\\end{equation*}\n\\end{defin}\n\\noindent As an easy consequence, if matrix $X$ satisfies $(2s,\\delta)$-RIP, then Euclidean inner-product is also approximately preserved, that is, $\\big|n^{-1}\\,\\langle Xu, Xv\\rangle - \\langle u,\\,v\\rangle \\big| \\leq \\delta\\,\\|u\\|\\cdot\\|v\\|$ holds for any two $s$-sparse vectors $u,\\,v\\in\\mathbb R^p$. \n\n\n\\noindent With these preparations, we make the following assumptions on the true parameter $\\beta^\\ast$, design matrix $X$, initialization parameter $\\alpha$ and step size $\\eta$ in Algorithm~\\ref{alg1}.\n\n\\paragraph{Assumption (A): } The true parameter $\\beta^\\ast$ is $s$-sparse, and $s=s_1+s_2$, that is, each nonzero signal in $\\beta^\\ast$ is either weak or strong. In addition, $\\kappa m \\lesssim1$. \n\n\\paragraph{Assumption (B):}  The design matrix $X$ satisfies $(s+1,\\delta)$-RIP condition with $\\delta\\lesssim   1 \/(\\kappa \\sqrt{s}\\log\\frac{p}{\\alpha})$.\n\n\\paragraph{Assumption (C):} The initialization parameter $\\alpha$ satisfies $0<\\alpha\\lesssim p^{-1}$, and the step size $\\eta$ satisfies $0<\\eta\\lesssim (\\kappa\\log\\frac{p}{\\alpha})^{-1}$.\n\nSome remarks are in order.\nFirst, our current proof heavily relies on the RIP condition as in Assumption (B), which is satisfied if the predictors are iid and $s\\log p\\ll n$. However, the extensive simulation studies in the next section provide a strong numerical evidence suggesting that our conclusions remain valid even when the RIP condition is violated. We leave the formal theoretical investigation as an open problem for our future studies. Second, Assumption (A) is made mainly for illustrating the possibility of achieving the fast parametric root $n$ rate of convergence when $s_1=0$. In fact, our current proof can still lead to the typical high-dimensional rate of $\\sqrt{s\\log p\/n}$ without introducing the notion of strong and weak signals. And due to space constraint we omit the details.\n\n\n\\subsection{Non-negative Signals}\nTo start with, we demonstrate the key ideas of our proofs and give an analysis of the non-negative case as a warm-up. More specifically, we consider the case when all components of true signal $\\beta^\\ast$ are non-negative. To exploit this non-negativeness, we may instead use the self-Hadamard product parametrization $\\beta = u^2=u\\circ u$ for $u\\in\\mathbb R^p$, and write $\\beta^\\ast=(u^*)^2=u^* \\circ u^*$. Now, we have the optimization problem:    \n\\begin{equation*}\n\\min_{u\\in \\mathbb{R}^{p}} f(u)=\\frac{1}{2n}\\,\\|Xu^2-y\\|^2,\n\\end{equation*}\nwith gradient descent updating formula $u_{t+1}=u_t-2\\eta\\,  u_t \\circ \\big[n^{-1}X^T(Xu_t^2-y)\\big]$. For technical simplicity, we instead focus on the deterministic initialization $u_0=\\alpha \\mathbf 1\\in\\mathbb R^p$. \nThis case is simpler for the analysis than the general case since components of $u_t$ will not change sign during the iterations, and will always stay away from saddle points.\nWe summarize our result in the following main theorem. Since the non-negative signal constraint resembles the positive semi-definiteness constraint in matrix sensing, our proof utilizes the proof strategy in \\cite{li2018algorithmic} for analyzing matrix factorized gradient descent for noiseless matrix sensing by dividing the convergence into two stages (more details are provided after the theorem).\n\\begin{thm}\\label{thm1}\n\tUnder the above assumptions (A), (B) and (C). Let $\\epsilon=\\max\\{  \\alpha^2, \\sigma^2 \\frac{  Ms_1}{n}, \\sigma^2 \\frac{s_2\\log p}{n}\\}$, $\\tau=\\max \\{\\delta \\alpha, \\sigma \\sqrt{\\frac{\\log p}{n}}\\}$ and any $M\\geq 1$. Then there exist positive constants $(c_1,\\,c_2,\\,c_3,\\,c_4,\\,c_5)$ such that for every time $t$ satisfying $c_1\\,  \\log(\\frac{p}{\\alpha})\/(\\eta m) \\leq t\\leq c_2 \/(\\eta \\tau)$, with probability at least $1-p^{-c_4}-e^{-c_5\\,Ms}$, the time $t$-iterate $u_t$ satisfies\n\t\\begin{align*}\n\t\\|u_t^2-\\beta^\\ast\\|^2\\leq  c_5 \\,\\epsilon ,\n\t\\end{align*}\n\\end{thm}\n This theorem tells us in high dimension linear regression, combining early stopping with implicit regularization can significantly improve the estimation accuracy. In particular, when all signals are strong ($s_1=s$ and $s_2=0$), the estimate $\\hat \\beta =u_t^2$ attains a parametric rate $\\sigma\\sqrt{s_1\/n}$ of convergence that is independent of the dimension $p$. In general when weak signals exist, then the overall rate $\\sigma \\sqrt{\\frac{s_1}{n}}+\\sigma  \\sqrt{\\frac{s_2\\log p}{n}}$ depends on the number of weak (strong) signals, which is still minimax-optimal \\citep{zhao2018pathwise}. The same remark also applies to our theory in the general case.\n\n\n\n\n\nOur proof strategy is to divide the convergence into two stages. Recall that $S=\\{j:\\,\\beta_j^\\ast\\neq 0\\}$ is the support set of true signal $\\beta^\\ast$, and $S_1 \\subset S$ corresponds to the subset of all strong signals.  In the first ``burn-in'' stage, we show that each component of the strong signal part $u_{t,S_1}$ increases at an exponential rate in $t$ until hitting $\\sqrt{m}\/2$, while the weak signal and error part $u_{t,{S_1}^c}$ remains bounded\nby $\\mathcal O(p^{-1})$. In the second stage, iterate $u_t$ enters a geometric convergence region where $u_t$ converges towards $u^\\ast$ at a linear rate up to some high-order error term, and then stay in a $O(\\epsilon)$ neighborhood of $u^\\ast$ up to the time $\\Theta(1\/\\tau)$. Therefore, the time interval $c_1\\,  \\log(\\frac{p}{\\alpha})\/(\\eta m) \\leq t\\leq c_2 \/(\\eta \\delta \\tau)$ would be the theoretical ``best solution region'' corresponding to the stabilized region in figure~\\ref{fig:31}.\n\n\nMore specifically, in the proof we consider the decomposition of $u_t$ into three parts: strong signal part $z_t$, weak signal part $d_t$ and error part $e_t$:\n\\begin{align*}\nu_t=z_t+d_t+e_t, \\quad\\mbox{with}\\quad\nz_t:=I_{S_1} u_t\\in\\mathbb R^p,  \\quad\nd_t:=I_{S_2} u_t \\in \\mathbb R^p \\quad\\mbox{and}\\quad\ne_t:=I_{S^c}u_t\\in\\mathbb R^p,\n\\end{align*} \nwhere recall that $I_E$ is the diagonal matrix with ones on the positions indexed by subset $E\\subset\\{1,\\ldots,p\\}$ and zero elsewhere. We use induction to prove the following results characterizing the gradient dynamics in the first stage. Recall that $\\theta_{s_1}(b)$ denote the $s_1$-th largest absolute component value of vector $b\\in\\mathbb R^p$ and $m$ is the minimal strength of the strong signals.\n\n\\begin{pro}[Stage one dynamics]\\label{pro2.1}\n\tUnder assumptions of theorem~\\ref{thm1}, there are constants $(c_7,c_8)$ and $(c_7',c_8')$, such that for each $t<T_1=c_7\\,\\log(\\frac{p}{\\alpha})\/(\\eta m)$, we have:\n\t\\begin{align*}\n\t\\mbox{strong signal dynamics:}& \\qquad   \\theta_{s}(z_{t+1})\\geq \\min\\Big\\{\\frac{\\sqrt{m}}{2},\\, \\Big(1+\\frac{\\eta m}{4}\\Big)\\,\\theta_{s}(z_t) \\Big\\},   \\\\\n\t&\\qquad \\|z_t\\|_\\infty\\leq c_7',\\quad\\mbox{and}\\quad \\|z_t^2-I_{S} \\beta^\\ast\\|\\leq c_7'\\,\\sqrt{s} ;\\\\[0.3em]\n\t\t\\mbox{weak signal dynamics:}& \\qquad \\|d_{t+1}\\|_{\\infty}\\leq \\big(1+c_8\\, \\eta\\, m\/\\log(p\\alpha)\\big)\\|d_t\\|_{\\infty} \\leq c_8' \/p.\\\\[0.3em]\n\t\\mbox{error dynamics:}& \\qquad \\|e_{t+1}\\|_{\\infty}\\leq \\big(1+c_8\\, \\eta\\, m\/\\log(p\\alpha)\\big)\\|e_t\\|_{\\infty} \\leq c_8' \/p.\n\t\\end{align*}\n\\end{pro}\n\nDefine $\\Theta_{LR}=\\{(z,\\, d, \\,e)\\in\\mathbb R^{3p}:\\, z_{S_1^c}=0, d_{S_2^c}=0,  \\theta_{s}(z) \\geq \\sqrt{m}\/2,\\,\\|z\\|_\\infty\\leq c_7' ,\\, \\|z^2-I_{S} \\beta^\\ast\\|\\leq c_7'\\,\\sqrt{s}, \\, \\|d\\|_\\infty \\leq c_8'\/p,\\, \\|e\\|_\\infty \\leq c_9'\/p\\}$ as a good parameter region for $(z_t,d_t,e_t)$. Proposition~\\ref{pro2.1} tells that for all $t<T_1$, iterate $(z_t, d_t, e_t)$ satisfies all constraints of $\\Theta_{LR}$ except $\\theta_{s}(z) \\geq \\sqrt{m}\/2$, but will enter this good region in at most $\\mathcal O(\\log(\\frac{m}{\\alpha^2})\/(\\eta m))$ iterations. The second stage starts when $(z_t,d_t,e_t)$ first enters $\\Theta_{LR}$. The next result summarizes the behavior of the gradient dynamics in the second stage---once it enters $\\Theta_{LR}$, it will stay in $\\Theta_{LR}$ up to time $t=\\Theta(1\/\\tau)$, where $u_t$ converges towards $u^\\ast$ at a linear rate up to the statistical error $\\epsilon$. \n\n\\begin{pro}[Stage two dynamics]\\label{pro2.4}\n\tUnder assumptions of theorem~\\ref{thm1}, there exists some constant $c_9$ such that if $(z_t,d_t,e_t) \\in \\Theta_{LR}$, then $(z_{t+1},d_{t+1},e_{t+1}) \\in \\Theta_{LR}$, and\n\t\\begin{equation*}\n\t\\|u_{t+1}^2-\\beta^\\ast\\|^2\\leq (1-\\eta m)\\,\\|u_t^2-\\beta^\\ast\\|^2+c_{9} \\,\\epsilon.\n\t\\end{equation*} \n\\end{pro}\nA combination of these two propositions, whose proofs are deferred to the supplementary material, leads to a proof of Theorem~\\ref{thm1}. \n\n\n\\subsection{General Signals}\nNow let us consider the general case where signs of nonzero components in signal $\\beta^\\ast$ are unknown. \nIn this case, we need the full Hadamard product parameterization $\\beta=g\\circ l$ and want to solve\n\\begin{equation*}\n\\min_{g,l\\in \\mathbb{R}^{p}} f(g,l)=\\frac{1}{2n}\\,\\|X(g\\circ l)-y\\|^2,\n\\end{equation*} \nwhere the associated gradient descent updating rule becomes:\n\\begin{equation*}\n\\begin{cases} \n{g}_{t+1}=g_t-\\eta \\,l_t \\circ \\big[n^{-1}X^{T}(Xg_t\\circ l_t-y)\\big]\\\\[0.2em] \n\\, {l}_{t+1}=l_t-\\eta \\,g_t \\circ \\big[n^{-1}X^{T}(Xg_t\\circ l_t-y)\\big]\n\\end{cases}\\ \\mbox{with initial condition}\\ \n\\begin{cases} \n{g}_{0}=\\alpha\\mathbf 1\\\\[0.2em] \n\\, {l}_{0}=\\mathbf 0\n\\end{cases},\n\\end{equation*}\nHere again for technical simplicity, we focus on a deterministic initialization as above.\nOur main result is summarized in the following theorem.\n\n\\begin{thm}\\label{thm2}\n\tUnder the above assumptions (A), (B) and (C). Let $\\epsilon=\\max\\{  \\alpha^2, \\sigma^2 \\frac{  Ms_1}{n}, \\sigma^2 \\frac{s_2\\log p}{n}\\}$, $\\tau=\\max \\{\\delta \\alpha, \\sigma \\sqrt{\\frac{\\log p}{n}}\\}$ and any $M\\geq 1$. Then there exist positive constants $(\\tilde c_1,\\,\\tilde c_2,\\,\\tilde c_3,\\,\\tilde c_4,\\,\\tilde c_5)$ such that for every time $t$ satisfying $\\tilde c_1 \\, \\log(\\frac{p}{\\alpha})\/(\\eta  m) \\leq t\\leq \\tilde c_2\/(\\eta \\tau)$,  with probability at least $1-p^{-\\tilde c_4}-e^{-\\tilde c_5\\,Ms}$, the time $t$-iterate $\\beta_t=g_t\\circ l_t$ satisfies\n\t\\begin{align*}\n\t\\|\\beta_t-\\beta^\\ast\\|^2\\leq \\tilde c_5 \\,\\epsilon .\n\t\\end{align*}\n\\end{thm}\n\nThe proof strategy of this result is similar to the previous nonnegative signal case by considering two stages.\nHowever, due to the lack of knowledge on signs of nonzero components in signal $\\beta^\\ast$, the proof of Theorem~\\ref{thm2} has the extra component in showing that $\\beta_t$ tends to shoot towards the right direction (positive versus negative) during the first ``burn-in\" stage before entering the second geometric convergence region as in the proof for nonnegative signals in the previous subsection. To deal with this extra complication, we introduce another reparametrization in the proof:\n\\begin{equation*}\n{a}_{t}=(g_t+l_t)\/2\\in\\mathbb R^p,\\quad\\mbox{and}\\quad\n{b}_{t}=(g_t-l_t)\/2\\in\\mathbb R^p \\quad\\mbox{for}\\quad  t=0,1\\ldots.\n\\end{equation*}\nNotice that $\\beta_t=a_t^2-b_t^2$, implying that the sign of each component $\\beta_{t,j}$ is determined by the relative magnitude of $|a_{t,j}|$ and $|b_{t,j}|$. These new variables $(a_t,b_t)$ are especially convenient for understanding the dynamics of the sign of $\\beta_t$ due to a more tractable updating formula:\n\\begin{align*}\na_{t+1}=a_{t}-\\eta \\,a_{t}\\circ \\big[n^{-1}X^T(X \\beta_t-y)\\big],\\ \\mbox{and} \\\nb_{t+1}=b_{t}+\\eta\\, b_{t}\\circ \\big[n^{-1}X^T(X \\beta_t-y)\\big].\n\\end{align*} \n More precisely, we have the following results characterizing the gradient dynamics in the first stage---showing that depending on the sign of $\\beta_{j}^\\ast$ for $j\\in S_1$, either $a_{j,t}^2$ or $b_{j,t}^2$ will increase at an exponential rate while the other remaining small. Recall that $S_1$ denotes the support of strong signals of $\\beta^\\ast$.\n\n\n\\begin{pro}[Stage one dynamics]\\label{pro3.1}\n\tUnder assumptions of theorem~\\ref{thm2}, there are constants $(\\tilde c_7,\\tilde c_8,\\tilde c_9)$ and $(\\tilde c_7',\\tilde c_8',\\tilde c_9')$, such that for each $t<T_1=\\tilde c_7\\,\\log(\\frac{p}{\\alpha})\/(\\eta m)$, we have:\n\t\\begin{align*}\n\t&\\mbox{signal dynamics:} \\qquad \\mbox{sign}(\\beta^\\ast_{S_1})\\circ \\beta_{t,S_1} \\geq \\min\\Big\\{\\frac{m}{2},\\, \\Big(1+\\eta m\/4\\Big)^t\\,\\alpha^2 - \\tilde c_9'\\,\\alpha^2\\Big\\}\\, \\mathbf 1\\in\\mathbb R^{s_1}, \\\\\n\t&\\qquad \\max\\big\\{\\|a_{t,S_1}\\|_\\infty,\\, \\|b_{t,S_1}\\|_\\infty\\big\\}\\leq \\tilde c_7',\\quad\\mbox{and}\\quad \\|\\beta_{t,S_1}-\\beta^\\ast_{S_1}\\|\\leq \\tilde c_7'\\,\\sqrt{s_1};\\\\[0.3em]\n\t&\\mbox{weak signals and error dynamics:} \\qquad \\|a_{t+1,S_1^c}\\|_{\\infty}\\leq \\big(1+\\tilde c_8\\, \\eta\\,m\/\\log(p\/\\alpha)\\big)\\|a_{t,S_1^c}\\|_{\\infty} \\leq \\tilde c_8' \/p,\\\\\n\t&\\quad\\mbox{and} \\qquad \\|b_{t+1,S_1^c}\\|_{\\infty}\\leq \\big(1+\\tilde c_8\\, \\eta\\,m\/ \\log(p\/\\alpha) \\big)\\|b_{t,S_1^c}\\|_{\\infty} \\leq \\tilde c_8' \/p.\n\t\\end{align*}\n\\end{pro}\n\nSimilarly, let $\\Theta_{LR}^G=\\{(a,\\,b)\\in\\mathbb R^{2p}:\\, \\mbox{sign}(\\beta^\\ast_{S_1})\\circ (a_{S_1}^2- b_{S_1}^2) \\geq (m\/2) \\, \\mathbf 1,\\,\\max\\big\\{\\|a_{S_1}\\|_\\infty,\\, \\|b_{S_1}\\|_\\infty\\big\\}\\leq \\tilde c_7',\\, \\|a_{S_1}^2- b_{S_1}^2-\\beta^\\ast_{S_1}\\|\\leq \\tilde c_7'\\,\\sqrt{s_1},\\, \\max\\big\\{\\|a_{S_1^c}\\|_\\infty,\\,\\|b_{S_1^c}\\|_\\infty\\big\\} \\leq \\tilde c_8'\/p\\}$ as a good parameter region for $(a_t,b_t)$. Proposition~\\ref{pro3.1} tells that for all $t<T_1$, iterate $(a_t,b_t)$ satisfies all constraints of $\\Theta_{LR}^G$ except $\\mbox{sign}(\\beta^\\ast_{S_1})\\circ (a_{S_1}^2- b_{S_1}^2) \\geq m\/2$, but will enter this good region in at most $\\mathcal O(\\log(\\frac{m}{\\alpha^2})\/(\\eta m))$ iterations. The second stage starts when $(a_t,b_t)$ first enters $\\Theta_{LR}^G$. The next result summarizes the behavior of the gradient dynamics in the second stage---once it enters $\\Theta_{LR}^G$, it will stay in $\\Theta_{LR}^G$ for a long time $\\Theta(1\/\\tau)$, where $\\beta_t=a_t^2-b_t^2$ converges toward $\\beta^\\ast$ at a linear rate up to the statistical error $\\epsilon$. \n\n\\begin{pro}[Stage two dynamics]\\label{pro3.4}\n\tUnder assumptions of theorem~\\ref{thm2}, there exists some constant $\\tilde c_9$ such that if $(a_t,b_t) \\in \\Theta_{LR}^G$, then $(a_{t+1},b_{t+1}) \\in \\Theta_{LR}^G$, and $\\beta_{t}=a_t^2-b_t^2$ satisfies\n\t\\begin{equation*}\n\t\\|\\beta_{t+1}-\\beta^\\ast\\|^2\\leq (1-\\eta m)\\,\\|\\beta_t-\\beta^\\ast\\|^2+\\tilde c_9\\, \\epsilon.\n\t\\end{equation*} \n\\end{pro}\nA combination of these two propositions leads to a proof of Theorem~\\ref{thm2}.\n\nFinally, we provide a theorem regarding the strong signal selection consistency.\n\\begin{thm}[Variable selection consistency]\\label{thm5}\nUnder assumptions of theorem~\\ref{thm2}, for every time $t$ satisfying $\\tilde c_1 \\, \\log(\\frac{p}{\\alpha})\/(\\eta m) \\leq t\\leq \\tilde c_2\/(\\eta \\tau)$,  with probability at least $1-p^{-\\tilde c_3}-n^{-\\tilde c_4}$,  the following hard thresholded estimator $\\beta'_t$,\n\t\\begin{align*}\n\\beta'_{t,j} =\\begin{cases} g_{t,j} l_{t,j}, & \\mbox{if } |g_{t,j} l_{t,j}| \\geq \\lambda \\\\ 0, & \\mbox{if }  |g_{t,j} l_{t,j}| < \\lambda, \\end{cases},\n\\end{align*}\nfor any $\\lambda \\in  \\left[\\frac{\\tilde c_{10}}{p},  \\tilde{c}_{11}\\sigma \\sqrt{\\frac{\\log p}{n}}\\right] $ where $\\tilde c_{10}$ and $\\tilde c_{11}$ are positive constants,\nsatisfies variable selection consistency, that is, $\\{j:\\, \\beta'_{t,j} \\neq 0,\\,j=1,\\ldots,p\\} =S_1$.\n\\end{thm}\n\nThis theorem implies the variable selection consistency of our method is not sensitive to the choice of tuning parameter of noise level $\\lambda$. As shown in the toy example in section~\\ref{sec:2.5}, the Lasso suffers form false positives due to non-discoveries of weak signals when the cross validation error in prediction is applied as the criterion. However, even when we apply the same regularization to our method, since the tuning parameter is still in the region $\\left[\\frac{\\tilde c_{10}}{p},  \\tilde{c}_{11}\\sigma \\sqrt{\\frac{\\log p}{n}}\\right] $,  the variable selection consistency of our method can still achieve.  As we will illustrate in the numerical part below, our method simultaneously achieves the best false discovery control and attains the most accurate prediction.\n\n\n\\section{Simulations and Real Data Analysis}\nWe start with examples with noiseless data to show that under this setting, our gradient descent Algorithm~\\ref{alg1} truly converges to the least $\\ell_1$-norm global minimizer. These examples provide strong numerical evidences to our statements in Section~\\ref{Sec:Theory} even when the RIP condition is violated. After that, we consider the general noisy settings, and compare our methods with state-of-the-art approaches. This section ends up with a real data analysis.\n\n\n\\subsection{Simulations for Noiseless Case}\nIn this section, we conduct some numerical experiments to illustrate the performance of our method. In our first example, we study the performance as we change the initialization under settings where the RIP may or may not hold. In the second example, we study the behavior of the algorithm when the null space property \\citep{cohen2009compressed} is violated, under which the $\\ell_1$-norm minimizer in the basis pursuit problem differs from the sparsest solution (the $\\ell_0$-norm minimizer).\n\n\n\\begin{figure}[t]\\label{fig:20}\n\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\\includegraphics[height=5.5cm, width=7.2cm]{Estimation.jpg}\n\t\t\\caption{Estimation Error vs initialization}\n\t\\end{subfigure}\n\t\\quad \\quad\n\t~\n\n\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\\includegraphics[height=5.5cm, width=7.2cm]{corrlated.jpg}\n\t\t\\caption{Signal components trajectories}\n\t\t\\label{fig:tiger}\n\t\\end{subfigure}\n\t\\caption{Panel (a) is a log-log plot of estimation error $\\|\\widehat \\beta-\\beta^\\ast\\|_2$ versus initialization level $\\alpha$. Panel (b) shows trajectories of signal components ($\\beta_1$ to $\\beta_4$) at $(n,p)=(200,2000)$. In both plots, solid curves correspond to the independent (i) design, and dashed curves the correlated (c) design. }\\label{fig:200}\n\t\\vspace{-0.7em}\n\\end{figure}\n\n\\paragraph{Impact of initialization:}\nWe consider two classes of random designs $X\\in\\mathbb R^{n\\times p}$ with i.i.d.~rows $X^{(i)}\\overset{iid}{\\sim}\\mathcal N(0,\\,\\Sigma)\\in \\mathbb R^p$, where population p.s.d~covariance matrix $\\Sigma\\in\\mathbb R^{p\\times p}$ is\n\\begin{align*}\n\\mbox{independent design:}\\quad \\Sigma_{jk}=\\mathbb I(j=k), \\quad\\mbox{or}\\quad \\mbox{correlated design:}\\quad \\Sigma_{jk}=0.5+0.5\\,\\mathbb I(j=k),\n\\end{align*}\nfor $j,k=1,\\ldots,p$, where $\\mathbb I(A)$ denotes the indicator function of event $A$.\nWe choose sample size $n=200$, sparsity level $s=4$, and signal dimension $p\\in\\{500, 2000,10000\\}$. For the independent (correlated) design, the RIP is satisfied (fails) with high probability for some small $\\delta>0$. In both scenarios, we choose true signal $\\beta^\\ast=(-1,2,2,3)^T\\in\\mathbb R^p$, and set $y=X\\beta^\\ast$. When implementing gradient descent, we choose step size $\\eta=0.2\\ (0.1)$ for the independent (correlated) design, $\\alpha\\in\\{10^{-10},10^{-9},\\ldots,10^{1}\\}$, and stopping threshold $\\epsilon=0.01\\alpha$. Figure~\\ref{fig:200} shows the estimation error $\\|\\widehat \\beta-\\beta^\\ast\\|_2$ versus $\\alpha$ in log-log plots. As we can see, they tend to have a linear trend under the log-scale, which is consistent with our theoretical error bound estimate in Section 4.\nIn addition, in the correlated design scenario where the RIP does not hold, the algorithm is still able to recover $\\beta^\\ast$ as $\\alpha\\to0$, albeit under a slower convergence (due to a smaller allowable step size and a larger condition number of $X$). This observation provides evidence to the correctness of our informal statement made at the beginning of Section 3 even without RIP condition. We leave the proof of this conjecture open.\n\n\n\\paragraph{$\\ell_0$-norm minimizer differs from $\\ell_1$-norm minimizer:}\nIn this example, we study the empirical performance of the algorithm  when the least $\\ell_1$-norm in the basis pursuit problem~\\eqref{Eqn:CS} is not the sparsest solution of $X\\beta=y$ (the null space property is violated). In particular, we choose\n\\begin{align*}\nX=\\begin{bmatrix} 0.2 & 1 & 0 \\\\ 0.2 & 0 &-1 \\end{bmatrix},\\quad \\beta^\\ast= \\begin{bmatrix} 5\\\\ 0 \\\\0 \\end{bmatrix}, \\quad\\mbox{and}\\quad y=\\begin{bmatrix} 1\\\\ 1 \\end{bmatrix},\n\\end{align*}\nso that $X\\beta^\\ast =y$. It is easy to verify that for this example, the sparsest solution of $X\\beta=y$ is $\\beta^\\ast$, while the least $\\ell_1$-norm solution is $\\beta^\\dagger = [0,1,-1]^T$. We use the same setting as before for implementing the algorithm with $\\alpha\\in\\{10^{-10},10^{-5},10^{-3},10^{-1},10^0,10^1\\}$. Table~\\ref{tb1} reports final outputs $\\beta=(\\beta_1,\\beta_2,\\beta_3)^T$ of the algorithm. Due to our intuition in Section 3, as expected, the algorithm still converges to the least $\\ell_1$-norm solution $\\beta^\\dagger$ instead of the least $\\ell_0$-norm solution $\\beta^\\ast$. Again, the estimation error decreases as the initialization level $\\alpha$ decreases. We conjecture this phenomenon of convergence to the least $\\ell_1$-norm solution to be generally true, and leave a formal proof as a future direction.\n\n\n\\begin{table}[h]\\caption{Convergent point without null space property} \\label{tb1}\n\t\\begin{center}\n\t\t$\\begin{array}{|ccccccc|} \n\t\t\\hline\n\t\t\\alpha & 10^{-10} & 10^{-5} & 10^{-3} & 0.1 & 1 & 10 \\\\ \n\t\t\\hline\n\t\t\\beta_1 & 7.433e-13 & 5.703e-7 & 1.289e-4 & 2.884e-2 & 2.987e-1 & 8.823e-1 \\\\ \n\t\t\\hline\n\t\t1-\\beta_2 & 1.492e-13 & 1.141e-7 & 2.577e-5 & 5.769e-3 & 5.974e-2 & 1.765e-1\\\\\n\t\t\\hline\n\t\t1+\\beta_3 & 1.492e-13 & 1.141e-7 & 2.577e-5 & 5.769e-3 & 5.974e-2 & 1.765e-1 \\\\\n\t\t\\hline \n\t\t\\end{array}$\n\t\\end{center}\n\t\\vspace{-0.7em}\n\\end{table}\n\\subsection{Simulations for Noisy Case}\\label{sec:simu}\n\n\n\n\\paragraph{Comparison with other high dimensional estimators:} We further demonstrate the advantages of our algorithm by considering the following $8$ simulation settings, the sparsity level $s=4$, signals $-1,2,2,3$ and noise level $\\sigma$ with $\\sigma =0.15*\\|\\beta^\\ast\\|$. We generate $3 n$ observations independently and split into $3$ even parts, then use the first part for training, the second part for validation and the final part for testing. The evaluation metric is standardized estimation error $\\|\\widehat \\beta-\\beta^\\ast\\|^2_2\/\\|\\beta^\\ast\\|^2_2$ and mean prediction error $\\sqrt{\\|y-\\hat{y}\\|^2\/n}$  for the test data set. We compare the median of the standardized estimation errors and prediction errors with the Lasso, SCAD and MCP by repeating $50$ times. We implement the Lasso using \\textit{glmnet} R package \\citep{friedman2010regularization} while for SCAD and MCP, we use the R package $\\textit{ncvreg}$ \\citep{breheny2011coordinate}. The standard error of medians are calculated by bootstrapping the calculated errors $1000$ times. For our algorithm, we use the initialization $\\alpha=10^{-5}\\times \\+1$. The simulation results in Table~\\ref{table1} and~\\ref{table2} indicate that our methods consistently have the best performance over all explicit penalization-based competitors across all settings.\n\n\\begin{enumerate}\n\t\\item \\textbf{S1:}  $n=200$, $p=500$, $\\Sigma_{jk}=1$ for $j=k$ while $\\Sigma_{jk}=0$ for $j\\neq k$;\n\t\\item \\textbf{S2:}  $n=200$, $p=500$, $\\Sigma_{jk}=0.1^{|j-k|}$;\n\t\\item \\textbf{S3:}  $n=200$, $p=500$, $\\Sigma_{jk}=0.2^{|j-k|}$;\n\t\\item \\textbf{S4:}   $n=200$, $p=500$, $\\Sigma_{jk}=0.5^{|j-k|}$;\n\t\\item \\textbf{S5:}  $n=200$, $p=2000$, $\\Sigma_{jk}=1$ for $j=k$ while $\\Sigma_{jk}=0$ for $j\\neq k$;\n\t\\item \\textbf{S6:}  $n=200$, $p=2000$, $\\Sigma_{jk}=0.1^{|j-k|}$;\n\t\\item \\textbf{S7:}  $n=200$, $p=2000$, $\\Sigma_{jk}=0.2^{|j-k|}$;\n\t\\item \\textbf{S8:}   $n=200$, $p=2000$, $\\Sigma_{jk}=0.5^{|j-k|}$.\n\\end{enumerate}\n\n\n\n\\begin{table}[H]\n\t\\begin{tabular*}{\\textwidth}{|c@{\\extracolsep{\\fill}}cccccccc|}\n\t\t\n\t\t\\hline\n\t\t&S1 &S2 &S3 &S4 &S5 &S6 &S7 & S8 \\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{GD}  \n\t\t&\\textbf{0.520}\t&\\textbf{0.448}\t&\\textbf{0.510}\t&\\textbf{0.568}\t&\\textbf{0.385}\t&\\textbf{0.290} &\\textbf{0.465}\t\t&\\textbf{0.460}\n\t\t\n\t\t\\\\\n\t\t&(0.0428)\t&(0.0530)\t&(0.0607)\t&(0.0850)\t&(0.0533)\t&(0.0465)\t&(0.0858)\t&(0.0863) \n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{Lasso} \n\t\t&3.11\t&3.10\t&3.42\t&4.51\t&4.62\t&4.04\t&4.40\t&6.98\n\t\t\\\\\n\t\t&(0.219)\t&(0.173)\t&(0.242)\t&(0.274)\t&(0.279)\t&(0.205)\t&(0.306)\t&(0.452)\n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{SCAD}\n\t\t&0.613\t&0.533\t&0.650\t&0.691\t&0.519\t&0.401\t&0.595\t&0.646\n\t\t\\\\\n\t\t&(0.0464)\t&(0.0679)\t&(0.0702)\t&(0.103)\t&(0.0527)\t&(0.0574)\t&(0.0837)\t&(0.0776)\n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{MCP}\n\t\t&0.628\t&0.552\t&0.594\t&0.733\t&0.484\t&0.405\t&0.595\t&0.708\n\t\t\\\\\n\t\t&(0.0392)\t&(0.0779)\t&(0.0809)\t&(0.0902)\t&(0.0706)\t&(0.0597)\t&(0.0741)\t&(0.0680)\n\t\t\\\\\n\t\t\\hline\n\t\\end{tabular*}\n\t\\caption{Simulation result for median of standardized estimation error of each method, with standard derivation in the parenthesis under the median. There are $10^{-3}$ factors for all medians and standard derivations.\n\t\t\\label{table1}}\n\\end{table}\n\n\n\\begin{table}[H]\n\t\\begin{tabular*}{\\textwidth}{|c@{\\extracolsep{\\fill}}cccccccc|}\n\t\t\n\t\t\\hline\n\t\t&S1 &S2 &S3 &S4 &S5 &S6 &S7 & S8 \\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{GD}  \n\t\t&0.638\t&\\textbf{0.636}\t&0.640\t&\\textbf{0.634}\t&\\textbf{0.646}\t&\\textbf{0.651} &\\textbf{0.641}\t\t&\\textbf{0.642}\n\t\t\n\t\t\\\\\n\t\t&(0.0597)\t&(0.0753)\t&(0.0718)\t&(0.0709)\t&(0.0491)\t&(0.0510)\t&(0.0406)\t&(0.0498) \n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{Lasso} \n\t\t&0.676\t&0.672\t&0.671\t&0.685\t&0.693\t&0.699\t&0.696\t&0.708\n\t\t\\\\\n\t\t&(0.0899)\t&(0.0981)\t&(0.0932)\t&(0.0510)\t&(0.0568)\t&(0.0918)\t&(0.0615)\t&(0.0488)\n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{SCAD}\n\t\t&0.638\t&0.638\t&\\textbf{0.637}\t&0.637\t&0.650\t&0.654\t&0.643\t&0.647\n\t\t\\\\\n\t\t&(0.0530)\t&(0.0724)\t&(0.0716)\t&(0.0713)\t&(0.0470)\t&(0.0451)\t&(0.0402)\t&(0.0434)\n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{2}{3em}{MCP}\n\t\t&\\textbf{0.637}\t&0.639\t&0.638\t&0.637\t&0.650\t&0.652\t&0.644\t&0.647\n\t\t\\\\\n\t\t&(0.0516)\t&(0.0756)\t&(0.0684)\t&(0.0753)\t&(0.0475)\t&(0.0441)\t&(0.0435)\t&(0.0408)\n\t\t\\\\\n\t\t\\hline\n\t\\end{tabular*}\n\t\\caption{Simulation result for median of mean prediction error of each method, with standard derivation in the parenthesis under the median. There are $10^{-1}$ factors for all standard derivations.\n\t\t\\label{table2}}\n\t\n\t\n\\end{table}\n\n\\paragraph{Comparison in Variable Selection:}  Now we consider variable selection when there exists some weak signals. Suppose the simulation settings are similar with S3 above, with only the true signals changed. Let $s=20$, and the strength of first $4$ signals is $0.5 \\sigma \\sqrt{\\log p\/n}$, while the other $16$ are $ 5 \\sigma \\sqrt{ \\log p \/ n}$, where $\\sigma =1$. Clearly the first $4$ signals are too weak to be selected by all methods. However, since all methods are based minimizing the prediction error, the effect of these weak signals pertains, and may increase the false discovery rate. Under the above settings, we perform a model selection based on minimized prediction errors through $5$-fold cross validation. For our method, we use the same regularization parameter as the lasso to perform hard threshold after estimation. We repeat the process $50$ times and compare variable selection errors. From the figure~\\ref{fi:vse}, we can see our method is more robust to the enhancement of false detection due to failure on detecting weak signals: although the true negative errors of our method is $4$, which means all weak signals can not be detected, the false detections of our methods are closed to zero. For other methods, although sometimes weak signals can be detected, the risk of false detections is high. Overall, our methods perform consistent variable selection for strong signals, and achieve better estimation than the competitors.\n\\begin{figure}[h!]\n\t\\includegraphics[scale=0.7]{0vse.jpeg}\n\t\\caption{Variable selection errors for selected model based on minimized prediction cross validation errors. `fp' stands for false positive when the truth is zero but detected as signal; `tn' stands for true negative when the truth is nonzero but not detected.}\\label{fi:vse}\n\\end{figure}\n\n\n\\subsection{Real Data Analysis}\nWe compare our method with others to analyze the Riboflavin data set \\citep{buhlmann2014high}, which is available in \\textit{hdi} R package. The  dataset contains $71$ observations of log-transformed riboflavin production rate verses the logarithm of expression level of 4088 genes. Before estimation, we first perform independence screening \\citep{fan2008sure} based on the rank of the correlation strength for each predictor verses response to decrease the dimension of feature space into $500$. Then we normalize and add the intercept column into the design matrix. For evaluation, we split the observations into $50$ training samples and $21$ testing samples, with performing $10$-fold cross validation to select iteration steps and regularization parameters in the training data. Still, for our algorithm, we use the initial value $\\alpha=10^{-5} \\times \\+1$ for all training processes. We record the prediction errors for testing data set and repeat $50$ times. From the figure~\\ref{fi:box} below, our method also obtain the least prediction errors, which implies the estimation of this high dimensional linear regression problems can also have the least errors.\n\n\n\\begin{figure}[h!]\n\t\\includegraphics[scale=0.7]{realdata.jpeg}\n\t\\caption{Prediction errors on the test data of Riboflavin data set for each method. $x$-axis stands for the methods used for estimation, and $y$-axis stands for the testing prediction error $\\|y-\\hat{y}\\|$.}\\label{fi:box}\n\\end{figure}\n\n\nWe also perform variable selection on the whole Riboflavin data set with the same tuning parameter obtained through minimized estimation errors based on cross validation. For our algorithm, when we get the number of iterations from cross validation, we run gradient descent on the whole data set with the same initial values and step size until the corresponding number of iterations. We use the same regularization parameter obtained by the lasso as the the hard thresholding value on the absolute value of obtained the `post-estimation' selection. The comparison between different variable selection methods is given in Table~\\ref{tb:3}. From the table, we can see except one variable, all other variables detected by our method are also selected by other methods, illustrating that our methods tend to have lower false positive rate for variable selection without sacrificing estimation and prediction accuracy. \n\n\n\n\\begin{table}[H]\n\t\\begin{tabular*}{\\textwidth}{|c@{\\extracolsep{\\fill}}cccc|}\n\t\t\n\t\t\\hline\n\t\t&Lasso &SCAD &MCP &GD \\\\\n\t\t\\hline\n\t\t\\multirow{1}{3em}{Lasso}  \n\t\t& 33\t&  \t&\t&\n\t\t\n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{1}{3em}{SCAD} \n\t\t&11\t&14\t&\t&\n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{1}{3em}{MCP}\n\t\t&2\t&3\t&5\t&\n\t\t\\\\\n\t\t\\hline\n\t\t\\multirow{1}{3em}{GD}\n\t\t&8\t&9\t&3\t&11\n\t\t\\\\\n\t\t\n\t\t\\hline\n\t\t\\multirow{1}{3em}{Independent}\n\t\t&20\t&2\t&2\t&1\n\t\t\\\\\n\t\t\\hline\n\t\\end{tabular*}\n\t\\caption{Variable selection result for Riboflavin data set, each cell stands for the number of the same detected variables between row labels and column labels. `Independent' means the number of detected variables for the corresponding column method that are not detected by others. \\label{tb:3}\n\t}\n\\end{table}\n\n\\section{Discussion}\nIn this paper, we illustrated the phenomenon of implicit regularization through Hadamard product change of variables in high dimensional linear regression, and  demonstrated that a combination of implicit regularization with early stopping yields better estimation than state-of-the-art penalized approaches with explicit regularization. However, we still face several important open problems on implicit regularizations as our future directions. First, our theory heavily relies on the RIP condition, which is relatively strong comparing to the restricted eigenvalue condition as the minimal possible assumption on the design in the literature.\nIt would be interesting to investigate whether our results remain valid without the RIP condition.\nSecond, it is interesting to study whether any computationally efficient early stopping rule (rather than cross validation) based on certain date-driven model complexity measure can be applied and provably works.\n\n\n\\section{Proof of the results in the paper}\n\\subsection{Overview}\n\nIn this supplementary material, we provide proofs of the main theorems presented in the paper. \n\n\\subsection{Notation}\nRecall that $\\|v\\|=\\sqrt{\\sum_{j=1}^p v_j^2}$ and $\\|v\\|_\\infty=\\max_{j}|v_j|$ denote the vector $\\ell_2$-norm and $\\ell_\\infty$-norm, respectively. Moreover, $I$ is the identity matrix in $\\mathbb R^p$, and for any subset $S$ of $\\{1,\\ldots,p\\}$, $I_S$ is the diagonal matrix with $1$ on the $j$th diagonals for $j\\in S$ and $0$ elsewhere. We use bold letter $\\mathbf 1\\in\\mathbb R^p$ to denote an all-one vector. $\\theta_s(\\beta)$ denote the $s$-largest component of vector $\\beta\\in\\mathbb R^p$ in absolute value. We use notation $\\lesssim$ and $\\gtrsim$ to denote $\\leq$ and $\\geq$ up to some positive multiplicative constant, respectively. For two vectors $u$ and $v$ of the same dimension, we use $a\\geq b$ and $a\\leq b$ to denote element-wise $\\geq$ and $\\leq$. Denote $\\lambda_{\\max}(A)$ and $\\lambda_{\\min}(A)$ be the maximal and minimal eigenvalues of matrix $A$. Through this document, letters $c$, $c'$ and $c''$ denote some constants whose meaning may change from line to line.\n\n\\subsection{Some Useful Results}\nIn our proof, we will constantly deal with the Hadamard product $u\\circ v$ and the operation $(n^{-1}\\,X^TXu)\\circ v$ for two vectors $u,v\\in\\mathbb R^p$. \nTherefore, we collect some useful properties in this section, some of which are consequences of the RIP condition. \n\nThe first property regarding the Hadamard product is a direct consequence of the H\\\"{o}lder inequality.\n\\begin{lem}\\label{lem2}\n\tFor any two vectors $u$ and $v$ in $\\mathbb{R}^{p}$,  we have:\n\t\\begin{equation}\n\t\\|u\\circ v\\| \\leq\\|u\\|\\|v\\|_\\infty.\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tThis follows since $\\|u\\circ v\\|^2 = \\sum_{j}u_j^2v_j^2 \\leq \\|v\\|_\\infty^2 \\sum_{j}u_j^2=\\|u\\|^2\\|v\\|_\\infty^2$.\n\\end{proof}\nThe second lemma shows that under the RIP, the product $(n^{-1}\\,X^TXu)\\circ v$ can be well-approximated by $u\\circ v$ for all sparse vectors $u\\in\\mathbb R^p$ and any vector $v\\in\\mathbb R^p$. \n\\begin{lem}\\label{lem3}\n\tLet $X$ be a matrix in $\\mathbb{R}^{n\\times p}$ that satisfies $(s+1,\\delta)$-restricted isometry property (see Definition 2.1 in the paper). Then for any $s$-sparse vectors $u$ and any $v$ in $\\mathbb{R}^{p} $, we have:\n\t\\begin{equation}\n\t\\|(n^{-1}\\,X^TXu)\\circ v-u\\circ v\\|_{\\infty} \\leq \\delta\\|u\\|_2\\|v\\|_{\\infty}.\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tLet $D(v)$ be the diagonal matrix in $\\mathbb{R}^{n\\times p}$ with diagonal elements the same as components of $v$ correspondingly. Then $\\|(n^{-1}\\,X^TXu)\\circ v-u\\circ v\\|_{\\infty}$ can be represented as:\n\t\\begin{align*}\n\t\\max_{i=1,2,...,p} |e_i^T D(v) n^{-1}X^TXu-e_i^T D(v) u|,\n\t\\end{align*}\n\twhere $e_i$ is a $p$-dimensional vector whose $i$-th component is $1$ and $0$ elsewhere. Using the fact that $X$ satisfies $(s+1,\\delta)$-RIP and $e_i^T D(v)$ is $1$-sparse, we have (see the remark right after Definition 2.1 in the paper):\n\t\\begin{align*}\n\t\\|(n^{-1}\\,X^TXu)\\circ v-u\\circ v\\|_{\\infty}\\leq  \\max_{i=1,2,...,p} \\delta\\|e_i^T D(v)\\|\\|u\\|= \\delta\\|u\\|_2\\|v\\|_{\\infty}.\n\t\\end{align*}\n\\end{proof}\nOur third lemma considers the case when $u$ and $v$ are both arbitrary. \n\\begin{lem}\\label{lem4}\n\tLet $X$ be a matrix in $\\mathbb{R}^{n\\times p}$ that satisfies $(2,\\delta)$-restricted isometry property. Then for any vectors $u,\\, v\\in\\mathbb R^p$, we have: \n\t\\begin{equation}\n\t\\|(n^{-1}\\,X^TXu)\\circ v-u\\circ v\\|_{\\infty} \\leq \\delta\\|u\\|_1\\|v\\|_{\\infty}.\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tSince we can decompose $u=\\sum_j I_j u$, we have\n\t\\begin{align*}\n\t\\|(n^{-1}\\,X^TXu)\\circ v-u\\circ v\\|_{\\infty}&=\\max_{i=1,2,...,p} |e_i^T D(v)\\, n^{-1}X^TXu-e_i^T D(v) u|\\\\\n\t&\\leq \\sum_{j=1}^{p} \\max_{i=1,2,...,p} |e_i^T D(v)\\, n^{-1}X^TX \\, I_j u-e_i^T D(v)\\, I_j u| \\\\\n\t&\\leq \\sum_{j=1}^{p} \\max_{i=1,2,...,p} \\delta \\|e_i^T D(v)\\| \\|u_j\\| \\\\\n\t&\\leq \\delta\\|u\\|_1\\|v\\|_{\\infty}.\n\t\\end{align*}\n\\end{proof}\nOur fourth and fifth lemma consider the concentration behavior about the noise terms.\n\\begin{lem}\\label{lem6}\n\t$w \\sim \\mathcal{N}(0, \\sigma^2 I_{n \\times n})$, all $\\ell_2$ norm of column vectors of $X_{n \\times s}$ are normalized to $\\sqrt{n}$, $s<n$, then with probability $1-e^{-Ms}$ for any $M\\geq 1$, such that:\n\t\\begin{equation}\n\t\\frac{1}{n}\\|X^\\top w\\| \\leq 3 \\sigma  \\sqrt{\\frac{Ms\\,\\lambda_{\\max}(\\frac{X^\\top X}{n})}{n}}.\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tBy directly applying Hanson-Wright inequality:\n\t\\begin{equation}\n\t\\begin{aligned}\n\t\\mathbb{P}(\tw^\\top X X^\\top w \\geq 9 \\sigma^2 n \\,Ms \\,\\lambda_{\\max}(\\frac{X^\\top X}{n})) \n\t\\leq \\exp \\left(-\\frac{9 \\sigma^2 n \\,Ms\\, \\lambda_{\\max}(\\frac{X^\\top X}{n})}{8\\sqrt{2} \\sigma^2 \\lambda_{\\max}(X^\\top X)}\\right) \\leq  e^{-Ms}.\n\t\\end{aligned}\n\t\\end{equation}\n\\end{proof}\n\n\n\n\n\\begin{lem}\\label{lem5}\n\t$w \\sim \\mathcal{N}(0, \\sigma^2 I_{n \\times n})$, all $\\ell_2$ norm of column vectors of $X_{n \\times p}$ are normalized to $\\sqrt{n}$, then with probability $1-2 p^{-1}$ such that:\n\t\\begin{equation}\n\t\\frac{1}{n}\\|X^\\top w\\|_{\\infty} \\leq  \\sigma \\sqrt{\\frac{4 \\log p}{n}}.\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\t$\\frac{1}{n}\\|X^\\top w\\|_{\\infty}$ is the maximum over $p$ normal random variables distributed as $\\mathbb{N}(0,\\frac{\\sigma^2}{n})$, so considering the union bound:\n\t\\begin{align*}\n\t\\mathbb{P}(\\frac{1}{n}\\|X^\\top w\\|_{\\infty} \\geq t) \\leq 2 e^{\\frac{-nt^2}{2\\sigma^2}+\\log p},\n\t\\end{align*}\n\tthe result is obtained by taking $t=\\sigma \\sqrt{\\frac{4 \\log p}{n}}$.\n\\end{proof}\n\n\n\n\\subsection{Proofs of Lemmas, Propositions and Theorems in the Paper}\nWe prove the results in the paper one-by-one. For simplicity, we assume the largest signal strength is at most constant level, which implies $\\kappa m \\lesssim 1$.\n\n\n\\subsection{Proof of Lemma \\ref{Lem:global_min}}\nLet $\\beta = g\\circ l$. Since the least square problem $\\min_{\\beta\\in\\mathbb R^p} (2n)^{-1}\\|X\\beta -y\\|^2$ is a convex optimization, it does not have local maximum and any local minimum is a global minimum. \n\nNow suppose $f(g,l)$ has a local maximum $(\\tilde g, \\tilde l)$, meaning that there exists an open set $\\mathcal B\\in\\mathbb R^{2p}$ centering at $(\\tilde g, \\tilde l)$ such for any $(g,l)\\in \\mathcal B$, $f(g,l) \\leq f(\\tilde g, \\tilde l)$. It is easy to verify that the set $\\mathcal B_\\beta = \\{g\\circ l:\\, (g,l)\\in\\mathcal B\\}$ is also an open set in $\\mathbb R^p$ centering at $\\tilde \\beta = \\tilde g\\circ \\tilde l$. Consequently, $\\tilde\\beta$ is also a local maximum of $(2n)^{-1}\\|X\\beta -y\\|^2$, which is a contradiction.\n\nDue to the same argument, any local minimum $(\\bar g,\\bar l)$ of $f(g,l)$ corresponds to a local minimum $\\bar\\beta =\\bar g\\circ \\bar l$ of $(2n)^{-1}\\|X\\beta -y\\|^2$. Since all local minimums of $(2n)^{-1}\\|X\\beta -y\\|^2$ are global and satisfies $X^T\\big(X\\bar\\beta-y\\big) = 0$, $(\\bar g,\\bar l)$ must be a global minimum of $f(g,l)$ and satisfies \n$X^T\\big(X(\\bar g\\circ \\bar l\\,)-y\\big) = 0$.\n\nNow let us prove the last part concerning saddle points. According to the previous paragraph and the discussion after lemma \\ref{Lem:global_min}, a point $(g^\\dagger,l^\\dagger)$ is a saddle point (that is, neither local minimum nor local maximum) if and only if there exists some $j_0\\in\\{1,\\ldots,p\\}$ such that\n\\begin{align*}\ng^\\dagger_{j_0} = l^\\dagger_{j_0}=0\\qquad\\mbox{and}\\qquad R^\\dagger_{j_0}\\neq 0,\n\\end{align*}\nwhere $R^\\dagger = X^T\\big(X(g^\\dagger\\circ l^\\dagger) - y\\big)\\in\\mathbb R^p$ (since otherwise this saddle point would satisfy $X^T\\big(X(\\bar g\\circ \\bar l\\,)-y\\big) = 0$, which is the sufficient and necessary condition for a global minimum).\nFor any vector $u\\in\\mathbb R^p$, we use Diag$(u)$ to denote the $p$-by-$p$ diagonal matrix whose diagonals are components of $u$.\nBy direct calculations, we can express the $2p$-by-$2p$ Hessian matrix $\\nabla^2_{g,l} f(g^\\dagger,l^\\dagger)$ of $f(g,l)$ at $(g^\\dagger,l^\\dagger)$ as\n\\begin{align*}\n\\begin{pmatrix}\n\\mbox{Diag}(l^\\dagger) & 0\\\\\n0 & \\mbox{Diag}(g^\\dagger)\n\\end{pmatrix}\n\\begin{pmatrix}\nn^{-1}X^TX & n^{-1}X^TX\\\\\nn^{-1}X^TX & n^{-1}X^TX\n\\end{pmatrix}\n\\begin{pmatrix}\n\\mbox{Diag}(l^\\dagger) & 0\\\\\n0 & \\mbox{Diag}(g^\\dagger)\n\\end{pmatrix}\n\\ \\ +\\ \\\n\\begin{pmatrix}\n0 & \\mbox{Diag}(R^\\dagger)\\\\\n\\mbox{Diag}(R^\\dagger) & 0\n\\end{pmatrix}.\n\\end{align*}\nConsequently, it is easy to verify that \n\\begin{align*}\n\\big(e_{j_0}^T, \\  -\\mbox{sgn}(R^\\dagger_{j_0}) \\,e_{j_0}^T\\big) \\,\\nabla^2_{g,l} f(g^\\dagger,l^\\dagger) \n\\begin{pmatrix}\ne_{j_0}\\\\ -\\mbox{sgn}(R^\\dagger_{j_0})\\, e_{j_0}\n\\end{pmatrix} =-2 (R^\\dagger_{j_0})^2 <0,\n\\end{align*}\nwhere $e_{j}$ denotes the vector whose $j$-th component is one and zero elsewhere for $j=1,\\ldots,p$, and sgn$(\\cdot)$ is the sign function. Consequently, there exists some nonzero vector $u$ such that the evaluation of the quadratic form $u^T\\nabla^2_{g,l} f(g^\\dagger,l^\\dagger)u <0$, implying $\\lambda_{\\min}\\big(\\nabla^2_{g,l} f(g^\\dagger,l^\\dagger)\\big)<0$.\n\n\n\n\n\\subsection{Proof of Lemma~\\ref{Lem:GD_global_min}}\nAccording to lemma \\ref{Lem:global_min}, all saddle points of $f(g,l)$ are strict, and all local minimums are global. \nTherefore, we can apply Theorem 4 in \\cite{lee2016gradient} to finish the proof, which states that for any twice continuously differentiable function, if it has only strict saddle points,\nthen gradient descent with a random initialization and sufficiently small constant step size almost surely converges to a local minimizer. \n\n\n\\subsection{Proof of Proposition \\ref{pro2.1}}\nRecall the decomposition of $u_t$ as\n\\begin{align*}\nu_t=z_t+d_t+e_t, \\quad\\mbox{with}\\quad\nz_t:=I_{S_1} u_t\\in\\mathbb R^p \\\\\n\\quad\\mbox{and}\\quad d_t:= I_{S_2} u_t\\in\\mathbb R^p, \\quad\ne_t:=(I-I_{S})u_t\\in\\mathbb R^p.\n\\end{align*}\nMoreover, we have the updating rule $u_{t+1} =u_t-2\\eta \\, n^{-1}X^T(Xu_t^2-y)\\circ u_t$. Therefore, the corresponding updating rules for strong signal component $z_t$, weak signal component $d_t$ and error component $e_t$ are\n\\begin{align*}\nz_{t+1} = z_t - 2\\eta\\,n^{-1}X^T(Xu_t^2-y)\\circ z_t,\\\\\nd_{t+1} = d_t - 2\\eta\\,n^{-1}X^T(Xu_t^2-y)\\circ d_t,\\\\\ne_{t+1} = e_t - 2\\eta\\,n^{-1}X^T(Xu_t^2-y)\\circ e_t.\n\\end{align*}\nAlso recall the following conditions on the step size $\\eta$, and RIP constant $\\delta$: \n$\\eta\\lesssim 1\/(\\kappa \\log(p\/\\alpha))$, and $\\delta \\lesssim 1\/(\\kappa s^{1\/2} \\log (p\/\\alpha))$. All the following analysis is based on the conditional events: \n\\begin{equation}\n\\frac{1}{n}\\|X_{S_1}^\\top w\\| \\leq c \\sigma  \\sqrt{\\frac{Ms_1}{n}} \\quad \\mbox{and} \\quad \\frac{1}{n}\\|X^\\top w\\|_{\\infty} \\leq  \\sigma \\sqrt{\\frac{4 \\log p}{n}} ,\n\\end{equation}  \nwhere the first concentration is based on lemma~\\ref{lem6} and the RIP of $X^\\top X\/n$.\nWe will use induction to prove that for all time $t<T_1=\\Theta(\\frac{ \\log (p\/\\alpha)}{\\eta m})$ in the first stage,\n\\begin{align}\n\\|e_t\\|_{\\infty}\\lesssim 1\/p, \\quad   \\|d_t\\|_{\\infty}\\lesssim 1\/p \\label{eq2.1.1},  \\\\\n\\|z_t\\|_{\\infty}\\lesssim 1 \\label{eq2.1.2},\\\\\n\\|z_t^2-I_{S_1} \\beta^\\ast\\|\\lesssim \\sqrt{s} \\label{eq2.1.3},\n\\end{align}\nwhere recall that $m$ is the smallest absolute value of components of $I_{S_1} \\beta^\\ast$, $s_1$ is the sparsity of $I_{S_1} \\beta^\\ast$, and $\\kappa$ is the ratio of largest and smallest of $I_{S_1} \\beta^\\ast$, $\\kappa m \\lesssim 1$ for simplicity.\nWhen $t=0$, we have $u_0=\\alpha \\mathbf{1}$, $z_0=\\alpha I_S \\mathbf{1}$, and $e_0=\\alpha I_{S^c} \\mathbf{1}$. Therefore, by choosing $\\alpha\\leq c\/p$ for some sufficiently small constant $c>0$, we can ensure $\\|e_0\\|_{\\infty}\\lesssim 1\/p$, and $\\|z_0\\|_{\\infty}\\lesssim 1$. Now suppose for time $t<T_1=\\Theta(\\frac{ \\log (p\/\\alpha)}{\\eta m})$, we have \\eqref{eq2.1.1}-\\eqref{eq2.1.3}. We will show that these inequalities also hold for time $t+1$.\n\nOur strategy is to show both signal part $\\|z_t\\|_{\\infty}$, $\\|d_t\\|_{\\infty}$ and error part $\\|e_t\\|_{\\infty}$ increases in an exponential rate when $\\theta_{s_1}(z_t)\\leq \\frac{\\sqrt{m}}{2}$. In addition, if $t+1<T_1$, then we show that induction hypothesis equations \\eqref{eq2.1.1}--\\eqref{eq2.1.3} still hold. \nIn particular, we divide the proof into 3 steps: \n\n\n\\subsubsection*{Step 1: Analyzing Error Dynamics and Weak Signals}\nThe proposition below gives us the dynamics of the error terms, we can use this dynamics to see the infinity norm of the error terms remains in the order of $O(1\/p)$ in the first stage when $t< T_1$.\n\\begin{pro}\\label{apro2.1}\n\tUnder assumptions in theorem~\\ref{thm1} and the induction hypothesis~\\eqref{eq2.1.1}-\\eqref{eq2.1.3} at $t$, when $t<T_1$ we have:\n\t\\begin{align*}\n\t\\|e_{t+1}\\|_\\infty&\\leq (1+c \\eta \\, m\/\\log(p\/\\alpha))\\|e_t\\|_\\infty, \\\\\n\t\\|d_{t+1}\\|_\\infty&\\leq (1+c \\eta \\, m\/\\log(p\/\\alpha))\\|d_t\\|_\\infty,\n\t\\end{align*}\n\twhen $\\|z_t^2-\\beta^\\ast\\| \\leq \\sqrt{\\epsilon}$, then let $\\tau=\\max\\{\\delta \\alpha, \\sigma \\sqrt{\\frac{\\log p}{n}}\\}$: \n\t\\begin{align*}\n\t\\|e_{t+1}\\|_\\infty&\\leq (1+c \\,\\eta \\tau)\\|e_t\\|_\\infty. \\\\\n\t\\|d_{t+1}\\|_\\infty&\\leq (1+c \\,\\eta \\tau)\\|d_t\\|_\\infty.\n\t\\end{align*}\n\\end{pro}\n\\begin{proof}\n\tFor $\\|e_{t+1}\\|_\\infty$,\twe bound the term $n^{-1}X^T(Xu_t^2-y) \\circ e_t$ in the updating formula:\n\t\\begin{align*}\n\t&\\|n^{-1}X^T(Xu_t^2-y) \\circ e_t\\|_{\\infty}= \\|e_t\\circ \\left(n^{-1}X^T\\left(X(u_t^2-\\beta^\\ast)-w\\right)\\right)\\|_\\infty \\\\\n\t&\\stackrel{(i)}{=} \\|e_t\\circ \\left( (n^{-1}X^TX(z_t^2+d_t^2-\\beta^\\ast))-(z_t^2+d_t^2-\\beta^\\ast)+n^{-1}X^TX(e_t^2)-n^{-1}X^T w \\right)\\|_\\infty \\\\\n\t&\\stackrel{(ii)}{\\leq} \\|e_t\\|_{\\infty} \\left(\\delta\\|z_t^2+d_t^2-\\beta^\\ast\\|+\\|(e_t^2)\\|_{\\infty}+\\delta \\|(e_t^2)\\|_1+ c \\sigma \\sqrt { \\frac { \\log  p  } { n } } \\right) \\\\\n\t&\\stackrel{(iii)}{\\leq}\\|e_t\\|_{\\infty}(c \\frac{m}{\\log (p\/\\alpha)}),\n\t\\end{align*} \n\twhere in step~$(i)$ we use the fact that $e_t\\circ (z_t^2+d_t^2-\\beta^\\ast)=0$; step~$(ii)$ follows by lemma~\\ref{lem3}:\n\t\\begin{align*}\n\t\\|e_t\\circ  \\left((n^{-1}X^TX(z_t^2+d_t^2-\\beta^\\ast))-(z_t^2+d_t^2-\\beta^\\ast)\\right)\\|_{\\infty} \\leq \\|e_t\\|_{\\infty} \\left(\\delta\\|z_t^2+d_t^2-\\beta^\\ast\\|\\right),\n\t\\end{align*}\n\tlemma~\\ref{lem4}:\t\n\t\\begin{align*}\n\t\\|e_t\\circ  \\left(n^{-1}X^TX(e_t^2)-e_t^2+e_t^2 \\right)\\|_{\\infty} \\leq \\|e_t\\|_{\\infty} \\left(\\delta\\|e_t^2\\|_1+\\|(e_t^2)\\|_{\\infty}\\right),\n\t\\end{align*}\n\tand lemma ~\\ref{lem5}:\n\t\n\t\\begin{align*}\n\t\\|e_t\\circ  \\left(n^{-1}X^Tw \\right)\\|_{\\infty} \\leq \\|e_t\\|_{\\infty} c\\sigma \\sqrt { \\frac { \\log  p  } { n } }.\n\t\\end{align*}\n\tIn step~$(iii)$, we applied the induction hypothesis~\\eqref{eq2.1.1}. Finally, note that $\\delta\\sqrt{s} \\kappa m \\lesssim \\frac{m}{\\log(p\/\\alpha)}$ and $\\sigma \\sqrt { \\frac { \\log  p  } { n } } \\lesssim   m\/\\log(p\/\\alpha)$ by the minimal strength of $m$, then an application of the triangle inequality leads to the claimed bound on $\\|e_{t+1}\\|_\\infty$.\\par \n\tSimilarly, for $\\|d_{t+1}\\|_\\infty$, \n\t\\begin{align*}\n\t&\\|n^{-1}X^T(Xu_t^2-y) \\circ d_t\\|_{\\infty}= \\|d_t\\circ \\left(n^{-1}X^T\\left(X(u_t^2-\\beta^\\ast)-w\\right)\\right)\\|_\\infty \\\\\n\t&\\stackrel{(i)}{=}\\|d_t\\circ \\left((n^{-1}X^TX(z_t^2+d_t^2-\\beta^\\ast))-(z_t^2+d_t^2-\\beta^\\ast)+(d_t^2-I_{S_2}\\beta^{\\ast})+n^{-1}X^TX(e_t^2)-n^{-1}X^T w \\right)\\|_\\infty \\\\\n\t&\\stackrel{(ii)}{\\leq} \\|d_t\\|_{\\infty} \\left(\\delta\\|z_t^2+d_t^2-\\beta^\\ast\\|+\\|d_t^2-I_{S_2}\\beta^{\\ast}\\|_{\\infty}+\\delta \\|(e_t^2)\\|_1+ c\\sigma \\sqrt { \\frac { \\log  p  } { n } } \\right) \\\\\n\t&{\\leq}\\|d_t\\|_{\\infty}(c \\, m\/\\log(p\/\\alpha)),\n\t\\end{align*} \n\twhere step~$(i)$ follows by $d_t \\circ (z_t^2- I_{S_1} \\beta^\\ast)=0$; step~$(ii)$ follows from the orthogonality between $d_t$ and $e_t$. The last inequality follows from the induction hypothesis. \\par\n\tThen, when $\\|z_t^2+d_t^2-\\beta^\\ast\\| \\leq \\sqrt{\\epsilon}$:\n\t\\begin{align*}\n\t\\|n^{-1}X^T(Xu_t^2-y) \\circ e_t\\|_{\\infty}&\\leq\\|e_t\\|_{\\infty} \\left(\\delta\\|z_t^2+d_t^2-\\beta^\\ast\\|+\\|(e_t^2)\\|_\\infty+\\delta \\|(e_t^2)\\|_1+ c\\sigma \\sqrt { \\frac { \\log  p  } { n } } \\right) \\\\\n\t&\\leq c \\tau \\|e_t\\|_{\\infty},\n\t\\end{align*} \n\twhere the last inequality follows from $\\delta \\cdot \\max \\{ \\alpha, \\sigma \\sqrt{\\frac{s \\log p}{n}}\\} \\leq \\max \\{ \\delta  \\alpha, \\sigma \\sqrt{\\frac{\\log p}{n}} \\}$ and $\\delta p \\alpha^2 \\lesssim \\delta \\alpha$.\n\t\\begin{align*}\n\t\\|n^{-1}X^T(Xu_t^2-y) \\circ d_t\\|_{\\infty}&\\leq \\|d_t\\|_{\\infty} \\left(\\delta\\|z_t^2+d_t^2-\\beta^\\ast\\|+\\|d_t^2-I_{S_2}\\beta^{\\ast}\\|_{\\infty}+\\delta \\|(e_t^2)\\|_1+ \\sigma \\sqrt { \\frac { \\log  p  } { n } } \\right) \\\\\n\t&\\leq c \\tau \\|d_t\\|_{\\infty},\n\t\\end{align*} \n\twhich is also based on $\\|d_t^2-I_{S_2}\\beta^{\\ast}\\|_{\\infty} \\leq c \\max\\{\\alpha^2,\\sigma \\sqrt { \\frac { \\log  p  } { n } }\\} $, since $\\|d_t\\| \\leq c \\alpha$ and the signals in set $S_2$ are weak.\n\\end{proof}\n\n\n\n\n\\subsubsection*{Step 2: Analyzing Strong Signal Dynamics}\nThe proposition below tells us the first-stage convergence of the strong signal terms, which are much faster than the growing of the weak signal and error terms, under the condition on $\\delta$ in Theorem~\\ref{thm1}. \n\n\\begin{pro}\\label{apro2.2}\n\tUnder assumptions in theorem~\\ref{thm1} and the induction hypothesis~\\eqref{eq2.1.1}-\\eqref{eq2.1.3} at $t$, when $\\theta_{s_1}(z_t)\\leq \\frac{\\sqrt{m}}{2}$ and $t<T_1$, we have\n\t\\begin{align*}\n\t\\theta_{s_1}(z_{t+1})&\\geq (1+\\frac{\\eta m}{4})\\theta_{s_1}(z_t)\n\t\\end{align*}\n\\end{pro}\n\\begin{proof}\n\tFirst,  applying lemma~\\ref{lem3} with $u= z_t^2-I_{S_1} \\beta^\\ast$ (which is $s_1$-sparse) and $v=\\mathbf 1\\in\\mathbb R^p$, we have\n\t\\begin{align}\n\t&\\|n^{-1}X^T\\left(Xu_t^2-y\\right)-(z_t^2-I_{S_1}\\beta^\\ast)\\|_{\\infty} \\nonumber\\\\\n\t=&\\,\\|n^{-1}X^TX(z_t^2-I_{S_1}\\beta^\\ast)-(z_t^2-I_{S_1}\\beta^\\ast) + n^{-1}X^TX (d_t^2+e_t^2-I_{S_2}\\beta^*)-n^{-1}X^T w\\|_\\infty\\nonumber\\\\\n\t\\leq&\\,\\delta\\|z_t^2-I_{S_1} \\beta^\\ast\\| +\\|n^{-1}X^TX (d_t^2-I_{S_2}\\beta^*)\\|_{\\infty}+  \\|n^{-1}X^TX e_t^2\\|_{\\infty}+\\|n^{-1}X^T w\\|_\\infty \\nonumber \\\\\n\t\\stackrel{(i)}{\\lesssim}&  m\/\\log(p\/\\alpha) \\label{eq2.3.1},\n\t\\end{align}\n\twhere step $(i)$ follows from induction hypothesis~\\eqref{eq2.1.1} and ~\\eqref{eq2.1.3}, and our condition on $\\delta$ and $m$. Let $z_{t+1}^+$ be the element-wise pseudo-inverse (meaning $1\/0=0$) of $z_{t+1}$. \n\t\n\tFirst, we show $\\|z_{t+1}^+\\circ z_t\\|_{\\infty}\\lesssim 1$. \n\tBy rewriting the updating rule of $z_t$ as\n\t\\begin{align}\\label{Eqn:Zt}\n\tz_{t+1}&= z_t\\circ(\\mathbf{1}-2\\eta \\,n^{-1}X^T\\left(X(u_t^2-\\beta^\\ast)-w\\right)), \n\t\\end{align}\n\twe have the element-wise lower bound \n\t\\begin{align*}\n\tz_{t+1}&\\geq z_t \\circ(\\mathbf{1}-2\\eta (z_t^2-I_{S_1}\\beta^\\ast)) -c\\,\\eta\\, \\frac{m}{\\log(p\/\\alpha)} |z_t|,\n\t\\end{align*}\n\twhere we applied \\eqref{eq2.3.1}.\n\tSince $\\eta \\leq 1\/4$ and $m\\leq 1$ under the assumptions of the proposition, the preceding display implies that each component of $z_{t+1}$ will larger than $1\/4$ times the corresponding component of $z_{t}$ in absolute value, which implies $\\|z_t \\circ z_{t+1}^+\\|_{\\infty}\\leq 4\\lesssim 1$.\n\t\n\tUsing the updating rule~\\eqref{Eqn:Zt} for $z_t$, equation~\\eqref{eq2.3.1}, and the induction hypothesis~\\eqref{eq2.1.2}, we have:\n\n\n\n\n\t\\begin{equation}\n\t\\|z_{t+1}-z_t \\circ(\\mathbf{1}-2\\eta (z_t^2-I_{S_1}\\beta^\\ast))\\|_{\\infty}\\lesssim \\eta \\frac{m}{\\log(p\/\\alpha)} |z_t|. \\label{eq2.3.2}\n\t\\end{equation} \n\tSince $\\|z_t \\circ z_{t+1}^+\\|_{\\infty}\\leq 4$, by conducting Hadamard product with $z_{t+1}^+$ on both sides, we obtain:\n\t\\begin{equation*}\n\t\\|\\mathbf{1}-(\\mathbf{1}-2\\eta (z_t^2-I_{S_1}\\beta^\\ast))\\circ z_{t+1}^+\\circ z_t\\|_{\\infty}\\lesssim \\eta \\frac{m}{\\log(p\/\\alpha)}.\n\t\\end{equation*}\n\t\\begin{equation*}\n\t\\|\\mathbf{1}-(\\mathbf{1}-2\\eta z_t^2)(\\+1+2\\eta I_{S_1} \\beta^\\ast))\\circ z_{t+1}^+\\circ z_t\\|_{\\infty}\\lesssim \\eta \\frac{m}{\\log(p\/\\alpha)} +\\eta^2 \\kappa^2 m^2.\n\t\\end{equation*}\n\tWe write \n\t\\begin{equation*}\n\tz_{t+1}-z_t \\circ(\\mathbf{1}-2\\eta z_t ^2)(\\mathbf{1}+2\\eta I_{S_1}\\beta^\\ast)=\\xi_t z_{t+1},\n\t\\end{equation*}\n\twhere $\\xi_t=\\mathbf{1}-(\\mathbf{1}-2\\eta z_t^2)(I+2\\eta I_{S_1}\\beta^\\ast)\\circ z_{t+1}^+\\circ z_t $, where we drop the $\\eta^2$ term since $\\eta \\kappa \\lesssim \\frac{1}{\\log(p\/\\alpha)}$ and $\\kappa m \\lesssim 1$, using the display after equation~\\eqref{eq2.3.2}, we have the following bound on $\\xi_t$,\n\t\\begin{equation*}\n\t\\|\\xi_t\\|_{\\infty} \\lesssim \\eta \\frac{m}{\\log(p\/\\alpha)}.\n\t\\end{equation*}\n\t\n\tConsider the function $f(x)=x(1-2\\eta \\,x^2)$, which is nondecreasing for $x\\in[-(6\\eta)^{-1}, (6\\eta)^{-1}]$. Therefore, when $\\|z_t\\|_{\\infty}\\lesssim 1$ and $\\eta  \\leq 1\/6$ (by choosing the constant $c_3$ in Theorem~\\ref{thm1} in the paper sufficiently small), we must have\n\t\\begin{equation} \\label{eqlem1}\n\t\\theta_{s_1}(z_t\\circ(\\mathbf{1}-2\\eta (z_t^2)))=\\theta_{s_1}(z_t)(1-2\\eta \\theta_{s_1}(z_t)^2).\n\t\\end{equation}\n\tThe last three displays together imply that when $\\theta_{s_1}(z_t)\\leq\\frac{\\sqrt{m}}{2}$, \n\t\\begin{align*}\n\t\\theta_{s_1}(z_{t+1})&\\stackrel{(i)}{\\geq} \\frac{(1+2\\eta m)\\theta_{s_1}(z_t)(1-2\\eta \\theta_{s_1}(z_t)^2)}{1+c\\eta \\frac{m}{\\log(p\/\\alpha)}} \\\\\n\t&\\geq (1+2\\eta m)(1-2\\eta \\theta_{s_1}(z_t)^2)\\theta_{s_1}(z_t)(1-c\\eta \\frac{m}{\\log(p\/\\alpha)}) \\\\\n\t&\\stackrel{(ii)}{\\geq} (1+\\frac{\\eta m}{4})\\theta_{s_1}(z_{t}), &&       \n\t\\end{align*}\n\twhere in step $(i)$ we used the definition of $m$ as $\\theta_{s_1}(\\beta^\\ast)$, and step $(ii)$ follows from the assumption $\\theta_{s_1}(z_t)\\leq\\frac{\\sqrt{m}}{2}$ and our conditions on $(\\eta,m)$, so that\n\t\\begin{align*}\n\t(1+2\\eta m)(1-\\frac{\\eta m}{2})&=(1+\\frac{3\\eta m}{2}-\\eta^2m^2)\n\t\\stackrel{(iii)}{\\geq} 1+\\eta m, \\quad\\mbox{and}\\\\\n\t(1+\\eta m)(1-c\\eta \\frac{m}{\\log(p\/\\alpha)})&=(1+\\eta m-c\\eta \\frac{m}{\\log(p\/\\alpha)}-c\\eta^2 \\frac{m^2}{\\log(p\/\\alpha)})\\\\\n\t&\\stackrel{(iv)}{\\geq} (1+\\frac{\\eta m}{2}-\\frac{\\eta^2 m^2}{2})\\stackrel{(v)}{\\geq} (1+\\frac{\\eta m}{4}), \n\t\\end{align*}\n\twhere step $(iii)$ follows from $\\eta m \\leq 1\/2$, step $(iv)$ by $\\frac{c}{\\log(p\/\\alpha)} \\leq 1\/2$, and in step $(v)$ we used $\\eta m \\leq1\/2$ and $\\frac{c}{\\log(p\/\\alpha)} \\leq 1\/2$ again. \\par\n\\end{proof}\n\n\n\n\n\\subsubsection*{Step 3: Prove Induction Hypothesis}\nOur last piece shows that the induction hypothesis is kept for $t+1$. This proposition combined with propositions~\\ref{apro2.1}-\\ref{apro2.2} leads to a proof for proposition \\ref{pro2.1} in the paper.\n\\begin{pro}\\label{apro2.3}\n\tUnder assumptions in theorem~\\ref{thm1} and the induction hypothesis~\\eqref{eq2.1.1}-\\eqref{eq2.1.3} at $t<T_1$, we have:\n\t\\begin{align*}\n\t&\\|e_{t+1}\\|_{\\infty}\\lesssim 1\/p, \\quad \\|d_{t+1}\\|_{\\infty}\\lesssim 1\/p, \\quad \\|z_{t+1}\\|_{\\infty}\\lesssim 1,  \n\t&\\quad\\mbox{and}\\quad \\|z_{t+1}^2-I_{S_1} \\beta^\\ast\\|\\lesssim \\sqrt{s}.\n\t\\end{align*}\n\\end{pro}\nWhen $T_1<t<T_2$, we also have:\n\\begin{align*}\n\\|e_{t+1}\\|_{\\infty} \\lesssim 1\/p, \\quad \\|d_{t+1}\\|_{\\infty}\\lesssim 1\/p.\n\\end{align*}\n\\begin{proof}\n\tFor error component $e_{t+1}$, we have:\n\t\\begin{align*}\n\t\\|e_{t+1}\\|_{\\infty}&\\leq (1+c\\,\\eta m\/\\log(p\/\\alpha))\\|e_t\\|_{\\infty} \n\t\\stackrel{(i)}{\\leq} (1+c\/T_1)^{T_1}\\|e_0\\|_{\\infty}  \\\\\n\t&\\leq e^c \\|e_0\\|_{\\infty} \\lesssim \\alpha \\lesssim 1\/p,\n\t\\end{align*}\n\twhere in step $(i)$ we apply $\\frac{m}{ \\log (p\/\\alpha)} \\lesssim 1\/(\\eta T_1)$. When $\\|z_t^2-\\beta^\\ast\\| \\leq \\sqrt{\\epsilon}$, since $T_2=c\/(\\eta \\tau)$, when $T_1<t<T_2$:\n\t\\begin{align*}\n\t\\|e_{t+1}\\|_{\\infty}&\\leq (1+c\\,\\eta \\tau )\\|e_t\\|_{\\infty} \n\t\\stackrel{(i)}{\\leq} (1+c\/T_2)^{T_2} \\cdot \\alpha \\\\\n\t&\\leq e^c \\cdot \\alpha \\lesssim \\alpha \\lesssim 1\/p.\n\t\\end{align*}\n\tSimilarly we can show the induction hypothesis for weak signals $\\|d_{t+1}\\|$.\n\tFor strong signal component $z_{t+1}$, we apply equation~\\eqref{Eqn:Zt} to get element-wise upper bound:\n\t\\begin{align*}\n\tz_{t+1,i}&\\leq z_{t,i} (1-2\\eta (z_{t,i}^2-\\beta^\\ast_i)) +c\\,\\eta\\, \\delta\\,\\sqrt{s}\\, z_{t,i} ,\n\t\\end{align*}\n\tSince $\\delta \\sqrt{s}\\lesssim 1$, and the function $f(x)=2\\eta x^3-c\\eta x$ is nonnegative when $x\\gtrsim 1$. Therefore, as long as $\\|z_t\\|_\\infty \\gtrsim 1$, we always have\n\t$\\|z_{t+1}\\|_{\\infty}\\leq \\|z_t\\|_{\\infty}$ , thus\n\t\\begin{align*}\n\t\\|z_{t+1}^2-\\beta^\\ast\\|&\\leq  \\|z_{t+1}^2\\|+\\|\\beta^\\ast\\| \\lesssim \\sqrt{s}.\n\t\\end{align*}\n\\end{proof}\n\n\n\n\\subsection{Proof of Proposition \\ref{pro2.4}}\n\\begin{proof}\n\tWe decompose $\\|u_{t+1}^2-\\beta^\\ast\\|^2=\\|(u_t-\\eta \\nabla f(u_t))\\circ(u_t-\\eta \\nabla f(u_t))-\\beta^\\ast\\|^2$ into 5 parts, according to the order of $\\eta$:\n\t\\begin{align*}\n\t\\|u_{t+1}^2-\\beta^\\ast\\|^2=\\|u_{t}^2-\\beta^\\ast\\|^2-4\\eta\\langle \\nabla f(u_t)\\circ u_{t},u_{t}^2-\\beta^\\ast\\rangle\\\\\n\t+4\\eta^2 \\langle \\nabla f(u_t)^2,u_{t}^2-\\beta^\\ast\\rangle+2\\eta^2 \\|f(u_t)\\circ u_{t}\\|^2\\\\\n\t-4\\eta^3 \\langle \\ \\nabla f(u_t)\\circ u_t, \\nabla f(u_t)^2\\rangle+\\eta^4 \\|\\nabla f(u_t)^2\\|^2.\n\t\\end{align*} \n\tIn addition, we have the decomposition,\n\t\\begin{align}\\label{decomp}\n\t\\|u_{t+1}^2-\\beta^\\ast\\|^2=\\|z_{t+1}^2-I_{S_1}\\beta^\\ast\\|^2 +\\|d_{t+1}^2-I_{S_2}\\beta^\\ast\\|^2+\\|e_t^2\\|^2 \\nonumber\\\\\n\t\\leq \\|z_{t+1}^2-I_{S_1}\\beta^\\ast\\|^2+ c s_2  \\frac{\\log p}{n}+c' p \\alpha^4,\n\t\\end{align}\n\twhere the rate are based on the dynamics for weak signals and error terms, so we only need to consider the estimation errors on the index set $S_1$. \\par\n\tBefore we bound each term, we need the following bound for the gradient:\n\t\\begin{align}\n\t\\|\\nabla f(u_t)\\|_{\\infty}&=\\|2u_t\\circ n^{-1}X^T(Xu_t^2-y)\\|_{\\infty} \\nonumber\\\\\n\t&\\leq c (\\|n^{-1}X^TX(u_t^2-\\beta^\\ast)\\|_{\\infty} +\\|X^Tw\\|_\\infty) \\nonumber \\\\\n\t&=c (\\|n^{-1}X^TX(z_t^2+d_t^2-\\beta^\\ast +e_t^2)\\|_{\\infty}+\\|X^Tw\\|_\\infty) \\nonumber\\\\\n\t&\\stackrel{(i)}{\\leq} c(\\|z_t^2+d_t^2-\\beta^\\ast\\|_{\\infty}+\\delta \\|z_t^2+d_t^2-\\beta^\\ast\\|+\\|n^{-1}X^TX (e_t)^2\\|_{\\infty}+\\|X^Tw\\|_\\infty) \\nonumber \\\\ &\\stackrel{(ii)}{\\lesssim} 1 \\label{eq2.4.1} ,\n\t\\end{align}\n\twhere (i) follows by lemma~\\ref{lem3} and lemma~\\ref{lem4} with $v=\\mathbf 1\\in\\mathbb R^p$, and (ii) follows by the assumption that $(z_t,d_t,e_t)\\in\\Theta_{LR}$, and\n\t\\begin{align} \\label{eq:15}\n\t&\\|n^{-1}X^TX (e_t)^2\\|_{\\infty}\\leq \\|e_t^2\\|_\\infty+\\delta \\|e_t^2\\|_1 \\leq c \\delta p \\alpha^2 \\nonumber \\\\\n\t&\\|n^{-1}(X^TX(d_t^2-I_{S_2} \\beta^*))\\|_\\infty \\leq  \\|d_t^2-I_{S_2} \\beta^*\\|_\\infty+\\delta \\|d_t^2-I_{S_2} \\beta^*\\|_2 \\leq  c\\alpha^2 +  c'\\sigma \\sqrt{\\frac{\\log p}{n}}           \\nonumber \\\\\n\t&\\|n^{-1}(X^TX(d_t^2-I_{S_2} \\beta^* +e_t^2)-X^Tw)\\|_{\\infty}\\leq c\\delta p \\alpha^2 +c'\\sigma \\sqrt{\\frac{\\log p}{n}}.\n\t\\end{align}\n\tNote that $z_t$ is $s_1\\leq s$ sparse, we have the core error term characterization:\n\t\\begin{equation} \\label{eq:16}\n\t\\begin{aligned}\n\t\\|z_t \\circ n^{-1}(X^TX (d_t^2-I_{S_2} \\beta^* +e_t^2)-X^Tw)\\|^2 &\\leq c (Ms_1) \\delta^2 p^2 \\alpha^4 +c'  \\sigma^2 \\frac{Ms_1}{n}\\\\\n\t&\\leq c(\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}),\n\t\\end{aligned}\n\t\\end{equation}\n\n\n\twhere we used $\\delta^2 s \\leq 1$, $\\|e_t\\|_{\\infty}\\lesssim \\alpha \\lesssim 1\/p$ and lemma~\\ref{lem4}  with $v=\\mathbf 1\\in\\mathbb R^p$ and lemma ~\\ref{lem6}. \n\t\n\tIn the rest of the proof, we will repeatedly apply lemma~\\ref{lem3}, \\ref{lem4} to get rid of the $n^{-1}X^TX$ term;  we will also repeatedly use lemma~\\ref{lem2} and the assumption $(z_t,d_t,e_t)\\in\\Theta_{LR}$ to bound  each term in the decomposition of $\\|u_{t+1}^2-\\beta^\\ast\\|^2$. For quantifying the errors, we will use equation~\\eqref{eq:15} and~\\eqref{eq:16}. Note that based on the decomposition~\\eqref{decomp}, we only need to care about terms contributed by the index set $S_1$.\n\t\n\tFirst, we need a lower bound for the order one term,\n\t\\begin{align*}\n\t&\\langle \\nabla f(u_t)\\circ u_{t},I_{S_1} (u_{t}^2-\\beta^\\ast)\\rangle = \\langle \\nabla f(u_t)\\circ z_{t},z_{t}^2-I_{S_1} \\beta^\\ast\\rangle \\\\\n\t&\\geq\\langle n^{-1}X^TX(z_t^2- I_{S_1}\\beta^\\ast)\\circ z_t^2,z_t^2- I_{S_1}\\beta^\\ast\\rangle \\\\\n\t&-\\|z_t \\circ n^{-1}(X^TX (d_t^2-I_{S_2} \\beta^* +e_t^2)-X^Tw)\\|\\|z_t \\circ (z_t^2-I_{S_1}\\beta^\\ast)\\|\\\\\n\t&\\geq \\|z_t \\circ (z_t^2-I_{S_1}\\beta^*)\\|^2-\\delta  \\|z_t^2-I_{S_1}\\beta^*\\|^2\\\\\n\t&-\\|z_t \\circ n^{-1}(X^TX (d_t^2-I_{S_2} \\beta^* +e_t^2)-X^Tw)\\|^2\/2-\\|z_t \\circ (z_t^2-I_{S_1}\\beta^\\ast)\\|^2\/2, \\\\\n\t&\\geq \\frac{1}{2}\\|z_t \\circ (z_t^2-I_{S_1}\\beta^*)\\|^2-\\delta    \\|z_t^2-I_{S_1}\\beta^*\\|^2-c(\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}), \n\t\\end{align*} \n\twhere the second inequality is based on Cauchy-Schwarz inequality and the last step follows by equation~\\eqref{eq:16}.  The rest of the terms are also based on the bound:\n\t\\begin{align*}\n\t\\|I_{S_1} \\nabla f(u_t)\\|^2 &= \\|z_t \\circ n^{-1}X^T(Xu_t^2-y)\\|^2 \\\\\n\t&\\stackrel{(i)}{\\leq} 2\\|z_t \\circ n^{-1}X^TX(z_t^2-I_{S_1} \\beta^\\ast)\\|^2 \n\t+ 2\\|z_t \\circ n^{-1}(X^TX (d_t^2-I_{S_2}\\beta^*+e_t^2)-X^Tw)\\|^2 \\\\ \n\t&\\stackrel{(ii)}{\\leq} c \\|z_t^2-I_{S_1}\\beta^\\ast\\|^2+c'(\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}),\n\t\\end{align*}\n\twhere in step (i) we use induction hypothesis and Cauchy-Schwarz inequality,  and step (ii) we use the bound for equation~\\eqref{eq:16}. \\\\\n\tFor the order two term, we apply a similar idea to bound $\\| \\langle I_{S_1}\\nabla f(u_t)^2,u_t^2-\\beta^\\ast\\rangle\\|$ and $\\|I_{S_1} f(u_t)^2\\|^2$. More precisely, since $\\|u_t^2-\\beta^\\ast\\|_{\\infty}\\lesssim 1$, we have\n\t\\begin{align*}\n\t\\langle \\nabla f(u_t)^2,I_{S_1} (u_{t}^2-\\beta^\\ast) \\rangle = \\langle \\nabla f(u_t)\\circ (u_{t}^2-\\beta^\\ast) ,\\nabla I_{S_1} f(u_t)\\rangle \\\\\\leq c\\|I_{S_1} \\nabla f(u_t)\\|^2 \\leq c \\|z_t^2-I_{S_1}\\beta^\\ast\\|^2+c'(\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tSimilarly $\\|u_t\\|_\\infty \\lesssim 1$,  we bound $\\|I_{S_1}\\nabla f(u_t)\\circ u_{t}\\|^2$ through\n\t\\begin{align*}\n\t\\|I_{S_1} \\nabla f(u_t)\\circ u_{t}\\|^2 \\leq c\\|I_{S_1}\\nabla f(u_t)\\|^2 \\leq c \\|z_t^2-I_{S_1} \\beta^\\ast\\|^2+c' (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*}\n\n\n\n\n\n\n\n\n\n\t\n\t\n\tFor the order three term, using the fact that $\\|u_{t}\\|_{\\infty}\\lesssim 1$ and $\\|\\nabla f(u_t)\\|_{\\infty}\\lesssim 1$, we have:\n\t\\begin{align*}\n\t\\langle I_{S_1} \\nabla  f(u_t)\\circ u_t, \\nabla f(u_t)^2\\rangle \\leq c \\|\\nabla f(u_t)\\|^2\n\t\\leq c\\|I_{S_1} \\nabla f(u_t)\\|^2 \\\\\n\t\\leq c \\|z_t^2-I_{S_1}\\beta^\\ast\\|^2+c'(\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*} \n\t\n\t\n\tFor the order four term, we have \n\t\\begin{equation*}\n\t\\|I_{S_1} \\nabla  f(u_t)^2\\|^2\\lesssim \\|I_{S_1} \\nabla f(u_t)\\|^2 \\lesssim c\\|z_t^2-I_{S_1}\\beta^\\ast\\|^2+c'(\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{equation*}\n\t\n\tPutting all pieces together, since $\\eta \\lesssim 1$ under our assumption on $\\eta$, we get\n\t\\begin{align*}\n\t\\|z_{t+1}^2-I_{S_1} \\beta^\\ast\\|^2 &\\leq \\|z_{t}^2-I_{S_1} \\beta^\\ast\\|^2- 2 \\eta \\|(z_t^2-I_{S_1} \\beta^\\ast)\\circ z_t\\|^2 \\\\\n\t&+c\\eta (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}) +\n\t4 \\delta \\eta\\|z_t^2-I_{S_1} \\beta^\\ast\\|^2,\n\t\\end{align*} \n\twith $\\|z_t^2-I_{S_1} \\beta^\\ast\\|^2\\cdot m\/4\\leq \\|(z_t^2-I_{S_1} \\beta^\\ast)\\circ z_t\\|^2$.\n\tSince $\\theta_{s}(z_t)\\geq \\sqrt{m}\/2$ for $(z_t,d_t,e_t)\\in\\Theta_{LR}$, and\n\t$\\delta \\kappa  \\leq \\frac{1}{4}$,\n\t\n\tunder our assumptions on $(\\eta,\\delta)$, we reach\n\t\\begin{equation*}\n\t\\|z_{t+1}^2-I_{S_1} \\beta^\\ast\\|^2\\leq(1-\\eta m)\\|z_{t}^2-I_{S_1} \\beta^\\ast\\|^2+c(\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}) .\n\t\\end{equation*} \n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm1}} \n\\begin{proof}\n\tAs mentioned after the theorem in the paper, we divide the convergence into two stages. In the first ``burn-in'' stage, we show that the smallest component of the strong signal part $z_t$ increases at an exponential rate in $t$ until hitting $\\sqrt{m}\/2$, \n\t\\begin{align*}\n\t\\theta_{s_1}(z_{t+1})&\\geq (1+\\frac{\\eta m}{4})\\theta_{s}(z_t), \\ \\text{when}\\  \\theta_{s_1}(z_t)\\leq \\sqrt{m}\/2.\n\t\\end{align*}\n\tIn the second stage, after iterate $u_t$ enters $\\Theta_{LR}$, we have the geometric convergence up to some high-order error term\n\t\\begin{align*}\n\t\\|z_{t+1}^2-I_{S_1} \\beta^\\ast\\|^2&\\leq(1-{\\eta m})\\|z_{t}^2-I_{S_1}\\beta^\\ast\\|^2+c (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}),  \\ \\text{when}\\  \\theta_{s_1}(z_t)\\geq \\sqrt{m}\/2,\n\t\\end{align*}\n\tWhen $t\\leq \\Theta(\\log\\frac{\\sqrt{m}}{\\alpha}\/(\\eta m))$, the convergence is in the first stage; and proposition~\\ref{apro2.1} implies that the iterate $u_t$ enters $\\Theta_{LR}$ (corresponding to the second stage) in at most $\\Theta(\\log\\frac{c}{p \\alpha^2}\/(\\eta m))$ iterations.  proposition~\\ref{apro2.1} and proposition~\\ref{pro2.4} together imply for any $t=\\Theta(\\log\\frac{p}{\\alpha}\/(\\eta m))$, \n\t\\begin{equation}\n\t\\|z_t^2-I_{S_1} \\beta^\\ast\\|^2 \\leq c (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{equation}\n\tFor any $T_2>t>T_1$, since $\\|e_t\\|, \\|d_t\\| \\lesssim 1\/p$ is still controlled, combined with bound~\\eqref{decomp}, we have:\n\t\\begin{align*}\n\t\\|u_t^2-\\beta^\\ast\\|^2 \\leq c (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n} + \\sigma^2 \\frac{s_2 \\log p}{n}).\n\t\\end{align*}\n\\end{proof}\n\n\n\n\\subsection{Proof of Proposition \\ref{pro3.1}}\n\nSimilar to the proof for the nonnegative case, we use induction to show that for each $t\\leq T_1$,\n\\begin{align}\n\\|a_{t,S_1^c}\\|_{\\infty}\\lesssim 1\/p,\\ &&\\|b_{t,S_1^c}\\|_{\\infty}\\lesssim 1\/p \\label{eq3.1.1},\\\\\n\\|a_{t,S_1}\\|_{\\infty}\\lesssim 1, &&\\|b_{t,S_1}\\|_{\\infty}\\lesssim 1 \\label{eq3.1.2},\\\\\n\\|\\beta_{t,S_1}-\\beta^\\ast_{S_1}\\|\\lesssim \\sqrt{s} \\label{eq3.1.3},\n\\end{align}\nwhere the set $S_1^c$ is the union of weak signals and errors. When $t=0$, we have $g_0=\\alpha \\mathbf{1}$, $l_0=0$. Therefore, under the assumption $\\alpha\\lesssim 1\/p$, we have $\\|a_{0,S_1^c}\\|_{\\infty}\\lesssim 1\/p $, $\\|a_{0,S_1}\\|_{\\infty}\\lesssim 1$, and similar bounds for $b$. Now suppose for time $t<T_1$, we have \\eqref{eq3.1.1}-\\eqref{eq3.1.3}. We divide into 3 steps to show the same bounds hold for time $t+1$. Note that the following analysis is still based on the conditional events: \n\\begin{equation}\n\\frac{1}{n}\\|X_{S_1}^\\top w\\| \\leq c \\sigma \\sqrt{\\frac{Ms_1}{n}} \\quad \\mbox{and} \\quad \\frac{1}{n}\\|X^\\top w\\|_{\\infty} \\leq  \\sigma \\sqrt{\\frac{4 \\log p}{n}} ,\n\\end{equation}  \nstill the first concentration is based on lemma~\\ref{lem6} and the RIP of $X^\\top X\/n$.\n\\subsubsection*{Step 1: Analyzing Error Dynamics}\n\\begin{pro}\\label{apro3.1}\n\tBased on RIP and the constant assumption in theorem~\\ref{thm2} and the induction hypothesis~\\eqref{eq3.1.1}-\\eqref{eq3.1.3}, when $t<T_1$ we have:\n\t\\begin{align*}\n\t\\|a_{t+1,S_1^c}\\|_{\\infty}&\\leq (1+c\\eta m\/\\log(p\/\\alpha))\\|a_{t+1,S_1^c}\\|_{\\infty},\\\\\n\t\\|b_{t+1,S_1^c}\\|_{\\infty}&\\leq (1+c\\eta m\/\\log(p\/\\alpha))\\|b_{t+1,S_1^c}\\|_{\\infty}.\n\t\\end{align*}\n\tWhen $\\|\\beta_{t,S_1}-\\beta_{S_1}^\\ast\\| \\leq c \\sqrt{\\epsilon}$, let $\\tau =\\max \\{\\sqrt{\\frac{\\log p}{n}},  \\delta \\alpha\\}$, then\n\t\\begin{align*}\n\t\\|a_{t+1,S_1^c}\\|_{\\infty}&\\leq (1+c\\eta\\tau)\\|a_{t+1,S_1^c}\\|_{\\infty},\\\\\n\t\\|b_{t+1,S_1^c}\\|_{\\infty}&\\leq (1+c\\eta\\tau)\\|b_{t+1,S_1^c}\\|_{\\infty}.\n\t\\end{align*}\n\\end{pro}\n\\begin{proof}\n\tBy the definition of $a_t$ and $b_t$, we have the their updating rules:\n\t\\begin{align*}\n\ta_{t+1,S_1^c}=a_{t,S_1^c}-\\eta a_{t,S_1^c}\\circ n^{-1}X^T(X \\beta_t-y),\\\\\n\tb_{t+1,S_1^c}=b_{t,S_1^c}+\\eta b_{t,S_1^c}\\circ n^{-1}X^T(X \\beta_t-y).\n\t\\end{align*} \n\tThe proofs of the two claimed bound follows exactly the same lines as in the nonnegative case, since we uses the triangle inequality $\\|a_{t+1,S_1^c}\\|_\\infty \\leq \\|a_{t,S_1^c}\\|_\\infty + \\|\\eta a_{t,S_1^c}\\circ n^{-1}X^T(X \\beta_t-y)\\|_\\infty$ at the beginning, which does not change when the sign becomes positive as in the second line of $b_{t+1,S^c}$.\n\tConsequently, the dynamics of weak signals and error terms $a_{t,S_1^c}$ and $b_{t,S_1^c}$ will behave like $e_t$ in the nonnegative case---the proof of proposition~\\ref{apro2.1} still works when $\\|a_{t,S_1^c}\\|_{\\infty}$, $\\|b_{t,S_1^c}\\|_{\\infty}\\lesssim 1\/p$.\n\\end{proof}\n\n\n\\subsubsection*{Step 2: Analyzing Signal Dynamics}\nThe proof becomes more complicated for the signal dynamics. Our goal is to show:\n\\begin{itemize}\n\t\\item  if the true signal $\\beta_i^\\ast$ is positive and $\\beta_{t,i}<\\beta_i^*\/2$, then $a_{t,i}^2$ increases and $b_{t,i}^2$ decreases, both in an exponential rate. Overall, the sign of $i$th component $\\beta_{t,i}=a_{t,i}^2-b_{t,i}^2$ of $\\beta_t$ tends to grow to positive in an exponential rate;\n\t\\item if the true signal $\\beta_i^\\ast$ is negative and $\\beta_{t,i}>\\beta_i^*\/2$, then $a_{t,i}^2$ decreases and $b_{t,i}^2$ increases, both in an exponential rate.  Overall, the sign of $i$th component $\\beta_{t,i}=a_{t,i}^2-b_{t,i}^2$ of $\\beta_t$ tends to fall to negative in an exponential rate.\n\\end{itemize}\nOur analysis will be based on the resemblance between update rules of $(a_t,b_t)$ and $u_t$ in the nonnegative case. Recall that we assumed that the RIP constant $\\delta\\lesssim 1\/(\\kappa \\sqrt{s}\\log\\frac{p}{\\alpha})$, step size $\\eta\\lesssim 1\/(\\kappa \\log\\frac{p}{\\alpha})$, and $T_1=\\Theta(\\frac{ \\log(p\/\\alpha)}{\\eta m})$.\n\\begin{pro}\\label{apro3.2}\n\tUnder assumptions in theorem~\\ref{thm2} and the induction hypothesis~\\eqref{eq3.1.1}-\\eqref{eq3.1.3} at $t<T_1$,  when $|\\beta_{t,i}-\\beta^\\ast_{t,i}|\\leq \\frac{1}{2}|\\beta_i^*|$, then for any $i\\in S$:\n\t\\begin{align*}\n\t\\beta_{t,i}\\geq (1+c\\,\\eta\\,\\beta^\\ast_{i})^t \\alpha^2-c'\\alpha^2,\\ \\ \\text{if} \\ \\beta^\\ast_{i}>0;\\\\\n\t\\beta_{t,i}\\leq (1-c\\,\\eta\\,\\beta^\\ast_{i})^t (-\\alpha^2)+c'\\alpha^2,\\ \\ \\text{if}\\  \\beta^\\ast_{i}<0.\n\t\\end{align*}\n\\end{pro}\n\\begin{proof}\n\tSimilar with the proof of proposition \\ref{pro2.2}, first we approximate $(n^{-1}X^TXu)\\circ v$ by $u\\circ v$ based on the RIP condition via lemmas~\\ref{lem3} and \\ref{lem4},\n\t\\begin{align}\n\t&\\|n^{-1}X^T(X\\beta_t-y)-(I_{S_1}\\beta_{t}-I_{S_1}\\beta^\\ast)\\|_{\\infty} \\nonumber\\\\\n\t\\leq&\\, \\|n^{-1}X^TX (I_{S_1^c}r_{t})\\|_{\\infty}+\\delta\\|r_{t,S_1}\\|  +\\|X^Tw\\|_\\infty \n\t\\lesssim \\frac{m}{\\log (p\/\\alpha)},    \\label{eq4.2.1}\n\t\\end{align}\n\timplying that under the condition $ m\\lesssim 1$, we have\n\t\\begin{align*}\n\t&\\|n^{-1}X^TX(\\beta_t-y)\\|_{\\infty}\\\\\n\t\\leq&\\, \\|\\beta_{t,S_1}-\\beta^\\ast_{S_1}\\|_{\\infty} +\n\t\\|n^{-1}X^T(X (I_{S_1^c}r_{t})-w)\\|_{\\infty}+\\delta\\|\\beta_{t,S_1}-\\beta^\\ast_{S_1}\\| \n\t\\lesssim 1,    \n\t\\end{align*}\n\twhere the last inequality uses $\\|\\beta_{t,S_1^c}\\|_{\\infty}\\lesssim 1\/p$ and $\\|\\beta_{t,S_1}-\\beta^\\ast_{S_1}\\|\\lesssim \\sqrt{s}$.\n\t\n\tIn order to analyze $\\beta_{t,S_1}=a_{t,S_1}^2-b_{t,S_1}^2$, let us focus on $a_{t,S_1}^2$ and $b_{t,S_1}^2$, separately. According to the updating rule of $a_{t,S_1}$, we have\n\t\\begin{align*}\n\t&a_{t+1,S_1}^2=a_{t,S_1}^2-2\\eta a_{t,S_1}^2 \\circ [n^{-1}X^T(X\\beta_{t}-y)]_{S_1}+\\eta ^2 a_{t,S_1}^2 \\circ [n^{-1}X^T(X\\beta_{t}-y)]_{S_1}^2,\n\t\\end{align*}\n\twhere recall that for a vector $a\\in\\mathbb R^p$, $a_{S_1}$ denote the sub-vector of $a$ with indices in $S_1$. Applying lemmas~\\ref{lem3} with $v=\\mathbf 1$, we obtain\n\t\\begin{align*}\n\t&\\|a_{t+1,S_1}^2-a_{t,S_1}^2-2\\eta a_{t,S_1}^2(\\beta_{t,S_1}-\\beta^\\ast_{S_1})\\|_{\\infty}\\lesssim \\eta\\frac{m}{\\log (p\/\\alpha)}+\\eta^2 \\kappa^2 m^2\\stackrel{(i)}{\\lesssim} \\eta\\frac{m}{\\log (p\/\\alpha)}.\n\t\\end{align*}\n\twhere in step $(i)$ we used $\\eta \\kappa m \\lesssim \\frac{m}{\\log (p\/\\alpha)}$ sand $\\kappa m \\lesssim 1$. Similar to the nonnegative case, since $\\eta  m \\leq 1\/2$, $\\frac{m}{\\log (p\/\\alpha)} \\leq 1\/2$, we have $a_{t,i}^2\/a_{t+1,i}^2 \\leq 4$ for $i\\in S_1$. Therefore, we can obtain an element-wise bound for $\\xi_t=(\\xi_{t,i})_{i\\in S_1}$,\n\t\\begin{equation*}\n\t\\xi_{t,i}:\\,=1-a_{t,i}^2\/a_{t+1,i}^2 \\circ(2\\eta ({\\beta}_{t,i}-\\beta^\\ast_i)), \n\t\\end{equation*} \n\tas $\\|\\xi_t\\|_{\\infty}\\lesssim \\eta\\frac{m}{\\log (p\/\\alpha)}$.\n\tEquivalently, we can write\n\t\\begin{align}\n\ta_{t+1,i}^2= a_{t,i}^2 (1-2\\eta ({\\beta}_{t,i}-\\beta^\\ast_i))+\\xi_{t,i} a_{t+1,i}^2. \\label{eq4.2.2}\n\t\\end{align} \n\tNow let us divide into two cases depending on the sign of $\\beta^\\ast_i,\\ i\\in S_1$:\n\t\n\t\\emph{Case $\\beta_i^* >0$:} When ${\\beta}_{t,i}-\\beta^\\ast_i\\leq -\\beta^\\ast_i\/2$, since $\\beta^\\ast_i \\geq m$, we have by equation~\\eqref{eq4.2.2},\n\t\\begin{align*}\n\ta_{t+1,i}^2 &\\geq \\frac{a_{t,i}^2(1+\\eta \\beta^\\ast_i)}{1+c\\eta\\frac{m}{\\log (p\/\\alpha)}} \\geq  a_{t,i}^2(1+\\eta \\beta^\\ast_i)(1-c\\eta\\frac{\\beta_i^*}{\\log (p\/\\alpha)} )\\geq  a_{t,i}^2(1+\\eta \\beta^\\ast_i\/4),\n\t\\end{align*}\n\twhere the last inequality follows since $1\/\\log(p\/\\alpha)\\leq 1\/2$ and $\\eta \\beta^\\ast_i \\leq 1\/2$. Similarly, we can analyze $b_{t,S}^2$ to get\n\t\\begin{align*}\n\tb_{t+1,i}^2 &\\leq \\frac{b_{t,i}^2(1-\\eta \\beta^\\ast_i)}{1-c\\eta\\delta\\sqrt{s}}\\leq  b_{t,i}^2(1-\\eta \\beta^\\ast_i)(1+c\\eta\\beta_i^*\/\\log(p\/\\alpha))\\leq  b_{t,i}^2(1-\\eta \\beta^\\ast_i\/4).\n\t\\end{align*}\n\tTherefore, $a_{t+1,i}^2$ increases at an exponential rate faster than the noise term $a_{t+1,S_1^c}$ while $b_{t+1,i}^2$ decreases to zero at an exponential rate, and when $a_{t+1,i}$ increases to $\\beta_{i}^\\ast\/2$, $b_{t+1,i}$ decreases to $O( \\alpha^4)$ correspondingly. A combination of these two leads to the first claimed bound for $\\beta_{i}^\\ast>0$.\n\t\n\t\\emph{Case $\\beta_i^* <0$:} The analysis for the case is similar: when ${\\beta}_{t,i}-\\beta^\\ast_i\\geq -\\beta^\\ast_i\/2$, we have:\n\t\\begin{align*}\n\ta_{t+1,i}^2 \\leq  a_{t,i}^2(1-\\eta \\beta^\\ast_i\/4),\\quad\\mbox{and}\\quad b_{t+1,i}^2 \\geq  b_{t,i}^2(1+\\eta \\beta^\\ast_i\/4),\n\t\\end{align*}\n\twhich leads to the second claimed bound for $\\beta_{i}^\\ast<0$.\n\\end{proof}\n\nAs a consequence of the proof in this step, after at most $T\\geq \\Theta ( \\frac{\\log(m\/\\alpha^2)}{\\eta m})$ iterations, we are guaranteed to have have $|{\\beta}_{T,i}|\\geq |\\beta^\\ast_i|\/2$ with $sign({\\beta}_{T,i})=sign(\\beta^\\ast_i)$ and $\\min\\{a_{T,i}^2, b_{T,i}^2\\} \\leq c \\alpha^4$.\n\n\\subsubsection*{Step 3: Prove Induction Hypothesis}\n\\begin{pro}\\label{apro3.3}\n\tUnder assumptions in theorem~\\ref{thm2},  the induction hypothesis~\\eqref{eq3.1.1}-\\eqref{eq3.1.3} at $t<T_1$, we have:\n\t\\begin{align*}\n\t&\\|a_{t+1,S_1^c}\\|_{\\infty}  \\lesssim 1\/p,\\quad \\|b_{t+1,S_1^c}\\|_{\\infty}\\lesssim 1\/p ,\\\\\n\t&\\|a_{t+1,S_1}\\|_{\\infty} \\lesssim 1, \\quad\\|b_{t+1,S_1}\\|_{\\infty}\\lesssim 1 ,\\\\\n\t&\\mbox{and}\\quad\\|{\\beta}_{t+1,S_1}-\\beta^\\ast_{S_1}\\|\\lesssim 1.\n\t\\end{align*}\n\tThen for $T_1<t<T_2$, we still have:\n\t\\begin{align*}\n\t\\|a_{t+1,S_1^c}\\|_{\\infty}  \\lesssim 1\/p,\\quad \\|b_{t+1,S_1^c}\\|_{\\infty}\\lesssim 1\/p.\n\t\\end{align*}\n\\end{pro}\n\\begin{proof}\n\tOur induction hypothesis \n\t\\begin{align*}\n\t&\\|a_{t,S_1^c}\\|_{\\infty}\\lesssim 1\/p,\\quad \\|b_{t,S_1^c}\\|_{\\infty}\\lesssim 1\/p,\\\\\n\t&\\|a_{t,S_1}\\|_{\\infty}\\lesssim 1, \\quad\\|b_{t,S_1}\\|_{\\infty}\\lesssim 1,\n\t\\end{align*} \n\timplies $\\|\\beta_{t,S_1^c}\\|_{\\infty}\\lesssim 1\/p^2$ and $\\|\\beta_{t,S_1^c}-\\beta^\\ast_{t,S_1^c}\\|\\lesssim \\sqrt{s}$. By updating rules:\n\t\\begin{align*}\n\ta_{t+1}=a_{t}-\\eta a_{t}\\circ n^{-1}X^T(X {\\beta}_t-y),\\\\\n\tb_{t+1}=b_{t}+\\eta b_{t}\\circ n^{-1}X^T(X {\\beta}_t-y),\n\t\\end{align*}\n\tand our condition $\\eta m\/\\log(p\/\\alpha) \\lesssim 1\/T_1$, we can prove along similar lines as in the proof of \\ref{pro2.3} that\n\t\\begin{align*}\n\t&\\|a_{t+1,S_1^c}\\|_{\\infty}\\lesssim 1\/p,\\quad \\|b_{t+1,S_1^c}\\|_{\\infty}\\lesssim 1\/p ,\\\\\n\t&\\|a_{t+1,S_1}\\|_{\\infty}\\lesssim 1, \\quad\\|b_{t+1,S_1}\\|_{\\infty}\\lesssim 1.\n\t\\end{align*}\n\tThese inequalities also imply $\\|\\beta_{t+1,S_1^c}\\|_{\\infty}\\lesssim 1\/p^2$ and $\\|\\beta_{t+1,S_1}-\\beta^\\ast_{t+1,S_1}\\|\\lesssim \\sqrt{s}$. Then similarly for $T_1<t<T_2$, we can also show:\n\t\\begin{align*}\n\t\\|a_{t+1,S_1}\\|_{\\infty}\\lesssim 1, \\quad\\|b_{t+1,S_1}\\|_{\\infty}\\lesssim 1.\n\t\\end{align*} \n\\end{proof}\n\n\n\n\\subsection{Proof of Proposition \\ref{pro3.4}}\n\\begin{proof}\n\tThe proof is similar to that of proposition \\ref{pro2.4}.\\\\\n\tIn particular, we can decompose $\\|\\beta_{t+1}-\\beta^\\ast\\|^2=\\|a_{t+1}^2-b_{t+1}^2-\\beta^\\ast\\|^2$ into 5 parts according to the order of $\\eta$:\n\t\\begin{align*}\n\t\\|\\beta_{t+1}-\\beta^\\ast\\|^2&=\\|(a_{t}-\\eta a_{t} \\circ n^{-1}X^T(X r_t-w))^2-(b_{t}+\\eta b_{t} \\circ n^{-1}X^T(X r_t-w))^2-\\beta^\\ast\\|^2, \\\\\n\t&=\\|\\beta_t-2\\eta(a_t^2+b_t^2)\\circ n^{-1}X^T(X r_t-w)+\\eta^2\\beta_t\\circ (n^{-1}X^T(X r_t-w))^2-\\beta^\\ast\\|^2 ,\\\\\n\t&=\\|\\beta_{t}^2-\\beta^\\ast\\|^2-4\\eta\\langle (a_t^2+b_t^2)\\circ n^{-1}X^T(X r_t-w),\\beta_t-\\beta^\\ast\\rangle,\\\\\n\t&\\quad+4\\eta^2 \\| (a_t^2+b_t^2)\\circ n^{-1}X^T(X r_t-w)\\|^2+2\\eta^2 \\langle \\beta_t\\circ (n^{-1}X^T(X r_t-w))^2 ,\\beta_{t}-\\beta^\\ast\\rangle, \\\\\n\t&\\quad-4\\eta^3 \\langle \\beta_t\\circ (n^{-1}X^T(X r_t-w))^2 ,(a_t^2+b_t^2)\\circ n^{-1}X^T(X r_t-w)\\rangle,\\\\\n\t&\\quad+\\eta^4 \\|\\beta_t\\circ (n^{-1}X^T(X r_t-w))^2\\|^2,\n\t\\end{align*} \n\twhere for notational simplicity, we denoted $\\beta_t-\\beta^\\ast$ by $r_t$.\n\tFirst look at the term $a_t^2+b_t^2$. We have the component-wise bound $a_{t,i}^2+b_{t,i}^2=g_{t,i}^2+l_{t,i}^2\\geq |g_{t,i} \\circ l_{t,i}|=|\\beta_{t,i}|$, implying\n\t\\begin{align}\\label{3.4.3}\n\ta_{t,i}^2+b_{t,i}^2-|\\beta_{t,i}|=2\\min\\{a_{t,i}^2,b_{t,i}^2\\}\\stackrel{(i)}{\\lesssim} \\alpha^4,\n\t\\end{align}\n\twhere (i) follows by proof of proposition \\ref{pro3.2}, since the smaller one of $\\{a_{t,i}^2,b_{t,i}^2\\}$ will be its initial value decreasing at an exponential rate until the rate $\\alpha^4$. Similarly, we have the decomposition:\n\t\\begin{align*}\n\t\\|\\beta_{t+1}-\\beta^\\ast\\|^2&=\\|\\beta_{t+1,S_1}-\\beta_{S_1}^\\ast\\|^2+\\|\\beta_{t+1,S_2}-\\beta_{S_2}^\\ast\\|^2+\\|\\beta_{t+1,S^c}\\|^2 \\\\\n\t&\\leq \\|\\beta_{t+1,S_1}-\\beta_{S_1}^\\ast\\|^2+ cs_2 \\frac{\\log p}{n}+ c' p \\alpha^4,\n\t\\end{align*}\n\twhich is based on the orthogonality between different sub vectors and the induction hypothesis, so we only need to calculate the errors induced by the strong signals.  \\\\\n\tFor the gradient, we have:\n\t\\begin{align}\n\t\\|n^{-1}X^T(Xr_t-w)\\|_{\\infty}\\leq &\\, \\|\\beta_{t,S}-\\beta^\\ast\\|_{\\infty}+\\delta\\|\\beta_{t,S}-\\beta^\\ast\\| \\nonumber \\\\\n\t&+\\|\\beta_{t,S^c}\\|_{\\infty}+\\delta p \\|\\beta_{t,S^c}\\|_{\\infty} +\\|X^Tw\\|_\\infty \\leq c; \\label{3.1.13}\\\\\n\t\\|n^{-1}X^T(X (I_{S_1^c} r_t) -w)\\|_{\\infty}\\leq&\\,  c \\delta p \\alpha^2+ c \\sigma \\sqrt{\\frac{\\log p}{n}}. \\label{eq3.1.15}\n\t\\end{align}\n\tNote that $I_{S_1} \\beta_t$ is $s_1$ sparse, so we have the following core error term  characterization:\n\t\\begin{align}\n\t\\|I_{S_1} \\beta_t \\circ n^{-1}X^T(X r_{t,S_1^c}-w)\\| \\leq&\\,  cs \\delta^2 p^2 \\alpha^4+ c' \\sigma^2 M s_1 \\frac{\\log p}{n} \\leq c (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}). \\label{eq3.1.16}\n\t\\end{align}\n\tSimilarly, our analysis for errors are based on equation~\\eqref{eq3.1.15} and~\\eqref{eq3.1.16}.\n\t\n\tFor the order one term, recall the bound\n\t\\begin{align*}\n\t&\\langle (a_t^2+b_t^2)\\circ n^{-1}X^T(X r_t-w),I_{S_1} r_{t}\\rangle \\\\\n\t& \\stackrel{(i)}{\\geq} \\langle |I_{S_1} \\beta_t|\\circ n^{-1}X^T (X r_t-w), I_{S_1} r_{t}\\rangle -s \\alpha^4 \\\\\n\t&=\\langle |I_{S_1} \\beta_t|\\circ n^{-1}X^T X (I_{S_1} r_t), I_{S_1} r_{t}\\rangle \n\t+\\langle |I_{S_1} \\beta_t|\\circ n^{-1}X^T (X (I_{S_1^c} r_t)-w), I_{S_1}r_{t}\\rangle\\\\\n\t&\\geq \\left\\|\\,r_{t,S_1}\\circ \\sqrt{|\\beta_{t,S_1}|}\\,\\right\\|^2-\\delta  \\|\\,r_{t,S_1}\\|^2\\\\\n\t&-\\|\\sqrt{|\\beta_{t,S_1}|} \\circ n^{-1}X^T (X (I_{S_1^c} r_t)-w)\\|^2\/2 -\\| \\sqrt{|\\beta_{t,S_1}|} \\circ \\,r_{t,S_1}\\|^2\/2 \\\\\n\t&\\geq \\frac{1}{2}\\left\\|\\,r_{t,S_1}\\circ \\sqrt{|\\beta_{t,S_1}|}\\,\\right\\|^2-\\delta  \\|\\,r_{t,S_1}\\|^2 -c (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\t\n\t\\end{align*} \n\twhere $(i)$ is based on Cauchy-Schwarz inequality and  we use the fact $(a_t,b_t)\\in\\Theta_{LR}^G$, lemma~\\ref{lem2}, equation~\\eqref{eq3.1.16}.\n\t\n\tFor the rest of terms, we can see they are all based on the bound: \n\t\\begin{align*}\\label{eq:3.1.17}\n\t&\\|I_{S_1} \\sqrt{\\beta_t}   \\circ n^{-1}X^T(Xr_t-w)\\|^2\\\\ \n\t&\\leq \\|I_{S_1} \\sqrt{\\beta_t}   \\circ n^{-1}X^TX (I_S r_t)\\|^2+\\|I_{S_1} \\sqrt{\\beta_t} \\circ n^{-1}X^T(I_{S_1^c} r_t-w)\\|^2 \\\\\n\t&\\lesssim  c \\|r_{t,S_1}\\|^2+c' (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*} \n\t\n\tFirst consider the order two terms $\\langle \\beta_t\\circ (n^{-1}X^TX r_t)^2 ,I_{S_1}(\\beta_{t}-\\beta^\\ast)\\rangle$ and $\\| I_{S_1}(a_t^2+b_t^2)\\circ n^{-1}X^TX r_t\\|^2$. For the former one, since $\\|r_t\\|_{\\infty}\\lesssim 1$, we have\n\t\\begin{align*}\n\t\\langle \\beta_t\\circ (n^{-1}X^TX r_t)^2 ,I_{S_1} r_{t}\\rangle &= \\langle I_{S_1}\\beta_t\\circ n^{-1}X^T(X r_t-w), n^{-1}X^T(X r_t-w) \\circ r_t)\\rangle\\\\\n\t&\\leq c \\|I_{S_1}\\sqrt{\\beta_t} \\circ n^{-1}X^T(X r_t-w)\\|^2 \\leq  c \\|r_{t,S_1}\\|^2+c' (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*}\n\twhere we use the condition $(a_t,b_t)\\in\\Theta_{LR}^G$. \n\tThe analysis of the latter one $\\| I_{S_1}(a_t^2+b_t^2)\\circ n^{-1}X^TX r_t\\|^2$  is similar based on $\\|a_t^2+b_t^2 - \\beta_t\\|_{\\infty}\\lesssim \\alpha^2$ and $\\|\\sqrt{\\beta_t} \\| \\lesssim 1 $: \n\t\\begin{align*}\n\t&\\quad\\|I_{S_1}(a_t^2+b_t^2)\\circ n^{-1}X^T(X r_t-w)\\|^2 \\\\\n\t&\\leq \\|(I_{S_1} |\\beta_t| \\circ n^{-1}X^T(X r_t-w)\\|^2 +s \\alpha^4 \\leq  c \\|r_{t,S_1}\\|^2+c'  (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*}\n\t\n\tFor the order three term, using inequality $\\|I_{S_1} (a_t^2+b_t^2)\\circ (n^{-1}X^TX r_t)\\|_{\\infty}\\lesssim 1$, we obtain:\n\t\\begin{align*}\n\t&\\quad\\langle \\beta_t\\circ (n^{-1}X^TX r_t)^2 ,I_{S_1} (a_t^2+b_t^2)\\circ (n^{-1}X^TX r_t)\\rangle\\\\\n\t&\\leq c \\|(I_{S_1}\\sqrt{\\beta_t}) \\circ n^{-1}X^T(X r_t-w)\\|^2 \\leq  c \\|r_{t,S_1}\\|^2+c'  (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*} \n\t\n\tFor the order four term, we have \n\t\\begin{align*}\n\t&\\|I_{S_1} \\beta_t\\circ (n^{-1}X^TX r_t)^2\\|^2\\leq c \\|I_{S_1} \\sqrt{\\beta_t} (n^{-1}X^TX r_t)\\|^2  \\leq  c \\|r_{t,S_1}\\|^2+c'  (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}).\n\t\\end{align*}\n\t\n\tPutting all pieces together, and using our condition $\\eta \\lesssim 1$, we can obtain\n\t\\begin{align*}\n\t\\|I_{S_1} r_{t+1}\\|^2 &\\leq \\|I_{S_1} r_{t}\\|^2-2\\eta \\,\\Big\\|\\,r_{t,S_1}\\circ \\sqrt{|\\beta_{t,S_1}|}\\,\\Big\\|^2 \\\\\n\t&+c  (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n})+4 \\eta \\delta \\|r_{t,S_1}\\|^2.\n\t\\end{align*} \n\tSince for $(a_t,b_t)\\in\\Theta_{LR}^G$, we have $\\|r_{t,S_1}\\circ \\sqrt{|\\beta_{t,S_1}|}\\|^2\\geq \\|r_{t,S_1}\\|^2\\cdot m\/2 $. Therefore, under our conditions on $(\\delta,\\eta)$ with $\\delta  \\kappa \\leq 1\/4$, we have:\n\t\\begin{equation}\n\t\\|\\beta_{t+1,S_1}-\\beta^\\ast_{S_1}\\|^2\\leq(1-\\eta m)\\|\\beta_{t,S_1}-\\beta^\\ast_{S_1}\\|^2+c  (\\alpha^2+\\sigma^2 \\frac{Ms_1}{n}) .\n\t\\end{equation} \n\\end{proof}\n\\subsection{Proof of Theorem~\\ref{thm2}} \nThe proof is the same as that of theorem~\\ref{thm1}, but now using propositions~\\ref{pro3.1} and \\ref{pro3.4}.\n\n\\subsection{Proof of Theorem~\\ref{thm5}} \nAccording to the proof of Theorem~\\ref{thm2} , we have $\\|\\beta_{t,S_1}-\\beta_{S_1}^*\\|_{\\infty} \\leq \\|\\beta_{t,S_1}-\\beta_{S_1}^*\\| \\leq \\sigma \\sqrt{s \/n} $, therefore $\\beta_{t,j}> \\beta^*_j - \\sigma \\sqrt{s \/n} \\geq \\lambda$ based on the definition of the strong signals for $j \\in S_1$, while for errors and weak signals $j \\in S_1^c$, we have $\\|\\beta_{t,S_1^c}\\|_{\\infty} \\leq c\\alpha^2 < \\lambda$. Consequently, after the component-wise hard thresholding operation at level $\\lambda$, all strong signals remains nonzero while all weak signals and errors become zero.\n\\bibliographystyle{apalike}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and Main Results}\n\nWe study slow-fast systems driven by fractional Brownian motions (fBm):\n\\begin{alignat}{4}\n  dX_t^\\varepsilon&=f(X_t^\\varepsilon,Y_t^\\varepsilon)\\,dt+g(X_t^\\varepsilon, Y_t^\\varepsilon)\\,dB_t, &\\qquad X_0^\\varepsilon&=X_0, \\label{eq:slow}\\\\\n  dY_t^\\varepsilon&=\\frac{1}{\\varepsilon}b(X_t^\\varepsilon,Y_t^\\varepsilon)\\,dt+\\frac{1}{\\varepsilon^{\\hat{H}}}\\sigma\\,d\\hat{B}_t, &\\qquad Y_0^\\varepsilon&=Y_0, \\label{eq:fast}\n\\end{alignat}\nwhere $B$ and $\\hat{B}$ are independent fBms on an underlying complete probability space $(\\Omega, {\\mathcal F},\\ensuremath\\mathbb{P})$ with Hurst parameters $H\\in(\\frac12,1)$ and $\\hat{H}\\in(1-H,1)$, respectively. Here, $g:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\Lin[m]{d}$ and $\\sigma\\in\\Lin{n}$ is non-degenerate. As the scale parameter $\\varepsilon>0$ is taken to $0$, one hopes that the \\emph{slow motion} $X^\\varepsilon$ is well approximated by an \\emph{effective dynamics} $\\bar{X}$. For $H=\\hat{H}=\\frac12$, this convergence has been studied by myriad authors since the seminal works of Bogolyubov-Mitropol{\\textquotesingle}ski\\u{\\i} \\cite{Bogolyubov1955} and Hasminskii \\cite{Hasminskii1968}, see e.g. the monographs and survey articles \\cite{Freidlin2012,Skorokhod2002,Pavliotis2008,Berglund2006,Liu2012,Li2018} and references therein for a comprehensive overview. It is still a very active research area \\cite{Liu2020,Roeckner2020,Roeckner2020a}.\n\nFor $H,\\hat{H}\\neq\\frac12$, the SDEs \\eqref{eq:slow}--\\eqref{eq:fast} provide a suitable model for economic, medical, and climate phenomena exhibiting a genuinely non-Markovian behavior in both the system and its environment. It is for example very well known that neglecting temporal memory effects in climate modeling by resorting to a diffusion model results in prediction notoriously mismatching observational data \\cite{Ashkenazy2003,Karner2002,Davidsen2010,Barboza2014}. It thus became widely popular to use fBm in climate modeling \\cite{Sonechkin1998,Yuan2014,Eichinger2020}.\n\n\nWhile slow-fast systems with fractional noise have seen a tremendous spike of interest in the last two years \\cite{Bourguin-ailus-Spiliopoulos-typical,Bourguin-Gailus-Spiliopoulos,Hairer2020,Pei-Inaham-Xu, Pei-Inaham-Xu2,Han2021}, all of these works resort to Markovian, strongly mixing fast processes by choosing $\\hat{H}=\\frac12$ in \\eqref{eq:fast}. The main contribution of this article is to establish the convergence $X^\\varepsilon\\to\\bar{X}$ even for a \\emph{non-Markovian} fast dynamics by allowing $\\hat{H}\\neq\\frac12$. It hardly comes as a surprise that this renders the analysis much more delicate and it is not clear at all if an averaging principle can even hold for a fractional, \\emph{non-mixing} environment. In fact, the usual assumption in the aforementioned works on Markovian averaging principles is a strong mixing condition with an algebraic rate \\cite{Heunis1994,Abourashchi2010}. This condition is essentially never satisfied for a fractional dynamics \\cite{Bai2016}.\n\nRecent work of Hairer and the first author of this article suggests the following ansatz for the effective dynamics:\n\\begin{equation}\\label{eq:effective_dynamics}\n  d\\bar{X}_t=\\bar{f}(\\bar{X}_t)\\,dt+\\bar{g}(\\bar{X}_t)\\,dB_t,\\qquad \\bar{X}_0=X_0,\n\\end{equation}\nwhere $\\bar{f}(x)\\ensuremath\\triangleq\\int f(x,y)\\,\\pi^x(dy)$ and similar for $\\bar{g}$ \\cite{Hairer2020}. For $\\hat{H}=\\frac12$, this work showed that the average is taken with respect to the unique invariant $\\pi^x$ of the fast dynamics with \\emph{frozen} slow input\n\\begin{equation}\\label{eq:frozen_fast}\n  dY_t^x=b(x,Y_t^x)\\,dt+\\sigma\\,d\\hat{B}_t.\n\\end{equation} \nFor $\\hat{H}\\neq\\frac12$, it is \\emph{a priori} not clear what $\\pi^x$ should be. We show that it is the one-time marginal of the unique stationary path space law $\\ensuremath\\mathbb{P}_{\\pi^x}\\in\\P\\big(\\ensuremath{\\mathcal{C}}(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)\\big)$, see \\cref{sec:physical_solution} for details. [Here and in the sequel, $\\P(\\ensuremath{\\mathcal{X}})$ denotes the set of Borel probability measures on a Polish space $\\ensuremath{\\mathcal{X}}$.]\n \n\nIn addition to standard regularity requirements ensuring well-posedness of the slow-fast system (see \\cref{cond:feedback} below), we shall impose a contractivity condition on the drift in \\eqref{eq:fast}:\n\\begin{definition}\\label{define-semi-contractive}\n  Let $\\lambda, R\\geq 0$ and $\\kappa>0$. We write $\\S(\\kappa, R, \\lambda)$ for the set of Lipschitz continuous functions $b:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}^n$ satisfying\n  \\begin{equation}\\label{eq:semicontractive}\n  \\Braket{b(x)-b(y),x-y}\\leq\\begin{cases}\n  -\\kappa|x-y|^2, & |x|,|y|\\geq R,\\\\\n  \\lambda|x-y|^2, &\\text{otherwise}.\\\\\n  \\end{cases}\n\\end{equation}\n\\end{definition}\n\nNote that $\\lambda$ may be smaller than $\\Lip{b}$, whence its prescription is not necessarily redundant. If $b=-\\nabla V$ is a gradient vector field with potential $V$, then \\eqref{eq:semicontractive} is equivalent to $V$ being at most $\\lambda$-concave on $|x|<R$ and $\\kappa$-convex on $|x|\\geq R$. If $V\\in\\ensuremath{\\mathcal{C}}^2(\\ensuremath{\\mathbb{R}}^n)$, these requirements are in turn equivalent to $\\nabla^2V\\geq-\\lambda$ and $\\nabla^2 V\\geq\\kappa$ on the respective sets.\n\n\\begin{theorem}[Fractional Averaging Principle]\\label{thm:feedback_fractional}\n  Consider the slow-fast system \\eqref{eq:slow}--\\eqref{eq:fast}. Suppose that $f,g\\in\\Cb{2}$ and $b$ satisfies \\cref{cond:feedback}. Let $\\alpha<H$ and $\\kappa,R>0$. Then there is a number $\\lambda_0>0$ such that, if $b(x,\\cdot)\\in\\S\\big(\\kappa, R,\\lambda_0\\big)$ for every $x\\in\\ensuremath{\\mathbb{R}}^d$, all of the following hold:\n  \\begin{itemize}\n    \\item For every $x\\in\\ensuremath{\\mathbb{R}}^d$, there exists a unique stationary path space law $\\ensuremath\\mathbb{P}_{\\pi^x}\\in\\P\\big(\\ensuremath{\\mathcal{C}}(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)\\big)$ for the frozen fast dynamics \\eqref{eq:frozen_fast}.\n    \\item Let $\\pi^x\\in\\P(\\ensuremath{\\mathbb{R}}^n)$ be the one-time marginal of $\\ensuremath\\mathbb{P}_{\\pi^x}$. If \n    \\begin{equation*}\n      x\\mapsto\\bar{g}(x)\\ensuremath\\triangleq\\int_{\\ensuremath{\\mathbb{R}}^n}g(x,y)\\,\\pi^x(dy)\\in\\ensuremath{\\mathcal{C}}_b^2\\big(\\ensuremath{\\mathbb{R}}^d,\\Lin[m]{d}\\big),\n    \\end{equation*}\n    then there is a unique pathwise solution to \\eqref{eq:effective_dynamics} and $X^\\varepsilon\\to\\bar{X}$ as $\\varepsilon\\to 0$ in $\\ensuremath{\\mathcal{C}}^\\alpha\\big([0,T],\\ensuremath{\\mathbb{R}}^d\\big)$ in probability for any $T>0$.\n  \\end{itemize} \n\\end{theorem}\n\nThe regularity of $\\bar{g}$ not only hinges on the regularity of $g$ but also on the fast dynamics. First we note that the requirement on $\\bar{g}$ clearly holds for a diffusion coefficient depending only on the slow motion $X^\\varepsilon$: \n\\begin{equation*}\n  dX_t^\\varepsilon=f(X_t^\\varepsilon,Y_t^\\varepsilon)\\,dt+g(X_t^\\varepsilon)\\,dB_t.\n\\end{equation*}\nAnother class of examples is provided by \\cref{cor:smooth} below. \n\nThe technical core of the proof of \\cref{thm:feedback_fractional} is a quantitative \\emph{quenched ergodic theorem} on the conditional evolution of the process \\eqref{eq:frozen_fast}. We prove this by means of a control argument, which is of independent interest. In fact, it allows us to improve recent work of Panloup and Richard \\cite{Panloup2020} by establishing geometric ergodicity for a class of SDEs driven by additive fractional noise. To our best knowledge, this is the first result achieving an exponential convergence rate for a fractional dynamics (excluding the trivial instance of an everywhere contractive drift).\n\nLet $\\TV{\\mu}\\ensuremath\\triangleq\\sup_A|\\mu(A)|$ denote the total variation norm, $\\ensuremath{\\mathcal{W}}^p$ be the $p$-Wasserstein distance, and $\\ensuremath{\\mathbb{W}}^p$ be the Wasserstein-like metric for generalized initial conditions introduced in \\cref{def:wasserstein}.\n \n\\begin{theorem}[Geometric Ergodic Theorem]\\label{thm:geometric}\nLet $(Y_t)_{t\\geq 0}$ be the solution to the SDE\n\\begin{equation}\\label{eq:sde_intro}\n  dY_t=b(Y_t)\\,dt+\\sigma\\,dB_t\n\\end{equation}\nstarted in the generalized initial condition $\\mu$, where $\\sigma\\in\\Lin{n}$ is non-degenerate and $B$ is an fBm with Hurst parameter $H\\in(0,1)$. Then, for any $p\\geq 1$ and any $\\kappa,R>0$, there exists a $\\Lambda=\\Lambda(\\kappa,R,p)>0$ such that, whenever $b\\in\\S\\big(\\kappa,R,\\Lambda\\big)$, there is a unique invariant measure $\\mathcal I_\\pi$ for \\eqref{eq:sde_intro} in the sense of \\cref{initial-condition}. Moreover, \n    \\begin{equation}\\label{eq:wasserstein_time_t}\n        \\ensuremath{\\mathcal{W}}^p(\\mathcal{L}(Y_t),\\pi)\\leq Ce^{-ct} \\ensuremath{\\mathbb{W}}^p\\big(\\mu,\\mathcal I_{\\pi}\\big) \\qquad \\forall\\, t\\geq 0\n    \\end{equation}\n    and\n    \\begin{equation}\\label{eq:tv_process}\n        \\TV{\\L(Y_{\\cdot+t})-\\ensuremath\\mathbb{P}_\\pi}\\leq Ce^{-ct}\\ensuremath{\\mathbb{W}}^1\\big(\\mu,\\mathcal I_\\pi\\big) \\qquad \\forall\\, t\\geq 0,\n    \\end{equation}\n    where $c,C>0$ are numerical constants independent of $t\\geq 0$ and $\\mu$.\n\\end{theorem}\n\n\nThe work \\cite{Hairer2005} already contained a result on the rate of convergence. There, the author assumed an off-diagonal contraction condition, see \\cref{cond:off_diagonal} below, and obtained an algebraic rate in \\eqref{eq:tv_process}. Very recently Panloup and Richard \\cite{Panloup2020} studied $b\\in\\S(\\kappa,R,0)$ for which they found a rate of order $e^{-Dt^\\gamma}$ for some $\\gamma<\\frac23$ in both \\eqref{eq:wasserstein_time_t} and \\eqref{eq:tv_process}. Albeit these works did not require a global Lipschitz condition on the drift for Hurst parameters $H<\\frac12$, we emphasize that they do impose this assumption for $H>\\frac12$ to obtain \\eqref{eq:tv_process}. This is due to the lack of regularity of a certain fractional integral operator. \\Cref{thm:geometric} thus provides a genuine ramification of the results of \\cite{Panloup2020} in the latter case. We note that similarly to the work of Panloup and Richard, the Wasserstein decay \\eqref{eq:wasserstein_time_t} also holds for more general Gaussian driving noises with stationary increments. We shall briefly comment on this in \\cref{sec:geometric_ergodicity}.\n\nWith the spiking interest in numerical methods based on the generalized Langevin equation with memory kernel \\cite{Chak2020,Leimkuhler2020}, \\cref{thm:geometric} and the quenched quantitative ergodic theorem underpinning it can give a better theoretical understanding. A first step would be to derive quantitative estimates on the constants $c$, $C$, and $\\Lambda$; a possible pathway is outlined in \\cref{rem:constant_xi} below. It is an interesting open question if there is indeed a finite threshold value of $\\Lambda$ beyond which the exponential rates \\eqref{eq:wasserstein_time_t}--\\eqref{eq:tv_process} no longer hold. As established by Eberle, such a transition from exponential to sub-exponential rates does not happen in case $H=\\frac12$ \\cite{Eberle2016}.\n\n\\begin{example}\n  Let us give an example of a drift not covered by the sub-exponential convergence theorems of \\cite{Panloup2020}. Consider the double-well potential\n  \\begin{equation*}\n      V(x)=\\alpha|x|^4-\\beta|x|^2\n  \\end{equation*}\n  for $\\alpha,\\beta>0$. We modify $V$ outside of a compact such that its Hessian is bounded. Set $b=-\\nabla V$. It is clear that $b\\notin\\bigcup_{\\kappa,R>0}\\S(\\kappa,R,0)$ as soon as $\\beta>0$. However, for $\\frac{\\beta}{\\alpha}$ sufficiently small, \\cref{thm:geometric} furnishes an exponential rate of convergence.\n\\end{example}\n\n\\paragraph{Outline of the article.} The next section features a brief overview of preliminary material. In \\cref{sec:convergence}, we prove the quantitative quenched ergodic theorem and deduce \\cref{thm:geometric}. The proof of \\cref{thm:feedback_fractional} is concluded in \\cref{sec:feedback}.\n\\paragraph{Acknowledgements.} We would like to thank the anonymous referees for their careful reading and helpful comments. Partial support from the EPSRC under grant no. EP\/S023925\/1 is also acknowledged.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\nRecall that one-dimensional fractional Brownian motion with Hurst parameter $H\\in(0,1)$ is the centered Gaussian process $(B_t)_{t\\geq 0}$ with\n\\begin{equation*}\n  \\Expec{(B_t-B_s)^2}=|t-s|^{2H},\\qquad s,t\\geq 0.\n\\end{equation*}\nTo construct $d$-dimensional fBm one lets the coordinates evolve as independent one-dimensional fBms with the same Hurst parameter. We will make frequent use of the following classical representation of one-dimensional fBm as a fractional integral of a two-sided Wiener process $(W_t)_{t\\in\\ensuremath{\\mathbb{R}}}$, which is due to Mandelbrot and van Ness \\cite{Mandelbrot1968}:\n\\begin{equation}\\label{eq:mandelbrot}\n B_t=\\alpha_H\\int_{-\\infty}^0 (t-u)^{H-\\frac12}-(-u)^{H-\\frac12}\\,dW_u+\\alpha_H\\int_0^t(t-u)^{H-\\frac12}\\,dW_u,\\qquad t\\geq 0.\n\\end{equation}\nHere, $\\alpha_H>0$ is some explicitly known normalization constant and we also write $B_t=\\bar B_t+\\tilde B_t$. \n\n\n\\subsection{Invariant Measures of Fractional SDEs} \\label{sec:physical_solution}\n\nAlbeit being certainly non-Markovian on its own, the solution to \\eqref{eq:sde_intro} can actually be cast as the marginal of an infinite-dimensional Feller process $Z_t\\ensuremath\\triangleq\\big(Y_t,(W_s)_{s\\leq t}\\big)$ with values in $\\ensuremath{\\mathbb{R}}^n\\times\\H_H$. Here, $W$ is the two-sided Wiener process driving the equation through \\eqref{eq:mandelbrot} and $\\H_H$ is a H\\\"older-type space of paths $\\ensuremath{\\mathbb{R}}_-\\to\\ensuremath{\\mathbb{R}}^n$ supporting the Wiener measure $\\ensuremath{\\mathsf{W}}$. More concretely, $\\H_H$ is the closure of the space $\\{f\\in\\ensuremath{\\mathcal{C}}_c^\\infty(\\ensuremath{\\mathbb{R}}_-,\\ensuremath{\\mathbb{R}}^n):\\,f(0)=0\\}$ in the norm\n\\begin{equation*}\n  \\|f\\|_{\\H_H}\\ensuremath\\triangleq\\sup_{s,t\\leq 0}\\frac{\\big|f(t)-f(s)\\big|}{|t-s|^{\\frac{1-H}{2}}\\sqrt{1+|t|+|s|}}.\n\\end{equation*} \nTo ensure that this construction actually furnishes a solution to \\eqref{eq:sde_intro}, we of course have to assume that the law of the second marginal of $Z$ coincides with $\\ensuremath{\\mathsf{W}}$ for each time $t\\geq 0$. This motivates the following definition:\n\\begin{definition}[\\cite{Hairer2005}]\\label{initial-condition}\nA measure $\\mu\\in\\P(\\ensuremath{\\mathbb{R}}^n\\times\\H_H)$ with $\\Pi_{\\H_H}^*\\mu=\\ensuremath{\\mathsf{W}}$ is called a \\emph{generalized initial condition}. A generalized initial condition $\\mathcal I_\\pi$, which is invariant for the Feller process $Z$ is called an \\emph{invariant measure} for the SDE \\eqref{eq:sde_intro}. We write $\\pi\\ensuremath\\triangleq\\Pi_{\\ensuremath{\\mathbb{R}}^n}^*\\mathcal I_\\pi$ for the first marginal and $\\ensuremath\\mathbb{P}_\\pi\\in\\P\\big(\\ensuremath{\\mathcal{C}}(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)\\big)$ for the law of the first coordinate of $Z$ when started in $\\mathcal I_\\pi$.\n\\end{definition} \n\nBy only adding the past of the driving noise to the auxiliary process $Z$, Hairer's framework rules out the existence of `unphysical' invariant measures, which frequently occur in the theory of random dynamical systems, see \\cite{Hairer2009} for details.\n\n\nThere are only a few examples for which the invariant measure can be written down explicitly:\n\\begin{example}\\label{ex:disintegration_fou}\n  Let $Y$ be the fractional Ornstein-Uhlenbeck process \\cite{Cheridito2003}, that is,\n  \\begin{equation*}\n    dY_t=-Y_t\\,dt+dB_t.\n  \\end{equation*}\n  Then it is well known that its invariant measure is given by\n  \\begin{equation*}\n    \\mathcal I_\\pi(dy,dw)=\\delta_{F(w)}(dy)\\ensuremath{\\mathsf{W}}(dw),\\qquad F(w)\\ensuremath\\triangleq-\\int_{-\\infty}^0 e^s D_Hw(s)\\,ds,\n  \\end{equation*}\n  where $D_H:\\H_H\\to\\H_{1-H}$ is a continuous linear operator switching between Wiener and fBm paths, see \\cite[Eq. (3.6)]{Hairer2005} for the precise definition. The first marginal of $\\mathcal I_\\pi$ and the stationary path space law are given by\n  \\begin{equation*}\n    \\pi=\\L\\left(\\int_{-\\infty}^0 e^s\\,dB_s\\right)\\quad\\text{and}\\quad\\ensuremath\\mathbb{P}_\\pi=\\L\\left(\\int_{-\\infty}^t e^s\\,dB_s\\right)_{t\\geq 0}.\n  \\end{equation*}\n  \n\\end{example}\n\n\\begin{remark}\n  The invariant measure of \\eqref{eq:sde_intro} is in general not of product form.\n\\end{remark}\n\nSince $\\sigma\\in\\Lin{n}$ is non-degenerate, one can show that there is an isomorphism between the strictly stationary solutions to \\eqref{eq:sde_intro} and the set of invariant measures (provided one quotients the latter by the equivalence relation identifying generalized initial initial conditions which generate the same evolution in the first marginal). It is also not hard to prove the following:\n\\begin{proposition}[\\cite{Hairer2005}]\\label{prop:existence_invariant_measure}\n  If $\\sigma\\in\\Lin{n}$ and $b\\in\\S(\\kappa,R,\\lambda)$ for some $\\kappa>0$, $R,\\lambda\\geq 0$, then there exists an invariant measure for \\eqref{eq:sde_intro} in the sense of \\cref{initial-condition}. Moreover, $\\mathcal I_\\pi$ has moments of all orders.\n\\end{proposition}\n\nThe conclusion of \\cref{prop:existence_invariant_measure} actually holds for a merely locally Lipschitz off-diagonal large scale contractive drift (see \\cref{cond:off_diagonal} below). See also \\cite{Hairer2007,Deya2019} for versions for multiplicative noise. Finally, we introduce a Wasserstein-type distance for generalized initial conditions:\n\\begin{definition}\\label{def:wasserstein}\nLet $\\mu$ and $\\nu$ be generalized initial conditions. Let $\\mathscr{C}_{\\Delta}(\\mu,\\nu)$ denote the set of couplings of $\\mu$ and $\\nu$ concentrated on the diagonal $\\Delta_{\\H_H}\\ensuremath\\triangleq\\{(w,w^\\prime)\\in\\H_H^2:\\,w=w^\\prime\\}$. For $p\\geq 1$, we set\n  \\begin{equation*}\n  \\ensuremath{\\mathbb{W}}^p(\\mu,\\nu)\\ensuremath\\triangleq\\inf_{\\rho\\in\\mathscr{C}_\\Delta(\\mu,\\nu)}\\left(\\int_{(\\ensuremath{\\mathbb{R}}^n\\times\\H_H)^2}|x-y|^p\\,\\rho(dx,dw,dy,dw^\\prime)\\right)^{\\frac1p}.\n\\end{equation*}\n\\end{definition}\n\nNote that clearly $\\ensuremath{\\mathcal{W}}^p\\big(\\Pi_{\\ensuremath{\\mathbb{R}}^n}^*\\mu,\\Pi_{\\ensuremath{\\mathbb{R}}^n}^*\\nu\\big)\\leq \\ensuremath{\\mathbb{W}}^p(\\mu,\\nu)$ and the inequality is strict in general.\n\n\\subsection{Large Scale Contractions}\n\nKnown ergodic theorems on \\eqref{eq:sde_intro} require either a Lyapunov-type stability or a large scale contractivity condition on the drift $b$. The former indicates that once far out, the solutions have the tendency to come back to a neighborhood of the origin. Under this condition, it is conceivable that two distinct solutions can come back from diverging routes, thus allowing to couple them. The Lyapunov stability condition was used in \\cite{Fontbona2017,Deya2019} for multiplicative noise. \n\nA large scale contraction on the other hand will force two solutions to come closer once they have left a ball $B_R$ of sufficiently large radius $R>0$. The following two conditions appeared in previous works:\n\n\\begin{condition}[Off-diagonal large scale contraction, \\cite{Hairer2005}]\\label{cond:off_diagonal}\nThere exist numbers $\\tilde \\kappa>0$ and $D,\\lambda\\geq 0$ such that  \n\\begin{equation}\\label{quasi-contr}\n  \\Braket{b(x)-b(y),x-y}\\leq \\big(D-\\tilde \\kappa|x-y|^2\\big)\\wedge\\big(\\lambda|x-y|^2\\big)\\qquad \\forall\\,   x,y\\in\\ensuremath{\\mathbb{R}}^n.\n\\end{equation}\n\\end{condition}\n\n\n\\begin{condition}[Large scale contraction, \\cite{Panloup2020}]\nThere exist numbers $R\\geq 0$ and $\\kappa>0$ such that \n  \\begin{equation}\\label{contractive}\n    \\Braket{b(x)-b(y),x-y}\\leq    -\\kappa|x-y|^2   \\qquad \\forall\\, x,y\\in \\ensuremath{\\mathbb{R}}^n\\setminus B_R.  \n  \\end{equation}\n\\end{condition}\n\n\\begin{example}\n  The function $b(x)=x-x^3$ is a large scale contraction. \n\\end{example}\n\nWe will later use the following standard result, a slightly weaker version of which was proven in \\cite[Lemma 5.1]{Panloup2020}.\n\\begin{lemma}\\label{lem:bigger_ball}\nIf $b$ is locally Lipschitz continuous and satisfies the large scale contraction condition \\eqref{contractive}, then for any $\\bar{\\kappa}\\in(0,\\kappa)$, there is an $\\bar{R}>0$ such that\n    \\begin{equation*}\n        \\braket{b(x)-b(y),x-y}\\leq -\\bar{\\kappa}|x-y|^2  \\qquad \\forall\\, y\\in\\ensuremath{\\mathbb{R}}^n,\\, |x|>\\bar{R}.\n    \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Since $\\braket{b(x)-b(y),x-y}\\leq  -{\\kappa}|x-y|^2$ for $x$ and $y$ outside of the ball $B_R$, \nwe only need to show that the required contraction holds for any $|y|\\leq R$ and $|x|>\\bar R$. \nFix such $x$ and $y$. \n\nWithout loss of generality, we may also assume that $\\bar{R}\\geq R+1$. Then there is a $\\beta\\in(0,1)$ such that $z_\\beta\\ensuremath\\triangleq (1-\\beta)x+\\beta y$ has norm $|z_\\beta|=R+1$. \nSince $x-y=\\ensuremath{\\frac} 1 \\beta(x-z_\\beta)$ and, since $x, z_\\beta$ are outside of $B_R$, \n    \\begin{equation*}\n        \\braket{b(x)-b(z_\\beta),x-y}\\leq-\\ensuremath{\\frac} 1 \\beta \\kappa|x-z_\\beta|^2=-\\kappa\\beta|x-y|^2.\n    \\end{equation*}\nLet $K\\ensuremath\\triangleq\\Lip[B_{R+1}]{b}$ denote the Lipschitz constant of $b$ on $B_{ R+1}$. Since $|z_\\beta-y|=(1-\\beta)|x-y|$, it holds that\n    \\begin{align*}\n        \\braket{b(x)-b(y),x-y}&=\\braket{b(x)-b(z_\\beta),x-y} +\\<b(z_\\beta)-b(y), x-y\\>\\\\\n        & \\leq -\\kappa\\beta|x-y|^2+K(1-\\beta)|x-y|^2\n    \\end{align*}\nSince $\\beta$ is the length of the proportion of the line segment outside of $B_{R+1}$,\nwe can choose it as close to $1$ as we like by choosing $\\bar R$ sufficiently large $\\big(\\beta=\\ensuremath{\\frac}{|x-z_\\beta|}{|x-y|}\\geq\\frac{|x|-R-1}{|x|+R}\\geq\\frac{\\bar{R}-R-1}{\\bar{R}+R}\\big)$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:large_scall_off_diagonal}\n\\leavevmode\n\\begin{enumerate}\n  \\item Let $b: \\ensuremath{\\mathbb{R}}^n\\to \\ensuremath{\\mathbb{R}}^n$ be a globally Lipschitz continuous function. Then the large scale contraction condition \\eqref{contractive} is equivalent to $b\\in\\bigcup_{\\lambda>0}\\S(\\kappa,R,\\lambda)$. In view of \\cref{lem:bigger_ball}, condition \\eqref{eq:semicontractive} also holds for a merely locally Lipschitz continuous $b$ at the cost of a smaller contractive rate and a bigger contractive range. In fact, choose $\\bar\\kappa\\in(0,\\kappa)$ and let $\\bar R>R$ be the corresponding radius furnished by \\cref{lem:bigger_ball}. This gives \\eqref{eq:semicontractive} with $\\kappa\\rightsquigarrow\\bar{\\kappa}$, $R\\rightsquigarrow\\bar{R}$, and $\\lambda\\rightsquigarrow \\Lip[B_{\\bar{R}}]{b}$.\n  \\item\\label{it:off_diagonal} The off-diagonal large scale contraction condition is weaker than the large scale contraction condition.  With the former, there may be no $\\kappa>0$ such that \\eqref{contractive} holds in the region $\\{|x-y| \\leq \\ensuremath{\\frac} D {2\\tilde \\kappa}\\} \\cap \\{|x|\\geq R, |y|\\geq R\\}$. On the other hand, if \\eqref{contractive} holds and $b$ is locally Lipschitz continuous, we can choose any $\\tilde \\kappa<\\kappa$. In fact, denoting the radius from \\cref{lem:bigger_ball} by $\\bar R>0$, one only needs to show \\eqref{quasi-contr} when both $x$ and $y$ are in $B_{\\bar R}$. To this end, we pick $\\lambda=\\Lip[B_{\\bar{R}}]{b}$ and $D\\geq\\sup_{x,y\\in B_{\\bar{R}}}(\\tilde\\kappa+\\lambda)|x-y|^2$.\n\\end{enumerate}\n\\end{remark}\n\n\n\\section{The Conditional Evolution of Fractional Dynamics}\\label{sec:convergence}\n\nTo derive strong $L^p$-bounds on the H\\\"older norm of the slow motion in \\cref{sec:feedback} below, we need to study the conditional distribution of the evolution \\eqref{eq:frozen_fast}. Unlike the Markovian case, the conditioning changes the dynamics and the resulting evolution may \\emph{no longer} solve the original equation. We will show that, in the limit $t\\to\\infty$, the law of the conditioned dynamics still converges to $\\pi^x$, the first marginal of the invariant measure for the fast dynamics with frozen slow input \\eqref{eq:frozen_fast}. The rate of convergence is however slower (only algebraic rather than exponential).\n\n\nLet us first state the regularity assumption imposed in \\cref{thm:feedback_fractional}. For this we introduce a convenient notation, which we shall frequently use in the sequel. We write $a\\lesssim b$ if there is a constant $C>0$ such that $a\\leq C b$. The constant $C$ is independent of any ambient parameters on which $a$ and $b$ may depend.\n\n\\begin{condition}\\label{cond:feedback}\n  The drift $b:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}^n$ satisfies the following conditions:\n  \\begin{itemize}\n    \\item \\emph{Linear growth:}\n    \\begin{equation*}\n      |b(x,y)|\\lesssim 1+|x|+|y|, \\qquad \\forall\\, x\\in\\ensuremath{\\mathbb{R}}^d,y\\in\\ensuremath{\\mathbb{R}}^n.\n    \\end{equation*}\n    \\item \\emph{Uniformly locally Lipschitz in the first argument:} For each $R>0$, there is an $L_R>0$ such that\n    \\begin{equation*}\n      \\sup_{y\\in\\ensuremath{\\mathbb{R}}^n}|b(x_1,y)-b(x_2,y)|\\leq L_R|x_1-x_2|, \\qquad \\forall\\, |x_1|,|x_2|\\leq R.\n    \\end{equation*}\n    \\item \\emph{Uniformly Lipschitz in the second argument:} There is an $L>0$ such that\n    \\begin{equation*}\n      \\sup_{x\\in\\ensuremath{\\mathbb{R}}^d}|b(x,y_1)-b(x,y_2)|\\leq L|y_1-y_2|, \\qquad \\forall\\, y_1,y_2\\in\\ensuremath{\\mathbb{R}}^n.\n    \\end{equation*}\n  \\end{itemize}\n\\end{condition}\n\nLet $({\\mathcal F}_t)_{t\\geq 0}$ be a complete filtration to which $\\hat{B}$ is adapted. For any continuous, $({\\mathcal F}_t)_{t\\geq 0}$-adapted, $\\ensuremath{\\mathbb{R}}^d$-valued process $X$ with continuous sample paths, and any $\\varepsilon>0$, the equation\n\\begin{equation}\\label{eq:general_flow}\n  d\\Phi_{t}^X=\\frac{1}{\\varepsilon}b\\big(X_t,\\Phi_{t}^X\\big)\\,dt+\\frac{1}{\\varepsilon^{\\hat{H}}}\\sigma\\,d\\hat{B}_t,\\qquad \\Phi_{t}^{X}=y,\n\\end{equation}\nhas a unique global pathwise solution under \\cref{cond:feedback}, see \\cref{lem:comparison} below. The flow $\\Phi_{s,t}^X(y)$ associated with \\eqref{eq:general_flow} is therefore well defined. An important special case of \\eqref{eq:general_flow} is when the extrinsic process is given by a fixed point $x\\in\\ensuremath{\\mathbb{R}}^d$. For this we reserve the notation $\\bar{\\Phi}^x$:\n\\begin{equation}\\label{eq:general_flow-fixed-x}\n  d\\bar{\\Phi}_t^x=\\frac{1}{\\varepsilon}b(x,\\bar{\\Phi}_t^x)\\,dt+\\frac{1}{\\varepsilon^{\\hat{H}}}\\sigma\\,d\\hat{B}_t,\\qquad \\bar{\\Phi}_0^x=y.\n\\end{equation}\nWe would like the reader to observe that the dependency of flows on the scale parameter $\\varepsilon>0$ is suppressed in our notation. Note that, by self-similarity, sending $\\varepsilon\\to 0$ in \\eqref{eq:general_flow-fixed-x} is equivalent to keeping $\\varepsilon=1$ fixed and taking $t\\to\\infty$. As the $\\varepsilon$-dependence of the flows \\eqref{eq:general_flow}--\\eqref{eq:general_flow-fixed-x} will play a key r\\^ole in \\cref{sec:feedback}, we choose to introduce a new notation in case $\\varepsilon=1$, which is used throughout the rest of this section: \n\\begin{definition}\\label{def:flow}\n  Let $\\ensuremath{\\mathfrak{h}}\\in\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)\\ensuremath\\triangleq\\big\\{f\\in\\ensuremath{\\mathcal{C}}(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n):\\,f(0)=0\\big\\}$ and $x\\in\\ensuremath{\\mathbb{R}}^d$. We denote the flow of the ordinary differential equation\n  \\begin{equation}\\label{eq:ode_solution}\n    dy_t=b(x,y_t)\\,dt+d\\ensuremath{\\mathfrak{h}}_t\n  \\end{equation}\n  by $\\Psi^x_{s,t}(y,\\ensuremath{\\mathfrak{h}})$, where $y\\in\\ensuremath{\\mathbb{R}}^n$ and $0\\leq s\\leq t$. It is given by the solution to the integral equation\n  \\begin{equation*}\n    \\Psi_{s,t}^x(y,\\ensuremath{\\mathfrak{h}})=y+\\int_s^t b\\big(x,\\Psi_{s,r}^x(y,\\ensuremath{\\mathfrak{h}})\\big)\\,dr+\\ensuremath{\\mathfrak{h}}_t-\\ensuremath{\\mathfrak{h}}_s.\n  \\end{equation*}\n   We also use the abbreviation $\\Psi^x_{t}\\ensuremath\\triangleq \\Psi^x_{0,t}$.\n\\end{definition}\n\nUnder \\cref{cond:feedback}, \\eqref{eq:ode_solution} is well posed and it follows that $\\Psi_{s,t}^x(y, \\ensuremath{\\mathfrak{h}})=\\Psi_{t-s}^x(y,\\theta_s \\ensuremath{\\mathfrak{h}})$ for each $0\\leq s\\leq t$ and $y\\in\\ensuremath{\\mathbb{R}}^n$, where $\\theta_sf=f(\\cdot+s)-f(\\cdot)$ is the Wiener shift operator on the path space. If $x\\in\\ensuremath{\\mathbb{R}}^d$, $y\\in\\ensuremath{\\mathbb{R}}^n$, or $\\ensuremath{\\mathfrak{h}}\\in\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)$ are random, we understand \\cref{def:flow} pathwise for each fixed sample $\\omega\\in\\Omega$. The solutions to \\eqref{eq:general_flow} and \\eqref{eq:general_flow-fixed-x} are also understood in this sense.\n\n\n\\subsection{Processes with a Locally Independent Increment Decomposition}\\label{sec:increment}\n\nThe derivation of the conditioned evolution relies on the following simple fact: For $t,h\\geq 0$, we have\n\\begin{equation}\\label{eq:increment_decomposition}\n  (\\theta_t\\hat{B})_h=\\hat{B}_{t+h}-\\hat{B}_t=\\bar{\\hat{B}}_h^t+\\tilde{\\hat{B}}_h^t,\n\\end{equation}\nwhere, in a slight abuse of notation (the integrand has to be multiplied by the identity matrix),\n\\begin{equation*}\n  \\bar{\\hat{B}}_h^t\\ensuremath\\triangleq \\alpha_{\\hat H}\\int_{-\\infty}^t\\left((t+h-u)^{\\hat{H}-\\frac12}-(t-u)^{\\hat{H}-\\frac12}\\right)\\,d\\hat{W}_u,\\quad \\tilde{\\hat{B}}_h^t\\ensuremath\\triangleq\\alpha_{\\hat H}\\int_t^{t+h} (t+h-u)^{\\hat{H}-\\frac12}\\,d\\hat{W}_u.\n\\end{equation*}\nThis decomposition is easily obtained by rearranging \\eqref{eq:mandelbrot}. For any $t\\geq 0$, the two components $\\bar{\\hat{B}}^t$ and $ \\tilde{\\hat{B}}^t$ are independent. We call $\\bar{\\hat{B}}^t$ the \\emph{smooth} part of the increment, whereas $\\tilde{\\hat{B}}^t$ is referred to as the \\emph{rough} part. This terminology is based on the fact that, away from the origin, the process $\\bar{\\hat{B}}^t$ has continuously differentiable sample paths and therefore the `roughness' of $\\hat{B}$ essentially comes from $\\tilde{\\hat{B}}^t$. Indeed, it is not hard to check that $\\tilde{\\hat{B}}^t$ is of precisely the same H\\\"older regularity as $\\hat{B}$. We also observe that $\\tilde{\\hat{B}}^t\\overset{d}{=}\\tilde{\\hat{B}}^0\\ensuremath\\triangleq\\tilde{\\hat{B}}$ for all $t>0$. \n\nThe process $\\tilde{\\hat{B}}$ is---up to a prefactor---known as Riemann-Liouville process (or type-II fractional Brownian motion) and was initially studied by L\\'evy \\cite{Levy1953}. Its use in modelling was famously discouraged in \\cite{Mandelbrot1968} due to its overemphasis of the origin and the `regularized' process \\eqref{eq:mandelbrot} was proposed instead. In fact as we shall see below, the lack of stationarity of the increments of $\\tilde{\\hat{B}}$ complicates the analysis of the conditioned evolution. \n\n\\begin{definition}\\label{def:ind_increment}\nLet  $({\\mathcal F}_t)_{t\\geq 0}$ be a complete filtration. An $({\\mathcal F}_t)_{t\\geq 0}$-adapted stochastic process $Z$ is said to have a \\emph{locally independent decomposition of its increments} with respect to $({\\mathcal F}_t)_{t\\geq 0}$ if for any $t\\geq 0$, there exists an increment decomposition of the form\n$$(\\theta_t Z)_h=\\tilde Z^t_h+\\bar Z^t_h, \\qquad h\\geq 0,$$\nwhere $\\bar Z^t \\in {\\mathcal F}_t$ and $\\tilde Z^t$ is independent of ${\\mathcal F}_t$. \n\\end{definition}\n\n\nAs seen in \\eqref{eq:increment_decomposition}, an fBm $\\hat{B}$ has a locally independent decomposition of its increments with respect to any filtration $({\\mathcal F}_t)_{t\\geq 0}$ \\emph{compatible} with $\\hat{B}$. By this we mean that $(\\hat{W}_s)_{s\\leq t}\\in{\\mathcal F}_t$ and $(\\theta_t\\hat{W}_s)_{s\\geq t}$ is independent of ${\\mathcal F}_t$ for any $t\\geq 0$. \n\n\n\\begin{example}\\label{example-1}\nLet us give some further examples, which will become important later on:\n\\begin{enumerate}\n\\item\\label{it:rough_decomposition} Let $(\\hat W_t)_{t\\geq 0}$ be a Wiener process and \n$ \\tilde{\\hat B}_t\\ensuremath\\triangleq\\alpha_{\\hat H}\\int_0^{t} (t-u)^{H-\\frac12}\\,d\\hat W_u$ be the Riemann-Liouville process. \nThen, for any $t\\geq 0$ and $h\\geq 0$, \n \\begin{align}\n    (\\theta_t\\tilde{\\hat B})_h&=\\alpha_{\\hat H}\\int_0^t \\Big( (t+h-u)^{\\hat H-\\frac12}-(t-u)^{\\hat H-\\frac12}\\Big)\\,d\\hat W_u+\\alpha_{\\hat H}\\int_t^{t+h}(t+h-u)^{\\hat H-\\frac12}\\,d\\hat W_u\\nonumber\\\\\n    &\\ensuremath\\triangleq Q^t_h+\\tilde{\\hat B}^t_h.\\label{eq:z_t}\n  \\end{align}\nThus, $\\tilde{\\hat B}$ admits a locally independent decomposition of its increments with respect to any filtration compatible with $\\hat{B}$.\n\n\\item Another example, given in \\cite{Gehringer-Li-2020, Gehringer-Li-2020-1}, is the stationary fractional Ornstein-Uhlenbeck process $Z_t=\\int_{-\\infty }^t e^{-(t-s)}\\,d\\hat{B}_s$. More generally, it is clear that $Z_t=\\int_{-\\infty }^t \\mathfrak{G}(s,t)\\,d\\hat{B}_s$ with a suitable kernel $\\mathfrak{G}$ also has this property.\n\n\\item\\label{it:smooth_decomposition} Albeit not being a direct instance of \\cref{def:ind_increment}, it is also interesting to observe a \\emph{fractal} property of $\\hat{B}$: The smooth part of the increment has an independent decomposition as $\\bar{\\hat B}_h^t=P_h^t+Q_h^t$, where $Q^t$ was defined in \\eqref{eq:z_t} and\n\\begin{equation*}\n  P_h^t\\ensuremath\\triangleq\\alpha_{\\hat H}\\int_{-\\infty}^0\\Big((t+h-u)^{\\hat H-\\frac12}-(t-u)^{\\hat H-\\frac12}\\Big)\\,d\\hat W_u.\n\\end{equation*}\n\\end{enumerate}\n\\end{example}\n\nOur argument for the quenched ergodic theorem will be based on a two step conditioning procedure making use of an explicit representation of the conditioned process. We state it for a general noise with locally independent increments:\n\n\\begin{lemma}\\label{lem:conditioning_general}\nLet $0\\leq s\\leq t<t+h$ and $(Z_t)_{t\\geq 0}$ be a continuous stochastic process admitting a locally independent decomposition $(\\theta_t Z)_{h}=\\tilde Z^t_h+\\bar Z^t_h$ with respect to $({\\mathcal F}_t)_{t\\geq 0}$. Let $X$ and $Y$ be ${\\mathcal F}_t$-measurable random variables. \nThen, for any $F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ bounded measurable, \n\\begin{equation*}  \n  \\Expec{F\\big(\\Psi^X_{s,t+h}(Y, Z)\\big)\\,\\Big|\\,{\\mathcal F}_t}\n  =\\Expec{F\\big(\\Psi_h^{x}(y,\\varsigma+\\tilde Z^t)\\big)}   \\bigg|_{\\substackal{x&=X,\\varsigma=\\bar Z^t,\\\\y&=\\Psi^X_{s,t}(Y,Z)}},\n\\end{equation*}\nwhere $\\Psi$ is defined in \\cref{def:flow}.\n\\end{lemma}\n\\begin{proof} \n  This in an immediate consequence of the flow property of the equation \\eqref{eq:ode_solution} and standard properties of conditional expectations.\n\\end{proof}\n\nComing back to the flow of the fast motion with frozen slow input \\eqref{eq:general_flow-fixed-x}, the following result is an easy consequence of \\cref{lem:conditioning_general}:\n\\begin{lemma}\\label{lem:conditioning}\nLet $({\\mathcal F}_t)_{t\\geq 0}$ be a filtration which is compatible with $\\hat{B}$. Fix $0\\leq s\\leq t<t+h$ and let $X,Y$ be ${\\mathcal F}_t$-measurable random variables. Then, for any $F:\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ bounded measurable,\n\\begin{equation*}  \n  \\Expec{F\\big(\\bar{\\Phi}^{X}_{s,t+h}(Y)\\big)\\,\\Big|\\,{\\mathcal F}_t}=\\Expec{F\\Big(\\Psi_{\\frac{h}{\\varepsilon}}^{x}\\big(y,\\varsigma+\\sigma\\tilde{\\hat{B}}\\big)\\Big)}\\bigg|_{\\substackal{x&=X,\\varsigma=\\varepsilon^{-\\hat{H}}\\sigma\\bar{\\hat{B}}^{t}_{\\varepsilon\\cdot},\\\\y&=\\bar{\\Phi}^X_{s,t}(Y)}},\n\\end{equation*}\nwhere $\\bar{\\hat{B}}^t_{\\varepsilon\\cdot}\\ensuremath\\triangleq\\big(\\bar{\\hat{B}}^t_{\\varepsilon h}\\big)_{h\\geq 0}$.\n\\end{lemma}\n\nWe now turn to the fine properties of the smooth part of the increment. For $\\alpha>0$ we define the set\n\\begin{equation}\\label{eq:omega}\n  \\Omega_{\\alpha}\\ensuremath\\triangleq\\Big\\{f\\in \\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)\\cap\\ensuremath{\\mathcal{C}}^2\\big((0,\\infty),\\ensuremath{\\mathbb{R}}^n\\big):\\limsup_{t\\to\\infty}\\left(t^{\\alpha}\\big|\\dot{f}(t)\\big|+t^{1+\\alpha}\\big|\\ddot{f}(t)\\big|\\right)<\\infty\\Big\\}.\n\\end{equation}\nThis space is equipped with the semi-norm \n\\begin{equation*}\n  \\|f\\|_{\\Omega_\\alpha}\\ensuremath\\triangleq\\sup_{t\\geq 1}t^{\\alpha}\\big|\\dot{f}(t)\\big|+\\sup_{t\\geq 1}t^{1+\\alpha}\\big|\\ddot{f}(t)\\big|.\n\\end{equation*}\nWe also set $\\Omega_{\\alpha-}\\ensuremath\\triangleq\\bigcap_{\\beta<\\alpha}\\Omega_{\\beta}$. The motivation for this definition stems from the following lemma:\n\n\\begin{lemma}\\label{lem:smooth_part_decay}\n  Let $\\varepsilon>0$ and $t\\geq 0$. Then $\\varepsilon^{-\\hat{H}}\\bar{\\hat{B}}^t_{\\varepsilon\\cdot}\\overset{d}{=}\\bar{\\hat{B}}^t\\overset{d}{=}\\bar{\\hat{B}}\\in\\Omega_{(1-\\hat{H})-}$ a.s. and $\\|\\bar{\\hat{B}}\\|_{\\Omega_\\alpha}\\in\\bigcap_{p\\geq 1} L^p$ for any $\\alpha<1-\\hat{H}$.\n\\end{lemma}\n\\begin{proof}\n  Let $\\delta\\in\\big(0,1-\\hat H\\big)$. It is enough to prove that there is a random variable $C>0$ with moments of all orders such that\n  \\begin{equation}\\label{eq:estimate_all_orders}\n    \\big|\\dot{\\bar{\\hat{B}}}_t\\big|\\leq \\frac{C}{t^{1-\\hat{H}-\\delta}},\\qquad\\big|\\ddot{\\bar{\\hat{B}}}_t\\big|\\leq\\frac{C}{t^{2-\\hat{H}-\\delta}}\n  \\end{equation}\n  for all $t\\geq 1$ on a set of probability one. This in turn easily follows from sample path properties of the standard Wiener process. Firstly, we have that\n  \\begin{equation*}\n    \\dot{\\bar{\\hat{B}}}_t=\\alpha_{\\hat{H}}\\left(\\hat{H}-\\frac12\\right)\\int_{-\\infty}^0(t-u)^{\\hat{H}-\\frac32}\\,dW_u=-\\alpha_{\\hat{H}}\\left(\\hat{H}-\\frac12\\right)\\left(\\hat{H}-\\frac32\\right)\\int_{-\\infty}^0 (t-u)^{\\hat{H}-\\frac52}W_u\\,du\n  \\end{equation*}\n  since $\\lim_{u\\to-\\infty}(t-u)^{\\hat{H}-\\frac32}W_u=0$. Therefore,\n  \\begin{align*}\n   \\big|\\dot{\\bar{\\hat{B}}}_t\\big|&\\lesssim \\left(\\sup_{-1\\leq s\\leq 0} |W_s| \\int_{-1}^0 (t-u)^{\\hat{H}-\\frac52}\\,du+\\sup_{s\\leq -1}\\frac{|W_s|}{(t-s)^{\\frac12+\\delta}}\\int_{-\\infty}^{-1} (t-u)^{\\hat{H}-2+\\delta}\\,du\\right)\\\\\n   &\\leq C\\left(t^{\\hat{H}-\\frac52}+(t+1)^{\\hat{H}-1+\\delta}\\right).\n  \\end{align*}\n  The fact that $C$ has moments of all order is an easy consequence of Fernique's theorem. In fact, the Wiener process defines a Gaussian measure on the separable Banach space\n  \\begin{equation*}\n    \\mathcal{M}^{\\frac12+\\delta}\\ensuremath\\triangleq\\left\\{f\\in\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n):\\,\\|f\\|_{\\mathcal{M}^{\\frac12+\\delta}}\\ensuremath\\triangleq\\sup_{u\\geq 0}\\frac{|f(u)|}{(1+u)^{\\frac12+\\delta}}<\\infty\\right\\}\n  \\end{equation*}\n  By Fernique's theorem, the random variable $\\|W\\|_{\\mathcal{M}^{\\frac12+\\delta}}$ has therefore Gaussian tails. The first estimate in \\eqref{eq:estimate_all_orders} follows. The bound on $\\big|\\ddot{\\bar{\\hat{B}}}_t\\big|$ is similar.\n\\end{proof}\n\n\n\\subsection{A Universal Control}\n\nLet $b\\in\\S(\\kappa,R,\\lambda)$, $\\varsigma\\in\\ensuremath{\\mathcal{C}}_0([0,1],\\ensuremath{\\mathbb{R}}^n)$, and $u\\in L^\\infty([0,1],\\ensuremath{\\mathbb{R}}^n)$. Let us consider the following controlled ordinary differential equation:\n\\begin{equation}\\label{eq:controlled_ode}\n    x^{\\varsigma,u}(t)=x_0+\\int_0^t b\\big(x^{\\varsigma,u}(s)\\big)\\,ds+\\varsigma(t)+\\int_0^t u(s)\\,ds,\\qquad t\\in[0,1].\n\\end{equation}\nWe think of $\\varsigma$ as an external `adversary' and of $u$ as a control. Since $b$ is Lipschitz continuous, it is standard that there is a unique global solution to \\eqref{eq:controlled_ode}. If $u\\equiv 0$, we adopt the shorthand $x^{\\varsigma}\\ensuremath\\triangleq x^{\\varsigma,0}$.\n\nThe aim of this section is to exhibit an $\\eta\\in(0,1)$ as large as possible so that the following holds: Given $\\bar{R}>0$, there is an $M>0$ such that, for any adversary $\\varsigma\\in\\ensuremath{\\mathcal{C}}_0([0,1],\\ensuremath{\\mathbb{R}}^n)$ and any initial condition $x_0\\in\\ensuremath{\\mathbb{R}}^n$, we can find a control $u\\in L^\\infty([0,1],\\ensuremath{\\mathbb{R}}^n)$ with $|u|_\\infty\\leq M$ ensuring that the occupation time of $x^{\\varsigma,u}$ of the set $\\ensuremath{\\mathbb{R}}^n\\setminus B_{\\bar{R}}$ is at least $\\eta$. It is important to emphasize that the sup-norm of the control $|u|_\\infty$ may neither depend on the adversary $\\varsigma$ nor on the initial condition $x_0$ (otherwise the construction of $u$ essentially becomes trivial). We shall actually choose $u$ as concatenation of the zero function and a \\emph{universal} control $\\hat u\\in L^\\infty([0,N^{-1}],\\ensuremath{\\mathbb{R}}^n)$ for a sufficiently large, but universal, $N\\in\\ensuremath{\\mathbb{N}}$.\n\nWe begin with a lemma:\n\n\\begin{lemma}\\label{lem:control_bound}\n    There is a constant $C>0$ independent of $\\varsigma$ and $u$ such that, for the solution of \\eqref{eq:controlled_ode},\n    \\begin{equation*}\n        |x^{\\varsigma,u}(t)-x^{\\varsigma}(t)|^2\\leq C(1+|u|^2_\\infty)t\n    \\end{equation*}\n    for all $t\\in [0,1]$.\n\\end{lemma}\n\\begin{proof}\n    Since $b$ is contractive on the large scale, there are constants $D,\\tilde{\\kappa}>0$ such that\n    \\begin{equation*}\n        \\braket{b(x)-b(y),x-y}\\leq D-\\tilde{\\kappa}|x-y|^2\n    \\end{equation*}\n    for all $x,y\\in\\ensuremath{\\mathbb{R}}^n$, see \\cref{rem:large_scall_off_diagonal} \\ref{it:off_diagonal}. Define now $f(t)\\ensuremath\\triangleq e^{\\tilde\\kappa t}\\big|x^{\\varsigma,u}(t)-x^{\\varsigma}(t)\\big|^2$, then\n    \\begin{equation*}\n        f^\\prime(t)=\\tilde\\kappa f(t)+2e^{\\tilde\\kappa t}\\Braket{b\\big(x^{\\varsigma,u}(t)\\big)-b\\big(x^{\\varsigma}(t)\\big)+u(t),x^{\\varsigma,u}(t)-x^{\\varsigma}(t)}\\leq 2D e^{\\tilde\\kappa}+\\frac{|u(t)|^2}{\\tilde\\kappa}\n    \\end{equation*}\n    for all $t\\in [0,1]$. Consequently, setting $C\\ensuremath\\triangleq \\max(2D,\\tilde\\kappa^{-1})$, we have\n    \\begin{equation*}\n      \\big|x^{\\varsigma,u}(t)-x^{\\varsigma}(t)\\big|^2\\leq C \\int_0^t e^{-\\tilde\\kappa(t-s)}\\left(1+|u(s)|^2\\right)\\,ds\n    \\end{equation*}\n    and the lemma follows at once.\n\\end{proof}\n\nFor a piecewise constant function $u:[0,1]\\to\\ensuremath{\\mathbb{R}}^n$, let $\\ensuremath{\\mathcal{D}}_u\\subset[0,1]$ denote the finite set of discontinuities. We then have the following control result:\n\\begin{proposition}\\label{prop:control}\n\tLet $\\eta<\\frac12$ and $\\bar{R}>0$. Then there is a value $M>0$ such that the following holds true: For each $\\varsigma\\in\\ensuremath{\\mathcal{C}}_0([0,1],\\ensuremath{\\mathbb{R}}^n)$ and each $x_0\\in\\ensuremath{\\mathbb{R}}^n$, we can find a piecewise constant control $u\\in L^\\infty([0,1],\\ensuremath{\\mathbb{R}}^n)$ with $|u|_\\infty+|\\ensuremath{\\mathcal{D}}_u|\\leq M$ such that the occupation time of $x^{\\varsigma,u}$ of the set $\\ensuremath{\\mathbb{R}}^n\\setminus B_{\\bar{R}}$ is greater than or equal to $\\eta$.\n\\end{proposition}\n\n\\begin{proof}\n  We prove that there exist an integer $N$ and a control $\\hat u\\in L^\\infty([0,N^{-1}])$ with at most two constant pieces independent of both the initial condition $x_0$ and the adversary $\\varsigma$ such that either\n  \\begin{equation*}\n    \\mathop{\\mathrm {Leb}} \\Big(\\Big\\{t\\in[0,N^{-1}]:|x^{\\varsigma}(t)|>\\bar{R} \\Big\\}\\Big)\\geq \\ensuremath{\\frac} \\eta N\\quad\\text{or}\\quad\\mathop{\\mathrm {Leb}} \\Big(\\Big\\{t\\in[0,N^{-1}]:|x^{\\varsigma,\\hat u}(t)|>\\bar{R}\\Big\\}\\Big)\\geq \\ensuremath{\\frac} \\eta N.\n  \\end{equation*}\n  In the former case, we of course choose $u\\equiv 0$, otherwise we let $u=\\hat u$. By the flow property of well-posed ordinary differential equations, the solution to \\eqref{eq:controlled_ode} restarted at time $N^{-1}$ solves a similar equation (with new adversary $\\tilde{\\varsigma}(\\cdot)=\\theta_{N^{-1}} \\varsigma \\in\\ensuremath{\\mathcal{C}}_0([0,1-N^{-1}],\\ensuremath{\\mathbb{R}}^n)$ and initial condition $x^{\\varsigma,u}(N^{-1})$). Upon constructing $\\hat u$, we can thus easily deduce the proposition by iterating this construction.\n\n  Suppose that the time spent by uncontrolled solution $(x_t^\\varsigma)_{t \\in [0,N^{-1}]}$ in $\\ensuremath{\\mathbb{R}}^n\\setminus B_{\\bar{R}}$ is strictly less than $\\frac{\\eta}{N}$. We let $A_{x_0,\\varsigma}$ be the set of times $t\\in[0,N^{-1}]$ at which $|x^\\varsigma(t)|\\leq \\bar{R}$. Note that $A_{x_0,\\varsigma}$ is the union of a countable number of closed, disjoint intervals. By assumption, we have $\\mathop{\\mathrm {Leb}}(A_{x_0,\\varsigma})>(1-\\eta)N^{-1}$. \n  \n  For $\\delta\\ensuremath\\triangleq (2N)^{-1}$ and $e$ any fixed unit vector, we define $\\hat u$ to be the piecewise constant function\n  \\begin{equation*}\n    \\hat u(t)=\\begin{cases}\n     \\frac{2\\bar{R}+1}{(1-2\\eta)\\delta}e, & t\\in [0,  \\delta],\\\\\n    -\\frac{2\\bar{R}+1}{(1-2\\eta)\\delta}e, & t\\in (\\delta, 2\\delta],\n    \\end{cases}\n  \\end{equation*}\n  so that\n  \\begin{equation*}\n    \\int_0^t \\hat u(s)\\,ds=\\begin{cases}\n     \\frac{2\\bar{R}+1}{(1-2\\eta)\\delta}te,   & t\\in [0,  \\delta],\\\\\n    \\frac{2\\bar{R}+1}{(1-2\\eta)\\delta}(2\\delta-t)e, & t\\in (\\delta, 2\\delta].\n    \\end{cases}\n  \\end{equation*}\n  We observe that\n  \\begin{equation}\\label{eq:lower_control}\n    |x^{\\varsigma,\\hat u}(t)|\\geq \\left  |\\int_0^t \\hat u(s)\\,ds \\right|-|x^\\varsigma(t)|-\\Lip{b}\\int_{0}^t\\big|x^{\\varsigma,\\hat u}(s)-x^\\varsigma(s)\\big|\\,ds.\n  \\end{equation}\n  Moreover, owing to \\cref{lem:control_bound}, we can bound\n  \\begin{equation}\\label{eq:lower_control_n}\n    \\phantom{\\leq}\\int_{0}^t\\big|x^{\\varsigma,\\hat u}(s)-x^\\varsigma(s)\\big|\\,ds\\leq\\sqrt{C}(1+|\\hat u|_\\infty)\\int_{0}^{2\\delta}\\sqrt{s}\\,ds=\\frac{2\\sqrt{C}}{3 N^{\\frac32}}\\left(1+\\frac{2(2\\bar{R}+1)N}{1-2\\eta}\\right)<\\Lip{b}^{-1},\n  \\end{equation}\n  provided we choose the integer $N=N(C,\\bar{R},\\eta,\\Lip{b})$ large enough. Define the set $B_{x_0,\\varsigma}\\ensuremath\\triangleq A_{x_0,\\varsigma}\\cap [(1-2\\eta)\\delta,(1+2\\eta)\\delta]$. Combining \\eqref{eq:lower_control} and \\eqref{eq:lower_control_n}, we then certainly have that $|x^{\\varsigma,\\hat u}(t)|>\\bar{R}$ for all $t\\in B_{x_0,\\varsigma}$. Since\n  \\begin{equation*}\n    \\mathop{\\mathrm {Leb}}(B_{x_0,\\varsigma})\\geq\\frac{(1-\\eta)}{N}-2(1-2\\eta)\\delta=\\frac{\\eta}{N}\n  \\end{equation*}\n  and $|\\hat u|_\\infty$ as well as $|\\ensuremath{\\mathcal{D}}_{\\hat{u}}|$ only depend on $N$ and $\\bar R$,  this finishes the proof.\n\\end{proof}\n\nWe conclude our study of the deterministic controlled ODE \\eqref{eq:controlled_ode} with the following stability result which is proven by a standard Gr\\\"onwall argument:\n\\begin{lemma}\\label{lem:cont_control}\n    Let $x^{\\varsigma,u}$ denote the solution to the controlled differential equation \\eqref{eq:controlled_ode} with initial condition $x_0\\in\\ensuremath{\\mathbb{R}}^n$ and control $u\\in L^\\infty([0,1],\\ensuremath{\\mathbb{R}}^n)$. Then, for any $w\\in\\ensuremath{\\mathcal{C}}_0([0,1],\\ensuremath{\\mathbb{R}}^n)$, we have the bound\n    \\begin{equation*}\n        |x^{\\varsigma,u}-\\tilde{x}|_\\infty\\leq e^{\\Lip{b}}\\left|\\int_0^\\cdot u(s)\\,ds-w\\right|_\\infty,\n    \\end{equation*}\n    where $\\tilde{x}$ is the unique solution to\n    \\begin{equation*}\n      \\tilde{x}(t)=x_0+\\int_0^t b\\big(\\tilde{x}(s)\\big)\\,ds+w(t)+\\varsigma(t),\\qquad t\\in[0,1].\n    \\end{equation*}\n\\end{lemma}\n\n\n\n\\subsection{Exponential Stability of the Conditional Evolution}\nWe now turn to the conditional evolution of \\eqref{eq:general_flow-fixed-x} derived in \\cref{lem:conditioning}. For brevity, we drop the hat on the driving fBm throughout this and the next section. Remember that we have to study SDEs driven by a Riemann-Liouville process \n\\begin{equation*}\n\\tilde{B}_t\\ensuremath\\triangleq\\alpha_H\\int_0^{t} (t-u)^{H-\\frac12}\\,dW_u,\n\\end{equation*}\nwhere $(W_t)_{t\\geq 0}$ is a standard Wiener process. Recall from \\cref{def:flow} that, for $\\varsigma\\in\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)$, $\\Psi_{s,t}(\\cdot, \\varsigma+\\sigma\\tilde B)$ denotes the solution flow to the equation \n\\begin{equation}\\label{eq:rl_sde}\n  dX_t=b(X_t)\\,dt+d\\varsigma_t+\\sigma\\,d\\tilde{B}_t.\n\\end{equation}\nFor brevity, let us henceforth set $\\Psi_{s,t}^{\\varsigma}(\\cdot)\\ensuremath\\triangleq\\Psi_{s,t}(\\cdot,\\varsigma+\\sigma\\tilde B)$.\n\nWe first prove that---starting from any two initial points---the laws of the solutions converge to each other with an exponential rate. This however does not yet imply the convergence of $\\L\\big(\\Psi_t^{\\varsigma}(x)\\big)$ to the first marginal of the invariant measure $\\pi$ of the equation $dX_t=b(X_t)\\,dt+\\sigma\\,dB_t$ since, even if we choose $X_0\\sim\\pi$, we have $\\L\\big(\\Psi_t^{\\varsigma}(X_0)\\big)\\neq\\pi$ for $t>0$ in general. \n\nAs a preparation, we let $\\big(\\ensuremath{\\mathcal{C}}_0([0,1],\\ensuremath{\\mathbb{R}}^n),\\ensuremath{\\mathcal{H}}_H,\\mu_H\\big)$ denote the abstract Wiener space induced by the Gaussian process $(\\tilde B_t)_{t\\in[0,1]}$. Recall that the Cameron-Martin space is given by $\\ensuremath{\\mathcal{H}}_H=\\mathscr{K}_H(H_0^1)$, where\n\\begin{equation*}\n  \\mathscr{K}_H f(t)\\ensuremath\\triangleq\\begin{cases}\n  \\displaystyle\\alpha_H\\int_0^t (t-s)^{H-\\frac32}f(s)\\,ds, & H>\\frac12,\\\\ \n  \\displaystyle\\alpha_H\\frac{d}{dt}\\int_0^t (t-s)^{H-\\frac12}f(s)\\,ds, & H<\\frac12,\n  \\end{cases}\\qquad t\\in[0,1],\n\\end{equation*}\nand \n\\begin{equation*}\n  H_0^1\\ensuremath\\triangleq\\left\\{f=\\int_0^\\cdot\\dot{f}(s)\\,ds:\\,\\dot{f}\\in L^2([0,1],\\ensuremath{\\mathbb{R}}^n)\\right\\}\n\\end{equation*}\nis the Cameron-Martin space of the standard Wiener process. The inner product on $\\ensuremath{\\mathcal{H}}_H$ is defined by $\\braket{\\mathscr{K}_H f,\\mathscr{K}_H g}_{\\ensuremath{\\mathcal{H}}_H}\\ensuremath\\triangleq\\braket{\\dot{f},\\dot{g}}_{L^2}$. \n\nWe shall make use of the following simple observation:\n\\begin{lemma}\\label{lem:cameron_martin_facts}\n  Let $f:[0,1]\\to\\ensuremath{\\mathbb{R}}^n$ be piecewise linear with $f(0)=0$. Then, for each $H\\in(0,1)$, $f\\in\\ensuremath{\\mathcal{H}}_H$ and\n  \\begin{equation}\\label{eq:cameron_martin_bound}\n    \\|f\\|_{\\ensuremath{\\mathcal{H}}_H}\\lesssim|\\dot{f}|_\\infty \\big(1+\\big|\\ensuremath{\\mathcal{D}}_{\\dot{f}}\\big|\\big).\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\n  It follows from \\cite[Theorem 5]{Picard2011} (see also \\cite{Samko1993}) that the inverse of $\\ensuremath{\\mathscr{K}}_H$ exists on the set of Lipschitz functions and there is a numerical constant $\\varrho_H>0$ such that $\\ensuremath{\\mathscr{K}}_H^{-1}=\\varrho_H\\ensuremath{\\mathscr{K}}_{1-H}$. Notice also that we have $\\frac{d}{dt}\\ensuremath{\\mathscr{K}}_H^{-1}f=\\ensuremath{\\mathscr{K}}_H^{-1}\\dot{f}$.\n  \n  Let us first consider the case $H<\\frac12$. The bound \\eqref{eq:cameron_martin_bound} is an immediate consequence of \n  \\begin{equation*}\n    \\left|\\frac{d}{dt}\\ensuremath{\\mathscr{K}}_H^{-1} f(t)\\right|\\leq\\varrho_H\\int_0^t (t-s)^{-H-\\frac12}\\big|\\dot{f}(s)\\big|\\,ds\\lesssim|\\dot{f}|_\\infty \\qquad\\forall\\,t\\in[0,1].\n  \\end{equation*}\n  For $H>\\frac12$ we let $\\tau_1,\\dots,\\tau_k$ denote the jump points of $\\dot{f}$ in the interval $[0,t)$. Notice that\n  \\begin{align*}\n    \\left|\\frac{d}{dt}\\ensuremath{\\mathscr{K}}_H^{-1} f(t)\\right|&\\leq\\varrho_H\\left|\\frac{d}{dt}\\left(\\sum_{i=1}^{k-1}\\int_0^{\\tau_1}(t-s)^{\\frac12-H}\\dot{f}(s)\\,ds+\\cdots+\\int_{\\tau_k}^t (t-s)^{\\frac12-H}\\dot{f}(s)\\,ds\\right)\\right| \\\\\n    &\\lesssim |\\dot{f}|_\\infty\\big(1+|\\ensuremath{\\mathcal{D}}_{\\dot{f}}|\\big)t^{\\frac12-H}.\n  \\end{align*}\n  Since $1-2H>-1$, we obtain\n  \\begin{equation*}\n    \\|f\\|_{\\ensuremath{\\mathcal{H}}_H}=\\left\\|\\frac{d}{dt}\\ensuremath{\\mathscr{K}}_H^{-1}f\\right\\|_{L^2}\\lesssim|\\dot{f}|_\\infty\\big(1+|\\ensuremath{\\mathcal{D}}_{\\dot{f}}|\\big),\n  \\end{equation*}\n  as required.\n  \n\\end{proof}\n\n\nThe next important lemma lifts the control result of \\cref{prop:control} to solutions of SDEs with additive noise:\n\\begin{lemma}\\label{lem:probabilistic_control}\n  Let $b\\in\\S(\\kappa, R,\\lambda)$ and $\\sigma\\in\\Lin{n}$ be invertible. Then, for any $\\bar R>0$ and any $\\eta\\in(0,\\frac12)$, there is constant $\\a_{\\eta,\\bar{R}}>0$ such that the following holds: For each $x\\in\\ensuremath{\\mathbb{R}}^n$ and each $\\varsigma\\in\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)$, we can find an event $\\ensuremath{\\mathscr{A}}_{x,\\varsigma}$ with $\\ensuremath\\mathbb{P}(\\ensuremath{\\mathscr{A}}_{x,\\varsigma})\\geq\\a_{\\eta,\\bar{R}}$ such that\n  \\begin{equation*}\n    \\int_0^1 \\mathbf 1_{\\big\\{t: \\big|\\Psi_{t}^{\\varsigma}(x)(\\omega)\\big|>\\bar R\\big\\}}(s)\\,ds > \\eta \\qquad  \\forall\\, \\omega \\in \\ensuremath{\\mathscr{A}}_{x,\\varsigma}.\n  \\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n  Let $u_{x,\\varsigma}\\in L^\\infty([0,1],\\ensuremath{\\mathbb{R}}^n)$ be the piecewise constant control furnished by \\cref{prop:control} such that the occupation time of $x^{\\varsigma,u_{x,\\varsigma}}$ of the set $\\ensuremath{\\mathbb{R}}^n\\setminus B_{\\bar{R}+1}$ is greater than $\\eta$. We set $U_{x,\\varsigma}\\ensuremath\\triangleq\\int_0^\\cdot u_{x,\\varsigma}(s)\\,ds$ and note that $U_{x,\\varsigma}$ is piecewise linear. \\Cref{lem:cont_control} allows us to choose an $\\varepsilon>0$ (independent of $x$ and $\\varsigma$) such that, on the event $\\ensuremath{\\mathscr{A}}_{x,\\varsigma}\\ensuremath\\triangleq\\big\\{\\big|U_{x,\\varsigma}-\\sigma\\tilde{B}\\big|_\\infty\\leq\\varepsilon\\big\\}$, the occupation time of $\\big(\\Psi^{\\varsigma}_{h}(x)\\big)_{h\\in[0,1]}$ of $\\ensuremath{\\mathbb{R}}^n\\setminus B_{\\bar R} $ exceeds $\\eta$. \n\n  It remains to show that $\\inf_{x,\\varsigma}\\ensuremath\\mathbb{P}(\\ensuremath{\\mathscr{A}}_{x,\\varsigma})>0$. To this end, we first note that $U_{x,\\varsigma}\\in\\ensuremath{\\mathcal{H}}_H$ by \\cref{lem:cameron_martin_facts}. By the Cameron-Martin formula (see e.g. \\cite{Bogachev1998}),  \n  \\begin{align*}\n    \\ensuremath\\mathbb{P}(\\ensuremath{\\mathscr{A}}_{x,\\varsigma})&\\geq\\ensuremath\\mathbb{P}\\big(\\big|\\sigma^{-1}U_{x,\\varsigma}-\\tilde{B}\\big|_\\infty\\leq|\\sigma|^{-1}\\varepsilon\\big)\\\\\n    &=\\exp\\left(-\\frac12\\|\\sigma^{-1}U_{x,\\varsigma}\\|_{\\ensuremath{\\mathcal{H}}_H}^2\\right)\\int_{\\{|x|_\\infty\\leq|\\sigma|^{-1}\\varepsilon\\}}e^{\\braket{x,U_{x,\\varsigma}}_{\\ensuremath{\\mathcal{H}}_H}}\\,\\mu_H(dx).\n  \\end{align*}\n  Consequently, Jensen's inequality and spherical symmetry give\n  \\begin{equation}\\label{eq:quant_lower_bound}\n    \\ensuremath\\mathbb{P}(\\ensuremath{\\mathscr{A}}_{x,\\varsigma})\\geq\\exp\\left(-\\frac12\\|\\sigma^{-1}U_{x,\\varsigma}\\|_{\\ensuremath{\\mathcal{H}}_H}^2\\right)\\ensuremath\\mathbb{P}\\big(|\\tilde{B}|_\\infty\\leq|\\sigma|^{-1}\\varepsilon\\big).\n  \\end{equation}\n  Combining \\cref{prop:control,lem:cameron_martin_facts}, we obtain that $\\sup_{x,\\varsigma}\\|U_{x,\\varsigma}\\|_{\\ensuremath{\\mathcal{H}}_H}\\lesssim M(1+M)$. This concludes the proof.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:conditional_initial_condition_wasserstein}\n Let $\\sigma\\in\\Lin{n}$ be invertible. Then, for any $\\kappa, R>0$ and any $p\\geq 1$, there exists a number $\\Lambda=\\Lambda(\\kappa,R,p)\\in(0,\\kappa)$ such that the following holds: If $b\\in\\S\\big(\\kappa,R, \\Lambda\\big)$, there are constants $c,C>0$ such that, for any $\\varsigma\\in\\ensuremath{\\mathcal{C}}_0([0,1],\\ensuremath{\\mathbb{R}}^n)$,\n  \\begin{equation*}\n    \\ensuremath{\\mathcal{W}}^p\\Big(  \\L\\big( {\\Psi}^{\\varsigma}_t(Y)\\big),  \\L\\big( {\\Psi}^{\\varsigma}_t(\\tilde Y)\\big)\\Big)\\leq C\\ensuremath{\\mathcal{W}}^p\\big(\\L(Y),\\L(\\tilde Y)\\big) e^{-c t}\n  \\end{equation*}\n   for all $t\\geq 0$.\n \\end{proposition}\n\n\\begin{proof} \nWrite $X_t\\ensuremath\\triangleq\\Psi^{\\varsigma}_{t}(Y)$ and $Z_t=\\Psi_{t}^{\\varsigma}(\\tilde Y)$. Let $\\mu_t\\ensuremath\\triangleq\\L(X_t)$ and $\\nu_t\\ensuremath\\triangleq\\L(Z_t)$, thus $(X_t,Z_t)$ is a synchronous coupling of $\\mu_t$ and $\\nu_t$. Our strategy for proving the exponential convergence of $t\\mapsto\\ensuremath{\\mathcal{W}}^p(\\mu_t,\\nu_t)$ is to show that, for any $t>0$, the evolution of $(X_s)_{s\\in[t,t+1]}$ conditional on ${\\mathcal F}_t$ spends a sufficient amount of time in the contractive region $\\{|x|>R\\}$. As noted in \\cref{example-1} \\ref{it:rough_decomposition}, there is an independent increment decomposition $(\\theta_t\\tilde{B})_{h}= Q^t_h+\\tilde{B}^t_h$ for the Riemann-Liouville process. Using this and the conditional evolution derived in \\cref{lem:conditioning_general}, we find\n  \\begin{align}\n    \\Expec{\\big|X_{t+1}-Z_{t+1}\\big|^p}&    =  \\Expec{\\Expec{\\big|\\Psi^{\\varsigma}_{t,t+1}(X_t)-\\Psi^{\\varsigma}_{t,t+1}(Z_t)\\big|^p\\,\\middle|\\,{\\mathcal F}_t}}\\nonumber\\\\\n    &=     \\Expec{\\Expec{\\Big|\\Psi_{1}\\big(X_t, \\theta_t\\varsigma+\\sigma\\theta_t\\tilde B\\big)-\\Psi_{1}\\big(Z_t, \\theta_t\\varsigma+\\sigma \\theta_t\\tilde B\\big)\\Big|^p\\,\\middle|\\,{\\mathcal F}_t}}\\nonumber\\\\\n    &=\\Expec{\\Expec{\\Big|\\Psi_{1}\\big(X_t, \\theta_t\\varsigma+\\sigma Q^t+\\sigma\\tilde{B}^t\\big)-\\Psi_{1}\\big(Z_t, \\theta_t\\varsigma+\\sigma Q^t+\\sigma\\tilde{B}^t\\big)\\Big|^p\\,\\middle|\\,{\\mathcal F}_t}} \\nonumber\\\\\n       &=     \\Expec{  \\Expec{\\Big|\\Psi^{\\theta_t\\varsigma+\\ell}_{1} (x)-\\Psi_{1}^{\\theta_t\\varsigma+\\ell}(z)\\Big|^p}\\bigg|_{\\substackal{x&=X_t,z=Z_t,\\\\\\ell&=\\sigma Q^t}}},\\label{two-initial-conditions}\n \\end{align}\n where in the last step we also used that $(\\tilde{B}^t_{h})_{h\\geq 0}\\overset{d}{=}(\\tilde{B}_{h})_{h\\geq 0}$.\n\nBy assumption, the drift $b$ does not expand by more than a factor of $\\Lambda$ on all of $\\ensuremath{\\mathbb{R}}^n$. We therefore have the pathwise estimate\n\\begin{equation}\\label{eq:rl_lipschitz}\n\\big|\\Psi_{s,t}^{\\theta_t\\varsigma+\\ell}(x)-\\Psi_{s,t}^{\\theta_t\\varsigma+\\ell}(z)\\big|^p\\leq e^{p(t-s)\\Lambda} |x-z|^p\n\\end{equation}\nfor all $0\\leq s< t\\leq 1$. Let $\\eta\\in(0,\\frac12)$ and $\\bar\\kappa\\in(0,\\kappa)$ be such that $\\Xi\\ensuremath\\triangleq\\bar\\kappa\\eta-\\Lambda(1-\\eta)>0$ (recall that we assume $\\Lambda<\\kappa$). Let $\\bar R>R$ be the corresponding radius furnished by \\cref{lem:bigger_ball}. For any $x\\in\\ensuremath{\\mathbb{R}}^n$ and any $\\varsigma,\\ell\\in\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)$, let $\\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell}$ be the event from \\cref{lem:probabilistic_control}. Recall that $\\ensuremath\\mathbb{P}(\\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell})\\geq\\a_{\\eta,\\bar{R}}>0$ and \n\\begin{equation*}\n  \\int_0^1 \\mathbf 1_{\\big\\{s: \\big|\\Psi_{s}^{\\theta_t\\varsigma+\\ell}(x)(\\omega)\\big|>\\bar R\\big\\}}(r)\\,dr > \\eta \\qquad  \\forall\\, \\omega \\in \\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell}.\n\\end{equation*}\nSince $\\Xi>0$, by possibly decreasing $\\Lambda$ we can also ensure that \n\\begin{equation}\\label{eq:lambda}\n  0<\\Lambda < \\frac1p\\log\\left(\\ensuremath{\\frac}  {1-\\a_{\\eta,\\bar{R}}e^{-p \\Xi} }{1-\\a_{\\eta,\\bar{R}}}\\right).\n\\end{equation}\nOwing to pathwise continuity of $h\\mapsto\\Psi_h^{\\theta_t\\varsigma+\\ell}(x)$, there are random times $t_1,\\dots,t_{2N(\\omega)}$ such that, for all $\\omega\\in \\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell}$,\n\\begin{itemize}\n  \\item $0\\leq t_1(\\omega)<\\cdots<t_{2N(\\omega)}(\\omega)\\leq 1$, \n  \\item $\\displaystyle\\sum_{i=1}^{N(\\omega)}\\big(t_{2i}(\\omega)-t_{2i-1}(\\omega)\\big)\\geq\\eta$, and\n  \\item $\\displaystyle\\bigcup_{i=1}^{N(\\omega)}\\big(t_{2i-1}(\\omega),t_{2i}(\\omega)\\big)\\subset\\big\\{h\\in[0,1]:\\,\\big|\\Psi_{h}^{\\theta_t\\varsigma+\\ell}(x)(\\omega)\\big|>\\bar R\\big\\}$.\n\\end{itemize}\nTogether with \\eqref{eq:rl_lipschitz} it follows that, on the event $\\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell}$,\n\\begin{align*}\n&\\phantom{\\leq}\\big|\\Psi_{1}^{\\theta_t\\varsigma+\\ell}(x)-\\Psi_{1}^{\\theta_t\\varsigma+\\ell}(z)\\big|^p\n= \\Big|\\Psi_{t_{2N},1}^{\\theta_t\\varsigma+\\ell}\\Big(\\Psi_{t_{2N}}^{\\theta_t\\varsigma+\\ell} (x)\\Big)-\\Psi_{t_{2N},1}^{\\theta_t\\varsigma+\\ell}\\Big(\\Psi_{t_{2N}}^{\\theta_t\\varsigma+\\ell} (z)\\Big)\\Big|^p\\\\\n&\\leq e^{p(1-t_{2N})\\Lambda} \\big|\\Psi_{t_{2N}}^{\\theta_t\\varsigma+\\ell}(x)-\\Psi_{t_{2N}}^{\\theta_t\\varsigma+\\ell}(z)\\big|^p\\\\\n&\\leq e^{p (1-t_{2N})\\Lambda} e^{-p (t_{2N}-t_{2N-1})\\bar\\kappa} \\big|\\Psi_{t_{2N-1}}^{\\theta_t\\varsigma+\\ell}(x)-\\Psi_{t_{2N-1}}^{\\theta_t\\varsigma+\\ell}(z)\\big|^p\\\\\n&\\leq\\cdots\\leq \\exp\\left[p\\left(\\Lambda\\sum_{i=0}^N(t_{2i+1}-t_{2i})-\\bar\\kappa\\sum_{i=0}^{N}  (t_{2i}-t_{2i-1}) \\right)\\right] |x-z|^p\\\\\n&\\leq e^{ -p\\Xi}|x-z|^p,\n\\end{align*}\nwhere we have set $t_{2N+1}\\ensuremath\\triangleq 1$ for convenience. On the complementary event $\\Omega\\setminus \\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell}$, we apply the trivial estimate \\eqref{eq:rl_lipschitz}. Inserting these bounds back into \\eqref{two-initial-conditions}, we conclude that\n\\begin{equation*}\n  \\Expec{\\big|X_{t+1}-Z_{t+1}\\big|^p}\\leq\\Big(\\big(1-\\a_{\\eta,\\bar R}\\big)e^{p\\Lambda}+\\a_{\\eta,\\bar R}e^{-p\\Xi}\\Big)\\Expec{|X_t-Z_t|^p}\\ensuremath\\triangleq\\rho\\Expec{|X_t-Z_t|^p}.\n\\end{equation*}\nObserve that $\\rho<1$ by \\eqref{eq:lambda}. Finally, a straight-forward induction shows that\n\\begin{equation}\\label{eq:to_minimize}\n  \\ensuremath{\\mathcal{W}}^p\\Big(\\L\\big( {\\Psi}^{\\varsigma}_t(Y)\\big),  \\L\\big( {\\Psi}^{\\varsigma}_t(\\tilde Y)\\big)\\Big)\\leq\n  \\big\\|X_t-Z_t\\big\\|_{L^p}\\leq e^{\\Lambda} \\rho^{[t]}\\big\\|Y-\\tilde Y\\big\\|_{L^p}\\leq\\frac{e^{\\Lambda}}{\\rho}e^{-|\\log\\rho|t}\\big\\|Y-\\tilde Y\\big\\|_{L^p},\n\\end{equation}\nwhere $[\\cdot]$ denotes the integer part. Minimize over the set of couplings of $\\L(Y)$ and $\\L(\\tilde{Y})$ to conclude the proof.\n\\end{proof}\n\nA more explicit expression for the threshold value $\\Lambda(\\kappa,R,p)$ can be derived by the method outlined in \\cref{rem:constant_xi} below. We abstain from including further details in this work. Let us however introduce the following notation:\n\\begin{definition}\n  Let $\\kappa,R>0$ and $p\\geq 1$. We abbreviate $\\S_p(\\kappa,R)\\ensuremath\\triangleq\\S\\big(\\kappa,R,\\Lambda(\\kappa,R,p)\\big)$ with the constant from \\cref{prop:conditional_initial_condition_wasserstein}.\n\\end{definition}\n\nBy \\cref{lem:conditioning}, the Wasserstein bound of \\cref{prop:conditional_initial_condition_wasserstein} lifts to bounds on the fast motion with frozen slow input \\eqref{eq:general_flow-fixed-x}. We obtain the following Lipschitz dependence of the flow $\\bar \\Phi$ on the initial value:\n\n\\begin{corollary}\\label{cor:fast_different_initial}\n  Let $({\\mathcal F}_t)_{t\\geq 0}$ be a filtration compatible with the fBm $\\hat{B}$. Let $0\\leq s\\leq t$ and let $X$, $Y$, and $\\tilde{Y}$ be ${\\mathcal F}_s$-measurable random variables. Suppose that there are $\\kappa,R>0$ such that $b(x,\\cdot)\\in\\S_1(\\kappa,R)$ for every $x\\in\\ensuremath{\\mathbb{R}}^d$. Then there is a constant $c>0$ such that, for any Lipschitz continuous function $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$,\n \\begin{equation*}\n    \\Big|\\Expec{h\\big(X,\\bar{\\Phi}_{s,t}^X(Y)\\big)-h\\big(X,\\bar{\\Phi}_{s,t}^X(\\tilde{Y})\\big)\\,\\middle|\\,{\\mathcal F}_s}\\Big|\\lesssim\\Lip{h}|Y-\\tilde Y|e^{-c\\frac{|t-s|}{\\varepsilon}}.\n  \\end{equation*}\n  If, in addition, $b(x,\\cdot)\\in\\S_p(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$, then also\n  \\begin{equation*}\n    \\Big\\|\\bar{\\Phi}_{s,t}^X(Y)-\\bar{\\Phi}_{s,t}^X(\\tilde{Y})\\Big\\|_{L^p}\\lesssim\\|Y-\\tilde Y\\|_{L^p}e^{-c\\frac{|t-s|}{\\varepsilon}}.\n  \\end{equation*}\n\\end{corollary}\n\\begin{proof}\n  The first estimate is an immediate consequence of \\cref{lem:conditioning} and Kantorovich-Rubinstein duality. The second bound follows from the fact that we used a synchronous coupling in the proof of \\cref{prop:conditional_initial_condition_wasserstein}.\n\\end{proof}\n\nThe proof of \\cref{prop:conditional_initial_condition_wasserstein} shows that its conclusion actually holds if $\\tilde B$ is replaced by another process $Z$ with similar properties:\n\\begin{remark}\\label{rem:Wasserstein-general}\nLet $Z$ be a process with locally independent increment decomposition $\\theta_t Z=\\bar Z^t+\\tilde Z^t$. Assume that\n\\begin{enumerate}\n\\item the ${\\mathcal F}_t$-adapted part $\\bar Z^t$ takes values in $\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^n)$ and\n\\item\\label{it:cm_dense} there is a unit vector $e\\in\\ensuremath{\\mathbb{R}}^n$ such that, for each $t\\geq 0$, $\\L\\big((\\tilde Z^t_h\\cdot e)_{h\\in[0,1]}\\big)$ is supported on all of $\\ensuremath{\\mathcal{C}}_0([0,1])$.\n\\end{enumerate} \nThen a statement similar to \\cref{prop:conditional_initial_condition_wasserstein} holds.\n\\end{remark}\n\n\\begin{example}\\label{ex:titmarsh}\n  Suppose that $\\tilde{Z}^t_h=\\int_t^{t+h}\\mathfrak{G}(t+h-s)\\,dW_s$ for some kernel $\\mathfrak{G}:\\ensuremath{\\mathbb{R}}_+\\to\\Lin{n}$ which is square integrable at the origin and continuous on $(0,\\infty)$. Then the requirement \\ref{it:cm_dense} in \\cref{rem:Wasserstein-general} holds if $\\int_0^t|\\mathfrak{G}(s)|\\,ds>0$ for each $t>0$. Indeed, this can be shown by a clever application of Titmarsh's convolution theorem as in \\cite[Lemma 2.1]{Cherny2008}.\n\\end{example}\n\nThe example shows that in particular an fBm of any Hurst parameter $H\\in(0,1)$ falls in the regime of \\cref{rem:Wasserstein-general}. Hence, we have the following corollary to \\cref{prop:conditional_initial_condition_wasserstein}:\n\\begin{corollary}\\label{conveergence-equilibrium}\n Let $p\\geq 1$ and suppose that $b\\in\\S_p(\\kappa,R)$ for some $\\kappa,R>0$. Let $(X_t)_{t\\geq 0}$ be the solution to \n \\begin{equation}\\label{eq:fbm_sde}\n   dX_t=b(X_t)\\,dt+\\sigma\\,dB_t\n \\end{equation}\n started in the generalized initial condition $\\mu$, where $(B_t)_{t\\geq 0}$ is an fBm with Hurst parameter $H\\in(0,1)$ and $\\sigma\\in\\Lin{n}$ is invertible. Then there is a unique invariant measure $\\mathcal I_\\pi\\in\\P(\\ensuremath{\\mathbb{R}}^n\\times\\H_H)$ for the equation \\eqref{eq:fbm_sde} in the sense of \\cref{initial-condition}. Moreover, writing $\\pi=\\Pi_{\\ensuremath{\\mathbb{R}}^n}^*\\mathcal I_\\pi$ for the first marginal, there are constants $c,C>0$ such that\n \\begin{equation}\\label{eq:wasserstein_fbm}\n    \\ensuremath{\\mathcal{W}}^p\\big(\\L( X_t),\\pi\\big)\\leq C\\ensuremath{\\mathbb{W}}^p(\\mu,\\mathcal I_\\pi) e^{-ct}\n  \\end{equation}\n  for all $t\\geq 0$.\n \\end{corollary}\n\\begin{proof}\n  By \\cref{prop:existence_invariant_measure}, we know that there is an invariant measure $\\mathcal I_\\pi$ to \\eqref{eq:fbm_sde} with moments of all orders. The Wasserstein estimate \\eqref{eq:wasserstein_fbm} then follows by the very same arguments as in \\cref{prop:conditional_initial_condition_wasserstein}. The only difference is that we now have to specify a generalized initial condition $\\nu\\in\\P\\big((\\ensuremath{\\mathbb{R}}^n\\times\\H_H)^2\\big)$ for the coupling $(X_t,Z_t)$, see \\cref{sec:physical_solution}. Unlike for the conditioned dynamics, we have $Z_t\\sim\\pi$ if we start $Z$ in the invariant measure $\\mathcal I_\\pi$. In order for our previous argument to apply, we need to ensure that the past of the noises in the synchronous coupling coincide. In \\eqref{eq:to_minimize} we can thus only minimize over couplings in the set \n  \\begin{equation*}\n    \\big\\{\\rho\\in\\P\\big((\\ensuremath{\\mathbb{R}}^n\\times\\H_H)^2\\big):\\,\\rho(\\ensuremath{\\mathbb{R}}^n\\times\\ensuremath{\\mathbb{R}}^n\\times\\Delta_{\\H_H})=1\\big\\},\n  \\end{equation*}\n  which precisely yields \\eqref{eq:wasserstein_fbm}.\n\\end{proof}\n \n\n\\subsection{Quenched Convergence to the Invariant Measure}\\label{quenched-convergence}\n\nThe other distance, which will play a r\\^ole in \\cref{sec:uniform_bounds} below, is between $\\L\\big(\\Psi_t^\\varsigma(Y)\\big)$ and the stationary law $\\pi$ of the equation \\eqref{eq:fbm_sde}. We stress that---contrarily to the proof of \\cref{conveergence-equilibrium}---we cannot simply start the process in the invariant measure. In fact, the measure $\\pi$ is not stationary for \\eqref{eq:rl_sde} since the increments of $\\tilde{B}$ are not stationary. It is therefore necessary to wait for a sufficient decay of the deterministic `adversary' $\\varsigma$, whence we only find an algebraic rate of convergence. Before we state the result, let us first illustrate that there is indeed no hope for an exponential rate:\n\\begin{example}\n  Let\n  \\begin{equation*}\n    dX_t=-X_t\\,dt+d\\tilde{B}_t,\\qquad dY_t=-Y_t\\,dt+dB_t.    \n  \\end{equation*}\n  If we start both $X$ and $Y$ in the generalized initial condition $\\delta_0\\otimes\\ensuremath{\\mathsf{W}}$, then $\\L(X_t)=N(0,\\Sigma_{t}^2)$ and $\\L(Y_t)=N(0,\\bar\\Sigma_{t}^2)$ where\n  \\begin{equation*}\n    \\Sigma_t^2=\\bar\\Sigma_t^2-\\Expec{\\left|\\int_0^te^{-(t-s)}\\dot{\\bar{B}}_s\\,ds\\right|^2}.\n  \\end{equation*}\n  In particular, $\\ensuremath{\\mathcal{W}}^2\\big(\\L(X_t),\\L(Y_t)\\big)=|\\Sigma_{t}-\\bar\\Sigma_t|\\gtrsim t^{-(1-\\hat{H})}$ uniformly in $t\\geq 1$. Since it is easy to see that $\\ensuremath{\\mathcal{W}}^2\\big(\\L(Y_t),\\pi\\big)\\lesssim e^{-t}$, it follows that $\\ensuremath{\\mathcal{W}}^2\\big(\\L(X_t),\\pi\\big)\\gtrsim t^{-(1-\\hat{H})}$.\n\\end{example}\n\n\\begin{proposition}\\label{prop:conditional_stationary_wasserstein}\n  Suppose that $b\\in\\S_p(\\kappa,R)$ for some $\\kappa,R>0$ and $\\sigma\\in\\Lin{n}$ is invertible. Let $p\\geq 1$, $\\varsigma\\in\\Omega_\\alpha$ for some $\\alpha>0$, and $Y$ be an ${\\mathcal F}_0$-measurable random variable. Then, for each $\\beta<\\min\\big(\\alpha,1-H\\big)$, there is a constant $C>0$ such that\n  \\begin{equation}\\label{eq:wasserstein_quenched}\n    \\ensuremath{\\mathcal{W}}^p\\big(\\L(\\Psi^\\varsigma_t(Y)), \\pi\\big)\\leq C\\frac{\\big(1+\\|\\varsigma\\|_{\\Omega_\\beta}\\big)\\big(1+\\ensuremath{\\mathcal{W}}^p(\\L(Y),\\pi)\\big)}{t^{\\beta}}\n  \\end{equation}\n  for all $t>0$.\n\\end{proposition}\n\n\\begin{proof}\n  Fix $t\\geq 1$, abbreviate $X\\ensuremath\\triangleq\\Psi_\\cdot^{\\varsigma}(Y)$, and let $Z$ be the stationary solution to the equation \\eqref{eq:fbm_sde}. We assume that $X$ and $Z$ are driven by the same Wiener process. Let us first consider the case $p\\geq 2$. Recall the following locally independent decompositions from \\cref{sec:increment}:\n    $$\\theta_t B=\\bar B^t+\\tilde B^t, \\qquad \\theta_t \\tilde B=Q^t+\\tilde B^t.$$\n  Remember also that the `smooth' part of the fBm increment can be further decomposed as $\\bar{B}^t=P^t+Q^t$, see \\cref{example-1} \\ref{it:smooth_decomposition}. Therefore, \n  \\begin{align}\n    \\Expec{\\big|X_{t+1}-Z_{t+1}\\big|^p}&    =  \\Expec{\\Expec{\\big|\\Psi_{t,t+1}(X_t,\\varsigma+ \\sigma\\tilde B)-\\Psi_{t,t+1}(Z_t,\\sigma B)\\big|^p\\,\\middle|\\,{\\mathcal F}_t}}\\nonumber\\\\\n        &=\\Expec{\\Expec{\\Big|\\Psi_{1}\\big(X_t, \\theta_t\\varsigma+\\sigma Q^t+\\sigma \\tilde B^t\\big)-\\Psi_{1}\\big(Z_t,\\sigma P^t+\\sigma Q^t+\\sigma\\tilde B^t\\big)\\Big|^p\\,\\middle|\\,{\\mathcal F}_t}}\\nonumber\\\\\n       &=\\Expec{\\Big|  \\Psi_1^{\\theta_t \\varsigma+ \\ell} (x) - \\Psi_1^{\\bar\\ell+\\ell} (z) \\Big|^p\\bigg|_{\\substackal{x&=X_t,z=Z_t,\\\\\\ell&=\\sigma Q^t,\\bar{\\ell}=\\sigma P^t}}}\\label{eq:expec_diff}\n  \\end{align}\n  Write $R_h\\ensuremath\\triangleq\\Psi_h^{\\theta_t \\varsigma+\\ell} (x)$ and $S_h\\ensuremath\\triangleq \\Psi_h^{\\bar\\ell+\\ell} (z) $. Notice that, since $\\varsigma$ and $\\bar\\ell$ are differentiable,\n  \\begin{align*}\n    \\frac{d}{dh}\\big|R_{h}-S_{h}\\big|^p&=p\\Braket{\\dot{\\varsigma}_{t+h}-\\dot{\\bar{\\ell}}_h+b\\big(R_{h}\\big)-b\\big(S_{h}\\big),R_{h}-S_{h}}\\big|R_{h}-S_{h}\\big|^{p-2}\\\\\n    &\\leq p(\\Lambda+\\gamma)\\big|R_{h}-S_{h}\\big|^p\n     +\\left(\\frac{p-1}{\\gamma p}\\right)^{p-1}\\left(|\\dot{\\varsigma}_{t+h}|+|\\dot{\\bar{\\ell}}_{h}|\\right)^p\n  \\end{align*}\n  for any $\\gamma>0$, where $\\Lambda=\\Lambda(\\kappa,R,p)$ is the expansion threshold derived in \\cref{prop:conditional_initial_condition_wasserstein}. It follows that, for any $0\\leq h_1\\leq h_2\\leq 1$,\n  \\begin{align}\n    &\\phantom{\\leq}\\big|R_{h_2}-S_{h_2}\\big|^p\\nonumber\\\\\n    &\\leq\\big|R_{h_1}-S_{h_1}\\big|^p e^{p(\\Lambda+\\gamma)(h_2-h_1)}+\\left(\\frac{p-1}{\\gamma p}\\right)^{p-1}\\int_{h_1}^{h_2}e^{p(\\Lambda+\\gamma)(h_2-s)}\\left(|\\dot\\varsigma_{t+s}|+|\\dot{\\bar{\\ell}}_{s}|\\right)^p\\,ds\\nonumber\\\\\n    &\\leq \\big|R_{h_1}-S_{h_1}\\big|^p e^{p(\\Lambda+\\gamma)(h_2-h_1)}+C_\\gamma(h_2-h_1),\\label{eq:estimate_waserstein_1}\n  \\end{align}\n  where we abbreviated\n  \\begin{equation*}\n    C_\\gamma\\ensuremath\\triangleq \\left(\\frac{p-1}{\\gamma p}\\right)^{p-1}\\left(\\frac{\\|\\varsigma\\|_{\\Omega_\\beta}}{t^{\\beta}}+|\\dot{\\bar{\\ell}}|_\\infty\\right)^p.\n  \\end{equation*}\n  We now argue similarly to \\cref{prop:conditional_initial_condition_wasserstein}: Pick $\\eta\\in(0,\\frac12)$ and $\\bar{\\kappa}\\in(0,\\kappa)$ such that $\\Xi\\ensuremath\\triangleq \\eta\\bar{\\kappa}-(1-\\eta)\\Lambda>0$.  Let $\\bar{R}>0$ be the corresponding constant of \\cref{lem:bigger_ball} and $\\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell}$ be the event furnished by \\cref{lem:probabilistic_control}. As before, we write $t_1,\\dots,t_{2N(\\omega)}$ for the random times characterizing the excursions of $(R_h)_{h\\in[0,1]}$ outside of $B_{\\bar R}$, see \\cref{prop:conditional_initial_condition_wasserstein}. By an argument similar to \\eqref{eq:estimate_waserstein_1},  \n  \\begin{equation}\\label{eq:estimate_waserstein_2}\n    \\big|R_{t_{2i}}-S_{t_{2i}}\\big|^p\\leq \\big|R_{t_{2i-1}}-S_{t_{2i-1}}\\big|^p e^{p(\\gamma-\\bar{\\kappa})(t_{2i}-t_{2i-1})}+C_\\gamma (t_{2i}-t_{2i-1})\n  \\end{equation}\n  for all $i=1,\\dots,N(\\omega)$ on the set $\\ensuremath{\\mathscr{A}}_{x,\\theta_t\\varsigma+\\ell}$. Combining \\eqref{eq:estimate_waserstein_1} and \\eqref{eq:estimate_waserstein_2}, we further find on this set\n  \\begin{align*}\n    \\phantom{\\leq}\\big|R_{1}-S_{1}\\big|^p&\\leq e^{p(\\Lambda+\\gamma)(1-t_{2k})}\\big|R_{t_{2k}}-S_{t_{2k}}\\big|^p+C_\\gamma(1-t_{2k})\\\\\n    &\\leq e^{p(\\Lambda+\\gamma)(1-t_{2k})}e^{p(\\gamma-\\bar{\\kappa})(t_{2k}-t_{2k-1})}\\big|R_{t_{2k-1}}-S_{t_{2k-1}}\\big|^p+C_\\gamma(1-t_{2k-1})\\\\\n    &\\leq\\cdots \\leq e^{p(\\Lambda+\\gamma)(1-\\eta) +p(\\gamma-\\bar \\kappa)\\eta}|x-z|^p+C_\\gamma \n    \\leq e^{-p\\big(\\Xi-\\gamma\\big)}|x-z|^p+C_\\gamma \n  \\end{align*}\n  Choose $\\gamma>0$ sufficiently small such that simultaneously $\\Xi-\\gamma>0$ and \n  \\begin{equation*}\n    \\rho\\ensuremath\\triangleq \\big(1-\\a_{\\eta,\\bar{R}}\\big)e^{p(\\Lambda+\\gamma)}+\\a_{\\eta,\\bar{R}}e^{-p(\\Xi-\\gamma)}<1.\n  \\end{equation*}\n  This shows that\n  \\begin{equation}\\label{eq:estimate_p_bigger}\n    \\Expec{\\big|R_{1}-S_{1}\\big|^p}\\leq\\rho|x-y|^p+C_\\gamma.\n  \\end{equation}\n  It is clear that the estimate \\eqref{eq:estimate_p_bigger} also holds for $p<2$ with the constant\n  \\begin{equation*}\n    C_\\gamma=\\frac{1}{(2\\gamma)^{\\frac{p}{2}}}\\left(\\frac{\\|\\varsigma\\|_{\\Omega_\\beta}}{t^{\\beta}}+|\\dot{\\bar{\\ell}}|_\\infty\\right)^p\n  \\end{equation*}\n   and a slightly increased $\\rho<1$. Since $P^t=\\bar{B}_{t+\\cdot}$, \\cref{lem:smooth_part_decay} and the identity \\eqref{eq:expec_diff} show that\n  \\begin{equation*}\n    \\Expec{\\big|X_{t+1}-Y_{t+1}\\big|^p}\\leq\\rho\\Expec{\\big|X_{t}-Y_{t}\\big|^p}+\\frac{C\\big(1+\\|\\varsigma\\|_{\\Omega_\\beta}^p\\big)}{t^{p\\beta}}\n  \\end{equation*}\n  for some numerical constant $C>0$ independent of $t$ and $\\varsigma$. Therefore, iterating this bound we find\n  \\begin{equation}\\label{eq:quenched_wasserstein}\n    \\Expec{\\big|X_{t}-Y_{t}\\big|^p}\\lesssim e^{-ct}\\Expec{|X_0-Y_0|^p}+C\\big(1+\\|\\varsigma\\|_{\\Omega_\\beta}^p\\big)\\sum_{i=0}^{[t]-2}\\frac{\\rho^i}{(t-1-i)^{p\\beta}}.\n  \\end{equation}\n  The last sum is easily seen to be $\\lesssim t^{-p\\beta}$ uniformly in $t\\geq 2$ and the claim follows at once.\n\\end{proof}\n\nBy a strategy inspired by \\cite[Section 7]{Panloup2020} (see also \\cite{Hairer2005}), we can lift \\cref{prop:conditional_stationary_wasserstein} to a total variation bound. Since the exposition of Panloup and Richard does not immediately transfer to the problem at hand, we choose to include the necessary details.  Consider the system\n\\begin{equation}\\label{eq:girsanov_coupling}\n  \\begin{aligned}[c]\n    dX_s&=b(X_s)\\, ds +d\\varsigma_s+\\sigma d\\tilde{B}_s,\\\\\n    dZ_s&=b(Z_s)\\, ds +\\sigma\\,dB_s+\\sigma\\varphi^t(s)\\,ds,\n    \\end{aligned}\n\\end{equation}\nwhere $X_0$ is an arbitrary initial condition and $Z$ is the stationary solution of the first equation. Our aim is to exhibit an adapted integrable function $\\varphi^t:[0,t+1]\\to\\ensuremath{\\mathbb{R}}^n$ which vanishes on $[0,t]$ and ensures that $X_{t+1}=Z_{t+1}$. To this end, we define\n\\begin{equation}\\label{eq:coupling_function}\n  \\varphi^t(s)\\ensuremath\\triangleq\\left\\{\n  \\begin{array}{ll}\n  \\left(2\\frac{|X_t-Z_t|^{\\frac12}}{|X_s-Z_s|^{\\frac12}}+\\lambda\\right)\\sigma^{-1}(X_s-Z_s)-\\dot{\\bar{B}}_s+\\sigma^{-1}\\dot{\\varsigma}_s, \\quad \\quad &s\\in [t,t+1],\\\\\n  0, \\qquad \\qquad &\\hbox{ otherwise.}\n  \\end{array}\\right.\n\\end{equation}\n\\begin{lemma}\\label{lem:girsanov}\n Let $t\\geq 1$,   $\\varsigma\\in\\Omega_\\alpha$, $b\\in\\S(\\kappa,R,\\lambda)$, \n   and consider the system \\eqref{eq:girsanov_coupling}  with $\\varphi^t$ defined in \\eqref{eq:coupling_function}. Then $X_{t+1}=Z_{t+1}$ and, for any $\\beta<\\alpha\\wedge(1-H)$,\n\\begin{equation}\\label{eq:phi_norm}\n  |\\varphi^t|_\\infty\\lesssim |X_t-Z_t|+\\frac{\\|\\varsigma\\|_{\\Omega_\\beta}+\\|\\bar{B}\\|_{\\Omega_\\beta}}{t^{\\beta}},\\qquad |\\dot{\\varphi}^t|_\\infty\\lesssim |X_t-Z_t|^{\\frac12}+|X_t-Z_t|+\\frac{\\|\\varsigma\\|_{\\Omega_\\beta}+\\|\\bar{B}\\|_{\\Omega_\\beta}}{t^{1+\\beta}},\n\\end{equation}\nwhere the derivative of $\\varphi^t$ is understood as right- and left-sided derivative at the boundaries $t$ and $t+1$, respectively.\n\\end{lemma}\n\\begin{proof}\n  The argument is a minor modification of \\cite[Lemma 5.8]{Hairer2005}: Abbreviate $f(s)\\ensuremath\\triangleq|X_s-Z_s|^2$, then\n  \\begin{equation*}\n    f^\\prime(s)=2\\braket{b(X_s)-b(Z_s)+\\dot{\\varsigma}_s-\\sigma\\dot{\\bar{B}}_s-\\sigma\\varphi^t(s),X_s-Z_s}\\leq -4|X_t-Z_t|^{\\frac12}f(s)^{\\frac34}\n  \\end{equation*}\n  since $b\\in\\S(\\kappa,R,\\lambda)$. It follows that\n  \\begin{equation*}\n    |X_s-Z_s|^{\\frac12}\\leq |X_t-Z_t|^{\\frac12}-(s-t)|X_t-Z_t|^{\\frac12}\\qquad\\forall\\, s\\in[t,t+1],\n  \\end{equation*}\n  whence $X_{t+1}=Z_{t+1}$. This also implies\n  \\begin{equation*}\n    \\left|\\frac{d}{ds}\\big(X_s-Z_s\\big)\\right|\\leq\\big(\\Lip{b}+2+\\lambda\\big)|X_t-Z_t|^{\\frac12}|X_s-Z_s|^{\\frac12}\n  \\end{equation*}\n  and consequently\n  \\begin{equation*}\n    \\left|\\frac{d}{ds}\\left(\\frac{X_s-Z_s}{|X_s-Z_s|^{\\frac12}}\\right)\\right|\\leq\\frac32\\frac{\\left|\\frac{d}{ds}\\big(X_s-Z_s\\big)\\right|}{|X_s-Z_s|^{\\frac12}}\\lesssim|X_t-Z_t|^{\\frac12}.\n  \\end{equation*}\n  The bounds \\eqref{eq:phi_norm} follow at once.\n\\end{proof}\n\\begin{remark}\n  We stress that the bound on $|\\dot{\\varphi}^t|_\\infty$ only holds for a Lipschitz continuous drift $b$.\n\\end{remark}\n\nIt is now easy to prove the following result:\n\\begin{proposition}\\label{prop:conditional_stationary_total}\n  Assume the conditions of \\cref{prop:conditional_stationary_wasserstein} for $p=1$. Then, for any $\\beta<\\alpha\\wedge(1-H)$, it holds that\n  \\begin{equation*}\n    \\TV{\\L\\big(\\Psi^\\varsigma_t(Y)\\big)-\\pi}\\lesssim {t^{-\\frac{\\beta}{3}}\\big(1+\\|\\varsigma\\|_{\\Omega_\\beta}\\big)\\big(1+\\ensuremath{\\mathcal{W}}^1(\\L(Y),\\pi)\\big)}  \\qquad \\forall\\, t>0.\n  \\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\n  Let $B$ and $B^\\prime$ be $H$-fBms built from underlying two-sided Wiener processes $W$ and $W^\\prime$, see \\eqref{eq:mandelbrot}. Recall that $\\tilde{B}$ is the Riemann-Liouville process associated with $B$. Let $X$ and $Z$ solve \n  \\begin{align}\\label{eq:proof_tv_equations}\n    \\begin{split}\n    dX_s&=b(X_s)\\,ds+d\\varsigma_s+\\sigma d\\tilde{B}_s,\\\\\n    dZ_s&=b(Z_s)\\,ds+\\sigma\\,dB^\\prime_s,\n    \\end{split}\n  \\end{align}\n  where $X_0\\overset{d}{=}Y$ and $Z$ is the stationary solution. \n    Fix $t>1$.\n    We shall use the bound\n  \\begin{align}\n    \\TV{\\L\\big(\\Psi^\\varsigma_{t+1}(Y)\\big)-\\pi}&=\\inf_{(\\tilde B,B^\\prime)}\\ensuremath\\mathbb{P}\\big(X_{t+1}\\neq Z_{t+1}\\big)\\leq\\inf_{(W,W^\\prime)}\\ensuremath\\mathbb{P}\\big(X_{t+1}\\neq Z_{t+1}\\big) \n    \\nonumber\\\\\n    &\\leq\\inf_{(W,W^\\prime)}\\ensuremath\\mathbb{P}\\big(X_{t+1}\\neq Z_{t+1},|X_t-Z_t|\\leq\\delta\\big)+ \\inf_{(W,W^\\prime)}\\ensuremath\\mathbb{P}\\big(|X_t-Z_t|>\\delta\\big).\\label{eq:tv_proof}\n  \\end{align}\n Taking $W$ and $W'$ equal, we are in the setting of \\cref{prop:conditional_stationary_wasserstein}. The estimate \\eqref{eq:quenched_wasserstein} thus shows that, for any $\\delta\\in(0,1]$,\n\\begin{equation}\n\\label{eq:tv_proof_2}\n\\inf_{(W,W^\\prime)}\\ensuremath\\mathbb{P}\\big(|X_t-Z_t|>\\delta)\\nonumber \\leq \\frac{C\\big(1+\\|\\varsigma\\|_{\\Omega_\\beta}\\big)\\big(1+\\ensuremath{\\mathcal{W}}^1(\\L(Y),\\pi)\\big)}{\\delta t^\\beta}.\n\\end{equation}\n To bound the first term in \\eqref{eq:tv_proof} we exploit the fact that $X_t$ and $Z_t$ are already close so that we can couple them at time $t+1$ with a controlled cost. \n Let $\\varphi^t$ be the function from \\cref{lem:girsanov}; in particular $\\varphi^t(s)=0$  for $s<t$.\n We observe that $B^\\prime=B+\\int_0^\\cdot\\varphi^t(s)\\,ds$ on $[0,t+1]$ if and only if $W^\\prime=W+\\int_{-\\infty}^\\cdot\\psi^t(s)\\,ds$ on $(-\\infty,t+1]$, where for a suitable constant $\\gamma_H\\in\\ensuremath{\\mathbb{R}}$,\n  \\begin{equation*}\n    \\psi^t(s)=\\begin{cases}\\displaystyle\\gamma_H\\frac{d}{ds}\\int_{t}^s (s-u)^{\\frac12-H}\\varphi^t(u)\\,du, & s\\in[t,t+1], \\\\\n    0, & s\\in(-\\infty,t),\n    \\end{cases}\n  \\end{equation*}\n  see \\cite[Lemma 4.2]{Hairer2005} for details. Let $\\ensuremath{\\mathcal{T}}$ be the linear transformation $\\ensuremath{\\mathcal{T}}(w)=w+\\int_{-\\infty}^\\cdot\\psi^t(s)\\,ds$, $w\\in\\ensuremath{\\mathcal{C}}\\big((-\\infty,t+1],\\ensuremath{\\mathbb{R}}^n\\big)$. We return to the equations \\eqref{eq:proof_tv_equations}. By the construction, we have $X_{t+1}=Z_{t+1}$ for those realizations for which $W^\\prime=\\ensuremath{\\mathcal{T}}(W)$.  In particular,\n\\begin{equation*}\n  \\inf_{(W,W^\\prime)}\\ensuremath\\mathbb{P}\\big(X_{t+1}\\neq Z_{t+1},|X_t-Z_t|\\leq \\delta\\big) \\leq \\inf_{(W, W')} \\ensuremath\\mathbb{P}\\big(W' \\not = \\ensuremath{\\mathcal{T}}(W), |X_t-Z_t|\\leq \\delta\\big).\n\\end{equation*}\nThroughout this proof, all stochastic processes are considered only up to time $t+1$. Let $\\hat \\ensuremath\\mathbb{P}(\\cdot)=\\ensuremath\\mathbb{P}\\big(\\cdot\\,\\big|\\, |X_t-Z_t|\\leq \\delta\\big)$. Let $\\mu$ and $\\nu$ denote the laws under $\\hat\\ensuremath\\mathbb{P}$ of $W$ and $\\ensuremath{\\mathcal{T}}(W)$, respectively.\nWe take an optimal coupling $(\\hat W, \\hat W^\\prime)$ achieving the total variation distance $ \\TV{\\mu-\\nu}$. Then the Pinsker-Csizsar inequality, a consequence of the Girsanov theorem, shows that\n \\begin{align*}\n    \\inf_{(W,W^\\prime)}\\ensuremath\\mathbb{P}\\big(X_{t+1}\\neq Z_{t+1},|X_t-Z_t|\\leq\\delta\\big)\n    &\\leq \\TV{\\mu-\\nu}\\ensuremath\\mathbb{P}\\big(|X_t-Z_t|\\leq \\delta \\big)\\\\\n &\\leq\\frac{1}{2} \\mathbb E_{\\hat \\ensuremath\\mathbb{P}}  \\left[ \\int_t^{t+1}|\\psi^t(s)|^2\\,ds\\right]^{\\frac12}\\ensuremath\\mathbb{P}\\big(|X_t-Z_t|\\leq\\delta\\big).   \n \\end{align*}\nOn integration by parts we find\n  \\begin{equation*}\n    \\int_t^{t+1}|\\psi^t(s)|^2\\,ds\\lesssim\\begin{cases}\n      |\\varphi^t|_\\infty^2,& H<\\frac12,\\\\\n      |\\varphi^t|^2_\\infty+|\\dot{\\varphi}^t|_\\infty^2,& H>\\frac12.\n    \\end{cases}\n  \\end{equation*}\n In either case, \\eqref{eq:phi_norm} yields\n  \\begin{equation*}\n    \\int_t^{t+1}|\\psi^t(s)|^2\\,ds\\lesssim \\delta+\\frac{\\|\\varsigma\\|^2_{\\Omega_\\beta}+\\|\\bar{B}\\|^2_{\\Omega_\\beta}}{t^{2\\beta}}\n  \\end{equation*}\n  on the event $\\{|X_t-Z_t|\\leq \\delta\\}$ and therefore\n  \\begin{align*}\n    &\\phantom{\\leq}\\inf_{(W,W^\\prime)}\\ensuremath\\mathbb{P}\\big(X_{t+1}\\neq Z_{t+1},|X_t-Z_t|\\leq\\delta\\big) \n    \\lesssim\\sqrt{\\delta}+\\frac{\\|\\varsigma\\|_{\\Omega_\\beta}+\\big\\|\\|\\bar{B}\\|_{\\Omega_\\beta}\\big\\|_{L^2}}{t^\\beta}.\n  \\end{align*}\nCombining this with \\eqref{eq:tv_proof} and \\cref{lem:smooth_part_decay}, we have proven\n  \\begin{equation}\\label{eq:tv_final}\n    \\TV{\\L\\big(\\Psi_{t+1}^\\varsigma(Y)\\big)-\\pi}\\lesssim \\big(1+\\|\\varsigma\\|_{\\Omega_\\beta}\\big)\\big(1+\\ensuremath{\\mathcal{W}}^1(\\L(Y),\\pi)\\big)\\left(\\sqrt{\\delta}+\\frac{1}{\\delta t^\\beta}\\right),\n  \\end{equation}\n  which is minimized for $\\delta=t^{-\\frac{2\\beta}{3}}$.\n\\end{proof}\n\n\nBy duality and \\cref{lem:conditioning}, we obtain the following ergodic theorem as a corollary to \\cref{prop:conditional_stationary_wasserstein,prop:conditional_stationary_total}. It provides the fundamental estimates for our proof of the averaging principle for the fractional slow-fast system with feedback dynamics.\n\\begin{corollary}\\label{cor:total_variation_conditional}\n  Let $0\\leq s\\leq t$ and let $X,Y$ be ${\\mathcal F}_s$-measurable random variables. Suppose that there are $\\kappa,R>0$ such that $b(x,\\cdot)\\in\\S_1(\\kappa,R)$ for every $x\\in\\ensuremath{\\mathbb{R}}^d$. Then, for any $\\zeta<1-\\hat{H}$ and\n  \\begin{enumerate}\n    \\item\\label{it:ergodicity_wasserstein} any Lipschitz function $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$, \n    \\begin{equation*}\n    \\Big|\\Expec{h\\big(X,\\bar{\\Phi}_{s,t}^X(Y)\\big)-\\bar{h}(X)\\,|\\,{\\mathcal F}_s}\\Big|\\lesssim\\Lip{h}\\Big(1+\\big\\|\\varepsilon^{-\\hat{H}}\\bar{\\hat{B}}_{\\varepsilon\\cdot}^s\\big\\|_{\\Omega_\\zeta}\\Big)\\big(1+|Y|\\big)\\left(1\\wedge\\frac{\\varepsilon^{\\zeta}}{|t-s|^{\\zeta}}\\right).\n  \\end{equation*}\n  \\item\\label{it:ergodicity_tv} any bounded measurable function $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$,\n  \\begin{equation*}\n    \\Big|\\Expec{h\\big(X,\\bar{\\Phi}_{s,t}^X(Y)\\big)-\\bar{h}(X)\\,|\\,{\\mathcal F}_s}\\Big|\\lesssim |h|_\\infty\\Big(1+\\big\\|\\varepsilon^{-\\hat{H}}\\bar{\\hat{B}}_{\\varepsilon\\cdot}^s\\big\\|_{\\Omega_\\zeta}\\Big)\\big(1+|Y|\\big)\\left(1\\wedge\\frac{\\varepsilon^{\\frac{\\zeta}{3}}}{|t-s|^{\\frac{\\zeta}{3}}}\\right).\n  \\end{equation*}\n  \\end{enumerate}\n  Here, as usual, $\\bar{h}(x)=\\int_{\\ensuremath{\\mathbb{R}}^n} h(x,y)\\,\\pi^x(dy)$.\n\\end{corollary}\n\n\n\n\n\\subsection{Geometric Ergodicity for SDEs Driven by Fractional Brownian Motion}\\label{sec:geometric_ergodicity}\n\nApplying the arguments of \\cref{prop:conditional_initial_condition_wasserstein,prop:conditional_stationary_total} to the equation \\begin{equation}\\label{eq:sde}\n  dY_t=b(Y_t)\\,dt+\\sigma\\,dB_t,   \n\\end{equation}   \nwe obtain an exponential rate of convergence improving the known results:\n\\begin{proof}[Proof of \\cref{thm:geometric}]\n In \\cref{conveergence-equilibrium} we have already proven the  Wasserstein decay \\eqref{eq:wasserstein_time_t}:\n    \\begin{equation*}\n        \\ensuremath{\\mathcal{W}}^p(\\mathcal{L}(Y_t),\\pi)\\leq Ce^{-ct}\\ensuremath{\\mathbb{W}}^p\\big(\\mu,\\pi\\big), \\qquad \\forall\\, t\\geq 0\n    \\end{equation*}\n    The total variation rate \\eqref{eq:tv_process} then follows by a similar Girsanov coupling as in the proof of \\cref{prop:conditional_stationary_total}. In fact, we now consider\n  \\begin{align*}\n    dX_s&=b(X_s)\\,ds+\\sigma dB_s,\\\\\n    dZ_s&=b(Z_s)\\,ds+\\sigma\\,dB_s +\\sigma\\varphi^t(s)\\,ds,\n  \\end{align*}\n  where $X$ is started in the generalized initial condition $\\mu$ and $Z$ is the stationary solution. Let us define\n  \\begin{equation*}\n    \\varphi^t(s)\\ensuremath\\triangleq -\\left(\\frac{4|X_t-Z_t|^{\\frac12}}{|X_s-Z_s|^{\\frac12}}+\\lambda\\right)\\sigma^{-1}(X_s-Z_s)\\mathbf 1_{[t,t+1]}(s).\n  \\end{equation*}\n  It can then be checked similarly to \\cref{lem:girsanov} that $X_{t+1}=Y_{t+1}$ and\n  \\begin{equation*}\n    |\\varphi^t|_\\infty\\lesssim |X_t-Z_t|,\\qquad |\\dot{\\varphi}^t|_\\infty\\lesssim |X_t-Z_t|^{\\frac12}+|X_t-Z_t|.\n  \\end{equation*}\n  Consequently, the estimate \\eqref{eq:tv_final} becomes\n  \\begin{equation}\\label{eq:geometric_time_t}\n    \\TV{\\L(Y_{t+1})-\\pi}\\lesssim\\ensuremath{\\mathbb{W}}^1\\big(\\mu,\\pi\\big)\\left(\\sqrt{\\delta}+\\frac{e^{-ct}}{\\delta}\\right)\n  \\end{equation}\n  and choosing $\\delta= e^{-\\frac{ct}{2}}$ shows a geometric decay of the total variation distance at a fixed time. To get asserted decay on the path space \\eqref{eq:tv_process}, we observe that, by the very same argument as in \\cite[Proposition 7.2 (iii)]{Panloup2020}, $\\varphi^t$ actually induces a coupling on the path space with a similar cost. Hence, $\\TV{\\L(Y_{t+\\cdot})-\\ensuremath\\mathbb{P}_\\pi}$ is still bounded by a quantity proportional to the right-hand side of \\eqref{eq:geometric_time_t} and \\eqref{eq:tv_process} follows at once. \n\\end{proof}\n\\begin{remark}\\label{rem:constant_xi}\n    The admissible repulsivity strength $\\Lambda(\\kappa,R,p)$ obtained in the proof of \\cref{thm:geometric} is certainly not optimal. We therefore abstain from deriving a quantitative upper bound. Let us however indicate one way to obtain such an estimate: Start from \\eqref{eq:quant_lower_bound} in the proof \\cref{lem:probabilistic_control} and recall a standard result (see e.g. \\cite[Theorem D.4]{Piterbarg2012}) saying that\n    \\begin{equation*}\n      \\ensuremath\\mathbb{P}\\big(|\\tilde{B}|_\\infty\\leq|\\sigma|^{-1}\\varepsilon\\big)\\geq 1-K\\big(|\\sigma|^{-1}\\varepsilon\\big)^{\\frac{1}{H}}e^{-H(|\\sigma|^{-1}\\varepsilon)^2}\n    \\end{equation*}\n    for a known numerical constant $K>0$. Finally optimize over all constants involved.\n\\end{remark}\n\nLet us finally sketch the main differences for a more general Gaussian driving noise $G$ in equation \\eqref{eq:sde}. We assume that $G$ has continuous sample paths and a moving average representation similar to \\eqref{eq:mandelbrot} with a kernel $\\mathfrak{G}:\\ensuremath{\\mathbb{R}}\\to\\Lin{n}$ which vanishes on $(-\\infty,0]$, is continuous on $(0,\\infty)$, and satisfies\n\\begin{equation*}\n\t\\int_{-\\infty}^t\\big|\\mathfrak{G}(t-u)-\\mathfrak{G}(-u)\\big|^2\\,du<\\infty\n\\end{equation*}\nfor each $t>0$. Then\n\\begin{equation*}\n\tG_t=\\int_{-\\infty}^t \\mathfrak{G}(t-u)-\\mathfrak{G}(-u)\\,dW_u,\\qquad t\\geq 0,\n\\end{equation*}\nhas the locally independent increment decomposition \n\\begin{equation*}\n\t\\big(\\theta_t G\\big)_h=\\int_{-\\infty}^t\\mathfrak{G}(t+h-u)-\\mathfrak{G}(t-u)\\,dW_u+\\int_t^{t+h}\\mathfrak{G}(t+h-u)\\,dW_u\\ensuremath\\triangleq\\bar{G}^t_h+\\tilde{G}^t_h\n\\end{equation*} \nwith respect to any compatible filtration. Moreover, we require that\n\\begin{equation*}\n  \\int_0^{\\delta} |\\mathfrak{G}(u)|\\,du>0\n\\end{equation*}\nfor each $\\delta>0$. We remark that (up to a time-shift) this is certainly implied by the assumptions of Panloup and Richard, see \\cite[Condition $\\boldsymbol{(\\mathrm{C}_2)}$]{Panloup2020}. As we have seen in \\cref{ex:titmarsh}, the Cameron-Martin space of $(\\tilde{G}_h)_{h\\in[0,1]}$ then densely embeds into $\\ensuremath{\\mathcal{C}}_0([0,1],\\ensuremath{\\mathbb{R}}^n)$. Thus \\cref{rem:Wasserstein-general} applies and we obtain a geometric rate in Wasserstein distance, provided that there is a stationary measure for the equation $dY_t=b(Y_t)\\,dt+\\sigma\\,dG_t$.\n\n\n\\section{The Fractional Averaging Principle}\\label{sec:feedback}\n\nLet us remind the reader of the setup of \\cref{thm:feedback_fractional}: We consider the slow-fast system\n\\begin{alignat}{4}\n  dX_t^\\varepsilon&=f(X_t^\\varepsilon,Y_t^\\varepsilon)\\,dt+g(X_t^\\varepsilon,Y_t^\\varepsilon)\\,dB_t,& \\qquad&X_0^\\varepsilon=X_0,\\label{eq:slow_feedback_sec}\\\\\n  dY_t^\\varepsilon&=\\frac{1}{\\varepsilon}b(X_t^\\varepsilon,Y_t^\\varepsilon)\\,dt+\\frac{1}{\\varepsilon^{\\hat{H}}}\\sigma\\,d\\hat{B}_t,&\\qquad &Y_0^\\varepsilon=Y_0,\\label{eq:fast_feedback_sec}\n\\end{alignat}\ndriven by independent $d$-dimensional and $n$-dimensional fractional Brownian motions $B$ and $\\hat{B}$ with Hurst parameters $H\\in(\\frac12,1)$ and $\\hat{H}\\in(1-H,1)$, respectively. We claim that $X_t^\\varepsilon$ converges to the solution of the na\\\"ively averaged equation \\eqref{eq:effective_dynamics} as $\\varepsilon\\to 0$. \n\nLet us also introduce the following filtrations for later reference:\n\\begin{equation*}\n  {\\mathcal G}_t\\ensuremath\\triangleq\\sigma(B_s,s\\leq t),\\quad\\hat{{\\mathcal G}}_t\\ensuremath\\triangleq\\sigma(\\hat{B}_s,s\\leq t),\\quad {\\mathcal F}_t\\ensuremath\\triangleq{\\mathcal G}_t\\vee\\hat{{\\mathcal G}}_t.\n\\end{equation*}\nTo be utterly precise, we actually use the right-continuous completion of ${\\mathcal F}$ in order to ensure that hitting time of an open sets by a continuous, adapted process is a stopping time. Observe that ${\\mathcal F}$ is compatible with the fBm $\\hat{B}$, see \\cref{sec:increment}.\n\nWe shall first convince ourselves that, under the conditions of \\cref{thm:feedback_fractional}, the pathwise solution of the  slow-fast system \\eqref{eq:slow_feedback_sec}--\\eqref{eq:fast_feedback_sec} exists globally. If the drift vector field $b:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}^n$ in \\eqref{eq:fast_feedback_sec} were \\emph{globally} Lipschitz continuous, this would be an easy consequence of the standard Young bound \\cite{Young1936}:\n\\begin{equation}\\label{eq:young}\n  \\left|\\int_s^t f_r\\,d\\ensuremath{\\mathfrak{h}}_r\\right|\\lesssim |f|_{\\ensuremath{\\mathcal{C}}^\\beta}|\\ensuremath{\\mathfrak{h}}|_{\\ensuremath{\\mathcal{C}}^\\alpha}|t-s|^{\\alpha+\\beta}+|f_s||\\ensuremath{\\mathfrak{h}}|_{\\ensuremath{\\mathcal{C}}^\\alpha}|t-s|^\\alpha,\n\\end{equation}\nprovided that $\\alpha+\\beta>1$. We shall also prove a bound on the moments of the H\\\"older norm of the solution for any fixed scale $\\varepsilon$. The main technical estimates in the proof of \\cref{thm:feedback_fractional} are delegated to \\cref{sec:uniform_bounds}, allowing us to easily conclude the argument in \\cref{sec:proof} by appealing to L\\^e's stochastic sewing lemma \\cite{Le2020}. \n\n\\subsection{A Solution Theory for the Slow-Fast System}\\label{sec:solution_theory}\n\n\nWe shall begin with a deterministic (pathwise) existence and uniqueness result. Fix a terminal time $T>0$ and let $\\ensuremath{\\mathfrak{h}}=(\\ensuremath{\\mathfrak{h}}^1,\\ensuremath{\\mathfrak{h}}^2)\\in\\ensuremath{\\mathcal{C}}^{\\alpha_1}([0,T],\\ensuremath{\\mathbb{R}}^{m})\\times\\ensuremath{\\mathcal{C}}^{\\alpha_2}([0,T],\\ensuremath{\\mathbb{R}}^{n})$, where $\\alpha_1>\\frac12$ and $\\alpha_2>1-\\alpha_1$. We consider the Young differential equation\n\\begin{equation}\\label{eq:ode}\n  z(t)=\\begin{pmatrix}z^1(t)\\\\z^2(t)\\end{pmatrix}=z_0+\\int_0^t \\begin{pmatrix}F_1\\big(z(s)\\big)\\{\\mathcal F}_2\\big(z(s)\\big)\\end{pmatrix}\\,ds+\\int_0^t G\\big(z(s)\\big)\\,d\\ensuremath{\\mathfrak{h}}_s.\n\\end{equation}\nWe impose the following assumptions on the data:\n\n\\begin{condition}\\label{cond:data_ode}\n\\leavevmode\n\\begin{enumerate}\n  \\item\\label{it:cond_ode_1} $F_1:\\ensuremath{\\mathbb{R}}^{d}\\times\\ensuremath{\\mathbb{R}}^{n}\\to\\ensuremath{\\mathbb{R}}^{d}$ is bounded and globally Lipschitz continuous.\n  \\item\\label{it:cond_ode_2} $F_2:\\ensuremath{\\mathbb{R}}^{d}\\times\\ensuremath{\\mathbb{R}}^{n}\\to\\ensuremath{\\mathbb{R}}^{n}$ is locally Lipschitz continuous and of linear growth, that is, $|F_2(z,x)|\\lesssim 1+|x|+|z|$ for all $x\\in\\ensuremath{\\mathbb{R}}^n$ and $z\\in\\ensuremath{\\mathbb{R}}^d$. Moreover, there are $\\kappa,D>0$ such that\n  \\begin{equation*}\n  \\Braket{F_2(z, x)-F_2(z,y),x-y}\\leq D- \\kappa|x-y|^2 \\qquad \\forall\\,   x,y\\in\\ensuremath{\\mathbb{R}}^n, \\forall \\,z\\in \\ensuremath{\\mathbb{R}}^d.\n\\end{equation*}\n  \\item\\label{it:cond_ode_3} $G:\\ensuremath{\\mathbb{R}}^{d}\\times\\ensuremath{\\mathbb{R}}^{n}\\to\\Lin[m+n]{d+n}$ is of the form $G=\\begin{pmatrix}G_1 & 0\\\\ 0 & G_2\\end{pmatrix}$ with $G_1\\in\\Cb{2}\\big(\\ensuremath{\\mathbb{R}}^{d}\\times\\ensuremath{\\mathbb{R}}^{n},\\Lin[m]{d}\\big)$ and $G_2\\in\\Lin{d}$ is constant.\n\\end{enumerate}\n\\end{condition}\n\nOur proof for the well-posedness of \\eqref{eq:ode} and the non-explosiveness is based on the following comparison lemma, versions of which will be of repeated use in the sequel:\n\\begin{lemma}\\label{lem:comparison}\n  Let $F_2:\\ensuremath{\\mathbb{R}}^{d}\\times\\ensuremath{\\mathbb{R}}^{n}\\to\\ensuremath{\\mathbb{R}}^{n}$ satisfy \\cref{cond:data_ode} \\ref{it:cond_ode_2} and let $\\varsigma\\in\\ensuremath{\\mathcal{C}}_0(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^{n})$, $z\\in\\ensuremath{\\mathcal{C}}(\\ensuremath{\\mathbb{R}}_+,\\ensuremath{\\mathbb{R}}^{d})$. \n  \\begin{enumerate}\n    \\item Then for any $x_0\\in\\ensuremath{\\mathbb{R}}^{n}$, there are unique global solutions to\n    \\begin{equation*}\n        x(t)=x_0+\\int_0^t F_2\\big(z(s),x(s)\\big)\\,ds+\\varsigma_t,\\qquad y(t)=x_0-\\int_0^ty(s)\\,ds+\\varsigma_t.\n      \\end{equation*}\n      Furthermore, on any finite time interval $[0,T]$, the difference of the solutions satisfies the bound\n      \\begin{equation}\\label{eq:solution_difference}\n        |x(t)-y(t)|^2\\lesssim\\int_0^t e^{-\\kappa(t-s)}\\big(1+|y(s)|+|z(s)|\\big)^2\\,ds\n      \\end{equation}\n      for all $t\\in[0,T]$. In particular,\n      \\begin{equation}\\label{eq:a_priori_sup}\n        |x|_\\infty\\lesssim 1+|x_0|+|\\varsigma|_\\infty+|z|_\\infty.\n      \\end{equation}\n\n      \\item  If, in addition, $\\varsigma\\in\\ensuremath{\\mathcal{C}}^\\alpha([0,T],\\ensuremath{\\mathbb{R}}^{n})$ for some $\\alpha>0$, then $x\\in\\ensuremath{\\mathcal{C}}^\\alpha([0,T],\\ensuremath{\\mathbb{R}}^{n})$ and the following bound holds:\n      \\begin{equation}\\label{eq:comparison_apriori} \n        |x|_{\\ensuremath{\\mathcal{C}}^{\\alpha}}\\lesssim 1+|x_0|+|z|_\\infty+|\\varsigma|_{\\ensuremath{\\mathcal{C}}^{\\alpha}}.\n      \\end{equation}\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}    \n  Since $F_2$ is locally Lipschitz, it is clear that uniqueness holds for the equation defining $x$. To see existence, first notice that $\\tilde{x}(t)\\ensuremath\\triangleq x(t)-\\varsigma_t$ solves\n  \\begin{equation*}\n    \\tilde{x}(t)=x_0+\\int_0^t F_2\\big(z(s),\\tilde{x}(s)+\\varsigma_s\\big)\\,ds.\n  \\end{equation*}\n  Set $\\Upsilon(s,x) =F_2\\big(z(s), x+\\varsigma_s\\big)$. This function is jointly continuous in $(s,x)$. Therefore, a local solution exists by the Carath\\'eodory theorem. \n\n  On the other hand, global existence and uniqueness of $y$ is standard. Consequently, the required non-explosion statement follows easily upon establishing \\eqref{eq:solution_difference}. To this end, we first observe that, for all $z\\in\\ensuremath{\\mathbb{R}}^{d}$ and all $x,y\\in\\ensuremath{\\mathbb{R}}^{n}$, the off-diagonal large scale contraction property and the linear growth of $F$ furnish the following bound:\n  \\begin{align*}\n    \\Braket{F_2(z,x)+y,x-y}&\\leq D-\\kappa|x-y|^2+\\<F_2(z,y)+y,x-y\\>\\\\\n    &\\leq D- \\frac{\\kappa}{2}|x-y|^2+\\frac{C}{\\kappa}\\big(1+|z|+|y|\\big)^2\n  \\end{align*}\n  for some uniform constant $C>0$, where we also used Young's inequality. Consequently, the function $h(t)\\ensuremath\\triangleq e^{\\kappa t}|x(t)-y(t)|^2$ satisfies \n  \\begin{equation*}\n    h^\\prime(t)\\lesssim e^{\\kappa t}\\big(1+|y(t)|+|z(t)|\\big)^2\n  \\end{equation*}\n  and \\eqref{eq:solution_difference} follows at once. \n\n  The bound \\eqref{eq:comparison_apriori} is an immediate consequence of \\eqref{eq:solution_difference} together with the fact that\n    \\begin{equation*}\n    |x|_{\\ensuremath{\\mathcal{C}}^{\\alpha}}\\lesssim|F_2(z,x)|_{\\infty}T^{1- \\alpha}+|\\varsigma|_{\\ensuremath{\\mathcal{C}}^{\\alpha}}\\lesssim \\big(1+|z|_\\infty+|x|_{\\infty}\\big) T^{1-\\alpha}+|\\varsigma|_{\\ensuremath{\\mathcal{C}}^{\\alpha}}.\\qedhere\n  \\end{equation*}\n\\end{proof}\n\nThe announced existence and uniqueness result for \\eqref{eq:ode} is as follows:\n\\begin{proposition}\\label{prop:abstract_ode}\n  Under \\cref{cond:data_ode}, for any $T>0$ and any $\\beta<\\alpha_1\\wedge\\alpha_2$, \\eqref{eq:ode} has a unique global solution in $\\ensuremath{\\mathcal{C}}^{\\beta}([0,T],\\ensuremath{\\mathbb{R}}^{d+n})$. \n\\end{proposition}\n\\begin{proof}\nOwing to \\cref{lem:comparison}, it is enough to derive an \\emph{a priori} bound on $|z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}$, $\\tilde{\\alpha}\\in[\\beta,\\alpha_1)$, to conclude with a standard Picard argument. \n\nLet $\\delta\\in(0,1)$. By the Young bound \\eqref{eq:young}, we see that\n  \\begin{align*}\n    |z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}&\\lesssim |F_1|_\\infty\\delta^{1-\\tilde{\\alpha}}+\\big(\\big|G_1(z^1,z^2)\\big|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}\\wedge\\alpha_2}}+|G_1|_\\infty\\big)|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}\\\\\n    &\\lesssim\\big(1+|z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}+|z^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}}\\big)\\big(1+|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\alpha_1}}\\big)\\delta^{\\alpha_1-\\tilde{\\alpha}},\n  \\end{align*}\n  where the prefactor is proportional to $M\\ensuremath\\triangleq|F_1|_\\infty+|G|_\\infty+\\Lip{G}$. We may apply \\cref{lem:comparison} to $z^2$ to further find\n \\begin{equation*}\n    |z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}\\lesssim\n    \\big(1+|z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}+|z_0|+|\\ensuremath{\\mathfrak{h}}^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}}\\big)\\big(1+|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\alpha_1}}\\big)\\delta^{\\alpha_1-\\tilde{\\alpha}}.\n  \\end{equation*}\n  Here, we take the H\\\"older norms of $z^1,z^2$ over the interval $[0,\\delta]$, whereas we use the full interval $[0,T]$ for $\\ensuremath{\\mathfrak{h}}^1$ and $\\ensuremath{\\mathfrak{h}}^2$. For $\\delta>0$ small enough, we therefore get\n  \\begin{equation}\\label{eq:iteration}\n    |z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}([0,\\delta])}\\lesssim\\big(1+|z_0|+|\\ensuremath{\\mathfrak{h}}^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}}\\big)\\big(1+|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\alpha_1}}\\big).\n  \\end{equation}\n  Combining this with \\cref{lem:comparison}, we can find a constant $C>0$ such that\n  \\begin{equation*}\n    |z(\\delta)|\\leq|z_0|+|z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}([0,\\delta])}+|z^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}([0,\\delta])}\\leq C\\big(1+|z_0|+|\\ensuremath{\\mathfrak{h}}^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}}\\big)\\big(1+|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\alpha_1}}\\big).\n  \\end{equation*}\n  This bound can now be easily iterated and together with \\eqref{eq:iteration} we see that there is a (increased) constant $C$ such that\n  \\begin{equation*}\n    |z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}([t,t+\\delta])}\\lesssim\\big(1+|z_t|+|\\ensuremath{\\mathfrak{h}}^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}}\\big)\\big(1+|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\alpha_1}}\\big)\\leq C^{\\left[\\frac{t}{\\delta}\\right]+1}\\big(1+|z_0|+|\\ensuremath{\\mathfrak{h}}^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}}\\big)\\big(1+|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\alpha_1}}\\big)^{\\left[\\frac{t}{\\delta}\\right]+2}\n  \\end{equation*}\n  for each $t\\in[0,T-\\delta]$. Since $|\\cdot|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}([0,T])}\\leq 2\\delta^{\\tilde{\\alpha}-1}\\sup_t |\\cdot|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}([t,t+\\delta])}$, we get that\n  \\begin{equation}\\label{eq:a_priori}\n    |z^1|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}([0,T])}\\leq\\frac{2C^{\\left[\\frac{t}{\\delta}\\right]+1}}{\\delta^{1-\\tilde\\alpha}}\\big(1+|z_0|+|\\ensuremath{\\mathfrak{h}}^2|_{\\ensuremath{\\mathcal{C}}^{\\alpha_2}}\\big)\\big(1+|\\ensuremath{\\mathfrak{h}}^1|_{\\ensuremath{\\mathcal{C}}^{\\alpha_1}}\\big)^{\\left[\\frac{T}{\\delta}\\right]+2}.\n  \\end{equation}\n\n  Local existence and uniqueness of a solution to \\eqref{eq:ode} is a classical consequence of the Young bound. Indeed, if we define \n  \\begin{equation*}\n    A_\\delta\\ensuremath\\triangleq\\left\\{f\\in\\ensuremath{\\mathcal{C}}^{\\beta}([0,\\delta],\\ensuremath{\\mathbb{R}}^{d+n}):\\,f(0)=z_0\\text{ and }|f|_{\\ensuremath{\\mathcal{C}}^{\\beta}}\\leq 1\\right\\},\n  \\end{equation*}\n  then, for $\\delta>0$ small enough, the operator $\\mathcal{A}_\\delta: A_\\delta\\to A_\\delta$,\n  \\begin{equation*}\n    (\\mathcal{A}_\\delta z)(t)\\ensuremath\\triangleq z_0+\\int_0^t\\begin{pmatrix}F_1\\big(z(s)\\big)\\{\\mathcal F}_2\\big(z(s)\\big)\\end{pmatrix}\\,ds+\\int_0^t G\\big(z(s)\\big)\\,d\\ensuremath{\\mathfrak{h}}_s,\n  \\end{equation*}\n   is contracting on a complete metric space. Abbreviating $\\gamma\\ensuremath\\triangleq\\alpha_1\\wedge\\alpha_2$, this in turn follows from the well-known bounds\n  \\begin{align*}\n    \\left|\\int_0^\\cdot G\\big(z(s)\\big)\\,d\\ensuremath{\\mathfrak{h}}_s\\right|_{\\ensuremath{\\mathcal{C}}^{\\beta}}&\\lesssim(\\Lip{G}+|G|_\\infty)(|z|_{\\ensuremath{\\mathcal{C}}^{\\beta}}+1)|\\ensuremath{\\mathfrak{h}}|_{\\ensuremath{\\mathcal{C}}^{\\gamma}}\\delta^{\\gamma-\\beta},\\\\\n    \\left|\\int_0^\\cdot G\\big(z(s)\\big)-G\\big(\\bar{z}(s)\\big)\\,d\\ensuremath{\\mathfrak{h}}_s\\right|_{\\ensuremath{\\mathcal{C}}^{\\beta}}&\\lesssim (\\Lip{G}+\\Lip{DG})|\\ensuremath{\\mathfrak{h}}|_{\\ensuremath{\\mathcal{C}}^{\\gamma}}\\delta^{\\gamma-\\beta}|z-\\bar{z}|_{\\ensuremath{\\mathcal{C}}^{\\beta}},\\\\\n    \\left|\\int_0^\\cdot \\begin{pmatrix}F_1\\big(z(s)\\big)\\{\\mathcal F}_2\\big(z(s)\\big)\\end{pmatrix}\\,ds\\right|_{\\ensuremath{\\mathcal{C}}^{\\beta}}&\\leq \\big(|F_1|_{\\infty;\\,B_{\\delta^{\\beta}}(z_0)}+|F_2|_{\\infty;\\,B_{\\delta^{\\beta}}(z_0)}\\big)\\delta^{1-\\beta},\\\\\n    \\left|\\int_0^\\cdot \\begin{pmatrix}F_1\\big(z(s)\\big)-F_1\\big(\\bar{z}(s)\\big)\\{\\mathcal F}_2\\big(z(s)\\big)-F_2\\big(\\bar{z}(s)\\big)\\end{pmatrix}\\,ds\\right|_{\\ensuremath{\\mathcal{C}}^{\\beta}}&\\leq \\big(\\Lip{F_1}+\\Lip[B_{\\delta^{\\beta}}(z_0)]{F_2}\\big)\\delta|z-\\bar{z}|_{\\ensuremath{\\mathcal{C}}^{\\beta}}\n  \\end{align*}\n  for all $z,\\bar{z}\\in A_\\delta$, where $|\\cdot|_{\\infty;\\,A}$ and $\\Lip[A]{\\cdot}$ denote the respective norms of the function restricted to the set $A$. Here, we also used that $\\max\\big(|z-z_0|_\\infty,|\\bar z-z_0|_\\infty\\big)\\leq\\delta^\\beta$ since $z,\\bar{z}\\in A_\\delta$ by assumption. Consequently, there is a unique solution to \\eqref{eq:ode} in $\\ensuremath{\\mathcal{C}}^{\\beta}([0,\\delta],\\ensuremath{\\mathbb{R}}^{d+n})$. Global existence and uniqueness follow from the \\emph{a priori} estimates \\eqref{eq:comparison_apriori} and \\eqref{eq:a_priori} by a standard maximality argument.\n\\end{proof}\n\nWe now bring the randomness back in the picture. To this end, let $\\alpha>0$, $p\\geq 1$, and $T>0$. We define the space\n\\begin{equation*}\n   {\\mathcal B}_{\\alpha,p}([0,T],\\ensuremath{\\mathbb{R}}^d)\\ensuremath\\triangleq\\left\\{X:[0,T]\\times\\Omega\\to\\ensuremath{\\mathbb{R}}^d:\\,X\\text{ is }({\\mathcal F}_t)_{t\\in[0,T]}\\text{-adapted and }\\|X\\|_{{\\mathcal B}_{\\alpha,p}([0,T],\\ensuremath{\\mathbb{R}}^d)}<\\infty\\right\\},\n\\end{equation*} \nwhere we introduced the semi-norm\n\\begin{equation*}\n  \\|X\\|_{{\\mathcal B}_{\\alpha,p}([0,T],\\ensuremath{\\mathbb{R}}^d)}\\ensuremath\\triangleq\\sup_{s\\neq t\\in[0,T]}\\frac{\\|X_t-X_s\\|_{L^p}}{|t-s|^\\alpha}.\n\\end{equation*}\nIf the terminal time $T$ and the dimension $d$ are clear from the context, we shall also write ${\\mathcal B}_{\\alpha,p}$ for brevity. By Kolmogorov's continuity theorem, we have the continuous embeddings\n\\begin{equation}\\label{eq:embeddings}\n  L^p\\big(\\Omega,\\ensuremath{\\mathcal{C}}^{\\alpha+\\delta}([0,T],\\ensuremath{\\mathbb{R}}^d)\\big)\\hookrightarrow{\\mathcal B}_{\\alpha,p}([0,T],\\ensuremath{\\mathbb{R}}^d)\\hookrightarrow  L^p\\big(\\Omega,\\ensuremath{\\mathcal{C}}^{\\alpha-\\delta-\\frac1p}([0,T],\\ensuremath{\\mathbb{R}}^d)\\big)\n\\end{equation}\nfor any $\\delta>0$. Finally, let us also introduce the Besov-type space\n\\begin{align*}\n  W_0^{\\alpha,\\infty}([0,T],\\ensuremath{\\mathbb{R}}^d)&\\ensuremath\\triangleq \\big\\{f:[0,T]\\to\\ensuremath{\\mathbb{R}}^d:\\,|f|_{\\alpha,\\infty}<\\infty\\big\\},\\\\\n  |f|_{\\alpha,\\infty}&\\ensuremath\\triangleq\\sup_{t\\in[0,T]}\\left(|f(t)|+\\int_0^t\\frac{|f(t)-f(s)|}{|t-s|^{\\alpha+1}}\\,ds\\right).\n\\end{align*}\nNualart and R\\u{a}s\\c{c}anu proved the following classical result:\n\\begin{proposition}[{\\cite[Theorem 2.1.II]{Rascanu2002}}]\\label{prop:nualart}\n   Let $f:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}^d$ be bounded Lipschitz continuous and $g:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\Lin[m]{d}$ be of class $\\ensuremath{\\mathcal{C}}_b^2$. Let $(Y_t)_{t\\in[0,T]}$ be a stochastic process with sample paths in $\\ensuremath{\\mathcal{C}}^{\\gamma}([0,T],\\ensuremath{\\mathbb{R}}^n)$ for some $\\gamma>1-H$ and let $B$ be an fBm with Hurst parameter $H>\\frac12$. Then there is a unique global solution to the equation\n   \\begin{equation*}\n     X_t=X_0+\\int_0^t f(X_s,Y_s)\\,ds+\\int_0^tg(X_s,Y_s)\\,dB_s\n   \\end{equation*}\n   and, provided that $X_0\\in L^\\infty$, we also have that\n   \\begin{equation*}\n     |X|_{\\alpha,\\infty}\\in\\bigcap_{p\\geq 1} L^p\n   \\end{equation*}\n   for each $\\alpha<\\frac12\\wedge\\gamma$.\n\\end{proposition} \n\n\\begin{corollary}\\label{cor:norm_bound_solution}\n  Fix the scale parameter $\\varepsilon>0$ and a terminal time $T>0$. Let $\\alpha<H\\wedge\\hat{H}$. There is a unique pathwise solution $(X^\\varepsilon,Y^\\varepsilon)\\in\\ensuremath{\\mathcal{C}}^{\\alpha}([0,T],\\ensuremath{\\mathbb{R}}^{d+n})$ to the slow-fast system \\eqref{eq:slow_feedback_sec}--\\eqref{eq:fast_feedback_sec}. Moreover, for any $p\\geq 1$ and any $\\beta<\\frac12\\wedge\\hat{H}$, we have that\n  \\begin{equation}\\label{eq:b_norm_bound}\n   \\|X^\\varepsilon\\|_{{\\mathcal B}_{\\beta,p}}<\\infty.\n  \\end{equation}\n\\end{corollary}\n\\begin{proof}\n  The first part is an immediate consequence of \\cref{prop:abstract_ode}. We stress that the bound \\eqref{eq:b_norm_bound} does not follow from our \\emph{a priori} estimate \\eqref{eq:a_priori} since, by Fernique's theorem, \n  \\begin{equation*}\n    \\Expec{\\exp\\left(a|B|_{\\ensuremath{\\mathcal{C}}^\\beta}^2\\right)}<\\infty\n  \\end{equation*} \n  if and only if $a>0$ is sufficiently small. Instead, we employ \\cref{prop:nualart}: Since $Y^\\varepsilon\\in\\ensuremath{\\mathcal{C}}^{\\hat H-}([0,T],\\ensuremath{\\mathbb{R}}^n)$ by \\cref{lem:comparison}, we see that, for each $\\alpha<\\frac12\\wedge\\hat H$, $|X^\\varepsilon|_{\\alpha,\\infty}\\in\\bigcap_{p\\geq 1} L^p$,  It is clear that\n\\begin{equation*}\n  W_0^{\\alpha,\\infty}([0,T],\\ensuremath{\\mathbb{R}}^d)\\hookrightarrow \\ensuremath{\\mathcal{C}}^{\\alpha-\\delta}([0,T],\\ensuremath{\\mathbb{R}}^d)\n\\end{equation*}\nfor any $\\delta>0$.  Combine this with the continuous embedding \\eqref{eq:embeddings} to conclude \\eqref{eq:b_norm_bound}.\n\\end{proof}\n\n\\begin{remark}\n  We finally record that \\cref{prop:abstract_ode,cor:norm_bound_solution} are the only places in the proof of \\cref{thm:feedback_fractional} which require a linear growth of the drift $b$, see \\cref{cond:feedback}. In fact, the remainder of the argument would still work, \\emph{mutatis mutandis}, under the weaker assumption of a polynomially growing drift, i.e., $|b(x,y)|\\lesssim 1+|x|^N+|y|^N$ for some $N\\in\\ensuremath{\\mathbb{N}}$. It is however unclear whether the solution to \\eqref{eq:slow_feedback_sec}--\\eqref{eq:fast_feedback_sec} exists globally in this case.\n\\end{remark}\n\n\n\n\\subsection{Uniform Bounds on the Slow Motions}\\label{sec:uniform_bounds}\n\nOur strategy in proving \\cref{thm:feedback_fractional} is as follows: The integrals in \\eqref{eq:slow_feedback_sec} are approximated by suitable Riemann sums, on which we then aim to establish uniform bounds. These estimates translate into bounds on the integrals in view of L\\^e's stochastic sewing lemma \\cite{Le2020}.\n\nFix a terminal time $T>0$ and let $\\mathcal{S}^p$ denote the set of adapted two-parameter processes on the simplex with finite $p^\\text{th}$ moments; in symbols:\n\\begin{equation*}\n  \\mathcal{S}^p\\ensuremath\\triangleq\\left\\{A:[0,T]^2\\times\\Omega\\to\\ensuremath{\\mathbb{R}}^d:\\,A_{s,t}=0\\text{ for }s\\geq t\\text{ and }A_{s,t}\\in L^p(\\Omega,{\\mathcal F}_t,\\ensuremath\\mathbb{P})\\text{ for all }s,t\\geq 0\\right\\}.\n\\end{equation*}\nGiven $\\eta,\\bar{\\eta}>0$, we define the spaces\n\\begin{align*}\n    H_\\eta^p&\\ensuremath\\triangleq\\left\\{A\\in\\mathcal{S}^p:\\,\\|A\\|_{H_\\eta^p}\\ensuremath\\triangleq\\sup_{0\\leq s<t\\leq T}\\frac{\\|A_{st}\\|_{L^p}}{|s-t|^\\eta}<\\infty\\right\\}, \\\\\n    \\bar{H}_{\\bar{\\eta}}^p&\\ensuremath\\triangleq\\left\\{A\\in\\mathcal{S}^p:\\,\\vertiii{A}_{\\bar{H}_{\\bar{\\eta}}^p}\\ensuremath\\triangleq\\sup_{0\\leq s<u<t\\leq T}\\frac{\\|\\expec{\\delta A_{sut}|\\mathcal{F}_s}\\|_{L^p}}{|s-t|^{\\bar{\\eta}}}<\\infty\\right\\},\n\\end{align*}\nwhere we have set $\\delta A_{s,u,t}\\ensuremath\\triangleq A_{s,t}-A_{s,u}-A_{u,t}$. With this notation we have the following version of the stochastic sewing lemma:\n\\begin{proposition}[{Stochastic Sewing Lemma \\cite[Theorem 2.1 and Propostion 2.7]{Le2020}}]\\label{prop:stochastic_sewing}\n    Let $p\\geq 2$, $\\eta>\\frac12$, and $\\bar{\\eta}>1$. Suppose that $A\\in H_\\eta^p\\cap\\bar{H}_{\\bar{\\eta}}^p$. Then, for every $0\\leq s\\leq t\\leq T$, the limit\n    \\begin{equation*}\n        I_{s,t}(A)\\ensuremath\\triangleq\\lim_{|P|\\to 0}\\sum_{[u,v]\\in P}A_{u,v}\n    \\end{equation*}\n    along partitions $P$ of $[s,t]$ with mesh $|P|\\ensuremath\\triangleq\\max_{[u,v]\\in P}|v-u|$ tending to zero exists in $L^p$. The limiting process $I(A)$ is additive in the sense that $I_{s,u}(A)+I_{u,t}(A)=I_{s,t}(A)$ for all $0\\leq s\\leq u\\leq t\\leq T$. Furthermore, there is a constant $C=C(p,\\eta,\\bar{\\eta})$ such that\n    \\begin{equation*}\n        \\|I_{s,t}(A)\\|_{L^p}\\leq C\\left(\\vertiii{A}_{\\bar{H}_{\\bar{\\eta}}^p}|t-s|^{\\bar{\\eta}}+\\|A\\|_{H_{\\eta}^p}|t-s|^\\eta\\right)\n    \\end{equation*}\n    for all $0\\leq s\\leq t\\leq T$. Moreover, if $\\|\\Expec{A_{s,t}\\,|\\,{\\mathcal F}_s}\\|_{L^p}\\lesssim|t-s|^{\\bar{\\eta}}$, then $I(A)\\equiv 0$.\n\\end{proposition}\nRecall our notation of the fast motion's flow from \\eqref{eq:general_flow} and \\eqref{eq:general_flow-fixed-x}, respectively. We are ultimately going to apply \\cref{prop:stochastic_sewing} with the two-parameter process\n\\begin{equation}\\label{eq:riemann_summands}\n  A_{s,t}^\\varepsilon\\ensuremath\\triangleq\\int_s^t \\left(g\\Big(X_s^\\varepsilon,\\bar{\\Phi}_{s,r}^{X_s^\\varepsilon}\\big(\\Phi_{0,s}^{X^\\varepsilon}(Y_0)\\big)\\Big)-\\bar{g}\\big(X_s^\\varepsilon\\big)\\right)\\,dB_r,\\quad 0\\leq s<t,\n\\end{equation}\nwhere, thanks to the conditional independence of the integrand and $\\sigma(B_r-B_s,r\\in[s,t])$ given ${\\mathcal F}_s$, the integral is well defined in the (mixed) Wiener-Young sense as detailed in \\cref{sec:integral_estimates} below. There we also show that the integral $I(A^\\varepsilon)$ constructed in \\cref{prop:stochastic_sewing} actually agrees with the Young integral in \\eqref{eq:slow_feedback_sec}. It will be clear that our bounds on $A^\\varepsilon$ also apply to the Riemann summands for the drift term in \\eqref{eq:slow_feedback_sec}, whence we exclude it from our considerations for now.\n\n\n\\subsubsection{A Priori Integral Estimates}\\label{sec:integral_estimates}\n\n\nWe will use the notion of \\emph{mixed Wiener-Young integrals}: If $F:(s,t]\\to\\Lin[m]{d}$ is a (sufficiently regular) random function \\emph{independent} of $(\\tilde{B}^s_r)_{r\\in[0,t-s]}$, we can make the definition\n\\begin{equation}\\label{eq:wiener_young}\n  \\int_s^t F_r\\,dB_r\\ensuremath\\triangleq\\int_s^t F_r\\dot{\\bar{B}}_{r-s}^s\\,dr+\\int_0^{t-s} F_{r+s}\\,d\\tilde{B}_r^s,\n\\end{equation}\nwhere the integral with respect to $\\tilde{B}^s$ is well defined in the Wiener sense (after all $\\tilde{B}^s$ is a Gaussian process). The H\\\"older norm of negative exponent $-\\kappa$, $\\kappa\\in[0,1]$, is defined by \n\\begin{equation*}\n  |F|_{-\\kappa}\\ensuremath\\triangleq\\sup_{u,v\\in(s,t]}\\frac{1}{|v-u|^{1-\\kappa}}\\left|\\int_u^v F_r\\,dr\\right|.\n\\end{equation*}\nNote that for $\\kappa=0$ we of course recover the usual sup-norm $|F|_\\infty$.\n\nIn terms of this norm, one can then prove the following fundamental estimate on \\eqref{eq:wiener_young}:\n\\begin{lemma}[{\\cite[Lemma 3.4]{Hairer2020}}]\\label{lem:wiener_integral_bound}\n  Let $2\\leq p<q$. Fix $\\kappa\\in\\big[0,H-\\frac12\\big)$ and $0\\leq s\\leq t\\leq T$. Suppose that $F:(s,t]\\to\\Lin[m]{d}$ is independent of $(\\tilde{B}^s_r)_{r\\in[0,t-s]}$ and $|F|_{-\\kappa}\\in L^q$. Then one has the bound\n  \\begin{equation*}\n    \\left\\|\\int_s^t F_r\\,dB_r\\right\\|_{L^p}\\lesssim \\||F|_{-\\kappa}\\|_{L^q}|t-s|^{H-\\kappa},\n  \\end{equation*}\n  where the prefactor is independent of $F$ and $0\\leq s\\leq t\\leq T$.\n\\end{lemma}\nWe also have the the following estimate, which is a simple consequence of \\cite[Lemmas 3.10 \\& 3.12]{Hairer2020}:\n\\begin{lemma}\\label{lem:stochastic_sewing_fbm}\n  Let $p\\geq 2$ and $\\alpha>1-H$. Let $X$ be an $({\\mathcal F}_t)_{t\\in[0,T]}$-adapted stochastic process with $\\alpha$-H\\\"older sample paths. Moreover assume that $X\\in{\\mathcal B}_{\\alpha,p}$. Let $f:\\ensuremath{\\mathbb{R}}^d\\to\\ensuremath{\\mathbb{R}}$ be a bounded Lipschitz continuous function. Then we have the following bound on the Young integral:\n  \\begin{equation*}\n    \\left\\|\\int_s^t f(X_r)\\,dB_r\\right\\|_{{\\mathcal B}_{H,p}}\\lesssim\\big(|f|_\\infty+\\Lip{f}\\big)\\big(1+\\|X\\|_{{\\mathcal B}_{\\alpha,p}}\\big),\n  \\end{equation*}\n  uniformly in $0\\leq s<t\\leq T$.\n\\end{lemma}\n\n\n\nIt is of course fundamental for our argument that the `integral' furnished by \\cref{prop:stochastic_sewing} indeed coincides with the Young integral. This is ensured by the two lemmas below.\n\n\\begin{lemma}\\label{lem:fast_process_moments}\n  Let $X=(X_t)_{t\\in[0,T]}$ be a continuous process with values in $\\ensuremath{\\mathbb{R}}^d$. Let $b:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}^n$ be of linear growth and satisfy\n  \\begin{equation*}\n   \t\\Braket{b(z, x)-b(z,y),x-y}\\leq D- \\kappa|x-y|^2 \\qquad\\forall\\, x,y\\in\\ensuremath{\\mathbb{R}}^n, \\forall \\,z\\in \\ensuremath{\\mathbb{R}}^d.\n  \\end{equation*} \n  Then, for any $p\\geq 2$ and any random variable $Y\\in L^p$, the following holds:\n  \\begin{equation*}\n\t\\sup_{\\varepsilon\\in (0,1]}\\sup_{0\\leq s\\leq t\\leq T}\\big\\|\\Phi^X_{s,t}(Y)\\big\\|_{L^p}\\lesssim 1+\\|Y\\|_{L^p}+ \\sup_{0\\leq t\\leq T} \\|X_t\\|_{L_p}.\n  \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n  It is clear that we can assume $s=0$ without loss of generality. Let $Z^\\varepsilon$ solve\n  \\begin{equation*}\n    Z_t^\\varepsilon=Y-\\frac{1}{\\varepsilon}\\int_0^t Z^\\varepsilon_s\\,ds+\\frac{1}{\\varepsilon^{\\hat{H}}}\\sigma\\hat{B}_t.\n  \\end{equation*}\nBy \\eqref{eq:solution_difference}, we have\n  \\begin{equation*}\n    \\big|\\Phi_{0,t}^X(Y)-Z_t^\\varepsilon\\big|^2\\lesssim\\int_0^{\\frac{t}{\\varepsilon}} e^{-\\kappa\\left(\\frac{t}{\\varepsilon}-s\\right)}\\big(1+|X_{\\varepsilon s}|+|Z_{\\varepsilon s}^\\varepsilon|\\big)^2\\,ds\n  \\end{equation*}\n  for all $t\\in[0,T]$. Since $(Z_{\\varepsilon h}^\\varepsilon)_{h\\geq 0}\\overset{d}{=}(Z_h^1)_{h\\geq 0}$ and $\\sup_{t\\geq 0}\\|Z^1_{t}\\|_{L^p}\\lesssim 1+\\|Y\\|_{L^p}$, the lemma follows at once.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:sewing_young}\n  Let $H>\\frac12$ and let $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n \\to\\ensuremath{\\mathbb{R}}$ be a Lipschitz continuous function. Let $p>2$ and $\\alpha>1-H$. Let $X$ be an $\\ensuremath{\\mathbb{R}}^d$-valued, $({\\mathcal F}_t)_{t\\in[0,T]}$-adapted process with $\\sup_{t\\in[0,T]}\\|X_t\\|_{L^p}<\\infty$ and sample paths in $\\ensuremath{\\mathcal{C}}^\\alpha([0,T],\\ensuremath{\\mathbb{R}}^d)$. Let $Y_0\\in L^p$. Define\n  \\begin{equation*}\n    A_{s,t}\\ensuremath\\triangleq\\int_s^t h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^{X}(Y_0)\\big)\\Big)\\,dB_r,\n  \\end{equation*}\n  where the integration is understood in the mixed Wiener-Young sense, see \\eqref{eq:wiener_young}. If $A\\in H_\\eta^2\\cap\\bar{H}_{\\bar{\\eta}}^2$ for some $\\eta>\\frac12$ and $\\bar{\\eta}>1$, then, for any $\\varepsilon>0$ and any $0\\leq s\\leq t\\leq T$,  \n  \\begin{equation*}\n  \\lim_{|P|\\to 0} \\sum_{[u,v]\\in P([s,t])} A_{u,v}=\\int_{s}^t h\\big(X_r,\\Phi_{0,r}^{X}(Y_0)\\big)\\,dB_r,\n  \\end{equation*} \n  where the right-hand side is the Young integral.\n\\end{lemma}\n\\begin{proof}\n  We first note that, by \\cref{lem:comparison}, the process $\\Phi_{0,\\cdot}^X(Y_0)$ takes values in $\\ensuremath{\\mathcal{C}}^\\beta([0,T],\\ensuremath{\\mathbb{R}}^d)$ for any $\\beta<\\hat{H}$. The pathwise Young integral $\\int h\\big(X_r,\\Phi_{0,r}^{X}(Y_0)\\big)\\,dB_r$ is thus well defined and is given by the limit of the Riemann sums of\n  \\begin{equation*}\n    \\tilde{A}_{s,t}\\ensuremath\\triangleq h\\big(X_s,\\Phi_{0,s}^X(Y_0)\\big)(B_t-B_s)\n  \\end{equation*}\n  along any sequence of partitions. By the last part of \\cref{prop:stochastic_sewing}, it now suffices to show that $\\|A_{s,t}-\\tilde{A}_{s,t}\\|_{L^2}\\lesssim |t-s|^{\\bar{\\eta}}$ for some $\\bar{\\eta}>1$. \n\n  To see this, we apply \\cref{lem:wiener_integral_bound} with $\\kappa=0$ to find that, for each $\\beta<\\hat{H}$,\n  \\begin{align*}\n    &\\phantom{\\leq}\\big\\|A_{s,t}-\\tilde{A}_{s,t}\\big\\|_{L^2}=\\left\\|\\int_s^t \\Big(h\\big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^{X}(Y_0)\\big)\\big)-h\\big(X_s,\\Phi_{0,s}^X(Y_0)\\big)\\Big)\\,dB_r\\right\\|_{L^2}\\\\\n    &\\leq\\Big\\|\\sup_{s\\leq r\\leq t}\\Big|h\\big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^{X}(Y_0)\\big)\\big)-h\\big(X_s,\\Phi_{0,s}^X(Y_0)\\big)\\Big|\\Big\\|_{L^p}|t-s|^H\\\\\n    &\\leq\\Lip{h}\\Big\\|\\Big|\\bar{\\Phi}_{s,\\cdot}^{X_s}\\big(\\Phi_{0,s}^{X}(Y_0)\\big)\\Big|_{\\ensuremath{\\mathcal{C}}^{\\beta}}\\Big\\|_{L^p}|t-s|^{H+\\beta}.\n  \\end{align*}\n  Since $H+\\hat{H}>1$, we can conclude with \\cref{lem:comparison,lem:fast_process_moments}.\n\\end{proof}\nOur interest in \\cref{lem:sewing_young} is of course in applying it to the slow motion \\eqref{eq:slow_feedback_sec} and the Riemann summands $A^\\varepsilon_{s,t}$ defined in \\eqref{eq:riemann_summands}. We have already seen in \\cref{cor:norm_bound_solution} that $X^\\varepsilon\\in\\bigcap_{p\\geq 1}{\\mathcal B}_{\\alpha,p}$ for any $\\alpha<\\frac12\\wedge\\hat{H}$. We are therefore left to check that $A^\\varepsilon\\in H_\\eta^p\\cap\\bar{H}_{\\bar{\\eta}}^p$ for some $\\eta>\\frac12$, $\\bar{\\eta}>1$, and $p\\geq 2$. Since these estimates are somewhat technically involved and require longer computations, we devote a subsection to each of the norms $\\|\\cdot\\|_{H_{\\eta}^p}$ and $\\vertiii{\\;\\cdot\\;}_{\\bar{H}_{\\bar{\\eta}}^p}$, respectively. \n\n\n\\subsubsection{Controlling the Increment $A^\\varepsilon_{s,t}$}\\label{sec:sewing_1}\n\nLet $h:\\ensuremath{\\mathbb{R}}^d\\times \\ensuremath{\\mathbb{R}}^n\\to \\ensuremath{\\mathbb{R}}^d$. Recall that write $\\bar h(x)=\\int h(x,y) \\pi^x(dy)$ for its average with respect to the first marginal of the invariant measure of the process $\\bar \\Phi^x$, see \\eqref{eq:general_flow-fixed-x} and \\cref{initial-condition}.\nThe following lemma exploits the convergence rates derived in \\cref{sec:convergence}. [The reader should observe that without further notice we assume that the conditions of \\cref{thm:feedback_fractional} on the drift $b:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}^n$ are in place.]\n\\begin{lemma}\\label{lem:sewing_helper_1}\n  Let $q>1$. Let $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be a bounded measurable function and let $X,Y\\in L^q$ be ${\\mathcal F}_s$-measurable random variables. Then, for any $0\\leq s\\leq t$, any $p\\geq 2$, and any $\\zeta<1-\\hat{H}$, we have that\n  \\begin{equation*}\n    \\left\\|\\int_s^t \\Big( h\\big(X,\\bar{\\Phi}_{s,r}^X(Y)\\big)-\\bar{h}(X)\\Big) \\,dr\\right\\|_{L^p}\\lesssim |h|_\\infty\\Big(1+\\|Y\\|_{L^q}^{\\frac{1}{p}}+\\|X\\|_{L^q}^{\\frac{1}{p}}\\Big)\\varepsilon^{\\frac{\\zeta}{3p}}|t-s|^{1-\\frac{\\zeta}{3p}}.\n  \\end{equation*}\n\\end{lemma}\n\\begin{proof}\nThere is no loss of generality in assuming that $\\bar{h}\\equiv 0$. Notice also that the trivial estimate $\\big\\|\\int_s^t h\\big(X,\\bar{\\Phi}_{s,r}^X(Y)\\big)\\,dr\\big\\|_{L^\\infty}\\leq|h|_\\infty|t-s|$. By interpolation, we can therefore restrict ourselves to the case $p=2$. Clearly,\n  \\begin{equation*}\n    \\Expec{\\left|\\int_s^t h\\big(X,\\bar{\\Phi}_{s,r}^X(Y)\\big)\\,dr\\right|^2}=2\\int_s^t\\int_s^v\\Expec{h\\big(X,\\bar{\\Phi}_{s,r}^X(Y)\\big)h\\big(X,\\bar{\\Phi}_{s,v}^X(Y)\\big)}\\,dr\\,dv.\n  \\end{equation*}\n  For $r<v$ we condition the integrand on ${\\mathcal F}_r$, and use \\cref{cor:total_variation_conditional} \\ref{it:ergodicity_tv} together with \\cref{lem:smooth_part_decay,lem:fast_process_moments} to find\n  \\begin{align*}\n    &\\phantom{\\lesssim}\\Big|\\Expec{h\\big(X,\\bar{\\Phi}_{s,r}^X(Y)\\big)h\\big(X,\\bar{\\Phi}_{s,v}^X(Y)\\big)}\\Big|\\leq |h|_\\infty\\,\\Expec{\\,\\Big|\\Expec{h\\big(X,\\bar{\\Phi}_{s,v}^X(Y)\\big)\\,\\middle|\\,{\\mathcal F}_r}\\Big|\\,}\\\\\n    &\\lesssim |h|_\\infty^2\\,\\Expec{\\Big(1+\\|\\varepsilon^{-\\hat{H}}\\bar{\\hat{B}}_{\\varepsilon\\cdot}^r\\|_{\\Omega_\\zeta}\\Big)\\Big(1+\\big|\\bar{\\Phi}_{s,r}^X(Y)\\big|\\Big)}\\left(\\frac{\\varepsilon}{v-r}\\right)^{\\frac{\\zeta}{3}}\\\\\n    &\\lesssim |h|_\\infty^2 \\big(1+\\|Y\\|_{L^q}+\\|X\\|_{L^q}\\big) \\left(\\frac{\\varepsilon}{v-r}\\right)^{\\frac{\\zeta}{3}}. \\qedhere\n  \\end{align*}\n\\end{proof}\n\nWe can now establish the required estimate on the $H_\\eta^p$-norm of $A_{s,t}$.\n\\begin{proposition}\\label{prop:sewing_1}\n   Let $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be bounded measurable and let $X$ be an $({\\mathcal F}_t)_{t\\in[0,T]}$-adapted, continuous process with $\\sup_{t\\in[0,T]}\\|X_t\\|_{L^q}<\\infty$ for some $q\\geq 1$. Define \n  \\begin{equation*}\n    A_{s,t}\\ensuremath\\triangleq\\int_s^t  \\left[h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)-\\bar{h}(X_s)\\right]\\,dB_r,\\quad 0\\leq s\\leq t\\leq T,\n  \\end{equation*}\n  in the mixed Wiener-Young sense, see \\eqref{eq:wiener_young}. Let $\\kappa\\in(0,H-\\frac12)$ and set $\\eta=H-\\kappa$. Then $A\\in H_\\eta^p$ for each $p\\geq 2$, and any $\\varepsilon>0$. Moreover, there is a $\\gamma>0$ such that \n  \\begin{equation*}\n    \\|A\\|_{H_{\\eta}^p}\\lesssim |h|_\\infty\\Big(1+\\sup_{0\\leq t\\leq T}\\|X_t\\|_{L^q}\\Big)\\varepsilon^{\\gamma}.\n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n  Again, we may assume that $\\bar{h}\\equiv 0$ without any loss of generality. Since $X$ is $({\\mathcal F}_t)_{t\\in[0,T]}$-adapted, we can use \\cref{lem:wiener_integral_bound} to obtain that, for $\\tilde{q}>p$ and $\\kappa\\in[0,H-\\frac12)$,\n  \\begin{equation*}\n    \\|A_{s,t}\\|_{L^p}\\lesssim\\left\\|\\left|h\\Big(X_s,\\bar{\\Phi}_{s,\\cdot}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)\\right|_{-\\kappa}\\right\\|_{L^{\\tilde{q}}}|t-s|^{H-\\kappa}.\n  \\end{equation*}\n By \\cref{lem:sewing_helper_1,lem:fast_process_moments}, we obtain \n    \\begin{equation*}\n \\left\\|\\int_u^v h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)\\,dr\\right\\|_{L^{\\tilde{q}}}\\lesssim |h|_\\infty\\bigg(1+\\|Y_0\\|^{\\frac{1}{\\tilde{q}}}_{L^q}+\\sup_{0\\leq r\\leq s}\\|X_r\\|_{L^q}^{\\frac{1}{\\tilde{q}}}\\bigg)\\varepsilon^{\\frac{\\zeta}{3\\tilde{q}}}|v-u|^{1-\\frac{\\zeta}{3\\tilde{q}}}\n  \\end{equation*}\n  for all $u,v\\in[s,t]$ and any $\\zeta<1-\\hat{H}$. Therefore, Kolmogorov's continuity theorem shows that\n  \\begin{equation*}\n    \\left\\|\\left|h\\Big(X_s,\\bar{\\Phi}_{s,\\cdot}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)\\right|_{-\\kappa}\\right\\|_{L^{\\tilde{q}}}\\lesssim |h|_\\infty\\bigg(1+\\|Y_0\\|^{\\frac{1}{\\tilde{q}}}_{L^q}+\\sup_{0\\leq t\\leq T}\\|X_t\\|_{L^q}^{\\frac{1}{\\tilde{q}}}\\bigg)\\varepsilon^{\\frac{\\zeta}{3\\tilde{q}}},\n  \\end{equation*}\n  provided that we choose $\\tilde{q}>\\kappa^{-1}\\left(1+\\frac{\\zeta}{3}\\right)$, and the final result follows.\n  \\end{proof}\n\n\n\\subsubsection{Continuity of the Invariant Measures}\nLet $\\varepsilon>0$ and $s<t$. We write \n\\begin{equation*}\n   P([s,t];\\varepsilon)\\ensuremath\\triangleq\\begin{cases}\n   \\left\\{s+k\\varepsilon:\\,k=0,\\dots,\\left[\\frac{t-s}{\\varepsilon}\\right]\\right\\}\\cup\\{t\\},& t-s-\\varepsilon\\left[\\frac{t-s}{\\varepsilon}\\right]\\geq\\frac{\\varepsilon}{2},\\\\\n   \\left\\{s+k\\varepsilon:\\,k=0,\\dots,\\left[\\frac{t-s}{\\varepsilon}\\right]-1\\right\\}\\cup\\{t-\\frac{\\varepsilon}{2},t\\},& t-s-\\varepsilon\\left[\\frac{t-s}{\\varepsilon}\\right]<\\frac{\\varepsilon}{2}.\n   \\end{cases}\n\\end{equation*} \nNotice that the distance between two subsequent points $(t_i,t_{i+1})\\in P([s,t];\\varepsilon)$ satisfies $|t_{i+1}-t_i|\\in[\\frac{\\varepsilon}{2},\\varepsilon]$. Recall from \\cref{cond:feedback} that the drift $b$ is assumed to be locally Lipschitz uniformly with respect to the second argument. We write\n\\begin{equation*}\n  \\Lip[K]{b}\\ensuremath\\triangleq\\sup_{\\substack{|x_1|,|x_2|\\leq K\\\\y\\in\\ensuremath{\\mathbb{R}}^n}}|b(x_1,y)-b(x_2,y)|\n\\end{equation*}\nfor $K>0$. In order to keep the statements of the next lemmas concise, we shall freely absorb quantities independent of $0\\leq s\\leq t$ and $\\varepsilon\\in(0,1]$ into the prefactor hidden beneath $\\lesssim$.\n\n\\begin{lemma}\\label{lem:continuity}\n  Let $p\\geq 1$ and suppose that $b(x,\\cdot)\\in\\S_p(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$. Let $X,\\bar{X}\\in L^\\infty$, and $Y\\in L^p$ be ${\\mathcal F}_s$-measurable random variables. Then\n  \\begin{equation*}\n    \\left\\|\\bar{\\Phi}_{s,t}^X(Y)-\\bar{\\Phi}^{\\bar{X}}_{s,t}(Y)\\right\\|_{L^p}\\lesssim\\|X-\\bar{X}\\|_{L^{p}}.\n  \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n  We abbreviate $\\Lambda\\ensuremath\\triangleq\\Lambda(\\kappa,R,p)$ and observe that, for any $s\\leq u\\leq r$,\n  \\begin{align*}\n    \\frac{d}{dr}\\Big|\\bar{\\Phi}_{u,r}^{X}(Y)-\\bar{\\Phi}_{u,r}^{\\bar{X}}(Y)\\Big|^2&=\\frac{2}{\\varepsilon}\\Braket{b\\big(X,\\bar{\\Phi}_{u,r}^{X}(Y)\\big)-b\\big(\\bar{X},\\bar{\\Phi}_{u,r}^{\\bar{X}}(Y)\\big),\\bar{\\Phi}_{u,r}^{X}(Y)-\\bar{\\Phi}_{u,r}^{\\bar{X}}(Y)}\\\\\n    &\\leq \\frac{2(\\Lambda+1)}{\\varepsilon}\\Big|\\bar{\\Phi}_{u,r}^{X}(Y)-\\bar{\\Phi}_{u,r}^{\\bar{X}}(Y)\\Big|^2+ \\frac{\\Lip[\\|X\\|_{L^\\infty}\\vee\\|\\bar{X}\\|_{L^\\infty}]{b}^2}{2\\varepsilon}|X-\\bar{X}|^2\n  \\end{align*}\n  with probability $1$. It follows that\n  \\begin{equation}\\label{eq:cont_interpolate_1}\n    \\Big|\\bar{\\Phi}_{u,r}^X(Y)-\\bar{\\Phi}^{\\bar{X}}_{u,r}(Y)\\Big|\\lesssim\\Lip[\\|X\\|_{L^\\infty}\\vee\\|\\bar{X}\\|_{L^\\infty}]{b}e^{(\\Lambda+1)\\frac{|r-u|}{\\varepsilon}}|X-\\bar{X}|.\n  \\end{equation}\n  This bound is of course only useful on a time interval with length of order $\\varepsilon$. We therefore expand\n  \\begin{equation*}\n    \\left\\|\\bar{\\Phi}_{s,t}^X(Y)-\\bar{\\Phi}^{\\bar{X}}_{s,t}(Y)\\right\\|_{L^p}\\leq\\sum_{(t_i,t_{i+1})\\in P([s,t];\\varepsilon)}\\left\\|\\bar{\\Phi}_{t_{i+1},t}^{\\bar{X}}\\big(\\bar{\\Phi}_{s,t_{i+1}}^X(Y)\\big)-\\bar{\\Phi}_{t_{i},t}^{\\bar{X}}\\big(\\bar{\\Phi}_{s,t_{i}}^X(Y)\\big)\\right\\|_{L^p}.\n  \\end{equation*}\n  \\Cref{cor:fast_different_initial} shows that\n  \\begin{align*}\n    \\left\\|\\bar{\\Phi}_{t_{i+1},t}^{\\bar{X}}\\big(\\bar{\\Phi}_{s,t_{i+1}}^X(Y)\\big)-\\bar{\\Phi}_{t_{i},t}^{\\bar{X}}\\big(\\bar{\\Phi}_{s,t_{i}}^X(Y)\\big)\\right\\|_{L^p}&\\lesssim\\Big\\|\\bar{\\Phi}_{s,t_{i+1}}^X(Y)-\\bar{\\Phi}_{t_{i},t_{i+1}}^{\\bar{X}}\\big(\\bar{\\Phi}_{s,t_{i}}^X(Y)\\big)\\Big\\|_{L^p} e^{-c\\frac{|t-t_{i+1}|}{\\varepsilon}}\\\\\n    & \\lesssim \\|X-\\bar{X}\\|_{L^p}e^{-c\\frac{|t-t_{i+1}|}{\\varepsilon}},\n  \\end{align*}\n  where the last inequality uses \\eqref{eq:cont_interpolate_1} together with $|t_{i+1}-t_i|\\asymp\\varepsilon$. Consequently,\n  \\begin{equation*}\n    \\left\\|\\bar{\\Phi}_{s,t}^X(Y)-\\bar{\\Phi}^{\\bar{X}}_{s,t}(Y)\\right\\|_{L^p}\\lesssim\\|X-\\bar{X}\\|_{L^p}\\sum_{(t_i,t_{i+1})\\in P([s,t];\\varepsilon)}e^{-c\\frac{|t-t_{i+1}|}{\\varepsilon}}\\lesssim\\|X-\\bar{X}\\|_{L^p}\n  \\end{equation*}\n  uniformly in $0\\leq s\\leq t$ and $\\varepsilon\\in(0,1]$. \n\\end{proof}\n\n\\Cref{lem:continuity} implies the local Lipschitz continuity of the invariant measure $\\pi^x$ in the parameter $x\\in\\ensuremath{\\mathbb{R}}^d$:\n\\begin{proposition}\\label{lem:wasserstein_holder}\n  Let $p\\geq 1$ and $K>0$. Suppose that $b(x,\\cdot)\\in\\S_p(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$. Then\n  \\begin{equation*}\n    \\ensuremath{\\mathcal{W}}^p(\\pi^{x_1},\\pi^{x_2})\\lesssim |x_1-x_2|,\n  \\end{equation*}\n  uniformly for $|x_1|,|x_2|\\leq K$.\n\\end{proposition}\n\\begin{proof}\n  Owing to \\cref{thm:geometric},  it follows that\n  \\begin{equation*}\n     \\ensuremath{\\mathcal{W}}^p(\\pi^{x_1},\\pi^{x_2})\\leq\\limsup_{\\varepsilon\\to 0}\\big\\|\\bar{\\Phi}^{x_1}_{0,1}(0)-\\bar{\\Phi}^{x_2}_{0,1}(0)\\big\\|_{L^p}\n   \\end{equation*} \n   and we conclude with \\cref{lem:continuity}.\n\\end{proof}\n\nThe simple proof of the following corollary is left to the reader.\n\\begin{corollary}\\label{cor:lipschitz_average}\nLet $h:\\ensuremath{\\mathbb{R}}^d\\times \\ensuremath{\\mathbb{R}}^n\\to \\ensuremath{\\mathbb{R}}^d$ be Lipschitz continuous. Then $\\bar{h}:\\ensuremath{\\mathbb{R}}^d\\to\\ensuremath{\\mathbb{R}}^d$ is locally Lipschitz.\n\\end{corollary}\n\\subsubsection{Controlling the Second Order Increment $\\delta A^\\varepsilon_{s,u,t}$}\\label{sec:sewing_2}\n\nUniform bounds on the second order increments are difficult to obtain even for the Markovian fast dynamic. The first technical estimate of this subsection is the following:\n\\begin{lemma}\\label{lem:sewing_helper_2}\n  Let $1\\leq p<q$ and suppose that $b(x,\\cdot)\\in\\S_p(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$. Let $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be a Lipschitz continuous function with $\\bar{h}\\equiv 0$. Suppose that $X,\\bar{X}\\in L^\\infty$ and $Y\\in L^q$ are $\\mathcal{F}_s$-measurable random variables. Then, for any $\\rho\\in(0,1)$, there is a $\\gamma>0$ such that \n    \\begin{equation*}\n      \\Big\\|\\Expec{h\\big(X,\\bar{\\Phi}^{X}_{s,t}(Y)\\big)-h\\big(\\bar{X},\\bar{\\Phi}^{\\bar{X}}_{s,t}(Y)\\big)\\,\\middle|\\,\\mathcal{F}_s}\\Big\\|_{L^p}\\lesssim\\Lip{h}\\big(1+\\|Y\\|_{L^q}\\big)\\|X-\\bar{X}\\|_{L^{p}}^{\\rho}\\left(1\\wedge\\frac{\\varepsilon^\\gamma}{|t-s|^\\gamma}\\right).\n    \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n  By \\cref{cor:total_variation_conditional} \\ref{it:ergodicity_wasserstein} and H\\\"older's inequality, we certainly have\n  \\begin{equation}\\label{eq:sewing_helper_2_interpolate}\n    \\Big\\|\\Expec{h\\big(X,\\bar{\\Phi}^{X}_{s,t}(Y)\\big)-h\\big(\\bar{X},\\bar{\\Phi}^{\\bar{X}}_{s,t}(Y)\\big)\\,\\middle|\\,\\mathcal{F}_s}\\Big\\|_{L^p}\\lesssim\\Lip{h}\\big(1+\\|Y\\|_{L^q}\\big)\\left(1\\wedge\\frac{\\varepsilon^{\\zeta}}{|t-s|^{\\zeta}}\\right).\n  \\end{equation}\n  On the other hand, by the continuity lemma (\\cref{lem:continuity}),\n  \\begin{align*}\n    &\\phantom{\\lesssim}\\Big\\|\\Expec{h\\big(X,\\bar{\\Phi}^{X}_{s,t}(Y)\\big)-h\\big(\\bar{X},\\bar{\\Phi}^{\\bar{X}}_{s,t}(Y)\\big)\\,\\middle|\\,\\mathcal{F}_s}\\Big\\|_{L^p}\\\\\n    &\\lesssim\\Lip{h}\\left(\\|X-\\bar{X}\\|_{L^p}+\\Big\\|\\bar{\\Phi}^{X}_{s,t}(Y)-\\bar{\\Phi}^{\\bar{X}}_{s,t}(Y)\\Big\\|_{L^p}\\right)\\lesssim\\Lip{h}\\|X-\\bar{X}\\|_{L^{p}}.\n  \\end{align*}\n  Finally, we interpolate this bound with \\eqref{eq:sewing_helper_2_interpolate}.\n\\end{proof}\n\nOur remaining task is to derive an estimate on the distance between $\\Phi^Z_{s,t}$ and $\\bar{\\Phi}_{s,t}^{Z_s}$. This is based on the following version of \\cref{lem:continuity}:\n\\begin{lemma}\\label{lem:continuity_path}\n  Let $p\\geq 1$ and suppose that $b(x,\\cdot)\\in\\S_p(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$. Let $Y\\in L^p$ be ${\\mathcal F}_s$-measurable and $Z$ be a continuous process. Assume that $|Z|_\\infty\\in L^\\infty$. Then\n  \\begin{equation*}\n    \\Big\\|\\bar{\\Phi}^{Z_s}_{s,t}(Y)-\\Phi_{s,t}^Z(Y)\\Big\\|_{L^p}\\lesssim\\Big\\|\\sup_{r\\in[s,t]}|Z_r-Z_s|\\Big\\|_{L^{p}}\n  \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n  The reader can easily check that the very same argument we gave at the beginning of the proof of \\cref{lem:continuity} also shows that, for $0\\leq s\\leq u\\leq r\\leq T$,\n  \\begin{align*}\n    \\Big|\\bar{\\Phi}_{u,r}^{Z_s}(Y)-\\Phi_{u,r}^Z(Y)\\Big|&\\lesssim\\Lip[\\||Z|_{\\infty}\\|_{L^\\infty}]{b}\\left(\\int_{\\frac{u}{\\varepsilon}}^{\\frac{r}{\\varepsilon}} e^{2(\\Lambda+1)\\left(\\frac{r}{\\varepsilon}-v\\right)}|Z_{\\varepsilon v}-Z_s|^2\\,dv\\right)^{\\frac12}\\\\\n    &\\lesssim \\sup_{v\\in[u,r]}|Z_v-Z_s| e^{(\\Lambda+1)\\frac{|r-u|}{\\varepsilon}}.\n  \\end{align*}\n  The asserted bound then follows along the same lines as \\cref{lem:continuity}.\n\\end{proof}\n\nThe following estimate is now an easy consequence:\n\\begin{lemma}\\label{lem:sewing_helper_3}\n  Let $p\\geq 1$ and suppose that $b(x,\\cdot)\\in\\S_p(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$. Let $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be Lipschitz continuous. Assume furthermore that $X$ and $Y$ are ${\\mathcal F}_u$- and ${\\mathcal F}_s$-measurable random variables, respectively. Moreover, let $Z\\in{\\mathcal B}_{\\alpha, p}([0,T],\\ensuremath{\\mathbb{R}}^d)$ for some $\\alpha>0$ and assume that $|Z|_\\infty\\in L^\\infty$. Then\n    \\begin{equation}\\label{eq:sewing_moderately_3}\n      \\left\\|\\Expec{h\\Big(X,\\bar{\\Phi}^{X}_{u,t}\\big(\\bar{\\Phi}_{s,u}^{Z_s}(Y)\\big)\\Big)-h\\Big(X,\\bar{\\Phi}^{X}_{u,t}\\big(\\Phi_{s,u}^{Z}(Y)\\big)\\Big)\\,\\middle|\\,\\mathcal{F}_u}\\right\\|_{L^{p}}\\lesssim \\Lip{h}\\|Z\\|_{{\\mathcal B}_{\\alpha,p}}|u-s|^{\\alpha}e^{-c\\frac{|t-u|}{\\varepsilon}}.\n    \\end{equation} \n\\end{lemma}\n\\begin{proof}\n  By \\cref{cor:fast_different_initial}, we have that\n  \\begin{align*}\n    &\\phantom{\\lesssim}\\Big\\|\\Expec{h\\Big(X,\\bar{\\Phi}^{X}_{u,t}\\big(\\bar{\\Phi}_{s,u}^{Z_s}(Y)\\big)\\Big)-h\\Big(X,\\bar{\\Phi}^{X}_{u,t}\\big(\\Phi_{s,u}^{Z}(Y)\\big)\\Big)\\,\\middle|\\,\\mathcal{F}_u}\\Big\\|_{L^p}\\\\\n    &\\lesssim \\Lip{h}\\Big\\|\\bar{\\Phi}_{s,u}^{Z_s}(Y)-\\Phi_{s,u}^{Z}(Y)\\Big\\|_{L^p}e^{-c\\frac{|t-u|}{\\varepsilon}}.\n  \\end{align*}\n  By \\cref{lem:continuity_path},\n  \\begin{equation*}\n    \\Big\\|\\bar{\\Phi}_{s,u}^{Z_s}(Y)-\\Phi_{s,u}^{Z}(Y)\\Big\\|_{L^p}\\lesssim\\|Z\\|_{{\\mathcal B}_{\\alpha,p}}|u-s|^{\\alpha}.\\qedhere\n  \\end{equation*}\n\\end{proof}\n\nFinally, we can establish the second estimate needed for the application of \\cref{prop:stochastic_sewing}:\n\\begin{proposition}\\label{prop:sewing_2}\n  Let $1\\leq p<q$ and suppose that $b(x,\\cdot)\\in\\S_q(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$. Let $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be a Lipschitz continuous function. Assume that $X\\in{\\mathcal B}_{\\alpha,p}$ for some $\\alpha>1-H$ and $|X|_\\infty\\in L^\\infty$. Define \n  \\begin{equation*}\n    A_{s,t}\\ensuremath\\triangleq\\int_s^t \\bigg(h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)-\\bar{h}(X_s)\\bigg)\\,dB_r,\n  \\end{equation*}\n  in the mixed Wiener-Young sense, see \\eqref{eq:wiener_young}. Then $A\\in\\bar{H}_{\\bar{\\eta}}^p$ for any $\\bar{\\eta}<\\alpha+H$ and any $\\varepsilon>0$. Moreover, there is a $\\gamma>0$ such that\n  \\begin{equation*}\n    \\vertiii{A}_{\\bar{H}_{\\bar{\\eta}}^p}\\lesssim\\Lip{h}\\big(1\\vee\\||X|_\\infty\\|_{L^\\infty}\\big)\\big(1\\vee\\|X\\|_{{\\mathcal B}_\\alpha,p}\\big)\\varepsilon^{\\gamma}.\n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n  Fix $1<\\bar{\\eta}<\\alpha+H$ and choose $\\rho\\in(0,1)$ such that $\\bar{\\eta}<H+\\alpha\\rho^2$ and $p\\leq\\rho q$. Since $|X|_\\infty\\in L^\\infty$, owing to \\cref{cor:lipschitz_average} we may assume that $\\bar{h}\\equiv 0$ without any loss of generality. Recall that\n  $\\delta A_{s,u,t}=A_{s,t}-A_{s,u}-A_{u,t}$, so\n  \\begin{equation*}\n    \\delta A_{s,u,t}=\\int_u^t \\bigg(h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)-h\\Big(X_u,\\bar{\\Phi}_{u,r}^{X_u}\\big(\\Phi_{0,u}^X(Y_0)\\big)\\Big)\\bigg)\\,dB_r.\n  \\end{equation*}\n  We condition on ${\\mathcal F}_u$ instead of ${\\mathcal F}_s$. This gives\n  \\begin{align*}\n    &\\phantom{\\leq}\\|\\Expec{\\delta A_{s,u,t}\\,|\\,\\mathcal{F}_s}\\|_{L^p}\\\\\n    &\\leq \\left\\|\\int_u^t \\Expec{h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)-h\\Big(X_u,\\bar{\\Phi}_{u,r}^{X_u}\\big(\\Phi_{0,u}^X(Y_0)\\big)\\Big)\\,\\middle|\\,\\mathcal{F}_u}\\dot{\\bar{B}}^u_{r}\\,dr\\right\\|_{L^p}\\\\\n    &\\leq\\rom{1}+\\rom{2}\n  \\end{align*}\n  with\n  \\begin{align*}\n    \\rom{1}&\\ensuremath\\triangleq\\left\\|\\int_u^t \\Expec{h\\Big(X_s,\\bar{\\Phi}^{X_s}_{s,r}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)-h\\Big(X_u,\\bar{\\Phi}^{X_u}_{u,r}\\big(\\bar{\\Phi}_{s,u}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\big)\\Big)\\,\\middle|\\,\\mathcal{F}_u} \\dot{\\bar{B}}^u_{r}\\,dr\\right\\|_{L^p},\\\\\n    \\rom{2}&\\ensuremath\\triangleq\\left\\|\\int_u^t \\Expec{h\\Big(X_u,\\bar{\\Phi}^{X_u}_{u,r}\\big(\\bar{\\Phi}_{s,u}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\big)\\Big)-h\\Big(X_u,\\bar{\\Phi}_{u,r}^{X_u}\\big(\\Phi_{0,u}^X(Y_0)\\big)\\Big)\\,\\middle|\\,\\mathcal{F}_u} \\dot{\\bar{B}}^u_{r}\\,dr\\right\\|_{L^p}.\n  \\end{align*}\n  These terms are now bounded individually. Let us begin with the bound on $\\rom{1}$. Thanks to \\cref{lem:fast_process_moments}, this term falls in the regime of \\cref{lem:sewing_helper_2}. By H\\\"older's inequality, we therefore find\n  \\begin{align*}\n    \\rom{1}&\\lesssim\\Lip{h}\\|X_s-X_u\\|_{L^\\infty}^{\\rho(1-\\rho)}\\|X_s-X_u\\|_{L^p}^{\\rho^2}\\int_u^t\\big\\|\\dot{\\bar{B}}_r^u\\big\\|_{L^{\\frac{p}{1-\\rho}}}\\left(1\\wedge\\frac{\\varepsilon^\\gamma}{|r-u|^\\gamma}\\right)\\,dr\\\\\n    &\\lesssim\\Lip{h}\\||X|_\\infty\\|_{L^\\infty}^{\\rho(1-\\rho)}\\|X\\|_{{\\mathcal B}_{\\alpha,p}}^{\\rho^2}\\varepsilon^\\delta|t-s|^{\\bar{\\eta}}\n  \\end{align*}\n  for $\\delta>0$ sufficiently small. Here, the last inequality used that, for any $p\\geq 1$, $\\big\\|\\dot{\\bar{B}}_r^u\\big\\|_{L^{p}}\\lesssim |r-u|^{H-1}$ together with the elementary fact\n  \\begin{equation*}\n    \\int_u^t\\frac{1}{|r-u|^{1-H}}\\left(1\\wedge\\frac{\\varepsilon^\\gamma}{|r-u|^\\gamma}\\right)\\,dr\\lesssim\\varepsilon^\\delta|t-u|^{H-\\delta}\n  \\end{equation*}\n  for any $\\delta\\in(0,\\gamma]$.\n\n  The term $\\rom{2}$ can be handled similarly in view of \\cref{lem:sewing_helper_3}.\n\\end{proof}\n\n\\subsection{Proof of \\cref{thm:feedback_fractional}}\\label{sec:proof}\n\nThe estimates of the previous two subsection furnish the following fundamental estimates:\n\\begin{proposition}\\label{prop:final_control}\n  Let $2\\leq p<q$ and suppose that $b(x,\\cdot)\\in\\S_q(\\kappa,R)$ for all $x\\in\\ensuremath{\\mathbb{R}}^d$. Let $h:\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n\\to\\ensuremath{\\mathbb{R}}$ be a bounded Lipschitz continuous function. Assume that there is an $\\alpha>1-H$ such that $X$ has $\\alpha$-H\\\"older sample paths and $X\\in{\\mathcal B}_{\\alpha,p}$. If, in addition, $|X|_\\infty\\in L^\\infty$, then, for any $\\eta<H$ and any $\\bar{\\eta}<\\alpha+H$, there is a $\\gamma>0$ such that\n  \\begin{equation}\n    \\left\\|\\int_0^\\cdot \\Big(h\\big(X_r,\\Phi_{0,s}^X(Y_0)\\big)-\\bar{h}(X_r)\\Big)\\,dB_r\\right\\|_{{\\mathcal B}_{\\eta,p}}\\lesssim\\big(|h|_\\infty+\\Lip{h}\\big)\\big(1+\\||X|_\\infty\\|_{L^\\infty}\\big)\\big(1+\\|X\\|_{{\\mathcal B}_{\\alpha,p}}\\big)\\varepsilon^{\\gamma},\\label{eq:combine_sewing_1}\n  \\end{equation}\n  and\n  \\begin{equation}\\label{eq:combine_sewing_2}\n    \\left\\|\\int_0^\\cdot h\\big(X_r,\\Phi_{0,r}^X(Y_0)\\big)\\,dB_r\\right\\|_{{\\mathcal B}_{\\eta,p}}\\lesssim\\big(|h|_\\infty+\\Lip{h}\\big)\\big(1+\\||X|_\\infty\\|_{L^\\infty}\\big)\\big(1+\\|X\\|_{{\\mathcal B}_{\\alpha,p}}\\big),\n  \\end{equation}\n  uniformly in $0\\leq s<t\\leq T$ and $\\varepsilon\\in(0,1]$. Here, the integrals are both taken in the Young sense.\n\\end{proposition}\n\\begin{proof}\n  First note that, by \\cref{lem:sewing_young}, the Young integrals in both \\eqref{eq:combine_sewing_1} and \\eqref{eq:combine_sewing_2} coincide with the processes $I(A^i)$ obtained by `sewing' the Riemann summands\n  \\begin{equation*}\n  \tA_{s,t}^1\\ensuremath\\triangleq\\int_s^t \\Big(h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)-\\bar{h}(X_s)\\Big)\\,dB_r,\\qquad A^2_{s,t}\\ensuremath\\triangleq\\int_s^t h\\Big(X_s,\\bar{\\Phi}_{s,r}^{X_s}\\big(\\Phi_{0,s}^X(Y_0)\\big)\\Big)\\,dB_r,\n  \\end{equation*}\n  where integration is now understood in the mixed Wiener-Young sense (see \\eqref{eq:wiener_young}), with the help of \\cref{prop:stochastic_sewing}. Consequently, the estimate \\eqref{eq:combine_sewing_1} follows immediately from combining \\cref{prop:sewing_1,prop:sewing_2}. Owing to \\cref{lem:stochastic_sewing_fbm,cor:lipschitz_average}, \\eqref{eq:combine_sewing_2} is then an easy consequence of the first bound. \n\\end{proof}\n\nFor $\\varepsilon>0$ and $M>0$, let us define the $({\\mathcal F}_t)_{t\\geq 0}$-stopping time $\\tau_M^\\varepsilon\\ensuremath\\triangleq\\inf\\{t\\geq 0:\\,|X_t^\\varepsilon|>M\\}$. Applying the previous proposition to the slow-fast system \\eqref{eq:slow_feedback_sec}--\\eqref{eq:fast_feedback_sec}, we can deduce relative compactness of the stopped slow motion $X^{\\varepsilon,M}\\ensuremath\\triangleq X^\\varepsilon_{\\cdot\\wedge\\tau_M^\\varepsilon}$:\n\n\\begin{corollary}\\label{cor:tightness}\n  Consider the slow-fast system \\eqref{eq:slow_feedback_sec}--\\eqref{eq:fast_feedback_sec} with \\cref{cond:feedback} in place. Let $\\beta<\\frac12\\wedge\\hat{H}$ and $p\\geq 2$. Suppose that there are $\\kappa,R>0$ and $q>p$ such that $b(x,\\cdot)\\in\\S_q(\\kappa,R)$ for each $x\\in\\ensuremath{\\mathbb{R}}^d$. Then, for any $M>0$,\n  \\begin{equation*}\n    \\sup_{\\varepsilon\\in(0,1]}\\big\\|X^{\\varepsilon,M}\\big\\|_{{\\mathcal B}_{\\beta,p}}<\\infty.\n  \\end{equation*}\n\\end{corollary} \n\\begin{proof}\n  Recall from \\cref{cor:norm_bound_solution} that, for each $\\varepsilon>0$, there is a unique global solution $X^\\varepsilon$ to \\eqref{eq:slow_feedback_sec} with values in $\\ensuremath{\\mathcal{C}}^{\\alpha}([0,T],\\ensuremath{\\mathbb{R}}^d)$ for some $\\alpha>1-H$. Moreover, since the H\\\"older norm of the stopped solution $X^{\\varepsilon,M}$ is controlled by the H\\\"older norm of $X^\\varepsilon$, the argument of \\cref{cor:norm_bound_solution} also shows that $\\big\\|X^{\\varepsilon,M}\\big\\|_{{\\mathcal B}_{\\beta,p}}<\\infty$ for each $\\beta<\\frac12\\wedge\\hat{H}$ and $p\\geq 1$. Employing \\cref{prop:final_control}, we obtain that, for any $\\gamma<H-\\beta$ and any $\\delta\\in(0,T]$,\n  \\begin{equation*}\n    \\big\\|X^{\\varepsilon,M}\\restriction_{[0,\\delta]}\\big\\|_{{\\mathcal B}_{\\beta,p}}\\lesssim \\big(|g|_\\infty+\\Lip{g}\\big)\\big(1+\\big\\|X^{\\varepsilon,M}\\restriction_{[0,\\delta]}\\big\\|_{{\\mathcal B}_{\\beta,p}}\\big)\\delta^\\gamma+|f|_\\infty\\delta^{1-\\beta},\n  \\end{equation*}\n  uniformly in $\\varepsilon\\in(0,1]$. Hence, choosing $\\delta>0$ sufficiently small, the proof is concluded by a standard iteration argument.\n\\end{proof}\n\nNow we can finish the proof of \\cref{thm:feedback_fractional} by localizing the argument of Hairer and Li. To this end, we rely on the following deterministic residue lemma:\n\\begin{lemma}[Residue Lemma]\\label{lem:residue_lemma}\n  Let $F:\\ensuremath{\\mathbb{R}}^d\\to\\ensuremath{\\mathbb{R}}^d$ be Lipschitz continuous, $G:\\ensuremath{\\mathbb{R}}^d\\to\\Lin[m]{d}$ be of class $\\ensuremath{\\mathcal{C}}_b^2$, and $\\ensuremath{\\mathfrak{h}}\\in\\ensuremath{\\mathcal{C}}^\\alpha([0,T],\\ensuremath{\\mathbb{R}}^n)$ for some $\\alpha>\\frac12$. Moreover, let $Z,\\bar{Z}\\in\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}([0,T],\\ensuremath{\\mathbb{R}}^d)$ for some $\\tilde{\\alpha}\\in(1-\\alpha,\\alpha]$ with $Z_0=\\bar{Z}_0$. Then there is a constant $C$ depending only on $F$, $G$, and the terminal time $T$ such that\n  \\begin{equation*}\n    |z-\\bar{z}|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}\\leq C\\exp\\left(C|\\ensuremath{\\mathfrak{h}}|_{\\ensuremath{\\mathcal{C}}^\\alpha}^{\\frac1\\alpha}+C|Z|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}^{\\frac{1}{\\tilde{\\alpha}}}+C|\\bar{Z}|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}}^{\\frac{1}{\\tilde{\\alpha}}}\\right)|Z-\\bar{Z}|_{\\ensuremath{\\mathcal{C}}^{\\tilde{\\alpha}}},\n  \\end{equation*}\n  where $z$ and $\\bar{z}$ are the solutions to the equations\n  \\begin{equation*}\n    z_t=Z_t+\\int_0^t F(z_s)\\,ds+\\int_0^t G(z_s)\\,d\\ensuremath{\\mathfrak{h}}_s,\\qquad\\bar{z}_t=\\bar{Z}_t+\\int_0^t F(z_s)\\,ds+\\int_0^t F(\\bar{z}_s)\\,d\\ensuremath{\\mathfrak{h}}_s.\n  \\end{equation*}\n\\end{lemma}\nAlbeit the statement of \\cref{lem:residue_lemma} is slightly stronger than \\cite[Lemma 2.2]{Hairer2020}, it is straight-forward to show that the very same proof still applies. We therefore omit the details and finally turn to the proof of the main result of this article:\n\n\\begin{proof}[{Proof of \\cref{thm:feedback_fractional}}]\n  First observe that, by the assumptions of the theorem and \\cref{cor:lipschitz_average}, there exists a unique global solution to the averaged equation \\eqref{eq:effective_dynamics}, see \\cite{Lyons1998,Lyons2002,Rascanu2002}. We fix $\\bar{\\alpha}\\in(\\alpha,H)$ with $(\\bar\\alpha-\\alpha)^{-1}<p$. Choose $\\beta\\in(1-H,\\hat{H}\\wedge\\frac12)$. By \\cref{cor:tightness}, $\\sup_{\\varepsilon\\in(0,1]}\\|X^{\\varepsilon,M}\\|_{{\\mathcal B}_{\\beta,p}}<\\infty$ for each $M>0$. Consequently, by \\cref{prop:final_control}, we deduce that\n  \\begin{align*}\n    \\left\\|\\int_0^\\cdot \\Big(g\\big(X_r^{\\varepsilon,M},\\Phi_{0,r}^{X^{\\varepsilon,M}}(Y_0)\\big)-\\bar{g}\\big(X_r^{\\varepsilon,M}\\big)\\Big)\\,dB_r\\right\\|_{{\\mathcal B}_{\\bar{\\alpha},p}}&\\lesssim\\varepsilon^\\gamma,\n    \\\\\n    \\left\\|\\int_0^\\cdot \\Big(f\\big(X_r^{\\varepsilon,M},\\Phi_{0,r}^{X^{\\varepsilon,M}}(Y_0)\\big)-\\bar{f}\\big(X_r^{\\varepsilon,M}\\big)\\Big)\\,dr\\right\\|_{{\\mathcal B}_{\\bar{\\alpha},p}}&\\lesssim\\varepsilon^\\gamma.\n  \\end{align*}\n  Therefore, $\\big\\|\\hat{X}^{\\varepsilon,M}-\\bar{X}^{\\varepsilon,M}\\big\\|_{{\\mathcal B}_{\\bar{\\alpha},p}}\\lesssim\\varepsilon^\\gamma$, where\n  \\begin{align*}\n    \\hat{X}^{\\varepsilon,M}_t&\\ensuremath\\triangleq X_0+\\int_0^t f\\big(X^{\\varepsilon,M}_r,\\Phi_{0,r}^{X^{\\varepsilon,M}}(Y_0)\\big)\\,dr+\\int_0^t g\\big(X^{\\varepsilon,M}_r,\\Phi_{0,r}^{X^{\\varepsilon,M}}(Y_0)\\big)\\,dB_r,\\\\\n    \\bar{X}^{\\varepsilon,M}_t&\\ensuremath\\triangleq X_0+\\int_0^t\\bar{f}\\big(X^{\\varepsilon,M}_r\\big)\\,dr+\\int_0^t\\bar{g}\\big(X^{\\varepsilon,M}_r\\big)\\,dB_r.\n  \\end{align*}\n  In particular, $\\big|\\hat{X}^{\\varepsilon,M}-\\bar{X}^{\\varepsilon,M}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha}\\to 0$ in probability by the embedding \\eqref{eq:embeddings}. Note also the decomposition\n  \\begin{equation*}\n    X_t^{\\varepsilon,M}=\\hat{X}_t^{\\varepsilon,M}-\\bar{X}_t^{\\varepsilon,M}+X_0+\\int_0^t\\bar{f}(X_r^{\\varepsilon})\\,dr+\\int_0^t \\bar{g}(X_r^{\\varepsilon})\\,dB_r,\\quad t\\in[0,\\tau_M^\\varepsilon\\wedge T],\n  \\end{equation*}\n  whence \\cref{lem:residue_lemma} furnishes the bound\n  \\begin{equation}\\label{eq:residue_bound}\n    \\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha([0,\\tau_M^\\varepsilon\\wedge T])}\\leq C\\exp\\left(C|B|_{\\ensuremath{\\mathcal{C}}^\\alpha}^{\\frac{1}{\\alpha}}+C\\big|\\hat{X}^{\\varepsilon,M}-\\bar{X}^{\\varepsilon,M}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha}^{\\frac{1}{\\alpha}}\\right)\\big|\\hat{X}^{\\varepsilon,M}-\\bar{X}^{\\varepsilon,M}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha}.\n  \\end{equation}\n  As we have seen above, for each $M>0$, the right-hand side goes to $0$ in probability as $\\varepsilon\\to 0$. Hence, we also have that $\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha([0,\\tau_M^\\varepsilon\\wedge T])}\\to 0$ in probability.\n\n  On the other hand, note that\n  \\begin{align}\n  \t\\ensuremath\\mathbb{P}(\\tau_M^\\varepsilon<T)&\\leq\\ensuremath\\mathbb{P}\\left(\\sup_{t\\in[0,\\tau_M^\\varepsilon]}\\big|X_t^{\\varepsilon}\\big|\\geq M,\\tau_M^\\varepsilon<T\\right)\\leq\\ensuremath\\mathbb{P}\\left(\\big|X^{\\varepsilon}\\big|_{\\ensuremath{\\mathcal{C}}^\\gamma([0,\\tau_M^\\varepsilon\\wedge T])}\\geq T^{-\\gamma}(M-\\|X_0\\|_{L^\\infty})\\right)\\nonumber\\\\\n  \t&\\leq\\ensuremath\\mathbb{P}\\left(\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\gamma([0,\\tau_M^\\varepsilon\\wedge T])}\\geq T^{-\\gamma}(M-\\|X_0\\|_{L^\\infty})-|\\bar{X}|_{\\ensuremath{\\mathcal{C}}^\\gamma([0,T])}\\right)\\nonumber\\\\\n  \t&\\leq\\ensuremath\\mathbb{P}\\left(\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\gamma([0,\\tau_M^\\varepsilon\\wedge T])}\\geq 1\\right)+\\ensuremath\\mathbb{P}\\left(\\big|\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\gamma([0,T])}>T^{-\\gamma}(M-\\|X_0\\|_{L^\\infty})-1\\right)\\label{eq:split}\n  \\end{align}\n  for each $\\gamma>0$. By \\cref{prop:nualart}, we know that $\\big|\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\gamma([0,T])}\\in L^1$ provided that $\\gamma<\\frac12$. We fix such a $\\gamma$.\n\n  It is now easy to finish the proof. Let $\\delta_1,\\delta_2\\in(0,1)$ be given. Then we can find a $M>0$ such that\n  \\begin{equation*}\n  \t\\ensuremath\\mathbb{P}\\left(\\big|\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\gamma([0,T])}>T^{-\\gamma}(M-\\|X_0\\|_{L^\\infty})-1\\right)\\leq\\frac{\\delta_2}{2}.\n  \\end{equation*}\n  For this $M$, we can also find an $\\varepsilon_0>0$ such that\n  \\begin{equation*}\n  \t\\ensuremath\\mathbb{P}\\left(\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha([0,\\tau_M^\\varepsilon\\wedge T])}>\\delta_1\\right)\\leq\\frac{\\delta_2}{4}\\qquad\\forall\\,\\varepsilon\\in(0,\\varepsilon_0).\n  \\end{equation*}\n  The estimate \\eqref{eq:split} therefore yields that\n  \\begin{align*}\n  \t\\ensuremath\\mathbb{P}\\left(\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha([0,T])}>\\delta_1\\right)&\\leq\\ensuremath\\mathbb{P}\\left(\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha([0,\\tau_M^\\varepsilon\\wedge T])}>\\delta_1,\\tau_M^\\varepsilon\\geq T\\right)+\\ensuremath\\mathbb{P}(\\tau_M^\\varepsilon<T)\\\\\n  \t&\\leq 2\\ensuremath\\mathbb{P}\\left(\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha([0,\\tau_M^\\varepsilon\\wedge T])}>\\delta_1\\right)+\\frac{\\delta_2}{2}\\leq\\delta_2\n  \\end{align*}\n  for all $\\varepsilon\\in(0,\\varepsilon_0)$. Hence, $\\big|X^{\\varepsilon}-\\bar{X}\\big|_{\\ensuremath{\\mathcal{C}}^\\alpha([0,T])}\\to 0$ in probability as $\\varepsilon\\to 0$, as required.\n\\end{proof}\n\n\\begin{remark}\n  The proof above shows that we can choose\n  \\begin{equation*}\n    \\lambda_0=\\inf_{x\\in\\ensuremath{\\mathbb{R}}^d}\\Lambda(\\kappa,R,p)\n  \\end{equation*}\n  for any $p>\\max\\big(2,(H-\\alpha)^{-1}\\big)$ in \\cref{thm:feedback_fractional}. Here, $\\Lambda$ is the constant from \\cref{prop:conditional_initial_condition_wasserstein}.\n\\end{remark}\n\n\n\\subsection{Smoothness of the Averaged Coefficients}\n\nLet us finally show that an \\emph{everywhere contractive} fast process falls in the regime of \\cref{thm:feedback_fractional}. While smoothness of $\\bar g$ also holds under less restrictive conditions, the proof becomes much more involved. To keep this article concise, we chose to report on these results in future work.\n\n\n\\begin{corollary}\\label{cor:smooth}\n  Suppose that\n  \\begin{itemize}\n    \\item $g\\in\\ensuremath{\\mathcal{C}}_b^3\\big(\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n,\\Lin[m]{d}\\big)$,\n    \\item there is a $\\kappa>0$ such that $b(x,\\cdot)\\in\\S(\\kappa,0,0)$ for every $x\\in\\ensuremath{\\mathbb{R}}^d$,\n    \\item $b\\in\\ensuremath{\\mathcal{C}}^3\\big(\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n,\\ensuremath{\\mathbb{R}}^d\\big)$ is globally Lipschitz continuous and there is an $N\\in\\ensuremath{\\mathbb{N}}$ such that, for each $i,j,k\\in\\{x,y\\}$,\n    \\begin{equation*}\n      |D^2_{i,j} b(x,y)|+|D^3_{i,j,k}b(x,y)|\\lesssim 1+|y|^N\\qquad\\forall\\,x\\in\\ensuremath{\\mathbb{R}}^d,\\,\\forall\\, y\\in\\ensuremath{\\mathbb{R}}^n.\n    \\end{equation*}\n  \\end{itemize}\n  Then the conclusion of \\cref{thm:feedback_fractional} holds.\n\\end{corollary}\n\\begin{example}\n  Let $V\\in\\ensuremath{\\mathcal{C}}^4(\\ensuremath{\\mathbb{R}}^d\\times\\ensuremath{\\mathbb{R}}^n)$. If $\\inf_{x,y}D_{y,y}^2 V(x,y)\\geq\\kappa$, $|D^2_{x,y}V|_\\infty+|D^2_{y,y}V|_\\infty<\\infty$, and, for each $i,j,k\\in\\{x,y\\}$,\n  \\begin{equation*}\n    |D^3_{i,j,y}V(x,y)|+|D^4_{i,j,k,y}V(x,y)|\\lesssim 1+|y|^N\\qquad\\forall\\,x\\in\\ensuremath{\\mathbb{R}}^d,\\,\\forall\\, y\\in\\ensuremath{\\mathbb{R}}^n,\n  \\end{equation*}\n  then $b=-D_y V$ falls in the regime of \\cref{cor:smooth}. To give a concrete example, we can choose $V(x,y)=\\big(2+\\sin(x)\\big)\\big(y^2+\\sin(y)\\big)$, which furnishes the drift $b(x,y)=-\\big(2+\\sin(x)\\big)\\big(2y+\\cos(y)\\big)$.\n\\end{example}\n\n\n\\begin{proof}[Proof of \\cref{cor:smooth}]\n  In order to apply \\cref{thm:feedback_fractional} it is enough to show that, for any $g\\in\\ensuremath{\\mathcal{C}}_b^3(\\ensuremath{\\mathbb{R}}^n)$, the function\n  \\begin{equation*}\n    \\bar{h}(x)\\ensuremath\\triangleq\\int_{\\ensuremath{\\mathbb{R}}^n}g(y)\\,\\pi^x(dy)\n  \\end{equation*}\n  is again of class $\\ensuremath{\\mathcal{C}}_b^2(\\ensuremath{\\mathbb{R}}^d)$. To this end, we define $h_t(x)\\ensuremath\\triangleq\\Expec{g(Y_t^x)}$ where $Y^x$ is the solution to the SDE\n  \\begin{equation*}\n    dY_t^x=b(x,Y_t^x)\\,dt+\\sigma\\,d\\hat{B}\n  \\end{equation*}\n  started in the generalized initial condition $\\delta_0\\otimes\\ensuremath{\\mathsf{W}}$. Note that $h_t\\to\\bar{h}$ pointwise as $t\\to\\infty$ by \\cref{thm:geometric}. Since $h_t\\in\\ensuremath{\\mathcal{C}}_b^2(\\ensuremath{\\mathbb{R}}^d)$ for each $t\\geq 0$, it thus suffices to show that \n  \\begin{equation}\\label{eq:derivative_bound}\n    \\sup_{t\\geq 0} \\left(|D h_t|_\\infty+|D^2 h_t|_\\infty\\right)<\\infty\n  \\end{equation} \n  and both $D h_t$ and $D^2 h_t$ converge locally uniformly along a subsequence. By a straight-forward `diagonal sequence' argument, we actually only need to prove uniform convergence on a fixed compact $K\\subset\\ensuremath{\\mathbb{R}}^d$.\n\n  Under the assumptions of the corollary, it is easy to see that the mapping $x\\mapsto Y_t^x$ is three-times differentiable for each $t\\geq 0$ and it holds that\n  \\begin{align*}\n    D_x Y_t^x&=\\int_0^tJ_{s,t}D_x b(x,Y_s^x)\\,ds, \\label{eq:first_derivative}\\\\\n    D^2_{x,x} Y_t^x(u\\otimes v)&=\\int_0^tJ_{s,t}\\Big(D_{x,x}^2 b(x,Y_s^x)(u\\otimes v)+2D_{x,y}^2 b(x,Y_s^x)\\big(u\\otimes D_x Y_s^x(v)\\big) \\nonumber\\\\\n    &\\phantom{=\\int_0^tJ_{s,t}}+D^2_{y,y}b(x,Y_s^x)\\big(D_x Y_s^x(u)\\otimes D_x Y_s^x(v)\\big)\\Big)\\,ds,\n  \\end{align*}\n  where $J_{s,t}$ solves the homogeneous problem\n  \\begin{equation*}\n    J_{s,t}=\\mathrm{id}+\\int_s^t D_yb(x,Y_r^x)J_{s,r}\\,dr.\n  \\end{equation*}\n  Since $b(x,\\cdot)\\in\\S(\\kappa,0,0)$, it is not hard to see that, for each $x\\in\\ensuremath{\\mathbb{R}}^d$ and $y\\in\\ensuremath{\\mathbb{R}}^n$, $D_yb(x,y)\\leq-\\kappa$ in the sense of quadratic forms. In particular, the operator norm of $J$ satisfies the bound\n  \\begin{equation*}\n    |J_{s,t}|\\leq e^{-\\kappa(t-s)}.\n  \\end{equation*}\n  By an argument similar to \\cref{lem:fast_process_moments}, it follows that, for any $p\\geq 1$,\n  \\begin{equation*}\n    \\sup_{t\\geq 0}\\sup_{x\\in\\ensuremath{\\mathbb{R}}^d}\\big\\|D_xY_t^x\\big\\|_{L^p}<\\infty\\quad\\text{and}\\quad \\sup_{t\\geq 0}\\sup_{x\\in\\ensuremath{\\mathbb{R}}^d}\\big\\|D_{x,x}^2Y_t^x\\big\\|_{L^p}<\\infty.\n  \\end{equation*}\n  Based on this, it is straight-forward to verify \\eqref{eq:derivative_bound}. Consequently, by the Arzela-Ascoli theorem, there is a subsequence of times along which $Dh$ converges uniformly on $K$. By a similar---albeit more tedious---computation, the reader can easily check that also \n  \\begin{equation*}\n    \\sup_{t\\geq 0}\\sup_{x\\in\\ensuremath{\\mathbb{R}}^d}\\big\\|D_{x,x,x}^3Y_t^x\\big\\|_{L^p}<\\infty.\n  \\end{equation*}\n  In particular, $D^3 h$ is uniformly bounded, whence we can pass to a further subsequence along which $D^2 h$ also converges uniformly on $K$. Therefore, $\\bar{h}\\in\\ensuremath{\\mathcal{C}}_b^2(\\ensuremath{\\mathbb{R}}^d)$ as required.\n\\end{proof}\n\n{\n\\footnotesize\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\t\t\t\n\t\tA \\emph{square} $S$ in a matrix $M= \\left( a_{i,j} \\right)$ is a $2 \\times 2$ submatrix of the form \\[S = \\left(\\begin{array}{cc}\n\t\t\ta_{i,j} & a_{i, j+s} \\\\\n\t\t\ta_{i+s,j} & a_{i+s, j+s}\n\t\t\\end{array}\\right).\n\t\t\\]\n\t\t\n\t\tIn 1996 Erickson \\cite{erickson1996introduction} asked for the largest $n$ such that there exists an $n \\times n$ binary matrix $M$ with no squares which have constant entries. An upper bound was first given by Axenovich and Manske \\cite{axenovich2008monochromatic}, before the answer 14 was determined by Bacjer and Eliahou in \\cite{bacher2010extremal}. \n\t\t\n\t\tRecently,  Ar\\'evalo, Montejano and Rold\\'an-Pensado \\cite{arevalo2020zero} initiated the study of a zero-sum variant of Erickson's problem. Here we wish to avoid \\emph{zero-sum squares}, squares with entries that sum to $0$.\n\t\t\n\t\tZero-sum problems have been well-studied since the classic Erd\\H{o}s-Ginsburg-Ziv Theorem in 1961 \\cite{erdos1961theorem}. Much of the research has been on zero-sum problems in finite abelian groups (see the survey \\cite{gao2006zero} for details), but problems have also been studied in other settings such as on graphs (see e.g. \\cite{caro2016ero, caro2019zero, caro2020zero, bialostocki1993zero}). Of particular relevance is the result of Balister, Caro, Rousseau and Yuster in \\cite{balister2002zero} on submatrices of integer valued matrices where the rows and columns sum to $0 \\mod p$, and the result of Caro, Hansberg and Montejano on zero-sum subsequences in bounded sum $\\{-1,1\\}$-sequences \\cite{caro2019zerosum}. \n\t\t\n\t\tGiven an $n \\times m$ matrix $M = \\left( a_{i,j} \\right)$ define the \\emph{discrepancy} of $M$ as the sum of the entries, that is\n\t\t\\[\\disc(M) = \\sum_{\\substack{1 \\leq i \\leq n\\\\1 \\leq j \\leq m}} a_{i,j}. \\]\n\t\tWe say a square $S$ is a \\emph{zero-sum square} if $\\disc(S) = 0$, or equivalently,\n\t\t\\[a_{i,j} + a_{i, j+s} + a_{i+s,j} + a_{i+s, j+s} = 0.\\]\n\t\t\n\t\tWe will be interested in $\\{-1,1\\}$-matrices $M$ which do not contain any zero-sum squares, and we shall call such matrices \\emph{zero-sum square free}. Clearly matrices with at most one $-1$ are zero-sum square free and, in general, there are many such matrices when the number of $-1$s is low. Instead, we will be interested in matrices which have a similar number of $1$s and $-1$s or, equivalently, matrices with small discrepancy (in absolute value).\n\t\t\n\t\tAn $n \\times m$ $\\{-1,1\\}$-matrix $M = \\left(a_{i,j}\\right)$ is said to be \\emph{$t$-diagonal} for some $0 \\leq t \\leq n +m -1$ if\n\t\t\\[a_{i,j} = \\begin{cases}\n\t\t\t1 & i + j \\leq t + 1,\\\\\n\t\t\t-1 & i + j \\geq t +2.\n\t\t\\end{cases}\\]\n\t\tWe say a matrix $M$ is \\emph{diagonal} if there is some $t$ such that a $t$-diagonal matrix $N$ can be obtained from $M$ by applying vertical and horizontal reflections. \n\t\tDiagonal matrices are of particular interest since they can have low discrepancy, yet they never contain a zero-sum square.\n\t\t\n\t\tAr\\'evalo, Montejano and Rold\\'an-Pensado \\cite{arevalo2020zero} proved that, except when $n \\leq 4$, every $n \\times n$ non-diagonal $\\{-1,1\\}$-matrix $M$ with $|\\disc(M)| \\leq n$ has a zero-sum square. They remark that it should be possible to extend their proof to give a bound of $2n$, and they conjecture that the bound $Cn$ should hold for any $C > 0$ when $n$ is large enough relative to $C$.\n\t\t\n\t\t\\begin{conjecture}[Conjecture 3 in \\cite{arevalo2020zero}]\n\t\t\tFor every $C > 0$ there is a integer $N$ such that whenever $n \\geq N$ the\n\t\t\tfollowing holds: every $n \\times n$ non-diagonal $\\{-1, 1\\}$-matrix $M$ with $|\\disc(M)| \\leq Cn$\n\t\t\tcontains a zero-sum square.\n\t\t\\end{conjecture}\n\t\t\n\t\tWe prove this conjecture in a strong sense with the following theorem.\n\t\t\n\t\t\\begin{theorem}\\label{thm:low-bound}\n\t\t\tLet $n \\geq 5$. Every $n \\times n$ non-diagonal $\\{-1,1\\}$-matrix $M$ with $|\\disc(M)| \\leq n^2\/4$ contains a zero-sum square.\n\t\t\\end{theorem}\n\t\n\t\tThe best known construction for a non-diagonal zero-sum square free matrix has discrepancy close to $n^2\/2$, and our computer experiments suggest that this construction is in fact optimal. Closing the gap between the upper and lower bounds remains a very interesting problem and we discuss it further in Section \\ref{sec:open-problems}. \n\t\t\t\t\n\t\t\n\t\t\\section{Proof}\n\t For  $p\\leq r$ and $q \\leq s$ define the \\emph{consecutive submatrix} $M[p:r, q:s]$ by\n\t\t\\[M[p:r, q:s] = \\left(\\begin{array}{cccc}\n\t\t\ta_{p,q} & a_{p, q+1} & \\dotsb & a_{p, s} \\\\\n\t\t\ta_{p+1, q} & a_{p+1, q+1} & \\dotsb & a_{p+1, s} \\\\\n\t\t\t\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\t\t\ta_{r, q} & a_{r+1, q} & \\dotsb & a_{r,s}\n\t\t\\end{array}\n\t\t \\right).\n\t\t \\]\n\t\t Throughout the rest of this paper, we will assume that all submatrices except squares are consecutive submatrices.\n\t\t\n\t\tWe start by stating the following lemma from \\cite{arevalo2020zero} which, starting from a small $t'$-diagonal submatrix $M'$, determines many entries of the matrix $M$. An example application is shown in Figure \\ref{fig:struct}.\n\t\t\n\t\t\n\t\n\t\t\\begin{lemma}[Claim 3 in \\cite{arevalo2020zero}]\n\t\t\t\\label{lem:struct}\n\t\t\tLet $M$ be an $n \\times n $ $\\{-1,1\\}$-matrix with no zero-sum squares, and suppose that there is a submatrix $M' = M[p: p+s, q: q+s]$ which is $t'$-diagonal for some $2 \\leq t' \\leq 2s - 3$. Let $t = t + p + q -2$ and suppose $t \\leq n$. \t\t\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item The submatrix \\[N = M[1: \\min(t + \\floor{t\/2}, n), 1:\\min(t + \\floor{t\/2}, n)]\\] is $t$-diagonal.\n\t\t\t\\end{enumerate} \n\t\t\t\n\t\t\tFurthermore, both $a_{i,j} = 1$ and $a_{j,i} = 1$ whenever $t + \\floor{t\/2} < j \\leq t + \\floor{t\/2} + t -2$ and one of the following holds:\n\t\t\t\\begin{enumerate}\n\t\t\t\t \\setcounter{enumi}{1}\n\t\t\t\t\\item $j - t \\leq i \\leq t + \\floor{\\frac{t}{2}}$;\n\t\t\t\t\\item $i \\leq \\floor{\\frac{t}{2}} - \\floor{\\frac{j - t - \\floor{t\/2} -1}{2}}$;\n\t\t\t\t\\item $i = j$.\n\t\t\t\\end{enumerate}\n\t\\end{lemma}\n\n\t\t\\begin{figure}\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale=0.5\\textwidth\/11cm]\n\t\t\t\n\t\t\t\n\t\t\t\\fill[yellow5] (0,10) rectangle +(5,1);\n\t\t\t\\fill[yellow5] (0,9) rectangle +(4,1);\n\t\t\t\\fill[yellow5] (0,8) rectangle +(1,1);\n\t\t\t\\fill[yellow5] (0,7) rectangle +(1,1);\n\t\t\t\\fill[yellow5] (0,6) rectangle +(1,1);\n\t\t\t\n\t\t\t\\fill[yellow7] (1,8) rectangle +(2,1);\n\t\t\t\\fill[yellow7] (1,7) rectangle +(1,1);\n\t\t\t\n\t\t\t\\fill[blue7] (3, 8) rectangle +(1,1);\n\t\t\t\\fill[blue7] (2, 7) rectangle +(2,1);\n\t\t\t\\fill[blue7] (1, 6) rectangle +(3,1);\n\t\t\t\n\t\t\t\\fill[blue5] (4, 6) rectangle +(1, 4);\n\t\t\t\\fill[blue5] (0, 4) rectangle +(7, 2);\n\t\t\t\\fill[blue5] (5, 6) rectangle +(2, 5);\n\t\t\t\n\t\t\t\\fill[blue3] (7, 10) rectangle +(3, 1);\n\t\t\t\\fill[blue3] (7, 9) rectangle +(2,1);\n\t\t\t\\fill[blue3] (0, 1) rectangle +(1, 3);\n\t\t\t\\fill[blue3] (1,2) rectangle +(1, 2);\n\t\t\t\n\t\t\t\\fill[lightblue5] (3, 3) rectangle +(4, 1);\n\t\t\t\\fill[lightblue5] (4, 2) rectangle +(3, 1);\n\t\t\t\\fill[lightblue5] (5, 1) rectangle +(2, 1);\n\t\t\t\n\t\t\t\\fill[lightblue5] (7,4) rectangle +(1, 4);\n\t\t\t\\fill[lightblue5] (8,4) rectangle +(1, 3);\n\t\t\t\\fill[lightblue5] (9,4) rectangle +(1, 2);\n\t\t\t\n\t\t\t\\fill[lightblue3] (7, 3) rectangle +(1,1);\n\t\t\t\\fill[lightblue3] (8, 2) rectangle +(1,1);\n\t\t\t\\fill[lightblue3] (9, 1) rectangle +(1,1);\n\t\t\t\n\t\t\t\\draw[black] (0,1) -- +(11,0);\n\t\t\t\\draw[black] (0,2) -- +(11,0);\n\t\t\t\\draw[black] (0,3) -- +(11,0);\n\t\t\t\\draw[black] (0,4) -- +(11,0);\n\t\t\t\\draw[black] (0,5) -- +(11,0);\n\t\t\t\\draw[black] (0,6) -- +(11,0);\n\t\t\t\\draw[black] (0,7) -- +(11,0);\n\t\t\t\\draw[black] (0,8) -- +(11,0);\n\t\t\t\\draw[black] (0,9) -- +(11,0);\n\t\t\t\\draw[black] (0,10) -- +(11,0);\n\t\t\t\\draw[black, very thick] (0,11) -- +(11,0);\n\t\t\t\n\t\t\t\\draw[very thick, black] (0,0) -- +(0,11);\n\t\t\t\\draw[black] (1,0) -- +(0,11);\n\t\t\t\\draw[black] (2,0) -- +(0,11);\n\t\t\t\\draw[black] (3,0) -- +(0,11);\n\t\t\t\\draw[black] (4,0) -- +(0,11);\n\t\t\t\\draw[black] (5,0) -- +(0,11);\n\t\t\t\\draw[black] (6,0) -- +(0,11);\n\t\t\t\\draw[black] (7,0) -- +(0,11);\n\t\t\t\\draw[black] (8,0) -- +(0,11);\n\t\t\t\\draw[black] (9,0) -- +(0,11);\n\t\t\t\\draw[black] (10,0) -- +(0,11);\n\t\t\t\n\t\t\t\\draw[very thick] (1,6) rectangle +(3,3);\n\t\t\t\n\t\t\t\n\t\t\t\\draw[|-|] (0, 11.5) -- +(5, 0);\n\t\t\t\\draw[-|] (5,11.5) -- +(2,0);\n\t\t\t\\draw[-|] (7, 11.5) -- +(3,0);\n\t\t\t\n\t\t\t\\draw node[above] at (2.5, 11.5) {$t$};\n\t\t\t\\draw node[above] at (6, 11.5) {$\\floor{t\/2}$};\n\t\t\t\\draw node[above] at (8.5, 11.5) {$t-2$};\n\t\t\t\n\t\t\t\n\t\t\\end{tikzpicture}\n\t\t\n\t\t\\caption{The entries known from applying Lemma \\ref{lem:struct}. The yellow squares represent $-1$s and the blue squares represent $1$s. The submatrix $M'$ is show in a darker shade.}\n\t\t\\label{fig:struct}\n\t\\end{figure}\n\n\tNote that we can apply this lemma even when it is a reflection of $M'$ which is $t$-diagonal; we just need to suitably reflect $M$ and potentially multiply by $-1$, and then undo these operations at the end. The matrix $N$ will always contains at least one of $a_{1,1}$, $a_{1,n}$, $a_{n,1}$ and $a_{n,n}$, and if $N$ contains two, then $M$ is diagonal.\n\t\n\tWe will also make use of the following observation. This will be used in conjunction with the above lemma to guarantee the existence of some additional $1$s, which allows us to show a particular submatrix has positive discrepancy. \n\n\t\\begin{observation}\n\t\t\\label{obs:oneof}\n\t\tLet $M$ be an $n \\times n $ $\\{-1,1\\}$-matrix with no zero-sum squares, and suppose that $a_{i,i} = 1$ for every $i \\in [n]$. Then at least one of $a_{i,j}$ and $a_{j,i}$ is 1. In particular, $a_{i,j} + a_{j,i} \\geq 0$ for all $1 \\leq i ,j \\leq n$.\n\t\\end{observation}\n\n\n\n\n\tThe final lemma we will need to prove Theorem \\ref{thm:low-bound} is a variation on Claims 1 and 2 from \\cite{arevalo2020zero}. The main difference between Lemma \\ref{lem:submatrix} and the result used by  Ar\\'evalo, Montejano and Rold\\'an-Pensado is that we will always find a square submatrix. This simplifies the proof of Theorem \\ref{thm:low-bound}.\n\t\n\t\\begin{lemma}\n\t\t\\label{lem:submatrix}\n\t\tFor $n \\geq 8$, every $n \\times n $ $\\{-1,1\\}$-matrix $M$ with $|\\disc(M)| \\leq n^2\/4$ has an $n' \\times n'$ submatrix $M'$ with $|\\disc(M')| \\leq (n')^2\/4$ for some $(n-1)\/2 \\leq n' \\leq (n+1)\/2$.\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tWe only prove this in the case $n$ is odd as the case $n$ is even is similar, although simpler.\n\t\tPartition the matrix $M$ into 9 regions as follows. Let the four $(n-1)\/2 \\times (n-1)\/2$ submatrices containing $a_{1,1}$, $a_{1,n}$, $a_{n,1}$ and $a_{n,n}$ be $A_1, \\dots, A_4$ respectively. Let the $(n-1)\/2 \\times 1$ submatrix between $A_1$ and $A_2$ be $B_1$ and define $B_2$, $B_3$ and $B_4$ similarly. Finally, let the central entry be $B_5$. The partition is shown in Figure \\ref{fig:regions-part}.\n\t\t\n\t\t\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale=\\textwidth\/10cm]\n\t\t\t\\draw[step=1, very thin, gray] (0,0) grid (9,9);\n\t\t\t\\draw[fill=none, stroke=black, very thick] (0,5) rectangle(4,9);\n\t\t\t\\draw[fill=none, stroke=black, very thick] (5,5) rectangle (9,9);\n\t\t\t\\draw[fill=none, stroke=black, very thick] (0,0) rectangle (4,4);\n\t\t\t\\draw[fill=none, stroke=black, very thick] (5,0) rectangle (9,4);\n\t\t\t\\draw node[fill=white] at (2, 7) {$A_1$};\n\t\t\t\\draw node[fill=white] at (7, 7) {$A_2$};\n\t\t\t\\draw node[fill=white] at (7, 2) {$A_3$};\n\t\t\t\\draw node[fill=white] at (2, 2) {$A_4$};\n\t\t\t\\draw[fill=none, stroke=black, very thick] (4,5) rectangle(5,9);\n\t\t\t\\draw[fill=none, stroke=black, very thick] (5,4) rectangle(9,5);\n\t\t\t\\draw[fill=none, stroke=black, very thick] (4,0) rectangle(5,4);\n\t\t\t\\draw[fill=none, stroke=black, very thick] (0,4) rectangle(4,5);\n\t\t\t\\draw node[fill=white, inner sep=1] at (4.5, 7) {$B_1$};\n\t\t\t\\draw node[fill=white, inner sep=1] at (7, 4.5) {$B_2$};\n\t\t\t\\draw node[fill=white, inner sep=1] at (4.5, 2) {$B_3$};\n\t\t\t\\draw node[fill=white, inner sep=1] at (2, 4.5) {$B_4$};\n\t\t\t\\draw node[fill=white, inner sep=1] at (4.5, 4.5) {$B_5$};\n\t\t\t\n\t\t\\end{tikzpicture}\n\t\t\\caption{}\n\t\t\\label{fig:regions-part}\n\t\\end{subfigure}\\hfil\n\t\\begin{subfigure}{0.45\\textwidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale=\\textwidth\/10cm]\n\t\t\t\\draw[step=1, very thin, gray] (0,0) grid (9,9);\n\t\t\t\n\t\t\t\\draw[fill=none, black, very thick] (4,0) rectangle (9,5);\n\t\t\t\\draw[fill=none, black, very thick] (0,4) rectangle(5,9);\n\t\t\t\\draw[fill=none, black, very thick] (0,0) rectangle(9,9);\n\t\t\t\\draw node[] at (2.5, 6.5) {$A'_1$};\n\t\t\t\\draw node[] at (6.5, 2.5) {$A'_3$};\n\t\t\t\n\t\t\\end{tikzpicture}\n\t\t\\caption{}\n\t\t\\label{fig:regions-overlap}\n\t\\end{subfigure\n\t\\caption{A subset of the regions used in the proof of Lemma \\ref{lem:submatrix}.}\n\t\\label{fig:regions}\n\\end{figure}\n\t\t\n\tAs these partition the matrix $M$, we have\n\t\\begin{equation}\\label{eqn:part}\n\t\t\\disc(M) = \\disc(A_1) + \\dotsb + \\disc(A_4) + \\disc(B_1) + \\dotsb + \\disc(B_5).\n\t\\end{equation}\n\t\n\tLet the overlapping $(n+1)\/2 \\times (n+1)\/2$ submatrices containing $a_{1,1}$, $a_{1,n}$, $a_{n,1}$ and $a_{n,n}$ be $A_1', \\dots, A_4'$, as indicated in Figure \\ref{fig:regions-overlap}. The submatrices $B_1, \\dots, B_4$ each appear twice in the $A_i'$ and $B_5$ appears four times and, by subtracting these overlapping regions, we obtain a second equation for $\\disc(M)$:\n\t\t\\begin{multline}\\label{eqn:part2}\n\t\t\t\\disc(M) = \\disc(A_1') + \\dotsb + \\disc(A_4')\\\\ - \\disc(B_1) - \\dotsb - \\disc(B_4) - 3 \\disc(B_5).\n\t\t\\end{multline}\n\t\n\tIf any of the $A_i$ or $A_i'$ have $|\\disc(A_i)| \\leq (n-1)^2\/16$ or $|\\disc(A_i')| \\leq (n+1)^2\/16$ respectively, we are done, so we may assume that this is not the case.\n\tFirst, suppose that $\\disc(A_i) > (n-1)^2\/16$ and $\\disc(A_i') > (n+1)^2\/16$ for all $i = 1,2,3,4$. Since $n - 1$ is even and $\\disc(A_i) \\in \\mathbb{Z}$, we must have $\\disc(A_i) \\geq (n-1)^2\/16 + 1\/4$, and similarly, $\\disc(A_i') \\geq (n+1)^2\/16 + 1\/4$. Adding the equations (\\ref{eqn:part}) and (\\ref{eqn:part2}) we get the bound\n\t\\[n^2\/2 \\geq 2 \\disc(M) \\geq (n+1)^2\/4 + (n-1)^2\/4 + 2 - 2 \\disc(B_5), \\]\n\twhich reduces to $\\disc(B_5) \\geq 5\/4$. This gives a contradiction since $B_5$ is a single square. Similarly we get a contradiction if, for every $i$, both $\\disc(A_i) < - (n-1)^2\/16$ and $\\disc(A_i') < - (n+1)^2\/16$. \n\t\n\tThis only leaves the case where two of the 8 submatrices have different signs. If $A_i' > (n+1)^2\/16$, then, for $n \\geq 8$, \\[A_i > (n+1)^2\/16 - n > -(n-1)^2\/16,\\] and either $|\\disc(A_i)| \\leq (n-1)^2\/16$, a contradiction, or  $\\disc(A_i) > 0$. By repeating the argument when $\\disc(A_i')$ is negative, it follows that $A_i$ and $A_i'$ have the same sign for every $i$. In particular, two of the $A_i$ must have different signs, and we can apply an interpolation argument as in \\cite{arevalo2020zero}.\n\t\n\tWithout loss of generality we can assume that $\\disc(A_1) > (n-1)^2\/16$ and $\\disc(A_2) < -(n-1)^2\/16$. Consider the sequence of matrices $N_0, \\dots, N_{(n+1)\/2} $ where \\[N_i = M[1: (n-1)\/2, 1 + i: i + (n-1)\/2].\\]\n\tWe claim that there is a $j$ such that $|\\disc(N_j)| \\leq (n-1)^2\/16$, which would complete the proof of the lemma. By definition, $N_0 = A_1$ and $N_{(n+1)\/2} = A_2$ so there must be some $j$ such that $\\disc(N_{j-1}) > 0$ and $\\disc(N_j) \\leq 0$. Since the submatrices $N_{j-1}$ and $N_{j}$ share most of their entries $|\\disc(N_{j-1}) - \\disc(N_j)| \\leq (n-1)$, and as $(n-1)^2\/8 > (n-1)$, it cannot be the case that $\\disc(N_{j-1}) > (n-1)^2\/16$ and $\\disc(N_j) < -(n-1)^2\/16$. This means there must be some $j$ such that $|\\disc(N_j)| \\leq (n-1)^2\/16$, as required. \n\t\\end{proof}\n\n\tArmed with the above results, we are now ready to prove our main result, but let us first give a sketch of the proof which avoids the calculations in the main proof.\n\t\n\t\\begin{proof}[Sketch proof of Theorem \\ref{thm:low-bound}]\t\n\t Assume we have an $n\\times n$ $\\{-1,1\\}$-matrix $M$ with $|\\disc(M)| \\leq n^2\/4$ which is zero-sum square free. We will prove the result by induction, so we assume that the result is true for $5 \\leq n' < n$. \n\t \n\t Applying Lemma \\ref{lem:submatrix} gives a submatrix $M'$ with low discrepancy. Since $M'$ must also be zero-sum square free, we know that it is diagonal by the induction hypothesis. Applying Lemma \\ref{lem:struct} then gives us a lot of entries $M$ and, in particular, a submatrix $N$ with high discrepancy. Since we are assuming that $M$ has low discrepancy, the remainder $M \\setminus N$ of $M$ not in $N$ must either have low discrepancy or negative discrepancy. In both cases we will find $B$, a submatrix of $M$ with low discrepancy. When the discrepancy of $M \\setminus N$ is low, we use an argument similar to the proof of Lemma \\ref{lem:submatrix}, and when the discrepancy of $M \\setminus N$ is negative, we find a positive submatrix using Observation \\ref{obs:oneof} and use an interpolation argument.\n\t \n\t By the induction hypothesis, $B$ must also be diagonal and we can apply Lemma \\ref{lem:struct} to find many entries of $M$. By looking at specific $a_{i,j}$, we will show that the two applications of Lemma \\ref{lem:struct} contradict each other.\n\t\\end{proof}\n\n\tWe now give the full proof of Theorem \\ref{thm:low-bound}, complete with all the calculations. To start the induction, we must check the cases $n < 30$ which is done using a computer. The problem is encoded as a SAT problem using PySAT \\cite{imms-sat18} and checked for satisfiability with the CaDiCaL solver. The code to do this is attached to the arXiv submission.\n\n\t\\begin{proof}[Proof of Theorem \\ref{thm:low-bound}]\n\t\tWe will use induction on $n$. A computer search gives the result for all $n < 30$, so we can assume that $n \\geq 30$ and that the result holds for all $5 \\leq n' < n$. \n\t\t\n\t\tSuppose, towards a contradiction, that $M$ is an $n \\times n$ matrix with no zero-sum squares and $|\\disc(M)| \\leq n^2\/4$ . \t\t\n\t\tBy Lemma \\ref{lem:submatrix}, we can find an $n' \\times n'$ submatrix $M' = M[p:p+s, q:q+s]$ with $(n-1)\/2 \\leq n ' \\leq (n+1)\/2$ and $|\\disc(M')| \\leq (n')^2\/4$. By the induction hypothesis and our assumption that $M$ doesn't contain a zero-sum square, the matrix $M'$ must be diagonal. By reflecting $M$ and switching $-1$ and $1$ as necessary, we can assume that the submatrix $M'$ is $t'$-diagonal for some $t'$, and that $t := t' + p + q -2 \\leq n$.\n\t\t\n\t\tWe will want to apply Lemma \\ref{lem:struct}, for which we need to check $2 \\leq t' \\leq 2s - 3$. If $t' \\leq 1$ or $t' \\geq 2s - 2$, then the discrepancy of $M'$ is \\[|\\disc(M')| \\geq  (n')^2 -1 > (n')^2\/4,\\] which contradicts our choice of $M'$. In fact, since $\\disc(M') \\leq (n')^2\/4$ and $\\disc(M') \\leq (n')^2 - t'(t'+1)$ we find\n\t\t\\begin{equation}\n\t\t\\label{eqn:tbound}\n\t\tt \\geq t' \\geq \\frac{1}{2} \\left( \\sqrt{3(n')^2 + 1} -1 \\right) \\approx 0.433 n.\n\t\t\\end{equation}\t\n\t\n\t\tIf $t + \\floor{t\/2} \\geq n$, the matrix $M$ is $t$-diagonal and we are done, so we can assume that this is not the case, and that $t \\leq 2n\/3$. We will also need the following bound on $2t + \\floor{t\/2} -2$, which follows almost immediately from (\\ref{eqn:tbound}).\n\t\t\n\t\t\\begin{claim}\\label{claim:tgeqnmins1}\n\t\t\tWe have\n\t\t\t\\[2t + \\floor{t\/2} -2 \\geq n - 1.\\]\n\t\t\\end{claim}\n\t\t\\begin{proof}\n\t\t\tSubstituting $n' \\geq (n-1)\/2$ into (\\ref{eqn:tbound}) gives the following bound on $t$.\n\t\t\t\\[t \\geq \\frac{1}{4} \\left( \\sqrt{3n^2 - 6n + 7} - 2 \\right)\\]\n\t\t\tWe now lower bound $\\floor{t\/2}$ by $(t-1)\/2$ to find\n\t\t\t\\begin{align*}\n\t\t\t\t2t + \\floor{t\/2} -2 &\\geq 2t + \\frac{t-5}{2}\\\\\n\t\t\t\t&\\geq  \\frac{5}{8} \\sqrt{3n^2 - 6n + 7} - \\frac{15}{4}\n\t\t\t\\end{align*}\n\t\t\tThe right hand side  grows like $\\frac{\\sqrt{75}}{8} n$ asymptotically, which is faster than $n$, so the claim is certainly true for large enough $n$. In fact, the equation $ \\frac{5}{8} \\sqrt{3n^2 - 6n + 7} - \\frac{15}{4} \\geq n -1$ can be solved explicitly to obtain the following the bound on $n$:\n\t\t\t\\[n \\geq \\frac{1}{11} \\left( 251 + 20 \\sqrt{166} \\right) \\approx 46.2.\\]\n\t\t\tThis still leaves the values $30 \\leq n \\leq 46$ for which the bounds above are not sufficient. These cases can be checked using a computer.\n\t\t\\end{proof}\n\t\t\n\t\t\n\n\t\t\n\t\t\n\t\tLet $k = \\ceil{5n\/6}$ and let $N$ be the $k \\times k$ sub-matrix in the top left corner which contains $a_{1,1}$ i.e. $N = M[1:k, 1:k]$. We will apply Lemma \\ref{lem:struct} and Observation \\ref{obs:oneof} to guarantee lots of 1s in $N$, and therefore ensure $N$ has large discrepancy. This will mean that the rest of $M$ which is not in $N$ must have low discrepancy, and we can find another diagonal submatrix, $B$.\n\t\t\n\t\t\\begin{claim}\\label{claim:B}\n\t\t\tThere is an $(n-k) \\times (n-k)$ submatrix $B$ which is disjoint from $N$ and with $|\\disc(B)| \\leq (n-k)^2\/4$.\n\t\t\\end{claim}\n\t\t\\begin{proof}\n\t\t\t\t\\begin{figure}\n\t\t\t\t\t\\centering\n\t\t\t\t\t\\begin{subfigure}{0.45\\textwidth}\n\t\t\t\t\t\t\\begin{tikzpicture}[scale=\\textwidth\/12cm]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\\fill[gray] (10, 0) rectangle (11, 2);\n\t\t\t\t\t\t\t\\fill[gray] (11, 13) rectangle (13, 12);\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\\draw[very thick] (0,0) rectangle (13, 13);\n\t\t\t\t\t\t\t\\draw[very thick] (0, 13) rectangle (11, 2);\n\t\t\t\t\t\t\t\\draw node at (5.5, 7.5) {N};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\\draw[very thick] (0, 0) rectangle (2, 2);\n\t\t\t\t\t\t\t\\draw node at (1,1) {$B_7$};\n\t\t\t\t\t\t\t\\draw[very thick] (2, 0) rectangle (4, 2);\n\t\t\t\t\t\t\t\\draw node at (3,1) {$B_8$};\n\t\t\t\t\t\t\t\\draw[very thick] (4, 0) rectangle (6, 2);\n\t\t\t\t\t\t\t\\draw node at (5,1) {$B_9$};\n\t\t\t\t\t\t\t\\draw[very thick] (6, 0) rectangle (8, 2);\n\t\t\t\t\t\t\t\\draw node at (7,1) {$B_{10}$};\n\t\t\t\t\t\t\t\\draw[very thick] (8, 0) rectangle (10, 2);\n\t\t\t\t\t\t\t\\draw node at (9,1) {$B_{11}$};\n\t\t\t\t\t\t\t\\draw[very thick] (11, 0) rectangle (13, 2);\n\t\t\t\t\t\t\t\\draw node at (12,1) {$B_1$};\n\t\t\t\t\t\t\t\\draw[very thick] (11, 2) rectangle (13, 4);\n\t\t\t\t\t\t\t\\draw node at (12,3) {$B_2$};\n\t\t\t\t\t\t\t\\draw[very thick] (11, 4) rectangle (13, 6);\n\t\t\t\t\t\t\t\\draw node at (12,5) {$B_3$};\n\t\t\t\t\t\t\t\\draw[very thick] (11, 6) rectangle (13, 8);\n\t\t\t\t\t\t\t\\draw node at (12,7) {$B_4$};\n\t\t\t\t\t\t\t\\draw[very thick] (11, 8) rectangle (13, 10);\n\t\t\t\t\t\t\t\\draw node at (12,9) {$B_5$};\n\t\t\t\t\t\t\t\\draw[very thick] (11, 10) rectangle (13, 12);\n\t\t\t\t\t\t\t\\draw node at (12,11) {$B_6$};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\\end{subfigure}\\hfil\n\t\t\t\t\\caption{The matrix $M$ with the submatrices $N$ and $B_1$, $\\dots$, $B_{11}$. The entries of $M$ which are not in any of the submatrices are shown in grey.}\\label{fig:b-part}\n\t\t\t\t\\end{figure}\n\t\t\t\n\t\t\t\tConsider the 11 $(n-k) \\times (n-k)$ disjoint submatrices of $M$ $B_1 , \\dots, B_{11}$ given by \n\t\t\t\t\\[ B_i = \\begin{cases}\n\t\t\t\t\tM[k: n, n - ik: n - (i-1)k ] & \\text{if $i \\leq 6$}\\\\\n\t\t\t\t\tM[(i- 7)(n-k) : (i-6) (n - k), k : n] & \\text{if $ i > 6$},\n\t\t\t\t\\end{cases}\\]\n\t\t\t\tand shown in Figure \\ref{fig:b-part}. The submatrix $B_1$ contains $a_{n,n}$ and sits in the bottom right of $M$, while the others lie along the bottom and right-hand edges of $M$.\n\t\t\t\t\n\t\t\t\tIf one of the $B_i$ satisfies $|\\disc(B_i)| \\leq (n-k)^2\/4$, we are done, so suppose this is not the case. \n\t\t\t\t\n\t\t\t\tWe start by using Observation \\ref{obs:oneof} to show that $\\disc(B_1) >  0$. Let the entries of $B$ be $b_{i,j}$ where $1 \\leq i,j \\leq n-k$. By Claim \\ref{claim:tgeqnmins1}, $2t + \\floor{t\/2} - 2 \\geq n -1$ and, applying Lemma $1$, $b_{i,i} = 1$ for all $i \\leq n- k - 1$. Further, by Observation \\ref{obs:oneof}, we have $b_{i,j} + b_{j,i} \\geq 0$ for all $1 \\leq i,j \\leq n -k -1$. This means \n\t\t\t\t\\[\\disc(B_1) \\geq (n-k - 1) - (2(n-k) - 1) = - (n-k)\\]\n\t\t\t\t For $(n- k) \\geq 5$, $(n-k) < (n-k)^2\/4$ so we must have  $\\disc(B_1) > (n-k)^2\/4$. \n\t\t\t\t \n\t\t\t\t As $\\disc(B_1) > 0 $, if $\\disc(B_i) < 0$ for any $i \\neq 1$, we can use an interpolation argument as in Lemma \\ref{lem:submatrix} to find the claimed matrix. The argument only requires \n\t\t\t\t \\[2(n-k) < \\frac{(n-k)^2}{2}\\]\n\t\t\t\t which is true for $(n-k) > 4$.\n\t\t\t\t \n\t\t\t\tWe must now be in the case where $\\disc(B_i) > (n-k)^2\/4$ for every $i$. The bulk of the work in this case will be bounding the discrepancy of the matrix $N$, and then the discrepancy of $M$. There are $2nk - 12(n-k)^2 \\leq 10(n-k)$ entries of $M$ in the gaps between the $B_i$ i.e. there are at most $10(n-k)$ entries $a_{i,j}$ which are not contained in either $N$ or one of the $B_i$. In particular, we have \n\t\t\t\t\n\t\t\t\t\\begin{align}\n\t\t\t\t\t\\disc(M) &\\geq \\disc(N) + \\disc(B_1) + \\dotsb + \\disc(B_{11})  - 10 (n-k) \\notag\\\\\n\t\t\t\t\t&> \\disc(N) + 11 (n-k)^2\/4 - 10(n-k)\\label{eqn:disc}\n\t\t\t\t\\end{align}\n\t\t\n\t\tLet $s = \\min\\left\\{ k, t + \\floor{t\/2} \\right\\}$ so that $M[1:s, 1:s]$ is $t$ diagonal, and let $r =  k - s$ be the number of remaining rows. Let $a_1, \\dots, a_4$ be the number of 1s in $N$ guaranteed by Lemma \\ref{lem:struct}, and let $a_5$ be the number of additional 1s guaranteed by also applying Observation \\ref{obs:oneof}. This guarantees that at least one of $a_{i,j}$ and $a_{j,i}$ is $1$ for all $(t+2)\/2 \\leq i , j \\leq r$, and $a_5 \\geq r(r-1)$.\n\t\t\n\t\tWe have the following bounds.\n\t\t\\begin{align*}\n\t\ta_1 &= s^2 - \\frac{t(t+1)}{2},\\\\\n\t\ta_2 &= 2 \\sum_{i=1}^r(t-i),\\\\\n\t\ta_3 &= 2 \\sum_{i=1}^r \\left( \\floor{\\frac{t}{2}} - \\floor{\\frac{i-1}{2}} \\right),\\\\\n\t\ta_4 &= r,\\\\\n\t\ta_5 &\\geq r(r-1).\n\t\t\\end{align*}\n\t\t\n\t\tLet us first consider the case where $s = k$, so that $N$ is $t$-diagonal.  In this case $a_2 = \\dotsb = a_5 = 0$, and we can easily write down the discrepancy of $N$ as $k^2 - t(t+1)$. Since $k \\geq 5n\/6$, we get the bound\n\t\t\\begin{align*}\n\t\t\t\\disc(N) &\\geq \\frac{25n^2}{36} - t(t+1).\n\t\t\t\\intertext{Substituting this into (\\ref{eqn:disc}) and using the bounds $(n-5)\/6\\leq n - k \\leq n\/6$ we get}\n\t\t\t\\disc(N) &> \\frac{25n^2}{36} - t(t+1) + \\frac{11}{4}\\left( \\frac{n-5}{6} \\right)^2  - \\frac{10n}{6}\\\\\n\t\t\t&= \\frac{1}{144} \\left( 111 n^2 - 350n - 144t^2 - 144t + 275\\right).\n\t\t\t\\intertext{For $n \\geq 4$, the righthand side is greater than $n^2\/4$ whenever}\n\t\t\tt &< \\frac{1}{12} \\left( \\sqrt{75 n^2 - 350n + 311} - 6 \\right) \\approx 0.721n + o(n).\n\t\t\t\\intertext{Since we have assumed $t \\leq 2n\/3$, we get a contradiction for all sufficiently large $n$. In fact, we get a contradiction for all $n \\geq 39$. The remaining cases need to be checked  using exact values for the floor and ceiling functions which we do with the help of a computer.}\n\t\t\\end{align*}\n\t\t\n\t\tNow we consider the case where $s = t + \\floor{t\/2}$ which is very similar, although more complicated. To be in this case, we must have $t + \\floor{t\/2} \\leq k$ which implies \n\t\t\\[ t + \\frac{t-1}{2} \\leq \\frac{5(n+1)}{6},\\]\n\t\tand $t \\leq (5n + 8)\/9 \\approx 0.556n$.\n\t\t\n\t\t\\begin{align*}\n\t\t\t\\intertext{Start by using the bounds $(t-1)\/2 \\leq \\floor{t\/2}$ and $\\floor{(i-1)\/2} \\leq (i-1)\/2$ to get}\n\t\t\ta_1 + \\dotsb + a_5 &\\geq \\left( t + \\frac{t-1}{2} \\right)^2 - \\frac{t(t+1)}{2} + r (2t -r - 1) + r(t - 1) \\\\&\\qquad - \\frac{r(r-1)}{2} + r + r(r-1)\\\\\n\t\t\t&= \\frac{7t^2}{4} - 2t - \\frac{r^2}{2} + 3rt - \\frac{5r}{2}  + \\frac{1}{4}.\n\t\t\t\\intertext{By definition, $r = k - t - \\floor{t\/2}$, so we get the bounds $5n\/6 - t - t\/2 \\leq r \\leq 5(n+1)\/6 - t - (t-1)\/2$, and substituing these in gives}\n\t\t\ta_1 + \\dotsb + a_5 &\\geq \\frac{7}{4} t^2 - 2t + \\frac{1}{4} - \\frac{1}{2} \\left( \\frac{5(n+1)}{6} - t - \\frac{t-1}{2}\\right)^2 + 3t \\left( \\frac{5n}{6} -t - \\frac{t}{2} \\right)\\\\&\\qquad - \\frac{5}{2} \\left( \\frac{5(n+1)}{6} - t - \\frac{t-1}{2} \\right)\\\\\n\t\t\t&= \\frac{1}{72} \\left( - 25n^2 + 270nt - 230n - 279 t^2 + 270t - 286 \\right) \n\t\t\t\\intertext{Plugging this into (\\ref{eqn:disc}) and using the bounds $5n\/6 \\leq k \\leq 5(n+1)\/6$ we get}\n\t\t\t\\disc(M) &>2 (a_1 + \\dotsb a_5) - \\left( \\frac{5(n+1)}{6} \\right)^2 + \\frac{11}{4} \\left( \\frac{n-5}{6} \\right)^2 - \\frac{10n}{6}\\\\\n\t\t\t&\\geq \\frac{1}{48} \\left( - 63n^2 + 360nt - 490n - 372t^2 + 360t - 323 \\right).\n\t\t\\end{align*}\n\t\t\t\tWhen $n \\geq 27$, this is greater than $n^2\/4$ whenever\n\t\\begin{align*}\n\t\t&\\frac{1}{186}\\left(90n + 90 - \\sqrt{1125 n^2 - 29370n - 21939}\\right) <\\\\\n\t\t&\\qquad t < \\frac{1}{186}\\left(90n + 90 + \\sqrt{1125 n^2 - 29370n - 21939}\\right),\n \t\\end{align*}\n or approximately,\n \\[0.304n < t < 0.664n.\\]\n We have the bounds \n \\[ \\frac{1}{4} \\left( \\sqrt{3n^2 - 6n + 7} - 2 \\right) \\leq  t \\leq \\frac{5n + 8}{9}, \\]\n and so, for $n \\geq 36$, $\\disc(M) > n^2\/4$.\n \n This again leaves a few cases which we check with the help of a computer (although they could feasibly be checked by hand).\t\t\n\\end{proof}\n\t\t\n\t\tGiven a submatrix $B$ as in the above claim we apply the induction hypothesis, noting that $(n-k) \\geq 5$ since $n \\geq 30$, to find that $B$ is diagonal. Let $C$ be the diagonal submatrix obtained from applying Lemma 4 to $B$, and let $C$ be $\\ell$-diagonal up to rotation. Note that $\\ell \\geq 3$ as $(n-k) \\geq 5$, and we can assume $\\ell \\leq 2n\/3$ as $M$ is not diagonal.\n\t\t\n\t\tHence, $C$ contains exactly one of $a_{1,1}$, $a_{1,n}$, $a_{n,1}$ and $a_{n,n}$, and we will split into cases based on which one $C$ contains. We will also sometimes need to consider cases for whether the entry is $1$ or $-1$, but in all cases we will find a contradiction.\n\t\t\n\t\tFrom Lemma \\ref{lem:struct} applied to $M'$ and Claim \\ref{claim:tgeqnmins1}, we already know some of the entries and we highlight some important entries in the following claim.\n\t\t\n\t\t\\begin{claim}\\label{claim:particular1s}\n\t\t\tWe have\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $a_{j,1} = a_{1, j} = \\begin{cases}\n\t\t\t\t\t1 & t + 1 \\leq j \\leq n-1,\\\\\n\t\t\t\t\t-1 & 1 \\leq j \\leq t,\n\t\t\t\t\\end{cases}$\n\t\t\t\t\\item $a_{2,t} = a_{t,2} = 1$,\n\t\t\t\t\\item $a_{i,i} = 1$ for all $(t+2)\/2 \\leq i \\leq n -1$.\n\t\t\t\\end{enumerate}\n\t\t\\end{claim}\n\t\t\n\t\tSuppose the submatrix $C$ contains $a_{1,1}$ so sits in the top-left corner. Since $M[1:t + \\floor{t\/2}, 1:t + \\floor{t\/2}]$ is $t$-diagonal, $C$ must also be $t$-diagonal. As $C$ was found by applying Lemma \\ref{lem:struct} to $B$, it must contain a $-1$ from $B$. Hence, $t \\geq 5n\/6$ which is a contradiction as we assumed that $t \\leq  2n\/3$.\n\t\t\n\t\tSuppose instead that $C$ contains $a_{1,n}$ so sits in the top-right corner. Since $\\ell \\geq 3$, if the corner entry is $-1$, so is the entry $a_{1,n-1}$, but this contradicts Claim \\ref{claim:particular1s}. Suppose instead the corner entry is $1$. Since $C$ is $\\ell$-diagonal up to rotation we have, for all $1 \\leq i, (n - j  + 1) \\leq \\ell + \\floor{\\ell\/2}$, \n\t\t\\begin{equation}\\label{eqn:C}\n\t\t\ta_{i,j} = \\begin{cases}\n\t\t\t\t-1 & i + (n -j + 1) \\geq \\ell + 2,\\\\\n\t\t\t\t1 & \\text{otherwise}.\n\t\t\t\\end{cases}\n\t\t\\end{equation}\n\t\t\n\t\tIf $n - \\ell > t$, then $a_{1, n-\\ell} = -1$ by (\\ref{eqn:C}) and $a_{1, n-\\ell} = 1$ by Claim \\ref{claim:particular1s}.\n\t\tSuppose $n - \\ell < t$. Then $a_{1, t} = 1$ by (\\ref{eqn:C}) and $a_{1, t} = -1$ as  $M[1:t, 1:t]$ is $t$-diagonal.\n\t\tFinally, when $n-\\ell = t$, we have $a_{2,t} = -1$ by (\\ref{eqn:C}) and $a_{2,t} = 1$ from Claim \\ref{claim:particular1s}. Some illustrative examples of these three cases are shown in Figure \\ref{fig:case-1-n}.\n\t\t\n\t\t\\begin{figure}\n\t\t\t\\centering\n\t\t\t\\begin{subfigure}{0.25\\textwidth}\n\t\t\t\t\\centering\n\t\t\t\t\\begin{tikzpicture}[scale=\\textwidth\/12cm]\n\t\t\t\t\t\n\t\t\t\t\t\\fill[fill=yellow5] (0, 11) rectangle(6, 12);\n\t\t\t\t\t\\fill[fill=blue5] (6, 11) rectangle(11, 12);\n\t\t\t\t\t\\fill[fill=blue5] (5, 10) rectangle(6,11);\n\t\t\t\t\t\\draw[very thin, gray] (0,0) grid (12,12);\n\t\t\t\t\t\\draw[thick] (6,12)\n\t\t\t\t\t\\foreach \\myvar in {6, 5, 4, ..., 0}\n\t\t\t\t\t\t-- (\\myvar, 6 + \\myvar) -- (\\myvar, 5 + \\myvar)\n\t\t\t\t\t} -- (0, 12) -- (6,12);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (9,12)\n\t\t\t\t\t\\foreach \\myvar in {9, 10, ..., 12}\n\t\t\t\t\t\t-- (\\myvar, 21 -  \\myvar) -- (\\myvar, 20 - \\myvar)\n\t\t\t\t\t};\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (8,8) rectangle (12, 12);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (8, 12) -- (9,11);\n\t\t\t\t\t\\draw[thick] (9, 12) -- (8,11);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[very thick] (0,0) -- (12,0) -- (12,12) -- (0,12) --(0,0);\n\t\t\t\t\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\\caption{$n- \\ell > t$}\n\t\t\t\\end{subfigure}\\hfil\n\t\t\t\\begin{subfigure}{0.25\\textwidth}\n\t\t\t\t\\centering\n\t\t\t\t\\begin{tikzpicture}[scale=\\textwidth\/12cm]\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\\fill[fill=yellow5] (0, 11) rectangle(6, 12);\n\t\t\t\t\t\\fill[fill=blue5] (6, 11) rectangle(11, 12);\n\t\t\t\t\t\\fill[fill=blue5] (5, 10) rectangle(6,11);\n\t\t\t\t\t\\draw[very thin, gray] (0,0) grid (12,12);\n\t\t\t\t\t\\draw[thick] (6,12)\n\t\t\t\t\t\\foreach \\myvar in {6, 5, 4, ..., 0}\n\t\t\t\t\t\t-- (\\myvar, 6 + \\myvar) -- (\\myvar, 5 + \\myvar)\n\t\t\t\t\t} -- (0, 12) -- (6,12);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (9,12)\n\t\t\t\t\t\\foreach \\myvar in {5, 6, ..., 12}\n\t\t\t\t\t\t-- (\\myvar, 17 -  \\myvar) -- (\\myvar, 16 - \\myvar)\n\t\t\t\t\t};\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (3,3) rectangle (12, 12);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (5, 12) -- (6,11);\n\t\t\t\t\t\\draw[thick] (6, 12) -- (5,11);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[very thick] (0,0) -- (12,0) -- (12,12) -- (0,12) --(0,0);\n\t\t\t\t\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\\caption{$n- \\ell < t$}\n\t\t\t\\end{subfigure}\\hfil\n\t\t\t\\centering\n\t\t\t\\begin{subfigure}{0.25\\textwidth}\n\t\t\t\t\\centering\n\t\t\t\t\\begin{tikzpicture}[scale=\\textwidth\/12cm]\n\t\t\t\t\t\n\t\t\t\t\t\\fill[fill=yellow5] (0, 11) rectangle(6, 12);\n\t\t\t\t\t\\fill[fill=blue5] (6, 11) rectangle(11, 12);\n\t\t\t\t\t\\fill[fill=blue5] (5, 10) rectangle(6,11);\n\t\t\t\t\t\\draw[very thin, gray] (0,0) grid (12,12);\n\t\t\t\t\t\\draw[thick] (6,12)\n\t\t\t\t\t\\foreach \\myvar in {6, 5, 4, ..., 0}\n\t\t\t\t\t\t-- (\\myvar, 6 + \\myvar) -- (\\myvar, 5 + \\myvar)\n\t\t\t\t\t} -- (0, 12) -- (6,12);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (9,12)\n\t\t\t\t\t\\foreach \\myvar in {6, 7, ..., 12}\n\t\t\t\t\t\t-- (\\myvar, 18 -  \\myvar) -- (\\myvar, 17 - \\myvar)\n\t\t\t\t\t};\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (3,3) rectangle (12, 12);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[thick] (5, 11) -- (6,10);\n\t\t\t\t\t\\draw[thick] (6, 11) -- (5,10);\n\t\t\t\t\t\n\t\t\t\t\t\\draw[very thick] (0,0) -- (12,0) -- (12,12) -- (0,12) --(0,0);\n\t\t\t\t\t\n\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\\caption{$n - \\ell =t$}\n\t\t\t\\end{subfigure\n\t\t\t\\caption{The three cases when $C$ contains $a_{1,n}$ and $a_{1,n} = 1$. The yellow squares represent some of the $a_{i,j}$ which are known to be $-1$ from Claim \\ref{claim:particular1s} and the blue squares those which are $1$. The square which gives the contradiction is marked with a cross.}\n\t\t\t\\label{fig:case-1-n}\n\t\t\\end{figure}\n\t\t\n\t\tThe case where $C$ contains $a_{n,1}$ is done in the same way with the rows and columns swapped.\n\t\t\n\t\tThis leaves the case where $C$ contains $a_{n,n}$. Since $\\ell \\geq 3$, if the entry $a_{n, n}$ equals $-1$, so does the entry $a_{n-1, n-1}$, and this contradicts Claim \\ref{claim:particular1s}. If instead $a_{n,n} = 1$, we consider the entry $a_{i, i}$ where $i = n  + 1 - \\ceil{(l+2)\/2}$, which must be $-1$. However, since $\\ell \\leq 2n\/3$, \\[n  + 1 - \\ceil{(l+2)\/2}\\geq n + - \\frac{n}{3} - \\frac{1}{2} > \\frac{n}{3} + 1 \\geq \\frac{t+2}{2},\\] and $a_{i,i} = 1$ by Claim \\ref{claim:particular1s}. This final contradiction is shown in Figure \\ref{fig:case-n-n}.\n\t\t\\begin{figure}\n\t\t\t\\centering\n\t\t\t\\begin{tikzpicture}[scale=0.4\\textwidth\/12cm]\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\foreach \\myvar in {4,...,11}\n\t\t\t\t\t\\fill[blue5] (\\myvar-1, 12-\\myvar) rectangle +(1,1);}\n\t\t\t\t\n\t\t\t\t\\draw[very thin, gray] (0,0) grid (12,12);\n\t\t\t\t\\draw[thick] (6,12)\n\t\t\t\t\\foreach \\myvar in {6, 5, 4, ..., 0}\n\t\t\t\t\t-- (\\myvar, 6 + \\myvar) -- (\\myvar, 5 + \\myvar)\n\t\t\t\t} -- (0, 12) -- (6,12);\n\t\t\t\t\n\t\t\t\t\\draw[thick] (7,0)\n\t\t\t\t\\foreach \\myvar in {7,8, ..., 12}\n\t\t\t\t\t-- (\\myvar, \\myvar - 7) -- (\\myvar, \\myvar - 6)\n\t\t\t\t};\n\t\t\t\t\n\t\t\t\t\\draw[thick] (5,0) rectangle (12, 7);\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\draw[thick] (8, 3) -- (9,4);\n\t\t\t\t\\draw[thick] (8, 4) -- (9,3);\n\t\t\t\t\n\t\t\t\t\\draw[very thick] (0,0) -- (12,0) -- (12,12) -- (0,12) --(0,0);\n\t\t\t\t\n\t\t\t\\end{tikzpicture}\n\t\t\t\\caption{The case where $C$ contains $a_{n,n}$ and $a_{n,n} = 1$. The square marked with a cross gives a contradiction.}\n\t\t\t\\label{fig:case-n-n}\n\t\t\\end{figure}\n\t\\end{proof}\n\n\tWe remark that it should be possible to improve the bound $n^2\/4$ using a similar proof provided one can check a large enough base case. Indeed, we believe that all the steps in the above proof hold when the bound is increased to $n^2\/3$, but only when $n$ is large enough. For example, Claim \\ref{claim:tgeqnmins1} fails for $n = 127$ and our proof of Claim \\ref{claim:B} fails for $n = 86$. Checking base cases this large is far beyond the reach of our computer check, and some new ideas would be needed here.\n\t \n\\section{Open problems}\\label{sec:open-problems}\n\n\t\tThe main open problem is to determine the correct lower bound for the discrepancy of a non-diagonal $\\{-1,1\\}$-matrix with no zero-sum squares. We have improved the lower bound to $n^2\/4$, but this does not appear to be optimal.\n\n\t\tThe best known construction is the following example by Ar\\'evalo, Montejano and Rold\\'an-Pensado \\cite{arevalo2020zero}. Let $M = \\left(a_{i,j}\\right)$ be given by \\[a_{i,j} = \\begin{cases}\n\t\t\t-1 & \\text{$i$ and $j$ are odd},\\\\\n\t\t\t1 & \\text{otherwise}.\n\t\t\\end{cases}\\]\n\t\tThis has discrepancy $n^2\/2$ when $n$ is even and $(n-1)^2\/2 - 1$ when $n$ is odd. With the help of a computer we have verified that this construction is best possible when $9 \\leq n \\leq 32$, and we conjecture that this holds true for all $n \\geq 9$. In fact, our computer search shows that the above example is the unique zero-sum square free non-diagonal matrix with minimum (in magnitude) discrepancy, up to reflections and multiplying by $-1$. \n\n\t\tWe note that the condition $n \\geq 9$ is necessary, as shown by the $8 \\times 8$ zero-sum square free $\\{-1, 1\\}$-matrix with discrepancy 30 given in Figure \\ref{fig:counter}.\n\n\\begin{restatable}{conjecture}{conj}\n\tLet $n\\geq9$. Every $n \\times n$ non-diagonal $\\{-1, 1\\}$-matrix $M$ with \\[|\\disc(M)| \\leq \\begin{cases}\n\t\t\\frac{n^2}{2} - 1 & \\text{$n$ is even}\\\\\n\t\t\\frac{(n-1)^2}{2} -2 & \\text{$n$ is odd}\n\t\\end{cases}\\]\n\tcontains a zero-sum square.\n\\end{restatable}\n\n\n\n\t\t\t\\begin{figure}\n\t\t\\centering\n\t\t\\begin{tikzpicture}[scale=0.5\\textwidth\/8cm]\n\t\t\t\n\t\t\t\n\t\t\t\\fill[blue5] (0,0) rectangle (1,1);\n\t\t\t\\fill[blue5] (1,0) rectangle (2,1);\n\t\t\t\\fill[yellow5] (2,0) rectangle (3,1);\n\t\t\t\\fill[blue5] (3,0) rectangle (4,1);\n\t\t\t\\fill[blue5] (4,0) rectangle (5,1);\n\t\t\t\\fill[blue5] (5,0) rectangle (6,1);\n\t\t\t\\fill[blue5] (6,0) rectangle (7,1);\n\t\t\t\\fill[blue5] (7,0) rectangle (8,1);\n\t\t\t\n\t\t\t\\fill[blue5] (0,1) rectangle (1,2);\n\t\t\t\\fill[blue5] (1,1) rectangle (2,2);\n\t\t\t\\fill[blue5] (2,1) rectangle (3,2);\n\t\t\t\\fill[blue5] (3,1) rectangle (4,2);\n\t\t\t\\fill[blue5] (4,1) rectangle (5,2);\n\t\t\t\\fill[blue5] (5,1) rectangle (6,2);\n\t\t\t\\fill[blue5] (6,1) rectangle (7,2);\n\t\t\t\\fill[blue5] (7,1) rectangle (8,2);\n\t\t\t\n\t\t\t\\fill[blue5] (0,2) rectangle (1,3);\n\t\t\t\\fill[blue5] (1,2) rectangle (2,3);\n\t\t\t\\fill[blue5] (2,2) rectangle (3,3);\n\t\t\t\\fill[blue5] (3,2) rectangle (4,3);\n\t\t\t\\fill[blue5] (4,2) rectangle (5,3);\n\t\t\t\\fill[blue5] (5,2) rectangle (6,3);\n\t\t\t\\fill[blue5] (6,2) rectangle (7,3);\n\t\t\t\\fill[blue5] (7,2) rectangle (8,3);\n\t\t\t\n\t\t\t\\fill[yellow5] (0,3) rectangle (1,4);\n\t\t\t\\fill[blue5] (1,3) rectangle (2,4);\n\t\t\t\\fill[blue5] (2,3) rectangle (3,4);\n\t\t\t\\fill[blue5] (3,3) rectangle (4,4);\n\t\t\t\\fill[blue5] (4,3) rectangle (5,4);\n\t\t\t\\fill[blue5] (5,3) rectangle (6,4);\n\t\t\t\\fill[blue5] (6,3) rectangle (7,4);\n\t\t\t\\fill[blue5] (7,3) rectangle (8,4);\n\t\t\t\n\t\t\t\\fill[yellow5] (0,4) rectangle (1,5);\n\t\t\t\\fill[yellow5] (1,4) rectangle (2,5);\n\t\t\t\\fill[blue5] (2,4) rectangle (3,5);\n\t\t\t\\fill[blue5] (3,4) rectangle (4,5);\n\t\t\t\\fill[blue5] (4,4) rectangle (5,5);\n\t\t\t\\fill[blue5] (5,4) rectangle (6,5);\n\t\t\t\\fill[blue5] (6,4) rectangle (7,5);\n\t\t\t\\fill[blue5] (7,4) rectangle (8,5);\n\t\t\t\n\t\t\t\\fill[yellow5] (0,5) rectangle (1,6);\n\t\t\t\\fill[yellow5] (1,5) rectangle (2,6);\n\t\t\t\\fill[yellow5] (2,5) rectangle (3,6);\n\t\t\t\\fill[blue5] (3,5) rectangle (4,6);\n\t\t\t\\fill[blue5] (4,5) rectangle (5,6);\n\t\t\t\\fill[blue5] (5,5) rectangle (6,6);\n\t\t\t\\fill[blue5] (6,5) rectangle (7,6);\n\t\t\t\\fill[yellow5] (7,5) rectangle (8,6);\n\t\t\t\n\t\t\t\n\t\t\t\\fill[yellow5] (0,6) rectangle (1,7);\n\t\t\t\\fill[yellow5] (1,6) rectangle (2,7);\n\t\t\t\\fill[yellow5] (2,6) rectangle (3,7);\n\t\t\t\\fill[yellow5] (3,6) rectangle (4,7);\n\t\t\t\\fill[blue5] (4,6) rectangle (5,7);\n\t\t\t\\fill[blue5] (5,6) rectangle (6,7);\n\t\t\t\\fill[blue5] (6,6) rectangle (7,7);\n\t\t\t\\fill[blue5] (7,6) rectangle (8,7);\n\t\t\t\n\t\t\t\\fill[yellow5] (0,7) rectangle (1,8);\n\t\t\t\\fill[yellow5] (1,7) rectangle (2,8);\n\t\t\t\\fill[yellow5] (2,7) rectangle (3,8);\n\t\t\t\\fill[yellow5] (3,7) rectangle (4,8);\n\t\t\t\\fill[yellow5] (4,7) rectangle (5,8);\n\t\t\t\\fill[blue5] (5,7) rectangle (6,8);\n\t\t\t\\fill[blue5] (6,7) rectangle (7,8);\n\t\t\t\\fill[blue5] (7,7) rectangle (8,8);\n\t\t\t\n\t\t\t\\draw[step=1, thin] (0,0) grid (8,8);\n\t\t\t\n\t\t\t\\draw[very thick] (0,0) rectangle(8,8);\n\t\t\\end{tikzpicture}\n\t\t\n\t\t\\caption{An $8 \\times 8$ $\\{-1,1\\}$-matrix with no zero-sum squares and discrepancy 30. The yellow squares represent a $-1$ and the blue squares represent a $1$.}\n\t\t\\label{fig:counter}\n\t\\end{figure}\n\n\n\n\tAr\\'evalo, Montejano and Rold\\'an-Pensado prove their result for both $n \\times n$ and $n \\times (n+1)$ matrices, and computational experiments suggest that Theorem \\ref{thm:low-bound} holds for $n \\times (n+1)$ matrices as well. More generally, what is the best lower bound for a general $n \\times m$ matrix when $n$ and $m$ are large?\n\t\n\t\\begin{problem}\n\t\tLet $f(n, m)$ be the minimum $d \\in \\mathbb{N}$ such that there exists an $n \\times m$ non-diagonal $\\{-1,1\\}$ matrix $M$ with $|\\disc(M)| \\leq d$. What are the asymptotics of $f(n,m)$?\n\t\\end{problem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAnti-de Sitter (AdS) backgrounds of supergravity are an essential part of the \nAdS\/CFT correspondence \\cite{Maldacena:1997re} and  have been studied\nin recent years from varying perspectives. On the one hand they can be constructed as compactifications of higher-dimensional supergravities as is the natural\nset up in the AdS\/CFT correspondence.\\footnote{See \\cite{Kehagias:1998gn,Morrison:1998cs} for earlier work and \ne.g.\\ \\cite{Polchinski:2010hw} and references therein for a more recent review.} Alternatively, one can investigate and, if possible, \nclassify their appearance directly in a given supergravity without relating it\nto any compactification.\n\nFor a given AdS background it is also of interest to study its properties\nand in particular its moduli space $\\mathcal{M}$, i.e.\\ the subspace of the scalar field space\nthat is spanned by flat directions of the AdS background.\nThis moduli space has been heavily investigated  \nin Minkowskian backgrounds of string theory as it prominently appears\nin its low energy effective theory.\nFor AdS backgrounds much less is known about $\\mathcal{M}$, partly because the defining equations are more involved and furthermore quantum corrections contribute unprotected.\n\n\n\nIn \\cite{deAlwis:2013jaa,Louis:2014gxa} supersymmetric $\\textrm{AdS}_{4}$ vacua \nand their classical supersymmetric moduli spaces\nwere studied  in four-dimensional ($d=4$) supergravities \nwith $\\mathcal{N}=1,2,4$ supersymmetry\nwithout considering their relation to higher-dimensional theories.\\footnote{Throughout this paper we only consider $\\textrm{AdS}$ backgrounds that preserve all supercharges of a given supergravity and furthermore only consider the subspace of the moduli space that preserves all these supercharges. This is what we mean by supersymmetric $\\textrm{AdS}$ backgrounds and supersymmetric moduli spaces.}\nFor $\\mathcal{N}=1$ it was found that  the supersymmetric moduli space is at best a real submanifold of the original K\\\"ahler field space.\nSimilarly, for $\\mathcal{N}=2$ the supersymmetric moduli space \nis at best a product of a real manifold times a K\\\"ahler manifold\nwhile  $\\mathcal{N}=4$ $\\textrm{AdS}$ backgrounds have no supersymmetric moduli space.\n This analysis was repeated for $\\textrm{AdS}_{5}$ vacua in  $d=5$ gauged supergravity\nwith 16 supercharges ($\\mathcal{N}=4$) in \\cite{Louis:2015dca} and for $\\textrm{AdS}_{7}$ vacua in  $d=7$ gauged supergravity with 16~supercharges in \\cite{Louis:2015mka}. For the $d=5,\\, \\mathcal{N}=4$ theories it was shown that the supersymmetric moduli space is\nthe coset $\\mathcal{M}=SU(1,m)\/(U(1)\\times SU(m))$ while in $d=7$ it was proven that again \nno supersymmetric moduli space exists.\n\n\nIn this paper we focus on supersymmetric $\\textrm{AdS}_{5}$ vacua in $d=5$ gauged \nsupergravities with eight supercharges ($\\mathcal{N}=2$)\ncoupled to an arbitrary number of vector-, tensor- and hypermultiplets. \nA related analysis was carried out in \\cite{Tachikawa:2005tq}\nfor the coupling of Abelian vector multiplets and hypermultiplets.\nWe confirm the results of \\cite{Tachikawa:2005tq} and generalize \nthe analysis by including tensor multiplets and\nnon-Abelian vector multiplets. \nIn particular, we show that also in this more general case \nthe unbroken gauge group has to \n be of the form $H\\times U(1)_{R}$\nwhere the $U(1)_R$-factor is gauged by the graviphoton.\nThis specifically forbids unbroken semisimple gauge groups in AdS\nbackgrounds.\n\nIn a second step\nwe study the supersymmetric moduli space $\\mathcal{M}$ \nof the previously obtained  $\\textrm{AdS}_{5}$ backgrounds\nand show that it necessarily is a K\\\"ahler submanifold of the quaternionic scalar field space $\\mathcal{T}_H$ spanned by all scalars in the hypermultiplets.\\footnote{This result was also obtained in  \\cite{Tachikawa:2005tq}. Our results is more general as we include tensor multiplets and non-Abelian vector multiplet in the analysis.}\nThis is indeed consistent with  the AdS\/CFT correspondence where the \nmoduli space $\\mathcal{M}$ is mapped to the conformal manifold of the dual \nsuperconformal field theory (SCFT). For the gauged supergravities considered here\nthe dual theories are $d=4,\\, \\mathcal{N}=1$ SCFTs.\nIn \\cite{Asnin:2009xx} it was indeed shown that \nthe conformal manifold of these SCFTs  is a K\\\"ahler manifold. \n\nThe organization of this paper is as follows. In section \\ref{sec:sugra} we briefly review gauged $\\mathcal{N}=2$ supergravities in five dimensions. This will then be used to study the conditions for the existence of supersymmetric $\\textrm{AdS}_{5}$ vacua and determine some of their properties in section~\\ref{sec:vacua}. Finally, in section \\ref{sec:moduli} we compute the conditions on the moduli space of these vacua and show that it is a K\\\"ahler manifold. \n\n\n\n\\section{Gauged $\\mathcal{N}=2$ supergravity in five dimensions}\\label{sec:sugra}\n\nTo begin with let us review five-dimensional gauged $\\mathcal{N}=2$ supergravity following  \\cite{Gunaydin:2000xk,Bergshoeff:2002qk,Bergshoeff:2004kh}.\\footnote{Ref.~\\cite{Bergshoeff:2004kh}\nconstructed the most general version of five-dimensional gauged $\\mathcal{N}=2$ supergravity.} The theory consists of the gravity multiplet with field content\n\\begin{equation}\n\\{g_{\\mu\\nu}, \\Psi_{\\mu}^{\\mathcal{A}}, A_{\\mu}^{0}\\}\\ , \\quad \\mu,\\nu=0,...,4\\ ,\\quad \n\\mathcal{A}=1,2\\ ,\n\\end{equation}\nwhere $g_{\\mu\\nu}$ is the metric of space-time, $\\Psi_{\\mu}^{\\mathcal{A}}$ is\nan $SU(2)_{R}$-doublet of symplectic Majorana gravitini and $A_{\\mu}^{0}$ is the graviphoton. In\nthis paper we consider theories that additionally contain $n_{V}$\nvector multiplets, $n_{H}$ hypermultiplets and $n_{T}$ tensor\nmultiplets. A vector multiplet $\\{A_{\\mu}, \\lambda^{\\mathcal{A}}, \\phi\\}$\ntransforms in the adjoint representation of the gauge group $G$ and contains a vector $A_{\\mu}$, a doublet of gauginos $\\lambda^{\\mathcal{A}}$ and a real scalar~$\\phi$. In $d=5$ a vector is Poincar\\'e dual  to an antisymmetric \ntensor field $B_{\\mu\\nu}$ which carry an arbitrary representation of $G$.  This gives rise to tensor multiplets which have the same field content as vector multiplets, but with a two-form instead of a vector. Since vector- and tensor multiplets mix in the Lagrangian, we label their scalars $\\phi^{i}$ by the same index $i,j=1,...,n_{V}+n_{T}$. Moreover, we label the vector fields (including the graviphoton) by $I,J=0,1,...,n_{V}$, the tensor fields by $M,N=n_{V}+1,...,n_{V}+n_{T}$ and also introduce a combined index $\\tilde{I}=(I,M)$. Finally, the $n_{H}$ hypermultiplets\n\\begin{equation}\n\\{q^{u}, \\zeta^{\\alpha}\\}, \\quad u=1,2,...,4n_{H}\\ , \\quad \\alpha=1,2,...,2n_{H}\\ , \n\\end{equation}\ncontain $4n_{H}$ real scalars $q^{u}$ and $2n_{H}$ hyperini $\\zeta^{\\alpha}$.\n\nThe bosonic Lagrangian of $\\mathcal{N}=2$ gauged supergravity in five dimensions reads\\footnote{\nNote that we set the gravitational constant $\\kappa=1$ in this paper.}\n\\cite{Bergshoeff:2004kh}\n\\begin{equation}\\label{eq:Lagrangian}\n\\begin{aligned}\ne^{-1}\\mathcal{L}&=\\tfrac{1}{2}\n-\\tfrac{1}{4}a_{\\tilde{I}\\tilde{J}}H^{\\tilde{I}}_{\\mu\\nu}H^{\\tilde{J}\\mu\\nu}-\\tfrac{1}{2}g_{ij}\\mathcal{D}_{\\mu}\\phi^{i}\\mathcal{D}^{\\mu}\\phi^{j}-\\tfrac{1}{2}G_{uv}\\mathcal{D}_{\\mu}q^{u}\\mathcal{D}^{\\mu}q^{v}-g^{2}V(\\phi,q)\\\\\n&+\\tfrac{1}{16g}e^{-1}\\epsilon^{\\mu\\nu\\rho\\sigma\\tau}\\Omega_{MN}B^{M}_{\\mu\\nu}\\left(\\partial_{\\rho}B^{N}_{\\sigma\\tau}+2gt_{IJ}^{N}A_{\\rho}^{I}F_{\\sigma\\tau}^{J}+gt_{IP}^{N}A_{\\rho}^{I}B_{\\sigma\\tau}^{P}\\right)\\\\\n&+\\tfrac{1}{12}\\sqrt{\\tfrac{2}{3}}e^{-1}\\epsilon^{\\mu\\nu\\rho\\sigma\\tau}C_{IJK}A_{\\mu}^{I}\\left[F_{\\nu\\rho}^{J}F_{\\sigma\\tau}+f_{FG}^{J}A_{\\nu}^{F}A_{\\rho}^{G}\\left(-\\tfrac{1}{2}F_{\\sigma\\tau}^{K}+\\tfrac{g^{2}}{10}f_{HL}^{K}A_{\\sigma}^{H}A_{\\tau}^{L}\\right)\\right]\\\\\n&-\\tfrac{1}{8}e^{-1}\\epsilon^{\\mu\\nu\\rho\\sigma\\tau}\\Omega_{MN}t_{IK}^{M}t_{FG}^{N}A_{\\mu}^{I}A_{\\nu}^{F}A_{\\rho}^{G}\\left(-\\tfrac{g}{2}F_{\\sigma\\tau}^{K}+\\tfrac{g^{2}}{10}f_{HL}^{K}A_{\\sigma}^{H}A_{\\tau}^{L}\\right)\n\\ .\n\\end{aligned}\n\\end{equation}\nIn the rest of this section we recall the various ingredients which\nenter this Lagrangian.\nFirst of all $H^{\\tilde{I}}_{\\mu\\nu}=(F_{\\mu\\nu}^{I}, B_{\\mu\\nu}^{M})$\nwhere\n$F_{\\mu\\nu}^{I}=2\\partial_{[\\mu}A_{\\nu]}^{I}+gf_{JK}^{I}A^{J}_{\\mu}A^{K}_{\\nu}$\nare the field strengths with $g$ being the gauge coupling constant.\nThe scalar fields in $\\mathcal{L}$ can be interpreted as coordinate charts from spacetime $M_{5}$ to a target space $\\mathcal{T}$,\n\\begin{equation}\\label{eq:target space}\n\\phi^{i} \\otimes q^{u}: M_{5} \\longrightarrow \\mathcal{T}.\n\\end{equation}\nLocally $\\mathcal{T}$ is a product $\\mathcal{T}_{VT} \\times \\mathcal{T}_{H}$ where the first\nfactor is a projective special real manifold $(\\mathcal{T}_{VT}, g)$ of\ndimension $n_{V}+n_{T}$. It is constructed as a hypersurface in an $(n_{V}+n_{T}+1)$-dimensional real manifold $\\mathcal{H}$ with local coordinates $h^{\\tilde{I}}$. This hypersurface is defined by \n\\begin{equation}\\label{eq:polynomial}\nP(h^{\\tilde{I}}(\\phi))=C_{\\tilde{I}\\tilde{J}\\tilde{K}}h^{\\tilde{I}}h^{\\tilde{J}}h^{\\tilde{K}}=1,\n\\end{equation}\nwhere $P(h^{\\tilde{I}}(\\phi))$ is a cubic homogeneous polynomial with $C_{\\tilde{I}\\tilde{J}\\tilde{K}}$ constant and completely symmetric. Thus $\\mathcal{T}_{VT}=\\{P=1\\}\\subset \\mathcal{H}$. \n\nThe generalized gauge couplings in \\eqref{eq:Lagrangian} correspond to a positive metric on the ambient space $\\mathcal{H}$, given by\n\\begin{equation}\\label{adef}\na_{\\tilde{I}\\tilde{J}}:=-2C_{\\tilde{I}\\tilde{J}\\tilde{K}}h^{\\tilde{K}}+3h_{\\tilde{I}}h_{\\tilde{J}}\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:hlower}\n h_{\\tilde{I}}= C_{\\tilde{I}\\tilde{J}\\tilde{K}}h^{\\tilde{J}}h^{\\tilde{K}}\\ . \n\\end{equation}\nThe pullback metric $g_{ij}$ is the (positive) metric on the hypersurface \n$\\mathcal{T}_{VT}$ and is  given by\n\\begin{equation}\\label{gpull}\ng_{ij}:=h_{i}^{\\tilde{I}}h_{j}^{\\tilde{J}}a_{\\tilde{I}\\tilde{J}}\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:hder}\nh_{i}^{\\tilde{I}}:=-\\sqrt{\\tfrac{3}{2}}\\,\\partial_{i}h^{\\tilde{I}}(\\phi)\\\n.\n\\end{equation}\nThese quantities satisfy (see Appendix C in \\cite{Bergshoeff:2004kh} for more details)\n\\begin{equation}\nh^{\\tilde{I}}h_{\\tilde{I}}=1\\ ,\\qquad\nh_{\\tilde{I}}h_{i}^{\\tilde{I}}=0\\ ,\\qquad\nh_{\\tilde{I}}h_{\\tilde{J}}+h_{\\tilde{I}}^{i}h_{\\tilde{J}i}=a_{\\tilde{I}\\tilde{J}} \\ ,\n\\label{eq:hmetric}\n\\end{equation}\nwhere we raise and lower indices with the appropriate metrics $a_{\\tilde{I}\\tilde{J}}$ or $g_{ij}$ respectively.\nThe metric $g_{ij}$ induces a covariant derivative which acts on the $h^{\\tilde{I}}_{i}$ via \n\\begin{equation}\\label{eq:covderh}\n\\nabla_{i}h^{\\tilde{I}}_{j}=-\\sqrt{\\tfrac{2}{3}}\\, (h^{\\tilde{I}}g_{ij}+T_{ijk}h^{\\tilde{I}k})\\ ,\n\\end{equation}\nwhere $T_{ijk}:=C_{\\tilde{I}\\tilde{J}\\tilde{K}}h_{i}^{\\tilde{I}}h_{j}^{\\tilde{J}}h_{k}^{\\tilde{K}}$ is a completely symmetric tensor. \n\nThe second factor of $\\mathcal{T}$ in (\\ref{eq:target space}) is a quaternionic K\\\"ahler manifold $(\\mathcal{T}_{H},G, Q)$ of real dimension $4n_{H}$ (see \\cite{Andrianopoli:1996cm} for a more extensive introduction). Here $G_{uv}$ is a Riemannian metric and $Q$ denotes a $\\nabla^{G}$ invariant rank three subbundle $Q\\subset \\text{End} (T\\mathcal{T}_H)$ that is locally spanned by a triplet $J^{n}$, $n=1,2,3$ of almost complex structures which satisfy $J^{1}J^{2}=J^{3}$ and $(J^{n})^{2}=-\\text{Id}$. Moreover the metric $G_{uv}$ is hermitian with respect to all three $J^{n}$ and one defines the associated triplet of two-forms $\\omega^{n}_{uv}:=G_{uw}(J^{n})^{w}_{v}$. In contrast to the K\\\"ahlerian case, the almost complex structures are not parallel but the Levi-Civita connection $\\nabla^{G}$ of $G$ rotates the endomorphisms inside $Q$, i.e. \n\\begin{equation}\\label{nabladef}\n\\nabla J^{n}:=\\nabla^{G}J^{n}-\\epsilon^{npq}\\theta^{p}J^{q}=0\\ .\n\\end{equation}\nNote that $\\nabla$ differs from $\\nabla^{G}$ by an\n$SU(2)$-connection with connection one-forms $\\theta^{p}$.\nFor later use let us note that the metric $G_{uv}$ can be expressed in terms of vielbeins $\\mathcal{U}^{\\alpha\\mathcal{A}}_{u}$ as\n\\begin{equation}\nG_{uv}= C_{\\alpha\\beta}\\epsilon_{\\mathcal{A}\\mathcal{B}}\\mathcal{U}^{\\alpha\\mathcal{A}}_{u}\\mathcal{U}^{\\beta\\mathcal{B}}_{v}\\ ,\n\\end{equation}\nwhere $C_{\\alpha\\beta}$ denotes the flat metric on $Sp(2n_{H},\\mathbb{R})$\nand the $SU(2)$-indices $\\mathcal{A},\\mathcal{B}$ are raised and lowered with $\\epsilon_{\\mathcal{A}\\mathcal{B}}$. \n\nThe gauge group $G$ is specified by the generators $t_{I}$ of its Lie algebra $\\mathfrak{g}$ and the structure constants $f_{IJ}^{K}$,\n\\begin{equation}\n[t_{I},t_{J}]=-f_{IJ}^{K}t_{K}\\ .\n\\end{equation}\nThe vector fields  transform in the adjoint representation of the gauge group, i.e.\\ $t_{IJ}^{K}=f_{IJ}^{K}$ while the tensor fields\ncan carry an arbitrary representation.\nThe most general representation for $n_{V}$ vector multiplets and $n_{T}$ tensor multiplets has been found in \\cite{Bergshoeff:2002qk} and is given by\n\\begin{equation}\\label{trep}\n t_{I\\tilde{J}}^{\\tilde{K}}=\n\\begin{pmatrix}\nf_{IJ}^{K} & t_{IJ}^{N}\\\\\n0 & t_{IM}^{N}\\\\\n\\end{pmatrix}.\n\\end{equation}\nWe see that the block matrix $t_{IJ}^{N}$ mixes vector- and tensor\nfields. However the $t_{IJ}^{N}$ are only nonzero if the chosen\nrepresentation of the gauge group is not completely reducible. This\nnever occurs for compact gauge groups but there exist non-compact\ngauge groups containing an Abelian ideal that admit representations\nof this type, see\n\\cite{Bergshoeff:2002qk}. There it is also shown that the construction\nof a generalized Chern-Simons term in the action for vector- and\ntensor multiplets requires the existence of an invertible and\nantisymmetric matrix $\\Omega_{MN}$. In particular, the $t_{I\\tilde J}^{N}$\nare of the form\n\\begin{equation}\\label{eq:Omega}\nt_{I\\tilde{J}}^{N}=C_{I\\tilde{J}P}\\Omega^{PN}\\ .\n\\end{equation}\n\nThe gauge group is realized on the scalar fields via the action of\nKilling vectors $\\xi_{I}$ for the vector- and tensor multiplets and\n$k_{I}$ for the hypermultiplets that satisfy the Lie\nalgebra~${\\mathfrak{g}}$~of~$G$, \n\\begin{equation}\\label{Killingc}\n\\begin{aligned}\n{}[\\xi_{I},\\xi_{J}]^{i}&:=\\xi_I^j\\partial_j \\xi^i_J-\\xi_J^j\\partial_j \\xi^i_I=\n-f_{IJ}^{K}\\, \\xi_{K}^{i}\\ ,\\\\\n[k_{I},k_{J}]^{u}&:=k_I^v\\partial_v k_J^u-k_J^v\\partial_v k_I^u=\n-f_{IJ}^{K}\\,k_{K}^{u}\\ .\n\\end{aligned}\n\\end{equation}\nIn the case of the projective special real manifold, one can obtain an explicit expression for the Killing vectors $\\xi_{I}^{i}$ given by \\cite{Bergshoeff:2004kh}\n\\begin{equation}\\label{eq:VTkilling}\n\\xi_{I}^{i}:= -\\sqrt{\\tfrac{3}{2}}\\,t_{I\\tilde{J}}^{\\tilde{K}}h^{\\tilde{J}}h^{i}_{\\tilde{K}}=-\\sqrt{\\tfrac{3}{2}}\\,t_{I\\tilde{J}}^{\\tilde{K}}h^{\\tilde{J}i}h_{\\tilde{K}}\\ .\n\\end{equation}\nThe second equality is due to the fact that \\cite{Gunaydin:1984ak}\n\\begin{equation}\\label{eq:representation0}\nt_{I\\tilde{J}}^{\\tilde{K}}\\,h^{\\tilde{J}}h_{\\tilde{K}}= 0\\ ,\n\\end{equation}\nand thus \n\\begin{equation}\n0=\\partial_{i}(t_{I\\tilde{J}}^{\\tilde{K}}h^{\\tilde{J}}h_{\\tilde{K}}) = t_{I\\tilde{J}}^{\\tilde{K}}h^{\\tilde{J}}\\partial_{i}h_{\\tilde{K}}+t_{I\\tilde{J}}^{\\tilde{K}}(\\partial_{i}h^{\\tilde{J}})h_{\\tilde{K}}\\ , \n\\end{equation}\nwhich implies\\footnote{Note that the derivative\n  $h_{\\tilde{I}i}=\\sqrt{\\tfrac{3}{2}}\\,\\partial_{i}h_{\\tilde{I}}$\nhas an additional minus sign  compared to \\eqref{eq:hder} which can be\nshown by lowering the index with $a_{\\tilde{I}\\tilde{J}}$ given in \\eqref{adef}.}\n\\begin{equation}\\label{eq:representation}\nt_{I\\tilde{J}}^{\\tilde{K}}h^{\\tilde{J}}h^{i}_{\\tilde{K}}=t_{I\\tilde{J}}^{\\tilde{K}}h^{\\tilde{J}i}h_{\\tilde{K}}\\ .\n\\end{equation}\n\nThe Killing vectors $k_{I}^u$ on the quaternionic K\\\"ahler\nmanifold $\\mathcal{T}_H$ \\cite{Andrianopoli:1996cm,Alekseevsky:2001if,Bergshoeff:2002qk} have to be triholomorphic which implies \n\\begin{equation}\\label{eq:Jinvariance}\n\\nabla_{u}\nk^{I}_{w}(J^{n})_{v}^{w}-(J^{n})_{u}^{w}\\nabla_{w}k^{I}_{v}=2\\epsilon^{npq}\\omega^{p}_{uv}\\mu^{Iq}\\\n.\n\\end{equation}\nHere $\\mu_{I}^{n}$ is a\ntriplet of moment maps which also satisfy\n\\begin{equation}\\label{eq:covdermomentmap}\n\\tfrac{1}{2}\\omega^{n}_{uv}k_{I}^{v}=-\\nabla_{u}\\mu_{I}^{n}\n\\ ,\n\\end{equation}\nand the equivariance condition\n\\begin{equation}\\label{eq:equivariance}\nf_{IJ}^{K}\\mu_{K}^{n}=\\tfrac{1}{2}\\omega_{uv}^{n}k_{I}^{u}k_{J}^{v}-2\\epsilon^{npq}\\mu_{Ip}\\mu_{Jq}\\ .\n\\end{equation}\nFurthermore the covariant derivative of the Killing vectors \nobeys \\cite{D'Auria:2001kv,Alekseevsky:2001if}\n\\begin{equation}\\label{eq:covderkilling}\n\\nabla_{u}k_{Iv} +\\nabla_{v}k_{Iu} = 0\\ ,\\qquad \\nabla_{u}k_{Iv} -\\nabla_{v}k_{Iu} = \\omega^{n}_{uv}\\mu_{nI}+L_{Iuv} \\ ,\n\\end{equation}\nwhere \nthe $L_{Iuv}$ are related to the gaugino mass matrix and commute with\n$J^{n}$.\nFor later use we define\n\\begin{equation}\\label{SLdef}\nS^{n}_{Iuv}:={L}_{Iuw}(J^{n})^{w}_{v}\\ ,\\qquad L_{uv}:=h^{I}L_{Iuv}\\\n,\\qquad S_{uv}^{n}:=h^{I}S_{Iuv}^{n}\\ ,\n\\end{equation}\nwhere the $S^{n}_{Iuv}$ are symmetric in $u,v$ \\cite{Alekseevsky:2001if}. \n\nBefore we proceed let us \nnote that for $n_{H}=0$, i.e.\\ when there are no hypermultiplets,\nconstant Fayet-Iliopoulos (FI) terms can exist which have to satisfy\nthe equivariance condition \\eqref{eq:equivariance}.  \nIn this case the first term on the right hand side of\n\\eqref{eq:equivariance}\nvanishes which implies that \n there\nare only two possible solutions \\cite{Bergshoeff:2004kh}. \nIf the gauge group contains an $SU(2)$-factor, the FI-terms have to be\nof the form\n\\begin{equation}\n\\mu_{I}^{n}= c e_{I}^{n}\\ ,\\quad c \\in \\mathbb{R}\\ ,\n\\end{equation}\nwhere the $e_{I}^{n}$ are nonzero constant vectors for $I=1,2,3$ of\nthe $SU(2)$-factor that satisfy\n\\begin{equation}\n \\epsilon^{mnp}e^{m}_{I}e^{n}_{J}=f_{IJ}^{K}e^{p}_{K}\\ .\n\\end{equation}\n The second solution has  $U(1)$-factors in the gauge group and the constant moment maps are  given by \n\\begin{equation}\\label{eq:AbelianFI}\n \\mu_{I}^{n}=c_{I}e^{n}\\ ,\\quad c_{I}\\in \\mathbb{R}\\ ,\n\\end{equation}\nwhere $e^{n}$ is a constant $SU(2)$-vector and\n$I$  labels  the $U(1)$-factors.  \n\nFinally, the covariant derivatives of the scalars in \\eqref{eq:Lagrangian} are given by\n\\begin{equation}\\label{eq:covderivatives} \n\\mathcal{D}_{\\mu}\\phi^{i} = \\partial_{\\mu}\\phi^{i} + gA_{\\mu}^{I}\\xi_{I}^{i}(\\phi)\\ , \\qquad \\mathcal{D}_{\\mu} q^{u} = \\partial_{\\mu}q^{u}+gA_{\\mu}^{I}k_{I}^{u}(q)\\ .\n\\end{equation}\nThe scalar potential\n\\begin{equation}\\label{eq:potential}\nV=2g_{ij}W^{i\\mathcal{A}\\mathcal{B}}W_{\\mathcal{A}\\mathcal{B}}^{j}+2g_{ij}\\mathcal{K}^{i}\\mathcal{K}^{j}+2N^{\\alpha}_{\\mathcal{A}}N_{\\alpha}^{\\mathcal{A}}-4S_{\\mathcal{A}\\mathcal{B}}S^{\\mathcal{A}\\mathcal{B}},\n\\end{equation}\nis defined in terms of the couplings\\footnote{Note that the $h^{M}$ in\n  the direction of the tensor multiplets do not appear\n  explicitly. Nevertheless, the couplings can implicitly depend on the\n  scalars in the tensor multiplet as they might appear in $h^{I}$\n  after solving \\eqref{eq:polynomial}.}\n\\begin{equation}\\label{eq:definitions}\n\\begin{aligned}\nS^{\\mathcal{A}\\mathcal{B}}&:=h^{I}\\mu_{I}^{n}\\sigma_{n}^{\\mathcal{A}\\mathcal{B}}\\ ,\\qquad\nW_{i}^{\\mathcal{A}\\mathcal{B}}:=h^{I}_{i}\\mu^{n}_{I}\\sigma_{n}^{\\mathcal{A}\\mathcal{B}}\\ ,\\\\\n\\mathcal{K}^{i}&:=\\tfrac{\\sqrt{6}}{4} h^{I}\\xi_{I}^{i}\\ ,\\qquad\nN^{\\alpha\\mathcal{A}}:=\\tfrac{\\sqrt{6}}{4} h^{I}k_{I}^{u}\\mathcal{U}_{u}^{\\alpha\\mathcal{A}}\\ .\n\\end{aligned}\n\\end{equation}\nHere $\\sigma^{n}_{\\mathcal{A}\\mathcal{B}}$ are the Pauli matrices with an index lowered by $\\epsilon_{\\mathcal{A}\\mathcal{B}}$, i.e.\n\\begin{equation}\n\\sigma^{1}_{\\mathcal{A}\\mathcal{B}}= \\begin{pmatrix}  1 & 0 \\\\  0 & -1  \\end{pmatrix}\\\n,\\quad\n\\sigma^{2}_{\\mathcal{A}\\mathcal{B}}= \\begin{pmatrix}  -i & 0 \\\\  0 & -i \\end{pmatrix}\\\n, \\quad\n\\sigma^{3}_{\\mathcal{A}\\mathcal{B}}= \\begin{pmatrix}  0 & -1 \\\\  -1 & 0 \\end{pmatrix}\\\n.\n\\end{equation}\nAs usual the  couplings \\eqref{eq:definitions}\nare related to the\nscalar parts of the supersymmetry variations of the fermions via\n\\begin{equation}\\label{susytrans}\n\\begin{aligned}\n\\delta_{\\epsilon}\\psi_{\\mu}^{\\mathcal{A}}&=D_{\\mu}\\epsilon^{\\mathcal{A}}-\\tfrac{ig}{\\sqrt{6}}S^{\\mathcal{A}\\mathcal{B}}\\gamma_{\\mu}\\epsilon_{\\mathcal{B}}+...\\ , \\\\\n\\delta_{\\epsilon}\\lambda^{i\\mathcal{A}}&=g\\mathcal{K}^{i}\\epsilon^{\\mathcal{A}}-gW^{i\\mathcal{A}\\mathcal{B}}\\epsilon_{\\mathcal{B}}+...\\ ,\\\\\n\\delta_{\\epsilon}\\zeta^{\\alpha}&=gN_{\\mathcal{A}}^{\\alpha}\\epsilon^{\\mathcal{A}}+...\\ .\n\\end{aligned}\n\\end{equation}\nHere $\\epsilon^{\\mathcal{A}}$ denote the supersymmetry parameters. This concludes our review of $d=5$ supergravity and we now turn to its possible supersymmetric $\\textrm{AdS}$ backgrounds.\n\n\n\n\\section{Supersymmetric $\\textrm{AdS}_{5}$ vacua}\\label{sec:vacua}\n\nIn this section we determine the conditions that lead to\n $\\textrm{AdS}_{5}$ vacua which preserve all eight supercharges.\nThis requires the vanishing of all fermionic \nsupersymmetry transformations, i.e.\n\\begin{equation}\n\\vev{\\delta_{\\epsilon}\\psi_{\\mu}^{\\mathcal{A}}}=\\vev{\\delta_{\\epsilon}\\lambda^{i\\mathcal{A}}}=\\vev{\\delta_{\\epsilon}\\zeta^{\\alpha}}=0 \\ \n\\end{equation}\nwhere $\\vev{\\ }$ denotes the value of a quantity\nevaluated in the background. Using the fact that $W^{i\\mathcal{A}\\mathcal{B}}$ and $\\mathcal{K}^{i}$ are linearly\nindependent \\cite{Gunaydin:2000xk} and \\eqref{susytrans}, this implies the following four conditions,\n\\begin{equation}\\label{eq:conditions}\n\\vev{W_{i}^{\\mathcal{A}\\mathcal{B}}}=0\\ , \\quad \\vev{S_{\\mathcal{A} \\mathcal{B}}}\\,\\epsilon^{\\mathcal{B}}=\\Lambda U_{\\mathcal{A}\\mathcal{B}}\\,\\epsilon^{\\mathcal{B}}\\ ,\\quad \\vev{N^{\\alpha\\mathcal{A}}}=0\\ ,\\quad \\vev{\\mathcal{K}^{i}}=0\\ .\n\\end{equation}\nHere $\\Lambda \\in \\mathbb{R}$ is related to the cosmological constant and\n$U_{\\mathcal{A}\\mathcal{B}}=v_{n}\\sigma_{\\mathcal{A}\\mathcal{B}}^{n}$  for $v\\in S^{2}$ is an $SU(2)$-matrix.\n$U_{\\mathcal{A}\\mathcal{B}}$ appears in the Killing spinor equation for $\\textrm{AdS}_{5}$ which reads \\cite{Shuster:1999zf}\n\\begin{equation}\n \\vev{D_{\\mu}\\epsilon_{\\mathcal{A}}}=\\tfrac{ia}{2}\\, U_{\\mathcal{A}\\mathcal{B}}\\,\\gamma_{\\mu}\\epsilon^{\\mathcal{B}}\\ , \\quad a\\in \\mathbb{R}\\ .\n\\end{equation}\nAs required for an $\\textrm{AdS}$ vacuum, the conditions \\eqref{eq:conditions}\ngive a negative background value for the scalar potential\n$\\vev{V(\\phi,q)}< 0$ which can be seen from (\\ref{eq:potential}).\nUsing the definitions (\\ref{eq:definitions}), we immediately see that \nthe four conditions \\eqref{eq:conditions} can also be formulated as \nconditions on the moment maps and Killing vectors,\n\\begin{equation}\\label{eq:backgroundmomentmaps}\n\\vev{h^{I}_{i}\\mu^{n}_{I}}=0\\ ,\\qquad\n\\vev{h^{I}\\mu^{n}_{I}}=\\Lambda v^{n}\\ ,\\qquad\n\\vev{h^{I}k_{I}^{u}}=0\\ , \\qquad \\vev{h^{I}\\xi_{I}^{i}}=0\\ .\n\\end{equation}\n Note that  due to \\eqref{eq:polynomial}, \\eqref{gpull} we need to have\n $\\vev{h^{I}}\\neq0$ for some $I$ and $\\vev{h^{\\tilde I}_i}\\neq0$ for every $i$ and some $\\tilde I$.\\footnote{\nIn particular this can also hold at the\n origin of the scalar field space $\\vev{\\phi^i}=0$, i.e.\\ for unbroken gauge groups.}\n\n\nIn order to solve \\eqref{eq:backgroundmomentmaps} we combine\nthe first two conditions  as\n\\begin{equation}\\label{eq:momentummaps}\n \\vev{\\begin{pmatrix}h^{I} \\\\ h^{I}_{i} \\end{pmatrix} \\mu_{I}^{n}} = \\begin{pmatrix}\\Lambda v^{n} \\\\ 0\\end{pmatrix}.\n\\end{equation}\nLet us enlarge these equations to the tensor multiplet indices by introducing $\\mu_{\\tilde{I}}^{n}$ where we keep in mind that $\\mu^{n}_{N}\\equiv0$. Then we use the fact that the matrix $(h^{\\tilde{I}},h^{\\tilde{I}}_{i})$ is invertible in special real geometry (see Appendix C of \\cite{Bergshoeff:2004kh}), so we can multiply (\\ref{eq:momentummaps}) with $(h^{\\tilde{I}},h^{\\tilde{I}}_{i})^{-1}$ to obtain a solution for both equations given by\n\\begin{equation}\n\\vev{\\mu_{\\tilde{I}}^{n}}=\\Lambda v^{n}\\vev{h_{\\tilde{I}}}\\ .\n\\end{equation}\nNote that this condition is non-trivial since it implies that the moment maps point in the same direction in $SU(2)$-space for all $I$.\nFurthermore, using the $SU(2)_{R}$-symmetry we can rotate the vector $v^{n}$\nsuch that $v^{n}=v\\delta^{n3}$ and absorb the constant $v\\in \\mathbb{R}$\ninto $\\Lambda$. \nThus only $\\vev{\\mu_{I}}:=\\vev{\\mu_{I}^{3}}\\neq 0$, $\\forall I$ in the above equation. Since by definition $\\vev{\\mu_{N}^{n}}= 0$, this implies\n\\begin{equation}\\label{eq:momentmapsvacuum}\n\\vev{\\mu_{I}}=\\Lambda \\vev{h_{I}}\\ , \\quad \\vev{h_{N}}= 0\\ .\n\\end{equation}\nIn particular, this means that the first two equations in \\eqref{eq:hmetric} hold in the vacuum for only the vector indices, i.e.\\\n\\begin{equation}\\label{eq:hmetricvacuum}\n\\vev{h^{I}h_{I}}=1\\ , \\quad \\vev{h_{I}h^{I}_{i}}=0\\ .\n\\end{equation}\nMoreover due to the explicit form of the moment maps in \\eqref{eq:momentmapsvacuum}, the equivariance condition \\eqref{eq:equivariance} reads in the background\n\\begin{equation}\\label{equivariancevacuum}\n f_{IJ}^{K}\\vev{\\mu_{K}}=\\tfrac{1}{2}\\vev{\\omega^{3}_{uv}k_{I}^{u}k_{J}^{v}}.\n\\end{equation}\n\n\nSince \\eqref{eq:potential} has to hold in the vacuum, $\\vev{h^{I}}\\neq 0$ for some $I$ and thus the background necessarily has non-vanishing moment maps due to \\eqref{eq:momentmapsvacuum}. This in turn implies that part of the $R$-symmetry is gauged, as can be seen from the covariant derivatives of the fermions which always contain a term of the form $A_{\\mu}^{I}\\vev{\\mu_{I}^{3}}$ \\cite{Bergshoeff:2004kh}. More precisely, this combination gauges the $U(1)_{R}\\subset SU(2)_{R}$ generated by $\\sigma^{3}$. From \\eqref{eq:momentmapsvacuum} we infer $A_{\\mu}^{I}\\vev{\\mu_{I}^{3}}=\\Lambda A_{\\mu}^{I}\\vev{h_{I}}$ which can be identified with the graviphoton \\cite{Gunaydin:1984ak}.\n\nWe now turn to the last two equations in \\eqref{eq:backgroundmomentmaps}. Let us first prove that the third equation $\\vev{h^{I}k_{I}^{u}}=0$ implies the fourth $\\vev{h^{I}\\xi_{I}^{i}}=0$. This can be shown by expressing $\\vev{\\xi_{I}^{i}}$ in terms of $\\vev{k_{I}^{u}}$ via the equivariance condition \\eqref{equivariancevacuum}. Note that we learn from \\eqref{eq:VTkilling} that the background values of the Killing vectors on the manifold $\\mathcal{T}_{VT}$ are given by\n\\begin{equation}\\label{xinA}\n \\vev{\\xi_{I}^{i}}=\n-\\sqrt{\\tfrac{3}{2}}\\,\\vev{t_{I\\tilde{J}}^{\\tilde{K}}h^{\\tilde Ji}h_{\\tilde K}}\n=-\\sqrt{\\tfrac{3}{2}}\\,\\vev{f_{IJ}^{K}h^{Ji}h_{K} + t_{IJ}^{N}h^{Ji}h_{N}}\n=-\\sqrt{\\tfrac{3}{2}}\\,\\vev{f_{IJ}^{K}h^{Ji}h_{K}}\n\\ ,\n\\end{equation}\nwhere we used \\eqref{trep} and \\eqref{eq:momentmapsvacuum}. Inserting \\eqref{eq:momentmapsvacuum}, \\eqref{equivariancevacuum}  into \\eqref{xinA} one indeed computes\n\\begin{equation}\\label{eq:Killingvacuum}\n\\vev{\\xi_{I}^{i}} =\n-\\sqrt{\\tfrac{3}{2}}\\tfrac{1}{2\\Lambda}\\,\\vev{h^{J}_{i}\\omega_{uv}^{3}k_{I}^{u}k_{J}^{v}}\\\n.\n\\end{equation}\nBut then $\\vev{h^{I}\\xi_{I}^{i}}=0$ is always satisfied if $\\vev{h^{I}k_{I}^{u}}=0$. Moreover this shows that $\\vev{\\xi_{I}^{i}}\\neq0$ is only\npossible for  $\\vev{k^u_I}\\neq0$. Note that the reverse is not true in general as can be seen from \\eqref{xinA}.\nWe are thus left with analyzing the third condition in \\eqref{eq:backgroundmomentmaps}.\n\nLet us first note that for $n_{H}=0$ there are no Killing vectors ($k_I^u\\equiv0$) and the third equation in \\eqref{eq:backgroundmomentmaps} is automatically satisfied.\nHowever \\eqref{eq:momentmapsvacuum} can nevertheless hold if the constant FI-terms discussed below \\eqref{SLdef} are of the form given in \\eqref{eq:AbelianFI} and thus only gauge groups with Abelian factors are allowed in this case.\n\nNow we turn to $n_{H}\\neq 0$.  Note that then $\\vev{h^{I}k_{I}^{u}}=0$ has two possible solutions:\n\\begin{equation}\\begin{aligned}\\label{twocases}\ni)& \\quad \\vev{k_{I}^{u}}=0\\ ,\\quad \\textrm{for all}\\ I\\\\\nii)&\\quad \\vev{k_{I}^{u}}\\neq0 \\ ,\\quad \\textrm{for some}\\ I \\ \\textrm{with}\\  \\vev{h^{I}}\\ \\textrm{appropriately tuned}.\n\\end{aligned}\\end{equation}\nBy examining the covariant derivatives (\\ref{eq:covderivatives}) of the scalars we see that in the first case there is no gauge symmetry breaking by the hypermultiplets while in the second case $G$ is spontaneously broken. \nNote that not all possible gauge groups can remain unbroken in the vacuum. In fact, for case $i)$ the equivariance condition \\eqref{equivariancevacuum} implies\n\\begin{equation}\n f_{IJ}^{K}\\vev{\\mu_{K}}=0\\ .\n\\end{equation}\nThis can only be satisfied if the adjoint representation of ${\\mathfrak{g}}$ has a non-trivial zero eigenvector, i.e.\\ if the center of $G$ is non-trivial (and continuous).\\footnote{For more details on Lie groups and their adjoint representation, see for example \\cite{O'Raifeartaigh:1986vq}.} In particular, this holds for all gauge groups with an Abelian factor but all semisimple gauge groups have to be broken in the vacuum.\n\nIn the rest of this section we discuss the spontaneous symmetry\nbreaking for case $ii)$ and the details\nof the Higgs mechanism.\nLet us first consider the case where only a set of Abelian factors in $G$\nis spontaneously broken, i.e.\\ $\\vev{k^u_I}\\neq0$ for $I$ labeling\nthese Abelian factors.\nFrom \\eqref{xinA} we then learn \n$\\vev{\\xi_{I}^{i}}=0$ and \nthus we only have spontaneous symmetry breaking in the hypermultiplet\nsector\nand the Goldstone bosons necessarily are recruited out of  these \nhypermultiplets.\nHence the vector multiplet corresponding to a broken Abelian factor in\n$G$ becomes massive by ``eating'' an entire hypermultiplet. \nIt forms a ``long'' vector multiplet containing the massive vector,\nfour gauginos and four scalars obeying the AdS mass relations.\n\nNow consider spontaneously broken non-Abelian factors of $G$,\ni.e.\\ $\\vev{k^u_I}\\neq0$ for $I$ labeling\nthese non-Abelian factors.\nIn this case we learn from \\eqref{eq:Killingvacuum} \nthat either $\\vev{\\xi_{I}^{i}}=0$ as before or $\\vev{\\xi_{I}^{i}}\\neq 0$.\nHowever the Higgs mechanism is essentially unchanged compared to the Abelian\ncase in that entire hypermultiplets are eaten and  all massive vectors\nreside in long multiplets.\\footnote{Note that short BPS vector\n  multiplets which exist in this theory cannot appear since the breaking\n  necessarily involves the hypermultiplets.} \n\nHowever there always has to exists at least one unbroken generator of\n$G$ which commutes with all other unbroken generators, i.e.\\ the\nunbroken gauge group in the vacuum is always of the form $H\\times\nU(1)_{R}$. To see this, consider the mass matrix $M_{IJ}$ of the gauge\nbosons $A^{I}_{\\mu}$. \nDue to \\eqref{eq:covderivatives} and \\eqref{eq:Killingvacuum}, this is given by\n\\begin{equation}\n M_{IJ} = \\vev{G_{uv}k_{I}^{u}k_{J}^{v}}+\\vev{g_{ij}\\xi_{I}^{i}\\xi_{J}^{j}}=\\vev{K_{uv}k^{u}_{I}k^{v}_{J}}\\ .\n\\end{equation}\nHere $K_{uv}$ is an invertible matrix which can be given in terms of $G_{uv}$ and $S_{uv}$ defined in \\eqref{SLdef} as\n\\begin{equation}\n K_{uv} = \\vev{\\left(\\tfrac{5}{8}G_{uv}-\\tfrac{6}{8\\Lambda}S_{uv}\\right)}\\ .\n\\end{equation}\nSince $\\vev{h^{I}k_{I}^{u}}=0$ the mass matrix $M_{IJ}$ has a zero\neigenvector given by $\\vev{h^{I}}$, i.e.\\ the graviphoton\n$\\vev{h^{I}}A_{I}^{\\mu}$ always remains massless in the vacuum. In the\nbackground the commutator of the corresponding Killing vector $h^{I}k_{I}^{u}$ with any other isometry $k_{J}$ is given by\n\\begin{equation}\n \\vev{[h^{I}k_{I}, k_{J}]^{u}} =\n \\vev{h^{I}(k_{I}^{v}\\partial_{v}k_{J}^{u}-k_{J}^{v}\\partial_{v}k_{I}^{u})}=\n -\\vev{h^{I}k_{J}^{v}\\partial_{v}k_{I}^{u}}\\ .\n\\end{equation}\nThis vanishes for $\\vev{k_{J}^{u}}=0$ and thus the $R$-symmetry\ncommutes with every other symmetry generator of the vacuum, i.e.\\ the\nunbroken gauge group is $H \\times U(1)_{R}$. In particular, every\ngauge group $G$ which is not of this form has to be broken $G \\rightarrow H\\times U(1)_{R}$.\n\nLet us close this section with the observation that the number of broken generators is determined by the number of linearly\nindependent $\\vev{k_{I}^{u}}$. This coincides with the number of\nGoldstone bosons $n_{G}$. In fact the $\\vev{k_{I}^{u}}$ form a basis in the\nspace of\nGoldstone bosons  $\\mathcal{G}$ and we have $\\mathcal{G}=\\text{span}_{\\mathbb{R}}\\{\\vev{k_{I}^{u}}\\}$ with $\\text{dim}(\\mathcal{G}) = \\rk \\vev{k_{I}^{u}} = n_{G}$.\n\nIn conclusion, we have shown that the conditions for maximally supersymmetric $\\textrm{AdS}_{5}$ vacua are given by\n\\begin{equation}\n \\vev{\\mu_{I}}=\\Lambda\\, \\vev{h_{I}}, \\quad \\vev{h_{M}}=0, \\quad \\vev{h^{I}k_{I}^{u}}=\\vev{h^{I}\\xi_{I}^{i}}=0\\ .\n\\end{equation}\nNote that the tensor multiplets enter in the final result only implicitly since the $h^{I}$ and its derivatives are functions of all scalars $\\phi^{i}$.\nThe first equation implies that a $U(1)_{R}$-symmetry is always gauged\nby the graviphoton while the last equation shows that the unbroken\ngauge group in the vacuum is of the form $H\\times U(1)_{R}$.  This\nreproduces the result of \\cite{Tachikawa:2005tq} that the $U(1)_{R}$\nhas to be unbroken and gauged in a maximally supersymmetric $\\textrm{AdS}_{5}$\nbackground. In the dual four-dimensional SCFT this $U(1)_{R}$ is\ndefined by a-maximization. Moreover we discussed that if the gauge\ngroup is spontaneously broken the  massive vector multiplets\nare long multiplets. \nFinally, we showed that space of Goldstone bosons is given by\n$\\mathcal{G}=\\text{span}_{\\mathbb{R}}\\{\\vev{k_{I}^{u}}\\}$ which will be used in the next section to compute the moduli space $\\mathcal{M}$ of these vacua.\n\n\n\\section{Structure of the moduli space}\\label{sec:moduli}\n\nWe now turn to the computation of the moduli space $\\mathcal{M}$ of the maximally supersymmetric $\\textrm{AdS}_{5}$ vacua determined in the previous section.\nLet us denote by $\\mathcal{D}$ the space of all possible deformations of the\nscalar fields $\\phi\\rightarrow \\vev{\\phi}+\\delta \\phi$, $q\\rightarrow\n\\vev{q}+\\delta q$ that leave the conditions\n\\eqref{eq:backgroundmomentmaps} invariant. However, if the gauge group\nis spontaneously broken the corresponding Goldstone bosons are among\nthese deformations but they should not be counted as moduli. Thus the\nmoduli space is defined as the space of deformations $\\mathcal{D}$ modulo the space of\nGoldstone bosons $\\mathcal{G}$, i.e.\\ $\\mathcal{M}=\\mathcal{D} \/ \\mathcal{G}$. \nIn order to determine $\\mathcal{M}$ we vary (\\ref{eq:backgroundmomentmaps})\nto linear order and characterize the space $\\mathcal{D}$ spanned by $\\delta \\phi$\nand $\\delta q$ that are not fixed.\\footnote{Since we consider the\n  variations of the vacuum equations \\eqref{eq:backgroundmomentmaps}\n  to first order in the scalar fields, this procedure only gives a\n  necessary condition for the moduli space.} We then show that the\nGoldstone bosons also satisfy the equations defining $\\mathcal{D}$ and\ndetermine the quotient $\\mathcal{D} \/ \\mathcal{G}$. \n\nLet us start by varying the second condition of (\\ref{eq:backgroundmomentmaps}). This yields\n\\begin{equation}\n\\vev{\\delta(h^{I}\\mu^{n}_{I})}= \\vev{(\\partial_{i}h^{I})\\,\\mu^{n}_{I}}\\,\\delta\\phi^{i}+\\vev{h^{I}\\nabla_{u}\\mu^{n}_{I}}\\,\\delta q^{u}=-\\tfrac{1}{2}\\vev{\\omega_{uv}^{n}h^{I}k_{I}^{v}}\\delta q^{u}\\equiv 0\\ ,\n\\end{equation}\nwhere we used (\\ref{eq:backgroundmomentmaps}) and\n(\\ref{eq:covdermomentmap}). \nSince this variation vanishes automatically, no conditions are imposed on the scalar field variation.\n\nThe variation of the first condition in (\\ref{eq:backgroundmomentmaps}) gives\n\\begin{equation}\\label{varone}\n\\begin{aligned}\n\\vev{\\delta(h_{i}^{I}\\mu^{n}_{I})}&=\\vev{(\\nabla_{j}h_{i}^{I})\\,\\mu^{n}_{I}}\\,\\delta\\phi^{j}+\\vev{h_{i}^{I}\\nabla_{u}\\mu_{I}^{n}}\\,\\delta q^{u}\\\\\n&=-\\sqrt{\\tfrac{2}{3}}\\vev{\\mu^{n}_{I}(h^{I}g_{ij}+h^{Ik}T_{ijk})}\\,\\delta \\phi^{j}-\\tfrac{1}{2}\\vev{h^{I}_{i}\\omega^{n}_{uv}k^{v}_{I}}\\,\\delta q^{u}\\\\\n&=-\\sqrt{\\tfrac{2}{3}}\\Lambda \\delta^{n3} \\delta\\phi_{i}-\\tfrac{1}{2}\\vev{h^{I}_{i}\\omega^{n}_{uv}k^{v}_{I}}\\,\\delta q^{u}=0\\ ,\n\\end{aligned}\n\\end{equation}\nwhere in the second step we used (\\ref{eq:covderh}), (\\ref{eq:covdermomentmap})\nwhile in the third  we used (\\ref{eq:backgroundmomentmaps}). \nFor $n=1,2$ \\eqref{varone} imposes\n\\begin{equation}\n\\langle h^{I}_{i}\\omega_{uv}^{1,2}k^{v}_{I}\\rangle\\, \\delta q^{u} = 0\\ , \\label{eq:12}\n\\end{equation}\nwhile \nfor $n=3$ the deformations $\\delta \\phi_{i}$ can be expressed in terms of $\\delta q^{u}$ as\n\\begin{equation} \\label{eq:deltaphi}\n \\delta \\phi_{i} = -\\sqrt{\\tfrac{3}{2}}\\tfrac{1}{2\\Lambda}\\vev{h_{i}^{I}\\omega_{uv}^{3}k_{I}^{v}}\\, \\delta q^{u}\\ .\n\\end{equation}\nThus all deformations $\\delta \\phi_{i}$ are fixed either to vanish or to be related to $\\delta q^{u}$. In other words, the entire space of deformations can be spanned by scalars in the hypermultiplets only, i.e.\\ $\\mathcal{D}\\subset \\mathcal{T}_{H}$. Note that this is in agreement with \\eqref{eq:Killingvacuum} and also $\\mathcal{G} \\subset \\mathcal{T}_{H}$.\n\nFinally, we vary the third condition in (\\ref{eq:backgroundmomentmaps}) to obtain\n\\begin{equation}\n\\vev{\\delta(h^{I}k_{Iu})}=\\vev{\\partial_{i}h^{I}k_{Iu}}\\,\\delta\\phi^{i}+\\vev{h^{I}\\nabla_{v}k_{Iu}}\\,\\delta q^{v}=0.\n\\end{equation}\nInserting \\eqref{eq:deltaphi} and using \\eqref{eq:hmetric}, (\\ref{eq:backgroundmomentmaps}) we find\n\\begin{equation}\\label{eq:Killing1}\n\\big(\\tfrac{1}{2\\Lambda}\\vev{k^{Iu}\\omega^{3}_{vw}k_{I}^{w}} + \\vev{h^{I}\\nabla_{v}k_{I}^{u}}\\big)\\,\\delta q^{v} = 0\\ .\n\\end{equation}\nThus we are left with the two conditions \\eqref{eq:12} and\n \\eqref{eq:Killing1} whose solutions determine $\\mathcal{D}$. For a generic supergravity we will not solve them here in general. However the conditions alone suffice to prove that the moduli space is a K\\\"ahler submanifold of $\\mathcal{T}_H$ as we will now show.\n\nAs a first step we prove that the Goldstone bosons satisfy \\eqref{eq:12} and  \\eqref{eq:Killing1}.\nWe know from section~\\ref{sec:vacua} that the Goldstone directions are\nof the form $\\delta q^{u} = c^I\\vev{k_{I}^{u}}$ where  $c^I$ are constants.\nInserted into \\eqref{eq:12} we find\n\\begin{equation}\nc^I\\vev{h_{i}^{J}\\omega_{uv}^{1,2}k^{u}_{I}k^{v}_{J}}=2c^I\\vev{h_{i}^{J}f_{IJ}^{K}\\mu_{K}^{1,2}}\n= 0\\ ,\n\\end{equation}\nwhere we used (\\ref{equivariancevacuum}) and the fact that $\\vev{\\mu_{K}^{1,2}}=0$.\nTo show that the Goldstone bosons also satisfy (\\ref{eq:Killing1})\nwe first  observe that\n\\begin{equation}\\label{eq:killingalgebra2}\n \\vev{h^{I}(\\nabla_{v}k_{I}^{u})k^{v}_{J}}= \\vev{h^{I}(\\partial_{v}k_{I}^{u})k_{J}^{v}-h^{I}(\\partial_{v}k_{J}^{u})k_{I}^{v}} = -\\vev{h^{I}[k_{I},k_{J}]^{u}} = \\vev{f_{IJ}^{K}h^{I}k_{K}^{u}}\\ ,\n\\end{equation}\nwhere \nin the first step we used \\eqref{eq:backgroundmomentmaps},\nadded a term which vanishes in the\nbackground\n and then in the second step used \\eqref{Killingc}.\nIn addition  we need to show\n\\begin{equation}\\label{eq:structureconstants}\n\\vev{f_{IJ}^{K}h^{I}k_{K}^{u}}=\\vev{f_{IJ}^{K}h_{K}k^{Iu}}\\ .\n\\end{equation}\nIndeed, using \\eqref{eq:hmetric} and $\\vev{h^{I}k_{I}^{u}}=0$ we find\n\\begin{equation}\n\\vev{f_{IJ}^{K}h^{I}k_{K}^{u}}=\\vev{f_{IJ}^{K}h^{I}k^{Lu}a_{KL}}=\\vev{f_{IJ}^{K}h^{I}k^{Lu}h_{K}^{i}h_{Li}}\\ .\n\\end{equation}\nInserting \\eqref{eq:representation} evaluated in the vacuum, i.e.\\\n$\\vev{f_{IJ}^{K}h^{J}h_{K}^{i}}=\\vev{f_{IJ}^{K}h^{Ji}h_{K}}$ and using\nagain \\eqref{eq:hmetric}\nwe obtain \n\\begin{equation}\n\\vev{f_{IJ}^{K}h^{I}k_{K}^{u}}\n=\\vev{f_{IJ}^{K}h^{Ii}k^{Lu}h_{K}h_{iL}}=\\vev{f_{IJ}^{K}h_{K}k^{Lu}\\delta^{I}_{L}}=\\vev{f_{IJ}^{K}h_{K}k^{Iu}}\\ ,\n\\end{equation}\nwhich proves \\eqref{eq:structureconstants} as promised.\n\nTurning back to \\eqref{eq:Killing1}, we insert $\\delta q^{u}= c^{I}\n\\vev{k_{I}^{u}}$ and use \\eqref{equivariancevacuum} and \\eqref{eq:killingalgebra2}\nto arrive at\n\\begin{equation}\\label{GBint}\n\\tfrac{1}{2\\Lambda}c^{I}\\vev{k^{Ju}\\omega^{3}_{vw}k_{J}^{w}k_{I}^{v}}+c^{I}\\vev{h^{J}\\nabla_{v}k_{J}^{u}k_{I}^{v}}=\\tfrac{1}{\\Lambda}c^{I}\\vev{k^{Ju}f_{IJ}^{K}\\mu_{K}}+c^{I}\\vev{f_{JI}^{K}h^{J}k_{K}^{u}}\\ .\n\\end{equation}\nUsing again that $\\vev{\\mu_{I}}=\\Lambda \\vev{h_{I}}$ and applying \\eqref{eq:structureconstants}, this yields\n\\begin{equation}\n\\tfrac{1}{\\Lambda}c^{I}\\vev{k^{Ju}f_{IJ}^{K}\\mu_{K}}+c^{I}\\vev{f_{JI}^{K}h^{J}k_{K}^{u}}=(f_{JI}^{K}+f_{IJ}^{K})c^{I}\\vev{h^{J}k_{K}^{u}}= 0\\ .\n\\end{equation}\nThus the Goldstone directions $\\delta q^{u}=c^{I}\\vev{k_{I}^{u}}$ leave the vacuum conditions \\eqref{eq:backgroundmomentmaps} invariant and hence $\\mathcal{G} \\subset \\mathcal{D}$.\n\nLet us now consider the moduli space $\\mathcal{M} = \\mathcal{D} \/ \\mathcal{G}$ and show that\n$J^{3}(\\mathcal{M})=\\mathcal{M}$, i.e.\\ $J^{3}$ restricts to an almost complex\nstructure on $\\mathcal{M}$. Concretely we show that the defining equations for the moduli space, \\eqref{eq:12} and \\eqref{eq:Killing1}, are invariant under $J^{3}$. For equations (\\ref{eq:12}) this follows from the fact that $J^{3}$ interchanges the two equations. This can be seen by substituting $\\delta q'^{u} = (J^{3})^{u}_{v}\\delta q^{v}$ and using that $J^{1}J^{2}=J^{3}$ on a quaternionic K\\\"ahler manifold. \n\nTurning to \\eqref{eq:Killing1}, we note that since only\n$\\vev{\\mu_{I}^{3}}\\neq 0$ the covariant derivative\n\\eqref{eq:Jinvariance} of the Killing vectors $k_{I}^{u}$  commutes\nwith $J^{3}$ in the vacuum, i.e.\n\\begin{equation}\n \\vev{\\nabla_{u}k^{I}_{w}(J^{n})_{v}^{w}-(J^{n})_{u}^{w}\\nabla_{w}k^{I}_{v}}=2\\epsilon^{npq}\\vev{\\omega^{p}_{uv}\\mu^{Iq}} = 0\\ .\n\\end{equation}\nThis implies that the second term in \\eqref{eq:Killing1} is invariant\nunder $J^{3}$ and we need to show that this also holds for the first\nterm. In fact, we will show in the following\nthat this term vanishes on the moduli space and is only \nnonzero for Goldstone directions.\n\n\nLet us first note that in general\n$\\rk{\\vev{k_{I}^{u}\\omega_{vw}^{3}k^{wI}}}\\leq\\rk{\\vev{k_{I}^{u}}}=n_{G}$.\nHowever, \n$\\vev{k_{I}^{u}\\omega_{vw}^{3}k^{wI}k_{J}^{v}} \\neq 0$ (as we\nsaw in \\eqref{GBint}) implies that the rank of the two matrices has\nto coincide. This in turn says that the first term in\n\\eqref{eq:Killing1}\ncan only be nonzero in the Goldstone directions and thus has to\nvanish\nfor the directions spanning $\\mathcal{M}$. Thus the whole equation \\eqref{eq:Killing1} is $J^{3}$-invariant on\n$\\mathcal{M}$. \nTherefore we have an almost complex structure\n$\\tilde{J}:=J^{3}\\vert_{\\mathcal{M}}$ and a compatible metric\n$\\tilde{G}:=G\\vert_{\\mathcal{M}}$ on $\\mathcal{M}$. Thus $(\\mathcal{M}, \\tilde{G}, \\tilde{J})$ is an almost hermitian submanifold  of the quaternionic K\\\"ahler manifold $(\\mathcal{T}_{H}, G, Q)$. \n\nIn the following we want to use theorem 1.12 of \\cite{Alekseevsky:2001om}: an almost Hermitian submanifold $(M, G, J)$ of a quaternionic K\\\"ahler manifold $(\\tilde{M}, \\tilde{G}, Q)$ is K\\\"ahler if and only if it is totally complex, i.e.\\ if there exists a section $I$ of $Q$ that anticommutes with $J$ and satisfies\n\\begin{equation}\nI(T_{p}M) \\perp T_{p}M \\quad \\forall p\\in M\\ .\n\\end{equation}\nIn particular, this condition is satisfied if the associated fundamental two-form $\\omega_{uw}=G_{uw}I_{v}^{w}$ on $M$ vanishes.\n\nNow let us show that the moduli space $\\mathcal{M}$ actually is totally\ncomplex and hence K\\\"ahler. To do so, we use \\eqref{eq:covderkilling}\nand \\eqref{SLdef} to \nnote that in the vacuum\n\\eqref{eq:momentmapsvacuum} \n$ \\vev{\\omega^{3}_{uv}}$ is given by \n\\begin{equation}\\label{eq:omega3}\n \\vev{\\omega^{3}_{uv}}=\\tfrac{2}{\\Lambda}\\vev{h^{I}\\nabla_{u}k_{Iv}-L_{uv}}\\ .\n\\end{equation}\nWe just argued that \n$\\vev{k_{I}^{u}\\omega_{vw}^{3}k^{wI}}$ vanishes on $\\mathcal{M}$\nand thus \\eqref{eq:Killing1} projected onto $\\mathcal{M}$ also implies\n\\begin{equation}\\label{eq:nablaG}\n \\vev{h^{I} \\nabla_{u}k_{vI}}\\vert_{\\mathcal{M}} = 0 \\ .\n\\end{equation}\nSince $\\vev{\\omega^{1}_{uv}}=-\\vev{\\omega^{3}_{uw}(J^{2})_{v}^{w}}$, we can multiply \\eqref{eq:omega3} with $-(J^{2})^{w}_{v}$ from the right and obtain\n\\begin{equation}\n\\vev{\\omega^{1}_{uv}}\\vert_{\\mathcal{M}} =\n\\tfrac{2}{\\Lambda}\\vev{S^{2}_{uv}-h^{I}\n  \\nabla_{u}k_{wI}(J^{2})_{v}^{w}}\\vert_{\\mathcal{M}}=0\\ ,\n\\end{equation}\nwhere in the first step we used \\eqref{SLdef}.\nThis expression  vanishes due to \n\\eqref{eq:nablaG} and the fact that $S^{2}_{uv}$ is symmetric while\n$\\omega^{1}_{uv}$ is antisymmetric.\nThus $\\mathcal{M}$ is totally complex and in particular $(\\mathcal{M}, \\tilde{G}, \\tilde{J})$ is a K\\\"ahler submanifold. \n\nAs proved in \\cite{Alekseevsky:2001om} a K\\\"ahler submanifold can\nhave at most half the dimension of the ambient quaternionic K\\\"ahler\nmanifold, i.e.\\ $\\text{dim}(\\mathcal{M}) \\leq 2n_{H}$.\\footnote{Applying the same method as in $d=4$, $\\mathcal{N}=2$ this can be checked explicitly \\cite{deAlwis:2013jaa}.}\nNote that in the case of an unbroken gauge group we have $\\mathcal{G} = \\{\\emptyset\\}$ and thus $\\mathcal{D}=\\mathcal{M}$. This is the case of maximal dimension of the moduli space. If the gauge group is now spontaneously broken then additional scalars are fixed by \\eqref{eq:12}. Since $\\mathcal{M}$ is $J^{3}$-invariant, every $\\delta q^{u} \\in \\mathcal{M}$ can be written as $\\delta q^{u} = (J^{3})_{v}^{u}\\delta q'^{v}$ for some $\\delta q'^{u}\\in \\mathcal{M}$. Combined with the fact that $J^{1}J^{2}=J^{3}$ this implies that the two conditions in \\eqref{eq:12} are equivalent on $\\mathcal{M}$. Furthermore we have $\\rk{\\vev{h^{I}_{i}\\omega_{uv}^{1}k_{I}^{v}}}=\\rk{\\vev{k_{u}^{I}}}=n_{G}$ and thus $n_{G}$ scalars are fixed by \\eqref{eq:12}. In conclusion, we altogether have\n\\begin{equation}\n\\text{dim}(\\mathcal{M})=\\text{dim}(\\mathcal{D})-\\text{dim}(\\mathcal{G})\\leq (2n_{H}-n_{G})-n_{G}\\ ,\n\\end{equation}\nso the moduli space has at most real dimension $2n_{H}-2n_{G}$.\n\n\n\n\n\\section*{Acknowledgments}\nThis work was supported by the German Science Foundation (DFG) under\nthe Collaborative Research Center (SFB) 676 ``Particles, Strings and the Early\nUniverse'', the Research Training Group (RTG) 1670 ``Mathematics\ninspired by String Theory and Quantum Field Theory'' and the Joachim-Herz Stiftung.\n\nWe have benefited from conversations and correspondence with David Ciupke, Peter-Simon Dieterich, Malte Dyckmanns, Jonathan Fisher, Severin L\\\"ust, Stefan Vandoren and Owen Vaughan.\n\n\n\n\n\n\n\n\n\\newpage \n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe theory of artificial neural networks (ANN) represents an open research field setting the stage for the implementation of a statistical mechanical approach in novel interdisciplinary problems, such as the modeling of the collective behavior of the human brain neurons. An important field of application of ANN is represented by the pattern recognition analysis \\cite{Egmont,Wang}, which has received an increasing interest in the literature, witnessed by the extensive application of ANN to tackle complex real-word problems, e.g. in medical diagnosis \\cite{Haya,Jiang,Shayea} and in biological sequences analysis \\cite{Ding,Qian, Condon,Cart}.\nRecent works, in this field, paved also the way to the systematic use of technical tools borrowed from Information Theory and Statistical Mechanics \\cite{McKay,Anand,Tkacik}.\\\\\nIn this paper, in particular, we adopt information theoretic methods \\cite{Kim,Vihn} to classify a sequence of hazelnuts images, and show how our approach allows for improving the performance of pattern recognition procedures performed via ANN algorithms.\nFrom a preliminary statistical analysis on the image histograms, we identify some relevant observables to be used in the implementation of a machine learning algorithm. A special focus of our approach is on the role of \\textit{fluctuations} of the histograms around the corresponding \\textit{mean} distribution. In particular, by making use of various notions of ``distance'' between histograms, we introduce two statistical scales, whose magnitude affects the performance of a machine learning algorithm in disentangling and extracting the distinctive features of the hazelnuts.\\\\\nThe paper is organized as follows.\\\\\nIn Sec. \\ref{sec:sec1} we introduce the two aforementioned statistical scales and discuss their dependence on a quantity referred to as the ``image resolution''. We comment on the need of a large separation between two such scales to obtain an efficient pattern recognition: the lack of a wide separation between them is due to large histograms fluctuations which blur the distinctive features of the hazelnuts, thus hindering a proper classification of the data. \\\\\nIn Sec. \\ref{sec:sec2} we test, then, the prediction of our statistical analysis by employing a machine learning algorithm, known as Support Vector Machines (SVM) \\cite{Haykin,Webb}. The numerical results we obtained not only confirm the relevance of the aforementioned scale separation, but also show that the predicted onset of an optimal scale of description can be recovered through the use of a SVM algorithm, provided that its performance is \\textit{averaged} over a sufficiently large set of training samples. \\\\\nConclusions are finally drawn in Sec. \\ref{sec:conc}.\\\\\nThe main results of this work can be summarized as follows:\n\\begin{itemize}\n\t\\item We introduce two typical statistical scales, whose magnitude critically affects the performance of a pattern recognition algorithm based on statistical variables;\n        \\item We describe the dependence of such scales on the scale of resolution, thus unveiling the onset of an optimal resolution at which the pattern recognition is favoured;\n        \\item We numerically recover the results of the statistical analysis by using a SVM algorithm, and also shed light on the role of \\textit{averaging} the performance of a SVM over sufficiently many training samples.\n\\end{itemize}\n\n\\section{The original set of hazelnut images: a statistical approach}\n\\label{sec:sec1}\n\nIn this work we consider the problem of pattern recognition applied to a sequence of hazelnut images, to be categorized into three different sets: ``good'' ($G$), ``damaged'' ($D$) and ``infected'' ($I$). In the sequel, we will use the shorthand notation $\\mathcal{S}=\\{G,D,I\\}$, and, for any $A \\in \\mathcal{S}$, we will also denote $N_A=card(A)$. Our database consists of a set of $800$ x-ray scanned images, cf. Fig. \\ref{hazelnuts}, with $N_G=750$, $N_D=25$ and $N_I=25$. The analysis outlined below is meant to provide a guiding strategy to assess, and possibly enhance, the performance of pattern recognition methods based on ANN algorithms. The prominent distinctive features of the three sets $G$, $D$ and $I$ are not detectable from a solely visual inspection of the x-ray images. Hence, in order to extract some valuable information, we relied on the computation of the histograms of the hazelnut images, shown in Fig. \\ref{hazelnuts2}.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{S_6.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{S_7.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{S_8.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{A_1.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{A_2.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{A_3.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{C_15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{C_16.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.15\\textwidth, height=0.15\\textwidth]{C_17.pdf}\n\\caption{X-ray scanned images of good hazelnuts (top row), damaged hazelnuts (middle row) and infected hazelnuts (bottom row).}\\label{hazelnuts}\n\\end{figure}\n\nTherefore, for any $A \\in \\mathcal{S}$, we computed the number of pixels, in the image pertaining to the $i$-th hazelnut belonging to the set $A$ (with $i= 1,...,N_{A}$), characterized by the shade of gray $j$ (conventionally running from the value $0$ - black - to $255$ - white). After normalizing wrt the total number of pixels forming the same image, we thus obtained the so-called image histogram $p_i^{(A)}(j)$. We could also compute, then, the mean histogram pertaining to $A$, denoted by $\\overline{p}_i^{(A)}(j)$, which was obtained by averaging over the $N_A$ histograms $p_i^{(A)}(j)$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{meanhistS.pdf}\n\\includegraphics[width=0.45\\textwidth]{meanhistA.pdf}\\\\\n\\includegraphics[width=0.45\\textwidth]{meanhistC.pdf}\n\\includegraphics[width=0.45\\textwidth]{meanhistTot.pdf}\n\\caption{Image histograms of good hazelnuts (top left), damaged hazelnuts (top right), infected hazelnuts (bottom left). The horizontal axis displays the shades of gray, conventionally running from $0$ to $255$. For each of the three sets, the figures display the (normalized) histograms of single hazelnuts as well as the (normalized) mean histogram. On the bottom right corner, the mean histograms of the three different sets are compared.}\\label{hazelnuts2}\n\\end{figure}\n\nA quantitative characterization of the images can be afforded by introducing various notions of ``distance'' between different histograms \\cite{Cha}: we considered, in particular, the norm in $L^1$, in $L^2$ (euclidean), in $L^\\infty$, the Squared $\\chi^2$ distance and the Jeffrey's divergence \\cite{SHC}. \nIt is worth briefly recalling some basic aspects concerning the latter two notions of distance, borrowed from probability theory.\nThe Squared $\\chi^2$ distance corresponds to the symmetrized version of the Pearson's $\\chi^2$ test \\cite{Plackett}, which, given a histogram $p(j)$ and a reference histogram $q(j)$,  defines their relative distance as:\n\\begin{equation}\nd_{\\chi^2}=\\sum_j\\frac{(p(j)-q(j))^2}{q(j)} \\label{chisq} \\quad .\n\\end{equation}\nThus, the quantity $d_{\\chi^2}$ in (\\ref{chisq}) resembles the standard euclidean distance between the two histograms, except that it introduces a weight corresponding to the inverse of the reference histogram.\\\\\nOn the other hand, the Jeffreys' divergence \\cite{Jeffreys} belongs to the Shannon entropy family \\cite{Beck}, and corresponds to the symmetrized version of the Kullback-Leibler (K-L) divergence (or \\textit{relative entropy}) \\cite{KL}, defined as:\n\\begin{equation}\nd_{K-L}(p\\|q)=\\sum_j \\left(p(j)\\log\\left(\\frac{p(j)}{q(j)}\\right) \\right)=H(p,q)-H(p) \\label{KL} \\quad ,\n\\end{equation}\nwhere $H(p,q)$ is the cross entropy of $p$ and $q$, and $H(p)$ is the entropy of $p$ \\cite{Kull,Jay}.\nMore in general, the K-L divergence (\\ref{KL}), is a member of the family of the so-called $f$-divergencies \\cite{Mori,Ali} and stems as a limiting case of the more general R\\'enyi's (or $\\alpha$-) divergence \\cite{Xu}. It is worth recalling its definition: given any two continuous distributions $p$ and $q$, over a space $\\Omega$, with $p$ absolutely continuous wrt $q$, the $f$-divergence of $p$ from $q$ is\n\\begin{equation}\nd_{f}(p\\|q)=\\int_\\Omega f\\left(\\frac{dp}{dq}\\right)dq \\quad ,\n\\end{equation}\nwhere $f$ is a convex function such that $f(1)=0$.\\\\\nThen, for any $A \\in \\mathcal{S}$, we considered the distance (or \\textit{fluctuation}), defined according to the various notions introduced above, between the histogram $p_i^{(A)}(j)$ and the corresponding mean $\\overline{p}_i^{(A)}(j)$. Next, by averaging over the set $A$, one obtains a characteristic ``statistical scale'' (still depending on the chosen notion of distance) characterizing the fluctuations within each set $A$. \nTo clarify the meaning of the entries in Tab. \\ref{normtot}, let us illustrate, for instance, the procedure to calculate the quantity $\\langle d \\rangle^{(A)}_2$. To this aim, we introduce the euclidean distance between the histograms $p_i^{(A)}(j)$ and $\\overline{p}_i^{(A)}(j)$:\n\\begin{equation}\nd_{2,i}^{(A)}=\\sqrt{\\sum_{j=1}^{N_g}|p_i^{(A)}(j)-\\overline{p}_i^{(A)}(j)|^2} \\label{eucl}\n\\end{equation}\n\n\\begin{table}[bth]\n\\centering\n\\begin{tabular}{c|c|c|c|c|c|}\n \n    & $\\langle d \\rangle^{(A)}_1$ & $\\langle d \\rangle^{(A)}_2$ & $\\langle d \\rangle^{(A)}_\\infty$ & $\\langle d \\rangle^{(A)}_{\\chi^2}$ & $\\langle d \\rangle^{(A)}_{J}$\\\\\n\\hline\n  $A =G$ & $0.2079$ & $0.0372$ & $0.0139$ & $0.0495$ & $0.0369$ \\\\\n\\hline\n  $A =D$ & $0.2485$ & $0.0488$ & $0.0162$ & $0.0776$ & $0.0477$ \\\\   \n\\hline\n  $A =I$ & $0.2097$ & $0.0379$ & $0.0145$ & $0.0435$ & $0.0401$ \\\\\n  \\hline\n\\end{tabular}\n                            %\n\\caption{Typical fluctuation of the histograms of the hazelnuts from the corresponding mean histogram, within each of the sets $G$, $D$, and $I$. The quantities $\\langle d \\rangle^{(A)}$ are evaluated by using different notions of distances: norm in $L^1$, in $L^2$ (euclidean), in $L^\\infty$, Squared $\\chi^2$ distance and Jeffreys divergence.}\n                            %\n          \\label{normtot}\n          \\end{table}\n\n\\begin{table}[bth]\n\\centering\n\\begin{tabular}{c|c|c|c|c|c|}\n \n    & $\\Delta^{(A,B)}_{1}$ & $\\Delta^{(A,B)}_{2}$ & $\\Delta^{(A,B)}_{\\infty}$ & $\\Delta^{(A,B)}_{\\chi^2}$ & $\\Delta^{(A,B)}_{J}$\\\\\n\\hline\n  $A=G, B=D$ & $0.0923$ & $0.0162$ & $0.0036$ & $0.0089$ & $0.0200$ \\\\\n\\hline\n  $A=D, B=I$ & $0.0533$ & $0.0090$ & $0.0028$ & $0.0021$ & $0.0030$ \\\\   \n\\hline\n  $A=G, B=I$ & $0.0526$ & $0.0115$ & $0.0051$ & $0.0044$ & $0.0124$ \\\\  \n  \\hline\n\\end{tabular}\n                            %\n\\caption{Average distances between between pairs of mean histograms referring to two different sets $A$ and $B$, evaluated, as in Tab \\ref{normtot}, using different notions of distance: norm in $L^1$, in $L^2$ (euclidean), in $L^\\infty$, Squared $\\chi^2$ distance and Jeffreys divergence.}\n                            %\n          \\label{Deltatot}\n          \\end{table}\n\nFrom the knowledge of $d_{2,i}^{(A)}$ in (\\ref{eucl}), the quantity $\\langle d \\rangle^{(A)}_2$, shown in Tab. \\ref{normtot}, is then computed by averaging over $A$:\n\\begin{equation}\n\\langle d \\rangle_2^{(A)} =\\frac{1}{N_{A}}\\sum_{i=1}^{N_{A}}d_{2,i}^{(A)} \\label{aver}\n\\end{equation} \nIt is worth noticing, from Tab. \\ref{normtot}, that, no matter of what notion of distance is adopted, the magnitude of the fluctuations is not significantly affected by $N_{A}$.\nThe scale $\\langle d \\rangle^{(A)}$, which, for any $A \\in \\mathcal{S}$, is of the order $\\langle d \\rangle^{(A)}\\simeq 10^{-2}$, can be thus regarded as an intrinsic statistical scale pertaining to the set $A$. \nIt is worth comparing such scale with another statistical scale, denoted by $\\Delta^{(A,B)}$, whose values are listed in Tab. \\ref{Deltatot}. The quantity $\\Delta^{(A,B)}$ is defined as the distance, computed by using the various notions of distance introduced above, between the pair of mean histograms relative to the sets $A$ and $B$, with $(A,B)\\in\\mathcal{S}$ and $A \\neq B$. The symmetric form of the distances introduced above entails, in particular, that $\\Delta^{(A,B)}=\\Delta^{(B,A)}$. \nA better interpretation of the meaning of the scales $\\langle d \\rangle^{(A)}$ and $\\Delta^{(A,B)}$ can be achieved by noticing that a large value of $\\langle d \\rangle^{(A)}$ mirrors the presence of a considerable amount of noise on top of the mean histogram $\\overline{p}_i^{(A)}(j)$, which thus blurs the distinctive features of the set $A$. On the contrary, a larger value of $\\Delta^{(A,B)}$ reflects a more significant separation between the mean histograms of the two sets $A$ and $B$, which instead favours the pattern recognition. In the sequel of this Section we will focus, therefore, on the ratio of two such scales.\nFrom an inspection of Tabs. \\ref{normtot} and \\ref{Deltatot}, we first observe that $\\Delta^{(A,B)}\\sim\\langle d \\rangle^{(A)}$. That is, the two scales are comparable: the fluctuations, within each set, are comparable with the typical distances between different sets. This entails, hence, that the histograms shown in Fig. \\ref{hazelnuts2} can not be regarded as a useful source of information to perform a pattern recognition.\nA different route can be pursued by just focusing on a selected portion of the original images. This approach is motivated by the assumption that the distinctive features of each of the three sets are mostly contained in the ``nuclei'' of the hazelnuts. We calculated, therefore, the histograms corresponding to the cropped portions of the original images, delimited by the tick red rectangles shown in Fig. \\ref{hazelnuts3}. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.31\\textwidth]{HistoS1.pdf}\n\\includegraphics[width=0.31\\textwidth]{HistoS6.pdf}\n\\includegraphics[width=0.31\\textwidth]{HistoS12.pdf}\\\\\n\\includegraphics[width=0.31\\textwidth]{HistoA1.pdf}\n\\includegraphics[width=0.31\\textwidth]{HistoA12.pdf}\n\\includegraphics[width=0.31\\textwidth]{HistoA13.pdf}\\\\\n\\includegraphics[width=0.31\\textwidth]{HistoC2.pdf}\n\\includegraphics[width=0.31\\textwidth]{HistoC6.pdf}\n\\includegraphics[width=0.31\\textwidth]{HistoC13.pdf}\n\\caption{Image histograms of good hazelnuts (top row), damaged hazelnuts (middle row) and infected hazelnuts (bottom row). Each image shows the histogram of the entire hazelnut (top histogram) and the histogram referring to the fraction of the image delimited by the thick red rectangles, characterized by $\\epsilon=80$ and $\\rho=2.5$.}\\label{hazelnuts3}\n\\end{figure}\n\nThe red rectangles in Fig. \\ref{hazelnuts3} are identified by the pair of parameters $\\{\\epsilon, \\rho\\}$, where $\\epsilon$, related to the image resolution, is defined as the number of pixels comprised along the horizontal length of the rectangles, while $\\rho$ is the ratio of the number of pixels along the vertical length to the corresponding number of pixels along the horizontal one.\nIn our simulations, the values of the parameters $\\{\\epsilon, \\rho\\}$ were kept constant when calculating the histograms relative to different hazelnut nuclei.\nFigure \\ref{hazelnuts3} refers, for instance, to the case corresponding to $\\epsilon=80$ and $\\rho = 2.5$. \nIn Figs. \\ref{nuclS},\\ref{nuclA} and \\ref{nuclC}, shown is the result of the image processing of the hazelnut nuclei, performed through a noise removal filter (adaptive Wiener filtering) and various edge-detector algorithms. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{NucleusS10.pdf}\n\\includegraphics[width=0.45\\textwidth]{NucleusS8.pdf}\\\\\n\\includegraphics[width=0.45\\textwidth]{NucleusS6.pdf}\n\\includegraphics[width=0.45\\textwidth]{NucleusS13.pdf}\n\\caption{Image processing of the hazelnut nuclei belonging to the set $G$, for $\\epsilon=100$ and $\\rho=1.5$, by means of edge-detection algorithms, respectively: Sobel's algorithm (top right figure) , Canny's algorithm (bottom left figure) and Roberts' algorithm (bottom right figure).}\\label{nuclS}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{NucleusA1.pdf}\n\\includegraphics[width=0.45\\textwidth]{NucleusA6.pdf}\\\\\n\\includegraphics[width=0.45\\textwidth]{NucleusA10.pdf}\n\\includegraphics[width=0.45\\textwidth]{NucleusA15.pdf}\n\\caption{Image processing of the hazelnut nuclei belonging to the set $D$, for $\\epsilon=100$ and $\\rho=1.5$, by means of edge-detection algorithms, respectively: Sobel's algorithm (top right figure) , Canny's algorithm (bottom left figure) and Roberts' algorithm (bottom right figure).}\\label{nuclA}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{NucleusC3.pdf}\n\\includegraphics[width=0.45\\textwidth]{NucleusC7.pdf}\\\\\n\\includegraphics[width=0.45\\textwidth]{NucleusC6.pdf}\n\\includegraphics[width=0.45\\textwidth]{NucleusC11.pdf}\n\\caption{Image processing of the hazelnut nuclei belonging to the set $I$, for $\\epsilon=100$ and $\\rho=1.5$, by means of edge-detection algorithms, respectively: Sobel's algorithm (top right figure) , Canny's algorithm (bottom left figure) and Roberts' algorithm (bottom right figure).}\\label{nuclC}\n\\end{figure}\n\nIn Fig. \\ref{hazelnutsnucl}, which is worth comparing with Fig. \\ref{hazelnuts2}, we plotted the mean histograms relative to the cropped images, with $\\epsilon=80$ and $\\rho =2.5$. The question arises, then, as to whether the separation between the two scales $\\langle d \\rangle^{(A)}$ and $\\Delta^{(A,B)}$ is amenable to be enhanced by tuning the two parameters $\\epsilon$ and $\\rho$. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{meanhistSnucl80r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistAnucl80r25.pdf}\\\\\n\\vspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistCnucl80r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl80r25.pdf}\n\\caption{Image histograms of the hazelnut nuclei belonging to the sets $G$ (top left), $D$ hazelnuts (top right), $I$ hazelnuts (bottom left). For each of the three sets, the figures display the histograms of single hazelnuts as well as the mean histogram in the corresponding set (mean histogram). On the bottom right corner, the mean histograms of the three sets are compared. All the histograms were obtained by setting $\\epsilon = 80$ and $\\rho=2.5$.}\\label{hazelnutsnucl}\n\\end{figure}\n\nWe thus studied the behaviour of the mean histograms, shown in Fig. \\ref{hazelnutsnucl}, as well as of the typical fluctuations occurring in each set, as a function of $\\epsilon$ and $\\rho$: in our simulations, $\\epsilon$ spans a broad range of values, whereas we let $\\rho$ attain the values $1.5$ and $2.5$, cf.  Fig. \\ref{rect}. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{rect.pdf}\n\\caption{Different values of the scale of resolution: $\\epsilon=80$ (red rectangle), $\\epsilon =60$ (magenta rectangle), $\\epsilon=40$ (blue rectangle), $\\epsilon=20$ (green rectangle). All the colored rectangles shown in the picture are obtained by setting $\\rho =2.5$.}\\label{rect}\n\\end{figure}\n\nIn Fig. \\ref{nuclhist1} and \\ref{nuclhist2}, the mean histograms of the sets $G$, $D$ and $I$ are shown for different values of $\\epsilon$, and for two different values of $\\rho$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl80r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl60r15.pdf}\\\\\n\\vspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl40r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl20r15.pdf}\n\\caption{Mean histograms of the hazelnut nuclei at different scales of resolution: $\\epsilon =80$ (top left), $\\epsilon =60$ (top right), $\\epsilon =40$ (bottom left) and $\\epsilon =20$ (bottom right), with $\\rho=1.5$.}\\label{nuclhist1}\n\\end{figure}\n\nWe focused, in particular, on the investigation of the dependence of the scales $\\langle d \\rangle^{(A)}(\\epsilon; \\rho)$ and $\\Delta^{(A,B)}(\\epsilon; \\rho)$ on the resolution $\\epsilon$. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl80r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl60r25.pdf}\\\\\n\\vspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl40r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{meanhistTotnucl20r25.pdf}\n\\caption{Mean histograms of the nuclei of the hazelnuts at different scales of description: $\\epsilon =80$ (top left), $\\epsilon =60$ (top right), $\\epsilon =40$ (bottom left) and $\\epsilon =20$ (bottom right), with $\\rho=2.5$.}\\label{nuclhist2}\n\\end{figure}\n\nFigures \\ref{norm1} and \\ref{mean1} illustrate the behaviour of $\\langle d \\rangle^{(A)}$ and $\\Delta^{(A,B)}$ vs. $\\epsilon$ for $\\rho=1.5$, whereas \nFigs. \\ref{norm2} and \\ref{mean2} show the analogous behaviour of $\\langle d \\rangle^{(A)}$ and $\\Delta^{(A,B)}$ vs. $\\epsilon$ for $\\rho=2.5$\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{FluctS15.pdf}\n\\includegraphics[width=0.45\\textwidth]{FluctA15.pdf}\n\\includegraphics[width=0.45\\textwidth]{FluctC15.pdf}\n\\caption{Behaviour of the distances $\\langle d \\rangle^{(A)}$ vs. $\\epsilon$, with $\\rho=1.5$.}\\label{norm1}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dAS15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{dSC15.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{dAC15.pdf}\n\\caption{Behaviour of the distances $\\Delta^{(A,B)}$ vs. $\\epsilon$, with $\\rho=1.5$.}\\label{mean1}\n\\end{figure}\n\nThe two plots \\ref{mean1} and \\ref{mean2} reveal that reducing $\\epsilon$ leads, on the one hand, to a remarkable increase of $\\Delta^{(A,B)}$, which attains an order of magnitude of about $\\Delta^{(A,B)} \\simeq 10^{-1}$. On the other hand, this effect is counterbalanced by the simultaneous increase of the scale $\\langle d \\rangle^{(A)}$, evidenced in Figs. \\ref{norm1} and \\ref{norm2}, which turns out to be, for both the considered values of $\\rho$, of the same order of magnitude of $\\Delta^{(A,B)}$. This is more clearly visible in Fig. \\ref{ratio}, which illustrates the behaviour of the ratio of $\\Delta^{(A,B)}$ to $\\langle d \\rangle^{(A)}$ and to $\\langle d \\rangle^{(B)}$, for different values of $\\epsilon$, obtained by setting $A = G$ and $B = D$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{FluctS25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{FluctA25.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{FluctC25.pdf}\n\\caption{Behaviour of the distances $\\langle d \\rangle^{(A)}$ vs. $\\epsilon$, with $\\rho=2.5$.}\\label{norm2}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{dAS25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{dSC25.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{dAC25.pdf}\n\\caption{Behaviour of the distances $\\Delta^{(A,B)}$ vs. $\\epsilon$, with $\\rho=2.5$.}\\label{mean2}\n\\end{figure}\n\nThe plots in Fig. \\ref{ratio} confirm that the two scales $\\Delta^{(A,B)}$ and $\\langle d \\rangle^{(A)}$ remain of the same order, also when reducing $\\epsilon$. \nOn the contrary, an efficient pattern recognition, based on the analysis of the image histograms, can be obtained if the ratio  $\\Delta^{(A,B)} \/\\langle d \\rangle^{(A)} \\gg 1$, i.e. when the mean statistical distance between different sets overwhelms the typical size of fluctuations characteristic of each set.\nThus, the study of the behaviour of the two latter scales allows one to predict a poor performance of a machine learning algorithm aiming at classifying the hazelnuts on the basis of the image histograms.\nNevertheless, an interesting aspect can be evinced from an inspection of Fig. \\ref{ratio}: despite the similarity of the magnitudes of the two statistical scales, the plot of their ratio vs. $\\epsilon$ yields a non monotonic function. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{ratioASAr15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{ratioASSr15.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{ratioASAr25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{ratioASSr25.pdf}\n\\caption{Behaviour of the ratio $\\Delta^{(G,D)}\/\\langle d \\rangle^{(G)}$ (left column) and $\\Delta^{(G,D)}\/\\langle d \\rangle^{(D)}$ (right column) vs. $\\epsilon$,  for $\\rho=1.5$ (upper row) and $\\rho=2.5$ (lower row).}\\label{ratio}\n\\end{figure}\n\nTo better evidence this point, we plotted, in Fig. \\ref{ratio2}, the ratio of the scale $\\Delta^{(A,B)}$ to the geometric mean $\\sqrt{\\langle d \\rangle^{(A)} \\langle d \\rangle^{(B)}}$, where we set $A=G, B=D$ (left panel) and $A=G, B=I$ (right panel). \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{ratioASr15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{ratioCSr15.pdf}\n\\caption{\\textit{Left panel:} Behaviour of the ratio $\\Delta^{(G,D)}\/\\sqrt{\\langle d \\rangle^{(G)} \\langle d \\rangle^{(D)}}$ vs. $\\epsilon$, for $\\rho=1.5$. \\textit{Right panel:} Behaviour of the ratio $\\Delta^{(G,I)}\/\\sqrt{\\langle d \\rangle^{(G)} \\langle d \\rangle^{(I)}}$ vs. $\\epsilon$, for $\\rho=1.5$.}\\label{ratio2}\n\\end{figure}\n\nIn Fig. \\ref{ratio3}, instead, for reasons to be further clarified in Sec. \\ref{sec:sec2}, we show the results, analogous to those portrayed in Fig. \\ref{ratio2}, obtained by merging the two sets $D$ and $I$ into one single set, labeled as $nG$ (``not good'' hazelnuts). The plot in Fig. \\ref{ratio3} shows that, for $\\rho=1.5$, the value $\\epsilon^*=70$ maximizes the ratio of the aforementioned statistical scales wrt almost all the various notions of ``statistical distance'' we considered. In Sec. \\ref{sec:sec2} we will show that such optimal value $\\epsilon^*$, here obtained by only relying on information theoretic methods, can be also recovered by using Support Vector Machines numerical algorithms, by averaging their performance over a set of training samples.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.70\\textwidth]{GnGr15.pdf}\n\\caption{\\textit{Left panel:} Behaviour of the ratio $\\Delta^{(G,nG)}\/\\sqrt{\\langle d \\rangle^{(G)} \\langle d \\rangle^{(nG)}}$ vs. $\\epsilon$, for $\\rho=1.5$. The plot evidences the onset of an optimal scale $\\epsilon^*$ at which the ratio of the statistical scales is maximized.}\\label{ratio3}\n\\end{figure}\n\n\n\\section{Support Vector Machines}\n\\label{sec:sec2}\n\nIn this Section, we discuss the results obtained by elaborating our data through a supervised learning method known as Support Vector Machines (SVM) \\cite{Haykin,Boser, Cortes,Vapnik95,Vapnik98}. The SVM constitute a machine learning algorithm which seeks a separation of a set of data into two classes, by determining the \\textit{best separating hyperplane} (BSH) (also referred to, in the literature, as the ``maximal margin hyperplane'' \\cite{Webb}), cf. Fig. \\ref{SVM}. It is worth recapitulating the basic notions underpinning the numerical algorithm we used.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{SVM.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM2.pdf}\n\\caption{\\textit{Left panel:} Example of a linear discriminant analysis based on the SVM algorithm. Shown are three different hyperplanes: $\\Pi_1$, which does not separate the two classes, $\\Pi_2$ which separates the classes but only with a small margin, and $\\Pi_3$, which corresponds to the best separating hyperplane. \\textit{Right panel:} Illustration of the best separating hyperplane (red straight line), the canonical hyperplanes (black dashed lines), the support vectors (magenta circles) and the margin of separation $\\xi$.}\\label{SVM}\n\\end{figure}\n\nLet $\\{\\textbf{x}\\}$ denote the set of data (input pattern) to be classified, with $\\textbf{x} \\in E\\subseteq\\mathbb{R}^N$, and consider a given training set $\\mathcal{T}=\\{\\textbf{x}_k,d_k\\}_{k=1}^{N_T}$, where $N_T$ denotes the dimensionality of $\\mathcal{T}$. Let, then, $d_k=\\{+1,-1\\}$ denote the \\textit{desired response} parameter corresponding to $\\textbf{x}_k$, whose value depends on which of the two classes $\\textbf{x}_k$ belongs to.\nThe equation of a hyperplane $\\Pi$ in $\\mathbb{R}^N$ reads:\n\\begin{equation}\n\\textbf{w}^T \\cdot \\textbf{x} + b=0 \\quad , \\nonumber\n\\end{equation}\nwith $\\textbf{w}$ and $b$ denoting, respectively, a $N$-dimensional adjustable weight vector and a bias. The BHS is the hyperplane characterized by the pair $(\\textbf{w}_o,b_o)$ which, for linearly separable patterns, fulfills the following conditions \\cite{Haykin}:\n\n\\begin{eqnarray}\n\\textbf{w}_o^T \\cdot \\textbf{x}_k+b_o\\ge 1 \\quad \\text{for $d_k = +1$} \\quad ,\\nonumber\\\\\n\\textbf{w}_o^T \\cdot \\textbf{x}_k+b_o\\le -1 \\quad \\text{for $d_k = -1$} \\quad .\\label{suppvec}\n\\end{eqnarray}\n\nThe data points, portrayed in magenta color in the right panel of Fig. \\ref{SVM}, for which Eqs. (\\ref{suppvec}) are satisfied with the equality sign, are called \\textit{support vectors}, and lie on the so-called \\textit{canonical hyperplanes} \\cite{Webb}, represented by the black dashed lines in the right panel of Fig. \\ref{SVM}. Figure \\ref{SVM} also illustrates the so-called \\textit{margin of separation}, defined as the distance $\\xi=1\/\\|\\textbf{w}_o\\|$ between the support vectors and the BSH.\nThe BSH, which maximizes $\\xi$ under the constraints (\\ref{suppvec}), can be found by determining the saddlepoint of the Lagrangian function $d\\mathcal{L}(\\textbf{w},b,\\lambda_1,...,\\lambda_{N_T})=0$, given by:\n\\begin{equation}\n\\mathcal{L}(\\textbf{w},b,\\lambda_1,...,\\lambda_{N_T})=\\frac{1}{2}\\textbf{w}^T\\cdot \\textbf{w}-\\sum_{k=1}^{N_T} \\lambda_k[d_k(\\textbf{w} \\cdot \\textbf{x}_k+b)-1] \\label{lagr} \\quad .\n\\end{equation} \nThe solution of such variational problem is easily found in the form \\cite{Haykin}:\n\\begin{equation}\n\\mathbf{w}_o=\\sum_{k=1}^{N_T} \\lambda_k d_k \\textbf{x}_k \\label{sol1}\n\\end{equation}\nwhere the Lagrange multipliers $\\lambda_k$ satisfy the conditions:\n\\begin{eqnarray}\n\\sum_{k=1}^{N_T} \\lambda_k d_k&=&0 \\quad ,\\nonumber\\\\\n\\lambda_k[d_k(\\textbf{w} \\cdot \\textbf{x}_k+b)-1]&=&0 \\quad \\text{for   $k=1,...,N_T$}  \\quad ,\\nonumber\n\\end{eqnarray}\n(the latter being known as the ``Karush-Kuhn-Tucker complementarity condition'' \\cite{Webb}) whereas $b_o$ can be determined, once $\\textbf{w}_o$ is known, using Eqs. (\\ref{suppvec}).\nWhen the two classes are not linearly separable, a possible strategy consists in introducing a suitable (nonlinear) function $\\Phi:E\\rightarrow F$ , which makes it possible to map the original pattern inputs into a \\textit{feature space} $F\\subseteq\\mathbb{R}^M$, in which a linear separation can be performed, cf. Fig. \\ref{nlSVM} \\cite{Webb}.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.75\\textwidth]{nlSVM.pdf}\n\\caption{Patterns which are not linearly separable can be mapped, via a function $\\Phi$, into a \\textit{feature space} where a linear separation of the classes can be achieved}\\label{nlSVM}\n\\end{figure}\nThus,by denoting as $\\boldsymbol\\Phi(\\textbf{x})=\\{\\Phi_j(\\textbf{x})\\}_{j=1}^M$ a set of nonlinear transformations from the original input space to the feature space, the corresponding variational problem leads now, in place of Eq. (\\ref{sol1}), to the expression:\n\\begin{equation}\n\\mathbf{w}_o=\\sum_{k=1}^{N_T} \\lambda_k d_k \\boldsymbol\\Phi(\\textbf{x}_k) \\label{sol2} \\quad .\n\\end{equation}\nIn our implementation of the SVM algorithm, we regarded the set $G$ as one of the two classes, whereas the other class, formerly introduced in Sec. \\ref{sec:sec1} and denoted by $nG$, was thought of as given by the union $nG = D \\cup I$.\nWe thus relied on the analysis of the histograms of the hazelnut nuclei, detailed in Sec. \\ref{sec:sec1}. Therefore, we introduced two variables to identify each hazelnut: we set $\\textbf{x}=(x_{mean},x_{max})$, where, for each histogram relative to an hazelnut nucleus, $x_{mean}$ and $x_{max}$ denote, respectively, the \\textit{average} shade of gray and the shade of gray equipped with the highest probability.\nTherefore, in the space spanned by the coordinates $x_{mean}$ and $x_{max}$, and parameterized by the values of $\\epsilon$ and $\\rho$, each hazelnut is represented by a single dot. The resulting distribution of dots, for different values of $\\epsilon$ and  $\\rho$, is illustrated in Figs. \\ref{raw3a} and \\ref{raw3b}, which evidence a clustering of points, for both the considered values of $\\rho$, around the bisectrix of the plane. This is readily explained by considering that, when reducing $\\epsilon$, the histograms attain a more and more symmetric shape. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_20r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_40r15.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_60r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_80r15.pdf}\n\\caption{Classification of the data in the 2D space spanned by the values of the observables $x_{mean}$ (horizontal axis) and $x_{max}$ (vertical axis), for $\\epsilon=20$ (top left), $\\epsilon=40$ (top right), $ \\epsilon=60$ (bottom left), and $ \\epsilon=80$ (bottom right), with $\\rho=1.5$.}\\label{raw3a}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_20r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_40r25.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_60r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Raw3_80r25.pdf}\n\\caption{Classification of the data in the 2D space spanned by the values of the observables $x_{mean}$ (horizontal axis) and $x_{max}$ (vertical axis), for $\\epsilon=20$ (top left), $\\epsilon=40$ (top right), $ \\epsilon=60$ (bottom left), and $ \\epsilon=80$ (bottom right), with $\\rho=2.5$.}\\label{raw3b}\n\\end{figure}\n\nFurthermore, an inspection of Figs. \\ref{raw3a} and \\ref{raw3b} reveals that the dots corresponding to the sets $D$ and $I$ are nested within the ensemble of points belonging to the set $G$: the classes $G$ and $nG$ are not amenable to be disentangled by a linear SVM regression, as also confirmed by the plots in Figs. \\ref{lin1} and \\ref{lin2}. \nIn each of the two latter figures, the left plot shows the elements of the adopted (randomly selected) training set: green and red symbols identify the elements of the two classes $G$ and $nG$, while the black circles indicate the support vectors. The black line indicates the boundary (best separating hyperplane) detected by the SVM, which sensibly depends on the chosen training set. The right plot, instead, displays all the available data (red and blue crosses represent, respectively, the elements of the classes $G$ and $nG$), complemented by the SVM test set output (red and blue circles). \nThe proper match between the colours of the circles and the crosses would indicate a successfully accomplished separation between the two classes, which, though, is not obtained with our data.\nFurthermore, no remarkable improvement is obtained by attempting a classification of the data by means of a nonlinear SVM algorithm, based on radial basis functions \\cite{Webb}, as shown in Figs. \\ref{nonlin1} and \\ref{nonlin2}. \nThe results of this Section, confirm, therefore, the predictions of the statistical analysis outlined in Sec. \\ref{sec:sec1}: the presence of a not linearly separable entanglement between points belonging to different classes can be thus traced back to the lack of a suitable statistical scales separation.\\\\\nThere is another relevant aspect, concerned with the implementation of the SVM algorithm, to be pointed out. \\\\\nWe remark, in fact, that each of the plots shown in Figs. \\ref{lin1} and \\ref{lin2} pertains to a specific training set of data $\\mathcal{T}$.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_20r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_40r15.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_60r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_80r15.pdf}\n\\caption{Classification of the data through a linear SVM algorithm. Shown is the 2D space spanned by the values of the observables $x_{mean}$ (horizontal axis) and $x_{max}$ (vertical axis), for $\\epsilon=20$ (top left), $\\epsilon=40$ (top right), $\\epsilon=60$ (bottom left), and $ \\epsilon=80$ (bottom right), with $\\rho=1.5$. In each left subfigure, shown are the training set of data (green and red crosses, denoting, respectively, the elements of the classes $G$ and $nG$), the support vectors (black circles) and the best separating hyperplane (black line). According to the SVM classification,the elements of the class $nG$ are expected to lie on the right of the boundary line. The right sub-figures, instead, display the 2D representation of all the available data (red and blue crosses, denoting, respectively, the elements of $G$ and those of $nG$) and the SVM output (red and blue circles).}\\label{lin1}\n\\end{figure}\n \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_20r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_40r25.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_60r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_Lineare_80r25.pdf}\n\\caption{Classification of the data with a linear SVM algorithm, as in Fig. \\ref{lin1}, but with $\\rho=2.5$.}\\label{lin2}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_20r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_40r15.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_60r15.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_80r15.pdf}\n\\caption{Classification of the data through a nonlinear SVM algorithm (based on radial basis functions) for $\\rho=1.5$ (cf. the caption of Fig. \\ref{lin1}).}\\label{nonlin1}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_20r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_40r25.pdf}\\\\\n\\vspace{2mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_60r25.pdf}\n\\hspace{1mm}\n\\includegraphics[width=0.45\\textwidth]{SVM_nonLineare_80r25.pdf}\n\\caption{Classification of the data with a nonlinear SVM algorithm (based on radial basis functions) for $\\rho=2.5$ (cf. the caption of Fig. \\ref{lin1}).}\\label{nonlin2}\n\\end{figure}\n\nWe can introduce, then, the quantity $\\Psi_\\ell(\\mathcal{T}_\\ell;\\epsilon,\\rho)$, relative to the specific training set $\\mathcal{T}_\\ell$, and defined as the ratio of the number of hazelnuts, belonging to the class $G$ and mistakenly classified as belonging to the class $nG$, to the total number of hazelnuts in the database, given by $N_G+N_{nG}$. The function $\\Psi_\\ell$ is an indicator of the performance of the SVM algorithm, and sensibly depends on the structure of the training sample considered in the simulation. \nThus, while the behaviour of $\\Psi_\\ell$, pertaining to single training samples, yields no indication about the onset of an optimal scale $\\epsilon^*$, the average $\\langle \\Psi \\rangle$, given by\n\\begin{equation}\n\\langle \\Psi \\rangle(\\epsilon,\\rho) = \\frac{1}{N_c}\\sum_{\\ell=1}^{N_c}\\Psi_\\ell(\\mathcal{T}_\\ell;\\epsilon,\\rho) \\quad , \\nonumber\n\\end{equation}\nand computed over a sufficiently large number $N_c$ of training samples, attains a minimum precisely at $\\epsilon^*=70$, cf. Fig. \\ref{Psi}. The latter value of $\\epsilon$ corresponds, in fact, to the scale of resolution maximizing the two statistical scales introduced in Sec. \\ref{sec:sec1}, cf. Fig. \\ref{ratio3}. The plot in Fig. \\ref{Psi} confirms, hence, that the onset of an optimum scale $\\epsilon^*$ can be numerically evinced also by means of SVM algorithms, provided that performance of the SVM is \\textit{averaged} over a sufficiently large number of training samples.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.65\\textwidth]{psi.pdf}\n\\caption{Behavior of $\\langle \\Psi \\rangle$, averaged over $N_c=500$ training samples, vs. $\\epsilon$, with $\\rho=1.5$, with error bars (in red).}\\label{Psi}\n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:conc}\n\nIn this work we performed a statistical analysis on the histograms of a set of hazelnut images, with the aim of obtaining a preliminary estimate of the performance of a machine learning algorithm based on statistical variables. We shed light, in Sec. \\ref{sec:sec1}, on the relevance of two statistical scales, which need to be widely separated to accomplish a successful pattern recognition. The intrinsic lack of such scale separation in our data was also evidenced by the numerical results reported in Sec. \\ref{sec:sec2}, revealing that no exhaustive classification can be achieved through SVM algorithms.\nMoreover, the analysis outlined in Sec. \\ref{sec:sec1} also unveiled the onset of an optimal resolution $\\epsilon^*$, which is expected to optimize the pattern recognition. This observation was also corroborated by the results discussed in Sec. \\ref{sec:sec2}, where the same value $\\epsilon^*$, maximizing the performance of the SVM algorithm, is recovered by averaging over a sufficiently large number of training samples.\nOur results, thus, strengthen the overall perspective that a preliminary estimate of the intrinsic statistical scales of the data constitute a decisive step in the field of pattern recognition and, moreover, pave the way for the further implementation of statistical mechanical techniques aimed at the development of a generation of more refined neural networks algorithms.\n\n\\newpage\n\n{\\bf Acknowledgments}\n\n\\vskip 5pt\n\nWe would like to thank Ferrero and Soremartec for their long-standing support of our research activity. We also thank Dr. A. Boscolo and Dr. L. Placentino, for providing us with the set of x-ray images used in this work.\nThis study was funded by ITACA, a project financed by the European Union, the Italian Ministry of Economy and Finance and the Piedmont Region.\n\n\\vskip 10pt\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe rotation frequencies\n$\\nu$ of pulsars generally decrease slowly in time, but occasionally experience sudden increases $\\Delta\\nu$ that are usually accompanied by increases in the absolute value of their spin-down rates, $\\dot{\\nu}$ \\citep{rm69,rd69,sl96}. \nThese spin-up events, known as glitches, are infrequent, not periodic, and cover a wide range of sizes \\citep[from $\\Delta \\nu\/\\nu \\sim 10^{-11}$ to $\\Delta \\nu\/\\nu \\sim 10^{-5}$;][]{elsk11,ymh+13}. \nThe mechanism that generates these events is not completely understood, but they are believed to be caused by angular momentum transfer from an internal neutron superfluid\nto the rest of the neutron star \\citep{ai75}.\n\nThanks to the few long-term monitoring campaigns that keep operating, some since the 1970s \\citep[e.g.][]{hlk+04,ymh+13}, the number of detected glitches has slowly increased, thereby improving the significance of statistical studies in pulsar populations. \n\\citet{ml90}, \\citet{lsg00}, and \\citet{elsk11} showed that the glitch activity $\\dot{\\nu}_{\\rm{g}}$ (defined as the mean frequency increment per unit of time due to glitches) correlates linearly with $|\\dot\\nu|$. \nThey also found that young pulsars (using the characteristic age, $\\tau_c=-\\nu\/2\\dot{\\nu}$, as a proxy for age), which also have the highest $|\\dot\\nu|$, exhibit glitches more often than older pulsars, with rates varying from about one glitch per year to one per decade among the young pulsars. \nUsing a larger and unbiased sample, \\cite{fer+17} confirmed that the size distribution of all glitches in a large and representative sample of pulsars is multi-modal \\citep[recently also seen by][]{ka14b,apj17}, with at least two well-defined classes of glitches: large glitches in a relatively narrow range $\\Delta \\nu \\sim (10-30)\\, \\rm{\\mu Hz}$, and small glitches with a much wider distribution, from $\\sim 10\\,\\mathrm{\\mu Hz}$ down to at least $10^{-4}\\,\\mathrm{\\mu Hz}$. \nFurther, \\cite{fer+17} found that a constant ratio $\\dot\\nu_{\\rm{g}}\/|\\dot\\nu| = 0.010 \\pm 0.001$ is consistent with the behaviour of nearly all rotation-powered pulsars and magnetars.\nThe only exception are the (few) very young pulsars, which have the highest spin-down rates, such as the Crab pulsar (PSR B0531$+$21) and PSR B0540$-$69. \n\nBecause glitches are rare events, the number of known glitches in the vast majority of pulsars is not enough to perform robust statistical analyses on individual bases.\nThis has made people focus on the few objects that have the largest numbers of detected glitches (about 10 pulsars).\nThe statistical distributions of glitch sizes and times between consecutive glitches (waiting times), for the nine pulsars with more than five known glitches at the time, were studied by \\citet{mpw08}.\nThey found that seven out of the nine pulsars exhibited power-law-like size distributions and exponential waiting time distributions.\nThe distributions of the other two (PSRs J0537$-$6910 and B0833$-$45, the Vela pulsar) were\nbetter described by Gaussian functions, setting preferred sizes and time scales.\nThese results have been further confirmed by \\citet{fmh17} and \\citet{hmd18}, who also found that there are at least two main behaviours among the glitching pulsars.\n\nCorrelations between glitch sizes and the times to the nearest glitches, either backward or forward, are naturally expected.\nWe know that glitch activity is driven by the spin-down rate \\citep{fer+17}, which suggests that glitches are the release of some stress that builds up  at a rate determined by $|\\dot{\\nu}|$.\nIf the stress is completely released at each glitch, then one should expect a correlation between size and the time since the last glitch.\nConversely, if glitches occur when a certain critical state is reached, one should expect a correlation between size and the time to the next glitch, as longer times would be needed to come back to the critical state after the largest glitches.\nMoreover, if both assumptions were indeed correct, glitches would all be of equal sizes and occur periodically. \nHowever, with the exception of PSR J0537$-$6910 (see below), no other pulsars have shown significant correlations between glitch sizes and the times to the nearest events \\citep[e.g.][]{wmp+00,ywml10,mhf18}.\nThis may be partly due to small-number statistics and might improve in the future, provided a substantial number of pulsars continue to be monitored for glitches.\n\nThe case of PSR J0537$-$6910, however, is very clear.\nWith more than 40 glitches detected in $\\sim 13$\\,yr, the statistical conclusions about its behaviour are much more significant than for any other pulsar.\nAs first reported by \\citet{mmw+06}, its glitch sizes exhibit a strong correlation with the waiting time to the following glitch \\citep[see also][who confirmed the correlation using twice as much data]{aeka18,fagk18}.\n\n\\citet{aeka18} interpret\nthis behaviour as an indication that glitches in this pulsar occur only once some threshold is reached.\nMoreover, this behaviour would imply that not necessarily all the stress is released in the glitches, thereby giving rise to the variety of (unpredictable) glitch sizes observed and the lack of backward time correlation.\n\nIn this work we study the sequence of glitches in the pulsars with at least ten detected events, by characterizing their distributions of glitch sizes and waiting times between successive glitches. Also, we test two hypotheses to explain\nwhy most pulsars do not show a correlation between glitch size and time to the following glitch: the effects of undetected small glitches and the possibility that two different classes of glitches are present in each pulsar.\n\n\\section{Pulsars with at least ten detected glitches} \n\\label{s1}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{fig1.pdf}\n\\caption{Upper part of the $P-\\dot{P}$ diagram for all known pulsars. \n\tThe pulsars in our sample have at least ten detected glitches and are labeled with different symbols.  \n\tLines of constant spin-down rate $\\dot{\\nu}$ are shown and labeled. \n\t$P$ and $\\dot{P}$ values were taken from the ATNF pulsar catalog \\protect\\footnotemark.\n}\\label{fig1}\n\\end{figure}\n\\footnotetext{\\url{http:\/\/www.atnf.csiro.au\/research\/pulsar\/psrcat}}\nTo date, there are eight pulsars with at least 10 detected glitches (Fig. \\ref{fig1}).\nPSRs J0205$+$6449, B0531$+$21 (the Crab pulsar), B1737$-$30, B1758$-$23, and J0631$+$1036 have been observed regularly by the Jodrell Bank Observatory \\citep[JBO,][]{hlk+04}.\nPSR B1338$-$62 has been observed by the Parkes telescope, and  the Vela pulsar has been observed by several telescopes, including Parkes, the Jet Propulsion Laboratory, and others in Australia and  Southafrica \\citep[e.g.][]{downs81,mkhr87,ymh+13,buc13}. \nPSR J0537$-$6910 is the only object in our sample not detected in the radio band and was observed for 13 years by the \\textit{Rossi X-ray Timing Explorer} \\citep[RXTE,][]{aeka18,fagk18}. \nGlitch epochs and sizes were taken from the JBO online glitch catalog \\footnote{\\url{http:\/\/www.jb.man.ac.uk\/pulsar\/glitches\/gTable.html}}, where more information and the appropriate references for each measurement can be found.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{fig_2.pdf}\n\\caption{Logarithm (base 10) of glitch sizes $\\Delta\\nu$ (with $\\Delta\\nu$ measured in $\\mu$Hz) as a function of the glitch epoch for the pulsars in the sample. \n\tThe gray areas mark periods of time in which there were no observations for more than 3 months. \n\t$N_g$ is the number of glitches detected in the respective pulsar, until 20 April 2019 (MJD 58593). \n\tTo build a continuous sample, in the analyses of the Crab pulsar, we only use the 25 glitches after MJD 45000, when daily observations started \\citep{eas+14}. All panels share the same scale, in both axes.\n\t} \\label{fig2}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{fig_3.pdf}\n\\caption{Distribution of $\\log \\Delta \\nu$ (with $\\Delta \\nu$ measured in $\\rm{\\mu Hz}$) for the pulsars in our sample. The orange areas indicate that glitches with $\\Delta\\nu<0.01\\,\\rm{\\mu Hz}$ could be missing due to detectability issues.} \\label{fig3}\n\\end{figure*}\n\nFigures \\ref{fig2} and \\ref{fig3} show that the Vela pulsar and PSR J0537$-$6910 produce glitches of similar sizes, particularly large glitches ($\\Delta \\nu > 10$ $\\mu$Hz), and in fairly regular time intervals. The absence of smaller glitches in these pulsars is not a selection effect, as it is quite unlikely that a considerable amount of glitches with sizes up to $\\Delta \\nu \\sim 10\\,\\rm{\\mu Hz}$, far above the detection limits reported in the literature \\citep[see ][and text below]{wxe+15}, could have gone undetected. \nOn the other hand, the rest of the pulsars exhibit irregular waiting times and cover a wider range of sizes ($\\Delta \\nu \\sim 10^{-3}-10$ $\\mu$Hz). \n\nThe cadence of the timing observations varies considerably from pulsar to pulsar (and even with time for individual pulsars), and the sensitivity of the  observations, from which the glitch measurements were performed, are also different between different pulsars.\nThis means that the chances of detecting very small glitches are different for each pulsar and that the completeness of the samples towards small events might also be different \\citep{eas+14}.\nNonetheless, in this study we use a single value to represent the glitch size below which samples are likely to be incomplete due to detectability issues.\nFor an observing cadence of 30 days and a rotational noise of 0.01 rotational phases, glitch detection is severely compromised below sizes $\\Delta\\nu \\sim 10^{-2}\\, \\rm{\\mu Hz}$, especially if their frequency derivative steps are larger than $|\\Delta\\dot{\\nu}|\\sim 10^{-15} \\, \\rm{Hz\\, s^{-1}}$ \\citep[see][]{wxe+15}. We use the above numbers to characterize the glitch detection capabilities in this sample of pulsars, but we note that such cadence and rotational noise are rather pessimistic values in some cases.\n\n\\section{Distributions of glitch sizes and times between glitches} \n\\label{distris}\n \nIn the following, we model the distributions of glitch sizes ($\\Delta\\nu$, measured in $\\mu$Hz) and the distributions of times between successive glitches ($\\Delta \\tau$, measured in yr) for each pulsar in our sample. \nFour probability density distributions are considered: Gaussian,\n\\begin{equation}\nM(x|\\mu,\\sigma) = C_{\\rm{Gauss}}\\,\\exp\\left[\\frac{-(x-\\mu)^2}{2\\sigma^2}\\right]\\text{,}\n\\end{equation}\npower-law,\n\\begin{equation}\nM(x|\\alpha) = \\dfrac{\\alpha - 1}{x_{\\rm{min}}}\\left(\\dfrac{x}{x_{\\rm{min}}}\\right)^{-\\alpha}\\text{,}\n\\end{equation}\nlog-normal,\n\\begin{equation}\nM(x|\\mu_{\\rm{L-N}},\\sigma_{\\rm{L-N}}) = \\dfrac{C_{\\rm{L-N}}}{x}\\,\\exp\\left[\\frac{-(\\ln x-\\mu_{\\rm{L-N}})^2}{2\\sigma_{\\rm{L-N}}^2}\\right]\\text{,}\n\\end{equation}\nand exponential,\n\\begin{equation}\nM(x|\\lambda) = \\lambda\\, \\exp\\left[-\\lambda(x-x_{\\rm{min}})\\right]\\text{.}\n\\end{equation}\n\nThe set $\\{\\mu,\\sigma, \\alpha, \\mu_{\\rm{L-N}},\\sigma_{\\rm{L-N}}, \\lambda\\}$ are the fitting parameters.  \nAll the distributions are normalized in the range $x_{\\rm{min}}$ to $\\infty$. Formally, $x_{\\rm{min}}$ is given by detection limits. \nHowever, it is not simple to define precise values for $\\Delta \\nu_{\\rm{min}}$ and $\\Delta \\uptau_{\\rm{min}}$ for each pulsar.\nThus we use $\\Delta \\nu_{\\rm{min}} = 10^{-2}\\, \\mu$Hz for the glitch sizes (see previous section), and  the smallest interval of time between glitches in each pulsar as $\\Delta \\uptau_{\\rm{min}}$.\n\nFor the Gaussian and log-normal distributions the normalization constants $C_{\\rm{Gauss}}$ and $C_{\\rm{L-N}}$ were found numerically.\nWe use the maximum likelihood technique to obtain the parameters of the models that describe best the data, and use the Akaike Information Criterion \\citep[AIC,][]{aka74} to compare the different models \\citep[see also the Appendix in][]{fer+17}.\n\n\\begin{figure*}\n\\includegraphics[width=18cm]{fig_4.pdf}\n\\caption{Cumulative distribution of glitch sizes and model fits. The best-fitting models are indicated by thicker curves.}\\label{fig4}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=18cm]{fig_5.pdf}\n\\caption{Cumulative distribution of waiting times between successive glitches and model fits. The best-fitting models are indicated by thicker curves.}\\label{fig5}\n\\end{figure*}\n\n\\begin{table*\n\\centering\n\\caption{Distributions of glitch sizes: results of the fits and the AIC weights for each model; using glitches with $\\Delta\\nu \\geq 0.01\\, \\rm{\\mu Hz}$.}\n\\label{Table_sizes}\n\\begin{tabular}{@{}lcccccccccc@{}}\n\\toprule \\toprule\nPSR Name & $w^{\\rm{Gauss}}$ & $w^{\\textrm{Power law}}$ & $w^{\\textrm{L-N}}$ & $w^{\\rm{Exp}}$ & $\\hat{\\mu}$ & $\\hat{\\sigma}$ & $\\hat{\\alpha}$ & $\\hat{\\mu}_{\\rm{L-N}}$ & $\\hat{\\sigma}_{\\rm{L-N}}$ & $\\hat{\\lambda}$ \\\\\n & & & & & $\\rm{\\mu Hz}$ & $\\rm{\\mu Hz}$ &  &  &  & $(\\rm{\\mu Hz})^{-1}$\\\\\n\\midrule\nJ0205$+$6449 & $10^{-8}$ & $\\mathbf{0.66}$ & 0.33 & $10^{-5}$ & 15(5) & 20(4) & 1.27(6) & 0.7(7) & 2.5(3) & 0.07(6)\\\\\n\nB0531$+$21 & $10^{-17}$ & 0.02 & $\\mathbf{0.97}$ & $10^{-7}$ & 1.2(5) & 3(1) & 1.4(1) & -1.3(3) & 1.5(2) & 0.8(7)\\\\\n\nJ0537$-$6910 & $\\mathbf{0.96}$ & $10^{-24}$ & $10^{-8}$ & 0.03 & 15(1) & 9.9(9) & 1.19(5) & 2.2(2) & 1.3(2) & 0.063(6)\\\\\n\nJ0631$+$1036 & $10^{-12}$ & $\\mathbf{0.94}$ & 0.05 & $10^{-8}$ & 1(1) & 3(1) & 1.4(1) & -1.9(6) & 2.1(4) & 0.61(4)\\\\\n\nB0833$-$45 & $\\mathbf{0.997}$ & $10^{-13}$ & $10^{-6}$ & 0.002 & 21(2) & 9(1) & 1.2(4) & 2.7(2) & 1.2(4) & 0.05(1)\\\\\n\nB1338$-$62 & $10^{-5}$ & 0.07 & $\\mathbf{0.53}$ & 0.4 & 2.5(5) & 2.7(3) & 1.36(5) & -0.1(3) & 1.6(1) & 0.4(1)\\\\\n\nB1737$-$30 & $10^{-14}$ & $\\mathbf{0.82}$ & 0.17 & $10^{-7}$ & 0.6(2) & 1.0(2) & 1.38(6) & -2.0(3) & 1.9(1) & 1.5(8)\\\\\n\nB1758$-$23 & 0.06 & 0.004 & 0.07 & $\\mathbf{0.866}$ & 0.6(1) & 0.51(8) & 1.3(2) & -1.2(4) & 1.5(3) & 1.7(6)\\\\\n\\bottomrule\n\\end{tabular}\n\\tablefoot{$w^m$ denotes the Akaike weight of the model $m$. \n$\\hat \\mu$ and $\\hat \\sigma$ are the mean and the standard deviation of the Gaussian model, and $\\hat\\alpha$ is the power-law index. \n$\\hat \\mu_{\\rm{L-N}}$ and $\\hat \\sigma_{\\rm{L-N}}$ are the mean and the standard deviation of the log-normal model, respectively. $\\hat \\lambda$ is the rate parameter of the exponential distribution. \nThe values in parentheses correspond to the uncertainty in the last quoted digit and were calculated using the usual bootstrap method. We marked in bold the values of $w^m$ for the best models.}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{Distributions of waiting times between successive glitches: results of the fits and the AIC weights for each model.}\\label{Table_times}\n\\begin{tabular}{@{}lcccccccccc@{}}\n\\toprule \\toprule\nPSR Name & $w^{\\rm{Gauss}}$ & $w^{\\rm{Power law}}$ & $w^{\\rm{L-N}}$ & $w^{\\rm{Exp}}$ & $\\hat{\\mu}$ & $\\hat{\\sigma}$ & $\\hat{\\alpha}$ & $\\hat{\\mu}_{\\rm{L-N}}$ & $\\hat{\\sigma}_{\\rm{L-N}}$ & $\\hat{\\lambda}$\\\\\n& & & & & yr & yr &  &  &  & yr$^{-1}$\\\\\n\\midrule\n\nJ0205$+$6449 & $0.001$ & $0.40$ & 0.16 & $\\mathbf{0.43}$ & 1.3(4)& 1.4(4) & 1.7(1) & -$0.2(3)$ & 1.0(1) & 0.9(5)\\\\\n\nB0531$+$21 & $10^{-4}$ & $10^{-5}$ & 0.15 & $\\mathbf{0.84}$ & 1.3(2) & 1.3(2) & 1.4(1) & -$0.2(2)$ & 1.0(1) & 0.8(2)\\\\\n\nJ0537$-$6910 & $\\mathbf{0.72}$ & $10^{-10}$ & 0.07 & 0.2 & 0.28(2) & 0.15(1) & 1.64(8) & -1.44(9) & 0.65(6) & 4.3(4)\\\\\n\nJ0631$+$1036  & $10^{-4}$ & $10^{-5}$ & 0.20 & $\\mathbf{0.79}$ & 1.4(4) & 1.7(6) & 1.3(2) & -0.3(3) & 1.2(2) & 0.7(3)\\\\\n\nB0833$-$45  & $\\mathbf{0.993}$ & $10^{-10}$ & $10^{-4}$ & 0.006 & 2.5(2) & 1.2(1) & 1.3(3) & 0.7(2) & 0.9(2) & 0.41(9)\\\\\n\nB1338$-$62  & 0.25 & $10^{-3}$ & 0.20 & $\\mathbf{0.54}$ & 0.88(9) & 0.42(4) & 1.9(2) & -0.3(1) & 0.51(5) & 1.7(3)\\\\\n\nB1737$-$30  & $10^{-5}$ & $10^{-6}$ & 0.17 & $\\mathbf{0.82}$ & 0.9(1) & 0.9(1) & 1.44(7) & -0.6(1) & 1.0(1) & 1.2(2)\\\\\n\nB1758$-$23  & 0.04 & 0.16 & 0.08 & $\\mathbf{0.72}$ & 2.4(4) & 1.4(2) & 2.1(2) & 0.7(1) & 0.61(8) & 0.6(2)\\\\\n\\bottomrule\n\\end{tabular}\n\\tablefoot{$w^m$ denotes the Akaike weights of the model $m$. $\\hat \\mu$ and $\\hat \\sigma$ are the mean and the standard deviation of the Gaussian model, and $\\hat\\alpha$ is the power-law index. $\\hat \\mu_{\\rm{L-N}}$ and $\\hat \\sigma_{\\rm{L-N}}$ are the mean and the standard deviation of the log-normal model, respectively.  $\\hat \\lambda$ is the rate parameter of the exponential distribution. The values in parentheses correspond to the uncertainties in the last digit, and were calculated by using the bootstrap method. We marked in bold the values of $w^m$ for the best models.}\n\\end{table*}\n\nFigures \\ref{fig4}-\\ref{fig5} and Tables \\ref{Table_sizes}-\\ref{Table_times} summarize the results of fitting these distributions to each pulsar. There is no single distribution type that can simultaneously describe all the pulsars satisfactorily, for either sizes or waiting times.\nThe size distributions present a large variety (as also found in the model of \\citealt{cm19}): the log-normal distribution gives the best fit for the Crab pulsar and PSR B1338$-$62, power-law for PSRs J0631$+$1036, B1737$-$30, and J0205$+$6449, and exponential for PSRs B1758$-$23.\n\n\nWe also note that PSR J0205+6449 and PSR B1758$-$23 are the pulsars with the fewest recorded glitches in the sample (both have 13 glitches detected), hence we ought to wait and confirm this result once more events are detected.\n\nIn the case of PSRs J0537$-$6910 and B0833$-$45 (Vela), the best fit for both size and waiting time distributions are Gaussian functions.\nTheir size distributions are centered at large sizes $\\Delta \\nu \\approx 15$  and $20\\, \\rm{\\mu Hz}$, respectively, consistent with the peak of large glitches in the combined distribution for all pulsars \\citep{fer+17}. \n\nThe distributions of times between successive glitches offer more homogeneous results. Besides the case of PSR J0537$-$6910 and the Vela pulsar (best modelled by Gaussian functions), the waiting time distributions for all the other pulsars are best represented by exponential functions.\nThese results are in agreement with \\citet{mpw08,wwty12}, and \\citet{hmd18} for almost all the pulsars studied. \nThe only exception is PSR B1338$-$62, for which \\cite{hmd18} reported a local maximum in the distribution and classified this pulsar as a quasi-periodic glitcher.\n\nIf $\\Delta\\nu_{\\rm{min}}$ is set to the size of the smallest detected glitch in each pulsar (rather than to $10^{-2}\\, \\mu$Hz), the results of the fits are very similar, and give parameters within the uncertainties presented in Table \\ref{Table_sizes}.\n\n\n\\section{Time series correlations: Glitch size and time to the next glitch} \n\\label{s2}\n\nDifferent studies have shown that for PSR J0537$-$6910 the glitch magnitudes $\\Delta \\nu_k$ are strongly correlated with the waiting times to the following glitch $\\Delta \\uptau_{k+1}$ \\citep[][and see Fig. \\ref{fig6}]{mmw+06,aeka18,fagk18}. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{fig_6.pdf}\n\\caption{Time to next glitch, $\\Delta \\uptau_{k+1}$, as a function of glitch size, $\\Delta \\nu_k$, for all the pulsars in the sample.} \n\\label{fig6}\n\\end{figure*}\n\nWe test whether this correlation is also present in the other pulsars of the sample, and show the results in Table \\ref{dnu_dt_next} and Fig. \\ref{fig6} \\citep[this is fairly consistent with][though we note that the samples of glitches are not exactly the same]{mhf18}. None of them exhibits a correlation as clear as PSR J0537$-$6910. \nHowever, for PSRs J0205$+$6449, J0631$+$1036, B1338$-$62, and B1758$-$23, the Pearson correlation coefficients are larger than $0.5$ and the $p$-values are $\\sim 10^{-3}$, or less. \nTherefore, at $95\\%$ confidence level ($p$-values $ < 0.05$), we can reject the null hypothesis that $\\Delta \\nu_k$ and $\\Delta \\uptau_{k+1}$ are uncorrelated in these pulsars.\nSince the Pearson coefficient can be dominated by outliers, we also compute the Spearman rank correlation coefficient, obtaining similar or even stronger correlations, except for PSR J0631$+$1036.\n\n\\begin{table}\n\\centering\n\\caption{Correlation coefficients between $\\Delta \\nu_k$ and $\\Delta \\uptau_{k+1}$.} \n\\begin{tabular}{@{}lccccc@{}}\n\\toprule \\toprule\nPSR Name & $N_{\\mathrm{g}}$ & $r_p$ & $p_p$ & $r_s$ & $p_s$ \\\\\n\\midrule\nJ0205$+$6449 & 13 & 0.88 & 0.0002 & $0.76$ & 0.004 \\\\\nB0531$+$21 & 25 & -0.10 & 0.62 & -0.12 & 0.57 \\\\\nJ0537$-$6910 & 45 & 0.95 & $10^{-22}$  & 0.95 & $10^{-23}$ \\\\\nJ0631$+$1036 & 17 & 0.93 & $10^{-7}$ & 0.20 & 0.45 \\\\\nB0833$-$45 & 20 & 0.24 & 0.31 & 0.31 & 0.21 \\\\\nB1338$-$62 & 23 & 0.59 & 0.003 & 0.70 & 0.0002\\\\\nB1737$-$30 & 36 & 0.29 & 0.09 & 0.29 & 0.08 \\\\\nB1758$-$23 & 13 & 0.76 & 0.003 & 0.80 & 0.001 \\\\\n\\bottomrule\n\\end{tabular}\n\\tablefoot{The first and second columns contain the names of the pulsars and the respective number of glitches detected, respectively. \nThe third and fourth columns correspond to the Pearson linear correlation coefficient $r_p$ and the respective $p$-value $p_p$. \nThe last two columns correspond to the Spearman correlation coefficient $r_s$ and the respective $p$-value $p_s$.\n\t}\\label{dnu_dt_next}\n\\end{table}\n\nIt is also interesting to note that not only for PSR J0537$-$6910, but for all pulsars in the sample except the Crab, both the Pearson and Spearman correlation coefficients are positive. \nThe probability of finding at least six out of seven pulsars having the same sign as our reference case, just by chance, is rather low. \nThe probability of getting exactly $k$ successes among $n$ trials, with $1\/2$ success probability in each trial, is $P(k\\,|\\,n) = {n\\choose k}(1\/2)^n$. \nThus, the probability of getting at least 6 successes in 7 trials is\n\n\\begin{equation}\nP(\\geq 6\\,|\\,7)=P(6\\,|\\,7)+P(7\\,|\\,7)=\\frac{1}{16}=0.0625\\,.\n\\end{equation}\n\nThis low probability suggests that the waiting time to the following glitch is at least partially regulated by the size of the previous glitch.\n\nIn order to explain why the correlation for all other pulsars is much less clear than\nfor PSR J0537$-$6910, we explore two hypotheses, both of which are motivated by noting that most glitches in PSR J0537$-$6910 are large:\n\n\n\\begin{itemize}\n\\item[(I)] The correlation is intrinsically present in the full population of glitches of each pulsar, but glitches below a certain size threshold are not detected, thereby increasing by random amounts the times between the detected ones and worsening the correlation.\\\\\n\n\n\\item[(II)] There are two classes of glitches: glitches above a certain threshold size that follow the correlation, and glitches below the same threshold that are uncorrelated. \n\n\\end{itemize}\n\n\n\n\\subsection{Hypothesis I: Incompleteness of the sample}\n\nIn order to test the first hypothesis, we simulate a hypothetical pulsar with 100 glitches that follow a perfect correlation between $\\Delta\\nu_{k}$ and $\\Delta \\uptau_{k+1}$.\nThe events smaller than a certain value are then removed to understand the effect of their absence in the correlation. \nThe procedure is the following:\n\n\n\\begin{enumerate}\n\n\\item\nGlitch sizes are generated from a power-law distribution given by $dN\/d \\Delta\\nu \\propto \\Delta\\nu^{-\\alpha}$, with power-law index $\\alpha>1$. \nWe choose a power-law distribution because it mainly produces small events, and we want to see the effect of removing a substantial fraction of them.\nSeveral different choices for $\\alpha$ were considered. \nHere we only show the results for $\\alpha = 1.2$ and 1.4, as they generate distributions that resemble some of the ones observed.\n\nThe distributions do not have an upper cutoff, and the lower limit was varied so that, after reducing the sample of glitches (as we explain in step 3 below), the resulting sample covers the typical observed range of glitch sizes ($10^{-2} - 10^2\\, \\rm{\\mu Hz}$).\n\n\\item\nThe time to the next glitch $\\Delta \\uptau_{k+1}$ is computed in terms of the glitch size $\\Delta \\nu_k$ as:\n\\begin{equation}\n\\Delta\\uptau_{k+1}= C\\Delta\\nu_{k}\\, . \n\\label{eq_corr}\n\\end{equation}\nThe value of the proportionality constant $C$ is irrelevant in this case, since we are simulating a generic pulsar.\n\\\\\n\n\\item \nSteps (1) and (2) are repeated until a sequence of 100 glitches is reached. \nThen the 80 smallest are removed, thereby leaving a reduced sample of 20 to be analyzed, which is comparable to the number of glitches observed in each of our 8 pulsars.\nThe lower limit for the distribution is computed analytically so that, after reducing the sample of glitches, the final sample covers the typical observed range of glitch sizes ($10^{-2} - 10^2\\, \\rm{\\mu Hz}$).\\\\\n\n\\item \nFinally, we calculate the time interval between each pair of successive glitches in the reduced sample,  and determine both the Spearman and Pearson correlation coefficients between $\\Delta \\nu_k$ and $\\Delta \\uptau_{k+1}$.\n\n\\end{enumerate}\n\n\n\nAfter simulating $10^4$ cases, it was found that removing all glitches smaller than a certain value has a minor effect on the correlation. \nRepresentative realizations are shown in Fig. \\ref{hyp1}, where the correlation between $\\Delta \\nu_k$ and $\\Delta \\uptau_{k+1}$ is plotted in log-scale to show more clearly the dispersion produced by the removal of the smallest glitches. \nWe observe that missing small glitches does not substantially worsen the correlation: more than $90\\%$ of the realizations give correlation coefficients $\\geq 0.95$ (both Pearson and Spearman).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{fig_h1_v1.pdf}\n\\caption{Reduced samples of simulated glitches from an assumed parent distribution $dN\/d\\Delta\\nu\\propto \\Delta\\nu^{-\\alpha}$ with a perfect correlation $\\Delta\\tau_{k+1}=C\\Delta\\nu_k$, with $C=0.21\\, \\mathrm{yr\\, \\mu Hz^{-1}}$.\nTop: Resulting correlation between $\\Delta \\nu_k$ and $\\Delta \\uptau_{k+1}$. \nBottom: The corresponding distributions of $\\log \\Delta \\nu$ for the reduced samples of glitches. For both panels, each color (and point marker) represents a typical realization in the simulations, for different power-law exponents as shown in the legends.}\\label{hyp1}\n\\end{figure}\n\nFor $\\alpha>1.4$ the distribution becomes narrower, accumulating towards the lower limit. \nSince a large fraction of the simulated glitches have very similar sizes, after removing the 80 smallest glitches the correlation does worsen, and yields correlation coefficients between $0.4$ and $0.9$, which are similar to those exhibited by the real data.\nHowever, in these cases the distributions of glitch sizes differ strongly from those observed for the pulsars in our sample.\n\nFrom these simulations, we conclude that it is unlikely that the non-detection of all the glitches below a certain detection limit is the explanation for the low observed correlations in pulsars other than PSR J0537$-$6910.\n\n\\subsection{Hypothesis II: Two classes of intrinsically different glitches}\n\nThe second hypothesis states that pulsars exhibit two classes of glitches: larger events, which follow a linear correlation between $\\Delta \\nu_k$ and $\\Delta \\uptau_{k+1}$; and smaller events, for which these variables are uncorrelated.\nWe allow the point of separation between large and small glitches to be different for each pulsar.\n\nTo visualize whether this hypothesis works, correlation coefficients (for the same pair of variables, $\\Delta \\nu_k$ and $\\Delta \\uptau_{k+1}$) were calculated for sub-sets of glitches of the original sample. \nThe sub-sets are defined as all glitches with sizes larger or equal to a given $\\Delta\\nu_\\textrm{min}$. \nCorrelation coefficients as a function of $\\Delta\\nu_\\textrm{min}$ are plotted in Fig. \\ref{r_df0_min} for each pulsar.\nVisual inspection of the plots immediately tells us that by removing small glitches no pulsar reaches the level of correlation observed for PSR J0537$-$6910, for both correlation tests.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=180mm]{r_df0_min.pdf}\n\\caption{Pearson (orange squares) and Spearman (blue dots) correlation coefficients for glitches larger or equal than $\\Delta \\nu_\\textrm{min}$. Each panel represents a pulsar in our sample. For each pulsar, the last point in the plot was calculated with its five largest glitches. Note that some pulsars are shown in log-scale for a better visualization.} \n\\label{r_df0_min}\n\\end{figure*}\n\n\n\nIn the following we explore the curves in Fig. \\ref{r_df0_min} in some more detail.\nFor that purpose, Monte Carlo simulations of pulsars with correlated and uncorrelated glitches were performed.\nSince the underlying glitch size distributions of the pulsars in the sample are unknown, we use the measured values of a given pulsar.\nThe following is the procedure for one realization:\n\n\\begin{enumerate}\n\\item The glitches larger than a certain value $\\Delta\\nu^{\\star}$ are chosen in random order and assigned epochs according to their size.\nThe first one is set at an arbitrary epoch and the epochs of the following ones are assigned according to \n\\begin{equation}\n\\Delta\\uptau_{k+1}=\\Delta\\nu_{k}\\cdot 10^{x}\\, ,\n\\label{eq_hyp2}\n\\end{equation}\nwhere $x$ is drawn from a Gaussian distribution centred at $\\bar{x}=\\log(C)$ and with a standard deviation equal to $\\sigma_{\\bar{x}}$. \nThe latter allows us to introduce a dispersion in the correlation of the simulated glitches. \nThe distribution of $\\log(\\Delta\\uptau_{k+1}\/\\Delta\\nu_k)$ for all glitches with $\\Delta\\nu>5\\,\\mu$Hz in PSR J0537$-$6910 can be well modelled by a Gaussian distribution with standard deviation $\\sigma_{0537}=0.085$ (in logarithmic scale, if $\\Delta\\uptau_{k+1}$ is measured in days and $\\Delta\\nu_k$ is measured in $\\mu$Hz). \nIn the simulations, $\\sigma_{\\bar{x}}$ was set either to zero (i.e. $x=\\log(C)$, perfect correlation) or to multiples of $\\sigma_{0537}$. \\\\\n\n\\item The glitches smaller than $\\Delta\\nu^{\\star}$ are distributed randomly over the time span between the first and the last correlated glitches. \nThe resulting waiting times of all, correlated and uncorrelated glitches are then multiplied by a factor that ensures that their sum equals the time in between the first and the last observed glitches. \\\\\n\n\\item Steps 1 and 2 were repeated $10^4$ times for each considered value of $\\Delta \\nu^{\\star}$. \n\n\\end{enumerate}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{fig_explaining_sim2_v3.pdf}\n\\caption{Correlation coefficients $r_p$ (orange) and $r_s$ (blue) versus $\\Delta\\nu_\\textrm{min}$ for simulated glitches under hypothesis II, and for three $\\Delta\\nu^{\\star}$ cases: left, when all glitches are correlated ($\\Delta\\nu^{\\star}\\sim0$); middle, about half of them are correlated ($\\Delta\\nu^{\\star}=12.39\\,\\mu$Hz); right, none of them is correlated ($\\Delta\\nu^{\\star}=40\\,\\mu$Hz).\nShaded regions represent the values of the $70\\%$ closer to the median of all realizations.\nThe dashed lines show particular realizations.\nThese simulations used the glitch sizes of PSR J0537$-$6910 and $\\sigma_{\\bar{x}}=\\sigma_{0537}$.\nIn all cases the last points in the plots were calculated using the five largest glitches.\n}\n\\label{examples}\n\\end{figure*}\n\n\nThe plots in Fig. \\ref{examples} show the results of simulations using the glitch sizes of PSR J0537$-$6910 and $\\sigma_{\\bar{x}}=\\sigma_{0537}$ for three values of $\\Delta \\nu^{\\star}$.\nThe results are shown via curves of $r$ versus $\\Delta\\nu_\\textrm{min}$, to compare with Fig. \\ref{r_df0_min}.\nThe shaded areas represent the $70\\%$ of the correlation coefficients closer to the median of all realizations.\nWe visually inspected the distributions of $r_p$ and $r_s$ for all possible $\\Delta\\nu_\\textrm{min}$  values, and for many $\\Delta \\nu^{\\star}$ cases.\nIt was verified that the median is sufficiently close to the maximum of the distribution in most cases.\nThough, this tends to fail for the largest $\\Delta\\nu_\\textrm{min}$ values, where the $r_p$ and $r_s$ distributions are rather flat.\nBut this is irrelevant because any conclusion pointing to a case in which only a few glitches are correlated (large $\\Delta\\nu_\\textrm{min}$) would have little statistical value, regardless of the above.\nThus we are confident that the shaded areas effectively cover the most possible outcomes of series of glitches under the assumptions considered.\n\nWe now use the plots in Fig. \\ref{examples} to understand the curves of the correlation coefficients as functions of $\\Delta\\nu_\\textrm{min}$ in Fig. \\ref{r_df0_min}, in the frame of hypothesis~II:\n\n\n\\begin{itemize}\n\n\\item[(a)] If all glitches were correlated, which is the case shown in the leftmost plot in Fig. \\ref{examples}, the correlation coefficients would decrease gradually as $\\Delta\\nu_\\textrm{min}$ increases. \nThis is because a progressive reduction of the sample, starting from the smallest events (i.e. increasing the remaining waiting times by small random amounts), will gradually kill the correlation.\nNote that the correlation coefficients of the simulated glitches start at values just below $1.0$ for the smallest $\\Delta\\nu_\\textrm{min}$, just like the observations of PSR J0537$-$6910.\nThis is because $\\sigma_{\\bar{x}}=\\sigma_{0537}$ in those simulations.\nOnly for $\\sigma_{\\bar{x}}=0$ the simulations would start at correlation coefficients equal to $1.0$.\n\n\\item[(b)] If only glitches above a certain size $\\Delta \\nu^{\\star}$ were correlated, the correlation coefficients would improve as small glitches are eliminated, and the remaining sub-set approaches the one in which all glitches are correlated (as in the middle plot of Fig. \\ref{examples}).\nOne would expect a maximum correlation for $\\Delta\\nu_\\textrm{min}\\sim\\Delta \\nu^{\\star}$, and a gradual decrease as $\\Delta\\nu_\\textrm{min}$ increases beyond $\\Delta \\nu^{\\star}$.\n\n\\item[(c)] If there were no correlated glitches, we should expect a rather flat curve of low correlation coefficients oscillating around zero (rightmost plot in Fig. \\ref{examples}).\n\n\\end{itemize}\n\nThe behaviours just described correspond to the general trends exhibited by the shaded areas in Fig. \\ref{examples}, which evolve smoothly with $\\Delta\\nu_\\textrm{min}$. \nHowever, particular realizations show abrupt variations, of both signs, just as the observations do in Fig. \\ref{r_df0_min}.\n\nClearly, PSR J0537$-$6910 is best represented by case (a).\nIndeed, both correlation coefficients for this pulsar are maximum (and very similar) when all glitches are included and they decrease gradually as the smallest glitches are removed (Fig. \\ref{r_df0_min}).\nNonetheless, we note that $r_p$ stays above $0.9$ (and $p_p<3\\times10^{-12}$) for $\\Delta\\nu_\\textrm{min}\\leq7\\,\\mu$Hz, hence it is possible that the smallest glitches are not correlated. \nAnother indication for this possibility is that the six glitches below $5\\,\\mu$Hz fall to the right of the  distribution of $\\log(\\Delta\\uptau_{k+1}\/\\Delta\\nu_k)$ for all glitches, and the width of the distribution is reduced considerably (from more than 2 decades to a half decade) when they are removed.\nIn other words, the straight line that best fits the ($\\Delta\\uptau_{k+1}$, $\\Delta\\nu_k$) points passes closer to the origin \\citep[a more physically motivated situation,][]{aeka18}, and the data exhibit a smaller dispersion around this line, when the smallest glitches are not included.\n\n\n\n\nThe pulsars B1338$-$62, and B1758$-$23 may in principle also correspond to case (a).\nAs mentioned at the beginning of section \\ref{s2}, they present mildly significant correlations when all their glitches are considered, and both their $r_p$ and $r_s$ curves in Fig. \\ref{r_df0_min} decrease as $\\Delta\\nu_\\textrm{min}$ increases.\nBy performing simulations with $\\Delta \\nu^{\\star}=0$, and for different values of $\\sigma_{\\bar{x}}$, we find that the correlation coefficients of PSR B1758$-$23 are within the range of $70\\%$ of the possible outcomes if $\\sigma_{\\bar{x}}$ is set to 5-6 times $\\sigma_{0537}$.\n\nFor PSR B1338$-$62 the situation is less clear because the amplitudes of the variations of both $r_p$ and $r_s$ for $\\Delta\\nu_\\textrm{min}<1\\,\\mu$Hz are rather high.\nOne possible interpretation is that all glitches are correlated and the variations are due to the correlation not being perfect (i.e. $\\sigma_{\\bar{x}}\\neq0$).\nWe find that only for $\\sigma_{\\bar{x}}\\geq10\\times\\sigma_{0537}$ the simulations can reproduce such behaviour and the observed values. \nAnother possibility is that $\\Delta \\nu^{\\star}\\sim0.2\\,\\mu$Hz, which could explain the local maxima of $r_p$ and $r_s$ around that value.\nThe maxima and subsequent values can indeed be reproduced with lower levels of noise, $\\sigma_{\\bar{x}}=5\\times\\sigma_{0537}$. \nBut for smaller values of $\\Delta\\nu_\\textrm{min}$ most realizations ($>70\\%$) give correlation coefficients below $0.5$, thus they fail at reproducing the observed $0.6$-$0.7$ at $\\Delta\\nu_\\textrm{min}=0$.\n\nIt is clear that Hypothesis II does not apply to this pulsar directly, and that the observations are not consistent with a set of uncorrelated glitches either.\nBased on the lack of glitches with sizes equal or less than $0.1\\,\\mu$Hz after MJD $\\sim$ 50400 (Fig. \\ref{fig2}), we speculate that the sample might be incomplete for glitches smaller than this size after this date\\footnote{This would be a more extreme case than those considered for the Hypothesis I because $0.1\\,\\mu$Hz is a rather high limit.}.\n\n\n\nThe pulsars J0205$+$6449 and J0631$+$1036 also exhibit significant Pearson correlations when all their glitches are considered.\nHowever, their $r_s$ curves tend to increase with $\\Delta\\nu_\\textrm{min}$ rather to decrease.\nAs mentioned before, the Pearson test can be affected by outliers, hence the behaviour we see for $r_p$ is likely due to the very broad size and waiting times distributions and the low numbers of events towards the high ends of the distributions, which produce outlier points for both pulsars (Fig.\\ref{fig6}).\nIt is therefore difficult to conclude anything for PSR J0631+1036. \nMoreover, the observed behaviour is very hard to reproduce by the simulations, even for high levels of noise (we tried up to $\\sigma_{\\bar{x}}=12\\times\\sigma_{0537}$).\nPerhaps its largest glitches ($\\Delta\\nu\\geq0.1\\,\\mu$Hz) are indeed correlated, but the statistics are too low to conclude anything.\n\nFor J0205, however, the Spearman coefficients $r_s$ are rather high ($>0.55$ for all $\\Delta\\nu_\\textrm{min}$) and both coefficients become similar and even higher for $\\Delta\\nu_\\textrm{min}>1\\,\\mu$Hz. \nIt is possible that glitches above this size are correlated in this pulsar.\nWe find that the observed $r_p$ and $r_s$, and their evolution with $\\Delta\\nu_\\textrm{min}$, are within the $70\\%$ of simulations with $\\Delta \\nu^{\\star}=1.3\\,\\mu$Hz and for $\\sigma_{\\bar{x}} = 2\\times\\sigma_{0537}$. \nWe note, however, that in this case the correlation coefficients observed for  $\\Delta\\nu_\\textrm{min}\\leq0.1\\,\\mu$Hz are higher than the vast majority of the realizations.\nPerhaps the small glitches are also correlated and follow their own relation, though we did not simulate such scenario.\nWe conclude that the Hypothesis II does not fully explain this pulsar, although the 8 glitches above $1\\,\\mu$Hz appear to be well correlated indeed.\n\n\n\n\nThe Vela pulsar is the only pulsar in the sample that seems well represented by case (b).\nThe highest $r_p=0.68$ has a probability $p_p=0.003$ and is obtained for $\\Delta\\nu_\\textrm{min}\\sim2\\,\\mu$Hz.\nBoth $r_p$ and $r_s$ decline monotonically for larger $\\Delta\\nu_\\textrm{min}$ values. \nThis behaviour suggests that glitches of sizes above $\\sim2\\,\\mu$Hz might indeed be correlated, but the correlation is somewhat noisy.\nThe observed correlation coefficients fall within the middle 70$\\%$ of the realizations if $\\sigma_{\\bar{x}}=2\\times\\sigma_{0537}$ and for $\\Delta\\nu^{\\star}=2$-$10\\,\\mu$Hz.\nThe case $\\Delta\\nu^{\\star}=9.35\\,\\mu$Hz is presented in Fig. \\ref{vela_h2}.\nWe prefer this case because simulations for $\\Delta\\nu^{\\star}=2\\,\\mu$Hz tend to fail at reproducing the low correlation coefficients ($\\leq0.4$) observed for the smallest $\\Delta\\nu_\\textrm{min}$.\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{vela_2_hip2.pdf}\n\\caption{Observations and simulations of the Vela pulsar.\nLeft: Shaded regions indicate the values obtained by the $70\\%$ closer to the median of all realizations. \nThe observations are overlaid using dashed lines.\nCentre: comparison of observations (dashed) and one particular realization.\nRight: $\\Delta\\uptau_{k+1}$ versus $\\Delta\\nu_k$ for the same realization (red triangles) and for the observations (grey dots).\nOrange represents $r_p$ values and blue represents $r_s$ values in all panels.\nThe simulations were performed using $\\sigma_{\\bar{x}}=2\\times\\sigma_{0537}$ and $\\Delta\\nu^{\\star}=9.35\\,\\mu$Hz.}\n\\label{vela_h2}\n\\end{figure*}\n\n\n\nFinally, the cases of PSRs B0531+21 (the Crab) and B1737$-$30 are rather inconclusive.\nThe Crab pulsar is perhaps the pulsar for which case (c) applies the best. \nBoth correlation coefficients are negative or positive, and in both cases stay at relatively low absolute values, which leads to the conclusion that there are no correlated glitches in the Crab pulsar.\nWe note that the high $r_p$ and $r_s$ values observed for $\\Delta\\nu_\\textrm{min}\\sim0.6\\,\\mu$Hz are obtained with the 5-6 largest events and that a linear fit to their $\\Delta \\nu_k - \\Delta\\uptau_{k+1}$ does not pass close to the origin.\n\n \nThe case of B1737$-$30 is more complex.\nThe observations show two $\\Delta\\nu_\\textrm{min}$ values, $0.0015$ and $0.03\\,\\mu$Hz, after which the correlation coefficients decrease with the removal of more small glitches (Fig. \\ref{r_df0_min}). \nThis behaviour is hard to reproduce under Hypothesis II, unless the dispersion of the correlation is increased considerably, to $10\\times\\sigma_{0537}$ or more.\nWe conclude that Hypothesis II does not apply to this pulsar directly and that there is some extra complexity, as the data are also inconsistent with a set of purely uncorrelated glitches.\n\n\n\nSurprisingly, even though no pulsar complies perfectly with Hypothesis II, and the only way in some cases is to increase the dispersion of the correlation ($\\sigma_{\\bar{x}}\\gg\\sigma_{0537}$), there is no pulsar in the sample that is well represented by case (c) (only the Crab, to some extent).\n\nTherefore, the sizes of at least some glitches must be positively correlated with the times to the next glitch in the available datasets.\nThe question is why this correlation is much stronger in PSR J0537$-$6910 than in all other pulsars of our sample.\nCould this be an effect of its particularly high spin-down rate? Or the fact that most of its glitches are large?\nIt could be that the correlations are indeed there, as stated in Hypothesis II, but for some reason exhibit high $\\sigma_{\\bar{x}}$ values. \nMaybe the fact that the glitches in PSR J0537$-$6910 occur so frequently ensures that the relationship stays pure.\nBut it could also be that reality was more complex.\nFor instance, it could be that both small and large glitches were correlated, but each of them followed a different law.\n\n\n\n\\section{Other correlations} \n\\label{s3}\n\nWe looked for other correlations between the glitch sizes and the times between them.\nSpecifically, we tried  $\\Delta \\nu_k$ vs $\\Delta \\uptau_{k}$ (size of the glitch versus the time since the preceding glitch), and $\\Delta \\nu_k$ vs $\\Delta \\nu_{k-1}$ (size of the glitch versus the size of the previous glitch).\nNo pulsar shows a significant correlation between these quantities (Table \\ref{others_correlations}). \n\n\n\\begin{table*}\n\\caption{Correlation coefficients for the pairs of variables $(\\Delta \\nu_k,\\,\\Delta \\uptau_{k})$, and $(\\Delta \\nu_k,\\, \\Delta \\nu_{k-1})$.} \\label{others_correlations}\n\\small\n     \\begin{subtable}{0.47\\textwidth}\n     \\begin{tabular*}{\\linewidth}{@{}l \n        @{\\extracolsep{\\fill}} SS\n        S[table-format=2.2(2)]\n        S[table-format=2.2(2)]@{}}\n          \\toprule\n          \\phantom{Var.} &  \n          \\multicolumn{4}{c}{$\\Delta \\nu_k$ vs $\\Delta \\uptau_{k}$}\\\\\n          \\cmidrule{1-5}\n          {PSR Name}&  {$r_p$} & {$p_p$} & {$r_s$} & {$p_s$}\\\\\n          \\midrule\n          J0205$+$6449\\hspace{0.5cm}  & 0.16 & 0.60 & 0.44 & 0.15\\\\\n          B0531$+$21   & -0.02 & 0.90 & 0.40 & 0.05\\\\\n          J0537$-$6910   & -0.08 & 0.60  & -0.12 & 0.41 \\\\[1ex]\n          J0631$+$1036 & -0.10 & 0.68 & -0.18 & 0.49 \\\\\n          B0833$-$45 &  0.55 & 0.01 & 0.27 & 0.24 \\\\\n          B1338$-$62 &  -0.30 & 0.16 & -0.18 & 0.41 \\\\[1ex]\n          B1737$-$30 &  -0.02 & 0.89 & -0.10 & 0.56 \\\\\n          B1758$-$23 &  -0.02 & 0.94 & -0.04 & 0.89 \\\\\n          \\bottomrule\n     \\end{tabular*}%\n    \n     \\end{subtable}%\n     \\hspace*{\\fill}%\n     \\begin{subtable}{0.47\\textwidth}\n     \\begin{tabular*}{\\linewidth}{@{}l \n        @{\\extracolsep{\\fill}} SS\n        S[table-format=2.2(2)]\n        S[table-format=2.2(2)]@{}}\n          \\toprule\n          \\phantom{Var.}\n          &  \\multicolumn{4}{c}{$\\Delta \\nu_k$ vs $\\Delta \\nu_{k-1}$}\\\\\n          \\cmidrule{1-5}\n          {PSR Name}&  {$r_p$} & {$p_p$} & {$r_s$} & {$p_s$}\\\\\n          \\midrule\n          J0205$+$6449\\hspace{0.5cm}  & -0.06 & 0.83 & 0.25 & 0.42\\\\\n          B0531$+$21  & -0.10 & 0.61 & -0.15 & 0.47\\\\\n          J0537$-$6910  & -0.13 & 0.38 & -0.16  & 0.29\\\\[1ex]\n          J0631$+$1036 & -0.12 & 0.65 & 0.32 & 0.21 \\\\\n          B0833$-$45 & -0.08 & 0.71 & -0.12 & 0.59\\\\\n          B1338$-$62 & -0.33 & 0.13 & -0.13 & 0.55\\\\[1ex]\n          B1737$-$30 & -0.11 & 0.50 & 0.03 & 0.85\\\\\n          B1758$-$23 & -0.02 & 0.92 & -0.04 & 0.89\\\\\n          \\bottomrule\n     \\end{tabular*}%\n    \n     \\end{subtable}\n\\tablefoot{The first column contains the names of the pulsars considered in the sample. $r_{n}$ and $p_{n}$ correspond to a correlation coefficient and its $p$-value, respectively. The sub-index $n = p$ denotes the Pearson correlation, and  $n=s$ denotes the Spearman correlation.}\n\\end{table*}\n\n\nNearly all pulsars in our sample show negative correlation coefficients (both, Pearson and Spearman) for $\\Delta \\nu_k$ vs $\\Delta \\uptau_{k}$. \nThe only exceptions are the Vela pulsar and PSR J0205$+$6449 (see Table \\ref{others_correlations}). Our results are in general agreement with \\cite{mhf18},\nwho also found a lack of correlation between $\\Delta \\nu_k$ vs $\\Delta \\uptau_{k}$ for individual pulsars.\n\nFor $\\Delta \\nu_k$ vs $\\Delta \\nu_{k-1}$, in most cases the correlation coefficients are close to zero and the $p$-values are larger than $0.2$, i.e., no individual pulsar shows a significant correlation. However, the results could still be meaningful for the sample as a whole because all the pulsars have negative correlation coefficients, except for the Spearman coefficients for PSRs J0631$+$1036 and B1737$-$30). \nThe probability of getting all Pearson's correlations coefficients of the same sign just by chance, regardless of whether the sign is positive or negative, is $2\\times p_{\\mathrm{binom}}(8|8) = 0.007$. This could establish an interesting constraint on the glitch mechanism: Smaller glitches are somewhat more likely to be followed by larger ones, and vice-versa.\nHowever, this statement has to be confirmed with more data in the future.\n\n\n\n\\section{Discussion} \n\\label{disc}\n\n\\citet{fer+17} found that all pulsars (with the strong exception of the Crab pulsar and PSR B0540$-$69)\nare consistent with a constant ratio between the glitch activity, $\\dot{\\nu}_{\\rm g}$, and the spin-down rate,\n$\\dot\\nu_{\\rm{g}}\/|\\dot\\nu| = 0.010 \\pm 0.001$, i.e., $\\approx 1\\%$ of their spin-down is recovered by the glitches. This fraction has been interpreted as the fraction of the moment of inertia in a superfluid component that transfers its angular momentum to the rest of the star in the glitches  \\citep{lel99,aghe12}.\n\\citet{fer+17} used the observed bimodal distribution of glitch sizes to distinguish between large and small glitches, with the boundary at $\\Delta\\nu=10\\, \\mathrm{\\mu Hz}$, and argued that the constant ratio is determined by the large glitches, whose rate, $\\dot N_\\ell$ is also proportional to $|\\dot\\nu|$. In\nthis scenario, the\nmuch lower (sometimes null) glitch activities measured in many low-$|\\dot{\\nu}|$ pulsars are due to \ntheir observation time spans\nnot being long enough to include any large glitches (or any glitch at all).\nInterestingly, the pulsars in our sample (except the Crab) are quite consistent with the constant ratio (Fig. \\ref{fig_discussion}), even those, like PSRs B1338$-$62, B1737$-$30, and B1758$-23$, which do not have any large glitches contributing to their activities.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{discussion.pdf}\n\\caption{$\\dot{\\nu}_g\/|\\dot{\\nu}|$ versus $|\\dot{\\nu}|$ for pulsars in our sample.\nThe dashed-line with the blue region correspond to the constant ratio $\\dot{\\nu}_g\/|\\dot{\\nu}| = 0.010 \\pm\n0.001$, determined by \\citet{fer+17}.\nThe error bars were calculated as described in the latter paper.}\n\\label{fig_discussion}\n\\end{figure}\n\n\nOn the other hand, pulsars with higher spin-down rates also have a larger fraction of large glitches. At the highest spin-down rates ($|\\dot{\\nu}|\\geq 10^{-11}$\\,Hz\\,s$^{-1}$), the production of large glitches becomes comparable and sometimes higher than the production of small glitches, again with the notorious exception of the Crab and PSR B0540$-$69.\nThis trend is also followed by the pulsars in our sample: all large glitches (but one in PSR J0631+1036), are concentrated in PSRs J0205$+$6449, J0537$-$6910, and the Vela pulsar, which are (together with the Crab) the ones with largest $|\\dot{\\nu}|$ values (see Fig. \\ref{fig1} and \\ref{fig_discussion}). \n\n\nThus, it seems to be the case that both large and small glitches draw from the same angular momentum reservoir (for all but the very young, Crab-like pulsars), but have different trigger mechanisms, the large ones being produced once a critical state\nis reached, whereas small ones occur in a more random fashion. \nFor reasons still to be understood, the glitch activity of relatively younger, high $|\\dot\\nu|$, Vela-like pulsars is dominated by large glitches, whereas for smaller $|\\dot \\nu|$ the large glitches become less frequent, both in absolute terms and relative to the small ones \\citep{wmp+00,elsk11}. \nIn this context, it is interesting to note that recent long-term braking index measurements\nindicate that Vela-like pulsars move  towards the region where PSRs J0631+1036, B1737$-$30, and B1758$-$23 are located on the $P$--$\\dot{P}$ diagram \\citep[][]{els17}.\n\n\n\n\n\n\n\\section{Summary and Conclusions} \n\\label{conc}\n\nWe studied the individual glitching behaviour of the eight pulsars that today have at least ten detected glitches.\nOur main conclusions are the following:\n\n\n\\begin{enumerate}\n\n\\item \nWe confirm the previous result by \\cite{mpw08} and \\cite{hmd18} that, for Vela and PSR J0537$-$6910, the distributions of both their glitch sizes and waiting times are best fitted by Gaussians, indicating well-defined scales for both variables. For all other pulsars studied, the waiting time distribution is best fitted by an exponential (as would be expected for mutually uncorrelated events), but they have a variety of best-fitting size distributions: a power law for PSR J0205+6449, J0631+1036, and B1737$-$30, a log-normal for the Crab and PSR B1338$-$62, and an exponential for PSR B1758$-$23.\n\n\\item \nAll pulsars in our sample, except for the Crab, have positive Spearman and Pearson correlation coefficients for the relation between the size of each glitch, $\\Delta\\nu_k$, and the waiting time to the following glitch, $\\Delta\\tau_{k+1}$. For each coefficient, the probability for this happening by chance is $1\/16=6.25\\%$. \nBoth coefficients also stay positive as the small glitches are removed\n(see Fig. \\ref{r_df0_min}).\n\n\n\\item \nPSR J0537$-$6910 shows by far the strongest correlation between glitch size and waiting time until the following glitch ($r_p=r_s=0.95$, $p$-values $\\lesssim 10^{-22}$). \nAnother three pulsars, PSRs J0205$+$6449, B1338$-$62, and B1758$-$23, have quite significant correlations ($p$-values $\\leq 0.004$ for both coefficients).\n\n\\item\nOur first hypothesis to explain the much weaker correlations in all other pulsars compared to PSR J0537$-$6910, namely missing glitches that are too small to be detected, is very unlikely to be correct. Our Monte Carlo simulations show that, for reasonable glitch size distributions, it cannot produce an effect as large as observed.\n\n\\item \nOur alternative hypothesis, namely that there are two classes of glitches, large correlated ones and small uncorrelated ones, comes closer to reproducing the observed relations; notably for PSRs J0205$+$6449 and Vela.\nThe resulting correlations for both pulsars present dispersions that are twice the one observed for PSR J0537$-$6910.\nFor the other pulsars, the required dispersions to accommodate this hypothesis are much larger.\n\n\n\\item\nThe correlation coefficients between the sizes of two successive glitches, $\\Delta\\nu_{k-1}$ and $\\Delta\\nu_k$, as well as between the size of a glitch, $\\Delta\\nu_k$ and the waiting time since the previous glitch, $\\Delta\\tau_k$, are generally not significant in individual pulsars, but they are negative for most cases, suggesting some (weaker) relation also among these variables.\n\n\\item\nExcept for the Crab, all pulsars in our sample are consistent with the constant ratio between glitch activity and spin-down rate, $\\dot\\nu_\\mathrm{g}\/|\\dot\\nu|=0.010\\pm 0.001$ \\citep{fer+17}. This includes cases dominated by large glitches, as well as others with only small glitches. \n\n\\item\nThe previous results suggest that large and small glitches draw their angular momentum from a common reservoir, although they might be triggered by different mechanisms. Large glitches, which dominate at large $|\\dot\\nu|$ (except for the Crab and PSR B0540$-$69), might occur once a certain critical state\nis reached, while small glitches, dominating in older pulsars with lower $|\\dot\\nu|$, occur at essentially random times.\n\n\n\n\n\\end{enumerate}\n\nAll the above is based on the behaviour of the pulsars with the most detected glitches. \nEven though we have shown before that the activity of all pulsars appears to be consistent with one single trend, these pulsars could still be outliers among the general population. \nOnly many more years of monitoring will clarify the universality of these results.\n\n\n\\begin{acknowledgements}\nWe thank Vanessa Graber and Simon Guichandut for valuable comments on the first draft of this article. We are also grateful to Wilfredo Palma for conversations that guided us at the beginning of this work. We also thank Ben Shaw for information regarding the detection of recent glitches and for keeping the glitch catalog up to date.\nThis work was supported in Chile by CONICYT, through the projects ALMA31140029, Basal AFB-170002, and FONDECYT\/Regular 1171421 and 1150411.\nJ.R.F. acknowledges\tpartial support by an NSERC Discovery Grant awarded to A. Cumming at McGill University.\nC.M.E. acknowledges support by the Universidad de Santiago de Chile (USACH).\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nDiscrete statistical models with latent variables and graphical components are widely used in statistics, machine learning, and many real-world applications. Examples include noisy-or Bayesian networks in medical diagnosis \\citep{shwe1991qmrdt, halpern2013noisyor}, binary latent skill models in educational cognitive diagnosis \\citep{chen2015qmat, xu2017rlcm, gu2021jmle}, and restricted Boltzmann machines and their variants in machine learning \\citep{hinton2006fast, goodfellow2016deep}.\nGenerally, incorporating latent variables into graphical models can greatly enhance the flexibility of a model. But such flexibility comes at a cost of the increasing model complexity and statistical subtlety, including the identifiability as a fundamental and challenging issue.\nRecognizing the potential non-identifiability caused by latent variables in graphical models, a body of works \\cite[e.g.][]{foygel2012half, evans2014markovian} project away the latent variables and study the property of the induced mixed graph among the observed variables.\nIn some applications, however, the latent variables themselves carry important substantive meanings and inferring the parameters involving the latent is of paramount importance and practical interest \\citep[e.g.][]{bing2020overlap, bing2020detecting}.\nIn this work, we propose a general algebraic technique to investigate identifiability of discrete models with complicated latent and graphical components, characterize the minimal identifiability requirements for a class of such models motivated by diagnostic test applications, and along the way reveal a new geometry about multidimensional latent structures -- the blessing of dependence on identifiability.\n\n\n\nStatistically, a set of parameters for a family of models are said to be identifiable, if distinct values of the parameters correspond to distinct joint distributions of the observed variables.\nIdentifiability is a fundamental prerequisite for valid statistical estimation and inference. \nIn the literature, identifiability of discrete statistical models with latent variables is known to be challenging to study, partly due to their inherent complex nonlinearity.\nFor example, Latent Class Models \\citep[LCMs;][]{lazarsfeld1968latent} are a simplest form of discrete models with latent structure, which assumes a univariate discrete latent variable renders the multivariate categorical responses conditional independent.\nDespite the seemingly simple structure and the popularity of LCMs in social and biomedical applications, their identifiability issues had eluded researchers for decades.\n\\cite{goodman1974} investigated several specific small-dimensional LCMs, some being identifiable and some not.\n\\cite{gyllenberg1994non} proved LCMs with binary responses are not \\textit{strictly identifiable}. \\cite{carreira2000practical} empirically showed the so-called practical identifiability of LCMs using simulations. \nAnd finally, \\cite{allman2009} provided a rigorous statement about the \\textit{generic identifiability} of LCMs, whose proof leveraged Kruskal's Theorem from \\cite{kruskal1977three} on the uniqueness of three-way tensor decompositions. \n\n\n\n\nTo be concrete, \\textit{strict identifiability} means model parameters are identifiable everywhere in some parameter space $\\mathcal T$.\nA slightly weaker notion, \\textit{generic identifiability}\nformalized and popularized by \n\\cite{allman2009}, allows for a subset $\\mathcal N\\subseteq\\mathcal T$ where non-identifiability may occur, requiring $\\mathcal N$ to have Lebesgue measure zero in $\\mathcal T$; such $\\mathcal N$s are basically zero sets of polynomials of model parameters \\citep{allman2009}.\nIn some cases, these measure-zero subsets may be trivial, such as simply corresponding to the boundary of the parameter space. In some other cases, however, these subsets may be embedded in the interior of the parameter space, or even carries rather nontrivial geometry and interesting statistical interpretation (as is the case in this work under the minimal conditions for generic identifiability).\nA precise characterization of the measure-zero subset where identifiability breaks down is essential to performing correct statistical analysis and hypothesis testing \\citep{drton2009lrt}. But it is often hard to obtain a complete understanding of such sets or to derive sharp conditions for identifiability in complicated latent variable models. These issues become even more challenging when graphical structures are present in latent variable models.\n\n\n\n\nIn this work, we present a general algebraic technique to study the identifiability of discrete statistical models with latent and graphical components, and use it to investigate an interesting class of such models. \nIn the literature, pioneered by \\cite{allman2009}, many existing identifiability results for models involving discrete latent structures leveraged Kruskal's Theorem\n\\citep[e.g.,][]{allman2008covarion, fang2019, culpepper2019ordinal, chen2020slcm, fang2020bifactor}; these studies cover models ranging from phylogenetics in evolutionary biology to psychometrics in education and psychology.\nThese identifiability proofs using Kruskal's Theorem often rely on certain global rank conditions of the tensor formulated under the model.\nIn contrast, we \ncharacterize a useful transformation property of the Khatri-Rao tensor products of arbitrary discrete variables' probability tables.\nWe then use this property to investigate how any specific parameter impacts the zero set of \npolynomials induced by the latent and graphical constraints.\nThis general technique covers as a special case the one in \\cite{xu2017rlcm} for restricted latent class models with binary responses.\nOur approach will unlock  possibilities to study identifiability\nat the {finest} possible scale (rather than checking global rank conditions of tensors), and hence help obtain sharp conditions and characterize the aforementioned measure-zero non-identifiable sets.\nIn particular, we will study settings where Kruskal's theorem {does not apply}, demonstrating the power of this technique. \n\n\n\n\n\nWe provide an overview of our results.\nMotivated by epidemiological and educational diagnosis tests, we focus on discrete models with multiple binary latent variables, where\nthe latent-to-observed measurement graph is a forest of star trees. Namely, each latent variable can have several observed noisy proxy variables as children.\nWe allow the binary latent variables to have arbitrary dependencies among themselves for the greatest possible modeling flexibility. \nCall this model the \\textit{Binary Latent cliquE Star foreSt (BLESS)} model. \nWe characterize the necessary and sufficient graphical criteria for strict and generic identifiability, respectively, of the BLESS model; this includes identifying both the discrete star-forest structure and the various continuous parameters.\nUnder the minimal conditions for generic identifiability that each latent variable has \\emph{exactly two} observed children, we show that the measure-zero set $\\mathcal N$ in which identifiability breaks down is the independence model of the latent variables.\nThat is, our identifiability condition delivers a deep and somewhat surprising geometry of \\textit{blessing-of-dependence} -- the statistical dependence between latent variables can help restore identifiability.\nMore broadly, this blessing-of-dependence phenomenon has nontrivial connections to and implications on the uniqueness of matrix and tensor decompositions. \nBuilding on the blessing of dependence, we propose a formal statistical hypothesis test of identifiability in the boundary case. In fact, in this case testing identifiability amounts to testing the marginal dependence of the latent variables' observed children. \n\n    \nOur results have practical relevance on statistical modeling and real-world applications employing multidimensional latent structures.\nIn many applications, it is intrinsically natural and interpretable to conceptualize each latent construct as presence or absence of some underlying trait.\nExamples include diagnosing the presence\/absence of multiple unobserved disease pathogens of a patient in epidemiology \\citep{wu2016partially, wu2017nested, o2019causes}, and determining the mastery\/deficiency of multiple latent skills of a student in educational testing \\citep{von2005, henson2009, dela2011, george2015cdm}.\nStatistically, such an appealing conceptualization leads to statistical models with\nmultidimensional binary latent variables. \nIn addition, such models are also widely used in machine learning and deep learning\nas building blocks of deep generative models\n\\citep{hinton2006reducing, hinton2006fast, salakhutdinov2009deep}.\nThe models mentioned above often possess unique and curious algebraic structures.\nUnderstanding the statistical properties caused by these algebraic structures will provide valuable insight into scientific and statistical learning practices. This work contributes a new tool and new understanding in this regard.\n\n\n\nThe rest of this paper is organized as follows.\nSection \\ref{sec-setup} introduces the formal setup of the BLESS model and several relevant identifiability notions.\nSection \\ref{sec-main} presents the main theoretical results of identifiability and overviews our general proof technique.\nSection \\ref{sec-test} proposes a statistical hypothesis test of identifiability of the BLESS model under minimal conditions for generic identifiability.\nSection \\ref{sec-prac} presents two real-world examples and Section \\ref{sec-disc} concludes the paper.\n\n\n\n\\section{Model Setup and Identifiability Notions}\\label{sec-setup}\n\\subsection{Binary Latent cliquE Star foreSt (BLESS) model}\nWe next introduce the setup of the BLESS model, the focus of this study.\nWe first introduce some notation. \nFor an integer $m$, denote $[m]=\\{1,\\ldots,m\\}$. For a $K$-dimensional vector $\\boldsymbol  x=(x_1,\\ldots, x_K)$ and some index $k\\in[K]$, denote the $(K-1)$-dimensional vector by $\\boldsymbol  x_{-k} = (x_1,\\ldots,x_{k-1},x_{k+1},\\ldots,x_K)$.\nConsider discrete statistical models with $K$ binary latent variables $\\alpha_1,\\ldots,\\alpha_K\\in\\{0,1\\}$ and $p$ categorical observed variables $y_1,\\ldots,y_p \\in[d]$.\nHere $d\\geq 2$ is the number of categories of each observed variable.\nBoth the latent vector $\\boldsymbol \\alpha = (\\alpha_1,\\ldots,\\alpha_K) \\in \\{0,1\\}^K$ and the observed vector $\\boldsymbol  y=(y_1,\\ldots,y_p) \\in [d]^K$ are subject-specific random quantities, and have their realizations for each subject $i$ in a random sample.\nFor two random vectors (or variables) $\\boldsymbol  x$ and $\\boldsymbol  y$, denote by $\\boldsymbol  x \\perp\\!\\!\\!\\perp \\boldsymbol  y$ if $\\boldsymbol  x$ and $\\boldsymbol  y$ are statistically independent, and denote by $\\boldsymbol  x \\not\\! \\perp\\!\\!\\!\\perp \\boldsymbol  y$ otherwise.\n\n\\begin{figure}[h!]\n\\centering\n\\resizebox{0.4\\textwidth}{!}{\n\\begin{tikzpicture}\n\\def5 {5}\n\\def2.8cm {1.4cm}\n\n\\foreach \\s in {1,...,5}\n{\n\\node (\\s)[draw, circle, minimum size=20pt, inner sep=0pt] at ({90+360\/5 * (\\s - 1)}:2.8cm) {$\\alpha_{\\s}$};\n}\n\\def 11 {10}\n\\def 2.8cm {2.8cm}\n\n\\foreach \\ss in {1,...,11}\n{\n\\node (y\\ss)[draw, circle, minimum size=20pt, inner sep=0pt, fill=black!10] at ({72+360\/11 * (\\ss - 1)}:2.8cm) {$y_{\\ss}$};\n}\n\n\\draw[dotted, thick] (1) -- (2);\n\\draw[dotted, thick] (1) -- (3);\n\\draw[dotted, thick] (1) -- (4);\n\\draw[dotted, thick] (1) -- (5);\n\\draw[dotted, thick] (2) -- (3);\n\\draw[dotted, thick] (2) -- (4);\n\\draw[dotted, thick] (2) -- (5);\n\\draw[dotted, thick] (3) -- (4);\n\\draw[dotted, thick] (3) -- (5);\n\\draw[dotted, thick] (4) -- (5);\n\n\\draw[->, thick] (1) -- (y1);\n\\draw[->, thick] (1) -- (y2);\n\\draw[->, thick] (2) -- (y3);\n\\draw[->, thick] (2) -- (y4);\n\\draw[->, thick] (3) -- (y5);\n\\draw[->, thick] (3) -- (y6);\n\\draw[->, thick] (4) -- (y7);\n\\draw[->, thick] (4) -- (y8);\n\\draw[->, thick] (5) -- (y9);\n\\draw[->, thick] (5) -- (y10);\n\\end{tikzpicture}}\n\n\n\\caption\nBLESS model with $K=5$ latent variables and $p=10$ observed variables.\nAll nodes are discrete random variables, with $\\alpha_k\\in\\{0,1\\}$ latent and $y_j\\in\\{1,\\ldots,d\\}$ observed. Directed edges form the measurement graph and can be equivalently represented as a $10 \\times 5$ graphical matrix $\\mathbf G$. Undirected dotted lines between latent variables $\\alpha_k$'s indicate arbitrary dependence in the latent part.\n}\n\\label{fig-graph1}\n\\end{figure}\n\n\n\nA key structure in the BLESS model is the latent-to-observed \\emph{measurement graph}. \nThis is a bipartite graph with directed edges from the latent $\\alpha_k$'s to the observed $y_j$'s indicating direct statistical dependence.\nThe BLESS model posits that the measurement graph is a forest of star trees; namely,\neach latent variable can have multiple observed proxy variables as \\emph{children}, but each observed variable has exactly one latent \\emph{parent}. \nOn the latent part, we allow the binary latent variables to have arbitrary dependencies for the greatest possible modeling flexibility. Figure \\ref{fig-graph1} provides a graphical illustration of the BLESS model, where we draw latent variables as white nodes and observed variables as gray nodes.\nAlthough assuming each observed variable having exactly one latent parent may appear to be restrictive, we point out that the arbitrary dependence between the latent variables indeed allows the observables to have extremely flexible and rich joint distributions.\nIn Figure \\ref{fig-graph1}, the solid directed edges from the latent to the observed variables form a star-forest-shaped measurement graph, and the dotted undirected edges between all pairs of latent variables indicate arbitrary possible dependence among them. \n\n\nEquivalently, we can represent the bipartite measurement graph from the $K$ latent variables to the $p$ observed children in a $p\\times K$ \\emph{graphical matrix} $\\mathbf G=(g_{j,k})$ with binary entries, where $g_{j,k}=1$ indicates $\\alpha_k$ is the latent parent of $y_j$ and $g_{j,k}=0$ otherwise.\nEach row of $\\mathbf G$ contains exactly one entry of ``1'' due to the star-forest graph structure.\nStatistically, the conditional distribution of $y_j\\mid \\boldsymbol \\alpha$ equals that of $y_j\\mid\\alpha_{k}$ if and only if $g_{j,k}=1$.\nWe can therefore denote the conditional distribution of $y_j$ given the latent variables as follows,\n\\begin{align*}\n\\forall c_j \\in[d],\\quad \\mathbb P(y_j = c_j \\mid \\boldsymbol \\alpha, \\mathbf G) = \n\\mathbb P(y_j = c_j \\mid \\alpha_{k},~ g_{j,k}=1)\n=\n\\begin{cases}\n\\theta^{(j)}_{c_j\\mid 1}, & \\text{if } \\alpha_{k} = 1;\\\\[3mm]\n\\theta^{(j)}_{c_j\\mid 0}, & \\text{if } \\alpha_{k} = 0.\n\\end{cases}\n   \n   \n   \n\\end{align*}\nTo avoid the somewhat trivial non-identifiablility issue associated with the sign flipping of each binary latent variable ($\\alpha_k$ flipping between 0 and 1), we assume\n\\begin{align}\\label{eq-flip}\n    \\theta^{(j)}_{c_j\\mid 1} > \\theta^{(j)}_{c_j\\mid 0},\\quad c_j=1,\\ldots,d-1,\n\\end{align} \nfor all $j\\in[p]$; this could be understood as fixing the interpretation of $\\alpha_k$ to that possessing the underlying latent trait always increases the response probability to the first $d-1$ non-baseline categories.\nFixing any other orders equally works for our identifiability arguments.\n\n\n\nTo complete the model specification, we need to describe the distribution of the latent variables $\\boldsymbol \\alpha=(\\alpha_1,\\ldots,\\alpha_K)$. As mentioned before, we do not impose any restrictions on the dependence structure among the latent variables, but just adopt the most flexible saturated model. That is, we give each possible binary latent pattern $\\boldsymbol \\alpha\\in\\{0,1\\}^K$ a population proportion parameter $\\nu_{\\boldsymbol \\alpha}=\\mathbb P(\\boldsymbol  a_i = \\boldsymbol \\alpha)$, where $\\boldsymbol  a_i$ denoting the latent profile of a random subject $i$ in the population. The only constraint on $\\boldsymbol\\nu = (\\nu_{\\boldsymbol \\alpha})$ is $\\nu_{\\boldsymbol \\alpha}>0$ and $\\sum_{\\boldsymbol \\alpha\\in\\{0,1\\}^K} \\nu_{\\boldsymbol \\alpha}=1$.\nTherefore, we obtain the following probability mass function of the response vector $\\boldsymbol  y$ under the commonly adopted local independence assumption (i.e., observed variables are conditionally independent given the latent),\n\\begin{align}\\label{eq-model}\n    \\mathbb P(\\boldsymbol  y = \\boldsymbol  c\\mid \\mathbf G,  \\boldsymbol  \\theta, \\boldsymbol\\nu)\n    =\n    \\sum_{\\boldsymbol \\alpha\\in\\{0,1\\}^K} \\nu_{\\boldsymbol \\alpha} \\prod_{j=1}^p \n   \n    \\prod_{k=1}^{K}\n    \\left[\n    \\left(\\theta^{(j)}_{c_j\\mid 1}\\right)^{\\alpha_{k}} \n    \\cdot\n    \\left(\\theta^{(j)}_{c_j\\mid 0}\\right)^{1-\\alpha_{k}}\\right]^{g_{j,k}},\n   \n\\end{align}\nwhere $\\boldsymbol  c=(c_1,\\ldots,c_p)^\\top\\in\\times_{j=1}^p [d]$ is an arbitrary response pattern.\nThe name \\textit{Binary Latent cliquE Star foreSt} (BLESS) model is suggested as Equation \\eqref{eq-model} does not assume any conditional or marginal independence relations among latent variables a priori, and hence the graph among the latent can be viewed as a ``clique'' a priori in the graph terminology.\n\n\n\n\n\n\nIn real-world applications, the BLESS model can be useful in educational assessments, epidemiological diagnostic tests, and social science surveys, where the presence\/absence of multiple latent characteristics are of interest and there are several observed proxies measuring each of them.\nFor instance, in disease etiology in epidemiology \\citep{wu2017nested}, we can use each $\\alpha_k$ to denote the presence\/absence of a pathogen, and for each pathogen a few noisy diagnostic measures $y_j$'s are observed as the children variables of $\\alpha_k$.\nSee Section \\ref{sec-prac} for two real-world examples.\nIn addition, our BLESS model is interestingly connected to a family of models used in causal discovery and machine learning, the \\emph{pure-measurement} models in \\cite{silva2006latent}. Those are linear models of continuous variables, where the latent variables are connected in an acyclic causal graph; the commonality with the BLESS model is that each observed variable has at most one latent parent. The BLESS model can be thought of as a discrete analogue of such a pure-measurement model in \\cite{silva2006latent}, and indeed more general in terms of the latent dependence structure. This is because we do not constrain the $\\alpha_k$'s to follow a acyclic graph distribution but rather allow them to be arbitrarily dependent. Our identifiability conclusions always hold under this general setup.\n\n\n\n\n\n\n\n\n\n\\subsection{Strict, Generic, and Local Identifiability}\n\nWe first define strict identifiability in the context of the BLESS model. \nAll the model parameters are included in the identifiability consideration, including the continuous parameters: the conditional probabilities $\\boldsymbol \\theta = \\left\\{\\theta^{(j)}_{c_j\\mid 1}, \\theta^{(j)}_{c_j\\mid 1}\\right\\}$ and the proportions $\\boldsymbol\\nu$; and the discrete measurement graph structure $\\mathbf G$.\n\n\n\\begin{definition}[Strict Identifiability]\n\\label{def-str}\n\tThe BLESS model is said to be strictly identifiable under certain conditions, if for valid parameters $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$ satisfying these conditions, the following equality holds if and only if $(\\overline{\\mathbf G}, \\overline\\boldsymbol \\theta,  \\overline\\boldsymbol\\nu)$ and $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$ are identical up to a common permutation of $K$ latent variables:\n\t\\begin{align}\\label{eq-id}\n\t\t\\mathbb P(\\boldsymbol  y = \\boldsymbol  c\\mid \\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)\n\t\t=\n\t\t\\mathbb P(\\boldsymbol  y = \\boldsymbol  c\\mid \\overline{\\mathbf G}, \\overline\\boldsymbol \\theta,  \\overline\\boldsymbol\\nu),\n\t\t\\quad\n\t\t\\forall \\boldsymbol  c\\in \\times_{j=1}^p [d].\n\t\\end{align}\n\\end{definition}\n\nThe statement of ``identifiable up to latent variable permutation'' in Definition \\ref{def-str} is an inevitable property of any latent variable model.\nWe next define generic identifiability in the context of the BLESS model. Generic identifiability is a concept proposed and popularized by \\cite{allman2009}.\nGiven a graphical matrix $\\mathbf G$ and some valid continuous parameters $(\\boldsymbol \\theta,\\boldsymbol\\nu)$ under the BLESS model,\ndefine the following subset of the parameter space as\n\\begin{align}\\label{eq-ns}\n\\mathcal N^{\\mathbf G} = \n&~\\{(\\boldsymbol \\theta, \\boldsymbol\\nu)\\text{ are associated with some graphical matrix } \\mathbf G:\n~~\n\\exists~ (\\overline\\boldsymbol \\theta, \\overline\\boldsymbol\\nu)~\\text{associated}\n\\\\ \\notag\n&~~\\text{with some graphical matrix}~\\overline{\\mathbf G}~\n\\text{such that}~ \\mathbb P(\\boldsymbol y\\mid\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu) = \\mathbb P(\\boldsymbol y\\mid\\overline\\mathbf G, \\overline\\boldsymbol \\theta, \\overline\\boldsymbol\\nu)\\}.\t\n\\end{align}\n\n\\begin{definition}[Generic Identifiability]\\label{def-genid}\n\tUnder a BLESS model, parameters $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$ are said to be generically identifiable under certain conditions, if for valid parameters $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$ satisfying these conditions, the set $\\mathcal N^{\\mathbf G}$ defined in \\eqref{eq-ns} has measure zero with respect to the Lebesgue measure on the parameter space of $(\\boldsymbol \\theta,\\boldsymbol\\nu)$ under the identifiability conditions.\n\\end{definition}\n\n\nIt is believed that generic identifiability often suffices for data analyses purposes \\citep{allman2009}.\nFinally, we define local identifiability of continuous parameters in the model.\n\n\\begin{definition}[Local Identifiability]\nUnder a BLESS model, a continuous parameter $\\mu$ (e.g., some entry of $\\boldsymbol \\theta$ or $\\boldsymbol\\nu$) is said to be locally identifiable, if there exists an open neighborhood $\\mathcal S$ of $\\mu$ in the parameter space such that there does not exist any alternative parameter $\\overline\\mu\\in\\mathcal S\\setminus \\{\\mu\\}$ leading to the same distribution of the response vector $\\boldsymbol  y$.\n\\end{definition}\n\n\nThe lack of local identifiability usually has severe consequences in practice, because in an arbitrarily small neighborhood of the true parameter, there exist infinitely many alternative parameters that give rise to the same observed distributions. This would render any estimation and inference conclusions invalid. \n\n\n\n\n\\section{Main Theoretical Results}\\label{sec-main}\n\n\\subsection{Theoretical Results of Generic Identifiability and Their Illustrations}\n\\label{sec-mainsub}\n\nIn this subsection we will present sharp identifiability conditions and the blessing-of-dependence geometry for the BLESS model. The later Section \\ref{sec-overview} will provide an overview of the general algebraic proof technique used to derive the identifiability results.\nThroughout this work, assume $\\nu_{\\boldsymbol \\alpha}>0$ for any latent pattern $\\boldsymbol \\alpha\\in\\{0,1\\}^K$; i.e., all the possible binary latent patterns exist in the population with nonzero proportions.\nThis is the \\emph{only} assumption imposed on the distribution of the latent variables, simply requiring the proportion parameters not to be on the boundary of the probability simplex. \nThere is {no assumption} on whether or how the latent variables should depend on each other in the BLESS model. \n\n\n\nIt may be expected that each latent variable needs to have at least one observed child (i.e., $\\sum_{j=1}^p g_{j,k}\\geq 1$) to ensure identifiability of the BLESS model.\nWhat may not be expected at first is that such a condition is insufficient even for generic identifiability or local identifiability to hold.\nOur first conclusion below shows the condition that each latent variable has at least two observed children is necessary for generic identifiability or local identifiability.\n\n\n\\begin{proposition}[Necessary Condition for Generic Identifiability: $\\geq 2$ children]\n\\label{prop-nece}\nThe following two conclusions hold.\n\\begin{itemize}\n    \\item[(a)] If some binary latent variable has only one observed variable as child, then the model parameters are \\textbf{not} generically identifiable and \\textbf{not} locally identifiable.\n    \n    \\vspace{2mm}\n    \\item[(b)] Specifically, suppose $\\alpha_k$ has only one observed $y_j$ as child, then any of the $\\theta^{(j)}_{c\\mid 0}$ and $\\theta^{(j)}_{c\\mid 1}$ for $c\\in[d]$, and $\\nu_{\\boldsymbol \\alpha}$ for $\\boldsymbol \\alpha\\in\\{0,1\\}^K$ can not be generically or locally identifiable. In an arbitrarily small neighborhood of any of these parameters, there exist alternative parameters that lead to the same distribution of the observables as those given by the truth. \n\\end{itemize}\n\\end{proposition}\n\n\n\n\\begin{remark}\\label{rmk-tree}\nProposition \\ref{prop-nece} also has an interesting implication on a seemingly unrelated problem: learning tree models from noisy data. \\cite{nikolakakis2021ising} considered learning hidden tree-structured Ising models, which essentially can be reformulated as the BLESS model where each  node in the latent tree has exactly one child (its noisy observed proxy) and the responses are binary, i.e.,  $\\mathbf G=\\mathbf I_K$ and $d=2$.\n\\cite{nikolakakis2021ising} derived the sample complexity under the assumption that the noise level at each node is homogeneous and known. \nProposition \\ref{prop-nece} implies that when the noise level is unknown and potentially heterogeneous, then these node-wise noise parameters (analogous to our $\\boldsymbol \\theta$) are not even generically or locally identifiable, no matter what structure the tree graph among the latent nodes is.\n\\end{remark}\n\n\n\nThe conclusion of ``{not even generically identifiable or locally identifiable}'' in Proposition \\ref{prop-nece} has quite severe consequences in parameter interpretation or estimation. \nThere will be one-dimensional continuum of each of $\\theta^{(j)}_{c\\mid 0}$ and $\\theta^{(j)}_{c\\mid 1}$ for $c\\in[d]$, and $\\nu_{\\boldsymbol \\alpha}$ for $\\boldsymbol \\alpha\\in\\{0,1\\}^K$, that lead to the same probability mass function of the response vector $\\boldsymbol  y$.\nAs revealed in part (b) of Proposition \\ref{prop-nece}, the parameter space will have ``flat regions'' where identifiability is no hope, hence any statistical analysis in this scenario will be meaningless.\n\n\nIn Figure \\ref{fig-prop1}, we provide a numerical example to illustrate and corroborate Proposition \\ref{prop-nece}. Consider $p=5$ binary response variables, $K=3$ binary latent variables, and a $5\\times 3$ graphical matrix $\\mathbf G= (1 0 0;~ 0 1 0;~ 0 0 1;~ 0 1 0;~ 0 0 1)$. This $\\mathbf G$ indicates that the first latent variable $\\alpha_1$ only has one observed child $y_1$, violating the necessary condition for generic identifiability in Proposition \\ref{prop-nece}.\nIn the left panel of Figure \\ref{fig-prop1}, the horizontal axis records nine continuous parameters in the model, including one conditional probability $\\theta^{(1)}_{1\\mid 1}$ and $2^K=8$ proportion parameters for the binary latent pattern; the black solid line represents one set of true parameters, while the 150 colored lines represent those alternative parameters in a neighborhood of the truth constructed based on the proof of Proposition \\ref{prop-nece}.\nTo see the non-identifiablility, we calculate the probability mass function of the $5$-dimensional binary response vector $\\boldsymbol  y$, which has $2^p = 32$ entries, and plot it under the true and alternative parameters in the right panel of Figure \\ref{fig-prop1}. In particular, the horizontal axis in the plot presents the indices of the response patterns $\\boldsymbol  c\\in\\{0,1\\}^5$, and the vertical axis presents the values of $\\mathbb P(\\boldsymbol  y = \\boldsymbol  c\\mid \\mathbf G,\\boldsymbol \\theta,\\boldsymbol\\nu)$, where the ``$+$'' symbols correspond to response probabilities given by the true parameters and the ``${\\bigcirc}$'' represents those given by the 150 sets of alternative parameters. The marginal response probabilities of the observables given by all the alternative parameters perfectly equal those under the truth.\nThis illustrates the severe consequence of lack of local identifiability.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{one_attr_nogid2.png}\n    \\caption{Illustrating Proposition 1, severe consequence of the lack of local identifiability. The $\\mathbf G_{5\\times 3} = (1 0 0;~ 0 1 0;~ 0 0 1;~ 0 1 0;~ 0 0 1)$. On the left panel, the black line represents the true set of parameters and each colored line corresponds to an alternative set of parameters. On the right panel, the marginal probability mass functions of the observed  $\\boldsymbol  y \\in \\{0,1\\}^5$ are plotted for all the parameter sets, ``$+$'' for the true set and circles ``${\\bigcirc}$'' for alternative sets.}\n    \\label{fig-prop1}\n\\end{figure}\n\n\n\n\n\n\n\nNow that each latent variable has to have $\\geq 2$ observed children for generic identifiability to possibly hold, next we focus on this scenario.\nIn fact, our next result shows that such a condition is sufficient for identifying discrete structure $\\mathbf G$ of the BLESS model. Such a result is technically quite nontrivial, and in fact can not be derived using existing techniques such as Kruskal's Theorem.\n\n\\begin{theorem}[Identifiability of the Latent-to-observed Star Forest $\\mathbf G$]\\label{thm-graph}\nIn the BLESS model,\nif each latent variable has at least two observed variables as children (i.e., $\\sum_{j=1}^p g_{j,k} \\geq 2$), then the latent-to-observed star forest structure $\\mathbf G$ is identifiable.\n\\end{theorem}\n\n\nWe have the following main theorem on the identifiability of all the parameters in the BLESS model, which reveals the ``blessing of dependence'' phenomenon.\n\n\\begin{theorem}[Blessing of Latent Dependence and Generic Identifiability]\\label{thm-main}\nIn the BLESS model,\nsuppose each latent variable has exactly two observed variables as children. \nThe following conclusions hold.\n\n\\begin{itemize}\n    \\item[(a)] The model with parameters $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$ is generically identifiable.\n    \n    \\item[(b)] Specifically, any valid set of model parameters ($\\boldsymbol \\theta$, $\\boldsymbol\\nu$) are identifiable \\textbf{if and only if} the $K$ latent variables are \\textbf{not} independent according to $\\boldsymbol\\nu$.\n\\end{itemize}\n\\end{theorem}\n\n\n\n\n\\normalfont{We provide a numerical example to illustrate the blessing-of-dependence phenomenon and corroborate Theorem \\ref{thm-main}. Consider the BLESS model with each observed variable having $d=3$ categories and $\\mathbf G=(\\mathbf I_2;\\; \\mathbf I_2)^\\top$.\nWe first randomly generate $M=100$ sets of true parameters of the BLESS model, from which we further generate the observed datasets.\nGiven a fixed sample size $N=10^4$, for each of the $M=100$ parameter sets we further generate $L=200$ independent datasets each with $N$ data points. We further use an EM algorithm (presented as Algorithm 1 in the Supplementary Material) to compute the maximum likelihood estimators (MLE) of the model parameters for each dataset; here we focus on estimating continuous parameters $(\\boldsymbol \\theta,\\boldsymbol p)$ with $\\mathbf G$ fixed, because $\\mathbf G$ is guaranteed to be identifiable by Theorem \\ref{thm-graph}.\nTen random initializations are taken for the EM algorithm and we keep the one with the largest log likelihood value as the MLE. After collecting the MLEs, we calculate the Mean Squares Errors (MSEs) of continuous parameters calculated based on the 200 datasets for each of the 100 true parameter sets.}\n\n\\begin{figure}[h!]\n     \\centering\n         \\includegraphics[width=0.45\\textwidth]{indep_K2_ov3.png}\n         \\hfill\n         \\includegraphics[width=0.45\\textwidth]{indep_K2_ov2.png}\n         \\caption{Corroborating Theorem \\ref{thm-main}. Latent probability simplex $\\boldsymbol \\Delta^{2^2-1}$ for the proportion parameters of  $\\boldsymbol \\alpha=(\\alpha_1, \\alpha_2) \\in\\{0,1\\}^2$, where the saddle surface corresponds to the independence model of two latent variables $\\alpha_1\\perp\\!\\!\\!\\perp\\alpha_2$.\n         Black balls correspond to those parameter sets which have the largest 20\\% MSEs across the 100 sets, while blue balls correspond to the remaining 80\\% parameter sets.\n         MSEs are calculated based on sample size $N=10^4$.}\n         \\label{fig-mse}\n\\end{figure}\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{indep_noid_band.png}\n    \\caption{Illustrating Theorem \\ref{thm-main}. The $\\mathbf G_{6\\times 3} = (\\mathbf I_3;~\\mathbf I_3)$. The true parameters $\\boldsymbol\\nu_{\\text{\\normalfont{true}}}$ falls on the independence surface of $\\alpha_1 \\perp\\!\\!\\!\\perp \\boldsymbol \\alpha_{2:3}$.\n    On the right panel, the marginal probability mass functions of the observed  $\\boldsymbol  y \\in \\{0,1\\}^5$ are plotted for all the parameter sets, ``$+$'' for the true set and circles ``${\\bigcirc}$'' for 150 alternative sets.\n    }\n    \\label{fig-indep}\n\\end{figure}\n\n\\normalfont{Figure \\ref{fig-mse} plots the 100 sets of values of the proportion parameters $\\boldsymbol\\nu=(\\nu_{\\boldsymbol \\alpha},\\; \\boldsymbol \\alpha\\in\\{0,1\\}^2)$ inside the latent probability simplex $\\Delta^{2^K-1} = \\Delta^3$; such a simplex takes the shape of a polyhedron in three dimensions where $x$-, $y$-, $z$-axes correspond to $\\nu_{00}, \\nu_{01}, \\nu_{11}$, respectively. \nFor reference, we also plot the subset of the $\\Delta^3$ that corresponds to the case of independent $\\alpha_1$ and $\\alpha_2$ inside the simplex.\nSuch a subset is a smooth surface determined by $\\nu_{00}\\nu_{11} - \\nu_{01}\\nu_{10} = 0$, which can be equivalently written as $\\nu_{00}\\nu_{11} - \\nu_{01}(1-\\nu_{00}-\\nu_{11} - \\nu_{01}) = 0$ in terms of the three coordinates $(\\nu_{00}, \\nu_{01}, \\nu_{11})$. This subset takes the shape of a saddle surface embedded in the interior of the polyhedron, as shown in Figure \\ref{fig-mse}.\nEach dataset is plotted as a solid ball inside the latent simplex. In particular, we plot those parameter sets with the largest 20\\% MSEs as black balls and plot the remaining parameters as blue balls. The two views shown in Figure \\ref{fig-mse} clearly show that the black balls are closer to the saddle surface of $\\alpha_1\\perp\\!\\!\\!\\perp\\alpha_2$.\nThe simulation results demonstrate that the independence submodel of the latent variables defines a singular subset well within the \\emph{interior} of the parameter space (rather than on the boundary of it). Parameter estimation under the model becomes harder when true parameters are closer to this singular subset of $\\alpha_1\\perp\\!\\!\\!\\perp\\alpha_2$.}\n\n\nWe next provide another example to illustrate the other part of Theorem \\ref{thm-main} -- that when the latent variables are indeed independent, each with two observed children, then the BLESS model parameters are unidentifiable.\nIn Figure \\ref{fig-indep}, we further use the proof of Theorem \\ref{thm-main} to construct such indistinguishable sets of parameters.\nSpecifically, the true proportion parameters $\\boldsymbol\\nu^{\\text{\\normalfont{true}}}$ are constructed such that it implies $\\alpha_1\\perp\\!\\!\\!\\perp (\\alpha_2, \\alpha_3)$; in particular, this means the two subvectors $(\\nu_{000}^{\\text{\\normalfont{true}}}, \\nu_{001}^{\\text{\\normalfont{true}}}, \\nu_{010}^{\\text{\\normalfont{true}}}, \\nu_{011}^{\\text{\\normalfont{true}}})$ and $(\\nu_{100}^{\\text{\\normalfont{true}}}, \\nu_{101}^{\\text{\\normalfont{true}}}, \\nu_{110}^{\\text{\\normalfont{true}}}, \\nu_{111}^{\\text{\\normalfont{true}}})$ are linearly dependent; the true parameters are plotted with black ``$+$''s on the left panel of Figure \\ref{fig-indep}. We then follow the proof of Theorem \\ref{thm-main} to construct 150 alternative sets of parameters and plot each set as a colored line with circles. Then similarly to that in Figure \\ref{fig-prop1}, we calculate the marginal response probabilities of the observed vector $\\boldsymbol  y$ under the true and all the alternative parameter sets, respectively, and plot them in the right panel of Figure \\ref{fig-indep}. These $2^J$-dimensional marginal probability vectors are exactly equal under all the parameter sets, confirming the nonidentifiability.\nCombining Figure \\ref{fig-mse} and Figure \\ref{fig-indep}, we have demonstrated that in the boundary case where each latent variable has two observed children, the parameters are generically identifiable and will become nonidentifiable when the latent variables are independent.\nThese observations corroborate Theorem \\ref{thm-main}.\n\n\n\nWe next present a more nuanced statement about identifiability implied by the proof of the main result Theorem \\ref{thm-main}.\nIn the BLESS model, denote by $\\text{\\normalfont{Child}}(\\alpha_k) \\;\\big |\\; \\alpha_k$ the conditional distribution of all the child variables of $\\alpha_k$ given $\\alpha_k$; hence $\\text{\\normalfont{Child}}(\\alpha_k) = \\{y_j:\\; g_{j,k}=1\\}$.\nSpecifically, the parameters associated with $\\text{\\normalfont{Child}}(\\alpha_k) \\;\\big |\\; \\alpha_k$ are the follwoing conditional probabilities:\n\\begin{align}\\label{eq-cond}\n    \\left\\{\\boldsymbol \\theta^{(j)}:\\; y_j\\in\\text{Child}(\\alpha_k)\\right\\}\n    = \\left\\{\\theta^{(j)}_{1:d\\mid 0},\\; \\theta^{(j)}_{1:d\\mid 1}:\\; g_{j,k}=1\\right\\}.\n\\end{align}\nOur proof of Theorem \\ref{thm-main} gives the following fine-grained identifiability arguments regarding each latent variable.\n\n\n\n\\begin{corollary}[Blessing of Latent Dependence for Each Latent Variable]\\label{cor-fine}\nFor each latent variable $\\alpha_K$ that has exactly two observed variables as children, the following two statements are equivalent, in that (S1) is true if and only if (S2) is true.\n\\begin{itemize}\n    \\item[(S1)] \n    $\\alpha_k \\not\\!\\perp\\!\\!\\!\\perp (\\alpha_1,\\ldots,\\alpha_{k-1}, \\alpha_{k+1}, \\ldots, \\alpha_K)$ holds;\n    \n    \\item[(S2)] the parameters associated with the conditional distributions $\\text{\\normalfont{Child}}(\\alpha_k) \\;\\big |\\; \\alpha_k$ defined in \\eqref{eq-cond} are identifiable.\n\\end{itemize} \n\n\\end{corollary}\n\n\n\n\\begin{remark}\nWe would like to emphasize that it is impossible to obtain the conclusions of Theorem \\ref{thm-graph}, Theorem \\ref{thm-main}, and Corollary \\ref{cor-fine} by applying Kruskal's theorem.\nIn fact, the observed probability tensor $\\boldsymbol \\Pi = (\\pi_{c_1,\\ldots,c_p})$ in the minimal generically identifiable case cannot be concatenated in any way as in \\cite{allman2009} to satisfy Kruskal's rank conditions for unique three-way tensor decompositions.\nThe proofs of Theorem \\ref{thm-graph} and Theorem \\ref{thm-main} are in fact quite nontrivial.\nIn Section \\ref{sec-overview} we will give an overview of the general algebraic technique used to prove these results.\n\\end{remark}\n\n\n\\begin{figure}[h!]\n\\centering\n\\begin{minipage}[c]{0.47\\textwidth}\\centering\n\\resizebox{0.8\\textwidth}{!}{\n\\begin{tikzpicture}\n\\def5 {5}\n\\def2.8cm {1.4cm}\n\n\\foreach \\s in {1,...,5}\n{\n\\node (\\s)[draw, circle, minimum size=20pt, inner sep=0pt] at ({90+360\/5 * (\\s - 1)}:2.8cm) {$\\alpha_{\\s}$};\n}\n\\def 11 {10}\n\\def 2.8cm {2.8cm}\n\n\\foreach \\ss in {1,...,11}\n{\n\\node (y\\ss)[draw, circle, minimum size=20pt, inner sep=0pt, fill=black!10] at ({72+360\/11 * (\\ss - 1)}:2.8cm) {$y_{\\ss}$};\n}\n\n\\draw[dotted, thick] (1) -- (2);\n\\draw[dotted, thick] (1) -- (3);\n\\draw[dotted, thick] (1) -- (4);\n\\draw[dotted, thick] (1) -- (5);\n\\draw[dotted, thick] (2) -- (3);\n\\draw[dotted, thick] (2) -- (4);\n\\draw[dotted, thick] (2) -- (5);\n\\draw[dotted, thick] (3) -- (4);\n\\draw[dotted, thick] (3) -- (5);\n\\draw[dotted, thick] (4) -- (5);\n\n\\draw[->, thick] (1) -- (y1);\n\\draw[->, thick] (1) -- (y2);\n\\draw[->, thick] (2) -- (y3);\n\\draw[->, thick] (2) -- (y4);\n\\draw[->, thick] (3) -- (y5);\n\\draw[->, thick] (3) -- (y6);\n\\draw[->, thick] (4) -- (y7);\n\\draw[->, thick] (4) -- (y8);\n\\draw[->, thick] (5) -- (y9);\n\\draw[->, thick] (5) -- (y10);\n\\end{tikzpicture}}\n\n(a) CPTs for $\\text{\\normalfont{Child}}(\\alpha_1) \\mid \\alpha_1$ identifiable,\\\\\nthanks to blessing of dependence\n\\end{minipage}\n\\hfill\n\\begin{minipage}[c]{0.47\\textwidth}\\centering\n\\resizebox{0.8\\textwidth}{!}{\n\\begin{tikzpicture}\n\\def 5 {5}\n\\def 2.8cm {1.4cm}\n\n\\foreach \\s in {1,...,5}\n{\n\\node (\\s)[draw, circle, minimum size=20pt, inner sep=0pt] at ({90+360\/5 * (\\s - 1)}:2.8cm) {$\\alpha_{\\s}$};\n}\n\\def 11 {10}\n\\def 2.8cm {2.8cm}\n\n\\foreach \\ss in {1,...,11}\n{\n\\node (y\\ss)[draw, circle, minimum size=20pt, inner sep=0pt, fill=black!10] at ({72+360\/11 * (\\ss - 1)}:2.8cm) {$y_{\\ss}$};\n}\n\\draw[dotted, thick] (2) -- (3);\n\\draw[dotted, thick] (2) -- (4);\n\\draw[dotted, thick] (2) -- (5);\n\\draw[dotted, thick] (3) -- (4);\n\\draw[dotted, thick] (3) -- (5);\n\\draw[dotted, thick] (4) -- (5);\n\n\\draw[->, thick, dashed] (1) -- (y1);\n\\draw[->, thick, dashed] (1) -- (y2);\n\\draw[->, thick] (2) -- (y3);\n\\draw[->, thick] (2) -- (y4);\n\\draw[->, thick] (3) -- (y5);\n\\draw[->, thick] (3) -- (y6);\n\\draw[->, thick] (4) -- (y7);\n\\draw[->, thick] (4) -- (y8);\n\\draw[->, thick] (5) -- (y9);\n\\draw[->, thick] (5) -- (y10);\n\\end{tikzpicture}}\n\n(b) CPTs for $\\text{\\normalfont{Child}}(\\alpha_1) \\mid \\alpha_1$ nonidentifiable, due to lack of dependence of $\\alpha_1$ and $\\boldsymbol \\alpha_{2:5}$\n\\end{minipage}\n\n\\bigskip\n\\begin{minipage}[c]{0.47\\textwidth}\\centering\n\\resizebox{0.8\\textwidth}{!}{\n\\begin{tikzpicture}\n\\def 5 {5}\n\\def 2.8cm {1.4cm}\n\n\\foreach \\s in {1,...,5}\n{\n\\node (\\s)[draw, circle, minimum size=20pt, inner sep=0pt] at ({90+360\/5 * (\\s - 1)}:2.8cm) {$\\alpha_{\\s}$};\n}\n\\def 11 {11}\n\\def 2.8cm {2.8cm}\n\n\\foreach \\ss in {4,...,11}\n{\n\\node (y\\ss)[observed] at ({360\/10 * (\\ss)}:2.8cm) {$y_{\\ss}$};\n}\n\n\n\\node (y1) [observed, above left = 1 cm of 1] {$y_1$};\n\\node (y2) [observed, above = 1 cm of 1] {$y_2$};\n\\node (y3) [observed, above right = 1 cm of 1] {$y_3$};\n\n\n\n\\draw[dotted, thick] (2) -- (3);\n\\draw[dotted, thick] (2) -- (4);\n\\draw[dotted, thick] (2) -- (5);\n\\draw[dotted, thick] (3) -- (4);\n\\draw[dotted, thick] (3) -- (5);\n\\draw[dotted, thick] (4) -- (5);\n\n\\draw[->, thick] (1) -- (y1);\n\\draw[->, thick] (1) -- (y2);\n\\draw[->, thick] (1) -- (y3);\n\n\\draw[->, thick] (2) -- (y4);\n\\draw[->, thick] (2) -- (y5);\n\\draw[->, thick] (3) -- (y6);\n\\draw[->, thick] (3) -- (y7);\n\\draw[->, thick] (4) -- (y8);\n\\draw[->, thick] (4) -- (y9);\n\\draw[->, thick] (5) -- (y10);\n\\draw[->, thick] (5) -- (y11);\n\\end{tikzpicture}}\n\n(c) CPTs for $\\text{\\normalfont{Child}}(\\alpha_1) \\mid \\alpha_1$ identifiable\n\\end{minipage}\n\n\n\\caption\nCPTs refer to Conditional Probability Tables.\nAll nodes are discrete random variables, with $\\alpha_k\\in\\{0,1\\}$ latent and $y_j\\in\\{1,\\ldots,d\\}$ observed. The parameters corresponding to the dashed directed edges in (b) are unidentifiable, because $\\alpha_1$ is indepedent of $\\boldsymbol \\alpha_{2:5} = (\\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5)$.\n}\n\\label{fig-graph}\n\\end{figure}\n\n\n\n\n\nIt is useful to adopt a graphical perspective to our identifiability results of the BLESS model.\nFigure \\ref{fig-graph}(a)--(b) provide  graphical illustrations of generic identifiability conclusions and the blessing of dependence phenomenon.\nWith $K=5$ latent variables each having two observed variables as children (i.e., $\\mathbf G = (\\mathbf I_K;\\; \\mathbf I_K)^\\top$), the parameters corresponding to Figure \\ref{fig-graph}(a) are identifiable due to the dependence indicated by the dotted edges between $\\alpha_1,\\ldots,\\alpha_5$; \nwhile the parameters corresponding to Figure \\ref{fig-graph}(b) are not identifiable due to the lack of dependence between $\\alpha_1$ and $\\boldsymbol \\alpha_{-1}:=(\\alpha_2,\\ldots,\\alpha_5)$.\nSuch identifiability arguments guaranteed by Corollary \\ref{cor-fine} are of a very fine-grained nature, revealing that the dependence between a specific latent variable and the remaining ones exactly determines the identifiability of the conditional probability tables given this particular latent variable.\n\nThe identifiability results in Theorem \\ref{thm-graph} and Theorem \\ref{thm-main} yield the following observations. \nFirst, a notable fact is that our proof reveals that the blessing of latent dependence always holds regardless of the sign of the dependence. Either positive dependence or negative dependence helps deliver model identifiability.\nSecond, the easiest scenario for the star forest structure $\\mathbf G$ to be identifiable seems to be the hardest one for the continuous parameters to be identifiable.\nTo see this, consider the extreme case where all the $K$ latent variables are perfectly dependent\\footnote{Note that in this extreme case, many latent patterns $\\boldsymbol \\alpha$'s will have population proportions zero and hence does not satisfy our only assumption on $\\boldsymbol\\nu$. Therefore the fact that $\\mathbf G$ is unidentifiable in this extreme case does not contradict our result on the identifiability of $\\mathbf G$ in Theorem \\ref{thm-graph}.}, then the star forest structure cannot be recovered, because it is impossible to tell apart which observed variables are children of which latent ones. \nGenerally, the more independent the latent variables are, the easier it should be to identify the measurement graph.\nOn the other hand, however, according to our conclusions in Theorem \\ref{thm-main} and Corollary \\ref{cor-fine}, having the latent variables independent is the hardest, and in fact impossible, scenario for the continuous parameters $\\boldsymbol \\theta$ and $\\boldsymbol\\nu$ to be identifiable when each $\\alpha_k$ has two children.\nThis perhaps counterintuitive phenomenon shows the complexity and surprising geometry of discrete graphical models with latent variables.\n\n\n\n\n\n\n\nInterestingly, the case of each latent variable having two children forms  the exact boundary for the blessing of dependence to play a role.\nIn fact, as long as each latent variable has at least three observed variables as children, the Kruskal's theorem \\citep{kruskal1977three} on the uniqueness of three-way tensor decompositions can ``kick in'' to guarantee identifiability. In particular, we can use an argument similar to that in \\cite{allman2009} to establish this conclusion, by concatenating certain observed variables into groups and transforming the underlying $p$-way probability tensor into a three-way tensor. The following proposition formalizes this statement.\n\n\\begin{proposition}[Kruskal's Theorem Kicks in for the $\\geq 3$ Children Case]\\label{prop-3chi}\nConsider model \\eqref{eq-model} with \\eqref{eq-flip} satisfied.\nIf each latent variable has three or more observed variables as children (i.e., $\\sum_{j=1}^p g_{j,k} \\geq 3$), then the model is always strictly identifiable, regardless of the dependence between latent variables.\n\\end{proposition}\n\n\n\n\\begin{example}\\label{exp-g73}\nConsider the BLESS model with $K=3$, $p=7$ and the following $7\\times 3$ graphical matrix\n\\begin{align*}\n    \\mathbf G=\\begin{pmatrix}\n    & \\mathbf I_3 & \\\\\n    & \\mathbf I_3 & \\\\\n    1 & 0 & 0\n    \\end{pmatrix}.\n\\end{align*}\nThen by Theorem \\ref{thm-graph}, the $\\mathbf G$ matrix itself is identifiable from the joint distribution of the observed variables. And by Theorem \\ref{thm-main}, the continuous parameters are generically identifiable. \nFurther, since $\\alpha_1$ has three children with $\\text{\\normalfont{Child}}(\\alpha_1) = \\{y_1, y_4, y_7\\}$, by Proposition \\ref{prop-3chi}, the conditional probability tables $\\boldsymbol \\theta^{(1)}, \\boldsymbol \\theta^{(4)}, \\boldsymbol \\theta^{(7)}$ are strictly identifiable regardless of the dependence between the variable $\\alpha_1$ and the other latent variables $(\\alpha_1, \\alpha_2)$. \nBy Corollary \\ref{cor-fine}, since $\\alpha_2$ has two children $\\text{\\normalfont{Child}}(\\alpha_2) = \\{y_2, y_5\\}$, the $\\boldsymbol \\theta^{(2)}, \\boldsymbol \\theta^{(5)}$ are identifiable if and only if $\\alpha_2 \\not\\! \\perp\\!\\!\\!\\perp (\\alpha_1, \\alpha_3)$. Similarly, parameters $\\boldsymbol \\theta^{(3)}, \\boldsymbol \\theta^{(6)}$ are identifiable if and only if $\\alpha_3 \\not\\! \\perp\\!\\!\\!\\perp (\\alpha_1, \\alpha_2)$.\n\\end{example}\n\nSummarizing all the above conclusions in this section, we have the following conclusions. \n\n\n\\begin{corollary}\\label{cor-ns} Consider the BLESS model. The following statements hold.\n\n\\begin{itemize}\n\\item[(a)]\nThe condition that each binary latent variable has $\\geq 2$ observed variables as children is \\textbf{necessary and sufficient} for the generic identifiability of the model parameters.\n\\item[(b)]\nThe condition that each binary latent variable has $\\geq 3$ observed variables as children is \\textbf{necessary and sufficient} for the strict identifiability of the model parameters.\n\\end{itemize}\n\\end{corollary}\n\n\n\nCorollary \\ref{cor-ns} describes the minimal conditions for strict identifiability and those for generic identifiability of the BLESS model, respectively.\nThe conclusions in Corollary \\ref{cor-ns} are immediate consequences of Theorem \\ref{thm-main} and Proposition \\ref{prop-3chi}.\nIt is worth noting that both the minimal conditions for strict identifiability and those for generic identifiability only concern the discrete structure in the model -- the measurement graph $\\mathbf G$, but not on the specific values of the continuous parameters $\\boldsymbol \\theta$ or $\\boldsymbol\\nu$.\nTherefore, these identifiability conditions as graphical criteria are easily checkable in practice.\n\n\n\nThe blessing of dependence phenomenon when each latent variable has two children has nontrivial connections to and implications on the uniqueness of matrix and tensor decompositions. \n\\emph{On one hand}, if each latent variable has two children with $p=2K$ and $y_k$, $y_{K+k}$ are children of $\\alpha_k$ for each $k\\in[K]$, then we can group the first $y_1,\\ldots, y_K$ and define a surrogate categorical variable $Z_1 = (y_1,\\ldots, y_K) \\in [d]^K$ with $d^K$ latent states, and similarly group the $y_{K+1},\\ldots,y_{2K}$ to define $Z_2 \\in [d]^K$. The joint contingency table of $(Z_1, Z_2)$ can then be expressed as a two-way table of size $d^K \\times d^K$, where each entry in the table corresponds to the probability of a response pattern of the original vector $\\boldsymbol  y$ and all these probabilities sum up to one.\nThis setting can be considered as a reduced rank model for two-way contingency table (matrix) \\citep{de1991reduced}, where the rank of the matrix is $|\\{0,1\\}^K|=2^K$, equal to the number of states the latent vector $\\boldsymbol \\alpha$ can take. \nIt is well-known that such a matrix factorization generally can not be unique. \n\\emph{On the other hand}, if each latent variable has three children with $p=3K$ and $y_k$, $y_{K+k}$, $y_{2K+k}$ being children of $\\alpha_k$, then we can define $Z_1=(y_1,\\ldots,y_K)$, $Z_2=(y_{K+1},\\ldots,y_{2K})$, and $Z_3=(y_{2K+1},\\ldots,y_{3K})$. The joint contingency table of $Z_1, Z_2, Z_3$ is a three-way tensor. Due to the conditional independence of $Z_1, Z_2, Z_3$ given the latent $\\boldsymbol \\alpha$, such a tensor has a CP decomposition \\citep{koldabader2009} of rank $2^K$. By Kruskal's Theorem, this three-way decomposition is identifiable under mild conditions.\nOur results reveal that between the well-known unidentifiable matrix factorization and well-known identifiable tensor decomposition, there is a special middle ground where the dependence between multiple binary latent variables helps restore identifiability.\n\n\n\n\n\n\n\n\\subsection{Overview of the Proof Techniques and Its Usefulness}\\label{sec-overview}\nIn this subsection we provide an overview of the general proof techniques used to derive the identifiability results.\nFor the ease of understanding, we next describe the technique in the context of multidimensional binary latent variables; we will later explain that these techniques are generally applicable to discrete models with latent and graphical components.\nWith $K$ binary latent variables, define the binary vector representations of integers $1,\\ldots,2^K$ by $\\boldsymbol \\alpha_1,\\boldsymbol \\alpha_2,\\ldots,\\boldsymbol \\alpha_{2^K}$; that is, for a $K$-dimensional vector $\\boldsymbol  v=(2^{K-1}, 2^{K-2}, \\cdots, 2^0)^\\top$ there is\n$$\\boldsymbol \\alpha_{\\ell}^\\top \\boldsymbol  v = \\ell+1,\\quad \\ell=1,2,\\ldots,2^K.$$\nEach $\\boldsymbol \\alpha_\\ell$ represents a binary latent pattern describing the presence or absence of the $K$ latent variables and $\\{\\boldsymbol \\alpha_1,\\ldots,\\boldsymbol \\alpha_{2^K}\\} = \\{0,1\\}^K$.\nWith $p$ discrete observed variables $y_1,\\ldots,y_p$, generally denote the conditional distribution of each $y_j$ given latent pattern $\\boldsymbol \\alpha_\\ell$ by \n$$\n\\theta_{c\\mid \\boldsymbol \\alpha_\\ell}^{(j)} = \\mathbb P(y_j=c\\mid \\boldsymbol  a = \\boldsymbol \\alpha_\\ell), \\quad j\\in[p],~ c\\in[d],~\\ell\\in[2^K].\n$$\nNote that under the BLESS model, the $\\theta_{c\\mid \\boldsymbol \\alpha_\\ell}^{(j)}$ is a reparametrization of the probabilities $\\theta_{c\\mid 1}^{(j)}$ and $\\theta_{c\\mid 0}^{(j)}$.\nAccording to the star-forest measurement graph structure, whether $\\theta_{c\\mid \\boldsymbol \\alpha_\\ell}^{(j)}$ equals $\\theta_{c\\mid 1}^{(j)}$ or $\\theta_{c\\mid 0}^{(j)}$ depends only on whether or not the pattern $\\boldsymbol \\alpha_\\ell$ possesses the  latent parent of $y_j$.\nMathematically, since vector $\\boldsymbol g_j$ summarizes the parent variable information of $y_j$, we have that\n\\begin{align}\\label{eq-thetaeq}\n    \\theta_{c\\mid \\boldsymbol \\alpha_\\ell}^{(j)} =\n   \n   \n   \n   \n    \\begin{cases}\n    \\theta_{c\\mid 1}^{(j)}, & \\text{if } \\alpha_{\\ell,k}=1 \\text{ for the $k$ where } g_{j,k}=1;\\\\[2mm]\n    \\theta_{c\\mid 0}^{(j)}, & \\text{if } \\alpha_{\\ell,k}=0 \\text{ for the $k$ where } g_{j,k}=1.\n    \\end{cases}\n\\end{align}\nIn the above expression, the $\\alpha_{\\ell,k}$ denotes the $k$th entry of the binary pattern $\\boldsymbol \\alpha_\\ell$.\nFor each observed variable index $j\\in[p]$, define a $d\\times 2^K$ matrix $\\boldsymbol \\Phi^{(j)}$ as\n\\begin{align*}\n    \\boldsymbol \\Phi^{(j)} \n    &= \n    \\begin{pmatrix}\n     \\mathbb P(y_j=1\\mid \\boldsymbol  a = \\boldsymbol \\alpha_1) &~ \\cdots &~  \\mathbb P(y_j=1\\mid \\boldsymbol  a = \\boldsymbol \\alpha_{2^K}) \\\\[2mm]\n     \\vdots &~ \\vdots &~ \\vdots \\\\[2mm]\n     \\mathbb P(y_j=d\\mid \\boldsymbol  a = \\boldsymbol \\alpha_1) &~ \\cdots &~  \\mathbb P(y_j=d\\mid \\boldsymbol  a = \\boldsymbol \\alpha_{2^K})\n    \\end{pmatrix}\\\\[2mm]\n    &=\n    \\begin{pmatrix}\n       \n       \n       \n        \\theta^{(j)}_{1\\mid \\boldsymbol \\alpha_1} &~  \\cdots &~~ \\theta^{(j)}_{1\\mid \\boldsymbol \\alpha_{2^K}} \\\\[2mm]\n        \\vdots &~ \\vdots &~~ \\vdots \\\\[2mm]\n        \\theta^{(j)}_{d\\mid \\boldsymbol \\alpha_1} &~  \\cdots &~~ \\theta^{(j)}_{d\\mid \\boldsymbol \\alpha_{2^K}}\n    \\end{pmatrix},\n\\end{align*}\nthen $\\boldsymbol \\Phi^{(j)}$ is the conditional probability table of variable $y_j$ given $2^K$ latent patterns. Each column of $\\boldsymbol \\Phi^{(j)}$ is indexed by a pattern $\\boldsymbol \\alpha_\\ell$ and gives the conditional distribution of variable $y_j$ given the latent pattern $\\boldsymbol \\alpha_\\ell$.\nNote that many entries in $\\boldsymbol \\Phi^{(j)}$ are equal due to \\eqref{eq-thetaeq}; we deliberately choose this overparameterized matrix notation to facilitate further tensor algebra. The equality of the many parameters in each $\\boldsymbol \\Phi^{(j)}$ will later be carefully exploited when examining identifiability conditions.\n\n\n\nDenote by $\\bigotimes$ the Kronecker product of matrices and denote by $\\bigodot$ the Khatri-Rao product \\citep[][]{koldabader2009}. The Khatri-Rao product is a column-wise Kronecker product of matrices, and\nfor two matrices with the same number of columns $\\mathbf A=(a_{i,j})=(\\boldsymbol  a_{\\boldsymbol{:},1}\\mid\\cdots\\mid\\boldsymbol  a_{\\boldsymbol{:},k})\\in\\mathbb R^{n\\times k}$,\n$\\mathbf B=(b_{i,j})=(\\boldsymbol  b_{\\boldsymbol{:},1}\\mid\\cdots\\mid\\boldsymbol  b_{\\boldsymbol{:},k})\\in\\mathbb R^{\\ell\\times k}$, their Khatri-Rao product\n$\\mathbf A\\bigodot \\mathbf B \\in\\mathbb R^{n \\ell\\times k}$ still has the same number of columns and can be written as\n\\begin{align*}\n\t\\mathbf A\\bigodot \\mathbf B\n\t=\n\t\\begin{pmatrix}\n\t\t\\boldsymbol  a_{\\boldsymbol{:},1}\\bigotimes\\boldsymbol  b_{\\boldsymbol{:},1}\n\t\t~\\mid~ \\cdots ~\\mid~\n\t\t\\boldsymbol  a_{\\boldsymbol{:},k}\\bigotimes\\boldsymbol  b_{\\boldsymbol{:},k}\n\t\\end{pmatrix}.\n\\end{align*}\nUnder the considered model, all the $d^p$ marginal response probabilities form a $p$-way tensor $$\\boldsymbol \\Pi=(\\pi_{c_1,\\cdots,c_p}), \\quad c_j\\in[d],$$ \nwhere each entry $\\pi_{c_1,\\cdots,c_p} = \\mathbb P (y_1=c_1,\\ldots,y_p=c_p\\mid \\text{star-forest structure and parameters})$ denotes the marginal probability of observing the response pattern $\\boldsymbol y=\\boldsymbol  c$ under the latent variable model.\nWith the above notation, the probability mass function (PMF) of vector $\\boldsymbol  y$ under the BLESS model in \\eqref{eq-model} can be equivalently written as\n\\begin{align}\\label{eq-kreq}\n    \\text{\\normalfont{vec}}(\\boldsymbol \\Pi)\n    = \\Big(\\bigodot_{j=1}^p \\boldsymbol \\Phi^{(j)}\\Big) \\cdot \\boldsymbol\\nu,\n\\end{align}\nwhere $\\text{\\normalfont{vec}}(\\boldsymbol \\Pi)$ denotes the vectorization of the tensor $\\boldsymbol \\Pi$ into a vector of length $d^p$. The Khatri-Rao product of $\\boldsymbol \\Phi^{(j)}$ in the above display results from the basic local independence assumption in \\eqref{eq-model}.\nWe next state a useful technical lemma.\nThe following lemma characterizes a fundamental property of the transformations of Khatri-Rao product of matrices.\n\n\n\n\\begin{lemma}\\label{lem-poly}\nConsider an arbitrary set of conditional probability tables $\\{\\boldsymbol \\Phi^{(j)}: j\\in[p]\\}$, where $\\boldsymbol \\Phi^{(j)}$ has size $d_j\\times 2^K$ with each column summing to one.\nGiven any set of vectors $\\{{\\boldsymbol \\Delta}_j:\\,{j\\in[p]}\\}$ with $\\boldsymbol \\Delta_j = (\\Delta_{j,1},\\ldots,\\Delta_{j,d_j-1}, 0)^\\top \\in \\mathbb R^{d_j\\times 1}$, \nthere exists a $\\prod_{j=1}^p d_j \\times \\prod_{j=1}^p d_j$ \\textbf{invertible} matrix $\\mathbf B:=\\mathbf B(\\{\\boldsymbol \\Delta_j:\\,{j\\in[p]}\\})$ determined entirely by $\\{\\boldsymbol \\Delta_j:\\,{j\\in[p]}\\}$ such that \n\\begin{align}\\label{eq-algebra}\n\t\\bigodot_{j\\in[p]} \\Big(\\boldsymbol \\Phi^{(j)}-\\boldsymbol \\Delta_j\\boldsymbol\\cdot\\mathbf 1^\\top_{2^K} \\Big)\n\t&= \\mathbf B\\left(\\{\\boldsymbol \\Delta_j:\\,{j\\in[p]}\\}\\right) \\boldsymbol\\cdot \\Big(\\bigodot_{j\\in[p]} \\boldsymbol \\Phi^{(j)}\\Big),\n\\end{align}\nwhere $\\boldsymbol \\Delta_j\\boldsymbol\\cdot\\mathbf 1^\\top_{2^K}$ is a $d_j\\times 2^K$ matrix, of the same dimension as $\\boldsymbol \\Phi^{(j)}$.\n\nIn addition, replacing the index $j\\in[p]$ in \\eqref{eq-algebra} by $j\\in S$ where $S$ is an arbitrary subset of $[p]$ on both hand sides still makes the equality holds.\n\\end{lemma}\n\n\n\nLemma \\ref{lem-poly} covers as special case a result in \\cite{xu2017rlcm} for restricted latent class models with binary responses.\nInstead of exclusively considering moments of binary responses as \\cite{xu2017rlcm}, our Lemma \\ref{lem-poly} here characterizes a general algebraic property of Khatri-Rao products of conditional probability tables of multivariate categorical data. This property together with the model formulation in \\eqref{eq-kreq} will enable us to exert various transformations on the model parameters to investigate their identifiability. \nWe provide a proof of Lemma \\ref{lem-poly} below, because it is concise and delivers an insight into our technique's usefulness.\n\n\\begin{proof}[Proof of Lemma \\ref{lem-poly}]\nConsider an arbitrary subset $S\\in[p]$.\nFirst note that the sum of all the entries in each column of $\\boldsymbol \\Phi^{(j)}$ is one because each column vector is a conditional probability distribution of $y_j$ given a particular latent pattern. Therefore with $\\boldsymbol \\Delta_j = (\\Delta_{j,1},\\ldots,\\Delta_{j,d-1}, 0)^\\top$, we have\n\\begin{align*}\n    \\boldsymbol \\Phi^{(j)}-\\boldsymbol \\Delta_j\\boldsymbol\\cdot\\mathbf 1^\\top_{2^K}\n    &= \n    \\begin{pmatrix}\n        \\theta^{(j)}_{1\\mid \\boldsymbol \\alpha_1}-\\Delta_{j,1} &~  \\cdots &~~ \\theta^{(j)}_{1\\mid \\boldsymbol \\alpha_{2^K}}-\\Delta_{j,1} \\\\[2mm]\n        \\vdots &~ \\vdots &~~ \\vdots \\\\[2mm]\n       \n        \\theta^{(j)}_{d-1\\mid \\boldsymbol \\alpha_1}-\\Delta_{j,d-1} &~  \\cdots &~~ \\theta^{(j)}_{d-1\\mid \\boldsymbol \\alpha_{2^K}}-\\Delta_{j,d-1} \\\\[4mm]\n        \\theta^{(j)}_{d\\mid \\boldsymbol \\alpha_1} &~  \\cdots &~~ \\theta^{(j)}_{d\\mid \\boldsymbol \\alpha_{2^K}}\n    \\end{pmatrix}\n    \\\\\n   \n    &=\n    \\begin{pmatrix}\n    1 &~ 0  &~ \\cdots &~ 0 & -\\Delta_{j,1}\\\\\n    0 &~ 1  &~ \\cdots &~ 0 & -\\Delta_{j,2}\\\\\n    \\vdots &~ \\vdots &~ \\ddots &~ 0 & \\vdots\\\\\n    0 &~ 0  &~ \\cdots &~ 1 &\\quad -\\Delta_{j,d-1}\\\\\n    -1 &~ -1  &~ \\cdots & -1 & 1\n    \\end{pmatrix}\n    \\boldsymbol\\cdot\n    \\begin{pmatrix}\n    \\theta^{(j)}_{1\\mid \\boldsymbol \\alpha_1} &~  \\cdots &~~ \\theta^{(j)}_{1\\mid \\boldsymbol \\alpha_{2^K}} \\\\[2mm]\n        \\vdots &~ \\vdots &~~ \\vdots \\\\[2mm]\n       \n        \\theta^{(j)}_{d-1\\mid \\boldsymbol \\alpha_1} &~  \\cdots &~~ \\theta^{(j)}_{d-1\\mid \\boldsymbol \\alpha_{2^K}} \\\\[4mm]\n        1 &~  \\cdots &~~ 1\n    \\end{pmatrix}\n    \\\\\n   \n    &=\n    \\underbrace{\\begin{pmatrix}\n    1 &~ 0  &~ \\cdots &~ 0 & -\\Delta_{j,1}\\\\\n    0 &~ 1  &~ \\cdots &~ 0 & -\\Delta_{j,2}\\\\\n    \\vdots &~ \\vdots &~ \\ddots &~ 0 & \\vdots\\\\\n    0 &~ 0  &~ \\cdots &~ 1 &\\quad -\\Delta_{j,d-1}\\\\\n    -1 &~ -1  &~ \\cdots & -1 & 1\n    \\end{pmatrix}}_{d\\times d\\text{ matrix, denoted by }\\widetilde{\\boldsymbol \\Delta}_j}\n    \\boldsymbol\\cdot\n    \\underbrace{\\begin{pmatrix}\n    1 &~ 0  &~ \\cdots &~ 0 &~ 0\\\\\n    0 &~ 1  &~ \\cdots &~ 0 &~ 0\\\\\n    \\vdots &~ \\vdots &~ \\ddots &~ \\vdots &~ \\vdots \\\\\n    0 &~ 0  &~ \\cdots &~ 1 &~ 0\\\\\n    1 &~ 1  &~ \\cdots &~ 1 &~ 1\n    \\end{pmatrix}}_{d\\times d\\text{ matrix, denoted by }\\mathbf C}\n    \\boldsymbol\\cdot~ \\boldsymbol \\Phi^{(j)}\\\\\n    &=:\n    \\widetilde{\\boldsymbol \\Delta}_j \\mathbf C \\boldsymbol \\Phi^{(j)}.\n\\end{align*}\nIt is easy to see that both matrix $\\widetilde{\\boldsymbol \\Delta}_j$ and matrix $\\mathbf C$ have full rank $2^K$, so their product $\\widetilde{\\boldsymbol \\Delta}_j \\mathbf C$ also has full rank $2^K$. Then\n\\begin{align*}\n    \\bigodot_{j\\in S} \\Big(\\boldsymbol \\Phi^{(j)}-\\boldsymbol \\Delta_j\\boldsymbol\\cdot\\mathbf 1^\\top_{2^K} \\Big) \n    &= \n    \\bigodot_{j\\in S} \\Big(\\widetilde{\\boldsymbol \\Delta}_j \\mathbf C  \\boldsymbol \\Phi^{(j)} \\Big)\\\\\n    &=\n    \\bigotimes_{j\\in S} (\\widetilde{\\boldsymbol \\Delta}_j \\mathbf C ) \\boldsymbol\\cdot \\bigodot_{j\\in S}  \\boldsymbol \\Phi^{(j)},\n\\end{align*}\nwhere the last equality above follows from basic properties of the Kronecker and Khatri-Rao products and can be verified by checking corresponding entries in the products. Now define\n$$\n\\mathbf B\\left(\\{\\boldsymbol \\Delta_j:\\,{j\\in S}\\}\\right) : =\n\\bigotimes_{j\\in S} (\\widetilde{\\boldsymbol \\Delta}_j \\mathbf C ),\n$$\nthen $\\mathbf B\\left(\\{\\boldsymbol \\Delta_j:\\,{j\\in S}\\}\\right)$ is a $d^{|S|} \\times d^{|S|}$ matrix and it is invertible because it is the Kronecker product of $|S|$ invertible matrices $\\widetilde{\\boldsymbol \\Delta}_j \\mathbf C$. This proves  Lemma \\ref{lem-poly}. \n\\end{proof}\n\nRecall that many entries in $\\boldsymbol \\Phi^{(j)}$ are constrained equal under the graphical matrix $\\mathbf G$; in fact, the $\\boldsymbol \\Phi^{(j)}$ is entirely determined by $\\mathbf G$ and $\\boldsymbol \\theta$ and also the structure of $\\mathbf G$ and $\\boldsymbol \\theta$ can be read off given the $\\boldsymbol \\Phi^{(j)}$.\nNow suppose an alternative graphical matrix $\\bar\\mathbf G \\in\\{0,1\\}^{p\\times K}$ and some associated alternative parameters $(\\bar\\boldsymbol \\theta, \\bar\\boldsymbol\\nu)$ lead to the same distribution as $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$. \nThen by \\eqref{eq-kreq}, the following system of $d^{|S|}$ equations about the alternative parameters $\\overline{\\boldsymbol \\Phi}$ and $\\overline{\\boldsymbol\\nu}$ must hold for an arbitrary subset $S\\subseteq[p]$,\n\\begin{align*}\n    \\Big(\\bigodot_{j\\in S} \\boldsymbol \\Phi^{(j)}\\Big) \\cdot \\boldsymbol\\nu =\n    \\Big(\\bigodot_{j\\in S} \\overline{\\boldsymbol \\Phi}^{(j)}\\Big) \\cdot \\overline\\boldsymbol\\nu.\n\\end{align*}\nOur goal is to study under what conditions on the true parameters, the alternative $(\\bar\\mathbf G, \\bar\\boldsymbol \\theta, \\bar\\boldsymbol\\nu)$ must be identical to the true $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$.\nBy Lemma \\ref{lem-poly}, for arbitrary $\\{\\boldsymbol \\Delta_j\\}$, we have\n\\begin{align}\\notag\n    \\Big(\\bigodot_{j\\in S} \\boldsymbol \\Phi^{(j)} -\\boldsymbol \\Delta_j\\boldsymbol\\cdot\\mathbf 1^\\top_{2^K} \\Big) \\cdot \\boldsymbol\\nu \n    &= \\mathbf B\\left(\\{\\boldsymbol \\Delta_j:\\,{j\\in S}\\}\\right) \\boldsymbol\\cdot \\Big(\\bigodot_{j\\in S} \\boldsymbol \\Phi^{(j)}\\Big) \\cdot \\boldsymbol\\nu \n    \\\\ \\notag\n    &= \\mathbf B\\left(\\{\\boldsymbol \\Delta_j:\\,{j\\in S}\\}\\right) \\boldsymbol\\cdot \\Big(\\bigodot_{j\\in S} \\overline{\\boldsymbol \\Phi}^{(j)}\\Big) \\cdot \\overline\\boldsymbol\\nu \\\\ \\label{eq-trans}\n    &= \\Big(\\bigodot_{j\\in S} \\overline{\\boldsymbol \\Phi}^{(j)} -\\boldsymbol \\Delta_j\\boldsymbol\\cdot\\mathbf 1^\\top_{2^K} \\Big) \\cdot \\overline{\\boldsymbol\\nu}.\n\\end{align}\nWe next give a high-level idea of our proof procedure.\nEq.~\\eqref{eq-trans} will be frequently invoked for various subsets $S\\subseteq[p]$ when deriving the identifiability results.\nFor example, suppose we want to investigate whether a specific parameter $\\theta^{(j)}_{c\\mid\\boldsymbol \\alpha_\\ell}$ is identifiable under certain conditions.\nExploiting the fact that $\\overline\\mathbf G$ induces many equality constraints on entries of $\\overline{\\boldsymbol \\Phi}^{(j)}$, \nwe will construct a set of vectors $\\{\\boldsymbol \\Delta_j; j\\in S\\}$, which usually has the particular $\\bar\\theta^{(j)}_{c\\mid\\boldsymbol \\alpha_\\ell}$ as an entry. These vectors $\\{\\boldsymbol \\Delta_j; j\\in S\\}$ are purposefully constructed such that the right hand side of Eq.~$\\eqref{eq-trans}=0$ for some polynomial equation out of the $\\prod_{j\\in S} d_j$ ones.\nThis implies a polynomial involving parameters $(\\mathbf G, \\boldsymbol \\theta, \\boldsymbol\\nu)$ and the constructed vectors $\\{\\boldsymbol \\Delta_j; j\\in S\\}$ is equal to zero.\nWe will then carefully inspect under what conditions this equation gives $\\theta^{(j)}_{c\\mid\\boldsymbol \\alpha_\\ell}$'s identifiability; \nnamely, inspect whether $\\theta^{(j)}_{c\\mid\\boldsymbol \\alpha_\\ell} = \\overline{\\theta}^{(j)}_{c\\mid\\boldsymbol \\alpha_\\ell}$ must hold under the considered conditions. \n\n\n\n\nWe emphasize here that our algebraic technique described above can be generally useful beyond the BLESS model. \nEssentially, our proof technique exploits the following two key model properties.\n\\emph{First}, observed variables are conditionally independent given the (potentially multiple) latent variables. This property makes it possible to write the joint distribution of the observed variables as the product of (a) Khatri-Rao product of individual conditional probability tables and (b) the vector of the joint probability mass function of latent variables.\n\\emph{Second}, there exist rich graphical structures involving the latent variables and observed variables. The graph will induce many equality constraints on the conditional probability table $\\boldsymbol \\Phi^{(j)}$ of each observed variable given the configurations of the latent.\nThe first property above about conditional independence is an extremely prevailing assumption adopted in many other latent variable models, and it is often called ``local indenpendence'' in the literature. The second property above about graph-induced constraints is also frequently encountered across various directed and undirected graphical models \\citep{lauritzen1996graphical}.\nBecause of these two facts, we expect our techniques will be generally useful to find identifiability conditions for other complicated discrete models with multidimensional latent and graphical structures, e.g., discrete Bayesian networks with latent variables with application to causal inference \\citep{allman2015dag, mealli2016causal}, \nmixed membership models \\citep{erosheva2007aoas}, and overlapping community models for networks \\citep{todeschini2020exchangeable}.\n\n\n\n\\subsection{Discussing Connections to and Differences from Related Works}\n\n\n\n\n\n\n\nIt is worth connecting the BLESS model to discrete Latent Tree Models \\citep[LTMs;][]{choi2011learning, mourad2013survey}, which are popular tools in machine learning and have applications in phylogenetics in evolutionary biology. \nSome deep results about the geometry and statistical properties of LTMs are uncovered in \\cite{zwiernik2012tree}, \\cite{zwiernik2016semialg}, and \\cite{shiers2016gltm}.\nConceptually, the BLESS model is more general than LTMs because in the former, the latent variables can have entirely flexible and arbitrarily complex dependence structure according to the definition in Eq.~\\eqref{eq-model} (also implied by the word ``clique'' in the name of the BLESS model). Namely, the BLESS model only requires the latent-to-observed graph to be a tree and the latent graph can be a general clique; in contrast, LTMs require the entire graph among all the latent and observed variables to be a tree. \nAs a result, the identifiability and geometry of the BLESS model are more complicated than those of the LTMs.\nGeometry and identifiability of Bayesian networks with hidden variables have also been investigated in\n\\cite{settimi2000geometry}, \\cite{allman2015dag} and \\cite{anandkumar2013bn}. But these works often either consider a small number of variables, or employ certain specific (rather than entirely flexible) assumptions on the dependence of latent variables.\n\n\n\nNotably, in real-world applications in education and psychology, the aforementioned formulation of arbitrarily dependent latent variables has been widely employed in an emerging family of diagnostic models \\citep[3.g.,][]{chen2015qmat, xu2018q, gu2021jmle, von2019handbook}. This is because the binary latent variables in those applications have semantic meanings such as specific skills or mental disorders, and it is usually unsuitable to restrict the dependence graph between these latent constructs to be a tree. Rather, the latent variables may exhibit quite rich dependencies because of the complicated cognitive processes underlying learning or behaviors.\nBeing able to derive sharp identifiability results without assuming any specific dependence structure among latent variables shows the power of the general algebraic technique we employ in this work.\n\n\n\n\nA generic identifiability statement related to our work appeared in \\cite{gu2021idq} in the form of a small toy example for the aforementioned cognitive diagnostic models. More specifically, these are models where test items are designed to measure the presence\/absence of multiple latent skills and binary item responses of correct\/wrong answers are observed for each subject.\nIn the special case with two binary latent skills each measured by two binary observed variables, \\cite{gu2021idq} proved  the parameters are identifiable if and only if the two latent variables are not independent.\nIn this work, we investigate the fully general case of the BLESS model where there are (a) an arbitrary number of binary latent variables, (b) arbitrary dependence between these variables, and (c) the observed variables have an arbitrary number of categories. Under this general setup, we characterize a complete picture of the generic identifiability phenomenon with respect to the latent dependence in Section \\ref{sec-mainsub}. \n\n\n\n\n\n\n\n\n\n\\section{Statistical Hypothesis Test of Identifiability in the Boundary Case}\n\\label{sec-test}\nConsider the minimal conditions for generic identifiability of the BLESS model, where certain (all or a subset of) latent variables have only two children. \nIn this boundary scenario, a natural question of interest is whether one can decide whether the parameters are identifiable or not. To this end, it would be desirable to develop a formal statistical hypothesis test of identifiability. Our identifiability theory of blessing of dependence indeed provides a basis for such a simple testing approach.\nUnder the star-forest measurement graph in the BLESS model, we have the following proposition.\n\n\n\n\\begin{proposition}\\label{prop-depdep}\nUnder the BLESS model defined in \\eqref{eq-model}, consider two different latent variables $\\alpha_{k_1}$ and $\\alpha_{k_2}$. \nThe two groups of observed variables\n$\\{y_j=c_j:\\; g_{j,k_1}=1\\}$ and  $\\{y_m=c_m:\\; g_{m,k_2}=1\\}$ \nare independent if and only if $\\alpha_{k_1}$ and $\\alpha_{k_2}$ are independent.\n\\end{proposition}\n\n\nProposition \\ref{prop-depdep} states that under the BLESS model, the dependence\/independence of latent variables is exactly reflected in the dependence\/independence of their observed proxies (i.e., observed children variables). This fact is apparent from the graphical representation of the BLESS model in Figure \\ref{fig-graph}; it can also be formally proved using the model definition in \\eqref{eq-model}.\nA nice implication of Theorem \\ref{thm-main} and Proposition \\ref{prop-depdep} is that,\nwe can test the marginal dependence between certain observed variables to determine model identifiability, before even trying to fit a potentially unidentifiable model to data.\n\n\n\nFormally, in the boundary case (i.e., under minimal conditions for generic identifiability) where some latent variable $\\alpha_k$ only has two observed children, if one wishes to test the following hypothesis\n$$\nH_{0k}:~ \\text{Parameters associated with }~ \\text{\\normalfont{Child}}(\\alpha_k)\\mid\\alpha_k ~\\text{ are not identifiable},\n$$\nthen it is equivalent to testing the hypothesis $H_{0k}':~ \\alpha_k \\perp\\!\\!\\!\\perp \\boldsymbol \\alpha_{-k}$. \nFurther, to test $H_{0k}'$ it suffices to test the marginal independence between the following observed variables,\n$$\nH_{0k}':~ \\text{\\normalfont{Child}}(\\alpha_k) \\perp\\!\\!\\!\\perp \\text{\\normalfont{Child}}(\\boldsymbol \\alpha_{-k}).\n$$\nSince $\\text{\\normalfont{Child}}(\\alpha_k)$ and $\\text{\\normalfont{Child}}(\\boldsymbol \\alpha_{-k})$ are fully observed given the measurement graph, the above hypothesis $H_{0k}'$ can be easily tested.\nNote that $\\text{\\normalfont{Child}}(\\alpha_k)$  can be regarded as a categorical variable with $d^{|\\text{\\normalfont{Child}}(\\alpha_k)|}$ categories and that $\\text{\\normalfont{Child}}(\\boldsymbol \\alpha_{-k})$  can be regarded as another categorical variable with $d^{|\\text{\\normalfont{Child}}(\\boldsymbol \\alpha_{-k})|}$ categories. So the simple $\\chi^2$ test of independence between two categorical variables can be employed for testing $H_{0k}'$.\nIf the null hypothesis of independence is not rejected, then caution is needed in applying the BLESS model because some parameters may not be identifiable.\nIf, however, the hypothesis of independence is rejected, then this is statistical evidence supporting the identifiability of the BLESS model. In this case one can go on to fit the model to data, interpret the estimated parameters, and conduct further statistical analysis.\nIn fact, if $\\text{\\normalfont{Child}}(\\boldsymbol \\alpha_{-k})$ consists of many observed variables, one can start with a small subset $S\\subseteq  \\text{\\normalfont{Child}}(\\boldsymbol \\alpha_{-k})$ and testing whether $ \\text{\\normalfont{Child}}(\\alpha_{k}) \\perp\\!\\!\\!\\perp S$; the rejection of this hypothesis would already provide evidence for identifiability of parameters (see Section \\ref{sec-timss} for such an example).\nWe also point out that if multiple latent variables $\\alpha_{k_1},\\ldots,\\alpha_{k_m}$ each has only two observed children, then one can test the $m$ hypotheses simultaneously $\\{H_{0k}':~ \\text{\\normalfont{Child}}(\\alpha_k) \\perp\\!\\!\\!\\perp \\text{\\normalfont{Child}}(\\boldsymbol \\alpha_{-k});~ k=1,\\ldots,m\\}$, and then use the Bonferroni correction to reach the final conclusion about the overall model identifiability. \n\n\n\n\\begin{example}\nContinue to consider Example \\ref{exp-g73} where $\\mathbf G_{7\\times 3}=(\\mathbf I_3; \\; \\mathbf I_3;\\; 1~ 0 ~0)^\\top$. Recall that $\\{\\boldsymbol \\theta^{(2)}, \\boldsymbol \\theta^{(5)}\\}$ are identifiable if and only if $\\alpha_2 \\not\\! \\perp\\!\\!\\!\\perp (\\alpha_1, \\alpha_3)$, and $\\{\\boldsymbol \\theta^{(3)}, \\boldsymbol \\theta^{(6)}\\}$ are identifiable if and only if $\\alpha_3 \\not\\! \\perp\\!\\!\\!\\perp (\\alpha_1, \\alpha_2)$. In order to test the hypothesis\n\\begin{align*}\n    H_{01}:~ \\text{Parameters associated with }~ \\text{\\normalfont{Child}}(\\alpha_2)\\mid\\alpha_2 ~~(\\normalfont{\\text{i.e.}}~\\boldsymbol \\theta^{(2)}, \\boldsymbol \\theta^{(5)}) \\text{ are not identifiable},\n\\end{align*}\nit suffices to test $H_{01}':~ \\alpha_2 \\perp\\!\\!\\!\\perp (\\alpha_1, \\alpha_3)$. Because of the form of the $\\mathbf G$ matrix, the test $H_{01}'$ of latent independence can be further reduced to the test of the following hypothesis of the observed variables,\n$$H_{01}'':~ (y_2, y_5) \\perp\\!\\!\\!\\perp  (y_1, y_3, y_4, y_6).$$\nSimilarly, in order to test\n\\begin{align*}\n    H_{02}:~ \\text{Parameters associated with }~ \\text{\\normalfont{Child}}(\\alpha_3)\\mid\\alpha_3 ~~(\\normalfont{\\text{i.e.}}~\\boldsymbol \\theta^{(3)}, \\boldsymbol \\theta^{(6)}) \\text{ are not identifiable},\n\\end{align*}\nit suffices to test the following hypothesis about the observed variables\n$$H_{02}'':~ (y_3, y_6) \\perp\\!\\!\\!\\perp (y_1, y_2, y_4, y_5).$$\nThe tests of $H_{01}''$ and $H_{02}''$ can be carried out simply by testing the dependence between two concatenated categorical variables, one with $d^2$ categories and the other with $d^4$ categories. \n\\end{example}\n\n\nNote that our hypothesis test of identifiability is performed without fitting the BLESS model to data, and can serve as a first-step sanity check in real data analysis.\nIn a similar spirit but for a different purpose when studying the Gaussian Latent Tree Models, \\cite{shiers2016gltm} proposed to test certain covariance structures of variables to determine the goodness of fit before fitting the model to data. \nTo the author's best knowledge, there has not been previous formal statistical approaches to directly \\emph{testing the identifiability} of multidimensional latent variable models. \nOur test of identifiability of the BLESS model is enabled by the discovery of the nontrivial blessing of dependence phenomenon and may inspire future relevant hypothesis testing approaches in other latent variable models.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Examples of Real-world Applications}\\label{sec-prac}\n\nThis section presents two real-world examples where our new theory can bring new insights potentially -- one in educational assessment and the other in social science surveys.\n\n\\subsection{Educational assessment example}\n\\label{sec-timss}\n\nThe Trends in International Mathematics and Science Study (TIMSS) is a series of international assessments of the mathematics and science knowledge of middle school students around the world.\nTIMSS assesses fourth and eighth grade students and it has been held every four years since 1995 in over 50 countries.\nThe so-called cognitive diagnosis models have been used to analyze a subset of the Austrian TIMSS 2011 data in \\cite{george2015cdm}; this dataset is available in the R package \\texttt{CDM}.\nThe dataset involves fourth grade students' correct\/wrong (binary) responses to a set of TIMSS questions in mathematics.\nAccording to psychometricians, these questions were designed to measure the presence\/absence (binary) statuses of $K=3$ content-based latent skills of students: ($\\alpha_1$) Data,  ($\\alpha_2$) Geometry, and  ($\\alpha_3$) Numbers. \nEach test question targets exactly one content-based skill, which means the latent-to-observed measurement graph satisfies the assumption of the BLESS model.\nThis original Austrian TIMSS dataset in the \\texttt{CDM} package contains 1010 students' responses to a total number of 47 questions but has many missing data, \nbecause the 47 items were\ndivided up into three booklets and only two\nof the three booklets are presented to each student; such missingness is common to large-scale educational assessments \\citep{george2015cdm}.\nTo avoid dealing with the missing data issue in our example of identifiability considerations, here we focus on the first booklet containing the first $p = 21$ questions, and consider the $N=341$ students who answered all these 21 questions. \nTable \\ref{tab-timss} summarizes the dependence of these 21 questions on the three underlying latent skills, i.e., the $\\mathbf G$ matrix structure in our notation, which is also provided in the R package \\texttt{CDM}.\n\n\n\n    \n\n\n    \n\n\n\n\\begin{table}[h!]\n    \\centering\n    \\caption{Educational assessment example of the Austrian TIMSS 2011 data. Latent-to-observed measurement graph structure between the first $p=21$ questions and $K=3$ content-based latent skills, constructed using the information available in the R package \\texttt{CDM}.}\n    \\label{tab-timss}\n    \n    \\begin{tabular}{lll}\n    \\toprule\n    & Content-based latent skill  &  Indices of questions that measure the skill\\\\\n    \\midrule\n      $\\alpha_1$ & Data  & 20, 21 \\\\\n      $\\alpha_2$ & Geometry  & 7, 8, 16, 17, 18, 19 \\\\\n      $\\alpha_3$ & Numbers  & 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 15 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \n\\end{table}\n\n\n\nTable \\ref{tab-timss} shows that the first skill ``Data'' is measured by only two questions (questions 20 and 21), hence satisfying the minimal conditions for generic identifiability.\nSo according to our new results, whether the model parameters are identifiable would depend on whether there exists underlying dependence between latent variables.\nWe carry out a hypothesis test of identifiability of the BLESS model. \nIn particular, we want to test \n\\begin{align*}\nH_{0,\\text{Data}}:~ &\\text{The latent skill ``Data'' is independent of the two remaining skills ``Geometry'' and}\\\\ \n&\\text{``Numbers''};\n\\end{align*}\nand based on the $\\mathbf G$ matrix structure in Table \\ref{tab-timss}, we can test whether the questions targeting the ``Data'' skill are independent with  those targeting other two skills.\nIn particular, here we consider all the two-question-combinations consisting of one question targeting ``Geometry'' and one question targeting ``Numbers'', and then test whether this combination of questions are independent of those two ``Data'' questions; namely, we test\n\\begin{align*}\nH_{0,\\text{Data}}^{j_1, j_2}:~ (y_{20}, y_{21}) \\text{~are independent of~} (y_{j_1}, y_{j_2}),\\quad\nj_1 \\text{~targets Geometry,~} j_2 \\text{~targets Numbers}.\n\\end{align*}\nUsing the standard $\\chi^2$ test of independence between two categorical variables each of $2^{2}=4$ categories, each test statistic under the null hypothesis $H_{0,\\text{Data}}^{j_1, j_2}$ asymptotically follows the $\\chi^2$ distribution with $df = (2^2 - 1) \\cdot (2^2 - 1) = 9$ degrees of freedom. \nOut of the $6\\times 13 = 78$ such test statistics, we found 73 of them are greater than the 95\\% quantile of the reference distribution $\\chi^2(df, 0.95) = 16.92$, where we reject the null hypothesis of independence between $(y_{20}, y_{21})$ and $(y_{j_1}, y_{j_2})$. We point out that the rejection of any of these tests $H_{0,\\text{Data}}^{j_1, j_2}$ already indicates one should reject the original null $H_{0,\\text{Data}}$.  \nThanks to the blessing of dependence theory we have established, the test results provide statistical evidence to reject the original null hypothesis of non-identifiability, and hence support the identifiability of model parameters.\nThis provides a statistical conclusion of identifiability for the first time in such applications in educational cognitive diagnosis modeling.\n\n\n\n\\subsection{Prevention science survey example}\n\\label{sec-prev}\nAn influential paper in prevention science \\cite{lanza2013} used the latent class model (LCM; with a unidimensional latent variable) to analyse the treatment effects on different latent subgroups, and illustrated the method using a dataset extracted\nfrom the National Longitudinal Survey of Adolescent Health (NLSAH). \nObserved data for each subject are $p=6$ dichotomized characteristics: household poverty; single-parent status;\npeer cigarette use; peer alcohol use; neighborhood unemployment; and neighborhood poverty. \nThese observables actually measure three risks, with the first two measuring ($\\alpha_1$) \\emph{household risk}, the middle two measuring \n($\\alpha_2$) \\emph{peer risk}, and the last two measuring \n($\\alpha_3$) \\emph{neighborhood risk}.\nAccording to the estimated conditional probability tables of the observed variables given the five latent classes, \\cite{lanza2013} interpreted the latent classes as (a) Overall low risk, (b) Peer risk, (c) Household \\& neighborhood (economic) risk, (d) Household \\& peer risk, and (e) Overall high (multicontext) risk. \nInterestingly, we note that the analysis in \\cite{lanza2013} lends itself to a reformulation using the BLESS model, and we argue that such a reformulation provides an interpretable graphical modeling alternative to plain latent class analysis.\nSpecifically, if viewing the three underlying risks as three latent variables, then the latent-to-observed measurement graph indeed takes a star-forest shape; see Table \\ref{tab-prev1} for details of the $\\mathbf G$ matrix.\nMore importantly, the aforementioned five latent classes can be nicely formulated as five different binary configurations of the three latent risks, as $(0,0,0)$, $(1,0,0)$, $(1,0,1)$, $(1,1,0)$, and $(1,1,1)$, respectively.\nHere $\\alpha_k=1$ indicates the higher risk group while $\\alpha_k=0$ indicates the lower risk group.\nSee Table \\ref{tab-prev2} for the multidimensional binary configurations of latent classes.\n\n\\begin{table}[h!]\n    \\caption{Prevention science survey example reformulated using the BLESS model. Latent-to-observed measurement graph structure $\\mathbf G_{6\\times 3}$.}\n    \\label{tab-prev1}\n    \n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{llccc}\n    \\toprule\n     & \\multirow{3}{*}{Item Content} & & Fine-grained Latent Risks & \\\\\n     \\cmidrule(lr){3-5}\n     & & $\\alpha_1$ & $\\alpha_2$ & $\\alpha_3$ \\\\\n     & & Household risk & Peer risk & Neighborhood risk \\\\\n     \\midrule\n       1 & Household poverty & 1 & 0 & 0 \\\\\n       2 & Single-parent status & 1 & 0 & 0\\\\\n       3 & Peer cigarette use & 0 & 1 & 0 \\\\\n       4 & Peer alcohol use & 0 & 1 & 0 \\\\\n       5 & Neighborhood unemployment & 0 & 0 & 1 \\\\\n       6 & Neighborhood poverty & 0 & 0 & 1 \\\\\n       \\bottomrule\n    \\end{tabular}\n    }\n\\end{table}\n\n\n\\begin{table}[h!]\n    \\caption{Prevention science survey example reformulated using the BLESS model. Five latent classes obtained and explained in \\cite{lanza2013}, and reformulated in the interpretable multidimensional-binary latent variable format.}\n    \\label{tab-prev2}\n    \n    \\centering\n   \n   \n   \n   \n   \n   \n   \n   \n   \n\n\\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{llccc}\n    \\toprule\n     & \\multirow{3}{*}{Latent Class Explanation} & & Fine-grained Latent Risks & \\\\\n     \\cmidrule(lr){3-5}\n     & & $\\alpha_1$ & $\\alpha_2$ & $\\alpha_3$ \\\\\n     & & Household risk & Peer risk & Neighborhood risk \\\\\n     \\midrule\n       1 & Overall low risk  & 0 & 0 & 0 \\\\\n       2 & Peer risk & 1 & 0 & 0\\\\\n       3 & Household \\& neighborhood risk & 1 & 0 & 1 \\\\\n       4 & Household \\& peer risk & 1 & 1 & 0 \\\\\n       5 & Overall high risk & 1 & 1 & 1 \\\\\n    \\bottomrule\n    \\end{tabular}\n}\n    \n\\end{table}\n\nBecause $\\mathbf G$ shows that each latent risk has exactly two observed children characteristics, this example analysed in \\cite{lanza2013} can be exactly regarded as satisfying the minimal conditions for generic identifiability of the BLESS model.\nAs \\cite{lanza2013} did not include the original dataset that they analyzed which is extracted and sampled from the NLSAH survey, we do not perform the test here but point out the testing procedure is just the same as what we conducted in Section \\ref{sec-timss} for the TIMSS data.\nSpecifically, one could simply test the hypothesis of identifiability by testing the marginal independence of the three groups of binary characteristics falling under the household risk, peer risk, and neighborhood risk, respectively.\nOne plausible conjecture is these three risks are likely interdependent due to the interactions of an adolescent's household, peers, and neighborhood.\nIn such a case, the BLESS model would be identifiable when applied to the survey dataset, and one could use the BLESS model as a more fine-grained and interpretable graphical modeling alternative to plain latent class analysis.\n\n\n\n\n\n\n\n\n\\section{Concluding Remarks}\\label{sec-disc}\nThis work reveals an interesting and highly nontrivial phenomenon, blessing of latent dependence on identifiability, for the BLESS model, a class of discrete statistical models with multiple binary latent variables.\nWe have proved that under the minimal conditions for generic identifiability that each latent variable has two observed children, the model parameters are identifiable if and only if there exists dependence between the latent variables.\nUsing two real-world examples in education and prevention science, we have shown how our sharp identifiability results can be applied in practice and guide the use of interpretable, graphical, and more fine-grained latent variable modeling approaches.\n\nThe blessing of dependence phenomenon between latent variables is perhaps a bit surprising, partly because the independence assumption of latent variables is predominant in many latent variable modeling approaches. For example, in the traditional and popular factor analysis model, the latent factors are often assumed independent with a diagonal covariance matrix \\citep{anderson1956fa}.\nIn practice, however, especially in confirmatory latent variable analysis widely seen in education, psychology, and epidemiology, each latent construct of interest carries a substantive meaning (see the examples in Section \\ref{sec-prac}). \nSo it is highly likely that such latent constructs postulated by domain experts are dependent on each other.\nFrom this perspective, our theoretical result in this work provides reassurance that the dependence of latent variables can be a blessing, rather than a curse.\nIn the future, it would be interesting to explore whether similar blessing-of-dependence phenomenon is present in other types of graphical latent variable models.\n\n\nFinally, in a study of the geometry of the simplest discrete latent variable model, the latent class model, and in the special case involving only a total number of $p=2$ observed variables, \\cite{fienberg2009} made the following remark, ``\\textit{The study of higher dimensional tables is still an open area of research. The mathematical machinery required to handle larger dimensions is considerably more complicated}''. \nIndeed, due to the complex nonlinearity of discrete models with latent structures, previous studies about identifiability either draw on Kruskal's Theorem or focus on small number of variables \\citep[e.g.][]{allman2015dag}.\nIn contrast, this work provides a new algebraic technique useful to study the identifiability and geometry of general $p$-dimensional tables.\nThis technique has proved to be more powerful than Kruskal's theorem when applied to the BLESS model considered in this work, and we are able to use it to derive sharp identifiability results which Kruskal's Theorem cannot obtain, in addition to revealing the new geometry.\nUsing the new technique to study other properties (beyond identifiability) of discrete graphical latent variable models and exploring its connection to other algebraic statistical techniques would be an interesting future direction.\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*{Supplementary material}\n\\label{SM}\nThe Supplementary Material contains the proofs of all the theoretical results and the details of the EM algorithms.\n\n\\spacingset{1}\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor every natural number $n\\geqslant 1$ the \\textbf{\\emph{Harmonic Number}}, $H_{n}$ is the $n$th partial sum of the harmonic series:\\begin{equation}\n\\fbox{$\\displaystyle H_{n}:=1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}.$}\n\\end{equation}\n\nAlthough the asymptotics of $H_{n}$ were determined by \\textsc{Euler}, (see \\cite{K}), in his famous  formula: \\begin{equation}\n\\fbox{$\\displaystyle H_{n}\\sim \\ln n +\\gamma +\\frac{1}{2n}-\\frac{1}{12n^{2}}+\\frac{1}{120n^{4}}-\\left[\\cdots\\right],$}\n\\end{equation}where $\\gamma=0.57721\\cdots$ is \\textsc{Euler}'s constant and each summand in the asymptotic expansion is of the form $\\dfrac{B_{k}}{n^{k}}$, where $B_{k}$ denots the $k$th \\textsc{Bernoulli} number, mathematicians have continued to offer alternate approximative formulas to \\textsc{Euler}'s. We cite the following formulas, which appear in order of increasing accuracy:\\begin{align}\nH_{n}    &\\approx \\ln n +\\gamma +\\frac{1}{2n+\\frac{1}{3}}    \\\\\n    &\\approx\\ln\\sqrt{n(n+1)}+\\gamma +\\frac{1}{6n(n+1)+\\frac{6}{5}}\\\\\n    &\\approx\\ln\\left(n+\\frac{1}{2}\\right)+\\gamma +\\frac{1}{24\\left(n+\\frac{1}{2}\\right)^{2}+\\frac{21}{5}}.  \n\\end{align}The formula (3) is the \\textsc{T\\'oth-Mare} approximation, (see \\cite{TM}), and it\\textbf{\\emph{underestimates}} the true value of $H_{n}$ by terms of order $\\dfrac{1}{72n^{3}}$; the second, (4), is the \\textsc{Lodge-Ramanujan} approximation, and it \\textbf{\\emph{overestimates}} the true value of $H_{n}$ by terms of order $ \\dfrac{19}{3150\\left[n(n+1)\\right]^{3}}$, (see \\cite{Vill}); and the last, (5), is the \\textsc{DeTemple-Wang} approximation, and it \\textbf{\\emph{overestimates}} the true value of $H_{n}$ by terms of order $\\dfrac{2071}{806400\\left(n+\\frac{1}{2}\\right)^{6}},$ (see \\cite{D}).\n\n\nIn 2003, \\textsc{Chao-Ping Chen} and \\textsc{Feng Qi}, (see \\cite{CQ}), published a proof of the following sharp form of the \\textsc{T\\'oth-Mare} approximation:\n\n\\begin{thm} For any natural number $n\\geqslant 1$, the following inequality is valid:\n\\begin{equation}\n\\fbox{$\\displaystyle \\frac{1}{2n+\\frac{1}{1-\\gamma}-2}\\leqslant H_{n}-\\ln n -\\gamma <\\frac{1}{2n+\\frac{1}{3}}.$}\n\\end{equation}The constants $\\frac{1}{1-\\gamma}-2=.3652721\\cdots$ and $\\frac{1}{3}$ are the best possible, and equality holds only for $n=1.$\n\n\\end{thm}\nThe first \\emph{statement} of this theorem had been announced ten years earlier by the editors of the ``Problems\" section of the \\emph{American Mathemtical Monthyly}, Vol 99, No. 7, (Jul-Aug, 1992), p 685, as  part of a commentary on the solution of Problem 3432, but they did not publish the proof.  So, the first \\emph{published} proof is apparently that of \\textsc{Chen} and \\textsc{Qi}.\n\nIn this paper we will prove sharp forms of the \\textsc{Lodge-Ramanujan} approximation and the \\textsc{DeTemple-Wang} approximation.\n\n\\begin{thm} For any natural number $n\\geqslant 1$, the following inequality is valid:\n\\begin{equation}\n\\fbox{$\\displaystyle \\frac{1}{6n(n+1)+\\frac{6}{5}}< H_{n}-\\ln\\sqrt{n(n+1)}-\\gamma \\leqslant\\frac{1}{6n(n+1)+\\frac{12\\gamma -11-12\\ln 2}{1-\\gamma-\\ln\\sqrt{2}}}.$}\n\\end{equation}The constants $\\frac{12\\gamma -11-12\\ln 2}{1-\\gamma-\\ln\\sqrt{2}}=1.12150934\\cdots$ and $\\frac{6}{5}$ are the best possible, and equality holds only for $n=1.$\n\\end{thm} \n\n\\noindent and\n\n\\begin{thm} For any natural number $n\\geqslant 1$, the following inequality is valid:\n\\begin{equation}\n\\fbox{$\\displaystyle\\frac{1}{24\\left(n+\\frac{1}{2}\\right)^{2}+\\frac{21}{5}} \\leqslant H_{n}-\\ln n -\\gamma <\\frac{1}{24\\left(n+\\frac{1}{2}\\right)^{2}+\\frac{54\\ln\\frac{3}{2}+54\\gamma-53}{1-\\ln\\frac{3}{2}-\\gamma}}.$}\n\\end{equation}The constants $\\frac{54\\ln\\frac{3}{2}+54\\gamma-53}{1-\\ln\\frac{3}{2}-\\gamma}=3.73929752\\cdots\\cdots$ and $\\frac{21}{5}$ are the best possible, and equality holds only for $n=1.$\n\\end{thm}\n\nAll three theorems are corollaries of the following stronger theorem:\n\\begin{thm}For any natural number $n\\geqslant 1$, define $f_{n}$, $\\lambda_{n}$, and $d_{n}$ by:\\begin{align}\nH_{n}    &:= \\ln n +\\gamma +\\frac{1}{2n+f_{n}}    \\\\\n    &:=\\ln\\sqrt{n(n+1)}+\\gamma +\\frac{1}{6n(n+1)+\\lambda_{n}}\\\\\n    &:=\\ln\\left(n+\\frac{1}{2}\\right)+\\gamma +\\frac{1}{24\\left(n+\\frac{1}{2}\\right)^{2}+d_{n}},  \n\\end{align}respectively. Then for any natural number $n\\geqslant 1$ the sequence $\\{f_{n}\\}$  is \\textbf{monotonically decreasing} while the sequences $\\{\\lambda_{n}\\}$ and $\\{d_{n}\\}$ are \\textbf{monotonically increasing}.\n\\end{thm}\\textsc{Chen} and \\textsc{Qi}, (see \\cite{CQ}), proved that the sequence $\\{f_{n}\\}$ \\textbf{\\emph{decreases}} monotonically.  In this paper we will prove the monotonicity of the sequences $\\{\\lambda_{n}\\}$ and $\\{d_{n}\\}$.\n\\section{Lemmas}\n\nOur proof is based on inequalities satisfied by the \\textbf{digamma} function, $\\Psi(x)$:\n\\begin{equation}\n\\fbox{$\\displaystyle \\Psi(x):=\\frac{d}{dx}\\ln\\Gamma(x)\\equiv\\frac{\\Gamma'(x)}{\\Gamma(x)}\\equiv -\\gamma-\\frac{1}{x}+x\\sum_{n=1}^{\\infty}\\frac{1}{n(x+n)} ,$}\n\\end{equation}which is the generalization of $H_{n}$ to the real variable $x$ since $\\Psi(x)$ and $H_{n}$ satisfiy the equation:\\begin{equation}\\Psi(n+1)=H_{n}-\\gamma.\\end{equation} \n\n\\begin{lemma}For every $x>0$ there exist numbers $\\theta_{x}$ and $\\Theta_{x}$, with $0<\\theta_{x}<1$ and $0<\\Theta_{x}<1$, for which the following equations are true:\\begin{align}\n \\Psi(x+1)   &=\\ln x+\\frac{1}{2x}-\\frac{1}{12x^{2}}+\\frac{1}{120x^{4}}-\\frac{1}{252x^{6}}+\\frac{1}{240x^{8}}\\theta_{x}, \\\\\n \\Psi'(x+1)   &=\\frac{1}{x}-\\frac{1}{2x^{2}}+\\frac{1}{6x^{3}}-\\frac{1}{30x^{5}}+\\frac{1}{42x^{7}}-\\frac{1}{30x^{9}}\\Theta_{x}. \\\\\n      \\end{align}\n\\end{lemma}\n\\begin{proof}\n\nBoth formulas are well-known.  See, for example, \\cite{Ed}, pp 124-125.\n\n\\end{proof}\n\n\\begin{lemma}The following inequalities are true for $x>0$:\\begin{multline}\n\\frac{1}{3x(x+1)}-\\frac{1}{15x^{2}(x+1)^{2}}< 2\\Psi(x+1)-\\ln\\{x(x+1)\\}\\\\\n<\\frac{1}{3x(x+1)}-\\frac{1}{15x^{2}(x+1)^{2}}+\\frac{8}{315x^{3}(x+1)^{3}}  ,   \\end{multline}\\begin{multline}\n  \\frac{1}{x^{2}}-\\frac{1}{x(x+1)}-\\frac{1}{3x^{3}}+\\frac{1}{15x^{5}}-\\frac{1}{18x^{7}}<\\frac{1}{x}+\\frac{1}{x+1}-2\\Psi'(x+1)\\\\  <\\frac{1}{x^{2}}-\\frac{1}{x(x+1)}-\\frac{1}{3x^{3}}+\\frac{1}{15x^{5}}. \n   \\end{multline}\n\n\n\n\n\n\n\n\n\\end{lemma}\n\\begin{proof}\n\nThe inequalities (17) were proved in our paper, (see\\cite{Vill}), for integers $n$ instead of the real variable $x$.  But the proofs are valid for real $x$.\n\nFor (18) we start with (15) of \\textbf{Lemma 1.}  We conclude that $$\\frac{1}{2x^{2}}-\\frac{1}{6x^{3}}+\\frac{1}{30x^{5}}-\\frac{1}{36x^{7}}<\\frac{1}{x}-\\Psi'(x+1)<\\frac{1}{2x^{2}}-\\frac{1}{6x^{3}}+\\frac{1}{30x^{5}}.$$Now we multiply  to all three components of the inequality by 2 and add $\\dfrac{1}{x+1}-\\dfrac{1}{x}$ to them.\n\n\\end{proof}\n\n\\begin{lemma}The following inequalities are true for $x>0$:\\begin{multline}\n\\frac{1}{\\left(x+\\frac{1}{2}\\right)}-\\frac{1}{x}+\\frac{1}{2x^{2}}-\\frac{1}{6x^{3}}+\\frac{1}{30x^{5}}-\\frac{1}{42x^{7}}< \\frac{1}{x+\\frac{1}{2}}-\\Psi'(x+1)\\\\\n<\\frac{1}{\\left(x+\\frac{1}{2}\\right)}-\\frac{1}{x}+\\frac{1}{2x^{2}}-\\frac{1}{6x^{3}}+\\frac{1}{30x^{5}},     \\end{multline}\n\\begin{multline}\n\\frac{1}{24x^{2}}-\\frac{1}{24x^{3}}+\\frac{23}{960x^{4}}-\\frac{1}{160x^{5}}-\\frac{11}{8064x^{6}}-\\frac{1}{896x^{7}}< \\Psi(x+1)-\\ln\\left(x+\\frac{1}{2}\\right)\\\\\n<\\frac{1}{24x^{2}}-\\frac{1}{24x^{3}}+\\frac{23}{960x^{4}}-\\frac{1}{160x^{5}}-\\frac{11}{8064x^{6}}-\\frac{1}{896x^{7}}+\\frac{143}{30720x^{8}} .    \\end{multline}\n\\end{lemma}\n\\begin{proof} Similar to the proof of \\textbf{Lemma 2.}\n\\end{proof}\n\\section{Proof for the Lodge-Ramanujan approximation}\n\\begin{proof}\nWe solve (10) for $\\lambda_{n}$ and use (13) to obtain $$\\lambda_{n}=\\frac{1}{\\Psi(n+1)-\\ln\\sqrt{n(n+1)}}-6n(n+1).$$Define \\begin{equation}\n\\fbox{$\\displaystyle \\Lambda_{x}:=\\frac{1}{2\\Psi(x+1)-\\ln x(x+1)}-3x(x+1). $}\n\\end{equation}for all $x>0$.  Observe that $2\\Lambda_{n}=\\lambda_{n}.$\n\\\\\n\\\\\n\\emph{We will show that} $\\Lambda_{x}'>0$ for $x>5.$  Computing the derivative we obtain$$\\Lambda_{x}'=\\frac{\\frac{1}{x}+\\frac{1}{x+1}-\\Psi'(x+1)}{\\{ 2\\Psi(x+1)-\\ln\\{x(x+1)\\}^{2}}-(6x+3)$$ and therefore \\begin{align*}\n\\{ 2\\Psi(x+1)-\\ln\\{x(x+1)\\}^{2}\\Lambda_{x}'&=\\frac{1}{x}+\\frac{1}{x+1}-\\Psi'(x+1)-(6x+3)\\{ 2\\Psi(x+1)-\\ln\\{x(x+1)\\}^{2}.\\end{align*}  By \\textbf{Lemma 2}, this is greater than\\begin{align*}\n&\\frac{1}{x^{2}}-\\frac{1}{x(x+1)}-\\frac{1}{3x^{3}}+\\frac{1}{15x^{5}}-\\frac{1}{18x^{7}}\\\\\n&-(6x+3)\\left\\{\\frac{1}{3x(x+1)}-\\frac{1}{15x^{2}(x+1)^{2}}+\\frac{8}{315x^{3}(x+1)^{3}}\\right\\}^{2}\\\\\n&=\\frac{1071x^{6}+840x^{5}-17829x^{4}-49266x^{3}-502999x^{2}-22178x-3675}{66150x^{7}(x+1)^{6}}\\\\\n&=\\frac{(x-5)\\left(x^{5}+\\frac{295}{51}x^{4}+\\frac{628}{51}x^{3}+\\frac{784}{51}x^{2}+\\frac{32021}{1071}x\n+\\frac{137927}{1071}\\right)+\\frac{685960}{1071}}{\\frac{1051}{17}x^{7}(x+1)^{6}}\n\\end{align*}which is obviously\\emph{ positive} for $x>5.$  \n\nFor $x=1, \\ 2, \\ 3, \\ 4, \\ 5,$ we compute directly:\\begin{align*}\n\\label{}\n   \\Lambda_{1} &=.56075467\\cdots   \\\\\n   \\Lambda_{2} &=.58418229\\cdots   \\\\\n   \\Lambda_{3} &=.59158588\\cdots   \\\\\n   \\Lambda_{4} &=.59481086\\cdots   \\\\\n   \\Lambda_{5} &=.59649019\\cdots   \\\\\n\\end{align*}Therefore, the sequence $\\{\\Lambda_{n}\\}$, $n \\geqslant 1$, is a strictly increasing sequence, and therefore so is the sequence $\\{\\lambda_{n}\\}$.\n\nMoreover, in \\cite{Vill}, we proved that $$\\lambda_{n}=\\frac{6}{5}-\\Delta_{n},$$where $0<\\Delta_{n}<\\dfrac{38}{175n(n+1)}$. Therefore $$\\lim_{n\\rightarrow\\infty}\\lambda_{n}=\\frac{6}{5}.$$  This completes the proof.\n\\end{proof}\n\n\\section{Proof for the DeTemple-Wang Approximation}\n\n\\begin{proof}\nFollowing the idea in the proof of the  \\textsc{Lodge-Ramanujan} approximation we solve (11) for $d_{n}$ and define the corresponding real-variable version.  Let\n\\begin{equation}\n\\fbox{$\\displaystyle d_{x}:=\\frac{1}{\\Psi(x+1)-\\ln\\left(x+\\frac{1}{2}\\right)}-24\\left(x+\\frac{1}{2}\\right)^{2}$}\n\\end{equation}We compute the derivative, ask\\emph{ when it is \\textbf{positive}}, clear the denominator and observe that we have to solve the inequality:$$\\left\\{\\frac{1}{x+\\frac{1}{2}}-\\Psi'(x+1)\\right\\}-48\\left(x+\\frac{1}{2}\\right)\\left\\{\\Psi(x+1)-\\ln\\left(x+\\frac{1}{2}\\right)\\right\\}^{2}>0.$$By \\textbf{Lemma 3}, the left hand side of this inequality is\\begin{align*}\n &>\\frac{1}{\\left(x+\\frac{1}{2}\\right)}-\\frac{1}{x}+\\frac{1}{2x^{2}}-\\frac{1}{6x^{3}}+\\frac{1}{30x^{5}}-\\frac{1}{42x^{7}}-48\\left(x+\\frac{1}{2}\\right)\\\\\n &\\left(\\frac{1}{24x^{2}}-\\frac{1}{24x^{3}}+\\frac{23}{960x^{4}}-\\frac{1}{160x^{5}}-\\frac{11}{8064x^{6}}-\\frac{1}{896x^{7}}+\\frac{143}{30720x^{8}}\\right)^{2}   \\\\\n \\end{align*}for all $x>0.$ This last quantity is equal to\n\\begin{align*}&(-9018009-31747716 x-14007876 x^2+59313792 x^3+\n11454272 x^4-129239296 x^5+119566592 x^6\\\\\n&+65630208 x^7-701008896 x^8-534417408 x^9+\n178139136 x^{10})\/(17340825600 x^{16} (1+2 x))\\end{align*}\n\n\n\n\\noindent The denominator, $$17340825600 x^{16} (1+2 x),$$ is evidently \\emph{positive} for $x>0$  and the  \\emph{numerator} can be written in the form $$p(x)(x-4)+r$$ where \\begin{align*}\np(x)&=548963242092+137248747452 x+34315688832 x^2\n+8564093760 x^3+2138159872 x^4\\\\&+566849792 x^5\n+111820800 x^6+11547648 x^7+178139136 x^8+178139136 x^9\n\\end{align*}with remainder $r$ equal to\n$$r=2195843950359.$$\n\nTherefore, the numerator is clearly \\emph{positive} for $x>4,$ and therefore, the derivative, $d_{x}\\ '$, too, is \\emph{postive} for $x>4.$  Finally\\begin{align*}\n d_{1} &=3.73929752\\cdots   \\\\\n d_{2} &=4.08925414\\cdots   \\\\  \n d_{3} &=4.13081174\\cdots   \\\\\n d_{4} &=4.15288035\\cdots\n       \\end{align*}Therefore $\\{d_{n}\\}$ is an \\textbf{\\emph{increasing}} sequence for $n\\geqslant 1.$ \n\nNow, if we expand the formula for $d_{n}$ into an asymptotic series in powers of $\\dfrac{1}{\\left(n+\\frac{1}{2}\\right)}$, we obtain$$d_{n}\\sim \\frac{21}{5}-\\frac{1400}{2071\\left(n+\\frac{1}{2}\\right)}+\\cdots$$and we conclude that $$\\lim_{n\\rightarrow\\infty}d_{n}=\\frac{21}{5}.$$This completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe antiferromagnetic Heisenberg model for $S=1\/2$ spins interacting on the triangular lattice represents the simplest example in which \nquantum fluctuations give rise to strong modifications of the classical picture, where the minimum energy configuration shows $120^\\circ$ \norder. Indeed, this was the first microscopic model that has been proposed for the realization of the so-called resonating valence-bond \nstate~\\cite{anderson1973,fazekas1974}. Within this approach, the ground state is described by a superposition of an exponentially large \nnumber of singlet coverings of the lattice, generalizing the concept of resonance introduced and developed by Rumer~\\cite{rumer1932} and\nPauling~\\cite{pauling1933} to describe the chemical bond. Even though recent numerical investigations~\\cite{capriotti1999,chernyshev2007} \nhave shown that the ground state possesses a finite magnetization in the thermodynamic limit, the results confirmed large deviations \nfrom classical and semiclassical limits. In addition, small perturbations on top of the nearest-neighbor Heisenberg model have shown to \ndrive the system into magnetically disordered phases~\\cite{zhu2018,iaconis2018}. By keeping the spin SU(2) symmetry, a natural way to \ninduce further magnetic frustration is to include a next-nearest-neighbor super-exchange coupling, leading to the following \nHamiltonian:\n\\begin{equation}\\label{eq:hamj1j2}\n{\\cal H} = J_1 \\sum_{\\langle i,j \\rangle} {\\bf S}_i \\cdot {\\bf S}_j +\nJ_2 \\sum_{\\langle\\langle i,j \\rangle\\rangle} {\\bf S}_i \\cdot {\\bf S}_j,\n\\end{equation}\nwhere $\\langle \\dots \\rangle$ and $\\langle \\langle \\dots \\rangle\\rangle$ indicate nearest-neighbor and next-nearest-neighbor sites\nin the triangular lattice; ${\\bf S}_i=(S_i^x,S_i^y,S_i^z)$ is the spin-$1\/2$ operator at the site $i$ and, finally, $J_1$ and $J_2$ \nare the antiferromagnetic coupling constants. This model has been intensively investigated in the past, from the semi-classical \napproaches of the early days~\\cite{jolicoeur1990,chubukov1992} to the recent numerical approaches~\\cite{zhu2015,hu2015,iqbal2016}.\nThe latter ones indicated a rather fragile $120^\\circ$ magnetic order, which is melted for $J_2\/J_1 \\approx 0.07(1)$ (a value that\nis in very good agreement among these calculations). For larger values of the frustrating ratio $J_2\/J_1$ the nature of the \nnon-magnetic phase is not settled down, with evidences for either a gapped~\\cite{zhu2015,hu2015} or a gapless~\\cite{iqbal2016} \nspin liquid. \n\nAn important information about the physical properties is given by the features of the low-energy spectrum. In particular, the dynamical \nstructure factor $S({\\bf q},\\omega)$ gives a direct probe to assess the nature of the relevant excitations. These can be divided in two \nbroad classes: standard gapless magnons (or gapped triplons), which exist in magnetically ordered phases (or valence-bond solids), and \nmore exotic (gapped or gapless) spinons, which exist in deconfined spin liquids. In addition to spinons, another kind of excitation is \npresent, due to the emergence of gauge fluctuations in the low-energy effective theory of spin liquids~\\cite{savary2016}.\n\nFor the Heisenberg model with only nearest-neighbor couplings on the triangular lattice, semi-classical approaches, based upon the\nlarge-$S$ expansion, suggested that the excitation spectrum obtained within the leading order (i.e., within the linear spin-wave \napproximation) is subjected to significant corrections when interactions between spin waves are taken into account~\\cite{starykh2006}. \nThis fact is mainly due to the non-collinearity of the magnetization, which allows for three-magnon interactions. Then, despite the \npresence of long-range order, the Goldstone modes are not stable but they may decay in a large part of the Brillouin zone (see \nFig.~\\ref{fig:latt}); in particular, the existence of more than one Goldstone mode, with different velocities, immediately causes that \nmagnons may be kinematically unstable, decaying into two magnons with lower energy~\\cite{chernyshev2006,chernyshev2009}. A detailed \nanalysis, which includes interactions among spin waves, corroborated this outcome, also showing roton-like minima at $M=(0,2\\pi\/\\sqrt{3})$ \nand symmetry-related points (i.e., midpoints of the edges of the Brillouin zone)~\\cite{chernyshev2006,chernyshev2009,zhitomirsky2013}.\nThe latter aspect shares similarities with the Heisenberg model on the square lattice, where minima of the magnon dispersion are present \naround $(\\pi,0)$ and $(0,\\pi)$~\\cite{singh1995,zheng2005}. As far as the triangular lattice is concerned, aspects of the strong \nrenormalization of the magnon dispersion at high energies have been confirmed by series expansions~\\cite{zheng2006}. Moreover, within \nthese numerical calculations, a huge downward renormalization of the one-magnon excitations is recovered, leading to a relatively \ndispersionless mode.\n\nWhile there are a number of materials whose low-energy behavior can be well described by the $S=1\/2$ Heisenberg model on the square \nlattice (among them, we just mention La$_2$CuO$_4$ for its relevance to cuprate superconductors~\\cite{coldea2001a}), until very \nrecently there were no compounds that could be well approximated by the same model on the equilateral triangular lattice. For example,\nin Cs$_2$CuCl$_4$ the super-exchange couplings are not isotropic in the nearest-neighbor bonds, one out of the three being much \nstronger than the other ones (thus defining weakly-coupled zig-zag chains)~\\cite{coldea2001b}. Here, inelastic neutron scattering \nmeasurements have shown the existence of a very broad continuum, which has been associated to spin fractionalization and spin-liquid\nbehavior~\\cite{coldea2001b}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1a.pdf}\n\\includegraphics[width=0.4\\columnwidth]{fig1b.pdf}\\hspace{0.05\\columnwidth}\n\\includegraphics[width=0.4\\columnwidth]{fig1c.pdf}\n\\caption{\\label{fig:latt}\nUpper-left panel: the classical spin configuration (in the $XY$ plane) that is determined by the fictitious magnetic field $h$ in \nthe Hamiltonian~(\\ref{eq:auxham}) with ${\\bf Q}=(2\\pi\/3,2\\pi\/\\sqrt{3})$. Upper-right panel: pattern for the sign structure of the \nnearest-neighbor hopping $s_{i,j}$ of Eq.~(\\ref{eq:auxham}), $s_{i,j}=+1$ ($-1$) for solid (dashed) lines; notice the the amplitude \nfor the kinetic terms is chosen to be $t>0$. Lower-left panel: the path in the Brillouin zone that is used to plot the results of \nthe dynamical structure factor of the $30\\times30$ triangular lattice (blue arrows), see Figs.~\\ref{fig:not}, \\ref{fig:AF120}, \n\\ref{fig:dispersions}, \\ref{fig:007}, and~\\ref{fig:125}. Lower-right panel: the path in the Brillouin zone that is used to plot the \ndynamical structure factor of the $84 \\times 6$ cylinder (blue arrows), see Fig.~\\ref{fig:cylin}. In both lower panels\nthe orange shaded area corresponds to the region of the Brillouin zone in which magnon decay is predicted by the spin-wave \napproximation~\\cite{chernyshev2006,chernyshev2009} and the dashed line delimits the magnetic Brillouin zone.}\n\\end{figure}\n\nRecently, measurements on Ba$_3$CoSb$_2$O$_9$ have been reported, providing evidence that it can be described by a $S=1\/2$ Heisenberg \nmodel on the undistorted triangular lattice with predominant nearest-neighbor super-exchange couplings (a small easy-plane anisotropy \nis present, in addition to a small interlayer coupling)~\\cite{shirata2012}. The initial interest was aimed at the study of the \nmagnetization curve and the stabilization of magnetization plateaux~\\cite{shirata2012,suzuki2013}, and the proximity to a spin liquid\nphase~\\cite{zhou2012}. Later, inelastic neutron scattering measurements have been performed, in order to clarify the nature of the\nmagnetic excitations on top of the ground state~\\cite{ma2016,ito2017}. Even though Ba$_3$CoSb$_2$O$_9$ possesses long-range magnetic \norder (with $120^{\\circ}$ ordering), several aspects of the magnon dispersion and the multi-magnon continuum reveal an unconventional\nbehavior, which can only be partly explained within semi-classical approaches. First of all, at low-energies, the magnon dispersion \nis strongly renormalized with respect to the linear spin-wave approximation; an anomalous line broadening has also been detected, \nleading to the conclusion that magnon decay may be plausible; finally, the continuum presents unexpected dispersive features at high \nenergies. It should be noticed that, since neutron scattering data are sensitive to the full dynamical spin structure factor, three\ncopies of the magnon dispersion (translated by the ordering vectors) are visible in the spectrum. Experimental investigations have \nbeen also performed to infer the nature of the magnon excitations on top of the gapped phase that is stabilized at the one-third \nmagnetization plateau~\\cite{kamiya2018}. In this case, the situation seems to be more conventional, with the experimental results \nin relatively good agreement with theoretical predictions.\n\nMotivated by these experimental findings, there have been a few attempts to investigate the Heisenberg model (also including small\nperturbations) with both analytical and numerical tools~\\cite{ghioldi2015,ghioldi2018,verresen2018,chen2018}. In particular, by using \ndensity-matrix renormalization group (DMRG) calculations, Verresen and collaborators~\\cite{verresen2018} claimed that the magnon \ndecay does not take place, because of the strong coupling interactions between quasi-particles (i.e., magnons) in the Heisenberg \nmodel~\\cite{noteverre}. As a result of the avoided decay, the midpoint of the edge of the {\\it magnetic} Brillouin zone (dubbed \n$Y_1$) displays a minimum of the magnon dispersion, possibly explaining the high-energy features seen around the $M$ point in \nRef.~\\cite{ito2017}.\n\nWithin this context, also the discovery of YbMgGaO$_4$~\\cite{li2015} and, more recently, NaYbO$_2$~\\cite{ding2019} will give a further \nimpetus to study (generalized) spin models on the triangular lattice. In both cases, no signatures of magnetic order appear down to \nvery low temperatures, suggesting the existence of a quantum spin liquid. While both materials host effective $J=1\/2$ spin degrees of \nfreedom, the actual low-energy Hamiltonian may be more complicated than the $SU(2)$-invariant one of Eq.~(\\ref{eq:hamj1j2}); still, the \nphysical properties can share many similarities with the ground state of the $J_1-J_2$ model, as suggested in Ref.~\\cite{zhu2018}.\n\nIn this work, we employ a dynamical variational Monte Carlo approach~\\cite{li2010} to compute the out-of-plane dynamical spin structure \nfactor for the Heisenberg model on the triangular lattice, also in presence of a next-nearest-neighbor coupling $J_2$. First of all, we \nfocus our attention on the model with $J_2=0$ for which we confirm huge corrections from the linear spin-wave calculations. Our results\nsupport the idea that the magnon excitations are stable in the whole Brillouin zone; indeed, even though a {\\it discrete} set of\nexcitations is obtained within our numerical method, the lowest-energy state for each momentum ${\\bf q}$ appears to be rather well \nseparated from the rest of the spectrum at higher energies, suggesting the existence of a faint continuum just above the magnon branch.\nThe second part of this work deals with the $J_1-J_2$ model, to highlight the modifications in the dynamical structure factor that take \nplace when entering the spin-liquid phase (which, according to our variational approach, is gapless~\\cite{iqbal2016}). Here, the spectrum \nshows gapless excitations at $M$ points; in addition, a strong signal at low energies is present in correspondence of the corners of the \nBrillouin zone, i.e., $K=(2\\pi\/3,2\\pi\/\\sqrt{3})$ and $K^\\prime=(4\\pi\/3,0)$. While the former aspect can be easily understood by inspecting \nthe non-interacting spinon band structure, the latter one is a genuine feature that emerges from the Gutzwiller projector, which includes \ninteractions between spinons and gauge fields. Indeed, while the non-interacting wave function corresponds to a mean-field approximation, \nin which gauge fields are completely frozen, the Gutzwiller projection has the effect of inserting back the temporal fluctuations of \nthose fields~\\cite{wenbook}. In this respect, it is worth mentioning that a recent field-theoretical analysis indicated the existence of \nlow-energy (triplet) monopole excitations at the zone corners, which are expected to contribute to the dynamical structure \nfactor~\\cite{song2018}.\n\n\\section{Dynamical variational Monte Carlo}\\label{sec:method}\n\nThe dynamical structure factor, which is directly measured within inelastic neutron scattering experiments, can be used to unveil the \nnature of the elementary excitations of the models\/materials under investigation. In its spectral form, this quantity reads as\n\\begin{equation}\\label{eq:dsf}\nS^{a}({\\bf q},\\omega) = \\sum_{\\alpha} |\\langle \\Upsilon_{\\alpha}^q | S^{a}_q | \\Upsilon_0 \\rangle|^2 \\delta(\\omega-E_{\\alpha}^q+E_0),\n\\end{equation}\nwhere $|\\Upsilon_0\\rangle$ and $\\{|\\Upsilon_{\\alpha}^q\\rangle\\}_\\alpha$ are the ground state and the set of all excited states with \nmomentum $q$, whose corresponding energies are $E_0$ and $\\{E_{\\alpha}^q\\}_{\\alpha}$, respectively. In this work, we evaluate the dynamical \nstructure factor of the spin model~(\\ref{eq:hamj1j2}) by directly constructing accurate variational {\\it Ansatze} for its ground state and \na few low-energy excited states. Our variational approach is based on the so-called {\\it parton} construction, in which the spin degrees of \nfreedom of the model are rewritten in terms of auxiliary fermionic operators~\\cite{savary2016,wen2002}. The fermionic language constitute a \nversatile framework to define variational wave functions for both magnetically ordered and disordered phases of matter. The present Section \nis dedicated to the introduction of the fermionic wave functions for spin models and to the description of the variational Monte Carlo \nmethod employed for the calculation of the dynamical structure factor.\n\n\\subsection{Gutzwiller-projected fermionic wave functions for the ground state}\\label{sec:wavefunctions}\n\nHere, for the sake of generality, we consider a generic $SU(2)$ model for frustrated spin systems, which consists of a set of spin-$1\/2$ \ndegrees of freedom sitting on the sites of a lattice and interacting through the Heisenberg exchange couplings $J_{i,j}$:\n\\begin{equation}\\label{eq:generic_heis}\n{\\cal H} = \\sum_{i,j} J_{i,j} {\\bf S}_i \\cdot {\\bf S}_j.\n\\end{equation}\nThe interplay of the different interactions can lead to the stabilization of different phases of matter. In absence of frustration, i.e.,\nwhen no competing couplings are present, the ground state may develop some kind of magnetic order, which minimizes the classical energy of \nthe model. On the contrary, when different interactions compete with each other, magnetically disordered phases can arise, such as spin \nliquids. \n\nThe first attempt to describe spin-liquid states dates back to the resonating valence-bond approach, where a variational wave function is\ndefined in terms of a linear superposition of singlet coverings of the lattice~\\cite{anderson1973}. More recently, Wen~\\cite{wen2002} \ndeveloped a general approach to classify and construct spin-liquid states, which satisfy all the symmetries of a given lattice model. \nThis method is built upon the introduction of auxiliary Abrikosov fermions, which form a projective representation of $S=1\/2$ spin operators: \n\\begin{equation}\\label{eq:Sabrikosov}\n{\\bf S}_i = \\frac{1}{2} \\sum_{\\alpha,\\beta} c_{i,\\alpha}^\\dagger \n\\boldsymbol{\\sigma}_{\\alpha,\\beta} c_{i,\\beta}^{\\phantom{\\dagger}}.\n\\end{equation}\nHere $c_{i,\\alpha}^{\\phantom{\\dagger}}$ ($c_{i,\\alpha}^\\dagger$) destroys (creates) a fermion with spin $\\alpha=\\uparrow,\\downarrow$ on site $i$, and the \nvector $\\boldsymbol{\\sigma}=(\\sigma_x,\\sigma_y,\\sigma_z)$ is the set of Pauli matrices. The anticommutation relations among fermions ensure \nthat the Abrikosov representation yields the correct commutation relations among different spin components. Still, in order to faithfully \nreproduce the Hilbert space of the original spin model, only configurations with one fermion per site must be considered, which implies that \nthe Abrikosov fermions must satisfy the constraint:\n\\begin{equation}\\label{eq:Gutz_constraint1}\nc^\\dagger_{i,\\uparrow}c^{\\phantom{\\dagger}}_{i,\\uparrow}+c^\\dagger_{i,\\downarrow}c^{\\phantom{\\dagger}}_{i,\\downarrow}=1,\n\\end{equation}\nor equivalently:\n\\begin{equation}\\label{eq:Gutz_constraint2}\nc^\\dagger_{i,\\uparrow} c^\\dagger_{i,\\downarrow}=0,\n\\end{equation}\n\nBesides constant terms, the Hamiltonian of Eq.~(\\ref{eq:generic_heis}) can be rewritten in terms of Abrikosov fermions as follows:\n\\begin{equation}\\label{eq:quartic_ham}\n {\\cal H} = -\\frac{1}{2}\\sum_{i,j} \\sum_{\\alpha,\\beta} J_{i,j} \\left(\n c_{i,\\alpha}^\\dagger c_{j,\\alpha}^{\\phantom{\\dagger}} c_{j,\\beta}^\\dagger c_{i,\\beta}^{\\phantom{\\dagger}} \n + \\frac{1}{2} c_{i,\\alpha}^\\dagger c_{i,\\alpha}^{\\phantom{\\dagger}} c_{j,\\beta}^\\dagger c_{j,\\beta}^{\\phantom{\\dagger}} \\right).\n\\end{equation}\nAt this stage, the Hamiltonian~(\\ref{eq:quartic_ham}) with the constraints of Eqs.~(\\ref{eq:Gutz_constraint1}) and~(\\ref{eq:Gutz_constraint2}) \ngive an {\\it exact} representation of the original model. In order to tackle the above interacting fermionic system, one possibility is to \nperform a mean-field decoupling~\\cite{wen2002}. For the purpose of studying spin-liquid phases, we keep only the mean-field terms that do not \nbreak the $SU(2)$ symmetry of the original spins. The result is a quadratic Hamiltonian:\n\\begin{eqnarray}\\label{eq:generic_mf}\n {\\cal H}_{0} = \\sum_{i,j} \\sum_{\\sigma} t_{i,j} c_{i,\\sigma}^\\dagger c_{j,\\sigma}^{\\phantom{\\dagger}} +\n \\sum_{i,j}  \\Delta_{i,j} c_{i,\\uparrow}^\\dagger c_{j,\\downarrow}^\\dagger + h.c. \\nonumber \\\\\n +\\sum_{i} \\sum_{\\sigma} \\mu_i c_{i,\\sigma}^\\dagger c_{i,\\sigma}^{\\phantom{\\dagger}} +\n \\sum_{i} \\zeta_{i} c_{i,\\uparrow}^\\dagger c_{i,\\downarrow}^\\dagger + h.c.,\n\\end{eqnarray}\nwhich contains a hopping term $t_{i,j}$ and a singlet pairing term $\\Delta_{i,j}$, which are related to the expectation values \n$\\langle c_{j,\\sigma}^\\dagger c_{i,\\sigma}^{\\phantom{\\dagger}}\\rangle$ and $\\langle c_{i,\\sigma} c_{j,-\\sigma}\\rangle$, respectively. In addition, the \none-fermion-per-site constraint of the parton construction is enforced in a {\\it global} fashion by including a chemical potential $\\mu_i$ \nand an onsite-pairing $\\zeta_{i}$ as Lagrange multipliers in ${\\cal H}_{0}$~\\cite{wen2002}. Within the mere mean-field approach, the \nparameters of ${\\cal H}_{0}$ are computed self-consistently and define a low-energy effective theory for the spin model under investigation. \nHowever, the ground state of ${\\cal H}_{0}$, named $|\\Phi_0 \\rangle$, satisfies the constraints of Eqs.~(\\ref{eq:Gutz_constraint1}) \nand~(\\ref{eq:Gutz_constraint2}) only on average and, therefore, does not represent a valid wave function for spins. Within this approach, \na full treatment of the original spin model requires the inclusion of all fluctuations of the parameters around the mean-field solution. \nSince this task is in general unfeasible, an alternative approach can be pursued, in which the Hamiltonian ${\\cal H}_{0}$ is exploited \nas a starting point for the definition of a variational wave function for the initial spin model. Indeed, the one-fermion-per-site \nconstraint can be enforced exactly by applying the Gutzwiller projector,\n\\begin{equation}\n \\mathcal{P}_G= \\prod_i (n_{i,\\uparrow}-n_{i,\\downarrow})^2,\n\\end{equation}\nto the ground state wave function of ${\\cal H}_{0}$. We emphasize that in general the Gutzwiller projection cannot be treated analytically, \ndue to its intrinsic many-body character, however it can be considered within Monte Carlo sampling. At variance with the mean-field treatment, \nin the variational approach the parameters of ${\\cal H}_{0}$ are not computed self-consistently, but are optimized in order to minimize the \nenergy of the Gutzwiller-projected {\\it Ansatz} $\\mathcal{P}_G|\\Phi_0 \\rangle$.  \n\nThe artificial enlargement of the Hilbert space introduced by the parton construction gives rise to a {\\it gauge redundancy} in the \nrepresentation of the spin degrees of freedom. Specifically, the mapping~(\\ref{eq:Sabrikosov}) is invariant under {\\it local} $SU(2)$ \ntransformations of the Abrikosov fermions operators~\\cite{wen2002}. As a consequence, all physical properties of the spins are independent on \nthe gauge choice for fermions. For example, whenever we perform $SU(2)$ transformations to the unprojected Hamiltonian ${\\cal H}_{0}$, the\nvariational wave function with the Gutzwiller projector remains invariant. Exploiting this gauge redundancy, it is possible to classify all \nthe quadratic Hamiltonians ${\\cal H}_{0}$ whose Gutzwiller-projected ground states fulfill the symmetries of the lattice model. \nThis procedure, known as projective symmetry group analysis~\\cite{wen2002}, provides a recipe to construct all the distinct spin liquid \n{\\it Ansatze} for a given spin model. From a variational point of view, the spin-liquid wave function with the lowest variational energy is \nthe one which better describes the true ground state of the model.\n\nIn general, the variational {\\it Ansatze} defined by Gutzwiller-projecting the ground state of Eq.~(\\ref{eq:generic_mf}) do not display any \nmagnetic order~\\cite{li2013}. For the purpose of defining suitable wave functions for magnetically ordered phases, an additional term can be \nadded to ${\\cal H}_{0}$:\n\\begin{equation}\\label{eq:magnfield}\n {\\cal H}_{0} \\mapsto {\\cal H}_{0} + h \\sum_{i}  \n             \\left ( e^{i \\mathbf{Q} \\cdot \\mathbf{R}_i} c_{i,\\uparrow}^\\dagger c_{i,\\downarrow}^{\\phantom{\\dagger}}\n            + e^{-i \\mathbf{Q} \\cdot \\mathbf{R}_i} c_{i,\\downarrow}^\\dagger c_{i,\\uparrow}^{\\phantom{\\dagger}} \\right ).\n\\end{equation}\nHere, $h$ is a {\\it fictitious} magnetic field which lies in the $XY$ plane and displays a periodic pattern defined by the pitch vector \n$\\mathbf{Q}$. Since the ground-state wave function of the Hamiltonian~(\\ref{eq:magnfield}) tends to overestimate the magnetic \norder~\\cite{becca2011}, further transverse quantum fluctuations are added through the application of a spin-spin Jastrow factor,  \n\\begin{equation}\n\\mathcal{J}_s=\\exp \\left ( \\frac{1}{2} \\sum_{i,j} v_{i,j} S^z_i S^z_j \\right ),\n\\end{equation}\nto the Gutzwiller-projected state. Specifically, the complete form of the variational wave functions employed in this work is\n\\begin{equation}\\label{eq:wf}\n|\\Psi_0\\rangle= \\mathcal{P}_{S_z} \\mathcal{J}_s \\mathcal{P}_G |\\Phi_0 \\rangle,\n\\end{equation}\nwhere in addition to the Gutzwiller projection and the Jastrow factor, we apply a projector enforcing zero value for the $z$-component of the \ntotal spin ($\\mathcal{P}_{S_z}$).\n\nBy using this approach, the variational phase diagram for the $J_1-J_2$ model on the triangular lattice has been obtained in Ref.~\\cite{iqbal2016}:\nthe system undergoes a phase transition between a magnetically ordered phase to a gapless spin liquid at ${J_2\/J_1 \\approx 0.08}$. For this \nmodel, the optimal variational wave functions are obtained by considering only a hopping term (no pairing) \nand the fictitious magnetic field in the quadratic Hamiltonian:\n\\begin{eqnarray}\\label{eq:auxham}\n\\mathcal{H}_0 &=& t \\sum_{\\langle i,j \\rangle} s_{i,j} c_{i,\\sigma}^\\dagger c_{j,\\sigma}^{\\phantom{\\dagger}} \\nonumber \\\\\n&+& h \\sum_{i}  \\left ( e^{i \\mathbf{Q} \\cdot \\mathbf{R}_i} c_{i,\\uparrow}^\\dagger c_{i,\\downarrow}^{\\phantom{\\dagger}}\n+ e^{-i \\mathbf{Q} \\cdot \\mathbf{R}_i} c_{i,\\downarrow}^\\dagger c_{i,\\uparrow}^{\\phantom{\\dagger}} \\right ).\n\\end{eqnarray}\nHere $t$ is a first-neighbor hopping with a non-trivial sign structure ($s_{i,j} = \\pm 1$) which generates a pattern of alternating $0$ \nand $\\pi$ fluxes through the triangular plaquettes of the lattice, see Fig.~\\ref{fig:latt}; $h$ is a fictitious magnetic field which displays \nthe classical $120^\\circ$ order with ${\\bf Q}=(2\\pi\/3,2\\pi\/\\sqrt{3})$, see Fig.~\\ref{fig:latt} (considering ${\\bf Q}=(4\\pi\/3,0)$ would not \nchange the physical content of the ground state wave function). All the parameters included in $\\mathcal{H}_0$ and the pseudopotential \n$v_{i,j}$ (one parameter for each distance $|{\\bf R}_i-{\\bf R}_j|$ in the translational invariant lattice) entering the Jastrow factor can \nbe optimized to minimize the variational energy. While in the magnetic phase of the system the optimal value for the ratio $h\/t$ is finite, \nfor $J_2\/J_1 \\gtrsim 0.08$ the system enters the spin liquid phase and the magnetic field parameter vanishes in the thermodynamic \nlimit~\\cite{iqbal2016}. The values of the fictitious magnetic field as a function of $J_2\/J_1$ can be found in Ref.~\\cite{iqbal2016}.\n\nIn this work we compute the dynamical structure factor for the $J_1-J_2$ model on the $30 \\times30$ triangular lattice. For $J_2=0$, we first\nconsider the crudest approximation for the ground state, which consists in setting the hopping term $t$ to zero. The resulting wave function \nis equivalent to the state of Ref.~\\cite{huse1988} with only a two-body Jastrow factor. Much more accurate results are then obtained by restoring\nthe hopping term in the Hamiltonian and optimizing all the variational parameters, for the cases $J_2=0$ and $J_2\/J_1=0.07$. On the other hand, \nwhen the system is in the spin liquid regime ($J_2\/J_1=0.09$ and $J_2\/J_1=0.125$), the fictitious magnetic field is vanishing and the Jastrow \nfactor is not considered, because of its negligible effects on the variational results. According to the projective symmetry group classification, \nthe wave function obtained by considering only the hopping term in $\\mathcal{H}_0$ is a fully symmetric $U(1)$ spin liquid~\\cite{lu2016}.\n\n\\subsection{Dynamical structure factor}\\label{sec:dynamical}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2.pdf}\n\\caption{\\label{fig:not}\nDynamical structure factor of the nearest-neighbor Heisenberg model on the triangular lattice obtained by using the variational wave function \nof Eq.~(\\ref{eq:wf}) and~(\\ref{eq:auxham}) with $t=0$ on the $30 \\times 30$ cluster. The path along the Brillouin zone is shown in \nFig.~\\ref{fig:latt}. A Gaussian broadening of the spectrum has been applied ($\\sigma=0.02J_1$). The spin-wave energies of the magnon branch \n($\\epsilon_q$), on the same cluster size, are represented by the white dots connected with a solid line. The dashed line corresponds to the \nbottom of the continuum within linear spin waves, i.e.  $E_q=\\min_{k} \\{ \\epsilon_{q-k} + \\epsilon_{k} \\}$. Notice that $E_{q}<\\epsilon_{q}$ \nin most of the Brillouin zone, as obtained in Ref.~\\cite{chernyshev2006,chernyshev2009}.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig3.pdf}\n\\caption{\\label{fig:AF120}\nThe same as Fig.~\\ref{fig:not} but for the optimal variational wave function with both hopping $t$ and fictitious magnetic field $h$. \nThe path along the Brillouin zone is shown in Fig.~\\ref{fig:latt}. The dotted line denotes the bottom of the continuum \n$E_{q}=\\min_{k} \\{E_{0}^{q-k}+E_{0}^{k}\\}$, where $E_{0}^{q}$ is the lowest energy for a given momentum ${\\bf q}$ obtained within our \nvariational approach. Since the spectrum is gapless at the $\\Gamma$ point, we exclude the cases ${\\bf k}=(0,0)$ and ${\\bf k}={\\bf q}$ \nin the search of the minimum, because the resulting $E_{q}$ would simply coincide with the energy of the magnon branch $E_{0}^{q}$ \nall over the Brillouin zone. The purpose of this kinematic analysis is to show that no magnon decay can yield an energy $E_{q}$\nwhich is lower than the one of the magnon branch $E_{0}^{q}$ (in constrast with spin wave results).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig4.pdf}\n\\caption{\\label{fig:dispersions}\nEnergies of the magnon branch for the nearest-neighbor Heisenberg model on the triangular lattice obtained with different methods. The path \nalong the Brillouin zone is shown in Fig.~\\ref{fig:latt}. The black line corresponds to linear spin wave, the blue squares to series \nexpansion~\\cite{zheng2006}, and the orange circles to our variational results (on the $30 \\times 30$ cluster).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig5.pdf}\n\\caption{\\label{fig:dispersion}\nDispersion relation of the magnon branch (i.e., the lowest-energy excitation) as obtained within our variational approach (on the $30\\times 30$ \ncluster). The linear spin-wave results are also reported for comparison. Dashed lines represent the edges of the magnetic Brillouin zone. The \npresence of the roton minima at the $M$ and $Y_1$ points in the variational spectrum is evident.}\n\\end{figure}\n\nAs already mentioned, the dynamical structure factor of the $J_1-J_2$ model is computed by constructing variational {\\it Ansatze} to approximate \nthe low-energy excited states of the system. Here we limit ourselves to the calculation of the out-of-plane component $S^z({\\bf q},\\omega)$, \nand we employ the technique outlined in Ref.~\\cite{li2010,ferrari2018a,ferrari2018b}, which is briefly summarized in the following.\n\nFirst, we find the optimal variational \\textit{Ansatz} for the ground state of the model, which has the form of Eq.~(\\ref{eq:wf}), by \nminimizing the variational energy. The resulting wave function is employed as a reference state to construct a set of projected particle-hole \nexcitations with a given momentum $q$:\n\\begin{equation}\\label{eq:qRstate}\n|q,R\\rangle =  \\mathcal{P}_{S_z} \\mathcal{J}_s \\mathcal{P}_G \n\\frac{1}{\\sqrt{N}} \\sum_{i}\\sum_{\\sigma} e^{i {\\bf q} \\cdot {\\bf R}_i} \\sigma c^\\dagger_{i+R,\\sigma}c^{\\phantom{\\dagger}}_{i,\\sigma} |\\Phi_0\\rangle.\n\\end{equation}\nThese states are labelled by $R$, which runs over all lattice vectors. We approximate the low-energy excited states of the model by using \nlinear combinations of the elements of the basis set $\\{|q,R\\rangle\\}_R$:\n\\begin{equation}\\label{eq:psinq}\n |\\Psi_n^q\\rangle=\\sum_R A^{n,q}_R |q,R\\rangle.\n\\end{equation}\nFor a certain momentum {\\bf q}, we consider the Schr{\\\"o}dinger equation for the $J_1-J_2$ Hamiltonian restricting the form of its eigenvectors \nto the one of Eq.~(\\ref{eq:psinq}),  i.e. ${ {\\cal H}|\\Psi_n^q\\rangle = E_n^q |\\Psi_n^q\\rangle }$. Expanding everything in terms of \n$\\{|q,R\\rangle\\}_R$, we arrive to the following generalized eigenvalue problem\n\\begin{equation}\\label{eq:general_eig_prob}\n\\sum_{R^\\prime} \\langle q,R|{\\cal H}|q,R^\\prime \\rangle  A^{n,q}_{R^\\prime} = E_n^q \\sum_{R^\\prime} \\langle q,R|q,R^\\prime \\rangle \nA^{n,q}_{R^\\prime},\n\\end{equation}\nwhich is solved to find the expansion coefficients $A^{n,q}_R$ and the energies $E_n^q$ of the excitations. All the matrix elements,\n$\\langle q,R|{\\cal H}|q,R^\\prime \\rangle$ and $\\langle q,R|q,R^\\prime \\rangle$, are evaluated within the Monte Carlo procedure, by \nsampling according to the variational ground-state wave function. Finally the dynamical structure factor is computed by:\n\\begin{equation}\\label{eq:Szz_practical}\nS^{z}({\\bf q},\\omega) = \\sum_n |\\langle \\Psi_{n}^q | S^{z}_q | \\Psi_0 \\rangle|^2 \\delta(\\omega-E_{n}^q+E_0^{\\rm var}),\n\\end{equation}\nwhere $E_0^{\\rm var}$ is the variational energy of $|\\Psi_0 \\rangle$.\n\n\\section{Results}\\label{sec:results}\n\nIn this section, we present the numerical calculations for the dynamical structure factor $S({\\bf q},\\omega)$ obtained by the \nvariational approach described in the previous section. First, we discuss the case of the Heisenberg model with only nearest-neighbor\nsuper-exchange $J_1$, also comparing our results with recent DMRG calculations~\\cite{verresen2018}. Then, we include the \nnext-nearest-neighbor coupling $J_2$ to increase frustration and melt the magnetic order. In this way, a gapless spin-liquid regime \nis reached for $J_2\/J_1 \\approx 0.08$~\\cite{iqbal2016}.\n\n\\subsection{The nearest-neighbor model with $J_2=0$}\n\nLet us start our analysis by considering the case in which the ground-state wave function only contains the fictitious magnetic field,\ni.e., $t=0$. In this case, the Abrikosov fermions are completely localized (e.g., the eigenvalues of the auxiliary Hamiltonian define \nflat bands) and the wave function corresponds to the Jastrow state of Ref.~\\cite{huse1988} with only a two-body Jastrow factor. The \nresults for the dynamical structure factor on the $30 \\times 30$ cluster are shown in Fig.~\\ref{fig:not}. Here, the spectrum consists \nof a {\\it single} mode, which is identified as the magnon excitation (no continuum is visible). Notice that only one magnon branch is \nvisible, related to the magnon dispersion $\\epsilon_{q}$, since we consider the out-of-plane dynamical structure factor (the {\\it folded}\nbranches $\\epsilon_{q \\pm K}$ do not contribute to the signal). Remarkably, the dispersion of the magnon branch is possible thanks to \nthe Jastrow factor, since the wave function without it would give rise to a trivially flat (gapped) excitation spectrum, reflecting \nthe non-interacting band structure of fermions. By contrast, the long-range Jastrow term is able to produce a reasonable magnon mode, \nwhich agrees fairly well with the spin-wave calculations. In paticular, the spectrum is gapless at $\\Gamma=(0,0)$ (with a vanishingly \nsmall weight). Instead, in constrast to spin waves, which correctly predict gapless magnons at $K$ and $K^\\prime$ due to the coplanar\n$120^\\circ$ order, this simple wave function leads to a gapped spectrum at the corners of the Brillouin zone. In connection to that, \nthe out-of-plane static structure factor $S^z({\\bf q})=\\int d\\omega S^z({\\bf q},\\omega)$ does not diverge at $K$ or $K^\\prime$ when \n$L \\to \\infty$, showing only a maximum.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig6.pdf}\n\\caption{\\label{fig:cylin}\nThe dynamical structure factor for the nearest-neighbor Heisenberg model on a cylindrical geometry ($84 \\times 6$), to make a close\ncomparison with DMRG calculations by Verresen and collaborators~\\cite{verresen2018}. We apply a Gaussian broadening to the spectrum \nwhich is equivalent to the one of the aforementioned DMRG result ($\\sigma=0.077J_1$). The path in the Brillouin zone is shown in the \ninset and in Fig.\\ref{fig:latt} (the point $A$ lies at $1\/4$ of the $\\Gamma-K^{\\prime\\prime}$ line, where \n$K^{\\prime\\prime}=(-2\\pi\/3,2\\pi\/\\sqrt{3})$; the point $B$ lies at $1\/4$ of the $K-K^\\prime$ line). The dashed line denotes the bottom \nof the continuum, which is evaluated by taking $E_{q}=\\min\\{E_{0}^{q-K}+E_{0}^{K},E_{0}^{q+K}+E_{0}^{-K}\\}$, where $E_{0}^{q}$ is \nthe lowest energy for a given momentum $q$ obtained within our variational approach and $K=(2\\pi\/3,2\\pi\/\\sqrt{3})$.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig7a.pdf}\n\\includegraphics[width=\\columnwidth]{fig7b.pdf}\n\\caption{\\label{fig:007}\nThe dynamical structure factor for the $J_1-J_2$ Heisenberg model on the $30 \\times 30$ cluster with $J_2\/J_1=0.07$ (above) and\n$J_2\/J_1=0.09$ (below). The path along the Brillouin zone is shown in Fig.~\\ref{fig:latt} and a Gaussian broadening of the spectrum \nhas been applied ($\\sigma=0.02J_1$).}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{fig8a.pdf}\\hfill\n\\includegraphics[width=\\columnwidth]{fig8b.pdf}\n\\caption{\\label{fig:125}\nThe dynamical structure factor for the $J_1-J_2$ Heisenberg model on the $30 \\times 30$ cluster with $J_2\/J_1=0.125$. The variational results \n(left panel) are compared to the ones obtained from the unprojected Abrikosov fermion Hamiltonian $\\mathcal{H}_0$ of Eq.~(\\ref{eq:auxham}) with \n$t=1$ and $h=0$ (right panel). The path along the Brillouin zone is shown in Fig.~\\ref{fig:latt}. We applied a Gaussian broadening of $\\sigma=0.02J_1$ \nto the variational results. Notice that, for the unprojected data, the energy scale is given by the hopping amplitude $t$ of the unprojected \nHamiltonian~(\\ref{eq:auxham}), instead of $J_1$. In addition, the broadening has been rescaled in order to account for the larger bandwidth of the \nspectrum.}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{fig9a.pdf}\\hfill\n\\includegraphics[width=\\columnwidth]{fig9b.pdf}\n\\caption{\\label{fig:square}\nThe dynamical structure factor for the $J_1-J_2$ Heisenberg model on the square lattice ($22 \\times 22$) with $J_2\/J_1=0.55$. The \nvariational results (left panel) are compared to the ones obtained from the unprojected Abrikosov fermion Hamiltonian $\\mathcal{H}_0$ \n(right panel), which contains a flux-phase hopping (of strength $t$) and a $d_{xy}$ pairing (see Ref.~\\cite{ferrari2018b} for details). \nWe applied a Gaussian broadening of $\\sigma=0.02J_1$ to the variational results. Notice that, for the unprojected data, the energy scale \nis given by the hopping amplitude $t$ of the unprojected Hamiltonian of Ref.~\\cite{ferrari2018b}, instead of $J_1$. In addition, the broadening \nhas been rescaled in order to account for the larger bandwidth of the spectrum.}\n\\end{figure*}\n\n\nA much more realistic spectrum is obtained when considering a finite fermion hopping $t$ (with the $\\pi$-flux pattern shown in \nFig.~\\ref{fig:latt}), as well as the optimized value of the fictitious magnetic field $h$ (and the Jastrow factor). The results for\nthe $30 \\times 30$ lattice are reported in Fig.~\\ref{fig:AF120}. In this case, there are several excitations with a finite weight for\neach momentum, thus reproducing the existence of a broad continuum, which extends up to relatively large energies. We would like to\nmention that, with respect to the square lattice~\\cite{ferrari2018b,dallapiazza2015,yu2018}, here many more excitations for each momentum\npossess a visible spectral weight. Within this calculation, we identify the lowest-energy excitation $E_{0}^{q}$ as the magnon peak. This\nassumption is corroborated by the results shown in Fig.~\\ref{fig:dispersions}, where the variational energies $E_{0}^{q}$ closely follow \nthe magnon branch obtained by series expansions. Instead, identifying the lowest-energy peak as the bottom of the continuum is not very \nplausible, since a much broader signal should be present in this case. In this regard, the basis set that is used here for the excited \nstates is made of particle-hole spinon excitations on top of the ground state of the auxiliary Hamiltonian of Eq.~(\\ref{eq:auxham}), before \nGutzwiller projection. For this reason, we argue that, in general, our approach is particularly suited to capture (i) two-spinon excitations \nor (ii) bound states of spinons, e.g., magnons. Multi-magnon excitations are expected to show up with a reduced intensity. In order to discuss \nthe issue of magnon decay, we apply a kinematic argument (as done both in the linear spin-wave approach~\\cite{chernyshev2006,chernyshev2009} \nand within DMRG~\\cite{verresen2018}) and we consider all the possible two-magnons decays, which fulfill the conservation of momenta, i.e., \n$E_{q}= \\min_k \\{ E_{0}^{q-k}+E_{0}^{k}\\}$. For this purpose, we computed the spectrum $E_{0}^{k}$ for all the $k$-vectors in the Brillouin \nzone on the $30 \\times 30$ lattice. The outcome is that the bottom of the two-magnon continuum, defined by the kinematic analysis, lies above \nthe magnon branch. These results clearly indicate an avoided decay in a large part of the Brilloiun zone, as suggested by DMRG calculations,\nwhich considered certain (high-energy) parts of the magnon dispersion~\\cite{verresen2018}. Still, we cannot exclude the existence of small \nregions where the magnon decay may persist, especially close to the gapless points. In this respect, within the linear spin-wave approach, \nthe different velocities of the excitation spectrum at $\\Gamma$ and $K$ immediately lead to an unstable magnon branch close to the $\\Gamma$ \npoint~\\cite{chernyshev2006,chernyshev2009}. Should this aspect be a genuine feature of the model, the magnon would be unstable in a small \npart around the center of the Brillouin zone. Unfortunately, given the finiteness of the cluster used in our numerical calculations, we cannot \nreliably estimate the slope of the magnon spectrum at $\\Gamma$ and $K$ and, therefore, make definitive statements for this issue.\n\nHere, we would like to notice the strong renormalization of the magnon branch with respect to spin-wave calculations, see \nFig.~\\ref{fig:dispersions}. Most importantly, we emphasize that, within this most accurate calculation, the magnon branch shows a roton-like \nminimum not only at $M$, but also at $Y_1$, i.e., the midpoint of the edge of the magnetic Brillouin zone (see also Fig.~\\ref{fig:dispersion}), \nas already detected by neutron scattering measurements in Ba$_3$CoSb$_2$O$_9$~\\cite{ito2017}. This feature was not captured by the previous\nseries expansion calculations~\\cite{zheng2006} but, instead, has been observed also by recent DMRG calculations on an infinitely long cylinder \n(with a small circumference $L=6$)~\\cite{verresen2018} and has been interpreted as the hallmark for the absence of magnon decay. In order to \nmake a closer comparison with DMRG data, we perform the variational calculations on a long cylinder ($84 \\times 6$) along the same path in the \nBrillouin zone as the one that has been considered in Ref.~\\cite{verresen2018}. The results are shown in Fig.~\\ref{fig:cylin}. Here, the large \nnumber of lattice points along the cylinder allows us to have a detailed resolution of the magnon branch, which closely follows the one obtained \nby DMRG. In particular, we can estimate the bottom of the continuum by evaluating $E_{q}=\\min \\{ E_{0}^{q-K}+E_{0}^{K}, E_{0}^{q+K}+E_{0}^{-K} \\}$, \nwhere we consider the possible decays involving a magnon at $K$ and $-K$. In doing so, we find that the lowest-energy excitation $E_{0}^{q}$ is \nalways below $E_{q}$, indicating that well defined branch exists and magnon decay is avoided. We finally remark that a roton minimum is detected \nalong the same path as the one studied by Verresen and collaborators~\\cite{verresen2018}, strongly suggesting that this is a genuine feature of \nthe Heisenberg model.\n\n\\subsection{The $J_1-J_2$ model}\n \nWe now move to the case where also a next-nearest-neighbor coupling $J_2$ is present. Within our variational approach, a gapless\nspin-liquid phase is stabilized for $0.08 \\lesssim J_2\/J_1 \\lesssim 0.16$; here, the fictitious magnetic field vanishes in the\nthermodynamic limit and the best wave function only contains fermionic hopping (with $\\pi$-flux threading half of the triangular\nplaquettes)~\\cite{iqbal2016}. On a finite size, a small value of $h$ can be stabilized, as well as a tiny Jastrow pseudopotential.\nStill, we verified that these ingredients do not cause sensible differences in the dynamical structure factor. In Fig.~\\ref{fig:007},\nwe show the results for the $30 \\times 30$ cluster and for two values of $J_2\/J_1$, which are very close to the transition point, one \nstill inside the magnetic phase ($J_2\/J_1=0.07$) the other one in the spin-liquid region ($J_2\/J_1=0.09$). By approaching the quantum \nphase transition, the major modification of the spectrum comes from the softening of the magnon excitation at the $M$ points. This \nfeature closely resembles the case of the frustrated $J_1-J_2$ model on the square lattice, previously studied with the same numerical \ntechnique~\\cite{ferrari2018b}, where a softening is clearly detected for ${\\bf q}=(\\pi,0)$ [and $(0,\\pi)$]. In this latter case, this \nfact has been connected to the progressive deconfinement of spinons that have gapless (Dirac) points at ${\\bf q}=(\\pm \\pi\/2,\\pm \\pi\/2)$. \nWe would like to mention that the possibility to have (gapped) almost-deconfined spinon in the unfrustrated Heisenberg model has been \nsuggested by a recent quantum Monte Carlo calculation~\\cite{shao2017}; moreover, clear signatures for deconfined spinons at the transition \nbetween an antiferromagnetically ordered phase and a valence-bond crystal have been reported in the so-called $J-Q$ model~\\cite{ma2018}. \nOn the triangular lattice, the softening of the spectrum at the $M$ points is a direct consequence of the Dirac points at \n${\\bf q}=(0,\\pm \\pi\/\\sqrt{3})$ in the spinon band structure. Therefore, we expect both $M$ and $K$ points to be gapless at the transition \n(as well as $Y_1$, which can be obtained by combining $M$ and $K$ vectors). Indeed, this is necessary for a continuous phase transition, \nas the one that appears in the $J_1-J_2$ Heisenberg model, according to ground-state calculations~\\cite{iqbal2016}.\n\nIn Fig.~\\ref{fig:125}, we report the dynamical structure factor for $J_2\/J_1=0.125$. The \nspin-liquid state is characterized by a broad continuum that extends up to relatively large energies. In particular, around the $M$ \npoints, the magnon roton-like minima of the ordered phase fractionalize into an incoherent set of excitations at low energies. This \nfeature is compatible with the existence of Dirac points in the unprojected spectrum of the auxiliary Hamiltonian $\\mathcal{H}_0$, \nsee Fig.~\\ref{fig:125}. By contrast, a strong signal in the lowest-energy part of the spectrum is detected around the $K$ points, where \nthe unprojected spinon spectrum is instead gapped. In this respect, the Gutzwiller projection is fundamental to include interaction\namong spinons in a non-perturbative way and give a drastic modification of the low-energy features. This is a distinctive aspect of \nthe triangular lattice, since, on the square lattice, all the low-energy (gapless) points observed in presence of the Gutzwiller \nprojector [i.e. ${\\bf q}=(0,0)$, $(\\pi,\\pi)$, $(\\pi,0)$ and~$(0,\\pi)$] already exist in the non-interacting picture~\\cite{hu2013}, \nsee Fig.~\\ref{fig:square}. We would like to emphasize that, in contrast to the magnetically ordered phase, where no visible spectral \nweight is present right above the magnon branch (see Fig.~\\ref{fig:AF120}), in the spin-liquid phase the continuum is not separated \nfrom the lowest-energy excitation. This outcome corroborates the fact of having deconfined spinons in the magnetically disordered phase. \nThe intense signal at $K$ points immediately implies strong (but short-range) antiferromagnetic correlations in the variational wave \nfunction, which are absent in the unprojected $\\pi$-flux state (by contrast, on the square lattice, the $\\pi$-flux state has already \nsignificant antiferromagnetic correlations built in it).\n\nThe presence of low-energy spectral weight at the corners of the Brillouin zone could be ascribed to the existence of critical monopole \nexcitations, as suggested by the analysis of Ref.~\\cite{song2018}. In fact, the Gutzwiller projector, which imposes single occupacy\non each lattice site, introduces temporal fluctuations of the gauge fields that are completely frozen within the non-interacting\npicture (i.e., within the unprojected wave function). Even though we cannot exclude a more conventional picture where a bound state of \nspinons is responsible for the intense signal around $K$, it is plausible that this feature originates from the existence of gauge\nfields, which emerge in the field-theoretical description of spin liquids~\\cite{savary2016}. While gauge fields are known to predominantly\ncontribute to spectral functions of specific Kitaev spin liquids with $\\mathcal{Z}_2$ magnetic fluxes~\\cite{knolle2014}, our calculations \nsuggest that monopole excitations may give some relevant signature in the spin-liquid phase of the $J_1-J_2$ Heisenberg model on the \ntriangular lattice. Remarkably, on the $30 \\times 30$ cluster, the lowest-energy excitation at $K$ is slightly higher inside the \nspin-liquid phase (i.e., for $J_2\/J_1=0.125$) than close to the critical point (i.e., for $J_2\/J_1 \\approx 0.08$), see Figs.~\\ref{fig:007}\nand~\\ref{fig:125}. This fact may suggest the possibility that this kind of excitation may be slightly gapped in the spin-liquid region, \nwhile being gapless at the critical point. We finally highlight the existence of an unexpected high-energy dispersing mode, which bends \nfrom the $\\Gamma$ point down into the continuum, being seemingly connected to the low-energy excitation at $K$. A comparison with other \nnumerical techniques will be needed to clarify whether this feature is a genuine aspect of the model or an artifact of the present \nvariational approach.\n\n\\section{Conclusions}\n\nIn this work, we performed variational Monte Carlo calculations to estimate the dynamical structure factor of the $J_1-J_2$\nHeisenberg model on the triangular lattice. The results for $J_2=0$ are consistent with the existence of a well-defined magnon branch\nin the whole Brillouin zone, in agreement with recent DMRG calculations~\\cite{verresen2018}. This outcome contrasts the \nsemiclassical predictions~\\cite{chernyshev2006,chernyshev2009}, which suggested the presence of magnon decay in a large portion \nof the Brillouin zone. When a finite $J_2$ super-exchange is included and the spin-liquid phase is approached, a clear softening of\nthe spectrum is detected around the $M$ points, in close similarity to what happens on the square lattice~\\cite{ferrari2018b}. \nRemarkably, the low-energy physics of the spin liquid phase cannot be fully described by the unprojected spinon picture, since, \nbesides gapless excitations at $M$ and $M^\\prime$, there are anomalously low-energy states appearing around the $K$ points.  \nOur numerical calculations provide an indisputable evidence of the fact that the non-interacting (i.e., unprojected) spinon spectrum\nis not sufficient to fully explain the low-energy spectrum detected by the dynamical structure factor. In light of the recent \nfield-theoretical analysis~\\cite{song2018}, the natural interpretation of the spectral features around the corners of the Brillouin\nzone comes from the existence of low-energy monopole excitations. This outcome is particularly important, since it would give a \ndirect signature of the fact that these theoretical approaches correctly capture the nature of the spin-liquid phase. \nWe hope that the present results will motivate future investigations in this direction.\n\n\\acknowledgements\nWe are particularly indebted to T. Li, for pointing out interesting aspects of the problem, and A. Chernyshev, for highlighting some \naspects of his results. We also acknowledge C. Batista, Y.-C. He, A. Parola, F. Pollmann, R. Verresen, and A. Vishwanath for useful \ndiscussions.\n\n\\bibliographystyle{apsrev4-1}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nUnconventional superconductors usually refer to the superconductors\nthose can not be understood within the conventional\nBardeen-Cooper-Schrieffer (BCS) theory. Notable examples are\nhigh-$T_c$ cuprates \\cite{Lee06}, heavy fermion superconductors\n\\cite{Stewart84, Lohneysen07, Stockert12}, organic superconductors\n\\cite{Powell11} and iron based superconductors \\cite{Paglione10,\nHirschfeld11, Stewart11, Chubukov12}. The unconventional\nsuperconductivity is usually driven by strong electron-electron\ninteraction, and the electron pairing mechanism usually has a\nmagnetic origin. In the past decades, identifying the precise gap\nsymmetry of unconventional superconductors has attracted great\ntheoretical and experimental efforts since such efforts may lead to\nimportant progress in seeking the microscopic pairing mechanism.\n\n\nMany unconventional superconductors are believed to have a $d$-wave\nenergy gap, which is different from that of isotropic $s$-wave\nsuperconductors. However, it is not an easy task to determine the\nprecise $d$-wave gap symmetry. A powerful and frequently used\napproach is to probe the angular dependence of various observable\nquantities, such as upper critical field \\cite{Won94, Takanaka95,\nKoike96, Naito01, Metlushko97, Won04, Weickert06, Vieyra11},\nspecific heat \\cite{Vorontsov07a, Vorontsov10, An10, Kittaka12}, and\nthermal conductivity \\cite{Vorontsov10,Vorontsov07b,Kasahara08,\nKasahara09}. In this paper, we are mainly interested in the\nbehaviors of in-plane upper critical field $H_{c2}$ in heavy fermion\nsuperconductors. This issue has recently been addressed with the aim\nto identify the precise gap symmetry of some heavy fermion\ncompounds, especially CeCoIn$_5$ \\cite{Weickert06} and\nCeCu$_{2}$Si$_2$ \\cite{Vieyra11}. Despite the intensive theoretical\nand experimental efforts, it remains unclear whether the gap symmetry\nof these compounds is $d_{x^2-y^2}$-wave\nor $d_{xy}$-wave. These two gaps are different from\neach other primarily in the positions of gap nodes. In principle,\ntheir positions can be clarified by measuring the angular dependence\nof $H_{c2}$. Unfortunately, experimental studies have not yet reached\na consensus on this issue. In the case of CeCoIn$_{5}$, currently most of\nexperiments suggest that the gap symmetry should be $d_{x^2-y^2}$-wave\\cite{An10,Allan13,Zhou13},\nhowever there is currently still an experimental discrepancy in the\nconcrete angular dependence of $H_{c2}$ in CeCoIn$_{5}$: some experiments find that the maxima of\n$H_{c2}$ are along the [100] direction \\cite{Settai01, Bianchi03,\nWeickert06}, whereas other experiment observes the maxima along the\n[110] direction \\cite{Murphy02}. This discrepancy is still a open puzzle which need to\nbe resolved\\cite{Das13}. In the case of\nCeCu$_{2}$Si$_{2}$, many earlier experiments suggest a $d_{x^2 -\ny^2}$-wave gap\\cite{Stockert08, Eremin08}. Nevertheless, a recent\nmeasurement \\cite{Vieyra11} observes the maxima of $H_{c2}$ along\nthe $[100]$ direction, which is argued to infer a $d_{xy}$-wave gap according the\ncorresponding theoretical analysis\\cite{Vieyra11}. Apparently, more research efforts are\ncalled for to solve these puzzles, which have motivated us to revisit\nthis issue more systematically.\n\n\nNow suppose an external magnetic field is introduced to a\nsuperconductor. In principle, this field can couple to the charge\nand spin degrees of freedom of the electrons via the orbital and\nZeeman mechanisms respectively. The former mechanism is described by\nthe minimal coupling between the momentum of electrons and the\nvector potential, and can lead to the well-known Abrikosov mixed\nstate in type-II superconductors. The latter mechanism, usually\ncalled Pauli paramagnetic or Pauli limiting effect, is known to be\nimportant in some heavy fermion compounds \\cite{Vieyra11, Bianchi02,\nBianchi08, Kenzelmann08}. Which one of these two effects plays a\ndominant role is determined by a number of physical factors. When\nboth of them are important, novel and interesting properties may\nemerge.\n\n\nSince the middle of 1990s, the in-plane $H_{c2}$ has been applied to\nidentify the gap symmetry in layered unconventional superconductors\n\\cite{Won94, Takanaka95, Koike96, Naito01, Metlushko97, Won04,\nWeickert06, Vieyra11}. Early theoretical calculations have showed\nthat the in-plane $H_{c2}$ exhibits a fourfold oscillation in\n$d$-wave superconductors \\cite{Won94, Takanaka95}. The presence of\nsuch a fourfold oscillation has already been verified in many\nunconventional superconductors, including high-$T_c$ cuprate superconductors\n\\cite{Koike96, Naito01}, LuNi$_{2}$B$_{2}$C \\cite{Metlushko97},\nheavy fermion compounds CeCoIn$_5$ \\cite{Weickert06} and\nCeCu$_{2}$Si$_2$ \\cite{Vieyra11}.\n\n\n\n\\begin{figure}[htbp]\n\\center \\subfigure{\n\\includegraphics[width=3in]{dx2y2.eps}}\n\\\\\n\\vspace{-0.5cm} \\subfigure{\n\\includegraphics[width=3in]{dxy.eps}}\n\\caption{Shapes of $d_{x^2-y^2}$-wave and $d_{xy}$-wave gaps.}\n\\vspace{-0.5cm} \\label{Fig:Shaped}\n\\end{figure}\n\n\n\nIn the early calculations of Won \\emph{et. al.} \\cite{Won94} and\nTakanaka \\emph{et. al.} \\cite{Takanaka95} who solely considered the\norbital effect, $H_{c2}$ is found to exhibit its maxima along the\nantinodal directions where the $d$-wave superconducting gap is\nmaximal. The subsequent analysis of Weickert \\emph{et. al.}\n\\cite{Weickert06} includes both the orbital and Pauli paramagnetic\neffects, but still finds the maxima of $H_{c2}$ along the antinodal\ndirections. A similar conclusion is drawn in a recent work\n\\cite{Vorontsov10}, where the authors also show that increasing the\nPauli effect reduces the difference in $H_{c2}$ between nodal and\nantinodal directions. There seems to be a priori hypothesis in the\nliterature that a larger gap necessarily leads to a larger magnitude\nof $H_{c2}$, which means $H_{c2}$ and $d$-wave gap should always\nhave their maxima and minima at exactly the same azimuthal angles.\nIf such a hypothesis is correct, it would be straightforward to\nidentify the precise gap symmetry. For instance, if the\nexperimentally observed $H_{c2}$ displays its maxima along the [100]\ndirection, the gap possesses a $d_{x^2-y^2}$ symmetry. On the other\nhand, if the maxima are observed along the direction [110], the gap\nsymmetry should be $d_{xy}$-wave. To make a comparison, we show the\nangular dependence of $d_{x^2-y^2}$- and $d_{xy}$-wave gaps in\nFig.~\\ref{Fig:Shaped}.\n\n\nIt is necessary to emphasize that the above hypothesized connection\nbetween in-plane $H_{c2}$ and $d$-wave gap, though intuitively\nreasonable, is actually not always correct. When there is only\norbital effect, the maxima of $H_{c2}$ and $d$-wave gap are along the\nsame directions in all cases. In the presence of Pauli paramagnetic\neffect, however, there is indeed no guarantee that such a connection\nis valid. In order to clarify the detailed connection between the\nprecise gap symmetry and the angular dependence of $H_{c2}$, we will\nconsider the influence of the interplay of orbital and Pauli effects\non $H_{c2}$ more systematically. This problem is important because\nin-plane $H_{c2}$ has recently played a significant role in the\ndetermination of the gap symmetries of CeCoIn$_{5}$ and CeCu$_{2}$Si$_{2}$.\n\n\n\n\n\n\nIn this paper, motivated by the recent progress and the existing\ncontroversy, we analyze the angular dependence of in-plane $H_{c2}$\nand its connection with the $d$-wave gap symmetry by considering the\ninterplay of orbital and Pauli effects in the contexts of heavy\nfermion compounds. After carrying out systematical calculations, we\nwill show that the maxima of angle-dependent $H_{c2}(\\theta)$ are\nnot always along the antinodal directions when both the orbital and\nPauli effects are important. The concrete fourfold oscillation\npattern of $H_{c2}(\\theta)$ is determined by a number of physical\nparameters, including temperature $T$, critical temperature $T_{c}$,\ngyromagnetic ratio $g$, fermion velocity $v_0$, and two parameters\nthat characterize the shape of the underlying Fermi surface. Each of\nthese parameters can strongly affect the angular dependence of\n$H_{c2}$. Among the above six relevant parameters, the temperature $T$ is\nparticularly interesting, due to that in any given compound $t$ is the\nonly free parameter and all the other parameters are fixed at\ncertain values. If we vary temperature $T$ but fix all the\nrest parameters, $H_{c2}(\\theta)$ is found to exhibit its maxima\nalong the nodal directions at lower temperatures and along the\nantinodal directions at higher temperatures. This means the\nangle-dependent $H_{c2}(\\theta)$ is shifted by $\\pi\/4$ as\ntemperature increases across certain critical value.\n\n\nOur results can be used to clarify the aforementioned experimental\npuzzle about the angular dependence of in-plane $H_{c2}$. Since\n$H_{c2}(\\theta)$ shifts by $\\pi\/4$ as some of the relevant\nparameters are changed, the seemingly contradictory experimental\nresults reported in Refs.\\cite{Settai01, Bianchi03, Weickert06} may\nbe well consistent. On the other hand, since the concrete behavior\nof $H_{c2}(\\theta)$ is very sensitive to the specific values of\nseveral parameters, one should be extremely careful when judging the\ngap symmetry by measuring $H_{c2}$.\n\n\nIn Sec.\\ref{Sec:Derive}, we derive the equation for $H_{c2}$ after\nincluding both the orbital and Pauli paramagnetic effects. In\nSec.\\ref{Sec:NumResults}, we present numerical results for $H_{c2}$\nin three cases, i.e., pure orbital effect, pure Pauli paramagnetic\neffect, and interplay of both orbital and Pauli effects. We show\nthat $H_{c2}$ displays complicated angle dependence due to interplay\nof orbital and Pauli effects. In Sec.\\ref{Sec:Discussion}, we\ndiscuss the physical implications of our results and make a\ncomparison with some relevant experiments.\n\n\n\n\\section{Equation for in-plane upper critical field $H_{c2}$ \\label{Sec:Derive}}\n\n\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=3in]{FS.eps}\n\\caption{Schematic diagram for a rippled Fermi\nsurface.}\\label{Fig:FS}\n\\end{figure}\n\n\n\nHeavy fermion compounds are known to have a layered structure, which\nis analogous to cuprates. However, the inter-layer coupling is not\nas weak as that in cuprates. It is convenient to consider a rippled\ncylinder Fermi surface, schematically shown in Fig.~\\ref{Fig:FS}. The fermion\nvelocity has three components $k_{x,y,z}$, where $k_{x,y}$ denote\nthe two components in the superconducting plane. Here, we use\n$t_{c}$ to represent the inter-layer hoping parameter and $c$ the\nunit size along $z$-direction, and then write the dispersion as\n\\cite{Thalmeier05, Vorontsov07a, Vorontsov07b,Vorontsov10}\n\\begin{eqnarray}\n\\varepsilon(\\mathbf{\\mathbf{k}}) = \\frac{k_{x}^{2}+k_{y}^{2}}{2m} -\n2t_{c}\\cos(k_{z}c).\n\\end{eqnarray}\nIntroducing a constant magnetic field $H$ to the system leads to\nfruitful behaviors. For type-II superconductors, the field $H$\nweaker than lower critical field $H_{c1}$ cannot penetrate the\nsample due to the Meissner effect. As $H$ exceeds $H_{c1}$ and\nfurther increases, the superconducting pairing is gradually\ndestructed by the orbital effect. The superconductivity is entirely\nsuppressed once $H$ reaches the upper critical field $H_{c2}$, which\ncan be obtained by solving  the corresponding linearized gap\nequation. In some superconductors, the Pauli paramagnetic effect can\nalso close the gap by breaking spin singlet pairs, and may even be\nmore important than the orbital effect \\cite{Weickert06, Vieyra11}.\nIn order to make a general analysis, we consider both of these two\neffects in the following.\n\n\nTo proceed, it is useful to rewrite the in-plane magnetic field\n$\\mathbf{H}$ in terms of a vector potential $\\mathbf{A}$. Let us\nchoose the $a$-axis as $x$-coordinate and $b$-axis as\n$y$-coordinate, and then write down a vector potential\n\\begin{eqnarray}\n\\mathbf{A} = \\left(0,0,H(-x\\sin\\theta+y\\cos\\theta)\\right),\n\\end{eqnarray}\nwhere $\\theta$ denotes the angle between a-axis and field\n$\\mathbf{H}$. For conventional $s$-wave superconductors, the gap is\nisotropic and the upper critical field $H_{c2}$ is certainly\n$\\theta$-independent. In the case of $d$-wave superconductors,\nhowever, the gap is strongly anisotropic, thus $H_{c2}$ becomes\n$\\theta$-dependent. Now the field $\\mathbf{H}$ takes the form\n\\begin{eqnarray}\n\\mathbf{H} &=& \\mathbf{\\nabla}\\times\\mathbf{A} =\n\\left(H\\cos\\theta,H\\sin\\theta,0\\right).\n\\end{eqnarray}\nOne can write the generalized derivative operator as\n\\begin{eqnarray}\n\\mathbf{\\Pi}(\\mathbf{R}) &=& -i\\mathbf{\\nabla}_{\\mathbf{R}} +\n2e\\mathbf{A}(\\mathbf{R}) \\nonumber \\\\\n&=& -i\\partial_{x}\\mathbf{e}_{x} - i\\partial_{y}\\mathbf{e}_{y}\n\\nonumber \\\\\n&& + \\left(-i\\partial_{z} + 2eH\\left(-x\\sin\\theta +\ny\\cos\\theta\\right)\\right)\\mathbf{e}_{z}. \\nonumber\n\\end{eqnarray}\n\n\nFollowing the general methods presented in Refs.~\\cite{Helfand66,\nWerthamer66, Scharnberg80, Lukyanchuk87, Shimahara96,\nShimahara97,Suginishi06,Shimahara09}, we obtain the following\nlinearized gap equation:\n\\begin{eqnarray}\n-\\ln(\\frac{T}{T_{c}})\\Delta(\\mathbf{R}) &=&\n\\int_{0}^{+\\infty}d\\eta\\frac{\\pi T}{\\sinh(\\pi T\\eta)}\n\\int_{-\\pi}^{\\pi}\\frac{d\\chi}{2\\pi}\\int_{0}^{2\\pi}\\frac{d\\varphi}{2\\pi}\n\\nonumber \\\\\n&& \\times \\gamma_{\\alpha}^2(\\hat{\\mathbf{k}}) \\left\\{1-\n\\cos\\left[\\eta\\left(h'+\\frac{1}{2} \\mathbf{v}_{F}(\\hat{\\mathbf{k}})\n\\right.\\right.\\right. \\nonumber \\\\\n&&\\left.\\left.\\left.\\cdot \\mathbf{\\Pi}(\\mathbf{R})\\right)\\right]\n\\right\\}\\Delta(\\mathbf{R}), \\label{eqn:GapL}\n\\end{eqnarray}\nwhere $\\chi = k_{z}c$. The function $\\Delta(\\mathbf{R})$ is\n\\begin{eqnarray}\n\\Delta(\\mathbf{R}) = \\left(\\frac{2eH}{\\pi}\\right)^{\\frac{1}{4}}\ne^{-eH\\left(x\\sin\\theta-y\\cos\\theta\\right)^{2}}.\n\\end{eqnarray}\nHere we do not include Landau level mixing \\cite{Weickert06,\nVieyra11, Lukyanchuk87} for simplicity, which will not affect our\nconclusion. For the chosen Fermi surface, the Fermi velocity vector\nis \\cite{Thalmeier05}\n\\begin{eqnarray}\n\\mathbf{v}_{F}(\\hat{\\mathbf{k}})=v_{a}\\cos\\varphi \\mathbf{e}_{x}\n+v_{a}\\sin\\varphi \\mathbf{e}_{y}+v_{c}\\sin\\chi \\mathbf{e}_{z}.\n\\label{eqn:FermionV}\n\\end{eqnarray}\nThe Fermi velocity component along the $c$-axis is $v_{c} =\n2t_{c}c$. The two-component in-plane velocity vector has a constant\nmagnitude $v_{a}$, defined as $v_{a} = v_{0} \\sqrt{1 +\n\\lambda\\cos(\\chi)}$, where $v_{0} = \\frac{k_{F0}}{m}$ with the Fermi\nmomentum $k_{F0}$ being related to the Fermi energy $\\epsilon_{F}$\nby $k_{F0} = \\sqrt{2m\\epsilon_{F}}$. The shape of rippled cylinder\nFermi surface is characterized by a velocity ratio $v_{c}\/v_{0} =\n\\lambda\\gamma$, where $\\lambda = 2t_{c}\/\\epsilon_{F}$ and $\\gamma =\nck_{F0}\/2$. As will shown below, both $\\lambda$ and $\\gamma$ can\nstrongly affect the behavior of $H_{c2}$. Moreover, we define $h' =\n-\\frac{g\\mu_{B}H}{2}$, where $\\mu_B$ is Bohr magneton and $g$ is the\ngyromagnetic ratio. The orbital effect of magnetic field is\nreflected in the factor $\\mathbf{v}_{F}(\\mathbf{k}) \\cdot\n\\Pi(\\mathbf{R})$, whereas the Pauli paramagnetic effect is reflected\nin the factor $h'$. The concrete behavior of $H_{c2}$ is determined\nby the interplay of these two effects.\n\n\nIn Eq.(\\ref{eqn:GapL}), the influence of gap symmetry is reflected\nin the function $\\gamma_{\\alpha}(\\mathbf{k})$. For isotropic\n$s$-wave pairing, $\\gamma_{s}(\\hat{\\mathbf{k}}) = 1$; for $d_{x^2 -\ny^2}$-wave pairing, $\\gamma_{d}(\\hat{\\mathbf{k}}) =\n\\sqrt{2}\\cos(2\\varphi)$; for $d_{xy}$-wave pairing,\n$\\gamma_{d}(\\hat{\\mathbf{k}}) = \\sqrt{2}\\sin(2\\varphi)$.\n\n\n\n\\begin{figure}[htbp]\n\\includegraphics[width=3.1in]{Hc2tNP.eps}\n\\caption{Fourfold oscillation of $\\theta$-dependent $H_{c2}$ at two\nrepresentative temperatures $t = 0.1$ and $t = 0.9$.}\n\\label{Fig:Hc2tNP}\n\\end{figure}\n\n\n\nAlthough the linearized gap equation Eq.(4) is formally general and\nvalid in many superconductors, its solution is determined by a\nnumber of physical effects and associated parameters. For instance,\nthe behavior of $H_{c2}$ may be strongly influenced by the concrete\nshapes of the Fermi surface. The Fermi surface has different spatial\ndependence in various superconductors, which naturally leads to\ndifferent forms of fermion dispersion and Fermi velocity\n$\\mathbf{v}_{F}$. Such a difference certainly affects the equation\nof $H_{c2}$. For spherical Fermi surface, Fermi velocity\n$\\mathbf{v}_{F}$ depends on the azimuthal angle $\\varphi$ within the\nbasal plane and the angle between $z$-axis and $\\mathbf{v}_{F}$.\nTherefore, the equation of $H_{c2}$ contains the integrations over\nthese two variables \\cite{Shimahara96, Suginishi06, Shimahara09}.\nFor cylindrical Fermi surface, the direction of vector\n$\\mathbf{v}_{F}$ solely depends on the azimuthal angle $\\varphi$, so\nthere is only the integration over angle $\\varphi$ in the equation\nof $H_{c2}$ \\cite{Shimahara97}. For rippled Fermi surface, the\ndirection of $\\mathbf{v}_{F}$ depends on the azimuthal angle\n$\\varphi$ and the coordinate $\\chi$ along $z$-axis, then the\nintegrations over $\\varphi$ and $\\chi$ enter into the equation of\n$H_{c2}$, as shown in Eq.~(\\ref{eqn:GapL}). In addition, there are\ntwo independent parameters $\\lambda$ and $\\gamma$ which can\ncharacterize the rippled Fermi surface in Eq.~(\\ref{eqn:GapL}).\nNotice that once $\\lambda = 0$, the rippled cylindrical Fermi\nsurface reduces to the cylindrical Fermi surface. The influence of\nFermi surface on $H_{c2}$ is rarely studied in the literature. In\nthis paper, we adopt rippled Fermi surface and show that\n$H_{c2}$ can exhibit different behaviors under different parameters.\n\n\n\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=3.1in]{Hc2tCPNP.eps}\n\\caption{$t$-dependence of $H_{c2}$ with $T_{c}=1K$, $v_{0} =\n3000m\/s$, $\\lambda = 0.5$, and $\\gamma=1$.} \\label{Fig:Hc2tCPNP}\n\\end{figure}\n\n\n\nTo facilitate analytical computation, we can choose the direction of\nfield $\\mathbf{H}$ as a new $z'$-axis and define\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\mathbf{e}_{x}'=\\mathbf{e}_{x}\\sin\\theta-\\mathbf{e}_{y}\\cos\\theta\n\\\\\n\\mathbf{e}_{y}'=-\\mathbf{e}_{z}\n\\\\\n\\mathbf{e}_{z}'=\\mathbf{e}_{x}\\cos\\theta+\\mathbf{e}_{y}\\sin\\theta\n\\end{array}\\right..\n\\end{eqnarray}\nIn the coordinate frame spanned by $(\\mathbf{e}_{x}',\n\\mathbf{e}_{y}', \\mathbf{e}_{z}')$, we have a new velocity vector\n\\begin{eqnarray}\n\\mathbf{v}_{F}(\\hat{\\mathbf{k}}) &=& v_{a}\\sin(\\theta-\\varphi)\n\\mathbf{e}_{x}' - v_{c}\\sin(\\chi)\\mathbf{e}_{y}' \\nonumber \\\\\n&& + v_{a}\\cos(\\theta-\\varphi)\\mathbf{e}_{z}',\\label{eqn:FermionV2}\n\\end{eqnarray}\nand a new generalized derivative operator\n\\begin{eqnarray}\n\\mathbf{\\Pi}(\\mathbf{R}) &=& \\sqrt{eH}\n\\left[\\left(a_{+}+a_{-}\\right)\\mathbf{e}_{x}' -\ni\\left(a_{+}-a_{-}\\right)\\mathbf{e}_{y}' \\right.\\nonumber\n\\\\\n&&\\left.+\\sqrt{2}a_{0}\\mathbf{e}_{z}'\\right], \\label{eqn:PiDef}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\na_{\\pm} &=& \\frac{1}{2\\sqrt{eH}}\\left[-i\\sin\\theta\\partial_{x} +\ni\\cos\\theta\\partial_{y} \\mp \\partial_{z}\\right. \\nonumber \\\\\n&&\\left.\\pm 2ieH(x\\sin\\theta-y\\cos\\theta)\\right], \\\\\na_{0} &=& \\frac{1}{\\sqrt{2eH}}\\left[-i\\partial_{x}\\cos\\theta -\ni\\partial_{y}\\sin\\theta \\right],\n\\end{eqnarray}\nwhich satisfy\n\\begin{eqnarray}\n[a_{-},a_{+}] = 1, [a_{\\pm},a_{0}] = 0.\n\\end{eqnarray}\n\nIn the following analysis, we take $d_{x^2-y^2}$-wave pairing as an\nexample and assume that $\\gamma_{d}(\\hat{\\mathbf{k}}) =\n\\sqrt{2}\\cos(2\\varphi)$. The results in the case of $d_{xy}$-wave\npairing can be obtained analogously. It is easy to examine that the\nqualitative conclusion will be not changed. After averaging over the\nground state $\\Delta_{0}(\\mathbf{R})$ on both sides of\nEq.(\\ref{eqn:GapL}) and inserting the $d_{x^2-y^2}$-wave gap\n$\\gamma_{d}(\\hat{\\mathbf{k}}) = \\sqrt{2}\\cos(2\\varphi)$, we obtain\nthe following integral equation for $H_{c2}$,\n\\begin{eqnarray}\n-\\ln t &=& \\int_{0}^{+\\infty}\\frac{du}{\\sinh\\left(u\\right)}\n\\left\\{1-\\cos\\left(hu\\right)\\int_{-\\pi}^{\\pi}\\frac{d\\chi}{2\\pi}\n\\int_{0}^{2\\pi}\\frac{d\\varphi}{2\\pi}\\right. \\nonumber \\\\\n&&\\times \\left[1+\\cos(4\\theta)\\cos(4\\varphi)\\right] \\nonumber \\\\\n&&\\times \\exp\\left[-\\rho u^2\n\\left(\\lambda^2\\gamma^2\\sin^{2}(\\chi)\\right.\\right. \\nonumber \\\\\n&&\\left.\\left.\\left.+\\left(1+\\lambda\\cos(\\chi)\\right)\\sin^{2}(\\varphi)\n\\right)\\right] \\right\\},\\label{eqn:Hc2Expression}\n\\end{eqnarray}\nwhere $t=\\frac{T}{T_{c}}$, $h=\\frac{g\\mu_{B}H_{c2}}{2\\pi k_{B}T}$\nand $\\rho=\\frac{v_{0}^{2}eH_{c2}}{8\\pi^2 k_{B}^{2}T^2}$. One can\nanalyze the detailed behavior of $H_{c2}$, especially its dependence\non various physical parameters, systematically by solving this\nintegral equation. This will be done in the next section.\n\n\n\n\\begin{figure}[htbp]\n\\includegraphics[width=3.6in]{PauliOnly.eps}\n\\caption{$t$-dependence of $H_{c2}$ with $T_{c}=1K$, $\\lambda=0.5$,\n$\\gamma = 1$, and $g = 1$.} \\label{Fig:PauliOnly}\n\\end{figure}\n\n\n\n\\section{Numerical results of $H_{c2}$ and physical implications\n\\label{Sec:NumResults}}\n\n\nIn this section, we first present the numerical solutions of\nEq.~(\\ref{eqn:Hc2Expression}), then discuss the physical\nimplications of the results, and finally compare our results with\nsome recent experiments. From Eq.~(\\ref{eqn:Hc2Expression}), we know\nthe behavior of $H_{c2}(\\theta)$ is determined by six physical\nparameters:\n\\begin{eqnarray}\n&& T_{c}: \\mbox{Zero-field critical temperature},\n\\\\\n&& t = T\/T_{c},\n\\\\\n&& v_{0} = \\sqrt{2\\epsilon_{F}\/m},\n\\\\\n&& g: \\mbox{gyromagnetic ratio},\n\\\\\n&& \\lambda = 2t_{c}\/\\epsilon_{F},\n\\\\\n&& \\gamma = k_{F0}c\/2.\n\\end{eqnarray}\nAmong this set of parameters, $\\lambda$ and $\\gamma$ are related to\nthe shape of rippled cylinder Fermi surface. We notice that the\ninfluence of these two parameters are rarely investigated in\nprevious works on $H_{c2}$. The critical temperature $T_c$ and the\ngyromagnetic ratio $g$ will be taken as varying parameters.\n\n\nThe detailed behavior of $H_{c2}$ can be clearly seen from its\nangular dependence. In addition, it is also interesting to analyze\nthe difference of $H_{c2}$ between its values obtained at $\\theta =\n45^{\\degree}$ and $\\theta = 0^{\\degree}$:\n\\begin{eqnarray}\n\\Delta H_{c2} = H_{c2}(\\theta = 45^{\\degree}) - H_{c2}(\\theta =\n0^{\\degree}),\n\\end{eqnarray}\nsince the maxima and minima of $H_{c2}$ always appear at these two\nangles. $H_{c2}$ exhibits its maxima at $\\theta = 45^{\\degree}$ if\n$\\Delta H_{c2} > 0$ and at $\\theta = 0^{\\degree}$ if $\\Delta H_{c2}\n< 0$.\n\n\nIn order to demonstrate the influence of the orbital effect and that\nof the Pauli paramagnetic effect on the angular dependence of\nin-plane $H_{c2}$, we find it helpful to consider three cases\nseparately: pure orbital effect; pure Pauli effect; interplay of\norbital and Pauli effects.\n\n\n\n\\begin{figure}[htbp]\n\\includegraphics[width=3in]{Hc2t.eps}\n\\caption{Angular dependence of $H_{c2}$ at $t = 0.1$ and $t = 0.9$\nwith $T_{c} = 1K$, $v_{0} = 3000m\/s$, $\\lambda=0.5$, $\\gamma = 1$,\nand $g=1$. The fourfold oscillation patterns are apparently\ndifferent at low and high temperatures.} \\label{Fig:Hc2t}\n\\end{figure}\n\n\n\n\\subsection{Pure orbital effect\\label{subsec:PureOrbital}}\n\n\nFirst, we consider only the orbital effect by setting the\ngyromagnetic factor $g = 0$. In this case, the factor $\\cos(hu)$\nappearing in Eq.(\\ref{eqn:Hc2Expression}) is equal to unity, $\\cos(hu) = 1$. We assume\nthat $T_c = 1K$, $v_{0} = 3000m\/s$, $\\lambda=0.5$, and $\\gamma=1$,\nwhich are suitable parameters in heavy fermion compounds.\n\n\nAfter carrying out numerical calculations, we plot the angular\ndependence of $H_{c2}(\\theta)$ in Fig.~\\ref{Fig:Hc2tNP} at two\nrepresentative temperatures $t = 0.1$ and $t = 0.9$. It is easy to\nsee from Fig.~\\ref{Fig:Hc2tNP} that $H_{c2}(\\theta)$ exhibits a\nfourfold oscillation pattern. Moreover, the maxima of $H_{c2}$ are\nalways along the antinodal directions for any values of the relevant\nparameters, which means the angular dependence of orbital\neffect-induced $H_{c2}$ is exactly the same as that of $d$-wave gap.\nThis is consistent with the original theoretical predictions of Won\n\\emph{et. al.} \\cite{Won94} and Takanaka \\emph{et. al.}\n\\cite{Takanaka95}. An important feature that needs to be emphasized\nis that the positions of peaks are temperature independent, as\nclearly manifested in both Fig.~\\ref{Fig:Hc2tNP} and\nFig.~\\ref{Fig:Hc2tCPNP}.\n\n\nThe above properties can also be elaborated by the detailed\n$t$-dependence of $H_{c2}$ and $\\Delta H_{c2}$ are presented in\nFig.~\\ref{Fig:Hc2tCPNP}. $H_{c2}$ is an monotonously decreasing\nfunction of parameter $t$, valid for all values of $\\theta$. This is\neasy to understand since the magnitude of the superconducting gap\nalways decreases monotonously with growing temperature. Moreover,\nthe difference $\\Delta H_{c2}$ is negative for all values of $t$.\n\n\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=3in]{Hc2tCP.eps}\n\\caption{$t$-dependence of $H_{c2}$ with $T_{c} = 1K$, $v_{0} =\n3000m\/s$, $\\lambda = 0.5$, $\\gamma=1$, and $g = 1$.}\n\\label{Fig:Hx2tCP}\n\\end{figure}\n\n\n\n\\subsection{Pure Pauli paramagnetic effect}\n\n\nWe next consider the effects of pure Pauli paramagnetic effect by\nsetting $v_{0}=0$, which leads to\n\\begin{eqnarray}\n-\\ln(t) = \\int_{0}^{+\\infty}du\\frac{1 - \\cos(hu)}{\\sinh(u)},\n\\end{eqnarray}\nwhich is completely independent of $\\theta$. The $t$-dependence of\n$H_{c2}$ is shown in Fig.~\\ref{Fig:PauliOnly}. Different from pure\norbital effect, $H_{c2}$ is not a monotonous function: it rises\ninitially with growing $t$, but decreases as $t$ is larger than\ncertain critical value $t_c$, which is roughly $0.5t$ under the\nchosen set of parameters.\n\n\n\n\\subsection{Interplay of orbital and Pauli effects}\n\n\nWe now turn to the general and interesting case in which both the\norbital and Pauli paramagnetic effects are important. This case is\nbroadly believed to be realized in several heavy fermion compounds,\nsuch as CeCoIn$_5$ and CeCu$_{2}$Si$_2$. As aforementioned, the\nconcrete behaviors of $\\theta$-dependent $H_{c2}$ are influenced by\na number of parameters. In order to illustrate the numerical results\nand their physical implications, we vary one particular parameter\nwhile fixing all the rest parameters. In most of the following\ncalculations, the gyromagnetic factor is taken to be $g = 1$. The\ninfluence of various values of $g$ will be analyzed separately.\n\n\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=3in]{Hc2Tc.eps}\n\\caption{$T_c$-dependence of $H_{c2}$ with $t = 0.5$, $v_{0} =\n3000m\/s$, $\\lambda=0.5$, $\\gamma=1$, and $g=1$.} \\label{Fig:Hc2Tc}\n\\end{figure}\n\n\n\nAs shown in Fig. \\ref{Fig:Hc2t}, under the currently chosen\nparameters, the maxima of $H_{c2}$ locates along the antinodal\ndirections at a relatively higher temperature $t = 0.9$. This\nbehavior is very similar to that in the case of pure orbital effect.\nHowever, at a relatively lower temperature $t = 0.1$, the maxima of\n$H_{c2}$ are along the nodal directions where the $d_{x^2 - y^2}$-wave\ngap vanishes. Two conclusions can be immediately drawn: $H_{c2}$\ndoes not always exhibit its maxima at the angles where the\nsuperconducting gap reaches its maximal value; the fourfold\noscillation curves of $H_{c2}$ is shifted by $\\pi\/4$ as temperature\ngrows in the range of $0 < T < T_c$.\n\n\nFrom Fig. \\ref{Fig:Hx2tCP}(a), we see that $H_{c2}$ first arises\nwith growing $t$ and then decreases rapidly once $t$ exceeds a\ncritical value. Apparently, such a non-monotonous $t$-dependence of\n$H_{c2}$ is a consequence of the interplay of both orbital and Pauli\nparamagnetic effects. On the other hand, the difference $\\Delta\nH_{c2}$ shown in Fig. \\ref{Fig:Hx2tCP}(b) is positive for small\nvalues of $t$ but becomes negative for larger values of $t$.\n\n\nAddition to temperature $t$, the concrete angular dependence of\n$H_{c2}$ is also strongly influenced by a number of other physical\nparameters, including critical temperature $T_c$, fermion velocity\n$v_0$, gyromagnetic factor $g$, and two Fermi surface factors\n$\\lambda$ and $\\gamma$. Indeed, different values of these parameters\ncan lead to very different behaviors of $H_{c2}$. In the following,\nwe show how $H_{c2}$ and $\\Delta H_{c2}$ are changed as these\nparameters are varying. To simplify the analysis, we vary one\nparticular parameter and fix all the other parameters in each\nfigure.\n\n\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=3in]{Hc2v0.eps}\n\\caption{$v_{0}$-dependence of $H_{c2}$ with $t=0.1$, $T_{c}=1K$,\n$\\lambda=0.5$, $\\gamma=1$, and $g=1$.} \\label{Fig:Hc2V0}\n\\end{figure}\n\n\n\n$T_c$: First, we consider the influence of critical temperature\n$T_c$ on $H_{c2}$ and $\\Delta H_{c2}$, and show the results in Fig.\n\\ref{Fig:Hc2Tc}. It is well-known that $T_c$ of heavy fermion\ncompounds is actually quite low, especially when compared with\ncuprates and iron-based superconductors. To cover all possible heavy\nfermion compounds, we assume $T_{c}$ varies in the range of\n$[0,3K]$. All the other parameters are fixed. $H_{c2}$ rises\nmonotonously with growing $T_c$, which is obviously owing to the\nmonotonous increase of the superconducting gap. Moreover, if $T_{c}$\nis smaller than some critical value, $\\Delta H_{c2}$ is negative,\nwhich means the maxima of $H_{c2}$ are along the antinodal directions.\nFor larger $T_{c}$, $\\Delta H_{c2}$ becomes positive and the maxima\nof $H_{c2}$ are shifted to the nodal directions. Apparently, $T_c$\nhas important impacts on the concrete angular dependence of\n$H_{c2}$. In passing, we point out that the maxima of $H_{c2}$ will\nbe shifted back to the antinodal directions for even higher $T_c$\n(not shown in the figure).\n\n\n$v_0$: We then consider the influence of fermion velocity $v_0$ on\n$H_{c2}$ and $\\Delta H_{c2}$, and show the results in Fig.\n\\ref{Fig:Hc2V0}. In the limit $v_{0} = 0$, the orbital effect is\nactually ignored and the Pauli effect entirely determines $H_{c2}$.\nIn such a limit, $H_{c2}$ is angle independent, so $\\Delta H_{c2} =\n0$. For finite $v_0$, $H_{c2}$ becomes angle dependent and exhibits\nfourfold oscillation, as a consequence of the interplay between\norbital and Pauli effects. As $v_0$ is growing, $H_{c2}$ first\nincreases and then decreases, which indicates that the enhancement\nof orbital effect does not necessarily suppress $H_{c2}$ once the\nPauli paramagnetic effect is present. However, as already discussed\nearlier, $H_{c2}$ deceases monotonously with growing $v_{0}$ when\nthe Pauli effect is completely neglected. Furthermore, $\\Delta\nH_{c2}$ is negative for both small and large values of $v_{0}$, but\nis positive for intermediate values of $v_{0}$. Therefore, the\nconcrete angular dependence of $H_{c2}$ is very sensitive to the\nvalues of fermion velocity.\n\n\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=3.04in]{Hc2g.eps}\n\\caption{$g$-dependence of $H_{c2}$ with $t = 0.1$, $v_{0} =\n3000m\/s$, $T_{c} = 1K$, $\\lambda = 0.5$, and $\\gamma = 1$.}\n\\label{Fig:Hc2g}\n\\end{figure}\n\n\n\n$g$: We next consider the influence of the gyromagnetic factor $g$,\nwhich characterizes the effective strength of Pauli paramagnetic\neffect. The dependence of $H_{c2}$ and $\\Delta H_{c2}$ on $g$ is\ngiven in Fig. \\ref{Fig:Hc2g}. First of all, taking $g = 0$ simply\nleads to the known results obtained in the case of pure orbital\neffect presented in Sec.\\ref{subsec:PureOrbital}. Second, $H_{c2}$\ndecreases monotonously with growing $g$. An immediate indication of\nthis behavior is that increasing the Pauli paramagnetic effect\nalways tends to suppress $H_{c2}$ in the presence of orbital effect.\nFinally, it is easy to observe that $\\Delta H_{c2}$ is negative if\n$g$ takes very small values and positive when $g$ becomes larger\nthan certain critical value. Therefore, the gyromagnetic factor $g$\nalso plays a crucial role in the determination of the concrete angle\ndependence $H_{c2}$.\n\n\n$\\lambda$: $\\lambda$ represents the ratio of inter layer coupling\n$2t_{c}$ and the Fermi energy $E_{F}$. If $t_{c}=0$, the\ncorresponding $\\lambda=0$, then the rippled cylindrical Fermi\nsurface reduce to the cylindrical Fermi surface. The dependence of\n$H_{c2}$ and $\\Delta H_{c2}$ on $\\lambda$ is as depicted  in Fig.\n\\ref{Fig:Hc2lambda}. $H_{c2}$ deceases monotonously with the growing\n$\\lambda$. For given values of other parameters shown in\nFig.~\\ref{Fig:Hc2lambda}, the maxima of $H_{c2}$ are along the nodal\ndirections for small $\\lambda$, but along the antinodal directions for\nlarge values of $\\lambda$.\n\n\n$\\gamma$: $\\gamma$ represents the ratio of two momentum $k_{F0}$ and\n$1\/2c$, $c$ is the unit cell size along third direction. The\ndependence of $H_{c2}$ and $\\Delta H_{c2}$ on $\\gamma$ is shown in\nFig.~\\ref{Fig:Hc2gamma}. $H_{c2}$ deceases monotonously with the\ngrowing $\\gamma$. For given values of other parameters shown in\nFig.~\\ref{Fig:Hc2gamma}, the maxima of $H_{c2}$ are along the nodal\ndirections for small $\\gamma$, but along the antinodal directions for\nlarge values of $\\gamma$.\n\n\n\n\\begin{figure}[htbp]\n\\center\n\\includegraphics[width=3in]{Hc2lambda.eps}\n\\caption{Variation with $\\lambda$ fixing $t=0.5$, $v_{0}=5000m\/s$,\n$T_{c}=1K$, $\\gamma=1$ and $g=1$.} \\label{Fig:Hc2lambda}\n\\end{figure}\n\n\n\nFrom all these results, we see that both the magnitudes and the\ndetailed angular dependence of in-plane $H_{c2}$ are significantly\ninfluenced by a number of physical parameters. A particularly\ninteresting feature is the fourfold oscillation pattern of angle\ndependent $H_{c2}$ can be shifted by $\\pi\/4$ if one varies any one\nof these parameters. $H_{c2}$ may exhibit its maxima along either\nnodal or antinodal directions, depending on the specific values of\nrelevant parameters, which is apparently in sharp contrast in the\nnaive notion that $H_{c2}$ always displays the same angle dependence\nof the $d$-wave superconducting gap.\n\n\n\n\\subsection{Comparison with recent experiments}\n\n\nAs aforementioned, in the last several years the in-plane $H_{c2}$\nhas been widely investigated with the aim to identify the precise\nsuperconducting gap symmetry in two heavy fermion compounds\nCeCoIn$_5$ and CeCu$_{2}$Si$_2$. In this subsection, we make a\ncomparison between our theoretical analysis and some recent\nexperiments of $H_{c2}$. Our results are valuable to theoretical and\nexperimental research of $H_{c2}$ in two main aspects.\n\n\n\n\\begin{figure}[htbp]\n\\includegraphics[width=3in]{Hc2gamma.eps}\n\\caption{$\\gamma$-dependence of $H_{c2}$ with $t = 0.5$, $v_{0} =\n3000m\/s$, $T_{c} = 1K$, $\\lambda = 0.5$, and $g = 1$.}\n\\label{Fig:Hc2gamma}\n\\end{figure}\n\n\n\nFirst, one should be very careful when fitting theoretical\ncalculations with experimental data. In the current literature, it\nis often taken for granted that the in-plane $H_{c2}$ always\nexhibits exactly the same angular dependence as that of the\nsuperconducting gap. In other words, the maxima of in-plane\n$H_{c2}$ are believed to be always along the antinodal directions\nwhere the $d$-wave gap is maximal. According to this seemingly\ncorrect relationship, the superconducting gap symmetry is simply\nidentified as $d_{x^2 - y^2}$-wave ($d_{xy}$-wave) if\n$H_{c2}(\\theta)$ is found to exhibit its maxima at $\\theta =\n0^{\\degree}$ ($45^{\\degree}$). However, as showed in our extensive\ncalculations, such a relationship is not always correct. In a\nPauli-limited \\emph{d}-wave superconductor, the maxima of $H_{c2}$\nmay be along either the nodal or the antinodal direction, depending\non the specific values of a number of physical parameters, as a\nconsequence of the delicate interplay between orbital and Pauli\neffects. Inaccurate and even incorrect conclusions might be drawn if\nsome of these parameters are not properly chosen. In order to\nidentify the precise gap symmetry of CeCoIn$_5$ or\nCeCu$_{2}$Si$_{2}$, one should first choose suitable values for all\nthe relevant parameters before probing the angular dependence of\n$H_{c2}$ and deducing the gap symmetry from experimental data.\n\n\n\n\nAmong the above six relevant parameters, the temperature $t$ is\nparticularly interesting, because in any given compound $t$ is the\nonly free parameter and all the other parameters are fixed at\ncertain values. Our extensive calculations show that there is always\na $\\pi\/4$ difference between $H_{c2}$ and $d$-wave gap at small $t$\nand that $H_{c2}$ and $d$-wave gap always exhibit exactly the same\nangular dependence once $t$ exceeds certain critical value, provided\nthat the gyromagnetic factor $g$ is sufficiently large. It appears\nthat the impact of Pauli effect on $H_{c2}$ is much more important\nat low temperatures than at high temperatures. If one attempts to\ndeduce the precise gap symmetry by fitting experiments of $H_{c2}$,\nit would be better to measure $H_{c2}$ at a series of very different\ntemperatures. Otherwise, incorrect results might be obtained.\n\n\n\nSecond, our results may help to resolve some current controversies\nwith regard to the precise gap symmetry of heavy fermion compounds.\nThe gap symmetry of CeCoIn$_{5}$ has been investigated extensively\nby means of various experimental techniques. Settai \\emph{et. al.}\n\\cite{Settai01} reported that the maxima of in-plane $H_{c2}$ are along\n[100] direction through de Haas-van Alphen oscillation signal at\n$40$mK. The cantilever magnetometer measurements at $20$mK of Murphy\n\\emph{et. al.} \\cite{Murphy02} observed that the maxima of $H_{c2}$\nare along [110] direction. Bianchi et. al. \\cite{Bianchi03} measured the\nspecific heat and found the maxima of $H_{c2}$ along [100] direction\nat temperatures higher than $1$K. After measuring the magnetic field\ndependence of electric resistivity at $100$mK, Weickert \\emph{et.\nal.}\\cite{Weickert06} revealed that the maxima of\n$H_{c2}$ are along [100] direction. Obviously, there seems to be a\ndiscrepancy among the experimental results about the detailed\nangular dependence of $H_{c2}$, which is considered as an open\npuzzle \\cite{Das13} and complicates the search for the precise gap\nsymmetry.\n\n\nAccording to our results, however, probably such a discrepancy does\nnot exist at all, since the maxima of $H_{c2}$ may be along either\n[100] or [110] direction when some of the relevant physical\nparameters are moderately changed. In particular, the maxima of\n$H_{c2}$ can shift by $\\pi\/4$ as the temperature increases beyond\ncertain critical value. Notice that the measurements of\nRef.~\\cite{Murphy02} were performed at a temperature as low as\n$20$mK. There is a good possibility that the position of the maxima\nof $H_{c2}$ are shifted from [110] direction at low temperatures to\n[100] direction at higher temperatures. Although this possibility\nneeds to be further examined, it should be safe to say that the\nseemingly contradictory experimental results about in-plane $H_{c2}$\nof CeCoIn$_{5}$ may be well consistent with each other. More careful\nand more systematical research are required to completely solve this\nproblem.\n\n\n\\section{Discussion and Conclusion\\label{Sec:Discussion}}\n\n\nIn this paper, we have performed a detailed and systematical\nanalysis of the unusual behaviors of in-plane upper critical field\n$H_{c2}$ in the contexts of Pauli-limited heavy fermion compounds.\nWe show that the concrete angular dependence of $H_{c2}$ is\ndetermined by a delicate interplay of the orbital and Pauli\nparamagnetic effects. The most interesting result is that $H_{c2}$\ndoes not necessarily exhibit the same fourfold oscillation pattern\nas the $d$-wave superconducting gap, which is often taken for\ngranted in the literature. For certain values of a series of\nphysical parameters, $H_{c2}$ may display its maxima along the nodal\ndirections where the superconducting gap vanishes. We also have\ncompared our theoretical analysis with some current measurements of\nin-plane $H_{c2}$ in two heavy fermion compounds CeCoIn$_5$ and\nCeCu$_{2}$Si$_2$.\n\n\nThe theoretical results presented in this paper impose an important\nrestraint on the determination of the precise gap symmetry of Pauli\nlimited $d$-wave heavy fermion superconductors by means of measuring\nthe in-plane $H_{c2}$. One has to be extremely careful when trying\nto deduce the gap symmetry from experiments of $H_{c2}$.\n\n\nThe authors are supported by the National Natural Science Foundation\nof China under grants No. 11074234 and No. 11274286.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt has been shown that neural networks are vulnerable to adversarial examples~\\citep{szegedy2016rethinking,goodfellow2015explaining,carlini2017towards,athalye2018obfuscated}. \nGiven a victim neural network model and a correctly classified example, an adversarial attack aims to compute a small perturbation such that with this perturbation added, the original example will be misclassified. \nMany adversarial attacks have been proposed in the literature. \nMost of them consider the white-box setting, where the attacker has full knowledge about the victim model, and thus gradient based optimization can be used for attack. Popular Examples include \nC\\&W~\\citep{carlini2017towards} and PGD~\\citep{madry2017towards} attacks. \nOn the other hand, \nsome more recent attacks have considered the probability black-box  setting where the attacker does not know the victim model's structure and weights, but can iteratively query the model and get the corresponding probability output.\nIn this setting, although gradient (of output probability to the input layer) is not computable, it can still be estimated using finite differences, and \nalgorithms many attacks are based on this~\\citep{chen2017zoo, ilyas2018black,tu2018autozoom,jun2018adversarial}.\n\nIn this paper, we consider the most challenging and practical attack setting -- hard-label black-box setting -- where the model is hidden to the attacker and the attacker can only make  queries and get the corresponding hard-label decisions (e.g., predicted labels) of the model.  A commonly used algorithm proposed in this setting, also called Boundary attack~\\citep{brendel2017decision}, is based on random walks on the decision surface, but it does not have any convergence guarantee. More recently, \\cite{cheng2018queryefficient} showed that finding the minimum adversarial perturbation in the hard-label setting can be reformulated as  another optimization problem (we call this Cheng's formulation in this paper). This new formulation enjoys the benefit of having a smooth boundary in most tasks and the function value is computable using hard-label queries. Therefore, the authors of~\\citep{cheng2018queryefficient}\nare able to use standard zeroth order optimization to solve the new formulation.  Although their algorithm converges quickly, it still requires large number of queries (e.g., 20,000) for attacking a single image since every function evaluation of Cheng's formulation has to be computed using binary search requiring tens of queries. \n\nIn this paper, we follow the same optimization formulation of~\\citep{cheng2018queryefficient} which has the advantage of smoothness, but instead of using finite differences to estimate the magnitude of directional derivative,\nwe propose to evaluate its sign using only {\\bf a single query}. With this single-query sign oracle, we design novel algorithms for solving the Cheng's formulation, and we theoretically prove and empirically demonstrate the significant reduction in the number of queries required for hard-label black box attack. \n\nOur contribution are summarized below: \n\\begin{compactitem}\n \\item  {\\bf Novelty in terms of adversarial attack.} We elucidate an efficient approach to compute the sign of directional derivative of Cheng's formulation using a single query, and based on this technique we develop a novel optimization algorithm called Sign-OPT for hard-label black-box attack.  \n    \\item {\\bf Novelty in terms of optimization.}\n Our method \ncan be viewed as a new zeroth order optimization algorithm that features fast\nconvergence of signSGD. Instead of directly taking the sign of gradient estimation, our algorithm utilizes the scale of\nrandom direction. This make existing analysis inappropriate to our case, and we provide a new recipe to prove the convergence of this new optimizer. \n    \n    \n    \\item We conduct comprehensive experiments on several datasets and models. We show that the proposed algorithm consistently reduces the query count by 5--10 times across different models and datasets, suggesting a practical and query-efficient robustness evaluation tool. Furthermore, on most datasets our algorithm can find an adversarial example with smaller distortion compared with previous approaches. \n\\end{compactitem}\n\n\\section{Related Work}\n\\label{sec:related}\n\\vspace{-5pt}\n\\paragraph{White-box attack}\nSince it was firstly found that neural networks are easy to be fooled by adversarial examples \\citep{goodfellow2015explaining}, a lot of work has been proposed in the white-box attack setting, where the classifier $f$ is completely exposed to the attacker. For neural networks, under this assumption, back-propagation can be conducted on the target model because both network structure and weights are known by the attacker. \nAlgorithms including \\citep{goodfellow2015explaining, kurakin2016adversarial, carlini2017towards, chen2018ead, madry2017towards} are then proposed based on gradient computation. \nRecently, the BPDA attack introduced by \\cite{athalye2018obfuscated} bypasses some models with obfuscated gradients and is shown to successfully circumvent many defenses. In addition to typical attacks based on small $\\ell_p$ norm perturbation, non-$\\ell_p$ norm perturbations such as scaling or shifting have also been considered~\\citep{zhang2019limitations}.\n\n\\paragraph{Black-box attack}\nRecently, black-box setting is drawing rapidly increasing attention. In black-box setting, the attacker can query the model but\nhas no (direct) access to any internal information inside the model. Although there are some works based on transfer attack \\citep{papernot2017practical}, we consider the query-based attack in the paper.  Depending on the model's feedback for a given query, an attack can be classified as a soft-label or hard-label attack. In the soft-label setting, the model outputs a probability score for each decision.\n\\cite{chen2017zoo} uses a finite difference in a coordinate-wise manner to approximately estimate the output probability changes and does a coordinate descent to conduct the attack. \\cite{ilyas2018black} uses Neural evolution strategy (NES) to approximately estimate the gradient directly. Later, some variants \\citep{ilyas2018prior,tu2018autozoom} were proposed to utilize the side information to further speed up the attack procedure. \\cite{alzantot2019genattack} uses a evolutionary algorithm as a black-box optimizer for the soft-label setting. \nRecently, \\cite{al2019there} proposes SignHunter algorithm based on signSGD \\citep{pmlr-v80-bernstein18a} to achieve faster convergence in the soft-label setting.\nThe recent work  \\citep{al2019there} proposes SignHunter algorithm to achieve a more query-efficent sign estimate when crafting black-box adversarial examples through soft-label information.\n\nIn the hard-label case, only the final decision, i.e. the top-1 predicted class, is observed. As a result, the attacker can only make queries to acquire the corresponding hard-label decision instead of the probability outputs. \\cite{brendel2017decision} first studied this problem and proposed an algorithm based on random walks near the decision boundary. By selecting a random direction and projecting it onto a boundary sphere in each iteration, it aims to generate a high-quality adversarial example.  Query-Limited attack~\\citep{ilyas2018black} tries to estimate the output probability scores with model query and turn the hard-label into a soft-label problem. \\cite{cheng2018queryefficient} instead re-formalizes the hard-label attack into an optimization problem that finds a direction which could produce the shortest distance to decision boundary.\n\nThe recent arXiv paper \\citep{chen2019boundary} applied the zeroth-order sign oracle to improve Boundary attack, and also demonstrated significant improvement. The major differences to our algorithm are that we propose a new zeroth-order gradient descent algorithm,  provide its algorithmic convergence guarantees, and aim to improve the query complexity of the attack formulation proposed in~\\citep{cheng2018queryefficient}.  For completeness, we also compare with this method in Section \\ref{ssec:hsja-sign-comp}. \nMoreover, \\citep{chen2019boundary} \n uses one-point gradient estimate, which is unbiased but may encounter larger variance compared with the gradient estimate in our paper. Thus, we can observe in Section \\ref{ssec:hsja-sign-comp} that although they are slightly faster in the initial stage, Sign-OPT will catch up and eventually lead to a slightly better solution. \n\n\n\n\\section{Proposed Method}\n\\label{sec:proposed}\n\nWe follow the same formulation in\n\\citep{cheng2018queryefficient} and consider the hard-label attack as the problem of finding the direction with shortest distance to the decision boundary. \nSpecifically, for a given example $\\bx_0$, true label $y_0$ and the hard-label black-box function $f: \\RR^d \\rightarrow \\{1, \\dots, K\\}$, the objective function $g: \\RR^d \\rightarrow \\RR$ (for the untargeted attack) can be written as:\n\\begin{equation}\n    \\min_{\\btheta} g(\\btheta) \\ \\text{ where } \\ g(\\btheta) = \\arg\\min_{\\lambda>0} \\bigg( f(x_0+\\lambda\\frac{\\btheta}{\\|\\btheta\\|})\\neq y_0 \\bigg).\n    \\label{eq:g_theta}\n\\end{equation}\nIt has been shown that this objective function is usually smooth and the objective function $g$ can be evaluated by a binary search procedure locally. At each binary search step, we query the function $f(\\bx_0 + \\lambda \\frac{\\btheta}{\\|\\btheta\\|})$ and determine whether the distance to decision boundary in the direction $\\btheta$ is greater or smaller than $\\lambda$\nbased on the hard-label prediction\\footnote{Note that binary search only works in a small local region; in more general case $g(\\btheta)$  has to be computed by a fine-grained search plus binary search, as discussed in  \\cite{cheng2018queryefficient}.}. \n\nAs the objective function is computable, the directional derivative of $g$ can be estimated by finite differences: \n\\begin{align}\n    \\hat{\\nabla} g(\\btheta;\\ub):= \\frac{g(\\btheta+\\epsilon \\ub)-g(\\btheta)}{\\epsilon}\\ub \n    \\label{eq:grad_test}\n\\end{align}\nwhere $\\bu$ is a random Gaussian vector and $\\epsilon > 0$ is a very small smoothing parameter. This is a standard zeroth order oracle for estimating directional derivative and based on this we can apply many different zeroth order optimization algorithms to minimize $g$. \n\\begin{wrapfigure}{r}{0.4\\textwidth}\n    \\centering\n    \\includegraphics[width=0.3\\textwidth]{figures\/illu.pdf}\n    \\caption{Illustration}\n    \\label{fig:illustration}\n\\end{wrapfigure}\nFor example, \\cite{cheng2018queryefficient} used the Random Derivative Free algorithm~\\cite{nesterov2017random} to solve problem~\\eqref{eq:g_theta}. \nHowever, each computation of  \\eqref{eq:grad_test} \nrequires many hard-label queries due to binary search, so  \\cite{cheng2018queryefficient} still requires a huge number of queries despite having fast convergence.\n\nIn this work, we introduce an algorithm that hugely improves the query complexity over~\\cite{cheng2018queryefficient}. Our algorithm is based on the following key ideas: (i) one does not need very accurate values of directional derivative in order to make the algorithm converge, and (ii) there exists an {\\bf imperfect but informative estimation} of directional derivative of $g$ that can be computed by a single query. \n\n\\subsection{A single query oracle}\n\n\n\n\nAs mentioned before, the previous approach requires computing $g(\\btheta+\\epsilon \\bu) - g(\\btheta)$ which consumes a lot of queries. \nHowever, based on the definition of $g(\\cdot)$, \nwe can compute the sign of this value $\\text{sign}(g(\\btheta + \\epsilon \\bu)-g(\\btheta))$ \nusing a single query. Considering the untargeted attack case, the sign can be computed by\n\\begin{equation}\n\\text{sign}(g(\\btheta+\\epsilon \\ub)-g(\\btheta))=\n\\begin{cases}\n+1, & f(x_0 + g(\\btheta)\\frac{(\\btheta+\\epsilon \\ub)}{\\|\\btheta+\\epsilon \\ub\\|})=y_0 ,\\\\\n-1, & \\text{Otherwise.}\n\\end{cases}   \n\\label{eq:est_sign}\n \\end{equation}\nThis is illustrated in Figure \\ref{fig:illustration}. \nEssentially, for a new direction $\\btheta+\\epsilon\\ub$, we test whether a point at the original distance $g(\\btheta)$ from $x_0$ in this direction lies inside or outside the decision boundary,\ni.e. if the produced perturbation will result in a wrong prediction by classifier. If the produced perturbation is outside the boundary i.e. $f(x_0 + g(\\btheta)\\frac{(\\btheta+\\epsilon \\ub)}{\\|\\btheta+\\epsilon \\ub\\|})\\neq y_0$, the new direction has a smaller distance to decision boundary, and thus giving a smaller value of $g$. It indicates that $\\ub$ is a descent direction to minimize $g$. \n\\subsection{Sign-OPT attack}\nBy sampling random Gaussian vector $Q$ times, we can estimate the imperfect gradient by \n\\begin{equation}\n     \\hat{\\nabla} g(\\btheta)\\approx \\hat{\\bg}:= \\sum\\nolimits_{q=1}^Q\\text{sign}(g(\\btheta+\\epsilon \\ub_q) - g(\\btheta))\\ub_q,\n     \\label{eq:our_estimator}\n\\end{equation}\nwhich only requires $Q$ queries. \nWe then use this imperfect gradient estimate to update our search direction $\\btheta$ as $\\btheta \\leftarrow \\btheta-\\eta \\hat{\\gb}$ with a step size $\\eta$ and use the same search procedure to compute $g(\\btheta)$ up to a certain accuracy. The detailed procedure is shown in Algorithm \\ref{alg:signopt}.\n\n\\begin{algorithm}[t]\n\n\\SetAlgoLined\n\\textbf{Input}: Hard-label model $f$, original image $x_0$, initial $\\btheta_0$ \\;\n\\For{$t=1, 2, \\dots, T$}{\nRandomly sample $u_1, \\dots, u_Q$ from a Gaussian or Uniform distribution\\;\nCompute $\\hat{\\bg} \\leftarrow \\frac{1}{Q}\\sum_{q=1}^Q \\text{sign}(g(\\btheta_t+\\epsilon \\ub_q) - g(\\btheta_t))\\cdot \\ub_q$ \\;\nUpdate $\\btheta_{t+1} \\leftarrow \\btheta_t - \\eta \\hat{\\bg}$ \\;\nEvaluate $g(\\btheta_t)$ using the same search algorithm in \\cite{cheng2018queryefficient} \\;\n}\n\\caption{Sign-OPT attack}\n\\label{alg:signopt}\n\\end{algorithm}\nWe note that \\cite{liu2018signsgd}\ndesigned a Zeroth Order SignSGD \nalgorithm for soft-label black box attack (not hard-label setting).\nThey use $\\hat{\\nabla} g(\\btheta)\\approx \\hat{\\bg}:= \\sum_{q=1}^Q\\text{sign}(g(\\btheta+\\epsilon \\ub_q) - g(\\btheta)\\ub_q)$ and \nshows that it could achieve a comparable or even better convergence rate than zeroth order stochastic gradient descent by using only sign information of gradient estimation. \nAlthough it is possible to combine ZO-SignSGD with our proposed single query oracle for solving hard-label attack, \ntheir estimator will take sign of the whole vector and thus ignore the direction of $\\ub_q$, which leads to slower convergence in practice (please refer to Section 4.4 and Figure 5(b) for more details).\n\nTo the best of our knowledge, no previous analysis can be used to prove convergence of Algorithm~\\ref{alg:signopt}. \nIn the following, we show that Algorithm~\\ref{alg:signopt} can in fact converge and furthermore, with similar convergence rate compared with~\\citep{liu2018signsgd} despite using a different gradient estimator.\n\\begin{assumption}\nFunction $g(\\theta)$ is L-smooth with a finite value of L.\n\\end{assumption}\n\\begin{assumption}\nAt any iteration step t, the gradient of the function g is upper bounded by $\\|\\nabla g(\\btheta_t)\\|_2 \\leq \\sigma$.\n\\end{assumption}\n\\begin{theorem}\nSuppose that the conditions in the assumptions hold, and the distribution of gradient noise is\nunimodal and symmetric. Then, Sign-OPT attack with learning rate $\\eta_t = O(\\frac{1}{Q\\sqrt{dT}})$ and $\\epsilon = O(\\frac{1}{dT})$ will give following bound on $\\EE[\\|\\nabla g(\\btheta)\\|_2]$:\n\\begin{align*}\n     \\EE[\\|\\nabla g(\\btheta)\\|_2] = O(\\frac{\\sqrt{d}}{\\sqrt{T}} + \\frac{d}{\\sqrt{Q}}). \n\\end{align*}\n\\end{theorem}\nThe proof can be found in \\autoref{ssec:proof}. The main difference with the original analysis provided by~\\cite{liu2018signsgd} is that they only only deal with sign of each element, while our analysis also takes the magnitudes of each element of $\\bu_q$ into account. \n\n\n\n\\subsection{Other gradient estimations}\n\nNote that the value  $\\text{sign}(g(\\btheta+\\epsilon \\bu)-g(\\btheta))$ computed by our single query oracle is actually the sign of the directional derivative: \n\\begin{equation*}\n    \\text{sign}(\\langle \\nabla g(\\btheta), \\bu\\rangle ) = \\text{sign}(\\lim_{\\epsilon\\rightarrow \\infty} \\frac{g(\\btheta+\\epsilon \\bu)-g(\\btheta)}{\\epsilon}) = \\text{sign}(g(\\btheta+\\epsilon \\bu) - g(\\btheta))  \\text{ for a small $\\epsilon$.}\n\\end{equation*}\nTherefore, we can use this information to estimate the original gradient. The Sign-OPT approach in the previous section uses $\\sum_{q} \\text{sign}(\\langle \\nabla g(\\btheta), \\bu_q \\rangle) \\bu_q$ as an estimation of gradient. Let $y_q := \\text{sign}(\\langle \\nabla g(\\btheta), \\bu_q \\rangle)$, \na more accurate gradient estimation can be cast as the following constraint optimization problem: \n\\begin{align*}\n\\text{Find a vector $\\bz$  such that } \\text{sign}(\\langle \\bz, \\bu_q\\rangle) = y_q \\ \\ \\forall q=1, \\dots, Q. \n\\end{align*}\nTherefore, this is equivalent to a hard constraint SVM problem where each $\\bu_q$ is a training sample and $y_q$ is the corresponding label. The gradient can then be recovered by solving\nthe following quadratic programming problem: \n\\begin{equation}\n    \\min_{\\bz} \\ \\bz^T \\bz \\ \\ \\text{ s.t. } \\ \\  \\bz^T \\bu_q \\geq y_q,\\ \\  \\forall q=1,\\dots, Q.\n    \\label{eq:svm}\n\\end{equation}\nBy solving this problem, we can get a good estimation of the gradient. As explained earlier, each $y_q$ can be determined with a single query. Therefore, we propose a variant of Sign-OPT, which is called SVM-OPT attack. The detailed procedure is shown in Algorithm \\ref{alg:svmopt}. We will present an empirical comparison of our two algorithms in \\autoref{ssec:svm-sign-comp}.\n\\begin{algorithm}[h]\n\\SetAlgoLined\n\\textbf{Input}: Hard-label model $f$, original image $\\bx_0$, initial $\\btheta_0$ \\; \n\\For{$t=1, 2, \\dots,T$}{\nSample $\\bu_1, \\dots, \\bu_Q$ from Gaussian or orthogonal basis \\;\nSolve $\\bz$ defined by \\eqref{eq:svm} \\;\nUpdate $\\btheta_{t+1} \\leftarrow \\btheta_t - \\eta \\bz$ \\;\nEvaluate $g(\\btheta_t)$ using search algorithm in \\citep{cheng2018queryefficient} \\;\n}\n\\caption{SVM-OPT attack}\n\\label{alg:svmopt}\n\\end{algorithm}\n\n\n\\vspace{-8pt}\n\n\\section{Experimental Results}\n\\label{sec:exp}\n\\vspace{-6pt}\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[b]{\\textwidth}\n         \\centering\n         \\includegraphics[width=0.9\\textwidth]{figures\/example_mnist.pdf}\n       \n     \\end{subfigure}\n   \n   \n   \n   \n   \n     \\begin{subfigure}[b]{0.9\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/example_cifar_sign.pdf}\n        \n     \\end{subfigure}\n   \n   \n   \n   \n   \n     \\begin{subfigure}[b]{0.9\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures\/example_imgnt_car.pdf}\n       \n     \\end{subfigure}\n  \\captionsetup{justification=centering}\n  \\caption{Example of Sign-OPT targeted attack. $L_2$ distortions and queries used are shown above and below the images. First two rows: Example comparison of Sign-OPT attack and OPT attack. Third and fourth rows: Examples of Sign-OPT attack on CIFAR-10 and ImageNet}\n  \\label{fig:untarget-example}\n \n\\end{figure}\n\nWe evaluate the SIGN-OPT algorithm for attacking black-box models in a hard-label setting on three different standard datasets - MNIST~\\citep{lecun1998gradient}, CIFAR-10~\\citep{cifar10} and ImageNet-1000~\\citep{deng2009imagenet} and compare it with existing methods. For fair and easy comparison, we use the CNN networks provided by \\citep{carlini2017towards}, which have also been used by other previous hard-label attacks as well. Specifically, for both MNIST and CIFAR-10, the model consists of nine layers in total - four convolutional layers, two max-pooling layers and two fully-connected layers. Further details about implementation, training and parameters are available on \\citep{carlini2017towards}. As reported in \\citep{carlini2017towards} and \\citep{cheng2018queryefficient}, we were able to achieve an accuracy of 99.5\\% on MNIST and 82.5\\% on CIFAR-10. We use the pretrained Resnet-50 \\citep{he2016deep} network provided by torchvision \\citep{torchvision} for ImageNet-1000,  which achieves a Top-1 accuracy of 76.15\\%.\n\nIn our experiments, we found that Sign-OPT and SVM-OPT perform quite similarly in terms of query efficiency. Hence we compare only Sign-OPT attack with previous approaches and provide a comparison between Sign-OPT and SVM-OPT in \\autoref{ssec:svm-sign-comp}. We compare the following attacks:\n\\begin{itemize}\n    \\item \\textbf{Sign-OPT attack} (black box): The approach presented in this paper.\n    \\item \\textbf{Opt-based attack} (black box): The method proposed in \\cite{cheng2018queryefficient} where they use Randomized Gradient-Free method to optimize the same objective function. \n    We use the implementation provided at {\\small \\url{https:\/\/github.com\/LeMinhThong\/blackbox-attack}}.\n    \\item \\textbf{Boundary attack} (black box):\n    The method proposed in \\cite{brendel2017decision}. This is compared only in $L_2$ setting as it is designed for the same. We use the implementation provided in Foolbox ({\\small\\url{https:\/\/github.com\/bethgelab\/foolbox}}).\n    \n    \\item \\textbf{Guessing Smart Attack} (black box): The method proposed in \\citep{brunner2018guessing}. This attack enhances boundary attack by biasing sampling towards three priors. Note that one of the priors assumes access to a similar model as the target model and for a fair comparison we do not incorporate this bias in our experiments. We use the implementation provided at {\\small \\url{https:\/\/github.com\/ttbrunner\/biased_boundary_attack}}. \n    \\item \\textbf {C\\&W attack} (white box):\n    One of the most popular methods in the white-box setting proposed in\n     \\cite{carlini2017towards}. We use C\\&W $L_2$ norm attack as a baseline for the white-box attack performance. \n    \n   \n\\end{itemize}\n\nFor each attack, we randomly sample 100 examples from validation set and generate adversarial perturbations for them. For untargeted attack, we only consider examples that are correctly predicted by model and for targeted attack, we consider examples that are already not predicted as target label by the model. \nTo compare different methods, \nwe mainly use \\textit{median distortion}\nas the metric. \nMedian distortion for $x$ queries is the median adversarial perturbation of all examples achieved by a method using less than $x$ queries. \nSince all the hard-label attack algorithms will start from an adversarial exmample and keep reduce the distortion, if we stop at any time they will always give an adversarial example and medium distortion will be the most suitable metric to compare their performance. \nBesides, we also show \n\\textit{success rate (SR)} for $x$ queries for a given threshold ($\\epsilon$), which is the percentage of number of examples that have achieved an adversarial perturbation below $\\epsilon$ with less than $x$ queries. \nWe evaluate success rate on different thresholds which depend on the dataset being used. \nFor comparison of different algorithms in each setting, we chose the same set of examples across all attacks.\n\n\n\\textbf{Implementation details}: To optimize \\autoref{alg:signopt}, we estimate the step size $\\eta$ using the same line search procedure implemented  in~\\cite{cheng2018queryefficient}. At the cost of a relatively small number of queries, this provides significant speedup in the optimization. Similar to \\cite{cheng2018queryefficient}, $g(\\theta)$ in last step of \\autoref{alg:signopt} is approximated via binary search. The initial $\\theta_0$ in \\autoref{alg:signopt} is calculated by evaluating $g(\\theta)$ on 100 random directions and taking the best one.\nWe provide our implementation publicly\\footnote{https:\/\/github.com\/cmhcbb\/attackbox}.\n\\vspace{-5pt}\n\\subsection{Comparison between Sign-OPT and SVM-OPT}\n\\label{ssec:svm-sign-comp}\n\\vspace{-5pt}\nIn our experiments, we found that the performance in terms of queries of both these attacks is remarkably similar in all settings (both $L_2$\/$L_\\infty$ \\& Targeted\/Untargeted) and datasets. We present a comparison for MNIST and CIFAR-10 ($L_2$ norm-based) for both targeted and untargeted attacks in\n\\autoref{fig:svm_sign_comparison}. We see that the median distortion achieved for a given number of queries is quite on part for both Sign-OPT and SVM-OPT.\n\n\\textbf{Number of queries per gradient estimate}: In \\autoref{fig:svm_sign_comparison}, we show the comparison of Sign-OPT attack with different values of $Q$. Our experiments suggest that $Q$ does not have an impact on the convergence point reached by the algorithm. Although, small values of $Q$ provide a  noisy gradient estimate and hence delayed convergence to an adversarial perturbation. Large values of $Q$, on the other hand, require large amount of time per gradient estimate. After fine tuning on a small set of examples, we found that $Q=200$ provides a good balance between the two. Hence, we set the value of $Q=200$ for all our experiments in this section. \n\n\\subsection{Untargeted attack}\n\\vspace{-5pt}\n\\begin{figure}\n     \\centering\n     \\includegraphics[width=0.9\\textwidth]{figures\/svm-param.pdf}\n    \\captionsetup{justification=centering}\n    \\caption{Median $L_2$ distortion vs Queries. First two: Comparison of Sign-OPT and SVM-OPT attack for MNIST and CIFAR-10. Third: Performance of Sign-OPT for different values of $Q$.}\n    \\label{fig:svm_sign_comparison}\n\\end{figure}\n\nIn this attack, the objective is to generate an adversary from an original image for which the prediction by model is different from that of original image. \\autoref{fig:untargeted_distortion} provides an elaborate comparison of different attacks for $L_2$ case\nfor the three datasets. Sign-OPT attack consistently outperforms the current approaches in terms of queries. Not only is Sign-OPT more efficient in terms of queries, in most cases\nit converges to a lower distortion than what is possible by other hard-label attacks. Furthermore, we observe Sign-OPT  converges to a solution comparable with C\\&W white-box attack (better on CIFAR-10, worse on MNIST, comparable on ImageNet). This is significant for a hard-label attack algorithm since we are given very limited information. \n\nWe highlight some of the comparisons of Boundary attack, OPT-based attack and Sign-OPT attack ($L_2$ norm-based) in \\autoref{u-l2-table}. Particularly for ImageNet dataset on ResNet-50 model, Sign-OPT attack reaches a median distortion below 3.0 in less than $30k$ queries while other attacks need more than $200k$ queries for the same. \n\n\n\\begin{figure}\n     \\centering\n         \\includegraphics[width=\\textwidth]{figures_iclr\/u_l2.pdf}\n        \n    \n   \n   \n   \n   \n   \n        \\caption{Untargeted attack: Median distortion vs Queries for different datasets. }\n        \\label{fig:untargeted_distortion}\n\\end{figure}\n\n\\begin{figure}\n     \\centering\n         \\includegraphics[width=\\textwidth]{figures_iclr\/fig5.pdf}\n        \n    \n   \n   \n   \n   \n   \n        \\caption{(a) Targeted Attack: Median distortion vs Queries of different attacks on MNIST and CIFAR-10. (b) Comparing Sign-OPT and ZO-SignSGD with and without single query oracle (SQO).}\n        \\label{fig:targeted_distortion}\n\\end{figure}\n\n    \n\n    \n\n\n\\subsection{Targeted attack}\n\\vspace{-5pt}\nIn targeted attack, the goal is to generate an adversarial perturbation for an image so that the prediction of resulting image is the same as a specified target. For each example, we randomly specify the target label, keeping it consistent across different attacks. We calculate the initial $\\theta_0$ in \\autoref{alg:signopt} using 100 samples in target label class from training dataset and this $\\theta_0$ is the same across different attacks. \\autoref{fig:untarget-example} shows some examples of adversarial examples generated by Sign-OPT attack and the Opt-based attack. The first two rows show comparison of Sign-OPT and Opt attack respectively on an example from MNIST dataset. The figures show adversarial examples generated at almost same number of queries for both attacks. Sign-OPT method generates an $L_2$ adversarial perturbation of 0.94 in $\\sim6k$ queries for this particular example while Opt-based attack requires $\\sim35k$ for the same. \\autoref{fig:targeted_distortion} displays a comparison among different attacks in targeted setting. In our experiments, average distortion achieved by white box attack C\\&W for MNIST dataset is 1.51, for which Sign-OPT  requires $\\sim 12k$ queries while others need $> 120k$ queries. We present a comparison of success rate of different attacks for CIFAR-10 dataset in \\autoref{fig:success_rate} for both targeted and untargeted cases.\n\n\n\n\n\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[b]{0.24\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures_iclr\/success_u_l2_e1.pdf}\n        \n     \\end{subfigure}\n    \n     \\begin{subfigure}[b]{0.24\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures_iclr\/success_u_l2_e2.pdf}\n        \n     \\end{subfigure}\n     \\begin{subfigure}[b]{0.24\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures_iclr\/success_t_l2_e1.pdf}\n        \n     \\end{subfigure}\n     \\begin{subfigure}[b]{0.24\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{figures_iclr\/success_t_l2_e2.pdf}\n        \n     \\end{subfigure}\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n        \\captionsetup{justification=centering}\n        \\caption{Success Rate vs Queries for CIFAR-10 ($L_2$ norm-based attack). First two and last two depict untargeted and targeted attacks respectively. Success rate threshold is at the top of each plot.}\n        \\label{fig:success_rate}\n       \n\\end{figure}\n\n\\subsection{The power of single query oracle}\n\\vspace{-5pt}\nIn this subsection, we conduct several experiments to prove the effectiveness of our proposed single query oracle in hard-label adversarial attack setting. ZO-SignSGD algorithm  \\citep{liu2018signsgd} is proposed for soft-label black box attack and we extend it into hard-label setting. \nA straightforward way is simply applying ZO-SignSGD to solve the hard-label objective proposed in~\\cite{cheng2018queryefficient}, estimate the gradient using binary search as \\citep{cheng2018queryefficient} and take its sign. In Figure 5(b), we clearly observe that simply combining ZO-SignSGD and \\cite{cheng2018queryefficient} is not efficient. \nWith the proposed single query sign oracle, we can also reduce the query count of this method, as demonstrated in Figure 5(b). This \nverifies the effectiveness of single query oracle, which can universally improve many different optimization methods in the hard-label attack setting. To be noted, there is still improvement on Sign-OPT over ZO-SignSGD with single query oracle because instead of directly taking the sign of gradient estimation, our algorithm utilizes the scale of random direction $u$ as well. In other words, signSGD's gradient norm is always 1 while our gradient norm takes into account the magnitude of $u$. Therefore, our signOPT optimization algorithm is fundamentally different \\citep{liu2018signsgd} or any other proposed signSGD varieties. Our method can be viewed as a new zeroth order optimization algorithm that features fast convergence in signSGD.\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\vspace{-6pt}\nWe developed a new and ultra query-efficient algorithm for adversarial attack in the hard-label black-box setting. Using the same smooth reformulation in \\cite{cheng2018queryefficient}, we design a novel zeroth order oracle that can compute the sign of directional derivative of the attack objective using single query. Equipped with this single-query oracle, we design a new optimization algorithm that can dramatically reduce number of queries compared with \\cite{cheng2018queryefficient}. We prove the convergence of the proposed algorithm and show our new algorithm is overwhelmingly better than current hard-label black-box attacks. \n\n\\begin{table}[tbp]\n \n  \\centering\n  \\captionsetup{justification=centering}\n    \\caption{$L_2$ Untargeted attack - Comparison of average $L_2$ distortion achieved using a given number of queries for different attacks. SR stands for success rate.}\n    \\centering\n \n  \\resizebox{\\textwidth}{!}{\n  \\begin{tabular}{lccc|ccc|ccc}\n    \\toprule\n    \\cmidrule(r){1-2}\n    &  \\multicolumn{3}{c}{MNIST} &\\multicolumn{3}{c}{CIFAR10} & \\multicolumn{3}{c}{ImageNet (ResNet-50)}\\\\\n    &\\#Queries &Avg $L_2$& SR($\\epsilon=1.5$)&  \\#Queries  & Avg $L_2$ & SR($\\epsilon=0.5$)&\\#Queries & Avg $L_2$ & SR($\\epsilon=3.0$)\\\\\n    \\midrule \n    \\multirow{3}{*}{Boundary attack}   \n    &4,000      &4.24 &1.0\\%    &4,000    &3.12 &2.3\\%    &4,000   &209.63     &0\\% \\\\\n    &8,000      &4.24 &1.0\\%    &8,000    &2.84 &7.6\\%    &30,000    &17.40     &16.6\\%\\\\\n    &14,000     &2.13 &16.3\\%   &12,000    &0.78 &29.2\\%  &160,000    &4.62     &41.6\\%\\\\\n    \\hline\n    \\multirow{2}{*}{OPT attack} \n    &4,000  &3.65 &3.0\\%    &4,000  &0.77   &37.0\\% &4,000      &83.85     &2.0\\%\\\\\n    &8,000  &2.41 &18.0\\%   &8,000  &0.43   &53.0\\% &30,000     &16.77     &14.0\\%\\\\\n    &14,000 &1.76 &36.0\\%   &12,000 &0.33   &61.0\\% &160,000    &4.27     &34.0\\%\\\\\n   \\hline\n    \\multirow{2}{*}{Guessing Smart} \n    &4,000  &1.74 &41.0\\%    &4,000  &0.29   &75.0\\% &4,000      &16.69     &12.0\\%\\\\\n    &8,000  &1.69 &42.0\\%   &8,000  &0.25   &80.0\\% &30,000     &13.27     &12.0\\%\\\\\n    &14,000 &1.68 &43.0\\%   &12,000 &0.24   &80.0\\% &160,000    &12.88     &12.0\\%\\\\\n   \\hline\n    \\multirow{2}{*}{\\textbf{Sign-OPT attack}}\n    &4,000  &1.54  &46.0\\%  &4,000  &0.26  &73.0\\%  &4,000      &23.19  &8.0\\%\\\\\n    &8,000  &1.18  &84.0\\%  &8,000  &0.16  &90.0\\%  &30,000     &2.99   &50.0\\%\\\\\n    &14,000 &1.09  &94.0\\%  &12,000 &0.13  &95.0\\%  &160,000    &1.21   &90.0\\%\\\\\n   \\hline\n      C\\&W (white-box)\n    &-   &0.88      &99.0\\%  &-    &0.25 &85.0\\%  &- &1.51& 80.0\\%   \\\\\n    \\bottomrule\n  \\end{tabular}\n}\n\n  \\label{u-l2-table}\n  \n\\end{table}\n\n\n\n\n\n\\section*{Acknowledgement}\nThis work is based  upon  work  supported by the Department of Energy National Energy Technology Laboratory under Award Number DE-OE0000911 and by NSF under IIS1719097.\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe abundance of Li has attracted much attention, especially since the Li gap\nhas been discovered in the Hyades for stars with \\teff $\\sim 6600$~K. In the\ncontext of radiative diffusion, it is interesting to examine the atmospheric\nabundance of lithium in stars where such a mechanism is known to be at work from\nthe abundances of other elements, such as calcium, i.e. in the Am stars.\nSuch studies have been carried out especially by Burkhart \\& Coupry \\citep{bc91}\nand Burkhart et al. \\citep{bcfg05}.\nTheir conclusion was that in general, the Li abundance\nof Am stars is close to the cosmic value ($\\log N(Li)\\sim 3.0$ in the scale\nwhere $\\log N(H)= 12.0$), although a small proportion of them are deficient.\nThe latter seem in general to be either evolved stars or, as recently suggested\nby Burkhart et al. \\citep{bcfg05}, to lie on the red side of the Am domain,\namong the $\\rho$ Puppis--like stars.\n\nIn this poster, we present Li abundances obtained for 31 Am stars and 36 normal\nA and F stars in the field, all having Hipparcos parallaxes. This sample had\nbeen defined before the Hipparcos era, on purely photometric criteria, but with\nthe purpose of testing how far the Li abundance depends on the evolutionary\nstate, i.e. on the surface gravity $\\log g$. The Hipparcos data which became\navailable later showed that the photometric luminosity calibrations of Am stars\nwere not very satisfactory (North et al. 1997), but allowed to determine\n$\\log g$ in a more fundamental way. Furthermore, the sample has the advantage\nof presenting no bias against large rotational velocities.\n\n\n\\section{Observations and analysis}\nAll stars were observed at OHP with the Aur\\'elie spectrograph attached to the\n1.5m telescope, in April 1993 and in October 1993 and 1994. The grating No 7 was\nused, giving a resolving power $R=40000$ in the spectral range $6640-6760$~\\AA .\nThe typical exposure times were between 40 and 60 minutes, the resulting\nsignal-to-noise ratio being between 250 and 400. The spectra were\nreduced during the observing runs with the IHAP package, and were later\nnormalized to the continuum in an interactive way.\n\nThe analysis was made by comparison of the observed spectra with synthetic ones\nconvoluted with an assumed gaussian instrumental profile and with an appropriate\nrotational profile. The Synspec code (Hubeny et al. 1995) and Kurucz model\natmospheres were used to produce the synthetic spectra. The line parameters were\ntaken from Kurucz's $gfiron$ list, except of course the parameters for the Li\ndoublet. The effective temperatures were computed from Geneva photometry, while\nthe surface gravities were computed from the Hipparcos parallaxes, by combining\nthem with theoretical evolutionary tracks from Schaller et al. \\citep{ssmm92},\nas explained by North \\citep{n98}, assuming standard evolution.\nThe microturbulent velocity was either computed from the formula proposed by\nEdvardsson et al. \\citep[eqn 9]{e93}, for \\teff $< 7000$~K, or estimated from the\nFig.~1 of Coupry \\& Burkhart \\citep{cb92}, for \\teff $\\geq 7000$~K.\nThe abundance of\nFe, Ca and a few other elements (in cases of sharp lined stars) were first\nestimated by visual fits. Then, the Li abundance and the projected rotational\nvelocities were obtained by minimizing the $\\chi^2$ between observed and\nsynthetic spectra having various values of these parameters.\n\\begin{figure}\n  \\includegraphics[width=10cm]{north_hr.eps}\n  \\caption{HR diagram of the Am (black dots) and normal (white dots) stars of\nour sample. Black triangles (with error bars typical of the whole\nsample) are stars from Burkhart et al. (2005, Table~3) not in\ncommon with our sample. The error bars were drawn assuming a $\\pm 200$~K error\non \\teff and include, on the vertical axis, the parallax error of Hipparcos.\n}\n\\label{hr}\n\\end{figure}\n\\section{Results}\nFig.~\\ref{hr} shows the distribution of Am stars (full dots) and of normal A-F stars\n(open dots) in the HR diagram. Evolutionary tracks and isochrones from\nSchaller et al. \\citep{ssmm92} are shown for 4 masses ($1.5$ to $2.5~M_\\odot$)\nand for 3\nages ($\\log t = 8.7$ to $9.3$) respectively. The stars are well distributed\non the whole main sequence band. The lack of Am stars below \\teff $\\sim 7000$~K\nis the well-known limit due to the onset of convection.\n\nFig.~\\ref{LiTelg} (left) shows the lithium abundance as a function of \\teff\nfor Am stars (full dots) and for normal A--F stars (open dots).\nThe most striking feature of this diagram is\nthe bimodal distribution of the Li abundance for \\teff $\\lesssim 7500$~K, which\nis reminiscent of a similar distribution of F-type dwarfs in the range $5900\n<$ \\teff $< 6600$~K reported by Lambert et al. \\citep[Fig.~4]{lhe91}.\nThus, our data\ncomplement that of Lambert et al. as well as the larger sample of Chen et al.\n\\citep{cnbz01} by extending the results to higher \\teff. We have\nverified that duplicity cannot account for the low apparent Li abundances (even\nthough this might hold for some isolated cases). Restricting the diagram to\nthose stars with\n\\vsini $< 80$~\\kms, the upper branch almost disappears (there are only two\nnormal stars left around \\teff $\\sim 6500$~K), while the lower one remains\nintact. This is related to the fact that the upper branch is populated only with\nnormal stars, which rotate more rapidly than the Am stars, while the lower\nbranch is a mix of normal and Am stars. Thus, below $7500$~K, all Am stars of\nour sample are Li deficient. The black triangles refer to the 4 stars of\nBurkhart et al. \\citep[their Table~3]{bcfg05} which are not common to our\nsample. Their positions are in perfect agreement with the general picture.\n\nFig.~\\ref{LiTelg} (right) displays the Li abundance as a function of surface gravity. There is\nno strong trend, but one can notice that those stars (either Am or normal) which\nare strongly deficient in Li are {\\bf all} at least slightly evolved. There is one\nunevolved star (HD 18769) for which only an upper limit to its Li abundance\ncould be obtained, but this is due to its high \\teff ($8420$~K) and moderately\nbroad lines, and the upper limit is close to the ``cosmic'' Li abundance, so\nthis is not a significant exception. Thus, we confirm the suggestion made by\nBurkhart \\& Coupry that Li-deficient Am stars are evolved objects, although it\nseems that all evolved Am stars are not necessarily deficient.\n\\begin{figure}\n  \\includegraphics[width=10.3cm]{north_LiTelg.eps}\n  \\caption{{\\bf Left:} Li abundance (on the scale $\\log N(H)=12$) of Am\nstars (black\ndots) and normal A--F-type stars (white dots) versus effective temperature.\nUpper limits to the Li abundance are indicated by vertical arrows.\nBlack triangles are from Burkhart et al. \\citep{bcfg05}.\n{\\bf Right:} Li abundance versus surface gravity derived from Hipparcos\nparallaxes. The leftmost arrow refers to the Am star HD 18769, which has\n\\teff $=8420$~K and \\vsini $= 46$~\\kms, so that only an upper limit\nto its Li abundance can be obtained. The typical error on $\\log g$ is $0.1$~dex,\nwhile that on $\\log N(Li)$ vary from better than $0.1$~dex to more than\n$0.3$~dex, depending on \\teff and \\vsini.}\n\\label{LiTelg}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}    \n\n\n\nThere is a voluminous literature on second order analysis of distribution functions $F_N(z) = P(Z_N\\leq z)$ of statistics $Z_N = \\zeta_N(X_1,X_2,\\dots, X_N)$ that are functions of i.i.d. random variables $X_1,X_2,...$. The results obtained are generally refinements of the central limit theorem. Suppose  that $Z_N$ is asymptotically standard normal, that is $\\sup_z |F_N(z)-\\Phi (z)|\\rightarrow0 $ as $N\\rightarrow \\infty$, where $\\Phi $  denotes the standard normal distribution function. Then second order results are concerned with the speed of  this convergence, or with attempts to increase this speed by replacing the limit $\\Phi $ by a series expansion $\\Psi_N$ that provides a better approximation. Results of the first kind are called theorems of Berry-Esseen type and assert that for all $N$,\n$$\\sup_z|F_N(z)-\\Phi(z)|\\leq CN^{-\\frac12},$$\nwhere $C$ is a constant that depends on the particular statistic $Z_N$ and the\ndistribution of the variables $X_i$, but not on $N$. Such results are often\nvalid under mild restrictions such as the existence of a third moment of\n$X_i$. The original Berry-Esseen theorem dealt with the case where $Z_N$ is a\nsum of i.i.d. random variables, \n\\citet*{Esseen:1942},\n\\citet*{Berry:1941}. \nFor a more general version see \n\\citet*{vanZwet:1984}. \n\n\nResults of the second kind concern so-called Edgeworth expansions. These are series expansions such as \n\\begin{equation} \n        \\begin{split}\n         &\\Psi_{ N,1}(z) = \\Phi(z)+\\varphi(z)N^{-\\frac12} Q_1(z), \\quad \\text{or} \\\\ \n        &\\Psi_{N,2}(z)  =\\Phi(z)+\\varphi(z)\\left[ N^{-\\frac12}  Q_1(z)+ N^{-1}Q_2(z)\\right],\n        \\end{split}\n\\end{equation}  \n\n\\noindent\nwhere $ \\varphi $ is the standard normal density and $Q_1$ and $Q_2$ are polynomials depending on low moments of $X_i$ . One then shows that \n\\begin{equation} \n        \\begin{split}\n                &\\sup_z|F_N(z)-\\Psi_{N,1}(z)|\\leq CN^{-1}, \\quad \\text{or}  \\\\  \n                &\\sup_z|F_N(z)-\\Psi_{N,2}(z)|\\leq CN^{-\\frac32} .\n        \\end{split}\n\\end{equation} \n\n\\noindent\nFor this type of result the restrictions are more severe. Apart from moment\nassumptions one typically assumes that $Z_N$ is not a lattice random\nvariable. For the case where $Z_N$ is a sum of i.i.d. random variables a good\nreference is \n\\citep[chap. XVI]{Feller:1965\/2}. \nThere are numerous papers devoted to special types of statistics. For a\nsomewhat more general result we refer to \n\\citet*{Bickel-Goetze-vanZwet:1986}\n and \n\\citet*{bentkus-goetze-vanzwet:1997} .\n\nFor the case where $Z_N$ assumes its values on a lattice, say the integers, an alternative approach it to generalize the local central limit theorem and provide an expansion for the point probabilities  $P(Z_N=z)$ for values of $z$ belonging to the lattice. A typical case is the binomial distribution for which local expansions are well known. It is obvious that for the binomial distribution one can not obtain Edgeworth expansions as given in $(1.1)$  for which $(1.2)$ holds. The reason is that out of the $N$ possible values for a binomial $(N,p)$ random variable, only $cN^{\\frac12}$ values around the mean $Np$ really count and each of these has probability of order $N^{-\\frac12}$. Hence the distribution function has jumps of order $N^{-\\frac12}$ and can therefore not be approximated by a continuous function such as given in $(1.1)$ with an error of smaller order than $N^{-\\frac12}$. \n\nIn a sense the binomial example is an extreme case where the ease of\nthe approach through local expansions for $P(Z_N=z)$ precludes the one through\nexpansions of Edgeworth type for $P(Z_N\\leq z)$. In \n\\citet*{Albers-Bickel-vanZwet:1976}\n the authors found somewhat to their surprise that for the Wilcoxon statistic which is a pure lattice statistic, an Edgeworth expansion with remainder $O(N^{-\\frac32})$ for the distribution function is perfectly possible. In this case the statistic ranges over $N^2$ possible integer values, of which the central $N^{\\frac32}$ values have probabilities of order $N^{-\\frac32}$ so that one can approximate a distribution function with such jumps by a continuous one with error $O(N^{-\\frac32})$.\n \nOn the basis of these examples one might guess that the existence of an Edgeworth  expansion with error $O(N^{-p})$ for the distribution function $F_N(z) = P(Z_N\\leq z)$ would merely depend on the existence of some moments of $Z_N$ combined with the requirement that $F_N$ does not exhibit jumps of large order than $N^{-p}$. But one can envisage a more subtle problem if $F_N$ would assign a probability of larger order than $N^{-p}$ to an interval of length $N^{-p}$. Since Edgeworth expansions have bounded derivative, this would also preclude the existence of such an expansion with error $O(N^{-p})$.\n  \nLittle seems to be known about the case where $Z_N$ has a discrete but non-lattice distribution. Examples abound if one considers a lattice random variable with expectation $0$ and standardized by dividing by its sample standard deviation. As a simple example, one could for instance consider Student's $t$-statistic $\\tau_N = N^{-1\/2}\\sum_i X_i \/ \\sqrt{\\sum_i\\left( X_i -m \\right)^2\/(N-1) }$  with $m= \\sum_i X_i\/N $ and $X_1,X_2,\\dots$ i..i.d. random variables with a lattice distribution. Since we are not interested in any particular statistic, but merely in exploring what goes on in a case like this, we shall simplify even further by deleting the sample mean m and considering the statistic\\smallskip\n\n\\noindent\n\\begin{equation}   \nW_N = \\sum_{i=1}^N \\frac{X_i}{\\sqrt{\\sum_{i=1}^N X_i^2}} ,\n\\end{equation}\n\n\\noindent\nwith \\smallskip\n\n\\noindent\n\\begin{equation}\n X_1,X_2,\\dots i.i.d.\\mbox{ with } P(X_i=-1)= P(X_i=0)= P(X_i=1)= \\frac13 .\n\\end{equation}\n\n\nWe should perhaps point out that for $w>0$\n\\begin{align*}\nP(0<\\tau_N\\leq w)&=P\\left(0<W_N\\leq \\frac{\\sqrt{N\/(N-1)}x}{1+x^2\/(N-1)}\\right) \\\\\n&= P(0<W_N\\leq x + O(x(1+x^2)\/N)),\n\\end{align*}\nand since both $\\tau_N$ and $W_N$ have distributions that are symmetric about the origin, Theorem 1.1 ensures that under this model, the expansions for the distributions of $\\tau_N$ and $W_N$ are identical up to order $O(N^{-1})$. Hence the fact that we discuss an expansion for $W_N$ rather than for Student's statistic $\\tau_N$ is not an essential difference, but merely a matter of convenience.\n\n\nNotice that in (1.3) $ \\sum X_i^2$ equals the number of non-zero $X_i$ and that each of these equals $-1$ or $+1$ with probability $\\frac12$. Hence we may also describe the model given by (1.3) and (1.4) as follows.\n\nFor $N=1,2,\\dots, T_N$ has a binomial distribution with parameters $N$ and $\\frac23$. Given $T_N, S_N$ has a binomial distribution with parameters $T_N$ and $\\frac12$ . Define $D_N=2 S_N-T_N$ and notice that $D_N$ and $T_N$ are either both even or both odd. We consider the statistic \\smallskip\n\n\\noindent\n \\begin{equation}W_N = \\begin{cases} \n                       0  \\quad \\text{ if } T_N=0 \\\\\n                       \\frac{D_N}{\\sqrt{T_N} } \\quad \\text{ otherwise.}                                           \n                    \\end{cases}\n        \\end{equation}\n\n\\noindent\nNotice that unconditionally $S_N$ has a binomial distribution with parameters $N$ and $\\frac13$.\n\n \n  Let $F_N$ be the distribution function of  $W_N$.  Obviously $W_N$ has\n  expected value $EW_N=0$ and variance $\\sigma^2 (W_N)=1$, and is\n  asymptotically normal. By an appropriate Berry-Esseen theorem $\\sup_x\n  |F_N(x)-\\Phi(x)| = O(N^{-\\frac12})$, where $\\Phi$ denotes the standard\n  normal distribution function as before, see\n  e.g.\n\\citet*{Bentkus-Goetze:1996}.\n A curious thing about the distribution of $W_N$ is that $ P(W_N =0)=P(D_N=0)=P(S_N=\\frac12T_N)$ which is obviously of exact order $N^{-\\frac12}$, but all other point probabilities $P(W_N =w)$ for $w\\not= 0$ are clearly of smaller order, and we shall see that these are actually $O(N^{-1})$. Hence the following question arises: if we remove the point probability at the origin, to what order of magnitude can we approximate the distribution function of $W_N$  by an expansion?\n\nLet $\\mbox{frac}(x)$ denote the fractional part of a number $x\\geq 0$, i.e. $\\mbox{frac} (x)=x-\\lfloor x\\rfloor $ if $\\lfloor x\\rfloor $ is the largest integer smaller than or equal to $x$. For $N=1,2,\\dots$, define a function $\\Psi_N$ on $(-\\infty,\\infty)$ as follows.\\\\ For $w\\geq 0$,\\smallskip\n\n\\noindent \n\\begin{equation*}\n\\Psi_N(w) = \\Phi(w) + N^{-1\/2} \\Lambda_N(w),\\ {\\rm with}\n\\end{equation*}          \n\\begin{equation}\n\\Lambda_N(w) = -\\sqrt{ \\frac{3}{2}}\\, \\varphi(w)  \\sum_{0\\leq n\n           \\leq N} \\frac{3}{\\sqrt{\\pi N}} \\,e^{    -\\frac{9}{N}\\left(n-\\frac\n             N3\\right)^2  }  \n\\left(\\mbox{frac} \\left(  w \\sqrt{2n}  \\right)-\\frac12 \\right)\n\\end{equation}\nand\n\\begin{equation*}\n\\Psi_N(-w)= 1-\\Psi_N(w-).\n\\end{equation*} \nHere the argument $w-$ denotes a limit from the left at $w$. $\\Psi_N$ is of bounded variation and for sufficiently large $N$ it is a probability distribution function. It has upward jump discontinuities of order $O(N^{-1})$ at points $w$ where $w\\sqrt{2n}$ assumes an integer value $k\\not=0$. At the origin it has a jump discontinuity of magnitude \n\\begin{equation} \n        \\Psi_N(0)- \\Psi_N(0-) \\sim \\sqrt{\\frac{3}{2N}}\\varphi(0)=\\sqrt{\\frac{3}{4\\pi N}}.\n\\end{equation}\n \n\\noindent\n{\\bf Theorem 1.1.} \\quad {\\it As} $N\\rightarrow \\infty ,$\n\\begin{equation*}\n P(W_N=0)\\sim \\,\\,\\sqrt{\\frac{3}{(4\\pi N)} }\n=O\\left( \\frac{1}{\\sqrt{N}}\\right),\n\\end{equation*}\n\\begin{equation}\nP(W_N = w) =  \\,\\,O\\left(\\frac 1N\\right),\n\\end{equation}\n{\\it for $w \\not= 0$, and}          \n\\begin{equation*} \n\\sup_{w} |F_N(w)-\\Psi_N(w)| = \\,\\,O\\left(\\frac 1N\\right).\n\\end{equation*}\n{\\it Moreover the distribution $\\Psi_N$ - and hence $F_N$ -  assigns probability $O(N^{-1})$ to any closed interval of length $O(N^{-1})$ that does not contain the origin. }\n\nThe reader may want to compare this result with the expansion obtained in \n\\citet*{Brown-Cai-DasGupta:2002} \nfor the distribution of a normalized binomial variable $(Y_n-np)\/\\sqrt{np(1-p)}$, where $Y_n$ has a binomial distribution with parameters $n$ and $p$. For $p=1\/2$ this distribution coincides with the conditional distribution of $W_N$ given $N=n$, but of course this is a pure lattice distribution.\n\n\nSince $\\sum_n 3(\\pi N)^{-1\/2} \\exp\\{-(9\/N)(n-N\/3)^2\\}$ is the sum of the density of a normal $N(N\/3,N\/18)$ distribution taken at the integer values $n \\in (-\\infty,\\infty)$, it is asymptotic to 1 as $N \\to \\infty$, and hence bounded for all $N$. Because $|\\mbox{frac}(x)-1\/2|\\leq 1\/2$ for all $x>0$, $\\Lambda_N(w)$ is bounded and $N^{-1\/2}\\Lambda_N(w) = O(N^{-1\/2})$ uniformly in $w$. At first sight there is a striking similarity between the expansion $\\Psi_N$ in Theorem 1.1 and the two term Edgeworth expansion $\\Psi_{N,1}$ in (1.1). However, the term $\\phi(z)N^{-1\/2}Q_1(z)$ of order $O(N^{-1\/2})$ in the Edgeworth expansion is a skewness correction that vanishes for a symmetric distribution $F_N$. As we are dealing with a symmetric case, such a term is not present and for the continuous case the Edgeworth expansion with remainder $O(N^{-1})$ is simply $\\Phi(z)$. The origin of the term $N^{-1\/2}\\Lambda_N(w)$ is quite different. It arises from the fact that we are approximating a discrete distribution function by a continuous one, and as such it is akin to the classical continuity correction.\n\nTo make sure that the term $N^{-1\/2}\\Lambda_N(w)$ is not of smaller order than $N^{-1\/2}$, we shall bound $|\\Lambda_N(w)|$ from below  by the absolute value \nof the following series. Assume that $N$ is divisible by $3$ and let\n  \\begin{equation}\n        \\begin{split}\n\\lambda_N(w):=& \\sqrt{\\frac 3 2}\\4 \\varphi(w)\\sum_{k=1}^{M} \\frac 1 {\\pi \\4 k} \nf_{N,k}\\exp\\bigl(- \\frac{\\pi^2}6\\4 k^2\\4  w^2\\4)\\4 \n\\sin\\bigl(2 \\4 \\pi\\4 \\4 k \\4 w\\4 \\sqrt{\\frac {2 N} 3}\\bigr)\\\\\n& \\, \\quad + O\\bigl(N^{-1\/2}(\\log N)^5\\bigr), \\quad M := \\lfloor \\log N \\rfloor ,\n \\end{split}\n        \\end{equation}\nwhere  $f_{N,k}=1 + O((k\/M)^2)$ is defined in $(3.9)$. Thus $\\lambda_N(w)$ \nis a rapidly converging Fourier series, (illustrated in\nFigure  1. below)  the modulus of which is larger than a positive  constant $c(w)>0$,\nprovided that $4\\4 w\\4 \\sqrt{\\frac{2 N} 3} $ is an odd integer.\n\n\n\n\\begin{figure}[H]         \n\\psfig{file=oscillatory.eps,width=18cm, height=4cm}\n\\caption{$\\lambda_{100}(w)$:\\,\\, $0.05 \\le w \\le 2.34,\\,\\, M=10$, $f_{100,k}:=\\exp[-(k\/M)^{2\/3}]$\\label{fig1}} \n\\end{figure}  \n Hence, we shall prove\n\\noindent\n{\\bf Theorem 1.2.} {\\it For any $N$ divisible by $3$, we have} \n\\begin{equation}\n                 \\sup_{w>0} |F_N(w)-\\Phi(w)| \\ge  \\,\\,\n\\sup_{w\\ge 1}N^{-\\frac 12}|\\lambda_N(w)| + O\\bigl(N^{-1} (\\log N)^5\\bigr)\n>\\,\\, \\frac c {\\sqrt N},\n\\end{equation}\n{\\it for some absolute constant $c>0$.} \n\n\\bigskip\\noindent\nThe proof of Theorem 1.1 is given in Section 2. In Section 3 we investigate \nthe oscillatory part of $\\Psi_N$ in (1.6), relating it to the\nFourier series $\\lambda_N(w)$ above and thus proving  Theorem 1.2.\n\n\\noindent\n{\\bf Acknowledgment.}\n The authors would like to thank G.Chistyakov  for a careful reading of the\n manuscript\n and Lutz Mattner for his comments on the  current ArXiv version.\n \n\n\\section{Proof of Theorem 1.1}\n\nThe event $W_N = 0$ occurs iff $D_N = 0$.  Let $Z_1,Z_2,\\dots, Z_N$ be i.i.d. random variables assuming the values $0$, $-1$ and $+1$, each with probability $\\frac 13$. Then $D_N$ is distributed as  $ \\sum Z_i$, which has mean $0$ and variance $\\frac{2N}{3}$. By the local central limit theorem  $P(\\sum Z_i =0) \\sim (2\\pi)^{-\\frac12} \\left(\\frac{2N}{3} \\right)^{-\\frac12} = \\sqrt{\\frac{3}{4\\pi N}}$ which proves the first statement of Theorem 1.1. Because the distribution of $W_N$ is symmetric about the origin, this implies that in the remainder of the proof we only need to consider positive values of $W_N$. Hence we suppose that $w>0$ throughout and this implies that we need only be concerned with positive values of  $D_N$ also.\n\nHoeffding's inequality ensures that for all $N\\geq 2$,\n \\[P\\left(|D_N|\\geq  \\sqrt{6N \\log N} \\right)\\leq   \\frac{2}{N^{3}}   \\]\nand \n\\[P\\left(|T_N -2N\/3|\\geq \\sqrt{2 N \\log N}\\right)  \\leq  \\frac{2}{N^{3}}.\\] \n\\noindent\nSince the joint distribution of $T_N$ and $D_N$ assigns positive probability to at most $N^2$ points and events with probability $O(N^{-1})$ are irrelevant for the remainder of the proof,  we may at any point restrict attention to values $D_N=d$ and $T_N=t$ with $|d| \\leq t$ and satisfying \\smallskip\n\n\\noindent\n          \\begin{equation}\n                 |d|< \\sqrt{6N\\log N} \\quad \\mbox{ and }\\quad \\left|t-\\frac{2N}{3}\\right|< \\sqrt{2N\\log N}. \n             \\end{equation}\n\nFor positive integer  $m\\leq n$  we have\n\\begin{eqnarray*}\n                    P(D_N =2m, T_N =2n)&=& P(S_N =m+n, T_N =2n)  \\\\\n                                       &=& \\frac{N!}{3^{N}(n+m)! (n-m)! (N-2n)!} .\n\\end{eqnarray*}\n\\noindent\nIf  $d=2m$ and $t=2n$ satisfy $(2.1)$, then $(n+m)$, $(n-m)$ and  $(N-2n)$ are of exact order $N$ and we may apply Stirling's formula to see that\n$$          P(D_N=2m,T_N =2n)$$ \n$$      = \\frac{N^{N+\\frac12}   \\left(1+O\\left(\\frac 1N\\right)\\right)    }{ 2\\pi  3^{N} (n+m)^{(n+m+\\frac12)}(n-m)^{(n-m+\\frac12)}(N-2n)^{(N-2n+\\frac12)}}  $$\n$$      = \\frac{3^{\\frac 32} \\left(1+O\\left(\\frac 1N\\right)\\right) } { 2\\pi N\\left(\\frac{ 3(n+m)}{N}\\right)^{(n+m+\\frac12)}\\left( \\frac{3(n-m)}{N}\\right)^{(n-m+\\frac12)} \\left(\\frac{3(N-2n)}{N}\\right)^{(N-2n+\\frac12)}} $$\n$$      = \\frac{3^{\\frac32} \\left(1+O\\left( \\frac 1N \\right)\\right)} { 2\\pi N } \\, \\exp \\Bigg\\{ -\\left(n+m+\\frac12\\right)\\log\\left( 1+\\frac 3N \\left(n+m-\\frac N3\\right) \\right)  $$\n$$ -\\left(n-m+\\frac12\\right)  \\log\\left( 1+\\frac 3N\\left(n-m-\\frac N3\\right) \\right)$$\n$$ -\\left(N-2n+\\frac12 \\right)\\log\\left(1+\\frac 3N\\left(\\frac{2N}{3}-2n\\right)\\right)\\Bigg\\}.$$\n\n\\noindent\nNext we expand the logarithms in the exponent. For the first order terms we obtain\n$$-\\frac 3N \\bigg[\\left(n+m+\\frac 12\\right)\\left(n+m-\\frac N3\\right)+\\left(n-m+\\frac12\\right)\\left(n-m-\\frac N3\\right)+ $$\n$$\\left(N-2n+\\frac 12\\right)\\left(\\frac{2N}{3}-2n\\right)\\bigg]  $$ \n$$= -\\frac 3N \\left[\\left(n+m-\\frac N3\\right )^2 + \\left(n-m-\\frac N3 \\right)^2+\\left( \\frac{2N}{3}-2n\\right)^2\\right] $$\n$$= -\\frac{3}{N}\\left( 6\\,\\tilde{n}^2+2m^2\\right),$$\nwhere $ \\tilde{n}:= \\left(n-\\frac N3\\right)$.\\\\\n\n\\noindent\nThe second order terms yield\n$$ \\frac 12 \\left(\\frac 3N\\right)^2\\bigg[ \\left(n+m+\\frac 12\\right)\\left(n+m-\\frac N3\\right)^2+\\left(n-m+\\frac 12\\right)\\left(n-m-\\frac N3\\right)^2$$\n$$+\\left(N-2n+\\frac12\\right)\\left( \\frac{2N}{3}-2n\\right)^2 \\bigg] $$\n$$= \\frac12 \\left(\\frac 3N \\right)^2 \\left[ -6\\tilde{n}^3+(2N+3)\\tilde{n}^2+6\\tilde{n}m^2 +\\left( \\frac{2N}{3}+1 \\right)m^2 \\right]  $$\n$$= \\frac 3N \\left( 3\\tilde{n}^2+m^2 \\right) + \\frac{27}{N^{2}}\\left(-\\tilde{n}^3+\\tilde{n}m^2 \\right) +O\\left( \\frac{ \\tilde{n}^2+m^2}{ N^{2}}\\right).$$\n\\noindent\nThe third order terms contribute\n$$-\\frac 13\\left(\\frac 3N \\right)^3 \\bigg[\\left(n+m+\\frac12\\right)\\left(n+m-\\frac N3\\right)^3+\\left(n-m+\\frac12\\right)\\left(n-m-\\frac N3\\right)^3$$\n$$+\\left(N-2n+\\frac12\\right)\\left(\\frac{2N}{3}-2n\\right)^3\\bigg]$$\n$$= \\frac{18(\\tilde{n}^3-\\tilde{n}m^2)}{N^{2}}+O\\left(\\frac{\\tilde{n}^4+m^4}{N^{3}}\\right) $$ \n\\noindent\nAs $d=2m$ and $t=2n$ satisfy $(2.1)$, the contribution of the remaining terms is dominated by that of the fourth order terms and equals\n$$O\\left(\\frac{\\tilde{n}^4+m^4}{N^{3}}\\right). $$\n\nCollecting the results of these computations we arrive at \\smallskip\n\n\\noindent\n\\begin{equation} \n         \\begin{split}\n           & \\qquad \\qquad \\qquad P(D_N=2m,T_N =2n) =  \\frac{3^{\\frac 32}}{  2\\pi N}\\,\\\\ \n                 &\\times \\exp\\bigg\\{ -\\frac{3(3\\tilde{n}^2+m^2)}{N}-\\frac{9(\\tilde{n}^3 \n            -\\tilde{n}m^2)}{N^2}  +O\\left(\\frac1N+ \\frac{\\tilde{n}^4+m^4}{N^3}\\right) \\bigg\\} , \n\\end{split} \n\\end{equation}\n\n\\noindent\nprovided $m\\leq n$ are integers between $1$ and $\\frac12 N$ satisfying $m<\\sqrt{2N\\log N}$ and \n$|\\tilde n|=\\left|n-\\frac N3\\right|<\\sqrt{N\\log N}$. However, we shall also use $(2.2)$ if these inequalities do not hold, since in that case both left- and right-hand members of $(2.2)$ are negligible for our purposes.\n\n  \nBy Taylor expansion of the integrand about $x=m$, we find that for integer $0<m\\leq n$ with $m<\\sqrt{2N\\log N} $ and $|\\tilde n| = \\left|n-\\frac N3 \\right|<\\sqrt{N\\log N}$,\n\\begin{eqnarray*}\n           \\int_{[m-\\frac12,m+\\frac12) } \\exp\\bigg\\{ -\\frac{3}{N}x^2 +\\frac{9}{N^2}\\tilde{n}x^2  \n                                           +O\\left( \\frac1N+\\frac{1}{N^3}\\left(\\tilde{n}^4+x^4\\right)\\right) \\bigg\\}dx \\\\[3mm]\n        = \\exp\\bigg\\{-\\frac{3}{N}m^2 + \\frac{9}{N^2}\\tilde{n}m^2+  \n        O\\left(\\frac1N+ \\frac{1}{N^3}\\left(\\tilde{n}^4+m^4\\right)\\right)\\bigg\\}.      \n\\end{eqnarray*}\n\\noindent\nIt follows that for integers $0<m\\leq n$ with $m<\\sqrt{2N\\log N} $ and $|\\tilde n| = \\left|n-\\frac N3 \\right|<\\sqrt{N\\log N}$,\n\\begin{eqnarray*}\n&&  P(2\\leq D_N\\leq 2m,T_N =2n) \\\\ \n&&=\\frac{ 3^{\\frac32}}{ 2\\pi N}\\int_{[\\frac12,m+\\frac12)} e^{-\\left\\{\\frac{3}{N}x^2 -\\frac{9}{N^2}\\tilde{n}x^2 +O\\left(\\frac{1}{N^3}x^4\\right)\\right\\}}dx \\,\ne^{-\\left\\{ \\frac{9}{N}\\tilde{n}^2 +\\frac{9}{N^2}\\tilde{n}^3  +O\\left(\\frac1N+ \\frac{1}{N^3}\\tilde{n}^4 \\right)\\right\\} }\\\\ \n&& = \\frac{3^{\\frac32}}{ 2\\pi N}\\int_{[\\frac 12,m+\\frac12)} e^{-\\left\\{\\frac{3}{N}x^2 -\\frac{9}{N^2}\\tilde{n}x^2 \\right\\}}dx \\, \n e^{-\\left\\{ \\frac{9}{N}\\tilde{n}^2 + \\frac{9}{N^2}\\tilde{n}^3  + O\\left(\\frac 1N+\\frac{1}{N^3}\\tilde{n}^4\\right)\\right\\}}, \n\\end{eqnarray*}\n\\noindent\nand again we may use this for all integers $m$ and $n$ with $0<m\\leq n\\leq \\frac12 N$ with impunity.\n\nFor real $r>\\frac12 $ we write $r=m+\\theta$ where $m=\\lfloor r\\rfloor$ and $\\theta= \\mbox{frac} (r)=r-\\lfloor r \\rfloor \\in[0,1)$ denote the integer and fractional parts of r respectively. Then for $r<\\sqrt{2N\\log N}$ and $|\\tilde n| = \\left|n-\\frac N3\\right|<\\sqrt{N\\log N}$,\n\\begin{eqnarray*}\n P(2\\leq D_N\\leq 2r,T_N =2n) = P(2\\leq D_N \\leq 2m,T_N =2n)=  \\qquad \\\\[3mm] \n \\frac{3^{\\frac32}}{ 2\\pi N}e^{-\\left\\{\\frac{9\\tilde{n}^2}{N}+ \\frac{9\\tilde{n}^3}{N^2}+ O\\left(\\frac1N+\\frac{\\tilde{n}^4}{N^3}\\right)\\right\\}}  \\bigg[ \\int\\limits_{[\\frac12,r)}e^{\\frac{-3x^2}{N}+\\frac{9\\tilde{n}x^2}{N^2}}dx  \n+   \\int\\limits_{[r,m+\\frac12)} e^{\\frac{-3x^2}{N}+\\frac{9\\tilde{n}x^2}{N^2}} dx \\bigg]  .\n\\end{eqnarray*}\n\\noindent\nEvaluating the second integral by expanding the integrand about the point $x=m+\\frac12 $, we arrive at\n\\begin{eqnarray*}\n&& P(2\\leq D_N\\leq 2r,T_N =2n)  = \n\\frac{3^{\\frac32}}{ 2\\pi N}e^{-\\left\\{ \\frac{9}{N}\\tilde{n}^2 +\\frac{9}{N^2}\\tilde{n}^3+\n O\\left(\\frac1N+\\frac{1}{N^3}\\tilde{n}^4  \\right)\\right\\}} \\\\ && \\quad  \\quad\\times \n \\left[ \\int_{[\\frac12,r)} e^{-\\frac{3}{N}x^2 +\\frac{9}{N^2}\\tilde{n}x^2}dx -\n e^{-\\frac{3}{N}r^2 +\\frac{9}{N^2}\\tilde{n}r^2 } \\left(\\mbox{frac} (r)-\\frac12+O\\left(\\frac rN\\right)\\right) \\right]. \n\\end{eqnarray*}\n\\noindent\nAgain we may use this for all $r>0$ and integer $n\\leq\\frac12 N$.\n\nChoose $w>0$ and $r=w\\sqrt{\\frac n2}$. We have\n\\begin{eqnarray*}\n&& P(0<W_N\\leq w, T_N \\mbox{ is even }) = \\sum_{1\\leq n\\leq \\frac N2 } P(2\\leq D_N\\leq 2r, T_N=2n)\\\\ \n&&= \\sum_{1\\leq n\\leq \\frac N2} \\frac{3^{\\frac32}}{2\\pi N}e^{-\\left\\{\\frac{9}{N}\\tilde{n}^2  +\\frac{9}{N^2}\\tilde{n}^3\n+O\\left(\\frac1N+\\frac{1}{N^3}\\tilde{n}^4\\right)\\right\\}}\\\\\n && \\qquad \\times  \n\\left[ \\int_{[\\frac12,r)}e^{-\\frac{3}{N}x^2 +\\frac{9}{N^2}\\tilde{n}x^2 }dx-\ne^{-\\frac{3}{N}r^2 +\\frac{9}{N^2}\\tilde{n}r^2 }  \\left( \\mbox{frac} (r)-\\frac12 +O\\left(\\frac rN \\right)\\right)         \\right].  \n\\end{eqnarray*}\n\n\\noindent\nAs $|\\tilde n| = |n- \\frac N3|\\leq \\frac N3$, the expression between square brackets is of order $\\sqrt{N}$. Next, comparison with the normal $N(\\frac N3,\\frac {N}{18})$ distribution with mean \n$\\frac N3$ and variance $\\frac {N}{18}$ shows that  \n\n\\begin{equation*}\n\\frac{3}{\\sqrt{\\pi N}} \\sum_{1\\leq n\\leq \\frac N2 } e^{-\\left\\{\\frac{9}{N}\\tilde{n}^2 +\\frac{9}{N^2}\\tilde{n}^3+ O\\left(\\frac1N+\\frac{1}{N^3}\\tilde{n}^4 \\right)\\right\\}}      \n   = \\frac{3}{\\sqrt{\\pi N}} \\sum_{1\\leq n\\leq \\frac N2} e^{-\\frac{9}{N}\\tilde{n}^2 } + O\\left(\\frac 1N\\right),  \n\\end{equation*}\n\\noindent\nand hence \n\n\\noindent\n        \\begin{equation}\n                \\begin{split}\n                 P(0<W_N \\leq w, T_N \\mbox{ is even}) \n                 = O\\left( \\frac 1N\\right) + \\frac12 \\sum_{1 \\leq n \\leq \\frac N2 }\\frac{3}{\\sqrt{\\pi N}}e^{-\\frac{9}{N}\\tilde{n}^2 }\\sqrt{\\frac{3}{\\pi N}} \n\\\\ \\times \\left[ \\int\\limits_{[\\frac12 ,r) }e^{-\\frac{3}{N}x^2 +\\frac{9}{N^2}\\tilde{n}x^2  }dx- \n                e^{-\\frac{3}{N}r^2 +  \\frac{9}{N^2}\\tilde{n}r^2  }   \\left(\\mbox{frac} (r)-\\frac12 +O\\left(\\frac rN\\right)\\right)\\right] \\\\ \n             \\end{split}\n\\end{equation}\n\n\\noindent\nSince $\\sqrt{ \\frac{3}{\\pi N} }e^{ -\\frac{3}{N}x^2}$ is the density of the $N\\left(0,\\frac N6\\right)$ distribution, we see that\n$$ \\sqrt{ \\frac{3}{\\pi N} } \\int_{[\\frac12,r)} e^{-\\frac{3}{N}x^2+\\frac{9}{N^2}\\tilde{n}x^2 }dx =  \n\\sqrt{ \\frac{3}{\\pi N} } \\int_{[\\frac12,r)} e^{-\\frac{3}{N}x^2}dx + \\frac{3 \\tilde{n} g(r)}{2N} + O\\left(\\frac 1N\\right),  $$\n\\noindent\nwhere\n$$ g(r)= \\frac{1}{\\sqrt{2\\pi} } \\int_B x^2 e^{-\\frac12 x^2}dx,  $$\\smallskip\n\\noindent\nwith $B=\\left( \\frac12\\sqrt{\\frac 6N},r\\sqrt{\\frac 6N} \\right)$. Now \n$$r\\sqrt{\\frac 6N}=w\\sqrt{\\frac{3n}{N}} = w+\\frac{3w\\tilde{n}}{2N}+O\\left(w\\left(\\frac{\\tilde{n}}{N}\\right)^2\\right), $$\nand splitting the integral in one over $\\left(\\frac12 \\sqrt{\\frac 6N},w\\right)$ and one over $\\left(w,r\\sqrt{\\frac 6N} \\right)$ and expanding the latter around the point $w$, we obtain\n$$g(r)= h(w,N)+O\\left(\\frac{\\tilde{n}}{N} \\right),$$\nfor a bounded function $h$. It follows that\n\n\n$$ \\sqrt{ \\frac{3}{\\pi N} }  \\int_{[\\frac12 ,r)} e^{-\\frac{3}{N}x^2 +\\frac{9}{N^2}\\tilde{n}x^2}dx =   $$\n\n$$ =  \\sqrt{ \\frac{3}{\\pi N} } \\int_{[\\frac12,r)} e^{-\\frac{3}{N}x^2}dx + \\frac{3\\tilde{n}\\, h(w,N)}{2N} + O\\left( \\frac 1N+ \\left(\\frac{\\tilde{n}}{ N}\\right)^2 \\right).  $$\nSubstituting this in $(2.3)$ and comparing the distribution of n once more with $N\\left( \\frac N3,\\frac{N}{18} \\right)$, we see that the last two terms above contribute only $O(N^{-1})$. Finally the term containing $N^{-\\frac12}e^{-\\frac{3}{N}r^2}$. $O\\left(\\frac rN\\right)$ is clearly $O(N^{-1})$. Hence we have reduced $(2.3)$ to \n\\begin{equation}\n \\begin{split}\n         P(0<W_N\\leq w, T_N \\mbox{ is even}) = \n          \\frac12 \\sum_{1\\leq n\\leq \\frac N2} \\frac{3}{\\sqrt{\\pi N}} e^{-\\frac{9}{N}\\tilde{n}^2 } \\sqrt{\\frac{3}{\\pi N}} \\\\[3mm]\n          \\times [ \\int_{[\\frac12 ,r)} e^{-\\frac{3}{N}x^2}dx -e^{-\\frac{3}{N}r^2+\\frac{9}{N^2}\\tilde{n}r^2 } \n        \\left( \\mbox{frac} (r)-\\frac12 \\right) +O\\left(\\frac 1N \\right).  \n\\end{split}\n\\end{equation}\n\\noindent\nNow\n$$ \\frac{1 }{\\sqrt{N}}e^{-\\frac{3}{N}r^2 +\\frac{9}{N^2}\\tilde{n}r^2 } = \\frac{1}{\\sqrt{N}}e^{-\\frac12 w^2n \\left(\\frac3N-\\frac{9}{N^2}\\tilde{n} \\right)} =  \\frac{1}{\\sqrt{N}} e^{ -\\frac12 w^2\\left( 1- \\frac{9}{N^2}\\tilde{n}^2\\right) }$$\n$$ =\\frac{1}{\\sqrt{N}}e^{-\\frac12w^2}\\left(1+O\\left( \\frac{9}{N^2}\\tilde{n}^2\\right)\\right) $$\nand the remainder term will give rise to another $O(N^{-1})$ term in $(2.4)$. We obtain\n$$  P(0<W_N\\leq w, T_N \\mbox{ is even}) =O\\left(\\frac 1N\\right) +$$\n$$ \\frac12 \\sum_{1\\leq n\\leq \\frac N2} \\frac{3}{\\sqrt{\\pi N}} e^{-\\frac{9}{N}\\tilde{n}^2 } \\sqrt{\\frac{3}{\\pi N}}   \\left[ \\int_{[\\frac12,r)} e^{-\\frac{3}{N}x^2 }dx- e^{-\\frac12 w^2}\n\\left\\{\\mbox{frac}\\left(w\\sqrt{\\frac{n}{2}}\\right) - \\frac 12\\right\\} \\right]. $$\n\\medskip\n\nAs $r=w\\sqrt{n\/2}$ ,we have\n$$ \\sqrt{\\frac{3}{\\pi N}}  \\int_{[\\frac12,r)} e^{-\\frac{3}{N}x^2}dx = \\Phi\\left(w\\sqrt{\\frac{3n}{N}} \\right) -\\Phi\\left( \\sqrt{ \\frac {3}{2N} } \\right) $$\n$$= \\Phi(w)-\\frac12 - \\sqrt{ \\frac{3}{4\\pi N} }+ \\frac{3\\tilde{n}w\\varphi (w)}{2N}+ O\\left( \\left(\\frac{\\tilde{n}}{N}\\right)^2 \\right),  $$\nwhere $\\Phi$  and $\\varphi$ denote the standard normal distribution function and its density.  Obviously, the linear term containing $\\tilde{n}$ as well as the remainder term contribute $O(N^{-1})$ to $(2.4)$. Since\n$$ \\sum_{1\\leq n\\leq \\frac N2} \\frac{3}{\\sqrt{\\pi N}} e^{-\\frac{9}{N}\\tilde{n}^2 }= 1+O\\left(\\frac1N\\right) ,  $$\nwe find \n\\begin{equation}\n         \\begin{split}\n          &       P(0<W_N \\leq w, T_N \\mbox{ is even}) =  \n\\frac12 \\left(\\Phi (w)-\\frac12 -\\sqrt{ \\frac{3}{4\\pi N}}\\right) \\\\\n &\\quad -\\sqrt{\\frac{3}{2N}}\\varphi(w)\\sum_{1\\leq n\\leq \\frac N2} \\frac{3}{\\sqrt{\\pi N}} e^{-\\frac{9}{N}\\tilde{n}^2 }      \\left\\{\\mbox{frac}\\left(w\\sqrt{\\frac{n}{2}}\\right) - \\frac 12\\right\\} +O\\left(\\frac 1N \\right) \n              \\end{split}\n\\end{equation}\n\n\n  An almost identical computation produces an asymptotic expression for \nthe case where $T_N$ is odd. We have \n\\begin{equation}\n        \\begin{split}\n                &P(0<W_N \\leq w, T_N \\mbox{ is odd} ) = \\frac 12 \\left(\\Phi(w)-\\frac12 \\right)  - \\sqrt{\\frac{3}{2N}} \\varphi(w)\\  \\times \\\\\n                &\\sum_{0\\leq n\\leq \\frac{N-1}{2}} \\frac{3}{\\sqrt{\\pi N}} e^{-\\frac{9}{N}(n+\\frac12 -\\frac N3)^2 } \\left\\{\\mbox{frac}\\left(\\frac{1}{2}w\\sqrt{2n+1} + \\frac{1}{2}\\right) - \\frac 12\\right\\} +O\\left(\\frac1N \\right). \n        \\end{split}\n\\end{equation}\nA few minor modifications of $(2.5)$ and $(2.6)$ are in order. In both formulas we may replace the sum by a sum over $0 \\leq n \\leq N$ since the terms added are exponentially small in $N$. Next, in $(2.6)$,  we may change the factor $\\left(n+\\frac12 -\\frac N3\\right)^2$ in the exponent to $\\tilde n^2 = (n- \\frac N3)^2$ because the additional factor $e^{O\\left( (n-\\frac N3)\/N \\right)}=1+O\\left(\\tilde n\/N\\right)$ only produces an additional remainder term of order $N^{-1}$. Also in $(2.6)$, we may replace $\\mbox{frac} \\left( \\frac 12 w \\sqrt{2n+1}+\\frac12\\right)$ by $\\mbox{frac}\\left(\\frac12 w \\sqrt{2n}+\\frac12 \\right) =\\mbox{frac}\\left( w\\sqrt{\\frac n2}+\\frac12\\right)$. This is because $(\\frac 12 w\\sqrt{2n+1} + \\frac 12 ) - (\\frac 12 w\\sqrt{2n} + \\frac 12)$ is of exact order $\\frac{w}{\\sqrt n}$ and hence $\\frac{w}{\\sqrt N}$, so $\\mbox{frac}\\left( \\frac12 w \\sqrt{2n+1} +\\frac12 \\right) - \\mbox{frac}\\left( \\frac12 w \\sqrt{2n}+\\frac12\\right)$ is roughly equal to $-1$ for only one out of every $\\frac{c\\sqrt{N}}{w}$ values of $n$, i.e.\\ for $O\\left(w\\sqrt{N}\\right)$ values of $n$ at distances of exact order $\\sqrt{N}\/w$. For all other $n$, this difference is $O(N^{-\\frac12})$ and this implies that the substitution only adds another $O(N^{-1})$ remainder term. Hence\n\\begin{equation}\n        \\begin{split}\n                P\\left(0 < W_N\\leq w, T_N \\mbox{ is odd} \\right) = \\frac 12 \\left(\\Phi(w)- \\frac12\\right) - \\sqrt{ \\frac{3}{2N}} \\varphi(w) \\\\\n\\times  \\sum_{0\\leq n\\leq N} \\frac{3}{\\sqrt{\\pi N}} e^{-\\frac 9N \\tilde n^2}\\left[\\mbox{frac}\\left(w\\sqrt{\\frac n2}  +\\frac 12\\right)-\\frac12 \\right]+O\\left(\\frac{1}{N}\\right).\n        \\end{split}\n\\end{equation}\nIf $0\\leq \\mbox{frac}\\left( w\\sqrt{\\frac n2} \\right)<\\frac12$, then $\\mbox{frac}\\left( w\\sqrt{\\frac n2}\\right)+\\mbox{frac}\\left(w\\sqrt{\\frac n2}+\\frac12\\right) = 2\\mbox{frac}\\left( w\\sqrt{\\frac n2}\\right)+\\frac12 = \\mbox{frac}\\left(2w\\sqrt{\\frac n2}\\right)+ \\frac12 $. On the other hand, if $\\frac12 \\leq \\mbox{frac}\\left( w\\sqrt{\\frac n2}\\right)<1$, then $\\mbox{frac}\\left( w\\sqrt{\\frac n2}\\right) +\\mbox{frac}\\left( w\\sqrt{\\frac n2}+\\frac12 \\right) = 2\\mbox{frac}\\left( w\\sqrt{\\frac n2}\\right)-\\frac12 = \\mbox{frac}\\left( 2w\\sqrt{\\frac n2}\\right)+ \\frac12 $ . Hence $\\mbox{frac}\\left( w\\sqrt{\\frac n2}\\right) +\\mbox{frac}\\left( w\\sqrt{\\frac n2}+\\frac12\\right) = \\mbox{frac} \\left( 2w\\sqrt{\\frac n2}\\right)+ \\frac12 = \\mbox{frac}\\left( w\\sqrt{2n} \\right)+ \\frac12$  in both cases. Hence we may combine $(2.5)$ and $(2.7)$ to obtain for $w>0$,\n\\begin{equation}\n        \\begin{split}\n                P\\left( 0<W_N\\leq w \\right) = \\Phi(w)-\\frac12 -\\frac12 \\sqrt{ \\frac{3}{4\\pi N}} - \\sqrt{\\frac{3}{2N}} \\varphi(w) \\\\\n\\times                \\sum_{ 0\\leq n\\leq N}  \\frac{3}{\\sqrt{\\pi N}} e^{-\\frac 9N \\left(n-\\frac{N}{3}\\right)^2 } \\left[ \\mbox{frac}\\left(w \\sqrt{2n}\\right)-\\frac12 \\right]+O\\left( \\frac{1}{N} \\right) .\n        \\end{split}\n\\end{equation}\n \nBecause the distribution of $W_N$ is symmetric about the origin and $P(W_N=0)= \\sqrt{\\frac{3}{4\\pi N}}+ O(N^{-1})$, this determines the expansion for the distribution function $F_N$ of $W_N$. The expansion is uniform in $w>0$. Since it is identical to $(1.6)$, this proves the third statement of the theorem.\n\n\nIt remains to prove that any closed interval of length $O(N^{-1})$ that does not contain the origin has probability $O(N^{-1})$  under $\\Psi_N$ and hence $F_N$. Clearly, this will imply the second statement of the theorem. Obviously, the only term in $(2.8)$ that we need to consider is \n\\begin{equation*}\nR(w) = \\frac{\\Lambda_N(w)}{\\sqrt{N}}\n\\end{equation*}\n\\begin{equation}\n= -\\sqrt{\\frac{3}{2N}}  \\varphi (w) \\sum_{0\\leq n\\leq N} \\frac{3}{\\sqrt{\\pi N} }e^{-\\frac 9N\\left(n - \\frac N3\\right)^2}  \n        \\left(\\mbox{frac} \\left(w\\sqrt{2n} \\right)-\\frac12 \\right) ,\n\\end{equation}\nas the remainder of the expansion obviously has bounded derivative. \n\n We begin by noting that if for a given $w>0$, $w\\sqrt{ 2n}$ is an integer for some $1\\leq n\\leq  N$, then $\\mbox{frac} \\left(w\\sqrt{ 2n}\\right)$ and hence $R$ has a jump discontinuity at this value of $w$. In the range where\n$|n- \\frac N3|= x\\sqrt{N}$ for $|x|\\leq y$, there can be a most $wy$ such integer values of $n$. To see  this, simply note that if $w\\sqrt{ 2n}=k$ and $w\\sqrt{2n'}=k+1$ , then $|n'-n| \\geq \\frac{ 2\\sqrt{N}}{w}$ , so there can be only \n$\\frac{2y\\sqrt{N}}{ \\frac{2\\sqrt{N}}{w} }=wy$ values of $n$ in the required interval. Such a value of n contributes an amount $O\\left(N^{-1}\\varphi(w)e^{-9x^2}\\right)$ to the jump discontinuity at $w$, and hence $R(w)-R(w-0)= O(N^{-1})$ at such a point $w$. Incidentally, this proves the second part of Theorem 1.1.\n\n Choose $\\epsilon>0$ and consider two such jump points $w\\not=w'$ in $[\\epsilon , \\infty)$ with $w\\sqrt{  2n}=k$ and $w'\\sqrt{2n'}=k'$ for integers $k, k', n$ and $n'$ with $(n-\\frac N3)=x\\sqrt{N},\\ \\left(n'-\\frac N3\\right)=x'\\sqrt{N} $ and $|x|\\vee | x'|\\leq y$. Suppose that $(w'-w)=O(N^{-1})$ and hence $\\frac{w'-w}{w}=O(N^{-1})$ since $w\\geq \\epsilon$. For given $w$, $n$ and $k$, we ask how many integer values of $n'$ satisfy these conditions.\n\nFirst we note that, for some positive $c$ there are only at most $cw(y+1)$ possible choices for $k'$ since $\\sqrt{2n} = \\sqrt{2\\frac N3 +2x\\sqrt{N}} = \\sqrt{\\frac{2N}{3}} +\\sqrt{\\frac32}x+O\\left( \\frac{y^2}{\\sqrt{N}} \\right), \\sqrt{2n'} =\\sqrt{\\frac23 N} +\\sqrt{\\frac 32}x'+O\\left(\\frac{y^2}{\\sqrt{N}}\\right)$ and hence $|k'-k|\\leq 2wy + O\\left( w\\frac{y^2}{\\sqrt N}+|w'-w|\\sqrt{N}\\right)\\leq \\left(\\frac c2 \\right)w(y+1)$. For each choice of $k'$, the corresponding $n'$ satisfies $n' =\\frac12 \\left(\\frac{k'}{w'}\\right)^2$ for some admissible $w'$, and since $w,w'\\geq \\epsilon$ and $(w'-w)=O(N^{-1})$, this leaves a range of order $O\\left( \\left(\\frac{k'}{w'}\\right)^2 N^{-1} \\right)=O(1)$ for $n'$. Hence, for some $C>0$, there are at most $Cw(y+1)$ possible values of $n'$ for which there exists an integer $k'$ with $(w'-w) = O(N^{-1})$. By the same argument as above, the total contribution of discontinuities to$|R(w')-R(w) |$ is $O(N^{-1})$ as long as $|w-w'|= O(N^{-1})$. As any closed interval of length $O(N^{-1})$ that does not contain the origin is bounded away from $0$, this holds for the sum of the discontinuities in such an interval.\n\n At all other points $w>0$, $R$ is differentiable and the derivative of $\\mbox{frac} \\left(w\\sqrt{ 2n}\\right)$ equals $\\sqrt{2 n}$. Hence the derivative of $R$ is $O(1)$ and its differentiable part contributes at most $O(N^{-1})$ to the probability of any interval of length $O(N^{-1})$. This completes the proof of the Theorem 1.1.\n\n\n\n\\section{Evaluation of the oscillatory term}\n\nLet $W$ denotes a r.v. with non negative c.f. $\\psi(t)\\ge 0$ of\nsupport contained in $[-1,1]$ and exponential decay of density \nof type $\\exp\\{-|x|^{2\/3}\\},\\,x\\to\\infty$, \n\\citep[see e.g.] [p. 85] {Bhattacharya-Rao:1986\/2}.\nIntroduce  r.v. $w_N := w +N^{-1\/2}(\\log N)^{-1}\\4 W, \\, w >0$ and \nlet $c>0$ denote  an positive absolute constant.   \nThen we  may bound   the normal approximation\nerror in $(1.6)$ using  similar arguments as in the proof of  \nthe well-known smoothing inequality, (see Lemma 12.1 of  Bhattacharya and Rao),\n obtaining, for $w\\ge 1$,\n\\begin{equation}\n\\begin{split}\n\\,N^{-1\/2}\\bigl|\\mathbf E \\Lambda_N(w_N)\\bigr| \\le & \n\\, \\bigl|\\mathbf E \\bigl(F_N(w_N) -\\Phi(w_N)\\bigr)\\bigr| +\ncN^{-1}\\\\ \n& \\,\\le \\sup_{x\\in[w-1\/2,w+1\/2]}\\bigl|F_N(x) -\\Phi(x)\\bigr|+cN^{-1}, \n\\end{split}\n\\end{equation} \nwhere\n$$\\Lambda_N(w):=  \\, -\\varphi(w)\\4 \n\\sum_{1\\le n\\le N}\\frac {3^{3\/2}}{\\sqrt{2 \\pi N}}\n\\exp\\{-\\frac 9 N(n - \\frac N 3)^2\\}(\\mbox{frac}(w \\sqrt {2n})- 1\/2 ).$$\nWe start with the following Fourier series expansion\n\\begin{eqnarray} \n\\tau(x):= frac(x)-1\/2 = -\\sum_{k=1}^{\\infty} 2\\4  \\frac{ sin(2 \\pi\\4 k\\4 x)}\n{2\\4 k\\4 \\pi}, \n\\end{eqnarray*}\nwhich holds for all nonintegral $x$. \n\nNote that by the properties of $W$ (i.e. the vanishing of Fourier coefficients)\n$$ \n\\mathbf E \\tau(w_N\\sqrt{2n})=  -\\sum_{k=1}^{M_n}  \\mathbf E \\frac{ sin(\n  2\\pi\\4 k\\4 \\sqrt{2n}(w+N^{-1\/2}(\\log N)^{-1}\\4 W))} { k\\4 \\pi},\n$$\nwhere   $M_n:=[\\sqrt N\\log N\/(2\\pi\\sqrt {2n})]+1$, i.e. $M_n = O(\\log N)$ for \n$|n-N\/3|<\\sqrt{N\\log N}$.\n\nRewriting $\\Lambda_N(w)$ in $(3.1)$ in the form\n\\begin{equation}\n\\Lambda_N(w) := \\, -\\frac {3^{3\/2}} {(2 \\4 \\pi\\4 N)^{1\/2}} \\varphi(w) \n\\sum_{n=1}^{N} exp\\{-9\\4 (\\tilde{n}^2\/N\\}\\4 \n\\tau\\bigl(w\\4(2 \\4 n)^{1\/2}\\bigr),\n\\end{equation}\nwhere $\\tilde{n}:=n-N\/3$,\nwe get \n\\begin{equation}\n\\begin{split}\n\\mathbf E \\Lambda_N(w_N) =& \\, \\sqrt{\\frac 3 2}\\pi^{-1} \n\\sum_{k=1}^{M} \\frac 1 k \n\\lambda_{N,k}+O(N^{-3}), \\,\\, \n\\text{where}\\,\\, M:=[\\log N]\\,\\, \\text{and}\\\\\n\\lambda_{N,k}  :=& \\, \\frac 3{\\sqrt{\\pi \\4 N}} \\mathbf E \\varphi(w_N)\\sum_{n=1}^{N} \nexp\\{-9\\4 \n\\tilde{n}^2\/N\\}\\4  sin(2\\pi\\4 k\\4 w_N\\4  \\sqrt{2n}).\n\\end{split}\n\\end{equation}\nIn the arguments of the $\\sin$ function we\n use a Taylor expansion, for $|n-N\/3|<\\sqrt{N\\log N}$, \n$$\n\\sqrt{n} = \\sqrt{N\/3} + \\sqrt 3 \\tilde{n}\/(2\\sqrt N)+\nO\\bigl(\\tilde{n}^2\/N^{3\/2} \\bigr).\n$$\nThus, for $|\\tilde{n}|<\\sqrt{N\\log N}$,  \n\\begin{equation}\n sin(2\\pi\\4 k\\4 w_N\\4 \\sqrt{2n})=  sin\\bigl(d_0 + \\4 \\pi\\4 d_1 \\tilde{n} \\bigr) \n+  O\\bigl(k\\4 w_N N^{-3\/2}\\4 \\tilde{n}^2\\bigr),\n\\end{equation}\nwhere $ d_0 :=2\\pi\\4k \\4 w_N\\4 (\\frac 2 3)^{1\/2} \\sqrt{N}$, \n$d_1 := k\\4 \\4 w_N\\4\n(\\frac 3 2)^{1\/2}\\4\/ \\sqrt N$. %\nHence  we may write\n\\begin{equation}\n\\begin{split}\n \\lambda_{N,k} =& \\frac 3 {\\sqrt{\\pi \\4 N}} \\mathbf E \\varphi(w_N) \n\\sum_{n\\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} exp\\{-9\\4 \n\\tilde{n}^2\/N\\}\\4\n\\sin\\bigl(d_0 + \\4 2\\pi \\4d_1 \\4\\tilde{n}\\bigr)\\\\\n&\\quad  + O(k  N^{-1\/2} \\4 \\log N ). \n\\end{split}\n\\end{equation}\nWe shall now evaluate the theta sum on the left hand side using\nPoisson's formula, \n\\citep[see e.g.][p. 189]{Mumford:1983}.\n\\begin{equation}\n\\sum_{m \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} \\exp\\{-z\\4 m^2 +i2\\pi\\4 m \\4 b\\} = \\pi^{1\/2}z^{-1\/2}\n\\sum_{l \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }}\n\\exp\\{- \\pi^2 \\4 z^{-1}(l- b)^2\\},\n\\end{equation}\nwhere $b \\in \\mathbb R$, $\\Re z >0$ and $z^{1\/2}$ denotes the branch with \npositive real part.\nWriting $sin(x)=(\\exp[i\\4 x] - \\exp[-i\\4 x])\/2$ in (3.6) \nand assuming for simplicity $N\/3 \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }$ we may replace  summation over n by\nsummation over $ m:=\\tilde{n}= n - N\/3 \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }$ in (3.6). \nApplying now (3.7) we have  to bound the imaginary part of \n expectations of  theta functions of type \n\\begin{eqnarray}\nI_k:=\\frac 3 {\\sqrt{\\pi\\4 N}} \\mathbf E \\varphi(w_N)\\exp\\{\\4 i\\4 d_0 \\}\n\\sum_{m\\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} \\exp\\{-\n9\\4 m^2 \\4 N^{-1} + i\\4 2\\pi\\4 d_1\\4m\\}.\n\\end{eqnarray*}\nWe obtain for $ k \\le M =[\\log N]$ that $|d_1| \\le 2\n \\4N^{-1\/2}(\\log N) |w_N| \\le 4 \\4N^{-1\/2}(\\log N)^2 $ with probability\n $1- O(N^{-3\/2})$ by the assumption $w \\le \\log N$.\n   Hence the dominant term\nin (3.9) below  is the term with $l=0$ and we obtain with \n$c_{N,k}:=  \\exp\\{ \\4 2\\pi\\4 i\\4 \\4k \\4 w_N\\4 (\\frac 2 3)^{1\/2} \\sqrt{N}\\}$ \n\\begin{equation}\n\\begin{split}\nI_k   =& \\,  \\mathbf E c_{N,k}\\varphi(w_N)\\4 \\sum_{l \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} \n\\exp\\{ - N \\4( l- d_1)^2\\4 \\pi^2\/9\\}\n\\\\\n   = & \\,  \\mathbf E  c_{N,k}\\4 \\varphi(w_N)\\exp\\{-  N\\4 d_1^2\\4 \\pi^2\/9\\} + O\\bigl(N^{-3\/2}\\bigr)\n\\\\ \n =& \\,\\4 f_{N,k}\\4 \\varphi(w) \\exp\\{- \\pi^2 k^2 \\4  w^2\/6  \n+ \\4 i\\4 \\42\\pi k \\4 w\\4 (\\frac 2 3)^{1\/2} \\sqrt{N}\\}\n+  O\\bigl(N^{-1\/2}(\\log N)^4\\bigr),\n\\end{split}\n\\end{equation}\nwhere $f_{N,k}:= \\psi\\bigl(2\\pi\\4 (\\frac 2 3)^{1\/2} \\4 \\frac k {\\log\n  N}\\bigr)= 1+ O\\bigl((k\/\\log N)^2\\bigr)$. \nUsing the equation (3.9) in (3.4)  we get\n\\begin{equation}\n\\begin{split}\n\\mathbf E \\Lambda_N(w_N) =& \\, \\sqrt\\frac 3 2 \n\\varphi(w)\\Im \\sum_{k=1}^M \\frac {f_{N,k}} {k\\pi}\n\\exp\\bigl\\{- \\frac{\\pi^2}6\\4 k^2\\4  w^2\\4+ \n2\\pi\\4 i\\4 \\4k \\4 w\\4 \\sqrt{\\frac{2N}3}\\bigr\\}\\\\\n& \\, \\quad + O\\bigl(N^{-1\/2}(\\log N)^5\\bigr).\n\\end{split}\n\\end{equation}\n\nHence, there exists a constant $c_0(w) >0$ such that\n\\begin{equation}\n|\\mathbf E \\Lambda_N(w_N)| > c_0(w) >0,\n \\end{equation} \nprovided that  $4\\4 w\\4 \\sqrt{\\frac{2 N} 3} $ is an odd integer, which\nproves the assertion $(1.10)$. \n\n\\renewcommand\\bibsection{\\section*{REFERENCES}}\n\\bibliographystyle{ims}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA hypergraph $H = (V,E)$ is said to be bipartite or 2-colorable\nif the vertex set $V$ can be partitioned into two disjoint sets $V_1$ and $V_2$\nsuch that every edge $e\\in E$ has non-empty intersections with both the partitions.\nIn the case of graphs, one can easily find the two partitions from any given instance of\n$H$ by breadth first search.\nHowever, the problem turns out to be notoriously hard if edges of size more than 2 are present.\nIn fact, in the case of bipartite 3-uniform and 4-uniform hypergraphs,\nit is well known that the problem is NP-hard~\\cite{Dinur_2005_jour_Combinatorica,Khot_2014_conf_SODA}.\n\nIn general, finding a proper 2-coloring is relatively easy if the hypergraph is sparse. \nIn an answer to a question asked by Erd\\\"os~\\cite{Erdos_1963_jour_NordikMat} on 2-colorability of uniform hypergraphs, it is now known that for large $m$,\nany $m$-uniform hypergraph on $n$ vertices with at most \n$2^m0.7\\displaystyle\\sqrt{\\frac{m}{\\ln m}}$ edges is 2-colorable~\\cite{Radhakrishnan_1998_conf_FOCS}. As pointed in~\\cite{Radhakrishnan_1998_conf_FOCS}, the result can also be extended to \nnon-uniform hypergraphs with minimum edge size $m$. \nHowever, it is much worse if the restriction on the minimum edge size and the\nnumber of hyperedges is not imposed. Even when a hypergraph is 2-colorable, the best\nknown algorithms~\\cite{Alon_1996_jour_NordicJComput,Chen_1996_conf_IPCO}\nrequire $O\\left((n\\ln n)^{1-1\/M}\\right)$ colors to properly color the hypergraph\nin polynomial time,\nwhere $M$ is the maximum edge size, also called dimension, of the hypegraph.\nIn recent years, 2-colorability of random hypergraphs has also received considerable attention.\nThrough a series of works~\\cite{Achlioptas_2008_conf_FOCS,CojaOghlan_2012_conf_SODA,Panangiotou_2012_conf_STOC},\nit is now established that random uniform hypergraphs are 2-colorable only when\nthe number of edges are at most $Cn$, for some constant $C>0$.\nThus, it is evident that coloring relatively dense hypergraphs is difficult unless the \nhypergraph admits a ``nice\" structure.\n\nIn spite of the hardness of the problem,\nthere are a number of applications that require hypergraph coloring algorithms.\nFor instance, such algorithms have been used for approximate DNF counting~\\cite{Lu_2004_jour_SIAMJDiscMath}, as well as in various resource allocation and scheduling\nproblems~\\cite{Capitanio_1995_jour_IJPP,Ahuja_2002_conf_APPROX}.\n The connection between ``Not-All-Equal\" (NAE) SAT\nand hypergraph 2-coloring also demonstrate its significance in context of satisfiability problems. \nAmong the various approaches studied in the literature, perhaps\nthe only known non-probabilistic instances of efficient 2-coloring are in the cases \nwhere the hypergraph is $\\alpha$-dense, 3-uniform and bipartite~\\cite{Chen_1996_conf_IPCO},  \nor where the hypergraph is $m$-uniform and its every edge has equal number of vertices of either colors~\\cite{McDiarmid_1993_jour_CombProbComput}.\n\nIn this paper, we consider the problem of coloring random non-uniform hypergraphs of dimension $M$,\nthat has an underlying planted bipartite structure. We present a polynomial time algorithm\nthat can properly 2-color instances of the random hypergraph with high probability whenever\nthe expected number of edges in at least $dn\\ln n$ for some constant $d>0$.\nTo the best of our knowledge, such a model has been only considered \nby Chen and Frieze~\\cite{Chen_1996_conf_IPCO}, who extended a graph coloring \napproach of Alon and Kahale~\\cite{Alon_1997_jour_SIAMJComput} to \npresent an algorithm for \n 2-coloring of 3-uniform bipartite hypergraphs with $dn$ number of edges.\nTo this end, our work generalizes the results  of \\cite{Chen_1996_conf_IPCO} to\nnon-uniform hypergraphs, and it is the first algorithm that is guaranteed to properly color\nnon-uniform bipartite hypergraphs using only two colors. We also discuss the possible extension \nof our approach to the case of non-uniform $k$-colorable hypergraphs.\n\n\\subsection*{The Main Result}\nBefore stating the main result of this paper, we present the planted model under \nconsideration, which \nis based on the model that is studied in~\\cite{Ghoshdastidar_2015_arxiv}.\nThe random hypergraph $H_{n,(p_m)_{m=2,\\ldots,M}}$ is generated\non the set of vertices $V = \\{1,2,\\ldots,2n\\}$, which is arbitrarily split into \ntwo sets, each of size $n$, and the sets are colored with two different colors.\nGiven a integer $M$, and $p_2,\\ldots,p_M\\in[0,1]$, the edges of the hypergraph\nare randomly added in the following way. All the edges \nof size at most $M$ are added independently, and for any $e\\subset V$, \n\\begin{align*}\n \\P(e\\in E) = \\left\\{ \\begin{array}{ll}\n     p_m & \\text{if } e \\text{ is not monochromatic and } {|e|=m}, \\\\\n     0\t     & \\text{otherwise}.\\\\\n                      \\end{array}\\right.\n\\end{align*}\nWe prove the following result.\n\\begin{theorem}\n\\label{thm_spec_color}\n Assume $M=O(1)$. There is a constant $d>0$ such that if \n \\begin{equation}\n \\sum\\limits_{m=2}^M p_m \\binom{2n}{m} \\geq {dn\\ln n}, \n \\end{equation}\n then with probability \n $(1-o(1))$, Algorithm~\\ref{alg} (presented in next section) finds a proper 2-coloring of the random non-uniform bipartite hypergraph $H_{n,(p_m)_{m=2,\\ldots,M}}$. \n\\end{theorem}\nIt is easy to see that the expected number of edges in the hypergraph is \n$\\Theta\\left(\\sum_{m=2}^M p_m\\binom{2n}{m}\\right)$, and so the condition may be stated\nin terms of expected number of edges.\n\n\\subsection*{Organization of this paper}\nThe rest of the paper is organized in the following manner.\nIn Section~\\ref{sec_algorithm}, we present our coloring algorithm, followed by a proof of \nTheorem~\\ref{thm_spec_color} in Section~\\ref{sec_proof}. In the concluding remarks in \nSection~\\ref{sec_conclusion}, we provide discussions about the key assumptions made in this work,\nand also the possible extensions of our results to $k$-coloring and strong coloring of non-uniform\nhypergraphs.  The appendix contains proofs of the lemmas mentioned in Section~\\ref{sec_proof}.\n\n\\section{Spectral algorithm for hypergraph coloring}\n\\label{sec_algorithm}\n\nThe coloring algorithm, presented below, is similar \nin spirit to the spectral methods of~\\cite{Alon_1997_jour_SIAMJComput,Chen_1996_conf_IPCO},\nbut certain key differences exist, which are essential to deal with\nnon-uniform hypergraphs. \n\nGiven a hypergraph $H = (V,E)$, \nan initial guess of the color classes is formed by exploiting the spectral properties of a certain matrix\n$A\\in\\mathbb{R}^{|V|\\times|V|}$ defined as \n\\begin{align}\n A_{ij} = \\left\\{ \\begin{array}{rl}\n \\displaystyle\\sum_{e\\in E: e\\ni i,j} \\frac{1}{|e|}\t& \\text{if } i\\neq j, \\text{ and} \\\\\n \\displaystyle\\sum_{e\\in E: e\\ni i} \\frac{1}{|e|}\t& \\text{if } i= j. \n \\end{array}\\right.\n \\label{eq_defnA}\n\\end{align}\nThe above matrix has been used in the literature to construct the Laplacian of a \nhypergraph~\\cite{Bolla_1993_jour_DiscreteMath,Ghoshdastidar_2015_arxiv},\nand is also known to be related to the affinity matrix of the star expansion of \nhypergraph~\\cite{Agarwal_2006_conf_ICML}. \nThe use of matrix $A$ is in contrast to the adjacency based graph construction of~\\cite{Chen_1996_conf_IPCO} that is likely to\nresult in a complete graph if the hypergraph is dense.\n\nThe later stage of the algorithm considers an iterative procedure that\nis similar \nto~\\cite{Alon_1997_jour_SIAMJComput,Chen_1996_conf_IPCO}, but uses a  \nweighted summation of neighbors. Such weighting is crucial while\ndealing with the edges of\ndifferent sizes.\n\n\\begin{varalgorithm}{COLOR}\n\\caption {-- Colors a non-uniform hypergraph $H$:}\n\\label{alg}\n\\begin{algorithmic}[1]\n \\STATE Define the matrix $A$ as in~\\eqref{eq_defnA}.\n \\STATE Compute\n $x^A = \\underset{\\Vert x \\Vert_2 = 1}{\\textup{arg min~}} x^TAx$.\n \\STATE Let $T = \\lceil \\log_2 n\\rceil$, $V_1^{(0)} = \\{ i\\in V: x_i^A \\geq 0\\}$ and \n $V_2^{(0)} = \\{ i\\in V: x_i^A < 0\\}$.\n \\FOR {$t = 1,2,\\ldots, T$}\n \\STATE Let \n $V_1^{(t)} = \\left\\{ i\\in V: \\sum\\limits_{j\\in V_1^{(t-1)}\\backslash\\{i\\}} A_{ij} <\n \\sum\\limits_{j\\in V_2^{(t-1)}\\backslash\\{i\\}} A_{ij} \\right\\}$, \n \\newline and $V_2^{(t)} = V\\backslash V_1^{(t)}$.\n \\ENDFOR\n \\IF {{$\\exists e\\in E$} such that $e\\subset V_1^{(T)}$ or $e\\subset V_2^{(T)}$}\n \\STATE Algorithm FAILS.\n \\ELSE\n \\STATE 2-Color $V$ according to the partitions \n $V_1^{(T)},V_2^{(T)}$.\n \\ENDIF\n\\end{algorithmic}\n\\end{varalgorithm}\n\n\\section{Proof of Main Result}\n\\label{sec_proof}\n\nWe now prove Theorem~\\ref{thm_spec_color}.\nWithout loss of generality, assume that the true color classes in $V$ are $\\{1,2,\\ldots,n\\}$ and $\\{n+1,\\ldots,2n\\}$.\nAlso, let $W^{(t)}$, $t=0,1,\\ldots,T$, denote the incorrectly colored\nvertices after iteration $t$,\nwith $W^{(0)}$ being the incorrectly colored nodes after initial spectral step.\nWe prove Theorem~\\ref{thm_spec_color} by showing with probability $(1-o(1))$, \nthe size of $W^{(T)} <1$, which implies that all nodes are correctly colored, and hence, the hypergraph must be \nproperly colored.\n\nThe first lemma bounds the size of $W^{(0)}$, \\textit{i.e., } the error incurred at the initial spectral step.\n\\begin{lemma}\n\\label{lem_spectral}\nWith probability $(1-o(1))$,\n$|W^{(0)}| \\leq \\displaystyle\\frac{n}{M^22^{2M+4}}$.\n\\end{lemma}\nNext, we analyze the iterative stage of the algorithm to make the following claim,\nwhich characterizes the vertices that are correctly colored after iteration $t$.\n\\begin{lemma}\n\\label{lem_iteration_charac}\n Let $\\eta = \\displaystyle\\frac{1}{2^{M+2}}\\sum\\limits_{m=2}^M\\frac{p_m(n-1)}{m}\\binom{n-2}{m-2}$. \n For any $t\\in\\{1,\\ldots,T\\}$,\n if $\\sum\\limits_{j\\in W^{(t-1)}\\backslash\\{i\\}} A_{ij} < \\eta$ for any $i\\in V$, then \n $P(i\\in W^{(t)})\\leq n^{-\\Omega(d)}$.\n\\end{lemma}\nNote that there are only $T=\\lceil \\log_2 n\\rceil$ iterations, and $|V| =2n$. \nCombining the result of Lemma~\\ref{lem_iteration_charac} with union bound, we can conclude\nthat with probability $(1-o(1))$, for all iterations $t=1,2,\\ldots,T$, \nthere does not exist any $i\\in V$ such that\n$\\sum\\limits_{j\\in W^{(t-1)}\\backslash\\{i\\}} A_{ij} < \\eta$.\nWe also make the following observation, where $\\eta$ is defined in Lemma~\\ref{lem_iteration_charac}.\n\\begin{lemma}\n\\label{lem_iteration_size}\nWith probability $(1-o(1))$, there does not exist $C_1,C_2\\subset V$ such that $|C_1|\\leq\\frac{n}{M^22^{2M+4}}$,\n$|C_2| = \\frac12 |C_1|$ and for all $i\\in C_2$, $\\sum\\limits_{j\\in C_1\\backslash\\{i\\}} A_{ij} \\geq \\eta$.\n\\end{lemma}\nWe now use the above lemmas to proceed with the proof of Theorem~\\ref{thm_spec_color}.\nLemma~\\ref{lem_spectral} shows that $|W^{(0)}|\\leq \\frac{n}{M^22^{2M+4}}$ with probability $(1-o(1))$.\nConditioned on this event, and due to the conclusion of Lemma~\\ref{lem_iteration_charac},\none can argue that Lemma~\\ref{lem_iteration_size} is violated unless\n$|W^{(t)}| < \\frac12 |W^{(t-1)}|$ for all iteration $t$ with probability $(1-o(1))$. \nThus, in each iteration,\nthe number of incorrectly colored vertices are reduced by at least half. Hence, after \n$T=\\lceil \\log_2 n\\rceil$ iterations, $|W^{(T)}| <1$, which implies that all vertices are correctly colored.\n\n\\section{Discussions and Concluding remarks}\n\\label{sec_conclusion}\nIn this paper, we showed that a random non-uniform bipartite hypergraph of dimension $M$ \nwith balanced partitions can be properly 2-colored with \nprobability $(1-o(1))$ by a polynomial time algorithm.\nThe proposed method uses a spectral approach to form initial guess of the color classes,\nwhich is further refined iteratively.\nTo the best of our knowledge, this is the first work on 2-coloring bipartite non-uniform hypergraphs.\nPrevious works~\\cite{Chen_1996_conf_IPCO,Krivelvich_2003_jour_JAlgo} \nhave only restricted to the case of uniform hypergraphs.\n\n\\subsection*{A note on the assumptions in Theorem~\\ref{thm_spec_color}}\n\nThe key assumptions made in this paper are the following:\n\\begin{enumerate}\n\\item $M = O(1)$, and \n\\item $p_2,\\ldots,p_M$ are such that\nthe expected number of edges is larger than $dn\\ln n$, where $d>0$ is a large constant.\n\\end{enumerate}\nThe assumption $M = O(1)$ is crucial, particularly in Lemma~\\ref{lem_spectral},\nand helps to ensure that $d$ can be chosen to be a constant. This can be avoided \nif $d$ is allowed to increase with $n$ appropriately. We note that a previous \nwork on spectral hypergraph partitioning~\\cite{Ghoshdastidar_2015_arxiv} allows\n$M$ to grow with $n$, but imposes an additional restriction so that the number of \nedges of larger size decay rapidly.\n\nThe second assumption is stronger than the one in \\cite{Chen_1996_conf_IPCO},\nwhere it was shown that a random bipartite 3-uniform hypergraph can be properly\n2-colored with high probability if the expected number of edges is $dn$.\nThis is due to the use of matrix Bernstein inequality~\\cite{Tropp_2012_jour_FOCM}\nin Lemma~\\ref{lem_spectral} that does not provide useful bounds in the most sparse \ncase. On the other hand, Chen and Frieze~\\cite{Chen_1996_conf_IPCO}\nuse the techniques of Kahn and Szemeredi~\\cite{Friedman_1989_conf_STOC}\nthat allows them to work in the most sparse regime. \nHowever, it is not clear how the \nsame techniques can be extended even to uniform hypergraphs of higher order.\nThus, it remains an open problem whether a similar result can be proved when the  number of edges in the hypergraph grows linearly with $n$.\n\n\\subsection*{$k$-coloring of hypergraphs}\nThough Algorithm~\\ref{alg} has been presented only for the hypergraph 2-coloring problem,\none may easily extend the approach to achieve a $k$-coloring,\nwhere the objective is to color the vertices of the hypergraph with $k$ colors such that no edge \nis monochromatic.\nA possible extension of Algorithm~\\ref{alg} is as follows:\n\\begin{enumerate}\n \\item\n In Step~2, compute the eigenvectors corresponding to the $(k-1)$ smallest eigenvalues of $A$. \n \\item\n Use $k$-means algorithm~\\cite{Ostrovsky_2013_jour_JACM} to cluster rows of the eigenvector matrix into $k$ groups,\n and define the initial guess for the color classes $V_1^{(0)},\\ldots,V_k^{(0)}$ in Step~3 according\n to the above clustering.\n \\item\n The iterative computation in Step~6 is modified by defining\n \\begin{displaymath}\n \\qquad\n V_l^{(t)} = \\left\\{ i\\in V: \\sum\\limits_{j\\in V_l^{(t-1)}\\backslash\\{i\\}} A_{ij} <\n \\sum\\limits_{j\\in V_{l'}^{(t-1)}\\backslash\\{i\\}} A_{ij} \\text{ for all } l'\\neq l\\right\\}\n \\end{displaymath}\n for $l=1,2,\\ldots,(k-1)$, and $V_k^{(t)} = V\\backslash \\left(\\bigcup_{l<k} V_l^{(t)}\\right)$.\n\\end{enumerate}\nIn the above modification, we borrow the popular idea of using $k$-means on the rows of \neigenvector matrix to find $k$ planted partitions in a graph or \nhypergraph~\\cite{Lei_2015_jour_AnnStat,Ghoshdastidar_2015_arxiv}.\n\nWe believe that the result in Theorem~\\ref{thm_spec_color} can be extended to this setting,\nwhere the random model allows for $k$ planted color classes in the hypergraph\nwith non-monochromatic edges generated in the aforementioned manner.\nAssuming $k=O(1)$ and $k$-means algorithm always provides a near optimal solution,\none can follow the arguments of~\\cite{Ghoshdastidar_2015_arxiv}\nto prove a result similar to Lemma~\\ref{lem_spectral}. \nOn the other hand, Lemmas~\\ref{lem_iteration_charac} and~\\ref{lem_iteration_size} should hold\nfor an appropriate choice of $\\eta$. Hence, one can comment that the algorithm\nachieves a proper $k$-coloring with probability $(1-o(1))$.\n\nWe also note that Algorithm~\\ref{alg} can be used for finding solutions of NAE-SAT problems.\nThe extension of~\\ref{alg} is also applicable for strong coloring of hypergraphs, which finds \napplications in design of communication networks~\\cite{Wu_2015_jour_TIT}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $X$ be a symmetric space of noncompact type, and let\n$G = \\Isom(X)^0$.\nPick a basepoint $x_0 \\in X$, with isotropy group $K \\subset G$.\nThen $X \\cong G\/K$.\nSuppose that $G=\\mathbf G({\\mathbb R})^0$, where $\\mathbf G$ is a connected almost simple\nalgebraic group defined over ${\\mathbb Q}$,\nand that $\\Gamma$ is an arithmetic subgroup of $G$. The associated locally\nsymmetric space $\\Gamma\\backslash X$ is often not compact.\nFor example let\n$X=\\SL_n({\\mathbb R})\/\\SO_n$, the space of positive definite quadratic forms in $n$\nvariables of determinant $1$.  The arithmetic subgroup $\\Gamma=\\SL_n({\\mathbb Z})$\nacts on $X$ and the associated locally symmetric space $\\Gamma\\backslash X$\nis the moduli space of unimodular lattices in ${\\mathbb R}^n$ up to isometries in\n$\\SO_n$. Noncompact arithmetic locally symmetric spaces $\\Gamma\\backslash X$\nadmit several different compactifications such\nas the Satake compactifications, the Borel-Serre compactification and the\nreductive Borel-Serre compactification.  If $\\Gamma\\backslash X$ is a\nHermitian locally symmetric space, then one of the Satake\ncompactifications, called the Baily-Borel or Satake-Baily-Borel\ncompactification, is a projective variety.\n\nThe cohomology and homology groups of locally symmetric spaces\n$\\Gamma\\backslash X$ and of their compactifications have been intensively\nstudied because of their relation to the cohomology of $\\Gamma$ and to\nautomorphic forms.\nOur interest here is the fundamental group.  Looking at some results on the\nfundamental group of the Satake-Baily-Borel compactification of particular\nHermitian locally symmetric spaces, we noticed that the calculations were\nvery similar to the calculations of the congruence subgroup kernel of the\nunderlying algebraic group (see \\citelist{\\cite{hk} \\cite{kn} \\cite{hs}\n  \\cite{ge} \\cite{Gro} \\cite{gro2} \\cite{san}}). This lead us to wonder\nabout a possible connection between the two.  In this paper we deal with\narithmetic locally symmetric spaces in general (not necessarily Hermitian)\nand their compactifications.\n\nIt turns out that a more natural compactification of $\\Gamma\\backslash X$\nto consider is the reductive Borel-Serre compactification.  We will see\nthat its fundamental group is closely related to the congruence subgroup\nkernel and this gives a topological interpretation of the congruence\nsubgroup kernel.  Usually, the congruence subgroup problem is considered\nnot only for arithmetic groups but for the larger class of $S$-arithmetic\ngroups such as $\\SL_n(\\mathbb Z[1\/p])$, where $p$ is a prime number.  These\ngroups are not discrete subgroups of Lie groups, but rather discrete\nsubgroups of products of real Lie groups and $p$-adic Lie groups.  For\nexample, $\\SL_n(\\mathbb Z[1\/p])$ is a discrete subgroup of $\\SL_n(\\mathbb\nR) \\times \\SL_n(\\mathbb Q_p)$.  Therefore, they act properly on products of\nsymmetric spaces and Bruhat-Tits buildings.  Another goal of this paper is\nto introduce analogues of Satake compactifications and the reductive\nBorel-Serre compactification for $S$-arithmetic quotients of products of\nsymmetric spaces and Bruhat-Tits buildings, to calculate their fundamental\ngroups, and to precisely investigate the connection between their\nfundamental groups and the congruence subgroup kernels.  See\n\\S\\ref{sectFundGrpArithmetic} (in particular Theorem\n~\\ref{thmMainArithmetic}) for precise statements.\n\nThe authors would like to thank Alex Lubotzky and Gopal Prasad for helpful\ndiscussions and correspondence.\n\nIn the remainder of this section we carefully introduce arithmetic and\n$S$-arithmetic groups, the congruence subgroup problem, and our results.\n\n\\subsection{Arithmetic groups and congruence subgroups}\n\nLet $k$ be a number field and let $S$ be a finite set of places of $k$ which\ncontains the infinite places $S_\\infty$.  For $v\\in S$ let $k_v$ denote the\ncompletion of $k$ with respect to a norm associated to $v$.  Denote by\n${\\mathcal O}$ the ring of \\emph{$S$-integers}\n$$ \n{\\mathcal O} = \\{\\, x\\in k \\mid \\operatorname{ord}_v (x) \\ge 0 \\text{ for all\n  $v\\notin   S$}\\,\\} \\ .\n$$\nThe corresponding group of units ${\\mathcal O}^\\times$\nis finite if and only if $|S|=1$.\n\nLet $\\mathbf G$ be an algebraic group defined over $k$ and fix a faithful\nrepresentation\n\\begin{equation*}\n\\rho\\colon \\mathbf G \\longrightarrow \\GL_N\n\\end{equation*}\ndefined over $k$.  For any nonzero ideal ${\\mathfrak a}\n\\subseteq {\\mathcal O}$, set\n\\begin{equation*}\n\\Gamma({\\mathfrak a}) = \\{\\,\\gamma \\in \\mathbf G(k) \\mid  \\text{$\\rho(\\gamma) \\in \\GL_N({\\mathcal O})$, \n$ \\rho(\\gamma) \\equiv I \\pmod{{\\mathfrak a}}$}\\,\\}\\ .\n\\end{equation*}\nIn particular, $\\mathbf G({\\mathcal O})=\\Gamma({\\mathcal O})$ is the set of elements of $\\mathbf G(k)$ that\nmap to $\\GL_N({\\mathcal O})$ under $\\rho$. Note that $\\mathbf G({\\mathcal O})$ and the family of\nsubgroups $\\{ \\Gamma({\\mathfrak a}) \\}_{{\\mathfrak a}\\subseteq {\\mathcal O}}$ depends on the representation\n$\\rho$.\n\nA subgroup $\\Gamma \\subset \\mathbf G(k)$ is an \\emph{$S$-arithmetic subgroup} if it\nis commensurable with $\\mathbf G({\\mathcal O})$; an $S_\\infty$-arithmetic subgroup is\nsimply called an \\emph{arithmetic subgroup}.  This definition is\nindependent of the choice of $\\rho$. Note that if $S_1 \\subset S_2$, then\nan $S_1$-arithmetic group is not necessarily an $S_2$-arithmetic\ngroup. (For example, $\\SL_2({\\mathbb Z})$ is of infinite index in $\\SL_2({\\mathbb Z}[1\/p])$\nfor any prime $p$ and in particular they are not commensurable.)  However\nif $\\Gamma$ is an $S_2$-arithmetic subgroup and $K_v$ is a compact open\nsubgroup of $\\mathbf G(k_v)$ for each $v\\in S_2\\setminus S_1$, then $\\Gamma\\cap\n\\bigcap_{v\\in S_2\\setminus S_1} K_v$ is an $S_1$-arithmetic subgroup.\n\nA subgroup $\\Gamma \\subset \\mathbf G(k)$ is an \\emph{$S$-congruence subgroup} if it\ncontains $\\Gamma({\\mathfrak a})$ as a subgroup of finite index for some ideal ${\\mathfrak a}\n\\subseteq {\\mathcal O}$; again if $S=S_\\infty$ this is simply called a\n\\emph{congruence subgroup}.  This definition is also independent of the\nchoice of $\\rho$.\n\n\\subsection{The congruence subgroup problem}\n\nIn its simplest form, the {\\em congruence subgroup problem} asks whether\nevery $S$-arithmetic subgroup of $\\mathbf G(k)$ is an $S$-congruence subgroup.\nThis has been studied for $\\SL_2$ over $k = {\\mathbb Q}$ since the work of Fricke\nand Klein on elliptic modular functions (see \\cite{wo}). It is well-known that\n$\\SL_2({\\mathbb Z})$ contains many non-congruence\nsubgroups; examples can be found in \\cite{MurtyRamakrishnan} and \\cite{wo}.\n\nThe congruence subgroup kernel is a quantitative measure of how close the\nquestion above is to being true.  We briefly outline the definition\n\\cite{SerreBourbaki} (see also \\citelist{\\cite{HumphreysArithmetic}\n  \\cite{Ra2}*{pp.\\ 75--76}}).  Define a topology ${\\mathcal T}_c$ on\n$\\mathbf G(k)$ by taking the set of $S$-congruence subgroups to be a fundamental\nsystem of neighborhoods of $1$.  Similarly define a topology ${\\mathcal\n  T}_a$ by using the set of $S$-arithmetic subgroups.  Let $\\widehat G(c)$\nand $\\widehat G(a)$ denote the completions of $\\mathbf G(k)$ in these topologies.\nThe corresponding completions of $\\mathbf G({\\mathcal O})$ are denoted $\\widehat G({\\mathcal O},c)$\nand $\\widehat G({\\mathcal O},a)$.  Since every $S$-congruence subgroup is also\n$S$-arithmetic, ${\\mathcal T}_a$ is in general finer than ${\\mathcal T}_c$\nand we have the following diagram\n\\begin{equation*}\n\\begin{CD}\n\\widehat G(a) @>>> \\widehat G(c) \\\\\n@AAA               @AAA \\\\\n\\widehat G({\\mathcal O},a) @>>> \\widehat G({\\mathcal O},c).\n\\end{CD}\n\\end{equation*}\nOne can also view $\\widehat G({\\mathcal O},c)$ as the projective limit of\n$\\mathbf G({\\mathcal O})\/\\Gamma({\\mathfrak a})$ over nonzero ideals ${\\mathfrak a}\\subseteq {\\mathcal O}$ and similarly for\n$\\widehat G({\\mathcal O},a)$.  Thus they are profinite (and hence compact) groups,\nwhile $\\widehat G(c)$ and $\\widehat G(a)$ are locally compact.  It is then\neasy to see that the two horizontal maps are surjective and have the same\nkernel which is  called the \\emph{congruence subgroup kernel} $C(S, \\mathbf G)$. \n\nFrom a more general perspective, the congruence subgroup problem is the\ndetermination of $C(S,\\mathbf G)$.  The case when $C(S,\\mathbf G)=1$ is equivalent to\nevery $S$-arithmetic subgroup being an $S$-congruence subgroup.\n\n\\subsection{Reductions}\n\nThe congruence subgroup problem admits a number of reductions.  The functor\n$\\mathbf G\\to C(S,\\mathbf G)$ satisfies a weak form of exactness outlined in\n\\cite{Ra1}*{Introduction}.  Since $C(S,\\mathbf G)=1$ when $\\mathbf G$ is finite or the\nadditive group $\\mathbf G_a$, this implies that $C(S,\\mathbf G) = C(S, \\mathbf G^0\/\\mathbf\nN_{\\mathbf G})$, where $\\mathbf N_{\\mathbf G}$ is the unipotent radical of $\\mathbf G$.  We thus\nmay assume that $\\mathbf G$ is connected and reductive.  A theorem of Chevalley\n\\cite{Chevalley} based on class field theory implies that $C(S,\\mathbf\nT)=1$ for $\\mathbf T$ a $k$-torus.  Together with the weak exactness\nproperty, this implies that $C(S,\\mathbf G) = C(S,\\mathpsscr D\\mathbf G)$ where\n$\\mathpsscr D\\mathbf G$ is the derived group (see also\n\\cite{PlatonovSaromet}).  It thus suffices to assume that $\\mathbf G$ is\nconnected and semisimple.\n\nIf $\\mathbf G$ is not simply connected then $C(S,\\mathbf G)$ can be infinite.\nSpecifically let $\\widetilde{\\mathbf G}$ be the simply connected covering group of\n$\\mathbf G$ and let $\\mathbf B = \\Ker (\\widetilde{\\mathbf G}\\to \\mathbf G)$.  If all $k$-simple\ncomponents $\\mathbf H$ of $\\mathbf G$ satisfy $k_v\\text{-rank}\\: \\mathbf H > 0$ for some\n$v\\in S$, then $\\Coker(C(S,\\widetilde{\\mathbf G}) \\to C(S,\\mathbf G))$ will contain an\nisomorphic copy of the infinite group $\\mathbf B({\\mathbb A}_{k,S})\/\\mathbf B(k)$,\nwhere ${\\mathbb A}_{k,S}$ denotes the $S$-adeles of $k$\n\\citelist{\\cite{SerreBourbaki} \\cite{Ra1}}.  Thus we will make the\nassumption that $\\mathbf G$ is simply connected.\n\nAny simply connected group is a direct product of almost $k$-simple groups,\nso we may assume $\\mathbf G$ is almost $k$-simple.  We may then write $\\mathbf G =\\Res_{k'\/k}\n\\mathbf G'$, where $\\mathbf G'$ is an absolutely almost simple group over a finite\nextension $k'$ over $k$.  Since $C(S,\\mathbf G) = C(S',\\mathbf G')$ where $S'$ consists\nof all places of $k'$ lying over places of $S$, we may assume that $\\mathbf G$ is\nconnected, simply connected and absolutely almost simple.\n\n\\subsection{Some known results}\n\\label{ssectKnownResults}\n\nThe congruence subgroup kernel has been considered extensively by many\nauthors; see the survey \\cite{PrasadRapinchukSurvey}.  In particular, Bass,\nMilnor, and Serre \\cite{BMS} proved that $C(S, \\mathbf G)$ is finite for the\ngroups $\\SL_n$, $n\\ge 3$, and $\\SP_{2n}$, $n\\ge 2$; in fact they prove that\n$C(S,\\mathbf G)$ is trivial unless $k$ is totally imaginary and $S=S_\\infty$ in\nwhich case $C(S,\\mathbf G)\\cong \\mu(k)$, the roots of unity in $k$.  Serre\n\\cite{se3} later treated the case $\\SL_2$ and obtained the same\ndetermination of $C(S,\\mathbf G)$ if $|S|\\ge 2$; if $|S|=1$ he proves that\n$C(S,\\mathbf G)$ is infinite.\n\nLet $S\\text{-rank}\\: \\mathbf G = \\sum_{v\\in S} k_v\\text{-rank}\\: \\mathbf G$.\nFor a global field $k$ (that is, a number field or a function field of an\nalgebraic curve over a finite field) Serre \\cite{se3} has conjectured%\n\\footnote{The hypothesis that $k_v\\text{-rank}\\: \\mathbf G>0$ for all $v\\in S\\setminus\n S_\\infty$ was not included in \\cite{se3} but is necessary\n\\cite{Ra1}*{p.~109 and (6.2)}.}\nthat if $\\mathbf G$ is simply connected and absolutely almost simple, then\n\\begin{equation}\n\\label{eqnSerreConjecture}\n\\text{$C(S,\\mathbf G)$ is finite if  $S\\text{-rank}\\: \\mathbf G \\geq 2$ and $k_v\\text{-rank}\\: \\mathbf G>0$ for\n  all $v\\in S\\setminus S_\\infty$.}\n\\end{equation}\nWhen $k$ is a number field, the main theorems in Raghunathan's papers\n\\citelist{\\cite{Ra1} \\cite{Ra2}} established the conjecture when $k\\text{-rank}\\: \\mathbf G\n> 0$ (see also \\cite{PrasadOnRaghunathan}).  For a general global field,\nPrasad and Raghunathan \\cite{pr}*{Theorem~ 2.6} established the conjecture\nwhen $k\\text{-rank}\\: \\mathbf G > 0$ provided $C(S,\\mathbf G)$ is central in $\\widehat G(a)$; in\nfact they showed \\cite{pr}*{Theorems~ 2.9, 3.4} then that $C(S,\\mathbf G)$ is a\nquotient of $\\mu(k)$ provided in addition that the Kneser-Tits conjecture%\n\\footnote{Let $\\mathbf G(k)^+$ denote the subgroup of $\\mathbf G(k)$ generated by\n  $k$-rational points of the unipotent radicals of the parabolic\n  $k$-subgroups of $\\mathbf G$.  The Kneser-Tits conjecture states that if $\\mathbf G$ is\n  simply connected, almost $k$-simple, with $k\\text{-rank}\\: \\mathbf G>0$, then\n  $\\mathbf G(k)^+=\\mathbf G(k)$.}\nholds for global fields.  The centrality of $C(S,\\mathbf G)$ was proved when\n$k\\text{-rank}\\:\\mathbf G>0$ by Raghunathan \\citelist{\\cite{Ra1} \\cite{Ra2}} (again\nassuming that the Kneser-Tits conjecture holds) and the Kneser-Tits\nconjecture for global fields has since been demonstrated \\cite{Gille}.\nThus \\eqref{eqnSerreConjecture} holds for global fields when $k\\text{-rank}\\: \\mathbf G >\n0$; for the progress on groups with $k\\text{-rank}\\: \\mathbf G=0$ see the survey by\nRapinchuk \\cite{R2}.\n\nSerre \\cite{se3} also conjectures that $C(S,\\mathbf G)$ is infinite if $S\\text{-rank}\\: \\mathbf G\n= 1$ and verifies this for $\\mathbf G = \\SL_2$.  In fact for $\\SL_2$ over ${\\mathbb Q}$,\n$C(S,\\mathbf G)$ is a free profinite group on a countable number of\ngenerators \\cite{Melnikov2}, and over a quadratic imaginary\nfield it has a finite index subgroup of this type \\cite{Lubotzky0}.\n\n\\subsection{Connection with elementary matrices}\n\\label{ssectElementaryMatrices}\n\nOur goal is a topological interpretation of the congruence subgroup kernel.\nFor this we will use the relationship of $C(S,\\mathbf G)$ with\n``elementary'' matrices.  More precisely, for any $S$-arithmetic subgroup\n$\\Gamma$ let\n\\begin{equation*}\nE\\Gamma \\subset \\Gamma\n\\end{equation*}\nbe the subgroup generated by the elements of $\\Gamma$ belonging to the\nunipotent radical of any parabolic $k$-subgroup of $\\mathbf G$.  As $\\Gamma$ runs\nthrough the family of\n$S$-congruence subgroups $\\Gamma({\\mathfrak a})$, we obtain  a family\n$\\{E\\Gamma({\\mathfrak a})\\}_{{\\mathfrak a}\\subseteq {\\mathcal O}}$ of normal subgroups of $\\mathbf G({\\mathcal O})$ which define a\ntopology ${\\mathcal T}_e$ on $\\mathbf G({\\mathcal O})$. We denote by $\\widehat G({\\mathcal O},e)$ the\ncompletion of $\\mathbf G({\\mathcal O})$ in the topology ${\\mathcal T}_e$. For any ideal ${\\mathfrak a}\n\\subseteq {\\mathcal O}$ consider the exact sequence\n\\begin{equation*}\n1 \\ \\rightarrow\\ \\Gamma({\\mathfrak a})\/E\\Gamma({\\mathfrak a})\\ \\rightarrow\n\\ \\mathbf G({\\mathcal O})\/E\\Gamma({\\mathfrak a})\\ \\rightarrow\\ \\mathbf G({\\mathcal O})\/\\Gamma({\\mathfrak a})\n\\ \\rightarrow\\ 1\\ .\n\\end{equation*}\nTaking projective limits over the ideals ${\\mathfrak a}$ we obtain\n\\begin{equation*}\n1 \\ \\rightarrow\\ CG(e,c)\\ \\rightarrow\n\\widehat{G}({\\mathcal O},e)\\ \\rightarrow\\ \\widehat{G}({\\mathcal O},c) \\ \\rightarrow\\ 1\\ ,\n\\end{equation*}\nwhere $CG(e,c)$ is defined to be the kernel of the map on the right and\nRaghunathan's Main Lemma is used to prove that this map is surjective\n\\cite{Ra1}*{(1.21)}.\n\nAssume now that $k\\text{-rank}\\: \\mathbf G > 0$ and $S\\text{-rank}\\: \\mathbf G \\ge 2$.  Then $E\\Gamma({\\mathfrak a})$ is\n$S$-arithmetic \\citelist{\\cite{Margulis} \\cite{Ra2}*{Theorem~ A, Corollary~\n    1}} (see also \\cite{Venkataramana}) and any $S$-arithmetic subgroup\n$\\Gamma$ contains $E\\Gamma({\\mathfrak a})$ for some ${\\mathfrak a}\\neq 0$ \\cite{Ra1}*{(2.1)}.  So\nunder this condition, the topologies ${\\mathcal T}_e$ and ${\\mathcal T}_a$\nare the same,\n\\begin{equation*}\n\\widehat{G}({\\mathcal O},e)\\ \\cong\\ \\widehat{G}({\\mathcal O},a)\\ ,\n\\end{equation*}\nand thus\n\\begin{equation}\n\\label{eqnCongruenceKernel}\nC(S,\\mathbf G)\\ \\cong\\ CG(e,c)\\ \\cong\\ \\varprojlim_{\\mathfrak a}\n\\Gamma({\\mathfrak a})\/E\\Gamma({\\mathfrak a})\\ .\n\\end{equation}\nThis characterization of $C(S,\\mathbf G)$ will enable us to give a topological\nrealization.\n\n\\subsection{A topological realization of $C(S,\\mathbf G)$}\n\nIn this paper, our aim is to show that the algebraically and arithmetically\ndefined group $C(S,\\mathbf G)$ also has a topological interpretation as the\nfundamental group of certain compactifications of a locally symmetric\nspace.  More precisely, we consider a connected, absolutely almost simple,\nsimply connected algebraic group $\\mathbf G$ defined over $k$. Let $\\mathbf H$\ndenote the restriction of scalars $\\operatorname{Res}_{k\/{\\mathbb Q}} \\mathbf G$ of $\\mathbf G$;\nthis is a group defined over ${\\mathbb Q}$ with ${\\mathbb Q}\\text{-rank}\\: \\mathbf H = k\\text{-rank}\\: \\mathbf G$.\nLet $X_\\infty =\\mathbf H({\\mathbb R})\/K$ be the symmetric space associated to\n$\\mathbf H$, where $K$ is a maximal compact subgroup of $\\mathbf H({\\mathbb R})$,\nand for $v\\in S\\setminus S_\\infty$, let $X_v$ be the Bruhat-Tits building\nof $\\mathbf G(k_v)$.\n\nConsider $X = X_\\infty \\times \\prod_{v\\in S\\setminus S_\\infty} X_v$.  By\ngeneralizing the work of Borel and Serre \\citelist{\\cite{Borel-Serre}\n\\cite{BS2}} and of Zucker \\cite{Zu1}, we define in\n\\S\\S\\ref{subsectRBSarith}, \\ref{subsectRBSSarith} the reductive Borel-Serre\nbordification $\\overline{X}^{RBS}$ of $X$.  For an $S$-arithmetic\nsubgroup $\\Gamma$ of $\\mathbf G(k)$, the action of $\\Gamma$ on $X$ by left translation\nextends to $\\overline{X}^{RBS}$ and the quotient\n$\\Gamma\\backslash\\overline{X}^{RBS}$ is a compact Hausdorff topological\nspace, called the \\emph{reductive Borel-Serre compactification} of\n$\\Gamma\\backslash X$.  Our main result (Theorem ~\\ref{thmMainArithmetic}) is the\ncomputation of the fundamental group of\n$\\Gamma\\backslash\\overline{X}^{RBS}$.  Under the mild condition that $\\Gamma$\nis a neat $S$-arithmetic group, we show (Corollary ~\\ref{corNeat}) that\n\\begin{equation}\n\\pi_1(\\Gamma\\backslash\\overline{X}^{RBS}) \\cong  \\Gamma \/ E\\Gamma\n\\end{equation}\nIf  $k\\text{-rank}\\: \\mathbf G >0$ and $S\\text{-rank}\\: \\mathbf G \\ge 2$ this is finite and we\nconclude from \\eqref{eqnCongruenceKernel} that \n\\begin{equation}\nC(S,\\mathbf G) \\cong \\varprojlim_{\\mathfrak a}  \\pi_1(\\Gamma({\\mathfrak a})\\backslash\\overline{X}^{RBS}).\n\\end{equation}\nIn fact we show (Corollary ~\\ref{corIdentifyCSG}) that $C(S,\\mathbf G)$ is\nprecisely $\\pi_1(\\Gamma^*({\\mathfrak a})\\backslash\\overline{X}^{RBS})$ for ${\\mathfrak a}$ small,\nwhere $\\Gamma^*({\\mathfrak a})$, defined by Raghunathan \\cite{Ra1}, is the smallest\n$S$-congruence subgroup containing $E\\Gamma({\\mathfrak a})$.\n\nFrom the point of view of identifying the congruence subgroup kernel $C(S,\n\\mathbf G)$, we see that $\\Gamma\\backslash \\overline{X}^{RBS}$ is the most natural\ncompactification of $\\Gamma\\backslash X$.  On the other hand, the Satake\ncompactifications of the locally symmetric space $\\Gamma\\backslash X_\\infty$\nare important as well, as mentioned at the beginning of this introduction.\nIn \\S\\ref{subsectSatakeSArith} we define compactifications $\\Gamma\\backslash\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ of $\\Gamma \\backslash X$ which generalize the\nSatake compactifications of $\\Gamma\\backslash X_\\infty$ and in\n\\S\\ref{sectFundGrpArithmetic} we calculate that their fundamental groups\nare a certain quotient of $\\pi_1(\\Gamma\\backslash\\overline{X}^{RBS})$.\n\n\\subsection{Connection to bounded generation}\nAlthough not directly addressed by this paper, we close this introduction\nby mentioning the relation of the congruence subgroup problem to the notion\nof bounded generation.  A fundamental result of Borel and Harish-Chandra\n\\cite{BorelHarishChandra} is that arithmetic subgroups of algebraic groups\nare finitely generated. The proof of Borel and Harish-Chandra is in fact\nconstructive, and Grunewald and Segal \\cite{GrunewaldSegal1} have shown how\nto use it to find generators.  If one assumes that the algebraic group is\nreductive then this result extends to $S$-arithmetic subgroups and in fact\n$S$-arithmetic subgroups of reductive algebraic groups are even finitely\npresented \\citelist{\\cite{BS2}*{Th\\'eor\\`eme~6.2} \\cite{GrunewaldSegal2}}.\nNote that $S$-arithmetic subgroups of a general algebraic group need not be\neven finitely generated. For example, ${\\mathbb Z}[1\/p]$ is a\n$\\{p,\\infty\\}$-arithmetic subgroup of $\\mathbf G_a$ over ${\\mathbb Q}$ and is not finitely\ngenerated.\n\nA finitely generated group $\\Gamma$ has \\emph{bounded generation} if there\nexist elements $\\gamma_1,\\gamma_2,\\dots,\\gamma_m\\in \\Gamma$ (not necessarily\ndistinct) such that any $\\gamma\\in \\Gamma$ can be written in the form\n\\begin{equation*}\n\\gamma= \\gamma_1^{k_1}  \\dots  \\gamma_m^{k_m}\n\\end{equation*}\nwith $k_1 , \\dots , k_m \\in {\\mathbb Z}$.  The least possible value of $m$ is called\nthe \\emph{degree of bounded generation}.\n\nA free group on more than one generator does not have bounded generation.\nSince $\\SL_2({\\mathbb Z})$ contains a free group of finite index on two generators\n(for example, the commutator subgroup), it follows that it does not have\nbounded generation \\cite{Murty}*{\\S5}.  Rapinchuk \\cite{R1} conjectures that\nif $\\mathbf G$ is simple and the $S$-rank of $\\mathbf G$ is $\\geq 2$, then $\\mathbf G({\\mathcal O})$\nhas bounded generation.\n\nThe relation between bounded generation and the congruence subgroup problem\nhas been clarified by recent work of Platonov and Rapinchuk\n\\cite{PlatonovRapinchuk2} and independently by Lubotzky \\cite{Lubotzky}.\nLet $T$ be the (finite) set of primes $v$ where $\\mathbf G(k_v)$ is anisotropic\nand assume that $S\\cap T=\\emptyset$.  Suppose every non-central normal\nsubgroup of $\\mathbf G({\\mathcal O})$ is the inverse image of an open normal subgroup\nunder the map\n\\begin{equation*}\n\\mathbf G (k) \\to \\prod_{v\\in T} \\mathbf G(k_v) \\ .\n\\end{equation*}\nThen if $\\mathbf G({\\mathcal O})$ has bounded generation they prove that $C(S,\\mathbf G)$ is\nfinite.\n\nThus another way to establish that $C(S,\\mathbf G)$ is finite is to show that\n$\\mathbf G({\\mathcal O})$ has bounded generation.  For example, Tavgen\\cprime\\\n\\cite{tavgen} has established that $\\mathbf G({\\mathcal O})$ has bounded generation for\n$k$-simple groups $\\mathbf G$ which are quasi-split over $k$ with $k$-rank $\\ge 2$\n(except possibly for type ${}^6D_4$).  In another direction, if $|S|$ is\nassumed sufficiently large (depending only on $[ k:{\\mathbb Q} ]$), Murty and\nLoukanidis have proved bounded generation for $\\SL_n({\\mathcal O})$, $n\\ge 2$, and\n$\\SP_{2n}({\\mathcal O})$, $n\\ge 1$; this work is announced in \\cite{Murty} and\npartially included in the thesis of Loukanidis \\cite{Loukanidis}.  The\nproof, which uses analytic number theory, actually gives an explicit bound\non the degree of bounded generation depending only on $[k:{\\mathbb Q}]$; bounds on\nthe degree which depend also on the discriminant of $k$ have been obtained\npreviously by other authors.\n\n\\subsection{Other directions}\n\n\\subsubsection{Infinite $C(S,\\mathbf G)$}\n\nThis paper has focused on the case $S\\text{-rank}\\: \\mathbf G \\ge 2$ where Serre's\nconjecture says that $C(S,\\mathbf G)$ is finite.  It would be interesting to\ninvestigate topological interpretations in the case $S\\text{-rank}\\: \\mathbf G = 1$ and\n$C(S,\\mathbf G)$ is infinite.\n\n\\subsubsection{Function fields}\n\nUsually the congruence subgroup problem is considered for algebraic groups\ndefined over global fields, not just algebraic number fields as considered\nhere.  As noted in \\S\\ref{ssectKnownResults}, for $k$ a global field, the\ncongruence subgroup kernel $C(S, \\mathbf G)$ is finite for $\\mathbf G$ simply connected,\nabsolutely almost simple with  $k\\text{-rank}\\: \\mathbf G >0$ and $S\\text{-rank}\\: \\mathbf G\\ge 2$.  A\nnatural question is to give a topological interpretation in this case as\nwell.  Here there are no infinite places so it seems plausible to consider\nthe fundamental group of suitable compactifications of an $S$-arithmetic\nquotient of the product of Bruhat-Tits buildings $\\prod_{v\\in S} X_{v}$.\nSeveral compactifications of Bruhat-Tits buildings have been considered: the\nBorel-Serre compactification in which the spherical Tits building is placed\nat infinity \\cite{BS2}; a polyhedral compactification due to Landvogt\n\\cite{Landvogt}; and compactifications\nassociated to linear representations  \\citelist{\\cite{Werner}\n\\cite{RemyThuillierWernerI} \\cite{RemyThuillierWernerII}}.  These last \ncompactifications  are analogous to the Satake\ncompactifications of symmetric spaces and recover Landvogt's\ncompactification as a special case for the generic representation; thus\nLandvogt's compactification is analogous to the maximal Satake\ncompactification.  It would be interesting to see if there is an analogy of\nSatake's theory of rational boundary components which would lead to\ncorresponding compactifications of the $S$-arithmetic quotients.\n\n\\section{The reductive Borel-Serre and Satake compactifications: the\n  arithmetic case}\n\\label{sectCompactificationsArithmetic}\n\nIn order to establish notation and set the framework for later proofs, we\nrecall in \\S\\S\\ref{ssectBSarith}--\\ref{subsectSatakeArith} several natural\ncompactifications of the locally symmetric space $\\Gamma\\backslash X_\\infty$\nassociated to an arithmetic group $\\Gamma$; in each case a bordification of\n$X_\\infty$ is described on which $\\mathbf G(k)$ acts.  We also examine the\nstabilizer subgroups of points in these bordifications.  The case of\ngeneral $S$-arithmetic groups will be treated in\n\\S\\ref{sectCompactificationsSArithmetic}.  Throughout the paper, $\\mathbf G$ will\ndenote a connected, absolutely almost simple, simply connected algebraic\ngroup defined over a number field $k$.\n\n\\subsection{Proper and discontinuous actions}\n\\label{ssectProperDiscontinuousActions}\nRecall \\cite{BourbakiTopologiePartOne}*{III, \\S4.4, Prop.~7} that a\ndiscrete group $\\Gamma$ acts \\emph{properly} on a Hausdorff space $Y$ if and\nonly if for all $y$, $y'\\in Y$, there exist neighborhoods $V$ of $y$ and\n$V'$ of $y'$ such that $\\gamma V\\cap V'\\neq \\emptyset$ for only finitely\nmany $\\gamma \\in \\Gamma$.  We will also need the following weaker condition on\nthe group action:\n\n\\begin{defi}[\\cite{Gro}*{Definition~1}]\n\\label{defnDiscontinuous}\nThe action of a discrete group $\\Gamma$ on a topological space $Y$ is\n\\emph{discontinuous} if\n\\begin{enumerate}\n\\item\\label{itemDiscontinuousTwoPoints} for all $y$, $y'\\in Y$ with\n  $y'\\notin \\Gamma y$ there exists neighborhoods $V$ of $y$ and $V'$ of $y'$\n  such that $\\gamma V\\cap V' =\\emptyset$ for all $\\gamma\\in \\Gamma$, and\n\\item\\label{itemDiscontinuousOnePoint} for all $y\\in Y$ there exists a\n  neighborhood $V$ of $y$ such that $\\gamma V\\cap V = \\emptyset$ for\n  $\\gamma \\notin \\Gamma_y$ and $\\gamma V = V$ for $\\gamma \\in \\Gamma_y$.\n\\end{enumerate}\n\\end{defi}\n\nIt is easy to check that a group action is proper if and only if it is\ndiscontinuous and the stabilizer subgroup $\\Gamma_y$ is finite for all $y\\in\nY$.\n\n\\subsection{The locally symmetric space associated to an arithmetic subgroup}\nLet $S_{\\infty}$ be the set of all\ninfinite places of $k$.  For each $v\\in S_\\infty$, let $k_{v}$ be the\ncorresponding completion of $k$ with respect to a norm associated with $v$;\nthus either $k_{v}\\cong {\\mathbb R}$ or $k_{v}\\cong {\\mathbb C}$.  For each $v\\in\nS_{\\infty}$, $\\mathbf G(k_{v})$ is a (real) Lie group.\n\nDefine $G_{\\infty}=\\prod_{v\\in S_{\\infty}}\\mathbf G(k_{v})$, a semisimple Lie\ngroup with finitely many connected components.  Fix a maximal compact\nsubgroup $K$ of $G_{\\infty}$.  When endowed with a $G$-invariant metric,\n$X_\\infty = G_{\\infty}\/K$ is a Riemannian symmetric space of noncompact\ntype and is thus contractible.  Embed $\\mathbf G(k)$ into $G_\\infty$ diagonally.\nThen any arithmetic subgroup $\\Gamma\\subset \\mathbf G(k)$ is a discrete subgroup of\n$G_\\infty$ and acts properly on $X_\\infty$.  It is known that the quotient\n$\\Gamma\\backslash X_\\infty$ is compact if and only if the $k$-rank of $\\mathbf G$ is\nequal to 0.  In the following, we assume that the $k$-rank of $\\mathbf G$ is\npositive so that $\\Gamma\\backslash X_\\infty$ is noncompact.\n\nSince the theory of compactifications of locally symmetric spaces is\nusually expressed in terms of algebraic groups defined over ${\\mathbb Q}$, let\n$\\mathbf H=\\operatorname{Res}_{k\/{\\mathbb Q}}\\mathbf G$ be the algebraic group defined over\n${\\mathbb Q}$ obtained by restriction of scalars; it satisfies\n\\begin{equation}\n\\label{eqnPointsOfRestrictionScalars}\n\\mathbf H({\\mathbb Q})=\\mathbf G(k) \\quad\\text{and}\\quad \\mathbf H(\\mathbb R)=G_{\\infty}\\ .\n\\end{equation}\nThe space $X_{\\infty}$ can be identified with the symmetric space of\nmaximal compact subgroups of $\\mathbf H(\\mathbb R)$, $X_{\\infty}=\\mathbf\nH(\\mathbb R)\/K$, and the arithmetic subgroup $\\Gamma\\subset \\mathbf G(k)$ corresponds\nto an arithmetic subgroup $\\Gamma\\subset \\mathbf H({\\mathbb Q})$.  Restriction of\nscalars yields a one-to-one correspondence between parabolic $k$-subgroups\nof $\\mathbf G$ and parabolic ${\\mathbb Q}$-subgroups of $\\mathbf H$ so that the analogue of\n\\eqref{eqnPointsOfRestrictionScalars} is satisfied.\n\n\\subsection{The Borel-Serre compactification}\n\\label{ssectBSarith}\n(For details see the original paper \\cite{Borel-Serre}, as well as\n\\cite{Borel-Ji}.)  For each parabolic ${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$,\nconsider the Levi quotient $\\mathbf L_{\\P} = \\P\/\\mathbf N_{\\P}$ where $\\mathbf N_{\\P}$ is\nthe unipotent radical of $\\P$.  This is a reductive group defined over\n${\\mathbb Q}$.  There is an almost direct product $\\mathbf L_{\\P} = \\mathbf S_{\\P}\n\\cdot \\mathbf M_{\\P}$, where $\\mathbf S_{\\P}$ is the maximal ${\\mathbb Q}$-split\ntorus in the center of $\\mathbf L_{\\P}$ and $\\mathbf M_{\\P}$ is the\nintersection of the kernels of the squares of all characters of $\\mathbf\nL_{\\P}$ defined over ${\\mathbb Q}$.  The real locus $L_P= \\mathbf L_\\P({\\mathbb R})$ has a\ndirect product decomposition $A_P \\cdot M_P$, where $A_P = \\mathbf\nS_\\P({\\mathbb R})^0$ and $M_P = \\mathbf M_\\P({\\mathbb R})$.  The dimension of $A_P$ is called\nthe \\emph{parabolic ${\\mathbb Q}$-rank} of $\\P$.\n\nThe real locus $P=\\P({\\mathbb R})$ has a Langlands decomposition\n\\begin{equation}\\label{rationalLanglands}\nP=N_{P} \\ltimes (\\widetilde A_P \\cdot \\widetilde M_ P),\n\\end{equation}\nwhere $N_{P}= \\mathbf N_{\\P}({\\mathbb R})$ and $\\widetilde A_P \\cdot \\widetilde M_ P$ is\nthe lift of $A_P \\cdot M_P$ to the unique Levi subgroup of $P$ which is\nstable under the Cartan involution $\\theta$ associated with $K$.\n\nSince $P$ acts transitively on $X_\\infty$, the Langlands decomposition induces a\nhorospherical decomposition\n\\begin{equation}\\label{horo}\nX_\\infty \\cong  A_P\\times N_{P}\\times X_P,\\quad u\\tilde a\\tilde mK \\mapsto\n(\\tilde a,u,\\tilde m(K\\cap \\widetilde M_P),\n\\end{equation}\nwhere \n\\begin{equation*}\nX_P= \\widetilde M_P \/ (K \\cap \\widetilde M_P) \\cong  L_P\/(A_P\\cdot K_P)\n\\end{equation*}\nis a symmetric space (which might contain an Euclidean factor) and is\ncalled the \\emph{boundary symmetric space associated with $\\P$}.  The\nsecond expression for $X_P$ is preferred since $\\mathbf L_\\P$ is defined\nover ${\\mathbb Q}$; here $K_P\\subseteq \\mathbf M_\\P({\\mathbb R})$ corresponds to $K \\cap\n\\widetilde M_P$\n\nFor each parabolic ${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$, define the\nBorel-Serre boundary component\n\\begin{equation*}\ne(P)=N_{P}\\times X_P\n\\end{equation*}\nwhich we view as the quotient of $X_\\infty$ obtained by collapsing the\nfirst factor in \\eqref{horo}.  The action of $P$ on $X_\\infty$ descends to\nan action on $e(P)=N_{P}\\times X_P$ given by\n\\begin{equation}\n\\label{eqnBoundaryAction}\np\\cdot (u, y) = (pu\\tilde m_p^{-1}\\tilde a_p^{-1} , \\tilde a_p \\tilde m_p\ny), \\qquad \\text{for $p=u_p \\tilde a_p \\tilde m_p\\in P$.}\n\\end{equation}\nDefine the Borel-Serre partial compactification $\\overline{X}_\\infty^{BS}$ (as a\n\\emph{set}) by\n\\begin{equation}\n\\label{BSPartialCompactification}\n\\overline{X}_\\infty^{BS}=X_\\infty\\cup \\coprod_{\\P\\subset \\mathbf H} e(P).\n\\end{equation}\n\nLet $\\Delta_P$ be the simple ``roots'' of the adjoint action of $A_P$ on\nthe Lie algebra of $N_P$ and identify $A_P$ with $({\\mathbb R}^{>0})^{\\Delta_P}$ by\n$a \\mapsto (a^{-\\alpha})_{\\alpha\\in\\Delta_P}$.  Enlarge $A_P$ to the\ntopological semigroup $\\overline A_P \\cong ({\\mathbb R}^{\\ge0})^{\\Delta_P}$ by\nallowing $a^\\alpha$ to attain infinity and define\n\\begin{equation*}\n\\overline A_P(s) = \\{\\, a\\in \\overline A_P\\mid a^{-\\alpha} < s^{-1} \\text{\n  for all $\\alpha\\in \\Delta_P$}\\,\\}\\cong [0,s^{-1})^{\\Delta_P}\\ ,\\qquad\n  \\text{for $s>0$}\\ .\n\\end{equation*}\nSimilarly enlarge the Lie algebra ${\\mathfrak a}_P \\subset \\overline{\\mathfrak a}_P$.  The\ninverse isomorphisms $\\exp\\colon {\\mathfrak a}_P \\to A_P$ and $\\log\\colon A_P \\to\n{\\mathfrak a}_P$ extend to isomorphisms\n\\begin{equation*}\n\\overline A_P \\overset{\\log}{\\longrightarrow} \\overline {\\mathfrak a}_P\n\\qquad\\text{and} \\qquad \\overline {\\mathfrak a}_P \\overset{\\exp}{\\longrightarrow}\n\\overline A_P.\n\\end{equation*}\n\nTo every parabolic ${\\mathbb Q}$-subgroup $\\mathbf Q\\supseteq \\P$ there corresponds\na subset $\\Delta_P^Q \\subseteq \\Delta_P$ and we let $o_Q\\in\n\\overline A_P$ be the point with coordinates $o_Q^{-\\alpha} =1$\nfor $\\alpha\\in \\Delta_P^Q$ and $o_Q^{-\\alpha} =0$ for\n$\\alpha\\notin \\Delta_P^Q$.  Then $\\overline A_P = \\coprod_{\\mathbf Q\n\\supseteq \\P} A_P \\cdot o_Q$ is the decomposition into\n$A_P$-orbits.\n\nDefine the \\emph{corner associated to $\\mathbf P$} to be\n\\begin{equation}\n\\label{Pcorner}\nX_\\infty(P) = \\overline A_P \\times e(P) = \\overline A_P \\times N_P \\times X_P.\n\\end{equation}\nWe identify $e(Q)$ with the subset $ (A_P\\cdot o_Q) \\times N_P\\times X_P$.\nIn particular, $e(P)$ is identified with the subset $\\{o_P\\}\\times\nN_P\\times X_P$ and $X_\\infty$ is identified with the open subset $A_P \\times\nN_P\\times X_P \\subset X_\\infty(P)$ (compare \\eqref{horo}).  Thus we have a\nbijection\n\\begin{equation}\n\\label{strataPcorner}\nX_\\infty(P) \\cong X_\\infty \\cup \\coprod_{\\P \\subseteq \\mathbf Q \\subset\n  \\mathbf H} e(Q).\n\\end{equation}\n\nNow give $\\overline X_\\infty^{BS}$ the finest topology so that for all\nparabolic ${\\mathbb Q}$-subgroups $\\P$ of $\\mathbf H$ the inclusion of\n\\eqref{strataPcorner} into \\eqref{BSPartialCompactification} is a\ncontinuous inclusion of an open subset.  Under this topology, a sequence\n$x_n\\in X$ converges in $\\overline X_\\infty^{BS}$ if and only if there\nexists a parabolic ${\\mathbb Q}$-subgroup $\\P$ such that if we write $x_n=(a_n, u_n,\ny_n)$ according to the decomposition of \\eqref{horo}, then $(u_n,y_n)$\nconverges to a point in $e(P)$ and $a_n^\\alpha\\to \\infty$ for all\n$\\alpha\\in \\Delta_P$.  The space $\\overline X_\\infty^{BS}$ is a manifold\nwith corners.  It has the same homotopy type as $X_\\infty$ and is thus\ncontractible \\cite{Borel-Serre}.\n\nThe action of $\\mathbf H({\\mathbb Q})$ on $X_\\infty$ extends to a continuous action\non $\\overline{X}_\\infty^{BS}$ which permutes the boundary components:\n$g\\cdot e(P) = e(gPg^{-1})$ for $g\\in \\mathbf H({\\mathbb Q})$.  The normalizer of\n$e(P)$ is $\\P({\\mathbb Q})$ which acts according to \\eqref{eqnBoundaryAction}.\n\nIt is shown in \\cite{Borel-Serre} that the action of $\\Gamma$ on\n$\\overline{X}_\\infty^{BS}$ is proper and the quotient $\\Gamma\\backslash\n\\overline{X}_\\infty^{BS}$, the \\emph{Borel-Serre compactification}, is a compact\nHausdorff space.  It is a manifold with corners if $\\Gamma$ is torsion-free.\n\n\\subsection{The reductive Borel-Serre compactification}\n\\label{subsectRBSarith}\nThis compactification was first constructed by Zucker \\cite{Zu1}*{\\S4} (see also\n\\cite{GHM}).  For each parabolic ${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$, define\nits reductive Borel-Serre boundary component $\\hat{e}(P)$ by\n\\begin{equation*}\n\\hat{e}(P)=X_P\n\\end{equation*}\nand set\n\\begin{equation*}\n\\overline{X}_\\infty^{RBS}=X_\\infty\\cup \\coprod_{\\P} \\hat{e}(P).\n\\end{equation*}\nThe projections $p_P\\colon e(P) = N_P\\times X_P \\to \\hat e(P) = X_P$ induce\na surjection $p\\colon \\overline{X}_\\infty^{BS} \\to\n\\overline{X}_\\infty^{RBS}$ and we give $\\overline{X}_\\infty^{RBS}$ the\nquotient topology.  Its topology can also be described in terms of\nconvergence of interior points to the boundary points via the horospherical\ndecomposition in equation \\eqref{horo}.  Note that $\\overline{X}^{RBS}$ is\nnot locally compact, although it is compactly generated (being a Hausdorff\nquotient of the locally compact space $\\overline{X}^{BS}$).  The action of\n$\\mathbf H({\\mathbb Q})$ on $\\overline{X}_\\infty^{BS}$ descends to a continuous\naction on $\\overline{X}_\\infty^{RBS}$.\n\n\\begin{lem}\n\\label{lemStabilizersRBS}\nLet $\\P$ be a parabolic ${\\mathbb Q}$-subgroup of $\\mathbf H$.\nThe stabilizer $\\mathbf H({\\mathbb Q})_z= \\mathbf G(k)_z$ of $z\\in X_P$ under the action of\n$\\mathbf H({\\mathbb Q})$ on $\\overline{X}^{RBS}_\\infty$ satisfies a short exact sequence\n\\begin{equation*}\n1 \\to \\mathbf N_{\\P}({\\mathbb Q}) \\to \\mathbf H({\\mathbb Q})_z \\to \\mathbf L_{\\P}({\\mathbb Q})_z \\to 1\n\\end{equation*}\nwhere $\\mathbf L_{\\P}({\\mathbb Q})_z$ is the stabilizer of $z$ under the action of\n$\\mathbf L_{\\P}({\\mathbb Q})$ on $X_P$.\n\\end{lem}\n\\begin{proof}\nThe normalizer of $X_P$ under the action of $\\mathbf H({\\mathbb Q})$ is $\\P({\\mathbb Q})$\nwhich acts via its quotient $\\mathbf L_{\\P}({\\mathbb Q})$. \n\\end{proof}\n\nBy the lemma, the action of $\\Gamma$ on $\\overline{X}_\\infty^{RBS}$ is not\nproper since the stabilizer of a boundary point in $X_P$ contains the\ninfinite group $\\Gamma_{N_P} = \\Gamma\\cap N_P$.  Nonetheless\n\\begin{lem}\n\\label{lemRBSDiscontinuous}\nThe action of an arithmetic subgroup $\\Gamma$ on $\\overline{X}_\\infty^{RBS}$\nis discontinuous and the arithmetic quotient\n$\\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ is a compact Hausdorff space.\n\\end{lem}\n\n\\begin{proof}\nWe begin by verifying Definition\n~\\ref{defnDiscontinuous}\\ref{itemDiscontinuousOnePoint}.  Let $x\\in X_P\n\\subseteq \\overline{X}_\\infty^{RBS}$.  Set $\\Gamma_P = \\Gamma\\cap P$ and\n$\\Gamma_{L_P} = \\Gamma_P\/\\Gamma_{N_P}$.  Since $\\Gamma_{L_P}$ acts properly on $X_P$\nthere exists a neighborhood $O_x$ of $x$ in $X_P$ such that $\\bar \\gamma\nO_x \\cap O_x \\neq \\emptyset$ if and only if $\\bar \\gamma \\in \\Gamma_{L_P,x}$,\nin which case $\\bar \\gamma O_x = O_x$.  We can assume $O_x$ is relatively\ncompact.  Set $V=p(\\overline{A}_P(s)\\times N_P \\times O_x)$, where we chose\n$s$ sufficiently large so that that only identifications induced by $\\Gamma$\non $V$ already arise from $\\Gamma_P$ \\cite{Zu3}*{(1.5)}.  Thus $\\gamma V\\cap\nV\\neq \\emptyset$ if and only if $\\gamma \\in \\Gamma_P$ and $\\gamma \\Gamma_{N_P}\n\\in \\Gamma_{L_P,x}$; by Lemma ~\\ref{lemStabilizersRBS} this occurs if and only\nif $\\gamma \\in \\Gamma_x$ as desired.\n\nTo verify Definition\n~\\ref{defnDiscontinuous}\\ref{itemDiscontinuousTwoPoints} we will show the\nequivalent condition that $\\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ is\nHausdorff (compare \\cite{Zu1}*{(4.2)}).  Compactness will follow since it\nis the image of a compact space under the induced projection $p'\\colon\n\\Gamma\\backslash \\overline{X}_\\infty^{BS} \\to \\Gamma\\backslash\n\\overline{X}_\\infty^{RBS}$.  Observe that $p'$ is a quotient map and that\nits fibers, each being homeomorphic to $\\Gamma_{N_P}\\backslash N_P$ for some\n$\\P$, are compact. For $y\\in \\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ and\n$W$ a neighborhood of $p'^{-1}(y)$, we claim there exists $U\\ni y$ open\nsuch that $p'^{-1}(U)\\subseteq W$.  This suffices to establish Hausdorff,\nfor if $y_1\\neq y_2 \\in \\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ and $W_1$\nand $W_2$ are disjoint neighborhoods of the compact fibers $p'^{-1}(y_1)$\nand $p'^{-1}(y_2)$, there must exist $U_1$ and $U_2$, neighborhoods of\n$y_1$ and $y_2$, such that $p'^{-1}(U_i) \\subseteq W_i$ and hence $U_1\\cap\nU_2 =\\emptyset$.\n\nTo prove the claim, choose $x\\in X_P$ such that $y=\\Gamma x$.  Let $q\\colon\n\\overline{X}_\\infty^{BS} \\to \\Gamma\\backslash \\overline{X}_\\infty^{BS} $ be\nthe quotient map.  The compact fiber $p'^{-1}(y)$ may be covered by\nfinitely many open subsets $q(\\overline A_P(s_\\mu)\\times C_{P,\\mu} \\times\nO_{P,\\mu}) \\subseteq W$ where $C_{P,\\mu} \\subseteq N_P$ and $x\\in\nO_{P,\\mu}\\subseteq X_P$.  Define a neighborhood $V$ of the fiber by\n\\begin{equation*}\np'^{-1}(y) \\subset V = q(\\overline A_P(s)\\times C_{P}\n\\times O_{P}) \\subseteq W\n\\end{equation*}\nwhere $s = \\max \\,s_\\mu$, $O_P = \\bigcap O_{P,\\mu}$, and $C_P = \\bigcup C_{P,\\mu}$.\nSince $\\Gamma_{N_P}C_P = N_P$, we see $V=p'^{-1}(U)$ for some $U\\ni y$ as\ndesired.\n\\end{proof}\n\n\\subsection{Satake compactifications}\n\\label{subsectSatakeArith}\nFor arithmetic quotients of $X_\\infty$, the Satake compactifications\n$\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ form an important family of\ncompactifications.  When $X_\\infty$ is Hermitian, one example is the Baily-Borel\nSatake compactification. The construction has three steps.\n\\begin{enumerate}\n\\item Begin%\n\\footnote{Here we follow \\cite{Cass} in beginning with a spherical\n  representation. Satake's original construction \\cite{sat1} started with a\n  non-spherical representation but then constructed a spherical\n  representation by letting $G_\\infty$ act on the space of self-adjoint\n  endomorphisms of $V$ with respect to an admissible inner product. See\n  \\cite{sap2} for the relation of the two constructions.}\nwith a representation $(\\tau,V)$ of $\\mathbf H$ which has a nonzero\n$K$-fixed vector $v\\in V$ (a \\emph{spherical representation}) and which is\nirreducible and nontrivial on each noncompact ${\\mathbb R}$-simple factor of\n$\\mathbf H$.  Define the Satake compactification $\\overline{X}_\\infty^{\\tau}$\nof $X$ to be the closure of the image of the embedding $X_\\infty \\hookrightarrow\n\\mathbb P(V)$, $gK \\mapsto [ \\tau(g) v]$.  The action of $G_\\infty$ extends\nto a continuous action on $\\overline{X}_\\infty^{\\tau}$ and the set of points\nfixed by $N_P$, where $\\P$ is any parabolic ${\\mathbb R}$-subgroup, is called a\n\\emph{real boundary component}.  The compactification\n$\\overline{X}_\\infty^{\\tau}$ is the disjoint union of its real boundary\ncomponents.\n\n\\item Define a partial compactification\n  ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}\\subseteq \\overline{X}_\\infty^{\\tau}$\n  by taking the union of $X_\\infty$ and those real boundary components that\n  meet the closure of a Siegel set.  Under the condition that\n  $\\overline{X}_\\infty^{\\tau}$ is \\emph{geometrically rational}\n  \\cite{Cass}, this is equivalent to considering those real boundary\n  components whose normalizers are parabolic ${\\mathbb Q}$-subgroups; call these the\n  \\emph{rational boundary components}.  Instead of the subspace topology\n  induced from $\\overline{X}_\\infty^{\\tau}$, give\n  ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ the Satake topology \\cite{sat2}.\n\n\\item Still under the condition that $\\overline{X}_\\infty^{\\tau}$ is\n  geometrically rational, one may show that the arithmetic subgroup $\\Gamma$\n  acts continuously on ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ with a\n  compact Hausdorff quotient, $\\Gamma\\backslash\n  {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$.  This is the \\emph{Satake\n  compactification} of $\\Gamma\\backslash X_\\infty$.\n\\end{enumerate}\n\nThe geometric rationality condition above always holds if the\nrepresentation $(\\tau,V)$ is rational over ${\\mathbb Q}$ \\cite{sap2}.  It also holds\nfor the Baily-Borel Satake compactification \\cite{BB}, as well as most\nequal-rank Satake compactifications including all those where ${\\mathbb Q}\\text{-rank}\\:\n\\mathbf H >2$.\n\nWe will now describe an alternate construction of\n${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ due to Zucker \\cite{Zu2}.  Instead of\nthe Satake topology, Zucker gives ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ the\nquotient topology under a certain surjection $\\overline{X}_\\infty^{RBS}\n\\to {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ described below.  It is this\ntopology we will use in this paper.  Zucker proves that the resulting two\ntopologies on $\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ coincide.\n\nLet $(\\tau,V)$ be a spherical representation as above.  We assume that\n$\\overline{X}_\\infty^{\\tau}$ is geometrically rational.  For any parabolic\n${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$, let $X_{P,\\tau}\\subseteq \\overline\nX_\\infty^{\\tau}$ be the real boundary component fixed pointwise by $N_P$;\ngeometric rationality implies that $X_{P,\\tau}$ is actually a rational\nboundary component.  The transitive action of $P$ on $X_{P,\\tau}$ descends\nto an action of $L_P = P\/N_P$.  The geometric rationality condition ensures\nthat there exists a normal ${\\mathbb Q}$-subgroup $\\mathbf L_{\\P, \\tau} \\subseteq\n\\mathbf L_{\\P}$ with the property that $L_{P,\\tau}= \\mathbf L_{\\P,\n  \\tau}({\\mathbb R})$ is contained in the centralizer\n$\\operatorname{Cent}(X_{P,\\tau})$ of $X_{P,\\tau}$ and\n$\\operatorname{Cent}(X_{P,\\tau})\/L_{P,\\tau}$ is compact.  Then $X_{P,\\tau}$\nis the symmetric space associated to the ${\\mathbb Q}$-group $\\mathbf H_{\\P,\n\\tau} = \\mathbf L_{\\P} \/ \\mathbf L_{\\P,\\tau}$.   There is an\nalmost direct product decomposition\n\\begin{equation}\n\\label{eqnSatakeLeviDecomposition}\n\\mathbf L_{\\P} = \\widetilde {\\mathbf H}_{\\P, \\tau} \\cdot \\mathbf L_{\\P,\n  \\tau}\\ ,\n\\end{equation}\nwhere $\\widetilde {\\mathbf H}_{\\P, \\tau}$ is a lift of $\\mathbf H_{\\P,\n  \\tau}$; the root systems of these factors may be described using the\nhighest weight of $\\tau$.  We obtain a decomposition of symmetric spaces\n\\begin{equation}\n\\label{eqnBoundaryDecomposition}\nX_P = X_{P,\\tau} \\times  W_{P,\\tau}\\ .\n\\end{equation}\n\nDifferent parabolic ${\\mathbb Q}$-subgroups can yield the same rational boundary\ncomponent $X_{P,\\tau}$; if $\\P^\\dag$ is the maximal such parabolic\n${\\mathbb Q}$-subgroup, then $P^\\dag=\\P^\\dag({\\mathbb R})$ is the normalizer of $X_{P,\\tau}$.\nThe parabolic ${\\mathbb Q}$-subgroups that arise as the normalizers of rational\nboundary components are called \\emph{$\\tau$-saturated}.  For example, all\nparabolic ${\\mathbb Q}$-subgroups are saturated for the maximal Satake\ncompactification, while only the maximal parabolic ${\\mathbb Q}$-subgroups are\nsaturated for the Baily-Borel Satake compactification when $\\mathbf H$ is\n${\\mathbb Q}$-simple.  In general, the class of $\\tau$-saturated parabolic\n${\\mathbb Q}$-subgroups can be described in terms of the highest weight of $\\tau$.\n\nDefine \n\\begin{equation*}\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}=X_\\infty\\cup \\coprod_{\\text{$\\mathbf Q$\n    $\\tau$-saturated}} X_{Q,\\tau}\\ .\n\\end{equation*}\nA surjection $p\\colon \\overline{X}_\\infty^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ is obtained by mapping $X_P$ to\n$X_{P,\\tau} = X_{P^\\dag,\\tau}$ via the projection on the first factor in\n\\eqref{eqnBoundaryDecomposition}.  Give ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$\nthe resulting quotient topology; the action of $\\mathbf H({\\mathbb Q})$ on\n$\\overline{X}_\\infty^{RBS}$ descends to a continuous action on\n${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$. \n\nLet $\\P_\\tau$ be the inverse image of $\\mathbf L_{\\P,\\tau}$ under\nthe projection $\\P \\to \\P\/\\mathbf N_{\\P}$.\n\\begin{lem}\n\\label{lemStabilizersSatake}\nLet $\\P$ be a $\\tau$-saturated parabolic ${\\mathbb Q}$-subgroup of $\\mathbf H$.  The\nstabilizer $\\mathbf H({\\mathbb Q})_z = \\mathbf G(k)_z$ of $z\\in X_{P,\\tau}$ under the\naction of $\\mathbf H({\\mathbb Q})$ on \n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ satisfies a short exact sequence\n\\begin{equation*}\n1 \\to  \\P_{\\tau}({\\mathbb Q}) \\to \\mathbf H({\\mathbb Q})_z \\to  \\mathbf H_{\\P,\\tau}({\\mathbb Q})_z \\to 1,\n\\end{equation*}\nwhere $\\mathbf H_{\\P,\\tau}({\\mathbb Q})_z$ is the stabilizer of $z$ under the action\nof $\\mathbf H_{\\P,\\tau}({\\mathbb Q})$ on $X_{P,\\tau}$.\n\\end{lem}\n\\begin{proof}\nAs in the proof of Lemma ~\\ref{lemStabilizersRBS}, the normalizer of\n$X_{P,\\tau}$ is $\\P({\\mathbb Q})$ which acts via its quotient $\\P({\\mathbb Q})\/\\P_\\tau({\\mathbb Q}) =\n\\mathbf H_{\\P,\\tau}({\\mathbb Q})$.\n\\end{proof}\n\nSimilarly to $\\overline{X}^{RBS}$, the space\n${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ is not locally compact and $\\Gamma$ does\nnot act properly.  Nonetheless one has the\n\\begin{lem}\n\\label{lemSatakeDiscontinuous}\nThe action of an arithmetic subgroup $\\Gamma$ on ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$\nis discontinuous and the arithmetic quotient\n$\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ is a compact Hausdorff space.\n\\end{lem}\n\nThe proof is similar to Lemma ~\\ref{lemRBSDiscontinuous} since the fibers\nof $p'$ are again compact, being reductive Borel-Serre compactifications of\nthe $W_{P^\\dag,\\tau}$.  The \\emph{Satake compactification} of\n$\\Gamma\\backslash X_\\infty$ associated to $\\tau$ is $\\Gamma\\backslash\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$.\n\nIn the case when the representation $\\tau$ is generic one obtains the\nmaximal Satake compactification $\\overline{X}_\\infty^{\\max}$.  This is\nalways geometrically rational and the associated\n${}_{\\mathbb Q}\\overline{X}_\\infty^{\\max}$ is very similar to\n$\\overline{X}_\\infty^{RBS}$.  Indeed in this case $X_P = X_{P,\\tau} \\times\n({}_{\\mathbb R} A_{P}\/A_{P})$, where ${}_{\\mathbb R} A_{P}$ is defined like $A_P$ but using a\nmaximal ${\\mathbb R}$-split torus instead of a maximal ${\\mathbb Q}$-split torus, and the\nquotient map simply collapses the Euclidean factor ${}_{\\mathbb R} A_{P}\/A_{P}$ to a\npoint.  In particular, if ${\\mathbb Q}\\text{-rank }\\mathbf H = {\\mathbb R}\\text{-rank }\n\\mathbf H$, then $\\Gamma\\backslash {}_{\\mathbb Q}\\overline{X}_\\infty^{\\max} \\cong\n\\Gamma\\backslash\\overline{X}_\\infty^{RBS}$.\n\n\\section{The Bruhat-Tits buildings}\n\\label{sectBruhatTitsBuildings}\n\nFor a finite place $v$, let $k_{v}$ be the completion of $k$ with respect\nto a norm associated with $v$. Bruhat and Tits \\citelist{\\cite{BruhatTits1}\n  \\cite{BruhatTits2}} constructed a building $X_{v}$ which reflects the\nstructure of $\\mathbf G(k_{v})$.  The building $X_{v}$ is made up of subcomplexes\ncalled \\emph{apartments} corresponding to the maximal $k_{v}$-split tori in\n$\\mathbf G$ and which are glued together by the action of $\\mathbf G(k_{v})$.  We give an\noutline of the construction here together with the properties of $X_{v}$\nwhich are needed in the sections below; in addition to the original papers,\nwe benefited greatly from\n\\citelist{\\cite{ji}*{\\S3.2}\\cite{Landvogt}\\cite{Tits}}.\n\nIn this section we fix a finite place $v$ and a corresponding discrete\nvaluation $\\omega$.\n\n\\subsection{The apartment}\n\nLet $\\Split$ be a maximal $k_{v}$-split torus in $\\mathbf G$ and let\n$X^{*}(\\Split)= \\Hom_{k_{v}}(\\Split, \\mathbf G_{m})$ and $X_{*}(\\Split)\n=\\Hom_{k_{v}}(\\mathbf G_{m}, \\Split)$ denote the $k_v$-rational characters and\ncocharacters of $\\Split$ respectively.  Denote by $\\Phi \\subset\nX^{*}(\\Split)$ the set of $k_{v}$-roots of $\\mathbf G$ with respect to\n$\\Split$. Let $\\N$ and $\\Cent$ denote the normalizer and the centralizer,\nrespectively, of $\\Split$; set $N=\\N(k_{v})$, $Z=\\Cent(k_{v})$. The  Weyl\ngroup $W =\nN\/Z$ of $\\Phi$ acts on the real vector space\n\\begin{equation*}\nV = X_{*}(\\Split) \\otimes_{{\\mathbb Z}}{\\mathbb R} = \\Hom_{{\\mathbb Z}}(X^{*}(\\Split) , {\\mathbb R})\n\\end{equation*}\nby linear transformations; for $\\alpha\\in\\Phi$, let $r_\\alpha$ denote the\ncorresponding reflection of $V$.\n\nLet $A$ be the affine space underlying $V$ and let $\\Aff(A)$ denote the\ngroup of invertible affine transformations.  We identify $V$ with the\ntranslation subgroup of $\\Aff(A)$.  There is an action of $Z$ on $A$ via\ntranslations, $\\nu\\colon Z\\rightarrow V \\subset \\Aff(A)$, determined by\n\\begin{equation*}\n\\chi(\\nu(t)) = -\\omega(\\chi(t))\\ , \\quad t\\in Z,\\ \\chi\\in X^{*}(\\Cent)\\ ;\n\\end{equation*}\nnote that $V = \\Hom_{{\\mathbb Z}}(X^{*}(\\Cent), {\\mathbb R})$ since\n$X^{*}(\\Cent) \\subseteq X^{*}(\\Split)$ is a finite index subgroup. \n\nWe now extend $\\nu$ to an action of $N$ by affine transformations.  Let $H\n= \\ker\\nu$, which is the maximal compact subgroup of $Z$. Then $Z\/H$ is a\nfree abelian group with rank $= \\dim_{\\mathbb R} V = k_{v}\\text{-rank}\\: \\mathbf G$.  The group $W'\n= N\/H$ is an extension of $W$ by $Z\/H$ and there exists an affine action of\n$W'$ on $A$ which makes the following diagram commute\n\\cite{Landvogt}*{1.6}:\n\\begin{equation*}\n\\begin{CD}\n1 @>>> Z\/H @>>> W' @>>> W @>>> 1 \\\\\n@. @VVV @VVV @VVV \\\\\n1 @>>> V @>>> \\Aff(A) @>>> \\mathrm{GL}(V) @>>> 1\\rlap{\\ .}\n\\end{CD}\n\\end{equation*}\nThe action of $W'$ lifts to the desired extension $\\nu\\colon N \\to \\Aff(A)$.\n\nFor each $\\alpha \\in \\Phi$, let $U_{\\alpha}$ be the $k_v$-rational points\nof the connected unipotent subgroup of $\\mathbf G$ which has Lie algebra spanned\nby the root spaces $\\mathfrak g_\\alpha$ and (if $2\\alpha$ is a root)\n$\\mathfrak g_{2\\alpha}$.  For $u\\in U_{\\alpha}\\setminus \\{1\\}$, let $m(u)$\nbe the unique element of $N\\cap U_{-\\alpha}uU_{-\\alpha}$\n\\cite{Landvogt}*{0.19}; in $\\SL_2$, for example,\n$m\\left(\\left(\\begin{smallmatrix} 1 & x \\\\ 0\\vphantom{x^{-1}} &\n  1 \\end{smallmatrix}\\right)\\right) = \\left(\\begin{smallmatrix} 0 & x\n  \\\\ -x^{-1} & 0 \\end{smallmatrix}\\right)$.  The element $m(u) \\in N$ acts\non $A$ by an affine reflection $\\nu(m(u))$ whose associated linear\ntransformation is $r_\\alpha$.  The hyperplanes fixed by these affine\nreflections for all $\\alpha$ and $u$ are the \\emph{walls} of $A$.  The\nconnected components of the complement of the union of the walls are called\nthe \\emph{chambers} of $A$; since we assume $\\mathbf G$ is almost simple, these\nare (open) simplices.  A \\emph{face} of $A$ is an open face of a\nchamber.  The affine space $A$ is thus a simplicial complex (with the open\nsimplices being faces) and the action of $N$ is simplicial.\n\nFor convenience we identify $A$ with $V$ by choosing a ``zero'' point $o\\in\nA$.  For $\\alpha \\in \\Phi$, define $\\phi_\\alpha\\colon U_{\\alpha} \\to {\\mathbb R}\n\\cup \\{\\infty\\}$ by setting $\\phi_\\alpha(1)=\\infty$ and requiring for\n$u\\neq 1$ that the function $x\\mapsto \\alpha(x) + \\phi_\\alpha(u)$ vanishes\non the wall fixed by $\\nu(m(u))$.  For $\\ell \\in {\\mathbb R}$, let\n\\begin{equation*}\nU_{\\alpha,\\ell} = \\{\\, u\\in U_{\\alpha} \\mid \\phi_\\alpha(u) \\ge \\ell\\,\\}\\ .\n\\end{equation*}\nThese are compact open subgroups and define a decreasing exhaustive and\nseparated filtration of $U_{\\alpha}$ which has ``jumps'' only for $\\ell$ in\nthe discrete set $\\phi_\\alpha( U_{\\alpha}\\setminus \\{1\\})$.  The affine\nfunction $\\alpha + \\ell$ is called an \\emph{affine root} if for some $u\\in\nU_{\\alpha}\\setminus \\{1\\}$, $\\ell = \\phi_\\alpha(u)$ and (if $2\\alpha$ is a\nroot) $\\phi_\\alpha(u)= \\sup \\phi_\\alpha(u U_{2\\alpha})$; let\n$r_{\\alpha,\\ell} = \\nu(m(u))$ be the corresponding affine reflection.  Note\nthat the zero set of an affine root is a  wall of\n$A$ and every wall of $A$ arises in this fashion.\n\nDenote the set of affine roots by $\\Phi_{\\mathrm{af}}$; it is an \\emph{affine root\n  system} in the sense of \\cite{Macdonald}.  The Weyl group $W_{\\mathrm{af}}$ of the\naffine root system $\\Phi_{\\mathrm{af}}$ is the group generated by $r_{\\alpha,\\ell}$\nfor $\\alpha + \\ell \\in \\Phi_{\\mathrm{af}}$; it is an affine Weyl group in the sense\nof \\cite{Bourbaki}*{Ch.~VI, \\S2} associated to a reduced root system (not\nnecessarily $\\Phi$).  Since we assume $\\mathbf G$ is simply connected, $W_{\\mathrm{af}} =\n\\nu(N) \\cong W'$.\n\nThe \\emph{apartment} associated to $\\Split$ consists of the affine\nsimplicial space $A$ together with the action of $N$, the affine root\nsystem $\\Phi_{\\mathrm{af}}$, and the filtration of the root groups,\n$(U_{\\alpha,\\ell})_{\\substack{\\alpha\\in\\Phi \\\\ \\ell\\in {\\mathbb R}}}$.\n\n\\subsection{The building}\n\\label{ssectBuilding}\n\nFor $x\\in A$, let $U_x$ be the group generated by $U_{\\alpha,\\ell}$ for all\n$\\alpha + \\ell \\in\\Phi_{\\mathrm{af}}$ such that $(\\alpha + \\ell)(x) \\ge 0$.  The\n\\emph{building} of $\\mathbf G$ over $k_v$ is defined \\cite{BruhatTits1}*{(7.4.2)}\nto be\n\\begin{equation*}\nX_v =  (G\\times A ) \/ \\!\\sim \\ ,\n\\end{equation*}\nwhere $(gnp,x) \\sim (g, \\nu(n)x)$ for all $n\\in N$ and $p \\in H U_x$.  We\nidentify $A$ with the subset of $X_v$ induced by $\\{1\\}\\times A $.\n\nThe building $X_v$ has an action of $\\mathbf G(k_v)$ induced by left\nmultiplication on the first factor of $G\\times A$.  Under this action, $N$\nacts on $A\\subset X_v$ via $\\nu$ and $U_{\\alpha,\\ell}$ fixes the points in\nthe half-space of $A$ defined by $\\alpha +\\ell\\ge 0$.  The simplicial\nstructure on $A$ induces one on $X_v$ and the action of $\\mathbf G(k_v)$ is\nsimplicial.  The subcomplex $gA\\subset X_v$ may be identified with the\napartment corresponding to the maximal split torus $g\\Split g^{-1}$.\n\nChoose an inner product on $V$ which is invariant under the Weyl group $W$;\nthe resulting metric on $A$ may be transferred to any apartment by using the\naction of $\\mathbf G(k_v)$.  These metrics fit together to give a well-defined\nmetric on $X_{v}$ which is invariant under $\\mathbf G(k_{v})$\n\\cite{BruhatTits1}*{(7.4.20)} and complete \\cite{BruhatTits1}*{(2.5.12)}.\nGiven two points $x$, $y\\in X_v$, there exists an apartment $gA$ of $X_v$\ncontaining them \\cite{BruhatTits1}*{(7.4.18)}.  Since $gA$ is an affine\nspace we can connect $x$ and $y$ with a line segment, $t \\mapsto tx +\n(1-t)y$, $ t \\in [0,1]$; this segment is independent of the choice of\napartment containing the two points and in fact is the unique geodesic\njoining $x$ and $y$.\n\n\\begin{prop}[\\cite{BruhatTits1}*{(7.4.20)}]\nThe mapping $t \\mapsto tx + (1-t)y$ of $[0,1] \\times X_{v} \\times X_{v}\n\\rightarrow X_{v}$ is continuous and thus $X_{v}$ is contractible.\n\\end{prop}\n\nIn fact it follows from \\cite{BruhatTits1}*{(3.2.1)} that $X_v$ is a\n$\\CAT(0)$-space.  (Recall that a $\\CAT(0)$-space is a metric space where\nthe distance between any two points is realized by a geodesic and every\ngeodesic triangle is thinner than the corresponding triangle of the same\nside lengths in the Euclidean plane; see \\cite{BH} for a comprehensive\ndiscussion of $\\CAT(0)$-spaces.)  Besides affine buildings such as $X_v$,\nanother important class of $\\CAT(0)$-spaces are the simply connected,\nnon-positively curved Riemannian manifolds such as $X_\\infty$.\n\n\\subsection{Stabilizers}\n\\label{ssectStabilizersBuilding}\n\nFor $\\Omega \\subset X_{v}$, let $\\mathbf G(k_{v})_{\\Omega}$ be the subgroup that\nfixes $\\Omega$ pointwise (the \\emph{fixateur} of $\\Omega$).  Suppose now\nthat $\\Omega \\subseteq A$ and set \\begin{equation*}\nU_{\\Omega} = \\langle \\, U_{\\alpha,\\ell} \\mid (\\alpha+\\ell)(\\Omega)  \\geq 0,\\,\n\\alpha+\\ell\\in \\Phi_{\\mathrm{af}}\\, \\rangle\\ .\n\\end{equation*}\nSince $\\mathbf G$ is simply connected and the valuation $\\omega$ is discrete,\n$\\mathbf G(k_{v})_{\\Omega} = HU_{\\Omega}$ (see \\cite{BruhatTits1}*{(7.1.10),\n  (7.4.4)}).  In particular, the stabilizer of $x\\in A$ is the compact\nopen subgroup $\\mathbf G(k_{v})_x = HU_x$.\n\nIf $F$ is a face of $A$ and $x\\in F$, then the set of affine roots which\nare nonnegative at $x$ is independent of the choice of $x\\in F$.  Thus\n$\\mathbf G(k_{v})_{F} = \\mathbf G(k_{v})_x$.  Note that an element of $\\mathbf G(k_v)$ which\nstabilizes $F$ also fixes the barycenter $x_F$ of $F$; thus $\\mathbf G(k_v)_F$ is\nthe stabilizer subgroup of $F$.  The stabilizer subgroups for the building\nof $\\SL_2$ (a tree) are calculated in \\cite{SerreTrees}*{II, 1.3}.\n\nLet $\\P$ be a parabolic $k_v$-subgroup which without loss of generality we\nmay assume contains the centralizer of $\\Split$; let $\\mathbf N_{\\P}$ be its\nunipotent radical.  Let $\\Phi_P = \\{\\, \\alpha \\in \\Phi \\mid U_\\alpha\n\\subseteq \\mathbf N_{\\P}(k_v) \\, \\}$ and set $E_P = \\{\\, v\\in V \\mid \\alpha(v) \\ge\n0, \\, \\alpha \\in \\Phi_P\\, \\}$; note that $\\Phi_P$ is contained in a positive\nsystem of roots and hence $E_P$ is a cone with nonempty interior.\n\n\\begin{lem}\n\\label{lemUnipotentsHaveFixedPoints}\nFor  $u \\in \\mathbf N_{\\P}(k_v)$ there exists $x\\in A$ such that  $x + E_P$ is\nfixed pointwise by $u$.  In particular, $u$ belongs to a compact open subgroup.\n\\end{lem}\n\n\\begin{proof}\nSince $\\mathbf N_{\\P}(k_v)$ is generated by $(U_\\alpha)_{\\alpha\\in\\Phi_P}$, there\nexists $\\ell\\in {\\mathbb R}$ such that $u$ belongs to the group generated by\n$(U_{\\alpha,\\ell})_{\\alpha\\in\\Phi_P}$.  Since $U_{\\alpha,\\ell}$ fixes the\npoints in the half-space of $A$ defined by $\\alpha +\\ell\\ge 0$,\nchoosing $x\\in A$ such that $\\alpha(x) \\ge -\\ell$ for all $\\alpha\\in\n\\Phi_P$ suffices.\n\\end{proof}\n\n\\section{The reductive Borel-Serre and Satake compactifications: the\n  $S$-arithmetic case}\n\\label{sectCompactificationsSArithmetic}\n\nWe now consider a general $S$-arithmetic subgroup $\\Gamma$ and define a\ncontractible space $X=X_S$ on which $\\Gamma$ acts properly.  If the $k$-rank\nof $\\mathbf G$ is positive, as we shall assume, $\\Gamma\\backslash X$ is noncompact\nand it is important to compactify it.  Borel and Serre \\cite{BS2} construct\n$\\ga\\backslash\\osp^{BS}$, the analogue of $\\Gamma\\backslash\\overline{X}_\\infty^{BS}$ from\n\\S\\ref{ssectBSarith}, and use it to study the cohomological finiteness of\n$S$-arithmetic subgroups.  In this section we recall their construction and\ndefine several new compactifications of $\\Gamma\\backslash X$ analogous to\nthose in \\S\\ref{sectCompactificationsArithmetic}.\n\n\\subsection{\\boldmath The space $\\Gamma\\backslash X$ associated to an\n$S$-arithmetic group}\n\nLet $S$ be a finite set of places of $k$ containing the infinite places\n$S_\\infty$ and let $S_f = S \\setminus S_\\infty$.  Define\n\\begin{equation*}\nG =G_{\\infty}\\times \\prod_{v\\in S_{f}} \\mathbf G(k_{v}),\n\\end{equation*}\nwhich is a locally\ncompact group, and\n\\begin{equation*}\nX =X_{\\infty}\\times \\prod_{v\\in S_{f}} X_{v}\\ ,\n\\end{equation*}\nwhere $X_v$ is the Bruhat-Tits building associated to $\\mathbf G(k_v)$ as\ndescribed in \\S\\ref{sectBruhatTitsBuildings}.  If we need to make clear the\ndependence on $S$, we write $X_S$.  $X$ is a locally compact\nmetric space under the distance function induced from the factors.  Since\neach factor is a $\\CAT(0)$-space and contractible (see\n\\S\\ref{ssectBuilding}), the same is true for $X$.\n\nThe group $G$ acts isometrically on $X$.  We view $\\mathbf G(k)\\subset G$ under\nthe diagonal embedding.  Any $S$-arithmetic subgroup $\\Gamma \\subset \\mathbf G(k)$ is\na discrete subgroup of $G$ and acts properly on $X$ \\cite{BS2}*{(6.8)}.\nIt is known that the quotient $\\Gamma\\backslash X$ is compact if and\nonly if the $k$-rank of $\\mathbf G$ is equal to 0.  In the following, we assume\nthat the $k$-rank of $\\mathbf G$ over $k$ is positive.  Then for every $v\\in\nS_{f}$, the $k_{v}$-rank of $\\mathbf G$ is also positive.\n\n\\subsection{The Borel-Serre compactification}\n\\label{ssectBorelSerreSarithmetic}\nDefine\n\\begin{equation*}\n\\overline{X}^{BS} = \\overline{X}_\\infty^{BS}\\times \\prod_{v\\in S_{f}}X_{v} \\ ,\n\\end{equation*}\nwhere $\\overline{X}_\\infty^{BS}$ is as in \\S\\ref{ssectBSarith}.  This space\nis contractible and the action of $\\mathbf G(k)$ on $X$ extends to a continuous\naction on $\\overline X^{BS}$.  The action of any $S$-arithmetic subgroup\n$\\Gamma$ on $\\overline{X}^{BS}$ is proper \\cite{BS2}*{(6.10)}.  When\n$S_f=\\emptyset$ this is proved in \\cite{Borel-Serre} as mentioned in\n\\S\\ref{ssectBSarith}; in general, the argument is by induction on $|S_f|$.\nThe key points are \\cite{BS2}*{(6.8)}:\n\\begin{enumerate}\n\\item The covering of $X_v$ by open stars $V(F)$ about the barycenters\n  of faces $F$ satisfies\n\\begin{equation*}\n\\gamma V(F)\\cap V(F) \\neq \\emptyset \\quad \\Longleftrightarrow \\quad\n\\gamma\\in\\Gamma_{F} = \\Gamma\\cap \\mathbf G(k_v)_F \\text{ , and}\n\\end{equation*}\n\\item For any simplex $F \\subset X_{v}$,  $\\Gamma_F$ is an\n  $(S\\setminus\\{v\\})$-arithmetic subgroup and hence by induction acts\n  properly on $\\overline{X}_{S\\setminus\\{v\\}}^{BS}$.\n\\end{enumerate}\nFurthermore $\\Gamma\\backslash \\overline{X}^{BS}$ is compact Hausdorff\n\\cite{BS2}*{(6.10)} which follows inductively from\n\\begin{enumerate}[resume]\n\\item There are only finitely many $\\Gamma$-orbits of simplices in $X_{v}$ for\n  $v\\in S_f$ and the quotient of  $\\overline{X}^{BS}_\\infty$ by an\n  arithmetic subgroup is compact.\n\\end{enumerate}\n\n\\subsection{The reductive Borel-Serre compactification}\n\\label{subsectRBSSarith}\nDefine\n\\begin{equation*}\n\\overline{X}^{RBS}= \\overline{X}_\\infty^{RBS}\\times \\prod_{v\\in S_f}X_{v} \\ .\n\\end{equation*}\nThere is a $\\mathbf G(k)$-equivariant surjection $\\overline{X}^{BS} \\to\n\\overline{X}^{RBS}$ induced from the surjection  in \\S\\ref{subsectRBSarith}.\n\\begin{prop}\n\\label{propDiscontinuousRBS}\nAny $S$-arithmetic subgroup $\\Gamma$ of $\\mathbf G(k)$ acts discontinuously on\n$\\overline{X}^{RBS}$ with a compact Hausdorff quotient $\\Gamma\\backslash\\overline{X}^{RBS}$.\n\\end{prop}\nThe proposition is proved similarly to the case of $\\ga\\backslash\\osp^{BS}$ outlined in\n\\S\\ref{ssectBorelSerreSarithmetic}; one replaces ``proper'' by\n``discontinuous'' and begins the induction with Lemma\n~\\ref{lemRBSDiscontinuous}.  The space $\\Gamma\\backslash\\overline{X}^{RBS}$ is the\n\\emph{reductive Borel-Serre compactification} of $\\Gamma\\backslash X$.\n\n\\subsection{Satake compactifications}\n\\label{subsectSatakeSArith}\nLet $(\\tau,V)$ be a spherical representation of\n$\\operatorname{Res}_{k\/{\\mathbb Q}}\\mathbf G$ as in \\S\\ref{subsectSatakeArith} and define\n\\begin{equation*}\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}=\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}\n\\times\\prod_{v\\in S_{f}} X_{v}\\ .\n\\end{equation*}\nThere is a $\\mathbf G(k)$-equivariant surjection $\\overline{X}^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ induced by  $\\overline{X}_\\infty^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ from \\S\\ref{subsectSatakeArith}.\n\n\\begin{prop}\n\\label{propDiscontinuousSatake}\nAssume that the Satake compactification $\\overline{X}_\\infty^{\\tau}$ is\ngeometrically rational.  Then any $S$-arithmetic subgroup $\\Gamma$ acts\ndiscontinuously on ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ with a compact Hausdorff\nquotient $\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$.\n\\end{prop}\n\nThe compact quotient $\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ is\ncalled the \\emph{Satake compactification} associated with $(\\tau,V)$.\n\n\\section{The fundamental group of the compactifications and applications to\nthe congruence subgroup kernel}\n\\label{sectFundGrpArithmetic}\n\nIn this section we state our main result, Theorem ~\\ref{thmMainArithmetic},\nwhich calculates the fundamental group of the reductive Borel-Serre and the\nSatake compactifications of $\\Gamma\\backslash X$.  We then apply the main result\nto identify the congruence subgroup kernel with certain fundamental groups.\nThe proof of Theorem ~\\ref{thmMainArithmetic} is postponed to\n\\S\\ref{sectProofArithmetic}.\n\nThroughout we fix a spherical representation $(\\tau,V)$ such that\n$\\overline{X}_\\infty^{\\tau}$ is geometrically rational.\n\n\\begin{defi}  Let $\\Gamma$ be a group acting continuously on a topological\n  space $Y$.  For each point $y\\in Y$, let $\\Gamma_{y} =\\{\\,g\\in\\Gamma\\mid\n  gy=y\\,\\}$ be the \\emph{stabilizer subgroup} of $y$ in $\\Gamma$.  The\n  \\emph{fixed subgroup} $\\Gamma_{f}$ is the subgroup generated by the\n  stabilizer subgroups $\\Gamma_{y}$ for all $y\\in Y$.  (The fixed subgroup is\n  obviously normal.)\n\\end{defi}\n\nIn our situation of an $S$-arithmetic subgroup $\\Gamma$ acting on $\\overline{X}^{RBS}$\nand ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$, we denote $\\Gamma_f$ by $\\Gamma_{f,RBS}$ and\n$\\Gamma_{f,\\tau}$ respectively.  The main result of this paper is the\nfollowing theorem.\n\n\\begin{thm}\n\\label{thmMainArithmetic}\nFor any $S$-arithmetic subgroup $\\Gamma$, there exists a commutative diagram\n\\begin{equation*}\n\\begin{CD}\n\\pi_{1}(\\ga\\backslash\\oX^{RBS}) @<\\cong<<  \\Gamma\/\\Gamma_{f,RBS} \\\\\n@VVV                       @VVV \\\\\n\\pi_{1}(\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}) @<\\cong<<\n\\Gamma\/\\Gamma_{f,\\tau}\n\\end{CD}\n\\end{equation*}\nwhere the horizontal maps are isomorphisms and the vertical maps are\nsurjections induced by the $\\Gamma$-equivariant projection $\\overline{X}^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ and the inclusion $\\Gamma_{f,RBS} \\subseteq\n\\Gamma_{f,\\tau}$.\n\\end{thm}\n\nThe proof of the theorem will be given in \\S\\ref{sectProofArithmetic}.  In\nthe remainder of this section we present some applications to the\ncongruence subgroup kernel.  To do this we first need to calculate\n$\\Gamma_{f,RBS}$ and $\\Gamma_{f,\\tau}$ which will require the information on\nstabilizers from \\S\\S\\ref{subsectRBSarith}, \\ref{subsectSatakeArith}, and\n\\ref{ssectStabilizersBuilding}.\n\nLet $\\P$ be a parabolic $k$-subgroup $\\P$ of $\\mathbf G$.  The $S$-arithmetic\nsubgroup $\\Gamma$ induces $S$-arithmetic subgroups $\\Gamma_{P}=\\Gamma\\cap\n\\P(k)\\subseteq \\P(k)$, $\\Gamma_{N_{P}} = \\Gamma\\cap \\mathbf N_{\\P}(k) \\subseteq\n\\mathbf N_{\\P}(k)$, and $\\Gamma_{L_{P}} = \\Gamma_{P}\/\\Gamma_{N_{P}} \\subseteq \\mathbf\nL_{\\P}(k)$, as well as $\\Gamma_{P_\\tau} = \\Gamma\\cap \\P_\\tau(k) \\subseteq\n\\P_\\tau(k)$ and $\\Gamma_{H_{P,\\tau}} = \\Gamma_P \/ \\Gamma_{P_\\tau} \\subseteq \\mathbf\nH_{\\P, \\tau}(k)$.\n\nLet $E\\Gamma\\subseteq \\Gamma$ be the subgroup generated by $\\Gamma_{N_{P}}$\nfor every parabolic $k$-subgroup $\\P$ of $\\mathbf G$. Since $\\gamma\n\\mathbf N_{\\P}\\gamma^{-1}=\\mathbf N_{\\gamma \\P\\gamma^{-1}}$ for $\\gamma \\in \\Gamma$,\n$E\\Gamma$ is clearly normal. Let $E_{\\tau}\\Gamma\\subseteq \\Gamma$ be the\nsubgroup generated by $\\Gamma_{P_\\tau} \\cap \\bigcap_{v\\in S_f} K_v$ for\nevery $\\tau$-saturated parabolic $k$-subgroup $\\P$ of $\\mathbf G$ and compact open\nsubgroups $K_v\\subset \\mathbf G(k_v)$.  As above, $E_{\\tau}\\Gamma$ is normal.\nSince $\\Gamma_{N_P}$ is generated by $\\Gamma_{N_P} \\cap \\bigcap_{v\\in S_f} K_v$\nfor various $K_v$ by Lemma~\\ref{lemUnipotentsHaveFixedPoints}, it is easy\nto see that $E\\Gamma \\subseteq E_\\tau\\Gamma$.\n\nA subgroup $\\Gamma\\subset \\mathbf G(k)$ is \\emph{neat} if the subgroup of ${\\mathbb C}$\ngenerated by the eigenvalues of $\\rho(\\gamma)$ is torsion-free for any\n$\\gamma\\in\\Gamma$.  Here $\\rho$ is a faithful representation $\\mathbf G\\to \\GL_N$\ndefined over $k$ and the condition is independent of the choice of $\\rho$.\nClearly any neat subgroup is torsion-free.  Any $S$-arithmetic subgroup has a\nnormal neat subgroup of finite index \\cite{Borel}*{\\S17.6}; the image of a\nneat subgroup by a morphism of algebraic groups is neat\n\\cite{Borel}*{\\S17.3}.\n\n\\begin{prop}\n\\label{propGammaFixedIsEGamma}\nLet $\\Gamma$ be an $S$-arithmetic subgroup. Then $E\\Gamma \\subseteq\n\\Gamma_{f,RBS}$ and $E_{\\tau}\\Gamma \\subseteq \\Gamma_{f,\\tau}$.  If $\\Gamma$\nis neat then equality holds for both.\n\\end{prop}\n\n\\begin{proof}\nWe proceed by induction on $\\vert S_{f}\\vert$. Suppose first that\n$S_{f}=\\emptyset$.  By Lemma ~\\ref{lemStabilizersRBS}, $\\Gamma_{N_P}$\nstabilizes any point of $X_{P} \\subseteq \\overline{X}^{RBS}_\\infty$ for any\nparabolic $k$-subgroup $\\P$, and hence $E\\Gamma \\subseteq \\Gamma_{f,RBS}$.\nLikewise by Lemma ~\\ref{lemStabilizersSatake}, $\\Gamma_{P_\\tau}$ stabilizes\nany point of $X_{P,\\tau} \\subset {}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ and so\n$E_{\\tau}\\Gamma \\subseteq \\Gamma_{f,\\tau}$.\n\nIf $\\Gamma$ is neat, then $\\Gamma_{L_{P}}$ and $\\Gamma_{H_{P,\\tau}}$ are\nlikewise neat and hence torsion-free.  The actions of $\\Gamma_{L_P}$ and\n$\\Gamma_{H_{P,\\tau}}$ are proper and hence $\\Gamma_{L_{P},z}$ and\n$\\Gamma_{H_{P,\\tau},z}$ are finite.  Thus these stabilizer subgroups must\nbe trivial.  It follows then from Lemmas ~\\ref{lemStabilizersRBS} and\n\\ref{lemStabilizersSatake} that $E\\Gamma = \\Gamma_{f,RBS}$ and\n$E_{\\tau}\\Gamma = \\Gamma_{f,\\tau}$.\n\nNow suppose that $v \\in S_{f}$ and let $S' = S \\setminus\\{v\\}$.  Write\n$\\overline{X}^{RBS} = \\overline{X}_{S'}^{RBS} \\times X_v$.  Suppose that\n$\\gamma\\in \\Gamma_{N_P}$ for some parabolic $k$-subgroup $\\P$.  By Lemma\n~\\ref{lemUnipotentsHaveFixedPoints}, $\\gamma \\in \\mathbf G(k_v)_y$ for some $y\\in\nX_{v}$.  Thus $\\gamma \\in \\Gamma' \\cap \\mathbf N_{\\P}(k)$, where $\\Gamma' = \\Gamma\\cap\n\\mathbf G(k_v)_y$.  Since $\\mathbf G(k_v)_y$ is a compact open subgroup, $\\Gamma'$ is an\n$S'$-arithmetic subgroup.  By induction $\\gamma = \\gamma_1 \\dots \\gamma_m$\nwhere $\\gamma_i\\in\\Gamma'_{x_i}$ with $x_i\\in \\overline{X}_{S'}^{RBS}$.  Since\neach $\\gamma_i\\in \\Gamma_{(x_i,y)} \\subset \\Gamma_{f,RBS}$, we see $E\\Gamma\n\\subseteq \\Gamma_{f,RBS}$.  The proof that $E_{\\tau}\\Gamma \\subseteq\n\\Gamma_{f,\\tau}$ is similar since if $\\gamma \\in \\Gamma_{P_\\tau} \\cap\n\\bigcap_{v\\in S_f} K_v$ then $\\gamma \\in \\mathbf G(k_v)_y$ for some $y\\in X_v$\n\\cite{BruhatTits1}*{(3.2.4)}.\n\nAssume that $\\Gamma$ is neat.  Let $(x,y) \\in \\overline{X}_{S'}^{RBS}\\times\nX_{v}$, and let  $F$ be a face of $X_v$ containing $y$.  As above,\n$ \\Gamma_F = \\Gamma\\cap \\mathbf G(k_v)_F$ is $S'$-arithmetic and, in this case, neat.\nSo by induction, $\\Gamma_{F,x} \\subseteq E(\\Gamma_F)\\subseteq E\\Gamma$.  But\nsince $\\mathbf G(k_{v})_{y}=\\mathbf G(k_{v})_{F}$, $\\Gamma_{(x,y)} =\n\\Gamma_{F,x}$. Therefore $\\Gamma_{f,RBS} \\subseteq E\\Gamma $. A similar\nargument shows that $\\Gamma_{f,\\tau} \\subseteq E_{\\tau}\\Gamma $.\n\\end{proof}\n\nWe now can deduce several corollaries of Theorem ~\\ref{thmMainArithmetic}\nand Proposition ~\\ref{propGammaFixedIsEGamma}.\n\n\\begin{cor}\n\\label{corNeat}\n$\\pi_{1}(\\ga\\backslash\\oX^{RBS})$ is a quotient of $\\Gamma\/ E\\Gamma$ and\n$\\pi_{1}(\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau})$ is a quotient of\n$\\Gamma\/ E_\\tau\\Gamma$.  If $\\Gamma$ is  neat, then $\\pi_{1}(\\ga\\backslash\\oX^{RBS}) \\cong\n\\Gamma\/ E\\Gamma$ and $\\pi_{1}(\\Gamma\\backslash\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}) \\cong \\Gamma\/ E_\\tau\\Gamma$.\n\\end{cor}\n\n\\begin{cor}\nIf $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$, $\\pi_{1}(\\ga\\backslash\\oX^{RBS})$ and\n$\\pi_{1}(\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau})$  are finite.\n\\end{cor}\n\\begin{proof}\nUnder the rank assumptions, $E\\Gamma$ is $S$-arithmetic\n\\citelist{\\cite{Margulis} \\cite{Ra2}*{Theorem~ A, Corollary~ 1}}.\n\\end{proof}\n\n\\begin{cor}\n\\label{corRankTwoAndUp}\nIf $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$, then $C(S,\\mathbf G) =\n\\varprojlim\\limits_{{\\mathfrak a}} \\pi_{1}(\\Gamma({\\mathfrak a})\\backslash\\overline{X}^{RBS})$, where ${\\mathfrak a}$\nranges over nonzero ideals of ${\\mathcal O}$.  These fundamental groups and the\nlimit are finite.\n\\end{cor}\n\\begin{proof}\nUnder the rank hypothesis, Raghunathan proves that the congruence kernel is\nthe projective limit of $\\Gamma({\\mathfrak a})\/ E\\Gamma({\\mathfrak a})$ (see\n\\eqref{eqnCongruenceKernel} in \\S\\ref{ssectElementaryMatrices}).\nFurthermore these groups are finite (see the discussion in\n\\S\\S\\ref{ssectKnownResults}, \\ref{ssectElementaryMatrices}).  Now apply\nCorollary ~\\ref{corNeat} and the fact that $\\Gamma({\\mathfrak a})$ is neat for ${\\mathfrak a}$\nsufficiently small.\n\\end{proof}\n\nSet $\\Gamma^*({\\mathfrak a}) = \\bigcap_{{\\mathfrak b}\\neq 0} E\\Gamma({\\mathfrak a})\\cdot \\Gamma({\\mathfrak b})$ where ${\\mathfrak b}$ runs\nover nonzero ideals of ${\\mathcal O}$.  Clearly\n\\begin{equation}\n\\label{eqnGammaStar}\nE\\Gamma({\\mathfrak a}) \\subseteq \\Gamma^*({\\mathfrak a}) \\subseteq \\Gamma({\\mathfrak a}).\n\\end{equation}\nBy Raghunathan's Main Lemma \\cite{Ra1}*{(1.17)}, for every nonzero ideal\n${\\mathfrak a}$ there exists a nonzero ideal ${\\mathfrak a}'$ such that $\\Gamma^*({\\mathfrak a})\\supseteq\n\\Gamma({\\mathfrak a}')$.  Thus $\\Gamma^*({\\mathfrak a})$ is the smallest $S$-congruence subgroup\ncontaining $E\\Gamma({\\mathfrak a})$.\n\n\\begin{cor}\n\\label{corIdentifyCSG}\nIf $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$, then $C(S,\\mathbf G) =\n\\pi_{1}(\\Gamma^*({\\mathfrak a})\\backslash\\overline{X}^{RBS})$ for any sufficiently small nonzero\nideal ${\\mathfrak a}$ of ${\\mathcal O}$.\n\\end{cor}\n\\begin{proof}\nSince $\\Gamma^*({\\mathfrak a})$ is an $S$-congruence subgroup,\nequations \\eqref{eqnCongruenceKernel} and \\eqref{eqnGammaStar} imply that\n\\begin{equation*}\nC(S,\\mathbf G) = \\varprojlim\\limits_{{\\mathfrak a}} \\Gamma({\\mathfrak a})\/ E\\Gamma({\\mathfrak a}) \\cong\n\\varprojlim\\limits_{{\\mathfrak a}} \\Gamma^*({\\mathfrak a}) \/ E\\Gamma({\\mathfrak a}).\n\\end{equation*}\nSince $C(S,\\mathbf G)$ is finite, the second limit will stabilize if we show\n\\begin{equation*}\n  \\Gamma^*({\\mathfrak b})\/ E\\Gamma({\\mathfrak b}) \\longrightarrow\n \\Gamma^*({\\mathfrak a}) \/ E\\Gamma({\\mathfrak a})\n\\end{equation*}\nis surjective for ${\\mathfrak b}\\subset {\\mathfrak a}$.  But this follows from Raghunathan's\nMain Lemma \\cite{Ra1}*{(1.17)} applied to ${\\mathfrak b}$ and the definition of\n$\\Gamma^*({\\mathfrak a})$.  Finally we note that that\n$\\pi_{1}(\\Gamma^*({\\mathfrak a})\\backslash\\overline{X}^{RBS}) \\cong \\Gamma^*({\\mathfrak a}) \/ E\\Gamma({\\mathfrak a})$ by\nCorollary ~\\ref{corNeat} and the fact that $E\\Gamma({\\mathfrak a}) = E\\Gamma^*({\\mathfrak a})$ (apply\n$E$ to \\eqref{eqnGammaStar}).\n\\end{proof}\n\n\\begin{rem}\nFrom the point of view of identifying the congruence subgroup kernel $C(S,\n\\mathbf G)$, Corollary ~\\ref{corIdentifyCSG} shows that the reductive Borel-Serre\ncompactification $\\ga\\backslash\\osp^{RBS}$ is the most natural compactification.  On\nthe other hand, the Satake compactifications are important as well.  In\nparticular, when $X=X_{\\infty}$ is Hermitian, the Baily-Borel\ncompactification is a normal projective variety and has played an important\nrole in algebraic geometry and number theory. In the cases considered in\n\\citelist{\\cite{hk} \\cite{kn} \\cite{hs} \\cite{ge} \\cite{Gro} \\cite{gro2}},\nthe fundamental group of the Baily-Borel compactification is shown to\nvanish.  The maximal Satake compactification is also special among the\nfamily of all Satake compactifications and important for various purposes.\nIn the general situation in this paper, the precise relations between\n$C(S,\\mathbf G)$ and $\\pi_{1}(\\Gamma(\\mathfrak a) \\backslash\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau})$ are not clear, even when $\\mathfrak a$ is a\nsufficiently small ideal, aside from the fact that $\\pi_{1}(\\Gamma^*(\\mathfrak\na) \\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau})$ is a quotient of $C(S,\\mathbf G)$ when\nthe $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$.\n\\end{rem}\n\n\\section{Proof of the main theorem}\n\\label{sectProofArithmetic}\nIn this section we give the proof of Theorem ~\\ref{thmMainArithmetic}.  The\nmain tool is Proposition ~\\ref{propGrosche}.  Part ~\\ref{itemGrosche} in\nthe proposition is used for the proof of the case where $\\Gamma$ is neat; it\nrequires the notion of an \\emph{admissible} map (Definition\n~\\ref{defiAdmissible}).  Part ~\\ref{itemArmstrong} is needed in addition to\ncomplete the general case.  In order to apply Proposition\n~\\ref{propGrosche} we must first verify that the spaces\n$\\overline{X}^{RBS}$ and ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ are simply connected\n(Proposition ~\\ref{propSimply}) and that the $\\Gamma$-actions are admissible\nin the neat case (Proposition ~\\ref{propAdmissibleNeatCase}).  Both of\nthese arguments depend on deforming paths to the boundary where the\ngeometry is simpler; this technique is formalized in Lemma\n~\\ref{lemAdmissibilityViaRetract}.\n\nHomotopy of paths $\\omega$ and $\\eta$ will always mean homotopy relative to\nthe endpoints and will be denoted $\\omega \\cong \\eta$.  An action of a\ntopological group $\\Gamma$ on a topological space $Y$ will always be a\ncontinuous action.\n\n\\begin{defi}\n\\label{defiAdmissible}\nA continuous surjection $p\\colon Y \\to X$ of topological spaces is\n\\emph{admissible} if for any path $\\omega$ in $X$ with initial point $x_0$\nand final point $x_1$\nand for any $y_0\\in p^{-1}(x_0)$, there exists a path $\\tilde{\\omega}$ in\n$Y$ starting at $y_0$ and ending at $y_1\\in p^{-1}(x_1)$ such that $p\\circ \\tilde \\omega$ is homotopic to\n$\\omega$ relative to the endpoints.w  An action of a group $\\Gamma$\non a topological space $Y$ is \\emph{admissible} if the quotient map $Y\\to\n\\Gamma\\backslash Y$ is admissible.\n\\end{defi}\n\n\\begin{prop}\n\\label{propGrosche}\nLet $Y$ be a simply connected topological space and $\\Gamma$ a discrete group\nacting on $Y$.  Assume that either\n\\begin{enumerate}\n\\item\\label{itemGrosche} the $\\Gamma$-action is discontinuous and admissible,\n  or that\n\\item\\label{itemArmstrong} the $\\Gamma$-action is proper and $Y$ is a locally\n  compact metric space.\n\\end{enumerate}\nThen the natural morphism $\\Gamma \\to \\pi_{1}(\\Gamma\\backslash Y)$ induces an\nisomorphism $\\Gamma\/\\Gamma_{f} \\cong \\pi_{1}(\\Gamma\\backslash Y)$.\n\\end{prop}\n\\begin{proof}\nSee \\cite{Gro}*{Satz~5} and \\cite{Armstrong} for hypotheses\n\\ref{itemGrosche} and \\ref{itemArmstrong} respectively .\n\\end{proof}\n\n\\begin{prop}\n\\label{propAdmissibilityImpliesSC}\nLet $p\\colon Y \\to X$ be an admissible continuous map of a simply\nconnected topological space $Y$ and assume that $p^{-1}(x_0)$ is\npath-connected for some $x_0\\in X$.  Then $X$ is simply connected.\n\\end{prop}\n\\begin{proof}\nLet $\\omega\\colon [0,1] \\to X$ be a loop based at $x_0$ and let\n$\\tilde\\omega$ be a path in $Y$ such that $p\\circ \\tilde\\omega \\cong\n\\omega$ (relative to the basepoint).  Let $\\eta$ be a path in\n$p^{-1}(x_0)$ from $\\tilde\\omega(1)$ to $\\tilde\\omega(0)$.  Then the\nproduct $\\tilde\\omega\\cdot \\eta$ is a loop in the simply connected space\n$Y$ and hence is null-homotopic.  It follows that $\\omega\\cong p\\circ\n\\tilde \\omega\\cong p\\circ(\\tilde\\omega\\cdot \\eta)$ is null-homotopic.\n\\end{proof}\n\n\\begin{lem}\n\\label{lemAdmissibilityIsLocal}\nA continuous surjection $p\\colon Y \\to X$ of topological spaces is\nadmissible if and only if $X$ can be covered by open subsets $U$\nsuch that $p|_{p^{-1}(U)}\\colon p^{-1}(U) \\to U$ is\nadmissible.\n\\end{lem}\n\\begin{proof}\nBy the Lebesgue covering lemma, any path $\\omega\\colon [0,1] \\to X$ is\nhomotopic to the product of finitely many paths, each of which maps into\none of the subsets $U$.  The lemma easily follows.\n\\end{proof}\n\n\\begin{lem}\n\\label{lemAdmissibilityViaRetract}\nLet $p\\colon Y \\to X$ be a continuous surjection of topological spaces.\nAssume there exist deformation retractions $r_t$ of $X$ onto a subspace\n$X_0$ and $\\tilde r_t$ of $Y$ onto $Y_0 = p^{-1}(X_0)$ such that $p\\circ\n\\tilde r_t = r_t \\circ p$.  Also assume for all $x\\in X$ that\n$\\pi_0(p^{-1}(x)) \\xrightarrow{\\tilde r_{0*}} \\pi_0(p^{-1}(r_0(x)))$ is\nsurjective.  Then $p$ is admissible if and only if $p|_{Y_0}\\colon Y_0\\to\nX_0$ is admissible.\n\\end{lem}\n\n\\setlength{\\pinch}{.002128769252056923\\textwidth}\n\\setlength{\\mim}{2.85427559055181102\\pinch}\n\\begin{figure}[h]\n\\begin{equation*}\n\\begin{xy}\n<0\\mim,-15\\mim>;<3\\mim,-15\\mim>:\n<24\\mim,-3\\mim>=\"c\"+<0\\mim,6\\mim>=\"cdmid\"+<0\\mim,6\\mim>=\"d\",\n(15,0)=\"adown\";(0,10)=\"bdown\" **[bordergrey]\\crv{(10,5)&(5,8)}?(.3)=\"xdown\",\n?(.25)=\"main3\",?(.6)=\"main6\",\n\"adown\"+\"c\";{\"bdown\"+\"c\"} **[verylightgrey]\\crv{ (10,5)+\"c\" & (5,8)+\"c\"},\n(15,10)=\"aup\";(0,20)=\"bup\" **[bordergrey]\\crv{(10,15)&(5,18)}\n?(.5)=\"xup\",\n\"aup\"+\"d\";{\"bup\"+\"d\"} **[bordergrey]\\crv{\"d\"+(10,15) & \"d\"+(5,18)},\n\"adown\",\\blownupslice{bordergrey}{bordergrey},\n\"bdown\",\\blownupslice{verylightgrey}{bordergrey},\n\"main6\",\\blownupslice{verylightgrey}{verylightgrey},\n\"bot\";p+<0\\mim,42\\mim>**\\dir{}?(.65)*\\dir{*}=\"y1\"*+!L{_{y_1}}=\"f3\",\n?(.25)*\\cir<1\\pinch>{}*\\frm{*}=\"f7\",?(.85)*\\cir<1\\pinch>{}*\\frm{*}=\"f1\",\n?(.75)*+{}=\"f2\",?(.35)*+{}=\"f6\",?(.45)*\\cir<1\\pinch>{}*\\frm{*}=\"f5\",\n?(.55)*+{}=\"f4\",?(.05)*+{}=\"f9\",\n\"top\"+<0\\mim,6\\mim>*++!DC\\txt<20\\mim>\\tiny{$ p^{-1}(x_1)$ (marked by\n  $\\scriptscriptstyle \\bullet$)\\\\$\\downarrow$},\n\"main6\";\"upper\" **[lightgrey]\\dir{-}?(.5)=\"r0y1\"*\\dir{*}*+!RD{_{\\tilde\n    r_0(y_1)}},\n?(.25)=\"eta1\"*\\dir{*}*+!UR{_{\\eta(1)}},\n\"lower\"-<0\\mim,6\\mim>*++!UC\\txt<20\\mim>\\tiny{$\\uparrow$\\\\$p^{-1}(r_0(x_1))$},\n;\"lower\"**[lightgrey]\\dir{--},\n\"upper\";\"upper\"+<0\\mim,6\\mim>**[lightgrey]\\dir{--},\n\"main3\",\\blownupslice{verylightgrey}{verylightgrey},\n\"main3\";\"bot\" **\\crv{~*\\dir{} \"ccp\"},?(.8)=\"x0bot\";p+<0\\mim,37.5\\mim>=\"x0top\"\n**\\dir{}?(.4)=\"y0\"*\\dir{*}*+!L{_{y_0}},\n\"main3\";\"upper\" **\\dir{}?(.35)=\"r0y0\"*\\dir{*}*+!UR{_{\\tilde r_0(y_0)}},\n;\"y0\" **\\dir{},?(.5)+\/u2.25\\pinch\/=\"mcp\",\n\"r0y0\";\"y0\" **\\crv{ \"mcp\"},?(.5)*+!U{_{\\tilde\\sigma_0}},*\\dir{>},\n\"r0y0\"+<0\\pinch,.5\\pinch>;\"y0\"+<0\\pinch,.5\\pinch> **\\crv{ \"mcp\"+<0\\pinch,.5\\pinch>},?(.5)*\\dir{>},\n\"r0y0\";\"eta1\" **\\dir{},?(.45)+\/r3.75\\pinch\/=\"cp1\",?(.55)+\/l2\\pinch\/=\"cp2\",\n\"eta1\" **\\crv{ \"cp1\" & \"cp2\" },?(.5)*+!UR{_\\eta},*\\dir{>},\n\"r0y0\"+<.35\\pinch,.35\\pinch>;\"eta1\"+<.35\\pinch,.35\\pinch> **\\crv{ \"cp1\"+<.35\\pinch,.35\\pinch> & \"cp2\"+<.35\\pinch,.35\\pinch> },?(.5)*\\dir{>},\n\"eta1\";\"r0y1\" **\\dir{-},?(.5)*+!R{_{\\psi}},*\\dir{>},\n\"eta1\"+<.5\\pinch,0\\pinch>;\"r0y1\"+<.5\\pinch,0\\pinch> **\\dir{-},?(.5)*\\dir{>},\n\"r0y1\";\"y1\" **\\dir{},?(.5)+\/d4.5\\pinch\/=\"mcp\",\n\"r0y1\";\"y1\" **\\crv{ \"mcp\"},?(.5)*+!U{_{\\tilde\\sigma_1}},*\\dir{>},\n\"r0y1\"+<0\\pinch,.5\\pinch>;\"y1\"+<0\\pinch,.5\\pinch> **\\crv{ \"mcp\"+<0\\pinch,.5\\pinch>},?(.5)*\\dir{>},\n\"adown\"+<-2\\mim,-5\\mim>*{_{Y_0\\quad\\qquad\\subseteq \\quad\\qquad Y}},\n{<79\\mim,15\\mim> \\ar _{p} @{>} <89\\mim,15\\mim>},\n<90\\mim,0\\mim>;<93\\mim,0\\mim>:\n<24\\mim,-3\\mim>=\"c\"+<0\\mim,6\\mim>=\"cdmid\"+<0\\mim,6\\mim>=\"d\",\n(15,0)=\"a\";(0,10)=\"b\" **[bordergrey]\\crv{(10,5)&(5,8)},\n?(.25)=\"main3\"*\\dir{*}*+!UR{_{r_0(x_0)}},?(.6)=\"main6\"*\\dir{*}*+!UR{_{r_0(x_1)}},,\n\"main3\",\\slice{verylightgrey}{verylightgrey},\n\"main3\";\"bot\" **\\crv{~*\\dir{} \"ccp\"},?(.8)=\"x0\"*\\dir{*}*+!U{_{x_0}},\n\"x0\" **\\crv{ \"ccp\"},?(.5)*+!U{_{\\sigma_0}},*\\dir{>},\n\"main3\"+<0\\pinch,.5\\pinch>;\"x0\"+<0\\pinch,.5\\pinch> **\\crv{ \"ccp\"+<0\\pinch,.5\\pinch>},?(.5)*\\dir{>},\n\"main6\",\\slice{verylightgrey}{verylightgrey},\n\"main6\"+\"cdmid\"-<1.5\\mim,0\\mim>=\"x1\"*\\dir{*}*+!DR{_{x_1}}, \n\"a\"+\"cdmid\";{\"b\"+\"cdmid\"} **\\crv{~*\\dir{} \"cdmid\"+(10,5) & \"cdmid\"+(5,8)},\n\"a\"+\"c\";{\"b\"+\"c\"} **[verylightgrey]\\crv{\"c\"+(10,5) & \"c\"+(5,8)},\n\"main6\";\"x1\" **\\dir{},?(.5)+\/d3\\pinch\/=\"mcp\",\n\"main6\";\"x1\" **\\crv{ \"mcp\"},?(.6)*+!U{_{\\sigma_1}},*\\dir{>},\n\"main6\"+<0\\pinch,.5\\pinch>;\"x1\"+<0\\pinch,.5\\pinch> **\\crv{ \"mcp\"+<0\\pinch,.5\\pinch>},?(.6)*\\dir{>},\n\"x0\";\"x1\" **\\dir{},?(.45)+\/r37.5\\pinch\/=\"cp1\",?(.55)+\/l27.5\\pinch\/=\"cp2\",\n\"x1\" **\\crv{ \"cp1\" & \"cp2\" },?(.5)*+!LD{_\\omega},*\\dir{>},\n\"x0\"+<.35\\pinch,.35\\pinch>;\"x1\"+<.35\\pinch,.35\\pinch> **\\crv{ \"cp1\"+<.35\\pinch,.35\\pinch> & \"cp2\"+<.35\\pinch,.35\\pinch> },?(.5)*\\dir{>},\n\"main3\";\"main6\" **\\crv{(10.2,4.4)}?(.6)*\\dir{>},  \n*+!UR{_{r_0\\circ\\omega}},\n\"main3\"+<.35\\pinch,.35\\pinch>;\"main6\"+<.35\\pinch,.35\\pinch> **\\crv{(10.2,4.4)+<.35\\pinch,.35\\pinch>}?(.6)*\\dir{>},  \n\"a\",\\slice{bordergrey}{bordergrey},\n\"b\",\\slice{verylightgrey}{bordergrey},\n\"a\"+\"d\";{\"b\"+\"d\"} **[bordergrey]\\crv{\"d\"+(10,5) & \"d\"+(5,8)},\n\"a\"+<10\\mim,-6\\mim>*{_{X_0\\quad\\subseteq \\quad X}}\n\\end{xy}\n\\end{equation*}\n\\caption{$p\\colon Y \\to X$ as in Lemma~\\ref{lemAdmissibilityViaRetract}}\n\\label{figAdmissibility}\n\\end{figure}\n\n\\begin{proof}\n(See Figure ~\\ref{figAdmissibility}.) Assume $p|_{Y_0}$ is admissible.  If\n  $\\omega$ is a path in $X$ from $x_0$ to $x_1$, then $\\omega \\cong\n  \\sigma_0^{-1} \\cdot (r_0 \\circ \\omega) \\cdot \\sigma_1$ where $\\sigma_i(t)\n  = r_t(x_i)$ for $i=0$, $1$.  Pick $y_0\\in p^{-1}(x_0)$ and let $\\eta(t)$\n  be a path in $Y_0$ starting at $\\tilde r_0(y_0)$ such that $p\\circ \\eta\n  \\cong r_0\\circ \\omega$.  By assumption there exists $y_1\\in p^{-1}(x_1)$\n  such that $\\tilde r_0(y_1)$ is in the same path-component of\n  $p^{-1}(r_0(x_1))$ as $\\eta(1)$; let $\\psi$ be any path in\n  $p^{-1}(r_0(x_1))$ from $\\eta(1)$ to $\\tilde r_0(y_1)$.  Set\n  $\\tilde\\omega = \\tilde\\sigma_0^{-1} \\cdot \\eta \\cdot \\psi \\cdot\n  \\tilde\\sigma_1$, where $\\tilde \\sigma_i(t) = \\tilde r_t(y_i)$.  Then\n  $p\\circ \\tilde\\omega \\cong \\sigma_0^{-1} \\cdot (r_0 \\circ \\omega) \\cdot\n  \\sigma_1$ and thus $p$ is admissible.\n\\end{proof}\n\nRecall the $\\mathbf G(k)$-equivariant quotient maps $\\overline{X}^{BS}\n\\xrightarrow{p_1} \\overline{X}^{RBS} \\xrightarrow{p_2}\n            {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ from \\S\\S\\ref{subsectRBSSarith},\n            \\ref{subsectSatakeSArith}.\n\n\\begin{prop}\n\\label{propSimply}\nThe spaces $\\overline{X}^{RBS}$ and ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ are\nsimply connected.\n\\end{prop}\n\\begin{proof}\nFor any finite place $v$, the building $X_{v}$ is contractible. So we need\nonly prove that $\\overline{X}^{RBS}_\\infty$ and\n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ are simply connected (the case that\n$S_{f} = \\emptyset$).  By Proposition ~\\ref{propAdmissibilityImpliesSC},\nLemma ~\\ref{lemAdmissibilityIsLocal}, and the fact that\n$\\overline{X}^{BS}_\\infty$ is contractible, it suffices to find a cover of\n$\\overline{X}^{RBS}_\\infty$ by open subsets $U$ over which $p_1$\n(resp. $p_2\\circ p_1$) is admissible.\n\nConsider first $\\overline{X}^{RBS}_\\infty$.  The inverse image\n$p_1^{-1}(X_Q)$ of a stratum $X_Q \\subseteq \\overline{X}^{RBS}_\\infty$ is\n$e(Q) = N_Q\\times X_Q \\subseteq \\overline{X}^{BS}_\\infty$.  Set $\\tilde U =\n\\overline A_Q(1) \\times N_Q \\times X_Q \\subseteq \\overline{X}^{BS}_\\infty$\n(compare \\eqref{Pcorner}) and  $U=p_1(\\tilde U)$,  a neighborhood of $X_Q$; note $p_1^{-1}(U)= \\tilde\nU$.  Define a deformation retraction of \n$\\tilde U$ onto\n$e(Q)$ by\n\\begin{equation*}\n\\tilde r_t(a,u,z) =\n\\begin{cases}\n(\\exp(\\frac{1}{t}\\log a), u, z) & \\text{for $t\\in (0,1]$,} \\\\\n(o_Q,                     u, z) & \\text{for $t=0$.}\n\\end{cases}\n\\end{equation*}\nThis descends to a deformation retraction $r_t$ of $U$ onto $X_Q$.  Since\n$p_1|_{e(Q)}\\colon N_Q\\times X_Q \\to X_Q$ is admissible and $N_Q$ is\npath-connected, Lemma ~\\ref{lemAdmissibilityViaRetract} shows that\n$p_1|_{\\tilde U}$ is admissible.\n\nNow consider ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ and a stratum $X_{Q,\\tau}$,\nwhere $\\mathbf Q$ is $\\tau$-saturated.  The inverse image $(p_2\\circ\np_1)^{-1}(X_{Q,\\tau})$ is $\\coprod_{\\P^\\dag = \\mathbf Q} e(P) \\subseteq\n\\overline{X}^{BS}_\\infty$; it is an open subset of the closed stratum\n$\\overline{e(Q)} = \\coprod_{\\P \\subseteq \\mathbf Q} e(P)$.  For each $\\P$\nsuch that $\\P^\\dag = \\mathbf Q$, we can write $e(P) = N_P\\times X_P = N_P\n\\times X_{Q,\\tau} \\times W_{P,\\tau}$ by \\eqref{eqnBoundaryDecomposition}.\nThus $(p_2\\circ p_1)^{-1}(X_{Q,\\tau}) = Z_Q\\times X_{Q,\\tau}$, where $Z_Q =\n\\coprod_{\\P^\\dag = \\mathbf Q} ( N_P \\times W_{P,\\tau})$.  Note that\n$N_Q\\times W_{Q,\\tau}$ is dense in $Z_Q$, so $Z_Q$ is path-connected.\n\nFor $X_{Q,\\tau}\\subset {}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$, the construction\nof $\\tilde U$ is more subtle than in the case of\n$\\overline{X}^{RBS}_\\infty$.  The theory of tilings \\cite{sap1}*{Theorem\n  ~8.1} describes a neighborhood in $\\overline{X}^{BS}_\\infty$ of the\nclosed stratum $\\overline{e(Q)}$ which is piecewise-analytically\ndiffeomorphic to $\\overline A_Q(1)\\times \\overline{e(Q)}$.  (Note however\nthat the induced decomposition on the part of this neighborhood in\n$X_\\infty(Q)$ does \\emph{not} in general agree with that of\n\\eqref{Pcorner}.)  We thus obtain a neighborhood $\\tilde U$ of $(p_2\\circ\np_1)^{-1}(X_{Q,\\tau}) = Z_Q \\times X_{Q,\\tau}$ in $\\overline{X}^{BS}_\\infty$ and a\npiecewise-analytic diffeomorphism $\\tilde U \\cong \\overline A_Q(1)\\times\nZ_Q \\times X_{Q,\\tau}$; let $U = p_2\\circ p_1(\\tilde U)$ and note \n$(p_2\\circ p_1)^{-1}(U) = \\tilde U$.  Since $Z_Q$ is\npath-connected, we proceed as in the $\\overline{X}^{RBS}_\\infty$ case.\n\\end{proof}\n\n\\begin{rem}\nIt is proved in \\cite{ji2} that every Satake compactification\n$\\overline{X}^{\\tau}_\\infty$ of a symmetric space $X_\\infty$ is a topological ball\nand hence contractible. Though the partial Satake compactification\n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ is contained in\n$\\overline{X}^{\\tau}_\\infty$ as a subset, their topologies are different and\nthis inclusion is not a topological embedding.  Hence, it does not follow\nthat ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ is contractible or that a path in\n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ can be retracted into the interior.  In\nfact, it is not known if ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ is weakly\ncontractible.\n\\end{rem}\n\n\\begin{prop}\\label{propAdmissibleNeatCase}\nFor any neat $S$-arithmetic subgroup $\\Gamma$, the action of $\\Gamma$ on\n$\\overline{X}^{RBS}$ and on ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ is admissible.\n\\end{prop}\n\n\\begin{proof}\nLet $Y = \\overline{X}^{RBS}$ or ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ and let\n$p\\colon Y \\to \\Gamma\\backslash Y$ be the quotient map, which in this case is\nopen.  It suffices to find for any point $x\\in Y$ an open neighborhood $U$\nsuch that $p|_U$ is admissible.  For then $p|_{\\Gamma U}$\nis admissible and hence, by Lemma ~\\ref{lemAdmissibilityIsLocal}, $p$ is\nadmissible.\n\nWe proceed by induction on $\\vert S_{f}\\vert$ and we suppose first that\n$S_{f}=\\emptyset$.\n\nSuppose $x$ belongs to the stratum $X_Q$ of $\\overline{X}^{RBS}_\\infty$.  Since\n$\\Gamma$ is neat, $\\Gamma_{L_Q}$ is torsion-free.  Thus we can choose a\nrelatively compact neighborhood $O_Q$ of $x$ in $X_Q$ so that\n$p|_{O_Q}\\colon O_Q \\to p(O_Q)$ is a homeomorphism. Let $U =\np_1(\\overline A_Q(s) \\times N_Q \\times O_Q) \\subseteq \\overline{X}^{RBS}_\\infty$\nwhere $s>0$; this is a smaller version of the set $U$ constructed in the\nproof of Proposition ~\\ref{propSimply}.  By reduction theory, we can choose\n$s$ sufficiently large so that the identifications induced by $\\Gamma$ on $U$\nagree with those induced by $\\Gamma_Q$ \\cite{Zu3}*{(1.5)}.  Since $\\Gamma_Q\n\\subseteq N_Q \\widetilde M_Q $, it acts only on the last two factors of\n$\\overline A_Q \\times N_Q \\times X_Q$.  Thus the deformation retraction\n$r_t$ of $U$ onto $O_Q$ (from the proof of Proposition ~\\ref{propSimply})\ndescends to a deformation retraction of $p(U)$ onto $p(O_Q)=O_Q$.\nNow apply Lemma ~\\ref{lemAdmissibilityViaRetract} to see that $p|_U$ is\nadmissible.\n\nFor $x$ in the stratum $X_{Q,\\tau}$ of ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$, we\nagain emulate the construction of $U$ from the proof of Proposition\n~\\ref{propSimply}.  Specifically let $U= (p_2\\circ p_1)(\\overline\nA_Q(s)\\times Z_Q \\times O_{Q,\\tau})$ where $O_{Q,\\tau}$ is a relatively\ncompact neighborhood of $x$ in $X_{Q,\\tau}$ such that\n$p|_{O_{Q,\\tau}}\\colon O_{Q,\\tau} \\to p(O_{Q,\\tau})$ is a homeomorphism;\nsuch a $O_{Q,\\tau}$ exists since $\\Gamma_{H_{Q,\\tau}}$ is neat and hence\ntorsion-free.  By \\cite{sap1}*{Theorem ~8.1}, the identifications induced\nby $\\Gamma$ on $U$ agree with those induced by $\\Gamma_Q$ and these are\nindependent of the $\\overline A_Q(s)$ coordinate.  Thus the deformation\nretraction $r_t$ descends to $p(U)$ and we proceed as above.\n\nNow suppose that $v \\in S_{f}$ and let $S' = S \\setminus\\{v\\}$.  We\nconsider $Y = \\overline{X}^{RBS}$ which we write as\n$\\overline{X}^{RBS}_{S'}\\times X_{v}$; the case  $Y =\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ is identical.  Following \\cite{BS2}*{(6.8)}, for\neach face $F$ of $X_{v}$ let $x_{F}$ be the barycenter of $F$ and let\n$V(F)$ be the open star of $x_{F}$ in the barycentric subdivision of\n$X_{v}$. The sets $V(F)$ form an open cover of $X_{v}$.  For any $\\gamma\n\\in \\Gamma$, $\\gamma V(F) = V(\\gamma F)$. If $F_{1} \\neq F_{2}$ are two\nfaces with $\\dim F_{1} = \\dim F_{2}$, then $V(F_{1}) \\cap V(F_{2}) =\n\\emptyset$.  It follows that\n\\begin{equation*}\n\\gamma V(F)\\cap V(F) \\neq \\emptyset \\quad \\Longleftrightarrow \\quad\n\\gamma\\in\\Gamma_{F}\\ ,\n\\end{equation*}\nwhere $\\Gamma_F = \\Gamma \\cap \\mathbf G(k_v)_F$.  It follows from\n\\S\\ref{ssectStabilizersBuilding} that $\\Gamma_F$ fixes $F$ pointwise (since\n$\\mathbf G(k_v)_F$ does) and is a neat  $S'$-arithmetic subgroup (since $\\mathbf G(k_v)_F$ is\na compact open subgroup of $\\mathbf G(k_v)$)\n\nLet $U = \\overline{X}^{RBS}_{S'}\\times V(F)$ for some open face $F$ of\n$X_{v}$.  Define a deformation retraction $r_t$ of $U$ onto\n$\\overline{X}^{RBS}_{S'}\\times F$ by $r_t(w,z) = (w, tz + (1-z)r_F(z))$,\nwhere $r_F(z)$ is the unique point in $F$ which is closest to $z\\in V(F)$.\nThe map $r_t$ is $\\Gamma_F$-equivariant since $\\Gamma_{F}$ fixes $F$\npointwise and acts by isometries.  So $r_{t}$ descends to a deformation\nretraction of $p(U)$ onto $(\\Gamma_F\\backslash\n\\overline{X}^{RBS}_{S'})\\times F$.  The remaining hypothesis of Lemma\n~\\ref{lemAdmissibilityViaRetract} is satisfied since $r_0(\\gamma w, \\gamma\nz) = r_0(\\gamma w,z)$ for $\\gamma \\in \\Gamma_F$.  Since\n$\\overline{X}^{RBS}_{S'}\\times F \\to (\\Gamma_F\\backslash\n\\overline{X}^{RBS}_{S'})\\times F$ is admissible by induction, the lemma\nimplies that $p|_U$ is admissible.\n\\end{proof}\n\nTheorem ~\\ref{thmMainArithmetic} holds if $\\Gamma$ is neat by combining \nPropositions ~\\ref{propGrosche}\\ref{itemGrosche},\n~\\ref{propDiscontinuousRBS}, ~\\ref{propDiscontinuousSatake},\n\\ref{propSimply}, and \\ref{propAdmissibleNeatCase}.\n\n\\begin{cor}\n\\label{corAdmissibleSubgroupNeatCase}\nFor any neat $S$-arithmetic subgroup $\\Gamma$, the actions of $E\\Gamma$ on\n$\\overline{X}^{RBS}$ and $E_\\tau\\Gamma$ on ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ are admissible.\n\\end{cor}\n\\begin{proof}\nBy Proposition ~\\ref{propGammaFixedIsEGamma} the action of $\\Gamma\/E\\Gamma$ on\n$E\\Gamma \\backslash \\overline{X}^{RBS}$ is free and by Proposition\n~\\ref{propDiscontinuousRBS} it is discontinuous.  It follows that $E\\Gamma\n\\backslash \\overline{X}^{RBS} \\to (\\Gamma\/E\\Gamma)\\backslash (E\\Gamma\\backslash\n\\overline{X}^{RBS}) = \\Gamma \\backslash \\overline{X}^{RBS}$ is a covering\nspace (in fact a regular covering space) and thus $E\\Gamma $ acts admissibly\nif and only if $\\Gamma$ acts admissibly.\nNow apply the proposition.  The case of  ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$  is\ntreated similarly.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem ~\\textup{\\ref{thmMainArithmetic}}]\nLet $\\Gamma'\\subseteq \\Gamma$ be a normal neat subgroup of finite index.  The\nidea in the general case is to factor $\\overline{X}^{RBS}\\to \\Gamma\\backslash\n\\overline{X}^{RBS}$ as\n\\begin{equation*}\n\\overline{X}^{RBS}\\to E\\Gamma'\\backslash \\overline{X}^{RBS} \\to\n(\\Gamma\/E\\Gamma')\\backslash (E\\Gamma'\\backslash \\overline{X}^{RBS}) = \\Gamma\\backslash\n\\overline{X}^{RBS}\n\\end{equation*}\nand apply  Proposition ~\n\\ref{propGrosche}\\ref{itemGrosche}  to the first map and Proposition ~\n\\ref{propGrosche}\\ref{itemArmstrong}  to the second map.\n\nBy Proposition ~\\ref{propGammaFixedIsEGamma}, $\\Gamma'_{f,RBS} = E\\Gamma'$ and\nhence $(E\\Gamma')_{f,RBS} = E\\Gamma'$.  Thus $E\\Gamma' \\backslash\n\\overline{X}^{RBS}$ is simply connected by Propositions\n~\\ref{propDiscontinuousRBS}, \\ref{propSimply},\n\\ref{propGrosche}\\ref{itemGrosche}, and Corollary\n\\ref{corAdmissibleSubgroupNeatCase}.  We now claim that $E\\Gamma' \\backslash\n\\overline{X}^{RBS}$ is locally compact.  To see this, note that $E\\Gamma'\n\\backslash \\overline{X}^{BS}$ is locally compact since it is triangulable\n\\cite{BS2}*{(6.10)}.  Furthermore the fibers of $p_1'\\colon E\\Gamma'\n\\backslash \\overline{X}^{BS} \\to E\\Gamma' \\backslash \\overline{X}^{RBS}$ have\nthe form $\\Gamma'_{N_P}\\backslash N_P$ which are compact.  The claim follows.\nWe can now apply Proposition ~\\ref{propGrosche}\\ref{itemArmstrong} to\n$\\Gamma\\backslash \\overline{X}^{RBS} = (\\Gamma\/E\\Gamma' )\\backslash (E\\Gamma'\n\\backslash\\overline{X}^{RBS})$ and find that $\\pi_1(\\Gamma\\backslash\n\\overline{X}^{RBS}) \\cong (\\Gamma\/E\\Gamma' ) \/ (\\Gamma\/E\\Gamma' )_{f,RBS} \\cong \\Gamma \/\n\\Gamma_{f,RBS}$ as desired.  Furthermore the proof shows that the isomorphism\nis induced by the natural morphism $\\Gamma \\to \\pi_1(\\Gamma\\backslash\n\\overline{X}^{RBS})$.\n\nA similar proof using $E_\\tau\\Gamma'$ instead of $E\\Gamma'$ treats the case of\n$\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$; one only needs to observe\nthat the fibers of $p_2'\\colon E_\\tau\\Gamma' \\backslash \\overline{X}^{RBS} \\to\nE_\\tau\\Gamma' \\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ have the form\n$\\Gamma'_{L_{P,\\tau}} \\backslash \\overline{W}_{P,\\tau}^{RBS}$ which are\ncompact.\n\\end{proof}\n\n\\begin{bibdiv}\n\\begin{biblist}\n\\bib{Armstrong}{article}{\n   author={Armstrong, M. A.},\n   title={The fundamental group of the orbit space of a discontinuous group},\n   journal={Proc. Cambridge Philos. Soc.},\n   volume={64},\n   date={1968},\n   pages={299--301},\n}\n\\bib{BB}{article}{\n   author={Baily, W. L., Jr.},\n   author={Borel, A.},\n   title={Compactification of arithmetic quotients of bounded symmetric\n   domains},\n   journal={Ann. of Math. (2)},\n   volume={84},\n   date={1966},\n   pages={442--528},\n   issn={0003-486X},\n}\n\\bib{BMS}{article}{\n   author={Bass, H.},\n   author={Milnor, J.},\n   author={Serre, J.-P.},\n   title={Solution of the congruence subgroup problem for ${\\rm\n   SL}\\sb{n}\\,(n\\geq 3)$ and ${\\rm Sp}\\sb{2n}\\,(n\\geq 2)$},\n   journal={Inst. Hautes \\'Etudes Sci. Publ. Math.},\n   volume={33},\n   date={1967},\n   pages={59--137},\n   issn={0073-8301},\n}\n\\bib{Borel}{book}{\n   author={Borel, Armand},\n   title={Introduction aux groupes arithm\\'etiques},\n   series={Actualit\\'es Scientifiques et Industrielles, No. 1341},\n   publisher={Hermann},\n   place={Paris},\n   date={1969},\n   pages={125},\n}\n\\bib{BorelHarishChandra}{article}{\n   author={Borel, Armand},\n   author={Harish-Chandra},\n   title={Arithmetic subgroups of algebraic groups},\n   journal={Ann. of Math. 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Helv.},\n   volume={48},\n   date={1973},\n   pages={436--491},\n   issn={0010-2571},\n}\n\n\\bib{BS2}{article}{\n   author={Borel, A.},\n   author={Serre, J.-P.},\n   title={Cohomologie d'immeubles et de groupes $S$-arithm\\'etiques},\n   journal={Topology},\n   volume={15},\n   date={1976},\n   number={3},\n   pages={211--232},\n   issn={0040-9383},\n}\n\\bib{Bourbaki}{book}{\n   author={Bourbaki, N.},\n   title={\\'El\\'ements de math\\'ematique. Fasc. XXXIV. Groupes et alg\\`ebres\n   de Lie. Chapitre IV: Groupes de Coxeter et syst\\`emes de Tits. Chapitre\n   V: Groupes engendr\\'es par des r\\'eflexions. Chapitre VI: syst\\`emes de\n   racines},\n   series={Actualit\\'es Scientifiques et Industrielles, No. 1337},\n   publisher={Hermann},\n   place={Paris},\n   date={1968},\n   pages={288 pp. (loose errata)},\n}\n\\bib{BourbakiTopologiePartOne}{book}{\n   author={Bourbaki, N.},\n   title={\\'El\\'ements de math\\'ematique. Topologie g\\'en\\'erale. 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Sch\\'emas en groupes.\n   Existence d'une donn\\'ee radicielle valu\\'ee},\n   journal={Inst. Hautes \\'Etudes Sci. Publ. Math.},\n   volume={60},\n   date={1984},\n   pages={197--376},\n   issn={0073-8301},\n}\n\\bib{Cass}{article}{\n   author={Casselman, W. A.},\n   title={Geometric rationality of Satake compactifications},\n   conference={\n      title={Algebraic groups and Lie groups},\n   },\n   book={\n      series={Austral. Math. Soc. Lect. Ser.},\n      volume={9},\n      publisher={Cambridge Univ. Press},\n      place={Cambridge},\n   },\n   date={1997},\n   pages={81--103},\n}\n\\bib{Chevalley}{article}{\n   author={Chevalley, Claude},\n   title={Deux th\\'eor\\`emes d'arithm\\'etique},\n   journal={J. Math. Soc. Japan},\n   volume={3},\n   date={1951},\n   pages={36--44},\n}\n\\bib{ge}{book}{\n   author={van der Geer, Gerard},\n   title={Hilbert modular surfaces},\n   series={Ergebnisse der Mathematik und ihrer Grenzgebiete (3)},\n   volume={16},\n   publisher={Springer-Verlag},\n   place={Berlin},\n   date={1988},\n   pages={x+291},\n   isbn={3-540-17601-2},\n}\n\\bib{Gille}{article}{\n   author={Gille, Philippe},\n   title={Le probl\\`eme de Kneser-Tits},\n   part={Expos\\'e 983},\n   book={\n     series={Ast\\'erisque},\n     volume={326},\n     date={2009},\n     title={S\\'eminaire Bourbaki},\n     subtitle={Volume 2007\/2008, Expos\\'e 982--996},\n    },\n   pages={39--82},\n}\n\\bib{GHM}{article}{\n   author={Goresky, M.},\n   author={Harder, G.},\n   author={MacPherson, R.},\n   title={Weighted cohomology},\n   journal={Invent. 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+{"text":"\n\\section{Derivation of cooperative manager gradients} \\label{sec:derivation}\n\nIn this section, we derive an analytic expression of the gradient of the manager policy in a two-level goal-conditioned hierarchy with respect to both the losses associated with the high level and low level policies. In mathematical terms, we are trying to derive an expression for the weighted summation of the derivation of both losses, expressed as follows:\n\\begin{equation}\n    \\nabla_{\\theta_m} J_m' = \\nabla_{\\theta_m} \\left( J_m + \\lambda J_w \\right) = \\nabla_{\\theta_m} J_m + \\lambda \\nabla_{\\theta_m} J_w\n   \n\\end{equation}\nwhere $\\lambda$ is a weighting term and $J_m$ and $J_w$ are the expected returns assigned to the manager and worker policies, respectively. More specifically, these two terms are:\n\\begin{equation}\n    \\resizebox{!}{12pt}{$\n    J_m = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^{T\/k} \\gamma^t r_m(s_{kt}) \\right] = \\int_{\\mathcal{S}} \\rho_0(s_t) V_m(s_t) ds_t\n    $}\n\\end{equation}\\\\[-25pt]\n\\begin{equation}\n    \\resizebox{!}{12pt}{$\n    J_w = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^k \\gamma^t r_w(s_t, g_t,\\pi_w(s_t,g_t)) \\right] = \\int_{\\mathcal{S}} \\rho_0(s_t) V_w(s_t, g_t) ds_t \n    $}\n\\end{equation}\nHere, under the actor-critic formulation we replace the expected return under a given starting state with the value functions $V_m$ and $V_w$ This is integrated over the distribution of initial states $\\rho_0(\\cdot)$.\n\nFollowing the results by \\cite{silver2014deterministic}, we can express the first term in Eq.~\\eqref{eq:connected-gradient} as:\n\\begin{equation}\n    \\nabla_{\\theta_m} J_m = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\nabla_a Q_m (s,a)|_{a=\\pi_m(s)}\\nabla_{\\theta_m} \\pi_m(s) \\right]\n\\end{equation}\n\nWe now expand the second term of the gradient into a function of the manager and worker actor ($\\pi_m$, $\\pi_w$) and critic ($Q_m$, $Q_w$) policies and their trainable parameters.\nIn order to propagate the loss associated with the worker through the policy parameters of the manager, we assume that the goals assigned to the worker $g_t$ are not fixed variables, but rather temporally abstracted outputs from the manager policy $\\pi_m$, and may be updated in between decisions by the manager via a transition function $h$. Mathematically, the goal transition is defined as: \n\\begin{equation}\n    g_t(\\theta_m) = \n    \\begin{cases}\n        \\pi_m(s_t) & \\text{if } t \\text{ mod } k = 0 \\\\\n        h(s_{t-1}, g_{t-1}(\\theta_m), s_t) & \\text{otherwise}\n    \\end{cases}\n\\end{equation}\nFor the purposes of simplicity, we express the manager output term as $g_t$ from now on.\n\nWe begin by computing the partial derivative of the worker value function with respect to the parameters of the manager:\n\\begin{equation}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        \\nabla_{\\theta_m} V_w(s_t, g_t) &=\\nabla_{\\theta_m} Q_w (s_t, g_t, \\pi_w(s_t, g_t)) \\\\\n        &= \\nabla_{\\theta_m} \\bigg( r_w(s_t, g_t, \\pi_w(s_t,g_t)) +\\int_{\\mathcal{G}} \\int_{\\mathcal{S}} \\gamma p_w(s', g'| s_t,g_t, \\pi_w(s_t,g_t)) V_w(s',g')ds'dg' \\bigg) \\\\\n        &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{G}}\\int_{\\mathcal{S}} p_w(s',g'| s_t, g_t, \\pi_w(s_t,g_t)) V_w(s',g')ds'dg'\n    \\end{aligned}\n    $}\n    \\label{eq:gradient_p1}\n\\end{equation}\nwhere $\\mathcal{G}$ and $\\mathcal{S}$ are the goal and environment state spaces, respectively, and $p_w(\\cdot, \\cdot | \\cdot, \\cdot, \\cdot)$ is the probability distribution of the next state from the perspective of the worker given the current state and action.\n\nExpanding the latter term, we get:\n\\begin{equation}\n    \\begin{aligned}\n    &p_w(s',g'|s_t,g_t,\\pi_w(s_t,g_t)) = p_{w,1} (g'| s', s_t,g_t,\\pi_w(s_t,g_t)) p_{w,2} (s'| s_t,g_t,\\pi_w(s_t,g_t))\n    \\end{aligned}\n    \\label{eq:pw_decompose}\n\\end{equation}\nThe first element, $p_{w1}$, is the probability distribution of the next goal, and is deterministic with respect to the conditional variables. Specifically:\n\\begin{equation}\n    p_{w,1} (g'| s_t,g_t,\\pi_w(s_t,g_t)) = \n    \\begin{cases}\n        1 & \\text{if } g' = g_{t+1} \\\\\n        0 & \\text{otherwise}\n    \\end{cases}\n    \\label{eq:pw1}\n\\end{equation}\n\nThe second element, $p_{w,2}$, is the state transition probability from the MDP formulation of the task, i.e.\n\\begin{equation}\n    p_{w,2}(s'| s_t,g_t,\\pi_w(s_t,g_t)) = p (s'| s_t,\\pi_w(s_t,g_t))\n    \\label{eq:pw2}\n\\end{equation}\n\nCombining Eq.~\\eqref{eq:pw_decompose}-\\eqref{eq:pw2} into Eq.~\\eqref{eq:gradient_p1}, we get:\n\\begin{equation} \\label{eq:simplified-next-step-value}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        \\nabla_{\\theta_m} V_w(s_t,g_t) &=\\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) \\\\\n        &\\quad + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{G}}\\int_{\\mathcal{S}}\\bigg( p_{w,1} (g'| s', s_t,g_t,\\pi_w(s_t,g_t)) p_{w,2} (s'| s_t,g_t,\\pi_w(s_t,g_t)) V_w(s',g') ds'dg'\\bigg) \\\\\n        %\n        &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) \\\\\n        &\\quad + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{G}\\cap \\{g_{t+1}\\}}\\int_{\\mathcal{S}} 1 \\cdot p (s'| s_t,\\pi_w(s_t,g_t)) V_w(s',g') ds'dg' \\\\\n        &\\quad + \\gamma \\nabla_{\\theta_m} \\int_{(\\mathcal{G}\\cap \\{g_{t+1}\\})^c}\\int_{\\mathcal{S}} 0 \\cdot p (s'| s_t,\\pi_w(s_t,g_t)) V_w(s',g') ds'dg' \\\\\n        %\n        &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{S}} p(s'| s_t,\\pi_w(s_t,g_t)) V_w(s',g_{t+1})ds'\n    \\end{aligned}\n    $}\n\\end{equation}\n\nContinuing the derivation of $\\nabla_{\\theta_m}V_w$ from Eq.~\\eqref{eq:simplified-next-step-value}, we get,\n\\begin{equation} \\label{eq:continue-derivatione}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        \\nabla_{\\theta_m} V_w(s_t,g_t) &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) +\\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{S}} p(s'| s_t,\\pi_w(s_t,g_t)) V_w(g_{t+1}, s')ds' \\\\\n        %\n        &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) +\\gamma \\int_{\\mathcal{S}} \\nabla_{\\theta_m} p(s'| s_t,\\pi_w(s_t,g_t)) V_w(g_{t+1}, s')ds' \\\\\n        %\n        &= \\nabla_{\\theta_m} g_t \\nabla_g r_w(s_t,g,\\pi_w(s_t,g_t))|_{g=g_t} \\\\\n        &\\quad + \\nabla_{\\theta_m}g_t \\nabla_g \\pi_w (s_t,g)|_{g=g_t} \\nabla_a r_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)} \\\\\n        &\\quad +\\gamma\\int_\\mathcal{S} \\bigg(V_w(s',g_{t+1})\\nabla_{\\theta_m} g_t \\nabla_g \\pi_w(s_t,g)|_{g=g_t} \\nabla_a p(s'\\vert s_t,a)|_{a=\\pi_w(s_t,g_t)}ds'\\bigg)\\\\\n        &\\quad +\\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\\\\\n        %\n        &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) \\\\\n        &\\quad \\quad \\quad \\quad \\quad \\quad + \\pi_w (s_t,g) \\nabla_a r_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)} \\vphantom{\\int} \\\\\n        &\\quad \\quad \\quad \\quad \\quad \\quad + \\gamma\\int_\\mathcal{S} V_w(s',g_{t+1}) \\pi_w(s_t,g) \\nabla_a p(s'\\vert s_t,a)|_{a=\\pi_w(s_t,g_t)}ds' \\bigg) \\bigg\\rvert_{g=g_t}\\\\\n        &\\quad +\\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\\\\\n        %\n        &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) \\\\\n        &\\quad \\quad \\quad \\quad \\quad \\quad + \\pi_w (s_t,g) \\nabla_a \\bigg( r_w(s_t,g_t,a) + \\gamma\\int_\\mathcal{S} V_w(s',g_{t+1}) p(s'\\vert s_t,a)ds' \\bigg)\\bigg\\rvert_{a=\\pi_w(s_t,g_t)} \\bigg) \\bigg\\rvert_{g=g_t}\\\\\n        &\\quad + \\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(g_t, s_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\\\\\n        &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t}\n        \\\\\n        &\\quad + \\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\n    \\end{aligned}\n    $}\n\\end{equation} \n\nIterating this formula, we have,\n\\begin{equation}\n    \\resizebox{.9\\hsize}{!}{$\n     \\begin{aligned}\n        \\nabla_{\\theta_m} V_w(s_t,g_t) &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t}\\\\\n        &\\quad +\\gamma\\int_\\mathcal{S}p(s_{t+1}\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s_{t+1},g_{t+1}) ds_{t+1} \\\\\n        %\n        &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t}\n        \\quad \\\\\n        &\\quad +\\gamma\\int_\\mathcal{S}p(s_{t+1}\\vert s_t,\\pi_w(s_t,g_t)) \\nabla_{\\theta_m} g_{t+1} \\nabla_g \\bigg(r_w(s_{t+1},g,\\pi_w(s_{t+1},g_{t+1})) \\vphantom{\\int} \\\\\n        &\\quad \\quad \\quad \\quad \\quad + \\pi_w (s_{t+1},g) \\nabla_a Q_w(s_{t+1},g_{t+1},a)|_{a=\\pi_w(s_{t+1},g_{t+1})}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_{t+1}}ds_{t+1} \\\\\n        & \\quad +\\gamma^2 \\int_\\mathcal{S}\\int_\\mathcal{S} \\bigg( p(s_{t+1}\\vert s_t,\\pi_w(s_t,g_t)) p(s_{t+2}\\vert s_{t+1},\\pi_w(g_{t+1}, s_{t+1}))\\\\\n        &\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\nabla_{\\theta_m} V_w(s_{t+2},g_{t+2}) ds_{t+2}   ds_{t+1} \\bigg)\\\\\n       \n        & \\hspace{45mm} \\vdots\\\\\n        &= \\sum_{n=0}^{\\infty} \\gamma^n \\underbrace{\\int_\\mathcal{S} \\cdots \\int_\\mathcal{S}}_{n \\text{ times}} \\left(\\prod_{k=0}^{n-1} p(s_{t+k+1}|s_{t+k},\\pi_w(s_{t+k},g_{t+k})) \\right) \\\\\n        &\\quad \\quad \\quad \\quad \\times \\nabla_{\\theta_m} g_{t+n} \n        \\nabla_g \\bigg(r_w(s_{t+n},g,\\pi_w(s_{t+n},g_{t+n})) \\\\\n        & \\quad \\quad \\quad +\\pi_w (s_{t+n},g) \\nabla_a Q_w(s_{t+n},g_{t+n},a)|_{a=\\pi_w(s_{t+n},g_{t+n})}\\bigg)\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_{t+n}} ds_{t+n}\\cdots ds_{t+1}\n    \\end{aligned}\n    $}\n\\end{equation}\n\nTaking the gradient of the expected worker value function, we get,\n\\begin{small}\n\\begin{equation}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        \\nabla_{\\theta_m} J_w &= \\nabla_{\\theta_m} \\int_{\\mathcal{S}} \\rho_0(s_0) V_w(s_0, g_0) ds_0 \\\\\n        %\n        &= \\int_{\\mathcal{S}} \\rho_0(s_0) \\nabla_{\\theta_m} V_w(s_0, g_0) ds_0 \\\\\n        %\n        &= \\int_{\\mathcal{S}} \\rho_0(s_0) \\sum_{n=0}^{\\infty} \\gamma^n \\underbrace{\\int_\\mathcal{S} \\cdots \\int_\\mathcal{S}}_{n \\text{ times}} \\Bigg[\\left(\\prod_{k=0}^{n-1} p(s_{k+1}|s_k,\\pi_w(s_k,g_k)) \\right) \\nabla_{\\theta_m} g_n \\\\\n        &\\quad \\quad \\quad \\quad \\times \\nabla_g \\bigg(r_w(s_n,g,\\pi_w(s_n,g_n))\\vphantom{\\int} + \\pi_w (s_n,g) \\nabla_a Q_w(s_n,g_n,a)|_{a=\\pi_w(s_n,g_n)}\\vphantom{\\int} \\bigg)\\Bigg] \\bigg\\rvert_{g=g_n} ds_n\\cdots ds_0 \\\\\n        %\n        &= \\sum_{n=0}^{\\infty} \\underbrace{\\int_\\mathcal{S} \\cdots \\int_\\mathcal{S}}_{n+1 \\text{ times}} \\gamma^n p_{\\theta_m, \\theta_w, n}(\\tau) \\nabla_{\\theta_m} g_n\n        \\nabla_g \\bigg(r_w(s_n,g,\\pi_w(s_n,g_n))\\vphantom{\\int}\\\\\n        &\\quad \\quad \\quad \\quad + \\pi_w (s_n,g) \\nabla_a Q_w(s_n,g_n,a)|_{a=\\pi_w(s_n,g_n)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_n} ds_n\\cdots ds_0 \\\\\n       \n        &= \\mathbb{E}_{\\tau \\sim p_{\\theta_m, \\theta_w}(\\tau)} \\bigg[ \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t} \\bigg]\n    \\end{aligned}\n    $}\n\\end{equation}\n\\end{small}\nwhere $\\tau=(s_0, a_0, s_1, a_1, \\dots, s_n)$ is a trajectory and $p_{\\theta_m, \\theta_w, n}(\\tau)$ is the (improper) discounted probability of witnessing a trajectory a set of policy parameters $\\theta_m$ and $\\theta_w$.\n\nThe final representation of the connected gradient formulation is then:\n\\begin{equation}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        \\nabla_{\\theta_m} J_m' &= \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\nabla_a Q_m (s,a)|_{a=\\pi_m(s)}\\nabla_{\\theta_m} \\pi_m(s) \\right] \\\\\n        & \\quad + \\mathbb{E}_{\\tau \\sim p_{\\theta_m, \\theta_w}(\\tau)} \\bigg[ \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t} \\bigg]\n    \\end{aligned}\n    $}\n\\end{equation}\n\n\n\\section{Cooperative HRL as goal-constrained optimization}\n\\label{sec:constrained-hrl}\n\nIn this section we will derive a constrained optimization problem that motivates cooperation between a meta policy $\\pi$ and a worker policy $\\omega$. We will derive an update rule for the finite horizon reinforcement learning setting, and then approximate the derivation for stationary policies by dropping the time dependencies from the meta policy, worker policy, and the cooperative $\\lambda$. Our goal is to find a hierarchy of policies $\\pi$ and $\\omega$ with maximal expected return subject to a constraint on minimum expected distance from goals proposed by $\\pi$. Put formally, \n\\begin{gather}\n    \\max_{\\pi_{0:T}, \\omega_{0:T}} \\sum_{t = 0}^{T} \\mathbb{E} \\left[ r (s_{t}, a_{t}) \\right] \\;\\text{s.t.}\\; \\sum_{i = t}^{T} \\mathbb{E} \\left[ \\left\\| s_{i + 1} - g_{i} \\right\\|_{p} \\right] \\leq \\delta \\; \\forall t\n\\end{gather}\n\nwhere $\\delta$ is the desired minimum expected distance from goals proposed by $\\pi$. The optimal worker policy $\\omega$ without the constraint need not be goal-reaching, and so we expect the constraint to be tight in practice---this seems to be true in our experiments in this paper. The hierarchy of policies at iteration $t$ may only affect the future, and so we can use approximate dynamic programming to solve for the optimal hierarchy at the last timestep, and proceed backwards in time. We write the optimization problem as iterated maximization,\n\\begin{gather}\n    \\max_{\\pi_{0}, \\omega_{0}} \\mathbb{E} \\left[ r (s_{0}, a_{0}) + \\max_{\\pi_{1}, \\omega_{1}} \\mathbb{E} \\left[ \\cdots + \\max_{\\pi_{T}, \\omega_{T}} \\mathbb{E} \\left[ r (s_{T}, a_{T}) \\right]  \\right] \\right]\n\\end{gather}\n\nsubject to a constraint on the minimum expected distance from goals proposed by $\\pi$. Starting from the last time step, we convert the primal problem into a dual problem. Subject to the original constraint on minimum expected distance from goals proposed by $\\pi_{T}$ at the last timestep,\n\\begin{gather}\n    \\max_{\\pi_{T}, \\omega_{T}} \\mathbb{E} \\left[ r (s_{T}, a_{T}) \\right] = \\min_{\\lambda_{T} \\geq 0} \\max_{\\pi_{T}, \\omega_{T}} \\mathbb{E} \\left[ r (s_{T}, a_{T}) \\right] + \\lambda_{T} \\delta - \\lambda_{T} \\sum_{i = T}^{T} \\mathbb{E} \\left[ \\left\\| s_{i + 1} - g_{i} \\right\\|_{p} \\right]\n\\end{gather}\n\nwhere $\\lambda_{T}$ is a Lagrange multiplier for time step $T$, representing the extent of the cooperation bonus between the meta policy $\\pi_{T}$ and the worker policy $\\omega_{T}$ at the last time step. In the last step we applied strong duality, because the objective and constraint are linear functions of $\\pi_{T}$ and $\\omega_{T}$. Solving the dual problem corresponds to CHER, which trains a meta policy $\\pi_{T}$ with a cooperative goal-reaching bonus weighted by $\\lambda_{T}$. The optimal cooperative bonus can be found by performing minimization over a simplified objective using the optimal meta and worker policies.\n\\begin{gather}\n    \\min_{\\lambda_{T}\\geq 0} \\lambda_{T} \\delta - \\lambda_{T} \\sum_{i = T}^{T} \\mathbb{E}_{g_{i} \\sim \\pi^{*}_{T} (g_{i} | s_{i}; \\lambda_{T}), a_{i} \\sim \\omega^{*}_{T} (a_{i} | s_{i}, g_{i}; \\lambda_{T}) } \\left[ \\left\\| s_{i + 1} - g_{i} \\right\\|_{p} \\right]\n\\end{gather}\n\nBy recognizing that in the finite horizon setting the expected sum of rewards is equal to the meta policy's Q function and the expected sum of distances to goals is the worker policy's Q function for deterministic policies, we can separate the dual problem into a bi-level optimization problem first over the policies. \n\\begin{gather}\n    \\max_{\\pi_{T}, \\omega_{T}} Q_{m}(s_{T}, g_{T}, a_{T}) - \\lambda_{T} Q_{w}(s_{T}, g_{T}, a_{T})\\\\\n    \\min_{\\lambda_{T}\\geq 0} \\lambda_{T} \\delta + \\lambda_{T} Q_{w}(s_{T}, g_{T}, a_{T})\n\\end{gather}\n\nBy solving the iterated maximization backwards in time, solutions for $t<i\\leq T$ are a constant $c_{t:T}$ with respect to meta policy $\\pi_{t}$, worker policy $\\omega_{t}$ and Lagrange multiplier $\\lambda_{t}$. Dropping the time dependencies gives us an approximate solution to the dual problem for the stationary policies used in practice, which we parameterize using neural networks. \n\\begin{gather}\n    \\max_{\\pi, \\omega} Q_{m}(s_{t}, g_{t}, a_{t}) - \\lambda Q_{w}(s_{t}, g_{t}, a_{t}) + c_{t:T}\\\\\n    \\min_{\\lambda\\geq 0} \\lambda \\delta + \\lambda Q_{w}(s_{t}, g_{t}, a_{t}) + c_{t:T}\n\\end{gather}\n\nThe final approximation we make assumes that, for a worker policy that is maximizing a mixture of the meta policy reward, and the worker goal-reaching reward, the goal-reaching term tends to be optimized the most strongly of the two, leading to the following approximation.\n\\begin{gather}\n    \\max_{\\pi} Q_{m}(s_{t}, g_{t}) + \\lambda \\max_{\\omega} Q_{w}(s_{t}, g_{t}, a_{t}) + c_{t:T}\\\\\n    \\min_{\\lambda\\geq 0} \\lambda \\delta + \\lambda Q_{w}(s_{t}, g_{t}, a_{t}) + c_{t:T}\n\\end{gather}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{\\methodName Algorithm}\n\nFor completeness, we provide the \\methodName algorithm below.\n\n\n\\begin{algorithm}[H]\n\\captionsetup{}\n\\caption{\\methodName}\n\\label{alg:training}\n\\begin{algorithmic}[1]\n\\State $\\text{Initialize policy parameters}~\\theta_w, \\theta_m,~\\text{memory}~\\mathcal{D},~\\text{and cooperative term}~\\lambda$\n\\State $\\text{For dynamic updates of}~\\lambda,~\\text{initialize}~\\delta$\n\\While{True}\n\\ForEach{$t = 0,\\dots,T$}\n    \\State $g_{t} \\sim \\pi_{\\theta_{m}}(g_t | s_t)$ \\algorithmiccomment{Sample manager action}\n\n    \\State $a_{t} \\sim \\pi_{\\theta_{w}}(a_t| s_t, g_{t})$ \\algorithmiccomment{Sample worker action}\n    \n    \\State $s_{t+1}, r^{m}_{t} \\gets \\text{env}.\\text{step} \\; (a_t)$\n    \n    \\State $r^{w}_{t} \\gets r_{w}(s_{t}, g_{t}, s_{t + 1})$  \\algorithmiccomment{Worker reward}\n\\EndForEach\n\n\\State $\\theta_w \\gets \\theta_w + \\alpha \\nabla_{\\theta_w} J_w$  \\algorithmiccomment{Train worker}\n\n\\State $\\theta_m \\gets \\theta_m + \\alpha \\nabla_{\\theta_m} (J_m + \\lambda J_w)$  \\algorithmiccomment{Train manager}\n    \n\\If{$\\delta$ initialized}\n    \\State $\\lambda \\gets \\lambda + \\alpha \\nabla_{\\lambda}\\left( \\lambda \\delta - \\lambda Q_w \\right)$ \\algorithmiccomment{Train $\\lambda$}\n\\EndIf\n\\EndWhile\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Environment details} \\label{sec:environment-details}\n\nIn this section, we highlight the environmental setup and simulation software used for each of the environments explored within this article.\n\n\\subsection{Environments} \\label{sec:environments}\n\n\\paragraph{\\antgather}\nIn this task shown in Figure~\\ref{fig:antgather-env}, an agent is placed in a $20\\times20$ space with $8$ apples and $8$ bombs. The agent receives a reward of $+1$ or collecting an apple (denoted as a green disk) and $-1$ for collecting a bomb (denoted as a red disk); all other actions yield a reward of $0$. Results are reported over the average return from the past $100$ episodes.\n\n\\paragraph{\\antmaze}\nFor this task, immovable blocks are placed to confine the agent to a U-shaped corridor, shown in Figure~\\ref{fig:antmaze-sim}. The agent is initialized at position $(0,0)$, and assigned an $(x,y)$ goal position between the range $(-4,-4)\\times(20,20)$ at the start of every episode. The agent receives reward defined as the negative $L_2$ distance from this $(x,y)$ position. The performance of the agent is evaluated every $50,000$ steps at the positions $(16,0)$, $(16,16)$, and $(0,16)$ based on a ``success'' metric, defined as achieved if the agent is within an $L_2$ distance of $5$ from the target at the final step. This evaluation metric is averaged across $50$ episodes.\n\n\\paragraph{\\antfourrooms}\nAn agent is placed on one corner of the classic four rooms environment~\\citep{sutton1999between} and attempts to navigate to one of the other three corners. The agent is initialized at position $(0,0)$ and assigned an $(x,y)$ goal position corresponding to one of the other three corner ($(0, 20)$, $(20, 0)$, or $(20, 20)$). The agent receives reward defined as the negative $L_2$ distance from this $(x,y)$ position. ``Success'' in this environment is achieved if the agent is within an $L_2$ distance of $5$ from the target at the final step. The success rate is reported over the past $100$ episodes.\n\n\n\\paragraph{\\bipedalsoccer}\nA bipedal agent is placed in an open field with a soccer ball. The agent is rewarded for moving to the ball, and additionally dribbling the ball to the target. The reward function is the distance of the ball from a desired goal position shown in Figure~\\ref{fig:biped-sim}. This reward is positive to discourage the agent from falling prematurely~\\citep{DBLP:journals\/corr\/abs-1804-06424}. Results are reported over the average return from the past $100$ episodes.\n\n\n\\paragraph{\\ringroad}\nThis environment, see Figure~\\ref{fig:ring-env}, is a replication of the environment presented by~\\cite{wu2017flow}.\nA total of 22 vehicles are place in a ring road of variable length (220-270m). In the absence of autonomy, instabilities in human-driven dynamics result in the formation of stop-and-go traffic~\\citep{sugiyama2008traffic}. During training, a single vehicle is replaced by a learning agent whose accelerations are dictated by an RL policy. The vehicle perceives its speed, as well as the speed and gap between it immediate leader and follower from the five most recent time steps (resulting in a observation space of size 25). In order to maximize the throughput of the network, the agent is rewarded via a reward function defined as: $r_\\text{env} = \\frac{0.1}{n_v}\\left[\\sum_{i=1}{n_v}v_i\\right]^2$, where $n_v$ is the total number of vehicles in the network, and $v_i$ is the current speed of vehicle $i$. Results are reported over the average return from the past $100$ episodes.\n\n\\paragraph{\\highwaysingle}\nThis environment, see Figure~\\ref{fig:highwaysingle-env}, is an open-network extension of the \\ringroad environment. In this problem, downstream traffic instabilities resulting from a slower lane produce congestion in the form of stop-and-go waves, represented by the red diagonal lines in Figure~\\ref{fig:ts-control}. The states and actions are designed to match those presented for the \\ringroad environment, and is concatenated across all automated vehicles to produce a single centralized policy~\\citep{kreidieh2018dissipating}. Results are reported over the average return from the past $100$ episodes.\n\n\\subsection{Simulation details} \\label{sec:simulation-details}\n\nThe simulators and simulation horizons for each of the environments are as follows:\n\\begin{itemize}[noitemsep,leftmargin=1.5em]\n    \\item The \\antmaze, \\antfourrooms, and \\antgather environments are simulated using the MuJoCo physics engine for model-based control~\\citep{todorov2012mujoco}. The time horizon in each of these tasks is set to 500 steps, with dt = 0.02 and frame skip = 5.\n    \\item The \\bipedalsoccer environment environment using the TerrainRL simulator~\\citep{DBLP:journals\/corr\/abs-1804-06424}. The time horizon for this task is set to 512 steps.\n    \\item The \\ringroad and \\highwaysingle environments are simulated using the Flow~\\citep{wu2017flow} computational framework for mixed autonomy traffic control. During resets, the simulation is warmed up for 1500 steps to allow for regular and persistent stop-and-go waves to form. The horizon of these environments are set to 1500 steps. Additional simulation parameters are for each environment are:\n    \\begin{itemize}\n        \\item \\ringroad: simulation step size of 0.2 seconds\/step\n        \\item \\highwaysingle: simulation step size of 0.4 seconds\/step\n    \\end{itemize}\n\\end{itemize}\n\nFinally, both the \\antgather and \\bipedalsoccer environments are terminated early if the agent falls\/dies, defined as the z-coordinate of the agent being less than a certain threshold. If an agent dies, it receives a return of 0 for that time step.\n\n\\section{Algorithm details}\n\n\\subsection{Choice of hyperparamters} \\label{sec:hyperparams}\n\n\\begin{itemize}[noitemsep,leftmargin=1.5em]\n    \\item Network shapes of (256, 256) for the actor and critic of both the manager and worker with \\texttt{ReLU} nonlinearities at the hidden layers and \\texttt{tanh} nonlinearity at the output layer of the actors. The output of the actors are scaled to match the desired action space.\n    \\item Adam optimizer; learning rate: 3e-4 for actor and critic of both the manager and worker\n    \\item Soft update targets $\\tau =$ 5e-3 for both the manager and worker.\n    \\item Discount $\\gamma = 0.99$ for both the manager and worker.\n    \\item Replay buffer size: 200,000 for both the manager and worker.\n    \\item Lower-level critic updates 1 environment step and delay actor and target critic updates every 2 steps.\n    \\item Higher-level critic updates 10 environment steps and delay actor and target critic updates every 20 steps.\n    \\item Huber loss function for the critic of both the manager and worker.\n    \\item No gradient clipping.\n    \\item Exploration (for both the manager and worker): \n    \\begin{itemize}\n        \\item Initial purely random exploration for the first 10,000 time steps\n        \\item Gaussian noise of the form $\\mathcal{N}(0, 0.1 \\times \\text{action range})$ for all other steps\n    \\end{itemize}\n    \\item Reward scale of 0.1 for \\antmaze and \\antfourrooms, 10 for \\antgather, and 1 for all other tasks.\n    \\item Number of candidate goals = 10\\footnote{This hyperparameter is only used by the HIRO algorithm.}\n    \\item Subgoal testing rate = 0.3\\footnote{This hyperparameter is only used by the HAC algorithm.}\n    \\item Cooperative gradient weights ($\\lambda$):\n    \\begin{itemize}\n        \\item fixed $\\lambda$: We perform a hyperparameter search for $\\lambda$ values between $[0.005, 0.01]$, and report the best performing policy. This corresponds to $\\lambda=0.01$ for the \\antgather, \\ringroad, \\highwaysingle, and \\bipedalsoccer environments, and $\\lambda=0.005$ for the \\antmaze and \\antfourrooms environments.\n        \\item dynamic $\\lambda$: We explore varying degrees of cooperation by setting $\\delta$ such then it corresponds to $25\\%$, $50\\%$, or $75\\%$ of the worker expected return when using standard HRL, and report the best performing policy. This corresponds to $25\\%$ for the \\antgather environment, and $75\\%$ for all other environments.\n    \\end{itemize}\n\\end{itemize}\n\n\\subsection{Manager goal assignment}\n\nAs mentioned in Section~\\ref{sec:hyperparams}, the output layers of both the manager and worker policies are squashed by a \\texttt{tanh} function and then scaled by the action space of the specific policy. The scaling terms for the worker policies in all environments are provided by the action space of the environment.\n\nFor the \\antmaze, \\antfourrooms, and \\antgather environments, we follow the scaling terms utilized by~\\cite{nachum2018data}. The scaling term are accordingly $\\pm$10 for the desired relative x,y; $\\pm$0.5 for the desired relative z; $\\pm$1 for the desired relative torso orientations; and the remaining limb angle ranges are available from the ant.xml file.\n\nFor the \\bipedalsoccer environment, we limit goal assignment to the root positions and velocities of the agent, as well as the orientation of the agent's right and left feet. The scaling term are [0,1.5] for root height, $\\pm1$ for the root pose, $\\pm2$ for the positions of the right and left feet, $[\\pm2,\\pm1,\\pm2]$ in root velocity. The action space of the higher-level policy is accordingly:\n\nFinally, for the \\ringroad and \\highwaysingle environments, assigned goals represent the desired speeds by controllable\/automated vehicles. The range of desired speeds is set to 0-10 m\/s for the \\ringroad environment and 0-20 m\/s for the \\highwaysingle environment.\n\n\\subsection{Egocentric and absolute goals} \\label{sec:ego-goals}\n\n\\cite{nachum2018data} presents a mechanism for utilizing egocentric goals as a means of scaling hierarchical learning in robotics tasks. In this settings, managers assign goals in the form of desired changes in state. In order to maintain the same absolute position of the goal regardless of state change, a goal-transition function $h(s_t,g_t,s_{t+1}) = s_t + g_t - s_{t+1}$ is used between goal-updates by the manager policy. The goal transition function is accordingly defined as:\n\\begin{equation} \\label{eq:goal-transition}\n    g_{t+1} = \n    \\begin{cases}\n        \\pi_m(s_t) & \\text{if } t \\text{ mod } k = 0\\\\\n        s_t + g_t - s_{t+1} & \\text{otherwise}\n    \\end{cases}\n\\end{equation}\n\nFor the mixed-autonomy control tasks explored in this article, we find that this approach produces undesirable goal-assignment behaviors in the form of \\emph{negative} desired speeds, which limit the applicability of this approach. Instead, for these tasks we utilize a more typical \\emph{absolute} goal assignment strategy, defined as:\n\\begin{equation} \\label{eq:goal-transition}\n    g_{t+1} = \n    \\begin{cases}\n        \\pi_m(s_t) & \\text{if } t \\text{ mod } k = 0\\\\\n        g_t & \\text{otherwise}\n    \\end{cases}\n\\end{equation}\nIn this case, the goal-transition function is simply $h(s_t,g_t,s_{t+1}) = g_t$.\n\n\\subsection{Worker intrinsic reward}\n\\label{sec:intrinsic-reward}\n\nWe utilize an intrinsic reward function used for the worker that serves to characterize the goals as desired relative changes in observations, similar to~\\cite{nachum2018data}. When the manager assigns goals corresponding to \\emph{absolute} desired states, the intrinsic reward is:\n\\begin{equation}\n    r_w(s_t, g_t, s_{t+1}) = -||g_t - s_{t+1}||_2\n\\end{equation}\n\nMoreover, when the manager assigns goals corresponding to \\emph{egocentric} desired states, the intrinsic reward is:\n\\begin{equation}\n    r_w(s_t, g_t, s_{t+1}) = -||s_t + g_t - s_{t+1}||_2\n\\end{equation}\n\n\n\\subsection{Reproducibility of the HIRO algorithm} \\label{sec:hiro-reproducibility}\n\nOur implementation of HIRO, in particular the use of egocentric goals and off-policy corrections as highlighted in the original paper, are adapted largely from the original open-sourced implementation of the algorithm, as available in \\url{https:\/\/github.com\/tensorflow\/models\/tree\/master\/research\/efficient-hrl}. Both our implementation and the original results from the paper exhibit similar improvement when utilizing the off-policy correction feature, as seen in Figure~\\ref{fig:hiro-comparison}, thereby suggesting that the algorithm was successfully reproduced. We note, moreover, that our implementation vastly outperforms the original HIRO implementation on the \\antgather environment. Possible reasons this may be occurring could include software versioning, choice of hyperparameters, or specific differences in the implementation outside of the underlying algorithmic modification. More analysis would need to be performed to pinpoint the cause of these discrepancies.\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[b]{0.45\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/hiro-antgather.png}\n    \\caption{\\antgather} \n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.45\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/hiro-antmaze.png}\n    \\caption{\\antmaze} \n\\end{subfigure}\n \\caption{Training performance of the original implementation of the HIRO algorithm with ours. The original performance of HIRO, denoted by the right figure within each subfigure, is adapted from the original article, see~\\cite{nachum2018data}. While the final results do not match exactly, the relative evolution of the curves exhibit similar improvements, as seen within the regions highlighted by the red ovals.}\n\\label{fig:hiro-comparison}\n\\end{figure}\n\n\\subsection{Hierarchical Actor-Critic with egocentric goals}\n\\label{sec:hac-egocentric}\n\nTo ensure that the hindsight updates proposed within the Hierarchical Actor-Critic (HAC) algorithm~\\citep{levy2017hierarchical} are compared against other algorithms on a level playing field, we modify the non-primary features of this algorithm to result in otherwise similar training performance. For example, while the original HAC implementation utilizes DDPG as the underlying optimization procedure, we use TD3. Moreover, while HAC relies on binary intrinsic rewards that significantly sparsify the feedback provided to the worker policy, we on the other hand use distance from the goal.\n\nWe also extend the HAC algorithm to support the use of relative position, or egocentric, goal assignments as detailed in Appendix~\\ref{sec:ego-goals}. This is done by introducing the goal-transition function $h(\\cdot)$ to the hindsight goal computations when utilizing hindsight goal and action transitions. More concretely, while the original HAC implementation defines the hindsight goal $\\tilde{g}_t$ at time $t$ as $\\tilde{g}_t = \\tilde{g}_{t+1} = \\dots = \\tilde{g}_{t+k} = s_{t+k}$, the hindsight goal under the relative position goal assignment formulation is $\\tilde{g}_{t+i} = s_{t+k} - s_{t+i}, \\ i=0,\\dots,k$. This function results in the final goal $\\tilde{g}_{t+k}$ before a new one is issued by the manager to equal zero, thereby denoting the worker's success in achieving its allocated goal. These same goals are used when computing the intrinsic (worker) rewards in hindsight.\n\n\\section{Additional results}\n\n\n\n\n\n\n\\subsection{\\highwaysingle} \\label{appendix:highway-results}\n\n\\begin{figure*}[h!]\n\\begin{subfigure}[b]{0.075\\textwidth}\n\\begin{tikzpicture}\n    \\newcommand \\laneheight {2.50cm}\n    \\newcommand \\laneoffset {0.25cm}\n    \\newcommand \\lanewidth  {0.25cm}\n\n   \n    \\draw (0, -0.35cm) node {};\n\n   \n\n    \\filldraw [black!40!white] \n    (\\laneoffset, 0) rectangle \n    (\\laneoffset + \\lanewidth, \\laneheight);\n\n    \\draw \n    (\\laneoffset, 0) -- \n    (\\laneoffset, \\laneheight);\n\n    \\draw \n    (\\laneoffset + \\lanewidth, 0) -- \n    (\\laneoffset + \\lanewidth, \\laneheight);\n\n   \n    \n    \\node[inner sep=0pt] at \n    (\\laneoffset + 0.5 * \\lanewidth, 0.1 * \\laneheight)\n    {\\includegraphics[angle=90,origin=c,width=0.15cm]{figures\/car.png}};\n    \\node[inner sep=0pt] at \n    (\\laneoffset + 0.5 * \\lanewidth, 0.25 * \\laneheight)\n    {\\includegraphics[angle=90,origin=c,width=0.15cm]{figures\/car.png}};\n    \\node[inner sep=0pt] at \n    (\\laneoffset + 0.5 * \\lanewidth, 0.5 * \\laneheight)\n    {\\includegraphics[angle=90,origin=c,width=0.15cm]{figures\/car.png}};\n    \\node[inner sep=0pt] at \n    (\\laneoffset + 0.5 * \\lanewidth, 0.9 * \\laneheight)\n    {\\includegraphics[angle=90,origin=c,width=0.15cm]{figures\/car.png}};\n\n   \n\n    \\draw [->] \n    (0, 0.25 * \\laneheight) -- \n    (0, 0.75 * \\laneheight);\n\n\\end{tikzpicture}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.30\\textwidth}\n    \\centering\n    \\includegraphics[height=2.5cm]{figures\/ts-baseline.png}\n    \\caption{Human baseline}\n    \\label{fig:ts-no-control}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.30\\textwidth}\n    \\centering\n    \\includegraphics[height=2.5cm]{figures\/ts-control.png}\n    \\caption{\\methodName controller}\n    \\label{fig:ts-control}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.30\\textwidth}\n    \\centering\n    \\includegraphics[height=2.5cm]{figures\/HighwaySingle-avg-speed.pdf}\n    \\caption{Average speed for each method}\n    \\label{fig:avg-speed}\n\\end{subfigure}\\\\[-8pt]\n\\caption{Traffic flow performance of vehicles within the \\highwaysingle environment.\n\\textbf{a)} In the absence of control, downstream traffic instabilities result in the propagation of congestion in the form of stop-and-go waves, seen as the red diagonal streaks. \\textbf{b)} The control strategy generated via \\methodName results in vehicles forming gaps that prematurely dissipate these waves. \\textbf{c)} This behavior provides significant improvements to traveling speed of vehicles; in contrast, other methods are unable to improve upon the human-driven baseline.}\n\\label{fig:highwaysingle-results}\n\\end{figure*}\n\n\n\\subsection{Visual \\antmaze} \\label{sec:visual-antmaze}\n\nShown by figure~\\ref{fig:visual_ant_maze_total_steps}, CHER performs comparably to HIRO when trained on an image-based variant of our \\antmaze environment. In this particular environment, the XY position of the agent is removed from its observation. Instead, a top down egocentric image is provided to the agent, colored such that the agent can recover the hidden XY positions as a nonlinear function of image pixels. Perhaps more interesting, while CHER achieves competitive performance with HIRO, CHER trains moderately faster than HIRO, which is shown in Figure~\\ref{fig:visual_ant_maze_duration}. This is likely due to not requiring goal off policy relabelling at every step of gradient descent for the manager policy.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_out_eval_0.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_out_eval_1.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_out_eval_2.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_reward_eval_0.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_reward_eval_1.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_reward_eval_2.png}\n    \\caption{This figure shows the success rates, measured when the agent's center of mass enters within 5 units to the goal position, and an average return, calculated as the sum of negative distances from the agent's center of mass to the goal position. Total steps indicates the number of samples taken from the environment.}\n    \\label{fig:visual_ant_maze_total_steps}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_out_eval_0_duration.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_out_eval_1_duration.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_out_eval_2_duration.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_reward_eval_0_duration.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_reward_eval_1_duration.png}\n    \\quad\n    \\includegraphics[width=0.25\\linewidth]{figures\/visual_ant_maze_reward_eval_2_duration.png}\n    \\caption{This figure shows the success rates, measured when the agent's center of mass enters within 5 units to the goal position, and an average return, calculated as the sum of negative distances from the agent's center of mass to the goal position. Unlike figure \\ref{fig:visual_ant_maze_total_steps}, the x-axis in these plots is the duration of the experiment, measured by the wall clock time in seconds from start to 2.5 million environment steps.}\n    \\label{fig:visual_ant_maze_duration}\n\\end{figure}\n\n\n\n\n\\section{Background} \\label{sec:background}\n\n\nRL problems are generally studied as a \\emph{Markov decision problem} (MDP) \\citep{bellman1957markovian}, defined by the tuple: $(\\mathcal{S}, \\mathcal{A}, \\mathcal{P}, r, \\rho_0, \\gamma, T)$, where $\\mathcal{S} \\subseteq \\mathbb{R}^n$ is an $n$-dimensional state space, $\\mathcal{A} \\subseteq \\mathbb{R}^m$ an $m$-dimensional action space, $\\mathcal{P} : \\mathcal{S} \\times \\mathcal{A} \\times \\mathcal{S} \\to \\mathbb{R}_+$ a transition probability function, $r : \\mathcal{S} \\to \\mathbb{R}$ a reward function, $\\rho_0 : \\mathcal{S} \\to \\mathbb{R}_+$ an initial state distribution, $\\gamma \\in (0,1]$ a discount factor, and $T$ a time horizon. \n\nIn a MDP, an \\textit{agent} is in a state $s_t \\in \\mathcal{S}$ in the environment and interacts with this environment by performing actions $a_t \\in \\mathcal{A}$. The agent's actions are defined by a policy $\\pi_\\theta : \\mathcal{S} \\times \\mathcal{A} \\to \\mathbb{R}_+$ parametrized by $\\theta$. The objective of the agent is to learn an optimal policy: $\\theta^* := \\text{argmax}_\\theta J(\\pi_\\theta)$, where $J(\\pi_\\theta) = \\mathbb{E}_{p\\sim \\pi_\\theta} \\left[ \\sum_{i=0}^T \\gamma^i r_i\\right]$ is the expected discounted return.\n\n\n\\subsection{Hierarchical reinforcement learning} \\label{sec:background-hrl}\nIn HRL, the policy is decomposed into a high-level policy that optimizes the original task reward, and a low-level policy that is conditioned on latent goals from the high-level and executes actions within the environment. The high-level controller is decoupled from the true MDP by operating at a lower temporal resolution and passing goals~\\citep{dayan1993feudal} or options~\\citep{sutton1999between} to the lower-level. This can reduce the credit assignment problem from the perspective of the high-level controller, and allows the low-level policy to produce action primitives that support short time horizon tasks as well~\\citep{sutton1999between}.\n\nSeveral HRL frameworks have been proposed to facilitate and\/or encourage the decomposition of decision-making and execution during training~\\citep{dayan1993feudal, sutton1999between, parr1998reinforcement, dietterich2000hierarchical}. In this paper, we consider a two-level goal-conditioned arrangement~\\cite{vezhnevets2017feudal, nachum2018data, nasiriany2019planning} (see Figure~\\ref{fig:hrl-model}). This network consists of a high-level, or manager, policy $\\pi_m$ that computes and outputs goals $g_t \\sim \\pi_m(s_t)$ every $k$ time steps, and a low-level, or worker, policy $\\pi_w$ that takes as inputs the current state and the assigned goals and is encouraged to perform actions $a_t \\sim \\pi_w(s_t, g_t)$ that satisfy these goals via an intrinsic reward function $r_w(s_t, g_t, s_{t+1})$ (see Appendix~\\ref{sec:intrinsic-reward}).\n\n\\subsection{Hierarchical policy optimization} \\label{sec:no-cooperation}\n\nGoal-conditioned HRL considers a concurrent training procedure for the manager and worker policies. In this setting, the manager receives a reward based on the original environmental reward function: $r_m(s_t)$. The objective function from the perspective of the manager is then:\\\\[-3pt]\n\\begin{equation} \\label{eq:manager-objective}\n\\resizebox{!}{12pt}{$\n    J_m = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^{T\/k} \\left[ \\gamma^t r_m(s_t) \\right] \\right]\n$}\n\\end{equation}\nConversely, the worker policy is motivated to follow the goals set by the manager via an intrinsic reward $r_w (s_t, g_t, s_{t+1})$ separate from the external reward. The objective function from the perspective of the worker is then:\\\\[-6pt]\n\\begin{equation} \\label{eq:worker-objective}\n\\resizebox{!}{12pt}{$\n    J_w = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^k \\gamma^t r_w(s_t, g_t,\\pi_w(s_t,g_t)) \\right]\n$}\n\\end{equation}\nNotably, no gradients are shared between the manager and worker policies. \nAs discussed in Section~\\ref{sec:introduction} and depicted in Figure~\\ref{fig:goaldists}, the absence of such feedback often results in the formation of non-cooperative goal-assignment behaviors that further complicate the learning process.\n\n\\section{Conclusions and future work} \\label{sec:conclusions}\n\nIn this work, we propose connections between multi-agent and hierarchical reinforcement learning that motivates our novel method of inducing cooperation in hierarchies. We provide a derivation of the gradient of a manager policy with respect to its workers for an actor-critic formulation as well as introducing a $\\lambda$ weighting term for this gradient which controls the level of cooperation. We find that using \\methodName results in consistently better-performing policies, that have lower empirical non-stationarity than prior work, particularly for more difficult tasks.\n\nFor future work, we would like to apply this method to discrete action environments and additional hierarchical models such as the options framework. Potential future work also includes extending the cooperative HRL formulation to multi-level hierarchies, in which the multi-agent nature of hierarchical training is likely to be increasingly detrimental to training stability.\n\n\n\n\n\\section{Introduction} \\label{sec:introduction}\n\nTo solve interesting problems in the real world agents must be adept at planning and reasoning over long time horizons. For instance, in robot navigation and interaction tasks, \nagents must learn to compose lengthy sequences of actions to achieve long-term goals. In other environments, such as mixed-autonomy traffic control settings~\\citep{wu2017emergent, vinitsky2018benchmarks}, \nexploration is delicate, as individual actions may not influence the flow of traffic until multiple timesteps in the future. RL has had limited success in solving long-horizon planning problems such as these without relying on task-specific reward shaping strategies that limit the performance of resulting policies~\\citep{wu2017flow} or additional task decomposition techniques that are not transferable to different tasks~\\citep{sutton1999between, kulkarni2016hierarchical, 2017-TOG-deepLoco, florensa2017stochastic}.\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n\\centering\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {1.0}\n    \\newcommand \\boxheight {\\textwidth}\n    \\newcommand \\boxoffset {0}\n\n   \n    \\draw (0, -0.15) node {};\n\n    \\filldraw [limegreen] (0.305 * \\textwidth,\\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.32 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Apple ($+1$)};\n\n    \\filldraw [red] (0.46 * \\textwidth,\\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.475 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Bomb ($-1$)};\n\n    \\filldraw [blue] (0.615 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.63 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize High-level goal};\n\n    \\draw [dashed] \n    (0.790 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n    (0.825 * \\textwidth, \\boxwidth + \\boxoffset - 0.5); \n    \\draw [anchor=west]\n    (0.83 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Agent trajectory};\n\\end{tikzpicture}\n\\end{subfigure} \\\\ \\vspace{-0.4cm}\n\\begin{tikzpicture}\n    \\draw[->] (0, 0) -- (0.06*\\textwidth, 0);\n    \\draw[anchor=west] (0.065*\\textwidth, 0) node {\\scriptsize training iteration};\n\n    \\draw[->] (0.52*\\textwidth, 0) -- (0.58*\\textwidth, 0);\n    \\draw[anchor=west] (0.585*\\textwidth, 0) node {\\scriptsize training iteration};\n    \\draw (\\textwidth, 0) {};\n\\end{tikzpicture}\\\\[-15pt]\n\\begin{subfigure}[b]{0.48\\textwidth}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/100000.pdf}\n    \\end{subfigure}\n   \n   \n   \n   \n    \\hfill\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/500000.pdf}\n    \\end{subfigure}\n   \n   \n   \n   \n    \\hfill\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/900000.pdf}\n    \\end{subfigure}\n    \\caption{Standard HRL}\n    \\label{fig:goaldists-standrd}\n\\end{subfigure}\n\\hfill\n\\begin{tikzpicture}\n    \\draw [dashed] (0,0) -- (0,3.25);\n\\end{tikzpicture}\n\\hfill\n\\begin{subfigure}[b]{0.48\\textwidth}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/100000.pdf}\n    \\end{subfigure}\n   \n   \n   \n   \n    \\hfill\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/500000.pdf}\n    \\end{subfigure}\n   \n   \n   \n   \n    \\hfill\n    \\begin{subfigure}[b]{0.3\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/900000.pdf}\n    \\end{subfigure}\n    \\caption{Cooperative HRL}\n    \\label{fig:goaldists-cooperative}\n\\end{subfigure}\n\\caption{Here the learned goal proposal distributions are shown for normal HRL and our method. Our agents develop more reasonable goal proposals that allow low-level policies to learn goal reaching skills quicker. The additional communication also allows the high level to better understand why proposed goals failed.}\n\\label{fig:goaldists}\n\\end{figure*}\n\nConcurrent learning methods in hierarchical RL can improve the quality of learned hierarchical policies by flexibly updating both goal-assignment and goal-reaching policies to be better adapted to a given task~\\citep{levy2017hierarchical,nachum2018data,li2019sub}. \nThe process of simultaneously learning diverse skills and exploiting these skills to achieve a high-level objective, however, \nis an unstable and non-stationary optimization procedure that can be difficult to solve in practice.\nIn particular, at the early stages of training lower-level policies are unable to reach most goals assigned to them by a higher-level policy, and instead must learn to do so as training progresses. This inability of the lower-level to be able to reach assigned goals exacerbates the credit assignment problem from the perspective of the higher-level policy. It causes the higher-level policy to be unable to identify whether a specific goal under-performed as a result of the choice of goal or the lower-level policy's inability to achieve it. In practice, this results in the highly varying or random goal assignment strategies in~Figure~\\ref{fig:goaldists-standrd} that require a large number of samples~\\citep{li2019sub} and some degree of feature engineering~\\citep{nachum2018data} to optimize over.\n\nIn this work, we show how adding cooperation between internal levels within a hierarchy\\footnote{The connection of this problem to cooperation in multiagent RL is discussed in Section~\\ref{sec:algorithm}.}, and subsequently introduce mechanisms that promote variable degrees of cooperation in HRL. Our method named \\emph{Cooperative HiErarchical RL}, \\methodName, improves cooperation by encouraging higher-level policies to specify goals that lower-level policies can succeed at, thereby disambiguating under-performing goals from goals that were unachievable by the lower-level policy. In Figure~\\ref{fig:goaldists}(b) we show how \\methodName changes the high-level goal distributions to be within the capabilities of the agent. This approach results in more informative communication between the policies. \nThe distribution of goals or tasks the high-level will command of the lower level expands over time as the lower-level policy's capabilities increase.\n\n\nA key finding in this article is that regulating the degree of cooperation between HRL layers can significantly impact the learned behavior by the policy. Too little cooperation may introduce no change to the goal-assignment behaviors and subsequent learning, while excessive cooperation may disincentivize an agent from making forward progress. We accordingly introduce a constrained optimization that serves to regulate the degree of cooperation between layers and ground the notion of cooperation in HRL to quantitative metrics within the lower-level policy. This results in a general and stable method for optimizing hierarchical policies concurrently.\n\nWe demonstrate the performance of \\methodName on a collection of standard HRL environments and two previously unexplored mixed autonomy traffic control tasks. \n    For the former set of problems, we find that our method can achieve better performance compared to recent sample efficient off-policy and HRL algorithms. For the mixed autonomy traffic tasks, the previous HRL methods struggle while our approach subverts overestimation biases that emerge in the early stages of training, thereby allowing the controlled (autonomous) vehicles to regulate their speeds around the optimal driving of the task as opposed to continuously attempting to drive as fast as possible.\n    When transferring lower-level policies between tasks, we find that policies learned via inter-level cooperation perform significantly better in new tasks without the need for additional training. This highlights the benefit of our method in learning generalizable policies.\n\n\n\n\n\\section{Background} \\label{sec:background}\n\nRL problems are generally studied as a \\emph{Markov decision problem} (MDP) \\citep{bellman1957markovian}, defined by the tuple: $(\\mathcal{S}, \\mathcal{A}, \\mathcal{P}, r, \\rho_0, \\gamma, T)$, where $\\mathcal{S} \\subseteq \\mathbb{R}^n$ is an $n$-dimensional state space, $\\mathcal{A} \\subseteq \\mathbb{R}^m$ an $m$-dimensional action space, $\\mathcal{P} : \\mathcal{S} \\times \\mathcal{A} \\times \\mathcal{S} \\to \\mathbb{R}_+$ a transition probability function, $r : \\mathcal{S} \\to \\mathbb{R}$ a reward function, $\\rho_0 : \\mathcal{S} \\to \\mathbb{R}_+$ an initial state distribution, $\\gamma \\in (0,1]$ a discount factor, and $T$ a time horizon. \n\nIn a MDP, an \\textit{agent} is in a state $s_t \\in \\mathcal{S}$ in the environment and interacts with this environment by performing actions $a_t \\in \\mathcal{A}$. The agent's actions are defined by a policy $\\pi_\\theta : \\mathcal{S} \\times \\mathcal{A} \\to \\mathbb{R}_+$ parametrized by $\\theta$. The objective of the agent is to learn an optimal policy: $\\theta^* := \\text{argmax}_\\theta J(\\pi_\\theta)$, where $J(\\pi_\\theta) = \\mathbb{E}_{p\\sim \\pi_\\theta} \\left[ \\sum_{i=0}^T \\gamma^i r_i\\right]$ is the expected discounted return.\n\n\\subsection{Hierarchical reinforcement learning} \\label{sec:background-hrl}\nIn HRL, the policy is decomposed into a high-level policy that optimizes the environment task reward, and a low-level policy that is conditioned on latent goals from the high-level and executes actions within the environment. The high-level controller is decoupled from the true MDP by operating at a lower temporal resolution and passing goals~\\citep{dayan1993feudal} or options~\\citep{sutton1999between} to the lower-level. This can reduce the credit assignment problem from the perspective of the high-level controller, and allows the low-level policy to produce action primitives that support short time horizon tasks as well~\\citep{sutton1999between}.\n\nSeveral HRL frameworks have been proposed to facilitate and\/or encourage the decomposition of decision-making and execution during training~\\citep{dayan1993feudal, sutton1999between, parr1998reinforcement, dietterich2000hierarchical}. In this article, we consider a two-level goal-conditioned arrangement~\\citep{2017-TOG-deepLoco, vezhnevets2017feudal, nachum2018data, nasiriany2019planning} (see Figure~\\ref{fig:hrl-model}). This network consists of a high-level, or manager, policy $\\pi_m$ that computes and outputs goals $g_t \\sim \\pi_m(s_t)$ every $k$ time steps, and a low-level, or worker, policy $\\pi_w$ that takes as inputs the current state and the assigned goals and is encouraged to perform actions $a_t \\sim \\pi_w(s_t, g_t)$ that satisfy these goals via an intrinsic reward function $r_w(s_t, g_t, s_{t+1})$ (see Appendix~\\ref{sec:intrinsic-reward}).\n\n\\begin{figure*}\n\\centering\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {0.65}\n    \\newcommand \\boxoffset {0.5}\n    \\newcommand \\envwidth {7.75}\n    \\newcommand \\envheight {0.65}\n    \\newcommand \\layerskip {1.4}\n\n   \n    \\fill [orange!30!white] (0,0) rectangle (\\envwidth, \\envheight);\n    \\draw [brown] (0,0) -- (\\envwidth,0);\n    \\draw [brown] (0, \\envheight) -- (\\envwidth, \\envheight);\n    \\draw (\\envwidth\/2, \\envheight\/2) node {\\small Environment};\n\n   \n    \\filldraw [black!40!white, 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\\speedometerRadius * \\cosTen,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinTen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosTen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinTen);\n    \\draw [line width=0.35mm]\n    (\\speedometerCenterX + 0.75 * \\speedometerRadius * \\cosEleven,\n     \\speedometerCenterY + 0.75 * \\speedometerRadius * \\sinEleven) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosEleven,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinEleven);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosTwelve,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinTwelve) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosTwelve,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinTwelve);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosThirteen,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinThirteen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosThirteen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinThirteen);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosFourteen,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinFourteen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosFourteen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinFourteen);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosFifteen,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinFifteen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosFifteen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinFifteen);\n    \\draw [line width=0.35mm]\n    (\\speedometerCenterX + 0.75 * \\speedometerRadius * \\cosSixteen,\n     \\speedometerCenterY + 0.75 * \\speedometerRadius * \\sinSixteen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosSixteen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinSixteen);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosSeventeen,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinSeventeen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosSeventeen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinSeventeen);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosEighteen,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinEighteen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosEighteen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinEighteen);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosNineteen,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinNineteen) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosNineteen,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinNineteen);\n    \\draw [line width=0.25mm]\n    (\\speedometerCenterX + 0.90 * \\speedometerRadius * \\cosTwenty,\n     \\speedometerCenterY + 0.90 * \\speedometerRadius * \\sinTwenty) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosTwenty,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinTwenty);\n    \\draw [line width=0.35mm]\n    (\\speedometerCenterX + 0.75 * \\speedometerRadius * \\cosTwentyOne,\n     \\speedometerCenterY + 0.75 * \\speedometerRadius * \\sinTwentyOne) --\n    (\\speedometerCenterX + 1.00 * \\speedometerRadius * \\cosTwentyOne,\n     \\speedometerCenterY + 1.00 * \\speedometerRadius * \\sinTwentyOne);\n\n   \n    \\draw [line width=0.5mm, color=blue]\n    (\\speedometerCenterX + 0.0 * \\speedometerRadius * \\cosGoal,\n     \\speedometerCenterY + 0.0 * \\speedometerRadius * \\sinGoal) --\n    (\\speedometerCenterX + 1.0 * \\speedometerRadius * \\cosGoal,\n     \\speedometerCenterY + 1.0 * \\speedometerRadius * \\sinGoal);\n    \\draw \n    (\\speedometerCenterX + 1.15 * \\speedometerRadius * \\cosGoal + 0.3,\n     \\speedometerCenterY + 1.15 * \\speedometerRadius * \\sinGoal) node {\\small \\textcolor{blue}{goal}}; \n\n   \n    \\draw [line width=0.5mm, color=red]\n    (\\speedometerCenterX + 0.0 * \\speedometerRadius * \\cosSpeed,\n     \\speedometerCenterY + 0.0 * \\speedometerRadius * \\sinSpeed) --\n    (\\speedometerCenterX + 1.0 * \\speedometerRadius * \\cosSpeed,\n     \\speedometerCenterY + 1.0 * \\speedometerRadius * \\sinSpeed);\n    \\draw \n    (\\speedometerCenterX + 1.15 * \\speedometerRadius * \\cosSpeed + 1,\n     \\speedometerCenterY + 1.15 * \\speedometerRadius * \\sinSpeed) node {\\small \\textcolor{red}{current speed}}; \n\n   \n    \\filldraw (\\speedometerCenterX,\\speedometerCenterY) circle (0.05 * \\speedometerRadius);\n\n   \n   \n   \n   \n   \n    \n    \\draw [dashed] \n    (0.788 * \\textwidth, 0.0) -- \n    (0.788 * \\textwidth, 3.2); \n    \\draw [dashed] \n    (0.788 * \\textwidth, 3.2) -- \n    (\\textwidth, 3.2); \n\n    \\hspace*{0.8\\textwidth}\\includegraphics[width=0.2\\textwidth]{figures\/ant-goal.png}\n\n\\end{tikzpicture}\n\\caption{An illustration of the studied hierarchical model. \\textbf{Left:} A manager network $\\pi_m$ issues commands (or goals) $g_t$ over $k$ consecutive time steps to a worker $\\pi_w$. The worker then performs environment actions $a_t$ to accomplish these goals. \\textbf{Top\/right:} The commands issued by the manager denote desired states for the worker to traverse. For the AV control tasks, we define the goal as the desired speeds for each AV. Moreover, for agent navigation tasks, the goals are defined as the desired position and joint angles of the agent.}\n\\label{fig:hrl-model}\n\\end{figure*}\n\n\\subsection{Hierarchical policy optimization} \\label{sec:no-cooperation}\n\nWe consider a concurrent training procedure for the manager and worker policies. In this setting, the manager receives a reward based on the original environmental reward function: $r_m(s_t)$. The objective function from the perspective of the manager is then:\\\\[-3pt]\n\\begin{equation} \\label{eq:manager-objective}\n\\resizebox{!}{12pt}{$\n    J_m = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^{T\/k} \\left[ \\gamma^t r_m(s_t) \\right] \\right]\n$}\n\\end{equation}\nConversely, the worker policy is motivated to follow the goals set by the manager via an intrinsic reward $r_w (s_t, g_t, s_{t+1})$ separate from the external reward. The objective function from the perspective of the worker is then:\\\\[-6pt]\n\\begin{equation} \\label{eq:worker-objective}\n\\resizebox{!}{12pt}{$\n    J_w = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^k \\gamma^t r_w(s_t, g_t,\\pi_w(s_t,g_t)) \\right]\n$}\n\\end{equation}\nNotably, no gradients are shared between the manager and worker policies. As discussed in Section~\\ref{sec:introduction} and depicted in Figure~\\ref{fig:goaldists}, the absence of such feedback often results in the formation of non-cooperative goal-assignment behaviors that strain the learning process.\n\n\n\n\n\\section{Cooperative hierarchical reinforcement learning} \\label{sec:method}\n\n\\methodName promotes cooperation by propagating losses that arise from random or unachievable goal-assignment strategies. As part of this algorithm, we also present a mechanism to optimize the level of cooperation and ground the notion of cooperation in HRL to measurable variables. \n\n\\subsection{Promoting cooperation via loss-sharing} \\label{sec:algorithm}\n\n\\citet{tampuu2017multiagent} explored the effects of reward (or loss) sharing between agents on the emergence of cooperative and competitive in multiagent two-player games. Their study highlights two potential benefits of loss-sharing in multiagent systems: 1) emerged cooperative behaviors are less aggressive and more likely to emphasize improved interactions with neighboring agents, 2) these interactions reduce overestimation bias of the Q-function from the perspective of each agent. We develop a method to gain similar benefits by using loss-sharing paradigms in goal-conditioned hierarchies. In particular, we focus on the emergence of collaborative behaviors from the perspective of goal-assignment and goal-achieving policies In line with prior work, we promote the emergence of cooperative behaviors by incorporating a weighted form of the worker's expected return to the original manager objective $J_m$. The manager's new expected return is:\n\\begin{equation} \\label{eq:connected-return}\n    J_m' = J_m + \\lambda J_w\n\\end{equation}\nwhere $\\lambda$ is a weighting term that controls the level of cooperation the manager has with the worker.\n\nIn practice, we find that this addition to the objective serves to promote cooperation by aligning goal-assignment actions by the manager with achievable trajectories by the goal-achieving worker. This serves as a soft constraint to the manager policy: when presented with goals that perform similarly, the manager tends towards goals that more closely match the worker states. The degree to which the policy tends toward achievable states is dictated by the $\\lambda$ term. The choice of this parameter accordingly can have a significant effect on learned goals. For large or infinite values of $\\lambda$, for instance, this cooperative term can eliminate exploration by assigning goals that match the worker's current state and prevent forward movement (this is highlighted in Section~\\ref{sec:cg-weight}). In Section~\\ref{sec:method-constrained-hrl} we detail how we mitigate this issue by dynamically controlling the level of cooperation online.\n\n\\subsection{Cooperative gradients via differentiable communication}\n\n\\begin{wrapfigure}{R}{0.42\\textwidth}\n\\centering\n\\begin{minipage}[b]{0.95\\textwidth}\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {0.52}\n    \\newcommand \\boxoffset {0.4}\n    \\newcommand \\envwidth {6.2}\n    \\newcommand \\envheight {0.52}\n    \\newcommand \\layerskip {1.12}\n\n   \n    \\fill [orange!30!white] (0,0) rectangle (\\envwidth, \\envheight);\n    \\draw [brown] (0,0) -- (\\envwidth,0);\n    \\draw [brown] (0, \\envheight) -- (\\envwidth, \\envheight);\n    \\draw (\\envwidth\/2, \\envheight\/2) node {\\small Environment};\n\n   \n    \\filldraw [black!40!white, draw=black!90!white] \n    (2*\\boxoffset, 3.5*\\envheight) rectangle \n    (2*\\boxoffset + \\boxwidth, 3.5*\\envheight + \\boxwidth);\n    \\draw \n    (2.0*\\boxoffset + 0.5*\\boxwidth, 3.5*\\envheight + 0.5*\\boxwidth) node {\\scriptsize $\\pi_m$};\n\n   \n    \\filldraw [black!5!white, draw=black!15!white] \n    (4.5*\\boxoffset, 2*\\envheight) rectangle \n    (4.5*\\boxoffset + \\boxwidth, 2*\\envheight + \\boxwidth);\n\n    \\filldraw [black!5!white, draw=black!15!white] \n    (6.0*\\boxoffset + 1.0*\\boxwidth, 2*\\envheight) rectangle \n    (6.0*\\boxoffset + 2.0*\\boxwidth, 2*\\envheight + \\boxwidth);\n\n    \\filldraw [black!20!white, draw=black!80!white] \n    (9.0*\\boxoffset + 2.0*\\boxwidth, 2*\\envheight) rectangle \n    (9.0*\\boxoffset + 3.0*\\boxwidth, 2*\\envheight + \\boxwidth);\n    \\draw\n    (9.0*\\boxoffset + 2.5*\\boxwidth, 2*\\envheight + 0.5*\\boxwidth) node {\\scriptsize $\\pi_w$};\n\n   \n\n    \\draw [red] [->] \n    (8.6*\\boxoffset + 3.0*\\boxwidth, 3.25*\\envheight) --\n    (8.6*\\boxoffset + 3.0*\\boxwidth, 4.00*\\envheight);\n    \\draw \n    (13*\\boxoffset + \\boxwidth, 3.5*\\envheight) node {\\small \\color{red}{$\\nabla \\theta_m J_w$}}; \n\n   \n\n    \\draw [->] \n    (1.25 * \\boxoffset, \\envheight) -- (1.25 * \\boxoffset, 4 * \\envheight) -- (2*\\boxoffset, 4 * \\envheight);\n    \\draw [black!15!white] [->] \n    (1.25 * \\boxoffset, 2.5*\\envheight) -- (4.5*\\boxoffset, 2.5*\\envheight);\n    \\draw \n    (1.25 * \\boxoffset + 0.25, \\envheight + 0.2) node {\\small $s_t$}; \n\n    \\draw\n    (2*\\boxoffset + \\boxwidth, 4*\\envheight) -- \n    (5.5*\\boxoffset + 2.5*\\boxwidth, 4*\\envheight); \n    \\draw [black!15!white]\n    (4.5*\\boxoffset + 0.5*\\boxwidth, 4*\\envheight) -- \n    (4.5*\\boxoffset + 0.5*\\boxwidth, 3*\\envheight);\n    \\draw [black!15!white]\n    (4.5*\\boxoffset + 2.5*\\boxwidth, 4*\\envheight) -- \n    (4.5*\\boxoffset + 2.5*\\boxwidth, 3*\\envheight);\n    \\draw \n    (7.5*\\boxoffset + 2.0*\\boxwidth, 4.0*\\envheight) node {$\\cdots$}; \n    \\draw [->]\n    (8.5*\\boxoffset + 2.0*\\boxwidth, 4*\\envheight) -- \n    (9.0*\\boxoffset + 2.5*\\boxwidth, 4*\\envheight) --\n    (9.0*\\boxoffset + 2.5*\\boxwidth, 3*\\envheight); \n    \\draw \n    (2*\\boxoffset + \\boxwidth + 0.3, 4 * \\envheight + 0.2) node {\\small $g_t$}; \n\n    \\draw [black!15!white] [->] \n    (4.5*\\boxoffset + \\boxwidth, 2.5*\\envheight) --\n    (5.0*\\boxoffset + \\boxwidth, 2.5*\\envheight) --\n    (5.0*\\boxoffset + \\boxwidth, 1.0*\\envheight);\n    \\draw\n    (5.0*\\boxoffset + \\boxwidth - 0.25, \\envheight + 0.2) node {\\small \\color{black!15!white}{$a_t$}}; \n\n    \\draw [black!15!white] [->] \n    (5.5*\\boxoffset + \\boxwidth, 1.0*\\envheight) --\n    (5.5*\\boxoffset + \\boxwidth, 2.5*\\envheight) --\n    (6.0*\\boxoffset + \\boxwidth, 2.5*\\envheight);\n    \\draw\n    (5.5*\\boxoffset + \\boxwidth + 0.4, \\envheight + 0.2) node {\\small \\color{black!15!white}{$s_{t+1}$}}; \n\n    \\draw [black!15!white] [->] \n    (6.0*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) --\n    (6.5*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) --\n    (6.5*\\boxoffset + 2.0*\\boxwidth, 1.0*\\envheight);\n    \\draw\n    (6.5*\\boxoffset + 2.0*\\boxwidth + 0.40, \\envheight + 0.2) node {\\small \\color{black!15!white}{$a_{t+1}$}}; \n\n    \\draw (7.5*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) node {\\color{black!15!white}{$\\cdots$}}; \n\n    \\draw [->] \n    (8.5*\\boxoffset + 2.0*\\boxwidth, 1.0*\\envheight) --\n    (8.5*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) --\n    (9.0*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight);\n    \\draw\n    (8.5*\\boxoffset + 2.0*\\boxwidth + 0.40, \\envheight + 0.2) node {\\small $s_{t+i}$}; \n\n    \\draw [->] \n    (9.0*\\boxoffset + 3.0*\\boxwidth, 2.5*\\envheight) --\n    (9.5*\\boxoffset + 3.0*\\boxwidth, 2.5*\\envheight) --\n    (9.5*\\boxoffset + 3.0*\\boxwidth, 1.0*\\envheight);\n    \\draw\n    (9.5*\\boxoffset + 3.0*\\boxwidth + 0.40, \\envheight + 0.2) node {\\small $a_{t+i}$};\n\\end{tikzpicture}\n\\end{minipage}\n\\caption{An illustration of the policy gradient procedure. To encourage cooperation, we introduce a gradient that propagates the losses of the worker policies through the manager.\n}\n\\label{fig:hrl-connected}\n\\end{wrapfigure}\n\nTo compute the gradient of the additional weighted expected return through the parameters of the manager policy, we take inspiration from similar studies in differentiable communication in MARL. The main insight that enables the derivation of such a gradient is the notion that goal states $g_t$ from the perspective of the worker policy are structurally similar to communication signals in multi-agent systems. As a result, the gradients of the expected intrinsic returns can be computed by replacing the goal term within the reward function of the worker with the direct output from the manager's policy. This is depicted in Figure~\\ref{fig:hrl-connected}. The updated gradient is defined in Theorem~\\ref{theorem} below.\n\n\\begin{theorem} \\label{theorem}\nDefine the goal $g_t$ provided to the input of the worker policy $\\pi_w(s_t,g_t)$ as the direct output from the manager policy $g_t$ whose transition function is:\n\\begin{equation}\n    g_t(\\theta_m) = \n    \\begin{cases}\n        \\pi_m(s_t) & \\text{if } t \\text{ mod } k = 0 \\\\\n        h(s_{t-1}, g_{t-1}(\\theta_m), s_t) & \\text{otherwise}\n    \\end{cases}\n\\end{equation}\nwhere $h(\\cdot)$ is a fixed goal transition function between meta-periods (see Appendix~\\ref{sec:intrinsic-reward}). Under this assumption, the solution to the deterministic policy gradient~\\citep{silver2014deterministic} of Eq.~\\eqref{eq:connected-return} with respect to the manager's parameters $\\theta_m$ is:\n\\begin{equation} \\label{eq:connected-gradient}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        \\nabla_{\\theta_m} J_m' &= \\mathbb{E}_{s\\sim p_\\pi} \\big[ \\nabla_a Q_m (s,a)|_{a=\\pi_m(s)} \\nabla_{\\theta_m} \\pi_m(s)\\big] \\\\\n        %\n        &\\quad + \\lambda \\mathbb{E}_{s\\sim p_\\pi} \\bigg[ \\nabla_{\\theta_m} g_t \\nabla_g \\big(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\big) \\bigg\\rvert_{g=g_t} \\bigg]\n    \\end{aligned}\n    $}\n\\end{equation}\nwhere $Q_m(s,a)$ and $Q_w(s,g,a)$ are approximations for the expected environmental and intrinsic returns, respectively.\n\\end{theorem}\n\n\\vspace{-6pt}\n\n\\textit{Proof.} See Appendix~\\ref{sec:derivation}.\n\nThis new gradient consists of three terms. The first and third term computes the gradient of the critic policies $Q_m$ and $Q_w$ for the parameters $\\theta_m$, respectively. The second term computes the gradient of the worker-specific reward for the parameters $\\theta_m$. This reward is a design feature within the goal-conditioned RL formulation, however, any reward function for which the gradient can be explicitly computed can be used. We describe a practical algorithm for training a cooperative two-level hierarchy implementing this loss function in Algorithm~\\ref{alg:training}.\n\n\\subsection{Cooperative HRL as constrained optimization}\n\\label{sec:method-constrained-hrl}\n\nIn the previous sections, we introduced a framework for inducing and studying the effects of cooperation between internal agents within a hierarchy. The degree of cooperation is defined through a hyperparameter ($\\lambda$), and if properly defined can greatly improve training performance in certain environments. The choice of $\\lambda$, however, can be difficult to specify without a priori knowledge of the task it is assigned to. We accordingly wish to ground the choice of $\\lambda$ to measurable terms that can be reasoned and adjusted for. To that end, we observe that the cooperative $\\lambda$ term acts equivalently as a Lagrangian term in constrained optimization~\\citep{bertsekas2014constrained} with the expected return for the lower-level policy serving as the constraint. \nOur formulation of the HRL problem can similarly be framed as a constrained optimization problem, denoted as:\n\\begin{equation}\n   \n    \\begin{aligned}\n        &\\max_{\\pi_m} \\left[ J_m + \\min_{\\lambda\\geq 0} \\left( \\lambda \\delta - \\lambda \\min_{\\pi_w} J_w \\right) \\right]\n    \\end{aligned}\n   \n\\end{equation}\nwhere $\\delta$ is the desired expected discounted \\emph{intrinsic} returns. The derivation of this equation and practical implementations are provided in Appendix~\\ref{sec:constrained-hrl}.\n\nIn practice, this updated form of the objective provides two meaningful benefits: 1) As discussed in Section~\\ref{sec:cg-weight}, it introduces bounds for appropriate values of $\\delta$ that can then be explored and tuned, and 2) for the more complex and previously unsolvable tasks, we find that this approach results in more stable learning and better performing policies.\n\n\n\n\n\\section{Related Work} \\label{sec:related-work}\n\nThe topic explored in this article takes inspiration in part from studies of communication in multiagent reinforcement learning (MARL)~\\citep{thomas2011conjugate, thomas2011policy, DBLP:journals\/corr\/SukhbaatarSF16, DBLP:journals\/corr\/FoersterAFW16a}. In MARL, communication channels are often shared among agents as a means of coordinating and influencing neighboring agents. Challenges emerge, however, as a result of the ambiguity of communication signals in the early staging of training, with agents forced to coordinate between sending and interpreting messages~\\citep{mordatch2018emergence, eccles2019biases}. Similar communication channels are present in the HRL domain, with higher-level policies communicating one-sided signals in the form of goals to lower-level policies. The difficulties associated with cooperation, accordingly, likely (and as we find here in fact do) persist under this setting. The work presented here serves to make connections between these two fields, and will hopefully motivate future work on unifying the challenges experienced in each.\n\nOur work is most motivated by the Differentiable Inter-Agent Learning (DIAL) method proposed by~\\citet{DBLP:journals\/corr\/FoersterAFW16a}. Our work does not aim to learn an explicit communication channel; however, it is motivated by a similar principle - that letting gradients flow across agents results in richer feedback. Furthermore, we differ in the fact that we structure the problem as a hierarchical reinforcement learning problem in which agents are designed to solve dissimilar tasks. This disparity forces a more constrained and directed objective in which varying degrees of cooperation can be defined. This insight, we find, is important for ensuring that shared gradients can provide meaningful benefits to the hierarchical paradigm.\n\nAnother prominent challenge in MARL is the notion of \\emph{non-stationarity}~\\citep{busoniu2006multi, weinberg2004best, foerster2017stabilising}, whereby the continually changing nature of decision-making policies serve to destabilize training. This has been identified in previous studies on HRL, with techniques such as off-policy sample relabeling~\\citep{nachum2018data} and hindsight~\\citep{levy2017hierarchical} providing considerable improvements to training performance. Similarly, the method presented in this article focuses on non-stationary in HRL, with the manager constraining its search within the region of achievable goals by the worker. Unlike these methods, however, our approach additionally accounts for ambiguities in the credit assignment problem from the perspective of the manager. As we demonstrate in Section~\\ref{sec:experiments}, this cooperation improves learning across a collection of complex and partially observable tasks with a high degree of stochasticity. These results suggest that multifaceted approaches to hierarchical learning, like the one proposed in this article, that account for features such as information-sharing and cooperation in addition to non-stationarity are necessary for stable and generalizable HRL algorithms.\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{figures\/highway-single-env.png}\n    \\caption{\\footnotesize \\highwaysingle}\n    \\label{fig:highwaysingle-env}\n\\end{subfigure} \\\\ \\vspace{0.4cm}\n\\begin{subfigure}[b]{0.28\\textwidth}\n    \\centering\n    \\includegraphics[height=2.7cm]{figures\/ring-env.png}\n    \\caption{\\label{fig:ring-env} \\footnotesize \\ringroad}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.2\\textwidth}\n    \\centering\n    \\includegraphics[height=2.7cm]{figures\/AntGather-env.png}\n    \\caption{\\label{fig:antgather-env} \\footnotesize \\antgather}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.2\\textwidth}\n    \\centering\n    \\includegraphics[height=2.7cm]{figures\/AntFourRooms-env.png}\n    \\caption{\\label{fig:antfourrooms-env} \\footnotesize \\antfourrooms}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.2\\textwidth}\n    \\centering\n    \\includegraphics[height=2.7cm]{figures\/AntMaze-env.png}\n    \\caption{\\label{fig:antmaze-sim} \\footnotesize \\antmaze}\n\\end{subfigure}\n\\caption{\nTraining environments explored within this article. We compare the performance of various HRL algorithms on two mixed-autonomy traffic control task (a,b) and three ant navigation tasks (c,d,e). A description of each of these environments is provided in Section~\\ref{sec:environments}.\n}\n\\label{fig:envs}\n\\end{figure*}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\nIn this section, we detail the experimental setup and training procedure and present the performance of our method over various continuous control tasks. These experiments aim to analyze three aspects of HRL training: \n    (1) How does cooperation in HRL improve the development of goal-assignment strategies and the learning performance?\n    (2) What impact does automatically varying the degree of cooperation between learned higher and lower level behaviors have?\n    (3) Does the use of communication results in a more structured goal condition lower-level policy that transfers better to other tasks?\\\\[-20pt]\n\n\\subsection{Environments}\n\nWe explore the use of hierarchical reinforcement learning on a variety of difficult long time horizon tasks, see Figure~\\ref{fig:envs}. These environments vary from agent navigation tasks (Figures~\\ref{fig:antgather-env}),\nin which a robotic agent tries to achieve certain long-term goals, to two mixed-autonomy traffic control tasks (Figures~\\ref{fig:ring-env}~to~\\ref{fig:highwaysingle-env}), in which a subset of vehicles (seen in red) are treated as automated vehicles and attempt to reduce oscillations in vehicle speeds known as stop-and-go traffic. Further details are available in Appendix~\\ref{sec:environments}.\n\nThe environments presented here pose a difficulty to standard RL methods for a variety of reasons. We broadly group the most significant of these challenges into two categories: temporal reasoning and delayed feedback.\n\n\\noindent \\textbf{Temporal reasoning.} \\ \\\nIn the agent navigation tasks, the agent must learn to perform multiple tasks at varying levels of granularity. At the level of individual timesteps, the agent must learn to navigate its surroundings, moving up, down, left, or right without falling or dying prematurely. More macroscopically, however, the agent must exploit these action primitives to achieve high-level goals that may be sparsely defined or require exploration across multiple timesteps. For even state-of-the-art RL algorithms, the absence of hierarchies in these settings result in poor performing policies in which the agent stands still or follows sub-optimal greedy behaviors.\n\n\\noindent \\textbf{Delayed feedback.} \\ \\\nIn the mixed autonomy traffic tasks, meaningful events occur over large periods of time, as oscillations in vehicle speeds propagate slowly through a network. As a result, actions often have a delayed effect on metrics of improvement or success, making reasoning on whether a certain action improved the state of traffic a particularly difficult task. The delayed nature of this feedback prevents standard RL techniques from generating meaningful policies without relying on reward shaping techniques which produce undesirable behaviors such as creating large gaps between vehicles~\\citep{wu2017flow}.\n\n\\begin{figure*}\n\n\n\n\n\n\n\n\\begin{subfigure}[b]{0.22\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/ring-v0-rewards.pdf}\n    \\caption{\\footnotesize \\ringroad}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.22\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/highway-v2-rewards.pdf}\n    \\caption{\\footnotesize \\highwaysingle}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.205\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather-rewards.pdf}\n    \\caption{\\footnotesize \\antgather}\n    \\label{fig:antgather-rewards}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.22\\textwidth}\n    \\centering\n    \\begin{tikzpicture}\n        \\newcommand \\boxwidth {1.0}\n        \\newcommand \\boxheight {\\textwidth}\n        \\newcommand \\boxoffset {0}\n    \n       \n        \\draw (0.25, -2.0) node {};\n    \n        \\draw [color=color1]\n        (0.25, \\boxwidth + \\boxoffset - 0.5) -- \n        (1.00, \\boxwidth + \\boxoffset - 0.5); \n        \\draw [anchor=west] \n        (1.05, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize TD3};\n    \n        \\draw [color=color2]\n        (0.25, \\boxwidth + \\boxoffset - 0.9) -- \n        (1.00, \\boxwidth + \\boxoffset - 0.9); \n        \\draw [anchor=west] \n        (1.05, \\boxwidth + \\boxoffset - 0.9) node {\\scriptsize HRL};\n    \n        \\draw [color=color3] \n        (0.25, \\boxwidth + \\boxoffset - 1.3) -- \n        (1.00, \\boxwidth + \\boxoffset - 1.3); \n        \\draw [anchor=west] \n        (1.05, \\boxwidth + \\boxoffset - 1.3) node {\\scriptsize HIRO};\n    \n        \\draw [color=color4] \n        (0.25, \\boxwidth + \\boxoffset - 1.7) -- \n        (1.00, \\boxwidth + \\boxoffset - 1.7); \n        \\draw [anchor=west] \n        (1.05, \\boxwidth + \\boxoffset - 1.7) node {\\scriptsize HAC};\n    \n        \\draw [color=color5] \n        (0.25, \\boxwidth + \\boxoffset - 2.1) -- \n        (1.00, \\boxwidth + \\boxoffset - 2.1); \n        \\draw [anchor=west] \n        (1.05, \\boxwidth + \\boxoffset - 2.1) node {\\scriptsize \\methodName (fixed)};\n    \n        \\draw [color=color6] \n        (0.25, \\boxwidth + \\boxoffset - 2.5) -- \n        (1.00, \\boxwidth + \\boxoffset - 2.5); \n        \\draw [anchor=west] \n        (1.05, \\boxwidth + \\boxoffset - 2.5) node {\\scriptsize \\methodName (dynamic)};\n    \\end{tikzpicture}\n   \n   \n   \n\\end{subfigure} \\\\ \\vspace{0.2cm}\n\\begin{subfigure}[b]{0.22\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntFourRooms-rewards.pdf}\n    \\caption{\\footnotesize \\antfourrooms}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.22\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntMaze-0-rewards.pdf}\n    \\caption{\\footnotesize \\antmaze [16,0]}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.22\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntMaze-1-rewards.pdf}\n    \\caption{\\footnotesize \\antmaze [16,16]}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.22\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntMaze-2-rewards.pdf}\n    \\caption{\\footnotesize \\antmaze [0,16]}\n\\end{subfigure}\n\\caption{Training performance of the different algorithms on various environments. All results are reported over $10$ random seeds. Details on the choice of hyperparameters are provided in Appendix~\\ref{sec:hyperparams}.\n}\n\\label{fig:learning-curves}\n\\end{figure*}\n\n\\subsection{Baseline algorithms}\n\nWe evaluate our proposed algorithm against the following baseline methods:\\\\[-15pt]\n\\begin{itemize}[leftmargin=*,noitemsep]\n    \\item \\textit{TD3}: To validate the need for goal-conditioned hierarchies to solve our tasks, we compare all results against a fully connected network with otherwise similar network and training configurations.\n    \\item \\textit{HRL}: This baseline consists of the naive formulation of the hierarchical reinforcement learning optimization scheme (see Section~\\ref{sec:no-cooperation}). This algorithm is analogous to the \\methodName algorithm with $\\lambda$ set to $0$.\n    \\item \\textit{HIRO}: Presented by~\\citet{nachum2018data}, this method addresses the non-stationarity effects between the manager and worker policies by relabeling the manager's actions (or goals) to render the observed action sequence more likely to have occurred by the current instantiation of the worker policy. The details of this method are discussed in Appendix~\\ref{sec:hiro-reproducibility}.\n    \\item \\textit{HAC}: The HAC algorithm~\\citep{levy2017hierarchical} attempts to address non-stationarity in off-policy learning by relabeling sampled data via hindsight action and goal transitions as well as subgoal testing transitions to prevent learning from a restricted set of subgoal states. Implementation details are provided in Appendix~\\ref{sec:hac-egocentric}.\\\\[-15pt]\n\\end{itemize}\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {1.0}\n    \\newcommand \\boxheight {\\textwidth}\n    \\newcommand \\boxoffset {0}\n\n   \n    \\draw (0, -0.15) node {};\n\n    \\filldraw [blue] (0.615 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.63 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize High-level goal};\n\n    \\draw [dashed] \n    (0.790 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n    (0.825 * \\textwidth, \\boxwidth + \\boxoffset - 0.5); \n    \\draw [anchor=west]\n    (0.83 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Agent trajectory};\n\\end{tikzpicture}\n\\end{subfigure} \\\\ \\vspace{-0.4cm}\n\\begin{subfigure}[b]{0.08\\textwidth}\n    \\small{Standard\\\\ HRL} \\\\[10pt]\n    \n    \\textcolor{white}{.}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.42\\textwidth}\n    \\centering\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_hrl.png}\n    \\hfill\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_hrl.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antgather_hrl.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\includegraphics[width=\\linewidth]{figures\/antfourrooms_hrl.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antmaze_hrl.pdf}\n\\end{subfigure}\n\\\\[5pt]\n\\begin{subfigure}[b]{0.08\\textwidth}\n    \\small{\\methodName\\\\ (ours)} \\\\[25pt]\n    \n    \\textcolor{white}{.}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.42\\textwidth}\n    \\centering\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_cher.png}\n    \\hfill\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_cher.pdf}\n    \\caption{\\ringroad}\n    \\label{fig:ring-traj}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antgather_cher.pdf}\n    \\caption{Gather}\n    \\label{fig:antgather-traj}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\includegraphics[width=\\linewidth]{figures\/antfourrooms_cher.pdf}\n    \\caption{Four Rooms}\n    \\label{fig:antfourrooms-traj}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antmaze_cher.pdf}\n    \\caption{\\footnotesize Maze}\n    \\label{fig:antmaze-traj}\n\\end{subfigure}\n\\caption{Illustration of the agent and goal trajectories for some of the environments studied here. The high-level goals learned via \\methodName more closely match the agents' trajectories as they traverse the various environments. This results in improved dynamical behaviors by the agent in a majority of the tasks.}\n\\label{fig:goal-trajectories}\n\\end{figure*}\n\\subsection{Comparative analysis}\n\nFigure~\\ref{fig:learning-curves} depicts the training performance of each of the above algorithms and \\methodName on the studied tasks. We find that \\methodName performs comparably or outperforms all other studied algorithms for the provided tasks. Interestingly, \\methodName particularly outperforms other algorithms in highly stochastic and partially observable tasks. For the \\antgather and traffic control tasks, in particular, the objective (be it the positions of apples and bombs or the density and driving behaviors within a traffic network) varies significantly at the start of a new rollout and is not fully observable to the agent from its local observation. As such, the improved performance by \\methodName within these settings suggests that it is more robust than previous methods to learning policies in noisy and unstable environments.\n\nThe improvements presented above emerge in part from more informative goal-assignment behaviors by \\methodName. Figure~\\ref{fig:goal-trajectories} depicts these behaviors for a large number of tasks. We describe some of these behaviors and the performance of the policy below.\n\nFor the agent navigation tasks, the \\methodName algorithm produces goals that more closely match the agent's trajectory, providing the agent with a more defined path to follow to achieve certain goals. This results in faster and more efficient learning for settings such as \\antfourrooms, and in stronger overall policies for tasks such as \\antgather. An interesting corollary that appears to emerge as a result of this cooperative approach is more stable movements and actuation commands by the worker policy. For settings in which the agent can fall prematurely, this appears in the form of fewer early terminations as a result of agents attempting to achieve difficult or highly random goals. The absence of frequent early terminations allows the policy to explore further into its environment during training; this is a benefit in settings where long-term reasoning is crucial.\n\nIn the \\ringroad environment, we find standard HRL techniques fail to learn goal-assignment strategies that yield meaningful performative benefits. Instead, as seen in Figure~\\ref{fig:ring-traj} (top), the manager overestimates its worker's ability to perform certain tasks and assign maximum desired speeds. This strategy prevents the policy from dissipating the continued propagation of vehicle oscillations in the traffic, as the worker policy is forced to assign large acceleration to match the desired speeds thereby contributing to the formation of stop-and-go traffic, seen as the red diagonal streaks in Figure \\ref{fig:ring-traj}, top-left. \nFor these tasks, we find that inducing inter-level cooperation serves to alleviate the challenge of overestimating certain goals. In the \\ringroad environment, for instance, \\methodName succeeds in assigning meaningful desired speeds that encourage the worker policy to reduce the magnitude of accelerations assigned while near the free-flow, or optimal, speed of the network. This serves to eliminate the formation of stop-and-go traffic both from the perspective of the automated and human-driven vehicles, as seen in Figure \\ref{fig:ring-traj}, bottom. We also note that when compared to previous studies of a similar task~\\citep{wu2017flow}, our approach succeeds in finding a solution that does not rely on the generation of large or undesirable gaps between the AV and its leader. Similar results for the \\highwaysingle environment are provided in Appendix~\\ref{appendix:highway-results}.\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{0.25\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather-effects-worker-qvals.pdf}\n    \\caption{Worker expected returns}\n    \\label{fig:cg-delta-worker-qvals}\n\\end{subfigure}\n\\qquad\n\\begin{subfigure}[b]{0.25\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather-effects-cg-delta.pdf}\n    \\caption{Evolution of dynamic $\\lambda$}\n    \\label{fig:cg-delta-lambda}\n\\end{subfigure}\n\\qquad\n\\begin{subfigure}[b]{0.25\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather-cg-delta.pdf}\n    \\caption{Effects of cooperation}\n    \\label{fig:cg-delta-rewards}\n\\end{subfigure}\\\\ \\vspace{0.5cm}\n\\begin{subfigure}[b]{\\textwidth}\n\\centering\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {1.0}\n    \\newcommand \\boxheight {\\textwidth}\n    \\newcommand \\boxoffset {0}\n\n   \n    \\draw (0, -0.15) node {};\n\n    \\filldraw [limegreen] (0.305 * \\textwidth,\\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.32 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Apple ($+1$)};\n\n    \\filldraw [red] (0.46 * \\textwidth,\\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.475 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Bomb ($-1$)};\n\n    \\filldraw [blue] (0.615 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.63 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize High-level goal};\n\n    \\draw [dashed] \n    (0.790 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n    (0.825 * \\textwidth, \\boxwidth + \\boxoffset - 0.5); \n    \\draw [anchor=west]\n    (0.83 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Agent trajectory};\n\\end{tikzpicture}\n\\end{subfigure} \\\\ \\vspace{-0.4cm}\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/0.pdf}\n    \\caption{0\\%} \\label{fig:antgather-0p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-300\/0.pdf}\n    \\caption{50\\%} \\label{fig:antgather-50p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-200\/0.pdf}\n    \\caption{66.7\\%} \\label{fig:antgather-66.7p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-150\/0.pdf}\n    \\caption{75\\%} \\label{fig:antgather-75p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-100\/0.pdf}\n    \\caption{83.3\\%} \\label{fig:antgather-83.3p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-50\/0.pdf}\n    \\caption{91.7\\%} \\label{fig:antgather-91.7p}\n\\end{subfigure}\n\\caption{Effect of varying degrees of cooperation for the \\antgather environment. We plot the performance of the optimal policy and sample trajectories for various cooperation ratios as defined in Section~\\ref{sec:cg-weight} (see the subcaptions for d-i). Increasing degrees of cooperation improve the agent's ability to assign desired goals that match the agent's trajectory, which subsequently improves the performance of the policy. Large degrees of cooperation, however, begin to disincentivize the agent from moving.}\n\\label{fig:effect-of-cg-weights}\n\\end{figure*}\n\n\\subsection{Cooperation tradeoff} \n\\label{sec:cg-weight}\nIn this section, we explore the effects of varying degrees of cooperation on the performance of the resulting policy. Figure~\\ref{fig:effect-of-cg-weights} depicts the effect of increasing cooperative penalties on the resulting policy in the \\antgather environment. \nIn \nthis \ntask, we notice that expected intrinsic returns for the worker in the standard HRL approach converge to a value of about $-600$ (see Figure~\\ref{fig:cg-delta-worker-qvals}). As a result, we promote cooperation by assigning values of $\\delta$\nthat are progressively larger than $-600$ to determine what level of cooperation leads to the best performance.\n\nFigures~\\ref{fig:cg-delta-rewards}~to~\\ref{fig:antgather-91.7p} depict the effect of varying levels of cooperation on the performance of the policy\\footnote{We define the cooperative ratio in these figures as the ratio of the assigned $\\delta$ constraint between the standard hierarchical approach and the maximum expected return. The intrinsic rewards used here are non-positive meaning that the largest expected return is $0$. For example, a $\\delta$ value of $-450$ is equated to a cooperation ratio of $(-600 + 450) \/ (-600 - 0) = 0.25$, or $25\\%$.}.\nAs expected, we see that as the level of cooperation increases, the goal-assignment behaviors increasing consolidate near the path of the agent. This produces optimal behaviors in this setting for cooperation levels in the vicinity of 75\\% (see Figures~\\ref{fig:cg-delta-rewards}~and~\\ref{fig:antgather-75p}). For levels of cooperation nearing 100\\%, however, this consolidation begins to disincentivize forward movement, and subsequently exploration, by assignment goals that align with the current position of the agent (Figure~\\ref{fig:antgather-91.7p}), thereby deteriorating the overall performance of the policy.\n\nFigures~\\ref{fig:cg-delta-worker-qvals}~and~\\ref{fig:cg-delta-lambda} depict the agent's ability to achieve the assigned $\\delta$ constraint and the dynamic $\\lambda$ terms assigned to achieve these constraints, respectively. For most choices of $\\delta$, \\methodName succeeds in defining dynamic $\\lambda$ values that match the constraint, demonstrating the efficacy of the designed optimization procedure. For very large constraints, in this case for $\\delta=-50$, no choice of $\\lambda$ can be assigned to match the desired constraint, thereby causing the value of $\\lambda$ to grow exponentially. This explosion in the relevance of the constraint term likely obfuscates the relevance of the environment expected return to the manager gradients, resulting in the aforementioned disincentive for forward movement.\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{\\textwidth}\n    \\begin{tikzpicture}\n        \\newcommand \\boxwidth {1.0}\n        \\newcommand \\boxheight {\\textwidth}\n        \\newcommand \\boxoffset {0}\n    \n       \n        \\draw (0, -0.15) node {};\n    \n        \\draw [color2] \n        (0.33 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n        (0.39 * \\textwidth, \\boxwidth + \\boxoffset - 0.5);\n        \\draw [anchor=west]\n        (0.40 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize HRL worker};\n\n        \\draw [color5] \n        (0.56 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n        (0.62 * \\textwidth, \\boxwidth + \\boxoffset - 0.5);\n        \\draw [anchor=west]\n        (0.63 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize \\methodName worker};\n    \n        \\draw [color3] \n        (0.79 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n        (0.85 * \\textwidth, \\boxwidth + \\boxoffset - 0.5); \n        \\draw [anchor=west]\n        (0.86 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize HIRO worker};\n    \\end{tikzpicture}\n    \\end{subfigure} \\\\ \\vspace{-0.5cm}    \\begin{subfigure}[c]{0.36\\textwidth}\n        \\begin{subfigure}[c]{0.31\\textwidth}\n            \\includegraphics[height=1.9cm]{figures\/AntGather-env.png}\n        \\end{subfigure}\n        \\begin{subfigure}[c]{0.31\\textwidth}\n            \\begin{tikzpicture}\n                \\draw (1, 0.15) node {\\scriptsize Transfer $\\pi_w$};\n                \\draw (1, -0.15) node {\\scriptsize Retrain $\\pi_m$};\n                \\draw [->] (0.4, -0.5) -- (1.75, -1.1);\n                \\draw [->] (0.4, 0.5) -- (1.75, 1.1);\n            \\end{tikzpicture}\n        \\end{subfigure}\n        \\begin{subfigure}[c]{0.32\\textwidth}\n            \\includegraphics[height=1.9cm]{figures\/AntMaze-env.png} \\\\[10pt]\n            \\includegraphics[height=1.9cm]{figures\/AntFourRooms-env.png}\n        \\end{subfigure}\n    \\end{subfigure}\n    \\ \\\n    \\begin{subfigure}[c]{0.17\\textwidth}\n        \\caption*{\\ \\ \\ \\ \\ \\ \\ [16,0]}\n        \\includegraphics[height=2cm]{figures\/AntMaze_transfer_0.pdf}\n        \\caption*{\\ \\ \\ \\ \\ \\ \\ [20,0]}\n        \\includegraphics[height=2cm]{figures\/AntFourRooms_transfer_0.pdf}\n    \\end{subfigure}\n    \\quad\n    \\begin{subfigure}[c]{0.17\\textwidth}\n        \\caption*{\\ \\ \\ \\ \\ \\ \\ [16,16]}\n        \\includegraphics[height=2cm]{figures\/AntMaze_transfer_1.pdf}\n        \\caption*{\\ \\ \\ \\ \\ \\ \\ [0,20]}\n        \\includegraphics[height=2cm]{figures\/AntFourRooms_transfer_1.pdf}\n    \\end{subfigure}\n    \\quad\n    \\begin{subfigure}[c]{0.17\\textwidth}\n        \\caption*{\\ \\ \\ \\ \\ \\ \\ [0,16]}\n        \\includegraphics[height=2cm]{figures\/AntMaze_transfer_2.pdf}\n        \\caption*{\\ \\ \\ \\ \\ \\ \\ [20,20]}\n        \\includegraphics[height=2cm]{figures\/AntFourRooms_transfer_2.pdf}\n    \\end{subfigure}\n    \\caption{An illustration of the transferability of policies learned via \\methodName. A policy is trained in the \\antgather environment and the worker policy is frozen and transferred to the \\antmaze and \\antfourrooms environments. The policies learned when utilizing the \\methodName worker policy significantly outperform the HIRO policy for a wide variety of evaluation points, highlighting the benefit of \\methodName in learning a more informative and generalizable policy representations.}\n    \\label{fig:transfer}\n\\end{figure}\n\n\\subsection{Transferability of policies between tasks}\n\nFinally, we explore the effects of promoting inter-level cooperation on the transferability of learned policies to different tasks. To study this, we look to the Ant environments in Figure~\\ref{fig:envs} and choose to learn a policy in one environment (\\antgather) and transfer the learned worker policy to two separate environments (\\antmaze and \\antfourrooms). The initial policy within the \\antgather environment is trained for 1 million samples as in Figure~\\ref{fig:antgather-rewards} utilizing either the HRL, HIRO, or \\methodName algorithms. To highlight the transferability of the learned policy, we fix the weights of the worker policy, and instead attempt to learn a new manager policy for the given task; this allows us to identify whether the original policy generalizes better in the zero-shot setting. We still, however, must learn a new higher-level policy as the observations and objectives for the manager differ between problems.\n\nFigure~\\ref{fig:transfer} depicts the transfer setup and performance of the policy for a set number of evaluation points. We find that worker policies learned via \\methodName significantly improve the efficacy of the overall agent when exposed to new tasks. This suggests that the policies learned via more structured and informative goal-assignment procedures result in policies that are more robust to varying goal-assignment strategies, and as such allow the learned behaviors to be more task-agnostic.\n\n\n\n\\section{Conclusions and future work} \\label{sec:conclusions}\n\nIn this work, we propose connections between multi-agent and hierarchical reinforcement learning that motivates our novel method of inducing cooperation in hierarchies. We provide a derivation of the gradient of a manager policy with respect to its workers for an actor-critic formulation as well as introducing a $\\lambda$ weighting term for this gradient which controls the level of cooperation.\nWe find that using \\methodName results in consistently better-performing policies, that have lower empirical non-stationarity than prior work, particularly for more difficult tasks.\n\nNext, we find that policies learned with a fixed $\\lambda$ term are at times highly sensitive to the choice of value, and accordingly derive a dynamic variant of the cooperative gradients that automatically updates the value of $\\lambda$, balancing goal exploration. We demonstrate that this dynamic variant further expands the scope of solvable tasks, in particular allowing us to generate highly effective mixed-autonomy driving behaviors in a very sample-efficient manner.\n\nFor future work, we would like to apply this method to discrete action environments and additional hierarchical models such as the options framework. Potential future work also includes extending the cooperative HRL formulation to multi-level hierarchies, in which the multi-agent nature of hierarchical training is likely to be increasingly detrimental to training stability.\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\nIn this section, we detail the experimental setup and training procedure and present the performance of our method results of various continuous control tasks. Through these experiments, we aim to determine two things: \\\\[-20pt]\n\\begin{enumerate}[leftmargin=*,noitemsep]\n    \\item How does cooperation in HRL improve the behavior of goal-assignment strategies and the performance of the resulting policy?\n    \\item How important is it to control the varying degree of cooperation between learned higher and lower level behaviors?\\\\[-20pt]\n\\end{enumerate}\n\n\n\n\nAll results within this section are reported over $10$ random seeds. Details on the choice of hyperparameters are provided in Appendix~\\ref{sec:hyperparams}.\n\n\\subsection{Environments}\n\nWe explore the use of hierarchical reinforcement learning on a variety of difficult long time horizon tasks, see Figure~\\ref{fig:envs}. These environments vary from agent navigation tasks (Figures~\\ref{fig:antgather-env}~to~\\ref{fig:biped-sim}), in which a robotic agent tries to achieve certain long-term goals, to two mixed-autonomy traffic control tasks (Figures~\\ref{fig:ring-env}~to~\\ref{fig:highwaysingle-env}), in which a subset of vehicles (seen in red) are treated as automated vehicles and attempt to reduce oscillations in vehicle speeds known as stop-and-go traffic. Further details are available in Appendix~\\ref{sec:environments}.\n\nThe environments presented here pose a difficulty to standard RL methods for a variety of reasons. We broadly group the most significant of these challenges into two categories: temporal reasoning and delayed feedback.\n\n\\textbf{Temporal reasoning} \\ \\\nIn the agent navigation tasks, the agent must learn to perform multiple tasks at varying levels of granularity. At the level of individual timesteps, the agent must learn to navigate its surroundings, moving up, down, left, or right without falling or dying prematurely. More macroscopically, however, the agent must exploit these action primitives to achieve high-level goals that may be sparsely defined or require exploration across multiple timesteps. For even state-of-the-art RL algorithms, the absence of hierarchies in these settings result in poor performing policies in which the agent stands still or follows sub-optimal greedy behaviors.\n\n\\textbf{Delayed feedback} \\ \\\nIn the mixed autonomy traffic tasks, meaningful events occur over large periods of time, as oscillations in vehicle speeds propagate slowly through a network. As a result, actions often have a delayed effect on metrics of improvement or success, making reasoning on whether a certain action improved the state of traffic a particularly difficult task. The delayed nature of this feedback prevents standard RL techniques from generating meaningful policies without relying on reward shaping techniques which produce undesirable behaviors such as creating large gaps between vehicles~\\cite{wu2017flow}.\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {1.0}\n    \\newcommand \\boxheight {\\textwidth}\n    \\newcommand \\boxoffset {0}\n\n   \n    \\draw (0, -0.15) node {};\n\n    \\filldraw [blue] (0.705 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.72 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize High-level goal};\n\n    \\draw [dashed] \n    (0.840 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n    (0.875 * \\textwidth, \\boxwidth + \\boxoffset - 0.5); \n    \\draw [anchor=west]\n    (0.88 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Agent trajectory};\n\\end{tikzpicture}\n\\end{subfigure} \\\\ \\vspace{-0.4cm}\n\\begin{subfigure}[b]{0.08\\textwidth}\n    \\small{Standard\\\\ HRL} \\\\[10pt]\n    \n    \\textcolor{white}{.}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.42\\textwidth}\n    \\centering\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_hrl.png}\n    \\hfill\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_hrl.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antgather_hrl.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\includegraphics[width=\\linewidth]{figures\/antfourrooms_hrl.pdf}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antmaze_hrl.pdf}\n\\end{subfigure}\n\\\\[5pt]\n\\begin{subfigure}[b]{0.08\\textwidth}\n    \\small{\\methodName\\\\ (ours)} \\\\[25pt]\n    \n    \\textcolor{white}{.}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.42\\textwidth}\n    \\centering\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_cher.png}\n    \\hfill\n    \\includegraphics[width=0.48\\linewidth]{figures\/ring_cher.pdf}\n    \\caption{\\ringroad}\n    \\label{fig:ring-traj}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antgather_cher.pdf}\n    \\caption{Gather}\n    \\label{fig:antgather-traj}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\includegraphics[width=\\linewidth]{figures\/antfourrooms_cher.pdf}\n    \\caption{Four Rooms}\n    \\label{fig:antfourrooms-traj}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/antmaze_cher.pdf}\n    \\caption{\\footnotesize Maze}\n    \\label{fig:antmaze-traj}\n\\end{subfigure}\\\\[-8pt]\n\\caption{Illustration of the agent and goal trajectories for some of the environments studied here. The high-level goals learned via \\methodName more closely match the agents' trajectories as they traverse the various environments. This results in improved dynamical behaviors by the agent in a majority of the tasks.}\n\\label{fig:goal-trajectories}\n\\end{figure*}\n\n\\subsection{Baseline algorithms}\nWe test our proposed algorithm against the following baseline methods:\\\\[-15pt]\n\\begin{itemize}[leftmargin=*,noitemsep]\n    \\item \\textit{TD3}: To validate the need for goal-conditioned hierarchies to solve our tasks, we compare all results against a fully connected network with otherwise similar network and training configurations.\n    \\item \\textit{HRL}: This baseline consists of the naive formulation of the hierarchical reinforcement learning optimization scheme (see Section~\\ref{sec:no-cooperation}). This algorithm is analogous to the \\methodName algorithm with $\\lambda$ set to $0$.\n    \\item \\textit{HIRO}: Presented by~\\citet{nachum2018data}, this method addresses the non-stationarity effects between the manager and worker policies by relabeling the manager's actions (or goals) to render the observed action sequence more likely to have occurred by the current instantiation of the worker policy. \n   \n   \n    The reproducibility of this algorithm is addressed in Appendix~\\ref{sec:hiro-reproducibility}.\n    \\item \\textit{HAC}: \n   \n    The HAC algorithm~\\citep{levy2017hierarchical} attempts to address non-stationarity in off-policy learning by relabeling sampled data via hindsight action and goal transitions\n   \n    as well as subgoal testing transitions to prevent learning from a restricted set of subgoal states. Implementation details \n   \n    are provided in Appendix~\\ref{sec:hac-egocentric}.\\\\[-15pt]\n\\end{itemize}\n\n\n\n\\subsection{Comparative analysis}\n\nFigure~\\ref{fig:learning-curves} depicts the training performance of each of the above algorithms on the studied tasks. We find that \\methodName performs comparably or outperforms all other studied algorithms for the provided tasks. Interestingly, \\methodName particularly outperforms other algorithms in highly stochastic and partially observable tasks. For the \\antgather and traffic control tasks, in particular, the objective (be it the positions of apples and bombs or the density and driving behaviors within a traffic network) varies significantly at the start of a new rollout and is not fully observable to the agent from its local observation. As such, the improved performance by \\methodName within these settings suggests that it is more robust than previous methods to learning policies that generalize to a wider set of problems as opposed to overfitting to a specific solution.\n\nThe improvements presented above emerge in part from more informative goal-assignment behaviors by the \\methodName algorithm. Figure~\\ref{fig:goal-trajectories} depicts these behaviors for a large number of tasks. We describe some of these behaviors and the subsequent performance of the policy below.\n\nFor the agent navigation tasks, the \\methodName algorithm produces goals that more closely match the agent's trajectory, providing the agent with a more defined path to follow to achieve certain goals. This results in faster and more efficient learning for settings such as \\antfourrooms, and in stronger overall policies for tasks such as \\antgather. An interesting corollary that appears to emerge as a result of this cooperative approach is more stable movements and actuation commands by the worker policy. For settings in which the agent can fall prematurely, this appears in the form of fewer early terminations as a result of agents attempting to achieve difficult or highly random goals. The absence of frequent early terminations allows the policy to explore further into its environment during training; this is undoubtedly a benefit in settings where long-term reasoning is crucial.\n\n\nIn the \\ringroad environment, we find standard HRL techniques fail to learn goal-assignment strategies that yield meaningful performative benefits. Instead, as seen in Figure~\\ref{fig:ring-traj} (top), the manager overestimates its worker's ability to perform certain tasks and assign maximum desired speeds. This strategy prevents the policy from dissipating the continued propagation of vehicle oscillations in the network, as the worker policy is forced to assign large acceleration to match the desired speeds thereby contributing to the formation of stop-and-go traffic, seen as the red diagonal streaks in Figure \\ref{fig:ring-traj}, top-left.\nFor these tasks, we find that inducing inter-level cooperation serves to alleviate the challenge of overestimating certain goals. In the \\ringroad environment, for instance, \\methodName succeeds in assigning meaningful desired speeds that encourage the worker policy to reduce the magnitude of accelerations assigned while near the free-flow, or optimal, speed of the network. This serves to eliminate the formation of stop-and-go traffic both from the perspective of the automated and human-driven vehicles, as seen in Figure \\ref{fig:ring-traj}, bottom. We also note that when compared to previous studies of a similar task~\\citep{wu2017flow}, our approach succeeds in finding a solution that does not rely on the generation of large or undesirable gaps between the AV and its leader. Similar results for the \\highwaysingle environment are provided in Appendix~\\ref{appendix:highway-results}.\n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{0.25\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather-effects-worker-qvals.pdf}\n    \\caption{Worker expected returns}\n    \\label{fig:cg-delta-worker-qvals}\n\\end{subfigure}\n\\qquad\n\\begin{subfigure}[b]{0.25\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather-effects-cg-delta.pdf}\n    \\caption{Evolution of dynamic $\\lambda$}\n    \\label{fig:cg-delta-lambda}\n\\end{subfigure}\n\\qquad\n\\begin{subfigure}[b]{0.25\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather-cg-delta.pdf}\n    \\caption{Effects of cooperation}\n    \\label{fig:cg-delta-rewards}\n\\end{subfigure}\\\\ \\vspace{0.5cm}\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/0.pdf}\n    \\caption{0\\%} \\label{fig:antgather-0p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-300\/0.pdf}\n    \\caption{50\\%} \\label{fig:antgather-50p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-200\/0.pdf}\n    \\caption{66.7\\%} \\label{fig:antgather-66.7p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-150\/0.pdf}\n    \\caption{75\\%} \\label{fig:antgather-75p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-100\/0.pdf}\n    \\caption{83.3\\%} \\label{fig:antgather-83.3p}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/AntGather\/delta=-50\/0.pdf}\n    \\caption{91.7\\%} \\label{fig:antgather-91.7p}\n\\end{subfigure}\\\\[-8pt]\n\\caption{Effect of varying degrees of cooperation for the \\antgather environment. We plot the performance of the optimal policy and sample trajectories for various cooperation ratios as defined in Section~\\ref{sec:cg-weight} (see the subcaptions for d-i). Increasing degrees of cooperation improve the agent's ability to assign desired goals that match the agent's trajectory, which subsequently improves the performance of the policy. Large degrees of cooperation, however, begin to disincentivize the agent from moving.}\n\\label{fig:effect-of-cg-weights}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\\subsection{Cooperation Tradeoff} \n\\label{sec:cg-weight}\nIn this section, we explore the effects of varying degrees of cooperation on the performance of the resulting policy.\n\nFig.~\\ref{fig:effect-of-cg-weights} depicts the effect of increasing cooperative penalties on the resulting policy in the \\antgather environment. We define the degree of cooperation in comparison to the expected worker (or intrinsic) returns experienced in the standard hierarchical approach. The level of cooperation $l_\\text{coop}$ is:\n\\begin{equation}\n    l_\\text{coop}(\\delta) = \\frac{Q_\\text{w,HRL} - \\delta}{Q_\\text{w,HRL} - Q_\\text{w,max}}\n\\end{equation}\nwhere $Q_\\text{w,HRL}$ is the (approximate) converged expected intrinsic returns by the standard HRL approach and $Q_\\text{w,max}$ is the largest possible expected intrinsic returns.\n\nIn the \\antgather task, we notice that expected intrinsic returns in the standard HRL approach converge to a value of about $-600$ (see Figure~\\ref{fig:cg-delta-worker-qvals}). As a result, we promote cooperation by assigning values of $\\delta$, or the desired expected intrinsic returns, that are progressively larger than $-600$ to determine what level of cooperation leads to the best performance. We define the degree of cooperation as the ratio of the $\\delta$ constraint between the standard hierarchical approach and the maximum expected return (the intrinsic rewards used here are non-positive meaning that the largest expected return is $0$). For example, a $\\delta$ value of $-450$ is equated to a level of cooperation $l_\\text{coop}(-450) = (-600 + 450) \/ (-600 - 0) = 0.25$, or $25\\%$.\n\n\nFigures~\\ref{fig:cg-delta-worker-qvals}~and~\\ref{fig:cg-delta-lambda} depict the agent's ability to achieve the assigned $\\delta$ constraint and the dynamic $\\lambda$ terms assigned to achieve these constraints, respectively. For most choices of $\\delta$, \\methodName succeeds in defining dynamic $\\lambda$ values that match the constraint. For very large constraints, in this case for $\\delta=-50$, no choice of $\\lambda$ can be assigned to match the desired constraint, thereby causing the value of $\\lambda$ to grow exponentially.\n\n\n\nFigures~\\ref{fig:cg-delta-rewards}~to~\\ref{fig:antgather-91.7p} depict the effect of varying levels of cooperation on the performance of the policy. As expected, we see that as the level of cooperation increases, the goal-assignment behaviors increasing consolidate near the path of the agent. This produces optimal behaviors in this setting for cooperation levels in the vicinity of 75\\% (see Figures~\\ref{fig:cg-delta-rewards}~and~\\ref{fig:antgather-75p}). For levels of cooperation nearing 100\\%, however, this consolidation begins to disincentivize forward movement, and subsequently exploration, by assignment goals that align with the current position of the agent (Figure~\\ref{fig:antgather-91.7p}), thereby deteriorating the overall performance of the policy.\n\n\n\n\n\n\n\\section{Introduction} \\label{sec:introduction}\n\nTo solve interesting problems in the real world agents must be adept at planning and reasoning over long time horizons. For instance, in robot navigation and interaction tasks (Figures~\\ref{fig:antgather-env}~to~\\ref{fig:biped-sim}), agents must learn to compose lengthy sequences of actions to achieve long-term goals. In other environments, such as mixed-autonomy traffic control settings~\\citep{wu2017emergent, vinitsky2018benchmarks} (Figures~\\ref{fig:ring-env},~\\ref{fig:highwaysingle-env}), exploration is delicate, as individual actions may not influence the flow of traffic until multiple timesteps in the future. RL has had limited success in solving long-horizon planning problems such as these without relying on task-specific reward shaping strategies that limit the performance of resulting policies~\\citep{wu2017flow} or additional task decomposition techniques that are not transferable to different tasks~\\citep{sutton1999between, kulkarni2016hierarchical, 2017-TOG-deepLoco, florensa2017stochastic}.\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{\\textwidth}\n\\centering\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {1.0}\n    \\newcommand \\boxheight {\\textwidth}\n    \\newcommand \\boxoffset {0}\n\n   \n    \\draw (0, -0.15) node {};\n\n   \n   \n   \n   \n   \n   \n\n    \\filldraw [limegreen] (0.455 * \\textwidth,\\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.47 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Apple ($+1$)};\n\n    \\filldraw [red] (0.58 * \\textwidth,\\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.595 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Bomb ($-1$)};\n\n    \\filldraw [blue] (0.705 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) circle (0.125);\n    \\draw [anchor=west]\n    (0.72 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize High-level goal};\n\n    \\draw [dashed] \n    (0.840 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) -- \n    (0.875 * \\textwidth, \\boxwidth + \\boxoffset - 0.5); \n    \\draw [anchor=west]\n    (0.88 * \\textwidth, \\boxwidth + \\boxoffset - 0.5) node {\\scriptsize Agent trajectory};\n\\end{tikzpicture}\n\\end{subfigure} \\\\ \\vspace{-0.4cm}\n\\begin{tikzpicture}\n    \\draw[->] (0, 0) -- (0.06*\\textwidth, 0);\n    \\draw[anchor=west] (0.065*\\textwidth, 0) node {\\scriptsize training iteration};\n\n    \\draw[->] (0.52*\\textwidth, 0) -- (0.58*\\textwidth, 0);\n    \\draw[anchor=west] (0.585*\\textwidth, 0) node {\\scriptsize training iteration};\n    \\draw (\\textwidth, 0) {};\n\\end{tikzpicture}\\\\[-15pt]\n\\begin{subfigure}[b]{0.48\\textwidth}\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/100000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/300000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/500000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/700000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/HRL\/900000.pdf}\n    \\end{subfigure}\n    \\caption{Standard HRL}\n    \\label{fig:goaldists-standrd}\n\\end{subfigure}\n\\hfill\n\\begin{tikzpicture}\n    \\draw [dashed] (0,0) -- (0,2.5);\n\\end{tikzpicture}\n\\hfill\n\\begin{subfigure}[b]{0.48\\textwidth}\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/100000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/300000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/500000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/700000.pdf}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.18\\textwidth}\n        \\includegraphics[width=\\linewidth]{figures\/AntGather\/CHER\/900000.pdf}\n    \\end{subfigure}\n    \\caption{Cooperative HRL}\n    \\label{fig:goaldists-cooperative}\n\\end{subfigure}\\\\[-8pt]\n\\caption{Here the learned goal proposal distributions are shown for normal HRL and our method. Our agents develop more reasonable goal proposals that allow low-level policies to learn goal reaching skills quicker. The additional communication also allows the high level to better understand why proposed goals failed.}\n\\label{fig:goaldists}\n\\end{figure*}\n\n\nConcurrent learning methods in hierarchical RL are promising approaches for automating the process of task decomposition to previously unexplored problems~\\citep{levy2017hierarchical,nachum2018data,li2019sub}. \nThe process of concurrently learning diverse skills and exploiting these skills to achieve a high-level objective, however, is a difficult task. In particular, at the early stages of training lower-level policies are unable to reach most goals assigned to them by a higher-level policy, and instead must learn to do so as training progresses.\nThis exacerbates the credit assignment problem from the perspective of the higher-level policy, as it is unable to identify whether a specific goal under-performed as a result of the choice of goal or the lower-level policy's inability to achieve it. In practice, this results in the highly varying or random goal assignment strategies in~Figure~\\ref{fig:goaldists-standrd} that require a large number of samples~\\cite{li2019sub} and some degree of feature engineering~\\cite{nachum2018data} to optimize over.\n\n\n\n\nIn this paper, we pose this problem presented above as a result of insufficient cooperation between internal levels within a\nhierarchy\\footnote{The connection of this problem to cooperation in multiagent RL is discussed in Section~\\ref{sec:algorithm}.}, and subsequently introduce mechanisms that promote variable degrees of cooperation in HRL.\nOur approach improves cooperation by encouraging higher-level policies to specify goals that lower-level policies can succeed at,\nthereby disambiguating under-performing goals from goals that were unachievable by the lower-level policy.\nIn Figure~\\ref{fig:goaldists}(b) we show how our method changes the high-level goal distributions to be within the capabilities of the agent. This approach results in more informative communication between the policies. \nConceptually our method shares similarities to recent methods on automatic curriculum generation~\\cite{wang2019paired, dennis2020emergent, held2017automatic} but between the evolution of two agents. The distribution of goals or tasks the high-level will command of the lower level expands over time as the lower-level policy's capabilities increase.\n\nTo promote cooperation in HRL, we take inspiration from\ndifferentiable communication~\\citep{DBLP:journals\/corr\/SukhbaatarSF16,DBLP:journals\/corr\/FoersterAFW16a} and emergent cooperation phenomena~\\citep{claus1998dynamics,panait2005cooperative} in multiagent RL.\nOur approach attempts to encourage cooperation between various levels of a hierarchy by redefining the objective of higher-level policies to directly account for losses experienced by lower-level policies. The gradients of these additions to the loss are then propagated through the parameters of the higher-level policy by replacing the goals policy during training with direct connections to the lower-level policy.\n\n\n\n\nA key finding in this paper is that regulating the degree of cooperation between HRL layers can significantly impact the learned behavior by the policy. Too little cooperation may introduce no change to the goal-assignment behaviors and subsequent learning, while excessive cooperation may disincentivize an agent from making forward progress. We accordingly introduce a constrained optimization that serves to regulate the degree of cooperation between layers and ground the notion of cooperation in HRL to quantitative metrics within the lower-level policy. This approach results in a general and stable method for optimizing hierarchical policies together.\n\nWe demonstrate the performance of our method on a collection of standard HRL environments and two previously unexplored mixed autonomy traffic control tasks. For the former set of problems, we find that our method can achieve better performance compared to recent sample efficient off-policy and HRL algorithms.\nFor the mixed autonomy traffic tasks, the previous HRL methods struggle while our approach subverts overestimation biases that emerge in the early stages of training, thereby allowing the controlled (autonomous) vehicles to regulate their speeds around the optimal driving of the task as opposed to continuously attempting to drive as fast as possible.\n\n\n\n\n\\section{Cooperative hierarchical reinforcement learning} \\label{sec:method}\n\nIn this section, we introduce a technique for \\emph{Cooperative HiErarchical RL}, which we call \\methodName. \\methodName promotes cooperation by propagating losses that arise from random or unachievable goal-assignment strategies. As part of this algorithm, we also present a mechanism to optimize the level of cooperation and ground the notion of cooperation in HRL to measurable variables. \n\n\\subsection{Promoting cooperation via loss-sharing} \\label{sec:algorithm}\n\n\\citet{tampuu2017multiagent} explored the effects of reward (or loss) sharing between agents on the emergence of cooperative and competitive in multiagent two-player games. Their study highlights two potential benefits of loss-sharing in multiagent systems: 1) emerged cooperative behaviors are less aggressive and more likely to emphasize improved interactions with neighboring agents, 2) these interactions reduce overestimation bias of the Q-function from the perspective of each agent.\n\n\n\nIn this article, we aim to explore the benefits of similar loss-sharing paradigms in goal-conditioned hierarchies. In particular, we focus on the emergence of collaborative behaviors from the perspective of goal-assignment and goal-achieving policies In line with prior work, we promote the emergence of cooperative behaviors by incorporating a weighted form of the worker expected return to the original manager objective $J_m$. Our new expected return is:\n\\begin{equation} \\label{eq:connected-return}\n    J_m' = J_m + \\lambda J_w\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n\\end{equation}\nwhere $\\lambda$ is a weighting term that controls the level of cooperation the manager has with the worker.\n\nIn practice, we find that this addition to the objective serves to promote cooperation by aligning goal-assignment actions by the manager with achievable trajectories by the goal-achieving worker. This serves as a regularizer to the manager policy: when presented with goals that perform similarly, the manager tends towards goals that more closely match the worker states. The degree to which the policy tends toward achievable states is dictated by the $\\lambda$ term. The choice of this parameter accordingly can have a significant effect on learned goals. For large or infinite values of $\\lambda$, for instance, this cooperative term can eliminate exploration by assigning goals that match the worker's current state and prevent forward movement (this is highlighted in Section~\\ref{sec:cg-weight}). In Section~\\ref{sec:method-constrained-hrl}, we provide intuitions for understanding the notion of cooperation in HRL by connecting it to measurable terms within the worker policy.\n\n\n\\subsection{Cooperative gradients via differentiable communication}\n\nTo compute the gradient of the additional weighted expected return through the parameters of the manager policy, we take inspiration from similar studies in differentiable communication in MARL. The main insight that enables the derivation of such a gradient is the notion that goal states $g_t$ from the perspective of the worker policy are structurally similar to communication signals in multi-agent systems. As a result, the gradients of the expected intrinsic returns can be computed\nby replacing the goal term within the reward function of the worker with the direct output from the manager's policy. This is depicted in Figure~\\ref{fig:hrl-connected}. \n\n\\begin{figure}\n\\centering\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\centering\n\\begin{minipage}[b]{0.95\\textwidth}\n\\begin{tikzpicture}\n    \\newcommand \\boxwidth {0.65}\n    \\newcommand \\boxoffset {0.5}\n    \\newcommand \\envwidth {7.75}\n    \\newcommand \\envheight {0.65}\n    \\newcommand \\layerskip {1.4}\n\n   \n   \n   \n   \n\n   \n    \\fill [orange!30!white] (0,0) rectangle (\\envwidth, \\envheight);\n    \\draw [brown] (0,0) -- (\\envwidth,0);\n    \\draw [brown] (0, \\envheight) -- (\\envwidth, \\envheight);\n    \\draw (\\envwidth\/2, \\envheight\/2) node {\\small Environment};\n\n   \n    \\filldraw [black!40!white, draw=black!90!white] \n    (2*\\boxoffset, 3.5*\\envheight) rectangle \n    (2*\\boxoffset + \\boxwidth, 3.5*\\envheight + \\boxwidth);\n    \\draw \n    (2.0*\\boxoffset + 0.5*\\boxwidth, 3.5*\\envheight + 0.5*\\boxwidth) node {\\scriptsize $\\pi_m$};\n\n   \n    \\filldraw [black!5!white, draw=black!15!white] \n    (4.5*\\boxoffset, 2*\\envheight) rectangle \n    (4.5*\\boxoffset + \\boxwidth, 2*\\envheight + \\boxwidth);\n\n    \\filldraw [black!5!white, draw=black!15!white] \n    (6.0*\\boxoffset + 1.0*\\boxwidth, 2*\\envheight) rectangle \n    (6.0*\\boxoffset + 2.0*\\boxwidth, 2*\\envheight + \\boxwidth);\n\n    \\filldraw [black!20!white, draw=black!80!white] \n    (9.0*\\boxoffset + 2.0*\\boxwidth, 2*\\envheight) rectangle \n    (9.0*\\boxoffset + 3.0*\\boxwidth, 2*\\envheight + \\boxwidth);\n    \\draw\n    (9.0*\\boxoffset + 2.5*\\boxwidth, 2*\\envheight + 0.5*\\boxwidth) node {\\scriptsize $\\pi_w$};\n\n   \n\n    \\draw [red] [->] \n    (8.6*\\boxoffset + 3.0*\\boxwidth, 3.25*\\envheight) --\n    (8.6*\\boxoffset + 3.0*\\boxwidth, 4.00*\\envheight);\n    \\draw \n    (13*\\boxoffset + \\boxwidth, 3.5*\\envheight) node {\\small \\color{red}{$\\nabla \\theta_m J_w$}}; \n\n   \n\n    \\draw [->] \n    (1.25 * \\boxoffset, \\envheight) -- (1.25 * \\boxoffset, 4 * \\envheight) -- (2*\\boxoffset, 4 * \\envheight);\n    \\draw [black!15!white] [->] \n    (1.25 * \\boxoffset, 2.5*\\envheight) -- (4.5*\\boxoffset, 2.5*\\envheight);\n    \\draw \n    (1.25 * \\boxoffset + 0.25, \\envheight + 0.2) node {\\small $s_t$}; \n\n    \\draw\n    (2*\\boxoffset + \\boxwidth, 4*\\envheight) -- \n    (5.5*\\boxoffset + 2.5*\\boxwidth, 4*\\envheight); \n    \\draw [black!15!white]\n    (4.5*\\boxoffset + 0.5*\\boxwidth, 4*\\envheight) -- \n    (4.5*\\boxoffset + 0.5*\\boxwidth, 3*\\envheight);\n    \\draw [black!15!white]\n    (4.5*\\boxoffset + 2.5*\\boxwidth, 4*\\envheight) -- \n    (4.5*\\boxoffset + 2.5*\\boxwidth, 3*\\envheight);\n    \\draw \n    (7.5*\\boxoffset + 2.0*\\boxwidth, 4.0*\\envheight) node {$\\cdots$}; \n    \\draw [->]\n    (8.5*\\boxoffset + 2.0*\\boxwidth, 4*\\envheight) -- \n    (9.0*\\boxoffset + 2.5*\\boxwidth, 4*\\envheight) --\n    (9.0*\\boxoffset + 2.5*\\boxwidth, 3*\\envheight); \n    \\draw \n    (2*\\boxoffset + \\boxwidth + 0.3, 4 * \\envheight + 0.2) node {\\small $g_t$}; \n\n    \\draw [black!15!white] [->] \n    (4.5*\\boxoffset + \\boxwidth, 2.5*\\envheight) --\n    (5.0*\\boxoffset + \\boxwidth, 2.5*\\envheight) --\n    (5.0*\\boxoffset + \\boxwidth, 1.0*\\envheight);\n    \\draw\n    (5.0*\\boxoffset + \\boxwidth - 0.25, \\envheight + 0.2) node {\\small \\color{black!15!white}{$a_t$}}; \n\n    \\draw [black!15!white] [->] \n    (5.5*\\boxoffset + \\boxwidth, 1.0*\\envheight) --\n    (5.5*\\boxoffset + \\boxwidth, 2.5*\\envheight) --\n    (6.0*\\boxoffset + \\boxwidth, 2.5*\\envheight);\n    \\draw\n    (5.5*\\boxoffset + \\boxwidth + 0.4, \\envheight + 0.2) node {\\small \\color{black!15!white}{$s_{t+1}$}}; \n\n    \\draw [black!15!white] [->] \n    (6.0*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) --\n    (6.5*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) --\n    (6.5*\\boxoffset + 2.0*\\boxwidth, 1.0*\\envheight);\n    \\draw\n    (6.5*\\boxoffset + 2.0*\\boxwidth + 0.40, \\envheight + 0.2) node {\\small \\color{black!15!white}{$a_{t+1}$}}; \n\n    \\draw (7.5*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) node {\\color{black!15!white}{$\\cdots$}}; \n\n    \\draw [->] \n    (8.5*\\boxoffset + 2.0*\\boxwidth, 1.0*\\envheight) --\n    (8.5*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight) --\n    (9.0*\\boxoffset + 2.0*\\boxwidth, 2.5*\\envheight);\n    \\draw\n    (8.5*\\boxoffset + 2.0*\\boxwidth + 0.40, \\envheight + 0.2) node {\\small $s_{t+i}$}; \n\n    \\draw [->] \n    (9.0*\\boxoffset + 3.0*\\boxwidth, 2.5*\\envheight) --\n    (9.5*\\boxoffset + 3.0*\\boxwidth, 2.5*\\envheight) --\n    (9.5*\\boxoffset + 3.0*\\boxwidth, 1.0*\\envheight);\n    \\draw\n    (9.5*\\boxoffset + 3.0*\\boxwidth + 0.40, \\envheight + 0.2) node {\\small $a_{t+i}$};\n\\end{tikzpicture}\n\\end{minipage}\\\\[-8pt]\n\\caption{An illustration of the \npolicy gradient procedure.\nTo encourage cooperation, we introduce a \ngradient that propagates the rewards associated with the worker policies through the manager.\n}\n\\label{fig:hrl-connected}\n\\end{figure}\n\nThe updated gradient is defined in Theorem~\\ref{theorem} below.\n\n\\begin{theorem} \\label{theorem}\nDefine the goal $g_t$ provided to the input of the worker policy $\\pi_w(s_t,g_t)$ as the direct output from the manager policy $g_t$ whose transition function is:\n\\begin{equation}\n    g_t(\\theta_m) = \n    \\begin{cases}\n        \\pi_m(s_t) & \\text{if } t \\text{ mod } k = 0 \\\\\n        h(s_{t-1}, g_{t-1}(\\theta_m), s_t) & \\text{otherwise}\n    \\end{cases}\n\\end{equation}\n\nwhere $h(\\cdot)$ is a fixed goal transition function between meta-periods (see Appendix~\\ref{sec:intrinsic-reward}). Under this assumption, the solution to the deterministic policy gradient~\\cite{silver2014deterministic} of Eq.~\\eqref{eq:connected-return} with respect to the manager's parameters $\\theta_m$ is:\n\\begin{equation} \\label{eq:connected-gradient}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        \\nabla_{\\theta_m} J_m' &= \\mathbb{E}_{s\\sim p_\\pi} \\big[ \\nabla_a Q_m (s,a)|_{a=\\pi_m(s)} \\nabla_{\\theta_m} \\pi_m(s)\\big] \\\\\n        %\n        &\\quad + \\lambda \\mathbb{E}_{s\\sim p_\\pi} \\bigg[ \\nabla_{\\theta_m} g_t \\nabla_g \\big(r_w(s_t,g,\\pi_w(s_t,g_t)) \\\\\n        & \\quad + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\big) \\bigg\\rvert_{g=g_t} \\bigg]\n    \\end{aligned}\n    $}\n\\end{equation}\nwhere $Q_m(s,a)$ and $Q_w(s,g,a)$ are approximations for the expected environmental and intrinsic returns, respectively.\n\\end{theorem}\n\n\\vspace{-6pt}\n\n\\textit{Proof.} See Appendix~\\ref{sec:derivation}.\n\nThis new gradient consists of three terms. The first and third term computes the gradient of the critic policies $Q_m$ and $Q_w$ for the parameters $\\theta_m$, respectively. The second term computes the gradient of the worker-specific reward for the parameters $\\theta_m$. This reward is a design feature within the goal-conditioned RL formulation, however, any reward function for which the gradient can be explicitly computed can be used. We describe a practical algorithm for training a cooperative two-level hierarchy implementing this loss function in Algorithm~\\ref{alg:training}.\n\n\\subsection{Cooperative HRL as constrained optimization}\n\\label{sec:method-constrained-hrl}\n\nIn the previous sections, we introduced a framework for inducing and studying the effects of cooperation between internal agents within a hierarchy. The degree of cooperation is defined through a hyperparameter ($\\lambda$), \nand if properly defined can greatly improve training performance in certain environments. The choice of $\\lambda$, however,\ncan be difficult to specify without a priori knowledge of the task it is assigned to.\nWe accordingly wish to ground the choice of $\\lambda$ to measurable terms that can be reasoned and adjusted for. To that end, we observe that the cooperative $\\lambda$ term acts equivalently as a Lagrangian term in constrained optimization~\\citep{bertsekas2014constrained} with the expected return for the lower-level policy serving as the constraint. \nOur formulation of the HRL problem can similarly be framed as a constrained optimization problem, denoted as:\n\\begin{equation}\n    \\resizebox{.9\\hsize}{!}{$\n    \\begin{aligned}\n        &\\max_{\\pi_m} \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^{T\/k} \\left[ \\gamma^t r_m(s_t) \\right] \\right] \\\\\n        &\\quad + \\min_{\\lambda\\geq 0} \\left( \\lambda \\delta - \\lambda \\min_{\\pi_w} \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^k \\gamma^t r_w(s_t, g_t,\\pi_w(s_t,g_t)) \\right] \\right)\n    \\end{aligned}\n    $}\n\\end{equation}\nwhere $\\delta$ is the desired expected discounted \\emph{intrinsic} returns. The derivation of this equation and practical implementations are provided in Appendix~\\ref{sec:constrained-hrl}.\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}[b]{0.24\\textwidth}\n    \\centering\n    \\includegraphics[height=2.45cm]{figures\/ring-env.png}\n    \\caption{\\label{fig:ring-env} \\footnotesize \\ringroad}\n\\end{subfigure}\n\\quad\n\\begin{subfigure}[b]{0.72\\textwidth}\n    \\centering\n    \\includegraphics[height=2.45cm]{figures\/highway-single-env.png}\n    \\caption{\\footnotesize \\highwaysingle}\n    \\label{fig:highwaysingle-env}\n\\end{subfigure} \\\\ \\vspace{0.4cm}\n\\begin{subfigure}[b]{0.21\\textwidth}\n    \\centering\n    \\includegraphics[height=2.75cm]{figures\/AntGather-env.png}\n    \\caption{\\label{fig:antgather-env} \\footnotesize \\antgather}\n\\end{subfigure}\n\\\n\\begin{subfigure}[b]{0.21\\textwidth}\n    \\centering\n    \\includegraphics[height=2.75cm]{figures\/AntFourRooms-env.png}\n    \\caption{\\label{fig:antfourrooms-env} \\footnotesize \\antfourrooms}\n\\end{subfigure}\n\\\n\\begin{subfigure}[b]{0.21\\textwidth}\n    \\centering\n    \\includegraphics[height=2.75cm]{figures\/AntMaze-env.png}\n    \\caption{\\label{fig:antmaze-sim} \\footnotesize \\antmaze}\n\\end{subfigure}\n\\\n\\begin{subfigure}[b]{0.34\\textwidth}\n    \\centering\n    \\includegraphics[height=2.75cm]{figures\/soccer_env.png}\n    \\caption{\\label{fig:biped-sim} \\footnotesize \\bipedalsoccer}\n\\end{subfigure}\\\\[-8pt]\n\\caption{\nTraining environments explored within this paper. We compare the performance of various HRL algorithms on two mixed-autonomy traffic control task (a,b), three ant navigation tasks (c,d,e), and a bipedal\/humanoid navigation task (d). A description of each of these environments is provided in Section~\\ref{sec:environments}.\n}\n\\label{fig:envs}\n\\end{figure*}\n\nIn practice, this updated form of the objective provides two meaningful benefits: 1) As discussed in Section~\\ref{sec:cg-weight}, it introduces bounds for appropriate values of $\\delta$ that can then be explored and tuned, and 2) for the more complex and previously unsolvable tasks, we find that this approach results in more stable learning and better performing policies.\n\n\n\\section{Related Work} \\label{sec:related-work}\n\n\n\nThe topic explored in this article takes inspiration in part from studies of communication in multiagent reinforcement learning (MARL)~\\citep{thomas2011conjugate, thomas2011policy, DBLP:journals\/corr\/SukhbaatarSF16, DBLP:journals\/corr\/FoersterAFW16a}. \nIn MARL, communication channels are often shared among agents as a means of coordinating and influencing neighboring agents.\nChallenges emerge, however, as a result of the ambiguity of communication signals in the early staging of training, with agents forced to coordinate between sending and interpreting messages~\\citep{mordatch2018emergence, eccles2019biases}. Similar communication channels are present in the HRL domain, with higher-level policies communicating one-sided signals in the form of goals to lower-level policies. The difficulties associated with cooperation, accordingly, likely (and as we find here in fact do) persist under this setting. The work presented here serves to make connections between these two fields, and will hopefully motivate future work on unifying the challenges experienced in each.\n\nOur work is most motivated by the Differentiable Inter-Agent Learning (DIAL) method proposed by~\\citet{DBLP:journals\/corr\/FoersterAFW16a}. Our work does not aim to learn an explicit communication channel; however, it is motivated by a similar principle - that letting gradients flow across agents results in richer feedback. Furthermore, we differ in the fact that we structure the problem as a hierarchical reinforcement learning problem in which agents are designed to solve dissimilar tasks. This disparity forces a more constrained and directed objective in which varying degrees of cooperation can be defined. This insight, we find, is important for ensuring that shared gradients can provide meaningful benefits to the hierarchical paradigm.\n\nAnother prominent challenge in MARL is the notion of \\emph{non-stationarity}~\\citep{busoniu2006multi, weinberg2004best, foerster2017stabilising}, whereby the continually changing nature of decision-making policies serve to destabilize training.\nThis has been identified in previous studies on HRL, with techniques such as off-policy sample relabeling~\\citep{nachum2018data} and hindsight~\\citep{levy2017hierarchical} providing considerable improvements to training performance. Similarly, the method presented in this article focuses on non-stationary in HRL, with the manager constraining its search within the region of achievable goals by the worker. Unlike these methods, however, our approach additionally accounts for ambiguities in the credit assignment problem from the perspective of the manager. As we demonstrate in Section~\\ref{sec:experiments}, this cooperation improves learning across a collection of complex and partially observable tasks with a high degree of stochasticity. These results suggest that multifaceted approaches to hierarchical learning, like the one proposed in this paper, that account for features such as information-sharing and cooperation in addition to non-stationarity are necessary for stable and generalizable HRL algorithms.\n\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\\linewidth]{FingerAndAttack.pdf} \n \\caption{Different attacks on the fingerprint recognition systems shown as photographs~\\cite{9}, and as fingerprints~\\cite{LivDet2015}).}\n \\label{fig:fingersandattacks}\n\\end{figure}\nBiometrics based authentication systems provide more security than traditional information security-based systems based on passwords\/Personal Identification Number (PINs), and keys\/cards~\\cite{1}. The primary limitations with traditional information security methods are that they lack good user experience, using the same security measure with multiple applications, and forgetting\/losing the password\/PINs~\\cite{2}. Especially for keys\/cards, they can be duplicated apart from the previously mentioned limitations. Since biometric systems are based on human characteristics such as the face, fingerprint, or iris, which are unique for every individual, they have a definite advantage over information security-based systems. Due to these advantages, biometric systems are widely deployed in smartphones, border control (both in automated, and attended scenarios), and national identity cards. However, biometric systems are vulnerable to Presentation Attacks (PA)~\\cite{3}, due to which some crimes have been reported in the media, where the biometric systems were spoofed~\\cite{4,5,6}. An attacker can perform the attack on the biometric system by presenting a biometric artefact or a Presentation Attack Instruments (PAIs) ~\\cite{RaghuSurvey}. PA can be performed in different biometric modalities, including the face, fingerprint, and iris. Since fingerprint recognition systems are widely deployed in critical security systems, it is essential to develop fingerprint PAD.\n\n\nPAIs for fingerprint can either be an artificial object such as a gummy finger (made from play-doh, silicone, or gelatine) or a 2D\/3D printed photo. In terms of implementation, PAD systems can be either a hardware-based or a software-based, whose main task is to distinguish between a real (bona fide) user or a malicious (imposter) attacker~\\cite{8}.\nA summary of existing fingerprint PAD methods can be found in  Marcel et al.~\\cite{10}, Marasco et al.~\\cite{RossSurveyFPAD}, Galbally et al.~\\cite{Galbally2019}, and Sousedik et al.~\\cite{16}. In the current scenario, the majority of the existing PAD methods consist of training a classifier to accurately model the characteristics of the PAI. However, such an approach suffers from the problem of generalization to detect unknown attacks~\\cite{10}. Thus, developing a reliable PAD technique for unknown attacks is a significant problem that can also be posed as anomaly (outlier) detection. Fingerprint recognition systems have been widely deployed, as mentioned earlier, and are prone to PA. Since the attacks cannot be listed in advance, detecting unknown attacks for the fingerprint is critical.\nOur survey on fingerprint Presentation Attack Detection (FPAD) presents the following:\n\\begin{itemize}\n    \\item Comprehensive survey of existing methods for FPAD  for unknown attacks.\n    \\item Categorization of existing methods for the FPAD of unknown attacks.\n    \\item Discussion on advantages\/disadvantages of existing methods for FPAD, especially for unknown attacks.\n    \\item Concluding remarks with future directions for the area of FPAD.\n\\end{itemize}\n\nIn the rest of the paper, a comparison between traditional PAD done in a supervised manner, and anomaly detection based FPAD in Section~\\ref{sec:2}, which is followed by Section~\\ref{sec:3} summarizing related work in FPAD  which includes their categorization, advantages, and disadvantages in terms of generalization, and finally we present conclusions \\& future directions for FPAD in Section~\\ref{sec:4}.\n\n\n\\section{Traditional PAD \\& Anomaly Detection based PAD}\n\\label{sec:2}\n\\begin{figure}[htbp!]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{TaxonomyFigure.pdf} \n \\caption{Illustration of taxonomy for fingerprint presentation attack detection.}\n \\label{fig:taxonomyfigure}\n\\end{figure}\n\\begin{table}[t!]\n\\centering\n\\resizebox{1\\linewidth}{!}\n{\\begin{tabular}{|p{2.0cm}|l p{4.3cm}|l p{4.3cm}|}\n\\hline\n & & {\\bf{Traditional PAD}} & & {\\bf{Anomaly detection based PAD}} \\\\\n\\hline\n{\\bf{Characteristics}} &-& Information about PAI is gathered and known in advance. & - & Establish profiles of normality features which are extracted from regular data.  \\\\\n &-& Look for PAIs' features each time a presentation occurs. & - & Compares the normality features of each new presentation against the established profiles. \\\\\n&-& Alerts for PA if any PAI is found to be in the new presentation.   & - & Alerts for PA if a deviation from normality is detected in the new presentation based on threshold.  \\\\\n\n\\hline\n{\\bf{Advantages}} &-& Possibility to detect known PAs. &-& Possibility to detect known and unknown PAs. \\\\\n &-& There is a possibility of using existing knowledge to recognize new forms of old PAs.  &-& Does not care about the used PAI during the attack. \\\\\n\n\\hline\n{\\bf{Drawbacks}} &-& For each novel PA, PAD methods should be updated and tested with the new PAI. &-& Hard to define a profile of normality features for each bona fide presentation. \\\\\n &-&\n\n As the number of PAs increases, and correspondingly PAIs increase, the complexity of PAD increases.\n \n &-&  Higher false-positive for PAs depending on accessibility or usability.  \\\\\n &-& Hard to detect previously unseen PAs. &-& Hard to set the optimal threshold value for PAD.\\\\\n &-& Simple changes to PAI in a known PA can be enough to miss the detection of the PA. &-& The size of normality feature can be very large, which leads to a high false-positive rate.\\\\\n &-& A leak of PAIs' list that a system maintains could help attackers bypass the system's PAD method. &&  \\\\\n\\hline\n\\end{tabular}}\n\\caption{Characteristics, advantages and disadvantages of Anomaly detection based PAD as compared to Traditional PAD for biometrics.\\label{tab1}}\n\\end{table}\nIn this section, we present a comparison between traditional PAD (a form of supervised classification) and anomaly detection (supervised\/unsupervised classification) based PAD, as shown in Figure~\\ref{fig:taxonomyfigure}. Since we are interested in unknown attack detection for fingerprint, this can be achieved by Anomaly detection \\cite{11}. We now briefly review Anomaly detection in the following subsection:\n\\subsection{Anomaly Detection}\nAnomaly Detection refers to the determination of irregularity in a dataset. The dataset contains a set of records (aka., instances, objects, or entities), where each record includes a set of attributes (aka., characteristics or features), as pointed out by Chandola et al.~\\cite{11}. In general, an anomaly detection method is provided with a record\/set of records as an input, where no information about either anomalies or regular classes is known to the detection method in advance~\\cite{14}.\nThe three modes of anomaly detection methods, according to Chandola et al.~\\cite{11} are as follows:\n\\begin{itemize}\n    \\item \\textit{\\bf{Supervised anomaly detection}}: \\\\\n    Anomaly methods that are based on a predictive model which is trained by a labeled dataset of two classes (i.e., normal and anomaly records). Any unseen record is then compared against the model to determine whether it is a normal or an anomaly record. This can be achieved by using publicly labeled fingerprint datasets for training. This form of anomaly detection is used in traditional PAD and in unknown attack detection where the sensor is known in advance for FPAD. \n    \\item \\textit{\\bf{Semi-supervised anomaly detection}}:\\\\\n    Anomaly methods that are based on a single classifier trained using only normal behavior records from a dataset, as only those are labeled. This form of anomaly detection is used for unknown attack detection both for the known sensor, \\& unknown sensor for FPAD.\n    \\item \\textit{\\bf{Unsupervised anomaly detection}}:\\\\\n    Anomaly methods do not require training data, but no records are labeled in the dataset if training is applied. This method is based on the assumption that regular records are far more frequent than anomalies in both training and testing datasets, and can lead to high false reject rate if this assumption is violated. This form of anomaly detection is used for unknown attack detection for the known sensor, and the unknown sensor for FPAD.\n\\end{itemize}\nTable~\\ref{tab1} shows a description  of traditional  and anomaly detection based PAD. Theoretically, the main advantage of anomaly-based PAD methods over traditional methods is capturing both the known and the unknown PAs. In contrast, the traditional PAD methods can detect known PAs, and maybe new forms of these attacks. For instance, if a traditional PAD method is trained only to detect gummy fingers of play-doh, it may not detect gummy fingers of other materials like silicone or gelatine. This requires the traditional PAD methods to make a long list of PAIs gathered from known PAs, and the methods should be updated and re-trained each time a new unknown PA is revealed. Consequently, the list of PAIs and known PAs can become long and hard to maintain. Moreover, if an attacker gets access to the list of PAIs used to train a biometric system, the attacker will be able to conduct a PA using a novel PAI that is not known to the systems.\n\nEven if anomaly PAD methods solve several drawbacks in traditional PAD methods, they come with high risks, implementation difficulties, and critical disadvantages. In general, it is difficult to define and extract all features of bona fide presentations (i.e., normality features), because these features can have a broad scope and thus become hard to use for an implementation of a PAD method. Moreover, the threshold used to distinguish between PAs and bona fide presentations is affected with accessibility or usability issues between the subject, and the capture device, which makes it hard to define. Thus, the size of the normality features will be large, and it may require prioritizing some features over others during the feature selection. Nevertheless, size reduction methods can be used to reduce the number of features normality. However, this will lead to more false-positive alarms as the normality features are not precise enough to distinguish between all the cases of PAs and bona fide presentations.\n\n\\section{Known \\& Unknown Presentation Attack Detection for fingerprints}\n\\label{sec:3}\n\\begin{table}[t!]\n    \\centering\n    \\resizebox{1\\linewidth}{!}\n    {\\begin{tabular}{|l|l|l|l|l|l|l|l|}\n        \\hline\n        \\textbf{Ref.} & \\textbf{S\/H\/W} & \\textbf{Dataset} & \\textbf{Pre-processing} & \\textbf{Post-processing} & \\textbf{\\# unknown PAs} & \\textbf{A. D. methods} & \\textbf{A. D. mode}\\\\\n        \\hline\n        \\cite{fPA10} & S & LivDet 2009 & - & - & 1 & - & S \\\\ \\hline\n        \n        \\cite{fPA12} & S & LivDet 2011 & GLCM, HOG, BSIF & - & 2 & SVM, & S \\\\ \n        &  &  & LPQ, LBP, BGP &  &  & Rule-based &  \\\\ \\hline\n        \n        \\cite{fPA13} & S & LivDet 2011 & LBP & Score fusion & 4 & SVM  & S \\\\ \\hline\n        \n        \\cite{fPA11} & S & LivDet 2011 & BSIF, LBP, LPQ & - & 3 & SVM & S \\\\ \\hline\n        \n        \\cite{fPA3} & S & LivDet 2011, & Image segmentation & - & 4 & CNN & S \\\\ \n        &  & LivDet 2013, & (part of CNN) &  &  &  &  \\\\ \n        &  & LivDet 2015 &  &  &  &  &  \\\\ \\hline\n        \n        \\cite{fPA5} & S \\& H & Own dataset & ROI segmentation & - & 6 & Pre-trained CNN & S \\\\ \\hline\n        \n        \\cite{fPA6} & S \\& H & Own dataset & ROI segmentation & Score fusion & 3 & SVM & S \\\\ \\hline\n        \n        \\cite{fPA7} & S \\& H & Own dataset & ROI segmentation & Score fusion & 5 & SVM, & S \\\\\n        &  &  &  &  &  & CNN, &  \\\\ \n        &  &  &  &  &  & Pre-trained CNN &  \\\\ \\hline\n        \n        \\cite{fPA9} & S \\& H & Own dataset, & ROI segmentation, & Score fusion & 5 & SVM, & S \\\\ \n        &  & LivDet 2017 & RGB image creation &  &  & CNN, &  \\\\ \n        &  &  &  &  &  & Pre-trained CNNs &  \\\\ \\hline\n        \n        \\cite{fPA2} & S & MSU-FPAD, & Minutiae detection, & Score fusion & 6 & Pre-trained CNN  & S \\\\ \n        &  & PBSKD & Patches creation, &  &  &  &  \\\\\n        &  &  & Patches alignment &  &  &  &  \\\\ \\hline\n        \n        \\cite{fPA1} & S & LivDet 2011, & Dense-SIFT & Score fusion & 8 $\\leq$ & SVM,  & S, U \\\\ \n         &  & LivDet 2013, &  &  &  & K-means, &  \\\\\n         &  & LivDet 2015, &  &  &  & PCA &  \\\\\n         &  & LivDet 2019 &  &  &  &  &  \\\\ \\hline\n        \n        \\cite{fPA8}, & W & MSU-FPAD v2, & Patches extraction & Score fusion & 3 & Pre-trained CNN & S \\\\ \n        \\cite{fPA8_1} &  & LivDet 2015 &  &  &  &  &  \\\\\n        &  & LiveDet 2017 &  &  &  &  &  \\\\ \\hline\n       \n       \n    \\end{tabular}}\n    \\caption{Overview of Fingerprint PAD using anomaly detection for unknown PAs. (where the abbreviations used are Anomaly Detection (A. D.), Software\/Hardware\/Wrapper (S\/H\/W),  Supervised (S), Semi-Supervised (SS), and Unsupervised (U))\\label{tab2}}\n\\end{table}\nWe now review the related work for FPAD in general, and specifically for unknown attack detection of fingerprints. Many software and hardware PAD methods are presented in the literature to detect PAs against fingerprint recognition systems. PAs can be conducted using PAIs in two fingerprint forms (e.g., overlays), and additionally using 3d printed fingers~\\cite{16}. Software approaches make use of features extracted by standard sensing technologies, which can further be divided into static (e.g., sweat pores and texture of ridges and valleys) and dynamic features (e.g., skin color change over time due to pressure). Software approaches are usually cheaper to implement (as no extra hardware is needed), and less intrusive to the user~\\cite{Galbally2019}. Hardware approaches introduce a new device to the sensing technology to capture more details than standard sensors (e.g., fingerprint sweat, blood pressure, or odor). Keeping in mind that hardware solutions are only used to capture data, and they usually have associated software solutions with them that distinguish between bona fide and PAs, which can either be inbuilt in the sensor or as stand-alone software. So, in theory, if two different hardware approaches as in \\cite{fPA7} and \\cite{EgySWIR} use Short Wave Infrared (SWIR) and Laser Speckle Contrast Imaging (LSCI) techniques respectively, they can still process each other datasets using the same software in their approaches. According to Galbally et al.~\\cite{Galbally2019} hardware-based approach introduces a higher fake detection rate than a software-based approach. This survey paper considers the type of approach (i.e., hardware and software) as a comparison factor, as shown in Table~\\ref{tab2}.\n\n\n\n\\subsection{Pre-processing techniques (Software-based)}\nWe now briefly review the pre-processing techniques in the literature attached to the PAD methods presented in Table~\\ref{tab2}. These can be texture-based descriptors such as Local Binary Pattern (LBP)~\\cite{LBPPaper}, Grey Level Co-occurrence Matrix (GLCM)~\\cite{GLCM}, Histogram of Oriented Gradients (HOG)~\\cite{HOGPaper}, Binary Statistical Image Features (BSIF)~\\cite{fPA12}, Local Phase Quantization~\\cite{LPQPaper}, Binary Gabor Patterns (BGP)~\\cite{LBPBGP}, Dense-SIFT~\\cite{fPA1} or techniques such as Image Segmentation, Region of Interest (ROI) Segmentation or Finger-print Minutae detection.\n\\subsection{Convolutional Neural Network (Software-based)}\nWe now briefly review the deep learning-based approaches; Park et al.~\\cite{fPA3} presented a supervised software-based approach using a convolution neural network (CNN), which did not use the PAD of unknown PAs. However, they tested the approach on the LivDet 2015 data sets that contains four unknown PAs~\\cite{LivDet2015}. The CNN network devised by them takes the full image of a fingerprint. It outputs a three-dimensional tensor that is used to determine the probability of the image being a bona fide or an attack presentation. The liveness probability is compared to an optimal threshold obtained from the training phase, where they achieved an average classification error of 1.5\\% for the unknown PAs. The usage of deep learning approaches has become a trend in the last decade, which is mainly due to the freely available pre-trained networks such as VGG~\\cite{VGGPaper}, GoogleNet~\\cite{GoogleNetPaper}, and ResNet~\\cite{ResnetPaper}. \nTolosana et al.~\\cite{fPA9} published a new experiment, where a PAD method relies on the use of SWIR and RGB images. Deep features from RGB images are extracted via two pre-trained CNNs, namely VGG19 and MobileNet, and a ResNet network trained from scratch. The features output by the CNNs is feed to an SVM. Additionally, handcrafted features as spectral signatures were extracted from SWIR images. For the final evaluation, a score fusion applied, and the reported D-EER for this experiment was 1.36\\%.\n\n\n\\subsection{Known Sensor \\& Known Attacks}\nMarasco et al.~\\cite{RossSurveyFPAD} provided an overview of PAD methods in the literature for fingerprint recognition systems, and they specifically point out that commercial fingerprint recognition systems can be spoofed. Most of these approaches test their performance on a test dataset with the same PAs as used during the training. Thus, these PAs are considered known to the PAD-method, which is a less realistic scenario than a real-world environment setup where additional PAIs may be used to conduct PAs (i.e., unknown attacks). \n\\subsection{Known Sensor \\& Unknown Attacks}\nTo the best of our knowledge, Tan et al.~\\cite{fPA00} were the first to point to the effect of environmental conditions and new PAI materials on PAD methods for fingerprints. They showed that new PAI to increases the error rate by at least 14\\% and up to 55,6 \\% on different fingerprint scanners as Identix, Crossmatch, and Digital Persona. Moreover, their experiment showed that the error rate drops back into an acceptable range once new PAIs are used in the training phase. This was later confirmed by Marasco et al. in~\\cite{fPA10}, in which they experimented the increase of spoof detection error rates of five fingerprint liveness detection methods (given by Marasco et al.~\\cite{1of5}, Moon et al.~\\cite{2of5}, Shankar et al.~\\cite{3of5}, Abhyankar et al.~\\cite{4of5}, and Tan et al.~\\cite{5of5}) when tested on the new PAIs that were not used during training. Marasco et al.~\\cite{fPA10} used the leave-one-out approach in their experiment, where only one PAI out of gelatine, play-doh, and silicone is used for testing, and the other two are used for training as they train the PAD methods using both PAs and bona fide presentations and can be classified as supervised anomaly detection approach.\nTo solve the problem of unknown PAIs, Rattani et al.~\\cite{fPA12} proposed a scheme for automatic detection and adoption of the liveness detector to new PAIs. Their liveness detection is a combination of a multiclass-SVM and rule-based approaches that form an AdaBoost-based classifiers\\cite{ada}. The Adaboost classifiers are used to detect novel PAs, and new PAIs used in each attack, followed by a binary classification SVM that corresponds to live and spoof classes, where the thresholds are maintained by multi-class SVM. In a case where a novel PA is presented to the detector, two rules apply to determine whether the PA is novel or already known. The first rule computes the maximum posterior probabilities for each known PA and bona fide. So, PA is considered novel if it overcomes a defined threshold else it is regarded as a known PAs and belongs to the corresponding class value. The second rule estimates the standard deviation of the posterior probabilities computed in the first rule. A low standard deviation value indicates doubt in classifying the PA as a known.\nAdditionally, they state the possibility of their PAD method to update the maintained binary classification SVM automatically, thus that it is always considered learned to known PA materials. This method is considered supervised because two out of four materials in the LiveDet 2011 dataset were used for training (i.e., two known PAIs and two unknown PAIs). The published results mentioned up to 46\\% improvements in detecting unknown PAIs. Rattani et al.~\\cite{fPA13} published a study where they tried to reduce the material-specific noise and apply a software-based method that learns the general artifacts in images from PAIs that correspond to different materials. This is done by using two SVMs that combine linear filtering and non-linear denoising using wavelet decomposition of PAIs on an LBP-based textural-analysis liveness detector. Their experimental results gained up to 44\\% improvements in detecting unknown PAs on LiveDet 2011 dataset. The training phase during the experiment is done using one material out of five. Thus, the method is tested on four unknown attacks. Rattani et al.~\\cite{fPA12} used Weibull-calibrated SVM (W-SVM) can be used both for the detection of liveness and spoofs, and discovery of new novel PAs and PAIs. Also, they claim W-SVM that supports interoperability between individual detectors. The results show 44\\% improvements in detecting novel materials on Livedet 2011 dataset. Tolosona et al. ~\\cite{fPA5} used a VGG pre-trained network as a PAD method in the finger recognition system. They use ShortWave Infrared Imaging (SWIR) images since the skin reflection within the SWIR spectrum of 900\u20131700 nm is independent of the skin tone as analyzed by the National Institute of Standards Technology (NIST). Thus, they used a hardware sensor approach to capture SWIR images of bona fide and PAs (i.e., own dataset), and a software-based approach for PAD. A total number of six unknown PAIs were detected by their PAD method, giving high convenience and secure, supervised PAD method. The same hardware developed by~\\cite{fPA5} is capable of capturing finger vein images (i.e., Visible Light Images, VIS) and speckle contrast images (LSCI) in addition to SWIR images.\nGomez-Barrero et al.~\\cite{fPA6} proposed a multi-modal finger PAD method where they use different ad-hoc approaches in parallel for each image type, and several SVM classifications are set to output a score of each ad-hoc approach where the final score is given by the weighted sum of all individual scores obtained. The evaluation in this approach is applied to both known and unknown PAIs (in total 35, three are unknown), resulting in a Detection Equal Error Rate (D-EER) of 2.7\\%. Gomez-Barrero et al. proposed another multi-modal approach~\\cite{fPA7}, in which the proposed PAD method relies on a weighted sum of two CNN networks based on SWIR images and textural and gradient information from averaged LSCI images. They applied a pre-trained VGG19 network and a ResNet network that was trained from scratch for the CNNs. The textural and gradient information extracted from averaged LSCI images is passed into three SVMs for classification. They used the dataset from~\\cite{fPA6}, increasing the number of unknown attacks to five PAIs, and reporting a decrease in the D-EER from 2.7\\% to 0.5\\%. Chugh et al.~\\cite{fPA2} proposed a software-based FPAD method with a generalization against PAIs not seen during training. They studied the characteristics of twelve different PAIs and bona fide presentations using deep features extracted by a pre-trained CNN, namely, MobileNetv1.\nFurther, they applied an agglomerative clustering based on the shared characteristics of PAIs. Thus, they concluded that a subset of PAIs, namely silicone, 2D paper, play-doh, gelatine, latex body paint, and monster liquid latex, are essential PAIs to include during the training to achieve a robust PAD. An android smartphone application is presented without a significant drop in performance from the original PAD method. They achieved a True Detection Rate (TDR) of 95.7\\% and False Detection Rate (FDR) of 0.2 \\% when the generalization set is used for training (i.e., the six PAIs). \n\n \n \n\\subsection{UnKnown Sensor \\& Unknown Attacks}\nRattani et al.~\\cite{fPA11} declared the need for fingerprint PAs detection to be considered as an open set recognition problem. Thus, incomplete knowledge about neither PAs nor PAIs is known to the PAD method during training. Therefore, they adopted W-SVM, which uses recent advances in extreme value theory statistics for machine learning to directly address the risk of the anomalies in an open set recognition problem. Ding et al.~\\cite{Ross2016} proposed the use of an ensemble of One-Class Support Vector Machines (OC-SVM) using bona fide samples to generate a hypersphere boundary which is refined by a small number of spoof samples, for classification of unknown PAIs.  Jain et al.~\\cite{JainOneClass19} developed a one-class classifier that is based on training on learning of bona fide samples using multiple GANs (Generative-Adversarial Networks) which can reject any PAI. Gonz{\\'a}lez-Soler et al.~\\cite{fPA1} proposed a software-based PAD method and achieved an overall accuracy by 96.17\\% on the LivDet2019 competition. This method relied on three image representation approaches, which combine both local and global information of the fingerprint, namely Bag-of-words (BoW)~\\cite{bag}, Fisher Vector (FV)~\\cite{fisher}, and Vector Locally Aggregated Descriptors (VLAD)~\\cite{vlad}. They computed Dense-SIFT descriptors at different scales, and the features are then encoded using a previously learned visual vocabulary using the previously mentioned image representation approaches. A linear SVM classifier is applied to classify the fingerprint descriptor in each method. A weighted sum computes the final decision score. BoW approach uses K-means clustering local features and presents it as a pyramid of spatial histograms. FV approach is based on statistical and spectral-based techniques, where the Gaussian Mixture Model (GMM) locates local features that lie under the same distribution. Then, these features are presented in a lower dimension via Principal Component Analysis (PCA). VLAD approach, on the other hand, relied on non-probabilistic techniques and is used to reduce the high-dimension image representation in BoW and FV. They experimented with both scenarios of their PAD method, namely supervised (i.e., known PAs) and unsupervised (i.e., unknown PAs) scenarios. Chugh et al.~\\cite{fPA8} and Gajawada et al.~\\cite{fPA8_1} present a wrapper that can be adopted by any fingerprint PAD method to improve the generalization performance of the PAD method against unknown PAIs. These approaches are based on synthesizing fingerprint images that correspond to unknown PAs and bona fide images as well. The goal is to transfer the characteristics into a deep feature space so that a more precise presentation helps the PAD method increase its generalization performance. The method is based on multiple pre-trained VGG19 CNNs that encode and decode the content and style loss of the synthesized images, as they can be further used to train the PAD method. They use the same PAD software method as done by Chugh et al.~\\cite{fPA2} to experiment with the wrapper. Moreover, this approach is a supervised method in which they use the leave-one-out technique on each PAI for MSU-FPADv2, where the other PAIs are known in training. On the other hand, in LivDet 2017 dataset, three PAIs were considered unknown.\n \n\n\n\n\n\n\n\\section{Conclusions \\& Future Directions}\n\\label{sec:4}\nThis survey paper presented unknown attack detection for fingerprints, including a survey of existing methods summarized \\& categorized in Table~\\ref{tab2}, additionally a taxonomy of FPAD is presented in Figure~\\ref{fig:taxonomyfigure}. Currently, most unknown attack detection methods for fingerprints are solving the problem of known sensors and unknown PAIs, and there are only a few methods which are unknown sensor, and unknown PAI, including cross-dataset. \\\\\nUnknown attack detection with unknown sensors is a relatively new area of research for FPAD and should be the focus area in near-future. The first approach to solving it is of synthesis, as done by Jain et al.~\\cite{JainOneClass19}. The second approach is to arrive at a common deep-feature representation, such as the one used by Gonz{\\'a}lez-Soler et al.~\\cite{fPA1}. The challenge in synthesis based approach is to do high-quality synthesis of the bona fide samples, and the difficulty in arriving at a common deep-feature representation is the degree of invariance it can provide to sensor type and PAI.\n\\balance\n{\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries}\n\n\n\n\n\nLet $[n]=\\{1,\\dots,n\\}$ be our underlying set. If $F\\subseteq [n]$,\nthen $\\overline{F}$ denotes the complement of $F$. Let\n$\\mathcal{F}$ be a family of subsets of $[n]$\n(i.e. $\\mathcal{F}\\subseteq 2^{[n]}$). Let $\\overline{\\cF}:=\\{F\\subset [n]: \\overline{F}\\in\\cF\\}$. A family is called \\emph{intersecting} if any\ntwo members have non-empty intersection. Intersecting families of sets have attracted a lot of researchers, see e.g. Chapter 2 of the book \\cite{book}. Let us start with a well-known and trivial statement.\n\n\\begin{prop} The maximum size of an intersecting family is\n$2^{n-1}$.\n\n\\end{prop}\n\nThe maximum size is achieved e.g. by the family of all subsets containing a given fixed element. A family is called \\emph{$k$-uniform}, if all its members have cardinality $k$. Let $\\mathcal{F}_k$ denote the subfamily of the\n$k$-element subsets in $\\mathcal{F}$:\\, $\\mathcal{F}_k=\\{F:\nF\\in\\mathcal{F}, |F|=k\\}$.\n\n\\begin{thm}[Erd\\H{o}s, Ko, Rado \\cite{ekr}] Let $k \\le n\/2$. Then the maximum size of a $k$-uniform intersecting family is $\\binom{n-1}{k-1}$.\n\n\\end{thm}\n\nLet us call an intersecting family \\emph{trivial} if all its members contain a given fixed element, and non-trivial otherwise. The maximum in the above theorem is again achieved by the largest trivial intersecting family.\n\n\\begin{thm}[Hilton, Milner \\cite{hm}] Let $k \\le n\/2$. Then the maximum size of a non-trivial $k$-uniform intersecting family is $1+\\binom{n-1}{k-1}-\\binom{n-k-1}{k-1}$.\n\n\\end{thm}\n\nThe maximum is given by the Hilton-Milner type family $HM(k)$, which we define next. $HM(k)$ contains $A=\\{2,\\dots,k+1\\}$ and every $k$-element set which contains $1$ and intersects $A$. Moreover, Hilton and Milner \\cite{hm} also showed that $HM(k)$ is the unique maximum if $3<k<n\/2$, up to isomorphism (i.e. in every maximum family there is a fixed point contained in all but one of the members). If $k=3$, then there is another extremal family $\\{F\\in \\binom{[n]}{3}: |F\\cap [3]|\\ge 2\\}$.\n\nWe will also use the following generalized Hilton-Milner type families. Let $B=[m]$. Then $HM(k,m)=\\{F \\subset [n]: |F|=k, n \\in F, | F \\cap B|\\ge 1\\}$. This is a $k$-uniform, trivial intersecting family, but if we add $B$, it becomes a non-trivial intersecting family.\n\nWe will use the following simple observation.\n\n\\begin{lem}\\label{observ} Let $m$ be the maximum cardinality of the members of a non-trivial intersecting family $\\mathcal{F}$, let $i <m$ and $i\\le n\/2$. Then $|\\mathcal{F}_i|\\le |HM(i,m)|$. Moreover, if $i<n\/2$, equality holds if and only if $\\cF_i$ consists of all the sets containing a fixed element $x$ and intersecting an $m$-element set $B'$ with $x\\not\\in B'$ (i.e. $\\cF_i$ is isomorphic to $HM(i,m)$).\n\n\\end{lem}\n\n\\begin{proof} If $\\mathcal{F}_i$ is non-trivial, then $\\mathcal{F}_i\\le |HM(i,i)|< HM(i,m)$. If $\\mathcal{F}_i$ is trivial, all its members contain a fixed element $x$. There is a set $F$ in $\\cF$ which does not contain $x$ because of the non-triviality, and the $i$-element sets of $\\cF$ also intersect $F$. One can easily see that $\\mathcal{F}_i$ is a subset of a generalized Hilton-Milner type family then, and $HM(i,m)$ is the largest of those.\n\\end{proof}\n\nWe will use the Kruskal-Katona theorem \\cite{kat,kru}. Given a $k$-uniform family $\\cF\\subset 2^{[n]}$, its \\emph{shadow} is \\[\\Delta\\cF:=\\{G\\subset [n]: \\,|G|=k-1, \\text{ there exists $F\\in\\cF$ with $G\\subset F$}\\}.\\]\nThe \\emph{shade} of $\\cF$ is $\\nabla\\cF:=\\{G\\subset [n]: \\,|G|=k+1, \\text{ there exists $F\\in\\cF$ with $G\\supset F$}\\}$.\nGiven two sets $F,G\\subset 2^{[n]}$, we say that $F$ is before $G$ in the \\emph{colexicographical order} or \\emph{colex order} if the largest element of the symmetric difference of $F$ and $G$ is in $G$. Let $\\cC_k^\\ell$ denote the family of the first $\\ell$ sets from $\\binom{[n]}{k}$ in the colex order. \n\n Given two positive integers $\\ell$ and $i$, there is a unique way to write $\\ell$ in the form $\\ell=\\binom{n_i}{i}+\\binom{n_{i-1}}{i-1}+\\dots+\\binom{n_j}{j}$ with $n_i>n_{1-i}>\\dots>n_j\\ge j\\ge 1$. This form is called the \\emph{cascade form} of $\\ell$. The cascade form can be found in a greedy way: we pick the largest $n_i$ such that $\\binom{n_i}{i}\\le \\ell$, then the largest $n_{i-1}$ such that $\\binom{n_i}{i}+\\binom{n_{i-1}}{i-1}\\le \\ell$, and so on.\n\nThe Kruskal-Katona shadow theorem \\cite{kat,kru} states that if $\\cF$ is a $k$-uniform family with $|\\cF|=\\ell$, then $|\\Delta\\cF|\\ge |\\Delta\\cC_k^\\ell|$. It is not hard to calculate the cardinality of $|\\Delta\\cC_k^\\ell|$: if\n$\\ell=\\binom{n_k}{k}+\\binom{n_k-1}{k-1}+\\dots+\\binom{n_j}{j}$, then $|\\Delta\\cC_k^\\ell|=\\binom{n_k}{k-1}+\\binom{n_k-1}{k-2}+\\dots+\\binom{n_j}{j-1}$.\n\nThere is a simpler version of the shadow theorem due to Lov\\'asz \\cite{lov}. It states that if $\\cF$ is a $k$-uniform family with $|\\cF|=\\binom{x}{k}$, then $|\\Delta\\cF|\\ge \\binom{x}{k-1}$. Here $x$ is not necessarily an integer and $\\binom{x}{k}$ is defined to\nbe $\\frac{x(x-1)\\ldots (x-k+1)}{k!}$. This is a weaker bound, but easier to use. We will use both versions of the shadow theorem later.\n\n\n\\subsection{Profile polytopes}\n\n\nThe profile polytopes were introduced by P.L. Erd\\H os, P. Frankl and\nG.O.H. Katona in \\cite{efk1}.\nRecall that $\\mathcal{F}_i$ denotes the subfamily of the\n$i$-element subsets in $\\mathcal{F}$. Its size $|\\mathcal{F}_i|$ is denoted\nby $f_i$. The vector ${\\bf p}(\\mathcal{F})=(f_0,f_1,\\dots,f_n)$ in\nthe $(n+1)$-dimensional Euclidian space $\\mathbb{R}^{n+1}$ is\ncalled the \\emph{profile} or \\emph{profile vector} of $\\mathcal{F}$.\n\n\n\nIf $\\Lambda$ is a finite set in $\\mathbb{R}^d$, its \\emph{convex\nhull} $\\conv(\\Lambda)$ is the set of all convex combinations of\nthe elements of $\\Lambda$. A point of $\\Lambda$ is an\n\\emph{extreme point} if it is not a convex combination of other\npoints of $\\Lambda$. It is easy to see that the convex hull of a\nset is equal to the convex hull of the extreme points of the set.\n\nLet $\\mathbf{A}$ be a class of families of subsets of $[n]$. We\ndenote by $\\Lambda(\\mathbf{A})$ the set of profiles of the\nfamilies belonging to $\\mathbf{A}$:\n\n\\[\\Lambda(\\mathbf{A})=\\{{\\bf p}(\\mathcal{F}):\n\\mathcal{F}\\in\\mathbf{A}\\}.\\]\n\nThe \\emph{profile\npolytope} of $\\mathbf{A}$ is $\\conv(\\Lambda(\\mathbf{A}))$.\nWe are interested in the extreme points of $\\Lambda(\\mathbf{A})$. We simply call them the extreme points of\n$\\mathbf{A}$.\n\n\n\n\n\n\n\n\nSuppose we are given a weight function\n$w:\\{0,\\dots,n\\}\\rightarrow\\mathbb{R}$, and the weight of a family\n$\\mathcal{F}$ is defined to be $\\sum_{F\\in\\mathcal{F}} w(|F|)$,\nwhich is equal to $\\sum_{i=0}^n w(i)f_i$. Usually we are\ninterested in the maximum of the weight of the families in a class\n$\\mathbf{A}$. So we want to maximize this sum, i.e. find a family\n$\\mathcal{F}_0 \\in \\mathbf{A}$ and an inequality $\\sum_{i=0}^n\nw(i)f_i=w(\\mathcal{F})\\le w(\\mathcal{F}_0)=c$. This is a linear\ninequality, and it is always maximized in an extreme point.\n\nGiven a class (or property) of families, the first natural question in extremal combinatorics is the maximum cardinality such a family can have. When it is answered, often some simple weight functions are considered and the maximum weight of such a family is studied. Determining the extreme points answers these questions for every (linear) weight function.\n\n\n\n\nP.L. Erd\\H os, P. Frankl and\nG.O.H. Katona \\cite{efk1} determined the\nextreme points of the intersecting Sperner\nfamilies. In their next paper \\cite{efk2}, the extreme points of\nthe profile polytope of the intersecting families were determined. Now we define these. Let coordinate $i$ of ${\\bf a}$ be $0$ if $i<n\/2$, $\\binom{n-1}{i-1}$ if $i=n\/2$ and $\\binom{n}{i}$ if $i>n\/2$. Let $k \\le n\/2$. Coordinate $i$ of ${\\bf a}_k$ is $0$ if $i<k$, $\\binom{n-1}{i-1}$ if $k \\le i \\le n-k$, and $\\binom{n}{i}$ if $i>n-k$. Let $\\Gamma_a$ be the set of vectors that we can get from any of the vectors ${\\bf a}_k$ and ${\\bf a}$, if we replace an arbitrary set of coordinates by $0$. Note that if $n$ is even, then ${\\bf a}={\\bf a_{n\/2}}$.\n\n\n\\begin{thm}[P.L. Erd\\H{o}s, Frankl, Katona \\cite{efk2}]\\label{metszo} The set of extreme points of the intersecting families is $\\Gamma_a$.\n\\end{thm}\n\nThe corresponding intersecting families are the following. $\\mathcal{A}_k$ consists of the sets which have sizes at least $k$ and contain the element $n$, and of every other set which has size greater than $n-k$. $\\mathcal{A}$ consists of all the sets with size greater than $n\/2$, and the sets which have sizes $n\/2$ and contain $n$. These families are obviously intersecting and their profile vectors are ${\\bf a}_k$ and ${\\bf a}$. We can delete full levels and the families are still intersecting; in the corresponding vectors some coordinates are changed to $0$.\n\nSince then several other classes of families have been considered, see e.g. \\cite{eng}, \\cite{gerbner}, generalizations have been studied \\cite{patk}, \\cite{patk2}, and profile polytopes were applied for counting subposets \\cite{gkp}. Note that most of the classes of families where the profile polytope has been studied are \\emph{hereditary}, i.e. if we remove some members of a family in the class, the resulting family still belongs to the class. It makes determining the extreme points easier, as we do not have to deal with negative weights, and all extreme points can be achieved by changing some coordinates of a few essential ones to $0$.\nHowever, in this paper we determine the extreme points of the non-trivial intersecting families, which is not a hereditary property. \n\nIn the next section we define what is needed to state our main theorem. We prove an important special case in Section \\ref{biz}, and finish the proof by a case analysis in Section \\ref{mainbiz}.\n\n\\section{The main theorem}\n\nLet us start with some simple observations. A non-trivial intersecting family cannot contain the empty set or a singleton. It might contain the full set, but that does not change the intersecting property, nor the nontrivial property. It means that for a weight function $w$ if $w(n)>0$, the maximum family contains the full set, if $w(n)<0$, it does not. Moreover, changing only $w(n)$ does not change the other parts of the maximum family, hence we can basically forget about $n$. More precisely, $(p_0,p_1, \\dots, p_{n-2}, p_{n-1},0)$ is an extreme point if and only if $(p_0,p_1, \\dots, p_{n-2}, p_{n-1},1)$ is an extreme point.\n\nNow we define several vectors, which are going to be the extreme points of the non-trivial intersecting families. Then we state our main theorem, and after that we show that these vectors indeed correspond to non-trivial intersecting families and are extreme points (note that for most classes of families where profile polytopes have been studied, these statements are trivial, but not for the non-trivial intersecting families). That part also makes it easier to understand where these definitions come from. All these vectors are in the $(n+1)$-dimensional Euclidean space, but coordinates 0,1 and $n$ are always 0. Let $H\\subset \\{2,3,\\dots,n-2,n-1\\}$ be a nonempty set of indices, $h$ be its smallest and $h'$ be its largest element.\n\nLet ${\\bf b}_H=(b_0,\\dots,b_n)$ with\n\\begin{displaymath}\nb_i=\n\\left\\{ \\begin{array}{l l}\n0 & \\textrm{if\\\/ $i\\not\\in H$},\\\\\n|HM(i,h')| & \\textrm{if\\\/ $i\\in H$ and $i< h'$},\\\\\n|HM(i,h')|+1 & \\textrm{if\\\/ $i=h'$}.\\\\\n\\end{array}\n\\right.\n\\end{displaymath}\nLet $\\Gamma_b=\\{ {\\bf b}_H: h+h' \\le n\\}$.\n\nLet ${\\bf c}_H=(c_0,\\dots,c_n)$ with  \n\\begin{displaymath}\nc_i=\n\\left\\{ \\begin{array}{l l}\n0 & \\textrm{if\\\/ $i\\not\\in H$},\\\\\n\\binom{n-1}{i-1} & \\textrm{if\\\/ $i\\in H$ and $i\\le n-h'$},\\\\\n\\binom{n}{i} & \\textrm{otherwise}.\\\\\n\\end{array}\n\\right.\n\\end{displaymath}\nLet $\\Gamma_c=\\{ {\\bf c}_H: h+h' > n\\}$.\n\nLet ${\\bf d}_H=(d_0,\\dots,d_n)$ with\n\\begin{displaymath}\nd_i=\n\\left\\{ \\begin{array}{l l}\n0 & \\textrm{if\\\/ $i\\not\\in H$},\\\\\n|HM(i,h')| & \\textrm{if\\\/ $i\\in H$ and $i< h'$},\\\\\n1 & \\textrm{if\\\/ $i=h'$}.\\\\\n\\end{array}\n\\right.\n\\end{displaymath}\nLet $\\Gamma_d=\\{ {\\bf d}_H: |H|>1, h+h''\\le n\\}$, where $h''$ is the second largest element of $H$.\n\nLet us consider the set $P$ of vectors $(e_0,\\dots,e_n)$ satisfying the following properties.\n\n1, Every $e_i$ is a non-negative integer, $e_0=e_1=e_n=0$.\n\n2, $x:=\\sum_{i=2}^{n-1}e_i \\ge 3$.\n\n3, $\\sum_{i=2}^{n-1} ie_i \\le (x-1)n$.\n\nNow we show the connection between $P$ and non-trivial intersecting families. For two vectors ${\\bf p}=(p_0,\\dots,p_n)$ and ${\\bf p'}=(p_0',\\dots,p_n')$, we say that ${\\bf p'} \\le {\\bf p}$ if $p_i\\le p_i'$ for every $0\\le i\\le n$.\n\n\\begin{lem}\\label{ujabb}\n\n\\textbf{(i)} If a non-trivial intersecting family does not contain $[n]$, its profile is in $P$.\n\n\\textbf{(ii)} If ${\\bf p} \\in P$ and there is no ${\\bf p'} \\in P$ different from ${\\bf p} $ with ${\\bf p'} \\le {\\bf p}$, then ${\\bf p}$ is the profile of a non-trivial intersecting family.\n\n\\end{lem}\n\n\\begin{proof}\n\nTo show \\textbf{(i)}, observe that for the profile of a non-trivial intersecting family obviously $e_0=e_1=0$ holds, and also we need at least three members in the family, as any two members trivially intersect. The third property is needed, otherwise an element of the underlying set would be covered $x$ times, i.e. by every set, contradicting the non-triviality.\n\nLet us prove now \\textbf{(ii)}. We are given a vector ${\\bf p}$ and we are going to construct a non-trivial intersecting family $\\cF$ with profile ${\\bf p}$. Observe that ${\\bf p}$ shows how many $k$-element sets must be in the family for every $k$. Let us denote the sizes of the sets by $a_1,\\dots,a_\\ell$ in decreasing order. We choose the first (the largest) set $F_1$ of size $a_1$ arbitrarily. Let $B_i$ be the set of vertices which are not covered by each of the first $i$ sets $F_1,\\dots,F_i$ (only by at most $i-1$ of them), then $B_1=\\overline{F_1}$ and $B_i\\supset B_{i-1}$ for every $i>1$. We choose the second set $F_2$ of size $a_2$ in such a way that $F_2$ intersects $F_1$ and also $F_2$ contains $B_1$, if possible.\n\nIf it is not possible, then we claim that we have $x=3$. Indeed, in that case we have $a_1+a_2\\le n$, thus they together with the next set $F_3$ of size $a_3$ have their profile in $P$, which means no other set can be in the family because of our assumption on the minimality of ${\\bf p}$.\nThen we pick $F_2$ of size $a_2$ such that it intersects $F_1$ in a single element, and then we pick $F_3$ of size $a_3$ such that it contains an element of $F_1\\setminus F_2$ and an element of $F_2\\setminus F_1$. This is doable as $a_3\\ge 2$. The resulting family is clearly non-trivial intersecting.\n\nIf $x>3$, we choose every $F_i$ of size $a_i$ in such a way that it contains $B_{i-1}$, if possible. Note that in this case it automatically intersects $F_1, \\dots, F_{i-1}$. Indeed, $F_i$ contains $B_1$, which is also contained in $F_2, \\dots, F_{i-1}$. $F_i$ also contains $B_2$, which intersects $F_1$ (we also use that $B_1$ and $B_2$ are not empty).\n\n\n\n\nNow assume that when we add a set $F_i$, it is too small to cover every vertex in $B_i$, i.e. $a_i< |B_i|$. Then $i=\\ell$, i.e. $F_i$ is the last set (as the resulting profile vector is in $P$). We have to choose $F_i$ in such a way that it intersects the other sets. As every vertex is covered at least $i-1$ times, all we have to do is to put an arbitrary vertex of $B_{i-1}$ in $F$, then the new set intersects all but one of the earlier sets, say $F_j$. We have to choose a vertex in $B_{i-1}$ contained in $F_j$, and then other vertices from $B_{i-1}$ arbitrarily. As only vertices in $B_{i-1}$ are used, no vertex is covered $i$ times, hence the family is non-trivial.\n\\end{proof}\n\nLet $\\Gamma_e$ be the set of the extreme points of $P$. Now we can state our main theorem.\n\n\n\n\n\n\n\n\n\\begin{thm}\\label{main}\n\nThe extreme points of the profile polytope of the non-trivial intersecting families are the elements of $\\Gamma_b\\cup\\Gamma_c\\cup\\Gamma_d\\cup\\Gamma_e$, and additionally the vectors we get from these if we change the last coordinate from $0$ to $1$.\n\n\n\n\\end{thm}\n\nTo prove this statement, we have to show that the points listed are indeed extreme points, and that there are no other extreme points. The first part is the easier task, and we will deal with it in the rest of this section.\nWe give an example non-trivial intersecting family for each of the vectors ${\\bf v}={\\bf v}_H\\in \\Gamma_b\\cup\\Gamma_c\\cup\\Gamma_d\\cup\\Gamma_e$ and also show that ${\\bf v}$ is an extreme point, by showing a weight function such that ${\\bf v}$ is the unique maximum. \n\nLet us describe first the general approach to find such a weight function.\nWe start by assuming that if $i\\not\\in H$, then $w(i)$ is negative, moreover, $w(i)$ so small compared to the other weights $w(j)$, that if a family contains even one $i$-element set, its total weight is negative. On the other hand, there is a $1<j<n$ with $w(j)>0$, thus there is a family of positive weight. This shows that\nno $i$-element sets can be in the family of maximum weight. Similarly, we can say that for some $i\\in H$ its weight is very large compared to the other weights. It implies that the family of maximum weight contains as many $i$-element sets as possible, i.e. $|HM(i,j)|$, where $j$ is the largest non-zero coordinate of ${\\bf v}_H$. We describe these ideas in more details in the proof of the following lemma.\n\n\n\n\\begin{lem}\\label{gammab} The elements of $\\Gamma_b$ are extreme points of the non-trivial intersecting families.\n\n\\end{lem}\n\n\\begin{proof} For ${\\bf b}_H\\in\\Gamma_b$ we have to show a family $\\mathcal{B}_H$ which has ${\\bf b}_H$ as profile, and a weight $w$ which is maximized at ${\\bf b}_H$. Let $\\mathcal{B}_H=[h']\\cup\\left(\\bigcup_{i\\in H}HM(i,h')\\right)$, i.e. the union of $HM(i,h')$ for every $i\\in H$, and additionally $[h']$. This family is obviously non-trivial intersecting, as each of its members except for $[h']$ contains $n$ and intersects $[h']$.\n\nNow we are going to show a weight function that is maximized only by families with profile ${\\bf b}_H$.\nLet $w$ be a weight such that if $i\\not\\in H$, then $w(i)=-2^{2n}$. It is going to be so small compared to the other weights, that no $i$-element sets can be in the maximum family $\\cF$. All other sets have weight at most $2^n$, and there are less than $2^n$ sets in $\\cF$, hence positive weight can only be achieved without these negative sets. Let $w(h)=2^n$, it is very large compared to the other positive weights (but still very small compared to the absolute value of the negative weights), and all other weights are 1. Then a single $h$-element set has larger weight than all the other sets with positive weight, thus the maximum family $\\cF$ contains as many $h$-element sets as possible. If $h<n\/2$, then by Lemma \\ref{observ} the maximum number of $h$-element sets is $|HM(h,h')|$, and the largest family of $h$-element sets is isomorphic to $HM(h,h')$. Without loss of generality we can assume that $\\cF_h$ is equal to $HM(h,h')$. \n\nObserve that the only set of size at most $h'$ that intersects every member of $HM(h,h')$ is $[h']$. Therefore, every other member of $\\cF$ should contain the fixed point $n$ except for $[h']$.\nAlso, every member of $\\cF$ should intersect $[h']$, hence $\\cF$ is a subfamily of $\\mathcal{B}_H$. Then $\\mathcal{B}_H$ is the unique maximum.\n\nFinally, if $h=n\/2$, then the only non-zero coordinate is $\\binom{n-1}{n\/2-1}$ at coordinate $n\/2$. This vector is an extreme point of the class of intersecting families, thus it is an extreme point of this smaller family as well.\n\\end{proof}\n\n\n\\begin{lem} The elements of $\\Gamma_c$ are extreme points of the non-trivial intersecting families.\n\n\\end{lem}\n\n\\begin{proof} These are the elements of $\\Gamma_a$ which correspond to non-trivial intersecting families. They are extreme points of the larger set (of all intersecting families), thus they are extreme points of the smaller set as well.\n\\end{proof}\n\n\n\\begin{lem} The elements of $\\Gamma_d$ are extreme points of the non-trivial intersecting families.\n\n\\end{lem}\n\n\\begin{proof}\n\nFor ${\\bf d}_H\\in \\Gamma_c$ we define the family $$\\mathcal{D}_H=\\cup_{i\\in H,\\, i\\neq h'} HM(i,h')\\cup \\{[h']\\}.$$ It is the same as $\\mathcal{B}_H$, except we removed most of the $h'$-element sets. Let $w$ be the weight function described in the proof of Lemma \\ref{gammab}. We use almost the same weight here, we set $w'(h')=-1$ and $w'(i)=w(i)$ for every $i \\neq h'$. Just as in  Lemma \\ref{gammab}, we need to have the largest number of $h$-element sets in the family $\\cF$ of maximum weight, that is $|HM(h,h')|$. Then we need an $h'$-element set in $\\cF$. Without loss of generality, $\\cF_h=HM(h,h')$ and $\\cF_{h'}\\supset\\{[h']\\}$. However, there is no point in having more $h'$-element sets in $\\cF$. For every other $i\\in H$, there are at most $|HM(i,h')|$ sets of size $i$ in $\\cF$, completing the proof.\n\\end{proof}\n\n\\begin{lem}\\label{negati} The elements of $\\Gamma_e$ are extreme points of the non-trivial intersecting families.\n\n\\end{lem}\n\n\\begin{proof}\nObserve first that if ${\\bf v}=(e_0,\\dots, e_n)\\in P$ and we increase $e_i$, the resulting vector is in $P$, as we cannot violate any of the properties. It means that the extreme points of $P$ are minimal. More precisely if ${\\bf e}$ is an extreme point and ${\\bf e'} \\ge {\\bf e}$ (with ${\\bf e'} \\neq {\\bf e}$), then ${\\bf e'}$ cannot be an extreme point, as it is a convex combination of the following two elements of $P$: ${\\bf e}$ and $2{\\bf e'}-{\\bf e}$. By Lemma \\ref{ujabb}, ${\\bf e}$ is the profile vector of a non-trivial intersecting family.\n\nWe need to show that the extreme points of $P$ are extreme points of the non-trivial intersecting families as well. Let $P'$ be the set of the profile vectors of those non-trivial intersecting families which do not contain $[n]$. Then $P' \\subset P$ by Lemma \\ref{ujabb}.\n\nAn extreme point ${\\bf p}$ of the larger set $P$ which also belongs to the smaller set $P'$ must be obviously extreme point of $P'$ as well. Thus there exists a weight function $w$ where it gives the maximum in $P'$. One can easily see that if we change $w(n)$ to a negative number, then ${\\bf p}$ has the largest weight among every profile vectors of non-trivial intersecting families.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The main lemma}\\label{biz}\n\n\n\n\n\n\n\n\n\n\nOur most important special case is when there are only two non-empty levels $1<i< m<n$ with $m> n\/2$ and $i+m \\le n$. For other values of $i$ and $m$, it is going to be easy to see that Theorem \\ref{main} holds (we do it inside the proof of the main theorem in Section \\ref{mainbiz}). Thus, the lemma below contains the most complicated part of the proof.\n\n\\begin{lem}\\label{ketto} Let $(f_0, f_1, f_2, \\dots, f_n)$ be the profile vector of a non-trivial intersecting family $\\mathcal{F}$. Let us assume that $m$ is the maximum cardinality in $\\mathcal{F}$, $m> n\/2$ and $i+m \\le n$. Then there is a $0\\le \\lambda\\le 1$ such that $(f_i,f_m) \\le \\lambda (0, \\binom{n}{m})+(1-\\lambda)(|HM(i,m)|,|HM(m,m)|)$.\n\\end{lem}\n\n\nWe will use the following simple observations.\n\n\n\n\\begin{prop}\\label{triv}\n\\textbf{(i)} If $x\\le y$, then $\\binom{x}{k-1}\/\\binom{x}{k}\\ge \\binom{y}{k-1}\/\\binom{y}{k}$.\n\n\\textbf{(ii)} Let $0\\le c'$, $0<\\alpha,a,b,c,b'$ with $bc'\\le cb'$, $b\/c\\le\\alpha$ and $c\\ge c'$. Then \n\\[\\frac{\\alpha a+b}{a+c}\\le \\frac{\\alpha a+b'}{a+c'}.\\]\n\\end{prop}\n\n\n\\begin{proof} The first statement easily follows from the definition of $\\binom{x}{k}$.\n\n\nBy rearranging the desired inequality of $\\textbf{(ii)}$, we obtain the equivalent form  $\\alpha ac'+ab+bc'\\le \\alpha ac+ab'+cb'$. Recall that we have $bc'\\le cb'$. The other terms can be rewritten as $\\frac{b-b'}{c-c'}\\le\\alpha$. We have $\\frac{b-b'}{c-c'}\\le\\frac{b-bc'\/c}{c-c'}=\\frac{b(c-c')\/c}{c-c'}=b\/c\\le\\alpha$.\n\\end{proof}\n\nNow we are ready to prove Lemma \\ref{ketto}.\n\n\\begin{proof}[Proof of Lemma \\ref{ketto}]\n\nWe use induction on $n-m-i$. Observe that for the base case $i+m=n$ we have that $HM(i,m)\\cup HM(m,m)$ consists of all the $i$-sets and $m$-sets containing $n$, except that it contains $[m]$ instead of its complement. Thus $HM(i,m)\\cup HM(m,m)$ has $\\binom{n}{i}$ members, just like any maximal non-trivially intersecting family on these two levels. Let us choose $\\lambda=\\frac{|HM(i,m)|-f_i}{|HM(i,m)|}$, then by definition $f_i\\le (1-\\lambda)|HM(i,m)|$, and we need \\[f_m\\le \\lambda\\binom{n}{m}+(1-\\lambda)|HM(m,m)|=\\binom{n}{m}-\\frac{f_i\\binom{n}{m}}{|HM(i,m)|}+\\frac{f_i|HM(m,m)|}{|HM(i,m)|}=\\binom{n}{m}-f_i.\\]\nThis holds for every intersecting family, even the trivial one. For non-trivial intersecting families, we have $f_i\\le |HM(i,m)|$ by Lemma \\ref{observ}, thus we have $\\lambda\\ge 0$, completing the proof of the base step.\n\nLet us continue with the induction step. Let us consider $\\nabla\\cF_i$, which is the shade of $\\cF_i$ and let $g_{i+1}=|\\nabla\\cF_i|$. Then $\\nabla\\cF_i\\cup\\cF_m$ is obviously non-trivially intersecting, thus by the induction hypothesis there is a $0\\le \\lambda\\le 1$ such that $(g_{i+1},f_m) \\le \\lambda (0, \\binom{n}{m})+(1-\\lambda)(|HM(i+1,m)|,|HM(m,m)|)$. We will show that the same $\\lambda$ works for $f_i$, i.e. $(f_i,f_m) \\le \\lambda (0, \\binom{n}{m})+(1-\\lambda)(|HM(i,m)|,|HM(m,m)|)$. As the values in coordinate $m$ do not change, all we need to prove is that $f_i\\le (1-\\lambda)|HM(i,m)|$ if $g_{i+1}\\le (1-\\lambda)|HM(i+1,m)|$. It is enough to show  that$f_i\/|HM(i,m)|\\le g_{i+1}\/|HM(i+1,m)|$, or equivalently $g_{i+1}\/f_i\\ge |HM(i+1,m)|\/|HM(i,m)|$. As $HM(i+1,m)=\\nabla HM(i,m)$,\nthe last of the above inequalities means that the size of the shade of $\\cF_i$ is proportionally the smallest if $\\cF_i$ is $HM(i,m)$.\n\nWe will use the Kruskal-Katona theorem. To use it in the form we have stated it, we will consider the complement family, as the shade of a family is the shadow of its complement.\n\nObserve that $\\overline{HM(i,m)}$ is an initial segment of the colex ordering if we reorder the elements of $[n]$. Indeed, members of $\\overline{HM(i,m)}$ completely avoid a given element $z$, and then we take all the $(n-i)$-sets but those that contain an $m$-element set $B$. By reordering, we can assume that $z=n$ and $B=\\{n-m,\\dots,n-1\\}$. The sets containing $n$ are the last in the colex order, and a superset $F$ of $B$ cannot be before a set $G\\in\\overline{HM(i,m)}$, as the largest element of $F\\setminus G$ is in $B$, while every element of $G\\setminus F$ is less than $n-m$.\n\n\nThe cascade form of $|\\overline{HM(i,m)}|$ is $\\binom{n-2}{n-i}+\\binom{n-3}{n-i-1}+\\binom{n-4}{n-i-2}+\\dots+\\binom{n-m}{n-i-m+2}=\\sum_{j=2}^m\\binom{n-j}{n-i-j+2}$. Let $\\cG$ be a non-empty $(n-i)$-uniform family \nwith $|\\cG|< |\\overline{HM(i,m)}|$ and cascade form $|\\cG|=\\sum_{j=2}^{m'}\\binom{n_j}{n-i-j+2}$. Observe that $n_2\\le n-2$. This implies that for any $h$, $n_h\\le n_h$.\n\n\n\n\n\nWe partition $\\overline{HM(i,m)}$ into $m-1$ parts: $\\cH_2$ consists of the first $\\binom{n-2}{n-i}$ sets of $\\overline{HM(i,m)}$ in the colex order, $\\cH_3$ consists of the next $\\binom{n-3}{n-i-1}$ sets, and so on. $\\cH_j$ for $j\\le m$ consists of $\\binom{n-j}{n-i-j+2}$ sets that come after $\\cH_2,\\dots, \\cH_{j-1}$, i.e. after the first $\\binom{n-2}{n-i}+\\binom{n-3}{n-i-1}+\\binom{n-4}{n-i-2}+\\dots+\\binom{n-m}{n-i-m+2}$ sets in the colex order.\nWe also partition $\\cG$ into $m-1$ parts: for $2\\le j<m$, $\\cG_j$ similarly consists of $\\binom{n_j}{n-i-j+2}$ sets of $\\cG$ that come after $\\cG_2,\\dots, \\cG_{j-1}$ in the colex order. Then $\\cG_m$ consists of all the remaining $\\sum_{j=m}^{m'}\\binom{n_j}{n-i-j+2}$ sets of $\\cG$. Let us note that $\\cG_j$ can be empty if $j>2$.\n\n\n\nLet us assume that $n_2=n-2$, $n_3=n-3$,...,$n_{h}=n-(h)$ and $n_{h+1}< n-h-1$. \nLet $\\cH^*=\\cup_{j=1}^h \\cH_j$, $\\cH^{**}=\\cup_{j=h+1}^m \\cH_j$, $\\cG^*=\\cup_{j=1}^m \\cG_j$, $\\cG^{**}=\\cup_{j=h+1}^{m'} \\cG_j$. Observe that we have $|\\cH^*|=|\\cG^*|$ and $|\\Delta\\cH^*|\\le |\\Delta \\cG^*|$ since $\\cH^*$ is an initial segment of the colex ordering. We also have $|\\cH^{**}|\\ge \\binom{n-h-1}{n-i-h+1}$ and $|\\cG^{**}|<\\binom{n-h-1}{n-i-h+1}$.\n\nLet $a:=|\\cH^*|$, $c:=|\\cH^{**}|$, $\\alpha=|\\Delta\\cH^*|\/|\\cH^*|$, $b:=|\\Delta\\cH^{**}\\setminus\\Delta\\cH^*|$, $b'=|\\Delta\\cG|-|\\Delta\\cG^*|$, $c':=|\\cG^{**}|$ and $\\alpha'=|\\Delta\\cG^*|\/|\\cG^*|$. Our goal is to apply \\textbf{(ii)} of Proposition \\ref{triv}. By the above, we have $c>c'$. Now we will show that the other conditions are satisfied as well.\n\nWe let $p_\\ell:=\\binom{n-\\ell}{n-i-\\ell+2}=|\\Delta\\cH_\\ell\\setminus\\Delta\\bigcup_{\\ell'=2}^{\\ell-1}\\cH_{\\ell'})|$, i.e. the number of sets added to the shadow of $\\bigcup_{\\ell'=2}^{\\ell}\\cH_{\\ell'}$ by $\\cH_\\ell$.\nObserve first that $p_\\ell\/|\\cH_\\ell|=(n-i-\\ell+2)\/(i-1)$, thus $p_\\ell\/|\\cH_\\ell|$ decreases as $\\ell$ increases. This implies that\n$p_\\ell\/|\\cH_\\ell|\\le p_{h+1}\/|\\cH_{h+1}|$ for every $\\ell>h+1$. Therefore, we have that \\begin{equation}\\label{ineq0} \\frac{b}{c}\\frac{|\\Delta\\cH^{**}\\setminus\\Delta\\cH^*|}{|\\bigcup_{\\ell=h+1}^{m} \\cH_\\ell|}=\\frac{\\sum_{\\ell=h+1}^{m}p_\\ell}{|\\bigcup_{\\ell=h+1}^{m} \\cH_\\ell|}\\le \\frac{\\frac{p_{h+1}}{|\\cH_{h+1}|}|\\bigcup_{\\ell=h+1}^{m} \\cH_\\ell|}{|\\bigcup_{\\ell=h+1}^{m} \\cH_\\ell|}=\\frac{p_{h+1}}{|\\cH_{h+1}|}.\n\\end{equation}\n\nSimilarly, we have that \\[\\alpha=\\frac{|\\Delta\\cH^*|}{|\\cH^*|}=\\frac{|\\Delta\\cup_{j=1}^h \\cH_j|}{|\\cup_{j=1}^h \\cH_j|}=\\frac{\\sum_{j=1}^{h}p_j}{|\\bigcup_{j=1}^{h} \\cH_j|}\\ge \\frac{\\frac{p_{h+1}}{|\\cH_{h+1}|}|\\bigcup_{j=1}^{h} \\cH_j|}{|\\bigcup_{j=1}^{h} \\cH_j|}=\\frac{p_{h+1}}{|\\cH_{h+1}|}\\ge\\frac{b}{c}, \\]\nwhere the last inequality uses (\\ref{ineq0}).\n\nLet $x< n-h-1$ be defined by $\\binom{x}{n-i-h+1}:=\\binom{n_{h+1}}{n-i-h+1}+\\binom{n_{h+2}}{n-i-h}+\\ldots+\\binom{n_{m'}}{n-i-m'+2}=|\\bigcup_{\\ell=h+1}^{m'} \\cG_\\ell|$. We have $|\\Delta\\cG|\\ge |\\Delta\\cH^*|+\\binom{n_{h+1}}{n-i-h}+\\binom{n_{h+2}}{n-i-h-1}+\\ldots+\\binom{n_{m'}}{n-i-m'+1}$ by the Kruskal-Katona theorem. We claim that \n\n\\begin{equation}\\label{ineq1} \\binom{n_{h+1}}{n-i-h}+\\binom{n_{h+2}}{n-i-h-1}+\\ldots+\\binom{n_{m'}}{n-i-m'+1}\\ge \\binom{x}{n-i-h}.\n\\end{equation}\n\nIndeed, the left hand side is the sharp lower bound on the size of the shadow of an $(n-i-h+1)$-uniform family of size $\\binom{x}{n-i-h+1}$ by the Kruskal-Katona theorem, while the right hand side is the not necessarily sharp lower bound on the size of the  same family by Lov\\'asz's version of the shadow theorem. We have $\\frac{b'}{c'}=\\frac{\\binom{n_{h+1}}{n-i-h}+\\binom{n_{h+2}}{n-i-h-1}+\\ldots+\\binom{n_{m'}}{n-i-m'+1}}{\\binom{x}{n-i-h+1}}\\ge\\frac{\\binom{x}{n-i-h}}{\\binom{x}{n-i-h+1}}\\ge \\frac{\\binom{n-h-1}{n-i-h}}{\\binom{n-h-1}{n-i-h+1}}=p_{h+1}\/|\\cH_{h+1}|\\ge \\frac{b}{c}$. In the inequalities here we used (\\ref{ineq1}) first, then \\textbf{(i)} of Proposition \\ref{triv} and finally (\\ref{ineq0}). \n\nNow we can apply \\textbf{(ii)} of Proposition \\ref{triv} to show that\n$\\frac{\\alpha a+b}{a+c}\\le \\frac{\\alpha a+b'}{a+c'}\\le \\frac{\\alpha' a+b'}{a+c'}$.\nThis means\n$|\\Delta\\overline{HM(i,m)}|\/|\\overline{HM(i,m)}|\\le|\\Delta \\cG|\/|\\cG|$. By taking the complements, we obtain that $|\\nabla HM(i,m)|\/|HM(i,m)|\\le |\\nabla \\cG'|\/|\\cG'|$ for any $i$-uniform family $\\cG'$ with $|\\cG'|\\le |HM(i,m)|$. In particular, $|\\nabla HM(i,m)|\/|HM(i,m)|\\le g_{i+1}\/f_i$, completing the proof.\n\n\n\\end{proof}\n\n\\section{Proof of the main theorem}\\label{mainbiz}\n\nIn this section we finish the proof of Theorem \\ref{main}.\nIt is easy to see that we can consider only families not containing $[n]$.\nIt is enough to show that if a profile vector ${\\bf p}$ of a non-trivial intersecting family $\\mathcal{F}$ gives the unique maximum for a weight function $w$, then ${\\bf p}\\in \\Gamma_b \\cup \\Gamma_c \\cup \\Gamma_d\\cup \\Gamma_e$.\n\n\n\nAn important observation is that if $F\\in\\cF$, $F \\subset G$ and $G$ has positive weight, then $G$ is in the maximum family (as adding it would not violate any of the properties). In the proof we often start with fixing the maximum size $m$ of members, it implies that larger sets (except possibly $[n]$) do not have positive weight. Note that if $w(m)>0$, then $\\cF_m$ is non-trivial intersecting. Indeed, if $\\cF_m$ is trivial, then all its members contain a given element $x$ and there is a set $F\\in\\cF$ of smaller size not containing $x$. But then all the $m$-element sets which contain $F$ are in $\\cF$, even those which do not contain $x$, a contradiction.\n\n\nWe continue the proof with a case analysis.\n\n{\\bf Case 1. $w(i)\\le 0$ for every $1<i<n$.}\n\n\\medskip\n\n{\\bf Case 1a. $w(i)< 0$ for every $1<i<n$.}\n\nObviously ${\\bf p \\in P}$, as in all the cases, by Lemma \\ref{ujabb}. But in this special case we will show that ${\\bf p}$ is also an extreme point of $P$, thus it is in $\\Gamma_e$. \n\nIf ${\\bf p}$ is not an extreme point of $P$, i.e. there is an element ${\\bf p'}$ of $P$ with larger weight, then either ${\\bf p'}$ also corresponds to a non-trivial intersecting family (a contradiction), or there is another ${\\bf p''}\\in P$ with ${\\bf p''} \\le {\\bf p'}$ by Lemma \\ref{ujabb}. But then ${\\bf p''}$ has even larger weight. As ${\\bf p''}$ cannot correspond to a non-trivial intersecting family, there is an even smaller vector in $P$ (with even larger weight). It cannot continue forever, as each coordinate is a non-negative integer. We arrive to a vector which does correspond to a non-trivial intersecting family, hence has larger weight than ${\\bf p}$, a contradiction.\n\n\\smallskip\n{\\bf Case1b. $w(i)=0$ for some $1<i<n$.}\n\nObviously $w({\\bf p})\\le 0$, and $w(HM(i))=0$. The profile of $HM(i)$ is in $\\Gamma_b$.\n\n\\bigskip\n\n{\\bf Case 2. $w(i)>0$ for some $1<i<n-1$.}\n\nLet $m$ be the maximum size in $\\mathcal{F}$. We will use Lemma \\ref{observ} several times.\n\n\\medskip\n\n{\\bf Case 2a. $w(m)<0$.}\n\nThere is an $m$-element set $F$ in $\\mathcal{F}$. Obviously the only reason it is in the family is that without it the family would be trivial, hence every other member of $\\mathcal{F}$ contains a fixed point. Then for every level $i$ the maximum weight is given either by the empty family (in case $w(i)\\le 0$), or $HM(i,m)$. Then we take the union of these uniform families (on every level $i$, the empty family of $HM(i,m)$), and we add $[m]$. The resulting family is non-trivial intersecting, and its profile vector is in $\\Gamma_d$.\n\n\\medskip\n\n{\\bf Case 2b. $w(m)\\ge 0$ and $m \\le n\/2$.}\n\nFor every level below $m$, the maximum of $w(\\mathcal{F}_i)$ is either $0$ or given by $HM(i,m)$. In particular, on level $m$ clearly $HM(m)=HM(m,m) \\cup \\{[m]\\}$ gives the maximum weight. This already makes sure the family is non-trivial intersecting, hence for every other level $j$ with $j<m$ we can choose $HM(j,m)$ or the empty family, depending on whether $w(j)$ is positive or not. The union of these uniform families is non-trivial intersecting, and has the largest possible weight on every level up to $m$. Its profile is in $\\Gamma_b$.\n\n\n\n\\medskip\n\n{\\bf Case 2c. $w(m)\\ge 0$ and $m>n\/2$.}\n\n\n\n\n\n\n\nLet $m_0$ be the size of the smallest member of the family $\\mathcal{F}$.\n\n\\smallskip\n\n{\\bf Case 2c1. $w(m)\\ge 0$, $m>n\/2$ and $m+m_0> n$.}\n\nLet us consider the following modified weight function. Let $w'(i)$ be the same as $w(i)$ if $m_0 \\le i \\le m$ and negative otherwise. Obviously the maximum non-trivial intersecting family for $w'$ is also $\\mathcal{F}$. Let us examine the intersecting family $\\cF'$ with maximum weight $w'$ now. One can easily see using Theorem \\ref{metszo} that the profile of $\\cF'$ can be obtained from ${\\bf a_{m_0}}$ by changing some coordinates to 0. If $w(m)=0$, then $\\cF'$ might contain no $m$-element sets, but even in this case we can add every $m$-element set to $\\cF'$ without decreasing the weight (and without ruining the intersecting property). The resulting family $\\mathcal{F}''$ is non-trivial intersecting, and $w'(\\mathcal{F}'')=w'(\\mathcal{F}') \\ge w'(\\mathcal{F}) \\ge w(\\mathcal{F})$, thus $\\mathcal{F}''$ must have the same profile as $\\mathcal{F}$. The profile of $\\mathcal{F}''$ is in $\\Gamma_c$.\n\n\\smallskip\n\n{\\bf Case 2c2. $w(m)\\ge 0$, $m>n\/2$ and $m+m_0 \\le n$.}\n\nLet $H$ be the set of non-empty levels. Recall that coordinate $i$ of ${\\bf a}$ is $0$ if $i<n\/2$, $\\binom{n-1}{i-1}$ if $i=n\/2$ and $\\binom{n}{i}$ if $i>n\/2$. Let ${\\bf a'}$ be the vector we get from ${\\bf a}$ when we change the coordinates not in $H$ to 0. We will show that ${\\bf p}={\\bf b}_H$, by showing that there is a $\\lambda$ such that $\\lambda {\\bf b}_H + (1-\\lambda){\\bf a'} \\ge {\\bf p}$. We have that ${\\bf b}_H$ and ${\\bf a'}$ are both 0 in the negative coordinates, thus the weight of either ${\\bf b}_H$, or ${\\bf a'}$ is at least as large as the weight of ${\\bf p}$. But that was the unique maximum, thus ${\\bf p}$ is equal to either ${\\bf b}_H$, or ${\\bf a'}$. As ${\\bf p}$ has a non-zero coordinate below $n\/2$, ${\\bf p}$ cannot be equal to ${\\bf a'}$.\n\nLet $i \\le n\/2$ be such that $f_i\/|HM(i,m)|=:\\lambda$ is maximal. Then $\\lambda {\\bf b}_H$ has at least $f_j$ in coordinate $j$ for every $j \\le n\/2$. \nLet us consider now a coordinate $k>n\/2$ with $w(k)>0$.\n\nIf the family $\\cF_i\\cup\\cF_k$ is trivially intersecting, then $f_k\\le |HM(k,m)|$, while ${\\bf b}_H$ and ${\\bf a'}$ both have at least $|HM(k,m)|$ in coordinate $k$, thus so does $\\lambda {\\bf b}_H + (1-\\lambda){\\bf a'}$, completing the proof.\n\nIf the family $\\cF_i\\cup\\cF_k$ is non-trivially intersecting, we can apply Lemma \\ref{ketto}. It implies that there is a $\\lambda'$ such that $(f_i,f_k)\\le ((1-\\lambda')|HM(i,k)|,\\lambda'\\binom{n}{k}+(1-\\lambda')|HM(k,k)|)$. Coordinate $i$ shows that $((1-\\lambda')|HM(i,k)|\\ge \\lambda |HM(i,m)|$. Since $|HM(i,m)|\\ge |HM(i,k)|$, this implies that $\\lambda \\le 1-\\lambda'$. Consider now coordinate $k$. We have \\begin{equation}\\label{equa}\\tag{$\\star$}\n    f_k\\le \\lambda'\\binom{n}{k}+(1-\\lambda')|HM(k,k)|\\le \\lambda'\\binom{n}{k}+(1-\\lambda')|HM(k,m)|.\n\\end{equation} Since $|HM(k,m)|\\le \\binom{n}{k}$, increasing $\\lambda'$ increases the right hand side of (\\ref{equa}). Since $\\lambda'\\le 1-\\lambda$, the right hand side is at most $(1-\\lambda)\\binom{n}{k}+\\lambda|HM(k,m)|$, which is coordinate $k$ of $\\lambda {\\bf b}_H + (1-\\lambda){\\bf a'}$, completing the proof.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection*{Search and NP Problems.} \\label{search}\n\nLet us compare the inversion problems with another type -- the search problems\nspecified by computable\\\\ in time $\\|x\\|^{O(1)}$ relations $P(x,w)$:\ngiven $x$, find $w$ s.t. $P(x,w)$. There are two parts to a search problem:\\\\\n (a) decision problem: decide if $w$ (called \\trm{witness}) exist, and (b) a\nconstructive problem: actually find $w$.\n\nAny inversion problem is a search problem and any search problem can be\nrestated as an inversion problem. E.g., finding a Hamiltonian cycle $C$ in a\ngraph $G$, can be stated as inverting a $f(G,C)$, which outputs $G,0\\ldots 0$\nif $C$ is in fact a Hamiltonian cycle of $G$. Otherwise, $f(G,C) = 0\\ldots 0$.\n\nSimilarly any search problem can be reduced to another one equivalent to\nits decision version.\\\\ For instance, factoring $x$ reduces to bounded\nfactoring: given $x,b$ find $p,q$ such that $pq=x$, $p\\le b$\\\\\n(where decisions yield construction by binary search).\n\n{\\bf Exercise:} Generalize the two above examples to reduce any search\nproblem to an inverting problem and to a decision problem.\n\nThe \\trm {language} of a problem is the set of all acceptable inputs.\nFor an inversion problem it is the range of $f$. For a search problem it is\nthe set of all $x$ s.t. $P(x,w)$ holds for some $w$. An \\trm {NP language} is\nthe set of all inputs acceptable by a P-time \\trm {non-deterministic} Turing\nMachine (sec.~\\ref{gm-reduce}). All three classes of languages -- search,\ninversion and NP -- coincide (NP $\\iff$ search is straightforward).\n\nInterestingly, polynomial {\\em space} bounded deterministic and\nnon-deterministic TMs have equivalent power. It is easy to modify TM to have\na unique accepting configuration. Any acceptable string will be accepted in\ntime $2^s$, where $s$ is the space bound. Then we need to check $A(x,w,s,k)$:\nwhether the TM can be driven from the configuration $x$ to $w$ in time $<2^k$\nand space $s$. For this we need for every $z$, to check $A(x,z,s,k{-}1)$\nand $A(z,w,s,k{-}1)$, which takes space $t_k\\le t_{k{-}1} + \\|z\\|+O(1)$.\nSo, $t_k= O(sk)= O(s^2)$ \\cite{Sv}.\n\nSearch problems are games with P-time transition rules and one move duration.\nA great hierarchy of problems results from allowing more moves\nand\/or other complexity bounds for transition rules.\n\n\\newpage\\subsection{Complexity of NP Problems.}\\label{compl}\n\nWe discussed the (equivalent) inversion, search, and NP types of problems.\nNobody knows whether {\\em all} such problems are solvable in P-time (i.e.\nbelong to P). This question (called P=?NP) is probably the most famous one\nin Theoretical Computer Science. All such problems are solvable in exponential\ntime but it is unknown whether any better algorithms generally exist. For many\nproblems the task of finding an efficient algorithm may seem hopeless,\nwhile similar or slightly modified problems have been solved. Examples:\n\n \\begin{enumerate} \\itemsep0pt\n \\item Linear Programming: Given integer $n\\times m$\n  matrix $A$ and vector $b$, find a rational vector $x$ with $Ax<b$.\n  Note, if $n$ and entries in $A$ have $\\le k$-bits\n  and $x$ exists then an $O(nk)$-bit $x$ exists, too.\n\nSolution: The Dantzig's \\trm {Simplex} algorithm finds $x$\nquickly for many $A$.\\\\ Some $A$, however, take exponential time.\nAfter long frustrating efforts, a worst case\\\\\nP-time Ellipsoid Algorithm was finally found in \\cite{YN}.\n\n\\item Primality test: Determine whether a given integer $p$ has a factor?\n\nSolution: A bad (exponential time) way is to try\nall $2^{\\|p\\|}$ possible integer factors of $p$.\\\\\nMore sophisticated algorithms, however, run fast (see section~\\ref{prime}).\n\n\\item Graph Isomorphism Problem: Are two given graphs $G_1,G_2$, isomorphic?\\\\\n I.e., can the vertices of $G_1$ be re-numbered so that it becomes equal $G_2$?\n\nSolution: Checking all $n!$ enumerations of vertices is impractical\\\\\n (for $n=100$, this exceeds the number of atoms in the known Universe).\\\\\n \\cite{Luks} found an $O(n^d)$ steps algorithm where $d$ is the degree.\n This is a P-time for $d = O(1)$.\n\n\\item Independent Edges (Matching):\\\\ Find a given number of\n independent (i.e., not sharing nodes) edges in a given graph.\n\nSolution: Max flow algorithm solves a bipartite graph case.\\\\ The general case\nis solved with a more sophisticated algorithm by J. Edmonds. \\end{enumerate}\n\nMany other problems have been battled for decades or centuries\nand no P-time solution has been found. Even modifications of\nthe previous four examples have no known answers:\n\n \\begin{enumerate}\\itemsep0pt\n \\item Linear Programming: All known solutions produce rational $x$.\\\\\n No reasonable algorithm is known to find integer $x$.\n \\item Factoring: Given an integer, find a factor.\n Can be done in about exponential time $n^{\\sqrt{n}}$.\\\\\n Seems very hard: Centuries of quest for fast algorithm were unsuccessful.\n \\item Sub-graph isomorphism: In a more general case of finding isomorphisms\n of a graph\\\\ to a part of another, no P-time solution has been found,\n even for $O(1)$-degree graphs.\n \\item Independent Nodes: Find $k$ independent\n (i.e., not sharing edges) nodes in a given graph.\\\\\n No P-time solution is known. \\end{enumerate}\n\n{\\bf Exercise:}\n Restate the above problems as inverting easily computable functions.\n\nWe learned the proofs that Linear Chess and some other games have exponential\ncomplexity. None of the above or any other search\/inversion\/NP problem,\nhowever, have been proven to require super-P-time.\nWhen, therefore, do we stop looking for an efficient solution?\n\n\\paragraph {NP-Completeness} theory is an attempt to answer this question.\\\\\nSee results by S.Cook, R.Karp, L.Levin, and others surveyed in \\cite{GJ,trakh}.\n\nA P-time function $f$ reduces one NP-predicate $p_1(x)$ to $p_2(x)$ iff $p_1(x)\n= p_2(f(x))$, for all $x$. $p_2$ is NP-complete if {\\em all} NP problems can\nbe reduced to it. Thus, each NP-complete problem is at least as worst-case\nhard as all other NP problems. This may be a good reason to give up on fast\nalgorithms for it. Any P-time algorithm for one NP-complete problem would yield\none for all other NP (or inversion, or search) problems. No such solution\nhas been discovered yet and this is left as a homework (10 years deadline).\n\nFaced with an NP-complete problem we can sometimes restate it, find a similar\none which is easier (possibly with additional tools) but still gives the\ninformation we really want. We will do this in Sec.~\\ref{prime} for factoring.\nNow we proceed with an example of NP-completeness.\n\n\\newpage\\subsection {An NP-Complete Problem: Tiling.}\\label{tile}\n\n\\parbox{32pc} {{\\bf Tiling Problem.} Invert the function which, given a tiled\nsquare, outputs its first row and the list of tiles used. A tile is one of\nthe $26^4$ possible squares containing a Latin letter at each corner. Two tiles\nmay be placed next to each other if the letters on the shared side match. \n(See an example at the right.) We now reduce to Tiling any search problem:\ngiven $x$, find $w$ satisfying a P-time computable property $P(x,w)$.}\n \\hspace{9pt} \\parbox{6pc} {\\mbox\n {\\fbox{\\parbox{2pc} {\\makebox[2pc] {a\\hfill x} \\makebox[2pc] {m\\hfill r}}}\n \\fbox{\\parbox{2pc} {\\makebox[2pc] {x\\hfill c} \\makebox[2pc] {r\\hfill z}}}}\n \\par \\vspace {3pt} \\mbox\n {\\fbox{\\parbox{2pc} {\\makebox[2pc] {m\\hfill r} \\makebox[2pc] {n\\hfill s}}}\n \\fbox{\\parbox{2pc} {\\makebox[2pc] {r\\hfill z} \\makebox[2pc] {s\\hfill z}}}}}\n\n\\paragraph {Padding Argument.} First, we need to reduce it to some\n``standard\" NP problem. An obvious candidate is the problem ``Is there $w:\nU(v,w)$ ?\", where $U$ is the universal Turing Machine, simulating $P(x,w)$ for\n$v = px$. But $U$ does not run in P-time, so we must restrict $U$ to $u$ which\nstops within some P-time limit. How to make this fixed degree limit sufficient\nto simulate any polynomial (even of higher degree) time $P$? Let the TM\n$u(v,w)$ for $v{=}00\\ldots01px$ simulate $\\|v\\|^2$ steps of $U(px,w){=}P(x,w)$.\nIf the \\trm {padding} of $0$'s in $v$ is sufficiently long, $u$ will have\nenough time to simulate $P$, even though $u$ runs in quadratic time, while\n$P$'s time limit may be, say, cube (of a shorter ``un-padded\" string).\nSo the NP problem $P(x,w)$ is reduced to $u(v,w)$ by mapping instances $x$\ninto $f(x)= 0\\ldots01px=v$, with $\\|v\\|$ determined by the time limit for $P$.\nNotice that program $p$ for $P$ is fixed. So, if some NP problem {\\em cannot}\nbe solved in P-time then neither can be the problem $\\exists? w: u(v,w)$.\nEquivalently, if the latter {\\em IS} solvable in P-time then so is {\\em any}\nsearch problem. We do not know which of these alternatives is true.\nIt remains to reduce the search problem $u$ to Tiling.\n\n\\paragraph {The Reduction.} We compute $u(v,w)$ (where $v = 00\\ldots01px$)\nby a TM represented as an array of 1-pointer cellular automata that runs\nfor $\\|v\\|^2$ steps and stops if $w$ does NOT solve the relation $P$.\nOtherwise it enters an infinite loop. An instance $x$ has a solution\niff $u(v,w)$ runs forever for some $w$ and $v = 0\\ldots01px$.\n\n\\noindent \\parbox{297pt} {Here is the space-time diagram of computation of\n$u(v,w)$. We set $n$ to $u$'s time (and space) $\\|v\\|^2$. Each row in this\ntable represents the configuration of $u$ in the respective moment of time.\n The solution $w$ is filled in at the second step below a special symbol \"?\".\nIf a table is filled in wrongly, i.e. doesn't reflect any actual computation,\nthen it must have four cells sharing a corner that couldn't possibly\nappear in the computation of $u$ on any input.}\n \\hspace{1pc} \\parbox{13pc} {\\vector(1,0){89}~space:~$n{=}\\|v\\|^2$\\\\\\mbox\n {\\raisebox{3pc}{\\vector(0,-1){50}}\\hspace{-10pt}\\raisebox{-2pc}{time} \\fbox\n {\\parbox{11pc}{\\makebox[11pc]{$v$ ?\\ldots? \\#\\ldots\\#\\hfill(init.config.)}\n\\makebox[11pc]{\\makebox[2pc]{$v$\\hfill$w$}\\hfill $T_1$\\hfill\\makebox[2pc]{}}\n \\makebox[11pc]{\\vdots\\hfill\\vdots\\hfill\\vdots} \\makebox[11pc]{$T_n$}}}}}\n\n\\paragraph {Proof.} As the input $v$ and the guessed solution $w$ are\nthe same in both the right and the wrong tables, the first 2 rows agree.\nThe actual computation starts on the third row. Obviously, in the first\nmismatching row a transition of some cell from the previous row is wrong.\nThis is visible from the state in both rows of this cell and the cell it points\nto, resulting in an impossible combination of four cells sharing a corner.\n\n\\noindent\\parbox{383pt} {\\hspace*{1pc} For a given $x$, the existence of $w$\nsatisfying $P(x,w)$ is equivalent to the existence of a table with the\nprescribed first row, no halting state, and permissible patterns of each four\nadjacent squares (cells). Converting the table into the \\trm{Tiling Problem}:\\\\\nThe cells in the table are separated by ``---\" ; the tiles by ``...\";\nCut each cell into 4 parts by a vertical and a horizontal lines\nthrough its center and copy cell's content in each part.\nCombine into a tile each four parts sharing a corner of 4 cells.\nIf these cells are permissible in the table, then so is the respective tile.}\n \\hspace{2pc} $\\raisebox{-18pt}{\\dashbox{1}(40,40){\\parbox{2.5pc}\n {\\makebox[2.5pc]{u\\hfill v}\\\\\\hspace*{2.5pc}\\\\\\makebox[2.5pc]{v\\hfill x}}}}\n\\hspace{-57pt}\\parbox{75pt} {\\framebox[3pc]{\\rule{0pt}{2pc}}\n\\framebox[3pc]{\\rule{0pt}{2pc}}\\vspace{3pt} \\framebox[3pc]{\\rule{0pt}{2pc}}\n\\framebox[3pc]{\\rule{0pt}{2pc}}}$\n\n\\vspace{6pt} So, any P-time algorithm extending a given first row\nto the whole table of matching tiles from a given set could be\nused to solve any NP problem by converting it to Tiling as shown.\n\n\\paragraph {Exercise:}\nFind a polynomial time algorithm for $n\\times\\log n$ Tiling Problem.\n\n\\newpage\\section {Probability in Computing.}\\label{prob}\n\n\\subsection {A Monte-Carlo Primality Tester.}\\label{prime}\n\nThe factoring problem seems very hard. But to test if a number has factors\nturns out to be much easier than to find them. It also helps if we supply\nthe computer with a coin-flipping device. See: \\cite{rabin,miller,so-st}.\nWe now consider a Monte Carlo algorithm, i.e. one that with high\nprobability rejects any composite number, but never a prime.\n\n\\paragraph{Residue Arithmetic.} $p|x$ means $p$ divides $x$.\n$x\\equiv y\\pmod p$ means $p|(x{-}y)$. $y=(x \\bmod p)$ denotes the residue of\n$x$ when divided by $p$, i.e. $x\\equiv y\\in[0,p{-}1]$. Residues can be added,\nmultiplied and subtracted with the result put back in the range $[0,p {-}1]$\nvia shifting by an appropriate multiple of $p$. E.g., $-x$ means $p{-}x$\nfor residues $\\bmod\\ p$. We use $\\pm x$ to mean either $x$ or $-x$.\n\nThe Euclidean Algorithm finds $\\gcd(x,y)$ -- the greatest (and divisible\nby any other) common divisor of $x$ and $y$: $\\gcd(x,0)=x$; $\\gcd(x,y)=\\gcd\n(y,(x\\bmod y))$, for $y>0$. By induction, $g{=}\\gcd(x,y){=}A*x{-}B*y$,\nwhere integers $A{=}(g\/x\\bmod y)$ and $B{=}(g\/y\\bmod x)$ are produced as\na byproduct of Euclid's Algorithm. This allows division $(\\bmod\\ p)$ by any $r$\n\\trm {coprime} with $p$, (i.e. $\\gcd(r,p){=}1$), and operations $+,-,*,\/$\nobey all usual arithmetical laws. We will need to compute $(x^q\\bmod p)$\nin polynomial time. We cannot do $q{>}2^{\\|q\\|}$ multiplications. Instead\nwe compute all numbers $x_i=(x_{i{-}1}^2\\bmod p)= (x^{2^i}\\bmod p),i<\\|q\\|$.\nThen we represent $q$ in binary, i.e. as a sum of\npowers of $2$ and multiply $\\bmod\\ p$ the needed $x_i$'s.\n\n\\paragraph{Fermat Test.} The Little Fermat Theorem for every\nprime $p$ and $x\\in[1,p{-}1]$ says: $x^{(p{-}1)}\\equiv1\\pmod p$.\\\\\n Indeed, the sequence $(xi\\bmod p)$ is a permutation of $1,\\ldots,p{-}1$.\n So, $1{\\equiv}(\\prod_{i<p}(xi))\/(p{-}1)!\\equiv x^{p-1} (\\bmod p)$.\n\n This test rejects typical composite $p$.\n Other composite $p$ can be actually factored by the following tests.\n\n\\paragraph {Square Root Test.} For each y and prime p,\n $x^2\\equiv y\\pmod p$ has at most one pair of solutions $\\pm x$.\n\n{\\bf Proof.} Let $x,x'$ be two solutions: $y\\equiv x^2\\equiv x'^2\\pmod\np$. Then $x^2-{x'}^2 = (x{-}x')(x{+}x')\\equiv0\\pmod p$.\\\\ So, $p$ divides\n$(x{-}x') (x{+}x')$ and, if prime, must divide either $(x{-}x')$ or\n$(x{+}x')$.\\\\ (Thus either $(x\\equiv x')$ or $(x\\equiv-x')$.) Otherwise\n $p$ is composite, and $\\gcd(p,x{+}x')$ {\\em actually gives} its factor.\n\n\\paragraph {Miller-Rabin Test} $T(x,p)$ completes the Fermat Test:\nit factors a composite $p$, given $d$ that \\\\\\trm {kills} Z$_p^*$\n(i.e. $x^d\\equiv\\gcd(x,p)^d\\pmod p$ for all $x$) and a random choice of $x$.\nFor prime $p$, $d=p{-}1$.\n\nLet $d{=}2^k q$, with odd $q$. $T$ sets $x_0=(x^q\\bmod p)$,\n$x_i=(x_{i{-}1}^2\\bmod p)=(x^{2^i q}\\bmod p)$, $i\\le k$. If $x_k{\\ne}1$ then\n$\\gcd(x,p){\\ne}1$ factors $p$ (if $d$ killed $x$, else Fermat test rejects\n$p{=}d{+}1$). If $x_0{=}1$, or one of $x_i$ is $-1$, $T$ gives up for this $x$.\n Otherwise $x_i{\\notin}\\{\\pm1\\}$ for some $i{<}k$, while\n $(x_i^2\\bmod p){=}x_{i+1}{=}1$, and the Square Root Test factors $p$.\n\nFirst, for each odd composite $p$, we show that $T$ succeeds with {\\em some}\n$x$, coprime with $p$. If $p{}=a^j,j{>}1$, then $x{=}(1{+}p\/a)$ works for Fermat\nTest: $(1{+}p\/a)^{p{-}1}{=}1{+}(p\/a)(p{-}1){+}(p\/a)^2(p{-}1)(p{-}2)\/2{+}\\ldots\n\\equiv 1{-}p\/a{\\not\\equiv}1\\pmod p$, since $p|(p\/a)^2$. Otherwise $p{=}ab,\n\\gcd(a,b){=}1{<}a{<}b$. Take the {\\bf greatest} $i$ such that $x_i{\\not\\equiv}1$\nfor some $x$ coprime with $p$. It exists: $(-1)^q\\equiv-1$ for odd $q$.\nSo, $(x_i)^2\\equiv 1\\not\\equiv x_i\\pmod p$. (Or $i{=}k$, so Fermat test works.)\nThen $x'{=}1{+} b(1\/b\\bmod a)(x{-}1)\\equiv1{\\equiv}x'_i\\pmod b$, while\n$x'_i{\\equiv}x_i\\pmod a$. So, either $x_i$ or $x'_i$ is $\\not\\equiv\\pm1\\pmod p$.\n\nNow, $T(y,p)$ succeeds with {\\em most} $y_i$, as it does with $x_i$ (or $x'_i$):\nthe function $y\\mapsto xy$ is 1-1 and $T$ cannot fail with both $y$ and $xy$.\nThis test can be repeated for many randomly chosen $y$. Each time $T$ fails, we\nare twice more sure that $p$ is prime. The probability of $300$ failures on a\ncomposite $p$ is $<2^{-300}$, its inverse exceeds the number of atoms in the\nknown Universe.\n\n\\newpage\\subsection\n {Randomized Algorithms and Random Inputs.}\\label{average}\n\n\\trm {Las-Vegas} algorithms, unlike Monte-Carlo, never give wrong answers.\nUnlucky coin-flips just make them run longer than expected. Quick-Sort is a\nsimple example. It is about as fast as deterministic sorters, but is popular\ndue to its simplicity. It sorts an array $a[1\\ldots n]$ of $n>2$ numbers by\nchoosing in it a random \\trm {pivot}, splitting the remaining array in two\nby comparing with the pivot, and calling itself recursively on each half.\n\nFor easy reference, rename the array entries with their positions $1,\\ldots,n$\nin the {\\em sorted output} (no effect on the algorithm). Denote $t(i)$ the\n(random) time $i$ is chosen as a pivot. Then $i$ will ever be compared with $j$\niff either $t(i)$ or $t(j)$ is the smallest among $t(i),\\ldots,t(j)$.\nThis has $2$ out of $|j{-}i|+1$ chances. So, the expected number of comparisons\nis $\\sum_{i,j>i} 2\/(1{+}j{-}i)= -4n+ (n{+}1)\\sum_{k=1}^n 2\/k= 2n(\\ln n-O(1))$.\n Note, that the expectation of the sum of variables is\n the sum of their expectations (not true, say, for product).\n\nThe above Monte-Carlo and Las-Vegas algorithms require choosing strings {\\em at\nrandom} with uniform distribution. We mentally picture that as flipping a coin.\n(Computers use \\trm {pseudo-random generators} rather than coins in hope,\nrarely supported by proofs, that their outputs have all the statistical\nproperties of truly random coin flips needed for the analysis of the algorithm.)\n\n\\paragraph {Random Inputs} to Deterministic Algorithms are analyzed similarly\nto algorithms that flip coins themselves and the two should not be confused.\nConsider an example: Someone is interested in knowing whether or not certain\ngraphs contain Hamiltonian Cycles. He offers graphs and pays \\$100 if we show\neither that the graph {\\em has} or that it {\\em has not} Hamiltonian Cycles.\nHamiltonian Cycle problem is NP-Complete, so it should be very hard for {\\em \nsome}, but not necessarily for {\\em most} graphs. In fact, if our patron chooses\nthe graphs uniformly, a fast algorithm can earn us the \\$100 {\\em most of the\ntime}! Let all graphs have $n$ nodes and, say, $d<\\ln n\/2$ mean degree and be\nequally likely. Then we can use the following (deterministic) algorithm:\\\\\nOutput ``{\\bf No} Hamiltonian Cycles\" and collect the \\$100, if the graph has\nan isolated node. Otherwise, pass on that graph and the money. Now, how often\ndo we get our \\$100. The probability that a given node $A$ of the graph is\nisolated is $(1-1\/n)^{dn}>(1-O(1\/n))\/\\sqrt n$. Thus, the probability that\n{\\em none} of $n$ nodes is isolated (and we lose our \\$100) is $O((1-1\/\\sqrt\nn)^n)= O(1)\/e^{\\sqrt n}$ and vanishes fast. Similar calculations can be made\nwhenever $r = \\lim (d\/\\ln n)<1$. If $r>1$, other fast algorithms can actually\nfind a Hamiltonian Cycle.\\\\ See: \\cite{jnsn,karp-pr,gu}. See also \\cite{vl}\nfor a proof that another graph problem is NP-complete even on average.\nHow do this HC algorithm and the above primality test differ?\n\n\\begin{itemize}\\item The primality algorithm works for {\\em all} instances.\nIt tosses the coin itself and can repeat it for a more reliable answer.\nThe HC algorithms only work for {\\em most} instances\n(with isolated nodes or generic HC). \\item In the HC algorithms, we must\ntrust the customer to follow the presumed random procedure.\\\\ If he cheats\nand produces rare graphs often, the analysis breaks down.\\end{itemize}\n\n\\paragraph {Symmetry Breaking.} Randomness comes into Computer Science\nin many other ways besides those we considered.\nHere is a simple example: avoiding conflicts for shared resources.\n\n{\\bf Dining Philosophers.} They sit at a circular table.\nBetween each pair is either a knife or a fork, alternating. The problem is,\nneighboring diners must share the utensils, cannot eat at the same time. How\ncan the philosophers complete the dinner given that all of them must act in the\nsame way without any central organizer? Trying to grab the knives and forks at\nonce may turn them into fighting philosophers. Instead they could each flip a\ncoin, and sit still if it comes up heads, otherwise try to grab the utensils.\\\\\nIf two diners try to grab the same utensil, neither succeeds.\nIf they repeat this procedure enough times,\\\\ most likely each philosopher\nwill eventually get both a knife and a fork without interference.\n\nWe have no time to actually analyze this and many other scenaria,\nwhere randomness is crucial.\\\\\nInstead we will take a look into the concept of Randomness itself.\n\n\\newpage\\subsection {Arithmetization:\n One-Player Games with Randomized Transition.}\\label{ip}\n\nThe results of section~\\ref{games} can be extended to \\trm {Arthur-Merlin}\ngames which have one player -- Merlin -- but a randomized transition function,\neffectively using a dummy second player -- Arthur -- whose moves are just\ncoin-flips. We will reduce generic games to games in which any Merlin's\nstrategy in any losing position has exponentially small chance to win.\n\nThe trick achieving this, called \\trm {arithmetization}, expresses\nthe boolean functions as low degree polynomials, and applies them\nto $\\Z_p$-tokens (let us call them \\trm {bytes}) instead of bits.\nIt was proposed in Noam Nisan's article widely distributed over email in the\nFall of 1989 and quickly used in a flood of follow-ups for proving relations\nbetween various complexity classes. We follow \\cite{shamir,fl}.\n\nLet $g$ be the (ATM-complete) game of 1d-Chess (\\ref{dc1}), $r(m,x)$ with\n$x{=}x_1\\ldots x_s$, $m,x_i{\\in}\\{0,1\\}$ be its transition rule. Configurations\ninclude $x$ and a remaining moves counter $c\\le2^s$. They are terminal if\n$c{=}0$, winning to the player $x_1$. Intermediate configurations $(m,x,y)$\nhave $y$ claimed as a prefix of $r(m,x)$.\n\nLet $t(m,x,y)$ be $1$ if $y{=}r(m,x)$, else $t{=}0$. 1d-Chess is simple, so $t$\ncan be expressed as a product of $s$ multilinear $O(1)$-sized terms, any\nvariable shared by at most two terms. Thus $t$ is a polynomial, quadratic in\neach $m,x_i,y_i$. Let $V_c(x)$ be $1$ if the active player has a strategy to\nwin in the $c$ moves left, i.e. $V_0(x)\\edf x_1$, $V_{c+1}(x)\\edf$\n$1{-}V_c(0,x,\\{\\})V_c(1,x,\\{\\})= 1{-}V_c(r(0,x))V_c(r(1,x))$, where\n$V_c(m,x,y)\\edf V_c(y)t(m,x,y)$ for $y=y_1\\ldots y_s$ or\n$V_c(m,x,y)\\edf V_c(m,x,y{\\circ}0){+}V_c(m,x,y{\\circ}1)$ for shorter $y$.\n($\\circ$ stands for concatenation.)\n\n$G$ will allow Merlin to prove $x$ is winning i.e., $V_c(x)=1$. Configurations\n$X=(m,x,y,v)$ of $G$ replace bits with $\\Z_p$ bytes and add $v{\\in}\\Z_p$\nreflecting Merlin's claimed $V$. The polynomial $V_c(m,x,y)$ is quadratic in\neach $x_i,m$, as $t(m,x,y)$ is. Then $V_c(y)$ has degree 4 in $y_i$ and\n$V_c(m,x,y)$ has degree 6 in $y_i$.\n\nMerlin starts with choosing a $2s$-bit prime $p$. Then at each step with\n$X=(m,x,y,v)$ Merlin gives an $O(1)$-degree polynomial $P$ with\n$P(1)P(0)=1{-}v$ for $s$-byte $y$ or $P(0){+}P(1)=v$ for shorter $y$. Arthur\nthen selects a random $r{\\in}\\Z_p$ and $X$ becomes $(r,y,\\{\\},P(r))$ for\n$s$-byte $y$ or $(m,x,y{\\circ}r,P(r))$ for shorter $y$.\n\nIf the original $v$ is correct, then Merlin's obvious winning strategy\nis to always provide the correct polynomial. If the original $v$ is wrong\nthen either $P(1)$ or $P(0)$ must be wrong, too, so they will agree only\nwith a wrong $P$. A wrong $P$ can agree with the correct one only\non few (bounded by degree) points. Thus it will give a correct value only to\nexponentially small fraction of random $r$. Thus the wrong $v$ will propagate\nthroughput the game until it becomes obvious in the terminal configuration.\n\n\\vspace{2pc}\nThis reduction of Section~\\ref{games} games yields a hierarchy of Arthur-Merlin\ngames powers, i.e. the type of computations that have reductions to $V_c(x)$ of\nsuch games and back. The one-player games with randomized transition rule $r$\nrunning in space linear in the size of initial configuration are equivalent to\nexponential time deterministic computations. If instead the running time $T$ of\n$r$ combined for all steps is limited by a polynomial, then the games are\nequivalent to polynomial space deterministic computations.\n\nAn interesting twist comes in one move games with polylog $T$, too tiny\nto examine the initial configuration $x$ and the Merlin's move $m$.\n But not only this obstacle is removed but the equivalence to NP is achieved\nwith a little care. Namely, $x$ is set in an error-correcting code, and $r$ is\ngiven $O(\\log\\|x\\|)$ coin-flips and random access to the digits of $x,m$.\nThen the membership proof $m$ is reliably verified by the randomized $r$.\\\\\n See \\cite{holo} for details and references.\n\n\\newpage\\section {Randomness}\\label{rand}\n\n\\subsection {Randomness and Complexity.}\\label{kolm}\n\nIntuitively, a random sequence is one that has the same properties as\na sequence of coin flips. But this definition leaves the question,\nwhat {\\em are} these properties? Kolmogorov resolved these problems with\na new definition of random sequences: those with no description noticeably\nshorter than their full length. See survey and history in \\cite{ku87,vitan}.\n\n\\paragraph {Kolmogorov Complexity} $K_A(x|y)$ of the string $x$ given $y$ is\nthe length of the shortest program $p$ which lets algorithm $A$ transform $y$\ninto $x$: $\\min\\{(\\|p\\|):A(p,y)=x\\}$. There exists a Universal Algorithm $U$\nsuch that, $K_U(x)\\le K_A(x)+O(1)$, for every algorithm $A$.\nThis constant $O(1)$ is bounded by the length of the program $U$ needs to\nsimulate $A$. We abbreviate $K_U(x|y)$ as $K(x|y)$, or $K(x)$ for empty $y$.\n\nAn example: For $A:x\\mapsto x$, $K_A(x)=\\|x\\|$,\n so $K(x)<K_A(x)+O(1)<\\|x\\|+O(1)$.\n\nCan we compute $K(x)$ by trying all programs $p,\\|p\\|{<}\\|x\\|{+}O(1)$\nto find the shortest one generating $x$? This does not work because\nsome programs diverge, and the halting problem is unsolvable.\\\\ In fact,\nno algorithm can compute $K$ or even any its lower bounds except $O(1)$.\n\nConsider the Berry Paradox expressed in the phrase:\n ``The smallest integer which cannot\\\\ be uniquely and clearly\n defined by an English phrase of less than two hundred characters.\"\\\\\n There are $< 128^{200}$ English phrases of $< 200$ characters.\n So there must be integers not expressible\\\\ by such phrases and the\n smallest one among them. But isn't it described by the above phrase?\n\nA similar argument proves that $K$ is not computable. Suppose an algorithm\n$L(x)\\ne O(1)$ computes a lower bound for $K(x)$. We can use it to compute\n$f(n)$ that finds $x$ with $n< L(x)\\le K(x)$, but $K(x)< K_f(x)+ O(1)$\nand $K_f(f(n))\\le \\|n\\|$, so $n< K(f(n))< \\|n\\|+O(1)= \\log O(n)\\ll n$: a\ncontradiction.\\\\ So, $K$ and its non-constant lower bounds are not computable.\n\nAn important application of Kolmogorov Complexity measures\nthe Mutual Information:\\\\ $I(x:y)= K(x)+ K(y)- K(x,y)$.\nIt has many uses which we cannot consider here.\n\n\\subsubsection*{Deficiency of Randomness.}\n\nSome upper bounds of $K(x)$ are close in some important cases.\nOne such case is of $x$ generated at random. Define its \\trm {rarity}\nfor uniform on $\\{0,1\\}^n$ distribution as $d(x)=n-K(x|n)\\ge-O(1)$.\n\nWhat is the probability of $d(x)>i$, for uniformly random $n$-bit $x$ ?\nThere are $2^n$ strings $x$ of length $n$.\\\\ If $d(x)>i$, then $K(x|n)< n-i$.\nThere are $<2^{n-i}$ programs of such length, generating $<2^{n-i}$ strings.\\\\\nSo, the probability of such strings is $<2^{n-i}\/2^n= 2^{-i}$ (regardless of\n$n$)! Even for $n= 1,000,000$,\\\\ the probability of $d(x)>300$ is absolutely\nnegligible (provided $x$ was indeed generated by fair coin flips).\n\nSmall rarity implies all other enumerable properties of random strings. Indeed,\nlet such property ``$x{\\not\\in}P$\" have a negligible probability and $S_n$\nbe the number of $n$-bit strings violating $P$, so $s_n=\\log(S_n)$ is small.\\\\\nTo generate $x$, we need only the algorithm enumerating $S_n$ and the $s_n$-bit\nposition of $x$ in that enumeration. Then the rarity $d(x)> n-(s_n{+}O(1))$\nis large. Each $x$ violating $P$ will thus also violate the ``small rarity\"\nrequirement. In particular, the small rarity implies unpredictability of bits\nof random strings: A short algorithm with high prediction rate would assure\nlarge $d(x)$. However, the randomness can only be refuted, cannot be confirmed:\nwe saw, $K$ and its lower bounds are not computable.\n\n\\paragraph {Rectification of Distributions.} We rarely have a source of\nrandomness with precisely known distribution. But there are very efficient ways\nto convert ``roughly\" random sources into perfect ones. Assume, we have such a\nsequence with weird unknown distribution. We only know that its long enough\n($m$ bits) segments have min-entropy $>k+i$, i.e. probability $<1\/2^{k+i}$,\ngiven all previous bits. (Without such $m$ we would not know a segment needed\nto extract even one not fully predictable bit.) No relation is required between\n$n,m,i,k$, but useful are small $m,i,k$ and huge $n=o(2^k\/i)$. We can fold $X$\ninto an $n\\times m$ matrix. We also need a small $m\\times i$ matrix $Z$,\nindependent of $X$ and {\\bf really} uniformly random (or random Toeplitz, i.e.\nwith restriction $Z_{a+1,b+1}=Z_{a,b}$). Then the $n\\times i$ product $XZ$ has\nuniform with accuracy $O(\\sqrt{n i\/2^k})$ distribution. This follows from\n\\cite{gl}, which uses earlier ideas of U. and V. Vazirani.\n\n\\newpage\\subsection {Pseudo-randomness.} \\label{pseudor}\n\nThe above definition of randomness is very robust, if not practical. True\nrandom generators are rarely used in computing. The problem is {\\em not} that\nmaking a true random generator is impossible: we just saw efficient ways to\nperfect the distributions of biased random sources. The reason lies in many\nextra benefits provided by pseudorandom generators. E.g., when experimenting\nwith, debugging, or using a program one often needs to repeat the exact same\nsequence. With a truly random generator, one actually has to record all its\noutcomes: long and costly. The alternative is to generate \\trm {pseudo-random}\nstrings from a short seed. Such methods were justified in \\cite{bm,yao}:\n\nFirst, take any one-way permutation $F_n(x)$ (see sec.~\\ref{crypt}) with a\n\\trm {hard-core} bit (see below) $B_p(x)$ which is easy to compute from $x,p$,\nbut infeasible to guess from $p,n,F_n(x)$ with any noticeable correlation.\\\\\n Then take a random \\trm {seed} of three $k$-bit parts $x_0,p,n$ and Repeat:\n($S_i{\\gets}B_p(x_i)$; $x_{i+1}{\\gets}F_n(x_i)$; $i{\\gets}i{+}1$).\n\nWe will see how distinguishing outputs $S$ of this generator\nfrom strings of coin flips would imply the ability to invert $F$.\nThis is infeasible if $F$ is one-way. But if P=NP (a famous open problem),\nno one-way $F$, and no pseudorandom generators could exist.\n\nBy Kolmogorov's standards, pseudo-random strings are not random: let $G$ be the\ngenerator; $s$ be the seed, $G(s) = S$, and $\\|S\\|\\gg k=\\|s\\|$. Then $K(S)\\le\nO(1)+k\\ll\\|S\\|$, thus violating Kolmogorov's definition.\\\\ We can distinguish\nbetween truly random and pseudo-random strings by simply trying all short\nseeds. However this takes time exponential in the seed length. Realistically,\npseudo-random strings will be as good as a truly random ones if they can't\nbe distinguished in feasible time. Such generators we call \\trm {perfect}.\n\n\\paragraph {Theorem:} \\cite{yao} Let $G(s)=S\\in\\{0,1\\}^n$ run in time $t_G$.\n Let a probabilistic algorithm $A$ in expected (over internal coin flips)\n time $t_A$ accept $G(s)$ and truly random strings with different by $d$\nprobabilities. Then, for random $i$, one can use $A$ to guess $S_i$\nfrom $S_{i+1},S_{i+2}, \\ldots$ in time $t_A+t_G$ with correlation $d\/O(n)$.\n\n\\paragraph {Proof.} Let $p_i$ be the probability that $A$ accepts $S=G(s)$\nmodified by replacing its first $i$ digits\\\\ with truly random bits.\nThen $p_0$ is the probability of accepting $G(s)$ and must differ by $d$ from\\\\\nthe probability $p_n$ of accepting random string. Then $p_{i-1}-p_i = d\/n$, for\nrandomly chosen $i$.\\\\ Let $P_0$ and $P_1$ be the probabilities of accepting\n$r0x$ and $r1x$ for $x=S_{i+1},S_{i+2},\\ldots$, and random $(i{-}1)$-bit $r$.\\\\\nThen $(P_1{+}P_0)\/2$ averages to $p_i$, while $P_{S_i}=P_0{+}(P_1{-}P_0)S_i$\naverages to $p_{i-1}$ and\\\\ $(P_1{-}P_0) (S_i{-}1\/2)$ to $p_{i-1}{-}p_i=d\/n$.\nSo, $P_1{-}P_0$ has the stated correlation with $S_i.\\qed$\n\nIf the above generator was not perfect, one could guess $S_i$ from the sequence\n$S_{i+1},S_{i+2},\\ldots$\\\\ with a polynomial (in $1\/\\|s\\|$) correlation.\n But, $S_{i+1}, S_{i+2}\\ldots$ can be produced from $p,n,x_{i+1}$.\\\\\n So, one could guess $B_p(x_i)$ from $p,n,F(x_i)$ with correlation $d\/n$,\n which cannot be done for hard-core $B$.\n\n\\paragraph {Hard Core.} The key to constructing a pseudorandom generator\nis finding a hard core for a one-way $F$. The following $B$ is hard-core\nfor any one-way $F$, e.g., for Rabin's OWF in sec.~\\ref{crypt}.\\\\\n\\cite{Knuth} has more details and references.\n\nLet $B_p(x)=(x\\cdot p)= (\\sum_ix_ip_i\\bmod2)$. \\cite{gl} converts\nany method $g$ of guessing $B_p(x)$ from $p,n,F(x)$ with correlation\n$\\varepsilon$ into an algorithm of finding $x$, i.e. inverting $F$\n(slower $\\varepsilon^2$ times than $g$).\n\n\\paragraph {Proof.} (Simplified with some ideas of Charles Rackoff.)\nTake $k=\\|x\\|=\\|y\\|$, $j=\\log(2k\/\\varepsilon^2)$, $v_i= 0^i10^{k-i}$.\nLet $B_p(x) =(x\\cdot p)$ and $b(x,p)=(-1)^{B_p(x)}$.\nAssume, for $y=F_n(x)$, $g(y,p,w)\\in\\{\\pm 1\\}$ guesses $B_p(x)$\nwith correlation $\\sum_p2^{-\\|p\\|}b(x,p) g_p >\\varepsilon$, where $g_p$\nabbreviates $g(y,p,w)$, since $w,y$ are fixed throughout the proof.\n\n$(-1)^{(x\\cdot p)}g_p$ averaged over ${>}2k\/\\varepsilon^2$ random pairwise\nindependent $p$ deviates from its mean (over all $p$) by ${<}\\varepsilon$\n(and so is ${>}0$) with probability $>1-1\/2k$. The same for\n$(-1)^{(x\\cdot[p+v_i])} g_{p+v_i}= (-1)^{(x\\cdot p)} g_{p+v_i} (-1)^{x_i}$.\n\nTake a random $k\\times j$ binary matrix $P$. The vectors $Pr$,\n$r{\\in}\\{0,1\\}^j\\setminus\\{0^j\\}$ are pairwise independent. So,\nfor a fraction $\\ge1-1\/2k$ of $P$, sign$(\\sum_r(-1)^{xPr}g_{Pr+v_i})=(-1)^{x_i}$.\nWe could thus find $x_i$ for all $i$ with probability $>1\/2$\nif we knew $z=xP$. But $z$ is short: we can try all its\n$2^j$ possible values and check $y=F_n(x)$ for each !\n\nSo the inverter, for a random $P$ and all $i,r$, computes $G_i(r)=g_{Pr+v_i}$.\nIt uses Fast Fourier on $G_i$ to compute $h_i(z)=\\sum_rb(z,r)G_i(r)$. The sign\nof $h_i(z)$ is the $i$-th bit for the $z$-th member of output list. $\\qed$\n\n\\newpage\\subsection {Cryptography.} \\label{crypt}\n\n\\paragraph {Rabin's One-way Function.} Pick random prime numbers $p,q,\\|p\\|=\n\\|q\\|$ with two last bits ${=}1$, i.e. with odd $(p{-}1)(q{-}1)\/4$. Then\n$n=pq$ is called a Blum number. Its length should make factoring infeasible.\n\n Let $Q_n=(Z^*_n)^2$ be the set of squares,\n i.e. \\trm {quadratic residues} (all residues are assumed $\\pmod n$).\n\n\\paragraph {Lemma.} Let $n=pq$ be a Blum number, $F: x\\mapsto x^2\\in Q_n$.\nThen (1) $F$ is a permutation on $Q_n$\\\\ and (2)\nThe ability to invert $F$ on random $x$ is equivalent to that of factoring $n$.\n\n\\vspace{-8pt}\\paragraph {Proof.} (1) $t{=}(p{-}1)(q{-}1)\/4$ is odd, so\n$u{=}(t{+}1)\/2$ is an integer. Let $x{=}F(z)$. Both $p{-}1$ and $q{-}1$\ndivide~$2t$. So, by Fermat's little theorem, both $p$, $q$ (and, thus $n$)\ndivide $x^t{-}1\\equiv z^{2t}{-}1$. Then $F(x)^u\\equiv x^{2u}=xx^t\\equiv x$.\n\n(2) The above $y^u$ inverts $F$. Conversely, let $F(A(y))=y$ for a fraction\n$\\varepsilon$ of $y\\in Q_n$.\\\\ Each $y\\in Q_n$ has $x,x'{\\ne}\\pm x$ with\n$F(x){=}F(x'){=}y$, both with equal chance to be chosen at random.\\\\\nIf $F(x)$ generates $y$ while $A(y)=x'$ the Square Root Test\n(\\ref{prime}) has both $x,x'$ for factoring $n.\\qed$\n\nSuch one-way permutations, called ``trap-door\", have many applications;\nwe look at cryptography below.\n\nPicking random primes is easy: they have density $1\/O(\\|p\\|)$.\nIndeed, one can see that $\\binom{2n}n$ is divisible by every prime\n$p{\\in}[n,2n]$ but by no prime $p{\\in}[\\frac23n,n]$ or prime power $p^i{>}2n$.\nSo, $(\\log\\binom{2n}n)\/ \\log n=2n\/\\log n-O(1)$ is an upper bound on\nthe number of primes in $[n,2n]$ and a lower bound on that in $[1,2n]$\n(and in $[3n,6n]$ as a simple calculation shows).\nAnd fast VLSI exist to multiply long numbers and check primality.\n\n\\paragraph {Public Key Encryption.}\n\nA perfect way to encrypt a message $m$ is to add it $\\bmod2$ bit by bit to a\nrandom string $S$ of the same length $k$. The resulting encryption $m \\oplus S$\nhas the same uniform probability distribution, no matter what $m$ is. So it is\nuseless for the adversary who wants to learn something about $m$, without\nknowing $S$. A disadvantage is that the communicating parties must share a\nsecret $S$ as large as all messages to be exchanged, combined. \\trm {Public\nKey} Cryptosystems use two keys. One key is needed to encrypt the messages and\nmay be completely disclosed to the public. The \\trm {decryption} key must still\nbe kept secret, but need not be sent to the encrypting party. The same keys may\nbe used repeatedly for many messages.\n\nSuch cryptosystem can be obtained \\cite{b-gw} by replacing the above random $S$\nby pseudorandom $S_i= (s_i\\cdot x)$; $s_{i+1} =(s_i^2\\ \\bmod n)$. Here a Blum\nnumber $n=pq$ is chosen by the Decryptor and is public, but $p,q$ are kept\nsecret. The Encryptor chooses $x\\in Z_2^{\\|n\\|},s_0\\in Z_n$ at random and sends\n$x,s_k, m{\\oplus} S$. Assuming factoring is intractable for the adversary, $S$\nshould be indistinguishable from random strings (even with known $x,s_k$).\nThen this scheme is as secure as if $S$ were random. The Decryptor\nknows $p,q$ and can compute $u,t$ (see above) and $v=(u^{k-1}\\bmod t)$.\nSo, he can find $s_1=(s_k^v\\bmod n)$, and then $S$ and $m$.\n\nAnother use of the intractability of factoring is digital signatures\n\\cite{rsa,bb-sg}. Strings $x$ can be released as authorizations\nof $y=(x^2\\bmod n)$. Verifying $x$, is easy but the ability of\nforging it for generic $y$ is equivalent to that of factoring $n$.\n\n\\vfill\\subsubsection* {Go On!}\n\nYou noticed that most of our burning questions are still open. Take them on!\n\nStart with reading recent results (FOCS\/STOC is a good source).\nSee where you can improve them.\\\\ Start writing, first notes just for\nyour friends, then the real papers. Here is a little writing advice:\n\nA well written paper has clear components: skeleton, muscles, etc.\\\\\nThe skeleton is an acyclic digraph of basic definitions and statements,\nwith cross-references.\\\\ The meat consists of proofs (muscles) each\n{\\em separately} verifiable by competent graduate students having to read\nno other parts but statements and definitions cited. Intuitive comments,\nexamples and other comfort items are fat and skin: a lack or excess will\nnot make the paper pretty. Proper scholarly references constitute clothing,\nno paper should ever appear in public without! Trains of thought\nwhich led to the discovery are blood and guts: keep them hidden.\nMetaphors for other vital parts, like open problems, I skip out of modesty.\n\n\\vfill\\paragraph {Writing Contributions.} {\\small\n Section~\\ref{models} was originally prepared by Elena Temin,\n Yong Gao and Imre Kifor (BU), others by Berkeley students:\n \\ref{compress} by Mark Sullivan,\n \\ref{win} by Eric Herrmann and Elena Eliashberg,\n \\ref{halt-gm} by Wayne Fenton and Peter Van Roy,\n \\ref{gm-reduce} by Carl Ludewig, Sean Flynn, and Francois Dumas,\n \\ref{invert} by Jeff Makaiwi, Brian Jones and Carl Ludewig,\n \\ref{compl} by David Leech and Peter Van Roy,\n \\ref{tile} by Johnny and Siu-Ling Chan, \\ref{average} by Deborah Kordon,\n \\ref{kolm} by Carl Ludewig, \\ref{pseudor} by Sean Flynn,\n Francois Dumas, Eric Herrmann, \\ref{crypt} by Brian Jones.}\n\n\\newpage\\section {References}\n \\renewcommand\\refname{}\n\n\\vspace*{-1pc}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe massive recent growth of the computational cost of accurate deep learning models, in particular large language models (LLMs), has motivated the development of several advanced {model compression} techniques~\\citep{hoefler2021sparsity, gholami2021survey}, encompassing unstructured and structured pruning, quantization, and knowledge distillation. \nIn this paper, we focus on the unstructured pruning, for which we follow the standard pipeline. Such models are first \\emph{pre-trained} on a large \\emph{upstream} corpus of unlabelled text. Then, they are \\emph{fine-tuned} in a supervised manner on a smaller \\emph{downstream} task, such as question-answering or text classification.  \nIn the context of compression, this pipeline led to two paradigms: 1) \\emph{upstream pruning}, followed by fine-tuning of the remaining weights on a downstream task, and 2) \\emph{downstream pruning}, pruning and fine-tuning directly on the downstream task.\n\nA tempting baseline approach in most settings is \\emph{gradual magnitude pruning (GMP)}~\\citep{hagiwara1994, zhu2017prune}, that is, periodically removing the smallest fraction of weights during training, possibly interspersed with fine-tuning steps designed to recover accuracy. \nGMP has been shown to be an extremely strong baseline in the context of computer vision~\\citep{gale2019state, hoefler2021sparsity}.\nHowever, the literature on pruning LLMs, and in particular BERT models~\\cite{sanh2020movement, chen2020lottery, zafrir2021prune}, clearly suggests that GMP \\emph{does not} perform well. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/squad_and_mnli_merged.pdf}\n    \\caption{Performance of state-of-the-art unstructured pruning methods relative to the dense $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ model at high sparsities and two tasks, SQuADv1.1 and MNLI.}\n    \\label{fig:overview}\n    \\vspace{-0.2in}\n\\end{figure}\n\n\\paragraph{Contribution.} In this paper, we re-examine this conclusion and investigate whether GMP can be a competitive baseline, once carefully tuned. \nSpecifically, we show that a well tuned variant which we call GMP$\\scriptstyle\\bigstar$\\,, can produce highly accurate and sparse language models in both upstream and downstream pruning regimes, matching or even  outperforming more complex methods. We explore effects of the crucial parameters for gradual pruning, and provide simple and intuitive guidelines on how to integrate them in a principled manner.  \n\nOur results are summarized in Figure~\\ref{fig:overview}, which presents performance of state-of-the-art unstructured pruning techniques on two benchmarks. Specifically, we compare GMP$\\scriptstyle\\bigstar$\\, with the Lottery Ticket approach~\\citep{chen2020lottery}, Movement Pruning (MvP)~\\citep{sanh2020movement} (as well as its GMP baseline $\\textnormal{GMP}_{\\textnormal{MvP}}$), upstream Prune OFA~\\citep{zafrir2021prune}, as well as the recently-proposed second-order pruning oBERT~\\citep{kurtic2022optimal}. \nWe observe that: 1) for both benchmarks, GMP$\\scriptstyle\\bigstar$\\, is only second to the more complex oBERT method; 2) GMP$\\scriptstyle\\bigstar$\\, in fact outperforms the highly competitive Prune OFA and MvP methods; and 3) GMP$\\scriptstyle\\bigstar$\\, outperforms both Lottery Tickets and $\\textnormal{GMP}_{\\textnormal{MvP}}$ by extremely wide margins. \n\n\\comment\n}\n\n\\paragraph{Prior Work.} \nFollowing the vast BERT-pruning literature, we focus on the unstructured pruning of the $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ model~\\citep{devlin2018bert}. As previously noted, upstream and downstream pruning paradigms exist, and methods are usually developed and specialized for only one of the two. For example, Movement Pruning (MvP)~\\citep{sanh2020movement, lagunas2021block} for downstream pruning and Prune Once for All (Prune OFA)~\\citep{zafrir2021prune} for upstream pruning. Simplicity and generality of the GMP makes it suitable for both paradigms, without any regime-specific modifications. New and more advanced pruning techniques, which are, contrary to GMP, able to leverage gradients~\\citep{sanh2020movement, lagunas2021block}, loss curvature~\\citep{kurtic2022optimal}, compute-intensive pre-training setup~\\citep{zafrir2021prune} are built on the premise that the simple magnitude-based GMP method falters when applied to BERT-pruning. In this work, contrary to what is currently available in the literature, we present empirical evidence that GMP, when tuned carefully, can produce very accurate sparse models which are competitive or even better than most state-of-the-art pruning techniques across both regimes (upstream and downstream). As can be seen from Figure \\ref{fig:overview} and our later results, we massively improve upon existing GMP-based pruning baselines, in some cases by even more than \\textbf{20 accuracy points}.\n\n\\section{Competitive Gradual Magnitude Pruning (GMP$\\scriptstyle\\bigstar$\\,)}\n\\paragraph{Experimental setup.}\nWe focus our attention on the standard $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ model, composed of embedding and encoder layers, which has approximately 110M parameters. All methods focus on pruning among approximately 85M weights of encoder layers and report sparsities with respect to that number. We evaluate models on the validation split of the respective dataset, and to improve confidence in the obtained results we perform multiple runs with different seeds and report mean performance.\n\\subsection{Downstream pruning}\nFollowing the literature, we consider three popular tasks: question-answering SQuADv1.1~\\citep{rajpurkar2016squad}, recognition of textual entailment MNLI ~\\citep{williams2017broad}, and duplicate question detection QQP~\\citep{iyer2017first}. Now, we reflect upon the most important constituents of the gradual pruning framework that enabled us to attain massive improvements.\n\n\\paragraph{Sparsity schedule.}\nIn all of our gradual runs, there is no pruning during the first two and the last two epochs. The former fine-tunes the pre-trained model, and the latter fine-tunes the sparse model with the fixed mask. In between the two, GMP$\\scriptstyle\\bigstar$\\, follows the cubic sparsity scheduler~\\citep{zhu2017prune} and prunes weights with the frequency of ten times per epoch. Motivated by the fact that $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ is heavily overparametrized for downstream tasks, we deviate from the standard cubic schedule by introducing a large first pruning step. This showed to be of a crucial importance when pruning the model to high target sparsities (e.g. 97\\%) as it leaves more time to recover from later pruning steps which are much more difficult. In Table~\\ref{tab:initsparsity_sweep} we report results from an ablation study with respect to the size of the initial step. For convenience, we visualize the sparsity scheduler in Figure~\\ref{fig:lr_and_spars}. Our preliminary experiments showed similar performance between uniform and global sparsity distributions, so we use the former. \n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/lr_and_sparsity.pdf}\n    \\vspace{-0.25in}\n    \\caption{Learning rate and sparsity schedules for the proposed gradual pruning framework.}\n    \\label{fig:lr_and_spars}\n    \\vspace{-0.2in}\n\\end{figure}\n\n\\paragraph{Learning rate schedule.} \nOur goal is to provide a simple baseline setup that works well across wide range of datasets without any additional task-dependent tuning. Currently, papers either report best results following an extensive hyperparameter search for each task, e.g.~\\citet{zafrir2021prune}, or they make use of carefully crafted schedulers for each setup independently which may include warm-up phases with and without rewinds~\\citep{sanh2020movement, kurtic2022optimal}. This may lead to high specialization to the target task\/model, which is undesirable in practice and makes it hard to distinguish benefits from the pruning technique itself. We propose to simply \\textit{replicate} the standard 2-epoch fine-tuning schedule~\\citep{devlin2018bert} by a certain factor and intertwine it with pruning steps. For a fair comparison with~\\citet{sanh2020movement} we replicate it by a factor of 5, reproducing their 10-epoch setup. And for a fair comparison with~\\citet{chen2020lottery} we replicate it by a factor of 15, reproducing their 30-epoch setup. For convenience, we visualize the learning rate schedule in Figure~\\ref{fig:lr_and_spars}. In appendix~\\ref{app:failed_lr}, we describe results with other schedulers that didn't work.\n\n\\paragraph{Knowledge Distillation (KD) Hardness.} We leverage KD~\\cite{hinton2015distilling} of outputs from a fine-tuned dense teacher. KD is a standard practice when pruning, e.g.~\\cite{sanh2020movement, zafrir2021prune, xu2021rethinking}. The loss function is formulated as a linear combination of the standard loss associated with the specific task (e.g. cross-entropy for classification $\\mathcal{L}_{CE}$) and the Kullback-Leibler divergence ($\\mathcal{L}_{KL}$) between output distributions of the dense (teacher) model and the sparse (student) model in the form: $\\mathcal{L}= (1-h) \\mathcal{L}_{CE} + h \\mathcal{L}_{KL}$. The ratio between the two is controlled with the \\textit{hardness} hyperparameter $h$. To determine its optimal value at high sparsities we run an ablation study reported in Table \\ref{tab:hardness_sweep}, and adopt the hardness $h=1$.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/output_dist_x2.pdf}\n    \\vspace{-0.2in}\n    \\caption{Teacher's output distribution at commonly used temperatures ${T \\in \\{1.0, 2.0\\}}$ and the proposed $T = 5.5$.}\n    \\label{fig:logits_dist}\n    \\vspace{-0.25in}\n\\end{figure}\n\n\\paragraph{Knowledge Distillation Temperature.} The temperature \\textit{T} is an additional KD-hyperparameter that requires proper tuning, as it controls the ``softness'' of the output distribution. \nIn the pruning literature, it is standard to use the ``stronger'' $T = 1 $ or $T = 2$ values \\citep{xu2021rethinking, zafrir2021prune, sanh2020movement, lagunas2021block, kurtic2022optimal}; we revisit this by visualizing teacher's output distributions to get an insight into what the sparse student is learning. In Figure~\\ref{fig:logits_dist}, we visualize generated distributions for randomly picked samples from the SQuADv1.1 task softened with three values of the temperature. As can be seen, teacher's high confidence in predicting the correct class at the commonly used temperatures $T \\in \\{1.0 , 2.0 \\}$ makes the knowledge distillation almost obsolete. Motivated by this observation, we run an ablation study for many higher temperatures and report a fraction of results in Table~\\ref{tab:temperature_sweep}. Given the results, we adopt the temperature $T = 5.5$.\n\n\\subsubsection{GMP$\\scriptstyle\\bigstar$\\, vs. other GMP-based baselines}\nDue to space constraints, we aggregate all the previously analyzed improvements in a \\textit{downstream pruning recipe} and present it in detail in Appendix~\\ref{app:down_recipe}. We compare our optimized GMP$\\scriptstyle\\bigstar$\\, with other GMP results reported in the pruning literature. For a fair comparison, we consider both setups, 10 and 30-epoch. In the 10-epoch setup, we compare against the GMP baselines reported in \\citet{sanh2020movement} and refer to them as GMP$_{\\small{\\textrm{MvP}}}$\\,. In the 30-epoch setup, we compare against the best reported results in \\citet{chen2020lottery}, obtained either via GMP or via Lottery Ticket (LTH) approach, and refer to them as GMP$_{\\small{\\textrm{LTH}}}$\\,. As can be seen from the Table~\\ref{tab:gmp_downstream}, our GMP$\\scriptstyle\\bigstar$\\, remarkably outperforms all other results; in some cases the improvements are more than \\textbf{20 points}!\n\n\\begin{table}[t]\n    \\caption{Downstream pruning comparison of GMP$\\scriptstyle\\bigstar$\\, with other GMP-based baselines.}\n    \\label{tab:gmp_downstream}\n    \\centering\n    \\small{\n      \\begin{tabular}{lccccc}\n        \\toprule\n        \\multirow{2}{*}{Method} & \\multirow{2}{*}{Spars.} & \\multirow{2}{*}{Ep.} & SQuAD & MNLI & QQP \\\\\n        & & & F1 & m-acc & acc \\\\\n        \\midrule \n        $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ & 0\\% & & 88.5 & 84.5 & 91.1 \\\\\n        \\midrule\n        GMP$_{\\small{\\textrm{MvP}}}$\\, & \\multirow{2}{*}{90\\%} & \\multirow{2}{*}{10} & 80.1 & 78.3 & 79.8 \\\\\n        GMP$\\scriptstyle\\bigstar$\\, &  &  & \\textbf{86.7} & \\textbf{81.9} & \\textbf{90.6} \\\\\n        \\midrule \n        GMP$_{\\small{\\textrm{MvP}}}$\\, & \\multirow{2}{*}{97\\%} & \\multirow{2}{*}{10} & 59.6 & 69.4 & 72.4 \\\\\n        GMP$\\scriptstyle\\bigstar$\\, &  &  & \\textbf{81.3} & \\textbf{79.1} & \\textbf{89.7} \\\\\n        \\midrule[1pt]\n        GMP$_{\\small{\\textrm{LTH}}}$\\, & \\multirow{2}{*}{90\\%} & \\multirow{2}{*}{30} & 68.0 & 75.0 & 90.0 \\\\\n        GMP$\\scriptstyle\\bigstar$\\, & & & \\textbf{87.9} & \\textbf{82.7} & \\textbf{90.8} \\\\\n        \\midrule \n        GMP$\\scriptstyle\\bigstar$\\, & 97\\% & 30 & 85.4 & 80.9 & 90.6 \\\\\n        \\bottomrule\n      \\end{tabular}\n     }\n      \\vspace{-0.15in}\n\\end{table}\n\n\\comment{ \n\\begin{table}\n  \\caption{Comparison of GMP$\\scriptstyle\\bigstar$\\, in pruning during fine-tuning (downstream) setup against GMP baselines from Movement Pruning (GMP$_{\\small{\\textrm{MvP}}}$\\,) \\cite{sanh2020movement} and Lottery Tickets (GMP$_{\\small{\\textrm{LTH}}}$\\,) \\cite{chen2020lottery}. For GMP$\\scriptstyle\\bigstar$\\,, we report mean performance from three runs with different seeds.}\n  \\label{tab:gmp_downstream}\n  \\centering\n  \\small{\n      \\begin{tabular}{lcccccccc}\n        \\toprule\n        \\multirow{2}{*}{Method} & \\multirow{2}{*}{Sparsity} & \\multirow{2}{*}{Epochs} & \\multicolumn{2}{c}{SQuAD} & \\multicolumn{2}{c}{MNLI} & \\multicolumn{2}{c}{QQP} \\\\\n        & & & F1 & EM & m-acc & mm-acc & acc & F1 \\\\\n        \\midrule \n        $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ & 0\\% & & 88.5 & 81.4 & 84.5 & 85.0 & 91.1 & 88.0 \\\\\n        \\midrule\n        GMP$_{\\small{\\textrm{MvP}}}$\\, & \\multirow{2}{*}{90\\%} & \\multirow{2}{*}{10} & 80.1 & 70.2 & 78.3 & 79.3 & 79.8 & 65.0 \\\\\n        GMP$\\scriptstyle\\bigstar$\\, &  &  & \\textbf{86.7} & \\textbf{78.7} & \\textbf{81.9} & \\textbf{82.1} & \\textbf{90.6} & \\textbf{87.4} \\\\\n        \\midrule \n        GMP$_{\\small{\\textrm{MvP}}}$\\, & \\multirow{2}{*}{97\\%} & \\multirow{2}{*}{10} & 59.6 & 45.5 & 69.4 & 70.6 & 72.4 & 57.8 \\\\\n        GMP$\\scriptstyle\\bigstar$\\, &  &  & \\textbf{81.3} & \\textbf{71.3} & \\textbf{79.1} & \\textbf{79.6} & \\textbf{89.7} & \\textbf{86.1} \\\\\n        \\midrule[1pt]\n        GMP$_{\\small{\\textrm{LTH}}}$\\, & \\multirow{2}{*}{90\\%} & \\multirow{2}{*}{30} & 68.0 & - & 75.0 & - & 90.0 & - \\\\\n        GMP$\\scriptstyle\\bigstar$\\, & & & \\textbf{87.9} & \\textbf{80.4} & \\textbf{82.7} & \\textbf{83.2} & \\textbf{90.8} & \\textbf{87.7} \\\\\n        \\midrule \n        GMP$\\scriptstyle\\bigstar$\\, & 97\\% & 30 & 85.4 & 77.1 & 80.9 & 81.2 & 90.6 & 87.3 \\\\\n        \\bottomrule\n      \\end{tabular}\n     }\n\\end{table}\n}\n\n\\subsubsection{GMP$\\scriptstyle\\bigstar$\\, vs. advanced pruning techniques}\nNow, we wish to compare our GMP$\\scriptstyle\\bigstar$\\, with methods that rely on higher-order information to make pruning decisions, like gradients in MvP \\cite{sanh2020movement} and the loss curvature in oBERT \\cite{kurtic2022optimal}. Both of these impose higher computational overhead compared to the magnitude-based pruning, but we still put our results in the context with respect to theirs to fully grasp the scope of improvements introduced by careful optimizations of GMP. As can be seen from results in Table~\\ref{tab:high_downstream}, GMP$\\scriptstyle\\bigstar$\\, is able to improve upon the performance of Movement Pruning in 4 out of 6 analyzed configurations, but unfortunately can't match the performance of the oBERT method. In addition to these comparisons, we make use of the open-source implementation of oBERT, current state-of-the-art BERT-pruning method, and run it with optimized hyperparameters from GMP$\\scriptstyle\\bigstar$\\, on the SQuADv1.1 task. We refer to these results as oBERT$\\scriptstyle\\bigstar$\\,. As can be seen from the Table \\ref{tab:high_downstream}, even the very competitive oBERT results benefit from the GMP$\\scriptstyle\\bigstar$\\, setup. For all GMP$\\scriptstyle\\bigstar$\\, runs, we report mean performance across three runs with different seeds, and additional metrics in Tables \\ref{tab:gmp_downstream2} and \\ref{tab:high_downstream2}.\n\n\\begin{table}[t]\n    \\caption{Downstream pruning comparison of GMP$\\scriptstyle\\bigstar$\\, with advanced pruning techniques.}\n    \\label{tab:high_downstream}\n    \\centering\n      \\small{\n      \\begin{tabular}{lccccc}\n        \\toprule\n        \\multirow{2}{*}{Method} & \\multirow{2}{*}{Spars.} & \\multirow{2}{*}{Ep.} & SQuAD & MNLI & QQP \\\\\n        & & & F1 & m-acc & acc \\\\\n        \\midrule \n        $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ & 0\\% & & 88.5 & 84.5 & 91.1 \\\\\n        \\midrule\n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{2}{*}{90\\%} & \\multirow{2}{*}{10} & \\textbf{86.7} & \\textbf{81.9} & \\textbf{90.6} \\\\\n        MvP & & & 84.9 & 81.2 & 90.2 \\\\\n        \\midrule \n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{2}{*}{97\\%} & \\multirow{2}{*}{10} & 81.3 & 79.1 & \\textbf{89.7} \\\\\n        MvP & & & \\textbf{82.3} & \\textbf{79.5} & 89.1 \\\\\n        \\midrule[1pt]\n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{3}{*}{90\\%} & \\multirow{3}{*}{30} & 87.9 & 82.7 & 90.8 \\\\\n        oBERT & & & 88.3 & \\textbf{83.8} & \\textbf{91.4} \\\\\n        oBERT$\\scriptstyle\\bigstar$ & & & \\textbf{88.6} & & \\\\\n        \\midrule \n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{3}{*}{97\\%} & \\multirow{3}{*}{30} & 85.4 & 80.9 & 90.6 \\\\\n        oBERT & & & 86.0 & \\textbf{81.8} & \\textbf{90.9} \\\\\n        oBERT$\\scriptstyle\\bigstar$ & & & \\textbf{86.6} & & \\\\\n        \\bottomrule\n      \\end{tabular}\n      \\vspace{-0.15in}\n  }\n\\end{table}\n\\comment{\n\\begin{table}\n  \\caption{Comparison of GMP$\\scriptstyle\\bigstar$\\, in pruning during fine-tuning (downstream) setup against more advanced techniques, Movement Pruning (MvP) \\cite{sanh2020movement} and The Optimal BERT Surgeon (oBERT) \\cite{kurtic2022optimal}. oBERT$\\scriptstyle\\bigstar$\\, stands for results we obtained by running the open-sourced oBERT implementation in the GMP$\\scriptstyle\\bigstar$\\, setup. For GMP$\\scriptstyle\\bigstar$\\,, we report mean performance from three runs with different seeds.}\n  \\label{tab:high_downstream}\n  \\centering\n  \\small{\n      \\begin{tabular}{lcccccccc}\n        \\toprule\n        \\multirow{2}{*}{Method} & \\multirow{2}{*}{Sparsity} & \\multirow{2}{*}{Epochs} & \\multicolumn{2}{c}{SQuAD} & \\multicolumn{2}{c}{MNLI} & \\multicolumn{2}{c}{QQP} \\\\\n        & & & F1 & EM & m-acc & mm-acc & acc & F1 \\\\\n        \\midrule \n        $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ & 0\\% & & 88.5 & 81.4 & 84.5 & 85.0 & 91.1 & 88.0 \\\\\n        \\midrule\n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{2}{*}{90\\%} & \\multirow{2}{*}{10} & \\textbf{86.7} & \\textbf{78.7} & \\textbf{81.9} & \\textbf{82.1} & \\textbf{90.6} & \\textbf{87.4} \\\\\n        MvP & & & 84.9 & 76.6 & 81.2 & 81.8 & 90.2 & 86.8 \\\\\n        \\midrule \n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{2}{*}{97\\%} & \\multirow{2}{*}{10} & 81.3 & 71.3 & 79.1 & 79.6 & \\textbf{89.7} & \\textbf{86.1} \\\\\n        MvP & & & \\textbf{82.3} & \\textbf{72.7} & \\textbf{79.5} & \\textbf{80.1} & 89.1 & 85.5 \\\\\n        \\midrule[1pt]\n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{3}{*}{90\\%} & \\multirow{3}{*}{30} & 87.9 & 80.4 & 82.7 & 83.2 & 90.8 & 87.7 \\\\\n        oBERT & & & 88.3 & 81.1 & \\textbf{83.8} & \\textbf{84.4} & \\textbf{91.4} & \\textbf{88.3} \\\\\n        oBERT$\\scriptstyle\\bigstar$ & & & \\textbf{88.6} & \\textbf{81.3} & & & & \\\\\n        \\midrule \n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{3}{*}{97\\%} & \\multirow{3}{*}{30} & 85.4 & 77.1 & 80.9 & 81.2 & 90.6 & 87.3 \\\\\n        oBERT & & & 86.0 & 78.1 & \\textbf{81.8} & \\textbf{82.0} & \\textbf{90.8} & \\textbf{87.7} \\\\\n        oBERT$\\scriptstyle\\bigstar$ & & & \\textbf{86.6} & \\textbf{78.8} & & & & \\\\\n        \\bottomrule\n      \\end{tabular}\n  }\n\\end{table}\n}\n\\subsection{Upstream pruning}\n\nTo validate the optimized GMP$\\scriptstyle\\bigstar$\\, setup introduced in the previous section, we apply it now to the pre-training phase of LLMs. This is a two-stage process. In the first stage, the $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ model is pruned during pre-training and then, in the second stage, the remaining weights are fine-tuned with the fixed mask on a specific downstream task to evaluate performance. Given the high costs of experimenting in the pre-training phase, we use the dense teacher open-sourced by~\\citet{kurtic2022optimal}. Due to the space constraints, we summarize all hyperparameters in an \\textit{upstream pruning recipe} and present it in detail in Appendix~\\ref{app:up_recipe}. In Table \\ref{tab:upstream} we present results obtained in this setup and compare against other methods that are utilizing the same approach. More specifically, we compare against the Lottery Ticket~\\citep{chen2020lottery}, Prune OFA~\\citep{zafrir2021prune}, and The Optimal BERT Surgeon (oBERT)~\\citep{kurtic2022optimal}. In addition to this, we report the GMP baselines obtained in the Prune OFA work and refer to them as GMP$_{\\small{\\textrm{Prune OFA}}}$\\,. As can be seen from the Table \\ref{tab:upstream}, the GMP$\\scriptstyle\\bigstar$\\, significantly outperforms GMP$_{\\small{\\textrm{Prune OFA}}}$\\,, Lottery Tickets and even the Prune OFA, and comes really close to the performance of oBERT. For all GMP$\\scriptstyle\\bigstar$\\, runs, we report mean performance across four runs with different seeds. These results confirm findings from the previous section and establish the GMP$\\scriptstyle\\bigstar$\\, as an extremely competitive baseline in all regimes.\n\n\\begin{table}\n  \\caption{Upstream pruning comparison of GMP$\\scriptstyle\\bigstar$\\, with other GMP-based baselines and more advanced pruning techniques.}\n  \\label{tab:upstream}\n  \\centering\n  \\small{\n      \\begin{tabular}{lcccc}\n        \\toprule\n        \\multirow{2}{*}{Method} & \\multirow{2}{*}{Sparsity} & SQuAD & MNLI & QQP \\\\\n        & & F1 & m-acc & acc \\\\\n        \\midrule \n        $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ & 0\\% & 88.5 & 84.5 & 91.1 \\\\\n        \\midrule\n        GMP$_{\\small{\\textrm{Prune OFA}}}$\\, & 85\\% & 86.2 & 82.5 & 90.9 \\\\\n        \\midrule\n        Lottery Ticket & \\multirow{4}{*}{90\\%} & 68.0 & 75.0 & 90.0 \\\\\n        Prune OFA & & 87.3 & 81.5 & 90.9 \\\\\n        GMP$\\scriptstyle\\bigstar$\\, & & 88.2 & 83.2 & 90.8 \\\\\n        oBERT & & \\textbf{88.5} & \\textbf{83.4} & \\textbf{91.0} \\\\\n        \\midrule\n        GMP$\\scriptstyle\\bigstar$\\, & \\multirow{2}{*}{97\\%} & 84.7 & 80.3 & 89.8 \\\\\n        oBERT & & \\textbf{84.9} & \\textbf{80.9} & \\textbf{90.3} \\\\\n        \\bottomrule\n      \\end{tabular}\n      \\vspace{-0.15in}\n  }\n\\end{table}\n\n\\section{Conclusion}\nIn this work, we presented a set of updates to the standard gradual pruning setup for BERT models which enabled us to achieve very competitive results with the simple magnitude pruner. These results outperformed, by significant margins, all magnitude-based results currently available in the pruning literature which have been used as baselines for development and benchmarking of the new and more advanced pruning techniques. We hope that these \\textit{new baselines} will help the community to start off from a competitive set of results when compressing large language models. Moreover, our GMP$\\scriptstyle\\bigstar$\\, has even outperformed some results obtained with more advanced and computationally heavier pruning techniques. At this point, we would like to {strongly emphasize} that these results should not be interpreted as evidence that magnitude pruning is better than other more advanced methods. Rather, they should be interpreted as evidence that their current results could significantly benefit from updates of the gradual setup presented on the GMP$\\scriptstyle\\bigstar$\\, use-case. To support this claim, we ran the state-of-the-art oBERT pruner with the GMP$\\scriptstyle\\bigstar$\\, setup and managed to improve its results by non-trivial margins.\n\n\\section{Limitations}\nAs any academic study, our work is not without its limitations. Following the literature, our extensive empirical studies were conducted only on the standard $\\textrm{BERT}_{\\tiny{\\textrm{BASE}}}\\,$ model, giving us opportunity to compare against a vast amount of different pruning techniques. Throughout the literature, this model emerged as a consistent benchmark for unstructured pruning methods. However, the current results don't directly imply that our findings will be generally applicable to other language models as well. To partially fill in this uncertainty gap, we conduct a few experiments on the three times larger $\\textrm{BERT}_{\\tiny{\\textrm{LARGE}}}\\,$ model and report results in the Appendix~\\ref{app:additional_models}. Another limitation which we aim to remove in future work is the focus on fine-grained unstructured sparsity type, and explore other variants such as semi-structured and structured pruning. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\cite{Broadbent89} identified a feature near G357.2$-$0.2 (G357.1$-$00.2\nin some references) as a candidate supernova remnant (SNR) because\nits $S_{\\rm 60\\,\\mu m}\/S_{\\rm 6\\,cm}$ flux-density ratio is lower\nthan that of Galactic H\\textsc{ii} regions and it is resolved at\n6~cm with the Parkes telescope 4\\arcmin\\ beam.  \\cite{Gray94} added\nthe 1\\arcmin\\ resolution 843\\,MHz Molonglo Observatory Synthesis\nTelescope image clearly resolving a sinuous structure for the first\ntime and indicating a nonthermal radio spectrum.  There is a diffuse\nhalo surrounding the fine scale structure.\n\n\\cite{Gray96} was the first to present and discuss high-resolution\n(13\\arcsec) Very Large Array images of this nebula.  The author\nalso noted the high polarization of the filaments at C band (5\\,GHz)\nand the low polarization at L band (1.5\\,GHz), indicating depolarization\nand rotation measure $\\mbox{RM}\\sim2000$\\,rad\\,m$^{-2}$.  On the\nbasis of the unusual morphology, \\cite{Gray96} deprecated the SNR\ninterpretation and mentioned a variety of possibilities, including\na pulsar wind nebula (PWN) and one more example of peculiar nonthermal\nphenomena near the Galactic center \\citep[e.g., the ``Tornado''\nonly $0\\fdg5$ away from G357.2$-$0.2,][]{Gaensler2003}.\n\n\\cite{Gray94} and \\cite{Gray96} note that the pulsar B1736$-$31 is\nin the vicinity of G357.2$-$0.2, in projection.  Its location outside\nthe nebula precludes any connection to a PWN interpretation, and\nits spin-down age of 0.5\\,Myr \\citep{Clifton92} also makes it too\nold to still have an associated visible SNR.\n\nH\\textsc{i} observations of G357.2$-$0.2 by \\cite{Roy2002} give a\ndistance of at least 6\\,kpc and place it either in front of, or\npartly embedded in, a cloud believed to be beyond the Galactic\ncenter; they conclude that it is Galactic.\n\nWe observed G357.2$-$0.2 with the MeerKAT radio telescope\\footnote{Operated\nby the South African Radio Astronomy Observatory (SARAO).} in the\nUHF and L bands (0.56--1.68\\,GHz) with 7\\arcsec\\ resolution and\nwith the eROSITA X-ray instrument.  The observations and analysis\nare described in Section \\ref{ObsAnalysis}, the imaging results are\npresented in Section \\ref{Results}, and a discussion of these results\nis in Section \\ref{Discussion} followed by a summary in Section\n\\ref{Summary}.\n\n\n\\section{Observations and Data Analysis \\label{ObsAnalysis}}\n\n\\subsection{MeerKAT Observations, Analysis, and Imaging}\n\nWe observed G357.2$-$0.2 in both ``L'' (886--1682\\,MHz) and UHF\n(563--1068\\,MHz) bands with the 64 antenna MeerKAT array pointed\nat J2000 $\\mbox{R.A.} = 17^{\\mathrm h}39^{\\mathrm m}39\\fs82$,\n$\\mbox{Dec.} = -31\\arcdeg27\\arcmin47\\farcs0$ (G357.176$-$0.235).\nThe integration time was 8\\,s, and each band was divided into 4096\nspectral channels.\n\nThe observations were in two sessions, L band on 2020 July 21 for\n8 hours with 59 antennas and UHF on 2020 August 18 for 8 hours with\n53 antennas.  PKS~B1934$-$638 was used as the flux density, band-pass\nand delay calibrator, 3C~286 as the polarization calibrator, and\nJ1830$-$3602 as the astrometric calibrator.  The observing sequence\ncycled between J1830$-$3602 (2 minutes) and G357.2$-$0.2 (20 minutes)\nwith a flux\/band-pass calibrator (10 minutes) every 2 hours.  Our\nflux-density scale is based on the \\cite{Reynolds94} spectrum of\nPKS~B1934$-$638: \\begin{equation}\n  \\log(S) = -30.7667 + 26.4908 \\bigl(\\log\\nu\\bigr) - 7.0977\n  \\bigl(\\log\\nu\\bigr)^2 $$ $$+0.605334 \\bigl(\\log\\nu\\bigr)^3,\n\\label{eq:pks1934} \\end{equation} where $S$ is the flux density in\nJy and $\\nu$ is the frequency in MHz.\n\n\n\\subsubsection{Analysis}\n\nData flagging and calibration were performed as described for L-band\ndata in \\cite{DEEP2} and \\cite{XGalaxy}.  The UHF session was\ncalibrated independently, and we have adopted the L-band procedure\nfor the UHF data with some band-specific modifications described\nbelow.\n\nFirst, we trimmed 144 channels from each edge of the UHF band to\naccount for the roll-off in receiver response, leaving a frequency\nrange 563--1069\\,MHz. We then used a UHF-specific mask to identify\nfrequency ranges that contain persistent and strong radio frequency\ninterference (RFI). This covers only 934--960\\,MHz, where cellular\ncommunication signals are present.  After combining our empirical\nmask with the editing steps described in \\cite{DEEP2} during\ncalibration, $\\sim 10\\%$ of the target data were flagged from the\ntrimmed UHF band.\n\nThe data were split into 8 sub-bands with equal frequency width and\nthese were calibrated independently. We used a UHF sky model\nextrapolated from the L-band model of the PKS~B1934$-$638 field\ncontaining the power-law spectra of sources appearing brighter than\n1\\,mJy\\,beam$^{-1}$ at 1.3\\,GHz within $1^\\circ$ of PKS~B1934$-$638.\nThe flux density of PKS~B1934$-$638 in each sub-band was obtained\nfrom equation~(1), and used to derive the amplitude spectrum of\nJ1830$-$3602. The amplitudes of the gains measured from J1830$-$3602\nwere scaled by a smooth model fitted to its measured flux densities\nin each sub-band, and the scaled amplitude and phase corrections\nwere interpolated in time and applied to the target data. The data\nwere reweighted using the root mean square (RMS) in the observed\nvisibilities in 10 minute intervals.\n\nThe above extrapolation does not account for sources towards the\nedge of the wider UHF field of view (FoV). However, we have compared\nthe above analysis to one that uses a preliminary model of the full\nUHF FoV of PKS~B1934$-$638, and find no appreciable difference in\nthe derived flux scales above 700\\,MHz. Below this frequency our\nderived flux densities are somewhat (up to 10--20\\%) overestimated.\n\nImaging used the wide-band, wide-field imager MFImage in the\n\\emph{Obit}\npackage\\footnote{\\url{http:\/\/www.cv.nrao.edu\/~bcotton\/Obit.html}}\n\\citep{Obit} as described in \\cite{DEEP2} and \\cite{XGalaxy}.\nMFImage \\citep[described in detail in][]{SourceSize} uses faceting\nto account for the non-coplanarity of the MeerKAT baselines and\nmultiple frequency bins which are imaged independently and CLEANed\njointly to account for frequency variations in the sky and the\nantenna pattern.  Imaging used Robust weighting ($-1.5$ in\n\\emph{AIPS}\/\\emph{Obit} usage) to down-weight the central condensation of\nantennas in the array and improve the resolution.\n\n\n\\subsubsection{Total-Intensity Imaging}\\label{StokesIimaging}\n\nThe data in the two frequency bands were imaged independently.  With\nthe large bandwidth covered by the data, the shortest baseline\nlength in wavelengths varied by a factor of three between the highest\nand lowest frequencies in the two bands.  Due to the large-scale\nemission in the field, if uncorrected, this will lead to a variable\nfraction of the total intensity recovered as a fraction of frequency\nand a frequency-dependent negative bowl around the extended emission.\nThis will artificially cause the spectrum to appear steeper than\nit actually is.  In order to counteract this, an inverted Gaussian\ntaper centered at the origin was applied to the weights of the\nshortest baselines with a Gaussian $\\sigma$ of 500 wavelengths to\nboth the UHF and the L-band data.  \nThis will suppress emission on scales larger than $\\sim$200\\arcsec;\nthis is similar to the spectral index analysis in \\cite{XGalaxy}.  \nA multi-resolution CLEAN was used to help recover the very extended\nemission in the field.   \n\nThe L-band total-intensity data were imaged to a radius of 1$^\\circ$\nplus outlier facets to a distance of $1\\fdg5$ to cover sources\nexpected to appear brighter than 1\\,mJy\\,beam$^{-1}$ based on the\nSUMSS catalog at 843\\,MHz \\citep{SUMSS}.  Three iterations of\nphase-only self-calibration were applied.  The total band-pass was\ndivided into 14 $\\times$ 5\\% fractional bandwidth bands giving\nunequal widths in frequency.  L-band total-intensity imaging used\n366,886 components stopping at a depth of 45\\,$\\mu$Jy\\,beam$^{-1}$\nwith a total flux density of 23.7\\,Jy; the off-source RMS noise is\n20\\,$\\mu$Jy\\,beam$^{-1}$.  The CLEAN restoring beam was an elliptical\nGaussian with FWHM axes $7\\farcs0 \\times 6\\farcs8$ at position angle\n0$^\\circ$.\n\nAt UHF a field of view with radius $2\\fdg5$ was imaged in 14 $\\times$ 5\\%\nfractional bands with phase self-calibration using 419,484 components\nto a minimum of 200\\,$\\mu$Jy\\,beam$^{-1}$ and a total flux density\nof 60.9\\,Jy.  Outliers were added up to $3\\fdg5$ from the pointing.\nThe off-source RMS was 89\\,$\\mu$Jy\\,beam$^{-1}$.  The CLEAN restoring\nbeam was $11\\farcs6 \\times 10\\farcs3$ at position angle $-20^\\circ$.\n\nFor both L band and UHF, the 8\\,s integrations and sub-bands used\nintroduce negligible time and bandwidth smearing ($<2\\arcsec$)\nacross the full imaged FoVs.\n\n\n\\subsubsection{Deconvolution of Stokes Q and U}\n\nOnly the L-band data had adequate polarization calibration and were\nimaged in Stokes Q and U.  In order to recover the polarimetry in\nthe presence of the large Faraday rotation of polarized emission,\na relatively high spectral resolution was used for Stokes Q and U\nimaging --- a 1\\% fractional bandwidth resulting in 68 sub-bands\nacross the band.  The deconvolution also used the joint polarization\nCLEAN described in \\cite{Condon2021}.  Linear polarization imaging\nused 50,000 CLEAN components to a depth of 54\\,$\\mu$Jy\\,beam$^{-1}$\nresulting in an off-source RMS of 10\\,$\\mu$Jy\\,beam$^{-1}$.\n\n\n\\subsection{eROSITA Observations and Analysis} \\label{sec:eROSITA}\n\nThe X-ray eROSITA \\citep[extended R\\\"ontgen Survey Imaging Telescope\nArray,][]{Predehl2020a} is one of two instruments on the Spectrum\nR\\\"ontgen-Gamma observatory \\citep{Sunyaev2021}.  It consists of\nseven aligned X-ray telescopes (TM1--TM7) which have an FoV of\n1\\degr.  All telescopes observe the same sky region simultaneously\nin the 0.2--8\\,keV band-pass.  In survey mode, the instrument's\nangular resolution is $26\\arcsec$. eROSITA started its first all-sky\nsurvey on 2019 December 13, with eight such surveys planned over 4\nyears \\citep[see][]{Predehl2020a}.\n\nThe X-ray data we report here were taken during the first four\neROSITA surveys, eRASS:4. By end 2021 the position of G357.2$-$0.2\nhad been observed with a total of 27 telescope passages during four\nepochs, 2020 March 27--28, 2020 September 28--30, 2021 March 24--25,\nand 2021 September 24--25, resulting in an un-vignetted averaged\nexposure time of 1048\\,s.\n\nThe data used in our analysis were processed by the eROSITA Standard\nAnalysis Software System (\\emph{eSASS}) pipeline and have the\nprocessing number $\\#946$. For the data analysis we used \\emph{eSASS}\nversion 201009\\footnote{See \\url{https:\/\/erosita.mpe.mpg.de\/}}.\nWithin the \\emph{eSASS} pipeline, X-ray data of the eRASS sky are\ndivided into 4700 partly overlapping sky tiles of $3\\fdg6 \\times\n3\\fdg6$ each. These are numbered using six digits, three each for\nR.A. and Dec., encoding the sky tile center position in degrees.\nThe majority of G357.2$-$0.2 falls into the eRASS tiles 266120 and\n266123, with the surrounding tile 263123 also required for a complete\ncoverage of G357.2$-$0.2.\n\n\n\\section{Results\\label{Results}}\n\nThe MeerKAT L-band total-intensity image of G357.2$-$0.2 is shown\nin Figure~\\ref{Heartworm_L}.  \nThe region imaged most prominently contains a complex of filamentary\n(worm-like) structures spanning $\\sim 8\\arcmin$, some of which appear\nto terminate in brighter knots; for the first time, some of these\nfilaments are resolved into striking double tails\n(Figure~\\ref{Worm_L}).  \nThere is no overall organization apparent and this fine-scale\nstructure, at least in projection, is embedded in larger-scale low\nbrightness emission which contains a large amount of flux density.  \n\n\\chg{Since the imaging used in Figures~\\ref{Heartworm_L} and\n\\ref{Worm_L} only used the L-band data and explicitly removed the\nshorter baselines, the most extended emission is attenuated.\nIn order to bring out this extended emission, the UHF data were\nreimaged with enhanced brightness sensitivity ($\\mbox{Robust}=-0.75$) and\nincluding the shorter baselines. \nThis is shown in Figure~\\ref{Heartworm_LoRes} emphasizing the lower\nbrightness regions.\n}\n\nSome of \\chg{the} larger-scale emission appears to be organized in a\npartial shell-like heart-shaped feature spanning $\\sim 18\\arcmin$,\nreported here for the first time. \nOn the basis of this combined morphology, we have nicknamed these\nfeatures the ``Heartworm'' Nebula. \n\n\\begin{figure*}\n\\plotone{fig01.eps}\n\\caption{Reverse gray-scale of the L-band \\chg{(886--1681\\,MHz)} Stokes~I image of G357.2$-$0.2\n(the Heartworm) in double log stretch with a scale-bar at the top\nlabeled in mJy\\,beam$^{-1}$.  The resolution is shown in the box\nat lower left. This rendering optimizes the display of the larger-scale\nlow brightness emission, including the shell-like heart-shaped\nfeature spanning $\\sim 18\\arcmin$ northwards from (R.A., Dec.)\n$\\approx$ ($17^{\\mathrm h}39^{\\mathrm m}15^{\\mathrm s}$,\n$-31\\arcdeg38\\arcmin$).  The central fine-scale features, considerably\nsaturated in this view, are best discerned in Figure~\\ref{Worm_L}.\n} \n\\label{Heartworm_L}\n\\end{figure*}\n\n\\begin{figure*}\n\\plotone{fig02.eps}\n\\caption{Zoom in on Figure~\\ref{Heartworm_L}, with a different\ncontrast (reverse gray-scale in double log stretch with scale-bar\nat the top labeled in mJy\\,beam$^{-1}$), to highlight the fine-scale\nfeatures of G357.2$-$0.2.  The resolution is shown in the box at\nlower left. Prominent knots of emission are labeled (see\nTable~\\ref{tab:knots}).  The bright point source at (R.A., Dec.) =\n($17^{\\mathrm h}39^{\\mathrm m}24^{\\mathrm s}$,\n$-31\\arcdeg31\\arcmin12\\arcsec$) is the pulsar PSR~B1736$-$31 =\nJ1739$-$3131.\n} \n\\label{Worm_L}\n\\end{figure*}\n\n\\begin{figure*}\n\\plotone{fig03.eps}\n\\caption{\\chg{The UHF band (563--1068\\,MHz) Heartworm enhanced brightness\nsensitivity image in reverse \ngray-scale with double log stretch; a scale-bar is shown at the top labeled in\nmJy\\,beam$^{-1}$.\nThe resolution is $12\\farcs4\\times11\\farcs8$ and is shown in the box at\nlower left.} \n}\n\\label{Heartworm_LoRes}\n\\end{figure*}\n\n\n\\subsection{Spectral Index}\n\nThe individual total-intensity frequency-bin images in the UHF and\nL-band images were convolved to a common resolution (that of the\nUHF image \\chg{described in Section~\\ref{StokesIimaging}}) and\ninterpolated to the grid of the L-band image.  After \nprimary beam correction using the frequency-dependent antenna beam\nshape of \\cite{DEEP2}, a spectrum was fitted in each pixel with the\nflux density at 1000\\,MHz $S_{\\rm 1\\,GHz}$ and the spectral index\n$\\alpha$. The spectral index image is displayed in\nFigure~\\ref{Heartworm_SI}.  The northern and western rim of the\nheart are shown in more detail in Figure~\\ref{Heart_SI}.\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig04.eps}\n}\n\\caption{The spectral index of the Heartworm (G357.2$-$0.2). Intensity\nis flux density at 1000\\,MHz with square root stretch and color is\nspectral index as given by the scale-bar at the top. PSR~B1736$-$31,\nwith a typical steep pulsar spectrum, corresponds to the prominent\nred point. See Figure~\\ref{Heartworm_SI_err} for the corresponding\nerror map.\n} \n\\label{Heartworm_SI}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig05.eps}\n}\n\\caption{Like Figure~\\ref{Heartworm_SI} but emphasizing the northern\nand western rim of the heart.\n} \n\\label{Heart_SI}\n\\end{figure}\n\nThe uncertainty in the spectral index depends on both the signal-to-noise\nratio of a feature across the observed band and any systematics\nsuch as the frequency dependent ``missing'' flux density from\nstrongly resolved extended emission (see Section~\\ref{StokesIimaging}).\nThe spectral index error image, based only on the statistical\nuncertainty, is displayed in Figure~\\ref{Heartworm_SI_err}.\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig06.eps}\n}\n\\caption{The error map for the spectral index of the Heartworm\n(G357.2$-$0.2) shown in Figure~\\ref{Heartworm_SI}.  Color represents\nthe statistical uncertainty on the spectral index as given by the\nscale-bar at the top.  This is based only on RMS noise, and does\nnot account for systematic errors related to missing flux (see\nSection~\\ref{StokesIimaging}) or calibration.\n} \n\\label{Heartworm_SI_err}\n\\end{figure}\n\n\n\\subsection{Polarimetry}\n\nThe imaging in Stokes Q and U used 68 $\\times$ 1\\% fractional\nband-pass image planes although many were completely blanked due\nto the editing of RFI.  A rotation measure (RM) fit was performed in each pixel by\ndoing a direct search in Faraday space.  The test Faraday rotation\nthat gives the highest averaged, unwrapped polarized intensity was\ntaken as the Faraday rotation at that pixel, the unwrapped polarization\nangle extrapolated to zero wavelength was taken as the intrinsic\npolarization angle, and the maximum polarized intensity taken as the\npolarized intensity in that pixel.  This is essentially taking the\npeak of the Faraday synthesis \\citep{RMSynthesis}.\n\nFractional polarization ``B'' vectors in the worm are shown in\nFigure~\\ref{Heartworm_PolVec} and the RMs in Figure~\\ref{Heartworm_RM}.\nPolarization was detected only in limited areas but with moderately\nhigh fractional polarization (20--30\\%) and with the magnetic field\nlargely along the linear features and with large and variable Faraday\nrotation.\nThe rotation measures shown in Figure~\\ref{Heartworm_RM} are much less\nthan the 2000\\,rad\\,m$^{-2}$ at $\\lambda = 6$\\,cm found by \\cite{Gray96},\nsupporting the suggestion in the \\chg{Figure~\\ref{Heartworm_RM}}\ncaption that at L band and \nUHF we are seeing only through gaps in the dense foreground screen. \n\n\\begin{figure}\n\\centerline{\n\\includegraphics[height=3.25in]{fig07.eps}\n}\n\\caption{Total intensity contours of the worm in G357.2$-$0.2, with\nsuperposed red fractional polarization ``B'' vectors from the L-band\ndata.  Contours are at 2, 4, 8, 12 and 16 $\\times$ 0.2\\,mJy\\,beam$^{-1}$,\nand a vector length of $10''$ corresponds to 28\\% polarization.\nThe resolution is shown in the box in the lower left corner.\n} \n\\label{Heartworm_PolVec}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig08.eps}\n}\n\\caption{Total intensity contours of the worm in G357.2$-$0.2, with\nsuperposed RM in color with scale-bar at the top in rad\\,m$^{-2}$.\nThe spotty and highly variable nature of the detected Faraday\nrotation suggests that the foreground screen is quite dense and we\nare seeing through gaps.  Contours are at 2, 4, 8, 12 and 16 $\\times$\n0.2\\,mJy\\,beam$^{-1}$.  The resolution is shown in the box in the\nlower left corner.\n} \n\\label{Heartworm_RM}\n\\end{figure}\n\n\n\\subsection{X-ray Image} \\label{sec:X-ray-image}\n\nFigure~\\ref{fig:eROSITA} depicts a three-color image of G357.2$-$0.2\nwhich has been coded according to the energy of the detected X-ray\nphotons. To produce it, we first created images for the three energy\nbands 0.2--0.7\\,keV, 0.7--1.2\\,keV, and 1.2--2.4\\,keV, using data\nfrom all seven telescopes.  The spatial binning in these images was\nset to $26\\arcsec$ to match eROSITA's FoV-averaged FWHM angular\nresolution during survey mode.  In order to enhance the visibility\nof diffuse emission in the three-color image while leaving point\nsources unsmoothed to the greatest possible extent, we applied the\nadaptive kernel smoothing algorithm of \\cite{2006MNRAS.368...65E}\nwith a Gaussian smoothing kernel of $1.5\\,\\sigma$.\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig09.eps}\n}\n\\caption{Three-color image of G357.2$-$0.2 as seen in the eROSITA\nall-sky surveys eRASS:4. Photons to produce the image were color\ncoded according to their energy (red for energies 0.2--0.7\\,keV,\ngreen for 0.7--1.2\\,keV, blue for 1.2--2.4\\,keV).  An adaptive\nkernel smoothing algorithm was applied to the images in each energy\nband. Radio contour lines (yellow) from the image in\nFigure~\\ref{Heartworm_L} are overlaid to outline G357.2$-$0.2.  The\ngreen circle with radius 240\\arcsec\\ encompasses the worm, with a\nfaint unrelated soft point source located towards the southwest,\nindicated by a circle of radius 60\\arcsec.\n}\n\\label{fig:eROSITA}\n\\end{figure}\n\nAs can be seen from Figure~\\ref{fig:eROSITA}, no significant diffuse\nemission was detected from G357.2$-$0.2 during eRASS:4. There is\nsome mixture of very faint soft- (red) to medium-band (green)\nemission overlapping with the radio contour lines within the large\ngreen circle, but its significance is estimated to be only at the\n$\\sim 2.5$--3\\,$\\sigma$ level. Such low level emission is seen at\nvarious other locations in the wider image of all the merged sky\ntiles, making it very speculative to associate this faint emission\nwith G357.2$-$0.2. The small circle in Figure~\\ref{fig:eROSITA}\nindicates the position of a weak soft point source, which seems\nunrelated to the radio features.\n\n\n\\section{Discussion\\label{Discussion}}\n\nThe H\\textsc{i} observations of \\cite{Roy2002} indicate that the\nworm in G357.2$-$0.2 is at a distance of at least 6\\,kpc, possibly\nbeyond the Galactic center, and likely of Galactic origin. Hereafter\nfor the purposes of discussion we assume a distance $d = 8.5$\\,kpc.\nHowever it is quite unlike any known class of Galactic object, with\nthe possible exception of PWNe. The worm has a diameter of\n$\\sim8\\farcm3$ which at the assumed distance is equivalent to\n$\\sim20$\\,pc.\n\n\\subsection{(Not) Star Formation}\n\nInfrared observations of the Heartworm indicate that the bulk of\nthe radio features are unlikely to be related to current star\nformation.\n\nThere are no extended far-infrared (FIR) features visible near the\nworm (Figure~\\ref{3colour}) that could be indicative of thermal\ndust emission. However, the brightest portion of the heart coincides\nwith strong FIR emission and may be an H\\textsc{ii} region unrelated\nto the rest of G357.2$-$0.2 (and hence of unconstrained distance).\nThis interpretation is supported by the flat radio spectrum of this\nregion seen in Figure~\\ref{Heartworm_SI}. A second smaller clump\nof FIR\/sub-mm emission may likewise be an unrelated H\\textsc{ii}\nregion (Figure~\\ref{3colour}).\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig10.eps}\n}\n\\caption{\nThree-color image of the dust emission around the Heartworm Nebula,\nin Galactic coordinates and with arbitrary units. Red and green in\nthe image are respectively coded to PACS 70\\,$\\mu$m and SPIRE\n250\\,$\\mu$m emission from the \\emph{Herschel} Hi-GAL survey\n\\citep{molinari2010}. Blue is coded to 850\\,$\\mu$m emission from\nthe SCUBA-2 Galactic center survey \\citep{parsons2018}. Contours\ntrace the MeerKAT L-band emission from the Heartworm, with levels\nchosen using a power-law fitting scheme to emphasise both low-level\nand bright emission \\citep{thompson2006}. The image shows that there\nis little thermal dust emission associated with the worm, although\nthere is a compact warm dust clump positionally coincident with the\nnorthern end of the heart, indicating a candidate H\\textsc{ii}\nregion, and another such clump and possible H\\textsc{ii} region to\nits west.\n}\n\\label{3colour}\n\\end{figure}\n\nThe strongest argument that the knots in the worm are not H\\textsc{ii}\nregions is based on the observation that they are fairly strong\nradio sources ($S_{\\rm 1\\,GHz}\\sim7$\\,mJy according to\nTable~\\ref{tab:knots}) but are not visible at all ($S_{\\rm 24\\,\\mu\nm} \\ll 5\\,\\sigma$) in the deep \\emph{Spitzer} Enhanced Data Products\n$24\\,\\mu$m image (Figure~\\ref{hworm24}) made with $6\\arcsec$ FWHM\nresolution. The $24\\,\\mu$m flux densities of Galactic H\\textsc{ii}\nregions are typically $30\\times$ their 1.4\\,GHz flux densities\n\\citep{Anderson2014} and the $5\\,\\sigma$ upper limits for sources\nsmaller than $10\\arcsec$ FWHM on the knot positions are $S_{\\rm\n24\\,\\mu m} \\le 1$\\,mJy. Even $A_V = 50$\\,mag of extinction would\nlower $S_{\\rm 24\\,\\mu m}$ by only a factor of 10 \\citep{Anderson2014},\nso $< 5\\%$ of the knot radio emission is likely to be thermal.\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig11.eps}\n}\n\\caption{\nThe \\emph{Spitzer} Enhanced Data Products $24\\,\\mu$m MIPS image\ncovering the Heartworm, with circles centered on the knot positions\nfrom Table~\\ref{tab:knots} (see also Figure~\\ref{Worm_L}). The\nintensity scale on the right has units MJy\\,sr$^{-1} \\approx$\nmJy\\,beam$^{-1}$.\n}\n\\label{hworm24}\n\\end{figure}\n\nThere is also scant indication of correspondence between the compact\nradio features in Figure~\\ref{Worm_L} and infrared emission at\nshorter wavelengths. Knot \\#3 is the closest to a near-\/mid-infrared\n(NIR\/MIR) source, with its peak 1\\farcs3\\ away from a 3.6 and\n8\\,$\\mu$m GLIMPSE-II source \\citep{Churchwell2009}.  This source\nis also detected in the VVV K$_{\\rm s}$ survey but not, as noted\nabove, in MIPSGAL 24\\,$\\mu$m.  The \\citet{Downes1986} P-statistic\nfor the possible association of this 8\\,$\\mu$m 9.259 magnitude\nsource (the probability of finding a brighter IR source closer to\nthe radio peak) is $2.3\\times10^{-3}$.  Nominally, we might thus\nexclude a chance association at the 3\\,$\\sigma$ level. However this\ndoes not account for MeerKAT astrometric errors, which may contribute\nat the $\\sim 1\\arcsec$ level \\citep{Knowles2022,Heywood2022}.  As\nfor the remaining six radio knots, there are no plausible NIR-MIR\ncounterparts.\n\n\n\\subsection{The Worm and the Heart}\n\nBoth the spectrum and polarized emission suggest that the worm emits\nby a nonthermal process, likely synchrotron.  However, the spectrum\nof the emission in much of the worm is relatively flat for synchrotron\nemission suggesting that the radiating electrons have been recently\naccelerated.\n\\chg{Furthermore}, ionization losses can flatten the spectrum by up to\n$\\Delta\\alpha = +0.5$.\n\nDue to the extended size of the heart, much larger than the $\\sim$200\\arcsec\\\nscale filtering in the imaging, much of the emission may be\nresolved out.\nThe rim of this structure survives the filtering of the interferometer\narray.  \nThe spectrum of the bulk of the heart,\nat least in the parts of the rim which are well imaged, is relatively\nsteep (Figures~\\ref{Heartworm_SI}--\\ref{Heartworm_SI_err}) indicating\nan aged relativistic electron population. This excludes the brightest\nand flattest-spectrum portion of the heart, which as noted above\nmay be an unrelated H\\textsc{ii} region (see Figure~\\ref{3colour}).\nOther than positional coincidence, there is no evidence that the\nheart and the worm are physically related.\n\nThe worm also shares the heart with the pulsar B1736$-$31 (bright\nred point in Figure~\\ref{Heartworm_SI}) although as already alluded\nto in Section~\\ref{sec:intro} there is no physical connection between\nthis pulsar and any of the nearby features. This is further supported\nby the RM of the pulsar --- we measure $43.5\\pm0.2$\\,rad\\,m$^{-2}$\n\\citep[compared to $32\\pm8$\\,rad\\,m$^{-2}$ in][]{Rand94} --- which\nis far smaller than that over most of the worm (Figure~\\ref{Heartworm_RM}).\n\n\n\\subsection{The Loopy and Knotty Worm}\n\nThe worm is remarkably complex. Much of its emission seen in\nFigure~\\ref{Worm_L} consists of filaments.  Many of these are either\npaired and connected to a flatter spectrum knot\n(Figure~\\ref{Heartworm_SI_Close}) or are loops.  Where the polarization\nwas detectable, the magnetic field appears to be along the filaments\n(Figure~\\ref{Heartworm_PolVec}) suggesting that they are magnetically\nconfined structures which have been dragged into their current\nconfiguration, possibly by what is causing the bright knots.  The\nflatter spectra near the knots (an example spectrum together with\na least squares fit is given in Figure~\\ref{Heartworm_SI_Point})\nsuggest that these are the locations at which electrons are\naccelerated. The identified knots have all very nearly the same\nflux densities and nonthermal spectra (Table~\\ref{tab:knots}), with\nno hint of a break or turnover in the frequency range observed.\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[width=3.25in]{fig12.eps}\n}\n\\caption{Like Figure~\\ref{Heartworm_SI} but a close up with a tighter\nrange of spectral index. Note that the region immediately surrounding\nthe worm appears to have a very steep spectrum ($\\alpha \\sim -1$),\nbut this may be affected by the negative bowl due to missing flux\n(see Section~\\ref{StokesIimaging} and also Figure~\\ref{Heartworm_SI_err}).\n} \n\\label{Heartworm_SI_Close}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\n\\includegraphics[height=3.25in,angle=-90]{fig13.eps}\n}\n\\caption{Spectrum of knot \\#6 in Figure~\\ref{Worm_L}; see also\nFigure~\\ref{Heartworm_SI_Close} and Table~\\ref{tab:knots}.  UHF\ndata are displayed as ``+'' and L band as ``*''.  The line is the\nfitted spectrum given in the figure, with the flux density provided\nfor a frequency of 1000\\,MHz. Note the match in flux densities\nindependently determined in the overlapping range $\\approx\n900$--1050\\,MHz.\n} \n\\label{Heartworm_SI_Point}\n\\end{figure}\n\n\\begin{deluxetable}{lrrcc}\n\\tablewidth{0pt}\n\\tablecolumns{5}\n\\tablecaption{Seven knots within the G357.2$-$0.2 nebula}\n\\tablehead{\n        \\colhead{\\#\\tablenotemark{a}} & \\colhead{R.A.} & \\colhead{Dec.} &\n        \\colhead{$S_{\\rm 1\\,GHz}$\\tablenotemark{b}} & \n\t\\colhead{$\\alpha$\\tablenotemark{b}} \\\\\n\t\\colhead{} & \\colhead{($^{\\mathrm h}~^{\\mathrm m}~^{\\mathrm s}$)} & \n\t\\colhead{$(\\arcdeg\\:\\arcmin\\:\\arcsec)$} & \\colhead{(mJy)} & \\colhead{} \n}\n\\startdata\n1 & 17 39 33.95 & $-31$ 30 29.9 & 7.2 & $-0.37$ \\\\\n2 & 17 39 34.95 & $-31$ 30 16.9 & 7.0 & $-0.36$ \\\\\n3 & 17 39 37.15 & $-31$ 24 38.3 & 6.1 & $-0.39$ \\\\\n4 & 17 39 39.56 & $-31$ 27 44.6 & 7.2 & $-0.43$ \\\\\n5 & 17 39 40.18 & $-31$ 28 01.2 & 7.5 & $-0.39$ \\\\\n6 & 17 39 43.03 & $-31$ 27 55.1 & 7.1 & $-0.37$ \\\\\n7 & 17 39 54.87 & $-31$ 28 21.7 & 5.9 & $-0.33$ \\\\\n\\enddata\n\\tablenotetext{a}{Knots are labeled as in Figure~\\ref{Worm_L}.}\n\\tablenotetext{b}{Flux density values at 1\\,GHz and spectral index\n$\\alpha$ are obtained from pixel-by-pixel fitting over the UHF and\nL bands. Uncertainties in $S_{\\rm 1\\,GHz}$ and $\\alpha$ (noise\ncomponents only) are $35\\,\\mu$Jy and 0.02 respectively for each\nknot.}\n\\label{tab:knots}\n\\end{deluxetable}\n\nThere is also a long, relatively straight filament appearing to\nconnect the center of the worm to the southwestern part of the\nheart, at least in projection (see \\chg{Figures~\\ref{Heartworm_L}--\\ref{Heartworm_LoRes}}). \nIt is unclear what connection if any this\nfilament might have to the overall features.\n\nThe spotty but high RMs seen in Figure~\\ref{Heartworm_RM} and the\nstrong depolarization reported by \\cite{Gray96} indicate that the\nemission is behind a relatively dense plasma.  \\cite{Gray96} shows\npolarized emission at 5\\,GHz over most of the worm (Fig.~2) but\nreports little polarization at 1.5\\,GHz.  The author infers\n$\\mbox{RM}\\sim2000$\\,rad\\,m$^{-2}$.  This value is substantially\nhigher than those seen in Figure~\\ref{Heartworm_RM}; however, our\nresolution is higher than that of \\cite{Gray96} at 1.5\\,GHz and we\nmay just be seeing through gaps in an otherwise dense Faraday screen.\nNearby sources, presumed to be background AGNs, have RMs ranging\nfrom $-120$ to +160 rad\\,m$^{-2}$ which is outside most of the range\nshown in Figure~\\ref{Heartworm_RM}, indicating that the bulk of the\nFaraday rotation in front of the worm is local to it.\n\nThe filamentary and tangled structure of the worm bears resemblance\nto some known PWNe. For instance, the PWN in the composite SNR~G0.9+0.1\n(Figure~\\ref{G0.9+0.1PWN}) displays a complex web of twisted filaments\n(without reported polarization measurements).  By contrast to the\nworm, however, no prominent knots of emission are seen in G0.9+0.1.\nConversely, its compact PWN is known to be powered by one of the\nmost energetic pulsars in the Galaxy \\citep{Camilo2009a}, while no\nsuch powering source has been identified for the worm.\n\n\\begin{figure}\n\\includegraphics[width=3.25in]{fig14.eps}\n\\caption{\nMeerKAT image at 1.28\\,GHz showing the PWN at the center of the\nSNR~G0.9+0.1. The torus and jet structure inferred from X-ray\nobservations \\citep{gaensler01}, and subsequently reported in radio\nimaging by \\citet{dubner09}, is revealed here to be a more complex\nweb of tangled filamentary structures surrounding a prominent central\npoint-like source \\citep[presumably the pulsar discovered\nby][]{Camilo2009a}. Compare to the G357.2$-$0.2 worm in\nFigure~\\ref{Worm_L}.  The angular resolution is 4\\arcsec, shown in\nthe lower left.  The reverse gray-scale is linear with scale-bar at\nthe top labeled in mJy\\,beam$^{-1}$.  Adapted from \\cite{Heywood2022}.\n} \n\\label{G0.9+0.1PWN}\n\\end{figure}\n\n\n\\subsection{Pulsar Wind Nebula?}\n\n\\subsubsection{The Heartworm as a Composite SNR} \\label{sec:g327}\n\nComposite SNRs manifest as a shell (possibly partial and\/or distorted)\nresulting from the supernova explosion shockwave interacting with\nthe interstellar medium, together with an interior PWN powered by\na suitably energetic pulsar. The PWNe in middle-aged or older\ncomposite SNRs are often complex in structure due to the fact that\nthey have been disrupted by the SNR reverse shock (RS). Particularly\nin cases for which the shockwave has evolved in a nonuniform medium,\nthis disruption can result in a complex structure in which the relic\nPWN becomes highly distorted \\citep{Blondin2001,Kolb2017}, and in\nwhich freshly injected particles and magnetic flux create a new\nextended structure near the pulsar. The worm in G357.2$-$0.2, while\nunique in some ways, shares several properties of the comparatively\nbright PWN in G327.1$-$1.1, which appears to be an example of a\nsystem that has undergone an interaction between the PWN and an\nasymmetric RS \\citep{Temim2009,Temim2015}.\n\nAustralia Telescope Compact Array images of G327.1$-$1.1 taken at\n3~cm show diffuse emission from the PWN along with a network of\nfilamentary structures accompanied by bright knots \\citep{Ma2016}.\nAccompanying polarization measurements at 6~cm show that the magnetic\nfield is largely aligned with the filaments.  G327.1$-$1.1 also has\na dense and variable Faraday screen with up to $-600$\\,rad\\,m$^{-2}$\nand an average of $-380$\\,rad\\,m$^{-2}$ \\citep{Ma2016}.  These\nfeatures are similar to what is seen in G357.2$-$0.2 in\nFigures~\\ref{Heartworm_PolVec} and \\ref{Heartworm_RM}.\n\nAn elongated structure in G327.1$-$1.1 also extends from the putative pulsar ---\nidentified as an X-ray source with spectral properties consistent\nwith a neutron star --- back into the relic nebula.  Hydrodynamical\nstudies show that this appears to be associated with the current\noutflow from the pulsar, swept into a tail-like structure by the\nRS.  More detailed MHD studies are required to assess whether finer\nfilamentary structures such as seen in the worm might be formed in\nthis type of RS\/PWN interaction.\n\nIf the larger heart structure in G357.2$-$0.2 is considered to be\nthe shell of an SNR, then assuming a Sedov solution \\citep[see,\ne.g.,][]{Matthews98} yields an age of about $21\\,d_{8.5}^{5\/2}\n(n_0\/E_{51})^{1\/2}$\\,kyr.  For such a solution, the RS would have\nalready propagated back to the central regions of the SNR. This is\nsimilar to the age estimate for G327.1$-$1.1 ($\\sim 17$\\,kyr) at a\ndistance of 9\\,kpc.  The radio spectral index for the entire nebula\nin G327.1$-$1.1 is $\\alpha \\sim -0.3$, typical of PWNe, although\nthe tail-like structure extending from the pulsar has a steeper\nspectrum with $\\alpha \\sim -0.6$, similar to the filamentary\nstructures in the worm.\n\n\n\\subsubsection{X-ray Limits}\n\nPulsars that power appreciable PWNe convert a fraction of their\nspin-down luminosity $\\dot E$ into nonthermal X-rays.  Here we\ninvestigate whether the limits on X-ray emission obtained from the\neROSITA image presented in Section~\\ref{sec:X-ray-image} are\nconsistent with a PWN interpretation for the worm in G357.2$-$0.2.\nIn what follows we assume that the absorbing hydrogen column to\nG357.2$-$0.2 is $N_{\\rm{H}}= 10^{22}$\\,cm$^{-2}$. This is the total\naverage column in the direction of the worm \\citep{HI4PI2016}, which\nwe use in the absence of other constraints.\n\nWe calculate limits separately for the presence of a point source,\nthe putative pulsar powering the PWN, as well as extended emission\nfrom the candidate PWN. In what follows we always report unabsorbed\nflux and luminosity limits, i.e., intrinsic to the source after\ncorrection for the assumed absorbing column. All limits are reported\nat the $3\\,\\sigma$ level.\n\nNo X-ray point source is detected in eRASS:4 within the bounds of\nthe presumed PWN, indicated by radio contours inside the large green\ncircle in Figure~\\ref{fig:eROSITA}. We considered two different\nemission free spots within this region and obtained a mean cumulative\nTM1--TM7 count rate for a putative point source of $<0.059$\\,cts\\,s$^{-1}$\nin the 0.2--8\\,keV band.\n\nPulsars detected in X-rays that power PWNe have power-law spectra\nwith photon index $\\Gamma_{\\rm psr}$ in the range 1.0--2.7 \\citep[see,\ne.g.,][]{Becker09}. Here we assume $\\Gamma_{\\rm psr} = 1.7$\n\\citep[e.g., applicable to PSR~J2021+3651 with $\\dot E =\n3\\times10^{36}$\\,erg\\,s$^{-1}$,][]{Hessels2004}.  For this spectrum,\nthe above count rate limit yields $f_x(0.2-8\\,\\mbox{keV}) <\n1.3\\times10^{-13}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$ for the unabsorbed\nenergy flux of an undetected point source. For comparison with a\nmore commonly referenced band, $f_x(0.2-2.4\\,\\mbox{keV}) <\n7.9\\times10^{-14}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$. Using the assumed\n$d=8.5$\\,kpc for G357.2$-$0.2, we estimate that the isotropic X-ray\nluminosity of the undetected putative neutron star is $L_{x, \\rm\npsr} = 4 \\pi d^2 f_x < 6.9 \\times 10^{32}$\\,erg\\,s$^{-1}$ within\nthe 0.2--2.4\\,keV band.\n\nThe observed nonthermal X-ray efficiency of rotation-powered pulsars\n($\\eta_{x, \\rm psr} \\equiv L_{x, \\rm psr} \/ \\dot E$) clusters around\n$10^{-3}$ in the 0.1--2.4\\,keV band\n\\citep[see][]{1997A&A...326..682B,Becker09}. The above point source\nlimit therefore nominally implies $\\dot{E} < 6.9\\times\n10^{35}$\\,erg\\,s$^{-1}$. Given the scatter in the $\\eta_{x, \\rm\npsr}$ relation, and the uncertainties in $N_{\\rm H}$ and $d$, this\nlimit does not exclude the existence of a pulsar of intermediate\n$\\dot E \\sim 10^{36}$\\,erg\\,s$^{-1}$ powering G357.2$-$0.2 and\nbeaming towards the Earth. Also, it is always possible that an\nunfavorable beaming geometry would preclude direct detection of\nnonthermal emission from a pulsar regardless of $\\dot E$ and\nsensitivity.  However, regardless of geometry a suitably energetic\npulsar should manifest itself via a diffuse PWN.\n\nTo constrain extended X-ray emission from G357.2$-$0.2, we derived\nthe count rate limit within the circle of radius 240\\arcsec\\ in\nFigure~\\ref{fig:eROSITA}, which encompasses most of the putative\nradio PWN, after subtracting the contribution from the faint\nsouthwestern point source. We obtained a cumulative count rate\n$<0.18$\\,cts\\,s$^{-1}$ in the 0.2--8\\,keV band.\n\nPWNe detected in X-rays have power-law spectra with $\\Gamma_{\\rm\npwn}$ in the range 1.0--2.2 \\citep[see, e.g.,][]{2008AIPC..983..171K}.\nHere we assume $\\Gamma_{\\rm pwn} = 2.0$ (e.g., applicable to the\nG327.1$-$1.1 PWN discussed in Section~\\ref{sec:g327}). For this\nspectrum, the above count rate limit gives $f_x(0.2-8\\,\\mbox{keV})\n< 4.1\\times10^{-13}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$.  In turn, with\n$d=8.5$\\,kpc we obtain $L_{x, \\rm pwn} = 4 \\pi d^2 f_x < 3.6 \\times\n10^{33}$\\,erg\\,s$^{-1}$ for the putative PWN in G357.2$-$0.2.\n\nThe observed X-ray efficiency of PWNe spans a wide range, with the\nbulk within $10^{-5} < \\eta_{x, \\rm pwn} < 10^{-2}$\n\\citep{2008AIPC..983..171K}. In any case, there are many instances\nof X-ray PWNe powered by pulsars with $\\dot E = 10^{36-37}$\\,erg\\,s$^{-1}$\n(e.g., PSR~J2021+3651 and Vela) that have $L_{x, \\rm pwn}$ below\nour limit for G357.2$-$0.2, and a few such instances powered by\npulsars with even higher $\\dot E$ \\citep[e.g.,\nPSR~J2229+6114,][]{Halpern2001}.\n\nTherefore, the current X-ray limits\\footnote{We have also analyzed\n\\emph{Swift} X-Ray Telescope observations of this region resulting\nin the concatenated image available at\n\\url{https:\/\/www.swift.ac.uk\/2SXPS\/Fields\/10000013359}.  No sources\nare detected and the limits at the location of G357.2$-$0.2 are 5\ntimes poorer than those from the eROSITA observations.} do not rule\nout that G357.2$-$0.2 may be powered by a pulsar of intermediate\n$\\dot E$, like many that power a variety of PWNe.\n\nFor completeness, we also searched the \\emph{Fermi}-LAT 4FGL catalog\n\\citep{Fermi4FGL} for a source coincident with G357.2$-$0.2 but\nthere are none. This is not constraining: while many energetic\npulsars emit in GeV $\\gamma$-rays, their $\\dot E\/d^2$ flux needs\nto be large \\citep{Fermi2PC}.\n\n\n\\section{Summary\\label{Summary}}\n\nG357.2$-$0.2 consists of two possibly related components, the\n``worm'', a series of filaments; and the ``heart'' which is an\nextended heart-shaped feature of which we may only see the rim.\nH\\textsc{i} observations of \\cite{Roy2002} show the worm to be of\nGalactic origin.  The pulsar B1736$-$31 appears inside the heart\nbut is a chance positional coincidence.  Part of the rim of the\nheart appears to be an unrelated H\\textsc{ii} region.\n\nThe spectrum and polarization of the emission indicate that the\nbulk of the emission from both the worm and the heart is nonthermal\nsynchrotron.  There is a dense plasma, possibly associated with the\nheart, that results in a large Faraday rotation and some depolarization\nof the emission from the filaments of the worm.  These appear to\nbe magnetic structures lit up by particle acceleration in knots\nwhich are associated with the filaments and which appear to be\ndragging the magnetic field tubes.  The nature of these knots is\nuncertain.\n\nThe structure of the worm at least superficially resembles some\nPWNe with much of the emission appearing in the form of tangled\nfilaments.  More sensitive X-ray observations are of particular\ninterest to further understand the nature of this source.  MeerKAT\nobservations at S band, with higher angular resolution and less\nsusceptible to depolarization, may also be instructive. In addition,\ndetailed hydrodynamical studies could be revealing.  An ultra-deep\nradio pulsar search might also be illuminating \\citep[see][]{Camilo2009b}.\nNevertheless, if close to the Galactic center, this $\\sim20$\\,pc\nstructure would be a very large PWN.  The possibility remains that\nthis is a more exotic object, perhaps sculpted in part by interaction\nwith outflows from the Galactic center region.\n\nThe radio imaging products presented here are made available with\nthis article\\footnote{\\url{https:\/\/doi.org\/10.48479\/q20r-hb79}},\n\\chg{including Stokes~I (L band, UHF+L band, UHF enhanced surface\nbrightness sensitivity),\nspectral index (UHF+L band), and Stokes Q and U L-band cubes.} Raw\nvisibility products are available from the MeerKAT data\narchive\\footnote{\\url{https:\/\/archive.sarao.ac.za}} under project\ncode SSV-20200720-SA-01.\n\n\\acknowledgments\n\\chg{We would like to thank the anonymous reviewer for helpful comments.}\nThe MeerKAT telescope is operated by the South African Radio Astronomy\nObservatory which is a facility of the National Research Foundation,\nan agency of the Department of Science and Innovation.\nThe National Radio Astronomy Observatory is a facility of the National\nScience Foundation, operated under a cooperative agreement by Associated\nUniversities, Inc.\nMAT acknowledges support from the UK's Science \\& Technology Facilities\nCouncil [grant number ST\/R000905\/1].\neROSITA is the primary instrument aboard SRG, a joint Russian-German\nscience mission supported by the Russian Space Agency (Roskosmos),\nin the interests of the Russian Academy of Sciences represented by\nits Space Research Institute (IKI), and the Deutsches Zentrum f\\\"ur\nLuft- und Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin\nAssociation (NPOL) and its subcontractors, and is operated by NPOL\nwith support from IKI and the Max Planck Institute for Extraterrestrial\nPhysics (MPE).  The development and construction of the eROSITA\nX-ray instrument was led by MPE, with contributions from the Dr.~Karl\nRemeis Observatory Bamberg \\& ECAP (FAU Erlangen-N\\\"urnberg), the\nUniversity of Hamburg Observatory, the Leibniz Institute for\nAstrophysics Potsdam (AIP), and the Institute for Astronomy and\nAstrophysics of the University of T\\\"ubingen, with the support of\nDLR and the Max Planck Society.  The Argelander Institute for\nAstronomy of the University of Bonn and the Ludwig Maximilians\nUniversit\\\"at Munich also participated in the science preparation\nfor eROSITA. The eROSITA data shown here were processed using the\neSASS\/NRTA software system developed by the German eROSITA consortium.\n\n\\facilities{MeerKAT, eROSITA}\n\n\n\\software{\\emph{Obit} \\citep{Obit}}\n\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe laws of Brownian motion, formulated first by Einstein more than\na century ago \\cite{Einstein}, have now found so many applications\nand generalizations in all quantitative sciences \\cite{Haw}. Many\nfractal structures in the nature can be derived from the sample\npaths of Brownian motion characterized by some appropriate fractal\ndimensions \\cite{Mandelbrot}.\n\n\nA $d$-dimensional Brownian motion is known to be recurrent, i.e.,\nthe particle returns to the origin, for $d\\leq$2 and escapes to\ninfinity for $d>$2. It is also known that the fractal (Hausdorff)\ndimension of the graph of a Brownian motion is equal to 3\/2 for\n$d=$1, and 2 for $d\\geq$2.\n\n\nThe scaling limit of interfaces in various critical 2$d$ lattice\nmodels are proven or conjectured to be described by the family of\nconformally invariant random curves i.e., Schramm-Loewner evolution\n(or SLE$_\\kappa$) \\cite{schramm} which is driven by a 1$d$ Brownian\nmotion of diffusivity $\\kappa$ \\cite{SLE}.\n\n\n\nOne of the most important invariance properties of planar Brownian\nmotion is conformal invariance. Although the scaling limit of 2$d$\nrandom walk, i.e., 2$d$ Brownian motion, because of self-crossing\nitself does not fall in the SLE category, variations of Brownian\nmotion are described by SLE. Loop erased random walk (LERW) where\nloops are removed along the way, is one of the examples which has\nbeen studied by Schramm and shown that can be described by\nSLE$_2$.\\\\The external perimeter of 2$d$ random walk is also a\nnon-intersecting fractal curve which can be defined by SLE.\nVerifying an earlier conjecture by Mandelbrot \\cite{Mandelbrot}, it\nhas been proven using SLE techniques \\cite{Lawler} that the fractal\ndimension of the Brownian perimeter is $d_f=$4\/3, i.e, the same as\nthe fractal dimension of self-avoiding random walk (SAW) and the\nexternal perimeter of the percolation hull.\n\nIn this paper, we investigate the statistical and fractal properties\nof a 3$d$ random walker which is attracted by a plane. We believe\nthat this study can provide useful intuitive extensions for many\nrelated physical phenomena including the problems with a discrete\ntime lattice walk \\cite{appl0, appl1}, relaxation phenomena\n\\cite{relax}, exciton trapping \\cite{trap} and diffusion-limited\nreactions \\cite{appl1, react}.\n\n\n\\section{The model}\n\nWe consider a random walker moving along the bonds of a cubic\nlattice with the \\emph{xy}-plane as an attractive plane. The\n'walker' source is considered to be the origin of the coordinate\nsystem. At each lattice point with $z\\neq0$, there are six\npossibilities for the random walker to select a link and move along.\nIn our model, the random walker prefers walking on and near the\nattractive plane, and thus the probability that the random walker\nchooses the link which approximates it to the attractive plane is\nset to be $\\alpha p$, and for remaining five links is considered to\nbe $p$, such that $\\alpha>1$ (and will be called \\emph{the strength\nof attraction}) and $p=\\frac{1}{\\alpha+5}$. For each lattice point\non the attractive plane with $z=0$, the probability that each of the\nfour links on the plane to be chosen is set to be $\\alpha p'$ and\nfor two other links perpendicular to the plane is considered to be\n$p'$, where $p'=\\frac{1}{4\\alpha+2}$. The single parameter $\\alpha$\nin our model, controls the strength of attraction. Note that in the\nlimiting case $\\alpha\\rightarrow\\infty$ our model reduces to the\npure 2$d$ random walk on the plane, and for $\\alpha=1$ the pure 3$d$\nrandom walk would be recovered.\\\\Thus there are four possible\nprobabilities. $\\alpha p'$ for links that are in the attractive\nplane, $p'$ for links from the attractive plane to either of the\nneighboring planes, $p$ for links in all of the neighboring planes\nor leading from them into the bulk, and $\\alpha p$ for links from\nall the neighboring planes to the attractive plane.\\\\By detailed\nbalance, in equilibrium at inverse temperature $\\beta$, the ratio\n$\\alpha p \/ p'$ of the probabilities onto and off the attractive\nplane, defines an attraction energy $\\beta\\epsilon = \\ln[2\\alpha\n(1+2\\alpha)\/(\\alpha +5)]$.\n\n\n\\section{Fractal dimension of the set of all visited sites and its level set}\n\n\\begin{figure}[b]\\begin{center}\n\\includegraphics[scale=0.39]{Fig0.eps}\n\\narrowtext\\caption{\\label{Fig0}(Color online) The average number of\ntotal lattice sites $M^{(3d)}$ visited (at least) once by the\nattracted random walker (ARW) (main panel), and those $M^{(2d)}$ on\nthe attractive plane (inset), as function of their average radius of\ngyration for two different values of the strength of attraction\n$\\alpha=$1.3 ($\\blacksquare$) and $\\alpha=$10 ($\\blacktriangle$).\nThe solid lines show the best fit to our data. The error bars are\nalmost the same size as the symbols.}\\end{center}\n\\end{figure}\n\nIn the cases of random walks, systems exhibit a \\emph{generic scale\ninvariance}, meaning that the systems can exhibit self-similarity\nand power laws without special tuning of parameters. This is why we\nalready expect that our model would exhibit rich fractal properties\nfor all values of $\\alpha$.\\\\Let us first look at the fractal\nspatial structure of the 3$d$ \\emph{attracted} random walk (ARW) and\nits intersection with the attractive plane. In order to estimate the\nfractal dimension $d_f$ of the set of points visited (at least) once\nby the random walker, we examine the scaling relation between the\naverage number of such points $M^{(3d)}$ and their corresponding\nradius of gyration $R_g$, i.e., $M^{(3d)}\\sim R_g^{d_f}$. Each\nensemble averaging for $M^{(3d)}$ (and also for $M^{(2d)}$ in the\nfollowing) and $R_g$ was taken over $5\\times 10^4$ independent\nsamples for a fixed number of random walk steps $N$. The\nmeasurements were done for $10^3\\leq N\\leq 10^5$ with the number\ninterval $\\delta N=2\\times 10^3$. We have also computed the fractal\ndimension of the total number of sites on the attractive plane\n(i.e., $M^{(2d)}$) visited by the random walker (in this case the\ncorresponding radius of gyration is computed for all set of distinct\nvisited sites only on the attractive plane $-$ see Fig.\n\\ref{Fig0}).\\\\ We find that the fractal dimensions have a remarkable\ncontinuous dependence on the parameter $\\alpha$. The results of\nthese fractal dimensions as function of the strength of attraction\n$\\alpha$ are illustrated in Fig. \\ref{Fig1}. As can be seen from\nfigure \\ref{Fig1}, for large values of $\\alpha$, since the problem\nreduces to the 2$d$ random walk on the attractive plane, these two\nfractal dimensions converge to the same value close to the value\n$\\sim1.83$ (this is comparable with the fractal dimension of the set\nof distinct sites visited by an 2$d$ RW on a square lattice, deduced\nfrom the results reported in \\cite{Lee}).\n\\\\All error bars in this paper are estimated by using the standard\nleast-squares analysis, and are almost of the same size as the\nsymbols used in the figures.\n\\\\For an ideal linearly self-similar fractal of dimension $d_f$, one\nexpects that the fractal dimension of the intersection being\n$d'_f=d_f-1$ \\cite{Mandelbrot}. But this is not apparently the case\nhere for $\\alpha\\neq 1$, since in our model, the attractive plane\nhas disturbed the homogeneity of the probability distribution in the\n\\emph{z}-direction. Only for $\\alpha=1$ where $d_f=2$ \\footnote{The\nrandom walk on a simple cubic lattice is a \\emph{transient} process,\nsince it has a finite escape probability $\\approx$ 0.66. Therefore,\nthe number of distinct visited sites by the random walker is almost\nthe same as the number of steps or equivalently the trajectory\nlength, and thus, it is expected for both to have a same fractal\ndimension 2.}, we find $d'_f=1=d_f-1$.\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[scale=0.39]{Fig1.eps}\n\\narrowtext\\caption{\\label{Fig1}(Color online) The fractal dimension\nof the set of all lattice points visited (at least) once by the\nattracted random walker (ARW) ($\\blacksquare$), and the set of all\nnumber of visited points on the attractive plane ($\\square$), as\nfunction of the strength of attraction $\\alpha$. The error bars are\nalmost the same size as the symbols.}\\end{center}\n\\end{figure}\n\n\\section{Cluster size distribution on the attractive plane}\n\nHenceforth we investigate the fractal and scaling properties of the\nset of all distinct sites visited by the 3$d$ ARW only on the\nattractive plane. Each of these sites is visited at least once by\nthe 3$d$ ARW and marked upon visiting (if not already).\\\\In this\nsection, rather than analyzing the  properties of the whole set,\nafter marking all visited sites on the plane, we identify each\ncluster-site as a set of all nearest-neighbor visited-sites on the\nlattice with a specific color. Two typical examples of such\nclustering are shown in Fig. \\ref{Fig2} for two different values of\nthe strength of attraction $\\alpha=2$ and $\\alpha=10$. As Fig.\n\\ref{Fig2} shows, for lower values of $\\alpha$, there exist many\nisolated clusters of different scales which are accessed by the ARW\nonly via the third dimension. By increasing the strength of the\nattraction, the number of isolated clusters decreases until\n$\\alpha\\rightarrow\\infty$ for which, there will be only one large\ncluster on the attractive plane.\n\n\n\\begin{figure}[h]\\begin{center}\n\\includegraphics[scale=0.28]{Fig2a.eps}\\hspace{0.5cm}\\includegraphics[scale=0.23]{Fig2b.eps}\n\\narrowtext\\caption{\\label{Fig2}(Color online) Typical samples of\nclusters of the visited sites on the attractive plane by a 3$d$ ARW\nof $N=10^6$ shown in different colors, for $\\alpha=2$ (left) and\n$\\alpha=10$ (right). }\\end{center}\n\\end{figure}\n\nTo examine possible scale invariance of cluster ensemble for rather\nsmall values of $\\alpha$, we compute the cluster size distribution\nand check whether it follows a power-law scaling. In the critical\nstatistical physics, the scaling properties of fractal clusters can\nbe described by the percolation theory \\cite{SA}, where the\nasymptotic behavior of cluster distribution $n_s(\\lambda)$ near the\ncritical point $\\lambda\\rightarrow \\lambda_c$ has the following\ngeneral form \\be\\label{Eq1}n_s(\\lambda)=\ns^{-\\tau}F[(\\lambda-\\lambda_c)s^{\\sigma}],\\ee where $\\sigma$ is an\ncritical exponent, and the scaling function $F(u)$ approaches to a\nconstant value for $|u|\\ll 1$ and decays rather fast for $|u|\\gg 1$.\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[scale=0.4]{Fig3.eps}\n\\narrowtext\\caption{\\label{Fig3}(Color online) Cluster size\ndistribution exponent $\\tau$ defined in Eq. (\\ref{Eq1}), as a\nfunction of the strength of attraction $\\alpha$. Inset: number\ndensity $n_s$ of clusters of the visited lattice sites of size $s$\non the attractive plane for three different values $\\alpha=1.2$, $4$\nand $8$. The solid lines show the power-law behavior in the scaling\nregion. The error bars are almost the same size as the\nsymbols.}\\end{center}\n\\end{figure}\n\nWe undertook simulations for several values of $\\alpha$ to measure\nthe distribution of the cluster sizes of the visited lattice sites\nby the 3$d$ ARW on the attractive plane (this is the probability\nthat a visited lattice site on the attractive plane belongs to a\ncluster of size $s$). We gathered ensembles of a number of\n$5\\times10^4$ (for smaller $\\alpha$) and $1.5\\times10^6$ (for larger\nvalues of $\\alpha$) independent samples of fractal patterns with\nmarked visited-sites on the attractive plane. The number of the\nrandom walk steps was chosen to be $N=4\\times10^6$ in all\nsimulations. The number density $n_s$ of the clusters of size $s$\nhas then been computed for each specific value of $\\alpha$ by\ncounting the number of clusters of size $s$ divided by the total\nnumber of all clusters.\\\\We find that for rather small and\nintermediate size scale clusters, the distribution shows a power law\nbehavior compatible with the scaling relation in Eq. (\\ref{Eq1}). As\ncan be seen in the inset of Fig. \\ref{Fig3}, the curves for\ndifferent values of $\\alpha$ exhibit a sharp drop-off, indicating\nindeed that they contain only small clusters. By increasing $\\alpha$\nthe interval for scaling region decreases and a peak appears which\nsignals the formation of large scale clusters.\\\\Our estimation of\nthe cluster size distribution exponent $\\tau$ in the scaling region\nas a function of $\\alpha$ is also shown in Fig. \\ref{Fig3}. One\nobserves that the exponent $\\tau$ has a significant dependence on\nthe strength of attraction $\\alpha$.\n\n\n\\begin{figure}[t]\\begin{center}\n\\includegraphics[scale=0.39]{Fig4.eps}\n\\narrowtext\\caption{\\label{Fig4}(Color online) The fractal dimension\nof the perimeter of a cluster of visited sites on the attractive\nplane by 3$d$ ARW, as a function of the strength of attraction\n$\\alpha$. Inset: the average length of the perimeter $l$ of a\ncluster versus its average radius of gyration $r_g$, for two\ndifferent strengths of attraction $\\alpha=1.2$ (upper graph) and\n$\\alpha= 16$ (lower graph). The solid lines show the power-law\nbehavior in the scaling region. The error bars are almost the same\nsize as the symbols. }\\end{center}\n\\end{figure}\n\n\\section{Fractal dimension of the cluster boundaries on the attractive plane}\n\nThe remainder of this paper is dedicated to investigate the fractal\nproperties of the boundaries of the visited-sites clusters on the\nattractive plane.\\\\Given a configuration of visited sites by the\n3$d$ ARW on the attractive plane, the first step is to identifying\ndifferent clusters as outlined before. After that, the boundary\ncurve of each isolated cluster has to be identified. However the\ndefinition of interfaces and cluster boundaries on a square lattice\ncan contain some ambiguities, there has been introduced a\nwell-defined \\emph{tie-breaking} rule in \\cite{Saberi} that\ngenerates non-intersecting cluster boundaries on a square lattice\nwithout any ambiguity.\\\\To define the hull for each identified\ncluster according to the algorithm defined in \\cite{Saberi}, a\nwalker (which, of course, has to be distinguished from the 3$d$ ARW)\nmoves clockwise along the edges of the dual lattice (which is also a\nsquare lattice) around the cluster starting from a given boundary\nedge on the cluster. The direction at each step is always chosen\nsuch that walking on the selected edge leaves a visited site on the\nright and an empty plaquette on the left of the walker. If there are\ntwo possible ways of proceeding, the preferred direction is that to\nthe right of the walker. The directions \\emph{right} and \\emph{left}\nare defined locally according to the orientation of the\nwalker.\\\\According to this procedure, we have generated an ensemble\nof cluster boundary loops for several different strengths of\nattraction in the range $1.1\\leq\\alpha\\leq16$. Using the scaling\nrelation $l\\sim r_g^{d_f}$, between the average length of the\nperimeter of the loops $l$, and their average radius of gyration\n$r_g$, we computed the fractal dimension $d_f$ of the cluster\nboundaries as a function of $\\alpha$. The results are shown in Fig.\n\\ref{Fig4}.\n\nThe fractal dimension shows again a significant dependence on the\nstrength of attraction $\\alpha$. In the limit\n$\\alpha\\rightarrow\\infty$ $d_f$ converges to the value\n$\\frac{4}{3}=1.3\\bar{3}$ which is the fractal dimension of the SAW\ni.e., the outer perimeter of the planar Brownian motion.\n\n\\section{conclusions}\n\nIn this paper, we have studied the scaling properties and the\nfractal structure of the visited lattice-sites by a Brownian\nparticle in 3$d$ which is attracted by a plane with the strength\n$\\alpha$. The fractal dimensions of the set of visited sites by the\n3$d$ random walker in both three dimensions and on the attractive\nplane are computed which both converge to the same value $\\sim1.83$\nfor large $\\alpha$. We also found that size distribution of the\ncluster of visited sites by the particle on the attractive plane,\nhas a scaling form characterized by an exponent that depends\nsignificantly on the strength of attraction.\\\\The fractal dimension\nof the surrounding loops of the clusters on the plane has been\ncomputed as a function of $\\alpha$. This also converges\nasymptotically to the expected value for SAW i.e., the external\nperimeter of a planar Brownian motion.\n\nThese results need however some theoretical framework and\nmathematical proof. The other interesting feature which can be\ninvestigated, is the possible conformal invariance of the cluster\nboundaries on the attractive plane, which can be treated using SLE\ntechniques (such study is already done only for the limiting case\n$\\alpha\\rightarrow\\infty$ where the problem reduces to a 2$d$ random\nwalk in the attractive plane whose boundary is described by\nSLE$_{8\/3}$). The fractal dimension of an SLE$_\\kappa$ curve is\ngiven by $d_f=1+\\kappa\/8$. In case of conformal invariance of\ncluster boundaries on the attractive plane in our model, they would\nbe defined by a diffusivity $\\kappa$ which depends on the strength\nof attraction.\n\n\n\\textbf{Acknowledgement.} I would like to thank H. Dashti-Naserabadi\nfor his helps on programming. This work is financially supported by\nthe National Elite Foundation of Iran, and INSF grant No. 87041917.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\t\\footnotetext[2]{Correspondence to: Adri\u00e0 Recasens (arecasens@google.com)}\n\t\\renewcommand*{\\thefootnote}{\\arabic{footnote}}\n\n\\label{sec:intro}\nOur perception of the world is inherently multimodal: humans and other animals effortlessly integrate many modalities to build their view of the world \\cite{ghazanfar2006neocortex, amedi2017task}. Although multimodal integration can help construct a richer perspective on reality \\cite{bavelier2002cross, shams2008benefits}, humans can easily process information and perform tasks even when only a single modality (e.g. sound, vision, or touch) is present \\cite{shimojo2001sensory, lacey2014visuo, bola2017task}. \nHowever, this flexibility is hard to find in perceptual computational models. Architectures for multimodal perception have typically been divided on early fusion, mid-fusion and late-fusion, but most of them need all modalities to be present in order to operate. With human flexibility as an inspiration, in this paper we introduce {\\em Zorro}, a multimodal Transformer architecture which is able to operate in both a single-modality and multi-modality setting. This property improves the overall performance of the model while opening the door to off-the-shelf self-supervised pre-training.\n\nOur key architectural innovation in Zorro\\xspace is to create separate unimodal and multimodal (fusion) representation streams within a single standard Transformer backbone. We achieve this without engineering the architecture, but instead by applying appropriate masks in all attention operations, resulting in some outputs that only capture  individual modalities and some outputs that capture multimodal information. This has the direct benefit that the model can be applied when a subset of modalities is absent, e.g.\\ a model trained on audio and video can be evaluated on audio alone.\n\n\nWhile most of the emphasis of novel developments in the supervised space is put on the architecture, the unimodal outputs can be further exploited by introducing additional self-supervised training schemes. In contrast to recent multimodal attention-based models~\\cite{mbt,perceiver} that entangle both modalities throughout the network, Zorro\\xspace supports self-supervised contrastive training in a single network without representation collapse, thanks to its unimodal outputs (see Figure~\\ref{fig:teaser}). In this work, we explore this possibility by pre-training our model with an audio-visual contrastive loss~\\cite{alayrac2020self}. Differently from previous work, we can do this pre-training without the necessity of separate backbones per modality. \n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures\/pull_figure.pdf}\n\t\\caption{\\small \n\t\tIn this paper we introduce the Zorro\\xspace multimodal architecture which enables both self-supervised contrastive learning and supervised learning. When used for self-supervision, the single-modality outputs are used together with a standard cross-modal self-supervised loss. When used for supervised learning, all outputs can be used for the final classification.\n\t} \n\t\\vspace*{-0.3cm}\n\t\\label{fig:teaser}\n\\end{figure*}\n\n\n\n\nThis paper presents four contributions: \\textbf{(a)} we introduce Zorro\\xspace, a novel set of Transformer-based multimodal architectures which enable both supervised and self-supervised training and, once trained, can be used for multimodal or unimodal inputs; \\textbf{(b)} we introduce three Zorro\\xspace-based architectures using state-of-the-art models such as ViT, SWIN and HiP; \\textbf{(c)} we show that Zorro\\xspace can be pre-trained on a large-scale audio-visual dataset in a self-supervised manner, and can also be pre-trained on unimodal datasets; and \\textbf{(d)} we benchmark our resulting models on AudioSet, VGGSounds, Kinetics-400 and ESC-50. \nThe model achieves state-of-the-art performance when compared with previous self-supervised learning techniques on most relevant benchmarks, while also achieving comparable performance with previous work for supervised training with labels. \n\n\n\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\n\\noindent \\textbf{Multimodal perception}: Multimodal perception is challenging as data from the various modalities can have different topologies, temporal frequencies and relative importances that depend on each task~\\cite{baltruvsaitis2018multimodal}. \nWith the emergence of convolutional neural networks, numerous works fused activations from intermediate tensors~\\cite{wang2020makes,fayek2020large,arandjelovic2018objects,simonyan2014,Feichtenhofer_2016_CVPR,carreira2017quovadis,xiao2020audiovisual}, but this required considerable engineering, as different modalities come in differently shaped feature grids and there are many different ways to combine them. \n\n\\vspace{2mm} \\noindent\\textbf{Self-supervised audio-visual learning}: Various methods have been used to employ the cross-modality similarity as a self-supervisory signal~\\cite{arandjelovic17look,arandjelovic2018objects,Senocak_2018_CVPR,owens2018audio,korbar2018cooperative,alwassel2019self,mandela2020datatrans,morgado20avid}. Most approaches rely on single-modality backbones which produce representations which are used in the self-supervised loss~\\cite{alwassel2019self,mandela2020datatrans,alayrac2020self,recasens2021broaden}. These techniques process different modalities with different sets of weights and restrict the ability to reason across modalities. Less common are approaches which learn self-supervised models with multiple modalities at once. One recent work in this direction is \\cite{shvetsova2021everything}, which learns representations using audio, video and text. However, to avoid the collapse of the self-supervised loss, they feed the modalities two at a time, increasing the amount of necessary forward passes. Instead, Zorro\\xspace masking can produce unimodal outputs without running the model multiple times. \n\n\\vspace{2mm} \\noindent\\textbf{Transformer architectures}: Inspired by ViT~\\cite{vit}, follow up work proposed single-modality processing for video~\\cite{vivit} and audio~\\cite{gong2021ast} using patch-based encodings. Transformer-based methods have also been proposed to tackle audio-visual classification. The closest to our method is MBT~\\cite{mbt}, which builds a multimodal architecture out of single-modality Transformers for video~\\cite{vit,vivit} and audio~\\cite{gong2021ast}. MBT merges modalities by creating an attention bottleneck which restricts communication between the audio and visual heads. Our method also regulates cross-modality communication, but by masking the latent connections we are able to obtain modality-specific heads while in MBT the representation is entirely multimodal.  Another relevant work is VATT~\\cite{VATT}, a Transformer-based architecture to model video, audio and text with a single backbone. Differently from our work, in VATT each modality is independently processed by the transformer. Finally, the Perceiver architecture~\\cite{perceiver} scales to a large number of inputs by cross-attending to a set of latent queries. In this work, we use the follow-up Hierarchical Perceiver~\\cite{carreira2022hierarchical} which splits inputs and outputs into groups to improve model efficiency. \n\n\\vspace{2mm} \\noindent\\textbf{Masking attention in Transformers}: The original transformer architecture~\\cite{vaswani2017attention} used attention-masking for language modelling. After the success of image-based architectures, alternatives have been proposed to use attention masking to alleviate computational requirements of the architecture. Swin~\\cite{liu2021swin} proposed the use of local windows, restricting the self-attention layers to only neighbour pixels. Furthermore, mask2former~\\cite{cheng2022masked}, also restricted the cross-attention to local regions, enabling the use of transformers for high dimensional output (e.g segmentation). \n\n\n\n\n\n\n\n\\label{sec:method}\n\t\\begin{figure*}[t!]\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{figures\/model_figure.pdf}\n\t\t\\caption{\\textbf{The Zorro\\xspace-ViT model architecture}: The input to our model are video frames and audio spectrograms. Each of those inputs is patched using a 2D convolution and projection to input dimension $D$. Both audio and video input tokens are concatenated with a set of learned fusion vectors and added a position embedding. Next, we process these inputs through $L$ Zorro\\xspace's self-attention layers, where the Zorro\\xspace masking is applied. Specifically, our masking strategy blocks the information to flow towards the unimodal hidden representation, while still allowing the fusion representation to access all modalities. By doing this, we ensure that the image and audio representations are gated access to (i.e. depend on) only the video and audio inputs respectively. To produce the outputs, we learn a set of queries that cross-attend (also using masked attention) to the unimodal and multi-modal representation. \n\t\t}\n\t\t\\label{fig:model_figure}\n\t\t\\vspace*{-0.05cm}\n\t\\end{figure*}\n\t\n\t\n\\section{Zorro\\xspace: the masked multimodal Transformer}\n\\label{sec:model}\nIn this paper, we introduce Zorro\\xspace, a multimodal architecture which enables both supervised and self-supervised training. In this section, we unpack how Zorro\\xspace accomplishes this using modality-aware masking and by repurposing the original transformers components to allow contrastive learning between modalities. The key innovation of the architecture is introducing separate latent allocations for the different modalities, leading to a final representation which is partially unimodal (part of the representation sees only a single modality) and partially multimodal (part of the representation can attend to all modalities). First, we will describe Zorro\\xspace applied to the ViT architecture. Second, we  extend Zorro\\xspace to two other state-of-the-art transformer architectures, Swin and HiP. Finally, we end this section by describing how to use Zorro\\xspace for self-supervised contrastive learning. \n\n\\subsection{Architecture}\n\\label{section:zorro}\n\\vspace{2mm} \\noindent \\textbf{Zorro\\xspace-ViT overview.}\nFigure~\\ref{fig:model_figure} depicts the Zorro\\xspace architecture, which consist of three main blocks. First, Zorro\\xspace processes the data in form of patches (similar to ViT~\\cite{vit}). In this stage, data from each modality is first converted into a 2D array of representations. This can be done by either (i) dividing the input tensor into sequential groups (either points or patches) and applying a linear projection, or (ii) applying domain-specific processing such as 1D\/2D\/3D convolutions and flattening. We use a 2D convolution to extract $16 \\times 16$ patches and project them to the input dimension $D$. Next, position embeddings are added to the projected vectors so that the model is able to localise and distinguish each embedded patch.\nLearned multimodal fusion vectors are then introduced.\nSecond, the resulting tokens are concatenated to form a single set and are then processed by $L$ layers of a Transformer~\\cite{vaswani2017attention} with Zorro\\xspace masking. \nFinally, to produce the final output we learn a set of queries that cross-attend to the output of the last self-attention layer similar to PerceiverIO~\\cite{perceiverIO}.\nWe utilise the standard cross-attention operation~\\cite{perceiverIO}, and produce 4 different outputs: an audio output, a video output, a fusion output (which only sees the multi-modal part of the representation) and a global output that sees the whole representation.\nThese three steps are described in more detail next.\n\n\n\\vspace{2mm} \\noindent \\textbf{Input pre-processing.} Let $x=(x_v,x_a)$ be a video sample consisting of frames $x_v \\in \\mathbb{R}^{N_f \\times H \\times W \\times 3}$ and audio spectrogram $x_a \\in \\mathbb{R}^{T \\times N_s}$ where $N_f$ is the number of frames, $N_s$ the dimensionality of the spectrogram, $H$ is the height of the frame, $W$ is the width of the frame and $T$ is the number of temporal steps in the spectogram. To downscale the input, we use a 2D convolution $f^\\textrm{patch}$ which yields $u = (u_v,u_a) = (f_v^{\\textrm{pre}}(x_v),f_a^{\\textrm{pre}}(x_a))$. Arrays $(u_v,u_a)$ are then flattened and absolute learned position encoding are added. Finally, we learn a set of $n_\\textbf{fusion}$ latent vectors which are concatenated to the audio and video input tokens. \n\n\\vspace{2mm} \\noindent \\textbf{Masked attention.} The key contribution of this paper is splitting the Transformer representation into specialised groups. Using masked attention we force part of the representation to attend only to itself, while other parts can attend to the whole representation. The main goal of this approach is to split the representation in three parts: a part which only focuses on video tokens, a part which focuses on audio tokens, and the remaining vectors which can attend to the whole representation. \n\nWe mask two parts of the model: the self-attention~\\cite{vaswani2017attention} and the decoding cross-attention~\\cite{perceiverIO}. \nBoth parts consist of the same underlying operation which takes keys $k$, values $v$ and queries $q$ to produce the final output $o$. To this end, \nwe introduce a masking binary tensor $m$ that specifies which vectors are connected to each other.\nEntries of the masking matrix are $m_{ij}=1$ if information can flow from latent $i$ to latent $j$. By setting $m_{ij}=0$, we indicate to the model that this connection should be omitted. This mask is applied to the standard attention output operation $o_{i} = \\sum_{j} a_{ij} \\cdot v_j $ which becomes $o_{i} = \\sum_{j} \\hat{a}_{ij} \\cdot v_j$ where:\n\n\n\\begin{equation}\n    \\label{eqn:masked}\n    \\hat{a}_{ij} = \\frac{m_{ij} \\exp ({\\frac{q_i^\\top k_j}{\\sqrt{D}}})}{\\sum\\limits_{\\{j', \\ m_{i{j'}} = 1\\}} \\exp ( { \\frac{q_i^\\top k_{j'}}{\\sqrt{D}}} ) }.\n\\end{equation}\nIn contrast to MBT~\\cite{mbt}, our modality-specific representation does not have access to the global representation, which prevents cross-modality information flows. Specifically, we set $m_{ij}=1$ if $j$ is a part of the fusion representation, otherwise we only set $m_{ij}=1$ if $i$ and $j$ are vectors of the same modality. \nBy doing this, we explicitly prevent information from the fusion stream leaking into the unimodal representation. \nThis is the key to preserving pure streams that correspond to single modalities. \n\n\\vspace{2mm} \\noindent \\textbf{Output space.}\nIn ViT architecture, a learnable CLS token is used to produce the output embedding vector. Instead, inspired by the PerceiverIO~\\cite{perceiverIO}, we learn a set of decoding vectors which are used to query the output from the Transformer to produce the final output. Each decoding vector cross attends to a subset of tokens to produce the final output vector. This decoding strategy can be used to produce as many outputs as desired, opening up the possibility for dense tasks such as segmentation or flow estimation. \n\nAs we are relying on having the Transformer representation split into specialised groups, we need to also apply Zorro\\xspace's masking to the output cross attention. Specifically, we found it beneficial to define four outputs for our model. The audio-specific output $o_A$, which only contains information coming from the audio input. The video-specific output $o_v$, which only includes information from the video modality. The fusion specific output $o_F$, which is computed by attending only to the fusion stream. And finally, a global output $o_G$, which attends to all the outputs in the model. Although $o_G$ and $o_F$ do contain similar information, we found it useful to still keep two different heads.\n\n\n\\subsection{Extending Zorro\\xspace for other architectures}\nIn this section, we propose variants of Zorro\\xspace for two state-of-the-art attention-based architectures, Swin and HiP. Differently from the ViT implementation, when building  Zorro\\xspace-Swin and Zorro\\xspace-HiP we use the specific architecture building block for each modality and the fusion stream while we join the modalities with a cross-attention operation. This is required as the ViT masking is not directly applicable to Swin and HiP, but the overall idea remains the same. \n\n\\noindent \\textbf{Zorro\\xspace-Swin}: \nSwin~\\cite{liu2021swin} is a ViT-inspired transformer architecture which has shown improved efficiency and performance. The main innovation versus the original ViT architecture is to apply the self-attention operations on nearby tokens instead of all tokens in the input image. This reduces computational requirement while allowing the model to perform bottom-up inference. In order to build Zorro\\xspace-Swin, our main modification to the original architecture is to process individual modalities using Swin transformers. At the end of each Swin block, we update the fusion representation by cross-attending to both the unimodal and multimodal representation. To process the fusion representation, we use the same self-attention as in Zorro\\xspace-ViT. Given this design, we are free to use different architectures to process each modality. We use the original 2D Swin~\\cite{liu2021swin} to process the audio spectrograms while our adaptation of the Swin architecture for video. Similarly to Zorro\\xspace-ViT, no multimodal information flows into the unimodal streams. Detailed description of Zorro\\xspace-Swin can be found in Section~\\ref{arch:details} in the Appendix.\n\n\\noindent \\textbf{Zorro\\xspace-HiP}: The hierarchical perceiver~\\cite{carreira2022hierarchical} extends the previously introduced Perceiver models~\\cite{perceiver, perceiverIO} models, by splitting the inputs into groups, and operating only within those groups. Through the hierarchical architecture, those groups fuse together in order to aggregate information and globally reason about the input. In our implementation of HiP, instead of using directly the pixels and audio signal as input, we create patches similarly to the ViT\/Swin implementation. In order to create Zorro\\xspace-HiP, we use HiP building blocks for each modality. Specifically, those blocks group the inputs into smaller sets, cross-attend using learned features and finally apply self-attention layers to the outputs of the cross attention operation (see~\\cite{carreira2022hierarchical} for more details). In order to update the fusion representation, we learn a set of queries which cross attend to both unimodal and multimodal representation per each layer. More details can be found in Section~\\ref{arch:details} in the Appendix.\n\n\n\\subsection{Contrastive learning with Zorro\\xspace}\n\\label{sec:selfsup}\n\nContrastive audio-visual methods learn representations by aligning audio and video into a common embedding space. As opposed to unimodal approaches, instead of producing multiple views of the data, they use different modalities as views. \nOne important requirement is for the two backbones to not share information. If information is shared across modalities, the self-supervised training can easily collapse or converge to a trivial solution.\n\nModels for multimodal perception typically produce a single output for the multiple inputs. This is sufficient for supervised applications, but prevents the use of these audio-visual contrastive techniques. We design Zorro\\xspace in order to process unimodal and multimodal outputs, with the intention of enabling the use of self-supervised contrastive losses. \n\n\\vspace{2mm} \\noindent \\textbf{Noise Contrastive Estimation}: For training with the standard noise-contrastive estimation loss, we follow the implementation of the audio-visual loss \nfrom~\\cite{alayrac2020self}. Given the audio output $o_a$ and the video output $o_v$, we apply a final linear projection (different per modality) $g_a$ and $g_v$ to yield the final embedding vectors: $z_a = g_a(o_a)$ and $z_v = g_v(o_v)$. We compute the similarity between $z_a$ and $z_v$ by taking a normalised dot product and dividing by a temperature parameter $\\tau$, $\\textrm{sim}(z_a,z_v) = \\exp (\\frac{\\hat{z_a} \\hat{z_v}}{\\tau})$. Finally we apply the NCE loss:\n\\begin{equation}\n    L_{\\textrm{NCE}}(z_a,z_v) =- \\sum_i \\log \\frac{\\textrm{sim}(z_a^i,z_v^i)}{\\sum_{j,k} \\textrm{sim}(z_a^k,z_v^j)}\n    \\label{eqn:nce}\n\\end{equation}\n\nEquation~\\ref{eqn:nce} introduces describes the loss for audio-visual contrastive training. However, this technique does not train any parameters specific to the fusion representation or output (e.g, the fusion cross-attention or the fusion weights if the model has separate weights per modality). In order to self-supervise the output of the fusion stream, we add a fusion-visual and fusion-audio contrastive loss. We define a self-supervised loss contrasting both unimodal representations (audio and video) separately with the multimodal one (fusion). With those changes, the new loss is:\n\\begin{equation}\n    \\label{eqn:fusion_contrastive}\n    \\small L_{\\textrm{NCE}} = L_{\\textrm{NCE}}(z_a,z_v)+L_{\\textrm{NCE}}(z_a,z_f)+L_{\\textrm{NCE}}(z_I,z_f)\n\\end{equation}\n\n\n\\section{Experiments}\n\\label{sec:experiments}\nIn this section, we evaluate the Zorro\\xspace architecture on multiple settings. We first present details of the training and evaluation procedures, as well as the main datasets we use. We evaluate the method against state-of-the-art models on three standard audiovisual benchmarks ( AudioSet~\\cite{gemmeke2017audio}, VGGSound~\\cite{chen2020vggsound} and Kinetics-400~\\cite{carreira2017quovadis}), one vision benchmarks (Kinetics-400~\\cite{carreira2017quovadis}) and one audio benchmark (ESC-50~\\cite{piczak2015dataset}). Finally, we ablate the main design decisions that drove our research and showcase Zorro\\xspace's flexibility. Specifically, we compare the different architectures, study the effect of missing modalities, pre-train Zorro\\xspace with unimodal data and explore alternative attention-masking strategies.\n\n\\subsection{Experimental details}\nIn order to showcase Zorro\\xspace's ability to reason across different modalities, we pre-train it using self-supervision as well as with standard supervision using class labels. \nIn this section, we provide the most important details of the training procedure. Additional details about inputs, architectures and training can be found in Section \\ref{arch:details} and \\ref{sec:training_details} in the Appendix.\n\n\\vspace{2mm} \\noindent \\textbf{Pre-training datasets}:  We utilise four datasets for pre-training: AudioSet~\\cite{gemmeke2017audio}, YouTube-8M, ACAV-100M~\\cite{lee2021acav100m} and ImageNet-21k~\\cite{ridnik2021imagenet}. AudioSet consist of $1.9$M videos which contain $527$ classes of annotated sounds. As the dataset is highly unbalanced, \\cite{mbt} proposed a smaller more balanced variant of the training set with $500$k examples. For the ablation experiments and training from scratch, we use the $1.9$M version while for fine-tuning we also use AudioSet-500k for fair comparison with the state-of-the-art. YouTube-8M~\\cite{abu2016youtube} consist of $8$M videos with audio and visual frames, annotated in a multi-label fashion with $3862$ different classes. Videos are representative of many activities, resulting a very natural distribution of data. ACAV-100M consist of $100$M videos with audio and visual frames without associated labels, which have been curated to contain a strong audio-visual correlation. We use $59M$ of those videos for self-supervised learning.  ImageNet-21k consist of $13M$ images annotated on $21k$ classes, and been typically used for large-scale pretraining of visual transformer models~\\cite{vit}.  \n\n\n\\vspace{2mm} \\noindent  \\textbf{Audio-visual evaluation benchmarks}: To evaluate the ability of Zorro\\xspace to learn and transfer multimodal representations, we evaluate on standard audio-visual benchmarks. Specifically, we evaluate Zorro\\xspace in AudioSet, VGGSound~\\cite{chen2020vggsound} and Kinetics-400~\\cite{kay2017kinetics}. VGGSound consists of $163,603$ training and $13579$ test samples drawn from 10-second YouTube videos which span $309$ single-label, mutually exclusive classes. It focuses on real life audio evaluation with audio-visual correspondence where sounds are visually evident in the video. \nKinetics-400 consists of $201$K training videos of everyday actions which are classified into $400$ unique classes. While some datasets have bias in audio or video modality, Zorro\\xspace is able to learn the extent to rely on each modality. \n\n\\vspace{2mm} \\noindent  \\textbf{Unimodal evaluation benchmarks}: Zorro\\xspace can be trained on multi-modal data but evaluated on unimodal data. To further show this we evaluate the multi-modal trained Zorro\\xspace models on unimodal fine-tunning tasks: Kinetics-400 for vision and ESC-50 for audio. ESC-50 dataset contains $2k$ clips classified into $50$ unique classes.\n\n\\vspace{2mm} \\noindent\\textbf{Zorro\\xspace inputs}: The inputs to our model are video and audio. The audio and video are synced and cover the same time span. Video consists of $8$ frames of size $224 \\times 224$. When training in AudioSet, we sample videos at $3.12$FPS which results on $2.56s$ of audio and video. Specific FPS per model and audio length for pre-training and fine-tuning is reported in Section~\\ref{sec:training_details} in the Appendix.\nDuring training, we use random cropping as well as color augmentation in frames. \nFor ESC-50, we match the lengths of the pre-trained model, looping over the audio sequence if required. Audio is sampled at $48kHz$, converted to spectrograms as inputs to our model using $128$ bins.\nTo augment the audio in training, we use SpecAugment~\\cite{park19specaug} and frequency jittering. During evaluation, we subsample the input video and audio into multiple equally spaced clips and averge their predictions. \n\n\n\\vspace{2mm} \\noindent\\textbf{Architectural details}: \nZorro\\xspace is based on unimodal transformer architectures (ViT, Swin and HiP), adapted for multimodal processing (similar to~\\cite{mbt}). Through all our experiments we use ViT-B\/16. For details on ViT, Swin and HiP architecture, see Section~\\ref{arch:details} in the Appendix.\n\n\\vspace{2mm} \\noindent\\textbf{Training details}:\nWe use the Adam optimiser with cosine decay learning rate schedule, weight decay and learning rate warmup. When fine-tuning, for Zorro\\xspace-ViT and Zorro\\xspace-Swin we find better to use SGD optimiser and momentum $0.9$. We train all models for $50$ epochs except for the ACAV-100M datasets where we train for $10$ epochs and the \\textit{input-level} and \\textit{bottleneck} baselines where we train for $25$ to prevent severe overfitting. We find best to use $n_\\textbf{fusion}=6$ in all models. For  AudioSet fine-tuning, we use mixup ($\\alpha=0.3$) and label smoothing. We use cross-entropy loss for uni-label datasets and binary sigmoid cross-entropy for multi-label. We train one classifier for each of the $4$ outputs of the model and average its predictions. For contrastive training, we follow the procedure outlined in Section~\\ref{sec:selfsup}.\n\n\n\n\\subsection{State-of-the-art comparison}\nNext, we evaluate Zorro\\xspace against state-of-the-art methods. We evaluate our audio-visual trained Zorro\\xspace on benchmarks for audio-visual classification, video classification and audio classification, showcasing the universality of the approach.\n\n\t\\begin{table}[t]\n\t\t\\centering\n\t\t\\caption{\\small {\\bf AudioSet-2M comparison: training from scratch.} We report the performance of our models trained on audio-visual data compared with the state-of-the-art when trained from scratch. We report the mean average precision on the AudioSet test set.\n        }\n\t\t\t\\begin{tabular}{c|cc|c} \\toprule\n\t\t\tModel  & Train Mod & Eval Mod & AudioSet \\\\\n\t\t\t \\hline\n\t\t\t HiP~\\cite{carreira2022hierarchical} &  A+V &  A+V & 43.8\\\\\n\t\t\t Perceiver~\\cite{perceiver} &  A+V &  A+V & 44.2\\\\\n\t\t\t ERANN~\\cite{verbitskiy2021eranns} & A &  A & 45.0 \\\\\n            \\hline\n\t\t\t Zorro\\xspace-ViT &A+V &  A+V & 45.1 \\\\ \n\t\t\t Zorro\\xspace-HiP & A+V &  A+V & 45.2 \\\\ \n\t\t\t \\textbf{Zorro\\xspace-Swin} & \\textbf{A+V} &  \\textbf{A+V} & \\textbf{46.5} \\\\ \n\n\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{tab:audioset_scratch}\n\t\\end{table}\n\n\n\n\t\\begin{table*}[t]\n\t\t\\centering\n\t\t\\caption{\\small {\\bf State-of-the-art results:} We compare Zorro\\xspace with the state-of-the-art in two settings: when labels are not used in pre-training or when labels are used. We report the mean average precision on the AudioSet test set and top-1 accuracy on K-400, VGGSound and ESC-50. IN-21k is ImageNet-21k~\\cite{ridnik2021imagenet}, YT8M is YouTube-8M~\\cite{abu2016youtube}, ACAV is ACAV-100M~\\cite{lee2021acav100m} and K-400 is Kinetics-400~\\cite{carreira2017quovadis}. \n        }\n\t\t\t\\begin{tabular}{c|ccc|ccc|c|c} \\toprule\n  \t\t\tModel & \\multicolumn{3}{c|}{Pre-Training} & \\multicolumn{3}{c|}{Eval: Video+Audio}   & Eval: Video & Eval: Audio  \\\\      \n            \\hline\n\t\t\t & Dataset & Sup\/SSL   & Mod & AS & VGGSound & K-400 & K-400 & ESC-50 \\\\\n\t\t\t \\hline\n\n  \t\t\t \\multicolumn{4}{l|}{\\textbf{No pre-training}} & &    &  & &   \\\\         \n\n\t\t\t SlowFast R101-NL~\\cite{feichtenhofer2019slowfast} & &  &  & & & \\bf 79.8 & \\bf 79.8 \\\\\n\t\t\t AVSlowFast~\\cite{xiao2020audiovisual}, R101 & & & & & & 78.8  &  \\\\\n\n\t\t\t AudioSlowFast~\\cite{kazakos2021slow} &  &  & & & 52.5 &  &  \\\\\n\t\t\t ERANN~\\cite{verbitskiy2021eranns}, R101 & & & & 45.0& &   & &  89.2 \\\\\n\n\t\t\t PlayItBack~\\cite{stergiou2022play}, R101 & & & & 47.7& 53.7&   &  \\\\\n             \\midrule\n\n  \t\t\t \\multicolumn{4}{l|}{\\textbf{Self-supervised pre-training}} & &    &  & &   \\\\         \n\n\t\t\t MaskSpec~\\cite{chong2022masked}, ViT & AS &SSL & A& 47.1& & &   &89.6  \\\\\n\n\t\t\t Zorro\\xspace-HiP  &ACAV & SSL&  A+V  &49.4 & 61.3 &67.9 &  64.6 & 88.4\\\\\n\t\t\t Zorro\\xspace-Swin  &ACAV & SSL  &  A+V  &49.4 & 61.1& 73.7 &  69.4 & 91.4 \\\\\n\n\t\t\t Zorro\\xspace-ViT  &ACAV & SSL  &  A+V  & \\bf 50.3  & \\bf  63.6 & 76.5 &74.1 & \\bf 93.6\\\\\n\n             \\midrule\n  \t\t\t \\multicolumn{4}{l|}{\\textbf{Supervised pre-training}} & &    &  & &   \\\\         \n\n\t\t\t \\spv{ViViT-Base~\\cite{vivit}} & \\spv{IN-21k} & \\spv{Sup.} & \\spv{V} &  & &\\spv{80.0} &  \\spv{80.0} & \\\\\n\t\t\t \\spv{MaskSpec~\\cite{chong2022masked}, ViT} & \\spv{AS} &\\spv{Sup} & \\spv{A}& & &   && \\spv{98.2}  \\\\\n\n\t\t\t \\spv{ PaSST}~\\cite{koutini2021efficient} & \\spv{IN} &\\spv{Sup.} &  \\spv{V}   & \\spv{49.6}  & &  & &\\spv{96.8} \\\\\n\t\t\t \\spv{ AST~\\cite{gong2021ast}} & \\spv{IN-21k} & \\spv{Sup.} &  \\spv{V}   & \\spv{45.9} & & &  &  \\spv{95.7}\\\\\n\n\t\t\t  \\spv{MBT~\\cite{mbt},ViT}  & \\spv{IN-21k} & \\spv{Sup.}  & \\spv{V} & \\spv{ 52.1} & \\spv{64.1} & \\spv{80.8} &\\spv{79.4}  \\\\\n\t\t\t \\spv{Zorro\\xspace-ViT} & \\spv{IN-21k} & \\spv{Sup.} & \\spv{V} &  \\spv{50.9} & \\spv{63.1} & \\spv{79.8}  &\\spv{77.6}  & \\spv{81.7}\\\\\n\n\t\t\t  \\spv{Zorro\\xspace-ViT} & \\spv{YT8M} & \\spv{Sup.} & \\spv{A+V} &  \\spv{51.5} & \\ \\spv{64.8} &\\spv{79.6} & \\spv{76.1}&\\spv{93.1}   \\\\\n\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{tab:sota_experiments}\n\t\\end{table*}\n\n\\vspace{2mm} \\noindent \\textbf{Training AudioSet-2M from scratch}: First, we evaluate Zorro\\xspace when trained from scratch on Audioset-2M using both the audio and visual modalities. Table~\\ref{tab:audioset_scratch} reports that Zorro\\xspace matches or overperforms other methods that directly trained on AudioSet-2M from scratch. Note that PlayItBack~\\cite{stergiou2022play} is not listed in Table~\\ref{tab:audioset_scratch} as it was trained with AudioSet-500k. This setting shows the model's ability to adapt to the multi-modal inputs without the need of pre-trained data. \n\n\\vspace{2mm} \\noindent \\textbf{Multi-modal comparison}: We train and evaluate our pre-trained models on AudioSet-500k (see \\cite{mbt} for details), VGGSound and Kinetics-400 where we use both the audio and visual inputs. Similar to \\cite{mbt}, for Zorro\\xspace-ViT we allocate different weights for the audio, video and fusion latents. We found this useful for improving the fine-tuning accuracy. \nTable~\\ref{tab:sota_experiments} reports the performance of our models. We divide the table into two different parts. First, we report the Zorro\\xspace performance when contrastive self-supervision is used for pre-training (no labels). Zorro\\xspace improves over all previous works on AudioSet and VGGSound. In AudioSet, our best-performing model on that setting is only $2\\%$ away from the supervised state-of-the-art, which demonstrates the ability of the self-supervised pre-training technique for learning general features. In VGGSound, Zorro\\xspace performs similarly with the supervised state-of-the-art when pre-trained only with self-supervision. Finally, for Kinetics-400, the resulting performance is not far from models with supervised pre-training. In the bottom part of the table we report the peformance of Zorro\\xspace when using supervised pre-training. We include the performance of the model when initialized with ViT pre-trained on ImageNet-21k. Even without multi-modal pretraining, Zorro\\xspace is able to perform comparably with SOTA models. When pre-trained on YouTube-8M, Zorro\\xspace also performs similarly to MBT~\\cite{mbt}. However, differently than Zorro\\xspace, MBT cannot perform unimodal inference when trained with multi-modal data. \nNote, we have not demonstrated it here, but Zorro\\xspace can also be trained using unimodal self-supervised methods such as MAE~\\cite{he2022masked} and DINO~\\cite{caron2021emerging} separately on the audio and visual streams. We discuss supervised unimodal training below.\n\n\\vspace{2mm} \\noindent \\textbf{Video comparison}: To showcase Zorro\\xspace's performance in the unimodal regime, we fine-tune our models (pre-trained on audio and video) on the task of video classification for Kinetics-400 using only video. Table~\\ref{tab:sota_experiments} reports the results. Our goal is not to show state-of-the-art performance on this setting, as we are aware of the improvements made on Transformer architectures to solve that task~\\cite{zhang2021co,liu2021swin,yan2022multiview}. Our goal is to provide an efficient mechanism for pre-training those architectures in order to improve the final performance on unimodal and multimodal inference. When Zorro\\xspace is pretrained using a contrastive loss and fine-tuned on Kinetics-400 (video only), Zorro\\xspace-ViT performs only $2.4\\%$ worse than when using audio-visual input. This shows the robustness of our model when reduced to using a single modality. Furthermore, when using the Zorro\\xspace model pre-trained on YT8M, our model is able to perform similarly to comparable architectures. Alternative to fine-tuning, we can also use the audio-visual trained model (column \\textit{Audio+Video}) and only feed the video. In that setting, our model trained on YouTube-8M performs at $76.3$ top-1, on par with the video only fine-tuned result. This unimodal inference on a multi-modal trained model is not possible with MBT, where retraining is needed. \n\n\\vspace{2mm} \\noindent \\textbf{Audio comparison}: To evaluate Zorro\\xspace's audio capabilities, we fine-tune our models on ESC-50 (audio-only dataset) and report results in Table~\\ref{tab:sota_experiments}. When pre-trained on YouTube-8M, Zorro\\xspace performs close to AST, an specialised audio transformer comparable in size. When using self-supervised pre-training, Zorro\\xspace improves performance over previous methods; Zorro\\xspace-ViT has an accuracy of $93.6\\%$, close to state-of-the-art supervised methods. \n\n\t\\begin{table*}[t]\n\t\t\\centering\n\t\t\\caption{\\small {\\bf Masking configurations and architectures:} We evaluate the different masking configurations by training Zorro\\xspace on AudioSet with a supervised loss and audio-visual contrastive loss. Specifically, we test the audio-visual trained models on a unimodal (Audio, Video) and multimodal setting. Our proposed configuration performs well across the board while providing additional unimodal outputs.\n        }\n\t\t\t\\begin{tabular}{ccc|ccc|ccc} \\toprule\n\t\t\t& & &\\multicolumn{3}{c}{Supervised (Audio+Video)}   & \\multicolumn{3}{c}{Self-Supervised (Audio+Video)}  \\\\\n\t\t\t\\hline\n\n\t\t\tArchitecture & Params &  Fusion & Video & Audio   & Audio+Video & Video & Audio & Audio+Video \\\\\n\t\t\t\\hline\n\n\t\t\tViT & 98M & Two Streams  &23.1 & 40.1&  42.2  &18.9 &32.3 &34.8  \\\\\n\t\t\tViT & 98M &  Input Level  &9.1 & 31.6&  42.2  &Collapse & Collapse&Collapse  \\\\\n\t\t\tViT & 98M & Bottleneck ~\\cite{mbt}  &9.7 & 32.6&  42.5  &Collapse & Collapse&Collapse  \\\\\n\t\t\tViT & 98M &  Zorro\\xspace & 22.5 & 39.7&  45.1  &17.8 &29.8 &33.6  \\\\\n\t\t\tHiP & 136M &   Zorro\\xspace &22.0  & 39.5& 45.2   &11.3 &21.9 & 26.5  \\\\\n\t\t\tSwin & 161M &   Zorro\\xspace & 25.4& 40.6& 46.5  &20.5 & 31.6& 35.7 \\\\\n\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{tab:ablation_experiments}\n\t\\end{table*}\n\\subsection{Architecture comparison}\nIn this section, we discuss the different architectures introduced in this paper. \nIn Table~\\ref{tab:ablation_experiments} we report comparison for those architectures in two settings: when trained from scratch and when pre-trained with an audio-visual contrastive loss followed by a linear layer on top, using Audioset-2M. \nWhen training from scratch, we observe Zorro\\xspace-Swin performs the best across the different models, both in the supervised and contrastive regimes. Although the number of parameters is larger than ViT, Swin trains $25\\%$ faster than ViT. HiP is the fastest of the three, while not losing much on accuracy. See Section~\\ref{arch:details} in the Appendix for model speed comparison. Furthermore, in Table~\\ref{tab:sota_experiments} we also present the results of fine-tuning these architectures after contrastive pre-training. It is important to note that for ViT, in this table we use one set of parameters per modality, which significantly increases the parameter count ($98$M to $267$M). In this regime, we observe how ViT is the best. However, Swin and HiP are faster and retain most of the performance. \n\n\\subsection{Zorro model flexibility}\n\n\\vspace{2mm} \\noindent \\textbf{Unimodal inference with a multimodal backbone}:\nHere we study the ability of audio-visual trained Zorro\\xspace to produce meaningful unimodal outputs when fed with unimodal data.\nTo achieve this we zero out the missing modality and only provide useful inputs for one modality, either video or audio. \nResults are reported in Table~\\ref{tab:ablation_experiments}. Models without unimodal output suffer significantly from one missing modality. In contrast, both Zorro\\xspace and using two separate modality streams achieve a high performance when only a single modality is provided. This is due to the fact that in those models, some capacity is allocated to each modality specifically and the model is able to produce unimodal outputs. \n\n\n\\vspace{2mm} \\noindent \\textbf{Unimodal pre-training for multi-modal fine-tuning}:\nThrough the paper, we assumed availability of large multi-modal dataset for training. However, in some situations we only have available large amounts of unimodal samples (e.g.\\ video or audio) and a small set of multi-modal data. To showcase the flexibility of our proposal, we run a single experiment where we train with two unimodal datasets and fine-tune on a smaller multi-modal dataset. We use only the audio signal from the AudioSet dataset and the videos from the Kinetics-400 dataset. When training, we mix batches with probability $0.5$ per dataset, and do not compute the loss for the missing modalities. For evaluation, we fine-tune the resulting model on VGGSound and compare its result to the model trained from scratch. The fine-tuned model performs at $59.2$ top-1 accuracy while the model trained from scratch performs at $54.4$. This experiment shows the flexibility of the Zorro\\xspace model to adapt to unimodal training while providing useful initialization for multi-modal fine-tuning. \n\n\t\n\\subsection{Masking configurations}\nIn this ablation, we study four different types of attention masking. First, we evaluate having data independent stream (\\textit{two streams}), where both models share weights but modalities are not connected. Secondly, we evaluate input level fusion, which consist of no masking in the model. This reduces the model to a vanilla ViT applied to the two concatenated modalities. Inspired by~\\cite{mbt}, we also evaluate \\textit{bottleneck masking} where the fusion tokens can attend to each modalities' tokens but each modality can also attend to the fusion tokens. We want to make clear that although this approach uses the main proposal from MBT, it is not a reproduction of their work. This configuration forces each stream to mostly concentrate on one modality, but information can flow across modalities through the fusion vectors. Finally, we compare all those masking strategies with our Zorro\\xspace masking. For each masking configuration we train a model in a supervised manner (keeping the same number of outputs for fairness, except for the Two Streams which has two outputs). We also train the model in a self-supervised way, where the audio and the video outputs are used to compute the contrastive loss. To report performance, we train a linear classifier on top of the contrastive representations.\n\nTable~\\ref{tab:ablation_experiments} reports the results. We extract two main conclusions. First, having modality independent streams is crucial for self-supervised training. Both the \\textit{input-level} and the \\textit{bottleneck} configurations immediately collapse as information can flow from one modality to the other. Performance for Zorro\\xspace and \\textit{two streams} is very similar as Zorro\\xspace when trained in a self-supervised manner reduces to the two stream architecture. Secondly, we find that having separate modality streams is useful also for supervised learning. Specially interesting is looking at the performances of \\textit{input-level}, \\textit{bottleneck} and Zorro\\xspace, where Zorro\\xspace performs better as the modality streams are more independently treated. We believe this is due to the ability of the model to keep modality-specific information through the network, which can be useful at later stages of processing. Finally, for self-supervised training of Zorro\\xspace, we use equation~\\ref{eqn:fusion_contrastive}, which trains also the fusion output. Although this produces a slight decrease on performance vs \\textit{two streams}, it's beneficial for downstream tasks. Alternatively, when Zorro\\xspace is trained using only audio and video outputs, it performs the same as \\textit{two streams} ($35.0$ vs $34.8$) as the two models are equivalent.\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusions}\nIn this paper, we introduced Zorro\\xspace, a novel Transformer masking configuration which enables simultaneous unimodal and multimodal training and inference, as well as contrastive pre-training. Different from previous approaches to multimodal perception, our proposed method is able to generate both unimodal and multimodal outputs. By splitting the information flow into unimodal and multimodal streams, we are able to improve performance when the architecture is trained with a supervised loss and show the ability of the model to be self-supervised with a contrastive loss. We evaluate our model on multimodal tasks, showing great flexibility and state-of-the-art performance.\n\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}%\nSpontaneous symmetry breaking is one of the important notions \nin modern physics. \nIn particular, spontaneous breaking of continuous global symmetries leads to Nambu-Goldstone (NG) modes~\\cite{Nambu:1961tp,Goldstone:1961eq,Goldstone:1962es}, which are gapless excitations that dominate low-energy physics.\nVarious gapless excitations can be identified as NG modes, such as phonons in superfluids, pions in hadron physics, axions or axionlike particles in particle physics and condensed matter physics, and so on.\nWhile the counting rule for the number of NG modes was originally formulated in relativistic systems possessing Lorentz invariance~\\cite{Nambu:1961tp,Goldstone:1961eq,Goldstone:1962es}, \nLorentz invariance is usually absent in realistic materials.\nThe extension to nonrelativistic systems without Lorentz invariance~\\cite{Nielsen:1975hm, Miransky:2001tw, Schafer:2001bq, Nambu:2004yia, Watanabe:2011ec} was also established~\\cite{Watanabe:2012hr, Hidaka:2012ym, Watanabe:2014fva}.\nThere, the number of NG modes, which may be reduced in the absence of Lorentz invariance, can be counted by using symmetry algebra.\n\nRecently, the notion of symmetries has been extended to higher-form symmetries~\\cite{Gaiotto:2014kfa} \n(see also Refs.~\\cite{Batista:2004sc,Pantev:2005zs,Pantev:2005wj,Pantev:2005rh,Nussinov:2006iva,Nussinov:2008aa,Nussinov:2009zz, Nussinov:2011mz,Banks:2010zn,Distler:2010zg,Kapustin:2014gua}),\n where charged objects are extended objects rather than pointlike operators for ordinary symmetries. \nIn particular, the Maxwell theory in $(3+1)$ dimensions can be understood as a broken phase of $U(1)$ 1-form symmetry whose charged object is a Wilson loop, and the photon can be identified \nas the NG mode~\\cite{Kugo:1985jc,Kovner:1992pu,Gaiotto:2014kfa,Lake:2018dqm}. The counting rule of the number of NG modes for higher-form symmetries has also been formulated recently~\\cite{Hidaka:2020ucc}.\n\nOn the other hand, there are cases where NG modes become unstable in the presence of background fields, which are beyond the conventional counting rule above. In fact, such instabilities have been discussed in various contexts, such as the instability of a photon with a time-dependent axion background in cosmology~\\cite{Carroll:1989vb,Garretson:1992vt,Anber:2006xt}\nor with a chirality imbalance known as the chiral plasma instability in cosmology and astrophysics~\\cite{Joyce:1997uy,Akamatsu:2013pjd}, and instability of an emergent dynamical axion with the background electric field in condensed matter physics~\\cite{Ooguri:2011aa}; see also  Ref.~\\cite{Nakamura:2009tf} for a related instability in the five-dimensional Maxwell-Chern-Simons theory with a constant electric field.\nOne can ask whether the existence of such instabilities may be understood as a universal property of NG modes dictated by some symmetry algebra and whether there may be a general counting rule for these unstable modes similar to that of usual NG modes.\n\nIn this paper, we derive a general counting rule of these unstable NG  modes for the spontaneous breaking of internal symmetries in the presence of background fields.\nWe show that the number of unstable NG modes is determined by the rank of the matrix in terms of the correlation functions of broken  symmetry generators; see \\er{main} for our main result. \nWe verify the validity of our formula for known examples of  instabilities.\n\n\n\n\\section{Comparison between type-B and unstable Nambu-Goldstone modes}\nBefore going to the detailed discussion, we briefly summarize the previous studies and our results on the classification of NG modes based on the dispersion relations.\nIn Lorentz-invariant systems, the dispersion relation of NG modes is always $\\omega = |\\boldsymbol{k}|$, and the number of NG modes is equal to the number of broken symmetry generators~\\cite{Nambu:1961tp,Goldstone:1961eq,Goldstone:1962es}. \nIn the presence of background fields where Lorentz symmetry is explicitly broken, it has been shown that there can be NG modes with the quadratic dispersion $\\omega \\sim |\\boldsymbol{k}|^2$ and gapped modes with $\\omega = {\\rm const} + O (|\\boldsymbol{k}|^2)$ for 0-form symmetries~\\cite{Nielsen:1975hm} and higher-form symmetries~\\cite{Yamamoto:2015maz,Sogabe:2019gif}. \nThe counting rules of these modes for 0-form symmetries and higher-form symmetries are proved in Refs.~\\cite{Watanabe:2012hr,Hidaka:2012ym} and Ref.~\\cite{Hidaka:2020ucc}, respectively. \nWithout fine-tuning of parameters of a theory, whether the dispersion relation of a NG mode is linear or quadratic dispersion is classified by a quantity $\\rho_{\\cal IJ}^0 \\equiv \\langle [Q_{\\cal I}, Q_{\\cal J}] \\rangle$ in the ground state. Here, $Q_{\\cal I}$ are broken symmetry generators whose independent degrees of freedom are parametrized by the index ${\\cal I}$ (see Ref.~\\cite{Hidaka:2020ucc} for the detailed definition), and the superscript ``0'' refers to the temporal direction in which commutators are defined: NG modes for $\\rho_{\\cal IJ}^0 = 0$ have linear dispersion and are called type A, while NG modes for $\\rho_{\\cal IJ}^0 \\neq 0$ have quadratic (or gapped) dispersion and are called type B.\n\nOur new insight in this paper is that there are generically additional modes with the dispersion $\\omega \\sim \\pm {\\rm i} |\\boldsymbol{k}|$ that can also be understood as NG modes dictated by the symmetry algebra. Although several examples of the modes with this dispersion relation with the positive imaginary part are already known simply as instabilities~\\cite{Carroll:1989vb,Garretson:1992vt,Joyce:1997uy,Ooguri:2011aa,Akamatsu:2013pjd,Anber:2006xt}, they have not been identified as NG modes so far.%\n\\footnote{The partner mode with the negative imaginary part, $\\omega \\sim - {\\rm i} |\\boldsymbol{k}|$, also has a remarkable feature: Although it is a damping mode, it does not involve any entropy production (somewhat similarly to Landau damping) unlike usual diffusion modes with the dispersion $\\omega \\sim - {\\rm i} |\\boldsymbol{k}|^2$. While it can also be understood as a new type of NG mode, below we will mostly focus on the unstable NG mode with the positive imaginary part, as the number of the former is simply equal to the number of the latter. This is similar to  the fact that type-B NG modes are accompanied by gapped modes, and their numbers are equal.}\nMoreover, such a mode for even the conventional 0-form symmetry has not been known to the best of our knowledge. \nThe purpose of this paper is to generalize the counting rule for the conventional NG modes to these unstable NG modes and to provide a new such example of 0-form symmetry. To this end, we will introduce a new quantity $\\rho^l_{\\cal IJ}$ in Eq.~(\\ref{rho_cal}) below, which is the matrix of the correlators of broken symmetry generators put on the planes perpendicular to the spatial $x^l$ direction. The comparison between type-B and unstable NG modes is summarized in Table~\\ref{tab:classification}. \n\n\\begin{table}\n    \\begin{tabular}{c|c|c}\n      & Dispersion & Condition  \\\\\n      \\hline \n      Type B & $\\omega \\sim  |\\boldsymbol{k}|^2$ & $\\rank \\rho^0_{\\cal IJ} \\neq 0$  \\\\\n      \\hline\n       Unstable & $\\omega \\sim {\\rm i} |\\boldsymbol{k}| $ & \n       $\\rank \\rho^l_{\\cal IJ} \\neq 0$ \n    \\end{tabular}\n    \\caption{Comparison between type-B and unstable NG modes. Here, $\\rho^0_{\\cal IJ}$ is the matrix of the equal time commutators of the broken symmetry generators put on spatial directions, and $\\rho^l_{\\cal IJ}$ is the matrix of the correlators of broken symmetry generators put on the planes perpendicular to the spatial $x^l$ direction.}\n    \\label{tab:classification}\n\\end{table}\n\nBefore going to a mathematical proof for a more generic case, we first provide a rough idea on when and how the type-B or unstable NG modes appear. In essence, the directions of background fields can be classified to spatial and temporal ones (whose precise definitions will be given in Sec.~\\ref{sec:counting} below). All previous works on the classification of NG modes implicitly assume that background fields are in the spatial direction.\nIn this case, the dispersion relation is modified by the linear term of $\\omega$ as $\\omega^2  = \\pm \\alpha \\omega + |\\boldsymbol{k}|^2 +\\cdots $ with $\\alpha$ being some real constant, leading to type-B NG modes and gapped modes. The number of type-B NG modes can be counted by the correlation of spatially extended symmetry generators.\n\nOn the other hand, if the directions of the background field strengths are temporal, the modification of the dispersion relation is given by the linear term of $|\\boldsymbol{k}|$ as $\\omega^2  = \\pm \\beta |\\boldsymbol{k}| + |\\boldsymbol{k}|^2$ with $\\beta$ being some real constant, leading to unstable NG modes and dumping modes. \nIn this case, the number of unstable NG modes can be counted by the correlation of temporally extended symmetry generators.\n\nIn the following, we put this argument on a more mathematical basis using effective field theories for higher-form symmetries.\n\n\n\\section{Effective theories}%\nWe consider low-energy effective theories for spontaneous breaking of continuous 0- and higher-form internal symmetries with couplings to \nbackground fields.\nTo discuss the dispersion relations in the low-energy region, it is sufficient to focus on the effective action up to second order in derivatives. \nNote that, up to second order in fields and derivatives, the effective action for the higher-form symmetries includes that of 0-form symmetries.\nWe consider $D$-dimensional Minkowski spacetime with the mostly plus metric $\\eta_{\\mu\\nu} = \\diag (-1,1,...,1)$.\n\nFor the 0-form symmetries, we assume that a continuous 0-form symmetry with a compact Lie group $G$ is spontaneously broken to its subgroup $H$. For the higher-form symmetries, we assume that $U(1)$ $p_I$-form symmetries ($I = 1,..., N$) are spontaneously broken, since higher-form symmetries are always Abelian~\\cite{Gaiotto:2014kfa}.\nWe introduce a charged object for the $p_I$-form symmetry, which is a Wilson loop $W(C_I) = \\exp \\left(\\mathrm{i} \\int_{C_I }a_I \\right)$ on a $p_I$-dimensional closed subspace $C_I$. Here, $a_I$ is a $p_I$-form field, which has a gauge redundancy under the transformation $ a_I \\to a_I + \\mathrm{d} \\lambda_I $ for a $U(1)$ $(p_I-1)$-form parameter $\\lambda_I$ satisfying $ \\int_{C_I} \\mathrm{d} \\lambda_I \\in 2\\pi \\mathbb{Z}$. \nFor a $U(1)$ symmetry, the Wilson loop is transformed as $W_I (C_I ) \\to \\mathrm{e}^{\\mathrm{i} \\alpha_I} W(C_I)$ with $\\mathrm{e}^{\\mathrm{i} \\alpha_I} \\in U(1)$, which is generated by a shift $a_I \\to a_I + \\epsilon_I$ with a $p_I$-form $\\epsilon_I$ satisfying $\\mathrm{d} \\epsilon_I =0$ on $C_I$ and  $\\int_{C_I} \\epsilon_I = \\alpha_I$.\nFor a 0-form symmetry with a non-Abelian group, the global transformation leads to a constant shift for the leading order of a field.\nThe spontaneous breaking of the higher-form symmetries can be characterized by the nonzero vacuum expectation value of the Wilson loop in the large volume limit, $\\vevs{W(C_I)} \\to 1$ up to the renormalization that can depend on the volume of $C_I$.\n\nWe now construct the effective action. To have the action preserving higher-form global symmetries, we use the field strength  $f_{I} = \\mathrm{d} a_I$. \nUp to the second order of the derivatives and fields, the effective action is given by%\n\\footnote{For a construction of the effective actions based on the nonlinear realization of higher-form symmetries, see Ref.~\\cite{Hidaka:2020ucc}.}\n\\begin{equation}\nS \n =\n- \\frac{1}{2} \\int F^2_{IJ} \\mathrm{d} a_I \\wedge *\\mathrm{d} a_J\n+\n\\frac{1}{2}\n\\int \\mathrm{d} a_I \\wedge \\mathrm{d} a_J \\wedge  A_{IJ} \\,.\n\\label{S}\n\\end{equation}\nHere, $F^2_{IJ} = F_{KI} F_{KJ}$ is expressed in terms of an invertible matrix $F_{IJ}$ that represents the matrix of decay constants, and \n$A_{IJ} = (-1)^{(p_I+1)(p_J+1) }A_{JI}$ is a $p_{IJ}$-form with $p_{IJ} = D- p_I -p_J -2$ that can depend on the spacetime coordinate.%\n\\footnote{The background field satisfies the quantization condition $\\int_{\\Sigma_{IJ}} \\mathrm{d} A_{IJ} \\in \\frac{1}{2\\pi} \\mathbb{Z}$ on a $(p_{IJ}+1)$-dimensional closed compact subspace $\\Sigma_{IJ}$.} \nThe first term in \\er{S} can be understood as a generalization of the Maxwell term, and the second term is a topological term with a possible background field $A_{IJ}$.\nThe matrix $F_{IJ}$ is nonzero only for $p_I = p_J$.\nIt would also be possible to regard $F_{IJ}^2$ as a $(p_J - p_I)$-form background field, but we assume that $F_{IJ}$ is a just constant 0-form for simplicity.\nThe presence of the background field $A_{IJ}$ breaks Lorentz invariance in general.\nNote that the assumption of the $p_I$-form global symmetries of the action excludes terms that break these symmetries explicitly, such as dynamical charged matter, which may be allowed to exist if we assume only $(p_I -1)$-form gauge invariance.\n\nIn passing, we remark that we can reproduce the nonrelativistic effective actions previously considered in Refs.~\\cite{Watanabe:2012hr, Hidaka:2012ym, Watanabe:2014fva} by choosing, e.g., $A_{IJ} = \\mu_{IJ} x^1 \\mathrm{d} x^2\\wedge \\cdots \\wedge \\mathrm{d} x^{D-1} $ with an appropriate constant $\\mu_{IJ}$ up to a total derivative.\nIt is possible to identify $A_{IJ}$ as a background gauge field whose symmetry is called composite or Chern-Weil global symmetry~\\cite{Brauner:2020rtz,Heidenreich:2020pkc}.\n\n\n\n\\section{Counting rule of unstable NG modes}%\n\\label{sec:counting}\nTo count the number of the unstable modes in the presence of the background field, we first derive the dispersion relations.\nHereafter, we assume that the translational invariance of the system is not broken.\nWe consider the configuration of the background field in the temporal direction where $\\mathrm{d} A_{IJ} = \\frac{1}{p_{IJ}!} E_{IJ,i_1...i_{p_{IJ}}} \\mathrm{d} x^0 \\wedge \\mathrm{d} x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d} x^{i_{p_{IJ}}}$ with a constant $E_{IJ,i_1...i_{p_{IJ}}}$.\nThe background field can be understood as a generalization of an ordinary background electric field.%\n\\footnote{If we introduce a background magnetic field or its generalization in the spatial direction instead of the electric field, some of the NG modes become gapped rather than unstable~\\cite{Yamamoto:2015maz,Brauner:2017mui,Sogabe:2019gif,Hidaka:2020ucc}.}\n\n\n\\subsection{Dispersion relations}\nThe equation of motion for $a_I$ is\n\\begin{equation}\nF^2_{IJ} \\mathrm{d} * f_J\n= (-1)^{p_J +1} \nf_J  \\wedge \\mathrm{d} A_{IJ}.\n\\label{EOM}\n\\end{equation}\nTo focus on physical degrees of freedom, we take the temporal gauge $a_{I,0 i_1...i_{p_I-1}} =0$ with the Gauss law constraint $\\partial^i f_{I,0 i i_1 ... i_{p_I-2}} =0$.\nIn momentum space,\n\\er{EOM} \ncan be written as\n\\begin{equation}\n(\\omega^2 - |\\boldsymbol{k}|^2)\n \\hat{a}_I^{i_1...i_{p_I}}\n = -\\frac{\\mathrm{i}}{p_J!}\n\\hat{M}^{ i_1...i_{p_I} l  j_1 ... j_{p_J}}_{IJ}\nk_l \\hat{a}_{J, j_1 ...j_{p_J}}\\,,\n\\label{EOM_momentum}\n\\end{equation}\nwhere we defined\n$\\hat{a}_I = F_{IJ} a_J $\nto simplify our notation.\nWe also introduced $\\hat{M}^{i_1...i_{p_I} l j_1 ... j_{p_J}}_{IJ}\n = \n F^{-1}_{KI} M^{i_1...i_{p_I} l j_1 ... j_{p_J}}_{KL} F_{LJ}^{-1}$, where\n\\begin{equation}\nM^{i_1...i_{p_I} l j_1 ... j_{p_J}}_{IJ}\n= \n\\frac{\\epsilon^{0  i_1... i_{p_I} l j_1 ..j_{p_J}\nk_1 ... k_{p_{IJ}}}}{p_{IJ}!\n} \nE_{IJ, k_1 ... k_{p_{IJ}}}\\,, \n\\end{equation}\nwith $\\epsilon_{\\mu_1...\\mu_D}$ being the totally antisymmetric tensor satisfying $\\epsilon_{01...D-1} = 1$.\nWe remark that $\\hat{M}^{ i_1...i_{p_I}l  j_1 ... j_{p_J}}_{IJ}$ \nis antisymmetric under the exchange \n${\\cal I} = (I, i_1,...,i_{p_I}) \\leftrightarrow {\\cal J} = (J, j_1,..., j_{p_J})$.\nUsing these collective indices, the equation of motion in \\er{EOM_momentum} can be simplified as\n\\begin{equation}\n(\\omega^2 - |\\boldsymbol{k}|^2)\n \\hat{a}_{\\cal I}\n = \n - \\mathrm{i} k_l \\hat{M}_{{\\cal IJ}}^l\n \\hat{a}_{\\cal J},\n\\label{EOM_collective}\n\\end{equation}\nwith the antisymmetric matrices \n$\\hat{M}_{{\\cal IJ}}^l = \\hat{M}^{i_1...i_{p_I} l j_1 ... j_{p_J}}_{IJ}$\nsatisfying \n$\\hat{M}_{\\cal JI}^l  = -   \\hat{M}_{\\cal IJ}^l$.\nHere, the indices $(i_1,...,i_{p_I})$ in ${\\cal I}$ are ordered as $i_1< \\cdots < i_{p_I}$ to avoid overcounting.\nThe number of degrees of freedom of ${\\cal I}$ is denoted as ${\\cal N}$.\n\nWe can now count the number of unstable modes in the background \n$\\hat{M}_{{\\cal IJ}}^l \\neq 0$. Equation~\\eqref{EOM_collective} implies that the dispersion relations may depend on the direction of the wave vector. \nWe thus consider the dispersion relation for each direction.\nIf we choose a wave vector along the $x^l$ direction, i.e., $\\boldsymbol{k} = (0,...0,\\underbrace{k}_{l\\text{-th}},0,...,0)$, \\er{EOM_collective} is further simplified as\n $(\\omega^2 - k^2)\n \\hat{a}_{\\cal I}\n = \n- \\mathrm{i} k \\hat{M}_{{\\cal IJ}}^l\n \\hat{a}_{\\cal J}$.\nSince the matrix $\\hat{M}^l = (\\hat{M}_{\\cal IJ}^l)$ is antisymmetric, we can transform this matrix by using an orthogonal matrix $P$ into the form\n\\begin{equation}\n\\begin{split}\n&\nP \\hat{M}^{l} P^T \n= \\mtx{ \\Lambda_{1}^l && \\\\ &\\ddots& \\\\ && \\Lambda_{n }^l \\\\ &&&\n0_{({\\cal N} - 2n)\\times ({\\cal N} -2n)}\n}\\,,\n\\\\\n& \n \\Lambda_{ m}^l  = \\mtx{ 0 & - \\lambda_{m}^l  \\\\ \\lambda_m^{l} & 0}\\,, \n\\quad \nn = \\frac{1}{2}\\rank (\\hat{M}^{l})\\,.\n\\end{split}\n\\end{equation}\nNote that $n$ is an integer because the rank of an antisymmetric matrix is always an even integer.\nIn this basis, we can solve \\er{EOM_collective} and obtain the dispersion relation for each $\\Lambda_{m }^l$:\n\\begin{equation}\n\\label{instability}\n\\omega^2 = k^2 \\pm |k\\lambda_{m}^l|,\n\\end{equation}\nwhich exhibits an instability in the region $|k| < |\\lambda_{m}^l|$.\nThe number of the unstable NG modes $N^l_{\\rm unst}$ in the $x^l$ direction is, therefore,\n\\begin{equation}\n    N^l_{\\rm unst} \n    =\n    \\frac{1}{2}\\rank (\\hat{M}_{{\\cal IJ} }^l)=\\frac{1}{2}\\rank (M_{{\\cal IJ} }^l)\\,.\n    \\label{Nunst}\n\\end{equation}\nWe remark that the Gauss law $k_i a^{i i_1 ...i_{p_I -1}} =0 $ is automatically satisfied for the counting of the unstable modes, since $ k_i M_{{\\cal IJ} }^i = 0$ if the direction of $k_i$ is the same as either of polarization directions, $i_1,...,i_{p_I}$ or $j_1,..., j_{p_J}$, due to the totally antisymmetric tensor.\nIn other words, the unstable NG modes are always transverse modes.\nNote that the number of unstable modes is determined for a given direction of the wave vector.\n\n\n\\subsection{Correlation function between symmetry generators}%\nHere, we show that the matrix $M_{{\\cal IJ} }^l$ can be expressed by the correlation function of broken symmetry generators, and, hence, the number of the unstable NG modes is equal to half of the rank of the matrix in terms of the correlation function (see the Appendix for detail).\n\nFrom the equation of motion in \\er{EOM}, we can define a conserved charge on a $(D - p_I - 1)$-dimensional closed subspace $\\Sigma_{I}$:\n\\begin{equation}\n Q_I (\\Sigma_{I})\n = \\int_{\\Sigma_{I}} \n\\left(- F^{2}_{IJ} *  f_J + \nf_J \\wedge {A}_{IJ}\\right) \n.\n\\end{equation}\nThe correlation function can be calculated in the path integral formulation by using Ward-Takahashi identity as\n\\begin{equation}\n\\mathrm{i} \\vevs{ Q_I (\\Sigma_{I})  Q_J (\\Sigma_{J})}\n= - \n\\int_{\\Sigma_I \\cap \\Omega_J}\n \\mathrm{d} A_{IJ}\\,,\n\\label{correlation}\n\\end{equation}\nwhere $\\Omega_J$ is a $(D-p_J)$-dimensional subspace satisfying $\\partial \\Omega_J = \\Sigma_J$. \nFrom the correlation function, we can extract the matrix $M^l_{\\cal IJ}$\nby using a spatial version of the equal-time commutation relation of symmetry generators, where the $x^l$ direction plays the role of the temporal direction.\nA $(D- p)$-dimensional plane localized at $x^{i_1} = \\cdots = x^{i_p} =0$ is denoted by $S^{i_1...i_p}$.\nTo have a commutation relation, we also introduce a plane $S^{i_1...i_p}_{x^{i_p} = c}$ localized at $x^{i_1} = \\cdots = x^{i_{p-1}} =0$, $x^{i_p} = c$ with $c$ being some constant.\nWe first take the large volume limit $\\Sigma_I \\to S^{i_1...i_{p_I} l} \\cup \\b{S}^{i_1...i_{p_I} l}_{x^l = - \\epsilon}$ for an infinitesimal positive parameter $\\epsilon$. Here, $S^{i_1...i_{p_I} l}$ intersects with $\\Omega_J$ while $\\b{S}^{i_1...i_{p_I} l}_{x^l = - \\epsilon}$ (which is $S^{i_1...i_{p_I} l}_{x^l = - \\epsilon}$ with an opposite orientation) does not.\nThen, we take the large volume limit \n$\\Sigma_J \\to \nS^{j_1...j_{p_J} l}_{x^l =\\frac{\\epsilon}{2} } \\cup \n\\b{S}^{j_1...j_{p_J} l}_{x^l = - \\frac{\\epsilon}{2} }$.\nIn these limits, we have \n$\\Sigma_I \\cap \\Omega_J \\to\nS^{i_1...i_{p_I} l} \\cap \nS^{j_1...j_{p_J} l} = \nS^{i_1...i_{p_I} l j_1...j_{p_J}}$,\nand, hence,\n\\begin{align}\n&\n\\frac{\\mathrm{i} \\vevs{ Q_I (\\Sigma_{I})  Q_J(\\Sigma_{J})}}{\\vol (\\Sigma_I \\cap \n\\Omega_J)} \n\\nonumber \\\\\n&\\to\n\\frac{\\mathrm{i} \\vevs{ Q_I (S^{i_1...i_{p_I} l} \\cup \n\\b{S}^{i_1...i_{p_I} l}_{x^l = - \\epsilon} )  Q_J(S^{j_1...j_{p_J} l}_{x^l =\\frac{\\epsilon}{2} } \\cup \n\\b{S}^{j_1...j_{p_J} l}_{x^l = - \\frac{\\epsilon}{2} })}}{\\vol (S^{i_1...i_{p_I} l j_1...j_{p_J}})} \n\\nonumber \\\\\n&\n=:\n\\rho^{i_1...i_{p_I} l j_1...j_{p_J}}_{IJ}\n\\,.\n\\label{rho}\n\\end{align}\nHere, $\\vol (\\Sigma)$ is the volume of a subspace $\\Sigma$ including the temporal direction.\nBy the explicit calculation of \\er{correlation}, we have\n\\begin{align}\n&\n\\rho^{i_1...i_{p_I} l j_1...j_{p_J}}_{IJ}\n=(-1)^{p_{IJ} (D-p_{IJ}-1)+1}\nM^{i_1...i_{p_I} l j_1...j_{p_J}}_{IJ}\\,.\n\\label{rhoM}\n\\end{align}\nCombining \\ers{Nunst} and \\eqref{rhoM}, we arrive at \n\\begin{equation}\n    N^l_{\\rm unst}=\\frac{1}{2}\\rank (\\rho^l_{\\cal IJ}) \\,,\n    \\label{main}\n\\end{equation}\nwhere we have defined \n\\begin{equation}\n\\label{rho_cal}\n\\rho^l_{\\cal IJ}  \n=\n\\rho^{i_1...i_{p_I} l j_1...j_{p_J}}_{IJ}.\n\\end{equation}\n\nNote that our counting rule in \\er{main} is similar to that of the type-B NG modes for the higher-form symmetries derived in Ref.~\\cite{Hidaka:2020ucc}.\nThe difference between the counting of type-B NG modes and unstable modes is whether the directions of the nonzero components of field strengths for the background fields are spatial directions $\\mathrm{d} A_{IJ} \\sim \\mathrm{d} x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d} x^{i_{p_{IJ}+1}} $ or temporal directions $\\mathrm{d} A_{IJ} = \\frac{1}{p_{IJ}!}E_{IJ, i_1...i_{p_{IJ}}} \\mathrm{d} x^0 \\wedge \\mathrm{d} x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d} x^{i_{p_{IJ}}}$.\n\nWe also note that the correlation function evaluated here indicates only the presence of instabilities rather than the presence of long-lasting ones.\nAlthough we assumed $\\mathrm{d} A_{IJ}$ to be a constant when evaluating the correlation function, the background field may evolve dynamically to relax the instabilities in realistic systems. It is indeed the case for the chiral plasma instability discussed in Refs.~\\cite{Joyce:1997uy, Akamatsu:2013pjd}.\n\n\n\n\\section{Examples}\nIn this section, we provide examples of unstable NG modes to verify the validity of our general counting rule in physical systems.\n\n\\subsection{Chiral plasma instability}%\nFirst, we consider electromagnetism in the background of an axion field:\n\\begin{equation}\n S\n =\n-\\frac{1}{2e^2} \\int \\mathrm{d} a \\wedge *\\mathrm{d} a  \n+ \n \\frac{1}{2}\\int \\Theta_{aa} \\mathrm{d} a \\wedge \\mathrm{d} a\\,. \n\\end{equation}\nHere, the photon is described by a 1-form gauge field $a = a_\\mu \\mathrm{d} x^\\mu$, which can be understood as a NG mode for the spontaneously broken $U(1)$ 1-form symmetry, $e$ is a coupling constant, and $\\Theta_{aa} = -2 C \\mu_5 x^0$ is the background axion field with $\\mu_5$ corresponding to the chiral chemical potential for $C = 1\/(4\\pi^2)$. In fact, the second term above can be rewritten as the effective Chern-Simons term $S_{\\rm CS} = -C \\int \\mu_5 \\epsilon^{0ijk} a_i \\partial_j a_k$~\\cite{Redlich:1984md} by integration by parts. Then, taking its variation with respect to $a_i$ reproduces the so-called chiral magnetic effect: $j^i := e^2\\delta S_{\\rm CS}\/\\delta a_i = -2C e^2\\mu_5 \\epsilon^{0ijk} \\partial_j a_k$~\\cite{Vilenkin:1980fu,Nielsen:1983rb,Fukushima:2008xe}.\n\nThe equation of motion for the photon is $\\frac{1}{e^2} \n\\square \na^i \n- \n2C \\epsilon^{0 i j k} \n\\mu_5  \\partial_j a_k=0,\n$\nwhere we have taken the temporal gauge $a_0 =0$ and the Gauss law constraint $\\partial_i a^i =0$. Under the plane wave ansatz, we have the equation of motion in momentum space:\n$\n(\\omega^2 - |\\boldsymbol{k}|^2) \\hat{a}^i \n+\n \\mathrm{i}\n \\hat{M}^{ilj} k_l\n \\hat{a}_j=0,\n$\nwhere we defined $\\hat{a}_\\mu = \\frac{1}{e}a_\\mu$ and \n$\\hat{M}^{ilj}\n=e^2 M^{ilj}  = \n-\n2 \\epsilon^{0ilj} C e^2 \\mu_5 $.\nFor instance, we choose the wave vector along the $x^1$ direction, $\\boldsymbol{k} = (k_1, 0,0)$.\nIn this case, the Gauss law constraint leads to $a_1=0$ for nonzero $k_1$.\nThen the equation of motion is simplified as\n\\begin{equation}\n \\mathrm{i}  k_1 \n\\mtx{\n0 &\n \\hat{M}^{213 } \n\\\\\n \\hat{M}^{312}\n& 0\n}\\mtx{\\hat{a}_2 \\\\ \\hat{a}_3}\n= \n(\\omega^2 - k_1^2)\n\\mtx{\\hat{a}_2 \\\\ \\hat{a}_3}.\n\\end{equation}\nThe dispersion relation is obtained as\n$\\omega^2 \n= k_1^2  \\pm \n| \\hat{M}^{213 }\n k_1|\n$.\nTherefore, we have one unstable mode in the $x^1$ direction (and similarly for the $x^{2,3}$ directions). \nThis is the chiral plasma instability~\\cite{Joyce:1997uy,Akamatsu:2013pjd}.\n\nNext, we discuss the relation between the matrix $M^{ilj} $ and the correlation function of symmetry generators.\nThe equation of motion for $a$ gives the conserved charge on a closed surface $\\Sigma_a$:\n\\begin{equation}\n Q (\\Sigma_a)\n = \n\\int_{\\Sigma_a}\\left(\n-\\frac{1}{e^2} * \\mathrm{d} a\n+\n\\Theta_{aa} \\mathrm{d} a \\right)\\,.\n\\end{equation}\nThe correlation function between two symmetry generators is \n\\begin{equation}\n \\mathrm{i} \\vevs{ Q (\\Sigma_a)  Q (\\Sigma'_a)}\n =\n-\n\\int_{\\Sigma_a \\cap \\Omega_a' }\\mathrm{d} \\Theta_{aa}\\,,\n\\end{equation}\nwhere $\\Omega_a'$ is a world volume whose boundary is a closed surface $\\Sigma_a'$.\nNow, we take the limit \n$\\Sigma_a \\to S^{il} \\cup \\b{S}^{il}_{x^l = -\\epsilon} $\nand then \n$\\Sigma_a' \\to S^{jl}_{x^l = \\epsilon\/2} \\cup \\b{S}^{jl}_{x^l = -\\epsilon\/2} $.\nWe have a temporally extended one-dimensional subspace \n$\\Sigma_a \\cap \\Omega_a' \\to  S^{ilj}$. \nThen, the matrix $\\rho^{ilj}$ in \\er{rho} becomes\n\\begin{equation}\n\\begin{split}\n& \n\\rho^{ilj}\n = \n\\frac{2C \\epsilon^{0i l j} \\int_{S^{ilj}} \\mu_5 \\mathrm{d} x^0 }{\\vol(S^{ilj})}\n= - M^{ilj}\\,. \n\\end{split}\n\\end{equation}\nThe number of the unstable NG modes along the $x^l$ direction coincides with $\\frac{1}{2}\\rank (\\rho^{ilj})$.\n\n\n\\subsection{Dynamical axion in electric field}%\nThe next example is the dynamical axion in the background electric field in $(3+1)$ dimensions~\\cite{Ooguri:2011aa}. We consider the effective action\n\\begin{equation}\n S = -\\frac{F^2_\\phi}{2} \\int |\\mathrm{d} \\phi|^2 -\\frac{1}{2e^2}\n\\int |\\mathrm{d} a|^2\n+ \n\\int \\mathrm{d}\\phi \\wedge \\mathrm{d} a \\wedge A_{\\phi a}\\,,\n\\end{equation}\nwhere $\\phi$ is a 0-form axion field, $F_\\phi$ is a decay constant of  the axion, and $\\mathrm{d} A_{\\phi a} = E_{\\phi a, i} \\mathrm{d} x^0 \\wedge \\mathrm{d} x^i$ with a constant $E_{\\phi a,i }$.\nIn the plane wave basis, the equations of motion for $\\phi$ and $a$ can be written as\n$  (\\omega^2 -|\\boldsymbol{k}|^2) \\hat{\\phi }\n+\n\\mathrm{i} k_l \\hat{M}^{lj}_{\\phi a}\n  \\hat{a}_j \n=0$\nand \n$(\\omega^2 -|\\boldsymbol{k}|^2) \\hat{a}^i \n+\n\\mathrm{i} k_l \n\\hat{M}^{i l}_{ a \\phi}\n\\hat\\phi \n=0$,\nrespectively.\nHere, we defined\n$\\hat\\phi = F_\\phi \\phi$,\n $\\hat{a}_\\mu = \\frac{1}{e} a_\\mu$, and \n $\\hat{M}^{li}_{\\phi a} = - \\hat{M}^{il}_{ a \\phi }\n  = \\frac{e} {F_{\\phi}}M^{li}_{\\phi a} = \n  \\epsilon^{0 lik}\\frac{e} {F_{\\phi}} E_{\\phi a,k }$.\nFor concreteness, we take our coordinate so that \n$E_{\\phi a,i } = (0,0,E_{\\phi a,3 })$,\nand we focus on the wave vector $\\boldsymbol{k} = (k_1, 0,0)$.\nWe then find the dispersion relations $ \\omega^2 = k_1^2$, $ k_1^2 \\pm |k_1 \\hat{M}_{\\phi a}^{12}|$, among which there is one unstable mode.\n\nWe can also find the nontrivial correlations between symmetry generators. The equations of motion for $\\phi$ and $a$ lead to the conserved charges\n\\begin{equation}\n\\begin{split}\n Q_{\\phi} (\\Sigma_\\phi )\n&\n=  \\int_{\\Sigma_\\phi} \\left(-F^2_\\phi *\\mathrm{d} \\phi +\n\\mathrm{d} a  \n\\wedge A_{\\phi a}\\right),\n\\\\\n Q_{a} (\\Sigma_a)\n&\n= \\int_{\\Sigma_a} \\left(-\\frac{1}{e^2}* \\mathrm{d} a + \n\\mathrm{d}\\phi \\wedge A_{\\phi a}\\right).\n\\end{split}\n\\end{equation}\nHere, $\\Sigma_\\phi$ is a three-dimensional closed subspace.\nThe correlation function between them is\n\\begin{equation}\n\\mathrm{i} \\vevs{  Q_{\\phi} (\\Sigma_\\phi)\nQ_{a} (\\Sigma_a)}\n =  - \n\\int_{\\Sigma_\\phi  \\cap \\Omega_a} \\mathrm{d} A_{\\phi a}\\,.\n\\end{equation}\nWe take the limit \n$\\Sigma_\\phi \\to S^{l} \\cup \\bar{S}^{l}_{x^l = - \\epsilon}$ \nand then the limit \n$\\Sigma_a \\to S^{jl}_{x^l =\\epsilon\/2} \\cup \\b{S}^{jl}_{x^l = -\\epsilon\/2}$.\nIn this case, we have \n$\\Sigma_\\phi  \\cap \\Omega_a\n\\to  S^{lj}$, and\n\\begin{equation}\n\\begin{split}\n&\n\\rho_{\\phi a}^{lj}\n= \n- M^{lj}_{\\phi a}\n\\,.\n\\end{split}\n\\end{equation}\nFor the above choice of $(E_{\\phi a , i})$, we have, e.g.,\n$\\frac{1}{2}\\rank (\\rho^1) = 1$, \nwhich matches the number of unstable NG modes propagating along the $x^1$ direction.\n\n\n\\subsection{Unstable NG modes from 0-form symmetry breaking}\nBased on our generic description of unstable NG modes above, we give the third example of an unstable NG mode for a conventional 0-form symmetry with a background vector field, which is simple yet new to the best of our knowledge.\n\nWe consider the following low-energy effective action for the spontaneous breaking of the 0-form symmetry $U(1)\\times U(1) \\to \\{1\\}$ in $(2+1)$ dimensions:\n\\begin{equation}\n S = -\\frac{F_\\phi^2}{2} \n\\int |\\mathrm{d}\\phi|^2 - \\frac{F_\\chi^2}{2} \\int |\\mathrm{d}\\chi|^2\n+ \n\\int \\mathrm{d}\\phi \\wedge \\mathrm{d}\\chi \\wedge A_{\\phi \\chi}\\,.\n\\label{S_0}\n\\end{equation}\nHere, $(\\mathrm{e}^{\\mathrm{i} \\phi }, \\mathrm{e}^{\\mathrm{i}\\chi})$ is the set of \nNG modes for the $U(1)\\times U(1)$ symmetry,\nand \n$A_{\\phi \\chi} = - A_{\\chi \\phi}\n=  A_{\\phi \\chi, \\mu} \n\\mathrm{d} x^\\mu \n= E_{\\phi \\chi, i } x^0 \\mathrm{d} x^i $\nis a background field with a constant $E_{\\phi \\chi, i}$.\nOne may understand $A_{\\phi \\chi}$ as a background gauge field whose conserved charge is the number of linked vortex loops for $\\phi $ and $\\chi$~\\cite{Brauner:2020rtz}.\nThe equations of motion for $\\phi$ and $\\chi $ in the plane wave basis  are\n$  (\\omega^2 - |\\boldsymbol{k}|^2) \\hat{\\phi}\n+\n\\mathrm{i} k_l \\hat{M}^l_{\\phi \\chi} \\hat\\chi \n=0$\nand\n$ (\\omega^2 - |\\boldsymbol{k}|^2) \n\\hat\\chi\n-\n\\mathrm{i} k_l \\hat{M}^l_{\\phi \\chi} \\hat\\phi  =0$,\nrespectively.\nHere, we defined \n$\\hat{\\phi} = F_\\phi \\phi$, \n$\\hat{\\chi} = F_\\chi \\chi$,\nand\n$ \n\\hat{M}_{\\phi \\chi}^l\n= \n\\frac{1}{F_\\phi F_\\chi} M_{\\phi \\chi}^l \n= \\frac{1}{F_\\phi F_\\chi}\n\\epsilon^{0  l i } E_{\\phi \\chi, i }$. \nWithout loss of generality, we choose the direction of the background field so that only $E_{\\phi \\chi , 2}$ is nonvanishing.\nIn this case, we have \n$M^1_{\\phi \\chi} \n=\n\\epsilon^{0 12} E_{\\phi \\chi, 2}\n=\n- E_{\\phi \\chi, 2}\n$\nand $M^2_{\\phi \\chi} =0$.\nWe also choose a wave vector \n$\\boldsymbol{k} = (k_1,0)$.\nThen, we have the dispersion relation\n$\\omega^2 =  k_1^2\n\\pm \n|k_1 \\hat{M}_{\\phi \\chi}^1|$, \nand there exists one unstable mode for\n$|k_1| < |\\hat{M}_{\\phi \\chi}^1|$ along the $x^1$ direction.\n\nNext, we relate the matrix $M_{\\phi \\chi}^l $ to the correlation function of symmetry generators. The conserved charges for the spontaneously broken 0-form symmetries are found from the equations of motion as\n\\begin{equation}\n\\begin{split}\n Q_\\phi (\\Sigma_\\phi)\n &= \\int_{\\Sigma_\\phi} \n (-F_\\phi^2 *\\mathrm{d}\\phi \n+\n\\mathrm{d}\\chi \\wedge A_{\\phi \\chi})\\,,\n\\\\\nQ_\\chi (\\Sigma_\\chi)\n &\n =\n \\int_{\\Sigma_\\chi} \n (-F_\\chi^2 *\\mathrm{d}\\chi \n-\n\\mathrm{d}\\phi \\wedge A_{\\phi \\chi} )\\,,\n\\end{split}\n\\end{equation} \nwhere $\\Sigma_{\\phi, \\chi}$ are closed surfaces. \nThe conserved charges satisfy the correlation function\n\\begin{equation}\n \\mathrm{i} \\vevs{Q_\\phi(\\Sigma_{\\phi}) Q_\\chi(\\Sigma_{\\chi})}\n = \n-\\int_{\\Sigma_\\phi \\cap \\Omega_\\chi}  \\mathrm{d} A_{\\phi \\chi} \\,.\n\\end{equation}\nHere,\n$\\Omega_\\chi$ is\na three-dimensional subspace whose boundary is $\\Sigma_\\chi$.\nTo have the matrix $M_{\\phi \\chi}^l$, we take the limit \n$\\Sigma_\\phi \\to S^l\\cup \\bar{S}^l_{x^l  = -\\epsilon}$\nand then the limit\n$\\Sigma_\\chi \\to \nS^l_{x^l  = \\epsilon\/2}\\cup \\bar{S}^l_{x^l  = -\\epsilon\/2}$.\nWe have \n$\\Sigma_\\phi \\cap \\Omega_\\chi \\to S^l $ and\n\\begin{equation}\n\\begin{split}\n\\rho_{\\phi \\chi}^l\n= M^l_{\\phi \\chi} \\,.\n\\end{split}\n\\end{equation}\nFor the $x^1$ direction, we have $\\frac{1}{2} \\rank (\\rho^1) = 1$, which \ncoincides with the number of unstable NG modes.\n\n\n\n\\sectaps{Discussions}%\nWe remark that our derivation is similar to that of the counting rule for the so-called type-B NG modes~\\cite{Watanabe:2012hr,Hidaka:2012ym,Watanabe:2014fva,Hidaka:2020ucc}. \nIn our derivation, the background fields with only spatial directions \n$\\mathrm{d} A_{IJ} \\sim \\mathrm{d} x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d} x^{i_{p_{IJ}+1}} $ of the latter is replaced by those with the temporal direction $\\mathrm{d} A_{IJ} = \\frac{1}{p_{IJ}!}E_{IJ, i_1...i_{p_{IJ}}} \\mathrm{d} x^0 \\wedge \\mathrm{d} x^{i_1} \\wedge \\cdots \\wedge \\mathrm{d} x^{i_{p_{IJ}}} $, and the commutation relation of symmetry generators, i.e., the time-ordered product of symmetry generators, is replaced by the product ordered in the spatial directions.\nOne main difference here is that there are various choices of spatial directions, which leads to the fact that the unstable NG modes depend on the direction of the wave vector.\nThe classification of conventional type-B NG modes and unstable NG modes in the presence of both spatial and temporal background fields is deferred to future work. \nOne may also be able to extend our counting rule for unstable NG modes to finite temperature in a way similar to the one for conventional NG modes of 0-form symmetries in Ref.~\\cite{Minami:2015uzo}. In such a case, dissipative effects could modify the dispersion relation of unstable NG modes; see, e.g., Refs.~\\cite{Joyce:1997uy,Akamatsu:2013pjd} in the case of chiral plasma instability.\nFinally, it would also be interesting to explore possible physical realizations of the theory considered in \\er{S_0} with straightforward extensions to $D$ dimensions by replacing $A_{\\phi \\chi}$ with a $(D-2)$-form field.\n\n\\section*{Acknowledgements}\nThis work is supported in part by the Keio Institute of Pure and Applied Sciences (KiPAS) project at Keio University and JSPS KAKENHI Grant No.~JP19K03852 (N.~Y.) and by JSPS KAKENHI Grants No.~JP21J00480 and No.~JP21K13928 (R.~Y.).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecommender systems research has employed item ratings, bookmarking actions and other user activities as primary sources of information to generate personalized suggestions because they provide evidence about user preferences. \nIn particular, User-to-User Collaborative Filtering \\cite{Desrosiers-Karypis:11} (henceforth, denoted as U2UCF) analyzes the ratings of items provided by users in order to identify ``like-minded'' people for preference prediction. However, the sparsity of the rating matrices affects recommendation performance. Thus, recent algorithms have been proposed to improve the recognition of preference similarity from rating data (e.g., Matrix Factorization algorithms \\cite{,Koren-Bell:11} such as SVD++ \\cite{Koren:08}), possibly combined with trust information derived from the establishment of social links among users; e.g., \\cite{Tang-etal:13,Yang-etal:17}. While these algorithms achieve good accuracy and coverage, they challenge the explanation of recommendation results because the policies applied to rank items can hardly be described in an intuitive way.\n\nIn the present work, we are interested in assessing whether U2UCF, which has nice explanation properties, can be improved by using other types of information that are complementary to rating data.\nSpecifically, we investigate whether the identification of frequently co-occurring interests in information search can be used to improve recommendation performance. \nWe start from the observation that, if the people who search for items tagged with a certain information category typically also search for items tagged with another category, the two categories might represent related interests.\nTherefore, even though we ignore the reasons behind this relatedness, we might leverage the strength of the association in preference estimation.\nIn this perspective, we propose to to build rich user profiles by extending the preferences for categories of items identified from rating behavior with frequently co-occurring interests for item categories, extracted from the logs of search engines. It can be noticed that interest co-occurrence can be learned by analyzing anonymous interaction sessions because it is aimed at describing general user behavior. Therefore, it can be applied to anonymized search logs, as long as search sessions can be identified.\n\nStarting from a category-based representation of user preferences, based on the analysis of ratings and on items categorization, we propose the following research question:  \n\n{\\em RQ: How does the integration of data about interest co-occurrence in information search influence the performance of a collaborative recommender system that manages category-based user profiles? }\n\nIn order to answer this question we start from a {\\em Simple Category-based Collaborative Filtering (SCCF)} algorithm which infers a user's preferences on the basis of the distribution of her\/his ratings on item categories: a category-based user profile provides a conceptual view on preferences, so that user similarity can be computed by abstracting from item ratings, thus contrasting data sparsity; see \\cite{Sieg-etal:07b,Sieg-etal:10b}.\nThen, we propose the {\\em Extended Category-based Collaborative Filtering (ECCF)} algorithm that enriches category-based user profiles with evidence about interests that frequently co-occur in information search. ECCF employs the extended user profiles for rating estimation.\n\nIn order to evaluate the recommendation performance of ECCF, we extract information about co-occurring interests by analyzing the query log of a largely used search engine. Then, we test our algorithm by applying it to the Yelp Dataset \\cite{Yelp-dataset}, which stores user ratings of various types of businesses.\n\nWe analyze a few settings of ECCF in order to integrate different amounts of information about co-occurring preferences with rating data. In our experiments, we evaluate performance by taking U2UCF and SCCF as baselines: these algorithms differ in neighbor identification but are based on the same rating estimation approach. Therefore, they are a good basis to assess the impact of extended category-based user profiles on preference prediction. We also compare these algorithms with SVD++ to evaluate whether preference extension challenges the capability of recommending relevant items.\nThe results of our experiments show that ECCS outperforms U2UCF and SCCF in accuracy, MRR, diversity of recommendations and user coverage; moreover it outperforms SVD++ in accuracy and diversity of the generated suggestion lists. We thus conclude that preference co-occurrence information can positively contribute to the identification of good neighbors for rating estimation.\n\nIn summary, the main contributions of this work are:\n\\begin{itemize}\n    \\item\n    The integration of data about frequently co-occurring information interests (inferred by observing general search behavior) with category-based user preferences, in order to acquire rich individual user profiles.\n    \\item The ECCF category-based recommendation algorithm, which extends User-to-User Collaborative Filtering to take both frequently co-occurring information interests and preference similarity into account in neighbor identification.\n    \\item Evaluation results aimed at proving the benefits of frequently co-occurring interests to Collaborative Filtering.\n\\end{itemize}\n\nIn the following,\nSection \\ref{sec:related} positions our work in the related one. Section \\ref{model} presents ECCF. Section \\ref{sec:validation} describes the experiments we carried out to validate ECCF and discusses the evaluation results. Section \\ref{sec:conclusions} concludes the paper and outlines our future work. \n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\subsection{Recommender Systems}\n\nCross-domain recommendation has received the researchers' attention as a way to employ multiple information sources to contrast data sparsity; e.g., \\cite{Fernandez-Tobias-etal:16}. Moreover, holistic user models have been developed that jointly analyze different types of user behavior to enhance the recognition of the user's needs; e.g., \\cite{Teevan-etal:05, Musto-etal:2018b}.\nHowever, the fusion of personal information from different applications is problematic, unless it is done within a tightly integrated software environment. For instance, most people operate anonymously \\cite{Greenstein-etal:17} or have multiple identities \\cite{Doychev-etal:14}; moreover, most user activity logs are anonymized for privacy preservation purposes. It is thus interesting to consider other types of knowledge integration that do not require user identification across applications. Our work investigates this path of research.\n\n\nCollaborative Filtering generates suggestions by analyzing item ratings to identify similar users or similar items.\nSeveral algorithms have been developed, from K-Nearest Neighbors (KNN) to more recent ones such as Matrix Factorization \\cite{Desrosiers-Karypis:11,Koren-Bell:11}. In our work we adopt KNN because it has nice explanation capabilities and has proved to achieve good performance in a comparison with other approaches  \\cite{Jannach-Ludewig:17,Ludewig-Jannach:18}. \n\nOntological user profiles model preferences at the semantic level. In \\cite{Sieg-etal:07b,Sieg-etal:10b}, Sieg et al. propose to exploit a taxonomy whose concepts represent item types, and to infer user interests on the basis of the observed ratings to the instances of such concepts. The neighborhood for rating estimation is then identified by measuring the semantic similarity between ontological user profiles. The category-based user similarity we propose is close to this approach. However, we go one step forward in the identification of preferences by extending the user profiles with frequently co-occurring information interests.\nThis type of extension also differentiates our work from that of Ronen et al., who propose to extend the preferences of the individual user by analyzing her\/his behavior in search logs \\cite{Ronen-etal:16}: that work assumes that the user's activities can be tracked across applications and extends the user profile by analyzing her\/his overall behavior. In contrast, we extend user preferences by analyzing anonymous data about general search behavior. \n\nSen et al. define tag-aware recommender systems as\n``recommender algorithms that predict user's preferences for tags''.\nIn \\cite{Sen-etal:09} they describe different signs of interest; e.g., searching or applying a tag, and so forth.\nOur work relates to tag-aware recommender systems because we analyze rating behavior on items associated to categories expressed as tags. However, we do not consider any other types of interaction with tags for estimating user preferences. \n\nIn \\cite{Gemmel-etal:12}, Gemmel et al. present a linear-weighted hybrid framework for resource recommendation that models different scenarios, among which tag-specific item recommendation. They propose to match users and items on the basis of their tag profiles. Differently, we match users on the basis of category-based profiles learned from rating behavior.\nThe same kind of difference holds between our work and the one of Nakamoto \\cite{Nakamoto:2007}.\n\nWhile TagiCoFi \\cite{Zhen:2009} employs user similarities defined from tagging information to regularize Matrix Factorization, we use tags in a KNN algorithm. \nIn \\cite{Tso-Sutter:2008} Tso and Sutter extend the ratings matrix using tagging information.\nThey reduce the three-dimensional correlations $<user, tag, item>$ to two-dimensional correlations $<user,tag>$, $<item,tag>$ and $<user, item>$. Then, they apply a fusion method to combine the correlations for rating prediction. Differently, we extend the rating matrix with the categories (tags) associated to the items rated by users and with further categories identified from general search behavior. \n\nRecently, rating information has been combined with other types of data to improve recommendation. For instance, item reviews are used, possibly in combination with ratings, in \\cite{Chen-etal:15,Musat-Faltings:15,Muhammad-etal:15,Lu-etal:18}.\nMoreover, trust relations and reputation are used to steer recommendation on the basis of the feedback on items provided by trusted parties; e.g., \\cite{Kuter-etal:07,Liu-Lee:10,Tang-etal:13,Alotaibi-Vassileva:16,Mcnally-etal:14,Du-etal:17,Yang-etal:17}.\nIn \\cite{Mauro-etal:19}, we investigate multi-faceted trust for personalized recommendation. \nHowever, in the present work we focus on rating information to assess the potential improvement of Collaborative Filtering, when combined with general preference co-occurrence. \n\n\n\\subsection{Analysis of Interaction Sessions}\nThe identification of interest co-occurrence we propose is related to a few works supporting query expansion, query reformulation and term suggestion in Information Retrieval. Some researchers propose to analyze session-based user behavior in order to detect co-occurrence relations useful to improve search queries, taking the search context into account. For instance, in \\cite{Cao-etal:08} Cao et al. suggest queries on the basis of the context provided by the user's recent search history, by clustering queries on the basis of the search results visited by users. Moreover, Huang et al. \\cite{Huang-etal:03} and Chen et al. \\cite{Chen-etal:08} detect term co-occurrence in search sessions to group sets of relevant words that can be mutually suggested. \nOur work is different because we adopt a linguistic interpretation approach (based on lemmatization and Word Sense Disambiguation) to find the concepts referenced in the queries; see \\cite{Mauro-Ardissono:17b}. \nTherefore, we extract information about {\\em concept co-occurrence}, which is more general than {\\em term co-occurrence}. \n\nIt is worth mentioning that our analysis of interaction sessions differs from session-based recommendation, which analyzes the user's behavior during an interaction session to identify relevant item(s) to suggest; e.g., see \\cite{Garcin-etal:13,Jannach-Ludewig:17,Greenstein-etal:17,Jannach-etal:17}. In fact, we mine interest co-occurrence by abstracting from the particular sequence of queries performed by the users. Moreover, as previously discussed, we mine concept associations. \n\n\n\n\\subsection{Graph-based Information Filtering}\nKnowledge graphs describe item features and relations among entities, supporting the analysis of item relatedness, as well as similarity for information filtering and top-N recommendation. \nIn several works these graphs are extracted from document pools and\/or from the Linked Data Cloud. For instance, CoSeNa \\cite{Candan-etal:09} employs keyword co-occurrence in the corpus of documents to be retrieved, and ontological knowledge about the domain concepts, to support the exploration of text collections using a keywords-by-concepts graph. Moreover, in \\cite{DiNoia-etal:16}, Di Noia et al. create a relatedness graph by analyzing external data sources such as DBpedia in order to support the evaluation of semantic similarity between items. Analogously, item features have been extracted from the Linked Data Cloud to improve recommendation performance in \\cite{Musto-etal:16,Ragone-etal:17,Musto-etal:17,Musto-etal:18}. \n\nSome works attempt to extend the relations among information items by integrating data derived from the observation of different types of user behavior. E.g., Google search engine manages the Knowledge Graph \\cite{GoogleKnowledgeGraph} to relate facts, concepts and entities depending on their co-occurrence in queries. Moreover, entity2rec learns user-item relatedness from knowledge graphs by analyzing data about users' feedback and item information from Linked Open Data \\cite{Palumbo-etal:17}. Furthermore, in \\cite{Oramas-etal:15} Oramas et al. propose a hybrid recommender that integrates users implicit feedback into a knowledge graph describing item information, enriched with semantic data extracted from external sources. Finally, in \\cite{Vahedian-etal:17}, Vahedian et al. generalize graph-based approaches by simultaneously taking into account multiple types of relations among entities: they introduce meta-paths to represent patterns of relations and apply random-walk along such paths to identify relevant entities to suggest.\n\nOur work has analogies to the above listed ones because we employ a graph-based type of knowledge representation. However, we work at the conceptual level: our knowledge graph relates item categories instead of individual users and\/or items. Moreover, we do not compute similarity or relatedness by means of the knowledge graph: we use the graph to extend category-based user profiles. In turn, those profiles are employed in neighborhood identification. The separation between how preferences are inferred and how they are used for recommendation makes it possible to extend both types of activities in a modular way.\n\n\n\n\\section{Extended Category-based Collaborative Filtering}\n\\label{model}\nWe describe ECCF incrementally, starting from U2UCF that provides the basic match-making approach for rating estimation. \n\n\\subsection{User-to-User Collaborative Filtering}\nIn \\cite{Ricci-etal:11}, Ricci et al. define U2UCF as follows: ``the simplest and original implementation of this approach recommends to the active user the items that other users with similar tastes liked in the past. The similarity in taste of two users is calculated based on the similarity in the rating history of the users\".\nGiven: \n\\begin{itemize}\n    \\item $U$ as the set of users and $I$ as the set of items;\n    \\item $r: U X I \\Rightarrow {\\rm I\\!R}$ as a map of ratings;\n    \\item $R \\in {\\rm I\\!R}^{U X I}$ as the users-items rating matrix, where each value is a rating $r_{ui}=R[u,i]$ given by a user $u \\in U$ to an item $i \\in I$. \n\\end{itemize}\nThe recommender system estimates $u$'s rating of $i$ ($\\hat{r}_{ui}$) as follows:\n        \\begin{equation}\n        \\label{eq:rmeancentering}\n        \\hat{r}_{ui} = \\bar{r}_u + \\frac{ \n        \t\\sum\\limits_{v\\in N_i(u)}\\sigma(u,v) (r_{vi} - \\bar{r}_v)\n        }{\n        \t\\sum\\limits_{v\\in N_i(u)}|\\sigma(u,v)|}\n    \\end{equation}\nwhere $N_i(u)$ is the set of neighbors of $u$ that rated item $i$ and $\\sigma(u,v)$ is the similarity between user $u$ and user $v$ ($v \\in N_i(u)$). The similarity among users is computed by applying a distance metric, e.g., Cosine or Pearson similarity, to their rating vectors. \n\n\n\\subsection{Simple Category-based Collaborative Filtering (SCCF)}\n\\label{category-based-CF}\nSCCF manages user profiles in which the user's interest in each item category is represented as a positive number; the higher is the value, the stronger is the interest.  \nWe define:\n\\begin{itemize}\n    \\item $U$, $I$, $r$ and $R$ as above; \n    \\item $C$ as the set of item categories;\n    \\item $f: U X C \\Rightarrow {\\rm I\\!N} $ as a map between users and categories;\n    \\item $UC\\in {\\rm I\\!N}^{U X C}$ as the Users-Categories matrix. For each $u \\in U$ and $c \\in C$, $UC[u,c]$ represents the interest of $u$ in $c$. We take as evidence of interest the {\\em frequency of exploration} of a category, i.e., the frequency of interaction of the user with items associated with the category. \n\\end{itemize}\nCategory exploration can be mapped to different types of user behavior; e.g., tagging items and searching for items by tag. We map exploration to rating behavior and we define $UC[u, c]$ as the number of ratings that $u$ has given to the items associated with $c$. \n\nSCCF computes user similarity on the basis of the estimated user preferences for item categories. Specifically, $\\sigma(u, v)$ is defined as the Cosine similarity of the users vectors in the $UC$ matrix and it is used in Equation (\\ref{eq:rmeancentering}) to estimate ratings. Thus, $\\hat{r}_{ui}$ is computed on the basis of the ratings $r_{vi}$ provided by the users $v \\in U$ whose preferences for categories are similar to those of $u$. \n\n\n\\subsection{Acquisition of Preferences Co-occurrence}\n\\label{graph}\nIn order to learn the strength of the associations between item categories in search behavior, we analyze their co-occurrence in the search sessions of a query log. By co-occurrence we mean the fact that two or more categories are referred by the queries belonging to the same session. \nIn the following we summarize the analysis of category co-occurrence; see \\cite{Mauro-Ardissono:18} for details.\n\nThe Category Co-occurrence Graph ($CCG$) represents category co-occurrence:\nin the $CCG$, nodes represent the data categories referenced in the analyzed queries and the weight of edges represents the co-occurrence frequency of the connected categories; i.e., how many times the categories have been identified within the same search sessions.\n\nWe retrieve the categories occurring in the queries by applying a Natural Language approach that identifies the referred concepts in a flexible way, by considering synonyms and by applying Word Sense Disambiguation to resolve the meaning of words; see \\cite{Ardissono-etal:16,Mauro-Ardissono:17b}. For Word Sense Disambiguation we use the Babelfy tool \\cite{Babelfy}.\n\nThe $CCG$ is built as follows:\ngiven two categories $x$ and $y$, the weight of the edge that connects them is defined as:\n\\begin{equation}\n\\label{eq1}\nw_{xy}=\\sum_{S\\in|Sessions|} Freq_{S_{xy}}\n\\end{equation}\nwhere $Freq_{S_{xy}}$ represents the evidence provided by session $S$ to the co-occurrence frequency of $x$ and $y$.\nGiven $S=\\{Q_1, \\dots, Q_n\\}$,\n$Freq_{S_{xy}}$ is computed as the maximum evidence of co-occurrence of $x$ and $y$ in $S$: \n\\begin{equation}\n\\label{eq2}\nFreq_{S_{xy}} = Max_{k=1}^{n}(Freq_{xy_{Q_k}}, ev_{xy_{Q_{k-1}}})\n\\end{equation}\nwhere $Freq_{xy_{Q_k}}$ is the co-occurrence evidence of $x$ and $y$  provided by query $Q_k$, and $ev_{xy_{Q_{k-1}}}$ is the one provided by $Q_1, \\dots, Q_{k-1}$. Similar to \\cite{Mauro-Ardissono:18}, we take the maximum, and not the sum of evidence because co-occurrence could derive either from query reformulation \\cite{Rieh-Xie:06}, or from the repetition of queries in click-through events of the log; see Section \\ref{sec:AOLlog} that describes the query log we used.\n\nA query $Q$ contributes to the estimation of co-occurrence as follows:\n\\begin{itemize}\n\\item \nIf $Q$ contains $k$ terms ($k>=0$), each one identifying a non-ambiguous category: \n$T_1 \\Rightarrow c_1, \\quad \\dots, \\quad T_k \\Rightarrow c_k$, then, for each category $c$ of $Q$:\n\\begin{itemize}\n    \\item The co-occurrence evidence between $c$ and every other category $d$ of $Q$ is $Freq_{cd_{Q}} = 1$. \n    \\item The co-occurrence evidence between $c$ and every other category $e$ identified in a non-ambiguous way in the other queries of $S$ is $Freq_{ce_{Q}} = 1$. \n    \\item The co-occurrence evidence between any other categories $w$ and $z$ identified in $S$ is $Freq_{wz_{Q}} = 0$.\n\\end{itemize}\n\\item \nIf $Q$ contains an ambiguous term $t$ that refers to $m$ categories, the particular category the user is focusing on cannot be identified. Therefore, the co-occurrence evidence brought by $t$ is computed as above, but the assigned evidence is $\\frac{1}{m}$ in order to consider the possible interpretations of $Q$, and divide evidence among ambiguous categories.\n\\end{itemize}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{model.pdf}\n    \\caption{Extension of Category-based User Profiles. }\n    \\label{fig:matrici}\n\\end{figure}\n\n\n\\subsection{Extended Category-based Collaborative Filtering (\\textit{ECCF})}\n\\label{sec:ECCF}\nIn this recommendation model we employ frequent co-occurring information interests to extend category-based user profiles.\nWe reinforce the preferences for item categories learned by analyzing rating behavior (stored in the Users-Categories matrix $UC$) with interest co-occurrence associations (stored in the $CCG$ graph) in order to acquire an extended set of user preferences for neighbor identification.\n\nThe idea behind preference extension is that, the more the user has appreciated the items of a category, the more interest\n\\linebreak\nco-occurrence makes sense. Therefore, starting from the category-based user profiles stored in the $UC$ matrix, we increment user preferences with the contribution of the strongest co-occurrence relations of the $CCG$ graph, depending on the number of positive ratings available in the users-items matrix $R$. \nThe output of this process is stored in the Extended Preferences matrix $EP$, which is used to compute $\\sigma(u,v)$ in Equation \\ref{eq:rmeancentering}.\n\nFigure \\ref{fig:matrici} provides a graphical view of the computation of $EP$: the information stored in $UC$ is combined with that stored in the $CCG$ to set the values of this matrix. In this process, the users-ratings matrix $R$ is used to limit the reinforcement of preferences to the categories of the positively rated items.\nMoreover, the $CCG$ is used to propagate preference information according to the strongest co-occurrence of interests. \nIn detail, we compute the values of $EP$ as follows:\n\\begin{itemize}\n    \\item \n    let $Cat_i$ be the set of categories associated to item $i$; \n    \\item\n    let $CatSet_i$ be the set of categories directly connected to any category $c \\in Cat_i$ in the $CCG$ through the heaviest outbound arcs. These are the categories which most frequently co-occur with some categories of $Cat_i$ in search sessions.\n\\end{itemize}\nThen:\n\\begin{equation}\n\\label{eq:pm}\nEP[u,c]=UC[u,c]+\\sum_{i\\in|I|} f(u,i,c)\n\\end{equation}\nwhere\n\\begin{equation}\n             f(u,i,c) =\n            \\begin{cases}\n                1 & \\quad \\text{if $R[u,i] \\in$} ~ \\text{\\textit{PositiveRatings}} ~ \\text{$\\wedge$} ~\\text{$c \\in CatSet_i$}\\\\\n                0 & \\quad \\text{otherwise}\n            \\end{cases}\n\\label{eq:f}\n\\end{equation}\nIn Equation \\ref{eq:f} $PositiveRatings$ denotes the set of ratings that are considered as positive in the dataset; e.g., \\{5\\}, or \\{4, 5\\} in a [1, 5] Likert scale.\n\n\n\\section{Validation of ECCF}\n\\label{sec:validation}\n\n\\subsection{Dataset of Item Ratings}\n\\label{sec:YELP}\nAs a source of rating data we exploit the Yelp Dataset \\cite{Yelp-dataset}, which contains information about a set of businesses, users and reviews and is available for academic purposes. In the dataset, item ratings take values in a [1, 5] Likert scale where 1 is the worst value and 5 is the best one. \nMoreover, each item is associated with a list of categories describing the kind of service it offers.\n\nThe full list of Yelp categories is available at \\url{www.yelp.com\/developers\/documentation\/v3\/category_list} and is organized in a taxonomy to specify businesses at different levels of detail. The taxonomy includes a large set of first-level categories, representing broad types of businesses; e.g., ``Active life'', ``Arts \\& entertainment'', ``Automotive'', \\dots, ``Food'', ``Restaurants'', and many others. In turn, the first-level categories are specialized into sub-categories; e.g., ``Restaurants'' includes many types of restaurants such as ``Indian'', ``Chinese'' and the like.  \nWe apply two filters to the dataset:\n\\begin{enumerate}\n    \\item \n    We select all the Yelp categories that are subclasses of ``Restaurants'' or ``Food'': e.g., ``Indian'', ``Chinese'', ``Cafes'', ``Kebab'', ``Pizza'', and so forth; the total number of categories is 254. \n    Then, we project the Yelp dataset on the set of items associated with at least one of these categories. In the rest of this paper we refer to this set of categories as {\\em CATS}. \n    \\item We further filter the dataset on the users who rated at least 20 items.\n\\end{enumerate}\n\n\n\\begin{table}[t]\n\\centering\n\\caption{Statistics about the Filtered Datasets}\n\\begin{tabular}{l|l|l}\n\\hline\nYelp & Number of users            & 26,600   \\\\\n& Number of businesses       & 76,317   \\\\ \n& Number of ratings          & 1,326,409 \\\\ \\hline\nAOL & Number of sessions & 1,248,803 \\\\\n& Number of queries & 2,136,029 \\\\\n\\hline\n\\end{tabular}\n\\label{t:dataset}\n\\end{table}\nThe higher portion of Table \\ref{t:dataset} summarizes the number of users, businesses and ratings of the filtered Yelp dataset.\n\n\n\\subsection{Dataset of Search Sessions}\n\\label{sec:AOLlog}\nFor the generation of the Category Co-occurrence Graph we use the AOL query log.\\footnote{The log is available at \\url{http:\/\/www.cim.mcgill.ca\/~dudek\/206\/Logs\/AOL-user-ct-collection\/}.}  \nEach line of the log represents either a query or a click-through event on one of the search results of a query. The line contains various fields, among which the submitted query and the submission date and hour.\n\nIn order to build a graph that is thematically related to the items of the filtered Yelp dataset, we select from the log the search sessions relevant to the categories $c \\in CATS$ enriched with the following two types of external knowledge. The enrichment is useful to abstract from the specific category names used in Yelp and to take into account semantically related information: \n\\begin{enumerate}\n    \\item \n    {\\em Lemmatized knowledge:} we enrich each element $c \\in CATS$ with a set of keywords and synonyms from WordNet \\cite{WordNet} lexical database. \n    \\item \n    {\\em Relevant terms from the Probase \\cite{Wu-etal:12} taxonomy:} \n    \\begin{itemize}\n        \\item\n        For each element $c \\in CATS$, we enrich $c$ with the \n        \\linebreak\n        $<concept, instance>$ pairs of ProBase such that $concept$ has at least 85\\% WordNet similarity with any term of the lemmatized knowledge of $c$, and the WordNet similarity between the two components of the pair is 85\\%.\n        \\item \n        ProBase, recently called Microsoft Concept Graph, is a large concept network harnessed from web pages and search logs. It is organized as a list of $<instance, concept>$ pairs related by a subclass relation and it contains \n        \\linebreak \n        5,376,526 classes and 12,501,527 instances.\n    \\end{itemize}\n\\end{enumerate}\nFor the selection of relevant search queries in the AOL log we match the lemmatized words occurring in the queries to the enriched categories of $CATS$. If there is at least one match between a term and a query, we consider the query as relevant and we include its parent session in the filtered log. \n\nWe identify the search sessions by aggregating the queries performed by the same user according to their temporal proximity, following the widely applied rule that two consecutive queries belong to different sessions if the time interval between them exceeds half an hour; see \\cite{White-etal:07}. \n \nThe lower portion of Table \\ref{t:dataset} shows the number of sessions and queries of the filtered AOL dataset.\n\nIt is worth noting that the AOL log was involved in an information leak issue but we decided to use it for two reasons. Firstly, our analysis is ethically correct because we study general search behavior to acquire aggregate data abstracting from the search histories of individual users. Secondly, to the best of our knowledge, the AOL log is the only available large dataset that reports textual search queries, and which can therefore be used for linguistic interpretation. We analyzed some public datasets but they did not meet our requirements. For instance, the Excite query dataset\\footnote{\\url{https:\/\/svn.apache.org\/repos\/asf\/pig\/trunk\/tutorial\/data\/}} contains about 1M queries while AOL log contains 20M queries. Moreover, in the Yahoo dataset\\footnote{\\url{https:\/\/webscope.sandbox.yahoo.com\/catalog.php?datatype=l\\&did=50}} the queries are coded; thus, it is not possible to extract any linguistic information to learn category co-occurrence.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.6\\linewidth]{graph-distribution.jpg}\n  \\caption{Distribution of the Weight of Edges in the $CCG$.}\\label{fig:graph-distribution}\n\\end{figure}\n\n\n\\subsection{Category Co-occurrence Graph}\nWe instantiate the {\\em CCG} with the interests that co-occur in the sessions of the filtered AOL dataset by applying the procedure described in Section \\ref{graph}.\nThe resulting graph is strongly connected: almost all of the categories are linked to each other by an edge having weight $>0$.\nHowever, the distribution of weights in the graph shows that there is a large number of weakly connected categories and a very small number of strongly associated ones. The ``heavy'' edges identify the interests that co-occur very frequently in search sessions and suggest to select the arcs having maximum weight in the {\\em CCG} for the extension of the user profiles, as done in Section \\ref{sec:ECCF}.\nFigure \\ref{fig:graph-distribution} shows this distribution; the x-axis represents the edges of the graph, and the y-axis represents their weights, which take values in [1, 272224].\n\n\\begin{table}[b]\n\\centering\n\\caption{Performance Evaluation @10; the Best Values Are in Boldface, the Worst Ones Are Strikethrough}\n\\begin{tabular}{l|l|l|l|l|l}\n\\hline \n\\textbf{Metrics} \n                             & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}U2UCF\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{SCCF}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{3,4,5\\}\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{4,5\\}\\end{tabular}}} & \\multicolumn{1}{c}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{5\\}\\end{tabular}}} \\\\ \\hline\n\\textbf{Precision}    & \\st{0.7823}   & \\textbf{0.786}  & 0.7857   & 0.7855  & 0.7859   \\\\\n\\textbf{Recall}       & \\st{0.7473}   & 0.7526   & 0.7536   & \\textbf{0.755}   & 0.7529  \\\\\n\\textbf{F1}           & \\st{0.7644}   & 0.7689   & 0.7693   & \\textbf{0.7699}  & 0.769   \\\\\n\\textbf{RMSE}         & \\st{1.0001}   & 0.9899   & 0.9897   & 0.9893 & \\textbf{0.9892}   \\\\\n\\textbf{MRR}          & \\st{0.733}    & 0.7367   & 0.737    & \\textbf{0.7391}  & 0.7384    \\\\ \n\\textbf{Diversity}  & \\st{0.3042}  & 0.3053 & \\textbf{0.3056} & 0.3053  & 0.3049 \\\\\n\\textbf{User cov.}   & \\st{0.8497}  & 0.8521  & 0.8526 & \\textbf{0.8542}  & 0.8534  \\\\\n\\hline\n\\end{tabular}%\n\\label{t:results@10}\n\\end{table}\n\n\\subsection{Test Methodology}\n\\label{experiments}\nWe evaluate the recommendation performance of ECCF by comparing it to U2UCF and SCCF, which we consider as baselines. Moreover, we compare these algorithms with SVD++ in order to assess the improvement in the suggestion of relevant items given by frequently co-occurring interests.\n\nThe SCCF and ECCF recommendation algorithms are developed by extending the Surprise library \\cite{Surprise}, while we use the default Surprise implementations of U2UCF and SVD++.\n\nWe test the algorithms by applying a 10-fold cross-validation on the filtered Yelp dataset, after having randomly distributed ratings on folds: we use 90\\% of the ratings as training set and 10\\% as test set. In all the tests, we configure the KNN algorithms to work with 50 neighbors.\n\nIn order to analyze the impact on recommendation performance of a looser, or stricter extension of user preferences with category co-occurrence, we validate ECCF on different settings of $PositiveRatings$ in Equation \\ref{eq:f}, i.e., on different interpretations of what is a good rating. For each fold\nwe generate three versions of the Extended Preferences matrix $EP$ having set $PositiveRatings$ to $\\{3,4,5\\}$, $\\{4,5\\}$, and $\\{5\\}$ respectively.  \n\nWe evaluate Top-k recommendation performance with k=10 and k=20 by taking the ratings observed in the Yelp dataset as ground truth. For the evaluation we consider the following metrics: Precision, Recall, F1, RMSE, MRR, Diversity and User Coverage.\n\nDiversity describes the mean intra-list diversity of items in the suggestion lists @k; see \\cite{Bradley-Smyth:01}. In this work, we interpret diversity from the viewpoint of item classification. Therefore, we measure the diversity of a recommendation list as follows:\n\\begin{equation}\n\\text{intra-list diversity@k}={\\frac  {\\sum _{{i=1}}^{k}\\sum _{{j=i}}^{k} (1 - sim(i, j))} {\\frac{k*(k+1)}{2}}}\n\\end{equation}\nwhere $sim(i, j)$ is the cosine similarity between the lists of categories associated to items $i$ and $j$ in the ratings dataset. \n\n\\subsection{Results}\n\\label{results}\nTable \\ref{t:results@10} shows the performance results of the KNN recommenders we compared, by taking into account a maximum of 10 suggested items (performance@10). \n\\begin{itemize}\n    \\item \\textbf{Precision:} similar to previous results described in \\cite{Sieg-etal:07b}, all of the category-based recommenders outperform U2UCF. This can be explained by the fact that the matrices describing preferences for item categories are denser than the ratings one. Thus, they improve recommendation by supporting a better identification of neighbors for Equation \\ref{eq:rmeancentering}.  \n    However, SCCF outperforms all of the ECCF variants. The second best recommender is ECCF$\\{5\\}$ that extends user profiles in the strictest way: it only considers as pivots for extension the categories associated to the items that the user has rated 5 stars. Notice also that the precision of ECCF decreases when $PositiveRatings$ is lax. The reason is that the extension of user profiles with frequently co-occurring interests can increase the estimated interest in some noisy categories with respect to the pure observation of ratings distribution on categories. In particular, noise grows when the policy applied to extend preferences is less restrictive. \n    \\item \\textbf{Recall:} ECCF outperforms the baselines in all the settings of $PositiveRatings$. Specifically, ECCF\\{4,5\\} achieves the best result, while recall is lower in ECCF\\{3,4,5\\} and further decreases in ECCF\\{5\\}.\n    We explain this finding as follows: an extension of user profiles based on the categories of highly rated items supports the identification of a richer set of user preferences, and a more efficacious identification of neighbors, than only considering rating distribution on categories. However, if we restrict $PositiveRatings$ too much, the user profiles are not extended enough to sensibly improve Recall. Moreover, as noticed for Precision, if $PositiveRatings$ is lax, noise in the estimation of user preferences challenges neighbor selection.\n    \n\\begin{table}[t]\n\\centering\n\\caption{Performance Evaluation @20}\n\\begin{tabular}{l|l|l|l|l|l}\n\\hline \n\\textbf{Metrics} \n                             & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}U2UCF\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{SCCF}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{3,4,5\\}\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{4,5\\}\\end{tabular}}} & \\multicolumn{1}{c}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{5\\}\\end{tabular}}} \\\\ \\hline\n\\textbf{Precision}  & \\st{0.7806} & \\textbf{0.7842}  & 0.7839   & 0.7838  & \\textbf{0.7842} \\\\ \n\\textbf{Recall}     & \\st{0.757}  & 0.7624  & 0.7634  & \\textbf{0.7649} & 0.7626  \\\\ \n\\textbf{F1}  & \\st{0.7686}  & 0.7731   & 0.7735  & \\textbf{0.7742}  & 0.7732  \\\\\n\\textbf{RMSE}  & \\st{0.9935}  & 0.9838 & 0.9835 & \\textbf{0.9832}  & \\textbf{0.9832} \\\\ \n\\textbf{MRR}  & \\st{0.733} & 0.7369   & 0.7372   & \\textbf{0.7391}  & 0.7384  \\\\ \n\\textbf{Diversity} & \\st{0.3059}  & 0.307  & \\textbf{0.3073}  & 0.307  & 0.3067  \\\\ \n\\textbf{User cov.}   & \\st{0.8497}  & 0.8521  & 0.8526 & \\textbf{0.8542}  & 0.8534  \\\\\n\\hline\n\\end{tabular}%\n\\label{t:results@20}\n\\end{table}\n\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{accuracy_10.jpg}\n \\caption{Graphical Representation of Accuracy@10.}\n \\label{fig:accuracy@10}\n\\end{figure}\n\n\n    \\item \\textbf{F1:} ECCF outperforms the baselines. In detail, ECCF\\{4,5\\} achieves the best F1 = 0.7691; moreover, F1 varies consistently with Recall, depending on $PositiveRatings$. \n    \\item \\textbf{RMSE:} SCCF reduces the mean error between estimated and observed ratings with respect to the baseline, showing the benefits of category-based user profiles. Moreover, consistently with the variation of Precision, the best results are obtained by ECCF\\{5\\}, i.e., with a strict extension of user profiles. RMSE progressively increases (i.e., gets worse) for $PositiveRatings=\\{4, 5\\}$ and \\{3, 4, 5\\}.\n    \\item \\textbf{MRR:} ECCF outperforms the baselines. Specifically, \n    \\linebreak\n    ECCF\\{4,5\\} obtains the best MRR = 0.7391. The second best value corresponds to a more selective extension of user profiles in ECCF\\{5\\}; moreover, if $PositiveItems=\\{3, 4, 5\\}$ results get worse.\n    \\item \\textbf{Diversity}: both SCCF and ECCF outperform U2UCF. In this case, the best results are obtained with a lax extension of user preferences (ECCF\\{3,4,5\\}) and Diversity decreases while the preference extension policy becomes stricter. We explain these findings with the fact that category-based user profiles improve the estimation of user preferences concerning a variegate set of item categories, with respect to a flat recommendation based on ratings. However, the stricter is the extension of user preferences, the less item categories are used in neighbor identification.\n     \\item \\textbf{User coverage:} ECCF outperforms the baselines, confirming the usefulness of preference extension. However, the selection of the ratings for the extension influences coverage: ECCF\\{4,5\\} achieves the best results by suggesting at least one relevant item to 85.42\\% of the users, against 84.97\\% of U2UCF. The second best is ECCF\\{5\\} and ECCF\\{3,4,5\\} has the worst results.\n    \\end{itemize}\nIn the described experiments the $EP$ Matrix is defined by only taking into account positive ratings. In order to get a broader view on the performance of ECCF, we also consider its application to all the user ratings; i.e., we set $PositiveRatings$ to $\\{1, \\dots, 5\\}$. With respect to the previous results, in this case the algorithm achieves similar Precision but lower Recall (0.7524), MRR (0.7369) and User coverage (0.8155).   \n\nTable \\ref{t:results@20} shows the results obtained by comparing \n\\linebreak\nperformance@20. \nThese results confirm the usefulness of category-based user profiles and of their extension with frequently\n\\linebreak\nco-occurring information interests: \n\\begin{itemize}\n    \\item Also in this case, ECCF\\{4,5\\} is the best recommendation algorithm. It outperforms the others in Recall, F1, MRR and User coverage. Moreover both ECCF\\{5\\} and ECCF\\{4,5\\} achieve the best RMSE in comparison with the other recommenders.  \n    \\item However, while SCCF has the best Precision@10, both SCCF and ECCF\\{5\\} achieve the best Precision@20.\n\\end{itemize}\nWith respect to k=10, Precision@20 is lower while Recall@20 and F1@20 take higher values; this makes sense because we are considering longer suggestion lists. Moreover, RMSE@20 is lower, which tells us that the longer lists contain proportionally less errors in the estimation of ratings. Differently, most algorithms obtain the same MRR for k=10 and k=20 (except for SCCF and ECCF\\{3,4,5\\}): this shows that the first relevant item is almost always placed in the first 10 positions of the suggestion lists.\nFurthermore, the Diversity@20 has the highest values for all the recommenders: this might be due to the fact that the longer suggestion lists have more chances to include items belonging to different categories. Finally, User coverage@10 = User coverage@20 because we interpret coverage as the percentage of users who receive at least one suggestion.\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{accuracy_20.jpg}\n \\caption{Graphical Representation of Accuracy@20.}\n \\label{fig:accuracy@20}\n\\end{figure}\n\nFigures \\ref{fig:accuracy@10} and \\ref{fig:accuracy@20} depict the accuracy @10 and @20:\n\\begin{itemize}\n    \\item \n    All of the category-based recommenders outperform U2UCF, confirming the benefits of the introduction of category-based preferences in KNN Collaborative Filtering. The conceptual representation of user preferences generally improves performance because the matrices describing user preferences ({\\em UC} and {\\em EP}) are denser than the users-items matrix storing ratings ({\\em R}). Therefore, better neighbors can be identified for the computation of Equation \\ref{eq:rmeancentering}.\n    \\item\n    A comparison between category-based algorithms shows that the best performance results are obtained by extending user profiles on the basis of the items that users have rated very well, i.e., with 4 or 5 stars in a [1, 5] Likert scale. If the items that received middle ratings are considered as well, accuracy decreases. \n    \\item The category-based representation of user profiles has positive impact on the Diversity of recommendation lists. Conversely, the extension of user profiles does not further help this aspect, unless user profiles are extended in a lax way. However, a lax extension is not convenient because it decreases other measures.\n\\end{itemize}\n\n\nIn order to assess the usefulness of preference extension in Top-k recommendation, we also compare the previously described algorithms with SVD++ \\cite{Koren:08}, which adopts Matrix Factorization to learn latent user and item factors, basing rating prediction on the sole analysis of user ratings. The comparison results show that:\n\\begin{itemize}\n    \\item \n    SVD++ is more accurate than U2UCF and SCC. On the filtered Yelp dataset, SVD++ obtains F1@10 = 0.7696. This finding shows that the management of category-based user profiles helps recommendation but it can be outperformed by a deeper understanding of the features of items and users. \n    \\item\n    SVD++ achieves similar accuracy results with respect to ECCF but it is outperformed by ECCF\\{4, 5\\}. Therefore, the extension of user profiles with frequently co-occurring information interests, integrated into a KNN recommender, improves accuracy and makes it comparable or higher than that of Matrix Factorization algorithms. \n    \\item\n    ECCF outperforms SVD++ as far as the diversity of the recommendation lists is concerned: SVD++ has Diversity@10 = 0.3041; this is comparable to the diversity achieved by U2UCF and lower than that of all the category-based recommenders we presented.\n    \\item\n    In contrast, SVD++ has the highest User coverage of all the algorithms (0.8709), showing its superior capability to contrast data sparsity. \n    \\end{itemize}\n    \n\\subsection{Discussion}\n\\label{discussion}\nIn summary, the evaluation results show that ECCF outperforms U2UCF, SCCF and SVD++ in accuracy and intra-list diversity. Moreover, it outperforms U2UCF and SCCF in MRR and user coverage, while SVD++ excels in the latter metric. The results also show that ECCF achieves the best results when applied to positive ratings, while its performance slightly decreases when the user profiles are extended by taking both positive and negative ratings.\n\nThese results support the hypothesis that preference extension, based on frequently co-occurring information interests, improves the accuracy of the suggestions generated by a KNN recommender system. However, research has to be carried out to improve other performance metrics, possibly also investigating the integration of preference co-occurrence in Matrix Factorization algorithms.\n\nIt might be questioned whether extending user profiles with general interest co-occurrence data might provide less personalized recommendations than, e.g., focusing the extensions on the user's neighborhood. In this respect, we point out that we aim at developing a model that does not depend on cross-domain user identification. However, an investigation of this issue can be interesting to deal with the cases in which user information can be shared among the applications, or public information about the users can be connected to the local profiles; e.g., public data on social networks.\n\nBefore closing this discussion, it is worth noting that, even though the AOL query log dates back to 2006, it can be considered as a good information source as long as it is analyzed from the viewpoint of the concepts expressed by the users. In other words, while the specific information items mentioned in the log might not exist any more, the topics referred in the queries are general and long-lasting. Of course, some new topics (e.g., new types of restaurants) might have emerged since 2006, and maybe new concept associations could exist now. However, the described performance results show that the co-occurring interests we identified are useful to improve recommendation performance; moreover, the methodology described in this paper can be applied to other more recent datasets, if available.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe investigated whether the identification of frequently\n\\linebreak \nco-occurring interests in information search can be used to improve the performance of KNN collaborative recommender systems. For this purpose, we defined a preference extension model that, applied to a category-based representation of user profiles, infers user preferences by exploiting frequently co-occurring information interests. Then, we implemented the model in the Extended Category-based Collaborative Filtering algorithm (ECCF). This is is variant of User-to-User Collaborative Filtering that works on category-based user profiles, enriched with preferences inferred from general search behavior.\nFor the analysis of user interests, we analyzed the query log of a largely used search engine. \n\nWe evaluated ECCF on a large dataset of item ratings, by applying different levels of strictness in the extension of user profiles. The evaluation showed that ECCF outperforms User-to-User Collaborative Filtering in accuracy, MRR, intra-list diversity and user coverage. Interestingly, ECCS also obtains higher accuracy and diversity than the SVD++ recommender system, based on Matrix Factorization; however, ECCS has lower user coverage than SVD++. \n\nIn our future work we will focus on the coverage aspect in order to improve the performance of KNN Collaborative Filtering.\nMoreover, we will carry out further experiments, considering (i) a broader domain than Restaurants and Food, on which we have focused our current work, and (ii) users who have provided few or zero ratings. \nWe will also analyze other datasets to check whether the performance results described in this article can be generalized. Finally, we will compare the performance of ECCF with a larger set of recommendation approaches based on preference extension.\n\n\\begin{acks}\nThis work was supported by the University of Torino through projects ``Ricerca Locale'', MIMOSA\n(MultIModal Ontology-driven query system for the heterogeneous data of a SmArtcity, ``Progetto di Ateneo Torino\\_call2014\\_L2\\_157'', 2015-17)\nand the Computer Science PhD program.\nWe are grateful to Zhongli Filippo Hu, who helped us filter the Yelp dataset.\n\\end{acks}\n\n\n  \\bibliographystyle{ACM-Reference-Format} \n \n\\balance\n\n\\section{Introduction}\nRecommender systems research has employed item ratings, bookmarking actions and other user activities as primary sources of information to generate personalized suggestions because they provide evidence about user preferences. \nIn particular, User-to-User Collaborative Filtering \\cite{Desrosiers-Karypis:11} (henceforth, denoted as U2UCF) analyzes the ratings of items provided by users in order to identify ``like-minded'' people for preference prediction. However, the sparsity of the rating matrices affects recommendation performance. Thus, recent algorithms have been proposed to improve the recognition of preference similarity from rating data (e.g., Matrix Factorization algorithms \\cite{,Koren-Bell:11} such as SVD++ \\cite{Koren:08}), possibly combined with trust information derived from the establishment of social links among users; e.g., \\cite{Tang-etal:13,Yang-etal:17}. While these algorithms achieve good accuracy and coverage, they challenge the explanation of recommendation results because the policies applied to rank items can hardly be described in an intuitive way.\n\nIn the present work, we are interested in assessing whether U2UCF, which has nice explanation properties, can be improved by using other types of information that are complementary to rating data.\nSpecifically, we investigate whether the identification of frequently co-occurring interests in information search can be used to improve recommendation performance. \nWe start from the observation that, if the people who search for items tagged with a certain information category typically also search for items tagged with another category, the two categories might represent related interests.\nTherefore, even though we ignore the reasons behind this relatedness, we might leverage the strength of the association in preference estimation.\nIn this perspective, we propose to to build rich user profiles by extending the preferences for categories of items identified from rating behavior with frequently co-occurring interests for item categories, extracted from the logs of search engines. It can be noticed that interest co-occurrence can be learned by analyzing anonymous interaction sessions because it is aimed at describing general user behavior. Therefore, it can be applied to anonymized search logs, as long as search sessions can be identified.\n\nStarting from a category-based representation of user preferences, based on the analysis of ratings and on items categorization, we propose the following research question:  \n\n{\\em RQ: How does the integration of data about interest co-occurrence in information search influence the performance of a collaborative recommender system that manages category-based user profiles? }\n\nIn order to answer this question we start from a {\\em Simple Category-based Collaborative Filtering (SCCF)} algorithm which infers a user's preferences on the basis of the distribution of her\/his ratings on item categories: a category-based user profile provides a conceptual view on preferences, so that user similarity can be computed by abstracting from item ratings, thus contrasting data sparsity; see \\cite{Sieg-etal:07b,Sieg-etal:10b}.\nThen, we propose the {\\em Extended Category-based Collaborative Filtering (ECCF)} algorithm that enriches category-based user profiles with evidence about interests that frequently co-occur in information search. ECCF employs the extended user profiles for rating estimation.\n\nIn order to evaluate the recommendation performance of ECCF, we extract information about co-occurring interests by analyzing the query log of a largely used search engine. Then, we test our algorithm by applying it to the Yelp Dataset \\cite{Yelp-dataset}, which stores user ratings of various types of businesses.\n\nWe analyze a few settings of ECCF in order to integrate different amounts of information about co-occurring preferences with rating data. In our experiments, we evaluate performance by taking U2UCF and SCCF as baselines: these algorithms differ in neighbor identification but are based on the same rating estimation approach. Therefore, they are a good basis to assess the impact of extended category-based user profiles on preference prediction. We also compare these algorithms with SVD++ to evaluate whether preference extension challenges the capability of recommending relevant items.\nThe results of our experiments show that ECCS outperforms U2UCF and SCCF in accuracy, MRR, diversity of recommendations and user coverage; moreover it outperforms SVD++ in accuracy and diversity of the generated suggestion lists. We thus conclude that preference co-occurrence information can positively contribute to the identification of good neighbors for rating estimation.\n\nIn summary, the main contributions of this work are:\n\\begin{itemize}\n    \\item\n    The integration of data about frequently co-occurring information interests (inferred by observing general search behavior) with category-based user preferences, in order to acquire rich individual user profiles.\n    \\item The ECCF category-based recommendation algorithm, which extends User-to-User Collaborative Filtering to take both frequently co-occurring information interests and preference similarity into account in neighbor identification.\n    \\item Evaluation results aimed at proving the benefits of frequently co-occurring interests to Collaborative Filtering.\n\\end{itemize}\n\nIn the following,\nSection \\ref{sec:related} positions our work in the related one. Section \\ref{model} presents ECCF. Section \\ref{sec:validation} describes the experiments we carried out to validate ECCF and discusses the evaluation results. Section \\ref{sec:conclusions} concludes the paper and outlines our future work. \n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\subsection{Recommender Systems}\n\nCross-domain recommendation has received the researchers' attention as a way to employ multiple information sources to contrast data sparsity; e.g., \\cite{Fernandez-Tobias-etal:16}. Moreover, holistic user models have been developed that jointly analyze different types of user behavior to enhance the recognition of the user's needs; e.g., \\cite{Teevan-etal:05, Musto-etal:2018b}.\nHowever, the fusion of personal information from different applications is problematic, unless it is done within a tightly integrated software environment. For instance, most people operate anonymously \\cite{Greenstein-etal:17} or have multiple identities \\cite{Doychev-etal:14}; moreover, most user activity logs are anonymized for privacy preservation purposes. It is thus interesting to consider other types of knowledge integration that do not require user identification across applications. Our work investigates this path of research.\n\n\nCollaborative Filtering generates suggestions by analyzing item ratings to identify similar users or similar items.\nSeveral algorithms have been developed, from K-Nearest Neighbors (KNN) to more recent ones such as Matrix Factorization \\cite{Desrosiers-Karypis:11,Koren-Bell:11}. In our work we adopt KNN because it has nice explanation capabilities and has proved to achieve good performance in a comparison with other approaches  \\cite{Jannach-Ludewig:17,Ludewig-Jannach:18}. \n\nOntological user profiles model preferences at the semantic level. In \\cite{Sieg-etal:07b,Sieg-etal:10b}, Sieg et al. propose to exploit a taxonomy whose concepts represent item types, and to infer user interests on the basis of the observed ratings to the instances of such concepts. The neighborhood for rating estimation is then identified by measuring the semantic similarity between ontological user profiles. The category-based user similarity we propose is close to this approach. However, we go one step forward in the identification of preferences by extending the user profiles with frequently co-occurring information interests.\nThis type of extension also differentiates our work from that of Ronen et al., who propose to extend the preferences of the individual user by analyzing her\/his behavior in search logs \\cite{Ronen-etal:16}: that work assumes that the user's activities can be tracked across applications and extends the user profile by analyzing her\/his overall behavior. In contrast, we extend user preferences by analyzing anonymous data about general search behavior. \n\nSen et al. define tag-aware recommender systems as\n``recommender algorithms that predict user's preferences for tags''.\nIn \\cite{Sen-etal:09} they describe different signs of interest; e.g., searching or applying a tag, and so forth.\nOur work relates to tag-aware recommender systems because we analyze rating behavior on items associated to categories expressed as tags. However, we do not consider any other types of interaction with tags for estimating user preferences. \n\nIn \\cite{Gemmel-etal:12}, Gemmel et al. present a linear-weighted hybrid framework for resource recommendation that models different scenarios, among which tag-specific item recommendation. They propose to match users and items on the basis of their tag profiles. Differently, we match users on the basis of category-based profiles learned from rating behavior.\nThe same kind of difference holds between our work and the one of Nakamoto \\cite{Nakamoto:2007}.\n\nWhile TagiCoFi \\cite{Zhen:2009} employs user similarities defined from tagging information to regularize Matrix Factorization, we use tags in a KNN algorithm. \nIn \\cite{Tso-Sutter:2008} Tso and Sutter extend the ratings matrix using tagging information.\nThey reduce the three-dimensional correlations $<user, tag, item>$ to two-dimensional correlations $<user,tag>$, $<item,tag>$ and $<user, item>$. Then, they apply a fusion method to combine the correlations for rating prediction. Differently, we extend the rating matrix with the categories (tags) associated to the items rated by users and with further categories identified from general search behavior. \n\nRecently, rating information has been combined with other types of data to improve recommendation. For instance, item reviews are used, possibly in combination with ratings, in \\cite{Chen-etal:15,Musat-Faltings:15,Muhammad-etal:15,Lu-etal:18}.\nMoreover, trust relations and reputation are used to steer recommendation on the basis of the feedback on items provided by trusted parties; e.g., \\cite{Kuter-etal:07,Liu-Lee:10,Tang-etal:13,Alotaibi-Vassileva:16,Mcnally-etal:14,Du-etal:17,Yang-etal:17}.\nIn \\cite{Mauro-etal:19}, we investigate multi-faceted trust for personalized recommendation. \nHowever, in the present work we focus on rating information to assess the potential improvement of Collaborative Filtering, when combined with general preference co-occurrence. \n\n\n\\subsection{Analysis of Interaction Sessions}\nThe identification of interest co-occurrence we propose is related to a few works supporting query expansion, query reformulation and term suggestion in Information Retrieval. Some researchers propose to analyze session-based user behavior in order to detect co-occurrence relations useful to improve search queries, taking the search context into account. For instance, in \\cite{Cao-etal:08} Cao et al. suggest queries on the basis of the context provided by the user's recent search history, by clustering queries on the basis of the search results visited by users. Moreover, Huang et al. \\cite{Huang-etal:03} and Chen et al. \\cite{Chen-etal:08} detect term co-occurrence in search sessions to group sets of relevant words that can be mutually suggested. \nOur work is different because we adopt a linguistic interpretation approach (based on lemmatization and Word Sense Disambiguation) to find the concepts referenced in the queries; see \\cite{Mauro-Ardissono:17b}. \nTherefore, we extract information about {\\em concept co-occurrence}, which is more general than {\\em term co-occurrence}. \n\nIt is worth mentioning that our analysis of interaction sessions differs from session-based recommendation, which analyzes the user's behavior during an interaction session to identify relevant item(s) to suggest; e.g., see \\cite{Garcin-etal:13,Jannach-Ludewig:17,Greenstein-etal:17,Jannach-etal:17}. In fact, we mine interest co-occurrence by abstracting from the particular sequence of queries performed by the users. Moreover, as previously discussed, we mine concept associations. \n\n\n\n\\subsection{Graph-based Information Filtering}\nKnowledge graphs describe item features and relations among entities, supporting the analysis of item relatedness, as well as similarity for information filtering and top-N recommendation. \nIn several works these graphs are extracted from document pools and\/or from the Linked Data Cloud. For instance, CoSeNa \\cite{Candan-etal:09} employs keyword co-occurrence in the corpus of documents to be retrieved, and ontological knowledge about the domain concepts, to support the exploration of text collections using a keywords-by-concepts graph. Moreover, in \\cite{DiNoia-etal:16}, Di Noia et al. create a relatedness graph by analyzing external data sources such as DBpedia in order to support the evaluation of semantic similarity between items. Analogously, item features have been extracted from the Linked Data Cloud to improve recommendation performance in \\cite{Musto-etal:16,Ragone-etal:17,Musto-etal:17,Musto-etal:18}. \n\nSome works attempt to extend the relations among information items by integrating data derived from the observation of different types of user behavior. E.g., Google search engine manages the Knowledge Graph \\cite{GoogleKnowledgeGraph} to relate facts, concepts and entities depending on their co-occurrence in queries. Moreover, entity2rec learns user-item relatedness from knowledge graphs by analyzing data about users' feedback and item information from Linked Open Data \\cite{Palumbo-etal:17}. Furthermore, in \\cite{Oramas-etal:15} Oramas et al. propose a hybrid recommender that integrates users implicit feedback into a knowledge graph describing item information, enriched with semantic data extracted from external sources. Finally, in \\cite{Vahedian-etal:17}, Vahedian et al. generalize graph-based approaches by simultaneously taking into account multiple types of relations among entities: they introduce meta-paths to represent patterns of relations and apply random-walk along such paths to identify relevant entities to suggest.\n\nOur work has analogies to the above listed ones because we employ a graph-based type of knowledge representation. However, we work at the conceptual level: our knowledge graph relates item categories instead of individual users and\/or items. Moreover, we do not compute similarity or relatedness by means of the knowledge graph: we use the graph to extend category-based user profiles. In turn, those profiles are employed in neighborhood identification. The separation between how preferences are inferred and how they are used for recommendation makes it possible to extend both types of activities in a modular way.\n\n\n\n\\section{Extended Category-based Collaborative Filtering}\n\\label{model}\nWe describe ECCF incrementally, starting from U2UCF that provides the basic match-making approach for rating estimation. \n\n\\subsection{User-to-User Collaborative Filtering}\nIn \\cite{Ricci-etal:11}, Ricci et al. define U2UCF as follows: ``the simplest and original implementation of this approach recommends to the active user the items that other users with similar tastes liked in the past. The similarity in taste of two users is calculated based on the similarity in the rating history of the users\".\nGiven: \n\\begin{itemize}\n    \\item $U$ as the set of users and $I$ as the set of items;\n    \\item $r: U X I \\Rightarrow {\\rm I\\!R}$ as a map of ratings;\n    \\item $R \\in {\\rm I\\!R}^{U X I}$ as the users-items rating matrix, where each value is a rating $r_{ui}=R[u,i]$ given by a user $u \\in U$ to an item $i \\in I$. \n\\end{itemize}\nThe recommender system estimates $u$'s rating of $i$ ($\\hat{r}_{ui}$) as follows:\n        \\begin{equation}\n        \\label{eq:rmeancentering}\n        \\hat{r}_{ui} = \\bar{r}_u + \\frac{ \n        \t\\sum\\limits_{v\\in N_i(u)}\\sigma(u,v) (r_{vi} - \\bar{r}_v)\n        }{\n        \t\\sum\\limits_{v\\in N_i(u)}|\\sigma(u,v)|}\n    \\end{equation}\nwhere $N_i(u)$ is the set of neighbors of $u$ that rated item $i$ and $\\sigma(u,v)$ is the similarity between user $u$ and user $v$ ($v \\in N_i(u)$). The similarity among users is computed by applying a distance metric, e.g., Cosine or Pearson similarity, to their rating vectors. \n\n\n\\subsection{Simple Category-based Collaborative Filtering (SCCF)}\n\\label{category-based-CF}\nSCCF manages user profiles in which the user's interest in each item category is represented as a positive number; the higher is the value, the stronger is the interest.  \nWe define:\n\\begin{itemize}\n    \\item $U$, $I$, $r$ and $R$ as above; \n    \\item $C$ as the set of item categories;\n    \\item $f: U X C \\Rightarrow {\\rm I\\!N} $ as a map between users and categories;\n    \\item $UC\\in {\\rm I\\!N}^{U X C}$ as the Users-Categories matrix. For each $u \\in U$ and $c \\in C$, $UC[u,c]$ represents the interest of $u$ in $c$. We take as evidence of interest the {\\em frequency of exploration} of a category, i.e., the frequency of interaction of the user with items associated with the category. \n\\end{itemize}\nCategory exploration can be mapped to different types of user behavior; e.g., tagging items and searching for items by tag. We map exploration to rating behavior and we define $UC[u, c]$ as the number of ratings that $u$ has given to the items associated with $c$. \n\nSCCF computes user similarity on the basis of the estimated user preferences for item categories. Specifically, $\\sigma(u, v)$ is defined as the Cosine similarity of the users vectors in the $UC$ matrix and it is used in Equation (\\ref{eq:rmeancentering}) to estimate ratings. Thus, $\\hat{r}_{ui}$ is computed on the basis of the ratings $r_{vi}$ provided by the users $v \\in U$ whose preferences for categories are similar to those of $u$. \n\n\n\\subsection{Acquisition of Preferences Co-occurrence}\n\\label{graph}\nIn order to learn the strength of the associations between item categories in search behavior, we analyze their co-occurrence in the search sessions of a query log. By co-occurrence we mean the fact that two or more categories are referred by the queries belonging to the same session. \nIn the following we summarize the analysis of category co-occurrence; see \\cite{Mauro-Ardissono:18} for details.\n\nThe Category Co-occurrence Graph ($CCG$) represents category co-occurrence:\nin the $CCG$, nodes represent the data categories referenced in the analyzed queries and the weight of edges represents the co-occurrence frequency of the connected categories; i.e., how many times the categories have been identified within the same search sessions.\n\nWe retrieve the categories occurring in the queries by applying a Natural Language approach that identifies the referred concepts in a flexible way, by considering synonyms and by applying Word Sense Disambiguation to resolve the meaning of words; see \\cite{Ardissono-etal:16,Mauro-Ardissono:17b}. For Word Sense Disambiguation we use the Babelfy tool \\cite{Babelfy}.\n\nThe $CCG$ is built as follows:\ngiven two categories $x$ and $y$, the weight of the edge that connects them is defined as:\n\\begin{equation}\n\\label{eq1}\nw_{xy}=\\sum_{S\\in|Sessions|} Freq_{S_{xy}}\n\\end{equation}\nwhere $Freq_{S_{xy}}$ represents the evidence provided by session $S$ to the co-occurrence frequency of $x$ and $y$.\nGiven $S=\\{Q_1, \\dots, Q_n\\}$,\n$Freq_{S_{xy}}$ is computed as the maximum evidence of co-occurrence of $x$ and $y$ in $S$: \n\\begin{equation}\n\\label{eq2}\nFreq_{S_{xy}} = Max_{k=1}^{n}(Freq_{xy_{Q_k}}, ev_{xy_{Q_{k-1}}})\n\\end{equation}\nwhere $Freq_{xy_{Q_k}}$ is the co-occurrence evidence of $x$ and $y$  provided by query $Q_k$, and $ev_{xy_{Q_{k-1}}}$ is the one provided by $Q_1, \\dots, Q_{k-1}$. Similar to \\cite{Mauro-Ardissono:18}, we take the maximum, and not the sum of evidence because co-occurrence could derive either from query reformulation \\cite{Rieh-Xie:06}, or from the repetition of queries in click-through events of the log; see Section \\ref{sec:AOLlog} that describes the query log we used.\n\nA query $Q$ contributes to the estimation of co-occurrence as follows:\n\\begin{itemize}\n\\item \nIf $Q$ contains $k$ terms ($k>=0$), each one identifying a non-ambiguous category: \n$T_1 \\Rightarrow c_1, \\quad \\dots, \\quad T_k \\Rightarrow c_k$, then, for each category $c$ of $Q$:\n\\begin{itemize}\n    \\item The co-occurrence evidence between $c$ and every other category $d$ of $Q$ is $Freq_{cd_{Q}} = 1$. \n    \\item The co-occurrence evidence between $c$ and every other category $e$ identified in a non-ambiguous way in the other queries of $S$ is $Freq_{ce_{Q}} = 1$. \n    \\item The co-occurrence evidence between any other categories $w$ and $z$ identified in $S$ is $Freq_{wz_{Q}} = 0$.\n\\end{itemize}\n\\item \nIf $Q$ contains an ambiguous term $t$ that refers to $m$ categories, the particular category the user is focusing on cannot be identified. Therefore, the co-occurrence evidence brought by $t$ is computed as above, but the assigned evidence is $\\frac{1}{m}$ in order to consider the possible interpretations of $Q$, and divide evidence among ambiguous categories.\n\\end{itemize}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{model.pdf}\n    \\caption{Extension of Category-based User Profiles. }\n    \\label{fig:matrici}\n\\end{figure}\n\n\n\\subsection{Extended Category-based Collaborative Filtering (\\textit{ECCF})}\n\\label{sec:ECCF}\nIn this recommendation model we employ frequent co-occurring information interests to extend category-based user profiles.\nWe reinforce the preferences for item categories learned by analyzing rating behavior (stored in the Users-Categories matrix $UC$) with interest co-occurrence associations (stored in the $CCG$ graph) in order to acquire an extended set of user preferences for neighbor identification.\n\nThe idea behind preference extension is that, the more the user has appreciated the items of a category, the more interest\n\\linebreak\nco-occurrence makes sense. Therefore, starting from the category-based user profiles stored in the $UC$ matrix, we increment user preferences with the contribution of the strongest co-occurrence relations of the $CCG$ graph, depending on the number of positive ratings available in the users-items matrix $R$. \nThe output of this process is stored in the Extended Preferences matrix $EP$, which is used to compute $\\sigma(u,v)$ in Equation \\ref{eq:rmeancentering}.\n\nFigure \\ref{fig:matrici} provides a graphical view of the computation of $EP$: the information stored in $UC$ is combined with that stored in the $CCG$ to set the values of this matrix. In this process, the users-ratings matrix $R$ is used to limit the reinforcement of preferences to the categories of the positively rated items.\nMoreover, the $CCG$ is used to propagate preference information according to the strongest co-occurrence of interests. \nIn detail, we compute the values of $EP$ as follows:\n\\begin{itemize}\n    \\item \n    let $Cat_i$ be the set of categories associated to item $i$; \n    \\item\n    let $CatSet_i$ be the set of categories directly connected to any category $c \\in Cat_i$ in the $CCG$ through the heaviest outbound arcs. These are the categories which most frequently co-occur with some categories of $Cat_i$ in search sessions.\n\\end{itemize}\nThen:\n\\begin{equation}\n\\label{eq:pm}\nEP[u,c]=UC[u,c]+\\sum_{i\\in|I|} f(u,i,c)\n\\end{equation}\nwhere\n\\begin{equation}\n             f(u,i,c) =\n            \\begin{cases}\n                1 & \\quad \\text{if $R[u,i] \\in$} ~ \\text{\\textit{PositiveRatings}} ~ \\text{$\\wedge$} ~\\text{$c \\in CatSet_i$}\\\\\n                0 & \\quad \\text{otherwise}\n            \\end{cases}\n\\label{eq:f}\n\\end{equation}\nIn Equation \\ref{eq:f} $PositiveRatings$ denotes the set of ratings that are considered as positive in the dataset; e.g., \\{5\\}, or \\{4, 5\\} in a [1, 5] Likert scale.\n\n\n\\section{Validation of ECCF}\n\\label{sec:validation}\n\n\\subsection{Dataset of Item Ratings}\n\\label{sec:YELP}\nAs a source of rating data we exploit the Yelp Dataset \\cite{Yelp-dataset}, which contains information about a set of businesses, users and reviews and is available for academic purposes. In the dataset, item ratings take values in a [1, 5] Likert scale where 1 is the worst value and 5 is the best one. \nMoreover, each item is associated with a list of categories describing the kind of service it offers.\n\nThe full list of Yelp categories is available at \\url{www.yelp.com\/developers\/documentation\/v3\/category_list} and is organized in a taxonomy to specify businesses at different levels of detail. The taxonomy includes a large set of first-level categories, representing broad types of businesses; e.g., ``Active life'', ``Arts \\& entertainment'', ``Automotive'', \\dots, ``Food'', ``Restaurants'', and many others. In turn, the first-level categories are specialized into sub-categories; e.g., ``Restaurants'' includes many types of restaurants such as ``Indian'', ``Chinese'' and the like.  \nWe apply two filters to the dataset:\n\\begin{enumerate}\n    \\item \n    We select all the Yelp categories that are subclasses of ``Restaurants'' or ``Food'': e.g., ``Indian'', ``Chinese'', ``Cafes'', ``Kebab'', ``Pizza'', and so forth; the total number of categories is 254. \n    Then, we project the Yelp dataset on the set of items associated with at least one of these categories. In the rest of this paper we refer to this set of categories as {\\em CATS}. \n    \\item We further filter the dataset on the users who rated at least 20 items.\n\\end{enumerate}\n\n\n\\begin{table}[t]\n\\centering\n\\caption{Statistics about the Filtered Datasets}\n\\begin{tabular}{l|l|l}\n\\hline\nYelp & Number of users            & 26,600   \\\\\n& Number of businesses       & 76,317   \\\\ \n& Number of ratings          & 1,326,409 \\\\ \\hline\nAOL & Number of sessions & 1,248,803 \\\\\n& Number of queries & 2,136,029 \\\\\n\\hline\n\\end{tabular}\n\\label{t:dataset}\n\\end{table}\nThe higher portion of Table \\ref{t:dataset} summarizes the number of users, businesses and ratings of the filtered Yelp dataset.\n\n\n\\subsection{Dataset of Search Sessions}\n\\label{sec:AOLlog}\nFor the generation of the Category Co-occurrence Graph we use the AOL query log.\\footnote{The log is available at \\url{http:\/\/www.cim.mcgill.ca\/~dudek\/206\/Logs\/AOL-user-ct-collection\/}.}  \nEach line of the log represents either a query or a click-through event on one of the search results of a query. The line contains various fields, among which the submitted query and the submission date and hour.\n\nIn order to build a graph that is thematically related to the items of the filtered Yelp dataset, we select from the log the search sessions relevant to the categories $c \\in CATS$ enriched with the following two types of external knowledge. The enrichment is useful to abstract from the specific category names used in Yelp and to take into account semantically related information: \n\\begin{enumerate}\n    \\item \n    {\\em Lemmatized knowledge:} we enrich each element $c \\in CATS$ with a set of keywords and synonyms from WordNet \\cite{WordNet} lexical database. \n    \\item \n    {\\em Relevant terms from the Probase \\cite{Wu-etal:12} taxonomy:} \n    \\begin{itemize}\n        \\item\n        For each element $c \\in CATS$, we enrich $c$ with the \n        \\linebreak\n        $<concept, instance>$ pairs of ProBase such that $concept$ has at least 85\\% WordNet similarity with any term of the lemmatized knowledge of $c$, and the WordNet similarity between the two components of the pair is 85\\%.\n        \\item \n        ProBase, recently called Microsoft Concept Graph, is a large concept network harnessed from web pages and search logs. It is organized as a list of $<instance, concept>$ pairs related by a subclass relation and it contains \n        \\linebreak \n        5,376,526 classes and 12,501,527 instances.\n    \\end{itemize}\n\\end{enumerate}\nFor the selection of relevant search queries in the AOL log we match the lemmatized words occurring in the queries to the enriched categories of $CATS$. If there is at least one match between a term and a query, we consider the query as relevant and we include its parent session in the filtered log. \n\nWe identify the search sessions by aggregating the queries performed by the same user according to their temporal proximity, following the widely applied rule that two consecutive queries belong to different sessions if the time interval between them exceeds half an hour; see \\cite{White-etal:07}. \n \nThe lower portion of Table \\ref{t:dataset} shows the number of sessions and queries of the filtered AOL dataset.\n\nIt is worth noting that the AOL log was involved in an information leak issue but we decided to use it for two reasons. Firstly, our analysis is ethically correct because we study general search behavior to acquire aggregate data abstracting from the search histories of individual users. Secondly, to the best of our knowledge, the AOL log is the only available large dataset that reports textual search queries, and which can therefore be used for linguistic interpretation. We analyzed some public datasets but they did not meet our requirements. For instance, the Excite query dataset\\footnote{\\url{https:\/\/svn.apache.org\/repos\/asf\/pig\/trunk\/tutorial\/data\/}} contains about 1M queries while AOL log contains 20M queries. Moreover, in the Yahoo dataset\\footnote{\\url{https:\/\/webscope.sandbox.yahoo.com\/catalog.php?datatype=l\\&did=50}} the queries are coded; thus, it is not possible to extract any linguistic information to learn category co-occurrence.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.6\\linewidth]{graph-distribution.jpg}\n  \\caption{Distribution of the Weight of Edges in the $CCG$.}\\label{fig:graph-distribution}\n\\end{figure}\n\n\n\\subsection{Category Co-occurrence Graph}\nWe instantiate the {\\em CCG} with the interests that co-occur in the sessions of the filtered AOL dataset by applying the procedure described in Section \\ref{graph}.\nThe resulting graph is strongly connected: almost all of the categories are linked to each other by an edge having weight $>0$.\nHowever, the distribution of weights in the graph shows that there is a large number of weakly connected categories and a very small number of strongly associated ones. The ``heavy'' edges identify the interests that co-occur very frequently in search sessions and suggest to select the arcs having maximum weight in the {\\em CCG} for the extension of the user profiles, as done in Section \\ref{sec:ECCF}.\nFigure \\ref{fig:graph-distribution} shows this distribution; the x-axis represents the edges of the graph, and the y-axis represents their weights, which take values in [1, 272224].\n\n\\begin{table}[b]\n\\centering\n\\caption{Performance Evaluation @10; the Best Values Are in Boldface, the Worst Ones Are Strikethrough}\n\\begin{tabular}{l|l|l|l|l|l}\n\\hline \n\\textbf{Metrics} \n                             & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}U2UCF\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{SCCF}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{3,4,5\\}\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{4,5\\}\\end{tabular}}} & \\multicolumn{1}{c}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{5\\}\\end{tabular}}} \\\\ \\hline\n\\textbf{Precision}    & \\st{0.7823}   & \\textbf{0.786}  & 0.7857   & 0.7855  & 0.7859   \\\\\n\\textbf{Recall}       & \\st{0.7473}   & 0.7526   & 0.7536   & \\textbf{0.755}   & 0.7529  \\\\\n\\textbf{F1}           & \\st{0.7644}   & 0.7689   & 0.7693   & \\textbf{0.7699}  & 0.769   \\\\\n\\textbf{RMSE}         & \\st{1.0001}   & 0.9899   & 0.9897   & 0.9893 & \\textbf{0.9892}   \\\\\n\\textbf{MRR}          & \\st{0.733}    & 0.7367   & 0.737    & \\textbf{0.7391}  & 0.7384    \\\\ \n\\textbf{Diversity}  & \\st{0.3042}  & 0.3053 & \\textbf{0.3056} & 0.3053  & 0.3049 \\\\\n\\textbf{User cov.}   & \\st{0.8497}  & 0.8521  & 0.8526 & \\textbf{0.8542}  & 0.8534  \\\\\n\\hline\n\\end{tabular}%\n\\label{t:results@10}\n\\end{table}\n\n\\subsection{Test Methodology}\n\\label{experiments}\nWe evaluate the recommendation performance of ECCF by comparing it to U2UCF and SCCF, which we consider as baselines. Moreover, we compare these algorithms with SVD++ in order to assess the improvement in the suggestion of relevant items given by frequently co-occurring interests.\n\nThe SCCF and ECCF recommendation algorithms are developed by extending the Surprise library \\cite{Surprise}, while we use the default Surprise implementations of U2UCF and SVD++.\n\nWe test the algorithms by applying a 10-fold cross-validation on the filtered Yelp dataset, after having randomly distributed ratings on folds: we use 90\\% of the ratings as training set and 10\\% as test set. In all the tests, we configure the KNN algorithms to work with 50 neighbors.\n\nIn order to analyze the impact on recommendation performance of a looser, or stricter extension of user preferences with category co-occurrence, we validate ECCF on different settings of $PositiveRatings$ in Equation \\ref{eq:f}, i.e., on different interpretations of what is a good rating. For each fold\nwe generate three versions of the Extended Preferences matrix $EP$ having set $PositiveRatings$ to $\\{3,4,5\\}$, $\\{4,5\\}$, and $\\{5\\}$ respectively.  \n\nWe evaluate Top-k recommendation performance with k=10 and k=20 by taking the ratings observed in the Yelp dataset as ground truth. For the evaluation we consider the following metrics: Precision, Recall, F1, RMSE, MRR, Diversity and User Coverage.\n\nDiversity describes the mean intra-list diversity of items in the suggestion lists @k; see \\cite{Bradley-Smyth:01}. In this work, we interpret diversity from the viewpoint of item classification. Therefore, we measure the diversity of a recommendation list as follows:\n\\begin{equation}\n\\text{intra-list diversity@k}={\\frac  {\\sum _{{i=1}}^{k}\\sum _{{j=i}}^{k} (1 - sim(i, j))} {\\frac{k*(k+1)}{2}}}\n\\end{equation}\nwhere $sim(i, j)$ is the cosine similarity between the lists of categories associated to items $i$ and $j$ in the ratings dataset. \n\n\\subsection{Results}\n\\label{results}\nTable \\ref{t:results@10} shows the performance results of the KNN recommenders we compared, by taking into account a maximum of 10 suggested items (performance@10). \n\\begin{itemize}\n    \\item \\textbf{Precision:} similar to previous results described in \\cite{Sieg-etal:07b}, all of the category-based recommenders outperform U2UCF. This can be explained by the fact that the matrices describing preferences for item categories are denser than the ratings one. Thus, they improve recommendation by supporting a better identification of neighbors for Equation \\ref{eq:rmeancentering}.  \n    However, SCCF outperforms all of the ECCF variants. The second best recommender is ECCF$\\{5\\}$ that extends user profiles in the strictest way: it only considers as pivots for extension the categories associated to the items that the user has rated 5 stars. Notice also that the precision of ECCF decreases when $PositiveRatings$ is lax. The reason is that the extension of user profiles with frequently co-occurring interests can increase the estimated interest in some noisy categories with respect to the pure observation of ratings distribution on categories. In particular, noise grows when the policy applied to extend preferences is less restrictive. \n    \\item \\textbf{Recall:} ECCF outperforms the baselines in all the settings of $PositiveRatings$. Specifically, ECCF\\{4,5\\} achieves the best result, while recall is lower in ECCF\\{3,4,5\\} and further decreases in ECCF\\{5\\}.\n    We explain this finding as follows: an extension of user profiles based on the categories of highly rated items supports the identification of a richer set of user preferences, and a more efficacious identification of neighbors, than only considering rating distribution on categories. However, if we restrict $PositiveRatings$ too much, the user profiles are not extended enough to sensibly improve Recall. Moreover, as noticed for Precision, if $PositiveRatings$ is lax, noise in the estimation of user preferences challenges neighbor selection.\n    \n\\begin{table}[t]\n\\centering\n\\caption{Performance Evaluation @20}\n\\begin{tabular}{l|l|l|l|l|l}\n\\hline \n\\textbf{Metrics} \n                             & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}U2UCF\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{SCCF}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{3,4,5\\}\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{4,5\\}\\end{tabular}}} & \\multicolumn{1}{c}{\\textbf{\\begin{tabular}[c]{@{}c@{}}ECCF\\\\ \\{5\\}\\end{tabular}}} \\\\ \\hline\n\\textbf{Precision}  & \\st{0.7806} & \\textbf{0.7842}  & 0.7839   & 0.7838  & \\textbf{0.7842} \\\\ \n\\textbf{Recall}     & \\st{0.757}  & 0.7624  & 0.7634  & \\textbf{0.7649} & 0.7626  \\\\ \n\\textbf{F1}  & \\st{0.7686}  & 0.7731   & 0.7735  & \\textbf{0.7742}  & 0.7732  \\\\\n\\textbf{RMSE}  & \\st{0.9935}  & 0.9838 & 0.9835 & \\textbf{0.9832}  & \\textbf{0.9832} \\\\ \n\\textbf{MRR}  & \\st{0.733} & 0.7369   & 0.7372   & \\textbf{0.7391}  & 0.7384  \\\\ \n\\textbf{Diversity} & \\st{0.3059}  & 0.307  & \\textbf{0.3073}  & 0.307  & 0.3067  \\\\ \n\\textbf{User cov.}   & \\st{0.8497}  & 0.8521  & 0.8526 & \\textbf{0.8542}  & 0.8534  \\\\\n\\hline\n\\end{tabular}%\n\\label{t:results@20}\n\\end{table}\n\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{accuracy_10.jpg}\n \\caption{Graphical Representation of Accuracy@10.}\n \\label{fig:accuracy@10}\n\\end{figure}\n\n\n    \\item \\textbf{F1:} ECCF outperforms the baselines. In detail, ECCF\\{4,5\\} achieves the best F1 = 0.7691; moreover, F1 varies consistently with Recall, depending on $PositiveRatings$. \n    \\item \\textbf{RMSE:} SCCF reduces the mean error between estimated and observed ratings with respect to the baseline, showing the benefits of category-based user profiles. Moreover, consistently with the variation of Precision, the best results are obtained by ECCF\\{5\\}, i.e., with a strict extension of user profiles. RMSE progressively increases (i.e., gets worse) for $PositiveRatings=\\{4, 5\\}$ and \\{3, 4, 5\\}.\n    \\item \\textbf{MRR:} ECCF outperforms the baselines. Specifically, \n    \\linebreak\n    ECCF\\{4,5\\} obtains the best MRR = 0.7391. The second best value corresponds to a more selective extension of user profiles in ECCF\\{5\\}; moreover, if $PositiveItems=\\{3, 4, 5\\}$ results get worse.\n    \\item \\textbf{Diversity}: both SCCF and ECCF outperform U2UCF. In this case, the best results are obtained with a lax extension of user preferences (ECCF\\{3,4,5\\}) and Diversity decreases while the preference extension policy becomes stricter. We explain these findings with the fact that category-based user profiles improve the estimation of user preferences concerning a variegate set of item categories, with respect to a flat recommendation based on ratings. However, the stricter is the extension of user preferences, the less item categories are used in neighbor identification.\n     \\item \\textbf{User coverage:} ECCF outperforms the baselines, confirming the usefulness of preference extension. However, the selection of the ratings for the extension influences coverage: ECCF\\{4,5\\} achieves the best results by suggesting at least one relevant item to 85.42\\% of the users, against 84.97\\% of U2UCF. The second best is ECCF\\{5\\} and ECCF\\{3,4,5\\} has the worst results.\n    \\end{itemize}\nIn the described experiments the $EP$ Matrix is defined by only taking into account positive ratings. In order to get a broader view on the performance of ECCF, we also consider its application to all the user ratings; i.e., we set $PositiveRatings$ to $\\{1, \\dots, 5\\}$. With respect to the previous results, in this case the algorithm achieves similar Precision but lower Recall (0.7524), MRR (0.7369) and User coverage (0.8155).   \n\nTable \\ref{t:results@20} shows the results obtained by comparing \n\\linebreak\nperformance@20. \nThese results confirm the usefulness of category-based user profiles and of their extension with frequently\n\\linebreak\nco-occurring information interests: \n\\begin{itemize}\n    \\item Also in this case, ECCF\\{4,5\\} is the best recommendation algorithm. It outperforms the others in Recall, F1, MRR and User coverage. Moreover both ECCF\\{5\\} and ECCF\\{4,5\\} achieve the best RMSE in comparison with the other recommenders.  \n    \\item However, while SCCF has the best Precision@10, both SCCF and ECCF\\{5\\} achieve the best Precision@20.\n\\end{itemize}\nWith respect to k=10, Precision@20 is lower while Recall@20 and F1@20 take higher values; this makes sense because we are considering longer suggestion lists. Moreover, RMSE@20 is lower, which tells us that the longer lists contain proportionally less errors in the estimation of ratings. Differently, most algorithms obtain the same MRR for k=10 and k=20 (except for SCCF and ECCF\\{3,4,5\\}): this shows that the first relevant item is almost always placed in the first 10 positions of the suggestion lists.\nFurthermore, the Diversity@20 has the highest values for all the recommenders: this might be due to the fact that the longer suggestion lists have more chances to include items belonging to different categories. Finally, User coverage@10 = User coverage@20 because we interpret coverage as the percentage of users who receive at least one suggestion.\n\n\\begin{figure}[b]\n \\centering\n \\includegraphics[width=0.9\\linewidth]{accuracy_20.jpg}\n \\caption{Graphical Representation of Accuracy@20.}\n \\label{fig:accuracy@20}\n\\end{figure}\n\nFigures \\ref{fig:accuracy@10} and \\ref{fig:accuracy@20} depict the accuracy @10 and @20:\n\\begin{itemize}\n    \\item \n    All of the category-based recommenders outperform U2UCF, confirming the benefits of the introduction of category-based preferences in KNN Collaborative Filtering. The conceptual representation of user preferences generally improves performance because the matrices describing user preferences ({\\em UC} and {\\em EP}) are denser than the users-items matrix storing ratings ({\\em R}). Therefore, better neighbors can be identified for the computation of Equation \\ref{eq:rmeancentering}.\n    \\item\n    A comparison between category-based algorithms shows that the best performance results are obtained by extending user profiles on the basis of the items that users have rated very well, i.e., with 4 or 5 stars in a [1, 5] Likert scale. If the items that received middle ratings are considered as well, accuracy decreases. \n    \\item The category-based representation of user profiles has positive impact on the Diversity of recommendation lists. Conversely, the extension of user profiles does not further help this aspect, unless user profiles are extended in a lax way. However, a lax extension is not convenient because it decreases other measures.\n\\end{itemize}\n\n\nIn order to assess the usefulness of preference extension in Top-k recommendation, we also compare the previously described algorithms with SVD++ \\cite{Koren:08}, which adopts Matrix Factorization to learn latent user and item factors, basing rating prediction on the sole analysis of user ratings. The comparison results show that:\n\\begin{itemize}\n    \\item \n    SVD++ is more accurate than U2UCF and SCC. On the filtered Yelp dataset, SVD++ obtains F1@10 = 0.7696. This finding shows that the management of category-based user profiles helps recommendation but it can be outperformed by a deeper understanding of the features of items and users. \n    \\item\n    SVD++ achieves similar accuracy results with respect to ECCF but it is outperformed by ECCF\\{4, 5\\}. Therefore, the extension of user profiles with frequently co-occurring information interests, integrated into a KNN recommender, improves accuracy and makes it comparable or higher than that of Matrix Factorization algorithms. \n    \\item\n    ECCF outperforms SVD++ as far as the diversity of the recommendation lists is concerned: SVD++ has Diversity@10 = 0.3041; this is comparable to the diversity achieved by U2UCF and lower than that of all the category-based recommenders we presented.\n    \\item\n    In contrast, SVD++ has the highest User coverage of all the algorithms (0.8709), showing its superior capability to contrast data sparsity. \n    \\end{itemize}\n    \n\\subsection{Discussion}\n\\label{discussion}\nIn summary, the evaluation results show that ECCF outperforms U2UCF, SCCF and SVD++ in accuracy and intra-list diversity. Moreover, it outperforms U2UCF and SCCF in MRR and user coverage, while SVD++ excels in the latter metric. The results also show that ECCF achieves the best results when applied to positive ratings, while its performance slightly decreases when the user profiles are extended by taking both positive and negative ratings.\n\nThese results support the hypothesis that preference extension, based on frequently co-occurring information interests, improves the accuracy of the suggestions generated by a KNN recommender system. However, research has to be carried out to improve other performance metrics, possibly also investigating the integration of preference co-occurrence in Matrix Factorization algorithms.\n\nIt might be questioned whether extending user profiles with general interest co-occurrence data might provide less personalized recommendations than, e.g., focusing the extensions on the user's neighborhood. In this respect, we point out that we aim at developing a model that does not depend on cross-domain user identification. However, an investigation of this issue can be interesting to deal with the cases in which user information can be shared among the applications, or public information about the users can be connected to the local profiles; e.g., public data on social networks.\n\nBefore closing this discussion, it is worth noting that, even though the AOL query log dates back to 2006, it can be considered as a good information source as long as it is analyzed from the viewpoint of the concepts expressed by the users. In other words, while the specific information items mentioned in the log might not exist any more, the topics referred in the queries are general and long-lasting. Of course, some new topics (e.g., new types of restaurants) might have emerged since 2006, and maybe new concept associations could exist now. However, the described performance results show that the co-occurring interests we identified are useful to improve recommendation performance; moreover, the methodology described in this paper can be applied to other more recent datasets, if available.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nWe investigated whether the identification of frequently\n\\linebreak \nco-occurring interests in information search can be used to improve the performance of KNN collaborative recommender systems. For this purpose, we defined a preference extension model that, applied to a category-based representation of user profiles, infers user preferences by exploiting frequently co-occurring information interests. Then, we implemented the model in the Extended Category-based Collaborative Filtering algorithm (ECCF). This is is variant of User-to-User Collaborative Filtering that works on category-based user profiles, enriched with preferences inferred from general search behavior.\nFor the analysis of user interests, we analyzed the query log of a largely used search engine. \n\nWe evaluated ECCF on a large dataset of item ratings, by applying different levels of strictness in the extension of user profiles. The evaluation showed that ECCF outperforms User-to-User Collaborative Filtering in accuracy, MRR, intra-list diversity and user coverage. Interestingly, ECCS also obtains higher accuracy and diversity than the SVD++ recommender system, based on Matrix Factorization; however, ECCS has lower user coverage than SVD++. \n\nIn our future work we will focus on the coverage aspect in order to improve the performance of KNN Collaborative Filtering.\nMoreover, we will carry out further experiments, considering (i) a broader domain than Restaurants and Food, on which we have focused our current work, and (ii) users who have provided few or zero ratings. \nWe will also analyze other datasets to check whether the performance results described in this article can be generalized. Finally, we will compare the performance of ECCF with a larger set of recommendation approaches based on preference extension.\n\n\\begin{acks}\nThis work was supported by the University of Torino through projects ``Ricerca Locale'', MIMOSA\n(MultIModal Ontology-driven query system for the heterogeneous data of a SmArtcity, ``Progetto di Ateneo Torino\\_call2014\\_L2\\_157'', 2015-17)\nand the Computer Science PhD program.\nWe are grateful to Zhongli Filippo Hu, who helped us filter the Yelp dataset.\n\\end{acks}\n\n\n  \\bibliographystyle{ACM-Reference-Format} \n \n\\balance\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{\\@startsection{subsection}{2}%\n  \\z@{.35\\linespacing\\@plus.7\\linespacing}{.25\\linespacing}%\n  {\\normalfont\\itshape}}\n\\makeatother\n\n\\numberwithin{equation}{section}\n\n\\theoremstyle{plain}\n\\newtheorem{lemma}{Lemma}[section]\n\n\\newaliascnt{proposition}{lemma}\n\\newtheorem{proposition}[proposition]{Proposition}\n\\aliascntresetthe{proposition} \n\n\\newaliascnt{theorem}{lemma}\n\\newtheorem{theorem}[theorem]{Theorem} \n\\aliascntresetthe{theorem}\n\\newtheorem*{theorem*}{Theorem}\n\n\\newaliascnt{corollary}{lemma}\n\\newtheorem{corollary}[corollary]{Corollary} \n\\aliascntresetthe{corollary}\n\n\\newtheoremstyle{citing\n  {3pt\n  {3pt\n  {\\itshape\n  {\n  {\\bfseries\n  {.\n  {.5em\n       \n  {\\thmnote{#3}\n\n\\theoremstyle{citing}\n\\newtheorem*{varthm}{\n\n\n\n\\def\\equationautorefname~#1\\null{(#1)\\null}\n\\def\\itemautorefname~#1\\null{(#1)\\null}\n\\def\\sectionautorefname~#1\\null{\\S#1\\null}\n\\def\\subsectionautorefname~#1\\null{\\S#1\\null}\n\n\\providecommand*{\\lemmaautorefname}{Lemma}\n\\providecommand*{\\corollaryautorefname}{Corollary}\n\\providecommand*{\\propositionautorefname}{Proposition}\n\\providecommand*{\\theoremautorefname}{Theorem}\n\n\n\\newcommand\\CA{{\\mathcal A}} \\newcommand\\CB{{\\mathcal B}}\n\\newcommand\\CC{{\\mathcal C}} \\newcommand\\CE{{\\mathcal E}}\n\\newcommand\\CF{{\\mathcal F}} \\newcommand\\CG{{\\mathcal G}}\n\\newcommand\\CH{{\\mathcal H}} \\newcommand\\CL{{\\mathcal L}}\n\\newcommand\\CM{{\\mathcal M}} \\newcommand\\CN{{\\mathcal N}}\n\\newcommand\\CO{{\\mathcal O}} \\newcommand\\CP{{\\mathcal P}}\n\\newcommand\\CQ{{\\mathcal Q}} \\newcommand\\CR{{\\mathcal R}}\n\n\\newcommand\\fb{{\\mathfrak b}} \\newcommand\\fc{{\\mathfrak c}}\n\\newcommand\\fg{{\\mathfrak g}} \\newcommand\\fh{{\\mathfrak h}}\n\\newcommand\\fp{{\\mathfrak p}} \\newcommand\\fq{{\\mathfrak q}}\n\\newcommand\\fu{{\\mathfrak u}} \\newcommand\\fl{{\\mathfrak l}}\n\\newcommand\\fm{{\\mathfrak m}} \\newcommand\\fn{{\\mathfrak n}}\n\\newcommand\\FN{{\\mathfrak N}} \\newcommand\\fr{{\\mathfrak r}}\n\\newcommand\\ft{{\\mathfrak t}} \\newcommand\\fv{{\\mathfrak v}}\n\n\\newcommand\\BBC{{\\mathbb C}} \\newcommand\\BBD{{\\mathbb D}}\n\\newcommand\\BBF{{\\mathbb F}} \\newcommand\\BBP{{\\mathbb P}}\n\\newcommand\\BBQ{{\\mathbb Q}} \\newcommand\\BBR{{\\mathbb R}}\n\\newcommand\\BBZ{{\\mathbb Z}}\n\n\\newcommand\\Bbar{\\overline{B}} \\newcommand\\Cbar{\\overline{C}}\n\\newcommand\\CObar{\\overline{\\mathcal O}} \\newcommand\\Gbar{\\overline{G}}\n\\newcommand\\Hbar{\\overline{H}} \\newcommand\\jtilde{{\\widetilde j}}\n\\newcommand\\ktilde{{\\widetilde{k}}} \\newcommand\\Lbar{\\overline{L}}\n\\newcommand\\ptilde{{\\tilde {p}}} \\newcommand\\Tbar{\\overline{T}}\n\\newcommand\\Xtilde{\\widetilde X}\n\\newcommand\\Ytilde{\\widetilde Y} \\newcommand\\Ztilde{{\\widetilde Z}}\n\n\\DeclareMathOperator{\\Ad}{Ad} \n\\DeclareMathOperator{\\Alt}{Alt}\n\\DeclareMathOperator{\\Aut}{Aut} \n\\DeclareMathOperator{\\gr}{gr}\n\\DeclareMathOperator{\\Hom}{Hom} \n\\DeclareMathOperator{\\Lie}{Lie}\n\\newcommand{\\red}{\\operatorname{r}} \n\\DeclareMathOperator{\\res}{res}\n\n\\newcommand\\id{{id}}\n\\newcommand\\inverse{^{-1}}\n\\renewcommand\\th{{^{\\text{th}}}}\n\n\\newcommand\\CHIJ{{\\mathcal H}^{IJ}}\n\\newcommand\\etaIJ{\\eta^{IJ}}\n\\newcommand\\etaEJ{\\eta^{\\emptyset J}}\n\\newcommand\\FNt{\\widetilde{\\FN}}\n\\newcommand\\IzJ{I\\cap{}^zJ}\n\\newcommand\\LIbar{\\overline{L}_I}\n\\newcommand\\LJbar{\\overline{L}_J}\n\\newcommand\\OIbar{{\\overline{\\CO}_I}} \n\\newcommand\\Osbar{{\\overline{\\CO}_{\\{s\\}}}} \n\\newcommand\\point{{\\operatorname {pt}}}\n\\newcommand\\SN{\\mathfrak N}\n\\newcommand\\SNt{\\widetilde{\\SN}}\n\\newcommand\\ssleq{{\\scriptscriptstyle \\leq}}\n\\newcommand\\sslt{{\\scriptscriptstyle <}}\n\\newcommand\\Waf{W_{\\operatorname{af}}}\n\\newcommand\\Wex{W_{\\operatorname{ex}}}\n\\newcommand\\WIJ{{W^{IJ}}} \n\\newcommand\\XEJ{X^{\\emptyset J}}\n\\newcommand\\XIE{{X^{I \\emptyset}}}\n\\newcommand\\XIJ{X^{IJ}}\n\\newcommand\\ZIbar{{\\overline{Z}_I}} \n\\newcommand\\Zsbar{{\\overline{Z}_{\\{s\\}}}} \n\n\\begin{document}\n\n\n\\title[Equivariant K-theory] {Equivariant K-theory of\n  generalized Steinberg varieties}\n\n\\author[J. M. Douglass]{J. Matthew Douglass} \n\\address{Department of  Mathematics\\\\University of North Texas\\\\Denton TX,\n  USA 76203}\n\\email{douglass@unt.edu} \\urladdr{http:\/\/hilbert.math.unt.edu}\n\n\\author[G. R\\\"ohrle]{Gerhard R\\\"ohrle}\n\\address{Fakult\\\"at f\\\"ur Mathematik\\\\Ruhr-Universit\\\"at Bochum\\\\D-44780\n  Bochum, Germany} \n\\email{gerhard.roehrle@rub.de}\n\\urladdr{http:\/\/www.ruhr-uni-bochum.de\/ffm\/Lehrstuehle\/Lehrstuhl-VI}\n\n\\subjclass[2010]{Primary 20G05; Secondary 14L30 20C08}\n\n\\keywords{Equivariant $K$-theory, Hecke algebra, Steinberg variety}\n\n\n\n\\begin{abstract}\n  We describe the equivariant $K$-groups of a family of generalized\n  Steinberg varieties that interpolates between the Steinberg variety of a\n  reductive, complex algebraic group and its nilpotent cone in terms of the\n  extended affine Hecke algebra and double cosets in the extended affine\n  Weyl group. As an application, we use this description to define\n  Kazhdan-Lusztig ``bar'' involutions and Kazhdan-Lusztig bases for these\n  equivariant $K$-groups.\n\\end{abstract}\n\n\\maketitle\n\n\\section{Introduction}\n\nSuppose $G(\\BBF_q)$ is a Chevalley group defined over the finite field\n$\\BBF_q$. A fundamental result in the classification of irreducible complex\nrepresentations of $G(\\BBF_q)$ is the classification of representations that\ncontain a vector fixed by a Borel subgroup $B(\\BBF_q)$ of $G(\\BBF_q)$. These\nrepresentations are completely determined, and their characters may be\ncomputed, using the centralizer ring or Hecke algebra $\\CH(G(\\BBF_q),\nB(\\BBF_q))$. Iwahori \\cite{iwahori:structure} conjectured that\n$\\CH(G(\\BBF_q), B(\\BBF_q))$ is isomorphic to the group algebra of the Weyl\ngroup $W$ of $G(\\BBF_q)$. An explicit isomorphism between $\\CH(G(\\BBF_q),\nB(\\BBF_q))$ and the group algebra of $W$ was constructed by Lusztig\n\\cite{lusztig:theorem}.  More generally, irreducible representations that\ncontain a vector fixed by a parabolic subgroup $P_I(\\BBF_q)$ of $G(\\BBF_q)$\nare determined by the parabolic Hecke algebra $\\CH(G(\\BBF_q), P_I(\\BBF_q))$.\nCurtis, Iwahori, and Kilmoyer \\cite{curtisiwahorikilmoyer:hecke} showed that\nthis last algebra is isomorphic to the Hecke algebra $\\CH(W, W_I)$, where\n$W_I$ is the corresponding parabolic subgroup of $W$, and Curtis\n\\cite{curtis:isomorphism} extended Lusztig's construction to obtain an\nexplicit isomorphism between $\\CH(G(\\BBF_q), P_I(\\BBF_q))$ and $\\CH(W,W_I)$.\n\nNow suppose $G(\\BBQ_p)$ is a Chevalley group defined over $\\BBQ_p$. In this\ncase, one important class of representations consists of those\nrepresentations that contain a vector fixed by an Iwahori subgroup of\n$G(\\BBQ_p)$. These representations are again classified by a Hecke algebra,\nthis time the extended affine Hecke algebra $\\CH$ of the complex dual group,\n$\\check G$.  Kazhdan and Lusztig \\cite{kazhdanlusztig:langlands} construct\nan isomorphism between $\\CH$ and the equivariant $K$-theory of the Steinberg\nvariety of $\\check G$. They then use this isomorphism to give a construction\nof the irreducible representations of $\\CH$. In this paper we extend their\nconstruction and explicitly describe the equivariant $K$-groups of the\ngeneralized Steinberg varieties $\\XIJ$ from~\\cite{douglassroehrle:geometry}\nin terms of the Hecke algebra $\\CH$.  When $I=J$, the subspace of $\\CH$ we\nconsider is, up to an involution, the extension to $\\CH$ of the subalgebra\n$\\CH(G(\\BBF_q), P_I(\\BBF_q))$ of $\\CH(G(\\BBF_q), B(\\BBF_q))$.\n\nIn another direction, it follows from \\cite[Theorem 5.1.3]\n{douglassroehrle:homology} and \\cite[Theorem 2.5]\n{douglassroehrle:steinberg} that the rational Borel-Moore homology of $\\XIJ$\nmay be computed algebraically as the space of $W_I\\times W_J$-invariants in\nthe smash (semidirect) product of the coinvariant algebra of $W$ with the\ngroup algebra of $W$. The results in this paper may be viewed as the\nextension of this computation to the more refined level of equivariant\n$K$-theory and the affine Hecke algebra.\n\nFrom now on, suppose that $G$ is a connected, reductive complex algebraic\ngroup such that the derived group of $G$ is simply connected. Set\n$\\fg=\\Lie(G)$. For $g\\in G$ and $x\\in \\fg$ write $gx$ instead of\n$\\Ad(g)(x)$, where $\\Ad$ is the adjoint representation of $G$. Define $\\FN$\nto be the cone of nilpotent elements in $\\fg$ and let $B$ be a fixed Borel\nsubgroup of $G$ with Lie algebra $\\fb$. Then, the Steinberg variety of $G$\nis the variety $Z$ of all triples $(x,gB,hB)$ in $\\FN\\times G\/B \\times G\/B$\nsuch that $g\\inverse x, h\\inverse x\\in \\fb$. Based on a construction of\nKazhdan and Lusztig \\cite{kazhdanlusztig:langlands}, Chriss and Ginzburg\n\\cite {chrissginzburg:representation} and Lusztig \\cite{lusztig:bases} have\nshown that there is an algebra structure on $K^{G\\times \\BBC^*}(Z)$, the\n$G\\times \\BBC^*$-equivariant $K$-group of $Z$, such that $K^{G\\times\n  \\BBC^*}(Z)$ is isomorphic to the extended, affine Hecke algebra $\\CH$\nassociated to $G$. Ostrik \\cite{ostrik:equivariant} used this isomorphism to\ndescribe $K^{G\\times \\BBC^*}(\\FN)$ in terms of $\\CH$ and to define a\nKazhdan-Lusztig ``bar'' involution, and a Kazhdan-Lusztig basis, of\n$K^{G\\times \\BBC^*}(\\FN)$. As indicated above, in this paper we describe the\nequivariant $K$-groups of the generalized Steinberg varieties $\\XIJ$ in\nterms of $\\CH$. These generalized Steinberg varieties interpolate between\n$Z=X^{\\emptyset \\emptyset}$ and $\\FN=X^{SS}$ ($S$ is the Coxeter generating\nset for $W$ determined by $B$).  We then use our description to define\nKazhdan-Lusztig ``bar'' involutions and Kazhdan-Lusztig bases of the\nequivariant $K$-groups $K^{G\\times \\BBC^*}(\\XIJ)$.\n\nThe proof of the main theorem in this paper (\\autoref{thm:main}) relies on\nOstrik's computation of $K^{G\\times \\BBC^*}(\\FN)$.  For a generalized\nSteinberg variety $\\XIJ$ we use a filtration of $K^{G\\times \\BBC^*}(\\XIJ)$\nindexed by $G$-orbits in the product of two partial flag varieties.  In the\nspecial case of $\\FN$, there is a single $G$-orbit, the filtration of\n$K^{G\\times \\BBC^*}(\\FN)$ is trivial, and Ostrik has computed $K^{G\\times\n  \\BBC^*}(\\FN)$ in terms of $\\CH$. In the general case, each associated\ngraded piece has the form $K^{L'\\times \\BBC^*}(\\FN')$, where $L'$ is a Levi\nfactor of a parabolic subgroup of $G$ and $\\FN'$ is the nilpotent cone in\n$\\Lie(L')$. Thus, each of these graded pieces is described using Ostrik's\ntheory.\n\nIn \\autoref{sec:main} we give the basic constructions and state the main\ntheorem relating the extended affine Hecke algebra and the equivariant\n$K$-theory of generalized Steinberg varieties. Assuming facts that are\nproved in subsequent sections, \\autoref{sec:proof} contains a proof\nof~\\autoref{thm:main}. The constructions of the ``standard basis,'' the\n``bar'' involution, and the Kazhdan-Lusztig basis are given in\n\\autoref{sec:kl}. In \\autoref{sec:kthy} we review the intersection\/Tor\nproduct construction in the form it is used in this paper. The final three\nsections contain proofs of the main ingredients used in the proof\nof~\\autoref{thm:main}.\n\n\\subsection{Notation and conventions}\nWith $G$ and $B$ as above, fix a maximal torus $T$ contained in $B$. Let\n$W=N_G(T)\/T$ be the Weyl group of $(G,T)$, let $S$ be the set of simple\nreflections in $W$ determined by the choice of $B$, and let $X(T)$ be the\ncharacter group of $T$. Then $X(T)$ is a free abelian group and $W$ acts on\n$X(T)$ as group automorphisms. We use additive notation for $X(T)$ and\nconsider the root system $\\Phi$ of $(G,T)$ as a subset of $X(T)$. The roots\ncorresponding to the root subgroups in $B$ determine a positive system\n$\\Phi^+$, and a base $\\Pi$, of $\\Phi$. If $s$ is in $S$, then $s=s_\\alpha$\nfor a unique $\\alpha$ in $\\Pi$.\n\nLet $H$ be a complex linear algebraic group.  We use the convention that a\nlowercase fraktur letter denotes the Lie algebra of the group denoted by the\nsame uppercase roman letter, so for example, $\\fh= \\Lie(H)$. Let $R(H)$\ndenote the representation ring of $H$ and let $X(H)$ denote the character\ngroup of $H$. Define $\\overline H$ to be the product of $H$ with the\none-dimensional complex torus $\\BBC^*$, so $\\overline H= H\\times\n\\BBC^*$. Let $v\\colon \\Hbar \\to \\BBC^*$ be the character defined by\n$v(h,z)=z$ for $h\\in H$ and $z\\in \\BBC^*$ and let $A=\\BBZ[v,v\\inverse]$ be\nthe subring of $R(\\Hbar)$ generated by $v$. With this notation, there is a\nnatural isomorphism $R(\\Hbar)\\cong A\\otimes_{\\BBZ} R(H)$.  In particular,\n\\[\nR(\\Tbar) \\cong A\\otimes_{\\BBZ} R(T)\\cong A[X(T)]\\quad\\text{and} \\quad\nR(\\Gbar)\\cong A\\otimes_{\\BBZ} R(G).\n\\]\n\nWhen the group $H$ acts on a quasiprojective variety $Y$, let $K^{H}(Y)$\ndenote the Grothendieck group of the abelian category of $H$-equivariant\ncoherent sheaves on $Y$. The group $K^{H}(Y)$ is naturally an $R(H)$-module.\n\nSuppose $C$ is a closed subgroup of $H$.  For a $C$-variety $F$, let\n$H\\times^C F$ denote the quotient of $H\\times F$ by the $C$-action given by\n$c\\cdot (h,y)= (hc\\inverse, cy)$ for $c\\in C$, $h\\in H$, and $y\\in F$. The\nimage of $(h,y)$ in $H\\times^C F$ is denoted by $h*y$. The group $H$ acts on\n$H\\times^C F$ by left multiplication and the projection $f\\colon H\\times^C\nF\\to H\/C$ given by $h*y\\mapsto hC$ is a well-defined $H$-equivariant\nmorphism. Conversely, suppose $Y$ is an $H$-variety and $f_Y\\colon Y\\to H\/C$\nis an $H$-equivariant morphism. Set $F=f_Y\\inverse(C)$. Then the map\n$m\\colon H\\times^C F\\to Y$ given by $h*y\\mapsto hy$ is a well-defined\n$H$-equivariant isomorphism such that $f=f_Ym$. Suppose $\\Cbar$ acts on $F$.\nThen $\\Hbar$ acts on both $\\Hbar\\times^{\\Cbar}F$ and $H\\times^CF$, and these\nvarieties are canonically isomorphic $\\Hbar$-varieties. It follows from work\nof Thomason \\cite[Proposition 6.2]{thomason:algebraic} that\n$K^{\\Hbar}(H\\times^CF)$ is naturally isomorphic to $K^{\\Cbar}(F)$, and that\nif $C_{\\red}$ is a reductive subgroup of $C$ such that $C\\cong\nC_{\\operatorname{u}} \\rtimes C_{\\red}$, where $C_{\\operatorname{u}}$ is the\nunipotent radical of $C$, then $K^{\\Cbar}(F)$ is isomorphic to\n$K^{\\Cbar_{\\red}}(F)$ (see \\cite[\\S5.2]{chrissginzburg:representation}). Let\n\\[\n\\res_F\\colon K^{\\Hbar}(H\\times^CF) \\xrightarrow{\\ \\cong\\ }\nK^{\\Cbar_{\\red}}(F)\n\\]\ndenote the composition of these two isomorphisms.\n\nSuppose $Y_1$ and $Y_2$ are $H$-varieties with $Y_1\\subseteq Y_2$. To\nsimplify the notation, if $Y_1$ is closed in $Y_2$, then we sometimes denote\nthe direct image map $K^{H}(Y_1) \\to K^H(Y_2)$ simply by $()_*$, and if\n$Y_1$ is open in $Y_2$, then we sometimes denote the restriction map\n$K^{H}(Y_2) \\to K^H(Y_1)$ by $()^*$.\n\nUnless otherwise indicated, we consider $\\fg$ as a $\\BBC^*$-module with the\naction of $\\BBC^*$ given by $z\\cdot x= z^{-2}x$ for $z$ in $\\BBC^*$ and $x$\nin $\\fg$. Then $\\Gbar$ acts on $\\FN$ by $(g,z) \\cdot x =z^{-2}g\\cdot x$. For\na subgroup $P$ of $G$, $\\Gbar$ acts on $G\/P$ by $(g,z)\\cdot hP= gh P$.\nDefine\n\\[\n\\FNt=\\{\\, (x, gB)\\in \\FN\\times G\/B\\mid g\\inverse x\\in \\fb\\,\\}.\n\\]\nAs above, the Steinberg variety of $G$ is\n\\[\nZ=\\{\\, (x, gB, hB) \\in \\FN \\times G\/B \\times G\/B \\mid g\\inverse x, h\\inverse\nx\\in \\fb \\,\\} \\cong \\FNt\\times_{\\FN} \\FNt .\n\\]\nThen $\\Gbar$ acts on $\\FNt$ and $Z$ via the diagonal action, and the\nprojections\n\\[\np_Z\\colon Z\\to \\FN,\\quad q_Z\\colon Z\\to G\/B \\times G\/B,\\quad p\\colon \\FNt\\to\n\\FN, \\quad\\text{and} \\quad q\\colon \\FNt\\to G\/B\n\\]\nare all $\\Gbar$-equivariant. \n\n\\section{Statement of the main theorem }\\label{sec:main} \n\n\\subsection{Generalized Steinberg varieties}\n\nFor $I\\subseteq S$, let $W_I=\\langle I \\rangle$ be the subgroup of $W$\ngenerated by $I$ and let $P_I$ be the parabolic subgroup of $G$ that\ncontains $B$ such that $N_{P_I}(T)\/T = W_I$. Let $U_I$ denote the unipotent\nradical of $P_I$ and let $L_I$ be the Levi factor of $P_I$ that contains\n$T$. Then $P_I=L_IU_I$ and $\\fp_I=\\fl_I + \\fu_I$ are Levi decompositions of\n$P_I$ and $\\fp_I$, respectively.  Define $\\Pi_I= \\{\\, \\alpha\\in \\Pi\\mid\ns_\\alpha\\in I\\,\\}$ and let $\\Phi_I$ be the intersection of $\\Phi$ with the\nspan of $\\Pi_I$. Then, with respect to the action of $T$, $\\Phi_I$ is the\nset of roots of $\\fl_I$, $\\Phi^+\\cup \\Phi_I$ is the set of roots of $\\fp_I$,\nand $\\Phi^+\\setminus \\Phi_I$ is the set of roots of $\\fu_I$. In the special\ncase when $I=\\emptyset$, $P_I=B$ and we define $U = U_{\\emptyset}$.\n\nEach pair of subsets $I, J\\subseteq S$ determines a \\emph{generalized\n  Steinberg variety}\n\\[\n\\XIJ= \\{\\, (x, gP_I, hP_J)\\in \\FN\\times G\/P_I \\times G\/P_J \\mid g\\inverse\nx\\in \\fp_I,\\ h\\inverse x\\in \\fp_J\\,\\}\n\\]\n(see \\cite[\\S2]{douglassroehrle:geometry}). Define\n\\[\n\\etaIJ\\colon Z\\to \\XIJ \\quad\\text{by}\\quad \\etaIJ(x, gB, hB)= (x, gP_I,\nhP_J).\n\\]\nThen $\\XIJ$ is a $\\Gbar$-variety ($\\Gbar$ acts diagonally on $\\XIJ$) and\n$\\etaIJ$ is a surjective, proper, $\\Gbar$-equivariant morphism.  Notice that\n\\begin{itemize}\n\\item if $I=J=\\emptyset$, then $\\XIJ=Z$ and $\\etaIJ$ is the identity, and\n\\item if $I=J=S$, then $\\XIJ \\cong \\FN$ and $\\etaIJ$ may be identified with\n  projection $p_Z\\colon Z\\to \\FN$.\n\\end{itemize}\n\n\n\\subsection{Hecke algebras}\n\nThe \\emph{Iwahori-Hecke algebra of $W$} is the $A$-algebra $\\CH_S$ with\n$A$-basis $T_w$, for $w$ in $W$, and multiplication satisfying\n\\begin{equation}\n  \\label{eq:std}\n  \\begin{cases}\n    T_w T_{w'}= T_{ww'}& \\text{if $\\ell(ww')= \\ell(w)+ \\ell(w')$, and} \\\\\n    T_s^2= v^2 T_1+(v^2-1)T_s&\\text{for $s$ in $S$,}\n  \\end{cases}\n\\end{equation}\nwhere $\\ell$ is the length function on $W$ determined by $S$ and the\nsubscript $1$ in $T_1$ denotes the identity in $W$ (see\n\\cite{kazhdanlusztig:coxeter}).\n\nThe \\emph{extended affine Hecke algebra of $W$} is the $A$-algebra $\\CH$\nwith generators $T_w$, $\\theta_\\lambda$, for $w$ in $W$ and $\\lambda$ in\n$X(T)$, and multiplication satisfying\n\\begin{equation}\n  \\label{eq:bernstein}\n  \\begin{cases}\n    T_w T_{w'}= T_{ww'}& \\text{if $\\ell(ww')= \\ell(w)+ \\ell(w')$,} \\\\\n    T_s^2= v^2 T_1+(v^2-1)T_s&\\text{for $s$ in $S$,} \\\\\n    \\theta_\\lambda \\theta_{\\mu}= \\theta_{\\lambda+\\mu}& \\text{for\n      $\\lambda, \\mu$ in $X(T)$,} \\\\\n    \\theta_\\lambda T_s -T_s \\theta_{s(\\lambda)}= (v^2-1) \\frac\n    {\\theta_\\lambda-\\theta_{s(\\lambda)}} {1-\\theta_{-\\alpha}} &\\text{for\n      $\\lambda$ in $X(T)$ and $s=s_\\alpha$ in $S$, and} \\\\\n    \\theta_0=T_1 &\\text{is the identity in $\\CH$ .}\n  \\end{cases}\n\\end{equation}\n(See \\cite[\\S1]{lusztig:bases}. Note that for $w$ in $W$, the generator\n$T_w$ in the preceding definition is related to the generator $\\tilde T_w$\nin \\cite[\\S1]{lusztig:bases} by $\\tilde T_w=v^{-\\ell(w)} T_w$.)\n\nWe identify the $A$-span, in $\\CH$, of $\\{\\,T_w\\mid w\\in W\\,\\}$ with the\nIwahori-Hecke algebra $\\CH_S$, and we identify the $A$-span, in $\\CH$, of\n$\\{\\,\\theta_\\lambda \\mid \\lambda \\in X(T)\\,\\}$ with the group algebra\n$A[X(T)]$ of $X(T)$. Then $\\CH_S$ and $A[X(T)]$ are subalgebras of $\\CH$\nthat contain the identity. The center of $\\CH$ is $A[X(T)]^W$ (see\n\\cite{lusztig:singularities}).  We identify $R(\\Gbar)$ with $A[X(T)]^W$, and\nhence with the center of $\\CH$, via the isomorphism $R(\\Gbar) \\cong\nA[X(T)]^W$ given by associating with a representation of $\\Gbar$ its\ncharacter in $A[X(T)]$.  The map $A[X(T)]\\otimes_A \\CH_S\\to \\CH$ given by\nmultiplication, $\\theta_\\lambda\\otimes T_w\\mapsto \\theta_\\lambda T_w$, is an\n$A$-module isomorphism. We call $\\{\\, \\theta_\\lambda T_w\\mid \\lambda\\in\nX(T), w\\in W\\,\\}$ the \\emph{Bernstein basis} of $\\CH$ because it arises from\nthe Bernstein presentation~\\eqref{eq:bernstein}.\n\nFor $\\lambda$ in $X(T)$, let $t_\\lambda$ denote translation by $\\lambda$ in\n$X(T)$. Then $\\{\\, t_\\lambda\\mid \\lambda\\in X(T)\\,\\}$ is a subgroup of\n$\\Aut(X(T))$ isomorphic to $X(T)$. Recall that $W$ acts faithfully on\n$X(T)$. Define $\\Wex$, the \\emph{extended affine Weyl group of $\\Phi$,} to\nbe the subgroup of $\\Aut(X(T))$ generated by the image of $W$ and $\\{\\,\nt_\\lambda\\mid \\lambda \\in X(T)\\,\\}$. Then $\\Wex$ is isomorphic to the\nsemi-direct product $X(T) \\rtimes W$. We frequently identify $W$ with its\nimage in $\\Aut(X(T))$ and consider $W$ as a subgroup of $\\Wex$.\n\nThe \\emph{affine Weyl group of $\\Phi$,} $\\Waf$, is the subgroup of\n$\\Aut(X(T))$ generated by the image of $W$ and $\\{\\, t_\\alpha\\mid \\alpha \\in\n\\Phi\\,\\}$. Then $\\Waf$ is a normal subgroup of $\\Wex$ and there is a finite\nabelian subgroup $\\Gamma$ of $\\Wex$ such that $\\Wex= \\Waf \\Gamma$ and\n$\\Waf\\cap \\Gamma =1$. The group $\\Waf$ is a Coxeter group with a Coxeter\ngenerating set $S_{\\textrm{af}}$ that contains $S$, and $\\Gamma$ acts on\n$\\Waf$ as Coxeter group automorphisms preserving $S_{\\textrm{af}}$. Extend\nthe length function $\\ell$ and Bruhat order $\\leq$ on $\\Waf$ to $\\Wex$ by\ndefining\n\\[\n\\ell(y\\gamma)= \\ell(y)\\qquad\\text{and} \\qquad \\text{$y\\gamma \\leq y'\\gamma'$\n  if and only if $\\gamma=\\gamma'$ and $y\\leq y'$}\n\\] \nfor $y,y'$ in $\\Waf$ and $\\gamma, \\gamma'$ in $\\Gamma$ (see~\\cite[\\S2]\n{lusztig:singularities}).\n\nThe algebra $\\CH$ has a \\emph{standard basis}, $\\{\\, T_w\\mid w\\in \\Wex\\,\\}$,\nsuch that the relations~\\eqref{eq:std} hold (see \\cite[\\S1]\n{lusztig:bases}). The ``bar'' involution of $\\CH$, $\\overline{\\phantom {x}\n}\\colon \\CH \\to \\CH$, is the ring automorphism of $\\CH$ defined by\n$\\overline v=v\\inverse$ and $\\overline{T_x}= T_{x\\inverse} \\inverse$ for $x$\nin $\\Wex$. As observed by Lusztig~\\cite{lusztig:singularities}, the argument\nin the proof of \\cite[Theorem 1.1] {kazhdanlusztig:coxeter} can be applied\nto show that for $x$ in $\\Wex$, there are unique elements $C_x$ and $C_x'$\nin $\\CH$ such that\n\\begin{equation}\n  \\label{eq:cx}\n  \\begin{cases}\n    \\overline{C_x}= C_x  \\\\\n    C_x=v_x\\inverse T_x+ \\sum_{y< x} \\epsilon_y\\epsilon_x v_x v_y^{-2}\n    \\overline{P}_{y,x} T_y\n  \\end{cases}\n\\end{equation}\nand\n\\begin{equation}\n  \\label{eq:cx'}\n  \\begin{cases}\n    \\overline{C_x'}= C_x' \\\\\n    C_x'=v_x\\inverse T_x+ \\sum_{y< x} v_x\\inverse P_{y,x}T_y,\n  \\end{cases}\n\\end{equation}\nwhere \n\\[\n\\epsilon_x= (-1)^{\\ell(x)}\\quad\\text{and}\\quad v_x= v^{\\ell(x)}\n\\]\nfor $x$ in $\\Wex$, and $P_{y,x}$ is a polynomial in $v$ of degree at most\n$\\ell(x)-\\ell(y) -1$. (See also~\\cite[\\S1.7, 1.8]{lusztig:bases}.) We call\n$\\{\\, C_x\\mid x\\in \\Wex\\,\\}$ and $\\{\\, C_x'\\mid x\\in \\Wex\\,\\}$\n\\emph{Kazhdan-Lusztig bases} of $\\CH$. A fundamental property of the\nKazhdan-Lusztig bases is that if $x$ is in $\\Wex$ and $s,t$ are in $S$ with\n$\\ell(tx)<\\ell(x)$ and $\\ell(xs)<\\ell(x)$, then\n\\begin{equation}\n  \\label{eq:cxts}\n  T_tC_x= C_xT_s=-C_x\\quad\\text{and}\\quad T_tC_x'= C_x'T_s=v^2 C_x'.\n\\end{equation}\n\nFor a subset $I$ of $S$ let $w_I$ be the longest element in $W_I$. Then\n\\[\nC_{w_I}= (-v)^{\\ell(w_I)} \\sum_{y\\in W_I} \\epsilon_y v_y^{-2} T_y \\quad\n\\text{and}\\quad T_sC_{w_I}= C_{w_I}T_s=-C_{w_I}\n\\]\nfor $s$ in $I$. For subsets $I$ and $J$ of $S$ we have the Kazhdan-Lusztig\nbasis elements $C_{w_I}$ and $C_{w_J}$ of $\\CH$. Define\n\\[\n\\CHIJ=C_{w_I} \\CH C_{w_J}\\quad \\text{and}\\quad \\chi^{IJ}\\colon \\CH\\to\n\\CHIJ\\text{ by } \\chi^{IJ}(h)= C_{w_I} h C_{w_J}.\n\\]\nObviously, $\\CHIJ$ is an $R(\\Gbar)$-submodule of $\\CH$ and $\\chi^{IJ}$ is a\nsurjective $R(\\Gbar)$-module homomorphism.\n\n\\subsection{The isomorphism \\texorpdfstring{$\\CHIJ \\cong\n    K^{\\Gbar}(\\XIJ)$}{}}  \n\nThe group $\\Gbar$ acts on the Steinberg variety $Z$ and so $K^{\\Gbar}(Z)$ is\nnaturally an $R(\\Gbar)$-module, and hence an $A$-module. Chriss and Ginzburg\n\\cite[\\S7.2]{chrissginzburg:representation} and Lusztig \\cite{lusztig:bases}\nhave shown that $K^{\\Gbar}(Z)$ has an $A$-algebra structure such that $\\CH\n\\cong K^{\\Gbar}(Z)$. Let $\\varphi\\colon \\CH\\to K^{\\Gbar}(Z)$ be the\n$A$-algebra isomorphism constructed by Lusztig \\cite[Theorem\n8.6]{lusztig:bases}. The main result in this paper is the following theorem.\n\n\\begin{theorem}\\label{thm:main}\n  For each pair of subsets $(I,J)$ of $S$ there is a unique\n  $R(\\Gbar)$-module isomorphism\n  \\[\n  \\psi^{IJ}\\colon \\CHIJ \\xrightarrow{\\ \\cong\\ } K^{\\Gbar}(\\XIJ)\n  \\]\n  such that the diagram\n  \\begin{equation}\n    \\label{eq:mainthm}\n    \\vcenter{\\vbox{\n        \\xymatrix{\\CH \\ar[d]^{\\chi^{IJ}} \\ar[rr]^-{\\varphi}_-{\\cong} &&\n          K^{\\Gbar}(Z) \\ar[d]^{\\etaIJ_*} \\\\\n          \\CHIJ \\ar[rr]^-{\\psi^{IJ}}_-\\cong && K^{\\Gbar}(\\XIJ) }\n      }}\n  \\end{equation}\n  commutes.\n\\end{theorem}\n\nIn~\\autoref{sec:kl} we use the isomorphism $\\psi^{IJ}$ to define a standard\nbasis, a ``bar'' involution, and a Kazhdan-Lusztig basis for\n$K^{\\Gbar}(\\XIJ)$.  As we explain next, the proof of~\\autoref{thm:main}\ngives significantly more information about the isomorphism $\\psi^{IJ}$ than\nis encoded in~\\eqref{eq:mainthm}.  This leads to a graded refinement of the\ntheorem that is given in~\\autoref{cor:ref}.\n\nThe preimages of the $G$-orbits on $G\/P_I \\times G\/P_J$ under the projection\non the second and third factors, say $\\{\\XIJ_z\\}$, form a partition of\n$\\XIJ$ into locally closed, equidimensional subvarieties indexed by\n$\\{W_IzW_J\\}$, the set of $(W_I, W_J)$-double cosets in $W$. The closures of\nthese subvarieties are the irreducible components of $\\XIJ$ (see\n\\cite[\\S3]{douglassroehrle:geometry}). Thus, the fundamental classes of the\nclosures of the subvarieties $\\XIJ_z$ form a basis of the top Borel-Moore\nhomology of $\\XIJ$. As explained in detail below, the contribution of each\nsubvariety $\\XIJ_z$ to $K^{\\Gbar}(\\XIJ)$ is not just a single homology\nclass, rather it is the full equivariant $K$-group of isomorphism classes of\n$\\LIbar\\cap {}^z \\LJbar$-equivariant coherent sheaves on the nilpotent cone\nof $\\fl_I\\cap z\\fl_J$. A basis of this $K$-group is indexed by the set of\n$(W_I,W_J)$-double cosets in $W_I\\backslash \\Wex\/W_J$ that project to the\ndouble coset $W_IzW_J$ in $W$. Taking the union over $z$ (in a suitable\nsense) gives rise to a basis of $K^{\\Gbar}(\\XIJ)$ indexed by the\n$(W_I,W_J)$-double cosets in $\\Wex$.\n\nIn case $I=J=\\emptyset$, $X^{\\emptyset \\emptyset}=Z$, and $\\{W_\\emptyset z\nW_\\emptyset\\}=W$. For $w$ in $W$, $X^{\\emptyset \\emptyset}_w=Z_w$ is the\nconormal bundle to the $G$-orbit in $G\/B\\times G\/B$ corresponding to $w$,\n$L_\\emptyset \\cap {}^w L_\\emptyset =T$, the cone of nilpotent elements in\n$\\fl_\\emptyset \\cap w\\fl_\\emptyset$ is $\\{0\\}$, and $K^{\\Tbar}(\\{0\\}) \\cong\nA[X(T)]$. Obviously, $\\{\\, t_\\lambda w\\mid \\lambda\\in X(T)\\,\\}$ parametrizes\nthe set of $(W_\\emptyset, W_\\emptyset)$-double cosets in $\\Wex$ that project\nto $\\{w\\}$. It follows from~\\autoref{thm:ost} that $\\{\\, t_\\lambda w\n\\mid \\lambda\\in X(T)\\,\\}$ parametrizes an $A$-basis of $K^{\\Gbar}(Z_w)$ and\nit follows from ~\\cite[\\S8.6]{lusztig:bases} that $\\{\\, t_\\lambda w \\mid\nw\\in W,\\ \\lambda\\in X(T)\\,\\}$ parametrizes an $A$-basis of $K^{\\Gbar}(Z)$.\n\nAt the other extreme, when $I=J=S$, there is a single $(W,W)$-double coset\nin $W$ (with representative $1$) and $\\{W_S z W_S\\}=\\{W\\}$. In this case,\n$X^{SS}_1=X^{SS}\\cong \\FN$, $L_S \\cap {}^1 L_S =G$, and the cone of\nnilpotent elements in $\\fl_S \\cap {}^1\\fl_S$ is $\\FN$. The\n$(W_S,W_S)$-double cosets in $\\Wex$ that project to $W$ are simply the\n$(W,W)$-double cosets in $\\Wex$. These are parametrized by $\\{\\,\nt_\\lambda\\mid \\lambda\\in X(T)^+\\,\\}$, where $X(T)^+$ is the set of dominant\nweights relative to the choice of $B$.  For $\\lambda$ in $X(T)$ let\n$\\CL_\\lambda$ be the $G$-equivariant line bundle on $G\/B$ such that $T$ acts\non the fibre over $B$ with character $-\\lambda$. We consider $\\CL_\\lambda$\nas a $\\Gbar$-equivariant line bundle on $G\/B$ via the natural projection\n$\\Gbar \\to G$. It follows from a result of Broer~\\cite{broer:line} that if\n$\\lambda$ is dominant, then $p_* q^* ([\\CL_\\lambda]) = [R^0p_*q^*\n\\CL_\\lambda]$ in $K^{\\Gbar}(\\FN)$.  Ostrik~\\cite[\\S2.2] {ostrik:equivariant}\nhas proved the following key lemma he attributes to R.~Bezrukavnikov.\n\n\\begin{lemma}\\label{lem:ostlem}\n  The set $\\{\\, p_*q^* ([\\CL_\\lambda]) \\mid \\lambda\\in X(T)^+ \\,\\}$ is an\n  $A$-basis of $K^{\\Gbar}(\\FN)$.\n\\end{lemma}\n\nIn~\\cite{ostrik:equivariant} Ostrik assumes that the group $G$ is\nsimple. The extension to reductive groups is straightforward. It follows\nfrom the lemma that $\\{\\,t_\\lambda \\mid \\lambda\\in X(T)^+\\,\\}$ parametrizes\nan $A$-basis of $K^{\\Gbar} (X^{SS}_1)= K^{\\Gbar}(\\FN)$.\n\nFor arbitrary $I,J\\subseteq S$, the results we prove below are an amalgam of\nthe two extreme cases. Let $\\WIJ$ denote the set of minimal length\n$(W_I,W_J)$-double coset representatives in $W$. For $z$ in $\\WIJ$, set\n\\[\nL_z=L_I\\cap {}^zL_J\n\\]\nand let $X(T)^+_{z}$ be the set of weights in $X(T)$ that are dominant for\n$L_z$. Then $L_z$ is a reductive group (see~\\cite[\\S69B]\n{curtisreiner:methodsII}). We show in~\\autoref{thm:ost} that there is an\nisomorphism $K^{\\Gbar}(\\XIJ_z)\\cong K^{\\Lbar_z}(\\SN_z)$, where $\\SN_z$ is\nthe nilpotent cone in $\\fl_z$ and we show in~\\autoref{lem:double} that $\\{\\,\nt_\\lambda z \\mid \\lambda\\in X(T)^+_{z}\\,\\}$ parametrizes the set of $(W_I,\nW_J)$-double cosets in $\\Wex$ that project to $W_IzW_J$. It follows\nfrom~\\autoref{thm:wideiso2} that $\\{\\, t_\\lambda z \\mid \\lambda\\in\ngX(T)^+_{z}\\,\\}$ parametrizes an $A$-basis of $K^{\\Gbar}(\\XIJ_z)$ and it\nfollows from~\\autoref{cor:linind} that $\\{\\, t_\\lambda z \\mid z\\in \\WIJ,\\\n\\lambda\\in X(T)^+_{z} \\,\\}$ parametrizes an $A$-basis of $K^{\\Gbar}(\\XIJ)$.\n\nIn summary, there is a filtration of $K^{\\Gbar}(\\XIJ)$ such that the direct\nsummands of the associated graded $A$-module are naturally indexed by\n$(W_I,W_J)$-double cosets in $W$. In addition, the summand indexed by $z$ in\n$\\WIJ$ is isomorphic to $K^{\\Lbar_z}(\\SN_z)$ and has a basis consisting of\nisomorphism classes of equivariant coherent sheaves canonically indexed by\nthe $(W_I,W_J)$-double cosets in $\\Wex$ that project to $W_IzW_J$.\n\n\\section{The proof of \\autoref{thm:main}}\\label{sec:proof}\n\nIn this section we prove~\\autoref{thm:main}. The proof proceeds in three\nsteps. The first step is show that $K^{\\Gbar}(\\XIJ)$ is a free $A$-module,\nthe second step is to show that the composition $\\etaIJ_* \\varphi$ factors\nthrough $\\chi^{IJ}$, and the third step is to show that the resulting map\n$\\psi^{IJ}$ is an isomorphism. In the course of the argument we construct\nexplicit $A$-bases of of $\\CHIJ$ and $K^{\\Gbar}(\\XIJ)$ that correspond under\n$\\psi^{IJ}$.\n\nTo show that $K^{\\Gbar}(\\XIJ)$ is a free $A$-module, we use the filtration\non $K^{\\Gbar}(\\XIJ)$ determined by the $G$-orbits on $G\/P_I \\times G\/P_J$.\nRecall that the rule $z\\mapsto G\\cdot (P_I,zP_J)$ defines a bijection\nbetween $\\WIJ$ and the set of $G$-orbits in $G\/P_I \\times G\/P_J$.  Let\n$q_{IJ}\\colon \\XIJ\\to G\/P_I \\times G\/P_J$ be the projection on the second\nand third factors, and for $z$ in $\\WIJ$ define $\\XIJ_z$ to be the preimage\nin $\\XIJ$ of the orbit $G\\cdot (P_I, zP_J)$. Then\n\\[\n\\XIJ_z=\\{\\, (x, gP_I, gzP_J)\\in \\FN\\times G\/P_I\\times G\/P_J \\mid g\\inverse\nx\\in \\FN\\cap \\fp_I\\cap z\\fp_J\\,\\}.\n\\]\nChoose a linear order on $\\WIJ$, say $\\WIJ=\\{\\, z_i\\mid 1\\leq i\\leq\n|\\WIJ|\\,\\}$, that extends the Bruhat order and define\n\\[\n\\XIJ_{\\ssleq i}=\\coprod_{j\\leq i} \\XIJ_{z_j}.\n\\] \nThen $\\XIJ_{\\ssleq i}= \\XIJ_{\\ssleq i-1} \\amalg \\XIJ_{z_i}$, where\n$\\XIJ_{\\ssleq i-1}$ is closed in $\\XIJ_{\\ssleq i}$ and $\\XIJ_{{z_i}}$ is\nopen in $\\XIJ_{\\ssleq i}$. It is shown in \\autoref{ssec:ex} that for\n$i\\geq1$ the sequence\n\\begin{equation}\n  \\label{eq:exz}\n  \\xymatrix{0 \\ar[r] & K^{\\Gbar}(\\XIJ_{\\ssleq i-1}) \\ar[r]^-{()_*} &\n    K^{\\Gbar}(\\XIJ_{\\ssleq i}) \\ar[r]^-{()^*} & K^{\\Gbar}(\\XIJ_{z_i}) \\ar[r]&\n    0 } \n\\end{equation}\nof $R(\\Gbar)$-modules is exact. It follows that the embedding $\\XIJ_{\\ssleq\n  i}\\hookrightarrow \\XIJ$ induces an injection $K^{\\Gbar}(\\XIJ_{\\ssleq i})\n\\hookrightarrow K^{\\Gbar}(\\XIJ)$ in equivariant $K$-theory. \n\nSuppose $z$ is in $\\WIJ$ and recall that $L_z=L_I\\cap {}^zL_J$ and that\n$\\SN_z=\\FN\\cap \\fl_z$ is the cone of nilpotent elements in $\\fl_z$. It is\nshown in \\autoref{ssec:xijz}, in the course of the proof\nof~\\autoref{thm:ost}, that $K^{\\Gbar}(\\XIJ_z) \\cong\nK^{\\Lbar_z}(\\SN_z)$. Thus, the next proposition follows from the exact\nsequence~\\eqref{eq:exz} by induction and~\\autoref{lem:ostlem}.\n\n\\begin{proposition}\n  The equivariant $K$-group $K^{\\Gbar}(\\XIJ)$ is a free $A$-module.\n\\end{proposition}\n\nNext, to show that $\\etaIJ_* \\varphi$ factors through $\\chi^{IJ}$, it is\nenough to show that for each generator $s$ in $J$, and each generator $t$ in\n$I$,\n\\begin{equation}\n  \\label{eq:4b}\n  \\etaIJ_*  \\varphi (hT_s)= - \\etaIJ_* \\varphi(h)= \\etaIJ_*  \\varphi (T_th)  \n\\end{equation}\nfor all $h$ in $\\CH$. Indeed, if this condition holds, then for all\n$h$ in $\\CH$, $x$ in $W_I$, and $y$ in $W_J$, $\\etaIJ_* \\varphi(T_x h\nT_y)= \\epsilon_x \\epsilon_y \\etaIJ_*\\varphi(h)$, and so\n$\\etaIJ_*\\varphi(C_{w_I} h C_{w_J}) = r_{IJ} \\etaIJ_*\\varphi(h)$,\nwhere $r_{IJ}$ is a non-zero element of $A$ that depends only on $I$\nand $J$. Then $\\psi^{IJ}\\colon \\CHIJ\\to K^{\\Gbar}(\\XIJ)$ by\n$\\psi^{IJ}(C_{w_I} h C_{w_J}) = \\etaIJ_*\\varphi(h)$ is well defined\nbecause if $C_{w_I} h C_{w_J}= 0$, then $r_{IJ} \\etaIJ_*\\varphi(h)=0$\nand so $\\etaIJ_*\\varphi(h)=0$.\n\nLet $\\pi_I\\colon \\XEJ \\to \\XIJ$ by $\\pi_I(x, gB,\nhP_J)= (x, gP_I, hP_J)$ and let $\\pi_J\\colon \\XIE \\to \\XIJ$ by $\\pi_J(x,\ngP_I, hB)= (x, gP_I, hP_J)$. Then the diagram\n\\begin{equation*}\n  \\label{eq:3b}\n  \\xymatrix{Z\\ar[r]^{\\eta^{\\emptyset J}} \\ar[d]_{\\eta^{I \\emptyset}}\n        \\ar[dr]^{\\etaIJ} &  \\XEJ \\ar[d]^{\\pi_I} \\\\ \n    \\XIE \\ar[r]_{\\pi_J} &\\XIJ}\n\\end{equation*}\ncommutes and so~\\eqref{eq:4b} follows from the equalities\n\\[\n\\eta^{\\emptyset J}_* \\varphi (hT_s)= - \\eta^{\\emptyset J}_*\n\\varphi(h)\\quad\\text{and}\\quad \\eta^{I \\emptyset}_* \\varphi (T_th)= -\n\\eta^{I \\emptyset}_* \\varphi(h),\n\\]\nfor $s\\in J$ and $t\\in I$, by applying $(\\pi_I)_*$ and $(\\pi_J)_*$,\nrespectively. Thus, by symmetry, it is enough to show that $\\eta^{\\emptyset\n  J}_* \\varphi$ factors through the projection $\\CH\\to \\CH C_{w_J}$ given by\nright multiplication by $C_{w_J}$:\n\\[\n\\xymatrix{ \\CH \\ar@{-->}[d] \\ar[r]^{\\varphi} & K^{\\Gbar}(Z)\n  \\ar[d]^-{\\eta^{\\emptyset J}_*} \\\\\n  \\CH C_{w_J} \\ar@{-->}[r] & K^{\\Gbar}(\\XEJ).} \n\\]\n\nThe intersection\/Tor-product construction described by Lusztig in\n\\cite[\\S6.4]{lusztig:bases} can be used to define a\n$K^{\\Gbar}(Z)$-module structure on $K^{\\Gbar}(\\XEJ)$, say\n\\[\n\\star_J\\colon K^{\\Gbar}(Z) \\times K^{\\Gbar}(\\XEJ) \\to K^{\\Gbar}(\\XEJ),\n\\]\nsuch that the map $\\eta^{\\emptyset J}_*\\colon K^{\\Gbar}(Z) \\to\nK^{\\Gbar}(\\XEJ)$ is $K^{\\Gbar}(Z)$-linear (see~\\autoref{pro:xlin}). Thus,\nfor all $h$ in $\\CH$ and all $s$ in $S$,\n\\[\n\\eta^{\\emptyset J}_* \\varphi(hT_s)= \\eta^{\\emptyset J}_*(\\varphi(h) \\star\n\\varphi(T_s)) = \\varphi(h) \\star_J \\eta^{\\emptyset J}_* (\\varphi(T_s)),\n\\]\nwhere $\\star$ is the convolution product in $K^{\\Gbar}(Z)$. Hence, it is\nenough to show that $\\eta^{\\emptyset J}_* (\\varphi(T_s))= -\\eta^{\\emptyset\n  J}_*( \\varphi(1))$ for all $s$ in $J$, or equivalently that\n$\\eta^{\\emptyset J}_*(\\varphi(T_s+1))=0$. For a simple reflection $s$ in\n$W$, let $\\mathbf a_s$ in $K^{\\Gbar}(Z)$ be defined as in\n\\cite[\\S7.20]{lusztig:bases} (see \\autoref{sec:as}). By~\\cite[\\S7.25]\n{lusztig:bases}, $\\varphi(T_s+1)=-v \\mathbf a_s$. Therefore, the existence\nof the map $\\psi^{IJ}$ in~\\autoref{thm:main} follows from the next theorem.\n\n\\begin{theorem}\\label{thm:as}\n  Suppose $s$ is in $J$. Then $\\etaEJ_*(\\mathbf a_s)= 0$ in\n  $K^{\\Gbar}(\\XEJ)$.\n\\end{theorem}\n\nThis theorem is proved in \\autoref{sec:as}.\n\nFinally, to show that $\\psi^{IJ}$ is an isomorphism, we use an $A$-basis of\n$\\CHIJ$ and the partition $\\XIJ= \\coprod_{z\\in \\WIJ} \\XIJ_z$ to define\ncompatible filtrations on $\\CHIJ$ and $K^{\\Gbar}(\\XIJ)$. We also need the\nanalogous constructions for $\\CH$ and $K^{\\Gbar}(Z)$. \n\nRecall that $q_{Z}\\colon Z\\to G\/B \\times G\/B$ is the projection on the\nsecond and third factors. For $w$ in $W$ define $Z_w$ to be the preimage in\n$Z$ of the orbit $G\\cdot (B, wB)$. Then\n\\[\nZ_w= \\{\\, (x, gB, gwB)\\in \\FN\\times G\/B\\times G\/B \\mid g\\inverse x\\in\n\\fu\\cap w\\fu\\,\\}.\n\\]\nDefine \n\\[\nZ_{\\ssleq w}=\\coprod_{y\\leq w} Z_y \\quad \\text{and} \\quad Z_{\\sslt w}=\nZ_{\\ssleq w}\\setminus Z_w= \\coprod_{y< w} Z_y,\n\\]\nwhere $\\leq$ is the Bruhat order on $W$. Similarly, for $z$ in $\\WIJ$ define\n\\[\n\\XIJ_{\\ssleq z} =\\coprod_{y\\leq z} \\XIJ_y \\quad \\text{and} \\quad \\XIJ_{\\sslt\n  z}= \\XIJ_{\\ssleq z}\\setminus \\XIJ_z =\\coprod_{y< z} \\XIJ_y,\n\\] \nwhere the unions are over $y$ in $\\WIJ$.\n\nSuppose $z$ is in $\\WIJ$ and let $\\eta^z$, $\\eta^{\\ssleq z}$, and\n$\\eta^{\\sslt z}$ be the restrictions of $\\etaIJ$ to $Z_z$, $Z_{\\ssleq z}$,\nand $Z_{\\sslt z}$, respectively.  It is shown in\n\\cite[\\S3]{douglassroehrle:geometry} that if $w_1$ is in $W_I$ and $w_2$ is\nin $W_J$, then $\\etaIJ(Z_{w_1zw_2})\\subseteq \\XIJ_z$, and that\n$\\etaIJ(Z_{w_1zw_2}) =\\XIJ_z$ if and only if $w_1zw_2=z$. It is shown in\n~\\cite[Lemma 2.2] {douglass:inversion} that if $z$ and $z'$ are in $\\WIJ$,\nthen $z\\leq z'$ if and only if there are elements $w$ in $W_IzW_J$ and $w'$\nin $W_Iz'W_J$ such that $w\\leq w'$. Therefore, $\\etaIJ(Z_{\\ssleq\n  z})\\subseteq \\XIJ_{\\ssleq z}$, and letting $r_z$ and $r_z^{IJ}$ denote the\ninclusions $Z_z\\to Z_{\\ssleq z}$ and $\\XIJ_z\\to \\XIJ_{\\ssleq z}$,\nrespectively, the square\n\\begin{equation}\n  \\label{eq:cart}\n  \\vcenter{\\vbox{\n      \\xymatrix{Z_z\\ar[r]^{r_z} \\ar[d]^{\\eta^z} & Z_{\\ssleq\n          z}\\ar[d]^{\\eta^{\\ssleq z}} \\\\\n        \\XIJ_z\\ar[r]^{r_z^{IJ}} & \\XIJ_{\\ssleq z}}\n    }}  \n\\end{equation}\nis cartesian. \n\n\\begin{lemma}\\label{lem:hyp1}\n  The map $\\eta^z\\colon Z_z\\to \\XIJ_z$ is a proper morphism.\n\\end{lemma}\n\n\\begin{proof}\n  Define $Z_{W_IzW_J} = \\coprod_{w\\in {W_IzW_J}} Z_w$. Then $Z_{{W_IzW_J}}=\n  (\\etaIJ)\\inverse \\big( \\XIJ_z \\big)$ and so by base change, the\n  restriction of $\\etaIJ$ to $Z_{{W_IzW_J}}$ is proper. Now $G\\cdot (B,zB)$\n  is closed in $\\coprod_{w\\in W_IzW_J} G\\cdot (B, wB)$ and hence $Z_z$ is\n  closed in $Z_{{W_IzW_J}}$. It follows that $\\eta^z$ is proper.  \\qed\n\\end{proof}\n\nSince $\\etaIJ(Z_{\\leq z}) \\subseteq \\XIJ_{\\leq z}$ for all $z$ in $\\WIJ$\nand~\\eqref{eq:cart} is cartesian, the proper morphisms $\\etaIJ$ and $\\eta^z$\ninduce a map of short exact sequences such that the following diagram\ncommutes:\n\\begin{equation}\n  \\label{eq:exwz}\n  \\vcenter{\\vbox{\n      \\xymatrix{0 \\ar[r] & K^{\\Gbar}(Z_{\\sslt z}) \\ar[r]^{()_*}\n        \\ar[d]^{\\eta^{\\sslt z}_*} &  K^{\\Gbar}(Z_{\\ssleq z}) \\ar[r]^{()^*}\n        \\ar[d]^{\\eta^{\\ssleq  z}_*}& K^{\\Gbar}(Z_{z})  \\ar[r] \\ar[d]^{\\eta^{\n            z}_*}& 0 \\\\ \n        0 \\ar[r] & K^{\\Gbar}(\\XIJ_{\\sslt z}) \\ar[r]^{()_*} &\n        K^{\\Gbar}(\\XIJ_{\\ssleq z}) \\ar[r]^{()^*} & K^{\\Gbar}(\\XIJ_{z})\n        \\ar[r]& 0 ,}   \n    }}\n\\end{equation}\nwhere $Z_{\\sslt 1}= \\XIJ_{\\sslt 1}= \\emptyset$.\n\nSuppose $z$ is in $\\WIJ$. Then $L_z = L_I\\cap {}^zL_J =L_{\\IzJ}$ is a Levi\nsubgroup of $P_I\\cap {}^zP_J$ (see~\\cite[\\S69B] {curtisreiner:methodsII}).\nSet\n\\[\nB_z=L_z\\cap B \\quad\\text{and} \\quad \\SNt_z= \\{\\, (x, hB_z)\\in \\SN_z\\times\nL_z\/B_z\\mid h\\inverse x\\in \\fb_z\\,\\},\n\\]\nand let \n\\[\np_z\\colon \\SNt_z\\to \\SN_z\\quad\\text{and}\\quad q_z\\colon \\SNt_z\\to L_z\/B_z\n\\]\nbe the projections. Then $B_z$ is a Borel subgroup of $L_z$ and $p_z$ is the\nSpringer resolution of $\\SN_z$.\n\n\\begin{theorem}\\label{thm:ost}\n  Suppose $z$ is in $\\WIJ$. Then there is a commutative diagram of\n  $R(\\Gbar)$-modules\n  \\begin{equation*}\n    \\xymatrix{ K^{\\Gbar}(Z_z) \\ar[r]_{\\cong} \\ar[d]^{\\eta^z_*} &\n      K^{\\Lbar_z}(\\SNt_z) \\ar[d]^{(p_z)_*} \\\\ \n      K^{\\Gbar}(\\XIJ_z) \\ar[r]_{\\cong} & K^{\\Lbar_z}(\\SN_z), }  \n  \\end{equation*}\n  where the horizontal maps are isomorphisms and the vertical maps are\n  surjections.\n\\end{theorem}\n\nThis theorem is proved in \\autoref{ssec:xijz}.\n\n\\begin{corollary}\\label{cor:surj}\n  The map $\\eta^{\\ssleq z}_*\\colon K^{\\Gbar}(Z_{\\ssleq z}) \\to\n  K^{\\Gbar}(\\XIJ_{\\ssleq z})$ is surjective for all $z$ in\n  $\\WIJ$. Therefore, the maps $\\etaIJ_*\\colon K^{\\Gbar}(Z) \\to\n  K^{\\Gbar}(\\XIJ)$ and $\\psi^{IJ}\\colon \\CHIJ\\to K^{\\Gbar}(\\XIJ)$ are\n  both surjective.\n\\end{corollary}\n\n\\begin{proof}\n  We show that $\\eta^{\\ssleq z}_*$ is surjective using induction on $z$ in\n  the Bruhat order on $\\WIJ$. It is shown\n  in~\\cite[\\S3.17]{kazhdanlusztig:langlands} that the image of\n  $K^{\\Gbar}(Z_{\\sslt z})$ in $K^{\\Gbar}(Z)$ coincides with the sum of the\n  images of the spaces $K^{\\Gbar}(Z_{\\ssleq y})$ for $y<w$. A similar\n  argument, using~\\autoref{lem:k1}, shows that the image of\n  $K^{\\Gbar}(\\XIJ_{\\sslt z})$ in $K^{\\Gbar}(\\XIJ)$ coincides with the sum of\n  the images of the spaces $K^{\\Gbar}(\\XIJ_{\\ssleq y})$ for $y$ in $\\WIJ$\n  with $y<z$. Thus, the natural maps\n  \\[\n  \\bigoplus_{y< z} K^{\\Gbar}(Z_{\\ssleq y}) \\xrightarrow{\\ f_1\\ }\n  K^{\\Gbar}(Z_{\\sslt z}) \\quad\\text{and} \\quad \\bigoplus_{\\substack{y\\in\n      \\WIJ \\\\ y< z}} K^{\\Gbar}(\\XIJ_{\\ssleq y}) \\xrightarrow{\\ f_2\\ }\n  K^{\\Gbar}(\\XIJ_{\\sslt z})\n  \\]\n  are surjective. The composition $\\bigoplus_{y< z} K^{\\Gbar}(Z_{\\ssleq y})\n  \\xrightarrow{\\ f_1\\ } K^{\\Gbar}(Z_{\\sslt z}) \\xrightarrow{\\eta^{\\sslt\n      z}_*} K^{\\Gbar}(\\XIJ_{\\sslt z})$ is equal the composition\n  $\\bigoplus_{y< z} K^{\\Gbar}(Z_{\\ssleq y}) \\xrightarrow{\\ \\ }\n  \\bigoplus_{\\substack{y\\in \\WIJ \\\\ y< z}} K^{\\Gbar}(\\XIJ_{\\ssleq y})\n  \\xrightarrow{\\ f_2\\ } K^{\\Gbar}(\\XIJ_{\\sslt z})$. By induction, and the\n  fact that the map $f_2$ is surjective, this last composition is\n  surjective. It follows that $\\eta^{\\sslt z}_*$ is surjective. Now the fact\n  that $\\eta^{\\ssleq z}_*$ is surjective follows from\n  diagram~\\eqref{eq:exwz} by the Five Lemma and~\\autoref{thm:ost}.\n\n  Let $z_0$ be the element in $\\WIJ$ with maximal length. Then $\\XIJ_{\\ssleq\n    z_0}= \\XIJ$ and so taking $z=z_0$ we see that the map\n  $K^{\\Gbar}(Z_{\\ssleq z_0})\\to K^{\\Gbar}(\\XIJ)$ induced by $\\etaIJ$ is a\n  surjection. This map is $\\etaIJ_* \\circ (j_{z_0})_*$, where $j_{z_0}\\colon\n  Z_{\\ssleq z_0} \\to Z$ is the inclusion. Therefore, $\\etaIJ_*$ is\n  surjective and so $\\psi^{IJ}$ is surjective. \\qed\n\\end{proof}\n\nTo define filtrations on $\\CH$ and $\\CHIJ$ compatible with\ndiagram~\\eqref{eq:exwz} we need to study $(W_I,W_J)$-double cosets in\n$\\Wex$. For $I\\subseteq S$ define\n\\[\nX(T)_I^+ =\\{\\, \\lambda\\in X(T) \\mid \\langle \\lambda, \\check \\alpha \\rangle\n\\geq0\\ \\forall \\alpha\\in \\Pi_I\\,\\},\n\\]\nwhere $\\langle \\,\\cdot\\,, \\,\\cdot\\,\\rangle$ is the usual pairing between\n$X(T)$ and $\\Hom(\\BBC^*, T)$, the group of co-characters of $T$, and $\\{\\,\n\\check \\alpha\\mid \\alpha\\in \\Phi\\,\\}$ is the dual root system in\n$\\Hom(\\BBC^*, T)$. Then $X(T)_{\\emptyset}^+=X(T)$ and $X(T)_S^+=X(T)^+$, the\nset of dominant weights.  Because $X(T)^+$ is a fundamental domain for the\naction of $W$ on $X(T)$, it follows that $\\{\\, t_\\lambda\\mid \\lambda\\in\nX(T)^+\\,\\}$ is a set of $(W,W)$-double coset representatives in $\\Wex$. The\nnext lemma is the analog of this fact for $(W_I,W_J)$-double cosets.\n\n\\begin{lemma}\\label{lem:double}\n  Every $(W_I,W_J)$-double coset in $\\Wex$ contains a unique representative\n  of the form $t_\\lambda z$, where $z$ is in $\\WIJ$ and $\\lambda$ is in\n  $X(T)_{\\IzJ}^+$.\n\\end{lemma}\n\n\\begin{proof}\n  If $\\lambda$ is in $X(T)$, $w$ is in $W$, $w_1$ is in $W_I$, and\n  $w_2$ is in $W_J$, then $w_1t_\\lambda ww_2= t_{w_1(\\lambda)}\n  w_1ww_2$. Thus, every $(W_I,W_J)$-double coset has a representative\n  of the form $t_\\lambda z$ where $z$ is in $\\WIJ$. Suppose that $z$\n  is in $\\WIJ$ and $\\lambda$ and $\\mu$ are in $X(T)$. Then $t_\\mu z=\n  w_1t_\\lambda zw_2$ for some $w_1$ in $W_I$ and $w_2$ in $W_J$ if and\n  only if $\\mu= w_1(\\lambda)$ and $z=w_1zw_2$. Therefore, $t_\\lambda\n  z$ and $t_\\mu z$ are in the same $(W_I,W_J)$-double coset if and\n  only if $\\mu= w_1(\\lambda)$ for some $w_1$ in $W_I\\cap\n  {}^zW_J$. Because $W_I\\cap {}^zW_J= W_{\\IzJ}$ (see\n  \\cite[\\S64C]{curtisreiner:methodsII}), we see that $t_\\lambda z$ and\n  $t_\\mu z$ are in the same $(W_I,W_J)$-double coset if and only if\n  $\\mu= w_1(\\lambda)$ for some $w_1$ in $W_{\\IzJ}$. Finally,\n  $X(T)_{\\IzJ}^+$ is a fundamental domain for the action of $W_{\\IzJ}$\n  on $X(T)$ and so every $(W_I,W_J)$-double coset in $\\Wex$ contains a\n  unique representative of the form $t_\\lambda z$, where $z$ is in\n  $\\WIJ$ and $\\lambda$ is in $X(T)_{\\IzJ}^+$. \\qed\n\\end{proof}\n\nFor $z$ in $\\WIJ$, set \n\\[\nX(T)_z^+=X(T)_{\\IzJ}^+.\n\\]\nThen $X(T)_z^+$ is the set of weights that are dominant for $L_{z}$ with\nrespect to the Borel subgroup $B_z$.\n\nLet $\\epsilon_I=\\sum_{w\\in W_I} \\epsilon_ww$ for $I\\subseteq\nS$. Then~\\autoref{lem:double} shows that $\\{\\, \\epsilon_I t_\\lambda z\n\\epsilon_J \\mid z\\in \\WIJ,\\, \\lambda \\in X(T)_{z} ^+ \\,\\}$ is a $\\BBZ$-basis\nof $\\epsilon_I \\BBZ[\\Wex] \\epsilon_J$. A similar statement is true for\n$\\CHIJ$.\n\n\\begin{theorem}\\label{thm:hijbasis}\n  The set $\\{\\, C_{w_I} \\theta_\\lambda T_z C_{w_J} \\mid z\\in \\WIJ,\\, \\lambda\n  \\in X(T)_{z} ^+ \\,\\}$ is an $A$-basis of $\\CHIJ$.\n\\end{theorem}\n\nThis theorem is proved in \\autoref{sec:basis}.\n\nRecall that $\\{\\, \\theta_\\lambda T_w\\mid \\lambda\\in X(T), w\\in W\\,\\}$ is the\nBernstein basis of $\\CH$. By analogy, we call $\\{\\, C_{w_I} \\theta_\\lambda\nT_z C_{w_J} \\mid z\\in \\WIJ,\\, \\lambda \\in X(T)_{z} ^+ \\,\\}$ the\n\\emph{Bernstein basis} of $\\CHIJ$. We will see that\n\\[\n\\{\\, \\psi^{IJ}(C_{w_I} \\theta_\\lambda T_z C_{w_J})\\mid z\\in \\WIJ,\\, \\lambda\n\\in X(T)_{z} ^+ \\,\\}\n\\]\nis an $A$-basis of $K^{\\Gbar}(\\XIJ)$.\n\nFor $w$ in $W$ and $z$ in $\\WIJ$ define\n\\begin{align*}\n  \\CH_{w}&= \\operatorname{span}_A\\{\\, \\theta_\\lambda T_w\\mid \\lambda \\in\n  X(T)\\,\\}, & && \\CH_{\\ssleq w}&= \\sum_{y\\leq w} \\CH_y, \\\\\n  \\CHIJ_{z}&= \\operatorname{span}_A\\{\\, C_{w_I} \\theta_\\lambda T_zC_{w_J}\n  \\mid \\lambda \\in X(T)\\,\\}, & &\\text{and}& \\CHIJ_{\\ssleq z}&= \\sum_{y\\leq\n    z} \\CHIJ_{y},\n\\end{align*}\nwhere the last sum is over $y$ in $\\WIJ$. The proof\nof~\\autoref{thm:hijbasis} shows that in fact $\\{\\, C_{w_I} \\theta_\\lambda\nT_zC_{w_J} \\mid \\lambda \\in X(T)^+_z\\,\\}$ is an $A$-basis of\n$\\CHIJ_z$. Extending the definition of $\\CL_\\lambda$ for $\\lambda$ in\n$X(T)$, let $\\CL^z_\\lambda$ be the canonical $\\Lbar_z$-equivariant line\nbundle on $L_z\/B_z$ determined by $\\lambda$. Also, let $\\chi^{\\ssleq\n  z}\\colon \\CH_{\\ssleq z}\\to \\CHIJ_{\\ssleq z}$ be the restriction of\n$\\chi^{IJ}$. We can now extend the diagram in~\\autoref{thm:ost} to include\n$\\CH_{\\ssleq z}$ and $\\CHIJ_{\\ssleq z}$.\n\n\\begin{theorem}\\label{thm:wideiso2}\n  Suppose $z$ is in $\\WIJ$. Then there is a commutative diagram of\n  $A$-modules\n  \\begin{equation} \n    \\label{eq:wideiso}\n    \\vcenter{\\vbox{    \n        \\xymatrix{%\n          \\CH_{\\ssleq z} \\ar[r]^-{}_-{\\cong} \\ar[d]^{\\chi^{\\ssleq z}}& \n          K^{\\Gbar}(Z_{\\ssleq z}) \\ar@{->>}[r]^-{r_z^*} \\ar[d]^{\\eta^{\\ssleq\n              z}_*}& K^{\\Gbar}(Z_z) \\ar[r]_-{\\cong} \\ar[d]^{\\eta^z_*} &\n          K^{\\Lbar_z}(\\SNt_z) \\ar[d]^{(p_z)_*} \\\\ \n          \\CHIJ_{\\ssleq z} \\ar[r]^-{}_-{\\cong} & K^{\\Gbar}(\\XIJ_{\\ssleq z})\n          \\ar@{->>}[r]^-{(r_z^{IJ})^*} & K^{\\Gbar}(\\XIJ_z) \\ar[r]_-{\\cong} &\n          K^{\\Lbar_z}(\\SN_z), }\n      }}   \n  \\end{equation}\n  where\n  \\begin{enumerate}\n  \\item the horizontal maps are surjective or bijective, as indicated, and\n    the vertical maps are surjective, \\label{i:wi1}\n  \\item if $f_1$ is the composition across the top row, then $\\CH_y$ is in\n    the kernel of $f_1$ for all $y<z$ and the restriction of $f_1$ to\n    $\\CH_{z}$ is the isomorphism that carries $\\theta_\\lambda T_z$ to\n    $\\epsilon_z q_z^*[\\CL^z_\\lambda]$ for $\\lambda$ in $X(T)$,\n    and \\label{i:wi2}\n  \\item if $f_2$ is the composition across the bottom row, then $\\CHIJ_y$ is\n    in the kernel of $f_2$ for all $y<z$ in $\\WIJ$ and the restriction of\n    $f_2$ to $\\CHIJ_{z}$ is the isomorphism that carries $C_{w_I}\n    \\theta_\\lambda T_zC_{w_J}$ to $\\epsilon_z (p_z)_*q_z^*[\\CL^z_\\lambda]$\n    for $\\lambda$ in $X(T)_{z}^+$. \\label{i:wi3}\n  \\end{enumerate}\n\\end{theorem}\n\nThis theorem is proved in~\\autoref{ssec:isoz1}\\,--\\,\\autoref{ssec:isoz3}.\n\nThe isomorphisms in the left-hand square in~\\eqref{eq:wideiso} may be viewed\nas the restrictions of $\\varphi$ and $\\psi^{IJ}$ to $\\CH_{\\ssleq z}$ and\n$\\CHIJ_{\\ssleq z}$, respectively in the sense that the inclusions\n$\\CH_{\\ssleq z}\\subseteq \\CH$ and $\\CHIJ_{\\ssleq z} \\subseteq \\CHIJ$, and\nthe closed embeddings $Z_{\\ssleq z} \\subseteq Z$ and $\\XIJ_{\\ssleq z}\n\\subseteq \\XIJ$, induce commutative diagrams\n\\[\n\\vcenter{\\vbox{ \\xymatrix{ \\CH_{\\ssleq z} \\ar[r]^-{}_-{\\cong} \\ar[d]&\n      K^{\\Gbar}(Z_{\\ssleq z}) \\ar[d]\\\\\n      \\CH \\ar[r]^-{\\varphi} & K^{\\Gbar}(Z)} }} \\quad\\text{and} \\quad\n\\vcenter{\\vbox{ \\xymatrix{\\CHIJ_{\\ssleq z} \\ar[r]^-{}_-{\\cong} \\ar[d] &\n      K^{\\Gbar}(\\XIJ_{\\ssleq z}) \\ar[d] \\\\\n      \\CHIJ \\ar[r]^-{\\psi^{IJ}} & K^{\\Gbar}(\\XIJ) } }}\n\\]\nwhere the vertical maps are injective.\n\nLet $z_0$ be the element in $\\WIJ$ with maximal length. Then $\\CHIJ_{\\ssleq\n  z_0} = \\CHIJ$ and $\\XIJ_{\\ssleq z_0}= \\XIJ$. Thus, taking $z=z_0$ in the\ntheorem we have that $\\psi^{IJ} \\colon \\CHIJ\\to K^{\\Gbar}(\\XIJ)$ is an\nisomorphism. This completes the proof of~\\autoref{thm:main}.\n\n\\begin{corollary}\\label{cor:linind}\n  The set $\\{\\, \\psi^{IJ}(C_{w_I} \\theta_\\lambda T_z C_{w_J})\\mid z\\in\n  \\WIJ,\\, \\lambda \\in X(T)_{z} ^+ \\,\\}$ is an $A$-basis of\n  $K^{\\Gbar}(\\XIJ)$.\n\\end{corollary}\n\nIt follows from~\\autoref{thm:wideiso2} that $\\psi^{IJ}(\\CHIJ_{\\ssleq z}) =\nK^{\\Gbar}(\\XIJ_{\\ssleq z})$ for $z$ in $\\WIJ$, and so the following\nrefinement of~\\autoref{thm:main} is a consequence of the constructions\nabove.\n\n\\begin{corollary}\\label{cor:ref}\n  The $A$-submodules $K^{\\Gbar}(\\XIJ_{\\ssleq z})$ determine a filtration of\n  $K^{\\Gbar}(\\XIJ)$ such that if $\\gr\\, K^{\\Gbar}(\\XIJ)$ is the associated\n  graded $A$-module, then there are $A$-module isomorphisms\n  \\[\n  \\gr\\, K^{\\Gbar}(\\XIJ) \\cong \\bigoplus_{z\\in \\WIJ} \\CHIJ_z \\cong\n  \\bigoplus_{z\\in \\WIJ} K^{\\Lbar_z} \\left(\\SN_{z} \\right).\n  \\]\n\\end{corollary}\n\n\n\\section{The ``bar'' involution and Kazhdan-Lusztig basis of\n  \\texorpdfstring{$\\CHIJ$}{}} \\label{sec:kl}\n\nFor $z$ in $\\WIJ$ and $\\lambda$ in $X(T)_z^+$, let $m_{\\lambda,z}$ be the\nelement in the double coset $W_It_\\lambda zW_J$ with minimal length and\ndefine\n\\[\nT_{\\lambda,z}=\\chi^{IJ}( T_{m_{\\lambda,z}} )=C_{w_I} T_{m_{\\lambda,z}}\nC_{w_J}\n\\]\nin $\\CHIJ$.\n\n\\begin{lemma}\\label{lem:sb}\n  The set $\\{\\, T_{\\lambda,z}\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+ \\,\\}$ is\n  an $A$-basis of $\\CHIJ$.\n\\end{lemma}\n\n\\begin{proof}\n  If $x$ is in $W_It_\\lambda zW_J$, then there are elements $w_1$ in $W_I$\n  and $w_2$ in $W_J$ so that $x=w_1m_{\\lambda,z} w_2$ and $\\ell(x)=\n  \\ell(w_1)+ \\ell(m_{\\lambda,z}) +\\ell(w_2)$. Thus, it follows\n  from~\\eqref{eq:cxts} that $C_{w_I} T_xC_{w_J}= \\pm C_{w_I}\n  T_{m_{\\lambda,z}} C_{w_J}$. Therefore $\\{\\, T_{\\lambda,z}\\mid z\\in \\WIJ,\\\n  \\lambda\\in X(T)_z^+ \\,\\}$ spans $\\CHIJ$. By~\\autoref{lem:double} $\\{\\,\n  \\epsilon_I m_{\\lambda,z} \\epsilon_J \\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+\n  \\,\\}$ is a $\\BBZ$-basis of $\\epsilon_I \\BBZ[\\Wex] \\epsilon_J$. It follows\n  that $\\{\\, T_{ \\lambda,z}\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+ \\,\\}$ is\n  linearly independent. \\qed\n\\end{proof}\n\nWe call $\\{\\, T_{\\lambda,z}\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+ \\,\\}$ the\n\\emph{standard basis} of $\\CHIJ$. It follows from \\eqref{eq:cx} that the\nring involution $\\overline{\\phantom {x} }\\colon \\CH \\to \\CH$ induces an\ninvolution of abelian groups, also denoted by $\\overline{\\phantom {x} }$,\nfrom $\\CHIJ$ to $\\CHIJ$ that we call the \\emph{bar involution} of $\\CHIJ$.\n\nNow consider the Kazhdan-Lusztig basis $\\{\\, C_x'\\mid x\\in \\Wex\\,\\}$ of\n$\\CH$. If $t$ is in $I$ and $x$ is in $\\Wex$ with $tx<x$, then\nby~\\eqref{eq:cxts}\n\\[\nC_{w_I} C_x'C_{w_J}= -C_{w_I}T_t C_x'C_{w_J}= -v^2 C_{w_I} C_x'C_{w_J},\n\\]\nand so $C_{w_I} C_x'C_{w_J}=0$. Similarly, if $s$ is in $J$ and $x$ is in\n$\\Wex$ with $xs<x$, then $C_{w_I} C_x'C_{w_J}=0$. It follows that $C_{w_I}\nC_x'C_{w_J}=0$ unless $x=m_{\\lambda,z}$ for some $\\lambda, z$. For $z$ in\n$\\WIJ$ and $\\lambda$ in $X(T)_z^+$, define\n\\[\nC_{\\lambda,z}'=\\chi^{IJ}(C_{m_{\\lambda,z}}') =C_{w_I} C_{m_{\\lambda,z}}'\nC_{w_J}\n\\]\nin $\\CHIJ$. As usual, an element $h$ in $\\CH$ or $\\CHIJ$ is said to be\n\\emph{self-dual} if $\\overline{ h} =h$.\n\n\\begin{proposition}\n  The set $\\{\\, C_{\\lambda, z}'\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+\\,\\}$ is\n  a self-dual basis of $\\CHIJ$ and the map $\\chi^{IJ}\\colon \\CH\\to \\CHIJ$\n  may be identified with projection of $\\CH$ onto the span of $\\{\\,\n  C_{m_{\\lambda, z}}'\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+\\,\\}$, that is,\n  \\[\n  \\chi^{IJ}(C_x')=\\begin{cases} C_{\\lambda,z}'& \\text{if $x=m_{\\lambda,z}$\n      for some $z\\in \\WIJ$, $\\lambda\\in X(T)_z^+$,} \\\\ 0&\\text{otherwise.}\n  \\end{cases}\n  \\]\n\\end{proposition}\n\n\\begin{proof}\n  We need to show that $\\{\\, C_{\\lambda, z}'\\mid z\\in \\WIJ,\\ \\lambda\\in\n  X(T)_z^+\\,\\}$ is a basis of $\\CHIJ$. Suppose $x$ is in $\\Wex$, then\n  applying $\\chi^{IJ}$ to both sides of the formula for $C_x'$ given\n  in~\\eqref{eq:cx'} we obtain\n  \\[\n  C_{w_I}C_x' C_{w_J}=\\sum_{y\\leq x} v_x^{-1} P_{y,x} C_{w_I}T_y C_{w_J}.\n  \\]\n  If the left-hand side is non-zero, then it is $\\chi^{IJ}( C_{m_{\n      \\lambda,z}}') = C_{\\lambda,z}'$ for some $z$ and $\\lambda$. We've\n  observed that $C_{w_I}T_y C_{w_J}= \\pm C_{w_I} T_{\\lambda', z'} C_{w_J}$\n  when $y$ is in $W_Im_{\\lambda', z'}W_J$. Moreover, it follows from the\n  definition of the Bruhat order on $\\Wex$ and~\\cite[Lemma 2.2]\n  {douglass:inversion} that if $y<x$ and $y$ is in $W_Im_{\\lambda', z'}W_J$,\n  then $m_{\\lambda', z'}< m_{\\lambda, z}$. Therefore,\n  \\begin{equation}\n    \\label{eq:mob}\n    C_{\\lambda, z}'= v_{m_{\\lambda,z}}\\inverse T_{\\lambda,z} +\n    \\sum_{m_{\\lambda',z'}< m_{\\lambda, z} } p_{\\lambda', z'} T_{\\lambda', z'}\n  \\end{equation}\n  for some polynomials $p_{\\lambda', z'}$ in $A$. It follows that the system\n  of equations~\\eqref{eq:mob} indexed by pairs $(z, \\lambda)$ can be solved\n  for $\\{\\, T_{\\lambda,z} \\mid z\\in\\WIJ,\\ \\lambda\\in X(T)^+_z\\,\\}$ and so\n  $\\{\\, C_{\\lambda, z}'\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+\\,\\}$ is a basis\n  of $\\CHIJ$. \\qed\n\\end{proof}\n\nWe call $\\{\\, C_{\\lambda, z}'\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+\\,\\}$ the\n\\emph{Kazhdan-Lusztig basis} of $\\CHIJ$. The elements of the Kazhdan-Lusztig\nbasis have the standard uniqueness property, analogous to~\\eqref{eq:cx'}.\nThe proof of~\\cite[Theorem 1.1] {kazhdanlusztig:coxeter} is easily adapted\nto prove the following (see~\\cite[Claim 2.4]{soergel:kazhdan}).\n\n\\begin{proposition}\n  For $z$ in $\\WIJ$ and $\\lambda$ in $X(T)_z^+$, $C_{\\lambda,z}'$ is the\n  unique element in $\\CHIJ$ such that\n  \\begin{equation*}\n    \\label{eq:cxij'}\n    \\begin{cases}\n      \\overline{C_{\\lambda,z}'}= C_{\\lambda,z}'\\quad \\text{and} \\\\\n      C_{\\lambda,z}'=v_{m_{\\lambda,z}}\\inverse T_{\\lambda,z}+\n      \\sum_{m_{\\lambda',z'} < m_{\\lambda,z}} v_{m_{\\lambda,z}} \\inverse\n      P_{(\\lambda', z'),(\\lambda,z)}T_{\\lambda',z'},\n    \\end{cases}\n  \\end{equation*}\n  where $P_{(\\lambda', z'),(\\lambda,z)}$ is a polynomial in $v$ of degree\n  at most $\\ell(m_{\\lambda,z})-\\ell(m_{\\lambda',z'}) -1$.\n\\end{proposition}\n\nTo summarize, we have the standard basis \n\\[\n\\{\\, T_{\\lambda,z}\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+ \\,\\} = \\{\\,\n\\chi^{IJ}(T_{m_{\\lambda,z}})\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+ \\,\\},\n\\]\nthe Bernstein basis\n\\[\n\\{\\, C_{w_I} \\theta_\\lambda T_z C_{w_J}\\mid z\\in \\WIJ,\\, \\lambda \\in\nX(T)_{z} ^+ \\,\\} = \\{\\, \\chi^{IJ}(\\theta_\\lambda T_z)\\mid z\\in \\WIJ,\\,\n\\lambda \\in X(T)_{z} ^+ \\,\\},\n\\]\nand the Kazhdan-Lusztig basis\n\\[\n\\{\\, C_{\\lambda, z}'\\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+\\,\\} = \\{\\,\n\\chi^{IJ}(C_{m_{\\lambda, z}}') \\mid z\\in \\WIJ,\\ \\lambda\\in X(T)_z^+\\,\\}\n\\]\nof $\\CHIJ$. Applying $\\psi^{IJ}$ to each of these bases we obtain standard,\nBernstein, and Kazhdan-Lusztig bases of $K^{\\Gbar}(\\XIJ)$.\n\n\\section{Intersection\/Tor products} \\label{sec:kthy}\n\nIn this section we formalize the intersection\/Tor product\nconstructions described in~\\cite[\\S6.7]{lusztig:bases} in the form\nthey are used in this paper.\n\n\\subsection{The basic construction}\n\nSuppose that $V$, $A$, $B$, and $C$ are $H$-varieties such that $V$ is\nsmooth. Let $V_1$ and $V_2$ be closed, $H$-stable subvarieties of $V$\nand let $p_{1}\\colon V_1\\to A$, $p_{2}\\colon V_2\\to B$, and\n$p_{12}\\colon V_1\\cap V_2\\to C$ be $H$-equivariant maps such that\n$p_1$ and $p_2$ are smooth and $p_{12}$ is proper. These spaces and\nmaps can be assembled in the commutative diagram\n\\begin{equation}\n  \\label{eq:tor} \n  \\vcenter{\\vbox{\n      {\\tiny\n        \\xymatrix{ C&& B \\\\\n          & V_1\\cap V_2\\ar[ul]_{p_{12}} \\ar[r]\\ar[d] &V_2 \\ar[u]^{p_{2}}\n          \\ar[d]  \\\\ \n          A  &V_1 \\ar[l]_{p_{1}} \\ar[r]&V,} \n      } \n    }}\n\\end{equation}\nwhere the unlabeled maps are inclusions.\n\nThe intersection\/Tor-product construction described by Lusztig\n\\cite[\\S6.4]{lusztig:bases} can be used to define an $R(H)$-bilinear map\n$\\star\\colon K^{H}(A) \\times K^H(B) \\to K^H(C)$ as in\n\\cite[\\S7]{lusztig:bases}. Precisely, for $\\CF$ and $\\CG$ in $K^H(A)$ and\n$K^H(B)$, respectively, define\n\\begin{equation*}\n  \\label{eq:star} \n  \\CF \\star \\CG = (p_{12})_* \\left(p_{1}^*\\,\\CF \\otimes_{V}^L p_{2}^*\\,\n    \\CG\\right) \\in K^{H}(C) ,\n\\end{equation*}\nwhere the Tor-product $\\otimes_{V}^L$ is with respect to the smooth\nvariety $V$ and its closed subvarieties $V_1$ and $V_2$.\n\nNext suppose that $V'$, $A'$, $B'$, $C'$, $V_1'$, $V_2'$, $p_{1}'$,\n$p_{2}'$, and $p_{12}'$ are as in~\\eqref{eq:tor}. Then we have a commutative\ndiagram analogous to~\\eqref{eq:tor} that we connote\nby~$\\text{\\eqref{eq:tor}}'$, and an $R(H)$-bilinear map $\\star'\\colon\nK^{H}(A') \\times K^H(B') \\to K^H(C')$.\n\nBy a \\emph{morphism} from~\\eqref{eq:tor} to~$\\text{\\eqref{eq:tor}}'$ we mean a\ntuple $(\\mu,f,g,h)$ of $H$-equivariant morphisms where $\\mu\\colon V\\to V'$,\n$f\\colon A\\to A'$, $g\\colon B\\to B'$, and $h\\colon C\\to C'$ satisfy the\nfollowing conditions.\n\\begin{itemize}\n\\item $\\mu$ is smooth, $V_1\\subseteq \\mu\\inverse(V_1')$, and $V_2\\subseteq\n  \\mu\\inverse(V_2')$.\n\n  \\noindent Let $\\mu_1$, $\\mu_2$, and $\\mu_{12}$ denote the restrictions of\n  $\\mu$ to maps $V_1\\to V_1'$, $V_2\\to V_2'$, and $V_1\\cap V_2\\to V_1' \\cap\n  V_2'$, respectively.\n\n\\item $fp_1= p_1'\\mu_1$, $gp_2= p_2'\\mu_2$, and $hp_{12}=\n  p_{12}'\\mu_{12}$. \n\\end{itemize}\nThen the following diagram commutes:\n\\begin{equation}\\label{eq:dmap} {\\tiny\n\\xymatrix@1@!@R=0em@C=0em{%\n&&&& B \\ar[dr]^{g}&\\\\\nC\\ar[dr]_{h}&&&&&B'\\\\\n&C'&V_1\\cap V_2 \\ar[ull]_{p_{12}} \\ar[rr] \\ar[dr]^{\\mu_{12}}\n\\ar[dd]|(.25)\\hole &&V_2 \\ar[uu]^{p_{2}} \\ar[dr]_{}^{\\mu_2}\n\\ar'[d][dd]^{}&\\\\ \n&&&V_1'\\cap V_2' \\ar[ull]^(.7){p_{12}'} \\ar[dd]\n\\ar[rr]&&V_2' \\ar[uu]_{p_{2}'} \\ar[dd]^(.35){}\\\\\nA\\ar[dr]_{f}&&V_1 \\ar[ll]_{p_{1}} %\n\\ar[rr]_(.35){}|!{[r];[r]}\\hole\\ar[dr]^{\\mu_1}&&V \\ar[dr]^{\\mu}&\\\\\n&A'&&V_1'\\ar[ll]^{p_{1}'} \\ar[rr]^{}&&V' .\n} }\n\\end{equation}\n \nIf $f$ is flat, $g$ and $h$ are proper, $V_1= \\mu\\inverse(V_1')$, and the\ndiagram\n\\[ \n\\small\n\\xymatrix{ B\\ar[r]^{g} & B' \\\\\n  V_2 \\ar[u]^{p_{2}} \\ar[r]^{\\mu_2} &V_2' \\ar[u]_{p_2'}}\n\\]\nis cartesian, then $(\\mu, f,g,h)$ is a \\emph{proper} morphism. If $f$, $g$,\nand $h$ are smooth, $V_1= \\mu\\inverse(V_1')$, $V_2= \\mu\\inverse(V_2')$, and\nthe diagram\n\\[\n\\small\n\\xymatrix@1{C \\ar[r]^-{h} & C' \\\\\n  V_1\\cap V_2 \\ar[r]^-{\\mu_{12}} \\ar[u]^-{p_{12}} &V_1'\\cap V_2'\n  \\ar[u]^-{p_{12}'} }\n\\]\nis cartesian, then $(\\mu, f,g,h)$ is a \\emph{smooth} morphism.\n\nThe next proposition is a restatement of \\cite[\\S6.5, 6.6, and\n6.7]{lusztig:bases}.\n\n\\begin{proposition}\\label{pro:convlin}\n  Suppose $\\CF'$ is in $K^H(A')$, $\\CG'$ is in $K^H(B')$, $\\CF$ is in\n  $K^H(A)$, and $\\CG$ is in $K^H(B)$.\n  \\begin{enumerate}\n  \\item If $(\\mu, f,g,h)$ is proper, then $h_* \\left( f^* \\CF' \\star \\CG\n    \\right) =\\CF'\\star' g_*\\,\\CG $ in $K^H(C')$. \\label{i:conv1}\n  \\item If $(\\mu, f,g,h)$ is smooth, then $h^* \\left( \\CF' \\star' \\CG'\n    \\right) =f^*(\\CF')\\star g^*(\\CG') $ in $K^H(C)$. \\label{i:conv2}\n  \\item If $B=V_2=V$, $p_2=\\id$, and $\\BBC_V$ is the trivial line bundle on\n    $V$, then $\\CF \\star [\\BBC_V]= (p_{12})_* p_1^* (\\CF) $ in\n    $K^{H}(C)$. \\label{i:conv3}\n  \\end{enumerate}\n\\end{proposition}\n\n\\subsection{Applications}\n\nIn this subsection we apply the product constructions above in the\nsetting of Steinberg varieties.\n\nAs a first application we consider the $K^{\\Gbar}(Z)$-module structure on\n$K^{\\Gbar}(\\XEJ)$ used in the construction of $\\psi^{IJ}$. If $Y$ is a\nsubvariety of $Z$, let\n\\[\n\\Ytilde=\\{\\, ((x, gB), (x, hB))\\mid (x, gB, hB)\\in Y\\,\\}\n\\]\ndenote the image of $Y$ in $\\FNt\\times \\FNt$ under the natural embedding\n$Z\\to \\FNt\\times \\FNt$.\n\nAs in~\\cite[\\S7.9]{lusztig:bases}, it is easy to see that taking $V=\\FNt^3$,\n$V_1= \\Ztilde\\times \\FNt$, $V_2= \\FNt \\times \\Ztilde$, and using the obvious\nprojections $V_1\\to Z$ and $V_2\\to Z$ in~\\eqref{eq:tor}, we obtain a\n``convolution product''\n\\[\n\\star\\colon K^{\\Gbar}(Z) \\times K^{\\Gbar}(Z) \\to K^{\\Gbar}(Z)\n\\]\nthat defines an $R(\\Gbar)$-algebra structure on $K^{\\Gbar}(Z)$.\n\nDefine $\\FNt^J=\\{\\, (x, gP_J)\\in \\FN \\times G\/P_J \\mid g\\inverse x\\in\n\\fp_J\\,\\}$ and\n\\[\n\\Xtilde^{\\emptyset J} =\\{\\, ((x, gB), (x, hP_J))\\in \\FNt \\times (\\fg \\times\nG\/P_J)\\mid (x, gB, hB)\\in \\XEJ\\,\\}.\n\\]\nThen taking $V= \\FNt^2 \\times (\\fg\\times G\/P_J)$, $V_1= \\Ztilde\\times (\\fg\n\\times G\/P_J)$, $V_2= \\FNt \\times \\Xtilde^{\\emptyset J}$, $p_{1}$ to be the\ncomposition $V_1\\to \\Ztilde \\to Z$, $p_{2}$ to be the composition $V_2\\to\n\\Xtilde^{\\emptyset J} \\to \\XEJ$, and defining $p_{12}\\colon V_1\\cap V_2\\to\n\\XEJ$ by $p_{12} ((x,gB), (x,hB), (x,kP_J))= (x, gB, kP_J)$\nin~\\eqref{eq:tor}, we obtain an $R(\\Gbar)$-module linear map\n\\[\n\\star_J\\colon K^{\\Gbar}(Z) \\times K^{\\Gbar}(\\XEJ) \\to K^{\\Gbar}(\\XEJ)\n\\]\nthat defines a $K^{\\Gbar}(Z)$-module structure on $K^{\\Gbar}(\\XEJ)$.\n\nDefine $\\eta\\colon \\FNt^3\\to \\FNt^2\\times (\\fg\\times G\/P_J)$ by \n\\[\n\\eta((x, gB), (y, hB), (z, kB)) = ((x, gB), (y, hB), (z, kP_J)).\n\\]\nThen as in diagram~\\eqref{eq:dmap}, the tuple $(\\eta, \\id, \\etaEJ, \\etaEJ)$\ndefines a proper morphism from the diagram defining the multiplication on\n$K^{\\Gbar}(Z)$ to the diagram defining the $K^{\\Gbar}(Z)$-module structure\non $K^{\\Gbar}(\\XEJ)$. The next proposition follows\nfrom~\\autoref{pro:convlin}\\,\\autoref{i:conv1}.\n\n\\begin{proposition}\\label{pro:xlin}\n  Suppose $J$ is a subset of $S$. There is a left $K^{\\Gbar}(Z)$-module\n  structure on $K^{\\Gbar}(\\XEJ)$ such that $\\etaEJ_* \\colon K^{\\Gbar}(Z) \\to\n  K^{\\Gbar}(\\XEJ)$ is $K^{\\Gbar}(Z)$-linear.\n\\end{proposition}\n\nAs a second application we consider $K^{\\Gbar}(Z_1)$-module structures\non $K^{\\Gbar}(Z_{\\ssleq w})$ and $K^{\\Gbar}(Z_{w})$ for $w$ in\n$W$. Notice that $Z_1=\\{\\, (x, gB, gB)\\in Z \\mid g\\inverse x\\in \\fu\\}$\nis a closed subvariety of $Z$ isomorphic to $\\FNt$. Thus, $Z_1$ is\nsmooth and so $K^{\\Gbar}(Z_1)$ has a natural ring structure given by\ntensor product.\n\nSuppose $w$ is in $W$. Taking $V= \\FNt^3$, $V_1= \\Ztilde_1\\times \\FNt$,\n$V_2= \\FNt \\times \\Ztilde_{\\ssleq w}$, and using the obvious projections\n$V_1\\to Z_1$ and $V_2\\to Z_{\\ssleq w}$ in~\\eqref{eq:tor}, we obtain a\n$K^{\\Gbar}(Z_1)$-module structure on $K^{\\Gbar}(Z_{\\ssleq w})$ that is\ndenoted by $\\star_w$,\n\\[\n\\star_w\\colon K^{\\Gbar}(Z_1) \\times K^{\\Gbar}(Z_{\\ssleq w}) \\to\nK^{\\Gbar}(Z_{\\ssleq w}).\n\\]\nNotice that when $w=w_0$ is the longest element in $W$ we get a\n$K^{\\Gbar}(Z_1)$-module structure on $K^{\\Gbar}(Z)$. To simplify the\nnotation slightly, set $\\star_1=\\star_{w_0}$.  It follows\nfrom~\\autoref{pro:convlin}\\,\\autoref{i:conv1} that this module structure is\ncompatible, via the inclusion $j_1\\colon Z_1\\to Z$, with the multiplication\n$\\star$ in $K^{\\Gbar}(Z)$, in the sense that\n\\begin{equation}\n  \\label{eq:star1}\n  \\CF\\star_1 \\CG= (j_1)_*\\CF \\star \\CG\n\\end{equation}\nfor $\\CF$ in $K^{\\Gbar}(Z_1)$ and $\\CG$ in $K^{\\Gbar}(Z)$.\n\nSimilarly, taking $V=\\FNt\\times \\Ztilde_w$, $V_1= (\\Ztilde_1\\times \\FNt)\n\\cap (\\FNt \\times \\Ztilde_{ w})$, $V_2= \\FNt \\times \\Ztilde_{ w}$, and using\nthe obvious projections $V_1\\to Z_1$ and $V_2\\to Z_w$, we obtain a\n$K^{\\Gbar}(Z_1)$-module structure on $K^{\\Gbar}(Z_w)$ that is\ndenoted by $\\star_w'$,\n\\[\n\\star_w'\\colon K^{\\Gbar}(Z_1) \\times K^{\\Gbar}(Z_w) \\to K^{\\Gbar}(Z_w).\n\\]\n\nNow suppose $y$ and $w$ are in $W$ with $y\\leq w$. Let $j_y^w\\colon\nZ_{\\ssleq y}\\to Z_{\\ssleq w}$ be the inclusion. In the special case when\n$w=w_0$ set $j_y=j_y^{w_0}$. Then $j_y$ is the inclusion of $Z_{\\ssleq y}$\nin $Z$. Recall that $r_w\\colon Z_w\\to Z_{\\ssleq w}$ is the inclusion. Notice\nthat $j_y^w$ is a closed embedding and that $r_w$ is an open embedding. The\nproof of the next lemma is a straightforward application\nof~\\autoref{pro:convlin}.\n\n\\begin{lemma}\\label{lem:starlin}\n   Suppose $y$ and $w$ are in $W$ with $y\\leq w$. Then the maps\n   \\[\n   (j_y^w)_*\\colon K^{\\Gbar}(Z_{\\ssleq y}) \\to K^{\\Gbar}( Z_{\\ssleq w})\n   \\quad \\text{and}\\quad r_w^* \\colon K^{\\Gbar}(Z_{\\ssleq w}) \\to K^{\\Gbar}\n   (Z_w)\n   \\]\n   are $K^{\\Gbar}(Z_1)$-module homomorphisms.\n\\end{lemma}\n\n\\section{The \\texorpdfstring{$K^{\\Gbar}(Z)$}{}-module homomorphism\n  \\texorpdfstring{$\\eta^{\\emptyset J}_* \\colon K^{\\Gbar}(Z)\\to\n    K^{\\Gbar}(\\XEJ)$}{}} \\label{sec:as}\n\nIn this section we prove~\\autoref{thm:as}, the assertion that\n$\\etaEJ_*(\\mathbf a_s)= 0$ for $s$ in $J$, and thus complete the argument\nthat there is an $R(\\Gbar)$-module homomorphism $\\psi^{IJ}\\colon \\CHIJ\\to\nK^{\\Gbar}(\\XIJ)$ such that $\\psi^{IJ} \\chi^{IJ}= \\etaIJ \\varphi$ as\nin~\\autoref{thm:main}.\n\nFor a subset $I$ of $S$, define $\\CO_I$ to be the orbit $G\\cdot (B, w_IB)$\nin $G\/B \\times G\/B$ and define $Z_I=Z_{w_I}$ in $Z$.\n \n\\begin{proposition}\n  Suppose $I$ is a subset of $S$.\n  \\begin{enumerate}\n  \\item The closure of $\\CO_I$ in $G\/B\\times G\/B$ is\n    \\[\n    \\OIbar= \\coprod_{w\\in W_I} \\CO_w = \\{\\, (gB, hB) \\mid g\\inverse h\\in\n    P_I\\,\\}.\n    \\] \\label{i:1}\n  \\item The closure of $Z_I$ in $Z$ is  \n    \\begin{align*}\n      \\ZIbar&=\\{(x, gB, hB)\\in Z\\mid (gB,hB)\\in \\OIbar,\\, g\\inverse x\\in\n      \\fu_I,\\, h\\inverse x\\in \\fu_I\\,\\}\\\\\n      &=\\{(x, gB, hB)\\in Z\\mid g\\inverse h\\in P_I,\\, g\\inverse x\\in\n      \\fu_I\\,\\}.\n    \\end{align*} \\label{i:2}\n  \\item There are isomorphisms $\\OIbar \\cong G\\times^{B} P_I\/B$ and $\n    \\ZIbar \\cong G\\times^{B} (\\fu_I\\times P_I\/B)$ such that the\n    diagram\n    \\begin{equation}\n      \\label{eq:new}\n      \\vcenter{\\vbox{\n          \\xymatrix{ G\\times^{B} (\\fu_I\\times P_I\/B)\\ar[r]_-{\\cong} \\ar[d] &\n            \\ZIbar \\ar[d]^{q_I} \\\\\n            G\\times^{B} P_I\/B \\ar[r]_-{\\cong} & \\OIbar }    \n        }}\n    \\end{equation}\n    commutes. Here, the left vertical map is the obvious one and $q_I$ is\n    the projection on the second and third coordinates. In particular, the\n    varieties $\\OIbar$ and $\\ZIbar$ are smooth and the projection $q_I\n    \\colon \\ZIbar\\to \\OIbar$ is a vector bundle with fibre\n    $\\fu_I$. \\label{i:3}\n  \\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n  The first equality in~\\autoref{i:1} is well known. For the second equality,\n  set\n  \\[\n  Y_1= \\{\\, (gB, hB) \\mid g\\inverse h\\in P_I\\,\\}.\n  \\] \n  Given $g$ in $G$, $(gB, gw_IB)$ is obviously in $Y_1$ so $\\CO_I\\subseteq\n  Y_1$. Thus $\\OIbar\\subseteq Y_1$. Conversely, if $(gB, hB)$ is in $Y_1$,\n  then there is a $w$ in $W_I$ so that $g\\inverse h$ is in $BwB$. Say\n  $g\\inverse h=b_1wb_2$. Then $(gB, hB)= (gB, gb_1 w B)$ is in $\\CO_w$. This\n  proves the first statement.\n\n  The second equality in~\\autoref{i:2} follows from~\\autoref{i:1}. Set\n  \\[\n  Y_2= \\{(x, gB, hB)\\in Z\\mid g\\inverse h\\in P_I,\\, g\\inverse x\\in\n  \\fu_I\\,\\}.\n  \\] \n  Then $Y_2$ is closed in $Z$ and the projection $Y_2\\to \\OIbar$ is a vector\n  bundle with fibre $\\fu_I$. It follows that $Y_2$ is irreducible. Since\n  \\[\n  Z_I=\\{\\, (x, gB, gw_IB) \\mid g\\inverse x\\in \\fu\\cap w_I\\fu\\,\\}\n  \\]\n  and $\\fu_I=\\fu\\cap w_I\\fu$, we see that $Z_I$, and hence $\\ZIbar$, is\n  contained in $Y_2$. Since $\\ZIbar$ is an irreducible component of $Z$ and\n  $Y_2$ is closed and irreducible we have $\\ZIbar= Y_2$. This proves the\n  second statement.\n\n  For each of the varieties $\\OIbar$ and $\\ZIbar$, projection on the\n  first factor of $G\/B$ is a $G$-equivariant surjection and induces\n  $G$-equivariant isomorphisms\n  \\[\n  \\OIbar \\cong G\\times^{B} P_I\/B\\qquad \\text{and} \\qquad \\ZIbar \\cong\n  G\\times^{B} (\\fu_I\\times P_I\/B).\n  \\]\n  It is straightforward to check that diagram~\\eqref{eq:new} commutes.  This\n  proves the last statement. \\qed\n\\end{proof}\n\nSuppose $s$ is in $S$ and consider the vector bundle $q_I\\colon \\ZIbar \\to\n\\OIbar$ in the case when $I=\\{s\\}$.  Let $i_s\\colon \\Osbar\\to G\/B\\times G\/B$\nand $k_s\\colon \\Zsbar\\to Z$ be the inclusions and set $q_s=q_{\\{s\\}}$. Then\nthe diagram of $\\Gbar$-equivariant morphisms\n\\begin{equation*}\n \n  \\vcenter{\\vbox{\n      \\xymatrix{ \\Zsbar \\ar[d]^{q_s} \\ar[r]^-{k_s} & Z \\ar[r]^-{\\ktilde} &\n        \\FNt \\times G\/B \\ar[d]^{q\\times \\id} \\\\  \n        \\Osbar \\ar[rr]^-{i_s} && G\/B \\times G\/B,}  \n    }}\n\\end{equation*}\nwhere $\\ktilde$ is the inclusion, commutes. Given $\\lambda', \\lambda''$ in\n$X(T)$, set\n\\[\n\\CL_{\\lambda', \\lambda''}= (k_s)_* q_s^* i_s^* \\left( [\\CL_{\\lambda'}\n  \\boxtimes \\CL_{\\lambda''}] \\right) = (k_s)_* k_s^*  \\ktilde^* (q\\times \\id)^*\n\\left( [\\CL_{\\lambda'} \\boxtimes \\CL_{\\lambda''}] \\right)\n\\]\nin $K^{\\Gbar}(Z)$.\n\n\\begin{proposition}\\label{pro:special}\n  Suppose $J\\subseteq S$, $\\alpha\\in \\Pi_J$, $s=s_\\alpha$, $\\lambda''\\in\n  X(T)$ with $\\langle \\lambda'', \\check \\alpha \\rangle =-1$, and\n  $\\lambda'\\in X(T)$. Then $\\etaEJ_* \\left( [\\CL_{\\lambda',\\lambda''}]\n  \\right) =0$ in $K^{\\Gbar}(\\XEJ)$.\n\\end{proposition}\n\n\\begin{proof}\n  Set $P_s=P_{\\{s\\}}$, $\\fu_s=\\fu_{\\{s\\}}$, and define\n  \\[\n  Y_s= \\eta^{\\emptyset \\{s\\}}(\\Zsbar)= \\{\\, (x, gB, gP_s)\\in \\FN\n  \\times G\/B \\times G\/P_s \\mid g\\inverse x\\in \\fu_{s} \\,\\}.\n  \\]\n  We have a commutative diagram of proper morphisms,\n  \\[\n  \\xymatrix{ G\\times^{B} (\\fu_{s} \\times P_{s}\/B) \\ar[r]_-{\\cong}\n    \\ar[d]&  \\Zsbar \\ar[d]^{\\eta'} \\ar[r]^-{k_s} & Z \\ar[d]^{\\etaEJ} \\\\\n    G\\times^{B} \\fu_{s}\\ar[r]_-{\\cong} & Y_s \\ar[r]^{\\eta^J} & \\XEJ,}\n  \\]\n  where the isomorphism $G\\times^{B} \\fu_s \\xrightarrow{\\ \\cong\\ }\n  Y_s$ is induced by projection on the second factor, $\\eta'(x, gB,\n  hB)= (x, gB, hP_s)$, and $\\eta^J(x, gB, hP_s)= (x, gB,\n  hP_J)$. Notice in particular that $\\eta'$ is a Zariski locally\n  trivial $\\BBP^1$-bundle.\n\n  Set $\\CL^s_{\\lambda', \\lambda''}= q_s^* i_s^* \\left( \\CL_{\\lambda'}\n    \\boxtimes \\CL_{\\lambda''} \\right)$. To prove the proposition it is\n  enough to show that $\\eta'_*([\\CL^s_{\\lambda', \\lambda''}])= 0$ in\n  $K^{\\Gbar}(Y_s)$. We show that $R^n \\eta'_* \\CL^s_{\\lambda',\n    \\lambda''}|_y=0$ for all $y$ in $Y_s$ and for all $n\\geq 0$. It\n  then follows that $R^n \\eta'_* \\CL^s_{\\lambda', \\lambda''}=0$ for\n  all $n\\geq 0$ and so $\\eta'_*([\\CL^s_{\\lambda', \\lambda''}])= 0$ by\n  the definition of $\\eta'_*$.\n\n  Suppose $y$ is in $Y_s$. Because $\\eta'$ is $G$-equivariant we may\n  assume $y= (x, B, P_s)$ for some $x$ in $\\fu_s$. Define\n  $E=(\\eta')\\inverse(y)= \\{\\, (x, B, pB)\\mid p\\in P_s\\,\\}$ and let\n  $f\\colon E\\to \\Zsbar$ be the inclusion mapping. Projection on the\n  third factor defines an isomorphism $E \\cong P_s\/B$ and so $E\\cong\n  \\BBP^1$.\n\n  The restriction of $\\CL^s_{\\lambda',\\lambda''}$ to $E$ is given by\n  \\[\n  \\CL^s_{\\lambda',\\lambda''} |_{E} = (i_sq_sf)^* (\\CL_{\\lambda'} \\boxtimes\n  \\CL_{\\lambda''})= (p_1i_sq_sf)^* \\CL_{\\lambda'} \\otimes (p_2i_sq_sf)^*\n  \\CL_{\\lambda''}\n  \\]\n  where $p_1$ and $p_2$ are the two projections of $G\/B \\times G\/B$\n  onto $G\/B$. The composition $p_1 i_s q_s f$ is a constant map, so\n  $\\CL_{\\lambda'}$ pulls back to the trivial line bundle\n  $\\CO_{\\BBP^1}(0)$ on $E$. The map $p_2 i_s q_s f$ restricts to an\n  isomorphism $E \\cong P_s\/B$, and because $\\langle \\lambda'', \\check\n  \\alpha \\rangle =-1$, $\\CL_{\\lambda''}$ pulls back to the\n  tautological bundle on $P_s\/B$.  Therefore,\n  $\\CL^s_{\\lambda',\\lambda''} |_{E}$ may be identified with the line\n  bundle $\\CO_{\\BBP^1}(-1)$ on $\\BBP^1$.\n\n  Finally, for $n\\geq 0$, $H^n( E, \\CL^s_{\\lambda', \\lambda''} |_{E}) \\cong\n  H^n( \\BBP^1, \\CO_{\\BBP^1}(-1)) =0$, and hence by Grauert's Theorem, $R^n\n  \\eta'_* \\CL^s_{\\lambda', \\lambda''}|_y\\cong H^n( E, \\CL^s_{\\lambda',\n    \\lambda''} |_{E}) =0$ as claimed. \\qed\n\\end{proof}\n\nWe can now complete the proof of~\\autoref{thm:as}. With $J$, $\\alpha$, and\n$s$ as above, suppose that $\\lambda'$ and $\\lambda''$ are in $X(T)$,\n$\\langle\\lambda ', \\check \\alpha \\rangle= \\langle\\lambda '', \\check \\alpha\n\\rangle= -1$, and $-\\alpha=\\lambda'+\\lambda''$. Define $\\mathbf a_s$ in\n$K^{\\Gbar}(Z)$ by\n\\[\n\\mathbf a_s= v\\inverse \\, \\CL_{\\lambda', \\lambda''} = v\\inverse (k_s)_*\nq_s^* i_s^* \\left( [\\CL_{\\lambda'} \\boxtimes \\CL_{\\lambda''}] \\right).\n\\]\nLusztig \\cite[\\S7.19, 7.25]{lusztig:bases} has shown that $\\mathbf a_s$ does\nnot depend on the choice of $\\lambda'$ and $\\lambda''$. It follows\nfrom~\\autoref{pro:special} that $\\etaEJ_*(\\mathbf a_s)= 0$, as asserted in\nthe statement of the theorem.\n\n\\section{An \\texorpdfstring{$A$}{}-basis of\n  \\texorpdfstring{$\\CHIJ$}{}} \\label{sec:basis}\n\nBy~\\autoref{lem:double}, $\\{\\, t_\\lambda z\\mid z\\in \\WIJ,\\, \\lambda \\in\nX(T)_{z} ^+ \\,\\}$ is a set of $(W_I,W_J)$-double coset representatives in\n$\\Wex$. In this section we prove~\\autoref{thm:hijbasis} and thus show that\n$\\CHIJ$ has an $A$-basis indexed by $W_I\\backslash \\Wex \/W_J$.\n\nWe need to show that the set $\\CE=\\{\\, C_{w_I} \\theta_\\lambda T_z C_{w_J}\\mid\nz\\in \\WIJ,\\, \\lambda \\in X(T)_{z} ^+ \\,\\}$ is an $A$-basis of $\\CHIJ$.  To\nsee that $\\CE$ is linearly independent, notice that the specialization\n$v\\mapsto 1$ defines an $A$-linear homomorphism\n\\[\n\\CHIJ \\to \\epsilon_I \\BBZ[\\Wex] \\epsilon_J,\n\\]\nwhere as above, $\\epsilon_I=\\sum_{w\\in\n  W_I}\\epsilon_ww$. By~\\autoref{lem:double}, $\\{\\, \\epsilon_I t_\\lambda z\n\\epsilon_J \\mid z\\in \\WIJ,\\, \\lambda \\in X(T)_{z} ^+ \\,\\}$ is linearly\nindependent in $\\epsilon_I \\BBZ[\\Wex] \\epsilon_J$ and so $\\CE$ is linearly\nindependent in $\\CHIJ$.\n\nTo prove that $\\CE$ spans $\\CHIJ$ we adapt an argument of Nelsen and Ram\n\\cite{nelsenram:kostka}.  It follows from the relation\n\\begin{equation}\n  \\label{eq:8}\n  \\theta_\\lambda T_s -T_s \\theta_{s(\\lambda)}= (v^2-1) \\frac\n  {\\theta_\\lambda-\\theta_{s(\\lambda)}} {1-\\theta_{-\\alpha}},    \n\\end{equation}\nfor $\\lambda$ in $X(T)$ and $s=s_\\alpha$ with $\\alpha$ in $\\Pi$ that\nif $w$ is in $W$, then $\\theta_\\lambda T_w$ is in the $A$-span of\n$\\{\\, T_y \\theta_\\mu\\mid y\\leq w,\\ \\mu\\in X(T)\\,\\}$. Therefore, if $w$\nis in $W$ and $w=w_1zw_2$ where $z$ is in $\\WIJ$, $w_1$ in $W_I$,\n$w_2$ in $W_J$, and $\\ell(w)= \\ell(w_1)+\\ell(z)+\\ell(w_2)$, then\n$\\theta_\\lambda T_{w_1zw_2}=\\theta_\\lambda T_{w_1} T_z T_{w_2}$ is an\n$A$-linear combination of elements of the form $T_y \\theta_\\mu T_z\nT_{w_2}$ where $y\\leq w_1$ is in $W_I$. Therefore, the elements\n$C_{w_I} \\theta_\\lambda T_z C_{w_J}$ for $z$ in $\\WIJ$ and $\\lambda$\nin $X(T)$ span $\\CHIJ$.\n\nFix $z$ in $\\WIJ$ and suppose that $s=s_\\alpha$ with $\\alpha$ in $\\Pi_I \\cap\nz\\Pi_J$. Then $T_sT_z=T_{sz}=T_{z z\\inverse sz}= T_z T_{s'}$, where\n$s'=z\\inverse sz$ is in $J$. Thus, multiplying~\\eqref{eq:8} on the right\nby $T_z$ and then taking the image in $\\CHIJ$, we have\n\\begin{equation}\n  \\label{eq:9}\n  C_{w_I} \\left( -\\theta_\\lambda T_z +\\theta_{s(\\lambda)}T_z \\right)\n  C_{w_J} = C_{w_I} \\left( (v^2-1) \\frac\n    {\\theta_\\lambda-\\theta_{s(\\lambda)}} {1-\\theta_{-\\alpha}} T_z \\right)\n  C_{w_J}  \n\\end{equation}\nfor any $\\lambda$ in $X(T)$. Replacing $\\lambda$ by $\\lambda-\\alpha$,\nsubtracting the result from~\\eqref{eq:9}, and simplifying, we get\n\\begin{equation}\n  \\label{eq:x4}\n  C_{w_I} \\theta_{s(\\lambda)} T_z C_{w_J}  = \n  C_{w_I} \\left( v^2 \\theta_{\\lambda} - \\theta_{\\lambda-\\alpha} + \n    v^2\\theta_{s(\\lambda) +\\alpha} \\right) T_z C_{w_J}. \n\\end{equation}\nThe derivation of~\\eqref{eq:x4} from~\\eqref{eq:8} is the same as in the\nproof of \\cite[Proposition 2.1]{nelsenram:kostka}. After relabeling the\nelements in $\\CH$ and replacing $t$ by $v^2$,~\\eqref{eq:x4} is the same as\nthe recursion formula at the end of the proof of \\cite[Proposition\n2.1]{nelsenram:kostka}. It follows that if $\\lambda$ is in $X(T)$ with\n$d=\\langle \\lambda, \\check \\alpha \\rangle \\geq 0$, then there are\npolynomials $p_0$, $p_1$, \\dots, $p_{\\lfloor d\/2 \\rfloor}$ in $\\BBZ[v^2]$\nsuch that\n\\begin{equation}\n  \\label{eq:x5}\n  C_{w_I} \\theta_{s(\\lambda)} T_zC_{w_J}  = C_{w_I}\n  \\bigg( \\sum_{j=0}^{\\lfloor d\/2\\rfloor} p_j \\theta_{\\lambda -j\\alpha} \n  \\bigg) T_z C_{w_J}. \n\\end{equation}\nThe polynomials $p_j$ do not depend on $\\lambda$ and are given by the\nformula in \\cite[Proposition 2.1]{nelsenram:kostka} with $t$ replaced by\n$v^2$.\n\nNow if $\\mu$ is in $X(T)$ and $\\langle \\mu ,\\check \\alpha \\rangle <0$, then\nusing~\\eqref{eq:x5} with $s(\\lambda)=\\mu$ we see that $C_{w_I} \\theta_\\mu\nT_z C_{w_J}$ may be rewritten as an $A$-linear combination of terms $C_{w_I}\n\\theta_{\\mu'} T_z C_{w_J}$ where $\\langle \\mu', \\check \\alpha \\rangle \\geq\n0$ and $\\langle \\mu' ,\\check \\beta \\rangle \\geq \\langle \\mu ,\\check \\beta\n\\rangle$ for all $s_\\beta$ with $\\beta$ in $\\Pi$. It follows that $\\CE$\nspans $\\CHIJ$, as claimed.\n\n\n\\section{Equivariant \\texorpdfstring{$K$}{}-theory of\n  \\texorpdfstring{$\\XIJ$}{}} \\label{sec:wideiso}\n\nIn this section we compute $K^{\\Gbar}(\\XIJ)$ in terms of $\\CHIJ$, using\n$\\CH$ and $K^{\\Gbar}(Z)$, and thus complete the proof\nof~\\autoref{thm:main}. We begin with $K^{\\Gbar}(\\XIJ_z)$.\n\n\\subsection{\\texorpdfstring{$K^{\\Gbar}(\\XIJ_z)$}{} and the Springer\n  resolution of \\texorpdfstring{$\\SN_z$}{}} \\label{ssec:xijz}\n\nIn this subsection we prove~\\autoref{thm:ost}.\n\nSuppose $z$ is in $\\WIJ$. We need to show that there is a commutative\ndiagram of $R(\\Gbar)$-modules\n\\begin{equation*}\n  \\xymatrix{ K^{\\Gbar}(Z_z) \\ar[r]_{\\cong} \\ar[d]^{\\eta^z_*} &\n    K^{\\Lbar_z}(\\SNt_z) \\ar[d]^{(p_z)_*} \\\\ \n    K^{\\Gbar}(\\XIJ_z) \\ar[r]_{\\cong} & K^{\\Lbar_z}(\\SN_z), }  \n\\end{equation*}\nwhere the horizontal maps are isomorphisms, $p_z$ is the Springer resolution\nof the nilpotent cone $\\SN_z$ of $\\fl_z$, and the vertical maps are\nsurjections.\n\nSet\n\\begin{equation*}\n \n  P_z=P_I\\cap {}^zP_J  \\quad \\text{and} \\quad V_z=(L_I\\cap {}^zU_J)\n  (U_I\\cap {}^zL_J) (U_I\\cap {}^zU_J).  \n\\end{equation*}\nThen \n\\begin{equation}\n  \\label{eq:pz}\n  P_z= L_z V_z=L_z (L_I\\cap {}^zU_J) (U_I\\cap {}^zL_J) (U_I\\cap {}^zU_J),  \n\\end{equation}\n$L_z$ is a Levi subgroup of $P_z$, and $V_z$ is the unipotent radical of\n$P_z$ (see~\\cite[\\S69B] {curtisreiner:methodsII}). The Lie algebra analog of\nthe factorization~\\eqref{eq:pz} is\n\\begin{equation}\n  \\label{eq:qf}\n  \\fp_z= \\fl_z +\\fv_z=\\fl_z+ (\\fl_I\\cap z\\fu_J) +(\\fu_I\\cap z\\fl_J)+\n  (\\fu_I\\cap z\\fu_J).  \n\\end{equation}\nSet $\\fu_z=\\fu\\cap \\fl_z$. It follows from~\\eqref{eq:qf} that $\\fu\\cap\nz\\fu =\\fu_z+\\fv_z$. \n\nThe rule $(x, gB, gzB)\\mapsto (x, gP_I, gzP_J)$ defines a surjective,\n$G$-equivariant morphism from $Z_z$ onto the $G$-orbit of $(P_I, zP_J)$ in\n$G\/P_I \\times G\/P_J$. The fibre over $(P_I, zP_J)$ is\n\\begin{align*}\n  F&=\\{\\, (x, pB, pzB)\\in Z_z \\mid p\\in P_z,\\ p\\inverse\n  x \\in \\fu \\cap z\\fu\\,\\} \\\\\n  &= \\{\\, (x, hB, hzB)\\in Z_z \\mid h\\in L_z,\\ h\\inverse\n  x \\in \\fu_z+\\fv_z\\,\\}.\n\\end{align*}\nThus, $Z_z\\cong G\\times^{P_z}F$. Projection onto the second and third\nfactors is a surjective, $P_z$-equivariant morphism from $F$ onto the\n$P_z$-orbit of $(B, zB)$ in $G\/B \\times G\/B$ with fibre over $(B, zB)$\nisomorphic to $\\fu_z+\\fv_z$. Thus, $F\\cong P_z\\times^{B\\cap {}^zB}\n(\\fu_z+\\fv_z)$.  Define\n\\[\n\\ptilde\\colon F\\to \\SNt_z\\quad \\text{by}\\quad \\ptilde(h(x+v),hB, hzB)=(hx,\nhB_z),\n\\]\nfor $x$ in $\\fu_z$, $v$ in $\\fv_z$, and $h$ in $L_z$, and define\n\\[\ni_z\\colon \\SNt_z\\to F\\quad\\text{by}\\quad i_z(x, hB_z)= (x, hB, hzB).\n\\]\nThen $\\ptilde$ and $i_z$ are well-defined morphisms that may be identified\nwith the morphisms $P_z\\times^{B\\cap {}^zB} (\\fu_z+\\fv_z)\\to L_z\\times^{B_z}\n\\fu_z$ and $L_z\\times^{B_z} \\fu_z \\to P_z\\times^{B\\cap {}^zB} (\\fu_z+\\fv_z)$\ngiven by $hu *(x+v)\\mapsto h*x$ and $h*x\\mapsto h*x$ for $h$ in $L_z$, $u$\nin $V_z$, $x$ in $\\fu_z$, and $v$ in $\\fv_z$. Thus, $\\ptilde \\colon F\\to\n\\SNt_z$ is an $\\Lbar_z$-equivariant vector bundle with fibre $\\fv_z$ and\nzero section $i_z$.\n\nNext, the projection of $\\XIJ_z$ onto the second and third factors is a\nsurjective, $G$-equivariant morphism onto the $G$-orbit of $(P_I, zP_J)$ in\n$G\/P_I \\times G\/P_J$. The fibre over $(P_I, zP_J)$ is\n\\begin{align*}\n  F^{IJ}&=\\{\\, (x, P_I, zP_J)\\in \\XIJ_z \\mid x \\in \\SN_z+\\fv_z \\,\\}.\n\\end{align*}\nThus, $\\XIJ_z \\cong G\\times^{P_z} (\\SN_z+\\fv_z)$ and $F^{IJ}\\cong\n\\SN_z+\\fv_z$. Define\n\\[\n\\ptilde^{IJ}\\colon F^{IJ}\\to \\SN_z\\quad \\text{by}\\quad \\ptilde^{IJ}(x+v,P_I,\nzP_J)=x,\n\\]\nfor $x$ in $\\SN_z$ and $v$ in $\\fv_z$, and define\n\\[\ni_z^{IJ}\\colon \\SN_z\\to F^{IJ} \\quad \\text{by}\\quad i_z^{IJ}(x)= (x, P_I,\nzP_J).\n\\]\nObviously the morphism $\\ptilde^{IJ}$ is an $\\Lbar_z$-equivariant vector\nbundle with fibre $\\fv_z$ and zero section $i_z^{IJ}$.\n\nFinally, let $\\tilde \\eta^z\\colon F\\to F^{IJ}$ be the restriction of\n$\\eta^z$. Then the diagram\n\\begin{equation}\n  \\label{eq:co1}\n  \\vcenter{\\vbox{\n      \\xymatrix{ \\SNt_z\\ar[r]^-{i_z} \\ar[d]^{p_z} & F\\ar@{^{(}->}[r]\n        \\ar[d]^{\\tilde \\eta^z} & Z_z\\ar[d]^{\\eta^z}\\\\ \n        \\SN_z\\ar[r]^-{i_z^{IJ}} & F^{IJ}\\ar@{^{(}->}[r] &\\XIJ_z}  \n    }}\n\\end{equation}\ncommutes. Moreover, one checks that the left-hand square is cartesian.\n\nNow applying $K^{\\Gbar}$ and $K^{\\Lbar_z}$ to~\\eqref{eq:co1} we get\n\\[\n\\xymatrix{K^{\\Gbar}(Z_z) \\ar[r]^-{\\res_z}\\ar[d]^{\\eta^z_*} &\n  K^{\\Lbar_z}(F) \\ar[r]^-{i_z^*} \\ar[d]^{\\tilde \\eta^z_*} &\n  K^{\\Lbar_z}(\\SNt_z) \\ar[d]^{(p_z)_*} \\\\\n  K^{\\Gbar}(\\XIJ_z) \\ar[r]^-{\\res_z^{IJ}} & K^{\\Lbar_z}(F^{IJ})\n  \\ar[r]^-{(i_z^{IJ})^*} & K^{\\Lbar_z}(\\SN_z), }\n\\]\nwhere $\\res_z=\\res_{F}$ is defined using the isomorphism $Z_z\\cong\nG\\times^{P_z}F$ and $\\res_z^{IJ}=\\res_{F^{IJ}}$ is defined using the\nisomorphism $\\XIJ_z\\cong G\\times^{P_z}F^{IJ}$. The left-hand square commutes\nby the naturality of $\\res$. For the right-hand square, the diagram\n\\[\n\\xymatrix{F \\ar[r]^-{\\tilde p} \\ar[d]^{\\tilde \\eta^z} & \\SNt_z\n  \\ar[d]^{p_z} \\\\\n  F^{IJ} \\ar[r]^-{\\tilde p^{IJ}}& \\SN_z}\n\\]\nis cartesian and $\\tilde p$ and $\\tilde p^{IJ}$ are vector bundles, so\n$\\tilde \\eta^z_* \\tilde p^* = (\\tilde p^{IJ})^* (p_z)_*$. By the Thom\nisomorphism in equivariant $K$-theory, $i_z^*= (\\tilde p^*)\\inverse$ and\n$(i_z^{IJ})^* = ((\\tilde p^{IJ})^*)\\inverse$, so $(i_z^{IJ})^* \\tilde\n\\eta^z_*= (p_z)_* i_z^*$. Finally, by~\\autoref{lem:ostlem} $(p_z)_*$ is a\nsurjection. Therefore, $\\tilde \\eta^z_*$ and $\\eta^z_*$ are surjections as\nwell.\n\n\n\\subsection{The sequence (\\ref{eq:exz}) is exact} \\label{ssec:ex}\n\n\\begin{proposition}\\label{pro:k1}\n  Suppose $H$ is a linear algebraic group and that $Y$ is an $H$-variety\n  such that $H$ acts on $Y$ with finitely many orbits. Then $K^H_1(Y)=0$.\n\\end{proposition}\n\n\\begin{proof}\n  If $H$ acts transitively with point stabilizer $H_0$, then $Y\\cong\n  H\/H_0$ and the result is known (see~\\cite[\\S1.3\n  (p)]{kazhdanlusztig:langlands}). In the general case, choose an open\n  orbit $\\CO$ in $Y$. Then there is an exact sequence\n  \\[\n  \\dotsm \\to K^H_1(Y\\setminus \\CO) \\to K^H_1(Y) \\to K^H_1(\\CO) \\to \\dotsm .\n  \\]\n  By induction on the number of orbits, $K^H_1(Y\\setminus \\CO) =0$. We have\n  already observed that $K^H_1(\\CO)=0$. Thus, $K^H_1(Y)=0$. \\qed\n\\end{proof}\n\n\\begin{lemma}\\label{lem:k1}\n  Suppose $I, J\\subseteq S$, and $z\\in \\WIJ$. Then $K^{\\Gbar}_1(\\XIJ_z)=0$.\n\\end{lemma}\n\n\\begin{proof}\n  The constructions used in the proof of~\\autoref{thm:ost} apply to the\n  functors $K_i^{\\Gbar}$ for $i\\geq 0$ (see \\cite[\\S5.2,\n  5.4]{chrissginzburg:representation}) and give isomorphisms\n  \\[\n  \\xymatrix{ K_1^{\\Gbar}(\\XIJ_z) \\ar[r]^-{\\res_z^{IJ}}_-{\\cong} &\n    K_1^{\\Lbar_z}(F^{IJ}) \\ar[r]^-{(i_z^{IJ})^*}_-{\\cong} &\n    K_1^{\\Lbar_z}(\\SN_z) .}\n  \\]\n  Because $\\Lbar_z$ acts on $\\SN_z$ with finitely many orbits, it follows\n  from~\\autoref{pro:k1} that $K^{\\overline{L_{z}}}_1 (\\SN_z)=0$. \\qed\n\\end{proof}\n\nThe fact that sequence~\\eqref{eq:exz} is exact follows immediately\nfrom~\\autoref{lem:k1} and the long exact sequence in equivariant $K$-theory.\n\n\\subsection{The isomorphism \\texorpdfstring{$\\CHIJ_{\\ssleq z}\\cong\n    K^{\\Gbar}(\\XIJ_{\\ssleq z})$}{}} \\label{ssec:isoz1}\n\nThe rest of this section is devoted to the proof\nof~\\autoref{thm:wideiso2}. We first define the maps in~\\eqref{eq:wideiso},\n\\[\n\\xymatrix{%\n  \\CH_{\\ssleq z} \\ar[r]^-{} \\ar[d]^{\\chi^{\\ssleq z}}& K^{\\Gbar}(Z_{\\ssleq\n    z}) \\ar[r]^-{r_z^*} \\ar[d]^{\\eta^{\\ssleq z}_*} & K^{\\Gbar}(Z_z) \\ar[r]\n  \\ar[d]^{\\eta^z_*} &  K^{\\Lbar_z}(\\SNt_z) \\ar[d]^{(p_z)_*} \\\\\n  \\CHIJ_{\\ssleq z} \\ar[r] & K^{\\Gbar}(\\XIJ_{\\ssleq z})\n  \\ar[r]^-{(r_z^{IJ})^*} & K^{\\Gbar}(\\XIJ_z) \\ar[r] &\n  K^{\\Lbar_z}(\\SN_z), }\n\\]\nand show that the diagram commutes.\n\nThe middle square is induced by the cartesian diagram~\\eqref{eq:cart} and\nthe right-hand square is as in~\\autoref{thm:ost}. Both of these squares\ncommute.\n\n\n\\subsection{The left-hand square in diagram\n  (\\ref{eq:wideiso})}\n\nWe observed after~\\eqref{eq:8} that if $\\lambda$ is in $X(T)$, $y$ is in\n$\\WIJ$, and $w$ is in the double coset $W_IyW_J$, then $C_{w_I}\n\\theta_\\lambda T_{w}C_{w_J}$ is in the span of $\\{\\, C_{w_I} \\theta_\\mu T_y\nC_{w_J} \\mid \\mu \\in X(T)\\,\\}$. Therefore,\n$\\chi^{IJ}(\\CH_{w})\\subseteq\\CHIJ_y$. It follows that $\\chi^{\\ssleq\n  z}(\\CH_{\\ssleq z})\\subseteq \\CHIJ_{\\ssleq z}$. In particular,\n$\\chi^{\\ssleq z}\\colon \\CH_{\\ssleq z}\\to \\CHIJ_ {\\ssleq z}$ is defined.\n\nConsider the commutative diagram\n\\begin{equation*}\n  \\xymatrix{%\n    \\CH_z\\ar@{^{(}->}[r] \\ar[d]^{\\chi^z} &\n    \\CH_{\\ssleq z} \\ar[d]^{\\chi^{\\ssleq z}}\\\\   \n    \\CHIJ_z\\ar@{^{(}->}[r]  & \\CHIJ_{\\ssleq z}, }  \n\\end{equation*}\nwhere the horizontal maps are the inclusions and $\\chi^z$ is the restriction\nof $\\chi^{IJ}$ to $\\CH_z$. It is clear that $\\chi^y(\\CH_y)=\\CHIJ_y$ for $y$\nin $\\WIJ$ and so $\\chi^{\\ssleq z}$ is surjective.\n\nThe left-hand square in diagram~\\eqref{eq:wideiso} is\n\\begin{equation*}\n  \\xymatrix{%\n    \\CH_{\\ssleq z} \\ar[r]^-{\\varphi_z} \\ar[d]^{\\chi^{\\ssleq z}} &\n    K^{\\Gbar}(Z_{\\ssleq z}) \\ar[d]^{\\eta^{\\ssleq  z}_*} \\\\  \n    \\CHIJ_{\\ssleq z} \\ar[r]^-{\\psi_z} & K^{\\Gbar}(\\XIJ_{\\ssleq z}), }   \n\\end{equation*}\nwhere $\\varphi_z$ and $\\psi_z$ are defined below. \n\nRecall that for $w$ in $W$ $j_w\\colon Z_{\\ssleq w}\\to Z$ is the\ninclusion. Similarly, let $j^{IJ}_z$ denote the inclusion $\\XIJ_{\\ssleq\n  z}\\to \\XIJ$ for $z$ in $\\WIJ$. The maps $\\varphi_z$ and $\\psi_z$ are the\nrestrictions of $\\varphi$ and $\\psi^{IJ}$, respectively, in the sense that\n$(j_z)_* \\circ \\varphi_z$ is the restriction of $\\varphi$ to $\\CH_{\\ssleq\n  z}$ and $(j^{IJ}_z)_* \\circ \\psi_z$ is the restriction of $\\psi^{IJ}$ to\n$\\CHIJ_{\\ssleq z}$. In order to prove~\\autoref{thm:wideiso2} we need a\nformula for $\\varphi_z$, and so we define $\\varphi_z$ explicitly, show that\n$(j_z)_* \\circ \\varphi_z$ is the restriction of $\\varphi$ to $\\CH_{\\ssleq\n  z}$, and then define $\\psi_z$.\n\nFor $w$ in $W$, let $q_{w,1} \\colon Z_w\\to G\/B$ by $q_{w,1}(x,gB, gwB)= gB$.\nThen $q_{w,1}$ is a $\\Gbar$-equivariant affine space bundle over $G\/B$ and\nso $q_{w,1}^*\\colon K^{\\Gbar}(G\/B) \\to K^{\\Gbar}( Z_w)$ is an\n$R(\\Gbar)$-module isomorphism.\n\n\\begin{theorem}\\label{thm:basis}\n  Suppose $w$ is in $W$. There is an $A$-module isomorphism\n  \\[\n  \\varphi_w\\colon \\CH_{\\ssleq w} \\to K^{\\Gbar}(Z_{\\ssleq w})\n  \\]\n  such that\n  \\begin{enumerate}\n  \\item $(j_w)_* \\varphi_w\\colon \\CH_{\\ssleq w}\\to K^{\\Gbar}(Z)$ is the\n    restriction of $\\varphi$ to $\\CH_{\\ssleq w}$, and \\label{it:phi1}\n  \\item for $\\lambda$ in $X(T)$, $r_w^* \\varphi_w(\\theta_\\lambda T_w)=\n    \\epsilon_w q_{w,1}^* [\\CL_\\lambda]$. \\label{it:phi2}\n  \\end{enumerate}\n  In particular, $\\{\\, r_w^* \\varphi_w(\\theta_\\lambda T_w)\\mid \\lambda\\in\n  X(T) \\,\\}$ is an $A$-basis of $K^{\\Gbar}(Z_w)$.\n\\end{theorem}\n\n\\begin{proof}\n  Lusztig has shown (see \\cite[Lemma 8.9]{lusztig:bases}) that there is a\n  unique element $\\xi_y$ in $K^{\\Gbar}(Z_{\\ssleq y})$ such that\n  $(j_y)_*(\\xi_y)=\\varphi(T_y)$.  Let $d_1\\colon \\FNt\\to Z_1$ be the\n  ``diagonal'' isomorphism given by $d_1(x, gB)= (x, gB, gB)$. With this\n  notation, using the $K^{\\Gbar}(Z_1)$-module structure $\\star_w$ on\n  $K^{\\Gbar}(Z_{\\ssleq w})$, define\n  \\[\n  \\varphi_w\\colon \\CH_{\\ssleq w}\\to K^{\\Gbar}(Z_{\\ssleq w})\n  \\quad\\text{by}\\quad \\varphi_w (\\theta_\\lambda T_y)= (d_1)_*\n  q^*[\\CL_\\lambda] \\star_w (j_y^w)_* (\\xi_y)\n  \\]\n  for $\\lambda$ in $X(T)$ and $y$ in $W$ with $y\\leq w$.\n\n  Set $d=j_1d_1\\colon \\FNt \\to Z$, so $d(x,gB)=(x, gB, gB)$. By the\n  definition of $\\varphi$, for $\\lambda$ in $X(T)$, $\\varphi(\n  \\theta_\\lambda)= d_* q^*[\\CL_\\lambda]$. Thus, using\n  equation~\\eqref{eq:star1} and~\\autoref{lem:starlin} we have\n  \\begin{align*}\n    (j_w)_* \\varphi_w (\\theta_\\lambda T_y) & = (j_w)_* \\big((d_1)_*\n    q^*[\\CL_\\lambda] \\star_w (j_y^w)_* \\xi_y\\big) \\\\\n    & = (j_1)_*(d_1)_* q^*[\\CL_\\lambda] \\star (j_w)_*(j_y^w)_* \\xi_y \n    = d_* q^*[\\CL_\\lambda] \\star (j_y)_* \\xi_y \\\\ & = \\varphi(\\theta_\\lambda)\n    \\star \\varphi(T_y) = \\varphi(\\theta_\\lambda T_y) .\n  \\end{align*}\n  This proves the first statement.\n\n  To prove~\\autoref{it:phi2}, let $\\BBC_{Z_w}$ be the trivial line bundle on\n  $Z_w$ and let\n  \\[\n  p_{w,1}\\colon Z_w\\to \\FNt\\quad\\text{by}\\quad p_{w,1} (x,gB, gwB)= (x, gB).\n  \\]\n  Then $q_{w,1}=q p_{w,1}$. Set $V_w=(\\Ztilde_1\\times \\FNt) \\cap (\\FNt\n  \\times \\Ztilde_{ w})$ and let $p_{12}'\\colon V_w \\to Z_1$ and\n  $p_{13}'\\colon V_w \\to Z_w$ be the obvious projections.  It is\n  straightforward to check that $p_{w,1}$ and $p_{12}'$ are smooth,\n  that $p_{13}'$ is an isomorphism, and that the diagram\n  \\[\n  \\xymatrix{ Z_w \\ar[r]^-{(p_{13}')\\inverse} \\ar[d]^{p_{w,1}} & V_w\n    \\ar[d]^{p_{12}'} \\\\ \n    \\FNt \\ar[r]^-{d_1} &Z_1}\n  \\]\n  is cartesian. Thus $(p_{12}')^* (d_1)_*= (p_{13}')\\inverse_* p_{w,1}^*$.\n  Using the $K^{\\Gbar}(Z_1)$-module structure $\\star_w'$ on\n  $K^{\\Gbar}(Z_{w})$, we have\n  \\begin{align*}\n    r_w^* \\varphi_w(\\theta_\\lambda T_w)&= r_w^* \\big( (d_1)_*\n    q^*[\\CL_\\lambda] \\star_w \\xi_w \\big)&& \\\\\n    &= (d_1)_* q^*[\\CL_\\lambda] \\star_w' r_w^*( \\xi_w) &&\n    \\text{\\autoref{lem:starlin}}  \\\\\n    &= (d_1)_* q^*[\\CL_\\lambda] \\star_w' \\epsilon_w [\\BBC_{Z_w}] &&\n    \\text{\\cite[\\S8.9]{lusztig:bases}}  \\\\\n    &= \\epsilon_w\\ (p_{13}')_*\\, (p_{12}')^* (d_1)_* q^*[\\CL_\\lambda] &&\n    \\text{\\autoref{pro:convlin}\\,\\autoref{i:conv3}}  \\\\\n    &= \\epsilon_w\\ p_{w,1}^*\\, q^*[\\CL_\\lambda] && \\\\\n    &= \\epsilon_w\\ q_{w,1}^*\\, [\\CL_\\lambda]. &&\n  \\end{align*}\n  This completes the proof of the second statement.\n\n  The last statement in the theorem follows from~\\autoref{it:phi2} and the\n  fact that $q_{w,1}^*$ is an isomorphism.\n\n  Choose a linear order on the interval $[1,w]$ in the Bruhat poset of $W$\n  that extends the Bruhat order. This linear order determines gradings on\n  $\\CH_{\\ssleq w}$ and $K^{\\Gbar}(Z_{\\ssleq w})$. As $\\{\\,\n  \\theta_\\lambda T_y\\mid \\lambda\\in X(T) \\,\\}$ is an $A$-basis of $\\CH_{y}$\n  and $\\{\\, r_y^* \\varphi_y(\\theta_\\lambda T_y)\\mid \\lambda\\in X(T) \\,\\}$ is\n  an $A$-basis of $K^{\\Gbar}(Z_y)$ for $y$ in $W$, the associated graded map\n  $\\gr \\varphi_w$ is an isomorphism. Thus, $\\varphi_w$ is an\n  isomorphism. \\qed\n\\end{proof}\n\nIt follows from the theorem that for $z$ in $\\WIJ$,\n\\[\n\\etaIJ_* \\varphi(\\CH_{\\ssleq z})= \\etaIJ_* (j_z)_*( K^{\\Gbar}(Z_{\\ssleq z})\n)= (j^{IJ}_z)_* \\eta^{\\ssleq z}_*( K^{\\Gbar}(Z_{\\ssleq z})) \\subseteq\n(j^{IJ}_z)_* ( K^{\\Gbar}(\\XIJ_{\\ssleq z})).\n\\]\nOn the other hand, we have seen that $\\etaIJ_* \\varphi =\\psi^{IJ} \\chi^{IJ}$\nand that $\\chi^{\\ssleq z}$ is surjective, so\n\\[\n\\etaIJ_* \\varphi(\\CH_{\\ssleq z})=\\psi^{IJ} \\chi^{IJ}(\\CH_{\\ssleq z}) =\n\\psi^{IJ} \\chi^{\\ssleq z} (\\CH_{\\ssleq z})= \\psi^{IJ} (\\CHIJ_{\\ssleq z}).\n\\]\nTherefore, $\\psi^{IJ} (\\CHIJ_{\\ssleq z}) \\subseteq (j^{IJ}_z)_* (\nK^{\\Gbar}(\\XIJ_{\\ssleq z}))$ and so there is an $A$-module\nhomomorphism $\\psi_z\\colon \\CHIJ_{\\ssleq z}\\to K^{\\Gbar}(\\XIJ_{\\ssleq\n  z})$ such that $\\eta^{\\ssleq z}_* \\varphi_z= \\psi_z \\chi^{\\ssleq\n  z}$. In particular, the maps $\\varphi_z$ and $\\psi_z$ in the\nleft-hand square in~\\eqref{eq:wideiso} are defined and the diagram\ncommutes.\n\n\\subsection{Proof of~\\autoref{thm:wideiso2}} \\label{ssec:isoz3}\n\nIn this subsection we complete the proof of~\\autoref{thm:wideiso2}. The\narguments above show that diagram~\\eqref{eq:wideiso} commutes.\n\nTo prove~\\autoref{thm:wideiso2}\\,\\autoref{i:wi2}, let $f_1\\colon \\CH_{\\ssleq\n  z} \\to K^{\\Lbar_z}(\\SNt_z)$ be the composition across the top row\nin~\\eqref{eq:wideiso}. Then $f_1= i_z^* \\res_z r_z^* \\varphi_z$, where\n$i_z^*$, $\\res_z$, and $\\varphi_z$ are isomorphisms and $r_z^*$ is\nsurjective. If $y<z$, then $\\varphi_w(\\CH_y)\\subseteq K^{\\Gbar}(Z_{\\ssleq\n  y})$ and so it follows from the exactness of the top row of\ndiagram~\\eqref{eq:exwz} that $\\varphi_z(\\CH_y)$ is in the kernel of\n$r_z^*$. Therefore, $\\CH_y$ is in the kernel of $f_1$.\n\nBy~\\autoref{thm:basis}\\,\\autoref{it:phi2}, $r_z^* \\varphi_z(\\theta_\\lambda\nT_z)=\\epsilon_z\\ q_{z,1}^* ([\\CL_\\lambda])$, and so\n\\[\nf_1(\\theta_\\lambda T_z)=\\epsilon_z\\ i_z^* \\res_z \\ q_{z,1}^*\n([\\CL_\\lambda]).\n\\]  \nLet $g_z\\colon F\\to Z_z$ be the inclusion and let $f_z\\colon L_z\/B_z\\to G\/B$\nbe the canonical embedding. Then the diagram\n\\[\n\\xymatrix{\\SNt_z\\ar[r]^-{i_z} \\ar[d]^{q_z} & F \\ar[r]^-{g_z} &\n  Z_z\\ar[d]^{q_{z,1}} \\\\\n  L_z\/B_z \\ar[rr]^{f_z} && G\/B }\n\\]\ncommutes. The restriction map $\\res_z\\colon K^{\\Gbar}(Z_z)\\to\nK^{\\Lbar_z}(F)$ is induced by the functor $g_z^*$ from $\\Gbar$-equivariant\ncoherent sheaves on $Z_z$ to $\\Lbar_z$-equivariant sheaves on\n$F$. Therefore,\n\\[\ni_z^* \\res_z \\ q_{z,1}^* ([\\CL_\\lambda])= [ i_z^* g_z^* q_{z,1}^*\n(\\CL_\\lambda)] = [q_z^* f_z^* (\\CL_\\lambda)] = [q_z^* (\\CL_\\lambda^z)]\n=q_z^* [\\CL_\\lambda^z]\n\\]\nand so $f_1(\\theta_\\lambda T_z)=\\epsilon_z q_z^* [\\CL_\\lambda^z]$.\n\nTo prove~\\autoref{thm:wideiso2}\\,\\autoref{i:wi3}, let $f_2\\colon\n\\CHIJ_{\\ssleq z} \\to K^{\\Lbar_z}(\\SN_z)$ be the composition across the\nbottom row in~\\eqref{eq:wideiso}. Then $f_2= (i_z^{IJ})^* \\res_z^{IJ}\n(r_z^{IJ})^* \\psi_z$ and $f_2 \\chi^{\\ssleq z}= (p_z)_* f_1$.  If $y$ is in\n$\\WIJ$ and $y<z$, then $\\psi_z(\\CHIJ_y)\\subseteq K^{\\Gbar}(\\XIJ_{\\ssleq y})$\nand so it follows from the exact sequence~\\eqref{eq:exz} that\n$\\psi_z(\\CHIJ_y)$ is in the kernel of $(r_z^{IJ})^*$. Therefore, $\\CHIJ_y$\nis in the kernel of $f_2$. Suppose $\\lambda$ is in $X(T)$. Then\n\\[\nf_2(C_{w_I} \\theta_\\lambda T_zC_{w_J})= f_2 \\chi^{\\ssleq z}(\n\\theta_\\lambda T_z)= (p_z)_* f_1 ( \\theta_\\lambda T_z)= \\epsilon_z (p_z)_*\nq_z^*([\\CL^z_\\lambda]).\n\\]\nAs observed before the statement of~\\autoref{thm:wideiso2}, $\\{\\, C_{w_I}\n\\theta_\\lambda T_zC_{w_J} \\mid \\lambda \\in X(T)^+_z\\,\\}$ is a $A$-basis of\n$\\CHIJ_z$, and by~\\autoref{lem:ostlem}, $\\{\\, \\epsilon_z (p_z)_*\nq_z^*([\\CL^z_\\lambda])\\mid \\lambda\\in X(T)^+_z \\,\\}$ is an $A$-basis of\n$K^{\\Lbar_z}(\\SN_z)$. It follows that $f_2$ is an isomorphism.\n\nFinally, to prove~\\autoref{thm:wideiso2}\\,\\autoref{i:wi1} it remains to show\nthat $\\eta^{\\ssleq z}_*$ is surjective and that $\\psi_{z}$ is an\nisomorphism. The fact that $\\eta^{\\ssleq z}_*$ is surjective was established\nin~\\autoref{cor:surj}. Since $\\psi_z \\chi^{\\ssleq z}= \\eta^{\\ssleq z}_*\n\\varphi_z$, we see that $\\psi_z$ is a surjection as well. Induction on the\nlength of $z$ and statement~(\\ref{i:wi3}) of the theorem show that\n\\[\n\\{\\, \\psi_z(C_{w_I} \\theta_\\lambda T_y C_{w_J}) \\mid y\\in \\WIJ,\\ y\\leq\nz,\\ \\lambda\\in X(T)_{y}^+ \\,\\}\n\\] \nis a linearly independent subset of $K^{\\Gbar}(\\XIJ_{\\ssleq z})$ and so\n$\\psi_z$ is an injection. This completes the proof\nof~\\autoref{thm:wideiso2}.\n\n\\bigskip\\noindent {\\bf Acknowledgments}: This work was partially supported\nby a grant from the Simons Foundation (Grant \\#245399 to J. Matthew\nDouglass).  The authors acknowledge the financial support of a DFG grant for\nthe enhancement of bilateral cooperation and the DFG-priority program\nSPP1388 ``Representation Theory.''  The authors would like to thank Victor\nOstrik for clarifying the assertions in \\cite{ostrik:equivariant}, William\nGraham and an anonymous referee for help with the proof\nof~\\autoref{pro:special}, and the same anonymous referee for numerous\nsuggestions that substantially improved the paper.\n \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \n\nIn this note we consider a fourth order invariant related to the classical \nYamabe invariant. \n\n\\subsection{Basic definitions} \n\nLet $n \\geq 5$ and let $(M,g)$ be a compact Riemannian \nmanifold without boundary, and define \n\\begin{equation} \\label{q_curv_defn} \nQ_g = -\\frac{1}{2(n-1)} \\Delta_g R_g - \\frac{2}{(n-2)^2} |\\operatorname{Ric}_g |^2 + \n\\frac{n^3 - 4n^2 + 16 n - 16}{8 (n-1)^2(n-2)^2} R_g^2 ,\n\\end{equation}  \nwhere $\\Delta_g$ is the Laplace-Beltrami operator, $R_g$ is the \nscalar curvature, and $\\operatorname{Ric}_g$ is the Ricci curvature. We simplify \nthe expression of of $Q_g$ in \\eqref{q_curv_defn} by introducing \nthe Schouten tensor \n\\begin{equation} \\label{schouten_defn} \nA_g = \\frac{1}{n-2} \\left ( \\operatorname{Ric}_g - \\frac{R_g g}{2(n-1)} \\right ) , \n\\qquad J_g = \\operatorname{tr}_g(A_g) = \\frac{R_g}{2(n-1)}, \n\\end{equation} \nso that \n\\begin{equation} \\label{q_curv_defn2} \nQ_g = - \\Delta_g J_g - 2 |A_g|^2 + \\frac{n}{2} J_g^2. \n\\end{equation} \n\nThe $Q$-curvature transforms nicely under a conformal change, \nnamely\n\\begin{equation} \\label{trans_rule1} \n\\widetilde g = u^{\\frac{4}{n-4}} g \\Rightarrow \nQ_{\\widetilde g} = \\frac{2}{n-4} u^{- \\frac{n+4}{n-4}} P_g(u), \n\\end{equation} \nwhere $P_g$ is the Paneitz operator \n\\begin{equation} \\label{paneitz_op_defn} \nP_g(u) = (-\\Delta_g)^2 (u) + \\operatorname{div} ( 4A_g (\\nabla u, \n\\cdot ) - (n-2) J_g \\nabla u) + \\frac{n-4}{2} Q_g u. \n\\end{equation} \nIn general the Paneitz operator transforms according to the \nrule \n\\begin{equation} \\label{trans_rule2}\n\\widetilde g = u^{\\frac{4}{n-4}} g \\Rightarrow P_{\\widetilde g}(v) \n= u^{-\\frac{n+4}{n-4}} P_g(uv). \n\\end{equation} \nOne recovers \\eqref{trans_rule1} by substituting $v=1$ \ninto \\eqref{trans_rule2}. \n\nS. Paneitz \\cite{Pan1} introduced the operator \\eqref{paneitz_op_defn} \nand explored some of its transformation properties. Afterwards \nT. Branson \\cite{Bran1, Bran2} extended Paneitz' definition and studied \nthe associated $Q$-curvature. The interested reader can find excellent \nsurveys in \\cite{BG, CEOY, HY}. \n\nThe transformation rules \\eqref{trans_rule1} and \\eqref{trans_rule2} \nmotivate us to define the energy \nfunction $\\mathcal{Q} : [g] \\rightarrow \\R$, where $[g] = \\{ \n\\widetilde g = u^{\\frac{4}{n-4}} g : u \\in \\mathcal{C}^\\infty (M) , \nu>0\\}$ is the conformal class of $g$, by \n\\begin{equation} \\label{q_energy} \n\\mathcal{Q} (\\widetilde g) = \\frac{\\int_M Q_{\\widetilde g} \nd\\mu_{\\widetilde g}}{ (\\operatorname{Vol}_{\\widetilde g} (M)\n)^{\\frac{n-4}{n}}} = \\frac{2}{n-4} \\frac{\\int_M u P_g(u) \nd\\mu_g} {\\left ( \\int_M u^{\\frac{2n}{n-4}} d\\mu_g \\right )^{\\frac{n-4}{n}}} .\n\\end{equation}\nWe then the conformal invariant \n\\begin{eqnarray} \\label{paneitz_inv1} \n\\mathcal{Y}_4^+ ([g],M) & = & \\inf_{\\widetilde g \\in [g]}\n\\mathcal{Q}(\\widetilde g) = \\inf \\left \\{ \\frac{\\int_M \nQ_{\\widetilde g} d\\mu_{\\widetilde g}} {(\\operatorname{Vol}\n_{\\widetilde g}(M))^{\\frac{n-4}{n}}} : \\widetilde g \\in [g] \\right \\} \n\\\\ \\nonumber \n& = & \\inf \\left \\{ \\frac{2}{n-4} \\frac{\\int_M u P_g(u) d\\mu_g}\n{\\left ( \\int_M u^{\\frac{2n}{n-4}} d\\mu_g\\right )^{\\frac{n-4}{n}}} : \nu \\in \\mathcal{C}^\\infty (M) , u>0 \\right \\} ,\n\\end{eqnarray} \nwhich is a fourth-order analog of the famous Yamabe invariant,\nand the differential invariant \n\\begin{equation} \\label{paneitz_inv2} \n\\mathbb{Y}_4^+(M) = \\sup_{[g] \\in \\mathfrak{c}} \n\\mathcal{Y}_4^+ ([g],M)  = \\sup_{[g] \\in \\mathfrak{c}}\n\\inf_{\\widetilde g \\in [g]} \\left \\{ \\frac{ \\int_M \nQ_{\\widetilde g} d\\mu_{\\widetilde g} } { ( \n\\operatorname{Vol}_{\\widetilde g} (M))^{\\frac{n-4}{n}}} \n\\right \\} , \n\\end{equation} \nwhere $\\mathfrak{c}$ is the space of conformal classes on the \nmanifold $M$. \nThe subscript $4$ in both $\\mathcal{Y}_4^+$ and in $\\mathbb{Y}_4^+$ \nrefers to the fact that the underlying differential operator is fourth-order, \nwhile the $+$ refers to the fact that we require all test functions in the \ninfimum for $\\mathcal{Y}_4^+$ must all be positive. Naturally one may \nalso define \n$$\\mathcal{Y}_4([g],M) = \\inf \\left \\{ \\frac{2}{n-4} \\frac{\\int_M u \nP_g(u) d\\mu_g} { \\left ( \\int_M |u|^{\\frac{2n}{n-4}} d\\mu_g \\right \n)^{\\frac{n-4}{n}} } : u \\in \\mathcal{C}^\\infty(M), u \\not \\equiv 0 \n\\right  \\},$$ \nand clearly $\\mathcal{Y}_4([g],M) \\leq \\mathcal{Y}_4^+([g],M)$. \n\n\\subsection{Scalar curvature and the Yamabe invariant} \n\nMuch of the work devoted to the Paneitz operator $P_g$ \nand its associated $Q$-curvature is motivated by results about the total \nscalar curvature functional and its associated Yamabe invariant. \n\nGiven a compact Riemannian manifold $(M,g)$ without boundary one \ndefines the total scalar curvature functional on the conformal class \n$[g]$ as\n\\begin{equation} \\label{tot_scal_curv1} \n\\mathcal{R} (\\widetilde g) = \\frac{\\int_M R_{\\widetilde g} \nd\\mu_{\\widetilde g}} {(\\operatorname{Vol}_{\\widetilde g} (M) \n)^{\\frac{n-2}{n}} }. \n\\end{equation} \nOne can simplify this expression using the transformation rule \n\\begin{equation} \\label{trans_rule3} \n\\widetilde g = u^{\\frac{4}{n-2}} g \\Rightarrow R_{\\widetilde g} \n= \\frac{n-2}{4(n-1)} u^{-\\frac{n+2}{n-2}} \\mathcal{L}_g(u),\n\\end{equation} \nwhere $\\mathcal{L}_g$ is the conformal Laplacian \n\\begin{equation} \\label{conf_lap} \n\\mathcal{L}_g = -\\Delta_g  + \\frac{4(n-1)}{n-2} R_g ,\n\\end{equation} \nwhich enjoys the transformation rule \n\\begin{equation} \\label{trans_rule4} \n\\widetilde g = u^{\\frac{4}{n-2}} g \\Rightarrow\n\\mathcal{L}_{\\widetilde g} (v) = u^{-\\frac{n+2}{n-2}} \n\\mathcal{L}_g (uv). \n\\end{equation} \nObserve that these transformation rules mean we can \nrewrite $\\mathcal{R}$ as \n\\begin{equation} \\label{tot_scal_curv2} \n\\mathcal{R}(u^{\\frac{4}{n-2}} g ) = \\frac{n-2}{4(n-1)} \n\\frac{\\int_M u \\mathcal{L}_g(u) d\\mu_g}{\\left ( \\int_M\nu^{\\frac{2n}{n-2}} d\\mu_g \\right )^{\\frac{n-2}{n}}}.\n\\end{equation} \n\nThe classical Yamabe invariants are \n\\begin{eqnarray} \\label{yam_inv1} \n\\mathcal{Y}([g],M) & = & \\inf_{\\widetilde g \\in [g]} \\mathcal{R}\n(\\widetilde g) = \\inf \\left \\{ \\frac{\\int_M R_{\\widetilde g} d\\mu_{\\widetilde g}}\n{(\\operatorname{Vol}_{\\widetilde g} (M))^{\\frac{n-2}{n}}} : \\widetilde g \n\\in [g] \\right \\} \\\\ \\nonumber \n& = & \\inf \\left \\{ \\frac{n-2}{4(n-1)} \\frac{\\int_M u \\mathcal{L}_g(u) \nd\\mu_g}{\\left ( \\int_M u^{\\frac{2n}{n-2}} d\\mu_g \\right )^{\\frac{n-2}{n}}}\n: u \\in \\mathcal{C}^\\infty(M) , u>0 \\right \\}\n\\end{eqnarray}\nand \n\\begin{equation} \\label{yam_inv2} \n\\mathbb{Y}(M) = \\sup_{[g] \\in \\mathfrak{c}} \\mathcal{Y}([g], M) \n= \\sup_{[g] \\in \\mathfrak{c}} \\inf_{\\widetilde g \\in [g]} \n\\frac{\\int_M R_{\\widetilde g} d\\mu_{\\widetilde g}}{ (\n\\operatorname{Vol}_{\\widetilde g} (M) )^{\\frac{n-2}{n}}} . \n\\end{equation} \nIn contrast to the fourth-order case, in this situation the \nmaximum principle implies \n$$\\mathcal{Y}([g],M) = \\inf  \\left \\{ \\frac{n-2}{4(n-1)} \\frac{\\int_M u \\mathcal{L}_g(u) \nd\\mu_g}{\\left ( \\int_M |u|^{\\frac{2n}{n-2}} d\\mu_g \\right )^{\\frac{n-2}{n}}}\n: u \\in \\mathcal{C}^\\infty(M) , u \\not \\equiv 0 \\right \\} . \n$$\nIn particular, minimizing the functional $\\mathcal{R}$ over \nall nontrivial functions in $W^{1,2}(M)$ will automatically yield \na positive minimizer. On the other hand, minimizers of $\\mathcal{Y}_4\n([g],M)$, if they exist, might change sign. \n\nYamabe \\cite{Y} first defined these two invariants while investigating \nthe the problem of finding a constant scalar curvature metric \nin a given conformal class. Aubin \n\\cite{Aub} proved that $\\mathcal{Y}([g],M) \\leq \\mathcal{Y}\n([g_0],\\Ss^n)$, where $g_0$ is the round metric on the sphere $\\Ss^n$, \nand proved that if $\\mathcal{Y}([g],M) < \\mathcal{Y}([g_0],M)$ there \nexists a smooth, constant scalar curvature metric $\\widetilde g \\in [g]$\nsuch that $\\mathcal{R}(\\widetilde g) = \\mathcal{Y}([g],M)$. \nIn \\cite{Sch} Schoen completed Yamabe's program, proving \nthat $\\mathcal{Y}([g], M) < \\mathcal{Y}([g_0], \\Ss^n)$ for each \nconformal class $[g] \\neq [g_0]$. In particular, $\\mathcal{Y}([g],M) \n< \\mathcal{Y}([g_0], \\Ss^n)$ whenever $M$ is not the sphere. \n\nOn the other hand, the equality $\\mathbb{Y}(M) = \\mathbb{Y}(\\Ss^n)$ \nmay occur even when $M$ is not the sphere. In particular, \nSchoen \\cite{Sch_var} found an explicit sequence of metrics \n$g_k$ on the product $\\Ss^1 \\times \\Ss^{n-1}$ such that \n$\\mathcal{Y}([g_k], \\Ss^1 \\times \\Ss^{n-1}) \\rightarrow \\mathcal{Y}\n([g_0], \\Ss^n)$ as \n$k \\rightarrow \\infty$, and so $\\mathbb{Y}\\mathbb(\\Ss^1 \n\\times \\Ss^{n-1}) = \\mathbb{Y}(\\Ss^n)$. As the underlying manifolds \nare not diffeomorphic, the equality above cannot be realized by \na smooth metric on $\\Ss^1 \\times \\Ss^{n-1}$.\n\n\\subsection{Previous results and our main theorem} \n\nWe summarize some previous theorems regarding the invariant \n$\\mathcal{Y}_4^+ ([g],M)$. Esposito \nand Robert \\cite{ER} showed that $\\mathcal{Y}_4^+([g],M)$ is \nfinite for each conformal class $[g]$ on $M$. \nTo state the next result we define \n$$\\mathcal{Y}_4^* ([g],M) = \\inf_{\\widetilde g \\in [g], R_{\\widetilde g} > 0}\n\\mathcal{Q} (\\widetilde g).$$\nGursky, Hang and Lin \\cite{GHL} proved that if $n= \\dim(M) \\geq 6$ and \nif $\\mathcal{Y}([g],M) > 0$ \nand $\\mathcal{Y}_4^*([g],M)>0$ then \n$$\\mathcal{Y}_4([g],M) = \\mathcal{Y}_4^+ ([g],M) = \\mathcal{Y}_4^* \n([g],M) .$$\nShortly thereafter Hang and Yang \\cite{HY} proved that if $\\mathcal{Y}([g],M)>0$ \nand $Q_g \\geq 0$ with $Q_g \\not \\equiv 0$ then \n$$\\mathcal{Y}_4([g],M) = \\mathcal{Y}_4^+([g],M) \\leq \\mathcal{Y}_4 \n([g_0], \\Ss^n),$$ \nand that equality in the last inequality implies $[g] = [g_0]$. Moreover, \nunder these hypotheses there exists a smooth, constant $Q$-curvature \nmetric $\\widetilde g \\in [g]$ such that $\\mathcal{Q}(\\widetilde g) = \n\\mathcal{Y}_4^+ ([g],M)$. \n\nOur main result is the following theorem. \n\\begin{thm} \\label{main_thm} \nThere exists a sequence of metrics $g_k$ on the product \n$\\Ss^1 \\times \\Ss^{n-1}$ such that $\\mathcal{Y}_4^+ ([g_k], \n\\Ss^1 \\times \\Ss^{n-1}) \n\\rightarrow \\mathcal{Y}_4^+ ([g_0], \\Ss^n)$, where $g_0$ is the standard \nround metric on $\\Ss^n$. As a consequence $\\mathbb{Y}_4^+ \n(\\Ss^1 \\times \\Ss^{n-1}) = \\mathbb{Y}_4^+ (\\Ss^n)$. \n\\end{thm}\n\n\\begin{rmk} The theorem of Hang and Yang \\cite{HY} \nreferenced above implies \nthe equality $\\mathbb{Y}_4^+ (\\Ss^{n-1} \\times \\Ss^1) = \\mathbb{Y}_4^+\n(\\Ss^n)$ cannot be realized by a smooth metric on $\\Ss^{n-1} \\times \n\\Ss^1$.  \n\\end{rmk}\n\nWe base our proof of Theorem \\ref{main_thm} on the explicit \nexamples of the Delaunay metrics recently discovered by \nFrank and K\\\"onig \\cite{FK}, following the example of \nR. Schoen \\cite{Sch_var}. \n\n\\section{Proof of our main theorem} \n\nIn this section we present a proof of Theorem \\ref{main_thm}, \nusing the Delaunay metrics of Frank and K\\\"onig as our sequence \nof metrics. We first present some preliminary facts we require \nin our proof, and then carefully describe the Delaunay metrics, \nverifying some of their properties. Finally we complete the \nproof of Theorem \\ref{main_thm}. \n\n\\subsection{Preliminaries} \n\nWe begin with the well-known variational characterization of \nconstant $Q$-curvature metrics. One can find the following \ncomputation in \\cite{Rob}, among other places, but we \ninclude it for the reader's convenience. \n\nIt will be convenient to let $p^\\# = \\frac{2n}{n-4}$, let $\\| \\cdot \\|_p$\ndenote the $L^p$-norm on $(M, d\\mu_g)$, and define the bilinear \nform $\\mathcal{E} (u,v) = \\int_M v P_g(u) d\\mu_g$. Observe that \n\\begin{eqnarray} \\label{paneitz_bilin_form} \n\\mathcal{E} (u,v) & = & \\int_M v P_g(u) d\\mu_g \\\\ \\nonumber \n& = & \\int_M v \\left ( \\Delta_g ^2 u+ 4 \\operatorname{div} (A_g(\\nabla u, \\cdot)\n- (n-2) \\operatorname{div} (J_g \\nabla u ) + \\frac{n-4}{2} Q_g u \\right ) d\\mu_g\n\\\\ \\nonumber \n& = & \\int_M \\Delta_g v \\Delta_g u - 4 A_g(\\nabla u, \\nabla v) + (n-2) J_g \n\\langle \\nabla u, \\nabla v \\rangle  + \\frac{n-4}{2} Q_g uv d\\mu_g \n\\\\ \\nonumber \n& = & \\int_M u \\left ( \\Delta_g ^2 v+ 4 \\operatorname{div} (A_g(\\nabla v, \\cdot)\n- (n-2) \\operatorname{div} (J_g \\nabla v ) + \\frac{n-4}{2} Q_g v \\right ) d\\mu_g\n\\\\ \\nonumber \n& = & \\mathcal{E} (v,u) ,\n\\end{eqnarray} \nand so $\\mathcal{E}$ is symmetric. We denote $\\mathcal{E}(u,u) \n= \\mathcal{E}(u)$. \n\n\\begin{lemma} \nLet $W^{2,2}_+(M)$ denote the subspace of (almost everywhere) \npositive functions in $W^{2,2}(M)$. The functional \n$$ W^{2,2}_+(M) \\ni \nu \\mapsto \\mathcal{Q}(u^{\\frac{4}{n-4}} g)  $$\nis differentiable and its total derivative is \n\\begin{equation} \\label{derivative_paneitz_func}\nD\\mathcal{Q}(u) (v) = \\frac{4}{(n-4) \\| u \\|_{p^\\#}^2} \\int_M\nv \\left ( P_g(u) - \\| u \\|_{p^\\#}^{-p^\\#} \\mathcal{E}(u) u^{\\frac{n+4}{n-4}}\n\\right ) d\\mu_g .\n\\end{equation} \n\\end{lemma}\n\n\\begin{proof} \nLet $u \\in W^{2,2}_+(M) \\cap \\mathcal{C}^0(M)$ and choose $v \\in W^{2,2}(M)$ such that \n$0<\\sup |v| < \\frac{1}{2} \\inf u$, which is possible because $M$ is a \ncompact manifold. In particular, $u+tv \\in W^{2,2}_+(M)$ for \n$0<t<3\/2$. Expanding \\eqref{paneitz_bilin_form} we obtain \n\\begin{eqnarray} \\label{derivative1} \n\\mathcal{E}(u+tv) & = & \\mathcal{E}(u) + 2t\\mathcal{E}(u,v) + t^2 \n\\mathcal{E}(v) \\\\ \\nonumber \n& = & \\int_M u P_g (u) d\\mu_g + 2t \\int_M v P_g(u) d\\mu_g \n+ t^2 \\int_M v P_g(v) d\\mu_g. \\end{eqnarray} \nOn the other hand we have \n\\begin{eqnarray} \\label{derivative2} \n\\left ( \\int_M (u+tv)^{\\frac{2n}{n-4}} d\\mu_g \\right )^{\\frac{4-n}{n}} & = & \n\\left ( \\int_M u^{\\frac{2n}{n-4}} d\\mu_g \\right )^{\\frac{4-n}{n}}  \\\\ \\nonumber \n&& -2t \\left ( \\int_M u^{\\frac{2n}{n-4}} d\\mu_g \\right )^{\\frac{4-2n}{n}} \n\\left ( \\int_M v u^{\\frac{n+4}{n-4}} d \\mu_g \\right ) + \n\\mathcal{O} (t^2 \\| v \\|_{W^{2,2}(M)}^2) \\\\ \\nonumber \n& = & \\| u \\|_{p^\\#}^{-2} \\left ( 1 - 2t \\| u \\|_{p^\\#}^{-p^\\#} \\int_M \nv u^{\\frac{n+4}{n-4}} d\\mu_g + \\mathcal{O} (t^2 \\| v \\|_{W^{2,2}(M)}) \n\\right ) . \n\\end{eqnarray} \nCombining \\eqref{derivative1} and \\eqref{derivative2} gives \n$$ \\mathcal{Q}(u+tv) = \\frac{2}{(n-4) \\| u \\|_{p^\\#}^2} \\left ( \\mathcal{E}(u) \n+ 2t \\int_M v \\left ( P_g(u) - \\mathcal{E}(u) \\| u \\|_{p^\\#}^{-p^\\#} u^{\\frac{n+4}{n-4}} \n\\right ) d\\mu_g  + \\mathcal{O} (t^2 \\| v\\|_{W^{2,2}(M)}^2 ) \\right ) , $$ \nwhich completes the proof.\n\\end{proof} \n\n\\begin{lemma} The metric $\\widetilde g = u^{\\frac{4}{n-4}} g$ is a critical \npoint of the functional $\\mathcal{Q}$ if and only if its $Q$-curvature \n$Q_{\\widetilde g}$ is constant. \n\\end{lemma} \n\n\\begin{proof} First suppose $\\widetilde g$ is a critical point of $\\mathcal{Q}$, \nwhich implies \n$$\\left . \\frac{d}{dt} \\right |_{t=0} \\mathcal{Q}((u+tv)^{\\frac{4}{n-4}} g) =0$$ \nfor each sufficiently small $v$, and so \\eqref{derivative_paneitz_func} gives \n$$ P_g(u) = \\frac{\\mathcal{E}(u) u^{\\frac{n+4}{n-4}}}{\\| u \\|_{p^\\#}^{p^\\#}}.$$\nSubstituting this value for $P_g (u)$ into \\eqref{trans_rule1} gives \n$$Q_{\\widetilde g} = \\frac{2}{(n-4) \\| u \\|_{p^\\#}^{p^\\#}} \\mathcal{E}(u),$$\nwhich is indeed a constant. \n\nOn the other hand suppose $Q_{\\widetilde g} = \\overline {Q}$ is a \nconstant, in which case \\eqref{trans_rule1} gives \n$$P_g(u) = \\frac{n-4}{2} \\overline{Q} u^{\\frac{n+4}{n-4}} \\Rightarrow \n\\mathcal{E}(u)  = \\frac{n-4}{2} \\overline{Q} \\| u \\|_{p^\\#}^{p^\\#}.$$\nSubstituting this value of $\\mathcal{E}(u)$ \ninto \\eqref{derivative_paneitz_func} gives \n\\begin{eqnarray*} \n\\left. \\frac{d}{dt} \\right |_{t=0} \\mathcal{Q} ((u+tv)^{\\frac{4}{n-4}} g) & = & \n\\frac{4}{(n-4) \\| u\\|_{p^\\#}^2} \\int_M v \\left ( P_g(u) - \\frac{n-4}{2} \n\\overline{Q} u^{\\frac{n+4}{n-4}} \\right ) d\\mu_g \\\\ \n& = & \\frac{4}{(n-4) \\| u \\|_{p^\\#}^2}\\int_M v \\left ( \\frac{n-4}{2} \n\\overline{Q} u^{\\frac{n+4}{n-4}} - \\frac{n-4}{2} \\overline{Q} \nu^{\\frac{n+4}{n-4}} \\right ) d\\mu_g \\\\ \n& = & 0 , \n\\end{eqnarray*} \nand so $\\widetilde g = u^{\\frac{4}{n-4}} g$ is indeed a critical point of \n$\\mathcal{Q}$. \n\\end{proof} \n\n\\subsection{Delaunay metrics} \nWe seek constant $Q$-curvature metrics of the form $g_v \n= v^{\\frac{4}{n-4}} (dt^2 + d\\theta^2)$ on the cylinder $\\R \n\\times \\Ss^{n-1}$, where $d\\theta^2$ is the standard round metric \non $\\Ss^{n-1}$. It will be convenient to normalize the $Q$-curvature to \nbe $\\frac{n(n^2-4)}{8}$, the same value as the $Q$-curvature of the \nround metric on $\\Ss^n$. By \\eqref{trans_rule1}, the condition \n$Q_{g_v} = \\frac{n(n^2-4)}{8}$ is equivalent to \nthe partial differential equation \n\\begin{eqnarray} \\label{paneitz_pde1} \n\\frac{n(n-4)(n^2-4)}{16} v^{\\frac{n+4}{n-4}} & = & \\frac{\\partial^4 v}\n{\\partial t^4} - \\left ( \\frac{n(n-4) + 8}{2} \\right ) \\frac{\\partial^2 v}\n{\\partial t^2} + \\frac{n^2(n-4)^2}{16} v \\\\ \\nonumber \n& = & \\Delta_\\theta^2 + 2 \\Delta_\\theta \\frac{\\partial^2 v}{\\partial t^2} \n- \\frac{n(n-4)}{2} \\Delta_\\theta v \n\\end{eqnarray} \nwhere $\\Delta_\\theta$ is the Laplace-Beltrami metric on the \nsphere $\\Ss^{n-1}$ with respect to the standard round metric. \n\nWe restrict our attention to rotationally invariant metrics, \nso that $v$ is  a function of $t$ alone and \\eqref{paneitz_pde1} \nbecomes \n\\begin{equation} \\label{paneitz_ode1} \n\\ddddot v - \\left ( \\frac{n(n-4)+8}{2} \\right ) \\ddot v + \n\\frac{n^2(n-4)^2}{16} v = \\frac{n(n-4)(n^2-4)}{16} v^{\\frac{n+4}{n-4}}. \n\\end{equation} \nWe immediately find two special solutions, namely the cylindrical \nand spherical solutions \n\\begin{equation} \nv_{cyl} = \\left ( \\frac{n(n-4)}{n^2-4} \\right )^{\\frac{n-4}{8}}, \\qquad \nv_{sph} = (\\cosh t )^{\\frac{4-n}{2}}. \n\\end{equation} \nObserve that, since $n>4$ we have $0 < v_{cyl} < 1$ and \n$$v_{sph} (0) = 1 = \\max (v_{sph} (t)), \\qquad \\dot v_{sph}(t) < 0 \n\\textrm{ for }t>0, \\qquad \\dot v_{sph} (t) >0 \\textrm{ for }t< 0.$$\n\nFrank and K\\\"onig recently classified all positive global solutions \nof the ODE \\eqref{paneitz_ode1}, proving there exists a periodic \nsolution $v_a$ for each $a \\in (v_{cyl},1)$ attaining its maximal \nvalue of $a$ when $t=0$. Moreover, they show any global, \npositive solution of \\eqref{paneitz_pde1} must either have the \nform $v(t) = v_a(t+T)$ or $v(t) = (\\cosh (t+T))^{\\frac{4-n}{2}}$ for \nsome $T \\in \\R$, \nor $v \\equiv \\left ( \\frac{n(n-4)}{n^2-4} \\right )^{\\frac{n-4}{8}}$. We \ncall $g_{v_a} = v_a^{\\frac{4}{n-4}} (dt^2 + d\\theta^2)$ the \n{\\bf Delaunay metric} with Delaunay parameter $a$. \n\nEach solution $v_a$ is periodic with period $T_a$, attains its maximal \nvalue at each integer multiple of $T_a$, attains its minimal value at \neach half-integer multiple of $T_a$, and is symmtric about each \nof its critical points. Moreover, the period $T_a$ is an increasing \nfunction of $a$ with $\\lim_{a \\nearrow 1} T_a = \\infty$ and \n$\\lim_{a \\searrow v_{cyl}} T_a = T_{cyl}$, where $T_{cyl}$ is \nthe formal period of $v_{cyl}$, given by \n\\begin{equation} \\label{cyl_period} \nT_{cyl} = \\frac{2\\pi}{\\mu}, \\qquad \\mu = \\frac{1}{2} \\sqrt{ \n\\sqrt{n^4 - 64 n + 64} - n(n-4) + 8}. \n\\end{equation} \nThe period $T_{cyl}$ is the fundamental period of the linearization \nof the operator $P_g$, linearized about the cylindrical solution (see \nSection 3.3 of \\cite{R}). One can also show $\\sup v_a(t) \n\\leq 1$ for each Delaunay parameter $a$. We \nlet $\\epsilon (a) = \\min_{t \\in \\R} v_a(t)$. \n\nWe define the energy \n\\begin{equation} \\label{del_energy} \n\\mathcal{H} (v) = - \\dot v \\dddot v + \\frac{1}{2} (\\ddot v)^2 + \\left ( \n\\frac{n(n-4)+8}{4} \\right ) \\dot v^2 - \\frac{n^2(n-4)^2}{32} v^2 + \n\\frac{(n-4)^2(n^2-4)}{32} v^{\\frac{2n}{n-4}}.\n\\end{equation} \nDifferentiating $\\mathcal{H}$ with respect to $t$ we find \n$$ \\frac{d}{dt} \\mathcal{H} = \n-\\dot v \\left ( \\ddddot v - \\left ( \\frac{n(n-4)+8}{2} \\right ) \\ddot v + \n\\frac{n^2(n-4)^2}{16} - \\frac{n(n-4)(n^2-4)}{16} v^{\\frac{n+4}{n-4}} \n\\right ) ,$$\nand so $\\mathcal{H}(v)$ is constant if $v$ satisfies \\eqref{paneitz_ode1}. \nEvaluating this energy on the cylindrical and spherical \nsolutions we find \n\\begin{equation} \\label{cyl_sph_energy} \n\\mathcal{H}_{cyl} = \\mathcal{H} (v_{cyl}) = - \\frac{n(n-4)^2}{8} \n\\left ( \\frac{n(n-4)}{n^2-4} \\right )^{\\frac{n-4}{4}} < 0, \\qquad \n\\mathcal{H}_{sph} = \\mathcal{H} (v_{sph}) = 0. \n\\end{equation} \n\nRestricting attention to the $(v,\\dot v)$ in phase space we see \nthat the level set $\\{ \\mathcal{H} = 0 \\} \\cap \\{ \\ddot v = 0, \n\\dddot v = 0\\}$ consists entirely of the solution curve of $v_{sph}$ \ntogether with the point $(0,0)$. For each \n$$0 < H < -\\mathcal{H}_{cyl} = \\frac{n(n-4)^2}{8} \n\\left ( \\frac{n(n-4)}{n^2-4} \\right )^{\\frac{n-4}{8}}$$ \nthe level set $\\{ \\mathcal{H} = - H\\} \\cap \\{ \\ddot v = 0, \\dddot v = 0\\}$ is \na closed curve associated to the Delaunay solution $v_a$ for \nsome $a \\in (v_{cyl}, 1)$. Combining Theorems 1, 2, and 3 \nof \\cite{vdB} we find that these solution curves do not cross and \nand that the energy level completely determines the Delaunay solution. \nIn particular, we see \nthat $\\lim_{a \\nearrow 1} \\epsilon(a) = 0$. We sketch some of these \nsolution curves in the phase plane in Figure \\ref{phase_plane_fig} \nbelow. \n\n\\begin{figure} [h] \n\\centering\n\\begin{tikzpicture}\n\\coordinate (0) at (-2,-2); \n\\draw[->] (0,1) -- (8,1) coordinate[label = {below:$v$}] (xmax);\n\\draw[->] (1,-2) -- (1,4) coordinate[label = {right:$\\dot v$}] (ymax);\n\\draw[thick, blue,->>] (1,1) .. controls (2,4) and (5.5,4) .. (6,1);\n\\draw[thick, blue, ->>] (6,1) .. controls (5.5,-2) and (2,-2).. (1,1);\n\\draw[thick,red,->>] (2,1) .. controls (2.5,3) and (4.5,3) .. (5,1);\n\\draw[thick,red,->>] (5,1) .. controls (4.5,-1) and (2.5,-1) .. (2,1); \n\\draw[thick, green,->>] (2.8,1) .. controls (3.1,2) and (4,2) .. (4.2,1); \n\\draw[thick,green,->>] (4.2,1) .. controls (4,0) and (3.1,0) .. (2.8,1); \n\\node at (3.5,1) {$*$};  \n\\node at (6.8,-0.8) {cylindrical solution}; \n\\draw [->] (6.5,-0.5) -- (3.55,1); \n\\draw [->] (7,4) -- (5.5,2.3); \n\\node at (7.1,4.2) {spherical solution}; \n\\end{tikzpicture} \n\\caption{This figure shows the level curves of $\\mathcal{H}$ \nin the $(v,\\dot v)$ phase-plane.} \\label{phase_plane_fig}\n\\end{figure}\n\n\\subsection{Completion of the proof}\n\nFor each $T>0$ we consider metrics of the form $g_v = \nv^{\\frac{4}{n-4}} (dt^2 + d\\theta^2)$ on $\\Ss^1_T \\times \n\\Ss^{n-1}$, identifying $\\Ss^1_T$ with the \ninterval $[-T\/2,T\/2]$. Observe that the metric $g_1 = dt^2 + \nd\\theta^2$ has constant positive scalar curvature equal to \n$(n-1)(n-2)$, as well as positive $Q$-curvature $\\frac{(n-1)((n-1)^2-4)}{8}$, \nand so we may apply the theorem of Hang and Yang to \nconclude \n$$\\mathcal{Y}_4^+ ([dt^2 + d\\theta^2], \\Ss^1_T \\times \\Ss^{n-1})\n= \\mathcal{Y}_4 ([dt^2+d\\theta^2], \\Ss^1_T \\times \\Ss^{n-1}) < \n\\mathcal{Y}_4^+ ( [g_0], \\Ss^n).$$\nEach critical point of $\\mathcal{Q}$ \nin the conformal class $[dt^2 + d\\theta^2]$ must be a \nconstant $Q$-curvature metric on $\\Ss^1 \\times \\Ss^{n-1}$. \nWe pull this constant $Q$-curvature metric on $\\Ss^1_T \\times \\Ss^{n-1}$ \nback to the universal cover $\\R \\times \\Ss^{n-1}$, obtaining a \nsmooth, positive. $T$-periodic function \n$v : \\R \\times \\Ss^n \\rightarrow (0, \\infty)$\nsatisfying \\eqref{paneitz_pde1}. As we discussed above, Frank and \nK\\\"onig classified these solutions as either the constant \ncylindrical solution $v_{cyl}$, translates of the spherical \nsolution $v_{sph}$, or translates of a Delaunay solution \n$v_a$ for some $a \\in (v_{cyl}, 1)$. \n\nThe number of constant $Q$-curvature metrics in the \nconformal class $[dt^2 + d\\theta^2]$ on $\\Ss^1_T \\times \n\\Ss^{n-1}$ depends on $T$ in the following way. As in \nour previous discussion, we normalize the value of the \n$Q$-curvature to be $\\frac{n(n^2-4)}{8}$. The cylindrical \nsolution $v_{cyl}$ is the only solution when $0 < \nT \\leq T_{cyl}$, where $T_{cyl}$ is given in \\eqref{cyl_period}. \nFor $T_{cyl} < T \\leq 2T_{cyl}$ we have two constant $Q$-curvature \nmetrics, namely the cylinder and the Delaunay metric with Delaunay \nparameter $a$ such that $T = T_a$. When $2T_{cyl} < T\\leq 3T_{cyl}$ \nwe obtain $3$ constant $Q$-curvature metrics, namely the \ncylindrical solution $v_{cyl}$, the Delaunay solution $v_a$ \nsuch that $T_a = T$, and the Delaunay solution $v_\\alpha$ \nsuch that $T_\\alpha = T\/2$. Continuing inductively, when \n$(k-1)T_{cyl} < T \\leq k T_{cyl}$ we obtain $k$ distinct  \nconstant $Q$-curvature metrics, namely the cylindrical \nsolution $v_{cyl}$ together with the Delaunay solution $v_{a_l}$\nwith $T_{a_l} = T\/l$ for each $l = 1,2,\\dots,k-1$. \n\nFor each $T> T_{cyl}$ the Delaunay solution $v_{a_1}$ such that \n$T_a = T$ solves the initial value problem $v_a(0) = a$, \n$\\dot v_a(0) = 0$. By the results in \\cite{vdB} these two initial \nconditions actually uniquely determine a solution \nof \\eqref{paneitz_ode1}. Combining this uniqueness of \nwith the fact that $\\lim_{a \\nearrow 1} T_a = \\infty$ we \nconclude $v_a \\rightarrow v_{sph} = (\\cosh t)^{\\frac{4-n}{2}}$ \nas $a \\nearrow \\infty$. \nMoreover, because each $\\| v_a \\|_\\infty \\leq 1$, this \nconvergence is uniform on compact subsets by the Arzela-Ascoli \ntheorem. \n\nNext we show that $v_{a_1}$ is the only stable critical \npoint of $\\mathcal{Q}$ among $\\{ v_{cyl}, v_{a_1}, v_{a_2}, \n\\dots, v_{a_{k-1}} \\}$. The function $w_{a_l} = \\dot v_{a_l}$ \nsatisfies $L_{a_l} (w_{a_l}) = 0$, where $L_a$ is the linearization \nof \\eqref{paneitz_pde1} about $v_{a_l}$. Observe that \n$$\\{ t \\in [-T\/2, T\/2] : w_{a_l} > 0\\} = \\bigcup_{j=-\\lfloor l\/2 \n\\rfloor}^{\\lfloor l\/2 \\rfloor}  \\left ( j T_{a_l} , \\left ( \\frac{2j+1}{2} \\right ) \nT_{a_l} \\right ),$$\nwhere $\\lfloor l\/2 \\rfloor$ denotes the greatest non-negative integer \nless than or equal to $l\/2$. When $l \\geq 2$ the number of nodal \ndomains combined with Strum-Liouville theory implies $-L_{a_l}$ \nhas at least $l$ negative eigenvalues, and so $v_{a_l}$ cannot be a \nstable critical point of $\\mathcal{Q}$.  Furthermore the function \n$w_0 = \\cos (\\mu t)$, where $\\mu$ is given by \\eqref{cyl_period}, \nsatisfies $L_{cyl} (w_0) =0$, where $L_{cyl}$ is the linearization \nof \\eqref{paneitz_pde1} about $v_{cyl}$. When $T> 2 T_{cyl}$ the \nfunction $w_0$ has at least $2$ disjoint regions on which it is \npositive, so $v_{cyl}$ cannot be a stable critical point of $\\mathcal{Q}$ \nfor large values of $T$. \n\nWe conclude that $v_a$ minimizes $\\mathcal{Q}$ over \nthe conformal class $[dt^2 + d\\theta^2]$ on $\\Ss^1_{T_a} \n\\times \\Ss^{n-1}$, and so \n\\begin{eqnarray} \\label{del_tot_q_curv}\n\\mathcal{Y}_4^+ ([dt^2 + d\\theta^2], \\Ss^1_{T_a} \n\\times \\Ss^{n-1}) & = & \n\\mathcal{Q}(g_{v_a}) = \\frac{2}{n-4} \\frac{\\int_{\\Ss^1 \n\\times \\Ss^{n-1}} Q_{g_{v_a}} \nd\\mu_{g_{v_a}} }{(\\operatorname{Vol}_{g_{v_a}}(\\Ss^1 \n\\times \\Ss^{n-1}) )^{\\frac{n-4}{n}}} \\\\ \\nonumber \n& = & \\frac{2}{n-4} \\cdot \\frac{n(n^2-4)}{8}  \\frac \n{\\operatorname{Vol}_{g_{v_a}}(\\Ss^1 \\times \\Ss^{n-1}))}\n{(\\operatorname{Vol}_{g_{v_a}}(\\Ss^1 \n\\times \\Ss^{n-1}) )^{\\frac{n-4}{n}}} \\\\ \\nonumber \n& = & \\frac{n(n^2-4)}{4(n-4)} \\frac{ \\int_{-T_a\/2}^{T_a\/2} \\int_{\\Ss^{n-1}}\nv^{\\frac{2n}{n-4}} d\\theta dt} { \\left ( \\int_{-T_a\/2}^{T_a\/2} \n\\int_{\\Ss^{n-1}} v^{\\frac{2n}{n-4}} d\\theta dt \\right )^{\\frac{n-4}{n}}} \n\\\\ \\nonumber  \n& = & \\frac{n(n^2-4)}{4(n-4)} |\\Ss^{n-1}|^{n\/4}\n\\left ( \\int_{-T_a\/2}^{T_a\/2} v_a^{\\frac{2n}{n-4}} dt \\right )^{n\/4} .\n\\end{eqnarray}\n\nFinally, we let $a \\nearrow 1$ in \\eqref{del_tot_q_curv} to see \n\\begin{eqnarray*} \n\\mathbb{Y}_4^+ (\\Ss^1 \\times \\Ss^{n-1}) & \\geq & \n\\lim_{a \\nearrow 1} \\mathcal{Y}_4^+ ([dt^2 + d\\theta^2], \n\\Ss^1_{T_a} \\times \\Ss^{n-1}) \\\\ \n& = & \\frac{n(n^2-4)}{4(n-4)} |\\Ss^{n-1}|^{n\/4} \n\\left ( \\int_{-\\infty}^\\infty (v_{sph} (t) )^{\\frac{2n}{n-4}} dt \n\\right )^{n\/4} \\\\ \n& = & \\frac{n(n^2-4)}{4(n-4)} |\\Ss^{n-1}|^{n\/4} \\left ( \n\\int_\\R (\\cosh t)^{-n} dt \\right )^{n\/4} \\\\ \n& = & \\frac{n(n^2-4)}{4(n-4)} |\\Ss^{n-1}|^{n\/4} \n\\left ( \\int_0^\\infty \\left ( \\frac{1+r^2}{2} \\right )^{-n} r^{n-1}\ndr \\right )^{n\/4}\\\\ \n& = & \\mathcal{Q} (g_0) = \\mathcal{Y}_4^+ ([g_0], \n\\Ss^n) = \\mathbb{Y}_4^+(\\Ss^n),\n\\end{eqnarray*}  \nwhere $r = e^{-t}$. 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Yang. {\\it $Q$-curvature on a class of manifolds \nwith dimension at least $5$.} Comm. Pure Appl. Math. {\\bf 69} (2016), 1452--1491. \n\n\\bibitem {Pan1} S. Paneitz. {\\it A quartic conformally covariant differential operator \nfor arbitrary pseudo-Riemannian manifolds.} SIGMA Symmetry Integrability Geom. \nMethods Appl. {\\bf 4} (2008), 3 pages (preprint from 1983). \n\n\\bibitem {R} J. Ratzkin. {\\it On constant $Q$-curvature metrics with \nisolated singularities.} preprint, {\\tt arXiv:2001.07984}. \n\n\\bibitem {Rob} F. Robert. {\\it Fourth order equations with critical \ngrowth in Riemannian geometry.} private notes, available at \n{\\tt http:\/\/www.iecl.univ-lorraine.fr\/$\\sim$Frederic.Robert\/}\n\n\\bibitem{Sch} R. Schoen. {\\it Conformal deformation of a Riemannian \nmetric to constant scalar curvature.} J. Diff. Geom. {\\bf 20} (1984), 479--495. \n\n\\bibitem{Sch_var} R. Schoen. {\\it Variational theory for the total scalar \ncurvature functional for Riemannian metrics and related topics.} \nin {\\it Topics in calculus of variations.} Lecture Notes in Math. {\\bf 1365}, \nSpringer-Verlag (1989), 120--154.  \n\n\\bibitem {Y} H. Yamabe. {\\it On the deformation of Riemannian structures \non a compact manifold.} Osaka Math. J. {\\bf 12} (1960), 21--37. \n\n\\end {thebibliography}\n\n\\end {document}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\tIn this paper, we are mainly concerned with the existence and nonexistence of positives solutions to the following semilinear elliptic inequalities\n\t\\begin{equation}\\label{ieq}\n\t\t\\Delta u+u^p\\left|\\nabla u\\right|^q\\leq0,\\quad\\mbox{on $G$},\n\t\\end{equation}\n\twhere $(p, q)\\in \\mathbb{R}^2$, $G=(V, E)$ is an infinite connected locally finite graph, and $V$\n\tis the collection of the vertices, and $E$ is the collection of edges. Throughout the paper,\n\tthere exists only at most one edge for any two\n\tdistinct vertices, and exists no any edge from a vertex to itself.\n\t\n\tIf there is an edge connecting $x, y\\in V$, we say\n\t$x\\sim y$. On each edge, let us define an edge weight $\\mu: E \\to(0,\\infty)$, which satisfies\n\t$\\mu_{xy} = \\mu_{yx}$, and $\\mu_{xy} > 0$ if $x\\sim y$. Here $\\mu$ can be understood as a map from\n\t$V\\times V\\to[0,\\infty)$ by adding that $\\mu_{xy}=0$ provided that there is no edge connecting $x, y$.\n\tSuch graph $G=(V, E, \\mu)$ with edge weight $\\mu$ is called a weighted graph. Sometimes, we use $(V, \\mu)$ to denote\n\tthe weighted graph $G$ for brevity.\n\t\n\tIn this paper, we say condition $(p_0)$ is is satisfied on $G$: if there exists a constant $p_0\\geq 1$ such that for any $x\\sim y$ in $V$,\n\\begin{equation}\n\t\\frac{\\mu_{xy}}{\\mu(x)}\\geq \\frac{1}{p_0}.\\tag{$p_0$}\n\\end{equation}\n\t\n\t\n\tFor each vertex  $x\\in V$, let us define vertex measure $\\mu(x)=\\sum\\limits_{x \\sim y} \\mu_{xy}$,\n\tand the Laplace operator $\\Delta$ on $G$ (see \\cite{G2}) as\n\t\\begin{equation}\\label{lap}\n\t\\Delta u(x)=\\sum\\limits_{y \\sim x} \\frac{\\mu_{xy}}{\\mu(x)}(u(y)-u(x)),\\quad\\mbox{for $u\\in \\mathcal{l}(V)$},\n\t\\end{equation}\n\tand define the gradient form $\\Gamma$ (see \\cite{BHLLMY}.) as\n\t$$\\Gamma(f,g)=\\sum\\limits_{y \\sim x}\\frac{\\mu_{xy}}{2\\mu(x)}(f(y)-f(x))(g(y)-g(x)), \\quad\\mbox{for $f,g\\in \\mathcal{l}(V)$},$$\nthen the norm of gradient is defined by\n\t\\begin{align}\\label{gra}\n\t\t|\\nabla u(x)|=\\sqrt{\\Gamma(u,u)}=\\sqrt{\\sum\\limits_{y \\sim x}\\frac{\\mu_{xy}}{2\\mu(x)}(u(y)-u(x))^2},\n\t\t\\quad\\mbox{for $u\\in \\mathcal{l}(V)$},\n\t\\end{align}\n\twhere $\\mathcal{l}(V)$ is the collection\n\tof all real functions on $V$.\n\t\n\tBesides, there exists graph distance $d(x,y)$ on $G$ which means the minimal\n\tnumber of edges in a path among all possible paths connecting $x$ and $y$ in $V$. Fix some referenced vertex $o\\in V$, and for integer $n\\geq1$, let\n\t$$B(o,n):=\\{x \\in V: d(o,x) \\leq n\\}$$\n\tbe the closed ball centered at $o$ with radius $n$, and the volume of $B(o, n)$ be\n\t$$\\mu(B(o, n))=\\sum_{x\\in B(o,n)}\\mu(x).$$\n\t\nRecently, the study on elliptic equation on weighted graphs has attracted a lot of attentions, see \\cite{CM} \\cite{GHJ}, \\cite{GLY2} \\cite{GLY3}\n\\cite{HSZ}, \\cite{HWY} \\cite{LiuY}.\n\tIn this paper we would like to solve the following problem: what kind of sharp\n\tassumptions on $\\mu(B(o,n))$ can suffice the nonexistence of  nontrivial positive solution $u$ to (\\ref{ieq})?\n\tThere are two folds in this problem: the first one is to find these volume growth and to prove the nonexistence\n\tresults; the second is to show these volume assumptions are sharp.\n\t\n\t\n\tBy using the volume assumption to obtain the nonexistence and existence of solution to elliptic differential inequalities\n\tis widely used in the literature. Recall the famous Nash-Williams' test (e.g. \\cite{W}): if\n\t\\begin{eqnarray}\\label{votest}\n\t\t\\sum^{\\infty}_{n=1}\\frac{n}{\\mu(B(o,n))}=\\infty,\n\t\\end{eqnarray}\n\tthen any nonnegative solution on $(V, \\mu)$ is identically equal to a constant, or equivalently to say, $(V, \\mu)$ is parabolic\n\tor recurrent.\n\t\n\n\t\n\tThe notion of parabolicity of graph can be regarded as a generalization of the parabolicity of manifolds,\n\tsee Cheng-Yau's paper \\cite{CY}, and  Grigor'yan \\cite{G85}, Karp\\cite{K}, Varopoulos \\cite{V} for further developments.\n\t\n\t\n\tRecently, Gu, Sun, and Huang in \\cite{GSH} proved that, for $p>1$, if condition $(p_0)$ is satisfied, and if\n\t\\begin{eqnarray}\n\t\t\\mu(B(o,n))\\lesssim n^{\\frac{2p}{p-1}}(\\ln n)^{\\frac{1}{p-1}},\n\t\\end{eqnarray}\n\tthen $\\Delta u+u^p\\leq0$ admits no positive solution. While for $0<p\\leq1$, $\\Delta u+u^p\\leq0$ admits no positive solution.\n\tHere the exponent $\\frac{2p}{p-1}$ and $\\frac{1}{p-1}$ are also showed to be sharp when $p>1$.\n\t\nGu-Sun-Huang's result can be considered as a discrete version obtained by Grigor'yan-Sun on manifold case in \\cite{GS1}, where they proved that\n\\begin{eqnarray*}\n\\mu(B(o,r))\\lesssim r^{\\frac{2p}{p-1}}(\\ln r)^{\\frac{1}{p-1}}, \\quad\\mbox{for all large enough $r$},\n\\end{eqnarray*}\nthen there exists no nonnegative solution to $\\Delta u+u^p\\leq0$ on geodesically complete noncompact manifolds.\tHere $\\mu$ is Riemannian measure\non manifolds.\n\n\tMotivated by these results, we would like to study problem (\\ref{ieq}) involving\n\tgradient terms on weighted graphs. To express our classification more clearly, let us divide $R^2$\n\tinto six parts (Figure \\ref{fig1}).\n\t\\begin{align*}\n\t\t&G_1=\\{(p,q)|p\\geq 0,1-p<q<2\\},\\quad G_2=\\{(p,q)|q\\geq 2\\} \\\\\n\t\t&G_3=\\{(p,q)|p<0,1<q<2\\}, \\qquad \\quad G_4=\\{(p,q)|p<0,q=1 \\}\\\\\n\t\t&G_5=\\{(p,q)|p+q=1,p\\geq0,q>0,\\;\\text{or}\\;p+q=1,q<0\\}\\\\\n\t\t&G_6=\\{(p,q)|p<1-q,q<1,\\;\\text{or}\\; (p,q)=(1,0)\\}\\\\\n\t\\end{align*}\n\n\t\\begin{figure}[h]\n\t\t\\begin{tikzpicture}[x={(0.8cm,0cm)},y={(0cm,0.8cm)}]\\label{fig1}\n\t\t\t\\draw[->] (-6,0)--(6,0) ;\n\t\t\t\\draw[->] (0,-4)--(0,6);\n\t\t\t\\fill[green,opacity=0.7] (-6,6)--(6,6)--(6,4)--(-6,4) ;\n\t\t\t\\fill[blue,opacity=0.7] (0,4)--(6,4)--(6,-4)--(0,2) ;\n\t\t\t\\fill[yellow,opacity=0.7] (0,4)--(-6,4)--(-6,2)--(0,2);\n\t\t\t\\fill[red,opacity=0.7] (0,2)--(-6,2)--(-6,-4)--(6,-4);\n\t\t\t\\draw[very thick,dashed]  (-6,2)--(0,2);\n\t\t\t\\draw[very thick,dotted]  (6,-4)--(2,0) (2,0)--(0,2);\n\t\t\t\\fill[blue,opacity=0.7] (7.7,5.4) rectangle(8.3,4.8);\n\t\t\t\\fill[green,opacity=0.7] (7.7,3.6) rectangle(8.3,3);\n\t\t\t\\fill[yellow,opacity=0.7] (7.7,1.8) rectangle(8.3,1.2);\n\t\t\t\\fill[red,opacity=0.7] (7.7,0) rectangle(8.3,-0.6);\n\t\t\t\\fill[red,opacity=1] (2,0) circle(0.05);\n\t\t\t\\draw[dashed]  (7.5,-1.9)--(8.5,-1.9);\n\t\t\t\\draw[dotted]  (7.5,-3.3)--(8.5,-3.3);\n\t\t\t\\node[above] at (0,6) {$q$};\n\t\t\t\\node[right] at (6,0) {$p$};\n\t\t\t\\node[below] at (-5,4) {\\tiny{$q=2$}};\n\t\t\t\\node[below] at (-5,2) {\\tiny{$q=1$}};\n\t\t\t\\node[below] at (1.7,0) {\\tiny{$(1,0)$}};\n\t\t\t\\node[below] at (3,-1.6) {\\tiny{$p+q=1$}};\n\t\t\t\\node[right] at (8.3,5.1) {\\small{$G_1$}};\n\t\t\t\\node[right] at (8.3,3.3) {\\small{$G_2$}};\n\t\t\t\\node[right] at (8.3,1.6) {\\small{$G_3$}};\n\t\t\t\\node[right] at (8.3,-0.3) {\\small{$G_6$}};\n\t\t\t\\node[right] at (8.5,-1.9) {\\small{$G_4$}};\n\t\t\t\\node[right] at (8.5,-3.3) {\\small{$G_5$}};\n\t\t\\end{tikzpicture}\n\t\\caption{}\n\t\\end{figure}\n\t\n\n\tOur main results are as follows:\n\t\\begin{theorem}\\label{thm1} \\rm{\n\t\t\tLet $G=(V,E, \\mu)$ be an infinite, connected, locally finite graph on which  condition $(p_0)$\n\t\t\tis satisfied. Fix some $o\\in V$.\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item[(I).]{Assume $(p,q)\\in G_1$. If\n\t\t\t\t\t\\begin{align}\\label{vol-1}\n\t\t\t\t\t\t\\mu(B(o,n)) \\lesssim n^{\\frac{2p+q}{p+q-1}}(\\ln{n})^{\\frac{1}{p+q-1}},\n\t\t\t\t\t\t\\quad\\mbox{for all $n>>1$},\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tthen (\\ref{ieq}) admits no nontrivial positive solution.}\n\t\t\t\t\n\t\t\t\t\\item[(II).]{Assume $(p,q)\\in G_2$. If\n\t\t\t\t\t\\begin{align}\\label{vol-2}\n\t\t\t\t\t\t\\mu(B(o,n)) \\lesssim n^{2}\\ln{n},\\quad \\mbox{for all $n>>1$},\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tthen (\\ref{ieq}) admits no nontrivial positive solution.}\n\t\t\t\t\n\t\t\t\t\\item[(III).]{Assume $(p,q)\\in G_3$. If\n\t\t\t\t\t\\begin{align}\\label{vol-3}\n\t\t\t\t\t\t\\mu(B(o,n)) \\lesssim n^{\\frac{q}{q-1}}(\\ln{n})^{\\frac{1}{q-1}},\\quad \\mbox{for all $n>>1$},\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tthen (\\ref{ieq}) admits no nontrivial positive solution.}\n\t\t\t\t\n\t\t\t\t\\item[(IV).]{Assume $(p,q)\\in G_4$.  For any given $\\alpha>0$, if\n\t\t\t\t\t\\begin{align}\\label{vol-4}\n\t\t\t\t\t\t\\mu(B(o,n)) \\lesssim n^{\\alpha}, \\quad \\mbox{for all $n>>1$},\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tthen (\\ref{ieq}) admits no nontrivial positive solution.}\n\t\t\t\t\n\t\t\t\t\\item[(V).]{Assume $(p,q)\\in G_5$. There exists some $k_0>0$, for any given $\\kappa$ satisfying $0<\\kappa<k_0$, if\n\t\t\t\t\t\\begin{align}\\label{vol-5}\n\t\t\t\t\t\t\\mu(B(o,n)) \\lesssim e^{\\kappa n},\\quad \\mbox{for all $n>>1$},\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tthen (\\ref{ieq})  admits no nontrivial positive solution.}\n\t\t\t\t\n\t\t\t\n\t\t\\end{enumerate}}\n\t\\end{theorem}\n\t\\begin{remark}\n\t\tIn Theorem \\ref{thm1} (II), by Nash-Williams's test, (\\ref{vol-2}) condition can be relaxed to (\\ref{votest}).\n\t\\end{remark}\n\n\n\t\\textbf{Notations.} In the above and below, the letters $C,C',C_0,C_1,c_0,c_1$... denote positive constants whose values are unimportant and may vary at different occurrences. $A\\lesssim B$ means that the quotient of $A$ and $B$ is bounded from the above, $A\\gtrsim B$ means that the quotient of $A$ and $B$ is bounded from below, and $A\\asymp B$ means both $A\\lesssim B$\n\tand $A\\gtrsim B$ hold.\n\t\n\\vskip1ex\n\nFor $(p,q)\\in G_6$, we have the following nonexistence result.\n\\begin{theorem}\\label{thm1-1} \\rm{\nLet $G=(V,E, \\mu)$ be an infinite, connected, locally finite graph.\nUnder any of the following two assumptions\n\\begin{enumerate}\n\\item[(1).]\n{Assume $(p_0)$ condition is satisfied, and $p<1-q, q<0$; }\n\\item[(2).]{If either $p<1-q, 0\\leq q<1$ or $(p,q)=(1,0)$;}\n\\end{enumerate}\nthen (\\ref{ieq}) admits no nontrivial positive solution.}\n\\end{theorem}\n\n\\begin{remark}\\rm{\nLet us compare our results with the one obtained by Sun-Xiao-Xu on manifolds in \\cite{SXX}.\nOn manifolds, $(p,q)\\in\\mathbb{R}^2$ are also divided into six parts:\n\\begin{eqnarray*}\n&G_1^{\\prime}=\\{p\\geq0,1-p<q<2\\}, \\quad &G_2^{\\prime}=\\{q\\geq2\\},\\\\\n&G_3^{\\prime}=\\{p<0, 1<q<2\\},\\quad & G_4^{\\prime}=\\{p<0, q=1\\},\\\\\n&G_5^{\\prime}=\\{p=1-q, q\\leq1\\},\\quad &G_6^{\\prime}=\\{p<1-q, q<1\\}.\n\\end{eqnarray*}\nClearly, $G_i=G_i^{\\prime}$ for $i=1,2,3,4$, $G_5=G_5^{\\prime}\\setminus(1,0)$, $G_6=G_6^{\\prime}\\cup (1, 0)$.\nWhen  $(p, q)\\in G_i$ for $i=1,2,3,4,5$,\nthe graph behaves quite similarly as manifolds, namely, under the volume growth of\nthe same forms, the nonexistence results hold. However, when $(p,q)\\in G_6^{\\prime}$, there exists some sharp exponentially volume growth, under which the nonexistence results hold, while on weighted graphs, there exists no such sharp volume growth to suffice nonexistence results.\nHence, it is very interesting to point out that the results on weighted graphs does not need to be accordance with these obtained on manifolds.\n\n}\n\\end{remark}\t\n\\vskip1ex\n\n\tWe also show that our volume growth conditions in Theorem \\ref{thm1} are sharp, roughly speaking, if the volume conditions in Theorem \\ref{thm1} are relaxed, then there exist a graph which admits a nontrivial positive solution to (\\ref{ieq}). More precisely, we show these sharpness on the so-called homogeneous tree $T_N$.\n\t\n\tFix $N\\geq2$, recall that a connected\n\tgraph $(V, E)$ is a tree if for any two distinct points $x, y \\in V $, there is only one path between x and y. A homogeneous tree $T_N$ is a tree whose all vertices share the same degree $N$. The following existence result is obtained on $T_N$.\n\t\\begin{theorem}\\label{thm2} \\rm{\n\t\t\tLet $(V,E)=T_N$.\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item[(I).]{Assume $(p,q)\\in G_1$. For any arbitrary small $\\epsilon>0$, there exists a weight $\\mu$ on $T_N$ satisfying\n\t\t\t\t\t\\begin{align}\\label{e-vol-1}\n\t\t\t\t\t\t\\mu(B(o,n)) \\asymp n^{\\frac{2p+q}{p+q-1}}(\\ln{n})^{\\frac{1}{p+q-1}+\\epsilon},\n\t\t\t\t\t\t\\quad\\mbox{for $n\\geq 2$,}\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tsuch that (\\ref{ieq}) admits a nontrivial positive solution on $(V, \\mu)$.}\n\t\t\t\t\n\t\t\t\t\\item[(II).]{Assume $(p,q)\\in G_2$. For any arbitrary small $\\epsilon>0$, there exists a weight $\\mu$ on $T_N$ satisfying\n\t\t\t\t\t\\begin{align}\\label{e-vol-2}\n\t\t\t\t\t\t\\mu(B(o,n)) \\asymp n^{2}(\\ln{n})^{1+\\epsilon},\\quad \\mbox{for $n\\geq 2$,}\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tsuch that (\\ref{ieq}) admits a nontrivial positive solution on $(V, \\mu)$.}\n\t\t\t\t\n\t\t\t\t\\item[(III).]{Assume $(p,q)\\in G_3$. For any arbitrary small $\\epsilon>0$, there exists a weight $\\mu$ on $T_N$ satisfying\n\t\t\t\t\t\\begin{align}\\label{e-vol-3}\n\t\t\t\t\t\t\\mu(B(o,n)) \\asymp n^{\\frac{q}{q-1}}(\\ln{n})^{\\frac{1}{q-1}+\\epsilon},\\quad \\mbox{for $n\\geq 2$,}\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tsuch that (\\ref{ieq}) admits a nontrivial positive solution on $(V, \\mu)$.}\n\t\t\t\t\n\t\t\t\t\\item[(IV).]{Assume $(p,q)\\in G_4$.  Given $\\lambda>0$, there exists a weight $\\mu$ on $T_N$ satisfying\n\t\t\t\t\t\\begin{align}\\label{e-vol-4}\n\t\t\t\t\t\t\\mu(B(o,n)) \\asymp e^{\\lambda n}, \\quad \\mbox{for $n\\geq 2$,}\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tsuch that (\\ref{ieq}) admits a nontrivial positive solution on $(V, \\mu)$,}\n\t\t\t\t\n\t\t\t\t\\item[(V).]{Assume $(p,q)\\in G_5$. Then there exist a weight $\\mu$ on $T_N$, and a positive constant $\\lambda$ satisfying\n\t\t\t\t\t\\begin{align}\\label{e-vol-5}\n\t\t\t\t\t\t\\mu(B(o,n)) \\asymp e^{\\lambda n}\\qquad \\mbox{for $n\\geq 2$,}\n\t\t\t\t\t\\end{align}\n\t\t\t\t\tsuch that (\\ref{ieq}) admits a nontrivial positive solution on $(V, \\mu)$.}\n\t\t\\end{enumerate}}\n\t\\end{theorem}\n\t\n\t\\section{Proof of Theorem \\ref{thm1} and \\ref{thm1-1}}\n\tBefore proceeding to the proof of Theorem \\ref{thm1}, we first introduce Lemma \\ref{lem1}\nand \\ref{lem2}, which play important roles in proof of Theorem \\ref{thm1}.\n\t\t\\begin{lemma}\\label{lem1}\\rm{\n\t\t\tLet $(V,\\mu)$ satisfies $(p_0)$ condition. If $u$ is a nonnegative solution to (\\ref{ieq}), then either $u\\equiv 0$ or $u>0$ and\n\t\t\t\\begin{align}\\label{ep}\n\t\t\t\t\\frac{1}{p_0}\\leq\\frac{u(x)}{u(y)} \\leq p_0, \\quad\\mbox{if $y\\sim x$}.\n\t\t\\end{align}}\n\t\\end{lemma}\n\t\\begin{proof}\n\t\tThe proof is similar to \\cite[Lemma 3.1]{GSH}.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\end{proof}\nIn the following, for brevity, we denote by\n $$\\nabla_{xy}f=f(y)-f(x),\\quad \\mbox{for $f\\in \\mathcal{l}(V)$}.$$\n\nLet $\\Omega$ be a non-empty subset of $V$, we say that $u$ satisfies\n\\begin{align}\\label{vieq}\n\t\\Delta u(x)\n\t+ u(x)^{p} |\\nabla u(x)|^{q}\\leq 0,\\quad\\mbox{ when $x\\in \\Omega $}.\\end{align}\nmeans that (\\ref{vieq}) holds only for these vertices $x\\in \\Omega$, where $\\Delta u$ and $|\\nabla u|$ are still defined by\n(\\ref{lap}) and (\\ref{gra}) respectively in $V$.\n\\begin{lemma}\\label{lem2}\\rm{\n\t\tAssume $p+q\\neq 1$,\n $(V,\\mu)$ satisfies $(p_0)$ condition,\n and $\\Omega$ is a non-empty subset of $V$.\n\t\tLet $u$ be a nontrivial positive function on $V$ which satisfies (\\ref{vieq}), and  $\\frac{1}{p_0}\\leq\\frac{u(x)}{u(y)} \\leq p_0$ for any $y\\sim x$. Furthermore, when $\\Omega\\not=V$, assume $u$ also satisfies $u(y)-u(x)\\geq 0$ for any $(x,y) \\in\\{(x,y)|y\\sim x, x\\in\\Omega\\mbox{ and }y\\in \\Omega^c\\}$.\n\t\tThen there exists a positive pair\n\t\t$(s,t)$ such that for any $0\\leq\\varphi\\leq 1$ with compact support in $\\Omega $, the following estimates hold:\n\t\t\\begin{align}\\label{est-1}\n\t\t\t&\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t}|\\nabla u(x)|^q\\varphi(x)^s \\nonumber\\\\\n\t\t\t\\leq& C_{p_0,t} (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}} t^{- \\frac{p+t(q-1)}{p+q-1}}\n\t\t\t\\left(\\sum_{\\substack {x,y\\in \\Omega \\\\ \\nabla_{xy} \\varphi \\neq 0}} \\mu_{xy}\\varphi(x)^s u(x)^{p-t}|\\nabla u(x)|^q\\right)^{\\frac{1-t}{p+q-t}}\\nonumber\\\\\n\t\t\t&\\times\\left(\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy} |\\nabla_{xy}\\varphi|^{\\frac{2p+q+t(q-2)}{p+q-1}}\\right)^{\\frac{p+q-1}{p+q-t}},\n\t\t\\end{align}\n\t\tand\n\t\t\\begin{align}\\label{est-2}\n\t\t\t\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^q \\varphi(x)^s\n\t\t\t\\leq& ( C'_{p_0,t})^{\\frac{p+q-t}{p+q-1}}  (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}} t^{-\\frac{p+t(q-1)}{p+q-1}}\\nonumber\\\\\n\t\t\t&\\times\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy} |\\nabla_{xy}\\varphi|^{\\frac{2p+q+t(q-2)}{p+q-1}},\n\t\t\\end{align}\n\t\twhere\n\t\t$C_{p_0,t}= \\frac{(\\sqrt{2p_0}(1+p^t_0))^{\\frac{p + t(q-1)}{p + q-t}+1}   (p_0^{t+1})^{\\frac{p + t(q-1)}{p + q-t}}}{4}$, $C'_{p_0,t} =( C_{p_0,t})^{\\frac{p+q-t}{p+q-1}}$,\n\t\tand $s,t$ satisfy\n\t\t\\begin{equation}\\label{st-cond}\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{lr}\n\t\t\t\t\\frac{2p + q + t(q-2)}{p + q-t} > 1, \\\\\n\t\t\t\t\\frac{p+q-t}{1-t} > 1,\\\\\n\t\t\t\ts > \\frac{2p+q+t(q-2)}{p+q-1}.\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\\end{equation}}\n\\end{lemma}\n\n\\begin{proof}\n\tFor  $ \\varphi \\in \\mathcal{l}(\\Omega) $ with compact support in $\\Omega $, define $\\psi=\\varphi^s u^{-t}$, where $(s,t)$ are to be chosen later.\n\t\n\tMultiplying both sides of (\\ref{vieq}) by $\\mu(x)\\psi(x)$ and summing up over all $x\\in \\Omega $, we obtain\n\t\\begin{equation*}\n\t\t\\sum\\limits_{x \\in \\Omega,y\\in V }\\mu_{xy}(\\nabla_{xy}u)\\psi(x)\n\t\t+\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p} |\\nabla u(x)|^{q}\\psi(x)\\leq 0,\n\t\\end{equation*}\nIt follows that\n\\begin{align}\\label{lem2-1}\n\t&\\sum\\limits_{x,y \\in \\Omega}\\mu_{xy}(\\nabla_{xy}u)\\psi(x)+\\sum\\limits_{x \\in \\Omega, y\\in \\Omega^c}\\mu_{xy}(\\nabla_{xy}u)\\psi(x)\\nonumber \\\\\n\t&+\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p} |\\nabla u(x)|^{q}\\psi(x)\\leq 0.\n\\end{align}\nSpecially, when $\\Omega=V$, we have $\\sum\\limits_{x \\in \\Omega, y\\in \\Omega^c}\\mu_{xy}(\\nabla_{xy}u)\\psi(x) =0$.\n\n\tNoting\n\t\\begin{equation*}\n\t\t\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy}(\\nabla_{xy}u)\\psi(x)=\n\t\t-\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy}(\\nabla_{xy}u)(\\nabla_{xy}\\psi),\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\t\\nabla_{xy}\\psi=\\nabla_{xy}(\\varphi^s u^{-t})=\n\t\tu(y)^{-t} \\nabla_{xy}(\\varphi^s)+ \\varphi(x)^s \\nabla_{xy}(u^{-t}),\n\t\\end{equation*}\n\twe obtain\n\t\\begin{align*}\n\t\n\t\t\t\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy}(\\nabla_{xy}u)\\psi(x)=&\n\t\t\t-\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(y)^{-t} (\\nabla_{xy}u) \\nabla_{xy}(\\varphi^s)\\nonumber\\\\\n\t\t\t&-\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega}\\mu_{xy} \\varphi(x)^s (\\nabla_{xy}u)\n\t\t\t\\nabla_{xy}(u^{-t}).\n\t\n\t\\end{align*}\n\tThen (\\ref{lem2-1}) is transformed to\n\t\\begin{align}\\label{lem2-2}\n\t\t&-\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} \\varphi(x)^s (\\nabla_{xy}u)\n\t\t\\nabla_{xy}(u^{-t})+\\sum\\limits_{x \\in \\Omega, y\\in \\Omega^c}\\mu_{xy}(\\nabla_{xy}u)\\varphi(x)^su(x)^{-t}\\nonumber \\\\\n\t\t& +\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s\n\t\t\\leq \\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(y)^{-t} (\\nabla_{xy}u) \\nabla_{xy}(\\varphi^s).\n\t\\end{align}\nUsing the mid-value theorem, we have some $\\xi$ which is between $u(y)$ and $u(x)$, such that\n\t\\begin{equation*}\n\t\t\\nabla_{xy}(u^{-t})=u(y)^t-u(x)^t=-t\\xi^{-t-1}(u(y)-u(x))\n\t\t=-t\\xi^{-t-1}\\nabla_{xy}u,\n\t\\end{equation*}\n\tBy $\\frac{1}{p_0}\\leq\\frac{u(x)}{u(y)} \\leq p_0$,\n\n\twe have\n\t$\\frac{u(x)}{p_0} \\leq \\xi \\leq u(x) p_0$, and\n\t\\begin{align}\\label{lem2-3-1}\n\t\t&-\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} \\varphi(x)^s (\\nabla_{xy}u)\n\t\t\\nabla_{xy}(u^{-t})\\nonumber\\\\\n\t\t&=\\frac{t}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} \\varphi(x)^s (\\nabla_{xy}u)^2\n\t\t\\xi^{-t-1}\\nonumber\\\\\n\t\t&\\geq \\frac{t}{2p_0^{t+1}}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} \\varphi(x)^su(x)^{-t-1} (\\nabla_{xy}u)^2,\n\t\\end{align}\nBy $0<\\frac{u(y)-u(x)}{u(x)}\\leq \\frac{p_0u(x)-u(x)}{u(x)}=p_0-1$, we obtain\n\\begin{align}\\label{lem2-3-2}\n\t&\\sum\\limits_{x \\in \\Omega, y\\in \\Omega^c}\\mu_{xy}(\\nabla_{xy}u)\\varphi(x)^su(x)^{-t}\n\t\\nonumber\\\\&\n\t\\geq \\frac{1}{p_0-1}\\sum\\limits_{x \\in \\Omega, y\\in \\Omega^c}\\mu_{xy}(\\nabla_{xy}u)^2\\varphi(x)^su(x)^{-t-1}\n\t\\nonumber\\\\&\n\t\\geq \\frac{t}{2p_0^{t+1}}\\sum\\limits_{x \\in \\Omega, y\\in \\Omega^c}\\mu_{xy}(\\nabla_{xy}u)^2\\varphi(x)^su(x)^{-t-1}.\n\\end{align}\nwhere we have used that if $\\Omega\\neq V$, $u(y)>u(x)$ for $x\\in\\Omega$, $y\\in\\Omega^c$, and $2p_0^{t+1}\\geq t(p_0-1)$ holds for all $t\\geq0$.\n\t\n\tCombining (\\ref{lem2-3-1}) with (\\ref{lem2-3-2}), we get\n\t\\begin{align}\\label{lem2-3}\n\t\t-\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega }&\\mu_{xy} \\varphi(x)^s (\\nabla_{xy}u)\n\t\t\\nabla_{xy}(u^{-t})+\\sum\\limits_{x \\in \\Omega, y\\in \\Omega^c}\\mu_{xy}(\\nabla_{xy}u)\\varphi(x)^su(x)^{-t}\n\t\t\\nonumber\\\\&\n\t\t\\geq  \\frac{t}{2p_0^{t+1}}\\sum\\limits_{x \\in \\Omega, y\\in V}\\mu_{xy}(\\nabla_{xy}u)^2\\varphi(x)^su(x)^{-t-1}\n\t\t\\nonumber\\\\&\n\t\t=\\frac{t}{p_0^{t+1}}\\sum\\limits_{x \\in \\Omega}\\mu(x)|\\nabla u(x)|^2\\varphi(x)^su(x)^{-t-1}.\n\t\\end{align}\nEspecially, when $\\Omega=V$, (\\ref{lem2-3}) can be deduced from (\\ref{lem2-3-1}) directly.\n\n\tBy the mid-value theorem, there is some $\\eta$ between $\\varphi(x)$ and $\\varphi(y)$ such that\n\t\\begin{equation}\\label{lem2-4}\n\t\t\\nabla_{xy}(\\varphi^s)=s\\eta^{s-1}(\\varphi(y)-\\varphi(x))\n\t\t=s\\eta^{s-1}\\nabla_{xy}\\varphi.\n\t\\end{equation}\n\tSubstituting (\\ref{lem2-4}) and (\\ref{lem2-3}) into (\\ref{lem2-2}), we have\n\t\\begin{align}\\label{lem2-5}\n\t\t&\\frac{t}{p_0^{t+1}}\\sum\\limits_{x \\in \\Omega }\\mu(x) \\varphi(x)^s      u(x)^{-t-1} |\\nabla u(x)|^2\n\t\t+\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s\\nonumber\\\\&\n\t\t\\leq \\frac{s}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(y)^{-t}\\eta^{s-1} (\\nabla_{xy}u)( \\nabla_{xy}\\varphi).\n\t\\end{align}\n\tObserving that $|\\nabla u(x)|^2=\\sum\\limits_{y \\in V }\\frac{\\mu_{xy}}{2\\mu(x)}(\\nabla_{xy}u)^2$, and\n\t$ \\frac{1}{2p_0} \\leq \\frac{\\mu_{xy}}{2\\mu(x)} \\leq \\frac{1}{2}$, we derive\n\t\\begin{align}\\label{grd}\n\t\t|\\nabla_{xy}u| \\leq \\sqrt{2p_0}|\\nabla u(x)|,\\quad \\mbox{for any $y\\sim x$.}\n\t\\end{align}\n\t\n\t Since $\\eta^{s-1} \\leq\\varphi(x)^{s-1}+\\varphi(y)^{s-1} $, $\\frac{u(x)}{p_0} \\leq \\xi \\leq u(x) p_0 $, and (\\ref{grd}), we have\n\t\\begin{align}\\label{lem2-6}\n\t\t\\frac{s}{2}&\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(y)^{-t}\\eta^{s-1} (\\nabla_{xy}u)( \\nabla_{xy}\\varphi)\\nonumber \\\\\n\t\t&\\leq \\frac{s}{2}\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(y)^{-t}(\\varphi(x)^{s-1}+\\varphi(y)^{s-1}) (\\nabla_{xy}u)( \\nabla_{xy}\\varphi)\\nonumber \\\\\n\t\t&\\leq \\frac{s}{2}(1+p^t_0)\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(x)^{-t}\\varphi(x)^{s-1} (\\nabla_{xy}u)( \\nabla_{xy}\\varphi)\\nonumber \\\\\n\t\t&\\leq \\frac{s}{2}(1+p^t_0)\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(x)^{-t}\\varphi(x)^{s-1} |\\nabla_{xy}u|| \\nabla_{xy}\\varphi|\\nonumber\\\\\n\t\t&\\leq\\frac{s}{2}\\sqrt{2p_0} (1+p^t_0)\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(x)^{-t}\\varphi(x)^{s-1} |\\nabla u(x)||\\nabla_{xy}\\varphi|.\n\t\\end{align}\n\tLet\n\t\\begin{equation}\\label{def-ab}\n\t\ta=\\frac{2p + q + t(q-2)}{p + q-t},\\quad b=\\frac{2p + q + t(q-2)}{p + t(q-1)},\n\t\\end{equation}\n\tand $t$ to be chosen later such that $a, b\\geq 1$.\n\t\n\tBy applying Young's inequality, we obtain\n\t\\begin{align}\\label{lem2-7}\n\t\t\\frac{s}{2}&\\sum\\limits_{x,y \\in \\Omega }\\mu_{xy} u(x)^{-t}\\varphi(x)^{s-1} |\\nabla u(x)||\\nabla_{xy}\\varphi|\n\t\t\\nonumber \\\\\n\t\t=&\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega }\\left(\\mu_{xy}^{\\frac{1}{b}}(\\frac{t}{2})^{\\frac{1}{b}}\n\t\tu(x)^{-\\frac{t+1}{b}} |\\nabla u(x)|^{\\frac{2}{b}} \\varphi(x)^{\\frac{s}{b}}\\right)\\nonumber \\\\\n\t\t&\\quad\\quad\\times \\left(\\mu_{xy}^{\\frac{1}{a}}s(\\frac{t}{2})^{-\\frac{1}{b}}\n\t\tu(x)^{-t+\\frac{t+1}{b}} |\\nabla u(x)|^{1-\\frac{2}{b}} \\varphi(x)^{s-1-\\frac{s}{b}} |\\nabla_{xy}\\varphi|\\right) \\nonumber\\\\\n\t\t\\leq& \\frac{\\epsilon t}{4}\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t-1}  |\\nabla u(x)|^2 \\varphi(x)^s\\nonumber \\\\\n\t\t&+\\epsilon ^{-\\frac{a}{b}}s^a 2^{a-1} t^{1-a}\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t+a-1} |\\nabla u(x)|^{2-a} \\varphi(x)^{s-a}|\\nabla_{xy}\\varphi|^a\\nonumber \\\\\n\t\t\\leq & \\frac{\\epsilon t}{4}\\sum\\limits_{x \\in \\Omega } \\mu(x) u(x)^{-t-1}  |\\nabla u(x)|^2 \\varphi(x)^s\\nonumber \\\\\n\t\t&+\\epsilon ^{-\\frac{a}{b}}s^a 2^{a-1} t^{1-a}\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t+a-1} |\\nabla u(x)|^{2-a} \\varphi(x)^{s-a}|\\nabla_{xy}\\varphi|^a.\n\t\\end{align}\nwhere we used $\\sum\\limits_{y \\in \\Omega} \\mu_{xy}\\leq \\mu(x)$, for any $x\\in \\Omega$.\n\n\tLetting $\\epsilon =\\frac{2}{\\sqrt{2p_0} (1+p^t_0) p_0^{t+1}}$, and substituting (\\ref{lem2-7}) into (\\ref{lem2-6}), we obtain\n\t\\begin{align}\\label{lem2-8}\n\t\t\\frac{s}{2}&\\sum\\limits_{x,y \\in V}\\mu_{xy} u(y)^{-t}\\eta^{s-1} (\\nabla_{xy}u)( \\nabla_{xy}\\varphi)\\nonumber \\\\\n\t\t\\leq& \\frac{t}{2p_0^{t+1}}\\sum\\limits_{x \\in \\Omega } \\mu(x) u(x)^{-t-1}  |\\nabla u(x)|^2 \\varphi(x)^s\\nonumber \\\\\n\t\t&+C_{p_0,t}(2s)^a  t^{1-a}\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t+a-1} |\\nabla u(x)|^{2-a} \\varphi(x)^{s-a} |\\nabla_{xy}\\varphi|^a,\n\t\\end{align}\n\twhere\n\t$$C_{p_0,t}= \\frac{(\\sqrt{2p_0}(1+p^t_0))^{\\frac{p + t(q-1)}{p + q-t}+1}   (p_0^{t+1})^{\\frac{p + t(q-1)}{p + q-t}}}{4}.$$\n\tCombining (\\ref{lem2-8}) with (\\ref{lem2-5}), we have\n\t\\begin{align*}\n\t\t&\\frac{t}{2p_0^{t+1}} \\sum\\limits_{x \\in \\Omega }\\mu(x)\\varphi(x)^s u(x)^{-t-1} |\\nabla u(x)|^2\n\t\t+\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s \\\\\n\t\t&\n\t\t\\leq C_{p_0,t}(2s)^a  t^{1-a}\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t+a-1} |\\nabla u(x)|^{2-a} \\varphi(x)^{s-a} |\\nabla_{xy}\\varphi|^a.\n\t\\end{align*}\n\tIt follows that\n\t\\begin{align}\\label{lem2-10}\n\t\t&\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s \\nonumber\\\\\n\t\t\\leq& C_{p_0,t}(2s)^a  t^{1-a}\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t+a-1} |\\nabla u(x)|^{2-a} \\varphi(x)^{s-a} |\\nabla_{xy}\\varphi|^a.\n\t\\end{align}\n\t\n\tDefining $\\gamma=\\frac{p+q-t}{1-t},\\rho=\\frac{p+q-t}{p+q-1}$, and choosing $t$ to make $\\gamma,\\rho>1$, and applying H\\\"{o}lder's inequality to RHS of (\\ref{lem2-10}), we obtain\n\t\\begin{align}\\label{lem2-11}\n\t\t&\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t+a-1} |\\nabla u(x)|^{2-a} \\varphi(x)^{s-a}|\\nabla_{xy}\\varphi|^a  \\nonumber\\\\&\n\t\t=\\sum\\limits_{x,y\\in \\Omega \\atop \\nabla_{xy} \\varphi \\neq 0} \\mu_{xy}\\left(u(x)^{-t+a-1} |\\nabla u(x)|^{2-a}\\varphi(x)^{\\frac{s}{\\gamma}}\\right)\\left(\\varphi(x)^{s-a-\\frac{s}{\\gamma}}|\\nabla_{xy}\\varphi|^a \\right) \\nonumber\\\\&\n\t\t\\leq \\left(\\sum\\limits_{x,y\\in \\Omega \\atop \\nabla_{xy} \\varphi \\neq 0} \\mu_{xy}u(x)^{p-t}\n\t\t|\\nabla u(x)|^q \\varphi(x)^s\\right)^{\\frac{1}{\\gamma}}\n\t\t\\left(\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy} \\varphi(x)^{s-a\\rho} |\\nabla_{xy}\\varphi|^{a\\rho}\\right)^{\\frac{1}{\\rho}},\n\t\\end{align}\n\t\n\t\n\tChoosing large enough $s$ to let $ s \\geq a\\rho$, and noticing $0 \\leq \\varphi \\leq 1$, we derive\n\t\\begin{align}\\label{lem2-12}\n\t\t&\\sum\\limits_{x,y \\in \\Omega } \\mu_{xy} u(x)^{-t+a-1} |\\nabla u(x)|^{2-a} \\varphi(x)^{s-a}|\\nabla_{xy}\\varphi|^a  \\nonumber\\\\&\n\t\t\\leq \\left(\\sum\\limits_{x,y\\in \\Omega \\atop \\nabla_{xy} \\varphi \\neq 0} \\mu_{xy}u(x)^{p-t}\n\t\t|\\nabla u(x)|^q \\varphi(x)^s\\right)^{\\frac{1}{\\gamma}}\n\t\t\\left(\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy} |\\nabla_{xy}\\varphi|^{a\\rho}\\right)^{\\frac{1}{\\rho}}.\n\t\\end{align}\nSubstituting (\\ref{lem2-12}) into (\\ref{lem2-10}), we get\n\t\\begin{align*}\n\t\t&\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s \\nonumber\\\\&\n\t\t\\leq C_{p_0,t}(2s)^a  t^{1-a} \\left(\\sum\\limits_{x,y\\in \\Omega \\atop \\nabla_{xy} \\varphi \\neq 0} \\mu_{xy}u(x)^{p-t}\n\t\t|\\nabla u(x)|^q \\varphi(x)^s\\right)^{\\frac{1}{\\gamma}}\n\t\t\\left(\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy} |\\nabla_{xy}\\varphi|^{a\\rho}\\right)^{\\frac{1}{\\rho}}.\n\t\\end{align*}\n\tCombining the above with (\\ref{def-ab}), we derive\n\t\\begin{align*}\n\t\t&\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s \\nonumber\\\\\n\t\t\\leq& C_{p_0,t}(2s)^{\\frac{2p + q + t(q-2)}{p + q-t}}  t^{-\\frac{p + t(q-1)}{p + q-t}}\\left(\\sum\\limits_{x,y\\in \\Omega \\atop \\nabla_{xy} \\varphi \\neq 0} \\mu_{xy}u(x)^{p-t}\n\t\t|\\nabla u(x)|^q \\varphi(x)^s\\right)^{\\frac{1-t}{p+q-t}} \\nonumber\\\\&\n\t\t\\times \\left(\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy}  |\\nabla_{xy}\\varphi|^{\\frac{2p+q+t(q-2)}{p+q-1}}\\right)^{\\frac{p+q-1}{p+q-t}}.\n\t\\end{align*}\n\tthen (\\ref{est-1}) follows.\n\t\n\tNoting $\\sum\\limits_{x \\in \\Omega }\\mu(x)u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s$ is finite and\n\t\\begin{align*}\n\t\t&\\sum\\limits_{x,y\\in \\Omega \\atop \\nabla_{xy} \\varphi \\neq 0} \\mu_{xy}u(x)^{p-t}\n\t\t|\\nabla u(x)|^q \\varphi(x)^s\\leq \\sum\\limits_{x\\in \\Omega,y\\in V} \\mu_{xy}u(x)^{p-t}\n\t\t|\\nabla u(x)|^q \\varphi(x)^s\\\\&\n\t\t=\\sum\\limits_{x \\in \\Omega }\\mu(x)u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s,\n\t\\end{align*}\nwe obtain\n\t\\begin{align*}\n&\\left( \\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s \\right)^{\\frac{p+q-1}{p+q-t}} \\nonumber\\\\&\n\\leq C_{p_0,t}(2s)^{\\frac{2p + q + t(q-2)}{p + q-t}}  t^{-\\frac{p + t(q-1)}{p + q-t}} \\left(\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy}  |\\nabla_{xy}\\varphi|^{\\frac{2p+q+t(q-2)}{p+q-1}}\\right)^{\\frac{p+q-1}{p+q-t}}.\n\\end{align*}\nHence we have\n\t\\begin{align*}\n\t\t\\sum\\limits_{x \\in \\Omega }\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^{q} \\varphi(x)^s\n\t\t\\leq&( C_{p_0,t})^{\\frac{p+q-t}{p+q-1}} (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}} t^{-\\frac{p+t(q-1)}{p+q-1}}\\nonumber\\\\\n\t\t&\\times\n\t\t\\sum\\limits_{x,y\\in \\Omega } \\mu_{xy} |\\nabla_{xy}\\varphi|^{\\frac{2p+q+t(q-2)}{p+q-1}},\n\t\\end{align*}\n\twhich implies (\\ref{est-2}). Hence, we complete the proof.\n\\end{proof}\n\n\n\\begin{remark}\\label{rem}\n\tIn Lemma \\ref{lem2}, since $s$ is only needed to be chosen large enough, it\n\tsuffices to verify that such $t$ exists. For our convenience, let us divide $R^2\\setminus \\{p+q=1\\}$ into four different parts $K_1$, $K_2$, $K_3$, $K_4$ (see figure \\ref{fig2})\n\t\\begin{align*}\n\t\t&K_1=\\{(p,q)|p<1-q, q\\leq1\\},\\quad K_2=\\{(p,q)|p\\geq0, 1-p<q\\leq 1\\} \\\\\n\t\t&K_3=\\{(p,q)|p>1-q, q>1\\}, \\quad K_4=\\{(p,q)|p<0,1<q<1-p\\}\n\t\\end{align*}\n\t\t\\begin{figure}[H]\n\t\\begin{tikzpicture}[x={(0.8cm,0cm)},y={(0cm,0.8cm)}]\n\t\t\\draw[->] (-6,0)--(4,0) ;\n\t\t\\draw[->] (0,-2)--(0,6);\n\t\t\\fill[green,opacity=0.7] (0,2)--(4,2)--(4,6)--(-4,6) ;\n\t\t\\fill[blue,opacity=0.7] (0,2)--(4,2)--(4,-2);\n\t\t\\fill[yellow,opacity=0.7] (0,2)--(-6,2)--(-6,6)--(-4,6);\n\t\t\\fill[red,opacity=0.7] (0,2)--(-6,2)--(-6,-2)--(4,-2);\n\t\t\\fill[red,opacity=0.7] (5.7,5.4) rectangle(6.3,4.8);\n\t\t\\fill[blue,opacity=0.7] (5.7,3.6) rectangle(6.3,3);\n\t\t\\fill[green,opacity=0.7] (5.7,1.8) rectangle(6.3,1.2);\n\t\t\\fill[yellow,opacity=0.7] (5.7,0) rectangle(6.3,-0.6);\n\t\t\\node[above] at (0,6) {$q$};\n\t\t\\node[right] at (4,0) {$p$};\n\t\t\\node[below] at (-5,2) {\\tiny{$q=1$}};\n\t\t\\node[below] at (-2,3.5) {\\tiny{$p+q=1$}};\n\t\t\\node[right] at (6.3,5.1) {\\small{$K_1$}};\n\t\t\\node[right] at (6.3,3.3) {\\small{$K_2$}};\n\t\t\\node[right] at (6.3,1.6) {\\small{$K_3$}};\n\t\t\\node[right] at (6.3,-0.3) {\\small{$K_4$}};\n\t\\end{tikzpicture}\n\t\\caption{}\n\t\\label{fig2}\n\\end{figure}\nAccording to the location of $(p, q)$, we choose $t$ in the following way\n\\begin{enumerate}\n\\item[1.]{When $(p, q)\\in K_1$, take $t>1$; }\n\\item[2.]{When $(p, q)\\in K_2$, take $0<t<1$;}\n\\item[3.]{When $(p, q)\\in K_3$, take $\\max\\{-\\frac{p}{q-1}, 0\\}<t<1$;}\n\\item[4.]{When $(p, q)\\in K_4$, take $1<t<-\\frac{p}{q-1}$.}\n\\end{enumerate}\t\n\n\\end{remark}\nNow we step into the proof of Theorem \\ref{thm1}.\n\\begin{proof}[\\rm{Proof of Theorem \\ref{thm1} (I)}]\n\nAssume u is a nontrivial positive solution to (\\ref{ieq}). By Lemma \\ref{lem1}, u satisfies $\\frac{1}{p_0}\\leq\\frac{u(x)}{u(y)} \\leq p_0$ for any $y\\sim x$.\n\n\tFor our convenience, let us denote $d(x)= d(o,x)$, and $B_i=\\{x\\in V:d(x)\\leq 2^i\\}$ for integer $i$.\n\tFix integer $n$, define $h_n$ on $V$ as\n\t\\begin{align}\\label{hn}\n\t\th_n(x)=\n\t\t\\left\\{\n\t\t\\begin{array}{lr}\n\t\t\t1, \\qquad \\qquad d(x)\\leq n,\\\\\n\t\t\t2-\\frac{d(x)}{n}, \\quad n<d(x)<2n,\\\\\n\t\t\t0,  \\qquad \\qquad  d(x)\\geq 2n,\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{align}\n\tand let\n\t\\begin{align}\\label{def-phi}\n\t\t\\varphi_i=\\frac{1}{i}\\sum\\limits^{2i-2}_{k=i-1}h_{2^k}(x).\n\t\\end{align}\nIt follows that $0\\leq\\varphi_i\\leq1$, and $\\varphi_i=1$ in $B_{i-1}$, $\\varphi_i=0$ in $(B_{2i-1})^c$. Moreover, for any $x\\in B_k-B_{k-1}$, if $k \\leq i - 2$ or\n\t$k \\geq 2i + 1$, then $\\nabla_{xy}\\varphi_i = 0$ for any $y \\sim x$; while if $i-1\\leq k \\leq 2i$, we have\n\t$$|\\nabla_{xy}\\varphi_i|\\lesssim \\frac{1}{i\\cdot 2^k}, \\quad\\mbox{for any $y\\sim x$}.$$\n\t\n\tLetting $\\Omega=V$ in Lemma \\ref{lem2}, and substituting $\\varphi=\\varphi_i$ in (\\ref{est-2}), we get\n\t\\begin{align}\n\t\t&\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\nonumber \\\\&\n\t\t\\lesssim C_{p_0,t}  (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}} t^{-\\frac{p+t(q-1)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t\\sum\\limits_{x\\in B_k-B_{k-1}}\n\t\t\\sum\\limits_{y \\sim x} \\mu_{xy} |\\nabla_{xy}\\varphi|^{\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim C_{p_0,t}  (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}} t^{-\\frac{p+t(q-1)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t\\sum\\limits_{x\\in B_k-B_{k-1}}\n\t\t\\mu(x) (\\frac{1}{i\\cdot 2^k})^{\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim C_{p_0,t}  (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}} \\frac{t^{-\\frac{p+t(q-1)}{p+q-1}}}{i^{\\frac{2p+q+t(q-2)}{p+q-1}}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t\\mu(B_k) 2^{-k\\frac{2p+q+t(q-2)}{p+q-1}},\n\t\\end{align}\n\twhere $C_{p_0,t}=\\frac{(\\sqrt{2p_0}(1+p^t_0))^{\\frac{p + t(q-1)}{p + q-t}+1}   (p_0^{t+1})^{\\frac{p + t(q-1)}{p + q-t}}}{4}$.\n\t\n\tSince $G_1\\subset K_2 \\cup K_3$, and by Remark \\ref{rem}, we choose\n\t$$t=\\frac{1}{i},$$\n\tand $s$ to be some large fixed constant.\n\tRecalling (\\ref{vol-1}), and noting that $p+q>1$ and $q<2$, we obtain\n \\begin{align}\\label{I-3}\n\t\t&\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-1\/i} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\nonumber \\\\&\n\t\t\\lesssim i^{-1-\\frac{1-1\/i}{p+q-1}} \\sum\\limits^{2i}_{k=i-1}\n\t\t2^{\\frac{k(2-q)}{i(p+q-1)}} k^{\\frac{1}{p+q-1}}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim i^{-1+\\frac{1\/i}{p+q-1}} \\sum\\limits^{2i}_{k=i-1}\n\t\t2^{\\frac{k(2-q)}{i(p+q-1)}}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim i^{\\frac{1\/i}{p+q-1}}.\n\t\\end{align}\nConsequently from letting $i \\rightarrow \\infty$ in (\\ref{I-3})\n\t$$\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p} |\\nabla u(x)|^q <\\infty. $$\n\n\n\tSubstituting $\\varphi=\\varphi_i$ and $t=\\frac{1}{i}$ into (\\ref{est-1}), and repeating the same procedures, we have\n\t$$\\lim_{i\\to\\infty}\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-1\/i} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t= 0,$$\n\tnamely\n\t$$\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p} |\\nabla u(x)|^q = 0,$$\n\twhich is a contradiction to the assumption that $u$ is nontrivial. Hence, the proof of Theorem \\ref{thm1} (I) is complete.\n\\end{proof}\n\\begin{proof}[\\rm\\textbf{Proof of Theorem \\ref{thm1} (II)}]\n\tLet us divide the proof into three cases:\n\t\\begin{enumerate}\n\t\t\\item[(II-1).]{$(p,q)\\in \\{p+q>1,q>2\\};$}\n\t\t\\item[(II-2).]{$(p,q)\\in \\{p+q=1,q>2\\};$}\n\t\t\\item[(II-3).]{$(p,q)\\in \\{p+q<1,q>2\\}.$}\n\t\\end{enumerate}\n\t\n\tIn case (II-1), it follows that $(p,q)\\in K_2$. Hence let\n\t\\begin{align*}\n\t\tt=1-\\frac{1}{i},\n\t\\end{align*}\n\tand $s$ be some large fixed constant.\n\t\n\tLetting $\\Omega=V$ in Lemma \\ref{lem2}, substituting $\\varphi=\\varphi_i$ from (\\ref{def-phi}) into (\\ref{est-2}), and using the same technique as in (\\ref{I-3}), we obtain\n\t\\begin{align}\\label{2-1}\n\t\t&\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\nonumber \\\\&\n\t\t\\lesssim C_{p_0,t}  (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}} \\frac{t^{-\\frac{p+t(q-1)}{p+q-1}}}{i^{\\frac{2p+q+t(q-2)}{p+q-1}}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t\\mu(B_k) 2^{-k\\frac{2p+q+t(q-2)}{p+q-1}},\n\t\\end{align}\n\tCombining with (\\ref{vol-3}) and (\\ref{2-1}), and noting that $ C_{p_0,t}  (2s)^{\\frac{2p+q+t(q-2)}{p+q-1}}t^{-\\frac{p+t(q-1)}{p+q-1}}$ is uniformly bounded\n\tfor $ i $, we obtain\n\t\\begin{align}\\label{2-2}\n\t\t&\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\nonumber \\\\&\n\t\t\\lesssim  i^{-\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t\\mu(B_k) 2^{-k\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim  i^{-\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1} 2^{k(2-\\frac{2p+q+t(q-2)}{p+q-1})}k\n\t\t\\nonumber \\\\&\n\t\t\\lesssim i^{1-\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1} 2^{\\frac{k(q-2)}{i(p+q-1)}}.\n\t\\end{align}\n\tSubstituting $ t=1-\\frac{1}{i}$ into (\\ref{2-2}), we get\n\t\\begin{align}\\label{2-3}\n\t\t&\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-1+\\frac{1}{i}} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\nonumber \\\\&\n\t\t\\lesssim i^{1-\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1} 2^{\\frac{k(q-2)}{i(p+q-1)}}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim i^{\\frac{q-2}{i(p+q-1)}}.\n\t\\end{align}\n\tLetting $i \\rightarrow \\infty$ in (\\ref{2-3}), we have\n\t$$\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-1} |\\nabla u(x)|^q <\\infty. $$\n\tSubstituting $\\varphi=\\varphi_i$ and $t=1-\\frac{1}{i}$ into (\\ref{est-1}), and repeating the same procedures as in the proof of Theorem \\ref{thm1} (I), we derive\n\t\\begin{align}\n\t\t\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-1} |\\nabla u(x)|^q = 0.\n\t\\end{align}\n\twhich yields a contradiction with the nontrivialness of $u$.\n\t\n\tIn case (II-2),\n\tdenote $\\Omega_k=\\{ x\\in V|0<u(x)<k \\}$.\n\tSince $u$ is a nontrivial positive solution, then there exists a constant $k_0$ such that $\\mu(\\Omega_k)>0 $ for any $k>k_0$. Now fix such $k$, let $v=\\frac{u}{k}$, and $v$ satisfies\n\t$$ \\Delta v+v^p\\left|\\nabla v\\right|^q\\leq0, \\quad\\mbox{ on $V$}.$$\n\tObiviously $0<v<1$ in $\\Omega_k$, we have\n\t\\begin{align}\\label{2-2-1}\n\t\t\\Delta v+v^{p+\\epsilon}\\left|\\nabla v\\right|^q\\leq0, \\quad\\mbox{ on $\\Omega_k$},\n\t\\end{align}\n\twhere $p'=p+\\epsilon$ and $\\epsilon>0$, thus $(p^{\\prime},q)\\in \\{ p^{\\prime}+q=p+q+\\epsilon>1,q>2 \\} $, consequently\n\t$(p^{\\prime},q)\\in$ (II-1).\n\t\n\tFrom the definition of $v(x)$ and $\\Omega_k$ , we know $\\frac{1}{p_0}\\leq\\frac{v(x)}{v(y)}=\\frac{u(x)}{u(y)} \\leq p_0$ and  $v(y)-v(x)\\geq 0$ when $x\\in \\Omega_k$, $y\\in \\Omega_k^c$ .\nHence by Lemma \\ref{lem2}, and by taking the same procedure as in case (II-1) except replacing $V$ with $\\Omega_k$, we arrive\n\t\\begin{align}\\label{2-2-4}\n\t\t\\sum\\limits_{x \\in \\Omega_k}\\mu(x)  v(x)^{p'-1} |\\nabla v(x)|^q = 0,\\quad\\mbox{ on $\\Omega_k$}.\n\t\\end{align}\n\tLet $k_i=\\max\\{u(x)|d(o,x)\\leq i\\}+k_0$, we have\n\t$B(o,i) \\subset \\Omega_{k_i} $. Taking $k=k_i$ in (\\ref{2-2-4}), we obtain that\n\t$v\\equiv cons.$ in $B(o,i)$, which implies that $u\\equiv cons.$ in $B(o,i)$.\n\t\n\tLetting $i\\rightarrow \\infty$, we get $u\\equiv cons.$ in $V$, which is a contradiction with that $u$ is nontrivial.\n\t\n\tIn case (II-3), by taking the same argument as in case (II-1) and letting\n\t$$t=1+\\frac{1}{i},$$\n\twe finish the proof of Theorem \\ref{thm1} (II).\n\\end{proof}\n\n\\begin{proof}[\\rm\\textbf{Proof of Theorem \\ref{thm1} (\\uppercase\\expandafter{\\romannumeral3})}]\n\tLet us divide the proof into three cases:\n\t\\begin{enumerate}\n\t\t\\item[(III-1).]{$(p,q)\\in G_3 \\cap \\{p+q>1\\}$;}\\\\\n\t\t\\item[(III-2).]{$(p,q)\\in G_3 \\cap \\{p+q=1\\};$}\\\\\n\t\t\\item[(III-3).]{$(p,q)\\in G_3 \\cap \\{p+q<1\\}.$}\\\\\n\t\\end{enumerate}\n\t\n\tIn case (III-1), since $(p,q)\\in K_3$, we take\n\t\\begin{align*}\n\t\tt=-\\frac{p}{q-1}+\\frac{1}{i}.\n\t\\end{align*}\n\tand $b$ to be some large fixed constant.\n\t\n\tLetting $\\Omega=V$ in lemma \\ref{lem2}, substituting $\\varphi=\\varphi_i$ to (\\ref{est-2}), and using the same procedure as before, we obtain\n\t\\begin{align*}\n\t\t\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\lesssim  i^{-\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t\\mu(B_k) 2^{-k\\frac{2p+q+t(q-2)}{p+q-1}}.\n\t\\end{align*}\n\tCombining with (\\ref{vol-3}), we obtain\n\t\\begin{align}\\label{3-1-10}\n\t\t&\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{\\frac{pq}{q-1}+\\frac{1}{i}} |\\nabla u|^q \\varphi_i(x)^s\n\t\t\\nonumber \\\\ &\n\t\t\\lesssim  i^{-\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t2^{k\\left(\\frac{q}{q-1}-\\frac{2p+q+t(q-2)}{p+q-1}\\right)}k^{\\frac{1}{q-1}}\n\t\t\\nonumber \\\\ &\n\t\t\\lesssim  i^{-\\frac{2p+q+t(q-2)}{p+q-1}}\n\t\t\\sum\\limits^{2i}_{k=i-1}\n\t\t2^{-\\frac{k(q-2)}{i(p+q-1)}}k^{\\frac{1}{q-1}}\n\t\t\\nonumber \\\\ &\n\t\t\\lesssim  i^{\\frac{1}{q-1}-\\frac{2p+q+t(q-2)}{p+q-1}+1}\n\t\t\\nonumber \\\\ &\n\t\t=i^{-\\frac{q-2}{i(p+q-1)}},\n\t\\end{align}\n\twhere we have used that\n\t$$\\frac{q}{q-1}-\\frac{2p+q+t(q-2)}{p+q-1}=-\\frac{q-2}{i(p+q-1)}.$$\n\tThen letting $i\\to \\infty$ in (\\ref{3-1-10}), we obtain\n\t$$\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{\\frac{pq}{q-1}} |\\nabla u(x)|^q <\\infty.$$\n\tRepeating the same procedure as in proof of Theorem \\ref{thm1} (I), we derive\n\t$$\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{\\frac{pq}{q-1}} |\\nabla u(x)|^q=0,$$\n\twhich contradicts with that $u$ is a nontrivial positive solution.\n\t\n\tIn case (III-2), we take the same procedure as in case (II-2) except letting\n\t$0<\\epsilon<-\\frac{p}{2}$, thus $ (p^{\\prime},q)=(p+\\epsilon,q) \\in$ (III-1).\n\t\n\tIn case (III-3), we repeat the same argument as in case (III-1) except\n taking\n\t\\begin{align*}\n\t\tt=-\\frac{p}{q-1}-\\frac{1}{i}.\n\t\\end{align*}\n\tHence, we complete proof of Theorem \\ref{thm1} (III).\n\\end{proof}\n\\begin{proof}[\\rm\\textbf{Proof of Theorem \\ref{thm1} (IV)}]\n\tSince here $p<0, q=1$, we choose\n\t$$t=l+\\frac{1}{i},\\qquad s=-\\frac{l}{p}+2+\\frac{1}{i},$$\n\twhere $l>1$ is to be chosen later.\n\t\n\tLetting $\\Omega=V$ in Lemma \\ref{lem2}, substituting $\\varphi=\\varphi_i$  into (\\ref{est-2}), and repeating the same procedure, we obtain\n\t\\begin{align*}\n\t\t&\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\nonumber \\\\&\n\t\t\\lesssim  i^{-\\frac{2p+q-t}{p}}\n\t\t\\sum\\limits^{2i}_{k=i-1} \\mu(B_k) 2^{-k\\frac{2p+q-t}{p}}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim  i^{-\\frac{2p+q-t}{p}}\n\t\t\\sum\\limits^{2i}_{k=i-1} 2^{k(\\alpha-\\frac{2p+q-t}{p})}.\n\t\\end{align*}\n\tLetting $l$ be a fixed large enough constant such that for all $i$\n\t$$\\alpha-\\frac{2p+q-t}{p}<0,$$\n\twe obtain\n\t\\begin{align}\\label{4-1-3}\n\t\t\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-t} |\\nabla u(x)|^q \\varphi_i(x)^s\n\t\t\\lesssim i^{1-\\frac{2p+q-t}{p}}.\n\t\\end{align}\n\tFurther, we require that $l$ satisfies\n\t$$1-\\frac{2p+q-t}{p}<c<0,$$\nby letting $i\\to \\infty$ in (\\ref{4-1-3}), we obtain\n\t$$\\sum\\limits_{x \\in V}\\mu(x)  u(x)^{p-l} |\\nabla u(x)|^q=0.$$\n\twhich contradicts with that $u$ is nontrivial. Hence we complete the proof for Theorem \\ref{thm1} (IV).\n\\end{proof}\n\\begin{proof}[\\rm\\textbf{Proof of Theorem \\ref{thm1} (V)}]\n\tWe divide the proof into two cases:\n\t\\begin{enumerate}\n\t\t\\item[(V-1).]{$(p,q)\\in \\{p+q=1,p\\geq 0,q>0\\};$}\n\t\t\\item[(V-2).]{$(p,q)\\in \\{p+q=1,p>1, q<0\\}$.}\n\t\\end{enumerate}\n\tFrom (\\ref{ieq}), we have\n\t$$\\sum\\limits_{y \\in V}\\frac{\\mu_{xy}}{\\mu(x)}u(y)-u(x)\n\t+u(x)^{p} |\\nabla u(x)|^{q}\\leq 0,$$\n\tthat is\n$$\\sum\\limits_{y \\in V}\\frac{\\mu_{xy}}{\\mu(x)}u(y)\\leq u(x)(1-u(x)^{p-1} |\\nabla u(x)|^{q}),$$\n\twhich implies\n\t\\begin{align}\\label{5-ieq}\n\t\tu(x)^{p-1} |\\nabla u(x)|^{q}\\leq 1.\n\t\\end{align}\n\t\n\tIn case (V-1), since $p+q=1$, and $q>0 $, we obtain\n\t\\begin{align}\\label{5-1}\n\t\t|\\nabla u(x)|\\leq u(x).\n\t\\end{align}\n\tCombining this with (\\ref{ieq}), noting $p\\geq0$, we derive\n\t\\begin{align*}\n\t\t\\Delta u(x)+(u(x)^{-p} |\\nabla u(x)|^{p}) u(x)^{p} |\\nabla u(x)|^{q}\\leq \\Delta u(x)+u(x)^{p} |\\nabla u(x)|^{q}\\leq 0,\n\t\\end{align*}\n\twhich is\n\t\\begin{align}\n\t\t\\Delta u(x)+\\left|\\nabla u(x)\\right|\\leq 0.\n\t\\end{align}\n\t\nSet $\\Omega'_k:=\\{ x\\in V|0<u(x)<k,|\\nabla u(x)|\\neq 0 \\}$.\n\t\tSince $u$ is a nontrivial positive solution, thus there exists some large enough $k$. Such that $\\mu( \\Omega'_k)>0 $. Now let $v=\\frac{u}{k}$, it follows that $v$ satisfies\n\t$$ \\Delta v(x)+\\left|\\nabla v(x)\\right|\\leq0.$$\nNoting $0<v<1$ on $\\Omega'_k$, from (\\ref{5-1}), we have\n\t$$\\left|\\nabla v(x)\\right|\\leq v(x)\\leq 1.$$\nIt follows that\n\t\\begin{align}\\label{5-2}\n\t\t\\Delta v(x)+\\left|\\nabla v(x)\\right|^{\\lambda}\\leq0, \\quad\\mbox{ on $\\Omega'_k$},\n\t\\end{align}\n\twhere $\\lambda>1$ is to be chosen later.\n\t\n\tFrom the definition of $\\Omega'_k$, we know when $\\mu_{xy}\\neq 0$, $y\\in V \\setminus \\Omega'_k$ $x\\in \\Omega'_k$,  we have\n\t\\begin{align*}\n\t\t\\left\\{\n\t\t\\begin{array}{lr}\n\t\t\tv(y)>v(x), \\quad \\mbox{when $v(y)\\geq1$,}\\\\\n\t\t\tv(y)=v(x), \\quad \\mbox{when $v(y)<1$, and $|\\nabla v(y)|=0$.}\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{align*}\nIn both cases, we have $v(y)-v(x)\\geq 0$.\n\n\tThen, for any $x\\in \\Omega'_k$, we obtain\n\t\\begin{align}\\label{5-3}\n\t\t\\Delta v(x)&=\\sum\\limits_{y\\in V} \\frac{\\mu_{xy}}{\\mu(x)}(v(y)-v(x))\\nonumber\\\\&\n\t\t=\\sum\\limits_{y\\in \\Omega'_k} \\frac{\\mu_{xy}}{\\mu(x)}(v(y)-v(x))+\n\t\t\\sum\\limits_{y\\in V \\setminus \\Omega'_k} \\frac{\\mu_{xy}}{\\mu(x)}(v(y)-v(x))\\nonumber\\\\&\n\t\t\\geq \\sum\\limits_{y\\in \\Omega'_k} \\frac{\\mu_{xy}}{\\mu(x)}(v(y)-v(x)).\n\t\\end{align}\n\tSimilarly\n\t\\begin{align}\\label{5-4}\n\t\t|\\nabla v(x)|&=\\sqrt{\\sum\\limits_{y \\in V}\\frac{\\mu_{xy}}{2\\mu(x)}(\\nabla_{xy}v)^2} \\nonumber\\\\&\n\t\t=\\sqrt{\\sum\\limits_{y \\in\\Omega'_k}\\frac{\\mu_{xy}}{2\\mu(x)}(\\nabla_{xy}v)^2+\n\t\t\t\\sum\\limits_{y \\in(\\Omega'_k)^c}\\frac{\\mu_{xy}}{2\\mu(x)}(\\nabla_{xy}v)^2}\n\t\t\\nonumber\\\\&\n\t\t\\geq \\sqrt{\\sum\\limits_{y\\in \\Omega'_k}\\frac{\\mu_{xy}}{2\\mu(x)}(\\nabla_{xy}v)^2}\n\t\t:=|\\nabla_{\\Omega'_k} v(x)|.\n\t\\end{align}\n\tIt is should be know that $|\\nabla_{\\Omega'_k} u(x)|$ is not the norm of gradient of $ u $ in $\\Omega'_k$, since $\\mu(x)$ there is still the measure of $x$ in $ V $, instead of in $\\Omega$. Noticing $\\Omega'_k$ is a  subset of $V$, we have $\\sum\\limits_{y\\in \\Omega}\\mu_{xy}\\leq\\sum\\limits_{y\\in V}\\mu_{xy}=\\mu(x)$.\n\t\n\tSubsitituting (\\ref{5-3}) and (\\ref{5-4}) into (\\ref{5-2}), we obtain\n\t\\begin{align}\\label{5-5}\n\t\t\\sum\\limits_{y\\in \\Omega'_k} \\frac{\\mu_{xy}}{\\mu(x)}(v(y)-v(x))+|\\nabla_{\\Omega'_k} v(x)|^{\\lambda}\\leq0, \\quad \\mbox{on $\\Omega'_k$}.\n\t\\end{align}\n\t\n\tMultiplying (\\ref{5-5}) by $\\mu(x)h_n^z$ in (\\ref{hn}) and summing up over $x \\in \\Omega'_k$, we have\n\t\\begin{align}\\label{5-6}\n\t\t\\sum\\limits_{x \\in \\Omega'_k}\\mu(x)|\\nabla_{\\Omega'_k} v(x)|^{\\lambda}h_n(x)^z\n\t\t&\\leq -\\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy} h_n(x)^z(\\nabla_{xy}v) \\nonumber \\\\&\n\t\t=\\frac{1}{2}\\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy} (\\nabla_{xy}h_n^z)(\\nabla_{xy}v) \\nonumber \\\\&\n\t\t=\\frac{z}{2}\\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy} \\eta^{z-1}(\\nabla_{xy}h_n)(\\nabla_{xy}v),\n\t\\end{align}\n\twhere $ \\eta>0$ is between $h_n(x)$ and $h_n(y)$.\n\t\n\tNoting that $|\\nabla_{\\Omega'_k} v(x)|^2=\\sum\\limits_{y\\in \\Omega'_k}\\frac{\\mu_{xy}}{2\\mu(x)}(\\nabla_{xy}v)^2$, and\n\t$ \\frac{1}{2p_0} \\leq \\frac{\\mu_{xy}}{2\\mu(x)} \\leq \\frac{1}{2}$, we derive\n\t\\begin{align}\\label{grd2}\n\t\t|\\nabla_{xy}v| \\leq \\sqrt{2p_0}|\\nabla_{\\Omega'_k} v(x)|\\qquad \\mbox{for any $y\\sim x$.}\n\t\\end{align}\n\t\n\tCombining (\\ref{grd2})  and $ \\eta^{z-1} \\leq max(h_n(x)^{z-1},h_n(y)^{z-1})\\leq h_n(x)^{z-1}+h_n(y)^{z-1} $ with (\\ref{5-6}),  and applying H\\\"{o}lder's inequality, we obtain\n\t\\begin{align}\n\t\t&\\sum\\limits_{x \\in \\Omega'_k}\\mu(x)|\\nabla_{\\Omega'_k} v(x)|^{\\lambda}h_n(x)^z\n\t\t\\nonumber \\\\&\n\t\t\\leq \\frac{z}{2}\\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}  (h_n(x)^{z-1}+h_n(y)^{z-1})(\\nabla_{xy}h_n)(\\nabla_{xy}v) \\nonumber \\\\&\n\t\t=z\\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}  h_n(x)^{z-1}(\\nabla_{xy}h_n)(\\nabla_{xy}v)\t\t\\nonumber \\\\&\n\t\t\\leq z\\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}  h_n(x)^{z-1}|\\nabla_{xy}h_n||\\nabla_{xy}v| \\nonumber \\\\&\n\t\t\\leq\\sqrt{2p_0}z\\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}  h_n(x)^{z-1}|\\nabla_{xy}h_n||\\nabla_{\\Omega'_k} v(x)| \\nonumber \\\\&\n\t\t\\leq \\sqrt{2p_0}z\\left( \\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}|\\nabla_{\\Omega'_k} v(x)|^{\\lambda}h_n(x)^{z}\\right)^{\\frac{1}{\\lambda}}\n\t\t\\left( \\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}|\\nabla_{xy}h_n|^{z}\\right)^{\\frac{\\lambda-1}{\\lambda}}\\nonumber \\\\&\n\t\t\t\\leq \\sqrt{2p_0}z\\left( \\sum\\limits_{x \\in \\Omega'_k} \\mu(x)|\\nabla_{\\Omega'_k} v(x)|^{\\lambda}h_n(x)^{z}\\right)^{\\frac{1}{\\lambda}}\n\t\t\\left( \\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}|\\nabla_{xy}h_n|^{z}\\right)^{\\frac{\\lambda-1}{\\lambda}},\n\t\\end{align}\n\twhere we take\n\t$$z=\\dfrac{\\lambda}{\\lambda-1}.$$\n\t\n\tBy the boundedness of $\\sum\\limits_{x \\in \\Omega'_k}\\mu(x)|\\nabla_{\\Omega'_k} v(x)|^{\\lambda}h_n^z$, and $h_n=1$ in $ B(o,n) $, and (\\ref{vol-5}), we obtain\n\t\\begin{align}\\label{5-7}\n\t\t&\\sum\\limits_{x \\in \\Omega'_k\\cap B(o,n) }\\mu(x)|\\nabla_{\\Omega'_k}v(x)|^{\\lambda}\n\t\t\\nonumber \\\\&\n\t\t\\leq (\\sqrt{2p_0}z)^z \\sum\\limits_{x,y \\in \\Omega'_k} \\mu_{xy}|\\nabla_{xy}h_n|^{z}\n\t\t\\nonumber \\\\&\n\t\t\\leq (\\frac{\\sqrt{2p_0}z}{n})^z \\mu(B(o,2n))\n\t\t\\nonumber \\\\&\n\t\t\\lesssim (\\frac{\\sqrt{2p_0}z}{n})^z e^{2\\kappa n},\n\t\\end{align}\n\twhere we have used that $|\\nabla_{xy}h_n|\\leq \\frac{1}{n}$, and $|\\nabla_{xy}h_n|=0$ for  $x,y\\in B(o,2n)^c$.\n\tSet\n\t\\begin{align}\\label{5-8}\n\t\tz=\\theta n,\n\t\\end{align}\n\twhere $\\theta$ is a fixed positive constant to be determined later. It is easy to see that $\\lambda \\to 1_+$ is equivalent to $n \\to \\infty$.\n\t\n\tNow let $\\lambda \\to 1_+$ in (\\ref{5-7}), by (\\ref{5-8}), we obtain\n\t\\begin{align}\\label{5-9}\n\t\t\\sum\\limits_{x \\in \\Omega'_k }\\mu(x)|\\nabla_{\\Omega'_k}v(x)|&\\leq \\lim_{\\lambda \\to 1_+}\\sum\\limits_{x \\in \\Omega'_k\\cap B(o,n) }\\mu(x)|\\nabla_{\\Omega'_k}v(x)|^{\\lambda}\n\t\t\\nonumber \\\\&\n\t\t\\lesssim  \\lim_{n\\to \\infty} (\\frac{\\sqrt{2p_0}z}{n})^z e^{2\\kappa n}\\nonumber \\\\&\n\t\t\\asymp \\lim_{n\\to \\infty} e^{n(2\\kappa+\\theta\\ln(\\sqrt{2p_0}\\theta))}.\n\t\\end{align}\n\tIf we have\n\t\\begin{align}\\label{5-10}\n\t\t2\\kappa+\\theta\\ln(\\sqrt{2p_0}\\theta)<0,\n\t\\end{align}\n\twhich is equivalent to\n\t$$e^{2\\kappa}<\\frac{1}{(\\sqrt{2p_0}\\theta)^\\theta}.$$\n\t\n\tSince $\\frac{1}{(\\sqrt{2p_0}\\theta)^\\theta}$ attains its maximum at $\\theta=\\frac{1}{\\sqrt{2p_0}e}$. Hence\n\tif $\\kappa$ satisfies\n\t\\begin{align}\n\t\t0<\\kappa<\\kappa_0=\\frac{1}{2\\sqrt{2p_0}e},\n\t\\end{align}\n\tthere always exists $\\theta>0$ such that (\\ref{5-10}) holds.\n\t\n\tUnder the above choice of $\\kappa$ and $\\theta$, from (\\ref{5-9}), we obtain\n\t\\begin{align}\\label{5-11}\n\t\t\\sum\\limits_{x \\in \\Omega'_k }\\mu(x)|\\nabla_{\\Omega'_k}v(x)|=0.\n\t\\end{align}\n\t\n\tHence, $x\\in \\Omega'_k$, $|\\nabla_{\\Omega'_k}v(x)|=0$. We calim that $|\\nabla v(x)|=0$ for any $x\\in \\Omega'_k$. Then (\\ref{5-11}) contradicts with the definition of $\\Omega'_k$.\n\t\n\tNow assume there exists some $x_0\\in \\Omega'_k$ satisfying $ |\\nabla_{\\Omega'_k}v(x_0)|=0 $ but $|\\nabla v(x_0)|\\not=0$.\n\t\n\tWe define $ U=\\{y| y\\sim x_0 \\mbox{ and } u(y) \\not= u(x_0)\\}$, it is easy to see $U\\subset (\\Omega'_k)^c$ since\n\t$ |\\nabla_{\\Omega'_k}v(x_0)|=0 $. For any point $y\\in U$ , we derrive $v(y)>v(x_0)$. Otherwise $v(y)<v(x_0)$ by the definition of $U$, noting  $|\\nabla v(y)|\\geq \\sqrt{\\frac{\\mu_{x_0y}}{2\\mu(y)}(v(y)-v(x_0))^2}>0$,\n\twe derrive that $y\\in \\Omega'_k$, which contradicts with $y\\in U$.\n\t\n\tNow we consider the laplacian of $v(x_0)$. Noticing $u(y)= u(x_0)$ for any $y\\in U^c$, we can obtain\n\t\\begin{align*}\n\t\t&\\Delta v(x_0)=\\sum\\limits_{y\\in V} \\frac{\\mu_{x_0y}}{\\mu(x_0)}(v(y)-v(x_0))\\nonumber\\\\&\n\t\t=\\sum\\limits_{y\\in U} \\frac{\\mu_{x_0y}}{\\mu(x_0)}(v(y)-v(x_0))+\n\t\t\\sum\\limits_{y\\in U^c} \\frac{\\mu_{x_0y}}{\\mu(x_0)}(v(y)-v(x_0))\\nonumber\\\\&\n\t\t=\\sum\\limits_{y\\in U} \\frac{\\mu_{x_0y}}{\\mu(x_0)}(v(y)-v(x_0))>0,\n\t\\end{align*}\n\twhich contradicts with (\\ref{5-2}).\n\t\n\tIn case (V-2), since $p+q=1$ $q<0$, from (\\ref{5-ieq}), we obtain\n\t\\begin{align}\\label{5-2-1}\n\t\tu(x)\\leq|\\nabla u(x)|.\n\t\\end{align}\n\t\n\tDefine $\\Omega_k=\\{ x\\in V|0<u(x)<k \\}$.\n\tSince u is a nontrivial positive solution, there exists a $k_0>0$ such that $\\mu(\\Omega_k)>0 $ for any $k>k_0$. Now fix $k$, let $v=\\frac{u}{k}$, which satisfies\n\t$$ \\Delta v+v^p\\left|\\nabla v\\right|^q\\leq0, \\quad\\mbox{ on $V$}.$$\n\tIt is easy to see that $0<v<1$ in $\\Omega_k$, hence\n\t\\begin{align*}\n\t\t\\Delta v+v^{p+\\epsilon}\\left|\\nabla v\\right|^q\\leq0, \\quad\\mbox{ on $\\Omega_k$},\n\t\\end{align*}\n\twhere $\\epsilon>0$ is to be determined later.\n\t\t\n\tSince $(p',q):=(p+\\epsilon,q)\\in K_2$, $p'+q=1+\\epsilon$, we fix $0<t<1$, take $s=\\frac{2p+q+t(q-2)+2\\epsilon}{\\epsilon}+1$ and set\n\t$$z=\\frac{2p+q+t(q-2)+2\\epsilon}{\\epsilon}$$\n\tIt is easy to see that $z\\to\\infty$ is equivalent to $\\epsilon\\to 0$.\n\t\n\tNoting $\\frac{1}{p_0}\\leq\\frac{v(x)}{v(y)} \\leq p_0$ and $v(y)-v(x)\\geq 0$ when $x\\in \\Omega_k$ and $y\\in \\Omega_k^c$, letting $\\Omega=\\Omega_k$ in Lemma \\ref{lem2}, and substituting $\\varphi=h_n$ in (\\ref{est-2}), we get\n\t\\begin{align}\\label{5-2-3}\n\t\t&\\sum\\limits_{x \\in \\Omega_k}\\mu(x)  v(x)^{p-t+\\epsilon} |\\nabla v(x)|^q h_n(x)^s\n\t\t\\nonumber \\\\&\n\t\t\\lesssim C'_{p_0,t}  (2s)^{\\frac{2p'+q+t(q-2)}{p'+q-1}} t^{-\\frac{p'+t(q-1)}{p'+q-1}}\n\t\t\\sum\\limits_{x,y\\in \\Omega_k}\n\t\t\\mu_{xy} |\\nabla_{xy}h_n|^{\\frac{2p'+q+t(q-2)}{p'+q-1}}\n\t\t\\nonumber \\\\&\n\t\t=C'_{p_0,t}  (2s)^{\\frac{2p+q+t(q-2)+2\\epsilon}{\\epsilon}} t^{-\\frac{p+t(q-1)+\\epsilon}{\\epsilon}}\n\t\t\\sum\\limits_{x,y\\in \\Omega_k}\n\t\t\\mu_{xy} |\\nabla_{xy}h_n|^{\\frac{2p+q+t(q-2)+2\\epsilon}{\\epsilon}}\n\t\t\\nonumber \\\\&\n\t\t=2^{\\frac{p+t(q-1)+\\epsilon}{2p+q+t(q-2)+2\\epsilon}}(C_{p_0,t})^{\\frac{p'+q-t}{2p+q+t(q-2)+2\\epsilon}}\n\t\t(2C_{p_0,t}(z+1))^{z} (\\frac{1}{t})^{z}\n\t\t\\sum\\limits_{x,y\\in \\Omega_k}\n\t\t\\mu_{xy} |\\nabla_{xy}h_n|^{z},\n\t\t\\end{align}\n\twhere $C_{p_0,t}=\\frac{(\\sqrt{2p_0}(1+p^t_0))^{\\frac{p+ t(q-1)+\\epsilon}{p + q-t+\\epsilon}+1}   (p_0^{t+1})^{\\frac{p + t(q-1)+\\epsilon}{p+ q-t+\\epsilon}}}{4}$, $C'_{p_0,t} =( C_{p_0,t})^{\\frac{p+q-t+\\epsilon}{p+q-1+\\epsilon}}$.\n\t\n\tIt is easy to see that $C_{p_0,t}\\to C_1$  and $2^{\\frac{p+t(q-1)+\\epsilon}{2p+q+t(q-2)+2\\epsilon}}(C_{p_0,t})^{\\frac{p'+q-t}{2p+q+t(q-2)+2\\epsilon}}\\to C_2$ as $\\epsilon\\to 0$, where $C_1=\\frac{(\\sqrt{2p_0}(1+p^t_0))^{\\frac{p+ t(q-1)}{p + q-t}+1}   (p_0^{t+1})^{\\frac{p + t(q-1)}{p+ q-t}}}{4}$ and $C_2$ are two positive constant depending on $p$, $q$ and $p_0$.\n\t\n\tCombining (\\ref{5-2-3}) with $h_n=1$ in $ B(o,n) $, we obtain\n\t\\begin{align}\\label{5-2-4}\n\t\t&\\sum\\limits_{x \\in \\Omega'_k\\cap B(o,n) }\\mu(x) v(x)^{p-t+\\epsilon} |\\nabla v(x)|^q\n\t\t\\nonumber \\\\&\n\t\t\\lesssim \t(\\frac{2C_1(z+1)}{t})^{z} \\sum\\limits_{x,y\\in \\Omega_k}\\mu_{xy} |\\nabla_{xy}h_n|^{n}\n\t\t\\nonumber \\\\&\n\t\t\\leq (\\frac{2C_1(z+1)}{tn})^{z} \\mu(B(o,2n))\n\t\t\\nonumber \\\\&\n\t\t\\lesssim (\\frac{2C_1(z+1)}{tn})^{z} e^{2\\kappa n}.\n\t\\end{align}\n\tNow let us connect z and n by defining\n\t\\begin{align}\n\t\tz=\\theta n,\n\t\\end{align}\n\twhere $\\theta$ is a fixed positive constant to be determined later.\n\t\n\tBy let $n\\to \\infty$ in (\\ref{5-2-4}), we obtain\n\t\\begin{align}\\label{5-2-5}\n\t\t\\sum\\limits_{x \\in \\Omega'_k}\\mu(x) v(x)^{p-t}  |\\nabla v(x)|^q &\n\t\t\\lesssim  \\lim_{n\\to \\infty} (\\frac{2C_1(\\theta n+1)}{tn})^{\\theta Nn} e^{2\\kappa n}\\nonumber \\\\&\n\t\t\\asymp \\lim_{n\\to \\infty} e^{n(2\\kappa+\\theta\\ln(\\frac{2\\theta C_1}{t}))}.\n\t\\end{align}\n\t\n\tSimilarly, if $\\kappa$ satisfies\n\t\\begin{align}\n\t\t0<\\kappa<\\kappa_{0,t}=\\frac{t}{4C_1e},\n\t\\end{align}\n\t\n\tthere always exists $\\theta>0$ such that $2\\kappa+\\theta\\ln(\\frac{2\\theta C_1}{t})<0$.\n\t\n\tNoticing $p+q=1$, since $\\frac{p+t(q-1)}{p+q-t}=1-q$, we have\n\t$$C_1=\\frac{(\\sqrt{2p_0}(1+p^t_0))^{\\frac{p+ t(q-1)}{p + q-t}+1}   (p_0^{t+1})^{\\frac{p + t(q-1)}{p+ q-t}}}{4}=\\frac{(\\sqrt{2p_0}(1+p^t_0))^{2-q}   (p_0^{t+1})^{1-q}}{4}.$$\n\t\n\tSince $\\frac{t}{4C_1e}=\\frac{t}{(e\\sqrt{2p_0}(1+p^t_0))^{2-q}   (p_0^{t+1})^{1-q}}$, $t\\in[0,1]$ attains its maximum at $t=1$. Then for any $0<\\kappa<\\kappa_0:=\\frac{1}{(e\\sqrt{2p_0}(1+p_0))^{2-q}   (p_0^{2})^{1-q}}$, there exists some $0<t_{\\kappa}<1$\nsuch that $\\kappa<\\kappa_{0,t}$ and $2\\kappa+\\theta\\ln(\\frac{2\\theta C_1}{t})<0$.\t\n\t\n\tUnder the above choice of $\\kappa$ and $\\theta$ , from (\\ref{5-2-5}), we obtain\n\t\\begin{align}\\label{5-2-6}\n\t\t\\sum\\limits_{x \\in \\Omega'_k}\\mu(x) v(x)^{p-t_{\\kappa}}  |\\nabla v(x)|^q=0,\n\t\\end{align}\n\twhich contradicts with that u is a nontrivial positive solution by (\\ref{5-2-1}).\n\tHence we complete the proof for Theorem \\ref{thm1} (V).\n\\end{proof}\n\n\\begin{proof}[\\rm\\textbf{Proof of Theorem \\ref{thm1-1}}]\n\tSince $u$ is nontrivial, there exists some $x_0\\in V$ such that $|\\nabla u(x_0)|\\neq 0$. Let $x_1$ be a neighboring vertex of $x_0$ such that\n$$u(x_1)=min_{y \\sim x_0}u(y).$$\n\t\n\tBy (\\ref{ieq}), and noting $u$ is nontrivial, we obtain $\\Delta u(x_0)<0$.\nSince $|\\nabla u(x_0)|\\neq 0$, we have\n\t$$u(x_0)> u(x_1), \\quad|\\nabla u(x_1)|\\neq 0,\\quad \\text{and}\\quad \\Delta u(x_1)<0.$$\n\tInductively, for $x_i$, we can find $x_{i+1}\\in V$ such that $u(x_{i+1})=\\min\\limits_{y \\sim x_i}u(y)$ and\n\t$$u(x_i)> u(x_{i+1}),\\quad|\\nabla u(x_{i+1})|\\neq 0,\\quad\\text{and}\\quad \\Delta u(x_{i+1})<0. $$\n\tCombining $u(x_{i+1})< u(x_i)<\\cdot\\cdot\\cdot< u(x_0)$ with $u(x)>0$, by Monotone Convergence theorem, we obtain there exists nonegative constant $u_0$ such that\n\t\\begin{align}\\label{6-lim}\n\t\t\\lim\\limits_{i\\to \\infty}u(x_i)=u_0.\n\t\\end{align}\n\tSince\n\\begin{equation}\\label{6-lap}\n\t0>\\Delta u(x_i)=\\sum\\limits_{y \\sim x_i} \\frac{\\mu_{x_iy}}{\\mu(x_i)}u(y)-u(x_i)\\geq u(x_{i+1})-u(x_i).\n\\end{equation}\n\tby Squeeze theorem, we know\n\t\\begin{align}\\label{6-lim-1}\n\t\t\\lim_{i\\to \\infty}\\Delta u(x_i)=0.\n\t\\end{align}\n\tIt follows from (\\ref{ieq}) that\n\t\\begin{align}\\label{6-lim-2}\n\t\t\\lim_{i\\to \\infty}u(x_i)^p|\\nabla u(x_i)|^q=0.\n\t\\end{align}\n\tApplying Jesnsen's inequality, we obtain\n\t\\begin{align*}\n\t\t|\\Delta u(x_i)|^2&=\\left| \\sum\\limits_{y \\sim x_i}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-y(x_i))\\right|^2\\nonumber \\\\&\n\t\t\\leq\\left( \\sum\\limits_{y \\sim x_i}\\frac{\\mu_{x_iy}}{\\mu(x_i)}|u(y)-y(x_i)|\\right)^2\\nonumber \\\\&\n\t\t\\leq \\sum\\limits_{y \\sim x_i}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-y(x_i))^2=2|\\nabla u(x_i)|^2,\n\t\\end{align*}\n\nConsequently\n\t\\begin{align}\\label{6-ieq}\n\t\t|\\Delta u(x_i)|\\leq\\sqrt{2}|\\nabla u(x_i)|.\n\t\\end{align}\n\tCombining (\\ref{ieq}) with (\\ref{6-ieq}), we derive\n\t\\begin{align*}\n\t\tu(x_i)^p|\\nabla u(x_i)|^q\\leq -\\Delta u(x_i)\\leq|\\Delta u(x_i)|\\leq\\sqrt{2}|\\nabla u(x_i)|.\n\t\\end{align*}\n\tNoticing $u(x_i)>0$ and $|\\nabla u(x_i)|\\not=0$, we obtain\n\t\\begin{align}\\label{6-ieq-2}\n\t\t1\\leq\t\\sqrt{2}u(x_i)^{-p}|\\nabla u(x_i)|^{1-q}.\n\t\\end{align}\n\nLet us finish the proof by dividing into the following cases:\n\t\\begin{enumerate}\n\t\\item[(1).]{$q=0$, $0<p\\leq1$}\n\t\\item[(2).]{$q=p=0$}\n\t\\item[(3).]{$q=0, p<0$}\n\t\\item[(4).]{$0<q<1, p<1-q$}\n\t\\item[(5).]{$q<0$, $0<p<1-q$}\n\t\\item[(6).]{$q<0, p=0$}\n\t\\item[(7).]{$q<0, p<0$}\n\\end{enumerate}\n\n\n\nIn case (1), since $q=0$, $0<p\\leq 1$, combining (\\ref{6-lim}) and (\\ref{6-lim-2}),\nwe obtain\n$$\\lim_{i\\to \\infty}u(x_i)=0.$$\nBy (\\ref{ieq}), we get\n\\begin{align*}\n\t0\\geq\\sum\\limits_{y \\sim x_i} \\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-u(x_i))+u(x_i)^{p}.\n\\end{align*}\nFor $0<p< 1$, we have\n\\begin{align*}\n\t0\\geq -u(x_i)+u(x_i)^{p},\n\\end{align*}\nwhich implies $u(x_i)\\geq1$, and contradicts with $\\lim\\limits_{i\\to \\infty}u(x_i)=0$.\n\nFor $p= 1$, we have\n\\begin{align*}\n\t0\\geq\\sum\\limits_{y \\sim x_i} \\frac{\\mu_{x_iy}}{\\mu(x_i)}u(y)-u(x_i)+u(x_i)=\\sum\\limits_{y \\sim x_i} \\frac{\\mu_{x_iy}}{\\mu(x_i)}u(y),\n\\end{align*}\nwhich contradicts with $u>0$.\n\nIn case (2), by (\\ref{ieq}), we have\n\\begin{align*}\n\t\\Delta u(x_i)+1\\leq 0,\n\\end{align*}\nwhich contradicts with (\\ref{6-lim-1})\n\nIn case (3), since $q=0$, $p<0$, we derive a contradiction from (\\ref{6-lim})-(\\ref{6-lim-2}).\n\nIn case (4), we divide the proof into two cases:\n\\begin{enumerate}\n\t\\item[(4-1).]{there exists some $k_0$, such that when $i>k_0$, $|\\nabla u(x_i)|\\leq \\lambda(u(x_i)-u(x_{i+1}))$, where $\\lambda=u(x_0)^{\\frac{1-p-q}{q}}$}.\n\t\\item[(4-2).]{there exists a series $\\{i_k\\}$, such that $i_k\\to\\infty$ as $k\\to \\infty$ and $|\\nabla u(x_{i_k})|> \\lambda(u(x_{i_k})-u(x_{i_k+1}))$.}\n\\end{enumerate}\n\nIn case (4-1), since $1-q>0$, by substituting $|\\nabla u(x_i)|\\leq \\lambda(u(x_i)-u(x_{i+1}))$ into (\\ref{6-ieq-2}), we obtain\n$$1\\leq \\sqrt{2}\\lambda^{1-q}u(x_i)^{-p}(u(x_i)-u(x_{i+1}))^{1-q},$$\nwhich implies by (\\ref{6-lim}) that $p>0$ and $\\lim\\limits_{i\\to \\infty}u(x_i)=u_0=0$.\n\nNoticing $u(x_i)-u(x_{i+1})<u(x_i)$ and $0<p<1-q$, we have\n$$1\\leq \\sqrt{2}\\lambda^{1-q}u(x_i)^{1-q-p}.$$\nSince $p<1-q$, by letting $i\\to \\infty$, we derive a contradiction.\n\nIn case (4-2), and by (\\ref{ieq}), we have\n$$\\Delta u(x_{i_k})+u(x_{i_k})^p|\\nabla u(x_{i_k})|^q\\leq 0,$$\nwhich is equivalent to\n$$|\\nabla u(x_{i_k})|\\leq \\left[ |\\Delta u(x_{i_k})|u(x_{i_k})^{-p}\\right]^{\\frac{1}{q}}. $$\n\nNoting $0<q<1$, by (\\ref{6-lap}), (\\ref{6-lim-1}) and $|\\nabla u(x_{i_k})|> \\lambda (u(x_{i_k})-u(x_{i_k+1}))$, we obtain\n$$\\lambda (u(x_{i_k})-u(x_{i_k+1}))\\leq |\\nabla u(x_{i_k})|\\leq (u(x_{i_k})-u(x_{i_k+1}))^{\\frac{1}{q}}u(x_{i_k})^{-\\frac{p}{q}},$$\nwhich implies\n$$\\lambda\\leq (u(x_{i_k})-u(x_{i_k+1}))^{\\frac{1}{q}-1}u(x_{i_k})^{-\\frac{p}{q}}\\leq u(x_{i_k})^{\\frac{1-p-q}{q}} .$$\nNoting $\\lambda=u(x_0)^{\\frac{1-p-q}{q}}$, and $u(x_i)<u(x_0)$ for $i\\geq1$, we derive a contradiction.\n\n\n\tFor case (5), (6), and (7), since\n\\begin{align*}\n\t0\\geq\\Delta u(x_i)=\\sum_{y\\sim x_i\\atop u(y)>u(x_i)}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-u(x_i))+\\sum_{y\\sim x_i\\atop u(y)\\leq u(x_i)}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-u(x_i)),\n\\end{align*}\nIt follows that\n\\begin{align*}\n\t\\sum_{y\\sim x_i\\atop u(y)>u(x_i)}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-u(x_i))\\leq\\sum_{y\\sim x_i\\atop u(y)\\leq u(x_i)}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(x_i)-u(y)).\n\\end{align*}\nHence, we know for any $y_0\\sim x_i$, and $u(y_0)>u(x_i)$, and using $(p_0)$ condition, we have\n\\begin{align}\\label{6-5}\n\t\\frac{1}{p_0}(u(y_0)-u(x_i))&\\leq\\frac{\\mu_{x_iy_0}}{\\mu(x_i)}((u(y_0)-u(x_i))\\nonumber\\\\\n\t&\\leq\t\\sum_{y\\sim x_i\\atop u(y>u(x_i))}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-u(x_i))\\nonumber\\\\\n\t&\\leq \\sum\\limits_{y \\sim x_i \\atop u(y)\\leq  u(x_i)}\\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(x_i)-u(y)) \\leq u(x_i)-u(x_{i+1}).\n\\end{align}\nHere we have also used that $u(x_{i+1})=\\min\\limits_{y\\sim x_i}u(y)<u(x_i)$.\n\nCombining (\\ref{6-5}) with the definition of $|\\nabla u(x_i)|$, we have\n\\begin{align*}\n\t|\\nabla u(x_i)|^2&=\\sum\\limits_{y \\sim x_i \\atop u(y)> u(x_i)}\\frac{\\mu_{x_iy}}{2\\mu(x_i)}(u(y)-u(x_i))^2+\\sum\\limits_{y \\sim x_i \\atop u(y)\\leq u(x_i)}\\frac{\\mu_{x_iy}}{2\\mu(x_i)}(u(y)-u(x_i))^2\n\t\\\\ &\n\t\\leq \\frac{p_0^2}{2}(u(x_{i+1})-u(x_i))^2+\\frac{1}{2}(u(x_{i+1})-u(x_i))^2\n\t\\\\ &\n\t= \\frac{1+p_0^2}{2}(u(x_{i+1})-u(x_i))^2,\n\\end{align*}\nwhich is\n\\begin{align}\\label{6-6}\n\t|\\nabla u(x_i)|\\leq \\sqrt{\\frac{1+p_0^2}{2}}(u(x_i)-u(x_{i+1})),\n\\end{align}\nIt follows that\n\\begin{align}\\label{6-lim-3}\n\t\\lim_{i\\to \\infty}|\\nabla u(x_i)|=0.\n\\end{align}.\n\n\tIn case (5), noticing $p>0$, $q< 0$ and combining (\\ref{6-lim}) (\\ref{6-lim-2}) and (\\ref{6-lim-3}), we derive\n$$\\lim_{i\\to \\infty}u(x_i)=0.$$\n\tBy (\\ref{ieq}), we have\n\t\\begin{align*}\n\t\t0&\\geq\\sum\\limits_{y \\sim x_i} \\frac{\\mu_{x_iy}}{\\mu(x_i)}(u(y)-u(x_i))+u(x_i)^{p}|\\nabla u(x_i)|^q\\\\&\n\t\t=\\sum\\limits_{y \\sim x_i} \\frac{\\mu_{x_iy}}{\\mu(x_i)}u(y)-u(x_i)(1-u(x_i)^{p-1}|\\nabla u(x_i)|^q),\n\t\\end{align*}\n\twhich implies\n\t\\begin{align}\\label{6-0}\n\t\t1-u(x_i)^{p-1}|\\nabla u(x_i)|^q\\geq 0.\n\t\\end{align}\n\tCombining (\\ref{6-0}) with (\\ref{6-6}), we obtain\n\t\\begin{align}\\label{6-7}\n\t\t1\\geq u(x_i)^{p-1}|\\nabla u(x_i)|^q\\geq (\\frac{p_0^2+1}{2})^{\\frac{q}{2}}u(x_i)^{p-1}(u(x_i)-u(x_{i+1}))^q.\n\t\\end{align}\n Since $q<0$, we have\n \\begin{align}\\label{6-8}\n \t1\\geq (\\frac{p_0^2+1}{2})^{\\frac{q}{2}}u(x_i)^{p-1}(u(x_i)-u(x_{i+1}))^q\\geq(\\frac{p_0^2+1}{2})^{\\frac{q}{2}} u(x_i)^{p+q-1}.\n \\end{align}\n\t Using $p<1-q$, and by letting $i\\to \\infty$, we obtain a contradiction from (\\ref{6-8}) with $\\lim\\limits_{i\\to \\infty}u(x_i)=0$.\n\t\n\t\n\tIn case (6), noticing $p=0$, $q<0$, we obtain a contradiction by (\\ref{6-lim-2}) and (\\ref{6-lim-3}).\n\t\n\tIn case (7), since $p<0$, $q< 0$, substituting (\\ref{6-lim}) and (\\ref{6-lim-3}) into\n\t(\\ref{6-lim-2}), we derive a contradiction.\n\tHence, we complete the proof of Theorem \\ref{thm1-1}.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{thm2}}\n\nBefore presenting the proof of Theorem \\ref{thm2}, for our convenience, let us introduce some notations. For fixed integer $n\\geq0$, let us denote by $D_n: = \\{x \\in T_N : d(o, x) = n\\}$ the collection of all the vertices with distance $n$ from $o$, and denote by $E_n$ the collection of all the edges from vertices in $D_n$ to vertices in $D_{n+1}$.\n\n\\begin{proof}[\\rm{Proof of Theorem \\ref{thm2} (I)}]\n\tWhen $(p, q)\\in G_1$, let us take $\\mu$ and $u$ as follows\n\t\\begin{align}\\label{mu1}\n\t\t&\\mu_{xy}=\\mu_n=\\frac{(n+n_0)^{\\frac{p+1}{p+q-1}}(\\ln{(n+n_0)})^{\\frac{1}{p+q-1}+\\epsilon}}{(N-1)^n},\\quad \\mbox{for any $(x,y)\\in E_n$, $n\\geq0$}.\n\t\\end{align}\n\t\\begin{align}\\label{u1}\n\t\t&u(x)=u_n=\\frac{\\delta}{(n+n_0)^{\\frac{2-q}{p+q-1}}(\\ln{(n+n_0)})^{\\frac{1}{p+q-1}}},\n\t\t\\quad \\mbox{for any $x\\in D_n$, $n\\geq0$}.\n\t\\end{align}\n\twhere $n_0 \\geq 2$  and $\\delta > 0$ are to be chosen later.\n\t\nFirst, under the above choice of $\\mu$,\nfor $n \\geq 2$, we obtain\n\t$$\\mu(B(o,n))=\\sum\\limits^n_{k=0}\\mu(D_k)\\asymp \\sum\\limits^n_{k=0}(N-1)^k \\mu_k\\asymp  n^{\\frac{2p+q}{p+q-1}}(\\ln{n})^{\\frac{1}{p+q-1}+\\epsilon},$$\nwhich implies that (\\ref{e-vol-1}) holds.\n\nNext we verify that (\\ref{ieq}) holds for the above choice of $\\mu$ and $u$, namely the following two inequalities hold:\n\t\\begin{align}\\label{en0}\n\t\tu_1-u_0+u_0^p\\left[\\frac{(u_0-u_1)^2}{2}\\right]^{\\frac{q}{2}} \\leq 0,\n\t\\end{align}\nand\n\t\\begin{align}\\label{en}\n\t\t&\\frac{(N-1)\\mu_n u_{n+1}+\\mu_{n-1} u_{n-1}}{(N-1)\\mu_n+\\mu_{n-1}}-u_n\\nonumber\\\\\n&+u_n^p\\left[\\frac{(N-1)\\mu_n(u_{n+1}-u_n)^2+\\mu_{n-1}(u_{n-1}-u_n)^2}{2(N-1)\\mu_n+2\\mu_{n-1}}\\right]^{\\frac{q}{2}} \\leq 0.\n\t\\end{align}\nFor brevity, we denote\n$$\\lambda=\\frac{p+1}{p+q-1},\\quad \\beta=\\frac{1}{p+q-1}, \\quad\\sigma=\\frac{2-q}{p+q-1}.$$\nNow let us deal with cases of $n = 0$ and $n\\geq1$.\n\t\n\t\\textbf{Case of $n = 0$}.  Combining with (\\ref{u1}) and (\\ref{mu1}), then (\\ref{en0}) is equivalent to\n\t\\begin{align*}\n\t\t&\\frac{\\delta}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}-\\frac{\\delta}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}+\\left( \\frac{\\delta}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}\\right) ^p \\nonumber \\\\&\n\t\t\\times \\left[ \\frac{1}{2}\\left( \\frac{\\delta}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{\\delta}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}\\right) ^2\\right] ^{\\frac{q}{2}} \\leq 0,\n\t\\end{align*}\n\tthe above is satisfied if we choose $\\delta \\leq \\delta_0$ with\n\t\\begin{align*}\n\t\t\\delta_0&= 2^{\\frac{q}{2(p+q-1)}} \\left[\\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{1}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}} \\right]^{\\frac{1}{p+q-1}} \\\\ &\n\t\t\\times\\left( n_0^{\\sigma}(\\ln{n_0})^{\\beta}\\right)^{\\frac{p}{p+q-1}} \\left[ \\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{1}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}\\right]^{-\\frac{q}{p+q-1}}.\n\t\\end{align*}\n\t\n\t\\textbf{Case of $n\\geq1$}. Combining with (\\ref{u1}) and (\\ref{mu1}), then (\\ref{en}) is equivalent to\n\t\\begin{align*}\n\t\t&\\frac{\\delta\\frac{(n+n_0)^{\\lambda}(\\ln{(n+n_0)})^{\\beta+\\epsilon}}{(n+n_0+1)^{\\sigma}(\\ln{(n+n_0+1)})^{\\beta}}+\\delta\\frac{(n+n_0-1)^{\\lambda}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}{(n+n_0-1)^{\\sigma}(\\ln{(n+n_0-1)})^{\\beta}}}{(n+n_0)^{\\lambda}(\\ln{(n+n_0)})^{\\beta+\\epsilon}+(n+n_0-1)^{\\lambda}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}\t\\nonumber \\\\ &\n\t\t-\\frac{\\delta}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}}+\\left(\\frac{\\delta}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}} \\right)^p \\nonumber \\\\ &\n\t\t\\times\\delta^q\\left[ \\frac{(n+n_0)^{\\lambda}(\\ln{(n+n_0)})^{\\beta+\\epsilon}\\left(\\frac{1}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}}-\\frac{1}{(n+n_0+1)^{\\sigma}(\\ln{(n+n_0+1)})^{\\beta}} \\right)^2 }{2(n+n_0)^{\\lambda}(\\ln{(n+n_0)})^{\\beta+\\epsilon}+2(n+n_0-1)^{\\lambda}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}\\right.\\nonumber \\\\& \\left.\n\t\t+\\frac{(n+n_0-1)^{\\lambda}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}\\left(\\frac{1}{(n+n_0-1)^{\\sigma}(\\ln{(n+n_0-1)})^{\\beta}}-\\frac{1}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}} \\right)^2 }{2(n+n_0)^{\\lambda}(\\ln{(n+n_0)})^{\\beta+\\epsilon}+2(n+n_0-1)^{\\lambda}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}\n\t\t\\right]^\\frac{q}{2}\\leq0,\n\t\\end{align*}\n\tThe above is equivalent to the following estimate holds  for all $n\\geq1$\n\\begin{align}\\label{3-1-1}\n\\delta^{p+q-1}\\leq \\Lambda_1(n+n_0),\n\\end{align}\nwhere\n\t\\begin{align}\\label{3-1-2}\n\t\t\\Lambda_1(n):=&(n^{\\sigma}(\\ln{n})^{\\beta})^{p-1}\\left[ 1-\\frac{n^{\\sigma}(\\ln{n})^{\\beta}\\left( \\frac{n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}}+\\frac{(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}\\right) }{n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}+(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}\\right] \\nonumber \\\\\n&\n\t\t\\times\\left[ \\frac{n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}\\left(\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}-\\frac{1}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}} \\right)^2 }{2n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}+2(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}\\right.\\nonumber \\\\& \\left.\n\t\t+\\frac{(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}\\left(\\frac{1}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}-\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}} \\right)^2 }{2n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}+2(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}\n\t\t\\right]^{-\\frac{q}{2}}.\n\t\\end{align}\nHence, we have\n\t\\begin{align}\\label{3-1-lim}\n\t\t\\lim_{n\\to \\infty} \\Lambda_1(n)=(\\frac{2}{\\sigma})^{\\frac{q}{2}-1} \\epsilon.\n\t\\end{align}\nThe detailed calculation of (\\ref{3-1-lim}) is as follows: by using the facts that\n$$\\frac{\\ln{(n-1)}}{\\ln{n}}=1-\\frac{1}{n\\ln{n}}-\\frac{1}{2n^2\\ln{n}}+o(\\frac{1}{2n^2\\ln{n}}),$$\nand\n\t$$(1-\\frac{1}{n})^{\\alpha}=1-\\frac{\\alpha}{n}-\\frac{\\alpha(\\alpha-1)}{2n^2}+O(\\frac{1}{n^3}).$$\nand let us first deal with term\n\t\\begin{align}\\label{A1}\n\t\tA_1(n):=&\\left( \\frac{n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}\\left(\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}-\\frac{1}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}} \\right)^2 }{2n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}+2(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}\\right.\\nonumber \\\\& \\left.\n\t\t+\\frac{(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}\\left(\\frac{1}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}-\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}} \\right)^2 }{2n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}+2(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}\n\t\t\\right)^{-\\frac{q}{2}},\n\t\\end{align}\n\twhich is equal to\n\t\\begin{align}\\label{3-1-3}\n\t\t2^{\\frac{q}{2}}\\left( \\frac{1+(1-\\frac{1}{n})^{\\lambda}(\\frac{\\ln{(n-1)}}{\\ln{n}})^{\\beta+\\epsilon}}{\\left(\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}-\\frac{1}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}} \\right)^2+(1-\\frac{1}{n})^{\\lambda}(\\frac{\\ln{(n-1)}}{\\ln{n}})^{\\beta+\\epsilon}\\left(\\frac{1}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}-\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}} \\right)^2}\\right) ^{\\frac{q}{2}}.\n\t\\end{align}\n\tNotice that\n\t\\begin{align}\\label{3-1-4}\n\t\t\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}-\\frac{1}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}} &=\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}\\left( 1-(1-\\frac{1}{n+1})^{\\sigma}(\\frac{\\ln{n}}{\\ln{(n+1)}})^{\\beta}\\right) \\nonumber\\\\&\n\t\t=\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}\\left(\\frac{\\sigma}{n+1} +\\frac{\\beta}{(n+1)\\ln{(n+1)}}+o(\\frac{1}{n\\ln{n}})\\right)\\nonumber\\\\&\n\t\t=\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}\\left(\\frac{\\sigma}{n} +o(\\frac{1}{n})\\right).\n\t\\end{align}\n\tSimilarly\n\t\\begin{align}\\label{3-1-5}\n\t\t\\frac{1}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}-\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}\n\t\t=\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}\\left(\\frac{\\sigma}{n} +o(\\frac{1}{n})\\right).\n\t\\end{align}\n\tCombining (\\ref{3-1-4}) and (\\ref{3-1-5}) with (\\ref{3-1-3}), we obtain\n\t\\begin{align}\\label{3-1-6}\n\t\tA_1(n)=(\\frac{2}{\\sigma})^{\\frac{q}{2}}(n^{\\sigma+1}(\\ln{n})^{\\beta})^q\\left( \\frac{2+o(1)}{2+o(1)}\\right)^{\\frac{q}{2}}\n\t\t\\asymp (\\frac{2}{\\sigma})^{\\frac{q}{2}}(n^{\\sigma+1}(\\ln{n})^{\\beta})^q.\n\t\\end{align}\n\tSubstituting (\\ref{3-1-6}) into (\\ref{3-1-2}), we have\n\t\\begin{align*}\n\t\t\\lim_{n\\to\\infty}\\Lambda_1(n) =&\\lim_{n\\to\\infty} (\\frac{2}{\\sigma})^{\\frac{q}{2}}(n^{\\sigma}(\\ln{n})^{\\beta})^{p+q-1}n^q  \\left( 1-\\frac{n^{\\sigma}(\\ln{n})^{\\beta}\\left( \\frac{n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}}+\\frac{(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}\\right) }{n^{\\lambda}(\\ln{n})^{\\beta+\\epsilon}+(n-1)^{\\lambda}(\\ln{(n-1)})^{\\beta+\\epsilon}}\\right)\n\t\t\\nonumber\\\\\n\t\t= &\\lim_{n\\to\\infty}(\\frac{2}{\\sigma})^{\\frac{q}{2}}(n^{\\sigma}(\\ln{n})^{\\beta})^{p+q-1}n^q  \\left( 1-\\frac{(1-\\frac{1}{n+1})^{\\sigma}(\\frac{\\ln{n}}{\\ln{(n+1)}})^{\\beta}\n+(1-\\frac{1}{n})^{\\lambda-\\sigma}(\\frac{\\ln{(n-1)}}{\\ln{n}})^{\\epsilon}}{1+(1-\\frac{1}{n})^{\\lambda}(\\frac{\\ln{(n-1)}}{\\ln{n}})^{\\beta+\\epsilon}}\\right)\n\\nonumber\\\\\n=&\\lim_{n\\to\\infty}(\\frac{2}{\\sigma})^{\\frac{q}{2}}n^{2}\\ln{n}\n\t\t\\left( 1-\\frac{(1-\\frac{1}{n+1})^{\\sigma}(\\frac{\\ln{n}}{\\ln{(n+1)}})^{\\beta}+(1-\\frac{1}{n})(\\frac{\\ln{(n-1)}}\n{\\ln{n}})^{\\epsilon}}{1+(1-\\frac{1}{n})^{\\lambda}(\\frac{\\ln{(n-1)}}{\\ln{n}})^{\\beta+\\epsilon}}\\right).\n\t\\end{align*}\nwhere we have used that $\\lambda=\\frac{p+1}{p+q-1}$, $\\beta=\\frac{1}{p+q-1}$, $\\sigma=\\frac{2-q}{p+q-1}=\\lambda-1$.\n\n\tApplying Taylor expansion technique, we obtain\n\t\\begin{align*}\n\t\t\\lim_{n\\to\\infty}\\Lambda_1(n)\n\t\t=(\\frac{2}{\\sigma})^{\\frac{q}{2}-1} \\epsilon,\n\t\\end{align*}\n\twhich yields (\\ref{3-1-lim}).\n\tThis implies that there exists some large enough $n_0$ such that for all $n\\geq 0$, the RHS of (\\ref{3-1-1}) is bounded from above.\n\n\tFinally, take $n_0$ as above and $\\delta=\\min\\{\\delta_0, \\delta_1\\}$, where we choose $\\delta_1=\\left[\\frac{(\\frac{2}{\\sigma})^{\\frac{q}{2}-1} \\epsilon}{2}\\right]^{\\frac{1}{p+q-1}}$.\nIt follows that when $(p, q)\\in G_1$, $u$ is a solution to (\\ref{ieq}). Hence we complete the proof for Theorem \\ref{thm2} (I).\n\\end{proof}\n\\vskip1ex\n\n\\begin{proof}[\\rm{Proof of Theorem \\ref{thm2} (II)}]\n\tWhen $(p, q)\\in G_2$, take $\\mu$ and $u$ as follows\n\t\\begin{align}\\label{mu2}\n\t\t&\\mu_{xy}=\\mu_n=\\frac{(n+n_0)(\\ln{(n+n_0)})^{1+\\epsilon}}{(N-1)^n},\\quad \\mbox{for any $(x,y)\\in E_n$, $n\\geq0$},\n\t\\end{align}\n\t\\begin{align}\\label{u2}\n\t\t&u(x)=u_n=\\frac{1}{(\\ln{(n+n_0)})^{\\frac{\\epsilon}{2}}}+1,\n\t\t\\quad \\mbox{for any $x\\in D_n$, $n\\geq0$}.\n\t\\end{align}\n\tFor $n \\geq 2$, it is easy to verify that\n\t$$\\mu(B(o,n))\\asymp  n^{2}(\\ln{n})^{1+\\epsilon},$$\nwhich yields (\\ref{e-vol-2}) holds.\n\n\nNow we need to verify that (\\ref{en0}) and (\\ref{en}) holds under the above choice of $\\mu$ and $u$.\n\t\n\t\\textbf{Case of $n = 0$}. Substituting (\\ref{u2}) and (\\ref{mu2}) into (\\ref{en0}), we obtain\n\t\\begin{align}\\label{3-2-1}\n\t\t&\\frac{1}{(\\ln{(n_0+1)})^{\\frac{\\epsilon}{2}}}-\\frac{1}{(\\ln{n_0})^{\\frac{\\epsilon}{2}}}+\\left( \\frac{1}{(\\ln{n_0})^{\\frac{\\epsilon}{2}}}+1\\right) ^p\\nonumber \\\\&\n\t\t\\times \\left[\\frac{1}{2} \\left( \\frac{1}{(\\ln{n_0})^{\\frac{\\epsilon}{2}}}-\\frac{1}{(\\ln{(n_0+1)})^{\\frac{\\epsilon}{2}}}\\right)^2 \\right] ^{\\frac{q}{2}} \\leq 0.\n\t\\end{align}\n\n\tIt is easy to verify that the above holds for $q\\geq 2$ when $n_0$ is large enough.\n\t\n\t\\textbf{Case of $n\\geq1$}. By substituting (\\ref{u2}) and (\\ref{mu2}), we know (\\ref{en}) is equivalent to\n\t\\begin{align*}\n\t\t&\\frac{\\frac{(n+n_0)(\\ln{(n+n_0)})^{1+\\epsilon}}{ (\\ln{(n+n_0+1)})^{\\frac{\\epsilon}{2}}}+\\frac{(n+n_0-1)(\\ln{(n+n_0-1)})^{1+\\epsilon}}{(\\ln{(n+n_0-1)})^{\\frac{\\epsilon}{2}}}}{(n+n_0)(\\ln{(n+n_0)})^{1+\\epsilon}+(n+n_0-1)(\\ln{(n+n_0-1)})^{1+\\epsilon}}\t\\nonumber \\\\ &\n\t\t-\\frac{1}{(\\ln{(n+n_0)})^{\\frac{\\epsilon}{2}}}+\\left(\\frac{1}{(\\ln{(n+n_0)})^{\\frac{\\epsilon}{2}}}+1 \\right)^p \\nonumber \\\\ &\n\t\t\\times \\left( \\frac{(n+n_0)(\\ln{(n+n_0)})^{1+\\epsilon}\\left(\\frac{1}{ (\\ln{(n+n_0)})^{\\frac{\\epsilon}{2}}}-\\frac{1}{(\\ln{(n+n_0+1)})^{\\frac{\\epsilon}{2}}} \\right)^2}{2(n+n_0)(\\ln{(n+n_0)})^{1+\\epsilon}+2(n+n_0-1)(\\ln{(n+n_0-1)})^{1+\\epsilon}}\\right.\\nonumber \\\\& \\left.\n\t\t+\\frac{(n+n_0-1)(\\ln{(n+n_0-1)})^{1+\\epsilon}\\left(\\frac{1}{(\\ln{(n+n_0-1)})^{\\frac{\\epsilon}{2}}}-\\frac{1}{(\\ln{(n+n_0)})^{\\frac{\\epsilon}{2}}} \\right)^2}{2(n+n_0)(\\ln{(n+n_0)})^{1+\\epsilon}+2(n+n_0-1)(\\ln{(n+n_0-1)})^{1+\\epsilon}}\n\t\t\\right)^\\frac{q}{2} \\leq0,\n\t\\end{align*}\n\tnamely\n\\begin{equation}\\label{3-2-2}\n1\\leq \\Lambda_2(n+n_0),\n\\end{equation}\nwhere\n\t\\begin{align}\\label{3-2-3}\n\t\t\\Lambda_2(n):=&\\left(1-\\frac{(\\ln{n})^{\\frac{\\epsilon}{2}}\\left( \\frac{n(\\ln{n})^{1+\\epsilon}}{(\\ln{(n+1)})^{\\frac{\\epsilon}{2}}}+\n\t\t\t\\frac{(n-1)(\\ln{(n-1)})^{1+\\epsilon}}{(\\ln{(n-1)})^{\\frac{\\epsilon}{2}}}\\right)}{n(\\ln{n})^{1+\\epsilon}+(n-1)(\\ln{(n-1)})^{1+\\epsilon}}\\right)\n\t\t\\left( \\frac{1}{(\\ln{n})^{\\frac{\\epsilon}{2}}}+1\\right)^{-p} ((\\ln{n})^{\\frac{\\epsilon}{2}})^{-1}\\nonumber \\\\ &\n\t\t\\times\\left(\n\t\t\\frac{2n(\\ln{n})^{1+\\epsilon}+2(n-1)(\\ln{(n-1)})^{1+\\epsilon}}{n(\\ln{n})^{1+\\epsilon}\\left(\\frac{1}{(\\ln{n})^{\\frac{\\epsilon}{2}}}-\\frac{1}{(\\ln{(n+1)})^{\\frac{\\epsilon}{2}}} \\right)^2+(n-1)(\\ln{(n-1)})^{1+\\epsilon}\\left(\\frac{1}{(\\ln{(n-1)})^{\\frac{\\epsilon}{2}}}-\\frac{1}{(\\ln{n})^{\\frac{\\epsilon}{2}}} \\right)^2}\n\t\t\\right)^{\\frac{q}{2}}.\n\t\\end{align}\nApplying the same argument as in proof of Theorem \\ref{thm2} (I), we have\n\t\\begin{align}\\label{3-2-lim}\n\t\t\\lim_{n\\to \\infty} \\Lambda_2(n)=\\infty.\n\t\\end{align}\n\tThis implies that there exists some large $n_0$ such that both of (\\ref{3-2-1}) and (\\ref{3-2-2}) hold, hence we complete the proof for Theorem \\ref{thm2} (II).\n\\end{proof}\n\\vskip1ex\n\n\\begin{proof}[\\rm{Proof of Theorem \\ref{thm2} (III)}]\n\tWhen $(p, q)\\in G_3$, take $\\mu$ and $u$ as follows\n\t\\begin{align}\\label{mu3}\n\t\t&\\mu_{xy}=\\mu_n=\\frac{(n+n_0)^{\\frac{1}{q-1}}(\\ln{(n+n_0)})^{\\frac{1}{q-1}+\\epsilon}}{(N-1)^n},\\quad \\mbox{for any $(x,y)\\in E_n$, $n\\geq0$}.\n\t\\end{align}\n\t\\begin{align}\\label{u3}\n\t\t&u(x)=u_n=\\frac{\\delta}{(n+n_0)^{\\frac{2-q}{q-1}}(\\ln{(n+n_0)})^{\\frac{1}{q-1}}}+1,\n\t\t\\quad \\mbox{for any $x\\in D_n$, $n\\geq0$},\n\t\\end{align}\n\twhere $n_0 \\geq 2$ and $0<\\delta <1 $ are to be chosen later.\n\t\nUnder the choice of $\\mu$ in (\\ref{mu3}),\nit is easy to verify that (\\ref{e-vol-3}) holds.\n\n For brevity, let us denote $\\beta=\\frac{1}{q-1}$, $\\sigma=\\frac{2-q}{q-1}$.\n\n Let us deal with cases of  $n = 0$ and $n \\geq1$.\n\n\t\\textbf{Case of $n = 0$}. Substituting (\\ref{u3}) and (\\ref{mu3}) into (\\ref{en0}), we have\n\t\\begin{align*}\n\t\t&\\frac{\\delta}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}-\\frac{\\delta}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}+\\left( \\frac{\\delta}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}+1\\right) ^p \\nonumber \\\\&\n\t\t\\times \\left( \\frac{1}{2}\\left( \\frac{\\delta}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{\\delta}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}\\right) ^2\\right) ^{\\frac{q}{2}} \\leq 0,\n\t\\end{align*}\n\tsuch $\\delta$ is accessible by letting $\\delta \\leq \\delta_0$, where\n\t\\begin{align*}\n\t\t\\delta_0&= 2^{\\frac{q}{2(q-1)}} \\left(\\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{1}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}\\right)^{\\frac{1}{q-1}} \\\\ &\n\t\t\\times \\left( \\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}+1\\right) ^{\\frac{-p}{q-1}}\\left( \\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{1}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}\\right)^{-\\frac{q}{q-1}},\n\t\\end{align*}\n\twhere we used $\\delta<1$, $p<0$.\n\t\n\t\\textbf{Case of $n\\geq1$}. By substituting (\\ref{u3}) and (\\ref{mu3}), (\\ref{en}) is equivalent to\n\t\\begin{align*} &\\frac{\\delta\\frac{(n+n_0)^{\\beta}(\\ln{(n+n_0)})^{\\beta+\\epsilon}}{(n+n_0+1)^{\\sigma}(\\ln{(n+n_0+1)})^{\\beta}}+\\delta\\frac{(n+n_0-1)^{\\beta}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}{(n+n_0-1)^{\\sigma}(\\ln{(n+n_0-1)})^{\\beta}}}{(n+n_0)^{\\beta}(\\ln{(n+n_0)})^{\\beta+\\epsilon}+(n+n_0-1)^{\\beta}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}\t\\nonumber \\\\ &\n\t\t-\\frac{\\delta}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}}+\\left(\\frac{\\delta}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}} +1\\right)^p \\nonumber \\\\ &\n\t\t\\times\\delta^q\\left( \\frac{(n+n_0)^{\\beta}(\\ln{(n+n_0)})^{\\beta+\\epsilon}\\left(\\frac{1}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}}-\\frac{1}{(n+n_0+1)^{\\sigma}(\\ln{(n+n_0+1)})^{\\beta}} \\right)^2 }{2(n+n_0)^{\\beta}(\\ln{(n+n_0)})^{\\beta+\\epsilon}+2(n+n_0-1)^{\\beta}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}\\right.\\nonumber \\\\& \\left.\t\t+\\frac{(n+n_0-1)^{\\beta}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}\\left(\\frac{1}{(n+n_0-1)^{\\sigma}(\\ln{(n+n_0-1)})^{\\beta}}-\\frac{1}{(n+n_0)^{\\sigma}(\\ln{(n+n_0)})^{\\beta}} \\right)^2 }{2(n+n_0)^{\\beta}(\\ln{(n+n_0)})^{\\beta+\\epsilon}+2(n+n_0-1)^{\\beta}(\\ln{(n+n_0-1)})^{\\beta+\\epsilon}}\n\t\t\\right)^\\frac{q}{2}\\leq0,\n\t\\end{align*}\n\twhich is  equivalent to\n\t\\begin{equation*}\\label{3-3-1}\n\t\t\\delta^{q-1}\\leq \\Lambda_3(n+n_0),\n\t\\end{equation*}\nwhere\n\\begin{eqnarray*}\\label{3-3-2}\n\t\\Lambda_3(n):=&\\left( 1-\\frac{n^{\\sigma}(\\ln{n})^{\\beta}\\left( \\frac{n^{\\beta}(\\ln{n})^{\\beta+\\epsilon}}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}}+\\frac{(n-1)^{\\beta}(\\ln{(n-1)})^{\\beta+\\epsilon}}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}\\right) }{n^{\\beta}(\\ln{n})^{\\beta+\\epsilon}+(n-1)^{\\beta}(\\ln{(n-1)})^{\\beta+\\epsilon}}\\right) \\\\ &\n\t\t\\times \\left(\\frac{1}{(n)^{\\sigma}(\\ln{(n)})^{\\beta}} +1\\right)^{-p}((n)^{\\sigma}(\\ln{(n)})^{\\beta})^{-1} \\\\ &\n\t\t\\times\\left( \\frac{n^{\\beta}(\\ln{n})^{\\beta+\\epsilon}\\left(\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}}-\\frac{1}{(n+1)^{\\sigma}(\\ln{(n+1)})^{\\beta}} \\right)^2 }{2n^{\\beta}(\\ln{n})^{\\beta+\\epsilon}+2(n-1)^{\\beta}(\\ln{(n-1)})^{\\beta+\\epsilon}}\\right.\\\\& \\left.\n\t\t+\\frac{(n-1)^{\\beta}(\\ln{(n-1)})^{\\beta+\\epsilon}\\left(\\frac{1}{(n-1)^{\\sigma}(\\ln{(n-1)})^{\\beta}}-\\frac{1}{n^{\\sigma}(\\ln{n})^{\\beta}} \\right)^2 }{2n^{\\beta}(\\ln{n})^{\\beta+\\epsilon}+2(n-1)^{\\beta}(\\ln{(n-1)})^{\\beta+\\epsilon}}\n\t\t\\right)^{-\\frac{q}{2}}.\n\t\\end{eqnarray*}\n\tBy the similiar computation of the proof of Theorem \\ref{thm2} (I), we have that\n\t\\begin{align*}\n\t\t\\lim_{n\\to \\infty} \\Lambda_3(n)=(\\frac{2}{\\sigma})^{\\frac{q}{2}-1} \\epsilon.\n\t\\end{align*}\n\n\nThis implies that there exists some large $n_0$ such that for all $n\\geq 0$, the RHS of (\\ref{3-3-1}) is bounded from above by $\\delta_1^p$, where $\\delta_1=\\left[\\frac{(\\frac{2}{\\sigma})^{\\frac{q}{2}-1} \\epsilon}{2}\\right]^{\\frac{1}{q-1}}$.\n\t\n\tFinally, choosing $n_0$ as above and $\\delta=\\min\\{\\delta_0, \\delta_1\\}$, we obtain that when $(p, q)\\in G_3$, $u$ is a solution to (\\ref{ieq}). Hence we complete the proof for Theorem \\ref{thm2} (III).\n\\end{proof}\n\\vskip1ex\n\n\\begin{proof}[\\rm{Proof of Theorem \\ref{thm2} (IV)}]\n\tWhen $(p, q)\\in G_4$, we take $\\mu$ and $u$ as follows\n\t\\begin{equation}\\label{mu4}\n\t\t\\mu_{xy}=\\mu_n=\\frac{\\lambda e^{\\lambda(n+n_0)}}{(N-1)^n},\\quad \\mbox{for any $(x,y)\\in E_n$, $n\\geq0$},\n\t\\end{equation}\n\t\\begin{equation}\\label{u4}\n\t\tu(x)=u_n=\\frac{\\delta}{(n+n_0)}+\\delta,\n\t\t\\quad \\mbox{for any $x\\in D_n$, $n\\geq0$},\n\t\\end{equation}\n\twhere $n_0 \\geq 2$  and $\\delta>0 $ are to be determined later.\n\t\n\tUnder the above choices of $\\mu$, it follows that (\\ref{e-vol-4}) holds.\n\n\t\n\t\\textbf{Case of $n = 0$}. By substituting (\\ref{u4}) and (\\ref{mu4}), (\\ref{en0}) is equivalent to\n\t\\begin{align*}\n\t\t\\frac{\\delta}{n_0+1}-\\frac{\\delta}{n_0}+\\left( \\frac{\\delta}{n_0}+\\delta\\right) ^p\n\t\t\\times \\left( \\frac{1}{2}\\left( \\frac{\\delta}{n_0}-\\frac{\\delta}{n_0+1}\\right) ^2\\right) ^{\\frac{1}{2}} \\leq 0,\n\t\\end{align*}\n\twhich is satisfied by choosing $\\delta \\geq\\delta_0$ with\n\t\\begin{align*}\n\t\t\\delta_0&= 2^{\\frac{1}{2p}} \\left(\\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{1}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}} \\right)^{\\frac{1}{p}} \\\\ &\n\t\t\\times \\left( \\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}+1\\right) ^{-1}\\left( \\frac{1}{n_0^{\\sigma}(\\ln{n_0})^{\\beta}}-\\frac{1}{(n_0+1)^{\\sigma}(\\ln{(n_0+1)})^{\\beta}}\\right)^{-\\frac{1}{p}}.\n\t\\end{align*}\n\t\n\t\\textbf{Case of $n\\geq1$}. By substituting (\\ref{u4}) and (\\ref{mu4}), (\\ref{en}) is equivalent to\n\t\\begin{align*}\n\t\t&\\frac{e^{\\lambda(n+n_0)}  \\frac{\\delta}{n+n_0+1}+e^{\\lambda(n+n_0-1)} \\frac{\\delta}{n+n_0-1}}{e^{\\lambda(n+n_0)} +e^{\\lambda(n+n_0-1)} }-\\frac{\\delta}{n+n_0}\\nonumber\\\\+(\\frac{\\delta}{n+n_0}+\\delta)^p(&\\frac{e^{\\lambda(n+n_0)} (\\frac{\\delta}{n+n_0+1}-\\frac{\\delta}{n+n_0})^2+e^{\\lambda(n+n_0-1)} (\\frac{\\delta}{n+n_0-1}-\\frac{\\delta}{n+n_0})^2}{2e^{\\lambda(n+n_0)} +2e^{\\lambda(n+n_0-1)} })^{\\frac{1}{2}} \\leq 0,\n\t\\end{align*}\n\twhich is  equivalent to that $\\delta$ satisfies\n\\begin{equation}\\label{3-4-1}\n\t\\delta^{p}\\leq \\Lambda_4(n+n_0),\n\\end{equation}\n\twhere\n\t\\begin{align*}\n\t\t\\Lambda_4(n):=&\\left( \\frac{1}{n}-\\frac{e^{\\lambda n}  \\frac{1}{n+1}+e^{\\lambda(n-1)} \\frac{1}{n-1}}{e^{\\lambda n} +e^{\\lambda(n-1)} }\\right)(\\frac{1}{n}+1)^{-p} \\nonumber\\\\&\n\t\t\\times\\left(\\frac{e^{\\lambda n} (\\frac{1}{n+1}-\\frac{1}{n})^2+e^{\\lambda(n-1)} (\\frac{1}{n-1}-\\frac{1}{n})^2}{2e^{\\lambda n} +2e^{\\lambda(n-1)} }\\right) ^{-\\frac{1}{2}}.\n\t\\end{align*}\n\tSince\n\t\\begin{equation*}\n\t\t\\lim_{n\\to\\infty}\\Lambda_4(n)= \\sqrt{2}\\frac{1-e^{-\\lambda}}{1+e^{-\\lambda}}.\n\t\\end{equation*}\n\twe obtain that there exists some large $n_0$ such that for all $n\\geq 0$, the RHS of (\\ref{3-4-1}) is bounded.\nLetting $\\delta=\\max\\{\\delta_0, \\delta_1\\}$,\nwe derive that when $(p, q)\\in G_4$, $u$ is a solution to (\\ref{ieq}), where $\\delta_1:=\\left[\\frac{\\sqrt{2}\\frac{1-e^{-\\lambda}}{1+e^{-\\lambda}}}{2}\\right]^{\\frac{1}{p}}$. Hence we complete the proof for Theorem \\ref{thm2} (IV).\n\\end{proof}\n\\vskip1ex\n\\begin{proof}[\\rm{Proof of Theorem \\ref{thm2} (V)}]\tWe divide the proof into two cases:\n\t\\begin{enumerate}\n\t\t\\item[(V-1).]{$(p,q)\\in \\{p+q=1,p\\geq 0,q>0\\};$}\n\t\t\\item[(V-2).]{$(p,q)\\in \\{p+q=1,p>1, q<0\\};$}\n\t\\end{enumerate}\n\tIn case (V-1), we take $\\mu$ and $u$ as follows\n\t\\begin{align}\\label{mu5-1}\n\t\t&\\mu_{xy}=\\mu_n=\\frac{\\lambda e^{\\lambda n}}{(N-1)^n},\\quad \\mbox{for any $(x,y)\\in E_n$, $n\\geq0$}\n\t\\end{align}\n\t\\begin{align}\\label{u5-1}\n\t\t&u(x)=u_n=e^{-\\frac{\\lambda}{4} n},\n\t\t\\quad \\mbox{for any $x\\in D_n$, $n\\geq0$}.\n\t\\end{align}\n\tUnder these choices of $\\mu$, we obtain that (\\ref{e-vol-5}) holds. We choose $\\lambda$ later.\n\t\n\t\n\t\\textbf{Case of $n = 0$}. Combining (\\ref{u5-1}) and (\\ref{mu5-1}), (\\ref{en0}) is equivalent to\n\t\\begin{align}\\label{3-5-1}\n\t\t1\\leq 2^{\\frac{q}{2}}(1-e^{-\\frac{\\lambda}{4}})^{-q}(1-e^{-\\frac{\\lambda}{4}})= 2^{\\frac{q}{2}}(1-e^{-\\frac{\\lambda}{4}})^{1-q},\n\t\\end{align}\n\t\n\t\\textbf{Case of $n\\geq1$}. Combining (\\ref{u5-1}) and (\\ref{mu5-1}), (\\ref{en}) is equivalent to\n\t\\begin{align}\\label{3-5-2}\n\t\t1&\\leq\n2^{\\frac{q}{2}}(1-e^{-\\frac{\\lambda}{4}})^{1-q}\\left(\\frac{1+e^{-\\lambda}}{1+e^{-\\frac{\\lambda}{2}}} \\right) ^{\\frac{q}{2}}\\left(\\frac{1-e^{\\frac{3\\lambda}{4}}}{1+e^{-\\lambda}} \\right).\n\t\\end{align}\n\tIt is easy to see that both of (\\ref{3-5-1}) and (\\ref{3-5-2}) hold by choosing some large $\\lambda$. Hence $u$ is a solution to (\\ref{ieq}).\n\t\n\tIn case (V-2), Fix an arbitrary vertex $o\\in T_N$ as the root, we choose a special vertex $p$, $p\\sim o$. Let us define\n\t\\begin{align*}\n\t\t&P:=\\{x\\in T_N|\\mbox{o is not in the path between x and p}\\},\\\\\n\t\t&D'_{-n}:=\\{x\\in  P|d(o, x)=n\\} \\qquad\\mbox{for $n\\geq 0$},\\\\\n\t\t&D'_{n}:=\\{x\\in T_N\\setminus P|d(o, x)=n\\}\\qquad\\mbox{for $n\\geq 0$}.\n\t\\end{align*}\n\tAt last, we denote by $E'_n$ the collection of all the edges from vertices in $D'_n$ to\n\tvertices in $D'_{n+1}$ for $n\\in Z$.\n\t\n\tTake $\\mu$ and $u$ as follows\n\t\\begin{align}\\label{mu5-2-1}\n\t\t&\\mu_{xy}=\\mu_n=\\frac{\\lambda e^{\\lambda n}}{(N-1)^n},\\quad \\mbox{for any $(x,y)\\in E'_n$, $n\\geq0$},\n\t\\end{align}\n\t\\begin{align}\\label{mu5-2-2}\n\t\t&\\mu_{xy}=\\mu_n=\\frac{\\lambda e^{\\lambda n}}{(N-1)^{-n-1}},\\quad \\mbox{for any $(x,y)\\in E'_n$, $n\\leq 1$},\n\t\\end{align}\n\t\\begin{align}\\label{u5-2}\n\t\t&u(x)=u_n=e^{-(\\lambda-1) n},\n\t\t\\quad \\mbox{for any $x\\in D'_n$, $n\\in Z$}.\n\t\\end{align}\n\tNoticing $D_n=D'_{-n}\\cup D'_{n}$, for $n \\geq 2$, we have\n\t$$\\mu(B(o,n))=\\sum\\limits^n_{k=0}\\mu(D_k)\\asymp \\sum\\limits^n_{k=0}(N-1)^k (\\mu_k+\\mu_{-k})\\asymp  e^{\\lambda n},$$\n\twhere $\\lambda>0$ is to be chosen later.\n\t\n\tThen we check that (\\ref{ieq}) holds for the $\\mu$ and $u$ given as above: for any $n\\geq 0$,\n\t\\begin{align}\\label{3-5-2-1}\n\t\t&\\frac{(N-1)\\mu_n u_{n+1}+\\mu_{n-1} u_{n-1}}{(N-1)\\mu_n+\\mu_{n-1}}-u_n\\nonumber\\\\&+u_n^p(\\frac{(N-1)\\mu_n(u_{n+1}-u_n)^2+\\mu_{n-1}(u_{n-1}-u_n)^2}{2(N-1)\\mu_n+2\\mu_{n-1}})^{\\frac{q}{2}} \\leq 0,\n\t\\end{align}\n\tand for any $n\\leq -1$,\n\t\\begin{align}\\label{3-5-2-2}\n\t\t&\\frac{\\mu_n u_{n+1}+(N-1)\\mu_{n-1} u_{n-1}}{\\mu_n+(N-1)\\mu_{n-1}}-u_n\\nonumber\\\\&+u_n^p(\\frac{\\mu_n(u_{n+1}-u_n)^2+(N-1)\\mu_{n-1}(u_{n-1}-u_n)^2}{2\\mu_n+2(N-1)\\mu_{n-1}})^{\\frac{q}{2}} \\leq 0,\n\t\\end{align}\nhold.\n\n\t\nCombining  (\\ref{mu5-2-1})-(\\ref{u5-2}), we know (\\ref{3-5-2-1}) and (\\ref{3-5-2-2}) are equivalent to\n\t\\begin{align*}\n\t\t&\\frac{e^{\\lambda n} e^{-(\\lambda-1) (n+1)}+e^{\\lambda (n-1)} e^{-(\\lambda-1) (n-1)}}{e^{\\lambda n} + e^{\\lambda (n-1)}}- e^{-(\\lambda-1) (n)} \\nonumber\\\\&+ e^{-(\\lambda-1) p n}\\left(\\frac{e^{\\lambda n} (e^{-(\\lambda-1) (n+1)}- e^{-(\\lambda-1)n})^2+ e^{\\lambda (n-1)} (e^{-(\\lambda-1) (n-1)}- e^{-(\\lambda-1) n})^2}{2e^{\\lambda n} +2 e^{\\lambda (n-1)} }\\right)^{\\frac{q}{2}}\\\\&\n\t\t\\leq 0,\\quad\\mbox{ for any $n\\in \\mathbb{Z}$},\n\t\\end{align*}\n\tnamely\n\t\\begin{align}\\label{3-5-2-3}\n\t\t1&\\leq\n\t\t(1-e^{-(\\lambda-1)})^{1-q}\\left(\\frac{2+e^{-\\lambda}}{1+e^{\\lambda-2}} \\right) ^{\\frac{q}{2}}\\left(\\frac{1-e^{-1}}{1+e^{-\\lambda}} \\right).\n\t\\end{align}\n\tNoticing $q<0$, we obtain RHS of (\\ref{3-5-2-3}) tends to infinity as $\\lambda\\to\\infty$. This implies that there exists some large $\\lambda$, such that $ u $ is a solution to (\\ref{ieq}).\n\tHence we complete the proof for Theorem \\ref{thm2} (V).\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\\label{}\n\n\n\n\n\n\n\\section{}\n\\label{}\n\n\n\n\n\n\n\\section{Introduction}\n\nThe electric daily peak load is the maximum of the electricity power demand curve over one day. Having an accurate forecast of the daily peak enables independent system operators (ISOs) and energy providers to better deliver electricity and optimise power plant schedules. The importance of such a forecast is increasing as the integration of intermittent renewable production sources progresses. In particular, renewable energy sources are at the bottom of the merit order curve which makes them (currently) the most economical source of energy used to serve the market. However, they are intermittent and provide time-varying levels of power generation, which are only partially under human control. If electricity demand is high and renewables cannot provide for it alone, ISOs have to deliver electricity from sources with higher marginal costs (e.g., gas-fired plants) for the stakeholders as well as for the environment in terms of CO2 emissions. In such a context, accurately forecasting the peak demand magnitude and timing is essential for determining the generation capacity that must be held in reserve. \n\nElectrical equipment is tailored to support a specific peak load. If the demand comes close or exceeds the network capacity, it can lead to distribution inefficiencies and ultimately power system failures, such as blackouts. With the increasing number of electric vehicles (EV) in circulation, a further source of stress is added to the electricity system. For instance, 46\\% of vehicles sold in Norway in 2019 were EVs \\citep*{international_energy_agency_global_2019}. The challenge posed by the additional EV demand must be met by more tailored management systems and policies, if expensive infrastructural works are to be avoided. Dynamic electricity pricing schemes, for example, the Triads in the UK or the Global Adjustment in Ontario, Canada, have been developed to reduce the system peak load. Consumers who can correctly estimate and cut their use during peak events can unlock great savings. Peak demand forecasts will thus be key for the development of such policies.\n\n\nTo account for the increasing demand for electricity and to prevent system failures, smart grid technologies and policies are being implemented to foster communication between the various stakeholders of the electricity supply chain to achieve a more efficient use of energy. One major objective is to maximise the load factor. The load factor is the average load over a specific time period divided by the peak load over the same period. Maximising it leads to a more even use of energy through time, thus preventing system failures and surges in electricity prices. One of the most common ways to achieve load factor maximisation is peak shaving (Figure \\ref{Shaving}), which refers to the flattening of electrical load peaks. Three major strategies have been proposed for peak shaving, namely integration of Energy Storage System (ESS), integration of Vehicle-to-Grid (V2G) and Demand Side Management (DSM) \\citep*{uddin_review_2018}. ESS and V2G integration provide ancillary sources to balance the grid through batteries while DSM shifts consumer demand to flatten the peak. To be activated adequately, all these strategies require accurate forecasts of the demand peak magnitude (DP) and of the instant at which it occurs (IP).\n\n\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{Shaving.jpg}\n    \\caption{Illustration of peak shaving}\n    \\label{Shaving}\n\\end{figure}\n\n\nThis article proposes novel methods to forecast the DP and the IP by leveraging information at different time resolutions. In particular, the multi-resolution approach proposed here is illustrated in the context of two model classes: Generalised Additive Models (GAMs) and Neural Networks (NNs). Both are state of the art predictive models, widely used to forecast electrical load in industry and academia. The performance of the multi-resolution framework under both model classes is assessed using aggregate UK electricity demand data from the National Grid \\citep*{nationalgrid_eso_2021}.\n\nThe rest of the paper is structured as follows: Section 2 presents a literature review of daily peak forecasting methodologies. Section 3 introduces multi-resolution modelling using GAMs and neural networks. Section 4 explains how the different models were set up in the high-resolution, low-resolution and multi-resolution settings. Section 5 analyses the results of the models described in Section 4, using UK demand data.\n\n\\section{Related work}\n\nThis section provides an extensive literature review of peak forecasting methods and was conducted to identify gaps in the field. It includes methods ranging from probabilistic approaches to deep learning.\n\nProbabilistic forecasts have been widely adopted in the context of load forecasting applications \\citep*[e.g.,][for an overview]{hong_probabilistic_2016}, but little has been done on probabilistic peak demand forecasting. Two probabilistic set-ups, commonly used for peak load forecasting, were outlined by \\cite*{jacob_forecasting_2020}. The first is block maxima (BM), where data is separated into time chunks of equal lengths and the maximum of each chunk is assumed to approximately follow a generalised extreme value (GEV) distribution. The second is peaks over threshold (POT), which approximates the distribution of the excess load over a threshold by a generalised Pareto distribution. While the POT and BM settings can be unified via point processes \\citep*{boano-danquah_analysis_2020}, in this work we are mainly interested in the BM case.\n\nIn a long-term forecasting setting, \\cite*{mcsharry_probabilistic_2005} used demand data at the daily resolution to forecast the magnitude and timing of the yearly peak (i.e., the day characterised by the largest total demand). They considered a forecasting lead time of one full year and obtained a probabilistic forecast by simulating year-long trajectories for the weather variables and plugging them into a deterministic linear regression model. Similarly, \\cite*{hyndman_density_2010} considered a long-term forecasting application, where the aim was to forecast the probability distribution of the annual and weekly peak electricity demand. They used semi-parametric additive models to capture the effect of covariates, such as temperature, on the demand and obtained a probabilistic forecast by adopting a simulation and scenario-based approach. \\cite*{elamin_quantile_2018} used quantile regression methods to forecast the DP one day ahead. Even though they used quantile regression to obtain an upper bound on demand, quantile estimates at several probability levels could be used to estimate the full peak demand distribution. Also \\cite*{gibbons_quantile_2014} modelled the DP via a quantile regression model, but their objective was post-processing daily estimates to forecast the annual demand peak, rather than modelling the DP probabilistically.\n\nMultivariate regression models using multivariate adaptive regression splines (MARS) were proposed by \\cite*{sigauke_daily_2010} to forecast the DP in South Africa. Explanatory variables including meteorological variables are aggregated at the daily resolution (e.g., average, minimum and maximum temperature). The model outperforms piecewise polynomial regression models with an autoregressive error term. \\cite*{sigauke_prediction_2011} studied time series of the DP and illustrated its heteroscedastic structure. A SARIMA\u2013GARCH errors model and a regression-SARIMA\u2013GARCH model are then proposed to forecast it at a short-term horizon. Results show that SARIMA-like models produce forecasts with an accuracy around 1.4 in mean absolute percentage error on a testing period. \n\n\\cite*{saxena_hybrid_2019} proposed a hybrid model to forecast whether the following day will be a peak load day for the billing period for customers subject to demand charge structure. They apply their model to optimise the electricity bill of an American University. Load data is provided every five minutes from January 2013 to April 2016. Here, the POT set-up was used with a threshold depending on a monthly average and variance of the daily load. An original combination of 4 forecasts was proposed. First, a linear model is used to forecast the maximum daily load at a monthly horizon which is then coupled to short-term load forecasting models (NN and ARIMA) to provide two forecasts. Two other forecasts were computed using binary classifiers (logistic regression and NN) and a synthetic minority over-sampling technique (SMOTE) was used to balance the classes. The authors demonstrated that their methods led to better statistical accuracy and to reduced electricity bills.\n\n\nNNs are one of the most popular algorithms for peak load forecasting tasks because of their strong performance in non-linear modelling. Their flexibility is remarkable, but it is difficult to pick the right architecture and hyper-parameters for a specific problem. One of the first papers proposing a NN peak load forecasting method was produced by \\cite*{dash_peak_1995}. According to the authors, NNs performed well on load forecasting problems, but they were much less performant on peak load forecasting tasks. A fuzzy NN was found to be more robust and accurate than a traditional NN structure. It involved an additional layer of fuzzification of the inputs before entering the only hidden layer of the network. \n\nIn a more traditional set-up, \\cite*{saini_artificial_2002} tested a Fully Connected Neural Network (FCNN) with different variants of back-propagation algorithms where training was conducted separately in four periods of time during a year. Their work was further developed by \\cite*{saini_peak_2008}, where numerous weather variables were included (e.g., temperature, rainfall, wind speed, evaporation per day, sunshine hours and associated statistics). Similarly, different optimisation procedures were considered and it was found that an adaptive learning method based on the learning rate and momentum was the most performant. \\cite*{amin-naseri_combined_2008} combined a self-organising map with a NN to find better clusters of training data to improve forecasting performance.  Some authors considered other form of networks. For instance, \\cite*{abdel-aal_modeling_2006} adopted abductive networks with the aim of obtaining a better intuition and a more automated way to address peak load forecasting. In particular, these networks split the overall problem into smaller and simpler ones along the network with abductive reasoning. It is based on an automated procedure which organises the data available into different chunks and deals with them separately. \n\nMore recently, recurrent Neural Networks (RNNs) have been used by \\cite*{yu_deep_2019} in the form of Gated Recurrent Units (GRU). In particular, a dynamic time warping (DTW) analysis was used to produce the GRU inputs. The DTW distance was used to find the most similar load curve to the one observed before the targeted load curve. Assuming that subsequent load curves are also very similar, they used the subsequent load curve from the training data to encode the inputs of the GRU network. A Long Short-Term Memory (LSTM) architecture has been used by \\cite*{ibrahim_lstm_2020} and was found to be more computationally efficient compared to FCNNs and other RNNs. Three statistical metrics were used to evaluate model performance: Mean Absolute Percentage Error (MAPE), Root-Mean Squared Error (RMSE) and mean bias error. In our work, statistical metrics including MAPE and RMSE will also be used to avoid introducing any bias towards a particular operational application. \n\nThe literature on deep learning peak load forecasting is sparse, but deep learning probabilistic load forecasting is much more common (e.g., \\citealp*{guo_deep_2018}, \\citealp*{yang_deep_2019} and \\citealp*{yang_bayesian_2020}). Such models do not explicitly focus on the DP or the IP as the objective functions used to estimate their parameters are based on demand observed at a higher frequency (intra-day). The high-frequency forecasts thus obtained can be post-processed to produce a forecast for the DP. \n\nSupport Vector Regression (SVR) is another popular class of load forecasting method, based on structural risk minimisation instead of empirical risk minimisation as in NNs. \\cite*{el-attar_forecasting_2009} used SVR in a local prediction framework. Recently, \\cite*{kim_peak-load_2020} used an ensemble forecasting approach with other Machine Learning algorithms such as boosting machines, tree-based methods and bagging techniques. A compensation process based on an isolation forest is later added by analysing the predicted values of the ensemble models to detect outliers in the peak data. SVR are compared to NNs by \\cite*{li_analysis_2018} for a control strategy of peak load and frequency regulation. LSTM NNs were used to forecast power load and improve the control strategy considered in this particular use case.\n\nFrom this literature review, it can be concluded that a wide range of methodologies have been adopted in peak load forecasting applications. In most short-term applications, model inputs are manually chosen features that are defined at the same (daily) time resolution as the peak demand, which is the variable to be forecasted. Conversely, in long-term applications, weather variables are simulated at the original (high) resolution to produce demand forecasts at the same resolution, which are then post-processed to obtain low resolution (e.g., yearly) peak forecasts. Hence, to the best of our knowledge, the existing literature on peak forecasting has not explored methods that are able to integrate both low- and high-resolution signals in a single model. However, in the field of functional data analysis, hybrid approaches have been used for clustering and forecasting functional data (e.g., \\citealp*{antoniadis_functional_2006} and \\citealp*{cho_modeling_2013}). Therefore, this paper aims to exploit functional methods to tackle multi-resolution problems. From a feature engineering point of view, the goal is to automate feature extraction of high-resolution signals, that is to let the model decide which hidden features to extract from the signal. This can be done with signal processing procedures such as tensor product decomposition, wavelets or Fourier transforms \\citep*{amin_feature_2015}. \n\nThe literature review also suggests that not much effort has been directed towards forecasting the IP, which is surprising because forecasting the IP is at least as important as forecasting the DP, for the purpose of short-term smart grid management and operational planning \\citep*{soman_peak_2020}. To fill this gap, the performance of multi-resolution methods will be illustrated in this paper on both a DP and an IP forecasting problem. \n\n\\section{Multi-resolution modelling}\n\nIn this section, the multi-resolution modelling approach is introduced with its general principles. It is then developed formally and illustrated with GAMs and NNs.\n\n\\subsection{General idea}\n\nThe main idea behind multi-resolution modelling is to build a parsimonious model that is able to handle input and output variables that are available at different resolutions.\nIn the context of DP load forecasting, low-resolution variables (e.g., day of the week, maximum daily temperature) are observed daily, while high-resolution variables (e.g., temperatures or raw demand) are updated every hour or half-hour. Such problems are usually handled by manually placing all variables at the same resolution. In particular, one option is to take a high-resolution approach, which consists in doing the modelling at the highest available resolution, which might require interpolating some of the low-resolution variables. Such an approach often lacks in parsimony, as the low-resolution variables are brought to the higher resolution, thus increasing the size of the data that needs to be processed, while adding no extra useful information. Another option is to take a low-resolution approach, that is to transform the high-resolution variables into a set of manually chosen daily summaries or features. In this approach, the size of the data is reduced, but feature engineering is time consuming and some of the information contained in the high-resolution variables is lost in the process. \n\nThe multi-resolution approach proposed here aims at capturing all the information contained in the high-resolution variable, while avoiding explicit feature engineering and retaining the parsimony of the low-resolution approach. To describe the multi-resolution idea more formally, let us consider $\\textbf{y}_i = \\{y_i(t)\\}_{t\\in\\{1,\\ldots,T\\}}$ the vector of electricity demand at each time step $t > 0$ of the day $i \\in \\mathbb{N}$ . $T$ is the total number of daily steps (e.g., T=48 for half-hourly steps). Then, the DP of day $i$ is $\\textrm{DP}_i = max(\\textbf{y}_i)$ and $\\textrm{IP}_i$ is the time step corresponding to $\\textrm{DP}_i$. Let $\\textbf{x}^{low}_i$ be the $i$-th vector of covariates observed daily and let $\\textbf{x}^{high}_i$ be the corresponding vector of covariates containing information at the intra-day resolution. The multi-resolution approach exploits both sets of covariates as model inputs to obtain the forecasts of the $\\hat{\\textrm{DP}}_i$ or the $\\hat{\\textrm{IP}}_i$, that is\n\\begin{align}\n  \\hat{\\textrm{DP}}_i &= \\psi_1(\\textbf{x}^{low}_i,\\textbf{x}^{high}_i)  \\\\\n  \\hat{\\textrm{IP}}_i &= \\psi_2(\\textbf{x}^{low}_i,\\textbf{x}^{high}_i) \n\\end{align}\nwhere $\\psi_1$ and $\\psi_2$ represent the model for, respectively, the DP and the IP.\nThis general definition does not specify how the high-resolution inputs should be dealt with in practice. Several approaches could be considered, the aim being to process the information contained in a (possibly high-dimensional) signal vector, while avoiding information loss and retaining computational efficiency. In this paper, two options are considered. In particular, a description of how high-resolution covariates can be handled within GAMs and NNs is given below. \n\n\\subsection{Particular instances of the multi-resolution approach}\n\nThe multi-resolution approach is detailed firstly for GAMs which, due to their performance and interpretability \\citep*{amato_forecasting_2021}, are widely used in industry for load forecasting. Then, the multi-resolution approach is extended to NNs, which often perform well on load forecasting problems and enable the flexible handling of heterogeneous model inputs \\citep*{gao_matrix_2017}. \n\n\\subsubsection{Generalised Additive Models}\n\n First introduced by \\cite*{hastie_generalized_1999}, GAMs are a semi-parametric extension of generalised linear models (GLMs) where the response variable, $y_i$, is assumed to follow a parametric probability distribution. That is, $y_i \\sim \\text{Dist}(\\mu_i, \\bm \\theta)$ where $\\mu_i$ and $\\bm \\theta$ are model parameters. While the elements of $\\bm \\theta$ do not depend on $i$, parameter $\\mu_i$ is modelled as follows \\citep*{wood_generalized_2017}:\n\\begin{equation}\n    g(\\mu_{i})=\\mathbf{x}_{i}^T \\bm{\\gamma}+\\sum_{j} f_{j}(\\mathbf{x}_{i})\n\\end{equation}\nwhere $g$ is a monotonic transformation, which is simply the identity function in this paper. Two separate terms can be distinguished on the right-hand side of this equation: a parametric part $\\mathbf{x}_{i}^T \\bm{\\gamma}$, where $\\mathbf{x}_{i}$ is a vector of covariates while $\\bm \\gamma$ is a vector of regression coefficients, and a non-parametric part $\\sum_{j} f_{j}(\\bm{x}_{i})$ which is a sum of smooth functions of covariates. The smooth effects are built via linear combinations of $K_j$ basis functions, while the corresponding basis coefficients are penalised via generalised ridge penalties. The strength of the penalties is controlled via smoothing hyper-parameters, which are selected using criteria such a generalised cross-validation.\n\nIn the context of forecasting $\\textrm{DP}_i$, it is interesting to consider for $\\text{Dist}(\\mu_i, \\bm \\theta)$ a generalised extreme value (GEV) distribution. In fact, the GEV model is asymptotically justified for block-maxima as $T \\rightarrow \\infty$ \\citep*{jacob_forecasting_2020}. Thus, when enough steps are available throughout the day, the GEV distribution is particularly attractive for modelling the DP. The scaled-T (a scaled version of Student's t) distribution provides an alternative, which is particularly suited for heavy tailed data such as peak load. The Gaussian distribution can be used as a baseline model. As for the IP, an ordered categorical (ocat) distribution based on a logistic regression latent variable is used. All of these distributions as well as GAM building and fitting methods are implemented in the \\textit{mgcv} R package \\citep*{wood_mgcv_2020}.\n\nWithin the additive structure of GAMs, $\\mathbf{x}^{low}_i$ and $\\mathbf{x}^{high}_i$ can be treated as inputs for different smooth functions. The elements of $\\mathbf{x}^{low}_i$ can be handled via separate standard smooth effects, which take scalars as inputs, while the joint effect of several elements of $\\mathbf{x}^{low}_i$ can be captured via standard multivariate smooth effects. However, the $\\mathbf{x}^{high}_i$ covariates have to be treated via functional smooth effects. The latter are smooth functions which take the vectors of high-resolution covariates as inputs and output a scalar. Therefore, functional GAMs permit the handling of each covariate at its original resolution, thus avoiding interpolation and guaranteeing parsimony.\n\nIn addition to the principle of parsimony, the goal is also to retain the time dependence of the covariates. In fact, it is important to ensure that the model is aware that each element of the high-resolution covariates has a different impact on the peak load distribution, as it belongs to a different time of day. A way to retain the time dependence of each high-resolution series of covariates is to make them interact with the time of day sequence via tensor product effects. Such effects can easily be integrated in GAMs, as explained in the following.\n\nIn continuous time, the smooth effect for a high-resolution (functional) covariate, $x_i(u)$, can be written as follows:\n\\begin{align}\nf(x_i) = \\int_{0}^{T}\\phi({x}_i(u),u)du\n\\end{align}\nwhere $\\phi$ is the time-dependent effect of the covariate, which needs to be estimated, while $u$ is the time of day. In practice, on the $i$-th day, ${x}_i(u)$ is observed at $F$ discrete instants $0 \\leq t_1 \\leq \\cdots \\leq t_F \\leq T$ and the corresponding values of ${x}_i(u)$ are stored in the vector $\\bm x_i$. Hence, approximating the integral with a summation and constructing $\\phi$ via a tensor product expansion leads to:\n\\begin{align}\n\\hat{f}(\\bm{x}_i) & = \\sum_{r=1}^{F}\\hat{\\phi}({x}_i(t_r),t_r) \\nonumber\\\\\n                      & = \\sum_{r=1}^{F} \\sum_{k=1}^{K} \\sum_{l=1}^{L} \\beta_{kl}a_k({x}_i(t_r))b_{l}(t_r)\n\\end{align}\nwhere $\\{a_{k}\\}_{(k)\\in\\{1,\\ldots,K\\}}$ and $\\{b_{l}\\}_{(l)\\in\\{1,\\ldots,L\\}}$ are known spline basis functions and $\\{\\beta_{kl}\\}_{(k,l) \\in\\{1,\\ldots,K\\} \\times \\{1,\\ldots,L\\}}$ are parameters to be estimated. By using such effects, high-resolution information can be parsimoniously incorporated into the model, while retaining the temporal information contained in the covariates. \n\n\\subsubsection{Neural Networks}\n\nNNs are convenient machine learning algorithms to implement a multi-resolution model. In fact, common architectures such as Convolutional Neural Networks (CNN) and RNNs already make use of inputs from different scales. Recent work was undertaken to make tensor inputs available for multi-layer perceptrons with MatNet \\citep*{gao_matrix_2017} which further shows their versatility. From scalars to tensors, the flexibility of NNs is hard for other machine learning models to compete with.\n\nA FCNN or CNN architecture, without its output layer, can be generally written as follows:\n\\begin{align}\n   H_k(\\mathbf{x},\\Theta) = h_k (\\ldots h_3(h_2(h_1(\\mathbf{x},\\theta_1),\\theta_2),\\theta_3)\\ldots, \\theta_k ) \n\\end{align}\nwhere k is the number of hidden layers of the NN, ${h_i}_{,i \\in \\{1 \\ldots k\\}}$ are the transformations made by the hidden layers (e.g., linear operation, activation and dropout) and $\\Theta = \\{\\theta_i\\}_{,i \\in \\{1 \\ldots k\\}}$ is the sequence of parameter vectors (weights and biases). In a multi-resolution approach, one part of the architecture will contain low-resolution information feeding a FCNN branch and the other one will contain the reshaped high-resolution data feeding a CNN or RNN branch. In this paper, only CNNs were considered in depth for this latter branch, with the lags of the response provided as model inputs. The CNN enables a very close replication of the tensor product construction used for GAMs, thus creating a consistent set-up for comparing both algorithms.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{MR_for_NN.jpg}\n    \\caption{Multi-resolution architecture for NNs with a Multi-Layer Perceptron (MLP) taking low-resolution inputs, and a CNN with high-resolution inputs}\n    \\label{MRNN}\n\\end{figure}\n\nEven though the CNN and FCNN branches do not have similarly shaped inputs and outputs, the unit shapes can be transformed along the network to interact and be brought together without losing consistency. This process consists in flattening the tensor shapes in order to bounce back on vectorial inputs within some layer of the network. It is precisely this flexibility that can be leveraged to build a multi-resolution architecture (Figure \\ref{MRNN}).\nMore precisely, the CNN branch contains one convolutional block for each of the high-resolution time series. In this way, each tensor product of the GAM formula can find its equivalent in the CNN branch of the network. In fact, the multi-resolution NN architecture can be concisely written as follows:  \n\\begin{align}\n    \\mathbf{\\mu}_i = F_j(H_k(\\mathbf{x}_{low},\\Theta), H'_l(\\mathbf{x}_{high},\\Theta'))\n\\end{align}\nIn (7), $H_k$ is the FCNN which handles low-resolution terms while $H'_l$ is the CNN which deals with the high-resolution information. Then, in the final part of the network, both outputs are concatenated (after flattening the CNN branch) and enter another FCNN $F_j$ which can be reduced to the output layer when $j = 1$. Here, $\\mu_i$ is the mean of the random output variable considered. This multi-resolution architecture is summarised in Figure \\ref{MRNN}.\n\n\\section{Experiments}\n\nOn the DP and the IP forecasting tasks, the multi-resolution approach is compared to two alternative modelling approaches: a high-resolution approach and a low-resolution approach (Figure \\ref{High-, low- and multi-resolution modelling setting}). The low-resolution approach uses inputs aggregated at the daily level (e.g., maximum daily temperature, day of the week) to forecast the DP and the IP separately. The high-resolution approach uses inputs at the half-hourly level to forecast the half-hourly demand and then it extracts the DP and the IP by taking the maximum of the half-hourly forecasted values and the corresponding time of day. Therefore, the high-resolution approach leverages all the information available by taking half-hourly inputs and outputs while the low-resolution approach directly models the variables of interest (DP and IP) with less parameters to be estimated. The multi-resolution approach can be seen as a compromise, aimed at integrating the advantages of both approaches, and the following experiments are designed to assess whether it can outperform them. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{Modelling-settings.jpg}\n    \\caption{The different modelling settings compared in this work}\n    \\label{High-, low- and multi-resolution modelling setting}\n\\end{figure}\n\nThe comparison includes baseline models: a naive persistence model, which simply consists of forecasting the DP and the IP based on the value taken by the target variable on the previous day; a low-resolution ARIMA (on daily peaks with horizon 1); a high-resolution ARIMA aggregated forecast composed of 48 ARIMA models, each fitted on the half-hourly load of a specific time of day with horizon 1. That is, the high-resolution ARIMA produces 48 forecasts at horizon 1 instead of one forecast at horizon 48. All ARIMA models are fitted using the \\cite*{hyndman_automatic_2007} algorithm without using exogenous information.\n\nThe performance metrics chosen for DP models are the mean absolute percentage error (MAPE), the mean absolute error (MAE) and the root mean squared error (RMSE). As for IP models, the same metrics are used except for the MAPE, which is substituted with a relaxed accuracy (R-Accuracy) metric in the form of a binary loss function (equal to 1 if the IP forecasted is more than 2 instants away from the observed IP and 0 if it is within 2 instants of the observed IP). While the R-Accuracy metric is also relevant in operational settings where it is crucial to know the IP within a small time window, the RMSE and the MAE penalise forecasts proportionally to their distance from the observed IP.\n\n\nA rolling-origin forecasting procedure is used to replicate a realistic short-term load forecasting set-up (Figure \\ref{Rolling Origin}). Model parameters are updated on a monthly basis with consolidated data since, in an operational setting, threats to data validity and computational constraints can emerge when refitting a model too often using real-time data.\n\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{Rolling-origin.jpg}\n    \\caption{Rolling-origin forecasting procedure}\n    \\label{Rolling Origin}\n\\end{figure}\n\nThe data used in the experiments is the half-hourly load consumption (total national demand) between 2011-07-01 00:00:00 and 2016-06-30 23:30:00, available via the UK \\cite*{nationalgrid_eso_2021} website. Temperature data at different locations (London, Sheffield, Manchester, Leeds, Cardiff, Bristol, Birmingham, Liverpool, Crosby and Glasgow) was downloaded from the \\cite*{noaa_national_2021} website. The temperature data is at an hourly resolution. It is interpolated (natural cubic spline interpolation) to obtain half-hourly data. Furthermore, demographic information $pop_s$ is compiled around each station $s$ and a weighted mean temperature is calculated as follows:\n\\begin{equation*}\n    \\mathrm{temp}(t) = \\frac{1}{\\sum_{a=1}^{10} pop_s} \\sum_{s=1}^{10} pop_s T_{s,t}\n\\end{equation*}\nwhere $T_{s,t}$ is the temperature recorded at time $t$ by station $s$ and $\\mathrm{temp}(t)$ is the weighted mean temperature which will be used in the modelling experiments. An exponentially smoothed version of the weighted mean will also be included in the model features. It was computed using a smoothing parameter equal to 0.95.\n\n\\subsection{High-resolution approach}\n\nForecasting the electricity hourly or half-hourly demand is a problem that has been extensively studied in the literature \\citep*{kuster_electrical_2017}. It is well known that a common driver of electrical load is weather and in particular temperature. In addition, calendar information can be used to explain the seasonal variation of the demand. Finally, lagged demand values are highly informative for the subsequent values. These variables are summarised in Table 1.\n\n\\begin{table}[H]\n\\caption{High-resolution model inputs}\n\\centering\n\\begin{tabular}{@{}cccc@{}}\n\\toprule\nType                      & Name   & Unit        & Description                                                \\\\ \\midrule\n\\multirow{2}{*}{Weather}  & temp   & {[}C\u00b0{]}    & Half-hourly temperature                                    \\\\ \\cmidrule(l){2-4} \n                          & temp95 & {[}C\u00b0{]}    & Half-hourly smoothed temperature                           \\\\ \\midrule\n\\multirow{3}{*}{Calendar} & dow    & Categorical & Day of the week                                            \\\\ \\cmidrule(l){2-4} \n                          & toy    & None        & Time of year (between 0 and 1)                             \\\\ \\cmidrule(l){2-4} \n                          & t    & Categorical & Time of day (between 0 and 47)                             \\\\ \\midrule\nLag                       & load24 & {[}$10^{1}$ GW{]}    & Half-hourly load on the previous day \\\\ \\midrule\nOutput                    & load   & {[}$10^{1}$ GW{]}    & Half-hourly load                                           \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table}\n\nThe GAM chosen to implement this approach is $y_i(t) \\sim N(\\mu_i(t), \\sigma^2)$ where the mean of the Gaussian distribution is modelled by:\n\\begin{align}\n   \\mu_{i}(t) = & \\, \\psi_{1}(\\mathrm{dow}_{i})+ \\psi_{2}(\\mathrm{t}) + f_{1}^{20}(\\mathrm{toy}_{i}(t)) + f_{2}^{20}(\\mathrm{temp}_{i}(t)) + f_{3}^{24}(\\mathrm{temp95}_{i}(t))  \\nonumber \\\\ \n               & +\\mathrm{ti}_{1}^{5,5}(\\mathrm{temp}_{i}, t)+\\mathrm{ti}_{2}^{5,5}(\\mathrm{temp95}_{i}(t), t)+\\mathrm{ti}_{3}^{5,5}(\\mathrm{load24}_{i}(t), t) \\\\ \\nonumber\n               & + \\mathrm{ti}_{4}^{5,5}(\\mathrm{toy}_{i}(t), t)\n\\end{align}\nIn (8), the $\\psi$ functions are parametric effects, while the $f$ functions are univariate smooth effects and the $\\mathrm{ti}$ functions are bivariate tensor product smooth interactions. The number of basis functions used is indicated in the exponents. For instance, $f_{1}^{20}$ uses 20 basis functions and $\\mathrm{ti}_{1}^{5,5}$ uses 5 basis functions for each marginal. Thin-plate spline bases are used to build all smooth effects \\citep*{wood_thin_2003}. The model structure (8) was decided on the basis of previous experience in the field and the statistical significance of each effect. \n\nThere are many NN architectures which could be considered for this problem. We want an architecture with the minimum number of layers possible and using the same model inputs as the GAM. Adding too many layers would lead to a drastic difference in degrees of freedom between the NN and the GAM which is not realistic in a short-term load forecasting scenario. Furthermore, as we are not in the big data regime, adding too many layers may actually worsen the performance of the network.\n\nGiven that the universal approximation theorem (\\citealp*{cybenko_approximation_1989} and \\citealp*{hornik_approximation_1991}) guarantees that a two-layer FCNN can approximate any measurable function on a compact support, a FCNN carefully built can approximate any non-linear function of the input variables with only one hidden layer. Therefore, a FCNN architecture was used to build an NN analogue of the high-resolution GAM baseline model.\n\nIn practice, there is no bound for the number of hidden units, which can lead to poor generalisation of the model when assessed on the test set. Therefore, a dropout layer was added after the hidden layer to foster the network generalisation. The outcome of the optimisation of hyperparameters led to the architecture shown in Figure \\ref{HRFCNN}, which contains 50 neurons in the hidden layer and a dropout layer with a 10\\% dropout rate.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{HRFCNN.jpg}\n    \\caption{High-resolution FCNN architecture (input variable names are detailed in Table 1)}\n    \\label{HRFCNN}\n\\end{figure}\n\nAfter obtaining the half-hourly demand forecast for the GAM and the NN, $\\hat{\\textrm{DP}}_i$ is estimated as the maximum daily value forecasted and $\\hat{\\textrm{IP}}_i$ is estimated as the half-hour of the day during which $\\hat{\\textrm{DP}}_i$ occurred.\n\n\\subsection{Low-resolution approach}\n\n\\begin{table}[H]\n\\caption{Low-resolution model inputs}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{@{}cccc@{}}\n\\toprule\nType                      & Name      & Unit        & Description                        \\\\ \\midrule\n\\multirow{4}{*}{Weather}  & tempMax   & {[}C\u00b0{]}    & Daily maximum temperature          \\\\ \\cmidrule(l){2-4} \n                          & temp95Max & {[}C\u00b0{]}    & Daily maximum smoothed temperature \\\\ \\cmidrule(l){2-4} \n                          & tempMin   & {[}C\u00b0{]}    & Daily minimum temperature          \\\\ \\cmidrule(l){2-4} \n                          & temp95Min & {[}C\u00b0{]}    & Daily minimum smoothed temperature \\\\ \\midrule\n\\multirow{2}{*}{Calendar} & dow       & Categorical & Day of the week                    \\\\ \\cmidrule(l){2-4} \n                          & toy       & None        & Time of year (between 0 and 1)     \\\\ \\midrule\n\\multirow{2}{*}{Lag}      & DP24      & {[}$10^{1}$ GW{]}   & Previous day peak demand           \\\\ \\cmidrule(l){2-4} \n                          & IP24      & Categorical & Previous day instant of peak       \\\\ \\midrule\nOutput                    & DP or IP        & {[}$10^{1}$ GW{]} or Categorical   & Daily demand peak or Daily instant of peak                  \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\end{table}\n\nIn the low-resolution approach, all input variables are at the daily resolution (Table 2). Here several distributions could be considered for GAMs. In particular, the scaled-T distribution, which is particularly suited for heavy tailed data, as well as the GEV family, which encompasses several extreme value distributions (Weibull, Gumbell and Fr\u00e9chet), are used to model the DP. For the IP forecasting task, the ordered-logit model implemented in the \\textit{mgcv} R package \\citep*{wood_mgcv_2020} is used. The low-resolution GAM can be written as follows:\n\\begin{align}\n   \\mu_{i} =& \\psi_{1}(\\mathrm{dow}_{i})+ f_{1}^{10}(\\mathrm{IP24}_{i}) + f_{2}^{20}(\\mathrm{toy}_{i}(t)) +   f_{3}^{20}(\\mathrm{DP24}_{i})   \\nonumber \\\\ \n               & + f_{4}^{20}(\\mathrm{tempMax}_{i}(t)) + f_{5}^{20}(\\mathrm{temp95Max}_{i}(t))  \\\\ \\nonumber\n               & + f_{6}^{20}(\\mathrm{tempMin}_{i}(t)) + f_{7}^{20}(\\mathrm{temp95Min}_{i}(t))\n\\end{align}\nFor the DP, $\\mu_{i}(t)$ is the location parameter of the distributions estimated, the other parameters are assumed to be constants. For the IP, $\\mu_{i}(t)$ is also the location parameter of a latent logistic distribution. Cut-off points are estimated in the course of model fitting and do not depend on the covariates. See \\cite*{wood_smoothing_2016} for details. \n\nThe same FCNN architecture as for the high-resolution approach was used (Figure \\ref{LRFCNN}). The only difference between them is the response variable which here is directly the DP or the IP. Furthermore, the hyperparameters chosen are different. In particular the number of epochs and the batch size are much larger. The response structure for the DP is 1 neuron with a ReLU activation while 48 neurons are used for the IP. Instead of the traditional softmax output used in classification problems, an ordinal output structure, more suited to model the IP, is implemented as formalised by \\cite*{jianlin_cheng_neural_2008}. The observed response is structured as a vector of 1 and 0. If the peak was observed at $t \\in \\{1,\\ldots,T\\}$ all neurons before and including the t-th one will be 1 and all neurons after will be 0. Therefore, sigmoidal activation functions are used.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{LRFCNN.jpg}\n    \\caption{Low-resolution FCNN architecture (input variable names are detailed in Table 2)}\n    \\label{LRFCNN}\n\\end{figure}\n\n\\subsection{Multi-resolution approach}\n\nThe multi-resolution GAMs leverage the same level of information for model inputs as in the high-resolution GAMs. In addition, the directly targets the DP response variable as in the low-resolution approach.\n\\begin{table}[H]\n\\caption{Multi-resolution model inputs}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{@{}cccc@{}}\n\\toprule\nType                      & Name     & Unit        & Description                                  \\\\ \\midrule\n\\multirow{2}{*}{Weather}  & matTem   & {[}C\u00b0{]}    & Vector of half-hourly temperatures           \\\\ \\cmidrule(l){2-4} \n                          & matTem95 & {[}C\u00b0{]}    & Vector of half-hourly smoothed temperatures  \\\\ \\midrule\n\\multirow{3}{*}{Calendar} & dow      & Categorical & Day of the week                              \\\\ \\cmidrule(l){2-4} \n                          & toy      & None        & Time of year (between 0 and 1)               \\\\ \\cmidrule(l){2-4} \n                          & matInt   & Categorical & Vector of time steps (between 0 and 47)      \\\\ \\midrule\nLag                       & matLag   & {[}$10^{1}$ GW{]}    & Vector of half-hourly load from previous day \\\\ \\midrule\nOutput                    & DP or IP        & {[}$10^{1}$ GW{]} or Categorical   & Daily demand peak or Daily instant of peak                               \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\end{table}\nTensor products defined in Section 3.2.1 are used to capture high-resolution information. The \\textit{mat} covariates presented in Table 3 are matrices of dimension $(N \\times 48)$, $N$ being the number of observations of the response variable DP. The multi-resolution GAM model is:\n\\begin{align}\n   \\mu_{i} = & \\, \\psi_{1}(\\mathrm{dow}_{i})+ f_{1}^{20}(\\mathrm{toy}_{i}) + ti_{1}^{15,10}(\\mathrm{matTem}_{i}, \\mathrm{matInt}_{i}) \\nonumber \\\\  \n   & + ti_{2}^{5,5}(\\mathrm{matTem95}_{i}, \\mathrm{matInt}_{i}) + ti_{3}^{5,5}(\\mathrm{matLag}_{i}, \\mathrm{matInt}_{i})\n\\end{align}\nUnlike previous approaches, IP and DP lags are not directly included as they can be captured by the model through the $ti_{3}$ tensor interaction. As for the low-resolution approach, Gaussian, scaled-T and GEV distributions are considered for the DP and the ordered categorical distribution for the IP. \n\nFor the multi-resolution NN, the tensor product interactions will be replaced by convolution layers. The mechanism looked for through these convolution layers is essentially the same as for tensor products: extracting high-resolution information to directly model the DP or the IP. The high-resolution (half-hourly) data will be passed on to the convolution layers while the low-resolution (daily) data will go through the same FCNN architecture used in the previous approaches. As shown in Figure \\ref{MRCNN}, these two sections of the architecture are then concatenated to produce the final forecast of the DP load. The output structure for the DP and the IP are the same as detailed in Section 4.3 with one neuron for the DP and 48 neurons for the IP. \n\nThe convolutions used for the high-resolution information are 1D convolutions on two channels. Usually, only one convolution funnel is used to capture interactions between all inputs. Here, each tensor product interaction will be replicated as a unique convolutional block. Thus, three convolution blocks will independently extract the three high-resolution terms: matTem, matTem95 and matLag. The second channel of each block is the matrix containing the vectors of time steps matInt.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{MRCNN.jpg}\n    \\caption{Multi-resolution CNN architecture (input variable names are detailed in Table 3)}\n    \\label{MRCNN}\n\\end{figure}\n\n\\section{Results}\n\nThe performance of the models for the DP and the IP forecasting tasks is evaluated using three statistical metrics. As a rolling-origin forecasting procedure was chosen, a transitional regime can be observed in the first few iterations, particularly for NNs, which usually perform better with a large amount of training data. Therefore, Table 4 (DP) and Table 5 (IP) present the models' performances on the last year of data, that is, from 2015-07-01 to 2016-06-30 included.\n\n\\begin{table}[H]\n\\caption{Performance on the last year of data for the DP (best model and associated metrics are in \\textbf{bold})}\n\\centering\n\\begin{tabular}{@{}ccccc@{}}\n\\toprule\n\\multirow{2}{*}{Resolution}       & \\multirow{2}{*}{Model} & \\multicolumn{3}{c}{Metrics} \\\\ \\cmidrule(l){3-5} \n                                  &                        & MAPE [\\%]     & MAE [MW] & RMSE [MW]        \\\\ \\midrule\nNA                                & Persistence            & 4.38         & 23.0   & 34.3         \\\\ \\midrule\n\\multirow{4}{*}{High}  \n                                  & ARIMA           & 4.08 & 21.0  & 27.8        \\\\ \\cmidrule(l){2-5} \n                                  & Gaussian GAM          & 2.43   & 13.0  & 15.5          \\\\ \\cmidrule(l){2-5} \n                                  & FCNN                   & 1.47        & 7.77  & 10.3           \\\\ \\midrule\n\\multirow{7}{*}{Low}   & ARIMA           & 3.85      & 20.0  & 26.7 \\\\ \\cmidrule(l){2-5} \n                                  & Scat GAM               & 1.92          & 10.5 & 12.9          \\\\ \\cmidrule(l){2-5} \n                                  & GEV GAM             & 2.67        & 14.5  & 16.9           \\\\ \\cmidrule(l){2-5} \n                                  & Gaussian GAM           & 2.26       & 12.3  & 14.4           \\\\ \\cmidrule(l){2-5} \n                                  & FCNN                   & 2.11          & 11.2  & 14.4          \\\\ \\midrule\n\\multirow{5}{*}{Multi} & GEV GAM             & 1.52         & 8.19  & 10.3           \\\\ \\cmidrule(l){2-5} \n                                  & \\textbf{Scat GAM}      & \\textbf{1.41} & \\textbf{7.55}  & \\textbf{9.59}   \\\\ \\cmidrule(l){2-5} \n                                  & Gaussian GAM           & 1.42         & 7.65  & 9.63         \\\\ \\cmidrule(l){2-5} \n                                  & CNN                    & 1.56         & 8.44 & 10.5        \\\\ \\bottomrule\n\\end{tabular}%\n\\end{table} \n\n\nWith the exception of the high-resolution FCNN, the multi-resolution models perform better than the alternatives across all metrics (Table 4). The relative strong performance of the high-resolution FCNN can be explained by the large amount of high-resolution data available, which suits the needs of NNs. Further, the FCNN contains more parameters to estimate and is thus more flexible than the high-resolution GAMs, which require the user to manually specify how the effect of each input variable should be modelled. Nevertheless, the best model on all metrics is the scaled-T GAM, built using the multi-resolution approach. The GEV GAM performed worse than the other distributions, which is surprising given that the GEV distribution is asymptotically justified for BM. Interestingly, the shape parameter estimated was found to be close to 0, under which value the GEV model is simply a Gumbel distribution. \n\n\\begin{table}[H]\n\\caption{Performance on last year of data for the IP (best model and associated metrics are in \\textbf{bold})}\n\\resizebox{\\textwidth}{!}{%\n\\centering\n\\begin{tabular}{@{}cccccc@{}}\n\\toprule\n\\multirow{2}{*}{Resolution}                        & \\multirow{2}{*}{Model}                  & \\multicolumn{3}{c}{Metrics}                     \\\\ \\cmidrule(l){3-5} \n                                  &                        & R-Accuracy [\\%]       & MAE [half-hour]    & RMSE [half-hour]        \\\\ \\midrule\nNA                     & Persistence        & 79.4              &  2.49      & 5.36  \\\\ \\midrule\n\\multirow{2}{*}{High}  & Gaussian GAM       & 82.6      & 2.01       & 4.59 \\\\ \\cmidrule(l){2-5} \n                       & FCNN               & 81.8      & 1.93      & 4.39 \\\\ \\midrule\n\\multirow{2}{*}{Low}   & Ocat GAM           & 79.1      &  2.11      & 4.22  \\\\ \\cmidrule(l){2-5} \n                       & FCNN               & 83.2      &  1.94      & 4.40 \\\\ \\midrule\n\\multirow{2}{*}{Multi} & Ocat GAM           & 79.4      &  2.01 & 4.08 \\\\ \\cmidrule(l){2-5} \n                       & \\textbf{CNN}       & \\textbf{83.5}    & \\textbf{1.70}  & \\textbf{3.85} \\\\ \\bottomrule\n\\end{tabular}%\n}\n\\end{table}\n\n\n\nIP multi-resolution models have a similar or better performance than high- and low-resolution alternatives within the same model class on the MAE and RMSE metrics (Table 5) and the multi-resolution CNN is the best model under all metrics.  However, the metrics are affected by high sampling variability. The reasons for this are detailed later in this section, where we also argue that the mediocre performance of ocat GAMs for IP forecasting is not fundamental, but attributable to the insufficient flexibility of the specific ocat parametrisation adopted here.\n\nTo quantify the variability of the performance metrics considered so far, we used block-bootstrap resampling. As described by \\cite*{forecast_eval}, for a test set of size $N$, we sample with replacement data blocks of fixed size $B=7$ (i.e., one week) to obtain an  evaluation sets of size $N$. Repeating this procedure $K$ times creates $K$ metric samples, which can be used to estimate the metric's sampling variability. In particular, Figure \\ref{boxplots} shows block-bootstrapped boxplots for all metrics and models on the last year of data. Figures \\ref{boxplots} (a-c) clearly demonstrate that the improvement obtained by adopting a multi-resolution approach is substantial and robust within the GAM model class. The HR-FCNN is competitive in terms of prediction but, as we discuss below, it is not easily interpretable and does not have the computational advantages of multi-resolution GAMs. For the IP problem, Figures \\ref{boxplots} (e-d) make clear that the sampling variability is substantial (reasons for this are discussed below).\n\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{tabular}{c|c}\n    \\textbf{DP} & \\textbf{IP} \\\\\n        \\includegraphics[width=0.45\\linewidth,keepaspectratio,page=1]{DP-block-bootstrap.pdf} & \\includegraphics[width=0.45\\linewidth,keepaspectratio,page=2]{IP-block-bootstrap.pdf} \\\\\n        (a) & (d)  \\\\[6pt]\n        \\includegraphics[width=0.45\\linewidth,keepaspectratio,page=2]{DP-block-bootstrap.pdf} &\n        \\includegraphics[width=0.45\\linewidth,keepaspectratio,page=3]{IP-block-bootstrap.pdf} \\\\\n         (b) & (e)   \\\\[6pt]\n        \\includegraphics[width=0.45\\linewidth,keepaspectratio,page=3]{DP-block-bootstrap.pdf} &\n        \\includegraphics[width=0.45\\linewidth,keepaspectratio,page=4]{IP-block-bootstrap.pdf} \\\\\n         (c) & (f)   \\\\[6pt]\n         \\multicolumn{2}{c}{\\includegraphics[width=0.65\\linewidth,keepaspectratio,page=7]{legend-block-bootstrap.pdf}}\n     \\end{tabular}\n\\caption{Block-bootstrap boxplots of the three metrics considered for the DP models (a), (b), (c) and IP models (d), (e), (f) on the last year of data}\n\\label{boxplots}\n\\end{figure}\n\n\nAs mentioned above, the rolling-origin forecasting setting may present a transitional regime during the first few training iterations. Figure \\ref{cumulative_DP} and \\ref{cumulative_IP} show the evolution of the different cumulative metrics calculated on the prediction signal updated on a monthly basis. Interestingly, the multi-resolution CNN for the DP (Figure \\ref{cumulative_DP}) starts off with a very bad prediction error on the first months. With more data, its performance rapidly improves across all metrics. The other models have a less dramatic performance trend, with the multi-resolution GAMs consistently performing better than the other models. The prediction error of these models oscillates during the first few months, which can be explained by the fact that the models did not have enough information to adequately estimate the yearly cycle, because they were fitted to only one year of data. After a year, the prediction errors has stabilised. \n\n\\begin{figure}[H]\n    \\centering\n    \\begin{tabular}{>{\\centering\\arraybackslash}m{0.45\\linewidth} >{\\centering\\arraybackslash}m{0.45\\linewidth} }\n        \\includegraphics[width=\\linewidth,keepaspectratio]{DP_MAPE.pdf} &\n        \\includegraphics[width=\\linewidth,keepaspectratio]{DP_MAE.pdf}  \\\\\n        (a) & (b)  \\\\[6pt]\n        \\includegraphics[width=\\linewidth,keepaspectratio]{DP_RMSE.pdf} &\n        \\includegraphics[width=\\linewidth,keepaspectratio,page=2]{legend-DP.pdf} \\\\\n        (c) & \\\\[6pt]\n    \\end{tabular}\n\\caption{Cumulative forecasting metrics evolution for each of the monthly updated DP models: (a) MAPE, (b) MAE, (c) RMSE}\n\\label{cumulative_DP}\n\\end{figure}\n\nFor the IP forecasting task, the different metrics evolve with similar patterns (Figure \\ref{cumulative_IP}), but the seasonal oscillations in performance persist beyond the first year. Figure \\ref{fig:ocat_issue_1} explains why predicting the IP is harder in summer than in winter. In particular, while winter daily demand profiles have a reliable evening peak, summer load profiles are flatter and on some days the peak distribution becomes bimodal. That is, the daily peak might occur in the morning and or in the evening with equal probability. This is shown also by the right plot in Figure \\ref{fig:ocat_issue_2}. Hence, it is clear that in the summer the IP point estimates might be unfairly penalised under the simple metrics considered here. This implies that a forecasting model might be better off providing an IP forecast that falls between the two peaks, as MR-CNN is occasionally doing (see Figure \\ref{fig:ocat_issue_1}). Such a forecast might improve the metrics but has little value in an operational setting. Note also that the ocat model struggles to capture an IP distribution that is unimodal or bimodal depending on the time of year. In particular, the ocat model used here is based on a standard ordered-logit parametrisation, which involves modelling the mean of a latent logistic random variable via an additive model. It is not possible to transform a unimodal distribution on the ordered categories (here, IP) into a bimodal one, simply by controlling a location parameter. Hence, a more flexible model (e.g., \\citealp*{peterson1990partial}) would be preferable.\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{tabular}{>{\\centering\\arraybackslash}m{0.45\\linewidth} >{\\centering\\arraybackslash}m{0.45\\linewidth} }\n        \\includegraphics[width=\\linewidth,keepaspectratio]{IP_R-ACC.pdf} &\n        \\includegraphics[width=\\linewidth,keepaspectratio]{IP_MAE.pdf}  \\\\\n        (a) & (b)  \\\\[6pt]\n        \\includegraphics[width=\\linewidth,keepaspectratio]{IP_RMSE.pdf} &\n        \\includegraphics[width=\\linewidth,keepaspectratio,page=2]{legend-IP.pdf} \\\\\n        (c) & \\\\[6pt]\n    \\end{tabular}\n\\caption{Cumulative forecasting metrics evolution for each of the monthly updated IP models: (d) R-accuracy, (e) MAE, (f) RMSE}\n\\label{cumulative_IP}\n\\end{figure}\n\n\\begin{figure}[H] \n    \\centering\n    \\includegraphics[width=\\linewidth,keepaspectratio]{ocat-1.pdf}\n    \\caption{Left: observed IP as a function of the day of year (black) and corresponding predictions from MR-CNN (red, shifted downward for visibility) and MR-ocat (blue, shifted upward). Right: same plot for HR-FCNN (red, downward) and HR-Gauss (blue, upward).}\n    \\label{fig:ocat_issue_1}\n\\end{figure}\n\n\\begin{figure}[H] \n    \\centering\n    \\begin{tabular}{>{\\centering\\arraybackslash}m{0.45\\linewidth} >{\\centering\\arraybackslash}m{0.45\\linewidth} }\n        \\includegraphics[width=\\linewidth,keepaspectratio,page=8]{IP-block-bootstrap.pdf} &\n        \\includegraphics[width=\\linewidth,keepaspectratio]{ocat-2.pdf}  \\\\\n    \\end{tabular}\n    \\caption{Left: Block-bootstrap boxplots of the d-RMSE metric for the IP problem. Right: daily demand profile curves during winter (shifted upward by 15 GW) and summer. The blue curves are profiles with a small absolute difference between the  morning and evening peak ($<$ 50 MW).}\n    \\label{fig:ocat_issue_2}\n\\end{figure}\n\nIt is interesting to verify the performance of each model for IP forecasting via a bespoke metric. In particular, let $t^{\\text{m}}_i$ be the observed IP on day $i$ and let $\\hat{t}^{\\text{m}}_i$ be the corresponding forecast. We propose the following metric:\n\\begin{equation*}\n\\text{d-RMSE} = \\left(\\frac{1}{n}\\sum_{i=1}^n(y_{t^{\\text{m}}_i} - y_{\\hat{t}^{\\text{m}}_i})\\right)^{1\/2}\n\\end{equation*}\nwhich is based on the difference between the daily peak demand and the demand at the predicted IP (the d stands for demand). This metric is more relevant to operations than MSE or MAE. For instance, in peak shaving applications, providing a forecast $\\hat{t}^{\\text{m}}_i$ very different from $t^{\\text{m}}_i$ might not be a problem if $y_{t^{\\text{m}}_i}$ and $y_{\\hat{t}^{\\text{m}}_i}$ are similar, which is what d-RMSE quantifies. Figure \\ref{fig:ocat_issue_2} shows a bootstrapped boxplot of d-RMSE for each model. Interestingly, high-resolution methods are best here, by a substantial margin in the case of HR-FCNN.\n\nThe results obtained so far do not provide reliable evidence in favour or against the adoption of a multi-resolution approach for IP forecasting. In fact, the poor forecasting performance of MR-ocat is arguably attributable to the particular ordered-logit parametrisation used here. MR-CNN does well using standard, statistically motivated losses but it is inferior to high-resolution approaches on an operationally relevant one (d-RMSE). It would be interesting to verify whether fitting the MR-CNN model by minimising d-RMSE directly (rather than MSE as done here) would lead to better results. We leave this, and the search for a more flexible distribution for ordered categorical responses, for future work.\n\nImplementing the multi-resolution approach on the DP forecasting problem is more straightforward, hence the results discussed so far are positive and reliable. We further verify their significance by performing \\cite*{diebold_comparing_1995} (DM) tests on the absolute and squared error losses . The null hypothesis of the tests is: ``both forecasts have the same expected loss''. The results of the DM tests are available on Figure \\ref{DMTEST} which confirms that, within the GAM class, the multi-resolution forecasts are significantly different to the low-resolution and high-resolution approaches under both metrics. \n\n\\begin{figure}[H]\n    \\centering\n    \\begin{tabular}{c}\n        \\resizebox{\\textwidth}{!}{%\n        \\begin{tabular}{lllllllllllll}\nModel    & HR-arima                 & HR-gauss                 & HR-FCNN                  & LR-arima                 & LR-gauss                     & LR-scat                  & LR-gev                   & LR-FCNN                      & MR-gauss                   & MR-scat                    & MR-gev                     & MR-CNN                       \\\\\nHR-arima & \\cellcolor[HTML]{C0C0C0} & 0                        & 0                        & \\textcolor{red}{0.116}                    & 0                            & 0                        & 0 & 0                            & 0                          & 0                          & 0                          & 0                            \\\\\nHR-gauss & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & 0                        & 0                        &  0.042 & 0                        & 0 & 0.001 & 0                          & 0                          & 0                          & 0                            \\\\\nHR-FCNN  & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & 0                        & 0                            & 0                        & 0 & 0     & \\textcolor{red}{0.729}                      & \\textcolor{red}{0.549}                      & \\textcolor{red}{0.250}                      &  0.029\\\\\nLR-arima & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & 0                            & 0                        & 0 & 0     & 0                          & 0                          & 0                          & 0     \\\\\nLR-gauss & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & 0                        & 0 & { 0.010} & 0                          & 0                          & 0                          & 0     \\\\\nLR-scat  & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & 0 & \\textcolor{red}{0.122}                             & 0                          & 0                          & 0                          & 0     \\\\\nLR-gev   & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & 0                            & 0                          & 0                          & 0                          & 0     \\\\\nLR-FCNN  & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & 0                          & 0                          & 0                          & 0     \\\\\nMR-gauss & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & \\textcolor{red}{0.063}                         & 0                          & { 0.001} \\\\\nMR-scat  & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0}   & \\cellcolor[HTML]{C0C0C0} & 0                          & 0                            \\\\\nMR-gev   & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0}   & \\cellcolor[HTML]{C0C0C0}   & \\cellcolor[HTML]{C0C0C0} & \\textcolor{red}{0.143}                         \\\\\nMR-CNN   & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0} & \\cellcolor[HTML]{C0C0C0}     & \\cellcolor[HTML]{C0C0C0}   & \\cellcolor[HTML]{C0C0C0}   & \\cellcolor[HTML]{C0C0C0}   & \\cellcolor[HTML]{C0C0C0}  \n\\end{tabular}} \\\\\n        (a) \\\\ [6pt]\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{lllllllllllll}\nModel     & HR-arima                                        & HR-gauss                                        & HR-FCNN                                         & LR-arima                                        & LR-gauss                                        & LR-scat                                         & LR-gev                                          & LR-FCNN                                         & MR-gauss                                        & MR-scat                                         & MR-gev                                          & MR-CNN                                          \\\\\nHR-arima  & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & 0                        & \\textcolor{red}{0.161}                    & 0                        & 0                        & 0                        & 0                        & 0                        & 0                        & 0                        & 0                        \\\\\nHR-gauss & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & 0                        & { 0.010}                    & 0                        & { 0.003}                    & \\textcolor{red}{0.118}                    & 0                        & 0                        & 0                        & 0                        \\\\\nHR-FCNN    & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & 0                        & 0                        & 0                        & 0                        & \\textcolor{red}{0.400}                    & \\textcolor{red}{0.362}                    & \\textcolor{red}{0.492}                    & { 0.016}                    \\\\\nLR-arima & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & 0                        & 0                        & 0                        & 0                        & 0                        & 0                        & 0                        \\\\\nLR-gauss & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & 0                        & \\textcolor{red}{0.938}                    & 0                        & 0                        & 0                        & 0                        \\\\\nLR-scat  & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & { 0.026}                    & 0                        & 0                        & 0                        & { 0.002}                    \\\\\nLR-gev   & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                   & 0                        & 0                        & 0                        & 0                        \\\\\nLR-FCNN    & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & 0                        & 0                        & 0                        \\\\\nMR-gauss & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\textcolor{red}{0.468}                    & 0                        & 0                        \\\\\nMR-scat  & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & 0                        & 0                        \\\\\nMR-gev   & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & { 0.015}                    \\\\\nMR-CNN     & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ } & \\cellcolor[HTML]{C0C0C0}{ }\n\\end{tabular}} \\\\\n        (b) \\\\[6pt]\n\n    \\end{tabular}\n\\caption{P-values from the Diebold-Mariano test for DP forecasts. The test used is from the \\textit{multDM} package in R \\citep*{drachal_multdm_2020}. In black, the null hypothesis is rejected at the 5\\% threshold and both forecasts are significantly different. In red, the null hypothesis is not rejected at the 5\\% threshold and both forecasts cannot be significantly differentiated; (a) absolute errors (b) squared errors.}\n\\label{DMTEST}\n\\end{figure}\n\n\n\n\n\n\nIt is interesting to quantify the complexity or parsimony of the models considered so far. AIC can be interpreted as a parsimony measure, but it requires computing the effective number of models parameters and we are not aware of any method that would allow estimating them across all the model classes considered here. Figure \\ref{AIC} shows the AICs of low- and multi-resolution GAMs. The multi-resolution approaches consistently have a smaller AIC than the low-resolution approaches. Furthermore, the slopes indicate that with more data the gap continues to increase. \n\nFor NNs, parsimony is highly dependent on the chosen architecture. In our case, the low-resolution and high-resolution NNs have a very similar architecture with only one hidden layer and a dropout layer (Figure \\ref{HRFCNN} and Figure \\ref{LRFCNN}). Only the input shapes and the number of observations vary. On the other hand, the multi-resolution NN (Figure \\ref{MRCNN}) requires the use of convolutional layers which are leveraged to extract the high-resolution information. The extraction process requires multiple layers which forces the multi-resolution CNN to have a larger number of parameters than the low-resolution and high-resolution NNs.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth,keepaspectratio]{AIC.pdf}\n    \\caption{AIC for the low-resolution and multi-resolution DP GAMs}\n    \\label{AIC}\n\\end{figure}\n\nThe results discussed in this section show that multi-resolution approaches are superior to low- and high-resolution alternatives for the DP forecasting problem. The forecasting performance of the high-resolution FCNN and the multi-resolution GAMs are not significantly different but, in an operational peak demand forecasting context, the multi-resolution GAM would be preferred because it can be decomposed into additive components, which can be more easily interpreted (and manually adjusted) by operational staff. In addition, note that adopting a multi-resolution approach can bring substantial computational advantages, which are easy to quantify within the GAM model class. In particular, the GAM model matrix $\\bf X$ in the multi-resolution case has $T$ times less rows than in the high-resolution case, where $T$ is the number of daily observations (i.e., $T=48$ for half-hourly data). Therefore, $T$ times less memory is used, and many computations frequently required during GAM model fitting (such as ${\\bf X}^T {\\bf W} {\\bf X}$, where $\\bf W$ is a diagonal matrix) will take less time. \n\n\\section{Conclusion}\n\nThis paper proposes a novel modelling approach, which uses both high-resolution and low-resolution information to forecast the daily electrical load peak magnitude and timing. The results demonstrate that this multi-resolution approach is flexible enough to be applied to different model classes and that it provides a competitive predictive performance. In particular, GAMs and NNs with similar input structures were used to implement the multi-resolution approach and to compare its performance that of low-resolution, high-resolution and persistence alternatives. On UK aggregate demand data, the multi-resolution models performed significantly better across all metrics when forecasting peak magnitude. In addition to improved predictions, adopting a multi-resolution approach enables faster computation via data compression and leads to more parsimonious models, as demonstrated by the consistently lower AIC scores achieved by multi-resolution models within the GAM model class. \n\nThe results on the peak timing forecasting problem are mixed, but interesting. A multi-resolution neural network does marginally better than the alternatives, when performance is assessed via standard statistical metrics. However, the corresponding forecast is occasionally inappropriate (falling between the morning and evening peaks) and inferior to high-resolution alternatives when assessed via an operationally motivated metric. The results suggest that the multi-resolution neural network should be fitted to data by minimising a problem specific performance metric directly. For instance, one could consider financial metrics on billing periods as done by \\cite*{saxena_hybrid_2019}. The multi-resolution GAM does poorly on the peak timing problem, but this is attributable to the insufficient flexibility of the ordered logit parametrisation used here. Obtaining stronger evidence in favour or against the use of multi-resolution methods for the peak timing problem would require solving the issues just mentioned, which could be the subject of further work.\n\nThe forecasting methods presented here could be extended in several ways. The set of models described in this paper could be used within an aggregation of experts or ensemble methods, which might lead to more accurate forecasts. The benefits of multi-resolution methods have been demonstrated in a context where covariates were available at different temporal resolutions, but they could be generalised to other multi-resolution settings, such as spatio-temporal data or individual customer data (see e.g., \\citealp*{fasiolo_qgam_2020} for an example application of functional quantile GAMs \\citealp*{fasiolo_fast_2020} to residential electricity demand data). Finally, this paper focused on day-ahead daily peak magnitude and time forecasting, but multi-resolution methods could be applied to other short-term windows (e.g., weekly). However,  estimating monthly or yearly peaks would require a different approach, because the number of observed demand peaks would be too low. \n\n\n\\section*{Acknowledgments}\nMatteo Fasiolo was partially funded by EPSRC grant EP\/N509619\/1. The datasets used in this paper are available on the National Grid and National Oceanic and Atmospheric Administration websites. The R code as well as the data prepared for the experiments in this paper are available at the following link: \\href{https:\/\/cutt.ly\/CYvgIP3}{https:\/\/cutt.ly\/CYvgIP3}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\subsection{Introduction.} Given a compact operator $T:H \\rightarrow H$ on a Hilbert space $H$, compactness implies that the inversion problem, i.e. reconstructing $x$ from $y$ in\n$$ Tx = y$$\nis ill-posed: small changes in $y$ may lead to arbitrarily large changes in $x$. The simplest example is perhaps that of integral operators on $L^2(\\mathbb{R})$ where integration acts as a smoothing\nprocess and makes inversion of the operator difficult. Of particular importance is the Hilbert transform\n$$ (Hf)(x) = \\frac{1}{\\pi}\\mbox{p.v.}\\int_{\\mathbb{R}}{\\frac{f(y)}{x-y}dy},$$\nwhich satisfies $\\| Hf\\|_{L^2(\\mathbb{R})} = \\|f\\|_{L^2(\\mathbb{R})}$. However, in practice, measurements\nhave to be taken from a compact interval and this motivates the definition of the truncated Hilbert transform: using $\\chi_I$ to\ndenote the characteristic function on an interval $I \\subset \\mathbb{R}$, the truncated Hilbert transform $H_T:L^2(I) \\rightarrow L^2(J)$ on the\nintervals $I,J \\subset \\mathbb{R}$ is given by\n$$ H_T = \\chi_{J} H(f \\chi_{I}).$$\nWhenever the intervals $I$ and $J$ are disjoint, the singularity of the kernel never comes into play and the operator is highly smoothing: indeed, if $I$ and $J$\nare disjoint, the operator becomes \\textit{severely} ill-posed and the singular values decay exponentially fast. The inversion problem is ill-behaved even on finite-dimensional subspaces: \\textit{every}  subspace $V \\subset L^2(I)$ contains some $0 \\neq f \\in V$ with\n$$ \\| H_T f\\|_{L^2(J)} \\leq c_1 e^{-c_2 \\dim(V)} \\| f \\|_{L^2(I)} \\qquad \\mbox{for some}~c_1, c_2 > 0~\\mbox{depending only on}~I,J.$$\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[xscale=9,yscale=1.1]\n\\draw [ultra thick, domain=0:1, samples = 300] plot (\\x, {-0.15269*sin(2*pi*\\x r)    +    0.4830*sin(3*pi*\\x r)  +  0.3084*sin(4*pi*\\x r)  + 0.80509*sin(5*pi*\\x r)}  );\n\\draw [thick, domain=0:1] plot (\\x, {0}  );\n\\filldraw (0,0) ellipse (0.006cm and 0.048cm);\n\\node at (0,-0.3) {0};\n\\filldraw (1,0) ellipse (0.006cm and 0.048cm);\n\\node at (1,-0.3) {1};\n\\end{tikzpicture}\n\\caption{A function $f$ on $[0,1]$ with $\\|Hf\\|^2_{L^2([2,3])} \\sim 10^{-7}\\|f\\|^2_{L^2([0,1])}$} \n\\end{center}\n\\end{figure}\n\nThis strong form of ill-posedness makes it very easy to construct bad examples: take any finite\northonormal set $\\left\\{\\phi_1, \\phi_2, \\dots, \\phi_n \\right\\} \\subset L^2(I)$, By linearity, we have for any scalar $a_1, \\dots, a_n$ that\n$$ \\left\\| H_T \\left(\\sum_{k=1}^{n}{a_k \\phi_k}\\right)\\right\\|_{L^2(J)}^2 = \\sum_{i, j = 1}^{n}{a_i a_j \\left\\langle H_T \\phi_i, H_T \\phi_j \\right\\rangle_{L^2(J)}}$$\nwhich is a simple quadratic form. Finding the eigenvector corresponding to the smallest eigenvalue of the Gramian $G = (\\left\\langle H_T \\phi_i, H_T \\phi_j \\right\\rangle)_{i,j=1}^{n}$\nproduces a suitable linear combination of  $\\left\\{\\phi_1, \\phi_2, \\dots, \\phi_n \\right\\}$ for which $\\|H_Tf\\|_{L^2(J)} \\ll \\|f\\|_{L^2(I)}$. The strong degree of ill-posedness guarantees that the smallest eigenvalue decays\nexponentially in $n$ independently of the orthonormal basis. Recently, Alaifari, Pierce and the second author \\cite{al} showed that it is nonetheless possible to guarantee some control by proving a new type of stability\nestimate for the  Hilbert transform: for disjoint intervals $I,J \\subset \\mathbb{R}$\n$$  \\|H f\\|_{L^2(J)} \\geq c_1 \\exp{\\left(-c_2\\frac{ \\|f_x\\|_{L^2(I)}}{\\|f\\|_{L^2(I)}}\\right)} \\| f \\|_{L^2(I)},$$\nwhere the constants $c_1, c_2$ depend only on the intervals $I,J$.\nThis estimate guarantees that the only way for $Hf$ to be substantially smaller than $f$ is the presence of oscillations. If one reconstructs data $f$ from\nmeasurements $g$ (the equation being $H_T f = g$), then a small error $f + h$ yields\n$$ H_T(f +h) = H_Tf + H_T h= g + H_T h.$$\nThe stability estimate implies that one can guarantee to distinguish $f$ from $f+h$ when $h$ has few oscillations.\nThe only existing result in this direction is \\cite{al} for the Hilbert transform.\n\n\n\n\n\\section{Main results}\n\nThe purpose of our paper is to combine the argument developed by Alaifari, Pierce and the second author \\cite{al} with classical results of Bertero \\& Gr\\\"unbaum \\cite{gru1}, Landau \\& Pollak \\cite{pr2, pr3}  and Slepian \\& Pollak \\cite{pr1}  to establish such stability estimate in three other cases: we give essentially sharp stability estimates for the Truncated Laplace Transform, the Adjoint Truncated Laplace Transform and the Truncated Fourier Transform. While this shows that this class of stability estimates exist in a wider context, the question of whether such results could be 'generically' true (i.e. for a wide class of integral operators) remains open.\n\n\n\n\\subsection{Truncated Laplace Transform} \n The truncated Laplace transform $\\mathcal{L}_{a,b}:L^2[a,b] \\rightarrow L^2[0,\\infty]$ is defined via\n$$\n (\\mathcal{L}_{a,b}f)(s) = \\int_{a}^{b}{e^{-s t} f(t) dt},\n$$\nwhere $0 < a < b < \\infty$. \nThe operator $\\mathcal{L}_{a,b}$ is compact and its image is dense in $L^2[0,\\infty]$. We show\nthat if $\\|\\mathcal{L}_{a,b} f\\|_{L^2[0, \\infty]} \\ll \\|f\\|_{L^2[a,b]}$,\nthen this is due to the presence of oscillations. \n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[xscale=9,yscale=1.1]\n\\draw [ultra thick, domain=1:2, samples = 300] plot (\\x, {-0.0707*sin(pi*\\x r)    - 0.421*sin(2*pi*\\x r)  +  0.2137*sin(3*pi*\\x r)  + 0.8783*sin(4*pi*\\x r)}  );\n\\draw [thick, domain=1:2] plot (\\x, {0}  );\n\\filldraw (1,0) ellipse (0.006cm and 0.048cm);\n\\node at (1,-0.3) {1};\n\\filldraw (2,0) ellipse (0.006cm and 0.048cm);\n\\node at (2,-0.3) {2};\n\\end{tikzpicture}\n\\caption{A function $f$ on $[1,2]$ with $\\| \\mathcal{L}_{1,2} f \\|^2_{L^2[0,\\infty]} \\sim 10^{-8}\\|f\\|^2_{L^2([1,2])}$.} \n\\end{center}\n\\end{figure}\n\n\\begin{theorem} There exist $c_1, c_2>0$, depending only on $a,b$, so that for all real-valued $f \\in H^1[a,b]$\n$$ \\| \\mathcal{L}_{a,b} f \\|_{L^2[0,\\infty]} \\geq c_1 \\exp{\\left(-c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)}\\|f\\|_{L^2[a,b]}.$$\n\\end{theorem}\nThe result is sharp up to constants: if $c_2$ is chosen sufficiently small, then for every $c_1 > 0$ there is an infinite orthonormal sequence of functions for which the inequality fails. The proof proceeds similarly as in \\cite{al} with a crucial ingredient for Laplace transforms coming from a  a 1985 paper of Bertero \\& Gr\\\"unbaum \\cite{gru1}.\n\n\n\n\n\\subsection{Adjoint Truncated Laplace Transform.} The adjoint operator $\\mathcal{L}_{a,b}^*:L^2[0,\\infty] \\rightarrow L^2[a,b]$ \n$$ (\\mathcal{L}_{a,b}^*f)(s) = \\int_{0}^{\\infty}{e^{-s t} f(t) dt}.$$\nis very different in structure. We seek a lower bound on $\\|\\mathcal{L}_{a,b}^*f\\|_{L^2[a,b]}$ in terms of $\\| f \\|_{L^2[0, \\infty]}$: if $f$ is supported far away from the\norigin, then the exponentially decaying kernel will induce rapid decay even if no oscillations are present (additional oscillations can, of course, further decrease the size of  $\\|\\mathcal{L}_{a,b}^*f\\|_{L^2[a,b]}$).\nAny lower bound will therefore have to incorporate where the function is localized and the natural framework for this are weighted estimates.\n\n\n\n\\begin{theorem} There exist $c_1, c_2$, depending only on $a,b$, so that for all real-valued $f \\in H^2[0, \\infty]$\n$$ \\| \\mathcal{L}_{a,b}^* f \\|_{L^2[a,b]} \\geq c_1 \\exp{\\left(-c_2\\frac{ \\|x f_{xx}\\|_{L^2[0,\\infty]}  + \\|x f_{x}\\|_{L^2[0,\\infty]} + \\|x f_{}\\|_{L^2[0,\\infty]} + \\| f_{}\\|_{L^2[0,\\infty]}   }{\\|f\\|_{L^2[0,\\infty]}}\\right)}\\|f\\|_{L^2[0,\\infty]}.$$\n\\end{theorem}\nThe result is again sharp in the sense that there are counterexamples for every $c_1 > 0$ if the constant $c_2$ is smaller than some fixed positive\nconstant depending on $a,b$.\n\n\n\n\\subsection{Truncated Fourier Transform} Let $\\mathcal{F}_T: L^2[-1,1] \\rightarrow L^2[-1,1]$ be given by\n$$ \\mathcal{F}_T = \\chi_{[-1,1]}\\mathcal{F}\\left(\\chi_{[-1,1]} f\\right)$$\nwhere, as usual, $\\mathcal{F}$ denotes the Fourier transform\n$$ (\\mathcal{F} f)(\\xi) = \\int_{\\mathbb{R}}^{}{f(x) e^{i  \\xi x}dx}.$$\nThe Fourier transform of a compactly supported function is analytic and cannot vanish on an open set. Since it does not vanish on any open set, this yields\n$$ \\int_{-1}^{1}{|\\widehat{f}(\\xi)|^2d\\xi} > 0$$\nfor every nonzero $f \\in L^2[-1,1]$. The expression can certainly be small because $\\widehat{f}$ can have all its $L^2-$mass far away from the origin: however, if $\\widehat{f}$ has its $L^2-$mass\nfar away from the origin, $f$ oscillates on $[-1,1]$. We give a quantitative description of this phenomenon.\n\n\\begin{theorem} There exist $c_1, c_2 > 0$ such that for all real-valued $f \\in H^1[-1,1]$\n$$ \\int_{-1}^{1}{|\\widehat{f}(\\xi)|^2d\\xi}\\geq c_1\\left(c_2 \\frac{ \\left\\| f_x \\right\\|_{L^2[-1,1]}}{\\|f\\|_{L^2[-1,1]}} \\right)^{-c_2\\frac{\\left\\| f_x \\right\\|_{L^2[-1,1]}}{\\|f\\|_{L^2[-1,1]}} } \\int_{-1}^{1}{|f(x)|^2dx}.$$\n\\end{theorem}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[xscale=4.5,yscale=1.1]\n\\draw [ultra thick, domain=-1:1, samples = 300] plot (\\x, {0.00055*cos(pi*\\x r)    +  0.0824*cos(2*pi*\\x r)  +  0.6196*cos(3*pi*\\x r)  + 0.7805*cos(4*pi*\\x r)}  );\n\\draw [thick, domain=-1.05:1.05] plot (\\x, {0}  );\n\\filldraw (-1,0) ellipse (0.012cm and 0.048cm);\n\\node at (-1,-0.3) {-1};\n\\filldraw (0,0) ellipse (0.012cm and 0.048cm);\n\\node at (0,-0.3) {0};\n\\filldraw (1,0) ellipse (0.012cm and 0.048cm);\n\\node at (1,-0.3) {1};\n\\end{tikzpicture}\n\\caption{A function $f$ on $[-1,1]$ with $\\| \\mathcal{F}_T f \\|^2_{L^2[-1,1]} \\sim 10^{-18}\\|f\\|^2_{L^2([-1,1])}$.} \n\\end{center}\n\\end{figure}\n\n\n\nWe are not aware of any such results in the literature,\nhowever, the result is certainly close in spirit to the question to which degree simultaneous localization in space and frequency is possible. An example is Nazarov's\nquantitative form \\cite{naz} of the Amrein-Berthier theorem \\cite{am} (see also \\cite{bene}): for any $S, \\Sigma \\subset \\mathbb{R}$ with\nfinite measure and any $f \\in L^2(\\mathbb{R})$ it is not possible for $f$ to be too strongly localized in $S$ and $\\widehat{f}$ to be too\nstrongly localized in $\\Sigma$\n$$ \\left\\|f \\chi_{\\mathbb{R} \\setminus S} \\right\\|^2_{L^2(\\mathbb{R})} + \\left\\|\\widehat{f} \\chi_{\\mathbb{R} \\setminus \\Sigma} \\right\\|^2_{L^2(\\mathbb{R})} \\geq \\frac{e^{-133 |S| |\\Sigma|}}{133} \\| f\\|^2_{L^2(\\mathbb{R})}.$$\nThe proof of Theorem 3 makes use of \\textit{prolate spheroidal wave functions} introduced by Landau, Pollak and Slepian \\cite{pr2, pr3, pr1, pr4, pr5}. They appear naturally in the\nLandau-Pollak uncertainty principle \\cite{pr3} which states that if\n$\\mbox{supp}(\\widehat{f}) \\subset [-1,1]$\nand\n$$ \\int_{|x| \\geq T}{|f(x)|^2 dx} \\leq \\varepsilon \\|f\\|_{L^2(\\mathbb{R})},~\\mbox{then}\n\\qquad \\|f - \\pi(f) \\|_{L^2} \\leq 49\\varepsilon^2 \\|f\\|_{L^2},$$\nwhere $\\pi$ is the projection onto a $(4\\left\\lfloor T \\right\\rfloor +1)-$dimensional subspace spanned by the first elements of a particular \\textit{universal} orthonormal basis $(\\phi_n)_{n \\in \\mathbb{N}}$ (these are the prolate spheroidal wave functions).  \\\\\n\n\n\\textbf{Outline of the paper.} \\S 3 gives a high-level overview of the argument and provides two easy inequalities for real functions that will be used in the proofs.  \\S 4 explains the underlying machinery specially required to prove Theorem 1 and gives the full proof. A very similar argument allows to prove Theorem 2 and we describe the necessary modifications in \\S 5. \\S 6 gives a proof of Theorem 3. $c_1, \\dots, c_5$  are positive constants, $\\sim$ denotes equivalence up to constants.\n\n\n\n\n\n\n\n\\section{Outline of the arguments}\n\n\n\\subsection{The overarching structure.}\nThe proofs (also for the result in \\cite{al}) have the same underlying structure: we use a $T^* T$ argument and the fact that\nthere is a differential operator $D$ whose eigenfunctions coincide with the eigenfunctions of $T^* T$. This allows us to exploit the structure of\nthe differential operator to analyze the decomposition of a generic function into the orthonormal basis of singular functions.\nMore precisely: we are interested in establishing lower bounds for an \ninjective operator between two Hilbert spaces $T:H_1 \\rightarrow H_2$. In all these cases, we assume that\n\\begin{enumerate}\n\\item we control the decay of the eigenvalues of $T^*T$ from below,\n\\item there is a differential operator $D:H_1 \\rightarrow H_1$ with the same eigenfunctions as $T^*T$\n\\item and we can control the growth of eigenvalues $\\lambda_n$ of $D$.\n\\end{enumerate}\nLet us denote the $L^2-$normalized eigenfunctions of $D$ (which are also eigenfunctions of $T^*T$) by $(u_n)_{n=1}^{\\infty}$. They form an orthonormal basis\nof $L^2$ in all situations that are of interest to us. Furthermore, we will use the spectral theorem\n$$ \\left\\langle D f, f\\right\\rangle = \\sum_{n=1}^{\\infty}{\\lambda_n \\left| \\left\\langle f, u_n \\right\\rangle \\right|^2}$$\nand explicit information on the growth of the eigenvalues $\\lambda_n$. We can furthermore, using integration by parts and the structure of $D$, control the action of $D$ in the Sobolev space $H^{s}$\n$$  \\left\\langle D f, f\\right\\rangle \\sim    \\| f\\|^2_{H^s}.$$\nThe useful insight is that this implies that the eigenfunction $(u_n)_{n=1}^{\\infty}$ explore the phase space in a way that is analogous to classical eigenfunctions of the Laplacian: low-energy eigenfunctions\nhave small derivatives. In particular, if $Df$ is small, then at least some of the projections $|\\left\\langle f, u_n \\right\\rangle|$ have to be big for $n$ somewhat small. Conversely, functions whose $L^2-$energy\nis mostly concentrated on high-frequency eigenfunctions $(u_n)_{n \\geq N}$ have $ |\\left\\langle D f, f\\right\\rangle|$ large. The next Lemma makes this precise.\n\n\n\\begin{lemma}[Low oscillation implies low frequency] If $\\lambda_n \\geq c_1  n^{2}$ and $|\\left\\langle D f, f\\right\\rangle| \\leq c_2 \\|f_x\\|_{L^2}^2$ for some $0 < c_1, c_2 < \\infty$, then there exists a constant $0 < c < \\infty$ such that\n$$ \\sum_{n \\leq c \\frac{ \\|f_x\\|_{L^2}}{\\|f\\|_{L^2}}  }^{}{ \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2} \\geq \\frac{ \\| f\\|^2_{L^2}}{2}.$$\n\\end{lemma}\n\\begin{proof} Both inequalities have the same scaling under the multiplication with scalars $f \\rightarrow \\lambda f$, so we can assume w.l.o.g. that $\\|f\\|_{L^2} = 1$. Trivially,\n\\begin{align*}\n  \\sum_{n \\geq c_3\\|f_x\\|_{L^2}}^{}{ \\lambda_n  \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2} &\\geq   \\sum_{n \\geq c_3 \\|f_x\\|_{L^2}}^{}{ c_1 n^{2}  \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2} \\\\\n&\\geq c_1 \\left(c_3 \\|f_x\\|_{L_2}\\right)^2   \\sum_{n \\geq c_3 \\|f_x\\|_{L^2}}^{}{  \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2} \n\\end{align*}\nHowever, we also clearly have that\n$$ \\sum_{n \\geq c_3 \\|f_x\\|_{L^2}}^{}{\\lambda_n  \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2}  \\leq   \\sum_{n=1}^{\\infty}{\\lambda_n  \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2}= |\\left\\langle D f, f\\right\\rangle| \\leq   c_2\\| f_x\\|^2_{L^2}.$$\nAs a consequence\n$$   \\sum_{n \\geq c_3 \\|f_x\\|_{L^2}}^{}{ \\left(c_3 \\|f_x\\|_{L^2}\\right)^{2}  \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2}  \\leq \\frac{c_2}{c_1 c_3^2},$$\nwhich can be made smaller than $1\/2$ for a suitable choice of $c_3$ (depending on $c_1,c_2$). Since the $(u_n)_{n=1}^{\\infty}$ form an orthonormal system\n$$ 1 = \\|f\\|_{L^2}^2 =  \\sum_{n=1 }^{\\infty}{\\left| \\left\\langle f, u_n \\right\\rangle\\right|^2},~\\mbox{we get} \\quad \\sum_{n \\leq \\sqrt{\\frac{2c_2}{c_1}} \\frac{ \\|f_x\\|_{L^2}}{   \\|f\\|_{L^2}  }  }^{}{ \\left| \\left\\langle f, u_n \\right\\rangle\\right|^2} \\geq \\frac{ \\| f\\|^2_{L^2}}{2}.$$\n\\end{proof}\n\n\n\n\nWe may not know the eigenfunctions $(u_n)_{n=1}^{\\infty}$ but we can ensure that for any function $f$ half\nof its $L^2-$mass of the expansion will be contained in the subspace\n$$ \\mbox{span}\\left\\{u_n: n \\leq c \\frac{ \\|f_x\\|_{L^2}}{\\|f\\|_{L^2}} \\right\\} \\subset H_1.$$\nThe second step of the argument invokes decay of the eigenvalues $\\mu_n$ of $T^* T$ via\n\\begin{align*}\n\\| T f\\|^2_{H_2} &= \\left\\langle Tf, Tf \\right\\rangle_{H_2} = \\left\\langle T^*Tf, f \\right\\rangle_{H_1} = \\sum_{n =1 }^{\\infty}{ \\mu_n |\\left\\langle f, u_n \\right\\rangle|^2}\n\\end{align*}\nand combining this with the previous argument to obtain\n$$ \\sum_{n =1 }^{\\infty}{ \\mu_n |\\left\\langle f, u_n \\right\\rangle|^2} \\geq \\sum_{n \\leq c \\frac{ \\|f_x\\|_{L^2}}{\\|f\\|_{L^2}}   }^{\\infty}{ \\mu_n |\\left\\langle f, u_n \\right\\rangle|^2} \\geq \n\\mu_{ c \\frac{ \\|f_x\\|_{L^2}}{\\|f\\|_{L^2}}  } \\sum_{n \\leq  c \\frac{ \\|f_x\\|_{L^2}}{\\|f\\|_{L^2}}  }^{\\infty}{|\\left\\langle f, u_n \\right\\rangle |^2} \\geq  \\frac{\\mu_{ c \\frac{ \\|f_x\\|_{L^2}}{\\|f\\|_{L^2} }} \\|f\\|^2_{L^2(H_1)} }{2}.  \n$$\n\\textit{Sharpness of results.} It is not difficult to see that these types of arguments are actually sharp (up to constant) if $f=u_n$. This will immediately imply sharpness of our results: if constants\nin the statement are chosen too small, then the inequality will fail for $(u_n)_{n \\geq N}$ for some $N$ sufficiently large. While this is not our main focus, there is quite\na bit of additional research on precise asymptotics of the constants and how they depend on the intervals (see \\cite{led0}).\n\n\\subsection{An easy inequality.} All our proofs will have a natural case-distinction: either the function changes sign on the interval $[a,b]$ or it does not. If it changes sign, then\nwe can use standard arguments to bound all arising terms by $\\|f_x\\|_{L^2[a,b]}$ which simplifes the expressions. \n\n\\begin{lemma} Let $[a,b] \\subset \\mathbb{R}$. If $f:[a,b]$ is differentiable and changes sign on $[a,b]$, then\n$$ \\|f\\|_{L^{\\infty}[a,b]} \\leq  \\sqrt{b-a} \\|f_x\\|_{L^2[a,b]}.$$\n\\end{lemma}\n\\begin{proof} Let us assume $f(x_0) = 0$ for some $x_0 \\in [a,b]$. Then, for every $x \\in [a,b]$, using Cauchy-Schwarz\n$$ |f(x)| = \\left| \\int_{x_0}^{x}{f'(z) dz} \\right| \\leq \\int_{x_0}^{x}{|f'(z)| dz} \\leq \\sqrt{b-a} \\|f_x\\|_{L^2[a,b]}.$$\n\\end{proof}\n\nIf $f$ does \\textit{not} change sign, then we cannot bound low-regularity terms like $\\|f\\|_{L^2}$ by high-regularity terms like $\\|f_x\\|_{L^2[a,b]}$. However, there is also no cancellation in\nthe integral operator and arguments specifically taylored to the integral operators will admit easy lower bounds in terms of the $L^1-$norm. The next inequality shows that the lower bounds we obtain in the Theorems are much smaller than the $L^1-$norm so that we may treat both cases at the same time.\n\n\\begin{lemma} Let $[a,b] \\subset \\mathbb{R}$. Then, for every $c_2 > 0$, there exists a $c_1 > 0$ (depending on $c_2, a, b$) such that for all nonnegative, differentiable $f:[a,b] \\rightarrow \\mathbb{R}_{+}$ \n$$  \\int_{a}^{b}{ f(x) dx}  \\geq  c_1 \\exp{\\left(-c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)}\\|f\\|_{L^2[a,b]}.$$\n\\end{lemma}\n\\begin{proof} Squaring both sides of the desired inequality and using\n$$ \\|f\\|_{L^2[a,b]}^2 = \\int_{a}^{b}{f(x)^2 dx} \\leq \\|f\\|_{L^{\\infty}} \\int_{a}^{b}{f(x) dx}$$\nshows that the desired statement is implied by the stronger inequality\n$$ \\|f\\|_{L^{\\infty}[a,b]} \\leq \\frac{1}{c_1^2} \\exp{\\left(c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)} \\int_{a}^{b}{ f(x) dx}.$$\nThe inequality is invariant under multiplication with scalars $f \\rightarrow c f$, which allows us to assume w.l.o.g. that $\\|f\\|_{L^{\\infty}[a,b]} = 1$.\nLet us now take $J \\subset [a,b]$ to be the largest possible interval such that $f$ assumes the value 1 on the boundary of $J$ and\nthe value $1\/2$ on the other boundary point. If no such interval exists, then the original inequality trivially holds with $c_1 = \\sqrt{b-a}\/2$ since\n$$  \\int_{a}^{b}{ f(x) dx} \\geq \\frac{b-a}{2}  \\geq \\frac{ \\sqrt{b-a}}{2} \\|f\\|_{L^2[a,b]} \\geq \\frac{ \\sqrt{b-a}}{2} \\exp{\\left(-c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)} \\|f\\|_{L^2[a,b]}.$$\nSuppose now that $J$ exists. Clearly, \n$$ \\int_{a}^{b}{f(x)dx} \\geq  \\int_{J}^{}{f(x)dx} \\geq \\frac{|J|}{2} \\qquad \\mbox{and} \\qquad \\|f\\|_{L^2[a,b]} \\leq \\sqrt{b-a}.$$\nIt remains to bound $\\|f_x\\|_{L^2[a,b]}$ from below. We use the trivial estimate $\\|f_x\\|_{L^2[a,b]} \\geq \\|f_x\\|_{L^2(J)}$ and argue that\namong all functions on the interval $J$ assuming the values 1 and $1\/2$ on the boundary, the linear function yields the smallest value for $\\|f_x\\|_{L^2(J)}$.\nThe existence of a minimizing function is obvious because of compactness. The minimizer $g$ has to satisfy the Euler-Lagrange equation, which simplifies to $g_{xx} = 0$.\nThis implies \n$$\\|f_x\\|_{L^2(J)} \\geq \\left\\| \\left(1 - \\frac{x}{2|J|}\\right)_{x}\\right\\|_{L^2[0, |J|]} =  \\frac{1}{2\\sqrt{|J|}}.$$\nAltogether, we have\n$$  \\frac{1}{c_1^2} \\exp{\\left(c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)} \\int_{a}^{b}{ f(x) dx} \\geq  \\frac{1}{c_1^2} \\exp{\\left(\\frac{c_2}{2 \\sqrt{|J|} \\sqrt{b-a} }\\right)} \\frac{|J|}{2}.$$\nHowever, for every choice of $a,b,c_2>0$ such that $a<b$, the function $h:\\mathbb{R}_{+} \\rightarrow \\mathbb{R}_{+}$ given by\n$$ h(x) =  \\exp{\\left(\\frac{c_2}{2 \\sqrt{x} \\sqrt{b-a} }\\right)} \\frac{x}{2}$$\nis uniformly bounded away from 0 with a lower bound that only depends on $a,b,c_2$. This allows us to pick $c_1$ in such a way that \n$$  \\frac{1}{c_1^2} \\exp{\\left(\\frac{c_2}{2 \\sqrt{|J|} \\sqrt{b-a} }\\right)} \\frac{|J|}{2} \\geq 1 = \\|f\\|_{L^{\\infty}[a,b]}$$\nindependently of the length of the interval $J$. This gives the result.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem 1}\n\n\\subsection{The Differential Operator.}\n\nLet $f \\in L^2[a,b]$ be arbitrary. We write\n$$\n \\| \\mathcal{L}_{a,b}f \\|_{L^2[0,\\infty]}^2 = \\left\\langle  \\mathcal{L}_{a,b}f,  \\mathcal{L}_{a,b}f \\right\\rangle_{L^2[0,\\infty]} =  \\left\\langle   \\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b}f, f \\right\\rangle_{L^2[0,\\infty]}.$$\nA brief computation shows that\n\\begin{align*}\n \\| \\mathcal{L}_{a,b}f \\|_{L^2[0,\\infty]}^2 = \\int_{0}^{\\infty}{  \\left(\\int_{a}^{b}{ e^{-s t} f(t) dt} \\right)^2 } &= \\int_{a}^{b}{ \\int_{a}^{b}{    f(t)f(r) \\left( \\int_{0}^{\\infty}{e^{-s t} e^{s r} ds} \\right) dt dr} }\\\\\n&=  \\int_{a}^{b}{ \\left( \\int_{a}^{b}{ \\frac{f(r)}{ r + t} dr} \\right) f(t) dt}\n \\end{align*} \nand thus\n$$ (\\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b} f)(t) = \\int_{a}^{b}{\\frac{f(s)}{t+s} ds}.$$ \n\n\\subsection{The differential operator.} The next ingredient, due to Bertero \\& Gr\\\"unbaum \\cite{gru1}, is crucial: they discovered that $\\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b}$ commutes with the differential operator \n$$\nD_t =  \\frac{d}{dt} \\left(  (t^2-a^2)(b^2-t^2) \\frac{d}{dt} \\right) - 2(t^2-a^2).\n$$\n\\begin{lemma}[Bertero \\& Gr\\\"unbaum \\cite{gru1}]\\label{lem:gru_commute1}\nFor $f \\in C^2[a,b]$\n$$\n\\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b} D_t f = D_t \\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b} f\n$$\n\\end{lemma}\n\\begin{proof} The proof is an explicit computation starting with\n$$ A = \\int_a^b \\frac{1}{t+s} \\frac{d}{ds} \\left(  (s^2-a^2)(b^2-s^2) \\frac{d}{ds} \\right) f(s) ds.$$\nIntegration by parts yields that\n\\begin{align*}\n A &= \\frac{1}{t+s}(s^2-a^2)(b^2-s^2) \\frac{d}{ds} f(s) \\big|_a^b - \\int_a^b  \\left( \\frac{d}{ds} \\frac{1}{t+s} \\right)  \\left(  (s^2-a^2)(b^2-s^2) \\frac{d}{ds} f(s) \\right) ds  \\\\\n&=  - \\int_a^b  \\left(  (s^2-a^2)(b^2-s^2) \\frac{d}{ds} \\frac{1}{t+s} \\right)  \\left(  \\frac{d}{ds} f(s) \\right)  ds \\end{align*}\nA second integration by parts gives that\n$$ A = \n\\int_a^b  \\left( \\frac{d}{ds} (s^2-a^2)(b^2-s^2) \\frac{d}{ds} \\frac{1}{t+s} \\right)    f(s) ds. $$\nSummarizing, we have just shown that\n$$ \\int_a^b \\frac{1}{t+s} \\frac{d}{ds} \\left(  (s^2-a^2)(b^2-s^2) \\frac{d}{ds} \\right) f(s) ds = \\int_a^b  \\left( \\frac{d}{ds} (s^2-a^2)(b^2-s^2) \\frac{d}{ds} \\frac{1}{t+s} \\right) f(s) ds,$$\nwhich we can also write as\n$$\n\\int_a^b  \\left( \\frac{1}{t+s} \\right) D_s f(s) ds = \\int_a^b  \\left( D_s \\frac{1}{t+s} \\right) f(s) ds.\n$$\n\nA simple computation shows that \n$$\n D_s \\frac{1}{t+s} =  D_t \\frac{1}{t+s}\n$$\nand thus, by linearity,\n$$ D_t \\int_a^b  \\frac{1}{t+s}  f(s) ds = \\int_a^b \\frac{1}{t+s}  D_s f(s) ds.$$\n\\end{proof}\n\nUsing this commutation property in combination with some additional considerations regarding multiplicity (to ensure there are no degeneracies),\none can obtain the following result.\n\\begin{thm}[Bertero \\& Gr\\\"unbaum, \\cite{gru1}]\nThe eigenfunctions  $(u_n)_{n=1}^{\\infty}$ of $D_t$ coincide with the eigenfunctions of $\\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b}$.\n\\end{thm}\nThis means we can now restrict ourselves to an analysis of the differential operator\n$$\nD_t =  -\\frac{d}{dt} \\left(  (t^2-a^2)(b^2-t^2) \\frac{d}{dt} \\right) + 2(t^2-a^2) \\qquad \\mbox{on}~[a,b],\n$$\nwhere we switched the sign to make it positive-definite.\nWe note that the basic bound $\\lambda_n \\sim n^2$ on the eigenvalues that follows immediately from standard spectral theory.\n\n\\begin{lemma}[Standard estimate, cf. \\cite{led0}]   There exists $ c_1 > 0$ depending on $a,b$ such that the eigenvalues of $D_t$ on $[a,b]$\nsatisfy\n$$ \\lambda_n \\geq c_3 n^2 .$$\n\\end{lemma}\n\n\\subsection{Proof of Theorem 1.} \n\\begin{proof}\nThe proof combines the various ingredients. We assume w.l.o.g. that $\\| f\\|_{L^2[a,b]} = 1$. Integration by part gives, for differentiable $f$,\n\\begin{align*}  \\left\\langle D f, f \\right\\rangle  &\\leq  \\int_{a}^{b}{ (t^2-a^2)(b^2-t^2) \\left( \\frac{d}{dt} f(t)\\right)^2 + 2(t^2-a^2) f(t)^2 dt } \\\\\n&\\leq (b^2-a^2)^2\\| f_x\\|^2_{L^2[a,b]} + 2(b^2-a^2) \\| f\\|^2_{L^2[a,b]}.\\end{align*}\nWe distinguish two cases: (1) $f$ has a root in $[a,b]$ or (2) $f$ has no roots in $[a,b]$.\nWe start with the first case. Then Lemma 2 implies\n$$ \\| f\\|^2_{L^2[a,b]} \\leq (b-a) \\| f\\|^2_{L^{\\infty}[a,b]} \\leq (b-a)^2 \\| f_x\\|^2_{L^2[a,b]}$$\nand thus\n\\begin{align*} \\left| \\left\\langle D f, f \\right\\rangle \\right| &\\leq (b^2-a^2)^2\\| f_x\\|^2_{L^2[a,b]} + 2(b^2-a^2) \\| f\\|^2_{L^2[a,b]} \\\\\n&\\leq \\left( (b^2-a^2)^2 + 2(b^2-a^2)(b-a)^2 \\right)\\|f_x\\|_{L^2[a,b]}^2.\\end{align*}\nAt the same time, since the eigenfunctions form a basis, we may also write\n\\begin{align*} \\left\\langle D f, f \\right\\rangle\n= \\sum_{n=1}^{\\infty}{\\lambda_n |\\left\\langle f, v_n \\right\\rangle|^2}\n  \\end{align*}\nAltogether, we have, using the lower bound $\\lambda_n \\geq c_3 n^2$ that\n$$   \\sum_{n=1}^{\\infty}{c_3  n^2 |\\left\\langle f, v_n \\right\\rangle|^2} \\leq    \\sum_{n=1}^{\\infty}{\\lambda_n |\\left\\langle f, v_n \\right\\rangle|^2}  = |\\left\\langle Df, f\\right\\rangle| \\leq c_4 \\| f_x\\|^2_{L^2[a,b]} .$$\nAs a consequence, we can use Lemma 1 to deduce that the Littlewood-Paley projection onto low frequencies contains a positive fraction of the $L^2-$mass\n$$   \\sum_{n \\leq c_5 \\|f_x\\|_{L^2[a,b]}}^{}{ |\\left\\langle f, v_n \\right\\rangle|^2} \\geq \\frac{1}{2}\\|f\\|_{L^2[a,b]}^2.$$ \nThe argument can now be concluded as follows: it is known that the eigenvalues of $\\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b}$ decay exponentially (for estimates, see \\cite{led0,led3})\n and we have also just established that a positive proportion of the $L^2-$mass lies at suitably small frequencies. We write \n\\begin{align*}\n\\| \\mathcal{L}_{a,b}  f\\|^2_{L^2[0, \\infty]} &= \\left\\langle \\mathcal{L}_{a,b} f, \\mathcal{L}_{a,b} f \\right\\rangle_{L^2[0, \\infty]} = \\left\\langle \\mathcal{L}_{a,b}^* \\mathcal{L}_{a,b} f, f \\right\\rangle_{L^2[a,b]} = \\sum_{n =1 }^{\\infty}{ \\mu_n |\\left\\langle f, u_n \\right\\rangle|^2},\n\\end{align*}\nwhere $(\\mu_n)_{n=1}^{\\infty}$ are the eigenvalues of $ \\mathcal{L}_{a,b}^* \\mathcal{L}_{a,b}: L^2[a,b] \\rightarrow L^2[a,b]$ and $(u_n)_{n=1}^{\\infty}$ is the associated sequence of eigenfunctions. We bound\n\\begin{align*}\n \\sum_{n =1 }^{\\infty}{ \\mu_n |\\left\\langle f, u_n \\right\\rangle|^2} &\\geq \\sum_{n \\leq c_5 \\|f_x\\|_{L^2[a,b]}  }^{\\infty}{ \\mu_n |\\left\\langle f, u_n \\right\\rangle|^2} \\\\\n&\\geq \\mu_{ c_5 \\|f_x\\|_{L^2[a,b]}}  \\sum_{n \\leq  c_5 \\|f_x\\|_{L^2[a,b]}}^{\\infty}{|\\left\\langle f, u_n \\right\\rangle |^2} \\\\\n&\\geq  \\frac{\\mu_{ c_5 \\|f_x\\|_{L^2[a,b]}}}{2}.\n\\end{align*}\nIt is well-known (see e.g. \\cite{led0}) that the singular values decay exponentially\n$$ \\mu_n \\geq c_1 e^{-c_2 n},$$\nwhere the constants $c_1, c_2$ only depend on the interval. This yields \n$$ \\| \\mathcal{L}_{a,b} f \\|_{L^2[0,\\infty]}^2 = \\left\\langle   \\mathcal{L}_{a,b}^*  \\mathcal{L}_{a,b}f, f \\right\\rangle \\geq c_1 \\exp{\\left(-c_2 \\|f_x\\|_{L^2[a,b]}\\right)}\\|f\\|^2_{L^2[a,b]}$$\nfor functions satisfying $\\|f\\|_{L^2[a,b]} = 1$ which, in turn, implies that for general $f \\in L^2[a,b]$ \n$$ \\| \\mathcal{L}_{a,b} f \\|_{L^2[0,\\infty]}^2  \\geq c_1 \\exp{\\left(-c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)}\\|f\\|^2_{L^2[a,b]}.$$\nIt remains to consider the second case. In that case, $f$ cannot change sign. We assume w.l.o.g. that it is always positive and bound\n\\begin{align*}\n\\| \\mathcal{L}_{a,b}f \\|_{L^2[0,\\infty]}^2 =  \\int_{a}^{b}{ \\left( \\int_{a}^{b}{ \\frac{f(r)}{ r + t} dr} \\right) f(t) dt} \\geq  \\int_{a}^{b}{ \\left( \\int_{a}^{b}{ \\frac{f(r)}{ b+b} dr} \\right) f(t) dt}\n= \\frac{1}{2b} \\left( \\int_{a}^{b}{ f(t) dt} \\right)^2.\n\\end{align*}\nHowever, here Lemma 3 immediately yields that for every $c_2 > 0$ and all $a<b$ there exists a $c_1$ (depending only on $a,b,c_2$) such that for all differentiable $f:[a,b] \\rightarrow \\mathbb{R}$ that\ndo not change sign\n$$  \\left(\\int_{a}^{b}{ f(t) dt} \\right)^2 \\geq  c_1 \\exp{\\left(-c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)}\\|f\\|^2_{L^2[a,b]}.$$\n\\end{proof}\nIt is not difficult to check that all the steps are sharp up to constants for $f=u_n$ and this guarantees the sharpness of our Theorem up to constants.\n\n\n\\section{Proof of Theorem 2}\n\n\\subsection{The differential operator.} The self-adjoint operator $\\mathcal{L}_{a,b}  \\mathcal{L}_{a,b}^*$ has the form\n$$  (\\mathcal{L}_{a,b}  \\mathcal{L}_{a,b}^*f)(s) = \\int_{0}^{\\infty}{\\frac{e^{-a(s+t)} - e^{-b(s+t)}}{s + t} f(t) dt}$$\nand its eigenfunctions $(u_n)_{n=1}^{\\infty}$ now correspond to a weighted fourth-order differential operator. \n\\begin{lemma}[Bertero \\& Gr\\\"unbaum, \\cite{gru1}]\nThe eigenfunctions $(u_n)$ form a basis in $L^2[0, \\infty]$. Moreover, if \n$$ \\hat{D}_t= -\\frac{d^2}{dt^2}\\left(t^2 \\frac{d^2}{dt^2}  \\right) + (a^2 + b^2)\\frac{d}{dt} \\left(t^2  \\frac{d}{dt} \\right) + (-a^2b^2 t^2 + 2a^2),$$\nthen the eigenfunctions of $D_t$ are also given by $(u_n)_{n=1}^{\\infty}$.\n\\end{lemma}\nThe argument is very similar to before, the crucial ingredient is the commutation relation\n$$ \\hat{D}_t   \\mathcal{L}_{a,b}  \\mathcal{L}_{a,b}^*   =    \\mathcal{L}_{a,b}  \\mathcal{L}_{a,b}^*    \\hat{D}_t   \\  $$\nwhich reduces, after several integration by parts, to\n$$ \\hat{D}_s \\frac{e^{-a(s+t)} - e^{-b(s+t)}}{s + t} =  \\hat{D}_t \\frac{e^{-a(s+t)} - e^{-b(s+t)}}{s + t}.$$\n\n\n\\subsection{Proof of Theorem 2}\n\n\\begin{proof} The overall structure mirrors that of Theorem 2, it suffices to record the differences. \nWe proceed as before and normalize to $\\|f\\|_{L^2} = 1$. Using integration by parts, we can bound\n$$ \\left\\langle \\hat{D}_t f, f\\right\\rangle = \\left\\langle  -\\frac{d^2}{dt^2}\\left(t^2 \\frac{d^2}{dt^2} f(t) \\right) + (a^2 + b^2)\\frac{d}{dt} \\left(t^2  \\frac{d}{dt} f(t) \\right) + (-a^2b^2 t^2 + 2a^2)f, f \\right\\rangle$$\nfrom above by\n\\begin{align*} J &\\leq \\int_{0}^{\\infty}{ t^2 \\left(\\frac{d^2}{dt^2} f\\right)^2 dt} +  (a^2+b^2) \\int_{0}^{\\infty}{ t^2 \\left(\\frac{d}{dt} f(t)\\right)^2 dt} +  (a^2+b^2) \\int_{0}^{\\infty}{( a^2 b^2 t^2+2a^2) f(t)^2 dt}\\\\\n&\\leq \\| x f_{xx} \\|^2_{L^2[0,\\infty]} + c_1  \\|x f_x\\|^2_{L^2[0,\\infty]} + c_2\\|x f\\|^2_{L^2[0,\\infty]} + c_3 \\|f\\|^2_{L^2[0,\\infty]}\n\\end{align*}\nThe direct spectral analysis of the operator $\\hat{D}_t$ seems trickier, however, we can use\n$$\\hat{D}_t(f)= \\mathcal{L}_{a,b} \\circ D_t \\circ (\\mathcal{L}_{a,b})^{-1}$$\nto conclude that $\\hat{D}_t$ has the same eigenvalues as $D_t$, where $D_t$ is the differential operator from the proof of Theorem 1. In particular, $\\lambda_n \\geq c_4 n^2$ for some $c_4 > 0$.\nTherefore \n\\begin{align*}\n  \\sum_{n=1}^{\\infty}{c_4  n^2 |\\left\\langle f, v_n \\right\\rangle|^2} \\leq  \\sum_{n=1}^{\\infty}{\\lambda_n |\\left\\langle f, v_n \\right\\rangle|^2}  \\leq &\\| x f_{xx} \\|^2_{L^2[0,\\infty]} + c_1  \\|x f_x\\|^2_{L^2[0,\\infty]} \\\\\n&+ c_2\\|x f\\|^2_{L^2[0,\\infty]} + c_3 \\|f\\|^2_{L^2[0,\\infty]}.\\end{align*}\nLet \n$$ J   = \\| x f_{xx} \\|^2_{L^2[0,\\infty]} + c_1  \\|x f_x\\|^2_{L^2[0,\\infty]} + c_2\\|x f\\|^2_{L^2[0,\\infty]} + c_3 \\|f\\|^2_{L^2[0,\\infty]}.$$\nUsing the argument from the proof of Lemma 1 in conjunction with\n$$ \\|f\\|_{L^2} = 1 = \\sum_{n=1}^{\\infty}{|\\left\\langle f, v_n \\right\\rangle|^2},$$\nwe can conclude the existence of a constant $ 0 < c_5 < \\infty$ depending only on $c_4$ such that\n$$   \\sum_{n \\leq c_5 \\sqrt{J}}^{}{ |\\left\\langle f, v_n \\right\\rangle|^2} \\geq \\frac{\\|f\\|^2_{L^2}}{2}.$$\nThe argument now follows from the exponential decay of the singular values (see \\cite{led0}) and the elementary inequality $(a^2+b^2+c^2+d^2)^{1\/2} \\leq a+b+c+d$ for positive $a,b,c,d \\in \\mathbb{R}_{\\geq 0}$\n $$ \\sqrt{J} \\leq  \\| x f_{xx} \\|_{L^2[0,\\infty]} + c_1  \\|x f_x\\|_{L^2[0,\\infty]} + c_2\\|x f\\|_{L^2[0,\\infty]} + c_3 \\|f\\|_{L^2[0,\\infty]}.$$\n\\end{proof}\n\n\n\\section{Proof of Theorem 3}\n\n\\subsection{The Differential Operator} \nConsider the self-adjoint operator\n $\\mathcal{F}_T: L^2[-1,1] \\rightarrow L^2[-1,1]$\n$$ (\\mathcal{F}_Tf)(x) =  \\int_{-1}^{1}{f(x) e^{i  \\xi x}dx}.$$\nThe crucial ingredient, which the monograph of Osipov, Rokhlin \\& Xiao \\cite{mono} ascribes to Landau \\& Pollak \\cite{pr2, pr3}  and Slepian \\& Pollak \\cite{pr1}, is that\nthe eigenfunctions of $\\mathcal{F}_T$ coincide with the eigenfunctions of a differential operator.\n\\begin{lemma}[\\cite{pr2, pr3, pr1}]  The eigenfunctions $(u_n)_{n=1}^{\\infty}$ of $\\mathcal{F}_T$ coincide with the eigenfunctions of \n$$D = -(1-x^2)\\frac{d^2}{dx^2} + 2x\\frac{d}{dx} +  x^2 \\qquad \\mbox{on} ~ [-1,1].$$\n\\end{lemma}\nIt is classical that the eigenvalues of the differential operator grow asymptotically as $\\lambda_n \\sim_{} n^2$, in particular, we have $\\lambda_n \\geq c_3 n^2$ for some $c_3 > 0$.\n\n\\subsection{Proof of Theorem 3} \n\\begin{proof} Let $f \\in H^1[-1,1]$ be arbitrary. We have\n$$\n \\sum_{n=1}^{\\infty}{c_3  n^2 |\\left\\langle f, u_n \\right\\rangle|^2} \\leq \\sum_{n=1}^{\\infty}{\\lambda_n |\\left\\langle f, u_n \\right\\rangle|^2} = \\left\\langle Df, f\\right\\rangle.\n$$\nRepeated integration by parts gives that\n\\begin{align*}  \\left\\langle Df, f\\right\\rangle &= \\int_{-1}^{1}{(1-x^2)f_x(x)^2 + x  \\frac{d}{dx}(f(x)^2) + x^2 f(x)^2 dx}\\\\\n&= f(1)^2- f(-1)^2 + \\int_{-1}^{1}{(1-x^2)f_x(x)^2 + (x^2-1) f(x)^2 dx} \\\\\n&\\leq \\left[ f(1)^2 - f(-1)^2 \\right] + \\int_{-1}^{1}{f_x(x)^2 + f(x)^2 dx}.\n  \\end{align*}\nWe again distinguish cases: either $f$ changes sign or it does not. If $f$ has a root somewhere, then with Lemma 2 we may conclude that\n$$ \\max \\left( f(1)^2, f(-1)^2, \\int_{-1}^{1}{f(x)^2 dx} \\right) \\leq 4 \\int_{-1}^{1}{f_x(x)^2 dx}$$\nand the result follows as before. The difference in the final result is a result of the different asymptotical behaviour of the eigenvalues (see e.g. Widom \\cite{widom}, a very\nprecise description of the asymptotic behavior can be found in Fuchs \\cite{fuchs})\n$$ \\log{ \\lambda_n} \\sim - n \\log{n}.$$\nIf $f$ does not change sign, we have to argue differently. Assume w.l.o.g. that $f \\geq 0$. Then\n\\begin{align*}\n\\| \\mathcal{F}_{T} f\\|_{L^2[-1,1]}^2 &= \\left\\langle \\mathcal{F}_T f, \\mathcal{F}_T f\\right\\rangle_{L^2[-1,1]}  =\n \\int_{-1}^{1}{  \\left( \\int_{-1}^{1}{ f(x) e^{i x \\xi} dx} \\right) \\overline{  \\left( \\int_{-1}^{1}{ f(x) e^{i x \\xi} dx} \\right) } d\\xi}\\\\\n&=  \\int_{-1}^{1}{   \\int_{-1}^{1}{   \\int_{-1}^{1}{  f(x) f(y) e^{i \\xi (x-y)} d\\xi} d x} dy} \\\\\n&= \\int_{-1}^{1}{   \\int_{-1}^{1}{ \\frac{2 \\sin{(x-y)}}{x-y} f(x) f(y) dx dy}} \\\\\n&\\geq \\frac{1}{2} \\int_{-1}^{1}{   \\int_{-1}^{1}{  f(x) f(y) dx dy}} = \\frac{1}{2} \\left(\\int_{-1}^{1}{f(x) dx}\\right)^2.\n\\end{align*}\nIt remains to show that for every $c_2 > 0$ there exists $c_1 > 0$ such that for all differentiable $f:[-1,1] \\rightarrow \\mathbb{R}$\nthat do not change sign\n$$ \\frac{1}{2} \\left(\\int_{-1}^{1}{f(x) dx}\\right)^2 \\geq c_1 \\exp{\\left(-c_2\\frac{ \\|f_x\\|_{L^2[a,b]}}{\\|f\\|_{L^2[a,b]}}\\right)}\\|f\\|^2_{L^2[a,b]}$$\nwhich follows from Lemma 3.\n\\end{proof}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\nWe use the concept of probability extensively in science, and very\nbroadly in everyday life.  Many probabilistic tools used to ``quantify our ignorance'' seem \nintuitive even to non-scientists.  For example, if we consider\nthe value of one bit which we know nothing about,\nwe are inclined to assign probabilities to each value.  Furthermore,\nit seems natural to give it a ``$50$-$50$'' chance of being $0$ or\n$1$.  This everyday \nintuition is often believed to have deep theoretical justification based in ``classical\nprobability theory'' (developed in famous works such as~\\cite{Laplace:1774zz}).  \n\nHere we argue that the success of such\nintuition is fundamentally rooted in specific physical properties of the\nworld around us.  In our view the things we call ``classical\nprobabilities'' can be seen as originating in the quantum probabilities that govern the microscopic\nworld, suitably propagated by physical processes so as to be\nrelevant on classical scales. From this perspective the validity of\nassigning equal probabilities to the two states of an unknown bit\ncan be quantified by understanding the particular physical processes\nthat connect quantum fluctuations in the microscopic world to that\nparticular bit. The fact that we have simple beliefs about how to\nassign probabilities that do not directly refer to complicated\nprocesses of physical propagation is simply a\nreflection of the intuition we have built up by living in a world\nwhere these processes behave in a particular\nway. Our position has implications for how we use probabilities in general,\nbut here we emphasize \napplications to cosmology which originally motivated our interest in\nthis topic. Specifically, we question a number of applications of\nprobabilities to cosmology that are popular today.\n\nMany physicists view\nclassical physics as something that emerges from a fundamentally\nquantum world under the right conditions (for example in systems\nlarge enough to have negligible quantum fluctuations and with suitable\ndecohering behavior) without the need for new fundamental physics\noutside of the quantum theory\\footnote{We personally take this ``fundamentally\n  quantum'' view but our arguments go\n  through for some (but not all) other interpretations of\n  quantum mechanics}. Taking that point of view does not make the\nclaims in this paper trivial ones.  Yes, in that picture ``all physics is\nfundamentally quantum'', but here we focus specifically on the origin of\nrandomness.  Consider a classical computer well engineered to prevent\nquantum fluctuations of its constituent particles from\naffecting the classical steps of the computation.  One could\nmodel a fluctuating classical system on such a computer (e.g.\na gas of perfect classical billiards), but the fluctuations in such an\nidealized classical gas would indeed be classical ones.  The appearance\nof a given fluctuation would reflect information already encoded in\nclassical features of the initial state of the computation and\nwould {\\em not} come from quantum fluctuations of the particles\nmaking up the physical computer. \nWe argue that the real physical world does not contain \nsuch perfectly isolated \nclassical systems and that quantum uncertainty, not ignorance of\nclassical information dominates probabilistic behavior we\nobserve. (For the computer example just given, the quantum\nuncertainties will enter when setting up the \ninitial state.)\n\nIn Bayesian language, the probability of a theory $T$ being true given\na dataset $D$ is computed by combining the probability of $D$ given\n$T$ (``$P(D|T)$'') with the ``prior probability'' ($P(T)$) assigned to\n$T$.  Often $P(T)$ will include other data combined in a\nsimilar way. Inputting new data over time produces a \nlist of updated probabilities. The start of such a list always\nrequires a ``model uncertainty'' (MU) prior that provides a personal\nstatement about which model(s) you prefer. Expressions for $P(D|T)$ can be tested by\nstatistical analysis of data and good scientists (discussing well\ndesigned experiments) should agree on how\nto compute $P(D|T)$.  The MU prior is a personal choice which is not\nbuilt from a scientifically  \nrigorous process. The quantity $P(D|T)$ describes randomness in\nphysical systems, whereas MU priors represent states of mind of\nindividual scientists.   This paper only treats $P(D|T)$ probabilities, not\nMU priors.  A further indication of the deep differences between\n$P(D|T)$ and MU priors is that the goal of science is to produced\nsufficiently high quality data (and sufficient consensus about the\ntheories) that which MU priors the community are willing to take is of no consequence to the\nresult. On the other hand, results will always depend strongly on at least some parts of\n$P(D|T)$. \n\n\n\\section{The Page Problem}\n\\label{Page}\nWe outline the relevance of this question to cosmology using a simple\ntoy model. It is commonplace in cosmology to\ncontemplate a ``multiverse'' (e.g. in the context of ``eternal\ninflation''~\\cite{Guth:2007ng}) in which many equivalent copies of a given observer\n appear in the theory. \n\n\n\n\n\nAs pointed\nout by Page~\\cite{Page:2009qe}, even if one knew the full wavefunction for\nsuch a theory it would be impossible to make predictions about\nfuture observations using probabilities derived from that\nwavefunction.  The problem arises because multiverse theories\nare expected to contain many copies of the observer (sometimes said to\nbe in different\n``pocket universes'') that are identical in terms of\nall current data, but which differ in details of their environments\nthat affect outcomes of future\nexperiments (e.g. experiments measuring neutrino masses or\ncosmological perturbations).  In these theories it is impossible\nto construct appropriate projection operators to describe measurements\nwhere one does not know which part of the Hilbert space (i.e. which copy of\nus and our world) is being measured. Thus, the outcomes of future\nmeasurements are ill-posed quantum questions which cannot be answered\nwithin the theory.\n\nTo illustrate this problem consider a\nsystem comprised of two two-state subsystems called ``$A$'' and ``$B$''.\nThe whole system is spanned by the four basis states constructed as\nproducts of basis states of the two subsystems: $\\left\\{ {{\\left| 1\n    \\right\\rangle }^{A}}{{\\left| 1 \\right\\rangle }^{B}},{{\\left| 1\n    \\right\\rangle }^{A}}{{\\left| 2 \\right\\rangle }^{B}},{{\\left| 2\n    \\right\\rangle }^{A}}{{\\left| 1 \\right\\rangle }^{B}},{{\\left| 2\n    \\right\\rangle }^{A}}{{\\left| 2 \\right\\rangle }^{B}} \\right\\}$.\nFor the whole system in state $\\left| \\psi  \\right\\rangle$, the\nprobability assigned to measurement outcome ``$i$'' can be\nexpressed as $\\left\\langle  \\psi  \\right|\\hat{P}_i\\left| \\psi\n\\right\\rangle $ for a suitably chosen projection operator\n$\\hat{P}_i$.  One can readily construct projection\noperators corresponding to measuring system ``$A$'' in the ``$1$'' state\n(regardless of the state of the ``$B$'' subsystem):\n\\begin{equation}\n\\hat{P}_{1}^{A}\\equiv \\left( {{\\left| 1 \\right\\rangle\n  }^{A}}{{\\left| 1 \\right\\rangle }^{B}}{}^{B}\\left\\langle  1\n\\right|{}^{A}\\left\\langle  1 \\right| \\right)+\\left( {{\\left| 1\n    \\right\\rangle }^{A}}{{\\left| 2 \\right\\rangle\n  }^{B}}{}^{B}\\left\\langle  2 \\right|{}^{A}\\left\\langle  1 \\right|\n\\right).\n\\end{equation} A similar operator $\\hat{P}_{1}^{B}$ represents\nmeasurements of only subsystem ``$B$''.  Operators such as $\\hat{P}_{12} \\equiv  {{\\left| 1 \\right\\rangle\n  }^{A}}{{\\left| 2 \\right\\rangle }^{B}}{}^{B}\\left\\langle  2\n\\right|{}^{A}\\left\\langle  1 \\right|$ represent measurements of {\\em\n  both} subsystems.\n\nThe problem arises because there is no projection operator that\ngives the probability of outcome~``$1$'' when the subsystem to be\nmeasured (``$A$'' or ``$B$'') is undetermined.  That is an ill-posed\nquestion in the quantum theory.  Page emphasizes that this\nkind of question apparently needs to be addressed in order to make\npredictions in the multiverse, where our lack of knowledge about which\npocket universe we occupy corresponds to ``$A$'' vs. ``$B$'' not being\ndetermined in the toy model. Such ill-posed quantum questions exist\nin laboratory situations as well.  We tend not to be concerned about\nthese questions however, since there are also plenty of well-posed problems\non which to focus our attention.  Also, in the laboratory one might\nresolve the problem by adding a measurable ``label'' to the setup that\ndoes identify ``$A$'' vs. ``$B$''.  But such a resolution is believed\nnot to be \npossible in many cosmological cases. \n\nA natural response to this issue is to appeal to classical\nideas about probabilities to ``fill in the gap''.  In\nparticular, if one could assign classical probabilities $p_A$ and\n$p_B$ \nfor the measurement to be made on the respective subsystems,\nthen one could answer the question posed above (the probability of the\noutcome ``$1$'' with the \nsubsystem to be measured undetermined) by giving:\n\\begin{equation}\n{{p}_{1}}={{p}_{A}}\\left\\langle  \\psi  \\right|\\hat{P}_{1}^{A}\\left|\n\\psi  \\right\\rangle +{{p}_{B}}\\left\\langle  \\psi\n\\right|\\hat{P}_{1}^{B}\\left| \\psi  \\right\\rangle.\n\\label{p1}\n\\end{equation}\nNote that the values of  $p_A$ and $p_B$ are {\\em not} determined from\n$\\left| \\psi  \\right\\rangle$, and instead provide additional\ninformation introduced to write\nEqn. \\ref{p1}. Although $p_1$ can be written as the expectation value\nof\n${{\\hat{P}}_{1}}={{p}_{A}}\\hat{P}_{1}^{A}+{{p}_{B}}\\hat{P}_{1}^{B}$,\nthe operator $\\hat{P}_1$ is not a projection operator\n($\\hat{P}_1\\hat{P}_1\\neq\\hat{P}_1$), confirming that $p_1$ does not give\nprobabilities of fully quantum origin.\n\nAuthors who apply expressions like Eqn. \\ref{p1} to\ncosmology~\\cite{Srednicki:2009vb,*Page:2012gh} do not claim this gives  \na quantum probability.  Instead they appeal to classical\nnotions of probability along \nthe lines we have discussed at the start of this paper. Surely one \nsuccessfully introduces classical probabilities such as $p_A$ and\n$p_B$ all the time in everyday situations to quantify our ignorance,\nso why should the same approach not be used in the cosmological case?\n\nOur view is that the two cases are completely different.  We\nbelieve that in every situation where we use ``classical''\nprobabilities successfully to describe physical randomness these probabilities could in principle be\nderived from a wavefunction describing the full \nphysical situation.  In this context classical probabilities are just ways to\nestimate quantum probabilities when calculating\nthem directly is inconvenient. Our\nextensive experience using classical probabilities in this way (really\nquantifying our {\\em quantum} ignorance) cannot \nbe used to justify the use of classical \nprobabilities in situations where quantum probabilities have been\nclearly shown to be ill-defined and uncomputable. Translating the\nformal framework from one situation to the other is not an extrapolation\nbut the creation of a brand new conceptual framework that needs\nto be justified on its own\\footnote{Cooperman~\\cite{Cooperman:2010zc}\n  has explored the interpretation of these matters in the context of\n  the Positive Operator Valued Measure (POVM) formalism. In our view\n  this does not really resolve the problem, since one has to introduce\n  new probabilities equivalent to $p_A$ and $p_B$ in an equally ad hoc\n  way.  We definitely do agree with the connections he draws to the\n  standard treatment of identical particles, which we find quite intriguing.}.\n\nWe are only challenging the ad hoc introduction\nof classical probabilities such as $p_A$ and $p_B$. We are not criticizing\nthe use of standard ideas from probability theory to manipulate and\ninterpret probabilities that have a physical origin.\nOf course we never know the wavefunction completely (and thus often\nwrite states as density matrices). Our claim is that probabilities are only\nproven and reliable tools if they have clear values determined from the quantum\nstate, despite our uncertainties about it.  \n\n\n\n\\section{Billiards}\n\\label{Billiards}\nWe next use simple calculations to argue that it is realistic\nto expect all probabilities we normally use to have a quantum origin.\nConsider a gas of idealized billiards with radius $r$, mean free path\n$l$,average speed ${\\bar v}$ and mass $m$. If two of these billiards\napproach each other with impact parameter $b$, the uncertainties in the\ntransverse momentum ($\\delta {{p}_{\\bot }}$) and position ($\\delta\n{{x}_{\\bot }}$) contribute to an uncertainty in the impact parameter given by:\n\\begin{equation}\n   \\Delta b  \n =\\delta {{x}_{\\bot }}+\\frac{\\delta {{p}_{\\bot}}}{m}\\Delta t\n =\\sqrt{2}\\left( a+\\frac{\\hbar }{2a}\\frac{l}{m\\bar{v}} \\right) \n\\label{Eqn:Deltab}\n\\end{equation}\nwhere the second equality is achieved using $\\Delta t = l\/{\\bar\n  v}$ and assuming a minimum uncertainty wavepacket of width $a$ in\neach transverse direction. The value of $\\Delta b$ is\nminimized by $a=\\sqrt{\\hbar l \/(2m{\\bar v})} \\equiv \\sqrt{l\n  \\lambdabar_{dB}\/2}$. We will show that $\\Delta b$ is\nsignificant even when minimized.\n\nThe local nature of subsequent collisions creates a distribution of entangled\nlocalized states reflecting the range of possible collision points\nimplied by $\\Delta b$. We estimate the width of this distribution as\nit fans out toward the next collision by classically propagating\ncollisions that occur at either side of the range $\\Delta\nb$. (Neglecting additional quantum effects increases the\nrobustness of our argument.)\nThe geometry of the collision amplifies uncertainties in a manner\nfamiliar from many chaotic \nprocesses~\\cite{Birk27a,Zurek:1994wd}.  The quantity \n$\\Delta b_{n} =\\Delta b( 1+(2l)\/r)^{n}$\ngives the uncertainty in $b$ after $n$ collisions.\n\nSetting $\\Delta b_n=r$ and solving for $n$ determines $n_Q$,\nthe number of collisions after which the quantum spread is so large that\nthere is significant quantum uncertainty as to which billiard takes\npart in the next collision:\n\\begin{equation}\n{{n}_{Q}}=-\\frac{\\log \\left( \\frac{\\Delta b}{r} \\right)}{\\log \\left(\n  1+\\frac{2l}{r} \\right)}.\n\\label{Eqn:nQ}\n\\end{equation}\n For Table\n\\ref{Table} we evaluated Eqn. \\ref{Eqn:nQ} with different input\nparameters chosen to represent various physical\nsituations.\\footnote{Raymond~\\cite{Raymond:1967aa} presents similar\n  result, applied only to actual billiards.  He also makes some\n  general points about the implications of his result that overlap\n  with some of the points we are making here.}\n\\begin{table*}[htbp]\n\n    \\begin{tabular}{l|r|r|r|r|r|r|r|}\n\n      & \\multicolumn{1}{|c|}{$r$ {\\it(m)}} \n      & \\multicolumn{1}{|c|}{$l$ {\\it(m)}} \n      & \\multicolumn{1}{|c|}{$m$ {\\it (kg)}} \n      & \\multicolumn{1}{|c|}{${\\bar v}$ {\\it (m\/s)}} \n      & \\multicolumn{1}{|c|}{$\\lambdabar_{dB}$ {\\it (m)}}\n      & \\multicolumn{1}{|c|}{$\\Delta b$ {\\it (m)}}\n      & \\multicolumn{1}{|c}{$n_Q$} \\\\ \\hline\n\n    Nitrogen at STP (Air) & $1.6 \\times 10^{-10}$ & $3.4\\times\n    10^{-07}$ & $4.7\\times 10^{-26}$ & $360$\n    &$ 6.2 \\times 10^{-12}$ & $2.9\\times 10^{-9}$ & $-0.3$ \\\\ \\hline\n    Water at body temp & $3.0\\times 10^{-10}$ & $5.4 \\times 10^{-10}$\n    & $3.0\\times 10^{-26}$ & $460$ &\n    $7.6\\times 10^{-12}$ & $1.3 \\times 10^{-10}$ & $0.6$ \\\\ \\hline\n    Billiards game& $0.029$ & $1$     & $0.16$  & $1$ & $6.6 \\times\n    10^{-34}$ & $5.1 \\times 10^{-17}$ & $8$\n    \\\\ \\hline\n    Bumper car ride & $1$     & $2$     & $150$   & $0.5$ & $1.4\\times\n    10^{-36}$ & $3.4\\times 10^{-18}$& $25$\n    \\\\ \\hline\n\n    \\end{tabular}%\n  \\caption{The number of collisions, ($n_Q$ from  Eqn. \\ref{Eqn:nQ})\n    before quantum uncertainty dominates, evaluated for physical\n    systems modeled as a ``gas'' of billiards with\n    different properties. Values  $n_Q < 1$\n    indicate that quantum fluctuations are so dominant that\n    Eqn. \\ref{Eqn:nQ} breaks down.  All randomness in\n    these quantum dominated systems is fundamentally quantum in nature.  \\label{Table}}\n\\end{table*}%\n\nTable \\ref{Table} shows that water and air are so dominated by quantum fluctuations\nthat $n_q < 1 $, indicating the breakdown of Eqn. \\ref{Eqn:nQ}, but\nall the more strongly supporting our view that {\\em all} randomness in\nthese systems is fundamentally quantum. This result strongly indicates\nthat if one were able to fully\nmodel the molecules in these macroscopic systems one would find that the\nintrinsic quantum uncertainties of the molecules, amplified by\nprocesses of the sort we just described,  would be fully\nsufficient to account for all the fluctuations.\nOne would not be required to ``quantify our ignorance'' using \nclassical probability arguments to fully understand the system.  For\nexample, the Boltzmann distribution for one of these systems in a\nthermal state should really be derivable as a feature dynamically\nachieved by the wavefunction without appeal to formal arguments about\nequipartition etc. \n\nThis argument that the randomness in collections of molecules in the world\naround us has a fully quantum origin lies at the core of our case.  We\nexpect that all practical applications of probabilities can be traced\nto this intrinsic randomness in the physical world.   As an \nillustration, we next trace the randomness of a coin\nflip to Brownian motion of polypeptides in the human nervous system. \n\n\\section{Coin Flip}\n\\label{Coin}\n\nRandomness in a coin flip comes from a lack of correlation between the\nstarting and ending coin positions.   The\nsignal triggering the flip travels along\nhuman neurons which have an intrinsic\ntemporal uncertainty  of $\\delta t_n \\approx 1ms$~\\cite{Faisal2008}. \nIt has been argued that fluctuations in the number of open neuron ion channels can account for the\nobserved values of $\\delta t_n$~\\cite{Faisal2008}.   These molecular fluctuations are due to random Brownian motion\nof polypeptides in their surrounding fluid.  Based on our assessment that the\nprobabilities for fluctuations in water are\nfundamentally quantum, we argue that the value of $\\delta t_n$\nrealized in a given situation is also fundamentally quantum. Quantum\nfluctuations in the water drive the motion \nof the polypeptides, resulting in different numbers of ion\nchannels being open or closed at a given moment in each instance\nrealized from the many quantum possibilities. \n\nConsider a coin flipped and caught at about the same height,\nby a hand moving at speed $v_h$ in the direction of \nthe toss and with a flip\nimparting an additional speed $v_f$ to the coin. A neurological\nuncertainty in the time of the flip, $\\delta t_n$, results in a\nchange in flight time $\\delta t_f = \\delta t_n \\times v_h\/(v_h+v_f)$.  A\nsimilar catch time uncertainty gives a total flight time uncertainty\n$\\delta t_t = \\sqrt{2} \\delta t_f$.  A coin flipped upward by an\nimpact at its edge has a rotation frequency \n$f=4v_f\/(\\pi d)$ where $d$ is the coin diameter. \nThe uncertainty in the\nnumber of spins is $\\delta N = f \\delta\nt_t$. Using $v_h=v_f=5m\/s$ and $d=0.01m$ (and $\\delta t_n = 1ms$) gives $\\delta N = 0.5$, \nenough to make the outcome of the coin toss completely dependent on\nthe time uncertainty in the neurological signal which we \nhave argued is fully quantum.\n\nNo doubt we have neglected significant factors in \nmodeling the coin flip. \nThe point here\nis that even with all our simplifications, we \nhave a plausibility argument that the outcome of a coin flip is truly\na quantum measurement (really, a Schr\\\"{o}dinger cat) and that the\n$50$--$50$ outcome of a coin toss may in principle be derived from\nthe quantum physics of a realistic coin toss with no reference to\nclassical notions of how we must ``quantify our ignorance''.\nEstimates such as this one illustrate how the quantum nature of\nfluctuations in the gasses and fluids around us \ncan lead to a fundamental quantum basis for probabilities we\ncare about in the macroscopic world. \n\n\\section{Digits of $\\pi$}\n\\label{Digits}\n\nThe view that all practical applications of probabilities are\nbased on physical quantum probabilities \nseems a challenging proposition to\nverify.  As we have illustrated with the coin flip, the path from\nmicroscopic quantum fluctuations  \nto macroscopic phenomena is complicated \nto track. \nAnd there are\nendless cases to check (rolling dice, \nchoosing a random card etc.), most \nalso too complicated to work through conclusively.  So arguing \nour position on a case-by-case basis is certainly an impractical\ntask. \n\nOn the other hand, our\nideas are very easy to falsify. All one needs is one illustration of a\ncase where classical notions of probability are useful in a physical\nsystem that is fully isolated from the quantum fluctuations. Once the\npractical value of purely classical \nprobabilities is established there is no reason \nit should not be applicable to other situations. One idea for such a counterexample was proposed\nby Carroll.\\footnote{S. Carroll at the {\\em PCTS\n  workshop on inflation} (Jan 2011).} One could place bets on, say,\nthe value of the millionth digit of $\\pi$.  Since the digits of\n$\\pi$ are believed to be random~\\cite{Bailey:1997xx} one should be\nable to use this apparently purely classical notion to win bets.\nWhile on the face of it this appears to be an \nideal counterexample, further scrutiny reveals an essential quantum role. \n\nLet's phrase this problem more systematically:  One expects that if you\nfinds someone who thinks the digits of $\\pi$ are not randomly\ndistributed, you can make money betting against them. Or\nequivalently, the expected payout $P_\\pi$ is zero if betting with someone who\n{\\em does} think the digits are random. A simple formula for such a\npayout is given by\n\\begin{equation}\n P_\\pi   =  \\lim_{N_{tot} \\to \\infty} {1 \\over N_{tot}} \\sum_{\\{i\\}} \\left( N_{\\pi}^i - 4.5\n\\right) = 0 \n\\label{BetOnPi}\n\\end{equation}\nwhere $\\{i\\}$ is the ensemble (of size $N_{tot}$) of the digits chosen and $N_\\pi^i$ is the actual\nvalue of the $i$th digit of $\\pi$. The result depends entirely on\nthe choice of ensemble.  With enough knowledge of $\\pi $ one can\ncome up with ensembles that give any answer you like (for example that\nonly ever select the digit ``$1$''), despite all the randomness\n``intrinsic'' to $\\pi$ (and in fact {\\em because} \nthe properties of $\\pi$ are classical and knowable).  Thus\nwe argue that the outcomes of such bets are all about the ensemble\nselected, and the choice of the ensemble is the only source of\nrandomness in the entire activity. \n\nThe reason the initial idea of betting on $\\pi$ is so\ncompelling is that no one ever thinks an ensemble will be chosen with\nattention to the actual values of the digits of $\\pi$. One can see\nhow quantum mechanics comes in by scrutinizing the process of coming\nup with ensembles. \nIt could be through the human neurons used in selecting a classical\nrandom number seed\\footnote{Similarly, the involvement of neurons etc. with the\ninitial setup prevents the classical computer example in\nSect. \\ref{Introduction} from being a counterexample.}, or through something\nmore systematic like a roulette wheel.  Again this falls in the\ncategory where one counterexample could ruin the argument, but so far\nwe have not found one.    The bet really is\nabout the lack of correlation between the digit selection and the\ndigit value and we argue it is quantum processes such as those discussed\nhere that are being counted on to create the lack of correlation that\nis crucial to the fairness of the bet.  \n\n\nOur analysis depends crucially on seemingly\n``accidental'' levels of quantum noise in the physical world. Our\npoint is that accidental or not, we count on\nthis quantum noise to produce the uncorrelated\nmicroscopic states that lie at the heart of our understanding of\nrandomness and probabilities in the world around us. Extending this\nunderstanding to domains where quantum noise cannot play this role is\nnot at all straightforward.  Discussions \nof the non-random behaviors of classical random number generators\n(such as in~\\protect\\cite{Press:1992zz}) underscore the difficulty\nof even imagining a classical source of randomness with the necessary\nlack of correlations. \n\n\\section{Toward a solution of cosmic measure problems}\n\\label{Applications}\nSo far we have used our ideas about probability to critique\nthe introduction of purely classical probabilities into cosmological\ntheories, which is an approach advocated by others~\\cite{Srednicki:2009vb,*Page:2012gh}. In this section we\nuse the ideas introduced here to work out our own \napproach to probabilities in the multiverse.   \nWe embrace the idea advocated above, \nthat fundamentally classical probabilities have no place in\ncosmological theories, and declare that questions that seem to\nrequire classical probabilities for answers simply are not answered in\nthat theory. We are basically advocating a more strict discipline\nabout which questions are actually addressed by a given theory.\\footnote{Although here we focus on cosmology, it appears that\n  our approach is relevant to other areas where there is confusion\n  about about how to assign probabilities, such as the ``sleeping beauty problem''\\cite{Elga01042000}.}\nThen one can ask if there are multiverse theories with sufficient\npredictive power to remain viable after this discipline is\nimposed. Our first assessment of this question suggests that imposing \nthis discipline may reduce or completely eliminate the notorious\nmeasure problems of eternal inflation and the multiverse. \n\nOne challenge one faces when exploring this matter is the fact that most\ndiscussions of eternal inflation and the multiverse are approached in\na semiclassical manner (for example assuming well-defined\nclassical spatial slices of infinite extent).  A more careful attempt\nto identify the full quantum nature of the picture may point to\nadditional ways proper quantum probabilities are assigned. We will not\ntry to address that aspect of the question here, and really just take a\nfirst look at the impact of hewing to our proposed probability\ndiscipline. \n\nA general point immediately becomes clear:  We are used to\nlinking counting with probabilities, but such connections are not \nalways direct or relevant.  Counting up the heads and tails in a long string of\ncoin flips {\\em is} connected with proper quantum\nprobabilities. Starting with our results of Sect.~\\ref{Coin} one can\nsee that a specific quantum probability is assigned to each different\npossible heads\/tails count, and thus counting can be tied in to\nwell-defined quantum probabilities for that system.  However, the fact that one\ncosmology may have $3$ pocket universes of type $A$, while another may\nhave $10^{100}$ does not make a difference, because as we discussed in\nSect.~\\ref{Page}, no quantum probabilities can be constructed to\ndetermine which among different (equivalent so far) observers you\nmight be.  While these numbers (by analogy with the flips of multiple\ncoins) may be linked to global properties of the state, they cannot by\nused to determine which among equivalent patches a given observer occupies.  \n\nThe insight that counting of observers in itself is insufficient to\nlead to proper probabilities leads to some interesting conclusions. \nOne is immediately drawn to the question of ``volume factors'' that give large volume regions more weight than\nsmall ones.  To the extent that volume factors are only a stand-in for\ncounting observers we regard such counting as meaningless because it \ncannot be related to true quantum probabilities. \n\nThis insight also relates to the ``young universe'' or ``end of time''\nproblem~\\cite{Bousso:2010yn,Guth:2011ie}, which can be sketched as follows:\nIf one regulates the \ncosmology with a time cutoff, inflation guarantees that most pocket\nuniverses will be produced close to the cutoff. Then the\ntime cutoff shows up at early times (relative to their time of\nproduction which is under strong pressure to happen late) for most\npocket universes.  This problem persists even as one pushes the time\ncutoff out to infinity.  But there is no evidence \nthat this counting has anything to do with probabilities\npredicted by the theory which are relevant to an observer.  There is\nno sign that such theories are able to assign a true quantum\nprobability to the time when a particular observer's pocket \nuniverse was produced.  One is simply looking at different pocket\nuniverses, and which one we occupy is not determined by the theory.\n\nOur position appears to offer significant implications for the Boltzmann Brain\nproblem~\\cite{Albrecht:2014eaa,Albrecht:2004ke,Page:2006ys}.  For our purposes here,\nthis problem is simply the case where pathological\nobservers, called Boltzmann Brains or BB's, vastly outnumber realistic\nones.  (The pathology of the BB's is that they match all the data we\nhave so far, but the next moment experience catastrophic breakdown\nof physicality, experiencing a rapid heat death.)\nAgain, we claim here that counting numbers of BB's vs realistic\nobservers cannot be related to quantum probabilities predicting which an observer\nis more likely to experience.\nThus, as long as there is at least one realistic pocket universe,\nthere will be no BB problem, no matter how many BB's are produced in\nthe theory. \n\nNow let us look at this matter from a slightly different point of\nview.  The real problem arises when one does not know which part of\nthe Hilbert space one is about to measure.  However, if one just takes\none piece of the Hilbert space in an eternally inflating universe,\nthat patch alone will have probabilities of tunneling into pocket\nuniverse $A$ or $B$, and perhaps many other outcomes as well. If one\nsimply traces out the rest of the Hilbert space, one will have a\ndensity matrix for what is going on in that patch.  With that one {\\em\n  can} take expectation values of operators, without introducing\nclassical probabilities to determine which pocket you are in. To the\nextent that the BB problem can be phrased in this way (in terms of a\nquantum branching into BB's vs realistic cosmologies in a given patch),\nwe expect the BB problem will remain if realistic cosmologies are\nsufficiently suppressed\\footnote{In \\cite{Albrecht:2014eaa} one of us (AA) treats\n  BB's in the traditional counting language in toy models.  However, we\nexpect that with a bit more realism the kind of quantum chaos discussed in\nthis paper would allow those BB discussions to go over nicely into\nthe (more legitimate) quantum branching form described here, without changing the\nconclusions in~\\cite{Albrecht:2014eaa}.}. And if\nall patches are the same (as may well be the case for highly symmetric\ntheories such as eternal inflation) then it does not really matter\nwhat patch you are in.  The answer will still be the same. \n\nWhile we\nhave yet to offer a rigorous demonstration, this set of ideas seem\npromising to us as a way out of the measure problems in cosmology.  A\nmore formal way to describe this picture is that if one does consider\na theory with multiple possible locations for the observer, one would\nbe obliged to give a ``prior'' on which location we occupy.  These\npriors would look very much the same as the classical probabilities\nthat show up for example in Eqn.~\\ref{p1}. However, by viewing these\nprobabilities as priors,\nour agenda would be to reach a point where their values do not matter\nto our answers\\footnote{Note that while formally these priors look the\n  same as the classical probabilities discussed\n  in~\\cite{Srednicki:2009vb,*Page:2012gh}, those authors emphasize\n  cases where results {\\em do} depend in a fundamental way on the values chosen\n  for the classical probabilities.  So they\n  are not really treating their classical probabilities\n  as prior probabilities, the values of which should ultimately not be\n  important.}. It would \nappear that for sufficiently \nsymmetric theories, independence from these priors would be easy to\nachieve.  Also, if certain obserables are sufficiently correlated, the\nmeasurement of one (which itself did not have a prediction for the\noutcome due to dependence on priors) could then lead to predictions\nfor the other observable. Both of these pictures outlined here could\nlead to a substantial level of predictive power, despite the\nrestrictions imposed by our probability discipline. \n\n\\section{Conclusions}\n\\label{Conclusions}\n\nIn summary, we have argued that all successful applications of\nprobability to describe nature can be traced to quantum origins.\nBecause of this, there has not been any systematic validation\nof purely classical probabilities, even though we appear to \nuse them all the time. These matters are of particular importance in \nmultiverse theories where truly classical probabilities are used\nto address critical\nquestions not addressed by the quantum theory.   \nSuch applications of classical probabilities need to be built\nsystematically on separate foundations and not be thought of as\nextensions of already proven ideas. \nWe have yet to see purely classical probabilities motivated and\nvalidated in a compelling way, and thus are skeptical of\nmultiverse theories that depend on classical probabilities for their\npredictive power. Fundamentally finite cosmologies~\\cite{Banks:2003pt,*Albrecht:2011yg} that\ndo not have duplicate observers do not require classical\nprobabilities. These seem to be a more promising path. \n\nWe are not the only ones who regard quantum\nprobabilities as most fundamental\n(e.g.~\\cite{Deutsch:1999gs,*Wallace:2010aa,*Zurek:2011zz,*Bousso:2011up}), but there\nare also opposing views\\footnote{There are also some papers \n  where the degree overlap is not so clear.  Vilenkin appears to focus on\n  quantum probabilities in~\\cite{Vilenkin:2013loa}, but then also\n  seems to embrace a fundamentally classical picture similar to that\n  advocated in~\\cite{Aguirre:2010rw}. Some aspects\n  of~\\cite{Nomura:2011rb} also seem to overlap, although other things\n  (such as the emphasis on holography) seem very different, so it is\n  hard to tell the overall degree of agreement}.  In addition to the\ncase already discussed where classical probabilities are introduced in\nmultiverse theories to enhance predictive \npower (such\nas in~\\cite{Srednicki:2009vb,Page:2012gh}),\nsome theories insert classical\nideas for other reasons, often in hopes of allaying\ninterpretational concerns\n(e.g. ~\\cite{'tHooft:2010zz,Weinberg:2011jg,Aguirre:2010rw}).\nThe arguments presented here make us generally \ndoubtful of such classical formulations, since our analysis reinforces the\nfundamental role of quantum theory in our overall understanding\nof probabilities. Perhaps some of these alternate theories integrate the\nclassical ideas sufficiently tightly with the quantum piece that the\neveryday tests we have discussed could just as well be regarded as\ntests of the classical ideas in the alternate theory.  However, such \nlogic seems overly complex to us, and we prefer the simpler\ninterpretation that the strong connection between all our\nexperiences with probabilities and the quantum world means the quantum\ntheory really is the defining physical theory of probabilities.  We\nhave offered suggestions that sticking only to quantum probabilities\nto make predictions in the multiverse may not be all that debilitating to\nthe predictive power of multiverse theories and may actually offer a\nsolution to the notorious measure problems of eternal inflation. \n\n\n\n\n\n\n\\acknowledgments\nWe thank E.~Anderes, A.~Arrasmith, S.~Carroll, J.~Crutchfield, D.~Deutsch, \nB.~Freivogel, A.~Guth, \nJ.~Hartle, T.~Hertog, T.~Kibble, L.~Knox, Z. Maretic, D.~Martin, J.~Morgan, J.~Preskill,\nR.~Singh, A.~Scacco, M.~Sredniki, A.~Vilenkin and W.~Zurek for\nhelpful conversations.  We were supported in part by\nDOE Grant DE-FG03-91ER40674.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\nThe recent development of micro-fluidic based devices has pushed the\nscientific community to reconsider classical issues in fluid dynamics\nsuch as the hydrodynamical behavior of micro-scale bodies\nin the creeping regime. For instance, propulsion of microswimmers,\nsee e.g.~\\cite{fauci2006biofluidmechanics,lauga2009hydrodynamics,guasto2012fluid,elgeti2014physics}, \nhas been addressed for the production of energy oriented to \nmicro-devices~\\citep{di2010bacterial,sokolov2010swimming} \nand for the self-propulsion of micro-robots~\\citep{pak2012micropropulsion,alouges2013optimally}. \nThe microswimmer's behavior in presence of confining surfaces is also \ncrucial for biofilm formation. Most studies~\\citep{pratt1998genetic, rusconi2010laminar}\nfocused on the biofilm formation at  solid-liquid interfaces.\nBiofilm formation near air-liquid interfaces is also a problem of significant concern, since\nthe liquid-air biofilm can be  advantageous for aerobic microorganisms, \nproviding them access to oxygen (from the air) and nutrients \n(from the liquid) at the same time~\\citep{constantin2009bacterial, koza2009characterization}.\nSeveral species of flagellated bacteria are known to form biofilms at liquid\/air interfaces, e.g. \nBacillus subtilis, Bacillus cereus and Pseudomonas fluorescens. \nFor instance, the preparation of the well known Natto from Japanese cooking, consisting of fermented soy beens, involves \nB. subtilis in significant concentrations~\\citep{chantawannakul2002characterization}. In contrast, some strains  of B. cereus are known to be harmful \nto humans and cause foodborne illness~\\citep{ehling2004bacillus}. P. fluorescens, instead, is able to contaminate heparinized saline flushes used in cancer therapy~\\citep{gershman2008multistate}.  Despite its biological implications, the mechanics  of flagellated swimmers close to  a liquid-air interface is much less explored than the case of a liquid-solid interface.\n\n\nAt micro-scales viscous effects overwhelm inertia leading to the development\nof apparently counterintuitive swimming strategies as proved by the \nScallop theorem~\\citep{purcell1977life,lauga2011life}. In a nutshell, \nif the swimmer deforms through a sequence of body configurations which are\nperiodic and time-reversible (reciprocal motion), its average motion is zero. \nThe reciprocal motion can be exploited for locomotion only when \nnon-linear or memory effects are relevant.\nExamples are the non-Newtonian behavior of\nthe fluid~\\citep{qiu2014swimming,lauga2009life,keim2012fluid} or motions occurring close to \ndeformable interfaces~\\citep{trouilloud2008soft}. Microorganisms developed \nmany different strategies based on nonreciprocal effects to overcome the above \nrestrictions. The \\textit{Spiroplasma} deforms its cytoskeleton by propagating \npairs of kinks~\\citep{trachtenberg2003bacterial,yang2009kinematics}.\nOther microswimmers exploit their cilia wavy motion~\\citep{maxey2011biomimetics}\nas done by \\textit{Paramecium}~\\citep{jana2012paramecium}.\n\nHowever, many microorganisms take advantage of single or multiple \nflagella, such as the \\textit{Caulobacter crescentus}, which has a single \n(right-handed) helical filament driven by a rotary motor~\\citep{li2006low} and the \n\\textit{Escherichia coli} that has multiple flagella~\\citep{berg2004coli}. \nThe flagellar motor which activates and controls the filament rotation is \nable to switch between both rotation directions~\\citep{wang2014switching}, \nand, as a first approximation, the torque applied to the filament is \nconstant~\\citep{berg2004coli}. \nIn the case of the\n\\textit{Escherichia coli} the flagella arrange in a bundle, characterized by \n a left-handed rotation of the motor\nwhich \nconfers to bacteria a smooth forward motion (\\textit{run} phase).\nThe flagella is also able to invert the rotation direction in \norder to let the bacteria at rest and change its direction \n(\\textit{tumble} phase).\t\nDue to their internal structure, the filaments can only assume twelve prescribed \nshapes (helical polymorphic states) but only one of those is the \n\\lq\\lq normal state\\rq\\rq, i.e. it is the most observed one during the \\textit{run} \nphase~\\citep{darnton2007force,vogel2010force}.\nHence, the flagella can be \nmodeled as a rigid single filament~\\citep{phan1987boundary,shum2010modelling} \nrotating around the bacterial head axis.\n\n\nThe main purpose of this work is an extensive analysis of the hydrodynamical\nbehavior of a flagellated microswimmer close to air-liquid interfaces, with particular \nattention to free surfaces. \nFor the sake of definiteness, we  focus on a simplified geometry modeling the configuration of E. coli whose swimming  is probably the most widely studied  from both experimental and numerical point of view. Anyhow, in order to explore the effect of different geometrical parameters, several modified configurations are also discussed.\n\n\nFlagellated swimmers close to a free-surface have been already addressed by simplified models, see e.g. \n~\\cite{crowdy2011two, di2011swimming, lopez2014dynamics} where two-dimensionality, resistive force theory, and multipole expansions techniques were exploited, respectively.\nTo the best of our knowledge, the present work  addresses the first fully three dimensional numerical simulation of a swimmer in presence of a liquid-air interface.\nGiven the linearity of\nthe Stokes equations which are the appropriate model for creeping flows, the numerical\napproach exploits the Boundary Element Method (BEM). The BEM \ncan easily handle the complex geometry of the microswimmer and account for\ndifferent boundary conditions. Moreover, a flat and \ninfinitely extended free surface and\/or solid wall (when needed for comparisons) \ncan be easily included by considering the appropriate Green's function, thus \navoiding any undesired effect of the numerical truncation of the domain.\n\nThere is a wide body of literature available dealing with the motion of \nmicroorganisms in free space or close to solid boundaries. \nSince the first studies by~\\cite{taylor1951analysis}, the \nattention was principally focused on microorganisms whose \nflagellum was modeled as an infinite cylindrical filament\nin an unbounded fluid endowed with small amplitude wavy motion. \nIn fact, it was initially believed that the flagella were only moved by  \nwave propagation. However~\\cite{berg1973bacteria} showed that bacteria could\nalso rotate their flagella in a corkscrew-like motion, moving the flagellar bundle as \na single filament.\nThe flagellum has a large aspect ratio, with length exceeding \nthickness by even more than two orders of magnitude~\\citep{lauga2009hydrodynamics}. \nThis particular geometry pushed the adoption of a \nSlender Body Theory (SBT) that has been exploited in several studies to evaluate \nthe translational velocity and\/or the torque applied by the \nswimmer~\\citep{hancock1953self}. \nSuccessively \\cite{higdon1979hydrodynamic,higdon1979hydrodynamics} \ntransformed the Stokes equations into a system of singular integral equations \naccounting for the swimmer translational and angular velocities. He added to the SBT \n a variable strength of the singularity along the flagellum centerline, \nthus modelling different centerline geometries. In this case both the planar \nsinusoidal motion and the rotation about the body axis were amenable to modelling.\nBy an appropriate system of images the SBT could also account for a spherical cell body and \nthe presence of a wall. However too many restrictions still confined the SBT application \nto extremely simplified configurations. \n\nIn recent years the increase of computational resources, led to a more \nextensive use of the BEM which overcomes many drawbacks of the SBT in dealing \nwith microswimmers both in free space and confined conditions. \\cite{phan1987boundary} \nused the BEM to study the motion of a microorganism in free space. \nSuccessively,~\\cite{ramia1993role} applied the BEM \nto the interaction between the swimmer and a solid wall, showing that, \nwhen swimming close to a solid wall, the swimmer exhibited a \ncircular motion. The BEM was also \nused to study the interactions between two neighboring flagellated microswimmers\nhighlighting the possibility of a coordination between their flagellar motion \nin order to maximize their velocities~\\citep{ramia1993role}. \n \nMore recently,~\\cite{lauga2006swimming} experimentally investigated the motion \nof an E. coli near a solid wall and found, as predicted by~\\cite{ramia1993role}, \na circular clockwise motion.  In the paper the authors also\nprovided a simple theoretical model which was able to explain their experimental\nobservations. Interestingly, the same model suggested that the bacteria\nshould reverse its rotation when swimming in proximity of a free surface. \nThe same authors also showed that the swimmer is attracted by a solid wall.\nThis behavior was deeply investigated a few years later \nby~\\cite{giacche2010hydrodynamic} in a numerical work which highlighted \nhow the bacteria could move at a stable distance from a wall. At the same \ntime~\\cite{shum2010modelling} investigated the motion of a microswimmer \nclose to solid walls by considering many geometrical configurations\nand relating the shape of the body to the propulsion efficiency and to the\npossibility of achieving different motions, i.e. to be attracted by the wall, \nto escape from the wall or to reach stable circular orbits at a given \nwall-normal distance.\nThe same authors~\\citep{shum2012effects} also focused on the flexible hooks \nlinking each flagellum to the cell. They studied the modifications in the microswimmer trajectories\nwhen changing the hook rigidities. They found that, within an intermediate range of rigidities, the \nswimmer behavior doesn't change too much with respect to the simpler model of \nrigid hook. The same work highlighted how, for particular values of relative hook stiffness, \nthere is a transient phase of periodic motion with constant average distance from the wall, \nleading to boundary accumulation. \nThe tendency of swimming microorganisms to accumulate near solid \nwalls~\\citep{li2009accumulation}, with particular emphasis on collisions with the surface \nand rotational Brownian motion, has also been investigated and linked \nto the swimming speed and the cell size~\\citep{li2011accumulation} of the microswimmer.\nRecent works extended the investigation about the motion of microswimmers \nto more complex surfaces: a clean fluid-fluid interface, a slipping rigid wall, and a fluid interface covered \nby surfactants~\\citep{lopez2014dynamics}.  \nThe authors used an asymptotic, far field approximation \nto represent the actual swimmer, \nretaining\ninformation about velocities and rotations.\nThe case of two fluid interfaces, in the limit of vanishing \nviscosity of one of them, allowed to describe a free surface. \nIn such conditions, the swimmer exhibited counter-clockwise motion.\nThis confirmed the results anticipated by~\\cite{lauga2006swimming}. \nAlso~\\cite{di2011swimming} supported by means of experimental \nobservations the theoretical prediction made through a simplified model based \non the method of images and the resistive force theory. Even if neglecting all the \ndynamics of the swimmer outside the interface plane, this work provided a simple \nexplanation for a counter-clockwise motion over a perfectly slipping surface, showing \na good agreement between the results of the simplified model and the experimental \nobservation. \nBriefly, ~\\cite{lauga2006swimming} explain that, when swimming above a no slip surface, a positive rotation rate of the swimmer head around its longitudinal axis produces a lateral force which is opposite to the one induced by the negative tail rotation rate. As a consequence, a net torque \nnormal to the wall is exerted on the body such that the swimmer follows a clockwise (CW) trajectory.\nOtherwise, when swimming close to a liquid-air interface,~\\cite{di2011swimming} devise a simplified model based on resistive force theory endowed with suitable symmetries to satisfy the free slip condition at the interface. \nIn this model, the swimmer moves under the effect of the velocity generated by its mirror image below the interface.\nThe counter-rotating image head produces on the swimmer head a lateral velocity which is opposite to the force that is exerted in the case of a solid wall.\nSuch relative velocity gives rise to a corresponding viscous force in the same direction. Since the same reasoning applies to the counter rotating tail, the overall torque on the microswimmer  is also opposite to the one experienced on a no-slip wall, hence\n a net CCW motion is produced. \nIn literature a clockwise motion \nhas been observed~\\citep{lemelle2010counterclockwise} \nalso in presence of a free surface. \nBased on experimental observations,~\\cite{morse2013molecular} attributed such a behavior to \nthe molecular adsorption (due to the presence of biological material in the growth medium) \naltering the rheological properties of the air\/water interface, thus determining the swimming \npattern of nearby cells. \n\nIn principle, the mechanical causes that can affect the swimming direction are: \n$i)$ the rotation direction of the flagellum bundle;\n$ii)$ the effective boundary condition at the planar surface.\nThis study focuses on the point $ii)$ through the numerical simulation of\nthe motion of a \\textit{E. coli}-like microswimmer   \nclose to free-slip and no-slip surfaces, assuming a standard left-handed arrangement for \nthe flagellum bundle. \nThe aim is investigating the behavior of the swimmer by \naddressing in full detail the complete three dimensional nature of the hydrodynamical \ninteraction between the swimmer and the surface. \nIt is worth noting that free-slip and no-slip are the limits \nof the more general Navier boundary condition that \nconnects the velocity at the liquid boundary with the tangential \nstress~\\citep{bazant2008tensorial}.\nFor liquid water moving on a solid surface, the actual slip is \nnegligible at micro-scale also for hydrophobic coatings \n\\citep{chinappi2010intrinsic,sega2013regularization} and only the presence \nof vapor bubbles trapped in the surface asperities (superhydrophobic surfaces) \nleads to significant slippage~\\citep{gentili2014pressure,bolognesi2013novel}, potentially\naltering the motion of particles close to the surface\n~\\citep{pimponi2014mobility,nizkaya2014flows}. Hence, the no-slip boundary is, for\nthe present purposes, a reliable model of a rigid wall. \nOn the other hand, the proper boundary conditions at the liquid-air interface \nare impermeability and continuity of the tangential stress \ncomponents, that reduces to free-slip (zero tangential stress) since \nair density is order of magnitude smaller than the liquid one.\n\nThe paper is organized as follows:\nthe geometrical model of the swimmer and the BEM is addressed in \n\\S~\\ref{sec:model}. \\S~\\ref{sec:FreeSurf} reports the salient results \nconcerning the motion of the swimmer in presence of a free surface.\nFinally, we discuss the major findings and point out the main conclusions \n(\\S~\\ref{sec:conclusions}), giving a perspective view of future work. \n\\section{Boundary integral formulation for the microswimmer}\\label{sec:model}\n\n\\subsection{Swimmer model}\\label{ssec:swim_mod}\n\nThis section concerns the modelling of a microswimmer \ninspired to \\textit{Escherichia coli}. This bacteria has \nbeen deeply investigated in the literature and a lot of  information about \nits geometry and propulsion mechanism is available. \\textit{E. coli} has a \ncell length which varies between $1.6$ and $3.9 \\, \\mu m$, \nthe cell width ranges between $0.9$ and $1.7 \\, \\mu m$, \nwith a resulting cell volume from \n$1.5$ to $4.4 \\, \\mu m^3$~\\citep{volkmer2011condition} and an\naverage length of flagella of about $7 \\mu m$~\\citep{lauga2006swimming}.\n\n\\begin{figure}\n  \\centerline{\\includegraphics[height=6cm,width=12cm]{fig1.pdf}}\n  \\caption{The model of the flagellated swimmer comprises an\n           ellipsoidal head (cell) and a tubular, helical, rigid, tail (flagellum).\n           The tail rotates about its axis $\\vec{e}_T$ of an angle $\\phi(t)$ . \n           The dimensionless semi-axis of the head are $a_1$ and $a_2$. \n           The head aspect ratio is kept constant, $AR=a_1\/a_2=2$, and the \n           equivalent radius (i.e. the radius $\\bar{a}$ of the sphere with same \n           volume) is the assumed reference length.\n           The dimensionless tail length is $L=7$, with a cross-section \n           radius $a_t=0.05$. The dimensionless helix amplitude and \n           wavelength are $A$ and $\\lambda$, respectively. \n           $\\{ \\vec{e_1},\\vec{e_2},\\vec{e_3} \\}$ are the orthonormal \n           vectors of the frame attached to the swimmer head (body frame). \n           $\\vec{e_1}$ is longitudinal and identifies the swimmer \n           orientation with respect to the unit vectors of the \n           fixed frame $\\{ \\vec{X},\\vec{Y},\\vec{Z} \\}$.\n  }\n\\label{fig:geometry}\n\\end{figure}\nFigure~\\ref{fig:geometry} sketches the simplified geometry comprising the \nellipsoidal axisymmetric cell and the corkscrew tail. Hereafter the equivalent \nradius of the ellipsoidal cell $\\bar{a}$ (the radius of the sphere having the same \nvolume) will be used as reference length-scale, i.e. the dimensionless cell volume is \n$V = 3 V'\/ {4 \\pi \\bar a}^3 = 1$, where the prime identifies dimensional \nquantities. The cell aspect ratio is $AR=a_1\/a_2=2$, being $2 a_1$ and $2 a_2$ the \nlongitudinal and the transversal (dimensionless) axis. \nFollowing~\\cite{shum2010modelling}, the tail bundle is modelled \nas a single helix with radius $a_T=0.05$. The tail \nrotates around its axis $\\vec{e_T}$ (see figure~\\ref{fig:geometry}). \nThe dimensionless axial length of the helix is \n$L$, $A$ denotes the helix amplitude and $\\lambda$ its wavelength. \nIn the present work we selected typical values \nfor the tail length and cell axes, namely $L = 7$, $a_1 = 1.6$ and $a_2 = 0.8$, \nwhile the amplitude $A$ and the wavelength $\\lambda$ of the tail are systematically \nvaried. With the above choices, the\nswimmer is rescaled into an actual \\textit{E. coli} \nby assuming $\\bar{a}=1 \\,\\, \\mu m$.\n\\begin{figure}\n  \\centerline{\\includegraphics[height=6cm,width=12cm]{fig2.pdf}}\n  \\caption{Sketch of the discretized swimmer near a planar surface.\n           The configuration is identified by three parameters,\n           namely, the distance $h$ of the reference point $\\vec{x}_J$ \n           from the plane and the pitch angle $\\Theta$. \n           The third parameter is the tail rotation angle $\\phi$ defined in\n           figure~\\ref{fig:geometry}. \n           }\n\\label{fig:pitch_angle}\n\\end{figure}\n\nIt is useful to introduce a body reference frame with orthonormal base vectors \n$\\{ \\vec{e_1},\\vec{e_2},\\vec{e_3} \\}$ \nwhere $\\vec{e_1}=-\\vec{e}_T$ is longitudinal (see figure~\\ref{fig:geometry}).\nThe subscript $H$ identifies the head and the subscript $T$ refers \nto the tail, which can rotate with respect to the head about its axis $\\vec{e_T}$. \nDuring the tail rotation around $\\vec{e}_T$, each point of the rigid tail  \ndescribes a circle in the plane normal to $\\vec{e}_T$, being $\\phi$ the \ncorresponding rotation angle (or flagellum phase).\nThe time derivative $\\dot{\\phi}(t)$ is the tail rotational velocity $\\Omega_T$. \nThe swimmer position is identified in a fixed reference frame \nwith base \n$\\{\\vec{X},\\vec{Y},\\vec{Z}\\}$ by the three coordinates of the cell-to-tail \njunction point $\\vec{x}_J$, see figure~\\ref{fig:pitch_angle}.\nThe swimmer translates with dimensionless velocity $\\vec{U}=\\vec{U'}\/v$\nand has angular velocity $\\vec{\\Omega}_H=\\vec{\\Omega'}_H \\bar{a}\/ v$,\nwhere $v \\simeq 20\\mu m \/s$ is a typical swimming velocity \ntaken as reference quantity.\nThe $\\{ \\vec{X},\\vec{Y} \\}$ coordinate plane \nis taken to coincide with the planar boundary, either\nthe solid wall or the free surface, see figure~\\ref{fig:pitch_angle}.\n\nThe kinematics of the swimmer is described by seven degrees of freedom, \nthe junction position $\\vec{x}_J(t)$, the flagellum phase $\\phi(t)$ and the \nthree parameters defining the rotation of \nthe head. We stress that the discretely evolved rotation matrix should\nbelong to the proper matrix subspace, namely the $\\rm SO(3)$ subgroup. \nThis is enforced through a description in terms \nof quaternions with unit norm, see e.g.~\\cite{diebel2006representing}. \nIt follows,\n\\begin{eqnarray}\n\\vec{\\dot{x}}_J &=& \\vec{U}(t) \\,\\, , \\label{eqn:kinematic1} \\\\\n\\dot{\\phi} &=& \\Omega_T (t) \\,\\, , \\label{eqn:kinematic2} \\\\\n\\vec{\\dot{q}} &=& \\frac{1}{2}\\vec{S}(\\vec{q}) \\, \\vec{\\Omega}_H(t)\\,\\, , \\label{eqn:kinematic3}\n\\end{eqnarray}\nwhere $\\vec{q}= (q_0 , q_1 , q_2 , q_3)$ is the quaternion vector\nwith $\\vert \\vec{q} \\vert =1$. When $\\vec{\\Omega}_H(t)$  \nis expressed in the \n$\\{\\vec{X},\\vec{Y},\\vec{Z}\\}$ base, $\\vec{S}$ reads \n\\begin{equation}\n  \\vec{S}(\\vec{q})=\n  \\left[ {\\begin{array}{rrr}\n   -q_1 & -q_2 & -q_3 \\\\\n    q_0 & -q_3 &  q_2 \\\\\n    q_3 &  q_0 & -q_1 \\\\\n   -q_2 &  q_1 &  q_0 \\\\\n  \\end{array} } \\right] \\, .\n\\label{eqn:w_def}\n\\end{equation}\nGiven the quaternion $\\vec{q}(t)$, the components of the body frame unit vectors\ncan be reconstructed as\n\\begin{eqnarray}\n\\nonumber \n\\vec{e_1}= \\left( q_0^2+q_1^2-q_2^2-q_3^2, 2 (q_1 q_2+q_0 q_3), 2 (q_1 q_3 - q_0 q_2) \\right) \\,\\, , \\\\\n\\vec{e_2}= \\left( 2 (q_1 q_2 - q_0 q_3), q_0^2-q_1^2+q_2^2-q_3^2, 2(q_2 q_3 + q_0 q_1)  \\right) \\,\\, , \\\\\n\\nonumber \n\\vec{e_3}= \\left( 2(q_1 q_3+q_0 q_2), 2(q_2 q_3 - q_0 q_1), q_0^2-q_1^2-q_2^2+q_3^2 \\right) \\,\\, . \n\\label{eqn:bodyunitvectors}\n\\end{eqnarray}\nTime integration of equations (\\ref{eqn:kinematic1}),(\\ref{eqn:kinematic2}),\n(\\ref{eqn:kinematic3}), allows  to track the swimmer trajectory once \n$\\vec{U}$, $\\vec{\\Omega}_H$ and $\\Omega_T$ are determined by the solution of the \nhydrodynamical interactions between the swimmer and the fluid, \nas discussed in the next section. Since the\nnumerical integration error in the equations for the quaternion may affect\nits norm, the quaternion is normalized to unit length at each time step.\nIt easily shown that the accuracy \nof the integration scheme is exactly preserved by this procedure.\n\n\\subsection{Hydrodynamic model}\\label{ssec:math_model}\n\nGiven the characteristic length scale $\\bar{a}$ and swimming velocity $v$ of \nthe microorganism, \nthe typical Reynolds number in water is \n$Re=\\rho \\bar{a} v \/ \\mu \\simeq 2*10^{-5}$ where $\\mu$ and $\\rho$ \nare the water viscosity and density, respectively.\nHence the inertial terms are negligible and, for Newtonian fluids, the\nflow in the domain ${\\cal D}$ is described by the Stokes equations, \n\\begin{subeqnarray}\n\\boldsymbol{\\nabla} \\cdot \\vec{u} &=& 0 \\,\\, , \\label{eqn:Continuity} \\\\\n\\nabla^2 \\vec{u}-\\boldsymbol{\\nabla} p &=& 0 \\,\\, ,\n\\label{eqn:Stokes}\n\\end{subeqnarray}\nwhere $\\vec{u}=\\vec{u'}\/v$ and $p=p' \\bar{a}\/(\\mu v)$.\n\nThe flow velocity $\\vec{u}$ is forced by the tension \n$\\vec{f}$ at the swimmer surface acting on the fluid.\nThe swimmer propulsion \nis due to the internal (constant) torque $\\tau_M = \\tau'_M\/(\\mu v \\bar{a}^2)$\nexchanged between head and tail.\nBeing a free body, the total force and torque on the swimmer (head and tail) \nis zero, \n\\begin{subeqnarray} \n\\int_{H \\cup T}{\\vec{f} dS} &=& 0 \\,\\, , \\\\\n\\int_{H \\cup T}{\\vec{r} \\wedge \\vec{f} dS}&=&0 \\,\\, .\n\\label{eqn:balances}\n\\end{subeqnarray}\nwhere $\\vec{r} = \\vec{x} - \\vec{x}_J$. \nThe torque $\\tau_M$ exerted on the tail is balanced by the torque \nproduced by the fluid stresses $\\vec{f}$ on the tail boundary, namely\n\\begin{equation}\n\\vec{e_T} \\cdot \\int_{T}{\\vec{r} \\wedge \\vec{f} dS}=-\\tau_M \\,\\, .\n\\label{eqn:Torque}\n\\end{equation}\n\nThe system ~(\\ref{eqn:Stokes}),\n~(\\ref{eqn:balances}) and ~(\\ref{eqn:Torque}) \nneeds proper boundary conditions \nat the swimmer surface and external boundaries, i.e. wall or free surface.\nOn the microswimmer surface the no-slip \ncondition yields\n\\begin{subeqnarray} \n\\vec{u}(\\vec{x}) &=& \\vec{U}+\\vec{\\Omega}_H \\wedge \\vec{r}, \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, (\\vec{x}\\in H) \\,\\,,  \\\\\n\\vec{u}(\\vec{x}) &=& \\vec{U}+(\\vec{\\Omega}_H + \\Omega_T \\vec{e}_T) \\wedge \\vec{r}, \\,\\,\\,\\,\\, (\\vec{x}\\in T) \\,\\, .\n\\label{eqn:swim_velocity}\n\\end{subeqnarray}\nNo-slip boundary condition is also used to model the solid wall.\nFor the free surface vanishing \nnormal fluid velocity (impermeability) and zero tangential stresses \n(free slip condition) are the appropriate prescriptions,\n\\begin{subeqnarray}\n\\vec{u} \\cdot \\vec{n} &=& 0 \\,\\, , \\label{eqn:impermeab} \\\\\n(\\left( \\vec{\\nabla}\\vec{u} + (\\vec{\\nabla} \\vec{u})^\\mathrm{T} \\right) \\cdot \\vec{n})(\\vec{I}-\n\\vec{n}\\otimes\\vec{n}) &=& 0 \\,\\, .\n\\label{eqn:noshear}\n\\end{subeqnarray}\n\n\\subsection{Boundary Element Method}\\label{ssec:num_solution}\n\nThe solution of the Stokes equations~(\\ref{eqn:Stokes}) can be \nrecast in integral form\n\\begin{equation}\nE(\\vec{x}_0) \\, u_i(\\vec{x}_0)\n=\n\\int_{\\partial {\\cal D}}\n\\left[ u_j(\\vec{x}) T_{ikj}(\\vec{x},\\vec{x}_0) n_k(\\vec{x}) - G_{ij}(\\vec{x},\\vec{x}_0) \nf_j(\\vec{x}) \\right] dS\n\\label{eqn:BIM}\n\\end{equation}\nwhere $E(\\vec{x_0})=1$ for points belonging to the interior \nof the domain. \n$G_{ij}(\\vec{x},\\vec{x}_0)$ is the \nfree-space Green function, i.e. the $i-$th component of the\nvelocity at $\\vec{x}$ induced by a Dirac $\\delta$ like force\nat $\\vec{x}_0$ acting in direction $j$, and \n$T_{ikj}(\\vec{x},\\vec{x_0})$\nthe associated stress tensor~\\citep{happel1983low}.\nEquation (\\ref{eqn:BIM}) expresses the $i$-th velocity component at \nthe collocation point $\\vec{x}_0$ in the fluid domain ${\\cal D}$.\nIt requires the knowledge of the velocity $u_j(\\vec{x})$ and the stresses $f_j(\\vec{x})$ \nat the boundary $\\partial {\\cal D}$.\nThe integral representation (\\ref{eqn:BIM}) can be turned \ninto a boundary integral equation in the limit $\\vec{x}_0 \\to \\partial {\\cal D}$.\nThe resulting expression can be written in the same form (\\ref{eqn:BIM}) now with\n$E(\\vec{x}_0) = 1\/2$ and the integral understood in the Cauchy principal \nvalue sense.\nWe emphasize that the boundary $\\partial {\\cal D}$ \nconsists of two disjointed parts: the swimmer surface and the planar interface.\nThe integral can be restricted to \nthe swimmer's surface by exploiting the symmetries \nassociated with the boundary condition at the planar interface. \nThis is tantamount to \nthe use of the appropriate  Green's function, see appendix~\\ref{appA}, for the free-slip case \nand \\cite{blake1971note} for the no-slip case.\n\n\nDue to the boundary condition~(\\ref{eqn:swim_velocity}) \nthe fluid velocity at the swimmer surface is \nexpressed in term of seven unknowns, namely, \nthe velocity of the junction point $\\vec{U}$, the angular velocity\n$\\vec{\\Omega}_H$, and the tail rotation velocity $\\Omega_T$,\nwhich together with the tension $\\vec{f}$ at \nthe swimmer surface complete the set of unknowns. \nThe system of equations consists of the vector boundary \nintegral equation (\\ref{eqn:BIM}), the global force and \ntorque balance (\\ref{eqn:balances}), and the \ntorque balance for the swimmer tail (\\ref{eqn:Torque}).\nIt can be shown that the solution exists and is unique\n\\citep{ladyzhenskaya1969mathematical}.\n \nThe above system is discretized by means of $N$ curved 6-point elements \n(typically $N = 3518$ total elements,  with $N_H =  512$ panels on the swimmer head), \nas sketched in figure~\\ref{fig:pitch_angle}, with piecewise constant shape \nfunctions~\\citep{pozrikidis1992boundary}.  By selecting the center of each element as \na collocation point $\\vec{x_0}$, the boundary integral equation~(\\ref{eqn:BIM}) with \n$E(\\vec{x_0})=1\/2$ is recast in a system of $3N$ scalar equations for the stresses.\nAs will be shown in \\S~\\ref{sec:FreeSurf} the swimmer may \napproach the interface. In this case some of the panels belonging to the swimmer and to \nits image may get very close. This potentially spoils the accuracy of the integrals\nproviding the influence coefficients appearing in the discrete equations.\nCare is taken in the simulations discussed below to have the diameter of \nthe typical panel smaller than the distance between the body and its image\nthereby preventing the undesired loss of accuracy.\nConcerning the solution of the algebraic system, inverting a matrix coming from \nthe discretization of boundary integral \nequations may require some care, since the algebraic system may be prone to severe \nill-conditioning. An example is provided by the piecewise constant  approximation of \na first kind Fredholm operator  \\citep{hsiao1973solution}. \nIn the context of the Stokes equations,  such operator arises when solving for \nthe stresses given the velocity.  \nHowever, the problem of the swimmer  addressed in the present paper \ndoes not involve  a pure first kind Fredholm operator, since the equations are augmented by  \nthe free-body constrains (zero net forces and torques) and by the equation enforcing the internal torque, with \nunknowns the stresses on the body plus the rigid body velocities and the relative  tail-body rotation rate. \nThe spectral properties of the ensuing matrix differ substantially from those of a pure first kind Fredholm equation. \nIn fact, the analysis based on the singular value decomposition, \\citep{golub2012matrix}, excludes matrix ill-conditioning.\nAs a consequence,  the complete system of $3N+7$ algebraic equations for the  $3N+7$ unknowns collectively denoted  $\\boldsymbol{\\chi}$, \n$\\vec{A} \\cdot \\boldsymbol{\\chi} = \\vec{b}$,  can be solved by standard techniques, like the Gauss-Jacobi method here implemented in \nan in-house MPI parallel  code.\nOnce the solution $\\boldsymbol{\\chi}$ is obtained, the integration of the kinematic  equations~(\\ref{eqn:kinematic1}),~(\\ref{eqn:kinematic2}),\n~(\\ref{eqn:kinematic3}),  performed via  a third order low-storage Runge-Kutta method, allows to track the swimmer trajectory.\n\n\\section{Swimming close to a free surface: results and discussion}\\label{sec:FreeSurf}\n\n \\begin{figure}\n  \\subfigure[]{\\includegraphics[width=0.495\\textwidth]{fig3a.pdf}} \n  \\subfigure[]{\\includegraphics[width=0.495\\textwidth]{fig3b.pdf}}\n  \\subfigure[]{\\includegraphics[width=0.495\\textwidth]{fig3c.pdf}}\n  \\subfigure[]{\\includegraphics[width=0.495\\textwidth]{fig3d.pdf}}\n  \\caption{\n  Panel~(\\textit{a}): phase plane $\\hat{\\Theta}-\\hat{h}$ for a microswimmer\n  close to a rigid no-slip wall. The curvature radius of the trajectory,\n  equation (\\ref{eqn:traj_radius}), is provided by the color map.\n  Due to the presence of the wall the blanked regions are forbidden to\n  the swimmer. Three typical trajectories $I$, $II$ and $III$ are\n  shown by the solid black lines. Trajectories\n  $I$ and $II$ converges to the stable point. The corresponding \n  trajectory in physical space is shown in panel~(\\textit{b}) \n  where the curvature radius in the $XY-$plane is colour coded.\n  Results for the free-slip interface are shown in\n  panels (\\textit{c}) and (\\textit{d}). \n  All trajectories eventually intersect the planar interface \n  and no stable orbit exists. \n  Note the opposite sign of the curvature radius in comparison with the\n  no-slip case.\n  }\n  \\label{fig:NS_maptraj}\n \\end{figure}\n\n\\subsection{Phase plane analysis and trajectories for a no-slip wall}\n\\label{ssec:PhField}\n\nThe interaction between the swimmer\nand a homogeneous planar surface is completely determined by three parameters:  \nthe distance $h$ between the junction and the surface, the flagellum rotation \nphase angle $\\phi$ and the pitch angle $\\Theta$ between the swimmer longitudinal \naxis $\\vec{e_1}$ and the surface plane, see figures~\\ref{fig:geometry} \nand~\\ref{fig:pitch_angle}. \n$\\phi$ is a fast variable, thus, following~\\cite{shum2010modelling}, \nthe kinematics of the system can be described in term of \n$\\phi -$averaged quantities, in the following denoted by a circumflex. \nThe kinematics in the reduced $(\\hat{\\Theta},\\hat{h})$\nspace is ruled by \n\\begin{eqnarray}\n\\dot{\\hat{\\Theta}} &=& \\hat{\\Omega}_2 \\,\\, ,\\\\\n\\dot{\\hat{h}} &=& \\hat{U}_Z \\,\\, , \n\\label{eqn:Theta_h_punto}\n\\end{eqnarray}\nwhere $\\hat{\\Omega}_2 (\\hat{\\Theta},\\hat{h})\n= \\hat{\\vec \\Omega}_H \\cdot {\\vec{e}_2}$ \nand $\\hat{U}_Z(\\hat{\\Theta},\\hat{h})= \\hat{\\vec{U}} \\cdot {\\vec{Z}}$. \nFigure~\\ref{fig:NS_maptraj}(\\textit{a}) shows the phase plane for a \nmicroswimmer close to a solid no-slip surface.\nThree reduced trajectories, $I$, $II$ and $III$ \nare traced. The trajectory $III$ hits the wall while\ntrajectories $I$ and $II$, although starting from different initial \nconfigurations, both converge to the same attractor. \nThe attractor is an \nasymptotically stable equilibrium point~\\citep{cencini2009chaos}, \n($\\hat{\\Omega}_2$ and $\\hat{U}_Z$ are zero), i.e. \nnearby trajectories converge to the equilibrium point.\n\nThe $\\phi -$averaged trajectory in 3D space corresponding to \n$I$ is obtained by integrating \nthe $\\phi -$averaged version of the \nkinematic equations~(\\ref{eqn:kinematic1}), (\\ref{eqn:kinematic3}) \nand is reported in figure~\\ref{fig:NS_maptraj}(\\textit{b}). Apparently, \nafter a transient, the swimmer stabilizes on a circular clockwise (CW)\norbit which corresponds to the  \nstable point in the reduced space $(\\hat{\\Theta},\\hat{h})$.\nFollowing~\\cite{shum2010modelling}, the curvature radius of the stable orbit \nis given by \n\\begin{equation}\nR = \\frac{|\\vec{\\hat{U}(\\hat{\\Theta},\\hat{h})}|}\n{\\hat{\\Omega}_Z -\\hat{\\Omega}_X \\tan{\\hat{\\Theta}}} \\,\\, ,\n\\label{eqn:traj_radius}\n\\end{equation}\nwhich yields $R=-22.1$. As expected this result \nmatches within numerical accuracy \nits direct measure $R=-21.9$ obtained from the 3D \ntrajectory, see figure~\\ref{fig:NS_maptraj}(\\textit{b}). \nNote that equation (\\ref{eqn:traj_radius}) differs in sign \nfrom the original equation given in~\\cite{shum2010modelling}\ndue to a different choice of the body reference frame. It is also worth \nnoting that equation (\\ref{eqn:traj_radius}) provides the exact curvature radius \nonly for the stable orbit. Nevertheless,\nit also gives a reasonable approximation\nin the other conditions here explored, as discussed below. \nIn figure~\\ref{fig:NS_maptraj}(\\textit{a}) \nthe color map refers to $R$ as estimated from eq.~(\\ref{eqn:traj_radius}). \nThe color code in figure~\\ref{fig:NS_maptraj}(\\textit{b}) \ncorresponds to the local curvature radius of the trajectory \nprojection in the $xy$-plane. This information is also reported along the curve \n$I$ in the phase plane, panel~\\ref{fig:NS_maptraj}(\\textit{a}), through the \ncolor of the open circles superimposed to the trajectory $I$. From the data it \nis apparent that the color differences between the circles\nand the color map in the background can hardly be appreciated. \n\n\\subsection{Swimming in presence of a free-slip interface}\\label{ssec:free_slip}\n\nThe phase plane analysis has been performed for a microswimmer moving \nclose to a free-slip interface, as illustrated in \nfigures~\\ref{fig:NS_maptraj}(\\textit{c}) and~\\ref{fig:NS_maptraj}(\\textit{d}).\nThe first significant result is the sign of the curvature \nradius, now positive, i.e. the swimmer exhibits a counter-clockwise (CCW) motion \nin contrast to what observed for the no-slip wall, compare the panels (\\textit{b}) \nand (\\textit{d}) in figure~\\ref{fig:NS_maptraj}.\nThe curvature radius is smaller with respect to the no-slip case\nat corresponding $(\\hat{\\Theta}, \\hat{h})$. \nThe curvature radius measured in our simulation favorably\ncompares with experimental observation of~\\cite{di2011swimming} where \na CCW motion with a curvature radius of $\\simeq 10 \\mu m$ is described. \nIndeed, assuming $\\bar{a}=1 \\mu m$, appropriate for an E.coli, \nthe typical radius of curvature found from the present numerics, $R \\simeq 7$,\ngives $R' = R \\bar{a} \\simeq 7\\mu m$.\n \nThe second significant result \nis that no stable trajectory exists when the \nswimmer moves close to a free surface. In fact, a swimmer similar to an E.coli moving\nalmost parallel to the free-surface (low $\\hat{\\Theta}$) is always attracted to it. \nThis result confirms the\nindication of the 2D model proposed by~\\cite{crowdy2011two}\nfor undeformable interfaces.\nThis conclusion also partially agrees with the results illustrated by~\\cite{lopez2014dynamics}, \nwhere the authors, basing on a multipole expansion technique, highlighted that the average surface-normal \nvelocity $\\hat{U}_z$ of the swimmer is always negative for $\\hat{\\Theta} = 0$. Our work adds that the surface \nis no more able to attract the swimmer if it swims above a certain height $h$, even with \nzero tilt angle. \n\n\\begin{figure}\n  \\includegraphics[width=0.47\\textwidth]{fig4a.pdf}\n  \\includegraphics[width=0.47\\textwidth]{fig4b.pdf}\n  \\caption{A wider view of the phase plane is displayed for\n  the microswimmer near the no-slip (left) and the free-slip (right)\n  surface. The range exploited in figure~\\ref{fig:NS_maptraj} \n  correspond to the small dashed yellow box.\n  Three regions of the phase plane are identified by colours:\n  a trajectory belonging to the\n  light blue region ends colliding with wall, one in the green\n  region escapes away, and trajectories in the orange region are \n  attracted to the stable orbit. \n  Note that attraction basin is absent in the free-slip case (right).\n  Unstable equilibrium points are marked by the red circles.\n  }\n\\label{fig:large_angles}\n\\end{figure}\n\nIn figure ~\\ref{fig:large_angles} we extend our analysis to a wider range \nof initial conditions, for both no-slip and free-slip surfaces. \nIn both cases, at high positive $\\hat{\\Theta}$ the microswimmer escapes from the \nsurface while, at high negative $\\hat{\\Theta}$, it eventually hits the boundary. \nOn the no-slip wall three cases are possible depending on the initial \nconditions, namely the swimmer achieves a stable orbit, it escapes \nfrom the wall or it collides with the wall. \nIn contrast, for a free-slip surface only \ntwo possibilities exist, namely, \nescaping from the surface or being attracted towards the interface.\nThe maps also highlight the presence of a further equilibrium point \n$(\\hat{\\Theta}_u,\\hat{h}_u)$\n where the time derivatives \nof both ${\\hat{h}}$ and $\\hat{\\Theta}$ vanish. However, as shown \nby the behavior of the streamlines in the neighbor of\n$(\\hat{\\Theta}_u,\\hat{h}_u)$, this \npoint is unstable for both no-slip and free-slip cases, i.e. \nnearby trajectories escape from the equilibrium point.\n\nIn order to extend the analysis, the dynamics of swimmers with different geometrical \ncharacteristics was also examined, focusing in particular on the shape of the head  and the tail (axial) length. \nFigure~\\ref{fig:many_geometries} shows the phase plane $\\hat{\\Theta}-\\hat{h}$\nwhen the tail length $L \\in (3,5,10,15)$ and the head aspect ratio $AR=a_1\/a_2 \\in (1,3,4,5)$. \nWhen changing the aspect ratio, the swimmer does not substantially modify its behavior. In particular,  \nthe orientation as described by the pitch angle $\\hat \\Theta$ does not qualitatively change and the regions of phase space where\nthe swimmer is attracted to the free-surface or where it escapes away from it remain very similar. The general trend is an increase of the curvature of the trajectory for given $\\left({\\hat \\Theta}, \\, {\\hat h}\\right)$ as $AR$ increases.\nOn the other hand,  when changing  the relative tail-to-head length,  the swimmer hits the wall with different pitch angles \n$\\hat \\Theta$.  Decreasing the tail length, the region of phase plane where the swimmer escapes  becomes larger while the region where it is attracted  to\nthe free-surface shrinks.  The curvature of the trajectory, for  given $\\left({\\hat \\Theta}, \\, {\\hat h}\\right)$,  is found to increase with the tail length.\n\\begin{figure}\n  \\includegraphics[width=\\textwidth]{fig5.pdf}\n  \\caption{Phase plane $\\hat{\\Theta}-\\hat{h}$ for a microswimmer\n  close to a to a free-slip interface for different geometries, namely \n  head aspect ratio $AR=a_1\/a_2 \\in (1,3,4,5)$ and  tail length $L \\in (3,5,10,15)$.\n  The color map corresponds to the curvature radius along trajectories (e.g. solid black lines).\n  }\n\\label{fig:many_geometries}\n\\end{figure}\n\n\\subsection{Drift along the trajectory}\\label{ssec:drifting}\n\nTypically the longitudinal axis of  the microswimmer is misaligned with the local velocity evaluated at the junction point.\nThis behavior was experimentally observed for \\textit{E.Coli}\nmoving close to a liquid-air interface by~\\cite{di2011swimming}. \nIn order to compare with experiments we define the drift angle $\\alpha$ as \nthe angle between the ${X,Y}$ projections of the body frame unit vector \n$\\vec{e}_1$ and the swimmer velocity $\\vec{U}$. \nThe sign of $\\alpha$ is taken to be positive when the swimmer points \noutward with respect to the trajectory, see figure~\\ref{fig:drift_traj}a.\n\nIn figure \\ref{fig:drift_traj}b and c, the $\\phi-$averaged \ndrift angle is reported for both no-slip and free-slip cases.\nIn the free-slip case the head points outside the trajectory\nand $\\hat{\\alpha}$ increases as the swimmer approaches the wall.\nThe spanned range of values, $\\hat{\\alpha} \\in (10^\\circ,30^\\circ)$, is \nin good agreement with the experimental observation~\\citep{di2011swimming}.\nIn contrast, in the no-slip case, $\\hat{\\alpha}$ is slightly \nnegative, $\\hat{\\alpha} \\simeq -2^\\circ$, meaning that the swimmer is\nalmost aligned with the trajectory of the reference point $\\vec{x}_J$. \n\n\\begin{figure} \n\\centering\n\\subfigure[]{\\includegraphics[width=0.9\\textwidth]{fig6a.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{fig6b.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{fig6c.pdf}}\n\\caption{Panel (\\textit{a}), sketch illustrating the drift angle\n$\\alpha$. Panels (\\textit{b}) and (\\textit{c}) report \nthe $\\phi-$averaged drift angle $\\hat{\\alpha}$ in the phase-plane (colour code),\nfor the no-slip and free-slip interface, respectively.\nEach inset show a representative trajectory with superimposed the\nlongitudinal unit vector $\\vec{e}_1$ (the color coding corresponds to the\nlocal value of $\\hat{\\alpha}$).\n}\n\\label{fig:drift_traj}\n\\end{figure}\n\nHere, to complete the discussion,  the effect of modifying the tail geometry is briefly addressed.\nIndeed, among the large number of different parameters\ndefining the flagellated geometry, probably the most \nuncertain ones concern the tail, specifically amplitude of the helix, $A$, and\nnumber of turns, $N_\\lambda$.\nFigure~\\ref{fig:imm_grande} reports the radius of curvature $R$ and the drift angle\n$\\hat\\alpha$ for three different tails. The comparison shows that the results discussed so far are generic. \n\n\\begin{figure} \n\\centering\n\\includegraphics[width=\\textwidth]{fig7.pdf}\n\\caption{Different tail geometries. This figure shows the radius of curvature $R$ and the \ndrift angle $\\hat\\alpha$ when modifying the tail amplitude $A$ or the \nnumber of turns $N_\\lambda$. Three cases are reported: \n$A=0.8$ and $N_\\lambda=1$ (left), \n$A=0.4$ and $N_\\lambda=3$ (center), \n$A=0.8$ and $N_\\lambda=5$ (right). \nThe hydrodynamic behavior is qualitative the same observed for the reference configuration\nin figures~\\ref{fig:NS_maptraj} and~\\ref{fig:drift_traj} with \nfew changes in the variables values.}\n\\label{fig:imm_grande}\n\\end{figure}\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nThe motion of a flagellated microswimmer close to a boundary, either a solid wall or \na free surface, is relevant to several applications spanning from micro-robots to biology and medicine, as concerning in particular biofilm formation. Its dynamics can in principle be affected by several physical phenomena occurring at \nthe microscale and, for biological applications, by the behavior of the microorganism.\nThe chemical and physical nature of the interface may play a role in the interaction between the\nswimmer and the surface, e.g. surface charges and chemicals adsorbed at the interface may have a significant influence. \nConcerning the specific case of E. coli, taken as representative of most flagellated, recent experimental studies reported a characteristic motion\nof the swimmer in the two extreme cases of a solid wall and a free surface.  In the first case, the experimental observation\nconsistently show that the microswimmer typically moves in circulatory orbits oriented in clockwise direction (CW)  \\cite{lauga2006swimming}.\nIn contrast, there is evidence that the orientation of the trajectory is reversed (counterclockwise, CCW) when swimming occurs near a free surface \\cite{di2011swimming}.\nHowever, the effect of the free surface is less neatly defined and, sometimes, CW motion is reported, probably due to\nthe presence of contaminants adsorbed at the interface  \\citep{lemelle2010counterclockwise}.\nTheoretical models proposed to explain this behavior typically exploit several form of approximation, e.g. multipole expansion, that may become\npartially inaccurate when the microswimmer gets very close to the interface.\n\nThe present paper provides a complete description of the motion, considering a reasonably realistic geometry of the flagellated in \npresence of a  free-surface modeled as a rigid, free-slip plane.\nThe resulting model removes any concurrent effect, retaining a full hydrodynamics description.\nThe results for the free-slip boundary were compared with those already analyzed in~\\cite{shum2010modelling} for the no-slip case.\nThe  data clearly indicate that the motion close to a  liquid-air interface is CCW, in agreement with  the experimental data~\\citep{di2011swimming}\nand with the theoretical results  obtained by using multipole expansions~\\citep{lopez2014dynamics} and resistive force theory~\\citep{di2011swimming}.\n\nOther available experimental information, namely the orientation of the bacteria with respect  to its trajectory~\\citep{di2011swimming},\nis satisfactorily reproduced by the present simulations, confirming that the head of the swimmer points outward.\nIn contrast, the bacteria is roughly aligned with its trajectory close to a no-slip wall.\nIn principle,  this observable can be used together with the rotation direction to interpret experimental results on the interaction of a \nmicroswimmer with an interface.\nA characteristic aspect of the motion near a solid surface is the occurrence of stable orbits \\citep{giacche2010hydrodynamic,shum2010modelling}.\nTo the contrary, no stable orbit has been presently found on a free-slip interface. \n\nIn conclusion, the boundary conditions are confirmed to deeply influence the hydrodynamical behavior of the swimmer.\nThis consideration paves the way to suitably textured surfaces which, properly engineered  to stably support a \nsuper-hydrophobic state, can be exploited to passively control the microswimmer motion. \nIndeed the ability of superhydrophobic surfaces to alter the flow has been recently used for passive particle separation\n\\citep{asmolov2015principles}. Under this respect fully resolved hydrodynamic simulations able to model complex physical surfaces, \nlike the BEM adopted here, could provide the required fundamental knowledge to extend passive control strategies to active suspensions.\n\\\\\n\nThe authors acknowledge the CINECA Iscra C Award (IscrC-BSM-LAI) for the availability of HPC resources\nand the ERC Grant  No. [339446] {\\bf BIC}: {\\sl Bubbles from Inception to Collapse}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nConsider a set $X$, a group $G$ and a positive integer $m$. An action $G\\times X\\to X$ is said to be $m$-transitive if it is transitive on ordered $m$-tuples of pairwise distinct points in $X$, and is infinitely transitive if it is $m$-transitive for all positive integers $m$.\n\nIt is easy to see that the symmetric group $S_n$ acts $n$-transitively on a set of order $n$, while the action of the alternating group $A_n$ is $(n-2)$-transitive. A generalization of a classical result of Jordan~\\cite{Jo} based on the classification of finite simple groups claims that there\nare no other $m$-transitive finite permutation groups with $m>5$.\n\nClearly, the group $S(X)$ of all permutations of an infinite set $X$ acts infinitely transitively on $X$. The first explicit example of an infinitely transitive and faithful action of the free group $F_n$ with the number of generators $n\\ge 2$ was constructed in~\\cite{McD}; see \\cite{FMS,HO} and references therein for recent results in this direction.\n\nInfinite transitivity on real algebraic varieties was studied in \\cite{HM,HM1,BM,KM}. For multiple transitive actions of real Lie groups on real manifolds, see~\\cite{Bo,Kram}.\n\nA classification of multiple transitive actions of algebraic groups on algebraic varieties over an algebraically closed field is obtained in~\\cite{Kn}. It is shown there that the only 3-transitive action is the action of $\\PGL(2)$ on the projective line $\\PP^1$. Moreover, for reductive groups the only 2-transitive action is the action of $\\PGL(m+1)$ on $\\PP^m$.\n\nIn this paper we consider highly transitive actions in the category of algebraic varieties over an algebraically closed field $\\KK$ of characteristic zero. By analogy with the full permutation group $S(X)$ it is natural to ask about transitivity properties for the full automorphism group $\\Aut(X)$ of an algebraic variety $X$. The phenomenon of infinite transitivity for $\\Aut(X)$ in affine and quasiaffine settings was studied in many works, see~\\cite{Re,KZ,AKZ,AFKKZ,AFKKZ1,FKZ,APS}. The key role here plays the special automorphism group $\\SAut(X)$.\n\nMore precisely, let $\\GG_a$ (resp. $\\GG_m$) be the additive (resp. multiplicative) group of the ground field $\\KK$. We let $\\SAut(X)$ denote the subgroup of $\\Aut(X)$ generated by all algebraic one-parameter unipotent subgroups of $\\Aut(X)$, that is, subgroups in $\\Aut(X)$ coming from all regular actions $\\GG_a\\times X\\to X$.\n\nLet $X$ be an irreducible affine variety of dimension at least 2 and assume that the group $\\SAut(X)$ acts transitively on the smooth locus $X_{\\text{reg}}$. Then \\cite[Theorem~0.1]{AFKKZ} claims that the action is infinitely transitive. This result can be extended to quasiaffine varieties; see \\cite[Theorem~2]{APS} and \\cite[Theorem~1.11]{FKZ}.\n\nWe address the question whether transitivity of $\\SAut(X)$ is the only possibility for the automorphism group $\\Aut(X)$ of an irreducible quasiaffine variety $X$ to act infinitely transitively on $X$. We show that 2-transitivity of the group $\\Aut(X)$ implies transitivity\nof the group $\\SAut(X)$ provided $X$ admits a nontrivial $\\GG_a$- or $\\GG_m$-action; (Theorem~\\ref{tmain} and Corollary~\\ref{ctrans}). We conjecture that the assumption on existence of a nontrivial $\\GG_a$- or $\\GG_m$-action on $X$ is not essential and 2-transitivity of $\\Aut(X)$ always implies transitivity of $\\SAut(X)$ and thus infinite transitivity of $\\Aut(X)$ (Conjecture~\\ref{conj}).\n\nThe quasiaffine case differs from the affine one at least by two properties: the algebra of regular functions $\\KK[X]$ need not be finitely generated and not every locally nilpotent derivation on $\\KK[X]$ gives rise to a $\\GG_a$-action on $X$. These circumstances require new ideas when transferring the proofs obtained in the affine case. Our interest in the quasiaffine case, especially when the algebra $\\KK[X]$ is not finitely generated, is motivated by several reasons.\nHomogeneous quasiaffine varieties appear naturally as homogeneous spaces $X=G\/H$ of an affine algebraic group $G$. By Grosshans' Theorem, the question whether the algebra $\\KK[G\/H]$ is finitely generated is crucial for the Hilbert's fourteenth problem, see~\\cite{Gr} and \\cite[Section~3.7]{PV}. The group $\\Aut(X)$ acts infinitely transitively on $X$ provided the group $G$ is semisimple \\cite[Proposition~5.4]{AFKKZ}. On the other hand, quasiaffine varieties, including the ones with not finitely generated algebra of regular functions, appear as universal torsors $\\widehat{X}\\to X$ over smooth rational varieties $X$ in the framework of the Cox ring theory, see e.g. \\cite[Propositions~1.6.1.6, 4.3.4.5]{ADHL}. By \\cite[Theorem~3]{APS}, for a wide class of varieties $\\widehat{X}$ arising in this construction, the special automorphism group $\\SAut(\\widehat{X})$ acts infinitely transitively on $\\widehat{X}$.\n\nLet us give a short overview of the content of the paper. In Section~\\ref{s1} we recall basic facts on the correspondence between $\\GG_a$-actions on an affine variety $X$ and locally nilpotent derivations of the algebra $\\KK[X]$. Proposition~\\ref{lndga} extends this correspondence to the case when $X$ is quasiaffine.\n\nIn Section~\\ref{s2} we generalize the result of \\cite{AG} on the automorphism group of a rigid affine variety to the quasiaffine case. Recall that an irreducible algebraic variety $X$ is called rigid if $X$ admits no nontrivial $\\GG_a$-action. Theorem~\\ref{trigid} states that the automorphism group of a rigid quasiaffine variety contains a unique maximal torus; the proof is an adaptation of the method of  \\cite[Section~3]{FZ1} to our setting.\n\nAlso we describe all affine algebraic groups which can be realized as a full automorphism group of a quasiaffine variety (Proposition~\\ref{pdref}); the list of such groups turns out to be surprisingly short.\n\nSection~\\ref{s3} contains our main results, Theorem~\\ref{tmain} and Corollary~\\ref{ctrans}. In Corollary~\\ref{cunirat} we observe that if an irreducible quasiaffine variety $X$ admits a nontrivial $\\GG_a$- or $\\GG_m$-action, the group $\\Aut(X)$ acts on $X$ with an open orbit $\\OO$, and the action of $\\Aut(X)$ is 2-transitive on $\\OO$, then $X$ is unirational. This result follows also from~\\cite[Corollary~3]{Po}.\n\nIn the last section we discuss some questions related to Conjecture~\\ref{conj}. We pose a problem on transitivity properties for the automorphism group on a quasiaffine variety with few locally finite automorphisms (Problem~\\ref{p1}) and ask about classification of homogeneous algebraic varieties (Problem~\\ref{p2}).\n\nThe author would like to thank Sergey Gaifullin, Alexander Perepechko, Andriy Regeta and Mikhail Zaidenberg for helpful comments and remarks. Also he is grateful to the anonymous referee for valuable suggestions.\n\n\n\\section{Locally nilpotent derivations and $\\GG_a$-actions} \\label{s1}\n\nIn this section we discuss basic facts on locally nilpotent derivations and $\\GG_a$-actions on quasiaffine varieties; see~\\cite[Section~1.1]{FKZ}, \\cite[Section~2]{APS}, and \\cite{DL} for related results.\n\nLet $A$ be a $\\KK$-domain and $\\partial\\colon A\\to A$ a derivation, i.e., a linear map satisfying the Liebniz rule $\\partial(ab)=\\partial(a)b+a\\partial(b)$ for all\n$a,b\\in A$. The derivation $\\partial$ is called locally nilpotent if for any $a\\in A$ there exists a positive integer $m$ such that $\\partial^m(a)=0$. Let us denote the set of all locally nilpotent derivations of $A$ by $\\LND(A)$. Clearly, if $\\partial\\in\\LND(A)$ and $f\\in\\Ker(\\partial)$, then $f\\partial\\in\\LND(A)$.\n\nEvery locally nilpotent derivation defines\na one-parameter subgroup $\\{\\exp(s\\partial),\\, s\\in\\KK\\}$ of automorphisms of the algebra $A$.\nThis subgroup gives rise to an algebraic action of the group $\\GG_a$ on the algebra $A$.\nThe latter means that every element $a\\in A$ is contained in a finite dimensional $\\GG_a$-invariant subspace $U$ of $A$, and the $\\GG_a$-module $U$ is rational. Conversely, the differential of an algebraic $\\GG_a$-action on $A$ is a locally nilpotent derivation; see \\cite[Section~1.5]{F} for details.\n\nAssume that the domain $A$ is finitely generated and $X=\\Spec(A)$ is the corresponding irreducible affine variety. The results mentioned above establish a bijection between locally nilpotent derivations on $A$ and algebraic actions $\\GG_a\\times X\\to X$. Moreover, the algebra of invariants $A^{\\GG_a}$ coincides with the kernel of the corresponding locally nilpotent derivation.\n\nIf $X$ is an irreducible quasiaffine variety, then again every action $\\GG_a\\times X\\to X$ defines a locally nilpotent derivation of $A:=\\KK[X]$. Since regular functions separate points on $X$,\nsuch a derivation determines a $\\GG_a$-action uniquely. At the same time, not every locally nilpotent derivation of $A$ corresponds to a $\\GG_a$-action on $X$. For example, the derivation\n$\\frac{\\partial}{\\partial x_2}$ of the polynomial algebra $\\KK[x_1,x_2]$ does not correspond to\na $\\GG_a$-action on $X:=\\AA^2\\setminus\\{(0,0)\\}$, while the derivation $x_1\\frac{\\partial}{\\partial x_2}$ does.\n\n\\smallskip\n\nThe following result seems to be known, but for lack of a precise reference we give it with a complete proof.\n\n\\begin{proposition} \\label{lndga}\nLet $X$ be an irreducible quasiaffine variety and $A=\\KK[X]$. Then\n\\begin{enumerate}\n\\item[(i)]\nfor every $\\partial\\in\\LND(A)$ there exists a nonzero $f\\in\\Ker(\\partial)$ such that the locally nilpotent derivation $f\\partial$ corresponds to a $\\GG_a$-action on $X$;\n\\item[(ii)]\nif $\\partial\\in\\LND(A)$ corresponds to a $\\GG_a$-action on $X$, then for every $f\\in\\Ker(\\partial)$ the derivation $f\\partial$ corresponds to a $\\GG_a$-action on $X$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nWe begin with~(i). Fix a derivation $\\partial\\in\\LND(A)$ and the corresponding $\\GG_a$-action on~$A$. Consider an open embedding $X\\hookrightarrow Z$ into an irreducible affine variety $Z$. Fix a finite dimensional $\\GG_a$-invariant subspace $U$ in $A$ containing a set of generators of\n$\\KK[Z]$. Let $B$ be the subalgebra in $A$ generated by $U$ and $Y$ be the affine variety\n$\\Spec(B)$. Since $B$ is $\\GG_a$-invariant, we have the induced $\\GG_a$-action on $Y$. The inclusion $B\\subseteq A$ defines an open embedding $X\\hookrightarrow Y$.\n\n\\begin{claim}\nEvery divisor $D\\subseteq Y$ contained in $Y\\setminus X$ is $\\GG_a$-invariant.\n\\end{claim}\n\n\\begin{proof}\nAssume that the variety $Y$ is normal and take a function $f\\in\\KK(Y)$ which has a pole along\nthe divisor $D$. Multiplying $f$ by a suitable function from $B$ we may suppose that $f$ has\nno pole outside $D$. Then $f$ is contained in $A$. If the divisor $D$ is not $\\GG_a$-invariant,\nthere is an element $g\\in\\GG_a$ such that $g\\cdot D$ intersects $X$. It shows that the function $g\\cdot f$ has a pole on $X$ and thus is not in $A$, a contradiction.\n\nIf $Y$ is not normal, we lift the $\\GG_a$-action to the normalization of $Y$ and apply the same arguments to integral closures of $A$ and $B$.\n\\end{proof}\n\n\\begin{claim}\nThere is an open $\\GG_a$-invariant subset $W\\subseteq Y$ which is contained in $X$.\n\\end{claim}\n\n\\begin{proof}\nLet $F$ be the union of irreducible components of $Y\\setminus X$ of codimension at least $2$.\nThen the closure $\\overline{\\GG_a\\cdot F}$ is a proper closed $\\GG_a$-invariant subset whose complement intersected with $X$ is the desired subset $W$.\n\\end{proof}\n\nLet $Y_0:=Y\\setminus W$. This is a closed $\\GG_a$-invariant subvariety in $Y$ and its ideal\n$I(Y_0)$ in $B$ is a $\\GG_a$-invariant subspace. Applying the Lie-Kolchin Theorem, we find a nonzero $\\GG_a$-invariant function $f\\in I(Y_0)$. Then $f\\in\\Ker(\\partial)$ and the $\\GG_a$-action on $Y$ corresponding to the derivation $f\\partial$ fixes all points outside $W$. In particular,\nthis action induces a $\\GG_a$-action on $X$. This proves~(i).\n\nNow we come to~(ii). Consider the action $\\GG_a\\times X\\to X$ corresponding to $\\partial$.\nBy~\\cite[Theorem~1.6]{PV}, there is an open equivariant embedding $X\\hookrightarrow Y$ into an affine variety $Y$. For any $f\\in\\Ker(\\partial)$, the orbits of the $\\GG_a$-action on $Y$ corresponding to $f\\partial$ coincide with the orbits of the original actions on $Y\\setminus \\{f=0\\}$, while all points of the set $\\{f=0\\}$ become fixed. In particular, this action leaves the set $X$ invariant. This completes the proof of Proposition~\\ref{lndga}.\n\\end{proof}\n\n\\begin{corollary} \\label{corcc}\nLet $X$ be an irreducible quasiaffine variety and $A=\\KK[X]$. The variety $X$ admits a nontrivial $\\GG_a$-action if and only if  there is a nonzero locally nilpotent derivation on $A$.\n\\end{corollary}\n\n\\section{Torus actions on rigid quasiaffine varieties} \\label{s2}\n\nIn this section we generalize the results of \\cite[Section~3]{FZ1} and \\cite[Theorem~1]{AG} to the case of a quasiaffine variety. Let us recall that an irreducible algebraic variety $X$ is called \\emph{rigid}, if it admits no nontrivial $\\GG_a$-action.\n\n\\begin{theorem} \\label{trigid}\nLet $X$ be a rigid quasiaffine variety. There is a subtorus $\\TT\\subseteq\\Aut(X)$ such that\nfor every torus action $T\\times X\\to X$ the image of $T$ in $\\Aut(X)$ is contained in $\\TT$. In other words, $\\TT$ is a unique maximal torus in $\\Aut(X)$.\n\\end{theorem}\n\nLet us begin with some preliminary results.\n\n\\begin{lemma} \\label{lemloc}\nLet $X$ be an irreducible quasiaffine variety and $T\\times X\\to X$ be an action of a torus. Then there is a $T$-semi-invariant $f\\in\\KK[X]$ such that the localization $\\KK[X]_f$ is finitely generated.\n\\end{lemma}\n\n\\begin{proof}\nBy \\cite[Theorem~1.6]{PV}, there exists an open equivariant embedding $X\\hookrightarrow Z$ into an irreducible affine $T$-variety $Z$. Let $I$ be the ideal of the subvariety $Z\\setminus X$ in $\\KK[Z]$. Since $I$ is $T$-invariant, there is a non-constant $T$-semi-invariant $f\\in I$.\nThe principal open subset $Z_f$ is contained in $X$. Since the algebra $\\KK[Z_f]$ is the localization $\\KK[Z]_f$ and $\\KK[X]$ is contained in $\\KK[Z_f]$, we conclude that the algebra\n$\\KK[X]_f=\\KK[Z]_f$ is finitely generated.\n\\end{proof}\n\nLet $A=\\oplus_{i\\in\\ZZ} A_i$ be a graded $\\KK$-algebra and $\\partial\\colon A\\to A$ a derivation. We define a linear map $\\partial_k\\colon A\\to A$ by setting $\\partial_k(a)$ to be\nthe homogeneous component $\\partial(a)_{\\deg(a)+k}$ of the element $\\partial(a)$ for every homogeneous element $a\\in A$. It is easy to check that $\\partial_k$ is a derivation for all $k\\in\\ZZ$. We call it the $k$th homogeneous component of the derivation $\\partial$.\n\n\\begin{proof}[Proof of Theorem~\\ref{trigid}]\nAssume that there are two torus actions $T_i\\times X\\to X$, $i=1,2$, such that the images of $T_i$ in $\\Aut(X)$ are not contained in some torus $\\TT$. The latter means that the actions do not commute. We may assume that $T_1$ and $T_2$ are one-dimensional. Let $A:=\\KK[X]$ and\n$$\nA=\\bigoplus_{u\\in\\ZZ} A_u \\quad \\text{and} \\quad A=\\bigoplus_{u\\in\\ZZ} A_u'\n$$\nbe gradings corresponding to the actions of $T_1$ and $T_2$, respectively. Consider semisimple derivations $\\partial$ and $\\partial'$ on $A$ defined by $\\partial(a)=ua$ for every $a\\in A_u$ and\n$\\partial'(b)=ub$ for every $b\\in A_u'$.\n\nLet $\\partial'_k$ be the $k$th homogeneous component of $\\partial'$ with respect to the first grading. We claim that there are only finitely many nonzero homogeneous components and thus\nthe sum\n$$\n\\partial'=\\sum_{k\\in\\ZZ} \\partial'_k\n$$\nhas only finite number of nonzero terms.\n\nConsider a localization $\\KK[X]_f$ from Lemma~\\ref{lemloc}, where $f$ is homogeneous with respect to the first grading. The algebra $\\KK[X]_f$ is generated by some elements $f_1,\\ldots,f_k\\in \\KK[X]$, which are homogeneous with respect to the first grading, and the element $\\frac{1}{f}$.\n\nSince $\\KK[X]$ is contained in $\\KK[X]_f$, every element $h\\in\\KK[X]$ is a linear combination of elements of the form\n$$\n\\frac{f_1^{a_1}\\ldots f_k^{a_k}}{f^a}\n$$\nand the image $\\partial'(h)$ is a linear combination of the elements\n$$\n\\sum_s\\frac{a_s\\partial'(f_s)f_1^{a_1}\\ldots f_s^{a_s-1}\\ldots f_k^{a_k}}{f^{a}}-\\frac{a\\partial'(f)f_1^{a_1}\\ldots f_k^{a_k}}{f^{a+1}}.\n$$\nIt shows that the shift of degree with respect to the first grading from $h$ to $\\partial'(h)$ does not exceed the maximal shift of degree for $f_1,\\ldots,f_k,f$. Hence the shift is bounded and we obtain the claim.\n\nLet $\\partial_m'$ be a nonzero homogeneous component of $\\partial'$ with maximal absolute value\nof the weight $m$. Since the derivations $\\partial$ and $\\partial'$ do not commute, we have $m\\ne 0$. Then $(\\partial_m')^r(a)$ is the highest (or the lowest) homogeneous component of the element $(\\partial')^r(a)$ for every homogeneous $a\\in A$. Since $a$ is contained in a finite dimensional $\\partial'$-invariant subspace in $A$, the elements $(\\partial')^r(a)$ cannot have nonzero projections to infinitely many components~$A_u$. Thus $(\\partial_m')^r(a)=0$ for $r\\gg 0$. We conclude that $\\partial_m'$ is a nonzero locally nilpotent derivation of the algebra $A$. By Corollary~\\ref{corcc}, we obtain a contradiction with the condition that $X$ is rigid.\n\\end{proof}\n\n\\begin{corollary}\nIn the setting of Theorem~\\ref{trigid}, the maximal torus $\\TT$ is a normal subgroup of $\\Aut(X)$.\n\\end{corollary}\n\nLet us finish this section with a description of affine algebraic groups which can be realized as automorphism groups of quasiaffine varieties. When this paper was already written, I found the same result in \\cite[Theorem~1.3]{Kr}, cf. also \\cite[Theorem~4.10~(a)]{LZ}.\n\n\\begin{proposition} \\label{pdref}\nLet $X$ be an irreducible quasiaffine variety. Assume that the automorphism group $\\Aut(X)$ admits a structure of an affine algebraic group such that the action $\\Aut(X)\\times X\\to X$ is a morphism of algebraic varieties. Then either $\\Aut(X)$ is finite, or isomorphic to a finite extension of a torus, or isomorphic to the linear group\n$$\nG=\\left\\{\n\\left(\n\\begin{array}{cc}\n1 & 0 \\\\\na & t\n\\end{array}\n\\right), \\ \\ a\\in\\KK, \\ t\\in\\KK^{\\times}\n\\right\\}.\n$$\n\\end{proposition}\n\n\\begin{proof} We assume first that $X$ is a rational curve. If $X=\\AA^1$ then $\\Aut(X)$ is isomorphic to the group $G$. If $X$ is $\\AA^1$ with one point removed, then $\\Aut(X)$ is an extension of 1-torus. If we remove more than one point from $\\AA^1$, the group $\\Aut(X)$ becomes finite. For a singular rational curve $X$, the automorphism group $\\Aut(X)$ lifts to normalization and preserves the preimage of the singular locus. Thus $\\Aut(X)$ is contained in an extension of  1-torus.\n\nIt follows from the description of the automorphism group of an elliptic curve and from Hurwitz's Theorem that the automorphism group of an affine curve $X$ of positive genus is finite.\n\nNow let us assume that $\\dim X\\ge 2$. If $X$ is rigid then the affine algebraic group $\\Aut(X)$\ncontains no one-parameter unipotent subgroup. It means that the unipotent radical and the semisimple part of $\\Aut(X)$ are trivial. Hence $\\Aut(X)$ is either finite or a finite extension of a torus.\n\nFinally, let $\\GG_a\\times X\\to X$ be a non-trivial action and $\\partial\\in\\LND(\\KK[X])$ the corresponding locally nilpotent derivation. By~\\cite[Principle~11]{F}, the transcendence degree\nof the algebra $\\Ker(\\partial)$ equals $\\dim(X)-1\\ge 1$. Let $U$ be a subspace in\n$\\Ker(\\partial)$. Proposition~\\ref{lndga},~(ii) implies that the automorphisms $\\exp(f\\partial)$, $f\\in U$, form a commutative unipotent subgroup in $\\Aut(X)$ of dimension $\\dim(U)$. Since  $\\dim(U)$ may be arbitrary, the group $\\Aut(X)$ does not admit a structure of an affine algebraic group.\n\\end{proof}\n\n\\begin{remark}\nMany examples of affine algebraic varieties whose automorphism group is a finite extension of a torus are provided by trinomial hypersurfaces, see~\\cite[Theorem~3]{AG}.\n\\end{remark}\n\n\\begin{remark}\nThe class of affine algebraic groups which can be realized as the automorphism groups of complete\nvarieties is much wider. For example, the automorphism group of a complete toric variety is always an affine algebraic group of type A. A description of such groups is given in \\cite{De,Cox}. Some other affine algebraic groups appear as the automorphism groups of Mori Dream Spaces; see e.g.\n\\cite[Theorem~7.2]{AHHL}. It is shown in~\\cite[Theorem~1]{Br} that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal projective variety.\n\\end{remark}\n\n\\section{Main results} \\label{s3}\n\nWe come to a characterization of transitivity properties for the automorphism group $\\Aut(X)$ in terms of the special automorphism group $\\SAut(X)$.\n\n\\begin{theorem} \\label{tmain}\nLet $X$ be an irreducible quasiaffine variety of dimension at least $2$. Assume that $X$ admits a nontrivial $\\GG_a$- or $\\GG_m$-action and the group $\\Aut(X)$ acts on $X$ with an open orbit $\\OO$. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item\nThe group $\\Aut(X)$ acts 2-transitively on $\\OO$.\n\\item\nThe group $\\Aut(X)$ acts infinitely transitively on $\\OO$.\n\\item\nThe group $\\SAut(X)$ acts transitively on $\\OO$.\n\\item\nThe group $\\SAut(X)$ acts infinitely transitively on $\\OO$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof} Let us prove implications $(1)\\Rightarrow (3)\\Rightarrow (4) \\Rightarrow (2) \\Rightarrow (1)$. Implications ${(4)\\Rightarrow (2)\\Rightarrow (1)}$ are obvious. Implication $(3)\\Rightarrow (4)$ is proved in \\cite[Theorem~2.2]{AFKKZ} for $X$ affine and in \\cite[Theorem~2]{APS}, \\cite[Theorem~1.11]{FKZ} for $X$ quasiaffine.\n\nIt remains to prove $(1)\\Rightarrow (3)$\\footnote{This is the only implication where we use the condition on $\\GG_a$- or $\\GG_m$-action.}. Assume first that there is a nontrivial $\\GG_a$-action on~$X$. Let us take two distinct points $x_1$ and $x_2$ in $\\OO$ on one $\\GG_a$-orbit. By assumption, for every distinct points $y_1,y_2\\in\\OO$ there exists an automorphism $\\varphi\\in\\Aut(X)$ with\n$\\varphi(x_i)=y_i$, $i=1,2$. Then the points $y_1$ and $y_2$ lie in the same orbit for the $\\GG_a$-action obtained from the initial one by conjugation with $\\varphi$. It means that the group $\\SAut(X)$ acts transitively on $\\OO$.\n\nNow assume that $X$ is rigid and admits a nontrivial $\\GG_m$-action. If the maximal torus $\\TT$ from Theorem~\\ref{trigid} acts transitively on $\\OO$, then $\\OO$ is isomorphic to the torus $\\TT$ and $\\Aut(X)$ acts on $\\OO$ transitively, but not 2-transitively. Indeed, let us fix an isomorphism between $\\OO$ and $(\\KK^{\\times})^n$. The group $\\Aut(\\OO)$ is isomorphic to a semidirect product of $\\TT$ and the group $\\GL_n(\\ZZ)$. It shows that the stabilizer in $\\Aut(\\OO)$ of the unit in $(\\KK^{\\times})^n$ preserves the set of points with rational coordinates. Consequently, the group $\\Aut(\\OO)$, and thus the group $\\Aut(X)$, cannot act 2-transitively on $\\OO$.\n\nNow assume that the action of $\\TT$ is not transitive on $\\OO$. Let us take points\n$x_1,x_2,x_3\\in\\OO$ such that $x_1\\ne x_2$ lie in the same $\\TT$-orbit and $x_3$ belongs to other $\\TT$-orbit. By Corollary~\\ref{corcc}, every automorphism of $X$ permutes $\\TT$-orbits on $X$ and thus there is no automorphism preserving $x_1$ and sending $x_2$ to $x_3$, a contradiction with 2-transitivity.\n\nThis completes the proof of Theorem~\\ref{tmain}.\n\\end{proof}\n\n\\begin{remark}\nImplication $(1)\\Rightarrow (3)$ for an affine variety $X$ admitting a nontrivial $\\GG_a$-action was observed earlier in~\\cite{BGT}.\n\\end{remark}\n\n\\begin{corollary} \\label{ctrans}\nLet $X$ be an irreducible quasiaffine variety of dimension at least $2$. Assume that $X$ admits a nontrivial $\\GG_a$- or $\\GG_m$-action. Then the following conditions are equivalent.\n\\begin{enumerate}\n\\item\nThe group $\\Aut(X)$ acts 2-transitively on $X$.\n\\item\nThe group $\\Aut(X)$ acts infinitely transitively on $X$.\n\\item\nThe group $\\SAut(X)$ acts transitively on $X$.\n\\item\nThe group $\\SAut(X)$ acts infinitely transitively on $X$.\n\\end{enumerate}\n\\end{corollary}\n\nWe recall that the \\emph{Makar-Limanov invariant} $\\text{ML}(A)$ of an algebra $A$ is the intersection of kernels of all locally nilpotent derivations on $A$. Using Proposition~\\ref{lndga}, one can easily show that the Makar-Limanov invariant $\\text{ML}(\\KK[X])$ of the algebra of regular functions on an irreducible quasiaffine variety $X$ coincides with the algebra of invariants $\\KK[X]^{\\SAut(X)}$ of the special automorphism group. We denote $\\text{ML}(\\KK[X])$ just by $\\text{ML}(X)$. Note that a~quasiaffine variety $X$ is rigid if and only if $\\text{ML}(X)=\\KK[X]$.\n\nIn \\cite{Lie}, a field version of the Makar-Limanov invariant is introduced. Namely,\nthe \\emph{field Makar-Limanov invariant} $\\text{FML}(X)$ of an irreducible quasiaffine variety $X$ is the subfield of $\\KK(X)$ consisting of all rational $\\SAut(X)$-invariants. The condition $\\text{FML}(X)=\\KK$ implies $\\text{ML}(X)=\\KK$, but the converse is not true in general. By \\cite[Corollary~1.14]{AFKKZ}, we have $\\text{FML}(X)=\\KK$ if and only if the group $\\SAut(X)$ acts on $X$ with an open orbit. In this case the variety $X$ is unirational \\cite[Proposition~5.1]{AFKKZ}. Together with Theorem~\\ref{tmain} this yields the following result.\n\n\\begin{corollary} \\label{cunirat}\nLet $X$ be an irreducible quasiaffine variety. Assume that $X$ admits a nontrivial $\\GG_a$- or $\\GG_m$-action and the group $\\Aut(X)$ acts on $X$ with an open orbit $\\OO$. If the group $\\Aut(X)$ is 2-transitive on $\\OO$, then $X$ is unirational.\n\\end{corollary}\n\n\\begin{remark}\nCorollary~\\ref{cunirat} is a particular case of \\cite[Theorem~5]{Po}. The latter theorem claims that if $X$ is an irreducible variety, the group $\\Aut(X)$ acts generically 2-transitive on $X$, and $\\Aut(X)$ contains a non-trivial connected algebraic subgroup, then $X$ is unirational. Moreover, if $X$ is irreducible, complete, and the group $\\Aut(X)$ acts generically 2-transitive on $X$, then $X$ is unirational \\cite[Corollary~3]{Po}.\n\\end{remark}\n\nLet us finish this section with the following conjecture.\n\n\\begin{conjecture} \\label{conj}\nConditions (1)-(4) of Theorem~\\ref{tmain} are equivalent for any irreducible quasiaffine variety $X$ of dimension at least $2$.\n\\end{conjecture}\n\n\\begin{remark}\nJelonek~\\cite{Je} has proved that every quasiaffine variety $X$ with an infinite automorphism\ngroup is uniruled, i.e., for a generic point in $X$ there exists a rational curve in $X$ through this point.\n\\end{remark}\n\n\\section{Concluding remarks and questions} \\label{s4}\n\nIn this section we discuss some results and questions related to Conjecture~\\ref{conj}.\nLet $\\phi$ be an automorphism of a quasiaffine variety $X$ and $\\phi^*$ be the induced\nautomorphism of the algebra $\\KK[X]$. We say that $\\phi$ is \\emph{locally finite} if every element\nof $\\KK[X]$ is contained in a finite dimensional $\\phi^*$-invariant subspace.\n\nThe following fact is well known to experts, but for the convenience of the reader we give it with a short proof.\n\n\\begin{proposition}\nLet $X$ be an irreducible quasiaffine variety and $\\phi$ an automorphism of~$X$. The following conditions are equivalent.\n\\begin{enumerate}\n\\item[(1)]\nThere exists a regular action $G\\times X\\to X$ of an affine algebraic group $G$ on $X$ such that $\\phi$ is contained in the image of $G$ in the group $\\Aut(X)$.\n\\item[(2)]\nThe automorphism $\\phi$ is locally finite.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFor implication $(1)\\Rightarrow (2)$, see e.g. \\cite[Lemma~1.4]{PV}. Conversely, assume that\n$\\phi$ is locally finite and let $U$ be a finite-dimensional $\\phi^*$-invariant subspace in $\\KK[X]$ which generates a subalgebra $A$ in $\\KK[X]$ such that the morphism $X\\to Z:=\\Spec(A)$ is an open embedding. Let $G$ be the subgroup of all automorphisms of $X$ that preserve the subspace~$U$. Since $U$ generates the field $\\KK(X)$, the group $G$ is a subgroup of the general linear group~$\\GL(U)$. Moreover, every element of $G$ induces an automorphism of $Z$. The subgroup $G'$ of all elements of $\\GL(U)$ which induce an automorphism of $Z$ is closed in $\\GL(U)$. The subgroup $G$ of $G'$ consists of automorphisms of $Z$ which preserve the (closed) subvariety $Z\\setminus X$. This proves that $G$ is an affine algebraic group.\n\\end{proof}\n\n\\begin{remark}\nFor further characterizations of automorphisms belonging to algebraic subgroups of $\\Aut(X)$, see~\\cite{Ra}.\n\\end{remark}\n\nClearly, every automorphism of finite order is locally finite. The condition that a quasiaffine variety $X$ admits no nontrivial actions of the groups $\\GG_a$ and $\\GG_m$ means that every locally finite automorphism of $X$ has finite order.\n\n\\begin{problem} \\label{p1}\nLet $X$ be an irreducible quasiaffine variety such that every locally finite automorphism of $X$ has finite order. Can the group $\\Aut(X)$ act transitively (2-transitively, infinitely transitively) on $X$?\n\\end{problem}\n\nLet us give examples of automorphisms which are not locally finite. Let $X$ be a 2-torus with the algebra of regular functions\n$\\KK[X]=\\KK[T_1,T_1^{-1},T_2,T_2^{-1}]$. Then the map\n$$\n\\phi\\colon (t_1,t_2) \\mapsto (t_1t_2,t_2)\n$$\nis an automorphism of $X$ and the function $T_1$ is not contained in a finite dimensional $\\phi^*$-invariant subspace of $\\KK[X]$.\n\nAn automorphism of the affine plane $\\AA^2$ which is not locally finite may be given as\n$$\n(x,y)\\mapsto (x+y^2, x+y+y^2).\n$$\n\nMore examples of automorphisms which are not locally finite can be found in~\\cite{BD}. The authors describe a family of rational affine surfaces $S$ such that the normal subgroup $\\Aut(S)_{\\text{alg}}$ of $\\Aut(S)$ generated by all algebraic subgroups of $\\Aut(S)$ is not generated by any countable family of such subgroups, and the quotient $\\Aut(S)\/\\Aut(S)_{\\text{alg}}$ contains a free group over an uncountable set of generators. A description of automorphisms in~\\cite{BD} is given in a purely geometric terms. It seems to be an important problem to find more methods for constructing automorphisms of quasiaffine varieties which are not locally finite.\n\nWorking with Conjecture~\\ref{conj}, one may wish to replace an arbitrary quasiaffine variety by a quasiaffine variety admitting a nontrivial $\\GG_a$- or $\\GG_m$-action. For example, let $X$ be an irreducible quasiaffine variety such that the group $\\Aut(X)$ is 2-transitive on $X$. Is it true that the group $\\Aut(X\\times\\AA^1)$ is 2-transitive on $X\\times\\AA^1$? This question is related to algebraic families of automorphisms in the sense of~\\cite{Ra}.\n\n\\smallskip\n\nLet us finish this section with a general problem on transitivity for algebraic varieties. We say that an algebraic variety $X$ is \\emph{homogeneous} if the group $\\Aut(X)$ acts transitively on~$X$. A wide class of homogeneous varieties form homogeneous spaces of algebraic groups. At~the same time, not every homogeneous variety is homogeneous with respect to an algebraic group; an example of a homogeneous quasiaffine toric surface which is not a homogeneous space of an algebraic group is given in~\\cite[Example~2.2]{AKZ}. More generally, it follows from ~\\cite[Theorem~2.1]{AKZ} that every smooth quasiaffine toric variety is homogeneous. We plan to describe all homogeneous toric varieties in a forthcoming publication.\n\n\\begin{problem} \\label{p2}\nDescribe all homogeneous algebraic varieties.\n\\end{problem}\n\nConjecture~\\ref{conj} can be considered as a first step towards the solution of this problem.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{introduction}\nDatabase systems (DBs) are largely embracing ML. With data volumes reaching unprecedented levels, ML can provide highly-accurate methods to perform central data management tasks more efficiently. Applications abound: AQP engines are leveraging ML to answer queries much faster and more accurately than traditional DBs \\cite{ma2019dbest,hilprecht2019deepdb,thirumuruganathan2020approximate,ma2021learned}.\nCardinality\/selectivity estimation, has improved considerably leveraging ML  \\cite{yang2019deep,yang2020neurocard,hasan2020deep,zhu2020flat,wang2020we}. Likewise for query optimization \n\\cite{marcus2019neo,kipf2018learned,marcus2021bao},\nindexes \\cite{kraska2018case,ding2020alex,nathan2020learning,ding2020tsunami}, cost estimation \\cite{zhi2021efficient, siddiqui2020cost}, workload forecasting \\cite{zhu2019novel}, DB tuning \\cite{van2017automatic,li2019qtune,zhang2019end}, synthetic data generation \\citep{xu2019modeling,choi2017generating,park2018data}, etc. \n\n\\subsection{Challenges}\\label{challenges}\nAs research in learned DB systems\nmatures, two key pitfalls are emerging. First, if the \"context\" (such as the data, the DB system, and\/or the workload) changes, previously trained models are no longer accurate. Second, training accurate ML models is costly. Hence, retraining from scratch when the context changes should be avoided whenever possible.\nEmerging ML paradigms, such as active learning, transfer learning, meta-learning, and zero\/few-shot learning are a good fit for such context changes and have been the focus of recent related works   \\cite{ma2020active, hilprecht2021one, wu2021unified}, where the primary focus is to glean what is learned from existing ML models (trained for different learning tasks and\/or DBs and\/or workloads), \nand adapt them for new tasks and\/or DBs, and\/or workloads, while avoiding the need to retrain models from scratch.\n\n{\\bf OOD Data insertions.} In analytical DBs data updates primarily take the form of new data insertions. New data may be OOD (representing new knowledge --  distributional shifts), rendering previously-built ML models obsolete\/inaccurate.\nOr, new data may not be OOD. In the former case, the model must be updated and it must be decided how the new data could be efficiently reflected in the model to continue ensuring accuracy.\nIn the latter case, it is desirable to avoid updating the model, as that would waste time\/resources.\nTherefore, it is also crucial to check (efficiently) whether the new data render the previously built model inaccurate. \nHowever, related research has not yet tackled this problem setting, whereby\n\\textit{models for the same learning tasks (e.g., AQP, DG, CE, etc.) trained on old data, continue to provide high accuracy for the new data state} (on old and new data, as queries now may access both old data and new data, old data, or simply the new data).\nRelated work for learned DB systems have a limited (or sometimes completely lack the) capability of handling such data insertions (as is independently verified in \\cite{wang2020we} and will be shown in this paper as well).\n\n{\\bf Sources of Difficulty and Baselines.} \nIn the presence of OOD, a simple solution is adopted by some of the learned DB components  like Naru \\cite{yang2019deep}, NeuroCard \\cite{yang2020neurocard}, DBest++ \\cite{ma2021learned}, and even the aforementioned transfer\/few-shot learning methods \\cite{wu2021unified, hilprecht2021one}. That is to \"fine-tune\" the original model $M$ on the new data. Alas, this is problematic. For instance, while a DBest++ model on the \"Forest\" dataset has a 95th percentile q-error of 2, updating it with an OOD sample using fine-tuning increases the 95th q-error to ~63. A similar accuracy drop occurs for other key models as well -- \\cite{wang2020we} showcases this for learned CE works.\nThis drastic drop of accuracy is due to the fundamental problem of \\textit{catastrophic forgetting}{} \\cite{mccloskey1989catastrophic}, where retraining a previously learned model on new tasks, i.e. new data, causes the model to lose the knowledge it had acquired about old data. To avoid \\textit{catastrophic forgetting}{}, Naru and DBest++ suggest using a smaller learning rate while fine-tuning with the new data. This, however, causes another fundamental problem, namely \\textit{intransigence}, \\cite{chaudhry2018riemannian} whereby the model resists fitting to new data, rendering queries on new data inaccurate.\n\nAnother simple solution to avoid these problems would be to aggregate the old data and new data and retrain the model from scratch. However, as mentioned, this is undesirable in our environment. As a concrete example, training Naru\/NeuroCard on the \"Forest\" dataset (with only 600k rows) on a 40-core CPU takes ca. 1.5 hours. Similarly high retraining overheads are typically observed for neural network models, for various tasks.\nAnd, retraining time progressively increases as the DB size increases. \n\nTherefore, more sophisticated approaches are needed, which can avoid \\textit{intransigence} and \\textit{catastrophic forgetting}{},\nupdate models only when needed and do so while ensuring much smaller training overheads than retraining from scratch and at the same time ensure high accuracy for queries on old and new data. While for some tasks, like CE, some researchers question whether achieving very high accuracy through learned models will actually help the end-task (query optimization) \\cite{marcus2021bao}, for tasks like AQP (which is itself the end-task) and for DG (with classification as the end-task) high accuracy is clearly needed, as shown here. Even for CE, with OOD data, accuracy can become horribly poor, as shown here, which is likely to affect query optimization.\n\n\\subsection{Contributions} \\label{contribution}\nTo the best of our knowledge, this work proposes the first updatability framework (DDUp) for learned DBs (in the face of new data insertions possibly carrying OOD data)\nthat can ensure high accuracy for queries on new and\/or old data. \nDDUp is also efficient and \nit can enjoy wide applicability, capable of being utilized for different NNs and\/or  different learning tasks (such as AQP, DG, CE, etc.). DDUp consists of a novel OOD detection and a novel model-update module. More specifically, the contributions of DDUp are:\n\n\\begin{itemize}[leftmargin=10pt]\n    \\item A general and principled two-sample test for OOD detection. Generality stems from it being based on the training loss function of the NNs. Compared to prior art, it introduces no extra costs and overheads, and could be used with different NNs, with different loss functions, in different applications. To further minimize detection time, it is divided into offline and online phases.\n    \\item A novel and general formulation of transfer-learning based on sequential self-distillation for model updating. This formulation allows a higher degree of freedom in balancing tasks w.r.t new and old data, can adapt to different models and tasks, and maximizes performance via self-distillation.\n    \\item Importantly, DDUp can be used by any pre-trained NN without introducing any assumptions on models or requiring additional components that might require to retrain models or incur more costs. Here, we instantiate it for three different tasks (namely, the CE task, using the Naru\/NeuroCard deep autoregressive network (DARN) models  \\cite{yang2019deep, yang2020neurocard},  the AQP task, using the DBEst++ mixture density network (MDN) model \\cite{ma2021learned}, and for the DG task, using the Tabular Variational AutoEncoder (TVAE) model \\cite{xu2019modeling}) each of which employs a different NN type. These are representative learning tasks and networks with evident importance in DBs and beyond. These instantiations are also novel, showing how to distil-and-update MDNs, DARNs, and TVAEs.\n    \\item Finally, DDUp is evaluated using six different datasets and the three instantiated learned DB components, for AQP, CE, and DG\n\\end{itemize}\n\n\n\\subsection{Limitations} \\label{limits}\nDDUp focuses only on data insertions, which are essential and dominant in analytical DBs, and not on updates in place and deletes, which are prevalent \nin transactional DBs.\nNonetheless, the latter touch upon an open problem in the ML literature, namely $\"unlearning\"$, \nwhere it typically concerns privacy (e.g., removing sensitive data from images in classification tasks) \n(e.g., \\citep{sekhari2021remember, golatkar2020eternal}).\nStudying unlearning for DB problem settings is a formidable task of its own and of high interest for future research.\n\n\nAlso, DDUp is designed for NN-based learned DB components. This is so as neural networks are a very rich family of models which have collectively received very large attention for learned DBs. Extending DDUp principles beyond NN models is also left for future research.\n\n\n\\section{The Problem and Solution Overview} \\label{problemdef}\n\\subsection{Problem Formulation} \\label{problemformulation}\nConsider a database relation \\(R\\) with attributes \\(\\{A_1, A_2, ..., A_m\\}\\). This can be a raw table or the result of a join query. Also consider a sequence of \\(N\\) insertion updates denoted by \\(I=\\{I_1,I_2,...I_N\\}\\). Each \\(I_t\\) is an insert operation which appends a data batch \\(D_t=\\{(A_1, A_2, ..., A_m)_t^{(i)}; i=1,..., n_t\\}\\) to \\(R\\), where \\(n_t\\) is the number of rows. Let \\(S_t\\) be a sufficient sample of \\(D_t\\) and \\(S^{\\leq}_{t-1}\\) be a sufficient sample from \\(\\cup_{j=0}^{t-1} D_j\\). We naturally assume that \\(|R|\\) is finite. \nAnd, due to the training restrictions of existing models, we also make the natural assumption:\n\\[\\forall A_i \\in R: supp(D_{t}(A_i)) \\subseteq supp(D_{t-1}(A_i)) \\]\nwhere \\(supp(D(A_i))\\) is the support of attribute \\(A_i\\) in dataset \\(D\\). This assumption satisfies the condition based on which the domain of each attribute is not violated in the upcoming update batches. \n\n\\textbf{Statistical test for data changes}. We define out-of-distribution detection as a two-sample hypothesis test between a sample of historical data and a sample of the new data. Let \\(S^{\\leq}_{t-1}\\) have a joint distribution of \\(P(A_1,\\dots, A1_m) \\equiv \\mathbb{P}\\) and \\(S_{t}\\) have a joint distribution of \\(Q(A_1,\\dots, A_m) \\equiv \\mathbb{Q}\\). We define the null hypothesis \\(H_0: \\mathbb{P}=\\mathbb{Q}\\) which asserts that \\(S_{t}\\) and \\(S^{\\leq}_{t-1}\\) are coming from a same distribution; and the alternative hypothesis \\(H_A: \\mathbb{P}\\neq \\mathbb{Q}\\) which declares that the two samples are generated by two different distributions. \n\n\\textbf{Incrementally updating the model}. Consider for \\(I_0\\) a model \\(M_{0}\\) is trained by minimizing a loss function \\(\\mathscr{L}(D_{0};\\Theta_0\\)). This model may be stale for \\(I_t; t>0\\). Ideally, the goal of incremental learning is: at time \\(t\\) train a model \\(M_{t}\\) that minimizes a function over \\(\\sum_{i=1}^{t} \\mathscr{L}(D_{i};\\Theta_i)\\). This new model should not forget  \\(\\{I_{i}; i=0,1,...,t-1\\}\\) and also learn \\(I_t\\).\n\n\n\\subsection{A High Level View of DDUp}\\label{highlevel} \nThe overall architecture of DDUp is depicted in \\autoref{fig:Arch}.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth, height=4cm]{figures\/Detailed-DDUp-Architecture.png}\n  \\caption{The overall structure of DDUp. DDUp uses the latest model and previous data to build a sampling distribution for the two-sample test, and updates the learned component based on the shift in the data distribution.}\n  \\label{fig:Arch}\n  \\vspace{-0.3cm}\n\\end{figure*}\n\nDDUp process batches of tuples at a time. Such batched handling of insertions is typical in analytical DBs. Furthermore, this takes into account that \nthe effect of single tuples is usually negligible for the overall large space modelled by NNs. And, for most tasks like CE, AQP and DG, the effect of single tuples in the final result is very small, considering the large sizes of tables. And batching amortizes detect-and-update costs over many insertion operations.\n\nUpon a new batch insertion, DDUp takes the latest model \\(M_{t-1}\\), and performs a bootstrapping sampling from the previous data to build the sampling distribution for the average loss values. DDUp uses this distribution to calculate a significance level corresponding to a confidence interval (e.g a 95th confidence interval). The general idea is that if the new data is similar to the previous data (IND in \\autoref{fig:Arch}), the loss values of \\(M_{t-1}\\) for this new data should lie within the threshold. This means that the new data has the same distribution and therefore the model could be left intact (updating maybe just the hyper-parameters of the system, including possible frequency tables and other table statistics. Alternatively, a simple fine-tuning can be performed to adapt the model to the new data.\n\nIf the loss values exceeded the threshold, this implies that the data distribution has significantly changed. DDUp will deploy a teacher-student transfer learning method based on knowledge distillation to learn this new distribution without forgetting the knowledge of the old data. In this framework, while the student directly learns the distribution of the new data, the teacher act as a regularizer to make the student also learn about the old distribution.\n\\vspace{-0.3cm}\n\\section{Out-of-Distribution Detection} \\label{driftdetect}\n\\subsection{Background} \\label{oodback}\nIn ML, OOD is typically addressed from a classification perspective. Formally, assume \\(D\\) is a dataset of \\((x,y)\\) pairs which are drawn from a joint distribution, \\(p(x,y)\\), where \\(x \\in \\mathcal{X} := \\{x_1, x_2, \\dots, x_n\\}\\) is the input (independent variable) consisting of \\(n\\) features, and \\(y \\in \\mathcal{Y} := \\{1,2, \\dots, k\\}\\) is the label corresponding to one of the \\(k\\) in-distribution classes. A sample \\((x,y)\\), that probably is generated by a different distribution than \\(p(x,y)\\), is called OOD, if \\(y \\notin \\mathcal{Y}\\), i.e it does not belong to any previously seen classes. \n \nA similar problem has previously been addressed in statistics as {\\it concept drift} detection, where different types of shifts are distinguished by expanding \\(p(x,y)\\) using the Bayes rule:\n\\begin{equation}\\label{bayesrule}\n p(x,y)=p(x)p(y|x)\n\\end{equation}\nBased on Eq. \\ref{bayesrule}, changes in \\(P(y|x)\\) are usually referred to as \\textit{Real drift}, while changes in \\(P(x)\\) are called \\textit{virtual drift} \\cite{gama2014survey}. In \\(X\\rightarrow y\\) problems the latter mostly is known as \\textit{covariate shift}.\nDeciding which drift to detect is dependent on the underlying models. For example, deep autoregressive networks (e.g., used by \\cite{yang2019deep}) learn the full joint distribution of a table. Hence, they are sensitive to \\textit{covariate shift} upon insertions. \nOn the other hand, mixture density networks (e.g., used by \\cite{ma2021learned}), model the conditional probability between a set of independent attributes and a target attribute. Hence, for these models, one would be interested in detecting \\textit{real shift}.\n\\vspace{-0.2cm}\n\\subsection{Loss based OOD Detection} \\label{llfordrift}\nThere are several challenges that make it difficult to simply adopt one of the OOD detection algorithms in the ML or statistical learning literature.\nFirst, DB tables are multivariate in nature and learned models are usually trained on multiple attributes. As a result, uni-variate two-sample tests like Kolmogorov\u2013Smirnov (KS) test are not suitable for this purpose. Second, the test should introduce low overheads to the system as insertions may be frequent. Therefore,  multivariate tests like kernel methods that require to learn densities and perform expensive inference computations are not desirable. Third, we aim to support different learning tasks for which different models might be used. Thus, most of OOD detection methods in ML that are based on probability scores (confidence) of classification tasks are not useful here. Moreover, the test should be able to adapt efficiently to the case where insertions occur within old data, that is, without having to recalculate baseline thresholds etc.\n\nAn efficient OOD detection method is now proposed that resolves all above issues by leveraging the underlying ML models themselves. Central to most learned data system components is the ability to derive from the underlying data tables a model for the joint or conditional data distribution like \\(p(x)\\) or \\(p(y|x)\\). A model usually achieves this by learning a set of parameters \\(\\Theta\\) that represent a function \\(f\\) by iteratively optimizing over a loss function as follows:\n\n\\begin{equation} \\label{generalopt}\n    f_\\Theta = \\argmin_{f \\in \\mathcal{F}} \\frac{1}{n} \\sum_{i=1}^n \\mathscr{L}(f(x);\\Theta) + \\Omega(f)\n\\end{equation}\n\nwhere, \\(\\Omega\\) is a regularizer term, \\(n\\) is the number of samples, and \\(f\\) could be the outputs of the model in the last layer (called \\textit{logits}), or the probabilities assigned by a \"softmax\" function.\n\n\nWe will later discuss different loss functions in more details when instantiating different models. In general, loss functions are usually highly non-convex with many local mimina. However, a good learning strategy will find the global minimum. Because of the large data sizes, training is usually done by iterating over mini-batches and a gradient descent algorithm updates the parameters based on the average of loss of the samples in each mini-batch per iteration. For the rest of the paper, when we mention 'loss value' we mean average of losses of the samples in a batch. Once the model is trained, i.e. the loss values have converged, the model can serve as a transformer to map (high-dimensional) input data to the one-dimensional loss functions space around the global minimum. Accordingly, the previous data (seen by the model) are closer to the global minimum compared to the out of distribution data.\n\nThe above discussion explains the possibility to compare in- and out-of distribution data just by relying on the underlying models without any further assumptions\/components, in a low-dimensional space. With these in hand, we can perform a statistical testing to compare the loss values of old data and new data. In the following we will explain a two-sample test for this purpose. \n\n\\subsection{A Two-Sample Test Procedure}\nThe steps for a two-sample hypothesis test are: \n1. Define the null, \\(H_0\\), and alternative hypothesis, \\(H_A\\). \n2. Define a test statistic \\(d\\) that tests whether an observed value is extreme under \\(H_0\\). \n3. Determine a significance level \\(\\delta\\in[0,1]\\) that defines the \\(type\\mhyphen1\\ error\\) (false positives) of the test. \n4. Calculate \\(p\\mhyphen value\\) which equals the probability that a statistical measure, e.g. distance between two distributions, will be greater than or equal to the probability of observed results.\n5. If \\(p\\mhyphen value <= \\delta\\) then the \\(p\\mhyphen value\\) is statistically significant and shows strong evidence to reject \\(H_0\\) in favor of \\(H_A\\). Otherwise, the test failed to reject \\(H_0\\). \n\nThe main challenge herein is how to calculate the test significance of the test statistic, i.e the \\(p\\mhyphen value\\). As explained in Section \\ref{problemdef}, we aim to detect if a new data that is inserted to the system at time \\(t\\) has a different distribution than the previous data. Consider \\(S_{t-1}^{\\leq}\\) be a sample of the previous data and \\(S_{t}\\) be a sample of the newly inserted data. Let \\(d(S_{t-1}^{\\leq},\\ S_{t})\\) be a distance function that measures the distance between the two samples. If \\(P_d\\) is the distribution that explains the test statistic \\(d\\) under the null hypothesis, then the test significance could easily be computed by \\(p\\mhyphen value=P(P_d < d | H_0)\\). Note that since we assume that our test statistic is a distance function, we would perform a one-side left-tail test.\n\n\\textbf{Choosing the test statistic}. The test statistic should reflect the similarity of new data to old data. According to our discussion in Section \\ref{llfordrift}, we use\nthe loss function values after convergence of the models. We use a linear difference between the loss values of the two samples as our test statistics as follows:\n\\begin{equation}\\label{teststatistic}\nd(S_{t-1}^{\\leq},S_{t}) = \\frac{1}{|S_{t-1}|}\\sum_{s\\in S_{t-1}}\\mathscr{L}(s;\\Theta) - \\frac{1}{|S_t|}\\sum_{s\\in S_{t}}\\mathscr{L}(s;\\Theta)\n\\end{equation}\n\nwhere \\(\\mathscr{L}\\) is a loss function achieved by training model \\(M\\) with parameters \\(\\Theta\\). From Eq. \\ref{teststatistic} follows that if the loss function is Negative Log Likelihood, and the likelihoods are exact, the test statistics will be the logarithm of the well-known \\textit{likelihood-ratio} test. Eq. \\ref{teststatistic} also gives intuition about the effect size:  the larger \\(d\\) is, the larger the difference between two data distributions would be.\nAlthough many of the learned DB models are trained by maximizing likelihood, some other models (e.g., regressions) are trained using a \\textit{Mean-Squared-Error} objective. It has been shown \\cite{watkins1992maximum} that MSE optimization maximizes likelihood at the same time. Therefore, the form of the distance function in Eq. \\ref{teststatistic} still holds. \nThe important consequence of Eq. \\ref{teststatistic} is that, under i.i.d assumptions for both samples, it can be shown that the central limit theorem holds for \\(P_d\\) \nhence, it has a normal limiting distribution with a mean at 0 and unknown standard deviation. The normality of \\(P_d\\) allows us to make inference based on the confidence intervals. To estimate the standard deviation (std), we perform a bootstrapping approach.\n\n\\subsection{Offline and Online Steps}\nThe main bottleneck of such an OOD detection is bootstrapping. Fortunately, this part could be performed offline before data insertion. In the offline phase, \nwe draw \\(n\\) bootstrap samples of size \\(|S^{\\leq}_{t-1}|\\) from \\(S^{\\leq}_{t-1}\\). (In practice, when we have access to the original data, we make $n$ bootstrap samples of size $|S_{t-1}^{\\leq}|$) from $D_{t-1}^{\\leq}$). We use the model \\(M_{t-1}\\) to compute the likelihoods (or other losses) of each sample and create a sampling distribution using them. Then, we calculate the standard deviation of the sampling distribution, \\(std\\), and use it to find the significance level. In the online phase, we make a sample of the new data, \\(S_{t}\\) and use the latest model, \\(M_{t-1}\\) to calculate the likelihood of \\(S_{t}\\). Finally we compare the test statistic with the threshold. If \\(d > 2\\times std\\) (equivalently \\(p\\mhyphen value \\leq \\delta\\) where \\(\\delta=0.05\\)) we declare a significant shift in data and reject the null hypothesis in the favor of the alternative hypothesis. Otherwise the test fails to reject the null hypothesis and signals \"in-distribution\". \n\n\\subsection{The Test Errors}\\label{testerrors}\nThere are two errors associated with a hypothesis testing. \\textit{type-1 error} is rejecting the null hypothesis when it should not.  \\textit{Type-2 error} is the error of accepting the null hypothesis when it should be rejected. The first one introduces false positives to the system and the second causes false negatives. \nFalse positives (FPs) are only a (rather small) performance concern only, spending time to update the model while accuracy is preserved. False negatives (FNs), however, can cause a loss of accuracy.  \nTherefore, the system can afford to be stricter with respect to the significance level, in order to reduce the risk of false negatives and accuracy loss.\n\nDDUp uses the loss of the trained NNs for OOD detection. Sometimes NNs could be over-confident \\cite{nguyen2015deep,ren2019likelihood,nalisnick2018deep} which may introduce bias. \nHowever, we have not witnessed it for our tasks here on tabular data.\nIf there were bias, the FP and FN rates discussed above would signal it. \nWe have evaluated DDUp with respect to FPs\/FNs in Section \\ref{oodeval} showing that this is not a concern.\n\n\\section{Model Update} \\label{KD}\nIn this section, we propose a transfer-learning based method that can retain previous knowledge of the model while adapt it to the new insertions. The OOD detection module will either output 'in-distribution' or 'out-of-distribution' signals.\n\n\\textbf{The in-distribution case}. When no drift occurs, the new data distribution is similar to that of the historical data and this distribution could be represented by a similar parameter space of the latest model, \\(M_{t}\\). \nHence, the learned component of the system could remain unchanged. More specifically, the framework can copy \\(M_{t}\\) to \\(M_{t+1}\\) and update the required meta-data associated with the system (such as the frequency tables in DBEst++, or table cardinalities in Naru\/NeuroCard).  Even if there are slight permutations in data, fine-tuning the latest model's parameters on the new data will adjust it to the general representation of both old and new data. \nWe will show that when knowing that data is not OOD, \\textit{fine-tuning}{} with a relatively small learning rate, can retain model performance. \nSpecifically, with an \\textbf{in-distribution} signal at time \\(t+1\\), \\(M_{t}\\) is retrained on \\(S_{t+1}\\) with a small learning rate, $lr$. This learning rate could be tuned, as a hyper-parameter.\nWe intuitively set \\(lr_{t} = \\frac{|D_{t+1}|}{|D_{t}^\\leq|}\\ \\times \\ lr_{0}\\) and experimentally show that it is a good choice. \n\n\\textbf{The OOD case}. With a distributional shift,\nby fine-tuning on new data, the model's parameters would bias toward the new data distribution. Even smaller learning rates cause tiny deviations from the previous parameter space which may yield large errors during inference. And, retraining using all the data from scratch is too time consuming. Thus, we propose an updating approach grounded on the transfer-learning paradigm. The general idea is to use the learned model \\(M_{t}\\) and incorporate it in training \\(M_{t+1}\\). To this end, we utilize the \\textit{knowledge distillation}{} principles, which help to transfer the previously learned knowledge to a new model. Our rationale for such a model updating approach is based on the following: \n\\begin{itemize}[leftmargin=*]\n    \\item Distillation has several benefits including: faster optimization, better generalization, and may even outperform the directly trained models. \\cite{yim2017gift}.\n    \\item It is accurate for queries on old  as well as new data.\n    \\item It allows us to control the weights for queries on new and old data with just a couple of parameters.\n    \\item It is efficient memory-wise as well as computationally-wise, compared to methods like Gradient Episodic Memory, or Elastic Weight Consolidation and PathInt (cf. Section \\ref{litraturere})\n    \\item It does not make any assumptions about the training of the underlying models. This property, is especially desirable since: a) we can use it to update different neural networks; b) it prevents the high costs of rebuilding base models; c) different pre-processings could be left intact. For instance, Naru, DBEst++ and TVAE all use completely different types of embedding\/encoding. DDUp can update the model regardless of these differences.\n\\end{itemize}\n\n\\subsection{General Knowledge Distillation (KD)}\nKD was first introduced in \\cite{hinton2015distilling} for $model \\  compression$ by transferring knowledge from an accurate and \"cumbersome\" model, called \\textit{teacher}, to a smaller model called \\textit{student}. In its basic form, instead of fitting the student model directly to the actual data \\textit{labels}, one would use the class probability distribution learned by the teacher to fit the student model. Hinton et al. \\cite{hinton2015distilling} argued that small probabilities in \"wrong\" label logits, known as \"soft labels\", include extra information called \"dark knowledge\" that result in better learning than actual \"hard labels\". Distillation has since been extensively studied. \\autoref{fig:kdfig} shows a general view of the principles of a distillation process. A small dataset referred to as \\textit{transfer-set} is fed into a pre-trained model (teacher) and a new model (student) to be trained. A $distillation \\ loss$ is calculated using the predictions of the pre-trained model instead of the actual labels. This loss and a typical loss using actual labels will be used to train the new model. \n\n\\begin{figure}[hb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/distillation-diagram.png}\n    \\vspace{-0.2cm}\n    \\caption{The knowledge distillation process.}\n    \\label{fig:kdfig}\n    \\vspace{-0.35cm}\n\\end{figure}\n\nTo formulate \\textit{knowledge distillation}{}, consider a model with parameters \\(\\Theta\\), representing a function \\(f_t\\) (\\(t\\) for teacher) which has been trained via Eq. \\ref{generalopt}. We would like to transfer knowledge from this teacher model to a student model with parameter \\(\\Theta'\\), representing a function \\(f_s\\). This new model could be trained as follows:\n\n\\begin{equation} \\label{distillopt}\n    f_{s\\Theta'} = \\argmin_{f \\in \\mathcal{F}} \\frac{1}{|tr|} \\sum_{i\\in tr} \\left[\\lambda\\mathscr{L}_d(f_s(i);f_t(i);\\Theta;\\Theta') + (1-\\lambda)\\mathscr{L}(f_s(i);\\Theta')\\right]\n\\end{equation}\n\\\\\nfor weight \\(\\lambda\\), distillation loss \\(\\mathscr{L}_d\\), and transfer-set \\(tr\\). \n\n\\subsection{DDUp: Updating By Knowledge Distillation}\\label{upbykd}\n\n\\cite{furlanello2018born,seq-self-distill} showed that, for classification tasks, if instead of having a compact student model, one uses the same architecture of the teacher, and repeat distillation sequentially for several generations, the student models in the later generations could outperform the teacher model. This approach is called {\\it sequential self-distillation}.\nInspired by this and anticipating that this will be valid for our learning tasks, DDUp also employs a sequential self-distillation approach.\n\nTo update a model using KD,\na copy of the previously trained model becomes the new student. Then,  the student is updated using a distillation loss (to be defined soon). After updating, the previous teacher is replaced with the new updated model. This cycle repeats with every new insertion batch.\n\\begin{comment}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/Sequential Distillation.png}\n    \\caption{Sequential updating in a self-distillation scheme}\n    \\label{fig:sequpdate}\n\\end{figure}\n\\end{comment}\n\nTo formulate our training loss function, we consider two aspects that we would like to have in our updating scheme. First, to have control over the the new data\/queries versus the old data\/queries. Second, to make it general so that different learned DB systems could adopt it. As such, we first write down the general form of the total loss function and then, use cross-entropy and mean-squared-error as the loss functions to instantiate different models. Training in each update step is as follows:\n\\\\\n\\begin{equation} \\label{totalloss}\n\\begin{split}\n    f_{s\\Theta'} = \\argmin_{f \\in \\mathcal{F}} & \\bigg(\\alpha \\times \\frac{1}{|tr|} \\sum_{x\\in tr} \\big[\\lambda\\mathscr{L}_d(f_s(x),f_t(x);\\Theta') \\\\ \n    & + (1-\\lambda)\\mathscr{L}(f_s(x);\\Theta')\\big] \\\\\n    & + (1-\\alpha) \\times \\frac{1}{|up|}\\sum_{x\\in up}\\mathscr{L}(f_s(x);\\Theta') \\bigg)\n\\end{split}\n\\end{equation}\n\\\\\nHere, \\(\\alpha\\) and \\(\\lambda\\) are the new data and the distillation weights, respectively. Also, \\(tr\\) and \\(up\\) are the transfer-set and the update batch. \nIn summary, the rationale for proposing this novel loss function is: \nThe transfer-set term acts as a regularizer to avoid overfitting on new data.\nThe same goal is also helped by self-distillation (when copying the teacher to the student). Additionally, as mentioned sequential self-distillation \\cite{seq-self-distill} may attain increasingly higher accuracy, even outperforming \"retrain from scratch\"\n(cf. Section 5.3).\n\nFor models that provide a conditional probability in the last layer of the network (e.g. using a Softmax function), an annealed cross-entropy loss will be employed. Otherwise, we utilize mean-squared-error using the logits from the last layer of the network. Eq. \\ref{cedistillloss} and Eq. \\ref{mseloss} show these two loss functions. \n\\\\\n\\begin{equation}\\label{cedistillloss}\n    \\mathscr{L}_{ce}(D_{tr};z_t,z_s) = - \\sum_{i\\in [k]} \\frac{exp(z_{t_i}\/T)}{\\sum_{j\\in [k]}exp(z_{t_j}\/T)} \\log \\frac{exp(z_{s_i}\/T)}{\\sum_{j\\in [k]}exp(z_{s_j}\/T)}\n\\end{equation}\n\\\\\n\\begin{equation} \\label{mseloss}\n    \\mathscr{L}_{mse}(D_{tr};z_t,z_s) = \\sum_{i\\in[|z_t|]}(z_{t_i} - z_{s_i})^2\n\\end{equation}\n\\\\\nwhere \\(D_{tr}\\) is the \\textit{transfer-set} , \\(T\\) is a temperature scalar to smooth the probabilities so that it produces \"softer\" targets, and \\([k]\\) is the vector \\([0,1,\\dots,n]\\) which are the class probabilities and \\([|z_t|]\\) indicates the logits of the network.\n\n\\subsection{Instantiating the Approach}\n\n\\textbf{Mixture Density Networks}. MDNs consist of an NN to learn feature vectors and a mixture model to learn the \\textit{probability density function} (pdf) of data. Ma et al. \\cite{ma2021learned} uses MDNs with Gaussian nodes to perform AQP. For the Gaussian Mixture, the last layer of MDN consists of three sets of nodes \\(\\{\\omega_i, \\mu_i, \\sigma_i\\}_{i=1}^m\\) that form the pdf according to Eq. \\ref{mdneq}. \n\n\\begin{equation}\\label{mdneq}\n\\hat{P}(y|x_1, ..., x_n) = \\sum_{i=1}^{m}\\omega_i.\\mathscr{N}(\\mu_i, \\sigma_i)    \n\\end{equation}\n\nwhere \\(m\\) is the number of Gaussian components, \\(y\\) is the dependent variable and \\((x_1, ..., x_n)\\) is a set of independent variables, \\(w_i\\) is the weight of the \\(i^{th}\\) Gaussian with a mean of \\(\\mu_i\\) and a standard deviation of \\(\\sigma_i\\).\nFor MDNs, we define distillation loss as follows:\n\n\\begin{equation} \\label{mdnkdloss}\n\\mathscr{L}_d = \\mathscr{L}_{ce}(D_{tr}, \\omega_{t}, \\omega_{s}) + \\mathscr{L}_{mse}(D_{tr}, \\mu_{t}, \\mu_{s}) + \\mathscr{L}_{mse}(D_{tr}, \\sigma_{t}, \\sigma_{s})\n\\end{equation}\n\nThis summation of terms help us retain both the shape of data distribution as well as the intensity levels. \n\n\\textbf{Deep Autoregressive Networks}. The Naru and NeuroCard cardinality estimators \\cite{yang2019deep, yang2020neurocard} use deep autoregressive networks (DARNs) to approximate a fully factorized data density. DARNs are generative models capable of learning full conditional probabilities of a sequence using a masked autoencoder via Maximum Likelihood. Once the conditionals are available, the joint data distribution could be represented by the product rule as follows:\n\\[\n\\hat{P}(A_1, A_2, \\dots, A_n) = \\hat{P}(A_1)\\hat{P}(A_2|A_1)\\dots \\hat{P}(A_n|A1,\\dots ,A_{n-1})\n\\]\n\nwhere \\(A_i\\) is an attribute in a relation \\(R\\). Naru and NeuroCard use cross-entropy between input and conditionals as the loss function. This allows us to formulate the distillation loss function using the conditionals of the teacher and the student networks. Also, in Naru and NeuroCard, each conditional is calculated using a set of logits, hence we average over all as follows:\n\\begin{equation} \\label{narukdloss}\n\\mathscr{L}_d = \\frac{1}{|A|}\\sum_{i=1}^{|A|}\\mathscr{L}_{ce}(D_{tr}, z_{s_i}, z_{t_i})\n\\end{equation}\n\nWhere \\(|A|\\) is the number of attributes corresponding to the number of conditionals.\n\n\\textbf{Variational Autoencoders}. VAEs have been used for a number of DB components: \\cite{thirumuruganathan2020approximate} for AQP, \\cite{hasan2020deep} for CE, and \\cite{xu2019modeling} for synthetic tabular data generation. \nThey are a type of autoencoders that instead of learning deterministic encoder, decoder, and compressed vector (known as bottleneck), they learn a probabilistic encoder, decoder, and a latent random variable instead of the compressed vectors. (For more details, see the seminal paper \\cite{kingma2013auto}). \nInterestingly, a VAE is trained using a different loss function, known as Evidence-Lower-Bound (ELBO) loss (which amounts to a lower bound estimation of the likelihoods).\nHere we shall use TVAE for learned synthetic tabular data generation (of particular importance in privacy-sensitive environments, or when data is scarce for data augmentation purposes, or when wishing to train models over tables and accessing raw data is expensive in terms of time or money).\n\nTo distill a VAE, one must cope with the random noise added to the input of the decoder by the latent variable. For that, the latent variable in the teacher network is removed, and we use the same noise generated by the student in the teacher. The reason for doing this is that distillation tries to teach the student to behave like the teacher for a specific observation or action. If there is randomness, the student might mimic the teacher's behaviour for a completely different observation. After this change, the corresponding logits of the encoder\/encoder of the student and the teacher are compared using MSE. Finally, the loss function is:\n\\\\\n\\begin{equation}\n    \\mathscr{L}_d = \\frac{1}{2}( \\mathscr{L}_{mse}(D_{tr}, z_t^{(e)}, z_s^{(e)}) + \\mathscr{L}_{mse}(D_{tr}, z_t^{(d)}, z_s^{(d)}) )\n\\end{equation}\n\\\\\nwhere, \\(e\\) and \\(d\\) correspond to the encoder and the decoder networks. \n\n\n\n\\subsection{An Example}\nWe create a simple synthetic dataset consisting of a categorical attribute, \\(x\\), with 10 \\(distinct\\mhyphen values=\\{1,2,3,\\dots,9,10\\}\\), and with each category having 1000 real values. The dataset is balanced and the real values for each category are generated by a \\textit{Mixture of Gaussians} (MoG) with five peaks. \\autoref{fig:toyexam}.a is the dataset corresponding to \\(x=1\\). We fit a \\textit{Mixture Density Network} with ten components on this dataset. \\autoref{fig:toyexam}.b shows a sample generated by this MDN which asserts that the model has perfectly learnt the data distribution. Next, we introduce an update batch generated by a MoG with two different means. \\autoref{fig:toyexam}.c shows the update batches in red color compared to the previous data in blue. We update the previously learned MDN with the proposed loss function in Eq. \\ref{mdnkdloss}. We repeat updates for 50 batches generated with the new MoG. \\autoref{fig:toyexam}.d shows the final distribution learnt by the MDN.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/toy_example.png}\n    \\caption{An example to show how DDUp learns new data without forgetting. 'a' is the histogram of synthetic data corresponding to $x=1$. 'b' is the sample generated by the learned MDN for $x=1$. 'c' shows a sample of an update batch coming from different Gaussians. 'd' is the sample generated by the MDN after being updated by the DDUp loss function. We have performed the update 50 times to see the effect of high frequency updates (This explains the higher frequencies around the last two peaks for 'd').}\n    \\vspace{-0.4cm}\n    \\label{fig:toyexam}\n\\end{figure}\n\n\n\\subsection{Handling Join Operations}\\label{joins}\nDDUp can operate either on raw tables or tables from join results.\nIf the old data $R$ is the result of a join, the new data batch needs to be computed, due to new tuples being inserted in any of the joined tables in $R$.\nConsider at time \\(t-1\\) a model \\(M_{t-1}\\) has been trained on  \\(R=\\bigcup_{j=0}^{t-1}{T_1^j} \\bowtie \\bigcup_{j=0}^{t-1}{T_2^j} \\dots \\bowtie \\bigcup_{j=0}^{t-1}{T_n^j}\\), where $T_r^j$ denotes the new batch for table $T_r$ at time $j$.\nWithout loss of generality, suppose a new insertion operation $I_t$ at time \\(t\\)  adds new data to table \\(T_i\\), denoted \\(T_i^t\\). The new data for DDUp in this setting is \\( D_t = (R\\ \\setminus \\  \\bigcup_{j=0}^{t-1}T_i^{j}) \\bowtie T_i^{t} \\), where $\\setminus$ denotes a (multi)set-difference operator. Therefore, for the detection module, \\(S^{\\leq}_{t-1}\\) is a sample of R, and \\(S_t\\) a sample from $D_t$. Furthermore, during updating the transfer-set is a sample from $R$ and the new data is $D_t$. \nPlease note that all this data preparation and how each model deals with joins is orthogonal to DDUp.\nTherefore, it can be done by either computing the actual joins above or using join samplers like \\cite{zhao2018random,shanghooshabad2021pgmjoins}, as is done in NeuroCard and compared against in Section \\ref{joinexp}.\n\n\\begin{comment}\n\\subsection{Putting it All Together}\nAt time \\(t=0\\) we have data \\(D_{0}\\). We create two samples from \\(D_{0}\\): \n\\( S^{\\leq}_{0}\\) for OOD detection and a sample as the \\textit{transfer-set}. \nWe use \\(D_{0}\\) to train a model \\(M_{0}\\). \nNext, at time \\(t=1\\), a new data batch, \\(D_{1}\\) arrives. \nDDUp follows the following steps:\n\\begin{enumerate}\\setlength\\itemsep{0.5em}\n    \\item  The OOD detection module uses \\(S^{\\leq}_{0}\\) and a sample of \\(D_{1}\\) to test for a significant change in data distribution and issues an 'in-distribution' or 'out-of-distribution' signal.\n    \\item  If 'in-distribution', go to step 3, otherwise, go to step 4.\n    \\item  The Model Update module deploys a baseline method to update \\(M_{0}\\) to \\(M_{1}\\). This baseline method is usually fine-tuning with a smaller learning rate. Finally, go to step 5. \n    \\item  Start the distillation procedure:\n    \\begin{itemize}\\setlength\\itemsep{0.1em}\n            \\item Make an exact copy of \\(M_{0}\\)\n            \\item Feed the transfer-set into \\(M_{0}\\) and \\(M_{1}\\) \n            \\item Use Eq. \\ref{totalloss} to update the copied model \\(M_{1}\\)\n    \\end{itemize}\n    \\item Set the updated model as \\(M_{1}\\), Drop \\(M_{0}\\), Update \\(S^{\\leq}_{0}\\) and \\textit{transfer-set}.\n    \\item End.\n\\end{enumerate}\n\\end{comment}\n\n\\section{Experimental Evaluation} \\label{eval}\nWe evaluate DDUp for three different models for learned DB components: (i) Naru\/NeuroCard \\cite{yang2019deep,yang2020neurocard} which use DARN models for CE; (ii) DBest++ \\cite{ma2021learned} that uses MDNs for AQP; and (iii) TVAE \\cite{xu2019modeling}, that uses variational autoencoders for DG.\nWe evaluate in terms of model accuracy and update time.\nWe use as reference points the baseline update approach provided for AQP and CE (TVAE provides no update approach).\nWe also add as reference points the accuracy when retraining from scratch and when leaving models stale. With respect to \\textit{OOD detection}, we investigate whether it can detect significant data shifts successfully and how this will contribute to the final performance of the underlying models in their specific application, CE, AQP, DG. \nUltimately, the experiments are to address the following questions:\n\n\\begin{itemize}[leftmargin=*]\n    \\item How to best evaluate DDUp? (Section \\ref{setup})\n    \\item Can DDUp accurately detect a distributional shift? (Section \\ref{oodeval})\n    \\item Is DDUp accurate under in- $and$ out-of- distribution settings? (Section \\ref{perfeval})\n    \\item How does DDUp compare to the baseline approaches in accuracy and update time? (Section \\ref{perfeval})\n    \\item What is the effect of distillation? (Section \\ref{distilleval})\n    \\item Is DDUp efficient? (Section \\ref{overheads})\n\\end{itemize}\n\n\\vspace{-0.3cm}\n\\subsection{Experimental Setup} \\label{setup}\nTo establish a dynamic setup, we make a copy of the base table and randomly sample 20\\% of its rows as new data. In this setting, new data follows the previous data distribution which we denote as \\textit{in-distribution}. We introduce distributional drift as is typically done for tabular data settings, say in \\cite{wang2020we}. As such, after making the copy, we sort every column of the copied table individually in-place to permute the joint distribution of attributes. Next, we shuffle the rows and randomly select \\(20\\%\\) of the rows - this now becomes the new data.\nWith these new data, we perform two types of experiments. First, we consider the whole 20\\% sample as a new data batch and update the model with it. Second, to show the updatability in incremental steps, we split the 20\\% data into 5 batches. \nIn general, the size of the transfer-set is a tunable parameter \\cite{hinton2015distilling}, influenced by the dataset complexity, the underlying model generalization ability, and the downstream tasks. \nAfter tuning, we used a 10\\% transfer-set for MDN and DARN and a 5\\% for TVAE, which could be further tuned with methods like Grid search.\n\nDDUp does not impose any further constraints to those of the underlying models. For DBest++ we use a query template with a range and an equality attribute. Also, we use one-hot encoding to encode categorical attributes and normalize the range attribute to \\([-1,1]\\). For Naru\/NeuroCard and TVAE, we use the same settings as explained in their code documentation. We use the learned hyper-parameters of the base model, i.e the model we build at time zero, for all subsequent updates. Furthermore, we intuitively set \\(\\alpha\\) parameter in Eq. \\ref{totalloss} to the fraction of update batch size to the original data size and tune \\(\\lambda\\) for values in \\([9\/10, 5\/6, 1\/4, 1\/2]\\). \n\n\\subsubsection{Datasets} \\label{datasets}\nWe have mainly used three real-world datasets (census, forest, DMV) \n(see \\autoref{tab:Datasets}). These datasets  \nhave been widely used in the learned DB literature.  \nFor CE, \\cite{wang2020we} uses also forest, census and DMV, while NeuroCard\/Naru use JOB\/DMV. For AQP DBEst++ uses TPCDS.  For DG, \\cite{xu2019modeling} uses census and forest. Thus, we have also used census, forest, DMV, and TPCDS (\\texttt{store sales} table, scaling factor of 1). Finally, for join queries, we have used JOB (on IMDB data) and TPCH benchmarks, which are also used in \\cite{yang2020neurocard, yang2019deep}.\n\n\\begin{table}[hb]\n  \\caption{Characteristics of datasets.}\n  \\vspace{-0.3cm}\n  \\label{tab:Datasets}\n  \\begin{tabular}{c c c c} \n    \\toprule\n     Dataset&Rows&Columns&Joint Domain\\\\\n    \\midrule\n     Census & 49K & 13 & $10^{16}$  \\\\\n     Forest & 581K & 10 & $10^{27}$  \\\\\n     DMV & 11.6M & 11 & $10^{15}$  \\\\\n     TPCDS & 1M & 7 & $10^{30}$ \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.5cm}\n\\end{table}\n\n\\subsubsection{Workload} \\label{workload}Each model is evaluated using 2,000 randomly generated queries. These queries are generated at time zero for each model and are used throughout the subsequent updates. When an update batch is performed, the ground truth of the queries will be updated. For Naru\/NeuroCard, we use their generator to synthesize queries: It randomly selects the number of filters per query (forest:[3,8], census: [5,12], TPCDS: [2,6], dmv: [5,12]). Then, it uniformly selects a row of the table and randomly assigns operators \\([=,>=,<=]\\) to the columns corresponding to the selected filters. Columns with a domain less than 10 are considered categorical and only equality filters are used for them. For DBest++, we select a \\(lower\\mhyphen bound\\) and a \\(higher\\mhyphen bound\\) for the range filter and uniformly select a category from the categorical column for the equality filter. Throughout the experiments, we discard queries with actual zero answer. The structure of a typical query in our experiments is:\n\n\\begin{lstlisting}[mathescape=true,\n    basicstyle=\\footnotesize,\n]\nSELECT AGG(y) FROM $T_1 \\bowtie T_2 \\dots \\bowtie T_n$ WHERE $F_1$ AND ... AND $F_{d}$\n\\end{lstlisting}\n\nwhere, \\(F_i\\) is a filter in one of these forms: \\([att_i = val, att_i >= val, att_i <= val]\\). Also, \\texttt{AGG} is an aggregation function like \\texttt{COUNT}, \\texttt{SUM}, \\texttt{AVG}. For DBest++, the query template contains one categorical attribute and one range attribute. As such, we select the following columns from each dataset: census:[\\texttt{age, country}]; forest:[\\texttt{slope, elevation}]; dmv:[\\texttt{body type, max gross weight}]; TPCDS:[\\texttt{ss quantity,ss sales price}]; IMDB:[\\texttt{info type id,production year}]; TPCH:[\\texttt{order date,total price}] where the first\/second attribute is categorical\/numeric. Furthermore, Naru could not train on the full TPCDS dataset as the encodings were too large to fit to memory. Hence, we selected the following columns [\\texttt{ss sold date sk}, \\texttt{ss item sk}, \\texttt{ss customer sk},\\texttt{ss store sk}, \\texttt{ss quantity}, \\texttt{ss net profit}], and made a 500k sample.\n\n\\subsubsection{Metrics}\nFor \\textit{count} queries, we use \\textit{q-error} as follows:\n\n\\begin{equation}\n    error = \\frac{max(pred(q), real(q))}{min(pred(q), real(q))}\n\\end{equation} \n\nFor \\textit{sum} and \\textit{avg} aggregates, we use \\textit{relative-error} as follows:\n\\begin{equation}\n    error = \\frac{|pred(q) - real(q)|}{real(q)}\\times100\n\\end{equation} \n\nAdditionally, Lopez et al. \\cite{lopez2017gradient} introduce the notions of Backward Transfer (BWT) and Forward Transfer (FWT) as new metrics in class incremental learning tasks. BWT is the average accuracy of the model on old tasks, and FWT is the average accuracy of the model on new tasks. Here, we re-frame BWT and FWT.\nWe generate the queries at time \\(0\\) and use them for all update steps. At each step \\(t\\), we calculate \\(diff = real_t(q) - real_{t-1}(q)\\) for each query, \\(q\\), which gives us three set of queries; \\(G_{fix}\\) with \\(diff=0\\), \\(G_{changed}\\) with \\(diff>0\\), and \\(G_{all} = G_{fix} \\cup G_{changed}\\). With these groups, we define three measures. \\(AT\\): average q-error over \\(G_{all}\\). \\(FWT\\): average q-error over \\(G_{changed}\\). \\(BWT\\): average q-error over \\(G_{fix}\\).\n\n\\subsubsection{Evaluating Variational Autoencoders}\nDG is an  interesting learned application which is recently supported using TVAE. Thus, we evaluate DDUp for TVAE. In TVAE, once the training is done, only the decoder network is kept and used, as this is the generator. Hence, we apply our distillation-update method to the decoder network. We evaluate TVAE via the accuracy of an XGboost classifier trained by the synthetic samples, as in \\cite{xu2019modeling}. \nWe hold-out 30\\% of table as the test set, and train two classifiers with original and synthetic data, then predict the classes of the held-out data. We report \\textit{micro f1-score} for classifiers. For census, forest and DMV, we use: \\textit{income}, \\textit{cover-type}, and \\textit{fuel-type}, as the target class, respectively.\nFor TVAE, we created a smaller DMV with 1m records, as training TVAE on the whole DMV is very time\/resource consuming (proving indirectly the need to avoid retraining).\n\n\\subsection{OOD Detection} \\label{oodeval}\n\n\\subsubsection{Loss Functions as Signals}\nWe first show the results of loss\/log-likelihoods when the detector receives samples from the same distributions or from different distributions. The results are shown in \\autoref{tab:avgll}. For Naru\/NeuroCard and DBEst++ we report the actual log-likelihood values (not negatives, so higher is better). For TVAE, we report the ELBO loss values (hence lower is better). \n\n\\begin{table}[hb]\n \\centering\n  \\caption{Average log-likelihood and ELBO loss values of data samples on a trained model. $S_{old}$ is a sample of the previous training data. \"IND\", is a 20\\% sample from a straight copy of the original table; \"OOD\", is a 20\\% sample from a permuted copy of the original table.}\n  \\vspace{-0.2cm}\n  \\label{tab:avgll}\n  \\resizebox{\\linewidth}{!}{%\n  \\begin{tabular}{c c c c  | c c c | c c c } \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multicolumn{3}{c|}{DBEst++} & \\multicolumn{3}{c|}{Naru\/NeuroCard} & \\multicolumn{3}{c}{TVAE} \\\\\n& $S_{old}$  & IND & OOD & $S_{old}$ & IND & OOD & $S_{old}$ & IND & OOD \\\\\n    \\midrule\n     Census & -0.362 & -0.361 & -0.366 & -20.99 & -20.87 & -36.95 & -15.21\t& -15.22 & 81.47  \\\\\n     Forest & -0.0194 & -0.0202 & -0.052 & -43.16 & -43.9 & -141.10 & -19.96 & -20.09 & 142.38 \\\\\n     DMV & 2.520 & 2.532 & 2.444 & -13.74 & -13.16 & -18.67 & 9.114 & 9.28 & 34.95 \\\\\n    \\bottomrule\n  \\end{tabular}}\n  \\vspace{-0.35cm}\n\\end{table}\n\n\\autoref{tab:avgll} shows that the loss function (log likelihood and ELBO in our cases) can reliably signal OOD data.\nInterestingly, this corroborates similar findings in \\cite{detectOOD-iclr17} for classification tasks in various vision and NLP tasks, where the NN outputs can be used to signal OOD. Here we show it for tabular data and for NNs developed for AQP, CE, and DG tasks.  \n\nIn Naru\/NeuroCard and TVAE, when permuting, all columns are sorted individually, hence the large difference in likelihoods. \nFor DBEst++, only the selected columns for a query template have been permuted, yielding a small difference in likelihoods.\n\n\\begin{comment}\n\\begin{table}[hb]\n\\centering\n  \\caption{Change of log-likelihood with the number of permuted columns for a trained autoregressive model. 0 means no columns has been sorted individually therefore the data sample is following the distribution of the training data}\n  \\label{tab:permlevel}\n  \\begin{tabular}{c c c c} \n    \\toprule\n     \\#columns & census & forest & DMV\\\\\n    \\midrule\n0&-20.992&-43.16048&-13.745\\\\\n1&-28.687&-44.673&-14.616\\\\\n2&-31.103&-115.736&-17.935\\\\\n3&-31.201&-127.560&-18.151\\\\\n4&-31.591&-129.549&-18.308\\\\\n5&-34.793&-127.916&-18.955\\\\\n6&-34.359&-127.054&-18.838\\\\\n7&-35.626&-140.589&-18.858\\\\\n8&-35.938&-143.223&-18.836\\\\\n9&-36.969&-141.106&-18.670\\\\\n10&-37.029&&\\\\\n11&-37.243&&\\\\\n12&-36.953&&\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\\end{comment}\n\n\\subsubsection{The two-sample test results} \n\\autoref{tab:driftdetect} shows  results for two-sample testing for OOD detection. The significance level of the test (threshold) is \\(2\\times variance\\) of the bootstrapping distribution, which was obtained by $>$1000 iterations. \nIn each iteration, we use a 1\\% sample with replacement from previous data and a 10\\% sample without replacement from new data to calculate the test statistic. The results show that when data is permuted, the test statistic is far away from the threshold. This means it appears at a great dissonance in the tails of the bootstrapping distribution. \nAnd since the critical value to test for OOD is found by bootstrapping over \\(S_{old}\\), i.e., \\(S^{\\leq}_{t}\\), it will adjust even to small differences when faced with OOD. \nCase in point, the DBEst++ OOD likelihood value for census (which is similar to IND\/$S_{old}$ in \\autoref{tab:avgll}) vs the corresponding test-statistic value in  \\autoref{tab:driftdetect}.\n\n\\begin{table*}[t]\n  \\caption{The test-statistic values. Threshold is  $2\\times variance$ and bs-mean is the mean of bootstrapping distribution. }\n  \\label{tab:driftdetect}\n  \\resizebox{\\textwidth}{!}{\n  \\begin{tabular}{c | c c c c  | c c c c | c c c c } \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multicolumn{4}{c|}{DBEst++} & \\multicolumn{4}{c}{Naru\/NeuroCard} & \\multicolumn{4}{c}{TVAE} \\\\\n& bs-mean & threshold & IND  & OOD & bs-mean & threshold & IND & OOD & bs-mean & threshold & IND & OOD \\\\\n    \\midrule\nCensus&-0.3524 & 0.007 & 0.001 & 0.05 & -21.0076 & 0.0529 & 0.032 & 16.0052 & -15.1834 & 0.6041 & 0.0419 & 100.5126 \\\\\nForest&-0.0228 & 0.0122 & 0.007 & 0.2315 & -41.35 & 0.0141 & 0.0084 & 72.5473 & -19.99 & 0.0868 & 0.0417 & 167.0502 \\\\\nDMV&2.52 & 0.1287 & 0.0145 & 4.5745 & -13.7674 & 0.0012& 0.0007& 5.1145 & 9.1209 & 0.0177 & 0.0015 & 25.1398 \\\\\n    \\bottomrule\n  \\end{tabular}}\n\\end{table*}\n\n\n\n\\subsubsection{FP and FN rates in OOD detection}\\label{fpfnrates}\n\nTo evaluate OOD detection, we measure FP and FN rates (FPR, FNR). \nWe created an OOD test-set and an IND test-set, each equaling half the original size of the table. The latter is just a random sample from the original table. The former is constructed as follows. The perturbed data is obtained by perturbing one or more of five columns of the table, say $C1, \\ ... \\ C5$. First we perturb $C1$ and take a sample of the resulting table of size $10\\%$ and append it to the OOD test-set. Then we perturb $C1$ and $C2$ and similarly sample and append it to the OOD test-set. We repeat this for perturbations on $C1, C2, C3$, on $C1, C2, C3, C4$, and on $C1, C2, C3, C4, C5$, ending up with an OOD test-set of size 50\\% of the original table. Note that this setup creates a more-challenging case, as the degree of perturbations (for OOD data) is finer-grained.\nThen, at each batch, we fed a random sample from the OOD test-set and of the IND test-set to the DDUp detector. For each batch, the detector would signal IND or OOD and we recorded and calculated FPR and FNR. The batch size was 2,000 and we repeated the experiment for 1,000 batches.\n\nWe used the same parameters for all datasets and models: the bootstrapping size is 32 and the threshold is \\(2 \\times std\\). For DBEst++, the results are reported in \\autoref{tab:fprfnr}.  FPR and FNR for Naru\/NeuroCard and TVAE were always zero. These results further confirm that the OOD detection algorithm is not biased.\n\n\\begin{table}[hb]\n  \\vspace{-0.3cm}\n \\centering\n  \\caption{FPR and FNR for DBEst++.}\n  \\vspace{-0.3cm}\n  \\label{tab:fprfnr}\n  \\begin{tabular}{c c c } \n    \\toprule\nDataset & FPR & FNR \\\\\n    \\midrule\n     Census & 0.15 & 0.01 \\\\\n     Forest & 0.10 & 0 \\\\\n     DMV & 0.01 & 0 \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.35cm}\n\\end{table}\n\nFurthermore, we studied the sensitivity on the batch size and varied it from a size of 1 to 2,000. Results are shown in \\autoref{fig:oodsens}, which clearly show that after a low-threshold batch size, FPR and FPN tend to zero. The same results hold for other models and datasets, and are omitted here for space reasons.\n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/dbest-forest-fprfnr.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/tvae-dmv-fpfn.png}}\n\\end{minipage} %\n\\vspace{-0.4cm}\n\\caption{Sensitivity of OOD detection vs batch size.}\n\\label{fig:oodsens}\n\\vspace{-0.55cm}\n\\end{figure}\n\n\\subsection{Accuracy Results} \\label{perfeval}\n\\subsubsection{When there is OOD data} \\label{whenood}\n\nFor Naru\/NeuroCard, DBEst++, and TVAE, and for each dataset, we compare 4 updating approaches against each other and against the base model before any new data is inserted. The 4 approaches are as follows: \n\"\\texttt{Retrain}\", retrains the model from scratch using both old and new data.  \"\\texttt{Baseline}\" is the baseline approach in Naru\/NeuroCard and DBest++ where a trained model is updated with new data by performing \\textit{SGD} with a smaller learning rate. \"\\texttt{DDUp}\" is the proposed method.\nFinally, in \"\\texttt{stale}\", the model is not updated -- this is a do-nothing approach.\nFor reference, we also include the numbers for $M_0$, i.e., the original model accuracy before any new data came.\n\\autoref{tab:qerror} and \\autoref{tab:aqpacc} show the accuracy results for CE and AQP (SUM and AVG operations), respectively.\nFor TVAE, the classification f1-scores are reported in \\autoref{tab:tvaef1}. Results of these three tables correspond to the case where the update sample is permuted. \nDDUp always performs better than the baseline approach. \nMost of the times, the performance of DBEst++ on DMV dataset is not as well as for the other datasets. This probably is due to the complexity of data (large scale and highly correlated attributes). Nevertheless, DDUp stands on the top of the underlying models and regardless of the model's performance, DDUp ensures that it will retain the accuracy.\nPlease note the DMV dataset results in \\autoref{tab:qerror} and \\autoref{tab:aqpacc} and, census and forest datasets in \\autoref{tab:tvaef1}, where, DDUp even outperforms retraining from scratch. \nInterestingly, this corroborates similar evidence for sequential self-distillation (for boosting embeddings for) classification tasks \\cite{seq-self-distill}. This was one of the reasons we adapted a self-distillation based approach.\nFinally, baseline methods have poor performance for 95th and 99th percentiles. \n\n\\begin{table*}[t]\n  \\caption{Results of updating a base model with a 20\\% permuted sample in terms of q-error. $M_{0}$ denotes the base model.}\n  \\label{tab:qerror}\n  \\centering\n  \\begin{tabular}{c c | c | c | c | c | c | c | c | c | c | c} \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{metric} & \\multicolumn{5}{c|}{DBEst++} & \\multicolumn{5}{c}{Naru\/NeuroCard} \\\\\n&&$M_{0}$&DDUp&baseline&stale&retrain&$M_{0}$&DDUp&baseline&stale&retrain \\\\\n    \\midrule\n\\multirow{4}{*}{census}&median&1.05&1.11&1.17&1.16&1.07&1.08&1.09&4&1.14&1.07\\\\\n&95th&2&2&2.20&2&2&2&2&471.80&2&2\\\\\n&99th&3&3&4&3&3&3&3&1534.69&3.16&3\\\\\n&max&5&7&11&10.50&5&5.25&7&8385&21.88&6\\\\\n\\midrule\n\\multirow{4}{*}{forest}&median&1.026&1.046&2&1.18&1.02&1.04&1.07&1.54&1.10&1.05\\\\\n&95th&2&2&63.40&2&1.64&2.48&3&41&2.50&2.75\\\\\n&99th&2&2.583&503.12&5.60&2&4&6&157.16&5.48&5\\\\\n&max&4&5.33&3470&90.85&5.33&27&65.66&1691&484&34.66\\\\\n\\midrule\n\n\\multirow{4}{*}{DMV}&median&1.20&1.143&3.48&1.88&1.34&1.02&1.04&2.57&1.16&1.02\\\\\n&95th&4.91&5.07&234.88&7.00&5.50&1.20&1.41&468.68&1.50&1.25\\\\\n&99th&9.65&10&3897.87&12.50&8&1.83&2.31&4734.62&2.84&2\\\\\n&max&18.83&19&65875&39&17&8&9.81&343761&9.49&5\\\\\n\n\\midrule\n\\multirow{4}{*}{TPCDS}&median&1.02&1.04&57&1.27&1.02&1.01&1.07&1.15&1.10&1.05\\\\\n&95th&1.16&1.26&269&1.58&1.18&2&2&29&2&2\\\\\n&99th&1.5&1.61&1266&2.72&1.5&3.01&3.01&239&4&3\\\\\n&max&3&3&4534&10.66&5.64&5&28&5100&28&24\\\\\n\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\begin{table}[t]\n  \\caption{mean-relative-error for SUM and AVG aggregation functions for DBEst++.}\n  \\label{tab:aqpacc}\n  \\centering\n  \\resizebox{\\linewidth}{!}{%\n  \\begin{tabular}{c c | c c c c c} \n    \\toprule\nDataset&function&$M_{0}$&DDUp&baseline&stale&retrain\\\\\n    \\midrule\n\\multirow{2}{*}{census}&SUM&13.05&17.30&65.88&21.36&13.60\\\\\n&AVG&1.89&2.36&8.15&2.37&1.97\\\\\n\\midrule\n\\multirow{2}{*}{forest}&SUM&10.11&15.51&88.73&24.59&10.14\\\\\n&AVG&0.76&1.04&3.90&1.35&0.79\\\\\n\\midrule\n\\multirow{2}{*}{TPCDS}&SUM&4.53&6.37&61.40&22.64&5.12\\\\\n&AVG&0.88&1.47&12&3.50&1.21\\\\\n\\midrule\n\\multirow{2}{*}{DMV}&SUM&76.73&85.29&423&97.00&110\\\\\n&AVG&6.4&6.9&15.9&8.6&7.3\\\\\n\n\\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\\begin{table}[t]\n  \\caption{Classification results for TVAE in terms of micro f1. 'r' stands for real data, 's' stands for synthetic data.}\n  \\label{tab:tvaef1}\n  \\centering\n  \\resizebox{\\linewidth}{!}{%\n  \\begin{tabular}{c | c c | c c | c c | c c | c c } \n    \\toprule\n\\multirow{2}{*}{Dataset}\n&\\multicolumn{2}{c}{$M_{0}$}&\\multicolumn{2}{c}{DDUp}&\\multicolumn{2}{c}{baseline}&\\multicolumn{2}{c}{stale}&\\multicolumn{2}{c}{retrain}\\\\\n&r&s&r&s&r&s&r&s&r&s\\\\\n    \\midrule\ncensus&0.67&0.63&0.77&0.73&0.77&0.55&0.77&0.56&0.77&0.72\\\\\nforest&0.84&0.69&0.89&0.78&0.89&0.63&0.89&0.60&0.89&0.74\\\\\nDMV&0.97&0.97&0.98&0.97&0.98&0.92&0.98&0.93&0.98&0.98\\\\\n\n\\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\\subsubsection*{Performance on old and new queries} To better illustrate the effects of \\textit{catastrophic forgetting}{} and \\textit{intransigence} we elaborate on performance on FWT and BWT. (As \\texttt{retrain} avoids be definition \\textit{catastrophic forgetting}{} and \\textit{intransigence}, it is omitted).\nThe results are shown in \\autoref{tab:mdntranfers}. \nNote that any insertion affects only a percentage of queries, shown in \n\\autoref{tab:querypercents}.\nComparing AT, FWT, and BWT in \\autoref{tab:qerror} and \\autoref{tab:mdntranfers} first note that fine-tuning always performs much better in terms of FWT compared to BWT (due to catastrophic forgetting).\nSecond, conversely, a stale model shows better BWT compared to FWT. \nFor DDUp, FWT and BWT remain close to each other, especially in terms of median q-error, showing that DDUP can ensure accuracy for queries on old and new data.\nOverall, DDUp enjoys high accuracy.\n\n\\subsubsection*{Incremental Steps} To show the updates in incremental steps, we have split the \\(20\\%\\) data into 5 equal-sized chunks and have performed an update incrementally for each batch. \\autoref{fig:incupdates2} compares the trend of accuracy during updates. As it is clear from the figures, DDUp remains very close to \\texttt{retrain}, while there is a drastic drop in accuracy using \\texttt{baseline}. Starting point \\(0\\) is where the base model \\(M_{0}\\) is built from scratch. (The same results hold for 95th, 99th percentiles and maximum q-error). \n\n\n\\begin{table*}[t]\n \\caption{Comparing q-error of different updating approaches in terms of FWT and BWT.}\n \\vspace{-0.2cm}\n  \\label{tab:mdntranfers}\n  \\begin{tabular}{c c | c | c c | c c | c c | c | c c | c c | c c } \n    \\toprule\n\\multirow{3}{*}{Dataset} & \\multirow{3}{*}{metric} & \\multicolumn{7}{c|}{DBEst++} & \\multicolumn{7}{c}{Naru\/NeuroCard} \\\\\n&&\\multicolumn{1}{c}{$M_{0}$}&\\multicolumn{2}{c}{DDUp}&\\multicolumn{2}{c}{baseline}&\\multicolumn{2}{c|}{stale}&\\multicolumn{1}{c}{$M_{0}$}&\\multicolumn{2}{c}{DDUp}&\\multicolumn{2}{c}{baseline}&\\multicolumn{2}{c}{stale} \\\\\n\n&&&FWT&BWT&FWT&BWT&FWT&BWT& &FWT&BWT&FWT&BWT&FWT&BWT\\\\\n\n\\midrule\n\\multirow{3}{*}{census}&median&1.05&1.06&1.12&1.06&1.20&1.05&1.16&1.08&1.11&1.09&1.83&6&1.20&1.13 \\\\\n&95th&2&1.66&2&1.56&2.33&3.30&2&2&1.64&2&4.63&530.80&3.18&2 \\\\\n&99th&3&4.94&3&4.10&4&8.90&2.75&3&3.08&3&9.98&1598.53&8.49&3\\\\\n\n\\midrule\n\\multirow{3}{*}{forest}&median &1.02&1.01&1.08&1.23&2.66&1.05&1.20&1.04&1.07&1.07&1.39&1.65&1.18&1.08\\\\\n&95th&2&1.181&2&2.87&146.38&2.85&2&2.489&1.88&3&3.13&43.02&7.55&2.33\\\\\n&99th&2&1.52&3&3.72&590.57&18.33&2.24&4&4.89&6&5.27&163.80&191.53&4.86\\\\\n\n\\midrule\n\\multirow{3}{*}{DMV}&median&1.20&1.28&1.13&2.20&4.36&1.66&1.54&1.02&1.02&1.07&1.06&12.85&1.26&1.19\\\\\n&95th&4.910&4.30&5.87&3.34&484.46&9.50&6.87&1.20&1.16&1.55&1.65&1015.81&3.30&1.40\\\\\n&99th&9.65&9&11.65&10.50&5894.21&12.12&10.80&1.83&1.47&3&3.35&8183.34&11.93&2.49\\\\\n\n\\midrule\n\\multirow{3}{*}{TPCDS}&median&1.02&1.03&1.04&1.20&1.51&1.16&1.21&1.01&1.06&1.08&1.19&1.11&1.10&1.10\\\\\n\n&95th&1.16&1.21&1.29&2.37&339&2.26&1.35&2&2&2&2.60&54&2&2\\\\\n&99th&1.5&1.37&1.66&4.27&1536&4.48&1.66&3.01&9.77&3&9.47&434&9.64&3.77\\\\\n\n\\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\begin{table}[t]\n  \\caption{The percentage of the queries (out of 2k queries) with changed actual results after inserting 20\\% new data.}\n  \\vspace{-0.2cm}\n  \\label{tab:querypercents}\n  \\begin{tabular}{c c c } \n    \\toprule\n     dataset & DBEst++ & Naru \\\\\n     \\midrule\n     census&14\\%&12\\% \\\\\n     forest&32\\%&9\\% \\\\\n     TPCDS&36\\%&36\\% \\\\\n     dmv&52\\%&45\\%\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.5cm}\n\\end{table}\n\nWe also have evaluated the models with respect to the \\textit{log-likelihood goodness-of-fit}. Log-likelihood is  widely used to evaluate NN models.\nUsing log-likelihood allows evaluation to be independent\nof underlying applications. \\autoref{fig:incll} shows changes in log-likelihood in consecutive update steps. At each step, we calculate the average of log-likelihoods over a sample of new data and a sample from historical data. In these figures we again see that updating with DDUp is fitting to the old and the new data very similarly to the \\texttt{retrain} case.  In general, when keep using \\texttt{stale}, the log-likelihood drops after the first update and then remains low. The reason is that all update batches have similar permutation and since we calculate unweighted averages, the log-likelihood stays fixed. While, for \\texttt{baseline}, i.e fine-tuning, we can see a gradual decrease of likelihood which means that the network is increasingly forgetting about previous data in each step. \n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/dbest_census.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/naru_census.png}}\n\\end{minipage} %\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:c}\\includegraphics[scale=.29]{figures\/dbest_forest.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:d}\\includegraphics[scale=.29]{figures\/naru_forest.png}}\n\\end{minipage}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:e}\\includegraphics[scale=.29]{figures\/dbest_dmv.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:f}\\includegraphics[scale=.29]{figures\/naru_dmv.png}}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\caption{Updating results over 5 consecutive updates.}\n\\label{fig:incupdates2}\n\\vspace{-0.4cm}\n\\end{figure}\n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:c}\\includegraphics[scale=.29]{figures\/naru_census_loglikelihood.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:d}\\includegraphics[scale=.29]{figures\/naru_dmv_loglikelihood.png}}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\caption{log-likelihood results over 5 consecutive updates.}  \n\\label{fig:incll}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\subsubsection{When data is not OOD}\nIn this case, simple fine-tuning update algorithms, such as \\texttt{baseline}, will likely avoid \\textit{catastrophic forgetting}{}. \nTo illustrate this, we have repeated the 5 batched incremental updates with data without permutation. The results are reported in \\autoref{fig:incupdatenodrift}. For space reasons, we only show the results for census. The results indicate that for in-distribution data, simple baselines can have a performance close to \\texttt{retrain}.\n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/dbest_census_nodrift.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/naru_census_nodrift.png}}\n\\end{minipage} %\n\\vspace{-0.3cm}\n\\caption{Updating results over 5 consecutive updates when data follows the same distribution as the historical data.}\n\\label{fig:incupdatenodrift}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\begin{table}[hb]\n\\vspace{-0.4cm}\n  \\caption{DDUp's speed up over \\texttt{retrain}, for two update sizes. For census, forest, and dmv, sp1: 20\\% of the original table. sp2, 5\\% of the original table. for IMDB and TPCH sp1: updating the first partition and sp2: updating the last partition.}\n  \\label{tab:times}\n\\vspace{-0.25cm}\n  \\begin{tabular}{c | c c | c c | c c } \n    \\toprule\n\\multirow{2}{*}{Dataset}  & \\multicolumn{2}{c|}{DBEst++} & \\multicolumn{2}{c|}{Naru} & \\multicolumn{2}{c}{TVAE} \\\\\n&sp1&sp2&sp1&sp2&sp1&sp2 \\\\\n    \\midrule\ncensus&5&5.5&3.5&4&3.4&5.7 \\\\\nforest&1.6&4&5&9.2&3.6&7 \\\\\nDMV&4&6.5&2.3&9.6&3.4&6.8 \\\\\nIMDB&4.5&18&3.5&5&NA&NA \\\\\ntpch&6.5&16&2&4&NA&NA \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\subsection{Evaluating DDUp for Join Queries} \\label{joinexp}\nAs mentioned, DDUp is unconcerned whether at a time $t$, \n\\(S^{\\leq}_{t-1}\\) (a sample of  \\(\\cup_{j=0}^{t-1} D_j\\)) and $D_t$ come from a raw table or from a join.\nFor this experiment, we have evaluated DDUp running 2,000 queries over two 3-table joins from the JOB and TPCH datasets.\nFor each, the 2,000 queries involve a join of the fact table with two dimension tables: \nSpecifically, the join of tables [\\texttt{title}, \\texttt{movie info idx}, \\texttt{movie companies}] for IMDB, and [\\texttt{orders}, \\texttt{customer}, \\texttt{nation}] for TPCH. For the update dynamics, we have split the fact table into 5 time-ordered equally-sized partitions. We have built \\(M_0\\) on the join (of the fact table's first partition with the 2 dimension tables) and updated it with each subsequent partition at a time. This is similar to the update setting in NeuroCard.\nResults for both CE and AQP are in \\autoref{fig:joins}.\n\nNeuroCard, unlike other models, natively supports joins, using\na \"fast-retrain\" - i.e., a light retraining where the model is retrained using a 1 percent sample of the full join result. We have included this policy here as \"fast-retrain\". \nDDUp always signalled OOD for the new update batches, except for TPCH data on DBest++, where update was not triggered. Therefore, in \\autoref{fig:joins}.d the accuracy of the stale model and fine-tuning is  close to retrain. This further confirms the significance of OOD detection.\n\n\\begin{figure}[htbp]\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/Naru-imdb-95th.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/Naru-tpch-95th.png}}\n\\end{minipage} \\\\\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:c}\\includegraphics[scale=.29]{figures\/DBest-imdb-sum-rel-error.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:d}\\includegraphics[scale=.29]{figures\/DBest-tpch-sum-rel-error.png}}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\caption{DDUp's performance on joined tables.}\n\\label{fig:joins}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\subsection{Effect of Transfer Learning} \\label{distilleval}\nWe now delve into the effects of transfer-learning in DDUp. How much DDUp's transfer-learning via knowledge distillation contributes to better accuracy? \nWe perform experiments where we remove the transfer-learning term of Eq \\ref{totalloss}. Therefore, we combine the sample from previous data known as the transfer-set with the new update batch and create a model\nwith the same configurations as the base model. \\autoref{fig:tleffect} shows the results.\nThe results assert that the performance of DDUp is not only related to the previous data sample, and in fact, distillation has a big effect on the improvement of the new models. \n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/transfer_learning_effect.png}\n    \\caption{Effect of transfer-learning on q-error. AggTrain, is the case where we aggregate the transfer-set with the new data and train a model similar to the base model.}\n    \\label{fig:tleffect}\n\\end{figure}\n\\vspace{-0.2cm}\n\\subsection{Overheads} \\label{overheads}\nWe report on the costs of each DDUp module separately. All the codes are written and executed in Python 3.8, on an Ubuntu 20 machine  with 40 CPU cores, two Nvidia GTX 2080 GPUs and 64GB memory. With respect to memory usage, DDUp performs regular feed-forward steps as in regular NN training. Therefore, DDUp does not increase memory footprints\nIn terms of time, DDUp has two computation costs namely, \\textit{OOD detection} and \\textit{model update}. OOD detection is split into offline and online phases. \\autoref{tab:offontime} shows these two times. The largest detection time is for the forest dataset on a Naru model which takes around 3 minutes. However, please note that in \nthe online phase only takes 1 second to detect a change in data. \n\n\\begin{table}[hb]\n  \\caption{online and offline times during OOD detection.}\n  \\label{tab:offontime}\n\\vspace{-0.25cm}\n  \\begin{tabular}{c | c c | c c | c c } \n    \\toprule\n\\multirow{2}{*}{Dataset}  & \\multicolumn{2}{c|}{DBEst++} & \\multicolumn{2}{c|}{Naru} & \\multicolumn{2}{c}{TVAE} \\\\\n&off&on&off&on&off&on \\\\\n    \\midrule\ncensus&2.44&0.02&111&1.8&310&5.5\\\\\nforest&28&0.04&174&0.92&433&8.8\\\\\nDMV&86&2&144&10&99&0.44\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\autoref{tab:times} shows DDUp's speed up over \\texttt{retrain} for OOD data, for different update sizes. When data is OOD, DDUp can be over 9$\\times$ faster than \\texttt{retrain}. Obviously, speedups will be higher for incremental steps. This fact is reflected in IMDB and TPCH datasets where after inserting the last partition DDUp is 18$\\times$ faster than \\texttt{retrain}. Note that the updating time is dependent on a few parameters including update size, transfer-set size, training batch size etc. During updates, we have used smaller training batch sizes. If one tunes the model for bigger batches, and smaller transfer-set sizes, the speed up would be higher.\n\n\\vspace{-0.2cm}\n\\subsection{Non neural network models}\\label{nonnn}\nFor the sake of completeness and as an additional reference point, we include results for updating a state-of-the-art non-NN model that natively supports data insertions, (DeepDB \\cite{hilprecht2019deepdb}) used for CE. \nWhen an update happens, DeepDB traverses its sum-product-network graph and updates the weights of the intermediate nodes and the histograms at the leaves. We have repeated the same experiment in \\autoref{tab:qerror} for DeepDB. The results are reported in \\autoref{tab:deepdb}.\n\n\\begin{table}[t]\n  \\caption{Performance of DeepDB updating vs. DDUp for Naru, for a CE task in terms of q-error.}\n    \\vspace{-0.35cm}\n  \\label{tab:deepdb}\n  \\centering\n  \\begin{tabular}{c c | c | c | c | c | c } \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{ metric} & \\multicolumn{3}{c|}{DeepDB} & \\multicolumn{2}{c}{Naru} \\\\\n&&$M_{0}$&update&retrain&$M_{0}$&DDUp \\\\\n    \\midrule\n\\multirow{3}{*}{census}&median&1.05&1.2&1.05&1.08&1.09\\\\\n&95th&3&4.18&3&2&2\\\\\n&99th&5.11&8&5&3&3\\\\\n\\midrule\n\\multirow{3}{*}{forest}&median&1.02&1.2&1.02&1.04&1.07\\\\\n&95th&7.5&10.5&7&2.48&3\\\\\n&99th&31&52&31&4&6\\\\\n\\midrule\n\\multirow{3}{*}{DMV}&median&1.06&1.25&1.1&1.02&1.04\\\\\n&95th&2.5&3.5&2.5&1.20&1.41\\\\\n&99th&22&37&21&1.83&2.31\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.3cm}\n\\end{table}\n\nFrom \\autoref{tab:deepdb} it can be observed that DeepDB's updating policy is under-performing, as was independently verified in \\cite{wang2020we}. \nDDUp (coupled in this experiment with Naru\/NeuroCard for CE) always performs better. Nonetheless, we wish to emphasize that the saving grace for DeepDB based on our experiments is that retraining from scratch is very efficient -- significantly faster compared to NNs. \n\n\\section{Related Work} \\label{litraturere}\n\\subsection{Learned Database Systems}\\label{ldbliterature}\nNN-based components to be used by DBs are emerging rapidly. Different works exploit different neural network models.\n\\cite{yang2019deep, yang2020neurocard, hasan2020deep} used generative neural networks to build learned selectivity estimators. Thirumuruganathan et al. \\cite{thirumuruganathan2020approximate} used VAEs for AQP. Ma et al. \\cite{ma2021learned} used mixture density networks for AQP. Database indexing research\nrecently has adopted neural networks to approximate cumulative density functions \\cite{kraska2018case,ding2020alex,nathan2020learning,ding2020tsunami}. Query optimization and join ordering are also benefiting from neural networks \\cite{marcus2019neo, kipf2018learned}. Other applications include auto-tuning databases \\cite{van2017automatic,li2019qtune,zhang2019end}, cost estimation \\cite{zhi2021efficient, siddiqui2020cost}, and workload forecasting \\cite{zhu2019novel}.\n\nAmong these, this work provides a solution for handling NN model maintenance in the face of  insertion-updates with OOD data, when the models need to continue ensuring high accuracy on new and old data and on tasks for which models were originally trained (such as AQP, CE, DG, etc.).\nWhile there has been related research on transfer learning for learned DBs such as \\cite{hilprecht2021one, wu2021unified} these target a different problem setting:\nThey study how to transfer knowledge from a model trained for one task, and\/or a DB, and\/or a system, and\/or a workload to a new task and\/or DB, and\/or system, and\/or workload. They do not study how to keep performing the original task(s) on evolving datasets with insertions carrying OOD data with high accuracy for queries on both old and new data. Simply using these methods by fine-tuning on new data will incur catastrophic forgetting. Nevertheless, since these models employ some sorts of knowledge transfer, they might be useful to support updates. However, it remains open whether and how the models in \\cite{wu2021unified, hilprecht2021one} can be utilized to solve efficiently the problems tackled in this paper.\nWhile some of non-neural-network models (e.g., DeepDB) can very efficiently retrain from scratch,\nNN-based models for the above problem setting either do not support insertion-updates or suffer from poor accuracy when facing OOD data, unless paying the high costs of retraining from scratch.\n\n\n\\subsection{OOD Detection}\nOOD detection has recently attracted a lot of attention and it has long been studied in statistics as concept drift (CD) detection, or novelty detection. In general, CD and OOD detection methods could be divided into two broad categories \\cite{gama2014survey,lu2018learning,wang2020few}: \nFirst, prediction-based methods, which use the predictions of the underlying models to test for a change. Recent ML models usually use the predictive probabilities of the classifiers as a confidence score to identify changes \\cite{jiang2018trust,ovadia2019can, wilson2020bayesian,ruff2021unifying}. Others may monitor the error of the underlying models and trigger an OOD signal when a significant change is captured \\cite{gama2006learning, baena2006early,savva2019aggregate,nehme2009self,lopez2016revisiting}. While these approaches are very efficient in time, they typically come with limiting assumptions depending on the underlying model or application. For example, most of them can only be utilized and are only studied for classification (supervised) tasks.\nThe second broad family of methods is distribution-based methods. Some of these methods try to find a distance measure that can best show the discrepancy between new data and old data distributions, using tests like Kolmogorov-Smirnov (KS), \\cite{kolmogorov1933sulla}, Wilcoxon \\cite{pereira2009machine}, and their multi-variate variants \\cite{fasano1987multidimensional, baringhaus2004new}. Others try to learn the density of the underlying data distribution test for a significant change, like kernel-density-based approaches  \\cite{kifer2004detecting,dasu2006information,gu2016concept,lu2014concept,bu2016pdf,song2007statistical}. More recent works utilize the estimated likelihoods of generative models \\cite{ren2019likelihood, morningstar2021density, xiao2020likelihood}. Other approaches rely on the inner representations of the networks \\cite{li2021cutpaste,hendrycks2019using,lee2018simple}. Nonetheless, this second family of OOD detection methods are usually expensive (esp. for multi-dimensional data) and involve fitting a separate density estimator. Hence, the main problem is that in an insertion scenario, the density estimators also need to be updated (typically via training from scratch, upon each insertion).\n\n\\vspace{-0.3cm}\n\\subsection{Incremental Learning (IL)}\nMost IL methods regularize the model in a way that it acquires knowledge from the new task while retaining the knowledge of old tasks. For example, \\textit{Elastic Weight Consolidation (EWC)} \\cite{kirkpatrick2017overcoming} adds a regularizer to control the learning speed around important weights of the network for old tasks while learning a new task. Similar works are developed around this idea \\cite{liu2018rotate, lee2020continual,titsias2019functional}, \\textit{Path Integral (PathInt)} \\cite{zenke2017continual} ,\\textit{Riemanian Walk (RWalk)} \\cite{chaudhry2018riemannian}. Other approaches exploit knowledge distillation to retain the knowledge of previous tasks \\cite{li2017learning}.\nAnother group of IL methods, save exemplars from past data \\cite{wu2019large, castro2018end, rebuffi2017icarl} or generate samples\/features using generative models \\cite{ostapenko2019learning, kemker2017fearnet} and involve them in learning new tasks. Lopez et al. \\cite{lopez2017gradient} has proposed \\textit{Gradient Episodic Memory} that consists of \\textit{M} blocks of memory to store examples from \\textit{T} tasks and uses the model's prediction on these examples as a constraining loss that inhibits the model to bias toward new task and forget past tasks. Lastly, some works try to completely keep previous models and create new models (or part of a model like a single layer) for each new task.  Aljundi et al. \\cite{aljundi2017expert} introduce \\textit{Expert Gate} with different models for each task and an autoencoder which learns the representations of each task to assign test-time tasks to the proper model. Instead of learning a whole new model, Rusu et al. \\cite{rusu2016progressive} introduce \\textit{Progressing Neural Networks} which add new columns to the previous network architecture and learns lateral connections between them. Most of the above methods, do not account for in- and out- of distribution updates and are not easily extendable to different learning tasks. \n\n\\vspace{-0.2cm}\n\\section{Conclusion} \\label{conclusion}\nLearned DB components can become highly  inaccurate when faced with new OOD data when aiming to ensure high accuracy for queries on old and new data for their original learning tasks.\nThis work proposes, to our knowledge, the first solution to this problem, coined DDUp.\nDDUp entails two novel components, for OOD detection and model updating.\nTo make detection widely applicable, OOD detection in DDUp exploits the output of the neural network (be it based on log-likelihood, cross-entropy, ELBO loss, etc.), and utilizes a principled two-sample test and a bootstrapping method to efficiently derive and use thresholds to signal OOD data.\nDDUp also offers a general solution for model updating based on sequential self-distillation and a new loss function which carefully accounts for \\textit{catastrophic forgetting} and \\textit{intransigence}.\nThis work showcases the wide applicability of DDUp model updating by instantiating the general approach to three important learned functions for data management, namely AQP, CE, and DG, whereby a different type of NN (MDNs, DARNs, VAEs) is used for each. In fact, to our knowledge, no prior work has shown how to \"distill-and-update\" MDNs, VAEs, and DARNs.\nComprehensive experimentation showcases that DDUp detects OOD accurately and ensures high accuracy with its updated models with very low overheads.\n\n\n\\section{Acknowledgement}\nThis work is partially sponsored by Huawei IRC and by EPSRC while doing a PhD at the University of Warwick.\n\n\\balance\n\n\\bibliographystyle{ACM-Reference-Format}\n\n\\section{Introduction} \\label{introduction}\nDatabase systems (DBs) are largely embracing ML. With data volumes reaching unprecedented levels, ML can provide highly-accurate methods to perform central data management tasks more efficiently. Applications abound: AQP engines are leveraging ML to answer queries much faster and more accurately than traditional DBs \\cite{ma2019dbest,hilprecht2019deepdb,thirumuruganathan2020approximate,ma2021learned}.\nCardinality\/selectivity estimation, has improved considerably leveraging ML  \\cite{yang2019deep,yang2020neurocard,hasan2020deep,zhu2020flat,wang2020we}. Likewise for query optimization \n\\cite{marcus2019neo,kipf2018learned,marcus2021bao},\nindexes \\cite{kraska2018case,ding2020alex,nathan2020learning,ding2020tsunami}, cost estimation \\cite{zhi2021efficient, siddiqui2020cost}, workload forecasting \\cite{zhu2019novel}, DB tuning \\cite{van2017automatic,li2019qtune,zhang2019end}, synthetic data generation \\citep{xu2019modeling,choi2017generating,park2018data}, etc. \n\n\\subsection{Challenges}\\label{challenges}\nAs research in learned DB systems\nmatures, two key pitfalls are emerging. First, if the \"context\" (such as the data, the DB system, and\/or the workload) changes, previously trained models are no longer accurate. Second, training accurate ML models is costly. Hence, retraining from scratch when the context changes should be avoided whenever possible.\nEmerging ML paradigms, such as active learning, transfer learning, meta-learning, and zero\/few-shot learning are a good fit for such context changes and have been the focus of recent related works   \\cite{ma2020active, hilprecht2021one, wu2021unified}, where the primary focus is to glean what is learned from existing ML models (trained for different learning tasks and\/or DBs and\/or workloads), \nand adapt them for new tasks and\/or DBs, and\/or workloads, while avoiding the need to retrain models from scratch.\n\n{\\bf OOD Data insertions.} In analytical DBs data updates primarily take the form of new data insertions. New data may be OOD (representing new knowledge --  distributional shifts), rendering previously-built ML models obsolete\/inaccurate.\nOr, new data may not be OOD. In the former case, the model must be updated and it must be decided how the new data could be efficiently reflected in the model to continue ensuring accuracy.\nIn the latter case, it is desirable to avoid updating the model, as that would waste time\/resources.\nTherefore, it is also crucial to check (efficiently) whether the new data render the previously built model inaccurate. \nHowever, related research has not yet tackled this problem setting, whereby\n\\textit{models for the same learning tasks (e.g., AQP, DG, CE, etc.) trained on old data, continue to provide high accuracy for the new data state} (on old and new data, as queries now may access both old data and new data, old data, or simply the new data).\nRelated work for learned DB systems have a limited (or sometimes completely lack the) capability of handling such data insertions (as is independently verified in \\cite{wang2020we} and will be shown in this paper as well).\n\n{\\bf Sources of Difficulty and Baselines.} \nIn the presence of OOD, a simple solution is adopted by some of the learned DB components  like Naru \\cite{yang2019deep}, NeuroCard \\cite{yang2020neurocard}, DBest++ \\cite{ma2021learned}, and even the aforementioned transfer\/few-shot learning methods \\cite{wu2021unified, hilprecht2021one}. That is to \"fine-tune\" the original model $M$ on the new data. Alas, this is problematic. For instance, while a DBest++ model on the \"Forest\" dataset has a 95th percentile q-error of 2, updating it with an OOD sample using fine-tuning increases the 95th q-error to ~63. A similar accuracy drop occurs for other key models as well -- \\cite{wang2020we} showcases this for learned CE works.\nThis drastic drop of accuracy is due to the fundamental problem of \\textit{catastrophic forgetting}{} \\cite{mccloskey1989catastrophic}, where retraining a previously learned model on new tasks, i.e. new data, causes the model to lose the knowledge it had acquired about old data. To avoid \\textit{catastrophic forgetting}{}, Naru and DBest++ suggest using a smaller learning rate while fine-tuning with the new data. This, however, causes another fundamental problem, namely \\textit{intransigence}, \\cite{chaudhry2018riemannian} whereby the model resists fitting to new data, rendering queries on new data inaccurate.\n\nAnother simple solution to avoid these problems would be to aggregate the old data and new data and retrain the model from scratch. However, as mentioned, this is undesirable in our environment. As a concrete example, training Naru\/NeuroCard on the \"Forest\" dataset (with only 600k rows) on a 40-core CPU takes ca. 1.5 hours. Similarly high retraining overheads are typically observed for neural network models, for various tasks.\nAnd, retraining time progressively increases as the DB size increases. \n\nTherefore, more sophisticated approaches are needed, which can avoid \\textit{intransigence} and \\textit{catastrophic forgetting}{},\nupdate models only when needed and do so while ensuring much smaller training overheads than retraining from scratch and at the same time ensure high accuracy for queries on old and new data. While for some tasks, like CE, some researchers question whether achieving very high accuracy through learned models will actually help the end-task (query optimization) \\cite{marcus2021bao}, for tasks like AQP (which is itself the end-task) and for DG (with classification as the end-task) high accuracy is clearly needed, as shown here. Even for CE, with OOD data, accuracy can become horribly poor, as shown here, which is likely to affect query optimization.\n\n\\subsection{Contributions} \\label{contribution}\nTo the best of our knowledge, this work proposes the first updatability framework (DDUp) for learned DBs (in the face of new data insertions possibly carrying OOD data)\nthat can ensure high accuracy for queries on new and\/or old data. \nDDUp is also efficient and \nit can enjoy wide applicability, capable of being utilized for different NNs and\/or  different learning tasks (such as AQP, DG, CE, etc.). DDUp consists of a novel OOD detection and a novel model-update module. More specifically, the contributions of DDUp are:\n\n\\begin{itemize}[leftmargin=10pt]\n    \\item A general and principled two-sample test for OOD detection. Generality stems from it being based on the training loss function of the NNs. Compared to prior art, it introduces no extra costs and overheads, and could be used with different NNs, with different loss functions, in different applications. To further minimize detection time, it is divided into offline and online phases.\n    \\item A novel and general formulation of transfer-learning based on sequential self-distillation for model updating. This formulation allows a higher degree of freedom in balancing tasks w.r.t new and old data, can adapt to different models and tasks, and maximizes performance via self-distillation.\n    \\item Importantly, DDUp can be used by any pre-trained NN without introducing any assumptions on models or requiring additional components that might require to retrain models or incur more costs. Here, we instantiate it for three different tasks (namely, the CE task, using the Naru\/NeuroCard deep autoregressive network (DARN) models  \\cite{yang2019deep, yang2020neurocard},  the AQP task, using the DBEst++ mixture density network (MDN) model \\cite{ma2021learned}, and for the DG task, using the Tabular Variational AutoEncoder (TVAE) model \\cite{xu2019modeling}) each of which employs a different NN type. These are representative learning tasks and networks with evident importance in DBs and beyond. These instantiations are also novel, showing how to distil-and-update MDNs, DARNs, and TVAEs.\n    \\item Finally, DDUp is evaluated using six different datasets and the three instantiated learned DB components, for AQP, CE, and DG\n\\end{itemize}\n\n\n\\subsection{Limitations} \\label{limits}\nDDUp focuses only on data insertions, which are essential and dominant in analytical DBs, and not on updates in place and deletes, which are prevalent \nin transactional DBs.\nNonetheless, the latter touch upon an open problem in the ML literature, namely $\"unlearning\"$, \nwhere it typically concerns privacy (e.g., removing sensitive data from images in classification tasks) \n(e.g., \\citep{sekhari2021remember, golatkar2020eternal}).\nStudying unlearning for DB problem settings is a formidable task of its own and of high interest for future research.\n\n\nAlso, DDUp is designed for NN-based learned DB components. This is so as neural networks are a very rich family of models which have collectively received very large attention for learned DBs. Extending DDUp principles beyond NN models is also left for future research.\n\n\n\\section{The Problem and Solution Overview} \\label{problemdef}\n\\subsection{Problem Formulation} \\label{problemformulation}\nConsider a database relation \\(R\\) with attributes \\(\\{A_1, A_2, ..., A_m\\}\\). This can be a raw table or the result of a join query. Also consider a sequence of \\(N\\) insertion updates denoted by \\(I=\\{I_1,I_2,...I_N\\}\\). Each \\(I_t\\) is an insert operation which appends a data batch \\(D_t=\\{(A_1, A_2, ..., A_m)_t^{(i)}; i=1,..., n_t\\}\\) to \\(R\\), where \\(n_t\\) is the number of rows. Let \\(S_t\\) be a sufficient sample of \\(D_t\\) and \\(S^{\\leq}_{t-1}\\) be a sufficient sample from \\(\\cup_{j=0}^{t-1} D_j\\). We naturally assume that \\(|R|\\) is finite. \nAnd, due to the training restrictions of existing models, we also make the natural assumption:\n\\[\\forall A_i \\in R: supp(D_{t}(A_i)) \\subseteq supp(D_{t-1}(A_i)) \\]\nwhere \\(supp(D(A_i))\\) is the support of attribute \\(A_i\\) in dataset \\(D\\). This assumption satisfies the condition based on which the domain of each attribute is not violated in the upcoming update batches. \n\n\\textbf{Statistical test for data changes}. We define out-of-distribution detection as a two-sample hypothesis test between a sample of historical data and a sample of the new data. Let \\(S^{\\leq}_{t-1}\\) have a joint distribution of \\(P(A_1,\\dots, A1_m) \\equiv \\mathbb{P}\\) and \\(S_{t}\\) have a joint distribution of \\(Q(A_1,\\dots, A_m) \\equiv \\mathbb{Q}\\). We define the null hypothesis \\(H_0: \\mathbb{P}=\\mathbb{Q}\\) which asserts that \\(S_{t}\\) and \\(S^{\\leq}_{t-1}\\) are coming from a same distribution; and the alternative hypothesis \\(H_A: \\mathbb{P}\\neq \\mathbb{Q}\\) which declares that the two samples are generated by two different distributions. \n\n\\textbf{Incrementally updating the model}. Consider for \\(I_0\\) a model \\(M_{0}\\) is trained by minimizing a loss function \\(\\mathscr{L}(D_{0};\\Theta_0\\)). This model may be stale for \\(I_t; t>0\\). Ideally, the goal of incremental learning is: at time \\(t\\) train a model \\(M_{t}\\) that minimizes a function over \\(\\sum_{i=1}^{t} \\mathscr{L}(D_{i};\\Theta_i)\\). This new model should not forget  \\(\\{I_{i}; i=0,1,...,t-1\\}\\) and also learn \\(I_t\\).\n\n\n\\subsection{A High Level View of DDUp}\\label{highlevel} \nThe overall architecture of DDUp is depicted in \\autoref{fig:Arch}.\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=0.9\\linewidth, height=4cm]{figures\/Detailed-DDUp-Architecture.png}\n  \\caption{The overall structure of DDUp. DDUp uses the latest model and previous data to build a sampling distribution for the two-sample test, and updates the learned component based on the shift in the data distribution.}\n  \\label{fig:Arch}\n  \\vspace{-0.3cm}\n\\end{figure*}\n\nDDUp process batches of tuples at a time. Such batched handling of insertions is typical in analytical DBs. Furthermore, this takes into account that \nthe effect of single tuples is usually negligible for the overall large space modelled by NNs. And, for most tasks like CE, AQP and DG, the effect of single tuples in the final result is very small, considering the large sizes of tables. And batching amortizes detect-and-update costs over many insertion operations.\n\nUpon a new batch insertion, DDUp takes the latest model \\(M_{t-1}\\), and performs a bootstrapping sampling from the previous data to build the sampling distribution for the average loss values. DDUp uses this distribution to calculate a significance level corresponding to a confidence interval (e.g a 95th confidence interval). The general idea is that if the new data is similar to the previous data (IND in \\autoref{fig:Arch}), the loss values of \\(M_{t-1}\\) for this new data should lie within the threshold. This means that the new data has the same distribution and therefore the model could be left intact (updating maybe just the hyper-parameters of the system, including possible frequency tables and other table statistics. Alternatively, a simple fine-tuning can be performed to adapt the model to the new data.\n\nIf the loss values exceeded the threshold, this implies that the data distribution has significantly changed. DDUp will deploy a teacher-student transfer learning method based on knowledge distillation to learn this new distribution without forgetting the knowledge of the old data. In this framework, while the student directly learns the distribution of the new data, the teacher act as a regularizer to make the student also learn about the old distribution.\n\\vspace{-0.3cm}\n\\section{Out-of-Distribution Detection} \\label{driftdetect}\n\\subsection{Background} \\label{oodback}\nIn ML, OOD is typically addressed from a classification perspective. Formally, assume \\(D\\) is a dataset of \\((x,y)\\) pairs which are drawn from a joint distribution, \\(p(x,y)\\), where \\(x \\in \\mathcal{X} := \\{x_1, x_2, \\dots, x_n\\}\\) is the input (independent variable) consisting of \\(n\\) features, and \\(y \\in \\mathcal{Y} := \\{1,2, \\dots, k\\}\\) is the label corresponding to one of the \\(k\\) in-distribution classes. A sample \\((x,y)\\), that probably is generated by a different distribution than \\(p(x,y)\\), is called OOD, if \\(y \\notin \\mathcal{Y}\\), i.e it does not belong to any previously seen classes. \n \nA similar problem has previously been addressed in statistics as {\\it concept drift} detection, where different types of shifts are distinguished by expanding \\(p(x,y)\\) using the Bayes rule:\n\\begin{equation}\\label{bayesrule}\n p(x,y)=p(x)p(y|x)\n\\end{equation}\nBased on Eq. \\ref{bayesrule}, changes in \\(P(y|x)\\) are usually referred to as \\textit{Real drift}, while changes in \\(P(x)\\) are called \\textit{virtual drift} \\cite{gama2014survey}. In \\(X\\rightarrow y\\) problems the latter mostly is known as \\textit{covariate shift}.\nDeciding which drift to detect is dependent on the underlying models. For example, deep autoregressive networks (e.g., used by \\cite{yang2019deep}) learn the full joint distribution of a table. Hence, they are sensitive to \\textit{covariate shift} upon insertions. \nOn the other hand, mixture density networks (e.g., used by \\cite{ma2021learned}), model the conditional probability between a set of independent attributes and a target attribute. Hence, for these models, one would be interested in detecting \\textit{real shift}.\n\\vspace{-0.2cm}\n\\subsection{Loss based OOD Detection} \\label{llfordrift}\nThere are several challenges that make it difficult to simply adopt one of the OOD detection algorithms in the ML or statistical learning literature.\nFirst, DB tables are multivariate in nature and learned models are usually trained on multiple attributes. As a result, uni-variate two-sample tests like Kolmogorov\u2013Smirnov (KS) test are not suitable for this purpose. Second, the test should introduce low overheads to the system as insertions may be frequent. Therefore,  multivariate tests like kernel methods that require to learn densities and perform expensive inference computations are not desirable. Third, we aim to support different learning tasks for which different models might be used. Thus, most of OOD detection methods in ML that are based on probability scores (confidence) of classification tasks are not useful here. Moreover, the test should be able to adapt efficiently to the case where insertions occur within old data, that is, without having to recalculate baseline thresholds etc.\n\nAn efficient OOD detection method is now proposed that resolves all above issues by leveraging the underlying ML models themselves. Central to most learned data system components is the ability to derive from the underlying data tables a model for the joint or conditional data distribution like \\(p(x)\\) or \\(p(y|x)\\). A model usually achieves this by learning a set of parameters \\(\\Theta\\) that represent a function \\(f\\) by iteratively optimizing over a loss function as follows:\n\n\\begin{equation} \\label{generalopt}\n    f_\\Theta = \\argmin_{f \\in \\mathcal{F}} \\frac{1}{n} \\sum_{i=1}^n \\mathscr{L}(f(x);\\Theta) + \\Omega(f)\n\\end{equation}\n\nwhere, \\(\\Omega\\) is a regularizer term, \\(n\\) is the number of samples, and \\(f\\) could be the outputs of the model in the last layer (called \\textit{logits}), or the probabilities assigned by a \"softmax\" function.\n\n\nWe will later discuss different loss functions in more details when instantiating different models. In general, loss functions are usually highly non-convex with many local mimina. However, a good learning strategy will find the global minimum. Because of the large data sizes, training is usually done by iterating over mini-batches and a gradient descent algorithm updates the parameters based on the average of loss of the samples in each mini-batch per iteration. For the rest of the paper, when we mention 'loss value' we mean average of losses of the samples in a batch. Once the model is trained, i.e. the loss values have converged, the model can serve as a transformer to map (high-dimensional) input data to the one-dimensional loss functions space around the global minimum. Accordingly, the previous data (seen by the model) are closer to the global minimum compared to the out of distribution data.\n\nThe above discussion explains the possibility to compare in- and out-of distribution data just by relying on the underlying models without any further assumptions\/components, in a low-dimensional space. With these in hand, we can perform a statistical testing to compare the loss values of old data and new data. In the following we will explain a two-sample test for this purpose. \n\n\\subsection{A Two-Sample Test Procedure}\nThe steps for a two-sample hypothesis test are: \n1. Define the null, \\(H_0\\), and alternative hypothesis, \\(H_A\\). \n2. Define a test statistic \\(d\\) that tests whether an observed value is extreme under \\(H_0\\). \n3. Determine a significance level \\(\\delta\\in[0,1]\\) that defines the \\(type\\mhyphen1\\ error\\) (false positives) of the test. \n4. Calculate \\(p\\mhyphen value\\) which equals the probability that a statistical measure, e.g. distance between two distributions, will be greater than or equal to the probability of observed results.\n5. If \\(p\\mhyphen value <= \\delta\\) then the \\(p\\mhyphen value\\) is statistically significant and shows strong evidence to reject \\(H_0\\) in favor of \\(H_A\\). Otherwise, the test failed to reject \\(H_0\\). \n\nThe main challenge herein is how to calculate the test significance of the test statistic, i.e the \\(p\\mhyphen value\\). As explained in Section \\ref{problemdef}, we aim to detect if a new data that is inserted to the system at time \\(t\\) has a different distribution than the previous data. Consider \\(S_{t-1}^{\\leq}\\) be a sample of the previous data and \\(S_{t}\\) be a sample of the newly inserted data. Let \\(d(S_{t-1}^{\\leq},\\ S_{t})\\) be a distance function that measures the distance between the two samples. If \\(P_d\\) is the distribution that explains the test statistic \\(d\\) under the null hypothesis, then the test significance could easily be computed by \\(p\\mhyphen value=P(P_d < d | H_0)\\). Note that since we assume that our test statistic is a distance function, we would perform a one-side left-tail test.\n\n\\textbf{Choosing the test statistic}. The test statistic should reflect the similarity of new data to old data. According to our discussion in Section \\ref{llfordrift}, we use\nthe loss function values after convergence of the models. We use a linear difference between the loss values of the two samples as our test statistics as follows:\n\\begin{equation}\\label{teststatistic}\nd(S_{t-1}^{\\leq},S_{t}) = \\frac{1}{|S_{t-1}|}\\sum_{s\\in S_{t-1}}\\mathscr{L}(s;\\Theta) - \\frac{1}{|S_t|}\\sum_{s\\in S_{t}}\\mathscr{L}(s;\\Theta)\n\\end{equation}\n\nwhere \\(\\mathscr{L}\\) is a loss function achieved by training model \\(M\\) with parameters \\(\\Theta\\). From Eq. \\ref{teststatistic} follows that if the loss function is Negative Log Likelihood, and the likelihoods are exact, the test statistics will be the logarithm of the well-known \\textit{likelihood-ratio} test. Eq. \\ref{teststatistic} also gives intuition about the effect size:  the larger \\(d\\) is, the larger the difference between two data distributions would be.\nAlthough many of the learned DB models are trained by maximizing likelihood, some other models (e.g., regressions) are trained using a \\textit{Mean-Squared-Error} objective. It has been shown \\cite{watkins1992maximum} that MSE optimization maximizes likelihood at the same time. Therefore, the form of the distance function in Eq. \\ref{teststatistic} still holds. \nThe important consequence of Eq. \\ref{teststatistic} is that, under i.i.d assumptions for both samples, it can be shown that the central limit theorem holds for \\(P_d\\) \nhence, it has a normal limiting distribution with a mean at 0 and unknown standard deviation. The normality of \\(P_d\\) allows us to make inference based on the confidence intervals. To estimate the standard deviation (std), we perform a bootstrapping approach.\n\n\\subsection{Offline and Online Steps}\nThe main bottleneck of such an OOD detection is bootstrapping. Fortunately, this part could be performed offline before data insertion. In the offline phase, \nwe draw \\(n\\) bootstrap samples of size \\(|S^{\\leq}_{t-1}|\\) from \\(S^{\\leq}_{t-1}\\). (In practice, when we have access to the original data, we make $n$ bootstrap samples of size $|S_{t-1}^{\\leq}|$) from $D_{t-1}^{\\leq}$). We use the model \\(M_{t-1}\\) to compute the likelihoods (or other losses) of each sample and create a sampling distribution using them. Then, we calculate the standard deviation of the sampling distribution, \\(std\\), and use it to find the significance level. In the online phase, we make a sample of the new data, \\(S_{t}\\) and use the latest model, \\(M_{t-1}\\) to calculate the likelihood of \\(S_{t}\\). Finally we compare the test statistic with the threshold. If \\(d > 2\\times std\\) (equivalently \\(p\\mhyphen value \\leq \\delta\\) where \\(\\delta=0.05\\)) we declare a significant shift in data and reject the null hypothesis in the favor of the alternative hypothesis. Otherwise the test fails to reject the null hypothesis and signals \"in-distribution\". \n\n\\subsection{The Test Errors}\\label{testerrors}\nThere are two errors associated with a hypothesis testing. \\textit{type-1 error} is rejecting the null hypothesis when it should not.  \\textit{Type-2 error} is the error of accepting the null hypothesis when it should be rejected. The first one introduces false positives to the system and the second causes false negatives. \nFalse positives (FPs) are only a (rather small) performance concern only, spending time to update the model while accuracy is preserved. False negatives (FNs), however, can cause a loss of accuracy.  \nTherefore, the system can afford to be stricter with respect to the significance level, in order to reduce the risk of false negatives and accuracy loss.\n\nDDUp uses the loss of the trained NNs for OOD detection. Sometimes NNs could be over-confident \\cite{nguyen2015deep,ren2019likelihood,nalisnick2018deep} which may introduce bias. \nHowever, we have not witnessed it for our tasks here on tabular data.\nIf there were bias, the FP and FN rates discussed above would signal it. \nWe have evaluated DDUp with respect to FPs\/FNs in Section \\ref{oodeval} showing that this is not a concern.\n\n\\section{Model Update} \\label{KD}\nIn this section, we propose a transfer-learning based method that can retain previous knowledge of the model while adapt it to the new insertions. The OOD detection module will either output 'in-distribution' or 'out-of-distribution' signals.\n\n\\textbf{The in-distribution case}. When no drift occurs, the new data distribution is similar to that of the historical data and this distribution could be represented by a similar parameter space of the latest model, \\(M_{t}\\). \nHence, the learned component of the system could remain unchanged. More specifically, the framework can copy \\(M_{t}\\) to \\(M_{t+1}\\) and update the required meta-data associated with the system (such as the frequency tables in DBEst++, or table cardinalities in Naru\/NeuroCard).  Even if there are slight permutations in data, fine-tuning the latest model's parameters on the new data will adjust it to the general representation of both old and new data. \nWe will show that when knowing that data is not OOD, \\textit{fine-tuning}{} with a relatively small learning rate, can retain model performance. \nSpecifically, with an \\textbf{in-distribution} signal at time \\(t+1\\), \\(M_{t}\\) is retrained on \\(S_{t+1}\\) with a small learning rate, $lr$. This learning rate could be tuned, as a hyper-parameter.\nWe intuitively set \\(lr_{t} = \\frac{|D_{t+1}|}{|D_{t}^\\leq|}\\ \\times \\ lr_{0}\\) and experimentally show that it is a good choice. \n\n\\textbf{The OOD case}. With a distributional shift,\nby fine-tuning on new data, the model's parameters would bias toward the new data distribution. Even smaller learning rates cause tiny deviations from the previous parameter space which may yield large errors during inference. And, retraining using all the data from scratch is too time consuming. Thus, we propose an updating approach grounded on the transfer-learning paradigm. The general idea is to use the learned model \\(M_{t}\\) and incorporate it in training \\(M_{t+1}\\). To this end, we utilize the \\textit{knowledge distillation}{} principles, which help to transfer the previously learned knowledge to a new model. Our rationale for such a model updating approach is based on the following: \n\\begin{itemize}[leftmargin=*]\n    \\item Distillation has several benefits including: faster optimization, better generalization, and may even outperform the directly trained models. \\cite{yim2017gift}.\n    \\item It is accurate for queries on old  as well as new data.\n    \\item It allows us to control the weights for queries on new and old data with just a couple of parameters.\n    \\item It is efficient memory-wise as well as computationally-wise, compared to methods like Gradient Episodic Memory, or Elastic Weight Consolidation and PathInt (cf. Section \\ref{litraturere})\n    \\item It does not make any assumptions about the training of the underlying models. This property, is especially desirable since: a) we can use it to update different neural networks; b) it prevents the high costs of rebuilding base models; c) different pre-processings could be left intact. For instance, Naru, DBEst++ and TVAE all use completely different types of embedding\/encoding. DDUp can update the model regardless of these differences.\n\\end{itemize}\n\n\\subsection{General Knowledge Distillation (KD)}\nKD was first introduced in \\cite{hinton2015distilling} for $model \\  compression$ by transferring knowledge from an accurate and \"cumbersome\" model, called \\textit{teacher}, to a smaller model called \\textit{student}. In its basic form, instead of fitting the student model directly to the actual data \\textit{labels}, one would use the class probability distribution learned by the teacher to fit the student model. Hinton et al. \\cite{hinton2015distilling} argued that small probabilities in \"wrong\" label logits, known as \"soft labels\", include extra information called \"dark knowledge\" that result in better learning than actual \"hard labels\". Distillation has since been extensively studied. \\autoref{fig:kdfig} shows a general view of the principles of a distillation process. A small dataset referred to as \\textit{transfer-set} is fed into a pre-trained model (teacher) and a new model (student) to be trained. A $distillation \\ loss$ is calculated using the predictions of the pre-trained model instead of the actual labels. This loss and a typical loss using actual labels will be used to train the new model. \n\n\\begin{figure}[hb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/distillation-diagram.png}\n    \\vspace{-0.2cm}\n    \\caption{The knowledge distillation process.}\n    \\label{fig:kdfig}\n    \\vspace{-0.35cm}\n\\end{figure}\n\nTo formulate \\textit{knowledge distillation}{}, consider a model with parameters \\(\\Theta\\), representing a function \\(f_t\\) (\\(t\\) for teacher) which has been trained via Eq. \\ref{generalopt}. We would like to transfer knowledge from this teacher model to a student model with parameter \\(\\Theta'\\), representing a function \\(f_s\\). This new model could be trained as follows:\n\n\\begin{equation} \\label{distillopt}\n    f_{s\\Theta'} = \\argmin_{f \\in \\mathcal{F}} \\frac{1}{|tr|} \\sum_{i\\in tr} \\left[\\lambda\\mathscr{L}_d(f_s(i);f_t(i);\\Theta;\\Theta') + (1-\\lambda)\\mathscr{L}(f_s(i);\\Theta')\\right]\n\\end{equation}\n\\\\\nfor weight \\(\\lambda\\), distillation loss \\(\\mathscr{L}_d\\), and transfer-set \\(tr\\). \n\n\\subsection{DDUp: Updating By Knowledge Distillation}\\label{upbykd}\n\n\\cite{furlanello2018born,seq-self-distill} showed that, for classification tasks, if instead of having a compact student model, one uses the same architecture of the teacher, and repeat distillation sequentially for several generations, the student models in the later generations could outperform the teacher model. This approach is called {\\it sequential self-distillation}.\nInspired by this and anticipating that this will be valid for our learning tasks, DDUp also employs a sequential self-distillation approach.\n\nTo update a model using KD,\na copy of the previously trained model becomes the new student. Then,  the student is updated using a distillation loss (to be defined soon). After updating, the previous teacher is replaced with the new updated model. This cycle repeats with every new insertion batch.\n\\begin{comment}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/Sequential Distillation.png}\n    \\caption{Sequential updating in a self-distillation scheme}\n    \\label{fig:sequpdate}\n\\end{figure}\n\\end{comment}\n\nTo formulate our training loss function, we consider two aspects that we would like to have in our updating scheme. First, to have control over the the new data\/queries versus the old data\/queries. Second, to make it general so that different learned DB systems could adopt it. As such, we first write down the general form of the total loss function and then, use cross-entropy and mean-squared-error as the loss functions to instantiate different models. Training in each update step is as follows:\n\\\\\n\\begin{equation} \\label{totalloss}\n\\begin{split}\n    f_{s\\Theta'} = \\argmin_{f \\in \\mathcal{F}} & \\bigg(\\alpha \\times \\frac{1}{|tr|} \\sum_{x\\in tr} \\big[\\lambda\\mathscr{L}_d(f_s(x),f_t(x);\\Theta') \\\\ \n    & + (1-\\lambda)\\mathscr{L}(f_s(x);\\Theta')\\big] \\\\\n    & + (1-\\alpha) \\times \\frac{1}{|up|}\\sum_{x\\in up}\\mathscr{L}(f_s(x);\\Theta') \\bigg)\n\\end{split}\n\\end{equation}\n\\\\\nHere, \\(\\alpha\\) and \\(\\lambda\\) are the new data and the distillation weights, respectively. Also, \\(tr\\) and \\(up\\) are the transfer-set and the update batch. \nIn summary, the rationale for proposing this novel loss function is: \nThe transfer-set term acts as a regularizer to avoid overfitting on new data.\nThe same goal is also helped by self-distillation (when copying the teacher to the student). Additionally, as mentioned sequential self-distillation \\cite{seq-self-distill} may attain increasingly higher accuracy, even outperforming \"retrain from scratch\"\n(cf. Section 5.3).\n\nFor models that provide a conditional probability in the last layer of the network (e.g. using a Softmax function), an annealed cross-entropy loss will be employed. Otherwise, we utilize mean-squared-error using the logits from the last layer of the network. Eq. \\ref{cedistillloss} and Eq. \\ref{mseloss} show these two loss functions. \n\\\\\n\\begin{equation}\\label{cedistillloss}\n    \\mathscr{L}_{ce}(D_{tr};z_t,z_s) = - \\sum_{i\\in [k]} \\frac{exp(z_{t_i}\/T)}{\\sum_{j\\in [k]}exp(z_{t_j}\/T)} \\log \\frac{exp(z_{s_i}\/T)}{\\sum_{j\\in [k]}exp(z_{s_j}\/T)}\n\\end{equation}\n\\\\\n\\begin{equation} \\label{mseloss}\n    \\mathscr{L}_{mse}(D_{tr};z_t,z_s) = \\sum_{i\\in[|z_t|]}(z_{t_i} - z_{s_i})^2\n\\end{equation}\n\\\\\nwhere \\(D_{tr}\\) is the \\textit{transfer-set} , \\(T\\) is a temperature scalar to smooth the probabilities so that it produces \"softer\" targets, and \\([k]\\) is the vector \\([0,1,\\dots,n]\\) which are the class probabilities and \\([|z_t|]\\) indicates the logits of the network.\n\n\\subsection{Instantiating the Approach}\n\n\\textbf{Mixture Density Networks}. MDNs consist of an NN to learn feature vectors and a mixture model to learn the \\textit{probability density function} (pdf) of data. Ma et al. \\cite{ma2021learned} uses MDNs with Gaussian nodes to perform AQP. For the Gaussian Mixture, the last layer of MDN consists of three sets of nodes \\(\\{\\omega_i, \\mu_i, \\sigma_i\\}_{i=1}^m\\) that form the pdf according to Eq. \\ref{mdneq}. \n\n\\begin{equation}\\label{mdneq}\n\\hat{P}(y|x_1, ..., x_n) = \\sum_{i=1}^{m}\\omega_i.\\mathscr{N}(\\mu_i, \\sigma_i)    \n\\end{equation}\n\nwhere \\(m\\) is the number of Gaussian components, \\(y\\) is the dependent variable and \\((x_1, ..., x_n)\\) is a set of independent variables, \\(w_i\\) is the weight of the \\(i^{th}\\) Gaussian with a mean of \\(\\mu_i\\) and a standard deviation of \\(\\sigma_i\\).\nFor MDNs, we define distillation loss as follows:\n\n\\begin{equation} \\label{mdnkdloss}\n\\mathscr{L}_d = \\mathscr{L}_{ce}(D_{tr}, \\omega_{t}, \\omega_{s}) + \\mathscr{L}_{mse}(D_{tr}, \\mu_{t}, \\mu_{s}) + \\mathscr{L}_{mse}(D_{tr}, \\sigma_{t}, \\sigma_{s})\n\\end{equation}\n\nThis summation of terms help us retain both the shape of data distribution as well as the intensity levels. \n\n\\textbf{Deep Autoregressive Networks}. The Naru and NeuroCard cardinality estimators \\cite{yang2019deep, yang2020neurocard} use deep autoregressive networks (DARNs) to approximate a fully factorized data density. DARNs are generative models capable of learning full conditional probabilities of a sequence using a masked autoencoder via Maximum Likelihood. Once the conditionals are available, the joint data distribution could be represented by the product rule as follows:\n\\[\n\\hat{P}(A_1, A_2, \\dots, A_n) = \\hat{P}(A_1)\\hat{P}(A_2|A_1)\\dots \\hat{P}(A_n|A1,\\dots ,A_{n-1})\n\\]\n\nwhere \\(A_i\\) is an attribute in a relation \\(R\\). Naru and NeuroCard use cross-entropy between input and conditionals as the loss function. This allows us to formulate the distillation loss function using the conditionals of the teacher and the student networks. Also, in Naru and NeuroCard, each conditional is calculated using a set of logits, hence we average over all as follows:\n\\begin{equation} \\label{narukdloss}\n\\mathscr{L}_d = \\frac{1}{|A|}\\sum_{i=1}^{|A|}\\mathscr{L}_{ce}(D_{tr}, z_{s_i}, z_{t_i})\n\\end{equation}\n\nWhere \\(|A|\\) is the number of attributes corresponding to the number of conditionals.\n\n\\textbf{Variational Autoencoders}. VAEs have been used for a number of DB components: \\cite{thirumuruganathan2020approximate} for AQP, \\cite{hasan2020deep} for CE, and \\cite{xu2019modeling} for synthetic tabular data generation. \nThey are a type of autoencoders that instead of learning deterministic encoder, decoder, and compressed vector (known as bottleneck), they learn a probabilistic encoder, decoder, and a latent random variable instead of the compressed vectors. (For more details, see the seminal paper \\cite{kingma2013auto}). \nInterestingly, a VAE is trained using a different loss function, known as Evidence-Lower-Bound (ELBO) loss (which amounts to a lower bound estimation of the likelihoods).\nHere we shall use TVAE for learned synthetic tabular data generation (of particular importance in privacy-sensitive environments, or when data is scarce for data augmentation purposes, or when wishing to train models over tables and accessing raw data is expensive in terms of time or money).\n\nTo distill a VAE, one must cope with the random noise added to the input of the decoder by the latent variable. For that, the latent variable in the teacher network is removed, and we use the same noise generated by the student in the teacher. The reason for doing this is that distillation tries to teach the student to behave like the teacher for a specific observation or action. If there is randomness, the student might mimic the teacher's behaviour for a completely different observation. After this change, the corresponding logits of the encoder\/encoder of the student and the teacher are compared using MSE. Finally, the loss function is:\n\\\\\n\\begin{equation}\n    \\mathscr{L}_d = \\frac{1}{2}( \\mathscr{L}_{mse}(D_{tr}, z_t^{(e)}, z_s^{(e)}) + \\mathscr{L}_{mse}(D_{tr}, z_t^{(d)}, z_s^{(d)}) )\n\\end{equation}\n\\\\\nwhere, \\(e\\) and \\(d\\) correspond to the encoder and the decoder networks. \n\n\n\n\\subsection{An Example}\nWe create a simple synthetic dataset consisting of a categorical attribute, \\(x\\), with 10 \\(distinct\\mhyphen values=\\{1,2,3,\\dots,9,10\\}\\), and with each category having 1000 real values. The dataset is balanced and the real values for each category are generated by a \\textit{Mixture of Gaussians} (MoG) with five peaks. \\autoref{fig:toyexam}.a is the dataset corresponding to \\(x=1\\). We fit a \\textit{Mixture Density Network} with ten components on this dataset. \\autoref{fig:toyexam}.b shows a sample generated by this MDN which asserts that the model has perfectly learnt the data distribution. Next, we introduce an update batch generated by a MoG with two different means. \\autoref{fig:toyexam}.c shows the update batches in red color compared to the previous data in blue. We update the previously learned MDN with the proposed loss function in Eq. \\ref{mdnkdloss}. We repeat updates for 50 batches generated with the new MoG. \\autoref{fig:toyexam}.d shows the final distribution learnt by the MDN.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/toy_example.png}\n    \\caption{An example to show how DDUp learns new data without forgetting. 'a' is the histogram of synthetic data corresponding to $x=1$. 'b' is the sample generated by the learned MDN for $x=1$. 'c' shows a sample of an update batch coming from different Gaussians. 'd' is the sample generated by the MDN after being updated by the DDUp loss function. We have performed the update 50 times to see the effect of high frequency updates (This explains the higher frequencies around the last two peaks for 'd').}\n    \\vspace{-0.4cm}\n    \\label{fig:toyexam}\n\\end{figure}\n\n\n\\subsection{Handling Join Operations}\\label{joins}\nDDUp can operate either on raw tables or tables from join results.\nIf the old data $R$ is the result of a join, the new data batch needs to be computed, due to new tuples being inserted in any of the joined tables in $R$.\nConsider at time \\(t-1\\) a model \\(M_{t-1}\\) has been trained on  \\(R=\\bigcup_{j=0}^{t-1}{T_1^j} \\bowtie \\bigcup_{j=0}^{t-1}{T_2^j} \\dots \\bowtie \\bigcup_{j=0}^{t-1}{T_n^j}\\), where $T_r^j$ denotes the new batch for table $T_r$ at time $j$.\nWithout loss of generality, suppose a new insertion operation $I_t$ at time \\(t\\)  adds new data to table \\(T_i\\), denoted \\(T_i^t\\). The new data for DDUp in this setting is \\( D_t = (R\\ \\setminus \\  \\bigcup_{j=0}^{t-1}T_i^{j}) \\bowtie T_i^{t} \\), where $\\setminus$ denotes a (multi)set-difference operator. Therefore, for the detection module, \\(S^{\\leq}_{t-1}\\) is a sample of R, and \\(S_t\\) a sample from $D_t$. Furthermore, during updating the transfer-set is a sample from $R$ and the new data is $D_t$. \nPlease note that all this data preparation and how each model deals with joins is orthogonal to DDUp.\nTherefore, it can be done by either computing the actual joins above or using join samplers like \\cite{zhao2018random,shanghooshabad2021pgmjoins}, as is done in NeuroCard and compared against in Section \\ref{joinexp}.\n\n\\begin{comment}\n\\subsection{Putting it All Together}\nAt time \\(t=0\\) we have data \\(D_{0}\\). We create two samples from \\(D_{0}\\): \n\\( S^{\\leq}_{0}\\) for OOD detection and a sample as the \\textit{transfer-set}. \nWe use \\(D_{0}\\) to train a model \\(M_{0}\\). \nNext, at time \\(t=1\\), a new data batch, \\(D_{1}\\) arrives. \nDDUp follows the following steps:\n\\begin{enumerate}\\setlength\\itemsep{0.5em}\n    \\item  The OOD detection module uses \\(S^{\\leq}_{0}\\) and a sample of \\(D_{1}\\) to test for a significant change in data distribution and issues an 'in-distribution' or 'out-of-distribution' signal.\n    \\item  If 'in-distribution', go to step 3, otherwise, go to step 4.\n    \\item  The Model Update module deploys a baseline method to update \\(M_{0}\\) to \\(M_{1}\\). This baseline method is usually fine-tuning with a smaller learning rate. Finally, go to step 5. \n    \\item  Start the distillation procedure:\n    \\begin{itemize}\\setlength\\itemsep{0.1em}\n            \\item Make an exact copy of \\(M_{0}\\)\n            \\item Feed the transfer-set into \\(M_{0}\\) and \\(M_{1}\\) \n            \\item Use Eq. \\ref{totalloss} to update the copied model \\(M_{1}\\)\n    \\end{itemize}\n    \\item Set the updated model as \\(M_{1}\\), Drop \\(M_{0}\\), Update \\(S^{\\leq}_{0}\\) and \\textit{transfer-set}.\n    \\item End.\n\\end{enumerate}\n\\end{comment}\n\n\\section{Experimental Evaluation} \\label{eval}\nWe evaluate DDUp for three different models for learned DB components: (i) Naru\/NeuroCard \\cite{yang2019deep,yang2020neurocard} which use DARN models for CE; (ii) DBest++ \\cite{ma2021learned} that uses MDNs for AQP; and (iii) TVAE \\cite{xu2019modeling}, that uses variational autoencoders for DG.\nWe evaluate in terms of model accuracy and update time.\nWe use as reference points the baseline update approach provided for AQP and CE (TVAE provides no update approach).\nWe also add as reference points the accuracy when retraining from scratch and when leaving models stale. With respect to \\textit{OOD detection}, we investigate whether it can detect significant data shifts successfully and how this will contribute to the final performance of the underlying models in their specific application, CE, AQP, DG. \nUltimately, the experiments are to address the following questions:\n\n\\begin{itemize}[leftmargin=*]\n    \\item How to best evaluate DDUp? (Section \\ref{setup})\n    \\item Can DDUp accurately detect a distributional shift? (Section \\ref{oodeval})\n    \\item Is DDUp accurate under in- $and$ out-of- distribution settings? (Section \\ref{perfeval})\n    \\item How does DDUp compare to the baseline approaches in accuracy and update time? (Section \\ref{perfeval})\n    \\item What is the effect of distillation? (Section \\ref{distilleval})\n    \\item Is DDUp efficient? (Section \\ref{overheads})\n\\end{itemize}\n\n\\vspace{-0.3cm}\n\\subsection{Experimental Setup} \\label{setup}\nTo establish a dynamic setup, we make a copy of the base table and randomly sample 20\\% of its rows as new data. In this setting, new data follows the previous data distribution which we denote as \\textit{in-distribution}. We introduce distributional drift as is typically done for tabular data settings, say in \\cite{wang2020we}. As such, after making the copy, we sort every column of the copied table individually in-place to permute the joint distribution of attributes. Next, we shuffle the rows and randomly select \\(20\\%\\) of the rows - this now becomes the new data.\nWith these new data, we perform two types of experiments. First, we consider the whole 20\\% sample as a new data batch and update the model with it. Second, to show the updatability in incremental steps, we split the 20\\% data into 5 batches. \nIn general, the size of the transfer-set is a tunable parameter \\cite{hinton2015distilling}, influenced by the dataset complexity, the underlying model generalization ability, and the downstream tasks. \nAfter tuning, we used a 10\\% transfer-set for MDN and DARN and a 5\\% for TVAE, which could be further tuned with methods like Grid search.\n\nDDUp does not impose any further constraints to those of the underlying models. For DBest++ we use a query template with a range and an equality attribute. Also, we use one-hot encoding to encode categorical attributes and normalize the range attribute to \\([-1,1]\\). For Naru\/NeuroCard and TVAE, we use the same settings as explained in their code documentation. We use the learned hyper-parameters of the base model, i.e the model we build at time zero, for all subsequent updates. Furthermore, we intuitively set \\(\\alpha\\) parameter in Eq. \\ref{totalloss} to the fraction of update batch size to the original data size and tune \\(\\lambda\\) for values in \\([9\/10, 5\/6, 1\/4, 1\/2]\\). \n\n\\subsubsection{Datasets} \\label{datasets}\nWe have mainly used three real-world datasets (census, forest, DMV) \n(see \\autoref{tab:Datasets}). These datasets  \nhave been widely used in the learned DB literature.  \nFor CE, \\cite{wang2020we} uses also forest, census and DMV, while NeuroCard\/Naru use JOB\/DMV. For AQP DBEst++ uses TPCDS.  For DG, \\cite{xu2019modeling} uses census and forest. Thus, we have also used census, forest, DMV, and TPCDS (\\texttt{store sales} table, scaling factor of 1). Finally, for join queries, we have used JOB (on IMDB data) and TPCH benchmarks, which are also used in \\cite{yang2020neurocard, yang2019deep}.\n\n\\begin{table}[hb]\n  \\caption{Characteristics of datasets.}\n  \\vspace{-0.3cm}\n  \\label{tab:Datasets}\n  \\begin{tabular}{c c c c} \n    \\toprule\n     Dataset&Rows&Columns&Joint Domain\\\\\n    \\midrule\n     Census & 49K & 13 & $10^{16}$  \\\\\n     Forest & 581K & 10 & $10^{27}$  \\\\\n     DMV & 11.6M & 11 & $10^{15}$  \\\\\n     TPCDS & 1M & 7 & $10^{30}$ \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.5cm}\n\\end{table}\n\n\\subsubsection{Workload} \\label{workload}Each model is evaluated using 2,000 randomly generated queries. These queries are generated at time zero for each model and are used throughout the subsequent updates. When an update batch is performed, the ground truth of the queries will be updated. For Naru\/NeuroCard, we use their generator to synthesize queries: It randomly selects the number of filters per query (forest:[3,8], census: [5,12], TPCDS: [2,6], dmv: [5,12]). Then, it uniformly selects a row of the table and randomly assigns operators \\([=,>=,<=]\\) to the columns corresponding to the selected filters. Columns with a domain less than 10 are considered categorical and only equality filters are used for them. For DBest++, we select a \\(lower\\mhyphen bound\\) and a \\(higher\\mhyphen bound\\) for the range filter and uniformly select a category from the categorical column for the equality filter. Throughout the experiments, we discard queries with actual zero answer. The structure of a typical query in our experiments is:\n\n\\begin{lstlisting}[mathescape=true,\n    basicstyle=\\footnotesize,\n]\nSELECT AGG(y) FROM $T_1 \\bowtie T_2 \\dots \\bowtie T_n$ WHERE $F_1$ AND ... AND $F_{d}$\n\\end{lstlisting}\n\nwhere, \\(F_i\\) is a filter in one of these forms: \\([att_i = val, att_i >= val, att_i <= val]\\). Also, \\texttt{AGG} is an aggregation function like \\texttt{COUNT}, \\texttt{SUM}, \\texttt{AVG}. For DBest++, the query template contains one categorical attribute and one range attribute. As such, we select the following columns from each dataset: census:[\\texttt{age, country}]; forest:[\\texttt{slope, elevation}]; dmv:[\\texttt{body type, max gross weight}]; TPCDS:[\\texttt{ss quantity,ss sales price}]; IMDB:[\\texttt{info type id,production year}]; TPCH:[\\texttt{order date,total price}] where the first\/second attribute is categorical\/numeric. Furthermore, Naru could not train on the full TPCDS dataset as the encodings were too large to fit to memory. Hence, we selected the following columns [\\texttt{ss sold date sk}, \\texttt{ss item sk}, \\texttt{ss customer sk},\\texttt{ss store sk}, \\texttt{ss quantity}, \\texttt{ss net profit}], and made a 500k sample.\n\n\\subsubsection{Metrics}\nFor \\textit{count} queries, we use \\textit{q-error} as follows:\n\n\\begin{equation}\n    error = \\frac{max(pred(q), real(q))}{min(pred(q), real(q))}\n\\end{equation} \n\nFor \\textit{sum} and \\textit{avg} aggregates, we use \\textit{relative-error} as follows:\n\\begin{equation}\n    error = \\frac{|pred(q) - real(q)|}{real(q)}\\times100\n\\end{equation} \n\nAdditionally, Lopez et al. \\cite{lopez2017gradient} introduce the notions of Backward Transfer (BWT) and Forward Transfer (FWT) as new metrics in class incremental learning tasks. BWT is the average accuracy of the model on old tasks, and FWT is the average accuracy of the model on new tasks. Here, we re-frame BWT and FWT.\nWe generate the queries at time \\(0\\) and use them for all update steps. At each step \\(t\\), we calculate \\(diff = real_t(q) - real_{t-1}(q)\\) for each query, \\(q\\), which gives us three set of queries; \\(G_{fix}\\) with \\(diff=0\\), \\(G_{changed}\\) with \\(diff>0\\), and \\(G_{all} = G_{fix} \\cup G_{changed}\\). With these groups, we define three measures. \\(AT\\): average q-error over \\(G_{all}\\). \\(FWT\\): average q-error over \\(G_{changed}\\). \\(BWT\\): average q-error over \\(G_{fix}\\).\n\n\\subsubsection{Evaluating Variational Autoencoders}\nDG is an  interesting learned application which is recently supported using TVAE. Thus, we evaluate DDUp for TVAE. In TVAE, once the training is done, only the decoder network is kept and used, as this is the generator. Hence, we apply our distillation-update method to the decoder network. We evaluate TVAE via the accuracy of an XGboost classifier trained by the synthetic samples, as in \\cite{xu2019modeling}. \nWe hold-out 30\\% of table as the test set, and train two classifiers with original and synthetic data, then predict the classes of the held-out data. We report \\textit{micro f1-score} for classifiers. For census, forest and DMV, we use: \\textit{income}, \\textit{cover-type}, and \\textit{fuel-type}, as the target class, respectively.\nFor TVAE, we created a smaller DMV with 1m records, as training TVAE on the whole DMV is very time\/resource consuming (proving indirectly the need to avoid retraining).\n\n\\subsection{OOD Detection} \\label{oodeval}\n\n\\subsubsection{Loss Functions as Signals}\nWe first show the results of loss\/log-likelihoods when the detector receives samples from the same distributions or from different distributions. The results are shown in \\autoref{tab:avgll}. For Naru\/NeuroCard and DBEst++ we report the actual log-likelihood values (not negatives, so higher is better). For TVAE, we report the ELBO loss values (hence lower is better). \n\n\\begin{table}[hb]\n \\centering\n  \\caption{Average log-likelihood and ELBO loss values of data samples on a trained model. $S_{old}$ is a sample of the previous training data. \"IND\", is a 20\\% sample from a straight copy of the original table; \"OOD\", is a 20\\% sample from a permuted copy of the original table.}\n  \\vspace{-0.2cm}\n  \\label{tab:avgll}\n  \\resizebox{\\linewidth}{!}{%\n  \\begin{tabular}{c c c c  | c c c | c c c } \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multicolumn{3}{c|}{DBEst++} & \\multicolumn{3}{c|}{Naru\/NeuroCard} & \\multicolumn{3}{c}{TVAE} \\\\\n& $S_{old}$  & IND & OOD & $S_{old}$ & IND & OOD & $S_{old}$ & IND & OOD \\\\\n    \\midrule\n     Census & -0.362 & -0.361 & -0.366 & -20.99 & -20.87 & -36.95 & -15.21\t& -15.22 & 81.47  \\\\\n     Forest & -0.0194 & -0.0202 & -0.052 & -43.16 & -43.9 & -141.10 & -19.96 & -20.09 & 142.38 \\\\\n     DMV & 2.520 & 2.532 & 2.444 & -13.74 & -13.16 & -18.67 & 9.114 & 9.28 & 34.95 \\\\\n    \\bottomrule\n  \\end{tabular}}\n  \\vspace{-0.35cm}\n\\end{table}\n\n\\autoref{tab:avgll} shows that the loss function (log likelihood and ELBO in our cases) can reliably signal OOD data.\nInterestingly, this corroborates similar findings in \\cite{detectOOD-iclr17} for classification tasks in various vision and NLP tasks, where the NN outputs can be used to signal OOD. Here we show it for tabular data and for NNs developed for AQP, CE, and DG tasks.  \n\nIn Naru\/NeuroCard and TVAE, when permuting, all columns are sorted individually, hence the large difference in likelihoods. \nFor DBEst++, only the selected columns for a query template have been permuted, yielding a small difference in likelihoods.\n\n\\begin{comment}\n\\begin{table}[hb]\n\\centering\n  \\caption{Change of log-likelihood with the number of permuted columns for a trained autoregressive model. 0 means no columns has been sorted individually therefore the data sample is following the distribution of the training data}\n  \\label{tab:permlevel}\n  \\begin{tabular}{c c c c} \n    \\toprule\n     \\#columns & census & forest & DMV\\\\\n    \\midrule\n0&-20.992&-43.16048&-13.745\\\\\n1&-28.687&-44.673&-14.616\\\\\n2&-31.103&-115.736&-17.935\\\\\n3&-31.201&-127.560&-18.151\\\\\n4&-31.591&-129.549&-18.308\\\\\n5&-34.793&-127.916&-18.955\\\\\n6&-34.359&-127.054&-18.838\\\\\n7&-35.626&-140.589&-18.858\\\\\n8&-35.938&-143.223&-18.836\\\\\n9&-36.969&-141.106&-18.670\\\\\n10&-37.029&&\\\\\n11&-37.243&&\\\\\n12&-36.953&&\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\\end{comment}\n\n\\subsubsection{The two-sample test results} \n\\autoref{tab:driftdetect} shows  results for two-sample testing for OOD detection. The significance level of the test (threshold) is \\(2\\times variance\\) of the bootstrapping distribution, which was obtained by $>$1000 iterations. \nIn each iteration, we use a 1\\% sample with replacement from previous data and a 10\\% sample without replacement from new data to calculate the test statistic. The results show that when data is permuted, the test statistic is far away from the threshold. This means it appears at a great dissonance in the tails of the bootstrapping distribution. \nAnd since the critical value to test for OOD is found by bootstrapping over \\(S_{old}\\), i.e., \\(S^{\\leq}_{t}\\), it will adjust even to small differences when faced with OOD. \nCase in point, the DBEst++ OOD likelihood value for census (which is similar to IND\/$S_{old}$ in \\autoref{tab:avgll}) vs the corresponding test-statistic value in  \\autoref{tab:driftdetect}.\n\n\\begin{table*}[t]\n  \\caption{The test-statistic values. Threshold is  $2\\times variance$ and bs-mean is the mean of bootstrapping distribution. }\n  \\label{tab:driftdetect}\n  \\resizebox{\\textwidth}{!}{\n  \\begin{tabular}{c | c c c c  | c c c c | c c c c } \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multicolumn{4}{c|}{DBEst++} & \\multicolumn{4}{c}{Naru\/NeuroCard} & \\multicolumn{4}{c}{TVAE} \\\\\n& bs-mean & threshold & IND  & OOD & bs-mean & threshold & IND & OOD & bs-mean & threshold & IND & OOD \\\\\n    \\midrule\nCensus&-0.3524 & 0.007 & 0.001 & 0.05 & -21.0076 & 0.0529 & 0.032 & 16.0052 & -15.1834 & 0.6041 & 0.0419 & 100.5126 \\\\\nForest&-0.0228 & 0.0122 & 0.007 & 0.2315 & -41.35 & 0.0141 & 0.0084 & 72.5473 & -19.99 & 0.0868 & 0.0417 & 167.0502 \\\\\nDMV&2.52 & 0.1287 & 0.0145 & 4.5745 & -13.7674 & 0.0012& 0.0007& 5.1145 & 9.1209 & 0.0177 & 0.0015 & 25.1398 \\\\\n    \\bottomrule\n  \\end{tabular}}\n\\end{table*}\n\n\n\n\\subsubsection{FP and FN rates in OOD detection}\\label{fpfnrates}\n\nTo evaluate OOD detection, we measure FP and FN rates (FPR, FNR). \nWe created an OOD test-set and an IND test-set, each equaling half the original size of the table. The latter is just a random sample from the original table. The former is constructed as follows. The perturbed data is obtained by perturbing one or more of five columns of the table, say $C1, \\ ... \\ C5$. First we perturb $C1$ and take a sample of the resulting table of size $10\\%$ and append it to the OOD test-set. Then we perturb $C1$ and $C2$ and similarly sample and append it to the OOD test-set. We repeat this for perturbations on $C1, C2, C3$, on $C1, C2, C3, C4$, and on $C1, C2, C3, C4, C5$, ending up with an OOD test-set of size 50\\% of the original table. Note that this setup creates a more-challenging case, as the degree of perturbations (for OOD data) is finer-grained.\nThen, at each batch, we fed a random sample from the OOD test-set and of the IND test-set to the DDUp detector. For each batch, the detector would signal IND or OOD and we recorded and calculated FPR and FNR. The batch size was 2,000 and we repeated the experiment for 1,000 batches.\n\nWe used the same parameters for all datasets and models: the bootstrapping size is 32 and the threshold is \\(2 \\times std\\). For DBEst++, the results are reported in \\autoref{tab:fprfnr}.  FPR and FNR for Naru\/NeuroCard and TVAE were always zero. These results further confirm that the OOD detection algorithm is not biased.\n\n\\begin{table}[hb]\n  \\vspace{-0.3cm}\n \\centering\n  \\caption{FPR and FNR for DBEst++.}\n  \\vspace{-0.3cm}\n  \\label{tab:fprfnr}\n  \\begin{tabular}{c c c } \n    \\toprule\nDataset & FPR & FNR \\\\\n    \\midrule\n     Census & 0.15 & 0.01 \\\\\n     Forest & 0.10 & 0 \\\\\n     DMV & 0.01 & 0 \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.35cm}\n\\end{table}\n\nFurthermore, we studied the sensitivity on the batch size and varied it from a size of 1 to 2,000. Results are shown in \\autoref{fig:oodsens}, which clearly show that after a low-threshold batch size, FPR and FPN tend to zero. The same results hold for other models and datasets, and are omitted here for space reasons.\n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/dbest-forest-fprfnr.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/tvae-dmv-fpfn.png}}\n\\end{minipage} %\n\\vspace{-0.4cm}\n\\caption{Sensitivity of OOD detection vs batch size.}\n\\label{fig:oodsens}\n\\vspace{-0.55cm}\n\\end{figure}\n\n\\subsection{Accuracy Results} \\label{perfeval}\n\\subsubsection{When there is OOD data} \\label{whenood}\n\nFor Naru\/NeuroCard, DBEst++, and TVAE, and for each dataset, we compare 4 updating approaches against each other and against the base model before any new data is inserted. The 4 approaches are as follows: \n\"\\texttt{Retrain}\", retrains the model from scratch using both old and new data.  \"\\texttt{Baseline}\" is the baseline approach in Naru\/NeuroCard and DBest++ where a trained model is updated with new data by performing \\textit{SGD} with a smaller learning rate. \"\\texttt{DDUp}\" is the proposed method.\nFinally, in \"\\texttt{stale}\", the model is not updated -- this is a do-nothing approach.\nFor reference, we also include the numbers for $M_0$, i.e., the original model accuracy before any new data came.\n\\autoref{tab:qerror} and \\autoref{tab:aqpacc} show the accuracy results for CE and AQP (SUM and AVG operations), respectively.\nFor TVAE, the classification f1-scores are reported in \\autoref{tab:tvaef1}. Results of these three tables correspond to the case where the update sample is permuted. \nDDUp always performs better than the baseline approach. \nMost of the times, the performance of DBEst++ on DMV dataset is not as well as for the other datasets. This probably is due to the complexity of data (large scale and highly correlated attributes). Nevertheless, DDUp stands on the top of the underlying models and regardless of the model's performance, DDUp ensures that it will retain the accuracy.\nPlease note the DMV dataset results in \\autoref{tab:qerror} and \\autoref{tab:aqpacc} and, census and forest datasets in \\autoref{tab:tvaef1}, where, DDUp even outperforms retraining from scratch. \nInterestingly, this corroborates similar evidence for sequential self-distillation (for boosting embeddings for) classification tasks \\cite{seq-self-distill}. This was one of the reasons we adapted a self-distillation based approach.\nFinally, baseline methods have poor performance for 95th and 99th percentiles. \n\n\\begin{table*}[t]\n  \\caption{Results of updating a base model with a 20\\% permuted sample in terms of q-error. $M_{0}$ denotes the base model.}\n  \\label{tab:qerror}\n  \\centering\n  \\begin{tabular}{c c | c | c | c | c | c | c | c | c | c | c} \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{metric} & \\multicolumn{5}{c|}{DBEst++} & \\multicolumn{5}{c}{Naru\/NeuroCard} \\\\\n&&$M_{0}$&DDUp&baseline&stale&retrain&$M_{0}$&DDUp&baseline&stale&retrain \\\\\n    \\midrule\n\\multirow{4}{*}{census}&median&1.05&1.11&1.17&1.16&1.07&1.08&1.09&4&1.14&1.07\\\\\n&95th&2&2&2.20&2&2&2&2&471.80&2&2\\\\\n&99th&3&3&4&3&3&3&3&1534.69&3.16&3\\\\\n&max&5&7&11&10.50&5&5.25&7&8385&21.88&6\\\\\n\\midrule\n\\multirow{4}{*}{forest}&median&1.026&1.046&2&1.18&1.02&1.04&1.07&1.54&1.10&1.05\\\\\n&95th&2&2&63.40&2&1.64&2.48&3&41&2.50&2.75\\\\\n&99th&2&2.583&503.12&5.60&2&4&6&157.16&5.48&5\\\\\n&max&4&5.33&3470&90.85&5.33&27&65.66&1691&484&34.66\\\\\n\\midrule\n\n\\multirow{4}{*}{DMV}&median&1.20&1.143&3.48&1.88&1.34&1.02&1.04&2.57&1.16&1.02\\\\\n&95th&4.91&5.07&234.88&7.00&5.50&1.20&1.41&468.68&1.50&1.25\\\\\n&99th&9.65&10&3897.87&12.50&8&1.83&2.31&4734.62&2.84&2\\\\\n&max&18.83&19&65875&39&17&8&9.81&343761&9.49&5\\\\\n\n\\midrule\n\\multirow{4}{*}{TPCDS}&median&1.02&1.04&57&1.27&1.02&1.01&1.07&1.15&1.10&1.05\\\\\n&95th&1.16&1.26&269&1.58&1.18&2&2&29&2&2\\\\\n&99th&1.5&1.61&1266&2.72&1.5&3.01&3.01&239&4&3\\\\\n&max&3&3&4534&10.66&5.64&5&28&5100&28&24\\\\\n\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\begin{table}[t]\n  \\caption{mean-relative-error for SUM and AVG aggregation functions for DBEst++.}\n  \\label{tab:aqpacc}\n  \\centering\n  \\resizebox{\\linewidth}{!}{%\n  \\begin{tabular}{c c | c c c c c} \n    \\toprule\nDataset&function&$M_{0}$&DDUp&baseline&stale&retrain\\\\\n    \\midrule\n\\multirow{2}{*}{census}&SUM&13.05&17.30&65.88&21.36&13.60\\\\\n&AVG&1.89&2.36&8.15&2.37&1.97\\\\\n\\midrule\n\\multirow{2}{*}{forest}&SUM&10.11&15.51&88.73&24.59&10.14\\\\\n&AVG&0.76&1.04&3.90&1.35&0.79\\\\\n\\midrule\n\\multirow{2}{*}{TPCDS}&SUM&4.53&6.37&61.40&22.64&5.12\\\\\n&AVG&0.88&1.47&12&3.50&1.21\\\\\n\\midrule\n\\multirow{2}{*}{DMV}&SUM&76.73&85.29&423&97.00&110\\\\\n&AVG&6.4&6.9&15.9&8.6&7.3\\\\\n\n\\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\\begin{table}[t]\n  \\caption{Classification results for TVAE in terms of micro f1. 'r' stands for real data, 's' stands for synthetic data.}\n  \\label{tab:tvaef1}\n  \\centering\n  \\resizebox{\\linewidth}{!}{%\n  \\begin{tabular}{c | c c | c c | c c | c c | c c } \n    \\toprule\n\\multirow{2}{*}{Dataset}\n&\\multicolumn{2}{c}{$M_{0}$}&\\multicolumn{2}{c}{DDUp}&\\multicolumn{2}{c}{baseline}&\\multicolumn{2}{c}{stale}&\\multicolumn{2}{c}{retrain}\\\\\n&r&s&r&s&r&s&r&s&r&s\\\\\n    \\midrule\ncensus&0.67&0.63&0.77&0.73&0.77&0.55&0.77&0.56&0.77&0.72\\\\\nforest&0.84&0.69&0.89&0.78&0.89&0.63&0.89&0.60&0.89&0.74\\\\\nDMV&0.97&0.97&0.98&0.97&0.98&0.92&0.98&0.93&0.98&0.98\\\\\n\n\\bottomrule\n  \\end{tabular}}\n\\end{table}\n\n\\subsubsection*{Performance on old and new queries} To better illustrate the effects of \\textit{catastrophic forgetting}{} and \\textit{intransigence} we elaborate on performance on FWT and BWT. (As \\texttt{retrain} avoids be definition \\textit{catastrophic forgetting}{} and \\textit{intransigence}, it is omitted).\nThe results are shown in \\autoref{tab:mdntranfers}. \nNote that any insertion affects only a percentage of queries, shown in \n\\autoref{tab:querypercents}.\nComparing AT, FWT, and BWT in \\autoref{tab:qerror} and \\autoref{tab:mdntranfers} first note that fine-tuning always performs much better in terms of FWT compared to BWT (due to catastrophic forgetting).\nSecond, conversely, a stale model shows better BWT compared to FWT. \nFor DDUp, FWT and BWT remain close to each other, especially in terms of median q-error, showing that DDUP can ensure accuracy for queries on old and new data.\nOverall, DDUp enjoys high accuracy.\n\n\\subsubsection*{Incremental Steps} To show the updates in incremental steps, we have split the \\(20\\%\\) data into 5 equal-sized chunks and have performed an update incrementally for each batch. \\autoref{fig:incupdates2} compares the trend of accuracy during updates. As it is clear from the figures, DDUp remains very close to \\texttt{retrain}, while there is a drastic drop in accuracy using \\texttt{baseline}. Starting point \\(0\\) is where the base model \\(M_{0}\\) is built from scratch. (The same results hold for 95th, 99th percentiles and maximum q-error). \n\n\n\\begin{table*}[t]\n \\caption{Comparing q-error of different updating approaches in terms of FWT and BWT.}\n \\vspace{-0.2cm}\n  \\label{tab:mdntranfers}\n  \\begin{tabular}{c c | c | c c | c c | c c | c | c c | c c | c c } \n    \\toprule\n\\multirow{3}{*}{Dataset} & \\multirow{3}{*}{metric} & \\multicolumn{7}{c|}{DBEst++} & \\multicolumn{7}{c}{Naru\/NeuroCard} \\\\\n&&\\multicolumn{1}{c}{$M_{0}$}&\\multicolumn{2}{c}{DDUp}&\\multicolumn{2}{c}{baseline}&\\multicolumn{2}{c|}{stale}&\\multicolumn{1}{c}{$M_{0}$}&\\multicolumn{2}{c}{DDUp}&\\multicolumn{2}{c}{baseline}&\\multicolumn{2}{c}{stale} \\\\\n\n&&&FWT&BWT&FWT&BWT&FWT&BWT& &FWT&BWT&FWT&BWT&FWT&BWT\\\\\n\n\\midrule\n\\multirow{3}{*}{census}&median&1.05&1.06&1.12&1.06&1.20&1.05&1.16&1.08&1.11&1.09&1.83&6&1.20&1.13 \\\\\n&95th&2&1.66&2&1.56&2.33&3.30&2&2&1.64&2&4.63&530.80&3.18&2 \\\\\n&99th&3&4.94&3&4.10&4&8.90&2.75&3&3.08&3&9.98&1598.53&8.49&3\\\\\n\n\\midrule\n\\multirow{3}{*}{forest}&median &1.02&1.01&1.08&1.23&2.66&1.05&1.20&1.04&1.07&1.07&1.39&1.65&1.18&1.08\\\\\n&95th&2&1.181&2&2.87&146.38&2.85&2&2.489&1.88&3&3.13&43.02&7.55&2.33\\\\\n&99th&2&1.52&3&3.72&590.57&18.33&2.24&4&4.89&6&5.27&163.80&191.53&4.86\\\\\n\n\\midrule\n\\multirow{3}{*}{DMV}&median&1.20&1.28&1.13&2.20&4.36&1.66&1.54&1.02&1.02&1.07&1.06&12.85&1.26&1.19\\\\\n&95th&4.910&4.30&5.87&3.34&484.46&9.50&6.87&1.20&1.16&1.55&1.65&1015.81&3.30&1.40\\\\\n&99th&9.65&9&11.65&10.50&5894.21&12.12&10.80&1.83&1.47&3&3.35&8183.34&11.93&2.49\\\\\n\n\\midrule\n\\multirow{3}{*}{TPCDS}&median&1.02&1.03&1.04&1.20&1.51&1.16&1.21&1.01&1.06&1.08&1.19&1.11&1.10&1.10\\\\\n\n&95th&1.16&1.21&1.29&2.37&339&2.26&1.35&2&2&2&2.60&54&2&2\\\\\n&99th&1.5&1.37&1.66&4.27&1536&4.48&1.66&3.01&9.77&3&9.47&434&9.64&3.77\\\\\n\n\\bottomrule\n  \\end{tabular}\n\\end{table*}\n\n\\begin{table}[t]\n  \\caption{The percentage of the queries (out of 2k queries) with changed actual results after inserting 20\\% new data.}\n  \\vspace{-0.2cm}\n  \\label{tab:querypercents}\n  \\begin{tabular}{c c c } \n    \\toprule\n     dataset & DBEst++ & Naru \\\\\n     \\midrule\n     census&14\\%&12\\% \\\\\n     forest&32\\%&9\\% \\\\\n     TPCDS&36\\%&36\\% \\\\\n     dmv&52\\%&45\\%\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.5cm}\n\\end{table}\n\nWe also have evaluated the models with respect to the \\textit{log-likelihood goodness-of-fit}. Log-likelihood is  widely used to evaluate NN models.\nUsing log-likelihood allows evaluation to be independent\nof underlying applications. \\autoref{fig:incll} shows changes in log-likelihood in consecutive update steps. At each step, we calculate the average of log-likelihoods over a sample of new data and a sample from historical data. In these figures we again see that updating with DDUp is fitting to the old and the new data very similarly to the \\texttt{retrain} case.  In general, when keep using \\texttt{stale}, the log-likelihood drops after the first update and then remains low. The reason is that all update batches have similar permutation and since we calculate unweighted averages, the log-likelihood stays fixed. While, for \\texttt{baseline}, i.e fine-tuning, we can see a gradual decrease of likelihood which means that the network is increasingly forgetting about previous data in each step. \n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/dbest_census.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/naru_census.png}}\n\\end{minipage} %\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:c}\\includegraphics[scale=.29]{figures\/dbest_forest.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:d}\\includegraphics[scale=.29]{figures\/naru_forest.png}}\n\\end{minipage}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:e}\\includegraphics[scale=.29]{figures\/dbest_dmv.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:f}\\includegraphics[scale=.29]{figures\/naru_dmv.png}}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\caption{Updating results over 5 consecutive updates.}\n\\label{fig:incupdates2}\n\\vspace{-0.4cm}\n\\end{figure}\n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:c}\\includegraphics[scale=.29]{figures\/naru_census_loglikelihood.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:d}\\includegraphics[scale=.29]{figures\/naru_dmv_loglikelihood.png}}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\caption{log-likelihood results over 5 consecutive updates.}  \n\\label{fig:incll}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\subsubsection{When data is not OOD}\nIn this case, simple fine-tuning update algorithms, such as \\texttt{baseline}, will likely avoid \\textit{catastrophic forgetting}{}. \nTo illustrate this, we have repeated the 5 batched incremental updates with data without permutation. The results are reported in \\autoref{fig:incupdatenodrift}. For space reasons, we only show the results for census. The results indicate that for in-distribution data, simple baselines can have a performance close to \\texttt{retrain}.\n\n\\begin{figure}\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/dbest_census_nodrift.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/naru_census_nodrift.png}}\n\\end{minipage} %\n\\vspace{-0.3cm}\n\\caption{Updating results over 5 consecutive updates when data follows the same distribution as the historical data.}\n\\label{fig:incupdatenodrift}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\begin{table}[hb]\n\\vspace{-0.4cm}\n  \\caption{DDUp's speed up over \\texttt{retrain}, for two update sizes. For census, forest, and dmv, sp1: 20\\% of the original table. sp2, 5\\% of the original table. for IMDB and TPCH sp1: updating the first partition and sp2: updating the last partition.}\n  \\label{tab:times}\n\\vspace{-0.25cm}\n  \\begin{tabular}{c | c c | c c | c c } \n    \\toprule\n\\multirow{2}{*}{Dataset}  & \\multicolumn{2}{c|}{DBEst++} & \\multicolumn{2}{c|}{Naru} & \\multicolumn{2}{c}{TVAE} \\\\\n&sp1&sp2&sp1&sp2&sp1&sp2 \\\\\n    \\midrule\ncensus&5&5.5&3.5&4&3.4&5.7 \\\\\nforest&1.6&4&5&9.2&3.6&7 \\\\\nDMV&4&6.5&2.3&9.6&3.4&6.8 \\\\\nIMDB&4.5&18&3.5&5&NA&NA \\\\\ntpch&6.5&16&2&4&NA&NA \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\subsection{Evaluating DDUp for Join Queries} \\label{joinexp}\nAs mentioned, DDUp is unconcerned whether at a time $t$, \n\\(S^{\\leq}_{t-1}\\) (a sample of  \\(\\cup_{j=0}^{t-1} D_j\\)) and $D_t$ come from a raw table or from a join.\nFor this experiment, we have evaluated DDUp running 2,000 queries over two 3-table joins from the JOB and TPCH datasets.\nFor each, the 2,000 queries involve a join of the fact table with two dimension tables: \nSpecifically, the join of tables [\\texttt{title}, \\texttt{movie info idx}, \\texttt{movie companies}] for IMDB, and [\\texttt{orders}, \\texttt{customer}, \\texttt{nation}] for TPCH. For the update dynamics, we have split the fact table into 5 time-ordered equally-sized partitions. We have built \\(M_0\\) on the join (of the fact table's first partition with the 2 dimension tables) and updated it with each subsequent partition at a time. This is similar to the update setting in NeuroCard.\nResults for both CE and AQP are in \\autoref{fig:joins}.\n\nNeuroCard, unlike other models, natively supports joins, using\na \"fast-retrain\" - i.e., a light retraining where the model is retrained using a 1 percent sample of the full join result. We have included this policy here as \"fast-retrain\". \nDDUp always signalled OOD for the new update batches, except for TPCH data on DBest++, where update was not triggered. Therefore, in \\autoref{fig:joins}.d the accuracy of the stale model and fine-tuning is  close to retrain. This further confirms the significance of OOD detection.\n\n\\begin{figure}[htbp]\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:a}\\includegraphics[scale=.29]{figures\/Naru-imdb-95th.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:b}\\includegraphics[scale=.29]{figures\/Naru-tpch-95th.png}}\n\\end{minipage} \\\\\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:c}\\includegraphics[scale=.29]{figures\/DBest-imdb-sum-rel-error.png}}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\subfloat[]{\\label{main:d}\\includegraphics[scale=.29]{figures\/DBest-tpch-sum-rel-error.png}}\n\\end{minipage}\n\\vspace{-0.3cm}\n\\caption{DDUp's performance on joined tables.}\n\\label{fig:joins}\n\\vspace{-0.3cm}\n\\end{figure}\n\n\\subsection{Effect of Transfer Learning} \\label{distilleval}\nWe now delve into the effects of transfer-learning in DDUp. How much DDUp's transfer-learning via knowledge distillation contributes to better accuracy? \nWe perform experiments where we remove the transfer-learning term of Eq \\ref{totalloss}. Therefore, we combine the sample from previous data known as the transfer-set with the new update batch and create a model\nwith the same configurations as the base model. \\autoref{fig:tleffect} shows the results.\nThe results assert that the performance of DDUp is not only related to the previous data sample, and in fact, distillation has a big effect on the improvement of the new models. \n\\begin{figure}[htbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/transfer_learning_effect.png}\n    \\caption{Effect of transfer-learning on q-error. AggTrain, is the case where we aggregate the transfer-set with the new data and train a model similar to the base model.}\n    \\label{fig:tleffect}\n\\end{figure}\n\\vspace{-0.2cm}\n\\subsection{Overheads} \\label{overheads}\nWe report on the costs of each DDUp module separately. All the codes are written and executed in Python 3.8, on an Ubuntu 20 machine  with 40 CPU cores, two Nvidia GTX 2080 GPUs and 64GB memory. With respect to memory usage, DDUp performs regular feed-forward steps as in regular NN training. Therefore, DDUp does not increase memory footprints\nIn terms of time, DDUp has two computation costs namely, \\textit{OOD detection} and \\textit{model update}. OOD detection is split into offline and online phases. \\autoref{tab:offontime} shows these two times. The largest detection time is for the forest dataset on a Naru model which takes around 3 minutes. However, please note that in \nthe online phase only takes 1 second to detect a change in data. \n\n\\begin{table}[hb]\n  \\caption{online and offline times during OOD detection.}\n  \\label{tab:offontime}\n\\vspace{-0.25cm}\n  \\begin{tabular}{c | c c | c c | c c } \n    \\toprule\n\\multirow{2}{*}{Dataset}  & \\multicolumn{2}{c|}{DBEst++} & \\multicolumn{2}{c|}{Naru} & \\multicolumn{2}{c}{TVAE} \\\\\n&off&on&off&on&off&on \\\\\n    \\midrule\ncensus&2.44&0.02&111&1.8&310&5.5\\\\\nforest&28&0.04&174&0.92&433&8.8\\\\\nDMV&86&2&144&10&99&0.44\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\\autoref{tab:times} shows DDUp's speed up over \\texttt{retrain} for OOD data, for different update sizes. When data is OOD, DDUp can be over 9$\\times$ faster than \\texttt{retrain}. Obviously, speedups will be higher for incremental steps. This fact is reflected in IMDB and TPCH datasets where after inserting the last partition DDUp is 18$\\times$ faster than \\texttt{retrain}. Note that the updating time is dependent on a few parameters including update size, transfer-set size, training batch size etc. During updates, we have used smaller training batch sizes. If one tunes the model for bigger batches, and smaller transfer-set sizes, the speed up would be higher.\n\n\\vspace{-0.2cm}\n\\subsection{Non neural network models}\\label{nonnn}\nFor the sake of completeness and as an additional reference point, we include results for updating a state-of-the-art non-NN model that natively supports data insertions, (DeepDB \\cite{hilprecht2019deepdb}) used for CE. \nWhen an update happens, DeepDB traverses its sum-product-network graph and updates the weights of the intermediate nodes and the histograms at the leaves. We have repeated the same experiment in \\autoref{tab:qerror} for DeepDB. The results are reported in \\autoref{tab:deepdb}.\n\n\\begin{table}[t]\n  \\caption{Performance of DeepDB updating vs. DDUp for Naru, for a CE task in terms of q-error.}\n    \\vspace{-0.35cm}\n  \\label{tab:deepdb}\n  \\centering\n  \\begin{tabular}{c c | c | c | c | c | c } \n    \\toprule\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{ metric} & \\multicolumn{3}{c|}{DeepDB} & \\multicolumn{2}{c}{Naru} \\\\\n&&$M_{0}$&update&retrain&$M_{0}$&DDUp \\\\\n    \\midrule\n\\multirow{3}{*}{census}&median&1.05&1.2&1.05&1.08&1.09\\\\\n&95th&3&4.18&3&2&2\\\\\n&99th&5.11&8&5&3&3\\\\\n\\midrule\n\\multirow{3}{*}{forest}&median&1.02&1.2&1.02&1.04&1.07\\\\\n&95th&7.5&10.5&7&2.48&3\\\\\n&99th&31&52&31&4&6\\\\\n\\midrule\n\\multirow{3}{*}{DMV}&median&1.06&1.25&1.1&1.02&1.04\\\\\n&95th&2.5&3.5&2.5&1.20&1.41\\\\\n&99th&22&37&21&1.83&2.31\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\vspace{-0.3cm}\n\\end{table}\n\nFrom \\autoref{tab:deepdb} it can be observed that DeepDB's updating policy is under-performing, as was independently verified in \\cite{wang2020we}. \nDDUp (coupled in this experiment with Naru\/NeuroCard for CE) always performs better. Nonetheless, we wish to emphasize that the saving grace for DeepDB based on our experiments is that retraining from scratch is very efficient -- significantly faster compared to NNs. \n\n\\section{Related Work} \\label{litraturere}\n\\subsection{Learned Database Systems}\\label{ldbliterature}\nNN-based components to be used by DBs are emerging rapidly. Different works exploit different neural network models.\n\\cite{yang2019deep, yang2020neurocard, hasan2020deep} used generative neural networks to build learned selectivity estimators. Thirumuruganathan et al. \\cite{thirumuruganathan2020approximate} used VAEs for AQP. Ma et al. \\cite{ma2021learned} used mixture density networks for AQP. Database indexing research\nrecently has adopted neural networks to approximate cumulative density functions \\cite{kraska2018case,ding2020alex,nathan2020learning,ding2020tsunami}. Query optimization and join ordering are also benefiting from neural networks \\cite{marcus2019neo, kipf2018learned}. Other applications include auto-tuning databases \\cite{van2017automatic,li2019qtune,zhang2019end}, cost estimation \\cite{zhi2021efficient, siddiqui2020cost}, and workload forecasting \\cite{zhu2019novel}.\n\nAmong these, this work provides a solution for handling NN model maintenance in the face of  insertion-updates with OOD data, when the models need to continue ensuring high accuracy on new and old data and on tasks for which models were originally trained (such as AQP, CE, DG, etc.).\nWhile there has been related research on transfer learning for learned DBs such as \\cite{hilprecht2021one, wu2021unified} these target a different problem setting:\nThey study how to transfer knowledge from a model trained for one task, and\/or a DB, and\/or a system, and\/or a workload to a new task and\/or DB, and\/or system, and\/or workload. They do not study how to keep performing the original task(s) on evolving datasets with insertions carrying OOD data with high accuracy for queries on both old and new data. Simply using these methods by fine-tuning on new data will incur catastrophic forgetting. Nevertheless, since these models employ some sorts of knowledge transfer, they might be useful to support updates. However, it remains open whether and how the models in \\cite{wu2021unified, hilprecht2021one} can be utilized to solve efficiently the problems tackled in this paper.\nWhile some of non-neural-network models (e.g., DeepDB) can very efficiently retrain from scratch,\nNN-based models for the above problem setting either do not support insertion-updates or suffer from poor accuracy when facing OOD data, unless paying the high costs of retraining from scratch.\n\n\n\\subsection{OOD Detection}\nOOD detection has recently attracted a lot of attention and it has long been studied in statistics as concept drift (CD) detection, or novelty detection. In general, CD and OOD detection methods could be divided into two broad categories \\cite{gama2014survey,lu2018learning,wang2020few}: \nFirst, prediction-based methods, which use the predictions of the underlying models to test for a change. Recent ML models usually use the predictive probabilities of the classifiers as a confidence score to identify changes \\cite{jiang2018trust,ovadia2019can, wilson2020bayesian,ruff2021unifying}. Others may monitor the error of the underlying models and trigger an OOD signal when a significant change is captured \\cite{gama2006learning, baena2006early,savva2019aggregate,nehme2009self,lopez2016revisiting}. While these approaches are very efficient in time, they typically come with limiting assumptions depending on the underlying model or application. For example, most of them can only be utilized and are only studied for classification (supervised) tasks.\nThe second broad family of methods is distribution-based methods. Some of these methods try to find a distance measure that can best show the discrepancy between new data and old data distributions, using tests like Kolmogorov-Smirnov (KS), \\cite{kolmogorov1933sulla}, Wilcoxon \\cite{pereira2009machine}, and their multi-variate variants \\cite{fasano1987multidimensional, baringhaus2004new}. Others try to learn the density of the underlying data distribution test for a significant change, like kernel-density-based approaches  \\cite{kifer2004detecting,dasu2006information,gu2016concept,lu2014concept,bu2016pdf,song2007statistical}. More recent works utilize the estimated likelihoods of generative models \\cite{ren2019likelihood, morningstar2021density, xiao2020likelihood}. Other approaches rely on the inner representations of the networks \\cite{li2021cutpaste,hendrycks2019using,lee2018simple}. Nonetheless, this second family of OOD detection methods are usually expensive (esp. for multi-dimensional data) and involve fitting a separate density estimator. Hence, the main problem is that in an insertion scenario, the density estimators also need to be updated (typically via training from scratch, upon each insertion).\n\n\\vspace{-0.3cm}\n\\subsection{Incremental Learning (IL)}\nMost IL methods regularize the model in a way that it acquires knowledge from the new task while retaining the knowledge of old tasks. For example, \\textit{Elastic Weight Consolidation (EWC)} \\cite{kirkpatrick2017overcoming} adds a regularizer to control the learning speed around important weights of the network for old tasks while learning a new task. Similar works are developed around this idea \\cite{liu2018rotate, lee2020continual,titsias2019functional}, \\textit{Path Integral (PathInt)} \\cite{zenke2017continual} ,\\textit{Riemanian Walk (RWalk)} \\cite{chaudhry2018riemannian}. Other approaches exploit knowledge distillation to retain the knowledge of previous tasks \\cite{li2017learning}.\nAnother group of IL methods, save exemplars from past data \\cite{wu2019large, castro2018end, rebuffi2017icarl} or generate samples\/features using generative models \\cite{ostapenko2019learning, kemker2017fearnet} and involve them in learning new tasks. Lopez et al. \\cite{lopez2017gradient} has proposed \\textit{Gradient Episodic Memory} that consists of \\textit{M} blocks of memory to store examples from \\textit{T} tasks and uses the model's prediction on these examples as a constraining loss that inhibits the model to bias toward new task and forget past tasks. Lastly, some works try to completely keep previous models and create new models (or part of a model like a single layer) for each new task.  Aljundi et al. \\cite{aljundi2017expert} introduce \\textit{Expert Gate} with different models for each task and an autoencoder which learns the representations of each task to assign test-time tasks to the proper model. Instead of learning a whole new model, Rusu et al. \\cite{rusu2016progressive} introduce \\textit{Progressing Neural Networks} which add new columns to the previous network architecture and learns lateral connections between them. Most of the above methods, do not account for in- and out- of distribution updates and are not easily extendable to different learning tasks. \n\n\\vspace{-0.2cm}\n\\section{Conclusion} \\label{conclusion}\nLearned DB components can become highly  inaccurate when faced with new OOD data when aiming to ensure high accuracy for queries on old and new data for their original learning tasks.\nThis work proposes, to our knowledge, the first solution to this problem, coined DDUp.\nDDUp entails two novel components, for OOD detection and model updating.\nTo make detection widely applicable, OOD detection in DDUp exploits the output of the neural network (be it based on log-likelihood, cross-entropy, ELBO loss, etc.), and utilizes a principled two-sample test and a bootstrapping method to efficiently derive and use thresholds to signal OOD data.\nDDUp also offers a general solution for model updating based on sequential self-distillation and a new loss function which carefully accounts for \\textit{catastrophic forgetting} and \\textit{intransigence}.\nThis work showcases the wide applicability of DDUp model updating by instantiating the general approach to three important learned functions for data management, namely AQP, CE, and DG, whereby a different type of NN (MDNs, DARNs, VAEs) is used for each. In fact, to our knowledge, no prior work has shown how to \"distill-and-update\" MDNs, VAEs, and DARNs.\nComprehensive experimentation showcases that DDUp detects OOD accurately and ensures high accuracy with its updated models with very low overheads.\n\n\n\\section{Acknowledgement}\nThis work is partially sponsored by Huawei IRC and by EPSRC while doing a PhD at the University of Warwick.\n\n\\balance\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Conclusions}\n\nIn this paper, we introduced the notion of statistically significant lexical co-occurrences. We\ndetected skews in span distributions of bigrams to assess significance and showed how our method\nallows classification of co-occurrences into different types. We performed experiments to assess the\nperformance of various frequency-based measures for detecting lexically signficant co-occurrences.\nWe believe lexical co-occurrence can play a critical role in several applications, including sense disambiguation, mutli-word spotting, etc. We will address some of these in our future work.\n\n\\section{Discussion}\n\nThreshold span needs discussion. Essentially we say that, it is reasonable to believe that all\noccurrences with span less than the threshold value can plausibly be motivated, while all\noccurrences with span greater than the threshold span are probably arbitrary. This notion applies to\nall co-occurrence based measures, not just the CSR\n\n\n\n\\subsection{Performance of different co-occurrence measures}\n\nWe now compare the performance of various frequency-based measures in the context of lexical\nsignificance. Given the large numbers of measures proposed in the literature~\\cite{pecina06}, we need to identify a subset of measures to compare.\nInspired by \\cite{ecologyMeasures} and \\cite{dataMiningTan} we identify three properties\nof co-occurrence measure which may be useful for language processing applications. First is {\\em Symmetry} - does the measure yield the same association score for (x,y) and (y,x)? Second is {\\em Null Addition} - does addition of data containing neither x nor y affect the association\nscore for (x,y)? And, finally, {\\em Homogenity} - if we replicate the corpus several times and merge them to construct a\nlarger corpus, does the association score for (x,y) remain unchanged? Note that the concept of homogenity conflicts\nwith the notion of statistical support, as support increases in direct proportion with the absolute amount of evidence.\nDifferent applications may need co-occurrence measures having different combinations of these properties. \n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|p{1.8cm}| p{2.95cm}|l|l|l|}   \n \\hline\nMethod & Formula & \\begin{sideways}Symm. \\end{sideways} & \\begin{sideways}Null Add. \\end{sideways}& \\begin{sideways}Homo. \\end{sideways}\\\\\n\\hline\nCSR (this work) & $Z \/ (E(Z) + Kt)$ & Y & Y &  N\\\\ \\hline \nCSA (this work) &  $\\frac{\\hat{f}(x,y)}{\\sqrt{K}} $  & Y & N &  Y\\\\ \\hline\nLLR~\\cite{llr} & ${\\displaystyle{\\sum_{x', y' }}}p(x',y')log\\frac{p(x',y')}{p(x')p(y')}$ & Y & Y  & Y \\\\ \\hline\nPMI~\\cite{churchHanks89} & $log\\frac{p(x,y)}{p(x)p(y)}$ & Y & N &  Y \\\\ \\hline\nSCI~\\cite{cwcd} & $\\frac{p(x,y)}{p(x)\\sqrt{p(y)}}$ & N & N  &  Y\\\\ \\hline\nCWCD~\\cite{cwcd} & $\\frac{\\hat{f}(x,y)}{p(x)}\\frac{1\/max\\left(p(x),p(y)\\right)}{M}$ & N & N  &  Y\\\\ \\hline\nPearson's $\\chi^2$ test & ${\\displaystyle\\sum_{x',y'}} \\frac{\\left(\\hat{f}(x',y')-E\\hat{f}(x',y')\\right)^2}{E\\hat{f}(x',y')}$ & Y & Y &  Y\\\\ \\hline\nT-test & $\\frac{\\hat{f}(x,y)-E\\hat{f}(x,y)}{\\sqrt{\\hat{f}(x,y)\\left(1-\\frac{\\hat{f}(x,y)}{N}\\right)}}$ & Y & N & Y\\\\ \\hline\nDice~\\cite{dice} & $\\frac{2\\hat{f}(x,y)}{f(x)+f(y)}$ & Y & N &  Y\\\\ \\hline\nOchiai~\\cite{ecologyMeasures} & $\\frac{\\hat{f}(x,y)}{\\sqrt{f(x)f(y)}}$ & Y & N  & Y\\\\ \\hline\nJaccard~\\cite{jaccard} & $\\frac{\\hat{f}(x,y)}{f(x)+f(y)-\\hat{f}(x,y)}$ & Y & N  & Y \\\\ \\hline\n\\end{tabular}\n{\\scriptsize\nTerminology: ($x' \\in \\{x,\\neg x\\}$ and\n$y' \\in\\{y,\\neg y\\}$) \\\\\n\\begin{tabular}{l l }\n$N$ & Total number of tokens in the corpus \\\\\n$f(x),f(y)$ & unigram frequencies of $x,y$ in the corpus \\\\\n$p(x),p(y)$ & $f(x)\/N,f(y)\/N $\\\\\n$\\hat{f}(x,y)$ & Span-constrained ($x,y$) bigram frequency\\\\\n$\\hat{p}(x,y)$ & $\\hat{f}(x,y)\/N $\\\\\n$M$ & Harmonic mean of the spans of $\\hat{f}(x,y)$ occurrences\\\\\n$E\\hat{f}(x,y)$ & Expected value of f(x,y) \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{ \\small Properties of selected co-occurrence measures }\n\\label{tab:methods}\n\\end{table}\n\n\n\nTable~\\ref{tab:methods} shows the characteristics of our chosen co-occurrence measures, which were selected from several domains like ecology,\npsychology, medicine, and language processing. Except Ochiai~\\cite{Ochiai}, \\cite{ecologyMeasures}, and the recently introduced measure CWCD~\\cite{cwcd}\\footnote{From various so-called windowless measures introduced in~\\cite{cwcd}, we chose the best-performing variant Cue-Weighted Co-Dispersion (CWCD) and implemented a window based version of it with harmonic mean. We note that any of windowless (or spanless) measure can easily be thought of as a special case of a window-based measure where the windowless formulation corresponds to a very large window (or span in our terminology).}, all other selected measures are well-known in the NLP community~\\cite{pecina06}.\n Based on our extensive study of theoretical and empirical properties of CSR, we also introduce a new bigram frequency based measure called CSA ({\\em Co-occurrence Significance Approximated}), which approximates the behaviour of CSR over a wide range of parameter settings.\n\nIn our experiments, we found that Ochiai and Chi-Square have almost identical performance, differing only in 3rd decimal digits.\nThis can be be explained easily. In our context, for any word $x$,  as defined in Table~\\ref{tab:methods},\n$f(x) << N$ and therefore $p(x) << 1$. With this, Chi-Square reduces to square of Ochiai. Similarly Jaccard and Dice coincide,\nsince $f(x,y) << f(x)$ and $f(x,y) << f(y)$. Hence we do not\nreport further results for Chi-Square and Jaccard.\n\n\n\nIn our first set of experiments, we compared the performance of various frequency-based\nmeasures in terms of their suitability for detecting lexically significant co-occurrences \n(cf.~{\\em Definition~\\ref{def:test1}}). \nA high Spearman correlation coefficient between the ranked list produced by a given measure and the list produced by CSR with respect to some choice of\n$\\epsilon$ and $\\delta$ would imply that the measure is effective in detecting the corresponding {\\em type} of\nlexically significant co-occurrences.\n\n\\begin{table}\n\\centering\n\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n\t\t\t\t\t\t& \t\t\t& \\multicolumn{3}{|c|}{Span Threshold} \\\\ \\hline\n\tMeasure\t\t\t\t&\tData\t&\t 5w\t\t&\t25w\t\t&\t50w\t\\\\ \\hline\n\\multirow{3}{*}{PMI}\t& sim\t\t& C\t\t\t& -\t\t\t& - \\\\\n\t\t\t\t\t\t& rel\t\t& -\t\t\t& -\t\t\t& -\t\\\\\n\t\t\t\t\t\t& essli\t\t& -\t\t\t& -\t\t\t& -\t\\\\ \\hline\n\\multirow{3}{*}{CWCD}\t& sim\t\t& -\t\t\t& -\t\t\t& - \\\\\n\t\t\t\t\t\t& rel\t\t& -\t\t\t& -\t\t\t& -\t\\\\\n\t\t\t\t\t\t& essli\t\t& -\t\t\t& -\t\t\t& -\t\\\\ \\hline\n\\multirow{3}{*}{CSA}\t& sim\t\t&A, B, C, D\t& A, B, C& A, B, C \\\\\n\t\t\t\t\t\t& rel\t\t&A, B, C, D\t& A, B, C\t& A, C\t\\\\\n\t\t\t\t\t\t& essli\t\t&A, B, C, D\t& A, B, C\t\t& A, C\t\\\\ \\hline\n\\multirow{3}{*}{Dice}\t& sim\t\t&A, B, C, D\t& A, B, C\t& A, B \\\\\n\t\t\t\t\t\t& rel\t\t&A, B, C, D\t& -\t\t\t& -\t\\\\\n\t\t\t\t\t\t& essli\t\t& -\t\t\t& -\t\t\t& -\t\\\\ \\hline\n\\multirow{3}{*}{Ochiai}\t& sim\t\t&A, B, C, D\t& A, B, C, D& A, B, C \\\\\n\t\t\t\t\t\t& rel\t\t&A, B, C, D\t& A, B, C\t& A, B, C\t\\\\\n\t\t\t\t\t\t& essli\t\t&A, B, C, D\t& A, B\t\t& A\t\\\\ \\hline\n\\multirow{3}{*}{LLR}\t& sim\t\t&A, B, C, D\t& A, B\t\t& A \\\\\n\t\t\t\t\t\t& rel\t\t&A, B, C, D\t& A\t\t\t& A\t\\\\\n\t\t\t\t\t\t& essli\t\t&A, B, C\t& A\t\t\t& A\t\\\\ \\hline\n\\multirow{3}{*}{TTest}\t& sim\t\t&A, B, C\t& A\t\t\t& - \\\\\n\t\t\t\t\t\t& rel\t\t&A, B, C\t& -\t\t\t& -\t\\\\\n\t\t\t\t\t\t& essli\t\t& -\t\t\t& -\t\t\t& -\t\\\\ \\hline\n\\multirow{3}{*}{SCI}\t& sim\t\t& -\t\t\t& -\t\t\t& - \\\\\n\t\t\t\t\t\t& rel\t\t& -\t\t\t& -\t\t\t& -\t\\\\\n\t\t\t\t\t\t& essli\t\t& -\t\t\t& -\t\t\t& -\t\\\\ \\hline\n\\end{tabular}\n\\caption{Types of lexical co-occurrences detected by different measures}\n\\label{tab:ABCDsummary}\n\\end{table}\n\n\\begin{figure*}\n  \\begin{center}\n\\resizebox{160mm}{!}\n{\\includegraphics{sim_all_0.pdf}} \\\\\n    \\caption{\\small Maximum correlation of various measures with various types of CSR for sim dataset}\n    \\label{fig:maxCorSim}\n  \\end{center}\n\\end{figure*}\n\nThe Table~\\ref{tab:ABCDsummary} lists\nfor each measure and for each data set, the different types of lexically significant co-occurrences that the\nmeasure is able to detect effectively -- if the corresponding Spearman\ncorrelation coefficient exceeds 0.90, we consider the measure to be effective for the given\ntype. Results are shown for three different span constraints --\nsmall span of 5 words (or 5w), medium span of 25 words (or 25w) and large span of 50 words (or 50w).\nFor example, the CSA and Ochiai measures are effective in detecting all 4 types of lexically significant\nco-occurrences (A, B, C and D) in all three data sets, when the span constraint is set to 5 words.\nFigure~\\ref{fig:maxCorSim} presents a detailed quantitative comparison of the best performance of each\nmeasure with respect to each type of co-occurrence for a range of different span constraints on the\nsim data set (Similar results were obtained on other data sets). The inferences we can draw are\nconsistent with the results of Table~\\ref{tab:ABCDsummary}.\n\n\\begin{table}\n\\scriptsize\n\\centering\n\n\\begin{tabular}{|l|l|l|l|l|l|}\n\\hline\n\t\t\t\t\t\t& \t\t\t& \\multicolumn{4}{|c|}{Parameters for best correlation} \\\\ \\hline\n\tMeasure\t\t\t\t&\tSpan\t&\t$\\epsilon$\t&\t$\\delta$\t& Type\t& Correlation\t\t\\\\ \\hline\n\\multirow{3}{*}{PMI}\t& 5w\t\t& 0.05\t\t& 1\t\t\t& C\t\t& 91.3 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.40\t\t& 1\t\t\t& D\t\t& 85.3\t\\\\\n\t\t\t\t\t\t& 50w\t\t& 0.50\t\t& 1\t\t\t& D\t\t& 82.0 \\\\ \\hline\n\\multirow{3}{*}{CWCD}\t& 5w\t\t& 0.99\t\t& 0.9\t\t& D \t& 83.6 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.50\t\t& 0.9\t\t& D\t\t& 76.0\t\\\\\n\t\t\t\t\t\t& 50w\t\t& 0.50\t\t& 0.9\t\t& D\t\t& 74.4\t\\\\ \\hline\n\\multirow{3}{*}{CSA}\t& 5w\t\t& 0.1\t\t& 0.0005\t\t& A \t& 98.9 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.05\t\t& 0.0005\t\t& A\t\t& 96.7\t\\\\\n\t\t\t\t\t\t& 50w\t\t& 0.1\t\t& 0.0005\t\t& A\t\t& 94.9\t\\\\ \\hline\n\\multirow{3}{*}{Dice}\t& 5w\t\t& 0.1\t\t& 0.005\t\t& A\t\t& 96.1 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.05\t\t& 0.005\t\t& A\t\t& 93.0\t\\\\\n\t\t\t\t\t\t& 50w\t\t& 0.1\t\t& 0.0005\t& A\t\t& 91.3 \\\\ \\hline\n\\multirow{3}{*}{Ochiai}\t& 5w\t\t& 0.1\t\t& 0.1\t\t& A\t\t& 97.4 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.1\t\t& 0.01\t\t& A\t\t& 95.5\t\\\\\n\t\t\t\t\t\t& 50w\t\t& 0.1\t\t& 0.005\t\t& A\t\t& 94.5 \\\\ \\hline\n\\multirow{3}{*}{LLR}\t& 5w\t\t& 0.05\t\t& 0.0005\t& A \t& 97.3 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.05\t\t& 0.0005\t& A\t\t& 94.8 \\\\\n\t\t\t\t\t\t& 50w\t\t& 0.1\t\t& 0.0005\t& A\t\t& 92.6 \\\\ \\hline\n\\multirow{3}{*}{TTest}\t& 5w\t\t& 0.05\t\t& 0.0005\t& A \t& 94.2 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.05\t\t& 0.0005\t& A\t\t& 90.9 \\\\\n\t\t\t\t\t\t& 50w\t\t& 0.1\t\t& 0.0005\t& A\t\t& 88.8 \\\\ \\hline\n\\multirow{3}{*}{SCI}\t& 5w\t\t& 0.05\t\t& 0.0005\t& A \t& 82.7 \\\\\n\t\t\t\t\t\t& 25w\t\t& 0.05\t\t& 0.0005\t& A\t\t& 75.9 \\\\\n\t\t\t\t\t\t& 50w\t\t& 0.1\t\t& 0.0005\t& A\t\t& 73.1 \\\\ \\hline\n\\end{tabular}\n\\caption{Best performing $(\\epsilon,\\delta)$-pairs for different measures on {\\em sim} data}\n\\label{tab:topsummarysim}\n\\end{table}\n\nIn our next experiment, we examine which of the four types of co-occurrences are best captured by each measure.\nResults for the sim data set are listed in Table~\\ref{tab:topsummarysim} (Similar results were obtained on the other data sets). For each\nmeasure and for each span constraint, the table describes the best performing parameters ($\\epsilon$\nand $\\delta$), the corresponding co-occurrence Type and the associated `best' correlation achieved with respect to\nthe test of {\\em Definition~\\ref{def:test1}} .\nThe results show that, irrespective of the span\nconstraint, most measures perform best on Type A co-occurrences. This is reasonable because\nType A essentially represents the strongest correlations in the data and one would expect the\nmeasures to capture the strong correlations better than weaker ones. There are however, two\nexceptions to this rule, namely PMI and CWCD, which instead peak at Types C or D. The best correlations for\nthese two measures are also typically lower than the other measures. We now summarize the main\nfindings from our study:\n\\begin{itemize}\n\n\\item The relatively obscure Ochiai, and the newly introduce CSA are the best performing measure, in terms of detecting all types\nof lexical co-occurrences in all data sets and for a wide range of span constraints.\n\n\\item Dice, LLR and TTest are the other measures that effectively track lexically significant\nco-occurrences (although, all three are less effective as the span constraints become larger).\n\n\\item SCI, CWCD, and the popular PMI measure\nare ineffective at capturing {\\em any} notion of lexically significant co-occurrences, even for small\nspan constraints. In fact, the best result for PMI is the detection of Type C co-occurrences in the\nsim data set. The low $\\epsilon$ and high $\\delta$ setting of Type C suggests that PMI does a poor\njob of detecting the strongest co-occurrences in the data, overlooking both strong document-level as\nwell as corpus-level cues for lexical significance. \n\n\n\\end{itemize}\n\n\\begin{table*}\n\\begin{center}\n \\scriptsize \\addtolength{\\tabcolsep}{-5pt}\n\t\\begin{tabular}{|c | c || c | c |l| c | c || c | c |l| c | c || c | c |}  \\cline{1-4} \\cline{6-9} \\cline{11-14}\n  \\multicolumn{4}{|c|}{sim} && \\multicolumn{4}{|c|}{rel} && \\multicolumn{4}{|c|}{esslli} \\\\ \\cline{1-4} \\cline{6-9} \\cline{11-14}\nPMI top 10 & R & Ochiai top 10 & R && PMI top 10 & R & Ochiai top 10 & R && PMI top 10 & R & Ochiai top 10 & R\\\\ \\cline{1-4} \\cline{6-9} \\cline{11-14}\nvodka-gin & 42 & football-soccer & 3 &  & money-laundering & 2 & soap-opera & 1 &  & nook-cranny & 91 & floyd-pink & 4 \\\\\nseafood-lobster & 59 & street-avenue & 5 &  & soap-opera & 1 & money-laundering & 2 &  & hither-thither & 104 & either-or & 1 \\\\ \nbread-butter & 13 & physics-chemistry & 2 &  & opec-oil & 8 & computer-software & 18 &  & sprain-ankle & 60 & election-general & 7 \\\\ \nvodka-brandy & 99 & television-radio & 6 &  & weather-forecast & 5 & television-film & 7 &  & blimey-cor & 147 & nook-cranny & 91 \\\\ \nmidday-noon & 79 & championship-tournament & 10 &  & psychology-cognition & 77 & jerusalem-israel & 16 &  & margarine-butter & 77 & twentieth-century & 2 \\\\ \nmurder-manslaughter & 19 & man-woman & 16 &  & decoration-valor & 73 & weather-forecast & 5 &  & tinker-tailor & 65 & bride-groom & 16 \\\\ \ncucumber-potato & 130 & vodka-gin & 42 &  & gender-equality & 11 & drug-abuse & 4 &  & ding-dong & 26 & you-me & 14 \\\\ \ndividend-payment & 61 & king-queen & 9 &  & tennis-racket & 20 & credit-card & 3 &  & bride-groom & 16 & north-south & 19 \\\\ \nphysics-chemistry & 2 & car-automobile & 43 &  & liability-insurance & 25 & game-series & 12 &  & jigsaw-puzzle & 30 & question-answer & 11 \\\\ \npsychology-psychiatry & 27 & harvard-yale & 11 &  & fbi-investigation & 10 & stock-market & 9 &  & bidder-auction & 76 & atlantic-ocean & 10 \\\\ \\cline{1-4} \\cline{6-9} \\cline{11-14}\n\\end{tabular}\n\\caption{Top 10 bigrams according to PMI and Ochiai rankings on \\emph{sim}, \\emph{rel}, and \\emph{esslli} datasets. 'R' denotes the bigrams rankings according to type-A CSR measure($\\epsilon=0.1, \\delta=0.1$). Span of 25 words is used for all the three measures. }\n\\label{tab:top10PMIOchiai}\n\\end{center}\n\\end{table*}\n\nNote that our results do not contradict the utility of PMI, SCI, or, CWCD as word-association\nmeasures. We only observe their poor performance in context of detecting lexical co-occurrences. Also, our notion of lexical co-occurrence is symmetric.\nIt is possible that asymmetric SCI may have competitive performance for certain asymmetric tasks compared to the better performing symmetric measures.\nFinally, to give a qualitative feel about the differences in the correlations preferred by different methods, \nin Table~\\ref{tab:top10PMIOchiai}, we show the top 10 bigrams picked by PMI and Ochiai for all three datasets.\n\n\n\n\n\n\\section{Relation between lexical co-occurrence and human judgements}\n\n\\begin{table*}\n\\scriptsize \\addtolength{\\tabcolsep}{-5pt}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nMethod & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\tabularnewline\n\\hline\nHuman & environment & maradona & opec & computer & money & jerusalem &\nlaw & weather & network & fbi\\tabularnewline\nJudgement & ecology (84)& football (53)& oil (8)& software (18)& bank (28)& israel (16)&\nlawyer (42)& forecast (5)& hardware (107)& investigation (10)\\tabularnewline\n\\hline\n\\multirow{2}{*}{CSR} & soap & money  & credit & drug  & weather & cup & television &\nopec & stock & fbi\\tabularnewline\n & opera (24) & laundering (129)& card (20)& abuse (69)& forecast (8)& coffee (82)& film (31)& oil (3)\n& market (19)& investigation (10)\\tabularnewline\n\\hline\n\\end{tabular}\n\\caption{ Top 10 word associations picked in rel dataset. The numbers in the brackets are the cross rankings: CSR rankings in the human row and human rankings in the CSR row. CSR parameters are same as that for Table~\\ref{tab:top10PMIOchiai}. }\n\\label{top10humanCSR}\n\\end{table*}\n\nWhile the focus of our work is on characterizing the statistically significant lexical co-occurrence, as illustrated in in Table~\\ref{top10humanCSR}, human judgement of word association is governed by many factors in addition to lexical co-occurrence considerations, and many non co-occurrence based measures have been designed to capture semantic word association. Notable among them are distributional similarity based measures \\cite{simRelDatasets,bollegalaMI07,chenLW06}\nand knowledge-based measures \\cite{wikiLinkMeasure,hughes_lexical_2007,Gabrilovich07computingsemantic,wikiwalk09,wikirelate,wordsim353,lsaEsslli08}. Since our focus is on frequency based measures alone, we do not discuss these other measures.\n\nThe lexical co-occurrence phenomenon and the human judgement of semantic association are related but different dimensions of relationships between words and different applications may\nprefer one over the other. For example, suppose, given one word (say {\\em dirt}), the task is to choose from among a number of\nalternatives for the second(say {\\em grime} and {\\em filth}). Human judgment scores for {\\em (dirt, grime)} and {\\em (dirt, filth)}\nare 5.4 and 6.1  respectively. However, their lexical co-occurrence scores (CSR) are 1.49 and 0.84 respectively. This is because {\\em filth} is often used in a moral context as well.  {\\em Grime} is usually used\nonly in a physical sense.  {\\em Dirt} is used mostly in a physical sense,\nbut is a bit more generic and may be used in a moral sense\noccasionally. Hence {\\em (dirt, grime)} is more correlated in corpus than {\\em (dirt, filth)}. This shows that human judgement is fallible and annotators may ignore the subtleties of meanings that may be picked up by a statistical techniques like ours.\n\nIn general, for association with a given word, all synonyms of a second word will be given similar semantic\nrelatedness score by human judges but they may have very different lexical association scores. \n\nFor  applications where the notion of statistical lexical\nco-occurrence is potentially more relevant than semantic relatedness, our method can be used to generate a gold-standard of lexical association (against which other association measures can be evaluated). In this context, it is interesting to note that contrary to the human judgement, each one of the co-occurrence measures studied by us finds {\\em (dirt, grime)} more associated than {\\em (dirt, filth)}.\n\n\nHaving explained that significant lexical co-occurrence is a fundamentally different notion than human judgement of word association, we also want to emphasize that the two are not completely different notions either and they correlate reasonably well with each-other.\nFor {\\em sim, rel}, and {\\em essli} datasets, CSR's best correlations with human judgment are 0.74, 0.65, and 0.46 respectively. Note that CSR is a symmetric notion and hence correlates far more with human judgement for symmetric {\\em sim} and {\\em rel} datasets than for the asymmetric {\\em essli} dataset.\nAlso, at first glance, it is little counter-intuitive that the notion\nof lexical co-occurrence yields better correlations with the sim (based on {\\em similarity}) data set when compared to\nthe rel(based on {\\em relatedness}) data set. This can essentially be explained by our observation that\nsimilar words tend to co-occur less frequently by-chance than the related words.\n\n\\begin{comment}\n\n\\subsection{Comparison with Previous Work}\n\nAs discussed earlier, any given measure of word association models certain dimension of the relationship\nbetween words, and an application-neutral comparison of correlation with human judgment may not be very meaningful. It is still instructive to compare select co-occurrence based measures with other knowledge-based and distributional similarity based measures of word association proposed in the literature. Since our work is focused on co-occurrence based measures, we do not give any details of these other measures but simply present their correlations in Table~\\ref{prevWorkTable}. \nOther researchers using wordsim353 dataset have not publicized actual association score for each word pair or even the relative rankings. Hence we cannot compute the Spearmen's correlation coefficient for other measures on sim203 and rel252 dataset. The table includes all the publicly available data.\n\nWe see that co-occurrence based measures compare favorably with other resource heavy measures across the range of datasets. Of course, as mentioned several times, one should not use these correlation numbers to claim that one method is superior to other. That can only be done in application specific settings where the needs of an application are well-understood.\n\n\\begin{table}\n\\tiny \\addtolength{\\tabcolsep}{-3pt}\n\\begin{tabular}{|p{3cm}| p{1.2cm}| l l p{.7cm}| l|}   \n \\hline\nMethod & Resource & sim & rel & wordsim & esslli\\\\\n\\hline\nPMI & Wikipedia corpus & 0.74 & 0.69 & 0.70 & 0.34 \\\\\nOchiai & Wikipedia corpus & 0.69 & 0.64 & 0.63 & 0.46 \\\\\nSignificance Ratio (CSR) & Wikipedia corpus & 0.70 & 0.65 & 0.65 & 0.46  \\\\ \\hline\nLatent Semantic Analysis~\\cite{lsaEsslli08} & Newspaper corpus & - & - & - & 0.38 \\\\\nGraph Traversal (WN30g) ~\\cite{simRelDatasets}) & Wordnet & 0.72 & 0.56 & 0.66 & - \\\\\nBag of Words based Distributional Similarity (BoW)~\\cite{simRelDatasets}) & Web corpus & 0.70 & 0.62 & 0.65 & - \\\\ \nContext Window based Distributional Similarity (CW)~\\cite{simRelDatasets}) & Web corpus & 0.77 & 0.46 & 0.60 & - \\\\ \nHyperlink Graph~\\cite{wikiLinkMeasure} & Wikipedia hyperlinks graph & -& -& 0.69 & - \\\\\nRandom Graph Walk~\\cite{hughes_lexical_2007} & WordNet & -& -& 0.55  & - \\\\\nExplicit Semantic Analysis~\\cite{Gabrilovich07computingsemantic} (reimplemented in~\\cite{wikiwalk09}) & Wikipedia concepts & -& -& 0.75 (0.71)  & - \\\\\nNormalized Path-length (lch)~\\cite{wikirelate} & Wikipedia category tree & -& -& 0.55  & - \\\\\nThesarus based~\\cite{jarmasz03} & Roget's & - & - & 0.55  & - \\\\\nLatent Semantic Analysis~\\cite{wordsim353} & Web corpus, & - & - & 0.56 & - \\\\\n\\hline\n\\end{tabular}\n\\caption{ \\small Comparison with previous work. Data for missing entries is not available.}\n\\label{prevWorkTable}\n\\end{table}\n\\end{comment}\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe notion of {\\em word association} is important for\nnumerous NLP applications, like, word sense disambiguation,\noptical character recognition, speech\nrecognition, parsing, lexicography,\nnatural language generation, and machine\ntranslation. Lexical co-occurrence is an important indicator of word association and this has\nmotivated several frequency-based measures for word association \\cite{churchHanks89,llr,dice,cwcd}.\nIn this paper, we present a theoretical basis for detection and classification of lexical\nco-occurrences\\footnote{Note that we are interested in co-occurrence, not collocation, i.e.,\npairs of words that co-occur in a document with an arbitrary number of intervening words. Also, we use the term bigram to mean\nbigram at-a-distance or spanned-bigram -- again, other words can occur in-between the constituents of\na bigram.}. In general, a lexical co-occurrence\ncould refer to a pair of words that occur in a large number of documents; or it could refer\nto a pair of words that, although appear only in a small number of documents, occur frequently very\nclose to each other within each document. We formalize these ideas and construct a significance\ntest for co-occurrences that will allow us to detect different kinds of co-occurrences within a\nsingle unified framework (a feature which is absent in current measures for co-occurrence). As a\nby-product, our framework also leads to a better understanding of  existing measures for word\nco-occurrence.\n\n\n\n\n\n\nAs pointed out in ~\\cite{kilgariff05language}, language is never random -\nwhich brings us to the question of what model of\nrandom chance can give us a good statistical test\nfor lexical co-occurrences.\nWe need a null hypothesis that can account for an\nobserved co-occurrence as a pure chance event and\nthis in-turn requires a corpus generation model.\nIt is often reasonable to assume that documents\nin the corpus are generated independent of each\nother. Existing frequecy-based association\nmeasures like PMI~\\cite{churchHanks89},\nLLR~\\cite{llr} etc. further assume that each document\nis drawn from a multinomial distribution\nbased on global unigram frequencies. The main\nconcern with such a null model is the overbearing influence of unigram\nfrequencies on the detection of word associations. For example, the association between {\\em anomochilidae} (dwarf pipe\nsnakes) and {\\em snake} would go undetected in our wikepedia corpus, since less than\n$0.1\\%$ of the pages containing {\\em snake} also contained\n{\\em anomochilidae}. Similarly, under current models, the expected {\\em span} (inter-word distance)\nof a bigram is also very sensitive to the associated unigram frequencies:\nthe expected span of a bigram composed of low frequency\nunigrams is much larger than that with\nhigh frequency unigrams. This is contrary to\nhow word associations appear in language, where\nsemantic relationships manifest with small inter-word\ndistances irrespective of the underlying unigram\ndistributions.\n\nThese considerations motivate our search for a\nmore direct relationship between words,\none that can potentially be detected using careful\nstatistical characterization of inter-word distances, while minimizing the influence of the\nassociated unigram frequencies. We focus on only the documents containing both the terms (of a\ncandidate bigram) since in NLP applications, we often have\nto chose from a set of alternatives for a given word. Hence, rather than ask the abstract\nquestion of whether words $x$ and $y$ are related, our approach is to ask, given that $y$ is a candidate for pairing with $x$,\nhow likely is it that $x$ and $y$ are lexically correlated. For example, probability that  {\\em\nanomochilidae} is found in the vicinity of {\\em snake} is higher if we knew that\n{\\em anomochilidae} and {\\em snake} appear in the same context.\n\n\n\nWe consider a null model that represents each document as a bag of words \\footnote{There can be many ways to\nassociate a bag of words with a document. Details of this association are not important for us,\nexcept that the bag of words provides some kind of quantitative summary of the words within the document.}.\nThen, a random permutation of\nthe associated bag of words gives a linear\nrepresentation for the document. An arbitrary relation between a pair\nof words will result in the locations\nof these words to be randomly distributed\nin the documents in which they co-occur.\nIf the observed span distribution of a bigram resembles that under\nthe (random permutation) null model, then the relation between the words is not strong enough\nfor one word to influence the placement of the other. However, if the words are\nfound to occur closer together than explainable by our\nnull model, then we hypothesize existence of a more direct association\nbetween these words.\n\n\n\n\nIn this paper, we formalize the notion of statistically significant lexical co-occurrences by introducing a\nnull model that can detect biases in span distributions of word associations, while being\nagnostic to variations in global unigram frequencies. Our framework has the fidelity to\ndistinguish different classes of lexical co-occurrences, based on strengths of the document\nand corpus-level cues of co-occurrence in the data.\nWe perform extensive  experiments on benchmark data sets to study the performance of various co-occurrence\nmeasures that are currently known in literature. We find that a relatively obscure measure called\nOchiai, and a newly introduced measure CSA, capture the notion of lexical co-occurrence best, followed next by LLR, Dice, and TTest, while\nanother popular measure, PMI, suprisingly, performs poorly in the context of lexical co-occurrence.\n\n\n\\begin{comment}\nWe perform extensive  experiments on benchmark data sets to study how several well-known co-occurrence\nmeasures correlate with our notion of significant lexical co-occurrence.\nWe also compare performance of these measures against human produced\ngold-standards. To the best of our knowledge, a comparison study\nof different co-occurrence measures on a large English dataset\nfor the word association problem has not been reported before. \nOur significance test competes well with knowledge and computation\nintensive distributional similarity and knowledge-based \nmeasures reported in \\cite{simRelDatasets}, \\cite{wikiwalk09},\n\\cite{wikiLinkMeasure}, \\cite{Gabrilovich07computingsemantic}, \\cite{wikirelate} and \\cite{jarmasz03}.\n\\end{comment}\n\n\n\n\n\n\n\\section{Experimental Results}\n\n\\subsection{Datasets and Text Corpus}\n\nSince similarity and relatedness are\ndifferent kinds of word associations \\cite{budanitskyHirst}, in ~\\cite{simRelDatasets}\ntwo different data sets, namely 203 words {\\em sim} (the\nunion of similar and unrelated pairs) and 252 words {\\em rel} (the union of related\nand unrelated pairs) datasets are derived from {\\em wordsim}~\\cite{wordsim353}.\nWe use these two data sets in our experiments. These datasets are symmetric in that the order of words in a pair is not expected to matter. As some of our chosen co-occurrence measures are asymmetric, we also report results on the asymmetric 272-words {\\em esslli} dataset\nfor the `free association' task at~\\cite{esslli08}. \n\n\nWe use the Wikipedia~\\cite{wikipedia} corpus in our experiments. It contains 2.7 million articles\nfor a total size of 1.24 Gigawords.  We did not pre-process the corpus - no lemmatization,\nno function-word removal. When counting document size in words, punctuation symbols were ignored.\nDocuments larger than 1500 words were partitioned keeping the size of each part to no greater\nthan 1500 words.\n\n\nIn Table~\\ref{tab:typeExamples}, we present some examples of\ndifferent types of co-occurrences observed in the data. \n\n\n\\begin{table*}\n\\centering\n{\\scriptsize\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\nDataset & Type A bigrams & Type B bigrams & Type C bigrams & Type D bigrams \\\\ \\hline\n\\multirow{2}{*}{sim} & announcement-news & forest-graveyard & lobster-wine & stock-egg \\\\\n & bread-butter & tiger-carnivore & lad-brother & cup-object \\\\ \\hline\n\\multirow{2}{*}{rel} & baby-mother & alcohol-chemistry & victim-emergency & money-withdrawal \\\\\n & country-citizen & physics-proton & territory-kilometer & minority-peace \\\\ \\hline\n\\multirow{2}{*}{esslli} & arrest-police & pamphlet-read & meditate-think & fairground-roundabout \\\\\n & arson-fire & spindly-thin & ramble-walk & \\\\\n\\hline\n\\end{tabular}\n\\label{tab:typeExamples}\n \\caption{Examples of Type A, B, C and D co-occurrences under a span constraint of 20 words.}\n}\n\\end{table*}\n\n\\begin{comment}\n\n\\begin{table*}\n\\scriptsize\n\\centering\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n\\hline\n& & & \\multicolumn{4}{|c|}{$\\epsilon=0.4$; $\\delta=0.1$} & \\multicolumn{4}{|c|}{$\\epsilon=0.1$; $\\delta=0.1$} \\\\ \\cline{2-11}\nWord pair & Data set & $K$   & $Z$   & $E(Z)$ & $Kt$  & CSR & $Z$  & $E(Z)$  & $Kt$  & CSR   \\\\ \\hline\nforest-graveyard & sim & 265 & 45 & 16.6 & 17.5 & 1.3 & 19 & 8.4 & 17.5 & 0.7 \\\\ \ntiger-carnivore & sim & 50 & 13 & 2.6 & 7.6 & 1.3 & 8 & 1.6 & 7.6 & 0.9 \\\\ \nalcohol-chemistry & rel & 702 & 77 & 47.3 & 28.4 & 1.0 & 37 & 23.1 & 28.4 & 0.7 \\\\ \nphysics-proton & rel & 547 & 76 & 46.6 & 25.1 & 1.1 & 31 & 17.0 & 25.1 & 0.7 \\\\ \npamphlet-read & esslli & 389 & 40 & 17.6 & 21.2 & 1.0 & 29 & 12.5 & 21.2 & 0.9 \\\\ \nspindly-thin & esslli & 25 & 13 & 3.0 & 5.4 & 1.6 & 4 & 0.4 & 5.4 & 0.7 \\\\ \n\\hline\n\\end{tabular}\n\\caption{Examples of Type B co-occurrences found in data under a span\nconstraint of 20 words.}\n\\label{tab:typeBillustrations}\n\\end{table*}\n\n\n\\begin{table*}\n\\scriptsize\n\\centering\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n\\hline\n                                       &                       &               & \\multicolumn{4}{|c|}{$\\epsilon=0.1$; $\\delta=0.4$}    &\n\\multicolumn{4}{|c|}{$\\epsilon=0.1$; $\\delta=0.1$} \\\\ \\cline{2-11}\nWord pair                       & Data set      & $K$   & $Z$   & $E(Z)$        & $Kt$  & CSR   & $Z$   &\n$E(Z)$  & $Kt$  & CSR   \\\\ \\hline\nlobster-wine & sim & 87 & 9 & 2.6 & 6.3 & 1.0 & 9 & 2.6 & 10.0 & 0.7 \\\\ \nlad-brother & sim & 306 & 22 & 10.1 & 11.8 & 1.0 & 22 & 10.1 & 18.8 & 0.8 \\\\ \nvictim-emergency & rel & 633 & 37 & 18.6 & 17.0 & 1.0 & 37 & 18.6 & 27.0 & 0.8 \\\\ \nterritory-kilometer & rel & 206 & 18 & 6.8 & 9.7 & 1.1 & 18 & 6.8 & 15.4 & 0.8 \\\\ \nmeditate-think & esslli & 70 & 8 & 2.2 & 5.7 & 1.0 & 8 & 2.2 & 9.0 & 0.7 \\\\ \nramble-walk & esslli & 40 & 6 & 1.7 & 4.3 & 1.0 & 6 & 1.7 & 6.8 & 0.7 \\\\ \n\\hline\n\\end{tabular}\n\\caption{Examples of Type C co-occurrences found in data under a span\nconstraint of 20 words.}\n\\label{tab:typeCillustrations}\n\\end{table*}\n\nRecall that our test\nstatistic for lexical significance is $[Z \\geq E(Z)+Kt]$. When $\\epsilon$ is relaxed keeping\n$\\delta$ fixed, $E(Z)$ increases while $Kt$ remains constant. This can be observed in\nTables~\\ref{tab:typeAillustrations} \\& \\ref{tab:typeBillustrations}. Similarly, when $\\delta$ is\nrelaxed keeping $\\epsilon$ fixed $E(Z)$ is constant and $Kt$ changes (See\nTable~\\ref{tab:typeCillustrations}). Notice how the patterns are significant (indicated by CSR $>1$) for both sets of\n$\\epsilon$ and $\\delta$ in Table~\\ref{tab:typeAillustrations}; this is unlike in Tables\n\\ref{tab:typeBillustrations} \\& \\ref{tab:typeCillustrations} where we have reported exclusively Type\nB and Type C patterns respectively.\n\n\\end{comment}\n\n\n\\section{Lexically significant co-occurrences}\n\\label{sec:significance-test}\n\nConsider a bigram $\\alpha$. Let $\\ensuremath{\\mathcal{D}}=\\{D_1,\\ldots,D_K\\}$ denote the set of \n$K$ documents (from out of the entire corpus) that contain at least one occurrence of $\\alpha$.  The {\\em frequency} of $\\alpha$ in\ndocument $D_i$, $f_i$, is the maximum number of {\\em non-overlapped occurrences} of $\\alpha$ in $D_i$. A set of occurrences of \na bigram are called non-overlapping if the words corresponding to one occurrence from the set do not appear\nin-between the words corresponding to any other occurrence from the set. \n\n\nThe {\\em span} of an occurrence of $\\alpha$ is the `unsigned distance' \nbetween the first and last textual units of interest associated with that occurrence.\nWe mostly use words as the unit of distance, but in general, distance can be measured in\nwords, sentences, or even paragraphs (e.g.~an occurrence comprising two adjacent words in a sentence has a word-span of one \nand a sentence-span of zero). Likewise, the size of a document $D_i$, denoted as $\\ell_i$, \nis correspondingly measured in units of words, sentences or paragraphs.\nFinally, let $\\ensuremath{\\widehat{f}}_i$ denote the maximum number of non-overlapped occurrences of\n$\\alpha$ in $D_i$ with span less than a given threshold $x$. We refer to\n$\\ensuremath{\\widehat{f}}_i$ as the {\\em span-constrained frequency} of $\\alpha$ in $D_i$. Note that \n$\\ensuremath{\\widehat{f}}_i$ cannot exceed $f_i$.\n\n\nTo assess the statistical significance of the bigram $\\alpha$\nwe ask if the span-constrained frequency $\\ensuremath{\\widehat{f}}_i$ (of $\\alpha$)\nis more than what we would expect for it in a document of size $\\ell_i$ containing $f_i$ `random' occurrences of $\\alpha$.\nOur intuition is that if two words are semantically related, they will often appear close to\neach other in the document and so the distribution of the spans will  typically exhibit a prominent bias\ntoward values less than a small $x$.\n\nConsider the null hypothesis that a document is generated as a random permutation of  the bag of words\nassociated with the document. Let $\\pi_x(\\ensuremath{\\widehat{f}},f,\\ell)$ denote the probability of\nobserving a span-constrained frequency (for $\\alpha$) of {\\em at least} $\\ensuremath{\\widehat{f}}$ in a document of length $\\ell$ that contains\na maximum of $f$ non-overlapped occurrences of $\\alpha$. Observe that $\\pi_x(0,f,\\ell)=1$ for any\n$x>0$; also, for $x\\geq \\ell$ we have $\\pi_x(f,f,\\ell)=1$ (i.e.~all $f$ occurrences will always have span\nless than $x$ for $x\\geq \\ell$). However, for typical values of\n$x$ (i.e.~for $x \\ll \\ell$) the probability $\\pi_x(\\ensuremath{\\widehat{f}},f,\\ell)$ decreases with increasing $\\ensuremath{\\widehat{f}}$. \nFor example, consider a document of length 400 with 4 non-overlapped\noccurrences of $\\alpha$. The probabilities of observing at least 4, 3, 2, 1 and 0 occurrences of\n$\\alpha$ within a span of 20 words are 0.007, 0.09, 0.41, 0.83, and 1.0 respectively. \nSince $\\pi_{20}(3,4,400)=0.09$, even if 3 of the 4 occurrences of $\\alpha$\n(in the example document) have span less than 20 words, \nthere is 9\\% chance that the occurrences\nwere a consequence of a random event (under our null model). As a result, if\nwe desired a confidence-level of at least 95\\%, we would have to declare $\\alpha$ as {\\em\ninsignificant}.\n\nGiven an $\\epsilon$ ($0< \\epsilon < 1$) and a span upper-bound $x$ ($\\geq 0$) \nthe document $D_i$ is said to {\\em support} the hypothesis ``$\\alpha$ is a $\\epsilon$-significant bigram'' if $\\pi_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell) < \\epsilon$. \nWe refer to $\\epsilon$ as the {\\em document-level} lexical co-occurrence of $\\alpha$.\nDefine indicator variables $z_i$, $i=1,\\ldots,K$ as:\n\\begin{equation}\nz_i = \\left\\{\\begin{array}{ll}\n1 & \\mbox{if\\ } \\pi_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell) < \\epsilon \\\\\n0 & \\mbox{otherwise}\\end{array} \\right.\n\\label{eq:zi}\n\\end{equation}\\vspace{-0.2in}\n\nLet $Z = \\sum_{i=1}^K z_i$; $Z$ models the number of documents (out of $K$) that\nsupport the hypothesis ``$\\alpha$ is a $\\epsilon$-significant bigram.''\nThe expected value of $Z$ is given by\n\\begin{eqnarray}\nE(Z) &=& \\sum_{i=1}^K E(z_i) \\label{eq:ez1}\\\\\n\t &=& \\sum_{i=1}^K \\pi_x(g_\\epsilon(f_i,\\ell_i), f_i,\\ell_i) \\label{eq:ez2}\n\\end{eqnarray}\nwhere $g_\\epsilon(f_i,\\ell_i)$ denotes the smallest $\\ensuremath{\\widehat{f}}$ for which we can get\n$\\pi_x(\\ensuremath{\\widehat{f}},f_i,\\ell_i)<\\epsilon$ (This quantity is well-defined since $\\pi_x(\\ensuremath{\\widehat{f}},f_i,\\ell_i)$ is\nnon-increasing with respect to $\\ensuremath{\\widehat{f}}$). For the example given earlier, $g_{0.2}(4,400)=3$\nand $g_{0.05}(4,400)=4$.  \n \nUsing Hoeffding's Inequality, for $t>0$,\n\\begin{equation}\nP[ Z \\geq E(Z) + Kt ] \\leq \\exp(-2Kt^2)\n\\label{eq:hoeffding}\n\\end{equation}\nTherefore, we can bound the deviation of the observed value of $Z$ from its expectation by chosing $t$ appropriately.\nFor example, in our corpus, the bigram ({\\em canyon,\\ landscape}) occurs in $K= 416$ documents. For\n$\\epsilon = 0.1$, we find that $Z=33$ documents (out of 416) have $\\epsilon$-significant occurrences,\nwhile $E(Z)$ is 14.34. Let $\\delta = .01$. By setting $t = \\sqrt{\\ln{\\delta}\/(-2K)}=.07$, we get\n$E(Z) + Kt=43.46$, which is greater than the observed value of  $Z$ (=33).\nThus, we cannot be 99\\% sure that the occurrences of ({\\em canyon,\\ landscape}) in the 33 documents\nwere a consequence of non-random phenomena. Hence, our test declares  ({\\em canyon,\\ landscape}) as\n{\\em insignificant} at $\\epsilon=.1, \\delta=.01$. We formally state the significance test for lexical\nco-occurrences next:\n\\begin{definition}\n[Significant lexical co-occurrence] \nConsider a bigram $\\alpha$ and a set of $K$ documents containing at least one occurrence of $\\alpha$.\nLet $Z$ denote the number of documents (out of $K$) that support the hypothesis ``$\\alpha$ is\nan $\\epsilon$-significant bigram (for a given $\\epsilon>0$, $x>0$)\". \nThe $K$ occurrences of the bigram $\\alpha$ are regarded $\\epsilon$-significant with\nconfidence $(1-\\delta)$ (for some user-defined $\\delta>0$) if we have $[Z \\geq E(Z) + Kt]$, where $t=\\sqrt{\\log{\\delta}\/\n(-2K)}$ and $E(Z)$ is given by Eq.~(\\ref{eq:ez2}). The ratio $[Z \/ (E(Z) + Kt)]$ is called the\nCo-occurrence Significance Ratio (CSR) for $\\alpha$.\n\\label{def:test1}\n\\end{definition}\n\n\nWe now describe how to compute $\\pi_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)$ for $\\alpha$ in $D_i$. Let $N(f_i,\\ell_i)$\ndenote the number of ways of embedding $f_i$ non-overlapped occurrences of $\\alpha$ in a document of\nlength $\\ell_i$. Similarly, let $N_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)$ denote the number of ways of embedding $f_i$\nnon-overlapped occurrences of $\\alpha$ in a document of length $\\ell_i$, in such a way that, at\nleast $\\ensuremath{\\widehat{f}}_i$ of the $f_i$ occurrences have span less than $x$. Recall that\n$\\pi_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)$ denotes the probability of observing a span-constrained frequency\n(for $\\alpha$) of at least $\\ensuremath{\\widehat{f}}_i$ in a document of length $\\ell_i$ that contains\na maximum of $f_i$ non-overlapped occurrences of $\\alpha$. Thus, we can assign the probability \n$\\pi_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)$ in terms of $N(f_i,\\ell_i)$ and $N_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)$ as follows:\n\\begin{equation}\n\\pi_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i) = \\left( \\frac{N_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)}{N(f_i,\\ell_i)} \\right)\n\\label{eq:pi}\n\\end{equation}\n\nTo compute $N(f_i,\\ell_i)$ and $N_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)$, we essentially need the histogram for $\\ensuremath{\\widehat{f}}$\ngiven $f$ and $\\ell$. Let $hist_{f,\\ell}[\\ensuremath{\\widehat{f}}]$ denote the number of ways to embed $f$ non-overlapped\noccurrences of a bigram in a document of length $\\ell$ in such a way that exactly $\\ensuremath{\\widehat{f}}$ of the $f$\noccurrences satisfy the span constraint $x$. We can obtain  $N(f_i,\\ell_i)$ and\n$N_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i)$ from $hist_{f_i,\\ell_i}$ using\n\\begin{eqnarray}\nN_x(\\ensuremath{\\widehat{f}}_i,f_i,\\ell_i) &=& \\sum_{k=\\ensuremath{\\widehat{f}}_i}^{f_i} hist_{f_i,\\ell_i}[k] \\\\\nN(f_i,\\ell_i) &=&\\sum_{k=0}^{f_i} hist_{f_i,\\ell_i}[k]\n\\end{eqnarray}\n\n\\begin{algorithm}\n\\small\n\\caption{$ComputeHist(f,\\ell)$}\n\\label{algo-wf}\n\\begin{algorithmic}[1]\n\n\\REQUIRE $\\ell$ - length of document; $f$ - number of non-overlapped occurrences to be embedded; \n$x$ - span constraint for occurrences\n\n\\ENSURE $hist_{f,\\ell}[\\cdot]$ - histogram of $\\ensuremath{\\widehat{f}}$ when $f$ occurrences are embedded in a document\nof length $\\ell$\n\n\\STATE Initialize $hist_{f,\\ell}[\\ensuremath{\\widehat{f}}] \\leftarrow 0$ for $\\ensuremath{\\widehat{f}}=0,\\ldots,f$\n\n\\IF{$f>\\ell$}\n\t\\STATE return $hist_{f,\\ell}$\n\\ENDIF\n\n\\IF{$f=0$}\n\t\\STATE $hist_{f,\\ell}[0] \\leftarrow 1$;\n\t\\STATE return $hist_{f,\\ell}$\n\\ENDIF\n\n\\FOR{$i \\leftarrow 1$ to $(\\ell-1)$}\n\t\\FOR{$j \\leftarrow (i+1)$ to $\\ell$}\n\t\t\\STATE $hist_{f-1,\\ell-j} \\leftarrow ComputeHist(f-1, \\ell-j)$\n\t\t\\FOR{$k \\leftarrow 0$ to $f-1$}\n\t\t\t\\IF{$(j-i) < x$}\n\t\t\t\t\\STATE $hist_{f,\\ell}[k+1] \\leftarrow hist_{f,\\ell}[k+1] + hist_{f-1,\\ell-j}[k]$\n\t\t\t\\ELSE\n\t\t\t\t\\STATE $hist_{f,\\ell}[k] \\leftarrow hist_{f,\\ell}[k] + hist_{f-1,\\ell-j}[k]$\n\t\t\t\\ENDIF\n\t\t\\ENDFOR\n\t\\ENDFOR\n\\ENDFOR\n\\STATE return $hist_{f,\\ell}$\n\n\\end{algorithmic}\n\\end{algorithm}\n\n{\\em Algorithm~\\ref{algo-wf}} lists the pseudocode for computing the histogram $h_{f,\\ell}$. It enumerates all possible ways of embedding $f$ non-overlapped\noccurrences of a bigram in a document of length $\\ell$. \nThe main steps in the algorithm involve selecting a start and end position for\nembedding the very first occurrence (lines 7-8) and then recursively calling\n$ComputeHist(\\cdot,\\cdot)$ (line 9). The $i$-loop selects a\nstart position for the first occurrence of the bigram, and the $j$-loop selects the end position. The task in\nthe recursion step is to now compute the number of ways to embed the remaining $(f-1)$ non-overlapped occurrences in the remaining\n$(\\ell-j)$ positions. Once we have $hist_{f-1,\\ell-j}$, we need to check whether the\noccurrence introduced at positions $(i,j)$ will contribute to the $\\ensuremath{\\widehat{f}}$ count. If $(j-i)<x$, whenever there are $k$ span-constrained\noccurrences in positions $(j+1)$ to $\\ell$, there will be $(k+1)$ span-constrained occurrences in\npositions 1 to $\\ell$. Thus, we increment $hist_{f,\\ell}[k+1]$ by the quantity\n$hist_{f-1,\\ell-j}[k]$ (lines 10-12). However, if $(j-i)>x$, there is no contribution\nto the span-constrained frequency from the $(i,j)$ occurrence, and so we increment $hist_{f,\\ell}[k]$\nby the quantity $hist_{f-1,\\ell-j}[k]$ (lines 10-11, 13-14).\n\nThis algorithm is exponential in $f$ and $l$, but it\ndoes not depend explicitly on the data. This allows us to populate the histogram off-line, and\npublish the $\\pi_x(\\ensuremath{\\widehat{f}},f,\\ell)$ tables for various $x$, $\\ensuremath{\\widehat{f}}$, $f$ and $\\ell$. \n(If the paper is accepted, we will make an interface to this table publicly available).\n \n\n \n\n\\begin{comment}\nIn light of the above, $\\epsilon$ can either be chosen by the user based on the application needs,\nor it can be left unspecified, in which case, we can derive a sound, traditional\ntwo-parameter test involving just the span constraint $x$ and the corpus-level confidence $\\delta$:\n\\begin{definition}\n[Significant lexical co-occurrences] \nOccurrences of a bigram $\\alpha$ are regarded as\n{\\em significant lexical co-occurrences} with span less than $x$ and confidence $(1-\\delta)$\n(for some user-defined $x>0$, $\\delta>0$) if there exists an $\\epsilon$ ($0<\\epsilon<1$)\nsuch that we have $[Z \\geq E(Z) + Kt]$, where $t=\\sqrt{\\log{\\delta}\/ (-2K)}$.\nThe Co-occurrence Significance Ratio (CSR) can be defined as the maximum value attained by\n$[Z \/ (E(Z) + Kt)]$ as $\\epsilon$ is varied between 0 to 1.\n\\label{def:test2}\n\\end{definition}\nWe can also show that for a given co-occurrence pattern, only finitely many $\\epsilon$ values need\nto be tried. Due to space constraints, we omit the theoretical details of this variant of our\nsignificance test. In practice, we found that the test is robust to small fluctuations in $\\epsilon$\n(which is important since, otherwise, significance results may yield unusable, fragile results).\nThroughout the rest of the paper, we stick to the notion of significance as described in Definition~\\ref{def:test1}.\n\\end{comment}\n\n\n\n\n\\section{Utility of CSR test}\\label{sec:discussion}\n\n\n\nEvidence for significant lexical co-occurrences can be gathered at two levels in the data --\ndocument-level and corpus-level. First, at the document\nlevel, we may find that a surprisingly high proportion of occurrences {\\em within} a\ndocument (of a pair of words)  have smaller spans than they would by random chance. Second, at the corpus-level, we may find\na pair of words appearing closer-than-random in an unusually high number of documents in the\ncorpus. The significance test of {\\em Definition~\\ref{def:test1}} is capable of gathering both kinds\nof evidence from data in carefully calibrated amounts. Prescribing $\\epsilon$ essentially fixes the strength of the document-level\nhypothesis in our test. A small $\\epsilon$ corresponds to a strong document-level hypothesis and\nvice-versa. The second parameter in our test, $\\delta$, controls the confidence of our decision\ngiven all the documents in the data corpus. A small $\\delta$\nrepresents a high confidence test (in the sense that there are a surprisingly large number of documents in\nthe corpus, each of which, individually have some evidence of relatedness for the pair of words). \nBy running the significance test with different values of $\\epsilon$ and $\\delta$, we can detect\ndifferent types of lexically significant co-occurrences. We illustrate the utility of\nour test of significance by considering the 4 types of lexical significant co-occurrences\n\n {\\em Type A}: These correspond to the strongest lexical co-occurrences in the data, with strong\ndocument-level hypotheses (low $\\epsilon$) as well as high corpus-level confidence (low $\\delta$). Intuitively, if a pair of\nwords appear close together several times within a document, and if this pattern is observed in a large\nnumber of documents, then the co-occurrence is of {\\em Type A}. \n\n\n{\\em Type B}: These are co-occurrences based on weak document-level hypotheses (high $\\epsilon$) \nbut because of repeated observation in a substantial number of documents in the corpus, we can still detect them with\nhigh confidence (low $\\delta$). We expect many interesting\nlexical co-occurrences in text corpora to be of\nType B \u2013 pairs of words that appear close to each\nother only a small number of times within a document,\nbut they appear together in a large number of documents.\n\n\n\\begin{comment}\nTo detect\nType B co-occurrences, we need to run our significance test (cf.~{\\em Definition~\\ref{def:test1}})\nwith high $\\epsilon$ and low $\\delta$. This essentially amounts to {\\em relaxing} the document-level hypothesis\nof Type A (while keeping $\\delta$ at the same level). Thus, the test will\nreturn {\\em all} Type A co-occurrences, plus some more. To detect Type B co-occurrences, we simply\nremove those that also belong to Type A and return only those that uniquely correspond to a\nhigh $\\epsilon$ and low $\\delta$. In our experiments we classify all the unique co-occurrences\ndetected by the test for $\\epsilon \\geq 0.4$ and $\\delta \\leq 0.1$ as Type B.\n\\end{comment}\n\n{\\em Type C}: Sometimes we may be interested in words that are\nstrongly correlated within a document, even if we observe the strong correlation only in a\nrelatively small number of documents in the corpus. These correspond to Type C co-occurrences. \nAlthough they are statistically weaker inferences than\nthose of Type A and Type B (since confidence $(1-\\delta)$ is lower) Type C co-occurrences represent an important class of relationships\nbetween words. If the document corpus contains a very small of number documents on some topic, then\nstrong co-occurrences (i.e. those found with low $\\epsilon$)  which are unique to that topic may not be\ndetected at low values of $\\delta$. By relaxing the confidence parameter $\\delta$, we may be able to detect\nsuch occurrences (possibly at the cost of some extra false positives). \n\n\n{\\em Type D}: These co-occurrences represent the weakest correlations found in\nthe data, since they neither employ a strong document-level hypothesis nor enforce a high \ncorpus-level confidence. In most applications, we expect Type D co-occurrences to be of little use, with their best case\nutility being to provide a baseline for disambiguating Type C co-occurrences. \n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|l|l|l|}   \n \\hline\nType\t&\t$\\epsilon$\t&\t$\\delta$ \\\\ \\hline\nA\t\t&\t$\\leq 0.1$\t& $\\leq 0.1$ \\\\\nB\t\t&\t$\\geq 0.4$\t& $\\leq 0.1$ \\\\\nC\t\t&\t$\\leq 0.1$\t& $\\geq 0.4$ \\\\\nD\t\t&\t$\\geq 0.4$\t& $\\geq 0.4$ \\\\ \\hline\n\\end{tabular}\n\\caption{4 types of lexical co-occurrences.}\n\\label{tab:edpairs}\n\\end{table}\n\nIn the experiments we describe later, we fix the $\\epsilon$ and $\\delta$ for the different Types as\nper Table~\\ref{tab:edpairs}. Finally, we note that Types B and C subsume Type A; similarly, Type D\nsubsumes all three other types. Thus, to detect co-occurrences that are exclusively of (say) Type B,\nwe would have to run the test with a high $\\epsilon$ and low $\\delta$ and then remove from the\noutput, those co-occurrences that are also part of Type A.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe presence of charged quantum vacuum fluctuations induces self-interactions\nof the electromagnetic field~\\cite{Heisenberg:1936qt}. In particular, light\npassing through a strong external magnetic field is expected to\ntravel at reduced velocity compared to the propagation through plain\nvacuum~\\cite{Baier,Dittrich:2000zu}.\n\n\nAs we argue in the following, the combination of ground-based\ngravitational-wave interferometers and strong pulsed magnetic fields forms an\ninstrument which is sensitive enough to demonstrate nonlinearities in the\npropagation of light and thereby contribute to the research of\nstrong-field QED \\cite{Dobrich:2009}. At the same time, it facilitates a search for light particles beyond our\ncurrent standard model of particle physics.\n\n\n\\section{Alternative goals for gravitational-wave interferometers}\n\nIn order to detect gravitational-waves by means of interferometry, two evacuated tubes of equal length $L$ are installed orthogonally with respect to each other. The respective tubes have a mirror installed at their ends and thus form a cavity for a laser beam which is directed through both tubes by means of a beam splitter. An incoming gravitational-wave will \ninduce a relative change $\\Delta L(t)$ among the lengths of the two arms as a function of time. Alternatively, an \\textit{apparent} change of optical path length $L$ can be caused by applying an external magnetic field $B(t)$ over a distance $x$ in one of the interferometer arms, as the light traveling through the magnetic field region will propagate at reduced velocity. Using natural units $\\hbar=c=1$, this implies a so-called \\textit{strain} in the interferometer\n\\begin{equation}\nh(t) = \\frac{\\Delta L}{L}(t) = \\frac{x}{L} (1-v(t)) \\ , \\label{eq:strain}\n\\end{equation}\nas first suggested by \\cite{Boer:2002zw}, cf. also \\cite{Denisov_Zavattini}. \n\nSince the sensitivity of the interferometer to the strain $h(t)$ is limited by diverse sources of noise, the temporal variation of $h(t)$ should be adapted to the region of highest sensitivity. Generically, gravitational-wave interferometers are most sensitive to variations at frequencies of about $\\mathcal{O}(100\\mathrm{Hz})$. More precisely, the specific sensitivity of each interferometer can be read off its spectral noise density function $S_h(f)$, see e.g.~\\cite{Blair:1991wd}. In conclusion, for the detection of nonlinear light propagation with the help of gravitational-wave interferometers one needs magnetic fields varying at the millisecond scale.\n\nIn fact, such pulsed fields are provided by several magnetic field laboratories around the world. Focussing on the ongoing research at the Dresden High-Magnetic-Field-Laboratory (HDL)~\\cite{wosnitza}, we consider the specifications of a technically feasible Helmholtz-coil setup with a coil diameter of $x=0.2\\mathrm{m}$. The need for a Helmholtz setup arises from the fact that no nonlinearities are induced for light traveling along the direction of the magnetic field lines. By contrast, for light traveling orthogonally to the magnetic field lines, the effect is maximized\\footnote{For this reason, also the drop-off in field strength perpendicular to the field lines which is generic for Helmholtz coils must be minimized.}, depending on the beam polarization.\n\nA feasible model for $N$ subsequent field pulses is a damped sinusoidal oscillation:\n\\begin{equation}\nB(t)=B_{0}\\sum_{i=0}^{N-1}\\theta(t-t_{i})\n\\sin(2 \\pi \\nu_{B}(t-t_{i}))\\exp(-\\gamma(t-t_{i}))\\ ,\n\\label{eq:model_pulse}\n\\end{equation} \nwith pulse frequency $\\nu_{B}$ and a damping constant $\\gamma$. For the following estimates, we assume $B_{\\mathrm{max}}=60\\mathrm{T}$ and $B_{\\mathrm{min}}=-6\\mathrm{T}$  which fixes the amplitude $B_{0}\\approx148\\mathrm{T}$ and relates the remaining parameters via $\\gamma=2 \\nu_{B} \\ln\\left|B_{\\mathrm{max}}\/B_{\\mathrm{min}}\\right|$.\n\nA meaningful measure for the visibility of the strain $h(t)$ is the signal-to-noise-ratio (SNR) $d$. Its value is a measure for the likeliness that the strain is induced by the external magnetic field rather than due to random noise fluctuations. Applying a matched filter (or ''Wiener filter'')~\\cite{Blair:1991wd}, the square of the SNR is given by\n\n\\begin{equation}\nd^{2}=2\\int_{0}^{\\infty}\\frac{|\\tilde{h}(f)|^{2}}{S_{h}(f)}\\mathrm{d}f\\\n,\\quad \\tilde{h}(f)=\\int_{-\\infty}^{\\infty}h(t)e^{-2\\pi ift}\\mathrm{d}t,\n\\label{eq:SNR_def}\n\\end{equation}\nwhere $\\tilde{h}(f)$ is the Fourier transform of the induced strain.\nA lever arm for the enhancement of this observable is provided by the fact that the setup for the field pulse is non-destructive and thus the pulse can be repeated after the magnet system has been re-cooled. Depending on the details of the setup, the re-cooling time of the magnet system is on the order of several minutes.\nTo good accuracy, $N$ subsequent pulses can enhance the SNR by a factor of $\\sqrt{N}$:\n\\begin{equation}\n d^{2}|_{N}\\approx N\\ d^{2}|_{1}. \\label{eq:SNR_N}\n\\end{equation}\n\n\n\\section{Discovery potential at GEO600 and advanced LIGO}\n\n\n\\begin{figure}\n\\begin{minipage}{0.49 \\linewidth}\n\\includegraphics[scale=0.285]{doebrich_babette.fig1.eps}\n\\end{minipage}\n\\begin{minipage}{0.49 \\linewidth}\n\\includegraphics[scale=0.285]{doebrich_babette.fig2.eps} \n\\end{minipage}\n\\caption{The figure on the left-hand side shows the discovery potential for spin-$\\frac{1}{2}$ minicharged particles (MCP), while the figure on the right-hand side applies to axion-like particles (ALP). Already a single pulse measurement at advanced LIGO can improve the best current laboratory bounds~\\cite{Zavattini:2005tm,Chou:2007zzc} in the respective coupling-mass planes.} \n\\label{fig:figure1}\n\\end{figure}\n\nWe start by computing the number of pulses required to achieve a total SNR of $\\mathcal{O}(1)$ for the strain induced by nonlinear QED. To maximize the effect, the laser beam should be polarized in parallel to the external magnetic field lines. The velocity shift then reads~\\cite{Baier,Dittrich:2000zu} $1-v=14B^{2}\\alpha^{2}\/(45 m^{4})$, where $\\alpha\\approx1\/137$ denotes the fine-structure constant and $m$ the electron mass. Together with the parameterization of the field  pulse, see Eq.\\eqref{eq:model_pulse}, the velocity shift can be translated into the SNR through Eqs.~\\eqref{eq:SNR_def} and~\\eqref{eq:strain}, while the number of required pulses $N$ enters through Eq.~\\eqref{eq:SNR_N}. We perform the calculation for the noise densities $S_h(f)$ of the advanced LIGO~\\cite{ligocurves}, which consists of interferometer arms of length $L=4000\\mathrm{m}$, and GEO600~\\cite{geocurves}, where $L=600\\mathrm{m}$. By a variation of the SNR with respect to the pulse frequency $\\nu_B$, we find that for the advanced LIGO $\\nu_B \\approx 47 \\mathrm{Hz}$ yields the greatest strain, while for GEO600 $\\nu_B\\approx 273 \\mathrm{Hz}$ is optimal. In terms of the number of required pulses, this would imply $N \\approx 2763$ at advanced LIGO, demanding a continuous operation over a few days, which appears reasonable.\n(The operation time at GEO600, however, would be several years since  $N \\approx 2.3 \\times 10^6$ pulses would be needed for an SNR of $\\mathcal{O}(1)$ from the QED induced strain).\n\nIn analogy to the vacuum polarization induced by the electron fluctuations, also hypothetical particles with a weak coupling to photons can induce a velocity shift in the interferometer~\\cite{Gies:2008wv}. In the following, we therefore deduce the accessible parameter space with respect to coupling and mass for axion-like particles (ALPs) and minicharged particles (MCPs).\n\nThe velocity shift induced by fluctuating MCPs~\\cite{Gies:2006ca,Ahlers:2006iz} with fractional charge $Q=\\epsilon e$ depends strongly on their mass $m_{\\epsilon}$. While for large masses, the scaling is analogous to the electromagnetic situation $(1-v)\\sim \\varepsilon^4 B^2\/m_\\varepsilon^4$, for low MCP masses the asymptotic limit reads $(1-v)\\sim-\\varepsilon^{8\/3} B^{2\/3}\/\\omega^{4\/3}$, where the laser frequency $\\omega=1.2 \\mathrm{eV}$ for the interferometers.\nWe consider only MCP masses with a Compton wavelength smaller than the separation of the Helmholtz coils  $\\sim\\mathcal{O}(1\\mathrm{cm})$, implying $m_{\\varepsilon}\\gtrsim 2\\times10^{-5}\\mathrm{eV}$. For smaller masses, the homogeneous-field assumption underlying the prediction for the velocity shift is no longer valid.\n\nUncharged scalar (S) and pseudo-scalar (P) ALPs couple to the $\\bot$ and the $\\parallel$ mode of the laser beam in the magnetic field, respectively. The corresponding velocity shifts read~\\cite{Maiani:1986md} $1-v_\\parallel^{\\text{P}}=1-v_\\bot^{\\text{S}} =B^{2}g^{2}\/\\left[2 m_{\\phi}^{2}\n\\left(1-\\sin(2y)\/2y \\right)\\right]$, where $y=xm_{\\phi}^{2}\/(4\\omega)$ with ALP mass $m_{\\phi}$ and coupling $g$.\n\n\nAs displayed in Fig.~\\ref{fig:figure1}, already a single-pulse measurement at advanced LIGO can improve the currently best laboratory bounds for MCPs~\\cite{Zavattini:2005tm,Chou:2007zzc} and ALPs~\\cite{Zavattini:2005tm,Ahlers:2006iz} in the upper mass ranges (comparable to results for $\\mathcal{O}(10^3)$ pulses at GEO600). Taking $N=2763$ pulses at advanced LIGO, as needed for the QED effect, current laboratory bounds can be improved almost in the entire mass range.\n\n\n\n\n\n\\section{Conclusions}\n\nPulsed magnetic fields such as provided by the Dresden High-Magnetic-Field-Laboratory can contribute to the research in the strong-field domain of QED for two reasons. Although they have generically a reduced field extent $x$ in comparison to dipole magnets, they can provide for extremely high field strengths $B$.  Since the velocity shifts induced by nonlinear QED, ALPs and the large mass regime of MCPs scale with $x B^2$, the reduced field extent can well be compensated for, see also~\\cite{battesti}. Secondly, their pulse frequency can be well matched to the region of highest sensitivity of gravitational-wave interferometers.\nFor these reasons, combining strong pulsed magnetic fields with the interferometric techniques provided by modern gravitational-wave interferometers can give access to an unexplored parameter regime of strong field QED and at the same time allow to search for particles of a hidden sector.\n\n\n\n\n\n\\section*{Acknowledgments}\nB.D. would like to thank the organizers of the 5th Patras Workshop in Durham for the opportunity to contribute to the workshop on the one hand and even more profit from it on the other. The authors acknowledge support from the DFG under GRK1523, SFB\/TR18, and Gi328\/5-1.\n \n\n\\begin{footnotesize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nOne of the main tasks of the LHC is to search for Supersymmetry\n(SUSY)~\\cite{mssm}. \nThe Minimal Supersymmetric Standard Model (MSSM) predicts two scalar\npartners for all Standard Model (SM) fermions as well as fermionic\npartners to all SM bosons. \nOf particular interest are the scalar partners of the heavy SM\nquarks, the scalar top quarks, $\\tilde{t}_i$ ($i = 1,2$) and scalar bottom\nquarks $\\tilde{b}_j$ ($j = 1,2$) due to their large Yukawa couplings. \nA scalar top quark $\\tilde{t}_i$ has many possible\ndecay modes, depending on the mass patterns of the SUSY particles. \nAmong  those decay modes are the decays to a scalar bottom quark, \n$\\tilde{b}_j$, and a charged Higgs boson, $H^+$, or\n  $W$~boson, $W^+$,  \n\\begin{align}\n\\label{stsbH}\n&\\tilde{t}_i \\to \\tilde{b}_j H^+ \\quad (i,j = 1,2)~, \\\\\n\\label{stsbW}\n&\\tilde{t}_i \\to \\tilde{b}_j W^+ \\quad (i,j = 1,2)~.\n\\end{align}\nIf these channels are kinematically allowed they can even be dominant if\n(most of) the other decay modes are kinematically forbidden.\nConsequently, these processes can constitute a large part of\nthe total stop decay width, and, in case of decays to a Higgs boson, they\ncan serve as a source of charged Higgs bosons in cascade decays at the LHC.\n                      \nFor a precise prediction of the partial decay widths corresponding to\n\\refeq{stsbH} and \\refeq{stsbW}, at least the one-loop level contributions\nhave to be taken into account.\nThis in turn requires a renormalization of the relevant sectors,\nespecially a simultaneous renormalization of the top and bottom\nquark\/squark sector. \nDue to the $SU(2)_L$ invariance of the left-handed scalar top and\nbottom quarks, these two sectors cannot be treated independently.\nWithin the framework of the MSSM with complex parameters (cMSSM) \nwe analyze various bottom quark\/squark sector renormalization\nschemes, while we \napply a commonly used on-shell renormalization scheme for the\ntop quark\/squark sector throughout all the investigations.\nSpecial attention is payed to ``perturbativity'', i.e.\\ the loop\ncorrections should not be enhanced by large counterterm contributions\nresulting from an inappropriate renormalization scheme.\nThis turns out to be a constraint that is very difficult to fulfill over\nthe whole cMSSM parameter range, where it is especially \ndifficult to achieve this simultaneously for small and large values of\n$\\tan \\beta$.\n\nHigher-order corrections to scalar fermion decays have been evaluated in\nvarious analysis over the last\ndecade. The simultaneous renormalization \nof the top and the bottom quark\/squark sector was taken\ninto account only in a relatively small subset.\nIn \\citeres{squark_q_V_als,stopsbot_phi_als} stop and sbottom decays,\nincluding the ones to charged Higgs and SM\ngauge bosons, have been \nevaluated at \\order{\\alpha_s} within the MSSM with real parameters\n(rMSSM). The numerical investigation \nwas restricted to relatively low $\\tan \\beta$ values. These\ncalculations are implemented in the program {\\tt SDECAY}~\\cite{sdecay}.\nA similar analysis in\n\\citere{sbot_stop_Hpm_altb} included electroweak one-loop corrections,\nwhere again only relatively low $\\tan \\beta$ values were considered. \nThe decays of Higgs bosons to scalar fermions, including the charged\nHiggs decays, at the full one-loop level within the rMSSM was presented in\n\\citeres{A_sferm_sferm_full,H_sferm_sferm_full}, indicating very\nlarge one-loop corrections \nfor large $\\tan \\beta$. An effective Lagrangian approach in the rMSSM \nfor these types of decays was given in \\citere{squark_eff}, with a\nnumerical analysis for $\\tan \\beta = 5$. \n\nThe renormalization of the top and bottom quark\/squark\nsector has been analyzed also in the context of other calculations \nin the past. A comparison of different renormalization schemes within \nthe rMSSM was performed in \\citeres{dissHR,mhiggsFDalbals}, focusing on\nlarge $\\tan \\beta$. One of the renormalization schemes considered therein had\nbeen used before within the calculation of the two-loop bottom\nquark\/squark contributions to the neutral Higgs boson masses~\\cite{sbotrenold}\nwhich are important for large $\\tan \\beta$~values. Within the cMSSM a\nrenormalization was presented in \\citere{dissTF}, however without an\nanalysis of its practicability. In \\citeres{dissHR,mhcMSSM2L} the top and\nbottom quark\/squark sector was renormalized within the cMSSM, but only\nthe QCD part needed for the presented calculation was considered. Thus\nno complete top and bottom quark\/squark sector renormalization has been\nperformed within the cMSSM. Recently a renormalization of nearly all\nsectors of the rMSSM appeared~\\cite{FawziRen}. In this analysis,\nhowever, the main focus has been on gauge parameter independence.\n\nComplex phases, as assumed here in the cMSSM, can be relevant for\ncollider observables and possibly extracted from experimental data. \nScalar top quark branching ratios at a linear collider are discussed in \n\\citere{Bartl:2003pd}. Concerning LHC measurements, triple products\ninvolving the decay of scalar \ntop or bottom quarks are analyzed in\n\\citeres{Bartl:2004jr,Ellis:2008hq,Deppisch:2009nj,Deppisch:2010nc}.\nFinally, rate asymmetries are examined in \\citere{Eberl:2009xe}.\nDepending on assumptions about the LHC performance it might be\npossible to extract information on the phases of $M_1$, $A_t$ and $A_b$\nat the LHC.\n\n\nIn this paper we analyze the renormalization of the full top\nand bottom quark\/squark sector in the cMSSM. We show analytically \n(and numerically) why\ncertain renormalization schemes \nfail for specific parts of the parameter space. Finally, we explore the\none-loop effects \nfor the decays (\\ref{stsbH},\\ref{stsbW}) for important parts of the\ncMSSM parameter space in the favored\nrenormalization scheme. We present numerical results showing the size\nof the one-loop corrections, especially including small and large $\\tan \\beta$.\nThe evaluation of the partial decay widths of the scalar top quarks are\nbeing implemented into the Fortran code \n{\\tt FeynHiggs}~\\cite{feynhiggs,mhiggslong,mhiggsAEC,mhcMSSMlong}. \nA numerical analysis of {\\em all} scalar top quark decay modes,\ninvolving a renormalization of {\\em all} relevant sectors will be \npresented elsewhere~\\cite{Stop2decay}.\n\n\n\n \n\n\n\n\n\n\\section{The generic structure of the quark\/squark sector}\n\\label{sec:generic}\n\nThe decay channels (\\ref{stsbH},\\ref{stsbW}) are calculated at the full\none-loop level (including hard QED and QCD radiation). This requires the\nrenormalization of several sectors of the cMSSM as discussed below. \nThe sectors not discussed in detail are renormalized as follows:\n\\begin{itemize}\n\\item \nThe gauge and Higgs sector renormalization has been performed following\n\\citere{mhcMSSMlong}. The gauge boson masses, $M_W$ and $M_Z$, as well\nas the mass of the \ncharged Higgs boson, $M_H^\\pm$, has been defined on-shell while the sine squared\nof the weak mixing\nangle, $s_{\\mathrm{w}}^2$, is defined via the gauge boson masses,\n    $s_{\\mathrm{w}}^2 = 1 - M_W^2\/M_Z^2$. The $Z$ factors for \nthe $W$~boson field are also determined within an on-shell scheme while\nthe $Z$ factors of the charged Higgs boson field are given by a linear\ncombination of the $\\overline{\\text{DR}}$ $Z$ factors of the Higgs\ndoublets (see \\citere{mhcMSSMlong}). An additional finite $Z$ factor is\nintroduced to fulfill on-shell conditions for the external charged\n$H^\\pm$~field. \n $\\tan \\beta$ is defined as $\\overline{\\text{DR}}$\nparameter. \n\\item\nThe Higgs mixing parameter $\\mu$ has been renormalized via an\non-shell (OS) procedure for the neutralino and chargino\nsector~\\cite{dissTF,diplTF}. \n\\item\nFor the renormalization of the electromagnetic charge we require that the\nrenormalized $ee\\gamma$-vertex in the Thomson limit is not changed by\nhigher order corrections with respect to the corresponding tree-level\nvertex~\\cite{denner}. \n\\end{itemize}\nA detailed description of our renormalization of all sectors will be\ngiven in \\citere{Stop2decay}.\n\n\\bigskip\nIn the following we focus on the top and bottom quark\/squark sector.\nThe bilinear part of the  Lagrangian with top and bottom squark fields,\n$\\tilde{t}$ and $\\tilde{b}$, \n\\begin{align}\n{\\cal L}_{\\tilde{t}\/\\tilde{b}\\text{ mass}} &= - \\begin{pmatrix}\n{{\\tilde{t}}_{L}}^{\\dagger}, {{\\tilde{t}}_{R}}^{\\dagger} \\end{pmatrix}\n\\matr{M}_{\\tilde{t}}\\begin{pmatrix}{\\tilde{t}}_{L}\\\\{\\tilde{t}}_{R}\n\\end{pmatrix} - \\begin{pmatrix} {{\\tilde{b}}_{L}}^{\\dagger},\n{{\\tilde{b}}_{R}}^{\\dagger} \\end{pmatrix}\n\\matr{M}_{\\tilde{b}}\\begin{pmatrix}{\\tilde{b}}_{L}\\\\{\\tilde{b}}_{R} \n\\end{pmatrix}~,\n\\end{align}\ncontains the stop and sbottom mass matrices\n$\\matr{M}_{\\tilde{t}}$ and $\\matr{M}_{\\tilde{b}}$,\ngiven by \n\\begin{align}\\label{Sfermionmassenmatrix}\n\\matr{M}_{\\tilde{q}} &= \\begin{pmatrix}  \n M_{\\tilde Q_L}^2 + m_q^2 + M_Z^2 c_{2 \\beta} (T_q^3 - Q_q s_\\mathrm{w}^2) & \n m_q X_q^* \\\\[.2em]\n m_q X_q &\n M_{\\tilde{q}_R}^2 + m_q^2 +M_Z^2 c_{2 \\beta} Q_q s_\\mathrm{w}^2\n\\end{pmatrix} \n\\end{align}\nwith\n\\begin{align}\\label{kappa}\nX_q &= A_q - \\mu^*\\kappa~, \\qquad \\kappa = \\{\\cot\\beta, \\tan\\beta\\} \n        \\quad {\\rm for} \\quad q = \\{t, b\\\n~.\n\\end{align}\n$M_{\\tilde Q_L}^2$ and $M_{\\tilde{q}_R}^2$ are the soft SUSY-breaking mass\nparameters. $m_q$ is the mass of the corresponding quark.\n$Q_{{q}}$ and $T_q^3$ denote the charge and the isospin of $q$, and\n$A_q$ is the trilinear soft SUSY-breaking parameter.\nThe mass matrix can be diagonalized with the help of a unitary\n transformation ${\\matr{U}}_{\\tilde{q}}$, \n\\begin{align}\\label{transformationkompl}\n\\matr{D}_{\\tilde{q}} &= \n\\matr{U}_{\\tilde{q}}\\, \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger = \n\\begin{pmatrix} m_{\\tilde{q}_1}^2 & 0 \\\\ 0 & m_{\\tilde{q}_2}^2 \\end{pmatrix}~, \\qquad\n{\\matr{U}}_{\\tilde{q}}= \n\\begin{pmatrix} U_{\\tilde{q}_{11}}  & U_{\\tilde{q}_{12}} \\\\  \n                U_{\\tilde{q}_{21}} & U_{\\tilde{q}_{22}}  \\end{pmatrix}~. \n\\end{align}\nThe scalar quark masses, $m_{\\tilde{q}_1}$ and $m_{\\tilde{q}_2}$, will always be mass\nordered, i.e.\\  \n$m_{\\tilde{q}_1} \\le m_{\\tilde{q}_2}$:\n\\begin{align}\nm_{\\tilde{q}_{1,2}}^2 &= \\frac{1}{2} \\left( M_{\\tilde{Q}_L}^2 + M_{\\tilde{q}_R}^2 \\right) \n       + m_q^2 + \\frac{1}{2} T_q^3 M_Z^2 c_{2\\beta} \\nonumber \\\\\n&\\quad \\mp \\frac{1}{2} \\sqrt{\\left[ M_{\\tilde{Q}_L}^2 - M_{{\\tilde{q}_R}}^2 \n       + M_Z^2 c_{2\\beta} (T_q^3 - 2 Q_q s_\\mathrm{w}^2) \\right]^2 + 4 m_q^2 |X_q|^2}~.\n\\label{MSbot}\n\\end{align}\n\n\\smallskip\nThe parameter renormalization can be performed as follows, \n\\begin{align}\n\\matr{M}_{\\tilde{q}} &\\to \\matr{M}_{\\tilde{q}} + \\delta\\matr{M}_{\\tilde{q}}\n\\end{align}\nwhich means that the parameters in the mass matrix $\\matr{M}_{\\tilde{q}}$ \nare replaced by the renormalized parameters and a counterterm. After the\nexpansion $\\delta\\matr{M}_{\\tilde{q}}$ contains the counterterm part,\n\\begin{align}\\label{proc1a}\n\\delta\\matr{M}_{\\tilde{q}_{11}} &= \\delta M_{\\tilde Q_L}^2 + 2 m_q \\delta m_q \n- M_Z^2 c_{2 \\beta}\\, Q_q \\, \\delta s_\\mathrm{w}^2 + (T_q^3 - Q_q s_\\mathrm{w}^2) \n  ( c_{2 \\beta}\\, \\delta M_Z^2 + M_Z^2\\, \\delta c_{2\\beta})~, \\\\\\label{proc1b}\n\\delta\\matr{M}_{\\tilde{q}_{12}} &= (A_q^*  - \\mu \\kappa)\\, \\delta m_q \n+ m_q (\\delta A_q^* - \\mu\\, \\delta \\kappa - \\kappa \\, \\delta \\mu)~, \\\\\\label{proc1c}\n\\delta\\matr{M}_{\\tilde{q}_{21}} &=\\delta\\matr{M}_{\\tilde{q}_{12}}^*~, \\\\\\label{proc1d}\n\\delta\\matr{M}_{\\tilde{q}_{22}} &= \\delta M_{\\tilde{q}_R}^2 \n+ 2 m_q \\delta m_q +  M_Z^2 c_{2 \\beta}\\, Q_q \\, \\delta s_\\mathrm{w}^2\n+ Q_q s_\\mathrm{w}^2 ( c_{2 \\beta}\\, \\delta M_Z^2+ M_Z^2\\, \\delta c_{2 \\beta})\n\\end{align}\nwith $\\kappa$ given in \\refeq{kappa}.\n\nAnother possibility for the parameter renormalization is to start out\nwith the physical parameters which corresponds to\nthe replacement:\n\\begin{align} \\label{proc2}\n\\matr{U}_{\\tilde{q}}\\, \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger &\\to\\matr{U}_{\\tilde{q}}\\, \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger + \\matr{U}_{\\tilde{q}}\\, \\delta \\matr{M}_{\\tilde{q}} \\, \n{\\matr{U}}_{\\tilde{q}}^\\dagger =\n\\begin{pmatrix} m_{\\tilde{q}_1}^2 & Y_q \\\\ Y_q^* & m_{\\tilde{q}_2}^2 \\end{pmatrix} +\n\\begin{pmatrix}\n\\delta m_{\\tilde{q}_1}^2 & \\delta Y_q \\\\ \\delta Y_q^* & \\delta m_{\\tilde{q}_2}^2\n\\end{pmatrix}~,\n\\end{align}\nwhere $\\delta m_{\\tilde{q}_1}^2$ and  $\\delta m_{\\tilde{q}_2}^2$ are the counterterms \n of the squark masses squared. $\\delta Y_q$ is the\n counter\\-term\\footnote{The unitary \n     matrix $\\matr{U}_{\\tilde{q}}$ can be expressed by a mixing angle\n     $\\theta_{\\tilde{q}}$ and\n     a corresponding phase $\\varphi_{\\tilde{q}}$. Then the\n     counterterm  $\\delta Y_q$ can be related to the counterterms of the\n     mixing angle and the phase (see \\citere{mhcMSSM2L}).}   to the squark\n mixing parameter $Y_q$ (which vanishes\n   at tree level, $Y_q = 0$, and corresponds to the \n off-diagonal entries in $\\matr{D}_{\\tilde{q}} =\\matr{U}_{\\tilde{q}}\\,\n \\matr{M}_{\\tilde{q}} \\,  \n{\\matr{U}}_{\\tilde{q}}^\\dagger$, see~\\refeq{transformationkompl}). Using\n\\refeq{proc2} \n one can express $\\delta\\matr{M}_{\\tilde{q}}$ by the counterterms $\\delta m_{\\tilde{q}_1}^2$,\n $\\delta m_{\\tilde{q}_2}^2$ and $\\delta Y_q$. Especially for $\\delta\\matr{M}_{\\tilde{q}_{12}}$\n one yields\n\\begin{align}\\label{dMsq12physpar}\n\\delta\\matr{M}_{{\\tilde{q}}_{12}} &=\nU^*_{\\tilde{q}_{11}} U_{\\tilde{q}_{12}}\n(\\delta m_{\\tilde{q}_1}^2 - \\delta m_{\\tilde{q}_2}^2) +\nU^*_{\\tilde{q}_{11}} U_{\\tilde{q}_{22}} \\delta Y_q + U_{\\tilde{q}_{12}}\nU^*_{\\tilde{q}_{21}} \\delta Y_q^*~.\n\\end{align}\nIn the following the relation given by \\refeq{proc1b} and\n\\refeq{dMsq12physpar} will be used to express either $\\delta Y_q$, $\\delta\nA_q$ or $\\delta m_q$ by the other counterterms.\n\nFor the field renormalization the following procedure is applied,\n\\begin{align}\n\\begin{pmatrix} \\tilde{q}_1 \\\\ \\tilde{q}_2 \\end{pmatrix} &\\to \n  \\left( \\id + \\frac{1}{2} \\delta\\matr{Z}_{\\tilde{q}} \\right) \n  \\begin{pmatrix} \\tilde{q}_1 \\\\ \\tilde{q}_2 \\end{pmatrix} \n  \\quad {\\rm with} \\quad\n\\delta\\matr{Z}_{\\tilde{q}} = \\begin{pmatrix} \n                   \\delta Z_{\\tilde{q}_{11}} & \\delta Z_{\\tilde{q}_{12}} \\\\\n                   \\delta Z_{\\tilde{q}_{21}} & \\delta Z_{\\tilde{q}_{22}} \n                   \\end{pmatrix}~.\n\\end{align}\n\nThis yields for the renormalized self-energies\n\\begin{align}\n\\hat{\\Sigma}_{\\tilde{q}_{11}}(k^2) &= \\Sigma_{\\tilde{q}_{11}}(k^2) \n  + \\frac{1}{2} (k^2 - m_{\\tilde{q}_1}^2) (\\dZ{\\tilde{q}_{11}} + \\dZ{\\tilde{q}_{11}}^*)\n  - \\dem_{\\tilde{q}_1}^2~, \\\\\n\\hat{\\Sigma}_{\\tilde{q}_{12}}(k^2) &= \\Sigma_{\\tilde{q}_{12}}(k^2)\n  + \\frac{1}{2} (k^2 - m_{\\tilde{q}_1}^2) \\dZ{\\tilde{q}_{12}}\n  + \\frac{1}{2} (k^2 - m_{\\tilde{q}_2}^2) \\dZ{\\tilde{q}_{21}}^* \n  - \\delta Y_q~, \\\\\n\\hat{\\Sigma}_{\\tilde{q}_{21}}(k^2) &= \\Sigma_{\\tilde{q}_{21}}(k^2)\n  + \\frac{1}{2} (k^2 - m_{\\tilde{q}_1}^2) \\dZ{\\tilde{q}_{12}}^*\n  + \\frac{1}{2} (k^2 - m_{\\tilde{q}_2}^2) \\dZ{\\tilde{q}_{21}} \n  - \\delta Y_q^*~, \\\\\n\\hat{\\Sigma}_{\\tilde{q}_{22}}(k^2) &= \\Sigma_{\\tilde{q}_{22}}(k^2) \n  + \\frac{1}{2} (k^2 - m_{\\tilde{q}_2}^2) (\\dZ{\\tilde{q}_{22}} + \\dZ{\\tilde{q}_{22}}^*)\n  - \\dem_{\\tilde{q}_2}^2~.\n\\end{align}\nIn order to complete the quark\/squark sector renormalization also for the\ncorresponding quark (i.e. its mass, $m_q$, and the quark\nfield, $q$) renormalization constants have to be introduced:\n\\begin{align}\nm_q &\\to m_q + \\delta m_q~,\\\\\n\\omega_{\\mp} \\,q  &\\to (1 + \\frac{1}{2}\\dZ{q}^{L\/R})\\, \\omega_{\\mp} \\,q~\n\\end{align}\nwith $\\delta m_q$ being the quark mass counterterm and $\\dZ{q}^L$ and\n$\\dZ{q}^R$ being the $Z$~factors of the left-handed and the right-handed\ncomponent of the quark field $q$, respectively. $\\omega_{\\mp} =\n\\frac{1}{2}(\\id \\mp \\gamma_5)$\nare the left- and right-handed projectors, respectively.\nThen the renormalized self energy, $\\Hat{\\Sigma}_{q}$, can be decomposed\ninto left\/right-handed and scalar left\/right-handed parts, \n${\\hat{\\Sigma}}_q^{L\/R}$ and ${\\hat{\\Sigma}}_q^{SL\/SR}$, respectively,\n\\begin{align}\\label{decomposition}\n \\Hat{\\Sigma}_{q} (k) &= \\not\\! k\\, \\omega_{-} \\Hat{\\Sigma}_q^L (k^2)\n                   + \\not\\! k\\, \\omega_{+} \\Hat{\\Sigma}_q^R (k^2)\n                   + \\omega_{-} \\Hat{\\Sigma}_q^{SL} (k^2) \n                   + \\omega_{+} \\Hat{\\Sigma}_q^{SR} (k^2)~,\n\\end{align}\nwhere the components are given by\n\\begin{align}\n\\Hat{\\Sigma}_q^{L\/R} (k^2) &= {\\Sigma}_q^{L\/R} (k^2) \n   + \\frac{1}{2} (\\dZ{q}^{L\/R} + {\\dZ{q}^{L\/R}}^*)~, \\\\\n\\Hat{\\Sigma}_q^{SL} (k^2) &=  {\\Sigma}_q^{SL} (k^2) \n   - \\frac{m_q}{2} (\\dZ{q}^L + {\\dZ{q}^R}^*) - \\delta m_q~,  \\\\\n\\Hat{\\Sigma}_q^{SR} (k^2) &=  {\\Sigma}_q^{SR} (k^2) \n   - \\frac{m_q}{2} (\\dZ{q}^R + {\\dZ{q}^L}^*) - \\delta m_q~.\n\\end{align}\nNote that $\\Hat{\\Sigma}_q^{SR} (k^2) = {\\Hat{\\Sigma}_q^{SL} (k^2)}^*$ \nholds due to ${\\cal CPT}$ invariance.\n\n\n\n\\section{Field renormalization of the quark\/squark sector}\n\nWe first discuss the field renormalization of the top and bottom\nquark\/squark sector and turn to the parameter renormalization in the next\nsection \\ref{sec:stop}.\n\nThe field renormalization, meaning the determination of the $Z$~factors,\nis done within an on-shell scheme for squarks and quarks. We impose\n  equivalent renormalization conditions for the top as well as for\n  the bottom quark\/squark sector: \n\\begin{itemize}\n\n\\item[(a)]\nThe diagonal $Z$~factors of the squark fields are determined such that\nthe real part of the residua of propagators is set to unity, \n\\begin{align}\n\\label{residuumStopOS}\n\\widetilde\\re \\frac{\\partial\\, \\hat{\\Sigma}_{\\tilde{q}_{ii}}(k^2)}{\\partial\\, k^2}\n      \\Big|_{k^2 = m_{\\tilde{q}_i}^2} &= 0 \\qquad (i = 1,2)~.\n\\end{align}\nThis condition fixes the real parts of the diagonal $Z$~factors to\n\\begin{align}\n\\mathop{\\mathrm{Re}}\\dZ{\\tilde{q}_{ii}} = - \\widetilde\\re \\frac{\\partial\\, \\Sigma_{\\tilde{q}_{ii}}(k^2)}{\\partial\\, k^2}\n                  \\Big|_{k^2 = m_{\\tilde{q}_i}^2} \\qquad (i = 1,2)~.\n\\end{align}\n$\\widetilde\\re$ above denotes the real part with respect to\ncontributions from the loop integral, but leaves the complex\ncouplings unaffected.\n\nThe imaginary parts of the diagonal $Z$~factors are so far undetermined\nand are set to zero, \n\\begin{align}\n\\mathop{\\mathrm{Im}} \\dZ{\\tilde{q}_{ii}} &= 0 \\qquad (i = 1,2)~.\n\\end{align}\nThis is possible since they do not contain divergences.\n\n\\item[(b)]\nFor the non-diagonal $Z$~factors of the squark fields we impose the\ncondition that for \non-shell squarks no transition from one squark to the other occurs, \n\\begin{align}\n\\widetilde\\re\\hat{\\Sigma}_{\\tilde{q}_{12}}(m_{\\tilde{q}_1}^2) &= 0~, \\\\\n\\widetilde\\re\\hat{\\Sigma}_{\\tilde{q}_{12}}(m_{\\tilde{q}_2}^2) &= 0~.\n\\end{align}\nThis yields\n\\begin{align}\n\\dZ{\\tilde{q}_{12}} &= + 2 \\frac{\\widetilde\\re\\Sigma_{\\tilde{q}_{12}}(m_{\\tilde{q}_2}^2) - \\delta Y_q}\n     {(m_{\\tilde{q}_1}^2 - m_{\\tilde{q}_2}^2)}~, \\nonumber \\\\\n\\dZ{\\tilde{q}_{21}} &= - 2 \\frac{\\widetilde\\re\\Sigma_{\\tilde{q}_{21}}(m_{\\tilde{q}_1}^2) - \\delta Y_q^*}\n     {(m_{\\tilde{q}_1}^2 - m_{\\tilde{q}_2}^2)}~.\n\\label{dZstopoffdiagOS}\n\\end{align}\nThe counterterm $\\delta Y_q$ is determined in the corresponding parameter\nrenormalization scheme. This means the non-diagonal $Z$~factors of the\nsquark fields do also depend on the choice of the parameter\nrenormalization scheme.\n\n\\item[(c)] \nThe quark fields are also defined via an on-shell condition. We impose\n\\begin{align}\\label{ZquarkOS}\n\\lim_{k^2\\rightarrow m_q^2}\\frac{ \\not\\! k + m_q}{k^2 - m_q^2} \\widetilde\\re\n\\Hat{\\Sigma}_{q} (k) u(k) &= 0~,\\quad\\ \\lim_{k^2\\rightarrow m_q^2} \\bar{u}(k)\\widetilde\\re\n\\Hat{\\Sigma}_{q} (k)\\frac{ \\not\\! k + m_q}{k^2 - m_q^2}  = 0~,\n\\end{align}\nwhere $u(k)$, $\\bar{u}(k)$ are the spinors of the external fields. \nThis yields\n\\begin{align}\n\\mathop{\\mathrm{Re}} \\dZ{q}^{L\/R} &= - \\widetilde\\re \\Big\\{ {\\Sigma}_q^{L\/R} (m_q^2)  \\\\ \n&\\quad  + m_q^2 \\left[ {{\\Sigma}_q^{L}}'(m_q^2) + {{\\Sigma}_q^{R}}'(m_q^2) \\right]\n              + m_q \\left[ {{\\Sigma}_q^{SL}}'(m_q^2) \n              + {{\\Sigma}_q^{SR}}'(m_q^2) \\right] \\Big\\}~,  \\nonumber \\\\\nm_{q} \\left( \\mathop{\\mathrm{Im}} \\dZ{q}^L - \\mathop{\\mathrm{Im}}\\dZ{q}^R \\right) &= \n    i\\, \\widetilde\\re\\left\\{ {\\Sigma}_q^{SR}(m_q^2) - {\\Sigma}_q^{SL}(m_q^2) \\right\\}\n    = 2 \\mathop{\\mathrm{Im}} \\left\\{ \\widetilde\\re {\\Sigma}_q^{SL}(m_q^2) \\right\\}~,\n\\end{align}\nwith $\\Sigma'(k^2) \\equiv \\frac{\\partial \\Sigma(k^2)}{\\partial k^2}$. \nChoosing also $\\mathop{\\mathrm{Im}} \\dZ{q}^L = - \\mathop{\\mathrm{Im}}\\dZ{q}^R$, the imaginary parts of the\n$Z$~factors can be expressed as\n\\begin{align}\n\\mathop{\\mathrm{Im}} \\dZ{q}^{L\/R} &= \\pm \\frac{i}{2\\, m_q} \n        \\widetilde\\re \\left\\{ {\\Sigma}_q^{SR}(m_q^2) - {\\Sigma}_q^{SL}(m_q^2) \\right\\}\n      = \\pm \\frac{1}{m_q} \\mathop{\\mathrm{Im}} \\left\\{ \\widetilde\\re {\\Sigma}_q^{SL}(m_q^2) \\right\\}~.\n\\end{align}\nNote that the renormalization condition \\refeq{ZquarkOS} can only be \nfully satisfied if the corresponding quark mass is defined as on-shell, \ntoo.\n\nThe $Z$~factors of the quark fields are not needed for the calculation\nof the considered decay modes of the scalar top quarks (see, however,\n\\citere{Stop2decay}). \n\n\\end{itemize}\n\n\n\\section{Parameter renormalization of the top and bottom \nquark\/squark sector}\n\\label{sec:stop}\n\n\nWithin the top and bottom quark\/squark sector nine real parameters are\ndefined: The real\nsoft SUSY-breaking parameters $M_{\\tilde Q_L}^2$, $M_{{\\tilde{t}}_R}^2$\nand $M_{{\\tilde{b}}_R}^2$, the complex trilinear couplings $A_t$ and\n$A_b$ and the top and bottom Yukawa couplings $y_t$ and $y_b$ which both\ncan be chosen to be real. ($\\mu$ and $\\tan \\beta$ as well as the\ngauge boson masses and the weak mixing angle are determined within other\nsectors, see the beginning of \\refse{sec:generic}). Note that the\nsoft SUSY-breaking parameter $M_{\\tilde Q_L}^2$ is the same in the top\nas well as in the bottom squark sector due to the \n$SU(2)_L$ invariance of the \nleft-handed fields.\nAs in \\citeres{dissHR,mhcMSSM2L}, instead of choosing the five\nquantities $M_{\\tilde Q_L}^2$, $M_{{\\tilde{t}}_R}^2$,\n$M_{{\\tilde{b}}_R}^2$ and $y_t$, $y_b$ the\nsquark masses $m_{\\tilde{t}_1}^2$, $m_{\\tilde{t}_2}^2$, $m_{\\tilde{b}_2}^2$ as well as the top and\nbottom quark masses $m_t$, $m_{b}$ were taken as independent parameters. \n\nIf a regularization scheme is applied which does not break the symmetries\nof the model, it is sufficient to use counterterms which respects the\nunderlying symmetries. Such counterterms are\n generated by multiplicative\nrenormalization of\nparameters and fields of the MSSM. The parameter counterterms can be fixed by\nas many renormalization conditions as independent parameters exist\n\\cite{MSSMrenormierung}.\nConcerning the top and bottom quark\/squark sector we have to set\nnine renormalization conditions to define all indepedent parameters.\n\n For the renormalization of the top quark\/squark sector we follow\n\\citeres{dissHR,mhcMSSM2L} but we also include electroweak contributions.%\n\nWe impose five renormalization conditions, (A)--(E), to fix the\nparameters of the top quark\/squark sector:\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(A)] The top-quark mass is determined via an on-shell condition,\n  yielding the one-loop counterterm $\\delta m_t$:\n\\begin{align}\\label{dmt}\n\\delta m_t &= \\frac{1}{2} \\widetilde\\re \\left\\{ \n    m_t \\left[\\Sigma_t^L (m_t^2) + \\Sigma_t^R (m_t^2) \\right]  \n  + \\left[ \\Sigma_t^{SL} (m_t^2) + \\Sigma_t^{SR} (m_t^2) \\right] \\right\\}~.\n\\end{align}\n\\item[(B), (C)]\nThe two top squark masses are also defined on-shell, yielding the real\ncounterterms \n\\begin{align}\n\\label{dmst}\n\\dem_{\\tilde{t}_i}^2 &= \\widetilde\\re\\Sigma_{\\tilde{t}_{ii}}(m_{\\tilde{t}_i}^2) \\qquad (i = 1,2)~.\n\\end{align}\n\n\\item[(D), (E)]\nFinally, the non-diagonal entry in the matrix of \\refeq{proc2} is fixed\nas  \n\\begin{align}\n\\delta Y_t &= \\frac{1}{2} \\widetilde\\re \n    \\left\\{ \\Sigma_{\\tilde{t}_{12}}(m_{\\tilde{t}_1}^2) + \\Sigma_{\\tilde{t}_{12}}(m_{\\tilde{t}_2}^2) \\right\\}~,\n\\end{align}\nwhich corresponds to two seperate conditions as $\\delta Y_t$ is complex.\n\\end{itemize}\n\\end{itemize}\n\n\nThe counterterm of the trilinear coupling $\\delta A_t$ is then given via the\nrelation of \\refeqs{proc1b} and \\eqref{dMsq12physpar} as:\n\\begin{align}\n\\delta A_t &= \\frac{1}{m_t}\\bigl[U_{\\tilde{t}_{11}} U_{\\tilde{t}_{12}}^*\n           (\\delta m_{\\tilde{t}_1}^2 - \\delta m_{\\tilde{t}_2}^2) \n        +  U_{\\tilde{t}_{11}} U_{\\tilde{t}_{22}}^* \\delta Y_t^*\n        + U_{\\tilde{t}_{12}}^* U_{\\tilde{t}_{21}} \\delta Y_t  \n        - (A_t - \\mu^* \\cot\\beta)\\, \\dem_t  \\bigr] \\nonumber \\\\\n&\\quad +  (\\delta\\mu^* \\cot\\beta - \\mu^* \\cot^2\\beta\\, \\delta\\!\\tan\\!\\beta\\,)~.\n\\end{align}\nThe definition of $\\delta\\!\\tan\\!\\beta\\,$ and $\\delta\\mu$ is indicated in\n\\refse{sec:generic}. \n\n\\bigskip\nFor the bottom quark\/squark sector we are left with four independent\nparameters which are not defined yet. We choose the following\nfour renormalization conditions, (i)--(iv):\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(i)] The $\\tilde{b}_2$~mass is defined on-shell:\n\\begin{align}\\label{sbotzOS}\n\\dem_{\\tilde{b}_2}^2 &= \\widetilde\\re\\Sigma_{\\tilde{b}_{22}}(m_{\\tilde{b}_2}^2)~.\n\\end{align}\n\\item[(ii)--(iv)] These three renormalization conditions are chosen\n  according to the different renormalization conditions listed in\n  \\refta{tab:RS} and to the corresponding subsections \n  \\ref{sec:OS}--\\ref{sec:AbOS_ReYbOS}. \n  They yield the counterterms $\\delta m_{b}$, $\\delta A_b$\n  and $\\delta Y_b$ where only three of these five real counterterms are\n  independent (counting each of the complex counterterms, $\\delta A_b$\n  and $\\delta Y_b$, as two real counterterms). The two dependent\n  counterterms can be expressed as a combination of the other ones.\n\\end{itemize}\n\\end{itemize}\nApplying these renormalization conditions fixes the counterterms\ngenerated by multiplicative renormalization which fulfill the\nsymmetry relations \\cite{MSSMrenormierung}.\n\n\\begin{table}[ht!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c||c|c|}\n\\hline\nscheme & $m_{\\tilde b_{1,2}}$ & $m_{b}$ & $A_b$ & $Y_b$ & Sect. & name \n                                                           \\\\ \\hline\\hline\n{\\small analogous to the $t\/\\tilde{t}$ sector:} \n ``OS'' & OS & OS & & OS  \n& \\ref{sec:OS} & RS1 \\\\ \\hline\n``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & OS & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & \n& \\ref{sec:mbDRbar_AbDRbar} & RS2 \\\\ \\hline\n``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & OS & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & & \\ensuremath{\\overline{\\mathrm{DR}}}\\ \n& \\ref{sec:mbDRbar_YbDRbar} & RS3 \\\\ \\hline\n``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'' & OS & \\ensuremath{\\overline{\\mathrm{DR}}}\\ & & OS\n& \\ref{sec:mbDRbar_YbOS} & RS4 \\\\ \\hline\n``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & OS & & \\ensuremath{\\overline{\\mathrm{DR}}} & $\\mathop{\\mathrm{Re}} Y_b$:\\, OS\n& \\ref{sec:AbDRbar_ReYbOS} & RS5 \\\\ \\hline\n``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & OS & & vertex & $\\mathop{\\mathrm{Re}} Y_b$:\\, OS\n& \\ref{sec:AbOS_ReYbOS} & RS6 \\\\ \\hline\n\\end{tabular}\n\\caption{Summary of the six renormalization schemes for the\n  $b\/\\tilde{b}$~sector investigated below. Blank entries indicate dependent\n  quantities. $\\mathop{\\mathrm{Re}} Y_b$ denotes that only the real part of \n  $Y_b$ is renormalized on-shell, while the imaginary part is a\n  dependent parameter. The rightmost columns indicates the section that\n  contains the detailed description of the respective renormalization\n  and the abbreviated notation used in our analysis.}\n\\label{tab:RS}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{center}\n\\end{table}\n\n\n\nWhile the $\\tilde{b}_2$~mass is defined on-shell, the\n$\\tilde{b}_1$~mass receives a shift due to the radiative corrections: \n\\begin{align}\nm_{\\tilde{b}_{1,{\\rm OS}}}^2 &= m_{\\tilde{b}_1}^2 + \\left(\n \\dem_{\\tilde{b}_1}^2 - \\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)\\right)~.\n\\end{align}\nThe term in parentheses is the shift from $m_{\\tilde{b}_1}^2$ to the on-shell mass\nsquared. The value of \n$m_{\\tilde{b}_1}^2$ is derived from the diagonalization of the \nsbottom mass matrix, see \\refeq{transformationkompl}, and $\\dem_{\\tilde{b}_1}^2$\nis defined as a dependent\nquantity~\\cite{hr,mhiggsFDalbals}. $m_{\\tilde{b}_{1,{\\rm OS}}}^2$ is the on-shell\n$\\tilde{b}_1$~mass squared. In ~\\citere{hr} the size of the shift was\nanalyzed while in~\\citere{mhiggsFDalbals} bottom squarks appeared only as\n``internal'' particles, i.e.\\ as particles inside the loop\ndiagrams. Concerning the scalar top quark decay, \\refeqs{stsbH} and\n\\eqref{stsbW}, we are now dealing with \nscalar bottom quarks as ``external'' particles, which are defined as\nincoming or outgoing particles. These ``external'' particles should\nfulfill on-shell properties. At this point there are two options to \nproceed:\n\\begin{itemize}\n\\item[(${\\cal O} 1$)] \n  The first option is to use different mass values, $m_{\\tilde{b}_1}$ and\n  $m_{\\tilde{b}_{1,{\\rm OS}}}$, for the ``internal'' and the ``external'' particles,\n  respectively, which can cause problems for charged particles as,\n  for instance, scalar bottom quarks (see below).\n\\item[(${\\cal O} 2$)] \n  The second option is to impose a further renormalization\n  condition which ensures that the $\\tilde{b}_1$~mass is on-shell:\n\\begin{align}\\label{sboteOS}\n \\dem_{\\tilde{b}_1}^2 &=\\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)~.\n\\end{align}\nIn this case the input has to be chosen such that the symmetry\nrelations are fulfilled at the one-loop level.\n\\end{itemize}\n\nAs mentioned above, the option~(${\\cal O} 1$) leads to a problem. \nThe IR-divergences originating from the loop diagrams involve the\n``inner'' (i.e.\\ tree-level) \nmass $m_{\\tilde{b}_1}$. These have to cancel with the real\nBremsstrahlung IR-divergences, which are evaluated with the help of the\n``external'' (i.e.\\ one-loop on-shell) mass $m_{\\tilde{b}_{1,{\\rm OS}}}$, which is inserted into \nthe tree-level diagram (the result can, as usual, be expressed with the\nhelp of the Soft Bremsstrahlung (SB) factor $\\delta_{\\rm SB}$:\n${\\cal M}_{\\rm tree} \\times \\delta_{\\rm SB}$, see \\citere{denner}). \nDue to the two different sets of masses the IR-divergences do not\ncancel. \nOne way out would be the use of tree-level masses in all diagrams\ncontributing to the part $2 \\mathop{\\mathrm{Re}} \\{{\\cal M}_{\\rm tree} {\\cal M}_{\\rm loop}\\}$,\ni.e.\\ in all loop diagrams and in the hard and soft Bremsstrahlung diagrams. \nHowever, this would lead to inconsistencies in the evaluation of the complete\nloop corrected amplitude squared\n$\\propto (|{\\cal M}_{\\rm tree}|^2 + 2 \\mathop{\\mathrm{Re}} \\{{\\cal M}_{\\rm tree} {\\cal M}_{\\rm loop}\\} )$ \ndue to the different masses entering the phase space evaluation.\nA consistent phase space integration requires the use of the same\n``external'' masses for all outgoing particles in all parts of \nthe calculation.\n\n\nTo circumvent the problem of the non-cancellation of IR-divergences we\nchoose the option~(${\\cal O} 2$) and impose the further renormalization \ncondition \\refeq{sboteOS}. This requires to choose an input that restores \nthe symmetries. Relating $(\\matr{M}_{\\tilde q})_{11}$ of \n\\refeq{Sfermionmassenmatrix} and \n$(\\matr{U_{\\tilde{q}}}^{\\dagger} \\matr{D}_{\\tilde{q}} \\matr{U_{\\tilde{q}}})_{11}$\nwith $\\matr{D}_{\\tilde{q}}$ of \\refeq{transformationkompl} yields an\nexpression for the soft SUSY-breaking parameter $M_{\\tilde{Q}_L}^2$\n(depending on the squark flavor),\n\\begin{align}\nM_{\\tilde{Q}_L}^2(\\tilde{q}) = |U_{\\tilde{q}_{11}}|^2 m_{\\tilde{q}_1}^2\n   + |U_{\\tilde{q}_{12}}|^2 m_{\\tilde{q}_2}^2 \n   - M_Z^2 c_{2\\beta} (T_q^3 - Q_q s_\\mathrm{w}^2) - m_{q}^2\n\\end{align}\nwith $\\tilde{q} = \\{\\tilde{t}, \\tilde{b}\\}$. Requiring the $SU(2)_L$ relation\nto be valid at the one-loop level induces the following shift in \n$M^2_{\\tilde{Q}_L}$ (see also\n\\citeres{squark_q_V_als,stopsbot_phi_als,dr2lA}): \n\\begin{align}\nM_{\\tilde{Q}_L}^2(\\tilde{b}) = M_{\\tilde{Q}_L}^2(\\tilde{t}) \n   + \\delta M_{\\tilde{Q}_L}^2(\\tilde{t}) - \\delta M_{\\tilde{Q}_L}^2(\\tilde{b})\n\\label{MSbotshift}\n\\end{align}\nwith\n\\begin{align}\n\\delta M_{\\tilde{Q}_L}^2(\\tilde{q}) &= |U_{\\tilde{q}_{11}}|^2 \\dem_{\\tilde{q}_1}^2\n   + |U_{\\tilde{q}_{12}}|^2 \\dem_{\\tilde{q}_2}^2\n   - U_{\\tilde{q}_{22}} U_{\\tilde{q}_{12}}^* \\delta Y_q\n   - U_{\\tilde{q}_{12}} U_{\\tilde{q}_{22}}^* \\delta Y_q^* - 2 m_{q} \\dem_{q} \\nonumber \\\\\n&\\quad  + M_Z^2\\, c_{2\\beta}\\, Q_q\\, \\delta s_\\mathrm{w}^2 \n        - (T_q^3 - Q_q s_\\mathrm{w}^2) (c_{2\\beta}\\, \\delta M_Z^2 + M_Z^2\\, \\delta c_{2\\beta})~.\n\\label{MSbotshift-detail}\n\\end{align}\nIn other words, everywhere in the calculation the masses and mixing\nmatrix elements coming \nfrom the diagonalization of the bottom squark mass matrix, see\n\\refeq{transformationkompl}, are used with $M_{\\tilde{Q}_L}^2(\\tilde{b})$\nincluding the above shift as in \\refeq{MSbotshift}. \nIn this way the problems concerning UV- and IR-finiteness are\navoided. (An exception is the field renormalization of the $W$-boson\nfield: In the corresponding selfenergies the $SU(2)_L$ relation is needed\nat tree-level to ensure UV-finiteness. In this case, tree-level bottom\nsquark masses are used.)\n\nThe various renormalization schemes, following the general\nchoice~(${\\cal O} 2$), are summarized in \\refta{tab:RS} and\noutlined in detail in the following subsections.\n\n\nComparing with the literature, several of the renormalization\nschemes (or variants of them) have been used to calculate higher-order\ncorrections to squark or Higgs decays.\nThe older calculations of the loop corrections have all been\nperformed in the rMSSM. \n\n\\begin{itemize}\n\n\\item\nA renormalization scheme employing an ``OS'' renormalization for\n$m_{b}$ and $Y_b$ was used in \n\\citeres{stopsbot_phi_als,sbot_stop_Hpm_altb} for the calculation of\nstop and sbottom decays. (The calculation of\n\\citere{stopsbot_phi_als} is also implemented in\n\\citere{sdecay}.) In order to check our\nimplementation given in \\refse{sec:OS} we calculated the decay \n$\\tilde{b}_{1,2} \\to \\tilde{t}_1 H^{-}$ (see \\refse{sec:calc} for our set-up) and\nfound good agreement with \\citere{stopsbot_phi_als}.\n\n\\item\nA renormalization scheme similar to the real version of RS2,\ni.e.\\ ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''\nhas been employed in \\citere{H_sferm_sferm_full} for the\ncalculation of Higgs decays to scalar fermions. In the scalar top and\nthe Higgs sector they apply an on-shell scheme \n(partially \nbased on \\citeres{sbot_top_cha_alt,sfermprod_alf}), \nwhich differs in some points from our renormalization scheme.\n\n\\item\nAn on-shell scheme was also used in \\citere{sferm_f_V_full} (based on\n\\citeres{sbot_top_cha_alt,squark_q_chi_full}) to evaluate the decay\n$\\tilde{f} \\to \\tilde{f}' V$ ($V = W^\\pm, Z$). \n\n\\item\nIn \\citere{stop_stop_H_alt}, as a starting point, an on-shell renormalization\nscheme was used for the calculation of the electroweak corrections to\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{t}_1 \\phi)$, ($\\phi = h, H, A$).\nTo improve the calculation, the parameters $m_{b}$, $m_t$, $A_t$ and $A_b$ have\nalso been used as running parameters.\n\n\n\n\\item\nOther ``early'' papers considered QCD corrections to various scalar\nquark decays~\\cite{squark_q_chi_als,squark_q_gl_als,stop_top_gl_als}. \nThey mostly employed an on-shell \nscheme for the quark\/squark masses and the squark mixing angle \n$\\theta\\kern-.15em_{\\tilde{q}}$, where the counterterm to the mixing angle is\n$\\delta \\theta\\kern-.15em_{\\tilde{q}} \\propto \\delta Y_q$.\n\n\\item\nThe renormalization scheme ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' is the\ncomplex version of the renormalization used in\n\\citeres{sbotrenold,mhiggsFDalbals} for the \n\\order{\\alpha_b\\alpha_s} corrections to the neutral Higgs boson self-energies\nand thus to the lightest MSSM Higgs boson mass, $M_h$.\n\n\\end{itemize}\n\n\n\\bigskip\nIn the following subsections we define in detail the various\nrenormalization schemes. As explained before and indicated in\n\\refta{tab:RS} the two bottom squark\nmasses are renormalized on-shell in all the schemes, as in \n\\refeqs{sbotzOS} and \\eqref{sboteOS}, and taking into account\nthe shift of $M_{\\tilde Q_L}^2(\\tilde{b})$ in \\refeq{MSbotshift}. \nWithin the subsections only the remaining conditions and\nrenormalization constants are defined explicitly\n(where $\\delta \\mu$ and $\\delta\\!\\tan\\!\\beta\\,$ are\ndefined within the chargino\/neutralino sector and the Higgs sector,\nrespectively, in all the different renormalization schemes and are not\ndiscussed any further).\n\n\n\n\n\\subsection{On-shell (RS1)}\n\\label{sec:OS}\n\nThis renormalization scheme is analogous to the OS scheme employed for\nthe top quark\/squark sector.\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined OS, yielding the one-loop \ncounterterm $\\delta m_{b}$:\n\\begin{align} \n\\label{dmb_OS}\n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]\n    + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SL} (m_{b}^2) \\right] \\right\\}~.\n\\end{align}\n\n\\item[(iii), (iv)]\nWe choose an OS renormalization condition for the non-diagonal\nentry in the matrix of \\refeq{proc2}, analogous to the one\napplied in the top quark\/squark sector, setting\n\\begin{align}\n\\delta Y_b =  \\frac{1}{2} \\widetilde\\re \\left\\{\n    \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) + \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\label{dYb_OS}\n\\end{align}\n\\end{itemize}\n\\end{itemize}\nThe conditions (i)--(iv) fix all independent parameters and their\nrespective counterterms. \nAnalogous to the calculation of the counterterm of the trilinear\ncoupling $A_t$, relating \\refeq{proc1b} and \\refeq{dMsq12physpar} yields\nthe following condition for $\\delta A_b$, \n\\begin{align}\\label{Ab_OS}\n\\deA_b &= \\frac{1}{m_{b}} \\bigl[U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n          (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \n        + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n        + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \n        - (A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} \\bigr] \\nonumber \\\\\n&\\quad  + (\\delta\\mu^* \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,)\n\\end{align} \nwith $\\dem_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$ given  in \\refeqs{sboteOS} and \n\\eqref{sbotzOS}, respectively. \n\n\n\n\n\\subsection{\\boldmath{$m_{b}$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$A_b$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } (RS2)}\n\\label{sec:mbDRbar_AbDRbar}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined \\ensuremath{\\overline{\\mathrm{DR}}}, yielding the one-loop\n  counterterm $\\delta m_{b}$:\n\\begin{align} \n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~.\n\\end{align}\nThe $|_{\\rm div}$ terms are the ones proportional to\n$\\Delta = 2\/\\varepsilon - \\gamma_{\\rm E} + \\log(4 \\pi)$, when using dimensional\nregularization\/reduction in $D = 4 - \\varepsilon$ dimensions; $\\gamma_{\\rm E}$ is\nthe Euler constant.\n\n\\item[(iii), (iv)]\nThe complex parameter $A_b$ is renormalized \\ensuremath{\\overline{\\mathrm{DR}}},\n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\Bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n         \\left(  \\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)|_{\\rm div} \n             -\\widetilde\\re\\Sigma_{\\tilde{b}_{22}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2}\\, U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \n          \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n             +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2}\\, U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \n          \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n             +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right)^* \\nonumber \\\\\n&\\quad - \\frac{1}{2}(A_b - \\mu^* \\tan \\beta)\\, \n\\widetilde\\re \\bigl\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div} \\nonumber \\\\\n&\\qquad + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm\n    div} \\bigr\\} \n \\Bigr]  \n       + \\delta\\mu^*|_{\\rm div} \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,~. \n\\end{align} \n\\end{itemize}\n\\end{itemize}\n\nAll independent parameters are defined by the conditions (i)--(iv) and\nthe corresponding counterterms are determined.\nSolving \\refeqs{proc1b} and\n(\\ref{dMsq12physpar}) for $\\delta Y_b$ yields\n\\begin{align}\n\\delta Y_b &= \\frac{1}{|U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2} \\Big[ \n           U_{\\tilde{b}_{11}} U_{\\tilde{b}_{21}}^* \n          (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \\nonumber \\\\\n&\\quad  + m_{b} \\Big( U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \n          \\left( \\deA_b^* - \\mu\\, \\delta\\!\\tan\\!\\beta\\, - \\tan \\beta\\, \\delta\\mu \\right) \n                  - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \n          \\left( \\deA_b - \\mu^* \\delta\\!\\tan\\!\\beta\\, - \\tan \\beta\\, \\delta\\mu^* \\right) \\Big) \\nonumber \\\\\n&\\quad  + \\left( U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (A_b^* - \\mu \\tan \\beta)\n            - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (A_b - \\mu^* \\tan \\beta) \\right)\\, \\dem_{b}\n\\Big]~,\n\\label{dYb_mbDRbar_AbDRbar}\n\\end{align}\nwhere $\\dem_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$ are given in  \\refeqs{sboteOS} and  \n\\eqref{sbotzOS}, respectively. \n\n\n\n\n\\subsection{\\boldmath{$m_{b}$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$Y_b$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } (RS3)}\n\\label{sec:mbDRbar_YbDRbar}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined \\ensuremath{\\overline{\\mathrm{DR}}}, yielding the one-loop\n  counterterm $\\delta m_{b}$:\n\\begin{align} \n\\label{dmb_mbDRbar_YbDRbar}\n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~.\n\\end{align}\n\n\\item[(iii), (iv)]\nThe complex counterterm  $\\delta Y_b$ is determined via a \\ensuremath{\\overline{\\mathrm{DR}}}\\\nrenormalization condition, setting\n\\begin{align}\n\\delta Y_b =  \\frac{1}{2} \\widetilde\\re \\left\\{\n \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div} + \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right\\}~.\n\\label{dYb_mbDRbar_YbDRbar}\n\\end{align}\n\\end{itemize}\n\\end{itemize}\n\nAs in Sect.~\\ref{sec:OS}, the renormalization conditions \n(ii), (iii) and (iv) fix the\ncounterterms $\\delta m_{b}$ and $\\delta Y_b$, respectively. Together with the\nrenormalization conditions for $\\dem_{\\tilde{b}_1}^2$ and\n$\\dem_{\\tilde{b}_2}^2$ (see \\refeq{sboteOS} and \\refeq{sbotzOS}, respectively),\n$\\delta A_b$ is given by the linear \ncombination of these counterterms as\n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n          (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \n          + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n          + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \n          - (A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} \\bigr] \\nonumber \\\\\n&\\quad + (\\delta\\mu^* \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,)~,\n\\label{dAb_mbDRbar_YbDRbar}\n\\end{align}\nwhich, of course, shows the same analytical dependence of the independent\ncounterterms as $\\deA_b$ in \\refeq{Ab_OS} in \\refse{sec:OS}.\n\n\n\n\\subsection{\\boldmath{$m_{b}$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$Y_b$} on-shell (RS4)}\n\\label{sec:mbDRbar_YbOS}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nThe bottom-quark mass is defined \\ensuremath{\\overline{\\mathrm{DR}}}, yielding the one-loop\n  counterterm $\\delta m_{b}$:\n\\begin{align} \n\\label{dmb_mbDRbar_YbOS}\n\\dem_{b} = \\frac{1}{2} \\widetilde\\re \\left\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~.\n\\end{align}\n\n\\item[(iii), (iv)]\nThe complex counterterm $\\delta Y_b$ is fixed by an on-shell \nrenormalization condition, as in Sect.~\\ref{sec:OS},\n\\begin{align}\\label{dYb_mbDRbar_YbOS}\n\\delta Y_b =  \\frac{1}{2} \\widetilde\\re \\left\\{\n  \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) + \\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\end{align}\n\\end{itemize}\n\\end{itemize}\nAs in Sect.~\\ref{sec:OS} and in \\refse{sec:mbDRbar_YbDRbar}, the\n  renormalization conditions (i)--(iv) fix the \ncounterterms $\\dem_{\\tilde{b}_2}^2$, $\\delta m_{b}$ and $\\delta Y_b$. The further\nrenormalization condition \\refeq{sboteOS} determines the counterterm\n$\\dem_{\\tilde{b}_1}^2$. Analogous to Sect.~\\ref{sec:OS} and to\n\\refse{sec:mbDRbar_YbDRbar}, \n$\\delta A_b$  can be expressed in terms of these counterterms, \n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n          (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \n          + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n          + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \n          - (A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} \\bigr] \\nonumber \\\\\n&\\quad + (\\delta\\mu^* \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,)~,\n\\end{align}\nwhich, of course, has the same form as in \\refeqs{Ab_OS} and\n\\eqref{dAb_mbDRbar_YbDRbar}. \n\n\n\n\n\\subsection{\\boldmath{$A_b$ \\ensuremath{\\overline{\\mathrm{DR}}}\\ } and \\boldmath{$\\mathop{\\mathrm{Re}} Y_b$} on-shell (RS5)}\n\\label{sec:AbDRbar_ReYbOS}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nIn the subsections \\ref{sec:OS}--\\ref{sec:mbDRbar_YbOS} the second\nrenormalization condition defines the bottom quark mass. In this scheme,\nwe choose an on-shell renormalization condition for the real part of the\ncounterterm $\\delta Y_b$ which determines $\\mathop{\\mathrm{Re}} \\delta Y_b$ as following\n\\begin{align}\n\\mathop{\\mathrm{Re}} \\delta Y_b = \\frac{1}{2} \\mathop{\\mathrm{Re}} \\left\\{\n              \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) +\n              \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\label{dReYb_AbDRbar_ReYbOS}\n\\end{align}\n\n\n\\item[(iii), (iv)]\nThe complex $A_b$ parameter is defined \\ensuremath{\\overline{\\mathrm{DR}}}\n\\begin{align}\n\\deA_b &= \\frac{1}{m_{b}} \\Bigl[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n         \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)|_{\\rm div} \n            -\\widetilde\\re\\Sigma_{\\tilde{b}_{22}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2} U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \n         \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n            +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right) \\nonumber \\\\\n&\\quad + \\frac{1}{2} U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \n         \\left( \\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2)|_{\\rm div}\n            +\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2)|_{\\rm div} \\right)^* \\nonumber \\\\\n&\\quad - \\frac{1}{2}(A_b - \\mu^* \\tan \\beta)\\, \n\\widetilde\\re \\bigl\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div} \\nonumber\\\\\n&\\qquad + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm\n    div} \\bigr\\} \n       + \\delta\\mu^*|_{\\rm div} \\tan \\beta + \\mu^*\\, \\delta\\!\\tan\\!\\beta\\,~. \n\\label{dAb}\n\\end{align} \n\\end{itemize}\n\\end{itemize}\n\nWith the conditions (i)--(iv) the independent counterterms $\\delta\nm_{\\tilde{b}_2}^2$, $\\mathop{\\mathrm{Re}} \\delta Y_b$ and $\\delta A_b$ are determined, \nand $\\dem_{\\tilde{b}_1}^2$ is given by \\refeq{sboteOS}. The missing\ncounterterms $\\delta m_{b}$ and $\\mathop{\\mathrm{Im}} \\delta Y_b$ can be expressed by the\nindependent counterterms. Relating \\refeq{proc1b}, here explicitly\nwritten as\n\\begin{align}\\label{proc1bexpl}\n(\\delta \\matr{M}_{\\tilde{b}})_{12} &= (A_b^* - \\mu\\tan \\beta)\\, \\dem_{b} \n                              + m_{b} \\left( \\deA_b^*\n                              - \\mu\\, \\delta\\!\\tan\\!\\beta\\, - \\delta\\mu \\tan \\beta \\right)~,\n\\end{align}\nand \\refeq{dMsq12physpar}, here with $\\delta Y_b$ explicitly split into a\nreal and an imaginary part\n\\begin{align}\\label{dMsq12physparsplit}\n(\\delta \\matr{M}_{\\tilde{b}})_{12} &= \n  U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \\nonumber \\\\\n&\\quad\n + U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} (\\mathop{\\mathrm{Re}} \\delta Y_b + i \\mathop{\\mathrm{Im}} \\delta Y_b)\n + U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mathop{\\mathrm{Re}} \\delta Y_b - i \\mathop{\\mathrm{Im}} \\delta Y_b)~,\n\\end{align}\nresults in the two equations\n\\begin{align}\n\\label{eq:dM12R}\n\\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b}\n&= - m_{b} \\mathop{\\mathrm{Re}} \\deA_b - \\mathop{\\mathrm{Re}} \\delta S \n   - \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n             - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\} \\mathop{\\mathrm{Im}} \\delta Y_b~, \\\\\n\\label{eq:dM12I}\n\\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b} \n&= + m_{b} \\mathop{\\mathrm{Im}} \\deA_b - \\mathop{\\mathrm{Im}} \\delta S \n   + \\mathop{\\mathrm{Re}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n             - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\} \\mathop{\\mathrm{Im}} \\delta Y_b\n\\end{align}\nwith\n\\begin{align}\n\\delta S &= - m_{b}\\, (\\mu\\,\\delta\\!\\tan\\!\\beta\\, + \\delta\\mu \\tan \\beta)\n         - U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} (\\delta m_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2) \\nonumber \\\\\n&\\quad   - \\left( U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n         + U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right) \\mathop{\\mathrm{Re}} \\delta Y_b~,\n\\end{align}\nwhere $\\delta m_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$ are given by \\refeq{sboteOS} and\n\\refeq{sbotzOS}. \n\n\\medskip\nThe above two equations, \n(\\ref{eq:dM12R}) and (\\ref{eq:dM12I}), \ncan be solved for $\\mathop{\\mathrm{Im}}\\delta Y_b$ and $\\dem_{b}$, yielding\\\\[1em]\n\\begin{align}\\label{dmb_AbDRbar_ReYbOS}\n\\delta m_{b} &= \\frac{b_r c_i - b_i c_r}{a_r b_i - a_i b_r}~,\\\\[2mm]\n\\label{dImYb_AbDRbar_ReYbOS}\n\\mathop{\\mathrm{Im}}\\delta Y_b &= \\frac{a_i c_r - a_r c_i}{a_r b_i - a_i b_r}\n\\end{align}\nwith\n\\begin{align}\na_r &= \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}~, \\\\\na_i &= \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}~, \\\\\nb_r &= + \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n                 - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\}~, \\\\\nb_i &= - \\mathop{\\mathrm{Re}} \\left\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{22}} \n                 - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\right\\}~, \\\\\nc_r &= + m_{b} \\mathop{\\mathrm{Re}} \\deA_b + \\mathop{\\mathrm{Re}}\\delta S~, \\\\\nc_i &= - m_{b} \\mathop{\\mathrm{Im}} \\deA_b + \\mathop{\\mathrm{Im}}\\delta S~.\n\\end{align}\n\n\n\n\n\\subsection{\\boldmath{$A_b$} via vertex and \\boldmath{$\\mathop{\\mathrm{Re}} Y_b$ on-shell} (RS6)}\n\\label{sec:AbOS_ReYbOS}\n\n\\begin{itemize}\n\\item[]\n\\begin{itemize}\n\\item[(ii)]\nAn on-shell renormalization condition is imposed for the real part of\nthe counterterm $\\delta Y_b$ which determines  $\\mathop{\\mathrm{Re}} \\delta Y_b$ as\n\\begin{align}\n\\mathop{\\mathrm{Re}} \\delta Y_b = \\frac{1}{2} \\mathop{\\mathrm{Re}} \\left\\{\n \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_1}^2) +\n \\widetilde\\re{\\Sigma}_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) \\right\\}~.\n\\label{dReYb_AbOS_ReYbOS}\n\\end{align}\n\n\\item[(iii), (iv)]\nThe renormalization conditions introduced here are analogous to \nthe prescriptions used in \\citeres{dissHR,sbotrenold,mhiggsFDalbals}, \nbut extended to the complex MSSM.\nThe complex parameter $A_b$ is renormalized via the vertex\n$A\\, \\tilde{b}_1^\\dagger \\tilde{b}_2$, denoting the renormalized vertex as \n$\\hat\\Lambda(p_{A}^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2)$, see \\reffi{fig:vertex}.\n\\vspace{-2ex}\n\\begin{figure}[htb!]\n\\begin{center}\n\\setlength{\\unitlength}{1pt}\n\\begin{picture}(350, 180)\n\\DashArrowLine(160,105)(195,125){5}\n\\DashArrowLine(195,055)(160,075){5}\n\\DashLine(80,90)(140,90){5}\n\\put(65,85){$A$}\n\\put(200,50){$\\tilde{b}_2$}\n\\put(200,125){$\\tilde{b}_1$}\n\\put(240,85){$\\Hat{=} \\quad\n             i\\, \\hat{\\Lambda}(p_{A}^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2)$}\\,\n\\GCirc(140,90){20}{.6}\n\\end{picture}\n\\vspace{-2ex}\n\\caption{The renormalized vertex \n$\\hat\\Lambda(p_{A}^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2)$.}\n\\label{fig:vertex}\n\\end{center}\n\\end{figure}\n\nThe tree-level vertex $A\\, \\tilde{b}_1^\\dagger \\tilde{b}_2$, denoted as $V_{A\\,\n  \\tilde{b}_1^\\dagger \\tilde{b}_2}$, is given  as \n\\begin{align}\\nonumber\nV_{A\\, \\tilde{b}_1^\\dagger \\tilde{b}_2} = \\frac{i e\\,m_{b}}{2M_Ws_\\mathrm{w} \\cos \\beta} \n\\Bigl[&\n    U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (\\mu \\cos \\be_{\\rm n} + A_b^* \\sin \\be_{\\rm n}) \n\\nonumber \\\\\n- & U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mu^* \\cos \\be_{\\rm n} + A_b \\sin \\be_{\\rm n})\n\\Bigr]~,\n\\end{align}\nwhere $\\be_{\\rm n}$ is the mixing angle of the ${\\cal CP}$-odd Higgs boson fields\nwith $\\be_{\\rm n} = \\beta$ at tree-level. Note that in our renormalization\nprescription we do not renormalize the mixing angles but only $\\tan\n\\beta$ appearing in the Lagrangian before the transformation of the \n${\\cal CP}$-odd Higgs boson fields into mass eigenstate fields is performed.\nThe renormalized vertex reads,\n\\begin{align}\n& \\hat\\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; = \\;\n   \\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; \n   + \\frac{i e\\,m_{b}}{2M_Ws_\\mathrm{w}} \\Bigg\\{ \\nonumber \\\\\n&\\qquad\\;   \\tan \\beta \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\deA_b^*\n                   - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\delta A_b \\right] \n              + \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta\\mu\n                   - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\delta \\mu^* \\right] \\nonumber \\\\\n&\\quad + \\left[  U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( \\mu + \\tbA_b^* \\right)\n             - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( \\mu^* + \\tbA_b \\right) \n         \\right] \\nonumber \\\\\n&\\qquad  \\times \\left[ \\frac{\\dem_{b}}{m_{b}} + \\frac{1}{2} (\\delta \\bar Z_{\\tilde{b}_{11}}^* \n                    + \\delta\\bar Z_{\\tilde{b}_{22}} + \\dZ{AA}) \n                    + \\sin \\beta\\,\\cos \\beta\\, \\delta\\!\\tan\\!\\beta\\, \\right] \\nonumber \\\\\n&\\quad + \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (\\mu + \\tbA_b^*)\n            - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mu^* + \\tbA_b)\n         \\right] \\left( \\dZ{e} - \\frac{\\deM_W^2}{2\\,M_W^2}\n            - \\frac{\\des_\\mathrm{w}}{s_\\mathrm{w}} \\right)  \\nonumber \\\\ \n&\\quad  + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^* \n                           \\left( \\mu + \\tbA_b^* \\right)\n                     \\right\\} \\dZ{\\tilde{b}_{12}}\n       + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{21}} U_{\\tilde{b}_{22}}^* \n                     \\left( \\mu + \\tbA_b^* \\right)\n               \\right\\} \\dZ{\\tilde{b}_{21}}^*  \\nonumber \\\\\n&\\quad  - \\frac{1}{2} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( A_b^* - \\mu \\tan \\beta \\right)\n                 - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( A_b - \\mu^* \\tan \\beta \\right)\n              \\right] \\dZ{AG}\n\\Bigg\\}~.\n\\end{align}\nThe off-diagonal $Z$~factors are determined according to\n\\refeq{dZstopoffdiagOS}, \n\\begin{align}\n\\dZ{\\tilde{b}_{12}} &= + 2\\, \n\\frac{\\widetilde\\re\\Sigma_{\\tilde{b}_{12}}(m_{\\tilde{b}_2}^2) - \\mathop{\\mathrm{Re}}\\delta Y_b - i \\mathop{\\mathrm{Im}} \\delta Y_b}\n     {(m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2)} \\; \n=: \\dZ{\\tilde{b}_{12}}^{\\rm c} - \\frac{2 i \\mathop{\\mathrm{Im}} \\delta Y_b}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}~, \\nonumber\\\\\n\\dZ{\\tilde{b}_{21}} &= - 2\\,\n\\frac{\\widetilde\\re\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2) - \\mathop{\\mathrm{Re}} \\delta Y_b + i \\mathop{\\mathrm{Im}} \\delta Y_b}\n     {(m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2)} \\;\n=: \\dZ{\\tilde{b}_{21}}^{\\rm c} - \\frac{2 i \\mathop{\\mathrm{Im}} \\delta Y_b}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}~.\n\\end{align}\nIntroducing appropriate abbreviations we get\n\\begin{align}\n& \\hat\\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; = \\;\n  \\Lambda(p_A^2, p_{\\tilde{b}_1}^2, p_{\\tilde{b}_2}^2) \\; + \\\\\n&\\quad\\;\\, \\frac{i e}{2M_Ws_\\mathrm{w}} \\Big\\{ \nm_{b} \\tan \\beta\\; (U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\deA_b^*  \n         - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\deA_b) + \\delta M \n         + i\\, U_Y\\, \\mathop{\\mathrm{Im}} \\delta Y_b \\nonumber \\\\\n&\\quad + \\left[  U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( \\mu + \\tbA_b^* \\right)\n             - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( \\mu^* + \\tbA_b \\right) \n         \\right] \n         (\\dem_{b} + \\delta Z_{\\rm d}) \\Big\\} + \\delta Z_{\\rm o} \\nonumber\n\\end{align}\n\nwith\n\n\\begin{align}\n\\delta M &= m_{b} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta\\mu \n                - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\delta\\mu^* \n             \\right]~, \\\\[2mm]\nU_Y &= \\frac{4\\, i\\, m_{b}}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}\n      \\mathop{\\mathrm{Im}} \\Big\\{ U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} \\left( \\mu^* + \\tbA_b \\right)\n          \\Big\\}~, \\\\[2mm]\n\\dZ{\\rm d} &= m_{b} \\left[ \\frac{1}{2} (\\delta \\bar Z_{\\tilde{b}_{11}}^* \n                             + \\delta \\bar Z_{\\tilde{b}_{22}} + \\dZ{AA} )\n                    + \\sin \\beta\\,\\cos \\beta\\, \\delta\\!\\tan\\!\\beta\\, \\right]~,\\\\[2em]\n\\dZ{\\rm o} &= \\frac{i e\\, m_{b}}{2M_Ws_\\mathrm{w}} \\Bigg\\{ \\nonumber \\\\\n&\\quad \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (\\mu + \\tbA_b^*)\n          - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (\\mu^* + \\tbA_b)\n       \\right]  \\left( \\dZ{e} - \\frac{\\deM_W^2}{2\\,M_W^2}\n                        - \\frac{\\des_\\mathrm{w}}{s_\\mathrm{w}} \\right)  \\nonumber \\\\\n&\\quad + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^* \n                          \\left( \\mu + \\tbA_b^* \\right)\n                    \\right\\} \\dZ{\\tilde{b}_{12}}^{\\rm c}\n       + i \\mathop{\\mathrm{Im}} \\left\\{ U_{\\tilde{b}_{21}} U_{\\tilde{b}_{22}}^* \n                     \\left( \\mu + \\tbA_b^* \\right)\n               \\right\\} \\dZ{\\tilde{b}_{21}}^{{\\rm c}\\,*} \\nonumber \\\\\n&\\quad - \\frac{1}{2} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\left( A_b^* - \\mu \\tan \\beta \\right)\n                 - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* \\left( A_b - \\mu^* \\tan \\beta \\right)\n              \\right] \\dZ{AG}\n\\Bigg\\}~. \n\\end{align}\nThe renormalization condition reads~\\cite{dissHR,sbotrenold}\n\\begin{align}\n\\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) + \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) = 0~,\n\\end{align}\nwhich corresponds to the two conditions\n\\begin{align}\n\\label{eq:ReLam}\n\\mathop{\\mathrm{Re}} \\left\\{ \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n        + \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right\\} = 0~, \\\\\n\\label{eq:ImLam}\n\\mathop{\\mathrm{Im}} \\left\\{ \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n        + \\widetilde\\re\\hat\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right\\} = 0~.\n\\end{align}\n\n\\end{itemize}\n\\end{itemize}\n\nThe conditions (i)--(iv), are sufficient to fix all independent parameters\nand their respective counterterms. As in Sect.~\\ref{sec:AbDRbar_ReYbOS},\nrelating \\refeqs{proc1bexpl} and \\eqref{dMsq12physparsplit}, one\nderives \\refeqs {eq:dM12R} and  \\eqref{eq:dM12I} which can also be\nwritten in the form\n\\begin{align}\n\\label{eq:dM12R_AbOS_ReYbOS}\n\\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b} \n&= - m_{b} \\mathop{\\mathrm{Re}} \\deA_b - \\mathop{\\mathrm{Re}} \\delta S + \\mathop{\\mathrm{Im}} U_+ \\mathop{\\mathrm{Im}} \\delta Y_b~, \\\\\n\\label{eq:dM12I_AbOS_ReYbOS}\n\\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\}\\, \\dem_{b} \n&= + m_{b} \\mathop{\\mathrm{Im}} \\deA_b - \\mathop{\\mathrm{Im}} \\delta S + \\mathop{\\mathrm{Re}} U_- \\mathop{\\mathrm{Im}} \\delta Y_b\n\\end{align}\nwith\n\\begin{align}\n\\delta S &= -m_{b} (\\mu\\,\\delta\\!\\tan\\!\\beta\\, + \\delta\\mu \\tan \\beta)\n         - U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} (\\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2)\n         - \\left( \\mathop{\\mathrm{Re}} U_+ - i \\mathop{\\mathrm{Im}} U_- \\right) \\mathop{\\mathrm{Re}} \\delta Y_b~, \\\\\nU_{\\pm} &= U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\pm U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^*~.\n\\end{align}\n$\\dem_{\\tilde{b}_1}^2$ and $\\dem_{\\tilde{b}_2}^2$\nare fixed by \\refeqs{sboteOS} and \\eqref{sbotzOS}. \n\n\\medskip\nThe above four equations (\\ref{eq:ReLam}), (\\ref{eq:ImLam}),\n(\\ref{eq:dM12R_AbOS_ReYbOS}) and \n(\\ref{eq:dM12I_AbOS_ReYbOS}), \ncan be solved for $\\mathop{\\mathrm{Re}}\\delta A_b$, $\\mathop{\\mathrm{Im}}\\delta A_b$, $\\mathop{\\mathrm{Im}}\\delta Y_b$ and\n$\\dem_{b}$. Though, we still consider $\\mathop{\\mathrm{Re}}\\delta A_b$ and $\\mathop{\\mathrm{Im}}\\delta A_b$ as\nindependent counterterms we first calculate  $\\mathop{\\mathrm{Im}}\\delta Y_b$ and\n$\\dem_{b}$ in dependence of  $\\mathop{\\mathrm{Re}}\\delta A_b$ and $\\mathop{\\mathrm{Im}}\\delta A_b$ for\neconomically solving the systems of equations. The solution for  $\\mathop{\\mathrm{Im}}\\delta\nY_b$ and $\\dem_{b}$ is\\\\[1em]\n\\begin{align}\n\\label{dmb_AbOS_ReYbOS}\n\\delta m_{b} &= \\frac{d_i f_r - d_r f_i}{e_r f_i - e_i f_r}~,\\\\[2mm]\n\\mathop{\\mathrm{Im}}\\delta Y_b &= \\frac{d_r e_i - d_i e_r}{e_r f_i - e_i f_r}\n\\end{align}\nwith\n\\begin{align}\nd_r &= 2 \\tan \\beta \\left( \\mathop{\\mathrm{Im}} U_+ \\mathop{\\mathrm{Im}} \\delta S - \\mathop{\\mathrm{Re}} U_- \\mathop{\\mathrm{Re}} \\delta S \\right) \\\\\n&\\quad + 2 \\mathop{\\mathrm{Re}} \\left[ \\frac{M_W s_\\mathrm{w}}{i\\,e} \n               \\left( 2 \\dZ{\\rm o} + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n                                + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right)\n                + \\delta M + \\dZ{\\rm d} U_m \\right]~, \\nonumber \\\\\nd_i &= -2 \\tan \\beta \\left( \\mathop{\\mathrm{Re}} U_+ \\mathop{\\mathrm{Im}} \\delta S + \\mathop{\\mathrm{Im}} U_- \\mathop{\\mathrm{Re}} \\delta S \\right) \\\\\n&\\quad + 2 \\mathop{\\mathrm{Im}} \\left[ \\frac{M_W s_\\mathrm{w}}{i\\,e} \n               \\left( 2 \\dZ{\\rm o} + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_1}^2, m_{\\tilde{b}_1}^2) \n                                + \\widetilde\\re\\Lambda(0, m_{\\tilde{b}_2}^2, m_{\\tilde{b}_2}^2) \\right)\n                + \\delta M + \\dZ{\\rm d} U_m \\right]~, \\nonumber \\\\\ne_r &= +2 \\tan \\beta \\left[ \\mathop{\\mathrm{Im}} U_+ \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n                  -\\mathop{\\mathrm{Re}} U_- \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n              \\right] + 2 \\mathop{\\mathrm{Re}} U_m~,\\\\\ne_i &= -2 \\tan \\beta \\left[ \\mathop{\\mathrm{Re}} U_+ \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n                  +\\mathop{\\mathrm{Im}} U_- \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu\\tan \\beta \\right\\} \n              \\right] + 2 \\mathop{\\mathrm{Im}} U_m~, \\\\\nf_r &= - 2 \\mathop{\\mathrm{Im}} U_Y~, \\\\\nf_i &= 2 \\tan \\beta \\left( |U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2 \\right)\n\\end{align}\nand\n\\begin{align}\n\\label{def:Um}\nU_m &=   U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* (A_b^* \\tan \\beta + \\mu)\n       - U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}}^* (A_b \\tan \\beta + \\mu^*)~.\n\\end{align}\nFrom the \\refeqs{eq:dM12R_AbOS_ReYbOS} and \n\\eqref{eq:dM12I_AbOS_ReYbOS} we immediately obtain $\\delta A_b$ as\n\\begin{align}\n\\mathop{\\mathrm{Re}}\\deA_b &= \\ed{m_{b}} \\left[ + \\mathop{\\mathrm{Im}}\\delta Y_b \\mathop{\\mathrm{Im}} U_+ - \\mathop{\\mathrm{Re}} \\delta S\n             - \\dem_{b} \\mathop{\\mathrm{Re}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\} \\right]~, \\\\\n\\mathop{\\mathrm{Im}}\\deA_b &= \\ed{m_{b}} \\left[ - \\mathop{\\mathrm{Im}}\\delta Y_b \\mathop{\\mathrm{Re}} U_- + \\mathop{\\mathrm{Im}} \\delta S\n             + \\dem_{b} \\mathop{\\mathrm{Im}} \\left\\{ A_b^* - \\mu \\tan \\beta \\right\\} \\right]~.\n\\end{align}\n\nFinally the $\\bar{Z}$~factors in $\\hat{\\Lambda}$ have to be\ndetermined. The following \ncondition is used \n\\begin{align}\n\\widetilde\\re \\hat{\\Sigma}_{\\tilde{b}_{ii}}(m_{\\tilde{b}_1}^2) - \n\\widetilde\\re \\hat{\\Sigma}_{\\tilde{b}_{ii}}(m_{\\tilde{b}_2}^2) = 0 \\qquad (i = 1,2)~.\n\\end{align}    \nThis condition results in the following $\\bar{Z}$~factors\n\\begin{align}\n\\delta\\bar{Z}_{\\tilde{b}_{ii}} = \n-\\frac{\\widetilde\\re \\Sigma_{\\tilde{b}_{ii}}(m_{\\tilde{b}_1}^2) - \n       \\widetilde\\re \\Sigma_{\\tilde{b}_{ii}}(m_{\\tilde{b}_2}^2)}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2} \n       \\qquad (i = 1,2)~,\n\\end{align}\nwhich guarantees the IR finiteness of the renormalized vertex\n$\\hat{\\Lambda}$~\\cite{sbotrenold}.\\\\ \n\n\n\\smallskip\nAnother subtlety has to be explained here:\ndue to the fact that we have infrared divergent $C$-functions\nat $p_1 = 0$ in $\\Lambda(p_1^2=0,p^2,p^2)$, we must deal with vanishing \nGram-determinants. \nTherefore we follow \\citere{cfunc} (and references therein) and \nreplace the corresponding $C$-functions by well behaving linear \ncombinations of $B$-functions. \nDetails can be found in the appendix.\n\n\n\n\\subsection{Parameter definition}\n\n\nThe input parameters in the $b\/\\tilde{b}$ sector have to correspond to the\nchosen renormalization scheme. We start by defining the bottom quark\nmass, where the \nexperimental input is the SM \\ensuremath{\\overline{\\mathrm{MS}}}\\ mass \\cite{pdg},\n\\begin{align}\n\\label{def:mbMB}\nm_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(m_{b}) & = 4.2 \\,\\, \\mathrm{GeV}~.\n\\end{align}\nThe value of $m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R)$ (at the renormalization scale \n$\\mu_R$) is calculated from $m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(m_{b})$ at the three loop\nlevel following the prescription given in~\\citere{RunDec}.\n\nAn ``on-shell'' mass is derived from the \\ensuremath{\\overline{\\mathrm{MS}}}\\ mass via\n\\begin{align}\nm_{b}^{\\mathrm{OS}} &= m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R) \\; \n   \\left[ 1 + \\frac{\\alpha_s^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R)}{\\pi} \n        \\left( \\frac{4}{3} + 2\\, \\ln \\frac{\\mu_R}{m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(\\mu_R)} \\right) \n   \\right]~.\n\\end{align}\nThe $\\ensuremath{\\overline{\\mathrm{DR}}}$ bottom quark mass is calculated iteratively from%\n\\footnote{\nIn case of complex $\\Delta_b$ the replacement $(1 + \\Delta_b) \\to |1 + \\Delta_b|$\nshould be performed~\\cite{komplexDb}.}\n\\begin{align}\n\\label{eq:mbDR}\nm_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} &= \\frac{m_{b}^{\\mathrm{OS}} (1 + \\Delta_b) + \\dem_{b}^{\\mathrm{OS}} - \\dem_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}}}\n            {1 + \\Delta_b}\n\\end{align}\nwith an accuracy of $|1 - (m_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}})^{(n)}\/(m_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}})^{(n-1)}| < 10^{-5}$\nreached in the $n$th step of the iteration.\nThe bottom quark mass of a special renormalization scheme is then obtained\nfrom \n\\begin{align}\\label{eq:mbcorr}\nm_{b} &= m_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} + \\dem_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} - \\dem_{b}~.\n\\end{align}\nHere we have used\n\\begin{align}\n\\dem_{b}^{\\mathrm{OS}} &= \\frac{1}{2} \\widetilde\\re \\left\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]\n    + \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SL} (m_{b}^2) \\right] \\right\\}~, \\nonumber \\\\[2mm]\n\\dem_{b}^{\\ensuremath{\\overline{\\mathrm{DR}}}} &= \\frac{1}{2} \\widetilde\\re \\left\\{\n  m_{b} \\left[ \\Sigma_b^L (m_{b}^2) + \\Sigma_b^R (m_{b}^2) \\right]_{\\rm div}\n+ \\left[ \\Sigma_b^{SL} (m_{b}^2) + \\Sigma_b^{SR} (m_{b}^2) \\right]_{\\rm div} \\right\\}~, \n\\end{align}\nand $\\dem_{b}$ as given in \\refses{sec:OS}--\\ref{sec:AbOS_ReYbOS}.\nThe quantity $\\Delta_b$~\\cite{deltab1,deltab2} resums the \\order{(\\alpha_s\\tan \\beta)^n}\nand \\order{(\\alpha_t\\tan \\beta)^n} terms and is given by \n\\begin{align}\n\\Delta_b &= \\frac{2\\alpha_s(m_t)}{3\\pi} \\, \\tan \\beta \\, M_3^* \\, \\mu^* \\,\n                          I(m_{\\tilde{b}_1}^2, m_{\\tilde{b}_2}^2, m_{\\tilde{g}}^2) \\;\n     + \\frac{\\alpha_t(m_t)}{4\\pi} \\, \\tan \\beta \\, A_t^* \\, \\mu^* \\, \n                          I(m_{\\tilde{t}_1}^2, m_{\\tilde{t}_2}^2, |\\mu|^2)\n\\end{align}\nwith\n\\begin{align}\nI(a, b, c) &= - \\frac{a b\\, \\ln(b\/a) + a c\\, \\ln(a\/c) + b c\\, \\ln(c\/b)}\n                     {(a - c) (c - b) (b - a)}~.\n\\end{align}\nHere $\\alpha_t$ is defined in terms of the top Yukawa coupling \n$y_t(m_t) = \\sqrt{2} m_t(m_t)\/v$ as\n$\\alpha_t(m_t) = y_t^2(m_t)\/(4\\pi)$ with \n$v = 1\/\\sqrt{\\sqrt{2}\\, G_F} = 246.218 \\,\\, \\mathrm{GeV}$\nand \n$m_t(m_t)\\approx m_t\/(1-\\frac{1}{2\\,\\pi} \\alpha_t(m_t) +\\frac{4}{3\\,\\pi}\\alpha_s(m_t))$.\n$M_3$ is the soft SUSY-breaking parameter\nfor the gluinos, with the gluino mass given as $m_{\\tilde{g}} := |M_3|$.\n\n\\newpage\n\n\n\n\n\n\n\n\\section{Renormalization scheme analysis}\n\\label{sec:RSana}\n\n\\subsection{Calculation of loop diagrams}\n\\label{sec:calc}\n\nIn this section we give the relevant details about the calculation of the\nhigher-order corrections to the decay channels (\\ref{stsbH},\\ref{stsbW}). \nSample diagrams are shown in \\reffis{fig:fdsbotHpm}, \\ref{fig:fdsbotW}. \nNot shown are the diagrams for real (hard or soft) photon and gluon\nradiation (which, however, can become numerically very important). \nThey are obtained from the corresponding tree-level diagrams\nby attaching a photon (gluon) to the electrically (color) charged\nparticles. The internal, in a generical way depicted particles in\n\\reffis{fig:fdsbotHpm}, \\ref{fig:fdsbotW} are labeled as follows:\n$F$ can be a SM fermion, a chargino or neutralino or a gluino, $S$\ncan be a sfermion \nor a Higgs boson, $V$ can be a photon $\\gamma$, a $Z$ or $W^\\pm$ boson or a\ngluon $g$. \nNot shown are the diagrams with a gauge boson (Goldstone $G^\\pm$)--Higgs\nselfenergy \ncontribution on the external Higgs boson leg that can appear in \nthe decay $\\tilde{t}_2 \\to \\tilde{b}_i H^+$.\nOn the other hand, in our calculation, the wave function corrections for\n$\\tilde{t}_2 \\to \\tilde{b}_i W^+$ vanish as all the external particle fields are\nrenormalized on-shell.\n\nThe diagrams and corresponding amplitudes have been obtained with the\nprogram {\\em FeynArts}~\\cite{feynarts}. \nThe further evaluation has been performed with \n{\\em FormCalc}~\\cite{formcalc}. As regularization scheme for the UV-divergences we\nhave used constrained differential renormalization~\\cite{cdr}, \nwhich has been shown to be equivalent to \ndimensional reduction~\\cite{dred} at the one-loop\\ level~\\cite{formcalc}. \nThus the employed regularization preserves SUSY~\\cite{dredDS,dredDS2}. \nIt was checked that all UV-divergences cancel in the final result.\n\nThe IR-divergences from diagrams with an internal photon or gluon have\nto cancel with the ones from the corresponding real soft radiation.\nIn the case of QED we have included the soft photon contribution\nfollowing the description given in \\citere{denner}. \nIn the case of QCD we have modified this prescription by replacing the\nproduct of electric charges by the appropriate combination of color\ncharges (linear combination of $C_A$ and $C_F$ times $\\alpha_s$).\nMore details will be given in \\citere{Stop2decay}.\nUsing the sbottom masses at the one-loop level, see \\refse{sec:stop},\nwe found cancellation beyond one-loop order\nof the related IR and UV divergences for\nthe decay  $\\tilde{t}_2 \\to \\tilde{b}_i H^+$, and a cancellation, as required, at the\none-loop level for the decay $\\tilde{t}_2 \\to \\tilde{b}_i W^+$.%\n\\footnote{Using tree-level masses yields a cancellation of IR divergences\nbeyond one-loop order also for $\\tilde{t}_2 \\to \\tilde{b}_i W^+$.}\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.90\\textwidth]{fdsbotHpm}\n\\caption{\nGeneric Feynman diagrams for the decay \n$\\tilde{t}_2 \\to \\tilde{b}_i H^+$ ($i = 1,2$).\n$F$ can be a SM fermion, a chargino or neutralino or a gluino, $S$\ncan be a \nsfermion or a Higgs boson, $V$ can be a $\\gamma$, $Z$, $W^\\pm$ or $g$. \nNot shown are the diagrams with a $W^+$--$H^+$ or $G^+$--$H^+$ transition\ncontribution on the external Higgs boson leg. \n}\n\\label{fig:fdsbotHpm}\n\\vspace{2cm}\n\\includegraphics[width=0.90\\textwidth]{fdsbotW}\n\\caption{\nGeneric Feynman diagrams for the decay \n$\\tilde{t}_2 \\to \\tilde{b}_i W^+$ ($i = 1,2$).\n$F$ can be a SM fermion, a chargino or neutralino or a gluino, $S$ can be a\nsfermion or a Higgs boson, $V$ can be a $\\gamma$, $Z$, $W^\\pm$ or $g$. \n}\n\\label{fig:fdsbotW}\n\\end{center}\n\\end{figure}\n\n\nFor completness we show here also the formulas that have been\nused to calculate the tree-level decay widths:\n\\begin{align}\n\\Gamma^{\\rm tree}(\\tilde{t}_2 \\to \\tilde{b}_i H^+) &= \\frac{|C(\\tilde{t}_2, \\tilde{b}_i, H^+)|^2\\,\n                                \\lambda^{1\/2}(m_{\\tilde{t}_2}^2,m_{\\tilde{b}_i}^2,M_{H^\\pm}^2)}\n                                {16\\, \\pi\\, m_{\\tilde{t}_2}^3}\\qquad (i = 1,2)~, \\\\\n\\Gamma^{\\rm tree}(\\tilde{t}_2 \\to \\tilde{b}_i W^+) &= \\frac{|C(\\tilde{t}_2, \\tilde{b}_i, W)|^2\\,\n                                \\lambda^{3\/2}(m_{\\tilde{t}_2}^2,m_{\\tilde{b}_i}^2,M_W^2)}\n                                {16\\, \\pi\\, M_W^2\\, m_{\\tilde{t}_2}^3}\\qquad (i = 1,2)~,\n\\end{align}\nwhere $\\lambda(x,y,z) = (x - y - z)^2 - 4yz$ and the couplings \n$C(a, b, c)$ can be found in the {\\em FeynArts}~model files~\\cite{feynarts-mf}.\nThe bottom-Yukawa couplings generically are enhanced with $\\tan \\beta$.\n\n\n\\newpage\n\\pagebreak\n\\clearpage\n\n\n\n\\subsection{Numerical examples for the six renormalization schemes}\n\\label{sec:numpar}\n\nWe start our analysis by showing some representative numerical\nexamples. We evaluate the tree-level results and the one-loop correction\nfor $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ including wave function corrections. \nThe parameters are chosen according to the two scenarios, S1\\ and S2, \nshown in \\refta{tab:para}.\\footnote{It should be noted that we do not\n  include any further \nshifts in the parameters than the one given in \\refeq{MSbotshift}. \nCorrespondingly, the values for the parameters $A_b$ and $M_{\\tilde{b}_R}$ in\n\\refta{tab:para} do not reflect the actual values for the input\nparameters with respect to the \nchosen renormalization scheme. For example, the\n$\\tilde{b}_2$~mass  --- though considered as an input in the\nrenormalization scheme and defined as on-shell mass --- receives a shift\ngoing from tree- to one-loop level when starting out with the values in\n\\refta{tab:para} and including only the shift  \\refeq{MSbotshift}. \nTo circumvent this shift of the $\\tilde{b}_2$~mass, additional shifts to the\ntree-level values of \n$A_b$ and $M_{\\tilde b_R}$ would be required (depending on the\nrenormalization scheme).\n\\vspace{1mm}}\n\n\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c||r|r|r|r|r|r|r|r|}\n\\hline\nScen.\\ & $M_{H^\\pm}$ & $m_{\\tilde{t}_2}$ & $\\mu$ & $A_t$ & $A_b$ & $M_1$ & $M_2$ & $M_3$ \n\\\\ \\hline\\hline\nS1 & 150 & 600 & 200 &  900 &  400 & 200 & 300 & 800 \n\\\\ \\hline\nS2 & 180 & 900 & 300 & 1800 & 1600 & 150 & 200 & 400  \n\\\\ \\hline\n\\end{tabular}\n\\caption{MSSM parameters for the initial numerical investigation; all\nparameters are in GeV. \nWe always set $m_{b}^{\\ensuremath{\\overline{\\mathrm{MS}}}}(m_{b}) = 4.2 \\,\\, \\mathrm{GeV}$.\nIn our analysis we use \n$M_{\\tilde Q_L}(\\tilde{t}) = M_{\\tilde{t}_R} = M_{\\tilde{b}_R} =: M_{\\rm SUSY}$, where $M_{\\rm SUSY}$\n  is chosen such that the above value of $m_{\\tilde{t}_2}$ is realized.\nFor the $\\tilde{b}$~sector the shift in $M_{\\tilde Q_L}(\\tilde{b})$ as defined in\n\\refeq{MSbotshift} is taken into account.\nThe parameters entering the scalar lepton sector and\/or the first two\ngenerations do not play a relevant role in our analysis.\nThe values for $A_t$ and $A_b$ are chosen such that charge- or\ncolor-breaking minima are avoided~\\cite{ccb}.\n}\n\\label{tab:para}\n\\end{center}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\nSo far we concentrate on the rMSSM: if a scheme shows deficiencies in\nthe rMSSM, the same problems occur in the cMSSM. The final numerical\nexamples in \\refse{sec:numex} will also show complex parameters as well\nas results for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_{1,2} W^+)$.\nIt should be noted that $\\tan \\beta \\lsim 9.6\\, (4.6)$\nis excluded for S1\\ (S2) \ndue to the MSSM Higgs boson searches at LEP~\\cite{LEPHiggsSM,LEPHiggsMSSM}. \nHowever, we are interested in the general behavior of the renormalization \nschemes. If certain features appear in the two numerical scenarios \n(S1\\ and S2) only for experimentally excluded $\\tan \\beta$ values, \nother parameter choices may exhibit these features also in unexcluded \nparts of the MSSM parameter space. \nConsequently, in order to investigate the various renormalization\nschemes on \ngeneral grounds, in the following we show the results for $\\tan \\beta > 1$. \nA similar reasoning applies to the limits on the MSSM parameter space \ndue to SUSY searches. Nevertheless, to avoid completely unrealistic spectra, \nthe following exclusion limits \\cite{pdg} hold in our two\nscenarios:\n\\begin{align}\nm_{\\tilde{t}_1} &> 95 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{b}_1} > 89 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{q}} > 379 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{e}_1} > 73 \\,\\, \\mathrm{GeV}, \\nonumber \\\\\n\\mneu{1} &> 46 \\,\\, \\mathrm{GeV}, \\;\n\\mcha{1} > 94 \\,\\, \\mathrm{GeV}, \\;\nm_{\\tilde{g}} > 308 \\,\\, \\mathrm{GeV} .\n\\end{align}\n\n\nA few examples of the scalar top and bottom quark masses \nat the one-loop level%\n\\footnote{For the scalar top quark masses the\n  tree-level and the one-loop values are the same (according to\n    our renormalization conditions).}%\n~(using \n$M_{\\tilde{Q}_L}^2(\\tilde{b})$ in \\refeq{MSbotshift} for the\none-loop result) \nin the scenarios S1\\ and S2\\ are \nshown in \\refta{tab:squark}. The values of $m_{\\tilde{t}_2}$ allow copious\nproduction of the heavier scalar top quark at the LHC. For other\nchoices of the \ngluino mass, $m_{\\tilde{g}} > m_{\\tilde{t}_2}$, which would leave no visible effect for\nmost of the decay modes of the $\\tilde{t}_2$, the heavier\nscalar top quark could also be\nproduced from gluino decays at the LHC. \nFurthermore, in S1\\ (even for the nominal value of $m_{\\tilde{t}_2}$ as given in\n\\refta{tab:para}) the production of $\\tilde{t}_2$ at the ILC(1000), i.e.\\ with \n$\\sqrt{s} = 1000 \\,\\, \\mathrm{GeV}$, via $e^+e^- \\to \\tilde{t}_2\\tilde{t}_1$ will be possible,\nwith the subsequent decay modes (\\ref{stsbH}) and (\\ref{stsbW})\nbeing open. The clean environment of the ILC would permit a detailed\nstudy of the scalar top quark decays.\nDepending on the combination of allowed decay\nchannels a determination of the branching ratios at the few per-cent\nlevel might be achievable in the high-luminosity running of the ILC(1000).\nMore details will be discussed elsewhere~\\cite{Stop2decay}.\n\n\n\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|c|c||r|r|r|r|}\n\\hline\nScen. & $\\tan \\beta$ & $m_{\\tilde{t}_1}$~~ & $m_{\\tilde{t}_2}$~~ & $m_{\\tilde{b}_1}$~~ & $m_{\\tilde{b}_2}$~~  \n\\\\ \\hline\\hline\n    &  2 & 293.391 & 600.000 & 441.987 & 447.168\n\\\\ \\cline{2-6}\nS1 & 20 & 235.073 & 600.000 & 418.824 & 439.226\n\\\\ \\cline{2-6}\n    & 50 & 230.662 & 600.000 & 400.815 & 449.638\n\\\\ \\hline\\hline\n    &  2 & 495.014 & 900.000 & 702.522 & 707.598\n\\\\ \\cline{2-6}\nS2 & 20 & 445.885 & 900.000 & 678.531 & 695.180\n\\\\ \\cline{2-6}\n    & 50 & 442.416 & 900.000 & 628.615 & 697.202\n\\\\ \\hline\n\\end{tabular}\n\\caption{The top and bottom squark masses \n  at the one-loop level (see text) in the\n  scenarios S1 and S2 and \n  at different $\\tan \\beta$ for the numerical investigation; \n  all masses are in GeV and rounded to one MeV.\n}\n\\label{tab:squark}\n\\end{center}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\nLater we will also analyze numerical results for complex input parameters.\nHere it should be noted that the results for physical observables are\naffected only \nby certain combinations of the complex phases of the \nparameters $\\mu$, the trilinear couplings $A_f$, \n$f = \\{u,c,t,d,s,b,e,\\mu,\\tau\\}$, the \ngaugino mass parameters $M_1$, $M_2$,\n$M_3$ and the Higgs soft SUSY breaking parameter\n$m_{12}^2$~\\cite{MSSMcomplphasen,SUSYphases}. \nIt is possible, for instance, to eliminate the phase $\\varphi_{M_2}$ and\nthe phase $\\varphi_{m_{12}^2}$.\nExperimental constraints on the (combinations of) complex phases \narise in particular from their contributions to electric dipole moments of\nheavy quarks~\\cite{EDMDoink}, of the electron and \nthe neutron (see \\citeres{EDMrev2,EDMPilaftsis} and references therein), \nand of the deuteron~\\cite{EDMRitz}. While SM contributions enter \nonly at the three-loop level, due to its\ncomplex phases the MSSM can contribute already at one-loop\norder.\nLarge phases in the first two generations of sfermions can only be \naccommodated if these generations are assumed to be very\nheavy~\\cite{EDMheavy} or large cancellations occur~\\cite{EDMmiracle},\nsee however the discussion in \\citere{EDMrev1,plehnix}.\nA recent review can be found in \\citere{EDMrev3}.\nAccordingly, using the convention that $\\varphi_{M_2} =0$ and\n$\\varphi_{m_{12}^2} =0$, as done in this paper, in particular \nthe phase $\\varphi_\\mu$ is tightly constrained~\\cite{plehnix}, \nwhile the bounds on the phases of the third generation\ntrilinear couplings are much weaker.\nThe phase of $\\mu$ enters in the combinations \n$(\\varphi_{A_{t,b}} + \\varphi_{\\mu} - \\varphi_{m_{12}^2})$. Setting\n$\\varphi_\\mu = 0$ (and $\\varphi_{M_2} =\\varphi_{m_{12}^2} =0$, see above) \nleaves us with $A_t$ and $A_b$ as complex valued\nparameters. Since we are interested in the renormalization of the\n$b\/\\tilde{b}$~sector, in our numerical analysis we will focus on a\ncomplex~$A_b$ and keep $A_t$ real (see, however, \\citere{Stop2decay}).\n\n\n\\begin{table}[t!] \n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{|cc||c||r|r|r||r|r|r|}\n\\hline\n\\multicolumn{3}{|c||}{$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ for S1} & \n\\multicolumn{3}{|c||}{$\\tan \\beta = 2$} &\n\\multicolumn{3}{|c|}{$\\tan \\beta = 50$} \\\\\n\\hline\n& renorm.\\ scheme & $\\mu_R$ & tree & loop & \n$m_{b}$ & tree & loop & $m_{b}$ \n\\\\\n\\hline \\hline\nRS1: &``OS'' & $m_{\\tilde{t}_2}$ & \n0.0017 & -0.0011 & 3.29 & 2.5930 & -53.3469 & 3.84 \\\\ \\hline\nRS2: & ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n0.0009 &  0.0002 & 2.38 & 0.9653 &  -0.0311 & 2.16 \\\\ \\hline\nRS3: & ``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n0.0009 &  0.0004 & 2.38 & 0.9484 &  -1.5404 & 2.16 \\\\ \\hline\nRS4: & ``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n0.0009 &  0.0000 & 2.38 & 0.9593 &  -0.3411 & 2.16 \\\\ \\hline\nRS5: & ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n------ & ------ & ------ & 0.9399 & -0.0481 & 2.13 \\\\ \\hline\nRS6: & ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n0.0007 &  0.0001 & 2.19 & 0.9390 &  -0.0347 & 2.13 \\\\ \\hline\n\\end{tabular}\\\\[3ex]\n\\begin{tabular}{|cc||c||r|r|r||r|r|r|}\n\\hline\n\\multicolumn{3}{|c||}{$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ for S2} & \n\\multicolumn{3}{|c||}{$\\tan \\beta = 2$} &\n\\multicolumn{3}{|c|}{$\\tan \\beta = 50$} \\\\\n\\hline\n& renorm.\\ scheme & $\\mu_R$ & tree & loop & \n$m_{b}$ & tree & loop & $m_{b}$ \n\\\\\n\\hline \\hline\nRS1: & ``OS'' & $m_{\\tilde{t}_2}$ & \n2.0928 & -0.0776 & 3.23 & 8.5163 & -106.9700 & 3.70 \\\\ \\hline\nRS2: & ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n2.2171 & -0.1449 & 2.33 & 1.8173 &   -0.5125 & 2.11 \\\\ \\hline\nRS3: & ``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'' & $m_{\\tilde{t}_2}$ & \n0.0077 &  0.0582 & 2.33 & 3.1409 &  -11.6833 & 2.11 \\\\ \\hline\nRS4: & ``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n2.2564 & -0.1031 & 2.33 & 2.9230 &   -4.5506 & 2.11 \\\\ \\hline\nRS5: & ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n2.2332 & -0.1004 & 2.45 & 2.3018 &  0.2924 & 1.84 \\\\ \\hline\nRS6: & ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' & $m_{\\tilde{t}_2}$ & \n2.2925 & -0.1067 & 2.14 & 2.3558 & -0.0710 & 1.86 \\\\ \\hline\n\\end{tabular}\n\\caption{Examples for tree-level and full one-loop contributions\n  (see text) to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ for S1\\ (upper table) and S2\\ (lower table); \n  all values are in GeV (no comparison of the renormalization\n    schemes, see text). \n  In S1 using RS5 a divergence is reached for $\\tan \\beta = |A_b|\/|\\mu| = 2$ and no\n  value can be computed (see text below). The different\n    renormalization schemes are listed in \\refta{tab:RS}.\n}\n\\label{tab:numex}\n\\end{center}\n\\renewcommand{\\arraystretch}{1.0}\n\\end{table}\n\n\n\\bigskip\nWe start our numerical examples with the evaluation of $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ in\nS1\\ and S2\\ for $\\tan \\beta = 2$ and $\\tan \\beta = 50$ as shown in\n\\refta{tab:numex}. The corresponding results as a continuous function of\n$\\tan \\beta$ can be seen in \\reffi{fig:st2sb1H.RS}.\nIt must be emphasized here that the table and the plots do not\nconstitute a {\\em comparison} of the various schemes, but ``only''\nindividual numerical examples that are used to exhibit certain problems\nof the various schemes. A numerical comparison of the schemes requires\nthat the input parameters are converted from one scheme into another,\nsee, for instance, \\citere{mhiggsFDalbals}, which is not performed\nwithin this analysis. In our numerical examples \nthe renormalization scale, $\\mu_R$, has been set to the mass of the\ndecaying particle, i.e.\\ $\\mu_R = m_{\\tilde{t}_2}$. \nIn \\refta{tab:numex} the two main columns, labeled ``$\\tan \\beta = 2$'' and \n``$\\tan \\beta = 50$'', are divided into three columns where ``tree'' contains the \ntree-level results and ``loop'' the one-loop \ncontribution. $m_{b}$ denotes the corrected bottom quark value\ncorresponding to the respective renormalization, see \\refeq{eq:mbcorr}.\n\nThe two values of $\\tan \\beta$ were chosen as an example of a very low \nand a very high value. It should be kept in mind that the low value is\npossibly already in conflict with MSSM Higgs boson \nsearches~\\cite{LEPHiggsSM,LEPHiggsMSSM}, but kept to show an\n``extreme'' example as explained above.\nIt can be seen that RS1, RS3, RS4 and RS5 yield \nrelatively large absolute values of loop contributions  with respect to\nthe tree-level \nresult, either for $\\tan \\beta = 2$ {\\em or} for $\\tan \\beta = 50$, at least in one of\nthe two numerical scenarios.\nThis simple example shows that (by choosing a specific scenario) \nalready all except two renormalization schemes fail in part\nof the parameter space.  \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RS.S1.st2sb1H.eps}\n\\hspace{-4mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RS.S2.st2sb1H.eps} \n\\end{tabular}\n\\caption{$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. \n  Full one-loop corrected partial decay widths for the different\n  renormalization schemes (no comparison, see text). \n  The parameters are chosen according to S1\\ \n  in the\n  left plot and S2\\ in the right plot. For S1\\ the grey region \n  and for S2\\ the dark grey region is excluded by LEP Higgs searches\n  (see text).}\n\\label{fig:st2sb1H.RS}\n\\end{center}\n\\end{figure}\n\n\n\n\nMore problems of the renormalization schemes RS1, RS3, RS4 and RS5\nbecome visible in \\reffi{fig:st2sb1H.RS}.\nIn the left (right) plot of  \\reffi{fig:st2sb1H.RS} we show the results of\nS1 (S2) as a function of $\\tan \\beta$. For S1\\ the grey region \nand for S2\\ the dark grey region at low values of $\\tan \\beta$ are excluded\nby LEP Higgs searches~\\cite{LEPHiggsMSSM}. \nIt can be seen in \\reffi{fig:st2sb1H.RS} that RS1 and RS3 deviate\nstrongly from the (see the end of \\refse{sec:calc})\nexpected behavior of increasing\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ with growing $\\tan \\beta$ that the other schemes exhibit. The\nsame is observed for RS4 in S2\\ for $\\tan \\beta \\gsim 35$.\nProblems in RS2 are discussed in \\refse{sec:mbAb}, problems in RS6\nhave been found for complex parameters, see \\refse{sec:problem_non_mb}.\nThe various spikes and dips can be understood as follows:\n\\begin{itemize}\n\n\\item\nFor RS3 in S2\\ a ``peak'' appears at $\\tan \\beta \\approx 4.6$ and at \n$\\tan \\beta \\approx 6.2$. This is discussed in \\refse{sec:problem_non_Ab} below.\n\n\\item\nFor RS5 in S1\\ a ``peak'' appears (not visible) at \n$\\tan \\beta = |A_b|\/|\\mu| = 2$.\nThis is caused by large corrections to the bottom quark mass as discussed \nfurther in \\refse{sec:problem_non_mb}. This is also the reason why \nthere is no entry in \\refta{tab:numex} for RS5, S1 at $\\tan \\beta = 2$.\n\n\\item\nFor RS5 in S2\\ a ``peak'' appears at $\\tan \\beta = |A_b|\/|\\mu| = 5.33$.\nThis is caused by large corrections to the bottom quark mass as \ndiscussed further in \\refse{sec:problem_non_mb}.\n\n\\end{itemize}\n\n\n\n\n\\subsection{Generic considerations for the \\boldmath{$b\/\\tilde{b}$} sector\n  renormalization (I)}\n\\label{sec:genericI}\n\nAs discussed in \\refse{sec:stop}, a bottom quark\/squark sector\nrenormalization scheme always contains dependent counterterms which can\nbe expressed by the independent ones.\nAccording to our six definitions, this can be\n$\\dem_{b}$, $\\deA_b$ or $\\delta Y_b$. \nA problem can occur when the MSSM parameters are chosen such that the\nindependent counterterms (nearly) drop out of the relation determining\nthe dependent counterterms.\nAs will be shown below, even\nrestricting to the two numerical examples, S1\\ and S2, it is \npossible to find a set of MSSM parameters which show this behaviour for\neach of the chosen\nrenormalization schemes.\nConsequently, it appears to be difficult {\\em by construction} to\ndefine a \nrenormalization scheme for the bottom quark\/squark sector (once\nthe top quark\/squark\nsector has been defined) that behaves well for the full MSSM parameter\nspace. One possible exception could be a pure \\ensuremath{\\overline{\\mathrm{DR}}}\\ scheme, which,\nhowever, is not well suited for processes with external top\nsquarks and\/or bottom squarks. \n\nAssuming that SUSY, and more specifically the MSSM, will be discovered\nat the LHC and its parameters will be measured, the problem will have\ndisappeared. For a specific set of MSSM parameters, renormalization\nschemes can (easily) be found that behave well. \nHowever, due to our ignorance about the actual values of the SUSY\nparameters, scans over large parts of the MSSM parameter space are\nperformed, see also \\refse{sec:numex}. For this kind of analysis a\ncareful choice of the renormalization scheme has to be made.\n\nIn the following subsections we will analyze in more detail, analytically\nand numerically, the deficiencies of the various schemes.\n\n\n\n\n\n\n\\subsection{Problems of the ``OS'' renormalization}\n\\label{sec:problem_OS}\n\nThe ``OS'' renormalization as described in \\refse{sec:OS} does not yield\nreasonable results in perturbative \ncalculations as shown \nalready, e.g., in \\citere{mhiggsFDalbals,sbotrenold}. \nFor the sake of completeness we briefly repeat the results. \nThe ``OS'' scheme of \\refse{sec:OS} is the renormalization scheme\nanalogous to the one used \nin the $t\/\\tilde{t}$ sector and thus would be the ``naive'' choice. It\nincludes an on-shell renormalization \ncondition on the sbottom mixing parameter $Y_b$ that contains the\ncombination $(A_b - \\mu^* \\tan \\beta)$.  \nIn parameter regions where $(\\mu\\tan \\beta)$ is much larger than $A_b$, the\ncounterterm $\\deA_b$ receives a very large finite shift when calculated\nfrom the counterterm $\\delta Y_b$. More specifically, $\\deA_b$ as given in\n\\refeq{Ab_OS} contains the contribution\n\\begin{align}\n\\label{dAb_OSproblem}\n\\deA_b = \\frac{1}{m_{b}} \\left[ -(A_b - \\mu^*\\tan \\beta)\\, \\dem_{b} + \\ldots \\right]\n\\end{align}\nthat can give rise to very large corrections to $A_b$.\nThis is also visible in \\reffi{fig:dAb} below, where we show the \nnumerical values of $\\deA_b$ as a function of $\\tan \\beta$ for various \nrenormalization schemes. \nIn \\citere{mhiggsFDalbals} it was shown that, because of\n\\refeq{dAb_OSproblem}, the ``OS'' renormalization yields\nhuge corrections to the lightest MSSM Higgs mass. \nAlso the numerical results shown in \\refta{tab:numex} and \n\\reffi{fig:st2sb1H.RS} show extremely large one-loop corrections \nfor $\\tan \\beta = 50$. \n\nThis problem is (more or less) avoided in the other renormalization\nschemes introduced in \\refta{tab:RS}, where the renormalization\ncondition is applied directly to $A_b$, rather than deriving $\\deA_b$\nfrom a renormalization condition fixing $\\delta Y_b$.\nAlso the renormalization schemes \nRS3 (``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'') and RS4 (``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS'')\navoid this severe problem by renormalizing the bottom quark mass \\ensuremath{\\overline{\\mathrm{DR}}}.  \n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dA.S1.eps}\n\\hspace{-4mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dA.S2.eps} \n\\end{tabular}\n\\caption{\n  Finite parts of $\\deA_b$ in various renormalization schemes. The\n  parameters are chosen according to S1\\\n  left plot and S2\\ right plot. \n  For S1\\ the grey region is excluded \n  and for S2\\ the dark grey region is excluded.}\n\\label{fig:dAb}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{Problems of non-\\boldmath{$A_b$} renormalization schemes}\n\\label{sec:problem_non_Ab}\n\nTwo of our schemes, besides the ``OS'' scheme (RS1), do not employ\na renormalization of $A_b$: \nRS3 (``$m_{b},\\, Y_b$~\\ensuremath{\\overline{\\mathrm{DR}}}'') and RS4 (``$m_{b}$~\\ensuremath{\\overline{\\mathrm{DR}}}, $Y_b$~OS''). \nAs argued in \\refse{sec:problem_OS} a huge contribution to $\\deA_b$ as\nevaluated in that section is avoided by the \\ensuremath{\\overline{\\mathrm{DR}}}\\ renormalization of\n$m_{b}$. However, following \\refeq{Ab_OS} with $\\dem_{\\tilde{b}_1}^2$, $\\dem_{\\tilde{b}_2}^2$,\n$\\delta Y_b$ and $\\delta m_{b}$  chosen according to the renormalization\nschemes RS3 and RS4, respectively, one finds for the finite parts of $\\deA_b$:\n\\begin{align}\n\\mbox{RS3} &:~ \\deA_b|_{\\text{fin}} = \\ed{m_{b}} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n          \\left( \\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2 \\right) \\right]_{\\text{fin}} \n          + \\ldots~, \\\\\n\\label{dAb_problem}\n\\mbox{RS4} &:~ \\deA_b|_{\\text{fin}} = \\ed{m_{b}} \\left[ U_{\\tilde{b}_{11}} U_{\\tilde{b}_{12}}^*\n          \\left( \\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2 \\right)\n          + U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}}^* \\delta Y_b^*\n          + U_{\\tilde{b}_{12}}^* U_{\\tilde{b}_{21}} \\delta Y_b \\right]_{\\text{fin}} + \\ldots~,\n\\end{align}\nwhere the ellipses denote contributions from $\\delta\\mu$ which, however, are\nnot relevant for our argument.\nIt can be seen that still $\\deA_b$ depends on parameters (diagonal and\noff-diagonal sbottom self-energies) that are independent of $A_b$. As\nan example, Higgs boson loops in the sbottom self-energy contain\ncontributions $\\sim \\mu \\tan \\beta$, which can become very large, independently\nof the value of $A_b$. This can be seen in the right plot of\n\\reffi{fig:dAb}, where \nwe show $\\deA_b$ as a function of $\\tan \\beta$ in S2. In both renormalization\nschemes, RS3 and RS4, $\\deA_b$ becomes very large and negative for\nlarge $\\tan \\beta$. This yields the very large and negative loop corrections to\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ shown in the right plot of \\reffi{fig:st2sb1H.RS}. \nIn S1\\ this problem is less pronounced, as can be seen in the left \nplot of \\reffi{fig:dAb} ($\\deA_b$) and \\reffi{fig:st2sb1H.RS} \n($\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$). \n\nBut also for lower $\\tan \\beta$ values, $\\tan \\beta \\lsim 10$, problems can occur. \nThe (finite) ``multiple spike structure'' in RS3 for S2\\ around \n$\\tan \\beta \\approx 5.33$ (for details see the small insert within the \nright plot of \\reffi{fig:dAb}) \nis due to an interplay of top\/chargino contributions to the two \ndiagonal sbottom self-energies, invalidating this scenario also for \nthis part of the parameter space. \n\n\n\n\n\n\\subsection{Problems of an \\boldmath{$m_{b}$}--\\boldmath{$A_b$} \n            renormalization}\n\\label{sec:mbAb}\n\nIf $m_{b}$ and $A_b$ are renormalized, the sbottom mixing parameter $Y_b$\nis necessarily a dependent \nparameter, see \\refta{tab:RS}. This situation is realized in the scheme\nRS2 (``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''), see \\refse{sec:mbDRbar_AbDRbar}.\n$\\delta Y_b$ enters prominently into $\\dZ{\\tilde{b}_{21}}$. For real parameters\nwe have,\n\\begin{align}\n\\dZ{\\tilde{b}_{21}} &=- 2\\, \\frac{\\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_2}^2) - \\delta Y_b}\n                             {m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2}~.\n\\label{dZSbot21}\n\\end{align}\nIn this way $\\delta Y_b$ (or the interplay between $\\delta Y_b$ and\n$\\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_2}^2)$) can induce large loop corrections to the\nscalar top quark decay width.\n$\\delta Y_b$ can be decomposed according to \\refeq{dYb_mbDRbar_AbDRbar}\n(concentrating again on the case of real parameters),\n\\begin{align}\n\\delta Y_b &= \\frac{U_{\\tilde{b}_{11}} U_{\\tilde{b}_{21}}}\n                {|U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2}\n           \\left( \\dem_{\\tilde{b}_1}^2 - \\dem_{\\tilde{b}_2}^2 \\right)\n           + \\ldots~,\n\\label{dYb_problem1}\n\\end{align}\nwhere the ellipses denote terms with only divergent\ncontributions (due to the chosen renormalization scheme RS2) as well as\nfinite contributions from $\\delta\\mu$, \nwhich, however, do not play a role for our argument.\nFor ``maximal sbottom mixing'', \n$|U_{\\tilde{b}_{11}}| \\approx |U_{\\tilde{b}_{12}}|$,\n$\\delta Y_b$ diverges, and the loop calculation does not yield a reliable\nresult. In our two parameter scenarios, S1\\ and S2, this is not the\ncase. Such a \nlarge sbottom mixing is often associated with large values of $|A_b|$\nthat may be in conflict with charge- or color-breaking minima~\\cite{ccb}.\n\nHowever, in order to show an example with a divergence in $\\delta\nY_b$ we use a modified version of S1\\ with $A_b = 1000 \\,\\, \\mathrm{GeV}$ \n(a value still allowed following \\citere{ccb}).\nIn this scenario at $\\tan \\beta \\approx 37$ we indeed find the case of\n``maximal mixing'' in the scalar bottom sector.\nAs expected this leads to a divergence in $\\delta Y_b$,\nas can be seen in the left plot of\n\\reffi{fig:mbAb}. This divergence propagates into $\\dZ{\\tilde{b}_{21}}$ as\nshown in the right plot of \\reffi{fig:mbAb}.%\n\\footnote{\nThe scalar bottom masses could receive large corrections via \n$M_{\\tilde{Q}_L}^2(\\tilde{b})$ in \\refeq{MSbotshift}, with $\\delta Y_b$ \nentering via \\refeq{MSbotshift-detail}.\n}%\n(Also $\\Sigma_{\\tilde{b}_{21}}$ exhibits a discontinuity due to a sign change\nin $U_{\\tilde{b}}$ for this extreme set of MSSM parameters.)\nThe $\\tan \\beta$ value for which this ``divergence'' occurs depends on the\nchoice of the other MSSM parameters. For (numerical) comparison we\nalso show $\\dZ{\\tilde{t}_{21}}$ for the two scenarios.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dY.eps}\n\\hspace{-4mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.dZ.eps} \n\\end{tabular}\n\\caption{\n  $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$.\n  Left plot: size of $\\delta Y_b$ and $\\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2)$, the\n  two contributions to $\\dZ{\\tilde{b}_{21}}$, in RS2.\n  Right plot: comparison of the size of $\\dZ{\\tilde{b}_{21}}$ in the scheme RS2\n  (``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''). For both plots the parameters are chosen\n  according to S1 (but here with $A_b = 1000$ GeV), S2\\ in \\refta{tab:para}.\n  For S1\\ the grey region is excluded and for S2\\ the\n  dark grey region is excluded via LEP Higgs searches (see text).}\n\\label{fig:mbAb}\n\\end{center}\n\\end{figure}\n\n\nFor the different choice of MSSM parameters in S2\\ (without a\nhigher $A_b$ value) this divergences\ndoes not occur. However, for $\\tan \\beta \\lsim 7$ one finds \n$\\delta Y_b \\gsim \\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2)$ \n(with $\\delta Y_b = \\mathop{\\mathrm{Re}}\\Sigma_{\\tilde{b}_{21}}(m_{\\tilde{b}_1}^2)$ for $\\tan \\beta \\approx 7.5$).\nIn this part of the parameter space we also find $m_{\\tilde{b}_1} \\approx m_{\\tilde{b}_2}$,\nyielding a relatively large value of $\\dZ{\\tilde{b}_{21}}$ according to\n\\refeq{dZSbot21}, as can be seen in the right plot of\n\\reffi{fig:mbAb}. This relatively large (negative) value of\n$\\dZ{\\tilde{b}_{21}}$ in \nturn induces relatively large corrections to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. \nHowever, the loop corrections do not exceed the tree-level value\nof $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ (for our choice of MSSM parameters).\nIn summary: while for S1\\ a divergence in $\\delta Y_b$ and thus in\n$\\dZ{\\tilde{b}_{12}}$ can appear for very large values of $|A_b|$\n(possibly in conflict with charge- or color-breaking minima),\ninvalidating the renormalization scheme~RS2 in this part of the \nparameter space, these kind of problems are not encountered in S2. Here\nonly moderate loop corrections to the respective tree-level values are\nfound, and RS2 can be applied safely.\n\n\n\n\n\\subsection{Problems of non-\\boldmath{$m_{b}$} renormalization schemes}\n\\label{sec:problem_non_mb}\n\nTwo of our schemes do not employ a renormalization condition for $m_{b}$:\nRS5 (``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'') and \nRS6 (``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS''). \nSince $A_b$ and $Y_b$ are complex, we chose to renormalize $A_b$ and the\nreal part of $Y_b$. \n\nWe start with the discussion of the (simpler) ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS''\nscheme. We will focus on the real case as a subclass of\nthe more general complex case. In this renormalization scheme the bottom\nquark mass counterterm has the following form for real parameters\n(compare to \\refeq{dmb_AbDRbar_ReYbOS}),\n\\begin{align}\n\\label{dmb_problem}\n\\dem_{b} &= - \\frac{m_{b}\\, \\deA_b + \\delta S}{(A_b - \\mu\\tan \\beta)}~.\n\\end{align}\nFor vanishing sbottom mixing one finds $(A_b - \\mu\\tan \\beta) \\to 0$.\nIn the ``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' scheme this yields a finite (and\nnegative) numerator in \\refeq{dmb_problem}, but a vanishing denominator.\n\nIn a numerical evaluation, starting out with a value for the bottom\nquark mass defined as \\ensuremath{\\overline{\\mathrm{DR}}}~parameter, the actual value of the bottom\nquark mass  receives a shift with respect to the \\ensuremath{\\overline{\\mathrm{DR}}}~bottom quark\nmass according to \\refeq{eq:mbcorr}. This shift corresponds to the\nfinite part of $\\delta m_{b}$ in \\refeq{dmb_problem}.\nConsequently, large positive or negative contributions to the bottom quark \nmass can occur, yielding possibly \nnegative values for the bottom quark mass and thus invalidating the\nrenormalization scheme for  \nthese parts of the parameter space.\nThis can be seen in the left plot of \\reffi{fig:mb.explanation}, \nwhere we show $m_{b}$ in RS5 (and RS6) for the two numerical scenarios \ngiven in \\refta{tab:para} as a function of $\\tan \\beta$. \n$m_{b}$ exhibits a strong upward\/downward shift around the pole reached\nfor $\\tan \\beta = A_b\/\\mu$ and consequently yields unreliable results in this\npart of the parameter space.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.TB.mb.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{RC.PhiAb.mb.eps} \n\\end{tabular}\n\\caption{\n  Left plot: $m_{b}$ in RS5 and RS6 for S1, S2. \n  For S1\\ the grey region is excluded and for S2\\ the dark grey region \n  is excluded.\n  Right plot: $m_{b}$ in RS6 for S1, S2\\ but both with $\\tan \\beta = 20$ and \n  $\\varphi_{A_b}$ varied. In S2\\ we used also $|\\mu| = 120 \\,\\, \\mathrm{GeV}$.}\n\\label{fig:mb.explanation}\n\\end{center}\n\\end{figure}\n\n\n\\bigskip\nWe now turn to the RS6 (``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'') scheme. \nFollowing the same analysis as for the\n``$A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}, $\\mathop{\\mathrm{Re}} Y_b$~OS'' scheme an additional term in the\ndenominator of the bottom quark mass counterterm\n$\\sim U_m\/U_-$ appears,\n\\begin{align}\n\\dem_{b} &= - \\frac{\\delta S + F}{(A_b - \\mu\\tan \\beta) - U_m\/(\\tan \\beta\\,U_-)}~,\n\\end{align}\nwhere $F$ denotes other (relatively small) additional contributions. \nWith the help of \\refeq{def:Um} one finds for real parameters\n\\begin{align}\n\\frac{U_m}{\\tan \\beta\\,U_-} &= \\frac{U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}} (A_b \\tan \\beta + \\mu)\n                             -U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}} (A_b \\tan \\beta + \\mu)}\n                             {\\tan \\beta (U_{\\tilde{b}_{11}} U_{\\tilde{b}_{22}} - \n                                   U_{\\tilde{b}_{12}} U_{\\tilde{b}_{21}})}\n = (A_b + \\mu\/\\tan \\beta)~,\n\\end{align}\nand therefore\n\\begin{align}\n\\label{dmb_problem2}\n\\dem_{b} &= \\frac{\\delta S + F}{\\mu\\, (\\tan \\beta + 1\/\\tan \\beta)}~.\n\\end{align}\nThe denominator of \\refeq{dmb_problem2} can go to zero only for \n$\\mu \\to 0$, which is experimentally already excluded. \nConsequently, the problem of (too) large contributions to $m_{b}$ is\navoided in this scheme. This can be seen in the left plot of\n\\reffi{fig:mb.explanation}, where RS6, contrary to RS5, does not\nexhibit any pole-like structure in $m_{b}$.\n\nIn the complex case the above argument is no longer valid, and larger\ncontributions to $\\dem_{b}$ can arise. In the limit of $\\tan \\beta \\gg 1$ and\n$\\mu$~real the denominator of $\\dem_{b}$ in \\refeq{dmb_AbOS_ReYbOS}\nreads\n\\begin{align}\n\\ed{\\dem_{b}} &\\sim 4\\,\\mu\\, \\tan^3\\beta\\, \\Big[\n         \\mathop{\\mathrm{Re}} U_- \\left( |U_{\\tilde{b}_{11}}|^2 - |U_{\\tilde{b}_{12}}|^2 \\right) \n         + \\mathop{\\mathrm{Im}} U_- \\frac{4\\, m_{b}}{m_{\\tilde{b}_1}^2 - m_{\\tilde{b}_2}^2} \n         \\mathop{\\mathrm{Im}} \\left( U_{\\tilde{b}_{11}}^* U_{\\tilde{b}_{12}} A_b \\right) \\Big]~. \n\\label{dmb_RS6_pole}\n\\end{align}\nDepending on $\\varphi_{\\Ab}$ this denominator can go to zero and thus yield\nunphysically large corrections to $m_{b}$ in RS6.\nIn the right plot of \\reffi{fig:mb.explanation} we show $m_{b}$ as \nfunction of $\\varphi_{\\Ab}$. At \n$\\varphi_{\\Ab} \\approx 41.5^\\circ$, $87.5^\\circ$, $272.5^\\circ$, $318.2^\\circ$ \nthe denominator in \\refeq{dmb_RS6_pole} goes to zero and changes its \nsign which explains the corresponding structures.\nThis divergence in $\\dem_{b}$ enters via \\refeq{eq:mbcorr} already\ninto the tree-level prediction.\nTo summarize: while in S1\\ the scheme RS6 is well-behaved and can\nbe safely applied (also for complex $A_b$), in S2\\ \n(with $|\\mu| = 120$ GeV) severe problems\n(divergences in the counterterms) arise once complex parameters are\ntaken into account. Consequently, for S2\\ the scheme RS6 cannot be applied.\n\nIt should be noted that \nthe ``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS'' (RS6) scheme is the complex version of the \nrenormalization scheme used in \\citeres{sbotrenold,mhiggsFDalbals} for the\n\\order{\\alpha_b\\alpha_s} corrections to the neutral Higgs boson self-energies\nand thus to the mass of the lightest MSSM Higgs boson, $M_h$. For\nreal parameters, no problems occured. \n Therefore,\nemploying this renormalization scheme in \\citeres{sbotrenold,mhiggsFDalbals} \nyields numerically stable results.\n\n\n\n\n\\subsection{Generic considerations for the \\boldmath{$b\/\\tilde{b}$} sector\n  renormalization (II)} \n\\label{sec:genericII}\n\nIn the previous subsections we have analyzed analytically \n(and numerically) the\ndeficiences of the various renormalization schemes. We have shown that\ndespite of the variety of schemes, even concentrating on the two sets of\nparameters, S1\\ and S2, severe problems can be encountered\nin all schemes.\n\nFor the further numerical evaluation of the partial stop quark decay\nwidths we choose \nRS2 as our ``preferred scheme''. According to our analyses in the\nprevious subsections, RS2 shows the ``relatively most stable'' behavior,\nproblems only occur for maximal sbottom mixing, \n$|U_{\\tilde{b}_{11}}| = |U_{\\tilde{b}_{12}}|$, where a divergence in $\\delta Y_b$\nappears. \nHaving $\\delta Y_b$ as a dependent counterterm induces \neffects in the field renormalization constants $\\dZ{\\tilde{b}_{12}}$ and\n$\\dZ{\\tilde{b}_{21}}$ and in\n$\\delta M_{\\tilde{Q}_L}^2(\\tilde{b})$ entering the scalar bottom quark masses.\nIn a process with only internal scalar bottom quarks, no problems occur\ndue to the field renormalization, but \ncounterterms to propagators, which\ninduce a transition from a $\\tilde{b}_1$ squark to a $\\tilde{b}_2$ squark contain\nalso the term $\\delta Y_b$. However, $\\delta Y_b$ appearing in counterterms of\n{\\em internal} scalar bottom quarks does not exhibit a problem, since in\nthis case these ``dangerous'' contributions cancel (which we have checked\nanalytically).\nOn the other hand,\nother schemes with $\\dem_{b}$ or $\\deA_b$ as dependent counterterms \nmay exhibit problems in larger parts of the parameter\nspace and may induce large effects, since $m_{b}$ (or the bottom Yukawa \ncoupling) and $A_b$ enter prominently into the various couplings of the\nHiggs bosons to other particles.\n\nWe are not aware of any paper dealing with scalar quark decays (or\ndecays into scalar quarks) that has employed exactly RS2 \n(or its real version), see our discussion in the beginning of \\refse{sec:stop}.\nVery recently a calculation of the scalar top decay width in the rMSSM\nusing a pure \\ensuremath{\\overline{\\mathrm{DR}}}\\ scheme for all parameters was\nreported~\\cite{HelmutLL2010}.\n\n\n\n\\section{Numerical examples for our favorite scheme}\n\\label{sec:numex}\n\nFollowing the discussion in \\refse{sec:RSana} we pick the renormalization \nscheme that shows the ``most stable'' behavior over the MSSM parameter space.\nWe choose the ``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''(RS2) scheme.\nTree-level values of the partial decay widths shown in this section have\nbeen obtained including a \nshift in $m_{b}$ according to \\refeq{eq:mbcorr}. \nWe will concentrate on the calculation of the  partial $\\tilde{t}_2$ decay widths\nincluding one scalar bottom quark in the final state. A calculation of\nthe respective branching ratios \nrequires the evaluation of {\\em all} partial scalar top quark decay\nwidths, which in turn \nrequires the renormalization of the full cMSSM. This is beyond the scope of\nour paper and will be presented elsewhere~\\cite{Stop2decay}.\n \n\n\\subsection{Full one-loop results}\n\\label{sec:full1L}\n\nWe start our numerical analysis with the upper left plot of\n\\reffi{fig:st2sb1H}, where we show the partial decay width \n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ as a function of $\\tan \\beta$. ``tree'' denotes the tree-level value\nand ``full'' is the decay width including {\\em all} one-loop \ncorrections as described in \\refse{sec:calc}.\nAs one can see, the full one-loop corrections are negative and rather small \nover the full range of $\\tan \\beta$, the largest size of the loop corrections\nis found to be $\\sim 28\\%$ of the tree-level value for $\\tan \\beta = 50$ in S2.%\n\\footnote{\nIt is interesting to note that at $\\tan \\beta = |A_b|\/|\\mu| = 2\\, (5.33)$ in\nS1\\ (S2) we get $U_{\\tilde{b}_{11,22}} = 1$ and $U_{\\tilde{b}_{12,21}} = 0$,\nand consequently $\\tilde{b}_{L,R} = \\tilde{b}_{1,2}$, respectively.\n}%\n~In S1\\ the grey region and in S2\\ the dark grey region is excluded due \nto too small values of the mass of the lightest MSSM Higgs boson,\n$M_h$.\n\nIn the upper right plot of \\reffi{fig:st2sb1H} we show the partial\ndecay width varying $|A_b|$ for $\\tan \\beta = 20$. \nIn S1\\ and S2\\ the full one-loop corrections grow with \n$A_b$, but never exceed $\\sim 25\\%$ of the tree-level result. \nNote, that for S1 $|A_b| > 1130 \\,\\, \\mathrm{GeV}$\n(grey region) and S2 $|A_b| > 1800 \\,\\, \\mathrm{GeV}$ (dark grey region) \nis excluded due to the charge- or color-breaking minima.\nOver the full parameter space the loop corrections are smooth and\nsmall with respect to the tree-level results.\n\nIn the lower left plot of \\reffi{fig:st2sb1H} we analyze \nthe partial decay width varying $|\\mu|$ for $\\tan \\beta = 20$. \nValues for $|\\mu| \\lsim 120 \\,\\, \\mathrm{GeV}$ are excluded due to \n$\\mcha1 < 94 \\,\\, \\mathrm{GeV}$~\\cite{pdg}.\nThe loop corrected predictions for the partial decay width show several dips and\nspikes. In S1\\ \nthe first dip at $|\\mu| \\approx 285 \\,\\, \\mathrm{GeV}$ is due to \n$|U_{\\tilde{b}_{11}}| \\approx |U_{\\tilde{b}_{12}}|$, see the discussion in\n\\refse{sec:mbAb}.\nThe second peak\/dip (already present in the tree-level prediction) at\n$|\\mu| = 300 \\,\\, \\mathrm{GeV}$ is due to the \nrenormalization of $\\mu$~\\cite{dissTF} and will be discussed in more\ndetail in \\citere{Stop2decay}.%\n\\footnote{\nThe chosen renormalization exhibits a divergence for $\\mu = M_2$. \n$\\delta \\mu$ enters via $\\delta Y_b$ into $\\delta M_{\\tilde{Q}_L}^2(\\tilde{b})$ \nand thus into the values of $m_{\\tilde{b}_i}$. Consequently, the dip is already\npresent in  \nthe tree-level result.}\n~The third dip at $|\\mu| \\approx 424 \\,\\, \\mathrm{GeV}$, which is hardly visible, is\ndue to the production threshold  $m_t + \\mneu{3} = m_{\\tilde{t}_2}$.\nThe fourth dip at $|\\mu| \\approx 873 \\,\\, \\mathrm{GeV}$ is the threshold\n$m_{\\tilde{t}_1} + M_{H^\\pm} = m_{\\tilde{b}_1}$ of the self energy $\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)$ \nin the renormalization constants $\\dZ{\\tilde{b}_{11}}$ and $\\dem_{\\tilde{b}_1}^2$.\nThe fifth dip at $|\\mu| \\approx 1107 \\,\\, \\mathrm{GeV}$ is the production threshold \n$m_{\\tilde{b}_2} + M_W = m_{\\tilde{t}_2}$.\nFor $|\\mu| > 790 \\,\\, \\mathrm{GeV}$ the value of $M_h$ drops strongly, and the \nscenario S1 is excluded by LEP Higgs searches as indicated by the \ngray shading.\nApart from the dips analyzed above the loop corrections are very\nsmall and do not exceed $\\sim 7\\%$ of the tree-level result, \nthe prediction for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$ is well under control.\nWe now turn to the scenario S2. Here, for growing $|\\mu|$, the squark mass\nsplitting in the $\\tilde{t}\/\\tilde{b}$ sector becomes very large, leading to\nlarge contributions to the electroweak precision observables. The dark\ngray region for $|\\mu| > 1060 \\,\\, \\mathrm{GeV}$ yields $W$~boson masses outside the\nexperimentally favored region at the $2\\,\\sigma$ level, \n$M_W \\gsim 80.445 \\,\\, \\mathrm{GeV}$~\\cite{lepewwgNEW}. \nSuch large $|\\mu|$ values are consequently disfavored.\nThe dip\/peak at $|\\mu| = 200 \\,\\, \\mathrm{GeV}$ in the tree and the loop contribution\nis due to $\\delta \\mu$, where $\\mu = M_2$ is reached, see above. \nThe second dip at $|\\mu| = 477 \\,\\, \\mathrm{GeV}$, which is hardly visible, \nis the threshold $m_t + \\mcha{2} = m_{\\tilde{b}_1}$ of the self energy \n$\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1}^2)$ in the renormalization constants \n$\\dZ{\\tilde{b}_{11}}$ and $\\dem_{\\tilde{b}_1}^2$.\nThe third dip at $|\\mu| = 725 \\,\\, \\mathrm{GeV}$ is the production threshold \n$m_t + \\mneu{3} = m_{\\tilde{t}_2}$.\nThe fourth dip at $|\\mu| = 850 \\,\\, \\mathrm{GeV}$ is again the threshold \n$m_{\\tilde{t}_1} + M_{H^\\pm} = m_{\\tilde{b}_1}$. \nIn S2\\ the one-loop corrections are negative and growing with $|\\mu|$.\nApart from the dips described above, also in this numerical evaluation\nthe loop corrections stay mostly relatively small \nwith respect to the tree-level result, \nreaching the largest relative contribution at the smallest \n$|\\mu|$ values, and are thus well under control.\n\n\nWe now turn to the case of complex parameters.  \nAs discussed in \\refse{sec:numpar} we consider only $A_b$ as a complex\nparameter.\nIn the lower right plot of \\reffi{fig:st2sb1H} we show \nthe partial decay width depending on $\\varphi_{\\Ab}$ for $\\tan \\beta = 20$.\nIn S1, the tree-level values and the loop corrections are well-behaved. \nThe latter ones stay relatively small for the whole parameter space, \nnot exceeding $\\sim 18\\%$ of the tree-level result.\nIn S2, the largest corrections occur for real positive values of \n$A_b$ and reach $\\sim 12\\%$ of the tree-level values. \nFor negative $A_b$, the tree-level result becomes very small \n($< 0.01\\,\\, \\mathrm{GeV}$) and here the size of the loop corrections can be as \nlarge as the tree-level values. \nA small (and barely visible) asymmetry in the one-loop corrections \nappears in the lower right plot of \\reffi{fig:st2sb1H}, \ndue to terms $\\sim U_{\\tilde{b}_{ij}} \\times C_{0,1,2}$-function.\nThe peak\/dip at $\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$ are again due to \n$|U_{\\tilde{b}_{11}}| \\approx |U_{\\tilde{b}_{12}}|$, see \\refse{sec:mbAb}.\nIt can be seen that the peaks due to this divergence are relatively\nsharp, i.e.\\ the region of parameter space that is invalidated remains\nrelatively small.\n\n\\smallskip\nIn \\reffi{fig:st2sb2H} we show the results for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ for \nthe same set and variation of parameters as above. \nConsequently, the same peak and dip structures are visible in \nthe lower plots of \\reffi{fig:st2sb2H}. \nIn the lower left plot of \\reffi{fig:st2sb2H} in S1\\ both lines \nend because the phase space closes, \n$m_{\\tilde{b}_2} + M_{H^\\pm} > m_{\\tilde{t}_2}$ for $|\\mu| > 300 \\,\\, \\mathrm{GeV}$.\nOverall the partial decay width is much smaller than for\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$, which can \npartially be attributed to the smaller phase space, see for instance\nthe results within S2\\ in\nthe upper left plot of \\reffi{fig:st2sb2H}, and partially to the \nsmallness of the tree-level coupling.\nOnly in S2\\ for $\\tan \\beta \\gsim 35$ we find $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+) \\gsim 1 \\,\\, \\mathrm{GeV}$.\nThe relative corrections become very large for $|A_b| \\gsim 1200 \\,\\, \\mathrm{GeV}$\nas shown in the upper right plot of \\reffi{fig:st2sb2H}, \nhowever these values are disfavored by the constraints from charge and \ncolor breaking minima as discussed above.\nThe smallness of $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ at the tree-level can lead sometimes to \na ``negative value at the loop level''. \nIn this case of (accidental) smallness of the\ntree-level partial decay width also $|{\\cal M}_{\\rm loop}|^2$ would have to\nbe taken into \naccount, yielding a positive value for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$.\nOverall, because of the smallness of the tree-level result due to the tree-level\ncoupling the {\\em relative} size of the loop corrections are a bit larger than\nfor $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. Nevertheless, apart from the peaks visible in\nthe lower plots of \\reffis{fig:st2sb2H}, the loop corrections are\nwell under control also for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ using the renormalization\nscheme RS2. \nAgain a small asymmetry in the one-loop corrections in\nthe lower right plot of \\reffi{fig:st2sb2H} can be observed, \nwhich is due to terms $\\sim U_{\\tilde{b}_{ij}} \\times C_{0,1,2}$-function.\n\n\\smallskip\nFinally we evaluate the partial decay width of a scalar top quark to a\nscalar bottom quark\nand a $W$~boson, $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$ and $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. Since the\n$W$~boson is relatively light, also the latter channel is open.\nIn \\reffi{fig:st2sb1W} the results for $\\tilde{t}_2 \\to \\tilde{b}_1 W^+$ are shown, \nin \\reffi{fig:st2sb2W} the ones for $\\tilde{t}_2 \\to \\tilde{b}_2 W^+$.\nThe divergences visible in the various plots are the same ones as found\nin the respective plot for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. \nAn additional (finite) dip is visible in the lower left plot of \n\\reffi{fig:st2sb1W} in S2\\ for $|\\mu| \\approx 521 \\,\\, \\mathrm{GeV}$,  \ndue to an interplay of $t\/\\cha{2}$ contributions to \n$\\Sigma_{\\tilde{b}_{11}}(m_{\\tilde{b}_1})$, similar to the structure discussed for \n\\reffi{fig:dAb}. In this part of the parameter space the results\n  calculated within the\n  renormalization scheme RS2 have to be discarded.\n\nOverall, the loop corrections to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$ calculated within the\nrenormalization scheme RS2 behave similar\nto the ones to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. The size is relatively small, i.e.\n$\\lsim 20\\%$ and $\\lsim 30\\%$ of the tree-level results in the upper left\n and  \n in the upper right plot of \\reffi{fig:st2sb1W}, respectively, \nfor the regions  which are not in conflict with charge- or color\nbreaking minima  \n(for $|A_b| = 2000 \\,\\, \\mathrm{GeV}$ a correction of $\\sim 70\\%$ of the tree-level\nresult can be observed in \nS1\\ due to the smallness of the tree-level value). \nWe find loop corrections of the size of $\\lsim 20\\%$ of the tree-level\nresults  in the lower left plot of \\reffi{fig:st2sb1W} \nexcept for very small values of $|\\mu|$ and in the lower right plot of\n\\reffi{fig:st2sb1W}. \nIn the latter plot for S2\\ the known divergences appear at \n$\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$, leading to larger loop\ncorrections for intermediate values of $\\varphi_{\\Ab}$.\nApart from the latter case the full one-loop corrections to\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$ are well under control employing the renormalization\nscheme RS2.\n\nSimilar observations hold for the decay $\\tilde{t}_2 \\to \\tilde{b}_2 W^+$, as shown in\n\\reffi{fig:st2sb2W}.\nIn the upper left plot of \\reffi{fig:st2sb2W} in the scenario S2\\ for \n$\\tan \\beta = |A_b|\/|\\mu| \\approx 5.3$, the tree-level partial decay width vanishes, \nleading to a ``negative value at the loop level''. \nAs discussed above, in this case also $|{\\cal M}_{\\rm loop}|^2$ would have to\nbe taken into account, yielding a positive value for $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$.\n(A similar situation is found in the lower left plot of\n\\reffi{fig:st2sb2W} for $|\\mu| \\approx 200 \\,\\, \\mathrm{GeV}$.) For somewhat\nlarger $\\tan \\beta$ values, loop corrections of $\\sim 50\\%$ of the tree-level\nvalues are reached, while\nin S1\\ they stay below $\\sim 23\\%$ of the tree-level results.\nIn the upper right plot of \\reffi{fig:st2sb2W} the loop corrections \nare smaller than $\\sim 40\\%$ of the tree-level values, depending on the\nsize of $|A_b|$, see above.\n The loop corrections shown in the lower left plot of \\reffi{fig:st2sb2W}\nyield  maximal $\\sim 9 (37) \\%$ of the tree-level results in S1\\\n(S2), apart from very small $\\mu$  \nvalues, where the tree-level partial decay width can become accidentally small.\n\nFinally, looking at the dependence on $\\varphi_{\\Ab}$ in\nthe lower right plot of \\reffi{fig:st2sb2W}, apart from \nthe known divergences in S2\\ around $\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$, \nthe loop corrections do not exceed $\\sim 6\\%$ and $\\sim 35\\%$ of\nthe tree-level values\nin S1\\ and in S2, respectively. Overall, except for the small\nparameter regions around $\\varphi_{\\Ab} \\approx 117^\\circ, 243^\\circ$, the full\none-loop corrections to $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$ are well under control employing the\nrenormalization scheme RS2.\n\n\\newpage\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb1H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb1H.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb1H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb1H.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$. Tree-level and full one-loop corrected partial decay widths \n  for the renormalization scheme RS2. The parameters are chosen according to\n  the scenarios S1\\ and S2\\ (see \\refta{tab:para}). \n  For S1\\ the grey region is excluded and for S2\\ the dark grey region \n  is excluded.\n  Upper left plot: $\\tan \\beta$ varied.\n  Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n  Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n  Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb1H}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb2H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb2H.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb2H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb2H.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$. Tree-level and full one-loop corrected partial decay widths \n  for the renormalization scheme RS2. The parameters are chosen\n  according to the scenarios S1\\ and S2\\ \n  (see \\refta{tab:para}). \n  For S1\\ the grey region is excluded and for S2\\ the dark grey region \n  is excluded.\n  Upper left plot: $\\tan \\beta$ varied.\n  Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n  Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n  Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb2H}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb1W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb1W.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb1W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb1W.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$. Tree-level and full one-loop corrected partial decay widths \n  for the renormalization scheme RS2. The parameters are chosen\n  according to the scenarios S1\\ and S2\\ \n  (see \\refta{tab:para}). \n  For S1\\ the grey region is excluded and for S2\\ the dark grey region \n  is excluded.\n  Upper left plot: $\\tan \\beta$ varied.\n  Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n  Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n  Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb1W}\n\\end{center}\n\\end{figure}\n\n\n\\newpage\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{TB.st2sb2W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{Ab.st2sb2W.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{MUE.st2sb2W.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{PhiAb.st2sb2W.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n$\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. Tree-level and full one-loop corrected partial decay widths \n  for the renormalization scheme RS2. The parameters are chosen\n  according to the scenarios S1\\ and S2\\ \n  (see \\refta{tab:para}). \n  For S1\\ the grey region is excluded and for S2\\ the dark grey region \n  is excluded.\n  Upper left plot: $\\tan \\beta$ varied.\n  Upper right plot: $\\tan \\beta = 20$ and $|A_b|$ varied.\n  Lower left plot: $\\tan \\beta = 20$ and $|\\mu|$ varied.\n  Lower right plot: $\\tan \\beta = 20$ and $\\varphi_{A_b}$ varied.\n}\n\\label{fig:st2sb2W}\n\\end{center}\n\\end{figure}\n\n\n\n \n\n\\subsection{Comparison with SQCD calculation}\n\\label{sec:sqcd}\n\nOften QCD corrections to SM or MSSM processes are considered as the\nleading higher-order contributions. However, it has also been observed\nfor SM processes (e.g.\\ in the case of $WH$ and $ZH$ production at the\nTevatron and LHC~\\cite{WHZH}, for $H + 2$\\,jet production at the\nLHC~\\cite{H2j}, or for the Higgs decay to four fermions in the\nSM~\\cite{HVV4f}) that the electroweak (EW) corrections can be of similar\nsize as the QCD corrections. \nTherefore, in the last step of our numerical evaluation, we show the size\nof the one-loop effects based on  SUSY QCD (SQCD) only. The size of the\nSQCD corrections \ncan then be compared to the full calculation presented in the previous\nsubsection. It should be kept in mind that, following\n\\refeq{MSbotshift}, also the masses of the scalar bottom quarks depend\non the order of the calculation. Consequently, we do not explicitly compare\nSQCD with the full one-loop calculation, but analyze only the size and\nthe sign of the pure SQCD corrections.\n\nIn \\reffi{fig:SQCD} we show the tree-level values and SQCD one-loop \ncorrected partial decay widths for \n$\\tilde{t}_2 \\to \\tilde{b}_1 H^+$, $\\tilde{t}_2 \\to \\tilde{b}_1 W^+$, $\\tilde{t}_2 \\to \\tilde{b}_2 H^+$, $\\tilde{t}_2 \\to \\tilde{b}_2 W^+$,\nrespectively. The renormalization scheme RS2 is used, \nand hard gluon radiation is taken into account.\nThe parameters are chosen according to S1\\ and S2\\ with $\\tan \\beta$ varied. \nFor S1\\ and S2\\ the grey and the dark grey region \nis excluded via LEP Higgs searches, respectively.\nIn the lower left plot of \\reffi{fig:SQCD} the curves in S1\\ \nend at $\\tan \\beta \\approx 27$ due to the closing of the phase space.\nThe size of the SQCD one-loop corrections reaches the highest values for\nlarge $\\tan \\beta$ \nin the case of $\\tilde{t}_2 \\to \\tilde{b}_{1,2} H^+$ and for intermediate $\\tan \\beta$\nin the case of $\\tilde{t}_2 \\to \\tilde{b}_{1,2} W^+$. \nThe relative size in percent of the tree-level values do not exceed \n$-8\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$, \n$+18\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$,\n$-24\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$ and\n$-6\\%$ in $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. \nThe absolute size of the SQCD corrections can be compared with\nthe upper left plots of \\reffis{fig:st2sb1H}--\\ref{fig:st2sb2W}, where\nthe full one-loop corrections are shown. It becomes obvious, especially\nin S2, that restricting an evaluation to the pure SQCD corrections would\nstrongly underestimate the full one-loop corrections. (Hard photon radiation\ncan be as relevant as hard gluon radiation.) Consequently, \nthe full set of one-loop corrections must be taken into account to yield\na reliable prediction of the scalar top quark decay width.\n\n\n\n\\begin{figure}[htb!]\n\\vspace{2em}\n\\begin{center}\n\\begin{tabular}{c}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb1H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb1W.eps} \n\\\\[4em]\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb2H.eps}\n\\hspace{-2mm}\n\\includegraphics[width=0.49\\textwidth,height=7.5cm]{SQCD.TB.st2sb2W.eps}\n\\end{tabular}\n\\vspace{2em}\n\\caption{\n  Tree-level and SQCD corrected partial decay widths for the\n  renormalization scheme RS2 with $\\tan \\beta$ varied.  \n  The parameters are chosen according to the scenarios S1\\ and S2\\\n  (see \\refta{tab:para}). \n  For S1\\ the grey region is excluded and for S2\\ the dark grey region \n  is excluded.\n  Upper left plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 H^+)$.\n  Upper right plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_1 W^+)$. \n  Lower left plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 H^+)$. \n  Lower right plot: $\\Gamma(\\tilde{t}_2 \\to \\tilde{b}_2 W^+)$. \n}\n\\label{fig:SQCD}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions}\n\nA scalar top quark can decay into a scalar bottom quark \nand a charged Higgs boson or a $W$~boson\nif the process is kinematically allowed.\nThese decay modes can comprise a large part of\nthe total stop decay width. The decay channels with a charged Higgs\nboson in the final state form a potentially important subprocess\nof cascade decays \nwhich are interesting for the search of charged Higgs bosons at the LHC.\nIn order to arrive at a precise prediction of these scalar top quark\npartial decay widths\nat least a (full) one-loop calculation has to be performed.\nIn such a calculation a renormalization procedure has to be applied\nthat takes into account the top quark\/squark as well as the bottom\nquark\/squark sector in the MSSM. These two sectors are connected via the soft\nSUSY-breaking mass parameter $M_{\\tilde{Q}_L}$ of the superpartners of the\nleft-handed quarks, \nwhich is the same in both sectors due to the $SU(2)_L$ invariance.\n\nWithin the MSSM with complex parameters (cMSSM)\nwe defined six different renormalization schemes for the bottom\nquark\/squark sector, while \nin the top quark\/squark sector we applied a commonly used on-shell\nrenormalization scheme, which is well suited for processes with external\ntop and stop quarks. \nIn our analysis we focused on the problem that, for\ncertain parameter sets, an applied renormalization scheme might fail\nand cause large counterterm contributions that enhance the loop\ncorrections to unphysically large values. \nWe have analyzed analytically the drawbacks and shortcomings of each of the\nsix renormalization schemes. Because of the relations between the\nparameters that have to be respected also at the one-loop level\n we did not find \nany renormalization scheme that results in reasonably small counterterm\ncontributions over {\\em all}\nthe cMSSM parameter space we have analyzed (we did not consider\na pure \\ensuremath{\\overline{\\mathrm{DR}}}~scheme which is not well suited to describe external particles). \nSome renormalization schemes (for instance, the ``on-shell'' scheme\nwhich is defined analogously to the one applied in the\ntop quark\/squark sector) fail over large parts of the parameter\nspace. Others fail only in relatively small parts where, \nfor instance, a divergence due to a vanishing denominator occurs.\nThe most robust schemes turn out to be the \n``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''(RS2) scheme and the\n``$A_b$~vertex, $\\mathop{\\mathrm{Re}} Y_b$~OS''(RS6) scheme.\nThese renormalization schemes appear to be most suitable for\nhigher-order corrections \ninvolving scalar top and bottom quarks.\n\nWe performed a detailed numerical analysis for the full one-loop result \nof the partial decay widths corresponding to the four processes $\\tilde{t}_2\n\\to \\tilde{b}_j H^+\/W^+$ ($j = 1,2$) in our ``preferred'' scheme,\n``$m_{b},\\,A_b$~\\ensuremath{\\overline{\\mathrm{DR}}}''.  \nThe higher-order corrections, besides the full set of one-loop diagrams, also\ncontain soft and hard QED and QCD radiation. \nWe evaluated the higher-order predictions of the four partial decay\nwidths as a function of $\\tan \\beta$, $\\mu$, $A_b$ and $\\varphi_{\\Ab}$.  \nWe found mainly modest corrections at the one-loop level.\nLarger corrections are mostly found in regions of the parameter space that are\ndisfavored by experimental constraints and\/or charge and color breaking\nminima. \nA comparison of the full one-loop calculation with a pure SQCD\ncalculation showed that the latter one can result in a very poor\napproximation of the full result and cannot be used for a reliable\nprediction.\n\nA full one-loop calculation of the corresponding branching ratios requires the\ncalculation of all possible partial decay widths of the scalar top quark\n(and consequently a\nrenormalization of the full cMSSM) and will be presented\nelsewhere~\\cite{Stop2decay}. \n\n\n\n\\subsection*{Acknowledgements}\n\nWe thank for helpful discussions:\nF.~Campanario, \nS.~Dittmaier,\nT.~Fritzsche, \nJ.~Guasch, \nT.~Hahn, \nW.~Hollik,\nL.~Mihaila,  \nF.~von der Pahlen, \nT.~Plehn, \nM.~Spira,\nD.~St\\\"ockinger\nand \nG.~Weiglein.\nThe work of S.H.\\ was partially supported by CICYT (grant FPA 2007--66387).\nWork supported in part by the European Community's Marie-Curie Research\nTraining Network under contract MRTN-CT-2006-035505\n`Tools and Precision Calculations for Physics Discoveries at Colliders'.\nH.R.\\ acknowledges  support by the Deutsche\nForschungsgemeinschaft via the Sonderforschungsbereich\/Transregio\nSFB\/TR-9 ``Computational Particle Physics'' and the Initiative and\nNetworking Fund of the Helmholtz Association, contract HA-101 ``Physics\nat the Terascale''.\n\n\n\\newpage\n\n\\begin{appendix}\n\\section*{Appendix: \\boldmath{$C$}-functions}\n\nAs explained in \\refse{sec:AbOS_ReYbOS}, \nin RS6 we have to deal with infrared divergent $C$-functions \n(appearing in $\\Lambda(p_1^2 = 0,p^2,p^2)$) with vanishing \nGram-determinants. This case is not implemented in \n{\\em LoopTools}~\\cite{formcalc}. \nTherefore we follow \\citere{cfunc} (and references therein) and \nreplace the corresponding $C$-functions by well behaving linear \ncombinations of $B$-functions \\footnote{FormCalc \\cite{formcalc} \nsorts the loop integrals with help of the masses. \nConsequently, any momentum can become zero, \nnot only $p_1$. Furthermore {\\em LoopTools} uses a different convention \nthan \\cite{cfunc}: $C_1 = C_{11} - C_{12}$, $C_2 = C_{12}$.}.\nFor sake of completeness we briefly review our implementation.\nThe class of $C$-functions with only one external momentum zero, can be\ncompletely reduced to $B$-functions. Having three different masses we\ncan use partial fraction decomposition:\n\\begin{align}\\label{C0}\nC_0(0,p,p,m_1,m_2,m_3) = \n   \\frac{B_0(p, m_1, m_3) - B_0(p, m_2, m_3)}{m_1^2 - m_2^2}~.\n\\end{align}\nWith only two different masses applying partial differentiation \n(l'Hospital) yields\n\\begin{align}\n\\label{dB0}\nC_0(0,p,p,m_1,m_1,m_3) &= \n   \\frac{\\partial\\, B_0(p,m_1,m_3)}{\\partial\\,(m_1^2)}~.\n\\end{align}\nWe also used symmetry relations and decompositions which can be found \nin \\cite{cfunc} and the following short hand notation:\n\\begin{align}\nD_{ij} &= p^4 + m_i^4 + m_j^4 - 2 (p^2 m_i^2 + p^2 m_j^2 + m_i^2 m_j^2)~.\n\\end{align}\nWe included the following replacements of $C_i$\nfunctions with $p_1 = 0$:\n\\begin{align}\nC_0(0, p, p, m_1, m_2, m_3) &\\to \n   \\frac{B_0(p, m_1, m_3) - B_0(p, m_2, m_3)}{m_1^2 - m_2^2}~, \\\\\nC_0(0, p, p, m_1, m_1, m_3) &\\to \n   \\frac{1}{D_{13}} \\Big[ (p^2 + m_3^2 - m_1^2) (2 - B_0(p, m_1, m_3)) \\nonumber \\\\\n   &\\qquad + (p^2 - m_3^2 - m_1^2) B_0(0, m_1, m_1)\n           + 2 m_3^2 B_0(0, m_3, m_3)\\Big]~, \\\\\nC_1(0, p, p, m_1, m_2, m_3) &\\to \n   \\frac{1}{3 (m_1^2 - m_2^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_2, m_3) \n   - B_0(p, m_1, m_3)) \\nonumber \\\\\n   &\\qquad + (m_1^2 - m_2^2) B_0(p, m_2, m_3) - m_1^2 + m_2^2 \\nonumber \\\\\n   &\\qquad + (3 m_1^2 - 2 m_2^2 - m_3^2 + p^2) B_1(p, m_2, m_3) \\nonumber \\\\\n   &\\qquad + (m_3^2 - m_1^2 - p^2) B_1(p, m_1, m_3) \\Big]~, \\\\\nC_2(0, p, p, m_1, m_2, m_3) &\\to \n   \\frac{B_1(p, m_1, m_3) - B_1(p, m_2, m_3)}{m_1^2 - m_2^2}~, \\\\\nC_2(0, p, p, m_1, m_1, m_3) &\\to \n   \\frac{1}{2 p^2} \\bigg\\{B_0(0, m_1, m_1) - B_0(p, m_1, m_3) \\nonumber \\\\\n   &\\qquad - \\frac{p^2 + m_1^2 - m_3^2}{D_{13}} \\Big[ \n           (p^2 - m_1^2 + m_3^2) (2 - B_0(p, m_1, m_3)) \\nonumber \\\\\n   &\\qquad + (p^2 - m_3^2 - m_1^2) B_0(0, m_1, m_1) \n           + 2 m_3^2 B_0(0, m_3, m_3)\\Big] \\bigg\\}~.\n\\end{align}\nIn the case of $p_2 = 0$, we used the following replacements:\n\\begin{align}\nC_0(p, 0, p, m_1, m_2, m_3) &\\to \n   \\frac{B_0(p, m_1, m_2) - B_0(p, m_1, m_3)}{m_2^2 - m_3^2}~, \\\\\nC_0(p, 0, p, m_1, m_2, m_2) &\\to\n   \\frac{1}{D_{12}} \\Big[ (p^2 - m_2^2 + m_1^2) (2 - B_0(p, m_1, m_2)) \\nonumber \\\\\n   &\\qquad + (p^2 - m_1^2 - m_2^2) B_0(0, m_2, m_2) \n           + 2 m_1^2 B_0(0, m_1, m_1)\\Big]~, \\\\\nC_1(p, 0, p, m_1, m_2, m_3) &\\to \n   \\frac{1}{3 (m_2^2 - m_3^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_1, m_2) \n   - B_0(p, m_1, m_3)) \\nonumber \\\\\n   &\\qquad + (m_2^2 - m_3^2) B_0(0, m_2, m_3) - m_3^2 + m_2^2 \\nonumber \\\\\n   &\\qquad - (3 m_3^2 - 2 m_2^2 - m_1^2 - p^2) B_1(p, m_1, m_2) \\nonumber \\\\\n   &\\qquad + (m_3^2 - m_1^2 - p^2) B_1(p, m_1, m_3) \\Big]~, \\\\\nC_2(p, 0, p, m_1, m_2, m_3) &\\to \n   \\frac{1}{3 (m_2^2 - m_3^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_1, m_3) \n   - B_0(p, m_1, m_2)) \\nonumber \\\\\n   &\\qquad - (m_2^2 - m_3^2) B_0(0, m_2, m_3) + m_3^2 - m_2^2 \\nonumber \\\\\n   &\\qquad - (3 m_2^2 - 2 m_3^2 - m_1^2 - p^2) B_1(p, m_1, m_3) \\nonumber \\\\\n   &\\qquad + (m_2^2 - m_1^2 - p^2) B_1(p, m_1, m_2) \\Big]~.\n\\end{align}\nFinally, for $p_3 = (p_1 + p_2) = 0$ we employed:\n\\begin{align}\nC_0(p, p, 0, m_1, m_2, m_3) &\\to \n   \\frac{B_0(p, m_1, m_2) - B_0(p, m_2, m_3)}{m_1^2 - m_3^2}~, \\\\\nC_0(p, p, 0, m_1, m_2, m_1) &\\to\n   \\frac{1}{D_{12}} \\Big[ (p^2 - m_1^2 + m_2^2) (2 - B_0(p, m_1, m_2)) \\nonumber \\\\\n   &\\qquad + (p^2 - m_2^2 - m_1^2) B_0(0, m_1, m_1) \n           + 2 m_2^2 B_0(0, m_2, m_2)\\Big]~, \\\\\nC_1(p, p, 0, m_1, m_2, m_3) &\\to \n   \\frac{B_0(p, m_2, m_3) + B_1(p, m_1, m_2) \n   + B_1(p, m_2, m_3)}{m_1^2 - m_3^2}~, \\\\\nC_1(p, p, 0, m_1, m_2, m_1) &\\to \n   \\frac{1}{2 p^2} \\bigg\\{ B_0(0, m_1, m_1) - B_0(p, m_1, m_2) \\nonumber \\\\\n   &\\qquad - \\frac{p^2 + m_1^2 - m_2^2}{D_{12}} \\Big[ \n           (p^2 - m_1^2 + m_2^2) (2 - B_0(p, m_1, m_2)) \\nonumber \\\\\n   &\\qquad + (p^2 - m_2^2 - m_1^2) B_0(0, m_1, m_1) \n           + 2 m_2^2 B_0(0, m_2, m_2) \\Big] \\bigg\\}~, \\\\\nC_2(p, p, 0, m_1, m_2, m_3) &\\to \n   \\frac{1}{3 (m_1^2 - m_3^2)^2} \\Big[ 2 m_1^2 (B_0(p, m_2, m_3) \n   - B_0(p, m_1, m_2)) \\nonumber \\\\\n   &\\qquad - (2 m_1^2 - m_2^2 - m_3^2 + p^2) B_0(p, m_2, m_3) \n           + m_3^2 - m_1^2 \\nonumber \\\\\n   &\\qquad - (3 m_1^2 - 2 m_3^2 - m_2^2 + p^2) B_1(p, m_2, m_3) \\nonumber \\\\\n   &\\qquad + (m_2^2 - m_1^2 - p^2) B_1(p, m_1, m_2) \\Big]~.\n\\end{align}\n\n\n\\end{appendix}\n\n\n\n\n\\newpage\n\\pagebreak\n\\clearpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection*{Introduction}\n\\label{submission}\nThe Census Organisation in India has been publishing the tabulated results of Census since the initiation of modern Census in 1872 in various book forms.  As the information given by any respondent is treated as confidential, this micro-data is not publicly available. \\cite{isi-census} However Agent-based models, such as infectious disease modelling for Covid-19 require individual level information \\cite{Bonabeau02} to model and simulate the behavior of the system's constituent units (the agents) and their interactions. Though, within the 12th Five Year Plan (2012-17) Indian Census set-up 18 Workstations country-wide for research on micro-data from Census, it lacked the flexibility which researchers  need in using such huge datasets. Thus it is crucial to have the ability to model population data and generate synthetic population at various administrative levels like, the Country, State, District, Sub-District, Town, Village or Ward in Town as the case may be. \n\nIn this paper, we use a variety of data sources to generate a population of individuals and households with demographic attributes that are statistically identical to real data. This population is generated using a hybrid \\footnote{Our model is open-sourced at \\url{https:\/\/github.com\/bhaveshneekhra\/synthpop} } of statistical methods and machine learning algorithms that are flexible enough to generate data at various administrative levels, ranging from small communities to states. The primary sources of data for these algorithms include the Census of India \\cite{census-2011}, the India Human Development Survey (IHDS) (\\cite{Desai18}, the National Sample Survey (NSS) \\cite{nss18}, and the Gridded Population of the World (GPW) \\cite{ciesin2016}.\n\nWhile the synthetic population should faithfully reproduce demographic statistics, it must also incorporate other realistic network structures, such as those appropriate to households and workplaces. \\footnote{Otherwise, we could end up, for example, with ``families'' composed entirely of toddlers, or workplaces with strange mixes of professions.} Because different kinds of data respond well to different techniques, a hybrid process is used to scale up these datasets. First, the data is cleaned to remove obvious inconsistencies. Next, a customized hybrid of Iterative Proportional Fitting (IPF) \\cite{beckman-1996, Deming1940}, Iterative Proportional Updating (IPU) \\cite{ipu2009}, and a specialized variant of a neural network, called Conditional-Tabular Generative Adversarial Network (CTGAN) \\cite{xu2019}, is used to generate new data. \n\nBriefly, Iterative Proportional Fitting finds a joint distribution that matches the marginals, while trying to stay as close to the sample distribution as possible. Iterative Proportional Updating is a heuristic iterative approach which can simultaneously match or fit to multiple distributions (constraints). Finally, Conditional-Tabular Generative Adversarial Networks is a method to model the tabular data distribution and sample rows from the distribution. A Generative Adversarial Network (GAN) \\cite{goodfellow14} uses two ``competing'' neural networks, the generator and the discriminator. The generator creates realistic samples with the goal that the discriminator should be unable to differentiate between a real sample and a generated sample. In this zero-sum game, capabilities of both the networks are enhanced iteratively. Critically, our techniques are designed to work seamlessly across data-scarce and data-rich areas; even if a particular area has error-prone or missing data, a synthetic population can still be generated, albeit of a lower quality.\n\n\n\n\n\\subsubsection*{The Population Generation Process}\n    \nWe use IPF to generate a base population, using census data for the demographics and the IHDS survey dataset for personal and household attributes. The base population  thus consists of individual data and household data. We assign each household to an administrative unit within a district. \n\nWe also experimented with CTGAN to generate a base population. The major advantage of IPU over CTGAN is that IPU is capable of matching individual level and household level characteristics of an individual while making sure that members of the household have a realistic age and gender joint distribution.\n\nTo assign job labels to individuals, the relevant data from the IHDS dataset is used. For the time-being, we classify individuals below the age of 18 as students, but could easily relax this assumption. A subset of the population is also assumed be home-bound. This subset consists of unemployed individuals, homemakers, infants and children under the age of 3 and elderly people over the retirement age. We use data from the NSS survey to determine the percentage of adult males and females in a city who are home-bound. A random independent sample is drawn from a Bernoulli distribution with this gender-based marginal value as a parameter in order to decide if an individual will be home-bound or not.\n\n\nEach student in the population is assigned a school. Similarly, each working individual is assigned a workplace based upon their job label. We generate a synthetic latitude and longitude pair for each home, school and workplace in our dataset using GADM grid population density data \\cite{j-hijmans-2018}. We select a subset of grid points that lie within a given geographical boundary and sample grid-points with replacement grid points from the subset, weighing each point by the population density in the associated grid. We add independent random noise drawn from a uniform distribution to the latitudes and longitudes, rejecting those samples which fall outside the given geographical boundary. We follow this process to generate synthetic geolocation data for households, schools and workplaces.\n\nTo assign an individual a school, we sample from the list of schools within that geographical boundary, weighing each school by the inverse of the euclidean distance between it and the individual's home. This weighting factor increases the probability of assigning an individual a school that is closer to their home \\cite{rte-2019}. We follow a similar method to assign workplaces to adults. Additionally, based on every individual's job label, a workplace is assigned at random from a suitable subset of allowed workplaces.\n\n\nA number of additional attributes are included in our synthetic population, including whether an individual uses public transport or whether an individual is an essential worker. These values are assigned using the individual's job label.\n\n\\begin{figure}[htb]\n\\vskip 0.2in\n\\begin{center}\n\n\\centerline{\\includegraphics[width=\\columnwidth]{histogram}}\n\\caption{Histogram: Comparing source population (left) with synthetic population for the city of Mumbai in India}\n\\label{hist-plot-comparison}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\\begin{figure}[htb]\n\\vskip 0.2in\n\\begin{center}\n\n\\centerline{\\includegraphics[width=\\columnwidth]{plot_3}}\n\\caption{Scatter plot: Comparing source population (left) with synthetic population for the city of Mumbai in India}\n\\label{scatter-plot-comparison}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\\subsubsection*{Population Verification Metrics}\nTo compare and verify the the generated synthetic population with the survey data, we used several methods. We used the Bhattacharya\ndistance to quantify the similarity of the joint age-height and age-weight distributions. In addition, apart from comparing the two\npopulations visually as seen in \\cref{hist-plot-comparison} and \\cref{scatter-plot-comparison}, we have also used a number of\nother metrics such as statistical likelihood techniques (CS-test, KS-test). We also visualise the geographical spread of the households, schools and the workplaces in the population as in \\cref{geo-plots}. The visual comparison shows the synthetic population resembles the real population.\n\n\n\n\\begin{figure}[H]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{gpd_1.png}}\n\\caption{Geographical distribution of Households, Schools and Workplaces, respectively, in the Synthetic Population}\n\\label{geo-plots}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection*{Conclusion}\nLack of data due to access and privacy issues results in poor model design. To tackle this issue, we propose a hybrid method to generate a synthetic population. We also provide a combination of metrics to verify the generated data. In ongoing work, we are generating the synthetic population for the entire country of India. As future work, we want to explore the possibility of modelling more nuanced and complex features for the synthetic population. \n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThis work was funded by the Mphasis foundation.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\nModels with softly broken supersymmetry\n(SUSY)~\\cite{Ramond,Golfand,Volkov,Wess,Fayet} predict superpartners\nof the standard model (SM) particles.  Experimental limits from the\nTevatron and LEP showed that superpartner particles, if they exist,\nare significantly heavier than their SM counterparts.  Proposed\nexperimental searches for $R$-parity conserving SUSY at the Large\nHadron Collider (LHC) have therefore focused on a combination of two\nSUSY signatures: multiple energetic jets and\/or leptons from the\ndecays of pair-produced squarks and gluinos, and large missing\ntransverse energy (\\ETm) from the two weakly interacting\nlightest superpartners (LSP) produced in separate decay chains.\n\nIn this article a new approach is presented that is inclusive not only\nfor SUSY but also in the larger context of physics beyond the standard\nmodel. The focal point for this novel \\textit{razor}\nanalysis~\\cite{rogan} is the production of pairs of heavy particles\n(of which squarks and gluinos are examples), whose masses are\nsignificantly larger than those of any SM particle. The analysis is\ndesigned to kinematically discriminate the pair production of heavy\nparticles from SM backgrounds, without making strong assumptions about\nthe \\ETm spectrum or details of the decay chains of these\nparticles. The baseline selection requires two or more reconstructed\nobjects, which can be calorimetric jets, isolated electrons or\nisolated muons. These objects are grouped into two\n  \\textit{megajets}. The razor analysis tests the consistency, event by\nevent, of the hypothesis that the two megajets represent the visible\nportion of the decays of two heavy particles.  This strategy is\ncomplementary to traditional searches for signals in the tails of the\n\\ETm distribution ~\\cite{:2007ww, Aaltonen:2008rv, Collaboration:2011xk,\n  daCosta:2011qk, Aad:2011hh, Aad:2011xm, RA2, alphaT, :2011wb, Chatrchyan:2011bz}\nand is applied to data collected with the Compact Muon Solenoid (CMS)\ndetector from pp collisions at $\\sqrt{s}=7\\TeV$ corresponding to an\nintegrated luminosity of 35\\pbinv.\n\n\\section{The CMS Apparatus}\n\n\nA description of the CMS detector can be found\nelsewhere~\\cite{:2008zzk}.\nA characteristic feature of the CMS detector is its superconducting\nsolenoid magnet, of 6~m internal diameter, providing a field of\n3.8~T. The silicon pixel and strip tracker, the crystal\nelectromagnetic calorimeter (ECAL) and the brass\/scintillator hadron\ncalorimeter (HCAL) are contained within the solenoid. Muons are\ndetected in gas-ionization chambers embedded in the steel return\nyoke. The ECAL has an energy resolution of better than 0.5\\,\\% above\n100\\GeV. The HCAL combined with the ECAL, measures the jet energy with\na resolution $\\Delta E\/E \\approx 100\\,\\% \/ \\sqrt{E\/\\GeV} \\oplus\n5\\,\\%$.\n\nCMS uses a coordinate system with the origin located at the nominal\ncollision point, the $x$-axis pointing towards the center of the LHC,\nthe $y$-axis pointing up (perpendicular to the LHC plane), and the\n$z$-axis along the counterclockwise beam direction. The azimuthal\nangle $\\phi$ is measured with respect to the $x$-axis in the $xy$\nplane and the polar angle $\\theta$ is defined with respect to the\n$z$-axis. The pseudorapidity is $\\eta = -\\ln [\\tan(\\theta \/ 2)]$.\n\n\n\\section{The Razor Analysis\\label{intro}}\n\nThe pair production of two heavy particles, each decaying to an unseen\nLSP plus jets, gives rise to a generic SUSY-like signal.  Events in\nthis analysis are forced into a dijet topology by combining all jets\nin the event into two megajets.\nWhen an isolated lepton is present, it can be included\nin the megajets or not, as described in Sections~\\ref{sec:es}\nand~\\ref{sec:be}.  To the extent that the pair of megajets accurately\nreconstructs the visible portion of the underlying parent particle\ndecays, the kinematic properties of the signal are equivalent to the pair production of,\nfor example, two heavy squarks $\\PSq_1$, $\\PSq_2$, with\n$\\PSq_i\\rightarrow {{\\rm{j}}_i} \\PSGcz_i $, for $i=1,~2$,\nwhere  ${\\rm{j}}_i$ and $\\PSGcz_i$  denote the visible and invisible\nproducts of the decays, respectively. In the approximation that the\nheavy squarks are produced at threshold and  their visible decay\nproducts are massless, the center of mass (CM) frame four-momenta are\n\n\\begin{eqnarray}\n&&\\hspace*{-50pt}\np_{\\rm j_1}  =  \\frac{M_\\Delta}{2}(1,\\hat{\\boldmath{u}}_1)  \\;,\\quad\np_{\\rm j_2}  =  \\frac{M_\\Delta}{2}(1,\\hat{\\boldmath{u}}_2)\\; , \\\\\n&&\\hspace*{-50pt}\np_{\\PSGc_{1}} =  \\frac{M_\\Delta}{2} \\left( \\frac{2M_{\\PSq}}{M_\\Delta}-1,-\\hat{\\boldmath{u}}_1\\right) \\; ,\\quad\np_{\\PSGc_{2}}  =  \\frac{M_\\Delta}{2} \\left(\n  \\frac{2M_{\\PSq}}{M_\\Delta}-1,-\\hat{\\boldmath{u}}_2 \\right) \\; ,\n\\end{eqnarray}\n\nwhere $\\hat{\\boldmath{u}}_i$ is the unit vector in the direction of $\\rm{j}_i$, and\n\n\\begin{eqnarray}\n&&\\hspace*{-50pt}\nM_\\Delta \\equiv  \\frac{M_{\\PSq}^{2}-M_{\\PSGc}^{2}}{M_{\\PSq}}~,~ \\;\n\\end{eqnarray}\n\nwhere $M_{\\PSq}$ and $M_{\\PSGc}$ are the squark and LSP\nmasses, respectively.\n\nIn events with two undetected particles in the partonic final state, it\nis not possible to reconstruct the actual CM frame. Instead, an\napproximate event-by-event  reconstruction is made assuming the dijet signal\ntopology, replacing the CM frame with the  \\textit{$R$ frame} \\cite{rogan},\ndefined as the longitudinally boosted frame that equalizes the magnitude\nof the two megajets' three-momenta. The $R$ frame would be the CM frame for\nsignal events, if the squarks were produced at threshold and if the CM\nsystem had no overall transverse momentum from initial-state\nradiation.  The longitudinal Lorentz boost factor is defined by\n\\begin{equation}\n\\beta_{R}\\equiv\\frac{E^{\\rm j_1}-E^{\\rm j_2}}{p^{\\rm j_1}_{z}-p^{\\rm j_2}_{z}}\\; ,\n\\end{equation}\nwhere $E^{\\rm j_1}$, $E^{\\rm j_2}$ and $p^{\\rm j_1}_{z}$, $p^{\\rm\n  j_2}_{z}$ are the megajet energies and longitudinal momenta ,\nrespectively.  To the extent that the $R$ frame matches the true CM\nframe, the maximum value of the scalar sum of the megajets'\ntransverse momenta ($\\PT^1,~\\PT^2$) is $M_\\Delta$ for signal\nevents. The maximum value of the \\ETm is also $M_\\Delta$.  A\ntransverse mass $M_T^{R}$ is defined whose maximum value for signal events is\nalso $M_\\Delta$ in the limit that the $R$ and CM frames coincide:\n\n\\begin{equation}\nM_T^{R}\\equiv \\sqrt{ \\frac{\\ETm(\\PT^{\\rm j_1}+\\PT^{\\rm j_2}) -\n    \\VEtmiss {\\mathbf \\cdot}\n    (\\vec{p}_T^{\\,\\rm j_1}+\\vec{p}_T^{\\,\\rm j_2})}{2}} \\; .\n\\end{equation}\n\nThe event-by-event estimator of $M_\\Delta$ is\n\\begin{equation}\nM_R\\equiv 2|\\vec{p}^{R}_{{\\rm j_1}}| = 2|\\vec{p}^{R}_{{\\rm j_2}}|\\; ,\n\\end{equation}\nwhere $\\vec{p}^{R}_{{\\rm j_1}}$ and $\\vec{p}^{R}_{{\\rm j_2}}$ are the\n3-momenta of the megajets in the $R$ frame.  For signal events in the\nlimit where the $R$ frame and the true CM frame coincide, $M_R$ equals\n$M_\\Delta$, and more generally $M_R$ is expected to peak around\n$M_\\Delta$ for signal events.\nFor QCD dijet and multijet events the only relevant scale is\n$\\sqrt{\\hat{s}}$, the CM energy of the partonic subprocess.  The\nsearch for an excess of signal events in a tail of a distribution is\nthus recast as a search for a peak on top of a steeply falling SM\nresidual tail in the $M_R$ distribution. To extract the peaking\nsignal, the QCD multijet background needs to be reduced to manageable\nlevels. This is achieved using the \\textit{razor} variable defined as:\n\\begin{equation}\n  R\\equiv \\frac{M_T^{R}}{M_R} \\; .\\end{equation}\n\nSince for signal events $M_T^R$ has a maximum value of $M_\\Delta$\n(i.e., a kinematic edge), $R$ has a maximum value of approximately 1\nand the distribution of $R$ for signal events peaks around 0.5. These\nproperties motivate the appropriate kinematic requirements for the\nsignal selection and background reduction. It is noted that, while $M_T^R$\nand $M_R$ measure the same scale (one as an end-point, the other as a\npeak), they are largely uncorrelated for signal events, as shown in\nFig.~\\ref{fig:MR_v_R}. In this figure, the $\\PW$+jets and $\\ttbar$+jets\nbackgrounds peak at $M_R$ values partially determined by the $\\PW$ and top quark masses,\nrespectively.\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{R-MR-QCD.pdf}\n\\includegraphics[width=0.49\\textwidth]{R-MR-Wjets.pdf}\n\\includegraphics[width=0.49\\textwidth]{R-MR-top.pdf}\n\\includegraphics[width=0.49\\textwidth]{R-MR-LM1.pdf}\n\\caption{ Scatter plot in the ($M_R$, $R$) plane for simulated events:\n  (top left) QCD multijet, (top right) $\\PW$+jets, (bottom left)\n  $\\ttbar$+jets, and (bottom right) the SUSY benchmark model LM1\n  \\cite{PTDR2} with $M_\\Delta = 597\\GeV$. The yields are normalized to\n  an integrated luminosity of 35\\pbinv. The bin size is (20\\GeV\n  $\\times$ 0.015). \\label{fig:MR_v_R}}\n\\end{center}\n\\end{figure*}\n\nIn this analysis the SM background shapes and normalizations are\nobtained from data. The backgrounds are extracted from control regions\nin the $R$ and $M_R$ distributions dominated by SM processes.\nInitial estimates of the background distributions in these regions are\nobtained from the individual simulated background components, but\ntheir  shapes  and normalizations are then corrected using data.\nThe analysis flow is as follows:\n\n\\begin{enumerate}\n\\item The inclusive data sets are collected using\n  the electron, muon,  and hadronic-jet  triggers.\n\\item These data sets are examined for the presence of a\n  well-identified isolated electron or muon, irrespective of the\n  trigger path. Based on the presence or absence of such a lepton,\n  each event is assigned to one of three disjoint event samples,\n  referred to as the electron, muon, and hadronic \\textit{boxes}.  These\n  boxes serve as controls of processes in the SM with leptons, jets,\n  and neutrinos, e.g. QCD multijet, $\\PW$+jets or $\\cPZ$+jets, and\n  $\\cPqt$+X. The diboson background is found to be negligible.  Exclusive\n  multilepton boxes are also defined but are not sufficiently\n  populated to be used in this analysis.\n\\item Megajets are constructed for events passing a baseline kinematic\n  selection, and the $R$ and $M_R$ event variables are computed. In\n  the electron box, electrons are clustered with jets in the\n  definition of the megajets. Jets matched to these electrons are\n  removed to avoid double-counting. In the muon box, muons are\n  included in the megajet clustering.\n\\item In order to characterize the distribution of the SM background\n  events in the ($M_R$, $R$) plane, a kinematic region is identified\n  in the lepton boxes that is dominated by $\\PW(\\ell \\nu)+$jets. Another\n  region is found that is dominated by the sum of the non-QCD\n  backgrounds.\n\\item Events remaining in the hadronic box primarily consist of QCD\n  multijet, $\\cPZ(\\nu\\bar{\\nu})$+jets, $\\PW(\\ell\\nu)$+jets, and $\\cPqt$+X\n  events that produce $\\ell$+jets+$\\ETm$ final states with\n  charged leptons that do not satisfy the electron or muon selections.\n  The shapes and normalizations of these non-QCD background processes\n  in the hadronic box are estimated using the results from the lepton\n  boxes in appropriate regions in the ($M_R$, $R$) plane.\n\\item The QCD background shape and normalization in each of the lepton\n  boxes is extracted by reversing the lepton isolation requirements to\n  obtain control samples dominated by QCD background.\n\\item The QCD background in the hadronic box is estimated using QCD\n  control samples collected with prescaled jet triggers.\n\\item The large-$R$ and high-$M_R$ regions of all boxes are signal candidate\n  regions not used for the background estimates.  Above a given\n  $R$ threshold, the $M_R$ distribution of the backgrounds observed in the\n  data is well modeled by simple exponential functions.  Having determined the $R$\n  and $M_R$ shape and normalization of the backgrounds in the control\n  regions, the SM yields are extrapolated to the large-$R$\n  and high-$M_R$ signal candidate regions for each box.\n\\end{enumerate}\n\n\\section{Event Selection\\label{sec:es}}\n\nThe analysis uses data sets recorded with triggers based on the\npresence of an electron, a muon, or on $H_T$, the uncorrected scalar\nsum of the transverse energy of jets reconstructed at the trigger\nlevel.  Prescaled jet triggers with low thresholds are used  for the\nQCD multijet background estimation in the hadronic box.\n\nThe  analysis is guided by studies of Monte Carlo  (MC)  event\nsamples generated with the {\\sc Pythia} \\cite{Sjostrand:2006za} and\n{\\sc Madgraph} \\cite{Maltoni:2002qb} programs, simulated using the CMS\n{\\sc Geant}-based \\cite{G4} detector simulation, and then processed by\nthe same software used to reconstruct real collision data.  Events\nwith QCD multijet, top quarks, and electroweak bosons were generated\nwith {\\sc Madgraph} interfaced with {\\sc Pythia} for parton showering,\nhadronization, and underlying event description. To generate Monte\nCarlo samples for SUSY, the mass spectrum was first calculated with\n{\\sc {Softsusy}} \\cite{softsusy} and the decays with {\\sc {Susyhit}}\n\\cite{Susyhit}. The {\\sc {Pythia}} program was used with the {\\sc SLHA}\ninterface \\cite{SLHA} to generate the events. The generator level\ncross section and the K factors for the next-to-leading order  (NLO) cross\nsection calculation were computed using {\\sc Prospino} \\cite{prospino}.\n\n\nEvents are required to have at least one good reconstructed\ninteraction vertex \\cite{TRK-10-005}. When multiple vertices are\nfound, the one with the highest associated $\\sum_{\\rm track}\\PT$ is\nused.  Jets are reconstructed offline from calorimeter energy deposits\nusing the infrared-safe anti-k$_{\\rm{T}}$~\\cite{antikt} algorithm with\nradius parameter $0.5$. Jets are corrected for the nonuniformity of\nthe calorimeter response in energy and $\\eta$ using corrections\nderived with the simulation and are required to have $\\PT> 30\\GeV$\nand $|\\eta| < 3.0$.  The jet energy scale uncertainty for these\ncorrected jets is $5\\%$ \\cite{JES}. The \\ETm is\nreconstructed using the particle flow algorithm \\cite{PFMET}.\n\nThe electron and muon reconstruction and identification criteria are\ndescribed in ~\\cite{EWK-PAS}. Isolated electrons and muons are\nrequired to have $\\PT>20\\GeV$ and $\\vert\\eta|<$ 2.5 and 2.1,\nrespectively, and to satisfy the selection requirements from\n\\cite{EWK-PAS}.  The typical lepton trigger and reconstruction\nefficiencies are 98\\% and 99\\%, respectively, for electrons and 95\\%\nand 98\\% for muons.\n\n\nThe reconstructed hadronic jets, isolated electrons, and isolated\nmuons are grouped into two megajets, when at least two such objects\nare present in the event.  The megajets are constructed as a sum of\nthe four-momenta of their constituent objects.  After considering all\npossible partitions of the objects into two megajets, the combination\nminimizing the invariant masses summed in quadrature of the resulting\nmegajets is selected among all combinations for which the $R$ frame is\nwell defined.\n\n\nAfter the construction of the two megajets the boost variable\n$|\\beta_{R}|$ is computed; due to the approximations mentioned above,\n$|\\beta_{R}|$ can fall in an unphysical region (${\\ge}1$) for signal or\nbackground events; these events are removed. The additional\nrequirement $|\\beta_{R}|\\le 0.99$ is imposed to remove events for\nwhich the razor variables become singular. This requirement is typically\n85\\%    efficient for simulated SUSY events. The azimuthal angular\ndifference between the megajets is required to be less than $2.8$ radians;\nthis requirement suppresses nearly back-to-back QCD dijet\nevents. These requirements define the inclusive baseline\nselection. After this selection, the signal efficiency in the\nconstrained minimal supersymmetric standard model\n(CMSSM)~\\cite{Chamseddine,Barbieri,Hall,Kane} parameter space for a gluino\nmass of  ${\\sim} 600\\GeV$ is over 50\\%.\n\n\\section{Background Estimation\\label{sec:be}}\n\nIn traditional searches for SUSY based on missing transverse energy, it\nis difficult to model the tails of the \\ETm\ndistribution and the contribution from events with spurious\ninstrumental effects. The QCD multijet production is an especially\ndaunting background because of its very high cross section and complicated\nmodeling of its high-\\PT and \\ETm tails. In this analysis\na cut on $R$ makes it possible to isolate the QCD multijet\nbackground in the low-$M_R$ region.\n\nApart from QCD multijet backgrounds, the remaining backgrounds in the\nlepton and hadronic boxes are processes with genuine \\ETm due to\nenergetic neutrinos and leptons from massive vector boson decays\n(including $\\PW$ bosons from top quark decays). After applying an $R$ threshold,\nthe $M_{R}$ distributions in the lepton and hadronic boxes are\nvery similar for these backgrounds; this similarity is exploited in\nthe modeling and normalization of these backgrounds.\n\n\\subsection{QCD multijet background\\label{sec:qcd}}\n\nThe QCD multijet control sample for the hadronic box is defined from\nevent samples recorded with prescaled jet triggers and passing the\nbaseline analysis selection for events without a well-identified\nisolated electron or muon. The trigger requires at least two jets with\nan average uncorrected $\\PT > 15\\GeV$.  Because of the low jet\nthreshold, the QCD multijet background dominates this sample for low\n$M_R$, thus allowing the extraction of the $M_R$ shapes with different\n$R$ thresholds for QCD multijet events. These shapes are corrected\nfor the $H_T$ trigger turn-on efficiency.\n\nThe $M_{R}$ distributions for events satisfying the QCD control box\nselection, for different values of the $R$ threshold, are shown in\nFig.~\\ref{fig:DATA_QCD_calo} (left). The $M_{R}$ distribution is\nexponentially falling, after a turn-on at low $M_{R}$ resulting from\nthe \\PT threshold requirement on the jets entering the megajet\ncalculation. After the turn-on which is fitted with an asymmetric\nGaussian, the exponential region of these distributions is fitted for\neach value of $R$ to extract the exponential slope, denoted by $S$.\nThe value of $S$ that maximizes the likelihood in the exponential fit\nis found to be a linear function of $R^{2}$, as shown in\nFig.~\\ref{fig:DATA_QCD_calo} (right); fitting $S$ to the form $S = a +\nbR^{2}$ determines the values of $a$ and $b$.\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{DATA_QCD_calo_slope_log.pdf}\n\\includegraphics[width=0.47\\textwidth]{DATA_QCD_calo_slopefit.pdf}\n\\caption{(Left) $M_{R}$ distributions for different values of the $R$\n  threshold for data events in the QCD control box. Fits of the\n  $M_{R}$ distribution to an exponential function and an asymmetric\n  Gaussian at low $M_R$, are shown as dotted black curves.  (Right)\n  The exponential slope $S$ from fits to the $M_{R}$ distribution, as\n  a function of the square of the $R$ threshold for data events in the\n  QCD control box.}\n\\label{fig:DATA_QCD_calo}\n\\end{center}\n\\end{figure*}\n\nWhen measuring the exponential slopes of the $M_{R}$ distributions as\na function of the $R$ threshold, the correlations due to events\nsatisfying multiple $R$ threshold requirements are neglected.  The\neffect of these correlations on the measurement of the slopes is\nstudied by using pseudo-experiments and is found to be negligible.\n\nTo measure the shape of the QCD background component in the lepton boxes, the corresponding\nlepton trigger data sets are used with the baseline selection and\nreversed lepton isolation criteria. The QCD background component in the lepton\nboxes is found to be negligible.\n\nThe $R$ threshold shapes the $M_R$ distribution in a simple therefore\npredictable way. Event selections with combined $R$ and $M_R$\nthresholds are found to suppress jet mismeasurements, including\nsevere mismeasurements of the electromagnetic or hadronic\ncomponent of the jet energy, or other anomalous calorimetric noise\nsignals such as the ones described in~\\cite{hcalnoise,ecalnoise}.\n\n\n\\subsection{\\texorpdfstring{\\PW+jets, \\cPZ+jets, and \\cPqt+X backgrounds}{W+jets, Z+jets, and t+X backgrounds}\\label{bg-prop}}\n\nUsing the muon (MU) and electron (ELE) control boxes defined in\nSection~\\ref{intro}, $M_R$ intervals dominated by $\\PW(\\ell\\nu)$+jets\nevents are identified for different $R$ thresholds.  In both simulated\nand data events, the $M_{R}$ distribution is well described by two\nindependent exponential components. The first component of\n$W(\\ell\\nu)$+jets corresponds to events where the highest \\PT\nobject in one of the megajets is the isolated electron or muon;\nthe second component consists of events where the leading object\nin both megajets is a jet, as is typical also for the $t$+X background\nevents. The first component of $W(\\ell\\nu)$+jets can be measured\ndirectly in data, because it dominates over all other backgrounds in a control region\nof lower $M_R$ set by  the $R$ threshold.\nAt higher values of $M_R$, the\nfirst component of $W(\\ell\\nu)$+jets falls off rapidly,\nand the remaining background is instead dominated by the\nsum of  $t$+X and the second component of $W(\\ell\\nu)$+jets;\nthis defines a second control region of intermediate $M_R$\nset by the $R$ threshold.\n\nUsing these two control regions in a given box,\na simultaneous fit determines both exponential slopes along\nwith the absolute normalization of the first component of $W(\\ell\\nu)$+jets\nand the relative normalization of the sum of the second\ncomponent of $W(\\ell\\nu)$+jets  with the other backgrounds.\nThe $M_R$ distributions as a function of $R$ are shown in\nFig.~\\ref{fig:DATA_MU_slopes} (left).  The slope parameters\ncharacterizing the exponential behavior of the first\n$W(\\ell\\nu)$+jets component are shown in Fig.~\\ref{fig:DATA_MU_slopes}\n(right); they are consistent within uncertainties between the electron\nand muon channels. The values of the parameters $a$ and $b$ that\ndescribe the $R^{2}$ dependence of the slope are in good agreement\nwith the values extracted from simulated $W(\\ell\\nu)$+jets events.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{DATA_MUBOX_slopes2.pdf}\n\\includegraphics[width=0.49\\textwidth]{DATA_MUBOX_slopefit.pdf}\\\\\n\\includegraphics[width=0.49\\textwidth]{DATA_EGBOX_slopes.pdf}\n\\includegraphics[width=0.49\\textwidth]{DATA_EGBOX_slopefit.pdf}\n\\caption{(Left) $M_{R}$ distributions for different values of the $R$\n  threshold from data events selected in the MU (upper) and ELE\n  (lower) boxes. Dotted curves show the results of fits using two\n  independent exponential functions and an asymmetric Gaussian at low\n  $M_R$. (Right) The slope $S$ of the first exponential component\n  as a function of the square of the $R$ threshold in the MU (upper)\n  and ELE (lower) boxes. The dotted lines show the results of the fits\n  to the form $S = a + bR^{2}$.}\n\\label{fig:DATA_MU_slopes}\n\\end{center}\n\\end{figure*}\n\nThe  data\/MC  ratios $\\rho(a)_1^{\\mathrm{data\/MC}}$,  $\\rho(b)_1^{\\mathrm{data\/MC}}$\nof the  first component slope parameters $a$, $b$  measured in the MU and ELE boxes are thus combined\nyielding\n\n\\begin{eqnarray}\n\\rho(a)_{1}^{\\mathrm{data\/MC}} = 0.97 \\pm 0.02 ~~;~~\n\\rho(b)_{1}^{\\mathrm{data\/MC}} = 0.97 \\pm 0.02~~,\n\\end{eqnarray}\nwhere the quoted uncertainties are determined from the fits.\n\nThe ratios\n$\\rho^{\\mathrm{data\/MC}}$\nare taken as correction factors for the shapes of the\n$Z$+jets and $t+$X backgrounds as extracted from simulated samples\nfor the MU and ELE boxes; the same corrections are used\nfor the shape of the first component of $W(\\ell\\nu)$+jets as\nextracted from simulated samples for the\nhadronic (HAD) box.\n\nThe data\/MC correction factors for\nthe $\\cPZ(\\nu\\bar\\nu)$+jets and $\\cPqt$+X backgrounds in the HAD box, as well as\nthe second component of $\\PW(\\ell\\nu)$+jets in the MU, ELE, and HAD boxes,\nare measured in the  MU and ELE boxes\nusing a \\textit{lepton-as-neutrino} treatment of leptonic events. Here\nthe electron or muon is excluded from the megajet reconstruction,\nkinematically mimicking the presence of an additional neutrino.  With\nthe lepton-as-neutrino treatment in the MU and ELE boxes only one\nexponential component is observed both in data and in\n$\\PW(\\ell\\nu)$+jets simulated events.  In the simulation, the value of\nthis single exponential component slope is found to agree with the\nvalue for the second component of $\\PW(\\ell\\nu)$+jets obtained in\nthe default treatment.\n\nThe combined data\/MC correction factors measured using this\nlepton-as-neutrino treatment are\n\\begin{eqnarray}\n\\rho(a)_{2}^{\\mathrm{data\/MC}} = 1.01 \\pm 0.02 ~~;~~\n\\rho(b)_{2}^{\\mathrm{data\/MC}} = 0.94 \\pm 0.07.\n\\end{eqnarray}\nFor the final background prediction the magnitude of the relative\nnormalization between the two $\\PW(\\ell\\nu)$+jets components, denoted\n$f^{\\PW}$,  is determined from a binned maximum likelihood fit\nin the region 200 $< M_{R} < $ 400\\GeV.\n\n\\section{Results}\n\\subsection{Lepton box background predictions \\label{sec:LEPBOX}}\n\nHaving extracted the $M_{R}$ shape of the $\\PW$+jets and $\\cPZ$+jets\nbackgrounds, their relative normalization is set from the $\\PW$ and $\\cPZ$\ncross sections measured by CMS in electron and muon final\nstates~\\cite{EWK-PAS}.\nSimilarly, the  normalization of the \\ccbar\nbackground relative to  $\\PW$+jets  is taken from the  \\ttbar cross section  measured by CMS in the dilepton channel~\\cite{top}. The measured values of these cross sections are summarized below:\n\\begin{eqnarray}\n\\sigma(pp \\to {\\PW}X) \\times \\mathrm{B}(\\PW \\to \\ell\\nu) &=& 9.951 \\pm\n0.073~(\\mathrm{stat}) \\pm 0.280~(\\mathrm{syst}) \\pm\n1.095~(\\mathrm{lum})~ \\mathrm{nb}~,~ \\nonumber \\\\\n\\sigma(\\Pp\\Pp \\to {\\cPZ}X) \\times \\mathrm{B}(\\cPZ \\to \\ell\\ell) &=&  0.931 \\pm 0.026~(\\mathrm{stat}) \\pm 0.023~(\\mathrm{syst}) \\pm 0.102~(\\mathrm{lum})~ \\mathrm{nb}~,~ \\\\\n\\sigma(\\Pp\\Pp \\to \\ttbar) &=& 194 \\pm 72~(\\mathrm{stat}) \\pm 24~(\\mathrm{syst}) \\pm 21~(\\mathrm{lum})~ \\mathrm{pb}~.~\\nonumber\n\\end{eqnarray}\n\nFor an $R > 0.45$ threshold the QCD background is virtually\neliminated.  The region 125~$< M_{R}< 175\\GeV$ where the QCD contribution is\nnegligible and the $\\PW(\\ell\\nu)$+jets component is dominant is used to\nfix the overall normalization of the total background prediction.\nThe final background prediction in the ELE and MU boxes for $R > 0.45$ is shown in Fig.~\\ref{fig:ELEMUBOX}.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{ELEBOX_45.pdf}\n\\includegraphics[width=0.49\\textwidth]{MUBOX_R45.pdf}\n\\caption{The $M_R$ distributions with $R > 0.45$  in the\n  ELE (left) and MU (right) boxes for data (points) and backgrounds\n  (curves). The bands show the uncertainties of the background predictions.}\n\\label{fig:ELEMUBOX}\n\\end{center}\n\\end{figure*}\nThe number of events with $M_{R}>500\\GeV$ observed in data and the\ncorresponding number of predicted background events are given in\nTable~\\ref{tab:ELEMUBOX} for the ELE and MU boxes.\nAgreement between the predicted and observed yields is found.  The\n$p$-value of the measurement in the MU box is 0.1, given the predicted\nbackground (with its statistical and systematic uncertainties) and the\nobserved number of\nevents.\nA summary of the uncertainties entering the background measurements is\npresented in Table~\\ref{tab:LEPSYS}.\n\\begin{table}[ht!]\n\\caption{The number of predicted background events in the  ELE and MU\n  boxes for $R>$0.45 and $M_{R}>500\\GeV$ and the number of events\n  observed in data. \\label{tab:ELEMUBOX}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n & Predicted  & Observed \\\\\n\\hline\n\\hline\nELE box &  0.63 $\\pm$ 0.23 & 0 \\\\\n\\hline\nMU box  &  0.51 $\\pm$ 0.20 & 3 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table*}[ht!]\n  \\caption{Summary of the uncertainties on the background predictions\n    for  the ELE and MU boxes and their relative magnitudes. The range\n    in the Monte Carlo uncertainties is owing to the different\n    statistical precisions of the simulated background samples.\\label{tab:LEPSYS}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c||c|c|c|}\n\\hline\nParameter  & Description & Relative magnitude \\\\\n\\hline\n\\hline\nSlope parameter $a$ &  systematic bias from correlations in fits & 5\\% \\\\\n\\hline\nSlope parameter $b$ &  systematic bias from correlations in fits & 10\\% \\\\\n\\hline\nSlope parameter $a$ &  uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\nSlope parameter $b$ &  uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\n$\\rho(a)^{\\mathrm{data\/MC}}$  &   data fit & 3\\% \\\\\n\\hline\n$\\rho(b)^{\\mathrm{data\/MC}}$  &   data fit  & 3\\% \\\\\n\\hline\nNormalization  &  systematic+statistical component & 3--8\\% \\\\\n\\hline\n$f^{\\PW}$  &  extracted from fit ($\\PW$ only) & 30\\% \\\\\n\\hline\n$PW\/\\ttbar$ cross section ratio  & CMS measurements (top only) & 40\\% \\\\\n\\hline\n$\\PW\/\\cPZ$ cross section ratio  & CMS measurements ($\\cPZ$ only) & 19\\% \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Hadronic box background predictions\\label{sec:yesyoucan}}\n\nThe procedure for estimating the total background predictions in the\nhadronic box is summarized as follows:\n\\begin{itemize}\n\\item Construct the non-QCD  background shapes in $M_{R}$ using measured values\n  of $a$ and $b$ from simulated events, applying correction\n  factors derived from data control samples, and taking into account the $H_T$ trigger turn-on\n  efficiency.\n\\item Set the relative normalizations of the $\\PW$+jets, $\\cPZ$+jets, and $\\cPqt$+X\n  backgrounds using the relevant inclusive cross section measurements from\n  CMS (Eq. 10).\n\\item Set the overall normalization by measuring the event yields in\n  the lepton boxes, corrected for lepton reconstruction and\n  identification efficiencies. The shapes and normalizations of all the\n  non-QCD backgrounds are now fixed.\n\\item The shape of the QCD background is extracted, as described in\n  Section ~\\ref{sec:qcd}, and its normalization in the HAD box is\n  determined from a fit to the low-$M_{R}$ region, as described below.\n\\end{itemize}\n\n\nThe final hadronic box background prediction is calculated  from a binned\nlikelihood fit of the total background shape to the data in the\ninterval 80~$<M_{R}<400\\GeV$ with all background normalizations and\nshapes fixed, except for the following free parameters: i) the $H_T$\ntrigger turn-on shapes, ii) $f^{\\PW}$ as introduced in\nSection~\\ref{bg-prop}, and iii) the overall normalization of the QCD\nbackground.  A set of pseudo-experiments is used to test the overall\nfit for coverage of the various floated parameters and for systematic\nbiases.  A 2\\% systematic uncertainty is assigned to the high-$M_{R}$\nbackground prediction that encapsulates systematic effects related to\nthe fitting procedure.  Figure~\\ref{fig:HADBOX} shows the final hadronic box\nbackground predictions with all uncertainties on this prediction\nincluded for $R > 0.5$.  The observed $M_{R}$ distribution is consistent with\nthe predicted one over the entire $M_{R}$ range. The predicted and\nobserved background yields in the high-$M_{R}$ region are summarized\nin Table~\\ref{table:PRED}. A summary of the uncertainties entering\nthese background predictions is listed in Table~\\ref{tab:HADSYS}. A\nlarger $R$ requirement is used in the HAD box analysis due to the\nlarger background.\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{HADBOX_R50_lin.pdf}\n\\includegraphics[width=0.85\\columnwidth]{HADBOX_R50_log.pdf}\n\\caption{The $M_R$ distributions with $R > 0.5$ in the HAD box for\n  data (points) and backgrounds (curves) on (top) linear and (bottom)\n  logarithmic scales.  The bands show the uncertainties of the\n  background predictions. The corresponding distributions for SUSY\n  benchmark models LM1 \\cite{PTDR2} with $M_\\Delta = 597\\GeV$ and LM0\n  \\cite{alphaT} with $M_\\Delta = 400\\GeV$ are overlaid. }\n\\label{fig:HADBOX}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[ht!]\n\\caption{Predicted and observed yields for $M_{R }$$>500\\GeV$  with $R > 0.5$ in the HAD box.\n\\label{table:PRED}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c||c|\nc|}\n\\hline\n$M_{R}$   &  Predicted & Observed \\\\\n\\hline\n\\hline\n\\hline\n$M_{R} > 500\\GeV$ &  5.5 $\\pm$ 1.4 & 7 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table*}[ht!]\n\\caption{Summary of uncertainties entering the background predictions for the HAD box.\n\\label{tab:HADSYS}}\n\\smallskip\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nParameter  & Description & Relative magnitude \\\\\n\\hline\n\\hline\nSlope parameter $a$ &  systematic bias from correlations in fits & 5\\% \\\\\n\\hline\nSlope parameter $b$ &  systematic bias from correlations in fits & 10\\% \\\\\n\\hline\nSlope parameter $a$ &  uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\nSlope parameter $b$ &  uncertainty from Monte Carlo & 1--10\\% \\\\\n\\hline\n$\\rho(a)^{\\mathrm{data\/MC}}$  &  data fit & 3\\% \\\\\n\\hline\n$\\rho(b)^{\\mathrm{data\/MC}}$  & data fit & 3\\% \\\\\n\\hline\nNormalization  &  systematic+statistical component & 8\\% \\\\\n\\hline\nTrigger parameters  & systematic from fit pseudo-experiments & 2\\% \\\\\n\\hline\n$f^{\\PW}$  &  extracted from fit ($\\PW$ only) & 13\\% \\\\\n\\hline\n$W\/\\ttbar$ cross section ratio  & CMS measurements (top only) & 40\\% \\\\\n\\hline\n$\\PW\/\\cPZ$ cross section ratio  & CMS measurements ($\\cPZ$ only) & 19\\% \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\\section{Limits in the CMSSM Parameter Space}\nHaving observed no significant excess of events beyond the SM\nexpectations, we extract a model-independent 95\\% confidence level (CL) limit on\nthe number of signal events. This limit is then\ninterpreted in the parameter spaces of SUSY models.\n\nThe likelihood for the number of observed events $n$ is modeled as a\nPoisson function, given the sum of the number of signal events ($s$)\nand the number of background events. A posterior probability\ndensity function $P(s)$ for the signal yield is derived using Bayes theorem,\nassuming a flat prior for the signal and a log-normal prior for the\nbackground.\n\nThe model-independent upper limit is derived by integrating the\nposterior probability density function between 0 and $s^*$ so that\n$\\int_0^{s^*}P(s)ds=0.95$.  The observed upper limit in the hadronic\nbox is $s^{*}=8.4$ (expected limit 7.2 $\\pm$ 2.7); in the muon box\n$s^{*}=6.3$ (expected limit 3.5 $\\pm$ 1.1); and in the electron box\n$s^{*}=2.9$ (expected limit 3.6 $\\pm$ 1.1).  For 10\\% of the\npseudo-experiments in the muon box the expected limit is higher than\nthe observed.  The stability of the result was studied with different\nchoices of the signal prior. In particular, using the reference priors\nderived with the methods described in Ref.~\\cite{refprior}, the\nobserved upper limits in the hadronic, muon, and electron boxes\nare 8.0, 5.3, and 2.9, respectively.\n\nThe results can be interpreted in the context of the CMSSM, which is a\ntruncation of the full SUSY parameter space motivated by the minimal\nsupergravity framework for spontaneous soft breaking of\nsupersymmetry. In the CMSSM the soft breaking parameters are reduced\nto five: three mass parameters $m_0$, $m_{1\/2}$, and $A_0$ being,\nrespectively, a universal scalar mass, a universal gaugino mass, and a\nuniversal trilinear scalar coupling, as well as $\\tan\\beta$, the\nratio of the up-type and down-type Higgs vacuum expectation values,\nand the sign of the supersymmetric Higgs mass parameter\n$\\mu$. Scanning over these parameters yields models which, while not\nentirely representative of the complete SUSY parameter space, vary\nwidely in their superpartner spectra and thus in the dominant\nproduction channels and decay chains.\n\nThe upper limits are projected onto the ($m_0$, $m_{1\/2}$) plane by\ncomparing them with the predicted yields, and excluding any model if\n$s(m_{0},m_{1\/2})>s^{*}$.  The systematic uncertainty on the signal\nyield (coming from the uncertainty on the luminosity, the selection\nefficiency, and the theoretical uncertainty associated with the cross\nsection calculation) is modeled according to a log-normal prior. The\nuncertainty on the selection efficiency includes the effect of\njet energy scale (JES) corrections, parton distribution function (PDF)\nuncertainties~\\cite{Bourilkov:2006cj}, and the description of\ninitial-state radiation (ISR). All the effects are summed in\nquadrature as shown in Table~\\ref{tab:syst}. The JES, ISR, and PDF\nuncertainties are relatively small owing to the insensitivity of the\nsignal $R$ and $M_R$ distributions to these effects.\n\n\\begin{table}[ht!]\n  \\caption{Summary of the systematic uncertainties on the signal yield\n    and totals for each of the event boxes. For the CMSSM scan the NLO signal cross section uncertainty is included.  \\label{tab:syst}}\n\\centering\n\\smallskip\n\\begin{tabular}{|lccc|}\n\\hline\nbox & MU & ELE & HAD \\\\\\hline\n\\multicolumn{4}{|c|}{Experiment}\\\\\\hline\nJES & 1\\% & 1\\% & 1\\% \\\\\\hline\nData\/MC $\\epsilon$& 6\\% & 6\\% & 6\\% \\\\\\hline\n$\\mathcal{L}$\\cite{lumi-moriond} & 4\\% & 4\\% & 4\\% \\\\\n\\hline\\hline\n\\multicolumn{4}{|c|}{Theory}\\\\\\hline\nISR & 1\\% & 1\\% & 0.5\\% \\\\\\hline\nPDF & 3--6\\% & 3--6\\% & 3--6\\% \\\\ \\hline\nSubtotal & 8--9\\% & 8--9\\% & 8--9\\% \\\\\n\\hline\\hline\n\\multicolumn{4}{|c|}{CMSSM}\\\\\\hline\nNLO  & 16--18\\% & 16--18\\% & 16--18\\% \\\\\\hline\nTotal & 17--19\\% & 17--19\\% & 17--19\\% \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\nThe observed limits from the ELE, MU, and HAD boxes are shown in\nFigs.~\\ref{fig:ELE_LIMIT},~\\ref{fig:test}, and~\\ref{fig:HAD_LIMIT},\nrespectively, in the CMSSM\n($m_{0}$, $m_{1\/2}$) plane for the values $\\tan\\beta = 3$, $A_{0} = 0$,\n$\\operatorname{sgn}(\\mu) = +1$,\n together with the 68\\% probability band for the\nexpected limits, obtained by applying the same procedure to an ensemble of\nbackground-only  pseudo-experiments. The band is computed\naround the median of the limit distribution.  Observed limits are also shown\nin Figs.~\\ref{fig:ELE_LIMIT10}\n--\\ref{fig:HAD_LIMIT10} in the CMSSM ($m_{0}$, $m_{1\/2}$) plane for the\nvalues $\\tan\\beta = 10$, $A_{0} = 0$,\n$\\operatorname{sgn}(\\mu) = +1$, and in\nFigs.~\\ref{fig:ELE_LIMIT50}--\\ref{fig:HAD_LIMIT50} for the values\n$\\tan\\beta = 50$, $A_{0} = 0$,\n$\\operatorname{sgn}(\\mu) = +1$.\n\nFigure ~\\ref{fig:SMS} shows the same result in terms of 95\\% CL upper\nlimits on the cross section as a function of the physical masses for two benchmark\nsimplified models~\\cite{Alwall-1,Alwall-2,Sanjay,RA2}: four-flavor\nsquark pair production  and gluino pair production. In\nthe former, each squark decays to one quark and the LSP, resulting in\nfinal states with two jets and missing transverse energy, while in the\nlatter each gluino decays directly to two light quarks and the LSP,\ngiving events with four jets and missing transverse energy.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_ELEBOX_tanB3.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=3$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the ELE box selection ($R >\n  0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits. The area labeled $\\tilde{\\tau}$=LSP\n  is the region of the parameter space where the LSP is a\n  $\\tilde{\\tau}$ and not the lightest neutralino.}\n\\label{fig:ELE_LIMIT}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_MUBOX_tanB3.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=3$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the MU box selection ($R >\n  0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits. }\n\\label{fig:test}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_HADBOX_tanB3.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=3$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the HAD box selection ($R >\n  0.5$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits.  }\n\\label{fig:HAD_LIMIT}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_ELEBOX_tanB10.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=10$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the ELE box selection ($R >\n  0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits.  }\n\\label{fig:ELE_LIMIT10}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_MUBOX_tanB10.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=10$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the MU box selection ($R >\n  0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits.  }\n\\label{fig:MU_LIMIT10}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_HADBOX_tanB10.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=10$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the HAD box selection ($R >\n  0.5$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits.  }\n\\label{fig:HAD_LIMIT10}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_ELEBOX_tanB50.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% CL\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=50$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the ELE box selection ($R >\n  0.45$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits.  }\n\\label{fig:ELE_LIMIT50}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.85\\columnwidth]{CMSSM_HADBOX_tanB50.pdf}\n\\caption{Observed (solid curve) and expected (dot-dashed curve) 95\\% L\n  limits in the ($m_{0}$, $m_{1\/2}$) CMSSM plane with $\\tan\\beta=50$,\n  $A_{0} = 0$, $\\operatorname{sgn}(\\mu) = +1$ from  the HAD box selection ($R >\n  0.5$, $M_{R} > 500\\GeV$). The $\\pm$ one standard deviation\n  equivalent variations in the uncertainties are shown as a band\n  around the expected limits.  }\n\\label{fig:HAD_LIMIT50}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth]{T2_limit.pdf}\n\\includegraphics[width=0.7\\columnwidth]{T1_limit.pdf}\n\\caption{Upper limits on two simplified models: di-squark production (top)\n  resulting in a 2-jet + \\ETm final state and di-gluino (lower)\n  production resulting in a 4-jet + \\ETm final state. The\n  shade scale indicates the value of the cross section excluded at 95\\%\n  CL  for each value of $m_{\\text{LSP}}$ and $m_{\\text{gluino}}$\n  or $m_{\\text{squark}}$. The solid and dashed contours indicate the\n  95\\% CL limits assuming the NLO cross section and its variations\n  up and down by a factor of three.}\n\\label{fig:SMS}\n\\end{center}\n\\end{figure}\n\\section{Summary}\nWe performed a  search for squarks and gluinos using a data\nsample of 35\\pbinv integrated luminosity from pp collisions at\n$\\sqrt{s} = 7\\TeV$, recorded by the CMS detector at the LHC.  The\nkinematic consistency of the selected events was tested against the\nhypothesis of heavy particle pair production using the dimensionless\nrazor variable $R$ related to the missing transverse energy \\ETm,\nand $M_R$, an event-by-event indicator of the heavy particle\nmass scale.  We used events with large $R$ and high $M_R$ in inclusive\ntopologies.\n\nThe search relied on predictions of the SM backgrounds determined from\ndata samples dominated by SM processes.  No significant excess over\nthe background expectations was observed, and model-independent upper\nlimits on the numbers of signal events were calculated.\nThe results were presented in the ($m_0$, $m_{1\/2}$) CMSSM parameter\nspace. For simplified models the results were given as  limits on the\nproduction cross sections as a function of the squark, gluino, and LSP\nmasses.\n\nThese results demonstrate the strengths of the razor analysis\napproach; the simple exponential behavior of the various SM\nbackgrounds when described in terms of the razor variables is useful\nin suppressing these backgrounds and in making reliable estimates from\ndata of the background residuals in the signal regions. Hence, the\nrazor method provides an additional powerful probe in searching for\nphysics beyond the SM at the LHC.\n\n\n\\section*{Acknowledgments}\n\n\n\n\\hyphenation{Bundes-ministerium Forschungs-gemeinschaft Forschungs-zentren} We wish to congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine. We thank the technical and administrative staff at CERN and other CMS institutes. This work was supported by the Austrian Federal Ministry of Science and Research; the Belgian Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport; the Research Promotion Foundation, Cyprus; the Estonian Academy of Sciences and NICPB; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucl\\'eaire et de Physique des Particules~\/~CNRS, and Commissariat \\`a l'\\'Energie Atomique et aux \\'Energies Alternatives~\/~CEA, France; the Bundesministerium f\\\"ur Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Office for Research and Technology, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Korea; the Lithuanian Academy of Sciences; the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Science and Innovation, New Zealand; the Pakistan Atomic Energy Commission; the State Commission for Scientific Research, Poland; the Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia, Portugal; JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); the Ministry of Science and Technologies of the Russian Federation, and Russian Ministry of Atomic Energy; the Ministry of Science and Technological Development of Serbia; the Ministerio de Ciencia e Innovaci\\'on, and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the National Science Council, Taipei; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.\n\nIndividuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Associazione per lo Sviluppo Scientifico e Tecnologico del Piemonte (Italy); the Belgian Federal Science Policy Office; the Fonds pour la Formation \\`a la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); and the Council of Science and Industrial Research, India.\n\\cleardoublepage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThere are several indications by now from angle-resolved photoemission spectroscopy (ARPES) that in underdoped cuprates a gap opens at the Fermi surface in the diagonal (nodal) direction~\\cite{Drachuck,AHarter,vishik2012,razzoli2013evolution,peng2013disappearance}. In La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) this nodal gap (NG) extends to $x=8\\%$. At doping around $x=12.5\\%$ samples develop a charge-density-wave (CDW) below $T\\approx100$~K \\cite{croft14}. Traces of antiferromagnetism (AFM) in the form of spin-density-waves (SDW) \\cite{matsuda2002} or spin-glass \\cite{Niedermayer98} appear at doping up to $x=12.5\\%$ and temperatures $T\\approx10$~K. It is therefore natural to speculate that one of these symmetry breaking phenomena is responsible for the opening of a nodal gap. In this work, we would like to clarify which one is the most likely. Our strategy is to carefully examine a sample which is known to at least have both AFM and SDW order, and opens a nodal gap at low temperatures. The sample is LSCO with $x=1.92\\%$~\\cite{Drachuck}.\n\nPrevious neutron diffraction measurements on LSCO $x=1.92\\%$ \\cite{Drachuck} showed a magnetic Bragg peak at the AFM wave vector $\\textbf{Q}_{AF}$ below $T=140$~K, and two satellites that stand for static SDW order (on top of the AFM one). The satellites appear below $T=30$~K. Like in Matsuda et al. ~\\cite{matsuda2002}, there are two domains in the sample. We focus on one of them, in which the AFM peak is observed when scanning near (1,0,0), with no contribution from SDW. In contrast, the SDW peaks are observed when scanning near (0,1,0), with no contribution from the AFM peak. Neutron scattering detects the component of spin fluctuations perpendicular to the momentum transfer $\\textbf{q}$ ~\\cite{squires2012introduction}. Hence, the SDW fluctuations are perpendicular to the AFM order. ARPES measurements on the same sample found that a nodal gap opens below $T_{NG}=45$~K~\\cite{Drachuck}. Even though there is a temperature mismatch between the NG and SDW appearance, the two phenomena might be related. Moreover, CDW in LSCO $x=1.92\\%$ is expected to be very weak \\cite{capati2015}, and indeed this sample is out of the CDW dome ~\\cite{croft14,Hucker14}. Therefore, \\textit{a priori}, CDW is not expected to generate the nodal gap.\n\nHere we add to the available ARPES and neutron diffraction data, inelastic neutron scattering (INS) and x-ray diffraction data on the same piece of LSCO $x=1.92\\%$. We show that the fluctuating SDW amplitude of the frequency where it is the strongest, decreases at a temperature equal to $T_{NG}$ within experimental error. In addition, we could not find any indications for CDW in our sample. We argue that these findings explain the previously measured $15$~K discrepancy between the SDW freezing and the opening of a NG, and tie the latter to fluctuating SDW.\n\n\n\\begin{figure*}[h!t]\n\t\\begin{center}\n\t\t\\includegraphics[trim=0cm 0cm 0.5cm 0cm ,clip=true,width=18cm]{dispersion.png}\n\t\\end{center}\n\t\\caption{\\textbf{SDW dispersion.}\n\t\tFalse color map of normalized intensity as a function of neutron energy transfer $\\hbar \\omega$ and momentum transfer $\\textbf{q}$ at $T=50$~K (a) and $T=2$~K (b). The raw data is interpolated. The black horizontal line in panel (b) demonstrates a constant energy cut along which the intensity is integrated and plotted in Fig.~\\ref{fig:edep}. Dashed white lines in panel (b) represent cuts along which the background is determined. The black symbols indicate the center of the $\\hbar \\omega =0$ and $2$~meV peaks demonstrated in Fig.~\\ref{fig:qdep}, and make the bottom part of an hourglass.}\n\t\\label{fig:dispersion}\n\\end{figure*}\n\n\nThe neutron experiment was performed at Rita-II, the cold neutrons triple axis spectrometer at the Paul Scherrer Institut. Throughout this paper, we work in orthorhombic notation, with cell parameters $a=5.344$~\\AA, $b=5.421$~\\AA\\  and $c=13.14$~\\AA\\  at $T=2$~K. In this notation, the tetragonal 2D $\\textbf{Q}_{AF}=\\left(1\/2,1\/2,0\\right)$ is equivalent to $\\left( 0,1,0\\right)$ in reciprocal lattice units (r.l.u) of $2\\pi\/a$. More information is available in the Methods section. In Fig.~\\ref{fig:dispersion} we present a false color map of neutron counts versus energy transfer $\\hbar \\omega$ and momentum transfer $\\textbf{q}$. The raw data, in this figure alone, is interpolated for presentation purpose. Data is presented at two temperatures, $2$~K and $50$~K, which are below and above the freezing temperature of the incommensurate magnetic order of $30$~K ~\\cite{Drachuck}. In both cases, strong intensity is observed at $\\hbar \\omega=0$. This is due to high order contamination of the incoming beam scattering from a nuclear Bragg peak at $(0,2,0)$, despite the use of Br filter. Around $(0,1,0)$, the intensity extends to energy transfers as high as 8 meV for both temperatures, in a cone shape, which is in fact a poorly-resolved bottom part of an hourglass. This will be demonstrated subsequently. The scattering intensity is stronger at elevated temperatures. Interestingly, at $T=2$~K spectral weight is missing at low energies, suggesting the presence of a soft gap for spin excitations. A similar spectrum, including the gap, was observed at the fully developed hourglass dispersion of La$_{1.875}$Ba$_{0.125}$CuO$_4$~\\cite{tranquada2004}, La$_{1.88}$Sr$_{0.12}$CuO$_4$ ~\\cite{matsuda2008,Romer13}, and La$_{1.6}$Sr$_{0.4}$CoO$_4$ ~\\cite{drees2013hour}. \n\nq-scans at specific constant energies at $T=2$~K are presented in Fig.~\\ref{fig:qdep}, showing the evolution of the SDW peaks with energy transfer. The intensities are shifted vertically for clarity. At $\\hbar \\omega=0.6$~meV, some intensity is detected around $(0,1,0)$ above the background. However, this could stem from the tail of the high order contamination. At $\\hbar \\omega=2$~meV two clear peaks appear \n\nFor fitting, the instrument was modeled using Popovici ResCal5 ~\\cite{popovici1975resolution}, and the resolution was calculated. Black horizontal lines in Fig.~\\ref{fig:qdep} represent the q-resolution at each energy. This was taken into account as a constant width Gaussian at each energy, which was convoluted with a Lorentzian (Voigt function). The fit with two Voigt functions is demonstrated in Fig.~\\ref{fig:qdep} by solid lines. The fit to the $\\hbar \\omega=2$~meV data indicates a peak separation of 0.04 r.l.u. The same separation is found in the elastic peaks ~\\cite{Drachuck}, as demonstrated in the inset. The peaks centers are illustrated in Fig.~\\ref{fig:dispersion}(b) by the solid points. The static and dynamic SDW correlation lengths, determined from the peaks width, are $85\\pm12~\\AA$ and $44\\pm5~\\AA$ respectively. With increasing energy to $4$~meV and then to $6$~meV, the two peaks are no longer resolved. However, the measured peak is asymmetric because of the two underlying incommensurate peaks coming closer together. At $8$~meV the intensity diminishes. This behavior reminds two ``legs\" dispersing downwards from some crossing energy as in the hourglass.\n\n\\begin{figure}[tbph]\n\t\t\\includegraphics[trim=1cm 2cm 1cm 2.5cm ,clip=true,width=\\columnwidth]{qdep.pdf}\n\n\t\\caption{\\textbf{Evolution of the SDW peaks with energy at $T=2$~K.}\n\t\tMomentum scan along k centered at (0,1,0) for different energy transfers at $T=2$~K. Scans are shifted consecutively by $2.5\\times10^{-5}$ counts\/monitor for clarity. Inset: SDW elastic peaks for the same $\\textbf{q}$ scan also at $T=2$~K. Background from higher temperature was subtracted. For energies of $\\hbar \\omega=0$ and $2$~meV, a sum of two Voigt functions is fitted to the data (solid black lines). The peak separation for $\\hbar \\omega=2$~meV is 0.04 r.l.u, as in the $\\hbar \\omega=0$ case (see inset). Black horizontal lines represents the instrumental resolution.   \n\t}\n\t\\label{fig:qdep}\n\\end{figure}\n\nTo further investigate the inelastic behavior, we sum the intensity over $\\textbf{q}$ at constant energy cuts. The horizontal line in Fig.~\\ref{fig:dispersion}(b) presents one such cut. Background contribution is estimated from the data along the dashed diagonal lines in Fig.~\\ref{fig:dispersion}(b), and subtracted. Fig.~\\ref{fig:edep} presents the background subtracted \\textbf{q}-integrated intensity versus energy transfer $ \\langle I \\rangle (\\omega)=\\sum_\\textbf{q} I(\\textbf{q},\\omega)$, starting from $\\hbar\\omega = 0.15$ meV to avoid the high intensity elastic peak. At $T=50$~K, $\\langle I \\rangle (\\omega)$ monotonically grows as the frequency decreases. In contrast, at $T=2$~K, $\\langle I \\rangle (\\omega)$ reaches a maximum at some $\\hbar\\omega_{max}$ between 2 and 3~meV, and drops towards $\\hbar\\omega=0$, although residual elastic scattering intensity is observed near $\\hbar\\omega=0$. Measurements on La$_{2-x}$Ba$_x$CuO$_4$ with $0.0125\\le x \\le 0.035$  which were limited to energies below 1meV agree with our results ~\\cite{Wagman2013}. This plot demonstrates more clearly the aforementioned soft gap in spin excitations which develops at low temperatures.\n\n\\begin{figure}[tbph]\n\n\t\t\\includegraphics[trim=1cm 1cm 1cm 1.2cm ,clip=true,width=\\columnwidth]{E_dependence.pdf}\n\n\t\\caption{\\textbf{q-integrated intensity vs. neutron energy transfer at low (2 K) and high (50 K) temperatures.}\n\t\tIntegrated intensity is calculated for each energy as sum of the counts over $\\textbf{q}$ along horizontal lines like the one shown in Fig.~\\ref{fig:dispersion}. Background is estimated from the counts along the two dashed lines shown in Fig.~\\ref{fig:dispersion}(a) and subtracted from the raw data.\n\t}\n\t\\label{fig:edep}\n\\end{figure}\n\n\nWe summarize the available data on LSCO $x=1.92\\%$ in Fig.~\\ref{fig:tdep}(a). In this figure we show the temperature dependence of the $\\textbf{q}$-integrated scattering intensity at three different energies. The data at $\\hbar \\omega=0$ is taken from Ref.~\\cite{Drachuck} and multiplied by $2\\times 10^{-3}$ for clarity. It shows that a long range static SDW appears at a temperature of $30$~K. The intensity at $\\hbar \\omega=0.6$~meV increases as the temperature is lowered, peaks at $38$~K, and then decreases. This result demonstrates that dynamically fluctuating SDW at $\\hbar\\omega >0$ diminishes upon cooling before long range static incommensurate order develops. The same effect, although less sharp, is observed for $\\hbar \\omega=2$~meV at $45$~K. \n\nFigure ~\\ref{fig:tdep}(b) depicts the temperature dependence of the nodal gap from Ref.~\\cite{Drachuck} as measured by ARPES. This gap opens at $T_{NG}=45$~K, which is the same temperature where the spectral density at $\\hbar\\omega_{max}$ begins to diminish. The maximum electronic gap value $\\Delta$ agrees with isolated dopant-hole bound state calculations~\\cite{sushkov2005}. We note that $\\hbar\\omega_{max}$ and $k_B T_{NG}$ are of the same order of magnitude. Our result indicates a strong link between the dynamically fluctuating SDW and the nodal gap. \n\n\n\\begin{figure}[tbph]\n\t\\begin{center}\n\t\t\\includegraphics[trim=0cm 3cm 0cm 1cm ,clip=true,width=\\columnwidth]{Tdep.pdf}\n\t\\end{center}\n\t\\caption{\\textbf{Temperature dependence of all experimental parameters.}\n\t\t(a) Elastic and inelastic incommensurate SDW intensities at different energies from neutron scattering. (b) ARPES measurement of the nodal gap at $k_F$ \\cite{Drachuck}. The dashed vertical line emphasizes the fact that the nodal gap opens when the amplitude of dynamic spin fluctuations at $\\hbar\\omega\\approx~2$meV decreases.\n\t}\n\t\\label{fig:tdep}\n\\end{figure}\n\nIn order to investigate whether CDW plays a role in the nodal gap \\cite{Berg08}, we conducted a search for CDW in this sample by two different methods: off resonance x-ray diffraction (XRD) and resonance elastic x-ray scattering (REXS). The experiments were done at PETRA III on the P09 beam-line and at BESSY on the UE46-PGM1 beam-line, respectively. In REXS, the background subtraction is not trivial, so we only present here our XRD data. Nonetheless, the final conclusion from both methods is the same.\n\n\nIn Fig.~\\ref{fig:cdw} we show results from LSCO samples with $x=1.92 \\%$, $x=6.0\\%$, and La$_{2-x}$Ba$_x$CuO$_4$ (LBCO) $x=12.5\\%$. The data sets are shifted vertically for clarity. The LBCO sample is used as a test case, since it has well established CDW and presents strong diffraction peaks. The measurements were taken at $7$~K and at $70$~K, which are below and above the CDW critical temperature of LBCO~\\cite{Tranquada2008cdw}. We performed two types of scans: a ``stripes\" scan along $(0,\\delta q,8.5)$ direction and a ``checkerboard\" scan along $(\\delta q,\\delta q,8.5)$ direction. We chose to work at $l=8.5$ to minimize contribution from a Bragg peak at $l=8$ or $l=9$. For LBCO at $T=7$~K, there is a clear CDW peak at $\\delta q=\\pm 0.24$ in the ``checkerboard\" scan, which is absent at high temperatures. In contrast, for the LSCO samples there is no difference between the signal at high and low temperatures. Since $\\delta q$ of the CDW peak depends on doping, in our sample it is expected to be close to $\\delta q=0$, where a tail of the Bragg peak could potentially obscure the CDW peak. Arrows in Fig.~\\ref{fig:cdw} show where we might expect the CDW peaks, should they appear, based on linear scaling with doping. These positions are out of the $\\delta q=0$ peak tail, and not obscured. Thus, although we are in experimental conditions appropriate to find a CDW, it is not observed within our sensitivity. In fact, CDW is even absent at higher doping as demonstrated by our experiment with LSCO $x=6\\%$ sample. We observed the same null-result with the REXS experiment. It is important to mention that hourglass excitations with no stripe-like CDW were observed previously \\cite{drees2013hour}.\n\nOur main results are as follows: we find the bottom part of an hourglass dispersion inside the AFM phase of LSCO. The hourglass does not start from zero energy, but has a soft gap from the static SDW order. A CDW order seems to be absent in our sample. Upon cooling the system, a nodal gap in electronic excitations opens just when the strongest spin excitations start to diminish. It is therefore sufficient for the SDW fluctuations to slow down without completely freezing out in order to modify the band structure.\n\n\n\\begin{figure}[tbph]\n\t\\begin{center}\n\t\t\\includegraphics[trim=1.5cm 1cm 3cm 0cm ,clip=true,width=\\columnwidth]{NoCDWL.pdf}\n\t\\end{center}\n\t\\caption{\\textbf{Hard x-ray diffraction on three different samples: }LSCO with $x=1.92\\%$ and $x=6\\%$, and LBCO $x=12.5\\%$. Scans are done in two different orientations and two different temperatures. CDW is detected only in LBCO. }\n\t\\label{fig:cdw}\n\\end{figure}\n\n\\section*{Methods}\n\nFor the Neutron scattering experiment, the sample was mounted on aluminum holder covered with Cd foils, and oriented in the (h,k,0) scattering plane. A Be filter was used to minimize contamination from high order monochromator Bragg reflections.  The scattered neutrons are recorded with a nine bladed graphite analyzer. All the blades are set to scatter neutrons at the same final energy of 5 meV, and direct the scattered neutrons through an adjustable radial collimator to different predefined areas on a position sensitive detector ~\\cite{bahl2006inelastic,lefmann2006realizing}. This monochromatic q dispersive mode allows for an efficient mapping of magnetic excitations with an excellent q resolution.\n\nTwo types of scans were used: I) energy scan, in which the incoming neutrons energy is swept, and the $\\textbf{q}$ information is embedded in the position of each blade. II) momentum scan, in which the incoming neutrons energy is fixed, the nine blades cover a small window in $\\textbf{q}$, and the entire window is scanned. The contribution to a given $\\textbf{q}$ is a weighted sum from the different blades.\n\nDespite the Be filter, some contribution from the nuclear structure is unavoidable. For elastic scattering, this contribution survives to higher temperature than does the magnetic part, and therefore can be easily subtracted. For inelastic scattering, the contribution from phonons could not be subtracted, but it is expected to vary slowly with temperature close to the magnetic phase transitions. Therefore, all features in this scattering experiment which show abrupt temperature dependence around and below $T=50$~K are associated with the electronic (magnetic) system.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\section{Introduction}\n\\subsection{The model}\nThe aim of the present paper is to study the ergodicity of a piecewise\ndeterministic Markov process (PDMP) linked to Lotka-Volterra type dynamics.\nThese lines can be seen as a companion paper to~\\cite{lotka} since we go one\nstep further in the description of different regimes of the process and the support of the\ninvariant measures. Let us  first provide  an overview of the \nmain results in~\\cite{lotka} before stating our contribution.\n\nFor a given set of positive parameters $\\mathcal{E}=(a,b,c,d,\\alpha,\\beta)$,\nconsider the Lotka-Volterra differential system in $\\dR_+^2$, given by \n\\begin{equation}\n  \\label{eq:lv}\n  \\begin{cases}\n  \\dot{x}=\\alpha x (1-ax-by) \\\\\n  \\dot{y}=\\beta y (1-cx-dy) \\\\\n  (x_0,y_0)\\in\\dR_+^2. \n  \\end{cases}\n\\end{equation}\nThis system modelizes the evolution of the populations of two species ($x$ of\ntype~$\\textbf{x}$ and $y$ of type~$\\textbf{y}$).  The populations\ngrow logistically --- as encoded by the $\\alpha x(1-ax)$ and $\\beta y(1-dy)$ terms --- and\ncompete with each other, which  gives rise to the cross terms $\\alpha b xy$\nand $\\beta c xy$.  We denote by $F_\\mathcal{E}$ the associated  vector\nfield: $(\\dot x,\\dot y)=F_{\\mathcal{E}}(x,y)$. In the sequel, the variable $z$\nstands for $(x,y)$ and we will sometimes write $F_{\\mathcal{E}}(z)$ instead of\n$F_{\\mathcal{E}}(x,y)$. This ODE system, taken alone,  is easy to analyze. \nIn particular it has only a finite number of equilibrium points, towards which \nthe dynamics converges. These equilibria may be on the coordinate axes --- meaning\nthat one of the species gets extinct --- or inside the positive quadrant. The \nposition and nature of the equilibria turn out to depend only on the signs of\n$c-a$ and $d-b$. A complete picture  will be given in Section~\\ref{sec:notation};\nlet us note already that when $a<c$ and $b<d$,\nthe point $(1\/a,0)$ attracts any path starting in $(0,+\\infty)^2$. We say \nin this case that\nthe environnement is \\emph{favorable to species $\\textbf{x}$}\\,; it leads to the\nextinction of species $\\textbf{y}$.    \n\nConsider now two such systems, labeled $0$ and $1$, with environments \n$\\mathcal{E}_i=(a_i,b_i,c_i,d_i,\\alpha_i,\\beta_i)_{i=0,1}$. We make the following\nstanding assumption: \n\\begin{ass} The environments $\\cE_0$ and $\\cE_1$ are both \n  favorable to $\\textbf{x}$: $a_i<c_i$ and $b_i<d_i$ for $i=0,1$. \n  In particular, taken alone, each system leads to the extinction of the second\n  species, that is, $y_t$ converges to $0$ and $\\limsup x_t>0$ as soon as\n  $x_0>0$.  \n\\end{ass}\n\nFinally, introduce the random process obtained by switching between \nthese two deterministic dynamics, at rates $(\\lambda_i)_{i=0,1}$. More precisely, \nwe consider the process $(Z,I)$ on $\\dR^2\\times\\BRA{0,1}$ driven by the \ninfinitesimal generator \n\\[\nLf(z,i)=F_{\\mathcal{E}_i}(z)\\cdot\\nabla_z f(z,i) +\\lambda_i (f(z,1-i)-f(z,i)).  \n\\]\nIn other words, $I$ jumps from $i$ to $1-i$ after a random time with an exponential distribution \nof parameter $\\lambda_i$, and while $I_t$ is equal to $i$,\n$Z$ evolves deterministically by $\\dot Z_t=F_{\\mathcal{E}_i}(Z_t)$. The coordinates \nof $Z_t$ are denoted by $X_t$ and $Y_t$. We refer to~\\cite{lotka} for a detailed biological \nmotivation.  \n\nIt is shown in \\cite{lotka} that, depending on the environments $\\cE_0$, $\\cE_1$ \nand the jump rates $\\lambda_0$, $\\lambda_1$, \none of the following four things occur almost surely: \n\\begin{itemize}\n\\item extinction of species $\\textbf{x}$: $X_t \\to 0$ and $\\limsup Y_t>0$,\n\\item extinction of species \\textbf{y}: $Y_t\\to 0$ and $\\limsup X_t>0$, \n\\item extinction of one of the two species,  chosen randomly,\n\\item persistence: the empirical occupation measure (and, in many cases, the\n distribution) of ${(X_t,Y_t)}_{t\\geq 0}$ \nconverges to a probability measure on $(0,+\\infty)^2$ that is absolutely continuous with \nrespect to the Lebesgue measure. \n\\end{itemize}\nMoreover, one or more of these regimes may occur when the jump \nrates $(\\lambda_0,\\lambda_1)$ vary, the environments $(\\mathcal{E}_0,\\mathcal{E}_1)$ \nbeing fixed. Similar surprising behaviors for switched processes have\nbeen studied for linear ODEs in \\cite{BLMZ1,mattingly}.\n\n\n\\subsection{The frequent jumps asymptotics and the averaged vector field}\nRecall that $\\lambda_0$, $\\lambda_1$ are the jump rates from one environment\nto the other. \nNote that the index process $(I_t)_{t\\geq 0}$ is Markov by itself, \nand its invariant measure is a Bernoulli distribution with parameter \n$\\lambda_0\/(\\lambda_0+\\lambda_1)$. As a consequence, \nit will be convenient to choose the alternative\nparametrization\n\\begin{equation}\\label{eq:para_st}\n(s,t)\\in [0,1]\\times(0,+\\infty)\\mapsto (st,(1-s)t) \n\\end{equation}\nfor the jump rates, that is, let $t$ be the sum $\\lambda_0+\\lambda_1$ \nand $s$ be the ratio $\\lambda_0\/(\\lambda_0+\\lambda_1)$. \n\\begin{rem}[Length of interjump times I]\\label{rem:st}\nNotice that the expectations of the interjump times are given by $(st)^{-1}$ and $((1-s)t)^{-1}$. \nIf $t$ is small, the jumps are rare and the jump times are large in average; as \n$t$ grows the jumps become more frequent and the jump times shorter on average. \n\\end{rem}\nAs the parameter $t$ goes to infinity --- the frequent jumps asymptotics --- it\ncan be shown that the stochastic process ${(Z_t)}_{t\\geq 0}$ converges to the \nsolution of \n\\[\n\\dot{z}_t=F_s(z_t) \\quad\\text{where}\\quad \nF_s=(1-s)F_{\\mathcal{E}_0}+sF_{\\mathcal{E}_1}. \n\\]\nAs noticed in \\cite{lotka}, for any $s\\in [0,1]$, the vector field \n$F_s$ is the Lotka-Volterra system associated to the \"averaged\" environment \n$\\mathcal{E}_s=(a_s,b_s,c_s,d_s,\\alpha_s,\\beta_s)$ with \n\\begin{align}\n\\alpha_s &= (1-s)\\alpha_0+s\\alpha_1,  \n&\na_s &= \\frac{(1-s)\\alpha_0a_0+s\\alpha_1a_1}{\\alpha_s},\n&\nb_s &= \\frac{(1-s)\\alpha_0b_0+s\\alpha_1b_1}{\\alpha_s},\\label{eq:def_alphas}\n\\\\\n\\beta_s &= (1-s)\\beta_0+s\\beta_1,\n&\nc_s &= \\frac{(1-s)\\beta_0c_0+s\\beta_1c_1}{\\beta_s},\n&\nd_s &= \\frac{(1-s)\\beta_0d_0+s\\beta_1d_1}{\\beta_s}.\\label{eq:def_betas}\n\\end{align}\nRecall that by our standing assumption, $a_i<c_i$ and $b_i<d_i$ for $i=0,1$. \nThe key point is that \\emph{these inequalities may be reversed} \nfor the averaged environment $\\cE_s$; in some situations \n$\\cE_s$ may even be \\emph{unfavorable to species~$\\textbf{x}\\,$!} \n\\begin{defi}[Critical parameter regions]\nTwo critical (possibly empty) parameter regions are defined in~\\cite{lotka} by: \n\\begin{equation}\\label{eq:def-I}\nI=(s_1,s_2)=\\BRA{ s\\in[0,1], a_s > c_s} \n\\quad\\text{and}\\quad \nJ=(s_3,s_4)=\\BRA{ s\\in[0,1], b_s > d_s}. \n \\end{equation}\n\\end{defi}\n The fact that $I$ and $J$ are indeed intervals is obvious from the definition\n of $a_s$, $b_s$, $c_s$ and $d_s$ by~\\eqref{eq:def_alphas}\n and~\\eqref{eq:def_betas}. As we have seen, the relevance of these\n intervals stems from the fact that they correspond to different types\n for the averaged environment $\\cE_s$. For example, \n the vector field $F_s$ always has two stationary points on the coordinate axes, \n but their nature vary: \n\\begin{itemize}\n\\item the stationary point $(1\/a_s,0)$ is a well if $s\\notin I$ and a saddle point if $s\\in I$, \n\\item the stationary point $(0,1\/d_s)$ is a saddle point if $s\\notin J$ and a well if $s\\in J$.  \n \\end{itemize}\n\\begin{rem}\\label{rem:adiff}\nNotice that if $a_0=a_1$ then the interval $I$ is empty. In the sequel we will focus on the \ncase when $a_0\\neq a_1$ and without loss of generality we will assume that $a_0<a_1$. \n \\end{rem}\nMore details and  a complete description of the phase portrait of the Lotka-Volterra system\n will be given below (see Section~\\ref{sec:notation}). \n\n \\subsection{Invasion rates}\n\nIt is shown in~\\cite{lotka} that the regime of the process $(Z,I)$ is determined by the \nthe signs of two quantities, called \\emph{invasion rates}, $\\Lambda_\\textbf{x}$ and \n$\\Lambda_\\textbf{y}$. These quantities are obtained by considering the \nsystem where one of the species, say $\\textbf{y}$,  is almost extinct: $y\\ll 1$. \nIn this case, remark two things: \n\\begin{itemize}\n  \\item if the current environment is $i$, the growth rate $\\frac{\\dot{y}}{y}$ \n    of species $\\textbf{y}$ is approximately $\\beta_i(1-c_ix)$, by \\eqref{eq:lv}; \n  \\item while $y$ stays negligible, the evolution of $(X,I)$ looks like \nthe dynamics of the Markov process on $[0,\\infty)\\times\\BRA{0,1}$ driven by the \ngenerator\n\\[\n  L_{\\textbf{x}}f(x,i)=\\alpha_i x(1-a_ix) \\partial_x f(x,i)+\\lambda_i(f(x,1-i)-f(x,i)),\n\\]\nwhich turns out to be ergodic with an invariant measure $\\mu_\\textbf{x}$ on $[0,\\infty)\\times\\BRA{0,1}$ \nthat depends on the jump rates $(\\lambda_0,\\lambda_1)=(st,(1-s)t)$.\n\\end{itemize}\nOver a long period of time, the growth rate of $\\textbf{y}$ is therefore\nobtained by averaging $\\beta_i(1-c_i x)$ with respect to the \ninvariant measure $\\mu_\\textbf{x}$: this leads to the definition \nof the \\emph{invasion rate} $\\Lambda_\\textbf{y}$ by\n\\[\n\\Lambda_\\textbf{y}(s,t)=\\int_{[0,\\infty)\\times\\{0,1\\}} \\beta_i(1-c_ix)\\mu_{\\textbf{x}}(dx,di).\n\\]\nWhen $\\Lambda_\\textbf{y}$ is positive (respectively negative) \nspecies $\\textbf{y}$ tends to increase (respectively decrease) from low density. Similarly, \none can define \n\\[\n\\Lambda_\\textbf{x}(s,t)=\\int_{[0,\\infty)\\times\\{0,1\\}} \\alpha_i(1-b_iy)\\mu_\\textbf{y}(dy,di). \n\\]\nwhere $\\mu_\\textbf{y}$ is the invariant probability measure of the stochastic process on \n$\\dR\\times\\BRA{0,1}$ with generator \n\\[\n  L_{\\textbf{y}}f(y,i)=\\beta_i y(1-d_iy) \\partial_ yf(y,i)+\\lambda_i(f(y,1-i)-f(y,i)),\n\\]\n\nThe main result in~\\cite{lotka} ensures that the long time behavior of the Markov \nprocess ($Z_t,I_t)$ is characterized by the sign of these invasion rates, as summed \nup in the following table (see Theorems~3.1, 3.3, 3.4 and 4.1 of \\cite{lotka} for precise \nstatements). \n\n\\begin{center}\n\\begin{tabular}{ccc}\n\\toprule\n  & $\\Lambda_\\textbf{y}>0$ & $\\Lambda_{\\textbf{y}}<0 $ \\\\\n\\hline \n$\\Lambda_\\textbf{x}>0$& persistence of the two species & extinction of species $\\textbf{y}$\\\\\n\\hline\n$\\Lambda_\\textbf{x}<0$ & extinction of species $\\textbf{x}$ & random extinction of one of the two species \\\\\n\\bottomrule\n\\end{tabular}\n\\label{tab:signs_of_lambda12}\n\\end{center}\n\n\\subsection{Our contribution}\nIn view of the previous result, the study of the model is reduced to \nfinding the sign of the invasion rates, depending on the parameters \nof the environment and on the jump rates. To state our results, \nwe need to introduce a second parametrization for the jump rates \n$(\\lambda_0,\\lambda_1)\\in (0,+\\infty)^2$ slightly different from~\\eqref{eq:para_st}:\n\\[\n(u,v)\\in [0,1]\\times(0,+\\infty) \\mapsto (\\alpha_0 uv, \\alpha_1 (1-u)v)\n\\]\nin such a way that \n\\begin{equation}\\label{eq:def_u_v}\nu = \\gamma_0\/(\\gamma_0+\\gamma_1)\n\\quad\\text{and}\\quad \nv=\\gamma_0+\\gamma_1\n\\quad\\text{where }\\gamma_i = \\lambda_i\/\\alpha_i\n\\quad\\text{for } i=0,1.  \n\\end{equation}\n\n\nThe change of parameters $(u,v)=\\xi(s,t)$ is triangular in the sense\nthat $u$ only depends on $s$: \n\\[\n(u,v)=\\xi(s,t)=\\PAR{\n  \\frac{s\\alpha_1}{(1-s)\\alpha_0 + s\\alpha_1},\n  \\frac{t}{\\alpha_0\\alpha_1}( (1-s)\\alpha_0 + s\\alpha_1)\n}.\n\\]\n\\begin{rem}[Length of interjump times II]\nNotice that the new parameter $v$ is proportional to $t$ when $u$ (or $s$) is fixed. \nAs a consequence, as in Remark \\ref{rem:st}, the interjump times are short when $v$ \nis large and large when $v$ is small.   \n\\end{rem}\n\n\\begin{defi}[Reparametrized invasion rates]\nThe invasion rates in the $(u,v)$ coordinates are denoted by\n\\[\n\\tilde \\Lambda_\\emph{\\textbf{x}}(u,v)=\\Lambda_\\emph{\\textbf{x}}(\\xi^{-1}(u,v))\n\\quad\\text{and}\\quad \n\\tilde \\Lambda_\\emph{\\textbf{y}}(u,v)=\\Lambda_\\emph{\\textbf{y}}(\\xi^{-1}(u,v)). \n\\]\nSimilarly,  $\\tilde{I}$ (resp. $\\tilde{J}$) is the image of $I$ (resp. $J$) \nfor the other parametrization. \n\\end{defi}\nNote that $\\tilde{I}$ and $\\tilde{J}$ still are \n(possibly empty) intervals. \n\n\n\n\\begin{rem}\n  The parameter $u$ is already implicitly considered in~\\cite{lotka}, where \n  it appears in the computations leading to the explicit conditions for the \n  non-emptyness of $I$ (which are equivalent to the positivity of a second \n  degree polynomial). \n\\end{rem}\n\nOur first result is an explicit formula for~$\\tilde \\Lambda_\\textbf{y}$, \nsuited both to fast numerical computations and theoretical study. \n\\begin{lem}[Expression of $\\tilde\\Lambda_\\textbf{y}$]\n  \\label{lem:exprLambda2}\n  Assume that $a_0<a_1$ (see Remark~\\ref{rem:adiff}). Let $(u,v)$ \n  given by~\\eqref{eq:def_u_v}. The quantity $\\tilde \\Lambda_\\emph{\\textbf{y}}$ can \n  be rewritten as:\n  \\[\n    \\tilde\\Lambda_\\emph{\\textbf{y}}(u,v) \n    = \\frac{1}{(a_1-a_0) \\PAR{\\frac{1}{\\alpha_0} (1-u) + \\frac{1}{\\alpha_1} u}}\n    \\esp{\\phi(U_{u,v})},\n  \\]\n  where $\\phi:[0,1]\\to \\dR$ is defined by\n  \\[\n    \\phi(y) = \\frac{1}{a_0+(a_1-a_0) y}P(a_0+(a_1-a_0) y)\n  \\]\n  for some explicit second degree polynomial $P$, and \n  $U_{u,v}$ is a Beta distributed $\\mathrm{Beta}\\PAR{uv,(1-u)v}$ random variable. \n    \n  Moreover, $\\phi$ has the following properties: \n  \\begin{itemize}\n      \\item If $\\tilde{I}=\\emptyset$ then $\\phi$ is nonpositive on $[0,1]$; \n      \\item If $\\tilde{I}=(u_1,u_2)\\neq \\emptyset$  then $\\phi$ is concave, \n      negative on $(0,u_1)\\cup(u_2,1)$  and positive on $\\tilde{I}$. \n  \\end{itemize}\n\\end{lem}\n\nRecall that a function $f$ is called quasi convex on $(a,b)$ if its\n  level sets $\\BRA{f\\leq t}$ are convex, that is, $f$ decreases\n  on $(a,c)$ and increases on $(c,b)$ for some $c\\in[a,b]$. \n\n    Our main result describes more precisely the region of positivity\n    for $\\Lambda_\\textbf{x}$ and $\\Lambda_\\textbf{y}$. \n\n\\begin{thm}[Shape of the regions]  \\label{thm:mainResult}\n  There exists a function $u\\mapsto v_\\emph{\\textbf{y}}(u)$ from $(0,1)\\to\n  [0,\\infty]$, with domain $\\tilde{I}$, such that $\\tilde \\Lambda_\\emph{\\textbf{y}}(u,v) <0$ \n  when $v<v_\\emph{\\textbf{y}}(u)$ and $\\tilde \\Lambda_\\emph{\\textbf{y}}(u,v) >0$ \n  when $v>v_\\emph{\\textbf{y}}(u)$. \n\n  Moreover $v_\\emph{\\textbf{y}}$ is  quasi-convex, continuous  on its domain $\\tilde{I}$, and \n  tends to $+\\infty$ on the endpoints of $\\tilde{I}$. \n\n  Similarly, there exists a function $s\\mapsto t_\\emph{\\textbf{y}}(s)\\in[0,\\infty]$, with\n  domain $I$, going to infinity at the endpoints of $I$, such that: \n  \\begin{itemize}\n\t\\item $\\Lambda_\\emph{\\textbf{y}}(s,t) < 0$ if $t<t_\\emph{\\textbf{y}(s)}$, \n\t\\item $\\Lambda_\\emph{\\textbf{y}}(s,t) = 0$ if $t=t_\\emph{\\textbf{y}(s)}$, \n\t\\item $\\Lambda_\\emph{\\textbf{y}}(s,t) > 0 $ if $t>t_\\emph{\\textbf{y}(s)}$. \n  \\end{itemize}\n\n  The same statement holds in the parameters $(s,t)$ for the function\n  $(-\\Lambda_\\emph{\\textbf{x}})$ with~$I$ replaced by~$J$ and\n  with a critical function $t_\\emph{\\textbf{x}}(s)$. \n\\end{thm}\n\n\\begin{rem}\\label{rem:convex}\n  Numerical computations suggest that both $v_\\emph{\\textbf{y}}$ \n  and~$t_\\emph{\\textbf{y}}$ are in fact smooth and convex on $\\tilde I$ and $I$ \n  respectively. \n\\end{rem}\n\n\\begin{rem}\n  This result is cited in \\cite[Proposition 2.5]{lotka}, since it\n  answers a conjecture that appeared in a preprint version of~\\cite{lotka}. \n\\end{rem}\nFor an illustration of Theorem~\\ref{thm:mainResult} and Remark~\\ref{rem:convex}, \nsee Figure~\\ref{fig:lambda_12}. \n\n\\begin{figure}\n \n  {\\centering\n  \\input{lambda1Et2.tex}\n\n  \\bigskip\n\n  \\input{lambda1Et2_bis.tex}\n\n}\n\n \n  {\\small\n    These plots represent the \"critical\" functions $t_\\textbf{y}$ and \n   $t_\\textbf{x}$ for different choices of the environments.  \n   Denoting environments by the couple \n   $\\PAR{\\begin{smallmatrix} \\alpha \\\\ \\beta\\end{smallmatrix}}$; \n   $\\PAR{\\begin{smallmatrix} a & b \\\\ c & d\\end{smallmatrix}}$,\n   the functions are plotted with\n   \\[\n     \\cE_0 = \n   \\begin{pmatrix} 1 \\\\ 5 \\end{pmatrix} ; \n   \\begin{pmatrix} 1 & 1 \\\\ 2 & 2\\end{pmatrix} \n   \\text{ (top plot);}\n   \\qquad\n   \\cE_0 = \n   \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix} ; \n   \\begin{pmatrix} 1 & 2\/3 \\\\ 2 & 4\/3\\end{pmatrix} \n   \\text{ (bottom plot);}\n   \\qquad \\cE_1 = \n   \\begin{pmatrix} 5 \\\\ 1 \\end{pmatrix} ; \n   \\begin{pmatrix} 3 & 3 \\\\ 4 & \\rho\\end{pmatrix}\n \\]\n for various values of the parameter $\\rho$ appearing in the definition of the environment $\\cE_1$. \n The black curve in both plots is $t_\\textbf{y}$, \n and does not depend on the value of $\\rho$. The colored curves are $t_\\textbf{x}$. \nThe respective domains of these curves are the intervals~$I$ and~$J$. All configurations\nare possible: $I\\cap J$ may be empty (bottom plot, $\\rho=6.8$), a strict subset of\n$I$ and $J$ (bottom plot, $\\rho=6.2$) or may be $I$ or $J$ itself (top plot). \n\nThanks to the results of \\cite{lotka} summarized in the table page\n\\pageref{tab:signs_of_lambda12}, these plots\ndescribe exactly what regimes are possible when the jump rates (parametrized by \n$s$ and $t$) vary, for a given choice of the environments. \n\nFor example the top plot for $\\rho=5.5$ has three regimes: extinction of $\\textbf{x}$ \n(above the red curve), persistence (between the red and the black curves) \nand extinction of $\\textbf{y}$ (below the black curve). For $\\rho=4.5$ there is an \nadditonal zone (above the yellow curve and below the black one) of extinction of\na random species. In particular, the knowledge of the relative positions\nof $I$ and $J$ is not enough to determine the possible regimes. \n\nAll these plots are computed by finding, for a fixed $s$, the zero of the function\n$t\\mapsto\\Lambda(s,t)$; this is done by a simple root finding \nalgorithm, using the explicit formula given in Lemma~\\ref{lem:exprLambda2}\nto evaluate $\\Lambda(s,t)$. \n}\n\n  \\caption{Shape of positivity regions for $\\Lambda_\\textbf{x}$ and $\\Lambda_{\\textbf{y}}$ }\n  \\label{fig:lambda_12}\n\\end{figure}\n\nFinally, our last results are dedicated to the support of the non-trivial invariant probability \nmeasure in the persistence regime. In~\\cite{lotka}, it is shown that this measure has a \ndensity with respect to the Lebesgue measure on the quadrant. Theorem~\\ref{thm:support}\nprovides a full description of its support when the set $I\\cap J$ is not empty. Since a precise \nstatement requires several notations introduced in Section~\\ref{sec:notation}, we postpone \nit to the last section of the document. \n\nThe remainder of the paper is organized as follows.  In\nSection~\\ref{sec:notation} we describe the various phase portraits for\nLotka-Volterra vector fields, and narrow down the choices of $\\cE_0$, $\\cE_1$\nthat lead to interesting behaviour.  In Section~\\ref{sec:proofLemma} we prove\nLemma~\\ref{lem:exprLambda2}; the main result is proved in\nSection~\\ref{sec:proofTheorem}. The final section is dedicated \nto the description of the support of invariant measures in the persistence \nregime. \n\n\n\\section{Deterministic picture}\\label{sec:notation}\n\\subsection{Phase portraits of Lotka-Volterra vector field}\n\n\nWe consider here the ODE \\eqref{eq:lv} in an environment $\\mathcal{E}=(a,b,c,d,\\alpha,\\beta)$\nand describe its possible qualitative behaviours. Much of this description\ncan be found in~\\cite{lotka}, we give it here for the sake of  clarity. \n\nBarring limit cases that we will not consider, there are essentially\nfour different phase portraits for the system, that are depicted in Figure~\\ref{fi:cases}. \nThese four regimes are obtained as follows. \n\nNotice first that the vector $F_{\\mathcal{E}}(x,y)$ is horizontal if $y=0$ or\n$cx+dy=1$: we call the line $cx+dy=1$ the horizontal isocline. Similarly\n$F_\\cE(x,y)$ is vertical if $x=0$ or if $(x,y)$ is on the vertical isocline\n$ax+by=1$. These isoclines are the bold straight lines in\nFigure~\\ref{fi:cases}. \n\nEach axis is invariant that why in the sequel we are only interested in initial \nconditions with positive coordinates. The three points $(0,0)$, $(0,1\/d)$ and \n$(1\/a,0)$ are  stationary for $F_{\\mathcal{E}}$. The origin is always a source.  \nThe nature of the other points and the existence of a fourth stationary point \ndepends on the parameters; this gives rise to the four types announced above. \n\n\\emph{Type 1.} If $a<c$ and $b<d$,  species $\\textbf{y}$ gets extinct.\n\t\\begin{itemize}\n\\item The point $(1\/a,0)$ is the unique global attractor: any solution of the\n  ODE starting from a point with positive coordinates converges to $(1\/a,0)$. \n\\item The point $(0,1\/d)$ is a hyperbolic saddle point.  Its stable manifold is\n  the vertical axis.  We denote by $\\Sigma$ the intersection of its unstable\n  manifold with the positive quadrant: $\\Sigma$ is the curve made of points $(x_0,y_0)$ such\n  that the solution $(x_t,y_t)$ starting at $(x_0,y_0)$ satisfies \n\\[\n(x_t,y_t)\\xrightarrow[t\\to-\\infty]{}(0,1\/d).\n\\]\nWe may compute explicitely some characteristics of $\\Sigma$: \nin particular, it  leaves $(0,1\/d)$ with a slope $-\\frac{\\alpha (d-b)+\\beta d}{\\beta c}$\nand ends in $(1\/a,0)$ with a possibly degenerate slope\n$\\min\\PAR{0,-\\frac{\\beta(a-c)+\\alpha a }{\\alpha b}}$. Moreover, $\\Sigma$ lies \nin between the horizontal and vertical isoclines. \n\\end{itemize}         \n\n\\emph{Type 2.} If $a>c$ and $b>d$, species $\\textbf{x}$ gets extinct. \n$(0,1\/d)$ is the unique sink and $(1\/a,0)$ is a saddle point. This is the same\nas Type 1 except that the two species $\\textbf{x}$ and $\\textbf{y}$ are\nswapped. \n\n\\emph{Type 3.} If $a>c$ and $b>d$, both species survive.  The points\n$(1\/a,0)$ and $(0,1\/d)$ are saddle points. The isoclines meet at\nthe sink $(\\bar x,\\bar y) = (ad-bc)^{-1} (a-c,b-d)$, which \nis the unique global attractor. \n\n\\emph{Type 4.} If $a>c$ and $b<d$, one of the two species gets extinct\ndepending on the starting point. The meeting point of the isoclines \n$(\\bar x,\\bar y)=(ad-bc)^{-1} (a-c,d-b)$ is now a saddle point, \nand (non trivial) trajectories converge to one of the\ntwo sinks on the axes, $(1\/a,0)$ and $(0,1\/d)$. \n\n\\begin{figure}\n\\begin{center}\n \\input{dessins}\n\\end{center}\n\n {\\small\n In each picture the bold lines are the horizontal and vertical isoclines, \n the gray lines are trajectories of the~ODE. Sources, sinks and saddle points\n are pictured respectively by white circles, black circles and crosses. \n\n Type~$1$ environments correspond to the upper left picture, Type~$2$ to \n the bottom right one. In Type~$3$ environments (bottom left), the intersection\n of the isoclines attracts the whole quadrant. In Type~$4$ environments (upper right)\n one species or the other gets extinct depending on the starting point. \n\n For fixed environments $\\cE_0$, $\\cE_1$ of Type~$1$, the mixed environment $\\cE_s$\n may be of any of the four types, depending on whether~$s$ belongs to the\n intervals~$I$ and~$J$. \n }\n\\begin{center}\n \\caption{The four possible deterministic regimes for a given environment.}\n \\label{fi:cases}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Relative positions of the two environments} \n  \nRecall that the vector fields $F_{\\mathcal{E}_0}$ and $F_{\\mathcal{E}_1}$ are assumed \nfavorable to species~\\textbf{x}: $a_i<c_i$ and $b_i<d_i$ for $i=0,1$. \nWithout loss of generality, we suppose that $a_0<a_1$. The switched \nsystem may present a surprising behavior if the interval $I$ defined in Equation~\\eqref{eq:def-I} \nis not empty. This requires that $c_0<a_1$. As a consequence, we impose \nin the sequel that \n\\[\na_0<c_0<a_1<c_1, \\quad b_0<d_0 \\quad\\text{and}\\quad b_1<d_1.\n\\]\n\n\\begin{lem}\\label{lem:parameters}\nIf the set $J$ is not empty then $d_1<b_0$ or $d_0<b_1$. Moreover, if $I\\cap J$ is non empty \n then \n\\begin{equation}\\label{eq:parameters}\na_0<c_0<a_1<c_1 \\quad\\text{and}\\quad b_0<d_0 < b_1<d_1. \n\\end{equation}\n\\end{lem}\n\\begin{proof}\nThe first point is clear from the definition of $J$. As a consequence, two configurations \nare compatible with $I$ and $J$ non empty: \\eqref{eq:parameters} and   \n\\[\na_0<c_0<a_1<c_1\\quad\\text{and}\\quad b_1<d_1<b_0<d_0.\n\\]\nSuppose, to derive a  contradiction, that we are in the latter configuration. \nDefine the points\n$C_i=(1\/c_i,0)$ and $D_i=(0,1\/d_i)$ for \n$i=0,1$. If $M=(\\tilde x,\\tilde y) $ is the intersection of the lines $(C_0D_0)$ \nand $(C_1D_1)$ then the set $[\\tilde x,+\\infty)\\times [0,\\tilde y]$ is strongly \npositively invariant under the action by $F_0$ and $F_1$. Furthermore, if $I\\cap J$ \nis not empty, this means that there exists $s_0\\in (0,1)$ such that $F_{\\mathcal{E}_{s_0}}$ is \nof Type 2. In particular, the point $(0,1\/d_s)$ is accessible. This is incompatible with the previous \nremark. As a consequence,  $I\\cap J$ is empty for this configuration.  \n\\end{proof}\n\n\n\\section{Expression of the invasion rate}\n\\label{sec:proofLemma}\n\nLet us first recall the expression of $\\Lambda_\\textbf{y}$ derived in~\\cite{lotka}. \nLetting $p_i = 1\/a_i$, $\\Lambda_\\textbf{y}$ is given by\n\\[\n  \\Lambda_\\textbf{y}=p_0p_1C\\int_{p_0}^{p_1}\\! P(x)\\theta(x)\\,dx  \n\\]\nwhere \n\\begin{align*}\nC^{-1} &= \\frac{p_1}{\\alpha_1}\n          \\int_{\\min(p_0,p_1)}^{\\max(p_0,p_1)}\n\t     \\!\\frac{\\ABS{x-p_0}^{\\gamma_0}\\ABS{p_1-x}^{\\gamma_1-1}}\n             {x^{\\gamma_0+\\gamma_1+1}}\\,dx\n        + \\frac{p_0}{\\alpha_0}\n\t  \\int_{\\min(p_0,p_1)}^{\\max(p_0,p_1)}\\!\n\t    \\frac{\\ABS{x-p_0}^{\\gamma_0-1}\\ABS{p_1-x}^{\\gamma_1}}\n            {x^{\\gamma_0+\\gamma_1+1}}\\,dx,\\\\\n\\theta(x)&=\n\\frac{\\ABS{x-p_0}^{\\gamma_0-1}\\ABS{p_1-x}^{\\gamma_1-1}}{x^{\\gamma_0+\\gamma_1+1}},\\\\\nP(x)&=\n\\SBRA{\\frac{\\beta_1}{\\alpha_1}(1-c_1x)(1-a_0x)-\\frac{\\beta_0}{\\alpha_0}(1-c_0x)(1-a_1x)}\n\\frac{a_1-a_0}{\\ABS{a_1-a_0}}.\n\\end{align*}\nThis quantity is obtained by averaging a growth rate of the second species\nwith respect to  the invariant measure \nof the one-dimensional PDMP $(X,I)$ on $[0,\\infty)\\times\\BRA{0,1}$ driven by \n\\[\n  Lf(x,i)=\\alpha_i x(1-a_ix) \\partial_x f(x,i)+\\lambda_i(f(x,1-i)-f(x,i)),\n\\]\nwhich corresponds to the dynamics of species \\textbf{x} when species $\\textbf{y}$ is\ngone.  In the sequel, we assume that $a_0<a_1$ and set $\\delta = a_1 - a_0$.\nIt is obvious that the recurrent set of $(X,I)$ is $[p_1,p_0]\\times\\BRA{0,1}$. \n\n\n\nIn~\\cite{lotka}, the continuous part $[p_1,p_0]$ of the state space \nis parametrized by a mapping $[0,1]\\ni s\\mapsto p_s$\n given by $p_s = 1\/a_s$, where\n$a_s = (s\\alpha_1 a_1 + (1-s)\\alpha_0 a_0) \/ (s\\alpha_1 + (1-s)\\alpha_0)$. \nIt is interesting to look at another\nparametrization $[0,1]\\ni u \\mapsto \\tilde{p}(u)$, where\n$\\tilde{p}(u) = \\tilde{a}(u)^{-1} = \\frac{1}{a_0 + \\delta u}$. \nThese parametrizations are summed up in the following diagram:\n\n\\begin{center}\n\\begin{tikzpicture}\n  \\matrix (m) [matrix of math nodes,column sep=8em,row sep = 3ex]{\n     & {[a_0,a_1]} &  \\\\\n     I\\subset [0,1] & & {[0,1] \\supset \\tilde{I} }\\\\\n     & {[p_1,p_0]} & \\\\\n    };\n    \\draw[->] (m-2-1)-- node[auto]      {$s\\mapsto a_s$}          (m-1-2);\n    \\draw[->] (m-2-3)-- node[auto,swap] {$u\\mapsto \\tilde{a}(u)$} (m-1-2);\n    \\draw[->] (m-2-1)-- node[auto,swap] {$s\\mapsto p_s$}          (m-3-2);\n    \\draw[->] (m-2-3)-- node[auto] {$u\\mapsto \\tilde{p}(u)$} (m-3-2);\n    \\draw[->] (m-1-2)-- node[auto] {$x\\mapsto 1\/x$}          (m-3-2);\n\\end{tikzpicture}\n\\end{center}\n\nThis parameter $u$ is  the one given in the introduction\nand\ncorresponds\nto a ratio of the $\\gamma$, when $s$ corresponds to a ratio of $\\lambda$, \nin the sense that:\n\\[ \n  \\tilde{p}\\PAR{\\frac{\\gamma_0}{\\gamma_0 + \\gamma_1}} = p\\PAR{\\frac{\\lambda_0}{\\lambda_0+\n\\lambda_1}}.\n\\]\n\\begin{rem}\n  As already mentioned above, the parameter $u$ and the interval $\\tilde{I}$\n  are used implicitly in~\\cite{lotka}: $u$ appears in Remark 1, and the map $S$\n  defined at the beginning of Section $4$ is given in our notation by $S(u) =\n  p^{-1}(\\tilde{p}(u))$. \n\\end{rem}\n\n\n\nLet us study the integral $\\int_{p_1}^{p_0} P(x)\\theta(x)dx$. \nSet  $y=\\tilde{p}^{-1}(x)$, so that:\n\\begin{align*}\n  x&= \\tilde{p}(y) = \\frac{1}{\\tilde{a}(y)} = \\frac{1}{a_0+\\delta y}, &\n  dx &= -\\delta \\tilde{p}(y)^2 dy \\\\\n  p_0 - x &= \\delta p_0  y\\tilde{p}(y), &\n  x - p_1 &=  \\delta p_1 (1-y)\\tilde{p}(y).\n\\end{align*}\nChanging variables in the integral yields:\n\\begin{align*}\n  \\int_{p_1}^{p_0} P(x) \\theta(x) dx \n  &= \\int_0^1 P(\\tilde{p}(y))\\PAR{\\delta p_0 y \\tilde{p}(y)}^{\\gamma_0 - 1}\n  \\PAR{\\delta p_1 (1-y) \\tilde{p}(y)}^{\\gamma_1 - 1} \n  \\tilde{p}(y)^{-\\gamma_0 - \\gamma_1 - 1}\n  \\delta \\tilde{p}(y)^2 dy \\\\\n  &= \n  \\delta^{\\gamma_0+\\gamma_1 - 1} p_0^{\\gamma_0-1}p_1^{\\gamma_1 - 1}\n  \\int_0^1 P(\\tilde{p}(y)) \\frac{1}{\\tilde{p}(y)} y^{\\gamma_0 -1} (1-y)^{\\gamma_1 - 1} dy \\\\\n  &= \n  \\delta^{\\gamma_0+\\gamma_1 - 1} p_0^{\\gamma_0-1}p_1^{\\gamma_1 - 1}\n  \\int_0^1 \\phi(y) y^{\\gamma_0 -1} (1-y)^{\\gamma_1 - 1} dy \\\\\n  &= \n  \\delta^{\\gamma_0+\\gamma_1 - 1} p_0^{\\gamma_0-1}p_1^{\\gamma_1 - 1}\n  B(uv,(1-u)v) \\esp{\\phi(U_{u,v})}.\n\\end{align*}\nsince $\\phi(y) = \\frac{1}{\\tilde{p}(y)} P (\\tilde{p}(y))$. \nA similar computation gives the exact formula\n\\[ C^{-1} = \\PAR{ \\delta^{\\gamma_0+\\gamma_1 - 1 } p_0^{\\gamma_0 - 1} p_1^{\\gamma_1 - 1}\n  B(uv,(1-u)v) \n}p_0p_1\\delta \\PAR{ \\frac{1}{\\alpha_0}(1- u) + \\frac{1}{\\alpha_1} u} \n\\]\nfor the normalization constant $C$. Therefore\n\\begin{align*}\n  \\Lambda_\\textbf{y}(\\gamma_0,\\gamma_1) \n  &= \\frac{1}{\\delta\\PAR{\\frac{1}{\\alpha_0} (1-u) + \\frac{1}{\\alpha_1} u}}\n  \\esp{\\phi(U_{u,v})}.\n\\end{align*}\n\n\nLet us study $\\phi$ more precisely. Since $P$ is a second-degree polynomial, \nlet us write it down as $P(x)=A_2x^2 + A_1x + A_0$. Then \n\\[ \\phi(y) = \\frac{A_2}{a_0 + \\delta y} + A_1 + A_0(a_0 + \\delta y).\\]\nThe second derivative is \n\\[ \\phi''(y) = \\frac{2A_2\\delta^2}{(a_0 + \\delta y)^3},\\]\nwhich has the sign of $A_2$ on $[0,1]$, so $\\phi$ is either strictly\nconvex or strictly concave. However, the proof of the first item\nof Proposition 2.2 in~\\cite{lotka} shows that (still in the case $a_0<a_1$)\n$P(p_s) = \\frac{\\beta_s}{\\alpha_1 s} (1 - a_0\/a_s)(1 - c_s\/a_s)$ \nhas the same sign as $a_s-c_s$, that is, $P(p_s)$ is positive iff $s\\in I$. \nIf $I$ is empty, so is $\\tilde{I}$, and $\\phi$ is negative on $]0,1[$. \nIf $I$ is not empty, so is $\\tilde{I}$, and $\\phi$ is positive on $\\tilde{I}$\n(and nonpositive outside $\\tilde{I}$), therefore $\\phi$ must be concave. \n\n\n\n\\section{Shape of the positivity region}\n\\label{sec:proofTheorem}\n\nWe begin with a lemma, which is proved in the next two sections.  The existence\nand  properties of $v_c$ and $t_c$ stated in Theorem~\\ref{thm:mainResult} are\ndeduced from this lemma in Section~\\ref{sec:reg}. \n\\begin{lem}\n  If $I$ is non empty, the map $(u,v)\\mapsto \\esp{\\phi(U_{u,v})}$ is (strictly)\n  increasing in $v$ and concave in~$u$. \n\\end{lem}\n\n\n\\subsection{\\texorpdfstring{Monotonicity in $v$}{Monotonicity in v}}\nWe wish to compare $\\esp{\\phi(U_{u,v})}$ for different values of $v$.  Since\nthe function $\\psi = (-\\phi)$ is convex, a natural idea is to compare the distributions\nof $U_{u,v}$ for various $v$ in the \\emph{convex order}. Let us first recall\nthe definition of this order. \n\\begin{defi}[Convex order]\n  Let $X$ and $Y$ be two random variables. If the inequality \n  \\[ \\esp{\\psi(X)} \\leq \\esp{\\psi(Y)}\\]\n  holds for all convex functions $\\psi$ such that the expectations exist,\n  $X$ is said to be \\emph{smaller than $Y$ in the convex order}. \n  We denote by $X\\leq_{\\mathrm{cvx}} Y$ this relation. \n\\end{defi}\nThe convex order admits the following  characterization\nin terms of cumulative distribution functions (\\cite[Theorem 3.A.1]{SS07}). \n\\begin{thm}[Convex order and distribution functions]\n  \\label{thm:convex_order_cdf}\n  The variables $X$ and $Y$ satisfy $X\\leq_{\\mathrm{cvx}} Y$ if and only if\n  $\\esp{X} = \\esp{Y}$ and, for all $x$, \n  \\begin{equation}\n    \\label{eq:ssd}\n    \\int_{-\\infty}^x F_X(t)dt \\leq \\int_{-\\infty}^x F_Y(t) dt < \\infty,\n  \\end{equation}\n  where $F_X$ and $F_Y$ are the cumulative distribution functions\n  of $X$ and $Y$. \n\\end{thm}\n\\begin{rem}[Terminology]\n  The convex order is one possible way of formalizing the idea\n  that $Y$ is \"more variable\/more spread out\" than $X$. Note \n  in particular if $X\\leq_{\\mathrm{cvx}} Y$ then $\\var{X}\\leq \\var{Y}$. \n  For details on the convex order,  a survey of other formalizations of\n  variability  and many other notions of stochastic order we refer\n  to~\\cite{SS07}. \n\n  Let us also note that, when \\eqref{eq:ssd} holds (without assuming\n  $\\esp{X} = \\esp{Y}$), $X$ is said to \\emph{second-order stochastically dominate\n    $Y$}; this is equivalent to asking the inequality $\\esp{\\psi(X)}\\leq\\esp{\\psi(Y)}$\n  for any \\emph{non-increasing} convex function $\\psi$. We refer to \n  \\cite[Appendix B.19]{MOA11} for a proof,  additional discussion, \n  and references to the literature. \n\\end{rem}\nWe will need the following easy fact. \n\\begin{thm}\n  \\label{thm:ordre_strict}\n  If $X\\leq_{\\mathrm{cvx}} Y$ and $\\esp{\\psi(X)} = \\esp{\\psi(Y)}$ for some \n  strongly convex $\\psi$, then $X$ and $Y$ have the same distribution. \n\\end{thm}\n\\begin{proof}\n  Suppose that $X$ and $Y$ satisfy the hypotheses. \n  By a classical characterization of the convex order\n  (\\cite[Theorem 3.A.4]{SS07})\n  there exists a couple $(X,Z)$ such that $\\esp{Z|X} = 0$ and $X+Z$ \n  has the same distribution as $Y$. Since $\\psi$ is strongly convex, \n  there exists an $m>0$ such that for all $t\\in[0,1]$, \n  \\[\n    \\psi(X+tZ) =  \\psi( (1-t)X + t(X+Z))\\leq (1-t)\\psi(X) + t\\psi(X+Z) - \\frac{mt(1-t)}{2}Z^2. \n  \\]\n  Taking expectations we get\n  \\[\n    \\esp{\\psi(X)} \\leq \\esp{\\psi(X+tZ)} \\leq (1-t)\\esp{\\psi(X)} + t\\esp{\\psi(Y)} \n    - \\frac{mt(1-t)}{2} \\esp{Z^2},\n  \\]\n  where the first inequality comes from Jensen's inequality and $\\esp{Z|X}=0$. \n  Since $\\esp{\\psi(Y)} = \\esp{\\psi(X)}$, $Z$ must be zero almost surely, \n  so $X$ and $Y$ have the same distribution. \n\\end{proof}\n\n\n\n\\begin{thm}[Orderings between Beta r.v.]\n  Let $X\\sim \\mathrm{Beta}(\\alpha,\\beta)$ and $X'\\sim\\mathrm{Beta}(\\alpha',\\beta')$. \n\n  If $\\alpha< \\alpha'$, $\\beta<\\beta'$ and $\\alpha\/(\\alpha+\\beta)  =  \\alpha'\/(\\alpha'+\\beta')$, \n  then $X' \\leq_{\\mathrm{cvx}} X$. \n\n \n\\end{thm}\n\\begin{proof}\n  Call $f_{\\alpha,\\beta}$, $f_{\\alpha',\\beta'}$ the densities of the distributions. \n  Compute their ratio:\n  \\[ \\frac{f_{\\alpha',\\beta'}(x)}{f_{\\alpha,\\beta}(x)} = C_{\\alpha,\\beta,\\alpha',\\beta'} x^{\\alpha'-\\alpha} (1-\nx)^{\\beta'-\\beta}.\\]\n  In the first case, \n  this ratio starts and ends in zero, is strictly increasing on $[0,x_0]$ and\n  strictly decreasing on $[x_0,1]$. Since the\n  two functions are densities, the ratio must cross $1$ exactly twice, say in $x_1$, $x_2$.  Therefore\n  \\[\n  d(x) = f_{\\alpha',\\beta'} - f_{\\alpha,\\beta}\n  \\]\n  is positive on $(x_1,x_2)$ and negative on $(0,x_1)$ and $(x_2,1)$. \n  Therefore\n  \\[ D(x) = F_{X'}(x) - F_{X}(x)\\]\n  starts at zero, decreases on $[0,x_1]$, increases on $[x_1,x_2]$ and decreases on $[x_2,1]$, so\n  $D(x)$ is negative on $[0,x_3]$ and positive on $[x_3,1]$ (since it ends at zero). \n  Integrating once more, \n  \\[ \\int_0^x D(t) dt\\]\n  starts and ends at zero (since $\\esp{X} = \\esp{X'}$) and is decreasing-increasing, therefore it is non-\npositive. Thanks to Theorem~\\ref{thm:convex_order_cdf}, this implies $X'\\leq_{\\mathrm{cvx}} X$. \n\\end{proof}\n\n\\begin{proof}[Proof of the monotonicity in $v$]\n   Suppose $v<v'$ and put $\\alpha= uv$, $\\alpha'=uv'$, $\\beta=(1-u)v$, $\\beta'=(1-u)v'$. The\n   theorem shows that $U_{u,v'}\\leq_{\\mathrm{cvx}} U_{u,v}$. Since \n   $\\psi = (-\\phi)$ is strongly convex, this implies\n   \\[ \\esp{\\phi(U_{u,v})} < \\esp{\\phi(U_{u,v'})}, \\]\n   by Theorem~\\ref{thm:ordre_strict},\n   so $v\\mapsto \\esp{\\phi(V_{u,v})}$ is (strictly) increasing. \n\\end{proof}\n\n\n\n\\subsection{\\texorpdfstring{Concavity in $u$}{Concavity in u}}\nEven though $\\phi$ is concave and the r.v. $U_{u,v}$, $U_{u',v}$ are\nstochastically comparable (for the usual stochastic order), this is not enough\nto show the concavity of $u\\mapsto \\esp{\\phi(U_{u,v})}$. \nWe prove it by elementary means. First, recalling the explicit expression\nof $\\phi$ we write:\n\\begin{align*}\n  \\esp{\\phi(U_{u,v})} \n  &= A\\esp{\\frac{1}{a_0 + \\delta U_{u,v}}} + B + C \\esp{a_0 + \\delta U_{u,v}} \\\\\n  &= A\\esp{\\frac{1}{a_0 + \\delta U_{u,v}}} + B + C(a_0 + \\delta u). \n\\end{align*}\nSince $A$ is negative (see the proof of Lemma~\\ref{lem:exprLambda2}) and\nthe last term is linear, it is enough to show that \n\\[\n  g: u \\mapsto \\esp{\\frac{1}{a_0+\\delta U_{u,v}}}\n\\]\nis convex. It is a bit easier to use the symmetry of Beta random \nvariables\n($U_{u,v}$ and $1-U_{1-u,v}$ have the same distribution) and look at\n\\[ \n  h: u\\mapsto \\esp{\\frac{1}{a_1 - \\delta U_{u,v}}}. \n\\]\nSince $g(u)=h(1-u)$, $g$ will be convex if $h$ is convex. \nNow, recalling that $\\delta = (a_1-a_0)< a_1$, we write\nthe series development:\n\\begin{align*}\n  h(u) = \\frac{1}{a_1} \\esp{\\sum_{k=0}^\\infty \\PAR{\\frac{\\delta}{a_1}}^k U_{u,v}^k} \n  = \\frac{1}{a_1} \\sum_{k=0}^\\infty           \\PAR{\\frac{\\delta}{a_1}}^k \\esp{U_{u,v}^k}\n\\end{align*}\nso $h$ is a mixture of the functions $h_k$ given by:\n\\[\n  h_k(u) =  \\esp{U_{u,v}^k} \n  = \\prod_{r=0}^{k-1} \\frac{ uv + r}{v+r}\n  = \\frac{1}{\\prod_{r=0}^{k-1}(v+r)} (uv)(uv+1)\\cdots (uv + k-1). \n\\]\nThis is a polynomial function of $u$ with positive coefficients, therefore\nit is convex on $[0,1]$. This concludes the proof. \n\n\\subsection{Properties of the frontier}\n\\label{sec:reg}\nWe have just shown that $v\\mapsto \\tilde \\Lambda_\\textbf{y}(u,v)$ is  \nincreasing.  From~\\cite[Proposition 2.2]{lotka} and the fact that in the change\nof variables $(s,t)\\leftrightarrow (u,v)$, $u$ only depends on $s$, and for\nfixed $u$, $v$ is an increasing function of $t$,  we know additionally that it\nadmits:\n\\begin{itemize}\n  \\item negative limits at $0$ and $\\infty$ if $u$ does not belong to the closure of $\\tilde{I}$, \n  \\item a negative limit at $0$ and a positive limit\n    at $\\infty$ if $u\\in \\tilde{I}$. \n\\end{itemize}\nSince $\\tilde \\Lambda_\\textbf{y}$ has the same sign as $\\esp{\\phi(U_{u,v})}$, \nthis justifies the existence of $v_c$, and we have\n\\[ \n  \\tilde \\Lambda_\\textbf{y}(u,v) = 0 \\iff u\\in\\tilde{I}, v=v_c(u). \n\\]\n\nLet us prove that $v_c$ is quasi-convex. \nLet $a<b<c$ be three points in $(0,1)$. If $a$ or $c$ are not in $\\tilde{I}$, \n$v_c(b)$ is less than $\\max(v_c(a), v_c(c)) = \\infty$. If $a$ and $c$ are\nin $\\tilde{I}$, let $M>\\max(v_c(a), v_c(c))$. Since $\\tilde \\Lambda_\\textbf{y}(u,\\cdot)$ \nis increasing, $\\tilde \\Lambda_\\textbf{y}(a,M)$ and $\\tilde \\Lambda_\\textbf{y}(c,M)$ are positive. \nSince $u\\mapsto \\dE\\phi(U_{u,v})$ is concave, $\\tilde \\Lambda_\\textbf{y}(b,M)$ is positive. \nTherefore $v_c(c) \\leq M$. Sending $M$ to $\\max(v_c(a), v_c(b))$ yields the \nquasi-convexity of $v_c$. \n\nLet us now show the regularity properties. Let $u_n$ be an increasing \nsequence in $\\tilde{I}$, converging to some $u\\in(0,1)$. Since\n$v_c$ is quasi-convex, $v_n = v_c(u_n)$ is eventually  monotone, \nso it converges to some  $v\\in [0,\\infty]$. If $v$ is finite, since \nthe zero set of $\\Lambda_\\textbf{y}$ is closed, by continuity, $v = v_c(u)$, so \n$u$ must be in $\\tilde{I}$. Conversely, \nif $u\\in \\tilde{I}$, $v_c$ is bounded on\na neighborhood of $u$ by quasi-convexity, so $v$ is finite. This shows\nthat $v_c$ is continuous on $\\tilde{I}$ and converges to $\\infty$ \nat the endpoints. \n\nThe properties of the change of variables $(s,t)\\leftrightarrow(u,v)$ \nshow that $v_c$ is well-defined and continuous with the correct limits. \n\n\\section{Support of the invariant measure} \\label{se:support}\n\nNote that the stochastic Lotka-Volterra process has at least two invariant \nprobability measures, supported on the coordinate axes. In the persistence regime, \nwe are interested in the third invariant measure, whose support \n$\\Gamma\\times \\BRA{0,1}$ is such that $\\Gamma$ \nhas non empty interior. Several properties of $\\Gamma$ are \nestablished in~\\cite{lotka} (see below). In this section, we aim at providing \na full description of $\\Gamma$. Its shape essentially depends \non the fact that $I\\cap J$ is empty or not. \n\n\\subsection{Persistence with \"full support\"}\n\nIn this subsection, we assume that $I\\cap J$ is not empty. According to \nLemma~\\ref{lem:parameters}, the vector fields $F_{\\mathcal{E}_0}$ and $F_{\\mathcal{E}_1}$\nare such that \\eqref{eq:parameters} holds.\nLet us denote by $\\Sigma_i$ the  intersection of $[0,\\infty)^2$ and the\nunstable manifold of $(0,1\/d_i)$ and $\\Gamma'$ the bounded subset of\n$[0,+\\infty)^2$ with border \n\\[\n\\Sigma_1\\cup\\BRA{(x,0)\\,:\\,1\/a_1\\leq x\\leq 1\/a_0}\n\\cup\\Sigma_0\\cup\\BRA{(0,y)\\,:\\,1\/d_1\\leq x\\leq 1\/d_0}.\n\\]\n\n\\begin{thm}\\label{thm:support}\nSuppose that $I\\cap J\\neq \\emptyset$. Then, for any $(s,t)\\in [0,1]\\times\n(0,\\infty)$ such that $\\Lambda_\\emph{\\textbf{x}}(s,t)>0$ and\n$\\Lambda_\\emph{\\textbf{y}}(s,t)>0$, then $\\Gamma'=\\Gamma$. \n\\end{thm}\n\n\\begin{figure}\n  {\n  \\centering\n  \\input{proof_support.tex}\n \n  \\par\n}\n\\small\nThe isoclines (straight lines) and unstable manifolds (curved lines)\nfor the three environments $\\cE_0$ (bottom left, in blue), $\\cE_s$ (middle,\nin purple) and $\\cE_1$ (upper right, in red). Note how the isoclines \nare \"swapped\" for $\\cE_s$, a Type 2 environment. \n\\caption{Full support case: isoclines and unstable manifolds}\n  \\label{fig:IinterJ_non_vide}\n\\end{figure}\n\n\\begin{figure}\n  {\\centering\n    \n    \\includegraphics[width=10cm]{support_away.png}\n   \\par\n  }\n\n  \\small\n  The outer curves are $\\Sigma_0$ and $\\Sigma_1$. The region \n  between these curves is positively invariant. The\n  inner curves are the two trajectories coming from the \n  unique point $z\\in T$: they form the boundary of the support. \n  The sample trajectory shows that the invariant measure is \n  in practice often concentrated on a smaller subset. \n   \\caption{Support away from the $y$ axis}\n   \\label{fig:away}\n\\end{figure}\n\n\\begin{proof}\nFirstly, notice that the set $\\Gamma'$ is positively invariant for each flow \nsince both vector fields $F_{\\mathcal{E}_0}$ and $F_{\\mathcal{E}_1}$ \npoint inside $\\Gamma'$. \n\nPick an $s\\in I\\cap J$. The isoclines and the unstable manifold of the \nsaddle point for the three environments $\\cE_0$, \n$\\cE_1$ and $\\cE_s$ are necessarily in the position depicted in\nFigure~\\ref{fig:IinterJ_non_vide}. Denote by $\\Sigma_s$ the \nintersection of the unstable manifold of $(1\/a_s,0)$ with the\nupper right quadrant. \n\nFirst step: the set $\\Sigma_s$ is contained in the support. \nIndeed, pick a point $(x,y)$ in the interior of the support (such\na point exists by \\cite[Remark 6]{lotka}). The loop formed \nby the trajectories starting from $(x,y)$ with both flows (converging\nto $A_1:(1\/a_1,0)$ and $A_0:(1\/a_0,0)$ and the line segment $[A_0,A_1]$\nis included in the support (by positive invariance). As a consequence, \nthe support must contain a closed half ball centered on $A_s$ --- \nlet us call it $\\cB$. Now pick a point $(x,y)\\in \\Sigma_s$: by definition its $\\cE_s$ \nflow converges for $t\\to-\\infty$ to $1\/a_s$. By continuity there exists a \npoint in the past of $(x,y)$ which is in $\\cB$. Running the time forward again, \nthe point $(x,y)$ must be in the support. \n\nSecond step: any point (strictly) between $\\Sigma_1$ and $\\Sigma_s$ is in $\\Gamma$. \nStarting from such a point $(x,y)$, run the $\\cE_1$ flow in reverse time. \nThe trajectory must cross $\\Sigma_s$. So $(x,y)$ is in the \nfuture of a point in $\\Sigma_s\\subset\\Gamma$, and $(x,y)\\in\\Gamma$ by positive invariance. \n\nThird step: any point between $\\Sigma_0$ and $\\Sigma_s$ is in $\\Gamma$. \nThis step is similar to the previous one and is omitted. \n\nSimilarly, any point between $\\Sigma_1$ and $\\Sigma_s$ is in $\\Gamma$.\n\\end{proof}\n\n\\subsection{Support away from the $y$ axis}\n\nWe suppose in the sequel that $I\\cap J$ is empty. Let us introduce the \nset where the two vector fields $F_0$ and $F_1$ are collinear:\n\\[\nC=\\BRA{z\\in\\dR_+^2\\, : \\, \\det(F_0(z),F_1(z))=0}.\n\\]\nThis set is the union of $\\BRA{(0,y)\\,:\\, y\\geq 0}$, $\\BRA{(x,0)\\,:\\, x\\geq 0}$, and \n\\[\n\\tilde C=\\BRA{(x,y)\\in\\dR_+^2\\, : \\,G(x,y)=0}\n\\]\nwhere $G$ is a polynomial of degree 2. As a consequence, the set~$\\tilde C$ is \na subset of a conic. It is easy to see that $\\tilde{C}$ is also the \nset of non-degenerate equilibrium points for the vector field $F_{\\cE_s}$, as\n$s$ varies from $0$ to $1$. When $s\\in I$, $\\cE_s$ is of Type~$3$ so the\nequilibrium point is stable and globally attractive. Therefore the \npart of $\\tilde{C}$ that corresponds to $s\\in I$ must be included in $\\Gamma$, as\nwell as all trajectories (for both flows) starting from it. \n\nNumerical experiments suggest that there is a unique \"extremal point\" \non this part of $\\tilde{C}$, such that the trajectories starting from \nthis point form the boundary of $\\Gamma$. See Figure~\\ref{fig:away}. \n\nTo describe it more precisely, consider the subset of $\\tilde C$ made of the points \nwhere $F_0$ (or $F_1$) is tangent to the curve $\\tilde C$. This set is given by \n\\[\nT=\\BRA{(x,y)\\in \\dR_+^2\\,:\\, G(x,y)=0\\text{ and } (F_0\\cdot \\nabla G)(x,y)=0}.\n\\]\nSince $G$ and $F_0\\cdot\\nabla G$ are polynomials with respective degrees 2 and 3, $T$ \nis made of at most six points according to Bezout's Theorem. \n\n\nFor any $z\\in T$ let us define $C(z)$ the bounded region enclosed\nby the Jordan curve\n\\[\n\\BRA{\\varphi^{0}_z(t)\\, :\\, t\\in [0,\\infty)}\\cup \\BRA{\\varphi^{1}_z(t)\\, :\\, t\\in [0,\\infty)} \n\\cup [1\/a_1,1\/a_0]\\times \\BRA{0}, \n\\]\nwhere $t\\mapsto \\varphi^{i}_z(t)$ is the flow associated to the vector field $F_i$ for $i=0,1$.    \n\n\\begin{conj}\nThe set $T$ is a singleton $\\BRA{z_0}$ and the support of the invariant measure \nwhich is not supported by one of the two axes is $C(z_0)\\times\\BRA{0,1}$. \n\\end{conj}\n\n\n\n\\paragraph*{Acknowledgements.}\nWe thank an anonymous referee for constructive remarks.\nWe acknowledge financial support from the French ANR project ANR-12-JS01-0006-PIECE.\nNumerical computations were done in Julia and graphics in TikZ. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\nSmall particles, such as laser-cooled atoms or dielectric nanospheres, are nowadays routinely trapped at submicron distances from solids. Structures currently investigated include photonic crystal waveguides~\\cite{thompson_coupling_2013,goban_superradiance_2015,magrini_near-field_2018}, optical nanofibers~\\cite{vetsch_optical_2010,goban_demonstration_2012,beguin_generation_2014,kato_strong_2015,lee_inhomogeneous_2015,corzo_large_2016}, single carbon-nanotubes~\\cite{gierling_cold-atom_2011,schneeweiss_dispersion_2012}, dielectric membranes~\\cite{diehl_optical_2018}, and even macroscopic prisms~\\cite{hammes_cold-atom_2002,bender_probing_2014}. The opportunities in research and application for systems combining atoms and solids are numerous, including the search for novel fundamental forces~\\cite{geraci_improved_2008,geraci_short-range_2010,arkanihamed_hierarchy_1998,dalvit_casimir_2011,klimchitskaya_casimir_2009}, the implementation of quantum metrology and sensing using collective atomic state entanglement~\\cite{beguin_observation_2018}, and integrated quantum memories for photons guided in nanoscale waveguides~\\cite{sayrin_storage_2015,gouraud_demonstration_2015,corzo_waveguide-coupled_2019}. A rich toolbox is already available for the cooling, trapping, positioning, and probing of atoms and nanoparticles. However, not all techniques commonly used in free-space traps for manipulating trapped particles are compatible with the presence of solid structures in their immediate proximity: Control laser beams, for instance, may be reflected or scattered in undesired ways. Moreover, additional effects such as van der Waals forces or coupling of the atoms or particles to thermal excitations in the solid have to be considered.\n\nFull control at the quantum level over the internal as well as external degrees of freedom of individual atoms coupled to a nanophotonic structure was achieved only recently~\\cite{meng_near-ground-state_2018}. A key challenge in this context is the  heating of the atomic motion observed in these systems~\\cite{reitz_coherence_2013,albrecht_fictitious_2016} which can reach rates of several hundred motional quanta per second -- about three orders of magnitude larger than in comparable free-space optical traps. Large cooling rates realized, for example, by ultrastrong spin-motion coupling~\\cite{schneeweiss_cold-atom-based_2018,dareau_observation_2018}, are required to overcome the heating and prepare atoms close to their motional ground state. In essence, the observed storage times of atoms in nanophotonic traps have fallen short of expectations, both for trapped cesium~\\cite{goban_demonstration_2012,beguin_generation_2014,kato_strong_2015,corzo_large_2016,goban_superradiance_2015} and rubidium~\\cite{lee_inhomogeneous_2015} atoms, ever since the first implementation of a nanofiber-based trap for laser-cooled atoms~\\cite{vetsch_optical_2010}. The origin of the strong heating and the corresponding low lifetimes has so far remained elusive. There is a range of conceivable causes, such as Raman scattering of the trapping light fields in the waveguide material \\cite{engelbrecht_nichtlineare_2015}, Brillouin scattering~\\cite{beugnot_brillouin_2014,florez_brillouin_2016}, or Johnson-Nyquist noise~\\cite{henkel_loss_1999}. However, estimates of their effect, provided as supplemental material %\n\\footnote{See supplemental material at the end of this article for estimates of the contribution of other mechanisms to the heating of nanofiber-trapped cold atoms}, %\ndemonstrate that these mechanisms fail to explain heating rates observed in experiments. Additionally, tapered optical fibers, as used for realizing nanofiber-based cold-atom traps, exhibit thermally driven high-$Q$ torsional mechanical resonances which have been considered as a likely candidate for explaining the large heating in these systems~\\cite{wuttke_optically_2013}. In contrast, optical traps that are based on the evanescent field above a prism surface seem to feature small heating rates which are compatible, for instance, with Bose-Einstein condensation of cesium atoms~\\cite{rychtarik_two-dimensional_2004}. Indeed, even at room temperature, one does not expect thermally excited phonon modes of the macroscopic prism to contribute to the heating of the trapped atoms~\\cite{henkel_heating_1999}.\n\nHere, we identify thermally populated flexural phononic modes of the nanoscopic waveguide as the dominant contributor to the large heating rates observed in nanofiber-based cold-atom traps. We give a concise description of the effect of mechanical modes on light guided in optical waveguides and provide a general theory of the resulting atom-phonon interaction in nanophotonic traps. Based on this formalism, we perform a case study for the cesium two-color nanofiber-based trap described in refs.~\\cite{vetsch_optical_2010,reitz_coherence_2013,albrecht_fictitious_2016}. Relying on independently measured system properties, we predict heating rates in excellent quantitative agreement with experimental observations. Surprisingly, the effect of the high-$Q$ torsional mechanical resonances that have previously been observed in this system~\\cite{wuttke_optically_2013} can be neglected, even if they coincide with the trap frequencies. We then use our model to numerically and analytically infer the scaling of the heating rates with system parameters such as the mechanical properties of the fiber, its temperature, or the trap frequencies. This systematic analysis allows us to outline strategies for minimizing the heating, thereby suggesting a solution to a long-standing problem of nanofiber-based cold-atom systems. While we formulate our theory in terms of atoms near nanofibers, it is indeed applicable to any kind of polarizable object trapped by conservative forces due to the light field surrounding a photonic structure. Building on the agreement obtained in the case study, our quantitative formalism might therefore be used for the faithful description of other nanophotonic cold-atom systems and, more generally, optomechanical systems with small particles, such as dielectric nanospheres \\cite{magrini_near-field_2018,chang_cavity_2010,romero-isart_toward_2010,li_millikelvin_2011,gieseler_subkelvin_2012,kiesel_cavity_2013,fonseca_nonlinear_2016,jain_direct_2016}, trapped in close vicinity to hot solid bodies.\n\nThis article is structured as follows: In \\cref{sec: framework}, we provide a general quantum theory describing atoms trapped in the optical near field of a vibrating photonic structure. In particular, we derive the general form of the atom-phonon interaction and discuss the resulting heating rates of the atomic motion. \\Cref{sec: case study} is dedicated to a case study of heating rates expected in a nanofiber-based two-color trap for laser-cooled atoms. In \\cref{sec: photon appendix}, we review the concept of photonic eigenmodes and summarize the modes of a nanofiber, while \\cref{sec: atom appendix} recapitulates the resulting forces acting on trapped atoms. In \\cref{sec: phonon appendix}, we review quantized linear elastodynamics and summarize the phononic eigenmodes of a nanofiber. In \\Cref{sec: interaction appendix}, we supply details on how to calculate the atom-phonon coupling constants based on the framework presented in \\cref{sec: photon appendix,sec: atom appendix,sec: phonon appendix}. The parameters of the experimental setup considered in the case study are listed in \\cref{sec: case study appendix}.\n\n\n\\section{Atoms Trapped near Vibrating Photonic Structures}\n\\label{sec: framework}\nMicro- and nanophotonic traps rely on the optical near fields surrounding a photonic structure to spatially confine laser-cooled atoms in high vacuum. The optical fields are detuned from resonances of the atom such that they do not drive transitions between its internal (electronic) states. Confinement is achieved through gradients in the electric field that result in optical forces acting on the atom, analogous to free-space optical dipole traps \\cite{grimm_optical_2000}. In contrast to free-space setups, a dielectric photonic structure is used to pattern laser light in a way that creates local minima suitable for trapping atoms in the optical potential \\cite{chang_colloquium_2018}.\nThe light can either be guided by the structure such that atoms interact with the evanescent fields surrounding it \\cite{mabuchi_atom_1994,dowling_evanescent_1996,vernooy_quantum_1997,le_kien_atom_2004,vetsch_optical_2010,christensen_trapping_2008,goban_demonstration_2012,hung_trapped_2013,goban_atom-light_2014}, or scattered by the structure \\cite{ovchinnikov_atomic_1991,le_kien_microtraps_2009,thompson_coupling_2013,goban_superradiance_2015}; see \\cref{sec: photon appendix}. In either case a fraction of the light is absorbed, which can lead to a bulk temperature of the dielectric of several hundred kelvins due to the weak thermal coupling to its environment \\cite{wuttke_thermalization_2013}. In consequence, mechanical modes of the photonic structure are thermally excited. These mechanical modes (phonons) are in turn coupled to the external (motional) state of trapped atoms through the optical forces and other forces acting between the atoms and the structure.\n\n\\medskip{}\n\nAn individual atom trapped in the optical near field surrounding a mechanically vibrating photonic structure suspended in high vacuum can be modeled by the Hamiltonian\n\\begin{equation}\\label{eqn: total Hamiltonian}\n  \\op{\\Hamilfunc} = \\op{\\Hamilfunc}_\\text{at}  + \\op{\\Hamilfunc}_\\text{phn} + \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational}~.\n\\end{equation}\nThe first term describes the dynamics of the trapped atom in the absence of phonons. Atoms are trapped at a distance of a few hundred nanometers from the surface of the structure because the near fields decay on a scale given by the optical wavelength. At such distances, corrections $\\op\\pot_\\text{ad}$ to the optical potential $\\op\\pot_\\text{opt}$ due to surface effects like dispersion forces become relevant \\cite{le_kien_atom_2004,buhmann_dispersion_2012}. Optical forces and dispersion forces are additive to first order \\cite{fuchs_nonadditivity_2018}; hence, the total potential experienced by the atom is $\\op\\pot_0 \\equiv \\op\\pot_\\text{opt} + \\op\\pot_\\text{ad}$. While the potential in general couples all atomic degrees of freedom \\cite{dareau_observation_2018,meng_near-ground-state_2018}, we focus on scenarios without coupling of electronic and motional states and assume that the atom does not change its internal state. In this case $\\op\\pot_0 = V_0(\\op\\atpos)$; that is, the center of mass of the atom is subject to a potential $V_0$ which depends on the internal state of the atom (see \\cref{sec: atom appendix}). Approximating the potential as harmonic for an atom close to its trapped motional ground state yields the atom Hamiltonian\n\\begin{equation}\\label{eqn: atom Hamiltonian harmonic approximation}\n  \\op{\\Hamilfunc}_\\text{at} \\equiv \\sum_i \\hbar \\trapfreq_{ i} \\hconj\\hat{a}_i\\hat{a}_i ~,\n\\end{equation}\nwhere $i$ labels the three orthogonal symmetry axes of the potential in harmonic approximation, $\\trapfreq_{ i} $ are the trap frequencies, $\\hbar$ is the reduced Planck constant, and $\\hat{a}_i$ and $\\hconj\\hat{a}_i$ are ladder operators for the harmonic motion of the trapped atom.\n\nThe second term $\\op{\\Hamilfunc}_\\text{phn}$ in \\cref{eqn: total Hamiltonian} describes the free evolution of the phonon field of the photonic structure. Vibrations at frequencies relevant to atom traps can be modeled by linear elasticity theory because the corresponding phonon wavelengths are sufficiently large not to resolve the microscopic structure of the solid. Linear elasticity theory describes the dynamics of elastic deformations of a continuous body around its equilibrium configuration \\cite{achenbach_wave_1973,auld_acoustic_1973-1,gurtin_linear_1984}. The deformations are described by the \\emph{displacement field} $\\vec{\\ufieldcomp}$, a real-valued vector field which indicates magnitude and direction of the displacement of each point of the body from equilibrium at a given time. A quantum formulation of linear elasticity theory can be obtained through canonical quantization based on phononic eigenmodes; see \\cref{sec: phonon appendix}. The eigenmodes can be labeled by a suitable multi-index $\\gamma$ which may contain both discrete and continuous indices. In terms of ladder operators $\\op{b}_\\gamma$ and $\\hconj\\op{b}_\\gamma$ of the phonon field, the resulting phonon Hamiltonian is\n\\begin{equation}\\label{eqn: Hamiltonian phonons}\n  \\op{\\Hamilfunc}_\\text{phn} \\equiv \\sum_\\gamma \\hbar\\omega_\\gamma \\hconj\\op{b}_\\gamma\\op{b}_\\gamma~,\n\\end{equation}\nwhere the sum symbolizes an integral in the case of the continuous index components.\n\nThe last term $\\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational}$ in the Hamiltonian \\cref{eqn: total Hamiltonian} describes the coupling between the atomic motion and the phonon field. In order to obtain explicit expressions for the atom-phonon coupling, it is necessary to know how the potential experienced by the atom is changed by vibrations. Here, we give an overview of how this dependence can be modeled, while further details as well as explicit expressions for the resulting coupling constants in the case of a nanofiber-based atom trap are provided in \\cref{sec: interaction appendix}. The coupling arises both because vibrations displace the photonic structure relative to the atom and because they change the electromagnetic properties of the structure in two ways \\cite{zoubi_optomechanical_2016}: First, vibrations deform the surface of the structure, as determined by the displacement field $\\vec{\\ufieldcomp}$. Second, they locally change the refractive index and introduce birefringence (photoelastic effect), as determined by the \\emph{strain tensor} $\\tens{\\straintenscomp}$. The strain tensor describes deformations of the solid and has components $S^{ij} \\equiv \\pare{\\partial_i u^j + \\partial_j u^i}\/2$, where $\\partial_i$ indicates a spatial derivative. Both effects modify the photonic eigenmodes and hence the optical trapping fields. The optical fields and surface forces adapt to changes caused by vibrations on a timescale that is fast compared to the motion of the trapped atom. We can therefore treat the total potential as a functional $V[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}](\\vec{\\atpossymbol})$ which, in the absence of vibrations, reduces to the potential $V[\\boldsymbol{0},\\boldsymbol{0}](\\vec{\\atpossymbol}) \\equiv V_0(\\vec{\\atpossymbol})$ included in $\\op{\\Hamilfunc}_\\text{at}$.\n\nThermal vibrations only weakly modify the atom trap. In consequence, it is justified to expand the potential to linear order around \\mbox{$\\vec{\\ufieldcomp} = \\boldsymbol{0}$} and \\mbox{$\\tens{\\straintenscomp} = \\boldsymbol{0}$}, and approximate $V[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}] \\simeq V_0 + \\frechetDV_{(\\boldsymbol{0},\\boldsymbol{0})}[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}]$. The first-order term is the functional derivative of $V[\\vec{\\ufieldcomp}',\\tens{\\straintenscomp}']$, evaluated at $(\\vec{\\ufieldcomp}', \\tens{\\straintenscomp}') = (\\boldsymbol{0},\\boldsymbol{0})$ and in direction $(\\vec{\\ufieldcomp}$, $\\tens{\\straintenscomp})$; see %\n\\footnote{%\n\\label{note: functional derivative}\nThe Fr\u00e9chet derivative $D F$ of a functional $F[\\vec{x}]$ evaluated at $\\vec{x} = \\vec{a}$ and in direction $\\vec{n}$ is defined as \\cite{yamamuro_differential_1974,werner_funktionalanalysis_2011} %\n$$ D F_{\\vec{a}}[\\vec{n}] \\equiv \\lim_{h \\to 0} \\cpare{ F[\\vec{a} + h\\vec{n}] - F[\\vec{a}] }\/h~.$$\nThe derivative is linear in $\\vec{n}$, and can be used in a Taylor \\mbox{expansion \\cite{werner_funktionalanalysis_2011}}. In particular, it is suitable for the linear-order approximation $F[\\vec{x}] \\simeq F[\\vec{a}] + D F_{\\vec{a}}[\\vec{x}]$. The partial Fr\u00e9chet derivative $\\delta_{\\vec{x}}G$ of a multivariate functional $G[\\vec{x},\\vec{y}]$ with respect to $\\vec{x}$ evaluated at $(\\vec{x},\\vec{y}) = (\\vec{a},\\vec{b})$ and in direction $\\vec{n}$ is defined as \\cite{yamamuro_differential_1974}\n$$\\delta_\\vec{x} G_{(\\vec{a},\\vec{b})}[\\vec{n}] \\equiv \\lim_{h \\to 0} \\cpare{ G[\\vec{a} + h\\vec{n},\\vec{b}] - G[\\vec{a},\\vec{b}] }\/h~.$$\nPartial derivatives can be used to express the total derivative $D$ of a multivariate functional \\cite{yamamuro_differential_1974}, for instance\n$$D G_{(\\vec{a},\\vec{b})}[\\vec{n},\\vec{m}] = \\delta_\\vec{x} G_{(\\vec{a},\\vec{b})}[\\vec{n}] + \\delta_\\vec{y} G_{(\\vec{a},\\vec{b})}[\\vec{m}]$$ in the case of a bivariate functional.}.\nThis term approximates phonon-induced variations of the potential and acts as the atom-phonon interaction Hamiltonian\n\\begin{equation}\\label{eqn: definition atom-phonon interaction Hamiltonian}\n  \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational} \\equiv \\frechetDV_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\ufield},\\op{\\straintens}](\\op\\atpos)~.\n\\end{equation}\nTruncating the expansion at linear order corresponds to assuming that the atom interacts only with single phonons at a time. Since the potential depends on both displacement and strain, there are two contributions to the interaction Hamiltonian, a \\emph{displacement coupling} (\\text{dp}) due to the direct dependence of the potential on $\\vec{\\ufieldcomp}$, and a \\emph{strain coupling} (\\text{st}) due to the dependence on $\\tens{\\straintenscomp}$:\n\\begin{equation}\\label{eqn: atom-phonon interaction Hamiltonian contributions}\n  \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational} = \\delta_{\\vec{\\ufieldcomp}}V_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\ufield}] + \\delta_{\\tens{\\straintenscomp}}V_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\straintens}]~,\n\\end{equation}\nHere, $\\delta$ is the partial functional derivative \\cite{Note1}. The interaction Hamiltonian is linear in $\\op{\\ufield}$ and $\\op{\\straintens}$ because the functional derivative is linear. By expanding displacement and strain in terms of phononic eigenmodes, the Hamiltonian can thus be expressed in terms of a position-dependent, complex-valued \\emph{coupling function} $g_{\\gamma}(\\vec{\\atpossymbol})$ for each phonon mode $\\gamma$,\n\\begin{equation}\\label{eqn: general form atom-phonon interaction Hamiltonian}\n    \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational} = \\sum_\\gamma\\spare{g_{\\gamma}(\\op\\atpos) \\op{b}_\\gamma + \\text{H.c.}}~,\n\\end{equation}\nwhere $g_{\\gamma}(\\vec{\\atpossymbol}) = g^\\text{dp}_{\\gamma}(\\vec{\\atpossymbol}) + g^\\text{st}_{\\gamma}(\\vec{\\atpossymbol})$.\nThe coupling function $g^\\text{dp}_{\\gamma}(\\vec{\\atpossymbol})$ derives from displacement coupling and $g^\\text{st}_{\\gamma}(\\vec{\\atpossymbol})$ from strain coupling.\n\nFurthermore, we approximate the phonon-induced forces acting on a trapped atom as linear in the atom position by  expanding \\cref{eqn: definition atom-phonon interaction Hamiltonian} to first order around the trap minimum $\\vec{\\atpossymbol}_0$. The interaction Hamiltonian then takes the form %\n\\footnote{%\nThe term $\\frechetDV_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\ufield},\\op{\\straintens}](\\vec{\\atpossymbol}_0)$ at order zero in the expansion describes a light-induced change in the mechanical equilibrium configuration of the photonic structure. We may safely neglect this constant shift, because it is small compared to the dimensions of a nanoscale structure and therefore only weakly modifies its photonic and phononic spectrum.%\n}\n\\begin{equation}\\label{eqn: linear force interaction Hamiltonians}\n  \\begin{split}\n    \\op{\\Hamilfunc}_{{\\atomic\\text{-}\\vibrational}} &\\simeq \\sum_{i\\gamma}  \\hbar (\\hat{a}_i + \\hconj\\hat{a}_i) (g_{\\gamma i}\\op{b}_\\gamma + \\cconjg_{\\gamma i} \\hconj\\op{b}_\\gamma)~,\n  \\end{split}\n\\end{equation}\nwhere the coupling constants are\n\\begin{equation}\\label{eqn: definition coupling constants}\n    g_{\\gamma i} \\equiv \\frac{\\Delta \\atiposb}{\\hbar} \\partial_i g_{\\gamma}(\\vec{\\atpossymbol}_0)~.\n\\end{equation}\nThe length $\\Delta \\atiposb \\equiv \\sqrt{\\hbar\/(2 M \\trapfreq_{ i})}$ is the zero-point motion of the atom of mass $M$ in the trap. The coupling constants quantify the interaction of each phonon mode $\\gamma$ with the motion of the atom in direction $i$. Analogous to the coupling function, there are contributions from both displacement and strain coupling, $g_{\\gamma i} = g^\\text{dp}_{\\gamma i} + g^\\text{st}_{\\gamma i}$.\n\nThe variation of the optical potential caused by displacement can in general be modeled by perturbatively calculating the new photonic eigenmodes in the presence of shifted boundaries of the nanostructure \\cite{johnson_perturbation_2002}. The displacement has two effects: First, it shifts the photonic structure, together with the electromagnetic fields surrounding it, relative to the trapped atom. Second, it deforms the surface of the structure, leading to new photonic eigenmodes and thereby also deforming the electromagnetic fields. The first effect scales with the ratio between the displacement of the surface and the size of the atom trap (the extent of the wave function of the atom). The second effect, on the other hand, scales with the ratio between the displacement and the dimensions of the structure. Since the trap is typically at least one order of magnitude smaller than the photonic structure (see \\cref{sec: case study}), we neglect the second effect and assume that both optical and surface potential are displaced as a whole together with the fiber surface \\cite{le_kien_phonon-mediated_2007}. This model is particularly useful for structures such as nanofibers which have a simple geometrical shape and highly symmetric mechanical modes. The resulting displacement coupling functions $g_{\\gamma}^\\text{dp}(\\vec{\\atpossymbol})$ for a nanofiber-based atom trap in particular are given in \\cref{sec: interaction appendix}.\n\nStrain leads to changes in the optical potential through the photoelastic effect, which can be modeled by a strain-dependent permittivity tensor $\\prtrbd\\tens{\\relpermitttenscomp}[\\tens{\\straintenscomp}]$ \\cite{nelson_theory_1971,narasimhamurty_photoelastic_2012,wuttke_optically_2013}. The modified permittivity is then in general neither homogeneous nor isotropic, and results in modified electric fields $\\prtrbd\\vec{\\Efieldcomp}$ surrounding the fiber and thus in a modified optical potential $V_\\text{opt}[\\prtrbd\\vec{\\Efieldcomp}]$. In consequence, the total potential $V[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}]$ depends on strain. We neglect the influence of strain on the surface forces because they arise from the interaction of the atom with charges in a thin slice at the surface of the fiber and are largely independent of changes in the interior of the fiber \\cite{buhmann_dispersion_2012}. The strain coupling function $g_{\\gamma}^\\text{st}(\\vec{\\atpossymbol})$ can then be obtained by perturbatively calculating the new photonic eigenmodes in the presence of a modified permittivity; see \\cref{sec: interaction appendix}.\n\n\n\\medskip{}\n\nHaving obtained the Hamiltonian of the coupled atom-phonon system, we can now describe the resulting evolution of the atomic motion. The cold atom can absorb kinetic energy from the thermally excited phonon field of the photonic structure (\\emph{heating} of the atomic motion). Provided that the atom-phonon coupling is weak compared to the trap frequencies and the coherence time of phonon excitations, the phonon field can be adiabatically eliminated. The effective evolution of the density matrix $\\densopc(t)$ describing the motional state of the atom is then governed by a master equation \\cite{cohen-tannoudji_atom-photon_1998,breuer_theory_2002}; see \\cref{sec: interaction appendix}. Heating of the atom is reflected in the increase of the expected number of motional quanta $n_i(t) \\equiv \\textrm{tr} [\\densopc(t) \\hconj\\hat{a}_i \\hat{a}_i]$ along a spatial direction $i$. The population grows linearly with heating rate $\\Gamma_i^\\text{th}$ for sufficiently short times,\n\\begin{equation}\\label{eqn: short time population evolution}\n n_i(t) \\simeq \\Gamma_i^\\text{th} t~,\n\\end{equation}\nassuming that the atom is in the motional ground state at $t=0$.\n\nThe phononic eigenmodes supported by the photonic structure can feature both discrete and continuous frequency spectra. Discrete spectra are observed for phonon modes with a spacing in frequency that is larger than their damping rates. In contrast, if a set of modes has frequency spacings much smaller than their damping rates (e.g., because the mechanical excitation is efficiently transmitted from the structure to its suspension), the discrete mechanical resonances are no longer discernible, and the spectrum is effectively continuous. Hence, we distinguish the contribution $\\Gamma^{\\text{d}}_{i}$ of discrete mechanical resonances from the contribution $\\Gamma^{\\text{c}}_{i}$ of a continuum of phonon modes:\n\\begin{equation}\\label{eqn: heating rate contributions}\n \\Gamma^{\\text{th}}_{i} = \\Gamma^{\\text{c}}_{i} + \\Gamma^{\\text{d}}_{i}~.\n\\end{equation}\n\nFor continuous phonon modes, Fermi's golden rule can be employed to calculate the heating rate $\\Gamma^{\\text{c}}_{i}$ \\cite{cohen-tannoudji_atom-photon_1998}:\n\\begin{equation}\\label{eqn: Fermis golden rule}\n  \\Gamma^{\\text{c}}_{i} = 2\\pi \\bar{n}_i \\sum_{\\gamma_i} \\rho_{\\gamma_i} |g_{\\gamma_i i}|^2~.\n\\end{equation}\nThe sum runs over the discrete set of continuous phonon modes $\\gamma_i$ that are resonant with the trap, $\\omega_{\\gamma_i} = \\trapfreq_{ i}$. The thermal occupation of the resonant phonon modes is $\\bar{n}_i \\equiv 1\/\\spare{\\exp\\pare{\\hbar \\trapfreq_{ i}\/k_BT}-1}$, where $T$ is the temperature of the photonic structure and $k_B$ is the Boltzmann constant \\cite{gerry_introductory_2005}. The phonon density of states is given by the inverse slope of the phonon dispersion relation (band structure), $\\rho_\\gamma \\equiv | d\\omega_\\gamma \/ dp |^{-1}$, where $p$ is the propagation constant along the fiber; see \\cref{sec: phonon appendix}.\n\nThe discrete resonances have finite lifetimes corresponding to decay rates $\\kappa_\\gamma$ due to internal losses and nonzero coupling to the suspension. Adiabatic elimination of these discrete mechanical modes in general leads to the heating rate $\\Gamma^{\\text{d}}_{i}$ given in \\cref{eqn: torsional heating general} in \\cref{sec: interaction appendix} \\cite{cirac_laser_1992,wilson-rae_cavity-assisted_2008}. There are two limiting cases that are of interest in \\cref{sec: case study}: In the case where the atom-trap frequency is smaller than the lowest-frequency phonon mode $\\gamma_1$, $\\trapfreq_{ i} < \\omega_{\\gamma_1}$, and detuned from resonance, $\\kappa_{\\gamma_1} \\ll |\\trapfreq_{ i} - \\omega_{\\gamma_1}|$, the ground-state  heating rate of the atom is\n\\begin{equation}\\label{eqn: heating rate discrete phonon detuned}\n  \\Gamma_{i}^\\text{d} \\simeq 2 \\bar{n} \\kappa_{\\gamma_1} |g_{\\gamma_1 i}|^2 \\frac{\\trapfreq_{ i}^2 + \\omega_{\\gamma_1}^2}{(\\trapfreq_{ i}^2 - \\omega_{\\gamma_1}^2)^2}~.\n\\end{equation}\nIn the case where the atom trap is resonant with a single phonon mode $\\gamma$, $\\kappa_{\\gamma} \\gg |\\trapfreq_{ i} - \\omega_{\\gamma}|$, the rate is\n\\begin{equation}\\label{eqn: heating rate discrete phonon}\n  \\Gamma_{i}^\\text{d} \\simeq \\frac{4 \\bar{n}|g_{\\gamma i}|^2}{\\kappa_{\\gamma}}~,\n\\end{equation}\nwhere we assume $\\bar{n} \\gg 1$.\n\n\\medskip{}\n\nThe theory of atom-phonon interaction outlined in this section applies to any optical atom trap that relies on a photonic structure to shape light fields. The explicit calculation of atom-phonon coupling constants requires modeling of the dependence of the potential that the atom experiences on the displacement and the strain caused by the mechanical eigenmodes of the structure. Once the mechanical modes and corresponding atom-phonon coupling constants of a particular structure are known, \\cref{eqn: Fermis golden rule,eqn: heating rate discrete phonon detuned,eqn: heating rate discrete phonon}, or more generally \\cref{eqn: torsional heating general}, can be used to predict the phonon-induced heating of the atomic motion. In the next section, we apply this theory to explain heating rates observed in nanofiber-based atom traps.\n\n\n\\section{Case Study of a Nanofiber-based Trap}\n\\label{sec: case study}\n\\begin{table*}\n  \\begin{tabularx}{\\textwidth}{c @{\\qquad} l @{\\quad}  l @{\\qquad} l @{\\quad} l @{\\qquad} l @{\\quad} l X}\n    \\toprule\n    Trap & \\multicolumn{2}{c}{$\\text{T}_{01}$} & \\multicolumn{2}{c}{$\\text{L}_{01}$} & \\multicolumn{2}{c}{$\\text{F}_{11}$} & ~ \\\\\n    \\cmidrule{2-8}\n    & $|g^\\text{dp}_{\\gamma i}|\/2\\pi$ (\\si{\\hertz}) & $|g^\\text{st}_{\\gamma i}|\/2\\pi$ (\\si{\\hertz})\n    & $|g^\\text{dp}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$) & $|g^\\text{st}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$)\n    & $|g^\\text{dp}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$) & $|g^\\text{st}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$) \\\\\n    \\midrule\n    $r$ & \\num{0} & \\num{5.47e-08} & \\num{3.08e-09} & \\num{1.56e-08} & \\num{3.93e-4} & \\num{2.18e-08} \\\\\n    $\\varphi$ & \\num{0} & \\num{7.81e-4} & \\num{0} & \\num{7.76e-11} & \\num{2.28e-4} & \\num{2.99e-10}\\\\\n    $z$ & \\num{0} & \\num{2.19e-12} & \\num{0} & \\num{1.05e-4} & \\num{0} & \\num{1.13e-10} \\\\\n    \\bottomrule\n  \\end{tabularx}\n    \\caption{Atom-phonon coupling constants. Listed are the contributions of displacement (\\text{dp}) and strain (\\text{st}) coupling to the coupling constants. The displacement coupling constants $g^\\text{dp}_{\\gamma i}$ are calculated according to \\cref{eqn: 3d trap radiation pressure coupling constants}. The strain coupling constants  $g^\\text{st}_{\\gamma i}$ are obtained from \\cref{eqn: definition coupling constants} with the coupling functions listed in \\cref{tab: strain coupling functions} in \\cref{sec: interaction appendix}. Coupling to modes on the continuous $\\text{L}_{01}$ and $\\text{F}_{11}$ bands is independent of the position of the trap site along the fiber axis. In contrast, the strain coupling constants to the discrete $\\text{T}_{01}$ modes depend on the position since the torsional modes form standing waves; see \\cref{sec: phonon appendix}. Listed here are the maximal coupling constants; for radial motion, the coupling is maximal at the end of the nanofiber ($z = 0,L$), while it is maximal at the center of the nanofiber ($z = L\/2$) for the azimuthal and axial motion.}\n    \\label{tab: coupling constants}\n\\end{table*}\n\n\\begin{table}\n\\newcolumntype{C}[1]{>{\\centering\\arraybackslash}p{#1}}\n\\newcommand{$\\ll$}{$\\ll$}\n  \\begin{center}\n    \\begin{tabular}{c C{2.1cm} C{2.1cm} C{2.1cm} }\n      \\toprule\n      Trap & $\\text{T}_{01}$ \\qquad & $\\text{L}_{01}$ & $\\text{F}_{11}$ \\\\\n       \\midrule\n      $r$ & $\\ll$ & $\\ll$ & \\SI{446}{\\hertz} \\\\\n      $\\varphi$ & $\\ll$ & $\\ll$ & \\SI{340}{\\hertz} \\\\\n      $z$ & $\\ll$ & \\SI{8.36e-2}{\\hertz} & $\\ll$ \\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption{Atom heating rates. Listed are the contributions of the relevant phonon modes $\\text{T}_{01}$, $\\text{L}_{01}$, and $\\text{F}_{11}$ to the heating rate $\\Gamma_i^\\text{th}$ of a trapped atom in direction $i \\in \\{r,\\varphi,z\\}$ calculated according to \\cref{eqn: Fermis golden rule,eqn: heating rate discrete phonon detuned}. Contributions below $10^{-4}\\,\\si{\\hertz}$ are indicated by \\lq{}$\\ll$.\\rq{} The  rates are independent of the position of the trap site along the fiber. The fiber temperature is assumed to be $T = \\SI{805}{\\kelvin}$, and the remaining parameters are specified in \\cref{sec: case study appendix}.}\n    \\label{tab: heating rates}\n  \\end{center}\n\\end{table}\n\nLet us now use the framework sketched in \\cref{sec: framework} to study the phonon-induced heating rates of the atomic motion in a nanofiber-based two-color atom trap. In particular, we consider a cesium atom trapped in the evanescent optical field surrounding a silica nanofiber \\cite{dowling_evanescent_1996,le_kien_atom_2004}. The nanofiber is formed by the waist of an optical fiber which has been heated and pulled \\cite{ward_optical_2014}. There have been several experimental realizations of this nanophotonic atom trap configuration \\cite{vetsch_optical_2010,goban_demonstration_2012,kato_strong_2015,lee_inhomogeneous_2015,corzo_large_2016,ostfeldt_dipole_2017,meng_near-ground-state_2018,albrecht_fictitious_2016}. We calculate atom heating rates for the setup described in \\cite{albrecht_fictitious_2016}, where a measured heating rate of $\\Gamma_\\varphi^\\text{th} = \\SI{340(10)}{\\hertz}$ in the azimuthal direction was reported. In order to explicitly calculate the phonon-induced heating rates, it is necessary to know the mechanical eigenmodes of the nanofiber close to resonance with the trap frequencies and to obtain the atom-phonon coupling constants. The latter calculation requires knowledge of the trap potential as well as the photonic eigenmodes of the nanofiber. \\Cref{sec: photon appendix} summarizes the photonic eigenmodes of a nanofiber, and \\cref{sec: atom appendix} provides details on the resulting trapping potential. \\Cref{sec: phonon appendix} summarizes the phononic eigenmodes. In \\cref{sec: interaction appendix}, we derive the resulting atom-phonon coupling constants for a nanofiber-based trap. The parameters of the particular setup considered in this \\namecref{sec: case study} are listed in \\cref{sec: case study appendix}.\n\n\nTrapping of atoms is achieved by means of two lasers, one red and the other blue detuned with respect to the $D$ lines of cesium. The lasers are guided as photonic $\\text{HE}_{11}$ spatial modes in the nanofiber region; see \\cref{sec: photon appendix}. \\Cref{fig: trap} in \\cref{sec: interaction appendix} shows the resulting trapping potential. The red-detuned laser is coupled into the fiber at both ends, leading to a standing wave that confines the atoms in the axial direction and creates a one-dimensional optical lattice. The laser beams are linearly polarized when coupled into the fiber, which leads to \\emph{quasilinearly} polarized fields with intensity maxima at opposite poles of the fiber cross section in the nanofiber region \\cite{le_kien_state-dependent_2013}. The corresponding electric field profiles are listed in \\cref{sec: case study appendix}. The red- and blue-detuned field have orthogonal polarizations to obtain stronger azimuthal confinement \\cite{vetsch_optical_2010}. There is an offset magnetic field oriented perpendicular to the fiber axis ($z$ axis) along $\\vec{z}_B = \\cos(\\phi) \\unitvec_\\xpos + \\sin(\\phi) \\unitvec_\\ypos$, with $\\phi = \\SI{66}{\\degree}$.  Atoms are initially prepared in the Zeeman substate $F=4$, $M_\\atF = -4$ of the hyperfine structure, where the offset magnetic field provides the quantization axis. The magnetic field causes a slight azimuthal shift of the trap sites. Nonetheless, the symmetry axes of the potential at the trap minimum are to a good approximation aligned with the radial, azimuthal, and axial unit vectors of a cylindrical coordinate system whose $z$ axis coincides with the nanofiber axis. We can therefore use $i\\in \\{r,\\varphi,z\\}$ for the atom trap directions in the atom Hamiltonian \\cref{eqn: atom Hamiltonian harmonic approximation}. The resulting frequencies of the atom trap are $(\\trapfreq_{\\rpos}, \\,\\trapfreq_{\\phipos}, \\,\\trapfreq_{\\zpos}) = 2\\pi \\times(\\num{123}, \\,\\num{71.8}, \\,\\num{193})\\, \\si{\\kilo\\hertz}$.\n\nAn infinitely long nanofiber supports three phonon bands which do not have a cutoff at low frequencies: the torsional $\\text{T}_{01}$ band, longitudinal $\\text{L}_{01}$ band, and flexural $\\text{F}_{11}$ band; see \\cref{sec: phonon appendix}.\n\\Cref{fig: phonon modes} shows the displacement of the nanofiber caused by phonon modes on each of these bands. The torsional band is linear and the longitudinal band asymptotically linear for low frequencies, with speeds of sound $c_\\trans$ and $c_h$ introduced in \\cref{sec: phonon appendix}, respectively. The flexural band has a quadratic asymptote. The dispersion relations describing these bands as functions of the propagation constant $p$ are\n\\begin{align}\\label{eqn: fundamental phonon bands}\n  \\omega_\\text{T} &= c_\\trans |p| & \\omega_\\text{L} &\\simeq c_h |p| & \\omega_\\text{F} &\\simeq \\frac{c_h R}{2} p^2~.\n\\end{align}\nHere, $R$ is the radius of the nanofiber. These three fundamental bands are the only candidates for phonon-induced heating of the atomic motion since all other bands have frequencies much larger than the trap frequencies.\n\nIn experiments, the optical nanofibers used for atom trapping are typically realized as the waist of a tapered optical fiber \\cite{vetsch_optical_2010}. The mechanical eigenmodes of this system -- including the nanofiber, the tapers, and the surrounding macroscopic fiber -- can be calculated either analytically or using finite-element methods \\cite{wuttke_thermal_2013}. Since the fiber is finite in length, the eigenmodes are standing waves and the spectrum consists of discrete mechanical resonances. The system can in general support the same kinds of excitations as an infinite cylinder: torsional, longitudinal, and flexural. For some modes, the tapers act as reflectors and strongly localize them in the nanofiber region. Others are transmitted through the tapers and are delocalized over the entire fiber \\cite{wuttke_thermal_2013}. In practice, all modes are damped. Dissipation occurs, among others, due to clamping losses \\cite{pennetta_tapered_2016}, friction with the background gas \\cite{wuttke_optically_2013}, material losses \\cite{wiedersich_spectral_2000}, and surface losses \\cite{penn_frequency_2006}. Depending on the magnitude of the damping $\\kappa$ of each mode compared to the free spectral range (FSR), the actual spectrum ranges from  discrete ($\\text{FSR} \\gg \\kappa$) to continuous ($\\text{FSR} \\ll \\kappa$). In the case of a discrete spectrum, standing waves of finite lifetime $1\/\\kappa$ are a useful description of the mechanical dynamics of the fiber. In the limit of a continuous spectrum, the idealized eigenmodes of the system are no longer faithful representations, since the phonons interact too strongly with other degrees of freedom and are dissipated before they can form standing waves. Instead, it is more useful to represent the phonons as propagating modes of an infinite structure which interact with the atom once and then never return (analogous to an atom interacting with fiber-guided or free-space photons). Some of the damping mechanisms can be modeled theoretically \\cite{penn_frequency_2006,wiedersich_spectral_2000}. However, more reliable results are obtained by measuring damping rates for the particular fiber in use. We perform measurements of the mechanical modes of the particular nanofiber setup considered here \\cite{albrecht_fictitious_2016}, similar to \\cite{wuttke_thermal_2013,fenton_spin-optomechanical_2018}. While torsional resonances are clearly visible, there is no indication of resonantly enhanced longitudinal or flexural nanofiber modes. The mode of lowest frequency is at $\\omega_\\text{T} = 2\\pi \\times \\SI{258}{\\kilo\\hertz}$ with a wavelength of $\\SI{14.6}{\\milli\\meter}$ and a decay rate of $\\kappa = 2\\pi \\times \\SI{48(1)}{\\hertz}$. The torsional modes can be modeled faithfully by imposing hard boundary conditions on an elastic cylinder; see \\cite{wuttke_thermal_2013} and \\cref{sec: phonon appendix}. The resulting spectrum is a discrete subset of the $\\text{T}_{01}$ band of an infinite cylinder. In keeping with the absence of discrete resonances corresponding to longitudinal and flexural modes, we model these modes as the propagating modes of an infinite cylinder, with a continuous dispersion relation given by the longitudinal and flexural bands \\cref{eqn: fundamental phonon bands}. The form of the longitudinal and flexural mechanical bands and the corresponding eigenmodes are then determined by the elastic mechanical properties of silica and the fiber radius alone. The wavelengths of the modes resonant with the azimuthal trap frequency, for instance, are $\\SI{80.0}{\\milli\\meter}$ for the $\\text{L}_{01}$ mode and $\\SI{0.251}{\\milli\\meter}$ for the $\\text{F}_{11}$ mode.\n\nThe theory derived in \\cref{sec: framework} allows us to calculate atom heating rates based on these physical parameters. The only parameter not provided by ref.~\\cite{albrecht_fictitious_2016} is the fiber temperature $T$. We choose the temperature such that the azimuthal heating rate $\\Gamma_\\varphi^\\text{th}$ observed in \\cite{albrecht_fictitious_2016} is reproduced. Agreement with the measurement in ref.~\\cite{albrecht_fictitious_2016} is achieved for $T = \\SI{805}{\\kelvin}$, which agrees well with the temperature of $T = \\SI{850\\pm150}{\\kelvin}$ measured independently in \\cite{wuttke_thermalization_2013} for a similar nanofiber at the given transmitted laser power. Heating in the azimuthal direction is dominantly caused by resonant flexural $\\text{F}_{11}$ modes. To our knowledge, this is the first time that a theoretical prediction of the atom heating rate based on measured parameters and in quantitative agreement with measured heating rates has been obtained. We are then able to calculate the phonon-induced heating rates of the atomic motion in the radial, azimuthal, and axial direction, accounting for both displacement and strain coupling. The predicted atom-phonon coupling constants are listed in \\cref{tab: coupling constants} and the resulting heating rates in \\cref{tab: heating rates}.\n\nThe predicted heating rate for the radial degree of freedom is of a magnitude similar to the rate for the azimuthal degree of freedom. The calculated radial heating rate is $\\Gamma_r^\\text{th} = \\SI{446}{\\hertz}$, which agrees with the heating rate assumed in \\cite{reitz_coherence_2013} to explain measured $T_2^\\prime$ decoherence rates for nanofiber-trapped atoms. Heating along the radial axis, like heating in the azimuthal direction, is dominated by coupling to the resonant flexural $\\text{F}_{11}$ modes. The coupling constants in \\cref{tab: coupling constants} reveal that the coupling is due to displacement of the fiber surface, while coupling due to strain is lower by several orders of magnitude. \\latin{A priori}, both longitudinal $\\text{L}_{01}$ and flexural $\\text{F}_{11}$ modes couple to the radial motion by displacement. However, the flexural modes lead to much higher heating rates for two reasons: First, flexural modes displace the fiber surface by a factor of $|w^r_\\text{F}\/w^r_\\text{L}| \\simeq \\sqrt{E\/(2 \\rho)}\/(\\trapfreq_{\\rpos} \\PoissonnuR) \\simeq 10^5$ more than the longitudinal modes which leads to larger displacement coupling constants. Here, $w^r_\\text{F}$ and $w^r_\\text{L}$ are the radial components of the displacement eigenmode for the flexural and longitudinal modes, respectively. The quantity $E$ is Young's modulus and $\\nu$ is the Poisson ratio; together, they describe the elastic properties of the nanofiber. The quantity $\\rho$ is the mass density of the nanofiber and $\\trapfreq_{\\rpos}$ the radial trap frequency. The second reason is that the density of states of the flexural modes is larger than the one of longitudinal modes by a factor of $\\rho_{\\text{F} r}\/\\rho_\\text{L} \\simeq \\sqrt{c_h\/(2 \\trapfreq_{\\rpos} R)} \\simeq 100$, and the heating rates are enhanced accordingly; see \\cref{eqn: Fermis golden rule}.\n\nHeating in the axial direction is predicted to be predominantly due to strain coupling to the resonant longitudinal $\\text{L}_{01}$ mode, with a rate much smaller than the heating rates in the radial and azimuthal direction. To the best of our knowledge, the heating rate in the axial direction has not been measured so far.\n\nOne might expect heating by near-resonant torsional modes to be dominant because they are tightly confined to the nanofiber region, leading to Purcell enhancement of the coupling strength \\cite{gerry_introductory_2005}. The strain induced by torsional modes causes a tilt of the quasilinear polarization of the light fields, see \\cref{fig: discrete T modes blue and red scalar coupling r phi} in \\cref{sec: interaction appendix}, which leads to coupling to the azimuthal motion of the atom in particular. In the present case, the contribution of torsional modes to the heating is negligible due to the large detuning between the torsional mode and trap frequencies compared to the phonon decay rate. However, we can use \\cref{eqn: heating rate discrete phonon} with the coupling constants given in \\cref{tab: coupling constants}  to obtain an estimate of the heating rates expected in the case when the torsional modes are resonant (e.g., in the case the nanofiber is longer). In this worst-case scenario, the predicted contribution to the heating rate in the azimuthal direction is $\\Gamma^\\text{d}_\\varphi = \\SI{17.8}{\\hertz}$, while heating in the other trap directions is still below $10^{-4}\\,\\si{\\hertz}$, despite the Purcell enhancement. For the hypothetical case in which the torsional modes are not reflected at the ends of the nanofiber, our model predicts even lower heating rates. Hence, torsional modes are not a relevant source of heating in \\cite{albrecht_fictitious_2016}, even if they are resonant with the trap frequencies.\n\nIn summary, the atom heating in the radial and azimuthal direction observed in experiments is well explained by the displacement coupling to the continuous $\\text{F}_{11}$ band alone. In this case, \\cref{eqn: Fermis golden rule} simplifies to the single equation\n\\begin{align}\\label{eqn: final heating formula}\n  \\Gamma^\\text{th}_i &\\simeq \\frac{1}{2\\sqrt{2}\\pi}\\frac{k_B}{\\hbar} T M\\sqrt{\\frac{\\trapfreq_{ i}}{R^5 \\sqrt{E \\rho^3}}} & i \\in \\{ r,\\varphi\\}~,\n\\end{align}\nwhere we use that $\\hbar \\trapfreq_{ i} \\ll k_B T$, such that the thermal occupation of the phonon modes is $\\bar{n}_i \\simeq k_B T\/\\hbar \\trapfreq_{ i}$. This simple formula agrees exceedingly well with calculations considering all phonon modes and both displacement and strain coupling.\n\\begin{figure*}\n  \\centering\n  \\setlength{\\widthFigA}{180.9393pt}\n  \\setlength{\\marginLeftFigA}{25.9393pt}\n  \\setlength{\\marginRightFigA}{7pt}\n  %\n  \\setlength{\\widthFigB}{162pt}\n  \\setlength{\\marginLeftFigB}{7pt}\n  \\setlength{\\marginRightFigB}{7pt}\n  %\n  \\setlength{\\widthFigC}{160.6pt}\n  \\setlength{\\marginLeftFigC}{5.6pt}\n  \\setlength{\\marginRightFigC}{7pt}\n  %\n  \\parbox[t]{\\widthFigA}{\\vspace{0pt}\\includegraphics[width = \\widthFigA]{figure1a.pdf}}\n  %\n  \\parbox[t]{\\widthFigB}{\\vspace{0pt}\\includegraphics[width = \\widthFigB]{figure1b.pdf}}\n  %\n  \\parbox[t]{\\widthFigC}{\\vspace{0pt}\\includegraphics[width = \\widthFigC]{figure1c.pdf}}\\\\\n  %\n  %\n  \\caption{Atom heating rate in the radial and azimuthal direction calculated using \\cref{eqn: final heating formula} as function of (a) the nanofiber radius, (b) the temperature of the nanofiber, and (c) the power of the blue-detuned trapping laser. The difference between \\cref{eqn: final heating formula} and the full theory \\cref{eqn: heating rate contributions} is not discernible at the given scales. In \\subcrefandb{fig: heating vs laser power}{a}{b}, all other parameters, in particular the trap frequencies, are unchanged. In \\subcrefb{fig: heating vs laser power}{c}, the ratio between the power of the red- and blue-detuned laser is kept constant, $P_\\text{b} \/ P_\\text{r} = 14.24$. The relation between the total laser power and temperature is modeled as $T(P) = m_0 + m_1P + m_2P^2$, with $m_0 = \\SI{400}{\\kelvin}$, $m_1 = \\SI{24}{\\kelvin\/\\milli\\watt}$, $m_2 = \\SI{-0.062}{\\kelvin\/\\milli\\watt^2}$ based on the measurements in \\cite{wuttke_thermalization_2013} for a nanofiber of radius $R = \\SI{250}{\\nano\\meter}$ and length $L = \\SI{5}{\\milli\\meter}$. The temperature then varies from $T = \\num{427}$ to $\\SI{2298}{\\kelvin}$ over the shown range of laser power. The trap frequencies simultaneously increase from $(\\trapfreq_{\\rpos},\\,\\trapfreq_{\\phipos}) = 2\\pi \\times (\\num{29.1},\\,\\num{23.9})\\,\\si{\\kilo\\hertz}$ to $2\\pi\\times (\\num{291},\\,\\num{168})\\,\\si{\\kilo\\hertz}$. The remaining parameters are specified in \\cref{sec: case study appendix}.}\n  \\label{fig: heating vs laser power}\n\\end{figure*}\n\\Cref{fig: heating vs laser power} shows the dependence of the predicted heating rates in the radial and azimuthal direction on individual parameters, keeping the remaining parameters unchanged. Most pronounced is the scaling with the nanofiber radius as $\\Gamma^\\text{th}_i \\propto R^{-5\/2}$, see \\subcref{fig: heating vs laser power}{a}. The strong dependence on the radius is mostly due to the increased mechanical stability of larger nanofibers which leads to smaller vibrational amplitudes, see \\cref{eqn: displacement F modes low-frequency limit}, in addition to a lower density of states. In contrast, the dependence on the fiber temperature is linear, see \\subcref{fig: heating vs laser power}{b}, since the thermal occupation of the resonant phonon modes increases linearly with the temperature. Comparison of \\subcref{fig: heating vs laser power}{a} and \\subcref{fig: heating vs laser power}{b} shows that increasing the nanofiber radius by $\\SI{150}{\\nano\\meter}$ to $R = \\SI{400}{\\nano\\meter}$ at constant temperature has an effect comparable to cooling the fiber down to room temperature if all other parameters of the setup could be kept unchanged. \\subCref{fig: heating vs laser power}{c} shows the dependence on the power of the blue-detuned laser, where the ratio of the power of the red- and blue-detuned lasers is kept constant. The temperature of the nanofiber increases with increased laser power since there is more absorption in the fiber \\cite{wuttke_thermalization_2013}; see caption for details. Moreover, higher intensities lead to a tighter confinement of the atoms. The observed increase of the heating rate when raising the laser power is therefore caused by an increase of both the fiber temperature and the trap frequencies. While Young's modulus $E$ also slightly changes with $T$ \\cite{spinner_elastic_1956}, the influence of this effect on the heating rate is negligible due to the weak dependence, $\\Gamma^\\text{th}_i \\propto E^{-1\/4}$.\n\n\\medskip{}\n\nLet us now discuss ways to reduce the atom heating caused by coupling to the continuous $\\text{F}_{11}$ band. Lowering the overall fiber temperature in order to reduce the heating rates is difficult even in cryogenic environments because thermal coupling of the fiber to its surroundings is very weak \\cite{wuttke_thermalization_2013}. However, based on the above analysis, different strategies to minimize the heating rates are conceivable. First of all, the fiber radius should be chosen as large as possible while maintaining the optical properties required for atom trapping. A second approach is to design the nanofiber such that it supports discrete, well-resolved resonances of flexural modes. While precise predictions of phonon linewidths are difficult, it may be possible to optimize the taper at both ends of the nanofiber and ensure that flexural modes are reflected and confined to $z\\in[0,\\len]$ with narrow linewidths, while the transmission of light is not reduced \\cite{pennetta_tapered_2016}. Such a resonator of length $\\len$ for the flexural modes would effectively break the $\\text{F}_{11}$ band into a discrete set of frequencies $\\omega_m$, and allow us to detune the atom trap from resonance with these mechanical modes.\n\\begin{figure*}[t]\n  \\raggedright\n  \\setlength{\\widthFigA}{256.0448pt}\n  \\setlength{\\marginLeftFigA}{29.0448pt}\n  \\setlength{\\marginRightFigA}{7pt}\n  %\n  \\setlength{\\widthFigB}{233pt}\n  \\setlength{\\marginLeftFigB}{7pt}\n  \\setlength{\\marginRightFigB}{6pt} %\n  %\n  \\parbox[t]{\\widthFigA}{\\vspace{0pt}\\includegraphics[width = \\widthFigA]{figure2a.pdf}}\n  %\n  \\parbox[t]{\\widthFigB}{\\vspace{0pt}\\includegraphics[width = \\widthFigB]{figure2b.pdf}}\\\\\n  %\n  %\n  \\caption{Atom heating rate in the radial direction due to flexural resonator modes as a function of (a) the resonator length and (b) the trap frequency. In both cases, an exemplary decay rate of $\\kappa = 2\\pi\\times \\SI{1.2}{\\hertz}$ is assumed for all resonator modes. The bold yellow line corresponds to the heating rate experienced by an atom trapped at the center of the resonator, $\\zpos_0 = \\len\/2$, calculated according to \\cref{eqn: general heating discrete flexural modes}. The thin blue line represents a position-independent upper bound obtained by pretending that the atom sits at an antinode of each phonon mode simultaneously: In consequence, no resonance between the atom and resonator is masked by a vanishing position-dependent coupling rate. This approach is useful, since in experiments an entire ensemble of atoms is trapped at various positions along the fiber. The dashed red lines show the approximations \\cref{eqn: heating off-resonant 1,eqn: heating off-resonant 2,eqn: resonant heating}. \\subCrefb{fig: reduce heating}{a} assumes a trap frequency of $\\trapfreq_{\\rpos} = 2\\pi\\times \\SI{123}{\\kilo\\hertz}$, and \\subcrefb{fig: reduce heating}{b} assumes a resonator length of $\\len = \\SI{600}{\\micro\\meter}$.}\n  \\label{fig: reduce heating}\n\\end{figure*}\nThe flexural eigenmodes are then standing waves (see \\cref{sec: phonon appendix}), with frequency spectrum\n\\begin{align}\\label{eqn: flexural resonator spectrum}\n  \\omega_m &\\equiv m^2 \\frac{\\pi^2 R}{2\\len^2}\\sqrt{\\frac{E}{\\rho}}~, & m &\\in \\mathds{N}~.\n\\end{align}\nThe heating rate in the radial and azimuthal direction due to these flexural resonator modes then depends on the position $z_0$ of the atom along the fiber axis; see \\cref{sec: interaction appendix}. \\Cref{fig: reduce heating} shows the dependence of the heating rate on the resonator length and trap frequency. Three regimes are clearly distinguishable: First, the trap is resonant with a flexural phonon mode. Second, the trap is off resonant and lies below the fundamental resonator frequency. Third, the trap is off resonant and lies above the fundamental resonator frequency. Assuming high thermal occupation of the phonon modes, $\\bar{n}_m \\gg 1$, simplified expressions for the heating rate can be obtained for each regime. If the trap frequency is below the fundamental phonon frequency but still much larger than the corresponding decay rate, $\\kappa_1 \\ll \\trapfreq_{ i} < \\omega_1$, as well as far detuned, $|\\trapfreq_{ i}-\\omega_1|\\gg \\kappa_1$, heating is dominated by off-resonant interaction with the fundamental phonon mode alone. In this case, the heating rate can be approximated as\n\\begin{equation}\\label{eqn: heating off-resonant 1}\n  \\begin{split}\n    \\Gamma_i^\\text{th} &\\simeq \\Gamma_{i<}^\\text{nres} \\sin^2(\\pi \\zpos_0\/\\len)\\\\\n    \\Gamma_{i<}^\\text{nres} &\\equiv \\frac{16}{\\pi^9}\\frac{k_B}{\\hbar}\\frac{T M \\rho \\kappa_1 \\trapfreq_{ i}^3 \\len^7}{E^2 R^6} ~.\n  \\end{split}\n\\end{equation}\n\nIf the trap has a frequency larger than the fundamental resonator frequency, $\\trapfreq_{ i}\\gg \\omega_1$, while still being off resonant, $|\\trapfreq_{ i}-\\omega_m|\\gg \\kappa_m$, heating is mainly due to the low-frequency phonon modes below the trap frequency. Assuming in addition that the phonon decay rate is the same for all relevant modes, $\\kappa_m \\simeq \\kappa$, an upper bound for the heating rate can be obtained:\n\\begin{equation}\\label{eqn: heating off-resonant 2}\n \\Gamma_i^\\text{th} \\lesssim \\Gamma_{i>}^\\text{nres} \\equiv \\frac{2}{45 \\pi} \\frac{k_B}{\\hbar}\\frac{TM \\kappa \\trapfreq_{ i} \\len^3}{E R^4}.\n\\end{equation}\nHere, we replace the sine in the coupling constant with $1$ for all modes, pretending the atom is located at an antinode of all modes simultaneously as a worst-case estimate. This approximation is useful because in experiments many atoms at different sites along the fiber axis are trapped at the same time.\n\nIf the trapped atom is resonant with a flexural phonon mode $m$, $|\\trapfreq_{ i}-\\omega_m|\\ll \\kappa_m$, and the contributions of the off-resonant modes can be neglected, the heating rate is\n\\begin{equation}\\label{eqn: resonant heating}\n  \\begin{split}\n  \\Gamma_i^\\text{th} &\\simeq \\Gamma_{i}^\\text{res} \\sin^2(p_m \\zpos_0)\\\\\n  \\Gamma_{i}^\\text{res} &\\equiv \\frac{2}{\\pi}\\frac{k_B}{\\hbar} \\frac{T M \\trapfreq_{ i}}{\\len \\rho \\kappa_m R^2}~.\n  \\end{split}\n\\end{equation}\nThe limiting expressions \\cref{eqn: heating off-resonant 1,eqn: heating off-resonant 2,eqn: resonant heating} are shown as dashed black lines in \\cref{fig: reduce heating}. Note that the dependence on decay rate and resonator length is inverted for off-resonant heating, \\cref{eqn: heating off-resonant 1,eqn: heating off-resonant 2}, compared to resonant heating, \\cref{eqn: resonant heating}. This inversion is expected, since large phonon linewidths $\\kappa_m$ assist off-resonant coupling, while small linewidths lead to a larger resonant enhancement. Small resonator lengths $\\len$ lead to higher coupling constants (Purcell enhancement), which increases resonant heating due to a single mode. In contrast, large resonator lengths result in a higher number of low-frequency modes and hence overcompensate the decrease in coupling strength and increase the heating due to off-resonant interaction.\n\n\nIn \\cref{fig: reduce heating}, we exemplarily assume a decay rate of $\\kappa_m = 2\\pi \\times \\SI{1.2}{\\hertz}$ for all relevant flexural modes. This corresponds to a quality factor of $\\omega_r\/\\kappa_m = 10^5$ at the frequency of the radial trap. Quality factors of this magnitude have been achieved for silica microspikes by optimization of the shape of the taper \\cite{pennetta_tapered_2016}. \\subCref{fig: reduce heating}{a} shows that a decrease of the radial heating rate below the value expected without a resonator for flexural modes (see \\cref{tab: heating rates}) is predicted for resonator lengths $\\len \\lesssim \\SI{3}{\\milli\\meter}$. A length of $\\len = \\SI{50}{\\micro\\meter}$ to the very left of \\subcref{fig: reduce heating}{a} can still be achieved for nanofibers, the calculated heating rate due to flexural phonon modes with the given decay rate is then as low as \\SI{0.1}{\\milli\\hertz}. \\subCref{fig: reduce heating}{b} assumes a resonator length of $\\len = \\SI{600}{\\micro\\meter}$, achieving heating rates of around $\\SI{1}{\\hertz}$ and shows the dependence on the trap frequency. The spacing between resonances is on the order of $2\\pi \\times \\SI{50}{\\kilo\\hertz}$, which would indeed render it possible to detune the radial and azimuthal trap from resonance.\n\nThese findings suggest that it may be possible to significantly reduce the heating rate of atomic motion in nanofiber-based traps by two orders of magnitude or more through optimization of the phononic properties of the fiber. Moreover, the scaling of the heating rate with the mass of the trapped particles as $\\Gamma^\\text{th}_i \\propto M$ is highly relevant for optomechanical experiments. Setups with levitated nanoparticles, for instance, may feature comparable trap frequencies for particles that are orders of magnitude heavier than a single atom \\cite{magrini_near-field_2018,diehl_optical_2018}. In order to stably trap heavier particles using nanophotonic structures and successfully cool their motion, it is imperative to carefully manage vibrations of the structure, for instance by improving the mechanical stability or by tuning mechanical modes out of resonance with the particle motion.\n\n\\section*{Conclusion}\nIn this article, we formulate a general theoretical framework for calculating the effect of phonons on guided optical modes and the resulting heating of atoms in nanophotonic traps. Our results are applicable to nanophotonic cold-atom systems~\\cite{chang_colloquium_2018} and can readily be extended to the heating of dielectric nanoparticles trapped close to surfaces~\\cite{magrini_near-field_2018,diehl_optical_2018}. In a case study for the example of cold cesium atoms in a two-color nanofiber-based optical trap, we predict heating rates of the atomic center-of-mass motion which are in excellent agreement with independently measured values~\\cite{albrecht_fictitious_2016,reitz_coherence_2013}. In this system, the dominant contribution to heating stems from thermally occupied flexural modes of the nanofiber. We find that the heating rate scales with the fiber radius as $R^{-5\/2}$. As a general design rule, this implies that structures of larger lateral dimensions are preferable regarding heating, albeit at the expense of smaller mode confinement and, hence, potentially lower atom-photon coupling strength. Given the fact that the heating rate is directly proportional to the temperature of the nanophotonic structure, reducing the absorption losses of the guided trapping light fields is advisable~\\cite{ravets_intermodal_2013}. Moreover, heating is expected to decrease for smaller trap frequencies, $\\Gamma \\propto \\sqrt{\\omega}$. In general, our case study shows that careful design of the phononic properties of the nanophotonic system and, in particular, of its mechanical resonances is an effective strategy for reducing the heating. Finally, by providing a coherent theoretical framework in a single source, our work is instrumental in calculating, understanding, and managing heating in a plethora of nanophotonic traps.\n\n\\begin{acknowledgments}\nWe thank Y.~Meng for the experimental characterization of the torsional mode resonances of the tapered optical fiber in the nanofiber-based two-color trap setup. Financial support by the European Research Council (CoG NanoQuaNt) and the Austrian Academy of Sciences (\u00d6AW, ESQ Discovery Grant QuantSurf) is gratefully acknowledged. We acknowledge support by the Austrian Federal Ministry of Science, Research, and Economy (BMWFW).\n\\end{acknowledgments}\n\n\n\n\\section*{Power fluctuations of the trapping laser fields}\nThe two-color optical dipole trap used in \\cite{albrecht_fictitious_2016} is formed by a blue-detuned running-wave light field and a red-detuned field which is in a standing-wave configuration. Beyond polarization fluctuations, which are already treated in the main text of our manuscript, only power fluctuations are relevant for the blue-detuned field and would lead to resonant heating (via a shift of the trap center) of the trap's radial DOF, and to parametric heating (via a change of the trap frequency) of the azimuthal DOF, while no coupling occurs for the axial DOF. For the red-detuned light field, power fluctuations would lead to resonant heating along the radial direction, as well as parametric heating along azimuthal and axial directions. We characterized the power fluctuations of the trap lasers used in experiment~\\cite{albrecht_fictitious_2016}. The measured power noise characteristics cannot explain the large heating rates we observe. We estimate that heating due this technical noise amounts to a heating rate of about~$\\SI{0.02}{quanta\/\\milli\\second}$.\n\n\n\\section*{Off-resonant scattering}\nIn the two-color dipole trapping scheme, the blue-detuned light field leads to an increase of the potential energy of the electronic atomic ground state while the red-detuned light field lowers this energy. The trap is formed as the sum of both potentials, and the trap depth is smaller than the magnitudes of the individual shifts. For this reason, the nanofiber-based two-color trap shows a larger off-resonant scattering rate for a given trap depth compared to, e.g., single-color free-space dipole traps. The expected heating rate due off-resonant scattering amounts to roughly $\\SI{2}{quanta\/s}$ (calculated for the z-axis), much less than what is experimentally observed.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{figureSupplement.png}\n\t\\caption{Measured Stokes Raman scattering spectrum of three pieces of Liekki Passive-6-125 optical fiber. The wavelength of the fiber-guided pump light field is $\\lambda=\\SI{780}{nm}$. The black arrow indicates the Cesium $D_2$ resonance at $\\SI{852}{nm}$.}\n\t\\label{fig:Raman}\n\\end{figure}\n\n\\section*{Raman scattering}\nThe trapping light fields propagating in the tapered optical fiber give rise to Raman scattering in the fiber material, which is fused silica. The contribution of the blue-detuned light field to the relevant wavelength-range of the Raman-scattered light is much larger than that of the red-detuned light field and, therefore, we focus the discussion on the effect of that laser only. We measured the Raman scattering induced by a fiber-guided laser field of $\\SI{780}{\\nano\\meter}$ wavelength and recorded the Stokes-scattered light using a spectrometer. The test fiber was a Liekki Passive-6-125, of the same type as the one used in~\\cite{albrecht_fictitious_2016}. Figure~\\ref{fig:Raman} shows the measured power spectral density of the Raman signal over the frequency difference from the pump laser frequency for three different pieces of fiber. The signal below $\\SI{600}{\\centi\\meter^{-1}}$ is blocked by two edge filters that are required to sufficiently extinguish the pump light. The $D_2$ line of Cesium corresponds to a Stokes shift of about $\\SI{1100}{\\centi\\meter^{-1}}$ where there is still a sizable Raman signal. However, while the Raman-scattered light is many THz wide, the absorption linewidth of cold Cesium atoms is only a few MHz. In order to get a precise estimate, we measure the Raman-scattered signal induced by the blue-detuned trap light field using a SPCM. The fiber including the tapered section as used in the experiment~\\cite{albrecht_fictitious_2016} has a length of about $\\SI{5}{\\meter}$. All the light exiting the fiber passes a filter that removes the trapping light field. It then passes a narrow (FWHM $\\SI{0.12}{\\nano\\meter}$) bandpass filtering stage centered around the Cesium's $D_2$ line, which isolated the fraction of the Raman-scattered light which is (near-)resonant with the atoms' dominant optical transition. From that and further system parameters, we estimate that a single nanofiber-trapped Cesium atom absorbs about $\\SI{50}{photons\/\\second}$ of Raman light, which yields a heating rate even lower than that arising from off-resonant scattering of trap light fields described above.\n\n\\section*{Brillouin scattering}\nGuided light fields such as these that generate the optical dipole trap in experiment~\\cite{albrecht_fictitious_2016} can experience Brillouin scattering. For this, a phase matching condition between the optical mode and an acoustic mode of the nanofiber has to be fulfilled. For a nanofiber of $\\SI{250}{nm}$ radius and a guided light field of $\\SI{780}{nm}$ wavelength, the first matching for Brillouin backscattering occurs for an acoustic wave with a frequency of about $\\SI{11}{GHz}$ (calculations analogous to \\cite{florez_brillouin_2016}. The next resonances occur at higher acoustic frequencies. The spectral width of the Brillouin-scattered light amounts to a few $\\SI{10}{MHz}$, i.e., much narrower than for Raman scattering. The Brillouin-scattered light together with the pump light can, in principle, drive stimulated Raman transitions between different internal states of the nanofiber-trapped cold atoms. When the trapping-potential has a dependence on the internal atomic state (as it can be the case for nanofiber-based traps, see \\cite{le_kien_state-dependent_2013}, this could lead to heating. The only internal atomic states with a comparable energy separation are the two hyperfine ground-state manifolds of the Cesium atom (HFS splitting: about $\\SI{9}{GHz}$). However, the two-photon detuning for this hyperfine state-changing Raman transition is still about $\\SI{2}{GHz}$. Such a large detuning, combined with the low power of the Brillouin-scattered light that we estimate to be at the sub-nW level, and given the narrow width of the Brillouin signal, suggests that Brillouin scattering in the nanofiber is irrelevant for the observations in the manuscript. A similar estimation can be made for the standard fiber part. Here, the Brillouin shift amounts to about $\\SI{22.3}{GHz}$, i.e., an even larger two-photon detuning. Again, a scattered power in the sub-nW level is found. In the view of these numbers, we consider also Brillouin scattering in the bulk fiber to have a negligible effect on heating.\n\nFor the discussion of another mechanism related to Brillouin scattering, we consider the following scenario: Each of the two red-detuned trapping light fields gives rise to a Brillouin scattered (BS) light field that also propagates in the fiber. These two BS fields form an additional standing wave (SW) that has a randomly fluctuating phase with respect to the original trapping SW. This SW formed from BS results in a stochastic modulation a) of the axial trapping frequency and b) of the position of the minima of the trapping potential along the fiber. The former gives rise to parametric heating while the latter leads to resonant heating.\n\nWe first consider parametric heating resulting from BS-induced fluctuations of the axial trap frequency, $\\omega_z$. When a potential minimum of the trapping SW coincides with a potential minimum of the SW formed from BS, the resulting trap frequency is increased. When the relative phase between the two SWs is such that a potential minimum of the trapping SW falls onto a potential maximum of the SW formed from BS, the resulting trap frequency is decreased. Due to the relatively small detuning of the BS light (about 10~GHz vs. about 300~THz), we can neglect the mismatch between the periodicity of the SW formed from BS and the trapping SW. The rate of parametric heating then depends on the power spectral density (PSD) of the relative phase fluctuations of the two SWs evaluated at twice the axial trap frequency, $2\\omega_z$. The fluctuations of the relative phase between the two BS light fields is determined by the individual spectral widths of the two BS fields. The spectrum of BS light for a situation comparable to ours has been published in \\cite{beugnot_brillouin_2014}. An approximately Gaussian spectrum with a FWHM of about $\\SI{25}{MHz}$ was found. In order to apply the formalism by~\\cite{savard_laser-noise-induced_1997} to the calculation of heating rates, we normalize the frequency-integrated PSD of the individual BS light fields to the root-mean square value of the relative intensity noise. For the latter, the ratio of the powers of the trapping and the BS light fields serves as an upper estimate. The PSD of the SW formed from BS then follows from the (appropriately normalized) convolution of the two PSDs of the two BS fields. Using this approach, we compute an exponential increase of the temperature with a time constant $\\tau \\approx 7 \\times 10^7\\,\\mathrm{s}$. This corresponds to a heating rate for the axial degree of freedom on the order of $\\SI{e-8}{quanta\/\\second}$, i.e., ten orders of magnitude smaller than the heating rate measured for the azimuthal degree of freedom.\n\nWe now consider resonant heating that originates from BS-induced position fluctuations of the trapping potential minimum. The heating rate depends on the PSD of the position fluctuations evaluated at the trap frequency. The displacement reaches a maximum value,  $\\Delta z_{\\rm max}$, when the SW formed from BS is shifted by a quarter of the wavelength of the trapping light with respect to the trapping SW. Again, we assume the PSD of the position fluctuations to have the above-mentioned Gaussian spectrum. We use $\\Delta z_{\\rm max}$ as a worst-case estimate for the RMS position fluctuation. For our setting, we estimate a maximal displacement of $\\Delta z_\\mathrm{max} \\approx 10^{-4}\\,\\mathrm{nm}$. This yields a heating rate in the axial direction on the order of $4 \\times 10^{-5}\\,\\mathrm{quanta\/s}$. Thus, we exclude BS-induced fluctuations of the axial trap frequency, or of the axial position of the trapping potential minimum, as mechanisms that currently limit the lifetime of atoms in our nanofiber-based trap.\n\n\\section*{Blackbody radiation and Johnson--Nyquist noise}\nBlackbody radiation and Johnson--Nyquist noise are fundamental processes which have been studied extensively in the context of atom chips. For example, the authors of \\cite{henkel_loss_1999} quantitatively estimate the expected heating rate for a spin confined in a harmonic potential with $\\SI{100}{kHz}$ trap frequency in close proximity to a material half space. While the surface-induced heating rates can be comparably large for atoms close to a conductive material such as copper (on the order of $\\SI{10}{quanta\/s}$), the rates are only on the order of $\\SI{e-14}{quanta\/s}$ for an atom $\\SI{200}{nm}$ away from a glass surface, mainly thanks to the low electrical conductivity of glass. Thus, these heating mechanisms are negligible for our experimental conditions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nLet us consider variational problems, consisting in the minimization of\n$$J \\,:\\, u \\mapsto \\int_\\Omega L(x,u(x),\\nabla u(x)) \\, \\text{\\normalfont d}\\mu(x) $$\nwhere the usual Lebesgue measure is replaced by a generic Borel measure~$\\mu$, under possible $\\mu$-a.e.\\@ or boundary constraints. In calculus of variations, the direct method consists in extracting a converging subsequence (in a suitable sense) from a minimizing sequence, thanks to a compactness result on the set of admissible functions, and to conclude by semi-continuity of the functional $J$. For our functional, two problems appear:\n\\begin{itemize}\n\\item which functional space should we consider in order to give a sense to the gradient $\\nabla u$? More precisely, if $\\mu$ is the Lebesgue measure or has a density $f$ bounded from above and below, we can work in the classical Sobolev space $H^1(\\Omega)$ (which is exactly the set of functions $u \\in L^2_\\mu$ having weak derivatives in $L^2_\\mu$), but it is not so clear if this assumption on $f$ does not hold or if $\\mu$ has a singular part.\n\\item does there exist a compactness result which allows to extract from a minimizing sequence a subsequence converging, in a suitable sense, to an admissible function? For instance, in the classical Sobolev space $H^1(\\Omega)$, the Rellich theorem allows to extract from any bounded sequence a strongly-convergent subsequence in $L^2(\\Omega)$ which is a.e.\\@ convergent on $\\Omega$.\n\\end{itemize}\n\nLet us fix more precisely the notations. Let $\\Omega$ be a bounded open set of $\\mathbb{R}^d$ and $f$ a measurable and a.e.\\@~positive function on $\\Omega$. If we assume $f$ to be bounded from below and above, it is obvious that the set\n$$ \\{u \\in L^2_\\mu(\\Omega) : \\nabla u \\text{ exists in the weak sense and belongs to } L^2_\\mu(\\Omega)^d \\} $$\nis exactly the classical Sobolev space $H^1(\\Omega)$, since the $L^2_\\mu$-norm is equivalent to the usual $L^2$-norm on $\\Omega$. If $f$ is only assumed to be positive, for $u \\in L^2(\\Omega)$, the Cauchy-Schwarz inequality gives\n$$ \\int_\\Omega |u(x)| \\, \\text{\\normalfont d} x = \\int_\\Omega \\left(|u(x)| \\sqrt{f(x)}\\right) \\frac{\\, \\text{\\normalfont d} x}{\\sqrt{f(x)}} \\leq \\left( \\int_\\Omega |u(x)|^2 f(x) \\, \\text{\\normalfont d} x \\right)^{1\/2} \\, \\left( \\int_\\Omega \\frac{\\, \\text{\\normalfont d} x}{f(x)} \\right)^{1\/2} $$\nthus, under the assumption\n\\begin{equation} 1\/f \\in L^1(\\Omega) \\label{f} \\end{equation}\nwe have the continuous embedding\n$$ L^2_f(\\Omega) := \\left\\{u : \\int_{\\Omega} |u(x)|^2 f(x) \\, \\text{\\normalfont d} x < +\\infty \\right\\} \\hookrightarrow L^1(\\Omega). $$\nIn this case, any function $f \\in L^2_f(\\Omega)$ has a gradient $\\nabla u$ in the weak sense, since it is locally integrable on~$\\Omega$, and we can define the weighted Sobolev space with respect to $f$\n$$H^1_f(\\Omega) = \\left\\{ u \\in L^2_f(\\Omega) \\; : \\; \\nabla u \\in L^2_f(\\Omega) \\right\\}. $$ \nMore generally, if $p \\in \\, ]1,+\\infty[ \\, $, the assumption\n$$ (1\/f)^{1\/(p-1)} \\in L^1_{loc}(\\Omega) $$\nis a well-known sufficient condition to define the weighted Sobolev space $W^{1,p}(\\Omega)$ with respect to $f$ (see~\\cite{kufner} for more details). For our problem, if the Lagrangian functional is quadratic with respect to the gradient, for instance\n$$J(u) = \\int_\\Omega |\\nabla u|^2 \\, \\text{\\normalfont d}\\mu, $$\nit means that the set of admissible functions is well-defined if $\\mu$ has a density $f$ such that $1\/f$ is integrable: it is the set of the elements of the weighted Sobolev space $H^1_f$ satisfying the constraints.\n\nIf $\\mu$ is absolutely continuous with density $f$, the weighted Lebesgue space with respect to $f$ is exactly the space $L^p_\\mu$, so that the space $H^1_f$ can be seen as a Sobolev space with respect to the measure $\\mu$. A natural generalization consists in defining the Sobolev space with respect to the measure $\\mu$, without condition on its density or when $\\mu$ is not anymore assumed to be absolutely continuous with respect to~$\\mathcal{L}^d$. There exists some general definitions of the Sobolev space in a generic metric measure space $(X,d,\\mu)$ (see \\cite{haj2}), but we will not enter to the details of this notions in this paper and we prefer to focus on the case of an open set of $\\mathbb{R}^d$.\n\nWe begin this paper by an overview of the definitions and already known results about this Sobolev spaces \\cite{bbs, fra, f-m, preiss, zhikov2, zhikov}, and present several new results: in particular, we give a precise description of the tangent space to any measure $\\mu$ on the real line. As a corollary of this result, we show a compactness result in $H^1_\\mu$, which states precisely that any bounded sequence admits a pointwise $\\mu$-a.e.\\@ convergent subsequence on the set of points where the tangent space is not null (this result is already known in any dimension under strong conditions on the measure $\\mu$, when the compact embedding of the Sobolev space $W^{1,p}_\\mu$ with respect to $\\mu$ into the Lebesgue space $L^p_\\mu$ still holds; see \\cite{bf, h-k}).\n\nThis is applied to a variational problem coming from optimal transportation: we consider the minimization of the functional\n$$ J(T) = \\int_\\Omega L(x,T(x),D_\\mu T(x)) \\, \\text{\\normalfont d}\\mu(x) $$\namong all the maps $T:\\Omega \\mapsto \\mathbb{R}^d$ which admit a Jacobian matrix $D_\\mu T$ with respect to $\\mu$ and under a constraint on the image measure $T_\\# \\mu$ (it corresponds to the classical Monge-Kantorovich optimal transportation problem \\cite{vil} if $L$ does not depend on its third variable, and is linked to minimization problems under volume-preservation or area-preservation constraints \\cite{tann}). In the one-dimensional case, we get the existence of a solution for any measure $\\mu$ (the optimal map is known if $\\mu$ is assumed to be the uniform measure on the interval, see \\cite{ls} for details). However, we are not able to give a precise description of the tangent space and to obtain the existence of solution to this transport problem in the most general case in any dimension.\n\n\\section{Sobolev spaces with respect to a measure}\n\nThis section is devoted to an overview of the definitions and already known results about tangent spaces to a generic Borel measure $\\mu$ and Sobolev spaces associated to this measure. First, let us recall that there exist some notions of Sobolev spaces in arbitrary metric measure spaces $(X,d,\\mu)$, for instance in the papers by Shanmugalingam \\cite{sha}, Haj\\l asz \\cite{haj} or Haj\\l asz and Koskela \\cite{h-k} (see \\cite{haj2} for a global summary of this notions). In our case, a usual method consists in defining the tangent space to $\\mu$ (which is a function defined $\\mu$-a.e.\\@ on $\\mathbb{R}^d$ and taking values in the set of linear subspaces of $\\mathbb{R}^d$), and the gradient with respect to $\\mu$ for a regular function $u$ through\n$$ \\nabla_\\mu u (x) = p_{T_\\mu(x)} (\\nabla u(x)) \\quad \\text{for }\\mu\\text{-a.e.\\@ } x \\in \\mathbb{R}^d, $$\nwhere $p_{T_\\mu(x)}$ is the orthogonal projection on $T_\\mu(x)$ in $\\mathbb{R}^d$. Then we consider for the Sobolev space $H^1_\\mu$ the closure of $C^\\infty\\left(\\overline{\\Omega}\\right)$ for the norm\n$$u \\in C^\\infty\\left(\\overline{\\Omega}\\right) \\mapsto ||u||_{L^p_\\mu} + ||\\nabla_\\mu u||_{L^p_\\mu}. $$\nThere exist several ways to define the tangent space of a generic measure $\\mu$. Preiss \\cite{preiss} gives a method based on the idea of blow-up: a $k$-dimensional subspace $P_\\mu$ is said to be an approximate tangent space of $\\mu$ at $x$ if we have, for some $\\theta > 0$, the following convergence in the vague topology of measure when $\\rho$~goes to $0$:\n$$ \\mu(x+ \\rho\\, \\cdot \\,) \\rightharpoonup \\theta \\mathcal{H}^k|_{P_\\mu}. $$\nIn order to examine variational problems, Bouchitt\\'e {\\it et al}.\\@ \\cite{bbs} have introduced a dual-formulation of the tangent space: it is the $\\mu$-ess.\\@ union (see \\cite{c-v} or later) $x \\mapsto Q_\\mu(x)$ of the tangent fields, {\\it i.e.} the vector fields belonging to\n$$ X^{p'}_\\mu = \\{ \\phi \\in (L^{p'}_\\mu)^d : \\div(\\mu \\phi) \\in L^{p'}_\\mu \\},  $$ \nwhere the operator $\\div(\\mu v)$ is defined in the distributional sense. Fragal\\`a and Mantegazza \\cite{f-m} have noticed that, with this notation, we have the inclusion $Q_\\mu(x) \\subseteq P_\\mu(x)$ for $\\mu$-a.e.\\@ of $\\mathbb{R}^d$ (see the PhD.\\@ thesis \\cite{fra} for a complete overview and more details about these definitions). \n\nWe are interested in another way to define tangent and Sobolev spaces, introduced by Zhikov \\cite{zhikov2, zhikov}. Let $\\Omega$ be a bounded open set of $\\mathbb{R}^d$ and $\\mu$ a finite positive measure on $\\Omega$ We will say that $u \\in L^2_\\mu$ belongs to the space $H^1_\\mu$ if it can be approximated by a sequence of regular functions whose gradients have a limit in the space $L^2_\\mu$:\n$$u \\in H^1_\\mu \\; \\Longleftrightarrow \\; \\exists (u_n)_n \\in C^\\infty(\\overline{\\Omega}), \\, v \\in (L^2_\\mu)^d : \\, \n\\left\\{\n\\begin{array}{l}\nu_n \\to u \\\\\nv_n \\to v\n\\end{array}\n\\right.\n\\; \\text{for the }L^2_\\mu\\text{-norm}.  $$\nThe set of these limits $v$ is denoted by $\\Gamma(u)$, and its elements are called gradients of $u$. In general, $u$ can have many gradients (see below the example of a measure supported on a segment of $\\mathbb{R}^2$), and it is obvious that $\\Gamma(u)$ is a closed affine subspace of $(L^2_\\mu)^d$ with direction $\\Gamma(0)$. The projection of $0$ onto this subspace (in the Hilbert space $(L^2_\\mu)^d$) is thus the unique element of $\\Gamma(u)$ with minimal $L^2_\\mu$-norm: we call it {\\it tangential gradient} of $u$ with respect to $\\mu$.\n\n\\noindent {\\bf Pointwise description of $\\nabla_\\mu u$ and tangent space to $\\mu$.} We define the tangent space to $\\mu$ as follows: the space $\\Gamma(0)$ can be seen as the set of vector-valued functions which are pointwise orthogonal to the measure $\\mu$. Let us denote by $(e_1, \\dots, e_d)$ the canonical basis of $\\mathbb{R}^d$, and set\n$$\\xi_i = p_{\\Gamma(0)}(e_i)$$\nwhere the projection is taken in the Hilbert space $L^2_\\mu$ (here $e_i$ is seen as a constant function on $\\Omega$). For $x \\in \\Omega$, we denote by\n$$ T_\\mu(x) = \\left(\\operatorname{Vect} (\\xi_1(x), \\dots, \\xi_d(x))\\right)^\\perp $$\nand call $T_\\mu(x)$ (which is defined for $\\mu$-a.e.\\@ $x \\in \\Omega$) the tangent space to $\\mu$ at $x$. Then, the following equivalence holds:\n$$ v \\in \\Gamma(0) \\quad \\Longleftrightarrow \\quad \\text{for $\\mu$-a.e.\\@ } x \\in \\Omega, \\; v(x) \\perp T_\\mu(x). $$\nThis result, combined to the orthogonality property of $\\nabla_\\mu u$ in $L^2_\\mu$, implies a pointwise description of the tangential gradient:\n\\begin{prop} Let $u \\in H^1_\\mu$. Then, for $v \\in \\Gamma(0)$, the function\n$$x \\in \\Omega \\mapsto p_{T_\\mu(x)}(v(x)) $$\nis independent of the function $v$ and only depends on $u$, and we have\n$$\\nabla_\\mu u(x) = p_{T_\\mu(x)} \\qquad \\text{for } \\mu\\text{-a.e.\\@ } x \\in \\Omega. $$\n\\end{prop}\n\n\\noindent {\\bf Some natural examples.} We can see that the words ``{\\it tangential} gradients'' are quite natural in the following cases:\n\\begin{itemize}\n\\item if $\\mu$ is the Lebesgue measure $\\mathcal{L}^1$ concentrated on the segment $I=[0,1]\\times\\{0\\} \\times \\dots \\times \\{0\\}$, then $T_\\mu$ is the line $\\mathbb{R} \\times \\{0\\} \\times \\dots \\times \\{0\\}$ a.e.\\@ on $I$ and\n$$H^1_\\mu = \\left\\{u \\in L^2_\\mu : \\frac{\\partial u}{\\partial x_1} \\in L^2_\\mu\\right\\} \\quad \\text{and} \\quad \\nabla_\\mu u = \\left(\\frac{\\partial u}{\\partial x_1},0,\\dots,0\\right); $$\n\\item more generally, if $\\mu$ is the uniform Hausdorff measure supported on a $k$-dimensional manifold $M$, then $T_\\mu$ is the tangent space to $M$ in the sense of the differential geometry.\n\\end{itemize}\nLet us remark that, if $v$ is a tangent field as defined above, {\\it i.e.}\\@ the operator $\\div(\\phi \\mu)$ is continuous for the $L^2_\\mu$-norm on $\\mathcal{D}(\\Omega)$, we have for any sequence $(u_n)_n$ of smooth functions having $0$ for limit in $L^2_\\mu$:\n$$ \\left|\\int_\\Omega \\nabla u_n \\cdot \\phi \\, \\text{\\normalfont d}\\mu \\right| \\leq C \\, ||u_n||_{L^2_\\mu} \\to 0.  $$\nThen, if $v \\in \\Gamma(0)$, we have $v \\cdot \\phi$ in $L^2_\\mu$ for any element $\\phi \\in X^2_\\mu$. We deduce that, with the above notations, the space $Q_\\mu(x)$ is pointwise orthogonal to $F_\\mu(x)$: it means that, up to a $\\mu$-negligible set, we have the inclusion between tangent spaces\n$$Q_\\mu(x) \\subseteq T_\\mu(x). $$\nWe are not able for the moment to prove the inverse inclusion, but the equality between this linear spaces holds for all the examples that we have~studied.\n \n\\section{Precise description and compactness result in one dimension}\n\n\\subsection{The main results}\n\nLet us now give a precise pointwise description of the tangent space $T_\\mu(x)$ when $d=1$ and $\\Omega$ is a bounded interval of $\\mathbb{R}$ (which we denote by $I$). In this case, there are only two options for $T_\\mu(x)$ which are of course $\\{0\\}$ and $\\mathbb{R}$, and the definitions of the tangent space give the following characterizations:\n\n\\noindent {\\bf Fact.} Let $B \\subseteq I$ be a Borel set with $\\mu(B) > 0$. We have the following implications:\n\\begin{enumerate}\n\\item if any $v \\in \\Gamma(0)$ is $\\mu$-a.e.\\@ null on $B$, then $T_\\mu = \\mathbb{R}$ $\\mu$-a.e.\\@ on $B$;\n\\item if, for any $u \\in H^1_\\mu$, there exists a gradient of $u$ which is $\\mu$-a.e.\\@ null on $B$, then $T_\\mu = 0$ $\\mu$-a.e.\\@ on $B$;\n\\item if there exists a gradient of $0$ which is positive $\\mu$-a.e.\\@ on $B$, then $T_\\mu$ = $0$ $\\mu$-a.e.\\@ on $B$.\n\\end{enumerate}\n\n\\noindent {\\bf Notations.} We denote by:\n\\begin{itemize}\n\\item $\\mu = \\mu_a + \\mu_s$ , where $\\mu_a$ and $\\mu_s$ are respectively the absolutely continuous and the singular part of $\\mu$ with respect to the Lebesgue measure;\n\\item $A$ a Lebesgue-negligible set on which is concentrated $\\mu_s$;\n\\item $f$ the density of $\\mu_a$, and\n$$ M = \\left\\{x \\in I : \\; \\forall \\varepsilon > 0, \\; \\int_{I \\cap B(x,\\varepsilon)} \\frac{\\, \\text{\\normalfont d} t}{f(t)} = +\\infty \\right\\}$$\nwhich is a closed set of $I$ verifying $1\/f \\in L^1_{loc}(I\\setminus M)$.\n\\end{itemize}\nNotice that if $\\mu$ is absolutely continuous with respect to $\\mathcal{L}^1$, the Sobolev space with respect to $\\mu$ (thus, to $f$) is well-defined exactly ``outside of the set $M$''. In our case, we find an analogous result:\n\n\\begin{theo}\nFor $\\mu$-a.e.\\@ $x \\in I$, the tangent space is given by\n$$ T_\\mu(x) =\n\\left\\{\n\\begin{array}{ll}\n\\{0\\} & \\text{if } x \\in M\\cup A \\\\\n\\mathbb{R} & otherwise.\n\\end{array}\n\\right.\n $$\n\\end{theo}\n\nLet us give a short comment of this result. Saying that the tangent space is $\\mathbb{R}$ on a set $B$ means exactly that, if $u \\in H^1_\\mu$ is given, all the gradients of $u$ are equals on $B$. In our case, let us denote by $V = I \\setminus (M \\cup A)$ and $U = I \\setminus M$. Notice that $U$ is an open subset of $I$ coinciding with $V$ up to the $\\mathcal{L}^1$-negligible set $A$. Let us fix $u \\in H^1_\\mu$. We will prove that the distributional derivative of $u|_U$ is well-defined, belongs to $L^2_f$ and that, if $v \\in \\Gamma(u)$, $u'=v$ $\\mu$-a.e.\\@ on $V$; therefore, $u'|_V$ is the only gradient of $u$ on the set $V$.\n\nFirst, let us recall that if $u$ is an element of $L^2_\\mu$, its restriction to $U$ belongs to $L^2_f(U)$, which is included into $L^1_{loc}(U)$ by definition of $M$. The weak derivative of $u|_U$ is thus well-defined. If $\\phi$ is a test function with support in $U$ and $(u_n)_n$ a sequence of regular functions such that $(u_n,u'_n) \\to (u,v)$ in $L^2_\\mu$, testing $v-u'$ against $\\phi$ gives\n$$|< v-u',\\phi>_{\\mathcal{D}'(U),\\mathcal{D}(U)}| = \\lim\\limits_{n \\to +\\infty} \\left| \\int_I (u_n-u) \\phi'\\right| \\leq ||u_n-u||_{L^2_f} \\left(\\int_I \\frac{(\\phi')^2}{f} \\right)^{1\/2} $$\nwhere the last inequality comes from the H\\\"o lder inequality, and the last term is finite since $\\phi'$ is bounded and $1\/f$ integrable on the support of $\\phi$. This proves that $u'_n \\to v$ in the sense of distributions on $U$. Then $v|_U$ is the weak derivative of $u$ on this set, and we know that $v|_U \\in L^2_f(U)$.\n\nFinally, any element $u \\in H^1_\\mu$ gives by restriction an element of the weighted Sobolev space $H^1_f(U)$ and, on $U$, $\\nabla_\\mu u$ and $u'$ are coinciding a.e.\\@ for the regular part $f \\, \\mathcal{L}^d$ of $\\mu$. To summarize, we have just proved the following:\n\n\\begin{prop} We denote by $V = I \\setminus (M\\cup A)$. Let us recall that $A$ is Lebesgue-negligible and that $V \\cup A$ is open; we still denote by $H^1_f(V)$ the weighted Sobolev space $H^1_f(V\\cup A)$. Then, a measurable function $u$ belongs to the Sobolev space $H^1_\\mu(I)$ if and only if the two following conditions are satisfied:\n$$u \\in L^2_\\mu(I) \\quad and \\quad u|_V \\in H^1_f(V) $$\nand in this case, its $\\mu$-Sobolev norm is given by\n$$||u||_{H^1_\\mu(I)}^2 = ||u||_{L^2_\\mu(I)}^2 + ||u'||_{L^2_f(V)}^2 $$\nwhere $u'$ is the weak derivative of $u|_V$.\n\\end{prop}\n\n\\noindent {\\bf Compactness result in $H^1_\\mu(I)$.} In order to examine variational problems in this Sobolev spaces, the following compactness result is useful (it is already known in the case of the Lebesgue measure, as a consequence of the Rellich theorem):\n\n\n\\begin{prop} Let $(u_n)_n$ be a bounded sequence of $H^1_\\mu(I)$. Then there exists a subsequence $(u_{n_k})_k$ which admits a pointwise limit $u$ on $\\mu$-a.e.\\@ every point on which $T_\\mu$ is $\\mathbb{R}$.\n\\end{prop}\n\n\\begin{proof} We know that $V$ is exactly (up to a $\\mu$-negligible set) the set of points where $T_\\mu$ is $\\mathbb{R}$. We still denote by $U = I \\setminus M$. $U$ is an open set and we have $U = V \\cup A$. We will show that $(u_n)_n$ admits a subsequence which is pointwise convergent on $\\mathcal{L}^1$-a.e.\\@ any point of $U$: it will be enough to conclude that this subsequence is $\\mu$-a.e.\\@ convergent on $V$, since $\\mu|_V$ is absolutely continuous with respect to the Lebesgue measure.\n\nThe sequence $(u_n)$ is bounded in $H^1_\\mu(I)$, thus the sequence $(u_n|_U)_n$ is bounded in the weighted Sobolev space $H^1_f$. But since $U$ is exactly the set of points around which $1\/f$ is integrable, we know that $L^2_f(U) \\hookrightarrow L^1_{loc}(U)$; this implies that the sequence of the weak derivatives of $u_n$ (which are functions of $L^2_f(U)$) is bounded in $L^1_{loc}(U)$. Then $(u_n)_n$ is bounded in the Sobolev space $W^{1,1}_{loc}(U)$, and admits a subsequence which is strongly convergent in $L^1(K)$, for any compact subset $K$ of $I$. We can again extract a subsequence which is pointwise convergent on $\\mu$-a.e.\\@ point of $I$; the proof is complete.\\end{proof}\n\n\n\\subsection{First part of the proof: the regular part, outside of the critical set}\n\nFirst, let us prove that $T_\\mu = \\mathbb{R}$ outside of $M \\cup A$. Using the first characterization of the tangent space, we take an element $g$ of $\\Gamma(0)$ and we want to show that $g = 0$ $\\mu$-a.e.\\@ outside of $M \\cup A$; by definition of~$A$, it is enough to show that $g = 0$ $\\mathcal{L}^1$-a.e.\\@ on $U$. As in the above remark, taking a sequence of regular functions $u_n \\to 0$ with $u'_n \\to g$ and a test function $\\varphi$ such that $1\/f$ is integrable on the support of $\\varphi$, we~obtain\n$$ \\left|\\int_U u_n' \\varphi\\right| = \\left|\\int_U u_n \\varphi'\\right| \\leq \\int_U \\left|u_n \\sqrt{f}\\right| \\left|\\frac{\\varphi'}{\\sqrt{f}}\\right| \\leq \\left( \\int_U u_n^2 f \\right)^{\\frac{1}{2}} \\left( \\int_U \\frac{\\varphi'^2}{f} \\right)^{\\frac{1}{2}} $$\nwhich goes to $0$ as $n \\to +\\infty$. The same computation gives $\\int_U u'_n \\varphi \\to \\int_U g\\varphi$. We deduce that $g = 0$~$\\mathcal{L}^1$-a.e.\n\n\\subsection{Second part: the singular part of the measure}\n\nSecond, we prove that $T_\\mu = \\{0\\}$ for the singular part of $\\mu$. We use the third characterization of the tangent space and build a sequence of $C^1$ functions $(u_n)_n$ such that\n$$u_n \\to 0 \\quad \\text{and} \\quad u'_n \\to \\mathds{1}_A \\quad \\text{in } L^2_\\mu.$$\nwhere $\\mathds{1}_A$ is the characteristic function of the set $A$; this will prove that $\\mathds{1}_A \\in \\Gamma(0)$ and imply the result.\n\nFor $n \\in \\mathbb{N}$, let $\\Omega_n$ be an open set such that $A \\subseteq \\Omega_n$ and $\\mu(\\Omega_n \\setminus A)+\\mathcal{L}^1(\\Omega_n) \\leq 1\/n$. By Lusin theorem, there exists a continuous function $v_n$ with $0 \\leq v_n \\leq 1$ on $I$ and\n$$ (\\mu+\\mathcal{L}^1)(\\{x \\in I : v_n(x) \\neq \\mathds{1}_{\\Omega_n}(x) \\}) \\leq 1\/n $$\nLet us consider $u(x) = \\int_a^x v_n(x) \\, \\text{\\normalfont d} x$, where $a$ is the lower bound of $I$. Then we have:\n\\begin{itemize}\n\\item for any $x \\in I$, \n$$|u_n(x)| \\leq \\int_I(|v_n-\\mathds{1}_{\\Omega_n}|(t)+\\mathds{1}_{\\Omega_n}(t)) \\, \\text{\\normalfont d} t \\leq \\mathcal{L}^1(\\{v_n \\neq \\mathds{1}_{\\Omega_n}\\}) + \\mathcal{L}^1(\\Omega_n) \\leq 2\/n $$\nthus $(u_n)_n$ goes to $0$ uniformly, and also in the space $L^2_\\mu$;\n\\item on the other hand, since $u'_n = v_n$ coincides with $\\mathds{1}_A$ outside of a set $E_n$ such that $\\mu(E_n) \\leq 1\/n$, we~have\n$$ \\int_I |u'_n(x)-\\mathds{1}_A(x)|^2 \\, \\text{\\normalfont d}\\mu(x) \\leq ||v_n-\\mathds{1}_A||_\\infty^2 \\, \\mu(E_n) \\leq 4\/n $$\nthus $u'_n \\to \\mathds{1}_A$ in $L^2_\\mu(I)$.\n\\end{itemize}\nWe obtain that $\\mathds{1}_A \\in \\Gamma(0)$, which guarantees that $T_\\mu = 0$ on $A$.\n\n\\subsection{Third part: the critical set}\n\nThis part is more difficult. Given a function $u \\in C^1\\left(\\overline{I}\\right)$, we build a sequence $(u_n)_n$ of regular functions (say, $C^1$) such that $u_n \\to u$ and $u'_n \\to v$ for the $L^2_\\mu$-norm, with $v=0$ on $M$. The strategy is the following:\n\\begin{itemize}\n\\item given a set $\\Omega_n$ which is ``almost'' $M$, we start from a function $u_n$ which coincides with $u$ outside of $\\Omega_n$ and is piecewise constant on $\\Omega_n$ (so that its derivative is null on $M$);\n\\item then, using the fact that the discontinuity points of $u_n$ belong to $M$, we regularize $u_n$ around this points so that its derivative stays small for the $L^2_\\mu$-norm.\n\\end{itemize}\n\nFirst, we build our set $\\Omega_n$:\n\n\\begin{lem} Let us denote by $(x_n)_n$ a sequence containing all the atoms of $\\mu$. For $n \\in \\mathbb{N}$, there exists a set $\\Omega_n$ such that:\n\\begin{itemize}\n\\item $\\Omega_n = \\bigcup\\limits_{i=1}^{p_n} \\, ]a_i,b_i[ \\, $, with $b_i < a_{i+1}$ for each $i$, and $\\, ]a_i,b_i[ \\,  \\cap M \\neq \\emptyset$;\n\\item $\\Omega_n \\supseteq M \\setminus \\{x_1,\\dots,x_n\\}$;\n\\item $\\mu(\\Omega_n \\setminus ( M \\setminus \\{x_1,\\dots,x_n\\})) \\leq 1\/n.$\n\\end{itemize}\n\\end{lem}\n\n\n\\begin{proof} Let $U_n$ be an open set such that $M \\subseteq \\Omega_n$ and $\\mu(U_n \\setminus M) \\leq 1\/n$ (such a set exists since $\\mu$ is regular from above); $U_n$ is a union of open intervals, and since $M$ is compact we can assume this union to be finite. We denote by $\\Omega_n = U_n \\setminus \\{ x_1,\\dots,x_n\\}$. It is still a finite union of open intervals, containing $M \\setminus \\{ x_1, \\dots, x_n\\}$ and with $\\mu(\\Omega_n \\setminus ( M \\setminus \\{x_1,\\dots,x_n\\})) \\leq 1\/n$. Moreover, we may assume that all these intervals contain an element of $M$: it is enough to remove from $\\Omega_n$ the intervals which do not contain any element of $M$ (if after that we obtain $\\Omega_n = \\emptyset$, it means that $M \\subseteq A$ and we already know that $T_\\mu = \\{0\\}$ on $A$, so there is nothing to prove). \\end{proof}\n\n\\par Let us thus take a sequence $(g_n)_n$ of piecewise constant functions such that $g_n \\to u$ in $L^2_\\mu$ (it is possible since $u$ is continuous, thus can be approximated uniformly on $I$ by a sequence of piecewise functions) and $||g_n||_\\infty \\leq C$, where $C$ only depends on $u$; we replace $g_n$ by $u$ outside of the set $\\Omega_n$ (the new function will still be called $g_n$), so that we have now\n\\begin{itemize}\n\\item $g_n \\to u$ in $L^2_\\mu$;\n\\item $g_n$ coincides with $u$ outside of $\\Omega_n$;\n\\item $g_n$ coincides on $\\Omega_n$ with a piecewise constant function.\n\\end{itemize}\n\n\\par We begin by regularizing $g_n$ around the endpoints of the intervals forming $\\Omega_n$. Let $\\varepsilon_n > 0$ be small enough so that:\n\\begin{itemize}\n\\item $a_i+\\varepsilon_n < b_i-\\varepsilon_n$, for each $i$ (we will set $a'_i = a_i+\\varepsilon_n$ and $b'_i = b_i-\\varepsilon_n$);\n\\item $]a'_i,b'_i[ \\, $ contains at least an element of $M$, for each $i$;\n\\item on $\\, ]a_i,b_i[ \\, $, $g_n$ has not any discontinuity point outside $\\, ]a'_i,b'_i[ \\, $;\n\\item if we denote by $\\Omega'_n$ the union of the intervals $\\, ]a'_i,b'_i[ \\, $, we have $\\mu(\\Omega_n \\setminus \\Omega'_n) \\leq 1\/n$.\n\\end{itemize}\n\n\\begin{lem} There exists a function $w_n$ coinciding with $g_n$ outside of $\\Omega_n \\setminus \\Omega'_n$, and such that, on each interval $\\, ]a_i,a'_i[ \\, $ and $\\, ]b'_i,b_i[ \\, $,\n\\begin{itemize}\n\\item $w_n$ and $w'_n$ are bounded by constants depending only on $u$ and $u'$;\n\\item $w_n(a_i) = u(a_i)$, $w'_n(a_i) = u'(a_i)$ and $w'_n=0$ on a (small) open interval having $a'_i$ for upper bound;\n\\item $w_n(b_i) = u(b_i)$, $w'_n(b_i) = u'(b_i)$ and $w'_n=0$ on a (small) open interval having $b'_i$ for lower bound.\n\\end{itemize}\n\\end{lem}\n\n\\begin{proof} It is enough to replace $g_n$ on the interval $\\, ]a_i,a_i+\\varepsilon_n[ \\, $ by the function $x \\mapsto Q(a_i+x)$ where\n$$Q(t) = -\\frac{u'(a_i)}{2\\varepsilon_n} t^2 + u'(a_i) t + u(a_i) $$\n(so that $Q(0)=u(a_i)$, $Q'(0)=Q'(a_i)$ and $Q'(\\varepsilon_n)=0$), to scale the new function on the interval $\\, ]a_i,a'_i[ \\, $ by replacing it by\n$$ x \\mapsto \\left\\{ \\begin{array}{ll}\nw_n(a_i+2(x-a_i)) & \\text{if } a_i \\leq x \\leq a_i+\\varepsilon_n\/2 \\\\\nw_n({a'_i}^-) & \\text{otherwise}\n\\end{array} \\right.$$\nand to make a similar construction on the interval $\\, ]b_i-\\varepsilon_n,b_i[$. \\end{proof}\n\nSince $g_n$ and $w_n$ are bounded uniformly in $n$ and coincide outside of the set $\\Omega_n \\setminus \\Omega'_n$, whose measure is at most $1\/n$, the sequence $(w_n)_n$ still converges to $u$ in $L^2_\\mu$; moreover, we have\n$$ ||w'_n||_{L^2_\\mu(\\Omega_n \\setminus \\Omega'_n)} \\leq (2\/n) ||u'||_\\infty $$\nand for any discontinuity point $y$ of $w_n$, $w_n$ is piecewise constant on a (small) neighborhood of $y$. We now have to regularize $w_n$ around its discontinuity points, which belong to $\\Omega'_n$; this is possible with a small cost only if these points belong to the set $M$. For this reason we are interested by the following ``displacement'' procedure of the discontinuity points:\n\n\\begin{lem}\nFor any $n$, there exists a function $v_n$ such that\n\\begin{itemize}\n\\item $v_n = w_n$ outside of $\\Omega'_n$;\n\\item $v_n$ is still piecewise constant on $\\Omega'_n$;\n\\item any discontinuity point of $v_n$ belongs to $M$;\n\\item $v_n \\to u$ in $L^2_\\mu$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{proof} We have to modify $w_n$ only on each interval $\\, ]a'_i,b'_i[ \\, $. On this interval, the number of discontinuity points of $w_n$ is finite; we denote these points by $a'_i \\leq x_1 < \\dots < x_n = b'_i$. We make the following construction:\n\\begin{itemize}\n\\item Let $m = \\inf([a'_i,b'_i] \\cap M)$. We define $v_n$ on the interval $[a'_i,m[ \\, $ (if it is nonempty) by setting $v_n = w_n({a'_i}^+)$.\n\\item Then we reiterate the construction starting from $m$:\n\\begin{itemize}\n\\item if $\\, ]m,b'_i[ \\,  \\cap M = \\emptyset$, we set $v_n = w_n({b'_i}^-)$ on this interval, and we are done;\n\\item otherwise, let $m'=\\inf(\\, ]m,b'_i[ \\,  \\cap M)$. We have naturally $m' \\geq m$. If $m \\geq x_n$, then we set $v_n = w_n({b'_i}^-)$ on $\\, ]m',b'_i[ \\, $,$w_n$ on $[m,m'[ \\, $ and we are done;\n\\item if $m = m' < x_n$, then we denote by $j$ the smallest index such that $x_j > m$, we set $v_n = w_n$ on $[m,x_j[ \\, $ and we reiterate this construction starting from $x_j$;\n\\item finally, if $m < m' < x_n$, we set $v_n = w_n$ on $[m,m'[ \\, $ and we reiterate this construction starting from $m'$.\n\\end{itemize}\n\\end{itemize}\n\nWith this construction, $w_n-v_n \\neq 0$ only on $\\Omega'_n$ and outside of the set $M$. Since $\\mu(\\Omega'_n \\setminus M) \\leq 1\/n $ and $w_n, v_n$ are uniformly bounded, we get $||v_n-w_n||_{L^2_\\mu} \\leq C\/n$, and we thus still have $v_n \\to u$. Moreover, by construction, $v_n$ is still piecewise constant on the set $\\Omega'_n$ and all its discontinuity points belong to $M$. \\end{proof}\n\nTo finish, we have to modify $v_n$ around each discontinuity point, so that the new function $u_n$ is regular and admits a derivative which is small for the $L^2_\\mu$-norm. This is possible thanks to the following result about embeddings between functional spaces:\n\n\\begin{lem} Let $J$ a bounded interval of $\\mathbb{R}$, and $\\mu$ a finite measure on $J$ with density $f > 0$. The following assertions are equivalent:\n\\begin{enumerate}\n\\item The function $1\/f$ belongs to $L^1(J)$\n\\item The space $L^2_\\mu(J)$ is continuously embedded into $L^1(J)$\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof} The direct implication is obvious and comes directly from the Cauchy-Schwarz inequality. For the converse one, let us assume that $\\int_J 1\/f = +\\infty$ and set\n$$E_n = \\left\\{t \\in J : \\frac{1}{n+1} \\leq f(t) < \\frac{1}{n} \\right\\} \\quad \\text{and} \\quad l_n = \\mathcal{L}^1(E_n). $$\nWe know that $\\sum\\limits_n l_n < +\\infty$ (it is the length of $J$) and\n$$ \\sum\\limits_n nl_n = \\sum\\limits_n \\int_J n \\, \\mathds{1}_{\\{n-1 \\leq 1\/f \\leq n\\}} \\geq \\sum\\limits_n \\int_J \\frac{1}{f} \\, \\mathds{1}_{\\{n-1 \\leq 1\/f \\leq n\\}} \\geq \\int_J \\frac{1}{f} = +\\infty $$\nthus $\\sum\\limits_n nl_n = +\\infty$. We will build a function $U$ which is constant on each set $E_n$, belongs to $L^2_\\mu$ and does not belong to $L^1$. If we denote by $u_n$ the value of $U$ on $E_n$, it is equivalent to find a sequence $(u_n)_n$ verifying\n$$ \\sum\\limits_n u_n^2 \\, (nl_n) < +\\infty \\quad \\text{and} \\quad \\sum\\limits_n |u_n| \\, l_n = +\\infty $$\nTo summarize, we want to prove the following statement: for any sequence $(l_n)_n$ of positive numbers such that $\\sum\\limits_n n l_n = +\\infty$ and $\\sum\\limits_n l_n < +\\infty$, there exists a sequence $(u_n)_n$ of positive numbers such that $\\sum\\limits_n u_n^2 (nl_n) < +\\infty$ and $\\sum\\limits_n nu_n = +\\infty$. By contraposition, it is equivalent to the following: for any sequence $(l_n)_n$ of positive numbers such that $\\sum\\limits_n l_n < +\\infty$, if the following implication holds:\n$$ \\left( \\sum\\limits_n u_n^2 \\,(n l_n) < + \\infty \\right) \\Rightarrow \\left(\\sum\\limits_n l_n |u_n| < + \\infty \\right)\n $$\nthen we have $\\sum\\limits_n n l_n < +\\infty$. This result can be seen as a corollary of the Banach-Steinhaus theorem. Denoting by $\\ell^2_{nl_n}$ the space of sequences $(u_n)_n$ such that $\\sum\\limits_n u_n^2 \\,(n l_n) < + \\infty$, the operator\n$$ T_N : u \\in \\ell^2_{nl_n} \\longmapsto \\sum\\limits_{n=0}^N l_n u_n  $$\nis linear continuous with norm $\\left(\\sum\\limits_{n=0}^N nl_n\\right)^{1\/2}$ and the assumption about $(l_n)_n$ is equivalent to\n$$\\forall u \\in \\ell^2_{nl_n} \\quad \\sup\\limits_{N \\in \\mathbb{N}} |T_N(u)| < + \\infty.$$\nBy Banach-Steinhaus theorem, we get $\\sup\\limits_{N \\in \\mathbb{N}} ||T_N|| < +\\infty$ and $\\sum\\limits_{n \\in \\mathbb{N}} nl_n < +\\infty$; the proof is complete.\n\\end{proof}\n\n\\noindent {\\bf End of the proof of Theorem 2.1.} Thanks to the two last lemmas, we are now able to transform the function $w_n$ into a $C^1$ function $u_n$, which will provide us our approximation of $u$. Let us recall that $v_n$ coincides with $u$ outside $\\Omega_n$, is piecewise constant on $\\Omega'_n$, all its discontinuity points are located in $M$ and each of this points admits a neighborhood where $v_n'$ is null. Denoting by $y_1 < \\dots < y_p$ the discontinuity points of $v_n$, we find $\\varepsilon_n$ such that, for each $j$, $v_n$ is constant on $\\, ]y_j-\\varepsilon_n,y_j[ \\, $ and $\\, ]y_j,y_j+\\varepsilon_n[ \\, $. Moreover, we have:\n$$ \\sum\\limits_{j=1}^p \\mu(\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, ) \\leq 2p\\varepsilon_n + \\sum\\limits_{j=1}^p \\mu(\\{y_j\\}). $$\nWe take $\\varepsilon_n$ small enough so that $2p \\varepsilon_n \\leq 1\/n$. On the other hand, since each $y_j$ does not belong to the set $\\{x_1,\\dots,x_n\\}$ of the ``big atoms'' of $\\mu$, we have\n$$ \\sum\\limits_{j=1}^p \\mu(\\{y_j\\}) \\leq \\sum\\limits_{k \\geq n} \\mu(\\{x_k\\}). $$\nTherefore,\n$$ \\mu \\left( \\bigcup\\limits_{j=1}^p \\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\,  \\right) \\xrightarrow[n \\to +\\infty]{} 0. $$\nOn the interval $\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, $, thanks to Lemma 2.4, $L^2_\\mu$ is not embedded into $L^1$, thus we can find a regular function $g_j$ such that\n$$ \\int_{y_j-\\varepsilon_n}^{y_j+\\varepsilon_n} g_j = v_n(y_j^+)-v_n(y_j^-) \\quad \\text{and} \\quad \\int_{y_j-\\varepsilon_n}^{y_j+\\varepsilon_n} g_j^2 \\, \\text{\\normalfont d}\\mu \\leq \\frac{1}{nq}. $$\nThen, we set\n$$ u_n(x) = \\left \\{ \\begin{array}{ll}\n\\tilde{v}_n(y_j-\\varepsilon_n) + \\int_{y_j-\\varepsilon_n}^x g_j & \\text{ if } y_j-\\varepsilon_n \\leq x \\leq y_j+\\varepsilon_n \\\\\nv_n(x) & \\text{ otherwise.}\n\\end{array}\n\\right. $$\nThis functions $u_n$ form our desired approximation of $u$:\n\n\\begin{prop} This sequence $(u_n)_n$ satisfies $u_n \\to u$ and $u'_n \\to v$ in the space $L^2_\\mu$, where\n\n$$v(x) =  \\left\\{ \\begin{array}{ll} u'(x) & \\text{if } x \\notin M \\text{ or is an atom of } \\mu  \\\\ 0 & \\text{otherwise.} \\end{array} \\right.$$\nConsequently, $T_\\mu = \\{0\\}$ $\\mu$-a.e.\\@ on $M$. \\end{prop}\n\n\\begin{proof} We know that $v_n \\to u$, thus $u_n \\to u$ in the space $L^2_\\mu$ outside of the intervals $\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, $. But since the total mass of these intervals goes to $0$ and $(u_n)_n$ is uniformly bounded, we get $u_n \\to u$. For the derivative, since $u_n=u$ outside of $\\Omega_n$, we have\n$$ ||u'_n-v||_{L^2_\\mu}^2 = ||u'_n-v||^2_{L^2_\\mu(\\Omega_n)}  = ||u'_n-v||^2_{L^2_\\mu(\\Omega_n \\setminus M)} + ||u'_n-v||^2_{L^2_\\mu(M \\setminus \\{x_1,...,x_n\\})} $$\nwhere the first term goes to 0 (since $(u_n)_n$ is uniformly bounded and $\\mu(\\Omega_n \\setminus M)$ goes to 0); for the second one, we have $v=0$ on $M$, thus it is enough to prove that $u'_n$ goes to $0$ for the $L^2_\\mu$-norm on $M \\setminus \\{x_1,\\dots,x_n\\}$; this term is bounded by\n$$ ||u'_n||_{L^2_\\mu(\\Omega'_n \\setminus \\{y_1,\\dots,y_p\\})}^2 + \\sum\\limits_{j=1}^p u'_n(y_j) \\mu(\\{y_j\\}).  $$\nSince $(u'_n)_n$ is uniformly bounded, we know that the second term goes to $0$, and since $u_n$ is constant outside of the intervals $\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, $ the first one is equal to\n$$\\sum\\limits_{j=1}^p \\int_{y_j-\\varepsilon_n}^{y_j+\\varepsilon_n} g_j^2 d\\mu$$\nwhich, by definition of $g_j$, is smaller than $1\/n$. This completes the proof. \\end{proof}\n\n\\section{Application to a transport problem with gradient penalization}\n\n\\subsection{Problem statement, and the easiest case}\n\nWe investigate the following problem, which is somehow intermediate between optimal transportation and elasticity theory:\n\n$$ \\inf\\left\\{ \\int_\\Omega (|T(x)-x|^2 + |\\nabla T(x)|^2) \\, \\text{\\normalfont d} \\mu(x) \\right\\} \\, , $$\nwhere the infimum is taken among all maps $T:\\Omega \\to \\mathbb{R}^d$ with prescribed image measure $T_\\# \\mu = \\nu$ and admitting a Jacobian matrix $\\nabla T$ in a suitable sense. Contrary to the Monge-Kantorovich optimal transport problem, if $\\mu$ has a density $f$ bounded from above and below, then the existence of a solution is obvious and comes from the direct method of the calculus of variations; more precisely:\n\n\\begin{prop} Let $f:\\Omega \\to \\mathbb{R}^d$ a measurable function such that $0 < c < f < C < +\\infty$ for some constants $c,C>0$. Let $\\nu \\in \\mathcal{P}(\\mathbb{R}^d)$. We assume that there exists at least one Sobolev transport map between $\\, \\text{\\normalfont d}\\mu = f \\cdot \\, \\text{\\normalfont d}\\mathcal{L}^d$ and $\\nu$. Then the problem\n$$ \\inf \\left\\{ \\int_\\Omega (|T(x)-x|^2 + |\\nabla T(x)|^2) f(x) \\, \\text{\\normalfont d} x : T \\in H^1(\\Omega), \\, T_\\#\\mu = \\nu \\right \\} $$\nadmits at least one solution.\n\\end{prop}\n\n\\begin{proof} Let $(T_n)_n$ be a minimizing sequence. We can extract from $(T_n)_n$ a sequence having, thanks to the Rellich theorem, a strong limit $T$ in $L^2$, and we also can assume that $T_n \\to T$ $\\mathcal{L}^1$-a.e.\\@ on $\\Omega$, thus $\\mu$-a.e.\\@ on $\\Omega$. Then for any function $\\phi \\in C_b(\\mathbb{R}^d)$ we have\n$$ \\forall n \\in \\mathbb{N} \\quad \\int_\\Omega \\phi(T_n(x)) \\, \\text{\\normalfont d}\\mu(x) = \\int_{\\mathbb{R}^d} \\phi(y) \\, \\text{\\normalfont d}\\nu(y). $$\nThanks to the pointwise $\\mu$-a.e.\\@ convergence of $(T_n)_n$, we can pass to the limit in the left-hand-side of this equality, which gives\n$$ \\forall \\phi \\in C_b(\\mathbb{R}^d) \\quad \\int_\\Omega \\phi(T(x)) \\, \\text{\\normalfont d}\\mu(x) = \\int_{\\mathbb{R}^d} \\phi(y) \\, \\text{\\normalfont d}\\nu(y) $$\nand $T$ satisfies the constraint on the image measure. Moreover, the functional that we consider is of course lower semicontinuous with respect to the weak convergence in $H^1(\\Omega)$, and $T$ minimizes our problem. \\end{proof}\n\n\\subsection{The general formulation, and the one-dimensional case}\n\nIf $\\mu$ is a generic Borel measure, we replace the term with the jacobian matrix of $T$ by $\\nabla_\\mu T$, so that our problem is now written\n\\begin{equation} \\inf \\left\\{ \\int_\\Omega (|T(x)-x|^2+|\\nabla_\\mu T(x)|^2) \\, \\text{\\normalfont d}\\mu(x) \\; : \\; T \\in H^1_\\mu(\\Omega) \\right\\}. \\label{1} \\end{equation}\n\nThe existence of solutions is not clear in general. In the case of the classical Sobolev space $H^1(\\Omega)$, we have seen that the key point to prove the existence is the following: from any minimizing sequence $(T_n)_n$ we can extract a sequence which converges $\\mathcal{L}^d$-a.e.\\@ on $\\Omega$, and this is enough to obtain that the limit is admissible. This is not possible in general, since we don't have any equivalent of Rellich compactness theorem for the Sobolev spaces with respect to a generic measure $\\mu$.\n\nIn the one-dimensional case, if $\\mu$ is the Lebesgue measure, it is known that the monotone transport map between $\\mathcal{L}^1$ and $\\nu$ is optimal for the problem \\eqref{1} (see \\cite{ls}). This result does not hold if we do not make any assumption of $\\mu$, but we can get an existence result thanks to the $\\mu$-a.e.\\@ compactness result of the second section:\n\n\\begin{theo} In dimension 1, the problem \\eqref{1} admits at least one solution. \\end{theo}\n\n\\begin{proof} Let us begin by rewriting precisely the functional that we consider in this case: we know that $T_\\mu = \\{0\\}$ on $M \\cup A$ and $\\mathbb{R}$ on $V$, so that we are now minimizing\n$$ J : U \\in H^1_\\mu(I) \\longmapsto \\int_{V} ((U(x)-x)^2+U'(x)^2) f(x) \\, \\text{\\normalfont d} x + \\int_{M \\cup A} (U(x)-x)^2 \\, \\text{\\normalfont d}\\mu(x). $$\nLet $(U_n)_n$ be a minimizing sequence. On the set $V$, which is exactly the set where $T_\\mu$ is~$\\mathbb{R}$, we can extract from $(U_n)_n$ a $\\mu$-a.e.\\@ (which means $\\mathcal{L}^1$-a.e.\\@ wherever $f \\neq 0$) pointwise convergent subsequence, whose limit is denoted by $U$; let us remark that $U$ is the weak limit of $(U_n)_n$ (up to a subsequence) in the space $H^1_f$, and by semicontinuity, we have\n$$ \\int_V ((U(x)-x)^2+U'(x)^2) f(x)\\, \\text{\\normalfont d} x \\leq \\liminf \\left(\\int_V ((U_n(x)-x)^2+U_n'(x)^2) f(x)\\, \\text{\\normalfont d} x \\right). $$\nMoreover, let us set, for $n \\in \\mathbb{N}$, $\\nu_n = (U_n)_\\#(\\mu|_{M \\cup A})$ and $\\tilde{U}_n$ the optimal transport map for the Monge-Kantorovich quadratic cost between the measures $\\mu|_{M \\cup A}$ and $\\nu_n$. It is well-known that $\\tilde{U}_n$ is the unique nondecreasing transport map between $\\mu|_{M \\cup A}$ and $\\nu_n$; because of compactness properties of nondecreasing maps, we can assume that $(\\tilde{U}_n)_n$ admits, for the $\\mu$-a.e.\\@ convergence, a limit $\\tilde{U}$. For any $n$, thanks to the optimality of $\\tilde{U}_n$, we have\n$$ \\int_{M \\cup A} (\\tilde{U}_n(x)-x)^2 \\, \\text{\\normalfont d}\\mu(x) \\leq \\int_{M \\cup A} (U_n(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) $$\nand by semicontinuity\n$$ \\int_{M \\cup A} (\\tilde{U}(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) \\leq \\liminf \\left(\\int_{M \\cup A} (U_n(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) \\right). $$\nThus, if we denote by\n$$ T_n(x) = \\left\\{ \\begin{array}{ll}\nU_n(x) & \\text{if } x \\in V \\\\\n\\tilde{U}_n(x) & \\text{if } x \\in M \\cup A\n\\end{array} \\right.\n\\qquad \\text{and} \\qquad\nT(x) = \\left\\{ \\begin{array}{ll}\nU_(x) & \\text{if } x \\in V \\\\\n\\tilde{U}(x) & \\text{if } x \\in M \\cup A\n\\end{array} \\right. $$\nwe have $T_n \\to T$ $\\mu$-a.e.\\@ on $I$, and\n$$ J(T) \\leq \\liminf \\left(\\int_V ((U_n(x)-x)^2+U_n'(x)^2) f(x)\\, \\text{\\normalfont d} \\right) +  \\liminf \\left(\\int_{M \\cup A} (U_n(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) \\right) = \\liminf J(U_n) $$\nwhere $(U_n)_n$ is a minimizing sequence for $J$ on the set of $H^1_\\mu$ transport maps between $\\mu$ and $\\nu$. Thus, it is enough to prove that $T$ satisfies the constraint on image measure to conclude. But for each $n$, by construction, $(T_n)_\\# \\mu = \\nu$ and the $\\mu$-a.e.\\@ convergence allows to obtain the same for the limit~$T$; the proof is complete. \\end{proof}\n\n\\noindent {\\bf Remark.} This result can be generalized to any functional $J: U \\mapsto \\int_\\Omega (L_1(x,U(x)) + L_2(\\nabla_\\mu U(x))) \\, \\text{\\normalfont d}\\mu(x)$, where $L_1$ and $L_2$ have one of the following forms:\n\\begin{itemize}\n\\item $L_1$ is a transport cost such that the nondecreasing map is optimal for the Monge-Kantorovich problem: it is the case if $L_1(x,u) = h(|x-u|)$, where $h$ is a convex function. Let us notice that in particular the statement holds if we study the problem of minimization of the norm of the gradient among all Sobolev transport maps (this corresponds to $L_1 = 0$). Of course we need to assume that the class\n$$ \\left \\{U \\in H^1_\\mu : U_\\# \\mu = \\nu \\text{ and } \\int_\\Omega L_1(x,U(x)) \\, \\text{\\normalfont d}\\mu(x) < +\\infty \\right \\} $$\nis nonempty (to guarantee that $J \\not\\equiv +\\infty$ on the set of admissible functions). Thanks to the quadratic structure of $H^1_\\mu$, this is automatically the case if $L_1$ is the quadratic cost and there exists a Sobolev transport map.\n\\item $L_2$ is ``quadratic'', so that the space where we study the problem is actually the Sobolev space $H^1_\\mu$. The natural cases are $L_2(\\nabla_\\mu U) = |\\nabla_\\mu U|^2$ or $|\\nabla_\\mu U - I_d|^2$, where $I_d$ is the identity matrix (in this last case, we can consider the functional $U \\mapsto ||U-\\text{id}||_{H^1_\\mu}$, which is a Sobolev version of the quadratic transport problem where we minimize $||U-\\text{id}||_{L^2_\\mu}$).\n\\end{itemize}\n\n\\subsection{Difficulties and partial results in any dimension}\n\nAs we said in the second section of this paper, we don't have a precise pointwise description of the $\\mu$-Sobolev space $H^1_\\mu(\\Omega)$ if $\\Omega$ is an open set of $\\mathbb{R}^d$, which was the key point for the compactness result. More precisely, the following results still hold in any dimension:\n\n\\begin{itemize}\n\\item Outside of the set\n$$ M = \\left\\{ x \\in \\Omega : \\forall \\varepsilon > 0, \\, \\int_{\\Omega \\cap B(x,\\varepsilon)} \\frac{\\, \\text{\\normalfont d} x}{f(x)} = +\\infty \\right\\} $$\nwe have $T_\\mu = \\mathbb{R}^d$, a.e.\\@ for the regular part of $\\mu$. The proof is identical to the one-dimensional case, based on the Cauchy-Schwarz inequality and the embedding $L^2_f(\\Omega \\setminus M) \\hookrightarrow L^1_{loc}$.\n\\item Of course, we don't have anymore $T_\\mu = 0$ for the singular part of $\\mu$: for instance, if $\\mu$ is uniform and supported on a segment, then $\\mu$ is singular, and we know that $\\operatorname{dim} T_\\mu = 1$ on any point. However, the tangent space on the atoms of $\\mu$ is known:\n\n\\begin{prop} If $x_0$ is an atom of $\\mu$, then $T_\\mu(x_0) = \\{0 \\}$. \\end{prop}\n\n\\begin{proof} Let us prove it if $x_0 = 0$. We want to build a sequence of functions $(u_n)_n$ such that\n$$ u_n \\to 0 \\quad \\text{and} \\quad \\nabla u_n \\to e \\, \\mathds{1}_{\\{0\\}} $$\nwhere $e$ is an arbitrary unit vector (this shows that any unit vector belongs to the space $T_\\mu(0)^\\perp$, thus $T_\\mu(0) = \\{0\\}$). For this goal, let us consider a smooth cutoff function $\\chi$ such that $0 \\leq \\chi \\leq 1$ and\n$$ \\chi_n(x) = 1 \\text{ if } 0 \\leq |x| \\leq 1 \\text{ and } \\chi_n(x) = 0 \\text{ if } |x| \\geq 2 $$\nand we denote by $\\chi_n(x) = \\chi(nx)$. We then set $u_n(x) = \\langle x,e \\rangle \\, \\chi_n(x)$ and show that $(u_n)_n$ is the function that we are looking for. First, noting that $u_n(0) = 0$, that $u_n$ is null outside of $B(0,2\/n)$ and that $0 \\leq u_n \\leq 1$ for any $n$, we have\n$$ ||u_n||_{L^2_\\mu} = ||u_n||_{L^2\\mu(\\Omega \\setminus x_0)} \\leq \\mu(B(0,2\/n))-\\mu(\\{0\\}) $$\nwhich goes to $0$ as $n \\to +\\infty$; this gives us $u_n \\to 0$ in $L^2_\\mu$. Second, for any~$n$, we have\n$$ \\nabla u_n(x) = \\chi_n(x) \\, e + \\langle e,x \\rangle \\, \\nabla \\chi_n(x) = \\chi_n(x) \\, e + \\langle e,x \\rangle \\, n \\nabla\\chi(nx). $$\nLet us notice that $\\nabla u_n(0) = e$ for any $n$, thus it is enough to prove that $||\\nabla u_n||_{L^2_\\mu (\\Omega \\setminus \\{0\\})} \\to 0 $. But $\\chi_n$ and $\\nabla \\chi_n$ are null outside of $B(0,2\/n)$ and if $0 < |x| \\leq 2\/n$ we have\n$$ |\\nabla u_n(x)| \\leq |e| |\\chi(nx)| + |\\langle e,x \\rangle| n |\\nabla \\chi(x)| \\leq C(1+n|x|) \\leq 3C $$\nwhere $C$ is an upper bound of $\\chi$ and $\\nabla \\chi$. Thus, $(\\nabla u_n)_n$ is uniformly bounded by a positive constant, and $\\nabla u_n - e\\, \\mathds{1}_{\\{0\\}}$ is supported on the set $B(0,2\/n) \\setminus \\{0\\}$ whose measure $\\mu$ goes to $0$. This completes the proof. \\end{proof}\n\\item Finally, we can prove that there exists absolutely continuous measure $\\mu$ such that $T_\\mu$ is neither $\\{0\\}$ nor $\\mathbb{R}^n$ on any point of $\\Omega$. We provide an explicit example:\n\\begin{prop} Let be $g : \\,]0,1[ \\,  \\to \\, ]0,+\\infty[ \\, $ such that $\\int_Jg = +\\infty$ for any open interval $J \\subseteq \\, ]0,1[ \\, $. Let $\\Omega = \\, ]0,1[ \\, ^2$, $f:(x,y) \\in \\Omega \\mapsto g(x)$, and $\\mu$ the measure with density $f$. Then the tangent space is the vertical line $\\mathbb{R} \\cdot e_2$ on $\\mu$-a.e.\\@ point of $\\Omega$.\\end{prop}\n\\begin{proof} We first show that $T_\\mu(x)$ is at most one-dimensional on $\\mu$-a.e.\\@ $x \\in \\Omega$. Let be $u(x,y)=x$. Since the tangent space of the measure $g(x) \\cdot \\mathcal{L}^1$ on $\\, ]0,1[ \\, $ is $\\{0\\}$, we can find a sequence of functions $(w_n)_n$ such that\n$$ \\int_{0}^1 |w_n(x)-x|^2 g(x) \\, \\text{\\normalfont d} x \\to 0 \\quad \\text{and} \\quad \\int_0^1 |w'_n(x)|^2 g(x) \\, \\text{\\normalfont d} x \\to 0 $$\nand we denote by $u_n(x,y)=w_n(x)$. It is clear that $\\nabla u_n \\to 0$ and $u_n \\to u$ in $L^2_\\mu$. Then, $\\nabla u = (1,0)$ but $\\nabla_\\mu u = (0,0)$ $\\mu$-a.e.\\@ on $\\Omega$. This would impossible if $T_\\mu$ was $\\mathbb{R}^2$ on a non-negligible set of $\\Omega$.\n\nWe now have to show that $\\mathbb{R} \\cdot e_2$ is included to $T_\\mu(x)$ for $\\mu$-a.e.\\@ $x$. For this, we prove that any element $v=(v_1,v_2)$ of $\\Gamma(0)$ satisfies $v_2 = 0$. Indeed, we have $v_2 = \\lim \\partial_2 u_n$ with $u_n \\to 0$. For any test function $\\phi$, integrating by parts with respect to $y$ (since the density of $\\mu$ depends only on $x$)~gives\n$$ \\int_\\Omega \\partial_2 u_n \\phi \\, \\text{\\normalfont d}\\mu = -\\int_\\Omega u_n \\partial_2\\phi \\, \\text{\\normalfont d}\\mu $$\nwhich goes to $0$ since $u_n \\to 0$ in $L^2_\\mu$. This gives $\\int_\\Omega v_2 \\phi \\, \\text{\\normalfont d}\\mu = 0$ for any $\\phi \\in \\mathcal{D}(\\Omega)$, thus $v_2 = 0$, and the proof is complete. \\end{proof}\n\nThis shows that we cannot hope to obtain a compactness result analogous to the one-dimensional case, where any bounded sequence in $H^1_\\mu$ has a subsequence which converges on $\\mu$-a.e.\\@ $x$ such that $T_\\mu(x) \\neq \\{0\\}$: it is enough to take a sequence of functions $(u_n)_n$ depending only on~$x$ and non-compact for the a.e.\\@ convergence.\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Intro}\n\nThe primary utilitarian motivation for supersymmetry being accessible\nto experiments at the electroweak scale, e.g., at the LHC, depends on\nits ability to alleviate the problem of fine-tuning of electroweak\nsymmetry breaking present in the Standard Model\n\\cite{EENZ,hep-ph\/0312378}.  A supplementary phenomenological\nmotivation for weak-scale supersymmetry is its ability to provide the\ncold dark matter required by astrophysics and cosmology \\cite{EHNOS,\nhep-ph\/9506380}. This is a natural feature of supersymmetric models\nthat conserve R parity, with the lightest neutralino $\\tilde{\\chi}^0_1$ being\nparticularly well-suited to provide the preferred amount of cold dark\nmatter if it is the lightest supersymmetric particle (LSP) and weighs\nless than about 1~TeV \\cite{EHNOS,etcEllis:1985jn}. Within the general\nsupersymmetric framework, one may find more plausible regions of the\nsupersymmetric parameter space that are less fine-tuned, in the sense\nthat the values of the model parameters chosen at some high input\nscale require less delicate adjustment in order to obtain the correct\nvalue of the electroweak scale \\cite{EENZ,Barbieri:1987fn}, as\nmeasured by $M_Z$, or the correct value of the cold dark matter\ndensity $\\Omega_{CDM}h^2$\n\\cite{Ellis:2001zk,hep-ph\/0603095,hep-ph\/0608135,King:2007vh}.\n\nIt is hard to make this type of plausibility argument at all rigorous:\nit is notoriously difficult to make probabilistic statements about the\nunique (by definition) Universe in which we live, it is largely a\nmatter of personal choice which derived quantity one should consider\nand which input parameters one wishes to avoid fine-tuning, it is\ndifficult to argue conclusively for the superiority of one measure of\nfine-tuning over any other, and even less easy to agree on a `pain\nthreshold' in the amount of fine-tuning one is prepared to tolerate\n\\cite{Barbieri:1987fn}. Nevertheless, within a given model framework\nwith its specific input parameters, it is legitimate to consider some\nimportant derived quantity such as $\\Omega_{CDM}h^2$, and compare the\namounts of fine-tuning required in different regions of its parameter\nspace, which frequently do not depend very sensitively on the specific\nsensitivity measure employed.\n\nMoreover, even if one does not accept that the less sensitive\nparameter regions are more plausible, measuring the dark matter\nfine-tuning may have other uses. For example, one hopes (expects) some\nday to discover supersymmetry and start to measure the values of its\nparameters. Unavoidably, these will have non-negligible measurement\nerrors, and these uncertainties propagate via the dark matter\nfine-tuning parameters into the calculation, e.g., of\n$\\Omega_{CDM}h^2$. One of the key features of supersymmetry is its\nability to provide a calculable amount of cold dark matter, and it is\ninteresting to know how accurately which of its parameters must be\nmeasured in order to calculate $\\Omega_{CDM}h^2$ with an accuracy\ncomparable to that quoted by astrophysicists and cosmologists\n\\cite{Ellis:2001zk,Battaglia:2003ab}. An accurate calculation of\n$\\Omega_{CDM}h^2$ might also reveal some deficiency of the\nsupersymmetric explanation of the cold dark matter, and possibly the\nneed for some other new physics in addition.\n\nIt is important to note that the parameters we refer to here are the\nGUT-scale soft supersymmetry-breaking masses and couplings, whereas\nexperiments would measure directly physical masses and mixings at much\nlower energies. Ideally, one would calculate the relic density\ndirectly from the low-energy measurements of MSSM parameters. However\nit will be difficult, if not impossible, to pin down all the key\nparameters using LHC data alone, except by making supplementary\nassumptions about the pattern of supersymmetry breaking at the GUT\nscale, as we do here. Assuming a structure of GUT-scale unification,\none may use experimental measurements to constrain these fewer\nhigh-energy parameters. The strength of the constraints will depend on\nthe magnitudes of these parameters and the experimental tools\navailable. Very likely some accelerator beyond the LHC will be needed,\nbut we do not yet know what will be available.  The fine-tuning\nmeasures we calculate here show clearly which of the high-energy\nparameters are most important for a precise calculation of the relic\ndensity, and hence contribute to the `wish list' for such an\naccelerator.\n\nFor these reasons, we make no further apologies for considering the\nfine-tuning of $\\Omega_{CDM}h^2$ in this paper, which we shall refer\nto as ``dark matter fine-tuning''\n\\cite{Ellis:2001zk,hep-ph\/0603095,hep-ph\/0608135,King:2007vh} to\ndistinguish it from the more commonly studied ``electroweak\nfine-tuning'' \\cite{EENZ,Barbieri:1987fn}, which we also consider for\ncompleteness. The issue of dark matter fine-tuning has been considered\npreviously in the context of several different models including the\nconstrained minimal supersymmetric extension of the Standard model\n(CMSSM) \\cite{Ellis:2001zk}, in which the soft supersymmetry-breaking\nscalar masses $m_0$, gaugino masses $m_{1\/2}$ and trilinear parameters\n$A_0$ are each assumed to be universal, a more general MSSM with\nnon-universal third family scalars and gaugino masses\n\\cite{hep-ph\/0603095}, a string-inspired non-universal model\n\\cite{hep-ph\/0608135} and SUSY GUTs with non-universal gaugino masses\n\\cite{King:2007vh}. Here we extend such considerations to models with\nnon-universal soft supersymmetry-breaking contributions to the Higgs\nmasses (NUHM).  Within this NUHM framework, the independent input\nparameters may be taken as \\cite{oldnuhm,Ellis:2002wv,hep-ph\/0210205}\n\\begin{equation}\n  a_{NUHM}=\\left\\{m_0,~m_{H_1},~m_{H_2},~m_{1\/2},~A_0,~\\tan\\beta,\n  ~\\text{sign}(\\mu)\\right\\},\n\\end{equation}\nand we take as our measure of dark matter fine-tuning the quantity\n\\begin{equation}\n  \\Delta_\\Omega \\; \\equiv \\; {\\rm Max}_i \\left| \\frac{a_i}{\\Omega_\\chi}\n  \\frac{\\partial \\Omega_\\chi}{\\partial a_i} \\right| .\n\\end{equation}\nOur objective will be three-fold: to compare the amount of dark matter\nfine-tuning required within the NUHM to that required within the\nCMSSM, to identify the regions of the NUHM parameter space that\nrequire relatively less (or more) dark matter fine-tuning, and thereby\nto quantify the accuracy in the determination of the GUT-scale NUHM\nparameters that would be needed in order to calculate $\\Omega_\\chi\nh^2$ with any desired accuracy.\n\nThe regions of the NUHM parameter space where $\\Omega_\\chi h^2$ falls\nwithin the range favoured by WMAP and other experiments has been\nstudied quite extensively, for example in \\cite{hep-ph\/0210205}.  It\nshares several features in common with the more restrictive CMSSM\nframework proposed in \\cite{Kane:1993td} and extensively studied in\n\\cite{Ellis:1999mm}.  For example, there are regions where $\\tilde{\\chi}^0_1$ -\nstau coannihilation is important, and others where $\\tilde{\\chi}^0_1$ pairs\nannihilate rapidly via direct-channel $H, A$ poles. However, other\npossibilities also occur. For example, there are regions where $\\tilde{\\chi}^0_1$\n- sneutrino coannihilation is dominant. Also there are regions where\nrapid-annihilation and bulk regions, which are normally separated by a\ncoannihilation strip, approach each other and may even merge. As we\ndiscuss below in more detail, the sneutrino coannihilation regions\nexhibit relatively high dark matter fine-tuning, whereas the `merger'\nregions may require significantly less dark matter fine-tuning.\n\nIn this work we provide a first calculation of the dark matter\nfine-tuning for the regions of the NUHM that are favoured by dark\nmatter measurements. In addition, we present a first calculation of\nthe electroweak fine-tuning within this model and update the parameter\nscans for the current measurement of the top mass.\n\nThe rest of the paper is laid out as follows. In\nSection~\\ref{sec:methods} we summarise the methods used in our\nnumerical studies. Next, in Section~\\ref{CMSSM} we review the familiar\ncase of the CMSSM, which serves as a baseline for later\ncomparison. Then, in Section~\\ref{NUHM} we study dark matter within\nthe NUHM model in which universality between the soft\nsupersymmetry-breaking masses of the sfermions (squarks and sleptons)\nand Higgs multiplets is broken. Finally, in Section~\\ref{Conc} we\npresent our conclusions.\n\n\\section{Methodology}\n\\label{sec:methods}\n\n\\subsection{Codes}\n\nIn order to study the low-energy phenomenology of the NUHM, we need a\ntool to run the mass spectrum from the GUT scale down to the\nelectroweak scale using the renormalisation group equations\n(RGEs)\\cite{Martin:1993zk}. For this purpose we use the RGE code {\\tt\nSoftSusy}~\\cite{hep-ph\/0104145}. This interfaces with the MSSM package\nwithin {\\tt micrOMEGAs}~\\cite{hep-ph\/0112278}, which we use to\ncalculate the dark matter relic density $\\Omega_{CDM}h^2$, $BR(b\\rightarrow s \\gamma)$ and $\\delta a_{\\mu}$. We\ntake $m_t=170.9$~GeV throughout.\n\n\\subsection{Theoretical, Experimental and Cosmological Bounds}\n\nAfter running the mass spectrum of any chosen model parameter set from\nthe GUT scale down to the electroweak scale, we perform a number of\nchecks on the phenomenological acceptability of the point chosen. A\npoint is ruled out if:\n\\begin{enumerate}\n\\item It does not provide radiative electroweak symmetry breaking\n  (REWSB). Such regions are displayed in light red in the subsequent\n  figures.\n\\item It violates the bounds on particle masses provided by the\n  Tevatron and LEP~2. Such regions are displayed in light\n  blue~\\footnote{The current LEP~2 bound on the lightest MSSM Higgs\n  stands at $114.4$~GeV. However, there is a theoretical uncertainty\n  of some $3$~GeV in the determination of the mass of the light\n  Higgs~\\cite{Allanach:2004rh}. Rather than placing a hard cut on the\n  parameter space for the Higgs mass, instead we plot a line at\n  $m_h=111$~GeV and colour the region in which $m_h <111$~GeV in very\n  light grey-blue.}.\n\\item It results in a lightest supersymmetric particle (LSP) that is\n  not the lightest neutralino. We colour these regions light green.\n\\end{enumerate}\n\nIn the remaining parameter space we display the 1- and 2-$\\sigma$\nregions for $\\delta a_{\\mu}$ and $BR(b\\rightarrow s \\gamma)$, as well as plotting the 2-$\\sigma$\nregion for the relic density allowed by WMAP and other observations.\n\n\\subsubsection{$\\delta a_{\\mu}$}\n\nPresent measurements of the anomalous magnetic moment of the muon\n$a_\\mu$ deviate from theoretical calculations of the SM contribution\nbased on low-energy $e^+ e^-$ data~\\footnote{There is a long-running\ndebate whether the calculation of the hadronic vacuum polarisation in\nthe Standard Model should be done with $e^+e^-$ data, or with $\\tau$\ndecay data. The weight of evidence indicates the $e^+e^-$ estimate is\nmore reliable so we use the $e^+ e^-$ value in our work.}. Taking the\ncurrent experimental world average and the state-of-the-art SM value\nfrom~\\cite{hep-ph\/0703049}, there is a discrepancy:\n\\begin{equation}\n  (a_\\mu)_{exp}-(a_\\mu)_{SM}=\\delta a_\\mu = (2.95\\pm\n  0.88)\\times 10^{-9},\n\\end{equation}\nwhich amounts to a 3.4-$\\sigma$ deviation from the SM value.  As\nalready mentioned, we use {\\tt micrOMEGAs} to calculate the SUSY\ncontribution to $(g-2)_\\mu$. The dominant theoretical errors in this\ncalculation are in the SM contribution, so in our analysis we neglect\nthe theoretical error in the calculation of the SUSY contribution.\n\n\\subsubsection{$BR(b\\rightarrow s \\gamma)$}\n\nThe variation of $BR(b\\rightarrow s \\gamma)$ from the value predicted by the Standard Model\nis highly sensitive to SUSY contributions arising from charged\nHiggs-top loops and chargino-stop loops. To date no deviation from the\nStandard Model has been detected. We take the current world average\nfrom~\\cite{hfag}, based on the BELLE~\\cite{hep-ex\/0103042},\nCLEO~\\cite{hep-ex\/0108033} and BaBar~\\cite{Aubert:2005cu}\nmeasurements:\n\\begin{equation}\n  BR(b\\rightarrow s \\gamma) = (3.55 \\pm 0.26) \\times 10^{-4}.\n\\end{equation}\nAgain, we use {\\tt micrOMEGAs} to calculate both the SM value of\n$BR(b\\rightarrow s \\gamma)$ and the SUSY contributions. It is hard to estimate the\ntheoretical uncertainty in the calculation of the SUSY contributions,\nbut note that there is an uncertainty of $10\\%$ in the NLO SM\nprediction of $BR(b\\rightarrow s \\gamma)$~\\cite{bsgNLO}~\\footnote{We recall that {\\tt\nmicrOMEGAs} calculates the SM contribution to $BR(b\\rightarrow s \\gamma)$ to NLO. A first\nestimate of the SM prediction of $BR(b\\rightarrow s \\gamma)$ to NNLO was presented\nin~\\cite{Misiak:2006zs}. This showed a decrease of around $0.4\\times\n10^{-4}$ in the central value of the SM prediction. The implementation\nof the NNLO contributions in the calculation is non-trivial and its\nimplementation in {\\tt micrOMEGAs} is currently underway. As a result\nwe do not include this decrease in the results we present, but instead\nnote that \\textit{positive} SUSY contributions to $BR(b\\rightarrow s \\gamma)$ look likely\nto be favoured in future. This would favour a negative sign of $\\mu$\nand thus cause tension with $(g-2)_\\mu$.}. As with $\\delta a_{\\mu}$, we plot the\n1-$\\sigma$ and 2-$\\sigma$ experimental ranges, and do not include a\ntheoretical error in the calculation.\n\n\\subsubsection{$\\Omega_{CDM}h^2$}\nEvidence from the cosmic microwave background, the rotation curves of\ngalaxies and other astrophysical data point to a large amount of cold\nnon-baryonic dark matter in the universe. The present\nmeasurements~\\cite{astro-ph\/0603449} indicate the following value for\nthe current cold dark matter density:\n\\begin{equation}\n  \\Omega_{CDM}h^2 = 0.106 \\pm 0.008.\n\\end{equation}\nWe calculate the relic dark matter density with {\\tt micrOMEGAs} using\nthe \\textit{fast} approximation. Given a low-energy mass spectrum,\n{\\tt micrOMEGAs} gives an estimated precision of $1\\%$ in the\ntheoretical prediction of the relic density. This is negligible\ncompared to the present observational error, so the 2-$\\sigma$ band\nplotted takes into account only the experimental error~\\footnote{We\nemphasize that the quoted $1\\%$ accuracy is for a given low-energy\nspectrum, which is obtained using {\\tt softsusy}.  However, there are\ndifferences in the details of the mass spectrum between\ncodes~\\cite{Allanach:2003jw}, for given high-energy inputs, and\ndifferent dark matter regions have different levels of sensitivity to\nthese variations: see~\\cite{Belanger:2005jk} for a detailed study. The\nresult of the discrepancies between codes is to move the dark matter\nregions slightly in the GUT scale parameter space. As we are\ninterested in broad features of these regions, rather than their\nprecise locations, these uncertainties are not important for our\npurposes.}.\n\nIn the following Sections, we calculate the dark-matter fine-tuning\nfor any point that lies within the $2\\sigma$ allowed region, and\nindicate the amount using colour coding. We also display electroweak\nfine-tuning contours over the different regions.\n\n\\section{The Constrained Minimal Supersymmetric Standard Model}\n\\label{CMSSM}\n\nWe first review the familiar Constrained Minimal Supersymmetric\nStandard Model (CMSSM) \\cite{Kane:1993td,Ellis:1999mm}. \nwhich serves as a standard to which we compare\nthe parameter space of the NUHM.\n\nThe CMSSM has a much simpler spectrum of soft masses than the full\nMSSM. First, all of the soft squark and slepton (mass)$^2$ matrices\nare chosen to be diagonal and universal at the GUT scale with the\ndiagonal entries equal to $m_0^2$. Secondly both the soft Higgs\n(mass)$^2$ are also set equal to $m_0^2$. Additionally, all the\ngaugino masses are assumed to be unified with a value $m_{1\/2}$ at the\nGUT scale. Finally, we take the trilinear coupling matrices to have\nonly one non-zero entry (the third-family dominance approximation) and\nassume that all these entries are equal to a common value\n$A_0$. Requiring that electroweak symmetry be broken radiatively to\ngive the observed electroweak boson masses, we trade the soft\nparameters $\\mu$ and $B$ for $\\tan\\beta$, the ratio of the Higgs vevs, and\nthe sign of $\\mu$. This results in a model with four free parameters\nand a sign:\n\\begin{equation}\n  a_{CMSSM} \\in \\left\\{\n  m_0,~m_{1\/2},~A_0,~\\tan\\beta,~\\text{sign}(\\mu)\\right\\}.\n\\end{equation}\nAlthough our main focus is the dark-matter fine-tuning, we also report\nthe required amounts of electroweak fine-tuning for\nspecific cases of interest.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/CMSSM,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small The $(m_{1\/2},m_0)$ plane of the CMSSM\n     with $A_0=0,~\\tan\\beta=10$ and sign$(\\mu)$\n    +ve.\\label{f:CMSSM,t10}}\n\\end{figure}\n\nIn Fig.~\\ref{f:CMSSM,t10} we show the $(m_0,m_{1\/2})$ plane of the\nCMSSM for $A_0=0,~\\tan\\beta=10$ and sign$(\\mu)$ positive. At low $m_0$ the\nparameter space is ruled out because $m_{\\tilde{\\tau}}<m_{\\tilde{\\chi}^0_1}$, resulting\nin a stau LSP (light green). Regions at low $m_{1\/2}$ are ruled out by\nLEP~2 bounds on the masses of the charginos and sleptons (light\nblue). A larger range of $m_0$ is ruled out by the absence of a light\nHiggs boson, and we shade the region with $m_h<111$~GeV (light grey\nwith a black boundary). Finally, the model fits the current $\\delta a_{\\mu}$\nmeasurement for low $m_0$ and $m_{1\/2}$. The $1\\sigma$ and $2\\sigma$\nbounds are shown as a short and long-dashed green lines\nrespectively. At larger $m_0$ and $m_{1\/2}$, the SUSY contribution\nbecomes small as the sparticles that contribute in loops become\nheavy. Therefore $\\delta a_{\\mu}\\neq 0$ favours relatively light soft masses.\n\nIn the remainder of the parameter space we find the regions that fit\nthe WMAP strip at $2\\sigma$, and for each such point we calculate the\nfine-tuning of the dark matter density. Each point is then plotted\nwith a colour that corresponds to the value of the tuning via the\nlog-scale on the right hand side.\n\n\\begin{table}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A1} &\n      \\multicolumn{2}{|c|}{A2}\\\\\n      \\cline{2-5}\n                & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$     & 60    & 0.62 & 100 & 5.7\\\\\n      $m_{1\/2}$ & 200   & 0.99 & 500 & 5.8\\\\\n      $\\tan\\beta$   & 10    & 0.13 & 10  & 1.5\\\\\n      \\hline\n      $\\Delta_\\Omega$ & & 0.99 &     & 5.8\\\\\n      \\hline\n      $\\Delta_{EW}$   & & 37   &     & 190\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small A summary of the properties of points A1 and A2,\n    shown in Fig.~\\ref{f:CMSSM,t10}, chosen as representatives of the\n    bulk region (A1) and stau-coannihilation region (A2). We also\n    present a breakdown of the dark-matter and electroweak\n    fine-tunings with respect to each parameter of the CMSSM, except\n    for $A_0$, which we fix: $A_0=0$ here and elsewhere in this\n    paper.\\label{t:CMSSM,t10}}\n\\end{table}\n\nThere are two distinct regions that fit the WMAP measurement of the\nrelic density. The first region lies at $m_{1\/2}\\approx 200$~GeV,\n$m_0\\approx 70$~GeV and contains the point A1. This is the bulk\nregion, which is adjacent to the light blue region that is ruled out\nby the LEP~2 slepton mass constraint. The lightness of the sleptons\nenhances neutralino annihilation via $t$-channel slepton exchange to\nthe extent that it allows bino dark matter to fit the WMAP relic\ndensity measurement. This process is relatively insensitive to the\nprecise masses of the neutralino and the sleptons. This is reflected\nin the fact that much of the bulk region is plotted in yellow,\nsignifying $\\Delta^{\\Omega}<1$. From the breakdown of the tunings of point A1\nin Table~\\ref{t:CMSSM,t10} it is clear that the tuning is mainly in\nthe parameters $m_{1\/2}$ and $m_0$. The sensitivity to $m_0$ is to be\nexpected, as $t$-channel slepton exchange is clearly dependent on the\nmass of the exchanged particle. This also explains the sensitivity to\n$m_{1\/2}$ since, although the masses of the sleptons are determined by\n$m_0$ at the GUT scale, the running to low energies is dominated by\n$m_{1\/2}$. Thus, in this region the masses of the light sleptons are\nsensitive to both $m_0$ and $m_{1\/2}$. The relic density is also\ndependent upon the mass of the LSP, in this case (mainly) a bino with\na mass determined primarily by $m_{1\/2}$. It is therefore apparent why\nthe sensitivity of the bulk regions lies primarily with $m_0$ and\n$m_{1\/2}$. The fact that no precise cancellations or balancing of\nparameters is required explains why the tuning is low. However, the\nbulk region lies inside a light grey region signifying a Higgs with a\nmass less than $111$~GeV. Therefore we do not consider this region\nfurther here.\n\nThe second region that fits the observed relic density is the band\nthat contains point A2. This band lies alongside the area that is\nruled out by a stau LSP (light green). Along the edge of the light\ngreen region, the mass of the stau is close to that of the lightest\nneutralino, resulting in comparable number densities of both particles\naround the time of freeze-out. Thus many more annihilation channels\nmust be considered, such as the annihilations of $\\tilde{\\tau}-\\tilde{\\tau}$ and\n$\\tilde{\\tau}-\\tilde{\\chi}^0_1$ in addition to $\\tilde{\\chi}^0_1-\\tilde{\\chi}^0_1$. This suppresses the number\ndensity of SUSY particles and thus reduces the resultant dark matter\nrelic density. The effect of coannihilation depends strongly upon the\nnumber density of NLSPs at freeze-out. This is in turn very sensitive\nto the mass difference between the NLSP and the LSP. Thus, it may at\nfirst sight seem surprising that the $\\tilde{\\tau}$ coannihilation band is\nplotted here in green and red, corresponding to relatively low dark\nmatter fine-tuning $\\Delta^{\\Omega}=3-11$. The reason, as discussed\nin~\\cite{hep-ph\/0603095}, is that in this region the RGE running\nresults in the mass of the right handed $\\tilde{\\tau}$ and the mass of the\nlightest neutralino both being dominantly dependent on $m_{1\/2}$, with\nthe stau having a smaller secondary dependence on $m_0$. As a result,\ntheir masses vary together as the soft masses are varied, and $\\Delta\nm$ is remarkably insensitive to $m_0$ or $m_{1\/2}$. This mitigates the\nnormal sensitivity of the coannihilation region. We take the point A2\nas a representative point in this band and provide a breakdown of the\nindividual tunings in Table~\\ref{t:CMSSM,t10}. The tuning is equally\ndependent upon $m_0$ and $m_{1\/2}$, as expected.\n\nAs well as the dark-matter fine-tuning, for reference we also\ncalculate the fine-tuning required to fit the electroweak boson\nmasses. This calculation is performed across the parameter space using\nthe same measure as we use for the dark matter fine-tuning. We plot\nthe tuning using blue dot-dashed contours in Fig.~\\ref{f:CMSSM,t10},\nand label each contour with their respective values of\n$\\Delta^{EW}=\\text{max}(\\Delta^{EW}_a)$. We also list the value of\n$\\Delta^{EW}$ in the last row of Table~\\ref{t:CMSSM,t10} for both points\nwe consider.  The $W$ boson mass is in general the result of a careful\nbalancing act between the soft masses: the larger the soft masses, the\nmore precisely they must cancel to give the $W$ boson mass. Therefore\nit is unsurprising that the amount of fine-tuning required to obtain\nthe correct electroweak symmetry breaking increases smoothly with the\nincreasing soft masses. The electroweak and dark matter fine-tunings\nare largely independent.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/CMSSM,t,50.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small The $(m_0,~m_{1\/2})$ plane of the\n    CMSSM parameter space for $A_0=0$, $\\tan\\beta=50$ and sign($\\mu$)\n    +ve.\\label{f:CMSSM}}\n\\end{figure}\n\nOther regions in which the CMSSM fits the observed relic density are\nseen when we consider larger values of $\\tan\\beta$. In Fig.~\\ref{f:CMSSM}\nwe consider a wider range in the $(m_0,m_{1\/2})$ plane and take\n$\\tan\\beta=50$.\n\nLow $m_0$ is once again ruled out because the stau is the LSP (light\ngreen). Equally low $m_{1\/2}$ and $m_0$ are ruled out by searches for\nthe lightest Higgs (light grey with a black boundary) or LEP~2 bounds\non the chargino and light sleptons (solid light blue). Finally large\n$m_0$ and low $m_{1\/2}$ are ruled out as they do not give REWSB (light\nred).  We plot the different regions that fit the relic density using\nthe same colour coding for dark-matter fine-tuning as was used in\nFig.~\\ref{f:CMSSM,t10}.\n\n\\begin{table}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A3} &\n      \\multicolumn{2}{|c|}{A4} &\n      \\multicolumn{2}{|c|}{A5} &\n      \\multicolumn{2}{|c|}{A6}\\\\\n      \\cline{2-9}\n                & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value\n& $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$     & 2030  & 27   & 540 & 5.0  & 277   & 19   & 1400& 13   \\\\\n      $m_{1\/2}$ & 500   & 18   & 600 & 8.0  & 350   & 12   & 250 & 22   \\\\\n      $\\tan\\beta$   & 50    & 2.0  & 50  & 76   & 50    & 48   & 50  & 8.2  \\\\\n      \\hline\n      $\\Delta_\\Omega$ & & 27   &     & 76   &       & 48   &     & 22   \\\\\n      \\hline\n      $\\Delta_{EW}$  &  & 150  &     & 230  &       & 92   &     & 48   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Here we summarize properties of the points A3-6,\n    shown in Fig.~\\ref{f:CMSSM}, which represent the higgsino\/bino\n    region (A3,6), the pseudoscalar Higgs funnel (A4) and the\n    stau-coannihilation region at large $\\tan\\beta$ (A5). We also present\n    a breakdown of the dark matter fine-tuning with respect to each\n    parameter, and also list the electroweak\n    fine-tuning.\\label{t:CMSSM}}\n\\end{table}\n\nWe now distinguish four regions in which the CMSSM can account for the\nobserved relic density of dark matter. The least interesting of these\nregions is the bulk region, which once again appears at low $m_0$ and\nlow $m_{1\/2}$, but requires a Higgs lighter than current search limits\nallow.  Moving on from the inaccessible bulk region, we again find the\ncoannihilation strip along the side of the region ruled out because\nthe stau is the LSP. In comparison to Fig.\\ref{f:CMSSM,t10}, it is\nthin and broken. This is a sampling artefact, arising because we have\nextended the range of the $m_0$ scan, and the resolution is not fine\nenough to resolve the full strip. In contrast to\nFig.\\ref{f:CMSSM,t10}, here the strip is plotted in purple,\ndesignating a dark matter fine-tuning in the range $\\Delta^{\\Omega}=30-100$.\n\nConsider now the representative point A5 and the corresponding\nfine-tuning. At large $\\tan\\beta$, the mass of the $\\tilde{\\tau}_R$ runs down\nmore than for low $\\tan\\beta$, and therefore a larger soft mass is\nnecessary to avoid a $\\tilde{\\tau}$ LSP. For low $\\tan\\beta$ the $\\tilde{\\tau}$\ncoannihilation region occurred when $m_0\\ll m_{1\/2}$, whereas here\n$m_0$ and $m_{1\/2}$ are much closer in magnitude. This reduces the\ndominance of $m_{1\/2}$ in the running and restores the need for a\nprecise balance of $m_0$ and $m_{1\/2}$ to keep the $\\tilde{\\tau}-\\tilde{\\chi}^0_1$ mass\ndifference small. However, it is the influence of $\\tan\\beta$ over the\nrunning that dominates the sensitivity.\n\nThe second region of interest shows up as two diagonal bands starting\nbeside the light green region ruled out because the stau is the\nLSP. These lines lie to either side of the line along which\n$2m_{\\tilde{\\chi}^0_1}=m_A$, and neutralino annihilation proceeds via the\nresonant production of a pseudoscalar Higgs boson. This region is\nknown as the pseudoscalar Higgs funnel region. The WMAP lines lie\nalong either side of this resonance, where there is just enough of an\nenhancement in the annihilation cross section to allow a bino LSP to\naccount for the observed relic density. As a result, the dark matter\nfine-tuning price of such a region is large. This is reflected in the\npurple and blue shading of the funnel region, corresponding to a dark\nmatter fine-tuning in the range $\\Delta^{\\Omega}=30-300$.\n\nWe consider the representative point A4 and break down the\nfine-tuning. The mass of the pseudoscalar Higgs is dependent upon the\ndetails of the running of the soft Higgs mass-squared terms, which also\ndetermine the Higgs vevs and thus $\\tan\\beta$. As we require REWSB and set\n$\\tan\\beta$ at the start, we can run this chain of logic back the other\nway. A fixed value of $\\tan\\beta$ requires a specific value of the soft\nHiggs mass-squared terms at the low-energy scale. Thus the value of\n$\\tan\\beta$ has a significant impact on the pseudoscalar Higgs boson\nmass. This explains the dominant sensitivity of the pseudoscalar Higgs\nfunnel region to $\\tan\\beta$. This is to be contrasted with the NUHM,\nwhich will allow access to the pseudoscalar Higgs funnel region\nwithout large $\\tan\\beta$, so that naively one might expect a more natural\nfunnel region.\n\nThe final WMAP strip lies at large $m_0$ in all panels. The corner of\nthe parameter space at large $m_0$ and low $m_{1\/2}$ (shown in light\nred) is ruled out as $\\mu^2<0$, signifying a failure of REWSB. Along\nthe perimeter of this region $\\mu\\approx 0$, and $\\mu$ decreases\nsteadily as one approaches the boundary. As $\\mu$ decreases, the\nhiggsino component of the LSP increases. Higgsino dark matter\nannihilates very efficiently resulting in $\\Omega_{CDM}h^2\\ll \\Omega^{WMAP}_{CDM}h^2$, whereas bino\ndark matter generally gives $\\Omega_{CDM}h^2\\gg\\Omega^{WMAP}_{CDM}h^2$. Along the WMAP strip, the\nhiggsino component of the $\\tilde{\\chi}^0_1$ is large enough that the\nannihilation cross section is enhanced sufficiently for the relic\ndensity to fit the WMAP data. This region is known as the focus-point\nregion.  It is sensitive to the composition of the lightest\nneutralino, and so depends upon the difference between $\\mu$ and $M_1$\nat the electroweak scale. The region is plotted in red and purple,\nwith a couple of green regions at low $m_{1\/2}$. This corresponds to a\ndark matter fine-tuning of $\\Delta^{\\Omega} \\approx 10$ at low $m_{1\/2}$,\nrising to a dark matter fine-tuning of $\\Delta^{\\Omega}\\approx 60$ at large\n$m_0$ and $m_{1\/2}$. The kink at $m_{1\/2}\\approx450~GeV$, where the\ntuning drops (signified by a change from purple to red) corresponds to\nthe top quark mass threshold where processes of the form $\\tilde{\\chi}^0_1\\neut\n\\rightarrow t\\overline{t}$ become kinematically allowed.\n\nWe take points A3 and A6 as representative points in this region and\nbreak the dark matter fine-tuning down into its individual\nelements. At point A3 the LSP is predominantly bino with a small, but\nsignificant, higgsino component. The higgsino component is determined\nby the relative size of $\\mu(EW)$ to $M_1(EW)$. As we set $\\mu(EW)$ by\nrequiring REWSB, $\\mu(EW)$ is determined by the running of the soft\nHiggs masses. Thus $\\mu(EW)$ is sensitive to the soft Higgs mass-squared\nterms at the GUT scale (set to $m_0^2$ in the CMSSM). It is also\nsensitive to strongly-interacting sparticle masses through the\nRGEs. This once again brings in a sensitivity to $m_0$, but also to\n$M_3$ through its strong influence on the stop quark mass. Therefore\n$\\mu(EW)$ is sensitive to both $m_0$ and $m_{1\/2}$. To achieve the\ncorrect balance of bino and higgsino components one needs to balance\n$\\mu(EW)$ and $M_1(EW)$. These both depend strongly upon $m_{1\/2}$,\nwith $\\mu$ also dependent upon $m_0$. This common dependence on\n$m_{1\/2}$ reduces the dark matter fine-tuning below what one would expect.\n\nAt point A6 we consider the kink at the bottom of the higgsino\/bino\nline, where the higgsino and bino components are almost equal. This\nwould normally result in extremely efficient annihilation of\nneutralinos in the early universe and give $\\Omega_{CDM}h^2\\ll\\Omega^{WMAP}_{CDM}h^2$. However at\nthis point the neutralino mass is $m_{\\tilde{\\chi}^0_1}=79.6$~GeV, disallowing\nthe annihilation of neutralinos to Higgs, $Z$ bosons and\n$t\\overline{t}$. It is also on the $W$ boson threshold, suppressing\nannihilation to $W$s. With these annihilation channels ruled out, the\nnormally efficient annihilation of an LSP with a substantial higgsino\nfraction is suppressed sufficiently to fit the WMAP relic density. As\nwith point A3, the determining factors for the annihilation cross\nsection are the relative sizes of $\\mu(EW)$ and $M_1(EW)$, resulting\nin a similar pattern and magnitude of dark matter fine-tuning measures.\n\nOnce again we also calculate the electroweak fine-tuning across the\nplane and plot the corresponding contours of $\\Delta^{EW}$. As in the\ncase of Fig.~\\ref{f:CMSSM,t10}, the degree of electroweak tuning\nincreases steadily with increasing $m_{1\/2}$. This is because the\ndominant soft term is $M_3$, the gluino mass. As $m_{1\/2}$ increases,\nthe gluino mass and squark masses increase, requiring more precise\ncancellations to reproduce the $W$ mass, and thus greater\nelectroweak fine-tuning.\n\n\\begin{table}[ht!]\n  \\begin{center}\n    \\begin{tabular}{|l|c|}\n      \\hline\n      {\\bf Region} & {\\bf Tuning Range} \\\\\n      \\hline\n      Pseudoscalar Higgs Funnel     & 60-1200+ \\\\\n      $\\tilde{\\tau}-\\tilde{\\chi}^0_1$ coannihilation (large $m_0,~m_{1\/2}$ or $\\tan\\beta$)  & 30-60\\\\\n      $\\tilde{\\tau}-\\tilde{\\chi}^0_1$ coannihilation (low $m_0,~m_{1\/2}$ and $\\tan\\beta$)  & 3-10\\\\\n      bino\/higgsino region          & 10-60\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Here we summarize the DM annihilation\n    channels present within the CMSSM and their associated\n    tunings. This provides a reference to which we will compare the\n    regions accessible within the NUHM.\\label{t:CMSSMTunings}}\n\\end{table}\n\nNow that we have considered the different dark matter regions present\nwithin the CMSSM, we summarize the typical tunings in each case. We\ndistinguish in Table \\ref{t:CMSSMTunings} the principal dark matter\nregions present within the CMSSM and the corresponding amounts of\ndark matter fine-tuning.\nWith this as our starting point, we now consider the case\nof the NUHM model in which the universality between the soft Higgs\nmasses and the soft sfermion masses is broken.\n\n\\section{The NUHM}\n\\label{NUHM}\n\nWe now consider the MSSM with non-universal Higgs soft masses\n(NUHM) \\cite{oldnuhm,Ellis:2002wv,hep-ph\/0210205}.\nAfter breaking the universality between the soft Higgs and\nsfermion masses of the CMSSM we have the following independent inputs:\n\\begin{equation}\n  a_{NUHM}=\\left\\{m_0,~m_{H_1},~m_{H_2},~m_{1\/2},~A_0,~\\tan\\beta,\n  ~\\text{sign}(\\mu)\\right\\},\n\\label{NUHMPar}\n\\end{equation}\nwhere $m_{H_1}$ and $m_{H_2}$ characterize the independent soft Higgs\nmasses.  These are subject to constraints arising from vacuum\nstability and cosmological considerations, and may be negative.  As\nlong as $m_{H_{1,2}}^2 + \\mu^2 > 0$ at the GUT scale, there is no\ndangerous high-scale vacuum state, but specifying the precise\nboundaries of the NUHM parameter space lies beyond the scope of this\nwork.\n\nAs with the CMSSM, the NUHM contains a finite number of distinct\nregions in which it can provide the observed dark matter relic\ndensity, which were catalogued in~\\cite{hep-ph\/0210205}. Here we\nfollow the approach of this previous work and reproduce the same\nregions of the parameter space. The plots we present here show the\nupdated parameter space for the current world average for the top\nmass, $m_t=170.9$~GeV, and include the current dark matter and $\\delta a_{\\mu}$\nconstraints. However, the primary goal of this work is rather to\nanalyse the fine-tuning of the dark-matter regions of the NUHM. To\nthis end we calculate and plot the dark-matter fine-tuning in the\nallowed parameter space, and also make some comments on the amount of\nelectroweak fine-tuning.\n\nAs the NUHM contains the CMSSM as a limiting case, all the dark-matter\nregions present in the CMSSM are present in the NUHM. In addition,\nthere are four new regions that are not present in the CMSSM:\n\n\\begin{itemize}\n\\item A pseudoscalar Higgs funnel at low $\\tan\\beta$.\n\\item A bulk region where $\\tilde{\\chi}^0_1$ annihilation is dominantly mediated\n  via $t$-channel $\\tilde{\\tau}$ exchange which does not violate Higgs mass\n  bounds.\n\\item A $\\tilde{\\nu}-\\tilde{\\chi}^0_1$ coannihilation region.\n\\item A mixed bino\/higgsino region at low $m_0$.\n\\end{itemize}\n\nWe shall be particularly interested in understanding how finely tuned\nthe NUHM parameters must be in each of these new regions.\n\n\\subsection{Comparison with the CMSSM}\n\nThe NUHM contains all the points in the CMSSM parameter\nspace. Therefore, we start by studying the tuning of the dark matter\npoints A1-6, presented in Tables~\\ref{t:CMSSM,t10},\\ref{t:CMSSM}, with\nrespect to the parameters of the NUHM.\n\n\\begin{table}[ht!]\n\\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A1} &\n      \\multicolumn{2}{|c|}{A2} &\n      \\multicolumn{2}{|c|}{A3} &\n      \\multicolumn{2}{|c|}{A4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 60    & 0.62 & 100    & 5.7 & 2030   & 200  & 540    & 8.1  \\\\\n      $m_{H_1}$   & 60    & 0.017& 100    & 0.26& 2030   & 14   & 540    & 28   \\\\\n      $m_{H_2}$   & 60    & 0.014& 100    & 0.26& 2030   & 230  & 540    & 30   \\\\\n      $m_{1\/2}$   & 200   & 0.99 & 500    & 5.8 & 500    & 18   & 600    & 8.0  \\\\\n      $\\tan\\beta$     & 10    & 0.13 & 10     & 1.5 & 50     & 2.0  & 50     & 76   \\\\\n      \\hline\n      $\\Delta_\\Omega$  &  & 0.99 &        & 5.8 &        & 230  &        & 76   \\\\\n      \\hline\n      $\\Delta_{EW}$    &  & 37   &        & 190 &        & 1300 &        & 230  \\\\\n      \\hline\n    \\end{tabular}\n    \\begin{tabular}{|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A5} &\n      \\multicolumn{2}{|c|}{A6}\\\\\n      \\cline{2-5}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$  \\\\\n      \\hline\n      $m_0$       & 277    & 23  & 1400   & 230  \\\\\n      $m_{H_1}$   & 277    & 1.5 & 1400   & 10  \\\\\n      $m_{H_2}$   & 277    & 2.5 & 1400   & 73  \\\\\n      $m_{1\/2}$   & 350    & 12  & 250    & 22  \\\\\n      $\\tan\\beta$     & 50     & 48  & 50     & 8.2 \\\\\n      \\hline\n      $\\Delta_\\Omega$ &    & 48  &        & 230 \\\\\n      \\hline\n      $\\Delta_{EW}$   &    & 92  &        & 600 \\\\\n      \\hline\n    \\end{tabular}\n    \\end{center}\\vskip -0.5cm\n  \\caption{\\small A re-analysis of the representative points A1-6 from\n    Figs.~\\ref{f:CMSSM,t10},\\ref{f:CMSSM}, calculating their tunings\n    with respect the NUHM rather than the CMSSM.\\label{t:CMSSMComp}}\n\\end{table}\n\nWe show the dark matter\nfine-tuning of these points with respect to the parameters\n$a_{NUHM}$ in Table~\\ref{t:CMSSMComp}. Point A1 represents the bulk\nregion of the CMSSM, which is inaccessible because the Higgs is\nlight. The primary annihilation channel is $t$-channel slepton\nexchange, and the sensitivity in the CMSSM is primarily due to $m_0$\nand $m_{1\/2}$ as they determine the neutralino and slepton\nmasses. This is also true in the NUHM, and the sensitivities to the\nHiggs soft masses are negligible.\n\nPoint A2 represents the low-$\\tan\\beta$ coannihilation region of the\nCMSSM, in which the primary sensitivities were to $m_{1\/2}$ and $m_0$,\nas these determine the stau mass and the neutralino mass. Once again,\nthis picture changes very little in the NUHM, with the sensitivity to\nthe soft Higgs masses being negligible.\n\nPoints A3-6 have large $\\tan\\beta$. We recall that A3 and A6 lie in the\nhiggsino-bino focus-point region. In the CMSSM the primary\nsensitivities were to $m_0$ and $m_{1\/2}$, as $m_0$ (and to a lesser\nextent $M_3$) determine the size of $\\mu$, and $m_{1\/2}$ determines\n$M_1(EW)$. Therefore these two parameters determine the mass and\ncomposition of the lightest neutralino, and the total CMSSM dark matter\nfine-tuning of the point in the CMSSM was $\\Delta^{\\Omega}=27$. In the NUHM we\nhave a very different picture. Here the total dark matter fine-tuning is\n$\\Delta^{\\Omega}=230$, and the primary sensitivities are to $m_0$ and\n$m_{H_2}$. This can be explained by the process of radiative\nelectroweak symmetry breaking. For electroweak symmetry breaking to\noccur, the Higgs (mass)$^2$ must become negative. By requiring this\nprocess to give the correct electroweak boson masses we set the size\nof $\\mu$, and thus the magnitude of the higgsino component of the\nlightest neutralino. Therefore to understand the sensitivity of a\nhiggsino-bino dark matter region, we must look for the terms that\ncontribute to the running of the Higgs mass-squared. First there is the\nsoft Higgs mass at the GUT scale, and then there are the running\neffects, primarily the contribution from the stop mass. In the CMSSM,\nthese two terms are coupled, reducing the dependence on either one\nindividually. Therefore even though the scalar masses are large, the\nsensitivity of $\\mu$ to $m_0$ remains small. In the NUHM there is no\nconnection between the soft sfermion masses and the soft Higgs masses,\ntherefore the sensitivity returns. Therefore one should not expect\nnatural bino-higgsino dark matter at large $m_0$ in the NUHM. The\nsignificant increase in the electroweak fine-tuning for these points\nis due to exactly the same physics.\n\nPoints A4 and A5 represent the pseudoscalar Higgs funnel and the\nstau-coannihilation band. At this value of $\\tan\\beta$, the primary\nsensitivity is to $\\tan\\beta$, a feature not altered by breaking the\nuniversality amongst the scalars.\n\n\\subsection{Detour: RGE behaviour with negative masses-squared}\n\nTo understand the dependence of the dark matter phenomenology on the\nNUHM GUT scale parameters we need to understand how the soft Higgs\nmasses affect the RGEs, and through them the low-energy\nparameters. Four low-energy parameters in particular are useful to\nconsider when we talk about dark matter: $\\mu$, $m_A$, and\n$\\tilde{\\tau}_{L,R}$. The higgsino component of the LSP is determined by\n$\\mu$, $m_A$ determines the position of the pseudoscalar Higgs funnel,\nand the lightest stau (a mixture of $\\tilde{\\tau}_{L,R}$) mediates the\nprevalent t-channel slepton exchange annihilation diagrams and\ndetermines the efficiency of $\\tilde{\\tau}$ coannihilation channels.\n\nAfter EW symmetry breaking we can write $\\mu$ as:\n\\begin{equation}\n  \\mu^2=\\frac{m_{H_1}^2-m_{H_2}^2\\tan^2\\beta}{\\tan^2\\beta-1} -\n  \\frac{1}{2}m_Z^2.\n\\label{e:musq}\n\\end{equation}\nClearly $\\mu$ depends on the soft Higgs mass-squared terms and $\\tan\\beta$, as\nwell as other soft parameters through the RGEs. It is also useful to\nconsider the limit of large $\\tan\\beta$ where we can approximate\n(\\ref{e:musq}) as:\n\\begin{equation}\n\\mu^2=-m_{H_2}^2+\\frac{m_{H_1}^2}{\\tan^2\\beta},\n\\end{equation}\nassuming $|m_{H_{1,2}}^2|\\gg m_Z^2$. Therefore for large $\\tan\\beta$, to\nachieve REWSB and have $\\mu^2>0$ we require either negative\n$m_{H_2}^2$, or very large positive $m_{H_1}^2$.\n\nThe pseudoscalar Higgs mass is determined after EWSB by the relation:\n\\begin{equation}\n  m_A^2=m_{H_1}^2+m_{H_2}^2+2\\mu^2.\n\\label{mAsq}\n\\end{equation}\nClearly $m_A^2$ is strongly dependent upon the soft Higgs mass-squared\nterms, $\\tan\\beta$ through its effect on $\\mu$, and other soft terms\nthrough their influence on the Higgs RGEs.\n\nWe now consider the explicit form of the soft Higgs mass-squared RGEs:\n\\begin{eqnarray}\n\\nonumber \\frac{d(m_{H_1}^2)}{dt}&=&\\frac{1}{8\\pi^2}\n\\left(-3g_2^2M_2^2-g_1^2M_1^2+h_\\tau^2 (m_{\\tilde{\\tau}_L}^2 +\nm_{\\tilde{\\tau}_R}^2 + m_{H_1}^2 + A_\\tau^2) \\right.\\\\\n&&\\left.+3h_b^2(m_{\\tilde{b}_L}^2 +\nm_{\\tilde{b}_R}^2+m_{H_1}^2+A_b^2) - 2S\\right),\\\\\n\\frac{d(m_{H_2}^2)}{dt}&=&\\frac{1}{8\\pi^2} \\left(-3g_2^2M_2^2-g_1^2M_1^2\n+3h_t^2(m_{\\tilde{t}_L}^2 + m_{\\tilde{t}_R}^2 + m_{H_2}^2 +\nA_t^2) + 2S\\right),\n\\end{eqnarray}\nwhere $S$ is definedly:\n\\begin{eqnarray}\n\\nonumber S&\\equiv&\\frac{g_1^2}{4}\\left(m_{H_2}^2-m_{H_1}^2 +\n2\\left(m_{\\tilde{Q}_L}^2-m_{\\tilde{L}_L}^2-2m_{\\tilde{u}_R}^2 +\nm_{\\tilde{d}_R}^2+m_{\\tilde{e}_R}^2\\right)\\right.\\\\\n&&+\\left.\\left(m_{\\tilde{Q}_{3L}}^2-m_{\\tilde{L}_{3L}}^2 -\n2m_{\\tilde{t}_R}^2 + m_{\\tilde{b}_R}^2+m_{\\tilde{\\tau}_R}^2\\right)\\right).\n\\end{eqnarray}\n\nThe only parameters in these RGEs that we are not free to set at the\nGUT scale are the Yukawa couplings $h_i$. These are set by the\nrequirement that the Higgs vevs should give the correct SM particle\nmasses:\n\\begin{equation}\nm_{\\tau,b}=\\frac{1}{\\sqrt{2}}h_{\\tau,b} v_1,\n~m_{t}=\\frac{1}{\\sqrt{2}}h_{t} v_2.\n\\end{equation}\nTherefore $\\tan\\beta$ influences the RGEs indirectly through its\ndetermination of the size of the Yukawa couplings. The Yukawa\ncouplings multiply the contribution to the RGEs from the soft squark\nand slepton mass-squared terms and the soft Higgs terms. Therefore varying\nthe Yukawa couplings has a large impact on the running. As we increase\n$\\tan\\beta$, we increase $v_2$ with respect to $v_1$, and so we must\ndecrease $h_t$ and increase $h_{\\tau,b}$. Therefore we reduce the\nYukawa contribution to the running of $m_{H_2}^2$, while increasing\nthe contribution to the running of $m_{H_1}^2$.\n\nNow consider the RGEs for the right and left handed stau masses:\n\\begin{eqnarray}\n\\frac{d(m_{\\tilde{L}_{3L}}^2)}{dt}&=&\\frac{1}{8\\pi^2}\n\\left(-3g_2^2M_2^2-g_1^2M_1^2+h_\\tau^2 \\left(m_{\\tilde{L}_{3L}}^2 +\nm_{\\tilde{\\tau}_R}^2+m_{H_1}^2+A_\\tau^2\\right) -2S\\right)\\\\\n\\frac{d(m_{\\tilde{\\tau}_R}^2)}{dt}&=&\\frac{1}{8\\pi^2} \\left(-4g_1^2M_1^2 +\n2h_\\tau^2 \\left(m_{\\tilde{L}_{3L}}^2 +\nm_{\\tilde{\\tau}_R}^2+m_{H_1}^2+A_\\tau^2\\right)+4S\\right)\n\\end{eqnarray}\nIn both cases $m_{H_1}^2$ provides a substantial contribution to the\nrunning, with a coefficient of $h_\\tau$. As we have seen, increasing\n$\\tan\\beta$ increases $h_\\tau$ and thus increases the impact of the Higgs\nmasses on the running of the staus. Therefore we expect any effects of\nnon-universal soft Higgs masses on the stau running to be amplified\nfor large $\\tan\\beta$. In the CMSSM, $m_{H_1}^2$ will remain positive from\nthe GUT scale to the EW scale. Indeed, it is harder to achieve REWSB\nif $m_{H_1}^2$ runs negative. Therefore generally this term provides a\npositive contribution to both the left and right handed stau RGE and\nacts to suppress the stau masses.\n\nIn the CMSSM this poses a problem. As we increase $\\tan\\beta$ we must\nincrease the soft stau mass to avoid it becoming the LSP. However as\nwe increase $m_0$ we are also increasing $m_{H_1}^2$, and thus\nincrease the effect on the running. This can be avoided in the\nNUHM. We can set $m_{H_1}^2$ small and so avoid a very light\n$\\tilde{\\tau}$.\n\nHowever, there is another more subtle effect. The interaction of the\nneutralinos with the stau also depends upon the composition of the\nlightest stau which is determined by the mixing between\n$\\tilde{\\tau}_{L,R}$. This mixing is increased if the two states are close in\nmass. In the CMSSM $S$ is negligible and so $d(m_{\\tilde{\\tau}_R}^2)\/dt \\gg\nd(m_{\\tilde{L}_{3L}}^2)\/dt$, resulting in the right handed stau always\nbeing considerably lighter than the left-handed stau. In the NUHM we\ncan avoid this by having a large negative $S$. This acts to suppress\nthe left handed stau mass while increasing the right handed stau\nmass. As we increase the component of the left-handed stau, we\nincrease the annihilation rate of neutralinos via t-channel stau\nexchange.\n\n\\subsection{Sample $(m_0,m_{1\/2})$ planes in the NUHM}\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,m12,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small The $(m_0,~m_{1\/2})$ plane of the NUHM\n  parameter space with $A_0=0$, $\\tan\\beta=10$ and sign$(\\mu)$ +ve. The\n  values of $\\mu$ and $m_A$ vary between the panels: (a)\n  $\\mu=400$~GeV, $m_A=400$~GeV, (b) $\\mu=400$~GeV, $m_A=700$~GeV, (c)\n  $\\mu=700$~GeV, $m_A=400$~GeV, (d) $\\mu=700$~GeV, $m_A=700$~GeV. This\n  figure can be compared directly to Fig.~2\n  in~\\protect\\cite{hep-ph\/0210205}. The Roman cross in panel (b)\n  indicates the single point where the parameter space makes contact\n  with the CMSSM.\\label{f:m0,m12,t,10}}\n\\end{figure}\n\nHaving analysed the CMSSM points from the perspective of the NUHM, we\nnow turn to a sampling of the full NUHM parameter space. In\nFig.~\\ref{f:m0,m12,t,10}, we show $(m_{1\/2}, m_0)$ planes for\n$\\tan\\beta=10$, $A_0=0$ and sign$(\\mu)$ positive. We set the\nelectroweak scale parameters $\\mu$ and $m_A$ to different discrete\nvalues in each panel as explained in the figure caption.\n\nAs we saw in the previous section, $\\mu$ and $m_A$ are not high-scale\ninputs into the theory, rather they are the low-energy numbers derived\nfrom a given set of the true input parameters. However, displaying\nresults as functions of these parameters can be more informative. As\nboth have a strong dependence on $m_{H_{1,2}}^2$, we can fit a\nparticular value of $\\mu$, $m_A$ with the correct choice of\n$m_{H_{1,2}}^2$ at the GUT scale. Therefore we use a code that varies\n$m_{H_{1,2}}^2$ across the parameter space to fit the designated\nlow-energy values of $\\mu$ and $m_A$. All fine-tunings are calculated\nin terms of the inputs of the NUHM as listed in (\\ref{NUHMPar}).\n\nBy starting with $(m_0, m_{1\/2})$ planes, we make contact with the\nparameter space of the CMSSM as displayed in\nFigs.~\\ref{f:CMSSM,t10},~\\ref{f:CMSSM}~\\footnote{ We note in panel (b)\nof Fig.~\\ref{f:m0,m12,t,10} the appearance of a CMSSM point, the only\npoint in any of these planes where full GUT-scale universality is\nrecovered.}.  As before, low $m_0$ is ruled out by a $\\tilde{\\tau}$ LSP\n(light green), and low $m_{1\/2}$ results in a Higgs with $m_h<\n111$~GeV (light grey with black boundary). As before, $\\delta a_{\\mu}$ favours\nlight sleptons, and thus low $m_0$ and $m_{1\/2}$.\n\n\\begin{table}[ht!]\n\\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{B1} &\n      \\multicolumn{2}{|c|}{B2} &\n      \\multicolumn{2}{|c|}{B3} &\n      \\multicolumn{2}{|c|}{B4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 500   & 32   & 500    & 8.6 & 500    & 4.6  & 500    & 12   \\\\\n      $m_{H_1}^2$ & -80249& 16   & -126930& 12  & -248480& 0.61 & 90625  & 2.3  \\\\\n      $m_{H_2}^2$ & 461380& 62   & 675760 & 25  & 1202900& 24   & 1194100& 60   \\\\\n      $m_{1\/2}$   & 435   & 39   & 540    & 19  & 750    & 18   & 750    & 38   \\\\\n      $\\tan\\beta$     & 10    & 5    & 10     & 3.2 & 10     & 1.1  & 10     & 2.9  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &  & 62   &        & 25  &        & 24   &        & 60   \\\\\n      \\hline\n      $\\Delta_{EW}$    &  & 150  &        & 220 &        & 390  &        & 390  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 400   & -    & 400 & -   & 400 & -   & 400 & -   \\\\\n      $m_A$       & 400   & -    & 400 & -   & 400 & -   & 700 & -   \\\\\n      \\hline\n    \\end{tabular}\n    \\\\\n    \\begin{tabular}{|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{B5} &\n      \\multicolumn{2}{|c|}{B6} &\n      \\multicolumn{2}{|c|}{B7}\\\\\n      \\cline{2-7}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 500    & 40  & 100    & 6.3 & 0      & 0   \\\\\n      $m_{H_1}^2$ & -416350& 110 & -400510& 12  & -79656 & 2.2 \\\\\n      $m_{H_2}^2$ & -24320 & 4.1 & -332200& 10  & -266010& 7.4 \\\\\n      $m_{1\/2}$   & 442    & 52  & 400    & 3.5 & 445    & 4.3 \\\\\n      $\\tan\\beta$     & 10     & 5.8 & 10     & 0.55& 10     & 1.9 \\\\\n      \\hline\n      $\\Delta_\\Omega$ &    & 110 &        & 12  &        & 7.4 \\\\\n      \\hline\n      $\\Delta_{EW}$   &    & 250 &        & 250 &        & 250 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 700 & -   & 700 & -   & 700 & -   \\\\\n      $m_A$       & 400 & -   & 400 & -   & 700 & -   \\\\\n      \\hline\n    \\end{tabular}\n    \\end{center}\\vskip -0.5cm\n  \\caption{\\small Analysis of the points B1-7, shown in\n    Fig.~\\ref{f:m0,m12,t,10}, which are representative of the\n    pseudoscalar Higgs funnel (B1,2,4,5), mixed bino-higgsino dark\n    matter (B3) and $\\tilde{\\tau}$ coannihilation regions (B6,7). We present a\n    breakdown of the dark matter fine-tuning with respect to each parameter\n    of the NUHM. We give the value of $m_{H_{1,2}}^2$, but the tunings\n    are calculated with respect to $m_{H_{1,2}}$.\\label{t:m0,m12,t,10}}\n\\end{table}\n\nThe dark matter phenomenology shows some similarities to and some\nmarked differences from the CMSSM. First, we see a familiar $\\tilde{\\tau}$\ncoannihilation region alongside the region with a $\\tilde{\\tau}$ LSP. As in\nthe CMSSM, this region is plotted in red and green, designating a\ntuning of $\\Delta^{\\Omega}=3-30$. The only new feature of the coannihilation\nregion here is that effects of the non-universal Higgs soft masses\nalter the running of the stau mass, which allows access to regions\nwith $m_0=0$. We can access $m_0=0$ with small negative $m_{H_1}^2$\nand larger negative $m_{H_2}^2$. The combination of a small Yukawa\ncontribution (due to low $\\tan\\beta$ along with small $|m_{H_1}^2|$) along\nwith negative $S$ results in the stau mass that increases as we run\ndown from the GUT scale, allowing an acceptable stau mass even with\n$m_0=0$.\n\nThe points B6 and B7 are representative of the $\\tilde{\\tau}$ coannihilation\nregion, and the breakdowns of their tunings are also shown in\nTable~\\ref{t:m0,m12,t,10}. The dependences on $m_0$ and $m_{1\/2}$ are\nsimilar to what was observed in the CMSSM. However, the dominant\nsensitivities are now to $m_{H_{1,2}}$. For both these points the soft\nHiggs mass-squared terms are large and negative at the GUT scale. As we\nhave seen, these soft parameters have a significant effect on the stau\nRGE. Therefore the coannihilation strip exhibits tuning with respect to\nthese parameters. The total sensitivity remains low suggesting that,\neven though the soft Higgs masses play a role in the running, the\ndominant contribution to the stau mass is still from $M_1$.\n\nMore distinctive deviations from the familiar CMSSM phenomenology\narise in the forms of the strong vertical dark matter regions at\nparticular values of $m_{1\/2}$. In panel (a) three vertical strips are\npresent. To understand these lines we need to consider the mass and\ncomposition of the lightest neutralino. The bino component of the\nlightest neutralino is determined by $M_1(EW)\\approx 0.4\nm_{1\/2}(GUT)$, whereas the wino component is determined by\n$M_2(EW)\\approx 0.8 m_{1\/2}(GUT)$. Hence, $M_2(EW)>M_1(EW)$ throughout\nthe NUHM parameter space, and we never have a large wino component in\nthe LSP. Of more importance is the higgsino component, determined by\n$\\mu(EW)$. When $\\mu(EW)\\approx M_1(EW)$, there will be a sizeable\nhiggsino component in the LSP. In panel (a) we have set $\\mu=400$~GeV\nand $m_A=400$~GeV. Therefore, when $m_{1\/2}\\approx 1000$~GeV,\n$M_1(EW)\\approx \\mu$ and the lightest neutralino will be a\nbino\/higgsino mixture.  However, for $m_{1\/2}\\gg 1000$~GeV the\nlightest neutralino is mainly a higgsino, with a mass\n$m_{\\tilde{\\chi}^0_1}\\approx 400$~GeV, whereas for $m_{1\/2}\\ll 1000$~GeV the\n$\\tilde{\\chi}^0_1$ is predominantly a bino and has a mass determined by\n$M_1(EW)$.\n\nWith this in mind, we can understand the vertical lines in panel (a)\nat particular values of $m_{1\/2}$. At $m_{1\/2}=500$~GeV, the lightest\nneutralino is a bino with a mass $m_{\\tilde{\\chi}^0_1}\\approx 200$~GeV. As the\npseudoscalar Higgs mass is $m_A=400$~GeV throughout, this results in\nresonant neutralino annihilation through the pseudoscalar Higgs. As a\nresult, the relic density is below the WMAP value across the region\n$450<m_{1\/2}<530$. On the edges of the resonance the relic density may\nfall within the narrow range favoured by astrophysics.  The lines are\nmostly plotted in purple and blue, corresponding to a large degree of\ndark matter fine-tuning $\\Delta^{\\Omega}=30-300$.\nHowever, it is interesting that the edge\nof the resonance at larger $m_{1\/2}$ is plotted in red. This is the\nfirst instance of an acceptable pseudoscalar Higgs resonance region\nwith relatively low dark-matter fine-tuning, thanks to the larger\nhiggsino fraction in the LSP at larger $m_{1\/2}$. Both the LSP mass\nand the mass of $m_A$ are sensitive to $\\mu$, so the mass of the LSP\nand $m_A$ are coupled. This mitigates the dark matter\ntuning of the pseudoscalar\nHiggs funnel to an extent.\n\nPoints B1 and B2 lie on either side of the pseudoscalar Higgs funnel,\nwhere it is interesting to break the dark matter\nfine-tuning measure down into its\ncomponent parts. Unsurprisingly, the dark matter\nfine-tuning is due to a balancing\nact between the mass of the pseudoscalar Higgs, as shown by the large\nsensitivity to $m_{H_2}$, and the neutralino mass, as shown by the\nsensitivity to $m_{1\/2}$.\n\nThe other vertical strip is a wide region around $m_{1\/2}\\approx\n750$~GeV, where the predominantly bino LSP acquires a sufficient\nhiggsino admixture to suppress the relic density to the observed\nvalue. This wide band is plotted in red and purple, representing a\ntuning of $10-30$, similar to that of the low-$m_0$ end of the\nbino\/higgsino region within the CMSSM. There is also a thin line of\ngreen at the base of this band where it meets the coannihilation\nstrip. This shows that the interaction of a stau coannihilation region\nwith a bino\/higgsino LSP reduces the overall tuning of either region\nalone.\n\nPoint B3 is representative of this region. The primary dark matter\nfine-tunings are clearly with respect to $m_{H_2}$ and $m_{1\/2}$. In\nthis case, the dark matter fine-tuning is related to the balance\nbetween the roles of these terms in determining $\\mu$ and $M_1$ at the\nelectroweak scale.  This is the first instance of a mixed\nbino-higgsino region at low $m_0$ that we find in the NUHM. It is\ntherefore interesting to see that, even with a TeV-scale value of\n$m_{H_2}$, the tuning remains relatively small. This is in contrast to\nthe point A3, at which a TeV-scale soft Higgs mass gave large\nfine-tuning. To understand the origin of both electroweak and dark\nmatter fine-tunings, it is useful to consider the analytic form of the\ndependence of the low-energy parameter $\\mu(EW)$ on the GUT-scale soft\ninputs. For $\\tan\\beta=10$, we have:\n\\begin{eqnarray}\n  \\nonumber \\frac{m_Z^2}{2}=&&-0.94\\mu^2+0.010m_{H_1}^2-0.19M_2^2\n  -0.0017M_1^2-0.63m_{H_2}^2+0.38m_{Q_3}^2\\\\\n  \\nonumber &&+0.38m_{U_3}^2\n  +0.093A_t^2-0.011A_tM_1-0.070A_tM_2-0.30A_tM_3\\\\\n  &&+2.51M_3^2+0.0059M_1M_2+0.028M_1M_3+0.195M_2M_3 ,\n  \\label{musq}\n\\end{eqnarray}\nfrom which we can see that, when $m_{Q_3}^2=m_{H_{1,2}}^2$, the terms\nfrom the scalars approximately cancel. This explains the jump in\nsensitivity when this universality is broken. On the other hand, in\norder to obtain a small value of $\\mu$, these soft scalar terms need\nto provide a large contribution to balance out the contribution from\n$M_3^2$. This explains why we find a higgsino\/bino only at large $m_0$\nin the CMSSM. In the NUHM, it is unnecessary to go to large $m_0$,\njust large $m_{H_2}^2$. By keeping $m_0$, and thus $m_{Q_3}^2$, small\none keeps the dark matter and electroweak fine-tunings associated with\nthese parameters under control. The remaining large electroweak\nfine-tuning associated with point B3 is due to $M_{1\/2}$ being quite\nsizeable.\n\nIn panel (b) we have $\\mu=400$~GeV and $m_A=700$~GeV. Here we see only\nthe lower edge of the pseudoscalar resonance. This band lies at\n$750$~GeV and is plotted in purple, once again showing the large dark\nmatter fine-tuning we expect of such resonances. The upper edge of the\nresonance never appears, because at larger $m_{1\/2}$ the lightest\nneutralino is dominantly higgsino. Therefore, at values of $m_{1\/2}$\nabove the resonance, the higgsino nature of the LSP suppresses the\nrelic density so that the relic density never rises enough to fit the\nWMAP measurement.\n\nPoint B4 illustrates the component dark matter fine-tunings at the\nresonance. As with point B1, the dark matter fine-tuning is large, and\nprimarily due to sensitivities to $m_{H_2}^2$ and $m_{1\/2}$. These are\ndue to their effects on the pseudoscalar Higgs mass and the mass of\nthe LSP, respectively.\n\nIn panel (c) we set $\\mu=700$~GeV and $m_A=400$~GeV. As the\npseudoscalar mass is the same, the pseudoscalar Higgs funnel is\ncentred in the same place as in panel (a). However, the higgsino\nfraction of the LSP has dropped, increasing the overall relic\ndensity. Therefore one must go closer to the resonance before the\nenhancement to the annihilation cross section is sufficient to fit the\nobserved relic density. Therefore the WMAP lines are closer to the\npeak of the resonance, and require more precise dark matter\nfine-tuning than for the lower value of $\\mu$. This is shown by\nexamining the component dark matter fine-tunings for point B5. The\ndark matter fine-tuning is large, and primarily due to $m_{H_2}$\nthrough its influence on the pseudoscalar Higgs mass. In panel (c)\nthere is no region in which the LSP is higgsino, due to the larger\nvalue of $\\mu$. One would have to go to $m_{1\/2}\\approx 1700$~GeV\nbefore the LSP acquires a significant higgsino fraction as\n$\\mu=700$~GeV.\n\nFinally, in panel (d) we take $\\mu=700$~GeV and $m_A=700$~GeV, and we\nsee the pseudoscalar Higgs resonance at $m_{1\/2}=800$~GeV as expected,\nand it remains finely tuned, as before. As in panel (c), the value of\n$\\mu$ is too large to find a region of the parameter space in which\nthe neutralino is higgsino.\n\nOverall, Fig.~\\ref{f:m0,m12,t,10} shows some similarities and some\nmarked deviations from the phenomenology of the CMSSM. A $\\tilde{\\tau}$\ncoannihilation band appears in roughly the same region of the\nparameter space and exhibits slightly larger tuning. This increase is\ndue to the effect of the soft Higgs masses squared on the $\\tilde{\\tau}$\nrunning. A further deviation from the CMSSM comes in the form of a\npseudoscalar Higgs funnel at low $\\tan\\beta$. In most cases this exhibits\ndark matter fine-tuning similar to the CMSSM, supporting the\nobservation that resonances require significant dark matter\nfine-tuning wherever they appear. The exception is when the LSP has a\nsignificant higgsino fraction. The sensitivity of both $m_A$ and\n$m_{\\tilde{\\chi}^0_1}$ to $\\mu$ lowers the required dark matter fine-tuning\nsignificantly.  There also appears a band of higgsino\/bino dark matter\nand we find that the tuning is of the same order as in the CMSSM,\nwhich is surprising. In the CMSSM the focus-point region has\nrelatively low dark matter fine-tuning because of the cancellation\namong the scalar masses. By breaking this universality, this\ncancellation is broken. However, it also allows us to access mixed\nbino\/higgsino regions at low $m_0$. This enhances other annihilation\nchannels, such as that via $t$-channel slepton exchange, and we\nrequire a smaller higgsino admixture to obtain a suitable dark matter\nrelic density. These regions recover the relatively low dark matter\nfine-tuning of the focus-point region of the CMSSM.\n\nFinally, we note that we have also calculated the electroweak\nfine-tuning across these planes in the NUHM parameter space, and find\nit to be very similar to that of the CMSSM planes studied previously.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/mu,400,mA,700.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample $(m_{1\/2}, m_0)$ planes of the\n  NUHM parameter space with $A_0=0$, $\\mu=400$~GeV, $m_A=700$~GeV and\n  sign$(\\mu)$ and the following values of $\\tan\\beta$: (a) $\\tan\\beta=10$, (b)\n  $\\tan\\beta=20$, (c) $\\tan\\beta=35$, (d) $\\tan\\beta=50$. The Roman cross in panel\n  (a) shows the single point where the NUHM makes contact with the\n  MSSM. \\label{f:mu,400,mA,700}}\n\\end{figure}\n\nWe now consider the behaviours of these regions as $\\tan\\beta$\nincreases. In Fig.~\\ref{f:mu,400,mA,700} we take $\\mu=400$~GeV,\n$m_A=700$~GeV, $A_0=0$ and sign($\\mu$) positive, and in panels (a)-(d)\nwe take successively larger values of $\\tan\\beta$~\\footnote{ Note the\nCMSSM point in panel (a) of Fig.~\\ref{f:mu,400,mA,700}.}.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{B8} &\n      \\multicolumn{2}{|c|}{B9}\\\\\n      \\cline{2-5}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 800     & 21   & 500    & 150 \\\\\n      $m_{H_1}^2$ & 1609600 & 30   & 892230 & 98  \\\\\n      $m_{H_2}^2$ & 1357100 & 75   & 379620 & 1.3 \\\\\n      $m_{1\/2}$   & 730     & 14   & 432    & 35  \\\\\n      $\\tan\\beta$     & 50      & 50   & 50     & 290 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &  & 75   &        & 290 \\\\\n      \\hline\n      $\\Delta_{EW}$    &  & 420  &        & 130 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 400   & -    & 400 & -   \\\\\n      $m_A$       & 700   & -    & 700 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points B8 and B9, shown in\n    Fig.~\\ref{f:mu,400,mA,700} exemplify the behaviour of the\n    pseudoscalar Higgs funnel (B8) and the stau-coannihilation region\n    (B9) at large $\\tan\\beta$ within the NUHM. We present a breakdown of\n    the dark matter fine-tuning with respect to each parameter of the\n    NUHM. We give the value of $m_{H_{1,2}}^2$, although the tunings\n    are calculated with respect to\n    $m_{H_{1,2}}$.\\label{t:mu,400,mA,700}}\n\\end{table}\n\nAs the value of $\\tan\\beta$ increases, the mass and composition of the\nneutralino across the $(m_{1\/2}, m_0)$ plane remains essentially\nunaltered. Therefore we find the lower edge of the pseudoscalar funnel\nin the same place in all panels. Studying the breakdown of the dark matter\nfine-tunings of the pseudoscalar Higgs funnel for large $\\tan\\beta$ at\npoint B8, we find that in this case all parameters show a significant\ntuning. The primary tunings come from $m_{H_2}$ and $\\tan\\beta$. From\n(\\ref{mAsq}),(\\ref{e:musq}) we can see that this comes from the\ndominant sensitivity of $m_A^2$, through $\\mu$, on both $\\tan\\beta$ and\n$m_{H_2}^2$.\n\nThe pseudoscalar Higgs funnel remains in the same place as $\\tan\\beta$\nincreases, whereas the $\\tilde{\\tau}$-coannihilation strip moves\nconsiderably. For larger $\\tan\\beta$ the Yukawa contribution to the stau\nrunning is enhanced. This suppresses the stau mass, requiring larger\n$m_0$ to avoid a stau LSP. For very low $m_0$ there is a brick red\nregion. This is not due to a failure of REWSB, but rather it is due to\na tachyonic stau mass.\n\nBy increasing $\\tan\\beta$ we reduce the dominance of $M_1$ in the running\nof the stau mass-squared. This breaks the link between the masses of the\n$\\tilde{\\tau}$ and the $\\tilde{\\chi}^0_1$, resulting in an increase of the tuning\nrequired for the coannihilation region at larger values of\n$\\tan\\beta$. Point B9 is representative of the $\\tilde{\\tau}$ coannihilation\nregion at large $\\tan\\beta$. Though there is large sensitivity to the soft\ngaugino mass $m_{1\/2}$, this is eclipsed by the sensitivities to\n$\\tan\\beta$ and $m_0$.\n\n\\subsection{Sample $(\\mu,m_A)$ planes}\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/mu,mA,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_A)$ planes with\n$A_0=0$, $\\tan\\beta=10$ and sign$(\\mu)$ positive, and different values of\n$m_0$ and $m_{1\/2}$: (a) $m_0=100$~GeV, $m_{1\/2}=300$~GeV, (b)\n$m_0=100$~GeV, $m_{1\/2}=500$~GeV, (c) $m_0=300$~GeV,\n$m_{1\/2}=300$~GeV, (d) $m_0=300$~GeV, $m_{1\/2}=500$~GeV. The Roman\ncrosses in each panel show where the NUHM meets the\nCMSSM.\\label{f:mu,mA,t,10}}\n\\end{figure}\n\nWe now discuss other planes so as to explore more features of the NUHM\nmodel. In Fig.~\\ref{f:mu,mA,t,10} we consider $(\\mu,~m_A)$ planes with\n$A_0=0$, $\\tan\\beta=10$ and sign$(\\mu)$ positive, taking different values of\n$m_0$ and $m_{1\/2}$ in the various panels. As before, though we plot\nresults in terms of $\\mu$ and $m_A$, the primary variables are\n$m_{H_{1,2}}^2$.\n\nWe first consider some overall features. Low $\\mu$ is ruled out in all\ncases by the appearance of a light chargino (light blue). In panels (a)\nand (c), the low value of $m_{1\/2}=300~\\text{GeV}$ results in a Higgs\nboson with $m_h<111$~GeV across much of the parameter space. This bound is\nvery sensitive to variations in the top mass, so regions that are excluded\nhere could be allowed with a higher top mass. On the other hand, in panels\n(b) and (d), only very low values of $m_A$ result in a problematically\nlight Higgs. In panels (a) and (b), the low value of $m_0=100~\\text{GeV}$\nresults in a light $\\tilde{\\tau}$. This gives a region at low $\\mu$ and $m_A$ in\nwhich the $\\tilde{\\tau}$ is the LSP and as such is ruled out (light green).\nFinally, in panel (a) $m_{1\/2}$ and $m_0$ are light enough to give light\nsneutrinos. For large $\\mu$ and $m_A$, one of the sneutrinos becomes the\nLSP, ruling out this corner of the parameter space~\\footnote{Sneutrinos\nare massive, neutral and weakly-interacting, and so could in principle\naccount for the dark matter. However, they generally give $\\Omega_{CDM}h^2\\ll\\Omega^{WMAP}_{CDM}h^2$.\nThis can be avoided in some models with right-handed neutrinos, but this\npossibility lies beyond the scope of the MSSM, so we do not consider it\nfurther here.}.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C1}&\n      \\multicolumn{2}{|c|}{C2}&\n      \\multicolumn{2}{|c|}{C3}&\n      \\multicolumn{2}{|c|}{C4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 0.99 & 100    & 4.7  & 100     & 13   & 100      & 2.7 \\\\\n      $m_{H_1}^2$ & 279530  & 0.38 & -27130 & 0.99 & -2289300& 79   & -966510  & 390 \\\\\n      $m_{H_2}^2$ & 188070  & 17   & 111220 & 3.5  & -3637400& 110  & -1248800 & 330 \\\\\n      $m_{1\/2}$   & 300     & 16   & 300    & 0.20 & 300     & 30   & 300      & 46  \\\\\n      $\\tan\\beta$     & 10      & 0.8  & 10     & 0.51 & 10      & 0.57 & 10       & 1.4 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 17   &        & 4.7  &         & 110  &          & 390 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 74   &        & 73   &         & 1300 &          & 500 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 190   & -    & 300 & -   & 1620  & -    & 1000& -   \\\\\n      $m_A$       & 600   & -    & 315 & -   & 600   & -    & 268 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C5}&\n      \\multicolumn{2}{|c|}{C6}&\n      \\multicolumn{2}{|c|}{C7}&\n      \\multicolumn{2}{|c|}{C8}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 3.3  & 100    & 5.6  & 100     & 4.7  & 300      & 0.64\\\\\n      $m_{H_1}^2$ & 319380  & 2.7  & -22696 & 0.59 & -1445500& 35   & -70487   & 6.7 \\\\\n      $m_{H_2}^2$ & 222330  & 76   & -25802 & 0.68 & -1440300& 34   & 246160   & 2.4 \\\\\n      $m_{1\/2}$   & 300     & 85   & 500    & 5.6  & 500     & 3.4  & 300      & 3.6 \\\\\n      $\\tan\\beta$     & 10      & 2.7  & 10     & 1.3  & 10      & 0.068& 10       & 1.2 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 85   &        & 5.6  &         & 35   &          & 6.7 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 74   &        & 210  &         & 590  &          & 79  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 120   & -    & 640 & -   & 1170  & -    & 235 & -   \\\\\n      $m_A$       & 600   & -    & 700 & -   & 200   & -    & 150 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C9}&\n      \\multicolumn{2}{|c|}{C10}&\n      \\multicolumn{2}{|c|}{C11}\\\\\n      \\cline{2-7}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 300     & 6.1  & 300    & 0.55 & 300     & 1.3  \\\\\n      $m_{H_1}^2$ & 401250  & 0.62 & -106330& 5.1  & -211970 & 0.067\\\\\n      $m_{H_2}^2$ & 568170  & 40   & 525900 & 2.5  & 497010  & 6.8  \\\\\n      $m_{1\/2}$   & 500     & 31   & 500    & 0.023& 500     & 6.5  \\\\\n      $\\tan\\beta$     & 10      & 1.5  & 10     & 1.0  & 10      & 0.30 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 40   &        & 5.1  &         & 6.8  \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 180  &        & 180  &         & 180  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 260   & -    & 320 & -   & 350   & -    \\\\\n      $m_A$       & 750   & -    & 325 & -   & 150   & -    \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points C1-11, shown in Fig.~\\ref{f:mu,mA,t,10},\n    illustrate the behaviour of mixed bino-higgsino dark matter\n    (C1,5,9), sneutrino coannihilation (C3), the pseudoscalar Higgs\n    funnel (C4), stau-coannihilation (C2,6,7,11), and the\n    pseudoscalar funnel region with a mixed bino-higgsino LSP\n    (C8,10). We present breakdowns of the dark matter fine-tuning with\n    respect to each parameter of the NUHM. We give the value of\n    $m_{H_{1,2}}^2$, but the fine-tunings are calculated with respect\n    to $m_{H_{1,2}}$.\\label{t:mu,mA,t,10}}\n\\end{table}\n\nBefore considering the dark matter regions, we first note the\ncomposition of the lightest neutralino in different regions of the\nparameter space. In panels (a) and (c) $M_1(EW)\\approx 120$~GeV,\nwhereas in panels (b) and (d) $M_1(EW)\\approx 200$~GeV. Therefore for\n$\\mu\\gg 120(200)$~GeV the LSP is predominantly a bino with a mass\n$m_{\\tilde{\\chi}^0_1}\\approx 120(200)$~GeV. Below these values of $\\mu$, the\nlightest neutralino acquires a significant higgsino fraction.\n\nAs before, the regions that fit the relic density favoured by WMAP are\ndisplayed with the corresponding dark matter fine-tuning colours.  The\nvertical dark matter band at low $\\mu$ in all panels features a mixed\nbino\/higgsino dark matter particle, and is plotted in red and purple,\ncorresponding to a dark matter fine-tuning $\\Delta^{\\Omega}\\approx 20-40$.\nPoints C1 and C9 are representative of the higgsino\/bino band, and\ntheir dark matter fine-tunings are broken down in\nTable~\\ref{t:mu,mA,t,10}. As one would expect, the fine-tuning is due\nto a balancing act between $m_{H_2}$ and $m_{1\/2}$, as these determine\nthe composition of the lightest neutralino, and thus its annihilation\nrate.\n\nPoint C5 also represents a bino\/higgsino region. It is unusual in\nthat, normally, the larger the higgsino component, the more efficient\nthe annihilation. However, in this case, as $\\mu$ is lowered from\npoint C1, the higgsino fraction increases and yet the dark matter\nrelic density rises again. Indeed, C5 is almost evenly split between\nhiggsino and bino components and yet it fits the WMAP relic density\nmeasurement. This is due to the fact that, as $\\mu$ drops, the mass of\nthe neutralino is lowered. At point C5 the LSP becomes lighter than\nthe $Z$, shutting off the annihilation channel $\\tilde{\\chi}^0_1 \\tilde{\\chi}^0_1\n\\rightarrow ZZ$. This reduces the annihilation cross section enough\nthat the relic density is acceptable once more.\n\nAnother region that is easy to pick out is the pseudoscalar Higgs\nfunnel along $m_A=240(400)$~GeV for $m_{1\/2}=300(500)$~GeV\nrespectively. As before, these funnels require significant dark matter\nfine-tuning and as such are predominantly plotted in blue and black,\nshowing a tuning $\\Delta^{\\Omega}>100$. Point C4 is a representative point\nwhose dark matter fine-tuning breakdown we display in\nTable~\\ref{t:mu,mA,t,10}. In previous pseudoscalar Higgs regions the\ndark matter fine-tuning was due primarily to $m_{H_{1,2}}$ and\n$m_{1\/2}$. Here we find that the sensitivity to the Higgs masses has\nincreased significantly. From (\\ref{mAsq}) this is easy to\nunderstand. If we increase $\\mu$ while keeping $m_A$ the same we must\ncarefully balance the large $m_{H_{1,2}}^2$ contributions to give the\nrequired $m_A$. This careful balancing manifests as a steadily\nincreasing sensitivity of $m_A$ to the Higgs soft masses as we\nincrease $\\mu$. This translates to a large sensitivity of the\npseudoscalar Higgs funnel.\n\nAt the other end of the spectrum, there is a region of the\npseudoscalar Higgs funnel at low $\\mu$ with remarkably low\ntuning. This occurs when there is a significant higgsino fraction in\nthe LSP, such as at points C8 and C10. In this region, both $m_A$ and\nthe neutralino mass are sensitive to $\\mu$. This results in the mass\nof the neutralino and the pseudoscalar being coupled, and reduces the\nsensitivity of the mass difference $\\Delta_m=m_A-2m_{\\tilde{\\chi}^0_1}$. At\npoints C8 and C10 the dominant annihilation channels are to heavy\nquarks via an $s$-channel pseudoscalar Higgs. Remarkably the total\ndark matter fine-tunings of the points are only 6.7 and 5.1\nrespectively.\n\nAs the $\\tilde{\\nu}_{e,\\mu}$ become the LSPs in the large $\\mu$, large\n$m_A$ region of panel (a), there is a corresponding sneutrino\ncoannihilation region lying parallel to its boundary, which is plotted\nin purple and blue indicating a dark matter fine-tuning\n$\\Delta^{\\Omega}>80$. Point C3 is a representative of this region, whose dark\nmatter fine-tuning breakdown is also displayed in Table\n\\ref{t:mu,mA,t,10}. The dark matter fine-tuning is large, and comes\nprimarily from the Higgs sector. It is the existence of large negative\n$m_{H_1}^2$ that allows for light sneutrinos. Thus the sneutrino\nmasses are very sensitive to the Higgs soft mass-squared parameters, and\nthis is reflected in the dark matter fine-tuning. There is also some\ndark matter fine-tuning with respect to $m_{1\/2}$ that is typical of\nthe need to balance the bino mass against that of a coannihilation\npartner with an uncorrelated mass.\n\nFinally, the light $\\tilde{\\tau}$ at low $\\mu$ and $m_0$ has an effect on the\ndark matter relic density. As the mass of the $\\tilde{\\tau}$ is reduced, the\nannihilation cross section is increased via $t$-channel slepton\nexchange. Also, as one approaches the region in which the stau is the\nLSP, there are additional contributions from $\\tilde{\\tau}-\\tilde{\\chi}^0_1$\ncoannihilation processes. These two effects combine to give dark\nmatter bands along the edges of the stau LSP region in panels (a) and\n(b). Points C2 in panel (a) and C6,7 in panel (b) are representative\npoints. At point C2 the annihilation proceeds through equal parts of\n$t$-channel $\\tilde{e}_R,~\\tilde{\\mu}_R,~\\tilde{\\tau}$ annihilation (15-20\\% each),\nannihilation to $b,\\overline{b}$ via off-shell pseudoscalar Higgs\nbosons (18\\%) and $\\tilde{\\tau}$ coannihilation (15\\%). Only the\ncoannihilation processes would be expected to exhibit a high\nsensitivity to the soft parameters, as $t$-channel processes are\nfairly insensitive and the point is far from the pseudoscalar\nresonance, reducing significantly the sensitivity of the $s$-channel\npseudoscalar process. As a result, we have a region that arises from a\nmixture of channels and exhibits low tuning. The subdominant role of\ncoannihilation explains why there is so little dark matter fine-tuning\nwith respect to $m_{1\/2}$. The role of the stau in both the\ncoannihilation and $t$-channel processes explains the dominant dark\nmatter fine-tuning with respect to $m_0$, and the dependence on\n$m_{H_2}$ appears from running effects.\n\nUnfortunately, point C2 results in a light Higgs with $m_{h}=110$~GeV,\nwhich is probably unacceptably low, even allowing for the theoretical\nuncertainty in the calculation of its mass. On the other hand, panel\n(b) has a larger value of $m_{1\/2}$ and hence Higgs mass.  However,\nthe masses of the LSP and the sleptons are also increased. This\ndecreases the slepton $t$-channel annihilation cross sections,\nrequiring larger contributions from processes that are finely tuned in\norder to fit the WMAP relic density, which is apparent at points C6\nand C7~\\footnote{ We note that there is a CMSSM point very close\nto C6}. At point C6, $t$-channel slepton annihilation only accounts\nfor 3\\% of the annihilation rate via each channel (9\\% overall). The\nremaining 91\\% is made up entirely of coannihilation processes,\ndominantly with $\\tilde{\\tau}$, but also $\\tilde{e}_R,\\tilde{\\mu}_R$. As this plane has low\n$m_0$, the slepton masses are predominantly determined by\n$m_{1\/2}$. Once again, there is the familiar pattern of dark matter\nfine-tunings for a low-$\\tan\\beta$, low-$m_0$ slepton coannihilation\nregion. The overall dark matter fine-tuning is low, and what\nfine-tuning does exists is due to $m_0$ and $m_{1\/2}$. Point C7 tells\na slightly different story. The pattern of annihilation channels is\nalmost identical to C6, and we see the typical dark matter\nfine-tunings of a coannihilation region in the sensitivity to $m_0$\nand $m_{1\/2}$. However, the dark matter fine-tuning with respect to\n$m_{H_{1,2}}$ has increased dramatically, due to the massive increase\nin $m_{H_{1,2}}^2$ between points C6 and C7. Now the stau running is\ndominated by the Higgs mass-squared terms rather than the gaugino\nmass, and the coannihilation region becomes fine-tuned once again.\n\nThere is one further interesting region. In panel (d) at low $m_A$\nthere is a kink in the higgsino\/bino region. The band moves to larger\n$\\mu$ and the dark matter fine-tuning drops dramatically. The band is\nplotted in green rather than purple, indicating a dark matter\nfine-tuning of less than 10. The kink in the band appears at\n$m_A=280~$GeV. Around this region the LSP is predominantly a bino with\na small but significant higgsino component, and the LSP has a mass of\naround 200~GeV. As the pseudoscalar mass drops, the masses of the\nheavy Higgs, $H$, and the charged Higgses, $H^\\pm$, also\ndecrease. Around $m_A=280$~GeV, the annihilation channels\n$\\tilde{\\chi}^0_1\\neut\\rightarrow hA,W^\\pm H^\\mp,ZH$ open up, which are\nkinematically forbidden at larger $m_A$. These can progress through\neither $t$-channel neutralino (chargino) exchange or $s$-channel Higgs\nand Z processes. They require a small higgsino component, but\nsignificantly less than the higgsino\/bino region represented by point\nC9. This balance of the higgsino and bino components of the LSP\nappears in the sensitivity of point C11 on $m_{1\/2}$ and\n$m_{H_2}$. Thus C11 represents a higgsino\/bino region with low dark\nmatter fine-tuning - something that does not exist in the CMSSM. This\nis because a large negative $m_{H_1}^2$ is needed to achieve low\n$m_{A,H,H^\\pm}$.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,100,m12,300.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_A)$ planes with\n  $A_0=0$, $m_0=100$~GeV, $m_{1\/2}=300$~GeV and sign$(\\mu)$ positive,\n  and the following values of $\\tan\\beta$: (a) $\\tan\\beta=10$, (b) $\\tan\\beta=20$,\n  (c) $\\tan\\beta=35$. We do not show a plot for $\\tan\\beta=50$ as the\n  parameter space is entirely excluded. The Roman crosses in each\n  panel show where the NUHM meets the CMSSM.\\label{f:m0,100,m12,300}}\n\\end{figure}\n\nWe now consider in Fig.~\\ref{f:m0,100,m12,300} the behaviours of these\nregions as $\\tan\\beta$ increases. We have set $m_0=100$~GeV,\n$m_{1\/2}=300$~GeV, $A_0=0$ and increase $\\tan\\beta$ in steps in each\npanel.\n\nWe note first the bulk features of the plane. As noted previously,\nincreasing $\\tan\\beta$ decreases the mass of the lightest stau. Thus plots\nat larger $\\tan\\beta$ have larger regions ruled out because the LSP is a\n$\\tilde{\\tau}$, and we do not show very large $\\tan\\beta$ because at $\\tan\\beta=50$\nthe stau mass becomes tachyonic across the entire plane. By $\\tan\\beta=35$\nthe light stau rules out all the parameter space below\n$\\mu=200$~GeV. The mass of the light Higgs is also sensitive to\n$\\tan\\beta$, and is in all cases very close to $m_h=111$~GeV, so it only\ntakes a small shift to cause a significant change in the area plotted\nin light grey.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C12}&\n      \\multicolumn{2}{|c|}{C13}&\n      \\multicolumn{2}{|c|}{C14}&\n      \\multicolumn{2}{|c|}{C15}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 1.0  & 100    & 15   & 100     & 7.2  & 100    & 8.7 \\\\\n      $m_{H_1}^2$ & 477840  & 1.4  & 858150 & 48   & -532000 & 6.5  & -14857 & 0.28\\\\\n      $m_{H_2}^2$ & 175680  & 17   & 96420  & 6.3  & -1263800& 12   & -2379  & 0.041\\\\\n      $m_{1\/2}$   & 300     & 16   & 300    & 32   & 300     & 10   & 300    & 4.6  \\\\\n      $\\tan\\beta$     & 20      & 0.56 & 20     & 21   & 20      & 8.7  & 20     & 6.5  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 17   &        & 48   &         & 12   &        & 8.7  \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 71   &        & 71   &         & 480  &        & 78   \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 185   & -    & 275 & -   & 1000  & -    & 400 & -   \\\\\n      $m_A$       & 700   & -    & 940 & -   & 700   & -    & 400 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points C12-15, shown in Fig.~\\ref{f:m0,100,m12,300},\n    are representative of the higgsino\/bino region (C12), the\n    sneutrino coannihilation region (C13) and the\n    stau-coannihilation\/bulk region (C15,14) with increasing $\\tan\\beta$\n    within the NUHM. We present a breakdown of the dark matter\n    fine-tuning with respect to each parameter of the NUHM. We give\n    the value of $m_{H_{1,2}}^2$ but the tunings are calculated with\n    respect to $m_{H_{1,2}}$.\\label{t:m0,100,m12,300}}\n\\end{table}\n\nThe most significant change in the dark matter phenomenology is due to\nthe varying $\\tilde{\\tau}$ mass. Between panels (a) and (b) the stau\nbulk\/coannihilation region moves to larger $\\mu$ and $m_A$. We also\nfind a significant stau region at large $m_A$. These features are\nrepresented by points C15 and C14 respectively~\\footnote{ We note that\nthere is a CMSSM point very close to C15.}. Comparing C15 directly\nto C2, we see from Table~\\ref{t:m0,100,m12,300} that the dark matter\nfine-tuning is due primarily to $m_0$ and $\\tan\\beta$. This is because\nthese parameters determine the mass of the lighter stau and this is\nthe primary source of sensitivity for bulk regions. There is also a\ndegree of sensitivity to $m_{1\/2}$, as there is a significant\ncoannihilation contribution that requires the LSP and stau mass to be\nbalanced. At point C14 one has similar degrees of dark matter\nfine-tuning with respect to $\\tan\\beta,~m_{1\/2}$ and $m_0$. However, there\nis now also large dark matter fine-tuning with respect to\n$m_{H_{1,2}}$, due to the larger magnitude of the soft higgsino\nmass-squared terms. The stau mass in this region becomes highly\nsensitive to $m_{H_2}^2$.\n\nThe other regions are little changed from before. Point C12\nexemplifies the mixed bino\/higgsino region at increasing $\\tan\\beta$. It\ncan be compared directly to point C1, and we see that the component\ndark matter fine-tunings are virtually identical.  Point C13 is\nrepresentative of the sneutrino coannihilation region and can be\ncompared to point C3. Once again the dark matter fine-tuning is due\nprimarily to the soft Higgs masses through their impacts on the\nrunning of the sneutrino masses.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,300,m12,500.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_A)$ planes with\n  $A_0=0$, $m_0=300$~GeV, $m_{1\/2}=500$~GeV, sign$(\\mu)$ positive and\n  different values of $\\tan\\beta$: (a) $\\tan\\beta=10$, (b) $\\tan\\beta=20$, (c)\n  $\\tan\\beta=35$, (d) $\\tan\\beta=50$. The Roman crosses in each panel show\n  where the NUHM meets the CMSSM.\\label{f:m0,300,m12,500}}\n\\end{figure}\n\nMuch of the low-$m_0$ parameter space is forbidden by a light Higgs\nand\/or a light stau. We now consider the effect of increasing $\\tan\\beta$\nin a more open part of the parameter space. We take\nFig.~\\ref{f:mu,mA,t,10}(d) with $m_0=300$~GeV, $m_{1\/2}=500$~GeV as a\nstarting point and increase $\\tan\\beta$ steadily, as seen in\nFig.~\\ref{f:m0,300,m12,500}.  In contrast to\nFig.~\\ref{f:m0,100,m12,300}, the bulk features remain fairly stable\nfor moderate values of $\\tan\\beta$. The first hint of a change appears in\npanel (c) at $\\tan\\beta=35$, where we see a small region at large $m_A$ in\nwhich the stau is the LSP. This expands to cut off low $\\mu$ for\n$\\tan\\beta=50$.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C16}&\n      \\multicolumn{2}{|c|}{C17}&\n      \\multicolumn{2}{|c|}{C18}&\n      \\multicolumn{2}{|c|}{C19}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 300     & 1.0  & 300    & 0.94 & 300     & 52   & 300    & 37  \\\\\n      $m_{H_1}^2$ & -47935  & 2.0  & -170090& 0.86 & 1021100 & 22   & -52957 & 4.4 \\\\\n      $m_{H_2}^2$ & 518240  & 3.8  & 475340 & 4.2  & 390800  & 0.58 & -281880& 2.3 \\\\\n      $m_{1\/2}$   & 500     & 3.1  & 500    & 3.7  & 500     & 25   & 500    & 20  \\\\\n      $\\tan\\beta$     & 20      & 3.0  & 20     & 0.21 & 35      & 86   & 50     & 49  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 3.8  &        & 4.2  &         & 86   &        & 49  \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 170  &        & 170  &         & 170  &        & 290 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 315   & -    & 360 & -   & 380   & -    & 780 & -   \\\\\n      $m_A$       & 325   & -    & 150 & -   & 950   & -    & 600 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points C16-19, shown in\n    Fig.~\\ref{f:m0,300,m12,500}, illustrate the behaviours of\n    the mixed bino-higgsino, the pseudoscalar Higgs funnel (C16) and\n    the stau-coannihilation\/bulk regions (C17,18,19) at increasing\n    values of $\\tan\\beta$ within the NUHM. We present a breakdown of the\n    dark matter fine-tuning with respect to each parameter of the NUHM. We\n    give the value of $m_{H_{1,2}}^2$, but the tunings are calculated\n    with respect to $m_{H_{1,2}}$.\\label{t:m0,300,m12,500}}\n\\end{table}\n\nThere are few dark matter surprises at larger $\\tan\\beta$. The\npseudoscalar Higgs funnel and mixed higgsino\/bino regions remain\nrelatively unaltered throughout. The interaction of the pseudoscalar\nHiggs funnel with the higgsino\/bino LSP continues to provide a\nfavourable degree of tuning in panels (b) and (c). We take point C16\nas a representative point, and break the tuning down in\nTable~\\ref{t:m0,300,m12,500}. As for point C10, the tuning is small\nand the annihilation is primarily due to annihilation to heavy quarks\nvia an $s$-channel pseudoscalar Higgs.\n\nPoint C17 exemplifies the behaviour of a predominantly bino LSP with a\nsmall higgsino admixture that can annihilate to $hA,~ZH$ and $W^\\pm\nH\\mp$. As with point C11, the dark matter fine-tuning is small and\nmostly due to the composition of the LSP, through $m_{1\/2}$ and\n$m_{H_2}$.\n\nPoint C18 is in the new stau coannihilation region that appears at\nlarge $\\tan\\beta$. For $m_0=300$~GeV, $m_{1\/2}=500$~GeV the staus are too\nheavy to contribute significantly to $t$-channel slepton exchange, so\nthis region is pure coannihilation. The stau mass is mainly determined\nby $m_0$ and $\\tan\\beta$, and must be balanced against a predominantly\nbino LSP. Therefore, the tuning is dominated by $\\tan\\beta$ and $m_0$ with\na secondary dependence on $m_{1\/2}$.  The coannihilation grows\nsignificantly by $\\tan\\beta=50$ and point C19 represents this trend.  As\nwith point C18, we find the tuning to be due to $m_0$ and $\\tan\\beta$,\nwith a secondary dependence on $m_{1\/2}$.\n\nThroughout all of these parameter scans we have also calculated the\nelectroweak fine-tuning and found it to be of the same order as that\nfound in the CMSSM for typical scales of soft masses considered.\n\n\\subsection{Sample $(\\mu,m_{1\/2})$ planes}\n\nFinally,we consider sample $(\\mu,~m_{1\/2})$ planes in the NUHM. These\nare interesting, e.g., because $\\mu$ and $m_{1\/2}$ are the parameters\nthat determine the mass and composition of the lightest\nneutralino~\\footnote{Note that in the following plots $m_{1\/2}$ is the\nGUT-scale soft mass, whereas $\\mu$ is the electroweak-scale Higgs\nterm. This is in contrast to the plots of~\\cite{hep-ph\/0210205} where\nthe plots were in terms of $M_2(EW)$ and $\\mu(EW)$.}.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/mu,m12,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_{1\/2})$ planes\n  with $A_0=0$, $\\tan\\beta=10$, sign$(\\mu)$ positive and varying $m_0$\n  and $m_A$: (a) $m_0=100$~GeV, $m_A=500$~GeV, (b)\n  $m_0=100$~GeV, $m_A=700$~GeV, (c) $m_0=300$~GeV, $m_A=500$~GeV, (d)\n  $m_0=300$~GeV, $m_A=700$~GeV. The Roman crosses in each\n  panel show where the NUHM meets the CMSSM.\\label{f:mu,m12,t,10}}\n\\end{figure}\n\nIn Fig.~\\ref{f:mu,m12,t,10} we set $A_0=0$, $\\tan\\beta=10$ and take\ndiscrete values of $m_0$ and $m_A$. We see that either low $\\mu$ or\nlow $m_{1\/2}$ results in a light chargino that violates particle\nsearches (light blue). Low $m_{1\/2}$ also results in problems with a\nlight Higgs (light grey with a black boundary). On the other hand,\nlarge $m_{1\/2}$ results in a neutralino with a mass above that of the\nstau (light green). The exception is low $\\mu$ where the neutralino is\na higgsino and $m_{\\tilde{\\chi}^0_1}$ is insensitive to $m_{1\/2}$. In panels (a)\nand (b) we have $m_0=100$~GeV. This, combined with low $m_{1\/2}$ and\nlarge $\\mu$ results in a region in which the LSP is a sneutrino (light\ngreen).\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D1}&\n      \\multicolumn{2}{|c|}{D2}&\n      \\multicolumn{2}{|c|}{D3}&\n      \\multicolumn{2}{|c|}{D4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 1.0  & 100    & 7.2  & 100     & 3.6  & 100     & 1.1 \\\\\n      $m_{H_1}^2$ & 160370  & 0.30 & -97736 & 3.3  & -2206600& 39   & -2147800& 360 \\\\\n      $m_{H_2}^2$ & 255000  & 18   & -20502 & 0.70 & -2345100& 41   & -2588200& 300 \\\\\n      $m_{1\/2}$   & 350     & 17   & 400    & 4.7  & 670     & 5.9  & 570     & 59  \\\\\n      $\\tan\\beta$     & 10      & 0.51 & 10     & 1.1  & 10      & 0.027& 10      & 0.070\\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 18   &        & 7.2  &         & 41   &         & 360 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 96   &        & 140  &         & 1100 &         & 1100\\\\\n      \\hline\n      \\hline\n      $\\mu$       & 210   & -    & 530 & -   & 1500  & -    & 1500& -   \\\\\n      $m_A$       & 500   & -    & 500 & -   & 500   & -    & 500 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D5}&\n      \\multicolumn{2}{|c|}{D6}&\n      \\multicolumn{2}{|c|}{D7}&\n      \\multicolumn{2}{|c|}{D8}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 13   & 300    & 7.1  & 300     & 0.38 & 300     & 7.4 \\\\\n      $m_{H_1}^2$ & -2963700& 110  & 109920 & 0.42 & -120550 & 5.7  & -216760 & 0.91\\\\\n      $m_{H_2}^2$ & -4438200& 140  & 490790 & 39   & 845460  & 5.1  & 1153100 & 9.0 \\\\\n      $m_{1\/2}$   & 310     & 32   & 450    & 30   & 680     & 4.7  & 800     & 2.6 \\\\\n      $\\tan\\beta$     & 10      & 0.52 & 10     & 1.1  & 10      & 1.9  & 10      & 0.70\\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 140  &        & 39   &         & 5.7  &         & 9.0 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 1600 &        & 160  &         & 300  &         & 400 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 1785  & -    & 240 & -   & 410   & -    & 430 & -   \\\\\n      $m_A$       & 500   & -    & 500 & -   & 500   & -    & 500 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D9}\\\\\n      \\cline{2-3}\n                  & value & $\\Delta^{\\Omega}$\\\\\n      \\hline\n      $m_0$       & 300     & 5.6  \\\\\n      $m_{H_1}^2$ & 291010  & 0.38 \\\\\n      $m_{H_2}^2$ & 661620  & 42   \\\\\n      $m_{1\/2}$   & 550     & 34   \\\\\n      $\\tan\\beta$     & 10      & 1.4  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 42   \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 210  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 280   & -    \\\\\n      $m_A$       & 700   & -    \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points D1-9, shown in\n    Fig.~\\ref{f:mu,m12,t,10}, are representative of bino-higgsino dark\n    matter (D1,6,9), stau-coannihilation (D2,3,8), the pseudoscalar\n    Higgs funnel (D4) and its interaction with mixed bino-higgsino\n    dark matter (D7), and sneutrino coannihilation (D5). We present\n    breakdowns of the dark matter fine-tuning with respect to each parameter\n    of the NUHM. We give the value of $m_{H_{1,2}}^2$ but the tunings\n    are calculated with respect to $m_{H_{1,2}}$.\\label{t:mu,m12,t,10}}\n\\end{table}\n\nWe see once again the familiar dark matter regions of the previous\nplots. The pseudoscalar Higgs funnel appears as a pair of horizontal\nlines and exhibits large dark matter fine-tuning, and is characterized\nby the point D4 in Table~\\ref{t:mu,m12,t,10}. Here we see that the\nlarge dark matter fine-tuning is due to the soft Higgs masses through\ntheir influence on $m_A$, and to $m_{1\/2}$ through its influence on\nthe neutralino mass.\n\nThe exception to this large dark matter fine-tuning is where the\npseudoscalar funnel interacts with a higgsino\/bino LSP and there is a\nsmall corner with low fine-tuning, as characterized by point D7. The\nannihilation here is mainly to heavy quarks via an $s$-channel\npseudoscalar Higgs, and yet the total tuning is only 5.7. As noted\npreviously, this relatively small dark matter fine-tuning comes from\nthe common sensitivity of $m_A$ and $m_{\\tilde{\\chi}^0_1}$ on $\\mu$.\n\nThere is also a $\\tilde{\\tau}$ coannihilation region in all four plots, which\nlies alongside the region ruled out due to a stau LSP. It exhibits\nsimilar tuning to the CMSSM. We break down the dark matter\nfine-tunings of this region at points D2 and D3, finding that at both\npoints the tuning with respect to $m_0$ and $m_{1\/2}$ is standard for\na stau coannihilation strip at low $m_0$~\\footnote{This is also true\nfor the CMSSM point seen in panel (b).}. Point D3 has larger tuning\nbecause this region of parameter space requires large negative soft\nHiggs masses, which now dominate the determination of the mass of the\nlight stau.\n\nThe sneutrino coannihilation region shows up alongside the sneutrino\nLSP region. Once again we find it to require significant dark matter\nfine-tuning, although this decreases steadily as one moves to lower\n$\\mu$. Point D5 is a representative point with, as before, large dark\nmatter fine-tuning that depends on the soft Higgs masses.\n\nEach plot also has a dark matter region at low $\\mu$ that lies along a\ndiagonal in the $(\\mu, m_{1\/2})$ plane, incorporating points\nD1,6,9. These regions are mixed bino\/higgsino regions. In all cases\nthe pseudoscalar Higgs and heavy Higgs bosons are sufficiently massive\nthat annihilation of the mixed LSP proceeds mainly through the\nchannels $\\tilde{\\chi}^0_1 \\tilde{\\chi}^0_1 \\rightarrow W^+W^-(ZZ)$, via $t$-channel\nchargino (neutralino) exchange. This process is very sensitive to the\ncomposition of the LSP and the masses of the exchanged\nparticles. Therefore there is significant dark matter fine-tuning with\nrespect to $m_{H_2}$ and $m_{1\/2}$ at all these points.\n\nFinally, we consider the point D8 where the coannihilation strip and\nthe mixed bino\/higgsino strips meet. The combination of annihilation\nchannels has a beneficial effect, with the overall dark matter fine-tuning\ndropping to 9.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,100,mA,500.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_{1\/2})$ planes\n  with $A_0=0$, $m_0=100$~GeV, $m_A=500$~GeV, sign$(\\mu)$ positive and\n  $\\tan\\beta$ varying: (a) $\\tan\\beta=10$, (b) $\\tan\\beta=20$, (c) $\\tan\\beta=35$. We\n  do not show a plane for $\\tan\\beta=50$, as this part of the parameter\n  space is entirely excluded. The Roman crosses in each panel show\n  where the NUHM meets the CMSSM.\\label{f:m0,100,mA,500}}\n\\end{figure}\n\nOnce again it is interesting to go beyond $\\tan\\beta=10$, to understand\nhow the phenomenology changes with $\\tan\\beta$. In\nFig.~\\ref{f:m0,100,mA,500} we consider $(\\mu, m_{1\/2})$ planes with\n$m_0=100$~GeV, $m_A=500$~GeV and steadily increasing values of\n$\\tan\\beta$. As we saw before, increasing $\\tan\\beta$ decreases the $\\tilde{\\tau}$\nmass, causing the stau LSP regions to encroach on the parameter\nspace. By $\\tan\\beta=35$ the light stau rules out all values of low $\\mu$.\nAs noted earlier, at such a low value of $m_0$, $\\tan\\beta=50$ has a\ntachyonic stau and so is not shown here.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D10} &\n      \\multicolumn{2}{|c|}{D11} &\n      \\multicolumn{2}{|c|}{D12} &\n      \\multicolumn{2}{|c|}{D13}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 8.8  & 100    & 2.6  & 100     & 5.0  & 100     & 0.69\\\\\n      $m_{H_1}^2$ & -21261  & 0.37 & -377   & 0.0  & -658190 & 15   & -2200200& 320 \\\\\n      $m_{H_2}^2$ & -73998  & 1.5  & -243830& 0.91 & -772520 & 0.27 & -2597000& 280 \\\\\n      $m_{1\/2}$   & 345     & 4.7  & 220    & 2.1  & 470     & 4.1  & 567     & 32  \\\\\n      $\\tan\\beta$     & 20      & 8.5  & 20     & 3.1  & 35      & 0.99 & 35      & 11  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 8.8  &        & 3.1  &         & 15   &        & 320 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 120  &        & 120  &         & 420  &        & 1100\\\\\n      \\hline\n      \\hline\n      $\\mu$       & 500   & -    & 500 & -   & 930   & -    & 1500& -   \\\\\n      $m_A$       & 500   & -    & 500 & -   & 500   & -    & 500 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Properties of points D10-13, shown in\n    Fig.~\\ref{f:m0,100,mA,500} which are representative of the\n    pseudoscalar Higgs funnel (D13) and the stau-coannihilation\/bulk\n    region (D10,11,12) at increasing $\\tan\\beta$ within the NUHM. We\n    present a breakdown of the dark matter fine-tuning with respect to\n    each parameter of the NUHM. We give the value of $m_{H_{1,2}}^2$\n    but the tunings are calculated with respect to\n    $m_{H_{1,2}}$.\\label{t:m0,100,mA,500}}\n\\end{table}\n\nThe change in the stau mass is the dominant factor that changes the\ndark matter phenomenology. With the lighter stau, the contribution to\nneutralino annihilation from $t$-channel stau exchange increases. We\nconsider two points D10 and D11 in panel (b). At point D10 the\nannihilation is still dominated by coannihilation effects, but the\ngrowing contribution from $t$-channel stau exchange helps to lower the\ndark matter tuning. The dark matter fine-tuning is predominantly due\nto $m_0$ and $\\tan\\beta$ through their influence on the mass of the\nlighter stau, with a subsidiary fine-tuning with respect to\n$m_{1\/2}$. In contrast, point D11 lies in a dark matter band where the\nannihilation of neutralinos is dominantly through $t$-channel slepton\nexchange. As a result the dark matter fine-tuning is small, and due\nprimarily to $m_0$ and $\\tan\\beta$ through their influence on the slepton\nmasses.\n\nAs we move to larger $\\tan\\beta$, the coannihilation and bulk regions\nmeet. In panel (c) we take point D12 as a representative of the\nmeeting of these two regions. However, by this stage one needs large\nsoft Higgs mass-squared parameters and the stau mass is sensitive to\nthese, rather than to $m_0$ and $m_{1\/2}$. Therefore there is large\nfine-tuning with respect to $m_{H_1}$.  Finally, point D13 is\nrepresentative point of the pseudoscalar Higgs funnel for large\n$\\tan\\beta$. As before, we find the dark matter fine-tuning to be large and\npredominantly due to the soft Higgs masses. This chimes with the\ngeneral behaviour of the pseudoscalar Higgs funnel throughout our\nstudy.\n\n\\section{Conclusions}\n\\label{Conc}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\n{\\bf Region} & {\\bf Typical $\\Delta^{\\Omega}$} \\\\\n\\hline\n$\\tilde{\\tau}$ bulk region                                          & 1-5\\\\\n$\\tilde{\\tau}-\\tilde{\\chi}^0_1$ coannihilation                                 & 4-80\\\\\nBino annihilation via pseudoscalar Higgs Funnel              & 30-1200+\\\\\nBino\/higgsino annihilation via pseudoscalar Higgs Funnel     & 3-10\\\\\nBino\/higgsino region, $m_{\\tilde{\\chi}^0_1}>m_{H,A}$                    & 30-300\\\\\nBino\/higgsino region, $m_{\\tilde{\\chi}^0_1}<m_{H,A}$                    & 2-10\\\\\n$\\tilde{\\nu}-\\tilde{\\chi}^0_1$ coannihilation                                & 15-200\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vskip -0.5cm \\caption{\\small A summary of the different dark matter\n  regions present within the NUHM and typical values of the\n  corresponding dark matter fine-tunings. We also note that\n  combinations from many channels decrease the overall\n  tuning. \\label{t:NUHMTunings}}\n\\end{table}\n\nWe summarize in Table~\\ref{t:NUHMTunings} the ranges of dark matter\nfine-tunings found in the different dark matter regions appearing in\nthe NUHM, which may be compared to those found previously in the\nCMSSM, as shown in Tables~\\ref{t:CMSSM,t10},\\ref{t:CMSSM} and\nsummarised in Table~\\ref{t:CMSSMTunings}. Comparing first the bulk\nregions, which require the smallest amounts of dark matter fine-tuning\nin both the CMSSM and the NUHM, we see that CMSSM point A2 has a low\namount of dark matter fine-tuning that is at the end of the range\nfound in the NUHM. However, point A2 has too small a value of $m_h$,\nand hence is not acceptable, whereas the NUHM can circumvent this\nrestriction. Thus, {\\it the NUHM provides access to dark matter which\nis less fine-tuned than in the CMSSM}. Turning then to the $\\tilde{\\tau}$\ncoannihilation regions, we see that the NUHM offers an option of lower\nfine-tuning than that found in the CMSSM at point A5. For reasons\nexplained earlier in the text, there are very large variations in the\namounts of dark matter fine-tuning required in the pseudoscalar Higgs\nfunnel region of the NUHM, and the amount found at the CMSSM point A4\nlies within this range. In addition, the NUHM contains a region in\nwhich a bino\/higgsino LSP annihilates via a pseudoscalar Higgs boson\nthat requires dramatically less fine-tuning than the pseudoscalar\nfunnel in the CMSSM. Likewise, NUHM analogues of the focus points A3\nand A6 shown in Table~\\ref{t:CMSSM} require dark matter fine-tunings\nthat are substantially less than in the CMSSM. One new region appears\nin the NUHM that has no CMSSM analogue, namely the $\\tilde{\\nu}-\\tilde{\\chi}^0_1$\ncoannihilation region.  At least in the examples studied, this\nrequires rather more dark matter fine-tuning than the $\\tilde{\\tau}-\\tilde{\\chi}^0_1$\ncoannihilation region.\n\nAlthough it was not the primary focus of this paper, we have also\ncalculated the electroweak fine-tuning across the NUHM parameter\nspace, and found it to be of the same order of magnitude as in other\nMSSM studies.\n\nGenerally speaking, the fact that the NUHM has more parameters offers\nmore possibilities to find regions with particularly small (or large)\ndark matter fine-tunings. The smaller amounts of dark matter\nfine-tuning generally occur in regions where several different\n(co)annihilation processes contribute to the final dark matter\ndensity, e.g., where a bino\/higgsino band meets a stau coannihilation\nregion. Conversely, there are instances where a tight correlation is\nnecessary between two {\\it a priori} independent MSSM parameters, such\nas the stau and neutralino masses along a coannihilation strip, which is\nimposed automatically in the lower-dimensional parameter space of the\nCMSSM, resulting in smaller amount of dark matter fine-tuning than\nmight otherwise have been expected.\n\nThere have been several studies of the implications of prospective LHC\nand\/or ILC measurements for the accuracy with which the astrophysical\ndark matter density could be calculated on the basis of collider\nmeasurements\\cite{Battaglia:2003ab}. These studies have emphasized\nrelatively favourable points in the CMSSM coannihilation region at low\n$m_{1\/2}$, where the dark matter fine-tuning is relatively low and the\nprospective collider measurements relatively accurate. Our analysis\noffers some prognosis as to which NUHM regions might be favourable for\nextensions of these analyses. Clearly, the presence of NUHM points\nwith low dark matter fine-tuning and relatively light sparticles is\nencouraging {\\it a priori}.  However, we note that in general the NUHM\nrelic density is sensitive to the separate and independent values of\nthe soft Higgs masses $m_{H_{1,2}}$. Detailed consideration of their\ndeterminations using collider data has not yet been given, as far as\nwe know, and we recall that the prospects for measuring directly the\nmasses of the heavier Higgs bosons at the LHC and ILC are limited,\nthough the prospects of the latter would be improved with the $\\gamma\n- \\gamma$ option, or by going to higher energies as at CLIC.\n\nThe extension of these collider dark matter studies to the NUHM is a\nlarge task that lies beyond the scope of this paper. However, this\nexploratory study has revealed some of the prospective opportunities\nand pitfalls.\n\n\\section*{Acknowledgements}\n\nJPR would like to thank Ben Allanach for useful advice. The work of JPR\nwas funded under the FP6 Marie Curie contract MTKD-CT-2005-029466.\nWe also acknowledge partial support from the following grants:\nPPARC Rolling Grant PPA\/G\/S\/2003\/00096;\nEU Network MRTN-CT-2004-503369;\nEU ILIAS RII3-CT-2004-506222\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:Intro}\nConsider the following first order system of differential equations\n\\begin{equation} \\label{eq:system}\n \\mathcal{L} y = -i B^{-1} y' + Q(x) y=\\l y, \\qquad y = \\col(y_1,y_2),\n \\qquad x \\in [0,1],\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:BQ}\n B = \\begin{pmatrix} b_1 & 0 \\\\ 0 & b_2 \\end{pmatrix},\n \\quad b_1 < 0 < b_2 \\quad \\text{and}\\quad\n Q = \\begin{pmatrix} 0 & Q_{12} \\\\ Q_{21} & 0 \\end{pmatrix} \\in \\LL{2}.\n\\end{equation}\n\nIf $B = \\begin{psmallmatrix} -1 & 0 \\\\ 0 & 1 \\end{psmallmatrix}$\nsystem~\\eqref{eq:system} is equivalent to the Dirac system (see the classical\nmonographs~\\cite{LevSar88},~\\cite{Mar77}).\n\nLet us associate linearly independent boundary conditions\n\\begin{equation} \\label{eq:cond}\n U_j(y) := a_{j 1}y_1(0) + a_{j 2}y_2(0) + a_{j 3}y_1(1) + a_{j 4}y_2(1)= 0,\n \\quad j \\in \\{1,2\\},\n\\end{equation}\nwith system~\\eqref{eq:system}, and denote as $L_U(Q) := L_U(Q)$ an operator, associated in $\\mathfrak{H} := \\LLV{2}$ with the boundary value problem (BVP)~\\eqref{eq:system}--\\eqref{eq:cond}. It is defined by differential expression $\\mathcal{L}$ on the domain\n\\begin{equation}\n \\dom(L_U(Q)) = \\{f \\in {\\rm AC}([0,1]; \\mathbb{C}^2) :\\ \\mathcal{L} f \\in \\mathfrak{H},\n \\ U_1(f) = U_2(f) = 0\\}.\n\\end{equation}\n\nTo the best of our knowledge, the spectral properties of the general $n \\times n$ system of the form~\\eqref{eq:system} with a nonsingular diagonal $n\\times n$ matrix $B$ with complex entries and a potential matrix $Q(\\cdot)$ of the form\n\\begin{equation} \\label{1.2}\n B = \\diag(b_1, \\ldots, b_n) \\in \\mathbb{C}^{n\\times n} \\quad \\text{and}\\quad\n Q(\\cdot) =: (q_{jk}(\\cdot))_{j,k=1}^n \\in C^1([0,1]; \\mathbb{C}^{n\\times n}).\n\\end{equation}\nhas first been investigated by G.D.~Birkhoff and R.E.~Langer~\\cite{BirLan23}. Namely, they introduced the concepts of \\emph{regular and strictly regular boundary conditions}~\\eqref{eq:cond} and investigated the asymptotic behavior of eigenvalues and eigenfunctions of the corresponding operator $L_{U}(Q)$. Moreover, they proved \\emph{a pointwise convergence result} on spectral decompositions of the operator $L_{U}(Q)$ corresponding to the BVP~\\eqref{eq:system}--\\eqref{eq:cond} with regular boundary conditions.\n\nThe completeness property in $L^2([0,1]; \\mathbb{C}^n)$ of the system of root vectors of BVP for general $n \\times n$ system of the form~\\eqref{eq:system} with matrices $B = \\diag(b_1, \\ldots, b_n)$ and $Q \\in L^1([0,1]; \\mathbb{C}^{n\\times n})$ was established for the first time by M.M.~Malamud and L.L.~Oridoriga in~\\cite{MalOri12} for a wide class of BVPs, although for $2 \\times 2$ Dirac system with $Q\\in C([0,1]; \\mathbb{C}^{2 \\times 2})$ it was proved earlier by V.A.~Marchenko in~\\cite[Chapter 1.3]{Mar77}. As a development of~\\cite{MalOri12},\nin~\\cite{AgiMalOri12,LunMal14IEOT,LunMal15JST,ALMO19}\ncompleteness conditions for non-regular and even degenerate boundary conditions were found with applications to dissipative and normal operators.\nIn the joint paper~\\cite{LunMal15JST} the author and M.M.~Malamud\nalso established the Riesz basis property with parentheses of the system of root vectors for different classes of BVPs for $n \\times n$ system with arbitrary $B$ of the form~\\eqref{1.2} and $Q \\in L^{\\infty}([0,1]; \\mathbb{C}^{n\\times n})$.\nNote also that BVP for $2m \\times 2m$ Dirac equation ($B=\\diag(-I_m, I_m)$) were investigated in~\\cite{MykPuy13} (Bari-Markus property for Dirichlet BVP with $Q \\in L^2([0,1]; \\mathbb{C}^{2m \\times 2m})$ and in~\\cite{KurAbd18,KurGad20} (Bessel and Riesz basis properties on abstract level).\n\nThe Riesz basis property in $\\LLV{2}$ of BVP~\\eqref{eq:system}--\\eqref{eq:cond}\nfor $2 \\times 2$ Dirac system ($b_2 = -b_1 = 1$) with various assumptions on the potential matrix $Q$ was investigated in numerous papers (see~\\cite{TroYam01,TroYam02,HasOri09,DjaMit10,Bask11,DjaMit12UncDir,DjaMit12Crit,LunMal14Dokl,SavShk14,LunMal16JMAA}\nand references therein). The case of separated boundary conditions and $Q \\in C^1([0,1]; \\mathbb{C}^{2 \\times 2})$ was treated in~\\cite{TroYam01,TroYam02} and later in~\\cite{HasOri09} for Dirac-type system. For Dirichlet and periodic boundary conditions the case $Q \\in \\LL{2}$ was first treated by P.~Djakov and B.~Mityagin~\\cite{DjaMit10} and later by A.~Baskakov, A.~Derbushev, A.~Shcherbakov~\\cite{Bask11}. Shortly after, P.~Djakov and B.~Mityagin~\\cite{DjaMit12UncDir} extended these results to general regular boundary conditions.\n\nThe most complete result on the Riesz basis property for $2\\times 2$ Dirac and Dirac-type systems, respectively, with $Q \\in \\LL{1}$ and strictly regular boundary conditions was obtained independently by different methods and at the same time by A.M.~Savchuk and A.A.~Shkalikov~\\cite{SavShk14}\nand by the author and M.M.~Malamud~\\cite{LunMal14Dokl,LunMal16JMAA} (in~\\cite{LunMal14Dokl} the sketches of the proof are given). The case of regular boundary conditions is treated in~\\cite{SavShk14} for the first time. Other proofs were obtained later on in~\\cite{SavSad15DAN,SavSad15,LunMal16JMAA} (see also recent survey~\\cite{SavSad20} and references therein).\n\nIn~\\cite{LunMal16JMAA} results for Dirac operator were also applied to Timoshenko beam model. In general one can show that dynamic generators of many physical models governed by systems of linear PDE are similar to certain first order differential operators. Hence spectral properties of such operators are of significant importance in the study of stability of solutions and corresponding $C_0$-semigroups of many physical models. In particular, in Section~\\ref{sec:damped.string} we establish explicit connection between $2 \\times 2$ Dirac type operators (with $b_1 \\ne -b_2$) on the one hand and certain \\textbf{non-canonical initial-boundary value problem for a damped string} on the other hand. This allows us to apply results on Bari and Riesz basis property (see Definition~\\ref{def:bases}) for Dirac type operators obtained here and in~\\cite{LunMal16JMAA} to the damped string equation.\n\nRecall, that boundary conditions~\\eqref{eq:cond} are called \\textbf{regular}, if and only if they are \\emph{equivalent} to the following conditions\n\\begin{equation} \\label{eq:cond.canon.intro}\n \\widehat{U}_{1}(y) = y_1(0) + b y_2(0) + a y_1(1) = 0,\\quad\n \\widehat{U}_{2}(y) = d y_2(0) + c y_1(1) + y_2(1) = 0,\n\\end{equation}\nwith certain $a,b,c,d \\in \\mathbb{C}$ satisfying $ad-bc \\ne 0$. Recall also that regular boundary conditions~\\eqref{eq:cond} are called \\textbf{strictly regular}, if the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the eigenvalues of the unperturbed ($Q=0$) BVP~\\eqref{eq:system}--\\eqref{eq:cond} (of the operator $L_U(0)$), is asymptotically separated. In particular, the eigenvalues $\\{\\l_n^0\\}_{|n| > n_0}$ are geometrically and algebraically simple.\n\nIt is well known that \\emph{non-degenerate separated} boundary conditions are always \\emph{strictly regular}. Moreover, conditions~\\eqref{eq:cond.canon.intro} \\emph{are strictly regular for Dirac operator if and only if} $(a-d)^2 \\ne -4bc$. In particular, antiperiodic ($a=d=0$, $b=c=1$) boundary conditions \\emph{are regular but not strictly regular} for Dirac system, while they \\emph{become strictly regular for Dirac-type system} if $-b_1, b_2 \\in \\mathbb{N}$ and $b_2 - b_1$ is odd.\n\nNote in this connection that \\emph{periodic and antiperiodic (necessarily non-strictly regular) BVP} for $2 \\times 2$ Dirac and Sturm-Liouville equations have also attracted certain attention during the last decade. For\\ instance, a criterion for the system of root vectors of the \\emph{periodic} BVP for $2 \\times 2$ Dirac equation to contain a Riesz basis (without parentheses!) was obtained by P. Djakov and B. Mityagin in~\\cite{DjaMit12Crit} (see also\nrecent papers~\\cite{Mak19},~\\cite{Mak20} by A.S.~Makin and the references therein). It is also worth mentioning that F.~Gesztesy and V.~Tkachenko~\\cite{GesTka09,GesTka12} for $q \\in L^2[0,\\pi]$ and P.~Djakov and B.~Mityagin~\\cite{DjaMit12Crit} for $q \\in W^{-1,2}[0,\\pi]$ established by different methods a \\emph{criterion} for the system of root vectors to contain a Riesz basis for Sturm-Liouville operator $-\\frac{d^2}{dx^2} + q(x)$ on $[0,\\pi]$ (see also survey~\\cite{Mak12}).\n\nLet us emphasize that the proof of the Riesz basis property in~\\cite{DjaMit10,Bask11,DjaMit12UncDir,MykPuy13} substantially relies on the Bari-Markus property: the quadratic closeness in $\\LLV{2}$ of the spectral projectors of the operators $L_U(Q)$ and $L_U(0)$.\nAssuming boundary conditions to be strictly regular, let $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{f_n^0\\}_{n \\in \\mathbb{Z}}$ be the systems of root vectors of the operators $L_U(Q)$ and $L_U(0)$, respectively. Then Bari-Markus property states the implication: $Q \\in L^2 \\Rightarrow \\sum_{n \\in \\mathbb{Z}} \\|f_n - f_n^0\\|_2^2 < \\infty$. Later, this property was generalized to the case $Q \\in \\LL{p}$, $p \\in [1,2]$, in~\\cite{SavShk14,Sad16,LunMal22JDE}. The most complete results in this direction were established\nin the joint paper~\\cite{LunMal22JDE} by the author and M.M.~Malamud. One of these results reads as follows.\n\\begin{theorem}[Theorem 7.15 in \\cite{LunMal22JDE}] \\label{th:ellp-close}\nLet $\\mathcal{K} \\in \\LL{p}$ be a compact set for some $p \\in [1,2]$, let $Q, \\widetilde{Q} \\in \\mathcal{K}$ and boundary conditions~\\eqref{eq:cond} be strictly regular. Then for some normalized systems of root vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{\\widetilde{f}_n\\}_{n \\in \\mathbb{Z}}$ of the operators $L_U(Q)$ and $L_U(\\widetilde{Q})$ the following uniform relations hold for $Q, \\widetilde{Q} \\in \\mathcal{K}$:\n\\begin{align}\n\\label{eq:sum.fn-fn0}\n & \\sum_{|n| > N} \\|f_n - \\widetilde{f}_n\\|_{\\infty}^{p'} \\le C\n \\|Q - \\widetilde{Q}\\|_p^{p'}, \\qquad p \\in (1,2], \\quad 1\/p'+1\/p=1, \\\\\n\\label{eq:sum.fn-fn0.hardy}\n & \\sum_{|n| > N} (1+|n|)^{p-2} \\|f_n - \\widetilde{f}_n\\|_{\\infty}^{p} \\le\n C \\|Q - \\widetilde{Q}\\|_p^p, \\qquad p \\in (1,2], \\\\\n\\label{eq:lim.fn-fn0.c0}\n & \\lim_{n \\to \\infty} \\sup_{Q, \\widetilde{Q} \\in \\mathcal{K}}\n \\|f_n - \\widetilde{f}_n\\|_{\\infty} = 0, \\qquad p = 1.\n\\end{align}\n\\end{theorem}\nHere and throughout the paper we denote by $\\|f\\|_s$ the $L^s$-norm of the element $f$ of a scalar, vector or matrix $L^s$-space.\n\nEmphasize, that the proof of the estimates~\\eqref{eq:sum.fn-fn0}--\\eqref{eq:sum.fn-fn0.hardy} is based on the deep Carleson-Hunt theorem. Note, however, that these estimates with $\\|\\cdot\\|_{p'}$-norm instead of $\\|\\cdot\\|_{\\infty}$-norm can be proved in a more direct way, which is elementary in character. Note also that these results substantially rely on transformation operators method that goes back to~\\cite{Mal94,Mal99,LunMal16JMAA}.\n\nRecall that the concepts of Riesz bases and bases quadratically close to the orthonormal bases were introduced by N.K.~Bari in~\\cite{Bari51}. Results of this fundamental paper can also be found in the classical monograph~\\cite{GohKre65} where a basis quadratically close to the orthonormal basis is called a Bari basis. Let us recall the definition of Riesz and Bari bases following~\\cite[Section IV]{GohKre65}.\n\\begin{definition} \\label{def:bases}\n\\textbf{(i)} A sequence of vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ in a separable Hilbert space $\\mathfrak{H}$ is called a \\textbf{Riesz basis} if it admits a representation $f_n = T e_n$, $n \\in \\mathbb{N}$, where $\\{e_n\\}_{n \\in \\mathbb{Z}}$ is an orthonormal basis in $\\mathfrak{H}$ and $T : \\mathfrak{H} \\to \\mathfrak{H}$ is a bounded operator with bounded inverse.\n\n\\textbf{(ii)} A sequence of vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ in a separable Hilbert space $\\mathfrak{H}$ is called a \\textbf{Bari basis} if it is quadratically close to some orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ in $\\mathfrak{H}$, i.e.\n\\begin{equation} \\label{eq:sum.fn-en}\n \\sum_{n \\in \\mathbb{Z}} \\|f_n - e_n\\|_{\\mathfrak{H}}^2 < \\infty.\n\\end{equation}\n\\end{definition}\nA.S.~Markus in~\\cite{Markus69} studied in detail bases of subspaces with the property similar to~\\eqref{eq:sum.fn-en}. Bari basis property for different classes of differential operators was studied in~\\cite{BDL00,Zhidkov02,Allah14}.\nNote, however, that to the best of our knowledge the question of whether system of root vectors of the operator $L_U(Q)$ forms \\emph{a Bari basis  has not been studied before}. Namely, results of papers~\\cite{DjaMit10,Bask11,DjaMit12UncDir,MykPuy13,SavShk14,LunMal22JDE} in the case of $Q \\in L^2$ and strictly regular boundary conditions establish quadratic closeness of systems of root vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{f_n^0\\}_{n \\in \\mathbb{Z}}$, but whether $\\{f_n\\}_{n \\in \\mathbb{Z}}$ is quadratically close to some orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ remained an open question. The goal if this paper is to close this gap.\nOne of our main results establishes the criterion for the system of root vectors of the operator $L_U(Q)$ to form a Bari basis and reads as follows.\n\\begin{theorem} \\label{th:crit.bari}\nLet boundary conditions~\\eqref{eq:cond.canon.intro} be strictly regular and let $Q \\in \\LL{2}$. Then some normalized system of root vectors of the operator $L_U(Q)$ is a Bari basis in $\\LLV{2}$ if and only if the operator $L_U(0)$ is self-adjoint. The latter holds if and only if the coefficients $a,b,c,d$ in boundary conditions~\\eqref{eq:cond.canon.intro} satisfy the following relations:\n\\begin{equation} \\label{eq:abcd.sa.intro}\n |a|^2 + \\beta |b|^2 = 1, \\qquad\n |c|^2 + \\beta |d|^2 = \\beta, \\qquad\n a \\overline{c} + \\beta b \\overline{d} = 0, \\qquad \\beta := -b_2\/b_1 > 0.\n\\end{equation}\nIn this case every normalized system of root vectors of the operator $L_U(Q)$ is a Bari basis in $\\LLV{2}$.\n\\end{theorem}\nCombining Theorem~\\ref{th:crit.bari} with the results of the previous papers\n\\cite{DjaMit10,Bask11,DjaMit12UncDir,MykPuy13,LunMal14Dokl,LunMal16JMAA,SavShk14}\nconcerning the Riesz basis property we get the following surprising result.\n\\begin{corollary} \\label{cor:not.bari}\nLet $Q \\in \\LL{2}$ and let boundary conditions~\\eqref{eq:cond.canon.intro} be strictly regular but not self-adjoint, i.e. the operator $L_U(0)$ is not self-adjoint. Then every normalized system of root vectors of the operator $L_U(Q)$ is \\textbf{a Riesz basis but not a Bari basis} in $\\LLV{2}$.\n\\end{corollary}\n\\section{Definitions and formulations of the main results}\nLet us recall the following abstract criterion for Bari basis property.\n\\begin{proposition}~\\cite[Theorem VI.3.2]{GohKre65} \\label{prop:crit.bari}\nA complete system $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$ of unit vectors in a separable Hilbert space $\\mathfrak{H}$ forms a Bari basis if and only if there exists a sequence $\\{g_n\\}_{n \\in \\mathbb{Z}}$ biorthogonal to $\\mathfrak{F}$ that is quadratically close to $\\mathfrak{F}$:\n\\begin{equation} \\label{eq:sum.fn-gn.2}\n\\sum_{n \\in \\mathbb{Z}} \\|f_n - g_n\\|_{\\mathfrak{H}}^2 < \\infty, \\qquad (f_n, g_m)_{\\mathfrak{H}} = \\delta_{nm}, \\quad n,m \\in \\mathbb{Z}.\n\\end{equation}\n\\end{proposition}\nBased on this abstract criterion we will introduce a generalization of Bari basis concept. Let $p \\in [1,2]$ and $p' = p\/(p-1) \\in [2,\\infty]$. It is well-known that for the dual space of $\\ell^p := \\ell^p(\\mathbb{Z})$ we have,\n\\begin{equation} \\label{eq:ellp*}\n(\\ell^p(\\mathbb{Z}))^* \\cong \\ell^{p'}(\\mathbb{Z}), \\quad p \\in (1,2],\n\\qquad\\text{and}\\qquad\n(\\ell^1(\\mathbb{Z}))^* \\cong c_0(\\mathbb{Z}).\n\\end{equation}\nFor simplicity we identify $(\\ell^p(\\mathbb{Z}))^*$ with $\\ell^{p'}(\\mathbb{Z})$ for $p \\in (1,2]$ and with $c_0(\\mathbb{Z})$ for $p=1$, respectively. E.g. $\\{a_n\\}_{n \\in \\mathbb{Z}} \\in (\\ell^p(\\mathbb{Z}))^*$ for $p>1$ means that $\\sum_{n \\in \\mathbb{Z}} |a_n|^{p'} < \\infty$.\nWith this in mind, we can extend Definition~\\ref{def:bases}(ii) using equivalence from Proposition~\\ref{prop:crit.bari} to more general concept of closeness of sequences $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{g_n\\}_{n \\in \\mathbb{Z}}$.\n\\begin{definition} \\label{def:bari.c0}\nLet $p \\in [1,2]$, let $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ be a complete minimal sequence of unit vectors in a separable Hilbert space $\\mathfrak{H}$ and let $\\mathfrak{G} := \\{g_n\\}_{n \\in \\mathbb{Z}}$ be its (unique) biorthogonal sequence: $(f_n, g_m)_{\\mathfrak{H}} = \\delta_{nm}$, $n, m \\in \\mathbb{Z}$. A sequence $\\mathfrak{F}$ is called a \\textbf{Bari $(\\ell^p)^*$-sequence} if it is ``($\\ell^p)^*$-close'' to its biorthogonal sequence $\\mathfrak{G}$, i.e. $\\curl{\\|f_n - g_n\\|_{\\mathfrak{H}}}_{n \\in \\mathbb{Z}} \\in (\\ell^p)^*$. In view of~\\eqref{eq:ellp*} it means that\n\\begin{equation} \\label{eq:sum.fn-gn}\n\\sum_{n \\in \\mathbb{Z}} \\|f_n - g_n\\|_{\\mathfrak{H}}^{p'} < \\infty \\quad\\text{if}\\quad\np \\in (1,2],\n\\quad\\text{and}\\quad\n \\lim_{n \\to \\infty} \\|f_n - g_n\\|_{\\mathfrak{H}} = 0\n \\quad\\text{if}\\quad p = 1.\n\\end{equation}\nFor brevity we will call \\emph{Bari $(\\ell^1)^*$-sequence} as \\textbf{Bari $c_0$-sequence} and \\emph{Bari $(\\ell^p)^*$-sequence} as \\textbf{Bari $\\ell^{p'}$-sequence} for $p \\in (1,2]$.\n\n\\end{definition}\nProposition~\\ref{prop:crit.bari} implies that the notion of Bari $\\ell^2$-sequence coincides with the notion of Bari basis. Note also that every Bari $(\\ell^p)^*$-sequence is Bari $c_0$-sequence. We specifically chose the word ``sequence'' because it is not clear if Bari $c_0$-sequence is a Riesz basis or even a regular basis in general case.\n\\begin{remark} \\label{rem:c0.bari.diff}\nNote that Bari $c_0$-property from definition~\\ref{def:bari.c0} is not equivalent to more conventional formulation of $c_0$-closeness of $\\{f_n\\}_{n \\in \\mathbb{Z}}$ to a certain orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ even if $\\{f_n\\}_{n \\in \\mathbb{Z}}$ is already a Riesz basis. Indeed, in this case $f_n = e_n + K e_n$, where $K$ and $(I+K)^{-1}$ are bounded operators in $\\mathfrak{H}$. Hence $\\|f_n - e_n\\|_{\\mathfrak{H}} = \\|K e_n\\|_{\\mathfrak{H}}$. It is easily seen that $g_n = \\((I+K)^{-1}\\)^* e_n = e_n - \\((I+K)^{-1}\\)^* K^* e_n$, and hence $\\|g_n - e_n\\| \\to 0$ as $n \\to \\infty$ is equivalent to $\\|K^* e_n\\| \\to 0$ as $n \\to \\infty$. If $K$ is not compact then $\\lim_{n \\to \\infty}\\|K e_n\\| = 0$ is generally not equivalent to $\\lim_{n \\to \\infty}\\|K^* e_n\\| = 0$ for a given orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$.\n\\end{remark}\nLet us also recall the notion of the system of root vectors of an operator with compact resolvent. Firt, we recall a few basic facts regarding the eigenvalues of a\ncompact, linear operator  $T \\in \\mathcal{B}_{\\infty}(\\mathfrak{H})$ in a\nseparable complex Hilbert space $\\mathfrak{H}$. The {\\it geometric\nmultiplicity}, $m_g(\\lambda_0,T)$, of an eigenvalue $\\lambda_0\n\\in \\sigma_p (T)$ of $T$ is given by\n$\nm_g(\\lambda_0,T) := \\dim(\\ker(T - \\lambda_0)).\n$\n\nThe {\\it root subspace} of $T$ corresponding to $\\lambda_0 \\in\n\\sigma_p(T)$ is given by\n\\begin{equation}\\label{root.subspace}\n\\mathcal{R}_{\\lambda_0}(T) = \\big\\{f\\in\\mathfrak{H}\\,:\\, (T - \\lambda_0)^k f = 0 \\\n\\ \\text{for some}\\ \\ k\\in\\mathbb N \\big\\}.\n\\end{equation}\nElements of $\\mathcal{R}_{\\l_0}(T)$ are called {\\it root vectors}.\nFor $\\lambda_0 \\in \\sigma_p (T) \\backslash \\{0\\}$, the set\n$\\mathcal{R}_{\\lambda_0}(T)$ is a closed linear subspace of\n$\\mathfrak{H}$ whose dimension equals to the {\\it algebraic\nmultiplicity}, $m_a(\\lambda_0,T)$, of $\\lambda_0$,\n$\nm_a(\\lambda_0,T) := \\dim\\big(\\mathcal R_{\\lambda_0}(T)\\big)<\\infty.\n$\n\nDenote by $\\{\\l_j\\}_{j=1}^{\\infty}$ the sequence of non-zero\neigenvalues of $T$ and let $n_j$ be the algebraic multiplicity\nof $\\l_j$. By the {\\it system of root vectors} of the operator\n$T$ we mean any sequence of the form\n$\n\\cup_{j=1}^{\\infty}\\{e_{jk}\\}_{k=1}^{n_j},\n$\nwhere $\\{e_{jk}\\}_{k=1}^{n_j}$ is a basis in $\\mathcal{R}_{\\l_j}(T)$,\n$n_j = m_a(\\lambda_j,T) < \\infty$. The system or root vectors of the operator $T$ is called \\emph{normalized} if $\\|e_{jk}\\|_{\\mathfrak{H}} = 1$, $j \\in \\mathbb{N}$, $k \\in \\{1, \\ldots, n_j\\}$.\n\nWe are particularly interested in the case where $A$ is a\ndensely defined, closed, linear operator in $\\mathfrak{H}$ whose\nresolvent is compact, that is,\n$\nR_A(\\l):=(A - \\l)^{-1} \\in \\mathcal{B}_{\\infty}(\\mathfrak{H}), \\ \\l \\in \\rho (A).\n$\nVia the spectral mapping theorem all eigenvalues of $A$\ncorrespond to eigenvalues of its resolvent $R_A(\\l)$, $\\l \\in\n\\rho (A)$, and vice versa. Hence, we use the same notions of\nroot vectors, root subspaces, geometric and algebraic\nmultiplicities associated with the eigenvalues of $A$, and the\nsystem of root vectors of $A$.\n\nNow we are ready to formulate the main result of this paper, which involve notions of Bari $(\\ell^p)^*$-sequences and $c_0$-sequences from Definition~\\ref{def:bari.c0} above.\n\\begin{theorem} \\label{th:crit.lp.bari}\nLet boundary conditions~\\eqref{eq:cond.canon.intro} be strictly regular and let $Q \\in \\LL{p}$ for some $p \\in [1,2]$. Then some normalized system of root vectors of the operator $L_U(Q)$ is a Bari $(\\ell^p)^*$-sequence in $\\LLV{2}$ if and only if the operator $L_U(0)$ is self-adjoint, i.e. when relations~\\eqref{eq:abcd.sa.intro} hold for the coefficients $a,b,c,d$ in boundary conditions~\\eqref{eq:cond.canon.intro}. In this case every normalized system of root vectors of the operator $L_U(Q)$ is a Bari $(\\ell^p)^*$-sequence in $\\LLV{2}$.\n\\end{theorem}\nAs an immediate consequence of Theorem~\\ref{th:crit.lp.bari} we get Theorem~\\ref{th:crit.bari}: the criterion of Bari basis property for Dirac-type operator $L_U(Q)$ with $L^2$-potential and strictly regular boundary conditions.\n\nLet us briefly comment on the proof of our main result, Theorem~\\ref{th:crit.lp.bari}.\nFirst, we apply Theorem~\\ref{th:ellp-close} to reduce the Bari $(\\ell^p)^*$-property of the system of root vectors of operator $L_U(Q)$ with strictly regular boundary conditions to a certain explicit condition in terms of the eigenvalues $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$, which reads as follows for the case $p=1$.\n\\begin{proposition} \\label{prop:c0.close.cond}\nLet $Q \\in \\LL{1}$ and boundary conditions~\\eqref{eq:cond} be strictly regular. Then some normalized systems of root vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(Q)$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only if:\n\\begin{equation} \\label{eq:lim1.lim2.intro}\n b_1 |c| + b_2 |b| = 0, \\qquad \\lim_{n \\to \\infty} \\Im \\l_n^0 = 0\n \\quad\\text{and}\\quad \\lim_{n \\to \\infty} z_n = |bc|,\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:zn.def.intro}\n z_n := \\(1 + d \\exp(- i b_2 \\l_n^0)\\)\\overline{\\(1 + a \\exp(i b_1 \\l_n^0)\\)},\n\\end{equation}\nand $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ is the sequence of the eigenvalues of the operator $L_U(0)$, counting multiplicity.\n\\end{proposition}\nWith condition~\\eqref{eq:lim1.lim2.intro} established, the main difficulty arises in reducing this condition to the desired explicit condition~\\eqref{eq:abcd.sa.intro}.\nIn this connection, recall that the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the eigenvalues of the operator $L_U(0)$ coincides with the sequence of zeros of characteristic determinant\n\\begin{equation} \\label{eq:Delta0.intro}\n \\Delta_0(\\l) = d + a e^{i (b_1+b_2) \\l} + (ad-bc) e^{i b_1 \\l}\n + e^{i b_2 \\l}.\n\\end{equation}\nIf $b_2 \/ b_1 \\in \\mathbb{Q}$ then the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ has a simple explicit form: it is the union of arithmetic progression that lie on the lines parallel to the real axis, which simplifies the problem a lot.\n\nThe case $b_2 \/ b_1 \\notin \\mathbb{Q}$ is much more complicated.\nNamely, if $|a|+|d|>0$ and $bc \\ne 0$ there is no explicit description of the spectrum of the operator $L_U(0)$. Nevertheless, we were able to establish equivalence of~\\eqref{eq:lim1.lim2.intro} and~\\eqref{eq:abcd.sa.intro} using Weyl's equidistribution theorem (see~\\cite[Theorem 4.2.2.1]{SteinShak03}). It implies the following crucial property of zeros of $\\Delta_0(\\cdot)$.\n\\begin{proposition} \\label{prop:nlim.inf.intro}\nLet $b_2\/b_1 \\notin \\mathbb{Q}$ and boundary conditions~\\eqref{eq:cond.canon.intro} be regular, i.e. $ad-bc \\ne 0$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of zeros of the characteristic determinant $\\Delta_0(\\cdot)$ counting multiplicity. Then each of the sequences $\\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\exp(i b_2 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points.\n\\end{proposition}\nThis result was key for proving equivalence of~\\eqref{eq:lim1.lim2.intro} and~\\eqref{eq:abcd.sa.intro}, which in turn implies our main result, Theorem~\\ref{th:crit.lp.bari}, and its main corollary, Theorem~\\ref{th:crit.bari}.\n\\section{Regular and strictly regular boundary conditions}\n\\label{subsec:regular}\nIn this section we recall known properties of BVP~\\eqref{eq:system}--\\eqref{eq:cond} subject to regular or strictly regular boundary conditions from~\\cite{LunMal16JMAA}.\nLet us set\n\\begin{align}\n A := \\begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n a_{21} & a_{22} & a_{23} & a_{24} \\end{pmatrix}, \\qquad\n\\label{eq:Ajk.Jjk}\n A_{jk} := \\begin{pmatrix} a_{1j} & a_{1k} \\\\ a_{2j} & a_{2k} \\end{pmatrix},\n \\quad J_{jk} := \\det (A_{jk}), \\quad j,k\\in\\{1,\\ldots,4\\}.\n\\end{align}\nLet\n\\begin{equation} \\label{eq:Phi.def}\n \\Phi(\\cdot, \\l) =\n \\begin{pmatrix} \\varphi_{11}(\\cdot, \\l) & \\varphi_{12}(\\cdot, \\l)\\\\\n \\varphi_{21}(\\cdot,\\l) & \\varphi_{22}(\\cdot,\\l)\n \\end{pmatrix} =: \\begin{pmatrix} \\Phi_1(\\cdot, \\l) & \\Phi_2(\\cdot, \\l)\n \\end{pmatrix}, \\qquad \\Phi(0, \\l) = I_2,\n\\end{equation}\nbe a fundamental matrix solution of the\nsystem~\\eqref{eq:system}, where $I_2 = \\begin{psmallmatrix} 1 & 0 \\\\ 0 & 1 \\end{psmallmatrix}$. Here $\\Phi_k(\\cdot, \\l)$ is the\n$k$th column of $\\Phi(\\cdot, \\l)$.\n\nThe eigenvalues of the problem~\\eqref{eq:system}--\\eqref{eq:cond} counting multiplicity\nare the zeros (counting multiplicity) of the characteristic determinant\n\\begin{equation} \\label{eq:Delta.def}\n \\Delta_Q(\\l) := \\det\n \\begin{pmatrix}\n U_1(\\Phi_1(\\cdot,\\l)) & U_1(\\Phi_2(\\cdot,\\l)) \\\\\n U_2(\\Phi_1(\\cdot,\\l)) & U_2(\\Phi_2(\\cdot,\\l))\n \\end{pmatrix}.\n\\end{equation}\nInserting~\\eqref{eq:Phi.def} and~\\eqref{eq:cond} into~\\eqref{eq:Delta.def}, setting $\\varphi_{jk}(\\l) := \\varphi_{jk}(1,\\l)$, and taking notations~\\eqref{eq:Ajk.Jjk} into account we arrive at the following expression for the characteristic determinant\n\\begin{equation} \\label{eq:Delta}\n \\Delta_Q(\\l) = J_{12} + J_{34}e^{i(b_1+b_2)\\l}\n + J_{32}\\varphi_{11}(\\l) + J_{13}\\varphi_{12}(\\l)\n + J_{42}\\varphi_{21}(\\l) + J_{14}\\varphi_{22}(\\l).\n\\end{equation}\nIf $Q=0$ we denote a fundamental matrix solution as $\\Phi^0(\\cdot, \\l)$. Clearly\n\\begin{equation} \\label{eq:Phi0.def}\n \\Phi^0(x, \\l)\n = \\begin{pmatrix} e^{i b_1 x \\l} & 0 \\\\ 0 & e^{i b_2 x \\l} \\end{pmatrix}\n =: \\begin{pmatrix}\n \\varphi_{11}^0(x, \\l) & \\varphi_{12}^0(x, \\l)\\\\\n \\varphi_{21}^0(x,\\l) & \\varphi_{22}^0(x,\\l)\n \\end{pmatrix}\n =: \\begin{pmatrix} \\Phi_1^0(x, \\l) & \\Phi_2^0(x, \\l) \\end{pmatrix},\n\\end{equation}\nfor $x \\in [0,1]$ and $\\l \\in \\mathbb{C}$. Here $\\Phi_k^0(\\cdot, \\l)$ is the $k$th column of $\\Phi^0(\\cdot, \\l)$. In\nparticular, the characteristic determinant $\\Delta_0(\\cdot)$ becomes\n\\begin{equation} \\label{eq:Delta0}\n \\Delta_0(\\l) = J_{12} + J_{34}e^{i(b_1+b_2)\\l}\n + J_{32}e^{ib_1\\l} + J_{14}e^{ib_2\\l}.\n\\end{equation}\nIn the case of Dirac system $(B =\\diag (-1,1))$ this formula is\nsimplified to\n\\begin{equation} \\label{eq:Delta0_Dirac}\n \\Delta_0(\\l) = J_{12} + J_{34} + J_{32}e^{-i\\l} + J_{14}e^{i\\l}.\n\\end{equation}\nLet us recall the definition of regular boundary conditions.\n\\begin{definition} \\label{def:regular}\nBoundary conditions~\\eqref{eq:cond} are called \\textbf{regular} if\n\\begin{equation} \\label{eq:J32J14ne0}\n J_{14} J_{32} \\ne 0.\n\\end{equation}\n\\end{definition}\nLet us recall one more definition (cf.~\\cite{Katsn71}).\n\\begin{definition} \\label{def:incompressible}\nLet $\\L := \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ be a sequence of complex numbers. It is\ncalled \\textbf{incompressible} if for some $d \\in \\mathbb{N}$ every rectangle\n$[t-1,t+1] \\times \\mathbb{R} \\subset \\mathbb{C}$ contains at most $d$ entries of the sequence,\ni.e.\n\\begin{equation} \\label{eq:card.incomp}\n \\card\\{n \\in \\mathbb{Z} : |\\Re \\l_n - t| \\le 1 \\} \\le d, \\quad t \\in \\mathbb{R}.\n\\end{equation}\n\\end{definition}\nRecall that $\\mathbb{D}_r(z) \\subset \\mathbb{C}$ denotes the disc of radius $r$ with a\ncenter $z$.\n\nLet us recall certain important properties from~\\cite{LunMal16JMAA} of the characteristic determinant $\\Delta(\\cdot)$ in the case of regular boundary conditions.\n\\begin{proposition}~\\cite[Proposition 4.6]{LunMal16JMAA} \\label{prop:sine.type}\nLet the boundary conditions~\\eqref{eq:cond} be regular. Then the characteristic determinant $\\Delta_Q(\\cdot)$ of the problem~\\eqref{eq:system}--\\eqref{eq:cond} given by~\\eqref{eq:Delta}\nhas infinitely many zeros $\\L := \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ counting multiplicities and\n\\begin{equation} \\label{eq:ln.in.Pih}\n|\\Im \\l_n| \\le h, \\quad n \\in \\mathbb{Z}, \\qquad\\text{for some}\\ \\ h \\ge 0.\n\\end{equation}\nMoreover, the sequence $\\L$ is incompressible\nand can be ordered in such a way that the following asymptotical formula holds\n\\begin{equation} \\label{eq:lam.n=an+o1}\n \\Re \\l_n = \\frac{2 \\pi n}{b_2 - b_1} (1 + o(1)) \\quad\\text{as}\\quad n \\to\\infty.\n\\end{equation}\n\\end{proposition}\nClearly, the conclusions of Proposition~\\ref{prop:sine.type} are valid for the characteristic determinant $\\Delta_0(\\cdot)$ given by~\\eqref{eq:Delta0}. Let $\\L_0 = \\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of its zeros counting multiplicity. Let us order the sequence $\\L_0$ in a (possibly non-unique) way such that $\\Re \\l_n^0 \\le \\Re \\l_{n+1}^0$, $n \\in \\mathbb{Z}$.\nLet us recall an important result from~\\cite{LunMal14Dokl,LunMal16JMAA}\nand~\\cite{SavShk14} concerning asymptotic behavior of the eigenvalues.\n\\begin{proposition}[Proposition 4.7 in~\\cite{LunMal16JMAA}]\n\\label{prop:Delta.regular.basic}\nLet $Q \\in \\LL{1}$ and let boundary conditions~\\eqref{eq:cond} be regular. Then the sequence $\\L = \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ of zeros of $\\Delta_Q(\\cdot)$ can be ordered in such a way that the following asymptotic formula holds\n\\begin{equation} \\label{eq:l.n=l.n0+o(1)}\n \\l_n = \\l_n^0 + o(1), \\quad\\text{as}\\quad n \\to \\infty, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\n\\end{proposition}\nTo define strictly regular boundary conditions we need the following definition.\n\\begin{definition} \\label{def:sequences}\n\\textbf{(i)} A sequence $\\L := \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ of complex numbers is\nsaid to be \\textbf{separated} if for some positive $\\tau > 0,$\n\\begin{equation} \\label{separ_cond}\n |\\l_j - \\l_k| > 2 \\tau \\quad \\text{whenever}\\quad j \\ne k.\n\\end{equation}\nIn particular, all entries of a separated sequence are distinct.\n\n\\textbf{(ii)} The sequence $\\L$ is said to be \\textbf{asymptotically\nseparated} if for some $N \\in \\mathbb{N}$ the subsequence $\\{\\l_n\\}_{|n| > N}$ is\nseparated.\n\\end{definition}\nLet us recall a notion of strictly regular boundary conditions.\n\\begin{definition} \\label{def:strictly.regular}\nBoundary conditions~\\eqref{eq:cond} are called \\textbf{strictly regular}, if they\nare regular, i.e. $J_{14} J_{32} \\ne 0$, and the sequence of zeros $\\l_0 =\n\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the characteristic determinant $\\Delta_0(\\cdot)$ is\nasymptotically separated. In particular, there exists $n_0$ such that zeros\n$\\{\\l_n^0\\}_{|n| > n_0}$ are geometrically and algebraically simple.\n\\end{definition}\nIt follows from Proposition~\\ref{prop:Delta.regular.basic} that the sequence $\\L = \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ of zeros of $\\Delta_Q(\\cdot)$ is asymptotically separated if the boundary conditions are strictly regular.\n\nAssuming boundary conditions~\\eqref{eq:cond} to be regular, let us rewrite them in\na more convenient form. Since $J_{14} \\ne 0$, the inverse matrix $A_{14}^{-1}$\nexists. Therefore writing down boundary conditions~\\eqref{eq:cond} as the vector\nequation $\\binom{U_1(y)}{U_2(y)} = 0$ and multiplying it by the matrix\n$A_{14}^{-1}$ we transform these conditions as follows\n\\begin{equation} \\label{eq:cond.canon}\n\\begin{cases}\n \\widehat{U}_{1}(y) = y_1(0) + b y_2(0) + a y_1(1) = 0, \\\\\n \\widehat{U}_{2}(y) = d y_2(0) + c y_1(1) + y_2(1) = 0,\n\\end{cases}\n\\end{equation}\nwith some $a,b,c,d \\in \\mathbb{C}$. Now $J_{14} = 1$ and the boundary conditions\n~\\eqref{eq:cond.canon} are regular if and only if $J_{32} = ad-bc \\ne 0$. Thus, the\ncharacteristic determinants $\\Delta_0(\\cdot)$ and $\\Delta(\\cdot)$ take the form\n\\begin{align}\n\\label{eq:Delta0.new}\n \\Delta_0(\\l) &= d + a e^{i (b_1+b_2) \\l} + (ad-bc) e^{i b_1 \\l}\n + e^{i b_2 \\l}, \\\\\n\\label{eq:Delta.new}\n \\Delta(\\l) &= d + a e^{i (b_1+b_2) \\l} + (ad-bc) \\varphi_{11}(\\l)\n + \\varphi_{22}(\\l) + c \\varphi_{12}(\\l) + b \\varphi_{21}(\\l).\n\\end{align}\n\\begin{remark} \\label{rem:cond.examples}\nLet us list some types of \\emph{strictly regular} boundary\nconditions~\\eqref{eq:cond.canon}. In all of these cases except 4b the set of zeros\nof $\\Delta_0$ is a union of finite number of arithmetic progressions.\n\n\\begin{enumerate}\n\n\\item Regular boundary conditions~\\eqref{eq:cond.canon} for Dirac operator ($-b_1 = b_2 = 1$) are\nstrictly regular if and only if $(a-d)^2 \\ne -4bc$.\n\n\\item Separated boundary conditions ($a=d=0$, $bc \\ne 0$) are always strictly regular.\n\n\\item Let $b_2 \/ b_1 \\in \\mathbb{Q}$, i.e. $b_1 = -n_1 b_0$, $b_2 = n_2 b_0$, $n_1, n_2 \\in \\mathbb{N}$, $b_0 > 0$ and $\\gcd(n_1,n_2)=1$. Since $ad \\ne bc$, $\\Delta_0(\\cdot) e^{-i b_1 \\l}$ is a polynomial in $e^{i b_0 \\l}$ of degree $n_1 + n_2$ with non-zero roots. Hence, boundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if this polynomial does not have multiple roots. Let us list some cases with explicit conditions.\n\n\\begin{enumerate}\n\n\\item~\\cite[Lemma 5.3]{LunMal16JMAA} Let $ad \\ne 0$ and $bc=0$. Then\nboundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:bc=0.crit.rat}\n b_1 \\ln |d| + b_2 \\ln |a| \\ne 0 \\quad\\text{or}\\quad\n n_1 \\arg(-d) - n_2 \\arg(-a) \\notin 2 \\pi \\mathbb{Z}.\n\\end{equation}\n\n\\item In particular, antiperiodic boundary conditions ($a=d=1$, $b=c=0$) are strictly regular if\nand only if $n_1 - n_2$ is odd. Note that these boundary conditions are not strictly regular in\nthe case of a Dirac system.\n\n\\item~\\cite[Proposition 5.6]{LunMal16JMAA} Let $a=0$, $bc \\ne 0$. Then\nboundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:a=0.crit.rat}\n n_1^{n_1} n_2^{n_2} (-d)^{n_1 + n_2} \\ne (n_1 + n_2)^{n_1 + n_2} (-b c)^{n_2}.\n\\end{equation}\n\n\\end{enumerate}\n\n\\item Let $\\alpha := -b_1 \/ b_2 \\notin \\mathbb{Q}$. Then the problem of strict regularity of boundary conditions is generally much more complicated. Let us list some known cases:\n\n\\begin{enumerate}\n\n\\item~\\cite[Lemma 5.3]{LunMal16JMAA} Let $ad \\ne 0$ and $bc=0$. Then\nboundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:bc=0.crit.irrat}\n b_1 \\ln |d| + b_2 \\ln |a| \\ne 0.\n\\end{equation}\n\n\\item~\\cite[Proposition 5.6]{LunMal16JMAA} Let $a=0$ and $bc, d \\in \\mathbb{R}\n\\setminus \\{0\\}$. Then boundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:a=0.crit}\n d \\ne -(\\alpha+1)\\(|bc| \\alpha^{-\\alpha}\\)^{\\frac{1}{\\alpha+1}}.\n\\end{equation}\n\n\\end{enumerate}\n\n\\end{enumerate}\n\\end{remark}\nIt is well-known that the biorthogonal system to the system of root vectors of the operator $L_U(Q)$ coincides with the system of root vectors of the adjoint operator $L_U^*(Q) := (L_U(Q))^*$ after proper normalization. In this connection we give the explicit form of the operator $L_U(Q)^*$ in the case of boundary conditions~\\eqref{eq:cond.canon}.\n\\begin{lemma}\n\\label{lem:adjoint}\nLet $L_{U}(Q)$ be an operator corresponding to the problem~\\eqref{eq:system}, \\eqref{eq:cond.canon}. Then the adjoint operator $L_U^*(Q)$ is given by the differential expression~\\eqref{eq:system} with $Q^*(x) = \\begin{pmatrix} 0 & \\overline{Q_{21}(x)} \\\\ \\overline{Q_{12}(x)} & 0 \\end{pmatrix}$ instead of $Q$ and the boundary conditions\n\\begin{equation} \\label{eq:cond*}\n\\begin{cases}\n U_{*1}(y) = \\overline{a} y_1(0) + y_1(1) + \\beta^{-1} \\overline{c} y_2(1) &= 0, \\\\\n U_{*2}(y) = \\beta \\overline{b} y_1(0) + y_2(0) + \\overline{d} y_2(1) &= 0,\n\\end{cases}\n\\end{equation}\nwhere as before $\\beta = - b_2\/b_1 > 0$. I.e. $L_U^*(Q) = L_{U*}(Q^*)$. Moreover, boundary conditions~\\eqref{eq:cond*} are regular (strictly regular) simultaneously with boundary conditions~\\eqref{eq:cond.canon}.\n\\end{lemma}\n\\begin{corollary} \\label{cor:sa.crit}\nThe operator $L_U(0)$ corresponding to the problem~\\eqref{eq:system},~\\eqref{eq:cond.canon} with $Q=0$ is selfadjoint if and only if\n\\begin{equation} \\label{eq:abcd.sa2}\na = \\overline{d} u, \\quad d = \\overline{a} u, \\quad b = -\\beta^{-1} \\overline{c} u,\n\\quad c = -\\beta \\overline{b} u, \\qquad u := ad-bc \\ne 0,\n\\end{equation}\nwhich in turn is equivalent to~\\eqref{eq:abcd.sa.intro}.\n\\end{corollary}\n\\begin{proof}\nBoundary conditions~\\eqref{eq:cond.canon} and~\\eqref{eq:cond*} can be rewriten in a matrix form as\n\\begin{equation}\n\\binom{y_1(0)}{y_2(1)} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\n\\binom{y_1(1)}{y_2(0)} = 0 \\quad \\text{and} \\quad\n\\begin{pmatrix} \\overline{a} & \\beta^{-1} \\overline{c} \\\\ \\beta \\overline{b} & \\overline{d} \\end{pmatrix} \\binom{y_1(0)}{y_2(1)} +\n\\binom{y_1(1)}{y_2(0)} = 0,\n\\end{equation}\nrespectively. Hence boundary conditions~\\eqref{eq:cond.canon} and~\\eqref{eq:cond*} are equivalent if and only if\n\\begin{equation}\n\\begin{pmatrix} \\overline{a} & \\beta^{-1} \\overline{c} \\\\ \\beta \\overline{b} & \\overline{d} \\end{pmatrix}  = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^{-1} =\n\\frac{1}{ad-bc}\\begin{pmatrix} d & -b \\\\ -c & a \\end{pmatrix} =\n\\frac{1}{u}\\begin{pmatrix} d & -b \\\\ -c & a \\end{pmatrix},\n\\end{equation}\nwhich is equivalent to~\\eqref{eq:abcd.sa2}.\n\nOn the other hand we can rewrite condtions~\\eqref{eq:cond.canon} as\n\\begin{equation} \\label{eq:Cy0+Dy1}\nC y(0) + D y(1) = 0, \\qquad\nC = \\begin{pmatrix} 1 & b \\\\ 0 & d \\end{pmatrix}, \\qquad\nD = \\begin{pmatrix} a & 0 \\\\ c & 1 \\end{pmatrix}.\n\\end{equation}\nAccording to~\\cite[Lemma 5.1]{LunMal14IEOT}\noperator $L_U(0)$ with boundary conditions rewritten as~\\eqref{eq:Cy0+Dy1} is selfadjoint if and only if $C B C^* = D B D^*$. Straightforward calculations show that\n\\begin{align}\n\\label{eq:CBC*}\n b_1^{-1} C B C^* = b_1^{-1} \\begin{pmatrix} b_1 + b_2 |b|^2 &\n b_2 b  \\overline{d} \\\\ b_2 \\overline{b} d & b_2 |d|^2 \\end{pmatrix} &=\n \\begin{pmatrix} 1 - \\beta |b|^2 & -\\beta b \\overline{d} \\\\ -\\beta \\overline{b} d &\n -\\beta |d|^2 \\end{pmatrix}, \\\\\n\\label{eq:DBD*}\n b_1^{-1} D B D^* = b_1^{-1} \\begin{pmatrix} b_1 |a|^2 & b_1 a \\overline{c} \\\\\n b_1 \\overline{a} c & b_1 |c|^2 + b_2 \\end{pmatrix} &= \\begin{pmatrix}\n |a|^2 & a \\overline{c} \\\\ \\overline{a} c & |c|^2 - \\beta \\end{pmatrix}.\n\\end{align}\nHence $C B C^* = D B D^*$ is equivalent\nto the condition~\\eqref{eq:abcd.sa.intro}. It is interesting to note that establishing equivalence of~\\eqref{eq:abcd.sa.intro} and~\\eqref{eq:abcd.sa2} directly is somewhat tedious.\n\\end{proof}\n\\section{Properties of the spectrum of the unperturbed operator} \\label{sec:unperturb}\nIn this section we obtain some properties of the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the characteristic determinant $\\Delta_0(\\cdot)$ in the case of regular boundary conditions~\\eqref{eq:cond.canon} that will be needed in Section~\\ref{sec:bari.c0} to study Bari $c_0$-property of the system of root vectors of the operator $L_U(0)$ (see Definition~\\ref{def:bari.c0}).\nRecall that $x_n \\asymp y_n$, $n \\in \\mathbb{Z}$, means that there exists $C_2 > C_1 > 0$ such that $C_1 |y_n| \\le |x_n| \\le C_2 |y_n|$, $n \\in \\mathbb{Z}$. We start the following simple property of zeros of $\\Delta_0(\\cdot)$.\n\\begin{lemma} \\label{lem:ln0.exp.asymp}\nLet boundary conditions~\\eqref{eq:cond.canon} be regular and $\\L_0 := \\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of zeros of $\\Delta_0(\\cdot)$ counting multiplicity. Set\n\\begin{equation} \\label{eq:ekn.def}\n e_{1n} := e_{1,n} := e^{i b_1 \\l_n^0}, \\qquad\n e_{2n} := e_{2,n} := e^{-i b_2 \\l_n^0},\n \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\n\n\\textbf{(i)} Let $bc \\ne 0$. Then\n\\begin{equation} \\label{eq:1+ae1.1+de2}\n 1 + a e_{1n} \\asymp 1,\n \\qquad 1 + d e_{2n} \\asymp 1, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\n\n\\textbf{(ii)} Let boundary conditions~\\eqref{eq:cond.canon} be strictly regular. Then\n\\begin{equation} \\label{eq:|1+de|+|1+ae|.asymp.1}\n \\abs{1 + a e_{1n}}^2 + \\abs{1 + d e_{2n}}^2\n \\asymp 1, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nNote that\n\\begin{equation}\n \\Delta_0(\\l)\n = \\(1 + a e^{i b_1 \\l}\\) \\(d + e^{i b_2 \\l}\\) - b c \\cdot e^{i b_1 \\l}\n = e^{i b_2\\l} \\(1 + a e^{i b_1 \\l}\\) \\(1 + d e^{-i b_2\\l}\\)\n - b c \\cdot e^{i b_1 \\l}, \\quad \\l \\in \\mathbb{C}.\n\\end{equation}\nSince $\\Delta(\\l_n^0) = 0$, $n \\in \\mathbb{Z}$, then with account of notation~\\eqref{eq:ekn.def} we have\n\\begin{equation} \\label{eq:Delta_0_in_roots}\n \\(1 + a e_{1n}\\) \\(1 + d e_{2n}\\)\n = b c e_{1n} e_{2n}, \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nAccording to Proposition~\\ref{prop:sine.type} the relation~\\eqref{eq:ln.in.Pih} holds.\nHence\n\\begin{equation} \\label{eq:e.bj.ln.asymp.1}\n e_{jn} \\asymp 1, \\qquad n \\in \\mathbb{Z}, \\quad j \\in \\{1, 2\\}.\n\\end{equation}\n\n\\textbf{(i)} Since $bc \\ne 0$, then combining~\\eqref{eq:Delta_0_in_roots} with~\\eqref{eq:e.bj.ln.asymp.1} yields the following estimate with some $C_3 > C_2 > C_1 > 0$,\n\\begin{equation} \\label{eq:||+||>=bc.e}\n C_3 > C_2 \\abs{1 + a e_{1n}}\n \\ge \\abs{(1 + a e_{1n})(1 + d e_{2n})}\n = 2 |bc| \\cdot \\abs{e_{1n} e_{2n}} > C_1, \\quad |n| \\in \\mathbb{Z},\n\\end{equation}\nwhich proves the first relation in~\\eqref{eq:1+ae1.1+de2}. The second relation is proved similarly.\n\n\\textbf{(ii)} If $bc \\ne 0$ then~\\eqref{eq:|1+de|+|1+ae|.asymp.1} is implied by~\\eqref{eq:1+ae1.1+de2}. Let $b c = 0$. In this case $a d \\ne 0$ and $\\Delta_0(\\l) = e^{i b_2\\l} \\(1 + a e^{i b_1 \\l}\\) \\(1 + d e^{-i b_2\\l}\\)$. It is clear that $\\L_0 = \\L_0^1 \\cup \\L_0^2$, where $\\L_0^1 = \\{\\l_{1,n}^{0}\\}_{n \\in \\mathbb{Z}}$ and $\\L_0^2 = \\{\\l_{2,n}^{0}\\}_{n \\in \\mathbb{Z}}$ are the sequences of zeros of the first and second factor, respectively. Clearly, these sequences constitute arithmetic progressions lying on the lines, parallel to the real axis. More precisely,\n\\begin{equation} \\label{eq:l1n.l2n.bc=0}\n \\l_{1,n}^{0} =\n \\frac{\\arg(-a^{-1}) + 2 \\pi n}{b_1} + i\\frac{\\ln|a|}{b_1},\n \\qquad\n \\l_{2,n}^{0} = \\frac{\\arg(-d) + 2 \\pi n}{b_2} - i\\frac{\\ln|d|}{b_2},\n\\end{equation}\nfor $n \\in \\mathbb{Z}$. Since boundary conditions~\\eqref{eq:cond.canon} are strictly regular, then the union of these arithmetic progressions $\\L_0 = \\L_0^1 \\cup \\L_0^2$ is asymptotically separated. It is easily seen that, in fact, $\\L_0$ is separated: if $b_2\/b_1 \\in \\mathbb{Q}$ then $\\L_0$ is periodic and if $b_2\/b_1 \\notin \\mathbb{Q}$ then arithmetic progressions $\\L_0^1$ and $\\L_0^2$ necessarily lie on different parallel lines.\nThis implies the following asymptotic relations:\n\\begin{equation} \\label{eq:1+de.1+ae.asymp.1}\n 1 + d e^{-i b_2 \\l_{2,n}^0} \\asymp 1, \\quad\n 1 + a e^{i b_1 \\l_{1,n}^0} \\asymp 1, \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nSince $\\L_0 = \\L_0^1 \\cup \\L_0^2$, relations~\\eqref{eq:1+de.1+ae.asymp.1} trivially imply~\\eqref{eq:|1+de|+|1+ae|.asymp.1}.\n\\end{proof}\nThroughout the rest of the section we will denote by $\\fr{x} := x - \\floor{x}$ the fractional part of $x \\in \\mathbb{R}$. To treat the tricky case of $\\beta = -b_2\/b_1 \\notin \\mathbb{Q}$, we need Weyl's equidistribution theorem (see~\\cite[Theorem 4.2.2.1]{SteinShak03}). More precisely, we need the following its consequence.\n\\begin{lemma} \\label{lem:weyl}\nLet $\\beta \\in \\mathbb{R} \\setminus \\mathbb{Q}$ and $0 \\le a < b \\le 1$. Then for any $\\varepsilon>0$ there exists $M_{a,b,\\varepsilon} > 0$ such that for $M \\in \\mathbb{N}$ we have:\n\\begin{equation}\n \\card\\{m \\in \\{-M, \\ldots, M\\} : \\fr{\\beta m} \\in [a,b]\\}\n \\le 2 (b - a + \\varepsilon) M, \\qquad M \\ge M_{a,b,\\varepsilon}.\n\\end{equation}\n\\end{lemma}\n\nFirst, let us recall some simple properties of the sequences that have a finite set of limit points. For brevity we denote the cardinality of the limit points set of a bounded sequence $\\{z_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ as $\\nlim\\{z_n\\}_{n \\in \\mathbb{Z}}$,\n\\begin{multline}\n \\nlim\\curl{z_n}_{n \\in \\mathbb{Z}} := \\card\\left\\{z \\in \\mathbb{C} :\n \\lim_{k \\to \\infty} z_{n_k} = z \\right. \\\\\n \\left. \\text{for some} \\ \\ \\{n_k\\}_{k \\in \\mathbb{N}} \\subset \\mathbb{Z} \\ \\ \\text{such that} \\ \\ n_j \\ne n_k\n \\ \\ \\text{for} \\ \\ j \\ne k \\right\\}.\n\\end{multline}\nIf the set of limit points is infinite we set $\\nlim\\curl{z_n}_{n \\in \\mathbb{Z}} := \\infty$.\n\\begin{lemma} \\label{lem:limit}\nThe following statements hold:\n\n\\begin{enumerate}\n\\item[(i)] Let $\\{a_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ be bounded, $f$ be continuous on $\\cup_{|n| > N} \\overline{\\mathbb{D}_{\\varepsilon}(a_n)}$ for some $\\varepsilon>0$ and $N > 0$, and $\\nlim \\{a_n\\}_{n \\in \\mathbb{Z}} = m \\in \\mathbb{N}$. Then $\\nlim \\{f(a_n)\\}_{n \\in \\mathbb{Z}} \\le m$.\n\\item[(ii)] Let $\\{a_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ and $\\{b_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ be bounded sequences and let $\\nlim \\{a_n\\}_{n \\in \\mathbb{Z}} = m_a$ and $\\nlim \\{b_n\\}_{n \\in \\mathbb{Z}} = m_b \\in \\mathbb{N}$. Then $\\nlim \\{a_n + b_n\\}_{n \\in \\mathbb{Z}} \\le m_a m_b$ and $\\nlim \\{a_n b_n\\}_{n \\in \\mathbb{Z}} \\le m_a m_b$.\n\\item[(iii)] Let $y_n \\in [0, 1)$, $n \\in \\mathbb{Z}$, and let $\\nlim \\{\\sin (2 \\pi y_n)\\}_{n \\in \\mathbb{Z}} = m \\in \\mathbb{N}$. Then $\\nlim \\{y_n\\}_{n \\in \\mathbb{Z}} \\le 2m+1$.\n\\item[(iv)] Let $a, b \\in \\mathbb{R}$, $\\{x_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{R}$ be bounded and $\\nlim \\{x_n\\}_{n \\in \\mathbb{Z}} = m \\in \\mathbb{N}$. Then $$\\nlim\\{\\fr{a x_n + b}\\}_{n \\in \\mathbb{Z}} \\le m+1.$$\n\\end{enumerate}\n\\end{lemma}\nThe following result of Diophantine approximation nature plays crucial role in treating the tricky case of $b_2\/b_1 \\notin \\mathbb{Q}$.\n\\begin{lemma} \\label{lem:sin.weyl}\nLet $b_1,b_2 \\in \\mathbb{R} \\setminus \\{0\\}$ and $b_2\/b_1 \\notin \\mathbb{Q}$. Further, let $\\{\\alpha_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{R}$ be an incompressible sequence such that\n\\begin{equation} \\label{eq:card.alpn>}\n\\card\\{n \\in \\mathbb{Z} : |\\alpha_n| \\le M\\} \\ge \\gamma M,\n\\qquad M \\ge M_0,\n\\end{equation}\nfor some $\\gamma, M_0 > 0$.\nThen one of the sequences $\\{\\sin(b_1 \\alpha_n)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\sin(b_2 \\alpha_n)\\}_{n \\in \\mathbb{Z}}$\nhas an infinite set of limit points.\n\\end{lemma}\n\\begin{proof}\nAssume the contrary. Namely, let\n$$\n\\nlim \\curl{\\sin(b_1 \\alpha_n)}_{n \\in \\mathbb{Z}} = m_1 \\in \\mathbb{N}\n\\qquad\\text{and}\\qquad\n\\nlim \\curl{\\sin(b_2 \\alpha_n)}_{n \\in \\mathbb{Z}} = m_2 \\in \\mathbb{N}.\n$$\nLet us set\n\\begin{equation} \\label{eq:psi1}\n b_1 \\alpha_n = 2 \\pi (k_n + \\delta_n), \\qquad\n k_n := \\floor{\\frac{b_1 \\alpha_n}{2 \\pi}} \\in \\mathbb{Z}, \\quad\n \\delta_n = \\fr{\\frac{b_1 \\alpha_n}{2 \\pi}} \\in [0, 1).\n\\end{equation}\nIt is clear that $\\sin (2 \\pi \\delta_n) = \\sin(b_1 \\alpha_n)$. Hence by Lemma~\\ref{lem:limit}(iii)\n\\begin{equation} \\label{eq:nlim.deltan}\n \\nlim \\{\\delta_n\\}_{n \\in \\mathbb{Z}} \\le 2m_1+1.\n\\end{equation}\nIt is clear from~\\eqref{eq:psi1} that\n$$\nb_2 \\alpha_n = 2 \\pi \\( \\beta k_n + \\beta \\delta_n\\),\n\\quad n \\in \\mathbb{Z}, \\qquad \\beta := b_2\/b_1 \\notin \\mathbb{Q}.\n$$\nThe same reasoning as above shows that\n$$\n\\nlim \\curl{u_n}_{n \\in \\mathbb{Z}} \\le 2m_2+1, \\qquad\nu_n := \\fr{\\beta k_n + \\beta \\delta_n}.\n$$\nFurther, combining~\\eqref{eq:nlim.deltan} with by Lemma~\\ref{lem:limit}(iv) implies that\n\\begin{equation}\n\\nlim\\curl{v_n}_{n \\in \\mathbb{Z}} \\le 2m_1+2, \\quad\\text{where}\\quad\nv_n := \\fr{\\beta \\delta_n}, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\nFinally, note that $\\fr{\\beta k_n} = \\fr{u_n - v_n}$, $n \\in \\mathbb{Z}$. Hence by parts (ii) and (iv) of Lemma~\\ref{lem:limit}\nthe sequence $\\curl{\\fr{\\beta k_n}}_{n \\in \\mathbb{Z}}$ has exactly $p \\le (2m_2+1)(2m_1+2)+1$ limit points $0 \\le x_1 < \\ldots < x_p \\le 1$.\n\nLet $\\varepsilon > 0$ be fixed. Then there exists $N_{\\varepsilon} \\in \\mathbb{N}$ such that\n\\begin{equation} \\label{eq:beta.kn.in.Ieps}\n \\fr{\\beta k_n} \\in \\mathcal{I}_{\\varepsilon} := [0,1) \\cap\n \\bigcup_{j=1}^p (x_j-\\varepsilon, x_j+\\varepsilon), \\qquad |n| \\ge N_{\\varepsilon}.\n\\end{equation}\nSince $\\beta \\notin \\mathbb{Q}$, Lemma~\\ref{lem:weyl} implies that\n\\begin{align} \\label{eq:card.JepsM}\n \\card(\\mathcal{J}_{\\varepsilon,M}) & \\le 6 p \\varepsilon M,\n \\qquad M \\ge M_{\\varepsilon}, \\quad M \\in \\mathbb{N}, \\quad\\text{where} \\\\\n\\label{eq:JepsM.def}\n \\mathcal{J}_{\\varepsilon,M} & :=\n \\{m \\in \\{-M, \\ldots, M\\} : \\fr{\\beta m} \\in \\mathcal{I}_{\\varepsilon}\\},\n \\quad M \\in \\mathbb{N},\n\\end{align}\nFor $M_{\\varepsilon} := \\max\\bigl\\{M_{x_j-\\varepsilon,x_j+\\varepsilon,\\varepsilon} : j \\in \\{1,\\ldots, p\\}\\bigr\\}$.\n\nLet $M \\in \\mathbb{N}$ and consider the set\n$$\n\\mathcal{K}_{\\varepsilon,M} := \\curl{|n| \\ge N_{\\varepsilon} : |k_n| \\le M} \\subset \\mathbb{Z},\n$$\nIt is clear from~\\eqref{eq:psi1} and inequality $|[x]| < |x|+1$ that\n$$\n\\mathcal{K}_{\\varepsilon,M} \\supset \\curl{|n| \\ge N_{\\varepsilon} : |\\alpha_n| \\le \\widetilde{M}},\n\\qquad \\widetilde{M} := \\frac{2 \\pi (M-1)}{|b_1|}.\n$$\nHence if $\\widetilde{M} \\ge M_0$ condition~\\eqref{eq:card.alpn>} implies that\n\\begin{equation} \\label{eq:card.KepsM>}\n \\card(\\mathcal{K}_{\\varepsilon,M}) \\ge \\gamma \\widetilde{M} - 2 N_{\\varepsilon} + 1 \\ge \\gamma_1 M, \\qquad M \\ge \\widetilde{M}_{\\varepsilon},\n\\end{equation}\nwith $\\gamma_1 := \\pi \\gamma |b_1^{-1}| > 0$ and some $\\widetilde{M}_{\\varepsilon} \\ge M_{\\varepsilon}$.\nCondition~\\eqref{eq:beta.kn.in.Ieps} and definition~\\eqref{eq:JepsM.def} of $\\mathcal{J}_{\\varepsilon,M}$ imply that for $n \\in \\mathcal{K}_{\\varepsilon,M}$ we have $k_n \\in \\mathcal{J}_{\\varepsilon,M}$. Since $\\curl{\\alpha_n}_{n \\in \\mathbb{Z}}$ is incompressible then so is $\\curl{k_n}_{n \\in \\mathbb{Z}}$. Hence multiplicities $d_m := \\card\\curl{n \\in \\mathbb{Z} : k_n = m}$ are bounded, $d_m \\le d$, $m \\in \\mathbb{Z}$, for some $d \\in \\mathbb{N}$. Hence for every $m \\in \\mathcal{J}_{\\varepsilon,M}$ there are at most $d$ values of $n \\in \\mathcal{K}_{\\varepsilon,M}$ for which $k_n = m$. Combining this observation with the estimate~\\eqref{eq:card.JepsM} we arrive at\n\\begin{equation} \\label{eq:card.KepsM<}\n \\card\\(\\mathcal{K}_{\\varepsilon,M}\\) \\le d \\card\\(\\mathcal{J}_{\\varepsilon,M}\\) \\le 6 d p \\varepsilon M.\n\\end{equation}\nNow picking $\\varepsilon > 0$ such that that $6 d p \\varepsilon < \\gamma_1$ and $M > \\widetilde{M}_{\\varepsilon}$ we see that cardinality estimates~\\eqref{eq:card.KepsM>} and~\\eqref{eq:card.KepsM<} contradict to each other, which finishes the proof.\n\\end{proof}\n\\begin{remark}\nIt is clear from the proof of Lemma~\\ref{lem:sin.weyl} that the statement remains valid if we relax condition~\\eqref{eq:card.alpn>} to only hold for $M \\in \\mathcal{M} \\subset \\mathbb{N}$, where $\\mathcal{M}$ is some fixed unbounded subset of $\\mathbb{N}$.\n\\end{remark}\nTo apply Lemma~\\ref{lem:sin.weyl} we first need to establish property~\\ref{lem:sin.weyl} for the sequence $\\{\\Re \\l_n^0\\}_{n \\in \\mathbb{Z}}$. It easily follows from the asymptotic formula~\\eqref{eq:lam.n=an+o1}.\n\\begin{lemma} \\label{lem:density}\nLet the boundary conditions~\\eqref{eq:cond} be regular. Then for every $\\varepsilon > 0$ there exists $N_{\\varepsilon} > 0$ such that\n\\begin{equation} \\label{eq:card.Re.ln0}\n\\card \\left\\{ n \\in \\mathbb{Z} : |\\Re \\l_n^0| \\le N \\right\\} \\ge \\frac{N}{\\sigma + \\varepsilon}, \\quad N \\ge N_{\\varepsilon},\n\\qquad \\sigma := \\frac{\\pi}{b_2-b_1} > 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nAsymptotic formula~\\eqref{eq:lam.n=an+o1} for $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ implies that $|\\Re \\l_n^0| \\le (2 \\sigma + \\varepsilon) |n|$, $|n| \\ge n_{\\varepsilon}$, for some $n_{\\varepsilon} \\in \\mathbb{N}$. Hence\n$$\n\\mathbb{Z} \\cap \\(\\[-\\frac{N}{2\\sigma + \\varepsilon}, -n_{\\varepsilon}\\] \\cup \\[n_{\\varepsilon}, \\frac{N}{2\\sigma + \\varepsilon}\\]\\) \\subset \\mathcal{I}_N := \\left\\{ n \\in \\mathbb{Z} : |\\Re \\l_n^0| \\le N \\right\\},\n$$\nfor $N \\ge (2\\sigma+\\varepsilon) n_{\\varepsilon}$. Taking cardinalities in this inclusion implies\n\\begin{equation}\n\\card \\mathcal{I}_N \\ge 2 \\(\\floor{\\frac{N}{2\\sigma + \\varepsilon}} - n_{\\varepsilon} + 1\\)\n\\ge \\frac{N}{\\sigma + \\varepsilon\/2} - 2 n_{\\varepsilon} \\ge \\frac{N}{\\sigma + \\varepsilon},\n\\qquad N \\ge N_{\\varepsilon},\n\\end{equation}\nwith $N_{\\varepsilon} := 2(\\sigma\/\\varepsilon+1)(2\\sigma+\\varepsilon) n_{\\varepsilon}$.\n\\end{proof}\nCombining two previous results leads to the following important property of zeros of characteristic determinant $\\Delta_0(\\cdot)$, which coincides with Proposition~\\ref{prop:nlim.inf.intro} and is formulated again for reader's convenient.\n\\begin{proposition} \\label{prop:nlim.inf}\nLet $b_2\/b_1 \\notin \\mathbb{Q}$ and boundary conditions~\\eqref{eq:cond.canon.intro} be regular, i.e. $u := ad-bc \\ne 0$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of zeros of the characteristic determinant $\\Delta_0(\\cdot)$ counting multiplicity. Then each of the sequences $\\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\exp(i b_2 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points.\n\\end{proposition}\n\\begin{proof}\n\\textbf{(i)} First, let $bc=0$. Then according to the proof of Lemma~\\ref{lem:ln0.exp.asymp}, zeros of the characteristic determinant $\\Delta_0(\\cdot)$ are simple and split into two separated arithmetic progressions $\\L_0^1 = \\{\\l_{1,n}^{0}\\}_{n \\in \\mathbb{Z}}$ and $\\L_0^2 = \\{\\l_{2,n}^{0}\\}_{n \\in \\mathbb{Z}}$ given by~\\eqref{eq:l1n.l2n.bc=0}. Let $k \\in \\{1,2\\}$ and $j=2\/k$. Since $E(z) = e^{2 \\pi i z}$ is periodic with period 1, we have for $n \\in \\mathbb{Z}$,\n\\begin{equation}\n\\exp(i b_k \\l_{j,n}^0) = \\exp\\(2 \\pi i n b_k\/b_j + \\omega_{k,j,a,d}\\)\n= \\exp\\(2 \\pi i \\fr{n b_k\/b_j} + \\omega_{k,j,a,d}\\),\n\\end{equation}\nwhere $\\omega_{k,j,a,d}$ is an explicit constant that can be derived from~\\eqref{eq:l1n.l2n.bc=0}. Since $b_k\/b_j \\notin \\mathbb{Q}$, then by the classical Kronecker theorem, the sequence $\\curl{\\fr{n b_k\/b_j}}_{n \\in \\mathbb{Z}}$ is everywhere dense on $[0,1]$. This implies that the sequence $\\{\\exp(i b_k \\l_{j,n}^0)\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points, which finishes the proof in this case.\n\n\\textbf{(ii)} Now, let $bc \\ne 0$ and assume the contrary: one of the sequences $\\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\exp(i b_2 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ has a finite set of limit points. For definiteness assume that\n\\begin{equation} \\label{eq:nlim.e1n}\n \\nlim \\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}} = m_1 \\in \\mathbb{N}.\n\\end{equation}\nRecall that $e_{1n} := \\exp(i b_1 \\l_n^0)$ and $e_{2n} := \\exp(-i b_2 \\l_n^0)$, $n \\in \\mathbb{Z}$, and also set\n\\begin{equation} \\label{eq:ln0=an+ibn}\n \\l_n^0 = \\alpha_n + i \\beta_n, \\qquad \\alpha_n := \\Re \\l_n^0,\n \\quad \\beta_n := \\Im \\l_n^0, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nIt is clear that\n\\begin{equation} \\label{eq:abs.e1n}\n |e_{1n}| = |\\exp(i b_1 \\l_n^0)| = \\exp(-b_1 \\Im \\l_n^0)\n = \\exp(-b_1 \\beta_n), \\qquad n \\in \\mathbb{Z}\n\\end{equation}\nIt follows from~\\eqref{eq:abs.e1n}, \\eqref{eq:nlim.e1n}, \\eqref{eq:e.bj.ln.asymp.1} and Lemma~\\ref{lem:ln0.exp.asymp}(i), applied with $f_1(z) = -b_1^{-1} \\log |z|$, that\n\\begin{equation} \\label{eq:nlim.Imln0}\n \\nlim \\{\\beta_n\\}_{n \\in \\mathbb{Z}} =\n \\nlim \\{-b_1^{-1} \\log |e_{1n}| \\}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nIn turn, since $e_{1n} = |e_{1n}| e^{i b_1 \\alpha_n} \\asymp 1$, $n \\in \\mathbb{Z}$,\nthen by Lemma~\\ref{lem:limit}(i) applied with $f_2(z) = \\Im z \/ |z|$ we have\n\\begin{equation} \\label{eq:nlim.sinb1}\n \\nlim \\curl{\\sin\\(b_1 \\alpha_n\\)}_{n \\in \\mathbb{Z}} =\n \\nlim \\curl{\\Im e_{1n} \/ |e_{1n}|}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nSince $bc \\ne 0$ and boundary conditions~\\eqref{eq:cond.canon} are regular, then Lemma~\\ref{lem:ln0.exp.asymp}(i) implies~\\eqref{eq:1+ae1.1+de2}. Recall that $u := ad-bc \\ne 0$. Since $\\Delta(\\l_n^0)=0$, $1 + a e_{1n} \\ne 0$ and $1 + d e_{2n} \\ne 0$, $n \\in \\mathbb{Z}$, it follows from~\\eqref{eq:Delta0.new} that for $n \\in \\mathbb{Z}$:\n\\begin{equation} \\label{eq:e1.via.e2}\n 1 + d e_{2n} + a e_{1n} + u e_{1n} e_{2n} = 0, \\qquad\n e_{2n} = - \\frac{1 + a e_{1n}}{d + u e_{1n}}, \\qquad\n e_{1n} = - \\frac{1 + d e_{2n}}{a + u e_{2n}}.\n\\end{equation}\nSince $e_{1n} \\asymp 1$, $e_{2n} \\asymp 1$, $n \\in \\mathbb{Z}$, relations~\\eqref{eq:1+ae1.1+de2} and~\\eqref{eq:e1.via.e2} imply that\n\\begin{equation} \\label{eq:a+ue2.d+ue1}\n d + u e_{1n} \\asymp 1, \\qquad a + u e_{2n} \\asymp 1, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nHence $f_3(z) := - \\frac{1 + a z}{d + u z}$ is continuous in the neighborhood of $\\{e_{1n}\\}_{n \\in \\mathbb{Z}}$. Combining this with Lemma~\\ref{lem:limit}(i), the second identity in~\\eqref{eq:e1.via.e2} and relation~\\eqref{eq:nlim.e1n} we arrive at\n\\begin{equation} \\label{eq:nlim.e2n}\n \\nlim \\{e_{2n}\\}_{n \\in \\mathbb{Z}} = \\nlim \\{f_3(e_{1n})\\}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nSimilarly to~\\eqref{eq:nlim.sinb1} we get\n\\begin{equation} \\label{eq:nlim.sinb2}\n \\nlim \\curl{\\sin\\(b_2 \\alpha_n\\)}_{n \\in \\mathbb{Z}} =\n \\nlim \\curl{\\Im e_{2n} \/ |e_{2n}|}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nSince boundary conditions~\\eqref{eq:cond.canon} are regular then Proposition~\\ref{prop:sine.type} implies that the sequence $\\{\\alpha_n\\}_{n \\in \\mathbb{Z}}$ is incompressible and Lemma~\\ref{lem:density} implies the estimate~\\eqref{eq:card.Re.ln0}, which in turn yields the estimate~\\eqref{eq:card.alpn>} for $\\{\\alpha_n\\}_{n \\in \\mathbb{Z}}$ with $\\gamma = \\frac{1}{2\\sigma} = \\frac{b_2-b_1}{2\\pi}$. Since $b_2\/b_1 \\notin \\mathbb{Q}$ then by Lemma~\\ref{lem:sin.weyl} one of sequences $\\curl{\\sin(b_1 \\alpha_n)}_{n \\in \\mathbb{Z}}$ and $\\curl{\\sin(b_2 \\alpha_n)}_{n \\in \\mathbb{Z}}$ has infinite set of limit points. This contradicts relations~\\eqref{eq:nlim.sinb1} and~\\eqref{eq:nlim.sinb2} and finishes the proof.\n\\end{proof}\n\\section{Bari $c_0$-property of the system of root vectors of the unperturbed operator} \\label{sec:bari.c0}\n\nIn this section assuming boundary conditions~\\eqref{eq:cond.canon} to be strictly regular, we show that the system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only if the operator $L_U(0)$ is selfadjoint. Since eigenfunctions of $L_U(0)$ in their ``natural form'' are not normalized we need the following simple practical criterion of Bari $c_0$-property.\n\\begin{lemma} \\label{lem:crit.bari.not.norm}\nLet $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$ be a complete minimal system of vectors in a Hilbert space $\\mathfrak{H}$. Let also $\\{g_n\\}_{n \\in \\mathbb{Z}}$ be ``almost biorthogonal'' to $\\mathfrak{F}$. Namely, $(f_n, g_m) = 0$, $n \\ne m$, $(f_n, g_n) \\ne 0$, $n, m \\in \\mathbb{Z}$. Then the normalized system\n$$\n\\mathfrak{F}' := \\curl{f_n'}_{n \\in \\mathbb{Z}}, \\quad f_n' := \\frac{1}{\\|f_n\\|} f_n,\n\\quad n \\in \\mathbb{Z},\n$$\nis a Bari $c_0$-sequence in $\\mathfrak{H}$ (see Definition~\\ref{def:bari.c0}) if and only if\n\\begin{equation} \\label{eq:sum.fn.gn-1}\n \\frac{\\|f_n\\|_{\\mathfrak{H}} \\cdot \\|g_n\\|_{\\mathfrak{H}}}{|(f_n, g_n)_{\\mathfrak{H}}|} \\to 1\n \\quad\\text{as}\\quad n \\to \\infty.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor brevity we set $\\|\\cdot\\| := \\|\\cdot\\|_{\\mathfrak{H}}$ and $(\\cdot,\\cdot) := (\\cdot,\\cdot)_{\\mathfrak{H}}$.\nIt is clear that for the system $\\mathfrak{G}' := \\{g_n'\\}_{n \\in \\mathbb{Z}}$ that is biorthogonal to $\\mathfrak{F}'$ we have,\n\\begin{equation} \\label{eq:fn'gm'}\n (f_n', g_m') = \\delta_{nm}, \\quad n,m \\in \\mathbb{Z}; \\qquad\n g_n' = \\frac{\\|f_n\\|}{(f_n, g_n)} \\cdot g_n, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\nRelations~\\eqref{eq:fn'gm'} imply that\n\\begin{equation} \\label{eq:|fn'-gn'|}\n \\|f_n' - g_n'\\|^2 = \\|f_n'\\|^2 - (f_n', g_n') - \\overline{(f_n', g_n')}\n + \\|g_n'\\|^2 = \\|g_n'\\|^2 - 1\n = \\frac{\\|f_n\\|^2 \\cdot \\|g_n\\|^2}{|(f_n, g_n)|^2} - 1.\n\\end{equation}\nHence, systems $\\mathfrak{F}'$ and $\\mathfrak{G}'$ are $c_0$-close if and only if condition~\\eqref{eq:sum.fn.gn-1} holds.\n\\end{proof}\nThe following simple property of compact operators with asymptotically simple spectrum will be also useful in the next section.\n\\begin{lemma} \\label{lem:some.every}\nLet $T$ be an operator with compact resolvent in a separable Hilbert space $\\mathfrak{H}$ and let $\\{\\l_n\\}_{n \\in \\mathbb{Z}}$ be a sequence of its eigenvalues counting multiplicities. Let also $p \\in [1,2]$.\nAssume that for some $N \\in \\mathbb{N}$ eigenvalues $\\l_n$, $|n| \\ge N$, are algebraically simple.\nThen if some normalized system of root vectors of the operator $T$ is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$ then every normalized system of root vectors of the operator $T$ is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$\nbe a normalized system of root vectors of the operator $T$, which is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$. By definition,\nthe system $\\mathfrak{F}$ is complete and minimal in $\\mathfrak{H}$. Let $\\mathfrak{G} = \\{g_n\\}_{n \\in \\mathbb{Z}}$ be its (unique) biorthogonal system. Further, let $\\mathfrak{F}' = \\{f_n'\\}_{n \\in \\mathbb{Z}}$ be any other normalized system of root vectors of the operator $T$. Since eigenvalue $\\l_n$, $|n| \\ge N$, is algebraically simple then $\\dim(\\mathcal{R}_{\\l_n}(T)) = 1$, $|n| \\ge N$. Hence $f_n' = \\alpha_n f_n$, $|n| \\ge N$, for some $\\alpha_n \\in \\mathbb{T} := \\{z \\in \\mathbb{C} : |z|=1\\}$. It is clear that $\\mathfrak{F}'$ is also complete and minimal and for its biorthogonal sysyem $\\mathfrak{G}' = \\{g_n'\\}_{n \\in \\mathbb{Z}}$ we have that $g_n' = \\overline{\\alpha_n^{-1}} g_n = \\alpha_n g_n$, $|n| \\ge N$, since $|\\alpha_n|=1$. Hence $\\|f_n'-g_n'\\|_{\\mathfrak{H}} = \\|\\alpha_n \\cdot (f_n-g_n)\\|_{\\mathfrak{H}} = \\|f_n-g_n\\|_{\\mathfrak{H}}$, $|n| \\ge N$. This implies that $\\curl{\\|f_n'-g_n'\\|_{\\mathfrak{H}}}_{n \\in \\mathbb{Z}} = \\curl{\\|f_n-g_n\\|_{\\mathfrak{H}}}_{n \\in \\mathbb{Z}} \\in (\\ell^p)^*$ and finishes the proof.\n\\end{proof}\n\\begin{remark} \\label{rem:sa.c0.bari}\nLet $A$ be a selfadjoint operator with compact resolvent. Then every its normalized system of root vectors is an orthonormal basis in $\\LLV{2}$ and coincides with its biorthogonal sequence. This implies that every normalized system of root vectors of the operator $A$ is a Bari $c_0$-sequence.\n\\end{remark}\nTo study norms $\\|f_n^0\\|_2$ and $\\|g_n^0\\|_2$ of the eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$, we first need to obtain some properties of simple integrals $\\int_0^1 |e^{\\pm 2 i b_j \\l x}| dx$, $j \\in \\{1,2\\}$, $\\l \\in \\mathbb{C}$.\n\\begin{lemma} \\label{lem:ejx.ej.Ej}\nDenote for $j \\in \\{1,2\\}$ and $\\l \\in \\mathbb{C}$:\n\\begin{equation} \\label{eq:Ejpm.def}\n E_{j}^{\\pm}(\\l) := \\int_0^1 \\abs{e^{\\pm 2 i b_j \\l x}} dx\n = \\int_0^1 e^{\\mp 2 b_j \\Im \\l x} dx\n = \\frac{e^{\\mp 2 b_j \\Im \\l} - 1}{\\mp 2 b_j \\Im \\l}.\n\\end{equation}\nThen the following estimate holds:\n\\begin{align}\n\\label{eq:Ej+Ej->1}\n & E_j^+(\\l) E_j^-(\\l) - 1 \\ge \\frac{(b_j \\Im \\l)^2}{3},\n \\qquad j \\in \\{1,2\\}, \\quad \\l \\in \\mathbb{C}.\n\\end{align}\nIn particular, $E_j^+(\\l) E_j^-(\\l) - 1 > 0$ if $\\Im \\l \\ne 0$.\n\\end{lemma}\n\\begin{proof}\nLet $h \\ge 0$. It is clear that\n\\begin{equation} \\label{eq:Ej=f}\n E_j^{\\pm}(\\l) = f(\\mp 2 b_j \\Im \\l), \\quad\\text{where}\\quad\n f(x) := \\frac{e^x - 1}{x} = 1 + \\frac{x}{2} + O(x^2), \\quad |x| < h.\n\\end{equation}\nIt follows from Taylor expansion of $e^x$ that for $x \\in \\mathbb{R}$:\n\\begin{equation} \\label{eq:fx.f-x>1}\n f(x)f(-x) = \\frac{e^x - 1}{x} \\cdot \\frac{e^{-x} - 1}{-x}\n = \\frac{e^x + e^{-x} - 2}{x^2}\n = 2 \\sum_{k=1}^{\\infty} \\frac{x^{2k-2}}{(2k)!}\n \\ge 1 + \\frac{x^2}{12}.\n\\end{equation}\nEstimate~\\eqref{eq:Ej+Ej->1} now immediately follows from~\\eqref{eq:Ej=f} and~\\eqref{eq:fx.f-x>1}.\n\n\\end{proof}\nFirst we establish the Bari $c_0$-property criterion in a special case $b=c=0$.\n\\begin{proposition} \\label{prop:crit.bari.period}\nLet boundary conditions~\\eqref{eq:cond.canon} be strictly regular with $b=c=0$, i.e. they are of the form\n\\begin{equation} \\label{eq:quasi.per.bc}\n y_1(0) + a y_1(1) = d y_2(0) + y_2(1) = 0, \\qquad a d \\ne 0.\n\\end{equation}\nThen some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ (see Definition~\\ref{def:bari.c0}) if and only if $|a| = |d| = 1$.\n\\end{proposition}\n\\begin{proof}\n\\textbf{(i)} If $|a|=|d|=1$ (and $b=c=0$) then by Corollary~\\ref{cor:sa.crit} the operator $L_U(0)$ with boundary conditions~\\eqref{eq:quasi.per.bc} is self-adjoint. Remark~\\ref{rem:sa.c0.bari} now finishes the proof.\n\n\\textbf{(ii)} Now assume that some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$. Since boundary conditions~\\eqref{eq:quasi.per.bc} are strictly regular then by definition, eigenvalues of the operator $L_U(0)$ are asymptotically simple.\nHence by Lemma~\\ref{lem:some.every} every normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$.\n\nAccording to the proof of Lemma~\\ref{lem:ln0.exp.asymp} the eigenvalues of the operator $L_U(0)$ are simple and split into two separated arithmetic progressions $\\L_0^1 = \\{\\l_{1,n}^{0}\\}_{n \\in \\mathbb{Z}}$ and $\\L_0^2 = \\{\\l_{2,n}^{0}\\}_{n \\in \\mathbb{Z}}$ given by~\\eqref{eq:l1n.l2n.bc=0}.\nIt is easy to verify that the vectors\n\\begin{equation}\n f_{1,n}^0(x) = \\binom{e^{i b_1 \\l_{1,n}^0 x}}{0}, \\qquad\n g_{1,n}^0(x) = \\binom{e^{i b_1 \\overline{\\l_{1,n}^0} x}}{0},\n \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nare the eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$ corresponding to the eigenvalues $\\l_{1,n}^0$ and $\\overline{\\l_{1,n}^0}$, and the vectors\n\\begin{equation}\n f_{2,n}^0(x) = \\binom{0}{e^{i b_2 \\l_{2,n}^0 x}}, \\qquad\n g_{2,n}^0(x) = \\binom{0}{e^{i b_2 \\overline{\\l_{2,n}^0} x}}, \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nare the eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$  corresponding to the eigenvalues $\\l_{2,n}^0$ and $\\overline{\\l_{2,n}^0}$ respectively. It is clear that\n\\begin{equation} \\label{eq:fjn.gkm=delta}\n \\(f_{j,n}^0, g_{k,m}^0\\)_2 = \\delta_{j,n}^{k,m}, \\qquad j,k \\in \\{1,2\\},\n \\quad n,m \\in \\mathbb{Z}.\n\\end{equation}\nThus the union system $\\mathfrak{F} := \\{f_{1,n}^0\\}_{n \\in \\mathbb{Z}} \\cup \\{f_{2,n}^0\\}_{n\n\\in \\mathbb{Z}}$ is the system of root vectors of the operator $L_U(0)$ and $\\mathfrak{G} :=\n\\{g_{1,n}^0\\}_{n \\in \\mathbb{Z}} \\cup \\{g_{2,n}^0\\}_{n \\in \\mathbb{Z}}$ is biorthogonal to it. Hence normalization of the system $\\mathfrak{F}$ is a Bari $c_0$-sequence in $L^2([0,1]; \\mathbb{C}^2)$. According to Lemma~\\ref{lem:crit.bari.not.norm} we have\n\\begin{equation} \\label{eq:sumj.sumn.alp}\n \\alpha_{j,n} := \\frac{\\bigl\\|f_{j,n}^0\\bigr\\|_2 \\cdot\n \\bigl\\|g_{j,n}^0\\bigr\\|_2}{\\abs{\\bigl(f_{j,n}^0, g_{j,n}^0\\bigr)_2}}\n \\to 1 \\quad\\text{as}\\quad n \\to \\infty, \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nLet $j=1$. Then taking into account Lemma~\\ref{lem:ejx.ej.Ej} and formula~\\eqref{eq:fjn.gkm=delta} we have\n\\begin{equation} \\label{eq:alp1n>Im}\n \\alpha_{1,n}^2 = \\bigl\\|f_{1,n}^0\\bigr\\|_2^2 \\cdot\n \\bigl\\|g_{1,n}^0\\bigr\\|_2^2 = E_1^+(\\l_{1,n}^0) E_1^-(\\l_{1,n}^0)\n \\ge 1 + \\frac13 \\bigabs{b_2 \\Im \\l_{1,n}^0}^2, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nIt follows from~\\eqref{eq:l1n.l2n.bc=0} that $b_1 \\Im \\l_{1,n}^0 = \\ln|a|$. Since $\\alpha_{1,n} \\to 1$ as $n \\to \\infty$, formula~\\eqref{eq:alp1n>Im} implies that $\\ln|a|=0$, or $|a|=1$.\n\nSimilarly considering the case $j=2$ we conclude that $|d|=1$, which finishes the proof.\n\\end{proof}\nIn the following intermediate result we reduce condition~\\eqref{eq:sum.fn.gn-1} of Bari $c_0$-property of the system of root vectors of the operator $L_U(0)$ to explicit condition in terms of eigenvalues $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$.\nRecall that $\\beta = -b_2\/b_1 > 0$.\n\\begin{proposition} \\label{prop:crit.bari.b.ne.0}\nLet boundary conditions~\\eqref{eq:cond.canon} be strictly regular and let one of the parameters $b$ or $c$ in them be non-zero, $|b|+|c|>0$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of the eigenvalues of the operator $L_U(0)$ counting multiplicities. Then some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ (see Definition~\\ref{def:bari.c0}) if and only if the following conditions hold\n\\begin{equation} \\label{eq:lim1.lim2}\n |c| = \\beta |b|, \\qquad \\lim_{n \\to \\infty} \\Im \\l_n^0 = 0\n \\quad\\text{and}\\quad \\lim_{n \\to \\infty} z_n = |bc|,\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:zn.def}\n z_n := \\(1 + d e^{- i b_2 \\l_n^0}\\)\\overline{\\(1 + a e^{i b_1 \\l_n^0}\\)}.\n\\end{equation}\n\n\\end{proposition}\n\\begin{proof}\nWithout loss of generality we can assume that $b \\ne 0$. By definition of strictly regular boundary conditions there exists $n_0 \\in \\mathbb{N}$ such that eigenvalues $\\l_n^0$ of $L_U(0)$ for $|n| > n_0$ are algebraically simple and separated from each other. According to the proof of Theorem~1.1 in~\\cite{LunMal16JMAA} vector-functions $f_n^0(\\cdot)$ and $g_n^0(\\cdot)$, $|n| > n_0$, of the following form:\n\\begin{equation} \\label{eq:fn0x.gn0x}\n f_n^0(x) := \\binom{b e^{i b_1 \\l_n^0 x}}{\n - (1 + a e^{i b_1 \\l_n^0}) e^{i b_2 \\l_n^0 x}}, \\qquad\n g_n^0(x) := \\overline{\\binom{(1 + d e^{-i b_2 \\l_n^0})\n e^{-i b_1 \\l_n^0 x}}{ - \\beta b e^{-i b_2 \\l_n^0 x}}},\n\\end{equation}\nare non-zero eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$ corresponding to the eigenvalues $\\l_n^0$ and $\\overline{\\l_n^0}$ for $|n| > n_0$, respectively. Let $f_n^0(\\cdot)$ and $g_n^0(\\cdot)$ be some root vectors of operators $L_U(0)$ and $L_U^*(0)$ corresponding to the eigenvalues $\\l_n^0$ and $\\overline{\\l_n^0}$ for $|n| \\le n_0$. Clearly $\\mathfrak{F} := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ is a system of root vectors of the operator $L_U(0)$ and $\\mathfrak{G} := \\{g_n^0\\}_{n \\in \\mathbb{Z}}$ is the corresponding system for the adjoint operator $L_U^*(0)$. Let us show that normalization of $\\mathfrak{F}$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only condition~\\eqref{eq:lim1.lim2} holds. Since eigenvalues of the operator $L_U(0)$ are asymptotically simple Lemma~\\ref{lem:some.every} will imply the statement of the proposition.\nClearly, $\\mathfrak{G}$ is almost biorthogonal to $\\mathfrak{F}$. Hence Lemma~\\ref{lem:crit.bari.not.norm} implies that normalization of $\\mathfrak{F}$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only if condition~\\eqref{eq:sum.fn.gn-1} holds.\n\nSet for brevity $E_{jn}^{\\pm} := E_j^{\\pm}(\\l_n^0)$, $j \\in \\{1,2\\}$, $n \\in \\mathbb{Z}$, where $E_j^{\\pm}(\\l)$ is defined in~\\eqref{eq:Ejpm.def}. With account of this notation and notation~\\eqref{eq:ekn.def} we get after performing straightforward calculations:\n\\begin{align}\n\\label{eq:|fn|2}\n \\|f_n^0\\|_2^2 &= |b|^2 E_{1n}^+ + |1 + a e_{1n}|^2 E_{2n}^+, \\\\\n\\label{eq:|gn|2}\n \\|g_n^0\\|_2^2 &= \\abs{1 + d e_{2n}}^2 E_{1n}^- + \\beta^2 |b|^2 E_{2n}^-, \\\\\n\\label{eq:fn.gn}\n (f_n^0, g_n^0)_2 &= b\\((1 + d e_{2n}) + \\beta (1 + a e_{1n})\\).\n\\end{align}\nSince boundary conditions~\\eqref{eq:cond.canon} are strictly regular, it follows from the proof of Theorem~1.1 in~\\cite{LunMal16JMAA} that the following estimate holds\n\\begin{equation}\n (f_n^0, g_n^0)_2 \\asymp \\Delta'(\\l_n^0) \\asymp 1, \\quad |n| > n_0.\n\\end{equation}\nHence condition~\\eqref{eq:sum.fn.gn-1} is equivalent to\n\\begin{equation} \\label{eq:fn.gn-fngn.to0}\n \\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)_2|^2\n \\to 0 \\quad\\text{as}\\quad n \\to \\infty.\n\\end{equation}\nWith account of~\\eqref{eq:|fn|2}--\\eqref{eq:fn.gn} we get\n\\begin{multline} \\label{eq:fn2.gn2-fn.gn2=tau.sum}\n \\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)_2|^2\n = \\(|b|^2 \\cdot E_{1n}^+ + |1 + a e_{1n}|^2 \\cdot E_{2n}^+\\) \\cdot\n \\(\\abs{1 + d e_{2n}}^2 E_{1n}^- + \\beta^2 |b|^2 E_{2n}^-\\) \\\\\n - |b|^2 \\abs{(1 + d e_{2n}) + \\beta (1 + a e_{1n})}^2\n = \\tau_{1,n} + \\tau_{2,n} + \\tau_{3,n}, \\qquad |n| > n_0,\n\\end{multline}\nwhere\n\\begin{align}\n\\label{eq:tau1}\n \\tau_{1,n} &:= |b|^2 \\cdot \\abs{1 + d e_{2n}}^2 \\cdot\n (E_{1n}^+ E_{1n}^- - 1), \\\\\n\\label{eq:tau2}\n \\tau_{2,n} &:= \\beta^2 |b|^2 \\cdot |1 + a e_{1n}|^2 \\cdot (E_{2n}^+ E_{2n}^- - 1), \\\\\n\\label{eq:tau3}\n \\tau_{3,n} &:= \\beta^2 |b|^4 E_{1n}^+ E_{2n}^- +\n |z_n|^2 \\cdot E_{2n}^+ E_{1n}^- -\n 2 \\beta |b|^2 \\cdot \\Re z_n,\n\\end{align}\nwhere $z_n = (1 + d e_{2n}) \\overline{(1 + a e_{1n})}$ is defined in~\\eqref{eq:zn.def}. According to Proposition~\\ref{prop:sine.type}, $|\\Im \\l_n^0| \\le h$, $n \\in \\mathbb{Z}$, for some $h \\ge 0$. Hence terms $|1 + d e_{2n}|$, $|1 + a e_{1n}|$, $|z_n|$, $E_{1n}^{\\pm}$ and $E_{2n}^{\\pm}$ are all bounded for $n \\in \\mathbb{Z}$.\n\nFirst assume that $\\Im \\l_n^0 \\to 0$ as $n \\to \\infty$. Then it is clear from~\\eqref{eq:Ej=f} that\n\\begin{equation} \\label{eq:ejnto1}\n |e_{jn}| \\to 1 \\quad\\text{and}\\quad\n E_{jn}^{\\pm} \\to 1  \\quad\\text{as}\\quad n \\to \\infty,\n \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nHence $\\tau_{1,n} + \\tau_{2,n} \\to 0$ as $n \\to \\infty$, while\n$\\tau_{3,n} \\to 0$ as $n \\to \\infty$ if and only if\n\\begin{equation} \\label{eq:tau4n.def}\n \\tau_{4,n} :=\n \\beta^2 |b|^4 + |z_n|^2\n - 2 \\beta |b|^2 \\cdot \\Re z_n = \\abs{z_n - \\beta |b|^2}^2 \\to 0 \\quad\\text{as}\\quad n \\to \\infty.\n\\end{equation}\nIt follows from~\\eqref{eq:Delta_0_in_roots} and~\\eqref{eq:ejnto1} that $|z_n| = |bc|\\cdot|e_{1n} e_{2n}| \\to |bc|$ as $n \\to \\infty$. Hence, since $b \\ne 0$, then\n\\begin{equation} \\label{eq:tau4n.to0}\n \\Bigl(\\tau_{4,n} \\to 0 \\ \\ \\text{as}\\ \\ n \\to \\infty\\Bigr)\n \\ \\ \\Leftrightarrow \\ \\\n \\Bigl(|c| = \\beta |b| \\ \\ \\text{and}\\ \\ z_n \\to |bc|\n \\ \\ \\text{as}\\ \\ n \\to \\infty\\Bigr).\n\\end{equation}\nNow if condition~\\eqref{eq:lim1.lim2} holds then~\\eqref{eq:tau4n.to0} and previous observations on $\\tau_{1,n}$, $\\tau_{2,n}$, $\\tau_{3,n}$, $\\tau_{4,n}$ imply the desired condition~\\eqref{eq:fn.gn-fngn.to0}.\n\nNow assume that condition~\\eqref{eq:fn.gn-fngn.to0} holds.\nIt follows from~\\eqref{eq:Ej+Ej->1} and~\\eqref{eq:Ej=f} that\n\\begin{equation} \\label{eq:Ej.Ej-1.asymp}\n 0 \\le E_{jn}^+ E_{jn}^- - 1 \\asymp |\\Im \\l_n^0|^2, \\quad n \\in \\mathbb{Z},\n \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nSince $b \\ne 0$ and $\\beta > 0$ relations~\\eqref{eq:tau1}--\\eqref{eq:tau2} and~\\eqref{eq:Ej.Ej-1.asymp}\ncombined with Lemma~\\ref{lem:ln0.exp.asymp} imply that\n\\begin{equation} \\label{eq:tau1+tau2.asymp}\n \\tau_{1,n} \\ge 0, \\quad \\tau_{2,n} \\ge 0, \\quad \\tau_{1,n} + \\tau_{2,n} \\asymp |\\Im \\l_n^0|^2, \\quad |n| > n_0.\n\\end{equation}\nWith account of~\\eqref{eq:Ej.Ej-1.asymp} we get for $n \\in \\mathbb{Z}$:\n\\begin{equation} \\label{eq:tau3.estim}\n \\beta^2 |b|^4 E_{1n}^+ E_{2n}^- + |z_n|^2\n \\cdot E_{2n}^+ E_{1n}^-\n \\ge 2 \\beta |b|^2 |z_n|\n \\sqrt{E_{1n}^+ E_{1n}^- \\cdot E_{2n}^- E_{2n}^+} \\nonumber \\\\\n \\ge 2 \\beta |b|^2 \\cdot \\Re z_n.\n\\end{equation}\nHence $\\tau_{3,n} \\ge 0$, $n \\in \\mathbb{Z}$. Since $\\tau_{1,n} + \\tau_{2,n} + \\tau_{3,n} = \\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)|_2^2 \\to 0$ as $n \\to \\infty$, then $\\tau_{j,n} \\to 0$ as $n \\to \\infty$, $j \\in \\{1,2,3\\}$. Then condition~\\eqref{eq:tau1+tau2.asymp} implies that $\\Im \\l_n^0 \\to 0$ as $n \\to \\infty$. Combining this with the fact that $\\tau_{3,n} \\to 0$ as $n \\to \\infty$, implies that $\\tau_{4,n} \\to 0$ as $n \\to \\infty$, where $\\tau_{4,n}$ is defined in~\\eqref{eq:tau4n.def}. Now, equivalence~\\eqref{eq:tau4n.to0} finishes the proof.\n\\end{proof}\nIn the next result we reduce part of the condition~\\eqref{eq:lim1.lim2} to an explicit condition on the coefficients $a, b, c, d$ in the boundary conditions~\\eqref{eq:cond.canon} in the difficult case $b_2\/b_1 \\notin \\mathbb{Q}$.\n\\begin{lemma} \\label{lem:lim.Imln}\nLet boundary conditions~\\eqref{eq:cond.canon} be regular, i.e. $u := ad-bc \\ne 0 $. Let also $\\beta = -b_2\/b_1 \\notin \\mathbb{Q}$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be a sequence of zeros of the characteristic determinant $\\Delta_0(\\cdot)$ counting multiplicity. Let\n\\begin{equation} \\label{eq:lim.Imln.zn}\n bc \\ne 0, \\qquad |a|+|d|>0, \\qquad \\Im \\l_n^0 \\to 0\n \\quad\\text{and}\\quad\n z_n \\to |bc|  \\quad\\text{as}\\quad n \\to \\infty,\n\\end{equation}\nwhere $z_n$ is defined in~\\eqref{eq:zn.def}.\nThen\n\\begin{equation} \\label{eq.ad-bc=d\/a}\n |a|=|d|>0, \\qquad u = ad - bc = d\/\\overline{a}\n \\qquad\\text{and}\\qquad ad\\overline{bc} < 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince $\\Im \\l_n^0 \\to 0$ as $n \\to \\infty$ then $|e_{1n}| \\to 1$ as $n \\to \\infty$.\nFurther, since boundary conditions are regular, $b_2\/b_1 \\notin \\mathbb{Q}$ and $bc \\ne 0$ then all considerations in the proof of Proposition~\\ref{prop:nlim.inf} are valid. Since $u-ad=-bc$, then the second relation in~\\eqref{eq:e1.via.e2} implies:\n\\begin{multline} \\label{eq:zn=frac.e1}\n z_n = (1 + d e_{2n}) \\cdot \\overline{(1+a e_{1n})}\n = \\(1 - d \\frac{1 + a e_{1n}}{d + u e_{1n}}\\) \\cdot\n (1 + \\overline{a} \\overline{e_{1n}}) \\\\\n = \\frac{-bc e_{1n} (1 + \\overline{a} \\overline{e_{1n}})}{d + u e_{1n}}\n = \\frac{-bc (e_{1n} + \\overline{a} |e_{1n}|^2)}{d + u e_{1n}},\n \\qquad n \\in \\mathbb{Z}.\n\\end{multline}\nRecall that $d + u e_{1n} \\asymp 1$, $n \\in \\mathbb{Z}$, as established in~\\eqref{eq:a+ue2.d+ue1}. Since $z_n \\to |bc|$ and $|e_{1n}| \\to 1$ as $n \\to \\infty$, then~\\eqref{eq:zn=frac.e1} implies that\n\\begin{equation}\n |bc| (d + u e_{1n}) + bc (e_{1n} + \\overline{a}) \\to 0\n \\quad\\text{as}\\quad n \\to \\infty,\n\\end{equation}\nor\n\\begin{equation} \\label{eq:f.e1.lim}\n (|bc| u + bc) e_{1n} + |bc| d + bc \\overline{a} \\to 0\n \\quad\\text{as}\\quad n \\to \\infty,\n\\end{equation}\nBut by Proposition~\\ref{prop:nlim.inf} the sequence $\\{e_{1n}\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points. Hence relation~\\eqref{eq:f.e1.lim} is possible only if\n\\begin{equation} \\label{eq:ubc.ua=d}\n |bc| u = -bc \\quad\\text{and}\\quad |bc| d = -bc \\overline{a}.\n\\end{equation}\nSince $|a|+|d|>0$ and $bc\\ne 0$, then the second relation in~\\eqref{eq:ubc.ua=d} implies that $|a| = |d| > 0$ and that $bc \\overline{ad} = - |bc| |d|^2 < 0$. This implies the first and the third relations in~\\eqref{eq.ad-bc=d\/a}. Further, combining both relations in~\\eqref{eq:ubc.ua=d} implies the second relation in~\\eqref{eq.ad-bc=d\/a}: $u = d\/\\overline{a} = -bc\/|bc|$, which finishes the proof.\n\\end{proof}\nNow we are ready to state the main result of this section.\n\\begin{theorem} \\label{th:crit.c0.bari}\nLet boundary conditions~\\eqref{eq:cond.canon} be strictly regular. Then some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ (see Definition~\\ref{def:bari.c0}) if and only the operator $L_U(0)$ is self-adjoint. The latter holds if and only if coefficients $a,b,c,d$ from boundary conditions~\\eqref{eq:cond.canon} satisfy the following relations:\n\\begin{equation} \\label{eq:abcd.sa}\n |a|^2 + \\beta |b|^2 = 1, \\qquad\n |c|^2 + \\beta |d|^2 = \\beta, \\qquad\n a \\overline{c} + \\beta b \\overline{d} = 0, \\qquad \\beta := -b_2\/b_1 > 0.\n\\end{equation}\nIn this case every normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$.\n\\end{theorem}\n\\begin{proof}\n\n\\textbf{(i)} If conditions~\\eqref{eq:abcd.sa} hold then by Corollary~\\ref{cor:sa.crit} the operator $L_U(0)$ with boundary conditions~\\eqref{eq:quasi.per.bc} is self-adjoint. Remark~\\ref{rem:sa.c0.bari} now finishes the proof.\n\n\\textbf{(ii)} Now assume that some normalized system of root vectors of the operator $L_U(0)$ forms a Bari basis in $\\LLV{2}$.\n\nIf $b=c=0$ then Proposition~\\ref{prop:crit.bari.period} yields that $|a|=|d|=1$, in which case operator $L_U(0)$ is self-adjoint. This finishes the proof in this case.\n\nNow let $|b|+|c| \\ne 0$. Proposition~\\ref{prop:crit.bari.b.ne.0} implies that relations~\\eqref{eq:lim1.lim2} take place. In particular, $|c| = \\beta |b|$. Consider three cases.\n\n\\textbf{Case A.} Let $b_1 \/ b_2 \\in \\mathbb{Q}$. In this case $b_1 = -m_1 b_0$, $b_2 = m_2 b_0$, where $b_0 > 0$, $m_1, m_2 \\in \\mathbb{N}$. Set $m = m_1 + m_2$. Since $ad \\ne bc$, then $\\Delta_0(\\l) e^{-i b_1 \\l}$ is a polynomial in $e^{i b_0 \\l}$ of degree $m$ with non-zero roots $e^{i\\mu_k}$, $\\mu_k \\in \\mathbb{C}$, $k \\in \\{1, \\ldots, m\\}$, counting multiplicities. Hence, the sequence of zeros $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of $\\Delta_0(\\cdot)$ is a union of arithmetic progressions $\\left\\{\\frac{\\mu_k + 2 \\pi n}{b_0}\\right\\}_{n \\in \\mathbb{Z}}$, $k \\in \\{1, \\ldots, m\\}$. Clearly $\\Im \\l_n^0 = \\Im \\mu_{k_n} \/ b_0$ for some $k_n \\in \\{1, \\ldots, m\\}$. It is clear, that if $\\Im \\mu_k \\ne 0$, for some $k \\in \\{1, \\ldots, m\\}$, then $\\Im \\l_n^0$ does not tend to $0$ as $n \\to \\infty$.\nHence $\\Im \\l_n^0 = 0$, $n \\in \\mathbb{Z}$. This implies that\n\\begin{equation} \\label{eq:Ejn=1}\n E_{jn}^{\\pm} = \\int_0^1 \\abs{e^{\\pm 2 i b_j \\l_n^0 x}} dx = 1,\n \\qquad n \\in \\mathbb{Z}, \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nIt is clear that $e^{-i b_2 \\l_n^0} = (e^{-i \\mu_{k_n}})^{m_2}$ for some $k_n \\in \\{1, \\ldots, m\\}$, $n \\in \\mathbb{Z}$, and $\\{k_n\\}_{n \\in \\mathbb{Z}}$ is a periodic sequence. Hence the sequence $\\{e^{-i b_2 \\l_n^0}\\}_{n \\in \\mathbb{Z}}$ is periodic. Similarly the sequence $\\{e^{i b_1 \\l_n^0}\\}_{n \\in \\mathbb{Z}}$ is periodic. Hence, the sequence\n\\begin{equation}\n\\{z_n\\}_{n \\in \\mathbb{Z}}, \\qquad z_n = \\(1 + d e^{- i b_2 \\l_n^0}\\)\\overline{\\(1 + a e^{i b_1 \\l_n^0}\\)} = \\(1 + d e_{2n}\\)\\overline{\\(1 + a e_{1n}\\)},\n\\end{equation}\nis periodic. Since $z_n \\to |bc|$ as $n \\to \\infty$ and $|c| = \\beta|b|$, it implies that $z_n = |bc| = \\beta|b|^2$, $n \\in \\mathbb{Z}$. It now follows from~\\eqref{eq:fn2.gn2-fn.gn2=tau.sum}--\\eqref{eq:tau4n.def} and~\\eqref{eq:Ejn=1} that\n\\begin{equation}\n\\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)|_2^2 = \\tau_{4,n} =\n\\abs{z_n - \\beta |b|^2}^2 = 0, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nTaking into account formula~\\eqref{eq:|fn'-gn'|} we see that the normalized eigenvectors $f_n'$ and $g_n'$ of the operators $L_U(0)$ and $L_U^*(0)$ corresponding to the common eigenvalue $\\l_n^0 = \\overline{\\l_n^0}$ are equal for all $n \\in \\mathbb{Z}$, which implies that $L_U(0) = L_U^*(0)$.\n\n\\textbf{Case B.} Let $a=d=0$.\nThen $z_n = 1$, $n \\in \\mathbb{Z}$. Since $z_n \\to |bc|$ as $n \\to \\infty$, then $|bc| = 1$. Combined with $|c| = \\beta|b|$, this implies the desired condition~\\eqref{eq:abcd.sa}, and finishes the proof in this case.\n\n\\textbf{Case C.} Finally, let $b_1 \/ b_2 \\notin \\mathbb{Q}$, $|a|+|d|>0$ and $bc \\ne 0$. Since $\\Im \\l_n^0 \\to 0$ and $z_n \\to |bc|$ as $n \\to \\infty$, then Lemma~\\ref{lem:lim.Imln} implies condition~\\eqref{eq.ad-bc=d\/a}. In particular, $|a|=|d|>0$ and $ad\\overline{bc} < 0$. Since, in addition, $|c| = \\beta |b| > 0$, then\n\\begin{equation} \\label{eq:adbc=b2d2}\n -ad\\overline{bc} = |ad \\cdot bc| = |d|^2 \\cdot \\beta |b|^2 = \\beta b \\overline{d} \\cdot d \\overline{b}.\n\\end{equation}\nSince $d \\overline{b} \\ne 0$, this implies that $a \\overline{c} + \\beta b \\overline{d} = 0$ and coincides with the third condition in~\\eqref{eq:abcd.sa}. Further, the second relation in~\\eqref{eq.ad-bc=d\/a}, combined with relations~\\eqref{eq:adbc=b2d2}, $|a|=|d|$ and $|c| = \\beta |b|$, implies that\n\\begin{multline} \\label{eq:0=a+b=c+d}\n 0 = \\(-ad + bc + d \/ \\overline{a}\\)\\overline{bc} =\n -ad\\overline{bc} + |bc|^2 + \\frac{ad\\overline{bc}}{|a|^2}\n = |ad|\\cdot|bc| + |bc|^2 - \\frac{|ad|\\cdot|bc|}{|a|^2} \\\\\n = |bc|(|ad| + |bc|-1)\n = |bc|(|a|^2 + \\beta|b|^2-1) = |bc|(|d|^2 + \\beta^{-1}|c|^2-1).\n\\end{multline}\nSince $bc \\ne 0$, relation~\\eqref{eq:0=a+b=c+d} implies the first and second relations in~\\eqref{eq:abcd.sa}, which finishes the proof.\n\\end{proof}\n\\section{The proof of the main result}\nThis section is devoted to the proof of the main result of the paper, Theorem~\\ref{th:crit.lp.bari}. Throughout the section we use the following notations:\n\\begin{equation}\n\\mathfrak{H} := \\LLV{2}, \\qquad\n\\|\\cdot\\| := \\|\\cdot\\|_2 = \\|\\cdot\\|_{\\mathfrak{H}} \\quad\\text{and}\\quad\n(\\cdot,\\cdot) := (\\cdot,\\cdot)_{2} = (\\cdot,\\cdot)_{\\mathfrak{H}}.\n\\end{equation}\n\nFirst we need the following trivial corollary from Theorem~\\ref{th:ellp-close}.\n\\begin{corollary} \\label{cor:every.SRV}\nLet $Q \\in \\LL{p}$ for some $p \\in [1,2]$ and boundary conditions~\\eqref{eq:cond} be strictly regular. Let $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ be a system of root vectors of the operator $L_U(Q)$ such that $\\|f_n\\| \\asymp 1$, $n \\in \\mathbb{Z}$. Then there exists a system of root vectors $\\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ such that $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$ and $\\|f_n\\| = \\|f_n^0\\|$, $n \\in \\mathbb{Z}$.\n\\end{corollary}\n\\begin{proof}\nCombining relations~\\eqref{eq:sum.fn-fn0} and~\\eqref{eq:lim.fn-fn0.c0} from Theorem~\\ref{th:ellp-close} applied with $\\widetilde{Q}=0$, implies existence of normalized systems of root vectors $\\widetilde{\\mathfrak{F}} := \\{\\widetilde{f}_n\\}_{n \\in \\mathbb{Z}}$ and $\\widetilde{\\mathfrak{F}}_0 := \\{\\widetilde{f}_n^0\\}_{n \\in \\mathbb{Z}}$ of the operators $L_U(Q)$ and $L_U(0)$, respectively, such that $\\curl{\\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|_{\\infty}}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Hence\n$\\curl{\\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. By Proposition~\\ref{prop:Delta.regular.basic} eigenvalues of $L_U(Q)$ are asymptotically simple. Hence vectors $f_n$ and $\\widetilde{f}_n$, $|n| \\ge N$, are proportional for some $N \\in \\mathbb{N}$, i.e. $f_n = \\alpha_n \\widetilde{f}_n$, $|n| \\ge N$, for some $\\alpha_n \\in \\mathbb{C}$.\nLet us set\n\\begin{equation}\n \\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}, \\qquad\n f_n^0 := \\begin{cases}\n  \\alpha_n \\widetilde{f}_n^0, & |n| \\ge N, \\\\\n  \\|f_n\\| \\widetilde{f}_n^0, & |n| < N. \\\\\n \\end{cases}\n\\end{equation}\nIt is clear that $\\mathfrak{F}_0$ is a system of root vectors of the operator $L_U(0)$ and $\\|f_n^0\\| = \\|f_n\\|$, $n \\in \\mathbb{Z}$. Moreover, $\\|f_n - f_n^0\\| = |\\alpha_n| \\cdot \\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|$, $|n| \\ge N$. Since $|\\alpha_n| = \\|f_n\\| \\asymp 1$, $|n| \\ge N$, and $\\curl{\\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, then $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, which finishes the proof.\n\\end{proof}\nNow we are ready to prove our main result on Bari $(\\ell^p)^*$-property.\n\\begin{proof}[Proof of Theorem~\\ref{th:crit.lp.bari}]\nRecall that $Q \\in \\LL{p}$ for some $p \\in[1,2]$. Also note that if $L_U(0)$ is selfadjoint then Theorem~\\ref{th:crit.c0.bari} implies conditions~\\eqref{eq:abcd.sa.intro} on the coefficients from boundary conditions~\\eqref{eq:cond.canon}.\n\n\\textbf{(i)} First assume that the operator $L_U(0)$ is selfadjoint and let $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ be some normalized system of root vector of the operators $L_U(Q)$. By Corollary~\\ref{cor:every.SRV} there exists normalized system of root vectors $\\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ such that $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Since $L_U(0)$ is selfadjoint, then $\\{f_n^0\\}_{n \\in \\mathbb{Z}}$ is an orthonormal basis in $\\mathfrak{H}$. If $p=2$ then the proof would be already finished since $\\mathfrak{F}$ is $\\ell^2$-close to the orthonormal basis $\\mathfrak{F}_0$. But as Remark~\\ref{rem:c0.bari.diff} shows for $p \\in [1,2)$, the $(\\ell^p)^*$-closeness to the orthonormal basis is not equivalent to the Bari $(\\ell^p)^*$-property.\n\nTo this end, let $\\mathfrak{G} := \\{g_n\\}_{n \\in \\mathbb{Z}}$ be the system of vectors in $\\mathfrak{H}$ that is biorthogonal to the system $\\mathfrak{F}$. We need to prove that $\\curl{\\|f_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Clearly, $\\mathfrak{G}$ is (not normalized) system of root vectors of the adjoint operator $L_U^*(Q)$. Since $L_U(0)$ is self-adjoint then by Lemma~\\ref{lem:adjoint} we have $L_U^*(Q) = L_U(Q^*)$. Using Corollary~\\ref{cor:every.SRV} in the ``opposite'' direction we can find a normalized system of root vectors $\\widetilde{\\mathfrak{G}} := \\{\\widetilde{g}_n\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(Q^*)$ such that $\\curl{\\|\\widetilde{g}_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Therefore, $\\curl{\\|f_n - \\widetilde{g}_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$.\nSince both systems $\\mathfrak{G}$ and $\\widetilde{\\mathfrak{G}}$ are root vector systems of the operator $L_U^*(Q) = L_U(Q^*)$ and eigenvalues of $L_U(Q^*)$ are asymptotically simple due to Proposition~\\ref{prop:Delta.regular.basic}, then vectors $g_n$ and $\\widetilde{g}_n$, $|n| \\ge N$, are proportional for some $N \\in \\mathbb{N}$. Since $(f_n, g_n) = 1$, $n \\in \\mathbb{Z}$, it follows that $\\widetilde{g}_n = (f_n, \\widetilde{g}_n) g_n$, $|n| \\ge N$.\nNote that if $f, g \\in \\mathfrak{H}$ and $\\|f\\| = 1$, then\n\\begin{multline} \\label{eq:f.g-1}\n |(f, g) - 1|^2 = |(f, g)|^2 + 1 - 2 \\Re (f, g) \\\\\n \\le \\|f\\|^2 \\|g\\|^2 + 1 - 2 \\Re (f, g)\n = \\|f\\|^2 + \\|g\\|^2 - 2 \\Re (f, g) = \\|f - g\\|^2.\n\\end{multline}\nSince $\\|f_n\\| = 1$, $n \\in \\mathbb{Z}$, then~\\eqref{eq:f.g-1} implies that $|(f_n, \\widetilde{g}_n) - 1| \\le \\|f_n - \\widetilde{g}_n\\|$, $n \\in \\mathbb{Z}$. Hence for $|n| \\ge N$ we have,\n\\begin{equation} \\label{eq:gn-wcgn}\n\\|\\widetilde{g}_n - g_n\\| = \\|(f_n, \\widetilde{g}_n) g_n - g_n\\| =\n|(f_n, \\widetilde{g}_n) - 1| \\cdot \\|g_n\\| \\le \\|f_n - \\widetilde{g}_n\\| \\cdot \\|g_n\\|.\n\\end{equation}\nBy the main result of\n\\cite{LunMal14Dokl,LunMal16JMAA,SavShk14}\nthe system $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$ is a Riesz basis in $\\mathfrak{H}$. Hence so is its biorthogonal system $\\mathfrak{G} = \\{g_n\\}_{n \\in \\mathbb{Z}}$. This in particular implies that $\\|g_n\\| \\asymp 1$, $n \\in \\mathbb{Z}$.\nSince $\\curl{\\|f_n - \\widetilde{g}_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$ and $\\|g_n\\| \\asymp 1$, $n \\in \\mathbb{Z}$, then inequality~\\eqref{eq:gn-wcgn} implies that $\\curl{\\|\\widetilde{g}_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, which in turn implies the desired inclusion $\\curl{\\|f_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$.\n\n\\textbf{(ii)} Now assume that some normalized system of root vectors $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(Q)$ is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$. By definition $\\curl{\\|f_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, where $\\mathfrak{G} := \\{g_n\\}_{n \\in \\mathbb{Z}}$ is a system biorthogonal to $\\mathfrak{F}$ in $\\mathfrak{H}$. Clearly, $\\mathfrak{G}$ is a system of root vectors of the adjoint operator $L_U^*(Q)$.\nBy Corollary~\\ref{cor:every.SRV} there exists normalized system of root vectors $\\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ such that $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Similarly there exists (possibly not normalized) system of root vectors $\\mathfrak{G}_0 := \\{g_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U^*(0) = L_{U*}(0)$ such that $\\curl{\\|g_n - g_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. It is clear, now that $\\curl{\\|f_n^0 - g_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$.\n\nLet $\\widetilde{\\mathfrak{G}}_0 := \\{\\widetilde{g}_n^0\\}_{n \\in \\mathbb{Z}}$ be a system biorthogonal to $\\mathfrak{F}_0$. As in part (i), $\\widetilde{\\mathfrak{G}}_0$ is a Riesz basis in $\\mathfrak{H}$ and $g_n^0 = (f_n^0, g_n^0) \\widetilde{g}_n^0$, $|n| \\ge N$. Since $\\|f_n^0\\| = 1$, $n \\in \\mathbb{Z}$, then~\\eqref{eq:f.g-1} implies that $|(f_n^0, g_n^0) - 1| \\le \\|f_n^0 -g_n^0\\|$, $n \\in \\mathbb{Z}$. Hence\n\\begin{equation} \\label{eq:gn0-wtgn0}\n\\|g_n^0 - \\widetilde{g}_n^0\\| = |(f_n^0, g_n^0) - 1| \\cdot\n\\|\\widetilde{g}_n^0\\| \\le \\|f_n^0 - g_n^0\\| \\cdot \\|\\widetilde{g}_n^0\\|, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nSince\n$\\widetilde{\\mathfrak{G}}_0$ is a Riesz basis, then $\\|\\widetilde{g}_n^0\\| \\asymp 1$, $n \\in \\mathbb{Z}$.\nThus, inequality~\\eqref{eq:gn0-wtgn0} and inclusion $\\curl{\\|f_n^0 - g_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$ imply that $\\curl{\\|f_n^0 - \\widetilde{g}_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, which means that the normalized root vectors system $\\mathfrak{F}_0 = \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ is a Bari $(\\ell^p)^*$-sequence and, in particular, is a Bari $c_0$-sequence. Theorem~\\ref{th:crit.c0.bari} now implies that the operator $L_U(0)$ is selfadjoint and finishes the proof.\n\\end{proof}\n\\section{Application to a non-canonical string equation}\n\\label{sec:damped.string}\nIn this section we show the connection of $2 \\times 2$ Dirac type operators with a non-canonical string equation with $u_{xt}$ term, and apply our results on Riesz and Bari basis property.\n\nConsider the following non-canonical hyperbolic equation on a complex-valued function $u(x,t)$ defined for $x \\in [0, 1]$ and $t \\in [0, \\infty)$:\n\\begin{equation} \\label{eq:string}\n u_{tt} - (\\beta_1 + \\beta_2) u_{xt} + \\beta_1 \\beta_2 u_{xx} + a_1(x) u_x + a_2(x) u_t = 0,\n\\end{equation}\nwith the boundary conditions\n\\begin{equation} \\label{eq:string.cond}\n u(0,t)=0, \\qquad h_0 u_x(0,t) + h_1 u_x(1,t) + h_2 u_t(1,t)=0, \\qquad t \\in [0, \\infty),\n\\end{equation}\nand initial conditions\n\\begin{equation} \\label{eq:string.init}\n u(x, 0) = u_0(x), \\qquad u_t(x, 0) = u_1(x), \\qquad x \\in [0,1].\n\\end{equation}\nHere $\\beta_1, \\beta_2$ are constants and\n\\begin{equation} \\label{eq:b1.b2.a1.a2.h}\n \\beta_1 < 0 < \\beta_2, \\quad a_1, a_2 \\in L^1[0,1],\n \\quad h_0, h_1, h_2 \\in \\mathbb{C}, \\quad |h_1| + |h_2| > 0.\n\\end{equation}\n\nIf $-\\beta_1 = \\beta_2 = \\rho^{-1} > 0$ and $h_0 = 0$, the initial-boundary value problem~\\eqref{eq:string}--\\eqref{eq:string.init} governs the small vibrations of a string of length $1$\nand density $\\rho$ with the presence of a damping coefficient $a_2(x)$;\nthe string is fixed at the left end ($x=0$), while the right end ($x=1$) is damped with the coefficient $h_2\/h_1 \\in \\mathbb{C} \\cup \\{\\infty\\}$. Functions $u_0$ and $u_1$ represent the initial position and velocity of the string, respectively.\n\nIf $-\\beta_1 \\ne \\beta_2$ one can use linear transform of the variables $x$ and $t$ to reduce it to a classical string equation, but with damping that depends on $t$ and non-classical initial and boundary conditions: initial condition will be on a segment non-parallel to the $x$-axis ($t=0$), while boundary conditions will be on the rays non-parallel to the $t$-axis ($x=0$).\n\nRecall that $W^{1,p}[0, 1]$, $p \\ge 1$, denotes the Sobolev space of absolutely continuous functions\nwith the finite norm\n\\begin{equation}\n \\|f\\|_{W^{1,p}[0,1]}^p\n := \\int_0^{1} \\bigl(|f(x)|^p + |f'(x)|^p\\bigr) dx < \\infty.\n\\end{equation}\nFor convenience, we introduce the following notations:\n\\begin{equation} \\label{eq:W1p0}\n\\widetilde{W}^{1,p}[0,1] := \\{f \\in W^{1,p}[0,1] :\n f(0) = 0\\}, \\qquad \\widetilde{H}^1_0[0,1] := \\widetilde{W}^{1,2}[0,1],\n\\end{equation}\nwhere $p \\in [1, \\infty]$.\n\nThe non-canonical initial-boundary value problem~\\eqref{eq:string}--\\eqref{eq:string.init} of a damped string can be transformed into an abstract Cauchy problem in a Hilbert space $\\mathfrak{H}$ of the form\n\\begin{equation} \\label{eq:string.fH.norm}\n \\mathfrak{H} :=\n \\widetilde{H}^1_0[0, 1] \\times L^2[0, 1], \\qquad\n\\end{equation}\nwith the inner product\n\\begin{equation} \\label{eq:scal.fH}\n \\scal{f,g}_{\\mathfrak{H}} := \\int_0^1\n \\(f_1'(x) \\cdot \\overline{g_1'(x)}\n + f_2(x) \\cdot \\overline{g_2(x)} \\)\\,dx,\n\\end{equation}\nwhere $f = \\col(f_1, f_2)$, $g = \\col(g_1, g_2) \\in \\mathfrak{H}$.\n\nNow the new representation of the problem~\\eqref{eq:string}--\\eqref{eq:string.cond} reads as follows:\n\\begin{equation} \\label{eq:Y'=LhY}\n Y'(t) = i \\mathcal{L} Y(t), \\quad Y(t) := \\binom{u(\\cdot,t)}{u_t(\\cdot,t)},\n \\quad t \\ge 0, \\qquad Y(0) = \\binom{u_0}{u_1},\n\\end{equation}\nwhere the linear operator $\\mathcal{L} : \\dom(\\mathcal{L}) \\to \\mathfrak{H}$ is defined by\n\\begin{equation} \\label{eq:Lh.def}\n \\mathcal{L} y = \\mathcal{L} \\binom{y_1}{y_2} = -i \\, \\binom{y_2}{-\\beta_1\n \\beta_2 y_1'' + (\\beta_1 + \\beta_2) y_2' - a_1 y_1' - a_2 y_2},\n\\end{equation}\nwhere $y = \\col(y_1, y_2) \\in \\dom(\\mathcal{L})$, with\n\\begin{equation} \\label{eq:dom.cL}\n \\dom(\\mathcal{L}) = \\{y = \\col(y_1, y_2) \\in \\mathfrak{H}:\n y_1' \\in W^{1,1}[0,1], \\quad\n \\mathcal{L} y \\in \\mathfrak{H}, \\quad\n h_0 y_1'(0) + h_1 y_1'(1) + h_2 y_2(1)=0\\}.\n\\end{equation}\nIt is clear from the definition of $\\mathcal{L}$ and $\\dom(\\mathcal{L})$ that for $y = \\col(y_1, y_2) \\in \\dom(\\mathcal{L})$ we have: $y_1 \\in \\widetilde{W}^{1,1}_0[0,1]$ and $y_2 \\in \\widetilde{H}^1_0[0,1]$. In particular, $y_1(0)=y_2(0)=0$.\n\nSpectral properties of the operator $\\mathcal{L}$ play important role in the study of stability of solutions of the corresponding string equation. For example, Riesz basis property of the root vectors system of $\\mathcal{L}$ guarantees the exponential stability of the corresponding $C_0$-semigroup. The Riesz basis property and behavior of the spectrum of the operator $\\mathcal{L}$ have been studied in numerous papers (see~\\cite{CoxZua94,CoxZua95,Shubov96IEOT,Shubov97,BenRao00,GesHol11,GomRze15,Rzep17} and references therein).\n\nLet us show that the operator $\\mathcal{L}$ is similar to a certain $2 \\times 2$ Dirac type operator $L_U(Q)$. Since many spectral properties are preserved under similarity transform, known spectral properties for $2 \\times 2$ Dirac type operators will translate to corresponding properties of the dynamic generator $\\mathcal{L}$.\n\nTo this end, we need to introduce some notations. Set\n\\begin{equation} \\label{eq:string.B.def}\n B := \\diag(b_1,b_2), \\qquad b_1 := \\beta_1^{-1},\n \\quad b_2 := \\beta_2^{-1},\n\\end{equation}\n\\begin{equation} \\label{eq:string.Q.def}\n Q(x) := \\frac{i}{b_2-b_1} \\begin{pmatrix}\n 0 & w(x) \\cdot \\(b_2^2 a_1(x) + b_2 a_2(x)\\) \\\\\n \\frac{-1}{w(x)} \\cdot \\(b_1^2 a_1(x) + b_1 a_2(x) \\) & 0 \\end{pmatrix}, \\end{equation}\nwhere\n\\begin{equation} \\label{eq:wx.def}\n w(x) := w_1(x) w_2(x),\n\\end{equation}\n\\begin{equation} \\label{eq:wj.def}\n w_j(x) := \\exp\\(\\frac{b_1 b_2}{b_2-b_1} \\int_0^x (b_j a_1(t) + a_2(t)) dt\\),\n \\qquad x \\in [0,1], \\quad j \\in \\{1,2\\}.\n\\end{equation}\nNote, that $w_1(\\cdot)$, $w_2(\\cdot)$ are well defined and $Q \\in L^1([0,1], \\mathbb{C}^{2 \\times 2})$ in view of condition~\\eqref{eq:b1.b2.a1.a2.h}. Finally let\n\\begin{align}\n\\label{eq:string.U1}\n U_1(y) &:= y_1(0) + y_2(0) = 0, \\\\\n\\label{eq:string.U2}\n U_2(y) &:= b_1 h_0 y_1(0) + b_2 h_0 y_2(0) + (b_1 h_1 + h_2) w^{-1}_1(1)y_1(1)\n + (b_2 h_1 + h_2) w_2(1) y_2(1) = 0,\n\\end{align}\nbe boundary conditions for a Dirac operator $L_U(Q)$. Here $w_1(\\cdot)$, $w_2(\\cdot)$ are given by~\\eqref{eq:wj.def}.\n\\begin{proposition} \\label{prop:Lh.simil}\nOperator $\\mathcal{L}$ is similar to the $2 \\times 2$ Dirac type operator $L_U(Q)$ with the matrix $B$ given by~\\eqref{eq:string.B.def}, the potential matrix $Q(\\cdot)$ given by~\\eqref{eq:string.Q.def} and boundary conditions $Uy=\\{U_1,U_2\\}y=0$ given by~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}.\n\\end{proposition}\n\\begin{proof}\nWe will transform the operator $\\mathcal{L}$ into the desired operator $L_U(Q)$ via series of similarity transformations.\n\n\\textbf{Step 1.} Define\n\\begin{equation}\n\\mathcal{V}_0 : \\mathfrak{H} \\to \\LLV{2} \\quad\\text{as}\\quad\n\\mathcal{V}_0 y := \\binom{y_1'}{y_2}, \\quad y = \\binom{y_1}{y_2} \\in \\mathfrak{H}.\n\\end{equation}\nSince $\\frac{d}{dx}$ isometrically maps $\\widetilde{H}_0^1[0,1] = \\{f \\in W^{1,2}[0,1] : f(0)=0\\}$ onto $L^2[0,1]$,\nthen the operator $\\mathcal{V}_0$ is bounded with bounded inverse. It is easy to verify that\n\\begin{equation} \\label{eq:L1def}\n L_1 y := \\mathcal{V}_0 \\mathcal{L} \\mathcal{V}_0^{-1} y\n = -i \\,\\binom{y_2'}{-\\beta_1\n \\beta_2 y_1' + (\\beta_1 + \\beta_2) y_2' - a_1 y_1 - a_2 y_2},\n\\end{equation}\nwhere\n\\begin{multline} \\label{eq:string.dom.wtL}\n y = \\binom{y_1}{y_2} \\in \\dom(L_1) := \\mathcal{V}_0 \\dom(\\mathcal{L}) = \\{y \\in W^{1,1}([0,1]; \\mathbb{C}^2) : \\\\\n L_1y \\in L^2([0,1]; \\mathbb{C}^2), \\quad y_2(0) = 0, \\ \\\n h_0 y_1(0) + h_1 y_1(1) + h_2 y_2(1)=0\\},\n\\end{multline}\nin view of~\\eqref{eq:dom.cL} and definition of $\\widetilde{H}_0^1[0,1]$. Thus, the operator $\\mathcal{L}$ is similar to the operator $L_1$,\n\\begin{equation*}\n L_1y = -i B_1 y' + Q_1(x)y,\n\\end{equation*}\nwith the domain $\\dom(L_1)$ given by~\\eqref{eq:string.dom.wtL}, and the matrices $B_1$, $Q_1(\\cdot)$, given by\n\\begin{equation}\n B_1 := \\begin{pmatrix} 0 & 1 \\\\ -\\beta_1 \\beta_2 & \\beta_1 + \\beta_2 \\\\\n \\end{pmatrix}, \\qquad\n Q_1(x) := \\begin{pmatrix} 0 & 0 \\\\ i a_1(x) & i a_2(x) \\end{pmatrix}.\n\\end{equation}\nNote, that $Q_1 \\in L^1([0,1], \\mathbb{C}^{2 \\times 2})$ in view of condition~\\eqref{eq:b1.b2.a1.a2.h}.\n\n\\textbf{Step 2.} Next we diagonalize the matrix $B_1$. To this end let\n\\begin{equation} \\label{eq:string.V1def}\n V_1 := \\begin{pmatrix} 1\/\\beta_1 & 1\/\\beta_2 \\\\ 1 & 1 \\end{pmatrix}\n = \\begin{pmatrix} b_1 & b_2 \\\\ 1 & 1 \\end{pmatrix},\n\\quad \\text{and so} \\quad\n V_1^{-1}\n = \\frac{1}{b_2-b_1} \\begin{pmatrix}\n -1 & b_2 \\\\ 1 & -b_1 \\end{pmatrix},\n\\end{equation}\nwhere $b_1$ and $b_2$ are defined in~\\eqref{eq:string.B.def}. We easily get after straightforward calculations that\n\\begin{equation} \\label{eq:string.V1B1V}\n V_1^{-1} B_1 V_1 = \\diag(\\beta_1, \\beta_2) =\n \\diag(b_1^{-1}, b_2^{-1}) = B^{-1},\n\\end{equation}\n\\begin{equation} \\label{eq:string.V1Q1V}\n V_1^{-1} Q_1(x) V_1 = \\frac{i}{b_2-b_1} \\begin{pmatrix}\n b_1 b_2 a_1(x) + b_2 a_2(x) &\n b_2^2 a_1(x) + b_2 a_2(x) \\\\\n - b_1^2 a_1(x) - b_1 a_2(x) &\n - b_1 b_2 a_1(x) - b_1 a_2(x)\n \\end{pmatrix} =: Q_2(x), \\qquad x \\in [0,1].\n\\end{equation}\nNote, that $Q_2 \\in L^1([0,1], \\mathbb{C}^{2 \\times 2})$ in view of condition~\\eqref{eq:b1.b2.a1.a2.h}. Introducing bounded operator $\\mathcal{V}_1 : y \\to V_1 y$ in $L^2([0,1]; \\mathbb{C}^2)$, noting that it has a bounded inverse, and taking into account~\\eqref{eq:string.V1B1V} and~\\eqref{eq:string.V1Q1V}, we obtain\n\\begin{align}\n\\nonumber\n L_2 y & := \\mathcal{V}_1^{-1} L_2 \\mathcal{V}_1 y\n = -i V_1^{-1} B_1 V_1 y' + V_1^{-1} Q_1(x) V_1 y \\\\\n\\label{eq:L2.def}\n & = -i B^{-1} y' + Q_2(x) y,\n \\qquad y \\in \\mathcal{V}_1^{-1} \\dom(L_1) =: \\dom(L_2),\n\\end{align}\nwhere\n\\begin{multline} \\label{eq:string.dom.L2}\n \\dom(L_2) = \\{y \\in W^{1,1}([0,1]; \\mathbb{C}^2) : \\\n L_2 y \\in L^2([0,1];\\mathbb{C}^2), \\\n y_1(0) + y_2(0) = 0, \\\\\n b_1 h_0 y_1(0) + b_2 h_0 y_2(0) + (b_1 h_1 + h_2) y_1(1)\n + (b_2 h_1 + h_2) y_2(1) = 0\\},\n\\end{multline}\nwith account of formula~\\eqref{eq:string.dom.wtL} for the domain $\\dom(L_1)$ and the formula~\\eqref{eq:string.V1def} for the matrix $V_1$.\n\n\\textbf{Step 3.} On this step we make potential matrix $Q_2$ to be off-diagonal. To this end, Let $\\widetilde{Q}_2$ be a diagonal of $Q_2$, i.e.\n$$\n \\widetilde{Q}_2(x) := \\frac{i}{b_2-b_1} \\begin{pmatrix}\n b_1 b_2 a_1(x) + b_2 a_2(x) & 0 \\\\\n 0 & - b_1 b_2 a_1(x) - b_1 a_2(x)\n \\end{pmatrix}.\n$$\nLet $V_2(\\cdot)$ be a solution of the initial value problem\n\\begin{equation} \\label{eq:V2.equ}\n -i B^{-1} V_2'(x) + \\widetilde{Q}_2(x) V_2(x) = 0,\n \\qquad V_2(0) = I_2.\n\\end{equation}\nIt is easily seen that\n\\begin{equation} \\label{eq:V2.def}\n V_2(x) := \\begin{pmatrix} w_1(x) & 0 \\\\\n 0 & w_2^{-1}(x) \\end{pmatrix}, \\qquad x \\in [0,1],\n\\end{equation}\nwhere $w_1(\\cdot)$, $w_2(\\cdot)$ are defined in~\\eqref{eq:wj.def}. Let us introduce operator $\\mathcal{V}_2 : y \\to V_2(x) y$ in $L^2([0,1]; \\mathbb{C}^2)$. Since $a_1, a_2 \\in L^1[0,1]$, the operator $\\mathcal{V}_2$ is bounded and has a bounded inverse. Combining relation~\\eqref{eq:V2.equ}, definition~\\eqref{eq:string.Q.def} of $Q$ and definition~\\eqref{eq:wx.def} of $w$, we get\n\\begin{align}\n\\nonumber\n L_3 y & := \\mathcal{V}_2^{-1} L_2 \\mathcal{V}_2 y \\\\\n\\nonumber\n & = -i [V_2(x)]^{-1} B^{-1} V_2(x) y'\n + [V_2(x)]^{-1} (-i B^{-1} V_2'(x) + Q_2(x) V_2(x)) y \\\\\n\\nonumber\n & = -i B^{-1} y'\n + [V_2(x)]^{-1} (Q_2(x) - \\widetilde{Q}_2(x)) V_2(x)) y \\\\\n\\label{eq:L3.def}\n & = -i B^{-1} y' + Q(x) y,\n \\qquad y \\in \\mathcal{V}_2^{-1} \\dom(L_2) =: \\dom(L_3).\n\\end{align}\nIt is clear from the definition of $\\mathcal{V}_2$ that $\\dom(L_3)$ coincides with $\\dom(L_U(Q))$ defined via~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}. Hence $L_3 = L_U(Q)$. Combining all the steps of the proof one concludes that $\\mathcal{L}$ is similar to $L_U(Q)$.\n\\end{proof}\nCombining Proposition~\\ref{prop:Lh.simil} with our previous results for $2 \\times 2$ Dirac type operators we obtain the Riesz basis property and analogous of Bari basis property for the dynamic generator $\\mathcal{L}$ of the non-canonical initial-boundary value problem~\\eqref{eq:string}--\\eqref{eq:string.init} for a damped string equation. The part (i) of the following result improves known results in the literature on the Riesz basis property for the operator $\\mathcal{L}$ in the case $-\\beta_1=\\beta_2$, $a_1 \\equiv 0$, $h_0=0$ (see~\\cite{CoxZua94,CoxZua95,Shubov96IEOT,Shubov97,BenRao00,GesHol11,GomRze15,Rzep17} and references therein). The part (ii) shows the application of one of our main results Theorem~\\ref{th:crit.bari}.\n\\begin{theorem} \\label{th:string.riesz}\n\\textbf{(i)} Let parameters of the damped string equation satisfy relaxed conditions~\\eqref{eq:b1.b2.a1.a2.h},\n\\begin{equation} \\label{eq:h2neh1}\n\\beta_2 h_2 + h_1 \\ne 0, \\qquad \\beta_2 h_2 + h_1 \\ne 0,\n\\end{equation}\nand in addition boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} are strictly regular. Then the system of root vectors of the operator $\\mathcal{L}$ \\ \\textbf{forms a Riesz basis} in $\\mathfrak{H} = \\widetilde{H}^1_0[0,1] \\times L^2[0,1]$.\n\n\\textbf{(ii)} Let in addition $a_1, a_2 \\in L^2[0,1]$. Let also $\\mathcal{V}_0$, $\\mathcal{V}_1$, $\\mathcal{V}_2$ be the operators defined in the steps of the proof of Proposition~\\ref{prop:Lh.simil}. Then the system of root vectors of the operator $\\mathcal{L}$ is quadratically close in $\\mathfrak{H}$ to a system of the form $\\{\\mathcal{V}_0^{-1} \\mathcal{V}_1 \\mathcal{V}_2 e_n\\}_{n \\in \\mathbb{Z}}$, where $\\{e_n\\}_{n \\in \\mathbb{Z}}$ is an orthonormal basis in $\\LL{2}$, if and only if boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} are self-adjoint, which is equivalent to the condition\n\\begin{equation} \\label{eq:h0=0.b1=-b2}\n h_0 = 0, \\quad \\beta_1 = -\\beta_2, \\quad \\int_0^{1} \\Im a_2(t) dt = \\beta_2 \\log \\abs{\\frac{\\beta_2 h_2 + h_1}{\\beta_2 h_2 - h_1}}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFirst, let us transform boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} to a canonical form~\\eqref{eq:cond.canon.intro} assuming condition~\\eqref{eq:h2neh1}. For this we multiply the first condition $U_1$ by $b_1 h_0$ and subtract from $U_2$ and then multiple the second condition $U_2$ by $(b_2 h_1 + h_2)^{-1} w_2^{-1}(1)$. Boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} will take the form\n\\begin{equation} \\label{eq:cond.canon.string}\n\\begin{cases}\n \\widehat{U}_{1}(y) = y_1(0) + y_2(0) = 0, \\\\\n \\widehat{U}_{2}(y) = d y_2(0) + c y_1(1) + y_2(1) = 0,\n\\end{cases}\n\\end{equation}\nwhere\n\\begin{equation}\nd = \\frac{(b_2-b_1) h_0}{(b_2 h_1 + h_2) w_2 (1)}, \\qquad\nc = \\frac{b_1 h_1 + h_2}{(b_2 h_1 + h_2) w(1)}.\n\\end{equation}\nHere $w, w_1, w_2$ are given by~\\eqref{eq:wx.def}--\\eqref{eq:wj.def}. In particular\n\\begin{equation} \\label{eq:wlen}\n w(1) := \\exp\\(\\frac{b_1 b_2}{b_2-b_1} \\int_0^{1} ((b_1 + b_2) a_1(t) + 2a_2(t)) dt\\).\n\\end{equation}\n\n\\textbf{(i)} Proposition~\\ref{prop:Lh.simil} implies that the operator $\\mathcal{L}$ is similar to the operator $L_U(Q)$ with the matrix $B$ given by~\\eqref{eq:string.B.def}, the potential matrix $Q(\\cdot)$ given by~\\eqref{eq:string.Q.def} and boundary conditions $Uy=\\{U_1,U_2\\}y=0$ given by~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}. Note that condition~\\eqref{eq:h2neh1} implies regularity of boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}. In addition they are strictly regular by the assumption. Hence operator $L_U(Q)$ has compact resolvent and by Proposition~\\ref{prop:Delta.regular.basic} its eigenvalues are asymptotically simple and separated. Moreover, Theorem 1.1 from~\\cite{LunMal16JMAA} implies that the system of root vectors of the operator $L_U(Q)$ forms a Riesz basis in $\\LLV{2}$. Similarity of $\\mathcal{L}$ and $L_U(Q)$ implies the same properties for $\\mathcal{L}$ in the space $\\mathfrak{H}$, which finishes the proof of part (i).\n\n\\textbf{(ii)} Since $a_1, a_2 \\in L^2[0,1]$ it follows that $Q \\in \\LL{2}$. Since boundary conditions~\\eqref{eq:cond.canon.string} are strictly regular then by Theorem~\\ref{th:crit.bari} (any and every) system of root vectors of the operator $L_U(Q)$ forms a Bari basis in $\\LLV{2}$ if only if boundary conditions~\\eqref{eq:cond.canon.string} are self-adjoint, which in turn is equivalent to conditions~\\eqref{eq:abcd.sa.intro}. Since $a=0$ and $b=1$ then~\\eqref{eq:abcd.sa.intro} is equivalent to\n\\begin{equation}\nd=0, \\quad b_1 = -b_2, \\quad |c| = 1.\n\\end{equation}\nSince $\\beta_1 = b_1^{-1}$ and $\\beta_2 = b_2^{-1}$, this in turn is equivalent to~\\eqref{eq:h0=0.b1=-b2}.\n\nLet us set $\\mathcal{V} := \\mathcal{V}_0^{-1} \\mathcal{V}_1 \\mathcal{V}_2$ and let $\\{f_n\\}_{n \\in \\mathbb{Z}}$ be some system of root vectors of the operator $L_U(Q)$. It follows from the proof of Proposition~\\ref{prop:Lh.simil} that $\\{\\mathcal{V} f_n\\}_{n \\in \\mathbb{Z}}$ is a system of root vectors of the operator $\\mathcal{L}$. Hence $\\{f_n\\}_{n \\in \\mathbb{Z}}$ is quadratically close to an orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ in $\\LLV{2}$ if and only if $\\{\\mathcal{V} f_n\\}_{n \\in \\mathbb{Z}}$ is quadratically close to $\\{\\mathcal{V} e_n\\}_{n \\in \\mathbb{Z}}$ in $\\mathfrak{H}$. This completes the proof.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWith the discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) \\cite{atlas:2012gk,cms:2012gu} particle physics has entered a new era. Both LHC collaborations, ATLAS and CMS, have confirmed the existence of a boson with a mass of about 126~GeV and properties consistent with those of the scalar CP-even particle predicted by the Standard Model \\cite{Aad:2013xqa,Chatrchyan:2012jja}. In order to fully establish the nature of the Higgs boson, a precise determination of its couplings to fermions and gauge bosons is essential \\cite{Zeppenfeld:2000td,Duhrssen:2004cv,LHCHiggsCrossSectionWorkingGroup:2012nn}.  \n\nA rather clean environment for such coupling measurements is provided by the vector-boson fusion (VBF) production mode \\cite{Rainwater:1998kj,Plehn:1999xi,Rainwater:1999sd,Kauer:2000hi,Rainwater:1997dg,Eboli:2000ze}, where the Higgs boson is produced via quark-scattering mediated by weak gauge boson exchange in the $t$-channel, $qq'\\to qq'H$. Because of the low virtuality of the exchanged weak bosons, the tagging jets emerging from the scattered quarks are typically located in the forward and backward regions of the detector, while the central-rapidity region exhibits little jet activity due to the color-singlet nature of the $t$-channel exchange. These features can be exploited to efficiently suppress QCD backgrounds with a priori large cross sections at the LHC. \nIn the context of central-jet veto (CJV) techniques, events are discarded if they exhibit one or more jets in between the two tagging jets. To quantitatively employ such selection strategies, a precise knowledge of the VBF cross section with an additional jet, i.e. the reaction $pp\\to H jjj$, is crucial. \n\nNext-to-leading order (NLO) QCD corrections to VBF-induced $Hjjj$ production have first been computed in \\cite{Figy:2007kv}, yielding results with only small residual scale uncertainties of order 10\\% or less. In particular, in that approach the survival probability for the Higgs signal has been estimated to exhibit a perturbative accuracy of about 1\\%. The calculation of \\cite{Figy:2007kv} is implemented in the {\\tt VBFNLO} package~\\cite{Arnold:2008rz,Arnold:2011wj,Arnold:2012xn} in the form of a flexible parton-level Monte-Carlo program.   \nMore recently, an NLO-QCD calculation for electroweak $Hjjj$ production has been presented~\\cite{Campanario:2013fsa}, where several approximations of Ref.~\\cite{Figy:2007kv} have been dropped. \n\nIn this work, we merge the parton-level calculation of \\cite{Figy:2007kv} with a parton-shower Monte-Carlo in the framework of the \\POWHEG{} formalism~\\cite{Nason:2004rx,Frixione:2007vw}, a method for the matching of an NLO-QCD calculation with a transverse-momentum ordered parton-shower program. For our implementation we are making use of version~2 of the  \\POWHEGBOX{}~\\cite{Alioli:2010xd,Nason:2013ydw}, a repository that provides the process-independent ingredients of the \\POWHEG{} method. The code we develop yields precise, yet realistic predictions for VBF-induced $Hjjj$ production at the LHC in a public framework that can easily be used by the reader for further phenomenological studies. \n\nThis article is organized as follows: In Sec.~\\ref{sec:tech} we describe some technical details of our implementation. Phenomenological results are presented in Sec.~\\ref{sec:pheno}. We conclude in Sec.~\\ref{sec:concl}.\n\\section{Technical details of the implementation}\n\\label{sec:tech}\nThe implementation of $Hjjj$ production via VBF in the context of the \\POWHEGBOX{} requires, as major building blocks, the matrix elements for all relevant partonic scattering processes at Born level and at next-to-leading order. These have first been calculated in \\cite{Figy:2007kv} and are publicly available in the {\\tt VBFNLO} package~\\cite{Arnold:2008rz}. We extracted the matrix elements from {\\tt VBFNLO} and adapted them to the format required by the \\POWHEGBOX. \n\nAt leading order (LO), processes of the type $qq'\\to qq'gH$ and all crossing-related channels are taken into account, if they include the exchange of a weak boson in the $t$-channel . Some representative Feynman diagrams are depicted in Fig.~\\ref{fig:lo-graphs}. \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{.\/figs\/Higgs3j_LO-crop.ps}\n \\caption{Representative tree-level diagrams for VBF $Hjjj$ production. }\n \\label{fig:lo-graphs}\n\\end{figure}\nThe gauge-invariant class of diagrams involving weak-boson exchange in the $s$-channel is considered as part of the Higgs-strahlung process, and disregarded in the context of our work on VBF-induced Higgs production. The interference of $t$-channel with $u$-channel diagrams in flavor channels with quarks of the same type is neglected. Once VBF-specific selection cuts are imposed, these approximations are well justified~\\cite{Ciccolini:2007ec}.\nThroughout, we assume a diagonal CKM matrix. \nWe refer to the electroweak $Hjjj$ production process at order $\\mathcal{O}(\\alpha_s\\alpha^3)$ within these approximations as ``VBF $Hjjj$ production''.  \n\nThe virtual corrections to this reaction comprise the interference of the Born amplitudes with one-loop diagrams where a virtual gluon is attached to a single fermion line [c.f.~Fig.~\\ref{fig:virt-graphs}~(a)--(g)], and diagrams where a virtual gluon is exchanged between the two different fermion lines, see~Fig.~\\ref{fig:virt-graphs}~(h),(i). \n\\begin{figure}[tp]\n\\vspace*{8cm}\n \\includegraphics[width=\\textwidth,bb=0 0 447 0]{.\/figs\/Higgs3j_virt-crop.ps}\n \\\\\n \\includegraphics[width=\\textwidth]{.\/figs\/Higgs3j_penta-crop.ps}\n \\caption{\n Representative one-loop diagrams for $qq'\\to qq'gH$, with the virtual gluon being attached to one single fermion line [graphs ($a$)-($g$)], or to the two different fermion lines  [graphs ($h$) and ($i$)]. }\n \\label{fig:virt-graphs}\n\\end{figure}\nAs discussed in some detail in \\cite{Figy:2007kv}, the \nlatter contributions are strongly suppressed by color factors and due to the VBF dynamics. They can be neglected, if the respective color structures of the real-emission contributions are disregarded as well, as these would serve to cancel the  infrared singularities of the pentagon and hexagon contributions that we drop. \n\nThe real-emission contributions involve subprocesses with four external (anti-)quarks and two gluons such as $qq'\\to qq'ggH $, as well as pure quark scattering processes of the type $qq'\\to qq'Q\\bar QH $, and all crossing-related channels with $t$-channel weak boson exchange. \nBecause of the approximations we have employed in the virtual contributions, where we dropped color-suppressed contributions giving rise to pentagon and hexagon integrals, the respective color structures have to be disregarded in the real-emission contributions as well. In practice this means to neglect interference terms between diagrams where a given gluon is emitted once from the upper and once from the lower quark line in the Feynman graphs. For example, interference terms like $2\\,\\mathcal{R}e\\left( {\\cal B}^3_4 {\\cal B^*}^4_3\\right)$, with ${\\cal B}^3_4$ as depicted in Fig.~\\ref{fig:real-gg-supp}, are dropped, \n\\begin{figure}[tp]\n \\includegraphics[width=1\\textwidth]{.\/figs\/Higgs3j_real3-crop.ps}\n \\caption{\nRepresentative diagrams of the color structure $\\mathcal{B}_4^3$ as introduced in Ref.~\\cite{Figy:2007kv} for the subprocess $qq' \\rightarrow  qq'\\, gg \\, H$. \n}\n \\label{fig:real-gg-supp}\n\\end{figure}\nwhile their squares or the squares of the topologies sketched in Fig.~\\ref{fig:real-gg} are fully considered. \n\\begin{figure}[tp]\n\\begin{center}\n \\includegraphics[width=1\\textwidth]{.\/figs\/Higgs3j_real1-crop.ps} \\\\\n \\vspace{0.5cm} \n \\hspace{1cm} \\includegraphics[width=0.66\\textwidth]{.\/figs\/Higgs3j_real2-crop.ps}  \n \\caption{Representative diagrams  of the color structure $\\mathcal{A}_{43}^{1a}$ as introduced in Ref.~\\cite{Figy:2007kv} for the subprocess $qq' \\rightarrow  qq'\\, gg \\, H$. }\n \\label{fig:real-gg}\n \\end{center}\n\\end{figure}\nRepresentative diagrams for a pure quark subprocess are depicted in Fig.~\\ref{fig:real-QQ}.  For this class of subprocesses we require that the $Q\\bar Q$ pair stems from a gluon. Contributions involving the hadronic decay of a weak boson, $V\\to Q\\bar Q$, such as graph~\\ref{fig:real-QQ}~(c),  are disregarded within our VBF~setup. \n\\begin{figure}[tp]\n \\centering\n \\includegraphics[width=1\\textwidth]{.\/figs\/Higgs3j_realQQ-crop.ps}\n \\caption{\n Representative diagrams for the subprocess $qq' \\to qq' \\, Q\\bar{Q}\\, H$. }\n \\label{fig:real-QQ}\n\\end{figure}\n\nWhile in \\cite{Figy:2007kv} soft and collinear singularities have been taken care of by a dipole subtraction procedure, the \\POWHEGBOX{} makes use of the so-called FKS subtraction scheme \\cite{Frixione:1995ms}. From the color- and spin-correlated amplitudes provided by the user, the \\POWHEGBOX{} internally constructs the counterterms that are needed to cancel soft and collinear singularities in the real-emission contributions. Because we are disregarding certain color-suppressed contributions in the virtual and real-emission amplitudes, we have to make sure that only the counterterms relevant for our setup are constructed. This is achieved by passing only those  color- and spin-correlated Born amplitudes to the \\POWHEGBOX{} that correspond to the color structures we consider within our approximations. \n\nWe have carefully tested that the counter terms constructed in this way approach the real-emission amplitudes in the soft and collinear limits. Additionally, we have compared the tree-level and real emission amplitudes for selected phase space points with code generated by {\\tt MadGraph} \\cite{Stelzer:1994ta} that has been adapted to match the approximations of our calculation. We found agreement at the level of more than ten digits. The virtual amplitudes have been compared to {\\tt VBFNLO}, again showing full agreement for single phase space points. We note that some care has been necessary in this latter check, as finite parts of the subtraction terms are included in the virtual amplitudes in the default setup of {\\tt VBFNLO}.  \nTo verify the entire setup of our code, we have compared cross sections and distributions for various sets of selection cuts at LO and NLO-QCD accuracy as obtained with the \\POWHEGBOX{} with respective results of {\\tt VBFNLO}. We found full agreement for all considered scenarios. \n\nWe note that special care is needed when performing the phase-space integration of VBF $Hjjj$\\;~production in the framework of the \\POWHEGBOX{}. In contrast to the VBF-induced $Hjj$ production cross section that is entirely finite at leading order, the inclusive VBF $Hjjj$ cross section diverges already at leading order when a pair of partons becomes collinear  or a soft gluon is encountered in the final state. While such divergent contributions disappear after phenomenologically sensible selection cuts are imposed, their presence considerably reduces the efficiency of the numerical phase space integration. This effect can be avoided by appropriate phase-space cuts at generation level, or by a so-called Born-suppression factor $F(\\Phi_n)$ that dampens the integrand whenever singular configurations in phase-space are approached. In order to \nensure that our results are independent of technical cuts in the phase-space integration, we recommend the use of a Born-suppression factor. In our  \\POWHEGBOX{} implementation we provide two alternative versions of Born-suppression factors: \n\\begin{itemize}\n\\item\nIn our first, multiplicative, approach, the factor is of the form \n\\begin{equation}\n\\label{eq:bsupp1}\nF(\\Phi_n) = \n\\prod_{i=1}^3\n\\left(\n\\frac{p_{T,i}^2}{p_{T,i}^2+\\Lambda_p^2}\n\\right)^2\n\\prod_{i,j=1;\\\\\nj\\neq i}^3\n\\left(\n\\frac{m_{ij}^2}{m_{ij}^2+\\Lambda_m^2}\n\\right)^2\\,,\n\\end{equation}\nwhere the $p_{T,i}$ and $m_{ij}=\\sqrt{(p_i+p_j)^2}$ respectively denote the transverse momenta and invariant masses of the three final-state partons of the underlying Born configuration. The $\\Lambda_p$ and $\\Lambda_m$ are cutoff parameters that are typically set to values of a few GeV.   %\n\\item\nFollowing the procedure suggested for the related case of trijet production in the framework of the \\POWHEGBOX{}~\\cite{Kardos:2014dua}, we use an exponential suppression factor of the form\n\\begin{equation}\nS_1 = \\exp \\left[ \n- \\Lambda_1^4 \\cdot \\left(\\sum_{i=1}^3 \\frac{1}{p^4_{T,i}} + \\sum_{i,j=1;\\\\\nj\\neq i}^3 \\frac{1}{q^2_{ij}}\\right) \n\\right]\\,,\n\\end{equation}\nwith \n\\begin{equation}\nq_{ij}=p_i \\cdot p_j \\, \\frac{E_i \\, E_j}{E_i^2 + E_j^2}\\,,\n\\end{equation}\nfor the suppression of infrared divergent configurations in the underlying Born kinematics, \naccompanied by a factor \n\\begin{equation}\n S_2 = \\left(\\frac{H_T^2}{H_T^2 + \\Lambda_2^2}\\right)^2,\n\\end{equation}\nwhere\n\\begin{equation}\nH_T = p_{T,1} + p_{T,2} + p_{T,3}.\n\\end{equation}\nThe factor $S_2$ serves to suppress configurations where all partons are having small transverse momenta, and at the same time increase the fraction of events generated with large transverse momenta. \nCombining $S_1$ with $S_2$, we construct\n\\begin{equation}\n\\label{eq:bsupp2}\nF(\\Phi_n) = S_1 \\cdot S_2\\,.\n\\end{equation}\n\\end{itemize}\nFor the generation of the phenomenological results shown below we are using a Born suppression factor of the form given in Eq.~(\\ref{eq:bsupp2}) with $\\Lambda_1 = 10 $~GeV and  $\\Lambda_2 = 30$~GeV, supplemented by a small generation cut on the transverse momenta of the three outgoing partons of the underlying Born configuration, $p_{T,i}^\\mathrm{gen} > 1$~GeV.  \n\nTo make sure our results do not depend on these technical parameters, in addition to our default setup we ran our code using the Born suppression factor of Eq.~(\\ref{eq:bsupp1}) with $\\Lambda=20$~GeV and, again, $p_{T,i}^\\mathrm{gen} > 1$~GeV. The results in the two setups are in full agreement with each other and, at fixed order, also with respective results obtained with {\\tt VBFNLO} that is using an entirely different phase-space generator. \n\\section{Phenomenological results}\n\\label{sec:pheno}\nOur implementation of VBF $Hjjj$\\; production at the LHC is made publicly available in version~2 of  the \\POWHEGBOX{}, and can be obtained as explained at the project webpage,  {\\tt http:\/\/powhegbox.mib.infn.it\/}. \n\nHere, we are providing phenomenological results for a representative setup at the LHC with a center-of-mass energy of $\\sqrt{s}=8$~TeV. \nWe are using the CT10 fixed-four-flavor set \\cite{Lai:2010vv} for the parton distribution functions of the proton as implemented in the {\\tt LHAPDF} library \\cite{Whalley:2005nh} and the accompanying value of the strong coupling, $\\alpha_s(m_Z)=0.1127$.  Jets are reconstructed via the anti-$k_T$ algorithm with a resolution parameter of $R=0.5$, with the help of the {\\tt FASTJET}~package~\\cite{Cacciari:2005hq,Cacciari:2008gp,Cacciari:2011ma}. As electroweak input parameters we are using the masses of the weak gauge bosons, $m_W=80.398$~GeV and $m_Z=91.188$~GeV, and the Fermi constant, $G_F=1.16639\\times 10^{-5}$~GeV$^{-1}$. Other electroweak parameters are computed thereof via tree-level relations. \nThe widths of the massive gauge bosons are set to $\\Gamma_W =  2.095$~GeV and $\\Gamma_Z=2.51$~GeV, respectively. For the Higgs boson, we are using $m_H = 126$~GeV and $\\Gamma_H=4.095$~MeV. \nThe renormalization and factorization scales are identified as $\\mu_\\mr{R}=\\mu_\\mr{F}=m_H\/2$. \nIn order to assess uncertainties that remain after matching the NLO calculation with a parton shower program, we consider three different tools:  \\PYTHIA{}~{\\tt 6.4.25} with the Perugia~0 tune~\\cite{Sjostrand:2006za}, \\HERWIGPP{}~{\\tt 2.7.0}~\\cite{Bahr:2008pv,Bellm:2013lba} with its default angular-ordered shower, and with a transverse-momentum ordered dipole shower~\\cite{Platzer:2011bc} which we tag as {\\tt PYT}, {\\tt HER}, and {\\tt DS++}, respectively. We note that wide-angle, soft radiation that is in principle needed when matching an NLO calculation with a parton-shower program using the \\POWHEG{} method, is missing in the default angular-ordered \\HERWIGPP{} shower. The impact of this missing piece on observables can only be estimated by a comparison with predictions obtained with transverse momentum ordered showers, such as the \\DSPP{} version of \\HERWIGPP{}. We do not consider hadronization, QED radiation, multiple parton interactions, and underlying event effects in this work. \n\nIn order to define a $Hjjj$ event, we demand at least three well-observable jets with \n\\begin{equation} \np_{T,j}>20 \\text{ GeV}\\,,\n\\quad\n|y_j| < 4.5\\,.\n\\end{equation}\nIn addition, we impose VBF-specific selection cuts.  The two hardest jets, referred to as ``tagging jets'', are required to fulfill \n\\begin{equation}\np_{T,j}^\\mathrm{tag}>30~\\mathrm{GeV}\\,,\\quad\n|y_j^\\mathrm{tag}|<4.5\\,,\n\\end{equation}\nand be well-separated from each other, \n\\begin{equation}\n|y_{j_1}^\\mathrm{tag}-y_{j_2}^\\mathrm{tag}|>4.0\\,, \n\\quad\ny_{j_1}^\\mathrm{tag}\\times y_{j_2}^\\mathrm{tag}<0\\,, \n\\quad\nm_{jj}^\\mathrm{tag}>500~\\mathrm{GeV}\\,.\n\\end{equation}\nThe kinematics of the Higgs boson is not restricted. \n\nWith these cuts, we obtain a cross section of $\\sigma^\\mathrm{NLO}=71.5 \\pm  0.4$~fb at fixed order, where the error is the statistical error of the Monte Carlo calculation. After matching the NLO result with a parton shower, some of the events fail to pass the cuts, resulting in slightly smaller cross sections of $\\sigma^{\\tt PYT}=65.8 \\pm  0.3$~fb, $\\sigma^{\\tt HER}=68.3 \\pm 0.3$~fb,  and $\\sigma^{\\tt DS++}=69.8\\pm 0.5$~fb, respectively. \nApart from this change in normalization the impact of the parton shower on observables related to the tagging jets is very mild, as  illustrated in Fig.~\\ref{fig:tag-jet}\n\\begin{figure}[t]\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/ptj1.eps}\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/mjj.eps}\n\\caption{\nTransverse momentum distribution of the hardest tagging jet (left) and invariant mass distribution of the two tagging jets (right) at NLO (black), and at NLO+PS level: \\PYT{}~(red), \\HERPP{}~(blue), \\DSPP{}~(cyan). \nThe lower panels show the NLO+PS results normalized to the pure NLO prediction together with its statistical uncertainty (yellow band). \n}\n\\label{fig:tag-jet}\n\\end{figure}\nfor the transverse momentum distribution of the hardest tagging jet and the invariant mass of the tagging jet pair. \n\nIn contrast to NLO calculations for VBF $Hjj$ production, where the third jet can be described only with LO accuracy, our calculation is NLO accurate in distributions related to the third jet. In Fig.~\\ref{fig:jet3}, NLO+PS results for the transverse momentum and the rapidity distribution of the third jet are shown for different parton shower programs together with the fixed-order NLO result. \n\\begin{figure}[t]\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/ptj3.eps}\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/etaj3.eps}\n\\caption{\nTransverse momentum and rapidity distributions of the third jet at NLO, and at NLO+PS level (line styles as in Fig.~\\ref{fig:tag-jet}). \n}\n\\label{fig:jet3}\n\\end{figure}\nFor all considered parton showers, the difference between the NLO and the NLO+PS results is small. However, \\PYTHIA{} tends to produce slightly more jets in the central-rapidity region of the detector, while \\HERWIGPP{} preferentially radiates in the collinear region between the two tagging jets and the beam axis. We will see below that this effect is more pronounced in the case of sub-leading jets. \n\nLarger differences between the fixed-order and the various matched predictions occur in distributions related to the fourth jet. \nIn the parton-level NLO calculation a fourth jet can only stem from the real-emission contributions, and can thus be described only at tree-level accuracy. Larger theoretical uncertainties are therefore expected for observables related to the fourth jet. Fig.~\\ref{fig:jet4} \n\\begin{figure}[t]\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/ptj4.eps}\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/y4star.eps}\n\\caption{\nTransverse momentum distribution of a fourth jet for our default setup with an extra cut of $p_{T,j_4}>1$~GeV~(left) and rapidity distribution of a fourth hard jet with $p_{T,j_4}>20$~GeV relative to the two tagging jets (right) at NLO, and at NLO+PS level (line styles as in Fig.~\\ref{fig:tag-jet}). \n}\n\\label{fig:jet4}\n\\end{figure}\nillustrates the effect of the \\POWHEG{}-Sudakov factor on the transverse momentum of the fourth jet and clarifies how extra radiation in the VBF setup is distributed by the different parton shower programs via the $y_4^\\star$ variable. This quantity\t is defined as\n\\begin{equation} \ny_4^\\star=y_{j_4} - \\frac{y_{j_1} + y_{j_2}}{2}\\,,\n\\label{eq:y4star}\n\\end{equation}\nin order to parameterize the rapidity of the fourth jet relative to the two hard tagging jets. The respective distribution shows, more pronouncedly than in the case of the third jet, that \\PYTHIA{} and \\HERWIGPP{} tend to produce radiation in different regions of phase space. The differences between the various NLO+PS curves can thus be considered as inherent uncertainty of the matched prediction. \n\\section{Conclusions}\n\\label{sec:concl}\nIn this work, we have presented an implementation of VBF $Hjjj$\\; production in version~2 of the \\POWHEGBOX{} repository. We have performed the matching of an existing NLO-QCD calculation with parton-shower programs using the \\POWHEG{} formalism and presented phenomenological results for a representative setup at the LHC. The code we developed is publicly available and can be adapted to the user's need in a straightforward manner. \n\nWe have shown that theoretical uncertainties associated with the description of the third jet by  genuinely different parton-shower programs are mild at NLO+PS level, contrary to what is observed in studies based on matrix elements for VBF $Hjj$ production that are only LO accurate in the third jet. Our implementation thus provides an important improvement in the theoretical assessment of central-jet veto observables that are crucial for VBF analyses at the LHC.  \n\\section*{Acknowledgments} \nWe are grateful to Carlo Oleari for help with implementing the code in the \\POWHEGBOX{} repository. \nThe work of\nB.~J.\\ is supported by the Institutional Strategy of the University of T\\\"ubingen (DFG, ZUK~63). \nF.~S.\\ is supported by the ``Karlsruher Schule f\\\"ur Elementar\\-teilchen- und Astroteilchenphysik: Wissenschaft und Technologie (KSETA)'' and D.~Z.\\ by the BMBF under ``Verbundprojekt 05H2012 -- Theorie''.  \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nLet ${{\\mathcal A}_g}$ be the moduli space of complex $g$-dimensional\nprincipally polarized abelian varieties (ppavs). Denote $\\pi':{\\mathcal X}_g\\rightarrow {{\\mathcal A}_g}$ the universal family of ppavs over it. Let $\\Theta_g\\subseteq {\\mathcal X}_g$, and $\\pi:\\Theta_g\\rightarrow{{\\mathcal A}_g}$ denote\nthe universal theta divisor. We shall omit the index $g$ when it is clear.\\smallskip\n\nWe can identify ${\\mathcal X}$ and ${\\mathcal A}$ respectively  with the spaces\n$$\n  \\operatorname{Sp}(g,\\Z)\\times{\\mathbb Z}^{2g}\\backslash{\\mathcal H}_g\\times{\\mathbb C}^g,\\quad \\operatorname{Sp}(g,\\Z)\\backslash{\\mathcal H}_g,\n$$\nwhere ${\\mathcal H}_g$ is the Siegel upper half-space. For $\\tau\\in{{\\mathcal H}}$,  we denote by $X_{\\tau}$, resp. $\\Theta_\\tau$, the fiber of $\\pi'$, resp. $\\pi$, over $\\tau$ (more precisely, over the image of $\\tau$ in ${\\mathcal A}$.\n\nA symmetric principal polarization $\\Theta$ is the zero set of the holomorphic function (where $^tn$ stands for the transpose of a vector $n$)\n$$\n \\vartheta(\\tau, z)=\\sum_{n\\in {\\mathbb Z}^g}\\exp(i\\pi ({}^tn\\tau n+2{}^tnz),\n$$\nand all the other principal polarization divisors symmetric under the $\\pm 1$ involution of $X_\\tau$ are obtained from this divisor by translating by points of order two $\\frac{\\tau\\epsilon+\\delta}{2}\\in X_\\tau[2]\\subset X_\\tau$, for some $\\epsilon,\\delta\\in ({\\mathbb Z}\/2{\\mathbb Z})^g$.\n\nWe denote by $\\T_{\\rm sing}\\subseteq\\Theta$  the singular  locus on $\\Theta$, i.e. the set of the classes of the points $(\\tau, z)$ defined by\n$$\n \\T_{\\rm sing}:=\\lbrace (\\tau,z)\\in{\\mathcal H}_g\\times{\\mathbb C}^g\\mid \\vartheta(\\tau, z)=\\partial_i\\vartheta(\\tau, z)=\\partial_i\\partial_j\\vartheta(\\tau, z) =0\\rbrace\n$$\n(where from here on we denote by $\\partial_i=\\frac{\\partial}{\\partial z_i}$ the partial derivative in the $z_i$ direction). Moreover we  denote by ${\\mathcal S}$  the union of the singular points of $\\Theta_{\\tau}$, i.e. the set of  the classes of the points $(\\tau, z)$ defined by\n$$\n {\\mathcal S}:=\\lbrace (\\tau,z)\\in{\\mathcal H}_g\\times{\\mathbb C}^g\\mid \\vartheta(\\tau, z)= \\partial_i\\vartheta(\\tau, z)=0\\rbrace.\n$$\nObviously we have\n$$\n  \\T_{\\rm sing}\\subset{\\mathcal S}\\subset \\Theta.\n$$\nThere has been great interest in the singularities of the theta divisor and the loci of ppavs for which the theta divisor is singular at least since the ground-breaking work of Andreotti and Mayer \\cite{am}, who defined what are now called the Andreotti-Mayer loci\n$$\n N_k:=\\pi_*({\\mathcal S})=\\{(X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid \\operatorname{dim}\\operatorname{Sing}\\Theta_{\\tau}\\ge k\\}.\n$$\nIt is known that $N_0\\subset{\\mathcal A}$ is a divisor, which has at most two components, cf. \\cite{mu,bea,deb}:\n$$\n \\theta_{\\rm null}:=\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid X_\\tau[2]^{\\rm even}\\cap\\operatorname{Sing}  \\Theta_{\\tau}\\ne\\emptyset\\}\n$$\n$$\n N_0' :={\\rm the\\ closure\\ of\\ }\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid (X_\\tau\\setminus X_\\tau[2]^{\\rm even})\\cap\\operatorname{Sing}  \\Theta_{\\tau}\\ne\\emptyset\\}\n$$\n(where $X_\\tau[2]^{\\rm even}$ denotes the even points of order two). The intersection of these two components was studied in \\cite{deb,sing}.\n\nIn general the dimensions of the loci $N_k$ are not known. They were studied in detail by Ciliberto and van der Geer in \\cite{cilvdg},\\cite{amsp}, who conjecture that within the locus of ppavs with endomorphism ring ${\\mathbb Z}$ (i.e. essentially with Picard group ${\\mathbb Z}$) the codimension of any component of $N_k$ is equal to at least $\\frac{(k+1)(k+2)}{2}$. They prove that for $g\\ge 4$ and $1\\le k\\le g-3$ one has $\\operatorname{codim} N_k\\ge k+2$.\n\nIn this paper we will be interested in the loci of ppavs for which the theta divisor has points of higher multiplicity. For points of multiplicity two the natural loci to consider are\n$$\n \\begin{aligned}G_k:=\n &\\{(X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid \\operatorname{dim}\\{z\\in X_\\tau : {\\rm mult}_z\\Theta_{\\tau}\\ge 2\\}\\ge k\\}\\\\\n=&\\{(X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid\n \\operatorname{dim} (X_\\tau\\cap\\T_{\\rm sing})\\ge k\\}\\end{aligned}\n$$\n(the two definitions are equivalent by the heat equation).\n\nAlready the locus $G_0$ is still a rather unknown object, and we will mostly concentrate on studying it. It has a natural subset\n$$\n (\\partial\\theta)_{\\rm null}:=\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid {\\mathcal X}_{\\tau}[2]^{\\rm odd}\\cap\\T_{\\rm sing}\\ne\\emptyset\\}.\n$$\nWe can further generalize this  to define\n$$\n (\\partial^k\\theta)_{\\rm null}:=\\{ (X_{\\tau}, \\Theta_{\\tau})\\in{{\\mathcal A}}\\mid \\exists x\\in {\\mathcal X}_{\\tau}[2]: {\\rm mult}_x\\Theta_\\tau- k\\in 2{\\mathbb Z}_{>0}\\}\n$$\nFor some computations it will be important to keep track of which dimension we are in. In this case we will use an upper index $(g)$ and write $\\T_{\\rm sing}^{(g)},(\\partial\\theta)_{\\rm null}^{(g)},$ etc.\n\n\\smallskip\nIn this paper we give equations for  $(\\partial\\theta)_{\\rm null}$ and $(\\partial^2\\theta)_{\\rm null}$  using modular forms and so we can say that we solve Schottky's problem  for them. We describe explicitly some irreducible components of $(\\partial^k\\theta)_{\\rm null}$, and thus obtain an estimate for their codimension in ${\\mathcal A}_g$. Moreover we shall prove that these loci are reducible for $1\\leq k\\leq g-4$ (from \\cite{el} it follows that $(\\partial^{g-2}\\theta)_{\\rm null}$ is irreducible). Finally  we shall give some evidence for the expected dimensions of  $(\\partial\\theta)_{\\rm null}$,  $G_0$ and $(\\partial^2\\theta)_{\\rm null}$.  Doing this we  will relate the dimension of these three  varieties with the dimension of $\\T_{\\rm sing}$.\n\nThe methods we use consist mostly of working with theta functions and their derivatives, computing and bounding the dimensions of the tangent spaces by the ranks of explicit matrices of derivatives. We use the heat equation in many places, and compute the intersections of the loci we are interested in with the boundary of the partial compactification of ${\\mathcal A}$. In \\cite{amsp} Ciliberto and van der Geer study primarily the dimensions of the loci $N_k$, while we are mostly interested in the dimensions of $G_k$ and $(\\partial^k\\theta)_{\\rm null}$. Perhaps uniting the two approaches may yield further insights into the geometry of the theta divisor.\n\n\\section{Notations}\nWe start by recalling  some basic facts about theta functions and modular forms. We denote ${\\mathcal H}_g$ the {\\it Siegel upper\nhalf-space}, i.e. the set of symmetric complex $g\\times g$ matrices\n$\\tau$ with positive definite imaginary part. Each such $\\tau$\ndefines a complex   abelian variety   ${\\mathbb C}^g\/\\tau{\\mathbb Z}^g+{\\mathbb Z}^g$. If $\\sigma=\\left(\\begin{matrix} a&b\\\\ c&d\\end{matrix}\\right)\\in\\operatorname{Sp}(g,\\Z)$ is a symplectic matrix in a $g\\times g$ block form, then its action on $\\tau\\in{\\mathcal H}_g$ is defined by $\\sigma\\cdot\\tau:=(a\\tau+b)(c\\tau+d)^{-1}$, and the moduli space of ppavs is the quotient ${\\mathcal A}_g={\\mathcal H}_g\/\\operatorname{Sp}(g,\\Z)$. A period matrix $\\tau$ is called {\\it decomposable} if there exists $\\sigma\\in\\operatorname{Sp}(g,\\Z)$ such that\n$$\n \\sigma\\cdot\\tau=\\left(\\begin{matrix} \\tau_1&0\\\\\n 0&\\tau_2\\end{matrix}\\right),\\quad\\tau_i\\in{\\mathcal H}_{g_i},\\ g_1+g_2=g, g_i>0;\n$$\notherwise we say that $\\tau$ is indecomposable.\n\nFor $\\epsilon,\\delta\\in ({\\mathbb Z}\/2{\\mathbb Z})^g$, thought of as vectors of zeros and ones,\n$\\tau\\in{\\mathcal H}_g$ and $z\\in {\\mathbb C}^g$, the {\\it theta function with\ncharacteristic $[\\epsilon,\\delta]$} is\n$$\n \\tt\\epsilon\\delta(\\tau,z):=\\sum\\limits_{m\\in{\\mathbb Z}^g} \\exp \\pi i \\left[\n ^t(m+\\frac{\\epsilon}{2})\\tau(m+\\frac{\\epsilon}{2})+2\\ ^t(m+\\frac{\\epsilon}{2})(z+\n \\frac{\\delta}{2})\\right].\n$$\nWe write  $\\vartheta(\\tau, z)$ for the theta function with\ncharacteristic $[0,0]$. Observe that\n$$\n \\tt{0}{0}\\left(\\tau,z+\\tau\\frac\\e2+\\frac\\de2\\right)= \\exp \\pi i\n \\left(-\\frac{^t\\epsilon}{2}\\,\\tau\\frac{\\epsilon}{2}\\,\\,-\\frac{^t\\epsilon}{2}\\,(z+\n \\frac{\\delta}{2})\\right)\\tt\\epsilon\\delta(\\tau,z),\n$$\ni.e. theta functions with characteristics are, up to some non-zero\nfactor, equal to $\\vartheta(\\tau,z)$ shifted by points of order two.\n\nA {\\it characteristic} $[\\epsilon,\\delta]$ is called {\\it even} or {\\it odd}\ndepending on whether  the scalar product\n$\\epsilon\\cdot\\delta\\in{\\mathbb Z}\/2{\\mathbb Z}$ is zero or one, respectively. The function\n$\\tt\\epsilon\\delta(\\tau,z)$ is even or odd as a function\nof $z$, according to the parity of the characteristic; thus a characteristic $[\\epsilon,\\delta]$ is even (resp. odd) if the multiplicity of the theta function $\\vartheta(\\tau,z)$ at the point $z=(\\tau\\epsilon+\\delta)\/2$ is even (resp. odd). We denote by $X_\\tau[2]^{\\rm even\/odd}$ the set of even\/odd points of order two on $X_\\tau$.\n\nA {\\it theta constant} is the evaluation at $z=0$ of a theta function. All odd theta constants of course vanish identically in $\\tau$, but their first derivatives at zero\ndo not vanish identically, and in fact transform in a nice way under the $\\operatorname{Sp}(g,\\Z)$ action.\n\n\\smallskip\nFor a finite index subgroup $\\Gamma\\subset\\operatorname{Sp}(g,\\Z) $ a multiplier system of weight $r\/2$ is a map $v:\\Gamma\\to {\\mathbb C}^*$, such that the map\n$$\n  \\sigma\\mapsto v(\\sigma)\\det(c\\tau+d)^{r\/2}\n$$\nsatisfies the cocycle condition for every $\\sigma\\in\\Gamma$ and\n$\\tau\\in{\\mathcal H}_g$.\n\n\\smallskip\nGiven a pair $\\rho=(\\rho_0,r)$, where $r$ is half integral, and $\\rho_0:{\\rm GL}(g,{\\mathbb C})\\to \\operatorname{End} V$ is an irreducible rational representation with the highest weight $(k_1,k_2,\\dots,k_g)$, $k_1\\geq k_2 \\geq\\dots\\geq k_g=0$, we use the notation\n$$\n  \\rho(A)=\\rho_0(A)\\det A^{r\/2}\\ .\n$$\n\nA map $f:{\\mathcal H}_g\\to V$ is called a modular form for $\\rho$,\nor simply a {\\it vector-valued modular form}, if the choice\nof $\\rho$ is clear, {\\it with multiplier $v$}, with respect to a\nfinite index subgroup $\\Gamma\\subset\\operatorname{Sp}(g,\\Z)$ if\n\\begin{equation}\\label{transform}\n  f(\\sigma\\cdot\\tau)=v(\\sigma)\\rho(c\\tau+d)f(\\tau)\\qquad\\forall\n  \\tau\\in{\\mathcal H}_g,\\forall\\sigma\\in\\Gamma,\n\\end{equation}\nand if additionally $f$ is holomorphic at all cusps of\n${\\mathcal H}_g\/\\Gamma$.\n\nWe define the {\\it level}  and {\\it Igusa}'s subgroups of the symplectic group to be\n$$\n  \\Gamma_g(n):=\\left\\lbrace \\sigma=\\left(\\begin{matrix} a&b\\\\ c&d\\end{matrix}\\right)\n  \\in\\operatorname{Sp}(g,\\Z)\\, |\\, \\sigma\\equiv\\left(\\begin{matrix} 1&0\\\\\n  0&1\\end{matrix}\\right)\\ {\\rm mod}\\ n\\right\\rbrace\n$$\n$$\n  \\Gamma_g(n,2n):=\\left\\lbrace \\sigma\\in\\Gamma_g(n)\\, |\\, {\\rm\n  diag}(a^tb)\\equiv{\\rm diag} (c^td)\\equiv0\\ {\\rm mod}\\\n  2n\\right\\rbrace.\n$$\nWhen $n$ is even, these are finite index normal subgroups of $\\operatorname{Sp}(g,\\Z)$.\n\n\\smallskip\n\\noindent Under the action of $\\sigma\\in\\operatorname{Sp}(g,\\Z)$ the theta functions transform\nas follows:\n$$\n  \\theta\\bmatrix \\sigma\\left(\\begin{matrix} \\epsilon\\\\ \\delta\\end{matrix}\\right)\\endbmatrix\n  (\\sigma\\cdot\\tau,\\,^{t}(c\\tau+d)^{-1}z)\\qquad\\qquad\\qquad\n$$\n$$\n  \\qquad\\qquad\\qquad=\\phi(\\epsilon,\\,\\delta,\\,\\sigma,\\,\n  \\tau,\\,z)\\det(c\\tau+d)^{\\frac{1}{2}}\\tt\\epsilon\\delta(\\tau,\\,z),\n$$\nwhere\n$$\n  \\sigma\\left(\\begin{matrix} \\epsilon\\cr \\delta\\end{matrix}\\right) :=\\left(\\begin{matrix} d&-c\\cr\n  -b&a\\end{matrix}\\right)\\left(\\begin{matrix} \\epsilon\\cr \\delta\\end{matrix}\\right)+ \\left(\\begin{matrix} {\\rm\n  diag}(c \\,^t d)\\cr {\\rm diag}(a\\,^t b)\\end{matrix}\\right),\n$$\nconsidered in $({\\mathbb Z}\/2{\\mathbb Z})^g$, and $\\phi(\\epsilon,\\,\\delta,\\,\\sigma,\\,\\tau,\\,z)$ is some complicated explicit function. For more details, we refer to\n\\cite{igbook}. Thus theta constants with characteristics are (scalar) modular forms of weight $1\/2$ with multiplier with respect to $\\Gamma_g(2)$, i.e. we have\n$$\n  \\tt\\epsilon\\delta(\\sigma\\cdot\\tau,0)=v(\\sigma, \\epsilon, \\delta) \\det(c\\tau+d)^{1\/2}\\tt\\epsilon\\delta(\\tau,0)\n  \\qquad \\forall \\sigma\\in\\Gamma_g(2).\n$$\nwhere the multiplier $v$ becomes trivial if we assume  $\\sigma\\in\\Gamma_g(4,8)$.\n\n\\smallskip\nWe call the {\\it theta-null divisor} $\\t_{\\rm null}\\subset{\\mathcal A}_g$ the\nzero locus of the product of all even theta constants. We  denote  by $\\operatorname{grad}\\tt\\epsilon\\delta(\\tau)$ the gradient of the theta function of characteristic $[\\epsilon,\\,\\delta]$  with respect to $ z_1,\\dots,z_g$ and evaluating at $z=0$. This gradient is not identically zero if and only if the characteristic is odd. The gradient is a vector valued modular form  for $\\rho=(St, 1\/2)$  with multiplier $v$, cf.\\cite{gs}.\n\nWe recall that theta functions  (and their derivatives) satisfy the heat equation, i.e.\n$$\n \\frac{\\partial^2\\tt\\epsilon\\delta(\\tau,z)}{\\partial z_j\\partial z_k} =2\\pi\n i(1+\\delta_{j,k})\\frac{\\partial\\tt\\epsilon\\delta(\\tau,z)}{\\partial\\tau_{jk}},\n$$\n(where $\\delta_{j,k}$ is Kronecker's symbol).\n\n\\smallskip\nThe   symmetric matrix  associated to the second derivatives\n\\begin{equation}\\label{2z}\n \\left(\\frac{\\partial^2\\tt\\epsilon\\delta(\\tau,z)}{\\partial z_j\\partial z_k}|_{z=0}\\right)\n\\end{equation}\nis a  vector valued modular form for $\\Gamma_g(2)$ if we restrict  to the locus $\\tt\\epsilon\\delta(\\tau,0)=0$.\nThis is a general fact: the derivative of a section of a line bundle is a section of the same bundle when restricted to the zero locus of the section --- the modularity of this particular gradient is discussed in \\cite{genus4}. Similarly note that third derivatives of a theta function with an odd characteristics for a modular form,\nwhen restricted to the  locus   $\\operatorname{grad}\\tt\\epsilon\\delta(\\tau,0)=0$.\n\n\\section{Equations for the loci $(\\partial^k\\theta)_{\\rm null}$}\nSimilarly to the case of $\\t_{\\rm null}$, we give vector valued (and alternatively, scalar valued) equations for the loci\n$(\\partial\\theta)_{\\rm null}$ and  $(\\partial^2\\theta)_{\\rm null}$. We will work on the level covers ${\\mathcal A}_g(2):={\\mathcal H}_g\/\\Gamma_g(2)$ or\n${\\mathcal A}_g(4, 8):={\\mathcal H}_g\/\\Gamma_g(4, 8)$, where the $\\t_{\\rm null}$ divisor decomposes into a union of components corresponding to the individual characteristics. Note also that  ${\\mathcal A}_g(4,8)$ is a smooth manifold cover of the stack\/orbifold ${\\mathcal A}_g$, so working on the level cover takes care of the stackiness.\n\nOn ${\\mathcal A}_g(2)$ for any odd $[\\epsilon, \\delta]$ the vector valued equation\n$\\operatorname{grad}\\tt \\epsilon\\delta(\\tau) =0$ defines a certain set of components of $(\\partial\\theta)_{\\rm null}.$ The (possibly reducible) loci $\\operatorname{grad}\\tt \\epsilon\\delta(\\tau) =0$ for various $\\epsilon,\\delta$ are conjugate under the action of $\\operatorname{Sp}(g,\\Z)$.\n\nFor any $[\\epsilon_1,\\,\\delta_1],\\dots,[\\epsilon_g,\\,\\delta_g]$ we define\n$$\n D([\\epsilon_1,\\,\\delta_1],\\dots,[\\epsilon_g,\\,\\delta_g])(\\tau):=\\operatorname{grad} \\tt{\\epsilon_1}{\\delta_1}\\wedge\\operatorname{grad}\\tt{\\epsilon_2}{\\delta_2}\\wedge \\dots\\wedge \\operatorname{grad}\\tt{\\epsilon_g}{\\delta_g}(\\tau)\n$$\nwhich is a scalar modular form with multiplier of weight $\\frac{g+2}{2}$ with respect to $\\Gamma_g(2)$, cf. \\cite{sm}.\n\nIt follows from Lefschetz theorem  for abelian varieties, cf. \\cite{gs}, that the $g\\times 2^{g-1}(2^g -1)$ matrix\n$$\n \\left(\\dots\\operatorname{grad}\\tt \\epsilon\\delta(\\tau)\\dots\\right)_{{\\rm all\\, odd} [\\epsilon,\\,\\delta]}\n$$\nhas maximal rank for all $\\tau$. Thus if all its minors including a fixed characteristic $[\\epsilon,\\delta]$ vanish, the corresponding gradient must be zero, and we get\n\\begin{prop}\nThe common zero locus of the scalar modular forms\n$$\n D([\\epsilon,\\,\\delta],[\\epsilon_2,\\,\\delta_2], \\dots([\\epsilon_g,\\,\\delta_g](\\tau)\n$$\nfor $[\\epsilon,\\delta]$ fixed, and all possible odd characteristics $[\\epsilon_2,\\,\\delta_2],\\dots, [\\epsilon_g,\\,\\delta_g]$,\nis equal to the locus $\\operatorname{grad}\\tt \\epsilon\\delta(\\tau) =0$.\n\\end{prop}\n\\begin{prop}\nSimilarly, if $[\\epsilon, \\delta]$ is an even characteristic,\nthe common zero locus of the scalar valued equation\n\\begin{equation}\\label{theta}\n  \\tt \\epsilon\\delta(\\tau)=0\n\\end{equation}\nand the vector valued equations\n\\begin{equation}\\label{theta2}\n \\left((1+\\delta_{i, j})\\frac{\\partial^2\\tt\\epsilon\\delta}{\\partial\\tau_{i\\,j}}\\tt\\alpha\\beta- (1+\\delta_{i,j})\\frac{\\partial^2\\tt\\alpha\\beta}{\\partial\\tau_{i\\,j}}\\tt\\epsilon\\delta\\right)(\\tau)=0\n\\end{equation}\nfor all even characteristics  $[\\alpha,\\, \\beta]$, defines a union of some irreducible components of  $(\\partial^2\\theta)_{\\rm null}$ on ${\\mathcal A}_g(2)$. (To see this, one notes that all even theta constants cannot vanish simultaneously, cf. \\cite{genus4}.)\n\\end{prop}\n\nNote that of course the components of $(\\partial^2\\theta)_{\\rm null}$ on ${\\mathcal A}_g(2)$ are given by\n\\begin{equation}\\label{theta2bis}\n \\left((1+\\delta_{i,j})\\frac{\\partial^2\\tt\\epsilon\\delta}{\\partial\\tau_{i\\,j}}\\right)(\\tau)=0;\n\\end{equation}\nhowever, this expression is a modular form only along the divisor $ \\tt \\epsilon\\delta(\\tau,0)=0$, cf. \\cite{genus4}.\n\nThere is also another method to produce scalar valued modular forms  vanishing exactly on the  component of  $(\\partial^2\\theta)_{\\rm null}$. This method is  similar to the method used for  $(\\partial\\theta)_{\\rm null}$. Indeed, the $2^{g-1}(2^g+1)\\times\\left(\\frac{g(g+1)}{2}+1\\right)$ matrix\n$$\n\\left(\\begin{array}{ccc}\n  \\dots &\\tt \\epsilon\\delta&\\dots \\\\\n\\dots&\\dots&\\dots\\\\\n \\dots& {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{i\\, j}}&\\dots\n  \\end{array}\\right)_{{\\rm all\\, even} [\\epsilon,\\,\\delta]}\n$$\nhas maximal rank, cf.\\cite{igbook}. Hence, setting $N=(1\/2)g(g+1)$, the form\n$$\n D^2([\\epsilon,\\,\\delta],[\\epsilon_1,\\,\\delta_1]\\dots[\\epsilon_N,\\,\\delta_N])(\\tau):=\n$$\n$$\n {\\rm det}\n \\left(\\begin{array}{llll}\n  \\ \\ \\tt \\epsilon\\delta&\\ \\ \\tt{\\epsilon_1}{\\delta_1}&\\dots&\\ \\ \\tt{\\epsilon_N}{\\delta_N}\\\\\n {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{1\\, 1}}&{\\partial\\tt{\\epsilon_1}{\\delta_1}}\/{{\\partial \\tau_{1\\, 1}}}&\\dots  &{\\partial\\tt{\\epsilon_N}{\\delta_N}}\/{{\\partial \\tau_{1\\, 1}}}\\\\\n\\dots&\\dots&\\dots&\\dots\\\\\n {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{i\\, j}}&{\\partial\\tt{\\epsilon_1}{\\delta_1}}\/{{\\partial \\tau_{i\\, j}}}&\\dots  &{\\partial\\tt{\\epsilon_N}{\\delta_N}}\/{{\\partial \\tau_{i\\, j}}}\\\\\n\\dots&\\dots&\\dots&\\dots\\\\\n {\\partial\\tt\\epsilon\\delta}\/{\\partial\\tau_{g\\, g}}&{\\partial\\tt{\\epsilon_1}{\\delta_1}}\/{{\\partial \\tau_{g\\, g}}}&\\dots  &{\\partial\\tt{\\epsilon_N}{\\delta_N}}\/{{\\partial \\tau_{g\\, g}}}\\\\\n  \\end{array}\\right)(\\tau) ,\n$$\nis a modular form with multiplier, of weight $g+1+(1\/2)(N+1)$ relatively to $\\Gamma_g(2)$, cf. \\cite{diff}, and similarly to the previous case we have\n\\begin{prop}\nThe modular forms\n$$\n D^2([\\epsilon,\\,\\delta],[\\epsilon_1,\\,\\delta_1]\\dots[\\epsilon_N,\\,\\delta_N])(\\tau)\n$$\nfor $[\\epsilon,\\delta]$ and all possible even characteristics $[\\epsilon_1,\\,\\delta_1],\\dots, [\\epsilon_N,\\,\\delta_N]$, vanish  simultaneously along the locus defined by (\\ref{theta}) and (\\ref{theta2bis}), and give scalar equations for one  component of the locus $(\\partial^2\\theta)_{\\rm null}$.\n\\end{prop}\n\\begin{rem}\nOne could also write more complicated equations for the locus $(\\partial^k\\theta)_{\\rm null}$ for $k\\geq 3$, using suitable vector-valued modular forms. Unfortunately our method for obtaining equations using scalar modular forms does not work in this case, since the jacobian matrices do not have maximal rank everywhere, cf. \\cite{diff}.\n\\end{rem}\n\n\\section{Some components of $(\\partial^k\\theta)_{\\rm null}$ within the locus of decomposable ppavs}\nIn this section we start our investigation of the irreducible components of  $(\\partial^k\\theta)_{\\rm null}$ and their possible dimensions.\n\nAs an immediate consequence of the results of \\cite{ko} or \\cite{el}, we have\n\\begin{prop}\n$$(\\partial^{k}\\theta)_{\\rm null}=\\emptyset \\quad {\\rm for}\\, k\\geq g-1$$\n$$(\\partial^{g-2}\\theta)_{\\rm null}={\\mathcal A}_1\\times\\dots\\times{\\mathcal A}_1 $$\n\\end{prop}\nFor lower values of $k$ we can describe some components:\n\\begin{thm}\\label{irred}\nFor $1\\leq k\\leq g-2$, the variety\n$\\theta_{g-k,\\,\\rm null}  \\times{\\mathcal A}_1\\times\\dots \\times A_1$  is an  irreducible components of $(\\partial^k\\theta)_{\\rm null}$.\nThe codimension of this subvariety gives the bound\n$$\n \\operatorname{codim}(\\partial^k\\theta)_{\\rm null} \\leq gk+1-(1\/2)(k^2+k)\n$$\n\\end{thm}\n\\begin{proof}\nWe perform the computation on ${\\mathcal A}_g(4,8)$ for the characteristic\n$\\epsilon=(\\alpha,1, \\dots,1)$ and $\\delta=(\\beta,1,\\dots,1)$, where $[\\alpha,\\beta]$ is a $(g-k)$-dimensional characteristic. Note that $[\\alpha,\\beta]$ is necessarily even. Consider then the set\n$$\n A_k([\\epsilon,\\delta]):=\\left\\{\\frac{\\partial^h\\tt\\epsilon\\delta(\\tau, z)}{\\partial z_{i_1}\\dots\\partial z_{i_h}}\\vert_{z=0}=0\\right\\}_{{\\rm for\\ all}\n \\ h\\leq k.}\n$$\nObviously $A_k$ contains the $\\Gamma_g(2)$ conjugates of the locus\n$$\n \\left(\\tt\\alpha\\beta(\\tau')=0\\right)\\times {\\mathcal H}_1\\times\\dots\\times{\\mathcal H}_1,\n$$\n(where $\\tau'\\in{\\mathcal A}_{g-k}$),\nwhich is of codimension $ gk+1-(1\/2)(k^2+k)$ in ${\\mathcal A}_g$. The only non-zero elements of the jacobian matrix of the equations defining $A_k$ are those involving derivatives of  order $k+2$. These form a matrix with $(1\/2)(g+1)g$ rows and the columns that can be indexed in the following three ways:\\smallskip\n\n1) the $k$ derivatives involve the indices $g-k+1,g-k+2,\\dots, g$\n\n2) the $k$ derivatives involve only one of the first $g-k$ indices and all, but one among $g-k+1,g-k+2,\\dots, g$\n\n3) the $k$ derivatives involve two ( even with multiplicity) of the first $g-k$ indices and all, but two among $g-k+1,g-k+2,\\dots, g.$\n\nThen we have to consider the derivative $\\partial\/\\partial \\tau_{a,b}$,  or equivalently, by the heat equation, $\\partial^2\/\\partial z_a \\partial z_b$. In  the first of the above cases we get a column with non-zero entries being\n$$\n \\frac{\\partial^2  \\tt\\alpha\\beta(\\tau')}{\\partial z_{a}\\partial z_{b} }\\prod_{i=1}^{k}  \\left(  \\tt 1 1^{'}(\\lambda_i)\\right)\\quad {\\rm with}\\,\\,1\\leq a\\leq b\\leq g-k.\n$$\nIn the second case we get $(g-k)k$ columns involving in the $(a,b)$ row\n$$\n  \\frac{\\partial^2  \\tt\\alpha\\beta(\\tau')}{\\partial z_{a}\\partial z_{j} }\\prod_{i=1}^{k}  \\left(  \\tt 1 1^{'}(\\lambda_i)\\right)\\quad {\\rm with}\\,\\,1\\leq a \\leq g-k<b\\leq g.\n$$\nIn the third case we get $1\/2)k(k-1)$ columns involving in the $(a,b)$ row\n$$\n \\frac{\\partial^2  \\tt\\alpha\\beta(\\tau')}{\\partial z_{l}\\partial z_{j} }\\prod_{i=1}^{k}  \\left(  \\tt 1 1^{'}(\\lambda_i)\\right)\\quad {\\rm with}\\,\\, g-k<a< b\\leq g,\n$$\nwhere we denoted $\\tt 1 1^{'}(\\lambda_i)$ the derivative of the one-dimensional theta function with odd characteristic evaluated at zero for the elliptic curve with periods 1 and $\\lambda_i=\\tau_{g-k+i,g-k+i}$.\n\nIt can be easily  checked that all these columns are linearly independent, and hence the Jacobian has maximal rank, so that the equations give an irreducible component of the locus $(\\partial^k\\theta)_{\\rm null}$, as claimed.\n\\end{proof}\n\n\n\\begin{prop}\nWhen $k\\leq g-4$, the locus $(\\partial^k\\theta)_{\\rm null}$ is reducible.\n\\end{prop}\n\\begin{proof}\nWe recall that  $\\theta_{2,\\,\\rm null} ={\\mathcal A}_1\\times {\\mathcal A}_1$ and $\\theta_{3,\\,\\rm null}$ is the hyperelliptic locus in ${\\mathcal A}_3$ (in particular a generic element of it is indecomposable). Thus the locus\n$$\n \\theta_{g-k-1,\\,\\rm null}  \\times{\\mathcal A}_2\\times{\\mathcal A}_1\\times\\dots\\times{\\mathcal A}_1 \\subset (\\partial^k\\theta)_{\\rm null},\n$$\nand must be contained in an irreducible component of $(\\partial^k\\theta)_{\\rm null}$ different from $\\theta_{g-k,\\,\\rm null}  \\times{\\mathcal A}_1\\times\\dots \\times A_1$.\n\\end{proof}\n\n\\begin{rem}\nWhen $k=g-3$,  for small $g$, we have that $(\\partial^{g-3}\\theta)_{\\rm null}$  is irreducible. In general this is not known. Repeating  the argument of the proof of corollary 2 in   \\cite{el}, one can show that if a principally polarized abelian variety in  $(\\partial^{g-3}\\theta)_{\\rm null}$  is decomposable, and thus necessarily lies in  $\\theta_{g-3,\\,\\rm null}  \\times{\\mathcal A}_{1}\\times\\dots\\times{\\mathcal A}_{1}$. It is interesting to know whether there are indecomposable ppavs in $(\\partial^{g-3}\\theta)_{\\rm null}$. See also remark \\ref{gminus3}\n\\end{rem}\n\n\\begin{rem}\nFor a generic $\\tau\\in\\theta_{g-k,\\,\\rm null}  \\times{\\mathcal A}_1\\times\\dots \\times A_1$ the theta divisor $\\Theta_{\\tau}$ has a unique point of order 2 having  multiplicity $k+2$. However, for a generic point in\n$\\theta_{g-k-n,\\,\\rm null}  \\times{\\mathcal A}_{i_1}\\times\\dots\\times{\\mathcal A}_{i_k}$, with $k<i_1+\\dots i_k=k+n\\leq g-2$, the theta divisor $\\Theta_{\\tau}$ has several points of order 2 and multiplicity $k+2$.\nAs a consequence we see that the latter product is contained in the    singular locus of  $(\\partial^k\\theta)_{\\rm null}\\subset{\\mathcal A}_g(2)$.\n\\end{rem}\n\nWe now investigate the reducibility of the locus $G_0$.\n\\begin{thm}\\label{reduc}\nThe loci $N_{g-1,\\,0} ' \\times{\\mathcal A}_1$ and $\\theta_{g-1,\\,\\rm null}  \\times{\\mathcal A}_1$ are irreducible components of $G_0$.\n\\end{thm}\n\\begin{proof}\nRecall that the locus $\\T_{\\rm sing}\\subset{\\mathcal X}_g$ is defined by the following equations (using the heat equation for the theta function)\n$$\n 0=\\vartheta(\\tau,z)=\\partial_i\\vartheta(\\tau,z)=\\partial_i\\partial_j\\vartheta(\\tau,z)\\quad\\forall 1\\le i,j\\le g.\n$$\nComputing the Jacobian matrix of these defining equations for $\\tau\\in{\\mathcal A}_{g-1}\\times{\\mathcal A}_1$ we get\n$$\n \\left(\\begin{array}{cccccccc}\n \\,\\partial_1\\vartheta&\\partial_1\\partial_1\\vartheta\n &\\dots&\\partial_1\\partial_g\\vartheta&\\partial_1\\partial_1\\partial_1\\vartheta&\\dots&\\partial_1\\partial_g\\partial_g \\vartheta\\\\\n \\,\\partial_2\\vartheta&\\partial_2\\partial_1\\vartheta\n &\\dots&\\partial_2\\partial_g\\vartheta&\\partial_2\\partial_1\\partial_1\\vartheta&\\dots&\\partial_2\\partial_g\\partial_g \\vartheta\\\\  \\dots&\\dots&\\dots&\\dots&\\dots&\\dots&\\dots \\\\\\\n \\,\\partial_g\\vartheta&\\partial_g\\partial_1\\vartheta&\\dots&\\partial_g\\partial_g\\vartheta& \\partial_g\\partial_1\\partial_1\\vartheta&\\dots&\\partial_g\\partial_g\\partial_g \\vartheta\\\\\n \\,\\partial_1\\partial_1\\vartheta&\\partial_1\\partial_1\\partial_1\\vartheta\n &\\dots&\\partial_1\\partial_1\\partial_g\\vartheta&\\partial_1\\partial_1\\partial_1\\partial_1\\vartheta&\\dots& \\partial_1\\partial_1\\partial_g\\partial_g\\vartheta\\\\\n   \\dots&\\dots&\\dots&\\dots&\\dots&\\dots \\\\\\\n  \\,\\partial_g\\partial_g\\vartheta&\\partial_g\\partial_g\\partial_1\\vartheta\n &\\dots&\\partial_g\\partial_g\\partial_g\\vartheta&\\partial_g\\partial_g\\partial_1\\partial_1\\vartheta&\\dots& \\partial_g\\partial_g\\partial_g\\partial_g\\vartheta\n \\end{array}\\right)\n$$\nIn both cases, let $$(X_{\\tau}, \\Theta_{\\tau})=(X_{\\tau'}, \\Theta_{\\tau'})\\times(E_{\\lambda}, \\Theta_{\\lambda})$$\nand let $x=(x', y)$ be the point of multiplicity at least 3 such that\n$x'$ is a singular point of  $\\Theta_{\\tau'}$ and $y= \\Theta_{\\lambda}$. We can assume that $x'$ is an ordinary double point, then we have that the jacobian  matrix at the point $(\\tau, x)\\in \\T_{\\rm sing}$ is of the form\n$$\n  \\left(\\begin{array}{ccc }\n  0&0&\\partial_i\\partial_j\\partial_k\\vartheta\\\\\n  0&{}^t(\\partial_i\\partial_j\\partial_k\\vartheta)&\\partial_i\\partial_j\\partial_k\\partial_l\\vartheta\n  \\end{array}\\right)\n$$\nHere the first column is in fact a column-vector, the entry (1,2) is the $g\\times g$ null matrix; $(\\partial_i(\\partial_j\\partial_k\\vartheta))$ is  the  $g\\times (1\/2)g(g+1)$ matrix of the third derivatives of $\\vartheta$, ${}^t(\\partial_i\\partial_j\\partial_k\\vartheta)$ is its transpose, and  the entry  (2,3)  is the $(1\/2)g(g+1)\\times (1\/2)g(g+1)$ matrix of the fourth derivatives of $\\vartheta$. An elementary computation gives that the rank of the matrix $(\\partial_i\\partial_j\\partial_k\\vartheta)$ is $g$, since $x'$ is an ordinary double point, hence the previous matrix has rank $2g$.\n\nHence  in a neighborhood of the point  $(\\tau, x)$ the subvariety $\\T_{\\rm sing}$ has codimension at least $2g$ in ${\\mathcal X}_g$, and thus its projection has codimension at least $g$ in ${\\mathcal A}_g$. Since ${N}_{g-1,\\,0} ' \\times{\\mathcal A}_1$ and $\\theta_{g-1,\\,\\rm null}  \\times{\\mathcal A}_1$ both have codimension $g$ in ${\\mathcal A}_g$, these are two irreducible components of $G_0$.\n\\end{proof}\n\nIn more generality, note that $(\\partial^k\\theta)_{\\rm null}\\times{\\mathcal A}_1\\subset (\\partial^{k+1}\\theta)_{\\rm null}$, and we will now consider the question of whether these give irreducible components. The result is the following generalization of theorem \\ref{irred}.\n\n\\begin{prop}\\label{codg}\nLet $X$ be an irreducible component of $(\\partial^k\\theta)_{\\rm null}^{(g)}$.\nThen the locus $X\\times {\\mathcal A}_1$  is an irreducible component of $ (\\partial^{k+1}\\theta)_{\\rm null}^{(g+1)}$ if and only if the rank of the\n$g \\times \\binom{g+k}{k+1}$ matrix\n$$\n L:=\\left(\\frac{\\partial\\tt\\epsilon\\delta (Z)}{\\partial z_{a}\\partial z_I}\\right)\n$$\nfor $1\\leq a \\leq g$ and $I\\subset\\{1,\\dots,g\\}$ with $\\#I=k+1$, is equal to $g$.\n\nMoreover, if $k=1$, the set $X\\times {\\mathcal A}_1$  is an irreducible component of $ (\\partial^{2}\\theta)_{\\rm null}$ if and only if the codimension of $X$ in ${\\mathcal A}_g$ is equal to $g$.\n\\end{prop}\n\\begin{proof}\nAs before, we will compute the Jacobian matrix of the defining equations at a smooth point $\\tau\\in X$, lifted to the level cover ${\\mathcal A}_g(4,8)$, so that we are talking about the multiplicity of the theta function with a given characteristic $[\\alpha,\\beta]$ at zero. The   $(1\/2)g(g+1)\\times \\binom{g+k-1}{k}$ Jacobian matrix is then\n$$\n  J=\\left(\\frac{\\partial^2  \\tt\\alpha\\beta(\\tau)}{\\partial z_{a}\\partial z_{b} \\partial z_I}\\right)\n$$\nwith $1\\leq a\\leq b\\leq g$, and $I\\subset\\{1,\\dots,g\\},$ with $\\#I=k$. Since this is the Jacobian matrix of the equations defining $X$, its rank is equal to the codimension of $X$ in ${\\mathcal A}_g$.\n\nThen for any $x\\in{\\mathcal A}_1$ the point $(\\tau,x)\\in {\\mathcal A}_g\\times {\\mathcal A}_1$ is a smooth point of $(\\partial^{k+1}\\theta)_{\\rm null}$, with the vanishing theta function with high multiplicity being $\\tt{\\alpha\\ 1}{\\beta\\ 1}$. The Jacobian matrix of the equations defining $\\partial^{k+1}\\theta)_{\\rm null}\\subset{\\mathcal A}_{g+1}$ is the same as $J$ together with the matrix $L$. The first statement of the proposition follows, while for the special case of $k=1$ we observe that $L$ is the transpose of $J$ and thus has the same rank.\n\\end{proof}\n\n\\section{Codimensions of $(\\partial\\theta)_{\\rm null},(\\partial^2\\theta)_{\\rm null}$, and $G_0$}\nNote that the dimension of a reducible variety means the maximum dimension of the components, and thus $\\operatorname{codim}$ means the minimal codimension of the components.\n\nIn the previous section we have computed some components of the loci $(\\partial^k\\theta)_{\\rm null}$ and thus know their dimensions. It is natural to conjecture the following.\n\\begin{conj}\\label{dthetanull}\nFor any $g\\geq 3$\n$$\n \\operatorname{codim}_{{\\mathcal A}_g} (\\partial\\theta)_{\\rm null}=\\operatorname{codim}_{{\\mathcal A}_g} G_0=g.\n$$\n\\end{conj}\n\nWe give some evidence for this conjecture by using the degeneration techniques. We want to consider the  intersection of $ (\\partial\\theta)_{\\rm null}$ with $\\partial{\\mathcal A}_{g+1}$, the corank one boundary component of any toroidal compactification of ${\\mathcal A}_{g+1}$. A point of $\\partial{\\mathcal A}_{g+1}$ is a semiabelian variety of torus rank one, i.e. a pair $(\\overline{G}, D)$, where $\\overline{G}$ is a non-normal compactification of the extension $1\\to{\\mathbb G}_m\\to G\\to B\\to 0$ for some $B\\in{\\mathcal A}_g$, and $D$ is an ample divisor that is the limit of the theta divisors. The restriction of the map\n\\begin{equation} \\label{fib}\n \\pi_{|\\partial{\\mathcal A}_{g+1}}:\\partial{\\mathcal A}_{g+1}\\to{\\mathcal A}_{g}\n\\end{equation}\nhas $B\/Aut (B, \\Xi)$ as fiber over $(B,\\Xi)$. This means that the\nfiber over a general $(B,\\Xi)\\in{\\mathcal A}_{g}$ is the Kummer variety\n$B\/\\pm 1$.\n\nStudying the boundary behavior of the universal theta divisor and using the Fourier-Jacobi expansion similarly to \\cite{mu}, we get the following\n\\begin{prop}\n\\begin{equation}\\label {inter0}\n (\\partial\\theta)_{\\rm null}^{(g+1)}\\cap\\partial{\\mathcal A}_{g+1}=\\left(\\bigcup_{ (\\overline{G},D)}2_B({\\mathcal S}^{(g)}\\cap\\pi^{-1}((B,\\Xi))\\right)\\cup \\left(\\pi^{-1} (\\partial\\theta)_{\\rm null}^{(g)}\\cap\\Theta_g\\right)\n \\end{equation}\n\\end{prop}\n\\begin{proof}\nAnalytically   we consider the Fourier-Jacobi expansion of\n$$\n \\operatorname{grad}\\tt\\epsilon\\delta(Z,0)\\quad{\\rm with}\\quad Z=\\left(\\begin{array}{cc}\n\\tau&z\\\\\nz'&w\n \\end{array}\\right)\n$$\nalong the boundary components. Since the symplectic group acts transitively on the boundary components and on the odd characteristics, considering the  Fourier-Jacobi expansion of  a fixed $\\operatorname{grad}\\tt\\epsilon\\delta$ at all boundary components is equivalent to considering the Fourier-Jacobi expansion of $\\operatorname{grad}\\tt\\epsilon\\delta$ for all possible $[\\epsilon,\\delta]$ at the standard boundary component where the period matrix element $\\tau_{g+1,g+1}$ goes to $i\\infty$. It is easy to check that the Fourier-Jacobi expansions are of two types depending on whether the $(g+1)$'th entry  of the characteristic is 0 and 1. If it is 1  one we get\n$$\n \\partial_i\\,  \\tt \\epsilon\\delta(Z, 0) =\\partial_i\\tt {\\alpha}{\\beta}(\\tau, z\/2)exp(i\\pi w\/4)+\\dots\\quad {\\rm when}\\,i=1,\\dots, g\n$$\n$$\n \\partial_{g+1}\\,  \\tt \\epsilon\\delta(Z, 0) =\\tt {\\alpha}{\\beta}(\\tau, z\/2)exp(i\\pi w\/4)+\\dots\n$$\nIf it is 0 we get\n$$\n  \\partial_i\\,  \\tt \\epsilon\\delta(Z, 0) =\\partial_i\\tt {\\alpha}{\\beta}(\\tau, 0)+\\dots\n$$\n$$\n  \\partial_{g+1}\\,  \\tt \\epsilon\\delta(Z, 0) =\\tt {\\alpha}{\\beta}(\\tau, z)exp(i\\pi w)+\\dots\n$$\nNow the intersection  of $\\operatorname{grad}\\tt\\epsilon\\delta=0$ with the blow up of the first boundary components gives $2_{X_{\\tau}}(\\operatorname{Sing} (\\Theta_{\\tau}))$ along each fiber, and $\\Theta_{\\tau}$ along $p^{-1}((\\partial\\theta)_{\\rm null}^{(g)})$\n\\end{proof}\n\nNotice that by results of Keel and Sadun \\cite{ke}, any component of $(\\partial\\theta)_{\\rm null}^{(g)}$, which is a priori of codimension at most $g$, must intersect the boundary of any toroidal compactification of ${\\mathcal A}_g$. We can handle the degenerations only for the case when the components intersect the torus rank one boundary component, i.e. when they intersect the boundary of the partial compactification.\n\\begin{thm}\\label{boun}\nIf all irreducible components of $(\\partial\\theta)_{\\rm null}^{(g)}$  meet $\\partial{\\mathcal A}_g$ (the boundary of the partial compactification),  and $\\operatorname{codim}_{{\\mathcal A}_{g-1}} (\\partial\\theta)^{(g-1)}_{\\rm null}=g-1$, then   $\\operatorname{codim}_{{\\mathcal A}_g} (\\partial\\theta)_{\\rm null}=g$, i.e. if for any $g$ all irreducible components of $(\\partial\\theta)_{\\rm null}^{(g)}$ meet $\\partial{\\mathcal A}_g$, then conjecture \\ref{dthetanull} holds.\n\\end{thm}\n\\begin{proof}\nObviously we  know that the codimension is less or equal to $g$.\nThe first case  in the intersection above gives a subvariety of codimension $g$. By induction the second case also gives a subvariety of codimension $g$, since $(\\partial\\theta)_{\\rm null}^{(g-1)}$ has codimension $g-1$ (and thus also its preimage under $\\pi^{-1}$), and we are then intersecting with a divisor.\n\\end{proof}\n\nOne can also conjecture the codimensions of the other loci we study.\n\\begin{conj} \\label{co1}\nFor any $g\\geq 3$\n$$\n \\operatorname{codim}_{{\\mathcal X}_g}\\T_{\\rm sing}=2g.\n$$\n\\end{conj}\n\n\\begin{conj} \\label{co2}\nFor any $g\\geq 4$\n$$\n  \\operatorname{codim}_{{\\mathcal A}_g}(\\partial^{2}\\theta)_{\\rm null}= 2g-2.\n$$\n\\end{conj}\n\nWe will give a degeneration argument relating these two conjectures.\n\\begin{thm}\\label{muthm}\n\\begin{equation}\\label {inter}\n (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}\\cap\\partial{\\mathcal A}_{g+1}=\\left(\\bigcup_{ (\\overline{G},D)}2_B(\\T_{\\rm sing}\\vert_{\\Xi})\\right)\\cup (\\pi^{-1}(\\partial^2\\theta)_{\\rm null}^{(g)}\\cap {\\mathcal S}^{(g)})\n \\end{equation}\n\\end{thm}\n\\begin{proof}\nThe proof is analogous to the proof of the identity (\\ref{inter0}), if one considers the second derivatives as well. There are no new technicalities or ideas involved, and we omit the details here.\n\\end{proof}\n\n\n\\begin{rem}\nThis intersection give us a rough estimate for the dimension of $ (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}$. In fact the intersection  of any irreducible components of  $ (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}$,  meeting  the first  boundary component $\\partial{\\mathcal A}_{g+1}$ have dimension less or equal to $\\operatorname{dim}\\,{\\mathcal S}^{(g)}=(1\/2)g(g+1)-1$. This implies that all\nirreducible components of  $ (\\partial^{2}\\theta)_{\\rm null}^{(g+1)}$  meeting  the first  boundary component have codimension at least\n$g+1$.\n\\end{rem}\n\nWe can relate the two conjectures by the following theorem\n\\begin{thm}\nIf for any $k\\in[4,g+1]$ all components of $(\\partial^2\\theta)_{\\rm null}^k$ intersect $\\partial{\\mathcal A}_{k}$, then conjecture $\\ref{co2}$ is true for all $k\\in[4,g+1]$ if and only if conjecture $\\ref{co1}$ is true for all $k\\in[3,g]$.\n\\end{thm}\n\\begin{proof}\nWe use the description of the boundary of the locus $(\\partial^2\\theta)_{\\rm null}$ from formula (\\ref{inter}). We know a priori that $\\T_{\\rm sing}^{(g)}$ has codimension at most $2g$ (since its projection to ${\\mathcal A}_g$  contains $(\\partial\\theta)_{\\rm null}$, which is of codimension at most $g$). Similarly we know a priori that $\\operatorname{codim}(\\partial^2\\theta)_{\\rm null}\\le 2g-2$, since this locus contains $\\t_{\\rm null}^{(g-2)}\\times{\\mathcal A}_1\\times{\\mathcal A}_1$.\n\nOne implication is easy: if conjecture \\ref{co2} holds for $k=g+1$, i.e. $\\operatorname{codim}_{{\\mathcal A}_{g+1}}(\\partial^2\\theta)_{\\rm null}^{(g+1)}=2g$, then $\\operatorname{codim}_{\\partial{\\mathcal A}_{g+1}}\\left((\\partial^2\\theta)_{\\rm null}^{(g+1)}\\cap \\partial{\\mathcal A}_{g+1}\\right)=2g$ (we know from (\\ref{inter}) that this intersection with an open divisor $\\partial{\\mathcal A}_{g+1}\\subset\\bar{{\\mathcal A}_{g+1}}$ is non-empty, so it has codimension one in $(\\partial^2\\theta)_{\\rm null}^{(g+1)}$), and thus the codimension of its component $2_B(\\T_{\\rm sing}^{(g)})$ cannot be less than $2g$.\n\nFor the other direction, assume that conjecture \\ref{co1} holds for $k=g$, i.e. that $\\operatorname{codim}_{{\\mathcal X}_g}(\\T_{\\rm sing})=2g$, and also inductively assume that we know that $\\operatorname{codim} (\\partial^2\\theta)_{\\rm null}^{(g)}=2g-2$. Since for any $\\tau\\in{\\mathcal A}_g$ we have $\\operatorname{codim}_{X_\\tau} {\\mathcal S}\\cap X_\\tau\\ge 2$ (a generic point of $\\Theta_{\\tau}$ is smooth). Thus we have $\\operatorname{codim}_{{\\mathcal X}_g} \\pi^{-1}(\\partial^2\\theta_{\\rm null}^{(g)}\\cap{\\mathcal S}^{(g)})\\ge 2g$, and thus $\\operatorname{codim}_{{\\mathcal A}_{g+1}}(\\partial^2\\theta)_{\\rm null}^{(g+1)}=2g$.\n\nFor the base of the induction, observe that the locus $\\T_{\\rm sing}^{(3)}$ consists of the family of odd points of order two over ${\\mathcal A}_1\\times{\\mathcal A}_1\\times{\\mathcal A}_1 $ and so has codimension $6$ in ${\\mathcal X}_3$, while $(\\partial^2\\theta)_{\\rm null}^{(4)}={\\mathcal A}_1\\times{\\mathcal A}_1\\times{\\mathcal A}_1\\times{\\mathcal A}_1=\\t_{\\rm null}^{(2)}\\times{\\mathcal A}_1\\times{\\mathcal A}_1$ also has codimension $6$ (see theorem \\ref{irred}).\n\\end{proof}\n\n\n\\begin{rem}\nIn lemma 9 of \\cite{sing} we proved and used  that $ (\\partial^{2}\\theta)_{\\rm null}$ has codimension greater than $2$ in ${\\mathcal A}_{g+1}$. Our proof quoted a result of  \\cite{amsp} the proof of theorem 8.11 in \\cite{amsp}, while it seems to us the statement in theorem 8.6(ii) of \\cite{amsp} yields this more directly, implying that $3\\le\\operatorname{codim} G_0\\le\\operatorname{codim} (\\partial^{2}\\theta)_{\\rm null}$.\n\\end{rem}\n\n\\medskip\nIn analogy with the situation for Andreotti-Mayer loci, it is natural to make the following\n\\begin{conj} \\label{co3}\nFor any $g\\geq 3$\n$$\n \\operatorname{codim} G_k \\ge g+k\n$$\n\\end{conj}\n\\begin{rem}\nNotice that we know that the locus $(\\partial\\theta)_{\\rm null}$ is by definition a subset of $G_0$; however, $(\\partial\\theta)_{\\rm null}$ is locally given by the vanishing of a gradient of an odd theta function with characteristic at zero, i.e. is given by $g$ equations, and thus we have $g\\le\\operatorname{codim} (\\partial\\theta)_{\\rm null}\\le\\operatorname{codim} G_0$, so the above conjecture implies that $\\operatorname{codim} G_0=\\operatorname{codim} (\\partial\\theta)_{\\rm null}=g$.\n\\end{rem}\n\n\\begin{rem}\nIn fact it seems likely that the locus $(\\partial\\theta)_{\\rm null}$ is of pure codimension $g$, while it seems less likely that $G_0$ is of pure dimension: the second derivatives of theta automatically vanish at odd points of order two, but not generically on ${\\mathcal X}_g$, and thus it is natural to expect that there exist components of $G_0\\setminus(\\partial\\theta)_{\\rm null}$ are of codimension higher than $g$.\n\\end{rem}\n\\begin{rem}\\label{gminus3}\nIn the notations of \\cite{amsp} the loci $N_{g,k,3}=G_k^{(g)}$ are studied, and the following results are know for them. Theorem 8.6(ii) in \\cite{amsp} then says that $\\operatorname{codim} G_k\\ge k+3$, while theorem 7.5(ii) in \\cite{amsp} in our notations is that $$G_{g-3}^{(g)}=\\mathop{\\bigcup}\\limits_{g_1+g_2+g_3=g,\n\\ g_1g_2g_3>0}{\\mathcal A}_{g_1}\\times{\\mathcal A}_{g_2}\\times{\\mathcal A}_{g_3},$$\nwhich is not purely dimensional, but has codimension $2g-3$ --- so that conjecture \\ref{co3} is true for $k=g-3$.\n\\end{rem}\n\nThe statement  about the codimension of $G_k$ seems to be   rather convincing, but we could not find a proof.  Note that the argument to prove $\\operatorname{codim} N_1>1$ given by Mumford in \\cite{mu} uses an involved heat equations argument, while Ciliberto and van der Geer in \\cite{cilvdg}, \\cite{amsp} show, with a lot of work, that $\\operatorname{codim} N_k\\ge k+2$. The basic question seems to be whether the locus $\\T_{\\rm sing}$ is pure-dimensional or not, and the relationship of the conjecture is the following\n\\begin{prop}\nConjecture \\ref{co1}, for a fixed $g$, is equivalent to conjecture \\ref{co3}, for the same $g$, and all $k$.\n\\end{prop}\n\\begin{proof}\nIndeed recall that by definition the map $\\pi$ restricted to $\\T_{\\rm sing}\\cap\\pi^{-1}(G_k)$ has fiber dimension $k$, and thus $\\operatorname{dim}\\T_{\\rm sing}\\ge k+\\operatorname{dim} G_k$. Thus if we know that $\\operatorname{codim}_{{\\mathcal X}_g}\\T_{\\rm sing}=2g$, it follows that $\\operatorname{codim}_{{\\mathcal A}_g}G_k\\ge g+k$. In the other direction, if $\\operatorname{codim} G_k\\ge g+k$ for all $k$, we have $\\operatorname{dim}\\T_{\\rm sing}=\\max_k(k+\\operatorname{dim} G_k)\\le \\max_k(k+\\operatorname{dim}{{\\mathcal A}_g}-g-k)$, and thus $\\operatorname{codim}\\T_{\\rm sing}\\ge 2g$, in which case it must be equal to $2g$.\n\\end{proof}\n\nNow we would like to give some evidence for the validity of these conjectures. Note that if $z=(\\tau\\epsilon+\\delta)\/2\\in X_\\tau[2]$ is an odd point, then $0=\\vartheta(\\tau,z)=\\partial_i\\partial_j\\vartheta(\\tau,z)$ automatically as the value and derivatives of an odd function, and thus the locus\n$$\n Y:=\\{(\\tau,z)\\in{\\mathcal X}_g \\mid \\operatorname{grad} \\vartheta(\\tau,z)=0;\\ z=(\\tau\\epsilon+\\delta)\/2\\}\n$$\nis a subset of $\\T_{\\rm sing}\\subset{\\mathcal X}_g$, while the projection $\\pi(Y)=(\\partial\\theta)_{\\rm null}\\subset{\\mathcal A}_g$ (notice that by definition $(\\partial\\theta)_{\\rm null}$ is the union of such projections over all odd $[\\epsilon,\\delta]$, but they are all conjugate under $\\operatorname{Sp}(g,\\Z)$, and thus have the same image on ${\\mathcal A}_g$). The expected codimension of $Y$ in ${\\mathcal X}_g$ is equal to $2g$ ($g$ for fixing a point on $X_\\tau$, and $g$ for the vanishing of the gradient).\n\\begin{thm}\\label{tch}\nLet $Z$ be a reduced irreducible component of $\\T_{\\rm sing}$ that is contained in $Y$ as above. Then $Z$ has codimension $2g$ in ${\\mathcal X}_g$ and $\\pi(Z)$ has codimension $g$ in ${\\mathcal A}_g$ (and is thus an irreducible component of $(\\partial\\theta)_{\\rm null}$).\n\\end{thm}\n\\begin{proof}\nWe apply  the jacobian criterion, in a smooth point  of $Z$. As in the theorem \\ref{reduc}, the dimension of the tangent space to $\\T_{\\rm sing}$ is $(1\/2)g(g+1)-g+2k$ if $rk (\\partial_i\\partial_j\\partial_k\\vartheta)=g-k .$\n\nOn the other hand, the normal space to $Y$ at the same point is given by the matrix\n$$\n  \\left(\\begin{array}{cc }\n  0&1_g\\\\\n  (\\partial_i\\partial_j\\partial_k\\vartheta)'&M(\\epsilon)\n  \\end{array}\\right)\n$$\nwith $M(\\epsilon)$  a $(1\/2)g(g+1)\\times g$   matrix  depending on $\\epsilon$. This matrix has rank $2g-k$, hence the dimension of the tangent space is $(1\/2)g(g+1)-g+k$. Since the  two computations    give the same result , we get $k=0$.  We get that the codimension of $\\pi(Y)$ is at least $g$. Since on a suitable covering of ${\\mathcal A}$ we have that this locus is defined by the equations\n$$\n \\partial_i \\tt \\epsilon\\delta(\\tau, 0)=0,\n$$\nWe have that the codimension is at most $g$, hence it is exactly $g$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThroughout the paper, a domain is a not necessarily commutative unital ring in which the zero element is the unique zero divisor.\nIn \\cite{claborn66}, Claborn showed that every abelian group $G$ is the class group of a commutative Dedekind domain. An exposition is contained in \\cite[Chapter III \\S14]{fossum73}.\nSimilar existence results, yielding commutative Dedekind domains which are more geometric, respectively number theoretic, in nature, were obtained by Leedham-Green in \\cite{leedham-green72} and Rosen in \\cite{rosen73,rosen76}.\nRecently, Clark in \\cite{clark09} showed that every abelian group is the class group of an elliptic commutative Dedekind domain, and that this domain can be chosen to be the integral closure of a PID in a quadratic field extension.\nSee Clark's article for an overview of his and earlier results.\nIn commutative multiplicative ideal theory also the distribution of nonzero prime ideals within the ideal classes plays an important role.\nFor an overview of realization results in this direction see \\cite[Chapter 3.7c]{ghk06}.\n\nA ring $R$ is a Dedekind prime ring if every nonzero submodule of a (left or right) progenerator is a progenerator (see \\cite[Chapter 5]{mcconnell-robson01}).\nEquivalently, $R$ is a hereditary Noetherian prime ring which is also a maximal order in its simple Artinian quotient ring.\nA Dedekind domain is a Dedekind prime ring $R$ which is also a domain (equivalently, $\\udim R_R = \\udim {}_R R = 1$).\nTo a Dedekind prime ring $R$ one can associate an (abelian) class group $G(R)$ in such a way that $K_0(R) \\cong G(R) \\times \\bZ$.\nEquivalently, $G(R)$ can also be interpreted as a group of stable isomorphism classes of essential right ideals of $R$.\nSince $K_0$ is Morita invariant, the same holds for the class group.\nEvery Dedekind prime ring is Morita equivalent to a Dedekind domain.\n\nRealization questions for class groups within the class of strictly noncommutative Dedekind prime rings have an easy answer.\nIf $R$ is a commutative Dedekind domain with class group $G(R)$ and $M$ is a finitely generated projective $R$-module, then $S=\\End_R(M)$ is a Dedekind prime ring with $G(S) \\cong G(R)$.\nHowever, $S$ is a PI ring, and thus in many aspects close to being commutative.\n\nOn the other hand, there exist Dedekind prime rings (and domains) of a very different nature.\nFor instance, the first Weyl algebra $A_1(K)$ over a field $K$ of characteristic $0$ is a simple Dedekind domain with trivial class group.\nThe ring $R=\\bR[X,Y]\/(X^2+Y^2-1)$ is a commutative Dedekind domain with $G(R) \\cong \\bZ\/2\\bZ$.\nIf $\\sigma \\in \\Aut(R)$ denotes the automorphism induced by the rotation by an irrational angle, then the skew Laurent polynomial ring $T = R[x,x^{-1};\\sigma]$ is a noncommutative Dedekind domain with $G(T) \\cong \\bZ\/2\\bZ$.\nSimilar constructions exist that show that $\\bZ^n$ for $n \\in \\bN_0$ appears as class group of a noncommutative Dedekind prime ring.\n(See \\cite[\\S7.11 and \\S12.7]{mcconnell-robson01} for details.)\n\nThe mentioned rings are not Morita equivalent to commutative Dedekind domains and they are all simple rings.\nIn fact, in \\cite{goodearl-stafford05}, a striking dichotomy is established: A Dedekind domain which is finitely generated as an algebra over $\\bC$ is commutative or simple.\nMore generally, if $K$ is a field and a $K$-algebra $R$ is a Dedekind prime ring such that $\\dim_K R < \\card{K}$ and $R \\otimes_K \\overline K$ is Noetherian, then $R$ is a PI ring or simple.\n\nIn \\cite[Problem 54.7]{levy-robson11}, Levy and Robson state it as an open problem to determine which abelian groups can appear as class groups of simple Dedekind prime rings.\nThe present paper answers this question by showing that any abelian group can be realized as the class group of a simple Dedekind domain.\nThe main theorem we prove is the following.\n\n\\begin{thm} \\label{t-main}\n  Let $G$ be an abelian group, $K$ a field, and $\\kappa$ a cardinal.\n  Then there exists a $K$-algebra $T$ which is a noncommutative simple Dedekind domain, $G(T)\\cong G$, and each class of $G(T)$ contains at least $\\kappa$ maximal right ideals of $T$.\n\\end{thm}\n\nSimple noncommutative Dedekind domains are canonically obtained either as skew Laurent polynomial rings $R[x,x^{-1};\\sigma]$ or as skew polynomial rings $R[x;\\delta]$, where $R$ is a commutative Dedekind domain and $\\sigma$ is an automorphism, respectively $\\delta$ a derivation.\nThe domains we construct are skew Laurent polynomial rings.\nIt is well understood how class groups behave under this extension.\nIn this way, the problem reduces to the construction of a commutative Dedekind domain $R$ with prescribed class group and automorphism $\\sigma$ of $R$.\nThe automorphism $\\sigma$ must be such that no proper nonzero ideal $\\fa$ is $\\sigma$-stable (that is, $\\sigma(\\fa) = \\fa$), but such that the induced automorphism on the class group of $R$ is trivial.\n\nThe actual construction is very conceptual in nature and proceeds through the following steps:\n\n\\begin{enumerate}\n  \\item Construct a commutative Krull monoid with class group $G$ and a monoid automorphism $\\tau$ of $H$ such that no nonempty proper divisorial ideal of $H$ is $\\tau$-stable.\n  \\item Extend $\\tau$ to $K[H]$.\n    The semigroup algebra $K[H]$ is a commutative Krull domain with class group isomorphic to $G$.\n    The crucial step lies in establishing that no nonzero proper divisorial ideal of $K[H]$ is $\\tau$-stable.\n  \\item A suitable localization $R=S^{-1}K[H]$ is a commutative Dedekind domain, has the same class group as $K[H]$, and $\\tau$ extends to $R$.\n    This is analogous to the same step in Claborn's proof.\n  \\item The skew Laurent polynomial ring $T=R[x,x^{-1};\\tau]$ is a noncommutative simple Dedekind domain with $G(T) \\cong G$.\n\\end{enumerate}\n\nThe methods work in greater generality.\nFor instance, the field $K$ can be replaced by a commutative Krull domain with suitable automorphism.\nThe full result is stated in \\cref{t-extend}.\n\\Cref{t-main} is an immediate consequence of \\cref{t-ex-mon-aut} and \\cref{t-extend}.\nThe actual construction is mostly commutative in nature.\nBefore giving the proofs in \\cref{s-proofs}, a number of preliminary results are recalled in \\cref{s-prelim}.\n\n\\begin{remark}\n  Let $R$ be a commutative Dedekind domain which is an affine algebra over a field $K$ of characteristic $0$.\n  Then the ring of differential operators $\\cD(R)$ is a simple Dedekind domain, and the inclusion $R \\hookrightarrow \\cD(R)$ induces an isomorphism $K_0(R) \\cong K_0(\\cD(R))$ (see \\cite[Chapter 15]{mcconnell-robson01}).\n  This induces an isomorphism $G(R) \\cong G(\\cD(R))$.\n  In \\cite{rosen73}, Rosen has shown that any finitely generated abelian group is the class group of a commutative Dedekind domain which is affine over a number field.\n  This gives a different way of showing that any finitely generated abelian group is the class group of a simple Dedekind domain.\n  Using the results from \\cite{clark09}, this can be extended to groups of the form $F\/H$ where $F$ is free abelian and $H$ is a finitely generated subgroup.\n\\end{remark}\n\n\\section{Background: Krull monoids and skew Laurent polynomial rings} \\label{s-prelim}\n\nAll rings and modules are unital.\nRing homomorphisms preserve the multiplicative identity.\nIf $X$ is a subset of a domain, we set $X^\\bullet=X \\setminus \\{0\\}$.\nA \\emph{monoid} is a cancellative semigroup with a neutral element.\nMonoid homomorphisms preserve the neutral element.\nIf $H$ is a monoid, $H^\\times$ denotes its group of units.\n$H$ is \\emph{reduced} if $H^\\times = \\{1\\}$.\nA commutative monoid is \\emph{torsion-free} if its quotient group is torsion-free.\n$\\bN$ denotes the set of positive integers and $\\bN_0$ the set of all nonnegative integers.\nFor sets $A$ and $B$, inclusion is denoted by $A \\subset B$ and strict inclusion by $A \\subsetneq B$.\n\nFor a set $P$, let $\\cF(P)$ denote the multiplicatively written free abelian monoid with basis $P$.\nThe quotient group $\\quo(\\cF(P))$ of $\\cF(P)$ is the free abelian group with basis $P$.\nEach $a \\in \\quo(\\cF(P))$ has a unique (up to order) representation of the form $a = p_1^{n_1}\\cdots p_r^{n_r}$ with $r \\in \\bN_0$, pairwise distinct $p_1$,~$\\ldots\\,$,~$p_r \\in P$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\nWe define $\\supp(a) = \\{p_1,\\ldots, p_r\\}$, $\\val_{p_i}(a)=n_i$ for $i \\in [1,r]$ and $\\val_q(a) = 0$ for all $q \\in P \\setminus \\supp(a)$.\n\n\\subsection{Commutative Krull monoids and commutative Krull domains}\nWe use \\cite[Chapter 2]{ghk06} as a reference for commutative Krull monoids and \\cite{fossum73} as reference for commutative Krull domains.\nFor semigroup algebras we refer to \\cite{gilmer84}.\n\nLet $(H,\\cdot)$ be a commutative monoid and let $\\tau \\in \\Aut(H)$.\nLet $\\quo(H)$ denote the quotient group of $H$.\nThe automorphism $\\tau$ naturally extends to an automorphism of $\\quo(H)$, which we also denote by $\\tau$.\nFor subsets $X$,~$Y \\subset \\quo(H)$ we define $\\cc{Y}{X} = \\{\\, a \\in \\quo(H) \\mid aX \\subset Y \\,\\}$.\nWe set $X^{-1} = \\cc{H}{X}$ and $X_v = (X^{-1})^{-1}$.\nThen $\\tau\\big({\\cc{Y}{X}}\\big) = \\cc{\\tau(Y)}{\\tau(X)}$ and hence $\\tau(X^{-1}) = \\tau(X)^{-1}$ and $\\tau(X)_v = \\tau( X_v )$.\n\nA subset $\\fa \\subset \\quo(H)$ is a \\emph{fractional ideal} of $H$ if $H\\fa \\subset \\fa$ and there exists a $d \\in H$ such that $d\\fa \\subset H$.\nIf in addition $\\fa \\subset H$, then $\\fa$ is an \\emph{ideal} of $H$.\nA fractional ideal $\\fa$ is \\emph{divisorial} if $\\fa = \\fa_v$.\nFor all $a \\in \\quo(H)$, $(aH)_v = aH$ and hence principal fractional ideals are divisorial.\nIf $\\fa$ and $\\fb$ are divisorial fractional ideals of $H$, their \\emph{divisorial product} is $\\fa \\cdot_v \\fb = (\\fa \\cdot \\fb)_v$.\nFor principal fractional ideals, the divisorial product coincides with the usual ideal product.\n\n$H$ is a \\emph{commutative Krull monoid} if it is $v$-Noetherian (i.e., satisfies the ascending chain condition on divisorial ideals) and completely integrally closed (i.e, whenever $x \\in \\quo(H)$ is such that there exists a $c \\in H$ such that $cx^n \\in H$ for all $n \\in \\bN$, then already $x \\in H$).\nFrom now on, let $H$ be a commutative Krull monoid.\nIf $\\fa$ is a nonempty divisorial fractional ideal of $H$, then $\\fa$ is invertible with respect to the divisorial product, i.e., $\\fa \\cdot_v \\fa^{-1} = H$.\nWe denote by $\\cF_v(H)^\\times$ the group of all nonempty divisorial fractional ideals, and by $\\cI_v^*(H)$ the monoid of all nonempty divisorial ideals.\nLet $\\mathfrak X(H)$ be the set of nonempty divisorial prime ideals.\nRecall that $\\mathfrak X(H)$ consists precisely of the prime ideals of height $1$.\n\nWith respect to the divisorial product, $\\cI_v^*(H)$ is the free abelian monoid with basis $\\mathfrak X(H)$, and $\\cF_v(H)^\\times$ is the free abelian group with basis $\\mathfrak X(H)$.\nHence, every $\\fa \\in \\cF_v(H)^\\times$ has a unique representation of the form\n\\[\n\\fa = \\fp_1^{n_1} \\cdot_v \\ldots \\cdot_v \\fp_r^{n_r}\n\\]\nwith $r \\in \\bN_0$, pairwise distinct $\\fp_1$,~$\\ldots\\,$,~$\\fp_r \\in \\mathfrak X(H)$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\nWe have $\\supp(\\fa) = \\{\\fp_1,\\ldots,\\fp_r\\}$ and $\\val_{\\fp_i}(\\fa)= n_i$ for $i \\in [1,r]$.\n\nThe principal fractional ideals form a subgroup of $\\cF_v(H)^\\times$. The \\emph{class group} of $H$ is the factor group\n\\[\n\\cC(H) = \\cF_v(H)^\\times \/ \\{\\, aH \\mid a \\in \\quo(H) \\,\\}.\n\\]\nWe use additive notation for $\\cC(H)$.\nIf $\\fa \\in \\cF_v(H)^\\times$, we write $[\\fa]=[\\fa]_H$ for its class in $\\cC(H)$.\nIf $\\fa$,~$\\fb \\in \\cF_v(H)^\\times$, then $[\\fa \\cdot_v \\fb] = [\\fa] + [\\fb]$.\n\nAny $\\tau \\in \\Aut(H)$ induces an automorphism $\\tau_*$ of $\\cF_v(H)^\\times$ by means of $\\tau_*(\\fa) = \\tau(\\fa)$.\nThen $\\tau_*(\\mathfrak X(H)) = \\mathfrak X(H)$, the restriction $\\tau_*=\\tau_*|_{\\cI_v^*(H)}$ is a monoid automorphism of $\\cI_v^*(H)$, and $\\tau_*(aH)=\\tau(a)H$ for all $a \\in \\quo(H)$.\nIn particular, $\\tau_*$ induces an automorphism of $\\cC(H)$, also denoted by $\\tau_*$, by means of $\\tau_*([\\fa])=[\\tau_*(\\fa)]$.\n\nA \\emph{commutative Krull domain} is a domain $D$ such that $D^\\bullet$ is a commutative Krull monoid.\nWe use similar notation for Krull domains as we have introduced for Krull monoids.\nIf $\\quo(D)$ denotes the quotient field of $D$ and $X \\subset \\quo(D)$, then $\\cc{D}{X}$ is always additively closed.\nThis implies that there exists an isomorphism\n\\[\n\\cF_v(D)^\\times \\to \\cF_v(D^\\bullet)^\\times,\\quad \\fa \\mapsto \\fa^\\bullet.\n\\]\nConcepts related to divisorial ideals on $D$ correspond to ones on $D^\\bullet$.\nWe make use of this without further mention.\nIn particular, $\\cC(D) \\cong \\cC(D^\\bullet)$ canonically, and we identify.\nA commutative domain $D$ is a Dedekind domain if and only if it is a Krull domain with $\\dim(D) \\le 1$.\nThen every nonzero fractional ideal of $D$ is invertible and hence divisorial.\nIn particular, $\\cC(D)$ is the usual ideal class group of the Dedekind domain.\n\nWe will construct Krull domains from Krull monoids using semigroup algebras.\nThe following result is essential.\n\n\\begin{prop}[{\\cite[Theorem 15.6 and Corollary 16.8]{gilmer84}}] \\label{p-krull-semigroup-domain}\n  Let $D$ be a commutative domain and $H$ a torsion-free commutative monoid.\n  The semigroup algebra $D[H]$ is a Krull domain if and only if $D$ is a Krull domain, $H$ is a Krull monoid, and $H^\\times$ satisfies the ascending chain condition on cyclic subgroups.\n  In this case $\\cC(D[H]) \\cong \\cC(D) \\times \\cC(H)$.\n\\end{prop}\n\nThe isomorphism between $\\cC(D) \\times \\cC(H)$ and $\\cC(D[H])$ is obtained naturally by extending representatives of the divisorial ideal classes in $D$, respectively $H$, to $D[H]$.\nIf $\\fa$ is a fractional ideal of $D$, let $\\fa[H]=\\fa D[H]$ be the extension of $\\fa$ to $D[H]$.\nIt consists of all elements all of whose coefficients are contained in $\\fa$.\nIf $\\fb$ is a fractional ideal of $H$, let $D[\\fb]= \\fb D[H]$ be the extension of $\\fb$ to $D[H]$.\nIt consists of all elements whose support is contained in $\\fb$.\nBy $\\fa[\\fb]$ we denote the fractional ideal whose support is contained in $\\fb$ and whose coefficients are contained in $\\fa$.\nThen $\\fa[\\fb] = \\fa[H] \\cdot D[\\fb]$.\nExplicitly, the isomorphism of class groups is given by\n\\[\n\\cC(D) \\times \\cC(H) \\to \\cC(D[H]), \\quad\n([\\fa]_D, [\\fb]_H)   \\mapsto \\big[ \\fa[\\fb] \\big]_{D[H]}.\n\\]\nLet $\\sigma \\in \\Aut(D)$, $\\tau \\in \\Aut(H)$ and let $\\varphi \\in \\Aut(D[H])$ be the extension of $\\sigma$ and $\\tau$ to $D[H]$ (i.e., $\\varphi|_D=\\sigma$ and $\\varphi|_H = \\tau$).\nUnder the stated isomorphism of the class groups, the automorphism $(\\sigma_*, \\tau_*)$ on $\\cC(D) \\times \\cC(H)$ corresponds to $\\varphi_*$ on $\\cC(D[H])$.\nFrom now on we identify $\\cC(D[H]) \\cong \\cC(D) \\times \\cC(H)$.\n\n\\begin{prop}[Nagata's Theorem, {\\cite[Corollary 7.2]{fossum73}}] \\label{p-nagata}\n  Let $D$ be a commutative Krull domain and $S \\subset D^\\bullet$ a multiplicative subset.\n  Then the localization $S^{-1}D$ is a Krull domain and the map $\\cI_v^*(D) \\to \\cI_v^*(S^{-1}D)$, $\\fa \\mapsto S^{-1}\\fa$ induces an epimorphism $\\cC(D) \\to \\cC(S^{-1}D)$ with kernel generated by those $\\fp \\in \\mathfrak X(D)$ with $\\fp \\cap S \\ne \\emptyset$.\n  In particular, if $S$ is generated by prime elements of $D$, then $\\cC(D) \\cong \\cC(S^{-1}D)$.\n\\end{prop}\n\nLet $D$ be a commutative Krull domain and let $S \\subset D^\\bullet$ be a multiplicative subset.\nThen $S^{-1}D$ is a Dedekind domain if and only if $\\dim(S^{-1}D) \\le 1$.\nThis is the case if and only if $S \\cap \\fP \\ne \\emptyset$ for all $\\fP \\in \\spec(D)$ with $\\height(\\fP) > 1$.\n\n\\subsection{Skew Laurent polynomial rings}\n\nLet $R$ be a ring and $\\sigma \\in \\Aut(R)$.\nBy $R[x,x^{-1};\\sigma]$ we denote the ring of \\emph{skew Laurent polynomials}.\n$R[x,x^{-1};\\sigma]$ consists of polynomial expressions in $x$ and $x^{-1}$ with coefficients in $R$ and subject to $ax=x\\sigma(a)$ for all $a \\in R$.\nLet $\\fa$ be an ideal of $R$.\nIf $\\sigma \\in \\Aut(R)$, then $\\fa$ is \\emph{$\\sigma$-stable} if $\\sigma(\\fa) = \\fa$.\nThe ring $R$ is \\emph{$\\sigma$-simple} if $\\mathbf 0$ and $R$ are the only $\\sigma$-stable ideals of $R$.\n\n\\begin{prop}[{\\cite[Theorem 1.8.5]{mcconnell-robson01}}]\n  Let $R$ be a ring, $\\sigma \\in \\Aut(R)$ and $T = R[x,x^{-1};\\sigma]$.\n  Then $T$ is a simple ring if and only if $R$ is $\\sigma$-simple and no power of $\\sigma$ is an inner automorphism.\n\\end{prop}\n\nIf $R$ is a commutative ring, the identity is the only inner automorphism of $R$.\nHence the second condition in the previous theorem reduces to $\\sigma$ having infinite order.\nIf $R$ is a $\\sigma$-simple commutative domain which is not a field, then $\\sigma$ has infinite order.\nFor suppose $\\sigma^n = \\id$ for some $n \\in \\bN$.\nLet $\\mathbf 0 \\ne \\fa \\subsetneq R$ be an ideal of $R$.\nThen $\\fa\\sigma(\\fa) \\cdots \\sigma^{n-1}(\\fa) \\ne \\mathbf 0$ is a proper ideal of $R$ which is $\\sigma$-stable.\n\nCombining our observations so far with \\cite[Theorem 7.11.2]{mcconnell-robson01}, we obtain the following.\n\\begin{prop}[{\\cite[Theorem 7.11.2]{mcconnell-robson01}}] \\label{p-nc-dedekind}\n  Let $R$ be a commutative Dedekind domain which is not a field, let $\\sigma \\in \\Aut(R)$, and let $T=R[x,x^{-1};\\sigma]$.\n  The following conditions are equivalent:\n  \\begin{equivenumerate}\n    \\item $T$ is simple.\n    \\item $T$ is hereditary.\n    \\item The Krull dimension of $T$ is $1$.\n    \\item $T$ is a noncommutative Dedekind domain.\n    \\item $R$ is $\\sigma$-simple.\n  \\end{equivenumerate}\n\\end{prop}\n\nThe behavior of the Grothendieck group $K_0$ under skew Laurent polynomial extensions is well understood.\nWe denote classes in $K_0$ using angle brackets.\nWe recall the result from \\cite[\\S12.5]{mcconnell-robson01}.\nLet $R$ be a ring and $\\sigma \\in \\Aut(R)$.\nLet $M$ be a right $R$-module.\nDefine a new right $R$-module $M^\\sigma$ as follows:\nAs a set, $M^\\sigma$ is in bijection with $M$, where the element of $M^\\sigma$ corresponding to $m \\in M$ is written as $m^\\sigma$.\nThe abelian group structure on $M^\\sigma$ is the one induced from $M$, i.e., $m^\\sigma + n^\\sigma = (m+n)^\\sigma$.\nThe right $R$-module structure on $M^\\sigma$ is defined by $(m^\\sigma) r = (m \\sigma^{-1}(r))^\\sigma$.\nA similar construction works for left modules: To a left module $M$ associate ${}^\\sigma M$ with $r ({}^\\sigma m) = {}^\\sigma (\\sigma(r) m)$.\nIn particular, ${}^\\sigma R$ with the usual right $R$-module structure is an $R$-bimodule, and $M \\otimes_R ({}^\\sigma R) \\cong M^\\sigma$ as right $R$-modules.\nNow, $\\sigma$ induces an automorphism $\\sigma_*$ of $K_0(R)$ by means of $\\kcls{M} \\mapsto \\kcls{M^\\sigma}$.\n\nLet $T = R[x,x^{-1};\\sigma]$.\nFor a finitely generated projective right $R$-module $M$, $M \\otimes_R T$ is a finitely generated projective right $T$-module.\nThis induces a homomorphism $\\alpha\\colon K_0(R) \\to K_0(T)$.\nA ring $R$ is right regular if each finitely generated right $R$-module has a projective resolution of finite length.\n\n\\begin{prop}[{\\cite[Theorem 12.5.6]{mcconnell-robson01}}] \\label{p-k0-ext}\n  Let $R$ be a right regular, right Noetherian ring.\n  Let $\\sigma \\in \\Aut(R)$ and $T = R[x,x^{-1};\\sigma]$.\n  Then the sequence\n  \\[\n  \\xymatrix@C=1.5cm{\n    K_0(R) \\ar[r]^{\\id-\\sigma_*} & K_0(R) \\ar[r]^{\\alpha} & K_0(T) \\ar[r] & \\mathbf 0\n  }\n  \\]\n  is exact.\n\\end{prop}\n\nLet $R$ be a Dedekind prime ring.\nEach finitely generated projective right $R$-module $P$ has a uniform dimension $\\udim_R (P) \\in \\bN_0$.\nThe uniform dimension is additive on direct sums and induces an epimorphism $\\udim_R \\colon K_0(R) \\to \\bZ$.\nThe \\emph{(ideal) class group} of $R$ is $G(R) = \\ker(\\udim_R\\colon K_0(R) \\to \\bZ)$.\nThe epimorphism $\\udim_R$ splits, hence $K_0(R) \\cong G(R) \\times \\bZ$.\nLet $G'$ denote the set of stable isomorphism classes of essential right ideals of $R$.\n$G'$ can be endowed with the structure of an abelian group by setting $[\\fa] + [\\fb] = [\\fc]$ if and only if $\\fa \\oplus \\fb \\cong R \\oplus \\fc$.\nThen $G'$ is isomorphic to $G(R)$ by means of $G' \\to G(R), [\\fa] \\mapsto \\kcls{\\fa} - \\kcls{R}$, and we identify.\nWhen we say that a class $g \\in G(R)$ contains an essential right ideal $\\fa$ of $R$, we mean $g=[\\fa]=\\langle \\fa \\rangle - \\langle R \\rangle$.\n\nIf $R$ is commutative, $G(R)$ is indeed isomorphic the usual ideal class group.\nThe isomorphism $\\cC(R) \\to G(R)$ is given by $[\\fa] \\mapsto \\kcls{\\fa} - \\kcls{R}$.\nIf $\\sigma$ is an automorphism of $R$, we note that under the stated isomorphism of $\\cC(R)$ and $G(R)$, the induced automorphism $\\sigma_* \\colon \\cC(R) \\to \\cC(R)$ corresponds to $\\sigma_* \\colon G(R) \\to G(R)$.\nThis is so, because for an ideal $\\fa \\subset R$, we have $\\fa^\\sigma \\cong \\sigma(\\fa)$ as right $R$-modules, via $a^\\sigma \\mapsto \\sigma(a)$.\n\nLet $R$ be a commutative Dedekind domain.\nSince $\\udim_T(P \\otimes_R T) = \\udim_R(P)$ for all finitely generated projective $R$-modules $P$, we obtain a commutative diagram\n\\[\n  \\xymatrix@C=1.5cm{\n    K_0(R) \\ar[r]^{\\id-\\sigma_*} \\ar[d] & K_0(R) \\ar[r]^{\\alpha} \\ar[d] & K_0(T) \\ar[r] \\ar[d] & \\mathbf 0 \\\\\n    G(R) \\times \\bZ \\ar[r]^{(\\id-\\sigma_*,\\id)} & G(R) \\times \\bZ \\ar[r]^{(\\alpha_0,\\id)} & G(T) \\times \\bZ \\ar[r] & \\mathbf 0 \\\\\n  }\n\\]\nwith the vertical arrows being isomorphisms induced by the splitting of $\\udim_R$, $\\udim_R$, and $\\udim_T$ respectively.\nHere $\\alpha_0$ is the map induced on $G(R) \\to G(T)$ by $\\alpha$.\nUsing the isomorphism $\\cC(R) \\cong G(R)$, we obtain a short exact sequence\n\\[\n  \\xymatrix@C=1.5cm{\n    \\cC(R) \\ar[r]^{\\id-\\sigma_*} & \\cC(R) \\ar[r]^{\\beta} & G(T) \\ar[r] & \\mathbf 0.\n  }\n\\]\nHere, $\\beta([\\fa]_R) = \\kcls{\\fa \\otimes_R T} - \\kcls{T} = [\\fa \\otimes_R T]_T \\in G(T)$.\n\n\\begin{remark}\n  \\begin{enumerate}\n  \\item\n    If $R$ and $S$ are Morita equivalent Dedekind prime rings, the Morita equivalence induces an isomorphism $K_0(R) \\cong K_0(S)$, which restricts to an isomorphism $G(R) \\cong G(S)$.\n  \\item\n  Let $R$ be a Dedekind prime ring.\n  If $\\fa$ and $\\fb$ are stably isomorphic essential right ideals of $R$, that is $[\\fa]=[\\fb]$ in $G(R)$, then, in general, it does not follow that $\\fa \\cong \\fb$.\n  However, if $\\udim_R R \\ge 2$, then $[\\fa]=[\\fb]$ does imply $\\fa \\cong \\fb$ (\\cite[Corollary 35.6]{levy-robson11}).\n  Note that $S=M_n(R)$, with $n \\ge 2$ is a Dedekind prime ring with $G(R) \\cong G(S)$ and $\\udim_S S =n \\ge 2$.\n\\end{enumerate}\n\\end{remark}\n\n\\section{Construction and main results} \\label{s-proofs}\n\n\\begin{lemma}\n  Let $H$ be a commutative Krull monoid, $\\tau \\in \\Aut(H)$, and $\\fa \\in \\cF_v^\\times(H)$.\n  Then $\\tau(\\fa) = \\fa$ if and only if $\\tau(\\fa) \\subset \\fa$.\n\\end{lemma}\n\n\\begin{proof}\n  Suppose that $\\tau(\\fa) \\subset \\fa$.\n  Then there exists $\\fb \\in \\cI_v^*(H)$ such that $\\tau(\\fa) = \\fa \\cdot_v \\fb$.\n  Let $\\fa = \\fp_1^{n_1} \\cdot_v \\ldots \\cdot_v \\fp_r^{n_r}$ with $r \\in \\bN_0$, $\\fp_1$,~$\\ldots\\,$,~$\\fp_r \\in \\mathfrak X(H)$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\n  Similarly, let $\\fb = \\fq_1^{m_1} \\cdot_v \\ldots \\cdot_v \\fq_s^{m_s}$ with $s \\in \\bN_0$, $\\fq_1$,~$\\ldots\\,$,~$\\fq_s \\in \\mathfrak X(H)$ and $m_1$,~$\\ldots\\,$,~$m_s \\in \\bN$.\n  Then\n  \\[\n  \\tau(\\fa) = \\tau(\\fp_1)^{n_1} \\cdot_v \\ldots \\cdot_v \\tau(\\fp_r)^{n_r} = \\fp_1^{n_1} \\cdot_v \\ldots \\cdot_v \\fp_r^{n_r} \\cdot_v \\fq_1^{m_1} \\cdot_v \\ldots \\cdot_v \\fq_s^{m_s}.\n  \\]\n  Then necessarily $n_1+\\cdots + n_r = n_1 + \\cdots + n_r + m_1+ \\cdots + m_s$.\n  Hence $s=0$ and $\\fb=H$.\n  Thus $\\tau(\\fa) = \\fa$.\n\\end{proof}\n\nOf course, the claim of the previous lemma does not hold for ideals which are not divisorial.\nFor a counterexample, let $K$ be a field, $H=K[...,X_{-1},X_0,X_1,\\ldots]^\\bullet$, $\\tau(X_i) = X_{i+1}$ with $\\tau|_K=\\id$, and $\\fa = (X_0,X_1,\\ldots)$.\n\n\\begin{lemma} \\label{l-simple}\n  Let $H$ be a commutative Krull monoid and let $\\tau \\in \\Aut(H)$.\n  The following statements are equivalent:\n  \\begin{equivenumerate}\n    \\item\\label{l-simple:fracideal} $\\tau(\\fa) \\ne \\fa$ for all $\\fa \\in \\cF_v(H)^\\times \\setminus \\{H\\}$.\n    \\item\\label{l-simple:ideal} $\\tau(\\fa) \\ne \\fa$ for all $\\fa \\in \\cI_v^*(H) \\setminus \\{H\\}$.\n    \\item\\label{l-simple:sqf} $\\tau(\\fa) \\ne \\fa$ for all squarefree $\\fa \\in \\cI_v^*(H) \\setminus \\{H\\}$.\n    \\item\\label{l-simple:prime} For all finite $\\emptyset \\ne X \\subset \\mathfrak X(H)$, it holds that $\\tau_*(X) = \\{\\, \\tau(\\fp) \\mid \\fp \\in X \\,\\} \\ne X$.\n    \\item\\label{l-simple:orbits} The induced permutation $\\tau_*$ of $\\mathfrak X(H)$ has no finite orbits.\n  \\end{equivenumerate}\n  If $\\cC(H)=\\mathbf 0$, then any of the above conditions is equivalent to\n  \\begin{equivenumerate}\n    \\setcounter{enumi}{5}\n    \\item\\label{l-simple:principal} For all $a \\in H\\setminus H^\\times$ and $\\varepsilon \\in H^\\times$, $\\tau(a) \\ne \\varepsilon a$.\n  \\end{equivenumerate}\n\n  In particular, if these equivalent conditions are satisfied and $\\emptyset \\ne A \\subset \\quo(H)$ is finite with $A \\not\\subset H^\\times$, then $\\tau(A) \\ne A$.\n\\end{lemma}\n\n\\begin{proof}\n  \\ref*{l-simple:fracideal}${}\\Rightarrow{}$\\ref*{l-simple:ideal}${}\\Rightarrow{}$\\ref*{l-simple:sqf}: Trivial.\n\n  \\ref*{l-simple:sqf}${}\\Rightarrow{}$\\ref*{l-simple:prime}:\n  By contradiction. Suppose that $\\emptyset \\ne X \\subset \\mathfrak X(H)$ is such that $\\tau_*(X)=X$.\n  Set $\\fa = ( \\prod_{\\fp \\in X} \\fp )_v$.\n  Then $\\fa \\in \\cI_v^*(H) \\setminus \\{H\\}$, $\\fa$ is squarefree, and $\\tau(\\fa) = \\tau\\big( ( \\prod_{\\fp \\in X} \\fp )_v \\big ) = ( \\prod_{\\fp \\in X} \\tau(\\fp) )_v = \\fa$.\n  This contradicts \\ref*{l-simple:sqf}.\n\n  \\ref*{l-simple:prime}${}\\Rightarrow{}$\\ref*{l-simple:orbits}: Clear.\n\n  \\ref*{l-simple:orbits}${}\\Rightarrow{}$\\ref*{l-simple:fracideal}:\n  Let $\\fa \\in \\cF_v^*(H) \\setminus \\{H\\}$.\n  Then $\\fa = \\fp_1^{n_1}\\cdot_v \\ldots \\cdot_v \\fp_r^{n_r}$ with $r \\in \\bN$, $\\fp_1$, $\\ldots\\,$,~$\\fp_r \\in \\mathfrak X(H)$ and $n_1$, $\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\n  Now $\\tau(\\fa) = \\tau(\\fp_1)^{n_1} \\cdot_v \\ldots \\cdot_v \\tau(\\fp_r)^{n_r}$ is the unique representation of $\\tau(\\fa)$ as divisorial product of divisorial prime ideals.\n  Suppose that $\\tau(\\fa)=\\fa$.\n  Then $\\tau^n(\\fa) = \\fa$ for all $n \\in \\bZ$.\n  Hence the $\\tau_*$-orbit of $\\fp_1$ is contained in $\\supp(\\fa) = \\{ \\fp_1,\\ldots,\\fp_r \\}$.\n  This contradicts \\ref*{l-simple:orbits}.\n\n  \\ref*{l-simple:ideal}${}\\Leftrightarrow{}$\\ref*{l-simple:principal}\n  Suppose that $\\cC(H)$ is trivial.\n  Then every divisorial ideal is principal.\n  The claim follows since $aH = bH$ for $a$,~$b \\in H$ if and only if there exists $\\varepsilon \\in H^\\times$ with $a = b \\varepsilon$.\n\n  We still have to show the final implication and do so by contradiction.\n  Let $\\emptyset \\ne A=\\{a_1, \\ldots, a_n\\} \\subset \\quo(H)$ with $A \\not\\subset H^\\times$.\n  Since $A \\not\\subset H^\\times$, the set $X = \\bigcup_{i=1}^n \\supp(a_i H) \\subset \\mathfrak X(H)$ is nonempty.\n  Thus $\\tau_*(X) \\ne X$ by \\ref*{l-simple:prime}, and hence $\\tau(A) \\ne A$.\n\\end{proof}\n\n\\begin{defi}\n  Let $H$ be a commutative Krull monoid and $\\tau \\in \\Aut(H)$.\n  $H$ is called \\emph{$\\tau$-$v$-simple} if the equivalent conditions of \\cref{l-simple} are satisfied.\n  If $D$ is a commutative Krull domain and $\\sigma \\in \\Aut(D)$, then $D$ is called \\emph{$\\sigma$-$v$-simple} if the commutative Krull monoid $D^\\bullet$ is $(\\sigma|_{D^\\bullet})$-$v$-simple.\n\\end{defi}\n\nA lemma analogous to \\cref{l-simple} holds for commutative Krull domains.\nSince there is a correspondence between divisorial ideals of $D$ and divisorial ideals of $D^\\bullet$, $D$ is $\\sigma$-$v$-simple if and only if $\\sigma(\\fa) \\ne \\fa$ for all divisorial ideals $\\fa$ of $D$, etc.\n\nWe first construct a reduced commutative Krull monoid $H$ with given class group $G$, as well as an automorphism of $H$ such that $H$ is $\\tau$-$v$-simple, and such that $\\tau_*$ acts trivially on the class group.\n\n\\begin{thm} \\label{t-ex-mon-aut}\n  Let $G$ be an abelian group and $\\kappa$ an infinite cardinal.\n  Then there exists a reduced commutative Krull monoid $H$ and an automorphism $\\tau$ of $H$ such that $\\cC(H) \\cong G$, $\\tau_*=\\id_{\\cC(H)}$, and $H$ is $\\tau$-$v$-simple.\n  Each class of $\\cC(H)$ contains $\\kappa$ nonempty divisorial prime ideals.\n\\end{thm}\n\n\\begin{proof}\nLet $(G,+)$ be an additive abelian group, and let $\\Omega$ be a set of cardinality $\\kappa$.\nLet $\\tau_0 \\colon \\Omega \\to \\Omega$ be a permutation such that $\\tau_0(X) \\ne X$ for all finite $\\emptyset \\ne X \\subset \\Omega$.\n(Such a permutation always exists. $\\Omega$ is in bijection with $\\Omega \\times \\bZ$, and the map $\\Omega \\times \\bZ \\to \\Omega \\times \\bZ$, $(x,n) \\mapsto (x,n+1)$ has the desired property.)\n\nLet $D = \\cF(\\Omega \\times G)$ be the free abelian monoid with basis $\\Omega \\times G$.\nThen $\\tau_0$ induces an automorphism $\\tau \\in \\Aut(D)$ with the property that $\\tau((x,g)) = (\\tau_0(x),g)$ for all $x \\in \\Omega$ and $g \\in G$.\nLet $\\psi\\colon D \\to G$ be the unique homomorphism such that $\\psi((x,g)) = g$ for all $x \\in \\Omega$ and $g \\in G$.\nSet $H = \\psi^{-1}(0_G)$.\nSince $\\psi(\\tau((x,g))) = g = \\psi((x,g))$, we find $\\tau(H) \\subset H$.\nHence $\\tau$ restricts to an automorphism of $H$, again denoted by $\\tau$.\n\nWe claim that $(H,\\cdot)$ is a reduced commutative Krull monoid with class group $G$, that $H$ is $\\tau$-$v$-simple, and that the induced automorphism $\\tau_*$ of $\\cC(H)$ is the identity.\nMoreover, each class of $\\cC(H)$ contains $\\card{\\Omega}$ divisorial prime ideals.\nThat $H$ is a reduced commutative Krull monoid with class group $G$ follows from \\cite[Proposition 2.5.1.4]{ghk06}.\nIt also follows that the inclusion $\\iota\\colon H \\hookrightarrow D$ is a divisor theory.\nHence, $\\mathfrak X(H) = \\{\\, (x,g)D \\cap H \\mid x \\in \\Omega,\\; g \\in G \\,\\}$.\nBy construction, $\\tau$ does not fix any finite nonempty subset of $\\mathfrak X(H)$, and hence $H$ is $\\tau$-$v$-simple.\nOn the other hand, $\\psi(\\tau(x,g)) = \\psi((x,g)) = g$, so that $\\tau_*$ acts trivially on $\\cC(H)$.\n\\end{proof}\n\n\\begin{remark}\n  Let $H$ be a reduced commutative Krull monoid.\n  We note that it is easy to determine $\\Aut(H)$.\n  Let $\\tau \\in \\Aut(H)$.\n  Then $\\tau$ induces an automorphism $\\tau_*$ of $\\cI_v^*(H)$ and further an automorphism of $\\cC(H)$, that we denote by $\\tau_*$ again.\n  For $g \\in \\cC(H)$, denote by $\\mathfrak X(H)(g) = \\{\\, \\fp \\in \\mathfrak X(H) \\mid [\\fp]=g \\,\\}$ the nonempty divisorial prime ideals in class $g$.\n  For all $\\fp \\in \\mathfrak X(H)$, it holds that $[\\tau_*(\\fp)] = \\tau_*([\\fp])$.\n  In particular, if $g$ and $h \\in \\cC(H)$ lie in the same $\\tau_*$-orbit, then $\\card{\\mathfrak X(H)(g)} = \\card{\\mathfrak X(H)(h)}$.\n  The automorphism $\\tau$ is uniquely determined by the induced $\\tau_* \\in \\Aut(\\cC(H))$ as well as the family of bijections $\\mathfrak X(H)(g) \\to \\mathfrak X(H)(\\tau_*(g))$ induced by $\\tau_*$.\n\n  Conversely, suppose that $\\alpha$ is an automorphism of $\\cC(H)$ such that for all $g \\in \\cC(H)$, $\\card{\\mathfrak X(H)(g)}=\\card{\\mathfrak X(H)(\\alpha(g))}$.\n  For each class $g \\in \\cC(H)$, let $\\beta_g \\colon \\mathfrak X(H)(g) \\to \\mathfrak X(H)(\\alpha(g))$ be a bijection.\n  Then there exists a (uniquely determined) automorphism $\\tau \\in \\Aut(H)$ with $\\tau_*(\\fp) = \\beta_g(\\fp)$ for all $g \\in \\cC(H)$ and $\\fp \\in \\mathfrak X(H)(g)$.\n\n  In particular, we obtain the following strengthening of \\cref{t-ex-mon-aut}:\n  If $H$ is a commutative Krull monoid such that each class contains either zero or infinitely many divisorial prime ideals, then there exists a $\\tau \\in \\Aut(H)$ such that $H$ is $\\tau$-$v$-simple and $\\tau_*$ is the identity on $\\cC(H)$.\n\\end{remark}\n\nLet $P$ be a set, $\\cF(P)$ the (multiplicatively written) free abelian monoid with basis $P$, and $G=\\quo(\\cF(P))$ the free abelian group with basis $P$.\nSince every element of $G\\cong \\bZ^{(P)}$ has finite support, any total order on $P$ induces a total order on $G$ by means of the lexicographical order and the natural total order on $\\bZ$.\nExplicitly, for $a \\in G \\setminus \\{1\\}$ we define $a \\ge 1$ if and only if $\\val_p(a) \\ge 0$ for $p = \\max\\supp(a)$.\nWith respect to any such order, $G$ is a totally ordered group.\nIf $(G,\\cdot,\\le)$ is a totally ordered group, we set $G_{>1} = \\{\\, a \\in G \\mid a > 1 \\,\\}$ and $G_{\\ge 1} = \\{\\, a \\in G \\mid a \\ge 1 \\,\\}$.\n\n\\begin{lemma} \\label{l-ex-order}\n  \\begin{enumerate}\n    \\item\\label{l-ex-order:set} Let $P$ be a set and let $\\tau\\colon P \\to P$ be a permutation having no finite orbits.\n      Then there exists a total order $\\le$ on $P$ such that $\\tau$ is order-preserving with respect to $\\le$.\n      Moreover, $\\tau(x) > x$ for all $x \\in P$.\n    \\item\\label{l-ex-order:group}\n      Let $(P,\\le_P)$ be a totally ordered set.\n      Let $\\tau\\colon P \\to P$ be a permutation such that $\\tau$ is order-preserving and $\\tau(x) >_P x$ for all $x \\in P$.\n      Let $G = \\quo(\\cF(P))$,  $\\overline\\tau \\in \\Aut(G)$ with $\\overline\\tau|_P=\\tau$, and let $\\le$ be the total order on $G$ induced by $\\le_P$.\n      Then $\\overline\\tau(a) > a$ for all $a \\in G_{>1}$.\n      In particular, $\\overline\\tau$ is order-preserving and $\\overline\\tau(G_{>1}) \\subset G_{>1}$.\n  \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n  \\ref*{l-ex-order:set}\n  For $x \\in P$, let $x^\\tau = \\{\\, \\tau^n(x) \\mid n \\in \\bZ \\,\\}$ be its $\\tau$-orbit.\n  Since $x^\\tau$ is infinite, it is naturally totally ordered by $\\tau^m(x) \\le \\tau^n(x)$ if and only if $m \\le n$.\n  Fix an arbitrary total order on the set of all $\\tau$-orbits.\n  For $x$, $y \\in P$, define $x \\le y$ if and only if either $x^\\tau < y^\\tau$, or if $x^\\tau=y^\\tau$ and there exists an $n \\in \\bN_0$ such that $y = \\tau^n(x)$.\n  Then $\\le$ is a total order on $P$, and $\\tau$ is order-preserving with respect to this order.\n  Moreover, $\\tau(x) > x$ for all $x \\in P$.\n\n  \\ref*{l-ex-order:group}\n  As already observed, $(G, \\cdot, \\le)$ is a totally ordered group.\n  Let $a \\in G$ with $a > 1$.\n  We show $\\overline\\tau(a) > a$.\n  We have $a = p_1^{n_1}\\cdots p_r^{n_r}$ with $r \\in \\bN$, pairwise distinct $p_1$,~$\\ldots\\,$,~$p_r \\in P$ and $n_1$,~$\\ldots\\,$,~$n_r \\in \\bZ^\\bullet$.\n  Using the total order on $P$, we may assume $p_1 > \\cdots > p_r$.\n  Since $a > 1$, we have $n_1 > 0$.\n  Now, $\\overline\\tau(a) = \\overline\\tau(p_1)^{n_1} \\cdots \\overline\\tau(p_r)^{n_r}$.\n  Since $\\overline\\tau$ is order-preserving with respect to $\\le_P$, we have $\\overline\\tau(p_1) > \\cdots > \\overline\\tau(p_r)$ and moreover $\\overline\\tau(p_1) > p_1$.\n  From the way we defined the total order on $G$, it follows that $\\overline\\tau(a) > a > 1$.\n  In particular, $\\overline\\tau$ is order-preserving and $\\overline\\tau(G_{>1}) \\subset G_{>1}$.\n\\end{proof}\n\nIf $\\cF(P)$ is a free abelian monoid and $\\overline\\tau$ is an automorphism of $\\cF(P)$ which has no finite orbits on $P$, then \\cref{l-ex-order} implies that the quotient group $\\quo(\\cF(P))$ admits the structure of a totally ordered group with respect to which $\\overline\\tau$ is order-preserving, etc.\nThe following is a strengthening of this result to quotient groups of reduced commutative Krull monoids.\n\n\\begin{prop} \\label{p-grp-order}\n Let $H$ be a reduced commutative Krull monoid and let $\\tau \\in \\Aut(H)$ be such that $H$ is $\\tau$-$v$-simple.\n Let $G$ denote the quotient group of $H$, and denote the extension of $\\tau$ to $\\Aut(G)$ again by $\\tau$.\n Then there exists an order $\\le$ on $G$ such that $(G,\\cdot,\\le)$ is a totally ordered group, $H \\subset G_{\\ge 1}$, and $\\tau(a) > a$ for all $a \\in G_{> 1}$.\n In particular, $\\tau$ is order-preserving on $G$ and $\\tau(G_{>1}) \\subset G_{>1}$.\n\\end{prop}\n\n\\begin{proof}\n  Since $H$ is a commutative Krull monoid, it has a divisor theory.\n  Because $H$ is reduced, this divisor theory can be taken to be an inclusion.\n  Thus, explicitly, there exists a set $P$ such that $\\iota\\colon H \\hookrightarrow F=\\cF(P)$, $G \\subset \\quo(F)$, and such that the inclusion $\\iota$ induces a monoid isomorphism\n  \\[\n  \\iota^*\\colon F \\to \\cI_v^*(H),\\quad  a \\mapsto aF \\cap H.\n  \\]\n  Moreover, $(\\iota^*)^{-1}(aH) = a$ for all $a \\in H$, and $\\iota^*|_P \\colon P \\to \\mathfrak X(H)$ is a bijection.\n  (See \\cite[Theorem 2.4.7.3]{ghk06}.)\n  Recall that $\\tau$ induces a monoid automorphism $\\tau_*\\colon \\cI_v^*(H) \\to \\cI_v^*(H)$ and that $\\tau_*(aH) = \\tau(a)H$ for all $a \\in H$.\n\n  We first show that $\\tau$ extends to an automorphism of $F$.\n  Through $\\iota^*$, we obtain an automorphism $\\overline\\tau = (\\iota^*)^{-1} \\circ \\tau_* \\circ \\iota^* \\in \\Aut(F)$.\n  But we also have $H \\subset F$ via the inclusion $\\iota$.\n  We claim that in fact $\\overline\\tau|_H = \\tau$.\n  Let $a \\in H$.\n  Then\n  \\[\n  \\overline\\tau(a) = (\\iota^*)^{-1} \\circ \\tau_* \\circ \\iota^*(a)\n           =  (\\iota^*)^{-1} \\circ \\tau_*(aH) = (\\iota^*)^{-1}(\\tau(a)H) = \\tau(a).\n  \\]\n  Moreover, $\\overline\\tau$ extends to an automorphism of $\\quo(F)$, again denoted by $\\overline\\tau$, and then also $\\overline\\tau|_G=\\tau$ on $G$.\n\n  The automorphism $\\overline\\tau$ induces a permutation on $P$, and \\subref{l-simple:orbits} implies that $\\overline\\tau$ does not have any finite orbits on $P$.\n  Thus, \\subref{l-ex-order:set} implies that there exists a total order $\\le_P$ on $P$ such that $\\overline\\tau|_P \\colon P \\to P$ is order-preserving and $\\overline\\tau(p) > p$ for all $p \\in P$.\n  Let $\\le$ denote the order on $\\quo(F)$ induced by $\\le_P$.\n  Then $(\\quo(F), \\cdot, \\le)$ is a totally ordered group.\n  By \\subref{l-ex-order:group}, $\\overline\\tau(a) > a$ for all $a \\in \\quo(F)_{>1}$.\n  Denote the restriction of $\\le$ to $G$ again by $\\le$.\n  Then $(G,\\cdot,\\le)$ is a totally ordered group, and $\\tau(a) > a$ for all $a \\in G_{>1}$.\n  Clearly $H \\subset G_{\\ge 1}.$\n\\end{proof}\n\nLet $(G,\\cdot,\\le)$ be a totally ordered group and $K$  a field.\nThe group algebra $K[G]$ is naturally $G$-graded.\nUsing the total order on $G$, it is easy to check that $K[G]$ is a domain.\nEvery unit of $K[G]$ is homogeneous, that is, $K[G]^\\times = \\{\\, \\lambda g \\mid \\lambda \\in K^\\times,\\; g \\in G \\,\\}$.\nIt follows that every nonzero principal ideal $\\fa$ of $K[G]$ has a uniquely determined generator of the form $1 + f$ with $\\supp(f) \\subset G_{>1}$.\nWe call $1+f$ the \\emph{normed generator} of $\\fa$.\n\n\\begin{prop} \\label{p-kg-simple}\n  Let $H$ be a reduced commutative Krull monoid, and let $\\tau \\in \\Aut(H)$ be such that $H$ is $\\tau$-$v$-simple.\n  Let $G$ denote the quotient group of $H$ and let $K$ be a field.\n  If $\\varphi \\in \\Aut(K[G])$ with $\\varphi|_H = \\tau$ and $\\varphi(K) \\subset K$, then $K[G]$ is $\\varphi$-$v$-simple.\n\\end{prop}\n\n\\begin{proof}\n  Denote the extension of $\\tau$ to $G$ again by $\\tau$.\n  Note that $\\varphi|_G = \\tau$.\n  By \\cref{p-grp-order}, there exists an order $\\le$ on $G$ such that $(G,\\cdot,\\le)$ is a totally ordered group and $\\tau(G_{> 1}) \\subset G_{> 1}$.\n  Since $G$ is a subgroup of a free abelian group, it satisfies the ascending chain condition on cyclic subgroups.\n  Hence, $K[G]$ is a commutative Krull domain with trivial class group by \\cref{p-krull-semigroup-domain}.\n  Thus, every divisorial ideal of $K[G]$ is principal.\n  To show that $K[G]$ is $\\varphi$-$v$-simple it therefore suffices to show $\\varphi(\\fa) \\ne \\fa$ for all principal ideals $\\fa$ of $K[G]$ with $\\fa \\notin \\{\\mathbf 0, K[G]\\}$.\n  Let $\\fa$ be such an ideal.\n  Let $f \\in K[G]$ with $\\supp(f) \\subset G_{>1}$ be such that $1+f$ is the normed generator of $\\fa$.\n  Since $\\fa \\ne K[G]$, we have $\\supp(f) \\ne \\emptyset$.\n  Now $\\varphi(1+f) = \\varphi(1) + \\varphi(f) = 1 + \\varphi(f)$.\n  Moreover, $\\supp(\\varphi(f)) = \\tau(\\supp(f)) \\subset G_{>1}$.\n  Hence $1+\\varphi(f)$ is the normed generator of $\\varphi(\\fa)$.\n  Since $H$ is $\\tau$-$v$-simple, $\\tau(\\supp(f)) \\ne \\supp(f)$ by \\cref{l-simple}.\n  Thus $1+\\varphi(f) \\ne 1 + f$, and $\\varphi(\\fa) \\ne \\fa$.\n\\end{proof}\n\n\\begin{thm} \\label{t-extend-simple}\n  Let $D$ be a commutative Krull domain and let $\\sigma \\in \\Aut(D)$ be such that $D$ is $\\sigma$-$v$-simple.\n  Let $H$ be a reduced commutative Krull monoid and let $\\tau \\in \\Aut(H)$ be such that $H$ is $\\tau$-$v$-simple.\n  Let $\\varphi\\in \\Aut(D[H])$ denote the extension of $\\sigma$ and $\\tau$ to $D[H]$, i.e., $\\varphi|_D=\\sigma$ and $\\varphi|_H=\\tau$.\n  Then $D[H]$ is $\\varphi$-$v$-simple.\n\\end{thm}\n\n\\begin{proof}\n  Let $\\varphi_*$ denote the permutation of $\\mathfrak X(D[H])$ induced by $\\varphi$.\n  There are injective maps\n  \\begin{align*}\n    \\iota_D^*\\colon&\n    \\begin{cases} \\mathfrak X(D) &\\to \\mathfrak X(D[H]), \\\\\n      \\fp &\\mapsto \\fp[H],\n    \\end{cases}\n    &\n    \\iota_H^*\\colon&\n    \\begin{cases} \\mathfrak X(H) &\\to \\mathfrak X(D[H]), \\\\\n      \\fp &\\mapsto D[\\fp].\n    \\end{cases}\n  \\end{align*}\n  The image of $\\iota_D^*$ consists of all $\\fp \\in \\mathfrak X(D[H])$ with $\\fp \\cap D^\\bullet \\ne \\emptyset$, while the image of $\\iota_H^*$ consists of all $\\fp \\in \\mathfrak X(D[H])$ with $\\fp \\cap H \\ne \\emptyset$.\n  Let $K=\\quo(D)$ and $G = \\quo(H)$.\n  The group algebra $K[G]$ is the localization of $D[H]$ by $H$ and $D^\\bullet$.\n  There is a bijection\n  \\[\n  \\iota_{K[G]}^* \\colon\n  \\begin{cases}\n      \\{\\, \\fp \\in \\mathfrak X(D[H]) \\mid \\fp \\cap (D^\\bullet \\cup H) = \\emptyset \\,\\} &\\to \\mathfrak X(K[G]), \\\\\n      \\fp           & \\mapsto    \\fp K[G], \\\\\n      \\fP\\cap D[H]  & \\mapsfrom  \\fP.\n  \\end{cases}\n  \\]\n  In particular,\n  \\[\n  \\mathfrak X(D[H]) = \\iota_D^*\\mathfrak X(D) \\;\\cup\\; \\iota_H^*\\mathfrak X(H) \\;\\cup\\; (\\iota_{K[G]}^*)^{-1}\\mathfrak X(K[G]).\n  \\]\n  All of this follows from \\cite[Chapter III, Sections 15 and 16]{gilmer84}, together with the fact that nontrivial essential discrete valuation overmonoids (overrings) of commutative Krull monoids (domains) bijectively correspond to nonempty (nonzero) divisorial prime ideals.\n  That $\\iota_D^*$ takes the stated form follows from \\cite[Theorem 15.3]{gilmer84}, and the corresponding fact for $\\iota_H^*$ is a consequence of \\cite[Theorem 15.7]{gilmer84}.\n  The stated decomposition of $\\mathfrak X(D[H])$ follows from \\cite[Corollary 15.9]{gilmer84}.\n\n  Since $\\varphi(H) = H$ and $\\varphi(D) = D$, each of the sets $\\iota_D^*\\mathfrak X(D)$, $\\iota_H^*\\mathfrak X(H)$, and $(\\iota_{K[G]}^*)^{-1}\\mathfrak X(K[G])$ is fixed by $\\varphi_*$.\n  To show $\\varphi_*(X) \\ne X$ for all finite $\\emptyset \\ne X \\subset \\mathfrak X(D[H])$, it therefore suffices to consider subsets of each of these three sets.\n  If $X \\subset \\iota_D^*\\mathfrak X(D)$, then $\\varphi_*(X) \\ne X$, since $D$ is $\\sigma$-$v$-simple.\n  If $X \\subset \\iota_H^*\\mathfrak X(H)$, then $\\varphi_*(X) \\ne X$, since $H$ is $\\tau$-$v$-simple.\n\n  Finally, consider the case where $X \\subset (\\iota_{K[G]}^*)^{-1}\\mathfrak X(K[G])$.\n  Since $\\varphi(H) \\subset H$ and $\\varphi(D^\\bullet) \\subset D^\\bullet$, $\\varphi$ extends to an automorphism of $K[G]$, which we again denote by $\\varphi$.\n  It now suffices to show that $K[G]$ is $\\varphi$-$v$-simple.\n  However, this follows from \\cref{p-kg-simple}.\n\\end{proof}\n\nThe following \\subref{l-loc:basic} is a slight reformulation of the original localization argument of Claborn, which can be found in  \\cite[Theorem 14.2]{fossum73} and \\cite[Theorem 7]{claborn66}.\nSince we need to observe some details in the argument, we give the proof anyway.\nRecall that if $D$ is a commutative domain and $a$,~$b \\in D$ are coprime (that is, $aD \\cap bD = abD$), then $aX + b$ is a prime element of $D[X]$ (see \\cite[Lemma 14.1]{fossum73}).\n\n\\begin{lemma} \\label{l-loc}\n  Let $D$ be a commutative Krull domain, and let $H$ be a commutative Krull monoid containing a countable set $P$ of non-associated prime elements, so that $H = H_0 \\times \\cF(P)$ with a commutative Krull monoid $H_0$.\n  Suppose that $D[H]$ is a Krull domain.\n  \\begin{enumerate}\n    \\item\\label{l-loc:basic} There exists a multiplicative subset $S \\subset D[H]^\\bullet$ such that $S$ is generated by prime elements of $D[H]$, $S \\cap D[H_0] = \\emptyset$, and $S^{-1}D[H]$ is a Dedekind domain but not a field.\n    \\item\\label{l-loc:auto} If $\\varphi \\in \\Aut(D[H])$ with $\\varphi(P) \\subset P$, then $S$ can be chosen in such a way that $\\varphi(S) \\subset S$.\n    \\item\\label{l-loc:prop}\n      Let $S$ be a multiplicative subset of $D[H]^\\bullet$ such that $S^{-1}D[H]$ is a Dedekind domain.\n      Let $\\varphi \\in \\Aut(D[H])$ be such that $D[H]$ is $\\varphi$-$v$-simple and  $\\varphi(S) \\subset (S^{-1}D[H])^\\times$.\n      Then $\\varphi$ extends to an automorphism $\\varphi_S \\in \\Aut(S^{-1}D[H])$ and $S^{-1}D[H]$ is $\\varphi_S$-simple.\n  \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n\\ref*{l-loc:basic}\nWe have $D[H]\\cong D[H_0][\\cF(P)] = D[H_0][\\ldots,X_{-1},X_0,X_1,\\ldots]$.\nLet $\\fP \\in \\spec(D[H])$ with $\\height(\\fP) > 1$, and let $a_\\fP \\in \\fP^\\bullet$.\nLet $\\fp_1$, $\\ldots\\,$,~$\\fp_r$ be the divisorial prime ideals of $D[H]$ that contain $a_\\fP$.\nBy prime avoidance, there exists an element $b_\\fP \\in \\fP \\setminus (\\fp_1 \\cup \\ldots \\cup \\fp_r)$.\nLet $X_\\fP \\in \\{ \\ldots, X_{-1},X_0,X_1, \\ldots \\}$ be such that $X_\\fP$ is not contained in the support of $a_\\fP$ or $b_\\fP$.\nThen $D[H] = R_0[X_\\fP]$ with $R_0=D[H_0][\\{\\, X_i \\mid i \\in \\bZ,\\; X_i \\ne X_\\fP \\,\\}]$ and $a_\\fP$,~$b_\\fP \\in R_0$ are coprime.\nHence $f_\\fP = a_\\fP X_\\fP + b_\\fP$ is a prime element of $D[H]$, and $f_\\fP \\in \\fP$.\nBy construction, $f_\\fP \\notin D[H_0]$.\n\nLet $Q = \\{\\, f_\\fP \\mid \\fP \\in \\spec(D[H]),\\, \\height(\\fP) > 1 \\,\\}$ and let $S$ be the multiplicative set generated by $Q$.\nSince $\\spec(S^{-1}D[H])$ is in bijection with $\\{\\, \\fp \\in \\spec(D[H]) \\mid \\fp \\cap S \\ne \\emptyset \\,\\}$, it follows that $S^{-1}D[H]$ is a Krull domain of dimension at most $1$, i.e., a Dedekind domain.\nMoreover, $S \\cap D[H_0] = \\emptyset$.\n\nIf $D[H_0]$ is not a field, then neither is $S^{-1}D[H]$.\nIt only remains to consider the, degenerate, special case where $D[H_0]$ is a field, i.e., $H_0$ is the trivial monoid and $D=K$ is a field.\nThen $D[H] = K[\\ldots,X_{-1},X_0,X_1,\\ldots]$ is a polynomial ring in countably many indeterminates.\nBy construction, $Q$ only contains elements with $X_i$-degree equal to $1$ for some $i \\in \\bZ$.\nHowever, $D[H]$ contains prime elements which are not of this form (e.g., $X_1^2 + X_0^2 X_1 + X_0$).\n\n\\ref*{l-loc:auto}\nIf $\\fP \\in \\spec(D[H])$ with $\\height(\\fP) > 1$, then also $\\height(\\varphi(\\fP))>1$.\nThus $\\varphi$ induces a permutation of prime ideals of height greater than $1$.\nDenote a set of representatives for the orbits by $\\Omega$.\nFor each $\\fP$ in $\\Omega$, choose $f_\\fP$ as in \\ref*{l-loc:basic}.\nFor all $n \\in \\bZ$, $\\varphi^n(f_\\fP)$ is a prime element contained in $\\varphi^n(\\fP)$.\nSince $\\varphi(P) \\subset P$, we have $\\varphi^n(f_\\fP) \\notin D[H_0]$.\nSet $Q = \\bigcup_{\\fP \\in \\Omega} \\bigcup_{n \\in \\bZ} \\varphi^n(f_\\fP)$, and let $S$ be the multiplicative subset of $D[H]^\\bullet$ generated by $Q$.\nSince $\\varphi(Q) \\subset Q$, also $\\varphi(S) \\subset S$.\nThus, $S$ has the stated properties.\n\n\\ref*{l-loc:prop}\nSince $\\varphi(S) \\subset (S^{-1}D[H])^\\times$, $\\varphi$ extends to an automorphism $\\varphi_S$ of $S^{-1}D[H]$.\nLocalization induces a bijection between $\\{\\, \\fp \\in \\mathfrak X(D[H]) \\mid \\fp \\cap S = \\emptyset \\,\\}$ and $\\mathfrak X(S^{-1}D[H])$.\nHence $S^{-1}D[H]$ is $\\varphi_S$-$v$-simple.\nSince $S^{-1}D[H]$ is a Dedekind domain, every ideal is divisorial.\nThus $S^{-1}D[H]$ is $\\varphi_S$-simple.\n\\end{proof}\n\n\\begin{remark}\n  Most of the technicalities in the previous proof can be avoided as long as $\\cC(H)$ is non-trivial and we are not picky about whether or not $S \\cap D[H_0] = \\emptyset$.\n  In this case, we take $S$ to be the multiplicative set generated by all prime elements of $D[H]$.\n  Claborn's argument shows that each $\\fP \\in \\spec(D[H])$ with $\\height(\\fP) > 1$ contains some prime element, so that indeed $\\dim(S^{-1}D[H]) \\le 1$.\n  We have $\\varphi(S) \\subset S$, since prime elements are mapped to prime elements by $\\varphi$.\n  And, finally, since $\\cC(H)$ is non-trivial, there must exist a non-principal divisorial prime ideal $\\fp \\in D[H]$.\n  Then $\\fp \\cap S= \\emptyset$, hence $\\dim(S^{-1}D[H]) = 1$.\n\\end{remark}\n\n\\begin{thm} \\label{t-extend}\n  Let $D$ be a commutative Krull domain and let $\\sigma \\in \\Aut(D)$ be such that $D$ is $\\sigma$-$v$-simple.\n  Let $H$ be a reduced commutative Krull monoid containing prime elements,\n  and let $\\tau$ be an automorphism of $H$ such that $H$ is $\\tau$-$v$-simple.\n  Let $\\varphi\\colon D[H]\\to D[H]$ denote the extension of $\\sigma$ and $\\tau$ to $D[H]$, that is, $\\varphi|_D = \\sigma$ and $\\varphi|_H = \\tau$.\n  Let $p \\in H$ be a prime element and let $p^\\tau = \\{\\, \\tau^n(p) \\mid n \\in \\bZ \\,\\}$ be its $\\tau$-orbit, so that $H = H_0 \\times \\cF(p^\\tau)$ for a Krull monoid $H_0$.\n  \\begin{enumerate}\n    \\item \\label{t-extend:exist} There exists a multiplicative subset $S$ of the semigroup algebra $D[H]$ such that $S$ is generated by prime elements, $S \\cap D[H_0] = \\emptyset$, $\\varphi(S) \\subset S$, and the localization $S^{-1}D[H]$ is a Dedekind domain but not a field.\n    \\item \\label{t-extend:clsgrp}\n      Let $S \\subset D[H]^\\bullet$ be a multiplicative subset such that $R=S^{-1}D[H]$ is a Dedekind domain but not a field and $\\varphi(S) \\subset R^\\times$.\n      Let $A$ be the subgroup of $\\cC(D) \\times \\cC(H) = \\cC(D[H])$ generated by classes of $\\fp \\in \\mathfrak X(D[H])$ with $\\fp \\cap S \\ne \\emptyset$.\n      Then $\\varphi$ extends to an automorphism $\\varphi_S$ of $R$, the skew Laurent polynomial ring $T=R[x,x^{-1};\\varphi_S]$ is a noncommutative simple Dedekind domain, and the following sequence of abelian groups is exact:\n      \\[\n      \\xymatrix@C=0.75cm{\n        \\cC(D) \\times \\cC(H)\/A \\ar[rr]^{\\id - (\\sigma_*,\\tau_*)}\n        & & \\cC(D) \\times \\cC(H)\/A \\ar[r]^{\\beta}\n        & G\\big(R[x,x^{-1};\\varphi_S]\\big) \\ar[r]\n        & \\mathbf 0.\n      }\n      \\]\n      Here, $(\\sigma_*,\\tau_*)$ is the automorphism of $\\cC(D) \\times \\cC(H)\/A$ induced by $\\sigma$ and $\\tau$.\n      The map $\\beta$ is induced as follows: If $\\fa \\in \\cF_v(D)^\\times$, the class of $\\fa$ is mapped to $\\kcls{\\fa[H] \\otimes_R T} - \\kcls{T}$.\n      If $\\fb \\in \\cF_v(H)^\\times$, the class of $\\fb$ is mapped to $\\kcls{D[\\fb] \\otimes_R T} - \\kcls{T}$.\n  \\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n  \\ref*{t-extend:exist}\n  Since $\\tau$ does not have finite orbits on $\\mathfrak X(H)$, the orbit $p^\\tau$ of $p$ consists of countably many non-associated prime elements.\n  Let $S$ be a multiplicative subset of $D[H]$ as in \\subref{l-loc:auto}, where we take $P=p^\\sigma$.\n\n  \\ref*{t-extend:clsgrp}\n  By \\cref{t-extend-simple}, $D[H]$ is $\\varphi$-$v$-simple.\n  By \\subref{l-loc:prop}, $\\varphi$ extends to an automorphism $\\varphi_S \\in \\Aut(R)$, and $R$ is $\\varphi_S$-simple.\n  By Nagata's Theorem and the identifications we have made, $\\cC(D) \\times \\cC(H)\/A \\cong \\cC(R)$ with the isomorphism given by $([\\fa]_D,[\\fb]_H) +A \\mapsto [S^{-1}\\fa[\\fb]]$.\n  Since $R$ is $\\varphi_S$-simple, $T = R[x,x^{-1};\\sigma]$ is a noncommutative simple Dedekind domain by \\cref{p-nc-dedekind}.\n  Under the identification of $\\cC(D) \\times \\cC(H) \/ A$ with $G(R)$, the automorphism $(\\sigma_*,\\tau_*)$ corresponds to $\\varphi_*$.\n  The exact sequence of class groups follows from \\cref{p-k0-ext} and the discussion that followed it.\n\\end{proof}\n\n\\begin{remark} \\label{r-primes-techn}\n  The technical condition that $H$ contains a prime element (and hence, since $\\tau$ does not have any finite orbits on $\\mathfrak X(H)$, infinitely many non-associated ones) is necessary so that $D[H]$ has the form $D[H_0][\\ldots,X_{-1},X_0,X_1,\\ldots]$ with $\\varphi$ acting by $\\varphi(X_i) = X_{i+1}$.\n  The countably many indeterminates are used to construct the prime elements which generate $S$, see \\cref{l-loc}.\n  (See \\cite[Proposition 14]{chang11} for a refinement that only needs one indeterminate.)\n  If $H$ does not contain a prime element, we may replace $H$ by $H'=H \\times \\cF(\\ldots,p_{-1},p_0,p_1,\\ldots)$ and extend $\\tau$ by $\\tau(p_i)=p_{i+1}$.\n  Then $H'$ satisfies the conditions of the theorem, and $\\cC(H)\\cong \\cC(H')$.\n  By formulating the theorem in the slightly more technical way, we avoid the need to enlarge $H$ if it already contains prime elements.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem 1.1]\n  We assume without restriction that $\\kappa$ is infinite.\n  Let $G$ be an abelian group.\n  \\Cref{t-ex-mon-aut} implies that there exist a reduced commutative Krull monoid with $\\cC(H) \\cong G$ and an automorphism $\\tau$ of $H$ such that $H$ is $\\tau$-$v$-simple, $\\tau_* \\colon \\cC(H) \\to \\cC(H)$ is the identity, and each divisorial ideal class of $H$ contains $\\kappa$ nonempty divisorial prime ideals.\n  Let $K$ be a field.\n  Then $K$ is simple, and hence $\\id_K$-$v$-simple.\n  Let $\\varphi\\colon K[H] \\to K[H]$ be the automorphism of $K[H]$ with $\\varphi|_H = \\tau$ and $\\varphi|_K=\\id_K$.\n  Let $P \\subset H$ be a countable set of prime elements such that $H\\setminus P$ still contains $\\kappa$ prime elements.\n  Then $H = H_0 \\times \\cF(P)$ with a Krull monoid $H_0$.\n  Each class of $\\cC(H_0)$ contains $\\kappa$ divisorial prime ideals.\n  Applying \\cref{t-extend}, we find a subset $S \\subset K[H]$ such that $S \\cap K[H_0] = \\emptyset$, the localization $R=S^{-1}K[H]$ is a commutative Dedekind domain but not a field, and $T = S^{-1}K[H][x,x^{-1};\\varphi_S]$ is a noncommutative simple Dedekind domain with $G(T) \\cong G$.\n\n  If $\\fp \\in \\mathfrak X(H)$, then $K[\\fp] \\in \\mathfrak X(K[H])$.\n  If $\\fp \\notin \\{\\, (p) \\mid p \\in P \\,\\}$, then $K[\\fp] \\cap S = \\emptyset$ by construction.\n  In this case, $S^{-1}\\fp$ is a nonzero prime ideal of $R$.\n  Thus, each ideal class of $R$ contains at least $\\kappa$ nonzero prime ideals.\n  If $\\fq$ is a nonzero prime ideal of $R$, then $\\fq T$ is a maximal right ideal of $T$ by \\cite[Lemma 6.9.15]{mcconnell-robson01}.\n  Since $T$ is flat over $R$, we have $\\fq T \\cong \\fq \\otimes_R T$.\n  If $\\fp \\in \\mathfrak X(H)$, the isomorphism $\\beta\\colon \\cC(H) \\to G(T)$ maps $[\\fp]$ to $\\kcls{K[\\fp] \\otimes_R T} - \\kcls{T}$.\n  It follows that each class of $G(T)$ contains at least $\\kappa$ maximal right ideals.\n\\end{proof}\n\n\\begin{remark}\n  \\begin{enumerate}\n    \\item We can only give a lower bound on the cardinality of maximal right ideals in each class.\n      Apart from the divisorial prime ideals of the form $K[\\fp]$ with $\\fp \\in \\mathfrak X(H)$, additional divisorial prime ideals arise from prime elements of $K[\\quo(H)]$.\n      In \\cite{chang11}, Chang has shown that if $D[H]$ is a commutative Krull domain and $H$ is non-trivial, then each divisorial ideal class contains a nonzero divisorial prime ideal.\n\n    \\item\n      A domain $D$ is \\emph{half-factorial} if every element of $D^\\bullet$ can be written as a product of irreducibles and the number of irreducibles in each such factorization is uniquely determined.\n      It is conjectured that every abelian group is the class group of a half-factorial commutative Dedekind domain.\n      See \\cite[\\S5]{gilmer06} for background.\n      The conjecture is equivalent to one purely about abelian groups (\\cite[Proposition 3.7.9]{ghk06}).\n      See \\cite{geroldinger-goebel03} for progress on this question.\n  \\end{enumerate}\n\\end{remark}\n\n\\noindent\n\\textbf{Acknowledgments.}\n\\phantomsection\n\\addcontentsline{toc}{section}{Acknowledgments}\nI thank Alfred Geroldinger for feedback on a preliminary version of this paper.\n\n\\bibliographystyle{hyperalphaabbr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAccelerating beams, i.e., electromagnetic fields that propagate along curved trajectories in free space without being subject to any external force, have been the subject of a thorough investigation in the last years. The most famous representative of such class of beams is, without doubts, the Airy beam. Firstly introduced in the context of quantum mechanics by Berry and Balazs as an exotic solution of the Schr\\\"odinger equation \\cite{berryAiry}, it was then introduced in optics in 2007 by Siviloglou and co-workers \\cite{siviloglou,siviloglou2}, as an exact solution of the paraxial equation propagating along a parabolic trajectory in free space. Due to their intriguing features, Airy beams were studied in different contexts, such as nonlinear optics \\cite{airy1}, particle manipulation \\cite{airy2}, and proposed as an efficient way to generate curved plasma channels \\cite{airy3}. \n\nInspired by these results, the last years witnessed the emergence of many different types of accelerating beams in different coordinate systems, such as parabolic \\cite{acc1} and Weber \\cite{acc2} beams. Moreover, beams of light capable to propagate along curved \\cite{acc3,curved1, curved2} and arbitrary \\cite{arbitrary1,arbitrary2} trajectories, has also been proposed. Recently, two new classes of accelerating beams have been introduced, namely angular \\cite{angularAcc, vettiOE} and radially self-accelerating beams \\cite{nostroPRL, nostroAPL}. While the former acquire angular acceleration during rotation around their optical axis \\cite{angularAcc},  the latter exhibit radial acceleration, a feature which makes them propagate along spiralling trajectories around their optical axis. \n\nRadially self-accelerating beams (RSABs) can be understood in terms of superpositions of Bessel beams, where each single component is characterised by an angular velocity proportional to the amount of orbital angular momentum it carries. This, ultimately, results in an electromagnetic field, whose transverse field or intensity distribution rotates around the propagation direction with a given constant angular velocity $\\Omega$ \\cite{nostroPRL}. Among the vast zoology of RSABs, in particular, helicon beams, i.e., a subclass of RSABs consisting of rotating diffraction-free beams based on the superposition of two Bessel beams with opposite orbital angular momentum, have attracted a lot of interest in the last decades \\cite{nostroAPL,helicon1,helicon2,helicon3,helicon4,helicon5,helicon6,helicon7,helicon8,helicon9}. Beyond helicon beams, RSABs have potentially significant  applications in different areas of physics, such as sensing \\cite{airy3}, material processing \\cite{matProc1,matProc2}, and particle manipulation \\cite{partMan1,partMan2}. \n\nDespite this broad interest, RSABs have only been defined within the scalar electromagnetic theory, and their vector nature, as well as the effect of focussing on their self-accelerating character, has not been yet investigated. In this work, therefore, we introduce vector RSABs, and study their vector properties, in terms of their linear and angular momentum content. Moreover, we carefully analyse what is the impact of focussing on the self-accelerating character of RSABs, and under which conditions the focussing process does not spoil this property.\n\nThis work is organised as follows: in Sect. 2 we briefly recall the definition of RSABs, and recall some of their main properties. In Sect. 3 we use the method of Hertz potentials to construct vector RSABs, and use these solutions as a model for focussed RSABs. Then, we derive a condition on the polarisation that a scalar RSAB must possess, in order to maintain its self-accelerating character upon focussing. Section 4 is then devoted to calculate the linear and angular momentum for paraxial, intensity rotating RSABs.  Conclusions are then drawn in Sect. 5.\n\n\\section{Radially Self-Accelerating  Beams}\nWe start our analysis by considering scalar, monochromatic, free space solutions of the Helmholtz equation \n\\begin{equation}\\label{eq1}\n\\left(\\nabla^2+k_0^2\\right)\\psi(\\vett{r})=0,\n\\end{equation}\nwhere $k_0$ is the vacuum wave vector. The most general solution of the above equation in cylindrical coordinates, can be given in terms of superposition of Bessel beams, i.e.,\n\\begin{equation}\\label{eq2}\n\\psi(\\vett{r})=\\sum_{m}\\,\\int\\,d\\xi\\,C_m(\\xi)\\text{J}_m(\\rho\\sqrt{1-\\xi^2} )e^{i(m\\theta+\\xi\\zeta)},\n\\end{equation}\nwhere $\\rho=k_0R$, and $\\zeta=k_0 z$ are normalised radial and longitudinal coordinates, $\\text{J}_m(x)$ is the Bessel function of the first kind \\cite{nist}, and the integration variable $\\xi=\\cos\\vartheta_0$  plays the role of the Bessel cone angle $\\vartheta_0$ \\cite{durnin}. \n\nFrom the above solution, it is possible to extract RSABs by applying the requirements that Eq. \\eqref{eq2} must fulfil, in order to be a RSAB \\cite{nostroPRL}. First, $\\psi(\\vett{r})$ must propagate freely, and not under the action of a certain potential. Then, there should exist a suitable reference frame, in which $\\psi(\\vett{r})$ is manifestly propagation invariant, i.e., no explicit $\\zeta$-dependence must appear. Finally, an observer at rest in such reference frame should experience a fictitious force, which, ultimately, is at the core of self-accelerating character of RSABs.\n\nWhle the first requirement is automatically met by the fact that we are considering free space propagation, the second one is very useful to define RSABs properly. Once it is fulfilled, in fact, it is not hard to show that the third requirement follows accordingly. We therefore require, that, after a suitable coordinate transformation $\\vett{r}'=S\\, \\vett{r}$,  the field $\\psi(\\vett{r}')$ in the new coordinate frame is manifestly propagation invariant, i.e., \n$\\partial\\psi(\\vett{r}')\/\\partial\\zeta=0$. To this aim, we introduce the co-rotating coordinate $\\Phi=\\theta+\\Lambda\\zeta$, and choose the expansion coefficient as  $C_m(\\xi)=D_m\\delta(\\xi-(m\\Lambda+\\beta))$, where $\\Lambda=\\Omega\/k_0>0$ is the normalised angular velocity of the RSAB, and $\\beta$ is a free (dimensionless) parameter, with the physical meaning of a normalised propagation constant. Substituting this Ansatz in Eq. \\eqref{eq2} we get the following result\n\\begin{equation}\\label{eq3}\n\\psi_{RSAB}(\\rho,\\Phi)=e^{i\\beta\\zeta}\\sum_{m\\in\\mathcal{M}}D_m\\text{J}_m(\\alpha_m\\rho)e^{im\\Phi},\n\\end{equation}\nwhere $\\alpha_m=\\sqrt{1-(m\\Lambda+\\beta)^2}$, and $\\mathcal{M}=\\{m\\in\\mathbb{N}: \\alpha_m>0\\}$. For $\\beta=0$, the above field is manifestly propagation invariant, as no explicit $\\zeta$-dependence is present. Moreover, its amplitude and phase both rotate with normalised angular velocity $\\Lambda$ during propagation. For $\\beta\\neq 0$, on the other hand, the field itself is not anymore propagation invariant, due to the presence of the global phase factor $\\exp{(i\\beta\\zeta)}$. Nevertheless, the intensity $|\\psi_{RSAB}(\\vettGreek{\\rho})|^2$ is propagation invariant also for $\\beta\\neq 0$. In this case, however, while both intensity and phase propagate describing spiralling trajectories, they are not synchronised anymore. These two classes of RSABs are called field rotating, and intensity rotating, respectively \\cite{nostroPRL}. \n\nIt is worth noticing, moreover, that while for $\\beta=0$ the set $\\mathcal{M}$ contains only positive integers, for $\\beta\\neq 0$ positive and negative values of $m$ are allowed. Thus, helicon beams, for example, are a particular case of intensity rotating RSABs, where only two Bessel beams are participating in the sum in Eq. \\eqref{eq3} An Example of both classes of RSABs is given in Fig. \\ref{figure1}.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure1.pdf}\n\\caption{Intensity and phase distribution for field rotating RSABs (top row) and intensity rotating RSABs (bottom row). Panels (a), (b), (e), and (f) correspond to the intensity and phase distributions at $z=0$, while panels (c), (d), (g), and (h) to $z=0.8(2\\pi\/\\Lambda)$. Moreover, for the top row, the intensity and phase distributions have been plotted in the region $0\\leq\\rho\\leq 10$, while for the lower row in the region $0\\leq\\rho\\leq 1200$ has been chosen. The difference in the plotting range for the normalised radial coordinate $\\rho$ reflects the paraxial (bottom) and nonparaxial (top) character of the plotted RSABs. In all these figures, $\\Lambda=10^{-5}$ (corresponding to an angular velocity of $\\Omega\\simeq 75 $ rad\/m at $\\lambda=800$ nm), $m_{max}=4$, and $D_m=1$ has been used. For the top row, $\\beta$ is set to zero, while for the bottom row $\\beta=1-m_{max}\\Lambda=0.99996$ (corresponding to a value of a global propagation constant $\\beta_0\\simeq 7.8$ $\\mu m^{-1}$ for $\\lambda=800$ nm) has been used. The white arrow in the intensity profiles show the direction of rotation.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nOf particular interest are RSABs with $\\Lambda\\ll 1$. Since $\\Lambda=\\Omega\/k_0$, this condition corresponds to RSABs, whose actual angular velocity $\\Omega$ is much smaller than the beam's wave vector $k_0$. This, ultimately, corresponds to experimentally realisable RSABs. In the rest of this manuscript, if not specified otherwise, we will always implicilty assume that $\\Lambda\\ll 1$ holds. This assumption, moreover, has different consequences for field and intensity rotating RSABs.\n \n In the former case (i.e., for $\\beta=0$),  $\\Lambda\\ll1$ implies that the (normalised) transverse momentum of each Bessel component is given by $\\alpha_m=\\sqrt{1-m^2\\Lambda^2}\\simeq 1+\\mathcal{O}(m^2\\Lambda^2)$. If we recall, that the transverse momentum of a Bessel beam is related to the Bessel cone angle by the relation $k_{\\perp}=k_0\\sin\\vartheta_0$, a value of the normalised transverse momentum $\\alpha_m\\simeq 1$ corresponds to $\\vartheta_0\\simeq\\pi\/2$, i.e., to a highly nonparaxial Bessel beam. \n \n Despite this fact, however, the nature of the resulting RSAB can be tuned at will between paraxial and nonparaxial, by simply changing the number of Bessel beams that participate to the sum in Eq. \\eqref{eq3}. To obtain nonparaxial RSABs, it is sufficient to limit the summation in Eq. \\eqref{eq3} to $m_{max}=\\text{max}\\{\\mathcal{M}\\}<\\ceil[\\big]{\\Lambda^{-1}}$. In this case, in fact, the transverse momentum of every Bessel component will be $\\alpha_m\\simeq 1$, and the resulting RSAB will be highly nonparaxial. \n \n On the other hand, if one includes only values of $m$, that are close to $\\ceil[\\big]{\\Lambda^{-1}}$, i.e., if $m\\in[m_{max}-\\bar{m},m_{max}]$ in Eq. \\eqref{eq3}\\footnote{with $\\bar{m}$ small compared to $m_{max}$, such that $\\bar{m}^2\\Lambda^2\\ll1$ still holds}, then $m\\Lambda\\simeq 1$, and, correspondingly, $\\alpha_m\\simeq 0$. In this case, all Bessel components will be paraxial (i.e., the correspondent cone angle will be $\\vartheta_0\\ll1$), and the resulting RSAB can also be interpreted as a paraxial beam.\n \n For intensity rotating RSABs (i.e., for $\\beta\\neq 0$), instead, $\\alpha_m$ can be made arbitrarily small, independently from the value of $\\Lambda$, by suitably tuning the parameter $\\beta$. In this case, then, the paraxial limit is simply obtained by choosing $\\beta$ such that $\\alpha_m\\simeq 0$, i.e., $\\beta=1-m_{max}\\Lambda$, with \n $m_{max}=\\text{max}\\{\\mathcal{M}\\}$. Notice, that with this choice of $\\beta$, $\\alpha_{m_{max}}=0$, and therefore the sum in Eq.\\eqref{eq3} extends to $m_{max}-1$, as $\\text{J}_{m_{max}}(\\alpha_{m_{max}}\\rho)=0$. \n \nThis extra flexibility in tuning the propagation constant $\\beta$ and the angular velocity $\\Lambda$ independently makes intensity rotating RSABs easier to generate and manipulate experimentally, than their field rotating counterparts \\cite{nostroPRL, nostroAPL}.  \n\n\n\\section{Vector Radially Self-Accelerating Beams}\\label{vectorialisation}\nThe solution presented in Eq. \\eqref{eq3} describes scalar RSABs. In many situations, however, a scalar representation of the electromagnetic field is not enough to fully describe its properties. A typical example is the focussing of a beam of light by means of a thick lens. On the focal plane of the lens, in fact, the scalar approximation given by Eq. \\eqref{eq3} would fail to describe the properties of a focussed RSAB, and a full vector theory should be instead employed. A simple way to retrieve a full vector solution of Maxwell's equations from a solution to the scalar Helmholtz equation is given by the method of Hertz potentials  \\cite{stratton, joptHertz}. First, one defines the Hertz potential $\\boldsymbol\\Pi(\\vett{r},t)=\\psi(\\vett{r})\\exp{(-i\\omega t)}\\uvett{f}$, where $\\uvett{f}$ is a suitable polarisation unit vector, and $\\psi(\\vett{r})$ is a solution of Eq. \\eqref{eq1}. Then, the vector electric and magnetic fields can then be retrieved from $\\vettGreek{\\Pi}(\\vett{r},t)$ as follows:\n\\begin{subequations}\\label{eq5}\n\\begin{align}\n\\vett{E}(\\vett{r},t) & =-\\frac{\\partial\\boldsymbol\\Pi(\\vett{r},t)}{\\partial t},\\\\\n\\vett{B}(\\vett{r},t) & =\\nabla\\times\\nabla\\times\\boldsymbol\\Pi(\\vett{r},t).\n\\end{align}\n\\end{subequations}\nIn the general case, both an electric ($\\boldsymbol\\Pi_e$) and magnetic ($\\boldsymbol\\Pi_m$) Hertz potential should be introduced, each accounting for the sources of electric and magnetic field, respectively. For free space propagation, however, no sources are present, and the electric and magnetic Hertz potential coincide (up to a global constant), i.e., $\\boldsymbol\\Pi_e=\\boldsymbol\\Pi_m\\equiv\\boldsymbol\\Pi$ \\cite{stratton}. According to the convention adopted by Jackson \\cite{jackson}, the electric and magnetic fields defined by Eqs. \\eqref{eq5} correspond to TE fields. The TM fields, however, can be obtained straightforwardly from the TE ones by setting $\\vett{E}_{TM}\\rightarrow\\vett{B}_{TE}$, and $\\vett{B}_{TM}\\rightarrow-\\vett{E}_{TE}\/c^2$.\n\nTo calculate the vector electric and magnetic fields corresponding to RSABs, we first rewrite Eq. \\eqref{eq3} as $\\psi_{RSAB}(\\vett{r})=\\sum_mD_m\\phi_m(\\vett{r})$, with $\\phi_m(\\vett{r})=\\exp{[i m\\theta+i(m\\Lambda+\\beta)\\zeta]}\\text{J}_m(\\alpha_m\\rho)$ being the usual Bessel beam \\cite{durnin}. This allows us to define the Hertz potentials for RSABs in terms of the Hertz potentials for ordinary Bessel beams, i.e.,\n\\begin{equation}\\label{eq6}\n\\boldsymbol\\Pi(\\vett{r},t) =\\sum_{m\\in\\mathcal{M}}D_m\\vett{P}^{(m)}(\\vett{r},t)\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq6bis}\n\\vett{P}^{(m)}(\\vett{r},t)=\\phi_m(\\vett{r})e^{-i\\omega t}\\uvett{f}\n\\end{equation}\nis the Hertz potential corresponding to a single Bessel beam $\\phi_m(\\vett{r})$, whose polarisation is defined by the unit vector $\\uvett{f}$. Then, using Eq. \\eqref{eq5}, we can first calculate the electric and magnetic vector fields for a single Bessel beam, namely\n\\begin{subequations}\\label{eq8}\n\\begin{align}\n\\vett{E}^{(m)}(\\vett{r},t) &=-\\frac{\\partial\\vett{P}^{(m)}(\\vett{r},t)}{\\partial t},\\\\\n\\vett{B}^{(m)}(\\vett{r},t) &=\\nabla\\times\\nabla\\times\\vett{P}^{(m)}(\\vett{r},t).\n\\end{align}\n\\end{subequations}\nThen, the electric and magnetic fields of vector RSABs can be written as follows:\n\\begin{subequations}\\label{eq9}\n\\begin{align}\n\\vett{E}(\\vett{r},t) & =\\sum_{m\\in\\mathcal{M}}D_m\\vett{E}^{(m)}(\\vett{r},t),\\\\\n\\vett{B}(\\vett{r},t) & =\\sum_{m\\in\\mathcal{M}}D_m\\vett{B}^{(m)}(\\vett{r},t).\n\\end{align}\n\\end{subequations}\n\\subsection{The Role of Polarisation of Hertz Potential in Determining the Properties of Vector RSABs}\nVector beams are frequently used as a model to describe focused light. From this perspective, the method of Hertz potential offers an intuitive and insightful perspective on the process of focussing of a beam of light by a lens, or an objective, for example. In fact, one can interpret the Hertz potential \n$\\boldsymbol\\Pi$ as the electromagnetic field before the focussing system, consisting of a scalar field distribution, and a given polarisation $\\uvett{f}$. The vectorialisation procedure described in Eqs. \\eqref{eq5}, then, represents the full vector field after the focussing process (for example, in the focal plane of a lens). Because of the structure of Eqs. \\eqref{eq5}, it is not difficult to see, that the initial polarisation $\\uvett{f}$ possessed by the field will contribute in determining all the components of the focussed field. \n\nFor the case of RSABs, it is interesting to see whether the vectorialisation procedure described above (i.e., the focussing process) preserves their self-accelerating character, or, in case it does not, under which conditions the self-accelerating character of RSABs is preserved. To do so, first we introduce the polarisation vector $\\uvett{f}=f_p\\uvett{x}+f_s\\uvett{y}$ (where $f_{p,s}\\in\\mathbb{C}$, and $|f_p|^2+|f_s|^2=1$). Then, we use Eqs. \\eqref{eq6bis} and \\eqref{eq8} to calculate the vector electric and magnetic fields corresponding to arbitrary polarised Bessel beams. Because of the intrinsic cylindrical symmetry of RSABs, we also introduce a (normalised) cylindrical reference frame $\\{\\uvettGreek{\\rho},\\uvettGreek{\\theta},\\uvettGreek{\\zeta}\\}$. In this reference frame, the electric and magnetic fields of a single vector Bessel component can be written as \n\\begin{subequations}\\label{eq10}\n\\begin{align}\n\\vett{E}^{(m)}(\\vettGreek{\\rho},t) &= e^{i(\\beta\\zeta-\\omega t+m\\Phi)}\\left[E^{(m)}_{\\rho}(\\vettGreek{\\rho})\\uvettGreek{\\rho}+E^{(m)}_{\\theta}(\\vettGreek{\\rho})\\uvettGreek{\\theta}+E^{(m)}_{\\zeta}(\\vettGreek{\\rho})\\uvettGreek{\\zeta}\\right],\\\\\n\\vett{B}^{(m)}(\\vettGreek{\\rho},t) &= e^{i(\\beta\\zeta-\\omega t+m\\Phi)}\\left[B^{(m)}_{\\rho}(\\vettGreek{\\rho})\\uvettGreek{\\rho}+B^{(m)}_{\\theta}(\\vettGreek{\\rho})\\uvettGreek{\\theta}+ B^{(m)}_{\\zeta}(\\vettGreek{\\rho})\\uvettGreek{\\zeta}\\right],\\\\\n\\end{align}\n\\end{subequations}\nwhere $\\Phi=\\theta+\\Lambda\\zeta$ is the co-rotating coordinate defined in the previous section, and the field components are given by\n\\begin{subequations}\\label{eq11}\n\\begin{align}\nE^{(m)}_{\\rho}(\\vettGreek{\\rho})&=\\omega(\\beta+m\\Lambda)(f_s\\cos\\theta-f_p\\sin\\theta)\\text{J}_m(\\alpha_m\\rho),\\\\\nE^{(m)}_{\\theta}(\\vettGreek{\\rho})&=-\\omega(\\beta+m\\Lambda)(f_p\\cos\\theta+f_s\\sin\\theta)\\text{J}_m(\\alpha_m\\rho),\\\\\nE^{(m)}_{\\zeta}(\\vettGreek{\\rho})&=\\frac{m\\omega}{\\rho}\\text{J}_m(\\alpha_m\\rho)(f_p\\cos\\theta+f_s\\sin\\theta)\\nonumber\\\\\n&+i\\omega\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_{m}(\\alpha_m\\rho)\\right](f_s\\cos\\theta-f_p\\sin\\theta),\n\\end{align}\n\\end{subequations}\nfor the electric field, and\n\\begin{subequations}\\label{eq12}\n\\begin{align}\nB^{(m)}_{\\rho}(\\vettGreek{\\rho})&=-\\frac{\\alpha_m}{\\rho}\\left[\\left(f_p-imf_s\\right)\\cos\\theta+\\left(f_s+imf_p\\right)\\sin\\theta\\right]\\text{J}_{m}^{'}(\\alpha_m\\rho)\\nonumber\\\\\n&-2im\\text{J}_m(\\alpha_m\\rho)\\left[\\left(f_s+imf_p\\right)\\cos\\theta-\\left(f_p-imf_s\\right)\\sin\\theta\\right],\\\\\nB^{(m)}_{\\theta}(\\vettGreek{\\rho})&=\\frac{im}{\\rho}\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\left[\\alpha_m\\text{J}_m^{'}(\\alpha_m\\rho)-\\text{J}_m(\\alpha_m\\rho)\\right]\\nonumber\\\\\n&+\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\left[(\\beta+m\\Lambda)^2\\text{J}_m(\\alpha_m\\rho)-\\alpha^2\\text{J}_m^{''}(\\alpha_m\\rho)\\right],\\\\\nB^{(m)}_{\\zeta}(\\vettGreek{\\rho})&=-\\frac{m(\\beta+m\\Lambda)}{2}\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\text{J}_m(\\alpha_m\\rho)\\nonumber\\\\\n&+i\\alpha_m(\\beta+m\\Lambda)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\text{J}_m^{'}(\\alpha_m\\rho),\n\\end{align}\n\\end{subequations}\nfor the magnetic field. In the equations above, $\\text{J}_{m}^{'}(\\alpha_m\\rho)$, and $\\text{J}_{m}^{''}(\\alpha_m\\rho)$  are the first and second derivative of the Bessel function with respect to their argument, respectively \\cite{nist}. The electric and magnetic fields of arbitrary polarised RSABs can be then constructed by substituting the expresisons above into Eqs. \\eqref{eq11}. Their explicit expression is reported in Appendix A, for completeness.\n\\subsection{Polarisation Constraint for Vector RSABs}\nIn Sect. 2, we have described the requirements that a scalar field must fulfill, in order to be a RSAB. In particular, the most important requirement is the existence of a suitable co-rotating reference frame, in which the field appears propagation invariant. If such reference frame exists, an observer at rest in such reference frame would then experience a fictitious centrifugal force. \n\nFor scalar fields, however, this condition is independent on polarisation, as it only applies to the field distribution, and not to the constant polarisation pattern possessed by the field. For vector beams, on the other hand, this assumption may not be valid anymore, as different polarisation states are focussed in different ways, thus resulting in a mixing of the various field components \\cite{bornWolf}. In this case, then, it is necessary to investigate under which condition the polarisation coefficients $f_p$ and $f_s$ preserve the self-accelerating character of vector RSABs. A natural way to prove this, is to impose that vector RSABs fulfill the same requirements described in Sect. 2. \n\nTo do so, we first define a suitable co-rotating reference frame, in which the electric and magnetic fields of a vector RSAB appear propagation invariant. If such reference frame exists, this automatically implies that the self-accelerating character has been preserved by the vectorialisation procedure. This means, that the electric (magnetic) field described by the first (second) of Eqs. \\eqref{eq9} must be propagation invariant in a co-rotating reference frame $\\vettGreek{\\rho}'=\\mathcal{S}\\vettGreek{\\rho}$ defined by  the following coordinate transformation\n\\begin{equation}\n\\left\\{\\begin{array}{ll}\n\\rho'=\\rho,\\\\\n\\Phi=\\theta+\\Lambda\\zeta,\\\\\n\\zeta'=\\zeta.\n\\end{array}\\right.\n\\end{equation}\nIn principle, one should check that both the electric and the magnetic field are independently propagation invariant in this reference frame. However, Maxwell's equation impose that if one field fulfills the requirement, the other must fulfill it too. For this reason, we limit or analysis to the electric field only. The same condition that we will derive for the polarisation coefficients $f_p$ and $f_s$ will apply to the magnetic field as well, and can be also derived using the same approach with the magnetic, rather than electric, field. \n\nWe then start by separating the electric field into its transverse and longitudinal parts, namely  $\\vett{E}(\\vettGreek{\\rho}')=\\vett{E}_{\\perp}(\\vettGreek{\\rho}')+\\vett{E}_{\\parallel}(\\vettGreek{\\rho}')$, and require that they are both propagation invariant, i.e., $\\partial\\vett{E}_{\\perp,\\parallel}(\\vettGreek{\\rho}')\/\\partial\\zeta=0$. Instead of dealing directly with this condition, however, we can require that the transverse and longitudinal intensities, rather than amplitudes, are propagation invariant. By doing this, we are formally requiring that only intensity rotating RSABs remain propagation invariant upon focussing. However, if the intensity of a field is independent from $\\zeta$, its amplitude will be $\\zeta$-independent as well, and the $\\zeta$ dependence can be at most contained into a phase factor. Once the condition on the intensity has been met, one could then look at the phase of the corresponding field, and check, whether it remains synchronised with its corresponding intensity profile.\n\nThe transverse  $\\left|\\vett{E}_{\\perp}(\\vettGreek{\\rho})\\right|^2=\\left|E_{\\rho}\\right|^2+\\left|E_{\\theta}\\right|^2$, and longitudinal $\\left|\\vett{E}_{\\parallel}(\\vettGreek{\\rho})\\right|^2=\\left|E_{\\zeta}\\right|^2$ intensities can be calculated using the expressions given in Appendix A, thus obtaining\n\\begin{subequations}\\label{eq14}\n\\begin{align}\n\\left|\\vett{E}_{\\perp}(\\vettGreek{\\rho})\\right|^2 &= \\sum_{m\\in\\mathcal{M}}\\left|\\mathcal{E}_m^{(1)}(\\rho)\\right|^2+2\\sum_{n\\neq m\\in\\mathcal{M}}\\mathcal{E}_m^{(1)}(\\rho)\\left[\\mathcal{E}_n^{(1)}(\\rho)\\right]^*\\cos\\left[\\left(m-n\\right)\\Phi\\right],\\\\\n\\left|\\vett{E}_{\\parallel}(\\vettGreek{\\rho})\\right|^2 &=G_1(\\rho,\\theta)+2\\sum_{n\\neq m\\in\\mathcal{M}}\\mathcal{E}_m^{(2)}(\\rho)\\left[\\mathcal{E}_n^{(3)}(\\rho)\\right]^*\\Big\\{a_pa_s\\cos^2\\theta\\sin\\left[\\left(m-n\\right)\\Phi-\\Delta\\right]\\nonumber\\\\\n&-(a_p^2-a_s^2)\\sin\\theta\\cos\\theta\\sin\\left[\\left(m-n\\right)\\Phi\\right]-a_pa_s\\sin^2\\theta\\sin\\left[\\left(m-n\\right)\\Phi+\\Delta\\right]\\Big\\}\\label{parallel}\n\\end{align}\n\\end{subequations}\nwhere we have rewritten the polarisation coefficients as $f_p=a_p$, $f_s=a_s\\exp{(i\\Delta)}$ (with $a_p, a_s, \\Delta \\in \\mathbb{R}$), being $\\Delta$ the relative phase between the two polarisation components,  and\n\\begin{eqnarray}\\label{G1}\nG_1(\\rho,\\theta)&=&\\sum_{m\\in\\mathcal{M}}\\Big\\{a_p^2\\left[\\cos^2\\theta \\left|\\mathcal{E}_m^{(2)}(\\rho)\\right|^2+\\sin^2\\theta \\left|\\mathcal{E}_m^{(3)}(\\rho)\\right|^2\\right]\\nonumber\\\\\n&+&a_s^2\\left[\\cos^2\\theta \\left|\\mathcal{E}_m^{(3)}(\\rho)\\right|^2+\\sin^2\\theta \\left|\\mathcal{E}_m^{(2)}(\\rho)\\right|^2\\right]\\nonumber\\\\\n&+&2a_pa_s\\left(\\left|\\mathcal{E}_m^{(2)}(\\rho)\\right|^2-\\left|\\mathcal{E}_m^{(3)}(\\rho)\\right|^2\\right)\\sin\\theta\\cos\\theta\\cos\\Delta\\Big\\}.\n\\end{eqnarray}\nEquations \\eqref{eq14} already contain an important information. No matter the polarisastion, the transverse intensity always remains propagation invariant, as no explicit $\\zeta$-dependence appears in the expression of $\\left|\\vett{E}_{\\perp}(\\vettGreek{\\rho})\\right|^2$. \n\nThe longitudinal part of the intensity, on the other hand, \ncontains terms that depend on $\\sin\\theta$ and $\\cos\\theta$. Once transformed in the co-rotating frame, these terms become $\\zeta$-dependent, as $\\theta=\\Phi-\\Lambda\\zeta$. To avoid this problem, the polarisation coefficients must be chosen in such a way to guarantee the propagation invariance of the longitudinal intensity as well.  The condition on $a_p$, $a_s$, and $\\Delta$ can be then found by requiring that \n\\begin{eqnarray}\\label{eq15}\n\\frac{\\partial\\left|\\vett{E}_{\\parallel}(\\vettGreek{\\rho})\\right|^2}{\\partial\\zeta} &=\\left(a_p^2-a_s^2\\right)\\Big\\{\\left[F_2(\\rho)-F_3(\\rho)\\right]\\sin2\\theta+F_4(\\rho)\\cos2\\theta\\Big\\}\\nonumber\\\\\n&+a_pa_s\\cos\\Delta\\left\\{F_4(\\rho)\\sin2\\theta-\\left[F_2(\\rho)-F_3(\\rho)\\right]\\cos2\\theta\\right\\}=0,\n\\end{eqnarray}\nwhere the functions $F_k(\\rho)$ (with $k=\\{1,2,3,4\\}$) can be determined from Eqs. \\eqref{parallel} and \\eqref{G1}. It is not difficult to see, that the above equation is satisfied if and only if $a_p=a_s$, and $\\Delta=\\pm\\pi\/2$. Moreover, since $|f_p|^2+|f_s|^2=1$, this condition implies that $f_p=1\/\\sqrt{2}$, and $f_s=\\pm i\/\\sqrt{2}$., which correspond to left-handed ($+$) and right-handed ($-$) circular polarisation, respectively\n\nThis is the main result of our work. Vector RSABs only maintain their self-accelerating character if the polarisation of the Hertz vector is chosen to be circular. In other words, when focussing polarised RSABs, only circular polarisation is allowed, in order to preserve the self-accelerating character of the focused RSABs.  \n\nA simple explanation of this result can be given by looking at the symmetry of the scalar and vector beams, respectively. In the scalar case, in fact, RSABs naturally possess cylindrical symmetry, due to their transverse profile. By virtue of this symmetry, the co-rotating coordinate can be chosen as a $\\zeta$-dependent azimuthal coordinate, namely $\\Phi=\\theta+\\Lambda\\zeta$. Upon focussing, the overall cylindrical symmetry must be preserved, in order for the vector RSAB to maintain its self-accelerating character. This, ultimately, constraints the polarisation to be chosen as circular.\n\\subsection{Vector Fields from Circularly Polarised RSABs}\nWe now apply the polarisation constraints derived above and investigate the form of the electric and magnetic fields generated by focussing circularly polarised RSABs. By substituting $f_p=1\/\\sqrt{2}$ and $f_s=i\\sigma\/\\sqrt{2}$ into Eqs. \\eqref{eq11} and \\eqref{eq12}, and using Eqs. \\eqref{eq9}, the electric and magnetic fields of a circularly polarised focussed RSAB can be written as\n\\begin{subequations}\\label{eq16}\n\\begin{align}\n\\vett{E}(\\vettGreek{\\rho},t)&=\\sum_{m\\in\\mathcal{M}}\\vett{e}_m(\\rho)\\,e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]},\\\\\n\\vett{B}(\\vettGreek{\\rho},t)&=\\sum_{m\\in\\mathcal{M}}\\vett{b}_m(\\rho)\\,e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]},\n\\end{align}\n\\end{subequations}\nwhere $\\sigma=\\pm 1$ is the helicity index, which distinguishes between left-handed ($+$) and right-handed ($-$) circular polarisation \\cite{mandelWolf}, $\\vett{e}_m(\\rho)$, and $\\vett{b}_m(\\rho)$ are  radially dependent vector field, whose explicit expression, is given by\n\\begin{subequations}\\label{eq16bis}\n\\begin{align}\n\\vett{e}_m(\\vettGreek{\\rho})&=\\sigma D_m\\omega\\left\\{ i(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho)\\,\\uvett{h}_{\\sigma}-\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\zeta}\\right\\},\\\\\n\\vett{b}_m(\\rho) &= \\frac{D_m}{\\sqrt{2}}\\Big\\{(1+m\\sigma)\\left[\\frac{\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)+2m\\sigma\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\rho}\\nonumber\\\\\n&+i\\left\\{\\frac{m}{\\rho}\\left[\\text{J}_m^{'}(\\alpha_m\\rho)-\\text{J}_m(\\alpha_m\\rho)\\right]+\\sigma\\left[(\\beta+m\\Lambda)^2\\text{J}_m(\\alpha_m\\rho)-\\alpha_m^2\\text{J}_m^{''}(\\alpha_m\\rho)\\right]\\right\\}\\uvettGreek{\\theta}\\nonumber\\\\\n&+i(\\beta+m\\Lambda)\\left[\\alpha_m\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m\\sigma}{2}\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\zeta}\\Big\\},\n\\end{align}\n\\end{subequations}\nwhere $\\uvett{h}_{\\sigma}=\\left(\\uvettGreek{\\rho}+i\\sigma\\uvettGreek{\\theta}\\right)\/\\sqrt{2}=\\left(\\uvett{x}+i\\sigma\\uvett{y}\\right)\\sqrt{2}$ is the helicity basis \\cite{mandelWolf}.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure2.pdf}\n\\caption{Intensity and phase distribution for the longitudinal component $E_{\\zeta}$ of the electric field described by Eq. \\eqref{eq17a}, for $\\sigma=1$.  Panels (a) and (b) correspond to the intensity and phase distributions at $z=0$, while panels (c), (d) to $z=0.8(2\\pi\/\\Lambda)$. These plots are made assuming $0\\leq\\rho\\leq 10$.  The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The white arrow in the intensity profiles show the direction of rotation.}\n\\label{figure2}\n\\end{center}\n\\end{figure}\nFor experimentally realisable RSABs, $\\Lambda\\ll 1$. Within this approximation, one should distinguish between field rotating, and intensity rotating vector RSABs. For the former, $\\beta=0$, and the radial and azimuthal components of the electric field, as well as the longitudinal component of the magnetic field, are $\\mathcal{O}(\\Lambda)$, and can therefore be neglected, leaving a purely longitudinal electric field, and a purely transverse magnetic field, namely\n\\begin{subequations}\\label{eq17}\n\\begin{align}\n\\vett{E}(\\vettGreek{\\rho},t) &\\simeq\\left(-\\frac{\\sigma\\omega}{\\sqrt{2}}\\right)\\sum_{m\\in\\mathcal{M}}D_m\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right]e^{i[(m+\\sigma)\\theta+m\\Lambda\\zeta-\\omega t]}\\uvettGreek{\\zeta},\\label{eq17a}\\\\\n\\vett{B}(\\vettGreek{\\rho},t)&\\simeq\\frac{1}{\\sqrt{2}}\\sum_{m\\in\\mathcal{M}}D_m\\Bigg\\{(1+m\\sigma)\\left[\\frac{\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)+2m\\sigma\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\rho}\\nonumber\\\\\n&+i\\left[\\frac{m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)-\\sigma\\alpha_m^2\\text{J}_m^{''}(\\alpha_m\\rho)\\right]\\uvettGreek{\\theta}\\Bigg\\}e^{i[(m+\\sigma)\\theta+m\\Lambda\\zeta-\\omega t]}.\n\\end{align}\n\\end{subequations}\nFor intensity rotating vector RSABs, and within the paraxial approximation, $(\\beta+m\\Lambda)\\simeq 1$, and therefore $\\alpha_m\\ll 1$. In this case, all three components  of the electric and magnetic field are nonzero, and assume the following, simplified, form:\n\\begin{subequations}\\label{eq19}\n\\begin{align}\n\\vett{E}(\\vettGreek{\\rho},t) &\\simeq\\sum_{m\\in\\mathcal{M}}\\sigma\\omega D_m\\left\\{i\\text{J}_m(\\alpha_m\\rho)\\,\\uvett{h}_{\\sigma}-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\uvettGreek{\\zeta}\\right\\}e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]}\\label{eq19a},\\\\\n\\vett{B}(\\vettGreek{\\rho},t)&\\simeq\\sum_{m\\in\\mathcal{M}}\\frac{D_m}{\\sqrt{2}}\\Bigg\\{2m(\\sigma+m)\\text{J}_m(\\alpha_m\\rho)\\uvettGreek{\\rho}+i\\left[\\frac{m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right]\\uvettGreek{\\theta}\\nonumber\\\\\n&-\\frac{i m\\sigma}{2}\\text{J}_m(\\alpha_m\\rho)\\uvettGreek{\\zeta}\\Bigg\\}e^{i[(m+\\sigma)\\theta+(\\beta+m\\Lambda)\\zeta-\\omega t]}.\n\\end{align}\n\\end{subequations}\nFrom the expressions above, it appears clear that, upon focussing, the electric field maintains the original circular polarisation (in the transverse plane) of the focussed beam, while the polarisation of the magnetic field gets mixed. This, however, is only a result of the fact that we only considered TE fields to start with. If one would repeat the above calculations for TM fields, in fact, the result would be the same, with the magnetic field retaining the original polarisation and the electric field being mixed up. In the most general case, where both TE and TM waves are present, each field has these two components of polarisation, thus resulting in a more complex polarisation pattern. \n\nThe intensities and phases of the electric field components for field rotating and intensity rotating RSABs are reported in Figs. \\ref{figure2}, and \\ref{figure3}, respectively.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure3.pdf}\n\\caption{Intensity and phase distribution for the radial (top row) and longitudinal (bottom row) components  of the electric field described by Eq. \\eqref{eq19a}, for $\\sigma=1$.  Panels (a), (b), (e), and (f) correspond to the intensity and phase distributions at $z=0$, while panels (c), (d), (g), and (h) to $z=0.8(2\\pi\/\\Lambda)$. These plots are made assuming $0\\leq\\rho\\leq 1200$.  The azimuthal component of the field is not shown, as its intensity profile is the same as the radial one [ see Eq. \\eqref{eq19a}]. The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The white arrow in the intensity profiles show the direction of rotation.}\n\\label{figure3}\n\\end{center}\n\\end{figure}\nAs it can be seen from Fig. \\ref{figure2}(d), upon focussing, field rotating vector RSABs lose their property, that intensity and phase profile are synchronised in rotation during propagation. This, ultimately is due to the fact that while the field intensity contains terms of the form $\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)\\right]$, the phase contains terms that oscillate like $\\cos\\left[m\\left(\\theta+\\Lambda\\zeta\\right)+\\sigma\\theta\\right]$. The presence of the extra term $\\sigma\\theta$ (which disappears in the intensity) is then responsible for the different evolution of amplitude and phase of the field, as it corresponds to a $\\zeta$-dependent term, once transformed in the co-rotating frame.\n\\section{Linear and Angular Momentum Densities of Vector RSABs}\nIn this section, we calculate the linear and angular momentum for intensity rotating vector RSABs. We limit ourselves to the paraxial case, as within this approximation, we can separate the angular momentum in its spin and orbital parts. This gives us the possibility to distinguish between intrinsic and extrinsic orbital angular momentum of vector RSABs, and to then isolate the extrinsic contribution given by the fact that the intensity rotates with angular velocity $\\Lambda$. \n\nFollowing Jackson, the linear and angular momentum of the electromagnetic field are defined as follows \\cite{jackson}:\n\\begin{subequations}\\label{eq21}\n\\begin{align}\n\\vett{P}=\\int\\,d^2\\rho\\,\\vett{p}(\\vettGreek{\\rho}),\\\\\n\\vett{J}=\\int\\,d^2\\rho\\,\\vett{j}(\\vettGreek{\\rho}),\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{subequations}\\label{eq20}\n\\begin{align}\n\\vett{p}(\\vettGreek{\\rho})=\\frac{\\varepsilon_0}{2}\\,\\operatorname{Re}\\left\\{\\vett{E}(\\vettGreek{\\rho})\\times\\vett{B}^*(\\vettGreek{\\rho})\\right\\},\\label{eq20a}\\\\\n\\vett{j}(\\vettGreek{\\rho})=\\frac{\\varepsilon_0}{2}\\,\\operatorname{Re}\\left\\{\\vettGreek{\\rho}\\times\\vett{p}(\\vettGreek{\\rho})\\right\\},\\label{eq20b}\n\\end{align}\n\\end{subequations}\nare the correspondent densities, $d^2\\rho=\\rho d\\rho d\\theta$, and the integrals are extended over the whole space. \n\nAs it can be seen from Eqs. \\eqref{eq3} and \\eqref{eq16}, RSABs are defined in terms of superpositions of Bessel beams. Therefore, as Bessel beams cary infinite energy, the above integrals diverge, and linear and angular momentum (as well as energy) are not well defined quantities for RSABs. This problem, however, can be overcome in different ways, by introducing different forms of regularisation. For example, one could limit the radial integration, up to a maximum radius. Alternatively, one could insert a regularisation function, such a Gaussian function, in the radial integrals to make them finite. Physically speaking, both regularisations can be implemented. The former, in fact, corresponds to use a pupil of a fixed diameter to filter the field. The latter, on the other hand, corresponds to describe RSABs in terms of Bessel-Gauss beams, which, de facto, are the closest approximation to Bessel beams that can be realised experimentally.\n\nIn the remaining of this section, we calculate the explicit expressions for both the momentum densities, and their integrated counterpart. For the sake of simplicity, however, we will not compute the radial integrals. These, in fact, only contribute to a multiplicative constant, and do not carry any valuable information for the purpose of investigating the properties of linear and angular momentum of RSABs.\n\\subsection{Linear Momentum}\nIf we substitute the expressions of the electric and magnetic fields of a paraxial RSABs as given by Eqs. \\eqref{eq16} into Eq. \\eqref{eq20a}, the linear momentum density can be written as follows:\n\\begin{eqnarray}\\label{eq22}\n\\vett{p}(\\vettGreek{\\rho}) &=& \\sum_{m,n\\in{\\mathcal{M}}}\\frac{|D_mD_n|\\omega\\varepsilon_0}{4}\\Big\\{P_{\\rho}^{(m,n)}(\\rho)\\sin\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\uvettGreek{\\rho}\\nonumber\\\\\n&+&P_{\\theta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\uvettGreek{\\theta}\\nonumber\\\\\n&+&P_{\\zeta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\uvettGreek{\\zeta}\\Big\\},\n\\end{eqnarray}\nwhere $\\phi_{m,n}=\\arg[D_{m,n}]$, and\n\\begin{subequations}\\label{eq23}\n\\begin{align}\nP_{\\rho}^{(m,n)}(\\rho) &=\\sigma\\text{J}_m(\\alpha_m\\rho)\\Bigg\\{\\left[\\frac{mn}{\\rho^2}-\\frac{m\\sigma(\\beta+n\\Lambda)^2}{\\rho}-\\frac{n(\\beta+m\\Lambda)(\\beta+n\\Lambda)}{2}\\right]\\text{J}_n(\\alpha_n\\rho)\\nonumber\\\\\n&-\\frac{mn}{\\rho^2}\\text{J}_n^{'}(\\alpha_n\\rho)\\Bigg\\},\\\\\nP_{\\theta}^{(m,n)}(\\rho) &=\\sigma\\left[\\frac{2mn(n+\\sigma)}{\\rho}+\\frac{n\\sigma(\\beta+m\\Lambda)(\\beta+n\\Lambda)}{2}\\right]\\text{J}_m(\\alpha_m\\rho)\\text{J}_n(\\alpha_n\\rho),\\\\\nP_{\\zeta}^{(m,n)}(\\rho) &=(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho)\\Bigg\\{\\left[(\\beta+n\\Lambda)^2+2n(n+\\sigma)-\\frac{n\\sigma}{\\rho}\\right]\\text{J}_n(\\alpha_n\\rho)\\nonumber\\\\\n&+\\frac{n}{\\rho}\\text{J}_n^{'}(\\alpha_n\\rho)\\Bigg\\},\n\\end{align}\n\\end{subequations}\nwhere terms of order $\\mathcal{O}(\\alpha_m)$ have been neglected, since in the paraxial regime $\\alpha_m\\ll 1$.  The components of the linear momentum density are shown in Fig. \\ref{figure5}. Notice, that the transverse part of the linear momentum presents an unusual characteristic. While it rotates clockwise along the propagation direction, as the intensity distribution of the correspondent RSAB does, the local orientation of the transverse momentum is purely azimuthal (despite $\\vett{p}(\\vettGreek{\\rho})$ has a nonzero radial component), and always directed in the opposite direction, with respect to the rotation direction of the RSAB, as it can be seen from the white arrows in Fig. \\ref{figure5}(a). This has an interesting consequence for applications such particle manipulation and material processing, where the local, rather than the global, behaviour of the momentum plays an important role. While the RSAB (and, with it, the transverse momentum density) rotates clockwise during propagation, a particle placed in the vicinity of a RSAB will experience a local momentum, that will tend to push it in the opposite direction. This effect, however, is purely local, and it disappears when considering the whole momentum. \n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure5.pdf}\n\\caption{Transverse (a) and longitudinal (b) components of the linear momentum density, as given by Eq. \\eqref{eq22}, in the plane $\\zeta=0$, for $\\sigma=1$. The white arrows in panel (a) represent the flow of the transverse component of the linear momentum density. As it can be seen, the transverse momentum density always points in the opposite direction with respect to the field rotation (red arrow). These plots are made assuming $0\\leq\\rho\\leq 1200$. The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The red arrow in both panels show the direction of rotation of the RSAB intensity.}\n\\label{figure5}\n\\end{center}\n\\end{figure}\nTo understand this, let us integrate Eq. \\eqref{eq22} over the transverse space. The linear momentum can be then written as follows\n\\begin{equation}\\label{momentumP}\n\\vett{P}=\\sum_{m\\in\\mathcal{M}}|D_m|^2\\Bigg[\\mathcal{P}^{(m)}_{\\theta}\\uvettGreek{\\theta}+\\mathcal{P}^{(m)}_{\\zeta}\\uvettGreek{\\zeta}\\Bigg],\n\\end{equation}\nwhere\n\\begin{equation}\\label{Plambda}\n\\mathcal{P}_{\\lambda}^{(m)}=\\frac{\\pi\\omega\\varepsilon_0}{2}\\int_0^{\\infty}\\,d\\rho\\,\\rho\\,P_{\\lambda}^{(m,m)}(\\rho),\n\\end{equation}\nwhere $\\lambda\\in\\{\\theta,\\zeta\\}$. Notice that the radial integrals (once regularised) amount to a positive constant. Moreover, there is no radial component of the momentum, since the radial part of $\\vett{p}(\\vett{\\rho})$ depends on $\\sin[(m-n)(\\theta+\\Lambda\\zeta)+\\phi_m-\\phi_n]$, which gives zero once integrated with respect to the azimuthal coordinate $\\theta$. \n\\subsection{Spin and Orbital Angular Momentum}\nTo calculate the spin and orbital angular momentum for intensity rotating, paraxial RSABs, we make use of the usual decomposition of the total angular momentum in its spin (SAM) and orbital (OAM) components, namely $\\vett{J}=\\vett{S}+\\vett{L}$ \\cite{libroOAM}. To do so, we first need to introduce the vector potential $\\vett{A}(\\vettGreek{\\rho})$ associated to the electric and magnetic fields defined above, since the decomposition assumes a rather simple form if expressed in terms of the vector potential. Looking at Eqs. \\eqref{eq5}, it is not difficult to see that $\\vett{A}=\\nabla\\times\\vettGreek{\\Pi}(\\vettGreek{\\rho},t)$. The explicit expression of $\\vett{A}$ for an intensity rotating RSAB is given in Appendix B. \n\nFollowing Ref. \\cite{libroOAM}, the angular momentum density then assumes the following form\n\\begin{equation}\\label{eq24}\n\\mathbf{j}(\\vettGreek{\\rho})=\\vett{s}(\\vettGreek{\\rho})+\\vett{l}(\\vettGreek{\\rho})=\\frac{\\varepsilon_0}{2}\\operatorname{Re}\\left\\{-i\\vett{A}^*\\times\\vett{A}\\right\\}+\\frac{\\varepsilon_0}{2}\\operatorname{Re}\\left\\{\\vett{A}^*\\cdot\\left(-i\\vettGreek{\\rho}\\times\\nabla\\right)\\vett{A}\\right\\},\n\\end{equation}\nwhere $-i\\vettGreek{\\rho}\\times\\nabla$ is the angular momentum operator \\cite{jackson} in the normalised cylindrical reference frame $\\{\\uvettGreek{\\rho},\\uvettGreek{\\theta},\\uvettGreek{\\zeta}\\}$. Using the expression for the vector potential given in Appendix B, the SAM and OAM of a paraxial, intensity rotating vector RSAB are given as follows:\n\\begin{subequations}\\label{eq25}\n\\begin{align}\n\\vett{s}(\\vettGreek{\\rho})&=\\frac{\\varepsilon_0}{2}\\sum_{m,n\\in\\mathcal{M}}\\Bigg\\{-S_{\\rho}^{(m,n)}\\sin\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\rho}\\nonumber\\\\\n&+S_{\\theta}^{(m,n)}\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\theta}\\nonumber\\\\\n&-S_{\\zeta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\zeta}\\Bigg\\},\\\\\n\\vett{l}(\\vettGreek{\\rho}) &=\\frac{\\varepsilon_0}{2}\\sum_{m,n\\in\\mathcal{M}}\\Bigg\\{L_{\\rho}^{(m,n)}(\\rho)\\sin\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\rho}\\nonumber\\\\\n&-L_{\\theta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\theta}\\nonumber\\\\\n&+L_{\\zeta}^{(m,n)}(\\rho)\\cos\\left[\\left(m-n\\right)\\left(\\theta+\\Lambda\\zeta\\right)+\\phi_m-\\phi_n\\right]\\,\\uvettGreek{\\zeta},\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{eqnarray}\\label{SAMdensity}\nS_{\\rho}^{(m,n)}(\\rho)&=&\\frac{\\mathcal{D}_{m,n}(\\rho)\\beta(m-n)}{\\rho},\\\\\nS_{\\theta}^{(m,n)}(\\rho)&=&\\sigma S_{\\rho}(\\rho),\\\\\nS_{\\zeta}^{(m,n)}(\\rho)&=&2\\sigma\\mathcal{D}_{m,n}(\\rho)(\\beta+m\\Lambda)(\\beta+n\\Lambda),\n\\end{eqnarray}\nare the components of the spin angular momentum density, while\n\\begin{eqnarray}\nL_{\\rho}^{(m,n)}(\\rho)&=&\\mathcal{D}_{m,n}(\\rho)(\\beta+m\\Lambda)\\left[\\sigma\\frac{n(m+\\sigma)}{\\rho}+\\rho(\\beta+m\\Lambda)(\\beta+n\\Lambda)\\right],\\\\\nL_{\\theta}^{(m,n)}(\\rho)&=&\\sigma L_{\\rho}(\\rho),\\\\\nL_{\\zeta}^{(m,n)}(\\rho)&=&\\mathcal{D}_{m,n}(\\rho)(\\beta+m\\Lambda)\\left[\\frac{m(\\beta+m\\Lambda)+n(m+\\sigma)}{\\rho}\\right],\n\\end{eqnarray}\nare the components of the orbital angular momentum density, and $\\mathcal{D}_{m,n}(\\rho)=|D_mD_n|\\text{J}_m(\\alpha_m\\rho)\\text{J}_n(\\alpha_n\\rho)\/2$. In the above expressions, terms of order $\\mathcal{O}(\\alpha_m)$ have been neglected, since, for paraxial fields, $\\alpha_m\\ll 1$.  The longitudinal and transverse SAM densities are plotted in Fig. \\ref{figure6}. As it can be seen, the SAM density can become negative. This means, that locally, the helicity of the vector RSAB can change sign. However, a close comparison between the SAM density distribution in Fig. \\ref{figure6} and the transverse and longitudinal intensity distributions depicted in Fig. \\ref{figure3} reveals, that regions of negative SAM density occur where the RSAB intensity is very low, or even zero. \n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\textwidth]{figure6.pdf}\n\\caption{Radial (a) and longitudinal (b) components of the SAM density, as given by Eqs. \\eqref{SAMdensity}, in the plane $\\zeta=0$, for $\\sigma=1$. The blue regions in both panels indicate areas of negative SAM density, where the helicity is oriented in the opposite direction, with respect to the propagation direction. The azimuthal component of the SAM density is not reported here, as it is, up to a constant, the same as the radial one. These plots are made assuming $0\\leq\\rho\\leq 1200$. The plot parameters are the same as the one chosen for Fig. \\ref{figure1}. The red arrow in both panels show the direction of rotation of the RSAB intensity.}\n\\label{figure6}\n\\end{center}\n\\end{figure}\n\nThe spin and orbital angular momenta of paraxial vector RSABs are then obtained by integrating Eqs. \\eqref{eq25} over the transverse space. By doing so we obtain\n\\begin{subequations}\\label{eq26}\n\\begin{align}\n\\vett{S}&=\\sigma\\sum_{m\\in\\mathcal{M}}|D_m|^2\\mathcal{S}^{(m)}_{\\zeta}\\uvettGreek{\\zeta}\\label{eq26a},\\\\\n\\vett{L}&=\\sum_{m\\in\\mathcal{M}}|D_m|^2\\Bigg[\\mathcal{L}^{(m)}_{\\theta}\\uvettGreek{\\theta}+\\mathcal{L}^{(m)}_{\\zeta}\\uvettGreek{\\zeta}\\Bigg],\\label{eq26b}\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{equation}\n\\mathcal{S}_{\\zeta}^{(m)}=\\frac{\\varepsilon_0}{4}\\int_0^{\\infty}\\,d\\rho\\,\\rho\\,S_{\\zeta}^{(m,m)}(\\rho),\n\\end{equation}\nand\n\\begin{equation}\n\\mathcal{L}_{\\lambda}^{(m)}=\\frac{\\varepsilon_0}{4}\\int_0^{\\infty}\\,d\\rho\\,\\rho\\,L_{\\zeta}^{(m,m)}(\\rho),\n\\end{equation}\nbeing $\\lambda\\in\\{\\theta,\\zeta\\}$. The total SAM is purely longitudinal, and given as the sum of the longitudinal components of the individual Bessel components. This is not surprising, since we are dealing with paraxial fields, for which the SAM is only directed along the propagation direction \\cite{libroOAM}.\n\nThe OAM, on the other hand, can be seen as the sum of two contributions: an intrinsic component relative to the intrinsic OAM carried by Bessel beams, and an extrinsic one, connected to the fact that the beam rotates around the $\\zeta$-axis during propagation. Their explicit expression read then as follows:\n\\begin{subequations}\\label{eq27}\n\\begin{align}\n\\vett{L}^{(int)}&=\\sum_{m\\in\\mathcal{M}}|\\tilde{D}_m|^2\\mathcal{L}_{int}^{(m)}\\left(\\uvettGreek{\\theta}+\\uvettGreek{\\zeta}\\right),\\\\\n\\vett{L}^{(ext)}&=\\sum_{m\\in\\mathcal{M}}|\\tilde{D}_m|^2\\left[\\sigma\\mathcal{L}^{(m)}_{ext,\\theta}\\uvettGreek{\\theta}+\\mathcal{L}_{ext,\\zeta}^{(m)}\\,\\uvettGreek{\\zeta}\\right],\n\\end{align}\n\\end{subequations} \nwhere $\\tilde{D}_m=D_m\\sqrt{\\beta+m\\Lambda}$, and\n\\begin{subequations}\\label{eq28}\n\\begin{align}\n\\mathcal{L}_{int}^{(m)}&=\\frac{\\varepsilon_0}{4}\\, m(m+\\sigma)\\int_0^{\\infty}\\,d\\rho\\,\\text{J}_m^2(\\alpha_m\\rho),\\\\\n\\mathcal{L}_{ext,\\theta}^{(m)}&=\\frac{\\varepsilon_0}{4}\\,(\\beta+m\\Lambda)^2\\int_0^{\\infty}\\,d\\rho\\,\\rho^2\\text{J}_m^2(\\alpha_m\\rho),\\\\\n\\mathcal{L}_{ext,\\zeta}^{(m)}&=\\frac{\\varepsilon_0}{4}\\,m(\\beta+m\\Lambda)\\int_0^{\\infty}\\,d\\rho\\,\\text{J}_m^2(\\alpha_m\\rho).\n\\end{align}\n\\end{subequations}\nThe intrinsic part of the OAM has the standard spin-orbit interaction form, through the mixed term $(m+\\sigma)$ \\cite{libroOAM}. The extrinsic part, on the other hand, depends on $(\\beta+m\\Lambda)$, which is, essentially, the angular velocity of the beam along the $\\zeta$-axis. The beam rotation, moreover, also induces a longitudinal OAM, which is, as well, proportional to the angular velocity $(\\beta+m\\Lambda)$.\n\\section{Conclusions}\nIn this work, we have analysed the properties of vector RSABs, generated by focussing a scalar, polarised RSAB. Using the method of Hertz potentials as a model for the focussing process, we have demonstrated that only circularly polarised scalar RSABs, when focussed, maintain their self-accelerating character. For this case, we have given explicit expressions of the TE vector electric and magnetic fields for both field and intensity rotating RSABs. In particular, we have shown, that the vectorialisation (focussing) process does not allow anymore the amplitude and phase of field rotating RSABs to rotate synchronously during propagation. Within the paraxial approximation, moreover, we have presented the explicit expressions for the linear and angular momentum densities of intensity rotating RSABs. For SAM, in particular, we have shown, that, locally, the SAM density can be negative, thus meaning a local inversion of the helicity axis. Moreover, for the case of OAM, we have distinguished between the intrinsic and extrinsic contributions, and shown how the rotation of the RSAB around the propagation axis is connected with the extrinsic OAM.\n\nOur work represents a useful guideline for investigating experimentally focussed RSABs and their properties. Moreover, the properties highlighted in this work represent a useful toolbox for studying the interaction of RSABs with matter and dielectric particles. In particular, the fact that locally the linear momentum density flows in the opposite direction, with respect to the overall beam rotation during propagation, could open new possibilities for particle manipulation\n\n\\section*{Acknowledgements}\nThe authors wish to thank the Deutsche Forschungsgemeinschaft (grant SZ 276\/17-1) for financial support.\n\n\\section*{Appendix A: Explicit Form of RSAB Electric and Magnetic Fields}\nThe vector electric and magnetic fields for single Bessel beams defined in Eqs. \\eqref{eq11} and \\eqref{eq12} can be used to write the expressions for the RSAB vector electric and magnetic fields explicitly. Substituting these expressions into  Eqs. \\eqref{eq9} we then get\n\\begin{eqnarray}\\label{electric}\n\\vett{E}(\\vettGreek{\\rho}) &=&e^{i(\\beta\\zeta-\\omega t)} \\sum_{m\\in\\mathcal{M}}e^{im\\Phi}\\Big\\{\\mathcal{E}_m^{(1)}(\\rho)\\Big[\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\uvettGreek{\\rho}\\nonumber\\\\\n&-&\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\uvettGreek{\\theta}\\Big]+\\Big[\\mathcal{E}_m^{(2)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{E}_m^{(3)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\zeta}\\Big\\},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\mathcal{E}_m^{(1)}(\\rho)&=&D_m\\omega(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho),\\\\\n\\mathcal{E}_m^{(2)}(\\rho)&=&D_m(m\\omega\/\\rho)\\text{J}_m(\\alpha_m\\rho),\\\\\n\\mathcal{E}_m^{(3)}(\\rho)&=&iD_m\\omega\\left[\\alpha_m\\text{J}_{m-1}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\right],\n\\end{eqnarray}\nare the radially dependent expansion coefficients for the electric field\n\\begin{eqnarray}\\label{magnetic}\n\\vett{B}(\\vettGreek{\\rho}) &=&e^{i(\\beta\\zeta-\\omega t)}\\sum_{m\\in\\mathcal{M}}e^{im\\Phi}\\Big\\{\\Big[\\mathcal{B}_m^{(1)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{B}_m^{(2)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\rho}+\\Big[\\mathcal{B}_m^{(3)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{B}_m^{(4)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\theta}+\\Big[\\mathcal{B}_m^{(5)}(\\rho)\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\nonumber\\\\\n&+&\\mathcal{B}_m^{(6)}(\\rho)\\left(f_s\\cos\\theta-f_p\\sin\\theta\\right)\\Big]\\uvettGreek{\\zeta}\\Big\\},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\mathcal{B}_m^{(1)}(\\rho) &=&D_m\\left[-\\frac{\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho) +2m^2\\text{J}_m(\\alpha_m\\rho)\\right],\\\\\n\\mathcal{B}_m^{(2)}(\\rho) &=&D_m\\left[\\frac{im\\alpha_m}{\\rho}\\text{J}_m^{'}(\\alpha_m\\rho)+2im\\text{J}_m(\\alpha_m\\rho)\\right],\\\\  \n\\mathcal{B}_m^{(3)}(\\rho) &=&\\frac{imD_m}{\\rho}\\left[\\alpha_m\\text{J}_m^{'}(\\alpha_m\\rho)-\\text{J}_m(\\alpha_m\\rho)\\right],\\\\\n\\mathcal{B}_m^{(4)}(\\rho) &=& D_m\\left[(\\beta+m\\Lambda)^2\\text{J}_m(\\alpha_m\\rho)-\\text{J}_m^{''}(\\alpha_m\\rho)\\right],\\\\\n\\mathcal{B}_m^{(5)}(\\rho) &=&-\\frac{mD_m(\\beta+m\\Lambda)}{2}\\text{J}_m(\\alpha_m\\rho),\\\\\n\\mathcal{B}_m^{(6)}(\\rho) &=&iD_m\\alpha_m(\\beta+m\\Lambda)\\text{J}_m^{'}(\\alpha_m\\rho),\n\\end{eqnarray}\nare the radially dependent expansion coefficients for the magnetic field.\n\\section*{Appendix B: Explicit Expression for the Vector Potential for RSABs}\nThe vector potential can be defined from the Hertz potential as $\\vett{A}(\\vettGreek{\\rho},t)=\\nabla\\times\\vettGreek{\\Pi}(\\vettGreek{\\rho},t)$ \\cite{stratton}. Using Eqs. \\eqref{eq6} and \\eqref{eq6bis} the vector potential for an arbitrary polarised vector RSAB is given, in cylindrical coordinates, as follows:\n\\begin{eqnarray}\\label{eqB1}\n\\vett{A}(\\vettGreek{\\rho},t) &=& \\sum_{m\\in\\mathcal{M}}D_me^{i[m(\\theta+\\Lambda\\zeta)+\\beta\\zeta-\\omega t]}\\Bigg\\{-i(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho)\\Bigg[(f_s\\cos\\theta-f_p\\sin\\theta)\\,\\uvettGreek{\\rho}\\nonumber\\\\\n&+&i(f_p\\cos\\theta+f_s\\sin\\theta)\\,\\uvettGreek{\\theta}\\Bigg]\\nonumber\\\\\n&+&\\Bigg[\\alpha_m\\left(f_p\\cos\\theta+f_s\\sin\\theta\\right)\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{im}{\\rho}\\Big(f_p\\cos\\theta\\nonumber\\\\\n&+&f_s\\sin\\theta\\Big)\\text{J}_m(\\alpha_m\\rho)\\Bigg]\\,\\uvettGreek{\\zeta}\\Bigg\\}.\n\\end{eqnarray}\nFor the case of circular polarisation, the above expression simplifies to\n\\begin{eqnarray}\\label{eqB2}\n\\vett{A}(\\vettGreek{\\rho},t) &=& \\sum_{m\\in\\mathcal{M}}\\frac{D_m}{\\sqrt{2}}e^{i[m(\\theta+\\Lambda\\zeta)+\\sigma\\theta+\\beta\\zeta-\\omega t]}\\Bigg[A_{\\rho}^{(m)}(\\rho)\\,\\uvettGreek{\\rho}+A_{\\theta}^{(m)}(\\rho)\\,\\uvettGreek{\\theta}\\nonumber\\\\\n&+&A_{\\rho}^{(m)}(\\zeta)\\,\\uvettGreek{\\zeta}\\Bigg],\n\\end{eqnarray}\nwhere\n\\begin{subequations}\\label{eqB3}\n\\begin{align}\nA_{\\rho}^{(m)}(\\rho) &=\\sigma(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho),\\\\\nA_{\\theta}^{(m)}(\\rho) &=i(\\beta+m\\Lambda)\\text{J}_m(\\alpha_m\\rho),\\\\\nA_{\\zeta}^{(m)}(\\rho) &=i\\Bigg[\\sigma\\alpha_m\\,\\text{J}_m^{'}(\\alpha_m\\rho)-\\frac{m}{\\rho}\\text{J}_m(\\alpha_m\\rho)\\Bigg].\n\\end{align}\n\\end{subequations}\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGavitational lensing is the phenomenon of deflection of light rays in a\ngravitational field, which has been successfully employed to explain the astronomical observations in the weak field approximation \\cite{Schneider_Ehlers_Falco_1992,Petters_Levine_Wambsganss_2001,Schneider_Kochanek_Wambsganss_2006, Bartelmann:2010fz} when deflection angle is small. \nWhen the light rays approach towards the photon sphere of black hole where the gravitational field is extremely strong, the deflection angle becomes so large that the weak field method is no longer valid. \nIt was first noticed by Darwin \\cite{Darwin} in 1959 that the light rays passing very close to a black hole would make complete one or more loops around it before falling into the event horizon, hence an infinite series of exotic images were produced.\nLater, strong gravitational lensing was regained wide attention \\cite{Atkinson, Luminet:1979nyg, Ohanian, Nemiroff:1993he}. \nThe  exact lensing equation with arbitary large value of deflection angle is obtained in 2000 \\cite{Frittelli:1999yf, Virbhadra:1999nm}. In 2001 Bozza et al. \\cite{Bozza:2001xd} developed a reliable and analytical method to obtain the deflection angle of Schwarzschild black hole in strong field region and they found the logarithmic divergence of the deflection angle in strong field limit. Later Bozza \\cite{Bozza:2002zj} extended the conclusion to a general asymptotically flat, static, and spherically symmetric spacetime.\nWith the help of strong gravitational lensing it is possible to compare alternative theories of gravity \\cite{Claudel:2000yi, Hasse:2001by, Iyer:2006cn, Virbhadra:2007kw, Bozza:2008ev, Bozza:2009yw, Ghosh:2010uw, Wei:2011nj, Chen:2009eu} and pick up information from different compact objects \\cite{Tsukamoto:2016qro, Bozza:2002af, Vazquez:2003zm, Bozza:2005tg, Bozza:2006nm, Chen:2010yx, Chen:2011ef, Cunha:2015yba, Cavalcanti:2016mbe, Gyulchev:2008ff, Sahu:2012er, Sahu:2013uya, Kuhfittig:2013hva, Nandi:2006ds, Tsukamoto:2012xs, Tsukamoto:2016jzh}.\nLast year, the first image of the supermassive black hole M87$^{*}$ at the center of the galaxy M87 has been captured by the Event Horizon Telescope (EHT)\\cite{Akiyama:2019cqa, Akiyama:2019fyp, Akiyama:2019eap}, which provides us the deeper understanding of the strong gravitational physics.\n\nOne of the simplest natural extension of Einstein's gravity by higher curvature correction is the Einstein-Gauss-Bonnet (EGB) gravity, the action of which in $D$-dimensional spacetime is given by\n\\begin{equation}\nS\n= \\frac{1}{16\\pi}\\int d^{D}x \\sqrt{-g}\n\\left[\\frac{M_{\\rm P}^2}{2}R+\\alpha\\mathcal{G}\\right],\n\\end{equation}\nwhere $\\alpha$ is the coupling constant of the Gauss-Bonnet (GB) term\n\\begin{equation}\n\\mathcal{G}= {R^{\\mu\\nu}}_{\\rho\\sigma} {R^{\\rho\\sigma}}_{\\mu\\nu}-\n4 {R^\\mu}_\\nu {R^\\nu}_\\mu + R^2 = \n6 {R^{\\mu\\nu}}_{[\\mu\\nu} {R^{\\rho\\sigma}}_{\\rho\\sigma]},\n\\end{equation}\nwith $R_{\\mu\\nu\\rho\\sigma}$ the Riemann tensor, $R_{\\mu\\nu}$ the Ricci tensor and $R$ the Ricci scalar.\nIn the  $4$-dimensional spacetime, GB term is a total derivative \\cite{Lanczos:1938sf}, so it has no contribution to the gravitational dynamics.\nHowever, the role of the GB term in $4$-dimensional gravity, in particular, holographic implications to the addition of it to the gravity action was studied in Ref. \\cite{Miskovic:2009bm}.\nNotice that standard thermodynamics for AdS black holes is recovered in this way.\nRecently, Glavan and Lin \\cite{Glavan:2019inb} reformulate the $D$-dimensional EGB gravity by rescaling the coupling $\\alpha\\rightarrow\\alpha\/(D-4)$. They obtain a novel $4$-dimensional EGB gravity theory in the limit $D\\rightarrow4$, where the GB term can give the nontrivial contribution of gravitational dynamics. They also have shown that it can bypass the Lovelock's theorem \\cite{Lovelock:1971yv,Lovelock:1972vz} and prevent Ostrogradsky instability \\cite{Woodard:2015zca}.\nThis idea of regularization can be traced back to Refs \\cite{Tomozawa:2011gp, Cognola:2013fva}, which gives the quantum corrections of Einstein's gravity.\nIn addition, a novel static spherically symmetric black hole solution  was obtained within this theory.\nNote that the black hole solution was found earlier in the gravity theories with conformal anomaly \\cite{Cai:2009ua} and quantum corrections \\cite{Tomozawa:2011gp, Cognola:2013fva}, and recently in regularized Lovelock gravity \\cite{Casalino:2020kbt}, respectively.\n\nThe novel $4$-dimensional EGB black holes are free from singularity problem. Their photon sphere and shadow, as well as the innermost stable circular orbit (ISCO) of a spinless test particle \\cite{Guo:2020zmf} and spinning test particle \\cite{Zhang:2020qew} around them, have been calculated. \nQuasinormal modes of bosonic fields \\cite{Konoplya:2020bxa} and fermionic fields \\cite{Churilova:2020aca} of these black holes have been investigated, and it is found that for the bosonic fields the damping rate is more sensitive than the real part of quasinormal modes by changing of the GB coupling constant $\\alpha$,\nwhile for the fermionic fields the damping rate usually decreases and the real part of the quasinormal modes increases with the increase of $\\alpha$. Konoplya and Zhidenko discussed the stability \\cite{Konoplya:2020juj} of spherically symmetric black holes in the novel EGB gravity. Moreover, other topics in this new theory including the charged black holes in AdS spaces \\cite{Fernandes:2020rpa}, the shadow of dS black holes \\cite{Roy:2020dyy}, the bending of light in dS black holes \\cite{Heydari-Fard:2020sib}, the rotating black holes \\cite{Wei:2020ght,Kumar:2020owy}, radiating black holes \\cite{Ghosh:2020vpc},  the structure of relativistic stars \\cite{Doneva:2020ped}, the thermodynamics of the black holes \\cite{Hegde:2020xlv, Singh:2020xju, Zhang:2020qam, HosseiniMansoori:2020yfj} and the accretion disk around the black hole \\cite{Liu:2020vkh} have also been studied.\nHowever, several problems, such as completeness, about the regularization procedure have been put foward in Refs \\cite{Ai:2020peo, Gurses:2020ofy, Shu:2020cjw, Hennigar:2020lsl, Mahapatra:2020rds, Tian:2020nzb,Ge:2020tid}, in the meantime some prescriptions have been suggested \\cite{Casalino:2020kbt, Hennigar:2020lsl, Lu:2020iav,  Kobayashi:2020wqy}.\tL\\\"u and Pang \\cite{Lu:2020iav} proposed a more rigorous way to regularize the EGB gravity by compactifying the $D$ dimensional EGB gravity on the $(D-4)$ dimensional maximally symmetric space and redefining the coupling constant as $\\alpha\/(D-4)$. \nIn accordance with the results of Ref. \\cite{Kobayashi:2020wqy}, a special scalar-tensor theory that belongs to the family of Horndeski gravity is obtained by this method.\nRef. \\cite{Hennigar:2020lsl} extends the method for obtaining the $D\\rightarrow2$ limit of general relativity \\cite{Mann:1992ar} to the $D\\rightarrow4$ limit of EGB gravity.\nAnyhow, in these regularised theories \\cite{Casalino:2020kbt, Hennigar:2020lsl, Lu:2020iav} the spherically symmetric 4D black hole solution obtained in Refs \\cite{Glavan:2019inb, Cognola:2013fva}  is still valid.\n\nOn the other hand, it is believed that there exists plasma fluid surrounding black holes and other compact objects. When the light moves towards the compact objects through the plasma, the trajectory of light is different from the vacuum case. The theory of the light propagation in a curved spacetime in the presence of an isotropic dispersive medium was considered in the classical book of Synge \\cite{Synge}. Synge used the general relativistic Hamiltonian approach to deal with the geometrical optics in a dispersive medium.\nFurthermore, the influence of a spherically symmetric and time-independent plasma on the light defection in Schwarzschild spacetime and Kerr spacetime was discussed in the book of Perlick \\cite{Perlick1}.\nThe effect of plasma on the shadows of black holes and wormholes has been investigated in \\cite{Bisnovatyi-Kogan:2017kii, Abdujabbarov:2015pqp, Perlick:2017fio, Abdujabbarov:2016efm, Huang:2018rfn}. \nGravitational lensing by the compact object in homogeneous and inhomogeneous plasma was considered in \\cite{BisnovatyiKogan:2008yg, BisnovatyiKogan:2010ar, Morozova, Er:2013efa, Atamurotov:2015nra, Rogers, Perlick:2015vta, Tsupko:2013cqa}.\n\nIn this work, we shall study the strong gravitational lensing by this novel $4$-dimensional EGB black hole in an unmagnetized homogeneous plasma medium. The rest of the paper is organized as follows.\nIn Sec.~II, we study the photon sphere radius and the critical value of impact parameter of this novel black hole in the presence of plasma and derive the expression for the deflection angle of light in Sec.~III.\nIn Sec.~IV, we investigate the effects of  plasma on the deflection angle, the coefficients and the observable quantities for gravitational lensing in the strong field limit. Finally, We end the paper with a summary in Sec.~V.\nThroughout this paper we use the units in which $G=c=1$.\n\n\\section{Photon sphere of an Einstein-Gauss-Bonnet black hole in the presence of plasma}\n\nLet us start from the line element of the EGB black hole spacetime \\cite{Glavan:2019inb}, which is given by\n\\begin{eqnarray}\\label{metric}\nds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)(d\\theta^{2}+\\sin^{2}\\theta d\\varphi^{2}),\n\\end{eqnarray}\nwhere the functions $A(r)$, $B(r)$ and $C(r)$ have respectively the following form,\n\\begin{eqnarray}\n&&A(r)= 1 + \\frac{r^2}{2\\alpha}\n\\Biggl( 1- \\sqrt{1+\\frac{8\\alpha M}{r^3}}\\Biggr),\\label{A}\\\\\n&&B(r)=\\Biggl[1 + \\frac{r^2}{2 \\alpha}\n\\Biggl( 1- \\sqrt{1+\\frac{8\\alpha M}{r^3}}\\Biggr)\\Biggr]^{-1},\\label{B}\\\\\n&&C(r)=r^2\\label{C}.\n\\end{eqnarray}\nIt has been shown that the metric is asymptotic flat by the expansion at large $r$.\nHere $M$ is the mass of the EGB black hole and the GB coupling constant $\\alpha$  is constrained in the range $-8\\le {\\alpha}\/{M^2}\\le 1$ \\cite{Guo:2020zmf}. For the case $0<{\\alpha}\/{M^2}\\le 1$, there are two horizons \n\\begin{equation}\nr_{\\pm}=M\\pm \\sqrt{M^2-\\alpha}.\n\\end{equation}\nWhile for the case $-8\\le{\\alpha}\/{M^2}< 0$, there is only one horizon $r_{+}$,\nwhere the singular short radial distances $r<\\sqrt[3]{-8\\alpha M}$ are concealed inside this outer horizon. We will take the region $-8\\le {\\alpha}\/{M^2}\\le 1$ for the coupling constant in this paper.\n \nWe assume that the spacetime is filled with a spherically symmetric distribution of plasma with electron plasma frequency\n\\begin{equation}\n\\omega_p(r)^2 = \\frac{4\\pi e^2}{m} N(r),\n\\end{equation}\nwhere $e$ and $m$ are the charge of the electron and the mass of the electron respectively. The number density of the electrons $N(r)$ is the function of the radius coordinate only.\nThe relation between the refraction index $n$ and the photon frequency $\\omega$ is given as\n\\begin{equation}\nn ^2 = 1 - \\frac{\\omega_p^2(r)}{\\omega^2}.\n\\end{equation}\nIt is found that when $\\omega > \\omega_p$, the photon can propagate through the plasma. On the other hand, when $\\omega < \\omega_p$, the photon motion is forbidden \\cite{BisnovatyiKogan:2010ar, Rogers}.\nNote that one has $n=1$ in the vacuum case. \n\nWe start to calculate the strong gravitational lensing of the EGB black hole surrounded by plasma. The trajectories of photons in a curved space-time with plasma mediums, were obtained by Synge \\cite{Synge}. The Hamiltonian for\nthe light rays around the black hole surrounded by plasma has the\nfollowing form \\cite{Kulsrud:1991jt}\n\\begin{equation}\\label{H}\nH(x,p) = \\frac{1}{2} \\left[ g^{\\mu\\nu} p_{\\mu} p_{\\nu} +\\omega_p^2(r)\n\\right] = 0,\n\\end{equation}\nwhere $p_{\\mu}$ is the four-momentum of the photon and \n$g^{\\mu\\nu}$ is the contravariant metric tensor.\nSubstituing (\\ref{metric}) into (\\ref{H}), we get the equation\n\\begin{equation}\\label{H1}\n0=-\\frac{p_t^2}{A(r)}+\\frac{p_r^2}{B(r)}+\\frac{p_{\\varphi}^2}{C(r)}+\\omega_p^2(r).\n\\end{equation}\nUsing the Hamiltonian (\\ref{H}) for the photon around\nthe EGB black hole, the paths of light rays are then described in terms of the\naffine parameter $\\lambda$ by\n\\begin{equation}\\label{dxp} \n\\frac{dx^{\\mu}}{d \\lambda} = \\frac{\\partial H}{\\partial\n\tp_{\\mu}},~ \\frac{dp_{\\mu}}{d \\lambda} = - \\frac{\\partial\n\tH}{\\partial x^{\\mu}}.\n\\end{equation}\nBecause of the spherical symmetry, we can confine the photon orbits in the equatorial plane by taking $\\theta=\\pi\/2$ without the loss of generality.\nThe coordinates $t$ and $\\varphi$ are cyclic, leading two costants of motions which are the energy $E$ and the angular\nmomentum $L$ of the photon \n\\begin{equation}\nE=-p_t=\\omega_{\\infty},~ L=p_\\varphi,\n\\end{equation}\nwhere $\\omega_{\\infty}$ is the photon frequency at infinity. From Eqs. (\\ref{metric}) and (\\ref{dxp}), the expression for \n${dr}\/{d\\lambda}, ~{d\\varphi}\/{d\\lambda}$ is obtained in terms of $p_r$ and $p_\\varphi$ \n\\begin{eqnarray}\n\\label{dr}&&\\frac{dr}{d\\lambda} =  \\frac{\\partial H}{\\partial p_r}= \n\\frac{p_r}{B(r)},\\\\\n\\label{dphi}&&\\frac{d\\varphi}{d\\lambda} =  \\frac{\\partial H}{\\partial p_{\\varphi}} =  \n\\frac{p_{\\varphi}}{C(r)}.\n\\end{eqnarray}\nUsing Eqs. (\\ref{H1}), (\\ref{dr}) and (\\ref{dphi}), we obtain the equation of trajectory for a photon which is similar to the formalism in Ref. \\cite{Tsukamoto:2016qro}\n\\begin{equation}\\label{drphi}\n\\left(\\frac{dr}{d\\varphi}\\right)^{2}=\\frac{R_p(r)C(r)}{B(r)}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n&&R_p=\\frac{E^2}{L^2}\\frac{C(r)}{A(r)}W(r)-1,\\label{Rp}\\\\\n&&W(r)=1 -\\frac{\\omega _p (r) ^2}{E^2}A(r).\\label{W}\n\\end{eqnarray}\nIn the case $\\omega_p(r)=0$ or equivalently, $W(r)=1$, Eq. (\\ref{drphi}) gives the motion of light ray in vacuum.\n\nWe are interested in a photon with a given enegy $E$ that comes in from infinity, reaches a closest distance $r=r_{0}$, and goes out to infinity.\nAs $r_0$ corresponds to the turning point of the path, $dr\/d\\varphi$ vanishes and $R_p(r_0)=0$. Hereafter subscript $0$ indicates the quantity at the closest distance $r=r_{0}$. \nFor a light ray initially in the asymptotically flat spacetime, the impact parameter can be represented as \n\\begin{equation}\\label{impact}\nb(r_{0})=\\frac{L}{E}=\\sqrt{\\frac{C_{0}W_{0}}{A_{0}}}. \n\\end{equation}\nWith the help of Eq. (\\ref{impact}), $R_p(r)$ can be rewritten as\n\\begin{equation}\\label{R2}\nR_p(r)= \\frac{A_{0}CW}{AC_{0}W_{0}}-1.\n\\end{equation}\nTo find the radius of photon sphere, which is the unstable circular photon orbit of static, spherically symmetric compact objects, one can introduce a function $h(r)$ given by Perlick \\cite{Perlick:2015vta}\n\\begin{equation}\\label{h}\nh(r)^2=\\frac{C(r)}{A(r)}W(r)=\\frac{C(r)}{A(r)}\\left[1 -\\frac{\\omega _p (r) ^2}{E^2}A(r)\\right]. \n\\end{equation}\nThe photon sphere radius $r_m$ is the biggest real root of the equation\n\\begin{equation}\\label{dh}\n\\frac{d}{dr}h(r)^2=0.\n\\end{equation}\nFrom Eq. (\\ref{dh}), we obtain\n\\begin{equation}\\label{Dp}\n\\frac{C'}{C}+\\frac{W'}{W}-\\frac{A'}{A}=0,\n\\end{equation}\nwhere prime denotes the differentiantion with respect to the radical coordinate $r$.\n\nNow we consider the EGB black hole surrounded by homogeous plasma, which has the following form\n\\begin{equation}\\label{beta0}\n\\frac{\\omega_p(r)}{E}=\\beta_0,\n\\end{equation}\nwhere $\\beta_0$ is a positive dimensionless constant. Then we rewrite Eq. (\\ref{Dp}) as\n\\begin{equation}\\label{eqrm}\nr\\left[\\beta _0 \\left(2 \\alpha ^2+r^4+2 \\alpha  r^2+4 \\alpha\nMr\\right)-2 \\alpha ^2 \\right]\\sqrt{\\frac{8 \\alpha M}{r^3}+1}=\\beta _0 \\left(16 \\alpha^2M+r^5+2 \\alpha  r^3+8 \\alpha M\nr^2\\right)-6 \\alpha ^2M.\n\\end{equation}\nWe can solve this equation numerically to get the radius of the photon sphere  which is plotted in Fig. \\ref{rm}.\nIn the left panels of Fig. \\ref{rm} we\nshow the function $r_m\/M$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively, \nand we demonstrate that the radius of the photon sphere of the EGB black hole decreases with the increase of $\\alpha\/M^2$ for fixed $\\beta_0$. \nIn the right panels of Fig. \\ref{rm} we\nshow the function $r_m\/M$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively, \nand we find that the radius of the photon sphere of the EGB black hole increases with the increase of $\\beta_0$ for fixed $\\alpha\/M^2$.\nIt is clear that the presence of coupling constant $\\alpha$ and the plasma parameter $\\beta_0$, affects the photon sphere radius significantly.\n In the absence of $\\beta_0$, from Eq. (\\ref{eqrm}), the largest real root has a form\n\\begin{equation}\nr_{m}=2 \\sqrt{3}M \\cos \\left[\\frac{1}{3} \\cos ^{-1}\\left(-\\frac{4 \\alpha }{3 \\sqrt{3}M^2}\\right)\\right],\n\\end{equation}\nwhich is the photon radius of the EGB black hole in vacuum \\cite{Guo:2020zmf}.\nOn the other hand, in the case $\\alpha= 0$, we can get the photon radius of Schwarzschild black hole with homogeneous plasma   \n\\begin{equation}\nr_{m}=\\frac{3-4 \\beta_0 +\\sqrt{9-8 \\beta _0}}{2\\left(1-\\beta _0\\right)}M,\n\\end{equation}\nwhich has been obtained in Ref. \\cite{Tsupko:2013cqa}.\n   \n\\begin{figure}\n\t\\includegraphics[width=80mm,angle=0]{rmEGB.eps}\\,\n\t\\includegraphics[width=80mm,angle=0]{rm1EGB.eps}\n\\caption{Left panel: The plot of the radius of the photon sphere $r_m\/M$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the radius of the photon sphere $r_m\/M$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\t\\label{rm}\n\\end{figure}\n\nWe define the critical value of the impact parameter $b_{c}$ for the light ray as \n\\begin{equation}\\label{cip}\nb_{c}\n\\equiv \\lim_{r_{0}\\rightarrow r_{m}} \\sqrt{\\frac{C_{0}W_{0}}{A_{0}}}.\n\\end{equation}\nThe strong deflection limit corresponds to the limit $r_0\\rightarrow r_m$ or $b\\rightarrow b_c$. From Eqs. (\\ref{A}), (\\ref{C}) and (\\ref{W}), the critical impact parameter is given by\n\\begin{eqnarray}\nb_{c}(r_{m})=\\sqrt{\\frac{\\beta _0 \\left[r_m^4 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-2 \\alpha r_m^2\\right]+2 \\alpha  r_m^2}{2 \\alpha +r_m^2 \\left(1-\\sqrt{\\frac{8 \\alpha\tM}{r_m^3}+1}\\right)}}.\n\\end{eqnarray}\nThe dependence of the critical impact parameter from  the coupling constant and the plasma parameters is shown in Fig. \\ref{bc}.\nThe left panels of Fig. \\ref{bc} presents the function $b_c\/M$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively, \nand it shows that the critical impact parameter of the EGB black hole decreases with the increase of $\\alpha\/M^2$ for fixed $\\beta_0$. \nThe right panels of Fig. \\ref{bc} presents the function $b_c\/M$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively, and it shows that the critical impact parameter of the EGB black hole decreases with the increase of $\\beta_0$ for fixed $\\alpha\/M^2$.\nWe found that both the coupling constant and the presence of plasma have remarkable influences on the critical impact parameter.\n\\begin{figure}\n\\includegraphics[width=80mm,angle=0]{bcEGB.eps}\\,\n\\includegraphics[width=80mm,angle=0]{bc1EGB.eps}\n\\caption{Left panel: The plot of the critical impact parameter $b_c\/M$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the critical impact parameter $b_c\/M$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\t\\label{bc}\n\\end{figure}\n\n\\section{Strong gravitational lensing of EGB black hole in homogeneous plasma}\n\nIn this section, we will calculate the deflection angle of a light ray in the strong deflction limit in the EGB black hole spacetime with plasma medium.\nFrom Eq. (\\ref{drphi}), the deflection angle $\\hat{\\alpha}_p(r_{0})$ for the photon coming from infinite to the EGB black hole in homogeneous plasma is given by\n\\begin{equation}\\label{alpha}\n\\hat{\\alpha}_p(r_{0})=I_p(r_{0})-\\pi,\n\\end{equation}\nwhere $I_p(r_{0})$ is defined as\n\\begin{equation}\\label{angle2}\nI_p(r_{0})\\equiv 2\\int^{\\infty}_{r_{0}}\\frac{1}{\\sqrt{\\frac{R_p(r)C(r)}{B(r)}}}dr.\n\\end{equation}\nIt is found that the deflection angle increases when the closest distance $r_0$ decreases, and for a special point, the deflection angle will arrive at $2\\pi$ which means the photon winds a complete loop around the black hole. Furthermore, when $r_{0}$ approach the radius of the photon sphere $r_m$ the  deflection angle will diverge \\cite{Virbhadra:1999nm}.\nTo discuss the divergence, following Ref.\\cite{Chen:2009eu}, we  introduce a new variable $z$ \n\\begin{equation}\\label{z1}\nz\\equiv 1-\\frac{r_{0}}{r}.\n\\end{equation}\nUsing Eqs. (\\ref{A})-(\\ref{C}), (\\ref{Rp}) and (\\ref{W}), we can rewrite $I_p(r_{0})$ as\n\\begin{equation}\nI_p(r_{0})=\\int^{1}_{0}f_p(z,r_{0})dz\n=\\int^{1}_{0}\\frac{2r_0}{\\sqrt{G_p(z,r_{0})}}dz,\n\\end{equation}\nwhere the function $G_p(z,r_{0})$ in the EGB spacetime is given by\n\\begin{eqnarray}\nG_p(z,r_{0})=&&\\frac{R_p(z,r_0)C(z,r_0)}{B(z,r_0)}(1-z)^4=r_0^2\\left((1-z)^2-\\frac{r_0^2 \\left(\\sqrt{1-\\frac{8 \\alpha  M (z-1)^3}{r_0^3}}-1\\right)}{2 \\alpha}\\right)\n\\\\\\nonumber\n&&\\left(\\frac{\\left(r_0^2\n\t\t\t\\left(\\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-1\\right)-2 \\alpha \\right) \\left(-\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{r_0^3-8 \\alpha  M\n\t\t\t\t\t(z-1)^3}{r_0^3}}-1\\right)-2 \\alpha  (z-1)^2\\right)-2 \\alpha  (z-1)^2\\right)}{(z-1)^2 \\left(2 \\alpha +\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{8 \\alpha\n\t\t\t\t\tM}{r_0^3}+1}-1\\right)-2 \\alpha \\right)\\right) \\left(2 \\alpha  (z-1)^2-r_0^2 \\left(\\sqrt{\\frac{r_0^3-8 \\alpha  M\n\t\t\t\t\t(z-1)^3}{r_0^3}}-1\\right)\\right)}-1\\right).\n\\end{eqnarray}\nWe can expand the above expression into a power series of z in the following form\n\\begin{equation}\nG_p(z,r_{0})=\\sum_{n=1}^{\\infty}c_n(r_0)z^n,\n\\end{equation}\nwhere $c_{1}(r_{0})$ and $c_{2}(r_{0})$ are given by\n\\begin{eqnarray}\nc_{1}(r_{0})=&&\\frac{\\beta _0 \\left(-4 \\alpha ^2 r_0^2 \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}+32 \\alpha ^2 M r_0+\\alpha  r_0^4 \\left(4-4 \\sqrt{\\frac{8 \\alpha \n\t\t\tM}{r_0^3}+1}\\right)-8 \\alpha  M r_0^3 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-2\\right)\\right)}{\\alpha  \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1} \\left(2\n\t\\alpha +\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-1\\right)-2 \\alpha \\right)\\right)}\n\\\\\\nonumber\n&&+\\frac{\\beta _0 r_0^6 \\left(2-2 \\sqrt{\\frac{8 \\alpha M}{r_0^3}+1}\\right)+4 \\alpha ^2 r_0 \\left(r_0 \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1}-3 M\\right)}{\\alpha  \\sqrt{\\frac{8 \\alpha  M}{r_0^3}+1} \\left(2 \\alpha +\\beta _0 \\left(r_0^2 \\left(\\sqrt{\\frac{8 \\alpha\tM}{r_0^3}+1}-1\\right)-2 \\alpha \\right)\\right)},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\\label{c2}\nc_{2}(r_{0})=&&\\frac{2 \\alpha ^2 {r_0}^2 \\left(96 \\alpha ^2 M^3 {r_0}+{r_0}^8 \\left(-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)+{r_0}^8+2 \\alpha \n\t{r_0}^6\\right)}{{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8\\alpha M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}}\n\\\\\\nonumber\n&&+\\frac{2 \\alpha ^2 M^2 {r_0}^2 \\left(128 \\alpha ^3+\\alpha  {r_0}^4 \\left(60-12 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)+\\alpha ^2 {r_0}^2\n\t\\left(64-88 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)}{{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}}\n\\\\\\nonumber\n&&+\\frac{2 \\alpha ^2 M {r_0}^2 \\left({r_0}^7 \\left(6-6 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)+\\alpha  {r_0}^5 \\left(16-28 \\sqrt{\\frac{8\n\t\t\t\\alpha  M}{{r_0}^3}+1}\\right)+32 \\alpha ^2 {r_0}^3\\right)}{{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}}\n\\\\\\nonumber&&+\\frac{-2 \\beta _0 {r_0}^2 \\left(8 \\alpha  M+{r_0}^3\\right) \\left(2 \\alpha ^2 {r_0}^3 \\left(\\alpha -24 M^2\\right)+32 \\alpha ^3 M^2 {r_0}+16\n\t\\alpha ^4 M+\\alpha ^2 {r_0}^5 \\left(\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-1\\right)\\right)}{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}\n\\\\\\nonumber&&+\\frac{-2 \\beta _0 {r_0}^2 \\left(8 \\alpha  M+{r_0}^3\\right) \\left(6 \\alpha  M {r_0}^6 \\left(3 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-5\\right)-8\n\t\\alpha ^3 M {r_0}^2 \\left(2 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}+1\\right)\\right)}{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}\n\\\\\\nonumber&&+\\frac{-2 \\beta _0 {r_0}^2 \\left(8 \\alpha  M+{r_0}^3\\right) \\left({r_0}^9 \\left(3 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-3\\right)+\\alpha \n\t{r_0}^7 \\left(5 \\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}-5\\right)+4 \\alpha ^2 M {r_0}^4 \\left(4 \\sqrt{\\frac{8 \\alpha \n\t\t\tM}{{r_0}^3}+1}-9\\right)\\right)}{\\alpha  \\left(8 \\alpha  M+{r_0}^3\\right)^2 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right) \\left(\\beta\n\t_0 \\left(2 \\alpha +{r_0}^2 \\left(1-\\sqrt{\\frac{8 \\alpha  M}{{r_0}^3}+1}\\right)\\right)-2 \\alpha \\right)}.\n\\end{eqnarray}\nIt is easy to get $c_{1}(r_{m})= 0$ in the limit $r_{0}\\rightarrow r_{m}$,  while $c_2(r_m)$ is complex in this limit. Furthermore, when $\\beta_0= 0$, i.e., in vacuum, the $c_2(r_m)$ term in the limit $r_{0}\\rightarrow r_{m}$ becomes\n\\begin{eqnarray}\nc_{2}(r_{m})=&&\\frac{96 \\alpha ^2 M^3 r_m^3+\\alpha  M^2 r_m^2 \\left(128 \\alpha ^2-12 r_m^4 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-5\\right)+8 \\alpha  r_m^2 \\left(8-11 \\sqrt{\\frac{8\n\t\t\t\\alpha  M}{r_m^3}+1}\\right)\\right)}{\\left(r_m^2 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-2 \\alpha \\right) \\left(r_m^3+8 \\alpha  M\\right)^2}\n\\\\\\nonumber\n&&+\\frac{M \\left(-6 r_m^9 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-4 \\alpha  r_m^7 \\left(7 \\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-4\\right)+32 \\alpha ^2\n\tr_m^5\\right)+r_m^{10} \\left(-\\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)\\right)+2 \\alpha  r_m^8}{\\left(r_m^2 \\left(\\sqrt{\\frac{8 \\alpha  M}{r_m^3}+1}-1\\right)-2 \\alpha \\right) \\left(r_m^3+8 \\alpha  M\\right)^2}.\n\\end{eqnarray}\nSince this expression is still intricate, for the sake of clarity, let's continue to look at the form under the limit $\\alpha\\rightarrow 0$.\nIn the case $\\beta_0=0$ and $\\alpha=0$, $r_m=3M$, and Eq. (\\ref{c2}) has a form\n\\begin{equation}\nc_{2}(r_{m})=(6 M-r_m) r_m=9M^2,\n\\end{equation}\nwhere the vacuum Schwarzschild solution is recovered.\nBy the discussion above, we can find that the leading term of the divergence in $f_p(z,r_{0})$ is $z^{-1}$ in the strong deflection limit, which implies $I_p(r_{0})$ diverges logarithmically.\n\nOne can separate $I_p(r_{0})$ into two parts which are the divergent\npart $I_{D}(r_{0})$ and the regular part $I_{R}(r_{0})$\n\\begin{equation}\nI_p(r_{0})=I_{D}(r_{0})+I_{R}(r_{0}).\n\\end{equation}\nThe divergent part $I_{D}(r_{0})$ is defined as\n\\begin{equation}\nI_{D}(r_{0})\\equiv \\int^{1}_{0}f_{D}(z,r_{0})dz,\n\\end{equation}\nwhere\n\\begin{equation}\nf_{D}(z,r_{0})\\equiv \\frac{2r_{0}}{\\sqrt{c_{1}(r_{0})z+c_{2}(r_{0})z^{2}}}.\n\\end{equation}\n$I_{D}(r_{0})$ can be integrated and the result is\n\\begin{equation}\\label{ID}\nI_{D}(r_{0})=\\frac{4r_{0}}{\\sqrt{c_{2}(r_{0})}}\\log \\frac{\\sqrt{c_{2}(r_{0})}+\\sqrt{c_{1}(r_{0})+c_{2}(r_{0})}}{\\sqrt{c_{1}(r_{0})}}.\n\\end{equation}\nThe regular part $I_{R}(r_{0})$ is defined as\n\\begin{equation}\nI_{R}(r_{0})\\equiv \\int^{1}_{0}f_{R}(z,r_{0})dz,\n\\end{equation}\nwhere\n\\begin{equation}\nf_{R}(z,r_{0}) \\equiv f(z,r_{0})-f_{D}(z,r_{0}).\n\\end{equation}\n\nUsing a similar derivation as in Ref. \\cite{Tsukamoto:2016jzh}, we obtain the deflection angle $\\hat{\\alpha}_p(b)$ in the strong deflection limit $r_{0}\\rightarrow r_{m}$ or $b \\rightarrow b_{c}$ in the EGB black hole with homogeneous plasma\n\\begin{equation}\\label{alpha1}\n\\hat{\\alpha}_p(b)= -\\bar{a}\\log \\left( \\frac{b}{b_{c}}-1 \\right) +\\bar{b}+O((b-b_{c})\\log (b-b_{c})).\n\\end{equation}\nThe coefficients $\\bar{a}$ and $\\bar{b}$ are obtained as\n\\begin{eqnarray}\\label{abar}\n&&\\bar{a}=\\sqrt{\\frac{2B_{m}}{C_{m}\\left[\\frac{(CW)_{m}^{''}}{(CW)_{m}}-\\frac{A_{m}^{''}}{A_{m}}\\right]}},\\\\\n&&\\bar{b}=\\bar{a}\\log \\left\\{r^{2}_{m}\\left[\\frac{(CW)_{m}^{''}}{(CW)_{m}}-\\frac{A_{m}^{''}}{A_{m}}\\right]\\right\\} +I_{Rp}(r_{m})-\\pi,\\label{bbar}\n\\end{eqnarray}\nwhere the subscript $m$ denotes the quantities at $r=r_m$.\nIn the vacuum case, i.e., $\\beta_0=0$, and the coupling constant $\\alpha= 0$,  $\\bar{a}$ and $\\bar{b}$ will reduce to the formalism in Ref. \\cite{Tsukamoto:2016jzh}, which are the cofficients of a Schwarzschild black hole without plasma.\n\nThe numerical results of the strong field limit coefficients $\\bar{a}$ and $\\bar{b}$ are shown in  Fig. \\ref{ab1} and Fig. \\ref{bb1}. \nThe left panels of Fig. \\ref{ab1} show the strong field limit coefficient $\\bar{a}$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nWe find that the strong field limit coefficient $\\bar{a}$ increases with the increase of the coupling constant $\\alpha\/M^2$ for fixed $\\beta_0$.\nFrom the right panels of Fig. \\ref{ab1}, which refers to the strong field limit coefficient $\\bar{a}$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively,\nwe find that the strong field limit coefficient $\\bar{a}$ increases with the increase of the plasma parameter $\\beta_0$ for fixed  $\\alpha\/M^2$.\nIn the left panels of Fig. \\ref{bb1}, we illustrate the strong field limit coefficient $\\bar{b}$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively, and show\nthat the strong field limit coefficient $\\bar{b}$ decreases with the increase of the coupling parameter $\\alpha\/M^2$ for fixed $\\beta_0$.\nFrom the right panels of Fig. \\ref{bb1}, which refers to the strong field limit coefficient $\\bar{b}$ for $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively, we find that the strong field limit coefficient $\\bar{b}$ increases with the increase of the plasma parameter $\\beta_0$ for fixed  $\\alpha\/M^2$.\nObviously, the strong field limit coefficients $\\bar{a}$ and $\\bar{b}$ are influenced by the choice of coupling constant $\\alpha$ and plasma parameter $\\beta_0$.\n\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{aEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{a1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the strong field limit coefficients $\\bar{a}$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the strong field limit coefficients $\\bar{a}$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\t\\label{ab1}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{bEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{b1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the strong field limit coefficients $\\bar{b}$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the strong field limit coefficients $\\bar{b}$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{bb1}\n\\end{figure}\n\n\\section{Observables in the strong deflection limit}\n\nIn this section we consider the observables of the strong gravitational lensing by the EGB black hole in the presence of a uniform plasma.\nWe are interested in the case where the observer, the lens and the source are nearly in alignment, and the source and the observer are very far from the lens. \nThe distance between the lens and the source, and the distance between the lens and the observer are represented by $D_{LS}$ and $D_{OL}$ respectively. \n$D_{OS}$ is the distance between the observer and the source, and $D_{OS}=D_{LS}+D_{OL}$.\n$\\beta$ denotes the angular position with respect to the optical axis of the source.\n$\\theta$ is the angular position with respect to the optical axis of the image and can be expressed as $\\theta=b\/D_{OL}$. \nThus the lens eqaution can be written as \\cite{Bozza:2002zj}\n\\begin{equation}\\label{beta}\n\\beta=\\theta-\\frac{D_{LS}}{D_{OS}}\\triangle\\alpha_{n},\n\\end{equation}\nwhere $\\triangle\\alpha_{n}=\\alpha-2n\\pi$ is the offset of deflection angle, and $n$ denotes the loop numbers of the light ray around the light sphere.\n\nThe angular position $\\theta_{n}$ between the lens and the $n$-th relativistic image and the magnification of the $n$-th relativistic image $\\mu_{n}$ can be obtained approximately as\n\\begin{eqnarray}\\label{theta}\n&&\\theta_{n}=\\theta^{0}_{n}+\\frac{b_{c}(\\beta-\\theta_{n}^{0})D_{OS}}{\\bar{a}D_{LS}D_{OL}}\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right),\\\\\n&&\\mu_{n}=\\frac{b_{c}^{2}D_{OS}}{\\bar{a}\\beta D^{2}_{OL}D_{LS}}\n\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right)\\left[1+\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right)\\right].\\label{mun}\n\\end{eqnarray}\nHere $\\theta_{n}^{0}$ is the angular position corresponding to the case that the light ray winds completely $2n\\pi$ and can be expressed as \n\\begin{equation}\\label{theta1}\n\\theta_{n}^{0}=\\frac{b_{c}}{D_{OL}}\\left[1+\\exp\\left({\\frac{\\bar{b}-2n\\pi}{\\bar{a}}}\\right)\\right].\n\\end{equation}\nIn the limit $n\\rightarrow\\infty$, we can find the relation between \nthe critical impact parameter $b_{c}$ and the asymptotic position $\\theta_{\\infty}$  approached by a set of images\n\\begin{equation}\\label{theta2}\n\\theta_{\\infty}=\\frac{b_{c}}{D_{OL}}.\n\\end{equation}\nSince the outermost relativistic image is the brightest, one can use the observable $s$ to describe the separation between this first image $\\theta_{1}$ and all the others packed images at $\\theta_{\\infty}$ \\cite{Bozza:2002zj}. The other observable $\\mathcal{R}$ represents the ratio of the received flux beteween this first image and all the others images \\cite{Bozza:2002zj}. \nUsing Eqs. (\\ref{theta}) and (\\ref{mun}), the angular separation $s$ and the ratio of the flux $\\mathcal{R}$ can be obtained as\n\\begin{eqnarray}\\label{sr}\n&&s=\\theta_{1}-\\theta_{\\infty}=\\theta_{\\infty} \\exp\\left({\\frac{\\bar{b}-2\\pi}{\\bar{a}}}\\right),\\\\\n&&\\mathcal{R}=\\frac{\\mu_{1}}{\\sum\\limits_{n=2}^{\\infty}\\mu_{n}}=\\exp\\left({\\frac{2\\pi}{\\bar{a}}}\\right).\n\\end{eqnarray}\nIf the observables  $s$, $\\theta_{\\infty}$ and $\\mathcal{R}$ are available, the coefficients $\\bar{a}$ and $\\bar{b}$ in the strong deflction limit and the critical impact parameter $b_{c}$ can be obtained easily by\n\\begin{eqnarray}\\label{ab2}\n&& \\bar{a}=\\frac{2\\pi}{\\log \\mathcal{R}},\\\\\n&& \\bar{b}=\\bar{a}\\log(\\frac{\\mathcal{R}s}{\\theta_{\\infty}}),\\\\\n&& b_{c}=\\theta_{\\infty}D_{OL}.\n\\end{eqnarray}\nThen one can numerically compute the above value by measuring  the observables  $s$, $\\theta_{\\infty}$ and $\\mathcal{R}$  and study their difference with the corresponding theoretical coefficients.\n\nLet's take the supermassive black hole M87$^{*}$ as an example.  The results from the EHT show that the angular diameter of the shadow of M87$^{*}$ is $(42\\pm 3) \\mu$as, and the observed shadow is almost circular which is supported by the fact that the axis ratio is smaller than $4\/3$ and the deviation from circularity is less than 10\\% \\cite{Akiyama:2019cqa, Akiyama:2019fyp, Akiyama:2019eap}.\nTherefore, the image of M87$^{*}$ is nearly circular due to the relatively small value of spin and low inclination angle $\\sim 17 ^\\circ$ of the source \\cite{Walker:2018muw}. So it is reasonable to choose spherically symmetric metric as  an approximation to discuss the strong gravitational lensing of M87$^{*}$.\nMeanwhile, the mass of M87$^{*}$ is estimated by the EHT collaboration as  $M=(6.5\\pm 0.7)\\times 10^{9}~M_{\\odot}$ \\cite{Akiyama:2019eap}. Note that the mass of M87$^{*}$ is also estimated to be $6.2^{+1.1}_{-0.5}\\times10^9$ $M_{\\odot}$  and $3.5^{+0.9}_{-0.3}\\times10^9$ $M_{\\odot}$   by the stellar dynamics \\cite{Gebhardt:2011yw} and gas dynamics measurements \\cite{Walsh:2013uua}, respectively.\nIn addition, the distance $D_{OL}$ of M87$^{*}$ from us is estimated to be $D_{OL}$=(16.8$\\pm$0.8) Mpc from stellar population measurements \\cite{Blakeslee:2009tc, Bird:2010rd, Cantiello:2018ffy}.\n\n\nIn the following the lens is supposed to be M87$^{*}$ which is described by the EGB black hole. For simplicity, we use the following data from M87$^{*}$, $D_{OL}$=16.8 Mpc and $M=$6.5$\\times10^9$ $M_{\\odot}$. With these data we can estimate the values of the  angular image position $\\theta_{\\infty}$, the angular image separation $s$ and the relative magnifications $r$ of the relativistic images which is defined as $r=2.5\\log_{10}\\mathcal{R}$.\nFigs. \\ref{th}-\\ref{dm} show the behaviors of these observables and the \ninfluences on them by the choince of coupling constant $\\alpha$ and plasma parameter $\\beta_0$.\nThe left panels of Fig. \\ref{th} show the value of $\\theta_{\\infty}$ as function of\n$\\alpha\/M^2$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively. As the coupling constant $\\alpha\/M^2$ increases,\nthe angular image position decreases for fixed $\\beta_0$. \nThe right panels of Fig. \\ref{th} show the value of $\\theta_{\\infty}$ as function of\n$\\beta_0$ for $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively. As the plasma parameter $\\beta_0$ increases,\nthe angular image position decreases for fixed $\\alpha\/M^2$.\n \nAs shown in the left panels of Fig. \\ref{s}, the value of $s$ is expressed as a function of $\\alpha\/M^2$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively. As the coupling constant $\\alpha\/M^2$ increases, the angular image separation increases for fixed $\\beta_0$.\nIt is shown that in the right panels of Fig. \\ref{s}, the value of $s$ is expressed as function of $\\beta_0$ for $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively. As the plasma parameter $\\beta_0$ increases, the angular image separation increases for fixed $\\alpha\/M^2$.\nInterestingly, as $\\alpha\/M^2\\rightarrow-8$ for different plasma parameter $\\beta_0$, the angular image separation converges to the value at $\\alpha\/M^2=-8$ in the vacuum case,\nwhich means plasma has little effect on the angular image separation. \n\nIn the left panels of Fig. \\ref{dm}, we show the value of relative magnifications as function of\n$\\alpha\/M^2$ for $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively. As the coupling constant $\\alpha\/M^2$ increases,\nthe relative magnifications  decreases for fixed $\\beta_0$. \nIn the right panels of Fig. \\ref{dm}, we show the value of relative magnifications as function of\n$\\beta_0$ for $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively. As the plasma parameter $\\beta_0$ increases,\nthe relative magnifications decreases for fixed $\\alpha\/M^2$.\nWe find that as $\\alpha\/M^2\\rightarrow 1$ for different plasma parameter $\\beta_0$, the relative magnifications converges to the value at $\\alpha\/M^2=1$ in the vacuum case,\nwhich means plasma has little influence on the relative magnifications.\n\nIn Table \\ref{tab1}, we list the numerical estimates of the observables as well as strong field limit coefficients of a EGB black hole in uniform plasma. \nThe parameter $\\beta_0=0$  corresponds to the case of the EGB black hole in vacuum and $\\alpha=0$ means the case of Schwarzschild\nblack hole in homogeneous plasma.\nFrom Table \\ref{tab1}, we can easily obtain the differences between the Schwarzschild black hole and the EGB  black hole, as well as the EGB  black hole with various plasma parameter.\n\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{thetaEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{theta1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the angular image position $\\theta_{\\infty}$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the angular image position $\\theta_{\\infty}$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{th}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{sEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{s1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the angular image separation $s$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the angular image separation $s$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{s}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\t\t\\includegraphics[width=80mm,angle=0]{dmEGB.eps}\\,\n\t\t\\includegraphics[width=80mm,angle=0]{dm1EGB.eps}\n\\end{center}\n\\caption{Left panel: The plot of the relative magnifications $r$ as a function of $\\alpha\/M^2$. The five curved lines are plotted when $\\beta_0=0$, $\\beta_0=0.1$, $\\beta_0=0.3$, $\\beta_0=0.5$ and $\\beta_0=0.7$ respectively.\nRight panel: The plot of the relative magnifications $r$ as a function of $\\beta_0$. The six curved lines are plotted when $\\alpha\/M^2=-8$, $\\alpha\/M^2=-4$, $\\alpha\/M^2=-2$, $\\alpha\/M^2=0$, $\\alpha\/M^2=0.4$ and $\\alpha\/M^2=1$ respectively.}\n\\label{dm}\n\\end{figure}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{cccccccc}\n\\hline\\hline\n$\\beta_0$ & $\\alpha\/M^2$ & $\\theta_{\\infty}$($\\mu\\text{as}$) & $s$($\\mu \\text{as}$) & $r(\\text{mag})$  & $b_{c}\/R_{s}$ & $\\bar{a}$ & $\\bar{b}$ \\\\\n\\hline\n & -8 & 52.10 & 0.000511 & 12.30 & 3.41 & 0.555 & -0.1151 \\\\\n & -4 & 47.39 & 0.00272  & 10.35 & 3.10 & 0.659 & -0.1558 \\\\\n & -2 & 44.21 & 0.00867  & 8.97  & 2.89 & 0.760 & -0.2087 \\\\\n0& 0  & 39.69 & 0.0497   & 6.82  & 2.60 & 1     & -0.4003 \\\\\n & 0.4& 38.41 & 0.0838   & 6.14  & 2.51 & 1.111 & -0.5264 \\\\\n & 1  & 35.85 & 0.253    & 4.51  & 2.35 & 1.511 & -1.2017 \\\\\\hline\n   & -8 & 50.82 & 0.000644 & 12.03 & 3.33 & 0.567 & -0.1088 \\\\\n   & -4 & 46.33 & 0.00320  & 10.16 & 3.03 & 0.671 & -0.1490 \\\\\n   & -2 & 43.31 & 0.00980  & 8.83  & 2.83 & 0.773 & -0.2024 \\\\\n0.1& 0  & 39.01 & 0.0530   & 6.74  & 2.55 & 1.012 & -0.3971 \\\\\n   & 0.4& 37.80 & 0.0878   & 6.08  & 2.47 & 1.123 & -0.5249 \\\\\n   & 1  & 35.38 & 0.254    & 4.50  & 2.32 & 1.515 & -1.1947 \\\\\\hline\n   & -8 & 48.08 & 0.000111 & 11.43 & 3.15 & 0.597 & -0.08858 \\\\\n   & -4 & 44.08 & 0.00473  & 9.73  & 2.89 & 0.701 & -0.1274 \\\\\n   & -2 & 41.39 & 0.00132  & 8.50  & 2.71 & 0.803 & -0.1815 \\\\\n0.3& 0  & 37.57 & 0.0622   & 6.55  & 2.46 & 1.041 & -0.3833 \\\\\n   & 0.4& 36.50 & 0.0991   & 5.93  & 2.39 & 1.150 & -0.5144 \\\\\n   & 1  & 34.38 & 0.261    & 4.47  & 2.25 & 1.527 & -1.1715 \\\\\\hline   \n   & -8 & 45.03 & 0.00220  & 10.70 & 2.95 & 0.638 & -0.04854 \\\\\n   & -4 & 41.58 & 0.00788  & 9.18  & 2.72 & 0.743 & -0.08515 \\\\\n   & -2 & 39.26 & 0.00196  & 8.08  & 2.57 & 0.845 & -0.1400 \\\\\n0.5& 0  & 35.97 & 0.0783   & 6.30  & 2.35 & 1.082 & -0.3494 \\\\\n   & 0.4& 35.05 & 0.119    & 5.73  & 2.29 & 1.190 & -0.4843 \\\\\n   & 1  & 33.25 & 0.278    & 4.41  & 2.18 & 1.548 & -1.1255 \\\\\\hline   \n   & -8 & 41.54 & 0.00574  & 9.72  & 2.72 & 0.702 &  0.04408 \\\\\n   & -4 & 38.73 & 0.0165   & 8.44  & 2.54 & 0.808 &  0.01158 \\\\\n   & -2 & 36.83 & 0.0352   & 7.50  & 2.41 & 0.910 & -0.04313 \\\\\n0.7& 0  & 34.14 & 0.113    & 5.95  & 2.24 & 1.146 & -0.2606 \\\\\n   & 0.4& 33.40 & 0.160    & 5.45  & 2.19 & 1.251 & -0.3993 \\\\\n   & 1  & 31.95 & 0.322    & 4.30  & 2.09 & 1.588 & -1.0194 \\\\\\hline\n\\hline\n\\end{tabular}\n\\caption{Numerical estimation for the observables and the strong deflection limit coefficients for EGB black holes supposed to describe the supermassive black hole M87$^{*}$. $R_S=2GM\/c^2$ is the Schwarzschild radius.}\\label{tab1}\n\t\\end{center}\n\\end{table}\n\n\\section{Conclusions}\n\nIn this work, we have investigated the strong gravitational lensing generated by a 4-dimensional Einstein-Gauss-Bonnet black hole in a plasma. \nIn the presence of plasma around the black hole, the trajectory of a photon differs from the null geodesic in vacuum, resulting in the changes of the deflection angle of light.\nUsing Hamilton's equation of the light ray in plasma with a frequency dependent refraction index, we have derived the equation of motion for light rays in the novel $4$-dimensional EGB black hole. \nFurthermore, we numerically obtained the theoretical strong field limit parameters for the lensing by the black hole in a uniform plasma. Among these parameters we found that the radius of the photon sphere $r_{m}$,\nthe critical impact parameter $b_c$ and the strong field limit coefficient $\\bar{b}$ \ndecrease monotonically, while the strong field limit coefficient $\\bar{a}$ \nincreases, with the increase of the coupling constant $\\alpha\/M^2$ for fixed value of plasma parameter $\\beta_0$.\nOn the other hand, for a fixed value of the coupling constant $\\alpha\/M^2$,   with the increase of the plasma parameter $\\beta_0$, $r_{m}$, $\\bar{a}$ and  $\\bar{b}$ increase, but  $b_c$ decreases monotonically.\nModelling the supermassive $\\mathrm{M}87^{\\ast}$ with this EGB black hole,\nwe have estimated the observables including the angular image position $\\theta_{\\infty}$, the angular image separation $s$ and the relative magnifications $r$ of the relativistic images in the uniform plasma.\nWe have shown that among these observables, when the coupling parameter $\\alpha\/M^2$ increases for fixed plasma parameter $\\beta_0$, the angular image position $\\theta_{\\infty}$ and the relative magnifications $r$ decrease,\nwhile the angular image separation $s$ increases. \nWhen the  plasma parameter $\\beta_0$ increases for fixed coupling constant $\\alpha\/M^2$, the angular image position $\\theta_{\\infty}$ and the relative magnifications $r$ decrease, but the angular image separation $s$ increases.\nAbove all, both the coupling constant $\\alpha$ and plasma parameter $\\beta_0$ have significant effects on the parameters and observables in strong gravitational lensing.\nInterestingly, it is found that  plasma has little effect on the angular image separation as $\\alpha\/M^2\\to -8$ and  the relative magnifications as $\\alpha\/M^2\\to 1$, respectively. \n\nTheoretically we can use the observations on strong gravitational lensing to test this modified gravity, although the relativistic images of strong gravitational lensing are so faint that it is hard to detect. However, with the improvement of technology, we wish observations in the future may provide the opportunity to distinguish the EGB black hole from those in general relativity. Finally, it is worth noting that in this paper we ignore the rotation of M87$^{*}$ and use the spherically symmetric metric to  provide some hints about its gravitational lensing signals. The gravitational lensing effect of rotating EGB black hole deserves a new work in the future.\n\n \n\\begin{acknowledgments}\n\t{{This work is supported in part by Science and Technology Commission of Shanghai Municipality under Grant No. 12ZR1421700 and Shanghai Normal University  KF201813.}}\n\\end{acknowledgments}\n\n\\textbf{Note added:} After this work is completed, we are  aware of a similar work by \nIslam et al. \\cite{Islam:2020xmy}, which appeared in arXiv a couple of days before.\nThey focus on the strong gravitational lensing in vacuum, in which the variation range of the coupling constant is $0\\le {\\alpha}\/{M^2}\\le 0.019$, while we discuss the strong gravitational lensing in homogeneous plasma, in which the variation range of coupling constant is $-8\\le {\\alpha}\/{M^2}\\le 1$.\nOur results agree with theirs where we overlap.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $f$ be a cusp form of positive real weight with multiplier system, and let $a(n)$ be its $n$-th Fourier coefficient. In \\cite{KKP,Pr} Knopp, Kohnen and Pribitkin, proved that the sequence $\\{a(n)\\}_{n\\in\\mathbb{N}}$, is oscillatory i.e., for each real number $\\phi\\in [0,\\pi)$, either the sequence $\\{{\\Re e\\,} (a(n)e^{-i\\phi})\\}_{n\\in\\mathbb{N}}$ changes sign infinitely often or is trivial. Geometrically speaking, this means that no matter how we slice the plane with a straight line going through the origin, there will always be infinitely many terms of $\\{a(n)\\}_{n\\in\\mathbb{N}}$ on either side of the line, unless all the terms are on the line itself, this fact motivates the following questions:\n\\begin{itemize}\n\\item[$\\bullet$] What is the proportion of integers for which the $a(n)$ lies in the same half-plane? \n\\item[$\\bullet$] If all the $a(n)$ are on a line, what is the proportion of integers for which the $a(n)$ lies in either side of the origin?\n\\end{itemize}\n\nThe latter question was asked in the particular case when $a(n)$ are real in \\cite[Section 6]{KKP}. In the case when $f$ is a newform of integral weight without complex multiplication (CM), the celebrate Sato-Tate conjecture suggests that no matter how we slice the plane with a straight line going through the origin, the proportion of primes for which the $a(p)$ lies in the same half-plane is equal to the half of the proportion of primes for which the $a(p)$ are not on the line. The questions that then naturally arise are: whether this is still true for newforms with CM? Can we infer similar results for $a(n)$ when $n$ runs through natural integers? Numerical calculations seem to suggest that the answer is positive. \n\nGoing further in this direction, in the case when $f$ is a cusp form of half-integral weight with real Fourier coefficients contained in the plus space, Kohnen and Bruinier \\cite{bruinier} gave the conjecture  \n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{n\\le x\\; :\\; a(n)\\gtrless 0\\}}{\\#\\{n\\le x \\; :\\; a(n)\\neq 0\\}}=\\frac{1}{2}.\n$$\nIn \\cite{IW2} Inam and Wiese partially verified this conjecture, more precisely they proved that for a fixed square-free integer $t$, the proportion of integers from $\\{tn^2\\}_{n\\in\\mathbb{N}}$ on which the $a(tn^2)$ are of the same sign is equal to the half of the proportion of integers from $\\{tn^2\\}_{n\\in\\mathbb{N}}$ on which the $a(tn^2)$ are non-zero.\n\nThis work was intended as an attempt to answer the questions mentioned above. However, it seems quite difficult to prove any general theorem here and we can only prove results that seem to point into the right direction. One of the motivations of this paper is an earlier work \\cite{Amri} of the author, in which he proved that the Fourier coefficients of a newform supported on prime powers have infinitely many ``angular changes\". Moreover, he established the ``angular changes\" of some subfamilies of Fourier coefficients of holomorphic cusp forms of half-integral weight reachable via the Shimura correspondence, using a robust analytic tool.\n\nThe outline of the paper is as follows. In Section \\ref{sec:1} we prove a Sato-Tate theorem for CM newforms with non-trivial nebentypus, which is  presumably well-known to experts. But, it seems that this result has not previously appeared in the literature. We also recall the Sato-Tate theorem for newforms without CM due to its importance in the sequel. In Section \\ref{sec:2}, we shall prove that for a given newform, the sub-sequences of its Fourier coefficients $\\{a(p^\\nu)\\}_{p,\\text{primes}}$ (for a fixed $\\nu\\in\\mathbb{N}$), and $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$ (for a fixed prime $p$) are oscillatory. Moreover, we calculate the proportion of primes and integers, respectively from $\\{p^\\nu\\}_{p,\\text{primes}}$, and $\\{p^\\nu\\}_{\\nu\\in\\mathbb{N}}$, on which $a(p^\\nu)$ lies in the same half-plane. In Section \\ref{sec:3} we study the oscillatory behavior results for the Fourier coefficients of eigenforms of half-integral weight which are accessible via the Shimura correspondence. Indeed, we prove that the sequences  $\\{a(tp^2)\\}_{p,\\text{primes}}$ and $\\{a(tp^{2\\nu})\\}_{\\nu\\in\\mathbb{N}}$, are oscillatory. Furthermore, we calculate the proportion of primes from $\\{tp^2\\}_{p,\\text{primes}}$ on which the $a(tp^2)$ lies in the same half-plane. Some conclusions are drawn in the final section.\n\n\n\n\\paragraph{\\textbf{Notations}}\nThroughout the paper, for any $k\\ge 2$, $N\\ge 1$ and any Dirichlet character $\\varepsilon\\pmod N$, we\ndenote by $r_\\varepsilon$ the order of $\\varepsilon$. We write $S_k^{\\mathrm{new}}(N,\\varepsilon)$ for the space of newforms of weight $k$ for the group $\\Gamma_0(N)$, with nebentypus $\\varepsilon$. If $4\\mid N$ we write $S_{k+1\/2}(N,\\varepsilon)$ for the space of half-integral weight cusp forms, when $k=1$, we shall work only with the orthogonal complement (with respect to the Petersson inner product) of the subspace of $S_{k+1\/2}(N,\\varepsilon)$ spanned by single-variable unary theta functions. The  letter $\\mathcal{H}$ stands for the upper half-plane, for $z\\in\\mathcal{H}$ we set $q:=e^{2\\pi iz}$.\n\nLet $\\mathbb{P}$ denote the set of all prime numbers. If $S$ is a subset of $\\mathbb{P}$,  we denote by  $\\delta(S)$ its natural density (if it exists), and we shall denote by $\\pi(x)$ the prime-counting function.\n\n\n\\section{Preliminaries}\\label{sec:1}\nIn this section, we mention some results about the equidistribution of eigenvalues of newforms with non-trivial nebentypus, which are crucial for our purpose. In the CM-case (newforms with CM), we provide a full proof, as we are not aware of any appropriate reference.\n\nLet  $f(z)=\\sum_{n\\ge1}a(n)q^n\\in S_k^{\\text{new}}(N,\\varepsilon)$ be a normalized newform. Fixing a root of unity $\\zeta$ such that $\\zeta^2\\in\\text{Im}(\\varepsilon)$, let $p$ be a prime number satisfying $\\varepsilon(p)=\\zeta^2$, it is clear that $\\frac{a(p)}{\\zeta}$ is real, hence in view of the Ramanujan-Petersson bound, we have \n\\begin{equation}\n\\frac{a(p)}{2 p^{(k-1)\/2}\\zeta}\\in[-1,1].\\label{eq1}\n\\end{equation} \n\nThe distribution of the sequence $\\left(\\frac{a(p)}{2 p^{(k-1)\/2}\\zeta}\\right)_p$ on $[-1,1]$ as $p$ varies over primes such that $\\varepsilon(p)=\\zeta^2$, depends on whether $f$ has complex multiplication (of CM-type) or not (not of CM-type), then we have to consider two cases. Let us first look at the CM-case. We shall prove.\n\\begin{thm}\\label{thmCMST}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform of CM-type, write\n$$\nf(z)=\\sum_{n\\ge1}a(n)q^n\\quad z\\in\\mathcal{H},\n$$\nfor its Fourier expansion at $\\infty$. Assume that the order of $\\varepsilon$ and the discriminant of the imaginary quadratic field by which $f$ has complex multiplication are coprime. Let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. Then the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ is equidistributed in $[-1,1]$ as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2,$ with respect to the measure $\\mu_{\\text{CM}}:=\\frac{1}{2\\pi}\\frac{dt}{\\sqrt{1-t^2}}+\\frac{1}{2}\\delta_{0}$, where $\\delta_0$ denotes the Dirac measure concentrated at zero. In particular, for any sub-interval $I\\subset [-1,1]$, we have \n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{p\\le x: \\varepsilon(p)=\\zeta^2,\\;\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\in I\\}}{\\#\\{p\\le x\\;:\\; \\varepsilon(p)=\\zeta^2\\}}=\\mu_{\\text{CM}}(I)=\\dfrac{1}{2\\pi}\\int_{I}\\dfrac{dt}{\\sqrt{1-t^2}}+\\dfrac{1}{2}\\delta_{0}(I).\n$$\n\\end{thm}\n\nNotice that if $\\varepsilon$ is trivial we find Deuring's equidistribution theorem \\cite{deuring}. Before we proceed to prove the theorem, we first recall some basic facts concerning newforms with complex multiplication, following the exposition given in \\cite[Section 2, pp.8-9]{Fite2015}.\n\nAssume that $f$ is of CM-type, by definition (cf. \\cite[Definition, pp. 34]{Ribet}) there exists a Dirichlet character $\\chi$ such that \n\\begin{equation}\\label{eq2}\na(p)=\\chi(p)a(p),\n\\end{equation}\nfor a set of primes $p$ of density $1$. We see that $\\chi$ here has to be a quadratic character and then it corresponds to a quadratic imaginary field say $F$, let $d_F$ denote its discriminant. \n\nAccording to \\cite[Corollary 3.5, Theorem 4.5]{Ribet} the newform $f$ should arise from an algebraic Hecke character $\\xi_f$ of $F$ of modulus $\\mathfrak{m}$ (integral ideal of $F$), in the sense that the Fourier expansion of $f$ at $\\infty$ can be written as\n$$\nf(z)=\\sum_{\\mathfrak{a}}\\xi_f(\\mathfrak{a})q^{\\mathcal{N}(\\mathfrak{a})},\n$$\nwhere $\\mathfrak{a}$ runs through all integral ideals of $F$ and $\\mathcal{N}=\\mathcal{N}_{F\/\\mathbb{Q}}$ is the norm relative to the extension $F\/\\mathbb{Q}$. \n\nLet $K_f$ denote the number field obtained by adjoining to $\\mathbb{Q}$ the Fourier coefficients of $f$.  Let $\\ell$ be prime.  For each prime ideal $\\lambda$  of $K_f$ lying above $\\ell$, there is an irreducible 2-dimensional Galois representation\n$$\n\\rho_{f,\\lambda} : \\mathrm{Gal}(\\overline{\\mathbb{Q}}\/\\mathbb{Q})\\rightarrow \\mathrm{GL}_2(K_{f,\\lambda})\n$$\nwhich is unramified outside $N\\ell$, and satisfies\n\\begin{equation}\\label{eq3}\n\\det(\\rho_{f,\\lambda})=\\varepsilon\\chi_{\\ell}^{k-1},\n\\end{equation}\nwhere $K_{f,\\lambda}$ is the completion of $K_{f}$ at $\\lambda$ and $\\chi_{\\ell}$ is the $\\ell$-adic cyclotomic character. \n\nLet $\\mathfrak{p}$ be a prime of $F$ lying above a prime $p$ not dividing $N\\ell$, and let $\\overline{\\mathfrak{p}}$ be its conjugate, then $\\rho_{f,\\lambda}(\\mathrm{Frob}_{\\mathfrak{p}})$ has a characteristic polynomial \n$$\nP_{\\mathfrak{p},\\rho_{f,\\lambda}}(T):=T^2-a(p)T+p^{k-1}\\varepsilon(p),\n$$ \nwhich can be factored into\n\\begin{equation}\\label{eq4}\nP_{\\mathfrak{p},\\rho_{f,\\lambda}}(T)=(T-\\xi_f(\\mathfrak{p}))(T-\\xi_f(\\mathfrak{\\overline{p}})).\n\\end{equation}\nTherefore, in view of \\eqref{eq3} we have\n\n\\begin{equation}\\label{eq5}\n\\xi_f(\\mathfrak{\\overline{p}})=\\varepsilon(\\mathcal{N}(\\mathfrak{p}))\\overline{\\xi_f(\\mathfrak{p})}.\n\\end{equation}\n\nNext, consider the commutative group\n$$\nG=\\left\\{\\left(\\begin{array}{cc}\nu & 0 \\\\ \n0 & \\zeta \\bar{u}\n\\end{array}\\right)\\; | \\; \\zeta\\in\\text{Im}(\\varepsilon), \\; u\\in\\mathbb{C}^*\\;\\;|u|=1 \\right\\}.$$\nLet $\\mu$ denote the Haar measure of $G$ and $X:=\\mathrm{conj}(G)$ the set of\nits conjugacy classes. Let $S$ be the set of primes of $F$ lying over $N\\ell.$ For every\n$\\mathfrak{p}\\notin S$, define the sequence\n$$x_{\\mathfrak{p}}:=\\left(\\begin{array}{cc}\n\\xi_f(\\mathfrak{p})\/\\mathcal{N}(\\mathfrak{p})^{(k-1)\/2} & 0 \\\\ \n0 & \\varepsilon(\\mathcal{N}(\\mathfrak{p}))\\overline{\\xi_f(\\mathfrak{p})}\/\\mathcal{N}(\\mathfrak{p})^{(k-1)\/2}\n\\end{array}\\right)\\in X,$$\nwith $\\mathfrak{p}$ runs over prime ideals of $F$ such that $\\text{Frob}_{\\mathfrak{p}}=\\mathcal{C}$, where $\\mathcal{C}$ is a certain conjugacy class of $\\mathrm{Gal}(F_{\\varepsilon}\/F)$, and $\\mathrm{Frob}_{\\mathfrak{p}}$ denotes the Frobenius element at $\\mathfrak{p}$ in the cyclic extension $F_{\\varepsilon}\/F$, where $F_{\\varepsilon}$ denotes the compositum of the field $\\mathbb{Q}(\\varepsilon)$ generated by the values of the character $\\varepsilon$ and $F$. \n\nAt this point we state the following proposition which can be proved by a similar argument to the one in \\cite[Proof of Proposition 3.6]{Fite2014}.\n\\begin{pro}\\label{pro1}\nFor any conjugacy class $\\mathcal{C}$ of $\\mathrm{Gal}(F_{\\varepsilon}\/F)$  the sequence $\\{x_{\\mathfrak{p}}\\}_{\\mathfrak{p}}$ is $\\mu$-equidistributed on $X$.\n\\end{pro}\n\nHaving disposed of this preliminary setup, we can now return to prove Theorem \\ref{thmCMST}.\n\\begin{proof}[Proof of Theorem \\ref{thmCMST}]\nLet $\\zeta$ be a fixed root of unity, such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. Let $p$ be a prime satisfying $\\varepsilon(p)=\\zeta^2$. We distinguish two cases.\n\nIf $p=\\mathfrak{p}\\overline{\\mathfrak{p}}$ splits in $F$, then from \\eqref{eq4} we have\n\\begin{equation}\\label{eq6}\na(p)=\\xi_{f}(\\mathfrak{p})+\\xi_{f}(\\overline{\\mathfrak{p}}).\n\\end{equation}\nConsider the map $\\vartheta : X \\to[-1,1]$, got by associating a given element of $X$ to its trace divided by $2\\zeta$. Altogether from \\eqref{eq5} and \\eqref{eq6} we see that the conjugacy class of $x_{\\mathfrak{p}}$ is mapped to $\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}$ by $\\vartheta$. From Proposition \\ref{pro1} and taking into account the isomorphism $\\mathrm{Gal}(F_{\\varepsilon}\/F)\\cong \\mathrm{Im}(\\varepsilon)$, it follows that the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ is equidistributed on $[-1,1]$ as $p$ varies over primes that split in $F$ and satisfying $\\varepsilon(p)=\\zeta^2$ with respect to the measure $\\frac{1}{\\pi}\\frac{dt}{\\sqrt{1-t^2}}$, which is the push-forward measure with respect to $\\vartheta$ of the Haar measure $\\mu$ on the Sato-Tate group $G$.\n\nIf $p$ remains inert in $F$, from \\eqref{eq2} we have $a(p)=0$, since $(r_{\\varepsilon},d_F)=1$ then the fields $F$ and $\\mathbb{Q}(\\varepsilon)$ are linearly disjoint over $\\mathbb{Q}$. Hence, by Chebotarev's density theorem, half of the primes $p$ for which $\\varepsilon(p)=\\zeta^2$ split in $F$ and the other half are inert in $F$. Consequently the sequence $\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}$ is equidistributed in $[-1,1]$ as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2$, with respect to the measure $\\mu_{\\text{CM}}=\\frac{1}{2\\pi}\\frac{dt}{\\sqrt{1-t^2}}+\\frac{1}{2}\\delta_{0}$, where we use the Dirac measure $\\delta_0$ to put half the mass at $0$ to account for the inert primes.\n\\end{proof} \n\nIn the non-CM situation the equidistribution of the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ in $[-1,1]$ as $p$ varies over primes such that $\\varepsilon(p)=\\zeta^2$, is given by case 3 of \\cite[Theorem B]{ST}.\n\\begin{thm}(Barnet-Lamb, Geraghty, Harris, Taylor)\\label{thmST}\nLet  $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform not of CM-type, write\n$$\nf(z)=\\sum_{n\\ge1}a(n)q^n\\quad z\\in\\mathcal{H},\n$$\nfor its Fourier expansion at $\\infty$. Let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. Then the sequence $\\left(\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\right)_p$ is equidistributed in $[-1,1]$ as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2,$ with respect to the Sato-Tate measure $\\mu_{\\text{ST}}:=\\frac{2}{\\pi}\\sqrt{1-t^2}dt$. In particular, for any sub-interval $I\\subset [-1,1]$ we have \n$$\\lim_{x\\to\\infty}\\dfrac{\\#\\{p\\le x: \\varepsilon(p)=\\zeta^2,\\;\\frac{a(p)}{2p^{(k-1)\/2}\\zeta}\\in I\\}}{\\#\\{p\\le x\\;:\\; \\varepsilon(p)=\\zeta^2\\}}=\\mu_{\\text{ST}}(I)=\\frac{2}{\\pi}\\int_{I}\\sqrt{1-t^2}dt.$$\n\\end{thm}\n\\section{Equidistribution of sign results for integral weight Newforms}\\label{sec:2}\nIn this section, we shall state two of our main results and shall give a proof of them. Throughout this section, we shall stick to the following notations. \n\n\\begin{hyp}\\label{hyp}\nLet \n$$\nf(z)=\\sum_{n\\ge1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H},\n$$\nbe a normalized newform of integral weight $k\\ge 2$ and level $N\\ge 1$, with Dirichlet character $\\varepsilon\\pmod N$. If $f$ has CM by a quadratic imaginary field $F$, with discriminant $d_F$, we suppose that $(d_F,r_\\varepsilon)=1$.\n\\end{hyp}\n\nConsider $\\zeta$ a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, if $p$ is a prime number satisfying $\\varepsilon(p)=\\zeta^2$, in view of \\eqref{eq1}, we may write the $p$-th Fourier coefficient of $f$ as follows\n\\begin{equation}\\label{eq7}\na(p)=2\\zeta \\cos\\theta_p,\n\\end{equation}\nfor a uniquely defined angle $\\theta_p\\in [0,\\pi]$. Notice that Theorem \\ref{thmCMST} and \\ref{thmST} are equivalent to say that the sequence $\\{\\theta_p\\}_p$  is equidistributed in $[0,\\pi]$, when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta^2$ with respect to the measure $\\mu$, where $\\mu=\\frac{1}{2\\pi}d\\theta+\\frac{1}{2}\\delta_{\\pi\/2}$ if $f$ has CM, and $\\mu=\\frac{2}{\\pi}\\sin^2\\theta d\\theta$ otherwise.  \n\nFor an integer $\\nu\\ge 1$, and a real $\\phi$ belonging to $[0,\\pi)$, we let\n$$P_{> 0}(\\phi,\\nu):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(p^\\nu)e^{-i\\phi})> 0\\},$$\n\n$$P_{< 0}(\\phi,\\nu):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(p^\\nu)e^{-i\\phi})< 0\\},$$\nand  \n$$P_{\\neq 0}(\\phi,\\nu):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\neq 0\\}.$$ \n\nHere is our first main theorem.\n\\begin{thm}\\label{thm:1}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform of integral weight $k\\ge 2$ and level $N\\ge 1$, with Dirichlet character $\\varepsilon\\pmod N$, satisfying Hypothesis \\ref{hyp}. Let\n$$\nf(z)=\\sum_{n\\ge 1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H},\n$$\nbe its Fourier expansion at $\\infty$. Let $\\nu$ be a positive odd integer. Then the sequence $\\{a(p^\\nu)\\}_{p\\in\\mathbb{P}}$ is oscillatory, and for each $\\phi\\in[0,\\pi)$ the sets $P_{>0}(\\phi,\\nu)$ and $P_{<0}(\\phi,\\nu)$ have equal positive natural density, that is, both are precisely half of the natural density of the set $P_{\\ne 0}(\\phi,\\nu)$.\n\\end{thm}\n\nBefore proving this theorem, we need the following preliminary lemmas.\n\\begin{lem}\\label{lem:1}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform of integral weight $k\\ge 2$ and level $N\\ge 1$, with Dirichlet character $\\varepsilon\\pmod N$, let\n$$\nf(z)=\\sum_{n\\ge 1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H},\n$$\nbe its Fourier expansion at $\\infty$. Let $p$ be a prime number, and $\\zeta$ be a root of unity such that $\\zeta^2\\in \\mathrm{Im}(\\varepsilon)$. If $\\varepsilon(p)=\\zeta^2$ then for any positive integer $\\nu$, the $p^\\nu$--th Fourier coefficient of $f$ is expressible by the trigonometric identity\n$$\na(p^\\nu)=\\frac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}\\zeta^\\nu,\n$$\nfor some $\\theta_p\\in (0,\\pi)$ and in the limiting cases when $\\theta_p=0$ and $\\theta_p=\\pi$ respectively we have $a(p^{\\nu})=(\\nu+1)\\zeta^\\nu$ and $a(p^{\\nu})=(-1)^\\nu(\\nu+1)\\zeta^\\nu$.\n\\end{lem}\n\\begin{rem}\nIt is worth pointing out that if the weight $k$ is even, then the cases when $a(p^{\\nu})=(\\nu+1)\\zeta^\\nu$ and $a(p^{\\nu})=(-1)^\\nu(\\nu+1)\\zeta^\\nu$ can happen for at most finitely many primes $p$ only. In fact, if we denote by $K_f$ the field generated by all the Fourier coefficients of $f$, and pick a prime $p$ satisfying $\\varepsilon(p)=\\zeta^2$, so that $\\theta_p=0$, or $\\pi$, then we should have $\\sqrt{p}\\in K_f$, which can happen for only finitely many primes, because $K_f$ is a number field.   \n\\end{rem}\n\\begin{proof}\nSince $f$ is a normalized newform, we have the following power series expansion\n$$\n\\sum_{\\nu\\ge0}a(p^\\nu)X^\\nu=\\dfrac{1}{1-a(p)X+p^{k-1}\\varepsilon(p)X^2}.\n$$\nSetting $X=x\\zeta^{-1}$, and write \n$$\n1-a(p)\\zeta^{-1}x+p^{k-1}x^2=(1-\\alpha_px)(1-\\beta_px),\n$$ \none sees\n$$\n\\sum_{\\nu\\ge0}a(p^\\nu)\\zeta^{-\\nu} x^\\nu=\\dfrac{1}{(\\alpha_p-\\beta_p)x}\\left(\\dfrac{1}{1-\\alpha_px}-\\dfrac{1}{1-\\beta_px}\\right).\n$$\nNow, expanding both geometric series, we deduce that the $p^\\nu$-th Fourier coefficient of $f$ is\n\\begin{equation}\na(p^\\nu)=\\dfrac{\\alpha^{\\nu+1}_p-\\beta^{\\nu+1}_p}{\\alpha_p-\\beta_p}\\zeta^{\\nu}.\\label{eq:7}\n\\end{equation}\nOn the other hand, since $\\frac{a(p)}{\\zeta}\\in\\mathbb{R}$, then $\\beta_p=\\overline{\\alpha_p}$ and by Deligne's theorem \\cite[Theorem 8.2]{Deligne} we have $|\\alpha_p|=|\\beta_p|=1$. Thus, we may write $\\alpha_p=e^{i\\theta_p}$ and $\\beta_p=e^{-i\\theta_p}$ for some $\\theta_p\\in[0,\\pi]$. Inserting this in \\eqref{eq:7} we obtain the desired identities. \n\\end{proof}\n\\begin{lem}\\label{lem:2}\nWe make the same assumptions as in Theorem \\ref{thm:1}, and let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, then $\\frac{a(p^\\nu)}{\\zeta^\\nu}$ is real if $\\varepsilon(p)=\\zeta^2$. With the notation\n$$\\mathbb{P}_{\\gtrless0}(\\zeta,\\nu):=\\left\\{p\\in\\mathbb{P}\\;:\\; \\varepsilon(p)=\\zeta^2,\\;\\frac{a(p^\\nu)}{\\zeta^\\nu}\\gtrless0\\right\\},$$ \nwe have\n$$\n\\delta(\\mathbb{P}_{>0}(\\zeta,\\nu))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is of CM-type},\n    \\end{array}\n\\right.\n$$\nand \n$$\n\\delta(\\mathbb{P}_{<0}(\\zeta,\\nu))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is of CM-type}.\n    \\end{array}\n\\right.\n$$\n\\end{lem}\n\\begin{proof}\nLet $p$ be a prime number such that $\\varepsilon(p)=\\zeta^2$. By the previous lemma, the $p^\\nu$--th Fourier coefficient $a(p^\\nu)$ of $f$ is expressible by the trigonometric identity\n$$\na(p^\\nu)=\\frac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}\\zeta^\\nu,\n$$\nwhere $\\theta_p\\in (0,\\pi)$. Since the set $\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta^2, \\theta_p=0\\;\\text{or}\\;\\pi\\}$ has density zero, we may assume that $\\theta_p$ is different from $0$ and $\\pi$. \n\nTherefore, the sign of $\\frac{a(p^\\nu)}{\\zeta^\\nu}$ is the same as the sign of $\\sin((\\nu+1)\\theta_p)$, it follows that\n$$\np\\in \\mathbb{P}_{>0}(\\zeta,\\nu)\\Longleftrightarrow\\varepsilon(p)=\\zeta^2,\\;\\theta_p\\in A_{>0}:=\\bigcup_{j=1}^{\\frac{\\nu+1}{2}}\\left(\\frac{(2j-2)\\pi}{\\nu+1},\\frac{(2j-1)\\pi}{\\nu+1}\\right),\n$$\nand\n$$\np\\in \\mathbb{P}_{<0}(\\zeta,\\nu)\\Longleftrightarrow \\varepsilon(p)=\\zeta^2,\\;\\theta_p\\in A_{<0}:=\\bigcup_{j=1}^{\\frac{\\nu+1}{2}}\\left(\\frac{(2j-1)\\pi}{\\nu+1},\\frac{2j\\pi}{\\nu+1}\\right).\n$$\n\nOn the other hand from \\cite[Proof of Theorem 1.1, odd case]{Meher2017}, we have\n$$\n\\mu(A_{>0})=\\mu(A_{<0})=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{4} & \\mbox{if}\\;\\; f\\;\\;  \\mbox{is of CM-type}.\n    \\end{array}\n\\right.\n$$\nTaking into account that\n$$\n\\delta\\left(\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta^2\\}\\right)=\\dfrac{1}{r_{\\varepsilon}},\n$$\nthe desired conclusion can be derived easily from Theorem \\ref{thmCMST} and \\ref{thmST}.\n\\end{proof}\n\nNow we are in the position to prove Theorem \\ref{thm:1}.\n\\begin{proof}[Proof of Theorem \\ref{thm:1}]\n\\sloppy Fix $\\phi\\in[0,\\pi)$. Let $\\zeta$ be a root of unity such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$. For the notational convenience throughout the proof we let $\\pi_{>0}(x,\\zeta):=\\#\\{p\\le x : p\\in\\mathbb{P}_{>0}(\\zeta,\\nu)\\}$ and similarly $\\pi_{<0}(x,\\zeta)$, where $\\mathbb{P}_{>0}(\\zeta,\\nu)$ and $\\mathbb{P}_{<0}(\\zeta,\\nu)$ be as in Lemma \\ref{lem:2}. \n\nLet us first examine the oscillatory behavior of the sequence $\\{a(p^\\nu)\\}_{p\\in\\mathbb{P}}$. We need to consider the following two cases.\n\\begin{description}\n\\item[\\textbf{Case 1}: $\\mathrm{arg}(\\zeta^\\nu)\\not\\equiv\\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.]\nThe sequence $\\{{\\Re e\\,}(a(p^\\nu) e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta^2}$ is not trivial. Moreover, by Lemma \\ref{lem:1} we have the trigonometric identity\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})=\\dfrac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})\\quad\\text{with}\\quad\\varepsilon(p)=\\zeta^2,\n$$\nfor some $\\theta_p\\in(0,\\pi)$. There is no loss of generality in assuming ${\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0$. Then the sign of the sequence ${\\Re e\\,}(a(p^\\nu)e^{-i\\phi})$ is the same as the sign of $\\sin((\\nu+1)\\theta_p)$, when $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta^2$. Thus we have\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})>0\\quad \\text{if and only if}\\quad \\theta_p\\in A_{>0},\n$$\nand\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})<0 \\quad \\text{if and only if}\\quad \\theta_p\\in A_{<0},\n$$\nwhere $A_{\\lessgtr 0}$ be defined as in the proof of Lemma \\ref{lem:2}. From Theorem \\ref{thmCMST} and \\ref{thmST}, we know that the sequence $\\{\\theta_p\\}_{p,\\varepsilon(p)=\\zeta^2}$ is equidistributed in $[0,\\pi]$ with respect to the measure $\\mu$. Thereby, there are infinitely many primes $p$ satisfying $\\varepsilon(p)=\\zeta^2$ such that $\\theta_p\\in A_{>0}$, and infinitely many primes $p$ satisfying $\\varepsilon(p)=\\zeta^2$ such that $\\theta_p\\in A_{<0}$. Hence the sequence $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta^2}$ changes sign infinitely often.\n\n\\item[\\textbf{Case 2}: $\\mathrm{arg}(\\zeta^\\nu)\\equiv \\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.] The sequence $\\{{\\Re e\\,}(a(p^\\nu) e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta^2}$ is trivial.\n\\end{description}\n\nSummarizing, we have thus proved that for any root of unity $\\zeta$ such that $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, the sequence $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{p}$ when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta^2$, either changes sign infinitely often or is trivial. Accordingly, for each $\\phi\\in [0,\\pi)$ either the sequence $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{p\\in\\mathbb{P}}$ is trivial or changes sign infinitely often.\n\nIt remains to calculate the natural density of the sets $P_{>0}(\\phi,\\nu)$ and $P_{<0}(\\phi,\\nu)$. Here we restrict ourselves to the case when $f$ is not of CM-type, as the argument is entirely similar to the CM situation. The key point here is to see that \n$$P_{>0}(\\phi,\\nu)=\\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0}}\\mathbb{P}_{>0}(\\zeta,\\nu)\\bigsqcup \\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})<0}}\\mathbb{P}_{<0}(\\zeta,\\nu),$$\nand\n$$P_{<0}(\\phi,\\nu)=\\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})<0}}\\mathbb{P}_{>0}(\\zeta,\\nu)\\bigsqcup \\coprod_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0}}\\mathbb{P}_{<0}(\\zeta,\\nu),$$\nup to finitely many primes, where $\\zeta=e^{\\frac{\\pi i j}{r_{\\varepsilon}}}$, is so chosen that $1\\le j \\le r_{\\varepsilon}$, when $r_{\\varepsilon}$ is even, (note that there is a unique choice of $\\zeta$ when $r_{\\varepsilon}$ is odd). \n\nThe above displayed formula combined with Lemma \\ref{lem:2}, gives\n\\begin{eqnarray*}\n\\delta\\left(P_{>0}(\\phi,\\nu)\\right) &=& \\!\\!\\lim_{x\\to\\infty}\\sum_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{ \\xi=\\zeta^2 ,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})>0}}\\!\\!\\!\\!\\!\\dfrac{\\pi_{>0}(x,\\zeta)}{\\pi(x)}+\\lim_{x\\to\\infty}\\sum_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{\\xi=\\zeta^2 ,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})<0}}\\!\\!\\!\\!\\!\\!\\!\\dfrac{\\pi_{<0}(x,\\zeta)}{\\pi(x)},\\\\\n  &=& \\!\\!\\frac{1}{2}\\sum_{\\mycom{\\xi,r_\\varepsilon\\text{-th root of unity}}{ \\xi=\\zeta^2,{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})\\neq 0}}\\frac{1}{r_{\\varepsilon}},\\\\\n  &=&\\!\\!\\frac{\\delta(P_{\\neq 0}(\\phi,\\nu))}{2}.\n\\end{eqnarray*}\nIn the same manner we can see that $\\delta\\left(P_{<0}(\\phi,\\nu)\\right) =\\frac{\\delta(P_{\\neq 0}(\\phi,\\nu))}{2}$, which concludes the proof.\n\\end{proof}\n\n\\sloppy Our next concern will be the oscillatory behavior of the sequence $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$, and the equidistribution of signs of $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{\\nu\\in\\mathbb{N}}$, for a fixed prime number $p$. \n\\begin{thm}\\label{thm:2}\nLet $f\\in S_k^{\\mathrm{new}}(N,\\varepsilon)$ be a normalized newform, and let\n$$\nf(z)=\\sum_{n\\ge 1}a(n)n^{(k-1)\/2}q^n\\quad z\\in\\mathcal{H}\n$$ \nbe its Fourier expansion at $\\infty$. Then there exists a set $S$ of primes of density zero, such that the following holds: For every prime $p\\notin S$, the sequence $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$ is oscillatory, and for any $\\phi\\in [0,\\pi)$ we have \n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x\\; :\\; {\\Re e\\,}\\{a(p^\\nu)e^{-i\\phi}\\}\\gtrless 0\\}}{\\#\\{\\nu\\le x \\; :\\; {\\Re e\\,}\\{a(p^\\nu)e^{-i\\phi}\\} \\neq 0\\}}=\\frac{1}{2}.\n$$\n\\end{thm}\n\\begin{rem}\n\\begin{enumerate}\n\\item The sequence $\\{a(p^\\nu)\\}_{\\nu\\in\\mathbb{N}}$ is oscillatory means that the sequence escape infinitely often from any half-plane. Hence, it improves the result of the author in \\cite[Theorem 2.1]{Amri}.\n\\item It is worth pointing out that the theorem holds for prime in $S$, under some further restrictions, which can be easily deduced from the techniques of our proof. \n\\end{enumerate}\n\\end{rem}\n\\begin{proof}[Proof of Theorem \\ref{thm:2}] Set\n$$\nS :=\\coprod_{\\xi\\in\\mathrm{Im}(\\varepsilon)}\\left\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\xi, \\theta_p=0\\:\\text{or}\\:\\pi\\right\\},\n$$\nwhere $\\theta_p$ is defined as in \\eqref{eq7}. By Theorem \\ref{thmCMST} and \\ref{thmST} we see that the set $S$ has density zero. Let $p$ be a prime outside $S$, then there exists a root of unity $\\zeta$ satisfying $\\zeta^2\\in\\mathrm{Im}(\\varepsilon)$, such that $\\varepsilon(p)=\\zeta^2$. From Lemma \\ref{lem:1}, one can write\n$$\n{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})=\\dfrac{\\sin((\\nu+1)\\theta_p)}{\\sin\\theta_p}{\\Re e\\,}(\\zeta^\\nu e^{-i\\phi}),\n$$ \nfor some $\\theta_p\\in(0,\\pi)$. We derive the oscillatory behavior of the sequence $\\{a(p^\\nu))\\}_{\\nu\\in\\mathbb{N}}$, from the well-known behavior of the sequence $\\left\\{\\sin((\\nu+1)\\theta_p)\\right\\}_{\\nu\\in\\mathbb{N}}$.\n\nWe are left with the task of studying the equidistribution of signs of $\\{{\\Re e\\,}(a(p^\\nu)e^{-i\\phi})\\}_{\\nu\\in\\mathbb{N}}$, to this end, write ${\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})=\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right)$, if $r_{\\varepsilon}$ is even, and ${\\Re e\\,}(\\zeta^\\nu e^{-i\\phi})=\\cos\\left(\\frac{2\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right)$, if  $r_{\\varepsilon}$ is odd, for some $1\\le j\\le r_{\\varepsilon}$.  We treat only the former case, the second one being completely similar. We need to distinguish the following two cases.\n\\begin{description}\n\\item[\\textbf{Case 1}: $\\frac{\\theta_p}{2\\pi}$ is irrational.] Write\n$$\n{\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})=(\\sin\\theta_p)^{-1}\\sin\\left(2\\pi \\left<\\!\\frac{(\\nu+1)\\theta_p}{2\\pi}\\!\\right>\\right)\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right),\n$$ \nwhere $\\left<\\!\\frac{(\\nu+1)\\theta_p}{2\\pi}\\!\\right>$ denotes the fractional part of $\\frac{(\\nu+1)\\theta_p}{2\\pi}$. Note that the sequence $\\left\\{\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right)\\right\\}_{\\nu\\in\\mathbb{N}}$ is $t_{\\varepsilon}$-periodic and takes only finitely many different values, with $t_{\\varepsilon}=2r'_{\\varepsilon}$ if $\\frac{j}{(r_{\\varepsilon},j)}$ is odd, and $t_{\\varepsilon}=r'_{\\varepsilon}$ if $\\frac{j}{(r_{\\varepsilon},j)}$ is even, where $r'_\\varepsilon=\\frac{r_{\\varepsilon}}{(r_{\\varepsilon},j)}$. So, one may split the total range for $\\nu$ into different arithmetic progressions $d\\pmod{ t_{\\varepsilon}}$ where $1\\le d \\le t_{\\varepsilon}$, so that when $\\nu$ runs through each of these arithmetic progressions the $\\cos$-factor becomes constant and takes always the same value, say $c_d$. Accordingly we may write\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})>0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)>0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)<0}}1,\n$$\nand\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})<0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)<0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)>0}}1.\n$$ \nOn the other hand we have\n\\begin{eqnarray*}\n\\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)>0}}1 &=& \\lim_{x\\to\\infty}\\frac{1}{x}\\displaystyle \\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)\\in [0,1]}} 1,\\\\\n &=&  \\lim_{x\\to\\infty}\\frac{1}{x}\\displaystyle \\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\in \\left[0,\\pi\\right]}} 1,\\\\\n  &=& \\frac{1}{2t_\\varepsilon},\n \\end{eqnarray*}\n\\sloppy where we used the fact that the sequence $\\left\\{\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right\\}_{\\nu\\in\\mathbb{N}}$ is uniformly distributed $\\pmod 1$ in $[0,1]$ in the last step, (Weyl's equidistribution theorem, see, e.g., \\cite{Kuipers}[Example 2.1, pp.8]). Similarly we have\n$$\n\\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(2\\pi\\left<\\frac{\\nu \\theta_p}{2\\pi}\\right>\\right)<0}}1=\\frac{1}{2t_{\\varepsilon}}.\n$$\nIt follows\n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{x}=\\sum_{\\mycom{d=1}{c_d\\ne 0}}^{t_{\\varepsilon}}\\frac{1}{2 t_{\\varepsilon}},\n$$\nand therefore \n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\ne 0\\}}=\\frac{1}{2}.\n$$\n\\item[\\textbf{Case 2}: $\\frac{\\theta_p}{2\\pi}=\\frac{n}{m}\\in (0,\\frac{1}{2})$ is rational, where $m$ and $n$ are coprime.] Write\n$$\n{\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})=(\\sin\\theta_p)^{-1}\\sin\\left(\\frac{2\\pi(\\nu+1) n}{m}\\right)\\cos\\left(\\frac{\\pi j\\nu}{r_{\\varepsilon}}-\\phi\\right).\n$$\nBy similar considerations as in the previous case, we have\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})>0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)>0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)<0}}1,\n$$\nand\n$$\n\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})<0\\}=\\sum_{\\mycom{d=1}{c_d>0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)<0}}1+ \\sum_{\\mycom{d=1}{c_d<0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)>0}}1.\n$$ \nThus, the study of the distribution of signs of ${\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})$, turned out to study the distribution of signs of the sequence $\\left\\{\\sin\\left(\\frac{2\\pi\\nu n}{m}\\right)\\right\\}_{\\nu}$ when $\\nu$ runs through an arithmetic progression $d\\pmod{t_\\varepsilon}$. Note that this sequence is $t_{\\varepsilon}m'$-periodic and takes only finitely many different values,  where $m'=\\frac{m}{(m,t_\\varepsilon)}$. Thus, we can split this arithmetic progression into $m'$ different sub-arithmetic progressions $d+t_{\\varepsilon}\\ell\\pmod{m't_\\varepsilon}$ where $1\\le\\ell\\le m'$, such that when $\\nu$ runs through each of these sub-arithmetic progressions the sequence becomes constant and has always the same value, say $s_{d,\\ell}$. Consequently\n\\begin{eqnarray*}\n\\nonumber \\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi \\nu n}{m}\\right)>0}}1 &=& \\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}>0}}^{m'}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\frac{\\nu-d}{t_\\varepsilon}\\equiv \\ell \\!\\!\\!\\!\\!\\pmod{m'}}}1, \\\\  \n&=& \\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}>0}}^{m'}\\sum_{\\mycom{\\nu\\le x}{\\nu\\equiv d+t_\\varepsilon\\ell \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}m'}}}1,\\\\ \n&=& \\sum_{\\mycom{\\ell=1}{s_{d,\\ell}>0}}^{m'}\\frac{1}{t_{\\varepsilon}m'}.\n\\end{eqnarray*}\nSimilarly, we obtain \n$$\n\\lim_{x\\to\\infty}\\frac{1}{x}\\sum_{\\mycom{\\nu\\le x, \\nu\\equiv d \\!\\!\\!\\!\\!\\pmod {t_{\\varepsilon}}}{\\sin\\left(\\frac{2\\pi \\nu n}{m}\\right)<0}}1=\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}<0}}^{m'}\\frac{1}{t_{\\varepsilon}m'}.\n$$\nTherefore\n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{x}=\\sum_{\\mycom{d=1}{c_d\\ne 0}}^{t_{\\varepsilon}}\\sum_{\\mycom{\\ell=1}{s_{d,\\ell}\\ne 0}}^{m'}\\frac{1}{2t_{\\varepsilon}m'},\n$$\nand hence\n$$\n\\lim_{x\\to\\infty}\\dfrac{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\gtrless 0\\}}{\\#\\{\\nu\\le x : {\\Re e\\,}(a(p^{\\nu})e^{-i\\phi})\\ne 0\\}}=\\frac{1}{2},\n$$\nas desired.\n\\end{description}\nThis completes the proof of Theorem \\ref{thm:2}.\n\\end{proof}\n\n\\section{Equidistribution of sign results of half-integral weight cuspidals eigenforms}\\label{sec:3}\nIn this section, we shall present and prove our results concerning the oscillatory behavior and signs equidistribution of Fourier coefficients of cuspidal Hecke eigenforms following a similar philosophy to that in the previous section. In order to state our results, we need to develop some notations and make some assumptions for this section. \n\n\\begin{hyp}\\label{hyp:2}\nLet $N\\ge 4$ be divisible by $4$ and $k\\ge 1$ be a natural number. Fix any Dirichlet character $\\varepsilon\\pmod N$. Let $$f(z)=\\sum_{n=1}^{\\infty}a(n)q^n\\quad z\\in\\mathcal{H},$$ be a non-zero cuspidal Hecke eigenform of half-integral weight $k+1\/2$ and level $N$ with Dirichlet character $\\varepsilon\\pmod N$, and let  $t$ be a square-free integer such that $a(t)\\ne 0$.  The Shimura correspondence \\cite{Shi} lifts $f$ to a Hecke eigenform $\\mathrm{Sh}_t(f)$ of weight $2k$ for the group $\\Gamma_0(N\/2)$ with character $\\varepsilon^2$. Let us write\n$$\n\\mathrm{Sh}_t(f)=\\sum_{n\\ge 1} A_t(n)q^n,\n$$\nfor its expansion at $\\infty$. For simplicity we assume that $a(t)=1.$ According to \\cite{Shi}, the $n$-th Fourier coefficient of $\\mathrm{Sh}_t(f)$ is given by \n\\begin{equation}\nA_t(n)=\\sum_{d|n}\\varepsilon_{t,N}(d)d^{k-1}a\\left(\\frac{n^2}{d^2}t\\right),\\label{eq8}\n\\end{equation}\nwhere $\\varepsilon_{t,N}$ denotes the character $\\varepsilon_{t,N}(d):=\\varepsilon(d)\\left(\\frac{(-1)^{k}N^{2}t}{d}\\right)$, we let $\\chi_0(d):=\\left(\\frac{(-1)^{k}N^{2}t}{d}\\right)$.  If $\\mathrm{Sh}_t(f)$ has complex multiplication by an imaginary quadratic field $F$ denote by $d_F$ its fundamental discriminant. We suppose that $(r_{\\varepsilon},d_F)=1$ and the fields $F_\\varepsilon$ and $\\mathbb{Q}(\\sqrt{(-1)^k t})$ are linearly disjoint over $\\mathbb{Q}$, where $F_{\\varepsilon}$ is the field obtained by adjoining to $F$ the values of $\\varepsilon$. We let $\\chi_F$ be the quadratic character associated to $F$.\n\\end{hyp}\n\nLet $\\zeta$ be a root of unity such that $\\zeta\\in\\mathrm{Im}(\\varepsilon)$, if $p$ is a prime number satisfying $\\varepsilon(p)=\\zeta,$ then we have\n$$B_\\zeta(p):=\\frac{A_t(p)}{2p^{k-1\/2}\\zeta}\\in [-1,1].$$\nBy \\eqref{eq8} we have $a(tp^2)=A_t(p)-\\varepsilon_{t,N}(p)p^{k-1}$, and hence \n\\begin{equation}\\label{eq9}\n\\frac{a(tp^2)}{2p^{k-1\/2}\\zeta}=B_{\\zeta}(p)-\\frac{\\chi_0(p)}{2\\sqrt{p}}.\n\\end{equation}\n\nFor abbreviation  we let $A_\\zeta(p)$ stand for $\\frac{a(tp^2)}{2p^{k-1\/2}\\zeta}$.\nSet\n$$P_{> 0}(\\phi):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(tp^2)e^{-i\\phi})> 0\\},$$\n\n$$P_{< 0}(\\phi):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(tp^2)e^{-i\\phi})< 0\\},$$\nand \n$$P_{\\neq 0}(\\phi):=\\{p\\in\\mathbb{P}\\; :\\; {\\Re e\\,}(a(tp^2)e^{-i\\phi})\\neq 0\\},$$\nwhere $\\phi$ is a real number belonging to $[0,\\pi)$.\n\\begin{thm}\\label{thm:3}\nLet $f\\in S_{k+1\/2}(N,\\varepsilon)$ be a cuspidal Hecke eigenform  satisfying Hypothesis \\ref{hyp:2}. Let us write \n$$\nf(z)=\\sum_{n\\ge 1}a(n)q^n\\quad z\\in\\mathcal{H},\n$$\nfor its Fourier expansion at $\\infty$. Then the sequence $\\{a(tp^2)\\}_{p\\in\\mathbb{P}}$ is oscillatory, and if moreover $\\mathrm{Sh}_t(f)$ is not of CM-type, or of CM-type and $\\chi_0\\ne \\chi_{\\mathrm{triv}},\\chi_F$, then for each $\\phi\\in[0,\\pi)$ the sets $P_{>0}(\\phi)$ and $P_{<0}(\\phi)$ have equal positive natural density, that is, both are precisely half of the natural density of the set $P_{\\ne 0}(\\phi)$. In the remaining cases we have\n$$\n\\delta(P_{>0}(\\phi))=\\left\\{\n    \\begin{array}{llll}\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+\\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})>0}} \\frac{1}{r_{\\varepsilon}} &\\mbox{if}\\;\\;\\chi_0 =\\chi_F, \\\\\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+ \\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})<0}} \\frac{1}{r_{\\varepsilon}} & \\mbox{if}\\;\\;  \\chi_0 =\\chi_{\\mathrm{triv}},\n    \\end{array}\n\\right.\n$$\nand\n$$\n\\delta(P_{<0}(\\phi))=\\left\\{\n    \\begin{array}{llll}\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+\\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})<0}} \\frac{1}{r_{\\varepsilon}} &\\mbox{if}\\;\\;\\chi_0 =\\chi_F, \\\\\n        \\dfrac{\\delta(P_{\\neq0}(\\phi))}{4}+ \\dfrac{1}{2}\\displaystyle \\sum_{\\mycom{\\zeta\\in\\mathrm{Im}(\\varepsilon)}{{\\Re e\\,}(\\zeta e^{-i\\phi})>0}} \\frac{1}{r_{\\varepsilon}} & \\mbox{if}\\;\\;  \\chi_0 =\\chi_{\\mathrm{triv}}.\n    \\end{array}\n\\right.\n$$\n\\end{thm}\n\nWe shall need the following lemma.\n\\begin{lem}\\label{lem:3}\nWe make the same assumptions as in Theorem \\ref{thm:3}, and let $\\zeta$ be a root of unity belonging to $\\mathrm{Im}(\\varepsilon)$. If we denote\n \n$$\\mathbb{P'}_{\\gtrless0}(\\zeta):=\\left\\{p\\in\\mathbb{P}\\;:\\; \\varepsilon(p)=\\zeta,\\;A_\\zeta(p)\\gtrless0\\right\\}.$$\nThen we have\n$$\n\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0\\ne \\chi_{\\mathrm{triv}},\\chi_F,\\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_{\\mathrm{triv}},\\\\\n        \\frac{3}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_F,\n    \\end{array}\n\\right.\n$$\nand\n$$\n\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\left\\{\n    \\begin{array}{ll}\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is not of CM-type}, \\\\\n        \\frac{1}{2r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0\\ne \\chi_{\\mathrm{triv}},\\chi_F,\\\\\n        \\frac{3}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_{\\mathrm{triv}},\\\\\n        \\frac{1}{4r_{\\varepsilon}} & \\mbox{if}\\;\\; \\mathrm{Sh}_t(f)\\;\\;  \\mbox{is of CM-type and}\\;\\; \\chi_0 =\\chi_F.\n    \\end{array}\n\\right.\n$$\n\\end{lem}\n\n\\begin{proof}\nDenote by $\\pi_{>0}(x,\\zeta):=\\#\\{p\\le x\\; :\\;p\\in\\mathbb{P'}_{>0}(\\zeta)\\}$ and similarly $\\pi_{\\ge 0}(x,\\zeta)$, $\\pi_{<0}(x,\\zeta)$ and $\\pi_{\\le0}(x,\\zeta)$. \n\nFirst assume that $\\mathrm{Sh}_t(f)$ is not of CM-type, we follow closely the method of \\cite{IW}. Let $p$ be a prime satisfying $\\varepsilon(p)=\\zeta$, from \\eqref{eq9} we have \n$$\nA_\\zeta(p)>0 \\Longleftrightarrow B_\\zeta(p)>\\frac{\\chi_0(p)}{2\\sqrt{p}}.\n$$\nIt follows that for any fixed $\\epsilon >0$, we have the following inclusion of sets\n$$\\{p\\leq x : \\varepsilon(p)=\\zeta, B_\\zeta(p)>\\epsilon\\}\\!\\subset\\!\\{p\\in\\mathbb P : p\\leq\\frac{1}{4\\epsilon^2}, \\varepsilon(p)=\\zeta\\}\\cup\\{p\\leq x : p\\in\\mathbb{P'}_{>0}(\\zeta)\\}.$$\nTherefore \n\\begin{equation}\\label{eq10}\n\\pi_{>0}(x,\\zeta)+\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)\\geq \\#\\{p\\leq x :\\varepsilon(p)=\\zeta,\\;\\; B_\\zeta(p)>\\epsilon\\},\n\\end{equation}\nwhere $\\pi_{\\zeta}(x):=\\#\\{p\\in\\mathbb{P}: p\\le x,\\;\\; \\varepsilon(p)=\\zeta\\}$. Now dividing \\eqref{eq10} by $\\pi_{\\zeta}(x)$ we obtain \n\\begin{equation}\\label{eq:10}\n\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}+\\dfrac{\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)}{\\pi_{\\zeta}(x)}\\geq \\dfrac{\\#\\{p\\leq x :\\varepsilon(p)=\\zeta,\\;\\; B_\\zeta(p)>\\epsilon\\}}{\\pi_{\\zeta}(x)}.\n\\end{equation}\nSince $\\pi_{\\zeta}(x)\\underset{x\\to\\infty}{\\sim}\\frac{x}{r_{\\varepsilon}\\log x}$, and the term $\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)$ is finite, it follows\n\\begin{equation}\\label{eq11}\n\\lim_{x\\to\\infty}\\dfrac{\\pi_{\\zeta}\\left(\\frac{1}{4\\epsilon^2}\\right)}{\\pi_{\\zeta}(x)}=0.\n\\end{equation}\nOn the other hand, by Theorem \\ref{thmST} we have \n\\begin{equation}\\label{eq12}\n \\lim_{x\\to\\infty}\\dfrac{\\#\\{p\\leq x : \\varepsilon(p)=\\zeta,\\;\\; B_\\zeta(p)>\\epsilon\\}}{\\pi_{\\zeta}(x)}=\\mu_{\\mathrm{ST}}([\\epsilon,1]).\n \\end{equation}\n\nTaking into account \\eqref{eq11} and \\eqref{eq12} a passage to the limit in \\eqref{eq:10} implies that\n\\begin{equation}\\label{eq13}\n\\liminf_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\geq \\mu_{\\mathrm{ST}}([\\epsilon,1]).\n\\end{equation}\nAs the inequality \\eqref{eq13} holds for all $\\epsilon>0$, we have\n$$\\liminf_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\geq \\frac{1}{2}.$$\nSimilarly we get $\\liminf\\limits_{x\\to\\infty}\\frac{\\pi_{\\le 0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\geq \\frac{1}{2}$ and in view of $\\pi_{\\le 0}(x,\\zeta)=\\pi_\\zeta(x)-\\pi_{>0}(x,\\zeta)$, one sees $\\limsup\\limits_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}\\le\\frac{1}{2}$. Consequently\n$$\n\\lim\\limits_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi_{\\zeta}(x)}=\\frac{1}{2}.\n$$\nSince $\\delta(\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta\\})=\\frac{1}{r_{\\varepsilon}}$, it follows\n $$\\delta(\\mathbb{P}'_{>0}(\\zeta))=\\lim_{x\\to\\infty}\\frac{\\pi_{>0}(x,\\zeta)}{\\pi(x)}=\\frac{1}{2r_{\\varepsilon}}.$$\nSimilarly we have $\\delta(\\mathbb{P}'_{<0}(\\zeta))=\\frac{1}{2r_{\\varepsilon}}$.\n\nNow, let us examine the CM situation, assume that $\\mathrm{Sh}_t(f)$ has CM by an imaginary quadratic field $F$. Set $I=(0,1]$, $J=[-1,0)$,\n$$\nT_{I}(\\zeta):=\\{p\\in\\mathbb{P}: \\varepsilon(p)=\n\\zeta, A_{\\zeta}(p)\\in I, B_\\zeta(p)\\ne 0\\}, S_{I}(\\zeta):=\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, B_{\\zeta}(p)\\in I\\}\n$$\nand \n$$\n T_{J}(\\zeta):=\\{p\\in\\mathbb{P}: \\varepsilon(p)=\\zeta, A_{\\zeta}(p)\\in J, B_\\zeta(p)\\ne 0\\}, S_{J}(\\zeta):=\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, B_{\\zeta}(p)\\in J\\}\n$$\n\nFrom Theorem \\ref{thmCMST} we have $\\delta(S_{I}(\\zeta))=\\delta(S_{J}(\\zeta))=\\frac{1}{4 r_\\varepsilon}$. Thus in view of \\cite[Theorem 4.2.1]{Arias} and Theorem \\ref{thmCMST} it follows that $$\\delta(T_{I}(\\zeta))=\\delta(T_{J}(\\zeta))=\\frac{1}{4 r_\\varepsilon}.$$\nNow by \\cite[Remark 4.2.2]{Arias} we may write\n\n$$\n\\mathbb{P'}_{>0}(\\zeta)= T_{I}(\\zeta)\\sqcup \n$$\n\\begin{equation}\\label{eq15}\n\\left(\\{p\\in\\mathbb{P}: \\varepsilon(p)=\\zeta,B_\\zeta(p)=0 \\}\\cap \\left\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, \\frac{-\\chi_0(p)}{2\\sqrt{p}}\\in I\\right\\}\\right),\n\\end{equation}\nand \n$$\n\\mathbb{P'}_{<0}(\\zeta)=T_{J}(\\zeta)\\sqcup\n$$\n\\begin{equation}\\label{eq16}\n\\left(\\{p\\in\\mathbb{P}: \\varepsilon(p)=\\zeta,B_\\zeta(p)=0 \\}\\cap \\left\\{p\\in\\mathbb{P} : \\varepsilon(p)=\\zeta, \\frac{-\\chi_0(p)}{2\\sqrt{p}}\\in J\\right\\}\\right).\n\\end{equation}\n\nIn order to calculate $\\delta(\\mathbb{P'}_{<0}(\\zeta))$ and $\\delta(\\mathbb{P'}_{>0}(\\zeta))$, we distinguish the following three cases\n\\begin{description}\n\\item[\\textbf{Case 1}: $\\chi_0\\ne \\chi_{\\mathrm{triv}}$ and $\\chi_F$.] By our hypothesis we have $(r_{\\varepsilon},d_F)=1$, and the fields $\\mathbb{Q}(\\sqrt{(-1)^k t})$ and $F_\\varepsilon$ are linearly disjoint over $\\mathbb{Q}$. Hence by Chebotarev's theorem the intersections in \\eqref{eq15} and \\eqref{eq16} have natural density $\\frac{1}{4r_{\\varepsilon}}$ and $\\frac{1}{4r_{\\varepsilon}}$ respectively. Consequently $\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\frac{1}{2r_{\\varepsilon}}$ and $\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\frac{1}{2r_{\\varepsilon}}$.\n\n\\item[\\textbf{Case 2}: $\\chi_0=\\chi_{\\mathrm{triv}}$.] Since $(r_{\\varepsilon},d_F)=1$ the intersections in \\eqref{eq15} and \\eqref{eq16} have natural density $0$ and $\\frac{1}{2r_{\\varepsilon}}$ respectively, it follows that $\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\frac{1}{4r_{\\varepsilon}}$ and \n$\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\frac{3}{4r_{\\varepsilon}}$.\n\n\\item[\\textbf{Case 3}: $\\chi_0=\\chi_F$.] By Hypothesis \\ref{hyp:2} we have $(r_{\\varepsilon},d_F)=1$. Thus, the intersections in \\eqref{eq15} and \\eqref{eq16} have natural density $\\frac{1}{2r_{\\varepsilon}}$ and $0$ respectively. It follows that $\\delta(\\mathbb{P'}_{>0}(\\zeta))=\\frac{3}{4r_{\\varepsilon}}$ and $\\delta(\\mathbb{P'}_{<0}(\\zeta))=\\frac{1}{4r_{\\varepsilon}}$.\n\\end{description}\nwhich finishes the proof of Lemma \\ref{lem:3}.\n\\end{proof}\n\nWe proceed now to prove Theorem \\ref{thm:3}.\n\\begin{proof}[Proof of Theorem \\ref{thm:3}]\nFix $\\phi\\in[0,\\pi)$, and pick $\\zeta$ a root of unity belonging to $\\mathrm{Im}(\\varepsilon)$. We first study the oscillatory behavior of the sequence $\\{a(tp^2)\\}_{p\\in\\mathbb{P}}$. To this end, two cases we shall need to consider.\n\\begin{description}\n\n\\item[\\textbf{Case 1}: $\\mathrm{arg}(\\zeta)\\not\\equiv  \\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.] The sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p,\\varepsilon(p)=\\zeta}$ is not trivial, and we may write\n$$\n{\\Re e\\,}(a(tp^2)e^{-i\\phi})=\\frac{a(tp^2)}{\\zeta}{\\Re e\\,}(\\zeta e^{-i\\phi}).\n$$\nBy Theorem \\ref{thmCMST} and \\ref{thmST}, it follows that the sequence $\\left(\\frac{a(tp^2)}{\\zeta}\\right)_{p,\\varepsilon(p)=\\zeta}$\nchanges sign infinitely often. Hence the sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p}$ changes sign infinitely often as $p$ varies over primes satisfying $\\varepsilon(p)=\\zeta$.\n\\item[\\textbf{Case 2}: $\\mathrm{arg}(\\zeta)\\equiv  \\phi\\pm\\frac{\\pi}{2}\\pmod{2\\pi}$.] The sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_p$ is trivial when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta$.\n\\end{description}\n\nWhat we have just proved is that for any root of unity $\\zeta$ such that $\\zeta\\in\\mathrm{Im}(\\varepsilon)$, the sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p}$ when $p$ runs over primes satisfying $\\varepsilon(p)=\\zeta$  either changes sign infinitely often or is trivial. Consequently, for each $\\phi\\in [0,\\pi)$ either the sequence $\\{{\\Re e\\,}(a(tp^2)e^{-i\\phi})\\}_{p\\in\\mathbb{P}}$\nis trivial or changes sign infinitely often. \n\nFor the purpose to calculate the density of the sets $P_{>0}(\\phi)$ and $P_{<0}(\\phi)$, we shall restrict ourselves to the case when $\\mathrm{Sh}_t(f)$ is not of CM-type, as the argument in the CM situation is entirely analogous. First note that\n$$P_{> 0}(\\phi)=\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{ \\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})>0}}\\mathbb{P'}_{>0}(\\zeta)\\bigsqcup\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{ \\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})<0}}\\mathbb{P'}_{<0}(\\zeta),$$\nand\n$$P_{< 0}(\\phi)=\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})<0}}\\mathbb{P'}_{>0}(\\zeta)\\bigsqcup\\coprod_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})>0}}\\mathbb{P'}_{<0}(\\zeta),$$\nup to finitely many primes. \n\nThese, together with Lemma \\ref{lem:3} yields\n\\begin{eqnarray*}\n\\delta\\left(P_{>0}(\\phi)\\right) &=& \\lim_{x\\to\\infty}\\sum_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})>0}}\\dfrac{\\pi_{>0}(x,\\zeta)}{\\pi(x)}+\\lim_{x\\to\\infty}\\sum_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\text{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})<0}}\\dfrac{\\pi_{<0}(x,\\zeta)}{\\pi(x)},\\\\\n  &=& \\frac{1}{2}\\sum_{\\mycom{\\zeta,\\text{root of unity}}{\\zeta\\in\\mathrm{Im}(\\varepsilon) ,{\\Re e\\,}(\\zeta e^{-i\\phi})\\neq 0}}\\frac{1}{r_{\\varepsilon}},\\\\\n  &=&\\frac{\\delta(P_{\\neq 0}(\\phi))}{2}.\n\\end{eqnarray*}\nIn a similar way, it can be shown that $\\delta\\left(P_{<0}(\\phi)\\right) =\\frac{\\delta(P_{\\neq 0}(\\phi))}{2}$, which concludes the proof.\n\\end{proof}\n\nOur next objective is to investigate the oscillatory behavior of the sequence $\\{a(tp^{2\\nu})\\}_{\\nu\\in\\mathbb{N}}$. We shall prove the following.\n\\begin{thm}\\label{thm:4}\nLet $f\\in S_{k+1\/2}(N,\\varepsilon)$ be a cuspidal Hecke eigenform of half integral weight, and \n$$\nf(z)=\\sum_{n\\ge 1}a(n)q^n\\quad z\\in \\mathcal{H}\n$$\nits expansion at $\\infty$. Let $t$ be a square-free integer such that $a(t)\\ne 0$. For all but finitely many  primes $p$ the sequence $\\{a(tp^{2\\nu})\\}_{\\nu\\in\\mathbb{N}}$ is oscillatory.\n\\end{thm}\n\\begin{proof}\nApplying the M\\\"obius inversion formula to \\eqref{eq8}, we derive that\n$$\na(tn^2)=\\sum_{d |n} \\mu(d)\\varepsilon_{t,N}(d)d^{k-1} A_t\\left(\\frac{n}{d}\\right).\n$$\nFor $n=p^{\\nu}$, with $\\nu\\in\\mathbb{N}$ ($p\\nmid N$ a prime), it follows that\n\\begin{equation}\\label{eq18}\na(tp^{2\\nu})=A_t(p^{\\nu})-p^{k-1}\\varepsilon_{t,N}(p) A_t(p^{\\nu-1})\n\\end{equation}\nDividing \\eqref{eq18} by $\\varepsilon(p)^\\nu$, we obtain\n$$\n\\dfrac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}=\\dfrac{A_t(p^{\\nu})}{\\varepsilon(p)^\\nu}-\\chi_0(p)p^{k-1}\\dfrac{A_t(p^{\\nu-1})}{\\varepsilon(p)^{\\nu-1}},\n$$\nhence $\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\in\\mathbb{R}$. Thus, we may write\n$$\n{\\Re e\\,}(a(tp^{2\\nu})e^{-i\\phi})=\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}{\\Re e\\,}(\\varepsilon(p)^{\\nu}e^{-i\\phi}),\n$$\nfor each $\\phi\\in[0,\\pi)$. We shall have established the theorem if we prove that  for all but finitely many primes $p$ the sequence $\\left(\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\right)_{\\nu\\in\\mathbb{N}}$, changes sign infinitely often. To this end, we shall follow \\cite[Proof of Theorem 2.2]{bruinier}.\n\n\nAssume, for the sake of contradiction that there exist infinitely many primes $p$ such that the sequence $\\left(\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\right)_{\\nu\\in\\mathbb{N}}$ does not changes sign infinitely often. Let $\\lambda_p$ denote the $p$-th Hecke eigenvalue of $f$. Since \n$$\nT(p)\\mathrm{Sh}_t(f)=\\mathrm{Sh}_t(T(p^2)f),\n$$\nit follows that the $p$-th Hecke eigenvalue of $\\mathrm{Sh}_t(f)$ is $\\lambda_p$, where $T(p^2)$ is the Hecke operator on $S_{k+1\/2}(N,\\varepsilon)$ and $T(p)$ is the Hecke operator\non $S_{2k}(N\/2,\\varepsilon^2)$. By \\cite[Corolary 1.8]{Shi} we have \n\\begin{equation}\\label{eq19}\n\\sum_{\\nu\\ge 0}a(tp^{2\\nu})X^\\nu=a(t)\\dfrac{1-\\varepsilon_{N,t}(p)p^{k-1}X}{1-\\lambda_pX+\\varepsilon(p)^2p^{2k-1}X^2},\n\\end{equation}\nwrite \n$$1-\\lambda_pX+\\varepsilon(p)^2p^{2k-1}X^2=(1-\\alpha_p X)(1-\\beta_p X).$$ \nReplacing $X=\\varepsilon(p)^{-1}p^{-s}$ ($s\\in\\mathbb{C}$) in \\eqref{eq19} we get\n\\begin{equation}\\label{eq20}\n\\sum_{\\nu\\ge0} a(tp^{2\\nu})\\varepsilon(p)^{-\\nu}p^{-s\\nu}=a(t)\\dfrac{1-\\chi_{0}(p)p^{k-1-s}}{(1-\\alpha_{p}' p^{-s})(1-\\beta_{p}'p^{-s})},\n\\end{equation}\nwhere  $\\alpha_{p}'=\\alpha_p \\varepsilon(p)^{-1}$ and $\\beta_{p}'=\\beta_p \\varepsilon(p)^{-1}$. \n\n Let $p$ be a prime such that $\\frac{a(tp^{2\\nu})}{\\varepsilon(p)^\\nu}\\ge 0$ for all but finitely many $\\nu \\ge 0$. Thus, by Landau's theorem \\cite[ pp. 697--699]{Landau}, the series in the left-hand side of \\eqref{eq20} either (a) converges for all $s\\in\\mathbb{C}$ or (b) has a singularity at the real point of its line of convergence. It is clear that the alternative (a) cannot occur, since the right-hand side of \\eqref{eq20} has a pole for $p^{s}=\\alpha'_p$ or $p^{s}=\\beta'_p.$\nThus the alternative (b) must hold, therefore $\\alpha'_p$ or $\\beta'_p$ must be real. On the other hand since\n $\\frac{\\lambda_p}{\\varepsilon(p)}\\in\\mathbb{R}$, it follows $\\alpha_p'=\\overline{\\beta_p'}$. Moreover, by Deligne's theorem \\cite[Theorem 8.2]{Deligne}\nwe have $|\\alpha'_p|=|\\beta'_p|=p^{k-1\/2}$. Consequently\n$$\\lambda_p=\\pm 2 p^{k-1\/2}\\varepsilon(p),$$\nhence $\\mathbb{Q}(\\sqrt{p})\\subset K_f$, where $K_f$ denotes the field generated   over $\\mathbb{Q}$ by $\\lambda_p$ ($p$ runs over primes numbers) and all the values of $\\varepsilon$. Therefore, by our hypothesis $K_f$ has infinitely many quadratic subfields, this is in contradiction with the fact that $K_f$ is a number field. Consequently, for all but finitely many primes $p$ and each $\\phi\\in [0,\\pi)$ the sequence $\\{{\\Re e\\,}(a(tp^{2\\nu})e^{-i\\phi}\\}_{\\nu\\in\\mathbb{N}}$ , changes sign infinitely often. \n\\end{proof}\n\\begin{rem}\nIt seems likely that we can prove similar results to \\cite[Theorem 3]{Kohnen} for the sequence $\\{{\\Re e\\,}(a(tp^{2\\nu})e^{-i\\phi})\\}_{\\nu\\in\\mathbb{N}}$. However, we have not checked this as yet.\n\\end{rem}\n\\section{Concluding Remarks}\nLet $k,N$ be natural numbers and $\\varepsilon$ be a Dirichlet character modulo $N$ assume that $N$ be an odd and square-free integer. We write $S^{+}_{k+1\/2}(4N,\\varepsilon)$ for the Kohnen's plus space (cf. \\cite{Kohnen1982}). \n\nLet $f$ be a cusp form of half integral weight belonging to $S^{+}_{k+1\/2}(4N,\\varepsilon)$, we let $a(n)$ to denote its $n$-th Fourier coefficient. Motivated by Theorem \\ref{thm:3} and numerical calculations, it seems reasonable to conjecture that, for each $\\phi\\in [0,\\pi)$ \n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{n\\le x\\; :\\; {\\Im m\\,}\\{a(n)e^{-i\\phi}\\}\\gtrless 0\\}}{\\#\\{n\\le x \\; :\\; {\\Im m\\,}\\{a(n)e^{-i\\phi}\\} \\neq 0\\}}=\\frac{1}{2},\n$$\n\n$$ \n\\lim_{x\\to\\infty}\\dfrac{\\#\\{n\\le x\\; :\\; {\\Re e\\,}\\{a(n)e^{-i\\phi}\\}\\gtrless 0\\}}{\\#\\{n\\le x \\; :\\; {\\Re e\\,}\\{a(n)e^{-i\\phi}\\} \\neq 0\\}}=\\frac{1}{2}.\n$$\n\nIt may be noted that one can get similar statements on the imaginary part in Theorem \\ref{thm:1}, \\ref{thm:2} and \\ref{thm:3} by a rotation around $\\pi\/2$.\n\nWe believe that these results (Theorem \\ref{thm:1}, \\ref{thm:2}, \\ref{thm:3} and the above conjecture) should extend to any totally real number field by the approach taken in the present paper.\n\n\n\\section*{Acknowledgments}\nThe author is greatly grateful to Francesc Fit\\'e for a helpful conversation. He also wishes to thank Gabor Wiese for his valuable comments on the first draft of this work as well as Ilker Inam for providing him with some data for numerical experiments. Thanks are also due to the referee for his careful reading and their helpful comments which improve the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\\subsubsection*{Acknowledgments}\nThe authors thank the ArTeCS Group from Universidad Complutense de Madrid for letting use their Xeon Phi KNL system.\n\n\n\\bibliographystyle{splncs03}\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nKNL is the second generation of Xeon Phi family and features new technologies in SIMD execution and memory access.\nIn this paper, we have evaluated a set of programming and optimization techniques for these processors taking the FW algorithm as a representative case study of graph and memory-bound applications. Among the main contributions of this research we can summarize:\n\\begin{itemize}\n\\item Blocking technique not only improved performance but also allowed us to apply a coarse-grain workload distribution in the parallel implementation.\n\n\\item SIMD exploitation was crucial to achieve top performance. In particular, the serial version run 2.9$\\times$, 6$\\times$ and 15.5$\\times$ faster using the SSE, AVX2 and AVX-512 extensions, respectively.\n\\item Aligning memory accesses and loop unrolling also showed significant speedups.\n\\item A single thread per core was enough to get maximal performance. In addition, \\emph{scatter} and \\emph{balanced} affinities provided extra performance.\n\\item Besides keeping portability, guided vectorization led to slightly better performance than the intrinsic counterpart, running upto 1.03$\\times$ faster. \n\\item MCDRAM usage demonstrated to be an efficient strategy to tolerate high-bandwidth demands with practically null programmer intervention, even when the dataset largely exceeded the MCDRAM size. In particular, it produced an average speedup of 9.8$\\times$ and a maximum speedup of 15.5$\\times$\n\\end{itemize}\n\nAs future work, we consider evaluating programming and optimization techniques in other cluster and memory modes as a way to extract more performance.\n\n\n\n\n\\section{Floyd-Warshall Algorithm}\n\\label{sec:floyd}\n\nThe FW algorithm uses a dynamic programming approach to compute the all-pairs shortest-paths\nproblem on a directed graph~\\cite{Floyd,Warshall}. \nThis algorithm takes as input a $N \\times N$ distance matrix $D$, where $D_{i,j}$ is initialized with the original distance from node \\emph{i} to node \\emph{j}. FW runs for $N$ iterations and at \\emph{k}-th iteration it evaluates all the possible paths between each pair of vertices from \\emph{i} to \\emph{j} through the intermediate vertex \\emph{k}. As a result, FW produces an updated matrix $D$, where $D_{i,j}$ now contains the shortest distance between nodes \\emph{i} and \\emph{j}. Besides, an additional matrix $P$ is generated when the reconstruction of the shortest path is required. $P_{i,j}$ contains the most recently added intermediate node between \\emph{i} and \\emph{j}. Figure~\\ref{fig:floyd_naive} exhibits the naive FW algorithm.\n\n\n\n\n\\begin{figure}[b]\n    \\centering\n    \\begin{minipage}{0.53\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{floyd_naive}\n        \\caption{\\label{fig:floyd_naive} Naive Floyd-Warshall Algorithm}\n    \\end{minipage}\\hfill\n    \\begin{minipage}{0.43\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{floyd_blocked}\n        \\caption{\\label{fig:floyd_blocked} Schematic representation of the blocked Floyd-Warshall Algorithm}\n    \\end{minipage}\n\\end{figure}\n\\section{Implementation}\n\\label{sec:implementation}\n\nIn this section, we address the optimizations performed on the Intel Xeon Phi KNL processor. First of all, we developed a serial implementation following the naive version described in Figure~\\ref{fig:floyd_naive}, as this implementation will work as baseline. Next, we optimized the serial version considering data locality and data level parallelism. Finally, we introduced thread level parallelism exploiting the OpenMP programming model to obtain a multi-threaded implementation.\n\n\\subsection{Data Locality}\n\\label{subsec:data_locality}\n\nTo improve data locality, the FW algorithm can be blocked~\\cite{floyd_blocked2}. Unfortunately, the three loops can not be interchanged in free manner due to the data dependencies from one iteration to the next in the \\emph{k}-loop (just \\emph{i} and \\emph{j} loops can be done in any order). However, under certain conditions, the \\emph{k}-loop can be put inside the \\emph{i}-loop and \\emph{j}-loop, making blocking possible. \nThe distance matrix  $D$ is partitioned into blocks of size $BS \\times BS$, so that there are $(N\/BS)^2$ blocks. The computations involve $R=N\/BS$ rounds and each round is divided into four phases based on the data dependency among the blocks:\n\\begin{enumerate}\n\\item Update the block \\emph{k,k} ($D^{k,k}$) because it is self-dependent.\n\\item Update the remaining blocks of the \\emph{k}-th row because each of these blocks depends on itself and the previously computed $D^{k,k}$.\n\\item Update the remaining blocks of the \\emph{k}-th column  because each of these blocks depends on itself and the previously computed $D^{k,k}$.\n\\item Update the rest of the matrix blocks as each of them depends on the \\emph{k}-th block of its row and the \\emph{k}-th block of its column.\n\\end{enumerate}\nIn this way, we satisfy all dependencies from this algorithm. Figure~\\ref{fig:floyd_blocked} shows a schematic representation of a round computation and the data dependences among the blocks while Figure~\\ref{fig:floyd_blocked_alg} presents the corresponding pseudo-code.\n\n\n\\begin{figure}[t]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{floyd_blocked_alg}\n\\par\\end{centering}\n\\caption{\\label{fig:floyd_blocked_alg} Blocked Floyd-Warshall algorithm.}\n\\end{figure}\n\n\\subsection{Data Level Parallelism}\n\\label{subsec:data_level_parallelism}\n\nThe innermost loop of FW\\_BLOCK code block from Figure~\\ref{fig:floyd_blocked_alg} is clearly the most computationally expensive part of the algorithm. In that sense, this loop is the best candidate for vectorization. The loop body is composed of an \\emph{if} statement that involves one addition, one comparison and (may be) two assign operations. Unfortunately, the compiler detects false dependencies in that loop and is not able to generate SIMD binary code. For that reason, we have explored two SIMD exploitation approaches: (1) guided vectorization through the usage of the OpenMP 4.0 \\emph{simd} directive  and (2) intrinsic vectorization employing the AVX-512 extensions. The guided approach simply consists of inserting the \\emph{simd} directive to the innermost loop of FW\\_BLOCK code block (line 4). On the opposite sense, the intrinsic approach consists of rewriting the entire loop body. Figures~\\ref{fig:fw_block_guided} and~\\ref{fig:fw_block_intrinsic} show the pseudo-code for FW\\_BLOCK implementation using guided and manual vectorization, respectively. In order to accelerate SIMD computation with 512-bit vectors, we have carefully managed the memory allocations so that distance and path matrices are 64-byte aligned. In the guided approach, this also requires adding the \\emph{aligned} clause to the \\emph{simd} directive. \n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{minipage}{0.43\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{FW_BLOCK_guided}\n        \\caption{\\label{fig:fw_block_guided} Pseudo-code for FW\\_BLOCK implementation using guided vectorization}\n    \\end{minipage}\\hfill\n    \\begin{minipage}{0.53\\textwidth}\n        \\centering\n        \\includegraphics[width=1\\textwidth]{FW_BLOCK_intrinsic}\n        \\caption{\\label{fig:fw_block_intrinsic} Pseudo-code for FW\\_BLOCK implementation using intrinsic vectorization}\n    \\end{minipage}\n\\end{figure}\n\n\n\n\n\\subsection{Loop Unrolling}\n\\label{subsec:loop_unrolling}\n\nLoop unrolling is another optimization technique that helped us to improve the code performance. Fully unrolling the innermost loop of FW\\_BLOCK code block was found to work well. Unrolling the \\emph{i}-loop of the same code block once was also found to work well.\n\n\\subsection{Thread Level Parallelism}\n\\label{subsec:thread_level_parallelism}\n\nTo exploit parallelism across multiple cores, we have implemented a multi-threaded version of FW algorithm based on OpenMP programming model. A \\emph{parallel} construct is inserted before the loop of line 13 in Figure~\\ref{fig:floyd_blocked_alg} to create a parallel block. To respect data dependencies among the block computations, the work-sharing constructs must be carefully inserted. At each round, phase 1 must be computed before the rest. So a \\emph{single} construct is inserted to enclose line 16. Next, phases 2 and 3 must be computed before phase 4. As these blocks are independent among them, a \\emph{for} directive is inserted before the loops of lines 18 and 22. Besides, \na \\emph{nowait} clause is added to the phase 2 loop to alleviate the thread idling. Finally, another \\emph{for} construct is inserted before the loop of line 26 to distribute the remaining blocks among the threads.\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe power consumption problem represents one of the major obstacles for Exascale systems design. As a\nconsequence, the scientific community is searching for different ways to improve power efficiency of High Performance Computing (HPC) systems~\\cite{Giles2014}. One recent trend to increase compute power and, at the same time, limit power consumption of these systems lies in adding accelerators, like NVIDIA\/AMD graphic processing units (GPUs), or Intel Many Integrated Core (MIC) co-processors. These manycore devices are capable of achieving better FLOPS\/Watt ratios than traditional CPUs. For example, the number of Top500~\\cite{Top500} systems using accelerator technology grew from 54 in June 2013 to 91 in June 2017. In the same period, the number of systems based on accelerators increased from 55 to 90 on the Green500 list~\\cite{Green500}.\n\nRecently, Intel has presented the second generation of its MIC architecture (branded Xeon Phi), codenamed Knigths Landing (KNL).  Among the main differences of KNL regarding its predecessor Knights Corner (KNC), we can find the incorporation of AVX-512 extensions, a remarkable number of vector units increment, a new on-package high-bandwidth memory (HBM) and the ability to operate as a standalone processor. Even though optimizing applications on CPUs, GPUs and KNC Xeon Phi's has been largely studied in the last years, accelerating applications on KNL processors is still a pending task due to its recent commercialization. In that sense, the new features in KNL processors require the revision of programming and optimization techniques for these devices. \n\nIn this work, we selected the Floyd-Warshall (FW) algorithm as a representative case study of graph and memory-bound applications. This algorithm finds the shortest paths between all pairs of vertices in a graph and occurs in domains of communication networking~\\cite{Floyd_networking}, traffic routing~\\cite{floyd_traffic}, bioinformatics~\\cite{floyd_bioinformatics}, among others. FW is both computationally and spatially expensive since it requires $O(n^3)$ operations and $O(n^2)$ memory space, where $n$ is the number\nof vertices in a graph. Starting from the default serial version, we show how data, thread and compiler level optimizations help the parallel implementation to reach 338 GFLOPS.\n\n\n\nThe rest of the present paper is organized as follows. Section~\\ref{sec:xeon_phi} briefly introduces the Intel Xeon Phi KNL architecture while Section~\\ref{sec:floyd} presents the FW algorithm. Section~\\ref{sec:implementation} describes our implementation. In Section~\\ref{sec:results} we analyze performance results while Section~\\ref{sec:related_works} discusses related works. Finally, Section~\\ref{sec:conclusion} outlines conclusions and future lines of work.\n\n\n\n\\section{Intel Xeon Phi Knights Landing}\n\\label{sec:xeon_phi}\n\nKNL is the second generation of the Intel Xeon Phi family and the first capable of operating as a standalone processor. The KNL architecture is based on a set of \\emph{Tiles} (up to 36) interconnected by a 2D mesh. Each Tile includes 2 cores based on the out-of-order Intel's Atom micro-architecture (4 threads per core), 2 Vector Processing Units (VPUs) and a shared L2 cache of 1 MB. These VPUs not only implement the new 512-bit AVX-512 ISA but they are also compatible with prior vector ISA's such as SSE\\emph{x} and AVX\\emph{x}. \nAVX-512 provides 512-bit SIMD support, 32 logical registers, 8 new mask registers for vector predication, and gather and scatter instructions to support loading and storing sparse data. As each\nAVX-512 instruction can perform 8 double-precision (DP) operations\n(8 FLOPS) or 16 single-precision (SP)  operations (16 FLOPS), the peak performance is over 1.5 TFLOPS in DP and 3 TFLOPS in SP, more than two times higher than that of the KNC. It is also more energy efficient than its predecessor~\\cite{KNLbook}.\n\nOther significant feature of the KNL architecture is the inclusion of an in-package HBM called MCDRAM. This special memory offers 3 operating modes: \\emph{Cache}, \\emph{Flat} and \\emph{Hybrid}. In Cache mode, the MCDRAM is used like an L3 cache, caching data from the DDR4 level\nof memory. Even though application code remains unchanged, the MCDRAM can suffer lower performance rates.  In Flat mode, the MCDRAM has a physical addressable space offering the highest bandwidth\nand lowest latency. However, software modifications may be required in order to use both the DDR and the MCDRAM in the same application. Finally, in the \\emph{Hybrid mode}, HBM is divided in two parts: one part in \\emph{Cache mode} and one in \\emph{Flat mode}~\\cite{knl_best_practice_guide}.\n\n\nFrom a software perspective, KNL supports  parallel programming models used traditionally on HPC systems such as OpenMP or MPI. This fact represents a strength of this platform since it simplifies code development and improves portability over other alternatives based on accelerator specific programming languages such as CUDA or OpenCL.\n However, to achieve high performance, programmers should attend to:\n\\begin{itemize}\n\\item the efficient exploitation of the memory hierarchy, especially\nwhen handling large datasets, and\n\\item how to structure the computations to take advantage of the VPUs.\n\\end{itemize}\nAutomatic vectorization is obviously the easiest programming way to exploit VPUs. However, in most cases the compiler is unable to generate SIMD binary code since it can not detect free data dependences into loops. In that sense, SIMD instructions are supported in KNL processors through the use of guided compilation or hand-tuned codification with intrinsic instructions~\\cite{KNLbook}.  On one hand, in guided vectorization,  the programmer indicates the compiler (through the insertion of tags) which loops are independent and their memory pattern access. In this way, the compiler is able to generate SIMD binary code preserving the program portability. On the other hand, intrinsic vectorization usually involves rewriting most of the corresponding algorithm. The programmer gains in control at the cost of losing portability. Moreover, this approach also suggests the inhibition of other compiler loop-level optimizations. Nevertheless, it is the only way to exploit parallelism in some applications with no regular access patterns or loop data dependencies which can be hidden by recomputing techniques~\\cite{Culler97}. \n\n\n\n\n\n\\section{Related Works}\n\\label{sec:related_works}\n\nDespite its recent commercialization, there are some works that evaluate KNL processors. In that sense, we highlight~\\cite{Rosales2016_mcdram} that presents a study of the performance differences observed when using the three MCDRAM configurations available in combination with the three possible memory access or cluster modes. Also, Barnes et al.~\\cite{KNL_NERSC} discussed the lessons learned from optimizing a number of different high-performance applications and kernels. Besides, Haidar et al.~\\cite{Haidar2016_KNL} proposed and evaluated several optimization techniques for different matrix factorizations methods on many-core systems.\n\nObtaining high-performance in graph algorithms is usually a difficult task since they tend to suffer from irregular dependencies and large space requirements. Regarding FW algorithm, there are many works proposed to solve the all-pairs shortest paths problem on different harwdare architectures. However, to the best of the authors knowledge, there are no related works with KNL processors.\nHan and Kang~\\cite{floyd_han} demonstrated that exploiting SSE2 instructions led to 2.3$\\times$-5.2$\\times$ speedups over a blocked version. Bondhugula et al.~\\cite{floyd_fpga} proposed a tiled parallel implementation using Field Programmable Gate Arrays. In the field of GPUs, we highlight the work of Katz and Kider~\\cite{floyd_katz_gpu}, who proposed a shared memory cache efficient implementation to handle graph sizes that are inherently larger than the DRAM memory available on the device. Also, Matsumoto et al.~\\cite{floyd_matsumoto_cpu_gpu} presented a blocked algorithm for hybrid CPU-GPU systems aimed to minimize host-device communication. Finally, Hou et al.~\\cite{floyd_knc} evaluated different optimization techniques for Xeon Phi KNC coprocessor. Just as this study, they found that blocking and vectorization are key aspects in this problem to achieve high performance. Also, guided vectorization led to better results than the manual approach, but with larger performance differences. Contrary to this work, their implementation benefited from using more than one thread per core. However, as stated before, there are significant architectural differences between these platforms that support this behavior.\n\n\n\n\n\\section{Experimental Results}\n\\label{sec:results}\n\n\\subsection{Experimental Design}\nAll tests have been performed on an Intel server running CentOS 7.2 equipped with a Xeon Phi 7250 processor 68-core 1.40GHz (4 hw thread per core and 16GB MCDRAM memory) and 48GB main memory. The processor was run in \\emph{Flat} memory mode and \\emph{Quadrant} cluster mode.\n\nWe have used Intel's ICC compiler (version 17.0.1.132) with the \\emph{-O3} optimization level. To generate explicit AVX2 and AVX-512 instructions, we employed the \\emph{-xAVX2} and \\emph{-xMIC-AVX512} flags, respectively. Also, we used the \\emph{numactl} utility to exploit MCDRAM memory (no source code modification is required). Besides, different workloads were tested: \\emph{N} = \\{4096, 8192, 16384, 32768, 65536\\}. \n\n\\subsection{Performance Results}\n\\label{sec:perf-knl}\n\nFirst, we evaluated the performance improvements of the different optimization techniques applied to the naive serial version, such as blocking (\\emph{blocked}), data level parallelism (\\emph{simd}, \\emph{simd (AVX2)} and \\emph{simd (AVX-512)}), aligned access (\\emph{aligned}) and loop unrolling (\\emph{unrolled}). Table~\\ref{tab:performance_serial} shows the execution time (in seconds) of the different serial versions when \\emph{N}=4096. As it can be observed, blocking optimization reduces execution time by 5\\%. \nRegarding the block size, 256 $\\times$ 256 was found to work best. In the most memory demanding case of each round (phase 4), four blocks are loaded into the cache (3 distance blocks and 1 path block). The four blocks requires 4 $\\times$ 256 $\\times$ 256 $\\times$ 4 bytes = 1024 KB = 1MB, which is exactly the L2 cache size.\n\nAs stated in Section~\\ref{subsec:data_level_parallelism}, the compiler is not able to generate SIMD binary code by itself in the blocked version. Adding the corresponding \\emph{simd} constructs to the blocked version reduced the execution time from 572.66 to 204.52 seconds, which represents a speedup of 2.8$\\times$. However, AVX-512 instructions can perform 16 SP operations at the same time. After inspecting the code at assembly level, we realized that the compiler generates SSE\\emph{x} instructions by default. As SSE\\emph{x} can perform 4 SP operations at the same time, the 2.8$\\times$ speedup has more sense since not all the code can be vectorized. Next, we re-compiled the code including the \\emph{-xAVX2} and \\emph{-xMIC-AVX512} flags to force the compiler to generate AVX2 and AVX-512 SIMD instructions, respectively. AVX2 extensions accelerated the blocked version by a factor of 5.8$\\times$ while AVX-512 instructions achieved an speedup of 15.5$\\times$. So, it is clear that this application benefits from larger SIMD width. In relation to the other optimization techniques employed, we have found that the \\emph{simd (AVX-512)} implementation runs 1.11$\\times$ faster when aligning memory accesses in AVX-512 computations (\\emph{aligned}). Additionally, applying the loop unrolling optimization to the \\emph{aligned} version led to higher performance, gaining a 1.45$\\times$ speedup. In summary, we achieve a 26.3$\\times$ speedup over the naive serial version through the combination of the different optimizations described.\n\n\\begin{table} [t!]\n\\centering\n\\caption{\\label{tab:performance_serial} Execution time (in seconds) of the different optimization techniques applied to the naive serial version when \\emph{N}=4096.}\n\\begin{tabular*}{12cm}{@{\\extracolsep{\\fill}}>{\\centering}p{1.2cm}>{\\centering}m{1.2cm}>{\\centering}m{1.4cm}>{\\centering}p{2.3cm}>{\\centering}m{2.7cm}>{\\centering}m{1.3cm}>{\\centering}m{1.4cm}}\n\\hline \n\\noalign{\\vskip0.2cm}\n\\emph{naive} & \\emph{blocked} & \\emph{simd} & \\emph{simd (AVX2)} & \\emph{simd (AVX-512)} & \\emph{aligned} & \\emph{unrolled}\\tabularnewline\n\\hline \n\\noalign{\\vskip0.2cm}\n602.8 & 572.66 & 204.52 & 100.47 & 36.95 & 33.28 & 22.95\\tabularnewline\n\\hline \n\\end{tabular*}\n\\end{table}\n\nTaking the optimized serial version, we developed a multi-threaded implementation as described in Section~\\ref{subsec:thread_level_parallelism}. Figure~\\ref{fig:performance_threads_affinity} shows the performance (in terms of GFLOPS) for the different affinity types used varying the number of threads when \\emph{N}=8192. As expected, \\emph{compact} affinity produced the worst results since it favours using all threads on a core before using other cores. \\emph{Scatter} and \\emph{balanced} affinities presented similar performances improving the \\emph{none} counterpart. As the KNL processor used in this study has all its cores in the same package, \\emph{scatter} and \\emph{balanced} affinities distribute the threads in the same manner when one thread per core is assigned.\nRegarding the number of threads, using a single thread per core is enough to get maximal performance (except in \\emph{compact} affinity). This behavior is opposed to the KNC generation where two or more threads per core where required to achieve high performance. However, it should not be a surprise since the KNL cores were designed to optimize single thread performance including out-of-order pipelines and two VPUs per core.\n\nIt is important to remark that, unlike the optimized serial version, the parallel implementation used a smaller block size  since it delivered higher performance. A smaller block size allowed a finer-grain workload distribution and decreased thread idling, especially when the number of threads was larger than the number of blocks in phases 2 and 3. Another reason to decrease block size was that the L2 available space is now shared between the threads in a tile, contrary to the single threaded case. In particular, \\emph{BS}=64 was found to work best.\n\n\\begin{figure}[t!]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{performance_threads_affinity}\n\\par\\end{centering}\n\\caption{\\label{fig:performance_threads_affinity}Performance for the different affinity types used varying the number of threads  when \\emph{N}=8192.}\n\\end{figure}\n\nFigure~\\ref{fig:performance_n_mcdram} illustrates performance evolution varying workload and MCDRAM exploitation for the different vectorization approaches. For small workloads (\\emph{N} = 8192), the performance improvement is little ($\\sim$1.1$\\times$). However, MCDRAM memory presents remarkable speedups for greater workloads, even when the dataset largely exceeds the MCDRAM size (\\emph{N} = 655536). In particular, MCDRAM exploitation achieves an average speedup of 9.8$\\times$ and a maximum speedup of 15.5$\\times$. In this way, we can see how MCDRAM usage is an efficient strategy for bandwidth-sensitive applications.\n\nIn relation to the vectorization approach, we can appreciate that guided vectorization leads to slightly better performance than the intrinsic counterpart, running upto 1.03$\\times$ faster. The best performances are 330 and 338 GFLOPS for the intrinsic and guided versions, respectively. After analyzing the assembly code, we realized that this difference is caused by the prefetching instructions introduced by the compiler when\nguided vectorization is used. Unfortunately, the compiler disables automatic prefetching when code is manually vectorized.\n\n\\begin{figure}[t]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{performance_n_mcdram}\n\\par\\end{centering}\n\\caption{\\label{fig:performance_n_mcdram}Performance evolution varying workload and the MCDRAM exploitation.}\n\\end{figure}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{SectionIntroduction}\n\nIn this paper we start the program of developing a general theory of rough PDEs aiming at extending classical PDE tools such as weak solutions, a priori estimates, compactness results, duality. This is a quite unexplored territory where few tools are available, so as a start, we will content ourselves in this work with the study of linear symmetric hyperbolic systems of the form\n\\begin{equation}\n\\label{eq:basic}\n\\partial_t f + a\\nabla f = 0,\n\\end{equation}\nwhere $f$ is an $\\RR^N$-valued space-time distribution on $\\RR_+\\times\\RR^d$, and $a :\\RR_+\\times \\RR^d \\to  \\textrm{L}(\\RR^d, \\RR^{N\\times N})$ is a $N\\times N$ matrix-valued family of time--dependent vector fields in $\\RR^d$. This setting includes as a particular case scalar transport equations. Moreover we restrict our attention to the case where the matrix-valued vector field $a$ is only a distribution in the time variable, rather than a regular bounded function. We however retain some smoothness assumption in the space variable, as expected from the fact that general transport equations do not possess the regularisation properties needed to drive them with space-time irregular signals. Even in the classical setting it is known that non-regular coefficients  can give rise to non-uniqueness of weak solutions~\\cite{FlandoliMilano}.\n\n\\smallskip\n\nWhen $a$ is only a distribution in the time variable the above weak formulation is not available since in the classical setting solutions are considered in spaces like $C([0,T],\\LL^2(\\RR^d))$ for general symmetric systems  or $L^\\infty(\\RR\\times \\RR^d)$ for scalar transport equation. In this case, the product $a\\nabla f$ is not well-defined, not even in a distributional setting. Rough paths have their origin in the need to handle such difficulties in the case of ordinary differential equations driven by distribution valued signals \\cite{Lyons98,Lyons2002,Lyons2007,Friz2010}. Controlled rough paths have been introduced in \\cite{Gubinelli04} as a setting for considering more general problems; they were used successfully in the study of some stochastic partial differential equations~\\cite{Bessaih2005,GT,Deya2012,Hairer2011,Gubinelli2012a,Hairer2012,Hairer2013},  including the remarkable solution by Hairer of the KPZ equation \\cite{HairerKPZ}. \n\n\\smallskip\n\nThese developments have ultimately lead to the notion of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski~\\cite{GIP} and to Hairer's general theory of regularity structures~\\cite{HairerRegularity},  providing a  framework for the analysis of non-linear operations on distributions. Despite their successes, these new tools and methods are somehow designed to deal with a prescribed class of singular PDEs which is far from exhausting the set of all interesting singular PDEs. \n\n\\smallskip\n\nPDEs with irregular signals have been studied using directly rough path methods also by Friz and co-authors~\\cite{Caruana2009, CaruanaFrizOberhauser, FrizOberhauser1, FrizOberhauser2, FrizGess}. They have developed an approach to some  fully non-linear PDEs and conservation laws driven by rough signals by interpreting these equations  as transformations of classical PDEs, generalising the method of characteristics. Subsequently a combination of rough path stability and PDE stability allows to go from smooth signal to the wider class of signals described by rough paths. Entropy solutions to scalar conservation laws with rough drivers have also been analysed also by P.-L. Lions, Souganidis and coauthors~\\cite{LionsPerthameSouganidis1,LionsPerthameSouganidis2, GessSouganidis}. A major drawback of this otherwise effective approach is that there is no intrinsic notion of solution to the PDE and that the study of the properties of the PDE has to be done on a global level.\n\n\\smallskip\n\nIn recent works intrinsic notions of weak solution to rough PDEs have been proposed by Tindel, Gubinelli and Torecilla~\\cite{GTT} for viscosity solutions,  Catellier~\\cite{CatellierTransport} for weak solutions to linear transport equations (see also Hu and Le~\\cite{HuLe} for classical solutions to transport equations) and more recently by Diehl, Friz and Stannat~\\cite{DiehlFrizStannat} for a general class of parabolic equations. All these notions are based on a weak formulation of the equation where the irregularity of some data is taken into account via the framework of controlled paths introduced in~\\cite{Gubinelli04}. However in all these papers explicit formulas involving the flow of rough characteristics play an important role, and this sets apart the study of rough PDEs with respect to the study of weak solutions to more regular PDEs. One of the main motivations of our investigations is an effort to understanding what kind of robust arguments can be put forward in the study of the a priori regularity of distributions satisfying certain rough PDEs formulated in the language of controlled paths. Extensions to regularity structures or paracontrolled distributions will be considered in forthcoming work. \n\n\\bigskip\n\nWe study equation \\eqref{eq:basic} by working in the technically easier setting of controlled paths. To motivate our formalism, note that a formal integration of the weak formulation \\eqref{eq:basic} over any time interval $[s,t]$, gives an equation of the form\n$$\nf_t = f_s + \\int_s^t V_r f_r dr, \n$$ \nwhere $V_r = a_r\\nabla$ is a matrix-valued vector-field and $f_r(x) = f(r,x)$, is a convenient notation of the distribution $f$ evaluated at time $r$, assuming this make sense. An expansion for the time evolution of $f$ is obtained by iterating the above equation, and reads\n\\begin{equation}\n\\label{eq:basic-increments}\nf_t = f_s + A^1_{ts} f_s + A^2_{ts} f_s + R_{ts},\n\\end{equation}\nwhere\n$$\nA^1_{ts} = \\int_s^t V_r  dr,\\qquad\\textrm{ and }\\qquad A^2_{ts} = \\int_s^t \\int_s^r V_r V_{r'} dr' dr,\n$$\nare respectively a first order differential operator (that is a vector field) and a second order differential operator, for each $s\\leq t$. As a function of $(s,t)$, they satisfy formally \\textit{Chen's relation}\n\\begin{equation}\n\\label{eq:operator-chen}\nA^2_{ts} =A^2_{tu}+ A^2_{us} + A^{1}_{tu}A^{1}_{us}\n\\end{equation}\nfor all $0\\leq s \\leq u \\leq t$. It is a key observation of rough path theory that equation \\eqref{eq:basic-increments} can be used as a replacement for the differential or integral formulation \nof equation \\eqref{eq:basic} if the remainder term can be shown to be sufficiently small as $t-s$ goes to $0$, in a suitable sense.\n\n\\medskip\n\nWe shall call a \\emph{rough driver} an operator-valued $2$-index maps ${\\bfA}_{ts}=\\big(A^1_{ts},A^2_{ts}\\big)$ satisfying the operator Chen relation \\eqref{eq:operator-chen} and some regularity assumptions. Building on the above picture, a path with values in some Banach space where the operators $A^1_{ts}$ and $A^2_{ts}$ act, will be said to solve the rough linear equation\n$$\ndf_s = {\\bfA}(ds)\\,f_s.\n$$\nif the Taylor expansion \\eqref{eq:basic-increments} holds.\n\n\\smallskip\n\nThere is a complete theory of such equations in the case where the equation is set in a Banach algebra and the operators $A^1_{ts}$ and $A^2_{ts}$ are given by left multiplication by some elements of the algebra. It is however natural, in the present PDE setting, to consider also unbounded operators $A^1, A^2$, which makes the use of rough paths ideas non-trivial, unless we work in the analytic category or in similar topologies. \n\n\\medskip\n \nWe lay out in this work a theory of such rough linear equation driven by unbounded drivers $\\bfA$, and obtain some a priori estimates that are used to study the well--posedness of some classes of linear symmetric systems in $\\LL^2$ and of the rough transport equation in $\\LL^\\infty$. The major difficulty which has to be overcome is the lack of a Gronwall lemma for rough equations and the main contribution to this paper is to develop suitable a priori estimates on weak controlled solutions that replace the use of Gronwall lemma in classical proofs. Along the way we refine the standard theory of controlled path by introducing weighed norms compatible with the sewing map and by revisiting the theory of linear rough differential equations in the context of bounded drivers. \n\n\\medskip\n\nAs a guide for the reader, here is how we have organised our work. Section~\\ref{SectionWeightedNorms} provides a refined version of the sewing lemma that allows to keep track of the growth in time of the additive function associated with an almost-additive $2$-index map. This result is used in Section~\\ref{SectionBoundedDrivers} in the proof of the well-posed character of linear differential equations driven by the bounded rough drivers defined there. \\emph{Unbounded rough drivers} are introduced in Section~\\ref{SectionUnboundedRoughDrivers}, where some fundamental a priori estimate is proved. An $\\LL^2$ theory of rough linear equations is developed for a class of unbounded drivers, that contains as a particular example the rough linear transport equation. Our main workhorse here is a novel renormalisation lemma obtained from a tensorization argument in the line of the \"doubling of variables\" method commonly used in the setting of transport equations or conservation laws. A complete $\\LL^\\infty$ theory of rough transport equations is given in Section~\\ref{SectionLInftyTheory}.\n\n\\bigskip\n\n\\noindent \\textbf{Acknowledgments} -- The authors would like to express their gratitude to Martina Hofmanova and Mario Maurelli who discovered a major error in the first version of the paper and hinted to a possible strategy to overcome it.\n\\bigskip\n\n\\noindent \\textbf{Notations} -- We gather here for reference a number of notations that will be used throughout the text.\n\\begin{itemize}\n   \\item We shall denote by E a generic Banach space. Given two Banach spaces E and F, we denote by $\\textrm{L}(\\textrm{E},\\textrm{F})$ the set of continuous linear maps from E to F. \\vspace{0.1cm}\n   \\item We shall denote by $c$ a constant whose value may change from place to place. \\vspace{0.1cm}\n   \\item Given two positive real numbers $\\alpha,\\beta$, we shall write $\\alpha \\lesssim \\beta$ to say that $\\alpha\\leq c\\beta$, for some positive constant $c$. To indicate that this constant $c$ depends on a parameter $\\lambda$, we write $\\alpha \\lesssim_\\lambda \\beta$. \\vspace{0.1cm}\n   \\item Denote by $\\|\\cdot\\|_{\\alpha\\,;\\,k}$ the $\\alpha$-H\\\"older norm of an $E_k$-valued path, for $k\\in\\ZZ$, and by $\\|\\cdot\\|_{\\alpha\\,;\\,\\textrm{E}}$ the $\\alpha$-H\\\"older norm of an E-valued path.\n\\end{itemize}\n\n\n\\bigskip\n\n\\section{Weighted norms}\n\\label{SectionWeightedNorms}\n\nWe introduce  some weighted norms that will be useful in getting a priori estimates on the growth of solutions to the linear differential equations studied in Section~\\ref{SectionBoundedDrivers}. These norms are modelled on Picard's well-known norms\n$$\n\\llparenthesis\\, f \\,\\rrparenthesis := \\underset{t\\geq 0}{\\sup}\\,e^{-\\lambda^{-1} t}\\big| f(t) \\big|,\n$$\nintroduced in the study of ordinary differential equations in order to provide a functional setting where to get well--posedness results on the whole time interval $[0,\\infty)$, as a direct consequence of the Banach fixed point theorem, and to get as a consequence a control on the growth of the size of solutions. \n\n\\medskip\n\nLet $T$ be a possibly infinite positive time horizon. As is common in rough paths theory, we shall work with Banach space-valued multi-index maps, mainly $2$ and $3$-index maps, defined on the simplexes\n$$\n\\Big\\{(s,t)\\in [0,T)^2\\,;\\,s\\leq t\\Big\\}\\quad \\textrm{ and } \\quad \\Big\\{(s,u,t)\\in [0,T)^3\\,;\\,s\\leq u\\leq t\\Big\\}.\n$$\nWith Picard's norm in mind, we introduce a norm on the set of $2$ and $3$-index maps which captures both their H\\\"older size and their growth at infinity. \nGiven $\\lambda >0$, an increasing non-negative function $g$ defined on $\\RR_+$ and a non-negative H\\\"older exponent $\\gamma$, we define the $(\\gamma,g)$-norm of a $2$-index map $a$, and a $3$-index map $b$, by the formulae\n$$\n\\llparenthesis\\, a \\,\\rrparenthesis_{\\gamma,g} := \\underset{\\stackrel{0\\leq s<t<T}{|t-s|\\le \\lambda}}{\\sup}\\;\\,\\frac{\\big| a_{ts}\\big|}{g\\big(\\lambda^{-1} t\\big)\\,|t-s|^\\gamma},\n$$\nand \n$$\n\\llparenthesis\\, b \\,\\rrparenthesis_{\\gamma,g} := \\underset{\\stackrel{0\\leq s< u<t<T}{{|t-s|\\le \\lambda}}}{\\sup}\\;\\,\\frac{\\big| b_{tus}\\big|}{g\\big(\\lambda^{-1} t\\big)\\,|t-s|^\\gamma};\n$$\nnote the following comparison: For $0\\leq \\gamma'\\leq \\gamma$, we have\n\\begin{equation}\n\\label{EqComparisonNorms}\n\\llparenthesis\\, \\cdot \\,\\rrparenthesis_{\\gamma',g} \\leq \\lambda^{\\gamma-\\gamma'} \\llparenthesis\\, \\cdot \\,\\rrparenthesis_{\\gamma,g}.\n\\end{equation}\n\n\\smallskip\n\nRecall that given an E-valued $2$-index map $a$, the \\textit{sewing lemma} \\cite{Gubinelli04, FdlP} asserts that if the inequality\n\\begin{equation}\n\\label{EqAlmostAdditivity}\n\\big| a_{ts} - (a_{tu}+a_{us})\\big| \\leq c|t-s|^\\zeta\n\\end{equation}\nholds for all $0\\leq s\\leq u\\leq t<T$, with $t-s\\leq 1$ say, for some exponent $\\zeta>1$ and some positive constant $c$, then there exists a unique map $A : [0,T)\\rightarrow \\textrm{E}$ whose increments $\\delta A_{ts} := A_t-A_s$ are well-approximated by $a_{ts}$, in the sense that \n$$\n\\big|\\delta A_{ts} - a_{ts} \\big| \\lesssim |t-s|^\\zeta,\n$$\nfor all $t-s\\leq 1$ say. Moreover, it $t_i$ denotes the times of a finite partition $\\pi_{ts}$ of an interval $(s,t)$, with mesh $\\big|\\pi_{ts}\\big|$, we have\n\\begin{equation}\n\\label{EqEstimateAlmostAdditiveFunctional}\n\\big|\\delta A_{ts} - \\sum a_{t_{i+1}t_i}\\big| \\lesssim |t-s|\\big|\\pi_{ts}\\big|^{\\zeta-1}.\n\\end{equation}\n\n\\smallskip\n\nThe \\textbf{sewing map} associates to the above $2$-index map $a$ the $2$-index map\n$$\n\\Lambda(a)_{ts} := \\delta A_{ts} - a_{ts}.\n$$\nFor a given function $g:\\RR\\to\\RR$ let \n$$\nG(t) = g(t) + \\int_0^t g(r)\\,dr,\n$$\nand write as usual  $\\delta a_{tus}$ for $a_{ts} - (a_{tu}+a_{us})$, for any $0\\leq s\\leq u\\leq t<T$. The following lemma provides an estimate of the weighted norm of $\\Lambda(a)$ in terms of the weighted norm of $\\delta a$; an estimate for the growth of $A$ follows as a consequence. \n\n\\begin{lem}\n\\label{LemmaSewingMap}\nThere exists a positive constant $c_\\zeta$, depending only on $\\zeta$, such that \n$$\n\\llparenthesis\\,\\Lambda(a)\\,\\rrparenthesis_{\\zeta,g} \\leq c_\\zeta \\, \\llparenthesis\\,\\delta a\\,\\rrparenthesis_{\\zeta,g}.\n$$\n\\end{lem}\n\n\\medskip\n\n\\begin{Dem}\nGiven $0\\leq s<t$ and $n\\geq 1$, set $t_i = s+i2^{-n}(t-s)$, and note that the telescopic sum\n\\begin{equation*}\n(\\star) := \\sum_{i=0}^{2^n-1}a_{t_{i+1}t_i}\\; - a_{ts} = \\sum_{k=0}^{n-1}\\;\\sum_{\\ell=0}^{2^{n-(k+1)}} \\Big(a_{(\\ell+2)2^k\\,(\\ell+1)2^k}+a_{(\\ell+1)2^k\\,\\ell2^k} - a_{(\\ell+2)2^k\\,\\ell2^k} \\Big)\n\\end{equation*}\nprovides a control of the quantity $(\\star)$ in terms of $\\delta a$ only. Sending $n$ to infinity, we see by identity \\eqref{EqEstimateAlmostAdditiveFunctional} that there exists a constant $c_\\zeta$ depending only on $\\zeta$, such that \n$$\n\\big| \\Lambda(a)_{ts} \\big| \\leq c_\\zeta \\, |t-s|^\\zeta \\,\\sup\\left\\{\\frac{\\big|\\delta a_{t'u's'}\\big|}{|t'-s'|^\\zeta}\\,;\\,0\\leq s<s'<u'<t'\\leq t\\right\\}.\n$$\nIf $|t-s| \\leq \\lambda$ the above inequality implies clearly that we have\n\\begin{equation}\n\\label{EqEstimateLambdaA}\n\\frac{\\big| \\Lambda(a)_{ts} \\big|}{g(\\lambda^{-1} t)\\,|t-s|^\\zeta} \\leq c_\\zeta \\, \\big|\\delta a\\big|_{\\zeta,g}.\n\\end{equation}\n\\end{Dem}\n\n\\medskip\n\nWe will consider only the particular choice of function $g(t) = e^{t}$ and we shall set for a path $f : [0,T)\\rightarrow \\textrm{E}$\n\\begin{equation}\n\\label{EqDefnNormFunction}\n\\llparenthesis\\, f_\\bullet \\,\\rrparenthesis :=  \\underset{t\\geq 0}{\\sup}\\,e^{-\\lambda^{-1} t}\\,\\big| f(t) \\big|,\n\\end{equation}\nand for a $2$-index map $a$, and a  $3$-index map $b$,\n\\begin{equation}\n\\label{EqDefnNormMultiIndexMaps}\n\\llparenthesis\\, a \\,\\rrparenthesis_\\gamma := \\underset{\\stackrel{0\\leq s<t<T}{|t-s|\\le \\lambda}}{\\sup} \\;\\, e^{-\\lambda^{-1} t} \\,\\frac{\\big| a_{ts}\\big|}{|t-s|^\\gamma}\\,, \\quad \\textrm{ and }\\quad \n\\llparenthesis\\, b \\,\\rrparenthesis_\\gamma := \\underset{\\stackrel{0\\leq s< u<t<T}{{|t-s|\\le \\lambda}}}{\\sup} \\;\\, e^{-\\lambda^{-1} t} \\,\\frac{\\big| b_{tus}\\big|}{|t-s|^\\gamma}\\,.\n\\end{equation}\nNote that these norms depend on a choice of parameter $\\lambda$, which may be tuned on demand. This will be particularly useful in the statement and proof of Theorem~\\ref{ThmIntegrationLinearBounded} below, giving the well-posed character of some linear rough differential equation. Note also that we can compare $\\llparenthesis\\, f_\\bullet \\,\\rrparenthesis$ and $\\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma$, as the inequality\n$$\n\\llparenthesis\\, f_\\bullet \\,\\rrparenthesis \\leq \\big|f_0\\big| + e \\lambda^\\gamma\\,\\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma\n$$\nholds for all $\\lambda>0$. This will be our starting point in the proof of the a priori estimate \\eqref{EqLinearBound} in Theorem~\\ref{ThmIntegrationLinearBounded} below. The proof is easy: for fixed $t$ let $\\{0=t_0< t_1 < \\cdots < t_n = t\\}$ be a partition of $[0,t]$ with intervals of at most size $\\lambda$. Then \n$$\n|f(t)| \\le |f(0)| + \\sum_{k=0}^{n-1}|f(t_k)-f(t_{k+1})| \\le  |f(0)| +  \\lambda^\\gamma  \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma \\sum_{k=0}^{n-1} \\exp(t_{k+1}\/\\lambda).\n$$\nBut now\n$$\n\\sum_{k=0}^{n-1} \\exp(t_{k+1}\/\\lambda)\n\\le   \\lambda^{-1}\\int_{t_{k}+\\tau}^{t_{k+1}+\\lambda} \\exp(s\/\\lambda) d s \\le  \\lambda^{-1} \\int_0^{t+\\lambda} \\exp(s\/\\lambda) d s \\le \\exp(t\/\\lambda + 1).\n$$\n\n\\smallskip\n\nLast, a $2$-index map $a$ such that $\\sup_{t-s\\leq 1}\\,\\frac{| a_{ts} |}{|t-s|^\\gamma}$ is finite will be called a $\\gamma$\\textbf{-H\\\"older map.}\n\n\n\\section{Linear differential equations with bounded rough drivers}\n\\label{SectionBoundedDrivers}\n\nLet $\\big(\\mcA,|\\cdot|\\big)$ be a Banach algebra with unit ${\\bf 1}_{\\mathcal{A}}$; one may think for instance to the space of continuous linear maps from some Hilbert space to itself, or to the truncated tensor algebra over some Banach space, equipped with a tensor norm and completed for that norm. We introduce in this section a notion of bounded rough driver in the Banach algebra $\\mcA$, and show that they generate some flows on the algebra. \n\n\\begin{defn}\nLet $\\frac{1}{3}<\\gamma\\leq \\frac{1}{2}$. A \\textbf{\\emph{bounded $\\gamma$-rough driver in $\\mcA$}} is a pair ${\\bf A}= \\big(A^1,A^2\\big)$ of $\\mcA$-valued $2$-index maps satisfying \\emph{Chen's relations}\n\\begin{equation}\n\\label{EqChenRelations}\n\\delta A^1=0, \\quad \\textrm{ and }\\quad \\delta A^2_{tus} = A^1_{tu}A^1_{us},\n\\end{equation}\nand such that $A^1$ is $\\gamma$-H\\\"older and $A^2$ is $2\\gamma$-H\\\"older. The norm of $\\bf A$ is defined by the formula\n$$\n\\| {\\bf A} \\| := \\sup_{0\\leq s<t<T\\,;\\,t-s\\leq 1}\\;\\frac{\\big|A^1_{ts}\\big|}{|t-s|^\\gamma}\\vee\\frac{\\big|A^2_{ts}\\big|}{|t-s|^{2\\gamma}}.\n$$\n\\end{defn}\n\n\\medskip\n\nAs an elementary example, think of $\\mcA$ as the truncated tensor algebra $\\bigoplus_{i=0}^N(\\RR^\\ell)^{\\otimes i}$ over $\\RR^\\ell$, for $N\\geq 2$, and consider a weak geometric $\\gamma$-H\\\"older rough path ${\\bfX}_{ts} = 1\\oplus X_{ts}\\oplus \\bbX_{ts}\\in\\bigoplus_{i=0}^N(\\RR^\\ell)^{\\otimes i}$, with $2\\leq \\gamma<3$. Left multiplication by $X_{ts}$ and $\\bbX_{ts}$ define operators $A^1$ and $A^2$ that are the components of a rough driver.\n\n\\medskip\n\nRecall the weighted norms $\\llparenthesis\\,\\cdot\\,\\rrparenthesis$ and $\\llparenthesis\\,\\cdot\\,\\rrparenthesis_\\gamma$ defined by identities \\eqref{EqDefnNormFunction} and \\eqref{EqDefnNormMultiIndexMaps}, respectively, depend on some parameter $\\lambda$. The proof of the following well--posedness result for linear rough differential equations driven by bounded $\\gamma$-rough drivers shows the interest of being flexible on the tuning of $\\lambda$.\n\n\\begin{thm}[Integration of bounded rough drivers]\n\\label{ThmIntegrationLinearBounded}\nGiven any initial condition $J_0\\in\\mcA$, there  exists a unique $\\gamma$-H\\\"older path $f_\\bullet$ starting from $f_0$, and such that the formula \n  \\begin{equation*}\n    \\delta f_{t, s} - \\Big(A^1_{t, s} +A^2_{t, s}\\Big) f_s \n  \\end{equation*}\ndefines a $3\\gamma$-H\\\"older $2$-index map $f^\\sharp$. Moreover, the following estimate holds\n  \\begin{equation}\n  \\label{EqLinearBound}\n    \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis \\leqslant 2\\,\\big| f_0 \\big|\n  \\end{equation}\nfor all $\\lambda$ greater than some $\\lambda_0$ depending only on $\\| \\bf A \\|$ and $\\gamma$. When $f_0  ={\\bf 1}_{\\mathcal{A}}$, we will use the notation $ f_t =e^{\\bf A}_{t,0}$; the flow property\n$$\ne^{\\bf A}_{t, s} = e^{\\bf A}_{t, u} e^{\\bf A}_{u, s},\n$$\nholds for all $0\\leq s\\leq u\\leq t<T$.\n\\end{thm}\n\n\\medskip\n\nApplied to the above example of rough driver in the truncated tensor product space, this well--posedness result provides a proof of Lyons' extension theorem \\cite{Lyons98}.\n\n\\medskip\n\n\\begin{Dem}\n{\\bf a) A priori estimate --} Let us prove the a priori bound \\eqref{EqLinearBound} first; the uniqueness claim in the theorem follows from this bound and the linear character of the problem. As mentioned above, we start from the inequality\n$$\n\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis \\leq | f_0 | + c \\lambda^\\gamma \\llparenthesis\\, \\delta f \\,\\rrparenthesis _\\gamma,\n$$\nand try and write $\\llparenthesis\\, \\delta f \\,\\rrparenthesis _\\gamma$ in terms of $\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis$; this can be done as follows.\n\nBy using Chen's relation \\eqref{EqChenRelations} and the definition of the remainder $f^\\sharp$, the identity\n$$\n- \\delta f^\\sharp_{t, u, s} = A^1_{t, u} \\Big(\\delta f_{u, s}  -A^1_{u, s} f_s\\Big) +A^2_{t, u} \\delta f_{u, s} = A^1_{t, u} \\Big(A^2_{t, s} f_s + f^{\\sharp}_{s, t}\\Big) + A^2_{t, u} \\delta f_{u, s}, \n$$\ngives us the estimate\n$$\n\\llparenthesis\\, \\delta f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\leq \\| {\\bf A} \\| ^2 \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis + \\| {\\bf A} \\| \\llparenthesis\\, f^{\\sharp} \\,\\rrparenthesis_{2 \\gamma} + \\| {\\bf A} \\| \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma.\n$$ \nBut since $3 \\gamma > 1$, we have \n$$\nf^{\\sharp} = \\Lambda \\delta f^{\\sharp},\n$$ \nso the inequality\n$$\n\\llparenthesis\\, f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_\\gamma \\| {\\bf A} \\|^2 \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis + \\| {\\bf A} \\| \\llparenthesis\\, f^{\\sharp} \\,\\rrparenthesis_{2 \\gamma} + \\| {\\bf A} \\| \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma\n$$\nfollows from Lemma~\\ref{LemmaSewingMap}. Using the inequality $\\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{2 \\gamma} \\leq \\lambda^\\gamma \\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma}$, emphasized in \\eqref{EqComparisonNorms}, the above equation gives \n$$\n\\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_\\gamma \\| {\\bf A} \\|^2\\, \\llparenthesis\\,  f_\\bullet \\,\\rrparenthesis  + \\lambda^\\gamma \\| {\\bf A} \\|\\,\\llparenthesis\\,  f^{\\sharp}\\,\\rrparenthesis_{3 \\gamma} + \\| {\\bf A} \\|\\, \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma.\n$$\nFor $\\lambda$ small enough so that $\\lambda^\\gamma \\| {\\bf A} \\| \\leq \\frac{1}{2}$, we obtain\n$$\n\\llparenthesis\\,  f^{\\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_{\\gamma} \\| {\\bf A} \\|^2\\, \\llparenthesis\\,  f_\\bullet \\,\\rrparenthesis  + \\| {\\bf A} \\|\\, \\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma,\n$$\nso, using again the definition of the remainder $f^\\sharp$, and the observation that \n$$\n\\llparenthesis\\, A^2 f\\,\\rrparenthesis_{\\gamma} \\lesssim_{\\gamma}  \\lambda^\\gamma \\llparenthesis\\, A^2 f\\,\\rrparenthesis_{2 \\gamma} \\lesssim_{\\gamma} \\lambda^\\gamma \\| {\\bf A} \\|\\, \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis,\n$$\nwe obtain the estimate\n\\begin{equation*}\n\\begin{split}\n\\llparenthesis\\, \\delta f \\,\\rrparenthesis_\\gamma &\\lesssim_{\\gamma} \\| {\\bf A} \\|\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis + \\llparenthesis\\, A^2 f\\,\\rrparenthesis_\\gamma + \\llparenthesis\\, f^{\\sharp}\\,\\rrparenthesis_\\gamma \\\\\n                            &\\lesssim_{\\gamma} \\Big\\{\\| {\\bf A} \\| \\big(1 + \\lambda^{2 \\gamma}\\| {\\bf A} \\|\\big) + \\lambda^\\gamma \\| {\\bf A} \\|\\Big\\} \\llparenthesis\\, f_\\bullet \\,\\rrparenthesis + \\lambda^{2 \\gamma}\\| {\\bf A} \\|\\,\\llparenthesis\\, \\delta f \\,\\rrparenthesis_{\\gamma}.\n\\end{split}\n\\end{equation*}\nTaking $\\lambda$ small enough, depending only on $\\| \\bf A \\|$, we eventually see that \n$$ \n\\llparenthesis\\,\\delta f \\,\\rrparenthesis_\\gamma \\lesssim_{\\gamma}\\Big\\{\\| {\\bf A} \\| \\big(1 +  \\lambda^{2 \\gamma} \\| {\\bf A} \\|\\big) + \\lambda^\\gamma  \\| {\\bf A} \\|\\Big\\} \\llparenthesis\\, f_\\bullet\\,\\rrparenthesis.\n$$\nThe a priori estimate $\\llparenthesis\\, f _\\bullet\\,\\rrparenthesis \\leqslant 2 | f_0 |$, follows now from a choice of sufficiently small  parameter $\\lambda$, since $\\llparenthesis\\,f_\\bullet \\,\\rrparenthesis \\leq | f_0 | + c \\lambda^\\gamma \\llparenthesis\\, \\delta f \\,\\rrparenthesis _\\gamma$.\n\n\\bigskip\n\n{\\bf b) Existence --} We can run a Picard iteration to prove the existence of a path satisfying the conditions of the theorem. Set first\n$$\nf^0_t = f_0,\\quad \\textrm{ and } \\quad f^1_t =A^1_{t0} f_0\n$$ \nfor all $t \\in \\RR$. Given the paths $f^{n-1}_\\bullet, f^n_\\bullet$, the $2$-index map \n$$\na^n_{ts} := A^1_{ts} f^n_s +A^2_{ts} f_s^{n - 1}\n$$\nsatisfies the almost-additivity condition \\eqref{EqAlmostAdditivity} with $\\zeta=3\\gamma>1$ here, so there is, by the sewing lemma, a unique $\\gamma$-H\\\"older path $f^{n + 1}_\\bullet$ for which the formula\n$$ \n\\delta f^{n + 1}_{ts} - \\Big(A^1_{ts} f^n_s +A^2_{ts} f_s^{n - 1}\\Big) \n$$\ndefines $2$-index $3\\gamma$-H\\\"older map $f^{n + 1, \\sharp}$. Setting $g^0_\\bullet := 0$ and\n$$\ng^{n + 1}_\\bullet := f^{n + 1}_\\bullet - f^n_\\bullet, \\quad  g^{n + 1, \\sharp} := f^{n + 1, \\sharp} - f^{n, \\sharp},\n$$ \nfor all $n \\geq 0$, we have\n$$\n\\delta g^{n + 1}_{ts} =A^1_{ts} g^n_s +A^2_{ts} g_s^{n - 1} + g^{n + 1, \\sharp}_{ts}. \n$$\nNote moreover that we have the identity\n\\begin{equation*}\n\\begin{split}\n- \\delta g^{n + 1, \\sharp}_{t, u, s} &= A^1_{t, u} \\Big(\\delta  g^n_{u, s} -A^1_{u, s} g^{n - 1}_s\\Big) +A^2_{t, u} \\delta g^{n - 1}_{u, s} \\\\\n&= A^1_{t, u} \\Big(A^2_{t, s} g^{n - 2}_s + g^{n, \\sharp}_{s, t}\\Big) +A^2_{t, u} \\delta g^{n - 1}_{u, s},\n\\end{split}\n\\end{equation*}\nso, proceeding as in the proof of the a priori bound, we see that the inequality\n$$\n\\llparenthesis\\, g^{n - 1}_\\bullet\\,\\rrparenthesis + \\llparenthesis\\, \\delta g^n \\,\\rrparenthesis_\\gamma + \\llparenthesis\\, g^{n + 1, \\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_{\\gamma, \\| {\\bf A} \\|} \\lambda^\\gamma \\Big\\{\\llparenthesis\\, g^{n - 2}_\\bullet \\,\\rrparenthesis + \\llparenthesis\\, \\delta g^{n - 1} \\,\\rrparenthesis_\\gamma + \\llparenthesis\\, g^{n, \\sharp} \\,\\rrparenthesis_{3 \\gamma}\\Big\\}\n$$  \nholds, by choosing $\\lambda$ small enough. The estimate\n$$\n\\llparenthesis\\, g^{n - 1}_\\bullet\\,\\rrparenthesis + \\llparenthesis\\, \\delta g^n \\,\\rrparenthesis_\\gamma + \\llparenthesis\\, g^{n + 1, \\sharp} \\,\\rrparenthesis_{3 \\gamma} \\lesssim_{\\gamma, \\| {\\bf A} \\|} \\lambda^{\\gamma n}\n$$\nfollows as a consequence, so the series $f^n = f^0 + \\sum_{n \\geqslant 1} g^n$ converges in the complete space of $\\mcA$-valued $\\gamma$-H\\\"older paths, and defines a path in $\\mathcal{A}$ satisfying the conditions of the theorem.\n\\end{Dem}\n\n\\smallskip\n\n\\begin{rems}\n\\begin{enumerate}\n   \\item Note that the proof given above gives back the known sharp growth rate $\\exp\\big((2\\| {\\bf A}\\|)^\\gamma\\,t\\big)$ for $\\big|f_t\\big|$; see \\cite{FrizOberhauser}. Bounded rough drivers can also be integrated by defining recursively the $(n\\gamma)$-H\\\"older $\\mcA$-valued $2$-index map $A^n$ using the formula\n$$\n\\delta A^n_{t, u, s} = \\sum_{k = 1}^{n - 1} A^{n - k}_{t, u} A^k_{u, s},\n$$\nand setting \n$$\ne^{\\bf A}_{t, s} = \\sum_{n = 0}^{\\infty} A^n_{t, s}.\n$$ \nStandard estimates on the sewing map \\cite{Gubinelli04} show that $\\delta A^n$ has $n \\gamma$-H\\\"older norm  no greater than $(n!)^{- \\gamma}$, so the above series converges in $\\mcA$ for all $0 \\leq s \\leq t$. The flow property is obtained by a direct calculation, and setting $f_t := e^{\\bf A}_{t, 0} f_0$, we see that the path $f_\\bullet$ solves the problem.   \\vspace{0.2cm}\n\n   \\item {\\bf Linear rough differential equations with a linear drift -- } The above theory extends easily to rough equations of the form\n\\begin{equation}\n\\label{EqRoughLinearDrift}\n\\delta f_{t, s} = \\int_s^t B_r f_r\\,dr + A^1_{t, s} f_s + A^2_{t, s} f_s + f^{\\natural}_{t, s}\n\\end{equation}\nwhere $B \\in \\LL^{\\infty} (\\RR; \\mcA)$ is a bounded measurable family of bounded operators. This equation is the rigorous meaning to give to solutions of the differential equation\n$$\n\\frac{d}{dt} f = (B_r + \\dot{A}_r) f_r\n$$\nwhere $\\dot{A}_t = \\partial_t A_{t0}^1$. In case $\\dot{A}_t \\in \\LL^{\\infty} (\\RR; \\mcA)$, the two formulations are equivalent provided ${\\bf A} = \\big(A^1, A^2\\big)$ is defined by the formula\n$$ \nA^1_{t, s} = \\int_s^t \\dot{A}_r \\,dr, \\quad A^2_{t, s} = \\int_s^t \\int_s^r \\dot{A}_u \\dot{A}_r \\,du dr.\n$$\nThe proof of Theorem~\\ref{ThmIntegrationLinearBounded} can be easily adapted in the present setting, and the final lower bound on $\\lambda$ gets an additional dependence on $\\| B \\|_\\infty$. It yields moreover the following Duhamel formula\n$$\nf_t = e^{\\bf A}_{t, 0} f_0 + \\int_0^t e^{\\bf A}_{t, r} B_r f_r\\,dr.\n$$\nIndeed, let $f_\\bullet$ be a function satifsying the above identity. If one computes the increment of the right hand side in the above equation, we get\n$$\n\\delta f_{t, s} = \\int_s^t e^{\\bf A}_{t, r} B_r f_r \\,dr - \\Big(e^{\\bf A}_{t, s} -\\textrm{\\emph{Id}}\\Big) f_s = \\int_s^t B_r f_r \\,dr + A^1_{t, s} f_s + A^2_{t, s} f_s + f^\\sharp_{t, s}\n$$\nwhere\n$$\nf^\\sharp_{s, t} = \\int_s^t \\Big(e^{\\bf A}_{t, r} -\\textrm{\\emph{Id}}\\Big) B_r f_r \\,dr + \\Big(e^{\\bf A}_{t, s} -\\textrm{\\emph{Id}} - A^1_{t, s} - A^2_{t, s}\\Big) f_s.\n$$\nUsing the bounds\n$$\n\\Big| e^{\\bf A}_{t, r} -\\textrm{\\emph{Id}}\\Big| \\lesssim_{\\|{\\bf A}\\|} | t -  r |^{\\gamma}, \\quad \\textrm{ and }\\quad  \\Big| e^{\\bf A}_{t, s}  -\\textrm{\\emph{Id}} - A^1_{t, s} - A^2_{t, s} \\Big| \\lesssim_{\\| {\\bf A} \\|} | t - s |^{3 \\gamma},\n$$\nthis allows to conclude that $\\big|f^{\\sharp}_{t, s}\\big| \\lesssim | t - s |^{3\\gamma}$, and that the path $f_\\bullet$ is indeed the unique solution to equation \\eqref{EqRoughLinearDrift}. \\vspace{0.2cm}\n\n   \\item Bounded rough drivers have also been introduced previously in the work \\cite{CoutinLejayLinear} of Coutin and Lejay, and studied in relation with the Magnus formula for what is called there the \\emph{resolvent operator} $e^{\\bf A}$. The above short proof of Theorem~\\ref{ThmIntegrationLinearBounded} can be considered an alternative proof of the main result of section 3 in \\cite{CoutinLejayLinear}. They also consider perturbed linear equations, with an a priori given drift of the more general form $C_{ts}$, rather than $\\int_s^t B_rf_r\\,dr$, with $C$ satisfying some regularity conditions. The pioneering work \\cite{FdlPM} of Feyel-de la Pradelle-Mokobodzki is also closely related to these questions.\n\\end{enumerate}\n\\end{rems}\n\n\n\\medskip\n\n\n\n\\section{Unbounded rough drivers and rough linear equations}\n\\label{SectionUnboundedRoughDrivers}\n\nThe above results apply in the particular case where $\\mcA$ is the Banach algebra of bounded operators on an Hilbert space $H$. We shall study in the remaining sections the integration problem \n\\begin{equation}\n\\label{EqLinearEquationHilbert}\n\\delta f_{ts} = \\Big(A^1_{ts} + A^2_{ts}\\Big) f_s +  f^\\sharp_{ts},\n\\end{equation}\nfor a particular class of drivers $\\bf A$ associated to a class of \\textit{unbounded} operators on $H$, or other Banach spaces, with in mind the model case of the \\textbf{rough transport equation}\n$$\n\\delta f(\\varphi)_{ts} = X_{ts}\\,f_s\\big(V^*\\varphi\\big) +  \\bbX_{ts}\\,f_s\\big(V^*V^*\\varphi\\big) + f^\\sharp_{ts}(\\varphi),\n$$\nwhere ${\\bfX} = (X,\\bbX)$ is an $\\ell$-dimensional $\\gamma$-H\\\"older rough path and $V = \\big(V_1,\\dots,V_\\ell\\big)$ is a collection of $\\ell$ vector fields on $\\RR^d$.\n\n\n\n\\subsection{Rough drivers}\n\\label{SubsectionRoughDrivers}\n\nTo make sense of this equation we need to complete the functional setting by the datum of a scale of Banach spaces $\\big(E_n,|\\cdot|_n\\big)_{n\\geq 0}$, with $E_{n+1}$ continuously embedded in $E_n$. For $n\\geq 0$, we shall denote by $E_{-n}=E_n^*$ the dual space of $E_n$, equipped with its natural norm, \n$$\n|e|_{-n} := \\underset{\\varphi\\in E_n,\\,|\\varphi|_n\\leq 1}{\\sup}\\; (\\varphi,e), \\quad\\quad e\\in E_{-n}.\n$$\nWe require that the following continuous inclusions \n\\begin{equation*}\nE_n\\subset \\cdots\\subset E_2\\subset E_1\\subset E_0 \\\\ \n\\end{equation*}\nhold for all $n\\geq 2$. One can think of $n$ as quantifying the 'regularity' of elements of some test functions, with the elements of $E_n$ being more regular with $n$ increasing. Denote by $\\|\\cdot\\|_{(b,a)}$ for the norm of a linear operator form $E_a$ to $E_b$. (Note that we use $(b,a)$ and not $(a,b)$ in the lower index for the norm.) We also assume the existence of a family $\\big(J^\\varepsilon\\big)_{0<\\varepsilon\\leq 1}$ of operators from $E_0$ to itself such that the estimates\n\\begin{equation}\n\\label{EqApproximationProperties}\n\\big\\|J^\\varepsilon - \\textrm{Id} \\big\\|_{(n+k,n)} \\leq c\\,\\varepsilon^k, \\quad \\big\\| J^\\varepsilon \\big\\|_{(n,n + k)} \\leq c\\,\\varepsilon^{- k}\n\\end{equation}\nhold for all $n, k \\geq 0$, for some positive constant $c$ independent of $\\varepsilon$. For $\\varphi\\in E_0$, the elements $\\varphi_\\varepsilon := J^\\varepsilon\\varphi$ are in particular 'smooth', that is in the intersection of all the spaces $E_n$, for $n\\geq 0$. \n\nWhenever we will work with Sobolev--like scales $E_n = W^{n,p}(\\RR^d)$ ($p\\ge 1$) we will take the operators $J^\\epsilon = \\big(\\textrm{I}-\\epsilon\\triangle\\big)^{-j_0}$, for $j_0$ big enough.\n\n\n\\smallskip\n\n\\begin{defn}\n\\label{DefnUnboundedRoughDriver}\nLet $\\frac{1}{3}<\\gamma\\leq \\frac{1}{2}$ be given. An \\textbf{\\emph{unbounded $\\gamma$-rough driver on the scales}} $\\big(E_n,|\\cdot|_n\\big)_{n\\geq 0}$, is a pair ${\\bf A} = \\big(A^1,A^2\\big)$ of $2$-index maps, with \n\\begin{equation}\n\\label{EqBoundednessAssumptionRoughDriver}\n\\begin{split}\n&A^1_{ts}\\in \\textrm{\\emph{L}}\\big(E_n,E_{n-1}\\big), \\textrm{ for } n\\in\\{-0,-2\\}, \\\\\n&A^2_{ts}\\in\\textrm{\\emph{L}}\\big(E_n,E_{n-2}\\big), \\textrm{ for } n\\in\\{-0,-1\\},\n\\end{split}\n\\end{equation}\nfor all $0\\leq s\\leq t<T$, which satisfies Chen's relations \\eqref{EqChenRelations}, and such that $A^1$ is $\\gamma$-H\\\"older and $A^2$ is $2\\gamma$-H\\\"older. \n\\end{defn}\n\n\\medskip\n\nEquation \\eqref{EqDefnSolutionRDE} below will make it clear that unbounded rough drivers can be thought of a some kind of (dual objects to some) multi-scales velocity fields, with two time scales given by $A^1$ and $A^2$. This is particularly clear on the following elementary example, where\n$$\nA^1_{ts} = \\sum_{i=1}^\\ell X^i_{ts}V_i, \\quad \\textrm{and } \\quad A^2_{ts} = \\sum_{j,k=1}^\\ell \\bbX^{jk}_{ts}V_jV_k,\n$$\nwhere ${\\bfX} = (X,\\bbX)$ is a weak geometric $\\gamma$-H\\\"older rough path over $\\RR^\\ell$, with $\\frac{1}{3}<\\gamma\\leq \\frac{1}{2}$, and $V_1,\\dots, V_\\ell$ are regular enough vector fields on $\\RR^d$ with values in $N\\times N$ matrices, understood here as first order differential operators. On can also take $V_i^* = -V_i - \\textrm{div}V_i$ in the above definition of a rough driver.\n\n\\smallskip\n\nIn a probabilistic setting, the above rough path $\\bfX$ could be the (Stratonovich) Brownian rough path. More generally, one could take as first level $A^1_{ts}$ a semimartingale velocity field of Le Jan-Watanabe-Kunita type (or its dual), where noise (the above $X_{ts}$) and dynamics (the above $V_i$) cannot be separated from one another. It is shown in the forthcoming work \\cite{BailleulRiedel} that these velocity fields can be lifted into a(n object very similar to a) rough driver under some mild conditions on the semimartingale structure. In the same spirit, Catellier has shown in \\cite{Catellierregularisation, CatellierTransport} the interest for the study of rough transport equations with irregular drift, of considering velocity fields in $\\RR^d$ given by the regularisation of some field $V$ (that may even be a distribution) along some irregular path $w$, such as a typical trajectory of a fractional Brownian motion with any Hurst index. One deals in this particular setting with the 'Young analogue' of the above $\\gamma$-rough drivers, corresponding to $\\frac{1}{2}<\\gamma\\leq 1$, with only one level, and given in this example by the formula\n$$\nA^1_{ts}(x) = \\int_s^t V(x+w_u)\\,du.\n$$\n\n\\medskip\n\nWe denote by $\\epsilon(e)$ the duality pairing between an element $\\epsilon$ of $E_{-n}$ and an element $e$ of $E_n$, for any $n\\in\\ZZ$. The dual of a continuous operator $A$ from $E_{-a}$ to $E_{-b}$ is denoted by $A^*$; this is a continuous operator from $E_b$ to $E_a$. We can now make sense of equation \\eqref{EqLinearEquationHilbert}. \n\n\\begin{defn}\n\\label{DefnSolutionEquation}\nAn $E_{-0}$-valued \\emph{\\textbf{path}} $f_\\bullet$ is said to \\emph{\\textbf{solve the linear rough differential equation}}\n\\begin{equation}\n\\label{EqRDE}\ndf_s = {\\bf A}(ds)f_s\n\\end{equation}\nif there exists an $E_{-2}$-valued $2$-index map $f^\\sharp$ such that one has\n\\begin{equation}\n\\label{EqDefnSolutionRDE}\n\\delta f_{ts}(\\varphi) = f_s\\Big(\\big\\{A^{1,*}_{ts}+A^{2,*}_{ts}\\big\\}\\varphi\\Big) + f^\\sharp_{ts}(\\varphi)\n\\end{equation}\nfor all $0\\leq s\\leq t<T$ and all $\\varphi\\in E_2$, and the map $f^\\sharp_{ts}(\\varphi)$ is $3\\gamma$-H\\\"older, for each $\\varphi\\in E_2$.\n\\end{defn}\n\n\\smallskip\n\nLet us insist on the fact that this notion of solution depends on the scale $(E_n)_{n\\geq 0}$. We define, for all $0\\leq s\\leq t<T$, an  $E_{-1}$-valued $2$-index map $f^\\flat$ setting, for $\\varphi\\in E_1$,\n\\begin{equation*}\nf^\\flat_{ts}(\\varphi) := \\delta f_{ts}(\\varphi) - f_s\\big(A_{ts}^{1,*} \\varphi\\big);\n\\end{equation*}\nfor $\\varphi\\in E_2$, one also has another expression for that quantity\n$$\nf^\\flat_{ts}(\\varphi) = f_s\\big(A^{2,*}_{ts}\\varphi\\big) + f^\\sharp_{ts}(\\varphi).\n$$\n\n\n\\bigskip\n\n\n\\subsection{A priori estimates}\n\\label{SubsectionAPrioriEstimates}\n\nWe show in this section that solutions $f_\\bullet$ of equation \\eqref{EqRDE} satisfy some a priori bounds that depend only on certains norms on the rough driver $A$ and on the uniform norm of $f$, when seen as a continuous path with values in some spaces of distributions. \n\n\\smallskip\n\nThe next lemma shows that the map $f^\\flat$ is actually $2\\gamma$-H\\\"older with values in a  space of distributions \"of order 2\" in a certain sense while $f^\\sharp$ is only expected to be $\\gamma$-H\\\"older from its definition. This \\textit{gain of time regularity, traded against a loss of 'space regularity'} may well be one of our main contribution, despite its elementary nature. It leads to a similar (and even better) result for $f^\\sharp$, as expressed in Theorem~\\ref{ThmGeneralRegularityGain}, that is the key technical result of this paper and which opens the road  to a quite complete theory of linear rough equation.\n\nRecall the notations $\\llparenthesis \\cdot\\rrparenthesis$ and $\\llparenthesis \\cdot\\rrparenthesis_\\gamma$ for the norms introduced in equations \\eqref{EqDefnNormFunction} and \\eqref{EqDefnNormMultiIndexMaps}. They depend implicitly on some positive parameter $\\lambda$ that we shall tune as needed along the way. Given $\\varphi\\in E_0$, set for all $n\\ge 0$\n\\begin{equation*}\n\\begin{split}\nK_n(\\varphi) := \\sup_{0 < \\varepsilon \\leq 1} \\inf_{\\varphi_1+\\varphi_2=\\varphi} \\Big\\{&   2\n (\\lambda^\\gamma \\varepsilon)^{0-n} \\, \\llparenthesis\\, f\\big(\\varphi_x\\big) \\,\\rrparenthesis\n  +  (\\lambda^\\gamma \\varepsilon)^{1-n} \\llparenthesis\\, f\\big(A^{1,*}\\varphi_x \\big) \\,\\rrparenthesis_\\gamma  \n \\\\ &\n + (\\lambda^\\gamma \\varepsilon)^{2-n}\\llparenthesis\\, f\\big(A^{2,*} \\varphi_x) \\,\\rrparenthesis_{2\\gamma} \n  +  (\\lambda^\\gamma \\varepsilon)^{3-n}\\,\\llparenthesis\\, f^\\sharp \\big(  \\varphi_x \\,) \\,\\rrparenthesis_{3\\gamma} \n  \\Big\\}\n\\end{split}\n\\end{equation*}\nwhere $\\varphi_x$ is $\\varphi_1$ if corresponds to a term of order $<n\\gamma$ or $\\varphi_2$ otherwise. \n\\smallskip\n\n\\begin{lem}\n\\label{LemmaGeneralImproveEstimates}\nWe have $\\llparenthesis\\, \\delta f(\\varphi) \\,\\rrparenthesis_\\gamma \\lesssim\\; K_1(\\varphi)$, and $\\llparenthesis\\,  f^\\flat(\\varphi) \\,\\rrparenthesis_{2\\gamma} \\lesssim\\; K_2(\\varphi)$.\n\\end{lem}\n\n\\medskip\n\n\\begin{Dem}\nWe start by an arbitrary decomposition of $\\varphi\\in E_1$ as $\\varphi = \\varphi_1 + \\varphi_2$ giving\n$$\n|\\delta f_{ts}(\\varphi)| \\leq |\\delta f_{ts}(\\varphi_1)| + |\\delta f_{ts}(\\varphi_2)|.\n$$\nTake $t,s$ such that ${|t-s|\\le \\lambda}$. The second term is bounded above by \n$$\ne^{-\\lambda^{-1}t} \\frac{|\\delta f_{ts}(\\varphi_2)|}{|t-s|^\\gamma} \\leq  2 \\lambda^{-\\gamma}\\varepsilon^{-1}\\llparenthesis\\, \\varphi_2 \\rrparenthesis ,\n$$ \nwhere  $\\varepsilon = \\lambda^{-\\gamma} | t - s |^\\gamma \\in(0,1]$ if $|t-s|\\le \\lambda$.\nThe first term can be estimated using properties \\eqref{EqApproximationProperties} and the defining identity \\eqref{EqDefnSolutionRDE}:\n$$\ne^{-\\lambda^{-1}t} \\frac{|\\delta f_{ts}(\\varphi_1)|}{|t-s|^\\gamma} \\leq\n\\llparenthesis  f(A^{1,*} \\varphi_1) \\rrparenthesis_\\gamma\n+ \\lambda^{\\gamma} \\varepsilon  \\llparenthesis f( A^{2,*} \\varphi_1) \\rrparenthesis_{2\\gamma}  \n+ \\lambda^{2\\gamma}\\varepsilon^2 \\llparenthesis f^\\sharp (  \\varphi_1) \\rrparenthesis_{3\\gamma}.\n$$ \n\n\\medskip\n\n$\\bullet$  This gives us an upper bound for $\\big|\\delta f_{ts}(\\varphi)\\big|$ depending on $\\varepsilon$ and $\\varphi_1, \\varphi_2$ over which we can optimise to obtain\n\\begin{equation*}\n\\llparenthesis \\delta f(\\varphi) \\rrparenthesis_\\gamma \\leq  K_1(\\varphi) .\n\\end{equation*}\n\n\\smallskip\n\n$\\bullet$ One can proceed in a similar way to estimate $\\big|f^\\flat_{ts}(\\varphi)\\big|$, starting from the inequality\n\\begin{equation*}\n\\begin{split}\n| f^\\flat_{ts}(\\varphi)| &\\leq | \\delta f_{ts}(\\varphi_1)| + | f_s(A^{1,*}_{ts}\\varphi_1)| + | f_s(A^{2,*}_{ts}\\varphi_2)| + |f^\\sharp_{ts}(\\varphi_2)|\n\\end{split}\n\\end{equation*}\nand optimizing as above we get the upper bound on $\\big|f^\\flat_{ts}(\\varphi)\\big|$.\n\\end{Dem}\n\n\\medskip\n\nThe estimates proved in Lemma~\\ref{LemmaGeneralImproveEstimates} are all we need to give an upper bound on the $3\\gamma$-H\\\"older norm of $f^\\sharp$ in terms of $f$ itself. As this result will be our key tool for proving a number of results in different situations, we formulate it here in some generality. Given a Banach space B and a B-valued path $m_\\bullet$, we set\n$$\n\\llparenthesis\\,m\\,\\rrparenthesis_{\\gamma\\,;\\,\\textrm{B}} := \\sup_{\\stackrel{0\\leq s<t\\leq T}{{|t-s|\\le \\lambda}}}e^{-\\lambda^{-1}t}\\,\\frac{\\big| \\delta m_{ts} \\big|_\\textrm{B}}{|t-s|^\\gamma},\n$$\nfor $\\gamma\\geq 0$. Let\n$$\nN({\\bfA}) := \\sup_\\varphi \\frac{\\|J_1(A^{1,*}\\varphi)\\|_\\gamma + \\|J_2(A^{2,*}\\varphi)\\|_{2\\gamma}}{|\\varphi|_F}\n$$ \nwith\n\\begin{equation*}\n\\begin{split}\nJ_n(\\varphi) :=\n\\sup_{0 < \\varepsilon \\leq 1} \\inf_{\\varphi_1+\\varphi_2=\\varphi} \\Big\\{&   2\n ( \\varepsilon)^{0-n} \\, | \\varphi_x|_E\n  +  (\\varepsilon)^{1-n} \\| A^{1,*}\\varphi_x \\|_{\\gamma;E}  \n + ( \\varepsilon)^{2-n}\\|A^{2,*} \\varphi_x\\|_{2\\gamma;E} \n \\\\ &\n  +  ( \\varepsilon)^{3-n}| \\varphi_x |_F \n  \\Big\\} .\n\\end{split}\n\\end{equation*}\n\n\\smallskip\n\n\\begin{thm}\n\\label{ThmGeneralRegularityGain} \nLet $\\textrm{\\emph{E}}\\subset E_0$ and $\\textrm{\\emph{F}}\\subset E_3$ be two Banach spaces of distributions such that the above quantity $N({\\bfA})$ is finite. Let  $\\lambda \\leq 1$ be sufficiently small so that $2\\lambda^{\\gamma} N({\\bfA}) \\leq 1$. Then any solution of the rough linear equation \\eqref{EqRDE} satisfies the a priori bound \n$$\n\\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma\\,;\\,F^*} \\leq 8 \\lambda^{-2\\gamma} N({\\bfA})\\,\\llparenthesis\\,f\\,\\rrparenthesis_{E^*}. \n$$\n\\end{thm}\n\n\\smallskip\n\n\\begin{Dem}\nGiven $\\varphi\\in E_3$, note that\n$$\n\\delta f_{tus} = f^\\flat_{us}\\big(A^{1,*}_{tu}\\varphi\\big) + \\delta f_{us}\\big(A^{2,*}_{tu}\\varphi\\big),\n$$\nis $3\\gamma$-H\\\"older by Lemma~\\ref{LemmaGeneralImproveEstimates}, so the sewing Lemma~\\ref{LemmaSewingMap} can be used, giving\n\\begin{equation*}\n\\begin{split}\n\\llparenthesis\\, f^\\sharp(\\varphi)\\,\\rrparenthesis_{3\\gamma} \n&\\lesssim \\llparenthesis\\,  f^\\flat \\big(A^{1,*}\\varphi\\big) \\,\\rrparenthesis_{3\\gamma} +  \\llparenthesis\\,  \\delta f (A^{2,*}\\varphi) \\,\\rrparenthesis_{3\\gamma}\\\\\n&\\lesssim  \\|  K_2\\big(A^{1,*}\\varphi\\big) \\|_\\gamma +  \\|  K_1\\big(A^{2,*}\\varphi\\big)\\|_{2\\gamma}\\\\\n\\end{split}\n\\end{equation*}\nNow\n\\begin{equation*}\n\\begin{split}\nK_n(\\varphi) \\leq \\lambda^{-n\\gamma}[(1+\\lambda^\\gamma+\\lambda^{2\\gamma})\\llparenthesis\\, f\\,\\rrparenthesis_{\\textrm{E}^*} + \\lambda^{3\\gamma} \\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma;\\textrm{F}^*}] J_n(\\varphi)\n\\end{split}\n\\end{equation*}\nso\n\\begin{equation*}\n\\begin{split}\n \\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma;\\textrm{F}^*}\n&\\leq  [4 \\lambda^{-2\\gamma}\\llparenthesis\\, f\\,\\rrparenthesis_{\\textrm{E}^*} + \\lambda^{\\gamma} \\llparenthesis\\, f^\\sharp\\,\\rrparenthesis_{3\\gamma;\\textrm{F}^*}] N({\\bfA})\n\\end{split}\n\\end{equation*}\nwhere we assumed  $\\lambda \\leq 1$, with no loss of generality, to get a simpler expression for the formula. \nTaking $\\lambda\\leq 1$ small enough so that\n$\n\\lambda^{\\gamma} N({\\bfA}) \\leq 1\/2,\n$\nwe obtain the  inequality\n$$\n\\llparenthesis\\, f^\\sharp \\,\\rrparenthesis_{3\\gamma\\,;\\,\\textrm{F}^*} \\leq  8 \\lambda^{-2\\gamma} \n\\llparenthesis\\, f\\,\\rrparenthesis_{\\textrm{E}^*} N({\\bfA}).\n$$\n\\end{Dem}\n\n\\bigskip\n\nTaking\n$$\n\\textrm{E} = E_0, \\quad \\textrm{ and }\\quad \\textrm{F} = E_3,\n$$ \nand assuming that \n$$\nA^{1,*}_{ts} \\in \\textrm{L}\\big(E_1,E_0\\big) \\cap \\textrm{L}\\big(E_3,E_2\\big)\n,\\quad\n\\text{and}\\quad \nA^{2,*}_{ts} \\in \\textrm{L}\\big(E_3 , E_1\\big) \\cap \\textrm{L}\\big(E_2,E_0\\big)\n$$ \nfor all $0\\leq s\\leq t\\leq T$, then \n$\nN({\\bfA}) \\lesssim C_0^2\n$ \nwhere\n\\begin{equation}\n\\label{EqDefnC0}\nC_0 := 1+\\big\\|A^1\\big\\|_{\\gamma\\,;\\,(-0,-1)}  + \\big\\|A^2\\big\\|_{2\\gamma\\,;\\,(-0,-2)} +  \\big\\|A^1\\big\\|_{\\gamma\\,;\\,(-2,-3)}  + \\big\\|A^2\\big\\|_{2\\gamma\\,;\\,(-1,-3)}  < \\infty .\n\\end{equation}\nIndeed in this case we can take the decomposition $\\varphi_2 = J^\\varepsilon \\varphi$ and $\\varphi_1 = (\\mathrm{Id} - J^\\varepsilon) \\varphi$ in the definition of $J_n(\\varphi)$ and obtain, for example,\n \\begin{equation*}\n\\begin{split}\nJ_1(\\varphi) \\leq\n\\sup_{0 < \\varepsilon \\leq 1} \\Big\\{&   2\n ( \\varepsilon)^{-1} \\, | (\\mathrm{Id} - J^\\varepsilon) \\varphi|_0\n  +   \\| A^{1,*}J^\\varepsilon \\varphi\\|_{\\gamma;0}  \n + ( \\varepsilon)^{1}\\|A^{2,*} J^\\varepsilon \\varphi\\|_{2\\gamma;0} \n \\\\ &\n  +  ( \\varepsilon)^{2}|J^\\varepsilon \\varphi|_3 \n  \\Big\\} \\lesssim  C_0 |\\varphi|_1\n\\end{split}\n\\end{equation*}\nand similarly $J_2(\\varphi) \\lesssim C_0 |\\varphi|_2$ from which is easy to conclude the bound $N({\\bfA}) \\lesssim C_0^2$.\n\n\n\\smallskip\n\nNote that it follows from Banach uniform boundedness principle that for a solution path $\\big(f,f^\\sharp\\big)$, we have\n$$\n\\big\\| f^\\sharp\\big\\|_{3\\gamma\\,;\\,-3} \\leq \\big\\| f^\\sharp\\big\\|_{3\\gamma\\,;\\,-2} < \\infty.\n$$\nTheorem~\\ref{ThmGeneralRegularityGain} shows that one has an explicit upper bound for $\\big\\| f^\\sharp_{ts}\\big\\|_{3\\gamma\\,;\\,-3}$ in terms of $ \\llparenthesis f\\rrparenthesis_{-0}$ and $C_0$ only. This fact is recorded in the following :\n\n\\begin{thm}\n\\label{ThmRegularityGain} \nFor any $\\lambda \\leq 1$ such that $\\lambda^\\gamma C_0^2 \\lesssim 1$ we have the a priori bounds\n\\begin{equation}\n\\label{EqFundamentalAPrioriEstimate}\n\\max\\Big\\{  \\llparenthesis\\, \\delta f \\,\\rrparenthesis_{\\gamma\\,;\\, - 1} , \\llparenthesis\\, f^\\flat \\,\\rrparenthesis_{2 \\gamma\\,;\\, - 2}, \\llparenthesis\\, f^\\sharp \\,\\rrparenthesis_{3 \\gamma\\,;\\, - 3} \\Big\\} \\lesssim_{\\gamma} C_0^2\\,\\lambda^{-2\\gamma}\\,\\llparenthesis\\, f\\,\\rrparenthesis_{-0} .\n\\end{equation}\n\\end{thm}\n\n\\medskip\n\n\n\\section{Integration of unbounded rough drivers in Hilbert spaces}\n\\label{SectionHilbertSpaceTheory}\n\nWe develop in this section the theory of integration of unbounded rough drivers in the Hilbert space $\\LL^2(\\RR^d)=E_0=E_{-0}$. We are able to give a rather complete theory a class of drivers that enjoys the same algebraic properties as the rough drivers ${\\bfA} = \\big(X\\,V,\\bbX\\,V\\,V\\big)$ involved in the rough transport equation, when the vector fields $V=\\big(V_1,\\dots,V_\\ell\\big)$ are divergence free. These drivers are called conservative. A general existence result for the rough linear equation $df_s = {\\bfA}(ds)f_s$, driven by conservative drivers is given in Section~\\ref{SubsectionIntegrationConservative}. To prove uniqueness of solutions to such equations, we develop in section \\eqref{SubsectionTensorization} a robust tensorization argument for a larger class of unbounded rough drivers that is the key to obtain some a priori bounds. These bounds imply uniqueness for rough linear equations driven by conservative drivers under a mild additional assumption, but they also lead to a complete $\\LL^2$-theory of rough transport equations, as illustrated in section \\eqref{SubsectionL2Transport}.\n\n\\smallskip\n\nNote that working in a Hilbert space setting, we have the continuous inclusions\n\\begin{equation}\n\\label{EqDoubleInclusion}\nE_n\\subset \\cdots \\subset E_1\\subset E_0 = E_{-0}\\subset E_{-1}\\subset \\cdots\\subset E_{-n}.\n\\end{equation}\n\n\\medskip\n\n\\subsection{Conservative drivers}\n\\label{SubsectionIntegrationConservative}\n\nWe start with the simple situation where the driver is \\emph{conservative} according to the following definition.\n\n\\begin{defn}\nA rough driver is said to be \\textbf{\\emph{conservative}} if we have \n\\begin{equation*}\n\\begin{split}\n&{\\bf (i)}\\quad A^{1,*}_{ts} +A^1_{ts} = 0, \\quad \\textrm{ on } E_1, \\\\\n&{\\bf (ii)}\\quad A^{2,*}_{ts} + A^2_{ts} + A_{ts}^{1,*}  A^1_{ts} = 0, \\quad \\textrm{ on } E_2,\n\\end{split}\n\\end{equation*}\nfor all $0\\leq s\\leq t<T$.\n\\end{defn}\nNotice that the conservative conditions make sense because of the above continuous inclusions \\eqref{EqDoubleInclusion}. As an elementary example of conservative unbounded driver, take as Banach spaces $E_n$ the Sobolev spaces $W^{n,2}(\\RR^d)$, with norm $|\\varphi|_n := \\sum_{k=0}^n \\big|\\nabla^k\\varphi\\big|_{\\LL^2}$, and consider the unbounded rough driver $\\bf A$  given by the formula\n\\begin{equation}\n\\label{EqRoughDriverRoughPath}\nA^1_{ts} = X^i_{ts}V_i,  \\quad\\quad A^2_{ts} = \\bbX^{jk}_{ts}V_jV_k,\n\\end{equation}\nfor some $\\ell$-dimensional geometric $\\gamma$-rough path, with $\\frac{1}{3}<\\gamma\\leq 1$, and some \\emph{divergence-free} vector fields $(V_i)_{i=1..\\ell}$ on $\\RR^d$, with the latter understood as first order differential operators. Condition {\\bf (ii)} partly plays the role in our setting that the notion of weak geometric rough path plays in a rough paths setting. Also, the antisymmetry condition {\\bf (i)} holds due to the fact that $V_i$ have null divergence, and condition {\\bf (ii)} holds as a consequence of the weak geometric character of $\\bfX$. Indeed, in this setting we have\n\\begin{equation*}\n\\begin{split}\n&A^{1,*}_{ts} = X^i_{ts}(V_i)^* = - X^i_{ts} V_i = -A^1_{ts}, \\\\\n&A^{2,*}_{ts} + A^2_{ts} =  \\bbX^{jk}_{ts}\\big(V_k^*V_j^*+V_j V_k\\big) = \\frac1 2  X^j_{ts}X^k_{ts} \\big(V_kV_j +V_j V_k\\big) = - A^{1,*}_{ts} A^1_{ts}\n\\end{split}\n\\end{equation*}\non $E_1$ and $E_2$ respectively. The boundedness assumptions \\eqref{EqBoundednessAssumptionRoughDriver} that $A^1_{ts}$ and $A^2_{ts}$ need to satisfy hold if, for instance, the vector fields $V_i$ are $\\mcC^2_b$. (Note that our setting is by no means restricted to working with vector fields. Working in the spaces $E_n = W^{kn,2}(\\RR^d)$, we may take $V_i$ in the above formula for $\\bfA$ the operators $V_i=W_i^k$, for some divergence-free vector fields $W_i$. Less trivial operators appear in a number of examples.)\n\n\\smallskip\n\nA general existence result holds for equations driven by conservative rough drivers under very mild conditions on the functional setting.\n\n\\begin{thm}\n\\label{ThmRDEHilbertGeneral}\nAssume one can associate to the scale $(E_n)_{n\\geq 0}$ a family of \\emph{self-adjoint} smoothing operators $\\big(J^\\epsilon\\big)_{0<\\epsilon\\leq 1}$, from $E_0$ to itself, satisfying the regularisation estimates \\eqref{EqApproximationProperties}, and let $\\bfA$ be a conservative unbounded $\\gamma$-rough driver on the scales $(E_n)_{n\\geq 0}$. Then for any $f_0\\in\\LL^2(\\RR^d)$, there exists an $\\LL^2(\\RR^d)$-valued path $f_\\bullet$, started from $f_0$, such that we have\n\\begin{equation*}\n\\delta f_{ts}(\\varphi) = f_s\\Big(A^{1,*}_{ts}\\varphi\\Big) + f_s\\Big(A^{2,*}_{ts}\\varphi\\Big) + f^\\sharp_{ts}(\\varphi)\n\\end{equation*}\nfor all $\\varphi\\in E_3$, with \n\\begin{equation*}\n\\big| f_t\\big|_0\\leq \\big| f_0\\big|_0,\n\\end{equation*}\nfor all $ t\\geq 0$, and we have, for each finite time horizon $T$,\n\\begin{equation}\n\\label{EqBoundRemainderX}\n\\Big| f^\\sharp_{ts}(\\varphi) \\Big| \\lesssim_{C_0,{\\bfA}, T,| f_0|_0}\\,\\big|\\varphi\\big|_3\\,|t-s|^{3\\gamma},\n\\end{equation}\nfor $0\\leq s\\leq t\\leq T$.\n\\end{thm}\n\n\\noindent The proof goes by approximating the unbounded rough driver $\\bfA$ by bounded rough drivers  ${\\bfA}^\\epsilon$, and by using the theory developed in Section~\\ref{SectionBoundedDrivers} to solve the equation\n\\begin{equation}\n\\label{EqApproximatedRDE}\n\\delta f^\\epsilon_{ts}(\\varphi) = f^\\epsilon_s\\Big(A^{\\epsilon,1,*}_{ts}\\varphi\\Big) + f^\\epsilon_s\\Big(\\big(A^{\\epsilon,2}_{ts}\\big)^*\\varphi\\Big) + f^{\\epsilon,\\sharp}_{ts}(\\varphi).\n\\end{equation}\nThe a priori bound \\eqref{EqFundamentalAPrioriEstimate} is used to pass to the limit as  $\\epsilon$ goes to $0$.\n\n\\medskip\n\n\\begin{Dem}\nMore concretly, let \n$$\nA^{\\epsilon,1}_{ts} := J^\\epsilon A^1_{ts}J^\\epsilon,\\quad\\quad A^{\\epsilon,2}_{ts} := J^\\epsilon A^{2,a}_{ts}J^\\epsilon - \\frac1 2 A^{\\epsilon,1}_{ts} A^{\\epsilon,1,*}_{ts},\n$$\nwith $J^\\epsilon$ understood as a continuous operator from $E_0$ to $E_{k_0+2}$, for $k_0$ big enough, and $A^{2,a}_{ts} = \\frac{1}{2}\\big(A^2_{ts}-A^{2,*}_{ts}\\big)$ stands for the antisymmetric part of $A^{2}_{ts}$. Both $A^{\\epsilon,1}$ and $A^{\\epsilon,2}$ are bounded linear operators from $E_0$ to itself, and by construction ${\\bfA}^\\epsilon = \\big(A^{\\epsilon,1},A^{\\epsilon,2}\\big)$ is a \\emph{bounded} conservative $\\gamma$-rough driver on the Banach algebra $\\textrm{L}(E_0)$. So one can denote by $M^\\epsilon_\\bullet$ the solution path in $\\textrm{L}(E_0)$ of the rough differential equation\n$$\n\\delta M^\\epsilon_{ts} = \\Big(A^{\\epsilon,1}_{ts} + A^{\\epsilon,2}_{ts}\\Big)M^\\epsilon_s + M^{\\epsilon,\\sharp}_{ts},\n$$\nprovided by the theory developed in Section~\\ref{SectionBoundedDrivers}. Given $f_0\\in E_0$, the path $f^\\epsilon_\\bullet = M^\\epsilon_\\bullet f_0$ solves the rough differential equation \\eqref{EqApproximatedRDE}. Whereas Theorem~\\ref{ThmIntegrationLinearBounded} provides an exponential control of the growth of the $E_0$-norm of $f^\\epsilon_t$, with an exponent $\\lambda^\\epsilon$ that may go to $\\infty$ as $\\epsilon$ goes to $0$, the conservative character of ${\\bfA}^\\epsilon$ ensures the uniform bound\n\\begin{equation*}\n\\big| f^\\epsilon_t\\big|_0 = \\big| f_0\\big|_0.\n\\end{equation*}\nIndeed, denoting by $(\\cdot,\\cdot)$ the scalar product on $E_0$, and setting $z^\\epsilon_t = \\big|f^\\epsilon_t\\big|^2_0$, one has\n\\begin{equation*}\n\\begin{split}\n\\delta z^\\epsilon_{ts} &= 2\\big(\\delta f^\\epsilon_{ts},f_s\\big) + \\big(\\delta f^\\epsilon_{ts}, \\delta f^\\epsilon_{ts}\\big) \\\\\n                                    &= 2\\Big(\\big(A^{\\epsilon,1}_{ts}+A^{\\epsilon,2}_{ts}\\big)f_s , f_s\\Big) + \\big(f^{\\epsilon,\\sharp}_{ts},f_s\\big) \\\\\n                                    &\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+ \\Big(\\big(A^{\\epsilon,1}_{ts}+A^{\\epsilon,2}_{ts}\\big)f_s , \\big(A^{\\epsilon,1}_{ts}+A^{\\epsilon,2}_{ts}\\big)f_s\\Big) + O\\big(|t-s|^{3\\gamma}\\big) \\\\\n                                    &= 2\\Big(A^{\\epsilon,2}_{ts}f_s , f_s\\Big) + \\Big(A^{\\epsilon,1}_{ts}f_s , A^{\\epsilon,1}_{ts}f_s\\Big) + O\\big(|t-s|^{3\\gamma}\\big)\n\\end{split}\n\\end{equation*} \nby the conservative character of $A^{\\epsilon,1}_{ts}$, which finally gives\n$$\n\\delta z^\\epsilon_{ts} = O\\big(|t-s|^{3\\gamma}\\big),\n$$\nsince the symmetric part of $A^{\\epsilon,2}_{ts}$ is $-\\frac{1}{2}A^{\\epsilon,1,*}_{ts}A^{\\epsilon,1}_{ts}$. The above equality shows that $z^\\epsilon$ is constant.\n\n\\smallskip\n\nRemark at that point that the constant \n\\begin{equation*}\nC^\\epsilon_0 := \\big\\|A^{\\epsilon,1}\\big\\|_{\\gamma\\,;\\,(-0,-1)}  + \\big\\|A^{\\epsilon,2}\\big\\|_{2\\gamma\\,;\\,(-0,-2)} +  \\big\\|A^{\\epsilon,1}\\big\\|_{\\gamma\\,;\\,(-2,-3)}  + \\big\\|A^{\\epsilon,2}\\big\\|_{2\\gamma\\,;\\,(-1,-3)}  < \\infty\n\\end{equation*}\nis not only finite but also uniformly bounded above, independently of $\\epsilon>0$. So we have, by Theorem~\\ref{ThmRegularityGain}  an $\\epsilon$-uniform upper bound on $\\big\\|f^{\\epsilon,\\sharp}\\big\\|_{3\\gamma\\,;\\,-3}$, of the form\n\\begin{equation}\n\\label{EqBondfSharpEpsilon}\n\\big\\|f^{\\epsilon,\\sharp}\\big\\|_{3\\gamma\\,;\\,-3} \\lesssim_{\\gamma,T,| f_0|_0}\\,1. \n\\end{equation}\nThese bounds ensure in particular that for each $\\varphi\\in E_3$, the functions $f^\\epsilon_\\bullet(\\varphi)$ form a bounded family of $\\gamma$-H\\\"older real-valued paths, so it has a subsequence converging uniformly to some $\\gamma$-H\\\"older real-valued function. Moreover, by weak-$\\star$ compactness, the uniform bound \\eqref{EqBondfSharpEpsilon} implies the existence of a sequence $(\\epsilon_n)_{n\\geq 0}$ converging to $0$, such that the sequence $f^{\\epsilon_n}$ converges weakly-$\\star$ in $\\LL^\\infty\\big([0,T],E_0\\big)$ to some limit $f\\in\\LL^\\infty\\big([0,T],E_0\\big)$. In particular, for each $\\varphi\\in E_3$, the sequence $f^{\\epsilon_n}(\\varphi)$ converges weakly-$\\star$ in $\\LL^\\infty\\big([0,T],\\RR\\big)$ to $f(\\varphi)$. As it has a uniformly converging subsequence, this shows the $\\gamma$-H\\\"older character of each function $f(\\varphi)$. \n\nAssuming $\\varphi\\in E_3$, it follows that one can pass to the limit in equation \\eqref{EqApproximatedRDE} in the three terms involving $f^\\epsilon_s$. The limit $f^\\sharp_{ts}(\\varphi)$ is defined as a consequence, and the bound \\eqref{EqBoundRemainderX} follows as a direct consequence of \\eqref{EqBondfSharpEpsilon}.\n\n\\smallskip \n\nIt is elementary to extend the above solution defined on $[0,T]$ to a globally defined solution satisfying the statement of the theorem.\n\\end{Dem}\n\n\\bigskip\n\nRather than working with a general scale of spaces satisfying some ad hoc conditions, we shall set \n$$\nE_n = W^{n,2}(\\RR^d),\n$$\nfor the remainder of this section on linear rough differential equation on Hilbert spaces. So we shall essentially be working from now on with rough drivers given by (at most) first order rough (pseudo-)differential operators. \n\n\n\\bigskip\n\n\n\\subsection{Tensorization}\n\\label{SubsectionTensorization}\n\nIn order to study the problem of uniqueness and further properties of solutions to general linear rough equations associated to unbounded rough drivers, we develop in this section a tensorization argument which can be seen as a rough version of the (differential) second quantisation functor in Hilbert spaces \\cite{ReedSimon}, or the variables doubling method commonly used in the theory of transport equations and conservation laws after the pioneering work of Kruzkhov \\cite{Kruzkhov}. As far as applications are concerned, we shall not restrict much our range in assuming that the rough drivers we are working with enjoy the following property. Given a bounded function $\\phi\\in W^{n_0,\\infty}$, denote by $M_\\phi$ the multiplication operator by $\\phi$; it is a bounded operator from $E_0 = \\LL^2(\\RR^d)$ to itself.\n\n\\begin{defn}\n\\label{DefnSymmetricDriver}\nAn unbounded rough driver $\\bfA$ is said to be \\textbf{\\emph{symmetric}} if the symmetric operators \n\\begin{equation*}\n\\begin{split}\n&{\\bf (i)}\\quad B^1_{{\\bfA},ts}(\\phi) = A^{1,*}_{ts}M_\\phi  + M_\\phi A^1_{ts} , \\\\\n&{\\bf (ii)}\\quad B^2_{{\\bfA},ts}(\\phi) = M_\\phi A^{2,*}_{ts}  +  A^2_{ts}M_\\phi + A_{ts}^1 M_\\phi A^{1,*}_{ts},\n\\end{split}\n\\end{equation*}\ndefine quadratic forms \n$$\ng\\mapsto \\Big(g,B^i_{{\\bfA},ts}(\\phi)g\\Big)\n$$\nthat are continuous on $E_0$, for all $0\\leq s\\leq t<T$, and for any test function $\\phi\\in W^{3,\\infty}$.\n\\end{defn}\n\nFor rough drivers ${\\bfA} = \\big(A^1,A^2\\big)$ of the form \n$$\nA^1_{ts} = X^i_{ts} V_i, \\quad\\quad A^2_{ts} = {\\bbX}^{jk}_{ts} V_jV_k,\n$$\nfor some vector fields $V_i$ on $\\RR^d$, and some weak geometric rough path $\\bfX$, the operator $B^1(\\phi)$ is the multiplication operator by $A^{1,*}_{ts}\\phi$, and the second quadratic form is of the form\n$$\n\\Big(g,B^2_{{\\bfA},ts}(\\phi)g\\Big) = \\Big(g^2 , h^\\phi_{ts}\\Big),\n$$\nfor some explicit function $h^\\phi_{ts}$ that turns $\\bfA$ into a symmetric unbounded rough driver if the vector fields $V_i$ are $\\mcC^2_b$ and $\\phi\\in W^{2,\\infty}$. \n\n\\bigskip \n \nLet now $\\bfA$ be a symmetric unbounded rough driver in the scale $(E_n)_{n\\geq 0}$, and let $f_\\bullet$ be a solution to the linear rough differential equation \n$$ \ndf_s = {\\bfA}(ds)f_s.\n$$\nConsider $f^{\\otimes 2} = f \\otimes f$, defined on $\\RR^d\\times\\RR^d$ by\n$\nf^{\\otimes 2}(x,y) = f(x)f(y)\n$; \nit satisfies the equation\n$$ \n\\delta f^{\\otimes 2} = \\Big( A^1  \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^1 \\Big) f^{\\otimes 2} + \\Big(A^2 \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^2 + \\tau (A^1 \\otimes A^1)\\Big) f^{\\otimes 2} + f^{\\otimes 2, \\sharp},\n$$\nwhere $\\mathbbm{I}$ stands for the identity map and\n$$\n\\tau (A^1 \\otimes A^1)_{t s} = A^1_{t s} \\otimes A^1_{t s}.\n$$\nSetting \n\\begin{equation*}\n\\begin{split}\n&\\Gamma_{\\bfA}^1 := A^1  \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^1, \\\\\n&\\Gamma_{\\bfA}^2 := A^2 \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes A^2 + \\tau (A^1 \\otimes A^1),\n\\end{split}\n\\end{equation*}\nand \n$$\n\\Gamma_{\\bfA} := \\big(\\Gamma_{\\bfA}^1, \\Gamma_{\\bfA}^2\\big), \n$$ \nwe have then \n$$\n\\delta \\Gamma_{\\bfA}^2 = \\big(A^1 A^1\\big) \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes \\big(A^1 A^1\\big) + A^1 \\otimes A^1 + \\sigma (A^1 \\otimes A^1),\n$$ \nwhere $\\sigma (v \\otimes w) = w \\otimes v$ is the exchange of the two factors in the tensor so that \n$$\n\\sigma (A^1 \\otimes A^1)_{t u s} = A^1_{us} \\otimes A^1_{tu} + A^1_{tu} \\otimes A^1_{us}.\n$$ \nIf follows that $\\Gamma_{\\bfA}$ satisfies Chen's relations \\eqref{EqChenRelations}. Endowing $E_n^{\\otimes 2}$ with its natural Hilbert space structure, we turn $\\Gamma_{\\bfA}$ into a $\\gamma$-H\\\"older unbounded rough driver in the scale $\\big(E_n^{\\otimes 2}\\big)_{n \\geq 0}$, for which $f^{\\otimes 2}$ happens to be a solution, in this scale of spaces, of the linear rough equation\n$$\ndf^{\\otimes 2}_s = \\Gamma_{\\bfA}(ds)f^{\\otimes 2}_s. \n$$\n\n\\smallskip \n\nThe goal of the doubling variable method is to use the dynamics of $f^{\\otimes 2}$ to gain some information of its behaviour near the diagonal of $\\RR^d \\times \\RR^d$ in order to control the trace of $f^{\\otimes 2}$ on the diagonal. \n\nThinking to this goal let us define a scale of spaces $\\mathcal{E}^\\nabla_n$ of test functions on  $\\RR^d \\times \\RR^d$, by requiring that they have finite $| \\cdot |^\\nabla_n$-norm. In order to introduce this norm let us define\n$$\n| \\Phi |^\\nabla_n :=\\sup_{0\\le k+l\\le n}  \\int_{\\RR^d} \\sup_{z\\in\\RR^d} \\big|\\partial_z^k \\partial_w^l\\Phi(z+w, z-w)\\big|\\, dw.\n$$ \nMoreover we introduce a blow-up transformation $T_\\varepsilon$ on test functions as follows:\n$$\nT_\\varepsilon \\Phi(x,y) =\\varepsilon^{-d} \\Phi\\big(x_+ + \\frac{x_-}{\\varepsilon}, x_+ - \\frac{x_-}{\\varepsilon}\\big) \n$$\nwhere $x_\\pm = \\frac{x \\pm y}{2}$ are coordinates parallel and transverse to the diagonal of $\\RR^d \\times \\RR^d$. Note that $T_\\varepsilon$ is a bounded operator in $\\mathcal{E}^\\nabla_0$ uniformly in $\\varepsilon$. The adjoint of this transformation for the $\\LL^2$ scalar product reads\n$$\nT^*_\\varepsilon F(x,y) = F\\big(x_+ + \\varepsilon x_- , x_+ -\\varepsilon x_-\\big).\n$$\nWe let  \n$$\n\\Gamma_{\\bfA,\\varepsilon} := T_\\varepsilon^{-1}\\Gamma_{\\bfA}T_\\varepsilon.\n$$ \nRecall the definition of the operators $ B^1_{{\\bfA},ts}(\\phi)$ and $ B^2_{{\\bfA},ts}(\\phi)$ used in the definition of a symmetric unbounded rough driver. We use below brackets $\\langle\\cdot,\\cdot\\rangle$ to denote the $\\LL^2$ scalar product. Note that $f_s$ is an element of $E_{-0} = E_0 = \\LL^2(\\RR^d)$, so $f_s^2$ is in $\\LL^1(\\RR^d)$. \n \n\\begin{defn}[Renomalizable drivers]\nThe unbounded $\\gamma$-rough driver $\\bfA$ is \\textbf{\\emph{renormalizable}} if $\\{\\Gamma_{\\bfA,\\varepsilon}\\}_{\\varepsilon>0}$  is a bounded family of unbounded $\\gamma$-rough drivers in the scale of spaces $\\big(\\mathcal{E}^\\nabla_n\\big)_{n\\geq 0}$.\n\\end{defn} \n\nThe following lemma gives flesh to the expresion 'Renormalizable driver', such as defined here. It needs to be understood in the light of di Perna-Lions' work on transport equation \\cite{diPernaLions} where a notion of renormalizable solution was first introduced. Definition~\\ref{DefnrenormalisedSolutions} will make that parallel clear in our study of the $\\LL^\\infty$ theory for the rough transport equation. Under some additional assumption stated in Definition~\\ref{DefnClosedDriver}, the present notion of renormalizable unbounded rough driver will provide in Theorem~\\ref{ThmRDEHilbertGeneralClosed} a general uniqueness result.  \n\n\\begin{lem}[Renormalisation]\n\\label{Lemmarenormalisation}\nLet $\\bfA$ be a symmetric rough driver and $f_\\bullet$ be a solution of the equation \n$$\ndf_s = {\\bfA}(ds)f_s,\n$$ \nin the initial scale of spaces $(E_n)_{n\\geq 0}$. If $\\bfA$ is renormalizable \nthen the $\\LL^1(\\RR^d)$-valued path $f^2_\\bullet$ satisfies, for all $\\phi \\in W^{3,\\infty}$, the equation\n\\begin{equation} \n\\label{EqPrimitiveGronwall}\n\\delta f^2(\\phi)_{ts} =  \\langle f_s , \\big(B^1_{{\\bfA},ts}(\\phi) + B^2_{{\\bfA},ts}(\\phi)\\big) f_s\\rangle  + O (| \\phi |_{W^{3,\\infty}} | t - s |^{3 \\gamma}) .\n\\end{equation}\n \\end{lem}\n\n\\smallskip\n \n\\noindent By polarisation the product $fg$ satisfies an equation analogous to equation \\eqref{EqPrimitiveGronwall} if both $f$ and $g$ are solutions of the equation $df_s = {\\bfA}(ds)f_s$, in the scale $(E_n)_{n\\geq 0}$.\n\n\\bigskip\n\n\\begin{Dem}\nNote that $f_\\bullet^{\\otimes 2}$ satisfies the equation\n\\begin{equation}\n\\label{EqFOtimes2}\n\\delta f^{\\otimes 2}_\\varepsilon (\\Phi)_{ts} = f^{\\otimes 2}_{\\varepsilon,s}((\\Gamma_{\\bfA,\\varepsilon}^{1,*})_{ts}\\Phi) +f^{\\otimes 2}_{\\varepsilon,s} ( (\\Gamma_{\\bfA,\\varepsilon}^{2,*})_{ts}\\Phi) +  f^{\\otimes 2,\\sharp}( T_\\varepsilon\\Phi)_{ts}\n\\end{equation}\nfor all smooth functions $\\Phi$ where $f^{\\otimes 2}_\\varepsilon = T^*_\\varepsilon f^{\\otimes 2}$.\nNote that if we show that $f^{\\otimes 2}_\\varepsilon$ is uniformly bounded in $\\mathcal{E}^\\nabla_{-0}$ then from the hypothesis that  $\\{\\Gamma_{\\bfA,\\varepsilon}\\}_{\\varepsilon>0}$ is a bounded family of unbounded $\\gamma$-rough drivers in the scale of spaces $\\big(\\mathcal{E}^\\nabla_n\\big)_{n\\geq 0}$ we also have\n $\\big\\|f^{\\otimes 2,\\sharp}(T_\\varepsilon \\Phi)\\big\\|_{3\\gamma}\\lesssim \\big| \\Phi\\big|^\\nabla_3$. As a consequence the $3\\gamma$-H\\\"older norm of the remainders $f^{\\otimes 2,\\,\\sharp}\\big(T_\\varepsilon \\Phi\\big)$ are bounded uniformly in  $\\varepsilon$ for fixed $\\Phi$. \n\nEquation \\eqref{EqPrimitiveGronwall} will come from taking in equation \\eqref{EqFOtimes2} some functions $\\Phi$ of the form\n$$\n\\Phi(x,y) = \\psi(x-y)\\,\\phi\\left(\\frac{x+y}{2}\\right),\n$$\nand by letting $\\varepsilon$ tend to $0$, after checking that some $\\varepsilon$-uniform estimates hold for the different terms in \\eqref{EqFOtimes2}.\n\n\\smallskip\n Cauchy--Schwartz inequality provides the  bound\n\\begin{equation*}\n\\begin{split}\n\\Big|f^{\\otimes 2}_s \\big(T_\\varepsilon \\Phi\\big)\\Big| &= \\left|\\int_{\\RR^d \\times \\RR^d} f_s(x_+ + \\varepsilon x_-) f_s(x_+ - \\varepsilon x_-) \\Phi(x_+ +  x_-,x_+ -  x_-)\\,dx_+ dx_- \\right|  \\\\\n&\\leq \\max_\\pm \\int_{\\RR^d \\times \\RR^d} |f_s(x_+ \\pm \\varepsilon x_-)|^2  |\\Phi(x_+ +  x_-,x_+ -  x_-)|\\,dx_+ dx_- \\\\\n&\\leq  \\max_\\pm   \\int_{\\RR^d \\times \\RR^d} |f_s(x_+ \\pm \\varepsilon x_-)|^2  \\sup_z |\\Phi(z +  x_-,z -  x_-)|\\,dx_+ dx_- \\\\\n&\\leq |f|_{\\LL^2}^2 \\int_{\\RR^d } \\sup_{z}| \\Phi(z +  w,z -  w)|\\,dw  \\\\\n&\\leq  |\\Phi|_{\\mathcal{E}^\\nabla_0} \\big|f_s\\big|_{\\LL^2}^2. \n\\end{split}\n\\end{equation*}\nwhich shows that $f^{\\otimes 2}_\\varepsilon$ is uniformly bounded in $\\mathcal{E}^\\nabla_{-0}$.\nNow, given a positive constant $\\delta$, the fact that for any smooth function $g$ which is $\\delta$-close in $\\LL^2(\\RR^d)$ of $f_s$, we have\n$$ \n\\Big|f^{\\otimes 2}_s \\big(T_\\varepsilon \\Phi\\big)- g^{\\otimes 2} \\big(T_\\varepsilon \\Phi\\big) \\Big| \\lesssim 2\\delta |f|_{\\LL^2} + \\delta^2,\n$$ \nuniformly in $\\varepsilon$, and\n$$\n\\lim_{\\varepsilon\\to 0} g^{\\otimes 2} \\big(T_\\varepsilon \\Phi) = \\int_{\\RR^d }\\big|g(x)\\big|^2 \\phi (x)\\,dx,\n$$\nshows that \n$$\nf^{\\otimes 2}_s \\big(T_\\varepsilon \\Phi\\big) \\underset{\\varepsilon\\to 0}{\\longrightarrow} f_s^2(\\phi). \n$$\nWe also have the convergence \n$$\ng^{\\otimes 2} \\left((\\Gamma_{\\bfA}^1)^*_{ts} T_\\varepsilon \\Phi\\right) = \\big( A^1_{ts} g \\otimes g + g \\otimes A^1_{ts} g, T_\\varepsilon \\Phi \\big) \\underset{\\varepsilon\\to 0}{\\longrightarrow} 2 \\int_{\\RR^d } g(x) \\big(A^1_{ts}g\\big)(x) \\phi (x)\\, dx,\n$$\nwhich we can rewrite as\n$$\ng^{\\otimes 2} \\left((\\Gamma_{\\bfA}^1)^*_{ts} T_\\varepsilon \\Phi\\right) \\underset{\\varepsilon\\to 0}{\\longrightarrow} 2\\Big( g , A^{1,*}_{ts} M_\\phi g \\Big) = \\Big(g,B^1_{{\\bfA},ts}(\\phi)g\\Big).\n$$\nUsing in addition the boundedness on $\\LL^2(\\RR^d)$ of the quadratic form associated to $B^1_{ts}(\\phi)$, one can then send $g$ to $f_s$, in $\\LL^2(\\RR^d)$, in the above convergence result and conclude that\n$$\nf^{\\otimes 2}_s \\big((\\Gamma_{\\bfA}^1)_{ts}^* T_\\varepsilon \\Phi\\big) \\to \\big( f_s , B^1_{{\\bfA},ts}(\\phi) f_s\\big). \n$$\nSimilarly, the boundedness in $\\mathcal{E}^\\nabla_0$ of the family $\\big((\\Gamma_A^2)^*_{ts}  T_\\varepsilon \\Phi\\big)_{0<\\varepsilon\\leq 1}$, together with the boundedness on $\\LL^2(\\RR^d)$ of the quadratic form associated with $B^2_{{\\bfA},ts}(\\phi)$, show that \n$$\nf^{\\otimes 2}_s \\big((\\Gamma_{\\bfA}^2)_{ts}^*\\Phi^\\epsilon\\big) \\to \\big( f_s , B^2_{{\\bfA},ts}(\\phi) f_s\\big);\n$$\nequation \\eqref{EqPrimitiveGronwall} follows, as we have the $\\varepsilon$-uniform bound $\\Big|f^{\\otimes 2,\\,\\sharp}_{ts}(T_\\varepsilon \\Phi)\\Big| \\leq  \\big|\\Phi\\big|_{3}^\\nabla \\leq \\big|\\phi\\big|_{W^{3,\\infty}}|t-s|^{3\\gamma}$.  \n\\end{Dem} \n\n\\bigskip\n\nThis result is sufficient to prove that rough linear equations driven by conservative drivers are unique if the driver is symmetric. \n\n\\begin{cor}\n\\label{CorClosedSymmCons}\nLet $\\bfA$ be a renormalizable symmetric conservative unbounded $\\gamma$-rough driver in the scale of spaces $(E_n)_{n\\geq 0}$. Then the rough linear equation\n$$\ndf_s = {\\bfA}(ds)f_s\n$$\nhas a unique solution in $\\LL^2(\\RR^d)$, started from any initial condition $f_0\\in E_0$; it satisfies $\\big|f_t\\big|_0 = \\big|f_0\\big|_0$, for all times $t$.\n\\end{cor}\n\n\\smallskip\n\n\\begin{Dem}\nIt suffices to notice that since the driver $\\bfA$ is conservative, we have $B^1_{{\\bfA},ts}({\\bf 1}) = B^2_{{\\bfA},ts}({\\bf 1}) = 0$, so it follows from equation \\eqref{EqPrimitiveGronwall} that any solution path $f_\\bullet$ has constant $\\LL^2$-norm, which proves the uniqueness claim. Existence was proved in Theorem~\\ref{ThmRDEHilbertGeneral}.\n\\end{Dem}\n\n\n\\bigskip\n\n\n\\subsection{A priori bounds for closed symmetric drivers}\n\\label{SymmetricDrivers}\n\nOne cannot use directly the renormalisation lemma to get some closed equation for $f^2$ when $\\bfA$ is non-conservative. We need for that purpose to assume that the symmetric unbounded rough driver $\\bfA$ enjoys the following property. Given that $\\bfA$ is symmetric, recall the definition of its associated family of symmetric operators $B^1_{{\\bfA},ts}(\\phi)$ and $B^2_{{\\bfA},ts}(\\phi)$ on $E_0$, indexed by $(s,t)$ and $\\phi\\in W^{3,\\infty}$, given in Definition~\\ref{DefnSymmetricDriver}.\n\n\\begin{defn}\n\\label{DefnClosedDriver}\nA \\emph{\\textbf{symmetric unbounded rough driver}} $\\bfA$ in the scale of spaces $\\big(W^{n,2}\\big)_{n\\geq 0}$, is said to be \\emph{\\textbf{closed}} if there exists some unbounded rough driver ${\\bf B}=\\big(B^1,B^2\\big)$ in the scale of spaces $\\big(W^{n,\\infty}\\big)_{n\\geq 0}$, such that we have  \n$$\n\\Big(g,B^1_{{\\bfA},ts}(\\phi)g\\Big) = \\Big(g^2,(B^1_{ts})^*\\phi\\Big), \\quad \\textrm{  and  } \\quad \\Big(g,B^2_{{\\bfA},ts}(\\phi)g\\Big) = \\Big(g^2,(B^2_{ts})^*\\phi\\Big),\n$$\nfor all $g\\in E_0$ and $\\phi\\in W^{3,\\infty}$. \n\\end{defn}\n\n\\smallskip\n\nAs an example, it is elementary to check that the unbounded rough driver ${\\bfA} = \\big(XV,\\bbX VV\\big)$ used in the rough transport equation\n\\begin{equation*}\n\\delta f_{ts} = X\\,Vf_s +  \\bbX\\,V V f_s + f^\\sharp_{ts}\n\\end{equation*}\nwith some $\\gamma$-H\\\"older weak geometric rough path ${\\bfX} = (X,\\bbX)$, is closed and symmetric if the vector fields $V = \\big(V_1,\\dots,V_\\ell\\big)$ are $C^3_b$, in which case $\\bfB = \\bfA$. Another interesting class of examples of closed symmetric unbounded rough driver in the scale of spaces $(E_n)_{n\\geq 0}$, is provided by the lift to rough drivers of $\\mcC^3_b$-semimartingale velocity fields, as given in the theory of stochastic flows. This kind of stochastic velocity fields appear in the study of Navier--Stokes equation. See the work \\cite{BailleulRiedel} for a thorough study of stochastic flows from this point of view. \n\n\\medskip\n\nBuilding on this notion of closed driver, the following statement provides amongst other things an a priori estimate on solutions of rough linear equations that plays in this setting the role played in the classical setting by a priori estimates obtained by any kind of Gronwall--type argument. The crucial point here is that no such Gr\\\"onwall machinery was available so far in a rough path--like setting; despite its elementary nature, this result may well be one of our main contributions. \n\n\\smallskip\n\n\\begin{thm}\n\\label{ThmRDEHilbertGeneralClosed}\nLet $\\bfA$ be a renormalizable  \\emph{closed symmetric} unbounded $\\gamma$-rough driver on the scales $\\big(W^{n,2}(\\RR^d)\\big)_{n\\in\\mathbb{N}}  \\newcommand{\\ZZ}{\\mathbb{Z}}  \\newcommand{\\QQ}{\\mathbb{Q}}  \\newcommand{\\HH}{\\mathbb{H}}$. Let $\\bfB$ be its associated driver, and assume that the inequality\n\\begin{equation}\n\\label{BoundPhi}\n\\Big|\\big(B^1_{t0}\\big)^*{\\bf 1} \\Big| \\vee \\Big|\\big(B^2_{t0}\\big)^*{\\bf 1} \\Big| \\leq c_t\n\\end{equation}\nholds for all times $t$, for some time-dependent mositive constant $c_t$ such that $e^{-\\lambda t}c_t$ tends to $0$ as $t$ goes to infinity, for any positive parameter $\\lambda$, so\n$$\n\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis < \\infty.\n$$\n Then, given any $f_0\\in\\LL^2(\\RR^d)$, there is at most one $\\LL^2(\\RR^d)$-valued solution path $f_\\bullet$ to the equation\n\\begin{equation}\n\\label{EqLinearRDEHilbert}\n\\delta f_{ts}(\\varphi) = f_s\\Big(A^{1,*}_{ts}\\varphi\\Big) + f_s\\Big(A^{2,*}_{ts}\\varphi\\Big) + f^\\sharp_{ts}(\\varphi),\n\\end{equation}\nand we have, for each finite time horizon $T$, \n\\begin{equation}\n\\label{EqBoundRemainder}\n\\Big| f^\\sharp_{ts}(\\varphi) \\Big| \\lesssim_{{\\bfB},T, | f_0|_0}\\,\\big|\\varphi\\big|_3\\,|t-s|^{3\\gamma},\n\\end{equation}\nfor all $\\varphi\\in W^{3,2}$, and all $0\\leq s\\leq t\\leq T$. It satisfies the upper bound\n\\begin{equation}\n\\label{EqAPrioriBoundNorm}\n\\big| f_t\\big|_0\\lesssim_{{\\bfB}, t} \\big| f_0\\big|_0\n\\end{equation}\n\\end{thm}\n\n\\smallskip\n \n\\begin{Dem}\n\\noindent Let $f_\\bullet$ be a solution to the rough linear equation \\eqref{EqLinearRDEHilbert} in the scale of spaces $\\big(W^{n,2}(\\RR^d)\\big)_{n\\in\\mathbb{N}}  \\newcommand{\\ZZ}{\\mathbb{Z}}  \\newcommand{\\QQ}{\\mathbb{Q}}  \\newcommand{\\HH}{\\mathbb{H}}$. Since $\\bfA$ is closed, the $\\LL^1(\\RR^d)$-valued path $f^2_\\bullet$ happens to be a solution to the rough linear equation\n\\begin{equation*} \n\\delta f^2(\\phi)_{ts} = f^2_s\\big((B^1_{ts})^*\\phi\\big) + f^2_s\\big((B^2_{ts})^*\\phi\\big) + (f^2)^\\sharp_{ts}(\\phi),\n\\end{equation*}\nin the scale of spaces $\\big(W^{n,\\infty}(\\RR^d)\\big)_{n\\in\\mathbb{N}}  \\newcommand{\\ZZ}{\\mathbb{Z}}  \\newcommand{\\QQ}{\\mathbb{Q}}  \\newcommand{\\HH}{\\mathbb{H}}$. Denote by $C_0^{\\bf B}$ the finite constant associated to the unbounded rough driver $\\bfB$, as defined by equation \\eqref{EqDefnC0}, with $\\bfB$ in the role of $\\bfA$. It follows from the general a priori estimates on solutions of rough linear equations proved in Theorem~\\ref{ThmGeneralRegularityGain}, and the fact that $f^2$ is in $\\LL^1(\\RR^d)$, that \n\\begin{equation}\n\\label{EqBoundf2}\n\\begin{split}\n\\llparenthesis\\, (f^2)^\\sharp \\,\\rrparenthesis_{3\\gamma\\,;\\,(W^{3,\\infty})^*} &\\lesssim_{\\gamma,\\lambda}C_0^{\\bf B}\\,\\llparenthesis\\, f^2 \\,\\rrparenthesis_{(\\LL^\\infty)^*} \n\\lesssim _{\\gamma,\\lambda} C_0^{\\bf B}\\,\\llparenthesis\\, f^2_\\bullet \\,\\rrparenthesis_{\\LL^1}.\n\\end{split}\n\\end{equation}\nBut since we have the identity \n$$\nf^2_t({\\bf 1}) = f_0^2\\Big({\\bf 1} + (B^1_{ts})^*{\\bf 1} + (B^2_{ts})^*{\\bf 1}\\Big) + (f^2)^\\sharp_{t0}({\\bf  1})\n$$\nand the bound \\eqref{BoundPhi}, we also have the estimate\n\\begin{equation*}\n\\begin{split}\n\\llparenthesis\\, f^2_\\bullet \\,\\rrparenthesis_{\\LL^1} = \\llparenthesis\\, f^2_\\bullet({\\bf 1}) \\,\\rrparenthesis &\\lesssim_{\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis} \\,\\big| f_0 \\big|_{\\LL^2} + \\llparenthesis\\, (f^2)^\\sharp_{\\bullet 0} \\,\\rrparenthesis_{(W^{3,\\infty})^*}  \\\\\n&\\lesssim_{\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis} \\, \\Big( \\big| f_0 \\big|_{\\LL^2} + \\lambda^{3\\gamma}\\,\\llparenthesis\\, (f^2)^\\sharp \\,\\rrparenthesis_{3\\gamma\\,;\\,(W^{3,\\infty})^*} \\Big).\n\\end{split}\n\\end{equation*}\n(Note that $(f^2)^\\sharp$, in the right hand side of the above inequality, is seen as a 2-index function.) Together with the bound \\eqref{EqBoundf2}, this gives the upper bound\n$$\n\\llparenthesis\\, f^2_\\bullet \\,\\rrparenthesis_{\\LL^1} \\lesssim_{\\llparenthesis\\,c_\\bullet\\,\\rrparenthesis,\\,C_0^{\\bf B}} \\,\\big| f_0 \\big|_{\\LL^2}\n$$\nfor $\\lambda$ small enough, which implies uniqueness.\n\\end{Dem}\n\n\n\\medskip\n\n\n\\subsection{Rough transport equation}\n\\label{SubsectionL2Transport}\n\nBuilding on Theorem~\\ref{ThmRDEHilbertGeneralClosed}, one can give a complete $\\LL^2$-theory of rough transport equations \n\\begin{equation*}\n\\delta f_{ts} = X\\,Vf_s +  \\bbX\\,V V f_s + f^\\sharp_{ts}\n\\end{equation*} \ndriven by non-divergence-free vector fields $V_i$ of class $W^{3,\\infty}$.\n\n\\medskip\n\n\\begin{lem}\n\\label{LemmaGammaA}\nLet $\\bfX$ be a geometric $\\gamma$-H\\\"older rough path on $\\RR^\\ell$, and $V_1,\\dots,V_\\ell$ be $W^{3,\\infty}$ vector fields on $\\RR^d$. Then the operator $\\Gamma_{\\bfA}$ associated with ${\\bfA} = \\big(XV, \\bbX VV\\big)$ is renormalizable in the scale of spaces $\\big(\\mathcal{E}^\\nabla_n\\big)_{n\\geq 0}$.\n\\end{lem}\n\n\\smallskip \n\n\\begin{Dem}\nFor a  geometric rough path $\\bfX = (X,\\bbX)$, the operator $\\Gamma_{\\bfA}$ takes the form $\\Gamma_{\\bfA} = \\big(X  \\Gamma_{V}^{1}, \\bbX \\Gamma_{V}^{2})$, with\n\\begin{equation*}\n\\begin{split}\n&\\Gamma_{V}^{1} := V  \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes V, \\\\\n&\\Gamma_{V}^{2} := V V \\otimes \\mathbbm{I}+\\mathbbm{I} \\otimes V V + 2 (V \\otimes V) = \\Gamma_{V}^{1} \\Gamma_{V}^{1}.\n\\end{split}\n\\end{equation*}\nSo  it is enough to show that the adjoints of these operators satisfy, uniformly in $\\varepsilon$, the inequalities\n\\begin{equation}\n\\label{eq:est-gamma-v}\n|\\Gamma_{V,\\varepsilon}^{1,\\ast} \\Phi|^\\nabla_{n} \\lesssim |V|_{W^{n+1,\\infty}} |\\Phi|^\\nabla_{n+1},\n\\qquad\n|\\Gamma_{V,\\varepsilon}^{2,\\ast} \\Phi|^\\nabla_{m} \\lesssim |V|_{W^{m+2,\\infty}}^2 |\\Phi|^\\nabla_{m+2}\n\\end{equation}\nfor $n=0,2,\\,m=0,1$, for smooth test functions $\\Phi$ where $\\Gamma_{V,\\varepsilon}^{j,\\ast} = T_\\varepsilon^{-1} \\Gamma_{V}^{j,\\ast}T_\\varepsilon $ for $j=1,2$.\n\n\\smallskip\n\nWrite $V = v_k \\partial_k$ where $(v_k)_{k=1,\\dots,d}$ are the coefficients of the vector fields in the canonical basis $(\\partial_k)_{k=1,\\dots,d}$ of derivations; with these notations, we have\n$$\nV^\\ast = - v_k \\partial_k - d,\n$$ \nwhere $d := \\textrm{div} v$, is the divergence of the vector field $V$, and\n$$\n\\Gamma_{V}^{1,\\ast} = v^+_k \\partial^+_k + v^-_k \\partial^-_k + d^+ + d^-,\n$$  \nwhere, for a real-valued function $h$ on $\\RR^d$, we denote by\n$$\nh^\\pm(x,y) := h(x) \\pm h(y)\n$$ \nits symmetric and antisymmetric lift to $\\RR^d \\times \\RR^d$ and\n$$\nh^\\pm_\\varepsilon(x,y) := h(x_+ +\\epsilon x_-) \\pm h(x_+ -\\epsilon x_-)\n$$ \ntheirs blowup according to the transformation $h^\\pm_\\varepsilon = T_\\varepsilon^{-1} h^\\pm T_\\varepsilon$.\nNote that\n$$\n\\partial^+_k T_\\varepsilon = T_\\varepsilon \\partial^+_k\n\\qquad\n\\partial^-_k T_\\varepsilon = \\varepsilon^{-1} T_\\varepsilon \\partial^-_k\n$$\nso that\n$$\n\\Gamma_{V,\\varepsilon}^{1,\\ast} = T_\\varepsilon^{-1} \\Gamma_{V}^{1,\\ast}T_\\varepsilon = v^+_{k,\\varepsilon} \\partial^+_k +\\frac{ v^-_{k,\\varepsilon}}{\\varepsilon} \\partial^-_k + d^+_\\varepsilon + d^-_\\varepsilon,\n$$  \n\n\nThe first estimate in \\eqref{eq:est-gamma-v} follows from the inequalities\n$$\n\\big| a^{+}_\\varepsilon \\nabla^{+} \\Phi\\big|^\\nabla_{n}+\\big| \\varepsilon^{-1} a^{-}_\\varepsilon \\nabla^{-} \\Phi\\big|^\\nabla_{n} \\lesssim ( |a|_{W^{n,\\infty}} + |\\nabla a|_{W^{n,\\infty}}) |\\Phi|^\\nabla_{n+1}.\n$$ \nThe second inequality in \\eqref{eq:est-gamma-v} is obtained by noting that we have \n\\begin{equation*}\n\\begin{split}\n|\\Gamma_{V}^{2,\\ast} \\Phi|^\\nabla_{m} =  |\\Gamma_{V}^{1,\\ast}\\Gamma_{V}^{1,\\ast} \\Phi|^\\nabla_{m} &\\lesssim |V|_{W^{m+1,\\infty}} |\\Gamma_{V}^{1,\\ast} \\Phi|^\\nabla_{m+1} \\\\\n&\\lesssim |V|_{W^{m+1,\\infty}} |V|_{W^{m+2,\\infty}} | \\Phi|^\\nabla_{m+2}.\n\\end{split}\n\\end{equation*}\n\n\\end{Dem}\n\n\\smallskip\n\n\\begin{thm}\n\\label{ThmTransportL2}\nLet $\\bfX$ be a geometric $\\gamma$-H\\\"older rough path on $\\RR^\\ell$, and $V_1,\\dots,V_\\ell$ be $W^{3,\\infty}$ vector fields on $\\RR^d$. Then the rough transport equation \n$$\n\\delta f_s = \\big(X_{ts}\\,V + \\bbX_{ts}\\,VV\\big)f_s + f^\\sharp_{ts}\n$$ \nis well-posed.\n\\end{thm}\n\n\\smallskip\n\n\\begin{Dem}\nNotice first that the regularity assumption on the $V_i$ puts us in a position to use the a priori bounds for symmetric closed drivers stated in Theorem~\\ref{ThmRDEHilbertGeneral}, with $\\bfA$ in the role of $\\bfB$. So uniqueness is a direct consequence of the a priori bound \\eqref{EqAPrioriBoundNorm} in Theorem~\\ref{ThmRDEHilbertGeneralClosed}.\n\n\\smallskip\n\nLet now $f_0\\in E_0$ be given. We prove the existence of a solution path to rough transport equation started from $f_0$, by a classical approximation-compactness argument, relying in a crucial way on the a priori bound \\eqref{EqAPrioriBoundNorm} on the $\\LL^2$-norm of the solution to the approximate problem, and on the uniform estimate \\eqref{EqBoundRemainderX} for the remainder. \n\n\\smallskip\n\nFix a finite time horizon $T$.\nGiven that $\\bfX$ is geometric, let $\\big({\\bfX}^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ be a family of rough path lifts of smooth paths which converge to $\\bfX$ is a rough paths sense over the time interval $[0,T]$. Let also $\\big(V^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ be a family of smooth vector fields that converge to $V$ in $W^{3,\\infty}$, and let $\\big({\\bf A}^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ be their associated rough driver, defined by formula \\eqref{EqRoughDriverRoughPath} with ${\\bfX}^\\epsilon$ and $V^\\epsilon$ in place of $\\bfX$ and $V$ respectively. One can choose $\\big({\\bfX}^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ in such a way that the constant $C_0^\\epsilon$ associated with ${\\bf A}^\\epsilon$ by formula \\eqref{EqDefnC0} satisfies the inequality $C_0^\\epsilon\\lesssim C_0$, independently of $0<\\epsilon\\leq 1$. Given the smooth character of the vector fields $V^\\epsilon$, one can solve uniquely the transport equation\n\\begin{equation*}\n\\delta f^\\epsilon_{ts}(\\varphi) = f^\\epsilon_s\\big(V^{\\epsilon,\\ast} \\varphi\\big)\\,X^\\epsilon_{ts} + f^\\epsilon_s\\big(V^{\\epsilon,\\ast}V^{\\epsilon,\\ast}\\varphi\\big)\\,\\bbX^\\epsilon_{ts} + f^{\\epsilon, \\sharp}_{ts}(\\varphi), \\quad\\quad \\textrm{for } \\varphi\\in E_2,\n\\end{equation*}\nby the method of characteristics, as the above equation is actually equivalent to the ordinary differential equation\n$$ \n\\frac{df^\\epsilon_t}{dt} = f^\\epsilon_t\\big(V^{\\epsilon,\\ast}\\varphi\\big)\\,{X^\\epsilon_t}'.\n$$\nThe solutions of this problem satisfy the uniform estimates\n\\begin{equation*}\n\\big| f^\\epsilon_t \\big|_0 \\lesssim_{C_0,T}\\,\\big| f_0 \\big|_0,\n\\end{equation*}\nfor all $0\\leq t\\leq T$, as a consequence of \\eqref{EqAPrioriBoundNorm}, and we also have the uniform bound\n\\begin{equation*}\n\\sup_{0<\\epsilon\\leq 1}\\;\\big\\| f^{\\epsilon, \\sharp} \\big\\|_{3 \\gamma\\,;\\, - 3} \\lesssim_{C_0, \\gamma, T,|f_0|_0} 1,\n\\end{equation*}\nby \\eqref{EqBoundRemainderX}. These two a priori estimates are all we need to finish the proof of the theorem following word by word the end of the proof of Theorem~\\ref{ThmRDEHilbertGeneralClosed}. \n\\end{Dem}\n\n\\medskip\n\nIt is perfectly possible to extend the present theory to deal with rough linear equations with a {\\bf drift}\n$$\ndf_s = Wf_s ds + {\\bf A}(ds)f_s,\n$$\nwhere $W\\in\\textrm{L}\\big(E_{-0},E_{-2}\\big)$, such as the Laplacian operator in the $W^{n,2}(\\RR^d)$ scale of spaces. We refrain from giving the details here as this is not our main point and this does not require the introduction of new tools or ideas. This provides an alternative road to some of the results of \\cite{DiehlFrizStannat} in a slightly different setting.\n\n\n\\bigskip\n \n\n\\section{The $\\LL^\\infty$ theory of rough transport equations}\n\\label{SectionLInftyTheory}\n\nWe develop in this section an $\\LL^\\infty$ theory of the rough transport equation \n\\begin{equation}\n\\label{EqTransportEq}\n\\delta f_{ts} = X\\,Vf_s +  \\bbX\\,V V f_s + f^\\sharp_{ts}\n\\end{equation}  \nand prove its well-posed character under the assumption that the vector fields be $\\mcC^3$, for some positive constant $\\nu$. We show for that purpose that all solutions are renormalised solutions, in the sense of Di~Perna--Lions, which classically leads to uniqueness and stability results in that setting. \n\n\\medskip\n\n\\subsection{A priori estimates and existence result}\n\\label{SubsectionAPrioriEstimatesLInfty}\n\nFor developping that $\\LL^\\infty$ theory, we shall be working in the scale of Sobolev spaces \n$$\nE_n = W^{n,1}(\\RR^d),\\quad\\textrm{for } n\\geq 0,\n$$\nwith norm denoted by $|\\cdot|_n$, and in which one has regularising operators $\\big(J^\\epsilon\\big)_{0<\\epsilon\\leq 1}$ for which estimates \\eqref{EqApproximationProperties} hold. Our minimal regularity assumptions on the vector fields will be the existence of a positive constant $C_1$ such that the inequalities\n\\begin{equation}\n\\label{EqContinuityConditionsVStar}\n\\big|V_i^*\\varphi\\big|_0 \\leq C_1 |\\varphi|_1, \\quad\\quad \\big|V_i^*V_j^*\\varphi\\big|_0 \\leq C_1 |\\varphi|_2\n\\end{equation}\nhold for all $1\\leq i,j\\leq\\ell$. These conditions hold for instance if the vector fields $V_i$ and $(V_iV_j)$ are all $\\mcC^1_b$; we write here $(V_iV_j)$ for $\\big(DV_j\\big)(V_i)$. One proves the following existence result by proceeding exactly as in the proof of Theorem~\\ref{ThmTransportL2}, using the a priori $\\LL^\\infty$-estimate\n$$\n\\big| f_t^\\epsilon \\big|_{\\LL^\\infty} = \\big|f_0^\\epsilon\\big|_{\\LL^\\infty},\n$$\nfor the regularised equation, and using Theorem~\\ref{ThmRegularityGain} to get an $\\epsilon$-uniform control on $\\big|f^{\\epsilon,\\sharp}\\big|_{3\\gamma\\,;\\,-3}$, in terms of $\\bfX$ and $\\big|f_0^\\epsilon\\big|_{\\LL^\\infty}$ only. It holds in particular if $V$ is $\\mcC^2_b$.\n\n\\smallskip \n \n\\begin{thm}[Existence for rough transport equations -- $\\LL^\\infty$ setting]\n\\label{ThmExistenceTransportEquationLInfty}\nUnder the continuity assumptions \\eqref{EqContinuityConditionsVStar} on the vector fields $V_i$, for any $f_0\\in\\LL^\\infty(\\RR^d)$, there exists an $\\LL^\\infty(\\RR^d)$-valued path $(f_t)_{t\\geq 0}$, started from $f_0$, satisfying the equation\n$$\n\\delta f_{ts}(\\varphi) = f_s\\big(V^*\\varphi\\big)\\,X_{ts} + f_s\\big(V^*V^*\\varphi\\big)\\,\\bbX_{ts} + f^\\sharp_{ts}(\\varphi)\n$$\nfor all $\\varphi\\in E_3$, and the bound\n$$ \n\\sup_{t\\geq 0} \\big| f_t\\big|_{\\LL^\\infty(\\RR^d)}\\leq \\big| f_0\\big|_{\\LL^\\infty(\\RR^d)},\n$$ \nwith a remainder $f^\\sharp(\\varphi)$ controlled by \n\\begin{equation}\n\\label{EqBoundRemainder-bis}\n\\big|f^\\sharp_{ts}(\\varphi) \\big| \\lesssim_{C_1,{\\bfX}, T, |f_0|_{\\LL^\\infty}} \\, |\\varphi|_3\\,|t-s|^{3\\gamma},\n\\end{equation}\nfor $0\\leq s\\leq t\\leq T$. \n\\end{thm} \n \n\n\\bigskip\n\n\n\\subsection{Renormalised solutions, uniqueness and stability}\n\\label{SubsectionrenormalisedSolutions}\n\nTo proceed one step further, we show that a mild strengthening of the regularity conditions imposed on the vector fields $V_i$ suffices to guarantee that all bounded solutions to the transport equation \\eqref{EqTransportEq} are actually renormalised solution, in the sense of the following definition. \n\n\\begin{defn}\n\\label{DefnrenormalisedSolutions}\nA solution $f_\\bullet$ to the transport equation \\eqref{EqTransportEq} in the scales $(E_n)_{n\\geq 0}$ is said to be a \\emph{\\bf renormalised solution} if for any function $H : \\RR\\rightarrow\\RR$, of class $\\mcC^3_b$, the path $h_\\bullet = H\\circ f_\\bullet$ is also a solution to equation \\eqref{EqTransportEq} in the same scale $(E_n)_{n\\geq 0}$.\n\\end{defn}\n\nAs expected, this property will lead below to uniqueness and stability results.\n\n\\begin{thm}\n\\label{ThmRenomalizedSolutions}\nAssume the vector fields $V_i$ are $\\mcC^3_b$. Then every solution to the transport equation \\eqref{EqTransportEq}, bounded in $ \\LL^\\infty(\\RR^d)$, is a renormalised solution.\n\\end{thm}\n \n\\medskip\n \n\\begin{Dem}  \nThe renormalisation Lemma~\\ref{Lemmarenormalisation} can be stated in the $\\LL^\\infty$ setting by chosing a slightly different scale $(\\mcF^\\nabla_n)_n$ of spaces of test functions,  with norms modelled on $\\LL^1$ \n\\begin{equation}\n\\label{EqNormNablaLInfty} \n\\big| \\varphi \\big|^\\nabla_n := \\sup_{0 \\leqslant k + \\ell \\leqslant n} \\int \\int \\Big|\\big(\\nabla^+\\big)^k \\big(\\nabla^-\\big)^\\ell \\varphi (x, y) \\Big|\\,dx dy  \n\\end{equation}\nrather than on an $\\LL^\\infty$ space used in Section~\\ref{SubsectionTensorization}. Identity \\eqref{EqPrimitiveGronwall} holds in that case with for functions $\\phi\\in W^{3,1}(\\RR^d)$, with an $O(\\cdot)$ term involving the $W^{3,1}$-norm of $\\phi$ rather than its $W^{3,\\infty}$-norm, as the proof of Lemma~\\ref{Lemmarenormalisation} works verbatim, provided we can prove that $\\Gamma_{\\bfA}$ is an unbounded rough driver in the scale of spaces $\\big(\\mcF_n^\\nabla\\big)_{n\\geq 0}$ associated with the norm \\eqref{EqNormNablaLInfty}.  (Note that we have in that case $\\big| f_s^{\\otimes 2}(T_\\varepsilon \\Phi)\\big| \\leq |\\Phi|_0^\\nabla \\big| f_s\\big|_{\\LL^\\infty}^2 = |\\Phi|_0^\\nabla \\big| f_s\\big|_{-0}^2$.)\n\n\\medskip\n\nThe proof that $\\Gamma_{\\bfA}$ is a unbounded rough driver renormalizable in the scale $(\\mcF^\\nabla_n)_n$ follows the same pattern as the proof given in Section~\\ref{SubsectionL2Transport}. We invite the reader to complete the details.\n\n\\smallskip\n\nSo it follows from the renormalisation lemma that if $f, g$ are two solutions the above argument also goes through and shows that $fg$ is also a solution, so any power $f^n$ of $f$ is also a solution, with a size of the remainder that depends only on the $\\LL^{\\infty}$ norm of $f^n$. By linearity the result can be extended to any polynomial of $f$, and by density to any continuous function $H (f)$, with $H$ defined on the interval $\\big[- \\| f \\|_{\\infty}, \\| f \\|_{\\infty}\\big]$. \n\\end{Dem}\n \n\\medskip\n\nWe can actually improve slightly this condition and require only a weak integrability for the third derivative of $V$; it provides a significant strengthening of the previous statement when the vector fields $V_i$ are divergence-free, giving some analogue of the traditional di~Perna--Lions conditions in the classical setting. Note that we do not have uniqueness of solutions for the associated rough differential equation under the conditions below.\n\n\\begin{thm}\n\\label{ThmRenomalizedSolutionsWeak}\nAssume that $V \\in \\mcC^2_b$, $\\nabla^3 V \\in L^1$ and $\\mathrm{div} V \\in \\mcC^2_b$. Then every solution to the transport equation \\eqref{EqTransportEq}, bounded in $ \\LL^\\infty(\\RR^d)$, is a renormalised solution.\n\\end{thm}\n \n\\medskip\n \n\\begin{Dem}  \nIn the proof of the renormalisation Lemma~\\ref{Lemmarenormalisation} we can use directly the general a priori estimate stated in Theorem~\\ref{ThmGeneralRegularityGain} applied to $\\Gamma_{\\bfA}$ with  $F=\\widetilde \\mathcal{E}^\\nabla_3$ and $E= \\mcF^\\nabla_0$ -- note the choice of function space for $F$. Here $\\widetilde \\mathcal{E}^\\nabla_n$ are spaces of test functions,  with norms modelled on $\\LL^\\infty$ like  $\\mathcal{E}^\\nabla_n$ but with a small change given by an additional averaging over the auxiliary variable $\\tau$ and a weight:\n\\begin{equation*}\n\\big| \\varphi \\big|^\\nabla_n := \\sup_{0 \\leqslant k + \\ell \\leqslant n} \\int  \\sup_{x\\in\\RR^d} \\Big[\\int_0^1 d\\tau \\Big|\\big(\\nabla^+\\big)^k \\big(\\nabla^-\\big)^\\ell \\varphi (x-\\tau w, x+(1-\\tau)w) \\Big|\\Big]\\, (1+|w|) dw \n\\end{equation*}\nthe reason of which will be clear below.\nIn this case we can  show that\n$$\nN_1(\\Gamma_{\\bfA,\\varepsilon}) \\lesssim (1+ |V|_{\\mcC^2_b})^2\n$$\nwhile\n$$\nN_2(\\Gamma_{\\bfA,\\varepsilon}) \\lesssim (1+|V|_{\\mcC^2_b}+|\\nabla^3 V|_{\\LL^1}+|\\mathrm{div} V|_{\\mcC^2_b})^3.\n$$\nIndeed apart from many contributions which can be estimated as in the $\\LL^2$ or as in the previous theorem, a difficult term come form the estimation of norms like $|\\Gamma_{V,\\varepsilon}^\\ast \\Gamma_{V,\\varepsilon}^\\ast  \\Gamma_{V,\\varepsilon}^\\ast  |_{\\textrm{L}(\\textrm{F},\\textrm{E})}$ of which the most singular contribution is given by $|\\Gamma_{V,\\varepsilon} \\Gamma_{V,\\varepsilon}  \\Gamma_{V,\\varepsilon}  |_{\\textrm{L}(\\textrm{F},\\textrm{E})}$. In this norm the contribution that requires more regularity to $V$ is due to  the first two vector fields $\\Gamma_{V,\\varepsilon}$ acting simultaneously on the third one giving terms of the form $\\varepsilon^{-1}|v_\\varepsilon^+ v_\\varepsilon^+ (\\nabla^2 v)^-_\\varepsilon \\nabla^- |_{\\textrm{L}(\\textrm{F},\\textrm{E})}$ and easier ones. Now expanding $(\\nabla^2 v_\\varepsilon)^-(x,y)= \\varepsilon \\int_0^1 d\\tau (\\nabla^3v)(x+\\varepsilon \\tau(y-x)) (y-x)$ we get\n$$\n\\varepsilon^{-1}|v^+_\\varepsilon v^+_\\varepsilon (\\nabla^2 v)^-_\\varepsilon \\nabla^- \\Psi|_E =\\varepsilon^{-1}  \\int \\int \\Big|(v^+_\\varepsilon v^+_\\varepsilon (\\nabla^2 v)^-_\\varepsilon  \\Psi) (x, y) \\Big| dx dy  $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 \\int_0^1 d\\tau \\int \\int |\\nabla^3 v(x+\\varepsilon \\tau w)| |\\Psi(x,x+w)| |w| dx dw \n $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 \\int_0^1 d\\tau \\int dx |\\nabla^3 v(x)| \\int dw  |\\Psi(x-\\tau w,x-\\tau w+w)| |w|\n $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 |\\nabla^3 V|_{\\LL^1} \\sup_{x\\in\\RR^d} \\int_0^1 d\\tau \\int dw  |\\Psi(x-\\tau w,x-\\tau w+w)| (1+|w|)\n $$\n$$\n \\lesssim  |V|_{\\LL^\\infty}^2 |\\nabla^3 V|_{\\LL^1}  |\\Phi|_{F}.\n $$\nGranted the bounds on $N_1(\\Gamma_{\\bfA})$ and $N_2(\\Gamma_{\\bfA})$ the proof continues as the proof of the previous theorem and gives the renormalisation result.\n\\end{Dem}\n \n\\medskip\n\nAs expected, Theorem~\\ref{ThmRenomalizedSolutions} on renormalised solutions to the transport equation \\eqref{EqTransportEq} comes with a number of important consequences, amongst which is an equivalent of the missing Gronwall lemma, as given by the a priori estimate \\eqref{EqGronwall} below.\n\n\\begin{thm}\n\\label{ThmUniqueness} \nAssume the vector fields $V_i$ are $\\mcC^3_b$. \\vspace{0.1cm}\n\\begin{enumerate}\n   \\item {\\bf Uniqueness --} Given an initial condition in $\\LL^\\infty(\\RR^d)$, there exists a unique solution to the transport equation which remains bounded in $\\LL^\\infty(\\RR^d)$.  \\vspace{0.2cm}\n   \\item {\\bf Stability --} Let the time horizon $T$ be finite. Let $\\Big(V^{(n)}_i\\Big)_{n\\geq 0}$, $i=1..\\ell$ and $\\Big(f^{(n)}_0\\Big)_{n\\geq 0}$ be a sequence of approximating sequences with $V^{(n)}_i$ converging to $V_i$ in $\\mcC^3_b$, and $f^{(n)}_0$ converging to $f_0$ in $\\LL(\\RR^d)$. Let also $\\Big({\\bfX}^{(n)}\\Big)_{n\\geq 0}$ be a sequence of weak geometric $\\gamma$-rough paths above smooth paths, that converge in a rough paths sense to $\\bfX$, over the time interval $[0,T]$. Then the solution paths $f^{(n)}_\\bullet$ to the transport equation associated with ${\\bfX}^{(n)}, V^{(n)}$ and $f^{(n)}_0$, converge weakly-$\\star$ in $\\LL^\\infty\\big([0,T],\\LL^\\infty(\\RR^d)\\big)$, and in $\\LL^1_\\textrm{\\emph{loc}}\\big([0,T],\\LL^\\infty(\\RR^d)\\big)$, to $f_\\bullet$.\n\\end{enumerate}\n\\end{thm}\n\n\\medskip\n\n\\begin{Dem}\n{\\bf Uniqueness --} We follow the same pattern of proof as that of Theorem~\\ref{ThmRDEHilbertGeneralClosed}. Let $f_\\bullet$ and $f'_\\bullet$ be two solution paths to equation \\eqref{EqTransportEq}, bounded in $\\LL^\\infty(\\RR^d)$, and started from the same initial condition. Let $H : \\RR\\rightarrow\\RR$ be a non-negative function, of class $\\mcC^3_b$, null at $0$ and positive elsewhere. Define the path \n$$\nh_\\bullet = H\\big(f_\\bullet - f'_\\bullet\\big);\n$$\nit is also a positive solution to the transport equation \\eqref{EqTransportEq} under the above regularity assumptions on the vector fields $V_i$, since all solutions are renormalised solution, by Theorem~\\ref{ThmRenomalizedSolutions}. Set $\\psi(x) = \\big(1+|x|^2\\big)^{-k_0}$, for $x\\in\\RR^d$, and some exponent $k_0>d$. That function satisfies \n\\begin{equation*}\n\\begin{split}\n&\\textrm{div}\\big(\\psi V\\big) = -V\\psi - (\\textrm{div}V)\\psi, \\\\\n&\\textrm{div}\\Big(\\textrm{div}\\big(\\psi V\\big)\\,V\\Big) = V^2\\psi + (\\textrm{div}V)V\\psi + \\big((V\\textrm{div}V)+(\\textrm{div}V)^2\\big)\\psi  \n\\end{split}\n\\end{equation*}\nwith \n\\begin{equation*}\n\\Big|V\\psi\\Big| \\,\\vee\\, \\Big|V^2\\psi + (\\textrm{div}V)V\\psi\\Big| \\lesssim \\psi\n\\end{equation*}\nas a consequence of the $\\mcC^1_b$ character of the vector fields $V_i$. \nDefine the scale of spaces \n$$\nE_n^\\psi := \\big\\{\\varphi = \\psi \\phi\\,;\\,\\phi \\in \\LL^\\infty\\big\\} \n$$\nwith norm \n$$\n|\\varphi|_{E_n^\\psi} := |\\phi|_{W^{n,\\infty}}\n$$ \nIt is not difficult to check that $(V X, VV \\mathbb{X})$ is a $\\gamma$-rough driver also in this scale of spaces. In this case however we have\n$$\n\\big|h_t(\\varphi)\\big| = \\big|h_t ( \\psi \\phi)\\big| \\leq |\\phi|_{\\LL^\\infty} \\big|h_t( \\psi)\\big| = |\\varphi|_{E^\\psi_0}\\big|z_t\\big|\n$$\nwhere \n$$\nz_t := h_t(\\psi), \n$$\nand so\n$$\n\\llparenthesis h_\\bullet\\rrparenthesis_{(E_0^\\psi)^*}\\leq \\llparenthesis z_\\bullet \\rrparenthesis\n$$\nBy the general a priori estimates we have that there exists $\\lambda$ and constants depending on $A$ such that\n$$\n\\llparenthesis h^\\sharp \\rrparenthesis_{3\\gamma\\,;\\,(E_3^\\psi)^*} \\lesssim \\llparenthesis z_\\bullet \\rrparenthesis\n$$\nBut now\n$$\nz_t = z_0 +  h_{0}(V^* \\psi)X_{0,t}+  h_{0}(V^*V^* \\psi)\\mathbb{X}_{0,t} + h^\\sharp_{0,t}(\\psi)\n$$\nso\n$$\n\\llparenthesis z_\\bullet \\rrparenthesis \\leq |z_0| \\big(1+ \\llparenthesis X_{0, \\bullet} \\rrparenthesis+ \\llparenthesis \\mathbb{X}_{0,\\bullet} \\rrparenthesis\\big) + \\llparenthesis h^\\sharp_{0,\\bullet} \\rrparenthesis_{(E_3^\\psi)^*}\n$$\nand since $  \\llparenthesis h^\\sharp_{\\bullet 0} \\rrparenthesis_{(E_3^\\psi)^*} \\leq \\lambda^{3\\gamma} \\llparenthesis h^\\sharp \\rrparenthesis_{3\\gamma\\,;\\,(E_3^\\psi)^*} \\leq \\lambda^{3\\gamma}  \\llparenthesis z_\\bullet \\rrparenthesis$, with $h^\\sharp$ considered as a 2-index map in its second occurence, we have for $\\lambda$ small enough\n\\begin{equation}\n\\label{EqGronwall}\n\\llparenthesis z_\\bullet \\rrparenthesis \\leq 2 |z_0| \\big(1+ \\llparenthesis X_{0, \\bullet} \\rrparenthesis+ \\llparenthesis \\mathbb{X}_{0, \\bullet} \\rrparenthesis\\big);\n\\end{equation}\nso $z_t = 0$, for all $t\\ge 0$, if $z_0 = 0$.\n\n\\bigskip\n  \n{\\bf Stability --} Denote by ${\\bfX}^{(n)}$ a smooth rough path converging to $\\bfX$ in the rough paths metric, and by $V^{(n)}_i$ a sequence of vector fields converging to $V_i$ in $\\mcC^3_b$. Let $f^{(n)}_0$ be a smooth function converging to $f_0$ in $(\\LL^1)^*$. One solves the transport equation associated with ${\\bfX}^{(n)}$ and $f^{(n)}_0$, using the elementary method of characteristics as the vector fields $V^{(n)}_i$ are sufficiently regular. It is elementary to use the uniform bound\n$$\n\\left|f^{(n)}_t\\right|_{\\LL^\\infty(\\RR^d)} \\leq \\left|f^{(n)}_0\\right|_{\\LL^\\infty(\\RR^d)} \\leq C<\\infty,\n$$ \nand the uniform a priori bound on $\\big|f^{(n),\\sharp}\\big|_{3\\gamma\\,;\\,-3}$ provided by Theorem~\\ref{ThmRegularityGain} and the convergence of ${\\bfX}^{(n)}$ to $\\bfX$, and $f^{(n)}_0$ to $f_0$, to get the existence of a subsequence of $f^{(n)}_\\bullet$ converging weakly-$\\star$ in $\\LL^\\infty\\big([0,T];\\LL^\\infty(\\RR^d)\\big)$ to some solution of the transport equation \\eqref{EqTransportEq}, bounded in $\\LL^\\infty(\\RR^d)$. Since this solution is unique, as proved above, the whole sequence $f^{(n)}_\\bullet$ converges weakly-$\\star$ to $f_\\bullet$ in $\\LL^\\infty\\big([0,T];\\LL^\\infty(\\RR^d)\\big)$. As the same conclusion holds for $\\Big(f^{(n)}_\\bullet\\Big)^2$ and $f^2$, by the renormalisation property, we classically get the convergence in $\\LL^1_\\textrm{loc}\\big([0,T];\\LL^\\infty(\\RR^d)\\big)$.\n\\end{Dem}\n\n\\bigskip\n\n\\begin{rem}\nIt may be tempting, in the light of the results exposed in Section~\\ref{SubsectionIntegrationConservative}, to try and develop an $\\LL^\\infty$ theory of differential equations driven by more general rough drivers ${\\bfA}_{ts}$ than those associated with the data of some vector fields $V_1,\\dots, V_\\ell$ and a weak geometric H\\\"older rough path over $\\RR^\\ell$, as in the transport equation \\eqref{EqTransportEq}. With a view towards the classical theory of stochastic flows, as developed by Le Jan-Watanabe, Kunita and many others, one may try, as a first step, to work with rough drivers whose first level are obtained as typical trajectories of semimartingale velocity fields. It is shown in \\cite{BailleulRiedel} that such random fields can be lifted into some objects very similar to rough drivers, under some mild regularity conditions on the semimartingale, and that the use of the approximate flow machinery introduced in \\cite{BailleulRMI} leads to some well--posedness result for some dual evolution equation \n$$\ndf_t(\\varphi) = f_t\\big({\\bf A}(dt)\\varphi\\big) .   \n$$ \n\\end{rem}\n\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOur aim in this paper is to take a fresh look at the logics of informational dependence and independence \\cite{HS89,HS96,Vaan}, and their compositional semantics due to Wilfrid Hodges \\cite{Hod97a,Hod97b}. We shall focus on Dependence Logic, introduced by the second author \\cite{Vaan}.\n\nThe main objective of Hodges' work was to provide a compositional model-theoretic semantics for the IF-logic of Hintikka and Sandu \\cite{HS89,HS96}, which matched their ``game-theoretical semantics''. This was achieved by lifting the standard Tarski semantics of first-order formulas, given  in terms of satisfaction in a structure with respect to an assignment to the free variables, to satisfaction by \\emph{sets of  assignments}.\n\nWe seek a deeper understanding of Hodges' construction:\n\\begin{itemize}\n\\item First and foremost, what is going on? Where does the Hodges construction come from? Is it canonical in any way? Why does it work? What structures are really at play here?\n\n\\item Because of the equivalence of Dependence Logic (or variants such as IF-logic) under this semantics to  (a significant fragment of) second-order logic, there is no hope for a completeness theorem. But we may get a useful completeness theorem with respect to a wider class of models. Understanding the general algebraic context for the semantics points the way to such a completeness notion.\n\n\\item We can  also look for \\emph{representation theorems}, with some infinitary ingredients.\n\\end{itemize}\nThe results of our investigation are quite surprising conceptually (at least to us). The main points can be summarized as follows.\n\\begin{itemize}\n\\item We find a general context for Hodges' construction. We shall not treat it in full generality here, as the general account is best stated in the language of categorical logic \\cite{Law69,Pitts}, and we wish to avoid undue technicalities. However, we will indicate the possibilities for a general algebraic semantics, as the basis for a useful completeness theorem.\n\\item We find that the natural propositional logic associated with the Hodges construction is the \\emph{logic of Bunched Implication} of Pym and O'Hearn \\cite{OHP99,Pym02}, which combines intuitionistic and multiplicative linear connectives. \n\\item This not only yields a more natural view of the strangely asymmetric notions of conjunction and disjunction in the Hodges semantics (one is intuitionistic, while ``disjunction'' is actually multiplicative \\emph{conjunction}!),  it also brings into prominence some connectives not previously considered in the setting of IF-logic or Dependence logic, in particular \\emph{intuitionistic implication}. This enables a novel analysis of the Dependence predicate of \\cite{Vaan}, as a Horn clause with respect to a more primitive predicate of single-valuedness. The well-known Armstrong axioms for functional dependence \\cite{Arm} then fall out as a standard axiomatization of intuitionistic (but not classical!) implication.\n\\item Intuitionistic implication also plays a natural r\\^ole in our version of a full abstraction theorem in the sense of Hodges.\n\\item The construction is shown to lift the interpretation of the standard quantifiers in a canonical way, so that \\emph{quantifiers are uniquely determined as the adjoints to substitution} \\cite{Law69}, just as in the standard Tarski semantics of first-order logic. This is also extended to characterizations of the dependence-friendly quantifiers of \\cite{Vaan} as adjoints.\n\\end{itemize}\n\n\\noindent The plan of the remainder of the paper is as follows. In the next section we provide background on branching quantifiers, IF-logic, dependence logic, and Hodges'  semantics. Then  in section~3 we show how the Hodges semantics is an instance of a general algebraic construction, in which the connectives of BI-logic arise naturally. In section~4, we show that the interpretation of the quantifiers in the Hodges construction is the canonical lift of the standard  interpretation of the quantifiers as adjoints, and hence is uniquely determined.  We also use the intuitionistic implication to show how the dependence-friendly quantifiers can be interpreted as certain adjoints. In section~5, we  show how the intuitionistic implication arises naturally in the proof of a full abstraction theorem. In section~6, we show how the dependence predicate can be analyzed in terms of a more primitive predicate of single-valuedness, using the intuitionistic implication. This turns the ``Armstrong axioms'' into standard theorems of intuitionistic implicational logic. The final section outlines some further directions.\n\n\\section{Dependence, Independence and Information Flow}\nWe begin with a standard example: the formal definition of continuity for a function $f : \\mathbb{R} \\longrightarrow \\mathbb{R}$ on the real numbers.\n\\[ \\forall x. \\, \\forall \\epsilon. \\, \\exists \\delta . \\, \\forall x'. \\, | x - x'| < \\delta \\; \\Rightarrow \\; | f(x) - f(x') | < \\epsilon \\, . \\]\nThis definition is often explained in current calculus courses in terms of an ``epsilon-delta game''.\\endnote{See e.g. online resources such as\\\\ \\texttt{http:\/\/library.wolfram.com\/infocenter\/MathSource\/4734\/}.} The Adversary proposes a number, $\\epsilon$, as a measure of how close we must stay to the value of $f(x)$; we must then respond with a number, $\\delta$, such that, whenever the input is within the interval $(x - \\delta, x+\\delta)$, the output does indeed pass the $\\epsilon$-test of closeness to $f(x)$. Clearly, the choice of $\\delta$ will depend on that of $\\epsilon$; the nesting of the quantifiers expresses this dependency.\n\nThis is the definition of \\emph{global continuity} of $f$, expressed in terms of \\emph{local continuity} at every point $x$. This means that the choice of $\\delta$ will depend, not only on $\\epsilon$, but on $x$ also. Now consider the definition of \\emph{uniform continuity}:\n\\[ \\forall \\epsilon. \\, \\exists \\delta . \\, \\forall x. \\, \\forall x'. \\, | x - x'| < \\delta \\; \\Rightarrow \\; | f(x) - f(x') | < \\epsilon \\, . \\]\nHere $\\delta$ still depends on $\\epsilon$, \\emph{but must be chosen independently of $x$}. This variation in dependency is tracked syntactically by the different order of the quantifiers.  Indeed,  it seems that it was only after the distinction between pointwise and uniform notions of continuity, and, especially, convergence, had been clarified in 19th-century analysis, that the ground was prepared for the introduction of predicate calculus.\n\nMore generally, dependence or independence of bounds on various parameters is an important issue in many results on estimates in number theory and analysis.\nHodges quotes a nice example from one of Lang's books \\cite{Lan64} in \\cite{Hod97a}.\n\nIntuitively, there is an evident relation between these notions and that of \\emph{information flow}. Dependence indicates a form of  information flow; independence is  the \\emph{absence} of information flow.\n\n\\subsection{Beyond first-order logic}\nIt turns out that mere rearrangement of the order of quantifiers in first-order formulas is not sufficient to capture the full range of possibilities for informational dependence and independence. This was first realized almost 50~years ago, with Henkin's introduction of  \\emph{branching quantifiers} \\cite{Hen61}. The simplest case is the eponymous \\emph{Henkin quantifier}:\n\\[ \\left( \\begin{array}{cc} \\forall x & \\exists y \\\\\n\\forall u & \\exists v\n\\end{array}   \\right) A(x, y, u, v) .\n\\]\nThe intention is that $y$ must be chosen depending on $x$, but \\emph{independently} of the choice of $u$; while $v$ must be chosen depending on $u$, but \\emph{independently} of the choice of $x$.\nThe meaning of this formula can be explicated by introducing \\emph{Skolem functions} $f$ and $g$: an equivalent formula will be\n\\[ \\exists f. \\, \\exists g. \\, \\forall x. \\, \\forall u. \\, A(x, f(x), u, g(u)) . \\]\nHere the constraints on dependencies are tracked by the dependence of the Skolem functions on certain variables, but not on others.\nNote that the Skolemized sentence is \\emph{second-order}; in fact, it belongs to the $\\Sigma_{1}^{1}$ fragment of second-order logic.\\endnote{This can be described as the fragment comprising formulas $\\exists f_{1} \\ldots \\exists f_{n}. \\, \\phi$,  where the $f_{i}$ are function variables, and $\\phi$ is a first-order formula over a signature extended by these function variables.} This second-order rendition of the meaning of the Henkin quantifier cannot be avoided, in the sense that  the Henkin quantifier strictly increases the expressive power of first-order logic, and in fact the extension of first-order logic with the Henkin quantifier is equivalent in expressive power to  the $\\Sigma_{1}^{1}$ fragment \\cite{Hen61}.\n\n\\paragraph{Examples}\n\\begin{enumerate}\n\\item Consider\n\\[ \\left( \\begin{array}{cc} \\forall x & \\exists y \\\\\n\\forall u & \\exists v\n\\end{array} \\right)  (A(x) \\rightarrow B(y)) \\; \\wedge \\; (B(u) \\rightarrow A(v)) \\; \\wedge \\; [(x = v) \\; \\leftrightarrow \\;  (y = u)].\n\\]\nThis expresses that $A$ and $B$ are equinumerous sets.\n\\item Now consider\n\\[   \\begin{array}{ll}\n\\exists v. \\, \\left( \\begin{array}{cc} \\forall x_{1} & \\exists y_{1} \\\\\n\\forall x_{2} & \\exists y_{2}\n\\end{array} \\right) (A(x_{1}) \\rightarrow A(y_{1}))   & \\wedge \\; [(x_{2} = y_{1}) \\; \\rightarrow \\;  (y_{2} = x_{1})] \\\\    & \\wedge \\; A(v)  \\; \\wedge \\;  (A(x_{1}) \\rightarrow (y_{1} \\neq v)) \\, .\n\\end{array}\n\\]\nThis expresses that $A$ is an infinite set.\n\\end{enumerate}\nThese examples show that the Henkin quantifier \\emph{is not expressible in first-order logic}.\n\n\\subsection{Further developments}\nThe next major development was the introduction of IF-logic (``inde\\-pendence-friendly logic'') by Jaakko Hintikka and Gabriel Sandu \\cite{HS89}.\nThe intention of IF-logic is to highlight informational dependence and independence. It provides a linear syntax for expressing branching quantification (and more), e.g.  the Henkin quantifier can be written in linear notation as:\n\\[ \\forall x. \\, \\exists y. \\, \\forall u. \\, (\\exists v\/x). \\, A(x, y, u, v) \\]\nThe ``slashed quantifier''  $(\\exists v\/x)$  has the intended reading ``there exists a $v$ \\emph{not depending on $x$}''. Note the strange syntactic form of this quantifier, with its ``outward-reaching'' scope for $x$.\n\n\\paragraph{Dependence Logic}\nA simplified approach was introduced by the second author, and developed extensively in the recent monograph \\cite{Vaan}.\nThe main novelty in the formulation of the logic is to use an atomic \\emph{dependence predicate\\endnote{In \\cite{Vaan}\nthe notation $=\\!\\!(x_{1}, \\ldots , x_{n}, x)$ for $D(x_{1}, \\ldots , x_{n}, x)$ is used.}} $D(x_{1}, \\ldots , x_{n}, x)$ which holds if \\emph{$x$ depends on $x_{1}, \\ldots , x_{n}$, and  only on these variables}.\nWe can then define ``dependence-friendly quantifiers'' as standard quantifiers guarded with the dependence predicate:\n\\[ (\\exists x \\setminus x_{1}, \\ldots , x_{n}). \\, \\phi \\;\\; \\equiv \\;\\; \\exists x. (D(x_{1}, \\ldots , x_{n}, x) \\; \\wedge \\; \\phi) \\, . \\]\nThis yields essentially the same expressive power as IF-logic.\n\n\\subsection{Compositionality: Hodges' Semantics}\n\nBut, what does it all mean?\nHintikka claimed that \\emph{a compositional semantics for IF logic could not be given} \\cite{Hin98}.\nInstead he gave a ``Game-Theoretical Semantics'', essentially reduction to Skolem form as above.\n\n\nWilfrid Hodges showed that it could \\cite{Hod97a,Hod97b}.\\endnote{Hintikka has apparently not conceded the point \\cite{Hin02}, although there is no argument as to the mathematical content of Hodges' results. As far as we are concerned, Hodges' semantics meets all the criteria for a compositional semantics, and is moreover fully abstract. Our concern here is to understand it better, as an interesting construction in its own right.}\n\nBefore giving Hodges' construction, it will be useful firstly to recall  Tarski's solution to the problem of how to define the truth of a sentence in a first-order structure $\\mathcal{M} = (A, \\ldots)$ with underlying set $A$.\\endnote{The classic reference is \\cite{Tar36}, but in fact the modern model-theoretic definition first appeared in \\cite{TV56}, as pointed out in Wilfrid Hodges' article on ``Tarski's Truth Definitions'' in the Stanford Encyclopedia of Philosophy, available online at \\texttt{http:\/\/plato.stanford.edu\/entries\/tarski-truth\/}, which gives an informative overview.} In order to do this, he had to deal with the more general case of open formulas. The idea was to define\n\\[ \\mathcal{M}, s \\models_{X} \\phi \\]\nwhere $X$ is a finite set of variables including those occurring free  in $\\phi$, and $s$ is an assignment of elements of $A$ to $X$.\\endnote{Explictly, an assignment is simply a function $s : X \\rightarrow A$. We write $A^{X}$ for the set of all such assignments. Older tradition was to define satisfaction relative to assignments to \\emph{all} variables, which were typically arrayed in infinite sequences. More recently, it has been understood, under the influence of categorical logic,  that to reveal the salient structure one should give the definition relative to a finite environment that  grows as quantifiers are stripped off in the recursive definition.}\nTypical clauses include:\n\\[ \\begin{array}{lcl}\n\\mathcal{M}, s \\models_{X} \\phi \\wedge \\psi & \\equiv & \\mathcal{M}, s \\models_{X} \\phi \\;\\; \\mbox{and} \\;\\; \\mathcal{M}, s \\models_{X} \\psi \\\\\n\\mathcal{M}, s \\models_{X} \\neg \\phi & \\equiv & \\mathcal{M}, s \\not\\models_{X} \\phi \\\\\n\\mathcal{M}, s \\models_{X} \\forall v. \\, \\phi & \\equiv & \\forall a \\in A. \\; \\mathcal{M}, s[v \\mapsto a] \\models_{X \\cup \\{v\\}} \\phi \\\\\n\\mathcal{M}, s \\models_{X} \\exists v. \\, \\phi & \\equiv & \\exists a \\in A. \\; \\mathcal{M}, s[v \\mapsto a] \\models_{X \\cup \\{v\\}} \\phi \\\\\n\\end{array}\n\\]\nHere $s[v\\mapsto a]$ is the assignment defined on $X \\cup \\{v\\}$ as follows: $s[v \\mapsto a](v) = a$, and $s[v \\mapsto a](w) = s(w)$ for $w \\neq v$.\n\nThe is the very prototype of a compositional semantic definition.\nVia Dana Scott, this idea led to the  use of \\emph{environments} in denotational semantics \\cite{Sco69}. Environments are nowadays ubiquitous in all forms of semantics in computer science \\cite{Win93,Mit96}.\n\n\\paragraph{Teams}\nHodges' key idea was to see that one must lift the semantics of formulas from single assignments to \\emph{sets of assignments}. Notions of dependence of one variable on others are only meaningful among a set of assignments. Hodges called these sets ``trumps''; we follow \\cite{Vaan} in calling them \\emph{teams}.\n\n\nWe consider the semantics of Dependence logic \\cite{Vaan}. Formulas are built up from standard atomic formulas and their negations and  the dependence predicates, by conjunction, disjunction, and universal and existential quantification. We shall distinguish between the usual atomic formulas (including equality statements) over the first-order signature we are working with, and the dependence formulas. In the case of the standard atomic formulas, we shall also allow their negations, and as usual refer to positive and negated atomic formulas collectively as \\emph{literals}. We shall \\emph{not} allow negations of dependence formulas; we will see later how to access negative information about dependence, using the new connectives we will introduce in the next section.\n\nThe set of all individual variables is denoted $\\mathcal{V}$.\nA \\emph{team on $X \\subseteq \\mathcal{V}$} is a set of Tarski assignments on $X$. We define the following operations on teams:\n\n\\begin{itemize}\n\\item If $T$ is a team on $X$ and $v \\in \\mathcal{V}$,  then $T[v \\mapsto A]$ is the team on $X \\cup \\{v\\}$ defined by:\n\n\\vspace{-.1in}\n\\[ T[v \\mapsto A] = \\{ t[v \\mapsto a] \\mid t \\in T \\; \\wedge \\; a \\in A \\} . \\]\n\n\n\\item If $T$ is a team on $X$, $v \\in \\mathcal{V}$, and $f : T \\longrightarrow A$, then $T[v \\mapsto f]$ is the team on $X \\cup \\{v\\}$ defined by:\n\\[ T[v \\mapsto f] = \\{ t[v \\mapsto f(t)] \\mid t \\in T \\} . \\]\n\\end{itemize}\n\n\n\n\\paragraph{The Satisfaction Relation}\n\nWe define a satisfaction relation\n\\[ \\mathcal{M}, T \\models_{X} \\phi \\]\nwhere the free variables of $\\phi$ are contained in $X$, and $T$ is a team on $X$. (In practice, we elide $\\mathcal{M}$).\n\n\nFirstly, for literals $L$  we have:\n\\[ T \\models_{X} L \\;\\; \\equiv \\;\\; \\forall t \\in T. \\, t \\models_{X} L  \\]\nwhere $t \\models_{X} L$ is the standard Tarskian definition of satisfaction of an atomic formula or its negation  in a structure with respect to an assignment.\n\n\n\n\\paragraph{Connectives and Quantifiers}\nThe  clauses for connectives and quantifiers are as follows:\n\n\\[ \\begin{array}{lcl}\nT \\models_{X} \\phi \\wedge \\psi & \\equiv & T \\models_{X} \\phi \\;\\; \\mbox{and} \\;\\; T \\models_{X} \\psi \\\\\nT \\models_{X} \\phi \\vee \\psi & \\equiv & \\exists \\, U, V. \\; ([U \\models_{X} \\phi \\;\\; \\mbox{and} \\;\\; V \\models_{X} \\psi] \\;\\; \\wedge \\;\\; [T = U \\cup V]) \\\\\nT \\models_{X} \\forall v. \\, \\phi & \\equiv & T[v \\mapsto A] \\models_{X \\cup \\{v\\}} \\phi \\\\\nT \\models_{X} \\exists v. \\, \\phi & \\equiv & \\exists f : T \\longrightarrow A. \\; T[v \\mapsto f] \\models_{X \\cup \\{v\\}} \\phi .\n\\end{array}\n\\]\n\n\\paragraph{Semantics of the dependence predicate}\nGiven a set of variables $X$ and $W \\subseteq X$, we define the following notions:\n\n\\begin{itemize}\n\\item An equivalence relation on  assignments on $X$:\n\\[ s \\simeq_{W} t \\;\\; \\equiv \\;\\;  \\forall w \\in W. \\, s(w) = t(w) . \\]\n\n\\item A function $f : A^{X} \\longrightarrow A$ \\emph{depends only on $W$}, written $f : A^{X} \\longrightarrow_{W} A$, if for some $g : A^{W} \\longrightarrow A$, $f = g \\circ p_{XW}$, where $p_{XW} : A^{X} \\longrightarrow A^{W}$ is the evident projection. Note that if such a $g$ exists, it is unique.\n\\end{itemize}\n\n\n\\noindent Now we can define:\n\\[ T \\models_{X} D(W, v) \\;\\; \\equiv \\;\\; \\forall s, t \\in T. \\, s \\simeq_{W} t \\; \\Rightarrow \\; s(v) = t(v) \\]\nNote that this expresses \\emph{functional dependence}, exactly as in database theory \\cite{Arm}.\n\n\n\\noindent An equivalent definition can be given in terms of the dependency condition on functions:\n\\[ T \\models_{X} D(W, v) \\;\\; \\equiv \\;\\; \\exists f : T \\longrightarrow_{W} A. \\, \\forall t \\in T. \\, t(v) = f(t) .  \\]\n\n\n\\noindent Strictly speaking, this is the ``positive part'' of the definition as given in \\cite{Vaan} following Hodges. There is also a negative part, which defines satisfaction for $\\phi$ as for the positive definition, but with respect to the De Morgan dual $\\phi^{d}$ of $\\phi$:\n\\[ (\\phi \\vee \\psi)^{d} = \\phi^{d} \\wedge \\psi^{d}, \\quad (\\exists v. \\, \\phi)^{d} = \\forall v. \\, \\phi^{d}, \\quad \\mbox{etc.} \\]\nThis allows for a ``game-theoretic negation'', which formally ``interchanges the r\\^oles of the players''. It is simpler, and from our perspective loses nothing, to treat this negation as a \\emph{defined operation}, and work exclusively  with formulas in negation normal form as above.\n\n\n\\noindent The theory of dependence logic: metalogical properties, connections with second-order logic, complexity and definability issues,  \\textit{et cetera}, is extensively developed in \\cite{Vaan}. However, as explained in the Introduction, many basic questions remain.\nWe shall now show how the Hodges semantics can be seen in a new light, as arising from a general construction.\n\n\\section{The Hodges construction revisited}\nAn important clue to the general nature of the construction is contained in the observation by Hodges \\cite{Hod97a} (and then in \\cite{Vaan}) that the sets of teams denoted by formulas of  IF-logic or Dependence logic are  \\emph{downwards closed}: that is, if $T \\models  \\phi $ and $S \\subseteq T$, then $S \\models \\phi$.\nThis is immediately suggestive  of well-known constructions on ordered structures.\n\n\\subsection{A general construction}\nWe recall a couple of definitions. A \\emph{commutative ordered monoid} is a structure $(M, {+}, 0, {\\leqslant})$, where $(M, {\\leqslant})$ is a partially ordered set, and $(M, {+}, 0)$ is a commutative monoid (a set with an associative and commutative operation $+$ with unit $0$), such that $+$ is monotone:\n\\[ x \\leqslant x' \\; \\wedge \\; y \\leqslant y' \\;\\; \\Rightarrow \\;\\; x+y \\leqslant x' + y' \\, . \\]\nThe primary example we have in mind is $\\pow{A^{X}}$, the set of all teams on a set of variables $X$, which we think of as the commutative ordered monoid $(\\pow{A^{X}}, {\\cup}, \\varnothing, {\\subseteq})$.\n\nA \\emph{commutative quantale} is a commutative ordered monoid where the partial order is a complete lattice, and $+$ distributes over all suprema: $m + \\bigvee_{i \\in I} m_{i} = \\bigvee_{i \\in I} (m + m_{i})$.\n\nLet $(M, {+}, 0, {\\leqslant})$ be a commutative ordered monoid. Then $\\mathcal{L}(M)$, the set of lower (or downwards-closed) sets of $M$, ordered by inclusion, is the \\emph{free commutative quantale} generated by $M$ \\cite{MS01}.\\endnote{More precisely, it is the left adjoint to the evident forgetful functor.}\n\nA downwards closed subset of a partially ordered set $P$ is a set $S$ such that:\n\\[ x \\leqslant y \\in S \\; \\Rightarrow \\; x \\in S \\, . \\]\nThus this notion generalizes the downwards closure condition on sets of teams.\n \n\\noindent The following notation will be useful.\nGiven $X \\subseteq P$, where $P$ is a partially ordered set, we define\n\\[ {\\downarrow}(X) = \\{ x \\in P \\mid \\exists y \\in X. \\, x \\leqslant y \\} \\, , \\]\nthe \\emph{downwards closure} of $X$. A set $S$ is downwards closed if and only if $S = {\\downarrow}(S)$.\n\nAs a commutative quantale, $\\mathcal{L}(M)$ is a model of intuitionistic linear logic (phase semantics \\cite{Yet,Ros90,Gir87}).\\endnote{It is also an instance of Urquhart's semilattice semantics for relevance logic \\cite{Urq72}. Mitchell and Simmons observe in \\cite{MS01} that in the case (such as ours) where the monoid is a boolean algebra, $\\mathcal{L}(M)$ is actually a model of \\emph{classical linear logic}. This does not seem apposite to our purposes here.}\nIn particular, we have\n\\[ \\begin{array}{rcl}\nA \\otimes B & = &  {\\downarrow}\\{ m + n \\mid m \\in A \\; \\wedge \\; n \\in B \\} \\\\\nA \\multimap B & = & \\{ m \\mid \\forall n. \\, n \\in A \\Rightarrow m + n \\in B \\}\n\\end{array} \\]\nWe note that when the definition of $\\otimes$, the multiplicative \\emph{conjunction}, is specialized to our concrete setting, it yields the definition of \\emph{disjunction} in the Hodges semantics!\n\nThe multiplicative implication $\\multimap$ has not been considered previously in the setting of IF-logic and Dependence logic. However, it is perfectly well defined, and is in fact uniquely specified as the adjoint of the linear conjunction:\n\\[ A \\otimes B \\leqslant C \\;\\; \\Longleftrightarrow \\;\\; A \\leqslant B \\multimap C \\, . \\]\nNote that linear implication automatically preserves downwards closure.\n\n\\subsection{What is the propositional logic of dependence?}\nIn fact, $\\mathcal{L}(M)$ carries a great deal of structure. Not\n\n only is it a commutative quantale (and hence carries an interpretation of linear logic), but it is also a \\emph{complete Heyting algebra}, and hence carries an interpretation of intuitionistic logic.\n\nWe have the clauses\n\\[ \\begin{array}{lcl}\nm \\models A \\wedge B & \\equiv & m \\models A \\; \\mbox{and} \\; m \\models B \\\\\nm \\models A \\vee B & \\equiv & m \\models A \\; \\mbox{or} \\; m \\models B \\\\\nm \\models A \\rightarrow B & \\equiv & \\forall n \\leqslant m. \\, \\mbox{if} \\; n \\models A \\; \\mbox{then} \\; n \\models B\n\\end{array}\n\\]\nThe situation where we have both intuitionistic logic and multiplicative linear logic coexisting is the setting for \\emph{BI logic}, the ``logic of Bunched Implications''  of David Pym and Peter O'Hearn \\cite{OHP99,Pym02}, which forms the basis for \\emph{Separation logic} (Reynolds and O'Hearn) \\cite{Rey02}, an increasingly influential logic for verification.\nThe construction $\\mathcal{L}(M)$ is exactly the way a ``forcing semantics'' for BI-logic is converted into an algebraic semantics as a ``BI-algebra'', \\textit{i.e.}~ a structure which is both a commutative quantale  \\emph{and} a complete Heyting algebra \\cite{POHY04}. $\\mathcal{L}(M)$ is in fact the free construction of a complete BI-algebra over an ordered commutative monoid.\n\nThis provides one reason for proposing BI-logic as the right answer to the question posed at the beginning of this subsection. The compelling further evidence for this claim will come from the natural\nr\\^ole played by the novel connectives we are introducing into the logic of dependence. This r\\^ole will become apparent in the subsequent developments in this paper.\n\n\\subsection{\\textrm{BID}-logic and its team semantics}\nWe shall spell out the extended logical language we are led to consider, and its concrete team semantics, extending the Hodges-style semantics already given in section~2.\n\nWe call the extended language \\textrm{BID}, for want of a better name. Formulas are built from atomic formulas and their negations, and  dependence formulas, by the standard first-order quantifiers, and the following propositional connectives: the intuitionistic (or ``additive'') connectives $\\wedge$, $\\vee$, $\\rightarrow$, and the multiplicative connectives $\\otimes$ and $\\multimap$.\n\n\\paragraph{Team Semantics for BI Logic}\nThe team semantics for \\textrm{BID}-logic is as follows:\n\\[ \\begin{array}{lcl}\nT \\models A \\wedge B & \\equiv & T \\models A \\; \\mbox{and} \\; T \\models B \\\\\nT \\models A \\vee B & \\equiv & T \\models A \\; \\mbox{or} \\; T \\models B \\\\\nT \\models A \\rightarrow B & \\equiv & \\forall U \\subseteq T. \\, \\mbox{if} \\; U \\models A \\; \\mbox{then} \\; U \\models B \\\\\nT \\models A \\otimes B & \\equiv & \\exists U, V. \\, T = U \\cup V \\; \\wedge \\;  U \\models A \\; \\wedge \\; V \\models B  \\\\\nT \\models A \\multimap B & \\equiv & \\forall U. \\, [U \\models A \\Rightarrow T \\cup U \\models  B ]\n\\end{array}\n\\]\nThe clauses for atomic formulas and their negations and for the dependence formulas and quantifiers are as given in section~2.\n\nAs already noted, the semantics of $\\wedge$ and $\\otimes$ coincide with those given for conjunction and disjunction in section~2. The connectives $\\vee$ and $\\rightarrow$, intuitionistic or additive disjunction and implication,  and the multiplicative implication $\\multimap$, are new as compared to IF-logic or Dependence logic.\n\n\\subsection{The semantics of sentences}\nIt is worth spelling out the semantics of sentences explicitly. By definition, sentences have no free variables, and there is only one assignment on the empty set of variables, which we can think of as the empty tuple $\\langle \\rangle$. In the Tarski semantics, there are only two possibilities for the set of satisfying assignments of a sentence, $\\varnothing$ and $\\{ \\langle \\rangle \\}$, which we can identify with \\emph{false} and \\emph{true} respectively. When we pass to the team semantics for \\textrm{BID}-logic, there are three possibilities for down-closed set of teams to be assigned to sentences: $\\varnothing$, $\\{ \\varnothing \\}$, or $\\{ \\varnothing, \\{ \\langle \\rangle \\} \\}$. Thus the semantics of sentences is \\emph{trivalent} in general.\n\nIn his papers, Hodges works only with non-empty teams, and has bivalent semantics for sentences. However, there is no real conflict between his semantics and ours. Let $\\mbox{\\textrm{BID}}^{-}$ be \\textrm{BID}-logic without the linear implication. Note that $\\mbox{\\textrm{BID}}^{-}$ properly contains Dependence logic, which is expressively equivalent to IF-logic \\cite{Vaan}.\n\n\\begin{proposition}\nEvery formula in $\\mbox{\\textrm{BID}}^{-}$-logic is satisfied by the empty team; hence in particular every sentence of $\\mbox{\\textrm{BID}}^{-}$-logic has either $\\{ \\varnothing \\}$ or $\\{ \\varnothing, \\{ \\langle \\rangle \\} \\}$ as its set of satisfying teams, and the semantics of sentences in $\\mbox{\\textrm{BID}}^{-}$-logic is bivalent.\n\\end{proposition}\n\\begin{proof}\nA straightforward induction on formulas of $\\mbox{\\textrm{BID}}^{-}$-logic.\n\\end{proof}\n\n\\noindent On the other hand, linear implication clearly violates this property. Note that the empty team satisfies $A \\multimap B$ if and only if  every team satisfying $A$ also satisfies $B$.\nWe obtain as an immediate corollary:\n\n\\begin{proposition}\nLinear implication is not definable in $\\mbox{\\textrm{BID}}^{-}$-logic, and \\textit{a fortiori}~is not definable in Dependence logic or IF-logic.\n\\end{proposition}\n\n\\subsection{The general Hodges construction}\n\nWe shall briefly sketch, for the reader conversant with categorical logic, the general form of the construction.\n\nThe standard Tarski semantics of first-order logic is a special case of Lawvere's notion of hyperdoctrine \\cite{Law69}. We refer to \\cite{Pitts} for a lucid expository account. Construing $\\mathcal{L}$ as a functor in the appropriate fashion, we can give a general form of the Hodges construction as  a functor from classical hyperdoctrines to BI-hyperdoctrines \\cite{BBTS05}.\nGiven a classical hyperdoctrine $\\mathbf{P} : \\op{\\CC} \\longrightarrow \\mathbf{Pos}$, we define a BI-hyperdoctrine  $\\mathcal{H}(\\mathbf{P})$  on the same base category by composition with the functor $\\mathcal{L}$:\n\\[ \\mathcal{H}(\\mathbf{P}) = \\mathcal{L} \\circ \\mathbf{P} : \\op{\\CC} \\longrightarrow \\mathbf{Pos} \\, . \\]\nNote that $\\mathbf{Pos}$ is an order-enriched category, and $\\mathcal{L}$ is an order-enriched functor, so it preserves adjoints, and hence in particular preserves the interpretations of the quantifiers.\nThis observation is spelled out in more detail in Proposition~\\ref{qadjprop}.\n\nThis exactly generalizes the concrete Hodges construction, which is obtained by applying $\\mathcal{H}$ to the standard Tarski hyperdoctrine.\n\nA full account will be given elsewhere.\n\n\\section{Quantifiers are adjoints in the Hodges construction}\nWe recall the team semantics for the quantifiers.\n\\[ \\begin{array}{lcl}\nT \\models_{X} \\forall v. \\, \\phi & \\equiv & T[v \\mapsto A] \\models_{X \\cup \\{v\\}} \\phi \\\\\nT \\models_{X} \\exists v. \\, \\phi & \\equiv & \\exists f : T \\longrightarrow A. \\; T[v \\mapsto f] \\models_{X \\cup \\{v\\}} \\phi .\n\\end{array}\n\\]\nWe may wonder what underlying principles dictate these definitions.\n\nTo answer this question, we firstly recall the fundamental insight due to Lawvere \\cite{Law69} that \\emph{quantifiers are adjoints to substitution}.\\endnote{See \\cite{DP02} for an introduction to adjunctions on posets.}\n\n\\subsection{Quantifiers as adjoints}\nConsider a function $f : X \\rightarrow Y$. This induces a function\n\\[ f^{-1} : \\pow{Y} \\longrightarrow \\pow{X} :: T \\mapsto \\{ x \\in X \\mid f(x) \\in T \\} . \\]\nThis function $f^{-1}$ has both a left adjoint $\\exists (f) : \\pow{X} \\longrightarrow \\pow{Y}$, and a right adjoint $\\forall (f) : \\pow{X} \\longrightarrow \\pow{Y}$. These adjoints are uniquely specified by the following conditions. For all $S \\subseteq X$, $T \\subseteq Y$:\n\\[ \\exists(f)(S) \\subseteq T \\;\\; \\Longleftrightarrow \\;\\; S \\subseteq f^{-1}(T), \\;\\; \\quad  f^{-1}(T) \\subseteq S \\;\\; \\Longleftrightarrow \\;\\;  T \\subseteq \\forall(f)(S) . \\]\nThe unique functions satisfying these conditions can be defined explicitly as follows:\n\\[ \\begin{array}{lcl}\n\\exists(f)(S) &:= & \\{ y \\in Y \\mid \\exists x \\in X. \\, f(x) = y \\; \\wedge \\; x \\in S \\}\\,,  \\\\\n\\forall(f)(S) &:= & \\{ y \\in Y \\mid \\forall x \\in X. \\, f(x) = y \\; \\Rightarrow \\; x \\in S \\}\\,.\n\\end{array}\n\\]\n\\noindent Given a formula $\\phi$ with free variables in $\\{ v_{1}, \\ldots , v_{n+1} \\}$, it will receive its Tarskian denotation $\\llbracket \\phi \\rrbracket$ in $\\mathcal{P}(A^{n+1})$ as the set of satisfying assignments:\n\\[ \\llbracket \\phi \\rrbracket = \\{ s \\in A^{n+1} \\mid s \\models_{X} \\phi \\} \\, . \\]\nWe have a projection function\n\\[ \\pi : A^{n+1} \\longrightarrow A^{n} \\; :: (a_{1}, \\ldots , a_{n+1}) \\mapsto (a_{1}, \\ldots , a_{n}) \\, . \\]\nNote that this projection is the Tarskian denotation of the tuple of terms $(v_{1}, \\ldots , v_{n})$.\nWe can characterize the standard quantifiers as \\emph{adjoints to this projection}:\n\\[ \\llbracket \\forall v_{n+1}. \\, \\phi \\rrbracket = \\forall(\\pi)(\\llbracket \\phi \\rrbracket), \\qquad  \\llbracket \\exists v_{n+1}. \\, \\phi \\rrbracket = \\exists(\\pi)(\\llbracket \\phi \\rrbracket) \\, . \\]\nIf we unpack the adjunction conditions for the universal quantifier, they yield the following bidirectional inference rule:\n\\[ \\infer=[\\quad X = \\{ v_{1}, \\ldots , v_{n} \\} \\, . ]{\\Gamma \\vdash_{X} \\forall v_{n+1}. \\, \\phi}{\\Gamma \\vdash_{X} \\phi} \\]\nHere the set $X$ keeps track of the free variables  in the assumptions $\\Gamma$. Note that\nthe usual ``eigenvariable condition'' is automatically taken care of in this way.\n\nSince adjoints are uniquely determined, this characterization completely captures the meaning of the quantifiers.\n\n\\subsection{Quantifiers in the Hodges semantics}\n\nWe shall now verify that the definitions of the  quantifiers in the Hodges semantics \\emph{are exactly the images under $\\mathcal{L}$ of their standard interpretations in the Tarski semantics}, and hence in particular that they are adjoints to substitution. Thus these definitions are \\emph{forced}.\n\n\nIt will be convenient to work with the semantic view of quantifiers, as operators on subsets. Consider formulas with free variables in $\\{ v_{1}, \\ldots , v_{n+1} \\}$. The Tarski semantics over a structure $\\mathcal{M} = (A, \\ldots)$ assigns such formulas values in $\\pow{A^{n+1}}$. We can regard the quantifiers $\\exists v_{n+1}$, $\\forall v_{n+1}$ as functions\n\\[ \\exists(\\pi), \\forall(\\pi) : \\pow{A^{n+1}} \\longrightarrow \\pow{A^{n}} \\]\n\\[ \\begin{array}{lcl}\n\\exists(\\pi)(S) & = & \\{ s \\in A^{n} \\mid \\exists a \\in A. \\, s[v_{n+1} \\mapsto a] \\in S \\} \\\\\n\\forall(\\pi)(S) & = & \\{ s \\in A^{n} \\mid \\forall a \\in A. \\, s[v_{n+1} \\mapsto a] \\in S \\}\n\\end{array}\n\\]\nFor any $m$, we define $\\mathcal{H}(A^{m}) = \\mathcal{L}(\\pow{A^{m}})$. Thus $\\mathcal{H}(A^{m})$ is the set of downwards closed sets of teams on the variables $\\{ v_{1}, \\ldots , v_{m} \\} $. This provides the corresponding ``space'' of  semantic values for formulas in the Hodges semantics.\nThe interpretation of quantifiers in that semantics is given by the following set operators:\n\\[ \\exists_{H}, \\forall_{H} : \\mathcal{H}(A^{n+1}) \\longrightarrow \\mathcal{H}(A^{n}) \\]\n\\[ \\begin{array}{lcl}\n\\exists_{H}(\\mathsf{U}) & = & \\{ T \\in \\pow{A^{n}} \\mid \\exists f : T \\rightarrow A. \\, T[v_{n+1}  \\mapsto f] \\in \\mathsf{U} \\} \\\\\n\\forall_{H}(\\mathsf{U}) & = & \\{ T \\in \\pow{A^{n}} \\mid T[v_{n+1}  \\mapsto A] \\in \\mathsf{U} \\}\n\\end{array}\n\\]\n\\noindent We extend the definition of $\\mathcal{L}$ to act on functions\\endnote{More precisely, homomorphisms of the appropriate kind. The reader familiar with category theory will see that we are really specifying the functorial action of $\\mathcal{L}$ in a particular case.} $h : \\pow{Y} \\longrightarrow \\pow{X}$:\n\\[ \\mathcal{L}(h) : \\mathcal{H}(Y) \\longrightarrow \\mathcal{H}(X) :: \\mathsf{U} \\mapsto  {\\downarrow} \\{ h(T) \\mid T \\in \\mathsf{U} \\} \\, . \\]\nIn the case that $h = f^{-1}$, where $f : X \\longrightarrow Y$, we write $\\mathcal{L}(h) = \\mathcal{H}(f)$.\n\n\\begin{proposition}\n\\label{lqprop}\nThe Hodges quantifiers are the image under $\\mathcal{L}$ of the Tarski quantifiers:\n\\[ \\exists_{H} = \\mathcal{L}(\\exists(\\pi)), \\qquad  \\forall_{H} = \\mathcal{L}(\\forall(\\pi)) \\, . \\]\n\\end{proposition}\n\\begin{proof}\nFirstly, we show that $ \\mathcal{L}(\\exists(\\pi))(\\mathsf{U}) \\subseteq \\exists_{H}(\\mathsf{U})$ for all $\\mathsf{U} \\in \\mathcal{H}(A^{n+1})$.\nSuppose that $T \\in \\mathsf{U}$. Let $T'  = \\exists(\\pi)(T)$. This means that\n\\[ \\forall t \\in T'. \\, \\exists a \\in A. \\, t[v_{n+1} \\mapsto a] \\in T \\, . \\]\nUsing the axiom of choice, there exists a function $f : T' \\longrightarrow A$ such that\n\\[ T'[v_{n+1} \\mapsto f] \\subseteq T \\in \\mathsf{U} \\, . \\]\nSince $\\mathsf{U}$ is downwards closed, this implies that $T' \\in \\exists_{H}(\\mathsf{U})$, as required.\n\n\\noindent The converse follows immediately from the fact that \n\\[ \\exists(\\pi)(T[v_{n+1} \\mapsto f]) = T \\, . \\]\nNext we show that $ \\mathcal{L}(\\forall(\\pi))(\\mathsf{U}) \\subseteq \\forall_{H}(\\mathsf{U})$. Since \n\\[ (\\forall(\\pi)(T))[v_{n+1} \\mapsto A] \\subseteq T \\, , \\]\nif $T \\in \\mathsf{U}$, then $\\forall(\\pi)(T) \\in \\forall_{H}(\\mathsf{U})$ by downwards closure. The converse follows similarly from $T \\subseteq \\forall(\\pi)(T[v_{n+1} \\mapsto A])$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{qadjprop}\nThe Hodges quantifiers are adjoints to substitution:\n\\begin{enumerate}\n\\item $\\exists_{H}$ is left adjoint to $\\mathcal{H}(\\pi)$:\n\\[ \\exists_{H}(\\mathsf{U}) \\subseteq \\mathsf{V} \\;\\; \\Longleftrightarrow \\;\\; \\mathsf{U} \\subseteq \\mathcal{H}(\\pi)(\\mathsf{V}) \\, . \\]\n\\item $\\forall_{H}$ is right adjoint to $\\mathcal{H}(\\pi)$:\n\\[ \\mathcal{H}(\\pi)(\\mathsf{V}) \\subseteq \\mathsf{U} \\;\\; \\Longleftrightarrow \\;\\; \\mathsf{V} \\subseteq \\forall_{H}(\\mathsf{U}) \\, .\\]\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nIt is straightforward to verify the adjunction conditions directly. We give a more conceptual argument.\nThere is a natural pointwise ordering on monotone functions between partially ordered sets, $h, k : P \\longrightarrow Q$:\n\\[ h \\leqslant k \\;\\; \\equiv \\;\\; \\forall x \\in P. \\, h(x) \\leqslant k(x) \\, . \\]\n$\\mathcal{L}$ is an \\emph{order-enriched functor} with respect to this ordering. Functoriality means that\n\\[ \\mathcal{L}(h \\circ g) = \\mathcal{L}(h) \\circ \\mathcal{L}(g), \\qquad \\mathcal{L}(\\id{M}) = \\id{\\mathcal{L}(M)} \\, , \\]\nwhile order-enrichment means that\n\\[ h \\leqslant k \\;\\; \\Rightarrow \\;\\; \\mathcal{L}(h) \\leqslant \\mathcal{L}(k) \\, . \\]\nThese properties imply that $\\mathcal{L}$ automatically preserves adjointness. That is, if we are given monotone maps\n\\[ f : P \\longrightarrow Q, \\qquad g : Q \\longrightarrow P \\]\nsuch that $\\id{P} \\leqslant g \\circ f$ and $f \\circ g \\leqslant \\id{Q}$, \\textit{i.e.}~ so that $f$ is left adjoint to $g$, then\n\\[ \\id{\\mathcal{L}(P)} = \\mathcal{L}(\\id{P}) \\leqslant \\mathcal{L}(g \\circ f) = \\mathcal{L}(g) \\circ \\mathcal{L}(f) \\, , \\]\nand similarly $\\mathcal{L}(f) \\circ \\mathcal{L}(g) \\leqslant \\id{\\mathcal{L}(Q)}$, so $\\mathcal{L}(f)$ is left adjoint to $\\mathcal{L}(g)$ (and of course $\\mathcal{L}(g)$ is right adjoint to $\\mathcal{L}(f)$). Combining this with Proposition~\\ref{lqprop} yields the required result.\n\\end{proof}\n\n\\subsection{The dependence-friendly quantifiers}\nWe shall also give characterizations of the dependence-guarded quantifiers as certain adjoints: this will be our first use of the intuitionistic implication.\n\nWe recall the definition of the dependence-friendly existential quantifier:\n\\[ (\\exists x \\setminus x_{1}, \\ldots , x_{n}). \\, \\phi \\;\\; \\equiv \\;\\; \\exists x. (D(x_{1}, \\ldots , x_{n}, x) \\; \\wedge \\; \\phi) \\, . \\]\nThere has not been a comparably natural notion of dependence-friendly universal quantification. According to our analysis, this is because the appropriate connective needed to express the right notion, namely intuitionistic implication, has not been available. Using it, we can define such a quantifier:\n\\[ (\\forall x \\setminus x_{1}, \\ldots , x_{n}). \\, \\phi \\;\\; \\equiv \\;\\; \\forall x. (D(x_{1}, \\ldots , x_{n}, x) \\; \\rightarrow \\; \\phi) \\, . \\]\nAs evidence for the naturalness of these quantifiers, we shall express them both as adjoints.\n\nFirstly, we recall that intuitionistic conjunction and implication are related by another fundamental adjointness \\cite{Law69}:\n\\begin{equation}\n\\label{impadjeq}\n\\mathsf{U} \\cap \\mathsf{V} \\subseteq \\mathsf{W} \\;\\; \\Longleftrightarrow \\;\\; \\mathsf{U} \\subseteq \\mathsf{V} \\rightarrow \\mathsf{W} \\, .\n\\end{equation}\nThis can be expressed as a bidirectional inference rule:\n\\[ \\infer={\\phi \\vdash \\psi \\rightarrow \\theta}{\\phi \\; \\wedge \\; \\psi \\vdash \\theta} \\, . \\]\nNext, we extend our semantic notation to the dependence-friendly quantifiers. Given $W \\subseteq \\{ v_{1}, \\ldots , v_{n} \\}$, we define $D_{W} \\in \\mathcal{H}(A^{n+1})$:\n\\[ D_{W} = \\{ T \\mid \\forall s, t \\in T. \\, s \\simeq_{W} t \\; \\Rightarrow \\; s(v_{n+1}) = t(v_{n+1}) \\} \\, . \\]\nNow we can define the semantic operators corresponding to the dependence-friendly quantifiers:\n\\[ \\exists_{W}, \\forall_{W} : \\mathcal{H}(A^{n+1}) \\longrightarrow \\mathcal{H}(A^{n}) \\]\n\\[ \\begin{array}{lcl}\n\\exists_{W}(\\mathsf{U}) & = & \\exists_{H}(D_{W} \\cap \\mathsf{U}) \\\\\n\\forall_{W}(\\mathsf{U}) & = & \\forall_{H}(D_{W} \\rightarrow \\mathsf{U})\n\\end{array}\n\\]\n\\begin{proposition}\nThe dependence-friendly existential $\\exists_{W}$ is left adjoint to the following operation:\n\\[ \\mathsf{V}  \\; \\mapsto \\; (D_{W} \\rightarrow \\mathcal{H}(\\pi)(\\mathsf{V})) \\, . \\]\nThe dependence-friendly universal $\\forall_{W}$ is right adjoint to the following operation:\n\\[  \\mathsf{V} \\; \\mapsto \\; (D_{W} \\cap \\mathcal{H}(\\pi)(\\mathsf{V})) \\, . \\]\n\\end{proposition}\n\\begin{proof}\nA direct verification is straightforward, but it suffices to observe that adjoints compose, and then to use Proposition~\\ref{qadjprop} and the adjointness~(\\ref{impadjeq}).\n\\end{proof}\n\nOf course, the analysis we have given in this sub-section  applies to any guarded quantifiers; the dependence predicates play no special r\\^ole here. The point is to show how the intuitionistic connectives round out the logic in a natural fashion. We shall apply them to a finer analysis of dependence itself in section~\\ref{depsec}.\n\n\\section{Full Abstraction}\nWe shall now prove a full abstraction result in the sense of Hodges \\cite{Hod97a}.\\endnote{As Hodges notes, he himself takes the term, and the concept, from Computer Science \\cite{Mil77,Plo77}.}\nThe point of this is to show that, even if we take sentences and their truth-values as primary,\nthe information contained in the semantics of formulas in general  is not redundant, since whenever two formulas receive different denotations, they make different contributions overall to the truth-values assigned to sentences.\n\nThe fact that such a result holds for $\\mbox{\\textrm{BID}}^{-}$-logic is notable, in that the logic is highly non-classical, while the semantics of sentences is bivalent. For \\textrm{BID}-logic, the set of possible truth values for open formulas is huge even in finite models \\cite{CH01}, while the semantics of sentences is trivalent.\n\nWhile our argument follows that of Hodges \\cite{Hod97a}, we find a  natural r\\^ole for the intuitionistic implication, and can give a very simple proof, while Hodges' argument goes through the correspondence with the game-theoretical semantics.\n\nTo formalize  full abstraction, we introduce the notion of a \\emph{sentential context} with respect to a set of variables $X$. This is a formula with an occurrence of a ``hole'' $[\\cdot]$ such that inserting a formula with free variables in $X$ into the hole yields a sentence.\nNow consider two formulas $\\phi$ and $\\psi$ of \\textrm{BID}-logic, with free variables in $X$. We say that the formulas are \\emph{semantically equivalent} if they have the same denotations, \\textit{i.e.}~ the same sets of satisfying teams, in all interpretations with respect to all structures. We say that $\\phi$ and $\\psi$ are \\emph{observationally equivalent} if for all sentential contexts $C[\\cdot]$ for $X$, $C[\\phi]$ and $C[\\psi]$ are assigned  the same truth values in all interpretations. The fact that semantic equivalence implies observational equivalence follows immediately from the compositional form of the semantics. The converse is \\emph{full abstraction}.\\endnote{While this notion is perfectly consistent with usage in Computer Science, one very important tensioning ingredient in the programming language context is missing, namely correspondence with an independently defined operational semantics \\cite{Mil77,Plo77}.}\n\n\\begin{proposition}\nThe team semantics is fully abstract for any sublanguage of \\textrm{BID}-logic containing universal quantification and intuitionistic implication.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\llbracket\\phi \\rrbracket \\setminus \\llbracket \\psi \\rrbracket$ in some interpretation contains a team $T$. Extend the language with a  relation symbol $R$, and the interpretation by assigning ${\\downarrow}(T)$ to $R$. Then use the context\n\\[ C[\\cdot] \\; \\equiv \\; \\forall v_{1}, \\ldots , \\forall v_{n}. \\, (R(v_{1}, \\ldots , v_{n}) \\rightarrow [\\cdot]) \\]\nwhere the free variables in $\\phi$ and $\\psi$ are contained in $\\{ v_{1}, \\ldots , v_{n} \\}$.\nThen $C[\\phi]$ is true (satisfied by the empty tuple), since for every team $T'$ satisfying $R(v_{1}, \\ldots , v_{n})$, $T' \\subseteq T$, and hence by assumption and downwards closure, $T'$ satisfies $\\phi$. This means that all teams  over  $\\{ v_{1}, \\ldots , v_{n} \\}$ satisfy the implication $R(v_{1}, \\ldots , v_{n}) \\rightarrow \\phi$, and hence $\\langle \\rangle $ satisfies $C[\\phi]$. On the other hand, $C[\\psi]$ is not satisfied by the empty tuple, since $T$ satisfies $R(v_{1}, \\ldots , v_{n})$, while $T$ does not satisfy $\\psi$ by assumption.\n\\end{proof}\nNote that the use of the intuitionistic implication in relativizing to those teams satisfying the precondition $R(v_{1}, \\ldots , v_{n})$ is exactly what is needed.\n\n\\section{Analyzing Dependence}\n\\label{depsec}\n\nWe now turn to the dependence predicate itself. Since it encapsulates the ``jump'' from  first-order to second-order semantics, we cannot be too hopeful about taming it axiomatically\\endnote{See \\cite{MR1867954} for details on this.}. But it turns out that we can give a finer analysis in \\textrm{BID}-logic.\n\nConsider the following ``trivial'' case of dependence:\n\\[ C(v) \\; \\equiv \\; D(\\varnothing, v) \\, . \\]\nThis expresses that $v$ depends on nothing at all, and hence has a fixed value --- functional dependency for the constant function. Semantically, this is the following simple special case of the semantics of dependence:\n\\[ T \\models_{X} C(v) \\; \\; \\equiv \\;\\; \\forall t_{1}, t_{2} \\in T. \\, t_{1}(v) = t_{2}(v) \\, . \\]\nUsing the intuitionistic implication, we can \\emph{define} the general dependence predicate from this special case:\n\\begin{equation}\n\\label{depeq}\nD(W, v) := \\left(\\bigwedge_{w \\in W} C(w) \\right) \\; \\rightarrow \\; C(v) \n\\end{equation}\n\n\\begin{proposition}\n\\label{depcprop}\nThe definition of $D$ from $C$ is semantically equivalent to the definition given previously:\n\\[ T \\models_{X} D(W, v) \\;\\; \\equiv \\;\\; \\forall s, t \\in T. \\, s \\simeq_{W} t \\; \\Rightarrow \\; s(v) = t(v) .) \\]\n\\end{proposition}\n\\begin{proof}\nThis is just an exercise in unwinding the definitions. Note that the intuitionistic implication lets us range over all subsets of the team which are in a single equivalence class under $\\simeq_{W}$, and require that $v$ is constant on those subsets.\n\\end{proof}\n\n\\subsection{Armstrong Axioms}\nThe current stock of plausible axioms for the dependence predicates is limited to the\n\\emph{Armstrong axioms} from database theory \\cite{Arm}. These are a standard complete set of axioms for functional   dependence. They can be given as follows.\n\n\\begin{center}\n\\begin{tabular}{ll}\n(1)\n & Always $D(x,x)$. \\\\\n(2) & If $D(x,y,z)$, then $D(y,x,z)$. \\\\\n(3) & If $D(x,x,y)$, then $D(x,y)$. \\\\\n(4) & If $D(x,z)$, then $D(x,y,z)$. \\\\\n(5) & If $D(x,y)$ and $D(y,z)$, then $D(x,z)$. \\\\\n\\end{tabular}\n\\end{center}\n\n\\noindent However, in the light of our analysis, the Armstrong axioms simply fall out as standard properties of  implication and conjunction.\\endnote{Formal connections between the Armstrong axioms and  propositional logic  were made by Fagin \\cite{Fag}. He only considered Horn clauses, so the distinction between intuitionistic and classical logic was not apparent. Nevertheless, the passage to two-element subsets in the ``Semantic proof of the Equivalence Theorem'' in \\cite{Fag} implicitly involves similar reasoning to Proposition~\\ref{depcprop}.}\nIf we  set $p = C(x)$, $q = C(y)$, $r = C(z)$, and use (\\ref{depeq}) to translate the Armstrong axioms into purely implicational form, we see that they  correspond to the following:\n\n\\begin{center}\n\\begin{tabular}{ll}\n(1)\n & $p \\rightarrow p$. \\\\\n(2) & $(p \\rightarrow q \\rightarrow r) \\rightarrow (q \\rightarrow p \\rightarrow r)$. \\\\\n(3) & $(p \\rightarrow p \\rightarrow q) \\rightarrow (p \\rightarrow q)$. \\\\\n(4) & $(p \\rightarrow r) \\rightarrow (p \\rightarrow q \\rightarrow r)$. \\\\\n(5) & $(p \\rightarrow q) \\rightarrow (q \\rightarrow r) \\rightarrow (p \\rightarrow r)$. \\\\\n\\end{tabular}\n\\end{center}\nThese are the well-known axioms $\\textbf{I, C, W, K, B}$ respectively\\endnote{Axiom (4) as given generalizes the standard \\textbf{K} axiom  $p \\rightarrow q \\rightarrow p$, which is obviously derivable from (1) and (4) by substitution and Modus Ponens.}  \\cite{CF58}  --- which form a complete axiomatization of \\emph{intuitionistic} (but not classical!) implication.\\endnote{Under the (Curry part of the) Curry-Howard correspondence, they correspond to a well-known functionally complete set of \\emph{combinators} \\cite{CF58}.}\nA standard example of a classically valid implicational formula which is \\emph{not} derivable from these axioms is \\emph{Peirce's law}: $((p \\rightarrow q) \\rightarrow p) \\rightarrow p$.\n\n\nThus we have reduced the understanding of the dependence predicate to understanding of the, \\textit{prima facie}~simpler, constancy predicate $C$.\n\n\\section{Further Directions}\nIn this final section, we shall sketch a number of further directions.\nDetailed accounts are under development, and  will appear elsewhere.\n\n\\subsection{Completeness}\nPredicate BI-logic is a well developed formalism, with a proof theory which is sound and complete relative to an algebraic semantics \\cite{Pym02}. Since \\textrm{BID}-logic is a special case, we have a sound ambient inference system. Of course this is not complete for the intended semantics for \\textrm{BID}-logic  --- and cannot be.\nWe may hope to obtain completeness for some smaller class of models, possibly on the lines of the Henkin completeness theorem for higher-order logic \\cite{Hen50}.\n\n\\subsection{Diagrams}\nNow fix a particular interpretation in a structure $\\mathcal{M}$  with universe $A$. Consider the following construction. We  introduce constants for each $a \\in A$, the usual first order diagram (all true atomic sentences), and the following  infinitary axiom:\n\\[ \\forall v. \\, \\bigotimes_{a \\in A} (v=a) \\, . \\]\nWe can \\emph{define} the predicate $C$ (and hence dependence $D$) by the following infinitary formula:\n\\[ C(v) := \\bigvee_{a \\in A} (v=a) \\, . \\]\nNote how the two different connectives (one additive, the other multiplicative) feature naturally.\n\nThis gives a logical (albeit infinitary) characterization of dependence.\n\n\\subsection{Representation}\nWe can also consider representation theory for the structures $\\mathcal{H}(X) = \\mathcal{L}(\\mathcal{P}(X))$. We seek lattice-theoretic properties of these structures which suffice to characterize them.\n\nFirstly, we note that the down-closures of single teams are exactly the \\emph{complete join-primes} of the lattice:\n\\[ a \\leqslant \\bigvee_{i} b_{i} \\;\\; \\Rightarrow \\;\\; \\exists i. \\, a \\leqslant b_{i} . \\]\nMoreover, these join-primes order generate, \\textit{i.e.}~ every element is the join of the join-primes below it.\nAll of this structure is in terms of the intuitionistic disjunction.\n\n\nNext, we note that the join-primes are closed under $\\otimes$, which is  moreover idempotent on the join-primes, endowing them with the structure of a semilattice.\nThis is very different to the semilattice structure given by intuitionistic disjunction: e.g.\n\\[ {\\downarrow}(T_{1}) \\vee {\\downarrow}(T_{2}) = {\\downarrow}(\\{T_{1}, T_{2}\\}) \\neq  {\\downarrow}(T_{1} \\cup T_{2}) = {\\downarrow}(T_{1}) \\otimes {\\downarrow}(T_{2}) \\, . \\]\nThe \\emph{double singletons} are exactly the complete atoms in this semilattice, which is complete atomic in the usual sense.\n\n\nSyntactically, assuming names for elements, we can describe these atomic join-primes in the lattice of propositions over variables $v_{1}, \\ldots , v_{n}$ as\n\\[  (v_{1} = a_{1}) \\; \\wedge \\; \\cdots \\; \\wedge \\;  (v_{n} = a_{n}). \\]\nThese are of course the tuples. (Downclosures of) arbitrary teams are then described by expressions\n$\\bigotimes_i A_i$,  where $A_i$ ranges over such atoms. Arbitrary elements are joins (intuitionistic disjunctions) of such elements. So there is a normal form for general elements:\n\\[ \\bigvee_{i} \\bigotimes_{ij} A_{ij} \\, . \\]\nMoreover, from the lattice-theoretic properties it is easily shown that the ordering between such normal forms agrees with the set inclusion ordering.\n\n\n\n\\subsection{Expressiveness}\n\\newcommand{\\rel}[1]{\\mbox{rel}(#1)}\nOne of the defining characteristics of Dependence Logic as well as IF-logic is that they can be expressed in Existential Second Order Logic, $\\Sigma^1_1$, and conversely, every $\\Sigma^1_1$ definable property of structures can be expressed with a sentence of Dependence Logic. Both are true even on finite structures. To see what this connection with $\\Sigma^1_1$ means let us adopt the notation that if $T$ is a team on a set $X$ of variables, then $\\rel{T}$ is the corresponding relation.\nHodges~\\cite{Hod97b} associates with every formula $\\phi$ of IF-logic (equivalently, of Dependence Logic) with free variables in the set $X=\\{x_1,...,x_n\\}$ an Existential Second Order sentence $\\tau_\\phi(R)$, with $R$ an $n$-ary predicate symbol, such that in any model $\\mathcal{M}$ and for any team $T$ on $X$ the following holds: \\begin{equation}\\label{sol}\\mathcal{M},T\\models_X\\phi\\iff (\\mathcal{M},\\rel{T})\\models\\tau_\\phi(R).\\end{equation} Conversely, if $\\Phi$ is any Existential Second Order sentence, then there is a sentence $\\phi$ of Dependence Logic such that the following holds for all models $\\mathcal{M}$: $$\\mathcal{M}\\models\\Phi\\iff\\mathcal{M},\\{\\langle\\rangle\\}\\models\\phi.$$\nVirtually all model theoretic properties of Dependence Logic follow from this relationship with $\\Sigma^1_1$, for example, the Compactness Theorem, the downward and upward L\\\"owenheim-Skolem Theorems, the Interpolation Theorem, and the fact that every sentence $\\phi$ in Dependence Logic for which there exists a ``negation\" $\\psi$ such that for all $\\mathcal{M}$ $$\\mathcal{M}\\models\\phi\\iff\\mathcal{M}\\not\\models\\psi,$$ is actually first order definable\\endnote{See e.g. \\cite{Vaan} for details.}. Also the interesting fact that the class of properties of finite structures expressible in Dependence Logic is exactly NP follows from this. Because of these connections it is quite interesting to ask whether the extensions $\\mbox{\\textrm{BID}}^{-}$ and $\\textrm{BID}$ can likewise be embedded in $\\Sigma^1_1$, the existential fragment of Second Order Logic.\n\nNow the question arises which semantics one should use. To be able to compare results with Dependence Logic and IF-logic, we use the full semantics familiar from \\cite{Hod97a} and \\cite{Vaan}.\n\n\\begin{proposition}\nThere is no translation of  any extension of  Dependence Logic containing either intuitionistic implication or linear implication into existential second order $\\Sigma^1_1$. The same is true on finite models, assuming NP$\\ne$co-NP.\n\\end{proposition}\n\n\\begin{proof} Let $\\phi(x_1)$ be a formula of Dependence Logic in the empty vocabulary such that for any team $T$: $\\mathcal{M},T\\models\\phi(x_1)$ if and only if $A$ is infinite\\endnote{The free variable $x_1$ plays no role in this.}. Let $\\bot$ denote a sentence in the empty vocabulary, only satisfied by the empty team, e.g. $\\forall x.x=x\\wedge\\neg x=x$.\nSuppose there were an Existential Second Order sentence $\\tau(R)$ such that\na model $\\mathcal{M}$ and a team $T$ on $\\{x_1\\}$ satisfy $\\phi(x_1)\\to\\bot$ if and only if $(\\mathcal{M},\\rel{T})$ satisfies $\\tau(R)$. If $\\mathcal{M}$ is any finite model and $T=\\{s\\}$, where $s(x_1)\\in M$, then  $\\mathcal{M},T\\models\\phi(x_1)\\to\\bot$, whence $(\\mathcal{M},\\{s(x_1)\\})\\models\\tau(R)\\wedge\\exists x_1R(x_1)$. By the Compactness Theorem of Existential Second Order Logic, $\\tau(R)\\wedge\\exists x_1R(x_1)$ has an infinite model $(\\mathcal{M}',\\rel{T'})$. Thus $\\mathcal{M}'$ and the team $T'$ satisfy $\\phi(x_1)\\to\\bot$. Moreover, $T'\\ne\\varnothing$. By the definition of the semantics of $\\to$, since $T'$ satisfies $\\phi(x_1)$ in $\\mathcal{M}'$, $T'$ must satisfy $\\bot$, a contradiction.\n\nLet us then consider finite models. It is easy to write down   a formula $\\phi(x_1)$ of Dependence Logic in the vocabulary of graphs such that for any team $T\\ne\\varnothing$: $\\mathcal{M},T\\models\\phi(x_1)$ if and only if $\\mathcal{M}$ is 3-colorable. Let $\\bot$ be as above. If $\\mathcal{M}$ is any graph that is not 3-colorable and $T=\\{s\\}$, where $s(x_1)\\in M$, then  $\\mathcal{M},T\\models\\phi(x_1)\\to\\bot$. On the other hand, suppose $\\mathcal{M}$ is 3-colorable, but $\\mathcal{M}$ and some team $\\{s\\}$ satisfy $\\phi(x_1)\\to\\bot$.  By the definition of the semantics of $\\to$, since $\\{s\\}$ satisfies $\\phi(x_1)$ in $\\mathcal{M}$, $\\{s\\}$ must satisfy $\\bot$, a contradiction. Thus a graph $\\mathcal{M}$ and a team $\\{s\\}$ satisfy $\\phi(x_1)\\to\\bot$ if and only if $\\mathcal{M}$ is not 3-colorable. Suppose now there were an Existential Second Order sentence $\\tau(R)$ such that\na graph $\\mathcal{M}$ and a team $T$ satisfy $\\phi(x_1)\\to\\bot$ if and only if $(\\mathcal{M},\\rel{T})$ satisfies $\\tau(R)$. Then we could check if a graph $\\mathcal{M}$ is not 3-colorable by checking if  $\\tau(R)$ is satisfied by $\\mathcal{M}$ and and a team $\\{s\\}$, where $s$ can be any assignment. The latter is NP, so we get NP=co-NP.\n\nThe same argument can be used to show that $\\multimap$ leads outside of $\\Sigma^1_1$: Suppose $\\phi(x_1)$ is as above and there is an Existential Second Order sentence $\\tau(R)$ such that\na model $\\mathcal{M}$ and a team $T$ satisfy $\\bot \\, \\wedge \\, (\\phi(x_1)\\multimap\\bot)$ if and only if $(\\mathcal{M},\\rel{T})$ satisfies $\\tau(R)$. If $\\mathcal{M}$ is any finite model and $T=\\varnothing$, then  $\\mathcal{M},T\\models\\bot \\, \\wedge \\, (\\phi\\multimap\\bot)$, whence $(\\mathcal{M},\\varnothing)\\models\\tau(R)$. By the Compactness Theorem of Existential Second Order Logic, $\\tau(R)$ has an infinite model $(\\mathcal{M}',\\rel{T'})$. Thus $\\mathcal{M}'$ and the team $T'$ satisfy $\\bot \\, \\wedge \\, (\\phi\\multimap\\bot)$. In particular, $T'=\\varnothing$ and $\\varnothing$ satisfies $\\phi\\multimap\\bot$. Since in this model any $\\{s\\}$ satisfies $\\phi$, by the definition of the semantics of $\\multimap$, $\\{s\\}$ satisfies $\\bot$, a contradiction.\n\\end{proof}\n\n\n\\noindent The proof actually shows that $\\textrm{BID}$ fails to satisfy the Compactness Theorem. A similar argument shows that $\\textrm{BID}$ fails to satisfy the Downward L\\\"owenheim Skolem Theorem. \n\n\\begin{proposition}\nThere is a translation of $\\textrm{BID}$ into Full Second Order Logic.\n\\end{proposition}\n\n\\begin{proof} We follow \\cite{Hod97b} (see also \\cite{Vaan}) and present only the additions needed over and above Dependence Logic and IF-logic:\n\\begin{eqnarray*}\n\\tau_{\\phi\\multimap\\psi}(R) &=&\\forall S(\\tau_\\phi(S)\\to\\forall U(\\forall \\vec{x}(U(\\vec{x})\\leftrightarrow(S(\\vec{x})\\vee\nR(\\vec{x})))\\to\\tau_\\psi(U)))\\\\\n\\tau_{\\phi\\to\\psi}(R) &=&\\forall S(\\forall \\vec{x}(S(\\vec{x})\\to\nR(\\vec{x}))\\to(\\tau_\\phi(S)\\to\\tau_\\psi(S))) \\, .\n\\end{eqnarray*}\\end{proof}\n\n\\noindent In conclusion, we may say that $\\mbox{\\textrm{BID}}^{-}$ and $\\textrm{BID}$ seem to have a more robust and uniform algebraic structure than Dependence Logic and IF-logic. We anticipate that this is reflected also in an effective proof theory, still to be developed. On the other hand the price of this seems to be that ``nice\" model theoretic properties are lost, at least in the full semantics. Perhaps there are some underlying, hitherto unidentified, reasons why logics developed for dependence cannot simultaneously have a ``nice\" model theory and effective proof theory. After all, we know from Lindstr\\\"om's Theorem (\\cite{MR0244013}) that there are intrinsic obstacles to having model-theoretically defined extensions of first order logic with both nice proof theory and nice model theory. However, we have a trivalent logic, unlike the setting considered by Lindstr\\\"om. So it is too early to say whether there are general reasons why $\\textrm{BID}$ does not satisfy Compactness and other model theoretic properties familiar from Dependence Logic, or whether we  have just not found the right concepts yet.\n\n\\theendnotes\n\n\\bibliographystyle{klunamed}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMicrostructure of materials are meticulously engineered to cater the ever progressing and more demanding needs. \nProperties, once thought to be irreconcilable, are increasingly combined through appropriate combination of phases in the microstructures~\\cite{jacques2001developments,huang2017multiphase,kumar2020carbon}.\nEven though microstructures characterised by more than one chemically-distinct phases have been around for a considerable period of time, the distribution of the constituent phases engenders a unique category of multiphase materials. \nFor instance, in polycrystalline pearlitic steel, the combination of ferrite and cementite, in a specific phase-fraction, is observed in all grains, despite being separated by the interfaces (grain boundaries)~\\cite{zhang2011microstructure,hwang2019microstructure}.\nSuch grains can be treated as chemically homogeneous, despite the presence of two distinct phases. \nHowever, on the other hand, there are specialised steels, wherein the grains are not chemically homogeneous but are exclusively associated with one of the constituent phases~\\cite{liljas2008development}. \nIn other words, instead of martensite and ferrite co-existing in all grains of the polycrystalline microstructure, individual grains assume a phase, thereby establishing a chemical inhomogeneity in the system~\\cite{armas2002mechanical}.\nPolycrystalline materials, characterised by these chemically-distinct grains, are qualified contextually as multiphase, and depending on number of constituent phases, these materials are referred to duplex, triplex and such. \n\nBesides steel, multiphase microstructures are established in wide-range of materials to achieve desired combination of properties~\\cite{filip2003effect,gollapudi2011microstructure}.  \nThe extensive applicability of two- and three-phase titanium alloys is primarily due to properties which is a direct consequence of their multiphase microstructure. \nThe prevalence of multiphase polycrystalline arrangement in high-entropy alloys is a principal reason for their characteristic behaviour~\\cite{tang2015tensile,liu2019fatigue}.  \nIn ceramics, mechanical properties including fracture toughness are noticeably enhanced by two-phase microstructure.\nDuplex system of alumina and silicon carbide is a prime example of such behaviour-enhanced ceramics~\\cite{jang1995effect,lutz1991k}. \nMoreover, it has been reported that the introduction of an additional phase, which institutes a triplex microstructure, exacts a more preferred response from ceramics to an imposed mechanical conditions~\\cite{neuman2017high,feng2020effect}. \nSimilar favourable effects of multiphase microstructure are observed in composite materials as well~\\cite{do2008microstructure}. \n\nIn addition to the characteristic feature of the phases associated with the multiphase materials, the emerging properties are also considerably influenced by the volume fraction of the constituent phases, $i.e$ phase-fraction, and average grain size. \nConsequently, to achieve desired behaviour, multiphase materials with varying phase-fraction ranging from minimal minor-phase (close to 1\\%) to equi-fraction (50-50 in duplex), and appropriate grain sizes, are fabricated~\\cite{sternitzke1997structural,fan1997computer}. \nWhile material-specific processing techniques are employed for introducing the necessary phase-fraction, the required grain sizes are obtained through a rather well-known microstructural transformation called grain growth~\\cite{yu2021high}. \nThe mechanism of grain growth in multiphase systems is significantly different single-phase polycrystalline microstructure.\nIn polycrystalline materials with chemically-homogeneous grains, the local diffusion of atoms across the grain boundaries, which consequently leads to its migration, ultimately governs the grain growth. \nIn other words, in single-phase systems, the grain growth is primarily dictated by the movement of the interface (or) grain boundaries. \nHowever, in multiphase microstructures, the association of grains to a specific constituent phase adds an additional facet to the conventional grain growth. \nBesides reducing the number of grains, grain growth in multiphase systems conserves the characteristic volume-fraction of the phases. \nTherefore, the curvature-driven transformation, which minimises the overall grain-boundary energy, in multiphase materials is, in principle, a combination of coarsening and grain growth~\\cite{cahn1991stability}. \nTypifying features of both coarsening and grain growth, which respectively are preservation of volume fraction and growth of larger grains at the expense of smaller ones, are simultaneously observed in multiphase materials. \nFor this reason, the energy-minimising curvature-driven transformation in multiphase systems is at times referred to as concurrent grain growth and coarsening. \nCorrespondingly, as opposed to interface migration, the long-range diffusion of atoms dictates grain growth in multiphase polycrystalline systems~\\cite{fan1997diffusion}. \nRegardless of the mechanism, owing to the influence of the grain size on the properties rendered by the materials, grain growth in multiphase systems have been extensively analysed~\\cite{guo2009microstructural,liu2015synergetic}. \nMoreover, it is conceivable, and indeed reported that, a change in the average grain-size of a multiphase material, while employed in an application, results in undesired behaviour. \nIn other words, besides the direct effect of grain-size on the properties of multiphase materials, its life in an application is dictated by the rate at which the grains grow.  \nSolid-oxide fuels with triplex microstructure are prime examples of this relation between the growth kinetics and life of a material~\\cite{lei2017phase}. \nConsidering this influence of grain size, investigations have been geared towards understanding the kinetics of grain growth in multiphase polycrystalline system.  \n\nExperimental techniques generally pose definite practical difficulties when adopted for comprehensive analyses of grain growth.\nTherefore, theoretical approaches have long since been adopted to complement, and extend, the existing understanding~\\cite{saito1992monte,kawasaki1989vertex,anderson1984computer}.\nThese theoretical studies, particularly one involves multiphase-field models, have been offering critical insights on mechanism and kinetics of grain growth in multiphase system, which are consistent with the experimental observation~\\cite{krill2002computer,perumal2017phase,mckenna2014grain}. \nExisting investigations, and resulting understanding, of grain-growth kinetics in multiphase polycrystalline systems can broadly be categorised into two. \nOne relates material-specific parameters, including diffusivities and grain-boundary energy anisotropies, to the rate of grain growth~\\cite{ravash2014three,ravash2017three2}, while the other focuses on the effect of microstructural features like phase-fractions~\\cite{fan1997computer,yadav2016effect}.\nWhile the material-specific studies are, in principle, inherently  bound by the choice parameter(s), the analyses involving phase-fraction offer more generalised insights that are relevant to a wide-range of multiphase polycrystalline systems. \n\nIn-keeping with the change in grain-growth mechanism, the investigations focusing on microstructural features unravel that the introduction of second phases significantly reduces the kinetics of overall evolution. \nMoreover, it is realised that the expression capturing the temporal change in the average radius of the polycrystalline system reflects coarsening kinetics, $\\bar{R}^n\\propto t$ with $n=3$, irrespective of the volume-fraction of the second phase~\\cite{fan1997diffusion,ohnuma1999computer}. \nEven though the major and minor phase(s) - categorised based on volume-fraction- , and the entire polycrystalline system as a whole, evolve at a rate that complies with the coarsening power-law, the kinetic-coefficient ($k$) varies with the  phases, particularly their volumes~\\cite{fan1997computer}.\nFor instances, in a duplex microstructure characterised by unequal volume-fraction of constituent phases, the grains of the major phase, with higher phase-fraction, grow at the noticeably faster rate when compared to the minor phase. \nThis disparity in the kinetics of the grain growth between the phases is indicated by the difference in the kinetic-coefficient ($k$).\nThe relatively sluggish growth of the minor phase(s) is attributed to the increased distance between the respective chemically-similar grains, and complex diffusion-pathways that facilitate the growth, when compared to the proximity of the major phases. \nFurthermore,  attempts have been quantify the effect of volume-fraction on the kinetic-coefficients of major- and minor-phases by comparing the evolution of the microstructures with varying phase-fractions~\\cite{yadav2016effect}. \nDespite these apparent advancements, there continues to exist certain aspects to the grain-growth kinetics that prevent a comprehensive understanding of the evolution of multiphase polycrystalline systems. \n\nIn addition to the characteristic nature, and volume-fraction, of the phases, the properties of multiphase systems depend on the \\textit{overall grain-size} of the polycrystalline microstructure.\nHowever, studies aimed at explicating the grain-growth rate in multiphase materials primarily report, and discuss, on the difference in the kinetics exhibited by the major- and minor-phases, while largely overlooking the evolution of the entire microstructure~\\cite{fan1997topological,jang1995effect}. \nMoreover, the collective influence of the growth rate of major- and minor-phases on the temporal change in overall average size of the grains is yet to be convincingly understood. \nBesides, in three (or more) phase systems, the interdependency between the grain-growth rate of different phases have not been thoughtfully considered so far. \nAmongst others, in the present work, attempts are made to address the aforementioned questions on the grain-growth kinetics of the overall multiphase microstructure through appropriate statistical analysis. \n\n\n\\section{Devising \\lq multidimensional\\rq \\thinspace dataset}\n\nPersuasive understanding on how the evolution of grains of an individual phase relates to the overall grain growth in a multiphase system, can hardly be gained from analysing of a few microstructures.  \nTo that end, a \\lq multidimensional\\rq \\thinspace dataset is developed by modelling grain growth exhibited by duplex and triplex microstructures in the multiphase-field framework~\\cite{perumal2018phase,amos2019understanding}. \nThermodynamical underpinnings of the adopted model is relatively well-established, and is often acclaimed to be a computationally-efficient alternate~\\cite{amos2020grand,amos2020distinguishing}. \nEven though the complete formulation of the present approach~\\cite{perumal2019concurrent,perumal2020quadrijunctions}, its ability to incorporate different modes of mass transfer, and recover sharp-interface solutions is exhaustively discussed elsewhere ~\\cite{amos2020multiphase,hoffrogge2021multiphase}, a brief outline focusing on the key aspects in rendered here. \n\n\\subsection{Multicomponent multiphase-field model}\n\nMultiphase-field approach is characterised by the introduction of scalar variable(s) which distinguishes the different phases in the system. \nConventionally, while modelling polycrystalline microstructures, these variables, called phase-field ($\\phi(\\vv{x},t)$), are employed to differentiate various grains, and their spatio-temporal evolution translates to grain growth. \nAcross the grain boundary separating two grains, phase-field assumes a value of $\\phi=0$ in one grain, while the other is realised by a constant non-zero value, which is generally $\\phi=1$. \nThe diffuse region, wherein the value of the phase-field gradually varies, $\\phi(\\vv{x},t)\\in(0,1)$, describes the grain boundary.  \nGiven that, in most cases, the grains in the polycrystalline systems are chemically homogeneous, phase-field typifying the various grains are introduced as a tuple, which is expressed as \n\\begin{align}\\label{eq:mp1a}\n\\mbox{{\\bm{$\\phi$}}}=\\left \\{\\phi_{1},\\phi_{2},\\dots,\\phi_{N}\\right \\},\n\\end{align}\nwhere $N$ denotes the total number of grains in the system. \nHowever, since the present approach models grain growth in multiphase systems comprising of $N$ phases, wherein numerous grains are associated each phase, the corresponding phase-field is extended and written as\n\\begin{align}\\label{eq:mp2}\n\\mbox{{\\bm{$\\phi$}}}=\\Big\\{ \\underbrace{ \\{ \\phi_{\\alpha}^{1},\\phi_{\\alpha}^{2}\\dots\\phi_{\\alpha}^{q_{\\alpha}} \\}}_{\\vv{\\phi_{\\alpha}}}, \\underbrace{ \\{ \\phi_{\\beta}^{1},\\phi_{\\beta}^{2}\\dots\\phi_{\\beta}^{q_{\\beta}} \\}}_{\\vv{\\phi_{\\beta}}}\\dots \\underbrace{ \\{ \\phi_{N}^{1},\\phi_{N}^{2}\\dots\\phi_{N}^{q_{N}} \\}}_{\\vv{\\phi_{N}}} \\Big\\}.\n\\end{align}\nMoreover, in this formulation, number of grains sharing a given phase is represented by $q_i$ where $i\\in\\{\\alpha,\\beta,\\dots,N\\}$.\nBy assigning appropriate concentration, grains of a given phase are distinguished from the rest, $\\{\\mbox{{\\bm{$\\phi$}}}(\\mbox{{\\bm{$c$}}})=\\mbox{{\\bm{$\\phi$}}}_\\alpha(\\mbox{{\\bm{$c$}}}) | \\mbox{{\\bm{$c$}}}=\\vv{c}_\\alpha\\}$.  \nSimilar to phase-field, in order to encompass the $K$-different chemical components, the concentration is introduced as a tuple.\nCorrespondingly, concentration of a random grain $m$ associated with phase-$\\alpha$ reads\n\\begin{align}\\label{eq:conc1}\n\\mbox{{\\bm{$c$}}}^{\\alpha}_{m}=\\left\\{c^{\\alpha}_{m:i},c^{\\alpha}_{m:j},\\dots,c^{\\alpha}_{m:K}\\right\\}.\n\\end{align}\nThe homogeneity in the chemical composition of the grains associated with a given phase, say $\\alpha$, yields\n\\begin{align}\\label{eq:conc2}\n\\mbox{{\\bm{$c$}}}^{\\alpha}_{1}=\\dots=\\mbox{{\\bm{$c$}}}^{\\alpha}_{m}=\\dots=\\mbox{{\\bm{$c$}}}^{\\alpha}_{q_{\\alpha}}\\equiv\\mbox{{\\bm{$c$}}}^{\\alpha}=\\left\\{c^{\\alpha}_{i},c^{\\alpha}_{j},\\dots,c^{\\alpha}_{k}\\right\\}.\n\\end{align}\nTo ensure that the volume-fraction of the phases remains unaltered despite the evolution, equilibrium composition is assigned to the corresponding phases.\nIn other words, since the current approach attempts to model grain growth in multiphase system, $\\mbox{{\\bm{$c$}}}^{\\alpha}$ represents a tuple of equilibrium composition that characterises phase-$\\alpha$.\n\nFollowing the conventional framework, the overall energy-density of the multiphase polycrystalline system is formulated as the combination interface (grain boundaries) and bulk (grain) contribution~\\cite{provatas2011phase}. \nCorrespondingly, by incorporating the appropriate phase-field and concentration, the energy-density of the system comprising of $N$-phases and $K$ chemical components is written as \n\\begin{align}\\label{eq:functional1}\n\\cal{F}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}}) & =\\cal{F}_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})+\\cal{F}_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}}) \\\\ \\nonumber\n&=\\int_{V}f_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})+f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})\\diff V,\n\\end{align}\nwhere $f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})$ and $f_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})$ are the respective energy-contribution of grain and grain boundary with $V$ representing the volume. \nThe interface contribution, $f_{\\text{int}}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})$, akin to most multiphase-field techniques, comprises of a gradient- and potential-energy term. \nGrain-boundary energy densities, and corresponding anisotropies, are introduced to the system through the interface-energy contribution~\\cite{tschukin2017concepts}. \nWhile multi-well potentials are generally employed to penalise phase-field, and ensure its bounds, in the present work, an obstacle-type potential operating in combination with Gibbs simplex is involved~\\cite{amos2018phase}. \nFurthermore, the energy contributions from the grains, $f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})$, which is reasonably assumed to be insignificant while modelling grain growth in single-phase system, is formulated as the interpolation of the energy contribution of the individual grains, $f(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})=\\sum_{\\alpha}^{N}\\sum_{m}^{q_{\\alpha}}f^{\\alpha}_{m}(\\mbox{{\\bm{$c$}}}^{\\alpha})h(\\phi_{\\alpha}^{m})$.\nGiven that the contribution of the individual grains is dictated by the characteristic equilibrium-composition of the associated phases, the volume of the phases during grain growth is preserved by the energy-density term, $f_{\\text{bulk}}(\\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})$~\\cite{amos2018globularization,amos2020limitations}.  \n\nThe spatio-temporal evolution of phase-field, which translates to grain growth, is formulated by considering phenomenological minimisation of the overall energy-density of the system.\nCorrespondingly, the evolution of a random grain $m$, which is associated with phase-$\\alpha$, is dictated by\n\\begin{align}\\label{eq:ph_evo1}\n\\tau\\epsilon\\frac{\\partial \\phi_{\\alpha}^{m}}{\\partial t} & = -\\frac{\\partial \\cal{F}(\\mbox{{\\bm{$\\phi$}}},\\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}},\\mbox{{\\bm{$c$}}})}{\\partial \\phi_{\\alpha}^{m}}\\\\ \\nonumber\n&=\\epsilon\\left[ \\vv{\\nabla} \\cdot \\frac{\\partial a(\\mbox{{\\bm{$\\phi$}}}, \\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})}{\\partial \\vv{\\nabla} \\phi_{\\alpha}^{m}} - \\frac{\\partial a(\\mbox{{\\bm{$\\phi$}}}, \\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})}{\\partial \\phi_{\\alpha}^{m}} \\right] - \\frac{1}{\\epsilon} \\left[ \\frac{\\partial w(\\mbox{{\\bm{$\\phi$}}})}{\\partial \\phi_{\\alpha}^{m}} \\right] - \\left[ \\frac{f^{\\alpha}_{m}(\\mbox{{\\bm{$c$}}}^{\\alpha},\\phi_{\\alpha}^{m})}{\\partial \\phi_{\\alpha}^{m}} \\right] - \\Lambda,\n\\end{align}\nwhere the Lagrange multiplier $\\Lambda$ is introduced to ensure that the summation of phase-fields at any point in the system is $1$.\nMoreover, in the above evolution equation, while $a(\\mbox{{\\bm{$\\phi$}}}, \\vv{\\nabla} \\mbox{{\\bm{$\\phi$}}})$ and $w(\\mbox{{\\bm{$\\phi$}}})$ correspond to the gradient and potential energy terms, the parameter dictating interface-width, and its stability during the migration, are denoted by $\\epsilon$ and $\\tau$, respectively.\n\nIn multiphase-field models, wherein the energy contribution of the bulk phases are described based on the dependent-concentration, the corresponding driving-force, which emerges from $\\frac{f^{\\alpha}_{m}(\\mbox{{\\bm{$c$}}}^{\\alpha},\\phi_{\\alpha}^{m})}{\\partial \\phi_{\\alpha}^{m}}$ in Eqn.~\\eqref{eq:ph_evo1}, and dictates the evolution of phase-field, can be viewed as the difference in the Legendre transform of the free-energy densities.\nThis understanding forms the basis of the grand-potential approach, and when consistently extended assumes chemical potential as the continuous and dynamic variable replacing phase-dependent concentration~\\cite{plapp2011unified}. \nGiven its computational efficiency, this approach is adopted in the present work, and the driving-force dictating phase-field evolution is formulated by treating chemical potential as the dynamic variable. \nThe temporal evolution of the chemical potential, which principally governs the bulk driving-force in phase-field evolution, is written as\n\\begin{align}\\label{chempot_ev}\n\\frac{\\partial \\mu_{i}}{\\partial t}=\\left \\{ \\vv{\\nabla}\\cdot\\left[ \\sum_{j=1}^{K-1} \\vv{M}(\\mbox{{\\bm{$\\phi$}}}) \\vv{\\nabla} \\mu_{j}\\right ] - \\sum_{\\alpha}^{N} \\sum_{m}^{q} c_{i}^{\\alpha}\\frac{\\partial \\phi_{\\alpha}^{m}}{\\partial t} \\right \\} \\left [ \\sum_{\\alpha}^{N} \\sum_{m}^{q} h(\\phi_{\\alpha}^{m}) \\frac{\\partial c_{m:i}^{\\alpha}}{\\partial \\mu_{j}} \\right ]_{ij}^{-1},\n\\end{align}  \nwhere $\\mu_{i}$ denotes the continuous chemical-potential of component-$i$. \nThe mobility of the migrating elements, in the multicomponent setup, is dictated by matrix $\\vv{M}(\\mbox{{\\bm{$\\phi$}}})$ which also facilitates the incorporation surface diffusion~\\cite{amos2020multiphase,hoffrogge2021multiphase}. \nThe phase-dependent concentration of component-$i$ in random grain $m$ belonging to phase-$\\alpha$, and the corresponding interpolation function, are represented by $c_{m:i}^{\\alpha}$ and $h(\\phi_{\\alpha}^{m})$, respectively.\nThe evolution of the different microstructures with varying phase-fractions, in this work, are modelled by solving the Eqns.~\\eqref{eq:ph_evo1} and ~\\eqref{chempot_ev}.\n\n\\subsection{Simulation setup}\n\nExisting studies unravel that, unlike Zener pinning~\\cite{fan1998numerical}, the overall trend in grain-growth kinetics exhibited by the individual phases, and entire microstructure, of the multiphase system are largely independent of the dimensionality of the simulation domain~\\cite{fan1997computer,poulsen2013three,yadav2016effect}.\nIn other words, in both two- and three-dimensional setup, similar disparity in the evolution kinetics of  major-, minor- and equifraction phases, in relation to overall growth rate, has been reported. \nTherefore, in the present work, the grain growth in various multiphase-polycrystalline systems are modelled in two-dimensional framework. \nMoreover, irrespective of the variation in phase-fraction, two-dimensional domains of similar configurations are adopted for all numerical studies.\n\nTwo-dimensional domains considered for the current investigations are uniformly discretised into $2048 \\times 2048$ cells of identical dimension, $\\Delta$x = $\\Delta$y = $5 \\times 10^{-7}$m, through the finite-difference scheme.\nPolycrystalline microstructures comprising of approximately $10000$ grains are instituted over the discretised domain through Voronoi tessellation. \nThese grains are associated to the constituent phases by assigning the characteristic chemical composition.\nSince the grains are equiaxed with almost similar size, the required phase-fraction in the microstructure is achieved by relating the appropriate number of randomly distributed grains to the corresponding phases.\nIn other words, duplex microstructure comprising of 33$\\%$ minor phase is devised, in the initial stages, by assigning the respective chemical composition to the one-third of the grains randomly. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{simplex}\n    \\end{tabular}\n    \\caption{  A simplex, analogous to ternary isotherm, depicting all-possible phase-fractions in duplex and triplex microstructures, along with single-phase systems (vertices). The section that encompasses the varied microstructures considered in this work is distinguished, and each  point referring to a specific system is highlighted and designated as $s1$, $s2$, and such. \n    \\label{fig:simplex}}\n\\end{figure}\n\nGiven that the principal focus of this work is to understand the evolution kinetics of  different phases in relation to each other, and to the overall grain-growth rate exhibited by the entire microstructure, a rather straightforward distinction is made between the phases. \nWhile a binary system with two chemical component, $\\tilde{i}$ and $\\tilde{j}$, is considered for establishing duplex microstructure, three-phase microstructure is construed in the framework of ternary system with components $\\tilde{i}$, $\\tilde{j}$ and $\\tilde{k}$.\nIn duplex system, $\\alpha$-phase is a $\\tilde{i}$-rich phase with equilibrium composition of $c^{\\alpha}_{\\tilde{i}:eq}=0.9$, and $c^{\\alpha}_{\\tilde{j}:eq}=0.9$ characterises matrix $\\gamma$-phase, wherein both concentrations are expressed in mole fraction.\nConcentration of the solvent in $\\alpha$- and $\\gamma$-phase remains unaltered in the triplex system, while the remnant content is equally partitioned between solutes, $\\{\\tilde{j},\\tilde{k}\\}$ and $\\{\\tilde{i},\\tilde{k}\\}$, respectively.\nThe $\\beta$-phase, exclusively introduced in the three-phase systems, is characterised by composition $c^{\\beta}_{\\tilde{i}:eq} = c^{\\beta}_{\\tilde{j}:eq} = 0.05$.\nSince the present investigation is primarily interested in microstructural features like phase-fractions, diffusivities of the components are assumed to be identical, and unit matrix is correspondingly incorporated in the formulation. \nMoreover, the energy-densities of the grain boundaries, irrespective of the chemical-composition of the grains they separate, is treated as isotropic and unity. \nThe length scale parameter, $\\epsilon$, is appropriately defined such that the diffuse interface is of constant thickness comprising of four cells~\\cite{mittnacht2021morphological}. \n\nThe temporal evolution of the dynamic variables, phase-field and chemical potential, that dictate the microstructural changes in the multiphase polycrystalline system, are solved over the homogeneous cells of the two-dimensional domain by forward-marching Euler's scheme. \nIn order to ensure that the computational resources are optimally used, the domain is decomposed into smaller segments, and dealt simultaneously, through Message Passing Interface (MPI). \nThe complexity of the numerical treatment is reduced by suitably non-dimensionalising the input parameters, and incorporating them as dimensionless values~\\cite{amos2020multiphase}.\n\n\\subsection{Varying phase-fractions}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{systems}\n    \\end{tabular}\n    \\caption{ Microstructure corresponding to each point in the section of the simplex that includes homogeneous systems along with duplex and triplex microstructures with varying volume-fraction of the constituent phases.  \n    \\label{fig:systems}}\n\\end{figure}\n\nIn applications, depending on the material need, multiphase systems with varying degree of phase-fractions are employed. \nThough investigating every combination of the phase-fraction would be redundant, convincing level of understanding can only be gained by considering relatively increased number of phase-fractions. \nParticularly, since the present study adopts statistical techniques to explicate the kinetic relation between the evolving phases, the accuracy of the outcome depends on the wealth of information (data) available. \nTo that end, in this work, grain growth in twenty different systems, which encompasses  one homogeneous, five duplex and fourteen triplex microstructures, is modelled, and \\lq multidimensional\\rq \\thinspace dataset is built by monitoring the temporal evolution of the grains. \n\nAs opposed to random consideration of different volume-fractions of phases, a systematic choice of various phase-fraction is made from a 2-simplex.\nThe simplex, in its entirety, along with the section focused for the current study is shown in Fig.~\\ref{fig:simplex}. \nThe points within, and on, the 2-simplex can be interpreted in a manner akin to the ternary isotherm.\nCorrespondingly, while the three vertices indicate the homogeneous microstructure of phase-$\\alpha$, -$\\beta$ and -$\\gamma$, the duplex microstructures are encapsulated by the edges joining the vertices. \nAny point within the simplex represents triplex system, with phase-fraction dictated by its position.\nAs illustrated in Fig.~\\ref{fig:simplex}, a section of the simplex emanating from the vertex characterising the homogeneous $\\gamma$-microstructure is considered for the present analyses. \nThis section of the simplex renders a wide-variety of polycrystalline systems ranging from single phase homogeneous to triplex with equifraction of constituent phases. \nMoreover, owing to the configuration of the section in Fig.~\\ref{fig:simplex}, $\\gamma$ -phase acts as the matrix for the duplex and triplex microstructures with unequal volume-fractions of phases. \nMultiphase microstructures corresponding to the different points of the simplex-section is collectively illustrated in Fig.~\\ref{fig:systems}. \nThough the volume-fraction of the minor-phases can be as low as 5$\\%$, the grains associated with these phases hardly occupy a position on the grain boundary. \nIn other words, despite the low volume-fraction and reduced size, the grains of the minor phases seldom render an influence analogous to the particles in Zener pinning. \nMoreover, the reduced size of the grains associated with minor phases, in the initial stages of the grain growth, is consistent with experimental observations~\\cite{liu2015synergetic,ritasalo2013microstructural,praveen2016exceptional} and existing theoretical studies~\\cite{fan1997computer,yadav2016effect}. \n\n\\section{Results and discussion}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.65\\textwidth]{duplex_validation}\n    \\end{tabular}\n    \\caption{  Temporal change in the average radius of the phase-$\\alpha$ and $-\\gamma$ grains, $\\bar{R}_\\alpha(t)$ and $\\bar{R}_\\gamma(t)$, and the entire microstructure,  $\\bar{R}(t)$, for homogeneous, and duplex systems characterised by $50\\%$ and $10\\%$ of phase-$\\alpha$. \n    \\label{fig:duplex_validation}}\n\\end{figure}\n\nBy monitoring the grain growth exhibited by the homogeneous and multiphase systems, a \\lq multidimensional\\rq \\thinspace dataset is devised which essentially comprises of temporal change in the average radius of the individual phase-associated grains, and the entire microstructure as a whole.\n(A dataset can truly be multidimensional only when its dependent variable is governed by more than one independent variable. \nConsidering that explicating the multivariate nature of the overall grain-growth kinetics in multiphase system is the ultimate outcome of the current study, the term \\lq multidimensional\\rq \\thinspace is written within quotes.)\nThis dataset is analysed through comfortably realised statistical techniques to unravel the effect of the individual phases on the evolution kinetics of the entire system.  \nSince the present study focuses primarily on the kinetics of the grain growth, the topological changes and the distribution of the grains are largely overlooked. \n\n\\subsection{Duplex microstructures}\n\nDuplex microstructure comprises of two distinct phases, and is characterised by grains associated with one of these constituent phases, $\\alpha$ and $\\gamma$. \nDespite the inhomogeneity in the concentration distribution, in duplex microstructures, given that the grain growth occurs in a continuum, the temporal evolution of one phase, and its corresponding kinetics, is inherently coupled with the other. \nThis is illustrated and discussed in Appendix.\nIn other words, the growth rate exhibited by the duplex microstructures with varying phase-fractions can be convincingly expressed by considering kinetics of the only one of the evolving phases. \nTherefore, the aim of the present investigation in duplex microstructure reduces to identifying which of the phases, major or minor, principally governs the kinetics of overall grain-growth. \n\n\\subsubsection{Comparison with single phase microstructure}\n\nBefore proceeding to realise the degree of influence rendered by the different phases on the overall growth-kinetics exhibited by the duplex microstructure, rather straightforward investigations are pursued to verify the outcomes of the present approach in relation to the existing reports~\\cite{fan1997computer,fan1997diffusion}. \nEven though,  the outcomes of the modelling technique in relation to the established theories and observations have already been reported elsewhere~\\cite{amos2020multiphase}, certain relevant aspects are analysed in a statistical framework here. \n\nIn Fig.~\\ref{fig:duplex_validation}, the progressive change in the average radius of homogeneous and two duplex microstructures with time are presented. \nMoreover, the temporal increase in the average radius of the phase-associated grains are monitored, and included in this illustration. \nWhile $\\bar{R}(t)$ represents the average radius of the entire polycrystalline microstructure, the corresponding parameter for the grains of  phase-$\\alpha$ and -$\\gamma$ in duplex microstructures is respectively denoted by $\\bar{R}_\\alpha(t)$ and $\\bar{R}_\\gamma(t)$.\nMoreover, the multiphase microstructures in this, and subsequent, discussions are described based on the volume fraction of the minor phase. \nFor instance, $\\alpha 10$ indicates duplex microstructure with $10\\%$ phase-$\\alpha$, while equifraction system are denoted $\\alpha 50$.\n\nIrrespective of the nature of the microstructure, homogeneous or otherwise, Fig.~\\ref{fig:duplex_validation} shows a continual increase in the average radius reflecting the grain growth exhibited by the system. \nWith the introduction of a second-phase in the microstructure, a significant decrease is observed in the rate at which the radius increases with time. \nThis noticeable change in the kinetics is predominantly due to the change in grain-growth mechanism, which is governed by the long-range diffusion of the chemical components in duplex microstructure.\nMoreover, in system with equal volume-fraction of phases, the temporal increase in average radius of the entire microstructure and individual phases are largely identical with marginal deviation. \nOn the other hand, in $\\alpha 10$, significant disparity is noticed in the rate at which the major-phase grains evolve when compared to the minor-phase. \nFig.~\\ref{fig:duplex_validation} illustrates that the increase in the average radius of the entire duplex microstructure lies in between the growth exhibited by the individual phases.\nThe difference on the growth kinetics between the phases, exclusively in the duplex microstructure characterised by unequal volume-fraction, is due to the corresponding distribution of the phases. \nOwing to its reduced volume, the grains of the minor phase in $\\alpha 10$ microstructure are considerably separated when compared to major-phase grains. \nTherefore, the diffusion path, which the chemical components need to transverse to achieve grain growth, is longer, and more convoluted.\nConsequently, the growth rate exhibited by the minor-phase grains is significantly lower the grains of phase-$\\gamma$. \nSo far, no convincing argument has been made on how the evolution kinetics of the entire duplex microstructure relates to the different growth-rates adhered-to by the major- and minor-phase grains, and to that end, this becomes the primary focus of the current analysis. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.85\\textwidth]{duplex_power}\n    \\end{tabular}\n    \\caption{  Correlation coefficient characterising the proportionality between time and the average radius of individual phase-grains and entire microstructure, raised to different exponents,  for homogeneous and duplex systems $\\alpha50$ and $\\alpha10$.  \n    \\label{fig:duplex_power}}\n\\end{figure}\n\n\\subsubsection{Grain growth kinetics}\n\nEven though the microstructures illustrated in Fig.~\\ref{fig:duplex_validation} render a progressive increase in the average radius with time, owing to the difference in the governing mechanism, the exponent of the power law capturing the growth kinetics vary depending on the nature of the system~\\cite{fan1997computer,fan1997diffusion}. \nWhile the exponent $n=2$ characterises grain growth in homogeneous system, evolution kinetics of the individual phases, and the duplex microstructure as a whole, largely follow relation $\\bar{R}^3(t) \\propto t$.\nIn order to affirm that the evolution of the entire microstructure, and its corresponding phases, adhere to the power law,  the temporally varying average radius is raised to different exponents, $n=\\{1,2,3,4\\}$, and related to time. \nThe correlation coefficient (Pearson) characterising the relation between the average radius with the various exponent and time is ascertained, and graphically represented in Fig.~\\ref{fig:duplex_power}. \nIn this illustration, $<\\bar{R}^n,t>$ denotes the correlation coefficient between the average radius raised to order n and time. \nFig.~\\ref{fig:duplex_power} shows that, in homogeneous system ( $\\alpha 0$), maximum correlation is observed when $n=2$, thereby indicating that the grain growth in this microstructure adheres to the power law, $\\bar{R}^2(t) \\propto t$.\nFurthermore, correlation coefficient relating the average radius of the individual phases, and entire duplex microstructures, with time, is higher when $n=3$, which implies that the evolution in the multiphase systems complies to the established power law~\\cite{fan1997diffusion}. \nEven though it might appear that,  in Fig.~\\ref{fig:duplex_power}, for equifraction duplex system ($\\alpha 50$) , the maximum correlation is observed in $n=4$.\nHowever, given the marginal difference when compared to $n=3$, such consideration leads to \\lq overfitting\\rq \\thinspace, thus returning to $\\bar{R}^3(t) \\propto t$ as the statistically sound relation. \n\n\\subsubsection{Ascertaining governing factor}\n\nAs shown in Fig.~\\ref{fig:duplex_validation}, the change in the average radius of the entire microstructure with time, in duplex systems with unequal volume-fraction of phases, invariably lies  between curves representing the increase in the radius of the individual phase-grains. \nMoreover, existing works unravel that, in radius versus time plot, the relative position of the overall microstructure curve varies with respect to the phase-associated grains curves, as the phase-fraction changes~\\cite{fan1997computer,yadav2016effect}.\nWhile the effect of phase-fraction on the evolution of the individual phases are convincingly elucidated, its influences on the growth rate of entire duplex microstructure is yet to be sufficiently addressed. \nIn order to identify this influence, more particularly, to answer the resulting question on evolution kinetics of which phase, major or minor, principally effects the growth rate of entire duplex microstructure, each system is statistically analysed.\nStatistical programming language R is employed for these, and all other relevant, investigations in this work. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.9\\textwidth]{rdq_duplex}\n    \\end{tabular}\n    \\caption{  Coefficient of determination, quantifying the effect of  growth kinetics of individual phase-grains on the overall grain-growth rate of the entire system, is  estimated using Eqn.~\\eqref {coeff_deter} for different duplex microstructure with varying phase-fraction.\n    \\label{fig:rdq_duplex}}\n\\end{figure}\n\nFrom the dataset comprising of  temporally varying average radius, the growth rate exhibited by the individual phase-grains ($d\\bar{R}_{\\alpha}\/dt$ or $d\\bar{R}_{\\gamma}\/dt$) and entire microstructure ($d\\bar{R}\/dt$), at every instance $(t)$, is determined for each duplex system. \nSubsequently, by treating the growth rate of the individual phase-grains, and entire microstructure, as \\textit{response} and \\textit{predictor} variable, respectively, the corresponding kinetics are related to the each other.\nFrom the emerging relation, the \\textit{coefficient of determination}, $\\chi$, for the combination of an individual phase-grains and overall microstructure is estimated.\nScatter plots that illustrates the dependency of $d\\bar{R}_{\\alpha}\/dt$ or $d\\bar{R}_{\\gamma}\/dt$ and $d\\bar{R}\/dt$ are included in \\textcolor{red}{Appendix2}.\n\nFor a given duplex system, following the conventional description, the coefficient of deterioration considering the growth kinetics of minor phase-$\\alpha$ ($d\\bar{R}_{\\alpha}\/dt$) and entire microstructure ($d\\bar{R}\/dt$) is calculated by \n\\begin{align}\\label{coeff_deter}\n \\chi_{\\alpha}=\\frac{\\chi^{\\text{SST}}_{\\alpha}-\\chi^{\\text{SSE}}_{\\alpha}}{\\chi^{\\text{SST}}_{\\alpha}},\n\\end{align}\nwhere $\\chi^{\\text{SST}}_{\\alpha}$ is estimated by treating the instantaneous growth-rate of overall microstructure as univariate parameter, and summing-up the squares of the disparity (error) between the individual values and the mean.  \nOn the other hand, $\\chi^{\\text{SSE}}_{\\alpha}$ represents the sum of the squared differences between the datapoints and regression line relating the instantaneous kinetics of the $\\alpha$-phase grains and overall duplex-microstructure. \nBased on the description of the coefficient of determination in Eqn.~\\eqref{coeff_deter}, $\\chi_{\\alpha}$ can be viewed as a parameter that quantifies the effect of $\\alpha$-grains growth-kinetics on the evolution rate of entire microstructure.  \nTherefore, in addition to $\\chi_{\\alpha}$, the corresponding parameter that realises the influence of the major-phase growth-kinetics ($d\\bar{R}_{\\gamma}\/dt$) on the overall evolution rate, $\\chi_{\\gamma}$ is appropriately determined for all the different duplex microstructures considered in this investigation. \n\nCoefficients of determination separately quantifying the role of $d\\bar{R}_{\\alpha}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$ in overall growth-rate exhibited by the microstructure, $\\chi_{\\alpha}$ and $\\chi_{\\gamma}$, is calculated for different duplex systems with varying phase-fractions and plotted in Fig.~\\ref{fig:rdq_duplex}. \nThe variation observed in the coefficients of determination, across the different duplex systems, unravels that the effect of individual phase-grains on the overall growth kinetics is primarily dependent on phase-fraction of the microstructure. \nIn a duplex system characterised equal volume-fraction of phases, identical coefficients of determination implies that both $\\alpha$- and $\\gamma$-grains similarly influence the evolution kinetics of the entire microstructure. \nOn the other hand, noticeable disparity between $\\chi_{\\alpha}$ and $\\chi_{\\gamma}$ is observed in duplex microstructures with varying volume-fraction of constituent phases. \nMoreover, Fig.~\\ref{fig:rdq_duplex} shows that this inequality in the coefficients of determination becomes more pronounced with increase in the difference  between the volume-fraction of the phases in duplex microstructure.\nWhile the coefficient of determination pertaining to major phase-$\\gamma$ exhibits a relatively marginal change, and continues to remain noticeably greater, $\\chi_{\\alpha}$ progressively decrease with reduction in the volume-fraction of the corresponding minor-phase grains. \nIn other words,  Fig.~\\ref{fig:rdq_duplex} unravels that, in duplex systems with unequal volume-fraction of phases, the overall growth rate of the entire microstructure ($d\\bar{R}\/dt$) is primarily influenced by the evolution kinetics of the major-phase grains ($d\\bar{R}_{\\gamma}\/dt$). \nFurthermore, it is evident from the illustration that the dominance of the major phase in effecting the overall growth kinetics becomes more definite with increase in the corresponding volume-fraction (or decrease the amount of minor phase).\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.9\\textwidth]{dupR_sep}\n    \\end{tabular}\n    \\caption{  Progressive increase in the average radius of phase-$\\alpha$ and -$\\gamma$ grains, $\\bar{R}_{\\alpha}^3$ and $\\bar{R}_{\\gamma}^3$,  with time for various duplex systems with characteristic phase-fraction.\n    \\label{fig:dupR_sep}}\n\\end{figure}\n\n\\subsubsection{Verifying the statistical claim}\n\nIn order to substantiate the understanding rendered by the analyses based on coefficient of determination, the temporal change in the average radius of the phase-associated grains ($\\bar{R}_{\\alpha}$ or $\\bar{R}_{\\gamma}$) and entire microstructure ($\\bar{R}$) are studied in a conventional manner.\nIn  Fig.~\\ref{fig:dupR_sep}, the progressive increase in the average radius of  major- and minor-phase grains with time, in duplex systems with varying phase-fractions, are cumulatively presented. \nSince the evolution of  different duplex microstructures are considered together, for the ease of distinction, temporal change in $\\bar{R}_{\\alpha}^3$ and $\\bar{R}_{\\gamma}^3$ is adopted for this illustration.\n\nConsistent with the mechanism of evolution, it is observed that the minor phase-grains in system with the minimal volume-fraction ($\\alpha 10$) grows at a least rate.\nHowever, the growth kinetics of the these grains noticeably increase as the corresponding phase gains more volume in the microstructure. \nAccordingly, in duplex systems with unequal phase-fractions, minor-phase grains of $\\alpha 40$ microstructure exhibits highest growth-rate, followed by $\\alpha 33$ and $\\alpha 16$. \nOn the other hand, the evolution kinetics of major-phase grains are minimal in  $\\alpha 40$ system, and significantly increase in $\\alpha 33$ and $\\alpha 16$ as the volume fraction of the phase-$\\alpha$ reduces. \nMoreover, the maximum growth-rate in $\\gamma-$phase grains is observed in microstructure with minimal volume of minor phase, $\\alpha 10$.\nOwing to the influence of volume fraction, which governs the kinetics through the diffusion paths, the disparity in the temporal change in average radius of the major- and minor-phase grains becomes more evident as the inequality in phase-fraction increases. \nIn other words, as shown in Fig.~\\ref{fig:dupR_sep}, the progressive change in $\\bar{R}_{\\alpha}$ and $\\bar{R}_{\\gamma}$ with time is notably far-apart in $\\alpha 10$ system when compared to the rest. \nNevertheless, this separation gets reduced with the increase in the volume-fraction of minor phase-$\\alpha$.\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.7\\textwidth]{dupR_conso}\n    \\end{tabular}\n    \\caption{ Temporal change in the average radius of entire duplex systems, $\\bar{R}^3$, with varying  phase-fractions during grain growth.\n    \\label{fig:dupR_conso}}\n\\end{figure}\n\nWhile Fig.~\\ref{fig:dupR_sep} can be appropriately discussed to affirm the consistency of the present approach, equipped with the understanding on how evolution of individual phase-grains relate to the growth rate of entire microstructure, the general trend in the overall growth kinetics can be predicted from it. \nAnalyses based on coefficient of determination, in Fig.~\\ref{fig:rdq_duplex}, unravels that the growth-rate of duplex microstructure is predominantly influenced by evolution kinetics of major-phase grains. \nMoreover, the effect of the minor-phase grains decreases as their corresponding volume-fraction reduces.\nAdopting these insights, and given that in Fig.~\\ref{fig:dupR_sep} $\\gamma$-grains of $\\alpha 10$ exhibit maximum growth rate, it can be predicted that the overall growth kinetics of the corresponding microstructure will be noticeably greater than the other duplex systems considered in this study. \nFurthermore, it can also be stated that, since the volume fraction of minor-phase continues to be significantly lower in $\\alpha 16$ and  $\\alpha 33$, the overall growth will be dominated by the $\\gamma$-phase grains, and their kinetics will correspondingly follow the $\\alpha 10$ microstructure. \nFinally, considering that the volume of phase-$\\alpha$ is close to major-phase in $\\alpha 40$,   based on Fig.~\\ref{fig:rdq_duplex}, it can be suggested that this duplex microstructure will exhibit the least rate of evolution.\n\nIn order to verify the accuracy of the above predictions, emerging from the understanding of coefficient of determination, the overall growth-rate exhibited by different duplex microstructures is cumulatively presented in Fig.~\\ref{fig:dupR_conso}.\nIt complete adherence to the prediction, it is observed that, in duplex microstructures with unequal phase-fraction, maximum and minimum growth-rate respectively pertains to $\\alpha 10$ and $\\alpha 40$ microstructure. \nAdditionally, the kinetics of evolution exhibited by $\\alpha 16$ and  $\\alpha 33$ lie in between the maximum and minimum, with the former noticeably greater than the later. \nUltimately, Fig.~\\ref{fig:dupR_conso} affirms that, in duplex systems characterised by unequal volume-fraction of constituent phases, the overall grain-growth kinetics is primarily governed by the evolution-rate of the major-phase grains.\nThis influence of the major-phase grains gets increasingly dominant with increase in its volume-fraction. \n\n\\subsubsection{Phase fraction and growth rate}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.7\\textwidth]{duplexK}\n    \\end{tabular}\n    \\caption{  Change in the kinetics coefficients that governs grain growth with the variation in the phase-fractions of evolving duplex microstructures. \n    \\label{fig:duplexK}}\n\\end{figure}\n\nHaving realised that, for a given duplex microstructure, the overall grain-growth rate ($d\\bar{R}\/dt$) is predominantly dictated by the kinetics adhere-to by the major-phase grains ($d\\bar{R}_{\\gamma}\/dt$), attempts are made to relate the varying phase-fraction to the observed growth-rate across different systems. \nIntroducing chemical-inhomogeneity amongst the grains, in a polycrystalline system, shifts the governing mechanism, irrespective of the volume-fraction of the phases. \nKinetics rendered by this mechanism in multiphase microstructure complies with the power-law characterised by the exponent $n=3$.\nConsidering that the exponent remains unaltered reflecting the governing mechanism, despite the varying phase-fraction in multiphase systems, the disparity in the rate of grain growth can only be understood from the kinetic coefficient, $k$, that relates the average radius to time. \n\nBy studying the evolution of different duplex systems considered in this study, the corresponding kinetic coefficients are ascertained. \nKinetic coefficients associated with duplex microstructures with varying phase-fractions are illustrated in Fig.~\\ref{fig:duplexK}. \nSince the growth rate of the duplex microstructure are principally dictated by the evolution of major-phase grains, the corresponding volume-fraction is considered for this representation. \nFig.~\\ref{fig:duplexK} indicates that with increase in the volume of a major phase in duplex system, grain growth in the system occurs at an higher rate. \nIn order definitively understand the influence of the $\\gamma$-fraction on the kinetic coefficient exhibited by the corresponding duplex microstructure, non-linear regression technique is adopted and a relation expressed as \n\\begin{align}\\label{Dpf_k}\nk_{\\text{dup}} = A_{\\text{dup}} + B_{\\text{dup}}\\exp(C_{\\text{dup}}\\dot V_{\\gamma}),\n\\end{align}\nis realised, where $V_{\\gamma}$ is the volume fraction of the major phase-$\\gamma$.\nThe constants $A_{\\text{dup}}$, $B_{\\text{dup}}$ and $C_{\\text{dup}}$, for the present consideration, respectively assume the value of $537.3$, $3.8\\times 10^{-3}$, and $0.13$. \nIn the above relation, it is vital to note that $k_{\\text{dup}}$ indicates the kinetic coefficient adhered-to by the entire duplex system during grain growth, not the evolving major- or minor-phases. \n\n\\subsection{Triplex microstructures}\n\nTriplex systems are characterised by the association of  individual grains, in the polycrystalline setup, to one of the three constituent phases. \nCorresponding microstructure, in this study, comprises of phases $\\alpha$, $\\beta$ and $\\gamma$, with $\\gamma$ largely acting as the matrix or major-phase. \nIn the existing works, unlike duplex systems, very few three-phase microstructures with varying phase-fractions are analysed~\\cite{ravash2017three2}.\nThis limited consideration of triplex microstructure can largely be attributed to the computational burden associated with it. \nMoreover, conventionally, the grain-growth kinetics of the triplex microstructure are discussed by focusing on the evolution of the individual phase-grains without sufficiently relating it the overall system.\nIn the present study, on the other hand, grain growth in fourteen different three-phase microstructures, with varying phase-fractions, are examined to elucidate with statistical certainty how the evolution of the individual phase-grains effects the growth kinetics of entire triplex microstructure. \n\n\\subsubsection{Grain growth kinetics}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.75\\textwidth]{triplex_power}\n    \\end{tabular}\n    \\caption{  Correlation coefficient characterising the relation between time and average radius of individual phase-grains and entire microstructure, raised to different power ($n=1,2,3$ and $4$), for two triplex systems, $\\alpha39 \\beta22$ and $\\alpha33\\beta33$. \n    \\label{fig:triplex_power}}\n\\end{figure}\n\nDespite the difference in the number of phases, grain growth in both duplex and triplex systems are fundamentally governed by the same mechanism. \nTherefore, the grain growth in three-phase systems, which is dictated by the diffusion of chemical components, adheres to the power law with the exponent, $n=3$. \nIn order to ensure that the grain growth in triplex systems is accurately modelled by the present approach, a validation technique adopted for duplex microstructures, is extended. \nCorrespondingly, the temporally-varying average radius of the individual phase-grains, $\\bar{R}_{\\alpha}$, $\\bar{R}_{\\beta}$ and  $\\bar{R}_{\\gamma}$, and overall microstructure, $\\bar{R}$, are raised to different powers ($n=\\{1,2,3,4\\}$) and related to time, $t$.\nCorrelation coefficient characterising the different relations are ascertained for two triplex microstructures, $\\alpha33\\beta33$ and $\\alpha39\\beta22$,  and are graphically illustrated in Fig.~\\ref{fig:triplex_power}. \n\nIt is evident in Fig.~\\ref{fig:triplex_power} that correlation coefficient relating the average radius to the time is highest when $n=3$ for both individual phases, and overall microstructure.\nThe maximum correlation exhibited by the cube of the different average radii, $\\bar{R}_{\\alpha}$, $\\bar{R}_{\\beta}$,  $\\bar{R}_{\\gamma}$ and $\\bar{R}$, with time implies that the growth of the individual phase-grains, and the entire microstructure, are predominantly governed by the long-range diffusion of the chemical components. \n\n\\subsubsection{Ascertaining governing factor}\n\nConsidering that grain growth in both duplex and triplex system are predominantly dictated by the diffusion of the chemical components, phase-fraction renders identical influence on the evolution of individual phases. \nIn other words, when certain phase(s) assumes minor volume-fraction in the three-phase microstructure, owing to relative increase in the length, and complexity, of the diffusion path, the growth of the corresponding grains are stunted. \nOn the other hand, the evolution kinetics is enhanced when the volume of the phase(s) is dominant in the multiphase systems. \nApart from these generalised understanding, existing report rarely offer any further insights on the grain-growth kinetics in triplex microstructures.\nParticularly, similar to duplex system, sufficient consideration has not been rendered to relate the growth kinetics of the individual phases to the evolution of the entire triplex system. \nTo that end, in this analysis, the impact of the growth rate of  individual phase-grains on that entire three-phase microstructure is examined by ascertaining the corresponding coefficient of determination. \n\nThe instantaneous growth rate for constituent phase-grains, $d\\bar{R}_{\\alpha}\/dt$, $d\\bar{R}_{\\beta}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$, along with the entire triplex microstructure, $d\\bar{R}\/dt$, is determined by monitoring temporal change in the respective parameter. \nThese instantaneous growth kinetics of the individual phase-grains are related to that of the entire microstructure, and the corresponding coefficient of determination is estimated through the Eqn.~\\eqref{coeff_deter}.\nFor each system, three distinct coefficients of determination, $\\chi_{alpha}$,  $\\chi_{beta}$ and $\\chi_{gamma}$, are estimated, reflecting the characteristic feature of the triplex microstructure. \nThese coefficients of determinations are related to the phase-fractions of the microstructure, and illustrated in  Fig.~\\ref{fig:triALL_Rsq}. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{triALL_Rsq}\n    \\end{tabular}\n    \\caption{  Change in coefficient of determination, typifying the influence of growth rate of individual phase-grains on the evolution kinetics of entire microstructure, with variation in the volume fraction of constituent phases.\n    \\label{fig:triALL_Rsq}}\n\\end{figure}\n\nIn the triplex systems considered in the present study, the volume-fraction of phase-$\\gamma$ reaches as low as 33\\% despite being the major phase. \nSuch volume-fraction of phase-$\\gamma$ is noticed in triplex microstructure characterised by equifraction of phases. \nFurthermore, in some system like $\\alpha 45 \\beta 10$, the phase-$\\alpha$ assumes a volume-fraction of $45\\%$, in spite of being one of the minor phases.\nThis, and similar, understanding of phase-fraction is vital to investigate the coefficients of determination presented in Fig.~\\ref{fig:triALL_Rsq}. \n\nFig.~\\ref{fig:triALL_Rsq} unravels that, even though the volume-fraction of the major-phase varies noticeably across different triplex systems, the corresponding coefficient of determination, $\\chi_{gamma}$, continues to remain the high. \nGiven that phase-$\\gamma$ stays as a major-phase, despite the change in phase-fractions, the high values of $\\chi_{gamma}$ can be attributed to the dominant volume of the respective grains.\nIn other words, analogous to the duplex microstructure, the evolution kinetics of major-phase grains offer relatively greater influence on the overall growth-rate exhibited by the entire triplex microstructure. \nFurthermore, Fig.~\\ref{fig:triALL_Rsq} suggests that  the coefficient of determination of the minor-phases, $\\alpha$ and $\\beta$, noticeably increases as their corresponding volume-fraction raises. \nParticularly, as the volume of phase-$\\alpha$ gets as dominant as $\\gamma$, in a triplex system, identical coefficient of determination is rendered, $\\chi_{alpha} = \\chi_{gamma}$.\nOn the other hand, when the volume of the phases are minimal, the respective coefficient of determination assumes least value. \nUltimately, it is evident from Fig.~\\ref{fig:triALL_Rsq} that the influence of the individual phase-grains on the overall growth kinetics depends largely on the corresponding volume-fraction. \nIn a triplex system with unequal volume-fraction of phases, the growth rate of the entire microstructure, $d\\bar{R}\/dt$, is predominantly governed by the evolution of the major-phase grains, $d\\bar{R}_{\\gamma}\/dt$.\nWhen volume of two phases are dominant in a three-phase system, the growth rate of both these phase-grains offer identical influence on evolution of the microstructure. \nThe contribution of a given phase-grains to the overall evolution kinetics, $d\\bar{R}\/dt$, becomes least, when its volume-fraction are minimal. \nBased on the understanding rendered by Fig.~\\ref{fig:triALL_Rsq},  as demonstrated for duplex systems (Fig.~\\ref{fig:dupR_conso}), the growth-kinetics of a triplex microstructure, in relation to others with varying phase-fractions, can be predicted from the temporal change in the average radius of the corresponding phase-associated grains. \n\n\\subsubsection{Interdependency in the evolving phases}\n\nIn duplex systems, since the grains of the polycrystalline microstructure are associated with either of the two  constituent phases, the evolution of a particular phase-grains, and its kinetics, are inherently bound to the other. \nHowever, the same interdependency cannot be expected in triplex systems, wherein the grains can be associated with one of the three possible phases. \nMoreover, in three-phase microstructures, level of influence offered by one evolving phase-grains on the rest of the phase-associated grains has not been conscientiously addressed yet.\nTherefore, by examining the temporal change in the average radius of a particular phase-grains in relation to the others, the interdependency exhibited between the phases, in triplex microstructures, during grain growth is elucidated. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{rate_corr}\n    \\end{tabular}\n    \\caption{  Correlation coefficient explicating the interdependency between the growth rate of different phase-grains, $d\\bar{R}_{\\alpha}\/dt$, $d\\bar{R}_{\\beta}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$, during grain growth of four different triplex system with varying phase-fractions. \n    \\label{fig:rate_corr}}\n\\end{figure}\n\nInstead investigating all the fourteen triplex microstructure to explicate the effect of one evolving phases on the other, the systems are categorised based on phase-fraction, and one microstructure from each category is analysed. \nSince the coefficient of determination, which quantifies the effect of  an evolving-phase grains on the growth of the overall triplex microstructure, depends on volume fraction, the phase-fraction based grouping is deemed reasonable. \nApart from the equifraction system, $\\alpha 33 \\beta 33$, wherein all the constituent phases largely occupy similar volume, the remaining systems can be categorised as \\lq equi-major \\rq \\thinspace, \\lq equi-minor \\rq \\thinspace and \\lq non-equifraction \\rq \\thinspace triplex microstructures. \nWhile, in equi-major system, volume of one of the minor phase is equal to that of the major-phase  ($V_\\alpha= V_\\gamma$), the volume fraction of minor phases are identical in equi-minor microstructures ($V_\\alpha= V_\\beta$).\nMoreover, the non-equifraction system stand in direct contrast to the equifraction microstructure, and characterised by totally unequal volume-fraction of the constituent phases ($V_\\alpha\\neq V_\\beta \\neq V_\\gamma$). \n\nIn order understand the degree of interdependency between the evolving phases, in addition to equifraction microstructure, grain growth exhibited systems  $\\alpha 36 \\beta 22$, $\\alpha 45 \\beta 10$ and $\\alpha08\\beta08$ pertaining to non-equifraction, equi-major and equi-minor, respectively, are analysed.\nGiven that the primary focus of  the present investigation is \\textit{not} to quantify the effect of an evolving-phase grains on the rest, but rather to qualitatively realise the degree of interaction between two phases, during grain growth in a triplex systems, coefficient of determination is not estimated. \nHowever, alternatively, the growth rate of different phase-associated grains are estimated, $d\\bar{R}_{\\alpha}\/dt$, $d\\bar{R}_{\\beta}\/dt$ and $d\\bar{R}_{\\gamma}\/dt$, and are related to  (or plotted against) each other.\nCorrelation coefficient characterising the relation between the growth rate of two phase-grains are realised for pre-determined triplex microstructures, and graphically illustrated in Fig.~\\ref{fig:rate_corr}. \n\nBefore elucidating level of interaction between two evolving phase-grain in a given triplex microstructure, based on Fig.~\\ref{fig:rate_corr}, it is exceedingly critical to realise the variation in the range of correlation coefficient across the different systems. \nParticularly, as opposed to the maximum value, which remains constant at unity, the least value of the correlation coefficient changes with phase-fraction. \nCorrespondingly, Fig.~\\ref{fig:rate_corr} unravels that the lowest correlation-coefficient in the equifraction system is maximum ($0.84$) when compared to the rest of the triplex microstructures, and it is respectively followed by non-equifraction ($\\alpha 36 \\beta 22$) and equi-major ($\\alpha 45 \\beta 10$) microstructures, with the absolute minimal exhibited by the equi-minor system ($\\alpha08\\beta08$). \nThis significant disparity in the lowest value of correlation coefficient emphasis the importance of considering the context, $i.e$ the correlation-coefficient range, while interpreting the interaction between two-phases in a given microstructure during grain growth.\nIn other words, apparently least interdependency between the two-phase grains in a equifraction system would translate to a strong interaction in the context of equi-minor triplex microstructure. \n\nThe graphical representation of correlation coefficient, in Fig.~\\ref{fig:rate_corr}, that indicates level of influence the growth rate of one evolving phase-grain has on the other,  during grain growth in triplex microstructures, unravels few similarities and dissimilarities across the systems with varying phase-fractions. \nIt is evident in this illustration that, irrespective of the nature of the three-phase system, the correlation coefficient relating the growth rate of $\\alpha$- and $\\beta$-phase grains are minimal, within a given microstructure. \nIn other words, during grain growth in a triplex system, the evolution kinetics of one minor-phase grains imposes the least effect on growth rate of other low-volume phase-grains.\nDespite being equifraction, largely owing the manner in which the triplex microstructure is initialised, such effect is also observed in $\\alpha 33 \\beta 33$ microstructure.\nHowever, it is vital to note that the least correlation in equifraction system is tantamount to noticeable interaction in relation to other triplex microstructures. \nTherefore, in  $\\alpha 33 \\beta 33$ system, the evolution kinetics of one-phase grains is generally interlinked with grains of the other phases, but this interaction is least between phase-$\\alpha$ and -$\\beta$.\n\nWithin a triplex system, during grain growth, Fig.~\\ref{fig:rate_corr} suggests that minimal interdependency following two minor-phases is observed between the minor-phase and the matrix grains, irrespective of the phase-fractions. \nThe only exception is the equi-minor system wherein the volume-fraction of the minor-phases are identical. \nFurthermore, in all triplex systems, the growth rate of minor-phase grains with relatively greater volume-fraction when compared to the other ($V_{\\alpha} > V_{\\beta}$ ) is strongly coupled with the evolution of the major-phase grains. \nUltimately, the study of interdependency between the rate of the evolving phase-grains during grain growth in triplex systems, using correlation coefficients, unravel that a general trend is observed in three-phase microstructures irrespective of the phase-fractions. \nIf we distinguish the constituent phases as minor- , inter- and major-phase depending on their corresponding volume fraction, which in the present study is $\\alpha$,$\\beta$ and $\\gamma$ respectively, then the least interaction during grain growth is exhibited by the minor- and inter-phase grains.\nWhile the growth rate of major-phase grains are considerably interlocked with the inter grains, the effect of the minor phase on the matrix is comparatively lower.\nIn other words, in a given triplex microstructure, the level of influence offered by the evolution rate of one phase-grains on the other, during grain growth, is primarily dictated by the their corresponding volume fractions. \nWhen the volume fraction of two phases are minimal, their degree of interaction is also minimal, whereas a considerable dependency is noticed when the volume of the two phases are dominant in view of the third. \n\nThis analysis on the interdependency of the kinetics of evolving phases in triplex system, during grain growth, unravels that, for expressing the overall growth rate of a three-phase microstructure, the two ideal variables, with least \\textit{multicollinearity}, are the evolution rate of the grains of minor phases. \n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=0.9\\textwidth]{triALL_fitting}\n    \\end{tabular}\n    \\caption{  Effect of phase-fraction on the kinetic coefficient governing the rate of grain growth in triplex systems.\n    \\label{fig:triALL_fitting}}\n\\end{figure}\n\n\\subsubsection{Phase fractions and growth rate}\n\nA principal insight rendered by the present study is that, in grain growth, the evolution rate of a multiphase system is primarily governed by the volume-fraction of its constituent phases through their respective growth kinetics. \nTherefore, by relating the phase-fraction of  fourteen different triplex microstructures, considered in this work, to its corresponding growth kinetics, an attempt to extract a generalised expression. \nUnlike in duplex system, since triplex microstructures are characterised by three constituent phases,  the corresponding evolution rate is dictated by two independent variables, $i.e$ volume fractions. \nBy directly relating the volume fraction, instead of the evolution rate of individual phase-grains, to the growth kinetics exhibited by the triplex systems, the question of multicollinearity is obviated.\n\nBased on the proportionality exhibited by the phase-fraction and growth kinetics in duplex microstructure, multiple non-linear regression is employed to relate the volume-fraction of constituent phases to the corresponding grain growth-rate exhibited by the entire three-phase microstructures. \nStrictly owing to the different phase-fractions considered in the present analysis, volume fraction phase-$\\alpha$ and -$\\gamma$ are considered as the independent variables. \nExpression rendered by multiple non-linear regression that relates phase-fractions to the kinetic coefficient governing the grain-growth rate of  entire three-phase microstructure reads\n\\begin{align}\\label{Tpf_k}\n k_{\\text{tri}}=A_{\\text{tri}} + B_{\\text{tri}} \\exp(C_{\\text{tri}}^{\\alpha}V_{\\alpha}+C_{\\text{tri}}^{\\gamma}V_{\\gamma}),\n\\end{align}\nwhere $V_{\\alpha}$ and $V_{\\gamma}$ correspond to the volume fraction of phase-$\\alpha$ and -$\\gamma$.\nMoreover, $A_{\\text{tri}}$, $B_{\\text{tri}}$, $C_{\\text{tri}}^{\\alpha}$ and $C_{\\text{tri}}^{\\gamma}$ are constants whose respective values are $268.2$, $2.2 \\times 10^{-7}$, $0.19$ and $0.22$. \nIn Fig.~\\ref{fig:triALL_fitting} kinetic coefficient adhered to different triplex microstructures during grain growth is plotted against the corresponding volume fraction of phase-$\\alpha$ and -$\\gamma$.\nThe curve reflecting Eqn.~\\eqref{Tpf_k} is included in this illustration.\nNoticeably good agreement between the numerical results and the curve indicates that Eqn.~\\eqref{Tpf_k} convincingly relates the kinetic coefficients characterising the grain growth in isotropic triplex systems to the phase-fractions.  \n\n\\section{Conclusion}\n\nGrain growth in polycrystalline systems can be desirous under certain conditions, while unwelcomed in others. \nFor instance, grain growth is induced during processing technique to establish required average grain-size, while noticeable measures are generally taken to avoid it during a given application. \nThe subjective role of grain growth extends beyond homogeneous polycrystalline system to multiphase microstructures as well. \nTherefore, it becomes vital to understand the grain-growth kinetics exhibited by highly-applicable multiphase polycrystalline microstructures associated with duplex and triplex systems. \nParticularly, generalised insights that aide in comprehending the growth rate of multiphase microstructures with varying phase-fraction are exceedingly critical, as they can be adopted for wide-range of systems and application. \nTo that end, in this study, the grain-growth kinetics of duplex and triplex systems are studied by employing approachable statistical techniques. \n\nConventionally, the grain-growth kinetics associated with multiphase systems are discussed by considering the evolution rate of individual phase-grains and entire-microstructure separately. \nSuch treatments rarely offer much insights on how the growth kinetics of individual phases relate to the overall evolution rate exhibited by the entire system. \nTherefore, in the present work, statistical tools are employed to realise the effect of growth rate of a given phase-associated grains on the overall kinetics of evolving microstructure. \nCorrespondingly, it is unraveled that, during grain growth in multiphase systems, the influence of the growth kinetics of a phase-grains on the evolution of entire system depends on its volume fraction. \nIn other words, in systems with varying volume fraction of constituent phases, the evolution kinetics of the major-phase grains predominantly govern the overall growth rate exhibited the microstructure. \nEven though, when focusing on a specific system, it might appear that the temporal change in the average radius of the entire multiphase system lies close the evolution of the corresponding radius of the minor-phase grains, when viewed in relation to microstructures of varying phase-fractions, the dominance of growth kinetics of the major-phase grains gets increasingly evident. \nThis holistic understanding that encompasses multiphase systems with varying phase-fractions, though arrived from  statistical treatments, in the present work, it is vindicated through conventional representation and corresponding discussions.\nMoreover, in addition to effecting the role of individual phase-grains on the evolution kinetics of entire multiphase system, phase fraction also dictates the interdependency between growth rates of different phase-grains. \nDuring grain growth, evolution kinetics of grains of two phases are largely independent when their corresponding volume-fractions in the multiphase system are minimal.\nOn the other hand, noticeable interaction is observed in the grain-growth rate of two phases that dominant the multiphase microstructure. \n\nUnderstanding rendered by the present investigation can be exploited for various purposes, however, one critical utilisation would be to alter the grain-growth rate of a given multiphase system, with a definite phase-fraction, by appropriately, and exclusively, varying the evolution kinetics of the major-phase grains. \nTo that end, in the upcoming works, attempts will be made to substantiate the approach of modifying the grain-growth rate in multiphase systems by employing the dominant influence of the major-phase grains.\n\n\\section*{Appendices}\n\n\\subsection*{Appendix 1: Interdependency in duplex systems}\\label{sec:app1}\n\n\\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{app1}\n    \\end{tabular}\n    \\caption{  The average radius of phase-$\\alpha$ grains, at a given instance, is plotted against the corresponding radius of matrix-phase grains for duplex microstructure with equal volume-fraction of phases. In the subplots, for the same system, the growth rate of individual phase-grains are related, and instantaneous average radius of phase-$\\alpha$ and -$\\gamma$ grains are plotted for duplex microstructure with $10\\%$ alpha.\n    \\label{fig:app1}}\n\\end{figure}\n\nIn a polycrystalline system, irrespective of its nature, the continuum is established by the multiple grains present it. \nDuring the grain growth, despite the continual disappearance of the grains, the continuum is sustained by the growth the surviving grains. \nThis characteristic feature of the grain growth introduces interdependency between the evolving grains. \nCorrespondingly, in duplex systems, wherein the grains are associated with one of the two-constituent phases, the evolution of phase-$\\alpha$ grains are inherently linked the grains of phase-$\\gamma$. \nEven though the interaction between the phase-associated grains, during grain growth of a duplex system, can be theoretically conceived, to explicate it with a statistical certainty, the average radius of $\\alpha$-grains, at a given time $t$, is plotted against corresponding of radius of $\\gamma$ grains in Fig.~\\ref{fig:app1} for two-phase microstructure with equal volume-fraction of phases, $\\alpha 50$.\nThe trend in this illustration indicates a inherent interlocking between the evolution of the phase-$\\alpha$ and -$\\gamma$ grains during the grain growth of equifraction duplex-microstructure.\nIn addition to the $\\alpha 50$ microstructure, the average radius of constituent phase-grains, at a given instance, are ascertained for duplex system with $10\\%$ minor phase-$\\alpha$.\nSimilar to equifraction system, these instantaneous average radii of phase-$\\alpha$ and -$\\gamma$ grains are plotted against each other, and illustrated in Fig.~\\ref{fig:app1} as a subplot.\nDespite the change in the phase-fraction, in $\\alpha 10$microstructure as well, a definite interaction between the radius of major- and minor-phase grains is evident. \n\nBesides the average radius of the phase-$\\alpha$ and -$\\gamma$ at a given instance, using the same approach, the relation between the kinetics of grain growth associated with these phases can be explicated. \nCorrespondingly, the evolution kinetics of the $\\alpha$-grains are related to that of the $\\gamma$ ones, for equifraction duplex microstructure, and are included as a subplot in Fig.~\\ref{fig:app1}. \nThis illustration unravels that even though there exists a perceivable interdependency between the instantaneous kinetics of phase-$\\alpha$ and -$\\gamma$ grains, it is not as straightforward as the average radius\n \n\\subsection*{Appendix 2: Effect of individual phase-grains kinetics on growth-rate of entire system}\\label{sec:app1}\n \n \\begin{figure}\n    \\centering\n      \\begin{tabular}{@{}c@{}}\n      \\includegraphics[width=1.0\\textwidth]{app2conso}\n    \\end{tabular}\n    \\caption{  The instances grain-growth rate exhibited by an individual phase-grains are plotted with respect to that of the entire duplex microstructure with equal volume fraction of phases. In (a) the kinetics of $\\alpha$-grains are related to the grain-growth rate of the entire microstructure, while $\\gamma$-grains evolution kinetics are considered (b). The corresponding outcomes for $\\alpha 10$ microstructure with $10\\%$ of minor phase-$\\alpha$ are included as subplots. \n    \\label{fig:app2conso}}\n\\end{figure}\n\nOne of the primary aim of the present investigation is to realise the effect of individual phases on grain-growth rate of entire two-phase microstructure. \nParticularly, the role of evolution kinetics of a given phase-grains on the overall growth rate of duplex system. \nTo that end, the kinetics of evolution exhibited by phase-$\\alpha$ grains, at a given instance, is related to the overall growth rate of equifraction duplex-system, and plotted in Fig.~\\ref{fig:app2conso}a. \nThe graphical illustration relating the evolution rate of  phase-$\\alpha$ grains and duplex microstructure with $10\\%$ minor phase is included as a subplot. \n\nFig.~\\ref{fig:app2conso}a unravels that the growth rate of phase-$\\alpha$ grains imposes a definite influence on the overall kinetics exhibited by the equifraction-duplex system, $\\alpha 50$, during grain growth. \nOn the other hand, the subplot of the corresponding illustration, which pertains to two-phase microstructure with $10\\%$ of phase-$\\alpha$, indicates a relation between $d\\bar{R}_{\\alpha}\/dt$ and $d\\bar{R}\/dt$, it is not as definite as one noticed in the equifraction microstructure.  \nIt is the degree of inter-relation between the kinetics of individual phase-grains, and overall growth-rate of a given duplex microstructure, which varies with the phase-fraction, is realised by \\textit{coefficient of determination}. \nIn other words, while the coefficient of determination relating the  $d\\bar{R}_{\\alpha}\/dt$ and $d\\bar{R}\/dt$ will be higher for equifraction duplex microstructure, in $\\alpha 10$ system it will assume a relatively low value reflecting a not so definite relation between the kinetics.\n\nIn Fig.~\\ref{fig:app2conso}b the grain-growth kinetics of phase-$\\gamma$ grains and entire duplex system, with equal volume-fraction of phases, is plotted against each other. \nAs a subplot, the corresponding relation between $d\\bar{R}_{\\gamma}\/dt$ and $d\\bar{R}\/dt$ for $\\alpha 10$ duplex microstructure with $10\\%$ of phase-$\\alpha$ is illustrated. \nUnlike the influence of phase-$\\alpha$ on overall growth-kinetics in Fig.~\\ref{fig:app2conso}a, a highly definite relation is observed between the evolution rate of phase-$\\gamma$ grains and entire microstructure in both equifraction and $\\alpha 10$ duplex microstructure. \nConsequently, the corresponding values of coefficient of determination will largely be independent of the phase-fraction.\n\n \n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe quest for the QCD chiral critical end point (CEP) in the phase diagram, \ntogether with the nature of the phase transition between hadron matter and quark \ngluon plasma (QGP), are open questions that have attracted the attention of the \nphysical community for some years \\cite{Halasz:1998qr}. Remarkable theoretical \nand experimental efforts \\cite{Brambilla:2014jmp} are being made to unveil the \nrich details of the QCD phase structures \\cite{Gupta:2011wh}. Experimentally, \none of the main goals of the heavy ion collision (HIC) program is the possible \nexistence and location of the CEP on the QCD phase diagram, with great \ndevelopments over the last years \n\\cite{Abelev:2009bw,Aggarwal:2010cw,Adamczyk:2013dal,Aduszkiewicz:2015jna}. \n\nIn relativistic HIC, the measurement of fluctuations of conserved \nquantities, such as baryon, electric charge, and strangeness number, play a \nmajor role in the experimental search for the CEP. Indeed, experimental \nmeasurements of cumulants of net-proton (proxy for net-baryon), net-charge, and \nnet-kaon (proxy for net-strangeness) are expected to carry significant amounts \nof information on the medium created by the collision (for a review, see \n\\cite{Friman:2011pf,Asakawa:2015ybt,Braun-Munzinger:2015hba,Luo:2017faz}).\nFluctuations are studied by measuring event-by-event fluctuations: a given \nobservable is measured on an event-by-event basis and its fluctuations are \nstudied for the ensemble of events \\cite{Braun-Munzinger:2015hba}.\n\nParticularly relevant are the cumulants of the net-baryon number because a \nsecond-order phase transition occurs at the CEP, resulting in divergences of\ncorrelation lengths for a static system of infinite size. \nThe cumulants of the baryon number thus diverge at the CEP \n\\cite{Stephanov:1998dy,Stephanov:1999zu}.\nThe study of the kurtosis \\cite{Stephanov:2011pb} and the skewness \n\\cite{Asakawa:2009aj} for the net-baryon number fluctuation distributions is \nessential as they are related to higher-order cumulants that can be extracted \nfrom event-by-event fluctuations in HIC experiments. \nOnce they are constituted by ratios of cumulants they are independent of the \nvolume of the system.\n\nThe study of fluctuations of conserved charges (baryon number, electric charge, \nand strangeness) at finite temperature and density has been done by using the \n(2+1) flavor Nambu$-$Jona-Lasinio (NJL) model in \n\\cite{Luo:2017faz,Chen:2015dra,Fan:2016ovc,Fan:2017kym}. \nBy using the (2+1) Polyakov$-$Nambu$-$Jona-Lasinio (PNJL) model, these \nfluctuations were investigated at finite temperature in \n\\cite{Fu:2009wy,Fu:2010ay,Bhattacharyya:2010ef,Bhattacharyya:2014uxa,Shao:2017yzv} \nand at finite temperature and density in \n\\cite{Fu:2010ay,Shao:2017yzv,Liu:2017qyc}.\n\nOther models have been employed to study higher-order baryon number \nsusceptibilities at finite temperature and density like the Polyakov-loop \nextended quark-meson model \\cite{Almasi:2017bhq}, where the influence of \nrepulsive vector-interactions on this fluctuations was also analyzed, the \nhybrid quark-meson-nucleon model \\cite{Marczenko:2017huu}, or the  SU(3) flavor \nparity-doublet quark-hadron model \\cite{Mukherjee:2016nhb} where the \nhigher-order baryon number susceptibilities near the chiral and the nuclear \nliquid-gas transitions were investigated.\n\nThe eventual location of the CEP can be affected by several conditions such as \nthe presence of external magnetic fields or the strangeness and isospin content \nof the medium \n\\cite{Costa:2013zca,Costa:2015bza,Rechenberger:2016gim}. \nThe study of the CEP location has been undertaken using different versions of \nthe NJL and PNJL models. In particular, it was shown that the presence of \nrepulsive vector interactions affects strongly the position of the CEP. The \nrole played by them were analyzed in detail in \n\\cite{Fukushima:2008wg,Costa:2015bza}. \nThe calibration of these models at high densities requires the existence of \nexperimental data or neutron star observables.\nParticularly relevant is the introduction of repulsive interactions, namely the\nvector-isoscalar terms, that seems to be necessary to describe $2M_\\odot$ \nhybrid stars \\cite{Pereira:2016dfg}.\n\nThe chiral restoration of strange quarks may play an important role \ninside neutron stars. \nIn particular, if this transition occurs at densities that can be found inside \ncompact stars, pure quark matter \\cite{Pereira:2016dfg}, or, exotic quark \nphases such as the color-flavor-locked (CFL) phase could be realized in their \ninterior \\cite{Alford:2001zr}. \nBesides, a phase transition could also have an important effect on the \nmean-free path of neutrinos in a protoneutron star as discussed in \n\\cite{Gulminelli:2013qr}. The cooling of protoneutron stars during the first \nseconds is essentially driven by the neutrinos that diffuse out of the star. \nA phase transition would give rise to a opalescence like phenomena reducing a \nlot the neutrino mean-free path, and, therefore, allowing for a much larger \ninteraction of neutrinos with matter.\n\n\nIn this work, we study the phase diagram using the (2+1)-flavor \nNambu--Jona-Lasinio model coupled to the Polyakov loop, designated as PNJL \nmodel, from the point of view of the kurtosis and skewness of net-baryon number \nfluctuations. \nIt is expected that in HIC the fireball evolves along isentropes, lines with \nconstant entropy per baryon, and, therefore, we analyze how these quantities, \nas well as the velocity of sound, behave along isentropes. \nOur main objective is to identify the similarities and differences of a QCD \nphase diagram which has a CEP or not, namely when a \nsufficiently strong repulsive vector interaction is taken into account.\n\nThe model is succinctly reviewed in Sec. \\ref{sec:model}, while the results are \ndiscussed in Sec. \\ref{sec:Results}.\nFinally we draw our conclusions in Sec. \\ref{sec:conclusions}. \n\n\\section{Model and formalism}\n\\label{sec:model}\n\nThe Lagrangian density for the Polyakov extended Nambu--Jona-Lasinio (PNJL) \nmodel reads\n\\begin{eqnarray}\n{\\cal L} &=& {\\bar{q}} \\left[i\\gamma_\\mu D^{\\mu}-\n\t{\\hat m}_c \\right ] q ~+~ {\\cal L}_\\text{sym}~+~{\\cal L}_\\text{det}~\n  +~{\\cal L}_\\text{vec} \\nonumber\\\\\n&+& \\mathcal{U}\\left(\\Phi,\\bar\\Phi;T\\right),\n\t\\label{Pnjl}\n\\end{eqnarray}\nwhere the quark field is represented by $q = (u,d,s)^T$ in flavor space, and \n${\\hat m}_c= {\\rm diag}_f (m_u,m_d,m_s)$ is the corresponding (current) mass \nmatrix.\nThe ${\\cal L}_\\text{sym}$ and ${\\cal L}_\\text{det}$ denote, respectively, \nthe scalar-pseudoscalar and the 't Hooft six-fermion interactions\n\\cite{Klevansky:1992qe,Hatsuda:1994pi},\n\\begin{align}\n\t{\\cal L}_\\text{sym}&= G_s \\sum_{a=0}^8 \\left [({\\bar q} \\lambda_ a q)^2 + \n\t({\\bar q} i\\gamma_5 \\lambda_a q)^2 \\right ] \\\\\n\t{\\cal L}_\\text{det}&=-K\\left\\{{\\rm det} \\left [{\\bar q}(1+\\gamma_5)q \\right] + \n\t{\\rm det}\\left [{\\bar q}(1-\\gamma_5)q\\right] \\right \\}.\n\\end{align}\nThe vector interaction is given by \\cite{Mishustin:2000ss}\n\\begin{equation} \n{\\cal L}_\\text{vec} = - G_V \\sum_{a=0}^8  \n\\left[({\\bar q} \\gamma^\\mu \\lambda_a q)^2 + \n ({\\bar q} \\gamma^\\mu \\gamma_5 \\lambda_a q)^2 \\right]. \n\\label{p1} \n\\end{equation}\nThe effective gluon field is given by \n$A^\\mu = g_{strong} {\\cal A}^\\mu_a\\frac{\\lambda_a}{2}$, where\n${\\cal A}^\\mu_a$ represents the SU$_c(3)$ gauge field.\nThe spatial components are neglected in Polyakov gauge at finite temperature,\ni.e., $A^\\mu = \\delta^{\\mu}_{0}A^0 = - i \\delta^{\\mu}_{4}A^4$. \nThe Polyakov loop value is defined as the trace of the Polyakov line,\n$ \\Phi = \\frac 1 {N_c} {\\langle\\langle \\mathcal{P}\\exp i\\int_{0}^{\\beta}d\\tau\\,\nA_4\\left(\\vec{x},\\tau\\right)\\ \\rangle\\rangle}_\\beta$,\nwhich is the order parameter of the $\\mathbb{Z}_3$ \nsymmetric\/broken phase transition in pure gauge.\nFor the pure gauge sector we use the following effective potential  \\cite{Roessner:2006xn},\n\\begin{eqnarray}\n\t& &\\frac{\\mathcal{U}\\left(\\Phi,\\bar\\Phi;T\\right)}{T^4}\n\t= -\\frac{a\\left(T\\right)}{2}\\bar\\Phi \\Phi \\nonumber\\\\\n\t& &\n\t+\\, b(T)\\mbox{ln}\\left[1-6\\bar\\Phi \\Phi+4(\\bar\\Phi^3+ \\Phi^3)\n\t-3(\\bar\\Phi \\Phi)^2\\right],\n\t\\label{Ueff}\n\\end{eqnarray}\nwhere \n$a\\left(T\\right)=a_0+a_1\\left(\\frac{T_0}{T}\\right)+a_2\\left(\\frac{T_0}{T}\\right)^2$, \n$b(T)=b_3\\left(\\frac{T_0}{T}\\right)^3$. \nIts parametrization values are $a_0 = 3.51$, $a_1 = -2.47$, $a_2 = 15.2$, \nand $b_3 = -1.75$ \\cite{Roessner:2006xn}, while the critical temperature is set \nto $T_0=210$ MeV.\nThe divergent ultraviolet sea quark integrals are regularized by a sharp cutoff \n$\\Lambda$ in three-momentum space.\nFor the NJL model parametrization, we consider:\n$\\Lambda = 602.3$ MeV, $m_u= m_d=5.5$ MeV, $m_s=140.7$ MeV, \n$G_s \\Lambda^2= 1.835$, and $K \\Lambda^5=12.36$ \\cite{Rehberg:1995kh}.\\\\\n\nFluctuations of conserved charges, such as the baryon number, provide vital \ninformation on the effective degrees of freedom and on critical phenomena.\nThey behave characteristically in a thermal equilibrium medium.\nIf there is a CEP in the phase diagram of strongly interacting matter, these \nfluctuations are then expected to provide characteristic signatures that, \nhopefully, can be experimentally observed.\nFor a static system of infinite size, the fluctuations of baryon number diverge \nat the CEP (second-order phase transition point). \nHowever, the created medium in HIC experiments has both finite size and \nlifetime that restricts its correlation length and, instead of divergent \nfluctuations, only moderate enhancements are expected.\nFluctuations of conserved charge are characterized by their cumulants or \nsusceptibilities.\nThe present work focuses on the baryon number charge susceptibilities. \nThe n$th$-order net-baryon susceptibility is given by\n\\begin{equation}\n \\chi_B^n(T,\\mu_B)= \\frac{\\partial^n\\pc{P(T,\\mu_B)\/T^4}}{\\partial(\\mu_B\/T)^n}.\n\\end{equation}\nDifferent susceptibility ratios $\\chi_B^n(T,\\mu_B)\/\\chi_B^m(T,\\mu_B)$\nare calculated in order to eliminate the volume dependence, allowing for a \ncomparison with experimental observables. \nIn this work, we analyze the following ratios\n\\begin{equation}\n\t\\frac{\\chi_B^4(T,\\mu_B)}{\\chi_B^2(T,\\mu_B)}=\\kappa\\sigma^2,\\quad\n\t\\frac{\\chi_B^3(T,\\mu_B)}{\\chi_B^1(T,\\mu_B)}=\\frac{S_B\\sigma^3}{M},\n\\label{eq:ratios}\n\\end{equation}\nwhere $M=VT^3\\chi_B^1$ is the mean, $\\sigma^2=VT^3\\chi_B^2$ the variance, $S_B$ \nthe skewness, and $\\kappa$ is the kurtosis of the net-baryon number \ndistribution. \n\n\\section{Results}\\label{sec:Results}\n\\begin{figure*}[t!]\n\t\\centering\n\n\t\\includegraphics[width=0.8\\linewidth]{1.pdf}\n\t\\caption{The light-quark condensate \n\t\t$\\ev{u\\bar{u}}(T,\\mu_B)\/\\ev{u\\bar{u}}(0,0)$ for $G_V=0$ (left) and \n\t\t$G_V=0.72G_s$ (right). \n\t\tThe following information is displayed: the CEP (dot), the chiral \n\t\tfirst-order phase transition boundary (solid line), and both the chiral \n\t\t(dashed line) and deconfinement (dotted line) crossover boundaries.} \n\t\\label{fig:1}\n\\end{figure*}\n\nWe analyze, herein, the net-baryon susceptibilities on the $(T,\\mu_B)$ plane.\nTwo PNJL models are analyzed: (i) a model with no vector interactions $G_V=0$,\nwhich predicts a CEP; \nand (ii) a model with vector interactions $G_V=0.72G_s$, which predicts no CEP.\nWe want to discuss what distinguishes these two scenarios. \nIn the following, symmetric quark matter is considered: \n$\\mu_u=\\mu_d=\\mu_s=\\mu_q=\\mu_B\/3$, where $\\mu_{i}$ are the chemical potential of \neach quark flavor and $\\mu_B$ is the baryonic chemical potential.\n\nThe hydrodynamical expansion of a HIC fireball is expected to follow  \ntrajectories of constant entropy per baryon, $s\/\\rho_B$, known as isentropes. \nThese trajectories contain important information on the adiabatic evolution of \nthe system. \nIt is thus interesting to analyze the susceptibility ratios \n[Eqs. (\\ref{eq:ratios})] along different isentropes \\cite{Costa:2010zw}.\nIt is important to note that while the net charge and the net strangeness are \nnot constrained in the present work; in a HIC, however, the ratio of electric \ncharge over baryon number is $Q\/\\rho_B\\simeq0.4$ and no net strangeness is \nproduced, $n_s=0$.\\\\\n\nThe phase diagrams for the chiral and deconfinement transitions are presented in \nFig. \\ref{fig:1}. The (normalized) light-quark condensate value\n$\\ev{u\\bar{u}}(T,\\mu_B)\/\\ev{u\\bar{u}}(0,0)$ is shown, where  \n$\\ev{u\\bar{u}}(0,0)$ is the vacuum value (due to isospin symmetry \n$\\ev{u\\bar{u}}=\\ev{d\\bar{d}}$).\nThe $G_V=0$ model predicts a CEP at \n$(T^{\\mbox{\\footnotesize{CEP}}},\\mu_B^{\\mbox{\\footnotesize{CEP}}})=(133\\, \\text{MeV},862\\,\\text{MeV})$, \nwhile the $G_V=0.72G_s$ model has no CEP, and the (approximate) \nchiral restoration is thus attained via an analytic transition (crossover) over \nthe whole phase diagram. \nThe chiral (dashed line) and deconfinement (dotted line) crossover boundaries \nare determined by the location $(T,\\mu_B)$ of the maximum of the order \nparameter susceptibilities (the point where fluctuations are largest).\nIt is interesting that the crossover boundaries show similar behavior for both \nmodels:\nthe gap between the deconfinement and chiral crossovers reduces with increasing \n$\\mu_B$ and becomes zero for some $\\mu_B$ values, which turns out to be near the \nCEP for $G_V=0$, above which they separate and follow distinct paths.\n\nBoth boundaries, the chiral  phase transition and the deconfinement phase \ntransition boundaries, are determined from the peaks of the susceptibility. The \ncrossing of the deconfinement and  chiral phase transitions has already been \nobserved before \\cite{Costa:2011fh} and it is possible to identify the crossing \nfrom the calculation of the susceptibility peaks at fixed temperatures: before \nand after the crossing they are two distinguishable peaks. At the crossing, \nthat stretches along a finite range of temperatures,  the two peaks overlap. \nThe crossing region includes part of the chiral crossover for both models, and, \nin the case of the model with a CEP, also the CEP, and part of the first-order \nphase transition.\n\nWe thus conclude that, due to the mixing between the gluonic and quarkionic \ndegrees of freedom, the chiral phase transition has a strong influence on \ndeconfinement transition. This is reflected on the behavior of the \ndeconfinement transition at the light quark and the strange quark chiral \ntransition. At the light quark transition, the crossing temperature is not much \naffected, but the crossing chemical potential is tightly connected with the \nposition of the chiral transition and the crossing follows, as referred above,  \nthe chiral crossover or both the chiral crossover and first-order transition. \nAs a consequence, the crossing occurs at a much larger chemical potential for \nthe $G_V=0.72 G_s$ model. A similar interconnection is observed at the strange \nchiral crossover in Fig. 2 and 3 in the $G_V=0$ model, where the deconfinement \ntransition presents a kink.\n\t\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[width=0.65\\linewidth]{2.pdf}\n\t\\caption{The net-baryon number susceptibilities \n\t\t$\\chi^3_B$ (top) and $\\chi^4_B$ (bottom) for $G_V=0$ (left) and \n\t\t$G_V=0.72G_s$ (right).\n\t\tThe following information is displayed: the CEP (black dot), the first-order\n\t\tphase chiral transition boundary (black solid line), both the chiral (black \n\t\tdashed line) and deconfinement (black dotted line) crossover boundaries,\n\t\tand the $s\/\\rho_B=\\{0.5,1,5,10,14\\}$ isentropic trajectories (dark green \n\t\tdotted-dashed lines) are also shown, which appear in the counterclockwise\n\t\tdirection, respectively.} \n\t\\label{fig:2}\n\\end{figure*}\nWe show  the $\\chi_B^3$ (top) and $\\chi_B^4$ (bottom) susceptibilities in Fig \n\\ref{fig:2}. \nTo a better understanding of their dependencies in the $(T,\\mu_B)$ plane, \nthe following features are also displayed:\nthe CEP (black dot), the first-order chiral phase transition boundary (black \nsolid line), and both the chiral (black dashed line) and the deconfinement \n(black dotted line) crossover boundaries.\nFurthermore, the isentropic trajectories (dark green dashed-dotted lines), \ni.e., paths along which the entropy density per baryon, $s\/\\rho_B$, is fixed, \nare also shown for $s\/\\rho_B=\\{0.5,1,5,10,14\\}$. \nThe two last trajectories cross the crossover line above the CEP of the $G_V=0$ \nmodel.\n\n\\begin{figure*}[!htbp]\n\t\\centering\n\t\\includegraphics[width=0.65\\linewidth]{3.pdf}\n\t\\caption{The net-baryon number susceptibility ratios\n\t\t$\\chi^4_B\/\\chi^2_B$ (top) and $\\chi^3_B\/\\chi^1_B$ (bottom) \n\t\tfor $G_V=0$ (left) and $G_V=0.72G_s$ (right).\n\t\tThe following information is displayed: the CEP (black dot),\n\t\tthe first-order phase chiral transition boundary (black solid line), both \n\t\tthe chiral (black dashed line) and deconfinement (black dotted line) \n\t\tcrossover boundaries, and the $s\/\\rho_B=\\{0.5,1,5,10,14\\}$ isentropic \n\t\ttrajectories (dark green dotted-dashed lines) are also shown, which appear\n\t\tin the counterclockwise direction, respectively.} \n\t\\label{fig:3}\n\\end{figure*}\n\nThe first three values allows us to discuss the phase diagram at low $T$ and \nhigh $\\mu_B$, where the (approximate) chiral restoration of the strange quark \noccurs.\nFor the $G_V=0$ model, the susceptibilities exhibit a nonmonotonic dependence \nnear the CEP, whose behavior strongly depends on the direction on which the CEP \nis approached. \nThe susceptibilities diverge at the CEP, with the divergence being stronger\nas higher susceptibilities orders are considered.\nAn interesting result is present at low $T$ and high $\\mu_B$. \nDespite the transition for the strange quark being just a crossover,\nand, therefore, without any nonanalytic behavior, a similar CEP \nstructure is seen at $\\mu_B\\approx1500$ MeV for the susceptibilities. \nThis indicates that a slight change on the model parametrization might induce \na first-order phase transition for the strange quark, and a corresponding CEP. \nThe $\\chi_B^3$ and $\\chi_B^4$ values for the $G_V=0.72G_s$ model show precisely \nthis behavior for the light quark sector: even though there is no CEP, and the \nchiral transition occurs via a crossover over the whole phase diagram, the \nnonmonotonic behavior of the susceptibilities is still present, as discussed \nwithin the NJL model \\cite{Fan:2017kym}.\nThe study of a scenario with a hypothetical negative \n$T^{\\mbox{\\footnotesize{CEP}}}$ for the light CEP, obtained by varying the \nvalue of the anomaly-induced six-fermion term, $K$, was done in \n\\cite{Chen:2016sxn}; it was shown that the magnitude of the susceptibilities \nalso changes significantly if a hypothetical negative temperature CEP is taken \ninto account.\n\nThe ratios $\\chi_B^4\/\\chi_B^2$ and $\\chi_B^3\/\\chi_B^1$ are shown in \nFig. \\ref{fig:3}. The sudden decrease near the deconfinement pseudocritical \ntemperature (dotted black line) indicates that both quantities are valuable \nsignatures of deconfinement transition.\nAs noted in \\cite{Fu:2009wy}, the statistical confinement, provided by the \nPolyakov loop (at low temperatures, when $\\Phi,\\bar{\\Phi}\\rightarrow 0$, \ncontributions coming from one- and two-quark states are suppressed, while \nthree-quark states are not \\cite{Hansen:2006ee}), is essential to obtain a \nlow-temperature limit for the susceptibility ratios that is consistent with \nthe hadron resonance gas model.\nThe results for the $G_V=0.72G_s$ model clearly show that the nonmonotonic \nbehavior of $\\chi_B^3$ and $\\chi_B^4$, which signals the presence of a\ncritical behavior, is still present even in the absence of a CEP. The \nnonmonotonic behavior persists, with a smaller intensity, up to almost the same \ntemperature as for the $G_V=0$ model. To make this feature clear, we show the \nnegative region of $\\chi^4_B\/\\chi^2_B$ in Fig. \\ref{fig:4}. \n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{4.pdf}\n\t\\caption{The region $\\chi^4_B\/\\chi^2_B<0$ for $G_V=0$ (blue) and \n\t$G_V=0.72G_s$ (red) models.\n\t\t} \n\t\\label{fig:4}\n\\end{figure}\nDespite the strong vector interaction used, it is remarkable that, for the \n$G_V=0.72G_s$ model (red), the $\\chi^4_B\/\\chi^2_B<0$ region extends from zero \nup to temperatures similar with the ones obtained for $G_V=0$ (blue). \nThis indicates that, at higher temperatures, both models are not discernible   \nexclusively from the sign change of $\\chi^4_B\/\\chi^2_B$. \nActually, a region with $\\chi^4_B\/\\chi^2_B<0$ is still present (at lower \ntemperatures though) even when the vector interaction strength is increased up \nto $G_V\\approx1.4G_s$. \nIf instead of looking at the whole negative region of $\\chi^4_B\/\\chi^2_B<0$, \none considers the stronger fluctuation region $\\chi^4_B\/\\chi^2_B<-200$ the \nfollowing pattern is seen:\nwhile this region extends to a range of $\\Delta \\mu_B\\approx100$ MeV and \n$\\Delta T\\approx20$ MeV for $G_V=0$, we get $\\Delta \\mu_B\\approx20$ MeV and \n$\\Delta T\\approx60$ MeV for $G_V=0.72G_s$.  \nThese different ranges on $T$ and $\\mu_B$ for the two different \nscenarios could help distinguish them, taking only into account the behavior of \nthe fluctuations. It should, however, be recalled that if the CEP exists, for \nmoderate temperatures and high enough baryonic density the line of first-order \ntransition could be crossed during the evolution of the fireball, giving rise \nto effects like multifragmentation \n\\cite{Mishustin:2006ka,Braun-Munzinger:2015hba}. \nThis would be a region where our no CEP model would present fluctuations \nsimilar to the ones existing in a model with CEP, above the CEP.\n\n\\begin{figure}[b!]\n\t\\centering\n\t\\includegraphics[width=0.80\\linewidth]{5.pdf}\n\t\\caption{Isentropic trajectories $s\/\\rho_B=\\{0.5,1,5,10,14\\}$ for \n\tthe $G_V=0$ (solid lines) and the $G_V=0.72G_s$ (dotted lines) models.} \n\t\\label{fig:5}\n\\end{figure}\n\nTo complete the discussion, in the following we analyze the  \nisentropic trajectories with a small $s\/\\rho_B$ within the two scenarios.\n\nThe comparison of the isentropic trajectories between $G_V=0$ (solid lines) \nand $G_V=0.72G_s$ (dashed lines) models is in Fig. \\ref{fig:5}.  \nThe trajectories differ for high values of $T$ and $\\mu_B$, i.e., as soon as \nthe system becomes denser enough for the vector interactions to set in.\nTwo features that distinguish the $G_V=0$ model from the $G_V=0.72G_s$\nis the behavior of the trajectories near the CEP and the existence of a \nunstable spinodal region. \nThe trajectories with low $s\/\\rho_B$ values get enclosed into the unstable \nspinodal region when crossing the first-order phase transition to the chiral \nbroken phase. \nAs the system enters into the unstable spinodal region, the rapid formation of \nfragments of high density matter that occur should enhance the baryon number \nfluctuations \\cite{Braun-Munzinger:2015hba}.\nDue to the absence of spinodal region for the $G_V=0.72G_s$ model, such effect does not occur and the susceptibilities have an analytic behavior.\n\n\nIn Fig. \\ref{fig:6}, we show the $\\chi_B^4\/\\chi_B^2$ (top) and \n$\\chi_B^3\/\\chi_B^1$ (bottom) values along the $s\/\\rho_B=14$ (left) and the \n$s\/\\rho_B=10$ (right) isentropes (these isentropic trajectories are shown in \nFig. \\ref{fig:2} and \\ref{fig:3}).\nAs the value $s\/\\rho_B$ of the isentropic trajectory decreases, we are covering \na higher $\\mu_B$ region on the phase diagram. \nAs we move from $s\/\\rho_B=14$ to $s\/\\rho_B=10$, we are then approaching a \nregion of higher baryon fluctuations that reflects the vicinity of a CEP.\nWhile the fluctuations of $\\chi_B^4\/\\chi_B^2$ and $\\chi_B^3\/\\chi_B^1$\ngrow with decreasing $s\/\\rho_B$,\nthey also become constrained to a smaller temperature region (this is clear \nthrough the shape of the blue region in Fig. \\ref{fig:4}). \nThe decreasing gap between the chiral and deconfinement transitions with \nincreasing $\\mu_B$, which vanishes at the CEP (see Figs. \\ref{fig:2} and \n\\ref{fig:3}), is also reflected in the fluctuations: for $s\/\\rho_B\\geq14$ a two \npeak structure is present on the left side of the fluctuation (for \n$s\/\\rho_B=14$, a small bump at $T\\approx 150$ MeV is barely seen).\n\n\nThis two peak structure, which reflects the deconfinement\/chiral restoration \ngap, is clearer when a vector interaction is included. \nThe fluctuations for the $G_V=0.72G_s$ model, over the same isentropic \ntrajectories, are shown in Fig. \\ref{fig:7}.\nThe fluctuations along the $s\/\\rho_B=14$ trajectory show a two peak structure \nagain on the left side of the fluctuation. \nDespite the existence of a sign change of $\\chi_B^4\/\\chi_B^2$ (top) for both \nmodels (also seen in Fig. \\ref{fig:4}), their intensity is weaker for the \n$G_V=0.72G_s$ model, allowing one to notice the effect of the deconfinement\ntransition on the fluctuations ratios.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{6.pdf}\n\t\\caption{The values of $\\chi_B^4\/\\chi_B^2$ (top)\n\tand $\\chi_B^3\/\\chi_B^1$ (bottom) as a function of temperature along the isentropes\n\t$s\/\\rho_B=14$ (left) and $s\/\\rho_B=10$ (right) for the $G_V=0$ model.} \n\t\\label{fig:6}\n\\end{figure}\n\nLet us now focus on the crossover region at low temperatures, i.e., low \n$s\/\\rho_B$ values, for the $G_V=0.72G_s$ model.\nIn Fig. \\ref{fig:8}, we display the values of $\\chi^4_B\/\\chi^2_B$ (red) and \n$\\chi^3_B\/\\chi^1_B$ (blue) along the isentropes $s\/\\rho_B=5$ (top) and \n$s\/\\rho_B=1$ (bottom). The $(T,\\mu_B)$ dependence of the isentropic \ntrajectories can be seen in Figs. \\ref{fig:2} and \\ref{fig:3}. \nThe fluctuations increase strongly as lower isentrope values are considered. \nThe large fluctuation of $\\chi^4_B\/\\chi^2_B$ for $s\/\\rho_B=1$ reflects the \ncrossing of the isentropic trajectory with the chiral crossover line at \n$T\\approx 40$ MeV.\nThe features obtained at these low $s\/\\rho_B$ values are similar with the ones \nof the model with CEP but at $s\/\\rho_B=10$ and $14$; i.e., we get similar \nfluctuation amplitudes for a much lower $T$.\n \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.98\\linewidth]{7.pdf}\n\t\\caption{The values of $\\chi_B^4\/\\chi_B^2$ (top)\n\tand $\\chi_B^3\/\\chi_B^1$ (bottom) as a function of temperature along the isentropes\n\t$s\/\\rho_B=14$ (left) and $s\/\\rho_B=10$ (right) for the $G_V=0.72G_s$ model.} \n\t\\label{fig:7}\n\\end{figure}\n\n\n\\begin{figure}[!b]\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{8.pdf}\n\t\\caption{The value of $\\chi^4_B\/\\chi^2_B$ (red) and $\\chi^3_B\/\\chi^1_B$ (blue) \n\t\tas a function of temperature along the isentropic trajectories $s\/\\rho_B=5$ (left) and $s\/\\rho_B=1$ \n\t\t(right) for the $G_V=0.72G_s$ model.} \n\t\\label{fig:8}\n\\end{figure}\n\n\\begin{figure*}[!tb]\n\t\\centering\n\t\\includegraphics[width=0.65\\linewidth]{9.pdf}\n\t\\caption{Sound velocity squared $v_s^2$ as a function \n\t\tof temperature (left) and baryon chemical potential (right) along the \n\t\tisentropic trajectories $s\/\\rho_B=10$ (top) and $s\/\\rho_B=1$ (bottom) for \n\t\t$G_V=0$ (blue) and $G_V=0.72G_s$ (red) models. \n\t\tThe squares and circles indicate the location of\n\t\tthe chiral and the deconfinement pseudocritical boundaries, respectively.} \n\t\\label{fig:9}\n\\end{figure*}\n\n\\begin{table*}[t!]\n\\centering\n\\begin{tabular}{|c||c|c|c||c|c|c||c|c|c|}\n\\hline\n\\multirow{2}{*}{$s\/\\rho_B$} & $T^{\\chi}$ & $\\mu_B^{\\chi}$ & \\multirow{2}{*}{$(v_s^2)^{\\chi}$} & $T^{\\Phi}$ & $\\mu_B^{\\Phi}$ & \\multirow{2}{*}{$(v_s^2)^{\\Phi}$} & $T^{min}$ & $\\mu_B^{min}$ & \\multirow{2}{*}{$(v_s^2)_{min}$} \\\\\n & [MeV] & [MeV] & & [MeV] & [MeV] & & [MeV] & [MeV] &  \\\\\n\\hline\n\\multicolumn{10}{|c|}{$G_V=0$ } \\\\\n\\hline\n8 & 133 & 861 & 0.018 & 133 & 863 & 0.018 & 133 & 863 & 0.018 \\\\\n\\hline\n10 & 145 & 790 & 0.036 & 143 & 796 & 0.032 & 141 & 803 & 0.031 \\\\\n\\hline\n12 & 153 & 727 & 0.053 & 148 & 738 & 0.040 & 147 & 738 & 0.039 \\\\\n\\hline\n14 & 161 & 667 & 0.070 & 153 & 672 & 0.044 & 150 & 668 & 0.042 \\\\\n\\hline\n20 & 176 & 525 & 0.10 & 161 & 504 & 0.045 & 150 & 668 & 0.042 \\\\\n\\hline\n\\multicolumn{10}{|c|}{$G_V=0.72G_s$}  \\\\\n\\hline\n0.1 & 5 & 1172 & 0.0049 & $-$ & $-$ & $-$ & 6 & 1173 & 0.0016 \\\\\n\\hline\n1 & 37 & 1165 & 0.019 & $-$ & $-$ & $-$ & 43 & 1166 & 0.016 \\\\\n\\hline\n10 & 144 & 931 & 0.088 & 140 & 906 & 0.069 & 134 & 859 & 0.060 \\\\\n\\hline\n12 & 153 & 870 & 0.10 & 147 & 819 & 0.067 & 141 & 768 & 0.059 \\\\\n\\hline\n14 & 161 & 814 & 0.11 & 152 & 737 & 0.064 & 174 & 689 & 0.056 \\\\\n\\hline\n20 & 185 & 554 & 0.18 & 162 & 512 & 0.048 & 159 & 494 & 0.042 \\\\\n\\hline\\end{tabular}\n\\caption{The sound velocity squared at the chiral \n\t\t$(v_s^2)^{\\chi}=(T^{\\chi},\\mu_B^{\\chi})$ and deconfinement \n\t\t$(v_s^2)^{\\Phi}=(T^{\\Phi},\\mu_B^{\\Phi})$ pseudocritical boundaries for\n\t\tseveral isentropes $s\/\\rho_B$. The minimum of $v_s^2$ [$(v_s^2)_{min}$] and\n\t\tits location $(T^{min},\\mu_B^{min})$ is also presented. }\n\\label{Table1}\n\\end{table*}\n\nFinally, we have determined the square of the sound velocity, \n$v_s^2=dP\/d{\\cal E}|_{s\/\\rho_B=\\mbox{const.}}$, along two isentropic trajectories. \nThe sound velocity plays a central role in the hydrodynamical evolution of \nmatter created in HIC being very different in the different stages of the \nexpansion. It affects, among others, the momentum distribution of the particles \noriginating from the fluid elements at the freeze-out stage \n\\cite{Mohanty:2003va}. \nThe values of square sound velocity are extracted from the widths of rapidity \ndistributions \\cite{Mohanty:2003va,Gao:2015sdb,Adam:2016ddh}. For example, from \nthe measured data on the widths of the pion rapidity spectra, $v_s^2$ in the \ndense stage of the reactions has been extracted \\cite{Steinheimer:2012bp}.\n\nIn Fig. \\ref{fig:9}, we show $v_s^2$ along $s\/\\rho_B=10$ (top) and \n$s\/\\rho_B=1$ (bottom) for $G_V=0$ (blue) and $G_V=0.72G_s$ (red) models.\nAs the isentropic trajectories follow specific paths, $(T,\\mu_B)$, on the phase \ndiagram (see Figs. \\ref{fig:2} and \\ref{fig:3}), we show the $v_s^2$ dependence \non temperature (left) and baryon chemical potential (right), for both \nisentropic trajectories.\nFor each isentropic, we give the values of $v_s^2(T,\\mu_B)$ at its minimum and \nat the chiral and deconfinement boundaries in Table \\ref{Table1}. \n\nConsidering in first place the $s\/\\rho_B=10$ trajectory as a function of \n$T$ (Fig. \\ref{fig:9}, left upper panel), we see that the minimum of \n$v_s^2(T,\\mu_B)$ occurs closer to the deconfinement pseudocritical boundary \n(circles) than to the chiral pseudocritical boundary (squares).\nWhile the temperature dependence of the $s\/\\rho_B=10$ isentrope is a \nsingle-valued function, the same does not hold for its $\\mu_B$ dependence \n(Fig. \\ref{fig:9}, right upper panel).\nThe loop behavior for the $G_V=0$ model (blue curve in upper right panel of \nFig. \\ref{fig:9}) \nrises from the bending effect towards the CEP that the $s\/\\rho_B=10$ isentrope \nundergoes when crossing into the chiral broken region (solid red line in Fig. \n\\ref{fig:5}). \nThis effect occurring in $v_s^2$ can then be seen as a signal for the vicinity \nof a CEP (if some kind of bending effect into the CEP exists), once this effect \nis not seen for the $G_V=0.72G_s$ model.\nFor $s\/\\rho_B=1$ (lower panels of Fig. \\ref{fig:9}), the $v_s^2$ shows negative \nvalues for $G_V=0$ model (blue curve), reflecting the first-order phase\ntransition that occurs at lower $T$.\nIt is interesting to note that, for small values of $s\/\\rho_B$, the local \nminimum of $v_s^2$ at $\\mu_B\\approx1480$ MeV is associated  with the crossover\nof the strange quark. \n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe have analyzed the net-baryon number fluctuations for three-flavor quark \nmatter within the Polyakov extended Nambu--Jona-Lasinio model.\nFor a strong enough vector interaction intensity, the model predicts no CEP\nin the phase diagram.\nFrom  the net-baryon number fluctuations one concludes that, even in the \nabsence of a CEP, the nonmonotonic behavior persists.\nTherefore, the existence of a CEP cannot be taken solely from the existence of \nnonmonotonic behavior on the net-baryon number susceptibilities.\n\nWe have analyzed further other possible properties that may\ndistinguish the two scenarios: for the no CEP model ($G_V=0.72G_s$),\nlarge fluctuations in the susceptibility ratios occur only at small $T$, and \nthe values of $v_s^2$ are almost unchanged at moderates $s\/\\rho_B$ values.\nThe values of the susceptibility ratios along two or three isentropic lines \nwould possibly allow us to distinguish both cases. Also, the value of the sound \nvelocity at the chiral transition for two or three isentropes would give some \nuseful information. \nFor the $G_V=0$ model, by going from $s\/\\rho_B=9$ to $14$, the value of $v_s^2$ \nincreases at least $50\\%$ for each step $\\Delta(s\/\\rho_B)=2$. \nInstead, for the no CEP model, the change is of the order of $10\\%$.\nIt should be noticed, however, that in the present work we have discussed \ninfinite size matter. For a finite system it is expected that the signals we \nhave discussed are less intense but still might allow to distinguish both \nscenarios. \n\nWe have shown that, for high chemical potentials and low temperatures,\na signature of the strange quark chiral symmetry restoration is observed \nin a decrease of the sound velocity and in a region with negative \n$\\chi^4_B\/\\chi^2_B$. \n\n\\vspace{0.25cm}\n{\\bf Acknowledgments}\nThis work was supported by ``Funda\u00e7\u00e3o para a Ci\u00eancia e Tecnologia,'' Portugal, \nunder Projects No. UID\/FIS\/04564\/2016 and No. POCI-01-0145-FEDER-029912 with\nfinancial support from POCI, in its FEDER component, and by the FCT\/MCTES \nbudget through national funds (OE), and under Grants No. SFRH\/BPD\/102273\/2014 \n(P.C.) and No. CENTRO-01-0145-FEDER-000014 (M.F.) through the CENTRO2020 \nprogram. Partial support comes from ``THOR'' (COST Action CA15213) and \n``PHAROS'' (COST Action CA16214).\n\n\\vspace{-0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFor more than a decade experiments at LEP (CERN) and SLC (SLAC) \ngathered  a wealth of high precision high energy hadronic data\nfrom electron-positron annihilation at a range of centre-of-mass \nenergies~\\cite{ALEPH-qcdpaper,lep,sld}. \nThis data provides one of the    \n cleanest\nways of probing our quantitative understanding of QCD. \nThis is particularly so because the strong interactions occur only in \nthe final state and are not entangled with the parton density functions associated \nwith beams of hadrons.\nAs the understanding of the strong interaction, and the capability of \nmaking more precise theoretical predictions, develops, \nmore and more stringent comparisons of theory and experiment are possible,\nleading to improved measurements\nof fundamental quantities such as the strong \ncoupling constant~\\cite{expreview}.\n\nIn addition\nto measuring multi-jet production rates, more specific information  about the\ntopology of the events can be extracted. To this end, many variables  have been\nintroduced which characterise the hadronic structure of an event. \nWith the precision data from LEP and SLC, experimental\ndistributions for such event shape variables have been extensively  studied and\nhave been compared with theoretical calculations based on next-to-leading order\n(NLO)  parton-level event generator  programs~\\cite{ERT,kunszt,event}, \n  improved by\nresumming kinematically-dominant leading and next-to-leading logarithms\n(NLO+NLL)~\\cite{ctwt}  and by the inclusion of  \nnon-perturbative models of power-suppressed hadronisation\neffects~\\cite{power}. \n\nThe precision of the strong coupling constant \ndetermined from event shape data has been limited up to now \nlargely by the scale\nuncertainty of the perturbative NLO calculation. We report here on the \nfirst calculation of NNLO corrections to event shape variables, and discuss \ntheir phenomenological impact.\n\n\n\\section{Event shape variables}\n\\label{sec:shapes}\n\nIn order to characterise hadronic final states in electron-positron\nannihilation, a variety of event shape variables have been proposed in \nthe literature, for a review see e.g.~\\cite{QCDbooks}. These variables can be categorised \ninto different classes, \naccording to the minimal number of final-state particles required for them \nto be non-vanishing: In the following we shall only consider three particle final states which are thus closely related to three-jet final states.\n\nAmong those shape variables,\nsix variables~\\cite{shapes}\n were studied in great detail: the thrust $T$, the\nnormalised heavy jet mass $\\rho$, \nthe wide and total jet\nbroadenings $B_W$ and $B_T$,  \nthe $C$-parameter and the transition from three-jet to \ntwo-jet final states in the Durham jet algorithm $Y_3$.\n\n\nThe perturbative expansion for the distribution of a \ngeneric observable $y$ up to NNLO at $e^+e^-$ centre-of-mass energy $\\sqrt{s}$, \nfor a renormalisation scale $\\mu^2$  is given by\n\\begin{eqnarray}\n\\frac{1}{\\sigma_{{\\rm had}}}\\, \\frac{\\hbox{d}\\sigma}{\\hbox{d} y} (s,\\mu^2,y) &=& \n\\left(\\frac{\\alpha_s{}(\\mu^2)}{2\\pi}\\right) \\frac{\\hbox{d} \\bar A}{\\hbox{d} y} +\n\\left(\\frac{\\alpha_s{}(\\mu^2)}{2\\pi}\\right)^2 \\left( \n\\frac{\\hbox{d} \\bar B}{\\hbox{d} y} + \\frac{\\hbox{d} \\bar A}{\\hbox{d} y} \\beta_0 \n\\log\\frac{\\mu^2}{s} \\right)\n\\nonumber \\\\ &&\n+ \\left(\\frac{\\alpha_s{}(\\mu^2)}{2\\pi}\\right)^3 \n\\bigg(\\frac{\\hbox{d} \\bar C}{\\hbox{d} y} + 2 \\frac{\\hbox{d} \\bar B}{\\hbox{d} y}\n \\beta_0\\log\\frac{\\mu^2}{s}\n\\nonumber \\\\ &&\n\\hspace{24mm} + \\frac{\\hbox{d} \\bar A}{\\hbox{d} y} \\left( \\beta_0^2\\,\\log^2\\frac{\\mu^2}{s}\n+ \\beta_1\\, \\log\\frac{\\mu^2}{s}   \\right)\\bigg)+ {\\cal O}(\\alpha_s{4})  \\;.\n\\label{eq:NNLOmu} \n\\end{eqnarray}\nThe dimensionless \nperturbative coefficients $\\bar A$, $\\bar B$ and $\\bar C$ depend only \non the event shape variable $y$. They are computed by a fixed-order \nparton-level calculation, which includes final states with three partons \nat LO, up to four partons at NLO and up to five partons at NNLO. \nLO and NLO corrections to event shapes have been available already for \na long time~\\cite{ERT,kunszt,event}. \n\n The calculation of the NNLO corrections is carried out using \na newly developed\nparton-level event generator programme {\\tt EERAD3} which contains \nthe relevant \nmatrix elements with up to five external partons~\\cite{3jme,muw2,V4p,tree5p}. \nBesides explicit infrared divergences from the loop integrals, the \nfour-parton and five-parton contributions yield infrared divergent \ncontributions if one or two of the final state partons become collinear or \nsoft. In order to extract these infrared divergences and combine them with \nthe virtual corrections, the antenna subtraction method~\\cite{ant} \nwas extended to NNLO level~\\cite{ourant} and implemented\nfor $e^+e^- \\to 3\\,\\mathrm{jets}$ and related event-shape variables~\\cite{eerad3}. The analytical cancellation of all \ninfrared divergences serves as a very strong check on the implementation. \n{\\tt EERAD3} yields the perturbative  $A$, $B$ and $C$ coefficients as \nhistograms for all infrared-safe event-shape variables \nrelated to three-particle \nfinal states at leading order. From these, \n $\\bar A$, $\\bar B$ and $\\bar C$ are computed by normalising to the total \nhadronic cross section.\nAs a cross check, the $A$ and $B$  coefficients have also been obtained from an independent integration~\\cite{event}\nof the NLO matrix elements~\\cite{ERT}, showing excellent agreement. \n\nFor small values of the event shape variable $y$, the fixed-order expansion, \neq.\\ (\\ref{eq:NNLOmu}), fails to converge, \nbecause the fixed-order coefficients are enhanced by powers of $\\hbox{ln}(1\/y)$.\nIn order to obtain reliable predictions\nin the region of $y \\ll 1$ it is necessary to resum entire sets of logarithmic terms at all orders in $\\alpha_s$. \nA detailed description of the predictions at next-to-leading-logarithmic approximation (NLLA) can\nbe found in Ref.\\ \\cite{as_theory-uncertainties}. \n\n\n\\section{NNLO results}\n\n\nThe precise size and shape of the NNLO corrections depend on the observable \nin question. Common to all observables is the divergent behaviour of \nthe fixed-order prediction in the two-jet limit, where soft-gluon effects \nat all orders become important, and where resummation is needed. For several \nevent shape variables \n (especially $T$ and $C$) the full kinematical range is not yet realised \nfor three partons, but attained only in the multi-jet limit. In this case,\nthe fixed-order description is also insufficient since it is  limited \nto a fixed multiplicity (five partons at NNLO). Consequently, the \nfixed-order description is expected to be reliable in a restricted \ninterval bounded by the two-jet limit on one side and the multi-jet \nlimit on the other side. \n\nIn this intermediate region, we observe that \ninclusion of  NNLO corrections (evaluated at the $Z$-boson mass, and \nfor fixed value of the strong coupling constant) typically increases \nthe previously available NLO prediction. \nThe magnitude of this increase differs considerably between \ndifferent observables\\cite{ourevent}, \nit is substantial for $T$ (18\\%), $B_T$ (17\\%)  and \n$C$ (15\\%), moderate for $\\rho$ and $B_W$ (both 10\\%) and small for \n$Y_3$ (6\\%). For all shape variables, we observe that the renormalisation\nscale uncertainty of the NNLO prediction is reduced by a factor 2 or more\ncompared to the NLO prediction. \nInclusion of the NNLO corrections modifies the shape of the event shape \ndistributions. We observe that \nthe NNLO prediction describes the shape of the measured event shape \ndistributions over a wider kinematical range than the NLO prediction, both \ntowards the two-jet and the multi-jet limit. To illustrate the \nimpact of the NNLO corrections, we compare the fixed-order predictions \nfor $Y_3$ to LEP2-data obtained by the ALPEH experiment in \nFigure~\\ref{fig:y23}, which illustrates especially the improvement\nin the approach to the two-jet region (large $-\\hbox{ln}(Y_3)$).   \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[angle=-90,width=10cm]{aleph.y23.ps}\n\\end{center}\n\\caption{\\small Perturbative fixed-order predictions \nfor the $Y_3$-distribution, compared to LEP2 data from ALEPH.} \n\\protect\\label{fig:y23}\n\\end{figure}\n\nThe information contained in the event shape distributions can be \nrestructured by computing individual moments. Moments of event shape \ndistributions have been studied \ntheoretically and experimentally in particular in view of \nunderstanding non-perturbative power corrections~\\cite{power}.\nConsequently, perturbative NNLO corrections will improve the discrimination \nbetween higher perturbative orders and genuine non-perturbative effects. \nFor the first moment $\\langle 1-T \\rangle$ of the thrust distribution, we find\nthe integrated coefficients\n\\begin{displaymath}\n{\\cal A} = 2.101\\,\\qquad {\\cal B} = 44.98\\,\\qquad \n{\\cal C} = 1095 \\pm 130\\;,\n\\end{displaymath}\nwhich yields for $\\sqrt{s}=\\mu=M_Z$:\n\\begin{displaymath}\n\\langle 1-T \\rangle(\\alpha_s(M_Z) = 0.1189) \n= 0.0398\\, ({\\rm LO})\\; +\\; 0.0146\\, ({\\rm NLO})  \\; \n+ \\; 0.0068 \\, ({\\rm NNLO})\\;.\n\\end{displaymath}\nWork on moments of the event shapes is ongoing.\n\n\\section{Determination of the strong coupling constant}\nUsing the newly computed NNLO corrections to event shape variables, we\nperformed\\cite{ouras} \na new extraction of $\\alpha_s$ from data on the standard set of \nsix event shape variables, measured \n by the ALEPH\\ collaboration \\cite{ALEPH-qcdpaper}\nat centre-of-mass energies of 91.2, 133, 161, 172, 183, 189, 200 and 206 GeV.\nThe combination of \nall NNLO determinations from all shape variables  yields \n\\begin{displaymath}\n    \\alpha_s(M_Z) = 0.1240 \\;\\pm\\; 0.0008\\,\\mathrm{(stat)}\n     \t\t\t\t\t \\;\\pm\\; 0.0010\\,\\mathrm{(exp)}\n                                   \\;\\pm\\; 0.0011\\,\\mathrm{(had)}\n                                   \\;\\pm\\; 0.0029\\,\\mathrm{(theo)} .\n \\end{displaymath}\nWe observe a clear improvement in the fit quality when going to\nNNLO accuracy. Compared to NLO the value of $\\alpha_s$ is lowered \nby about 10\\%, but still higher than for NLO+NLLA~\\cite{ALEPH-qcdpaper},\n which \nshows the obvious need for a matching of NNLO+NLLA for a fully reliable \nresult.  \n The scatter among the\n $\\alpha_s$-values extracted from different shape variables is \nlowered considerably, and the theoretical uncertainty is decreased by \na factor 2 (1.3) compared to NLO (NLO+NNLA). \n\nThese observations visibly illustrate the improvements gained from \nthe inclusion of the NNLO corrections, and highlight the need for \nfurther studies on the matching of NNLO+NLLA, and on the \nderivation of NNLLA resummation terms.\n\n\n\\section{Outlook}\nOur results for the NNLO corrections open up a whole \nnew range of possible \ncomparisons with the LEP data.\nThe potential of these studies is\nillustrated by the new determination of \n$\\alpha_s$ reported here, which can be \nfurther improved by the matching NLLA+NNLO, currently in progress. \nSimilarly, our results will also allow a renewed study of\npower corrections, now matched to NNLO. \n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNeural network models have recently contributed towards a great amount of progress in natural language processing. These models typically share a common backbone: recurrent neural networks (RNN), which have proven themselves to be capable of tackling a variety of core natural language processing tasks \\cite{hochreiter1997long,elman1990finding}.\nOne such task is language modeling, in which we estimate a probability distribution over sequences of tokens that corresponds to observed sentences (\\S\\ref{sec:background}). Neural language models, particularly models conditioned on a particular input, have many applications including in machine translation \\cite{bahdanau2016end}, abstractive summarization \\cite{chopra2016abstractive}, and speech processing \\cite{graves2013speech}. Similarly, state-of-the-art language models are almost universally based on RNNs, particularly long short-term memory (LSTM) networks \\cite{jozefowicz2016exploring,inan2016tying,merity2016pointer}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=.15,trim=1.2cm 0 0 0]{newlatticelm}\n    \\caption{Lattice decomposition of a sentence and its corresponding lattice language model probability calculation}\n    \\vspace{-2mm}\n    \\label{fig:latticelm}\n\\end{figure}\n\nWhile powerful, LSTM language models usually do not \\textit{explicitly} model many commonly-accepted linguistic phenomena. As a result, standard models lack linguistically informed inductive biases, potentially limiting their accuracy, particularly in low-data scenarios \\cite{adams2017,koehn}. In this work, we present a novel modification to the standard LSTM language modeling framework that allows us to incorporate some varieties of these linguistic intuitions seamlessly:\n\\textit{neural lattice language models} (\\S\\ref{sec:proposed}). Neural lattice language models define a lattice over possible paths through a sentence, and maximize the marginal probability over all paths that lead to generating the reference sentence, as shown in Fig. \\ref{fig:latticelm}. Depending on how we define these paths, we can incorporate different assumptions about how language should be modeled.\n\nIn the particular instantiations of neural lattice language models covered by this paper, we focus on two properties of language that could potentially be of use in language modeling: the existence of multi-word lexical units \\cite{zgusta1967multiword} (\\S\\ref{sec:multitoken}) and polysemy \\cite{ravin2000polysemy} (\\S\\ref{sec:polysemy}). Neural lattice language models allow the model to incorporate these aspects in an end-to-end fashion by simply adjusting the structure of the underlying lattices.\n\nWe run experiments to explore whether these modifications improve the performance of the model (\\S\\ref{sec:experiments}). Additionally, we provide qualitative visualizations of the model to attempt to understand what types of multi-token phrases and polysemous embeddings have been learned.\n\n\n\\section{Background}\n\\label{sec:background}\n\n\\subsection{Language Models}\n\nConsider a sequence $X$ for which we want to calculate its probability.\nAssume we have a vocabulary from which we can select a unique list of $|X|$ tokens $x_1,x_2,\\ldots,x_{|X|}$ such that $X = [x_1;x_2;\\ldots;x_{|X|}]$, i.e. the concatenation of the tokens (with an appropriate delimiter).\nThese tokens can be either on the character level \\cite{hwang2017character,DBLP:journals\/corr\/LingTDB15} or word level \\cite{inan2016tying,merity2016pointer}.\nUsing the chain rule, language models generally factorize $p(X)$ in the following way:\n\\begin{align}\n\\label{eq:regmarg}\np(X) &= p(x_1,x_2,\\ldots,x_{|X|}) \\nonumber \\\\\n     &= \\prod_{t=1}^{|X|}p(x_t\\mid x_1,x_2,\\ldots,x_{t-1})\n\\end{align}\n\nNote that this factorization is exact only in the case where the segmentation is unique.\nIn character-level models, it is easy to see that this property is maintained, because each token is unique and non-overlapping.\nIn word-level models, this also holds, because tokens are delimited by spaces, and no word contains a space.\n\n\\subsection{Recurrent Neural Networks}\n\nRecurrent neural networks have emerged as the state-of-the-art approach to approximating $p(X)$.\nIn particular, the LSTM cell \\cite{hochreiter1997long} is a specific RNN architecture which has been shown to be effective on many tasks, including language modeling \\cite{press2016using,jozefowicz2016exploring,merity2016pointer,inan2016tying}.%\n\\footnote{In this work, we utilize an LSTM with linked input and forget gates, as proposed by \\newcite{greff2016lstm}.}\nLSTM language models recursively calculate the hidden and cell states ($h_t$ and $c_t$ respectively) given the input embedding $e_{t-1}$ corresponding to token $x_{t-1}$:\n\\begin{align}\n\\label{eqn:lstm}\nh_t, c_t = \\text{LSTM}(h_{t-1},c_{t-1},e_{t-1},\\theta),\n\\end{align}\nthen calculate the probability of the next token given the hidden state, generally by performing an affine transform parameterized by $W$ and $b$, followed by a softmax:\n\\begin{align}\n\\label{eq:softmax}\np(x_t \\mid h_t) := \\text{softmax}(W * h_t + b).\n\\end{align}\n\n\n\n\n\\section{Neural Lattice Language Models}\n\n\\subsection{Language Models with Ambiguous Segmentations}\n\\label{sec:proposed}\n\nTo reiterate, the standard formulation of language modeling in the previous section requires splitting sentence $X$ into a unique set of tokens $x_1,\\ldots,x_{|X|}$.\nOur proposed method generalizes the previous formulation to remove the requirement of uniqueness of segmentation, similar to that used in non-neural $n$-gram language models such as \\newcite{dupont1997lattice} and \\newcite{goldwater2007distributional}.\n\nFirst, we define some terminology.\nWe use the term ``token'', designated by $x_i$, to describe any indivisible item in our vocabulary that has no other vocabulary item as its constituent part.\nWe use the term ``chunk'', designated by $k_i$ or $x_i^j$, to describe a sequence of one or more tokens that represents a portion of the full string $X$, containing the unit tokens $x_i$ through $x_j$: $x_i^j = [x_i,x_{i+1};\\ldots;x_j]$.\nWe also refer to the ``token vocabulary'', which is the subset of the vocabulary containing only tokens, and to the ``chunk vocabulary'', which similarly contains all chunks.\n\nNote that we can factorize the probability of any sequence of chunks $K$ using the chain rule, in precisely the same way as sequences of tokens:\n\\begin{align}\n\\label{eq:regmarg}\np(K) &= p(k_1,k_2,\\ldots,k_{|K|}) \\nonumber \\\\\n     &= \\prod_{t=1}^{|K|}p(k_t\\mid k_1,k_2,\\ldots,k_{t-1})\n\\end{align}\n\nWe can factorize the overall probability of a token list $X$ in terms of its chunks by using the chain rule, and marginalizing over all segmentations. \nFor any particular token list $X$, we define a set of valid segmentations $\\mathcal{S}(X)$, such that for every sequence $s \\in \\mathcal{S}(X)$, $X = [x_{s_0}^{s_1-1};x_{s_1}^{s_2-1};\\ldots;x_{s_{|s|-1}}^{s_{|s|}}]$.\nThe factorization is:\n\\small\n\\begin{align}\n\\label{eq:latmarg}\np(X) &= \\sum_S p(X, S) = \\sum_S p(X|S) p(S) = \\sum_{S \\in \\mathcal{S}(X)} p(S) \\nonumber \\\\\n     &= \\sum_{S \\in \\mathcal{S}(X)}\\prod_{t=1}^{|S|}p(x_{s_{t-1}}^{s_t-1}\\mid x_{s_0}^{s_1-1},x_{s_1}^{s_2-1},\\ldots,x_{s_{t-2}}^{s_{t-1}-1})\n\\end{align}\n\\normalsize\n\nNote that, by definition, there exists a unique segmentation of $X$ such that $x_1,x_2,\\ldots$ are all tokens, in which case $|S|=|X|$.\nWhen only that one unique segmentation is allowed per $X$, $\\mathcal{S}$ contains only that one element, so summation drops out, and therefore for standard character-level and word-level models, Eq.~(\\ref{eq:latmarg}) reduces to Eq.~(\\ref{eq:regmarg}), as desired. \nHowever, for models that license multiple segmentations per $X$, computing this marginalization directly is generally intractable.\nFor example, consider segmenting a sentence using a vocabulary containing all words and all 2-word expressions.\nThe size of $\\mathcal{S}$ would grow exponentially with the number of words in $X$, meaning we would have to marginalize over trillions of unique segmentations for even modestly-sized sentences.\n\n\\subsection{Lattice Language Models}\n\n\nTo avoid this, it is possible to re-organize the computations in a lattice, which allows us to dramatically reduce the number of computations required \\cite{dupont1997lattice,neubig2010learning}.\n\nAll segmentations of $X$ can be expressed as the edges of paths through a lattice over token-level prefixes of $X$: $x_{<1}, x_{<2}, \\ldots, X$. The infimum is the empty prefix $x_{<1}$; the supremum is $X$; an edge from prefix $x_{<i}$ to prefix $x_{<j}$ exists if and only if there exists a chunk $x_i^j$ in our chunk vocabulary such that $[x_{<i};x_i^j] = x_{<j}$.\nEach path through the lattice from $x_{<1}$ to $X$ is a segmentation of $X$ into the list of tokens on the traversed edges, as seen in Fig. \\ref{fig:latticelm}.\n\nThe probability of a specific prefix $p(x_{<j})$ is calculated by marginalizing over all segmentations leading up to $x_{j-1}$\n\\begin{equation}\n  p(x_{<j}) = \\sum_{S \\in \\mathcal{S}(x_{<j})} \\prod_{t=1}^{|S|} p(x_{s_{t-1}}^{s_t-1} \\mid x_{<s_{t-1}}),\n\\end{equation}\nwhere by definition $s_{|S|}=j$.\nThe key insight here that allows us to calculate this efficiently is that this is a recursive formula and that instead of marginalizing over all segmentations, we can marginalize over immediate predecessor edges in the lattice, $A_j$. Each item in $A_j$ is a location $i$ ($=s_{t-1}$), which indicates that the edge between prefix $x_{<i}$ and prefix $x_{<j}$, corresponding to token $x_i^j$, exists in the lattice. We can thus calculate $p(x_{<j})$ as\n\\begin{align}\n\\label{eq:neuralsum}\np(x_{<j}) = \\sum_{i \\in A_j}p(x_{<i})p(x_i^j \\mid x_{<i}).\n\\end{align}\n\nSince $X$ is the supremum prefix node, we can use this formula to calculate $p(X)$ by setting $j = |X|$. In order to do this, we need to calculate the probability of each of its $|X|$ predecessors. Each of those takes up to $|X|$ calculations, meaning that the computation for $p(X)$ can be done in O($|X|^2$) time. If we can guarantee that each node will have a maximum number of incoming edges $D$ so that $|A_j| \\leq D$ for all $j$, then this bound can be reduced to O($D|X|$) time.%\n\\footnote{Thus, the standard token-level language model where $D=1$ takes $O(|X|)$ computations.}\n\nThe proposed technique is completely agnostic to the shape of the lattice, and Fig. \\ref{fig:lattice} illustrates several potential varieties of lattices. Depending on how the lattice is constructed, this approach can be useful in a variety of different contexts, two of which we discuss in \\S\\ref{sec:instantiations}. \n\n\\subsection{Neural Lattice Language Models}\n\\label{sec:hs}\n\nThere is still one missing piece in our attempt to apply neural language models to lattices.\nWithin our overall probability in Eq.~(\\ref{eq:neuralsum}), we must calculate the probability $p(x_i^j \\mid x_{<i})$ of the next segment given the history.\nHowever, given that there are potentially an exponential number of paths through the lattice leading to $x_i$, this is not as straightforward as in the case where only one segmentation is possible.\nPrevious work on lattice-based language models \\cite{neubig2010learning,dupont1997lattice} utilized count-based $n$-gram models, which depend on only a limited historical context at each step making it possible to compute the marginal probabilities in an exact and efficient manner through dynamic programming.\nOn the other hand, recurrent neural models depend on the entire context, causing them to lack this ability. Our primary technical contribution is therefore to describe several techniques for incorporating lattices into a neural framework with infinite context, by providing ways to approximate the hidden state of the recurrent neural net.\n\n\n\\begin{figure}\n    \\centering\n        \\begin{tikzpicture}[->,auto,node distance=1cm,\n          thick,main node\/.style={circle,draw,font=\\sffamily\\Large\\bfseries}]\n        \n          \\node[main node] (1) {};\n          \\node[main node] (2) [right of=1, right=.3cm] {};\n          \\node[main node] (3) [right of=2, right=.3cm] {};\n          \\node[main node] (4) [right of=3, right=.3cm] {};\n          \\node[main node] (5) [right of=4, right=.3cm] {};\n        \n          \\path[every node\/.style={font=\\sffamily\\small}]\n            (1) edge node [below=.1cm] {the} (2)\n            (2) edge node [below=.1cm] {dog} (3)\n            (3) edge node [below=.1cm] {barked} (4)\n            (4) edge node [below=.25cm] {.} (5);\n\n          \\node[main node] (11) [below of=1, below=1cm] {};\n          \\node[main node] (12) [below of=2, below=1cm] {};\n          \\node[main node] (13) [below of=3, below=1cm] {};\n          \\node[main node] (14) [below of=4, below=1cm] {};\n          \\node[main node] (15) [below of=5, below=1cm] {};\n        \n          \\path[every node\/.style={font=\\sffamily\\small}]\n            (11) edge node [below=.1cm] {the} (12)\n            (12) edge node [below=.1cm] {dog} (13)\n            (13) edge node [below=.1cm] {barked} (14)\n            (14) edge node [below=.25cm] {.} (15)\n            (11) edge[bend left] node [above] {the\\_dog} (13)\n            (12) edge[bend left] node [above] {dog\\_barked\\_.} (15);\n\n          \\node[main node] (21) [below of=11, below=1cm] {};\n          \\node[main node] (22) [below of=12, below=1cm] {};\n          \\node[main node] (23) [below of=13, below=1cm] {};\n          \\node[main node] (24) [below of=14, below=1cm] {};\n          \\node[main node] (25) [below of=15, below=1cm] {};\n        \n          \\path[every node\/.style={font=\\sffamily\\small}]\n            (21) edge node [below=.1cm] {the} (22)\n            (22) edge node [below=.1cm] {dog} (23)\n            (23) edge node [below=.1cm] {barked} (24)\n            (24) edge node [below=.25cm] {.} (25)\n            (21) edge[bend left] node [above] {the\\_dog} (23)\n            (22) edge[bend left] node [above=.3cm] {dog\\_barked} (24)\n            (23) edge[bend left] node [above] {barked\\_.} (25);\n            \n            \n          \\node[main node] (31) [below of=21, below=.7cm] {};\n          \\node[main node] (32) [below of=22, below=.7cm] {};\n          \\node[main node] (33) [below of=23, below=.7cm] {};\n          \\node[main node] (34) [below of=24, below=.7cm] {};\n          \\node[main node] (35) [below of=25, below=.7cm] {};\n        \n          \\path[every node\/.style={font=\\sffamily\\small}]\n            (31) edge node [above=.1cm] {the} (32)\n            (32) edge[bend left] node [above=.1cm] {dog$_1$} (33)\n            (33) edge[bend left] node [above=.1cm] {barked$_1$} (34)\n            (34) edge node [above=.15cm] {.} (35)\n            (32) edge[bend right] node [below=.1cm] {dog$_2$} (33)\n            (33) edge[bend right] node [below=.1cm] {barked$_2$} (34);\n            \n        \\node (0) [left of=1] {(a)};\n        \\node (10) [left of=11] {(b)};\n        \\node (20) [left of=21] {(c)};\n        \\node (30) [left of=31] {(d)};\n        \n        \\end{tikzpicture}\n    \\caption{Example of (a) a single-path lattice, (b) a sparse lattice, (c) a dense lattice with $D = 2$, and (d) a multilattice with $D = 2$, for sentence ``the dog barked .''}\n    \\label{fig:lattice}\n\\end{figure}\n\n\n\\subsubsection{Direct Approximation}\n\\label{subsec:da}\n\nOne approach to approximating the hidden state is the TreeLSTM framework described by \\newcite{tai2015improved}.%\n\\footnote{This framework has been used before for calculating neural sentence representations involving lattices by \\newcite{DBLP:journals\/corr\/SuTXL16} and \\newcite{sperber2017neural}, but not for the language models that are the target of this paper.}\nIn the TreeLSTM formulation, new states are derived from multiple predecessors by simply summing the individual hidden and cell state vectors of each of them.\nFor each predecessor location $i \\in A_j$, we first calculate the local hidden state $\\tilde{h}$ and local cell state $\\tilde{c}$ by combining the embedding $e_i^j$ with the hidden state of the LSTM at $x_{<i}$ using the standard LSTM update function as in Eq.~(\\ref{eqn:lstm}):\n\\begin{align*}\n\\tilde{h}_i, \\tilde{c}_i = \\text{LSTM}(h_i,c_i,e_i^j,\\theta) \\text{ for } i \\in A_j\n\\end{align*}\n\nWe then sum the local hidden and cell states:\n\\begin{align*}\n\\begin{split}\nh_j = \\sum_{i \\in A_j} \\tilde{h}_i\n\\end{split}\n\\begin{split}\nc_j = \\sum_{i \\in A_j} \\tilde{c}_i\n\\end{split}\n\\end{align*}\n\nThis formulation is powerful, but comes at the cost of sacrificing the probabilistic interpretation of which paths are likely.\nTherefore, even if almost all of the probability mass comes through the ``true'' segmentation, the hidden state may still be heavily influenced by all of the ``bad'' segmentations as well.\n\n\\subsubsection{Monte-Carlo Approximation}\n\\label{subsec:af}\n\nAnother approximation that has been proposed is to sample one predecessor state from all possible predecessors, as seen in \\newcite{chan2016latent}. We can calculate the total probability that we reach some prefix $x_{<j}$, and we know how much of this probability comes from each of its predecessors in the lattice, so we can construct a probability distribution over predecessors in the lattice:\n\\begin{align}\n\\label{eqn:preddist}\nM(x_{<i} \\mid \\theta) = \\frac{p(x_{<i} \\mid \\theta)p(x_i^j\\mid x_{<i}; \\theta)}{p(x_{<j} \\mid \\theta)}\n\\end{align}\n\nTherefore, one way to update the LSTM is to sample one predecessor $x_{<i}$ from the distribution $M$ and simply set $h_j = \\tilde{h}_i$ and $c_j = \\tilde{c}_i$. However, sampling is unstable and difficult to train: we found that the model tended to over-sample short tokens early on during training, and thus segmented every sentence into unigrams. This is similar to the outcome reported by \\newcite{chan2016latent}, who accounted for it by incorporating an $\\epsilon$ encouraging exploration.\n\n\\subsubsection{Marginal Approximation}\n\\label{subsec:we}\n\nIn another approach, which allows us to incorporate information from all predecessors while maintaining a probabilistic interpretation, we can utilize the probability distribution $M$ to instead calculate the expected value of the hidden state:\n\\begin{align*}\nh_j = \\bm{E}_{x_{<i}\\sim M} [\\tilde{h_i}] = \\sum_{i \\in A_j} M(x_{<i} \\mid \\theta)\\tilde{h}_i \\\\\nc_j = \\bm{E}_{x_{<i}\\sim M} [\\tilde{c_i}] = \\sum_{i \\in A_j} M(x_{<i} \\mid \\theta)\\tilde{c}_i\n\\end{align*}\n\n\n\\subsubsection{Gumbel-Softmax Interpolation}\n\\label{subsec:gs}\n\nThe Gumbel-Softmax trick, or concrete distribution, described by \\newcite{jang2016categorical} and \\newcite{maddison2016concrete}, is a technique for incorporating discrete choices into differentiable neural computations. In this case, we can use it to select a predecessor.\nThe Gumbel-Softmax trick works by taking advantage of the fact that adding Gumbel noise to the pre-softmax predecessor scores and then taking the argmax is equivalent to sampling from the probability distribution.\nBy replacing the argmax with a softmax function scaled by a temperature $\\tau$, we can get this pseudo-sampled distribution through a fully differentiable computation:\n\\small\n\\begin{align*}\nN(x_{<i} \\mid \\theta) = \\frac{\\text{exp}((\\text{log}(M(x_{<i} \\mid \\theta)) + g_i)\/\\tau)}{\\sum_{k \\in A_j} \\text{exp}((\\text{log}(M(x_{<k} \\mid \\theta)) + g_k)\/\\tau)}\n\\end{align*}\n\\normalsize\n\nThis new distribution can then be used to calculate the hidden state by taking a weighted average of the states of possible predecessors:\n\\begin{align*}\n\\begin{split}\nh_j = \\sum_{i \\in A_j}^{j-1} N(x_{<i} \\mid \\theta)\\tilde{h}_i\n\\end{split}\n\\begin{split}\nc_j = \\sum_{i=j-L}^{j-1} N(x_{<i} \\mid \\theta)\\tilde{c}_i\n\\end{split}\n\\end{align*}\n\nWhen $\\tau$ is large, the values of $N(x_{<i} \\mid \\theta)$ are flattened out; therefore, all the predecessor hidden states are summed with approximately equal weight, equivalent to the direct approximation (\\S\\ref{subsec:da}). On the other hand, when $\\tau$ is small, the output distribution becomes extremely peaky, and one predecessor receives almost all of the weight. Each predecessor $x_{<i}$ has a chance of being selected equal to $M(x_{<i} \\mid \\theta)$, which makes it identical to ancestral sampling (\\S\\ref{subsec:af}). By slowly annealing the value of $\\tau$, we can smoothly interpolate between these two approaches, and end up with a probabilistic interpretation that avoids the instability of pure sampling-based approaches.\n\n\\section{Instantiations of Neural Lattice LMs}\n\\label{sec:instantiations}\n\nIn this section, we introduce two instantiations of neural lattice languages models aiming to capture features of language: the existence of coherent multi-token chunks, and the existence of polysemy.\n\n\\subsection{Incorporating Multi-Token Phrases}\n\\label{sec:multitoken}\n\n\\subsubsection{Motivation}\n\n\n\n\nNatural language phrases often demonstrate significant non-compositionality: for example, in English, the phrase ``rock and roll'' is a genre of music, but this meaning is not obtained by viewing the words in isolation. In word-level language modeling, the network is given each of these words as input, one at a time; this means it must capture the idiomaticity in its hidden states, which is quite roundabout and potentially a waste of the limited parameters in a neural network model. A straightforward solution is to have an embedding for the entire multi-token phrase, and use this to input the entire phrase to the LSTM in a single timestep. However, it is also important that the model is able to decide whether the non-compositional representation is appropriate given the context: sometimes, ``rock'' is just a rock.\n\nAdditionally, by predicting multiple tokens in a single timestep, we are able to decrease the number of timesteps across which the gradient must travel, making it easier for information to be propagated across the sentence.\nThis is even more useful in non-space-delimited languages such as Chinese, in which segmentation is non-trivial, but character-level modeling leads to many sentences being hundreds of tokens long.\n\nThere is also psycho-linguistic evidence which supports the fact that humans incorporate multi-token phrases into their mental lexicon. \\newcite{siyanova2011adding} show that native speakers of a language have significantly reduced response time when processing idiomatic phrases, whether they are used in an idiomatic sense or not, while \\newcite{bannard2008stored} show that children learning a language are better at speaking common phrases than uncommon ones.\nThis evidence lends credence to the idea that multi-token lexical units are a useful tool for language modeling in humans, and so may also be useful in computational models.\n\n\\subsubsection{Modeling Strategy}\n\nThe underlying lattices utilized in our multi-token phrase experiments are ``dense'' lattices: lattices where every edge (below a certain length $L$) is present (Fig. \\ref{fig:lattice}, c).\nThis is for two reasons.\nFirst, since every sequence of tokens is given an opportunity to be included in the path, all segmentations are candidates, which will potentially allow us to discover arbitrary types of segmentations without a prejudice towards a particular theory of which multi-token units we should be using.\nSecond, using a dense lattice makes minibatching very straightforward by ensuring that the computation graphs for each sentence are identical. If the lattices were not dense, the lattices of various sentences in a minibatch could be different; it then becomes necessary to either calculate a differently-shaped graph for every sentence, preventing minibatching and hurting training efficiency, or calculate and then mask out the missing edges, leading to wasted computation. Since only edges of length $L$ or less are present, the maximum in-degree of any node in the lattice $D$ is no greater than $L$, giving us the time bound O($L|X|$).\n\n\\subsubsection{Token Vocabularies}\n\nStoring an embedding for every possible multi-token chunk would require $|V|^L$ unique embeddings, which is intractable. Therefore, we construct our multi-token embeddings by merging compositional and non-compositional representations.\n\n\\paragraph{Non-compositional Representation}\n\nWe first establish a priori a set of ``core'' chunk-level tokens that each have a dense embedding. In order to guarantee full coverage of sentences, we first add every unit-level token to this vocabulary, e.g. every word in the corpus for a word-level model. Following this, we also add the most frequent n-grams (where $1 < n \\leq L$). This ensures that the vast majority of sentences will have several longer chunks appear within them, and so will be able to take advantage of tokens at larger granularities.\n\n\\paragraph{Compositional Representation}\n\n\nHowever, the non-compositional embeddings above only account for a subset of all $n$-grams, so we additionally construct compositional embeddings for each chunk by running a BiLSTM encoder over the individual embeddings of each unit-level token within it \\cite{dyer2016recurrent}. In this way, we can create a unique embedding for every sequence of unit-level tokens.\n\nWe use this composition function on chunks regardless of whether they are assigned non-compositional embeddings or not, as even high-frequency chunks may display compositional properties.\nThus, for every chunk, we compute the chunk embedding vector $x_i^j$ by concatenating the compositional embedding with the non-compositional embedding if it exists, or otherwise with an $\\textless$UNK$\\textgreater$ embedding.\n\n\\paragraph{Sentinel Mixture Model for Predictions}\n\nAt each timestep, we want to use our LSTM hidden state $h_t$ to assign some probability mass to every chunk with length less than $L$.\nTo do this, we follow \\newcite{merity2016pointer} in creating a new ``sentinel'' token $\\textit{\\textless s\\textgreater}$ and adding it to our vocabulary. At each timestep, we first use our neural network to calculate a score for each chunk $C$ in our vocabulary, including the sentinel token. We do a softmax across these scores to assign a probability $p_{\\textit{main}}(C_{t+1} \\mid h_t; \\theta)$ to every chunk in our vocabulary, and also to $\\textit{\\textless s\\textgreater}$. For token sequences not represented in our chunk vocabulary, this probability $p_{\\textit{main}}(C_{t+1} \\mid h_t; \\theta) = 0$.\n\nNext, the probability mass assigned to the sentinel value, $p_{\\textit{main}}(\\textit{\\textless s\\textgreater} \\mid h_t; \\theta)$, is distributed across all possible tokens sequences of length less than $L$, using another LSTM with parameters $\\theta_{\\textit{sub}}$. Similar to \\newcite{jozefowicz2016exploring}, this sub-LSTM is initialized by passing in the hidden state of the main lattice LSTM at that timestep. This gives us a probability for each sequence $p_{\\textit{sub}}(c_1,c_2,\\ldots,c_L \\mid h_t; \\theta_\\textit{sub})$.\n\nThe final formula for calculating the probability mass assigned to a specific chunk $C$ is:\n\\begin{align*}\np(C \\mid h_t; \\theta) = &p_{\\textit{main}}(C \\mid h_t; \\theta) + \\\\\n                        &p_{\\textit{main}}(\\textit{\\textless s\\textgreater} \\mid h_t; \\theta) p_{\\textit{sub}}(C \\mid h_t; \\theta_\\textit{sub}) \\nonumber\n\\end{align*}\n\n\\subsection{Incorporating Polysemous Tokens}\n\\label{sec:polysemy}\n\n\\subsubsection{Motivation}\n\nA second shortcoming of current language modeling approaches is that each word is associated with only one embedding.\nFor highly polysemous words, a single embedding may be unable to represent all meanings effectively.\n\nThere has been past work in word embeddings which has shown that using multiple embeddings for each word is helpful in constructing a useful representation. \\newcite{athiwaratkun2017multimodal} represented each word with a multimodal Gaussian distribution and demonstrated that embeddings of this form were able to outperform more standard skip-gram embeddings on word similarity and entailment tasks. Similarly, \\newcite{DBLP:journals\/corr\/ChenQJH15} incorporate standard skip-gram training into a Gaussian mixture framework and show that this improves performance on several word similarity benchmarks.\n\nWhen a polysemous word is represented using only a single embedding in a language modeling task, the multimodal nature of the true embedding distribution may causes the resulting embedding to be both high-variance and skewed from the positions of each of the true modes. Thus, it is likely useful to represent each token with multiple embeddings when doing language modeling.\n\n\\subsubsection{Modeling Strategy}\n\nFor our polysemy experiments, the underlying lattices are ``multilattices'': lattices which are also multigraphs, and can have any number of edges between any given pair of nodes (Fig. \\ref{fig:lattice}, d). Lattices set up in this manner allow us to incorporate multiple embeddings for each word. Within a single sentence, any pair of nodes corresponds to the start and end of a particular subsequence of the full sentence, and is thus associated with a specific token; each edge between them is a unique embedding for that token.\nWhile many strategies for choosing the number of embeddings exist in the literature \\cite{neelakantan2014efficient}, in this work, we choose a number of embeddings $E$ and assign that many embeddings to each word. This ensures that the maximum in-degree of any node in the lattice $D$, is no greater than $E$, giving us the time bound O($E|X|$).\n\nIn this work, we do not explore models that include both chunk vocabularies and multiple embeddings. However, combining these two techniques, as well as exploring other, more complex lattice structures, is an interesting avenue for future work.\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Data}\n\nWe perform experiments on two languages: English and Chinese, which provide an interesting contrast in linguistic features.%\n\\footnote{Code to reproduce datasets and experiments is available at: \\url{http:\/\/github.com\/jbuckman\/neural-lattice-language-models}}\n\nIn English, the most common benchmark for language modeling recently is the Penn Treebank, specifically the version preprocessed by \\newcite{mikolovptb}.\nHowever, this corpus is limited by being relatively small, only containing approximately 45,000 sentences, which we found to be insufficient to effectively train lattice language models.%\n\\footnote{Experiments using multi-word units resulted in overfitting, regardless of normalization and hyperparameter settings.}\nThus, we instead used the Billion Word Corpus \\cite{chelba2013one}.\nPast experiments on the BWC typically modeled every word without restricting the vocabulary, which results in a number of challenges regarding the modeling of open vocabularies that are orthogonal to this work. Thus, we create a preprocessed version of the data in the same manner as Mikolov, lowercasing the words, replacing numbers with $<$N$>$ tokens, and $<$UNK$>$ing all words beyond the ten thousand most common. Additionally, we restricted the data set to only include sentences of length 50 or less, ensuring that large minibatches could fit in GPU memory. Our subsampled English corpus contained 29,869,166 sentences, of which 29,276,669 were used for training, 5,000 for validation, and 587,497 for testing. To validate that our methods scale up to larger language modeling scenarios, we also report a smaller set of large-scale experiments on the full billion word benchmark in Appendix A.\n\nIn Chinese, we ran experiments on a subset of the Chinese GigaWord corpus.\nChinese is also particularly interesting because unlike English, it does not use spaces to delimit words, so segmentation is non-trivial.\nTherefore, we used a character-level language model for the baseline, and our lattice was composed of multi-character chunks. We used sentences from \\textit{Guangming Daily}, again $<$UNK$>$ing all but the 10,000 most common tokens and restricting the selected sentences to only include sentences of length 150 or less. Our subsampled Chinese corpus included 934,101 sentences for training, 5,000 for validation, and 30,547 for testing.\n\n\n\\subsection{Main Experiments}\n\nWe compare a baseline LSTM model, dense lattices of size 1, 2, and 3, and a multilattice with 2 and 3 embeddings per word. \n\nThe implementation of our networks was done in DyNet \\cite{neubig2017dynet}.%\n All LSTMs had 2 layers, each with hidden dimension of 200. Variational dropout \\cite{gal2016theoretically} of .2 was used on the Chinese experiments, but hurt performance on the English data, so it was not used. The 10,000 word embeddings each had dimension 256. For lattice models, chunk vocabularies were selected by taking the 10,000 words in the vocabulary and adding the most common 10,000 $n$-grams with $1 < n \\leq L$. The weights on the final layer of the network were tied with the input embeddings, as done by \\cite{press2016using,inan2016tying}. In all lattice models, hidden states were computed using weighted expectation (\\S\\ref{subsec:we}) unless mentioned otherwise. In multi-embedding models, embedding sizes were decreased so as to maintain the same total number of parameters. All models were trained using the Adam optimizer with a learning rate of .01 on a NVIDIA K80 GPU.\nThe results can be seen in Table \\ref{lm-results-en} and Table \\ref{lm-results-zh}.\n\nIn the multi-token phrase experiments, many additional parameters are accrued by the BiLSTM encoder and sub-LSTM predictive model, making them not strictly comparable to the baseline. To account for this, we include results for $L=1$, which, like the baseline LSTM approach, fails to leverage multi-token phrases, but includes the same number of parameters as $L=2$ and $L=3$.\n\nIn both the English and Chinese experiments, we see the same trend: increasing the maximum lattice size decreases the perplexity, and for $L=2$ and above, the neural lattice language model outperforms the baseline. Similarly, increasing the number of embeddings per word decreases the perplexity, and for $E=2$ and above, the multiple-embedding model outperforms the baseline.\n\n\\begin{table}[]\n\\centering\n\\caption{Results on English language modeling task}\n\\label{lm-results-en}\n\\begin{tabular}{|c|c|c|}\n\\hline\nModel             & Valid. Perp.     & Test Perp.  \\\\ \\hline \\hline\nBaseline          &  47.64           & 48.62       \\\\ \\hline \\hline\nMulti-Token ($L=1$)   &  45.69           & 47.21       \\\\ \\hline\nMulti-Token ($L=2$)   &  44.15           & 46.12       \\\\ \\hline\nMulti-Token ($L=3$)   &  45.19           & 46.84       \\\\ \\hline \\hline\nMulti-Emb ($E=2$)        &  44.80           & 46.32       \\\\ \\hline\nMulti-Emb ($E=3$)        &  \\textbf{42.76}  &  \\textbf{43.78}    \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\centering\n\\caption{Results on Chinese language modeling task}\n\\label{lm-results-zh}\n\\begin{tabular}{|c|c|c|}\n\\hline\nModel             & Valid. Perp.      & Test Perp.  \\\\ \\hline \\hline\nBaseline          &  41.46            & 40.72       \\\\ \\hline \\hline\nMulti-Token ($L=1$)   &  49.86            & 50.99       \\\\ \\hline\nMulti-Token ($L=2$)   &  38.61            & 37.22       \\\\ \\hline\nMulti-Token ($L=3$)   &  \\textbf{33.01}   & \\textbf{32.19}       \\\\ \\hline \\hline\nMulti-Emb ($E=2$) &  40.30            & 39.28       \\\\ \\hline\nMulti-Emb ($E=3$) &  45.72            & 44.40       \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Hidden State Calculation Experiments}\n\nWe compare the various hidden-state calculation approaches discussed in Section \\ref{sec:hs} on the English data using a lattice of size $L=2$ and dropout of .2. These results can be seen in Table \\ref{hs-results}.\n\nFor all hidden state calculation techniques, the neural lattice language models outperform the LSTM baseline. The ancestral sampling technique used by \\newcite{chan2016latent} is worse than the others, which we found to be due to it getting stuck in a local minimum which represents almost everything as unigrams. There is only a small difference between the perplexities of the other techniques.\n\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[scale=.5]{segmentations}\n    \\caption{Segmentation of three sentences randomly sampled from the test corpus, using $L=2$. Green numbers show probability assigned to token sizes. For example, the first three words in the first sentence have a 59\\% and 41\\% chance of being ``please let me'' or ``please let\\_me'' respectively. Boxes around words show greedy segmentation.}\n    \\label{fig:segmentations}\n\\end{figure*}\n\n\\begin{table}[]\n\\centering\n\\caption{Hidden state calculation comparison results}\n\\label{hs-results}\n\\small\n\\begin{tabular}{|c|c|c|}\n\\hline\nModel                & Valid. Perp.           & Test Perp.       \\\\ \\hline \\hline\nBaseline              &  64.18                & 60.67           \\\\ \\hline\nDirect (\\S\\ref{subsec:da}) &  59.74                & 55.98           \\\\ \\hline\nMonte Carlo (\\S\\ref{subsec:af}) &  62.97                & 59.08           \\\\ \\hline\nMarginalization (\\S\\ref{subsec:we}) &  \\textbf{58.62}                & \\textbf{55.06}           \\\\ \\hline\nGS Interpolation (\\S\\ref{subsec:gs}) &  59.19                & 55.73           \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Discussion and Analysis}\n\\label{sec:discussion}\n\nNeural lattice language models convincingly outperform an LSTM baseline on the task of language modeling.\nOne interesting note is that in English, which is already tokenized into words and highly polysemous, utilizing multiple embeddings per word is more effective than including multi-word tokens.\nIn contrast, in the experiments on the Chinese data, increasing the lattice size of the multi-character tokens is more important than increasing the number of embeddings per character. \nThis corresponds to our intuition; since Chinese is not tokenized to begin with, utilizing models that incorporate segmentation and compositionality of elementary units is very important for effective language modeling.\n\n\\begin{table*}[t!]\n\\centering\n\\caption{Comparison of randomly-selected contexts of several words selected from the vocabulary of the Billion Word Corpus, in which the model preferred one embedding over the other.}\n\\label{multi-viz}\n\\begin{tabular}{|l||l|}\n\n\\hline\n\\textbf{rock$_1$} & \\textbf{rock$_2$} \\\\ \\hline\n...at the $<$unk$>$ pop , \\textit{rock} and jazz... & ...including hsbc , northern \\textit{rock} and...                       \\\\ \\hline\n...a little bit $<$unk$>$ \\textit{rock} ,...        & ...pakistan has a $<$unk$>$ \\textit{rock} music scene...          \\\\ \\hline\n...on light \\textit{rock} and $<$unk$>$ stations... & ...spokesman for round \\textit{rock} , $<$unk$>$... \\\\ \\hline \\hline\n\n\\textbf{bank$_1$} & \\textbf{bank$_2$} \\\\ \\hline\n...being a \\textit{bank} holiday in... & ...the \\textit{bank} of england has...                       \\\\ \\hline\n...all the us \\textit{bank} runs and...        & ...with the royal \\textit{bank} of scotland...          \\\\ \\hline\n...by getting the \\textit{bank} 's interests... & ...development \\textit{bank} of japan and the... \\\\ \\hline \\hline\n\n\\textbf{page$_1$} & \\textbf{page$_2$} \\\\ \\hline\n...on \\textit{page} $<$unk$>$ of the... & ...was it front \\textit{page} news...                       \\\\ \\hline\n...a source told \\textit{page} six ....        & ...himself , tony \\textit{page} , the former ...          \\\\ \\hline\n...on \\textit{page} $<$unk$>$ of the... & ...sections of the \\textit{page} that discuss... \\\\ \\hline \\hline\n\n\\textbf{profile$_1$} & \\textbf{profile$_2$} \\\\ \\hline\n...( $<$unk$>$ : quote , \\textit{profile} , research )... & ...so $<$unk$>$ the \\textit{profile} of the city...                       \\\\ \\hline\n...( $<$unk$>$ : quote , \\textit{profile} , research )...        & ...the highest \\textit{profile} $<$unk$>$ held by...          \\\\ \\hline\n...( $<$unk$>$ : quote , \\textit{profile} , research )... & ...from high i , elite schools ,... \\\\ \\hline \\hline\n\n\\textbf{edition$_1$} & \\textbf{edition$_2$} \\\\ \\hline\n... of the second \\textit{edition} of windows... & ...of the new york \\textit{edition} . ...                       \\\\ \\hline\n... this month 's \\textit{edition} of$<$unk$>$ , the ...        & ...of the new york \\textit{edition} . ...          \\\\ \\hline\n...forthcoming d.c. \\textit{edition} of the hit... & ...of the new york \\textit{edition} . ... \\\\ \\hline \\hline\n\n\\textbf{rodham$_1$} & \\textbf{rodham$_2$} \\\\ \\hline\n...senators hillary \\textit{rodham} clinton and... &                     \\\\ \\hline\n...making hillary \\textit{rodham} clinton his...        &       \\\\ \\hline\n...hillary \\textit{rodham} clinton 's campaign has... &   \\\\ \\hline\n\n\\end{tabular}\n\\end{table*}\n\nTo calculate the probability of a sentence, the neural lattice language model implicitly marginalizes across latent segmentations. By inspecting the probabilities assigned to various edges of the lattice, we can visualize these segmentations, as is done in Fig. \\ref{fig:segmentations}. The model successfully identifies bigrams which correspond to non-compositional compounds, like ``prime minister'', and bigrams which correspond to compositional compounds, such as ``a quarter''. Interestingly, this does not occur for all high-frequency bigrams; it ignores those that are not inherently meaningful, such as ``$<$UNK$>$ in'', yielding qualitatively good phrases.\n\nIn the multiple-embedding experiments, it is possible to see which of the two embeddings of a word was assigned the higher probability for any specific test-set sentence. In order to visualize what types of meanings are assigned to each embedding, we select sentences in which one embedding is preferred, and look at the context in which the word is used. Several examples of this can be seen in Table \\ref{multi-viz}; it is clear from looking at these examples that the system does learn distinct embeddings for different senses of the word. What is interesting, however, is that it does not necessarily learn intuitive semantic meanings; instead it tends to group the words by the context in which they appear. In some cases, like \\textbf{profile} and \\textbf{edition}, one of the two embeddings simply captures an idiosyncrasy of the training data.\n\nAdditionally, for some words, such as \\textbf{rodham} in Table \\ref{multi-viz}, the system always prefers one embedding. This is promising, because it means that in future work it may be possible to further improve accuracy and training efficiency by assigning more embeddings to polysemous words, instead of assigning the same number of embeddings to all words.\n\n\n\\section{Related Work}\n\\label{related}\n\nPast work that utilized lattices in neural models for natural language processing centers around using these lattices in the encoder portion of machine translation. \\newcite{DBLP:journals\/corr\/SuTXL16} utilized a variation of the Gated Recurrent Unit that operated over lattices, and preprocessed lattices over Chinese characters that allowed it to effectively encode multiple segmentations. Additionally, \\newcite{sperber2017neural} proposed a variation of the TreeLSTM with the goal of creating an encoder over speech lattices in speech-to-text.\nOur work tackles language modeling rather than encoding, and thus addresses the issue of marginalization over the lattice.\n\nAnother recent work which marginalized over multiple paths through a sentence is \\newcite{ling2016latent}. The authors tackle the problem of code generation, where some components of the code can be copied from the input, via a neural network.\nOur work expands on this by handling multi-word tokens as input to the neural network, rather than passing in one token at a time.\n\nNeural lattice language models improve accuracy by helping the gradient flow over smaller paths, preventing vanishing gradients. Many hierarchical neural language models have been proposed with a similar objective \\newcite{koutnik2014clockwork,zhou2017chunk}.  Our work is distinguished from these by the use of latent token-level segmentations that capture meaning directly, rather than simply being high-level mechanisms to encourage gradient flow.\n\n\\newcite{chan2016latent} propose a model for predicting characters at multiple granularities in the decoder segment of a machine translation system. Our work expands on theirs by considering the entire lattice at once, rather than considering a only a single path through the lattice via ancestral sampling. This allows us to train end-to-end without the model collapsing to a local minimum, with no exploration bonus needed. Additionally, we propose a more broad class of models, including those incorporating polysemous words, and apply our model to the task of word-level language modeling, rather than character-level transcription.\n\nConcurrently to this work, \\newcite{van2017multiscale} have proposed a neural language model that can similarly handle multiple scales.\nOur work is differentiated in that it is more general: utilizing an open multi-token vocabulary, proposing multiple techniques for hidden state calculation, and handling polysemy using multi-embedding lattices.\n\n\\section{Future Work}\n\nIn the future, we would like to experiment with utilizing neural lattice language models in extrinsic evaluation, such as machine translation and speech recognition. Additionally, in the current model, the non-compositional embeddings must be selected a priori, and may be suboptimal. We are exploring techniques to store fixed embeddings dynamically, so that the non-compositional phrases can be selected as part of the end-to-end training.\n\n\\section{Conclusion}\n\n\nIn this work, we have introduced the idea of a neural lattice language model, which allows us to marginalize over all segmentations of a sentence in an end-to-end fashion. In our experiments on the Billion Word Corpus and Chinese GigaWord corpus, we demonstrated that the neural lattice language model beats an LSTM-based baseline at the task of language modeling, both when it is used to incorporate multiple-word phrases and multiple-embedding words.\nQualitatively, we observed that the latent segmentations generated by the model correspond well to human intuition about multi-word phrases, and that the varying usage of words with multiple embeddings seems to also be sensible.\n\n\n\n\\bibliographystyle{acl2012}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAnalyzing social media becomes a crucial task, due to the frequently usage of social media platforms. Sarcasm detection, the conflict of using the verbal meaning of a sentence and its intended meaning \\cite{clift_1999,doi:10.1207\/s15327868ms1301}, is an important task. Sarcasm detection is a challenge, since sarcastic contents are used to express the opposing of what is being said. Recently sarcasm detection has been studied from a computational perspective as one of classification problems that separates sarcastic from non-sarcastic contents\\cite{Reyes:2013:MAD:2447287.2447294,nayel-etal-2021-machine-learning}. \n\\par{}Arabic is an important natural language having an extensive number of speakers. The research in Natural Language Processing (NLP) for Arabic is continually increasing. However, there is still a need to handle the complexity of NLP tasks in Arabic. This complexity arises from various aspects, such as orthography, morphology, dialects, short vowels, and word order. Sarcasm detection in Arabic is a particularly challenging task \\cite{Alayba_2018}. \n\\par{}In this paper, we describe the system submitted to the iSarcasm detection shared task\\cite{abufarha-etal-2022-semeval}. The shared task aims at detecting the sarcasm contents in Arabic tweets. In this work, a machine learning framework has been developed and various machine learning algorithms have been implemented. Term Frequency-Inverse Document Frequency (TF-IDF) has been used as vector space model for tweet representation. The rest of this paper is organized as follows: in section 2, a background about sarcasm detection is given. Section 3 and section 4 overview the dataset and the system respectively. Experimental setup and results are given in section 5 and section 6 respectively. Finally, section 7 concludes the proposed work and suggests future work to be continued. \n\\section{Background}\nThe research work have been done on Arabic sarcasm detection were mainly focused on creating datasets and establish a baseline for each created dataset \\cite{10.1007\/978-3-030-45442-5_18}. \\citet{karoui2017soukhria} created a corpus of sarcastic Arabic tweets that are related to politics. Distant supervision has been used for the creation of corpus. The authors used keywords that are like sarcastic contents in Arabic to label the tweets as sarcastic tweets. They implemented different machine learning algorithms such as SVM, logistic regression, Na\u00efve Bayes, and other classifiers on the developed corpus.\n\\par{}An ensemble classifier of XGBoost, random forest and fully connected neural networks has been designed by \\citet{khalifa-2019-ensemble}. They extracted a set of features that consists of sentiment and statistical features, in addition to word $n$-grams, topic modelling features and word embeddings. \\citet{nayel-2019-benha} developed an ensemble-based system for irony detection in Arabic Tweets. A set of classification algorithms namely Random forests, multinomial Na\u00efve Bayes, linear, and SVM classifiers have been used as base-classifiers. In \\cite{nayel-etal-2021-machine-learning}, sarcasm detection has been formulated as a binary classification problem and SVM has been implemented. \n\\section{Dataset}\nA new data collection method has been introduced, where the sarcasm labels for texts are provided by the authors themselves. The author of each sarcastic text rephrased the text to convey the same intended message without using sarcasm. \\citet{leggitt2000emotional} defined a set categories of ironic speech namely; sarcasm, irony, satire, understatement, overstatement, and rhetorical question. Linguistic experts have been asked to further label each text into one of these categories. Each text in the Arabic dataset has the following information attached to it:\n\\begin{itemize}\n\\item a label specifying the text dialect;\n\\item a label specifying the nature of sarcasm (sarcastic or non-sarcastic), provided by its author;\n\\item a rephrase provided by its author that conveys the same message non-sarcastically.\n\\end{itemize}\n\\section{System Overview}\nIn this section, we review the main structure of the proposed model. The proposed system, as shown in figure \\ref{diagram}, consists of three phases namely; preprocessing, feature extraction and training the classification algorithms. Then, the resulted model used to predict the unseen test data. \n\\begin{figure}[htb!]\n\\centering\\includegraphics[height=2.8in,width=0.5\\textwidth]{model}\n\\caption{\\label{diagram}The structure of the proposed model}\n\\end{figure}\n\\subsection{Preprocessing}\nThe first stage of developing systems is preprocessing, where unwanted and uninformative piece of text has to be removed, it is also called text cleaning. We performed text cleaning by removing:\n\\begin{itemize}\n\\item special symbols, such as $\\{+, -, =, \\$,....\\}$;\n\\item repeated characters such as (\"\\emph{hhhhhhhh}\" will be normalized to \"\\emph{hh}\"); \n\\item non-Arabic words, such as English characters or any other language;\n\\item punctuations and Arabic diacritics.\n\\end{itemize}\n\\subsection{Features Extraction}\nTo prepare features to build classification model and before feeding the text into the classifier and after performing text cleaning,  Term Frequency-Inverse Document Frequency (TF-IDF) technique was used to change over content to vectors and all the algorithms to investigate the best performing algorithm.\\\\\n\\indent TF-IDF has been used to represent comments as vectors. If $ <\\!\\!\\!w_1,w_2, \\ldots, w_k\\!\\!\\!> $ are the tokenized words of a comment $ \\mathcal{T}_j$, the vector associated to the comment $ \\mathcal{T}_j$ will be represented as $<\\!\\!\\!v_{j1}, v_{j2},\\ldots ,v_{jk}\\!\\!\\!>$ where $v_{ji}$ is the weight of the token $w_i$ in tweet $ \\mathcal{T}_j$ which is calculated as:- \\[v_{ji} = tf_{ji} * \\log\\left(\\frac{N+1}{df_i+1}\\right)\\] where $tf_{ji}$ is the total number of occurrences of token $w_i$ in the comment $ \\mathcal{T}_j$, $df_i$ is the number of comments in which the token $w_i$ occurs and $N$ is the total number of comment.\\\\\n\\subsection{Methodology} \nWe explored various classification algorithms as well as ensemble approach by combining the output of these classifiers (also known as base classifiers) using hard voting. The base classifiers used in this work are listed below:\n\\begin{itemize}\n\\item{Support Vector Machines (SVMs)} \n\\item{Random Forest (RF)}\n\\item{K-Nearest Neighbours (KNN)}\n\\item{Multinomial Na\u00efve Bayes (M-NB)}\n\\item{Multi-Layer Perceptron (MLP)}\n\\item{Stochastic Gradient Descent (SGD)}\n\\item{AdaBoost Classifier}\n\\item{Voting Classifier}\n\\end{itemize}\n\\section{Experimental Setup}\nFor feature extraction phase we used unigram model. For the purpose of training the model, we have used 5-fold cross-validation technique to adjust the parameters.\\\\\n\\indent The Scikit-Learn library implementation of classification algorithms were used in the training phase. \\indent For SVM, two kernels have been tested: linear kernel and RBF with two parameters $\\gamma = 2 $ and $C = 1$. While, for SGD classifier the loss function used was Hinge and the maximum iteration was set at 10000 iterations.\\\\\n\\indent The number of nodes in the hidden layer of MLP was set at 20, logistic function was used as activation function and Adam solver was used. The maximum number of decision trees in random forests is set at 300.\n\\subsection{Evaluation Metrics}\n\\par{}F1-score has been used to evaluate the performance of all submissions. F1-score is a harmonic mean of Precision (P) and Recall (R) and calculated as follow: \n\\[ F\\!\\!-\\!\\!score = \\frac{2*P*R}{P + R }\\]\nF1-score for the sarcastic class (F1-sarcastic) has been used for final evaluation.\n\\section{Results}\nThe cross validation accuracy of all training classifiers for the training set is given in Table \\ref{tab1}. It is clear that MLP gives the best accuracy with moderate Standard Deviation (STD) for the five folds while development phase, so we decided to submit the output of this classifier.\\\\\n\\begin{table}\n\\centering\n\\caption{5-fold Cross-Validation accuracy for all classifiers in the training set}\\label{tab1}\n\\begin{tabular}{l|c|c}\\hline\nClassifier & Accuracy &  STD \\\\\\hline\n SVM-Linear&81.0\\% & 0.055  \\\\\n SVM-RBF&76.6\\% & 0.003  \\\\\n MNB&76.6\\% & 0.005\\\\\n SGD&80.0\\% & 0.045\\\\\n MLP&83.6\\% & 0.045\\\\\n RF&75.8\\% & 0.056\\\\ \n KNN &79.7\\% & 0.058\\\\\nAdaBoost &75.2\\% & 0.052\\\\\nVoting&80.4\\% & 0.043\\\\\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Results of MLP classifier for the test set }\\label{tab2}\n\\begin{tabular}{l|c|c}\\hline\nMeasure & Value &  Rank \\\\\\hline\nF-1 sarcastic & 0.3746 & 14  \\\\\nF-score & 0.6024 & 11  \\\\\nPrecision (P) & 0.5968 &15\\\\\nRecall (R) &0.6608 & 17\\\\\nAccuracy&0.7329 & 8\\\\\\hline\n\\end{tabular}\n\\end{table}\n\\indent Results for test set is given in Table \\ref{tab2}. The reported results show that, while training MLP gives better accuracy among implemented machine learning classifiers. Also, it gives better rank in accuracy for the unlabelled test set. While in other metrics, the performance was not satisfied. This may resulted because of using accuracy metric while comparing different classifiers in development phase.\\\\\n\\indent A good suggestion is to use different evaluation metrics while developing the system. In addition, using different word representation models such as word embeddings, which encompasses the semantic meaning of words could improve the performance. Complex models such as deep learning models is promising, but the challenge in such models is the availability of suitable resources.    \n\\section{Conclusion}\nIn this work, a classical machine learning framework has been designed for sarcasm detection in Arabic tweets. The proposed framework reported reasonable results. The future work may include applying complex framework such as deep learning structure. In addition, word representation is very important factor that can be used in different manner such as word embeddings and transformers. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\\subsection{ Problem Definition}\nLimbs are hugely valuable to many people, in that they improve mobility and the ability to manage daily activities, as well as provide the means to stay independent. It is costly (50K USD) and time-consuming for the manufacturers to design artificial limbs customized for one person, Designing intelligence prosthetics which deal with the large differences between humans (like human body dimensions, weights, height and walking styles) is so complicated by the large variability in response among many individuals. One key reason for this is that our understanding of the interactions between humans and prostheses is not well-understood, which limits our ability to predict how a human will adapt his or her movement. Physics-based, biomechanical simulations are well-positioned to advance this field as it allows for many experiments to be run at low cost.\n\\subsection{Environment}\nWe use OpenSim \"ProstheticsEnv\" environment, which models one human leg and prosthetic in another leg see in fig(1), OpenSim is a 3D human model simulator, which consists of observations of joints, muscles and tendons, 19 actions, and the reward $R_{t}$ is the negative distance from the desired velocity in eq(~\\ref{eq:reward}).\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{figures\/OpenSim1.png}\n\\label{fig:OpenSim ProstheticEnv Environment}\n\\caption {OpenSim ProstheticEnv Environment.}\n\\end{center}\n\\end{figure}\n\\begin{equation}\n{R_{t}=9-(3-V_{t})^2}\n\\label{eq:reward}\n\\end{equation}\nwhere $V_{t}$ is the horizontal velocity vector of the pelvis.\nOpenSim environment has a limitation it is very slow to run due to the high number of observations and state variables.\n\n\\subsection{Reinforcement Learning (RL) algorithms}\nReinforcement Learning (RL) will help prosthetics to calibrate with differences between humans and differences between walking environments \\cite{c1}, RL is a machine learning paradigm, where an agent learns the optimal policy for performing a sequential decision making without complete knowledge of the environment \\cite{c2}. The agent must explore the environment by taking action $A_{t}$ and edit the policy according to the reward function to maximize the reward $R_{t}$.\nWe use the DDPG algorithm \\cite{c3}, TRPO and PPO to train the agent. DDPG is a model-free, off-policy actor-critic algorithm using deep function approximators that can learn policies in high-dimensional, continuous action spaces.\n\\subsection{Imitation Learning}\nThe main problem with RL algorithms is the time needed to solve the problem -training time- because the algorithm must explore the environment and adapt its policy according to the reward at every timestep. Imitation learning is a specific subset of RL where the learner tries to mimic an expert's action in order to achieve the best performance. The main advantage of DAgger is that the expert teaches the learner how to recover from past mistakes \\cite{c4}, and we aim to leverage this to illustrate behaviour learning. \nThere are many ways to accelerate the learning process in RL, such as Cross-Domain Transfer \\cite{c2}, Inter-task Mapping via Artificial Neural Network (ANN) \\cite{c5}. We use Imitation learning to achieve that by implementing DAgger algorithm. The DAgger algorithm has shown to be able to achieve expert-level performance after a few data aggregation iterations \\cite{c6}.\nTo use imitation learning there are two assumptions:\n\\begin{enumerate}\n    \\item Similarity between the expert and the target agent in actions, observations space and the reward function.\n    \\item Environment must be described by a Markov Decision Process (MDP).\n\\end{enumerate}\nIn the standard DAgger algorithm, the target agent exploits the expert policy and stops exploring the environment. This may be a problem, as the target agent should balance between exploitation and exploration. We propose some improvements to the DAgger algorithm to encourage the exploration.\n\\section{Experiments}\nWe run the following experiments:\\footnote{codes available at \\url{github.com\/montaserFath\/Reinforcement-Learning-for-Prosthetics}}\n\\begin{enumerate}\n    \\item Run RL algorithms (DDPG, TRPO and PPO) in OpenSim ProstheticsEnv, 2,000 episodes and 1,000 timesteps in the episode to give an agent more time to walk or stand up.\n    \\item we trained a DDPG agent to achieve positive reward (around +100) in the standing up task.\n    \\item we use that agent as an expert to evaluate the DAgger algorithm.\n    \\item we modify DAgger so that the expert agent labels the target agent's actions based on the timestep reward, by comparing between the timestep reward of the expert agent and the target agent on a given timestep: if the expert agent has less reward than the target agent, the expert keeps the target agent's action and the opposite is true.\n    \\item we use the target action value instead of timestep reward to do the comparison, and we sum the timestep rewards from a given state and action pair until the end of the episode\n    \\item we used the epsilon-greedy method \\cite{c2}, where the algorithm has the choice to select between taking the target action with a probability $ 1-\\epsilon $ or the expert action with a probability $\\epsilon$.\n\\end{enumerate} \n\\section{Results}\nThe maximum reward mean achieved by TRPO (see table~\\ref{table:OpenSim_ProstheticsEnv} and fig~\\ref{fig:Number_of_Iterations}), but it takes more time comparing with PPO and DDPG because it need to find the inverse of matrix which takes time. Although of this reward the agent can not walk for more than one step and sometimes it falls before the first step.\nDagger algorithm achieved the best average reward comparing with other algorithms which balance between exploiting and exploring (see fig~\\ref{fig:imitation_learning}), we think the reasons behind that:\n\\begin{enumerate}\n    \\item The expert policy has a high reward.\n    \\item The high similarity between expert and naive agent.\n    \\item The naive agent needs more time to run by increasing the number of iterations.\n\\end{enumerate} \nThe main problem with timestep reward modification, it compares timestep reward (short-term) adding to this the large variation between timesteps. When the variation between episodes is small in Action-value.\nThe naive agent has gotten a reward greater than the expert agent, so roles can be exchanged, the expert can be naive and the naive can be an expert which will decrease training time significantly.\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{figures\/pro_mean.png}\n\\caption{Number of Iterations VS Reward Mean in OpenSim ProstheticsEnv Environment.}\n\\label{fig:Number_of_Iterations}\n\\end{center}\n\\end{figure}\n\\begin{table}[ht!]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n \\hline\n\\textbf{Algorithm} & \\textbf{Maximum reward} & \\textbf{Mean reward} \\\\  \\hline\n\\textbf{DDPG} &  113 & -42 \\\\  \\hline\n\\textbf{TRPO} & 194 & 43 \\\\  \\hline\n\\textbf{PPO} & 70 & -58 \\\\  \\hline\n\\end{tabular}\n\\caption{Comparison between algorithms in ProstheticsEnv.}\n\\label{table:OpenSim_ProstheticsEnv}\n\\end{center}\n\\end{table}\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{figures\/plots.png}\n\\caption{Comparison between DAagger algorithm, Timestep Reward, action value and Epsilon-greedy}\n\\label{fig:imitation_learning}\n\\end{center}\n\\end{figure}\n\\section{CONCLUSIONS}\nWe have applied imitation learning in a humanoid environment to accelerate the learning process. The naive agent reaches convergence within 5  iterations while the expert reaches it after 100 iterations which means reducing training time by 95\\%. \nThe DAgger algorithm achieved the best average reward comparing with other algorithms which balance between exploiting and exploring, these algorithms will work better when there is some degree of variation between expert and naive agent and this is what we are planning to do in the future by apply imitation learning from normal human legs to prosthetic, the main challenge will be how to figure out the differences and similarities between it.\n\\section{Research Limitations}\n\\begin{enumerate}\n    \\item The prosthetic model can not walk for large distances even can falls before completing the first step.\n    \\item Each experiment runs for one time, So we are planing to repeat each experiment number of times with different random seeds and take the average and variance.\n    \\item We used same hyperparameters for all algorithm to benchmark algorithms, we need to select the best hyperparameters for each algorithm and environment.\n    \\item We benchmarcked three RL algorithms only and from one library(ChainerRL). So we are planing to use different implementations.\n\\end{enumerate}\n\\addtolength{\\textheight}{-12cm}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nApproximate nearest neighbor (ANN) search plays a crucial and long standing role in various domains, including databases, computer vision and machine learning.\nSince collecting large amounts of data became easier, any algorithm involved in indexing and searching such data must be sufficiently scalable. Therefore, the creation of a scalable and efficient data structure for retrieving similar items has become an active research topic.\nEven when leveraging modern hardware, it remains impractical to perform an exhaustive search over billions of high-dimensional data points. This is especially true when tackling this problem under additional constrains such as a reasonable memory consumption and low latency.\nTo keep up with the scale of data, modern approaches use index structures that that are heavily tailored towards exploiting the massive parallelism of GPUs~\\cite{wieschollek2016efficient,johnson2019billion,chen2019vlq,harwood2016fanng} or custom hardware~\\cite{PQFPGA} compared to previous CPU-based methods~\\cite{flann_pami_2014,PQ,1BPQ,localPQ,optimizedPQ,invertedmultiindex,Babenko_2015_CVPR,Babenko_2014_CVPR}.\n\nDespite all recent advances, the only available method for guaranteed retrieval of the exact nearest neighbor is still exhaustive search due to the curse of dimensionality~\\cite{Weber1998AQA}. Instead, most popular methods relax the problem by searching for an entry that is likely to be the nearest neighbor, accepting a minimal loss in accuracy. The quality of the recall however heavily depends both on the choice of the search structure and the executed query. Structures based on quantization or hashing\/binning schemes~\\cite{PQ,1BPQ,localPQ,optimizedPQ,invertedmultiindex,Babenko_2015_CVPR,Babenko_2014_CVPR,wieschollek2016efficient,chen2019robustiq,johnson2019billion,Matsui2015PQTableFE} can be efficiently built but typically suffer from relatively low recall rates as enumerating and visiting neighboring cells is exhaustive in high dimensions. Better recall rates are recently achieved by graph-based methods~\\cite{harwood2016fanng,dong2011efficient,chen2009fast,Warashina2014EfficientKN,fu2016efanna,fu2019fast}, but building effective graph-based index structures requires global optimization or sequential updates on edges. Their construction time has to be measured in hours or days~\\cite{Matsui2015PQTableFE}.\n\nWe propose a novel hierarchical construction scheme that allows for fully parallel construction of a highly effective search structure, which is based on a kNN-graph with additional links.\nThe structure is designed to be built and traversed on the GPU, \\emph{e.g}\\onedot} \\def\\Eg{\\emph{E.g}\\onedot, the graph contains a fixed number of outgoing links per data point.\nCoarsened graphs, by selecting only a subset of the data base, are used to build a search tree.\nThe kNN-tree is built hierarchically via bottom-up merging of concurrently constructed sub-trees, where each sub-tree already provides a consistent kNN-graph for the contained data points at each layer.\nQueries from the top of the hierarchy are used to locate nearest neighbors in the other sub-trees to merge the trees layer by layer. A parallel local scheme optimizes for both very consistent local neighborhoods as well as additional links that are important for precise traversal (graph-diversification).\nThese symmetric links ensure that each data point inside a neighborhood can be found from its own neighbors or any query from any direction. The hierarchy is necessary to build up a high-quality kNN-graph in parallel. During the final query, our algorithm however\ndirectly descents\nfrom the top to the bottom layer, which turned out to be more efficient.\n\nAs evidenced by our empirical evaluation,\nthe presented scheme outperforms existing approaches concerning both the construction as well as the query time.\nAt the same time, the recall rate is consistently high and can be traded in for only even faster query or build time.\nWe present a multi-GPU scheme that is capable of achieving above 99\\% recall even for large data sets with billions of high-dimensional entries.\n\\section{Related Work}\nA large amount of literature exists on designing structures that accelerate a nearest neighbor search. Besides traditional approaches~\\cite{KD-tree, flann_pami_2014} most popular techniques rely either on data quantization in clusters~\\cite{PQ,1BPQ,localPQ,optimizedPQ,invertedmultiindex,Babenko_2015_CVPR,Babenko_2014_CVPR,wieschollek2016efficient,chen2019robustiq,johnson2019billion,Matsui2015PQTableFE} or building neighborhood graphs~\\cite{harwood2016fanng,dong2011efficient,chen2009fast,Warashina2014EfficientKN,fu2016efanna,fu2019fast}. To achieve peak performance, most of these methods compute a compressed representation for each entry as large datasets will not fit in fast memory. There exists several strategies to compute such a compression. While hashing methods~\\cite{LSH,Andoni2008,Korman2011} produce compact binary codes, quantization-based methods reuse centroids by assigning each data point a unique identifier based on the centroid to which they belong. It has been empirically shown, that quantization methods are more accurate than various hashing methods~\\cite{PQ,flann_pami_2014}.\n\n\\textit{Quantization methods} for nearest neighbor search using clustering methods were popularized by J{\\'e}gou~\\emph{et al}\\onedot~\\cite{PQ} while originally being introduced in~\\cite{originalPQ}. Such index structures like IVFADC~\\cite{PQ} partition the high-dimensional search space into disjoint Voronoi cells described by a set of centroids obtained by Vector Quantization (VQ)~\\cite{VQ}. The idea has been extended later by Babenko~\\emph{et al}\\onedot~\\cite{invertedmultiindex}, where the high-dimensional vector-space is factored in orthogonal subspaces. Hereby, each vector is assigned to a centroid independently for each subspace according to a separate codebook that resides there. Wieschollek~\\emph{et al}\\onedot~\\cite{wieschollek2016efficient} proposed a hierarchical representation of the codebook besides demonstrating superior performance using a GPU. Johnson~\\emph{et al}\\onedot~\\cite{johnson2019billion} ported IVFADC~\\cite{PQ} to the GPU in combination with a fast GPU-based implementation for k-selection, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot returning the $k$ lowest-valued elements from a given list (a crucial part of quantization based methods). They are the first employing multi-GPU parallelism by replication and sharding. Their work forms the library ``FAISS''. Eventually, Chen~\\emph{et al}\\onedot~\\cite{chen2019robustiq} proposed a GPU based method RobustiQ overcoming the memory limitations of FAISS by extending the idea of Line Quantization from ~\\cite{wieschollek2016efficient} in a hierarchical fashion. Still, the reported distances are only an approximation of the true distance.\n\nAll hashing and quantization-based indexing schemes share the same problem that they partition the space into cells.\nWhile the containing cell for a query might be found very efficiently, the exact nearest neighbor might be across the boundary to one of the neighboring cells. Determining and visiting all neighboring cells in high dimension is a problem severely limiting these approaches.\n\n\\textit{kNN-graph} based methods are another way to accelerate the query process. Our approach presented in this paper belongs to this category. The main idea is to link each point from the search space to $k$ of its nearby points. Each query will start at a random guess in the dataset. Then, the guess itself is refined by replacing it with a better point from the $k$ linked points. Chen~\\emph{et al}\\onedot~\\cite{chen2009fast} propose a fast divide and conquer strategy for computing such kNN-graphs. Done~\\emph{et al}\\onedot~\\cite{dong2011efficient} introduced NN-descent for using kNN-graphs to accelerate NN-search. Hereby, each point maintains a list of its own nearest neighbors and points where itself is considered as a nearest neighbor. This has been later extended~\\cite{Warashina2014EfficientKN} to make use of MapReduce. EFANNA as a multiple hierarchical index structure uses a truncated KD-tree to build a kNN-graph~\\cite{fu2016efanna}. In the ideal case, a kNN-graph augmented with additional links could guarantee that for an arbitrary start point the NN-descent will converge to the correct solution. Computing such a graph with additional links at scale is not practicable. Therefore, several methods exist to at least approximate such a graph~\\cite{fu2019fast,harwood2016fanng}. Fu~\\emph{et al}\\onedot~\\cite{fu2019fast} introduce NSG as an approximation.\nTo reduce the overall number of edges, their optimization tries to lower the out-degree individually per node.  Their method can scale beyond multiple cores, outperforming a GPU approach~\\cite{johnson2019billion} on a benchmark dataset.\nHarwood~\\emph{et al}\\onedot~\\cite{harwood2016fanng} suggest an alternative approach for constructing such a graph. Starting from a fairly dense graph, they remove ``shadowed edges'', which are redundant when considering traversing paths during a query. They showed promising results using the GPU but only on rather small datasets as their build time is rather high.\nMalkov~\\emph{et al}\\onedot~\\cite{malkov2018efficient} build a hierarchical graph structure to accelerate the nearest neighbor search.\n\nWhile ours uses a hierarchy as well to build the index structure, ours differs during the query process.\nWe will describe the usage of kNN-graphs in more detail in the next section.\n\\section{Background}\n\\label{sec:background}\nIn this section, we formally introduce the approximate nearest neighbor problem statement and used notations.\n\\subsection{ANN Search}\n\\label{subsec:annsearch}\nThe nearest neighbor problem retrieves a point from a dataset $x\\in\\mathcal{X}=\\left\\{x_1,\\ldots,x_n\\right\\}$ that has the smallest distance to a query $q$. For the sake of simplicity, we assume an Euclidean space ($\\mathcal{X}\\subseteq\\mathbb{R}^d$). Other distance metrics could easily be incorporated. The nearest neighbor $x^\\star\\in\\mathcal{X}$ of $q\\in\\mathbb{R}^d$ therefore is defined as\n\\begin{align}\n    x^\\star = \\argmin_{x\\in \\mathcal{X}} \\norm{q-x}_2, \\label{eq:nnsearch}\n\\end{align}\nwhere $\\norm{\\cdot}_2$ denotes the Euclidean distance.\nSimilarly, the $k$-nearest neighbor search retrieves the $k$ closest entries from $\\mathcal{X}$ for a given query. As finding the exact nearest neighbor might be costly, we accept points in $\\mathcal{X}$ which are close to $q$ and therefore deliver an approximate solution to Eq.~\\eqref{eq:nnsearch}.\n\\subsection{KNN Graph}\n\\label{subsec:knngraph}\nLet $x$ be a fixed but arbitrary point from the dataset $\\mathcal{X}$. We build a graph structure, where each point from $\\mathcal{X}$ represents a node in the graph. Further, we define $\\mathcal{N}_x\\subseteq \\mathcal{X}$ as a local neighborhood of $x$ and defer the details on how to construct $\\mathcal{N}_x$ to the next section.\nThe edges of the graph are then defined as $(u,v)$ where $v\\in \\mathcal{N}_u$. Note, the resulting graph is a directed graph $G=(\\mathcal{X}, E)$, where $E=\\left\\{(u,v)\\;|\\;u\\in\\mathcal{X}, v \\in \\mathcal{N}_u\\right\\}$.\n\nOne greedy algorithm to find the nearest neighbor for a query point $q$ is \\textit{NN-descent}~\\cite{dong2011efficient}.\nStarting from an initial guess $\\hat{x}\\in\\mathcal{X}$, the distance between $q$ and each point $y\\in \\mathcal{N}_{\\hat{x}}$ is computed.\nIf any $y\\in \\mathcal{N}_{\\hat{x}}$ is closer to $q$ than $\\hat{x}$, the guess $\\hat{x}$ is replaced by the closest point from $\\mathcal{N}_{\\hat{x}}$ to $q$. This process repeats until no point in $\\mathcal{N}_{\\hat{x}}$ has a smaller distance to $q$ than $\\hat{x}$.\nHowever, as the current $\\hat{x}$ might not provide an edge into the right search direction, this greedy algorithm might get stuck in a local optimum on a pure kNN-graph.\n\nBefore explaining our proposal of how to efficiently construct the kNN-graph $G$, we outline common pitfalls that impair the performance of NN-descent.\n\\paragraph{Common pitfalls.}\nSince the NN-descent is a greedy search, it offers no guarantee of finding the exact solution. Some of the reasons are listed below:\n\n\\textit{Connectivity:}\nAs a kNN-graph is a directed graph, $v$ might be directly connected to $u$ being the nearest neighbor, hence $v\\in\\mathcal{N}_u$. But this does not imply that the inverse link exists ($u\\in\\mathcal{N}_v$). Therefore, the construction of a kNN-graph has to deal with synchronizing outgoing and incoming (inverse) edges.\n\n\\textit{Gaps in high-dimensional spaces:} As each point is only linked to a finite number of local neighbors, there exist pathological cases (even in 2D), where close-by points are not directly connected at all.\nSuch a case is illustrated in Figure~\\ref{fig:slack}.\nDue to the gap, the true nearest neighbor will not be found.\nComputing an idealized monotonic relative neighborhood graph~\\cite{fu2019fast} (MRNG) would avoid this issue, but it would require a strongly varying connectivity, which is not suitable for parallel approaches -- besides the additional computational burden.\n\n\\textit{Degree of nodes:} There exists a trade-off when choosing the size of $\\mathcal{N}_{x}$ for any $x\\in\\mathcal{X}$. Too few edges amplify the previously described issues, but reduce the number of necessary comparisons. While many edges allow the greedy search to escape from local neighborhoods, they increase the cost for each iteration.\n\\newcommand{$k_\\text{nn}$}{$k_\\text{nn}$}\n\\newcommand{$k_\\text{sym}$}{$k_\\text{sym}$}\n\\SetStartEndCondition{ }{}{}\\SetKwProg{Fn}{Function}{:}{end}\n\\SetKwFunction{BT}{BuildGraph}\\SetKwFunction{Init}{init\\_tree}\\SetKwFunction{Merge}{merge}\\SetKwFunction{Sym}{sym}\\SetKwFunction{Query}{query}\\SetKwFunction{SymQuery}{sym\\_links}\\SetKwFunction{Stats}{compute\\_stats}\\SetKwFunction{Select}{select}\n\\SetKw{KwTo}{ to }\n\\SetKwFor{For}{for}{\\string:}{}\\SetKwFor{ParFor}{for all}{\\string do in parallel:}{}\\SetKwIF{If}{ElseIf}{Else}{if}{:}{elif}{else:}{}\\SetKwFor{While}{while}{:}{fintq}\\AlgoDontDisplayBlockMarkers\n\\SetAlgoNoEnd\n\\SetAlgoNoLine\n\\SetKwData{LT}{$l_{top}$}\n\\SetKwData{LB}{$l_{btm}$}\n\\SetKwData{Layer}{$l$}\n\\SetKwData{L}{$l$}\n\\SetKwData{Li}{$l_i$}\n\\SetKwData{K}{$k$}\n\\SetKwData{KL}{$k_{nn}$}\n\\SetKwData{KF}{$k_{sym}$}\n\\SetKwData{XIb}{$\\tau_{b}$}\n\\SetKwData{XIq}{$\\tau_{q}$}\n\\SetKwData{XIg}{$\\tau$}\n\\SetKwData{Base}{\\texttt{base}}\n\\SetKwData{Tree}{\\texttt{graph}}\n\\SetKwData{Buffer}{$b$}\n\\SetKwData{N}{$n$}\n\\SetKwData{Q}{$q$}\n\\SetKwData{X}{$x$}\n\\SetKwData{Sbf}{$\\mathbf{s}$}\n\\SetKwData{S}{$s$}\n\\SetKwData{Si}{$\\S_{i}$}\n\\SetKwData{Pbf}{$\\mathbf{{p}}$}\n\\SetKwData{P}{$p$}\n\\SetKwData{A}{$a$}\n\\SetKwData{Cache}{$cache$}\n\\SetKwData{Best}{$best$}\n\\SetKwData{PrioQ}{$prioQ$}\n\\SetKwData{Dist}{$dist$}\n\\SetKwFunction{TopSeg}{getTopSegment}\\SetKwFunction{StartPoint}{getStartPoint}\\SetKwFunction{Fetch}{fetch}\\SetKwFunction{Init}{init}\\SetKwFunction{Add}{add}\\SetKwFunction{Push}{push}\\SetKwFunction{Pop}{pop}\\SetKwFunction{Connected}{is\\_connected}\\SetKwFunction{Transform}{transform}\\SetKwFunction{Clear}{clear\\_visited}\\SetKwFunction{Skip}{skip\\_criteria}\\SetKwFunction{SkipSym}{skip\\_criteria\\_sym}\\SetKwFunction{Tlayer}{layer}\n\\section{Approximate Symmetric Nearest Neighborhood Graph Construction}\n\\label{sec:method}\n\\subsection{Overview}\nThe core of our method exploits kNN-graph structures to support high-quality recall queries.\nAs a pure unidirectional kNN-graph is not well suited for searching, it is augmented by two important link types. On the one hand, symmetric links are inserted where necessary to ensure that a query point which happens to lie between two data points is guaranteed to reach both independently from where the search started. The second ingredient is to support a hierarchy of coarsened sub-graphs similar to \\cite{malkov2018efficient} to ensure that all parts of the graph are connected and gaps in the data set are bridged. In addition, the hierarchy minimizes the number of hops between any pair of sub-graphs.\n\\begin{figure}[tb]\n\\centering\n \\includegraphics[width=.9\\columnwidth]{figures\/merging_tree.pdf}\n \\caption{Several steps of building the hierarchical structure.}\n \\label{fig:tree_construction}\n\\end{figure}\n\nBesides the quality of the results, our approach also focuses on the execution time, both for carrying out a query as well as for constructing the search graph. Both operations are supported on the GPU. Parallelization is enabled by a constant connectivity $k$ for all nodes. Constructing the hierarchy exploits very efficient distance computations between all points within a small batch to initialize kNN sub-graphs. The sub-graphs are recursively fused into sub-trees by selecting a few samples from each sub-graph to form a coarser top layer. Adding another layer is efficiently used to establish and update the correct NN-links between all points in the sub-layers by executing a query from the top of the currently built structure (see Figure~\\ref{fig:tree_construction}). This bottom-up construction creates a robust searchable kNN-graph for each merged tree. It can be parallelized on multiple GPUs to even support datasets with large memory requirements.\n\nThe hierarchy is mainly used to quickly construct a high-quality search structure. To answer a query in the refined graph, traversing the various layers is, however, too expensive. It is significantly faster to start a search directly at the bottom layer, given the right starting points.\n\\subsection{Query}\n\\label{subsec:query}\nSearching for $k$ nearest neighbors is the central operation both in answering a query as well as in building the kNN search graph. The graph consists of $n$ points $x$, for each of which a list of $k = k_\\text{nn} + k_\\text{sym}$ outgoing links are stored, representing the local neighborhood $\\mathcal{N}_x$.\nUp to $k_\\text{nn}$, these links represent the true nearest neighbors found so far while the remaining $k_\\text{sym}$ represent potential inverse or symmetric links to points which have $x$ in their nearest neighbor list (see Section~\\ref{subsec:symmetry}).\n\nGiven a starting point $x$ for query $q$, a greedy downhill search with backtracking is executed. Starting with $x$, the distance of $q$ to all neighbors of $x$, $y \\in \\mathcal{N}_x$, is computed and all points are inserted into a priority queue. The same procedure is repeated for the best point in the queue. A cache structure (Section~\\ref{subsec:caching}) stores which points already have been visited to avoid revisiting the same point over and over again. Finally, the best $k$ points found so far are returned.\n\n\n\\noindent \\textbf{Stopping Criterion.} In order to estimate when the $k$-closest point has been found, the search will terminate once the distance $d_\\text{next}$ to the best not yet visited point exceeds a threshold $\\xi$ beyond the distance to the $k$-closest known point $d_{\\text{best}_k}$, i.e.\\ if\n\\begin{equation}\n\\label{eq:stopping}\nd_\\text{next} > d_{\\text{best}_k} + \\xi =  d_{\\text{best}_k} + \\tau \\cdot \\min\\{d^+_{\\text{nn}_{1}}, d_{\\text{best}_1}\\}\n\\end{equation}\nholds, where $d^+_{\\text{nn}_{1}}$ denotes the maximum distance of the data points to their closest neighbor within a region of the graph (Figure~\\ref{fig:slack}).\nThe parameter $\\tau$ controls the size of the safety margin, while $d_{\\text{best}_1}$ serves as an estimate of the average local distance between points and $d^+_{\\text{nn}_{1}}$ provides a global limit.\n\\newline\n\\noindent \\textbf{Hierarchical Query.} Building up the search structure creates a tree of multiple layers. Starting at the coarsest layer, a query is carried out by brute force comparison against all nodes in the top layer segment. The best $k$ points found there will be the starting points on the next finer level, where a downhill search is executed as just explained. Again, the best $k$ points are used in the next finer layer. This is important to ensure that the final nearest neighbors are actually being approached from multiple sides to circumvent potential gaps. All points at a coarser level are also represented at a finer level, but at both levels with a different neighborhood. Thus, at coarser layers, the points are simply replicated.\nThe computed distances between points remain valid across levels.\n\\begin{figure}[tb]\n    \\centering\n    \\begin{subfigure}[t]{.45\\linewidth}\n        \\centering\n        \\includegraphics{figures\/slack}\n        \\caption{Adding a slack $\\xi$ allows to escape a local minimum $\\hat{x}$ eventually reaching the solution $x^\\star$ for a query $q$.}\n        \\label{fig:slack}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{.45\\linewidth}\n        \\centering\n        \\includegraphics{figures\/unsymmetric}\n        \\caption{Maintaining symmetric links. The edge $e$ is added to allow propagating nearest neighbor information between $z$ and $\\hat{x}$.}\n        \\label{fig:symmetric_links}\n    \\end{subfigure}\n    \\caption{Design options to prevent the greedy search algorithm from getting trapped in a local neighborhood.}\n\\end{figure}\n\\subsection{Symmetry}\n\\label{subsec:symmetry}\nIn order to reach any point from any direction, every edge of the graph, in principle, should be undirected.\nWhen every point should at least know its $k_\\text{nn}$ nearest neighbors, the total number of edges per point $x$ would vary significantly for an undirected graph as the number of points which have $x$ as their nearest neighbor will heavily depend on the geometry of the local neighborhood. However, to maintain a fixed number of edges $k$, our graph consists only of directed, \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot, outgoing edges (Section~\\ref{subsec:query}).\nThe list of links of point $x$ however is split into $k_\\text{nn}$ direct links to nearest neighbors of $x$ as well as $k_\\text{sym}$ potential inverse links to points which have $x$ in their nearest neighbor list.\nIn our implementation, the number of utilized inverse entries can vary, but it is guaranteed that that at least $k_\\text{nn} \\ge k\/2$ nearest neighbors are kept and will not be overwritten.\n\nAs the number of representable inverse links is limited, one has to determine which of the inverse links are necessary.\nHarwood and Drummond~\\cite{harwood2016fanng} introduce the concept of shadowed links, where a link from $x$ to $z$ is removed from the graph if a one-hop connection exists $(x \\rightarrow y \\rightarrow z)$ with $d(x,y) < d(x,z)$. Optimizing the entire graph to remove all potential shadow edges requires a complex global optimization. The same holds for the construction of monotonic relative neighborhood graphs~\\cite{fu2019fast} .\n\nInstead, we propose a relatively simple but effective, local criterion which enables faster construction times: It is clear, that an inverse link $x \\rightarrow z$ only needs to be considered if $z$ is not already in $x$'s $k_\\text{nn}$ nearest neighbor list (Figure~\\ref{fig:symmetric_links}).\nIn addition, the link is only considered to be included if there does not yet exist an indirect path back from $x$ to $z$ within the current graph structure.\nThis is checked by launching a query. If there is no path within the $\\xi$ range, the link is added.\n\nDuring construction, all points in parallel test for their $k_\\text{nn}$ neighbors if an inverse link should be added.\nFor the potentially $k_\\text{nn}$ queries, we maintain one local cache per point such that the cost of testing for indirect paths is really small as the query already starts within the direct vicinity and most distance calculations can be reused.\n\nInserting $z$ into $x$'s inverse list is done using an atomic increment on $k_\\text{sym}$. On average, less than $k\/4$ inverse links are necessary this way.\nIf $k_\\text{sym}$ is already full, the link is entered in the next best candidate along the path.\nIf no candidate is found the link is ignored. In our setting, this is, however, a very rare case and $z$ might potentially still be reachable through the hierarchy.\n\\subsection{Hierarchical Construction}\n\\label{subsec:hierarchy}\nIn order to cope with large-scale datasets or to allow for dynamic updates, there is a need to also scale the index construction according to modern hardware by a highly parallelizable approach.\n\nTypically, the construction of graph based-index structures is performed on CPUs \\citet{harwood2016fanng, fu2019fast, fu2016efanna, malkov2018efficient,li2019approximate}. Theses approaches rely on a global optimization schemes, where new edges are added sequentially (graph-diversification). While \\citet{malkov2018efficient} already leverage some sort of parallelization in the construction, their approach is still very dependent on global memory synchronization. Hence, their speed-up for each additional core is in logarithmic-scale. Additionally, all approaches work with dynamic edge lists for each point that vary heavily over the course of the construction. Thus, resulting in potentially high peak memory consumption and unconstrained access patterns.\n\nSuch graph-diversification methods already generate all relevant links ensuring global connectivity and avoiding local gaps, thus resulting in an effective search graph.\nUnfortunately, their construction time is generally very long -- even for million-scale point sets.\n\nThe goal of our parallel hierarchical construction scheme is to easily scale to large datasets.\nAs each sub-tree can be constructed independently, build-up times are within seconds for million-scale datasets and the scheme can easily be implemented via sharding in a multi-GPU setup. While flat, locally constructed kNN-based graphs might contain gaps impairing the query performance, the hierarchical structure will automatically provide bridges in these cases at some higher level.\n\nOur hierarchical construction procedure is illustrated in Figure~\\ref{fig:merging_new}.\nWe start with partitioning the complete dataset into $b$ batches of size $s$ and compute the brute-force kNN-graph for each batch.\nSince $s$ is generally small, e.g. $32$, this can be done quickly. After adding symmetric links, the batches form flat kNN-graphs that can also be seen as a hierarchical search graph of height $l=1$. First, $s$ points are selected equally from the current top layers of the $g$ sub-trees. For the top layer, a kNN-graph is constructed using brute-force distance calculations followed by symmetrization. Starting from the top, for each layer, one after the other, a query for all points is performed to find all nearest neighbors in that layer across all $g$ sub-trees and the kNN lists are updated accordingly. Afterwards, symmetric links are inserted. This way, all $g$ partitions of this layer are merged to form one consistent kNN-graph.\n\nAs the initial query into the $g$ sub-graphs might not always report the correct nearest neighbor at the first trial, one can run a couple of refinement iterations per layer. The quality of the initially imprecise kNN-graphs will improve with each iteration.\nAll layers are fused together top-down and a new search-tree emerges.\n\nEvery time the bottom layer is reached, we also save the distance to the first nearest neighbor $d_{nn_1}$ for each point and compute the mean and max over the dataset. This allows for an easy adaption of our stopping criterion (Eq.~\\eqref{eq:stopping}) to the given dataset. For selecting the points for the top layer, weighted reservoir sampling with $d_{nn_1}$ as weights is used with the method of \\citet{EFRAIMIDIS2006181}.\n\nThe entire merging approach for three sub-graphs is illustrated in Figure~\\ref{fig:tree_construction}.\n\n\n\\noindent \\textbf{Discussion.}\nThrough the bottom-up construction, an almost perfectly balanced tree is constructed. The number of points in each bottom layer batch might be  equally reduced if the total number of points does not match the tree geometry $(s\\times g)^l$  perfectly.\n\nThe bottom-up construction scheme creates an almost optimal search graph for each independent sub-tree.\nThus, during tree merging, the search query finds its true nearest neighbor with a very high probability in any of the search sub-trees. Since this means that all points in the bottom layer are updated with every new level, a high-quality search-graph is constructed. Using a sufficiently large branching factor $g$, the height $l$ of the tree is rather small even for huge datasets.\n\nPlease see the supplementary for pseudo code of the search and construction algorithms.\n\\subsubsection{Distance Caching on GPUs}\n\\label{subsec:caching}\nOne of the major deficits of GPUs is their limited memory. In particular, kNN-graphs constructions have a high demand for highly dynamic structures like unpredictably growing edge-lists, e.g.\\ a list of potential points to visit or that have already been visited. Our approach is to have a single query point per thread-block, where we use the shared memory as a multi-purpose cache. It consists out of three parts: 1) a best-list that stores the currently found best points and their distance to the query as a sorted list,  2) a priority queue that manages the points to be visited as a distance-sorted ring buffer, 3) a simple ring buffer that  holds the ids of points that have already been visited.\n\nWhen the next point to be visited is popped from the priority queue, the distances to all its $k$ neighbors have to be calculated and they need to be inserted into the three parts of the cache.\nBefore computing any distances, there is a parallelized check that tests in if any index is already known -- those points are discarded. For all unknown points, the distance to the query is computed. If the stopping criteria for the potential points is not exceeded, we perform a parallel insertion in the combined best-list and priority queue.\nPoints that drop out of the best-list, but are not visited yet are included in the priority queue buffer. Points that drop out of the priority queue buffer, which are already visited, are added to the ring buffer (3) to prevent cycles.\nThis approach enables very long paths while still being efficient with the resources a GPU has to offer.\n\\subsubsection{Multi-GPU}\nSince our approach is computing the correct distances on every comparison, it is important to always have fast access to the base vectors $\\mathcal{X}$ and the graph structure. They should reside on the GPU device memory.\nIn order to allow for large-scale datasets, we propose a very simple but effective multi-GPU scheme. $\\mathcal{X}$ is partitioned into shards that still fit onto the device. For each shard, an individual hierarchical search structure is assembled. The trees are kept separated. There is no need for merging the full kNN-graph on the bottom layer.\n\nFor a query point, each GPU computes the nearest neighbors in individually for its own sub-graph and reports the determined nearest neighbors. The results of the sub-graphs are finally merged together.\n\nThis scheme can also be carried out when the number of necessary shards exceeds the number of available GPUs. In this case, the GPUs load the data for each sub-graph sequentially.\n\\subsection{Starting Points}\n\\label{subsec:startingPoint}\nThe number of hops executed and the points visited and thus the number of distance calculations carried out does have major impact on the runtime of a query. Furthermore, the number of hops between any two points increases with the local point density. The effect is however less strong in high-dimensonal data sets.\n\nWith that in mind, we will discuss different strategies to select one or multiple starting points for a query:\n\n\\noindent\\textbf{Centroid.} Li~\\emph{et al}\\onedot \\cite{li2019approximate} start their query always from the centroid of the data set.\nAs the number of points grows, the distance to the centroid grows as well.\nIn addition, with a single starting point, any goal point will be approached from just one direction.\nThe search might be prone to gaps or linking problems inside the search-graph.\n\n\\noindent\\textbf{Hierarchical.} As explained above, during construction, we traverse the hierarchy from top to bottom, generating multiple seeds by fully exploring the top level.\nAlways starting with $k$ different points on the next layer allows for approaching any point from multiple directions.\nThis  results in very robust queries even if the search-graph is not yet perfect, which is the case during the construction phase.\nWhile in each layer there is only a small number of steps the complete the search visits a large number of points total. The distance calculations\nexceeds what has been reported in \\cite{harwood2016fanng}, leading to overall slower query times.\n\n\\noindent\\textbf{Jumping from Top to Bottom.}\nFor the final query, it turned out to be most efficient to first perform a brute-force search on the top level which allows for quick selection of multiple promising entry points,\nfollowed by a regular search on the bottom layer.\nSince the top layer consists of points which are also contained in the bottom layer, they can be used to initialize the priority queue as well as the cache.\nHaving multiple points of entry helps to make the query more robust against gaps in the graph as the query point is approached from multiple directions.\n\\section{Empirical Evaluation}\n\\label{sec:evalution}\nIn the following, we evaluate the performance of the proposed approach on several publicly available benchmark datasets and report qualitative and quantitative results in terms of timings and accuracy.\n\\paragraph{Datasets}\nWe ran experiments on SIFT1M~\\cite{PQ} and SIFT1B~\\cite{PQ} containing SIFT vectors of dimension 128.\nTo evaluate our approach on features extracted using convolutional neural networks, we ran experiments on 1 billion 96-dimensional feature representations of images in DEEP1B~\\cite{Babenko2016EfficientIO}.\nFor all evaluation on SIFT1M, we use a NVIDIA Titan RTX, whereas for SIFT1B and DEEP1B we use a machine with 8 NVIDIA Geforce GTX 1080 Ti.\n\\begin{table*}[tb]\n  \\caption{Comparison on different datasets. ${}^{(t)}$ indicates that this results is based on $t$ GPUs. For our proposed method (GGNN), we additionally report the required time to construct the index structure.}\n  \\label{tab:SIFT1M_comparision}\n  \\resizebox{\\textwidth}{!}{\\centering\n  \\begin{tabular}{\n  lS[table-format=3.2]ccccS[table-format=3.2]S[table-format=3.2]cccS[table-format=3.2]S[table-format=3.2]ccc\n  }\n  \\toprule\n  & \\multicolumn{4}{c}{SIFT1M} &\n  & \\multicolumn{4}{c}{SIFT1B} &\n  & \\multicolumn{4}{c}{DEEP1B}\\\\\n  \\cmidrule{2-5}\n  \\cmidrule{7-10}\n  \\cmidrule{12-15}\n  \\multirow{2}{*}{Approach}\n  & {Query time} & {Recall}  & {Recall} & {Recall} &\n  & {Query time} & {Recall}  & {Recall} & {Recall} &\n  & {Query time} & {Recall}  & {Recall} & {Recall}\n  \\\\\n  & {\\si{\\micro\\second}\/query} & {@1} & @10 & @100&\n  & {\\si{\\micro\\second}\/query} & {@1} & @10 & @100&\n  & {\\si{\\micro\\second}\/query} & {@1} & @10 & @100\n  \\\\\n  \\midrule\n     Exhaustive Search\n     & 23700~~ & 1.00 & {-} & {-} &\n     & {-} & {-} & {-} & {-}&\n     & {-} & {-} & {-} & {-}\n     \\\\\n  \\midrule\n     LOPQ~\\cite{localPQ}\n     & 51100 & 0.51 & 0.93 & 0.97 &\n     & 8000 & {-} & 0.20 & {-} &\n     & {-} & {-} & {-} & {-}\n     \\\\\n     IVFPQ~\\cite{PQ}\n     & 11200 & 0.28 & 0.70 & 0.93 &\n     & 74000 & 0.08 & 0.37 & 0.73 &\n     & {-} & {-} & {-} & -\n     \\\\\n     FLANN~\\cite{flann_pami_2014}\n     & 5320 & 0.97 & {-} & {-} &\n     & {-} & {-} & {-} & {-} &\n     & {-} & {-} & {-} & {-}\n     \\\\\n     Multi-D-ADC~\\cite{invertedmultiindex}\n     & {-} & {-}& {-}& {-}&\n     & 1600 & 0.33 & 0.80 & 0.97&\n     & 1500 & 0.36 & 0.71 & 0.91\n     \\\\\n  \\midrule\n     PQT~\\cite{wieschollek2016efficient}\n     & 20 & 0.51 & 0.83 & 0.86&\n     & 150 & 0.14 & 0.35 & 0.57&\n     & {-} & {-} & {-} & {-}\n     \\\\\n     FAISS~\\cite{johnson2019billion}\n     & 20 & 0.80 & 0.88& 0.95&\n     & 17.7 & {-} & 0.37 & {-} &\n     & 13 ${}^{(4)}$ & 0.45 & {-} & {-}\n     \\\\\n     PQFPGA~\\cite{PQFPGA}\n     & 20 & 0.88 & 0.94& 0.97&\n     & 20 & {-} & 0.55 & -&\n     & {-} & {-} & {-} & -\n     \\\\\n     RobustiQ~\\cite{chen2019robustiq}\n     & {-} & {-} & {-} & -&\n     & 33 & 0.33 & 0.76 & 0.90&\n     & 30 & 0.38 & 0.75 & 0.89\n     \\\\\n\\midrule\n  &  &  &  $\\tau$  & index &\n  &  &  &  $\\tau$  & index &\n  &  &  &  $\\tau$  & index\n  \\\\ \\cmidrule{4-5} \\cmidrule{9-10} \\cmidrule{14-15}\n     GGNN\n     & 4.2 & \\textbf{0.99} & 0.60 & 17.8s &\n     & 38${}^{(8)}$ & {~~~~\\textbf{0.99}} & 0.6 & 87 min &\n     & 90${}^{(8)}$ & {~~~~\\textbf{0.98}} & 0.6 & 101 min\n     \\\\\n     GGNN\n     & 1.1& 0.95& 0.42 & 17.8s &\n     & 20${}^{(8)}$ & 0.97 & 0.5 & 87 min &\n     & 48${}^{(8)}$ & 0.96 & 0.5 & 101 min\n     \\\\\n     GGNN\n     & 0.7 & 0.90 & 0.35 & 17.8s &\n     & 58${}^{(8)}$ & 0.98 & 0.7 & 34 min &\n     & 124${}^{(8)}$ & 0.97 & 0.7 & ~~40 min\n     \\\\\n  \\bottomrule\n  \\end{tabular}\n  }\n\\end{table*}\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/SIFT1M_comparison.pdf}\n    \\caption{Accuracy on SIFT1M for varying query time. Compared to other graph-based approaches at the same accuracy for the 10 nearest neighbors, our single-GPU based query is sped up by more than one order of magnitude.}\n    \\label{fig:sift1m_comparison}\n\\end{figure}\n\\subsection{Query Time and Recall}\nWhen comparing results to other approaches, one has to carefully look at the employed metric.\nThe query performance is typically measured in recall (R@k), \\emph{i.e}\\onedot} \\def\\Ie{\\emph{I.e}\\onedot the fraction of the nearest neighbors found in the first $k$ proposed vectors.\nAs our algorithm uses exact distance calculations, the true nearest neighbor is either reported as the first element in an answer or not found at all. Therefore, only R@1 is reported for our method.\nTable~\\ref{tab:SIFT1M_comparision} compares the query time and R@1 for recent CPU and GPU-based approaches on SIFT1M.\nBesides being significantly faster (around 1 $\\mu$s per query), our method can also achieve very high recall rates, close to perfect.\n\nSimilar accuracy can also be achieved with other graph-based approaches but their query time on the CPU is significantly larger (see Figure~\\ref{fig:sift1m_comparison}). CPU based reference implementations for the graph based methods \\cite{malkov2018efficient,li2019approximate,fu2019fast} are computed on a Intel Core i7-9700K.\nFANNG~\\cite{harwood2016fanng} report results for SIFT1M that come relatively close to ours in computation time and accuracy but we did not manage to reproduce their results with our reimplementation.\n\\subsection{Graph Construction and Quality}\n\\label{subsec:graph_quality}\nThe quality of our graphs can be controlled by the number of iterations used for refinement. We analyze both the quality of the construction in terms of finding the true $k_\\text{nn}$ nearest neighbors of the original data points as well as the influence on any other query.\n\nWhen looking at determining $k$ nearest neighbors, we report the following consensus measure\n\\begin{align}\n    C@k=\\frac{\\left\\vert \\mathcal{N}_x^\\text{gt}(k) \\cap \\mathcal{N}_x(k)\\right\\vert}{\\left\\vert \\mathcal{N}_x^\\text{gt}(k) \\right\\vert},\n\\end{align}\nwhich counts the overlap between the ground truth $k$ nearest neighbors and the reported $k$ points.\n\\begin{figure}[tb]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/construction_time_vs_query_time.pdf}\n\\caption{Trade-off between construction and query time on SIFT1M. Higher graph quality obtained by increasing $k$ and by varying the number of refinement iterations results in fewer steps during query.\n    However, increasing $k$ also increases the cost of each step.}\n    \\label{fig:build_vs_query}\n\\end{figure}\n\nOn SIFT1M, our algorithm achieves C@10 of 0.987 without refinement and 0.996 with 5 refinement iterations.\nIn Figure~\\ref{fig:build_vs_query}, the trade off between build time and query time for a fixed recall rate is visualized.\nThe build time is controlled by slack variable $\\tau$ and how many refinement iterations are carried out.\nThe longer the construction time, the more precise the resulting graph.\nA quickly assembled graph will have worse edges and a query might need to visit more nodes until fulfilling the stopping criterion.\n\nIn Table~\\ref{tab:SIFT1M_comparision}, the construction time of our method is listed for the standard datasets.\nFor comparison, FAISS~\\cite{johnson2019billion} report construction times between 4 and 24 hours for DEEP1B.\nThe hierarchical tree-merge algorithm speeds up the construction time by a large factor while the quality is similar or even improved.\n\nOur algorithm is the first graph-based approach that is able to construct an effective kNN-graph-based search structure even for the billion-scale data sets SIFT1B and DEEP1B for which it can reach up to 0.99 R@1.\n\\subsection{Query Behaviour}\n\\label{subsec:query_behaviour}\nThe runtime and the quality of a query can be controlled by adapting the stopping criterion through the slack variable $\\tau$. As can be seen in Table~\\ref{tab:SIFT1M_comparision}, by increasing $\\tau$, a larger safety margin is considered during query,\nresulting in better recall rates at the cost of visiting more points and consequently longer query time.\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figures\/distance_to_query_at_k.pdf}\n\\caption{Development of the distance to the 10th best point over the execution of multiple queries.\n    The marker indicates the iteration in which the last improvement is found. The plot visualizes the effectiveness of our termination criterion.}\n    \\label{fig:termination_criterion}\n\\end{figure}\n\nIt is instructive to look at the behaviour of the query over time. In Figure~\\ref{fig:termination_criterion}, the initial distance of the query to the starting point, i.e.\\ the closest point on the top layer, is drastically reduced already after very few iterations. Here, an iteration stands for fetching the best point from the priority queue and calculating the distance to all its neighbors. Most time is spent to carefully explore the neighborhood around the true nearest neighbor. The marks indicate that the last improvement in distance occurs relatively shortly before the search is terminated. This demonstrates that our termination criterion in Eq. \\eqref{eq:stopping} is effectively preventing too many iterations.\n\\section{Conclusion}\n\\label{sec:conclusion}\nOur algorithm accelerated the construction time for graph-based search structures by a large factor and represents the fastest kNN query technique while maintaining a recall rate between 0.9 and 1.\nDue to the parallel construction and merging of sub-graphs, our hierarchical construction scheme creates a high-quality kNN graph with graph-diversification links for very efficient traversal. It is easily deployed in multi-GPU systems to cope with very large-scale datasets. It is the first graph-based kNN method to report results on billion-scale data sets like SIFT1B and DEEP1B. In addition, it is the first ANN search method overall that achieves 0.99 Recall@1 for those, and it is by far the fastest.\nFor million-scale datasets, search-graphs of sufficient quality can be constructed within seconds, allowing for quick nearest-neighbor searches on intermediate data structures as a part of larger GPU-based algorithms.\nUpon acceptance, we will release an OpenSource\\footnote{\\url{https:\/\/github.com\/cgtuebingen\/ggnn}} implementation of our approach for easy inclusion in other projects.\n\nCurrently, our scheme computes the exact distance to all visited high-dimensional points.\nAs the number of distance calculations dominates the query time, a compressed representation of the data vectors could lead to further acceleration.\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nPlanet formation occurs in disks around young stellar objects. Interactions between planets and disks are very complex. Young planets are expected to cause rings, cavities, spirals, and disturbances in the velocity field and other features in the disk, which in turn may be used to infer the presence of these young planets. In the past few years, much evidence about this phase of planet formation has been accumulated because high-resolution images in the millimeter and sub-millimeter wavelength ranges have been provided by the Very Large Array (VLA) and the Atacama Large Millimeter Array (ALMA) (see e.g. the case of HL Tau: \\cite{ALMA2015}), and by high-contrast imagers such as the Gemini Planet Imager (GPI: \\cite{Macintosh2014}) and SPHERE (Spectro-\nPolarimetic High contrast imager for Exoplanets REsearch: (\\cite{Beuzit2008}; see, e.g., \\cite{Avenhaus2018}). The literature on indirect evidence of the presence of planets is now becoming very rich, and nearby young stars surrounded by gas-rich disks are intensively studied for this purpose. In most cases, available data cannot fully eliminate alternative hypotheses, or the data have ambiguous interpretations (see, e.g., \\cite{Bae2018} and \\cite{Dong2018}), although strong indirect evidence of the presence of planets from local disturbances of the velocity field have recently been considered for the case of HD~163296 (\\cite{Pinte2018, Teague2018}). In general, small grains are thought to be more strongly coupled with gas and are thus less sensitive to radial drift and concentration that can strongly affect large grains (see the discussion in \\cite{Dipierro2018}). For this reason, observations at short wavelengths provide an important complementary view of what can be seen with ALMA. On the other hand, a direct detection of still-forming planets embedded within primordial gas-rich disks, which is expected to be possible with high-contrast imaging in the near infrared (NIR), is still scarce; remarkable cases are LkCa-15 (\\cite{Kraus2012, Sallum2015}) and PDS-70 (\\cite{Keppler2018, Muller2018, Wagner2018}). In particular, in this second case, a clear detection of an accreting planet in the cavity between the inner and outer ring was obtained, making it an archetype for planet formation and planet-disk interactions. However, many cases remain ambiguous; a classical example is HD~100546 (see, e.g., \\cite{Quanz2013a, Currie2014, Quanz2015, Currie2015, Rameau2017, Sissa2018}).\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.8cm]{qphi.jpeg}\n  \n  \n\\caption{View of the surroundings of HD~169142 obtained from polarimetric observations: the left panel shows the Q$_\\Phi$\\ image in the J band acquired with SPHERE (\\cite{Pohl2017}) on a linear scale. The two rings are clearly visible. The right panel shows a pseudo-ADI image of the inner regions obtained by differentiating the Q$_\\Phi$\\ image (see \\cite{Ligi2017}, for more details). The white cross marks the position of the star, and the cyan circle shows the size of the coronagraphic mask. The other labels refer to the blobs we discuss in this paper that are also visible in these images. The color scale of the differential image is five times less extended to show the faint structures better. In both panels, N is up and E to the left; a segment represents 1 and 0.5 arcsec in the left and right panel, respectively. }\n\\label{f:pdi}\n\\end{figure*}\n\nHD~169142 is a very young Herbig Ae-Be star with a mass of 1.65-2~M$_\\odot$ and an age of 5-11~Myr (\\cite{Blondel2006, Manoj2007}) that is surrounded by a gas-rich disk ($i=13$~degree; \\cite{Raman2006}; $PA=5$~degree; \\cite{Fedele2017}) that is seen almost face-on. The parallax is $8.77\\pm 0.06$~mas (GAIA  DR2 2018). Disk structures dominate the inner regions around HD~169142 (see, e.g., \\cite{Ligi2017}). Figure~\\ref{f:pdi} shows the view obtained from polarimetric observations: the left panel shows the Q$_\\Phi$\\ image in the J band obtained by \\cite{Pohl2017} using SPHERE on a linear scale, and the two rings are clearly visible. The right panel shows a pseudo-ADI image of the inner regions obtained by differentiating the  Q$_\\Phi$\\ image (see \\cite{Ligi2017}, for more details). \\cite{Biller2014} and \\cite{Reggiani2014} discussed the possible presence of a point source candidate at small separation ($<0.2$~arcsec from the star). However, the analysis by Ligi et al. based on SPHERE data does not support or refute these claims; in particular, they suggested that the candidate identified by Biller et al. might be a disk feature rather than a planet. Polarimetric images with the adaptive optics system NACO at the Very Large Telescope (VLT) (\\cite{Quanz2013b}), SPHERE (\\cite{Pohl2017, Bertrang2018}) and GPI (\\cite{Monnier2017}) show a gap at around 36~au, with an outer ring at a separation $>$40~au from the star. This agrees very well with the position of the rings obtained from ALMA data (\\cite{Fedele2017}); similar results were obtained from VLA data (\\cite{Osorio2014, Macias2017}). We summarize this information about the disk structure in Table~\\ref{t:rings} and call the ring at 0.17-0.28 arcsec from the star Ring 1 and the ring at 0.48-0.64 arcsec Ring 2. We remark that in addition to these two rings, both the spectral energy distribution (\\cite{Wagner2015}) and interferometric observations (\\cite{Lazareff2017, Chen2018}) show  an inner disk at a separation smaller than 3 au. This inner disk is unresolved from the star in high-contrast images and consistent with ongoing accretion from it onto the young central star. While the cavities between the rings seem devoid of small dust, some gas is present there (\\cite{Osorio2014, Macias2017,  Fedele2017}). \\cite{Fedele2017} and \\cite{Bertrang2018} have suggested the possibility that the gap between Rings 1 and 2 is caused by a planet with a mass slightly higher than that of Jupiter. However, this planet has not yet been observed, possibly because it is at the limit of or beyond current capabilities of high-contrast imagers. On the other hand, \\cite{Bertrang2018} found a radial gap in Ring 1 at PA$\\sim 50$\\ degree that might correspond to a similar radial gap found by \\cite{Quanz2013b} at PA$\\sim 80$\\ degree. The authors noted that if this correspondence were real, then this gap might be caused by a planet at about 0.14 arcsec from the star. So far, this planet has not been unambiguously detected either.\n\n\\begin{table*}\n\\centering\n\\caption{Rings around HD169142 from the literature}\n\\begin{tabular}{lccc}\n\\hline\n\\hline\nInstrument & Source & Ring 1 & Ring 2 \\\\\n           &        & arcsec & arcsec \\\\\n\\hline\nALMA          & \\cite{Fedele2017}   & 0.17-0.28 & 0.48-0.64 \\\\\nVLA           & \\cite{Osorio2014}   & 0.17-0.28 & 0.48-     \\\\\nSUBARU-COMICS & \\cite{Honda2012}    &   0.16-   &           \\\\\nNACO          & \\cite{Quanz2013b}   & 0.17-0.27 & 0.48-0.55 \\\\\nSPHERE-ZIMPOL & \\cite{Bertrang2018} & 0.18-0.25 & 0.47-0.63 \\\\\nSPHERE-IRDIS  & \\cite{Pohl2017}     & 0.14-0.22 & 0.48-0.64 \\\\\nGPI           & \\cite{Monnier2017}  &    0.18   &    0.51   \\\\\n\\hline\n\\end{tabular}\n\\label{t:rings}\n\\end{table*}\n\nIn this paper, we pursue a new view on the subject through analyzing high-contrast images. In particular, we underline that while polarimetric observations in the NIR and millimeter observations are best to reveal the overall structure of the disk, pupil-stabilized NIR observations where angular differential imaging can be applied may reveal fainter structures on a smaller scale. The risk of false alarms inherent to the image-processing procedures used in high-contrast imaging can be mitigated by comparing different sets of observations taken at intervals of months or years. In the case of HD~169142, this is exemplified by the study of \\cite{Ligi2017}, who identified a number of blobs within Ring 1. We have now accumulated a quite consistent series of observations of this star with SPHERE that extends the set of data considered by Ligi et al. The observations have a comparable limiting contrast so that we may try to combine this whole data set to improve our knowledge of this system. The combination of different data sets acquired over a few years offers several advantages. In addition to verification of previous claims, we might try to detect persistent features around HD~169142 using a coincidence method to obtain a combined image that is deeper than the individual images and allows a quantitative discussion of the false-alarm probability of detected features. The expected orbital motion needs to be taken into account in this.\n\nIn Section 2 we describe observation and analysis methods. In Section 3 we present the main results about the blobs we detect around the star. in Section 4 we discuss the spiral arms within the disk and the possible connection to the blobs. Conclusions are given in Section 5.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.8truecm]{all_epochs.jpeg}\n  \n  \n\\caption{Signal-to-noise ratio maps for the individual epochs for IFS. The individual S\/N maps are obtained from the ASDI PCA algorithm using 50 components and making a median over the wavelength (see \\cite{Mesa2015}). In the upper row we show in the left panel JD=57180.17, in the middle panel JD=57201.12, and in the right panel JD=57499.34.  In the lower row we show in the  left panel JD=57566.15, in the middle panel JD=57873.30, and in the right panel JD=58288.19. In all panels, the central 0.1 arcsec is masked, the solid white line at the bottom represents 1 arcsec, the white cross represents the position of the star. N is up and E to the left.}\n\\label{f:individual}\n\\end{figure*}\n\n\n\\section{Observation and data analysis}\n\nData were acquired with the SPHERE high-contrast imager (\\cite{Beuzit2008}) at the ESO VLT Unit Telescope 3 within the guaranteed time observations used for the SHINE (SpHere INfrared survey for Exoplanets) survey (\\cite{Chauvin2017}). Data acquired up to 2017 have been described in \\cite{Ligi2017}. Here we add new data acquired in 2018 and study the system anew using different ways to combine different images. In these observations, we used SPHERE with both the Integral Field Spectrograph (IFS : \\cite{Claudi2008}) and the Infra-Red Dual Imaging and Spectrograph (IRDIS: \\cite{Dohlen2008, Vigan2010} simultaneously. IFS was used in two modes: Y-J, that is, with spectra from 0.95 to 1.35~$\\mu$m and a resolution of $R\\sim 50$; and Y-H, with spectra from 0.95 to 1.65~$\\mu$m and a resolution of $R\\sim 30$. When IFS was in Y-J mode, IRDIS observed in the H2-H3 narrow bands (1.59 and 1.66~$\\mu$m, respectively); when IFS was in Y-H mode, IRDIS observed in K1-K2 bands (2.09 and 2.25~$\\mu$m, respectively). Hereafter we mainly consider data acquired with the IFS;  IRDIS data are considered for the photometry in the K1-K2 bands. We considered the six best observations obtained for HD169142 (see Table~\\ref{t:obs}). Most of the epochs were obtained with an apodized Lyot coronagraph (\\cite{Boccaletti2008}: the field mask in YJH has a radius of 92~mas, and  of 120~mas for the K-band coronagraph). Two of the observations (obtained in better observing conditions) were acquired without the coronagraph in order to study the very central region around the star. The use of the coronagraph allows a better contrast at separation larger than $\\sim 0.1$~arcsec. For all data sets, the observations were acquired in pupil-stabilized mode. In addition to the science data, we acquired three kind of on-sky calibrations: (i) a flux calibration obtained by offsetting the star position by about 0.5 arcsec, that is, out of the coronagraphic mask (point spread function, PSF, calibration). (ii) An image acquired by imprinting a bidimensional sinusoidal pattern (waffle calibration) on the deformable mirror. The symmetric replicas of the stellar images obtained by this second calibration allow an accurate determination of the star centers even when the coronagraphic field mask is in place. These calibrations were obtained both before and after the science observation, and the  results were averaged. (iii) Finally, an empty field was observed at the end of the whole sequence to allow proper sky subtraction. This is relevant in particular for the K2 data sets.\n\nData were reduced to a 4D datacube (x, y, time, and $\\lambda$) at the SPHERE Data Center in Grenoble (\\cite{Delorme2017}) using the standard procedures in the SPHERE pipeline (DRH: \\cite{Pavlov2008}) and special routines that recenter individual images using the satellite spot calibration, and correct for anamorphism, true north, and filter transmission. Faint structures can be detected in these images using differential imaging. Various differential imaging procedures were run on these data sets. We used here results obtained with a principal component analysis (PCA; see \\cite{Soummer2012}) applied to the whole 4D datacubes, which combines both angular and spectral differential imaging in a single step (ASDI-PCA: see \\cite{Mesa2015}). The PCA algorithm we used is the singular-value decomposition that generates the eigenvectors and eigenvalues that are used to reconstruct the original data. A principal components subset was used to generate an image with the quasi-static noise pattern that can then be subtracted from the original image. Clearly, the larger the number of principal components, the better the noise subtraction, but this also means that the signal from possible faint companion objects is cancelled out more strongly. Most of the results were obtained using 50 modes, but we also considered other numbers of modes (10, 25, 100, and 150 modes). To avoid spectrum distortion characteristics of the ASDI-PCA, photometry was obtained using a monochromatic PCA with only two modes for each spectral channel. Photometry was obtained with respect to the maximum of the PSF calibration corrected for the attenuation inherent to the PCA. The final step  of the procedure was to obtain signal-to-noise ratio (S\/N) maps from the IFS images obtained by making a median over wavelengths. \n\nFinally, we also used the $Q_\\Phi$\\ image obtained by \\cite{Pohl2017} for astrometry, reduced as described in that paper and in \\cite{Ligi2017}. We note that this data set was obtained with the YJ field mask, whose radius is 72.5~mas.\n\n\\begin{table*}\n\\caption{Journal of observations}\n\\begin{centering}\n\\begin{tabular}{lccccccl}\n\\hline\n\\hline\nJD       & Mode & nDIT$\\times$DIT &   Angle & Seeing &  lim. cont  & coro & Ref\\\\\n         &      &   (sec)          & (degree)  & (arcsec) &   (mag)       &      &\\\\\n\\hline\n57145.   & Pol J & 3180 & Field & 0.90 & & YJ & \\cite{Pohl2017} \\\\\n57180.17 &  Y-J &   86$\\times$64  &  ~45.82 &  1.57 &   13.13 & YJH & \\cite{Ligi2017}\\\\        \n57201.12 &  Y-H &   65$\\times$64  &  ~36.42 &  1.00 &   13.52 & YJH & \\cite{Ligi2017}\\\\          \n57499.34 &  Y-J &   77$\\times$64  &  144.62 &  1.88 &   13.06 & YJH & \\cite{Ligi2017}\\\\          \n57566.15 &  Y-H &  322$\\times$2   &  147.33 &  0.67 &   13.64 &  no & \\cite{Ligi2017}\\\\  \n57873.30 &  Y-H &  192$\\times$2   &  ~98.82 &  0.62 &   14.07 &  no & \\cite{Ligi2017}\\\\\n58288.19 &  Y-H &   48$\\times$96  &  120.17 &  1.19 &   13.79 &  K  & This paper\\\\       \n\\hline\n\\end{tabular}\n\\end{centering}\n\\label{t:obs}\n\\end{table*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18.8cm]{mixed.jpeg}\n  \n  \n\\caption{Upper left panel: Median over time of the wavelength-collapsed images of HD169142 obtained with an ADI PCA algorithm, one mode per wavelength. Upper right panel: Image obtained by averaging the S\/N maps for the individual epochs for IFS. The individual S\/N maps are obtained from the ASDI PCA algorithm using 50 components and making a median of the wavelength (see \\cite{Mesa2015}), and they were rotated for Keplerian motion to the last image before making the median. Lower panel: Coincidence image obtained from the same data set. In all panels, the solid line at the bottom of each panel represents 1 arcsec, and a white cross shows the position of the star. N is up and E to the left.}\n\\label{f:sum}\n\\end{figure*}\n\n\\section{Results}\n\nThe following discussion is based on the application of differential imaging algorithms that allow detecting faint structure that is not otherwise easily detectable in the images. The typical contrast of Ring 1 that we were able to measure using simple subtraction of a reference image is about $1.5\\times 10^{-3}$. The structures we consider in this paper are more than an order of magnitude fainter. They represent small fluctuations of the signal that cannot be detected without differential imaging.\n\nFigure~\\ref{f:individual} shows the S\/N maps obtained by applying the PCA ASDI algorithm to the IFS data for the individual epochs. The images have a linear scale from S\/N=0 (dark) to S\/N=5 (bright). These figures clearly show a similar pattern of bright spots, as well as a rotation of these spots with time. This suggests that a combination of the images that takes into account a Keplerian motion around the star should improve detection of the real pattern present in the data. The full solution is quite complex, leaving many free parameters, and may be attempted using an approach such as that considered by K-stacker (\\cite{Nowak2018}). However, a simplified approach that greatly reduces the number of free parameters is to assume that the system is seen face-on and that the orbits are circular: if the distance is known, the only free parameter is the stellar mass. This appears to be a reasonable approximation for disk-related features around HD~169142 because in this case, we only consider a fraction of the orbit. On the other hand, observations spread over a few years enable separating static features that are due to radiative transfer effects from scattered-light fluctuations that are due to moving clouds or sub-stellar objects.\n\nThe upper panels of Figure~\\ref{f:sum} show images of HD169142 obtained by combining the six individual images, assuming the distance given by GAIA DR2, a mass of 1.7~M$_\\odot$\\ (see below), and circular orbits.\n\n\\subsection{Coincidence images}\n\nTo improve our ability of discerning faint signals, we combined data from different epochs using a coincidence map (see the lower panel of Figure~\\ref{f:sum}). The principle of this coincidence map is to start with S\/N maps for individual epochs. We used S\/N maps after correcting for the small-number-statistics effect using the formula by \\cite{Mawet2014}. The maps were then multiplied by each other. The S\/N maps average to zero, with both positive and negative values for individual pixels. Of course, this may result in a false-positive signal for an even number of negative signals in the individual S\/N maps. To avoid this problem, we arbitrarily set to negative the result for a given pixel when the signal for that pixel was negative for at least one epoch. Of course, this is not a realistic flux map: the aim is merely to identify consistent signals throughout all individual images.\n\nTo consider the orbital motion around the star over the three years covered by our observations, we divided the field into 65 rings, each one 2 IFS pixels wide (15 mas, i.e., about 1.8 au at the distance of HD~169142). For each ring, we rotated the S\/N maps obtained at different epochs with respect to the first reference image according to Kepler's third law. When the distance to the star was fixed, the only free parameter remaining in this model is the (dynamical) stellar mass. If there is a companion orbiting the star, the signal is maximized for a value of the mass that, if the assumptions made (circular motion seen face on) are correct, is the dynamical mass of the star. If these hypotheses are not correct, the estimate of the mass is incorrect, by a value that depends on the real orbital parameters. We adopted a mass of 1.7~$M_\\odot$, the GAIA parallax, and clockwise rotation, as indicated by the analysis of motion of disk features in \\cite{Ligi2017}; see also \\cite{Macias2017}. Figure~\\ref{f:sum} shows the coincidence map and a mean of the S\/N maps for the six epochs for this value of the mass.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{compdisk.jpeg}\n  \n  \n\\caption{Same as the lower panel of Fig~\\ref{f:sum}, but showing the edges of the two disk rings (Ring 1 in green, Ring 2 in magenta). The ring edges are drawn according to \\cite{Fedele2017}. The blobs are labeled. The white solid line represents 1 arcsec, and the white cross shows the position of the star. N is up and E to the left.}\n\\label{f:rings}\n\\end{figure}\n\n\\subsection{Blob detection}\n\nA quite large number of blobs can be found around HD~169142. Several of them are found consistently in all individual images and are also visible in the J-band $Q_\\Phi$ image seen in Figure~\\ref{f:pdi}; some of them have been identified and discussed by \\cite{Ligi2017}.  We fixed our attention on four of them (see Figure~\\ref{f:pdi} and Figure~\\ref{f:rings} for their definition). The two brightest blobs (B and C) are within Ring 1 and have been identified by \\cite{Ligi2017}; they called them blobs A and B, respectively. Our blob A is closer to the star than Ring 1. Blob D is between Rings 1 and 2. All of these blobs appear to be slightly extended. We verified in the individual images that this is not an artifact caused by combining individual images. For instance, when we consider the best set of data (the last set from June 2018), the FWHM of blobs B and C can be measured with reasonable accuracy at about 40 mas, which is significantly larger than expected for a point source at this separation (about 26 mas, after applying differential imaging). To better estimate the physical size of the blobs, we compared the FWHM measured in our differential images with the FWHMs obtained for fake blobs that are the result of convolving Gaussian profiles with the observed PSF inserted into the images at the same separation but at a different position angle, and processed through the same differential imaging procedure. We repeated this procedure for the images obtained considering 50 modes (best image for detection) and with a less aggressive image where only 25 modes were considered,  which better conserves the original shape of the blobs. In this way, we found that the FWHM of blobs B and C is the same as that of fake Gaussian blobs with an intrinsic FWHM of 42 and 30 mas for blobs B and C, respectively. However, these are average values for tangential and radial profiles (with respect to the star): both blobs appear elongated in the tangential (rotation) direction with axis ratios of 1.4 (blob B) and 1.9 (blob C)\\footnote{This is not as obvious from a simple visual inspection of the images because the ADI processing that is implicit in the PCA-ASDI procedure we used deforms the images.}. The uncertainty on this size estimate is of about 7 mas, as obtained by comparing the results obtained in individual images. This size corresponds to $\\sim 4-5$~au, with an uncertainty of about 1~au. This result should be considered with some caution because the light distribution of the blobs might be not well reproduced by Gaussians.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{zoomD.jpeg}\n  \n  \n\\caption{Zoom of a two-color image of the region around blob D. This image was constructed using the K2 observation of JD=2458288.19 (red) and the weighted sum of all the IFS images (collapsed against wavelength) and rotated for a Keplerian motion assuming that the star has a mass of 1.7~M$_\\odot$ (blue). This last image is for the same epoch as the K2 observation. For clarity, the region within 0.28 arcsec from the star (i.e., within the outer edge of Ring 1) was masked in the K2-band image. The green circle is centered on the position of the blob measured in the K2 image. We note the different aspect and small offset between the position of the blob in the K2 image with respect to that at shorter wavelength. The white solid line represents 1 arcsec. N is up and E to the left. The white cross marks the position of the star.}\n\\label{f:blobd}\n\\end{figure}\n\nIn Figure~\\ref{f:blobd} we show a zoom of the region around blob D in a two-color image. The blue structures visible in the image are obtained through IFS in the Y, J, and H bands. They could be interpreted as stellar light scattered by a (dusty) spiral structure around a protoplanet that is accreting material funnelled through the spiral arm from the disk. This interpretation agrees with the detection in the $Q_\\Phi$\\ image (see Figure~\\ref{f:pdi}). In the same image, the red structures are obtained through IRDIS observing in the K2 band. In particular, the structure in the green circle that appears to be much more similar to a point source might indicate a planetary photosphere. The position of the blob in the K2 image is Sep=332 mas, PA=34.9 degree, which is not the photocenter at shorter wavelengths. Even if this interpretation is speculative (there are other structures in this image that we consider as noise), various circumstantial arguments discussed below possibly support it. We return to this point in the next section.\n\nThere is of course some probability that these detections are spurious. In order to estimate the false-alarm probability (FAP), we proceeded as follows. First, we fixed the stellar mass at the value given by fitting isochrones (1.7~$M_\\odot$). With this assumption, the prediction for the orbital motion is fully independent of the SHINE data set. We derotated the individual images to the same epoch using the same approach as described above (ring by ring). We searched for signals in the final coincidence data set using the FIND procedure in IDL. We recovered the detection of the candidate. We ordered the different epochs according to the value of the S\/N at the candidate positions (separately for each candidate). We then used binomial statistics on the remaining epochs (i.e., excluding the reference with the highest S\/N), considering as number of trials the number of pixels with an S\/N higher than the S\/N measured in the candidate position in the image giving the highest S\/N value at this position. To estimate the probability in the binomial statistics, we considered the product $\\prod_c$\\ of the S\/N rankings in the pixel corresponding to the candidate position in the remaining images, and compared this product to a similar product $\\prod_r$ obtained from random extractions. We repeated the random extraction $10^7$ times, and assumed that the probability of success is given by the fraction of cases where $\\prod_r < \\prod_c$.\n\nWith this approach, we obtained the FAP values listed in the second column of Table~\\ref{t:blobrot}; values for blobs B, C, and D are highly significant.\n\n\n\\subsection{Blob astrometry}\n\n\\begin{figure*}\n\\begin{tabular}{cc}\n\\centering\n\\includegraphics[width=8truecm]{astro_bloba.png}&\n\\includegraphics[width=8truecm]{astro_blobb.png}\\\\\n\\includegraphics[width=8truecm]{astro_blobc.png}&\n\\includegraphics[width=8truecm]{astro_blobd.png}\\\\\n\\end{tabular}\n  \n  \n\\caption{Variation in PA of the blobs with time. Upper left panel: blob A. Upper right panel: blob B. Lower left panel: blob C. Lower right panel: blob D. The dashed lines are best-fit lines through the points. Dash-dotted lines are predictions for circular orbits assuming a mass of 1.85~M$_\\odot$\\ for the star.}\n\\label{f:astro}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{kepler_motion.png}\n    \\caption{Run of the angular speed as a function of separation for the four blobs around HD169142; these values are on the disk plane. Overimposed we show predictions for circular Keplerian motion for three different values for the stellar mass (1.5, 2.0, and 2.5 M$_\\odot$).}\n    \\label{f:kepler}\n\\end{figure}\n\nAll these blobs rotate around the star. This can be shown by measuring their position in the individual images (see Table~\\ref{t:blobastro}).  We used the IDL FIND algorithm that uses marginal Gaussian distributions to measure the position of the spot centers in the ASDI 50 components images obtained at the various epochs. In addition, we also measured the blob positions in the polarimetric image. We found that the rotational speed decreases with separation (see Table~\\ref{t:blobrot} and Figure~\\ref{f:astro}), as expected for Keplerian motions. Figure~\\ref{f:kepler} shows the run of the angular speed as a function of separation for the four blobs around HD169142; these values are on the disk plane. Overimposed are predictions for circular Keplerian motion for three different values for the mass of the star (1.5, 2.0, and 2.5 M$_\\odot$). \n\nWhen we interpret the observed angular motion as Keplerian circular orbits in the disk plane, we can determine the mass of HD~169142. When we use the three blobs B, C, and D, the mass of the star is $1.85\\pm 0.25$~M$_\\odot$ (we did not use blob A here because it has too few astrometric points). When we add the uncertainties that are due to parallax  (0.05 M$_\\odot$) and disk inclination (0.09 M$_\\odot$), the result is $1.85\\pm 0.27$~M$_\\odot$. \n\nThe mass estimated by this procedure might be underestimated because the photocenter of the blobs might be closer to the star than their center of mass and the mass determination depends on the cube of the separation. In particular, as noted above, we may interpret blob D as the accretion flows on a planet along a spiral arm (see the next section); in this case, the photocenter is dominated by the leading arm, which is at about 313 mas from the star, while the trailing arm is at 347 mas when it is deprojected on the disk plane. The putative planet would be in the middle of the two arms, that is, at 330 mas from the star, yielding a mass estimate that is $\\sim 10$\\% higher than listed in Table~\\ref{t:blobrot}. We note that the separation measured in the K2 band agrees very well with this interpretation.\n\nThe mass determined from blob motion is slightly higher than but in agreement within the error bars with the mass that fits photometry. To show this, we determined the stellar mass by minimizing the $\\chi^2$ with respect to the main-sequence values considered by \\cite{Pecaut2013}. We considered the GAIA DR2 parallax and included an absorption term $A_V$ multiplied for the reddening relation by \\cite{Cardelli1989}. We also left free the ratio between the stellar and the main-sequence radius. The best match is with an F0V star ($T_{\\rm eff}=7220$~K), with $A_V=0.25$~mag and a radius that is 0.97 times the radius of the main-sequence star. According to \\cite{Pecaut2013}, the mass of an F0V star is 1.59~M$_\\odot$. This spectral type compares quite well with the most recent determinations (A7V: \\cite{Dent2013}; A9V: \\cite{Vieira2003}; F0V: \\cite{Paunzen2001}; F1V: \\cite{Murphy2015}) and with the temperature determined by GAIA ($T_{\\rm eff}=7320\\pm 150$~K), but it is much later than the B9V spectra type proposed by \\cite{Wright2003}. \n\nFor comparison, other determinations of the mass of HD~169142 are 2.0~M$_\\odot$\\ (\\cite{Manoj2005}), 2.28~M$_\\odot$\\ (\\cite{Maaskant2013}), 1.8~M$_\\odot$\\ (\\cite{Salyk2013}), and 2.0~M$_\\odot$\\ (\\cite{Vioque2018}). The mass adopted by \\cite{Ligi2017} is 1.7~M$_\\odot$. We note that these values were obtained assuming distances different from the distance given by GAIA DR2: for instance, \\cite{Maaskant2013} adopted a distance of 145 pc, which is 27\\% larger than the GAIA DR2 value considered here. On the other hand, the value used by \\cite{Ligi2017} was taken from GAIA DR1 and it is only 3\\% longer than that from GAIA DR2.\nHereafter, we adopt a mass  of 1.7~M$_\\odot$ for HD~169142, which is the same value as was considered by \\cite{Ligi2017}.\n\nWe also note that the projected rotational velocity of the star $V~\\sin{i}=50.3\\pm 0.8$~km\/s determined from the HARPS spectra (see Appendix) is high when we consider that the star is likely seen close to the pole. This value agrees quite well with literature values ($V~\\sin{i}=55\\pm 2$~km\/s: \\cite{Dunkin1997a, Dunkin1997b}). When we assume that the stellar rotation is aligned with the disk, the equatorial rotational velocity is 224~km\/s, which is at the upper edge of the distribution for F0 stars. For a discussion, see \\cite{Grady2007}.\n\n\n\\begin{table*}\n\\caption{Blob astrometry}\n\\begin{centering}\n\\begin{tabular}{lcccccccc}\n\\hline\n\\hline\nJD       &\\multicolumn{2}{c}{Blob A}&\\multicolumn{2}{c}{Blob B}&\\multicolumn{2}{c}{Blob C}&\\multicolumn{2}{c}{Blob D}\\\\\n+2400000 & Sep & PA & Sep & PA & Sep & PA & Sep & PA \\\\\n         & (mas) & (degree) & (mas) & (degree) & (mas) & (degree) & (mas) & (degree) \\\\\n\\hline\n57145.   & $106\\pm 6$ & $247\\pm 3$ & $185.4\\pm 4.0$ & $22.3\\pm 1.0$ & $192.7\\pm 4.0$ & $315.8\\pm 2.0$ & $315.8\\pm 4.0$ & $43.8\\pm 0.7$ \\\\ \n57180.17 &            &            & $194.0\\pm 3.2$ & $24.8\\pm 1.0$ & $197.9\\pm 2.5$ & $316.0\\pm 0.7$ & $313.9\\pm 4.0$ & $40.3\\pm 0.7$ \\\\\n57201.12 &            &            & $188.3\\pm 3.2$ & $22.6\\pm 1.0$ & $202.8\\pm 2.5$ & $315.2\\pm 0.7$ & $314.8\\pm 4.0$ & $42.1\\pm 0.7$ \\\\\n57499.34 &            &            & $187.7\\pm 3.2$ & $21.0\\pm 1.0$ & $197.6\\pm 2.5$ & $313.2\\pm 0.7$ &                &               \\\\\n57566.15 & $125\\pm 6$ & $240\\pm 3$ & $188.7\\pm 3.2$ & $18.4\\pm 1.0$ & $203.7\\pm 2.5$ & $312.9\\pm 0.7$ & $315.5\\pm 4.0$ & $41.6\\pm 0.7$ \\\\\n57873.30 & $117\\pm 6$ & $230\\pm 3$ & $184.6\\pm 3.2$ & $14.0\\pm 1.0$ & $200.1\\pm 2.5$ & $307.5\\pm 0.7$ & $319.2\\pm 4.0$ & $40.2\\pm 0.7$ \\\\\n58288.19 &            &            & $189.7\\pm 3.2$ & $~9.8\\pm 1.0$ & $200.1\\pm 2.5$ & $299.7\\pm 0.7$ & $315.6\\pm 4.0$ & $34.9\\pm 0.7$ \\\\\n\\hline\n\\end{tabular}\n\\end{centering}\n\\label{t:blobastro}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Blob rotation}\n\\begin{centering}\n\\begin{tabular}{lcccccccc}\n\\hline\n\\hline\nBlob     &  FAP & a  &  a   & Period   & Period       & Rot. speed &  Mass      & Remark \\\\      \n         &      &    &      & Computed & Observed     &            &            &        \\\\\n         &      & (mas) &  (au)  &  (yr)      & (yr)           & (deg\/yr)     & (M$_\\odot$)  &        \\\\\n\\hline\n\nA   &   0.02  &  118 & 13.5 &   36.2   &  42.7$\\pm$5.6  & -11.9$\\pm$2.2 \n& 1.60$\\pm$0.98     & pol and nocoro images \\\\\nB   & $<1E-7$ &  188 & 21.4 &   72.5   &  78.3$\\pm$5.3  & -5.04$\\pm$0.38 & 2.30$\\pm$0.29 & \\\\\nC   & $<1E-7$ &  202 & 23.1 &   80.9   &  73.0$\\pm$4.3  & -4.48$\\pm$0.25 & 1.60$\\pm$0.23 & \\\\\nD   &  4E-6   &  319 & 36.4 &  160.5   & 173.8$\\pm$20.1 & -2.08$\\pm$0.25 & 1.34$\\pm$0.40 & \\\\\n\\hline\n\\end{tabular}\n\\\\\nNote: Semi-major axis a is obtained assuming circular orbits on the disk plane; the computed period is for a mass of 1.87~$M\\odot$;  the observed period is estimated from the angular speed on the disk plane; the mass is determined using Kepler's third law; the uncertainty here is due to the errors in the angular speed.\n\\end{centering}\n\\label{t:blobrot}\n\\end{table*}\n\n\\subsection{Blob photometry}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{photo_blob.png}\n\\caption{Contrast of blobs as a function of wavelength. Blob B: Asterisks and dotted line. Blob C: Triangles and solid line. Blob D: Diamonds and dashed line.}\n\\label{f:contrast}\n\\end{figure}\n\nWe measured the magnitude of the sources in various bands by weighting the results obtained from the different epochs according to the quality of the images. The magnitudes refer to a $3\\times 3$ pixel area centered on each object and are obtained by comparison with those of simulated planets inserted into the image at 0.2 and 0.3 arcsec from the star and run through the same differential imaging algorithm. The underlying assumption is that the blobs are point sources, while they are likely slightly extended. These results should then be taken with caution. Using the fake blob procedure described in Section 3.2, we estimated that the brightness is underestimated by about a factor of $\\sim 2.8$~because of this effect for blobs B and C, that is, these blobs are likely $\\sim 1.1$ magnitude brighter than estimated when we assume that they are point sources. The effect is likely slightly smaller for blob D because it is farther away from the star. We summarize the results in Table~\\ref{t:blobphot}; error bars are the standard deviation of the mean of the results obtained at different epochs. All the blobs have a rather flat, only slightly reddish contrast with respect to the star (see Figure~\\ref{f:contrast}). Results are consistent with stellar light scattered by grains with a size on the order of a micron or smaller if stellar light is extinguished between the star and blobs or the blobs and us. Under the hypothesis (not demonstrated) that they are optically thick, the albedo required to reproduce observations of blobs A, B, and C is about 0.1.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{photo_blob_d.png}\n\\caption{Absolute magnitude  of blob D in various bands (diamonds). The solid line is the prediction for a 3~M$_J$, 5~Myr old planet using dusty isochrones by \\cite{Allard2001}}\n\\label{f:photo_b}\n\\end{figure}\n\nBlob D is about two magnitudes fainter than expected from this consideration, suggesting that either it receives less light from the star (e.g., because of absorption by Ring 1) or it is not optically thick. For this blob, we obtained contrasts of $13.31\\pm 0.28$~mag in Y, $13.34\\pm 0.23$~mag in J band, $13.29\\pm 0.21$~mag in H band, $12.94\\pm 0.5$~mag in  K1 band, and $12.35\\pm 0.5$~mag in K2 band (the last two values being obtained from the IRDIS data set). As expected, the object is beyond the 5$\\sigma$\\ contrast limit in each individual image. However, we expect a detection with an S\/N in the range from 2.3 to 3.7 in the individual images, and at an S\/N$\\sim 6$\\ in the combination of the images. It is then not surprising that we detected it only by combining them. While error bars are quite large, the absolute K1 and K2 magnitude of $14.07\\pm 0.50$~mag and $13.48\\pm 0.50$~mag corresponds to a $\\sim 3$~M$_J$ object using dusty isochrones (\\cite{Allard2001}) with an age of 5 Myr, which is at the lower edge of the age range according to \\cite{Blondel2006} and \\cite{Manoj2007}. This model has an effective temperature of about 1260 K (see Figure~\\ref{f:photo_b}). Of course, the mass estimated from photometry depends on the model, the age used to derive it and the possible extinction, and it assumes that the object is in hydrostatic equilibrium, which may be incorrect for a very young planet. This result is then highly uncertain.\n\n\\begin{table*}\n\\caption{Blob photometry. These values are obtained assuming that the blobs are point sources; they may be as much as 1.1 mag brighter if their extension is taken into account}\n\\begin{centering}\n\\begin{tabular}{lccccc}\n\\hline\n\\hline\nBlob & \\multicolumn{5}{c}{Contrast (in magnitudes) } \\\\\n     &       Y      &     J       &      H      &     K1      &      K2      \\\\\n\\hline\nA    &      9.05    &    8.73     &    9.02\\\\\nB    & 10.06$\\pm$0.19 & 9.84$\\pm$0.01 & 9.67$\\pm$0.10 & 9.72$\\pm$0.21 & 9.54$\\pm$0.14  \\\\\nC    & 10.43$\\pm$0.15 &10.26$\\pm$0.14 &10.09$\\pm$0.06 & 9.90$\\pm$0.40 & 9.86$\\pm$0.40  \\\\\nD    & 13.31$\\pm$0.28 &13.34$\\pm$0.23 &13.29$\\pm$0.21 &12.94$\\pm$0.50 &12.35$\\pm$0.50  \\\\\n\\hline\n\\end{tabular}\n\\end{centering}\n\\label{t:blobphot}\n\\end{table*}\n\n\\subsection{Comparison with previous detection claims}\n\nWe note that none of these blobs coincides with either the sub-stellar companions proposed by \\cite{Biller2014} and \\cite{Reggiani2014}, nor with the structure observed by \\cite{Osorio2014}. More in detail, after taking into account their motion (see Section 3.3), the expected position angles for blobs A, B, and C at the observation epochs of Biller et al. and Reggiani et al. (both acquired at an epoch about 2013.5), are  276, 34, and 324 degree, respectively (blob D is much farther away from the star). For comparison, the object of Biller et al. is at PA=$0\\pm 14$~degree (separation of $110\\pm 30$\\  mas) and the object of  Reggiani et al. is at PA=$7.4\\pm 11.3$ degree (separation of $156\\pm 32$~mas). In addition, the objects proposed by Biller et al. and Reggiani et al., with a contrast of $\\Delta L\\sim 6.5$, are brighter than our blobs B and C, even after the finite-size correction is taken into account, see Section 3.4. However, the object proposed by Reggiani et al. might be the combination of blobs B and C, within the errors of their astrometry; the combination of their luminosity is also not that far from the value of Reggiani et al. We note that the resolution of their observation is lower than ours because they observed at much longer wavelength, and their object appears elongated (in the E-W direction, i.e., the direction expected at the epoch of their observation) in their published image beyond the diffraction limit.\n\nOn the other hand, the inner and brighter object detected by Biller et al. is too close to the star to coincide with any of the objects we observed, while a fainter object they found might be blob B, as discussed by \\cite{Ligi2017}. However, when we examine the image published by Biller et al, it seems that the two brightest sources have a relative separation and orientation that coincides with those of blobs B and C. In this case, the fainter object should be blob B (as discussed in \\cite{Ligi2017}) and the brighter object might coincide with our blob C. Of course, this would require that the stellar position in their images does not correspond with the position assumed in their paper.\n\n\n\\section{Spiral arms}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.8truecm]{polar.jpeg}\n\\caption{Median over time of the individual S\/N maps in polar coordinates. Each image has been rotated to the last image for the rotation angle of blob D before the median was made. Arrows mark the location of the primary (white), secondary (cyan), and tertiary arms (yellow). The location of the blobs is marked.}\n\\label{f:polar}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{nospiral.jpeg}\n\\includegraphics[width=\\columnwidth]{spiral.jpeg}\n  \n  \n\\caption{Upper panel: Median over time of the individual S\/N maps; each image has been rotated to the last image for the rotation angle of blob D before the median was made. Lower panel: Same as the upper panel, with a three-arm spiral design overplotted. The putative planet is at the position of blob D. In both panels, N is up and E to the left, and a white cross marks the stellar position.}\n\\label{f:spirals}\n\\end{figure}\n\nMost bright features in the coincidence images (including blobs B and D) can be reproduced by a three-arm spiral design (see below), while blob C differs slightly. A similar three-arm structure is predicted by models for not very massive companions (\\cite{Fung2015}) and it has been observed around other stars (see, e.g., the case of MWC758 recently published by \\cite{Reggiani2018}). While it is not at all obvious that spiral arms indicate a planet (see, e.g., \\cite{Dong2018}), we may interpret it as due to a planet in the location of blob D. The structure of this blob appears to resemble the structure expected for an accreting object with a leading and a trailing arm. If this hypothesis were correct, the radial separation between leading (sep=310 mas) and trailing (sep=343 mas) arms should be about twice the Hill radius (see \\cite{Machida2010}); the Hill radius would then be $16.5\\pm 5$~mas and the planet mass in the range 0.25-1.6~M$_J$, which is lower than the mass estimated with DUSTY isochrones. Because the Hill radius is not accurately estimated and the dependence of the mass on the Hill radius is strong, the error on the planetary mass is quite large. The photosphere of such an object would be too faint for detection in YJH, while it might be compatible with detection in K1 and K2 bands.\n\n\\subsection{Separation of spiral arms}\n\nWe may also estimate the mass of the object exciting the spiral design observed in Ring 1 by different criteria, using the calibration by \\cite{Fung2015} (their eq. (9)). After transforming into a polar coordinates system (see Figure~\\ref{f:polar}), we could identify the three spiral arms, which we may call primary, secondary, and tertiary, following the approach of Fung \\& Dong. The view in Cartesian coordinates is given in Figure~\\ref{f:spirals}. Position angle and separation of the arms in some reference positions are given in Table~\\ref{t:spiral}.\n\n\\begin{table}\n\\caption{Spiral position.}\n\\begin{tabular}{lccc}\n\\hline\n\\hline\n      &    PA   &     PA     &     PA    \\\\\nSep   & Primary &  Secondary &  Tertiary \\\\\n(mas) & (degree)  & (degree)   &   (degree)  \\\\\n\\hline\n157   &  196.7  &   321.0    &    38.9   \\\\\n172   &  203.0  &   335.9    &    50.9   \\\\\n194   &  244.2  &    16.0    &    79.0   \\\\\n209   &  268.3  &    28.0    &    91.6   \\\\\n\\hline\npitch &   15.3  &    16.3    &    20.8   \\\\\n\\hline\n\\end{tabular}\n\\label{t:spiral}\n\\end{table}\n\nThese arms may be density waves excited by a a planet at the location of blob D, which is indeed along the primary arm of the spiral design: the predicted PA at the separation of blob D, 334 mas, is $36\\pm 6$~degree, in very good agreement with the observed value of $\\sim 35$~degree (as measured in the K2 band).\n\nThe phase difference between the primary and secondary arm ($127.2\\pm 3.1$~degree) can be used to estimate the mass of the planet exciting the spiral design, using the calibration by Fung \\& Dong. We obtain a mass ratio of $q=0.0030\\pm 0.0004$, which translates into a mass of $M_p=5.1\\pm 1.1$~M$_J$, adopting the stellar mass derived above. The phase difference between secondary and tertiary arms ($69.9\\pm 3.9$~degree) agrees with the expectations by Fung \\& Dong given the pitch angle and the expected ratio for resonances 1:2 and 1:3.\n\n\\subsection{Pitch angle}\n\nThe pitch angle is the angle between a spiral arm and the tangent to a circle at the same distance from the star. \\cite{Zhu2015} showed that the pitch angle can be used to estimate the mass of the planet exciting the spiral design. We measured the pitch angle at a separation of 183 mas to be $17.5\\pm 1.7$~degree. This separation is about $r\/r_p=0.55$. This value agrees with the results they obtained from their simulations for a mass ratio of $q=0.006$, supporting the mass determination obtained from the separation of primary and secondary spiral arm; moreover, the larger pitch for the tertiary arm agrees with expectations from models.\n\n\\subsection{Disk gap}\n\nUsing the relation by \\cite{Kanagawa2016}, we expect that there is a planet at $\\sim 0.36$~arcsec from the star with a mass ratio with respect to the star of $q=2.1\\times 10^{-3}\\,(W\/R_p)^2\\,(h_p\/0.05~R_p)^{1.5}\\,(\\alpha\/10^{-3})^{0.5}$, where $h_p\/R_p$ is the disk thickness and $\\alpha$\\ is the disk viscosity. For $R_p=0.36$~arcsec, $W=0.2$~arcsec, $h_p\/R_p=0.05$, and $\\alpha=1E-3$, a value of $q=0.00044$\\ is obtained, which means a planet of 0.75~M$_J$. \n\n\\cite{Dong2017} considered the case of HD~169142 and concluded for a value of $q^2\/\\alpha=1.1E-4$\\ for $R_p=0.37$~arcsec, $W=0.17$\\ arcsec, and $h_p\/R_p$=0.079. For $\\alpha=1E-3$, their formula implies $q=0.00033,$\\  which suggests a 0.56~M$_J$\\ planet. We note that the formula by Dong \\& Fung produces planets that are smaller by a factor of 2.6 with respect to that by Kanagawa et al.; however, the value they suggest for $h_p\/R_p$\\ is higher than considered above. The value considered by Dong \\& Fung is similar to the value obtained by \\cite{Fedele2017} by modeling the ALMA observations ($h_p\/R_p$=0.07).\n\nThere are considerable uncertainties in these formulas that are due to the exact values to be adopted for $R_p$, $W$, $h_p\/R_p$, $\\alpha$, and the difference of a factor of 2.5 in the constant factor. While a mass around 1~M$_J$\\ seems favored, we cannot exclude values different by as  much as an order of magnitude. We conclude that a planet with about one Jupiter mass likely causes the gap seen in HD169142, but its mass is not yet well defined from the gap alone.\n\n\\begin{table}\n\\caption{Putative planet mass (sep=335 mas, PA=35 degree at JD=58288.19)}\n\\begin{tabular}{lcc}\n\\hline\n\\hline\nMethod          &   M$_J$   & Remark \\\\\n\\hline\nPhotometry      &    3      & Age dependent \\\\\nHill radius     &  0.25-1.6 & \\\\\nSpiral arm separation  &  4.0-6.2  & \\\\\nPitch angle     &    6      & \\\\\nDisk gap        &  0.06-6   & \\\\\n\\hline\n\\end{tabular}\n\\label{t:mass}\n\\end{table}\n\n\\subsection{Summary of mass determination}\n\nA summary of the mass determinations is given in Table~\\ref{t:mass}. All these estimates are quite uncertain. The higher values are given by the spiral arm parameters. If we make an harmonic mean of the various estimates, we would conclude for a planet with a mass of $2.2_{-0.9}^{+1.4}$~M$_J$. This mass seems lower than what we can detect with our SPHERE images (about 3~M$_J$\\ from photometry), but is within the error bar. This value is also within the range 1-10~M$_J$\\ suggested by \\cite{Fedele2017} to justify the dust cavity observed with ALMA between rings 1 and 2, and it is on the same order as the missing mass in the disk within the gap, as given by their disk model (4.3~M$_J$).\nFor comparison, we note that if we were to try to interpret the spiral arms of MWC~758 (\\cite{Reggiani2018}) using the same approach, we would conclude for a more massive faint companion because in that case the separation between the primary and secondary arm is much closer to 180 degree.\n \n\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe performed an analysis of faint structures around HD~169142 that are persistent among several data sets obtained with SPHERE and analyzed them using differential image techniques. We found a number of blobs that rotate around the star as well as spiral arms. These structures represent small fluctuations of the overall disk structure around this star. The blobs are found to consistently rotate around the star with Keplerian circular motion.\n\nAlthough we cannot exclude other hypotheses, blob D might correspond to a low-mass ($\\sim 1-4$~M$_J$, best guess of 2.2~M$_J$), 5 Myr old, and still-accreting planet at about 335 mas (38 au) from the star, causing the gap between Rings 1 and 2 and exciting the spiral arm design observed within Ring 1. The separation between the outer edge of Ring 1 and blob D is 55 mas, which is about twice the proposed value for the Hill radius of the planet. There is a clear excess of flux at short wavelengths with respect to the flux expected for a planetary photosphere (see Figure~\\ref{f:photo_b}). In our proposed scenario, the planetary photosphere is not detected in YJH band, where we only see the accreting material fueling through the spiral arm and reflecting star light (consistent with its detection in the $Q_\\Phi$\\ image), while it might have been detected in the K1 and K2 bands. A planet of 2.2~$M_J$\\ at 335~mas (38~au) from HD~169142 would have a Hill radius of about 25 mas (3.2 au). A disk around such an object would have an FWHM slightly larger than the resolution limit of SPHERE and may well reflect some $10^{-5}$ of the stellar light, which is required to justify the flux observed in the YJH bands. On the other hand, it is also possible that no other planet exists, and we merely observe a dust cloud. Detection of a planet could be confirmed by observations in the L' band. According to the AMES-dusty isochrones (\\cite{Allard2001}), a 5 Myr old planet of 2.2~$M_J$\\ should have an absolute L' magnitude of $\\sim 11$~mag. The contrast in the L' band should then be of 10.1 mag, which is 3.7 mag fainter than the objects proposed by \\cite{Biller2014} and \\cite{Reggiani2014} and likely too faint for a detection in their data set. However, a future deeper data set can solve this issue.\n\nThe location of blob B (and C to a lesser extent) suggests at first sight that the blobs might be related to the secondary and tertiary spiral arms (see, e.g., \\cite{Crida2017}). If this were the case, they would follow the same angular speed as the perturbing object, that is, the putative planet. However, we showed (along with Ligi et al.) that those blobs follow a Keplerian motion appropriate for their separation from the star.\n\nFinally, we note that \\cite{Ligi2017} proposed that blobs B and C could be vortices (\\cite{Meheut2012}). This explanation might very well be true. Another scenario might be suggested by the possibility that they are in 1:2 resonance with a putative planet related to blob D. It concerns planetesimals or asteroid giant impacts that generate dust clouds. This might be a manifestation of the general phenomenon of planetesimal erosion that is expected to follow the formation of giant planets (see, e.g., \\cite{Turrini2012, Turrini2018}). However, the probability of observing such clouds is low in a gas-rich disk such as that of HD~169142 because large planetesimals are required to generate clouds as large as blobs B and C, unless the impact occurs far from the disk plane. The debris cloud from an impact roughly expands until the debris sweeps a gas mass that is no more than an order of magnitude higher than the mass of the debris itself. Because the volume of the clouds is at least one hundredth of the total volume of ring 1, this requires that the mass of the interacting bodies is higher than 1\/1000 of the mass of the disk when we assume a disk gas-to-dust ratio of unity and that the impact occurs close to the disk plane. The impacting bodies should then have a mass on the order of that of Mars or at least the Moon. Since it is not likely that many such objects are present in the disk of HD~169142, the probability of observing one or even more similar debris clouds is likely very low.\n\n\n\n\\begin{acknowledgements}\nThe authors thank A. Pohl for allowing them to use the original reduction of the $Q_\\Phi$ data set and the ESO Paranal Staff for support for conducting the observations.  E.S., R.G., D.M., S.D. and R.U.C. acknowledge support from the \"Progetti Premiali\" funding scheme of the Italian Ministry of Education, University, and Research. E.R. and R.L. are supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk\u0142odowska-Curie grant agreement No 664931. This work has been supported by the project PRININAF 2016 The Cradle of Life - GENESIS-SKA (General Conditions in Early Planetary Systems for the rise of life with SKA). The authors acknowledge financial support from the Programme National de Plan\\'etologie (PNP) and the Programme National de Physique Stellaire (PNPS) of CNRS-INSU. This work has also been supported by a grant from the French Labex OSUG\\@2020 (Investissements d'avenir - ANR10 LABX56). The project is supported by CNRS, by the Agence Nationale de la Recherche (ANR-14-CE33-0018). This work is partly based on data products produced at the SPHERE Data Centre hosted at OSUG\/IPAG, Grenoble. We thank P. Delorme and E. Lagadec (SPHERE Data Centre) for their efficient help during the data reduction process. SPHERE is an instrument designed and built by a consortium consisting of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille, France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France), INAF Osservatorio Astronomico di Padova (Italy), Observatoire de Gen\u00e8ve (Switzerland), ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France) and ASTRON (Netherlands) in collaboration with ESO. SPHERE was funded by ESO, with additional contributions from CNRS (France), MPIA (Germany), INAF (Italy), FINES (Switzerland) and NOVA (Netherlands). SPHERE also received funding from the European Commission Sixth and Seventh Framework Programmes as part of the Optical Infrared Coordination Network for Astronomy (OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (2004-2008), grant number 226604 for FP7 (2009-2012), and grant number 312430 for FP7 (2013-2016).\n\\end{acknowledgements}\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\textbf{#1.}]\\ignorespaces\n}\n\n\n\\excludecomment{suggestion}\n\n\\excludecomment{pcidetail}\n\n\n\\definecolor{felix}{rgb}{0.2,0.2,1.0}\n\\definecolor{petru}{rgb}{0.7,0.1,0.1}\n\\definecolor{alternative}{rgb}{0.1,0.1,0.7}\n\\definecolor{detail}{rgb}{0.0,0.5,0.0}\n\n\\newcommand{\\color{black}}{\\color{black}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{black}}{\\color{black}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{black}}{\\color{felix}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{black}}{\\color{petru}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{alternative}}{\\color{alternative}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\\newcommand{\\color{detail}}{\\color{detail}}\n\\newcommand{\\color{black}}{\\color{black}}\n\n\n\\begin{document}\n\n\\title[The stochastic heat equation on polygonal domains]\n{On the regularity  of  the stochastic heat equation\n on polygonal domains in $\\bR^2$}\n\n\n\n\\author{Petru A. Cioica-Licht}\n\\thanks{The first named author has been partially supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand, and by a University of Otago Research Grant (114023.01.R.FO). The  research of the second and third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) \nfunded by the Ministry of Education (NRF-2017R1D1A1B03033255) and (NRF-2013R1A1A2060996), respectively.\nThe authors would like to thank Felix Lindner for his contribution at an early stage of this manuscript}\n\\address{Petru A. Cioica-Licht (n\\'e Cioica), Department of Mathematics and Statistics, University of Otago, PO Box~56, Dunedin 9054, New Zealand}\n\\email{pcioica@maths.otago.ac.nz}\n\\author{Kyeong-Hun Kim}\n\\address{Kyeong-Hun Kim, Department of Mathematics, Korea University, Anam-ro 145, Sungbuk-gu, Seoul, 02841, Republic of Korea}\n\\email{kyeonghun@korea.ac.kr}\n\\author{Kijung Lee}\n\\address{Kijung Lee, Department of Mathematics, Ajou University, Worldcup-ro 206, Yeongtong-gu, Suwon, 16499, Republic of Korea}\n\\email{kijung@ajou.ac.kr}\n\n\n\\subjclass[2010]{60H15; 35R60, 35K05}\n\n\\keywords{Stochastic partial differential equation,\nstochastic heat equation,\nweighted $L_p$-estimate,\nweighted Sobolev regularity,\nangular domain,\npolygon,\npolygonal domain,\nnon-smooth domain,\ncorner singularity}\n\n\\begin{abstract}\nWe establish existence, uniqueness and higher order weighted $L_p$-Sobolev regularity for the stochastic heat equation with zero Dirichlet boundary condition on angular domains and on  polygonal domains in $\\mathbb{R}^2$.\nWe use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable.\nIn this way we can capture the influence of both main sources for singularities: the incompatibility between noise and boundary condition on the one hand and the singularities of the boundary on the other hand.\nThe range of admissible powers of the distance to the vertexes is described in terms of the maximal interior angle and is sharp. \n\\end{abstract}\n\n\\maketitle\n\n\n\\mysection{Introduction}\\label{sec:Introduction}\n\nIn this article we continue the analysis started in~\\cite{CioKimLee+2018} towards a refined $L_p$-theory for second order stochastic partial differential equations (SPDEs, for short) on non-smooth domains.\nThe main challenges in the construction of such a theory come from two effects that are known to influence the regularity of the solution:\nOn the one hand, the incompatibility between noise and boundary condition results in blow-ups of the higher order derivatives near the boundary---even if the boundary is smooth. \nOn the other hand, the singularities of the boundary cause a similar effect in their vicinity---even if the forcing terms are deterministic.\nWe refer to the introduction of~\\cite{CioKimLee+2018} and the literature therein for details. \n\nThe well developed $L_p$-theory for second order SPDEs on smooth domains, \ncarried out within the analytic approach initiated by N.V.~Krylov,\nshows that the incompatibility between noise and boundary condition can be captured very accurately by using a system of weights based on the distance to the boundary, see, for instance,~\\cite{Kim2004, KimKry2004, Kry1994,KryLot1999, KryLot1999b}.\nMoreover, the results in~\\cite{CioKimLee+2018} indicate that a system of weights based on the distance to a corner of the underlying domain is suitable to describe the impact of this boundary singularity on the solution.\nThus, in order to capture both effects, a system based on a combination of appropriate powers of the distance to the boundary and of the distance to the boundary singularities suggests itself.\n\n\nOur primary goal in this article is to show how such a system of mixed weights can be used in order to provide higher order spatial weighted $L_p$-Sobolev regularity for second order SPDEs with zero Dirichlet boundary condition on angular domains and on polygonal domains in $\\bR^2$. \nFor the moment we restrict ourselves to the stochastic heat equation, since already the analysis of this equation involves many non-trivial steps and has been a persisting problem for a long time.\nAt the same time, we believe that in this way we can shade some light on the general strategy without getting lost in details.\nOur general setting is as follows: \nLet $(w^k_t)$, $k\\in\\bN$, be a sequence of independent real-valued standard Brownian motions on a probability space $(\\Omega,\\cF,\\ensuremath{\\mathbb P})$ and let  $T\\in (0,\\infty)$ be a finite time horizon.\nWe consider the stochastic heat equation \n\\begin{equation}\\label{eq:SHE:Intro:a}\n\\left.\n\\begin{alignedat}{3}\ndu \n&= \n\\grklam{ \\Delta && u + f^0+f^i_{x_i}}\\,dt\n+\ng^k\\,dw^k_t \\quad \\text{on } \\Omega\\times(0,T]\\times\\domain,\t\\\\\nu\n&=\n0 && \\quad \\text{on } \\Omega\\times(0,T]\\times\\partial\\domain,\t\\\\\nu(0)\n&=\n0 && \\quad \\text{on } \\Omega\\times\\domain,\n\\end{alignedat}\n\\right\\}\t\n\\end{equation}\non various types of domains $\\domain\\subseteq\\bR^d$.\nOur focus lies in particular on polygonal domains and on angular domains $\\domain \\subseteq\\bR^2$. \nNote that, as usual, here and in the sequel we use the Einstein summation convention on the repeated indexes $i$ and $k$.\n\n\nOur main results address the existence, uniqueness and higher order spatial regularity of the solution to Equation~\\eqref{eq:SHE:Intro:a} on angular domains and on polygonal domains $\\domain\\subset\\bR^2$. \nBy using a weight system based solely on the distance to the set of vertexes of $\\domain$, we establish existence and uniqueness of a solution to Equation~\\eqref{eq:SHE:Intro:a} with suitable weighted $L_p$-Sobolev regularity of order one with respect to the space variable; see Theorem~\\ref{thm:ex:uni:2DCone} (angular domains) and Theorem~\\ref{thm polygon main} (polygonal domains). \nThe lower bound of the range~\\eqref{eq:range:vertex} for the weight parameter $\\theta$, which corresponds to the best integrability property of the solution near the vertex, is sharp; see also the introduction of~\\cite{CioKimLee+2018}. \nMoreover, by using, in addition, appropriate powers of the distance to the boundary $\\partial \\domain$ we describe the behavior of higher order spatial derivatives of the solution; see Corollary~\\ref{high} (angular domains) and Theorem~\\ref{thm_polygons_1} (polygonal domains).  \n\nThe key estimate, which paves the way for all the results mentioned above, is presented in Theorem~\\ref{lem:estim:2DCone:Theta}. \nRoughly speaking, it shows which system of weights is suitable in order to be able to lift the spatial regularity of the solution of the stochastic heat equation~\\eqref{eq:SHE:Intro:a} on an angular domain with the regularity of the forcing terms. In short, it can be stated as follows:\nLet \n\\begin{equation}\\label{domain:angular}\n\\cD:=\\cD_{\\kappa_0}:=\\ggklam{x\\in \\bR^2: x=(r\\cos\\vartheta,r\\sin\\vartheta),\\; r>0,\\;\\vartheta\\in (0,\\kappa_0)},\n\\end{equation}\nbe an angular domain with vertex at the origin and angle~$\\kappa_0\\in(0,2\\pi)$.\nMoreover, let $\\dist(x):=\\dist_\\cD(x):=\\mathrm{dist}(x,\\partial\\cD)$ be the distance of a point $x\\in\\cD$ to the boundary $\\partial\\cD$ of $\\cD$. If $u$ is the solution to Equation~\\eqref{eq:SHE:Intro:a} on $\\cD$,\nthen, for arbitrary $m\\in \\bN$, $1<\\Theta<p+1$ and $\\theta\\in\\bR$,\nwe can estimate\n\\[\n{\\mathbb E}\\int_0^T \\ssgrklam{\\sum_{\\abs{\\alpha}\\leq m}\\int_\\cD \\Abs{ \\rho(x)^{\\abs{\\alpha}-1} D^\\alpha u(t,x)}^p\\abs{x}^{\\theta-2}\\ssgrklam{\\frac{\\rho(x)}{\\abs{x}}}^{\\Theta-2}\\,dx}\\,dt\n\\]\nby the weighted $L_p$-norm \n\\[\n{\\mathbb E}\\int_0^T\\int_\\cD \\Abs{ \\abs{x}^{-1} u(t,x)}^p \\abs{x}^{\\theta-2}\\ssgrklam{\\frac{\\rho(x)}{\\abs{x}}}^{\\Theta-2}\\,dx\\,dt\n\\]\nof the solution plus appropriate weighted $L_p$-Sobolev norms of the forcing terms $f^0$, $f^i$ and $g$, of order $(m-2)\\lor 0$, $m-1$, and $m-1$, respectively.\nIt is worth mentioning that the range for the parameter $\\Theta$ in this estimate is natural and sharp, see Remark~\\ref{explanation of key} for details.\n\n\n\nAs already mentioned above, there already exists a comprehensive $L_p$-regularity theory for second order SPDEs in weighted Sobolev spaces with weights based solely on the distance to the boundary, see also~\\cite{Kim2014} in addition to the reference given above. Typically, the solution $u$ to Equation~\\eqref{eq:SHE:Intro:a} on a domain $\\domain\\subseteq\\bR^d$ fulfills\n\\[\n{\\mathbb E}\\int_0^T \\ssgrklam{\\sum_{\\abs{\\alpha}\\leq m}\\int_\\domain \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p\\rho^{\\Theta-d}\\,dx}\\,dt<\\infty,\n\\]\nprovided that the domain $\\cO$ is sufficiently smooth and the free terms $f^0$, $f^i$ and $g$ are in corresponding weighted Sobolev spaces. More precisely,\non smooth domains, i.e., at least $\\ensuremath{\\mathcal{C}}^1$, such a theory is possible with\n\\[\nd-1<\\Theta<d+p-1,\n\\]\nsee, e.g.,~\\cite[Remark~2.7]{Kim2004}.\nHowever, on non-smooth domains, only $\\Theta\\in (d+p-2-\\varepsilon,d+p-2+\\varepsilon)$ is possible with a small $\\varepsilon>0$ that depends on the roughness of the boundary of the domain and is not explicitly given~\\cite{Kim2014}. In particular, for large $p>2$, $\\Theta=d$ is not admissible, see~\\cite[Example~2.17]{Kim2014} for a typical counterexample.\nOur results show that, on polygonal domains, if we use an appropriate power of the distance to the set of vertexes to control the behavior of the solution in their proximate vicinity, then $\\Theta=d=2$ is possible away from the vertexes.\n\nOur analysis takes place within the framework of the analytic approach.\nThe proofs of the main results rely on a mixture of Green function estimates on angular domains, suitable localization techniques and some delicate estimates for the stochastic heat equation on $\\ensuremath{\\mathcal{C}}^1$ domains.\nAlternatively, one could think of Equation~\\eqref{eq:SHE:Intro:a} as an abstract Banach space valued stochastic evolution equation and try to obtain a similar theory by using the extension of the semigroup approach for SPDEs to Banach spaces developed by J.M.A.M.~van Neerven, M.C.~Veraar and L.~Weis~\\cite{NeeVerWei2008, NeeVerWei2012, NeeVerWei2012b}. However, for this to succeed, one would have to (at least!) check whether the (properly defined) Dirichlet Laplacian on weighted Sobolev spaces has an appropriate functional calculus. \nMoreover, one would need a description of the domain of the square root of this operator in terms of suitable weighted Sobolev spaces.\nTo the best of our knowledge, both questions are not trivial and yet to be answered.\nIn this context it is worth mentioning that the recently developed Calder\\'on-Zygmund theory for singular stochastic integrals from~\\cite{LorVer2019+} together with the $L_p$-theory developed in~\\cite{CioKimLee+2018} lead to an $L_q(L_p)$-theory with $q\\neq p$ without making use of precise descriptions of the domains of fractional powers of the Laplacian nor of the existence of a bounded $H^\\infty$-calculus, see~\\cite[Example~8.12]{LorVer2019+}.\n\nThis article is organized as follows: In Section~\\ref{sec:2DCone} we present and prove the main results concerning existence, uniqueness (Theorem~\\ref{thm:ex:uni:2DCone}) and higher order regularity (Corollary~\\ref{high}) of the stochastic heat equation on angular domains.\nThe proofs rely on two key estimates, which are stated in Theorem~\\ref{lem:estim:2DCone:Theta} and Lemma~\\ref{lem 4.5.1} and proven in detail in Section~\\ref{sec:proof:lift} and Section~\\ref{4}, respectively. \nFinally, in Section~\\ref{sec:Polygons} we present our analysis of the stochastic heat equation on polygonal domains.\nBefore we start, we fix some notation.\n\n\\medskip\n\n\\noindent\\textbf{Notation.}  \nThroughout this article, $(\\Omega,\\cF,\\ensuremath{\\mathbb P})$ is a complete probability space and $\\nrklam{\\cF_{t}}_{t\\geq0}$ is an increasing filtration of $\\sigma$-fields $\\cF_{t}\\subset\\cF$, each of which contains all $(\\cF,\\ensuremath{\\mathbb P})$-null sets.\nWe assume that on $\\Omega$ we are given a family $(w_t^k)_{t\\geq0}$, $k\\in\\bN$, of independent one-dimensional Wiener processes relative to $\\nrklam{\\cF_{t}}_{t\\geq0}$. By $\\cP$ we denote the predictable $\\sigma$-algebra on $\\Omega\\times (0,\\infty)$ generated by $\\nrklam{\\cF_{t}}_{t\\geq0}$ and any of its trace $\\sigma$-algebras.\nMoreover, $T\\in(0,\\infty)$ is a finite time horizon and $\\Omega_T:=\\Omega\\times (0,T]$. \nFor a measure space $(A, \\cA, \\mu)$, a Banach space $B$ and $p\\in[1,\\infty)$, we write $L_p(A,\\cA, \\mu;B)$ for the collection of all $B$-valued $\\bar{\\cA}$-measurable functions $f$ such that \n$$\n\\|f\\|^p_{L_p(A,\\cA,\\mu;B)}:=\\int_{A} \\lVert f\\rVert^p_{B} \\,d\\mu<\\infty.\n$$\nHere $\\bar{\\cA}$ is the completion of $\\cA$ with respect to $\\mu$.  The Borel $\\sigma$-algebra on a topological space $E$ is denoted by $\\cB(E)$. We will drop $\\cA$ or $\\mu$ in $L_p(A,\\cA, \\mu;B)$  when the $\\sigma$-algebra $\\cA$ or the measure $\\mu$  are obvious from the context. \nFor functions $f$ depending on $\\omega\\in \\Omega$, $t\\geq 0$ and $x\\in\\bR^d$, we usually drop the argument $\\omega$, and denote them by $f(t,x)$. \nIf $\\domain\\subseteq\\bR^d$ is a domain in $\\bR^d$, we write $\\ensuremath{\\mathcal{C}}^{\\infty}_c(\\domain)$ for the space of infinitely differentiable functions with compact support in $\\domain$. Moreover, $\\ensuremath{\\mathcal{C}}^{2}_c(\\domain)$ is the space of twice continuously differentiable functions with compact support in $\\domain$.\nFor a function $f\\colon\\domain\\to\\bR$ and any multi-index $\\alpha=(\\alpha_1,\\ldots,\\alpha_d)$, $\\alpha_i\\in \\{0,1,2,\\ldots\\}$,   \n$$\nD^{\\alpha}f(x):=\\partial^{\\alpha_d}_d\\cdots\\partial^{\\alpha_1}_1u(x),\n\\quad x=(x^1,\\ldots,x^d),\n$$\nwhere $\\partial^{\\alpha_i}_i=\\frac{\\partial^{\\alpha_i}}{\\partial (x^i)^{\\alpha_i}}$ is the $\\alpha_i$ times (generalized) derivative with respect to the $i$-th coordinate;\n$f_{x^i}:=\\frac{\\partial}{\\partial x^i}u$.\nBy making slight abuse of notation, for $m\\in\\{0,1,2,\\ldots\\}$, we write $D^m f$ for any (generalized) $m$-th order derivative of $f$ and for the vector of all $m$-th order derivatives. For instance, if we write $D^mf\\in B$, where $B$ is a function space on $\\domain$, we mean $D^\\alpha\\in B$ for all multi-indexes $\\alpha$ with $\\abs{\\alpha}=m$.\nThe notation $f_x$ is used synonymously for $D^1f$, whereas $\\nnrm{f_x}{B}:=\\sum_i\\nnrm{f_{x^i}}{B}$.\nThroughout the article, the letter $N$ is used to denote a finite positive constant that may differ from one appearance to another, even in the same chain of inequalities.\nWhen we write $N=N(a,b,\\cdots)$, we mean that $N$ depends only on the parameters inside the parentheses.\n Moreover, $A\\sim B$ is short for `$A\\leq N B$ and $B\\leq N A$'. \n\n\n\n\n\n\\mysection{The  stochastic heat equation on  angular domains }\\label{sec:2DCone}\n\n\n\nIn this section we present our analysis for the stochastic heat equation \n\\begin{eqnarray}\\label{eq:SHE:Intro}\ndu=(\\Delta u+f^0+ f^i_{x^i})\\,dt+ g^k \\,dw^k_t, \\quad t\\in (0,T],\n\\end{eqnarray}\non angular domains $\\cD\\subseteq\\bR^2$ with zero Dirichlet boundary condition and vanishing initial value.\nWe establish existence and uniqueness (Theorem~\\ref{thm:ex:uni:2DCone}) as well as higher order spatial regularity of the solution (Corollary~\\ref{high}) within a framework of weighted Sobolev spaces. \nThe weights are products of appropriate powers of the distance to the vertex and of the distance to the boundary (two infinite edges and the vertex). \nThe key estimate, which enables us to describe the behavior of the higher order derivatives of $u$ near the boundary even if the forcing terms behave badly near the boundary but are sufficiently smooth inside the domain, is presented in Theorem~\\ref{lem:estim:2DCone:Theta}, see also Remark~\\ref{explanation of key}.\n\n\n\n To state our results, we first introduce appropriate function spaces.\nThe notation is mainly borrowed from~\\cite{CioKimLee+2018}. Throughout, $\\cD=\\cD_{\\kappa_0}$ is as defined in~\\eqref{domain:angular} with $\\kappa_0\\in(0,2\\pi)$ and $\\rho_\\circ(x):=\\abs{x}$ denotes the distance of a point $x\\in \\cD$ to the origin (the only vertex of $\\cD$).\nLet $p>1$ and $\\theta\\in\\bR$.\nWe write\n$$\nL^{[\\circ]}_{p,\\theta}(\\cD):=L_{p}(\\cD,\\cB(\\cD),\\rho_\\circ^{\\theta-2} dx;{\\mathbb R}}\\def\\RR {{\\mathbb R})\n\\quad\\text{and}\\quad\nL^{[\\circ]}_{p,\\theta}(\\cD; \\ell_2):=L_p(\\cD,\\cB(\\cD),\\rho_\\circ^{\\theta-2} dx;\\ell_2)\n$$\nfor the weighted $L_p$-spaces of real-valued and $\\ell_2$-valued functions with weight $\\rho_\\circ^{\\theta-2}$. For $n\\in\\{0,1,2,\\ldots\\}$ let\n\\[\n\\ensuremath{K}^n_{p,\\theta}(\\cD)\n=\n\\ssggklam{\nf : \\nnrm{f}{\\ensuremath{K}^n_{p,\\theta}(\\cD)} := \\ssgrklam{\\sum_{\\abs{\\alpha}\\leq n} \n\\gnnrm{\\rho_{\\circ}^{\\abs{\\alpha}} D^\\alpha f}{L_{p,\\theta}^{[\\circ]}(\\cD)}^p}^{1\/p}\n< \n\\infty\n},\n\\]\nand define $K^n_{p,\\theta}(\\cD;\\ell_2)$ accordingly.  Note that\n\\[\nK^0_{p,\\theta}(\\cD)=L^{[\\circ]}_{p,\\theta}(\\cD)\n\\quad\\text{and}\\quad\nK^0_{p,\\theta}(\\cD;\\ell_2)=L^{[\\circ]}_{p,\\theta}(\\cD;\\ell_2).\n\\]\nMoreover, we write $\\mathring{\\ensuremath{K}}^1_{p,\\theta}(\\cD)$ for the closure in $\\ensuremath{K}^1_{p,\\theta}(\\cD)$ of the space $\\ensuremath{\\mathcal{C}}^\\infty_c(\\cD)$ of test functions.\n\nThe weighted Sobolev spaces introduced above are classical examples of Kondratiev spaces. For their basic properties as well as their relevance in the analysis of elliptic partial differential equations on domains with conical singularities we refer to~\\cite[Part~2]{KozMazRos1997}, see also the pioneering works \\cite{Kon1967,Kon1970,KonOle1983, KufOpi1984}. \nIn the sequel, we will frequently use the following basic properties.\nThey are mainly a consequence of the fact that \nfor any multi-index $\\alpha$\n$$\n\\sup_{\\cD} \\left(\\rho^{|\\alpha|-1}_{\\circ} |D^{\\alpha}\\rho_{\\circ}|\\right)\\le N(\\alpha)<\\infty;\n$$\nthe proof is left to the reader. \n\\begin{lemma}\n\\label{lem 1}\nLet $p>1, \\theta\\in \\bR$ and  $n\\geq 1$.  If $\\alpha$ is a multi-index with \n$|\\alpha|\\leq n$, then\n$$\n\\|\\rho_{\\circ}^{|\\alpha|} D^{\\alpha} f\\|_{K^{n-|\\alpha|}_{p,\\theta}(\\cD)}+\\|D^{\\alpha}  (\\rho^{|\\alpha|}_{\\circ} f)\\|_{K^{n-|\\alpha|}_{p,\\theta}(\\cD)} \\leq  N \\|f\\|_{K^{n}_{p,\\theta}(\\cD)},\n$$\nand\n$$\n\\|f_{x}\\|_{K^{n-1}_{p,\\theta}(\\cD)}\\leq  N\\|f\\|_{K^{n}_{p,\\theta-p}(\\cD)},\n$$\nwith $N$ independent of $f$.\n\\end{lemma}\n\n To formulate our conditions on the different parts of the equations, we will use the $L_p$-spaces of predictable stochastic processes on $\\Omega_T:=\\Omega\\times(0,T]$ taking values in the weighted Sobolev spaces introduced above. \nFor $p>1$, $\\theta\\in{\\mathbb R}}\\def\\RR {{\\mathbb R}$, and $n\\in\\{0,1,2,\\ldots\\}$, we abbreviate \n$$\n\\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cD,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}},\\ensuremath{\\mathbb P}\\otimes dx;\\ensuremath{K}^{n}_{p,\\theta}(\\cD)),\n$$\n$$\n \\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cD,T;\\ell_2)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}},\\ensuremath{\\mathbb P}\\otimes dx;\\ensuremath{K}^{n}_{p,\\theta}(\\cD;\\ell_2)),\n$$\n$$\n\\bL^{[\\circ]}_{p,\\theta}(\\cD,T):=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cD,T), \\quad  \\bL^{[\\circ]}_{p,\\theta}(\\cD,T;\\ell_2)\n:=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cD,T;\\ell_2),\n$$\nand\n\\[\n\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta}(\\cD,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}},\\ensuremath{\\mathbb P}\\otimes dx;\\mathring{\\ensuremath{K}}^{1}_{p,\\theta}(\\cD)).\n\\]\n\nUsing these spaces we introduce the following classes of stochastic processes that are tailor-made for the analysis of Equation~\\eqref{eq:SHE:Intro} on $\\cD$.\n\n\\begin{defn}\n   \\label{defn sol}\n For $p\\geq 2$  and $\\theta\\in \\bR$\nwe write $u\\in\\mathcal{K}^1_{p,\\theta,0}(\\cD,T)$  if\n$u\\in\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cD,T)$\n and\nthere exist $f^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cD,T), f^i\\in  \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$, $i=1,2$, and  $g\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T;\\ell_2)$,\nsuch that \n\\begin{equation}\\label{eqn 28}\n du=(f^0+f^i_{x^i})\\, dt +g^k \\, dw^k_t,\\quad t\\in(0,T],\n\\end{equation}\n on $\\cD$  in the sense of distributions with $u(0,\\cdot)=0$, that is, for any $\\varphi \\in\n\\ensuremath{\\mathcal{C}}^{\\infty}_{c}(\\cD)$, with probability one, the equality\n\\begin{equation}\\label{eq:distribution}\n(u(t,\\cdot),\\varphi)=   \\int^{t}_{0}\n\\left[(f^0(s,\\cdot),\\varphi) -(f^i(s,\\cdot),\\varphi_{x^i}) \\right]ds + \\sum^{\\infty}_{k=1} \\int^{t}_{0}\n(g^k(s,\\cdot),\\varphi)\\,  dw^k_s\n\\end{equation}\nholds for all $t \\leq T$. \nIn this situation\nwe also write\n$$\n\\bD u:=f^0+f^i_{x^i}\\qquad\\text{and}\\qquad \\bS u :=g\n$$\nfor the deterministic part and the stochastic part, respectively.\n\\end{defn}\n\n\n\n\\begin{remark}\nThe spaces $\\cK^1_{p,\\theta,0}(\\cD,T)$ from Definition~\\ref{defn sol} coincide with the spaces $\\mathfrak{K}^1_{p,\\theta}(\\cD,T)$ introduced in~\\cite[Definition~3.4]{CioKimLee+2018}.\nThe only (apparent) difference is that in the definition of $\\mathfrak{K}^1_{p,\\theta}(\\cD,T)$ the deterministic part $\\bD u$ is required to be an element of $\\mathbb K^{-1}_{p,\\theta+p}(\\cD,T):=L_p(\\Omega_T;K^{-1}_{p,\\theta+p}(\\cD))$, where $K^{-1}_{p,\\theta+p}(\\cD)$ is the dual of $\\mathring{K}^1_{p',\\theta'-p'}(\\cD)$ with $1\/p+1\/p'=1$ and $\\theta\/p+\\theta'\/p'=2$.\nHowever, this is not really a difference, since \n$$\n\\mathbb K^{-1}_{p,\\theta+p}(\\cD,T)=\\ggklam{f^0+f^i_{x^i} : f^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cD,T), f^i \\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}.\n$$\nThis can be proven with a similar strategy as the analogous result for classical Sobolev spaces, see, e.g., \\cite[page~62ff.]{AdaFou2003}, and by using the fact that for reflexive Banach spaces $B$ with dual $B^*$, the dual of $L_{p'}(\\Omega_T;B)$ is isometrically isomorphic to $L_p(\\Omega_T;B^*)$, see, e.g., \\cite[Theorem~IV.1.1 and Corollary~III.2.13]{DieUhl1977}.\n\n\\end{remark}\n\n In this article, Equation~\\eqref{eq:SHE:Intro} has the following meaning on $\\cD$.\n\n\n\n\\begin{defn}\\label{def:solution:D}\nWe say that  $u$ is a solution to Equation~\\eqref{eq:SHE:Intro} on $\\cD$ in the class $\\mathcal{K}^{1}_{p,\\theta,0}(\\cD,T)$ \nif\n$u\\in \\mathcal{K}^{1}_{p,\\theta,0}(\\cD,T)$ with\n\\[\n\\bD u = \\Delta u + f^0+f^i_{x^i}=f^0+(f^i+u_{x^i})_{x^i}\n\\qquad\n\\text{and}\n\\qquad\n\\bS u = g.\n\\]\n\\end{defn}\n\n\nNow that we have specified the setting, we are ready to present our results. We start with the key estimate in this article. Its proof is given in Section~\\ref{sec:proof:lift}.\nRecall that $\\rho(x):=\\rho_{\\cD}(x):=\\mathrm{dist}(x,\\partial\\cD)$ denotes the distance of a point $x\\in\\cD$ to the boundary $\\partial \\cD$.\n\n\n\n    \n\\begin{thm}\\label{lem:estim:2DCone:Theta}\nLet $p\\ge 2$, $1<\\Theta<p+1$, $\\theta\\in \\bR$, and $m\\in\\{0,1,2,\\ldots\\}$.  Moreover, let $u\\in \\mathcal{K}^{1}_{p,\\theta,0}(\\cD,T)$ be a solution to Equation~\\eqref{eq:SHE:Intro} on $\\cD$. Then  \n\\begin{align*}\n{\\mathbb E}&\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+1} \\int_{\\cD} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\rho_{\\circ}^{\\theta-2}\\ssgrklam{\\frac{\\rho}{\\rho_{\\circ}}}^{\\Theta-2}\\,dx\\,dt\\\\\n&\\leq\nN\\,\n{\\mathbb E} \\int_0^T \\int_{\\cD} \\ssgrklam{\\Abs{\\rho_{\\circ}^{-1}\nu}^p \n+ \n\\sum_{\\abs{\\alpha}\\leq (m-1)\\vee 0}\\Abs{\\rho^{\\abs{\\alpha}+1}D^{\\alpha}f^0}^p+\n \\sum_i\\sum_{\\abs{\\alpha}\\leq m}\\Abs{\\rho^{\\abs{\\alpha}}D^{\\alpha}f^i}^p\n\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n+\n\\sum_{\\abs{\\alpha}\\leq m} \\Abs{\\rho^{\\abs{\\alpha}}D^\\alpha g}_{\\ell_2}^p}\\rho_{\\circ}^{\\theta-2}\\ssgrklam{\\frac{\\rho}{\\rho_{\\circ}}}^{\\Theta-2}\\,dx\\,dt,\n\\end{align*}\nwhere $N=N(p,\\theta,\\Theta,\\kappa_0,m)$.\nIn particular, $N$ does not depend on $T$.\n\\end{thm}\n\n\n\\begin{remark}\\label{explanation of key}\nAs mentioned in~\\cite[Remark~2.7]{Kim2004}, the restriction $1<\\Theta<p+1$ on the parameter $\\Theta$ from Theorem~\\ref{lem:estim:2DCone:Theta} is necessary in order to obtain  the corresponding  result for the stochastic heat equation on $\\ensuremath{\\mathcal{C}}^1$ domains with  $\\rho_0:=1$. \nTherefore, since a solution to Equation~\\eqref{eq:SHE:Intro:a} on $\\cD$ that vanishes near the vertex can be considered as a solution to the same equation on a suitable $\\ensuremath{\\mathcal{C}}^1$ domain, the range of $\\Theta$ in Theorem~\\ref{lem:estim:2DCone:Theta} is sharp.\n\n\n\n\\end{remark}\n\n\nIn the proof of Lemma~\\ref{lem:fi:estim:2DCone} below we first establish existence for equations with nice forcing terms and extend it to the general case with a limit argument.\nFor this step to succeed, in particular to make sure that we maintain equality `for all $t\\leq T$' and therefore the limit also fulfills the equation in the sense of distributions, see Definition~\\ref{defn sol}, we need the following lemma.\nIt also plays a crucial role in the proof of the existence result  for the stochastic heat equation on polygonal domains (Theorem~\\ref{thm polygon main}), as it is one of the main ingredients in the proof of Lemma~\\ref{lem for gronwall}, which in turn is used for a Gronwall argument to establish existence. \nIts proof is given in detail in Section~\\ref{4}.\n\n\n\\begin{lemma}\\label{lem 4.5.1}\nLet $p\\geq 2$ and $\\theta\\in \\bR$. Assume that $u\\in \\cK^1_{p,\\theta,0}(\\cD,T)$  satisfies\n\\begin{equation}\\label{disti}\ndu=(f^0+f^i_{x^i})\\,dt+g^k \\,dw^k_t, \\quad t\\in(0,T],\n\\end{equation}\nin the sense of distributions.  \nThen $u\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T];L^{[\\circ]}_{p,\\theta}(\\cD)))$ and\n\\begin{equation}\\label{eq:estim:sup:2DCone}\n\\begin{alignedat}{1}\n{\\mathbb E} \\sgeklam{\\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)}}\n\\leq\n& N  \n\\Big( \\|u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cD,T)}+\\|f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cD,T)}\\\\\n&\\qquad\\qquad+\n\\sum_i\\|f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}\n+\n\\|g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T;\\ell_2)}\\Big),\n\\end{alignedat}\n\\end{equation}\nwhere $N=N(p,\\theta,\\kappa_0,T)$.\n\\end{lemma}\n\n\n\nWe have now all ingredients needed to state and prove our main existence and uniqueness result for Equation~\\eqref{eq:SHE:Intro} on $\\cD$.\nThe representation formula therein uses the Green function for the heat equation on $\\cD=\\cD_{\\kappa_0}$ with zero Dirichlet boundary condition, which we denote by $\\Gamma$, see, e.g., \\cite[Section~1]{KozRos2012b} for a precise definition.\n\n\n\n\\begin{thm}[Existence and uniqueness\/angular domains]\\label{thm:ex:uni:2DCone}\nLet $p\\geq 2$ and let $\\theta\\in\\bR$ fulfill\n\\begin{equation}\\label{eq:range:vertex}\np \\left(1-\\frac{\\pi}{\\kappa_0}\\right)<\\theta<p\\left(1+\\frac{\\pi}{\\kappa_0}\\right).\n\\end{equation}\nAssume that $g\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T;\\ell_2)$, $f^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cD,T)$ and $f^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$, $i=1,2$.\nThen\n\\begin{align*}\nu(t,x)\n&:=\\int_0^t \\int_\\cD \\Gamma(t-s,x,y) f^0(s,y)\\,dy\\,ds-\\sum_{i}\\int_0^t \\int_\\cD \\Gamma_{y^i}(t-s,x,y) f^i(s,y)\\,dy\\,ds\n\\\\\n&\\qquad+\n\\sum_{k=1}^\\infty\n\\int_0^t \\int_\\cD \\Gamma(t-s,x,y) g^k(s,y)\\,dy\\,dw^k_s\n\\end{align*}\nis the unique solution to Equation~\\eqref{eq:SHE:Intro} on $\\cD$ in the class $\\cK^{1}_{p,\\theta,0}(\\cD,T)$.\nMoreover, \n\\begin{align}\n   \\label{main 1}\n\\|u\\|_{\\mathbb{K}^1_{p,\\theta-p}(\\cD,T)}\n\\leq\nN \\Big( \\|f^0\\|_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cD,T)}+\\sum_i \\|f^i\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}+\n\\|g\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T;\\ell_2)}\\Big),\n\\end{align}\nwhere $N=N(p,\\theta,\\kappa_0)$.\n In particular, $N$ does not depend on $T$.\n\\end{thm}\n\n\nIf $f^i=0$ for $i=1,2$, then this result has been already proven in \\cite[Theorem~3.7]{CioKimLee+2018}.  \nThe extension presented here is essential to treat Equation~\\eqref{eq:SHE:Intro} on polygons in Section~\\ref{sec:Polygons} even if the equation on the polygon does not contain terms of this type, see also Remark~\\ref{remark 8.24} for details. \n\nThe missing link between \\cite[Theorem~3.7]{CioKimLee+2018} and Theorem~\\ref{thm:ex:uni:2DCone} is presented in the following lemma.\nIts proof relies on Theorem~\\ref{lem:estim:2DCone:Theta} and Lemma~\\ref{lem 4.5.1}.  \n\n\n\\begin{lemma}\\label{lem:fi:estim:2DCone}\nLet $p\\geq 2$ and let $\\theta\\in \\bR$ fulfill~\\eqref{eq:range:vertex}.  Assume $f^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$ for $i=1,2$ and  define\n\\begin{equation*}\n    \\label{eqn 4.13.1}\nv(t,x):=-\\sum_i \\int_0^t \\int_\\cD \\Gamma_{y^i}(t-s,x,y) f^i(s,y) \\,dy\\,ds.\n\\end{equation*}\nThen $v$ is the unique solution in the class $\\cK^{1}_{p,\\theta,0}(\\cD,T)$ to  the equation\n\\begin{equation}\\label{eqn 4.2.1}\ndv=(\\Delta v + f^i_{x^i})\\,dt, \\quad t\\in (0,T],\n\\end{equation}\non $\\cD$.\nMoreover, there exists a constant $N=N(p,\\theta,\\kappa_0)$ such that \n\\begin{equation}\n    \\label{eqn 4.2.3}\n\\|v\\|_{\\mathbb{K}^1_{p,\\theta-p}(\\cD,T)}\n\\leq N \\sum_i \n\\nnrm{f^i}{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}.\n\\end{equation}\n In particular, $N$ does not depend on $T$.\n\\end{lemma}\n\n\n\\begin{proof}\n\\emph{Step 1.} Let $f^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$, $i=1,2$. By \\cite[Theorem~3.10]{KozNaz2014},  for any  $0<\\lambda<\\frac{\\pi}{\\kappa_0}$\n($=:\\lambda^{\\pm}_c$ in \\cite{KozNaz2014}),\n$$\n|\\Gamma_y(t-s,x,y)|\\leq N \\ssgrklam{\\frac{\\abs{x}}{\\abs{x}+\\sqrt{t-s}}}^{\\lambda} \\ssgrklam{\\frac{\\abs{y}}{\\abs{y}+\\sqrt{t-s}}}^{\\lambda-1} (t-s)^{-\\frac{3}{2}} e^{-\\frac{\\sigma |x-y|^2}{t-s}},\n$$\nwhere the constants $\\sigma,N>0$ depend only on $\\kappa_0$ and $\\lambda$.  \nSince $\\theta$ satisfies~\\eqref{eq:range:vertex}, we can take $\\lambda$ sufficiently large such that $1-\\lambda<\\theta\/p<1+\\lambda$. Then the kernel\n$$\n\\cT_1(t,s,x,y):=\\ensuremath{\\mathds 1}_{x\\in \\cD} \\ensuremath{\\mathds 1}_{y\\in \\cD} \\ensuremath{\\mathds 1}_{t>s} |x|^{-1} \\frac{|x|^{(\\theta-2)\/p}}{|y|^{(\\theta-2)\/p}}\\Gamma_y(t-s,x,y)\n$$\nsatisfies the algebraic conditions in \\cite[Proposition~A.5]{KozNaz2014} with $\\mu=(\\theta-2)\/p$, $\\lambda_1=\\lambda_2=\\lambda-1$ and $r=1$. Hence by this proposition, \n\\[\n\\|v\\|_{\\bL^{[\\circ]}_{p,\\theta-p}(\\cD,T)}=\\|\\rho_{\\circ}^{-1}v\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}\\leq N(p,\\theta,\\kappa_0)\\sum_i \\|f^i\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)}.\n\\]\n\n\\noindent\\emph{Step 2.} Assume the $f^i$s are sufficiently nice, say,  $f^i\\in L_p(\\Omega_T, \\ensuremath{\\mathcal{P}};  \\ensuremath{\\mathcal{C}}^2_c(\\cD))$. Then by  \\cite[Theorem~3.7]{CioKimLee+2018},\n$$\nv:=\\sum_i \\int_0^t \\int_\\cD \\Gamma (t-s,x,y) f^i_{x^i}(s,y)\\,dy\\,ds=-\\sum_i \\int_0^t \\int_\\cD \\Gamma_{y^i}(t-s,x,y) f^i(s,y)\\,dy\\,ds\n$$\nis the unique solution to Equation~\\eqref{eqn 4.2.1} in the class $\\cK^{1}_{p,\\theta,0}(\\cD,T)$, see also~\\cite{Naz2001, Sol2001}.\nThis, together with Step~1 and  Theorem~\\ref{lem:estim:2DCone:Theta} with $m=0$ and $\\Theta=2$ lead to \\eqref{eqn 4.2.3} for  $f^i\\in L_p(\\Omega_T, \\ensuremath{\\mathcal{P}};  \\ensuremath{\\mathcal{C}}^2_c(\\cD))$. \n\n\\noindent\\emph{Step 3.} General $f^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$, $i=1,2$. Uniqueness follows from the case $f^i=0$. Take a sequence $(f^i_n)_{n\\in\\bN}\\subset L_p(\\Omega_T, \\ensuremath{\\mathcal{P}}; \\ensuremath{\\mathcal{C}}^2_c(\\cD))$ such that\n $f^i_n \\to f^i$ in $\\bL^{[\\circ]}_{p,\\theta}(\\cD,T)$ for each $i$. Let $v_n \\in \\cK^{1}_{p,\\theta,0}(\\cD,T)$ be the solution to Equation~\\eqref{eqn 4.2.1}\n with $f^i_n$. Then  by Step~1 and Step~2, $(v_n)$ is a Cauchy sequence in $\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cD,T)$. \nLet $u:=\\lim_{n\\to \\infty} v_n$ in $\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cD,T)$. Fix $\\varphi \\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(\\cD)$. \nThen taking the limit in\n $$\n (v_n(t,\\cdot), \\varphi)=-\\sum_i \\int^t_0((v_n(s,\\cdot))_{x^i}+f^i_n(s,\\cdot), \\varphi_{x^i})ds, \\quad \\forall\\; t\\leq T, \\quad (\\ensuremath{\\mathbb P}\\textup{-a.s.})\n $$\nand using the continuity of $t\\mapsto (u(t),\\varphi)$ (due to Estimate~\\eqref{eq:estim:sup:2DCone} from Lemma~\\ref{lem 4.5.1}), we find that $du=(\\Delta u+f^i_{x^i})\\, dt$ in the sense of distributions. The integral representation formula for $u$ is due to the fact that by Step~1 we also know that $\\lim_{n\\to\\infty}v_n=v$ in $L_{p,\\theta-p}^{[\\circ]}(\\cD,T)$. Estimate~\\eqref{eqn 4.2.3}  follows by taking the limits in the estimates for $v_n$ proven in Step~2.  \n\\end{proof}\n\n\\begin{remark}\nSince Lemma~\\ref{lem:fi:estim:2DCone} addresses the deterministic heat equation, the restriction $p\\geq 2$ is obsolete. The result as well as the proof carry over to the case $p>1$ mutatis mutandis.\n\\end{remark}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:ex:uni:2DCone}]\nThis is now an immediate consequence of \\cite[Theorem~3.7]{CioKimLee+2018} and Lemma~\\ref{lem:fi:estim:2DCone} above.\n\\end{proof}\n\nTheorem \\ref{lem:estim:2DCone:Theta} with $\\Theta=2$ and Estimate~\\eqref{lem:estim:2DCone:Theta}  now lead to the following higher order regularity result of the solution depending on the regularity of the forcing terms $f^{0}$, $f^i$, and $g^k$. \n Recall that in this section $\\rho$ denotes the distance to the boundary of $\\cD$.\n\n\\begin{corollary}[higher order regularity\/angular domains]\n\\label{high}\n Given the setting of Theorem~\\ref{thm:ex:uni:2DCone}, let $u$ be the unique solution in the class $\\cK^{1}_{p,\\theta,0}(\\cD,T)$ to Equation~\\eqref{eq:SHE:Intro} on $\\cD$. Assume that  \n\\begin{align*}\nC(m,\\theta, f^i,f^0,g)\n&:=\n{\\mathbb E} \\int^T_0 \\int_{\\cD}\\ssgrklam{\\sum_{|\\alpha|\\leq  (m-1)\\vee 0} |\\rho^{\\abs{\\alpha}+1}D^{\\alpha}f^0|^p+\n \\sum_i\\sum_{|\\alpha|\\leq m} |\\rho^{|\\alpha|}D^{\\alpha}f^i|^p\\\\\n&\\quad\\quad\\qquad\\qquad\\qquad+|\\,\\rho_{\\circ} f^0|^p+\n\\sum_{\\abs{\\alpha}\\leq m} |\\rho^{\\abs{\\alpha}}D^\\alpha g|_{\\ell_2}^p }\\rho_{\\circ}^{\\theta-2}\\, dx\\,dt <\\infty\n\\end{align*}\nfor some $m\\in\\{0,1,2,\\ldots\\}$. Then\n$$\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+1} \\int_{\\cD}   \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\abs{x}^{\\theta-2}\\,dx\\,dt\\leq N\\,C(m,\\theta, f^i,f^0,g)<\\infty,$$\nwhere $N=N(p,\\theta,\\kappa_0,m)$.  In particular, $N$ does not depend on $T$.\n\n\\end{corollary}\n\n\nWe will need the following `general uniqueness' lemma  to handle the stochastic heat equation on polygons in Section~\\ref{sec:Polygons}.\n\n\\begin{lemma}\n   \\label{lem for uniqueness}  \nLet $2\\leq p_1\\leq p_2$ and let $\\theta_1, \\theta_2\\in\\bR$ satisfy \\eqref{eq:range:vertex} for  $p=p_1$ and $p=p_2$, respectively. Assume for both $j=1$ and $j=2$,\n\\[\nf^0 \\in \\bL^{[\\circ]}_{p_j,\\theta_j+p_j}(\\cD,T), \\quad f^i \\in \\bL^{[\\circ]}_{p_j,\\theta_j}(\\cD,T),\\,\\,i=1,2, \\quad g\\in \\bL^{[\\circ]}_{p_j,\\theta_j}(\\cD,T;\\ell_2)\n\\]\n and let $u\\in \\cK^{1}_{p_1,\\theta_1,0}(\\cD,T)$ be the solution to  Equation~\\eqref{eq:SHE:Intro}.   Then  $u\\in \\cK^{1}_{p_2,\\theta_2,0}(\\cD,T)$.     \\end{lemma}\n     \n\\begin{proof}\nThis follows from the integral representation formula of the solution in Theorem \\ref{thm:ex:uni:2DCone}, that is, the unique solutions in  \n$\\cK^{1}_{p_1,\\theta_1,0}(\\cD,T)$ and $\\cK^{1}_{p_2,\\theta_2,0}(\\cD,T)$ have the same representation formula.\n\\end{proof}\n\n\n\n\n\n\\begin{remark}\n\\label{remark cones}\n To keep the presentation short, the results in this section are formulated only for angular domains $\\cD\\subseteq\\bR^2$ with vertex at the origin and with one of the edges being the positive $x^1$-axis.\nHowever, since every angular domain in $\\bR^2$ can be seen as a translation of a rotation of such a domain, all results can be extended accordingly, as the Laplace operator is invariant under translations and rotations. \nMore precisely,  fix $a\\in (-\\pi, \\pi)$ and $x_0\\in \\bR^2$.  Let\n   \\begin{equation*}\n\\tilde{\\cD}:=\\tilde{\\cD}_{\\kappa_0}(x_0,a):=\\big\\{x\\in \\bR^2: x=x_0+(r\\cos\\vartheta,r\\sin\\vartheta),\\; r>0,\\;\\vartheta\\in (a,a+\\kappa_0)\\big\\}.\n\\end{equation*}\nReplacing $\\cD$ and $\\rho_\\circ$ by $\\tilde{\\cD}_{\\kappa_0}(x_0,a)$ and $\\tilde{\\rho}_\\circ(x):=|x-x_0|$, respectively, in the definitions of the weighted Sobolev spaces from above, we can define analogous spaces, such as $K^n_{p,\\theta}(\\tilde{\\cD})$, $\\mathbb K^n_{p,\\theta}(\\tilde{\\cD},T)$ and  \n$\\cK^1_{p,\\theta,0}(\\tilde{\\cD},T)$, on $\\tilde{\\cD}$.  \nThen,  the results in this section  hold with $\\tilde{\\cD}$ in place of $\\cD$.  Indeed,\nlet $Q=(q_{ij})_{1\\leq i,j\\leq 2}$ be the orthogonal matrix such that $\\tilde{\\cD}_{\\kappa_0}(x_0,a)=x_0+Q \\cD_{\\kappa_0}$. Then, since the Laplacian is invariant under the rotations and translations,  the statement that $u\\in \\cK^1_{p,\\theta,0}(\\tilde{\\cD},T)$  satisfies\n\\begin{equation}\n\\label{eqn 4.10.7}\ndu=(\\Delta u +f^0+f^i_{x^i})\\,dt+g^k dw^k_t,\n\\end{equation}\n in the sense of distribution (analogous meaning to Definition~\\ref{defn sol})  is the same as the statement that $v(t,x):=u(t,x_0+Qx)\\in   \\cK^1_{p,\\theta,0}(\\cD,T)$  satisfies\n\\[\ndv=(\\Delta v +\\tilde{f}^0+\\tilde{f}^i_{x^i})\\, dt+\\tilde{g}^k \\, dw^k_t,\n\\]\nwhere\n$\n\\tilde{f}^0(t,x)=f^0(t,x_0+Qx)$, $\\tilde{f}^i(t,x)=q_{1i}f^1(t,x_0+Qx)+q_{2i}f^2(t,x_0+Qx)$, $i=1,2$,\nand $\\tilde{g}(t,x)=g(t,x_0+Qx)$. \nHence, all existence and uniqueness results as well as all estimates  can be extended to general angular domains, since, obviously,\n$$\n\\|h(x)\\|_{K^n_{p,\\theta}(\\tilde{\\cD})} \\sim \\|h(x_0+Qx)\\|_{K^n_{p,\\theta}(\\cD)}\n$$\nfor any $h\\in K^n_{p,\\theta}(\\tilde{\\cD})$.  \n To extend Lemma~\\ref{lem 4.5.1}, formally set $\\Delta u=0$ in~\\eqref{eqn 4.10.7}.\n\\end{remark}\n\n\\mysection{Proof of Theorem  \\ref{lem:estim:2DCone:Theta}}\\label{sec:proof:lift}\n\n\nIn this section we give a detailed proof of the key estimate from Theorem~\\ref{lem:estim:2DCone:Theta}.\nOur proof is based on a suitable a-priori estimate for the stochastic heat equation on $\\ensuremath{\\mathcal{C}}^1$ domains, as presented in Lemma~\\ref{lem 10} below.\nWe use this result to establish an estimate for the solution on a subdomain of $\\cD$ which is bounded away from the vertex and from infinity (see Lemma~\\ref{lem:estim:2DCone:U1} below).\nThen we can prove Theorem~\\ref{lem:estim:2DCone:Theta} by using a dilation argument, as $\\cD$ is invariant under positive dilation.\nFor this strategy to succeed, it is crucial that the constant in Lemma~\\ref{lem 10} does not depend on the time horizon $T$.\n\n\n\n\nWe start with the definition of the weighted Sobolev spaces $H^n_{p,\\Theta}(G)$ on $\\ensuremath{\\mathcal{C}}^1$  domains $G\\subseteq\\bR^d$ ($d\\geq 1$), which we need for the statement of Lemma~\\ref{lem 10}. First we recall the definition of a $\\ensuremath{\\mathcal{C}}^1$ domain. \n\n\\begin{defn}\\label{definition domain}\n  Let $G$ be a  domain in $\\bR^d$, $d\\geq 1$.\nWe write  $\\partial G\\in \\ensuremath{\\mathcal{C}}^1_u$  and say that $G$ is a  $\\ensuremath{\\mathcal{C}}^1$ domain  if there  exist constants $r_0, K_0\\in(0,\\infty)$ such that \nfor any  $x_0 \\in \\partial G$ there exists\n a one-to-one continuously differentiable mapping $\\Psi$ of\n $B_{r_0}(x_0)$ onto a domain $J\\subset\\bR^d$ such that\n\\begin{enumerate}[align=right,label=\\textup{(\\roman*)}] \n\\item $J_+:=\\Psi(B_{r_0}(x_0) \\cap G) \\subset \\bR^d_+$ and\n$\\Psi(x_0)=0$;\n\n\\item $\\Psi(B_{r_0}(x_0) \\cap \\partial  G)= J \\cap \\{y\\in\n\\bR^d:y^1=0 \\}$;\n\n\\item $\\|\\Psi\\|_{\\ensuremath{\\mathcal{C}}^{1}(B_{r_0}(x_0))}  \\leq K_0 $ and\n$|\\Psi^{-1}(y_1)-\\Psi^{-1}(y_2)| \\leq K_0 |y_1 -y_2|$ for any $y_i\n\\in J$;\n\n\\item  $\\Psi_x$ is uniformly continuous in $B_{r_0}(x_0)$.\n\n \\end{enumerate}\n\\end{defn}\nThroughout this article, we assume that $G$ is  either $\\bR^d_+:=\\{x\\in \\bR^d\\colon x^1>0\\}$ or a bounded $\\ensuremath{\\mathcal{C}}^1$ domain in $\\bR^d$ ($d\\geq 1$). Note that in both cases, $G$ is of class $\\ensuremath{\\mathcal{C}}^1_u$ in the sense of \\cite[Assumption~2.1]{Kim2004}.\nRecall that $\\rho(x)=\\rho_G(x)=\\mathrm{dist}(x,\\partial G)$ for $x\\in G$;\n$\\rho(x)=x^1$ if $G=\\bR^2_+$. \nFor $p>1$ and $\\Theta\\in\\bR$, we write\n\\[\nL_{p,\\Theta}(G):=L_{p}(G,\\rho^{\\Theta-d} dx;{\\mathbb R}}\\def\\RR {{\\mathbb R}) \\quad \n\\text{and}\n\\quad\nL_{p,\\Theta}(G;\\ell_2):=L_p(G,\\rho^{\\Theta-d} dx;\\ell_2)\n\\]\nfor the weighted $L_p$-spaces of real-valued\/$\\ell_2$-valued functions with weight $\\rho^{\\Theta-d}$.\nFor $n\\in  \\{0,1,2,\\ldots\\}$, by $H^n_{p,\\theta}(G)$ we denote the space of all \n$f\\in L_{p,\\Theta}(G)$  such that\n\\begin{equation}\n     \\label{eqn 4.9.5}\n\\|f\\|^p_{H^n_{p,\\Theta}(G)}:=\\sum_{|\\alpha|\\leq n}  \\|\\rho^{\\abs{\\alpha}} D^\\alpha f\\|^p_{L_{p,\\Theta}(G)}<\\infty.\n\\end{equation}\nMoreover, we define the dual spaces\n\\[\nH^{-n}_{p,\\Theta}(G):=\\grklam{H^n_{p',\\Theta'}(G)}^*,\\qquad\\frac{1}{p}+\\frac{1}{p'}=1,\\quad \\frac{\\Theta}{p}+\\frac{\\Theta'}{p'}=d.\n\\]\nThe space $H^n_{p,\\Theta}(G;\\ell_2)$ is defined analogously for $n\\in \\bZ$.\n\nTo state the main properties of these spaces, we introduce some additional notation. \nFor $k\\in\\{0,1,2,\\ldots\\}$, let\n$$\n|f|^{(0)}_{k}:=|f|^{(0)}_{k,G} :=\\sup_{\\substack{x\\in G\\\\\n|\\beta| \\leq k}}\\rho^{|\\beta|}(x)|D^{\\beta}f(x)|.\n$$\nIf $G$ is bounded, let $\\psi$ be a bounded $\\ensuremath{\\mathcal{C}}^\\infty$ function defined in $G$ with $|\\psi|^{(0)}_k+|\\psi_x|^{(0)}_k<\\infty$ for any $k$, which is comparable to $\\rho$, i.e., $N^{-1}\\rho(x)\\leq \\psi(x)\\leq N \\rho(x)$ for some constant $N>0$; see, e.g., \\cite[ Section~2]{KimKry2004}.\nIt is known that, if $G$ is bounded, then  the map $\\Psi$ in Definition~\\ref{definition domain} can be chosen in such a way that $\\Psi$ is infinitely differentiable in $B_{r_0}(x_0)\\cap G$ and for any multi-index $\\alpha$\n\\begin{equation}\n                \\label{eqn 12.4.9}\n\\sup_i \\sup_{B_{r_0}(x_0) \\cap G} \\rho^{|\\alpha|}|D^{\\alpha}\\Psi_{x^i}|\\leq N(\\alpha)<\\infty;\n \\end{equation}\nsee, e.g., \\cite{KimKry2004} or the proof of \\cite[Lemma~4.9]{KimLee2011}.\nActually, after appropriate rotation and translation, one can take $\\Psi(x^1,x')=(\\psi(x),x')$.   By \\cite[Theorem~3.2]{Lot2000} and \\eqref{eqn 12.4.9} above,   if  $\\text{supp}\\, u\\subset B_{r}(x_0)\\cap \\overline{G}$ and $r<r_0\/{K_0}$,  then for any $\\nu, \\Theta \\in \\bR$ and $n\\in \\bZ$, we have\n \\begin{equation}\\label{eqn 12.7.1}\n \\|\\psi^{\\nu}u\\|_{H^{n}_{p,\\Theta}(G)} \\sim \\|(x^1)^{\\nu}(u\\circ\\Psi^{-1})\\|_{H^{n}_{p,\\Theta}(\\bR^d_+)}.\n \\end{equation}\n\n\nHere are some  other properties of the spaces $H^{n}_{p,\\Theta}(G)$ taken from~\\cite{Lot2000} (see also \\cite{KimKry2004,Kry1999c}). If $G=\\bR^d_+$, let $\\psi(x):=\\rho(x)=x^1$.\n\n\\begin{lemma}\\label{collection}\n\\begin{enumerate}[leftmargin=*,label=\\textup{(\\roman*)}, wide] \n\\item  $\\ensuremath{\\mathcal{C}}^\\infty_c(G)$ is dense in $H^n_{p,\\Theta}(G)$.\n\n\n\\item\\label{col:psiD} For any $n\\in \\bZ$ the operators $\\psi D, D \\psi: H^{n}_{p,\\Theta}(G)\\to\nH^{n-1}_{p,\\Theta}(G)$ are bounded linear operators. In fact, \nfor any $u\\in H^{n}_{p,\\Theta}(G)$,\n$$\n\\|u\\|_{H^{n}_{p,\\Theta}(G)} \\leq N\\|\\psi\nu_{x^i}\\|_{H^{n-1}_{p,\\Theta}(G)}+N\\|u\\|_{H^{n-1}_{p,\\Theta}(G)}\n\\leq N \\|u\\|_{H^{n}_{p,\\Theta}(G)},\n$$\n$$\n\\|u\\|_{H^{n}_{p,\\Theta}(G)} \\leq N\\|(\\psi\nu)_{x^i}\\|_{H^{n-1}_{p,\\Theta}(G)}+N\\|u\\|_{H^{n-1}_{p,\\Theta}(G)}\n\\leq N \\|u\\|_{H^{n}_{p,\\Theta}(G)}\n$$\nhold, where $N$ is independent of $u$ and  $i\\in\\{1,\\ldots,d\\}$.\n\n\\item\\label{col:psi:theta} For any $\\nu\\in \\bR$, $n\\in \\bZ$,\n$\\psi^{\\nu}H^{n}_{p,\\Theta}(G)=H^{n}_{p,\\Theta-p\\nu}(G)$\nand\n\\begin{equation}\n                            \\label{open}\n\\|u\\|_{H^{n}_{p,\\Theta-p\\nu}(G)} \\leq N\n\\|\\psi^{-\\nu}u\\|_{H^{n}_{p,\\Theta}(G)}\\leq\nN\\|u\\|_{H^{n}_{p,\\Theta-p\\nu}(G)}.\n\\end{equation}\n\n\\item\\label{col:multiplier}  For $\\Theta\\in \\bR$ and $n\\in \\bZ$,\n$$\n\\|a u\\|_{H^{n}_{p,\\Theta}(G)}\\leq\nN(d,n)|a|^{(0)}_{|n|}\\|u\\|_{H^{n}_{p,\\Theta}(G)}.\n$$\n\n\\item\\label{col:bounded:embedding} If $G$ is bounded and $\\Theta_1<\\Theta_2$, then $H^n_{p,\\Theta_1}(G)\\subset H^n_{p,\\Theta_2}(G)$ and\n$$\n\\|u\\|_{H^n_{p,\\Theta_2}(G)}\\leq N(n,d,\\Theta_1,\\Theta_2) \\|u\\|_{H^n_{p,\\Theta_1}(G)}.\n$$\n\n\\item\\label{col:cptsupport} Let $n\\in \\bZ$  and $u\\in H^{n}_{p,\\Theta}(G)$ with  $K:=\\mathrm{supp}\\, u  \\subset G$.  Then for some $N=N(d,p,\\Theta,K)>0$,\n$$\nN^{-1}\\|u\\|_{H^n_p(\\bR^d)}\\leq \\|u\\|_{H^n_{p,\\Theta}(G)}\\leq N\\|u\\|_{H^n_p(\\bR^d)},\n$$\nwhere $H^n_p(\\bR^d):=\\{u: D^{\\alpha}u\\in L_p(\\bR^d), \\, \\forall\\, |\\alpha|\\leq n\\}$ if $n\\geq 0$, and otherwise  it is the dual space of $H^{-n}_q(\\bR^d)$, where $\\frac1p+\\frac1q=1$.\n\\end{enumerate}\n\n\\end{lemma}\n\nNote that, by Lemma~\\ref{collection}\\ref{col:multiplier} and the properties of $\\psi$, $\\psi$ is a point-wise multiplier in $H^n_{p,\\Theta}(G)$   if $G$ is bounded.\n\n For the corresponding spaces of predictable $H^n_{p,\\Theta}(G)$\/$H^n_{p,\\Theta}(G;\\ell_2)$-valued stochastic processes we use the abbreviations\n$$\n\\bH^{n}_{p,\\Theta}(G,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};H^{n}_{p,\\Theta}(G))\n\\quad \n\\text{and}\\quad\\bL_{p,\\Theta}(G,T):=\\bH^0_{p,\\Theta}(G,T),\n$$\nas well as\n\\[\n\\bH^n_{p,\\Theta}(G,T;\\ell_2)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};H^{n}_{p,\\Theta}(G;\\ell_2))\\quad \n\\text{and}\\quad\\bL_{p,\\Theta}(G,T;\\ell_2):=\\bH^0_{p,\\Theta}(G,T;\\ell_2).\n\\]\n The following classes of stochastic processes are tailor-made for the analysis of Equation~\\eqref{eq:SHE:Intro} on $G$.\n\\begin{defn}\nFor $p\\geq 2$ and $\\Theta\\in\\bR$ we write $u\\in\\frH^n_{p,\\Theta,0}(G,T)$  if\n$u\\in\\bH^n_{p,\\Theta-p}(G,T)$\n and\nthere exist \n$f\\in \\bH^{n-2}_{p,\\Theta+p}(G,T)$ and\n $g\\in \\bH^{n-1}_{p,\\Theta}(G,T;\\ell_2)$\nsuch that \n\\begin{equation*}\\label{eqn 28_1}\n du=f\\, dt +g^k \\, dw^k_t,\\quad t\\in (0,T],\n\\end{equation*}\non $G$ in the sense of distributions with $u(0,\\cdot)=0$; see Definition~\\ref{defn sol} accordingly.  We denote \n$$\n\\bD u:=f\\qquad\\text{and}\\qquad \\bS u :=g.\n$$\n\n\\end{defn}\n\n In this article, Equation~\\eqref{eq:SHE:Intro} has the following meaning on $G$.\n\\begin{defn}\\label{defn sol:C1}\nWe say that  $u\\in\\bH^{n}_{p,\\Theta-p}(G,T)$ is\na solution to Equation~\\eqref{eq:SHE:Intro} on $G$ \nin the class $\\frH^{n}_{p,\\Theta,0}(G,T)$\nif\n$u\\in \\frH^{n}_{p,\\Theta,0}(G,T)$ with\n\\[\n\\bD u = \\Delta u + f^0+f^i_{x^i}\n\\qquad\n\\text{and}\n\\qquad\n\\bS u = g.\n\\]\n\\end{defn}\n\n\\begin{remark}\nAll definitions above are given only for $\\ensuremath{\\mathcal{C}}^1$ domains, as we say from the beginning that in this article $G$ is either a bounded $\\ensuremath{\\mathcal{C}}^1$ domain or the half plane. However,  all the spaces defined above as well as the solution concept make sense on any domain $\\domain\\subset\\bR^d$ with non-empty boundary.\n\\end{remark}\n\n\n Now we have all notions we need in order to state and prove the a-priori estimate for Equation~\\eqref{eq:SHE:Intro} on bounded $\\ensuremath{\\mathcal{C}}^1$ domains that we use to prove Lemma~\\ref{lem:estim:2DCone:U1} and therefore Theorem~\\ref{lem:estim:2DCone:Theta}.\n\n\n\\begin{lemma}\\label{lem 10}\nLet $G\\subset\\bR^d$ be a bounded $\\ensuremath{\\mathcal{C}}^1$ domain, $p\\geq2$, $n\\in\\{-1,0,1,\\ldots\\}$, and $d-1< \\Theta<d-1+p$. Moreover, let $f^0\\in \\bH^{n}_{p,\\Theta+p}(G,T)$, $f^i\\in \\bH^{n+1}_{p,\\Theta}(G,T)$, $i=1,2$, and $g\\in\n\\bH^{n+1}_{p,\\Theta}(G,T;\\ell_2)$. \nAssume  $u$ is a solution to Equation~\\eqref{eq:SHE:Intro} on $G$\nin the class $\\frH^{1}_{p,\\Theta_1,0}(G,T)$ for some $\\Theta_1\\in [\\Theta, d+p-1)$. Then\n$u\\in \\frH^{n+2}_{p,\\Theta,0}(G,T)$ and\n\\begin{equation}\\label{lem 10:estimate}\n\\begin{alignedat}{1}\n\\|u\\|_{\\bH^{n+2}_{p,\\Theta-p}(G,T)} \n&\\leq N \n\\Big(\n\\|u\\|_{\\bL_{p,\\Theta}(G,T)}\n+\n\\|f^0\\|_{\\bH^{n}_{p,\\Theta+p}(G,T)}\\\\\n&\\quad\\qquad\\qquad \\quad + \\sum_i\\|f^i\\|_{\\bH^{n+1}_{p,\\Theta}(G,T)}+\n\\|g\\|_{\\bH^{n+1}_{p,\\Theta}(G,T;\\ell_2)}\\Big), \n\\end{alignedat}\n\\end{equation}\nwhere $N=N(p,d, \\Theta, n,G)$.  In particular, $N$ does not depend on $T$.\n\\end{lemma}\n\n\\begin{proof}\n\\emph{Step 1.} First we prove that $u\\in \\frH^{n+2}_{p,\\Theta}(G,T)$. \nBy~\n\\cite[Theorem~2.9]{Kim2004}, under the given assumptions, there exists a solution $v\\in \\frH^{n+2}_{p,\\Theta}(G,T)$.  Since $G$ is bounded, by Lemma~\\ref{collection}\\ref{col:bounded:embedding}, \n$$\n\\bH^{n+2}_{p,\\Theta-p}(G,T)\\subset \\bH^{1}_{p,\\Theta-p}(G,T)\\subset\\bH^1_{p,\\Theta_1-p}(G,T),\n$$\nand therefore $v\\in \\frH^1_{p,\\Theta_1}(G,T)$.  \nBy the uniqueness part of \\cite[Theorem~2.9]{Kim2004} we get $u=v$ (in \n$\\frH^1_{p,\\Theta_1}(G,T)$).\n\n\n\n\n\\noindent\\emph{Step 2.}  We prove Estimate~\\eqref{lem 10:estimate}.  \nIn fact, by \\cite[Theorem~2.9]{Kim2004}, this estimate holds even without the term $\\|u\\|_{\\bL_{p,\\Theta}(G,T)}$ on the right hand side if we allow  the constant $N$ to depend on $T$. \nHowever, a close look at the proof of \\cite[Theorem~2.9]{Kim2004} reveals that, indeed, if we leave this term on the right hand side, the constant can be kept independent of $T$, since the dependence on $T$ comes in only in the very last step of the relevant part of the proof of \\cite[Theorem~2.9]{Kim2004}, when a Gronwall argument is used in order to get rid of the terms that depend on $u$ on the right hand side.\n\nInstead of reproving~\\cite[Theorem~2.9]{Kim2004}, we illustrate the relevant steps in the proof and the changes required to obtain independence of $T$.\nThe key estimate is (5.6) of ~\\cite{Kim2004}, which says that \n\\begin{equation}\\label{eqn 4.7.1}\n\\begin{alignedat}{1}\n  \\|u\\|_{\\bH^{n+2}_{p,\\Theta-p}(G,T)}&\\leq N\\sgrklam{\\|\\psi u_{x}\\|_{\\bH^n_{p,\\Theta}(G,T)}+\\|u\\|_{\\bH^n_{p,\\Theta}(G,T)}\\\\\n  &\\quad \\qquad\\qquad\\quad +\\|\\psi(f^0+f^i_{x^i})\\|_{\\bH^n_{p,\\Theta}(G,T)}+\\|g\\|_{\\bH^{n+1}_{p,\\Theta}(G,T;\\ell_2)} }. \n\\end{alignedat}\n\\end{equation}\nIn our setting, i.e., for the stochastic heat equation, the constant $N$ in this estimate does not depend on $T$.\nIndeed, as  explained in detail in the proof of \\cite[Theorem~2.9]{Kim2004}, by using a suitable partition of unity, Estimate~\\eqref{eqn 4.7.1} is obtained through a combination of an a-priori estimate on the half space (\\cite[Theorem~2.10]{Kim2004}) and its analogue on the entire space (\\cite[Theorem~5.1]{Kry1999}). \nThe former theorem does not add any dependence on $T$ as the constant therein is explicitly proven to be independent of $T$.\nThis is different for \\cite[Theorem~5.1]{Kry1999}: The constant therein may indeed depend on $T$. \nHowever, this dependence only occurs if we consider equations with variable coefficients. \nFor the stochastic heat equation we may use the a-priori estimate from~\\cite[Theorem~4.2]{Kry1999} instead, which holds with a constant that does not depend on $T$.\nNote that on the left hand side of the estimate in~\\cite[Theorem~4.2]{Kry1999}, we have the $L_p$-norm of the second order derivatives of the solution.\nHowever, this is not a problem since in the proof of~\\eqref{eqn 4.7.1} we only use this estimate for the solution of a stochastic heat equation with compact support in $\\bR^d$ and, due to Poincar\\'e's inequality, for a function $v$ with compact support in $\\bR^d$, the norms $\\|v\\|_{H^{n+2}_p}$, $\\sum_i\\|v_{x^i}\\|_{H^{n+1}_p}$ and $\\sum_{i,j}\\|v_{x^ix^j}\\|_{H^n_p}$ are all equivalent.\n\nTo derive~\\eqref{lem 10:estimate} from \\eqref{eqn 4.7.1} we argue as follows: Using  \\eqref{eqn 4.7.1} and the basic properties of the weighted Sobolev spaces $H^n_{p,\\Theta}(G)$ from Lemma~\\ref{collection}, we easily obtain\n\\begin{equation}\\label{eqn 4.7.2}\n\\begin{alignedat}{1}\n  \\|u\\|_{\\bH^{n+2}_{p,\\Theta-p}(G,T)}\n  &\\leq N \\sgrklam{\\|u\\|_{\\bH^{n+1}_{p,\\Theta}(G,T)}+\\|f^0\\|_{\\bH^n_{p,\\Theta+p}(G,T)}\\\\\n  &\\quad \\qquad\\qquad\\qquad\\quad +\\sum_i\\|f^i\\|_{\\bH^{n+1}_{p,\\Theta}(G,T)}+\\|g\\|_{\\bH^{n+1}_{p,\\Theta}(G,T;\\ell_2)} }. \n\\end{alignedat}\n\\end{equation}\nThus if  $n=-1$, then \\eqref{lem 10:estimate} is proved. If $n\\geq 0$, then another application of Lemma~\\ref{collection}\\ref{col:bounded:embedding} shows that~\\eqref{eqn 4.7.2} implies\n\\begin{equation*}\n\\begin{alignedat}{1}\n  \\|u\\|_{\\bH^{n+2}_{p,\\Theta-p}(G,T)}\n  &\\leq N \\sgrklam{\\|u\\|_{\\bH^{n+1}_{p,\\Theta-p}(G,T)}+\\|f^0\\|_{\\bH^n_{p,\\Theta+p}(G,T)}\\\\\n  &\\quad \\qquad\\qquad\\qquad\\quad +\\sum_i\\|f^i\\|_{\\bH^{n+1}_{p,\\Theta}(G,T)}+\\|g\\|_{\\bH^{n+1}_{p,\\Theta}(G,T;\\ell_2)} },\n\\end{alignedat}\n\\end{equation*}\nwhich means that we can control $\\|u\\|_{\\bH^{n+2}_{p,\\Theta-p}(G,T)}$ by $\\|u\\|_{\\bH^{n+1}_{p,\\Theta-p}(G,T)}$ and suitable norms of the free terms.\nAfter repeating this step for $n$ more times, we arrive at \n\\begin{equation*}\n\\begin{alignedat}{1}\n  \\|u\\|_{\\bH^{n+2}_{p,\\Theta-p}(G,T)}\n  &\\leq N \\sgrklam{\\|u\\|_{\\bH^{1}_{p,\\Theta-p}(G,T)}+\\|f^0\\|_{\\bH^n_{p,\\Theta+p}(G,T)}\\\\\n  &\\quad \\qquad\\qquad\\qquad\\quad +\\sum_i\\|f^i\\|_{\\bH^{n+1}_{p,\\Theta}(G,T)}+\\|g\\|_{\\bH^{n+1}_{p,\\Theta}(G,T;\\ell_2)} }.\n\\end{alignedat}\n\\end{equation*}\nEstimate~\\eqref{lem 10:estimate} follows by applying~\\eqref{eqn 4.7.2} with $n=-1$. Note that all constants in the estimates above are independent of $T$.\n  \\end{proof}\n\n\n\nNow we go back to $\\cD\\subset\\bR^2$ and  proceed to prove Theorem~\\ref{lem:estim:2DCone:Theta}. The key step is presented in Lemma~\\ref{lem:estim:2DCone:U1} below. It provides an estimate of suitable weighted $L_p$-norms of the derivatives of the solution in \n\\[\nU_1:=\\{x\\in\\cD\\,\\colon\\,1<\\abs{x}<4\\}\n\\]\nby appropriate weighted $L_p$-norms of $u$ and of the derivatives of the free terms on the slightly bigger domain\n\\[\nV_1:=\\{x\\in\\cD\\,\\colon\\, 1\/2<\\abs{x}<8\\}.\n\\]\nAs these domains are bounded away from the vertex, the estimate involves only the distance $\\rho$ to the boundary.\nIt is crucial that the constant in Lemma~\\ref{lem:estim:2DCone:U1} below does not depend on $T$. In the proof we use Lemma \\ref{lem 10} with $d=2$.\n\n\n\n\\begin{lemma}\\label{lem:estim:2DCone:U1}\n Let $p\\ge 2$, $1<\\Theta<p+1$, and $m\\in\\{0,1,2,\\ldots\\}$.  Moreover, let $u\\in \\mathcal{K}^{1}_{p,\\theta,0}(\\cD,T)$ be a solution to Equation~\\eqref{eq:SHE:Intro} on $\\cD$ for some $\\theta\\in\\bR$. Then  \n\\begin{equation}\\label{eqn 4.9.3}\n\\begin{alignedat}{1}\n{\\mathbb E}&\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+1} \\int_{U_1} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p\\rho^{\\Theta-2}\\,dx\\,dt\\\\\n&\\leq\nN\\,\n{\\mathbb E} \\int_0^T \\int_{V_1} \\ssgrklam{\\abs{\nu}^p \n+ \n\\sum_{\\abs{\\alpha}\\leq (m-1)\\vee 0}\\Abs{\\rho^{\\abs{\\alpha}+1}D^{\\alpha}f^0}^p+\n \\sum_{\\abs{\\alpha}\\leq m}\\sum_i\\Abs{\\rho^{\\abs{\\alpha}}D^{\\alpha}f^i}^p  \n\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n+\n\\sum_{\\abs{\\alpha}\\leq m} \\Abs{\\rho^{\\abs{\\alpha}}D^\\alpha g}_{\\ell_2}^p}\\rho^{\\Theta-2}\\,dx\\,dt,  \n\\end{alignedat}\n\\end{equation}\n where $N=N(p,\\Theta,\\kappa_0,m)$. In particular, $N$  does not depend on $T$.\n\\end{lemma}\n\n\n\\begin{proof}\nAssume that the integrals on the right hand side of~\\eqref{eqn 4.9.3} are finite (if not, the statement is trivial). \nFix a constant $\\varepsilon \\in (0,1\/4)$, and  for $k=1,2,3$, let\n$$U^k_1:=\\{x\\in \\cD: 2^{-k\\varepsilon}<|x|<2^{2+k\\varepsilon}\\}.\n$$\nChoose a $\\ensuremath{\\mathcal{C}}^{\\infty}$ radial non-negative  function $\\eta=\\eta(|x|)$ such that $\\eta(t)=1$ for $t\\in [1,4]$ and $\\eta(t)=0$ if $t\\not\\in [2^{-\\varepsilon}, 2^{2+\\varepsilon}]$. Also choose a $\\ensuremath{\\mathcal{C}}^{1}$ domain $G\\subset\\cD$ such that\n$$\nU^2_1 \\subset G \\subset U^3_1\\subset V_1.\n$$\nBy the choice of $\\eta$ and $G$, $u\\eta$ vanishes on the boundary of $G$ and there exists $N=N(\\varepsilon)$ such that for all $x\\in \\cD \\cap  \\text{supp}\\,\\eta$,\n\\begin{equation}\n \\label{rho}\nN^{-1} \\rho_{\\cD}(x)\\leq \\rho_G (x) \\leq N \\rho_{\\cD}(x).\n\\end{equation}\n Let $l:=2m+3$ and set $\\zeta:=\\eta^l$.  \nThen\n$$\nd(\\zeta u)=\\grklam{\\Delta (\\zeta u)+u\\Delta \\zeta-f^i\\zeta_{x^i}+f^0\\zeta+(-2u\\zeta_{x^i}+f^i\\zeta)_{x^i}}dt+\\zeta g^k dw^k_t, \\quad t\\in(0,T], \n$$\non $G$ in the sense of distributions.\n Moreover, $\\zeta u\\in\\frH^1_{p,2,0}(G,T)$ since $u\\in\\cK^1_{p,\\theta,0}(\\cD,T)$ solves Equation~\\eqref{eq:SHE:Intro} on $\\cD$ and since, by Hardy's inequality, \n\\begin{align*}\n\\nnrm{\\zeta u}{L_{p,2-p}(G)}\n\\leq\nN\\,\\nnrm{(\\zeta u)_x}{L_{p,2}(G)}\n\\leq\nN\\,\\nnrm{\\zeta u}{K^1_{p,\\theta-p}(\\cD)}\n\\leq\nN(p,\\theta,G,\\zeta)\\,\\nnrm{ u}{K^1_{p,\\theta-p}(\\cD)}.\n\\end{align*}\nThe second inequality above is due to $\\rho_0 \\sim 1$ on the support of $\\zeta$.\nThus, by an application of Lemma~\\ref{lem 10} with $n=m-1$, we obtain \n\\begin{equation*}\n\\begin{alignedat}{1}\n\\|\\zeta u\\|_{\\bH^{m+1}_{p,\\Theta-p}(G,T)}\n&\\leq\n N \\sgrklam{\\sum_i\\|\\zeta f^i\\|_{\\bH^{m}_{p,\\Theta}(G,T)}+\\sum_i\\|u\\zeta_{x^i}\\|_{\\bH^m_{p,\\Theta}(G,T)}\\\\\n&\\qquad\\qquad\\qquad + \\|\\zeta f^0\\|_{\\bH^{m-1}_{p,\\Theta+p}(G,T)} \n +\\|u\\Delta \\zeta\\|_{\\bH^{m-1}_{p,\\Theta+p}(G,T)}\\\\ \n&\\qquad\\qquad\\qquad\\qquad\\quad + \\|\\zeta g\\|_{\\bH^m_{p,\\Theta}(G,T;\\ell_2)}\n+ \\|\\zeta u\\|_{\\bL_{p,\\Theta}(G,T)} },\n\\end{alignedat}\n\\end{equation*}\n once we can prove that all norms on the right hand side are finite. \nThe norms that do not involve $u$ together with $\\|\\zeta u\\|_{\\bL_{p,\\Theta}(G,T)}$ can be estimated by\n\\begin{align*}\n\\|\\eta u\\|_{\\bL_{p,\\Theta}(G,T)}\n+\n\\|\\eta f^0\\|_{\\bH^{(m-1)\\vee 0}_{p,\\Theta+p}(G,T)} \n+\n\\|\\eta g\\|_{\\bH^m_{p,\\Theta}(G,T;\\ell_2)}\n+\n\\sum_i\\|\\eta f^i\\|_{\\bH^{m}_{p,\\Theta}(G,T)},\n\\end{align*}\nwhich is finite since the right hand side of~\\eqref{eqn 4.9.3} is finite (use~\\eqref{rho}, Lemma~\\ref{collection} (in particular, part~\\ref{col:multiplier}) and the properties of $\\eta$ to estimate the norms above by the right hand side of~\\eqref{eqn 4.9.3}).\nMoreover,  since  \n$$\n\\|u\\zeta_{x^i}\\|_{H^m_{p,\\Theta}(G)}\n=\nl \\,\\|u \\eta^{l-1}\\eta_{x^i}\\|_{H^{m}_{p,\\Theta}(G)}\\leq  N\\|u\\eta^{l-2}\\|_{H^m_{p,\\Theta}(G)},\n$$\nand \n$$\n\\|u \\Delta \\zeta\\|_{H^{m-1}_{p,\\Theta+p}(G)}\\leq N\\|u\\eta^{l-2}\\|_{H^m_{p,\\Theta}(G)},\n$$\nthe condition\n\\begin{equation}\n    \\label{eqn 4.15.1}\n\\|\\eta^{l-2} u\\|_{\\bH^m_{p,\\Theta}(G,T)}<\\infty\n\\end{equation}\nis sufficient in order be able to apply Lemma~\\ref{lem 10} and obtain\n\\begin{align*}\n\\|\\eta^l u\\|_{\\bH^{m+1}_{p,\\Theta-p}(G,T)}\n&\\leq\nN \\sgrklam{\n\\|\\eta^{l-2} u\\|_{ \\bH^m_{p,\\Theta}(G,T)}\n+ \\|\\eta u\\|_{\\bL_{p,\\Theta}(G,T)}  \\\\\n&\\quad\\,\\,+ \\sum_i\\|\\eta f^i\\|_{\\bH^{m}_{p,\\Theta}(G,T)}\n+ \\|\\eta f^0\\|_{\\bH^{m-1}_{p,\\Theta+p}(G,T)} \n+ \\|\\eta g\\|_{\\bH^m_{p,\\Theta}(G,T;\\ell_2)}}.\\nonumber\n\\end{align*}\nIn particular, \nthis shows that \n\\[\n\\nnrm{\\eta^{l-2} u}{\\bH^{m}_{p,\\Theta}(G,T)}<\\infty\n\\quad\\Rightarrow\\quad\n\\nnrm{\\eta^l u}{\\bH^{m+1}_{p,\\Theta-p}(G,T)}<\\infty.\n\\]\nIn order to prove that~\\eqref{eqn 4.15.1} holds, we argue as follows: \nSince $\\|\\eta^{l-2} u\\|_{H^m_{p,\\Theta}(G)}\\leq N \\|\\eta^{l-2} u\\|_{H^m_{p,\\Theta-p}(G)}$,\nwe can iterate the arguments above with $m$ replaced by $m-j$ and $l$ replaced by $l-2j$ successively for $j=1,\\ldots,m$. After finitely many steps we arrive at the statement that if \n$\n\\|\\eta u\\|_{\\bL_{p,\\Theta}(G,T)}<\\infty\n$, then\n\\begin{equation}\\label{eqn 4.9.3.2}\n\\begin{alignedat}{1}\n\\|\\eta^{2m+3} u\\|_{\\bH^{m+1}_{p,\\Theta-p}(G,T)}\n&\\leq\nN \\sgrklam{\n\\|\\eta u\\|_{\\bL_{p,\\Theta}(G,T)} + \\sum_i\\|\\eta f^i\\|_{\\bH^{m}_{p,\\Theta}(G,T)}  \\\\\n&\\quad\\quad\\qquad\n+ \\|\\eta f^0\\|_{\\bH^{(m-1)\\vee 0}_{p,\\Theta+p}(G,T)} \n+ \\|\\eta g\\|_{\\bH^m_{p,\\Theta}(G,T;\\ell_2)}}.\n\\end{alignedat}\n\\end{equation}\nBut, as already explained above, $\\|\\eta u\\|_{\\bL_{p,\\Theta}(G,T)}$ is indeed finite since the right hand side of~\\eqref{eqn 4.9.3} is assumed to be finite.\nTherefore, Estimate~\\eqref{eqn 4.9.3.2} holds.\nMoreover, it proves~\\eqref{eqn 4.9.3}, since \\color{black} due to \\eqref{rho} and that fact that  $\\eta=1$ on $U_1$,\n\\begin{equation*}\n\\begin{alignedat}{1}\n\\sum_{|\\alpha|\\leq m+1}\\int_{U_1}  \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p\\rho^{\\Theta-2}\\,dx \n&\\leq \n\\sum_{|\\alpha|\\leq m+1}\\int_{G}  \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha (u\\eta^{2m+3})}^p\\rho^{\\Theta-2}\\,dx \\\\\n&\\leq N\\sum_{|\\alpha|\\leq m+1}\\int_{G}  \\Abs{\\rho^{\\abs{\\alpha}-1}_GD^\\alpha (u \\eta^{2m+3})}^p\\rho^{\\Theta-2}_G \\,dx\\\\\n&=\nN\\nnrm{\\eta^{2m+3} u}{H^{m+1}_{p,\\Theta-p}(G)}^p,\n\\end{alignedat}\n\\end{equation*}\nand, as already mentioned above, the right hand side of~\\eqref{eqn 4.9.3.2} can be estimated from above by the right hand side of~\\eqref{eqn 4.9.3}.\nNote that all constants in the estimates above are independent of $T$.\n\\end{proof}\n\n\n\\vspace{0.1cm}\n\nNow we can prove Theorem \\ref{lem:estim:2DCone:Theta} by applying Lemma~\\ref{lem:estim:2DCone:U1} to $u_n(t,x):=u(2^{2n}t,2^nx)$ for each $n\\in\\bZ$ and summing up the resulting estimates.\n\n\n\\begin{proof}[Proof of Theorem \\ref{lem:estim:2DCone:Theta}]\nFor every $n\\in\\bZ$, let $u_n(t,x):=u(2^{2n}t, 2^nx)$, $x\\in \\cD$, $t\\leq 2^{-2n}T$. Since\n$u\\in\\cK^1_{p,\\theta,0}(\\cD,T)$ solves Equation~\\eqref{eq:SHE:Intro} on $\\cD$, for every $n\\in \\bZ$, $u_n\\in\\cK^1_{p,\\theta,0}(\\cD,2^{-2n}T)$ and\n\\begin{align*}\ndu_n\n= \\grklam{\\Delta u_n +2^{2n}f^0_n+2^n(f^i_{n})_{x^i} }\\,dt + 2^n g^k_n\\, d(2^{-n}w^k_{2^{2n}t}), \\qquad  t\\in (0, 2^{-2n}T],\n\\end{align*}\non $\\cD$ in the sense of distributions with $u_n(0,\\cdot)=0$ and\n\\[\nf^0_n(t,x)=f^0(2^{2n}t,2^nx), \\,\\, f^i_n(t,x)=f^i(2^{2n}t,2^nx),\\,\\, \\text{ and }\\,\\,g_n(t,x)=g(2^{2n}t,2^nx).\n\\]\nNote that $(2^{-n}w^k_{2^{2n}t})_{t\\geq 0}$, $k=1,2,\\ldots$, is a sequence of independent one-dimensional Wiener processes.\nBy Lemma \\ref{lem:estim:2DCone:U1} applied to $u_n$ for  $t\\leq 2^{-2n}T$, we have\n\\begin{align*}\n{\\mathbb E} &\\int^{2^{-2n}T}_0 \\;\\sum_{k=0}^{m+1} \\int_{U_1} |\\rho^{k-1}(x)2^{nk}D^{k}u(2^{2n}t,2^nx)|^p \\rho^{\\Theta-2}(x)\\,dx \\,dt\\\\\n&\\leq N \\;{\\mathbb E} \\int^{2^{-2n}T}_0 \\int_{V_1}\\ssgrklam{ \\; |u(2^{2n}t,2^nx)|^p  \\\\\n&  \\hspace{3.4cm}+\\sum_{k=0}^{(m-1)\\vee 0}|\\rho^{k+1}(x)2^{2n}2^{nk}D^kf^0(2^{2n}t,2^nx)|^p \\\\\n&\\hspace{3.4cm}+\\sum_{k=0}^m \\sum_i|\\rho^k(x) 2^n 2^{nk} D^kf^i(2^{2n}t,2^nx)|^p \\\\\n&\\hspace{3.4cm}+\\sum_{k=0}^m|\\rho^k(x) 2^n 2^{nk} D^kg(2^{2n}t,2^nx)|^p_{\\ell_2}\\;}\\rho^{\\Theta-2}(x)\\,     dx\\, dt.\n\\end{align*}\nThus, if for $n\\in\\bZ$,\n$$\nU_n:=\\{x\\in \\cD: 2^{n-1}<|x|<2^{n+1} \\}\\quad\\text{and}\\quad V_n:=\\{x\\in \\cD: 2^{n-2}<|x|<2^{n+2}\\},\n$$\nthen, multiplying both sides by $2^{n(\\theta-2-p)}$, changing variables $(2^{2n}t,2^n x)\\to (t,x)$, and using the relations $\\rho(2^{-n}x)=2^{-n}\\rho(x)$ and $|x|\\sim 2^n $ on $U_{n-1}$ and on $V_{n-1}$, we get\n\\begin{align*}\n{\\mathbb E} \\int^T_0 &\\sum_{k=0}^{m+1} \\int_{U_{n-1}} |\\rho^{k-1} D^{k}u|^p  \\rho_\\circ^{\\theta-2}  \\left(\\frac{\\rho}{\\rho_{\\circ}}\\right)^{\\Theta-2} \\,dx \\,dt\\\\\n&\\leq N\\; {\\mathbb E} \\int^T_0 \\int_{V_{n-1}}\\ssgrklam{|\\,\\rho_{\\circ}^{-1}u|^p+\\sum_{k=0}^{(m-1)\\vee 0}|\\rho^{k+1}D^kf^0|^p \\\\\n&\\hspace{3.2cm} +\\sum_{k=0}^m \\sum_i|\\rho^k D^kf^i|^p+ |\\rho^k D^kg|^p_{\\ell_2}} \\rho_\\circ^{\\theta-2}  \\ssgrklam{\\frac{\\rho}{\\rho_{\\circ}}}^{\\Theta-2}  \\, dx\\, dt.\n\\end{align*}\nBy summing up with respect to $n\\in \\bZ$, we obtain the desired result. \n\\end{proof}\n\n\n\nThe following uniqueness result on bounded $\\ensuremath{\\mathcal{C}}^1$ domains will be used in Section~\\ref{sec:Polygons} to treat the  stochastic heat equation on polygons.\n\n\n\n\\begin{lemma}\\label{lem for uniqueness2}     \nLet $G$ be a bounded $\\ensuremath{\\mathcal{C}}^1$ domain in $\\bR^d$. For $j=1,2$, let $p_j\\geq 2$ and $\\Theta_j\\in (d-1,d-1+p_j)$, and assume that $u\\in \\frH^1_{p_1,\\Theta_1,0}(G,T)$ is a solution to Equation~\\eqref{eq:SHE:Intro} on $G$ with $f^0$, $f^i$ and $g$ satisfying\n     $$f^0\\in \\bL_{p_j,\\Theta_j+p_j}(G,T)\\cap \\bL_{p_j,d+p_j}(G,T),\\quad f^i\\in \\bL_{p_j,\\Theta_j}(G,T) \\cap \\bL_{p_j,d}(G,T),\\;i=1,2, \n     $$\n     $$\n      g\\in \\bL_{p_j,\\Theta_j}(G,T;\\ell_2)\\cap \\bL_{p_j,d}(G,T;\\ell_2),\n     $$\nfor $j=1,2$.    \nThen $u\\in \\frH^1_{p_2,\\Theta_2,0}(G,T)$.\n    \n     \\end{lemma}\n     \n     \\begin{proof}\n    By \\cite[Theorem 2.9]{Kim2004}, we can define  $v_1$ and $v_2$ as the solution to the equation in $\\frH^1_{p_1,d,0}(G,T)$ and $\\frH^1_{p_2,d,0}(G,T)$ respectively. Denote $q_1=p_1 \\vee p_2$ and $q_2=p_1 \\wedge p_2$. Then by the uniqueness result in $\\frH^1_{q_2,d,0}(G,T)$ we conclude $v_1=v_2$ and it belongs to $\\frH^1_{q_1,d,0}(G,T)$ as $G$ is bounded. \nAlso, due to Lemma~\\ref{collection}(v) and the uniqueness in   $\\frH^1_{p_1,\\Theta_1 \\vee d,0}(G,T)$, we conclude $v_1=u$. Now let $v_3$ be the solution to the problem in $\\frH^1_{p_2,\\Theta_2.0}(G,T)$.  The same argument as above shows $v_3=v_2$. \n\\end{proof}\n\n\n\\mysection{Proof of  Lemma \\ref{lem 4.5.1}}\\label{4}\n\n\nIn this section we prove the second key auxiliary result of this article, Lemma~\\ref{lem 4.5.1}. \nThroughout, we take and fix a $\\ensuremath{\\mathcal{C}}^\\infty$ radial function $\\eta$ and a corresponding $\\ensuremath{\\mathcal{C}}^1$ domain $G\\subseteq \\cD$  as in the proof of Lemma~\\ref{lem:estim:2DCone:U1}.  Recall that $\\eta(t)=1$ for $1\\leq t\\leq 4$.  As a consequence, there exists a constant $c>0$ such that \n\\begin{equation}\\label{eqn 5.6.544}\n\\sum_{n=-\\infty}^{\\infty}\\eta(e^{n+t})>c>0, \\quad \\forall \\, t\\in \\bR.\n\\end{equation}\n\n\n\n\nOur proof of Lemma~\\ref{lem 4.5.1} relies on the the following characterization of the $L_{p,\\theta}^{[\\circ]}(\\cD)$-norm.\n\n\\begin{lemma} \n\\label{lem 3.1}\nLet  $p>1$ and $\\theta\\in \\bR$.  Let $u\\colon\\cD\\to\\bR$ be a measurable function. \n\\begin{enumerate}[leftmargin=*,label=\\textup{(\\roman*)}, wide] \n\\item\\label{lem 3.1.1} If $\\eta$ and $G$ are as above,  then\n$$\n\\|u\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} \\sim   \\sum_{n \\in \\bZ} e^{n\\theta} \\|\\eta(|x|) u(e^nx)\\|^p_{L_p(\\cD)}= \\sum_{n \\in \\bZ} e^{n\\theta} \\|\\eta(|x|) u(e^nx)\\|^p_{L_p(G)}.\n$$\n\\item\\label{lem 3.1.2} For any  function $\\xi  \\in C^{\\infty}_0((0,\\infty))$ we have\n$$\n\\sum_{n \\in \\bZ} e^{n\\theta} \\|\\xi(|x|) u(e^nx)\\|^p_{L_p(\\cD)}\\leq N(\\xi, \\eta, p,\\theta) \\|u\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)}.\n$$\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\\begin{proof}\nTo see~\\ref{lem 3.1.1}, it is enough to repeat the proof of \\cite[Remark~1.3]{Kry1999c}. Indeed, by the change of variables $e^nx \\to x$,\n$$\n\\sum_{n\\in \\bZ} e^{n\\theta}\\|\\eta(|x|)u(e^nx)\\|^p_{L_p(\\cD)}=\\int_{\\cD} \\zeta(x)|u(x)|^pdx,\n$$\nwhere\n$$\n\\zeta(x)=\\sum_{n\\in \\bZ} e^{n(\\theta-2)}\\eta^p(e^{-n}|x|) \\sim |x|^{\\theta-2},\n$$\n see \\cite[Remark~1.3]{Kry1999c}.  Moreover, since $\\mathrm{supp}\\,\\eta\\cap\\cD\\subset G$, the equality in~\\ref{lem 3.1.1} is also satisfied.  Part~\\ref{lem 3.1.2}\nholds since\n $$\n\\sum_{n\\in \\bZ} e^{n(\\theta-2)}\\xi^p(e^{-n}|x|) \\leq N(\\xi,\\eta,\\theta,p) |x|^{\\theta-2};\n$$\nsee \\cite[Lemma~1.4]{Kry1999c} for details.\n\\end{proof}\n\nIn addition to Lemma~\\ref{lem 3.1}, we also need the following counterpart of   Lemma~\\ref{lem 4.5.1} for the stochastic heat equation on bounded $\\ensuremath{\\mathcal{C}}^1$ domains.\nIn the proof, we are going to use the common abbreviations\n\\[\n\\bH^n_p(T):=L_p(\\Omega_T, \\cP;H^n_p(\\bR^d)) \\quad\\text{and}\\quad\n\\bL_p(T;\\ell_2):=L_p(\\Omega_T,\\cP;L_p(\\bR^d;\\ell_2)),\n\\]\nfor $n\\in\\bZ$.\n\n\n\\begin{lemma}\n\\label{lem krylov}\nLet $G$ be a bounded $\\ensuremath{\\mathcal{C}}^1$ domain, $\\Theta\\in \\bR$,  $p\\geq 2$, and $u\\in \\frH^{1}_{p,\\Theta,0}(G,T)$ with $du=fdt +g\\,dw^k_t$. Then $u\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_{p,\\Theta}(G))$, and for any $c>0$,\n\n\\begin{eqnarray*}\n{\\mathbb E} \\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L_{p,\\Theta}(G)}\\le N \\Big( c \\|u\\|^p_{\\bH^{1}_{p,\\Theta-p}(G,T)} \n + c^{-1}\\|f\\|^p_{\\bH^{-1}_{p,\\Theta+p}(G,T)}+\\|g\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)} \\Big),\n \\end{eqnarray*}\nwhere $N= N(d,p,\\theta,G,  T)$. In particular, if  $f=f^0+f^i_{x^i}$, then the right hand side above is bounded by a constant multiple of \n\\[\n c \\|u\\|^p_{\\bH^1_{p,\\Theta-p}(G,T)}+c^{-1}\\|f^0\\|^p_{\\bL_{p,\\Theta+p}(G,T)}+c^{-1}\\|f^i\\|^p_{\\bL_{p,\\Theta}(G,T)}\n +\\|g\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)}.\n\\]\n \\end{lemma}\n\n\n\n\\begin{proof}\nIntroduce a partition of unity  $\\zeta_0, \\zeta_1,\\zeta_2,\\cdots, \\zeta_M$\n of $G$ such that  $\\zeta_0\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(G)$ and\n $\\zeta_j\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(B_{r}(x_j))$ $(j=1,2,\\cdots,M)$, where\n  $x_j\\in \\partial G$ and $r<r_0\/K_0$.  \nFor any $\\Theta\\in \\bR$, $m\\in \\bZ$, and  $v\\in H^m_{p,\\Theta}(G)$, since $\\zeta_0$ has compact support in $G$, we can consider $\\zeta_0v$ as a function defined on the entire space, so that by Lemma~\\ref{collection}\\ref{col:cptsupport},\n\\begin{equation}\n         \\label{eqn 2018-12}\n\\|\\zeta_0 v\\|_{H^m_{p,\\Theta}(G)} \\sim \\|\\zeta_0v\\|_{H^m_p(\\bR^d)}.\n\\end{equation}\nAlso, by~\\eqref{eqn 12.7.1}, for $j\\geq 1$ and  any $\\Theta, \\nu\\in \\bR$,\n\\begin{equation}\n\\label{relation}\n\\|\\psi^{\\nu} \\zeta_j v\\|^p_{H^{m}_{p,\\Theta}(G)} \\sim \\|(x^1)^{\\nu}(\\zeta_j v)(\\Psi^{-1}_j)\\|^p_{H^{m}_{p,\\Theta}(\\bR^d_+)},\n\\end{equation}\nwhere $\\Psi_j$ is the corresponding mapping from Definition~\\ref{definition domain} related to $x_j\\in \\partial G$.\nThus\n\\begin{equation}\\label{eqv} \n\\begin{alignedat}{1}\n\\|v\\|^p_{H^{m}_{p,\\Theta}(G)}\n&= \\sgnnrm{\\sum_{j=0}^M \\zeta_j v}{{H^{m}_{p,\\Theta}(G)}}^p\n\\sim \\sum_{j=0}^M \\|\\zeta_j v\\|^p_{H^{m}_{p,\\Theta}(G)}\\\\\n& \\sim\n\\|\\zeta_0 v\\|^p_{H^{m}_p(\\bR^d)} + \\sum_{j= 1}^M \\|(\\zeta_j v)(\\Psi^{-1})\\|^p_{H^{m}_{p,\\Theta}(\\bR^d_+)}.  \n\\end{alignedat}\n\\end{equation}\nAs a consequence, \n\\begin{equation}\n  \\nonumber \\label{eqn 2018-15}\n{\\mathbb E} \\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L_{p,\\Theta}(G)} \n\\leq N \n\\ssgrklam{{\\mathbb E}\\sup_{t\\leq T} \\|\\zeta_0u\\|^p_{L_p(\\bR^d)} +\\sum_{j=1}^M {\\mathbb E} \\sup_{t\\leq T} \\|(\\zeta_ju)(\\Psi^{-1}_j)\\|^p_{L_{p,\\Theta}(\\bR^d_+)}}.\n\\end{equation}\nTherefore, in order to obtain the desired estimate and the continuity assertion, it is enough to estimate the terms on the right hand side appropriately and to prove continuity of $\\zeta_ju$, $j=0,\\ldots,M$. \nFor the first term, note that \n$$\nd(\\zeta_0 u)=\\zeta_0f \\,dt+\\zeta_0g\\,dw^k_t, \\quad  \\,t\\in (0,T],\n$$\non $\\bR^d$ and $\\zeta_0u\\in \\bH^1_p(T)$ due to~\\eqref{eqn 2018-12}.\nTherefore, by \\cite[Corollary~4.12]{Kry2001} (for $p>2$) and \\cite[Remark~4.14]{Kry2001} (for $p=2$), \n\\begin{equation}\\label{eq:cts:zeta0}\n\\zeta_0 u \\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_p(\\bR^d)),\n\\end{equation}\nand there exists a constant $N$, such that, for any $c>0$,\n\\begin{align*}\n{\\mathbb E} \\sup_{t\\leq T} \\|\\zeta_0u\\|^p_{L_p}\n&\\leq N c \\|\\zeta_0u\\|^p_{\\bH^1_p(T)}+\nNc^{-1}\\|\\zeta_0 f\\|^p_{\\bH^{-1}_p(T)}+N \\|\\zeta_0g\\|^p_{\\bL_p(T;\\ell_2)}\\\\\n&\\leq Nc \\|\\zeta_0u\\|^p_{\\bH^1_{p,\\Theta-p}(G,T)}+Nc^{-1}\\|\\zeta_0f\\|^p_{\\bH^{-1}_{p,\\Theta}(G,T)}\n+N\\|\\zeta_0g\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)}.\n\\end{align*}\nMoreover, for every $j\\in\\{1,\\ldots,M\\}$,\n\\[\nd((\\zeta_j u)(\\Psi^{-1}_j))\n=\n(\\zeta_j f)(\\Psi^{-1}_j)\\,dt+(\\zeta_jg^k)(\\Psi^{-1}_j) \\,dw^k_t =:F_j \\,dt+G^k_j \\,dw^k_t, \\quad t\\in (0,T],\n\\]\non $\\bR^d_+$ and due to~\\eqref{relation}, $(\\zeta_ju)(\\Psi^{-1}_j) \\in \\frH^1_{p,\\Theta}(\\bR^d_+,T)$ (see  Section~\\ref{sec:proof:lift}  for  notation). Therefore,\nby \\cite[Theorem~4.1]{Kry2001} (for $p>2$) and \\cite[Remark~4.5]{Kry2001} (for $p=2$),\n\\begin{equation}\\label{eq:cts:zetaj}\n(\\zeta_ju)(\\Psi^{-1}_j)\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_{p,\\Theta}(\\bR^d_+)),\n\\end{equation}\nand\n\\begin{align*}\n{\\mathbb E} &\\sup_{t\\leq T} \\|(\\zeta_ju)(\\Psi^{-1}_j)\\|^p_{L_{p,\\Theta}(\\bR^d_+)}\\\\\n&\\leq \nN c \\|(\\zeta_ju)(\\Psi^{-1}_j)\\|^p_{\\bH^1_{p,\\Theta-p}(\\bR^d_+,T)}\n+Nc^{-1}\\|F_j\\|^p_{\\bH^{-1}_{p,\\Theta+p}(\\bR^d_+,T)}+N\\|G_j\\|^p_{\\bL_{p,\\Theta}(\\bR^d_+,T;\\ell_2)}\\\\\n&\\leq N c \\|\\zeta_ju\\|^p_{\\bH^1_{p,\\Theta-p}(G,T)}\n+Nc^{-1}\\|\\zeta_jf\\|^p_{\\bH^{-1}_{p,\\Theta+p}(G,T)}+\nN\\|\\zeta_jg\\|^p_{\\bL_{p,\\Theta}(G,T;\\ell_2)}.\n\\end{align*}\nSumming up gives the desired estimate and $u\\in L_p(\\Omega;\\ensuremath{\\mathcal{C}}([0,T];L_{p,\\Theta}(G)))$ follows from~\\eqref{eq:cts:zeta0} and \\eqref{eq:cts:zetaj}, together with~\\eqref{eqv}.\nThe second assertion is due to the fact that, by Lemma~\\ref{collection}, \n\\[\n\\|f^i_{x^i}\\|_{H^{-1}_{p,\\Theta+p}(G)}\\leq N \\|\\psi f^i_{x^i}\\|_{H^{-1}_{p,\\Theta}(G)}\\leq N \\|f^i\\|_{L_{p,\\Theta}(G)}.\\qedhere\n\\]\n\\end{proof} \n\n\n\n\nWe have now all ingredients we need in order to prove Lemma~\\ref{lem 4.5.1}.\n\n\n\\begin{proof}[Proof of  Lemma~\\ref{lem 4.5.1}]\nWe first prove Estimate~\\eqref{eq:estim:sup:2DCone}.  By Lemma~\\ref{lem 3.1},\n\\begin{equation}\\label{eqn 2018-4}\n{\\mathbb E} \\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} \n\\leq \nN \\sum_{n\\in \\bZ} e^{n\\theta} \\,{\\mathbb E} \\sup_{t\\leq T} \\|u(t,e^nx) \\eta(x)\\|^p_{L_p(G)}.\n\\end{equation}\n For $n\\in\\bZ$, let  $v_n(t,x):=u(t,e^nx)\\eta(x)$. Then \n$$\ndv_n=[ e^{-n} (f^i(t,e^nx))_{x^i}\\eta(x)+f^0(t,e^nx)\\eta(x)]dt+g^k(t,e^nx)\\eta(x)dw^k_t, \\quad t\\in(0,T],\n$$\non $G$.\nNote that\n$$\n e^{-n} (f^i(t,e^nx))_{x^i}\\eta(x)=  e^{-n} [f^i(t,e^nx)\\eta(x)]_{x^i}-e^{-n}f^i(t,e^nx)\\eta_{x^i}(x),\n$$\nand\n\\begin{equation}\\label{eq_sec4_2}\n(v_n)_{x^i}=e^{n}u_{x^i}(t,e^nx)\\eta(x)-u(t,e^nx)\\eta_{x^i}(x).\n\\end{equation}\nObviously, $L_{p,2}(G)=L_p(G)$  and  by Hardy's inequality,\n\\begin{equation}\\label{eq_sec4_1}\n\\|v_n\\|_{H^1_{p,2-p}(G)}\n\\le \nN\\sgrklam{\\|\\rho^{-1}_G v_n\\|_{L_p(G)}+\\sum_i\\|(v_{n})_{x^i}\\|_{L_p(G)} }\n\\leq \nN \\|(v_{n})_{x}\\|_{L_p(G)}.\n\\end{equation}\nBy Lemma~\\ref{lem krylov} with $\\Theta=d=2$, \\eqref{eq_sec4_1}, and \\eqref{eq_sec4_2}, for any $c>0$\n\\begin{align*}\n&{\\mathbb E} \\sup_{t\\leq T} \\|v_n(t,\\cdot)\\|^p_{L_p(G)} \\\\\n&\\leq N \\Big(ce^{np}\\sum_i\\|u_{x^i}(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(G,T)}+\nc\\sum_i\\|u(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_{p,d}(G,T)} \\\\\n\\\\&\\quad \\quad +e^{-np}c^{-1}\\|f^i(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(G,T)}\n+e^{-np}c^{-1}\\sum_i\\| \\rho f^i(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_{p,d}(G,T)}\\\\\n&\\quad \\quad +c^{-1}\\|\\rho f^0(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(G,T)} +\\|\\eta g(\\cdot,e^n\\cdot)\\|^p_{\\bL_{p,d}(G,T;\\ell_2)} \\Big).\n\\end{align*}\nSince $\\rho$ is bounded in $G$, we can drop $\\rho$ above, so that, if we choose\n$c:=e^{-np}$ and use~\\eqref{eqn 2018-4}, we get\n\\begin{align*}\n{\\mathbb E} &\\sup_{t\\leq T} \\|u(t,\\cdot)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} \\\\\n&\\leq N \n\\ssgrklam{\\sum_n e^{n\\theta}\\|u_x(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T)}\n+ \\sum_n e^{n(\\theta-p)}\\sum_i\\|u(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_{p,d}(\\cD,T)}\\\\\n&\\quad\\quad\\quad+ \\sum_{n,i} e^{n\\theta}\\|f^i(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T)}\n+ \\sum_{n,i} e^{n\\theta} \\|f^i(\\cdot,e^n\\cdot)\\eta_{x^i}\\|^p_{\\bL_p(\\cD,T)}\\\\\n&\\quad\\quad\\quad+  \\sum_n e^{n(\\theta+p)}\\|f^0(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T)}\n+ N \\sum_n e^{n\\theta}\\|g(\\cdot,e^n\\cdot)\\eta\\|^p_{\\bL_{p,d}(\\cD,T;\\ell_2)}}.\n\\end{align*}\nTherefore,  due to Lemma~\\ref{lem 3.1}\\ref{lem 3.1.2}, Estimate~\\eqref{eq:estim:sup:2DCone} holds.\n\nTo prove the continuity assertion, we take a sequence of smooth functions $\\xi_n\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(\\bR^2)$ such that $\\xi_n=1$ if $3\/n<|x|<n$, $\\xi_n(x)=0$ if $|x|\\leq 2\/n$ or $|x|\\geq 2n$, and \n\\begin{equation}\n   \\label{eqn 4.9.9}\n\\sup_i\\sup_n \\sup_x |x|\\, |(\\xi_n)_{x^i}| < \\infty.\n\\end{equation}\nMoreover, for every $n\\in\\bN$, let $G_n\\subset\\cD$ be a bounded $\\ensuremath{\\mathcal{C}}^1$ domain such that  \n$$\n\\cD \\cap \\{x: 2\/n <|x|<2n\\} \\subset G_n \\subset \\cD \\cap \\{x: 1\/n <|x|<3n\\}.\n$$\nThen\n$$\nd(\\xi_n u)=\\left[ (\\xi_n f^i)_{x^i}-(\\xi_n)_{x^i} f^i+\\xi_nf^0\\right]dt+\\xi_n g^kdw^k_t, \\quad t\\in (0,T],\n$$\non $G_n$ and,  by the choice of $\\xi_n$, we also know that $\\xi_n u \\in \\frH^1_{p,2}(G_n,T)$ since $\\xi_n u\\in\\cK^1_{p,\\theta,0}(\\cD,T)\\subseteq \\mathring{\\ensuremath{K}}^1_{p,\\theta-p}(\\cD,T)$, see also~\\cite[Remark~3.2]{CioKimLee+2018}.  By Lemma~\\ref{lem krylov} with $\\Theta=2$, $\\xi_n u \\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_p(G_n))$.\n  Since $\\xi_n$ vanishes near the origin and toward infinity, we conclude  \n$$\n\\xi_n u \\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_{p,\\Theta}(\\cD)).\n$$\nApplying~\\eqref{eq:estim:sup:2DCone} to $\\xi_n u -\\xi_m u$ and using~\\eqref{eqn 4.9.9}, we find that $\\xi_n u$ is a Cauchy sequence in  \n$L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T]; L_{p,\\Theta}(\\cD))$, which converges in this space to a limit $v$. Moreover, applying~\\eqref{eq:estim:sup:2DCone} to $\\xi_n u-u$,  we find that $u=v$ in $L_p(\\Omega; L_{\\infty}([0,T]; L_{p,\\Theta}(\\cD))$. Therefore, $u(t)=v(t)$ for all $t\\leq T$ ($\\ensuremath{\\mathbb P}$-a.s.).  Thus $u$ has the desired version with continuous paths. \n\\end{proof}\n\n\n\\mysection{The stochastic heat equation on polygons}\\label{sec:Polygons}\n\n In this section we present our analysis for the stochastic heat equation~\\eqref{eq:SHE:Intro} on polygons in $\\bR^2$. We fix the following setting: Throughout, \nlet $\\cO \\subset \\bR^2$ be a polygon with vertexes $\\{v_1,v_2, \\ldots, v_M\\}\\subset\\bR^2$.  For $x\\in \\cO$, put\n$$\n\\tilde{\\rho}(x):=\\min_{1\\leq j\\leq M} |x-v_j|, \\quad  \\quad \\rho(x):=\\rho_\\domain(x):=\\text{dist}(x,\\partial \\cO),\n$$\nand for $j=1,\\ldots,M$ define\n$$\n\\kappa_j:=\\text{interior angle at}\\, v_j, \\quad \\quad \\kappa_0:=\\max_{1\\leq j\\leq M}\\kappa_j.\n$$\n\nMotivated by the analysis of the stochastic heat equation on angular domains from Section~\\ref{sec:2DCone}, we are going to use weighted Sobolev spaces with weights based on the distance $\\tilde{\\rho}$ to the set of vertexes in order to establish existence and uniqueness of a solution. More precisely,  for $\\theta\\in \\bR$, $p>1$ and $n\\in \\{0,1,2,\\ldots\\}$,  we define the spaces  $K^n_{p,\\theta}(\\cO)$, $K^n_{p,\\theta}(\\cO;\\ell_2)$,   $L^{[\\circ]}_{p,\\theta}(\\cO)$, and \n$L^{[\\circ]}_{p,\\theta}(\\cO;\\ell_2)$  in the same way as the corresponding spaces on $\\cD$ from Section~\\ref{sec:2DCone} with $\\rho_\\circ$ replaced by $\\tilde{\\rho}$, i.e., for instance,\n$$\n\\|u\\|^p_{K^n_{p,\\theta}(\\cO)}=\\sum_{|\\alpha|\\leq n} \\int_{\\cO} |\\tilde{\\rho}^{|\\alpha|} D^{\\alpha}u|^p \\tilde{\\rho}^{\\theta-2}dx.$$\nThe space $\\mathring{\\ensuremath{K}}^1_{p,\\theta}(\\cO)$ is  the closure of the space $\\ensuremath{\\mathcal{C}}^\\infty_c(\\cO)$ of test functions in $\\ensuremath{K}^1_{p,\\theta}(\\cO)$. \nIn analogy to Section~\\ref{sec:2DCone}, for the $L_p$-spaces of predictable stochastic processes with values in the weighted Sobolev spaces introduced above we use the  abbreviations \n\\[\n\\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cO,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};\\ensuremath{K}^{n}_{p,\\theta}(\\cO)),\n\\qquad \\ensuremath{\\mathbb{\\wso}}^{n}_{p,\\theta}(\\cO,T;\\ell_2)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}};\\ensuremath{K}^{n}_{p,\\theta}(\\cO;\\ell_2)),\n\\]\n\\[\n\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)\n:=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cO,T), \\quad\\quad\\quad  \n\\bL^{[\\circ]}_{p,\\theta}(\\cO,T,\\ell_2)\n:=\\ensuremath{\\mathbb{\\wso}}^0_{p,\\theta}(\\cO,T;\\ell_2),\n\\]\nand\n\\[\n\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta}(\\cO,T)\n:=\nL_p(\\Omega_T, \\ensuremath{\\mathcal{P}} ;\\mathring{\\ensuremath{K}}^{1}_{p,\\theta}(\\cO)).\n\\]\nMoreover, $\\cK^1_{p,\\theta,0}(\\domain,T)$ is defined the following way.\n\\begin{defn}\nLet $p \\geq 2$. We write $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$  if\n$u\\in\\mathring{\\ensuremath{\\mathbb{\\wso}}}^1_{p,\\theta-p}(\\cO,T)$\n and\nthere exist $f^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)$, $f^i\\in  \\bL^{[\\circ]}_{p,\\theta}(\\cO,T)$, $i=1,2$, and\n $g\\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)$\nsuch that \n\\begin{equation*}\\label{eqn 28_2}\n du=(f^0+f^i_{x^i})\\, dt +g^k \\, dw^k_t,\\quad t\\in(0,T],\n\\end{equation*}\non $\\domain$ in the sense of distributions with $u(0,\\cdot)=0$; see Definition~\\ref{defn sol} accordingly. In this situation\nwe also write\n$$\n\\bD u:=f^0+f^i_{x^i}\\qquad\\text{and}\\qquad \\bS u :=g.\n$$\n\\end{defn}\n\n\n\n\nIn this article, Equation~\\eqref{eq:SHE:Intro} has the following meaning on $\\domain$.\n\n\\begin{defn}\nWe say that  $u$ is a solution to Equation~\\eqref{eq:SHE:Intro} on $\\cO$ in the class $\\mathcal{K}^{1}_{p,\\theta,0}(\\cO,T)$ \nif\n$u\\in \\mathcal{K}^{1}_{p,\\theta,0}(\\cO,T)$ with\n\\[\n\\bD u = \\Delta u + f^0+f^i_{x^i}=f^0+(f^i+u_{x^i})_{x^i}\n\\qquad\n\\text{and}\n\\qquad\n\\bS u = g.\n\\]\n\\end{defn}\n\n\n\n\nBefore we look at Equation~\\eqref{eq:SHE:Intro} in detail, we first prove the following version of Lemma~\\ref{lem 4.5.1} for polygons.\nIt is a key ingredient in our existence and uniqueness proof below.\n\n\n\\begin{lemma}\n\\label{lem for gronwall}\nLet $p\\geq 2$ and $\\theta \\in \\bR$. Assume that $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$, such that $du=(f^0+f^i_{x^i})dt+g^kdw^k_t$ with\n\\[\nf^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T), \n\\quad f^i \\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T),\\,i=1,2, \\quad\\text{and } g\\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2).\n\\]\n Then $u\\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T];L^{[\\circ]}_{p,\\theta}(\\cO)))$ and \n \\begin{align*}\n{\\mathbb E} \\sup_{t\\leq T}& \\|u(t,\\cdot)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cO)} \\\\\n&\\leq N \\Big(\\|u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,T)}\n+\\|f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO, T)}+\\sum_i\\|f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO, T)}+\\|g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO, T; \\ell_2)} \\Big)\\\\\n&=:N \\,C(u,f^0,f^i,g, T),\n \\end{align*}\nwhere $N=N(d,p,\\theta,T)$ is a non-decreasing function of $T$.\nIn particular, for any $t\\leq T$,\n\\[\n\\|u\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,t)}\n\\leq \n\\int^t_0 {\\mathbb E}\\sup_{r\\leq s} \\|u(r)\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cD)} ds\\leq  N(d,p, \\theta,  T) \\int^t_0 C(u,f^0,f^i,g, s) \\,ds.\n \\]\n \\end{lemma}\n\n\\begin{proof}\nWe combine Lemma \\ref {lem 4.5.1} (see also Remark~\\ref{remark cones}) and Lemma~\\ref{lem krylov} as follows.\n\nFix a sufficiently small $r>0$ such that  $B_{3r}(v_j)$ contains only one vertex $v_j$  and intersects with only two edges   for each $j\\leq M$. \nChoose a function $\\xi\\in \\ensuremath{\\mathcal{C}}^{\\infty}_c(\\bR^2)$ such that $0<\\xi(x)\\leq 1$ for $|x|<2r$, $\\xi(x)=1$ for $|x|<r$, and $\\xi(x)=0$ for $|x|\\geq 2r$. \nLet $\\xi_j(x):=\\xi(x-v_j)$ and put $\\xi_0(x):=1-\\sum_{j=1}^M \\xi_j(x)$, $x\\in\\domain$.  \nNote that by the choice of $r$ and $\\xi$, the supports of the $\\xi_j$'s ($j\\geq 1$) do not overlap, and therefore $0\\leq \\sum_{j=1}^M \\xi^j \\leq 1$. Moreover, $\\xi_0(x)=1$ if  $x$ is not close to vertexes, that is,  if $\\tilde{\\rho}(x)=\\min_{1\\leq j\\leq M} |x-v_j|\\geq r$.\n For $j=1,2,\\ldots,M$, let $\\cD_j$ be the angular domain centered at $v_j$ with interior angle $\\kappa_j$ such that \n \\begin{equation}\n    \\label{eqn 4.10.10}\n \\cD_j \\cap B_{3r}(v_j)=\\cO \\cap B_{3r}(v_j).\n \\end{equation}\n Moreover, let $G$ be a $\\ensuremath{\\mathcal{C}}^1$ domain in $\\cO$ such that \n \\begin{equation}\n     \\label{eqn 4.10.12}\n \\xi_0(x)=0 \\quad \\text{for }\\, x\\in \\cO\\setminus G \\quad \\text{and}\\quad \\inf_{x\\in G} \\tilde{\\rho}(x)>c>0.\n \\end{equation}\n Note that by the choice of $\\xi_j$, $\\cD_j$, $j=0,1,\\ldots,M$, and $G$, for any $\\theta\\in\\bR$ \\color{black}  and $v\\in \\mathring{K}\\color{black}^1_{p,\\theta-p}(\\cO)$, \n\\begin{equation}\n \\label{eqn 4.11.1}\n\\|\\xi_0 v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)} \n\\sim \n\\|\\xi_0 v\\|_{L_p(G)}, \\quad \\quad   \n\\|\\xi_0v\\|_{K^1_{p,\\theta-p}(\\cO)}\\sim \\|\\xi_0v\\|_{H^1_{p,2-p}(G)},\n\\end{equation}\n\\begin{equation}\n \\label{eqn 4.11.2}\n\\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cO)}=\\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cD_j)} \\quad (j\\geq 1).\n \\end{equation}\nThe first and the third relation are trivial  and hold actually for arbitrary measurable $v\\colon\\domain\\to \\bR$, provided the expressions make sense. The second one is due  to \\eqref{eqn 4.10.12} and  Hardy's inequality as\n \\begin{align*}\n\\|\\xi_0v\\|_{K^1_{p,\\theta-p}(\\cO)}\n&\\leq \nN (\\|\\xi_0v\\|_{L_p(G)}+\\sum_i\\|(\\xi_0v)_{x^i}\\|_{L_p(G)}) \\\\\n& \\leq  N \\|\\xi_0 v\\|_{H^1_{p,2-p}(G)}\\leq N \\sum_i\\|(\\xi_0v)_{x^i}\\|_{L_p(G)} \n\\leq N \\|\\xi_0 v\\|_{K^1_{p,\\theta-p}(\\cO)}.\n \\end{align*}\n The three relations from~\\eqref{eqn 4.11.1} and~\\eqref{eqn 4.11.2} together imply, in particular, that \n\\[\n\\|v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n\\sim \n\\sum_{j=0}^M \\|\\xi_j v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}   \n\\sim \n\\|\\xi_0v\\|_{L_p(G)}+ \\sum_{j=1}^M \\|\\xi_j v\\|_{L^{[\\circ]}_{p,\\theta}(\\cD_j)},\n\\]\n\\begin{eqnarray}\n \\label{eqn 4.11.4}\n\\|v\\|_{K^1_{p,\\theta-p}(\\cO)} \n\\sim \n\\sum_{j=0}^M \\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cO)}\n\\sim\n\\|\\xi_0v\\|_{H^1_{p,2-p}(G)}\n+\n\\sum_{j=1}^M \\|\\xi_j v\\|_{K^1_{p,\\theta-p}(\\cD_j)}.\n\\end{eqnarray}\nAlso note that for  any multi-index $\\alpha$,\n\\begin{equation}\n   \\label{eqn 4.10.14}\n\\sum_{j=0}^M \\|v D^{\\alpha}\\xi_j\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n+  \n\\sum_{j=0}^M \\|v \\tilde{\\rho}D^{\\alpha}\\xi_j\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n\\leq N \n\\|v\\|_{L^{[\\circ]}_{p,\\theta}(\\cO)}.\n\\end{equation}\n\nUsing the preparations above, we can verify the assertion the following way. \nFor each $j\\in\\{1,2,\\ldots,M\\}$, $u^j:=\\xi_j u \\in \\cK^1_{p,\\theta,0}(\\cD_j,T)$ with\n\\begin{equation}\n     \\label{eqn 4.10.11}\n  du^j=\\big((\\xi_jf^i)_{x^i}+\\xi_jf^0-(\\xi_j)_{x^i}f^i\\big)\\,dt+ \\xi_jg^k\\, dw^k_t, \\quad t\\in(0,T],\n  \\end{equation}\non $\\cD_j$ in the sense of distributions.\nThus, by Lemma~\\ref{lem 4.5.1} (see also Remark~\\ref{remark cones}) and (\\ref{eqn 4.10.10}),\n  $ u^j \\in L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T];L^{[\\circ]}_{p,\\theta}(\\cO)))$, and \n\\begin{align}\n{\\mathbb E} \\sup_{t\\leq T} \\|\\xi_j u\\|^p_{L^{[\\circ]}_{p,\\theta}(\\cO)}\n\\leq &N\\Big( \\|\\xi_j f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}+\\sum_i\\|\\xi_j f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\\label{eqn 4.10.13}\\\\\n  &+ \\|\\xi_ju\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,T)}+\\|(\\xi_j)_{x^i}\\tilde{\\rho}f^i \\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}  +\n  \\|\\xi_j g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)} \\Big). \\nonumber\n  \\end{align}\nAlso, $u^0:=\\xi_0 u\\in \\frH^1_{p,2,0}(G,T)$ and \\eqref{eqn 4.10.11} holds with $j=0$. Thus, by Lemma~\\ref{lem krylov} and \\eqref{eqn 4.10.12}, $u^0\\in  L_p(\\Omega; \\ensuremath{\\mathcal{C}}([0,T];L^{[\\circ]}_{p,\\theta}(\\cO)))$, and \\eqref{eqn 4.10.13} holds with $j=0$. Therefore, by summing up all these estimates and using above relations, we get the desired result. \n\\end{proof}\n\n\nOur main existence and uniqueness result for the stochastic heat equation on polygons reads as follows.\nRecall that in this section $\\kappa_0$ denotes the maximum over all  interior angles of the polygon $\\cO$.\n\n\\begin{thm}[Existence and uniqueness\/polygons]\n    \\label{thm polygon main}\nLet $p\\geq 2$ and assume that $\\theta\\in\\bR$ satisfies \\eqref{eq:range:vertex}. Then for any \n\\[\nf^0\\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T), \\quad\nf^i\\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T),\\,i=1,2,\\quad\\text{and}\\quad \ng  \\in \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2),\n\\]\nEquation~\\eqref{eq:SHE:Intro} on $\\cO$ \nhas a unique solution $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$. \nMoreover, \n\\begin{equation}\n \\label{eqn polygon main}\n\\|u\\|_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,T)}\n\\leq \nN \\,\\sgrklam{\\|f^0\\|_{ \\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}\n+\n\\sum_i\\|f^i\\|_ {\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\n+\n\\|g\\|_{ \\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)}},\n\\end{equation}\nwhere $N=N(p,\\theta,\\kappa_0,T)$.\n\\begin{proof}\n\\emph{Step 1.}\nWe first prove that~\\eqref{eqn polygon main} holds given that a solution $u\\in\\cK^1_{p,\\theta,0}(\\cO,T)$ already  exists, by using corresponding results for the stochastic heat equation on angular domains and on $\\ensuremath{\\mathcal{C}}^1$ domains. This will, in particular, take care of the uniqueness. \n\n\n Let $r>0$, $\\xi_j$, $\\cD_j$, $j=1,\\ldots,M$, as well as $\\xi_0$ and $G$ be as in the proof of Lemma~\\ref{lem for gronwall}. \nA very similar reasoning as therein can be used to verify that $\\xi_0u\\in\\frH^{1}_{p,\\theta,0}(G,T)$, $\\xi_ju\\in\\cK^1_{p,\\theta,0}(\\cD_j,T)$ for $j\\geq 1$, and that for all $j\\in\\{0,1,\\ldots,M\\}$, \n\\begin{align*}\nd(\\xi_j u)\n&=\n\\grklam{\\Delta (\\xi_j u)+(-2u(\\xi_j)_{x^i}+\\xi_j f^i )_{x^i}\\\\\n&\\qquad\\qquad\\qquad\\qquad+u\\Delta \\xi_j-(\\xi_j)_{x^i}f^i+\\xi_jf^0}\\,dt+ \\xi_j g^k \\,dw^k_t, \\quad t\\in (0,T].\n\\end{align*}\nThus, by Theorem~\\ref{thm:ex:uni:2DCone} for $j\\geq 1$ and by Lemma~\\ref{lem 10} for $j=0$ (see also \\cite[Theorem~2.9]{Kim2004}), we obtain the estimate for \n$\\|\\xi_j u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,t)}$ for each $t\\leq T$. Then summing up over all $j$ and using \\eqref{eqn 4.11.4} and \\eqref{eqn 4.10.14}, yields that for each $t\\leq T$,\n\\begin{align}\n\\|u\\|_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,t)}\n&\\leq N \\sgrklam{\n\\|u\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,t)}\n+\\label{2018-7}\n\\|f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}\\\\\n&\\qquad\\qquad\\qquad\\qquad +\n\\sum_i\\|f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\n+\n\\|g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)} }. \\nonumber \n\\end{align}\nRecall that\n$$\ndu=(f^0+(f^i+u_{x^i})_{x^i})\\,dt+g^k\\,dw^k_t, \\quad t\\leq T,\n$$\nand $\\|u_x\\|_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,s)}\\leq N \\|u\\|_{\\ensuremath{\\mathbb{\\wso}}^1_{p,\\theta-p}(\\cO,s)}$. \nThus, by Lemma~\\ref{lem for gronwall} and \\eqref{2018-7}, for each $t\\leq T$,\n\\begin{align*} \n\\|u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,t)} \n&\\leq N \\int^t_0  \\|u\\|^p_{\\ensuremath{\\mathbb{\\wso}}^{1}_{p,\\theta-p}(\\cO,s)} \\,ds\\\\\n&\\qquad + N\\sgrklam{\\|f^0\\|^p_{\\bL^{[\\circ]}_{p,\\theta+p}(\\cO,T)}+ \\sum_i\\|f^i\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T)}\n  +  \\|g\\|^p_{\\bL^{[\\circ]}_{p,\\theta}(\\cO,T;\\ell_2)}}.\n\\end{align*}\nHence the desired estimate follows by Gronwall's inequality.\n\n\\smallskip\n \n\\noindent\\emph{Step 2.}  We prove existence as follows.\n Due to Lemma~\\ref{lem for gronwall} \nand the a-priori estimate obtained in Step~1, we may assume $f^0$, $f^i$, $i=1,2$, and $g$ are very nice  in the sense that they vanish near the boundary and \n\\[\nf^i, f^i_{x^i}, f^0\\in L_2(\\Omega_T,\\cP; L_2(\\cO)), \\quad \\text{and}\\quad g \\in L_2(\\Omega_T,\\cP; L_2(\\cO;\\ell_2)).\n\\]\nThen, by classical results (see, for instance, \\cite{Roz1990} or \\cite[Theorem~2.12]{Kim2014}), there exists a unique solution $u$ in $\\frH^1_{2,2,0}(\\cO,T)$, which satisfies, in particular,  \n\\begin{equation}\n   \\label{eqn 4.11.7}\n\\rho^{-1}u, u_{x^i} \\in L_2(\\Omega_T,\\cP;L_2(\\cO)),\\;i=1,2, \\;\\text{ and }\\; \\sup_x |u|\\leq N \\|u_{x}\\|_{L_2(\\cO)}.\n \\end{equation}\nNote that for each $j \\geq 1$,\n\\begin{equation}\n\\label{eqn 4.11.10}\nd(\\xi_j u)=\\grklam{\\Delta (\\xi_j u)+f^{j,i}_{x^i}+f^{j,0} }\\,dt+ \\xi_j g^k \\,dw^k_t, \\quad t\\in(0,T],\n\\end{equation}\non $\\cD_j$, where, due to \\eqref{eqn 4.11.7} and the fact that $(\\xi_j)_{x^i}=0$ near the vertex $v_j$,\n\\begin{equation}\n\\label{fji}\nf^{j,i}:=-2(\\xi_j)_{x^i} u+\\xi_jf^i \\in \\bL^{[\\circ]}_{p,\\theta}(\\cD_j,T) \\cap \\bL^{[\\circ]}_{2,2}(\\cD_j,T),\n\\end{equation}\n\\begin{equation}\n\\label{fj0}\nf^{j,0}:=u\\Delta \\xi_j+f^0\\xi_j  + \\sum_i (\\xi_j)_{x_i}f^i  \\in \\bL^{[\\circ]}_{p,\\theta+p}(\\cD_j,T) \\cap \\bL^{[\\circ]}_{2,2+2}(\\cD_j,T), \n\\end{equation}\nand \n\\[\n\\xi_j g \\in  \\bL^{[\\circ]}_{p,\\theta}(\\cD_j,T;\\ell_2) \\cap \\bL^{[\\circ]}_{2,2}(\\cD_j,T;\\ell_2).\n\\]\nSince $\\tilde{\\rho}(x) \\geq   \\rho(x)$,  it follows that for each for $j\\geq 1$ we have $\\xi_ju\\in \\cK^1_{2,2,0}(\\cO,T)$.  Thus, by  Lemma \\ref{lem for uniqueness}, we conclude $\\xi_ju\\in \\cK^1_{p,\\theta,0}(\\cO,T)$ if $j\\geq 1$. Similar arguments based on Lemma~\n\\ref{lem for uniqueness2}  yield that $\\xi_0 u\\in \\frH^1_{p,2,0}(G,T)$. Therefore, $\\xi_0 u\\in  \\cK^1_{p,\\theta,0}(\\cO,T)$ (see~\\eqref{eqn 4.11.1}), and consequently \n$u\\in  \\cK^1_{p,\\theta,0}(\\cO,T)$.\n\\end{proof}\n\\end{thm}\n\n\n\n\\begin{remark}\n\\label{remark 8.24}\n Note that even if we were to consider Equation~\\eqref{eq:SHE:Intro} on $\\domain$ with $f^i=0$, $i=1,2$, our proof strategy for Theorem~\\ref{thm polygon main} (and Theorem~\\ref{thm_polygons_1} below) requires that we are able to handle the localized equation on $\\cD_j$ with forcing term $((\\xi_j)_{x^i}u)_{x^i}$, which means that we have to be able to treat Equation~\\eqref{eq:SHE:Intro} on angular domains with $f^i\\neq 0$.  \nThis is why we need the extension of \\cite[Theorem~3.7]{CioKimLee+2018} presented in Theorem~\\ref{thm:ex:uni:2DCone} even for the proof of Theorem~\\ref{thm polygon main} with $f^i=0$, $i=1,2$.\n\\end{remark}\n\n\n\nWe conclude with our main higher order regularity result for the stochastic heat equation on polygons. \n\n\n\\begin{thm}[Higher order regularity\/polygons]\\label{thm_polygons_1}\n Given the setting of Theorem~\\ref{thm polygon main}, let $u$ be the unique solution in the class $\\cK^{1}_{p,\\theta,0}(\\cO,T)$ to Equation~\\eqref{eq:SHE:Intro} on $\\cO$.  \nAssume that \n\\begin{align*}\nC(m,\\theta, f^i,f^0,g)\n&:=\n{\\mathbb E} \\int^T_0 \\int_{\\cO} \\ssgrklam{\\sum_{|\\alpha|\\leq (m-1)\\vee 0\\color{black}} |\\rho^{\\abs{\\alpha}+1}D^{\\alpha}f^0|^p+ \\sum_i\\sum_{|\\alpha|\\leq m} |\\rho^{|\\alpha|}D^{\\alpha}f^i|^p\\\\\n&\\qquad\\qquad\\qquad\\qquad+|\\tilde{\\rho}f^0|^p+\n\\sum_{\\abs{\\alpha}\\leq m} |\\rho^{\\abs{\\alpha}}D^\\alpha g|_{\\ell_2}^p} \\tilde{\\rho}^{\\theta-2}\\, dx\\,dt <\\infty,\n\\end{align*}\n for some $m\\in\\{0,1,2,\\ldots\\}$.\nThen\n\\begin{equation}\n    \\label{eqn final}\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+1}\\int_{\\cO} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\\leq N\\, C(m,\\theta, f^i,f^0,g),\n\\end{equation}\nwhere $N=N(p,\\theta,\\kappa_0,m, T)$.\n\\end{thm}\n\n\n\n\\begin{proof}\nWe prove the statement by induction over $m$. As in the proof of the results above, we use a partition of unity and apply corresponding results for the stochastic heat equation on angular domains (Corollary~\\ref{high}) and on $\\ensuremath{\\mathcal{C}}^1$ domains (\\cite[Theorem~2.9]{Kim2004}) to estimate the solutions of the localized equations.\n\nLet $r>0$, $\\xi_j$, $\\cD_j$, $j=1,\\ldots,M$, as well as $\\xi_0$ and $G$ be as in the proof of Lemma~\\ref{lem for gronwall}.\nIn addition, assume that $G\\subset\\domain$ is chosen in such a way that\n\\[\n\\domain\\setminus\\bigcup_{j}B_{2r\/3}(v_j) \\subseteq G\\subseteq \\domain\\setminus\\bigcup_{j}B_{r\/3}(v_j).\n\\]\nAs a consequence, \n\\begin{equation}\\label{eq:equiv:dist}\n\\rho_G\\sim\\rho_\\cO\\quad \\text{and}\\quad \\tilde\\rho\\sim 1\\quad \\text{on}\\quad\\textup{supp}\\,\\xi_0\\cap\\domain.\n\\end{equation}\n  \n\\smallskip\n \n\\noindent\\emph{Step 1. The base case.} Let $m=0$. \nNote that in this case, the only difference in Estimate~\\eqref{eqn final} compared to~\\eqref{eqn polygon main} is the weight we put on $u$ on the left hand side of the inequality: $\\rho^{-p}\\tilde\\rho^{\\theta-2}$ in~\\eqref{eqn final} instead of the smaller $\\tilde\\rho^{\\theta-p-2}$ from~\\eqref{eqn polygon main}.\nBut to obtain this sharper estimate we argue in a very similar fashion as in the proof of the latter with two changes: \nWe use~Corollary~\\ref{high} instead of Theorem~\\ref{thm:ex:uni:2DCone} to estimate the solution in the vicinity of vertexes and we use the slightly modified choice of $G$ and~\\eqref{eq:equiv:dist} to replace $\\rho_G$ by $\\rho_\\domain$ after applying~\\cite[Theorem~2.9]{Kim2004} to estimate the solution away from the vertexes. \nIn detail, we argue as follows: The same reasoning as in the proof of Theorem~\\ref{thm polygon main} shows that $\\xi_0 u\\in \\frH^1_{p,\\theta,0}(G,T)$ and $\\xi_j u\\in\\cK^1_{p,\\theta,0}(\\cD_j,T)$ for $j\\geq 1$ satisfy~\\eqref{eqn 4.11.10} on $G$ and on $\\cD_j$, $j\\geq 1$, respectively. \nIn particular, if $1\\leq j\\leq M$, then \nby Corollary~\\ref{high} (see also Remark~\\ref{remark cones}), Estimate~\\eqref{eqn final} holds  with $\\xi_j u$ and $C(0, \\theta,f^{j,i}, f^{j,0},\\xi_jg)$ in place of $u$ and $C(0,\\theta,f^i,f^0, g)$, respectively. Here $f^{j,i}$ and $f^{j,0}$ are taken from \\eqref{fji} and \\eqref{fj0}.\nMoreover, by the corresponding result on $\\ensuremath{\\mathcal{C}}^1$ domains (see \\cite[Theorem~2.9]{Kim2004}) and \\eqref{eq:equiv:dist}, Estimate~\\eqref{eqn final} also holds for $\\xi_0u$ and  $C(0,\\theta, f^{0,i}, f^{0,0},\\xi_0g)$ in place of $u$ and $C(0,\\theta,f^i,f^0, g)$, respectively.  \n Summing up all these estimates and using the second relationship in~\\eqref{eq:equiv:dist} yields \n\\begin{align*}\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq 1} \\int_{\\cO} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\n&\\leq N \\sum_{j=0}^{M} C(0,\\theta, f^{j,i},f^{j,0},\\xi_j g)\\\\\n&\\leq N \\,\\|u\\|^p_{\\bL^{[\\circ]}_{p,\\theta-p}(\\cO)}+ N C(0,\\theta,f^i,f^0,g) \\\\\n&\\leq N\\, C(0,\\theta,f^i,f^0,g);\n\\end{align*}\nthe last inequality above is due to \\eqref{eqn polygon main}. The base case is proved.\n\n\\smallskip\n \n\\noindent\\emph{Step 2. The induction step.} Suppose  that \\eqref{eqn final} holds for some $m\\in\\{0,1,\\ldots\\}$ and $C(m+1,\\theta,f^i,f^0,g)<\\infty$. Then, by assumption,\n\\begin{equation}\\label{eqn 4.13.2.a}\n{\\mathbb E}\\int_0^T  \\sum_{\\abs{\\alpha}\\leq m+1}\\int_{\\cO} \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\\leq N C(m,\\theta, f^i,f^0,g).\n\\end{equation}\nUsing \\eqref{eqn 4.13.2.a}, one can easily check that\n $$\n \\sum_{j=0}^{M} C(m+1,\\theta,f^{j,i}, f^{j,0},\\xi_j g) \\leq N\\, C(m+1,\\theta,f^i,f^0,g).\n $$ \nTherefore, appropriate applications of Corollary~\\ref{high} (see also Remark~\\ref{remark cones}) and \\cite[Theorem~2.9]{Kim2004} yield suitable estimates of $\\sum_{|\\alpha|\\leq m+2} {\\mathbb E} \\int^T_0 |\\rho^{|\\alpha|-1}D^{\\alpha}(\\xi_j u)|^p \\tilde{\\rho}^{\\theta-2}dxdt$ for $j=0$ and $1\\leq j\\leq M$,  respectively, which, summed up, yield\n\\begin{align*}\n{\\mathbb E}\\int_0^T \\sum_{\\abs{\\alpha}\\leq m+2} \\int_{\\cO}& \\Abs{\\rho^{\\abs{\\alpha}-1}D^\\alpha u}^p \\tilde{\\rho}^{\\theta-2}\\,dx\\,dt\\\\\n& \\leq N \\sum_{j=0}^{M}\n C(m+1,\\theta,f^{j,i},f^{j,0},\\xi_j g)  \\leq N\\, C(m+1,\\theta,f^i,f^0,g). \n\\end{align*}\nThus the induction goes through and the theorem is proved.\n\\end{proof}\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFix a non-archimedean local field ${\\mathrm{k}}$ of characteristic zero, and\na continuous involution $\\tau$ on it. Denote by ${\\mathrm{k}}_0$ the fixed\npoints of $\\tau$. Then either ${\\mathrm{k}}={\\mathrm{k}}_0$ or ${\\mathrm{k}}$ is a quadratic\nextension of ${\\mathrm{k}}_0$. Let $\\epsilon=\\pm 1$ and let $E$ be an\n$\\epsilon$-hermitian space, namely it is a finite dimensional\n${\\mathrm{k}}$-vector space,  equipped with a non-degenerate ${\\mathrm{k}}_0$-bilinear\nmap\n\\[\n  \\langle\\,,\\,\\rangle_E:E\\times E\\rightarrow {\\mathrm{k}}\n\\]\nsatisfying\n\\[\n     \\langle u,v\\rangle_E=\\epsilon\\langle v,u\\rangle_E^\\tau, \\quad \\langle au,v\\rangle_E=a\\langle u,\n     v\\rangle_E,\\quad a\\in A,\\, u,v\\in E.\n\\]\nWrite $\\epsilon'=-\\epsilon$, and let $(E',\\langle\\,,\\,\\rangle_{E'})$ be an\n$\\epsilon'$-hermitian space. Then\n\\[\n   \\mathbf{E}:=E\\otimes_{\\mathrm{k}} E'\n\\]\nis a ${\\mathrm{k}}_0$-symplectic space under the form\n\\[\n  \\langle u\\otimes u', v\\otimes v'\\rangle_\\mathbf{E}:={\\mathrm{tr}}_{{\\mathrm{k}}\/{\\mathrm{k}}_0}(\\langle u, v\\rangle_E \\, \\langle\n  u',v'\\rangle_{E'}).\n\\]\n\nDenote by\n\\begin{equation}\\label{meta}\n   1\\rightarrow \\{\\pm 1\\}\\rightarrow\n   \\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\rightarrow\n   {\\mathrm{Sp}}(\\mathbf{E})\\rightarrow 1\n\\end{equation}\nthe metaplectic cover of the symplectic group ${\\mathrm{Sp}}(\\mathbf{E})$. Denote by\n\\[\n \\H:=\\mathbf{E}\\times {\\mathrm{k}}_0\n\\]\nthe Heisenberg group associated to $\\mathbf{E}$, whose multiplication is\ngiven by\n\\[\n   (u,t)(u',t'):=(u+u', t+t'+\\langle u,u'\\rangle_\\mathbf{E}).\n\\]\nThe group ${\\mathrm{Sp}}(\\mathbf{E})$ acts on $\\H$ as automorphisms by\n\\begin{equation}\\label{actsp0}\n  g.(u,t):=(gu,t).\n\\end{equation}\nIt induces an action of $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})$ on $\\H$, and further\ndefines a semidirect product $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\ltimes \\H$.\n\nFix a non-trivial character $\\psi$ of ${\\mathrm{k}}$, and denote by\n${\\omega}_\\psi$ the corresponding smooth oscillator representation\nof $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\ltimes \\H$. Up to isomorphism, this is the\nonly genuine smooth representation which, as a representation of\n$\\H$, is irreducible and has central character $\\psi$. Recall that\nin general, if $H$ is a group together with an embedding of $\\{\\pm\n1\\}$ in its center, a representation of $H$ is called genuine if the\nelement $-1\\in H$ acts via the scalar multiplication by $-1$.\n\n\nDenote by $G$ the group of all ${\\mathrm{k}}$-linear automorphisms of $E$\nwhich preserve the form $\\langle\\,,\\,\\rangle_E$. It is thus an orthogonal\ngroup, a symplectic group or a unitary group. The group $G$ is\nobviously mapped into ${\\mathrm{Sp}}(\\mathbf{E})$. Define the fiber product\n\\[\n  \\widetilde{G}:=\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})\\times_{{\\mathrm{Sp}}(\\mathbf{E})} G,\n\\]\nwhich is a double cover of $G$. Similarly, we define $G'$ and\n$\\widetilde G'$. As usual, the product group $\\widetilde{G}\\times\n\\tilde{G'}$ is mapped into $\\widetilde{{\\mathrm{Sp}}}(\\mathbf{E})$. {\\vspace{0.2in}}\n\n\nThe goal of this paper is to prove the following theorem, which is\nusually called the multiplicity preservation for theta\ncorrespondences, and is also called the Multiplicity One Conjecture\nby Rallis in \\cite{Ra84}. It is complementary to the famous Local\nHowe Duality Conjecture.\n\\begin{introtheorem}\\label{theorem}\nFor every genuine irreducible admissible smooth representation $\\pi$\nof $\\widetilde{G}$, and $\\pi'$ of $\\widetilde{G}'$, one has that\n\\[\n   \\dim {\\mathrm{Hom}}_{\\widetilde{G}\\times \\widetilde{G}'}(\\omega_\\psi,\n   \\pi\\otimes \\pi')\\leq 1.\n\\]\n\\end{introtheorem}\n\nWhen the residue characteristic of ${\\mathrm{k}}$ is odd, Theorem A is proved\nby Waldspurger in \\cite{Wa90}. The archimedean analog of Theorem A\nis proved by Howe in \\cite{Ho89}.\n\n\\section{A geometric result}\nWe continue with the notation of the Introduction. Following\n\\cite[Proposition 4.I.2]{MVW87}, we extend $G$ to a larger group\n$\\breve{G}$, which contains $G$ as a subgroup of index two, and\nconsists pairs $(g,\\delta)\\in{\\mathrm{GL}}_{{\\mathrm{k}}_0}(E)\\times \\{\\pm 1\\}$ such\nthat either\n\\[\n  \\delta=1 \\quad\\textrm{and}\\quad g\\in G,\n\\]\nor\n\\[\n\\label{dutilde}\n  \\left\\{\n   \\begin{array}{ll}\n     \\delta=-1,&\\medskip\\\\\n     g(au)=a^\\tau g(u),\\quad & a\\in {\\mathrm{k}},\\, u\\in E,\\quad \\textrm{ and}\\medskip\\\\\n     \\langle gu,gv\\rangle_E=\\langle v,u\\rangle_E,\\quad & u,v\\in E.\n   \\end{array}\n   \\right.\n\\]\nSimilarly, we define a group $\\breve{G'}$ and a group\n$\\breve{{\\mathrm{Sp}}}(\\mathbf{E})$, which extend $G'$ and ${\\mathrm{Sp}}(\\mathbf{E})$, respectively.\n\nIn general, if a group $\\breve H$ is equipped with a subgroup $H$ of\nindex two, we will associate on it the nontrivial quadratic\ncharacter which is trivial on $H$. We use $\\chi_H$ to indicate this\ncharacter.\n\nDenote the fiber product\n\\[\n  \\breve{\\mathbf{G}}:=\\breve{G}\\times_{\\{\\pm 1\\}} \\breve{G'}=\\{(g,g',\\delta)\\mid (g,\\delta)\\in \\breve{G},\\,(g',\\delta)\\in \\breve{G'}\\},\n\\]\nwhich contains\n\\[\n  \\mathbf{G}:=G\\times G'\n\\]\nas a subgroup of index two. Define a group homomorphism\n\\begin{equation}\\label{xi}\n   \\begin{array}{rcl}\n        \\xi:\\breve{\\mathbf{G}}&\\rightarrow &\\breve{{\\mathrm{Sp}}}(\\mathbf{E}),\\smallskip\\\\\n           (g,g',\\delta)&\\mapsto& (g\\otimes g',\\delta).\\\\\n   \\end{array}\n\\end{equation}\nLet $\\breve{{\\mathrm{Sp}}}(\\mathbf{E})$ act on the Heisenberg group $\\H$ as group\nautomorphisms by\n\\begin{equation}\\label{actsp}\n  (g,\\delta).(u,t):=(gu, \\delta t),\n\\end{equation}\nwhich extends the action (\\ref{actsp0}). By using the homomorphism\n$\\xi$, this induces an action of $\\breve{\\mathbf{G}}$ on $\\H$, and further\ndefines a semidirect product\n\\[\n  \\breve{\\mathbf{J}}:=\\breve{\\mathbf{G}}\\ltimes \\H,\n\\]\nwhich contains\n\\[\n  \\mathbf{J}:=\\mathbf{G}\\ltimes \\H\n\\]\nas a subgroup of index two.\n\nLet the group\n\\begin{equation}\\label{semid}\n  \\{\\pm 1\\}\\ltimes (\\breve{\\mathbf{G}}\\times \\breve{\\mathbf{G}})\n\\end{equation}\nact on $\\breve{\\mathbf{J}}$ by\n\\begin{equation}\\label{actionall}\n   (\\delta, \\breve{\\mathbf g}_1, \\breve{\\mathbf g}_2). \\breve{\\mathbf j}:=(\\breve{\\mathbf g}_1\\,\n   \\breve{\\mathbf j}\\, \\breve{\\mathbf g}_2^{-1})^\\delta,\n\\end{equation}\nwhere the semidirect product in (\\ref{semid}) is defined by the\naction\n\\[\n  -1.(\\breve{\\mathbf g}_1, \\breve{\\mathbf g}_2):=(\\breve{\\mathbf g}_2, \\breve{\\mathbf g}_1).\n\\]\nThe fibre product\n\\[\n   \\{\\pm 1\\}\\ltimes_{\\{\\pm 1\\}} (\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\n   \\breve{\\mathbf{G}})=\\{(\\delta,\\breve{\\mathbf g}_1, \\breve{\\mathbf\n   g}_2)\\mid \\chi_\\mathbf{G}(\\breve{\\mathbf g}_1)=\\chi_\\mathbf{G}(\\breve{\\mathbf\n   g}_2)=\\delta\\}\n\\]\nis a subgroup of (\\ref{semid}). It contains $\\mathbf{G}\\times \\mathbf{G}$ as a\nsubgroup of index two, and stabilizes $\\mathbf{J}$ under the action\n(\\ref{actionall}).\n\nWe prove the following proposition in the remaining of this section.\n\n\\begin{prpp}\\label{orbite1}\nEvery  $\\mathbf{G}\\times \\mathbf{G}$-orbit in $\\mathbf{J}$ is stable under the group $\\{\\pm\n1\\}\\ltimes_{\\{\\pm 1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\\breve{\\mathbf{G}})$.\n\\end{prpp}\n\n{\\vspace{0.2in}} Let $\\breve{\\mathbf{G}}$ act ${\\mathrm{k}}_0$-linearly on $\\mathbf{E}$  by\n\\begin{equation}\\label{acte}\n  (g,g',\\delta).u\\otimes u':=\\delta gu\\otimes g'u'.\n\\end{equation}\n\n\\begin{lemp}\\label{orbite2}\nEvery $\\mathbf{G}$-orbit in $\\mathbf{E}$ is $\\breve{\\mathbf{G}}$-stable.\n\\end{lemp}\n\nWe first prove\n\n\\begin{lemp}\\label{orbite3}\nLemma \\ref{orbite2} implies Proposition \\ref{orbite1}.\n\\end{lemp}\n\\begin{proof}\nNote that every $\\mathbf{G}\\times \\mathbf{G}$-orbit in $\\mathbf{J}$ intersect the subgroup\n$\\H$, and the subgroup\n\\[\n  \\{\\pm 1\\}\\times_{\\{\\pm 1\\}}(\\Delta(\\breve{\\mathbf{G}})) \\quad\\textrm{ of }\\quad \\{\\pm 1\\}\\ltimes_{\\{\\pm 1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm\n1\\}}\\breve{\\mathbf{G}}) \\] stabilizes $\\H$, where ``$\\Delta$\" stands for the\ndiagonal group. Therefore in order to prove Proposition\n\\ref{orbite1}, it suffices to show that every $\\Delta(\\mathbf{G})$-orbit in\n$\\H$ is $\\{\\pm 1\\}\\times_{\\{\\pm 1\\}}(\\Delta(\\breve{\\mathbf{G}}))$-stable.\nIdentify $\\{\\pm 1\\}\\times_{\\{\\pm 1\\}}(\\Delta(\\breve{\\mathbf{G}}))$ with\n$\\breve{\\mathbf{G}}$. Then as a $\\breve{\\mathbf{G}}$-space,\n\\[\n  \\H=\\mathbf{E}\\times {\\mathrm{k}},\n\\]\nwhere $\\mathbf{E}$ carries the action (\\ref{acte}), and ${\\mathrm{k}}$ carries the\ntrivial $\\breve{\\mathbf{G}}$-action. This finishes the proof.\n\\end{proof}\n\n\nLet $\\breve{\\mathbf{G}}$ act ${\\mathrm{k}}_0$-linearly on\n\\[\n  \\mathbf{E}':={\\mathrm{Hom}}_{\\mathrm{k}}(E,E')\n\\]\nby\n\\[\n  ((g,g',\\delta).\\phi)(u):=\\delta \\,g'(\\phi(g^\\tau u)),\n\\]\nwhere\n\\[\n  g^\\tau:=\\left\\{\n         \\begin{array}{ll}\n           g^{-1}, &\\quad\n           \\textrm{if } \\delta=1,\\medskip\\\\\n           \\epsilon g^{-1},& \\quad\n           \\textrm{if } \\delta=-1.\\\\\n         \\end{array}\n   \\right.\n\\]\nThen one checks that the ${\\mathrm{k}}_0$-linear isomorphism\n\\[\n  \\begin{array}{rcl}\n    \\mathbf{E}&\\rightarrow &\\mathbf{E}',\\\\\n    u\\otimes u'&\\mapsto&(v\\mapsto \\langle v,u\\rangle_E u')\n  \\end{array}\n\\]\nis $\\breve{\\mathbf{G}}$-intertwining. Therefore Lemma \\ref{orbite2} is\nequivalent to the following\n\\begin{lemp}\\label{orbite4}\nEvery $\\mathbf{G}$-orbit in $\\mathbf{E}'$ is $\\breve{\\mathbf{G}}$-stable.\n\\end{lemp}\n\n\nDenote by\n\\[\n  \\mathfrak g:=\\{x\\in {\\mathrm{End}}_{\\mathrm{k}}(E)\\mid \\langle xu,v\\rangle_E+\\langle u,xv\\rangle_E=0\\}\n\\]\nthe Lie algebra of $G$, and put\n\\[\n  \\tilde{\\mathfrak g}:=\\{(x,F)\\mid x\\in \\mathfrak g, F\\,\\textrm{ is a ${\\mathrm{k}}$-subspace of }E,\n  \\,x|_F=0\\}.\n\\]\nLet $\\breve G$ act on $\\tilde \\mathfrak g$ by\n\\[\n  (g,\\delta). (x,F):=(\\delta gxg^{-1}, gF).\n\\]\nThe action of $\\breve{\\mathbf{G}}$ on $\\mathbf{E}'$ induces an action of\n\\[\n  \\breve{G}=\\breve{\\mathbf{G}}\/G'\n\\]\non the quotient space $G'\\backslash\\mathbf{E}'$.\n\n\n\n\\begin{lemp}\\label{orbite5}\nThere is a $\\breve G$-intertwining embedding from $G'\\backslash\\mathbf{E}'$\ninto $\\tilde \\mathfrak g$.\n\\end{lemp}\n\\begin{proof}\nRecall that the map\n\\[\n  x\\mapsto\\langle x\\,\\cdot,\\,\\cdot\\rangle_E\n\\]\nestablishes a ${\\mathrm{k}}_0$-linear isomorphism form $\\mathfrak g$ onto the space of\n$\\epsilon'$-hermitian forms on the ${\\mathrm{k}}$-vector space $E$. Define a\nmap\n\\[\n   \\begin{array}{rcl}\n        \\Xi: \\mathbf{E}'={\\mathrm{Hom}}_{\\mathrm{k}}(E,E')&\\rightarrow &\\tilde \\mathfrak g,\\\\\n             \\phi&\\mapsto &(x,F),\n   \\end{array}\n\\]\nwhere $F$ is the kernel of $\\phi$, and $x$ is specified by the\nformula\n\\[\n  \\langle \\phi(u),\\phi(v)\\rangle_{E'}=\\langle xu,v\\rangle_E,\\quad u,v\\in E.\n\\]\nUse Witt's Theorem, one finds that two elements of $\\mathbf{E}'$ stay in the\nsame $G'$-orbit precisely when they have the same image under the\nmap $\\Xi$. Therefore $\\Xi$ reduces to an embedding\n\\[\n  G'\\backslash\\mathbf{E}'\\hookrightarrow\\tilde \\mathfrak g,\n\\]\nwhich is checked to be $\\breve G$-intertwining.\n\\end{proof}\n\n\nThe following lemma is stated in \\cite[Proposition 4.I.2]{MVW87}. We\nomit its proof.\n\n\\begin{lemp}\\label{geometry}\nFor every $(x,F)\\in \\tilde \\mathfrak g$, there is an element $(g,-1)\\in\n\\breve{G}$ such that \\[\n  gxg^{-1}=-x\\quad\\textrm{ and }\\quad gF=F.\n\\]\n\\end{lemp}\n\nIn other words, every element of $\\tilde \\mathfrak g$ is fixed by an element\nof $\\breve G\\setminus G$. Therefore every $G$-orbit in $\\tilde \\mathfrak g$\nis $\\breve G$-stable. Now Lemma \\ref{orbite5} implies that every\n$G$-orbit in $G'\\backslash \\mathbf{E}'$ is $\\breve G$-stable, or\nequivalently, every $\\mathbf{G}$-orbit in $\\mathbf{E}'$ is $\\breve \\mathbf{G}$-stable. This\nproves Lemma \\ref{orbite4}, and the proof of Proposition\n\\ref{orbite1} is now complete.\n\n\n\\section{Proof of Theorem \\ref{theorem}}\n\n\nWe first recall the notions of distributions and generalized\nfunctions on a t.d. group, i.e., a topological group whose\nunderlying topological space is Hausdorff, secondly countable,\nlocally compact and totally disconnected. Let $H$ be a t.d. group. A\ndistribution on $H$ is defined to be a linear functional on\n$\\textit{C}^\\infty_0(H)$, the space of compactly supported, locally\nconstant (complex valued) functions on $H$. Denote by\n$\\textit{D}^\\infty_0(H)$ the space of compactly supported distributions on\n$H$ which are locally scalar multiples of a fixed haar measure. A\ngeneralized function on $H$ is defined to be a linear functional on\n$\\textit{D}^\\infty_0(H)$.\n\n\n\nRecall the following version of Gelfand-Kazhdan criteria.\n\n\\begin{lem}\\label{gelfand}\nLet $S$ be a closed subgroup of a t.d. group $H$, and let $\\sigma$\nbe a continuous anti-automorphism of $H$. Assume that every\nbi-$S$-invariant generalized function on $H$ is $\\sigma$-invariant.\nThen for every irreducible admissible smooth representations $\\pi$\nof $H$, one has that\n\\begin{equation*}\n  \\dim {\\mathrm{Hom}}_{S}(\\pi, \\mathbb{C}) \\,\\cdot\\, \\dim {\\mathrm{Hom}}_{S}\n  (\\pi^{\\vee},\\mathbb{C})\\leq 1.\n\\end{equation*}\n\\end{lem}\nHere and henceforth, we use ``$^{\\vee}$\" to indicate the contragredient\nof an admissible smooth representation. Lemma \\ref{gelfand} is\nproved in a more general form in \\cite[Theorem 2.2]{SZ} for real\nreductive groups. The same proof works here and we omit the details.\n\n{\\vspace{0.2in}}\n\nNow we continue with the notation of the last section.\n\n\\begin{leml}\\label{invgeneralized}\nIf a generalized function on $\\mathbf{J}$ is $\\mathbf{G}\\times \\mathbf{G}$ invariant, then\nit is also invariant under the group  $\\{\\pm 1\\}\\ltimes_{\\{\\pm\n1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\\breve{\\mathbf{G}})$.\n\\end{leml}\n\\begin{proof}\nNote that the t.d. group $\\breve{\\mathbf{J}}$ is unimodular. Therefore we\nmay replace ``generalized function\" by ``distribution\" in the proof\nof the lemma. Then by \\cite[Theorem 6.9 and Theorem 6.15 A]{BZ76},\nthe lemma is implied by Proposition \\ref{orbite1}.\n\\end{proof}\n\n\\begin{leml}\\label{pmul}\nFor every irreducible admissible smooth representations $\\Pi$ of\n$\\mathbf{J}$, one has that\n\\begin{equation*}\n  \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\Pi, \\mathbb{C}) \\,\\cdot\\, \\dim {\\mathrm{Hom}}_{\\mathbf{G}}\n  (\\Pi^{\\vee},\\mathbb{C})\\leq 1.\n\\end{equation*}\n\\end{leml}\n\\begin{proof}\nThe lemma follows from Lemma \\ref{gelfand} and Lemma\n\\ref{invgeneralized} by noting that an element of the form\n\\[\n   (-1,\\breve{\\mathbf g}, \\breve{\\mathbf g})\\in \\{\\pm 1\\}\\ltimes_{\\{\\pm\n1\\}}(\\breve{\\mathbf{G}}\\times_{\\{\\pm 1\\}}\\breve{\\mathbf{G}})\n\\]\nacts as an anti-automorphism on $\\mathbf{J}$.\n\\end{proof}\n\n\nLet $\\omega_\\psi$, $\\pi$ and $\\pi'$ be as in Theorem A. As in the\nproof of \\cite[Lemma 5.3]{Sun08},\n$\\omega_{\\psi}\\otimes\\pi^{\\vee}\\otimes \\pi'^{\\vee}$ is an irreducible\nadmissible smooth representation of $\\mathbf{J}$. Therefore Lemma \\ref{pmul}\nimplies that\n\\begin{equation}\\label{pin}\n  \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\omega_{\\psi}\\otimes\\pi^{\\vee}\\otimes\n\\pi'^{\\vee}, \\mathbb{C}) \\,\\cdot\\, \\dim {\\mathrm{Hom}}_{\\mathbf{G}}\n  (\\omega_{\\psi}^{\\vee}\\otimes\\pi\\otimes\n   \\pi',\\mathbb{C})\\leq 1.\n\\end{equation}\nBy \\cite[Theorem 1.4]{Sun09}, the two factors in the left hand side\nof (\\ref{pin}) are equal. Therefore\n\\[\n   \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\omega_{\\psi}\\otimes\\pi^{\\vee}\\otimes \\pi'^{\\vee}, \\mathbb{C})\\leq\n   1,\n\\]\nand consequently,\n\\[\n   \\dim {\\mathrm{Hom}}_{\\mathbf{G}}(\\omega_{\\psi},\\pi\\otimes \\pi')\\leq\n   1.\n\\]\nThis finishes the proof of Theorem A.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe calculation of the dielectric tensor of a beam plasma system\nis a recurrent problem in plasma physics. Many efforts have been\ndedicated recently to such issue because of the Fast Ignition\nScenario for inertial thermonuclear fusion \\cite{Tabak,Tabak2005}.\nAccording to this view, the Deuterium Tritium target is first\ncompressed by means of some driver. Then, the compressed fuel is\nignited by a relativistic electron beam generated by a petawatt\nlaser shot. Such scenario implies therefore the interaction of a\nrelativistic electron beam with a plasma. This kind of\ninteraction, and its magnetized counterpart, is also relevant to\nastrophysics, in particular when investigating the relativistic\njets of microquasars \\cite{fender}, active galactic nuclei\n\\cite{zensus}, gamma ray burst production scenarios\n\\cite{Piran2004} or pulsar winds \\cite{gallant}. Theoretical works\non these subjects are usually focused on the instabilities of the\nsystem. Although many of them demands a kinetic treatment to be\nfully described, the fluid equations can set some very relevant\nguidelines, especially when the system is not too hot.\nFurthermore, it has been known for long that in the relativistic\nregime, instabilities with arbitrarily orientated wave vectors may\nbe essential\n\\cite{fainberg,Godfrey1975,Califano1,Califano2,Califano3}. One can\ntherefore figure out how some refined kinetic theory may lead to\nalmost unsolvable calculations whereas the fluid formalism is\nstill tractable. For example, a detailed description of the\ncollisional filamentation instability ($\\mathbf{k}\\perp$ beam)\nincluding the movement of the background ions plasma, and\naccounting for temperatures, was first performed through the fluid\nequations \\cite{Honda}. The very same equations were used to\nexplore the growth rate of unstable modes with arbitrarily\noriented wave vectors (with respect to the beam) when a\nrelativistic electron beam enters a plasma\n\\cite{fainberg,Califano1,Califano2,Califano3}. The results were\nfound crucial as it was demonstrated that the fastest growing\nmodes were indeed found for obliquely propagating waves, and the\nkinetic counterpart of these models has only been considered very\nrecently \\cite{BretPRE2004,BretPRE2005,BretPRL2005}. As far as the\nmagnetized case is concerned, the kinetic formalism has been\nthoroughly investigated for wave vectors parallel and normal to\nthe beam \\cite{Cary1981,Tautz2005,Tautz2006}. But\n the unstable oblique modes, which once again turn to be the most\n unstable in many cases, could only be explored through the fluid\n formalism \\cite{Godfrey1975}.\n\n It has been demonstrated that the fluid\n equations yield the same first order temperature corrections\n than the kinetic theory for oblique modes, and the roles of both\n beam and plasma parallel and perpendicular temperatures are\n retrieved \\cite{BretPoPFluide}. The fluid approximation is thus definitely a tool of\n paramount importance to deal with beam plasma instabilities.\n Additionally, it generally yields a polynomial dispersion\n equation for which numerical resolution is immediate.\n Nevertheless, even the fluid tensor can be analytically involved\n when considering arbitrarily oriented wave vectors, a guiding\n magnetic field, temperatures, and so on \\cite{BretPoPMagnet}. Indeed, on can think about\n any model based on whether the system is relativistic or\n not, collisional or not, magnetized or not, hot or cold\\ldots\n  Most of these models have not been implemented yet, and each one\n should leave a quite complicated dielectric tensor.\n\n This is why\n a \\emph{Mathematica} notebook has been developed which allows for the\n symbolic calculation of the fluid tensor, once the parameters of\n the system have been set. The basic system we study here is a\n cold relativistic electron beam entering a cold magnetized plasma with return current.\n As the reader shall check, the notebook is very easy to adapt the\n different scenarios (ion beam, temperatures, pair plasma...). The paper is\n structured as follow: we start introducing the theory leading to\n the fluid dielectric tensor in section \\ref{sec:theory}. The\n \\emph{Mathematica} notebook is then explained step by step in section\n \\ref{sec:notebook}, and we show how it can be modified to include temperatures or collisions before the comments and conclusion section.\n\n\n\\section{\\label{sec:theory}Theory}\nWe consider a beam of density $n_b$, velocity $\\mathbf{V}_b$ and\nrelativistic factor $\\gamma_b=1\/(1-V_b^2\/c^2)$ entering a plasma\nof density $n_p$. Ions from the plasma are considered as a fixed\nneutralizing background, and an electron plasma return current\nflows at velocity $\\mathbf{V}_p$ such as\n$n_p\\mathbf{V}_p=n_b\\mathbf{V}_b$. The system is thus charge and\ncurrent neutralized. We do not make any assumptions on the ratio\n$n_b\/n_p$ so that the return current can turn relativistic for\nbeam densities approaching, or even equalling, the plasma one. We\nset the $z$ axis along the beam velocity and align the static\nmagnetic field along this very axis. The wave vector investigated\nlies in the $(x,z)$ plan without loss of generality\n\\cite{Godfrey1975}, and we define the angle $\\theta$ between\n$\\mathbf{k}$ and $\\mathbf{V}_b\\parallel \\mathbf{B}_0\\parallel z$\nthrough $k_z=k\\cos\\theta$ and $k_x=k\\sin\\theta$. The dielectric\ntensor of the system is obtained starting with the fluid equations\nfor each species $j=p$ for plasma electrons and $j=b$ for the beam\nones,\n\\begin{equation}\\label{eq:conservation}\n  \\frac{\\partial n_j}{\\partial t}-\\nabla\\cdot (n_j\\mathbf{v}_j) =\n  0,\n\\end{equation}\n\\begin{equation}\\label{eq:force}\n \\frac{\\partial \\mathbf{p}_j}{\\partial t}+(\\mathbf{v}_j\\cdot\\nabla) \\mathbf{p}_j =\n q\\left(\\mathbf{E}+\\frac{\\mathbf{v}_j\\times \\mathbf{B}}{c}\\right),\n\\end{equation}\nwhere $\\mathbf{p}_j=\\gamma_j m\\mathbf{v}_j$, $m$ the electron mass\nand $q<0$ its charge. The equations are then linearized according\nto a standard procedure \\cite{Godfrey1975}, assuming small\nvariations of the variables according to $\\exp(i\\mathbf{k}\\cdot\n\\mathbf{r}-i\\omega t)$. With the subscripts 0 and 1 denoting the\nequilibrium and perturbed quantities respectively, the linearized\nconservation equation (\\ref{eq:conservation}) yields\n\\begin{equation}\\label{eq:conservationL}\n  n_{j1} = n_{j0} \\frac{\\mathbf{k}\\cdot \\mathbf{v}_{j1}}{\\omega -\\mathbf{k}\\cdot\n  \\mathbf{v}_{j0}},\n\\end{equation}\nand the force equation (\\ref{eq:force}) gives,\n\\begin{eqnarray}\\label{eq:forceL}\n  &&i m \\gamma_j (\\mathbf{k}\\cdot\n\\mathbf{v}_{j0}-\\omega)\\left(\\mathbf{v}_{j1}+\\frac{\\gamma_j^2}{c^2}(\\mathbf{v}_{j0}\\cdot\n\\mathbf{v}_{j1})\\mathbf{v}_{j0}\\right)\\nonumber\\\\\n   &=& q\\left(\\mathbf{E}_{1}+\\frac{(\\mathbf{v}_{j0}+\\mathbf{v}_{j1})\\times\n   \\mathbf{B}_0+\\mathbf{v}_{j0}\\times\n   \\mathbf{B}_1}{c}\\right),\n\\end{eqnarray}\nwhere $i^2=-1$. Through Maxwell-Faraday equations, the field\n$\\mathbf{B}_1$ is then replaced by\n$(c\/\\omega)\\mathbf{k}\\times\\mathbf{E}_1$ so that the perturbed\nvelocities $\\mathbf{v}_{j1}$ can be explained in terms of\n$\\mathbf{E}_1$ resolving the tensorial equations\n(\\ref{eq:forceL}). Once the velocities are obtained, the perturbed\ndensities can be expressed in terms of the electric field using\nEqs. (\\ref{eq:conservationL}). Finally, the linear expression of\nthe current is found in terms of $\\mathbf{E}_1$ through,\n\\begin{equation}\\label{eq:current}\n   \\mathbf{J} = q\\sum_{j=p,b}\n   n_{j0}\\mathbf{v}_{j1}+n_{j1}\\mathbf{v}_{j0},\n\\end{equation}\nand the system is closed combining Maxwell Faraday and Maxwell\nAmp\\`{e}re equations,\n\\begin{equation}\\label{eq:Maxwell}\n  \\frac{c^2}{\\omega^2}\\mathbf{k}\\times(\\mathbf{k}\\times \\mathbf{E_1})+\\mathbf{E_1} + \\frac{4\n  i\n  \\pi}{\\omega}\\mathbf{J} = 0.\n\\end{equation}\nInserting the current expression from Eq. (\\ref{eq:current}) into\nEq. (\\ref{eq:Maxwell}) yields an equation of the kind\n$\\mathcal{T}(\\mathbf{E_1})=0$, and the dispersion equation reads\ndet$\\mathcal{T}=0$.\n\nThe Mathematica notebook we describe in the next section performs\na symbolic computation of the tensor $\\mathcal{T}$ and the\ndispersion equation det$\\mathcal{T}=0$, in terms of the usual\n\\cite{Ichimaru} reduced variables of the problem\n\\begin{equation}\\label{eq:param}\n\\mathbf{Z}=\\frac{\\mathbf{k}V_b}{\\omega_p},~~x=\n   \\frac{\\omega}{\\omega_p},~~\\alpha=\\frac{n_b}{n_p},~~\\beta=\\frac{V_b}{c},~~\\Omega_B=\\frac{\\omega_b}{\\omega_p},\n\\end{equation}\nwhere $\\omega_p^2=4\\pi n_p q^2\/m$ is the electron plasma frequency\nand $\\omega_b=|q|B_0\/mc$ the electron cyclotron frequency.\n\n\\section{\\label{sec:notebook}\\emph{Mathematica} implementation}\nFor the most part, \\emph{Mathematica} is used to solve the\ntensorial equations (\\ref{eq:forceL}) for $\\mathbf{v}_{j1}$ and\nextract the tensors $\\mathcal{T}$ from Eqs.\n(\\ref{eq:current},\\ref{eq:Maxwell}). We start declaring the\nvariables corresponding to the wave vector, the  electric field,\nthe beam and plasma drift velocities and the  magnetic field,\n\n\\emph{In[1]:= }\\textbf{k = \\{kx, 0, kz\\}; E1 = \\{E1x, E1y, E1z\\};\nV0b = \\{0, 0, Vb\\}; V0p = \\{0, 0, Vp\\}; B0=\\{0, 0, $\\mathbf{m~ c~\n\\omega b\/q}$\\}; B1 = c Cross[k, E1]\/$\\mathbf{\\omega}$; vb1 =\n\\{vb1x, vb1y, vb1z\\}; vp1 = \\{vp1x, vp1y, vp1z\\};}\n\nNote that Maxwell Faraday's equation is already implemented in the\ndefinition of $\\mathbf{B1}$. The wave vector has no component\nalong the $y$ axis and the beam and plasma drift velocities only\nhave one along the $z$ axis. The guiding magnetic field is set\nalong $z$ and  defined in terms of the cyclotron frequency\n$\\omega_b$. This will be useful later when introducing the\ndimensionless parameters (\\ref{eq:param}).\n\nWe then have \\emph{Mathematica} solve Eqs. (\\ref{eq:forceL}) for\nthe beam and the plasma. The left hand side of the equation is not\nas simple as in the non-relativistic case because the $\\gamma$\nfactors of the beam and the plasma modify the linearization\nprocedure. We write this part of the equations in a tensorial form\nin \\emph{Mathematica} defining the tensors \\textbf{Mp} and\n\\textbf{Mb} such as ``left hand\nside''=\\textbf{Mj}$^{-1}$.\\textbf{vj1} with,\n\n\n\\emph{In[2]:= }\\textbf{Mb=\\{\\{$\\frac{\\mathbf{i}}{\\gamma\nb(\\omega-kz Vb)}$,0,0\\},\\{0,$\\frac{\\mathbf{i}}{\\gamma b(\\omega-kz\nVb)}$,0\\},\\{0,0,$\\frac{\\mathbf{i}}{\\gamma b^3(\\omega-kz\nVb)}$\\}\\}};\n\n\\emph{In[3]:= }\\textbf{Mp=\\{\\{$\\frac{\\mathbf{i}}{\\gamma\np(\\omega-kz Vp)}$,0,0\\},\\{0,$\\frac{\\mathbf{i}}{\\gamma p(\\omega-kz\nVp)}$,0\\},\\{0,0,$\\frac{\\mathbf{i}}{\\gamma p^3(\\omega-kz\nVp)}$\\}\\}};\n\nwhere $\\mathbf{i}^2=-1$. The reader will notice that relativistic\neffects are more pronounced in the beam direction due to the\n$\\gamma^3$ factors in the $zz$ component. We now have\n\\emph{Mathematica} solve the tensorial Eqs. (\\ref{eq:forceL}). For\nbetter clarity, we first define them\n\n\\emph{In[4]:=}\\textbf{EqVb=vb1-Dot[Mb,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\mathbf{E1}+\\frac{\\mathrm{Cross[\\mathbf{V0b+vb1,B0}]}}{\\mathbf{c}}\\right)$]-Dot[Mb,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\frac{\\mathrm{Cross[\\mathbf{V0b,B1}]}}{\\mathrm{\\mathbf{c}}}\\right)$];}\n\n\n\\emph{In[5]:=}\\textbf{EqVp=vp1-Dot[Mp,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\mathbf{E1}+\\frac{\\mathrm{Cross[\\mathbf{V0p+vp1,B0}]}}{\\mathbf{c}}\\right)$]-Dot[Mp,$\\frac{\\mathbf{q}}{\\mathbf{m}}\\left(\\frac{\\mathrm{Cross[\\mathbf{V0p,B1}]}}{\\mathrm{\\mathbf{c}}}\\right)$];}\n\nbefore we solve them,\n\n\\emph{In[6]:=}\\textbf{Vb1=FullSimplify[vb1\/.\nSolve[EqVb==0,vb1][[1]]];}\n\n\\emph{In[7]:=}\\textbf{Vp1=FullSimplify[vp1\/.\nSolve[EqVp==0,vp1][[1]]];}\n\nNote that the \\textbf{Vb}'s, with capital ``V'', store the\nsolutions of the equations whereas the \\textbf{vb}'s are the\nvariables. This is why the \\textbf{Vb}'s do not need to be defined\nat the beginning (see \\emph{In[1]}) of the notebook; they are\nimplicitly defined here.\n\nNow that we have the values of the perturbed velocities, we can\nderive the perturbed densities from Eqs. (\\ref{eq:conservationL}),\n\n\\emph{In[8]:=}\\textbf{Nb1=FullSimplify[$\\mathbf{\\omega\npb}^2\\frac{\\mathbf{m}}{4\\pi\n\\mathbf{q}^2}\\frac{\\mathbf{Dot[k,Vb1]}}{\\mathbf{\\omega-Dot[k,V0b]}}$];}\n\n\\emph{In[9]:=}\\textbf{Np1=FullSimplify[$\\mathbf{\\omega\npp}^2\\frac{\\mathbf{m}}{4\\pi\n\\mathbf{q}^2}\\frac{\\mathbf{Dot[k,Vp1]}}{\\mathbf{\\omega-Dot[k,V0p]}}$];}\n\n\n\nHere again, we prepare the introduction of the reduced variables\n(\\ref{eq:param}) by expressing the equilibrium beam and plasma\nelectronic densities in terms of the beam and plasma electronic\nfrequencies.\n\nWe can now have \\emph{Mathematica} calculate the current according\nto Eq. (\\ref{eq:current}),\n\n\\emph{In[10]:=}\n\n\\textbf{J=FullSimplify[q$\\left(\\mathbf{\\omega\npp}^2\\frac{\\mathbf{m}}{4\\pi\n\\mathbf{q}^2}\\mathbf{Vp1}+\\mathbf{\\omega\npb}^2\\frac{\\mathbf{m}}{4\\pi \\mathbf{q}^2}\\mathbf{Vb1} \\mathbf{+\nNp1 V0p+Nb1 V0b}\\right)$];}\n\nWe now have the symbolic expression of the current \\textbf{J}. In\norder to find the tensor $\\mathcal{T}$ yielding the dispersion\nequation, we need to explain first the current tensor. This is\nperformed through,\n\n\\emph{In[11]:=}\\textbf{M=}\n\\begin{displaymath}\n\\left(\n\\begin{array}{lll}\n  \\mathbf{Coefficient[J[[1]],E1x]} & \\mathbf{Coefficient[J[[1]],E1y]} & \\mathbf{Coefficient[J[[1]],E1z]} \\\\\n  \\mathbf{Coefficient[J[[2]],E1x]} & \\mathbf{Coefficient[J[[2]],E1y]} & \\mathbf{Coefficient[J[[2]],E1z]} \\\\\n  \\mathbf{Coefficient[J[[3]],E1x]} & \\mathbf{Coefficient[J[[3]],E1y]} & \\mathbf{Coefficient[J[[3]],E1z]} \\\\\n\\end{array}\n\\right)\\mathbf{;}\n\\end{displaymath}\n\nwhich just extract the tensor elements from the expression of\n\\textbf{J}. We now turn to Eq. (\\ref{eq:Maxwell}) where we explain\nthe tensor elements of the quantity\n$c^2\\mathbf{k}\\times(\\mathbf{k}\\times\n\\mathbf{E_1})+\\omega^2\\mathbf{E_1}$,\n\n\n\\emph{In[12]:=}\\textbf{M0=$\\mathbf{c}^2$\nCross[k,Cross[k,E1]]+$\\omega^2$E1 ;}\n\n\n\\emph{In[13]:=}\\textbf{M1=}\n\\begin{displaymath}\n\\left(\n\\begin{array}{lll}\n  \\mathbf{Coefficient[M0[[1]],E1x]} & \\mathbf{Coefficient[M0[[1]],E1y]} & \\mathbf{Coefficient[M0[[1]],E1z]} \\\\\n  \\mathbf{Coefficient[M0[[2]],E1x]} & \\mathbf{Coefficient[M0[[2]],E1y]} & \\mathbf{Coefficient[M0[[2]],E1z]} \\\\\n  \\mathbf{Coefficient[M0[[3]],E1x]} & \\mathbf{Coefficient[M0[[3]],E1y]} & \\mathbf{Coefficient[M0[[3]],E1z]} \\\\\n\\end{array}\n\\right)\\textbf{;}\n\\end{displaymath}\n\nWe can finally express the tensor $\\mathcal{T}$ defined by\n$\\mathcal{T}(\\mathbf{E})$=0 as\n\n\\emph{In[14]:=}\\textbf{T=M1+4 i $\\pi ~\\omega$ M;}\n\nAt this stage of the notebook, we could take the determinant of\nthe tensor to obtain the dispersion equation. Let us first\nintroduce the dimensionless variables (\\ref{eq:param}) through,\n\n\\emph{In[15]:=}\\textbf{T=T \/. \\{Vp $\\rightarrow -\\alpha$ Vb, kz\n$\\rightarrow \\omega \\mathbf{pp}$ Zz\/Vb, kx $\\rightarrow \\omega\n\\mathbf{pp}$ Zx\/Vb, $\\omega \\mathbf{pb}^2\\rightarrow \\alpha\n~\\omega \\mathbf{pp}^2$, $\\omega  \\rightarrow \\mathbf{x} ~\\omega\n\\mathbf{pp}$, $\\omega \\mathbf{b} \\rightarrow  \\Omega \\mathbf{b}\n~\\omega\\mathbf{pp}$\\}};\n\nand,\n\n\\emph{In[16]:=}\\textbf{T=T \/. \\{Vb $\\rightarrow \\beta$ c\\}}\n\n\\emph{Mathematica} leaves here some $\\omega \\mathbf{pp}$'s which\nshould simplify between each others. It is enough to perform\n\n\\emph{In[17]:=}\\textbf{T=T \/. \\{$\\omega\\mathbf{pp} \\rightarrow\n1$\\};}\n\n\nand a simple\n\n\\emph{In[18]:=}\\textbf{MatrixForm[FullSimplify[T]]}\n\ndisplays the result. The dispersion equation of the system is\neventually obtained through\n\n\\emph{In[19]:=}\\textbf{DisperEq=Det[T]}\n\nThe notebook evaluation takes 1 minute on a Laptop running a 1.5\nGHz Pentium Centrino under Windows XP Pro. This delay can be\nshortened down to 10 seconds by suppressing all the\n\\textbf{FullSimplify} routines while leaving a\n\\textbf{Simplify[T]} in entry \\emph{18}, but the final result is\nmuch less concise and readable.\n\n\\section{Comments and Conclusion}\nIn this paper, we have described a \\emph{Mathematica} notebook\nperforming the symbolic evaluation of the dielectric tensor of a\nbeam plasma system. Starting from the linearized fluid equations,\nthe notebook expresses the dielectric tensor, and eventually the\ndispersion equation, is terms of some usual dimensionless\nparameters. This notebook has been so far applied to the treatment\nof the temperature dependant non magnetized and magnetized\nproblems (see Refs \\cite{BretPoPFluide,BretPoPMagnet}). Indeed,\nthe procedure is very easy to adapt to different settings.\n\nWhen including beam or plasma temperatures, one  adds a pressure\nterm $-\\nabla P_j\/n_j$ on the right hand side of the force\nequations (\\ref{eq:force}). Setting then $\\nabla P_i = 3 k_BT_i\n\\nabla n_i$ \\cite{Honda,Kruer} if dealing only with electron\nmotion, one only needs to add to the notebook entries \\emph{4} and\n\\emph{5} the terms (\\textbf{i}$^2$=-1)\n\n\\textbf{-3i Tj k $\\frac{\\mathbf{Dot[k,\nvj1]}}{\\omega-\\mathbf{Dot[k, V0j]}}$},\n\nwhere \\textbf{j=p} for the plasma, and \\textbf{b} for the beam.\nWhen considering anisotropic temperatures \\cite{BretPoPFluide},\none just needs to define a temperature tensor \\textbf{Tj} for each\nspecies \\textbf{j}, and replace the scalar product \\textbf{Tj k}\nby the tensorial one \\textbf{Dot[Tj,k]} in both entries. Of\ncourse, a correct treatment of electromagnetic instabilities\ngenerally requires a kinetic formalism instead of a fluid one.\nHowever, kinetic calculations cannot be systematically entrusted\nto \\emph{Mathematica}, as is the case here. The reason why is that\nthe relativistic factors $\\gamma$ encountered in the kinetic\nquadratures are coupling the integrations along the three\ncomponents of the momentum. According to the distribution\nfunctions considered, the quadratures may be calculable through\nsome ad hoc change of variables, if they can be calculated at all.\nAt any rate, the process cannot be systematized enough for\n\\emph{Mathematica} to handle it.\n\nAs far as the magnetic field is concerned, its direction can be\nchanged from entry \\emph{1} without any modification of the next\nentries. When dealing with the motion of ions, or even with one of\nthese pair plasmas involved in the pulsar problems\n\\cite{GedalinPRL}, one just need to modify the conservation and\nforce equations according to the properties of the species\ninvestigated. It is even possible to add more equations to account\nfor more species because the resolution involves only the force\nand the conservation equations of one single specie at a time\nbefore the perturbed quantities merge together in entry \\emph{10}\nto compute the current \\textbf{J}.\n\nThe notebook can thus be easily adapted to different settings and\nallows for a quick symbolic calculation of the dielectric tensor\nand the dispersion equation, even for an elaborated fluid model.\n\n\n\\section{Acknowledgements}\nThis work has been  achieved under projects FTN 2003-00721 of the\nSpanish Ministerio de Educaci\\'{o}n y Ciencia and PAI-05-045 of\nthe Consejer\\'{i}a de Educaci\\'{o}n y Ciencia de la Junta de\nComunidades de Castilla-La Mancha.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nScientific portals such as PubMed, Google Scholar, Microsoft Academic Search, \nCiteSeer$^{\\tt x}$, and ArnetMiner provide access to scholarly publications and \ncomprise indispensable resources for researchers who search for literature \non specific subject topics. In addition, data mining applications such as \ncitation recommendation~\\cite{wsdm11he},  expert search~\\cite{ijcai07balog},\ntopic trend detection~\\cite{kdd06wang,cikm09he}, and author influence modeling~\\cite{ijcai11kataria}\ninvolve web-scale analysis of up-to-date research collections. While\nacademics and researchers\\footnote{\\scriptsize In this paper, we use \nthe terms ``researchers\/authors\/scholars\" and ``research documents\/papers\/publications\" interchangeably. \nWe also use (academic) homepages to refer to professional homepages maintained by scholars and {``Scholarly\/Academic Web\" to \nrefer to sections of the Web (for example, university websites and research centers) that cater to scholarly pursuits.}} \n continue to produce large numbers of scholarly documents worldwide, acquisition of \nresearch document collections becomes a challenging task for digital libraries.\n\nIn contrast with commercial portals \n(such as the ACM digital library)\nthat rely on clean and structured publishing sources for their collections, open-access, autonomous systems such as CiteSeer$^{\\tt x}$ and ArnetMiner \nacquire and index freely-available research articles on the Web~\\cite{infoscale06li,kdd08tang}. \nResearchers' homepages and paper repository URLs are crawled and processed \nperiodically for maintaining the research collections in these portals. Needless to say, \nthese repositories are incomplete since the crawl seed lists cannot \nbe comprehensive in face of the ever changing Scholarly Web. Not only do new authors and publication venues\nemerge, but also existing researchers may stop publishing or change universities resulting in\noutdated seed URLs. \\textit{Given this challenge, how can we automatically augment crawl seed lists\nfor a scientific digital library?} \n\n\nWeb search has been a constant topic of investigation for\nIR, ML, and AI research groups since several years. Current \nWeb search engines feature state-of-the-art\ntechnologies, ranking algorithms, syntax, personalization and localization features along\nwith efficient infrastructure and programmable APIs making them invaluable tools\nto access and process the otherwise intractable Web. Despite these attractive advancements, to the best of \nour knowledge, search-driven methods are yet to be investigated as alternatives\nto crawl-based approaches for acquiring documents in digital libraries. In this paper, \nwe address \nthis gap in the context of open-access, scientific digital libraries. We propose\na novel Search\/Crawl framework, describe its components and present experiments \nshowcasing its potential in acquiring research documents.\n\nTo motivate our framework, we recall how a\nWeb user typically searches for research papers or authors~\\cite{nips02richardson,cikm08serdyukov}. \nAs with regular document search, a user typically issues Web search queries \ncomprising of representative keywords or paper titles\nfor finding publications on a topic. Similarly, if the author is known, \na ``navigational query\"~\\cite{sigir02broder} may be employed to locate the homepage\nwhere the paper is likely to be hosted. Indeed, according to\nprevious studies, researchers provide access to their papers (when possible) to improve their visibility and \ncitation counts making researcher homepages {a likely hub for locating\nresearch papers}~\\cite{nature01lawrence,tweb15gollapalli}.\n\\begin{figure*}[!htp]\n\\centering\n\\hspace*{-0.65cm}\n\\includegraphics[scale=0.35]{anecdote_ijcai.eps}\n\\caption{\\small An anecdotal example for illustration (searches performed on Jan 26, 2016).}\n\\label{fig:anecdotalsearch}\n\\end{figure*}\n\nGiven previous knowledge \nin academic browsing, scholars are often able to accurately locate the correct research \npapers or academic homepages from the Web search results using hints\nfrom the titles, search summaries (or snippets) and the URL strings. To illustrate this process, Figure~\\ref{fig:anecdotalsearch} shows an anecdotal example of a \nsearch using Google for the title and authors of a paper published at IJCAI last year, ``Maximum Satisfiability using Cores and Correction Sets'' by Nikolaj Bjorner and Nina Narodytska. For the top-$5$ results shown for the paper title query (set 1), \nfour of the five results are research papers on the topic. The document\nat the Springer link is not available for free whereas the last document corresponds to\ncourse slides.\nFor the homepage URLs identified from author name search results (from sets 2 and 3), namely: \\\\\n\\begin{scriptsize}\n\\texttt{\\textbf{http:\/\/www.cse.unsw.edu.au\/~ninan\/}} \\\\\n\\texttt{\\textbf{http:\/\/research.microsoft.com\/en-us\/people\/nbjorner\/}} \\\\\n\\texttt{\\textbf{http:\/\/theory.stanford.edu\/people\/nikolaj\/}} \n\\end{scriptsize}\n\\\\\nwe found $55$ documents, $46$ of which correspond to research publications.\nThis anecdotal search example highlights the immense potential of Web search \nfor retrieving research papers and seed URLs that can be crawled for research papers.\n\nOur Search\/Crawl framework mimics precisely the above search and scrutinize approach\nadopted by Scholarly Web users. Freely-available information from the Web for specific subject disciplines\\footnote{\n\\scriptsize For example, from bibliographic listsings such as DBLP.}\nis used to frame title and author name queries in our framework. The two control flow paths\nfor obtaining research papers are highlighted in \nFigure~\\ref{fig:schematic}. Research paper titles are used as queries \nin \\textbf{Path 1}. The documents resulting from this search are classified \nwith a paper classifier based on Random Forests~\\cite{mlj01breiman}. Author names comprise the queries \nfor Web search in \\textbf{Path 2}, the\nresults of which are filtered using a \nhomepage identification module trained using the RankSVM algorithm~\\cite{kdd02joachims}.\nThe predicted academic homepages serve as seeds for \nthe crawler module that obtains all documents upto a depth $2$ starting from the seed URL. \nThe paper identification module is once again employed\nto retain only those documents relevant to a scientific digital library among the crawled documents.\nWe summarize our contributions below:\n\\begin{itemize}\n\\item We propose a novel framework based on search-driven methods to \nautomatically acquire research documents \nfor scientific collections. To the best of our knowledge, we are the first to\nuse ``Web Search\" to obtain seed URLs for initiating crawls \nin an open-access digital library.\n\\item Our Search\/Crawl framework interleaves several existing and new modules.\nWe extend existing research on academic document classification to identify research \npapers among documents. Next, we design a novel homepage identification module,\na crucial component\n for our framework, that\nuses several features based on webpage titles, URL strings, and terms in the \nresult snippet to identify researcher homepages from the results of {author name search}. The identified \nhomepages form seeds for our Web crawler.\n\\item We provide a thorough evaluation of both the paper and homepage identification components \nusing various publicly-available datasets. Our proposed features \nattain state-of-the-art performance on both these tasks.\n\\item Finally, we perform a large-scale, first-of-its-kind experiment using $43,496$ research paper \ntitles and $32,816$ author names from Computer Science. We not only recovered approximately $75$\\% of the papers\ncorresponding to the research paper title queries but were also able to collect about $0.665$ million \nresearch documents overall with our framework. These impressive yields showcase our \nWeb-search driven methods to be highly effective for obtaining and maintaining up-to-date\ndocument collections in open-access digital library portals.\n\\end{itemize}\n\\begin{figure}[!htbp]\n\\centering\n\\includegraphics[height=2.5in, width=3in]{schematic.eps}\n\\caption{\\small Schematic Diagram of our Search\/Crawl framework.}\n\\label{fig:schematic}\n\\end{figure}\n\nWe provide details of our paper and homepage identification modules in Section~\\ref{sec:methods}. In Section~\\ref{sec:expts}, we describe our experimental setup, results,\nand findings. We briefly summarize closely-related work in Section~\\ref{sec:related} and present\nconcluding remarks in Section~\\ref{sec:conclude}.\n\n\n\\section{AI Components in Our Framework}\n\\label{sec:methods}\nThe accuracy and efficiency our \nSearch\/Crawl framework is contingent\non the accuracies of two components: (1) the homepage identifier and (2)\nthe paper classifier. \n\n\\textbf{Homepage Identification}:\nAcademic\nhomepages, known to link to research papers~\\cite{nature01lawrence}, form potential seed URLs for initiating\ncrawls in digital libraries. For our Search\/Crawl framework to be effective and efficient, it is imperative to identify \nthese pages from the search results of author name queries. \nIdentifying researcher homepages among other types of webpages can be treated as an instance of the \nwebpage classification problem with the underlying classes: homepage\/non-homepage~\\cite{tweb15gollapalli}. \nHowever, given the Web search setting, the non-homepages retrieved in response to an author name query can be expected to be diverse with\nwebpages ranging from commercial websites such as LinkedIn, social media websites such as Twitter and Facebook,\npublication listings such as Google Scholar, Research Gate, and several more. To handle this aspect, we draw ideas from the recent developments in \nWeb search ranking and frame homepage identification as a ranking problem.\n\nGiven a set of webpages in response to a query, our objective\nis to rank homepages better, i.e., top ranks, relative to other types of webpages, capturing\nour preference among the webpages. For example, consider\na name query ``John Blitzer\" and let \nthe results in response to web search be:\\\\\n\\begin{small}\n\\begin{tabular}{ll}\n\\hline\nRank & URL \\\\\n\\hline\n1 & research.google.com\/pubs\/author14735.html \\\\\n2 & john.blitzer.com \\\\\n3 & https:\/\/www.linkedin.com\/pub\/john-blitzer\/5\/606\/425 \\\\\n4 & http:\/\/dblp.uni-trier.de\/pers\/hd\/b\/Blitzer:John \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\\\\n\nSuppose ``john.blitzer.com\" is known to be the correct homepage and we are not interested\nin other webpages. This desirable property can be expressed via\nthree preference pairs among the ranks:  $p_2>p_1, p_2>p_3, p_2>p_4$\nwhere $p_i$ refers to the webpage at rank $i$. Note that, we do not express preferences\namong the non-homepages $p_1$, $p_3$, and $p_4$. Preference information such as the\nabove is modeled through appropriate objective functions in \nlearning to rank approaches~\\cite{ftir09liu}. For example, a RankSVM minimizes the\nKendall's $\\tau$ measure based on the preferential ordering\ninformation present in training examples~\\cite{kdd02joachims}.\n\nLearning to rank methods were heavily investigated for capturing user\npreferences in clickthrough logs of search engines as well as in NLP tasks such as summarization and keyphrase extraction~\\cite{ltrbook11li}.\nNote that, unlike classification approaches that independently model both positive (homepage) and negative (non-homepage) \nclasses, \nwe are modeling instances in relation with each other \nwith preferential ordering~\\cite{kdd02joachims,icml05burges,ijcai15wan}.\nWe show that the ranking approach out-performs classification approaches for homepage identification\nin Section~\\ref{sec:expts}. We use the following feature types:\n\\begin{enumerate}\n\\item \\textbf{URL Features}: Intuitively, the URL strings of academic \nhomepages can be expected to contain or not contain certain\ntokens. For example, a homepage URL is less likely to be hosted on domains such as ``linkedin\" and ``facebook\".\nOn the other hand, terms such as\n``people\" or ``home\" can be expected to occur in the URL strings of homepages (example homepage URLs in Figure~\\ref{fig:anecdotalsearch}) .\nWe tokenize the URL strings based on the ``slash (\/)\" separator\nand the domain-name part of the URL based on the ``dot ($.$)\" separator to extract \nour URL and DOMAIN feature dictionaries. \n\\item \\textbf{Term Features}: Current-day search engines present Web search\nresults as a ranked list where each webpage is indicated by its HTML title, URL string as well\nas a brief summary of the content of the webpage (also known as the ``snippet\"). \nPrevious research has shown that\nusers are able to make appropriate ``click\" decisions during \nWeb searches based on this presented information~\\cite{nips02richardson,sigir04granka}. \nWe posit that users of Scholarly Web are able to identify homepages among the search results based on the term hints in titles\nand snippets (for example, ``professor\", ``scientist\", ``student\") and capture these keywords in TITLE and SNIPPET dictionaries.\n\\item \\textbf{Name-match Features}: These features capture the common observation that \nresearchers tend to use parts of their names in the URL strings of their homepages~\\cite{icdm07tang,tweb15gollapalli}.\nWe specify two types of match features: (1) a boolean feature that indicates whether any part of the author name matches a token \nin the URL string, and\n(2) a numeric feature that indicates the extent to which name tokens overlap with the (non-domain part of) URL string given by the fraction: $\\frac{\\#{\\tt matches}}{\\#{\\tt name tokens}}$.\nFor the example author name ``Soumen Chakrabarti\" and the URL string: \\texttt{\\small \\textbf{www.cse.iitb.ac.in\/$\\sim$soumen}}, \nthe two features have values ``true\" and $0.5$, respectively.\n\\end{enumerate}\n\nThe dictionary sizes for the above feature types based on our training datasets (Section~\\ref{sec:expts})\nare listed below: \\\\\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline\nFeature Type & Size\\\\\n\\hline\nURL+DOMAIN term features & 2025 \\\\\nTITLE term features & 19190 \\\\\nSNIPPET term features & 25280 \\\\\nNAME match features & 2 \\\\\n\\hline\n\\end{tabular}\t\n\\end{center}\n\n\\textbf{Paper Classification}: Recently, Caragea et al. \\shortcite{iaai16caragea} studied\nclassification of academic documents \ninto six classes: Books, Slides, Theses, Papers, CVs, and Others. \nThey experimented with\nbag-of-words from the textual content of the documents (BoW), tokens in the document URL string (URL),\nand structural features of the document (Str) and showed that a small \nset of structural features are highly indicative of \nthe class of an academic document. Their set of $43$ structural features\nincludes features such as size of the file, number of pages in the document, average number of words\/lines\nper page, phrases such as ``This thesis\", ``This paper\" and the relative \nposition of the Introduction and Acknowledgments sections.\\footnote{\\scriptsize\nWe refer\nthe reader to~\\cite{iaai16caragea} for a complete listing of features used\nfor training this classifier.}\n\nWe found that these structural features continue to perform\nvery well on our datasets (Section~\\ref{sec:expts}) with precision\/recall values in the ranges of $90+$.\nTherefore, we directly employ their features for training the \npaper classification module in our framework. However, since\nwe are not interested in other types of documents and because binary\ntasks are considered easier to learn than multiclass tasks~\\cite{mlboook06bishop}, \nwe re-train the classifiers for the two-class setting: papers\/non-papers. \n\n\\section{Datasets and Experiments}\n\\label{sec:expts}\nIn this section, we describe our experiments on homepage\nidentification and paper classification along with their performance\nwithin the Search\/Crawl \npaper acquisition framework.\nOur datasets are summarized in Table~\\ref{tab:dspaperauthor} and described below:\n\\begin{enumerate}\n\\item For evaluating homepage finding using author names, \nwe use the\nresearcher homepages from DBLP, the bibliographic reference\nfor major Computer Science publications.\\footnote{\\scriptsize http:\/\/dblp.uni-trier.de\/xml\/}\nIn contrast to previous works that use this dataset to train homepage classifiers\non academic websites~\\cite{tweb15gollapalli}, in\nour Web search scenario, \nthe non-homepages from the search results of a name query\nneed not be restricted to academic websites.\nExcept the true homepage, all other webpages \ntherefore correspond to negatives. We collected the DBLP dataset as follows: \nUsing the\nauthor names as queries, we perform Web search \nand scan the \ntop-$k$\nresults in response to each query.\\footnote{\\scriptsize We used the Bing API for \nall Web search experiments and retrieve the top-$10$ results. All queries are ``quoted\" to \nimpose exact match and ordering of tokens and the filetype syntax was used to \nretrieve PDF or HTML files as applicable.}\nIf the true homepage from DBLP is listed among\nthe top results, this URL and the others in the set of Web results can be used \nas training instances. We used RankSVM\\footnote{\\scriptsize http:\/\/svmlight.joachims.org\/}\nfor learning a ranking function for author name search. In this model,\nthe preference among the search results for a query can be indicated by simply assigning the\nranks ``1\" and ``2\" respectively to the true and remaining results. \nFor classification algorithms, we directly use \nthe positive and negative labels for these webpages. We were able to locate homepages for $4255$ authors in the top-$10$ results \nfor the author homepages listed in DBLP.\n\\begin{table}[htp]\n\\centering\n\\begin{small}\n\\begin{tabular}{|llr|}\n\\hline\n\\textbf{Dataset} & & \\\\ \n\\hline\nResearch Papers &(Train) &\t960(T) 472(+) \\\\\n\t\t&(Test)  & 959(T) 461(+) \\\\\n\\hline\nDBLP Homepages &\\multicolumn{2}{r|} {42,548(T) 4,255(+)} \\\\\n\\hline\nCiteSeer$^x$ & \\multicolumn{2}{l|} {43,496 (Titles), 32,816(Authors)} \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\caption{\\small Summary of datasets used in experiments. The numbers of total and positive instances are shown \nusing (T) and (+), respectively, for the labeled datasets.}\n\\label{tab:dspaperauthor}\n\\end{table}\n\\item Caragea et al. \\shortcite{iaai16caragea} randomly sampled\ntwo independent sets of approximately $1000$ documents each from the crawl\ndata of CiteSeer$^{\\tt x}$.\nThese sets, called Train and Test,  were manually labeled\ninto six classes: Paper, Book, Thesis, Slides, Resume\/CV, and Others. \nWe transform the documents' labels as the binary labels, Paper\/Non-paper, and use these datasets directly in our experiments.\n\\item For our third dataset,\nwe extracted research papers from the \npublication venues listed in Table~\\ref{tab:venues} from the\nCiteSeer$^{\\tt x}$ scholarly big dataset \\cite{ecir14caragea}, in which \npaper metadata (author names, venues, and paper titles) are mapped to entries in DBLP to ensure \na clean collection.\\footnote{\\scriptsize Machine learning-based modules are used for extracting titles, venues, and authors of a paper \nin CiteSeer$^{\\tt x}$ thus resulting in occasional erroneous metadata.} Overall, we obtained a set of $43,496$ paper titles,\nauthors ($32,816$ unique names) for evaluating  \nour Search\/Crawl framework at a large scale.\n\n\n\\end{enumerate}\n\n\\begin{table}[!htp]\n\\begin{center}\n\\begin{scriptsize}\n\\begin{tabular}{|l|}\n\\hline\t\n{Total \\# of research papers: 43,496, \\#authors (unique names): 32,816}\\\\\n\\hline\nNIPS (5211), IJCAI (4721), ICRA (3883), ICML (2979), \\\\ \nACL (2970) , VLDB (2594), CVPR (2373), AAAI (2201), \\\\ \nCHI (2030), COLING (1933), KDD (1595), SIGIR (1454), \\\\\nWWW (1451), CIKM (1408), SAC (1191), LREC (1128), SDM (1111), \\\\\nEMNLP (920), ICDM (891), EACL (760), HLT-NAACL (692) \\\\\n\\hline\n\\end{tabular}\n\\end{scriptsize}\n\\end{center}\n\\caption{\\small Conference venue\/\\#papers in the CiteSeer$^{\\tt x}$ dataset.\n}\n\\label{tab:venues}\n\\end{table}\n\nWe use the standard measures Precision, Recall, and F1 for summarizing the \nresults of author homepage identification and paper classification~\\cite{irbook08manning}. \nUnlike classification where we\nconsider the true and predicted labels for each instance (webpage), in RankSVM the prediction is \nper query~\\cite{kdd02joachims}. That is, the results with respect to a query are assigned ranks based on scores\nfrom the\nRankSVM and the result at rank-1 is chosen as the predicted\nhomepage.\nThe implementations in Weka~\\cite{weka}, Mallet~\\cite{mallet} and SVMLight~\\cite{svmlight} were used for models' training and evaluation.\n\n\\subsection{Author Homepage Finding}\nWe report the five-fold cross-validation performance of the homepage identification module\ntrained using various classification modules and RankSVM\nin Table~\\ref{tab:asresults}. \nThe best performance obtained with all features described in Section~\\ref{sec:methods}\non the DBLP dataset after tuning \nthe learning parameters (such as C for SVMs), is shown in this table.\nRankSVM captures the relative preferential ordering among search results and\nperforms the best in identifying the correct author homepage in response to a query. \nA possible reason for the lower performance of the classification approaches \nsuch as Binary SVMs, Na\\\"ive Bayes, and Maximum Entropy is \nthat they model the positive and negative instances\nindependently and not in relation to one another for a given query. Moreover, the\ndiversity in webpages among the negative class is ignored and they \nare modeled uniformly as a single class in these methods.\n\n\\begin{table}[!htp]\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n\\textbf{Method} & \\textbf{Precision} & \\textbf{Recall} & \\textbf{F1} \\\\\n\\hline\nNa\\\"ive Bayes & 0.4830 &   0.9239 &   0.63432 \\\\\nMaxEnt &  0.8207 &   0.8002 &   0.8102 \\\\\nBinary SVM & 0.8353 &   0.8149 &   0.8249 \\\\\nRankSVM & \\textbf{0.8900} & \\textbf{0.8900} & \\textbf{0.8900} \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\end{center}\n\\caption{\\small Classifier and RankSVM performances on DBLP dataset.}\n\\label{tab:asresults}\n\\end{table}\n\nWe point out that false positives are not very critical in our Search\/Crawl framework.\nIncluding an incorrectly predicted homepage as\na seed URL may result in crawling irrelevant documents and extra processing load. However, \nthese documents are subsequently filtered out by our paper classifier.\n\n\\begin{table}[!htp]\n\\centering\n\\begin{small}\n\\begin{tabular}{l r || l r}\n\\hline\n\\textbf{FeatureType} & \\textbf{Feature} & \\textbf{FeatureType} & \\textbf{Feature} \\\\ \\hline\nNAME & fracMatch & TITLE & university \\\\  \nDOMAIN & com & SNIPPET & computer \\\\ \nNAME & hasMatch & TITLE & homepage \\\\  \nTITLE & home & SNIPPET & university \\\\ \nTITLE & page & TITLE & linkedin \\\\  \nSNIPPET & professor & SNIPPET & science \\\\  \nDOMAIN & edu & SNIPPET & discover \\\\  \nSNIPPET & view & URL & author \\\\  \nSNIPPET & department & SNIPPET & linkedin \\\\ \nSNIPPET & profile & SNIPPET & professionals \\\\ \\hline \n\\end{tabular}\n\\end{small}\n\\caption{\\small The top-$20$ features ranked based on Information Gain.}\n\\label{tab:asigfeats}\n\\end{table}\nTable~\\ref{tab:asigfeats} shows the top features based on information gain values~\\cite{jmlr03forman03}.\nThese features make intuitive sense; for instance, a researcher homepage is likely to have parts of \nthe researcher name mentioned on it along with terms like ``home\" and ``page\" in the HTML title. \nSimilarly, webpages typically ending in ``.com\" or \nhaving ``linkedin\" in their description are unlikely to be homepages.\n\t\n\\subsection{Research Paper Identification}\nThe results of paper classification are summarized in Table~\\ref{tab:perfpaper}.\nWe directly used the feature sets proposed by Caragea et al. \\shortcite{iaai16caragea}\nand tested various classifiers including Na\\\"ive Bayes, Support Vector Machines and \nRandom Forests. All models are trained on the ``Train'' dataset. The parameters of each model are tuned \nthrough cross-validation on the ``Train\" dataset and the classification performance evaluated\non the ``Test\" dataset. {The results of various\nfeatures sets using a Random Forest for the ``paper\" class in the binary setting\nare shown in Table~\\ref{tab:perfpaper}. We also show\nthe performance on the ``paper\" class with the \nmulticlass setting and the weighted averages of all measures over all classes for\nboth the settings in this table. \nThe best classification performance is obtained using a Random Forest trained on\nstructural features with the overall performance being\nsubstantially better in the two-class setting rather than the multiclass setting.}\n\\begin{table}[htp]\n\\centering\n\\begin{small}\n\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n{Feature}&{Precision}&{Recall}&{F1}\\\\\n\\hline\nBoW (P)&  0.86  &    0.92 &  0.889   \\\\\nURL (P)& 0.729 & 0.729 & 0.729 \\\\\nStr\/Binary (P)& \\textbf{0.933}  &  \\textbf{0.967}  &  \\textbf{0.950} \\\\\nStr\/Multiclass (P) & 0.918 & 0.965 &  0.941   \\\\\n\\hline\nStr\/Binary (A)& \\textbf{0.952}  &   \\textbf{0.951}  & \\textbf{0.951}  \\\\\nStr\/Multiclass (A) & 0.893 &     0.902 &  0.892  \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\caption{\\small Classification performance on the test dataset. `P\/A' indicate performances for ``Paper\"\/``All\" classes.}\n\\label{tab:perfpaper}\n\\end{table}\n\n\\subsection{Search\/Crawl Experiments}\n\\begin{table*}[!htp]\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\#Queries & \\#PDFs & \\#Papers & \\#UniqueTitles & \\#Matches\\\\\n\\hline\n43,496 titles (Path 1)& 322,029 & 213,683 & 91,237 & 32,565 \\\\\n32,816 names (Path 2) & 665,661 & 452,273 & 204,014 & 17,627\\\\\n\\hline\n\\end{tabular}\n\\caption{\\small \\#Papers obtained through the two paths in our Search\/Crawl framework.}\n\\label{tab:csxresults}\n\\end{center}\n\\end{table*}\nFinally, we evaluate the two AI components in \npractice within our Search\/Crawl framework\nusing the CiteSeer$^{\\tt x}$ dataset. To this end, for \\textbf{Path 1}, we use\nthe $43,496$ paper titles as search queries. Structural features\nextracted \nfrom the resulting PDF documents of each search are used\nto identify research documents with our paper classifier. For \\textbf{Path 2}, the $32,816$ author names are used as queries. The\nRankSVM-predicted homepages from the results of each query are crawled for  PDF documents up to a depth of \n$2$.\\footnote{\\scriptsize We used the wget utility for our crawls (https:\/\/www.gnu.org\/software\/wget\/)} \nOnce again, the paper classifier is employed to\nidentify research documents among the crawled documents. \n\nThe number of PDFs and papers found through the two paths in \nour proposed Search\/Crawl framework are shown in Table~\\ref{tab:csxresults}. \nSince our dataset is based on CiteSeer$^{\\tt x}$, we removed all paper search results\nthat point to CiteSeer$^{\\tt x}$ URLs for a fair evaluation. The number of papers \nthat we could obtain from the original $43,496$ collection through both the paths \nare shown in the last column of this table. We use the title\nand author names available in the dataset to look up the first page of the PDF document\nfor computing this match.\n\nWe are able to obtain $75\\%$ ($\\frac{32565}{43496}$)\nof the original titles through \\textbf{Path 1} compared to the $40\\%$ \n($\\frac{17627}{43496}$) through \\textbf{Path 2} (column $5$ in Table~\\ref{tab:csxresults}).\n{In general,\ngiven that paper titles contain representative keywords~\\cite{emnlp14caragea,Litvak2008},\nif they are available online, a Web\nsearch with appropriate filetype filters\nis a successful strategy for finding them. The high percentage\nof papers found along  \\textbf{Path 2} confirms previous findings that\nresearchers tend to link to their papers via their homepages~\\cite{nature01lawrence,tweb15gollapalli}.}\n\nIntuitively, the overall yield can be expected to be higher through \\textbf{Path 2}.\nOnce an author homepage is reached, other research papers linked to this page \ncan be directly obtained. Indeed, as shown in columns $2$ and $3$ of Table~\\ref{tab:csxresults}, \nthe numbers of PDFs as well as classified papers are significantly higher along \\textbf{Path 2}. \nCrawling the predicted homepages of the $32,816$ authors we obtain approximately\n$14$ research papers per query on average ($\\frac{452273}{32816}=13.78$). In contrast,\nexamining only the top-$10$ search results along \\textbf{Path 1}, we obtain $5$ research \ndocuments per query ($\\frac{213683}{43496}=4.91$). \n\nWe used the CRF-based title extraction tool for research papers, ParsCit~\\footnote{\\scriptsize http:\/\/aye.comp.nus.edu.sg\/parsCit\/} \nto extract the titles of the research papers obtained from both the paths. \n{The number of extracted unique titles are shown in column $4$ of Table~\\ref{tab:csxresults}. \nThe overlap in the two sets of titles is $28,374$. Compared to the overall yields\nalong \\textbf{Path 1} and \\textbf{Path 2},\nthis small overlap indicates that the two paths are capable of reaching different sections of the Web and play\ncomplementary roles in our framework. For example, the top-$20$ domains of the URLs from which \nwe obtained research papers along \\textbf{Path 1} are shown in Table~\\ref{tab:topdomainsp1}. Indeed, \nvia Web search we are able to reach a wide range of domains. This is unlikely in crawl-driven methods\nwithout an exhaustive list of seeds since only\nlinks up to a specified depth from a given seed are explored~\\cite{irbook08manning}.}\n\nTo summarize, using about $0.076$ million queries ($43,496+32,816$) in our framework, we are able to build a\ncollection of $0.665$ million research documents ($213,683+452,273$) and \n$0.267$ million unique titles\n($91,237+204,014-28,374$). About $32-33\\%$ of the obtained documents\nare ``non-papers\" along both the paths. Scholarly Web is known to contain a variety of documents\nincluding project proposals, resumes, and course materials~\\cite{jis06ortega}. \nIndeed, some of these documents may include the exact paper titles and show up in paper search results as well\nas be linked to author homepages. In addition, \nusing incorrectly-predicted homepage as seeds may result in ``bad\" documents.\n\\begin{table}[!hp]\n\\centering\n\\begin{small}\n\\begin{tabular}{|l|}\n\\hline\nedu (71,139), org (47,272), net (20,552), com (19,178), de (5,424)\\\\\nuk  (5,065), fr (3,770), ca (3,651), it (2,647), gov (2,130), \\\\\nnl  (1,891), cn (1,777), jp (1,673), au (1,655), cc (1,489), \\\\ \nch (1,431), sg  (1,282), in (1,209), il (1,206), es (1,144) \\\\\n\\hline\n\\end{tabular}\n\\end{small}\n\\caption{\\small The top-20 domains from which papers were obtained along Path-1 of our framework.}\n\\label{tab:topdomainsp1}\n\\end{table}\n\n\\textbf{Sample Evaluation.} Given the size of the CiteSeer$^{\\tt x}$ dataset and the \nlarge number of documents obtained via the Search\/Crawl framework (Table~\\ref{tab:csxresults}), it is\nextremely labor-intensive to manually examine all documents resulting from this experiment. However,\nsince our classifiers and rankers are not $100\\%$ accurate and we only\nexamine the top-$k$ results from the search engine, we need an estimate of how many papers\nwe are able to obtain via our Search\/Crawl approach among those\nthat are actually obtainable on the Web. We randomly selected $10$ titles from the CiteSeer$^{\\tt x}$ dataset and their \nassociated set of $78$ authors and inspected \nall PDFs that can be obtained via our search\/crawl framework manually.\nThat is, we searched for the selected paper titles and manually examineed and annotated the resulting PDFs.\nSimilarly, the correct homepages of the $78$ authors were obtained by searching the Web\nand manually examining the resulting webpages. The correct homepages were crawled (to depth 2)\nfor PDFs and the resulting documents were manually labeled.\n\nWe were able to locate $49$ correct homepages of the $78$ authors in this manual experiment. Crawling these homepages resulted\nin $2116$ PDFs out of which $1418$ were found to be research papers.\nOur Search\/Crawl framework that crawls predicted homepages for the $78$ authors \nand uses paper classifier\npredictions to identify research papers was able to \nobtain $1291$ research papers. Out of these documents, $1104$ match with\nthe intended set of $1418$ papers. Thus, we are able to \nobtain approximately $\\textbf{78}\\%$ of the intended set of papers along with an additional $187$ new ones.\nPaper search using titles results in $59$ PDFs out of which $33$ are true papers.\nOur paper classifier obtains a precision\/recall of $\\textbf{84}$\\%\/$\\textbf{97}$\\%, predicting $32$ out of these $33$ papers correctly\nand $38$ papers overall. \n\n\\section{Related Work}\n\\label{sec:related}\nHomepage finding and document classification are \nvery well-studied problems. \nDue to space constraints, we refer the reader to \nthe TREC 2001 proceedings\\footnote{\\scriptsize http:\/\/trec.nist.gov\/proceedings\/proceedings.html}\nand the comprehensive\nreviews of the feature representations, methods, and results\nfor various text\/webpage classification problems~\\cite{cs02sebastiani,cs09qi}.\n\nThough homepage finding in TREC did not\nspecifically address researcher homepages, this track \nresulted in various state-of-the-art machine learning systems\nfor finding homepages~\\cite{spire02xi,tis03upstill,icadl06wang}. \nAmong works focusing specifically on researcher homepages, both Tang et al.~\\shortcite{icdm07tang} and Gollapalli et al.~\\shortcite{tweb15gollapalli}\ntreat homepage finding as a binary classification task and use\nvarious URL and content features. Ranking methods were \nexplored for homepage finding using the top terms\nobtained from topic models~\\cite{sigireos11gollapalli}.\n\nIn the context of scientific digital libraries,\ndocument classification into\nclasses related to subject-topics (for example, ``machine learning\", ``databases\")\nwas studied previously~\\cite{icml03lu,emnlp15caragea}.\nOften bag-of-words features as well as topics extracted using LDA\/pLSA are used to represent\nthe underlying documents in these works. Structural features, on the other\nhand, are popular in classifying and clustering semi-structured XML documents~\\cite{icpr08ghosh,aireview13asghari}. \n\nIn contrast with existing work,\nwe investigate features from web search engine results\nand formulate researcher homepage identification as a learning to rank task. In addition,\nwe are the first to interleave various AI components with \nexisting Web search and crawl modules to build an efficient paper acquisition framework.\n\n\\section{Conclusions}\n\\label{sec:conclude}\nWe proposed a search-driven framework for automatically acquiring research\ndocuments on the Web as an alternative to\ncrawl-driven methods adopted in current open-access digital libraries.\nOur framework crucially depends on accurate paper classification and researcher \nhomepage identification modules. To this end, we discussed features for \nthese modules and showed experiments illustrating their\nstate-of-the-art performance. In one experiment using a large collection\nof about $0.076$ million queries, our framework was able to automatically\nacquire a collection of approximately $0.665$ million research documents. These\nresults showcase the potential of our proposed framework in \nimproving scientific digital library collections.\n{For future work, apart from improving the accuracies of individual components in our framework, we will focus on \nincluding other document formats (for example, .ps and zipped files)\nas well as other document types (for example, course materials).}\n\n\\section*{Acknowledgments}\n\nWe are grateful to Dr. C. Lee Giles for the CiteSeerX data. We also thank Corina Florescu and Kishore Neppalli for their help with various dataset construction tasks. This research was supported in part by the NSF award \\#1423337 to Cornelia Caragea. Any opinions, findings, and conclusions expressed here are those of the authors and do not necessarily reflect the views of NSF.\n\n\\small\n\\bibliographystyle{named}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nAn interacting electron system could be in the strongly interacting collision-dominated hydrodynamic regime where frequent inter-particle collisions lead to local thermodynamic equilibrium so that the system should be thought of as manifesting a collective hydrodynamic behavior rather than the usual collisionless behavior of weak interactions where the standard independent quasiparticle picture applies. The analogy is to real liquids (e.g. water) whose macroscopic long wavelength flow properties hardly depend on the microscopic details of the molecular interactions which only serve to determine the macroscopic hydrodynamic parameters such as viscosity. Although the possibility of electron hydrodynamics was suggested a long time ago~\\cite{Gurzhi:1968hydrodynamic}, there has been a great deal of recent interest in the subject arising from the possibility of the experimental observation of electron hydrodynamics in solid state materials~\\cite{Lucas:2018hydrodynamics}. The current work focuses on a specific theoretical question regarding electron hydrodynamics. \n\n\nThe question we address is the interplay of the electronic plasmon mode with the hydrodynamic sound mode: How does the plasmon affect the hydrodynamic sound mode and does the electron system in the hydrodynamic regime undergo collective plasmon oscillations at all? \n\nThe theoretical question was recently addressed for 2D metals in a recent work~\\cite{Lucas:2018electronic}, and we generalize the theory to 3D metals. The 3D generalization is nontrivial and is of interest because of the true long-range nature of electronic Coulomb interaction in 3D metals, leading to a collective plasmon mode with a finite energy even in the long wavelength limit~\\cite{Bohm:1953collective}. This gapped massive nature of 3D plasmons arises from the 3D Coulomb coupling going as $1\/q^2$, where $q$ is the wavenumber (or momentum). By contrast, 2D Coulomb coupling goes as $1\/q$, leading to the 2D plasmon going at long wavelength as $q^{1\/2}$. Since the hydrodynamic sound mode (the so-called 'first sound' in the helium literature), which is the electron analog of the ordinary acoustic sound, goes linear in $q$ at long wavelength by definition, the interplay of a gapped 3D plasmon going as $\\mathcal{O}(q^0)$ with hydrodynamic sound going as $\\mathcal{O}(q)$ is more intriguing than the interplay between the hydrodynamic sound and the 2D plasmon, both of which vanish at long wavelength albeit with different momentum scaling. If the hydrodynamic sound mode in 3D metals develops a long wavelength gap by virtue of Coulomb interaction, this becomes reminiscent of the Higgs mechanism with a linearly dispersing Goldstone mode acquiring a mass by virtue of long-range coupling although the electron hydrodynamics problem does not involve any symmetry breaking (or an underlying Higgs field) in order to produce the gapped sound mode. In fact, this is precisely what happens in a metallic superconductor where the expected linearly dispersing Goldstone mode associated with the spontaneous breaking of the gauge symmetry acquires a long wavelength gap becoming effectively the plasmon mode in the presence of long-range Coulomb coupling instead of the usual zero sound acoustic mode as in a neutral superfluid where there is no long-range Coulomb coupling~\\cite{Anderson:1958random, Anderson:1963plasmons, Prange:1963dielectric, Fertig:1990collective, Fertig:1991collective}.\n\nWe show in this work that indeed 3D Coulomb coupling leads to a mass or a gap in the hydrodynamic sound mode in 3D metals, and the sound mode becomes the effective long wavelength plasmon mode in the hydrodynamic regime of 3D metals.  This is, however, true only in the leading order in momentum, and the next-to-leading-order dispersion corrections in wavenumber are different for 3D plasmons in the collisionless regime and the first sound in the hydrodynamic regime. We also study the damping or decay of the plasmon mode (which is akin to the zero sound mode in the helium literature) in the collisionless regime and the hydrodynamic first sound mode in the collision-dominated regime. Our terminology in the paper uses 'collisionless' to imply the non-hydrodynamic regime of weak inter-particle collisions (where the standard 'plasmon' or zero sound mode exists). By contrast, the collision-dominated regime is the hydrodynamic regime of rapid inter-particle collisions where the first sound mode exists. We study both regimes including effects of 3D long-range Coulomb interaction.\n\nThe hydrodynamic description applies when the momentum conserving inelastic electron-electron interaction is much stronger than any other momentum relaxing elastic scattering mechanisms which might be present in the system. In metals, such momentum non-conserving scattering processes arise from electron-impurity and electron-phonon scattering, which typically dominate at low and high temperatures respectively, making the hydrodynamic regime difficult to realize experimentally in the laboratory although, in principle, a very clean metal should manifest electron hydrodynamics at very low temperatures where the phonon scattering rate (in terms of resistivity) is suppressed as $T^5$ and the quasiparticle scattering rate goes as $T^2$, where $T$ is the temperature~\\cite{Ashcroft:1976solid}. Eventually, at low enough temperatures hydrodynamics is cut off in metals by any residual impurity scattering. In a metal with negligible impurity and phonon scattering, electron-electron interactions should produce hydrodynamic behavior at long wavelength and low frequency. \n\n\\section{Summary of the main results} \\label{sec:summary}\n\nWe extend the theory of~\\cite{Lucas:2018electronic} to three dimensions. Namely, we construct a solvable model that admits the exact calculation of sound modes in both the collisionless regime and hydrodynamic regime. The sound mode in the collisionless regime is the zero sound mode or the plasmon, and the sound mode in the hydrodynamic regime is the usual sound mode (i.e., the first sound). In this section, we represent the main results from the solvable model, including the effect of long-range Coulomb interaction on the sound modes. We leave the detailed derivation of the results to the following sections, but the exactly solvable model already demonstrates the physics clearly.\n\nIn this exactly solvable model consisting of spinless fermions, we assume spherical symmetry and consider the nonvanishing Landau parameter only in the s-wave channel. We denote the dimensionless Landau parameter $F_0$ and we consider repulsive interactions so that $F_0>0$. The sound mode is highly damped when the interaction is attractive, and may even lead to instabilities~\\cite{Nozieres:2018theory}. In any case, our interest is 3D metals in which the direct electron-electron interaction is repulsive. These considerations make the calculations analytically tractable, while maintaining the essential physics. \n\nWe first present the result in the clean limit where the sound mode is not damped by impurity scattering, $\\gamma_\\text{imp} = 0$.  $\\gamma_\\text{imp}$ is the scattering rate between quasiparticles and impurities. The zero sound in the collisionless regime $\\omega \\gg \\gamma$, where $\\gamma$ denotes the scattering rate between quasiparticles, is given by\n\\begin{eqnarray} \\label{eq:zeroFull}\n&& \\omega = \\pm v_0 q - \\mathfrak{i} \\gamma \\frac{(F_0+1)^2 - 2(F_0-1) (\\frac{v_0}{v_F})^2 - 3 (\\frac{v_0}{v_F})^4 }{F_0 [ F_0 +1 - (\\frac{v_0}{v_F})^2]}, \\\\  && \\frac{v_0}{v_F} \\text{arccoth} \\frac{v_0}{v_F} = 1 + \\frac1{F_0}.\n\\end{eqnarray}\nwhere the second equation determines the zero sound velocity. This equation is valid for all $F_0>0$. For the weakly interacting Fermi liquid, $F_0 \\ll 1$, and the zero sound velocity is nonperturbative in interaction strength, namely, $v_0 = v_F (1 + e^{-2\/F_0})$. It approaches the Fermi velocity as one should anticipate for the free electron gas. For $F_0 > 1$, the dispersion can be simplified as\n\\begin{eqnarray} \\label{eq:zero}\n\t\t\\omega = \\pm \\sqrt{\\frac{F_0}3 + \\frac35} v_F q - \\mathfrak{i} \\gamma \\frac{2(5F_0 + 21)}{5F_0(5F_0 + 3)},\n\\end{eqnarray}\nwhere $v_F$ is the Fermi velocity. Decreasing the frequency, the system enters the collision-dominated hydrodynamic regime where $ \\omega \\ll \\gamma$. Then the zero sound crosses over smoothly to the first sound,\n\\begin{eqnarray} \\label{eq:first}\n\\omega = \\pm \\sqrt{ \\frac{F_0}3 + \\frac13} v_F q - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2.\n\\end{eqnarray}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=6cm]{velocity} \n\t\\caption{\\label{fig:velocity}The zero sound and first sound velocities as a function of the Landau parameter, where the Fermi velocity is set to 1 as the unit. The zero sound velocity is larger than the first sound velocity for all $F_0>0$.}\n\\end{figure}\nThe linear term in momentum reveals the sound velocity. In Fig.~\\ref{fig:velocity}, one can see that the collisionless zero sound velocity is always larger than the hydrodynamic first sound velocity, $v_0>v_1$. For the asymptotically large Landau parameter, the following inequality holds true, too, i.e.,\n\\begin{eqnarray}\n\\frac{v_0}{v_1} = \\sqrt{\\frac{F_0 + 9\/5}{F_0+1}} > 1 . \n\\end{eqnarray}\nThe imaginary part leads to the damping of sound modes. In an interacting Fermi liquid, the scattering rate at low temperatures is given by $\\gamma \\propto T^2$. A crucial difference between the zero sound and the first sound is that the damping rate is proportional to the interaction scattering rate, $\\text{Im}(\\omega) \\propto\\gamma$, for zero sound while it is inversely proportional to the interaction scattering rate, $\\text{Im}(\\omega) \\propto \\gamma^{-1}$, for the first sound. These results are well known and also experimentally observed in normal He-3~\\cite{Abel:1966propagation}. Reproducing these results presents a consistency check of our solvable model.\n\nWhen the impurity scattering is strong $\\gamma_\\text{imp} > v_F q$, both sound modes will be damped by impurity scattering at small momentum. The propagating wave transitions to a quadratic diffusion mode\n\\begin{eqnarray}\n\t\\omega = - \\mathfrak{i} \\frac{v_1^2}{\\gamma_\\text{imp}}  q^2, \\quad v_1 = v_F \\sqrt{\\frac{1+F_0}3}\n\\end{eqnarray}\nand a fully gapped one $ \\omega = -\\mathfrak{i} \\gamma_\\text{imp}$. When the impurity scattering is weak $\\gamma_\\text{imp} < v_F q$, its effect is an additional damping of the zero sound mode (in addition to the damping induced by quasiparticle interactions $\\gamma$); that is, the correction is $ \\delta\\text{Im}(\\omega) = \\frac12 \\gamma_\\text{imp}$, also see~(\\ref{eq:weak_imp}).\n\nNow we discuss the effect of Coulomb interaction on the sound mode. It is well known that the plasmon is fully gapped in 3D metals because of Coulomb interaction~\\cite{Bohm:1953collective}. Since both the plasmon and the sound wave are density fluctuating collective modes in many-body systems, it is naturally expected that the sound mode should also develop a finite gap due to the Coulomb interaction. Since Coulomb interaction acts at the s-wave channel, one can make the following replacement,\n\\begin{eqnarray}\n\tF_0 \\rightarrow F_0 + \\frac{8\\pi \\alpha}{\\lambda_F^2 q^2},\n\\end{eqnarray}\nwhere $\\alpha = \\frac{e^2}{4\\pi v_F}$ is the effective finite structure constant for the 3D metal ($v_F$ is the Fermi velocity), and $\\lambda_F = \\frac{2\\pi}{k_F}$ is the Fermi wavelength.\nThe Coulomb interaction [the second term in (6)] should be understood as the effective one felt by quasiparticles, where the effective fine structure constant captures all possible renormalization effects in going from bare electrons to quasiparticles. This is a generalization from the neutral Fermi liquid theory to the Landau-Silin theory of a charged Fermi liquid~\\cite{Zala2001:interaction}.\nSince we are interested in the long wavelength limit, where the Coulomb interaction dominates over any short-range interaction, we focus on~(\\ref{eq:zero}) in the appropriate limit. After making this replacement, the zero sound becomes\n\\begin{eqnarray} \\label{eq:zeroPlas}\n\t\\omega = \\pm \\left( \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F} + \\frac{9\/5+F_0}{4\\sqrt{6\\pi \\alpha}} v_F \\lambda_F q^2 \\right) - \\mathfrak{i} \\gamma \\frac{\\lambda_F^2 q^2}{20 \\pi \\alpha},\n\\end{eqnarray}\nand the first sound becomes\n\\begin{eqnarray} \\label{eq:firstPlas}\n\t\\omega = \\pm \\left( \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F} + \\frac{1+F_0}{4\\sqrt{6\\pi \\alpha}} v_F \\lambda_F q^2 \\right) - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2.\n\\end{eqnarray}\n\nIt is instructive to compare the results with the plasmon mode. The 3D plasmon dispersion is well known and is determined by the following equation in RPA calculations,\n\\begin{eqnarray} \\label{eq:plasmon3D}\n\t\\omega^2 = \\omega_0^2 + \\frac35 \\frac{k_F^2}{m^2} \\frac{\\omega_0^2}{\\omega^2} q^2, \\quad \\omega_0^2 = \\frac{e^2 N_e}{m},\n\\end{eqnarray}\nwhere $N_e = \\frac{4\\pi}3 \\frac{k_F^3}{(2\\pi)^3} = \\frac{k_F^3}{6\\pi^2}$ is the electron density in 3D,  $m = \\frac{k_F}{v_F}$ is the effective mass of the quasiparticle, and $\\omega_0$ is the well-known plasmon frequency. (Note a conventional difference between the plasmon frequency in our~(\\ref{eq:plasmon3D}) and Eq.~(14) in Ref.~[4\n, which comes from the Coulomb potential we define as~(\\ref{eq:energy}) having an extra factor of $1\/4\\pi$ in real space as we use the rationalized unit instead of the Gauss unit.) If we expand the plasmon mode at long wavelength, its dispersion reads\n\\begin{eqnarray} \\label{eq:3Dplasmon}\n\t\\omega = \\omega_0 + \\frac35 \\frac{k_F^2}{m^2 \\omega_0} \\frac{q^2}2 =  \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F} + \\frac{9\/5}{4\\sqrt{6\\pi \\alpha}} v_F \\lambda_F q^2 , \\nonumber \\\\\n\\end{eqnarray}\nwhere in the second step, we change the parameters to better compare with sound mode results. Comparing~(\\ref{eq:3Dplasmon}) to the zero sound mode with Coulomb interaction~(\\ref{eq:zeroPlas}), we find a correction to the quadratic dispersion from the Landau parameter $F_0$ and a quadratic damping due to the collisions. Comparing~(\\ref{eq:3Dplasmon}) to the first sound mode with Coulomb interaction~(\\ref{eq:firstPlas}), we conclude that in the hydrodynamic regime, the plasmon dispersion gets corrected by a factor of $5\/9$ at the next-to-leading $q^2$ order because of Coulomb coupling. Again there is a damping effect in~(\\ref{eq:firstPlas}) inherited from the first sound mode.\n\nMore generally, we can consider a formal long-range interaction given by a power law defined by $q^{-2\\eta}$, where $\\eta=1$ for 3D Coulomb coupling, i.e.,\n\\begin{eqnarray}\n\tF_0 \\rightarrow F_0 + \\frac{8\\pi \\alpha}{(\\lambda_F q)^{2\\eta}},\n\\end{eqnarray}\nwhere $\\alpha$ is the effective interaction strength, and the parameter $\\eta$ defines the form of the interaction. In real space, this translates to a potential of the form $r^{2\\eta - 3}$. For $\\eta >1$ ($\\eta < 1$) it is stronger (weaker) than the Coulomb potential. This type of interaction leads to the zero sound mode\n\\begin{eqnarray}\n\\omega = \\pm \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F^\\eta} q^{1-\\eta} - \\mathfrak{i} \\gamma \\frac{(\\lambda_F q)^{2\\eta}}{20 \\pi \\alpha},\n\\end{eqnarray}\nand the first sound mode\n\\begin{eqnarray}\n\\omega = \\pm \\sqrt{\\frac{8\\pi \\alpha}{3}} \\frac{v_F}{\\lambda_F^\\eta} q^{1-\\eta} - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2.\n\\end{eqnarray}\nIn this case, the sound mode has an interesting dispersion resulting from the long-range interaction. Also note that the damping rate for the \"hydrodynamic plasmon\" does not change due to the specific form of the long-range interaction, remaining independent of the range parameter $\\eta$.\n\n\\section{Review of the Boltzmann equation in a 3D metal}\n\nThe collective mode is one of the fundamental excitations of a many-body system. It emerges from coherent interactions between quasiparticles, and is fundamentally different from single particle electron-hole type excitations. It is convenient to describe the collective mode by the distribution function $n(\\textbf k, \\textbf r, t)$ of the quasiparticle at given momentum $\\textbf k$ and position $\\textbf r$. As we are interested in the effect of Coulomb interaction on the sound mode, we will restrict ourselves to the spinless electron. Spin can be added straightforwardly at the cost of making the notations cumbersome---we emphasize that the modes we are discussing are charge density collective excitations which are independent of electron spin. The Boltzmann equation governing the dynamics of the distribution function is~\\cite{Nozieres:2018theory}\n\\begin{eqnarray}\n\t\\frac{d n(\\textbf k, \\textbf r, t)}{dt} = \\left( - \\frac{d \\textbf r}{dt} \\cdot \\frac{\\partial}{\\partial \\textbf r} - \\frac{d \\textbf k}{dt} \\cdot \\frac{\\partial}{\\partial \\textbf k} \\right) n(\\textbf k, \\textbf r, t) + \\mathcal{I}[n], \\nonumber \\\\\n\\end{eqnarray}\nwhere the first term on the right hand side is the drift term while the second term is the collision integral. The semiclassical equation of motion of a quasiparticle is well known\n\\begin{eqnarray}\n\t\\frac{d \\textbf k}{d t} &=& - \\frac{\\partial \\epsilon(\\textbf k, \\textbf r, t)}{\\partial \\textbf r}, \\\\\n\t\\frac{d \\textbf r}{d t} &=&  \\frac{\\partial \\epsilon(\\textbf k, \\textbf r, t)}{\\partial \\textbf k},\n\\end{eqnarray}\nwhere $\\epsilon(\\textbf k, \\textbf r, t)$ is the energy of the quasiparticle. Since we are interested in a conventional metal, it is sufficient to consider the semiclassical description without external electromagnetic field or Berry curvature. The quasiparticle energy should be determined self-consistently from the Boltzmann equation. We are going to solve the collective mode of a small variation from the Fermi-Dirac distribution function, namely,\n\\begin{eqnarray}\n\tn(\\textbf k, \\textbf r, t) = n_F(\\textbf k) + \\delta n(\\textbf k, \\textbf r, t), \\\\\n\tn_F(\\textbf k) \\equiv n_F[\\epsilon_0(\\textbf k)] = \\frac1{e^{\\beta(\\epsilon_0(\\textbf k) -\\mu)}+1},\n\\end{eqnarray}\nwhere $n_F(\\textbf k)$ is the Fermi-Dirac distribution at equilibrium, $\\epsilon_0(\\textbf k)$ is the bare energy of free electrons, $\\beta \\equiv 1\/T$ is the inverse temperature, and $\\mu$ denotes the chemical potential.\n\nThe total energy of the system with a small variation from equilibrium is\n\\begin{eqnarray} \n\t\\epsilon_{\\text{tot}}(t) &=& \\int d^3 \\textbf r \\int_{\\textbf k} \\epsilon_0(\\textbf k) \\delta n(\\textbf k, \\textbf r, t) \\nonumber\\\\\n\t&& + \\frac12 \\int d^3 \\textbf r \\int_{\\textbf k, \\textbf k'} \\delta n(\\textbf k, \\textbf r, t) f(\\textbf k, \\textbf k') \\delta n(\\textbf k', \\textbf r, t) \\nonumber\\\\\n\\label{eq:energy}\t&& + \\frac12 \\int d^3 \\textbf r d^3 \\textbf r' \\int_{\\textbf k, \\textbf k'} \\delta n(\\textbf k, \\textbf r, t) \\frac{e^2}{4\\pi}\\frac{1}{|\\textbf r - \\textbf r'|} \\delta n(\\textbf k', \\textbf r', t), \\nonumber\\\\\n\\end{eqnarray}\nwhere $\\int_{\\textbf k} \\equiv \\int \\frac{d^3 \\textbf k}{(2\\pi)^3}$ and $f(\\textbf k, \\textbf k')$ is the Landau parameter characterizing the short-range quasiparticle interactions, and the second line represents the long-range Coulomb interaction. \nThe presence of both short-range and long-range interactions is an important generalization to a charged Fermi liquid from the neutral Fermi liquid theory. \nThis generalization is also often referred to as Landau-Silin Fermi liquid theory.\nNote that throughout the paper, we use the rationalized unit, where the factor of $\\frac1{4\\pi}$ appears in the real space Coulomb potential.\nThus, by varying with respect to $\\delta n(\\textbf k, \\textbf r, t)$, we can get the quasiparticle energy\n\\begin{eqnarray}\n\t\\epsilon(\\textbf k, \\textbf r, t) =  \\epsilon_0(\\textbf k) + \\int_{\\textbf k'} f(\\textbf k, \\textbf k') \\delta n(\\textbf k', \\textbf r, t) \\nonumber \\\\\n\t+ \\int_{\\textbf k'} \\int d^3 \\textbf r' \\frac{e^2}{4\\pi} \\frac{1}{|\\textbf r - \\textbf r'|} \\delta n(\\textbf k', \\textbf r', t).\n\\end{eqnarray}\n\nTo get a wave-like collective mode, we assume in the usual manner that the variation takes the plane wave form\n\\begin{eqnarray}\n\t\\delta n(\\textbf k, \\textbf r, t) = \\delta n(\\textbf k) e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t}.\n\\end{eqnarray}\nActually, once we get the eigenmode from the plane wave expansion, we can construct any arbitrary mode using linear superposition and completeness. The energy for such a plane wave excitation is\n\\begin{eqnarray}\n\t\\epsilon(\\textbf k, \\textbf r, t) &=& \\epsilon_0(\\textbf k) + \\int_{\\textbf k'} f(\\textbf k, \\textbf k') \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t} \\nonumber\\\\ \n\t&& + \\int_{\\textbf k'} \\int d^3 \\textbf r' \\frac{e^2}{4\\pi}\\frac{1}{|\\textbf r - \\textbf r'|} \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r' - \\mathfrak{i} \\omega t} \\\\\n\t&=& \\epsilon_0(\\textbf k) + \\int_{\\textbf k'} \\left( f(\\textbf k, \\textbf k') +\\frac{e^2}{\\textbf q^2} \\right) \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t} , \\nonumber\\\\\n\\end{eqnarray}\nwhere in the second line, we have used the Fourier transform of the Coulomb potential in 3D. A simple derivation is given in Appendix~\\ref{append:Coulomb}. \n\nNow with the quasiparticle energy, the semiclassical equation of motion becomes\n\\begin{eqnarray}\n\t\\frac{d \\textbf k}{dt} &=& - \\mathfrak{i} \\textbf q \\int_{\\textbf k'} \\left( f(\\textbf k, \\textbf k') + \\frac{e^2}{\\textbf q^2}  \\right) \\delta n(\\textbf k') e^{\\mathfrak{i} \\textbf q \\cdot \\textbf r - \\mathfrak{i} \\omega t}, \\\\  \\frac{d \\textbf r}{dt} &=& \\frac{\\partial \\epsilon_0(\\textbf k)}{\\partial \\textbf k} \\equiv \\textbf v(\\textbf k).\n\\end{eqnarray}\nPutting this equation of motion into the Boltzmann equation, we arrive at\n\\begin{eqnarray}\n\t\\omega \\delta n(\\textbf k) = \\textbf q \\cdot \\textbf v(\\textbf k) \\Big( \\delta n(\\textbf k) - \\big[\\partial_\\epsilon\\big|_{\\epsilon = \\epsilon_0(\\textbf k)} n_F(\\epsilon) \\big] \\nonumber \\\\\n\t\\times \\int_{\\textbf k'} \\big(f(\\textbf k, \\textbf k') + \\frac{e^2}{\\textbf q^2} \\big) \\delta n(\\textbf k') \\Big) + \\mathfrak{i} \\mathcal{I}[n],\n\\end{eqnarray}\nwhere $\\partial_\\epsilon\\big|_{\\epsilon = \\epsilon_0(\\textbf k)}$ means taking the derivative with respect to $\\epsilon$ and then setting $\\epsilon= \\epsilon_0(\\textbf k)$. Since at low temperatures, the small variation $\\delta n(\\textbf k)$ concentrates near the Fermi surface according to the factor \n\\begin{eqnarray}\n\t\\lim_{\\beta \\rightarrow \\infty} -\\partial_\\epsilon\\big|_{\\epsilon = \\epsilon_0(\\textbf k)} n_F(\\epsilon) =\\delta\\big(\\epsilon_0(\\textbf k) - \\mu\\big),\n\\end{eqnarray}\nwhich tends to a delta function localized at the Fermi surface, it is easy to see that the solution to the equation has the form $\\delta n(\\textbf k) = - [\\partial_\\epsilon|_{\\epsilon = \\epsilon_0(\\textbf k)} n_F(\\epsilon)] \\delta n(\\sigma) = \\delta(\\epsilon_0(\\textbf k) - \\mu) \\delta n(\\sigma)$. Here, $\\sigma = (\\theta, \\phi) $ is determined by the vector $\\textbf k$ at the Fermi surface. Hence, it is convenient to change the variable from $\\textbf k$ to $\\big( \\epsilon_0(\\textbf k), \\sigma \\big) $. With the help of the identity,\n\\begin{eqnarray}\n\t\\int d^3 \\textbf k = \\int_0^\\infty d \\epsilon \\int_{\\epsilon_0(\\textbf k) = \\epsilon} \\frac{d\\sigma}{|\\textbf v(\\textbf k)|}, \n\\end{eqnarray}\nwhere $d \\sigma$ denotes the measure over the Fermi surface, and $d \\epsilon$ denotes the measure perpendicular to the Fermi surface, we arrive at\n\\begin{eqnarray} \\label{eq:Boltzmann}\n\t&& \\omega \\delta n(\\sigma) = \\textbf q \\cdot \\textbf v_F(\\sigma) \\Big( \\delta n(\\sigma) \\nonumber \\\\ \n\t&& + \\frac1{(2\\pi)^3} \\int \\frac{d\\sigma'}{|\\textbf v_F(\\sigma')|} \\big(f(\\sigma, \\sigma')  + \\frac{e^2}{\\textbf q^2} \\big) \\delta n(\\sigma') \\Big) + \\mathfrak{i} \\mathcal{I}[n],\n\\end{eqnarray}\nwhere due to the delta function, the integration is restricted to the Fermi surface, and we use $\\textbf v_F(\\sigma)$ to denote $\\textbf v(\\textbf k)$ when $\\textbf k$ is located at the Fermi surface which is the Fermi velocity, and we also use $f(\\sigma, \\sigma')$ to denote $f(\\textbf k, \\textbf k')$ when both $\\textbf k$ and $\\textbf k'$ are on the Fermi surface. The Coulomb interaction is independent of the angle variable and $\\textbf q$ is not a dynamical quantity, so one can regard the Coulomb interaction as a modification of the Landau parameter in the s-wave channel.\n\nThe Boltzmann equation~(\\ref{eq:Boltzmann}) is the central equation that governs the collective modes in a Fermi liquid, including sound modes. It can describe the situation with either short-range interaction or long-range interaction, treating the zero sound, first sound, and plasmon equivalently within one formalism. In the next section, we construct a simple model where the Boltzmann equation~(\\ref{eq:Boltzmann}) can be solved exactly.\n\n\n\\section{A solvable model}\n\nTo proceed, let us assume the Fermi liquid in question has spherical symmetry. This is the situation in simple 3D metals. As a result the Fermi velocity is independent of angle but $|\\textbf v_F(\\sigma)|= v_F$ and we can choose $\\textbf q = (0,0, q)$ pointing along the $k_z$ direction, and use the spherical coordinate $\\Omega=(\\theta, \\phi)$. Then (\\ref{eq:Boltzmann}) becomes\n\\begin{eqnarray} \\label{eq:Boltzmann_sph}\n\t&& \\omega \\delta n(\\Omega) = q v_F \\cos\\theta \\Big( \\delta n(\\Omega) \\nonumber\\\\\n\t&& + \\int \\frac{d\\Omega'}{4\\pi}\\left( F(\\Omega, \\Omega') + \\frac{8\\pi \\alpha}{\\lambda_F^2 \\textbf q^2} \\right) \\delta n(\\Omega') \\Big) + \\mathfrak{i} \\mathcal{I}[n],\n\\end{eqnarray}\nwhere we have used $\\int d \\sigma = k_F^2 \\int d\\Omega = k_F^2 \\int \\sin \\theta d\\theta d \\phi$, $k_F $ is the Fermi momentum $\\epsilon_0(k_F) =\\mu $ and $\\lambda_F$ is the corresponding Fermi wavelength $\\lambda_F = \\frac{2\\pi}{k_F}$. $\\alpha = \\frac{e^2}{4\\pi v_F}$ is the effective fine structure constant in the Fermi liquid defining the interaction coupling strength. \n$F(\\Omega, \\Omega') \\equiv \\frac{k_F^2 }{2\\pi^2 v_F} f(\\Omega, \\Omega')$ is the dimensionless Landau parameter arising from changing the variables from momenta to angles at the Fermi surface.\nAs we mentioned in the previous section, the Coulomb interaction is not a dynamical quantity in the Boltzmann equation. We can absorb the Coulomb interaction into the Landau parameter in the s-wave channel, and restore it back at the end of the calculation. \n\nIn the following, we assume that the only nonvanishing component of the Landau parameter is in the s-wave channel, namely, the Landau parameter is a constant $F(\\Omega, \\Omega') = F_0 $. We absorb the Coulomb interaction into the Landau parameter. This should be sufficient for our purpose to investigate the effect of the Coulomb interaction on the sound mode in a solvable mode. Without the collision integral, the Boltzmann equation reduces to the following eigen equation,\n\\begin{eqnarray} \\label{eq:Boltzmann_zero} \n\t(x_0 - \\cos \\theta) \\delta n(\\Omega) = F_0 \\int \\frac{d\\Omega'}{4\\pi}  \\delta n(\\Omega') , \\quad x_0 = \\frac{\\omega}{q v_F},\n\\end{eqnarray}\nwhich can solved~\\cite{Nozieres:2018theory} by \n\\begin{eqnarray} \\label{eq:n0}\n\\delta n(\\Omega) = \\frac{\\cos \\theta}{x_0 - \\cos \\theta}, \\quad x_0 \\text{arccoth} x_0  = 1+  \\frac1{F_0}.\n\\end{eqnarray}\nSince we ignore the collision integral, this solution represents, by definition, the zero sound solution in the collisionless regime. The second equation determines the velocity of the zero sound. When the $F_0 >1$, we get the approximate zero sound velocity given by $v_0 = \\sqrt{\\frac{F_0}3 + \\frac35} $. \n\n\nSince (\\ref{eq:Boltzmann_zero}) has spherical symmetry, we can study the eigen equation using the spherical harmonics. Any solution $\\delta n(\\Omega)$ can be expanded in the basis of spherical harmonics,\n\\begin{eqnarray}\n\t\\delta n (\\Omega) = \\sum_{l=0}^\\infty \\sum_{m=-l}^l \\delta n_{lm} Y_l^m(\\theta, \\phi),\n\\end{eqnarray}\nwhere $Y_l^m(\\theta, \\phi)$ is the spherical harmonic and $\\delta n_{lm}$ is the corresponding expansion coefficient. The sound modes are solutions with zero magnetic quantum number $m=0$. As the eigen equation conserves the magnetic quantum number, we consider the $m=0$ sector where the equation can be cast into\n\\begin{eqnarray}\n\tx_0 \\delta n_{l,0} &=& \\frac{l}{\\sqrt{4l^2-1}} \\delta n_{l-1,0} + \\frac{l+1}{\\sqrt{4(l+1)^2-1}} \\delta n_{l+1,0} \\nonumber\\\\ && + \\frac{F_0}{\\sqrt3} \\delta n_{0,0} \\delta_{l,1}.\n\\end{eqnarray}\nThe detailed derivation of this equation is given in Appendix~\\ref{append:math}. For $l \\ge 2$, it resembles a recurrence relation of a series. Indeed, it is not hard to check that for $l\\ge 2$, the following series of hypergeometric functions satisfies the recurrence relation,\n\\begin{eqnarray} \\label{eq:series}\n\ta_{l,m} (x) &=& 2\\pi \\delta_{m,0} \\sqrt{\\frac{2l+1}{4}}  \\frac{1}{(2x)^l} \\frac{\\Gamma(l+1)}{\\Gamma(l+ \\frac32)} \\nonumber\\\\\n\t&&\\times \\ _2F_1\\left( \\frac{l+1}2 , \\frac{l+2}2; l + \\frac32; \\frac1{x^2}\\right), ~ l \\ge 1.\n\\end{eqnarray}\nThis series is consistent with the eigenfunction (\\ref{eq:n0}). We present the details of obtaining the series in Appendix~\\ref{append:math}. Thus, we have $\\delta n_{l,m} = a_{l,m}(x_0)$ for $l \\ge 1$. The two equations for $l=1$ and $l=0$ read \n\\begin{eqnarray}\n\tx_0 \\delta n_{1,0} &=& \\frac{1+F_0}{\\sqrt 3} \\delta n_{0,0} + \\frac2{\\sqrt{15}} \\delta n_{2,0},  \\\\\n\tx_0 \\delta n_{0,0} &=& \\frac{1}{\\sqrt{3}} \\delta n_{1,0},\n\\end{eqnarray}\nwhich lead to the following solution,\n\\begin{eqnarray}\n\t\\delta n_{0,0} &=&  2\\sqrt \\pi (x_0 \\text{arccoth} x_0 -1), \\\\\n\t\\quad x_0 \\text{arccoth} x_0  &=& 1+  \\frac1{F_0}.\n\\end{eqnarray}\nThis is of course consistent with the previous eigen-solution~(\\ref{eq:n0}).\n\nTo access the hydrodynamic regime, where the collisionless zero sound crosses over to the hydrodynamic first sound, the collision integral plays an essential role. We take the collision integral to have the following form~\\cite{Lucas:2018electronic}\n\\begin{eqnarray}\n\t\\mathcal{I}(n) = -   \\sum_{l=0}^\\infty \\gamma_l \\delta n_{l,0}, \\quad \t\\gamma_l = \\begin{cases}\n\t\t\t\t0 & l = 0\\\\\n\t\t\t\t\\gamma_\\text{imp} & l = 1 \\\\\n\t\t\t\t\\gamma & l \\ge 2\n\t\t\t\t\\end{cases}.\n\\end{eqnarray}\nwhere $\\gamma$ is the collision rate from collisions between quasiparticles and $\\gamma_\\text{imp}$ is the collision rate between quasiparticles and impurities. Obviously, $\\gamma$ is the key hydrodynamic interaction parameter. Because collisions between quasiparticles conserve the particle number and their total momentum, the pure quasiparticle collision rate for $l=0, 1$ vanishes by virtue of conservation laws. On the other hand, elastic collisions between quasiparticles and quenched impurities relax the momentum, and, therefore, $\\gamma_\\text{imp}$ is nonzero for $l=1$. With this collision integral, the Boltzmann equation in the basis of spherical harmonics reduces to the following coupled equations,\n\\begin{eqnarray}\n\tx \\delta n_{l,0} &=& \\frac{l}{\\sqrt{4l^2-1}} \\delta n_{l-1,0} + \\frac{l+1}{\\sqrt{4(l+1)^2-1}} \\delta n_{l+1,0} , \\nonumber\\\\\n\t && \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad l \\ge 2, \\\\\n\tx_\\text{imp} \\delta n_{1,0} &=& \\frac{1 + F_0}{\\sqrt 3} \\delta n_{0,0} + \\frac2{\\sqrt{15}} \\delta n_{2,0},  \\\\\n\tx_0 \\delta n_{0,0} &=& \\frac{1}{\\sqrt{3}} \\delta n_{1,0}, \\\\\n\\label{eq:x}\tx_0 &=& \\frac{\\omega}{q v_F},~ x = \\frac{\\omega + \\mathfrak{i} \\gamma}{q v_F}, ~ x_\\text{imp} = \\frac{\\omega + \\mathfrak{i} \\gamma_\\text{imp}}{q v_F}.\n\\end{eqnarray}\nThe series~(\\ref{eq:series}) again solves the first recurrence relation, with $x$ replacing $x_0$. Thus for $l \\ge 1$, $\\delta n_{l,m} = a_{l,m}(x) $. The other two equations can be solved easily, namely, $\\delta n_{0,0} = 2 \\sqrt \\pi \\frac{x}{x_0} (x \\text{arccoth} x - 1)$ and\n\\begin{eqnarray} \\label{eq:eigen}\n (F_0 + 1 -3 x_0 x_\\text{imp}) (x \\text{arccoth} x - 1) \\nonumber\\\\ + x_0 [(3x^2-1) \\text{arccoth} x - 3x ] = 0.\n\\end{eqnarray}\nThis is the main result of our paper. The eigen dispersion of the sound mode is determined by~(\\ref{eq:eigen}), where the definitions of $x$'s are given by~(\\ref{eq:x}).\n\n\nTo investigate the sound mode, we can assume $\\gamma_\\text{imp} = 0$, otherwise the coherent propagating sound mode is damped. In the collisionless regime, $\\omega \\gg \\gamma$, we obtain the zero sound mode or the plasmon in~(\\ref{eq:zeroFull}) as\n\\begin{eqnarray}\n\t&& \\omega = \\pm v_0 q - \\mathfrak{i} \\gamma \\frac{(F_0+1)^2 - 2(F_0-1) (\\frac{v_0}{v_F})^2 - 3 (\\frac{v_0}{v_F})^4 }{F_0 [ F_0 +1 - (\\frac{v_0}{v_F})^2]}, \\\\ && \\frac{v_0}{v_F} \\text{arccoth} \\frac{v_0}{v_F} = 1 + \\frac1{F_0}.\n\\end{eqnarray}\nIn the case of strong repulsion $F_0 > 1$ (this is also the case for Coulomb interaction in the long wavelength limit), the sound mode can be simplified to\n\\begin{eqnarray}\n\t\\omega = \\pm \\sqrt{\\frac{F_0}3 + \\frac35} v_F q - \\mathfrak{i} \\gamma \\frac{2(5F_0 + 21)}{5F_0(5F_0 + 3)},\n\\end{eqnarray}\nwhich is nothing but~(\\ref{eq:zero}). On the other hand, in the collision-dominated hydrodynamic regime $\\omega \\ll \\gamma$ we get the first sound,\n\\begin{eqnarray}\n\t\\omega = \\pm \\sqrt{ \\frac{F_0}3 + \\frac13} v_F q - \\mathfrak{i} \\frac{2 v_F^2}{15 \\gamma} q^2,\n\\end{eqnarray}\nwhich is~(\\ref{eq:first}).\n\nWe can also consider the effect of the finite impurity scattering. For weak impurity scattering, the effect is to modify the damping defined by imaginary parts of the sound mode. For the zero sound, we have\n\\begin{eqnarray} \\label{eq:weak_imp}\n\\omega &=& \\pm v_0 q - \\mathfrak{i} \\gamma \\frac{(F_0+1)^2 - 2(F_0-1) (\\frac{v_0}{v_F})^2 - 3 (\\frac{v_0}{v_F})^4 }{F_0 [ F_0 +1 - (\\frac{v_0}{v_F})^2]} \\nonumber\\\\\n&& - \\mathfrak{i} \\gamma_\\text{imp} \\frac{3(\\frac{v_0}{v_F})^2[(\\frac{v_0}{v_F})^2-1]}{F_0(F_0 +1 -(\\frac{v_0}{v_F})^2)}.\n\\end{eqnarray}\nFor large $F_0 > 1$, this reduces to a simple correction $\\delta \\text{Im}(\\omega) = \\frac12 \\gamma_\\text{imp}$. \nFor strong impurity scattering $\\gamma_\\text{imp} > q$, both sound modes are over-damped into\n\\begin{eqnarray}\n\t\\omega = - \\mathfrak{i} \\frac{F_0+1}{3\\gamma_\\text{imp}} v_F^2 q^2, \\qquad \\omega = -\\mathfrak{i} \\gamma_\\text{imp}.\n\\end{eqnarray}\n\t\n \n\n\\section{Conclusions} \\label{sec:conclusion}\n\nWe have discussed electronic sound modes in 3D metals in the presence of long-range Coulomb coupling via the Boltzmann equation. A more microscopic approach to the collective mode, like plasmons, would be to start from the electron Hamiltonian with long-range interactions. Then the collective mode results from integrating out the electron fluctuations. In the lowest order, this is nothing but the RPA approach. Here, we recapitulate how it works for 3D metals. In the RPA approximation, the collective mode is determined by the following eigen equation,\n\\begin{eqnarray} \\label{eq:RPA}\n\t1- V(\\textbf q) \\Pi(\\textbf q, \\omega) = 0,\n\\end{eqnarray}\nwhere $V(\\textbf q)$ is the interaction at momentum $\\bf q$, and the dynamical electron polarization function is defined by\n\\begin{eqnarray}\n\t&& \\Pi(\\textbf q, \\omega) = \\frac{1}{\\beta} \\sum_n \\int_{\\textbf k} \\frac1{-\\mathfrak{i} \\Omega_n - \\frac\\omega2+ \\xi(\\textbf k + \\frac{\\textbf q}2)} \\frac1{-\\mathfrak{i} \\Omega_n + \\frac\\omega2+ \\xi(\\textbf k - \\frac{\\textbf q}2)}, \\nonumber \\\\\n\t\\\\\n\t&& \\Omega_n = \\frac{(2n+1)\\pi}{\\beta}, \\quad \\xi(\\textbf k) = \\epsilon_0(\\textbf k) - \\mu.\n\\end{eqnarray}\nwhere $\\Omega_n$ is the Matsubara frequency and $\\omega$ is the real frequency. To proceed, we assume a spherical Fermi surface with parabolic dispersion $\\epsilon_0(\\textbf k) = \\frac{\\textbf k^2}{2m}$. It is not hard to get the 3D polarization function at zero temperature, namely,\n\\begin{eqnarray} \\label{eq:polarization3D}\n\t\\Pi(\\textbf q, \\omega) = \\frac{N_e q^2}{m \\omega^2} \\left( 1 + \\frac35 \\frac{k_F^2}{m^2} \\frac{q^2}{\\omega^2} \\right), \\quad N_e = \\frac{4\\pi}3 \\frac{k_F^3}{(2\\pi)^3},\n\\end{eqnarray}\nwhere $q \\equiv |\\textbf q|$. Putting in the Coulomb potential given by $V_\\text{Cou}(q) = \\frac{e^2}{q^2}$ and the polarization function~(\\ref{eq:polarization3D}) into eigen equation~(\\ref{eq:RPA}), we obtain the conventional plasmon given in~(\\ref{eq:plasmon3D}). \n\nWe can also consider a short-range density-density interaction potential that is independent of momentum, namely,\n\\begin{eqnarray} \n\tV({\\bf q}) = \\frac{2\\pi^2 v_F}{ k_F^2} F_0,\n\\end{eqnarray}\nwhere the prefactor originates from how the Landau parameter is introduced via quasiparticle interactions at the Fermi surface, i.e., $\\int_{F.S.} \\frac{d^3 {\\bf q}}{(2\\pi)^3}  V({\\bf q})...= \\int \\frac{d\\Omega}{4\\pi} (\\frac{ k_F^2}{2\\pi^2 v_F} V(\\Omega))... $. For the s-wave channel, it reduces to the above equation.\n\nUsing this interaction potential to replace the Coulomb potential, we get the following linearly dispersing sound-like collisionless collective mode \n\\begin{eqnarray}\n\t\\omega^2 = \\frac{v_F F_0}{4\\pi k_F^2} \\frac{N_e q^2}{m} \\left( 1 + \\frac35 \\frac{k_F^2}{m^2} \\frac{q^2}{\\omega^2} \\right), \\quad \\omega \\approx \\pm \\sqrt{\\frac{F_0}3 + \\frac35} v_F q. \\nonumber \\\\\n\\end{eqnarray}\nThe second equation is exactly the zero sound mode in~(\\ref{eq:zero}). At zero temperature, within the RPA, there is no damping of the electronic collective mode by quasiparticle collisions since the collision rate vanishes as $T^2$, but finite impurity scattering would still contribute to the damping in the way we discussed earlier.  At finite temperatures, quasiparticle collisions would lead to Landau damping of the collective modes. \n\nThis simple calculation tells us that microscopically the interaction potential determines the dispersion of the electronic collective modes. The zero sound mode for short-range interactions becomes the gapped 3D plasmon mode in the presence of long-range Coulomb coupling. (The first sound mode also acquires a long wavelength gap as discussed earlier.) Indeed, from the perspective of symmetry, both sound modes and plasmons are density fluctuations that characterize the underlying particle number conservation. So they are actually the same collective mode, and it is not a surprise that they all develop long wavelength gaps because of the long-range Coulomb interaction. Thus, the hydrodynamic sound in 3D metals is not a sound mode at all since it has a finite energy at zero momentum defined by the 3D plasma frequency. We do note, however, that the sound modes differ from the plasmon in their next-to-leading-order dispersion corrections at finite momentum. \nAlthough the 2D case is clearly addressed in Ref.~[3],\nwe briefly discuss the RPA calculation in 2D for completeness. \nThe electron polarization function in two dimensions reads\n\\begin{eqnarray} \\label{eq:polarization2D}\n\t\\Pi({\\bf q}, \\omega) = \\frac{k_F^2 q^2}{4 \\pi m \\omega^2} \\left( 1 + \\frac34 \\frac{k_F^2}{m^2}\\frac{q^2}{\\omega^2} \\right).\n\\end{eqnarray}\nPutting the short-range potential given by $V({\\bf q}) = \\frac{2\\pi v_F}{ k_F} F_0$ and the polarization function~(\\ref{eq:polarization2D}) into the eigen equation~(\\ref{eq:RPA}), we recover the linear zero sound $\\omega = \\pm (\\frac{F_0}{2} + \\frac34 )^{1\/2}v_F q$.\nOn the other hand, using instead the 2D Coulomb potential given by $V_\\text{Cou}({\\bf q}) = \\frac12 \\frac{e^2}{|{\\bf q}|}$ (the factor of $\\frac12$ comes from the two dimensional Fourier transform of the usual Coulomb potential $V_\\text{Cou}({\\bf r}) = \\frac1{4\\pi} \\frac1{|\\bf r|}$) we obtain the following collective plasmon mode\n\\begin{eqnarray}\t\n\t\\omega = \\pm \\sqrt{\\frac{\\pi \\alpha v_F^2}{\\lambda_F} |\\bf q|}.\n\\end{eqnarray}\nTherefore, within the RPA calculation, one can already see that the linear sound wave is modified to be $\\omega \\propto \\sqrt{q}$ by replacing the short-range interaction by the long-range Coulomb interaction. Going beyond the RPA framework, it was shown in Ref.~[3]\nthat although both the first and second sound waves exhibit the same plasmon-like dispersion, the next-to-the-leading order momentum dependence differs.\n\nWith this in mind, we now briefly discuss the 1D case.  In 1D, the Boltzmann approach does not work simply because the Fermi liquid does not exist in the presence of any finite interaction~\\cite{Tomonaga:1950remarks, Luttinger:1963exactly}. \nThus, starting from the electron Hamiltonian including a single-particle dispersion (we take a parabolic dispersion for simplicity) and interaction potentials $V(\\bf q)$ like that given above is a good and simple way to look for sound or plasmon modes. \nThe electron polarization function is now given by~\\cite{Sarma:1985screening}\n\\begin{eqnarray}\n\t\\Pi(q, \\omega) = \\frac{m}{2\\pi q} \\ln \\left( \\frac{m^2 \\omega^2 - (k_F- \\frac{q}2)^2 q^2}{m^2 \\omega^2 - (k_F+ \\frac{q}2)^2 q^2} \\right).\n\\end{eqnarray}\nWe expect a sound wave-like (linear in momentum) mode when the interaction is short ranged. Indeed the short-range potential $V(q) = \\pi v_F F_0$ leads to the zero sound mode in 1D,\n\\begin{eqnarray}\n \t\\omega = \\pm v_F \\sqrt{F_0} q.\n\\end{eqnarray}\nHow does the Coulomb interaction affect this sound mode? The answer is simple, we just need to replace $V(q)$ with the 1D Coulomb potential~\\cite{Sarma:1985screening},\n\\begin{eqnarray}\n\tV_\\text{Cou}(q) = \\frac{e^2}{4\\pi} \\int dr \\frac{e^{\\mathfrak{i} q r}}{\\sqrt{r^2 + a^2}} = \\frac{e^2}{4\\pi} 2 K_0(a q),\n\\end{eqnarray}\nwhere $K_0(x)$ is the modified Bessel equation of the second kind and $a$ is a short-range cutoff introduced to make the integral converge in 1D ($a$ can be thought of as a lattice constant). Plugging the Coulomb potential and the polarization function in 1D into the eigen equation~(\\ref{eq:RPA}), the resultant long wavelength collective mode is \n\\begin{eqnarray}\n\t\\omega = \\pm \\frac{e}{\\pi} \\sqrt{\\frac{v_F}2} q \\sqrt{- \\ln \\frac{a q}2}, \\quad a q \\ll 1,\n\\end{eqnarray}\nwhich is nothing but the well-known plasmon mode in 1D. Although we consider zero temperature, we expect that the plasmon mode takes over the sound modes in 1D when the Coulomb interaction dominates since the dispersion is largely independent of temperature. A more physical argument is that from the symmetry perspective, sound modes and plasmons are the very same mode, and the different names just refer to whether the interaction potential is short-range or long-range. The curious thing is that in 3D systems, where the Coulomb coupling goes as $q^{-2}$, the sound mode is not acoustic at all since it acquires the plasmon mass at zero momentum.\n\n\nIn conclusion, we constructed a solvable model in 3D to obtain the sound modes in both the collisionless regime and the collision-dominated hydrodynamic regime. In particular, we discussed the effect of long-range Coulomb interaction on the sound modes. We found that in the presence of Coulomb interaction, both the zero sound and the first sound obtain a finite gap equal to the plasmon frequency, and a damping rate which is quadratic in momentum. We also discussed general long-range interactions that lead to unusual plasmon dispersions. Finally, we clarified the collective mode and sound mode dichotomy in 1D.\n\n\\section*{Acknowledgments}\n\nThis work is supported by the Laboratory for Physical Sciences. S.-K.J. is supported by the Simons Foundation via the It From Qubit Collaboration.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nPrecise and reliable control of a quantum system is an attractive and\nchallenging experimental issue in quantum physics. \nIn particular, the importance of its application to quantum information\nprocessing has been increasing recently. \nA promising way to achieve this is to employ geometric\nphases (or, more generally, non-Abelian\nholonomies)\\,\\cite{Nakahara2003,ChruscinskiJamiolkowski2004}, because \ngeometric phases are expected to be robust against noise and decoherence\nunder a proper condition\\,\\cite{BlaisTremblay2003,Ota;Kondo:2009}. \nA large number of studies for applying their potential\nrobustness to quantum computing have been done, e,g., \nphase-shift gates with Berry phases\\,\\cite{jones2000}, \nnonadiabatic geometric quantum \ngates\\,\\cite{Wang;Keiji:2001,Zhu;Wang:2002,zhu2003,Tianetal2004,\nDing;Wang:2005,ZhuZanardi2005,Das;Kumar:2005,Imai;Moriaga:2007}, \nholonomic quantum computing\\,\\cite{Zanardi;Rasetti:1999,Duan;Zoller:2001,\nNiskanen;Salomaa:2003,Solinas;Zanghi:2004,Karimipour;Majd:2004,\nTanimura;Hayashi:2005,Goto;Ichimura:2007,Ota;Nakahara:2008}, \nquantum gates with noncyclic geometric\nphases\\,\\cite{Friedenauer;Sjoqvist:2003}, and so on.\n \nFor physical realization of geometric quantum gates, elimination\nof a dynamical phase is a key point. \nJones {\\it et al.}\\,\\cite{jones2000} implemented a controlled phase-shift\ngate with a Berry phase\\,\\cite{Berry,Simon:1987} \nby quasi-statistically, or slowly,  \ncontrolling an effective field in a rotating frame. \nThey nulled dynamical phase effects using a conventional  \nspin echo approach\\,\\cite{levitt}.  \nZhu and Wang\\,\\cite{zhu2003} proposed a geometric quantum gate based on \nthe Aharonov-Anandan phase\\,\\cite{AharonovAnandan1987,Page:1987},\nwhich should be fast since a nonadiabatic process is employed. \nIn their proposal, elimination of a dynamical phase is achieved \nby a double-loop method, where a dynamical phase cancels out while\na geometric phase accumulates along two loops. \n\nAlthough several experimental techniques for the application of\ngeometric phases to quantum computation are\navailable\\,\\cite{Tianetal2004,Das;Kumar:2005,Imai;Moriaga:2007}, explicit \nimplementations of geometric phase gates have not been extensively\nstudied so far.  \nWithout explicit implementations, the often-cited advantage of the\nholonomic quantum gates is nothing more than a desk plan. \nThus, such explicit examples are highly desirable. \nIn this paper, we combine Zhu and Wang's approach with Jones \n{\\it et al.}'s one, employing an Aharonov-Anandan phase for fast gate operation and a\nspin echo technique for dynamical phase cancellation,  \nand demonstrate one-qubit gates with a commercial liquid-state  \nnuclear magnetic resonance (NMR) system. \nIn many experiments of nonadiabatic geometric quantum\ngates\\,\\cite{Tianetal2004,Das;Kumar:2005,Imai;Moriaga:2007}, \nthe gate operations in which the dynamical phase is arranged to\nvanish\\,\\cite{Ota;Kondo:2009,SuterMuellerPines1988} have been adopted. \nIn the present paper, we show that we may have another option for\nphysical realization of geometric quantum gates. \n \nThe next section summarizes Zhu and Wang's theoretical proposal and  \nour modifications for practical implementation in liquid-state NMR \nquantum computer. \nIn Sec.\\,\\ref{sec:experiments} we describe details of our experiments,\nwhere implemented gates are evaluated by performing quantum process\ntomography. \nSec.\\,\\ref{sec:summary} is devoted to summary. \n \n\\section{Theory} \n\\label{sec:theory}\n\\subsection{Quantum gates with orthonormal cyclic vectors} \n\\label{subsec:qg_ocv}\nThe Aharonov-Anandan phase is a geometric phase associated with nonadiabatic  \ncyclic time evolution of a quantum \nsystem\\,\\cite{AharonovAnandan1987,Page:1987}.  \nLet us write a state at $t$ ($0\\le t\\le \\tau$) as $|{\\psi(t)}\\rangle$ \nin the Hilbert space $\\mathcal{H}$ with dimension $n$.  \nThe time evolution of a system is given by the Schr\\\"odinger equation \nwith a time-dependent Hamiltonian $H(t)$. \nWe take the natural unit in which $\\hbar=1$. \nThe nonadiabatic cyclic time evolution of the quantum system means that \n\\begin{equation*}\n|{\\psi(\\tau)}\\rangle = U(\\tau)|{\\psi(0)}\\rangle\n = e^{i\\gamma}|{\\psi(0)}\\rangle, \n\\end{equation*} \nwhere $U(\\tau)$ is the time evolution operator and $\\gamma \\in \\mathbb{R}$.  \nLet us call $|{\\psi(0)}\\rangle$ a cyclic vector\\,\\cite{AharonovAnandan1987}. \nWe note that the dynamical phase $\\gamma_{\\rm d}$ associated with this\ntime evolution is \n\\begin{equation} \n\\gamma_{\\rm d} = - \\int_0^\\tau \\langle{\\psi(t)}|H(t)|{\\psi(t)}\\rangle dt, \n\\label{eq:dyn_phase} \n\\end{equation} \nwhile the geometric component is defined as \n\\begin{equation*}\n \\gamma_{\\rm g} = \\gamma - \\gamma_{\\rm d}. \n\\end{equation*}\nWe take a set of orthonormal cyclic vectors, \n$\\{ |{u_k(0)}\\rangle \\}$ so that \n\\begin{equation*}\n|{u_k(\\tau)}\\rangle =  e^{i \\gamma_k} |{u_k(0)}\\rangle,\n\\end{equation*}\nwhere $k = 1,\\ldots,n$. \nHereafter, we write $|u_k(0)\\rangle$  as $|u_k\\rangle$ for brevity. \nA general state $|\\Psi(0) \\rangle\\in\\mathcal{H}$ is expressed as \n\\(\n |{\\Psi}(0) \\rangle = \\sum_{k=1}^{n}a_{k}|{u_{k}}\\rangle\n\\). \nThen, we have \n\\(\n|{\\Psi(\\tau)}\\rangle = U(\\tau)|\\Psi(0) \\rangle= \n\\sum_{k=1}^{n}a_{k}e^{i\\gamma_{k}}|u_{k}\\rangle\n\\). \nWe denote a fixed basis in $\\mathcal{H}$ as $\\{|l\\rangle \\}$,\nwhich corresponds to the computational basis $\\{|0\\rangle,|1\\rangle\\}$\nfor the case $n=2$. \nIn terms of $\\{|l\\rangle\\}$, we have \n\\(\n |\\Psi(\\tau)\\rangle = \\sum_{l=0}^{n-1}c_{l}(\\tau)|l\\rangle\n\\). \nIt means that \n\\(\n a_{k} \n=\n\\langle u_{k}|\\Psi(0)\\rangle \n= \n\\sum_{l=0}^{n-1}c_{l}\\langle u_{k}|l\\rangle\n\\), \nwhere $c_{l}=c_{l}(0)$. \nTherefore, we find that \n\\begin{equation} \nc_{l}(\\tau) \n= \n\\sum_{l^{\\prime}=0}^{n-1} \n\\sum_{k=1}^{n}\\,e^{i\\gamma_{k}} \n\\langle{l}|u_{k}\\rangle \\langle{u_{k}}|l^{\\prime} \\rangle c_{l^{\\prime}} \n= \n\\sum_{l^{\\prime}=0}^{n-1}\\,V_{ll^{\\prime}}c_{l^{\\prime}}, \n\\label{eq:coeff_tau_cb}\n\\end{equation} \nwhere  \n\\begin{equation} \nV_{l  l^{\\prime}} = \\sum_{k=1}^{n} \ne^{i\\gamma_{k}} \\langle{l }|u_{k}\n\\rangle \\langle{u_{k}}|l ^{\\prime} \\rangle. \n\\label{eq:top_cb}\n\\end{equation} \nWe have \n\\(\n  \\gamma_{k} = \\gamma_{k,{\\rm d}} + \\gamma_{k,{\\rm g}} \n\\), \nwhere $\\gamma_{k,{\\rm d}}$ and $\\gamma_{k,{\\rm g}}$ are the dynamical\nand the geometric phases associated with $|u_{k}\\rangle$, respectively. \n\n\\subsection{Single-qubit case} \nLet us consider a single-qubit system.  \nWe choose the Hamiltonian $H_1(t)$ as  \n\\begin{equation*} \nH_{1}(t) = -\\frac{1}{2} {\\bm \\Omega}(t)\\cdot\\vsigma, \n\\end{equation*} \nwith NMR in mind, where \n\\(\n\\vsigma =\n(\\sigma_{x},\\,\\sigma_{y},\\,\\sigma_{z})\n\\) \nand\n\\begin{equation} \n {\\bm \\Omega}(t)  =\n\\left( \n\\omega_{1} \\cos(\\omega_{\\rm rf} t-\\phi),-\n\\omega_{1}\\sin(\\omega_{\\rm rf} t- \\phi), \n\\omega_{0} \n\\right).\n\\label{eq:field} \n\\end{equation}\nWe note that ${\\bm \\Omega(t)}$ is a time dependent  \nparameter corresponding to the external field and  \n$\\sigma_{\\alpha}$ ($\\alpha=x,\\,y,\\,z$) is the $\\alpha$th component of the\nPauli matrices. \nOne can freely control $\\omega_0$ by\ntaking a proper rotating frame. \nThe transformation to the rotating frame with the frequency\n$\\omega_{\\rm{rf}}$ is made possible by the unitary transformation \n$U = e^{i \\omega_{\\rm{rf}} \\sigma_z t\/2}$ \nand the Hamiltonian in the rotating frame is \n\\begin{equation}\nH_{1{\\rm r}} \n=  U^{\\dagger} H_1 U - i U^{\\dagger}\n\\frac{d}{dt} U \n= -\\frac{1}{2} \\Omega \\vm \\cdot \\vsigma,\n\\label{eq:nmr_rot_ham}\n\\end{equation}\nwhere \n\\(\n\\Omega = (\\omega_{1}^{2}+\\Delta^{2})^{1\/2}\n\\), \n\\(\n\\Delta = \\omega_{0}-\\omega_{\\rm rf}\n\\), \n\\(\n\\vm =\n( \\sin\\chi\\cos\\phi,\\,\\sin\\chi\\sin\\phi,\\,\\cos\\chi )\n\\), and \n\\(\n\\tan\\chi = \\omega_{1}\/\\Delta \n\\). \nThe solution of the Sch\\\"odinger equation is \n\\begin{eqnarray} \n|\\psi (t)\\rangle \n&=& \ne^{i\\omega_{\\rm rf} t\\sigma_{z}\/2}\\, \ne^{i\\Omega t \\vm\\cdot\\vsigma\/2}\\, \n|\\psi (0)\\rangle .\n\\label{eq:sol_1q} \n\\end{eqnarray} \n\\begin{figure}[tbp]\n\\centering\n\\scalebox{0.58}[0.58]{\\includegraphics{fig1a}}\n\\scalebox{0.58}[0.58]{\\includegraphics{fig1b}}\n\\vspace{-3mm}\n\\flushleft\n(a)\\hspace{40mm}(b)\n\\caption{Example of dynamics of a single-qubit cyclic vector. \n(a) A time-dependent external field\n $\\bm{\\Omega}(t)$ and (b) a closed trajectory on the Bloch\n sphere corresponding to a cyclic vector $|\\psi_{+}(t)\\rangle$, \n$0\\le t\\le \\tau=2\\pi\/|\\omega_{\\rm rf}|$. \nThe end point of each arrow represents the initial value. \nWe set the parameters $\\omega_{0}=2\\pi$, $\\omega_{1}=0.5\\times 2\\pi$,\n $\\omega_{\\rm rf}=0.8\\times 2\\pi$, and $\\phi=0$ in\n Eq.\\,(\\ref{eq:field}). }\n\\label{fig:singlel}\n\\end{figure}\nWe denote the eigenstates of $\\vm\\cdot \\vsigma$ with eigenvalues \n$\\pm 1$ as \n\\(\n |\\psi_{\\pm}\\rangle\n\\). \nTheir explicit forms are  \n\\begin{eqnarray*} \n |{\\psi_{+}}\\rangle  \n&=&  e^{-i\\phi\/2} \\cos\\frac{\\chi}{2} |{0}\\rangle \n +   e^{i\\phi\/2}  \\sin\\frac{\\chi}{2} |{1}\\rangle ,  \\\\ \n |{\\psi_{-}}\\rangle  \n&=& - e^{-i\\phi\/2} \\sin\\frac{\\chi}{2} |{0} \\rangle\n +    e^{i\\phi\/2}  \\cos\\frac{\\chi}{2} |{1} \\rangle,  \n\\end{eqnarray*} \nwhere $|{0}\\rangle$ and $|{1}\\rangle$ are the eigenstates of $\\sigma_z$\nwith eigenvalues $+1$ and $-1$, respectively, and taken as the\ncomputational basis.  \nThe corresponding Bloch vectors of $|{\\psi_{\\pm}}\\rangle$ are  \n\\begin{equation*} \n\\langle{\\psi_{\\pm}}| \\vsigma |{\\psi_{\\pm}}\\rangle = \\pm \\vm .  \n\\end{equation*} \nWe require that $|{\\psi_{\\pm}}\\rangle$ be cyclic vectors. \nSince $|\\psi_{+}\\rangle$ and $|\\psi_{-}\\rangle$ are mutually orthogonal,\nthey are identified as $\\{|u_{k}\\rangle\\}$ in Sec.\\,\\ref{subsec:qg_ocv}.  \nIt follows from Eq.\\,(\\ref{eq:sol_1q}) that the execution time $\\tau$\nmust satisfy the condition $|\\omega_{\\rm rf}| \\tau\/2 = \\pi$, i.e., \n$\\tau = 2\\pi\/|\\omega_{\\rm rf}|$.\nThen, $|\\psi_{\\pm}(\\tau) \\rangle$ is written as\n\\begin{eqnarray*} \n|{\\psi_{\\pm}(\\tau)}\\rangle = e^{i \\gamma_{\\pm}} |{\\psi_{\\pm}}\\rangle, \n\\end{eqnarray*} \nwhere \n\\(\n\\gamma_{\\pm} = \\pi \\pm  \\pi \\Omega\/|\\omega_{\\rm rf}|\n\\). \nFigure \\ref{fig:singlel} shows an example of $\\bm{\\Omega}(t)$ and the\nclosed trajectory on the Bloch sphere corresponding to $|\\psi_{+}\\rangle$. \nUsing Eq.\\,(\\ref{eq:dyn_phase}), we find that the dynamical phase is \n\\begin{equation*} \n\\gamma_{\\rm \\pm, d}  \n= \\pm \\frac{\\tau}{2}(\\omega_1 \\sin \\chi + \\omega_0 \\cos \\chi)  \n= \\pm \\frac{\\pi(\\omega_1^2+\\omega_0 \\Delta)}{|\\omega_{\\rm rf}| \\Omega},\n\\end{equation*} \nwhile the geometric phase is \n\\begin{equation*} \n\\gamma_{\\pm,{\\rm g}}  \n= \n\\gamma_{\\pm} -\\gamma_{\\pm, {\\rm d}} \n=  \\pi \\left(1 \n\\mp \\frac{\\omega_{{\\rm rf}}}{|\\omega_{{\\rm rf}}|}\\frac{\\Delta}{\\Omega} \n\\right).\n\\end{equation*} \nBased on Eqs.\\,(\\ref{eq:coeff_tau_cb}) and (\\ref{eq:top_cb}), we obtain\na unitary gate with the dynamical and the Aharonov-Anandan phases with respect to the\ncomputational basis $\\{|0\\rangle,|1\\rangle\\}$, \n\\begin{equation*} \nV(\\tau)  \n=\n\\left( \n\\begin{array}{cc} \n\\cos\\gamma-i\\sin\\gamma\\cos\\chi \n&  - i e^{-i \\phi}\\sin \\gamma \\sin \\chi \\\\ \n- i e^{i \\phi}\\sin \\gamma \\sin \\chi \n& \\cos\\gamma +i\\sin\\gamma\\cos\\chi\n\\end{array} \n\\right), \n\\end{equation*} \nwhere $\\gamma = 2\\pi -\\gamma_{+}=\\gamma_{-}$ has been used to simplify\nthe notation.  \n\n\\subsection{Cancellation of dynamical phases} \n\\label{subsec:cancellation}\nWe closely follow Zhu and Wang's proposal\\,\\cite{zhu2003} \nin order to eliminate a dynamical phase. \nThey proposed the use of two successive unitary operations, in which \na dynamical phase cancels out while a geometric phase accumulates along\nthese two operations.  \nEach unitary operation associated with a loop is characterized by the\ntime-dependent external field (\\ref{eq:field}). \nThe loop parameter corresponding to the $i$th loop is denoted by\n$\\bm{\\Omega}_{i}(t)$ ($i=1,2$). \nThus, in principle, we have four independent parameters in each\nloop, i.e., $\\omega_{i,1}$, $\\omega_{i,{\\rm rf}}$, $\\phi_{i}$, and\n$\\omega_{i,0}$.  \nWe note that they are not always tunable in a real experimental\nsituation. \n\\begin{figure}[tbp]\n\\begin{center}\n\\scalebox{0.5}[0.5]{\\includegraphics{fig2}}\n\\end{center}\n\\caption{Schematic diagram of double-loop method for dynamical phase\n cancellation according to the proposal by Zhu and\n Wang\\,\\cite{zhu2003}. \nTwo time-dependent magnetic fields are applied sequentially. \nThe first magnetic field (loop 1) rotates counter-clockwise,\n while the second one (loop 2) rotates clockwise in order to eliminate a\n dynamical phase. \n\\label{fig:loops}}\n\\end{figure}\n\nWe will search for the condition under which\nthe dynamical phases associated with the two loops cancel each other as\nshown in Fig.\\,\\ref{fig:loops}. \nWe focus on the case in which $\\phi_{i}=0$ and \n\\(\n\\omega_{1,{\\rm rf}}=\\omega_{2,{\\rm rf}}(\\equiv \\omega_{\\rm rf}) >0\n\\) \nfor simplicity. \nThe first loop (loop 1) is described by\n\\begin{equation} \n\\bm{\\Omega}_1(t) \n=\n\\left( \n\\omega_{1,1} \\cos \\omega_{\\rm rf} t,\\, \n-\\omega_{1,1} \\sin \\omega_{\\rm rf} t,\\, \n\\omega_{1,0} \n\\right),\n\\label{loop_1}\n\\end{equation}\nwhile the second loop (loop 2) by\n\\begin{equation} \n\\bm{\\Omega}_2(t)\n= \n- \n\\left( \n\\omega_{2,1} \\cos \\omega_{\\rm rf} t,\\, \n-\\omega_{2,1} \\sin \\omega_{\\rm rf} t,\\, \n\\omega_{2,0} \n\\right) \\bm{R}_{y}(\\theta). \n\\label{loop_2}   \n\\end{equation} \nLet $\\bm{R}_{y}(\\theta) \\in {\\rm SO}(3)$ represent a rotation around the\n$y$-axis by an angle $\\theta$.  \nThe rotation angle $\\theta$ is chosen so that the corresponding cyclic \nvectors $|{\\psi_{i,\\pm}}\\rangle$ for these two loops satisfies \n\\begin{equation}\n|\\psi_{1,\\pm}\\rangle = e^{i c}|\\psi_{2,\\pm}\\rangle, \n\\label{eq:connect_cond}\n\\end{equation} \nwhere $c \\in \\mathbb{R}$. \nIn other words, the Bloch vectors corresponding to\n$|\\psi_{1,\\pm}\\rangle$ coincide with those to \n$|\\psi_{2,\\pm}\\rangle$. \nUsing the notation in Eq.\\,(\\ref{eq:nmr_rot_ham}), we find that in the\nloop $i$ \n\\begin{equation*} \n\\bm{m}_{i} \n=\n\\bm{k}_{i}\\bm{O}_{i},\n\\quad\n\\bm{k}_{i}\n= (\\sin\\chi_i,0,\\cos\\chi_i),\n\\end{equation*} \nwhere \n\\(\n\\bm{O}_{1} = \\bm{I}_{3}\n\\), \n\\(\n\\bm{O}_{2} = \\bm{R}_{y}(\\theta)\n\\), and we write \nthe $3 \\times 3$ unit matrix as\n\\(\n\\bm{I}_{3}\n\\). \nThe rotation angle $\\chi_{i}$ is defined as  \n\\begin{eqnarray}\n&&\n\\tan \\chi_1 = \\frac{\\omega_{1,1}}{\\Delta_1}, \\quad\n\\Delta_1 =\\omega_{1,0} - \\omega_{\\rm rf}, \n\\label{eq:def_del1_omega1}\\\\ \n&&\n\\tan \\chi_2 = \\frac{\\omega_{2,1}}{\\Delta_2}, \\quad \n\\Delta_2 = \\omega_{2,0} + \\omega_{\\rm rf}. \n\\label{eq:def_del2_omega2}\n\\end{eqnarray} \nThe angle $\\theta$ is explicitly given as \n\\begin{equation*}\n\\theta = \\chi_{2} - \\chi_{1}.\n\\end{equation*} \nFigure \\ref{fig:exampleZW}(a) shows an example of the time-dependent\nexternal fields ${\\bm \\Omega}_{1}(t)$ and ${\\bm \\Omega}_{2}(t)$. \nThe corresponding closed trajectory on the Bloch sphere is drawn in\nFig.\\,\\ref{fig:exampleZW}(b), in which the initial point corresponds to\na cyclic vector $|\\psi_{1,+}\\rangle$. \nFigure \\ref{fig:exampleZW}(b) shows that $|\\psi_{1,+}\\rangle$ is not\nonly cyclic for loop 1 but also for the total process (i.e., loop $1$ and\nloop $2$) due to the connection condition (\\ref{eq:connect_cond}). \n\\begin{figure}[tp]\n\\centering\n\\scalebox{0.59}[0.59]{\\includegraphics{fig3a}}\n\\scalebox{0.59}[0.59]{\\includegraphics{fig3b}} \n\\vspace{-3mm}\n\\flushleft\n(a)\\hspace{45mm}(b)\n\\caption{(color online) Example of Zhu-Wang's double-loop method. \nThe time-dependent external fields $\\bm{\\Omega}_{1}(t)$ and\n $\\bm{\\Omega}_{2}(t)$ are shown in (a), while the closed trajectory \non the Bloch sphere corresponding to the cyclic vectors\n $|\\psi_{1,+}\\rangle$ and $|\\psi_{2,+}\\rangle$ in (b). \nWe note that these are connected and thus form one closed\n trajectory. \nWe set the loop parameters $\\omega_{1,1}=\\omega_{2,1}=2\\pi$, $\\omega_{\\rm rf}=0.7\\times 2\\pi$,\n $\\omega_{1,0}=0.27\\times 2\\pi$, and $\\omega_{2,0}=1.5\\times 2\\pi$ in\n Eqs.\\,(\\ref{loop_1}) and (\\ref{loop_2}). We note that these parameters are\n calculated on the basis of a condition for nulling dynamical phases in\n Ref.\\,\\cite{zhu2003}. \nIn this example, $\\Gamma=\\frac{1}{2}$ in Eq.\\,(\\ref{eq:ori_geop}).}\n\\label{fig:exampleZW}\n\\end{figure}\nIt is necessary to search for $\\omega_{i,a}$ and $ \\omega_{\\rm rf}$ \n($a=0,1$) so that  \n\\begin{eqnarray}\n&&\n\\gamma_{\\rm 1,d} +\\gamma_{\\rm 2,d} = 0, \n\\label{eq:ori_no_dynp}\\\\ \n&&\n\\gamma_{\\rm 1,g} + \\gamma_{\\rm 2,g} = \\Gamma \\pi . \n\\label{eq:ori_geop}\n\\end{eqnarray} \nWe write them more explicitly as follows: \n\\begin{eqnarray} \n \\frac{\\omega_{1,1}^2 + \\omega_{1,0} \\Delta_1}\n      {\\Omega_1}\n&=&  \n\\frac{\\omega_{2,1}^2 + \\omega_{2,0} \\Delta_2}{\\Omega_2}, \n\\label{eq:con_d}\\\\ \n\\frac{\\Delta_1}{\\Omega_1}+\\frac{\\Delta_2}{\\Omega_2} \n&=&2-\\Gamma, \n\\label{eq:con_g}\n\\end{eqnarray} \nwhere $\\Omega_i = (\\omega_{i,1}^2+\\Delta_i^2)^{1\/2}$. \nThere may be many combinations of $\\omega_{i,a}$ and $\\omega_{\\rm rf}$\nfor a given $\\Gamma$ which satisfy the conditions (\\ref{eq:con_d}) and \n(\\ref{eq:con_g})\\,\\cite{zhu2003}. \nWe note that a set of the parameters employed in\nFig.\\,\\ref{fig:exampleZW} is one example for the solution of\nEqs.\\,(\\ref{eq:con_d}) and (\\ref{eq:con_g}), in which\n$\\Gamma=\\frac{1}{2}$. \n\nAfter the elimination of a dynamical phase, we have a one-qubit\ngeometric quantum gate \n\\begin{equation}\n V_{\\rm ZW} \n= e^{i\\Gamma\\pi}|\\psi_{1,+}\\rangle\\langle \\psi_{1,+}|\n+\ne^{-i\\Gamma\\pi}|\\psi_{1,-}\\rangle\\langle \\psi_{1,-}|. \n\\label{eq:ZW_gate}\n\\end{equation}\n\n\\subsection{Spin-echo approach }\n\\label{subsec:spin_echo}\nZhu and Wang's proposal for eliminating a dynamical phase is not\nfeasible for a conventional commercial NMR system where a field along\nthe $z$-axis is strictly constant. \nIn other words, it is difficult to realize $\\bm{\\Omega}_2(t)$ in\nEq.\\,(\\ref{loop_2}). \n\\begin{figure}[tp]\n\\begin{center}\n\\scalebox{0.45}[0.45]{\\includegraphics{fig4}}\n\\end{center}\n\\caption{Schematic diagram of double-loop method for dynamical phase\n cancellation on the basis of a spin-echo approach.  \nTwo (four) soft (hard) square pulses are applied. \nWe note that $R_{y}(\\theta)=e^{-i\\theta\\sigma_{y}\/2}$, in which\n $\\theta=\\chi_{2}-\\chi_{1}$.  \n\\label{fig:pseqloop}}\n\\end{figure}\nIn the present paper, we propose an experimentally feasible method, in\nwhich the loop 2 is divided into three successive steps while the loop 1\nremains unchanged. \nThe separation of the loop 2 is motivated by the spin-echo technique\nfrequently employed in NMR experiments, in which the direction of time\nis reversed by an application of a pair of $\\pi$-pulses. \nThree successive operations are \n(a) a rotation around the $y$-axis by \n\\(\n\\theta (= \\chi_{2}-\\chi_{1})\n\\), \n(b) an operation corresponding to precession by a field \n\\(\n-\\left(\n\\omega_{2,1} \\cos \\omega_{\\rm rf}t,\\,\n\\omega_{2,1} \\sin \\omega_{\\rm rf}t,\\, \n\\omega_{2,0} \\right)\n\\)\nfor a duration $\\tau = 2\\pi\/|\\omega_{\\rm rf}|$, \nand (c) a rotation around the $y$-axis by $-\\theta$.  \nThe rotations $R_{y}(\\pm \\theta)$ correspond to the basis\nvector change and back as shown in Fig.\\,\\ref{fig:pseqloop}. \nRotation around the $y$-axis by $\\theta$ is easy to realize by\na radio-frequency (rf) pulse, which corresponds to the unitary operator\n\\begin{equation}\nR_{y}(\\theta) = e^{-i \\theta \\sigma_y\/2}. \n\\end{equation}\nWe emphasize here that \n\\(\n|\\psi_{2,\\pm}\\rangle = R_{y}(\\theta)|\\psi_{1,\\pm}\\rangle \n\\).   \nWe assume that the pulse duration is infinitely short for simplicity. \nIt should be noted that this operation does not generate a dynamical phase\nsince the $y$-axis is perpendicular to both $\\bm{k}_1$ and \n$\\bm{k}_2$\\,\\cite{Ota;Kondo:2009,SuterMuellerPines1988}. \nThe operation corresponds to a precession \nby a field \n\\(\n-\\left(\n\\omega_{2,1}\\cos\\omega_{\\rm rf}t,\\, \n\\omega_{2,1}\\sin\\omega_{\\rm rf}t,\\, \n\\omega_{2,0} \\right)\n\\)\nfor $\\tau = 2\\pi\/|\\omega_{\\rm  rf}|$ is given as \n\\begin{equation*} \nU_{2}^{\\prime} =\ne^{i\\Omega_{2} \\tau (-\\bm{k}_2) \\cdot\\vsigma\/2}, \n\\end{equation*} \nfrom Eq.\\,(\\ref{eq:sol_1q}). \nThe identity \n\\begin{equation*}\nR_{y}(-\\pi) e^{i\\theta \\bm{k}_{2} \\cdot\\vsigma\/2} \nR_{y}(\\pi) =\ne^{i\\theta (-\\bm{k}_{2}) \\cdot\\vsigma\/2}, \n\\end{equation*} \nimplies that $U_2^{\\prime}$ can be realized by a precession\nunder the field \n\\(\n \\left(\n\\omega_{2,1} \\cos\\omega_{\\rm rf}t,\\, \n-\\omega_{2,1} \\sin\\omega_{\\rm  rf}t,\\,\n\\omega_{2,0} \\right)\n\\)\nfor $\\tau = 2\\pi\/|\\omega_{\\rm rf}|$ sandwiched by a pair of \n$\\pm\\pi$-pulses\\,\\cite{levitt}. \nWe again assume that $R_{y}(\\pm\\pi)$ is implemented for an infinitely\nshort pulse for simplicity. \n\nSummarizing the above arguments, the total process is described by \n\\(\nR_{y}(-\\theta)R_{y}(-\\pi)U_{2}(\\tau)R_{y}(\\pi)R_{y}(\\theta)U_{1}(\\tau)\n\\)\\,\\cite{comment1}, which is equivalent to Eq.\\,(\\ref{eq:ZW_gate}) if\nthe dynamical phase is zero. \nThe unitary operator $U_{i}(\\tau)$ is defined as \n\\(\nU_{i}(\\tau) \n= \ne^{i\\omega_{\\rm rf}\\tau\\sigma_{y} \/2}\ne^{i\\Omega_{i}\\bm{k}_{i}\\cdot\\vsigma \/2}\n\\).  \nThe geometric gate which we are going to demonstrate takes the form \n\\begin{equation}\nU_{\\rm echo} \n= R_{y}(-\\pi)U_{2}(\\tau)R_{y}(\\pi)R_{y}(\\theta)U_{1}(\\tau). \n\\label{eq:gate_by_pulse}\n\\end{equation}\nAlthough the complete realization of Zhu and Wang's original proposal\n(\\ref{eq:ZW_gate}) requires $R_{y}(-\\theta)$ at the end of the process\n(\\ref{eq:gate_by_pulse}), we can omit it for constructing a geometric quantum gate since\n$R_{y}(-\\theta)$ does not generate any dynamical phase here. \nWe note that \n\\(\nV_{\\rm ZW} \n= R_{y}(-\\theta) U_{\\rm echo}\n\\). \nUnder the conditions (\\ref{eq:ori_no_dynp}) and (\\ref{eq:ori_geop}), the\nmatrix representation of Eq.\\,(\\ref{eq:gate_by_pulse}) in the\ncomputational basis $\\{|0\\rangle,|1\\rangle\\}$ is given by  \n\\begin{widetext}\n\\begin{equation}\nU_{\\rm echo}\n=\n\\left(\n\\begin{array}{cc}\n\\cos(\\Gamma \\pi)\\cos(\\theta\/2)-i\\sin(\\Gamma \\pi)\\cos\\Theta \n& \n-\\cos(\\Gamma \\pi)\\sin(\\theta\/2)-i\\sin(\\Gamma \\pi)\\sin\\Theta \n\\\\\n\\cos(\\Gamma \\pi)\\sin(\\theta\/2)-i\\sin(\\Gamma \\pi)\\sin\\Theta \n&\n\\cos(\\Gamma \\pi)\\cos(\\theta\/2)+i\\sin(\\Gamma \\pi)\\cos\\Theta \n\\end{array}\n\\right), \n\\label{eq:mrep_gate_pulse}\n\\end{equation}\n\\end{widetext} \nwhere \n\\(\n\\Theta = (\\chi_{2}+\\chi_{1})\/2\n\\). \nWe note that $\\Gamma\\pi$ is the total geometric phase. \nThe pulse sequence (\\ref{eq:gate_by_pulse}) leads to intuitive\nunderstanding of the cancellation mechanism of the dynamical phase. \nLet us consider the case of $\\chi_{1}=\\chi_{2}$, i.e., the two loop are\ncompletely identical, for simplicity. \nIf no $\\pi$-pulse is applied, the dynamical property of the loop 1 is\nthe same as that of the loop 2 and the total dynamical phase is the addition\nbetween the contributions from the loops 1 and 2. \nIt should be noted here that the $\\pm \\pi$-pulses induce the time-reversal\ndynamics in the form of $U^{\\prime}_{2}$ in the loop 2. \nUnder the time-reversal transformation, the sign of the dynamical phase\nassociated with the loop 2 is inverted and hence the dynamical phase is\ncompletely eliminated. \nIt is necessary to employ different processes between the loops 1 and 2 to\nprevent the cancellation of the geometric phase associated with the two loops.  \nThe matrix representation (\\ref{eq:mrep_gate_pulse}) implies that \n$U_{\\rm echo}$ contains three parameters $\\Gamma$, $\\theta$, and\n$\\Theta$. \nDue to the limitation in the control parameters, it may be\ndifficult to choose them independently in a standard liquid-state NMR. \nWe will show that $\\theta$ and $\\Theta$ should be regarded as functions \nof $\\Gamma$\nand $\\omega_{1}$ in Sec.\\,\\ref{subsec:implementation}, in order to\nsatisfy Eqs.\\,(\\ref{eq:ori_no_dynp}) and (\\ref{eq:ori_geop}) as shown in\nTable \\ref{table:sol}.  \nOn the other hand, we are still able to use the rf phase $\\phi$. \nThus, we have the necessary number of free parameters to express\narbitrary elements of SU(2). \n\n\\subsection{Implementation in liquid-state NMR}\n\\label{subsec:implementation}\nWe implement the double-loop scheme in liquid-state NMR.\nWe take different loop parameterization from that of Zhu and Wang for\nease of implementation.  \nWe consider the system in a rotating frame defined by $\\omega_{\\rm ref}$. \nHereafter, we will denote an angular frequency $x$ measured with respect\nto this rotating frame as $\\tilde{x}\\equiv x-\\omega_{\\rm ref}$. \nThus, one can explicitly understand which quantities are controllable by\nchoosing a proper rotating frame. \n\nWe take a common value \n\\begin{eqnarray*}\n\\tilde{\\omega}_{0}=\\omega_{0}-\\omega_{\\rm ref}< 0\n\\end{eqnarray*}\nto $\\omega_{1,0}$ and $\\omega_{2,0}$. \nThe value of $\\tilde{\\omega}_{0}$ in the experiment will be explained in\nSec.\\,\\ref{subsec:pulse_seq}.  \nSimilarly, we assume that $\\omega_{1,1}=\\omega_{1}$ and\n$\\omega_{2,1}=\\omega_{1}$. \nInstead of these simplification, we allow different values with respect\nto $\\omega_{{\\rm rf}}$ between the two\nloops, i.e. $\\omega_{\\rm rf}=\\omega_{1,{\\rm rf}}$ in \nEq.\\,(\\ref{loop_1}) and $\\omega_{\\rm rf}=\\omega_{2,{\\rm rf}}$ in\nEq.\\,(\\ref{loop_2}). \nThese changes do not alter\nthe basic strategy for searching parameters that satisfy \n$\\gamma_{\\rm 1,d} + \\gamma_{\\rm 2,d}=0$\nand \n$\\gamma_{\\rm 1,g} + \\gamma_{\\rm 2,g}= \\Gamma \\pi$. \nWe consider the two loops in the rotating frame in which the frequency \nis $\\tilde{\\omega}_{i,{\\rm rf}}$, the amplitude $\\omega_{1}$,\nand the duration $\\tau_i=2 \\pi \/|\\tilde{\\omega}_{i,{\\rm rf}}|$, i.e.\n\\begin{equation*}\n\\tilde{{\\bm \\Omega}}_i(t) = \n(\\omega_1 \\cos \\tilde{\\omega}_{i, \\rm rf}\\,t, \n-\\omega_1 \\sin \\tilde{\\omega}_{i, \\rm rf}\\,t,\n\\tilde{\\omega}_0)\n\\quad\n(0\\le t\\le \\tau_{i}).\n\\end{equation*}\nThe solutions $\\tilde{\\omega}_{i,{\\rm rf}}\/|\\tilde{\\omega}_0|$ are numerically\nobtained for given $\\epsilon(\\equiv \\omega_1\/|\\tilde{\\omega}_0|)$ and\n$\\Gamma$. \n\n\\begin{table}[bp]\n\\begin{tabular}{ccccccccc}\n\\hline \\hline\n$\\epsilon$ \n&& $\\tilde{\\omega}_{1, \\rm rf}\/\\tilde{\\omega}_0$ \n&& $\\tilde{\\omega}_{2, \\rm rf}\/\\tilde{\\omega}_0$ \n&& $\\theta$~[rad] && $\\Theta $~[rad]\\\\\n\\hline \\hline\n0.5 && -0.6815 && 0.7803 \n    && -0.7298  && -0.639 \\\\\n0.3 && -0.8221 && 1.105  \n    && -0.9571  && -0.589 \\\\\n0.1 && -0.9422 && 1.609  \n    && -1.008   && -0.542 \\\\ \n\\hline\n\\end{tabular}\n\\caption{\n\\label{table:sol}\nSolutions of Eqs.\\,(\\ref{eq:ori_no_dynp}) and (\\ref{eq:ori_geop}) for\n $\\Gamma = \\frac{1}{2}$ and \n$\\epsilon =\\omega_1\/|\\tilde{\\omega}_0| = 0.5,\\,0.3,\\,0.1$ in the rotating\n frame so that $\\tilde{\\omega}_{0}<0$. \nWe note that \n$ \\tau_i |\\tilde{\\omega}_{i, \\rm rf}| =  2\\pi$, \n$\\theta=\\chi_{2}-\\chi_{1}$, \nand \n$\\Theta=(\\chi_{1}+\\chi_{2})\/2$.}\n\\end{table}\n\nWe summarize our parameter choice. \nFirst of all, we adopt a common value to $\\omega_{1,0}$ and\n$\\omega_{2,0}$, i.e., $\\tilde{\\omega}_{0}$ in the rotating frame defined\nas the angular frequency $\\omega_{\\rm ref}$. \nThe value of $\\omega_{1}(=\\omega_{1,1}=\\omega_{1,2})$ is given by \n\\(\n\\omega_{1} = \\epsilon |\\tilde{\\omega}_{0}|\n\\), in which $\\epsilon$ is a positive number. \nFor a given $\\epsilon$ and an aimed geometric phase $\\Gamma$, we can\nnumerically find proper $\\tilde{\\omega}_{i,{\\rm rf}}$ so that \n\\(\n\\gamma_{1,{\\rm d}}+\\gamma_{2,{\\rm d}}=0\n\\) and \n\\(\n\\gamma_{1,{\\rm g}} + \\gamma_{2,{\\rm g}}=\\Gamma \\pi\n\\). \nThe results for $\\Gamma=\\frac{1}{2}$ and $\\epsilon =0.5$, $0.3$, and 0.1,\nfor example, are shown in Table \\ref{table:sol}. \nFrom the observation of Eqs.\\,(\\ref{eq:def_del1_omega1}) and\n(\\ref{eq:def_del2_omega2}), the sign of $\\tilde{\\omega}_{1, \\rm rf}$\nshould be opposite to the one of $\\tilde{\\omega}_{2, \\rm rf}$. \nIt should be noted that the parameters given in Table \\ref{table:sol} are\ncompatible with this requirement. \nThe resultant geometric quantum gate is\nEq.\\,(\\ref{eq:mrep_gate_pulse}). \nThe values of $\\theta(=\\chi_{2}-\\chi_{1})$ and\n$\\Theta(=(\\chi_{2}+\\chi_{1})\/2)$ are given in Table \\ref{table:sol}. \n\nWhen $\\Gamma=\\frac{1}{2}$, Eq.\\,(\\ref{eq:mrep_gate_pulse}) takes the form \n\\begin{eqnarray}\nU_{\\rm echo}(\\Theta)\n&=& \ne^{-i \\pi\\sigma_y\/2}\ne^{i\\Omega_2 \\tau_2 \\bm{k}_2 \\cdot\\vsigma\/2} \\nonumber \\\\\n&&\n\\quad\n\\times \ne^{-i (\\theta+\\pi)\\sigma_y\/2}\ne^{i\\Omega_1 \\tau_1 \\bm{k}_1 \\cdot\\vsigma\/2} \\nonumber \\\\\n&=& e^{-i \\pi\/2}\n\\left( \\begin{array}{cr}\n          \\cos \\Theta &  \\sin \\Theta \\\\\n          \\sin \\Theta & -\\cos \\Theta\n       \\end{array}\n\\right),\n\\label{eq:1q_gate}\n\\end{eqnarray} \nwhich we experimentally demonstrate in the next section.\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\subsection{Sample and spectrometer}\nWe implement a one-qubit gate described by Eq.\\,(\\ref{eq:1q_gate})\nwith a conventional commercial NMR system. \nWe employed a JEOL ECA-500 NMR spectrometer\\,\\cite{jeol} whose \nhydrogen Larmor frequency is approximately 500\\,MHz.\n$^{13}$C nucleus in a 0.6\\,ml, 0.2\\,M sample of ${}^{13}$C-labeled \nchloroform (Cambridge Isotope) in d-6 acetone is employed \nas a qubit, while protons are decoupled by a standard decoupling \ntechnique, called WALTZ\\,\\cite{levitt}. \nWe have chosen $^{13}$C-labeled chloroform for future experiments\ninvolving two-qubit gates. \nThe transverse and the longitudinal relaxation times are\n$T_2 \\sim 0.3\\,{\\rm s}$ and $ T_1 \\sim 5\\,{\\rm s}$, respectively.\nThe longitudinal relaxation time is shorten by adding a small amount of \nIron(I\\!I\\!I)-acetylacetonate so that a repetition rate \ncan be increased. $T_2$ and $T_1$ without Iron(I\\!I\\!I)-acetylacetonate\nare $\\sim 0.3\\,{\\rm s}$ and $\\sim 20\\,{\\rm s}$, respectively. \n\n\\subsection{Pulse sequence}\n\\label{subsec:pulse_seq}\nAs we discussed in the previous section, the gate (\\ref{eq:1q_gate}) \ncan be realized with two rotating magnetic fields and two hard \n(short) pulses. \nThe rotating fields are effectively obtained by two soft (long) pulses\nwhich are rotating with different frequencies \n\\mbox{$\\tilde{\\omega}_{i,\\rm rf} = \\omega_{i, \\rm rf} - \\omega_{\\rm ref}$} \n($i = 1, 2$) in the rotating frame with frequency $\\omega_{\\rm ref}$. \nThe first soft pulse (loop 1) is a usual square\npulse, while the second soft pulse (loop 2) is a\n(frequency) shifted laminar square pulse (SLP)\\,\\cite{ECA500}. \nThis SLP is employed in order to obtain the same phase $\\phi$ in\nEq.\\,(\\ref{eq:field}) for loop 2 as that for loop 1, i.e.,\n$\\phi_{1}=\\phi_{2}$.  \n\nWe take $|\\tilde{\\omega}_0| = 2 \\pi \\times 1000\\,{\\rm rad}\/{\\rm s}$\nand $\\phi_{i} = 0$ throughout the experiments. \nThe condition $\\phi_{i} = 0$ is taken for \nsimplicity as mentioned in the beginning of \nSec.\\,\\ref{subsec:cancellation}.\nWe independently calibrate the strengths of the soft and\nhard pulses in order to minimize a non-linearity error in setting the\nrf pulse amplitude. \nThe duration $t_{\\rm hp}$ of a hard $ \\pi $-pulse is set to \n$ 21.6\\,\\mu{\\rm s}$ throughout the experiments. \nWe ignore $t_{\\rm hp}$ in setting the phase of the second soft pulse, \nwhich is justified by the fact that \n$t_{\\rm hp} |\\tilde{\\omega}_0| \\ll 2\\pi$. \nThe precision of pulse duration control is 100\\,ns. \nThe durations $t_{i, \\rm sp}$ of two soft pulses are set to\n\\begin{eqnarray*}\nt_{i, \\rm sp} |\\tilde{\\omega}_{i, \\rm rf}|= 2 \\pi. \n\\end{eqnarray*}\nWe demonstrate three different gates with $\\epsilon=0.5,0.3$ and $0.1$. \nWe note that the phase of the second hard pulse \ncorresponding to $R_{y}(-\\pi)$ must be adjusted, \npresumably because the oscillator in the NMR spectrometer is \ndisturbed in generating a SLP. \nIt should be recalled that a SLP employs intensive phase modulation. \n\n\\subsection{Results}\nImplemented gates with $\\Gamma = \\frac{1}{2}$ are evaluated by performing  \nquantum process tomography\\,\\cite{qpt}. \nThe practical details are explained in Ref.\\,\\cite{kondo}. \nA quantum process ${\\mathcal E}$, such as a gate operation or relaxation\nprocess, is \n\\begin{equation*}\n\\rho \\mapsto \n{\\mathcal E}(\\rho) \n= \\frac{\\sum_{k} E_{k} \\rho E_{k}^\\dagger}{\\sum_{k}E^{\\dagger}_{k}E_{k}}\n\\end{equation*}  \nin the operator sum (or Kraus) representation\\,\\cite{Kraus:1983,op_sum_r}. \nWhen all $E_{k}$'s are determined, $\\mathcal{E}$ is considered \nto be identified. \nThis identification is called quantum process tomography. \n\n\\begin{figure}[tbp]\n\\centering\n\\scalebox{0.21}[0.21]{\\includegraphics{fig5a}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(a)\n\\end{flushleft} \n\\scalebox{0.21}[0.21]{\\includegraphics{fig5b}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(b)\n\\end{flushleft} \n\\scalebox{0.21}[0.21]{\\includegraphics{fig5c}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(c)\n\\end{flushleft} \n\\scalebox{0.21}[0.21]{\\includegraphics{fig5d}}\n\\vspace{-7mm}\n\\begin{flushleft}\n(d)\n\\end{flushleft} \n\\caption{\n\\label{fig:results}\nGate operations visualized. The Bloch sphere \nin (a) is mapped to the surfaces in (b), (c), and (d) under the gates with\n $\\epsilon = 0.5, 0.3$ and $0.1$, respectively. \nThe right surface in (a) is an expected Bloch sphere when \n$\\Theta =-\\pi\/4$, which corresponds to the Hadamard gate. \nEach left panel in (b), (c), and (d) corresponds to the theoretical\n final state. \nThe middle panels are the results for the single gate operation \n$U_{\\rm echo}(\\Theta)$. \nThe right panels are for the two-successive (double) gate\n operations. }\n\\end{figure}\n\n\\begin{table}[tbp]\n\\begin{tabular}{c|cc|cccc}\n\\hline \\hline\n  $\\epsilon$ \n& $F_{\\rm e}(I_{0}, {\\mathcal E}\\circ\\mathcal{U}_{\\rm echo}^{-1})$ \n& ${\\rm Tr}[{\\mathcal E}(I_{0})] $  &\n& $F_{\\rm e}(I_{0}, {\\mathcal E}^2)$ \n& ${\\rm Tr}[{\\mathcal E}^2(I_{0})] $ \\\\\n\\hline \\hline\n0.5 & 0.75 & 1.00 && 0.74 &  1.02 \\\\\n0.3 & 0.88 & 1.08 && 0.83 &  1.07 \\\\\n0.1 & 0.84 & 1.07 && 0.85 &  1.06 \\\\\\hline\n\\end{tabular}\n\\caption{\n\\label{tab:entangle}\nThe entanglement fidelities for single and double operations \nwith $\\epsilon = 0.5$, $0.3$, and $0.1$. }\n\\end{table}\nThe Bloch sphere in Fig.\\,\\ref{fig:results}(a) is mapped under the gate\noperations to the surfaces in Figs.~\\ref{fig:results}(b), (c) and\n(d), which correspond to $\\epsilon = 0.5, 0.3$ and $0.1$, respectively. \nIf the gate operations are perfect, the surfaces are the spheres of unit\nradius (i.e., the Bloch sphere). \nThe left panel of each row shows the theoretical final state. \nIn the middle panels, the results for the single gate operation are\nshown. \nFinally, the right panels are for the two-successive (double) gate\noperation. \nThe Hadamard gate obtained when $\\Theta = -\\pi\/4$ is, for comparison, shown \nin the right pannel of Fig.\\,\\ref{fig:results}(a). \nFrom these figures, we find that $U_{\\rm echo}(\\Theta)$ \nin Eq.\\,(\\ref{eq:1q_gate}) is implemented although it is not perfect.  \n\nWe numerically evaluated the fidelity of the implemented gate using the\nentanglement fidelity\\,\\cite{kondo,op_sum_r} given by \n\\begin{eqnarray*}\nF_{\\rm e}(I_{0},\\mathcal{E}\\circ\\mathcal{U}_{\\rm echo}^{-1})\n&=&\n\\frac{\\sum_{k}|{\\rm Tr}[E_{k}U_{\\rm echo}(-\\Theta)I_{0}]|^2}\n{{\\rm Tr}[\\sum_k E_k U_{\\rm echo}\n(-\\Theta)I_0 U_{\\rm echo}^\\dagger(-\\Theta) E_k^\\dagger ]} \\nonumber \\\\\n&=&\n\\frac{\\sum_{k}|{\\rm Tr}[E_{k}U_{\\rm echo}(-\\Theta)I_{0}]|^2}\n{{\\rm Tr}(\\sum_k E_k I_0 E_k^\\dagger )},\n\\end{eqnarray*}\nwhere $\\mathcal{U}_{\\rm echo}$ is a super operator corresponding to the\nunitary operator $U_{\\rm echo}(\\Theta)$ (i.e., \n\\(\n\\mathcal{U}_{\\rm echo}(\\rho) = U_{\\rm echo}(\\Theta)\\rho U^{\\dagger}_{\\rm echo}(\\Theta)\n\\)),  \n$I_{0} = \\openone\/2$, and $\\openone$ is the identity matrix of dimension $2$. \nOne can find that \n\\(\nF_{\\rm e}(I_{0}, {\\mathcal E}\\circ\\mathcal{U}_{\\rm echo}^{-1}) = 1\n\\)\nwhen the gate operation $\\mathcal{E}$ is perfect.  \nIn the case of two successive gate \noperation, $F_{\\rm e}(I_{0}, {\\mathcal E}^2)$ gives a measure of \nthe fidelity since $[U_{\\rm echo}(\\Theta)]^{2} = -\\openone$. \nThe entanglement fidelities corresponding to the gate operations \nare summarized in Table \\ref{tab:entangle}. \n\nThe fidelities of the demonstrated gates are not high. \nThis may be attributed to the inhomogeneous rf field. \nThe free induction decay signal of the thermal state \nafter a $5\\pi\/2$-pulse, which corresponds to the operation \n\\(\ne^{-5\\pi\\sigma_{x}\/4}\n\\) for example, reduces to about $85$\\% of that after \na $\\pi\/2$-pulse, which corresponds to the operation \n\\(\ne^{-\\pi\\sigma_{x}\/4} \n\\) for example. \nThis fact indicates that there is some\nrf field inhomogeneity which may account for most of the reduction in \nthe fidelities in Table \\ref{tab:entangle}. \nPulse sequences in usual NMR operations are designed so that the \nrf field inhomogeneity does not affect measurements, for example,\nby employing composite pulses. Such techniques are not available \nin our experiments. \n\n\\section{Summary}\n\\label{sec:summary}\nWe demonstrated the elimination of the dynamical phase and\nthe implementation of the quantum gates with pure nonadiabatic\ngeometric phases in a liquid-state NMR quantum computer, based on the\ndouble-loop method.  \nBy means of a spin echo technique, we modified the original proposal so\nthat quantum gates are implemented in a standard high precession NMR\nsystem for chemical analysis. \nWe have proposed and experimentally verified an alternative method to eliminate\ndynamical phase. \nThe extension of the present method to two-qubit operations is an\nimportant future work\\,\\cite{comment2}. \nWe believe that our work is the first step toward physical realization of\nworking geometric quantum gates and further efforts should be made for\nimprovement of the gates. \n\n\\begin{acknowledgments} \nThis work was supported by ``Open Research Center'' Project for \n Private Universities: Matching fund subsidy from MEXT (Ministry of \n Education, Culture, Sports, Science and Technology).  \nMN's work is supported in part by Grant-in-Aid for Scientific Research\n (C) from JSPS (Grant No. 19540422).  \n\\end{acknowledgments} \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe complexity of the constraint satisfaction problem (CSP) with a fixed target\nstructure is a well established field of study in combinatorics and computer\nscience (see~\\cite{nesetril} for an overview). In the last\ndecade, we have seen algebraic tools brought to bear on the question of CSP\ncomplexity, yielding major new results (see e.g.~\\cite{BJK},\n\\cite{larossetesson-introduction}, \\cite{libor}). \n\nIn the algebraic approach, it is customary to study relational structures that\ncontain all possible constants. If ${\\mathbb A}$ is such a structure and we are to\ndecide the existence of a homomorphism $f:{\\mathbb B}\\to {\\mathbb A}$ then the constant\nconstraints prescribe values for $f$ at some vertices of ${\\mathbb B}$. We are thus\ndeciding if some partial homomorphism\n$f_c:{\\mathbb B} \\to {\\mathbb A}$ can be extended to the whole ${\\mathbb B}$. Therefore, $\\operatorname{CSP}({\\mathbb A})$\nbecomes the \\emph{homomorphism extension problem} with target structure ${\\mathbb A}$,\ndenoted by $\\operatorname{EXT}({\\mathbb A})$. It is easy to see that $\\operatorname{CSP}({\\mathbb A})$ reduces to $\\operatorname{EXT}({\\mathbb A})$,\nsince in CSP we extend the empty partial homomorphism.\n\nIn \\cite{nesetril-random}, the authors prove that $\\operatorname{CSP}({\\mathbb A})$ is almost surely\nNP-complete for ${\\mathbb A}$ large random relational structure with at least one at\nleast binary relation and without loops. We show by a different method that\nthe same hardness result holds for $\\operatorname{EXT}({\\mathbb A})$ even if we allow loops.\n\n\n\\section{Preliminaries}\nA \\emph{relational structure} ${\\mathbb A}$ is any set $A$ together with a family of\nrelations $\\{R_i:i\\in I\\}$ where $R_i\\subset A^{n_i}$. We call the number\n$n_i$ the \\emph{arity} of $R_i$. The sequence $(n_i:i\\in I)$ determines the\n\\emph{similarity type} of ${\\mathbb A}$. We consider only finite structures (and\nfinitary relations) in this paper. We use the notation $[n]=\\{1,2,\\dots,n\\}$. \n\nLet ${\\mathbb A}=(A,\\{R_i:i\\in I\\})$ and ${\\mathbb B}=(B,\\{S_i:i\\in I\\})$ be two relational structures of the same\nsimilarity type. A mapping $f: A\\to B$ is a homomorphism if for every $i\\in I$\nand every $(a_1,\\dots,a_{n_i})\\in R_i$ we have $(f(a_1),\\dots,f(a_{n_i}))\\in\nS_i$. \n\nLet us fix some $p\\in(0,1)$ and let $A$ be a set. The relation $S\\subset A^l$\nis an \\emph{$l$-ary random relation} on $A$ if every possible $l$-tuple belongs to $S$\nwith probability $p$ (independently of other $l$-tuples). We will call any\nrelational structure with one or more random relations a \\emph{random\nrelational structure}. In particular, a random relational structure with just\none binary relation is a \\emph{random digraph}.\n\nThe \\emph{Constraint Satisfaction Problem} with the target structure ${\\mathbb A}$,\ndenoted by $\\operatorname{CSP}({\\mathbb A})$, consists of deciding whether a given input relational\nstructure ${\\mathbb B}$ of the same similarity type as ${\\mathbb A}$ can be homomorphically\nmapped to ${\\mathbb A}$. It is easy to come up with examples of ${\\mathbb A}$ such that\n$\\operatorname{CSP}({\\mathbb A})$ is NP-complete and this is in a sense typical behavior\nas proved in \\cite{nesetril-random}: If $R(n,k)$ is a $k$-ary random relation on the\nset $[n]$ (with $p=1\/2$) that does not contain any elements of the form $(a,a,\\dots,a)$ for\n$a\\in A$ then\n\\begin{align}\n\\forall k\\geq 2,\\,\\lim_{n\\to\\infty}\\operatorname{Prob}(\\operatorname{CSP}([n],R(n,k))\\text{ is\nNP-complete})=1,\\label{nesetril1}\\\\\n\\forall n\\geq 2, \\lim_{k\\to\\infty}\\operatorname{Prob}(\\operatorname{CSP}([n],R(n,k))\\text{ is\nNP-complete})=1.\\label{nesetril2}\n\\end{align}\n\nThere is a reason why the authors of \\cite{nesetril-random} disallow loops: If\n${\\mathbb A}$ has only one relation $R$ and $R$ contains a loop $(a,a,\\dots,a)$ then\nevery ${\\mathbb B}$ of the same similarity type as ${\\mathbb A}$ can be homomorphically mapped to\n${\\mathbb A}$ simply by sending everything to $a$, so $\\operatorname{CSP}(A)$ is very simple to solve.\n\nGiven a target structure ${\\mathbb A}$, the \\emph{Homomorphism Extension Problem} for\n${\\mathbb A}$, denoted by $\\operatorname{EXT}({\\mathbb A})$, consists of deciding whether a given input structure\n${\\mathbb B}$ and a given partial mapping $f:{\\mathbb B}\\to {\\mathbb A}$ can be extended to a\nhomomorphism from ${\\mathbb B}$ to ${\\mathbb A}$.\n\nLet ${\\mathbb A}$ be a set and $a\\in A$. The \\emph{constant relation} $c_a$ is the unary\nrelation consisting only of $a$, i.e. $c_a=\\{(a)\\}$. When searching for a\nhomomorphism to ${\\mathbb A}$, the relation $c_a$ prescribes a set of elements of $B$\nthat must be mapped to $a$. A little thought gives us that if ${\\mathbb A}$ contains \nconstant relations for each of its elements (as is usual in the algebraic\ntreatment of CSP) then $\\operatorname{CSP}({\\mathbb A})$ and $\\operatorname{EXT}({\\mathbb A})$ are\nessentially the same problem.\n\nSince the homomorphism extension problem is quite important to algebraists, it\nmakes sense to ask what is the typical complexity of $\\operatorname{EXT}({\\mathbb A})$. We will use\nthe phrase ``$\\operatorname{EXT}({\\mathbb A})$ is almost surely NP-compete for $n$ large'' as an\nabbreviation\nfor ``For each $n\\in{\\mathbb N}$, there exists a random relational structure ${\\mathbb A}_n$ (whose precise definition\nis obvious from the context) such that we have \n\\[\n\\lim_{n\\to\\infty}\\operatorname{Prob}(\\operatorname{EXT}({\\mathbb A}_n)\\text{ is NP-complete})=1.\\text{''} \n\\]\n\nBecause additional relations do not make $\\operatorname{CSP}$ easier to solve,\nthe limit (\\ref{nesetril1}) gives us that\nthat $\\operatorname{EXT}({\\mathbb A})$ is almost surely NP-complete if ${\\mathbb A}$ is a\nlarge random relational structure with no loops and at least one relation of arity greater than\none. In the remainder of the paper we show that we can allow loops without\nmaking the problem any easier.\n\n\\section{The problem $\\operatorname{EXT}$ for random digraphs}\nWe will begin by investigating random digraphs and then generalize our findings\nto all relational structures.\n\n\\begin{theorem}\\label{thmNPgraph}\nLet $G$ be a random digraph on $n$ vertices. Then $\\operatorname{EXT}(G)$ is almost surely\nNP-complete for $n$ large.\n\\end{theorem}\n\n\\begin{proof}\n\nLet $G=(V,E)$ be a digraph. Understand $G$ as a relational structure\nand add to $G$ every constant relation possible. Let $v_1,\\dots,v_l\\in V(G)$. Consider the set \n\\[\nF_{v_1,\\dots,v_l}=\\{u\\in V(G): \\forall i, (v_i,u)\\in E(G)\\}\n\\]\nWe will call this set a \\emph{subalgebra} of $G$. \n\nFor an interested reader, we note that sets $F_{v_1,\\dots,v_l}$ are \nsubalgebras in the universal algebraic sense and our technique can be greatly\ngeneralized to all primitive positive definitions (see \\cite{BJK}). For\nour proof, however, we need a lot less: Assume that for some choice of\n$v_1,\\dots, v_l$ the subalgebra $F_{v_1,\\dots,v_l}$ induces a loopless triangle\nin $G$.\nWe claim that we can then reduce graph 3-colorability\nto $\\operatorname{EXT}(G)$, making $\\operatorname{EXT}(G)$ NP-complete. \n\n\nLet $H$ be a graph whose 3-colorability we wish to test. We then understand $H$\nas a symmetric digraph and add to $H$ new vertices $w_1,\\dots,w_l$ and new\nedges $(w_i,u)$ for each $i\\in \\{1,\\dots,n\\}$ and all $u\\in V(H)$, obtaining\nthe digraph $H'$. \nOur $\\operatorname{EXT}(G)$ instance will then\nconsist of the digraph $H'$ along with the partial map $f$ which maps each\n$w_i$ to $v_i$. Now $f$ can be extended to a homomorphism if and only if $H$\ncan be homomorphically mapped into the triangle induced by $F_{v_1,\\dots,v_l}$\nwhich happens if and only if $H$ is 3-colorable.\n\n\n\nAll we need to do now is to show that $G$ almost surely contains a\nsubalgebra that induces a triangle. Our aim, roughly speaking, is to show that $G$ almost surely\ncontains many three element subalgebras because then there is a large chance\nthat at least one of these subalgebras will be a triangle.\n\nWe will partition $V(G)$ into two\nsets $A=\\{1,\\dots,\\lfloor n\/2\\rfloor\\}$ and $B=\\{\\lceil n\/2\n\\rceil,\\dots,n\\}$. \nWe will now use points of $A$ to define subalgebras lying in $B$.\nDenote by $S_k$ the event ``$G$ contains at least $k$ disjoint three-element\nsubalgebras of the form $F_{v_1,\\dots,v_l}\\subset B$ for some $v_1,\\dots,v_l\\in\nA$.''  We\ncan write \n$$S_k=\\bigcup_{\\substack{C_1,\\dots,C_k\\subset B\\\\\n\\forall i\\neq j,\\, C_i\\cap C_j=\\emptyset\\\\\n\\forall i,\\, |C_i|=3}} S_{C_1,\\dots,C_k},\n$$\nwhere $S_{C_1,\\dots,C_k}$ is the event ``The sets\n$C_1,\\dots,C_k$ are subalgebras of $G$''. Finally,\ndenote by $T_{C_1,\\dots,C_k}$ the event ``There exists an $i\\in\\{1,2.\\dots,k\\}$\nsuch that the set $C_i$ induces a triangle subgraph of $G$.''\n\nSince a probability that a fixed $C_i$ induces a triangle is $p^6(1-p^3)$, \nthe probability of the event $T_{C_1,\\dots,C_k}$ is (for $C_1,\\dots,C_k$\npairwise disjoint three element sets) \n$$\n\\operatorname{Prob}(T_{C_1,\\dots,C_k})=1-(1-p^6(1-p^3))^k,\n$$\nwhich tends to 1 when $k$ goes to infinity.\n\nObserve that the event $S_{C_1,\\dots,C_k}$ is independent from the event\n$T_{C_1,\\dots,C_k}$ for each choice of $C_1,\\dots,C_k\\subset B$ \nsince both events talk about disjoint sets of edges of $G$. \n\nAssume for a moment that for all $k\\in {\\mathbb N}$ the value of $\\operatorname{Prob}(S_k)$ tends to 1\nas $n$ tends to infinity. Then, given an $\\epsilon>0$, we choose $k$ so that\n$\\operatorname{Prob}(T_{C_1,\\dots,C_k})\\geq 1-\\epsilon$. When $n$ is large enough, \nthe digraph $G$ contains some $k$ pairwise disjoint\nthree element subalgebras $C_1,\\dots,C_k$\nwith probability at least $1-\\epsilon$. The probability that one of the sets\n$C_1,\\dots,C_k$ them induces a triangle is $T_{C_1,\\dots,C_k}\\geq\n1-\\epsilon$. Thus we get an NP-complete CSP problem with probability at least $(1-\\epsilon)^2>\n1-2\\epsilon$ and since $\\epsilon$ was arbitrary, we see that for large $n$ the\nhomomorphism extension problem is almost surely NP-complete.\n\nIt remains to show $\\lim_{n\\to\\infty}\\operatorname{Prob}(S_k)=1$ for all $k$. Fix the value\nof $k$. For each value of $n$, let $l$ be the integer satisfying $n p^l\\geq 1>n p^{l+1}$. We will now search\nfor the three element subalgebras of $B$ for $n$ large. We proceed in steps: Assume that after $i$\nsteps we have already found $m$ such subalgebras $C_1,\\dots,C_m$. In the $(i+1)$-th step, \nwe take the vertices $1+il,2+il,\\dots,l+il$\nof $A$ and consider the subalgebra $F_{1+il,2+il,\\dots,l+il}$. If this\nsubalgebra lies in $B$, has size three and is disjoint with all the sets $C_1,\\dots,C_m$, we\nlet $C_{m+1}=F_{1+il,2+il,\\dots,l+il}$, increase $m$ by one and continue with\nthe next step. Otherwise, $F_{1+il,2+il,\\dots,l+il}$ is not a good candidate for\n$C_{m+1}$, so we leave $m$ unchanged and continue with the next step.\n\nWhat is the probability that we find the $(m+1)$-th\nsubalgebra in a given step? Every vertex of $G$ is in $F_{1+il,2+il,\\dots,l+il}$ with the\nprobability $p^l$. The probability that $F_{1+il,2+il,\\dots,l+il}$ consists of\nthree yet-unused vertices of $B$ is then equal to\n\\[\nq={|B|-3\\cdot m \\choose 3}p^{3l}(1-p^l)^{n-3}\\geq\n\\frac{(n\/2-3m-3)^3}{6}p^{3l}(1-p^l)^n\n\\]\nIf $m\\geq k$, we have already won, so assume $m<k$:\n\\[\nq\\geq \\frac{(n\/2-3k)^3}{6}p^{3l}(1-p^l)^n=\\frac{(1\/2-3k\/n)^3}{6}\nn^3p^{3l}(1-p^l)^n\n\\]\nNow let $r=\\frac{(1\/2-3k\/n)^3}6$ and observe that $r>0$ for $n$ large\nenough. Using the the inequalities $n p^l\\geq 1>n p^{l+1}$ we have: \n\\[\nq \\geq r n^3 p^{3l}(1-p^l)^n \\geq r (1-p^l)^n> r \\left(1-\\frac{1}{pn}\\right)^n.\n\\]\nThe lower\nbound on $q$ tends to $r\/e^{1\/p}$ as $n$ tends to infinity, so there exists a $\\delta$ such that $q>\\delta>0$\nfor all $n$ large enough.\n\nTherefore, the probability of producing a new three-element subalgebra in a\ngiven step is at\nleast $\\delta>0$ and this bound does not depend on the number of subalgebras\nwe have already found. Now observe that $l$\nis approximately $\\log_{1\/p} n$ and therefore we have enough vertices in $A$ for\napproximately $s=\\frac{n}{2\\log_{1\/p}n}$ steps. If we choose $n$ large enough, we can have $s$ as\nlarge as we want and so the probability of finding at least $k$ subalgebras\ncan be arbitrarily close to 1. Therefore, $\\lim_{n\\to\\infty} \\operatorname{Prob}(S_k)=1$, concluding the\nproof.\n\\end{proof}\n\n\\section{Random relational structures}\nIt is easy to see that if ${\\mathbb A}$ is a relational structure with unary\nrelations only then $\\operatorname{EXT}({\\mathbb A})$ is always polynomial. We would now like to\ninvestigate the case of relations of arity greater than two. Intuition tells us\nthat greater arity means greater complexity. The intuition is right.\n\n\\begin{lemma}\nLet $l\\geq 2$, $n$ be large and let ${\\mathbb A}=([n],S)$ be a relational structure with\n$S$ a random $l$-ary relation. Then\nthe homomorphism extension problem $\\operatorname{CSP}({\\mathbb A})$ is almost surely NP-complete.\n\\end{lemma}\n\\begin{proof}\nWe have proven the result for $l=2$. If $l>2$, consider the binary relational structure ${\\mathbb B}=([n],R)$ where \n$R=\\{(x,y)\\in [n]^2: (x,y,1,1,\\dots,1)\\in S\\}$. It is easy to see that if $S$ is\na random $l$-ary relation then ${\\mathbb B}$ is a random \ndigraph where each edge exists with the probability $p$. From\nTheorem~\\ref{thmNPgraph} we see that\n$\\operatorname{EXT}({\\mathbb B})$ is almost surely NP-complete. We will now show how to reduce\n$\\operatorname{EXT}({\\mathbb B})$ to $\\operatorname{EXT}({\\mathbb A})$ in polynomial time, proving that $\\operatorname{EXT}({\\mathbb A})$\nis almost surely NP-complete.\n\nUsing algebraic tools, the reduction of $\\operatorname{EXT}({\\mathbb B})$ to $\\operatorname{EXT}({\\mathbb A})$ follows from the fact that\n$R$ is defined by a primitive positive formula that uses only $S$ and the constant $1$.\nHowever, we will provide an elementary reduction here: Let ${\\mathbb C}=(C,T)$ be a\nrelational structure with a single binary relation $T$ and let $f:C\\to [n]$ be a \npartial mapping. We add to $C$ a new element $e$, construct the\n$l$-ary relation $U=\\{(x,y,e,e,\\dots,e):(x,y)\\in T\\}$ and the partial mapping\n$g:C\\cup \\{e\\}\\to [n]$ so that $g_{|C}=f$ and $g(e)=1$. A little thought gives\nus that $g$ can be extended to a homomorphism $(C\\cup\\{e\\},U)\\to {\\mathbb A}$ if and\nonly if $f$ can be extended to a homomorphism $(C,T)\\to {\\mathbb B}$, concluding the\nproof.\n\\end{proof}\n\nAdditional relations in ${\\mathbb A}$ do not make $\\operatorname{EXT}({\\mathbb A})$ easier, so we have the most\ngeneral version of our NP-completeness result:\n\n\\begin{corollary}\nLet ${\\mathbb A}$ be the relational structure $([n],\\{R_i:i\\in I\\})$ where at least one\n$R_i$ is a random\nrelation of arity greater than one. Then $\\operatorname{EXT}({\\mathbb A})$ is \nalmost surely NP-complete for $n$ large.\n\\end{corollary}\n\nAs a final note, we will now prove the analogue of the limit~(\\ref{nesetril2})\nfor $\\operatorname{EXT}$.\n\n\\begin{corollary}\nLet us fix a set $A$ of at least two elements and let ${\\mathbb A}=(A,R)$ be \na relational structure with $R$\nrandom $k$-ary relation. Then $\\operatorname{EXT}({\\mathbb A})$ is almost surely NP-complete for $k$\nlarge.\n\\end{corollary}\n\\begin{proof}\nAssume first that $k$ is even and let $m=k\/2$.  \n\nConsider the relational structure ${\\mathbb B}=(A^m,S)$ with \n$$\nS=\\{((a_1,\\dots,a_m),(a_{m+1},\\dots,a_{2m})):(a_1,\\dots,a_{2m})\\in R\\}.\n$$\nIt is straightforward to prove that $S$ is a binary random relation on $A^m$ and therefore $\\operatorname{EXT}({\\mathbb B})$\nis almost surely NP-complete for large even $k$. What is more, $\\operatorname{EXT}({\\mathbb B})$ can\nbe easily reduced to $\\operatorname{EXT}({\\mathbb A})$: If ${\\mathbb C}=(C,T)$ is a relational\nstructure with $T$ binary and $f:C\\to A^m$ is a partial mapping, \nwe construct the structure ${\\mathbb C}'=(C',T')$ with \n\\begin{align*}\nC'&=\\{(c,i):c\\in C, i\\in \\{1,\\dots,m\\}\\},\\\\\nT'&=\\{((c,1),\\dots,(c,m),(d,1),\\dots,(d,m)):(c,d)\\in T\\}\n\\end{align*}\nand a partial mapping $g:C'\\to A$ such that $g(c,i)=a_i$ whenever $f(c)$ is defined and equal to\n$(a_1,\\dots,a_m)$.\n\nIt is easy to see that $g$ can be extended to a homomorphism from ${\\mathbb C}'$ to ${\\mathbb A}$\nif and only if $f$ can be extended to a homomorphism from ${\\mathbb C}$ to ${\\mathbb A}$. \n\nIn the case that $k=2m+1$, we fix an $e\\in A$, choose\n${\\mathbb B}=(A^m,S)$ with\n$$\nS=\\{((a_1,\\dots,a_m),(a_{m+1},\\dots,a_{2m}):(a_{1},\\dots,a_{2m},e)\\in R\\}\n$$\nand proceed similarly to the previous case. \n\nWe see that for a large enough $k$, no matter if it is odd or even, the problem\n$\\operatorname{EXT}({\\mathbb B})$ is almost surely NP-complete.\n\\end{proof}\n\n\\section{Conclusions}\nWe have shown that the homomorphism extension problem is almost surely\nNP-complete for large relational structures (assuming we have at least one\nnon-unary relation). In a sense, our result is not surprising since the relational\nstructures we consider are very dense, so it stands to a reason that we can \nfind hard instances most of the time.\n\nIt might therefore be interesting to see what is the complexity of CSP or EXT\nfor large structures obtained by other random processes, particularly when\nrelations are sparse. Such structures might better correspond to ``typical''\ncases of CSP or EXT encountered in practice. Some such results already exist;\nsee \\cite{nesetril-luczak-projective} for a criterion on the random graph\nprocess to almost surely produce projective graphs (if $G$ is\nprojective then $\\operatorname{EXT}(G)$ is NP-complete, see \\cite{BJK}). Our guess is that\nboth CSP and EXT will remain to be almost surely NP-complete in all the\nnontrivial cases.\n\n\n\\section{Acknowledgments}\nThe research was supported by the GA\\v CR project GA\\v CR 201\/09\/H012, by the Charles\nUniversity project GA UK 67410 and by the grant SVV-2011-263317. The \nauthor would like to thank Libor Barto for suggesting this problem.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the very beginning of the creation of quantum mechanics, the question of its formulation\nin terms of the distribution function on the phase space, like the classical kinetic theory, \nhas attracted the attention of many scientists, despite the fact that the Heisenberg uncertainty relation \nprohibits the existence  of the joint distribution function of the position and momentum in the quantum case.\n\nA great success in this connection was the introduction of the Wigner function \\cite{Wigner32}\nand the writing of the dynamic equation for it \\cite{Wigner32,Moyal1949}.\nFor the $N-$dimensional system the Wigner function is introduced as the Weyl symbol \\cite{Weyl1927} \nof the  density matrix $\\hat\\rho(t)$ of the state\n\\be\t\t\t\\label{WigDef}\nW(\\mathbf{q},\\mathbf{P},t)=\\frac{1}{(2\\pi\\hbar)^N}\n\\int\\left<\\left.\\mathbf{q}-\\frac{\\mathbf u}{2}\\right|\\hat\\rho(t)\\left|\\mathbf{q}+\\frac{\\mathbf u}{2}\\right.\\right>\n\\exp\\left(\\frac{\\rmi}{\\hbar}\\mathbf{u}\\mathbf{P}\\right)\\rmd ^Nu,\n\\ee\nwhere $\\mbf P$ is the generalized momentum \ncorresponding to the generalized momentum operator $\\hat{\\mbf P}=-\\rmi\\hbar\\partial\/\\partial{\\mbf q}$.\nThe transformation inverse to (\\ref{WigDef}) has the form:\n\\be\t\t\t\\label{RhoFromW}\n\\left<\\mbf q|\\hat\\rho(t)|\\mbf q'\\right>=\\int W\\left(\\frac{\\mbf q+\\mbf q'}{2},\\mbf P,t\\right)\n\\exp\\left(\\rmi\\mbf P\\frac{\\mbf q-\\mbf q'}{\\hbar}\\right)d^NP.\n\\ee\nDespite the fact that the Wigner function can take negative values, it is successfully applied\nin many applications since the 1950s (see, e.g., \\cite{Silin1, Silin2, Silin3, Silin4, Landau}) and to the present.\nThe properties of the Wigner function were considered, e.g., in \\cite{OConnellWignerPhysRep}.\n\nAt the same time, if we smoothly average the Wigner function on the scales of the hyper-volume\n$\\hbar^N$ of phase space, then it can be made nonnegative.\nThis way in the article \\cite{Husimi40} and later in \\cite{Kano1965} the Husimi function \n(the so called $Q-$function) was introduced  \n\\be\t\t\t\\label{QDef}\nQ(\\mathbf{q},\\mathbf{P},t)=\\frac{1}{(2\\pi\\hbar)^N}\\left<\\bm\\alpha|\\hat\\rho(t)|\\bm\\alpha\\right>,\n\\ee\nrepresenting up to the normalization factor the transition probability of a quantum system \nfrom the state $\\hat\\rho(t)$ into a coherent state\n$|\\bm\\alpha\\rangle$ \\cite{GlauberPhysRevLett1963,GlauberPhysRev1963}, where \n$\\bm\\alpha=(2\\hbar)^{-1\/2}(\\lambda^{1\/2}\\mbf q+\\rmi\\lambda^{-1\/2}\\mbf P)$, \n$\\lambda=m\\omega$. Up to the inessential phase factor the state $|\\bm\\alpha\\rangle$\nis defined in the position representation as follows:\n\\be\t\t\t\\label{CohStateDef}\n\\langle\\mbf q'|\\bm\\alpha\\rangle=\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{N\/4}\n\\exp\\left[-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q')^2-\\frac{\\rmi}{\\hbar}\\mbf P(\\mbf q-\\mbf q')\\right].\n\\ee\nAs is known, the state $|\\bm\\alpha\\rangle$ minimizes the Heisenberg uncertainty relation, \nand the remarkable property of the $Q-$function is its connection with the Wigner function \nby means of the formula: \n\\be\t\t\t\\label{WignerToHusimi}\nQ(\\mathbf{q},\\mathbf{P},t)=\\frac{1}{(\\pi\\hbar)^N}\\int\\exp\\left(-\\frac{\\lambda}{\\hbar}(\\mbf q-\\mbf q')^2\n-\\frac{1}{\\lambda\\hbar}(\\mbf P-\\mbf P')^2\\right)W(\\mbf q',\\mbf P',t)d^Nq'd^NP',\n\\ee\nrepresenting the averaging of the function $W(\\mbf q,\\mbf P,t)$ \nin the phase space with respect to a Gaussian distribution centered at the point $(\\mbf q,\\mbf P)$.\nFormula (\\ref{WignerToHusimi}) can be rewritten in the operator form,\n\\be\t\t\t\\label{WigToHusOperator}\nQ(\\mathbf{q},\\mathbf{P},t)=\\exp\\left(\\frac{\\hbar}{4\\lambda}\\partial_{\\mbf q}^2\n+\\frac{\\lambda\\hbar}{4}\\partial_{\\mbf P}^2\\right)W(\\mbf q,\\mbf P,t),\n\\ee\nand the inverse transform of (\\ref{WigToHusOperator}) is obviously has the shape:\n\\be\t\t\t\\label{HusToWigOperator}\nW(\\mathbf{q},\\mathbf{P},t)=\\exp\\left(-\\frac{\\hbar}{4\\lambda}\\partial_{\\mbf q}^2\n-\\frac{\\lambda\\hbar}{4}\\partial_{\\mbf P}^2\\right)Q(\\mbf q,\\mbf P,t).\n\\ee\nAlso the inverse transform of (\\ref{WignerToHusimi}) can be expressed\nas a repeated integral,\n\\be\t\t\t\\label{HusimiToWigner}\nW(\\mathbf{q},\\mathbf{P},t)=\\int\\frac{d^Nud^Nv}{(2\\pi)^{2N}}\\int\\exp\\left[\\frac{\\hbar\\mbf u^2}{4\\lambda}\n+\\rmi\\mbf u(\\mbf q-\\mbf q')+\\frac{\\lambda\\hbar\\mbf v^2}{4} +\\rmi\\mbf v(\\mbf P-\\mbf P')\\right]Q(\\mbf q',\\mbf P',t)d^Nq'd^NP'.\n\\ee \n\nExpansion of  formula (\\ref{WignerToHusimi}) to the classical (non-quantum) case\nallows to determine the Husimi function $Q_{\\mathrm{cl}}(\\mbf q,\\mbf P,t)$ of the state \nof the classical system having described by the classical distribution function\n$W_{\\mathrm{cl}}(\\mbf q,\\mbf P,t)$ as the overlap with a Gaussian distribution.\n\nConsider the motion of a quantum particle having a spin in the \nelectromagnetic field with the vector potential $\\mathbf{A}(\\mathbf{q},t)$\nand the scalar potential $\\varphi(\\mathbf{q},t)$. \nAs it is known, the Hamiltonian of such a system has the form \\cite{LandauIII}\n\\be\t\t\t\\label{Hamiltonian}\n\\hat H=\\frac{1}{2m}\\left(\\hat{\\mathbf P}-\\frac{e}{c}\\mathbf{A}\\right)^2\n+e\\varphi- \\hat{\\bm\\kappa}  \\mathbf{B},\n\\ee\nwhere $\\hat{\\mathbf P}=-i\\hbar\\partial\/\\partial\\mathbf{q}$ is a generalized momentum operator,\n$m$ and $e$ are mass and charge of the particle, $\\mathbf{B}=\\mathrm{rot}\\mathbf{A}$ \nis a magnetic field strength, $\\hat{\\bm\\kappa}$ is an operator \nof quantum-mechanical  magnetic moment\n\\be\t\t\t\\label{Moment}\n\\hat{\\bm\\kappa}=\\frac{\\kappa}{s}\\hat{\\mathbf{s}},\n\\ee\nwhere $s$ is a spin of the particle, $\\hat{\\mathbf{s}}$ is a spin operator,\nand $\\kappa$ is the value of the intrinsic magnetic moment of the particle.\n\nFrom the classical electrodynamics it is known that \npotentials of the field are defined only up to the gauge transformation\n\\cite{LandauII}\n\\be\t\t\t\\label{eq3}\n\\mathbf{A} ~~\\rightarrow ~~\\mathbf{A} +\\nabla \\chi,\n~~~~\n\\varphi~~\\rightarrow ~~\\varphi-\\frac{1}{c}\\frac{\\partial\\chi}{\\partial t},\n\\ee\nwhere $\\chi$ is an arbitrary function of spatial coordinates and time.\n\nSince the electric field intensity $\\mathbf{E}$ and the magnetic field strength\n$\\mathbf{B}$ are defined in terms of the potentials as:\n\\be\t\t\t\\label{eq4}\n\\mathbf{E}=-\\mathrm{grad}\\varphi-\\frac{1}{c}\\frac{\\partial}{\\partial t}\\mathbf{A},\n~~~~\n\\mathbf{B}=\\mathrm{rot}\\mathbf{A},\n\\ee\nthen the gauge transformation (\\ref{eq3}) does not affect the values \nof $\\mbf E$ and $\\mbf B$.\nTherefore the part of Hamiltonian (\\ref{Hamiltonian}) responsible for\nthe interaction of the spin with the magnetic field is independent \non the gauge transformation, and  we can restrict our considerations only to the case\nof $s=0$. The generalization to the non-zero spin particles is straightforward \n(see \\cite{Korarticle9,Korarticle14,Korarticle12}).\n\n\n\nThe requirement of invariance of the Schr\\\"odinger equation \nunder the gauge transformation simultaneously with the gauge-independence\nof ``probability density''\\, $|\\Psi|^2$ leads us to the form of the conversion\nof the wave function \\cite{LandauIII}:\n\\be\t\t\t\\label{eq5}\n\\Psi ~~\\rightarrow ~~\\exp\\left(\\frac{\\rmi e}{c\\hbar}\\chi \\right)\\Psi.\n\\ee\nAccordingly, the conversions of the density matrix of the state of the \nsystem under the gauge transformation acquires the form:\n\\be\t\t\t\\label{eq6}\n\\hat\\rho_{\\mathrm c}=\n\\exp\\left(\\frac{\\rmi e}{c\\hbar}\\chi \\right)\\hat\\rho\\,\\exp\\left(-\\frac{\\rmi e}{c\\hbar}\\chi \\right).\n\\ee\n\nIn \\cite{Stratonovich2} the gauge-independent Wigner function was constructed, and its \nevolution equation was derived in \\cite{Serimaa1986}. The gauge-invariance in the \ntomographic probability representation of quantum mechanics was considered in \\cite{Korarticle12}\n(see review articles \\cite{IbortPhysScr,MankoMankoFoundPhys2009} about the \ntomographic probability representation).\n\nThe aim of this work is introduction of\ngauge-independent Husimi function ($Q-$function) of states of charged quantum \nparticles in the electro-magnetic field, and  is derivation of the evolution equation for such function.\n\n\n\\section{\\label{Art08Section2}Gauge-independent Husimi function}\nFor the construction of quantum Husimi representations, \nin which the evolution equation would be gauge-independent, \nwe need to introduce gauge-independent quantum Husimi functions.\nThis can be done with the help of a gauge-independent Wigner function\nobtained in \\cite{Stratonovich2},\n\\be\nW_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t) \n=\\frac{1}{(2\\pi\\hbar)^{3}}\\int\n\\exp\\left(\\frac{\\rmi }{\\hbar}\\mathbf{u}\\left\\{\\mathbf{p}\n+\\frac{e}{c}\\int_{-1\/2}^{1\/2} \\rmd\\tau \\mathbf{A}(\\mathbf{q}+\\tau\\mathbf{u},\\,t)\\right\\}\\right)\n\\rho\\left(\\mathbf{q}-\\frac{\\mathbf u}{2},\\,\\mathbf{q}+\\frac{\\mathbf u}{2},\\,t\\right)\n\\rmd^3u,\n\t\t\t\\label{WigNew}\n\\ee\nwhere $\\mathbf{p}$ is a kinetic  momentum.\n\nThe gauge-independent Husimi function $ Q_\\mathrm{g}(\\mbf q,\\mbf p,t)$ should be introduced\nusing a formula similar to (\\ref{WignerToHusimi})\n\\be\t\t\t\\label{WignerGToHusimiG}\nQ_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=\\frac{1}{(\\pi\\hbar)^N}\\int\\exp\\left(-\\frac{\\lambda}{\\hbar}(\\mbf q-\\mbf q')^2\n-\\frac{1}{\\lambda\\hbar}(\\mbf p-\\mbf p')^2\\right)W_\\mathrm{g}(\\mbf q',\\mbf p',t)d^Nq'd^Np'.\n\\ee\nWith this definition the formulas (\\ref{WigToHusOperator},\\,\\ref{HusToWigOperator},\\,\\ref{HusimiToWigner})\nof the relations between $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ and $W_\\mathrm{g}(\\mbf q',\\mbf p',t)$ \nremain valid if we replace the generalized momentum $\\mbf P$ by the kinetic momentum $\\mbf p$ in them.\n\nCombining formulas (\\ref{WigNew}) and (\\ref{WignerGToHusimiG}) we can write\n\\be\t\t\t\\label{QgFromRho}\nQ_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=\\int \\langle\\mbf q_1|\\hat\\rho(t)|\\mbf q_2\\rangle \n\\langle\\mbf q_2|\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle d^3q_1d^3q_2,\n\\ee\nwhere we introduce the matrix element for the corresponding dequantizer operator\n\\bea\n\\langle\\mbf q_2|\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle&=&\n\\frac{1}{(2\\pi\\hbar)^3}\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{3\/2}\n\\exp\\Bigg\\{-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_2)^2-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_1)^2\n\\nonumber \\\\[3mm]\n&+&\\frac{\\rmi}{\\hbar}(\\mbf q_2-\\mbf q_1)\\left[\\mbf p\n+\\frac{e}{c}\\int_{-1\/2}^{1\/2}\\mbf A\\left(\\frac{\\mbf q_2+\\mbf q_1}{2}+\\tau(\\mbf q_2-\\mbf q_1),t\\right)d\\tau\\right]\\Bigg\\}.\n\t\t\t\\label{matrUQg}\n\\eea\nFrom (\\ref{matrUQg}) we can see that $\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ is Hermitian  operator, \nso the Husimi function $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ is real.\n\nThe explicit form of the operator $\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ can be written in the integral form\n\\be\t\t\t\\label{UQgint}\n\\hat U_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)=\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^6}\n\\exp\\left[-\\frac{\\lambda\\mbf u^2}{4\\hbar}-\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf u\\mbf p+\\mbf v\\mbf q)\\right]\n\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\left(\\hat{\\mbf P}-\\frac{e}{c}\\mbf A(\\hat{\\mbf q},t)\\right)\n+\\mbf v\\hat{\\mbf q}\\right]\\right\\}.\n\\ee \n\nThe transformation inverse to (\\ref{QgFromRho}) can be expressed using the matrix element of \nquantizer operator $\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\n\\be\t\t\t\\label{RhoFromQg}\n\\langle\\mbf q_1|\\hat\\rho(t)|\\mbf q_2\\rangle\n=\\int \\langle\\mbf q_1|\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_2\\rangle\nQ_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t) d^3qd^3p,\n\\ee\nwhere\n\\bea\n\\langle\\mbf q_1|\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_2\\rangle&=&\n\\int\\frac{d^3v}{(2\\pi\\hbar)^3}\\exp\\Bigg[\\frac{\\lambda(\\mbf q_2-\\mbf q_1)^2}{4\\hbar}\n+\\frac{\\mbf v^2}{4\\hbar\\lambda}-\\frac{\\rmi\\mbf v}{2\\hbar}(2\\mbf q-\\mbf q_1-\\mbf q_2)\n-\\frac{\\rmi\\mbf p}{\\hbar}(\\mbf q_2-\\mbf q_1) \\nonumber \\\\[3mm]\n&-&\\frac{\\rmi e}{\\hbar c}(\\mbf q_2-\\mbf q_1)\\int_{-1\/2}^{1\/2}d\\tau\n\\mbf A\\left(\\frac{\\mbf q_2+\\mbf q_1}{2}+\\tau(\\mbf q_2-\\mbf q_1),t\\right)\n\\Bigg].\n\t\t\t\\label{matrDg}\n\\eea\nIn these formulas it is assumed that $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)\\in S(\\mathbb{R}^6)$,\nwhere $S(\\mathbb{R}^6)$ is a Schwartz space, and first we take the integral over $d^3q$,\nand after that the integrals over $d^3p$ and $d^3v$ are taken.\n\nAssuming a special order of integration, the explicit form of the operator \n$\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ can be written as:\n\\be\t\t\t\\label{DQgint}\n\\hat D_{Q_\\mathrm{g}}(\\mbf q,\\mbf p)=\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^3}\n\\exp\\left[\\frac{\\lambda \\mbf u^2}{4\\hbar}+\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf{up}+\\mbf{vq})\\right]\n\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\left(\\hat{\\mbf P}-\\frac{e}{c}\\mbf A(\\hat{\\mbf q},t)\\right)\n+\\mbf v\\hat{\\mbf q}\\right]\\right\\}.\n\\ee\n\n\n\n\\section{\\label{Art08Section3}Evolution equation for the gauge-independent Husimi function}\nTo begin with, let us recall the Liouville equation in the electro-magnetic field \nfor the classical distribution function.\nFor the classical ensemble of non-interacting particles with \nmass $m$ and charge $e$ this equation in the phase space has the form:\n\\be\n\\left\\{\\partial_ t+\n\\frac{\\mathbf{p}}{m}\\partial_{\\mathbf{q}}\n+e\\left(\\mathbf{E}(\\mathbf{q},t)\n+\\frac{1}{mc}[\\mathbf{p}\\times\\mathbf{B}(\\mathbf{q},t)]\\right)\n\\partial_{\\mathbf{p}}\\right\\}W_\\mathrm{cl}(\\mathbf{q},\\mathbf{p},t)=0,\n\t\t\t\\label{Liouville}\n\\ee\nwhere $\\mathbf{p}$ is a kinetic momentum, $\\mathbf{E}(\\mathbf{q},t)$ and \n$\\mathbf{B}(\\mathbf{q},t)$ are electric and magnetic fields, \ndefined by formulas (\\ref{eq4}), $W_\\mathrm{cl}(\\mathbf{q},\\mathbf{p},t)$\nis a distribution function of non-interacting particles.\n\nThe distribution function $W_\\mathrm{cl}(\\mathbf{q},\\mathbf{p},t)$ is independent on \nthe gauge transformation \\cite{LandauII} \nbecause the Liouville equation (\\ref{Liouville}) includes \nonly gauge-independent intensities of the electro-magnetic field.\n\n\nGauge-independent Moyal equation for the Wigner function $W_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ has the form \\cite{Serimaa1986}:\n\\be\t\t\t\\label{EqWigNew}\n\\left\\{\\partial_t+\\frac{1}{m}\\left(\\mathbf{p}+\\triangle\\tilde{\\mathbf p}\\right)\\partial_{\\mathbf{q}}\n+ e\\left(\\tilde{\\mathbf{E}} +\\frac{1}{mc}\n\\left[\\left(\\mathbf{p}+\\triangle\\tilde{\\mathbf p}\\right)\\times\\tilde{\\mathbf{B}} \\right]\\right)\n\\partial_{\\mathbf p}\n\\right\\}W_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=0,\n\\ee\nwhere\n\\bdm\n\\triangle\\tilde{\\mathbf p}=-\\frac{e}{c}\\frac{\\hbar}{\\rmi }\n\\left[\n\\frac{\\partial}{\\partial\\mathbf{p}} \\times \\int_{-1\/2}^{1\/2}\n\\rmd\\tau\\,\\tau \n\\mathbf{B}\n\\left(\n\\mathbf{q}+\\rmi\\hbar\\tau\\frac{\\partial}{\\partial\\mathbf{p}},\\,t\n\\right)\n\\right],\n\\edm\n\\bdm\n\\tilde{\\mathbf E}=\\int_{-1\/2}^{1\/2}\n\\rmd\\tau\\,\n\\mathbf{E}\n\\left(\n\\mathbf{q}+\\rmi \\hbar\\tau\\frac{\\partial}{\\partial\\mathbf{p}},\\,t\n\\right),\n~~~~\n\\tilde{\\mathbf B}=\\int_{-1\/2}^{1\/2}\n\\rmd\\tau\\,\n\\mathbf{B}\n\\left(\n\\mathbf{q}+\\rmi \\hbar\\tau\\frac{\\partial}{\\partial\\mathbf{p}},\\,t\n\\right).\n\\edm\nThis equation in the classical limit $\\hbar\\to 0$ is converted into \nLiouville equation (\\ref{Liouville}).\n\nSince the relations between the functions $W_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ and\n$Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$ are the same as between $W(\\mathbf{q},\\mathbf{p},t)$ and\n$Q(\\mathbf{q},\\mathbf{p},t)$, then the correspondence rules between\noperators acting on the Wigner function and the Husimi function do not change.\nConsequently, we can write:\n\\be\t\t\\label{CorrespRulesWgQg}\n\\begin{array} {lcl} \n\\mbf q\\, W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_\\mbf{q}\\right)Q(\\mathbf{q},\\mathbf{p},t),\n\\\\\n\\mbf p\\, W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\left(\\mbf p+\\frac{\\hbar\\lambda}{2}\\partial_\\mbf{p}\\right)Q(\\mathbf{q},\\mathbf{p},t),\n\\\\\n\\partial_{\\mbf{q}} W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\partial_{\\mbf{q}}Q(\\mathbf{q},\\mathbf{p},t),\n\\\\\n\\partial_{\\mbf{p}} W_\\mathrm{g}(\\mathbf{q},\\mathbf{p}) &\\leftrightarrow &\n\\partial_{\\mbf{p}}Q(\\mathbf{q},\\mathbf{p},t).\n\\end{array}\n\\ee\nWith the help of (\\ref{CorrespRulesWgQg}) equation (\\ref{EqWigNew}) is transformed to\nthe evolution equation for the  gauge-independent Husimi function $Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)$\n\\bea\n&&\\bigg\\{\\partial_t+\\frac{1}{m}\\left(\\mbf p+\\frac{\\hbar\\lambda}{2}\\partial_{\\mbf p}\n+\\big[\\triangle\\tilde{\\mbf{p}}\\big]_Q\\right)\\partial_{\\mbf q} \\nonumber \\\\[3mm]\n&&~~~~~~+e\\left(\\big[\\tilde{\\mbf E}\\big]_Q\n+\\frac{1}{mc}\\left[\\left(\\mbf p+\\frac{\\hbar\\lambda}{2}\\partial_{\\mbf p}\n+\\big[\\triangle\\tilde{\\mbf{p}}\\big]_Q\\right)\\times\\big[\\tilde{\\mbf B}\\big]_Q\\right]\\right)\\partial_{\\mbf p}\n\\bigg\\}Q_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)=0,\n\t\t\t\\label{EqforQg}\n\\eea\nwhere\n\\bea\n&&\\big[\\triangle\\tilde{\\mbf{p}}\\big]_Q=-\\frac{e}{c}\\frac{\\hbar}{\\rmi}\\left[\\partial_{\\mbf p}\\times\n\\int_{-1\/2}^{1\/2}\\mbf B\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_{\\mbf{q}}\n+\\rmi\\hbar\\tau\\partial_{\\mbf{p}},t\\right)\\tau d\\tau\\right],\n\\nonumber \\\\[3mm]\n&&\\big[\\tilde{\\mbf E}\\big]_Q=\\int_{-1\/2}^{1\/2}\\mbf E\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_{\\mbf{q}}\n+\\rmi\\hbar\\tau\\partial_{\\mbf{p}},t\\right)d\\tau,\n\\nonumber \\\\[3mm]\n&&\\big[\\tilde{\\mbf B}\\big]_Q=\\int_{-1\/2}^{1\/2}\\mbf B\\left(\\mbf q+\\frac{\\hbar}{2\\lambda}\\partial_{\\mbf{q}}\n+\\rmi\\hbar\\tau\\partial_{\\mbf{p}},t\\right)d\\tau.\n\\eea\nAs it should be, equation (\\ref{EqforQg}) in the classical limit $\\hbar\\to0$ is converted \ninto the Liouville equation (\\ref{Liouville}).\n\n\\section{\\label{Art08Section4}Non-Stratonovich type of gauge-independent \\\\\nWigner and Husimi functions}\n\nIn the previous sections we considered the Wigner and Husimi functions  on the basis of the definition \nof Stratonovich \\cite{Stratonovich2}.\nHowever, it is possible to introduce a gauge-independent Wigner function and its corresponding Husimi function\naccording to (\\ref{WignerToHusimi}, \\ref{WigToHusOperator}) by other ways.\n\nLet us define the gauge-invariant dequantizer for the new Wigner function \n$\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$ as follows:\n\\bea\n\\hat U_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)&=&\\exp\\left[\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]\n\\nonumber \\\\[3mm]\n&\\times&\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^6}\n\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\left(\\hat{\\mbf P}-\\mbf p\\right)+\\mbf v(\\hat{\\mbf q}-\\mbf q)\\right]\\right\\}\n\\exp\\left[-\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]. \n\t\t\t\\label{newDequantW}\n\\eea\nSince for the Wigner function the dequantizer-quantizer scheme is self-dual, \nthen the following equality takes place for the corresponding quantizer:\n$\\hat D_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)=\n(2\\pi\\hbar)^3\\hat U_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$.\nCalculation of the matrix element of (\\ref{newDequantW}) yields\n\\bea\n\\langle\\mbf q_2|\\hat U_\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle&=&\n(2\\pi\\hbar)^{-3}\\delta\\left(\\mbf q-\\frac{\\mbf q_2+\\mbf q_1}{2}\\right) \n\\nonumber \\\\[3mm]\n&\\times&\\exp\\left[\\frac{\\rmi\\mbf p}{\\hbar}(\\mbf q_2-\\mbf q_1)\n+\\frac{\\rmi e}{\\hbar c}\\mbf q_2\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_2,t)\n-\\frac{\\rmi e}{\\hbar c}\\mbf q_1\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_1,t)\\right].\n\t\t\t\\label{matrnewDequantW}\n\\eea\nTaking into account (\\ref{WignerToHusimi}), we find the dequantizer for the corresponding\nHusimi function $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\n\\bea\n\\hat U_\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)&=&\\exp\\left[\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]\n\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^6}\n\\exp\\left[-\\frac{\\lambda\\mbf u^2}{4\\hbar}-\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf{up}+\\mbf{vq})\\right]\n\\nonumber \\\\[3mm]\n&\\times&\n\\exp\\left[\\frac{\\rmi}{\\hbar}(\\mbf u\\hat{\\mbf P}+\\mbf v\\hat{\\mbf q})\\right]\n\\exp\\left[-\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]. \n\t\t\t\\label{newDequantQ}\n\\eea\nThe matrix element (\\ref{newDequantQ}) is obviously equal to the following:\n\\bea\n\\langle\\mbf q_2|\\hat U_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_1\\rangle&=&\n\\frac{1}{(2\\pi\\hbar)^3}\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{3\/2}\n\\exp\\bigg[-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_2)^2-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q_1)^2\n\\nonumber \\\\[3mm]\n&+&\\frac{\\rmi}{\\hbar}(\\mbf q_2-\\mbf q_1)\\mbf p\n+\\frac{\\rmi e}{\\hbar c}\\mbf q_2\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_2,t)\n-\\frac{\\rmi e}{\\hbar c}\\mbf q_1\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_1,t) \\bigg].\n\t\t\t\\label{matrnewUQg}\n\\eea\nFrom  (\\ref{matrnewUQg}) it is obvious that up to the normalization factor the  dequantizer \n$\\hat U_\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$ is the projector of the considered state  \n$\\hat\\rho(t)$ onto the pure state $|\\Psi_{\\mbf{q,p}}\\rangle$, i.e. \n$\\hat U_\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)=(2\\pi\\hbar)^{-3}|\\Psi_{\\mbf{q,p}}\\rangle\\langle\\Psi_{\\mbf{q,p}}|$,\nwhere  we have up to the phase factor in the position representation:\n\\be\n\\langle\\mbf q'|\\Psi_{\\mbf{q,p}}\\rangle=\n\\left(\\frac{\\lambda}{\\pi\\hbar}\\right)^{3\/4}\n\\exp\\bigg[-\\frac{\\lambda}{2\\hbar}(\\mbf q-\\mbf q')^2\n-\\frac{\\rmi}{\\hbar}\\mbf p(\\mbf q-\\mbf q')\n+\\frac{\\rmi e}{\\hbar c}\\mbf q'\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q',t)\n\\bigg].\n\\ee\nCalculations of the matrix element of the quantizer for the function $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\ngive rise to the expression\n\\bea\n\\langle\\mbf q_1|\\hat D_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)|\\mbf q_2\\rangle&=&\n\\int\\frac{d^3v}{(2\\pi\\hbar)^3}\\exp\\bigg[\\frac{\\lambda(\\mbf q_2-\\mbf q_1)^2}{4\\hbar}\n+\\frac{\\mbf v^2}{4\\hbar\\lambda}-\\frac{\\rmi\\mbf v}{2\\hbar}(2\\mbf q-\\mbf q_1-\\mbf q_2)\n-\\frac{\\rmi\\mbf p}{\\hbar}(\\mbf q_2-\\mbf q_1) \\nonumber \\\\[3mm]\n&-&\\frac{\\rmi e}{\\hbar c}\\mbf q_2\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_2,t)\n+\\frac{\\rmi e}{\\hbar c}\\mbf q_1\n\\int_0^1d\\tau\\mbf A(\\tau\\mbf q_1,t)\n\\bigg].\n\t\t\t\\label{matrnewDg}\n\\eea\nWhen using quantizer (\\ref{matrnewDg}), it is assumed that\n$\\mathfrak{Q}_\\mathrm{g}(\\mathbf{q},\\mathbf{p},t)\\in S(\\mathbb{R}^6)$,\nwhere $S(\\mathbb{R}^6)$ is a Schwartz space, and first we take the integral over $d^3q$,\nand after that the integrals over $d^3p$ and $d^3v$ are taken.\nWith the same stipulation, the explicit form of the operator $\\hat D_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)$\ncan be written as:\n\\bea\t\t\t\\label{newDQgint}\n\\hat D_{\\mathfrak{Q}_\\mathrm{g}}(\\mbf q,\\mbf p)&=&\\exp\\left[\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right]\n\\int\\frac{d^3ud^3v}{(2\\pi\\hbar)^3}\n\\exp\\left[\\frac{\\lambda \\mbf u^2}{4\\hbar}+\\frac{\\mbf v^2}{4\\hbar\\lambda}\n-\\frac{\\rmi}{\\hbar}(\\mbf{up}+\\mbf{vq})\\right]\n\\nonumber \\\\[3mm]\n&\\times&\\exp\\left\\{\\frac{\\rmi}{\\hbar}\\left[\\mbf u\\hat{\\mbf P}\n+\\mbf v\\hat{\\mbf q}\\right]\\right\\}\\exp\\left[-\\frac{\\rmi e}{\\hbar c}\\hat{\\mbf q}\n\\int_0^1d\\tau\\mbf A(\\tau\\hat{\\mbf q},t)\\right].\n\\eea\n\nKnowing the quantizers and dequantizers for functions\n$\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$ and $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\none can find the evolution equations for them.\nSince $\\mathfrak{W_\\mathrm{g}}(\\mbf q,\\mbf p)$ and $\\mathfrak{Q_\\mathrm{g}}(\\mbf q,\\mbf p)$\ndo not depend on the gauge, then their evolution equations must also be gauge-independent.\n\n\n\\section{\\label{Art08Section5}Conclusion}\nIn conclusion, I point out the main results of the paper.\nThe gauge-independent Husimi function ($Q-$function) of states of charged quantum \nparticles in the electro-magnetic field was introduced using the gauge-independent\nStratonovich-Wigner function, \nthe corresponding dequantizer and quantizer operators transforming the density matrix\nof state to the such Husimi function and vice versa were found explicitly,\nand the evolution equation for such function was derived.\n\nAlso own non-Stratonovich gauge-independent Wigner function  and its \nHusimi function were suggested and their dequantizers and quantizers were obtained.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, dialogue systems based on KGs have become increasingly popular because of their wide range of applications from hotel bookings, customer-care to voice assistant services. Such dialogue systems can be realized using both goal and non-goal oriented methods. Whereas the former one is employed for carrying out a particular task, the latter is focused on performing natural (\"chit-chat\") dialogues. Both types of dialogue system can be implemented using a generative approach. In a generative dialogue system, the response is generated (usually word by word) from the domain vocabulary given a natural language user query, along with the previous dialogue context. Such systems can benefit from the integration of additional world knowledge~\\cite{eric2017key}. In particular, knowledge graphs, which are an abstraction of real world knowledge, have been shown to be useful for this purpose. Information of the real world can be stored in a KG in a structured (Resource Description Framework (RDF) triple, e.g., $<subject, relation, object>$) and abstract way (Paris is the capital city of France and be presented in $<Paris, capital \\: city, France>$).\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.95\\textwidth]{images\/intro_graph.png}\n    \\caption{Example of a knowledge grounded conversation.}\n    \\label{fig:kg_convo}\n\\end{figure}\nKG based question answering (KGQA) is already a well-researched topic~\\cite{chakraborty2019introduction}. However, generative dialogue systems with integrated KGs have only been explored more recently~\\cite{chaudhuri2019using,madotto2018mem2seq,eric2017key}.\nTo model the response using the KG, all current methods assume that the entity in the input query or a sub-graph of the whole KG, which can be used to generate the answer, is already known~\\cite{madotto2018mem2seq,wu2019global}.\nThis assumption makes it difficult to scale such systems to real-world scenarios, because the task of extracting sub-graphs or, alternatively, performing entity linking in large knowledge graphs is non-trivial~\\cite{rosales2018should}. An example of a knowledge graph based dialogue system is shown in Figure~\\ref{fig:kg_convo}. In order to generate the response \\textit{James Cameron is the director}, the system has to link the entity mentioned in the question in the first turn i.e.~\\textit{Titanic}, and identify the relation in the KG connecting the entities \\textit{Titanic} with \\textit{James Cameron}, namely \\textit{directed by}. Additionally, to obtain a natural dialogue system, it should also reply with coherent responses (eg. \"James Cameron is the director\") and should be able to handle small-talk such as greetings, humour etc. Furthermore, in order to perform multi-turn dialogues, the system should also be able to perform co-reference resolution and connect the pronoun (\\textit{he}) in the second question with \\textit{James Cameron}. \n\nIn order to tackle these research challenges, we model the dialogue generation process by jointly learning the entity and relation information during the dialogue generation process using a pre-trained BERT model in an end-to-end manner. The model's response generation is designed to learn to predict relation(s) from the input KG instead of the actual object(s) (intermediate representation).  Additionally, a graph Laplacian based method is used to encode the input sub-graph and use it for the final decoding process.\n\nExperimental results suggest that the proposed method improves upon previous state-of-the-art approaches for both goal and non-goal oriented dialogues. Our code is publicly available on Github~\\footnote{\\url{https:\/\/github.com\/SmartDataAnalytics\/kgirnet\/}}. Overall, the contributions of this paper are as follows:\n\n\n\n\n\n\n\\noindent \n\\begin{itemize}\n\t\\item A novel approach, leveraging the knowledge graph elements (entities and relations) in the questions along with pre-trained transformers, which helps in generating suitable knowledge grounded responses. \n\t\\item We have also additionally encoded the sub-graph structure of the entity of the input query with a Graph Laplacian, which is traditionally used in graph neural networks. This novel decoding method further improves performance.\n\n\n\t\\item An extensive evaluation and ablation study of the proposed model on two datasets requiring grounded KG knowledge: an in-car dialogue dataset\n\tand soccer dialogues\n\tfor goal and non-goal oriented setting, respectively.\n\tEvaluation results show that the proposed model produces improved knowledge grounded responses compared to other state-of-the-art dialogue systems w.r.t.~automated metrics, and human-evaluation for both goal and non-goal oriented dialogues.\n\n\\end{itemize}\n\n\\section{Model Description}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/model_diagram.png}\n    \\caption{KGIRNet model diagram.}\n    \\vspace{0.1mm}\n    \\label{fig:model_diagram}\n\\end{figure*}\n\nWe aim to solve the problem of answer generation in a dialogue using a KG as defined below.\n\n\n\\begin{definition}[Knowledge Graph]\nWithin the scope of this paper, we define a \\emph{knowledge graph} $KG$ as a labelled, undirected multi-graph consisting of a set $V$ of nodes and a set $E$ of edges between them. There exists a function, $f_l$ that maps the nodes and vertices of a graph to a string. The neighborhood of a node of radius $k$ (or $k$-hop) is the set of nodes at a distance equal to or less than $k$. \n\\label{def:kg}\n\\end{definition}\nThis definition is sufficiently generic to be applicable to knowledge graphs based on RDF\\footnote{\\url{https:\/\/www.w3.org\/RDF\/}} (Resource Description Framework) as well as property graphs~\\cite{gubichev2014graph}. The vertices $V$ of the $KG$ represent entities $e \\in V$, while the edges represent the relationships between those entities. A fact is an ordered triple consisting of an entity $e$ (or subject $s$), an object $o$ and the relation $r$ (a.k.a. predicate $p$) between them, denoted by ($s$, $p$, $o$).\n\nThe proposed model for dialogue generation, which we call KGIRNet, is quintessentially a sequence-to-sequence based model with a pre-trained transformer serving as its input as illustrated in Figure~\\ref{fig:model_diagram}. In contrast to previous works, we introduce an intermediate query representation using the relation information, for training. We also employ a Graph Laplacian based method for encoding the input sub-graph of the KG that aids in predicting the correct relation(s) as well as filter out irrelevant KG elements.\nOur approach consists of the following steps for which more details are provided below:\nFirstly, we encode the input query $q$ with a pre-trained BERT model (Section~\\ref{sec:bert}). Next we detect a core entity $e$ occurring in $q$ (Section~\\ref{sec:entitydetection}). Input query $q$ and generated output is appended to the dialogue context in every utterance. The input query, encoded using the BERT model is then passed through an LSTM encoder to get an encoded representation (Section~\\ref{query_enc}). This encoded representation is passed onto another LSTM decoder (Section~\\ref{dec}), which outputs a probability distribution for the output tokens at every time-step. Additionally, the model also extracts the $k$-hop neighbourhood of $e$ in the KG and encodes it using graph based encoding (Section~\\ref{sub_graph}) and perform a Hadamard product with token probability distribution from the decoder. The decoding process stops when it encounters a special token, $<EOS>$ (end of sentence). Dotted lines in the model diagram represent operations performed at a different time-step $t$ in the decoding process where solid lines are performed once for each utterance or input query. \n\nIn this work, we define complex questions as questions which require multiple relations to answer the given question. For example, for the following query: \"\\textit{please tell me the location, time and the parties that are attending my meeting}\", the model needs to use 3 relations from the KG for answering, namely location, time and parties. The answer given by the model could be : \"\\textit{you have meeting scheduled on friday at 10am with boss in conference\\_room\\_102 to go\\_over\\_budget}\". The model is able to retrieve  important relation information from the KG during decoding. However, the model is not able to handle questions which go beyond the usage of explicitly stored relations and require inference capabilities .\n\n\\subsection{Query Encoding}\n\\label{sec:bert}\n\nBERT is a pre-trained multi-layer, bi-directional transformer~\\cite{vaswani2017attention} model proposed in~\\cite{devlin2019bert}. It is trained on unlabelled data for two-phased objective: masked language model and next sentence prediction.  For encoding any text, special tokens [CLS] and [SEP] are inserted at the beginning and the end of the text, respectively. In the case of KGIRNet, the input query  $q = (q_1, q_2,...q_n)$ at turn $t_d$ in the dialogue, along with the context up to turn $t_{d}-1$ is first encoded using this pre-trained BERT model which produces hidden states ($T_1, T_2....T_n$) for each token and an aggregated hidden state representation $C$ for the [CLS] (first) token.\nWe encode the whole query $q$ along with the context, concatenated with a special token, $<EOU>$ (end of utterance).\n\n\\subsection{Entity Detection}\n\\label{sec:entitydetection}\n\nThe aggregated hidden representation from the BERT model $C$ is passed to a fully connected hidden layer to predict the entity $e_{inp} \\in V$  in the input question as given by\n\\begin{equation}\n    e_{inp} = softmax(w_{ent}C + b_{ent})\n\\end{equation}\n\n\\noindent Where, $w_{ent}$ and $b_{ent}$ are the parameters of the fully connected hidden layer.\n\n\n\\subsection{Input Query Encoder}\n\\label{query_enc}\nThe hidden state representations ($T_1, T_2....T_n$) of the input query $q$ (and dialogue context) using BERT is further encoded using an LSTM \\cite{hochreiter1997long} encoder which produces a final hidden state at the $n$-th time-step given by\n\n\\begin{equation}\n    h^e_n = f_{enc}(T_n, h^e_{n-1})\n\\end{equation}\n\n\\noindent $f_{enc}$ is a recurrent function and $T_n$ is the hidden state for the input token $q_n$ from BERT.\n\n\\noindent The final representation of the encoder response is a representation at every $n$ denoted by\n\n\\begin{equation}\n    H_{e} = (h^e_0, h^e_1....h^e_N)\n\\end{equation}\n\n\n\\subsection{Intermediate Representation}\n\\label{graph_ir}\n\nAs an intermediate response, we let the model learn the relation or edge label(s) required to answer the question, instead of the actual object label(s). In order to do this, we additionally incorporated the relation labels obtained by applying the label function $f_l$ to all edges in the KG into the output vocabulary set. If the output vocabulary size for a vanilla sequence-to-sequence model is $v_{o}$, the total output vocabulary size becomes $v_{od}$ which is the sum of $v_{o}$ and $v_{kg}$. The latter being the labels from applying the $f_l$ to all edges (or relations) in the KG.\n\nFor example, if in a certain response, a token corresponds to an object label $o_l$ (obtained by applying $f_l$ to $o$) in the fact $(e, r, o)$, the token is replaced with a KG token $v_{kg}$ corresponding to the edge or relation label $r_l$ of $r\\in E$ in the KG.\nDuring training, the decoder would see the string obtained by applying $f_l$ to the edge between the entities \\textit{Titanic} and \\textit{James Cameron}, denoted here as \\textit{r:directedBy}. Hence, it will try to learn the relation instead of the actual object. This makes the system more generic and KG aware, and easily scalable to new facts and domains.\n\nDuring evaluation, when the decoder generates a token from $v_{kg}$, a KG lookup is done to decode the label $o_l$ of the node $o \\in V$ in the KG ($V$ being the set of nodes or vertices in the KG). This is generally done using a SPARQL query.\n\n\\subsection{Decoding Process}\n\\label{dec}\nThe decoding process generates an output token at every time-step $t$ in the response generation process. It gets as input the encoded response $H_{e}$ and also the KG distribution from the graph encoding process as explained later. The decoder is also a LSTM, which is initialized with the encoder last hidden states and the first token used as input to it is a special token, $<SOS>$ (start of sentence). \nThe decoder hidden states are similar to that of the encoder as given by the recurrent function $f_{dec}$\n\n\\begin{equation}\n    h^d_n = f_{dec}(w_{dec}, h^d_{n-1})\n\\end{equation}\n\n\\noindent This hidden state is used to compute an attention over all the hidden states of the encoder following \\cite{luong2015effective}, as given by \n\n\\begin{equation}\n    \\alpha_t = softmax(W_s(tanh(W_c[H_{e}; h^d_t]))\n\\end{equation}\n\n\\noindent Where, $W_c$ and $W_s$ are the weights of the attention model.\nThe final weighted context representation is given by  \n\\begin{equation}\n    \\Tilde{h_t} = \\sum_t \\alpha_t h_t\n\\end{equation}\n\n \\noindent This representation is concatenated (represented by $;$) with the hidden states of the decoder to generate an output from the vocabulary with size $v_{od}$.\n \n \\noindent The output vocab distribution from the decoder is given by\n \n \\begin{equation}\n     O_{dec} = W_o([{h_t; \\Tilde{h^d_t}}])\n \\end{equation}\n \n \\noindent In the above equation, $W_o$ are the output weights with dimension $\\mathbf{R}^{h_{dim}Xv_{od}}$. $h_{dim}$ being the dimension of the hidden layer of the decoder LSTM. The total loss is the sum of the vocabulary loss and the entity detection loss.\n Finally, we use beam-search \\cite{tillmann2003word} during the the decoding method. \n \n\\subsection{Sub-Graph Encoding}\n \\label{sub_graph}\n\n In order to limit the KGIRNet model to predict only from those relations which are connected to the input entity predicted from step~\\ref{sec:entitydetection}, we encode the sub-graph along with its labels and use it in the final decoding process while evaluating.\n \n\\noindent The $k$-hop sub-graph of the input entity is encoded using Graph Laplacian \\cite{kipf2016semi} given by\n\n\\begin{equation}\n    G_{enc} = D^{-1}\\Tilde{A}f_{in}\n    \\label{graph_spec}\n\\end{equation}\n\n\\noindent Where, $\\Tilde{A} = A+I$. $A$ being the adjacency matrix, $I$ is the identity matrix and D is the degree matrix. $f_{in}$ is a feature representation of the vertices and edges in the input graph. \n$G_{enc}$ is a vector with dimensions $\\mathbf{R}^{ik}$ corresponding to the total number of nodes and edges in the $k$-hop sub-graph of $e$. An example of the sub-graph encoding mechanism is shown in Figure \\ref{fig:graph_lapl}.\n \n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{images\/KGIRNET-graphLapl.png}\n    \\caption{Sub-Graph Encoding using Graph Laplacian.}\n    \\vspace{-0.1mm}\n    \\label{fig:graph_lapl}\n\\end{figure*}\n\n\\noindent The final vocabulary distribution $O_{f} \\in R^{v_{od}}$ is a Hadamard product of this graph vector and the vocabulary distribution output from the decoder. \n\n\\begin{equation}\n    O_{f} = O_{dec} \\odot G_{enc}\n\\end{equation}\n\n\\noindent This step essentially helps the model to give additional importance to the relations connected at k-hop based on its similarity with the query also to filter(mask) out relations from the response which are not connected to it. For the query in \\ref{fig:graph_lapl} \\textit{who is the director of Avatar} and {how was it rated ?} The graph laplacian based encoding method using only relation labels already gives higher scores for the relations directed\\_by and rating, which are required by the question. This vector when multiplied with the final output vocabulary helps in better relation learning.\n\n\n\n\\section{Experimental Setup}\n\n\\subsection{Datasets}\n\nAvailable datasets for knowledge grounded conversations are the \\textit{in-car dialogue} data as proposed by~\\cite{eric2017key} and \\textit{soccer dialogues} over football club and national teams using a knowledge graph~\\cite{chaudhuri2019using}. The former contains dialogues for assisting the driver of a car with data related to weather, schedules and navigation, in a goal-oriented setting.\nThe soccer dataset contains non-goal oriented dialogues about questions on different soccer teams along with a knowledge graph consisting of facts extracted from Wikipedia about the respective clubs and national soccer teams. Both the datasets were collected using Amazon Mechanical Turk (AMT) by the respective authors~\\cite{eric2017key,chaudhuri2019using}. The statistics of the datasets are provided in Table ~\\ref{tab:stats}. As observed, the number of dialogues for both the dataset is low.\n\n\nTo perform a KG grounded dialogue as in KG based question-answering~\\cite{diefenbach2018core}, it is important to annotate the dialogues with KG information such as the entities and relations, the dialogues are about. Such information were missing in the soccer dataset, hence we have semi-automatically annotated them with the input entity $e_{inp}$ in the query $q$ and the relations in the $k$-hop sub-graph of the input entity required to answer $q$. For the domain of in-car it was possible to automatically extract the entity and relation information from the dialogues and input local KG snippets.\n\n\\vspace{-.8cm}\n\\begin{table}[]\n\\centering\n\\caption{Dataset statistics.}\n\\begin{tabular}{l|c|c|c|c|c|c}\n\\toprule\n     & \\multicolumn{3}{c|}{\\textbf{In-car dialogues}} & \\multicolumn{3}{c}{\\textbf{Soccer dialogues}} \\\\\\toprule\nNumber of triples, entity, relations &   \\multicolumn{3}{c|}{8561, 271, 36}    &  \\multicolumn{3}{c}{4301, 932, 30}\\\\\\midrule\n     & \\textbf{Train}     & \\textbf{Validation}      & \\textbf{Test}     & \\textbf{Train}     & \\textbf{Validation}     & \\textbf{Test}      \\\\\\midrule\nNumber of dialogues &     2011      &        242        &    256      &     1328      &     149     &      348          \\\\\nNumber of utterances &      5528     &        657        &    709      &    6523       &     737     &       1727         \\\\\nKG-grounded questions (\\%) &    44.95       &       33.94         &     43.84     &     6.53      &   4.61       &  3.88  \\\\\n\\bottomrule\n\\end{tabular}\n\\label{tab:stats}\n\\end{table}\n\\vspace{-.6cm}\n\\subsection{Evaluation Metrics}\n\nWe evaluate the models using the standard evaluation metrics BLEU~\\cite{papineni2002bleu} and Entity F1 scores as used in the discussed state-of-the-art models. However, unlike~\\cite{madotto2018mem2seq} we use Entity F1 scores based on the nodes $V$ of the KG, as inspired by previous works on KG based question answering~\\cite{Bordes2015LargescaleSQ}. Additionally, we use METEOR~\\cite{banerjee2005meteor} because it correlates the most with human judgement scores~\\cite{sharma2017relevance}.\n\n\n\\subsection{Model Settings}\n\\begin{table*}[ht]\n    \\caption{Evaluation on goal and non-goal oriented dialogues.}\n    \\centering \n   \\vspace{.1cm}\n    \\def1{1.2}\n\\begin{adjustbox}{width=0.8\\textwidth}\n\\begin{tabular} {c c|c|c|c|c|c|c}\n\\toprule\n\\multirow{2}{*}{\\textbf{Models}} &  \\multicolumn{3}{c}{\\textbf{In-Car Dialogues}} & \n\\multicolumn{3}{c}{\\textbf{Soccer Dialogues}} & Inference time\\\\\n\\cline{2-8}\n\n & \\textit{BLEU} & \\textit{Entity F1} & \\textit{METEOR} &\\textit{BLEU} & \\textit{Entity F1} & \\textit{METEOR} & \\textit{utterances\/sec}\\\\\n\\hline\nSeq2Seq & 4.96 & 10.70  & 21.20 & 1.37 & 0.0 & 7.8 & 133\\\\\nMem2Seq\\cite{madotto2018mem2seq} & 9.43 & 24.60 & 28.80 & 0.82 & 04.95 & 7.8 & 88 \\\\\nGLMP\\cite{wu2019global} & 9.15 & 21.28 & 29.10 & 0.43 & 22.40 & 7.7 & 136\\\\\nTransformer\\cite{vaswani2017attention} & 8.27 & 24.79 & 29.06 &0.45 & 0.0 & 6.7 & 7\\\\\nDialoGPT\\cite{zhang2019dialogpt} & 7.35 & 21.36 & 20.50 & 0.76 & 0.0 & 5.5 & 2\\\\\nKG-Copy\\cite{chaudhuri2019using} & - & - & - & \\textbf{1.93} & 03.17 & \\textbf{10.89} & 262\\\\\n\\midrule\n\nKGIRNet & \\textbf{11.76} & \\textbf{32.67} & \\textbf{30.02} & 1.51 & \\textbf{34.33} & 8.24 & 37\\\\\n\\bottomrule\n\n\\end{tabular}\n    \\label{tab:eval}\n    \\end{adjustbox}\n\\end{table*}\n\nFor entity detection we used a fully connected layer on top of CNN-based architecture. Size of the hidden layer in the fully connected part is 500 and a dropout value of 0.1 is used and ReLU as the activation function. In the CNN part we have used 300 filters with kernel size 3, 4 and 5. We use BERT-base-uncased model for encoding the input query. The encoder and decoder is modelled using an LSTM (long short term memory) with a hidden size of 256 and the KGIRNet model is trained with a batch size of 20. The learning rate of the used encoder is 1e-4. For the decoder we select a learning rate of 1e-3. We use the Adam optimizer to optimize the weights of the neural networks. For all of our experiments we use a sub-graph size of k=2. For calculating $f_{in}$ as in Equation~\\ref{graph_spec}, we use averaged word embedding similarity values between the query and the labels of the sub-graph elements. The similarity function used in this case is cosine similarity. We save the model with best validation Entity f1 score.\n\\vspace{-.8cm}\n\n\\begin{table}[ht]\n\\caption{Relation Linking Accuracy on SQB \\cite{wu-etal-2019-learning-representation} Dataset.}\n    \\centering \n    \n   \n\\begin{adjustbox}{width=0.6\\textwidth}\n\\begin{tabular} {p{20mm} | p{40mm} | p{20mm}}\n\\toprule\n\\textbf{Method} & \\textbf{Model} & \\textbf{Accuracy}\\\\\n\\midrule\n\\textbf{Supervised} & {Bi-LSTM \\cite{mohammed-etal-2018-strong}} & {38.5}\\\\ \n{} & {HR-LSTM \\cite{yu-etal-2017-improved}} & {64.3} \\\\ \n\\midrule\n\\textbf{Unsupervised} &{Embedding Similarity} & {60.1} \\\\ \n\n{} & {Graph Laplacian (this work)} & \\textbf{69.7}  \\\\ \n\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\n    \\label{tab:sq_rl}\n\\end{table}\n\\vspace{0.02cm}\n\\section{Results}\n\\label{results}\nIn this section, we summarize the results from our experiments. We evaluate our proposed model KGIRNet against the current state-of-the-art systems for KG based dialogues; namely Mem2Seq~\\cite{madotto2018mem2seq}, GLMP~\\cite{wu2019global}, and KG-Copy~\\cite{chaudhuri2019using}\\footnote{KG-Copy reports on a subset of the in-car testset hence it is not reported here}. To include a baseline method, the results from a vanilla Seq2Seq model using an LSTM encoder-decoder are also reported in Table~\\ref{tab:eval}, along with a vanilla transformer~\\cite{vaswani2017attention} and a pre-trained GPT-2 model, DialoGPT \\cite{zhang2019dialogpt} fine-tuned individually on both the datasets.\n\nWe observe that KGIRNet outperforms other approaches for both goal (in-car) and non-goal oriented (soccer) dialogues except for BLEU and METEOR scores on the soccer dataset.\nThe effect of knowledge groundedness is particularly visible in the case of soccer dialogues, where most models (except GLMP) produces feeble knowledge grounded responses. The Entity F1 score used here is adapted from \\cite{Bordes2015LargescaleSQ} and is defined as the average of F1-scores of the set of predicted objects, for all the questions in the test set.\n\nIn addition to evaluating dialogues, we also evaluate the proposed graph laplacian based relation learning module for the task of knowldge-graph based relation linking. Although, it is a well-researched topic and some systems claim to have solved the problem \\cite{petrochuk-zettlemoyer-2018-simplequestions}, but such systems are not able to handle relations which are not present in the training data \\cite{wu-etal-2019-learning-representation}. The latter also proposed a new, balanced dataset (SQB) for simple question answering which has same proportions of seen and unseen relations in the test or evaluation set. We have evaluated our unsupervised graph laplacian based method for relation linking on the SQB dataset against supervised methods namely Bi-LSTM \\cite{mohammed-etal-2018-strong}, hr-bilstm \\cite{yu-etal-2017-improved} and unsupervised method such as text embedding based similarity between the query and the relations connected to the subject entity with 1-hop. The results are reported in Table \\ref{tab:sq_rl}. As observed, \nGraph Laplacian performs better wrt. supervised methods on unseen relations and also better than shallow embedding based similarity. This is one of the motivation for using this simple method during KGIRNet's decoding process. We run all the inference on a setup with 32GB of RAM and 8GB of VGA and a Bandwidth of 256.3 GB\/s. \t\n\n\n\\section{Discussion}\n\\label{discussion}\n\nFor in-car dialogues, we train and evaluate on queries which require knowledge from the KG, hence we omit the scheduling conversations where the driver asks the car to schedule a meeting\/conference.\nIn order to get the knowledge graph candidate triples for all the other models (Mem2Seq and GLMP), we provide them with the 2-hop sub-graph of the correct input entity instead of the local knowledge as used in the respective models; this, represents a scenario closer to a real-world KG grounded conversation.\nFor the in-car dataset, the human baseline BLEU score as reported in~\\cite{eric2017key} is 13.5 (the KGIRNet score is 11.76).\nThe BLEU scores for soccer are low because non-goal oriented dialogues are more complicated to model due to large vocabulary sizes (more than 3 times the vocabulary size of the of in-car dataset). Also in the case of soccer dialogues, number of factual conversation is low (4\\%) compared to the total number of utterances and also the conversations are more generic and repetitive in nature.\n\\vspace{-.8cm}\n\\begin{table*}[ht]\n \\caption{Analyzing sample predictions.}\n    \\centering \n    \\vspace{.1cm}\n   \n\\begin{adjustbox}{width=\\textwidth}\n\\begin{tabular} {p{30mm}|p{25mm}|p{30mm}|p{35mm}|p{30mm}}\n\n\\toprule\n\\multirow{2}{*}{\\textbf{Input Query}} &\n\\multirow{2}{*}{\\textbf{True Response}} &\n\\multicolumn{3}{c}{\\textbf{Predicted Responses}} \\\\\n\\cline{3-5}\n\n & & \\textbf{GLMP} & \\textbf{Mem2Seq} & \\textbf{KGIRNet} \\\\%& \\textit{True} & \\textit{Predicted} \\\\\n\\hline\n{\\textbf{(S)} who is currently coaching bvb dortmund ? } & {lucien favre} & \\textit{the is the coach} & \\textit{yes , they have a good}  & \\textit{lucien favre is the coach of bvb dortmund} \\\\\n\\midrule\n{\\textbf{(S)} when did italy last win the world cup ?} & 2006 & \\textit{italy won the world cup in 2006} & \\textit{i think they have a good team} & \\textit{italy won the world cup in 2006} \\\\\n\\midrule\n\\midrule\n\n{\\textbf{(C)} what time is my doctorappointment?} & {your doctorappointment is scheduled for friday at 11am} & \\textit{your next is is at 1pm at 7pm} & \\textit{your doctorappointment is at 1pm} & \\textit{your doctorappointment is on friday at 11am} \\\\\n\\midrule\n\\textbf{(C)} i need gas &\nvalero is 4\\_miles away & \\textit{there is a valero away} & \\textit{chevron is gas\\_station away chevron is at away} & \\textit{there is a valero nearby} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n   \n    \\label{tab:resp_analysis}\n\\end{table*}\n\\vspace{-.8cm}\n\\subsection{Human Evaluation}\nWe  perform  a  human-based  evaluation  on  the  whole  test  dataset  of  the generated responses from the KGIRNet model and its closest competitors, i.e. Mem2Seq, GLMP and DialoGPT models. We asked 3 internal independent annotators who are not the authors of this paper (1 from CS and 2 from non-CS background) to rate the quality of the generated responses between 1-5 with respect to the dialogue context (higher is better). Note that for each dataset, we provide the annotators with 4 output files in CSV format (containing the predictions of each model) and the knowledge graph in RDF format. Each of the CSV files contains data in 5 columns: question, original response, predicted response, grammatical correctness, similarity between original and predicted response. In the provided files, the 4th (grammatically correctness) and 5th (similarity between original and predicted response) columns are empty and we ask the annotators to fill them with values (within a range of 1-5) according to their judgement. The measures requested to evaluate upon are correctness (Corr.) and human-like (human) answer generation capabilities. Correctness is measured by comparing the response to the correct answer object. Reference responses are provided besides the system generated response in order to check the factual questions. The results are reported in Table~\\ref{tab:humaneval}. Cohen's Kappa of the annotations is 0.55.\n\n\\subsection{Ablation Study}\nAs an ablation study we train a sequence-to-sequence model with pre-trained fasttext embeddings as input (S2S), and the same model with pre-trained BERT as input embedding (S2S\\_BERT). Both these models do not have any information about the structure of the underlying knowledge graph. Secondly, we try to understand how much the intermediate representation aids to the model, so we train a model (KGIRNet\\_NB) with fasttext embeddings as input instead of BERT along with intermediate relation representation. Thirdly, we train a model with pre-trained BERT but without the haddamard product of the encoded sub-graph and the final output vocabulary distribution from Step~\\ref{sub_graph}. This model is denoted as KGIRNet\\_NS. As observed, models that are devoid of any KG structure has very low Entity F1 scores, which is more observable in the case of soccer dialogues since the knowledge grounded queries are very low, so the model is not able to learn any fact(s) from the dataset. The proposed intermediate relation learning along with Graph Laplacian based sub-graph encoding technique observably assists in producing better knowledge grounded responses in both the domains; although, the latter aids in more knowledge groundedness in the domain of soccer dialogues (higher Entity F1 scores). We also did an ablation study on the entity detection accuracy of the end-to-end KGIRNet model in the domain of in-car and compared it with a standalone Convolutional neural network (CNN) model which predicts the entity from the input query, the accuracies are 79.69 \\% and 77.29\\% respectively. \n\n\\vspace{-.9cm}\n\\begin{table}[ht]\n\\caption{In-depth evaluation of KGIRNet model.}\n   \n        \\begin{subtable}{.48\\linewidth}\n    \\begin{tabular} {c|c|c|c|c}\n\\toprule\n\\multirow{2}{*}{\\textbf{Models}} &  \\multicolumn{2}{c|}{\\textbf{In-Car}} & \n\\multicolumn{2}{c}{\\textbf{Soccer}}\\\\\n\\cline{2-5}\n\n & \\textit{Corr.} & \\textit{Human} & \\textit{Corr.} & \\textit{Human} \\\\\n\\hline\nMem2Seq & 3.09 & 3.70 & 1.14 & 3.48  \\\\\nGLMP & 3.01 & 3.88 & 1.10 & 2.17 \\\\\nDialoGPT  & 2.32  & 3.43 & 1.32 & 3.88  \\\\\n\\midrule\nKGIRNet & 3.60 & 4.42 & 1.59 & 3.78 \\\\\n\n \\bottomrule\n\\end{tabular}\n    \\caption{Human evaluation.}\n    \\label{tab:humaneval}\n    \\end{subtable}\n   \n    \\begin{subtable}{.52\\linewidth}\n   \n   \n\\begin{adjustbox}{width=\\textwidth}\n\n\\begin{tabular} {lc|c|c|c}\n\\toprule\n\\multirow{2}{*}{\\textbf{Models}} &  \\multicolumn{2}{c|}{\\textbf{In-Car Dialogues}} & \n\\multicolumn{2}{c}{\\textbf{Soccer Dialogues}}\\\\\n\\cline{2-5}\n\n & \\textit{BLEU} & \\textit{EntityF1} & \\textit{BLEU} & \\textit{EntityF1} \\\\\n\\hline\nS2S & 4.96 & 10.70 & 1.49 & 0.0 \\\\\nS2S\\_BERT & 7.53 & 09.10 & 1.44 & 0.0 \\\\\nKGIRNet\\_NB & 9.52 & 29.03 & 0.91 & 29.85 \\\\\nKGIRNet\\_NS & 11.40 & 33.03 & 1.05 & 28.35 \\\\\n\\midrule\nKGIRNet & 11.76 & 32.67 & 1.51 & 34.32 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n    \\caption{Ablation study.}\n    \\label{tab:ablation}\n    \\end{subtable}\n\\vspace{0.1mm}    \n\n\\end{table}\n\\vspace{-.9cm} \n\n\\subsection{Qualitative Analysis}\nThe responses generated by KGIRNet are analyzed in this section. Responses from some of the discussed models along with the input query are provided in Table~\\ref{tab:resp_analysis} \\footnote{In the table \\textbf{(S)} and \\textbf{(C)} refers to example from Soccer and In-car dataset respectively.}. We compare the results with two other state-of-the-art models with the closest evaluation scores, namely Mem2Seq and GLMP. The first two responses are for soccer dialogues, while the latter two are for in-car setting. We inspect that  qualitatively the proposed KGIRNet model produces better knowledge grounded and coherent responses in both the settings. In the case of soccer dialogues, predicting single relation in the response is sufficient, while for the case of in-car dialogues, responses can require multiple relation identification. KGIRNet model is able to handle such multiple relations as well (e.g., $r$:date friday and $r$:time 11am for the third utterance).\n\n\\vspace{-.9cm}\n\\begin{table}[ht]\n\\caption{Analyzing fact-fullness of KGIRNet.}\n    \\centering \n    \\vspace{.1cm}\n   \n\n\\begin{adjustbox}{width=1\\textwidth}\n\\begin{tabular} {p{30mm}|p{28mm}|p{27mm}|p{35mm}}\n\n\\toprule\n{\\textbf{Input Query}} &  \\textbf{True Response} & \n\\textbf{Predicted Response} & \\textbf{Intermediate Response}\\\\\n\\midrule\n\n{who is senegal captain ?} & {cheikhou kouyate} & \\textit{sadio mane is the captain of senegal} & \\textit{r:captain is the captain of @entity} \\\\\n\\hline\n{who is the goalie for iceland ?} & hannes halldorsson & \\textit{runar alex runarsson} & \\textit{$r$\\:goalkeeper} \\\\\n\\bottomrule\n\\end{tabular}\n    \\label{tab:intermed}\n\\end{adjustbox}\n\n\\end{table}\n\\vspace{-.6cm}\nTo understand more about the advantages of KG grounded generation process, consider the examples from the soccer dataset below in Table~\\ref{tab:intermed}, in which the model has failed. We have also included the intermediate responses that are generated from the decoder before the KG lookup phase for the reference.\nIn the first case, while constructing the KG, the captain was \\textit{Sadio Mane}, whereas the during the dialogue collection phase the probable captain was \\textit{Checikhou Kouyate} (as per the annotator). These errors on temporal information can easily be tackled using Knowledge Grounded dialogue systems, where the system can potentially reflect current information with updated real world knowledge. For the second case, the current main goalkeeper for the Iceland national football team is \\textit{Hannes Halld\u00f3rsson}; however, \\textit{R\u00fanar Alex R\u00fanarsson} is also among the 3 goalkeepers for the team and the model has no additional information to disambiguate between them to produce the same response as the human annotator. KGIRNet is also able to infer conversational context implicitly in dialogues with the system (Table~\\ref{tab:kgirnet_convo}).\n\\vspace{-.9cm}\n\\begin{table}[ht]\n\\caption{A conversation with KGIRNet.}\n    \\centering \n    \\vspace{.2cm}\n   \n\\begin{adjustbox}{width=0.8\\textwidth}\n\\begin{tabular} {p{18mm} p{100mm}}\n\\toprule\n\\textbf{User:} & {what s the weather forecast for today and tomorrow ?}\\\\\n\\midrule\n\\textbf{KGIRNet:} & {what city do you want the weather for}\\\\ \n\\midrule\n\\textbf{User:} & {los angeles} \\\\ \n\\midrule\n\\textbf{KGIRNet:} & {it will be 40f - 60f on monday warm on tuesday windy on wednesday hot on thursday} \\\\ \n\\midrule\n\\textbf{User:} & {thanks}  \\\\ \n\\midrule\n\\textbf{KGIRNet:} & {you are welcome} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n    \n    \\label{tab:kgirnet_convo}\n\\end{table}\n\\vspace{-.6cm}\n\\subsection{Error Analysis}\n\nTo analyze more on the shortcomings of our proposed KGIRNet, we stated some sample erroneous responses from the model in Table~\\ref{tab:errors}. The first three are from soccer and the rest in-car dialogues. The model often produces generic responses for soccer as in the first example. Our model is only able to handle simple factoid queries and not able to answer queries requiring comparative reasoning such as \\textit{best player}, \\textit{youngest player} etc, as observable from the second and third cases. In the second case, the model answers with the coach's name and when asked for the best player for Argentina and it replies with the name of an Uruguayan player in the third case. The model also produces generic relations like coach name, number of world cup championship as response when any such keyword is present in the input. For the in-car dataset, the replies are less generic and more coherent; however, in cases like the fourth example, the model is not able to perform logical reasoning (negation). Moreover, the model fails in cases where the input entity is not recognized correctly as in case of the last example. Although, the name of the friend is fetched from the vocabulary, but the predicted entity is wrong hence the model is not able to decode the relation $r$:address from the KG during decoding.\n\\vspace{-.8cm}\n\\begin{table}[ht]\n\n \\caption{Error analysis of KGIRNet.}\n\n\\begin{adjustbox}{width=\\textwidth}\n\\begin{tabular} {p{42mm}|p{50mm}|p{50mm}}\n\n\\toprule\n{\\textbf{Input Query}} &  \\textbf{True Response} & \n\\textbf{Predicted Response}\\\\\n\\midrule\n\n{portugal win the last eurocup ?} & {yes they won the last europcup} & \\textit{portugal has won the world cup.} \\\\\n\\midrule\n{who is switzerland\\'s best player ?} & {xherdan shaqiri} & \\textit{petkovic is the coach of switzerland } \\\\ \n\\midrule\n{who is considered their best player ?} & {traditionally their top scorers.} & \\textit{luis suarez is the best player in argentina} \\\\ \n\\midrule\n{will it be warm in grand\\_rapids on saturday } & {it will not be warm in grand\\_rapids on saturday} & \\textit{it is not predicted to fall on tuesday in grand\\_rapids} \\\\ \n\\midrule\n{where does my friend live ?} & {tom lives at 580\\_van\\_ness\\_ave} & \\textit{tom s house is located at $r$:address} \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n       \\label{tab:errors}\n\\end{table}\n\\vspace{-.6cm}\nIn general, the model's entity detection fails in case of conversations with long dialogue contexts. Incorporating KG information into the entity detection process, where we consider the sub-graph of the entity in the first interaction can potentially help in further performance improvements.\n\n\\section{Related Work}\nLeveraging background information for dialogue system improvement is a well-researched topic, especially in goal-oriented setting~\\cite{bordes2016learning,dhingra2016towards,wen2016network}. \\cite{eric2017key} proposed the in-car dataset which uses a knowledge base for in-car conversation about weather, location etc. Recently, \\cite{madotto2018mem2seq} proposed memory-network based encoder-decoder architecture for integrating knowledge into the dialogue generation process on this dataset. Improved models in this task are proposed by\n\\cite{kassawat2019incorporating,wu2019global}. \n\\cite{chaudhuri2019using} proposed a soccer dialogue dataset along with a KG-Copy mechanism for non-goal oriented dialogues which are KG-integrated. In a slightly similar research line, in past years, we also notice the use of variational autoencoders (VAE)~\\cite{zhao-etal-2017-learning,li-etal-2020-improving-variational} and generative adversarial networks (GANs)~\\cite{olabiyi-etal-2019-multi,lopez2019differentiable} in dialogue generation. However, knowledge graph based dialogue generation is not well-explored in these approaches.\n\nMore recently, transformer-based~\\cite{vaswani2017attention} pre-trained models have achieved success in solving various downstream tasks in the field of NLP such as question answering~\\cite{shao2019transformer} \\cite{devlin2019bert}, machine translation~\\cite{wang2019learning}, summarization~\\cite{egonmwan2019transformer}. Following the trend, a hierarchical transformer is proposed by~\\cite{santra2020hierarchical} for task-specific dialogues. The authors experimented on MultiWOZ dataset~\\cite{budzianowski2018multiwoz}, where the belief states are not available. However, they found the use of hierarchy based transformer models effective in capturing the context and dependencies in task-specific dialogue settings. In a different work,~\\cite{oluwatobi2020dlgnet} experimented transformer-based model on both the task-specific and non-task specific dialogues in multi-turn setting. In a recent work, ~\\cite{zhang2019dialogpt} investigated on transformer-based model for non-goal oriented dialogue generation in single-turn dialogue setting. Observing the success of transformer-based models over the recurrent models in this paper we also employ BERT in the dialogue generation process which improves the quality of generated dialogues (discussed in section~\\ref{results} and~\\ref{discussion}).\n\nIn past years, there is a lot of focus on encoding graph structure using neural networks, a.k.a.~Graph Neural Networks (GNNs)~\\cite{bronstein2017geometric,velivckovic2017graph}. In the field of computer vision, Convolutional Neural Networks (CNNs) are used to extract the most meaningful information from grid-like data structures such as images. A generalization of CNN to graph domain, Graph Convolutional Networks (GCNs)~\\cite{kipf2016semi} has become popular in the past years. Such architectures are also adapted for encoding and extracting information from knowledge graphs \\cite{neil2018interpretable}. Following a similar research line, in this paper, we leverage the concept of Graph Laplacian~\\cite{kipf2016semi} for encoding sub-graph information into the learning mechanism.\n\n\\section{Conclusion and Future Work} \nIn this paper, we have studied the task of generating knowledge grounded dialogues. We bridged the gap between two well-researched topics, namely knowledge grounded question answering and end-to-end dialogue generation. We propose a novel decoding method which leverages pre-trained transformers, KG structure and Graph Laplacian based encoding during the response generation process. Our evaluation shows that out proposed model produces better knowledge grounded response compared to other state-of-the-art approaches, for both the task and non-task oriented dialogues. \n\nAs future work, we would like to focus on models with better understanding of text in order to perform better KG based reasoning. We also aim to incorporate additional KG structure information in the entity detection method. Further, a better handling of infrequent relations seen during training may be beneficial.\n\n\\section*{Acknowledgement}\nWe acknowledge the support of the excellence clusters ScaDS.AI (BmBF IS18026A-F), ML2R (BmBF FKZ 01 15 18038 A\/B\/C), TAILOR (EU GA 952215) and the  projects SPEAKER (BMWi FKZ 01MK20011A) and JOSEPH (Fraunhofer Zukunftsstiftung).\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Discussion}\n\nReal environments don't provide scalar reward signals to learn from. Instead, organisms have developed various internal drives based on either primary or secondary goals \\citep{Baldassarre13}. Here we examined intrinsic rewards based on features derived from other agents in the environment, in order to establish whether such social signals could enable the evolution of altruism to solve intertemporal social dilemmas. In accord with evolutionary theory \\citep{axelrod81, Nowak1560}, we found that na\\\"ively implementing natural selection via genetic algorithms did not lead to the emergence of cooperation. Furthermore, assortative matchmaking was sufficient to generate cooperative behavior in cases where honest signals were available. Finally, we proposed a new multi-level evolutionary paradigm based on shared reward networks that achieves cooperation in more general situations.\n\nWe demonstrated that the reward network weights evolved differently for Cleanup versus Harvest, indicating that the two tasks necessitate different forms of social cooperation for optimal performance. This highlights the advantage of evolving rather than hand-crafting the weighting between individual reward and group reward, as optimal weightings cannot necessarily be anticipated for all environments. Evolving such weightings thus constitutes a form of meta-learning, wherein an entire learning system, including intrinsic reward functions, is optimized for fast learning \\cite{singh2010intrinsically, fernando2018meta}. Here we have extended these ideas to the multi-agent domain.\n\nWhy does evolving intrinsic social preferences promote cooperation? Firstly, evolution ameliorates the intertemporal choice problem by distilling the long timescale of collective fitness into the short timescale of individual reinforcement learning, thereby improving credit assignment between selfish acts and their temporally displaced negative group outcomes \\citep{hughes2018inequity}. Secondly, it mitigates the social dilemma itself by allowing evolution to expose social signals that correlate with, for example, an agent's current level of selfishness. Such information powers a range of mechanisms for achieving mutual cooperation like competitive altruism \\citep{hardy2006nice}, other-regarding preferences \\citep{cooper2016other}, and inequity aversion \\citep{fehr1999}. In accord, laboratory experiments show that humans cooperate more readily when they can communicate \\citep{ostrom1992covenants, janssen2010lab}. \n\nThe shared reward network evolution model was inspired by multi-level selection; yet it does not correspond to the prototypical case of that theory since its lower level units of evolution (the policy networks) are constantly swapping which higher level unit (reward network) they are paired with. Nevertheless, there are a variety of ways in which we see this form of modularity arise in nature. For example, free-living  microorganisms occasionally form multi-cellular structures to solve a higher order adaptive problem, like slime mold forming a spore-producing stalk for dispersal \\citep{west2006social}, and many prokaryotes can incorporate plasmids (modules) found in their environment or received from other individuals as functional parts of their genome, thereby achieving cooperation in social dilemmas  \\citep{griffin2004cooperation, mc2011horizontal}. Alternatively, in humans a reward network may represent a shared ``cultural norm'', with its fitness based on cultural information accumulated from the groups in which it holds sway. In this way, the spread of norms can occur independently of the success of individual agents \\citep{boyd2009}.\n\nNote that in this work, we have assumed that agents have perfect knowledge of other agents' rewards, while in real-world systems this is not typically the case. This assumption was made in order to disentangle the effects of cultural evolution from the quality of the signals being evolved over. Natural next steps include adding partial observability or noise to this signal (to make it more analogous to, for instance, a smile\/frown or other locally observable social signals), identifiability across episodes, or even deception.\n\nThe approach outlined here opens avenues for investigating alternative evolutionary mechanisms for the emergence of cooperation, such as kin selection \\citep{griffin2002} and reciprocity \\citep{trivers1971evolution}. It would be interesting to see whether these lead to different weights in a reward network, potentially hinting at the evolutionary origins of different social biases. Along these lines, one might consider studying an emergent version of the assortative matchmaking model along the lines suggested by \\cite{henrich2003}, adding further generality and power to our setup. Finally, it would be fascinating to determine how an evolutionary approach can be combined with multi-agent communication to produce that most paradoxical of cooperative behaviors: cheap talk.\n\n\\begin{acks}\nWe would like to thank Simon Osindero, Iain Dunning, Andrea Tacchetti, and many DeepMind colleagues for valuable discussions and feedback, as well as code development and support. \n\\end{acks}\n\\section{Methods}\n\nWe varied and explored different combinations of parameters, namely: (1) environments \\{Harvest, Cleanup\\}, (2) reward network features \\{prospective, retrospective\\}, (3) matchmaking \\{random, assortative\\}, and (4) reward network evolution \\{individual, shared, none\\}. We describe these in the following sections.\n\n\n\\subsection{Intertemporal social dilemmas}\n\nIn this paper, we consider Markov games \\citep{littman1994markov} within a MARL setting. Specifically we study intertemporal social dilemmas \\citep{leibo17, hughes2018inequity}, defined as games in which individually selfish actions produce individual benefit on short timescales but have negative impacts on the group over a longer time horizon. This conflict between the two timescales characterizes the intertemporal nature of these games. The tension between individual and group-level rationality identifies them as social dilemmas (e.g. the famous Prisoner's Dilemma).  \n\nWe consider two dilemmas, each implemented as a partially observable Markov game on a 2D grid (see Figure \\ref{fig:gallery}), with $N=5$ players playing at a time. In the \\emph{Cleanup} game, agents tried to collect apples (reward ${+}1$) that spawned in a field at a rate inversely related to the cleanliness of a geographically separate aquifer. Over time, this aquifer filled up with waste, lowering the respawn rate of apples linearly, until a critical point past which no apples could spawn. Episodes were initialized with no apples present and zero spawning, thus necessitating cleaning. The dilemma occurred because in order for apples to spawn, agents must leave the apple field and clean, which conferred no reward. However if all agents declined to clean (defect), then no rewards would be received by any. In the \\emph{Harvest} game, again agents collected rewarding apples. The apple spawn rate at a particular point on the map depended on the number of nearby apples, falling to zero once there were no apples in a certain radius. There is a dilemma between the short-term individual temptation to harvest all the apples quickly and the consequential rapid depletion of apples, leading to a lower total yield for the group in the long-term. \n\nAll episodes last 1000 steps, and the total size of the playable area is 25$\\times$18 for Cleanup and 38$\\times$16 for Harvest. Games are partially observable in that agents can only observe via a 15$\\times$15 RGB window, centered on their current location. The action space consists of moving left, right, up, and down, rotating left and right, and the ability to tag each other. This action has a reward cost of 1 to use, and causes the player tagged to lose 50 reward points, thus allowing for the possibility of punishing free-riders \\citep{oliver1980rewards, gurerk2006competitive}. The Cleanup game has an additional action for cleaning waste.\n\n\n\\subsection{Modeling social preferences as intrinsic motivations}\n\nIn our model, there are three components to the reward that enter into agents' loss functions (1) total reward, which is used for the policy loss, (2) extrinsic reward, which is used for the extrinsic value function loss and (3) intrinsic reward, which is used for the intrinsic value function loss.\n\nThe {\\em total reward} for player $i$ is the sum of the extrinsic reward and an intrinsic reward as follows:\n\\begin{align}\\label{eq:totalreward}\n    r_i(s_i,a_i) = r_i^E(s_i,a_i) + u_i(\\mathbf{f}_i) \\, .\n\\end{align}\nThe {\\em extrinsic reward} $r^E_i(s, a)$ is the environment reward obtained by player $i$ when it takes action $a_i$ from state $s_i$, sometimes also written with a time index $t$.\nThe {\\em intrinsic reward} $u(\\mathbf{f})$ is an aggregate social preference across features $\\mathbf{f}$ and is calculated according to the formula,\n\\begin{align}\\label{eq:inequityextend}\nu_{i}(\\mathbf{f}_i | \\boldsymbol{\\theta}) = \\mathbf{v}^\\mathrm{T} \\sigma \\left ( \\mathbf{W}^\\mathrm{T} \\mathbf{f}_i  + \\mathbf{b} \\right )  \\,,\n\\end{align}\nwhere $\\sigma$ is the ReLU activation function, and $\\boldsymbol{\\theta} = \\{\\mathbf{W}, \\mathbf{v}, \\mathbf{b}\\}$ are the parameters of a 2-layer neural network with 2 hidden nodes. These parameters are evolved based on fitness (see Section \\ref{sec:arch_training}). The elements of $\\mathbf{v} = (v_1, v_2)$ approximately correspond to a linear combination of the coefficients related to advantagenous and disadvantagenous inequity aversion mentioned in \\cite{hughes2018inequity}, which were found via grid search in this previous work, but are here evolved. \n\n\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[width=1.\\textwidth]{figures\/arch_training2_NEW}\n    \\vspace{-1.0cm}\n\\caption{(a) Agent $A_j$ adjusts policy $\\pi_j(s,a|\\phi)$ using off-policy importance weighted actor-critic (V-Trace) \\citep{espeholt2018impala} by sampling from a queue with (possibly stale) trajectories recorded from 500 actors acting in parallel arenas. (b) The architecture (shown only for 1 agent) includes a visual encoder (1-layer convolutional neural net with 6 3x3 filters, stride 1, followed by two fully-connected layers with 32 units each), intrinsic and extrinsic value heads ($V^I$ and $V^E$), a policy head $\\pi$, and a long-short term memory (LSTM, with 128 hidden units), which takes last intrinsic and extrinsic rewards ($u(\\mathbf{f})$ and $r^E$) and last action as input. The reward network weights are evolved based on total episode return.}\n\\label{fig:arch}\n\\vspace{-.25cm}\n\\end{figure*} \n\n\n\n\\input{pseudocode}\n\n\nThe feature vector $\\mathbf{f}_i$ is a player-specific vector quantity that agents can transform into intrinsic reward via their reward network. It's composed of features $f_{ij}$ derived from all players \\footnote{Note that we use both $i$ and $j$ to index over the players, but $i$ makes reference to the player \\emph{receiving} the intrinsic reward, while $j$ indexes the players \\emph{sending} the features over which the intrinsic reward of player $i$ is defined.}, so that each player has access to the same set of features, with the exception that its own feature is demarcated specially (by always occupying the first element of the vector). The features themselves are a function of recently received or expected future (extrinsic) reward for each agent. In Markov games the rewards received by different players may not be aligned in time. Thus, any model of social preferences should not be overly influenced by the precise temporal alignment of different players' rewards.  Intuitively, they ought to depend on comparing temporally averaged reward estimates between players, rather than instantaneous values. Therefore, we considered two different ways of temporally aggregating the rewards. \n\n\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[width=0.85\\textwidth]{figures\/evolution_schematic}\n    \\vspace{-.5cm}\n\\caption{(a) Agents assigned and evolved with individual reward networks. (b) Assortative matchmaking, which preferentially plays cooperators with other cooperators and defectors with other defectors. (c) A single reward network is sampled from the population and assigned to all players, while 5 policy networks are sampled and assigned to the 5 players individually. After the episode, policy networks evolve according to individual player returns, while reward networks evolve according to aggregate returns over all players.}\n\\label{fig:evo}\n\\vspace{-.25cm}\n\\end{figure*} \n\n\nThe {\\em retrospective} method derives intrinsic reward from whether an agent judges that other agents have been actually (extrinsically) rewarded in the recent past. The {\\em prospective} variant derives intrinsic reward from whether other agents are expecting to be (extrinsically) rewarded in the near future.\\footnote{Our terms prospective and retrospective map onto the terms intentional and consequentialist respectively as used by \\cite{lerer17, peysakhovich2018}.} For the retrospective variant, $f_{ij} = e^t_j$, where the temporally decayed reward $e_j^t$ for the agents $j = 1,\\dots, N$ are updated at each timestep $t$ according to\n\\begin{equation}\n     e_j^t= \\eta \\,e_j^{t-1} + r_j^{E,t} \\, ,\n\\end{equation}\nand $\\eta = 0.975$. The prospective variant uses the value estimates $V^E_j$ (see Figure \\ref{fig:arch}b) for $f_{ij}$ and has a stop-gradient before the reward network module so that gradients don't flow back into other agents (as in for example DIAL from \\citep{foerster2016}).\n\n\n\\subsection{Architecture and Training}\n\\label{sec:arch_training}\n\n\nWe used the same training framework as in \\cite{jaderberg2018human}, which performs distributed asynchronous training in multi-agent environments, including population-based training (PBT) \\citep{jaderberg2017population}. We trained a population of $N=50$ agents\\footnote{Similar to as in \\citep{espeholt2018impala}, we distinguish between an \"agent\" which acts in the environment according to some policy, and a \"learner\" which updates the parameters of a policy. In principle, a single agent's policy may depend on parameters updated by several separate learners.} with policies $\\{\\pi_i\\}$, from which we sampled $5$ players in order to populate each of $500$ arenas (where \\emph{arena} is an instantiation of a single episode of the environment) running in parallel. Within each arena, an episode of the environment was played with the sampled agents, before resampling new ones. Agents were sampled using one of two matchmaking processes (described in more detail below). Episode trajectories lasted 1000 steps and were written to queues for learning, from which weights were updated using V-Trace (Figure \\ref{fig:arch}a).\n\nThe set of weights evolved included learning rate, entropy cost weight, and reward network weights $\\theta$\\footnote{We can imagine that the reward weights are simply another set of optimization hyperparameters since they enter into the loss.}. The parameters of the policy network $\\phi$ were inherited in a Lamarckian fashion as in \\citep{jaderberg2017population}.  Furthermore, we allowed agents to observe their last actions $a_{i,t-1}$, last intrinsic rewards ($u_{i,t-1}(\\mathbf{f}_i)$), and last extrinsic rewards ($r_{i,t-1}^E(s_i,a_i)$) as input to the LSTM in the agent's neural network. \n\nThe objective function was identical to that presented in \\cite{espeholt2018impala} and comprised three components: (1) the value function gradient, (2) policy gradient, and (3) entropy regularization, weighted according to hyperparameters baseline cost and entropy cost (see Figure \\ref{fig:arch}b). \n\nEvolution was based on a fitness measure calculated as a moving average of total episode return, which was a sum of apples collected minus penalties due to tagging, smoothed as follows:\n\\begin{equation}\n     F_i^{n}= (1-\\nu) F_i^{n-1} + \\nu R_i^{n} \\, ,\n\\end{equation}\nwhere $\\nu = 0.001$ and $R_i^n = \\sum_t r^{E,t}_{i}$ is the total episode return obtained on episode $n$ by agent $i$ (or reward network $i$ in the case of the shared reward network evolution (see Section \\ref{sec:multi_evolve} for details).\n\nTraining was done via joint optimization of network parameters via SGD and hyperparameters\/reward network parameters via evolution in the standard PBT setup. Gradient updates were applied for every trajectory up to a maximum length of 100 steps, using a batch size of 32. Optimization was via RMSProp with epsilon=$10{^{-5}}$, momentum=0, decay rate=0.99, and an RL discount factor of 0.99. The baseline cost weight (see \\citet{mnih2016}) was fixed at 0.25, and the entropy cost was sampled from LogUniform($2\\times10{^{-4}}$,0.01) and evolved throughout training using PBT. The learning rates were all initially set to $4\\times10{^{-4}}$ and then allowed to evolve.\n\nPBT uses evolution (specifically genetic algorithms) to search over a space of hyperparameters rather than manually tuning or performing a random search, resulting in an adaptive schedule of hyperparameters and joint optimization with network parameters learned through gradient descent \\cite{jaderberg2017population}. \n\nThere was a mutation rate of $0.1$ when evolving hyperparameters, using multiplicative perturbations of $\\pm 20\\%$ for entropy cost and learning rate, and additive perturbation of $\\pm 0.1$ for reward network parameters.  We implemented a burn-in period for evolution of $4\\times10^6$ agent steps, to allow network parameters and hyperparameters to be used in enough episodes for an accurate assessment of fitness before evolution.\n\n\n\n\\subsection{Random vs. assortative matchmaking}\n\nMatches were determined according to two methods: (1) random matchmaking and (2) assortative matchmaking. Random matchmaking simply selected uniformly at random from the pool of agents to populate the game, while cooperative matchmaking first ranked agents within the pool according to a metric of recent cooperativeness, and then grouped agents such that players of similar rank played with each other. This ensured that highly cooperative agents played only with other cooperative agents, while defecting agents played only with other defectors. For Cleanup, cooperativeness was calculated based on the amount of steps in the last episode the agent chose to clean. For Harvest, it was calculated based on the difference between the the agent's return and the mean return of all players, so that having less return than average yielded a high cooperativeness ranking. Cooperative metric-based matchmaking was only done with either individual reward networks or no reward networks (Figure \\ref{fig:evo}b). We did not use cooperative metric-based matchmaking for our multi-level selection model, since these are theoretically separate approaches.\n\n\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/rewevo_episode_reward_recolored-01.png}\n    \\vspace{-.5cm}\n\\caption{Total episode rewards, aggregated over players. (a), (b): Comparing retrospective (backward-looking) reward evolution with assortative matchmaking and PBT-only baseline in (a) Cleanup and (b) Harvest. (c), (d): Comparing prospective (forward-looking) with retrospective (backward-looking) reward evolution in (c) Cleanup and (d) Harvest. The black dotted line indicates performance from \\cite{hughes2018inequity}. The shaded region shows standard error of the mean, taken over the population of agents.}\n\\label{fig:results1}\n\\vspace{-.25cm}\n\\end{figure*}\n\n\n\\subsection{Individual vs. shared reward networks}\n\\label{sec:multi_evolve}\n\nBuilding on previous work that evolved either the intrinsic reward \\citep{jaderberg2018human} or the entire loss function \\citep{houthooft2018evolved}, we considered the reward network weights to be hyperparameters that could be evolved in parallel with the policy parameters (Figure \\ref{fig:evo}a). Distinct from these methods, we separately evolved the reward network within its own population, thereby allowing different modules of the agent to compete only with like components. This allowed for independent exploration of hyperparameters via separate credit assignment of fitness, and thus considerably more of the hyperparameter landscape could be explored compared with using only a single pool. In addition, reward networks could be randomly assigned to any policy network, and so were forced to generalize to a wide range of policies. In a given episode, $5$ separate policy networks were paired with the same reward network, which we term a {\\em shared reward network}. In line with \\citep{jaderberg2017population}, the fitness determining the copying of policy network weights and evolution of optimization-related hyperparameters (entropy cost and learning rate) were based on individual agent return. By contrast, the reward network parameters were evolved according to fitness based on total episode return across the group of co-players (Figure \\ref{fig:evo}c).\n\n\n\nThis contribution is distinct from previous work which evolved intrinsic rewards \\citep[e.g.][]{jaderberg2018human} because (1) we evolve over social features rather than a remapping of environmental events, and (2) reward network evolution is motivated by dealing with the inherent tension in ISDs, rather than merely providing a denser reward signal. In this sense it's more akin to evolving a form of communication for social cooperation, rather than learning reward-shaping in a sparse-reward environment. We allow for multiple agents to share the same components, and as we shall see, in a social setting, this winds up being critical. Shared reward networks provide a biologically principled method that mixes group fitness on a long timescale and individual reward on a short timescale. This contrasts with hand-crafted means of aggregation, as in previous work \\citep{chang2004all,mataric1994learning}.\n\n\n\\section{Results}\n\n\nAs shown in Figure \\ref{fig:results1}, PBT without using an intrinsic reward network performs poorly on both games, where it asymptotes to 0 total episode reward in Cleanup and 400 for Harvest (the number of apples gained if all agents collect as quickly as they can). \n\nFigures \\ref{fig:results1}a-b compare random and assortative matchmaking with PBT and reward networks using retrospective social features. When using random matchmaking, individual reward network agents perform no better than PBT at Cleanup, and only moderately better at Harvest.  Hence there is little benefit to adding reward networks over social features if players have separate networks, as these tend to be evolved selfishly. The assortative matchmaking experiments used either no reward network ($u(\\mathbf{f})$ = 0) or individual reward networks. Without a reward network, performance was the same as the PBT baseline. With individual reward networks, performance was very high, indicating that both conditioning the internal rewards on social features and a preference for cooperative agents to play together were key to resolving the dilemma. On the other hand, shared reward network agents perform as well as assortative matchmaking and the handcrafted inequity aversion intrinsic reward from \\citep{hughes2018inequity}, even using random matchmaking. This implies that agents didn't necessarily need to have immediate access to honest signals of other agents' cooperativeness to resolve the dilemma; it was enough to simply have the same intrinsic reward function, evolved according to collective episode return. Videos comparing performance of the PBT baseline with the retrospective variant of shared reward network evolution can be found at \\href{https:\/\/www.youtube.com\/watch?v=medBBLLM4c0}{https:\/\/youtu.be\/medBBLLM4c0} and \\href{https:\/\/www.youtube.com\/watch?v=yTjrlH3Ms9U}{https:\/\/youtu.be\/yTjrlH3Ms9U}.\n\nFigures \\ref{fig:results1}(c) and (d) compare the retrospective and prospective variants of reward network evolution. The prospective variant, although better than PBT when using a shared reward network, generally results in worse performance and more instability. This is likely because the prospective variant depends on agents learning good value estimates before the reward networks become useful, whereas the retrospective variant only depends on environmentally provided reward and thus does not suffer from this issue. Interestingly, we observed that the prospective variant does achieve very high performance if gradients are allowed to pass between agents via the value estimates $V^E_j$ (data not shown); however, this constitutes centralized learning, albeit with decentralized execution (see \\cite{foerster2016}). Such approaches are promising but less consistent with biologically plausible mechanisms of multi-agent learning which are of interest here and so were not pursued.\n\n\n\\begin{figure*}[htb]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{figures\/social_outcome_metrics_recolored-01.png}\n    \\vspace{-.25cm}\n\\caption{Social outcome metrics for (a) Cleanup and (b) Harvest. \\textit{Top:} equality, \\textit{middle:} total amount of tagging, \\textit{bottom:} sustainability. The shaded region shows the standard error of the mean.}\n\\label{fig:results2}\n\\vspace{-.25cm}\n\\end{figure*}\n\n\n\nWe next plot various social outcome metrics in order to better capture the complexities of agent behavior (see Figure \\ref{fig:results2}). \nEquality is calculated as $\\mathbb{E}(1-G(\\mathbf{R}))$, where $G(\\mathbf{R})$ is the Gini coefficient over individual returns. Figure \\ref{fig:results2}b demonstrates that, in Harvest, having the prospective version of reward networks tends to lead to lower equality, while the retrospective variant has very high equality. Equality in Cleanup is more unstable throughout training, since it's not necessarily optimal, but tends to be lower overall than for Harvest, even when performance is high, indicating that equality might be harder to achieve in different games.\nTagging measures the average number of times a player fined another player throughout the episode. The middle panel of Figure \\ref{fig:results2}b shows that there is a higher propensity for tagging in Harvest when using either a prospective reward network or an individual reward network, compared to the retrospective shared reward network. This explains the performance shown in Figure \\ref{fig:results1}, as being tagged results in a very high negative reward. Tagging in the Cleanup task is overall much lower than in Harvest.\nSustainability measures the average time step on which agents received positive reward, averaged over the episode and over agents. We see in the bottom panel of \\ref{fig:results2}b that having no reward network results in players collecting apples extremely quickly in Harvest, compared with much more sustainable behavior with reward networks. In Cleanup, the sustainability metric is not meaningful and so this was not plotted.\n\n\nFinally, we can directly examine the weights of the final retrospective shared reward networks which were best at resolving the ISDs. Interestingly, the final weights evolved in the second layer suggest that resolving each game might require a different set of social preferences. In Cleanup, one of the final layer weights $v_2$ evolved to be close to $0$, whereas in Harvest, $v_1$ and $v_2$ evolved to be of large magnitude but opposite sign. We can see a similar pattern with the biases $\\mathbf{b}$. We interpret this to mean that Cleanup required a less complex reward network: it was enough to simply find other agents' being rewarded as intrinsically rewarding. In Harvest, however, a more complex reward function was perhaps needed in order to ensure that other agents were not over-exploiting the apples. We found that the first layer weights $\\mathbf{W}$ tended to take on arbitrary (but positive) values. This is because of random matchmaking: co-players were randomly selected and thus there was little evolutionary pressure to specialize these weights.\n\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{figures\/compare_weights}\n\\caption{Distribution of layer 2 weights and biases of evolved retrospective shared reward network at $1.5\\times10{^{8}}$ training steps for (a) Cleanup, and (b) Harvest.}\n\\vspace{-0.3cm}\n\\label{fig:results3}\n\\end{figure}\n\\section{Introduction}\n\nNature shows a substantial amount of cooperation at all scales, from microscopic interactions of genomes and bacteria to species-wide societies of insects and humans \\citep{smith1997major}. This is in spite of natural selection pushing for short-term individual selfish interests \\citep{darwin1859}. In its purest form, altruism can be favored by selection when cooperating individuals preferentially interact with other cooperators, thus realising the rewards of cooperation without being exploited by defectors \\citep{hamilton-1964a, hamilton1964genetical, dawkins1976selfish, santos2006cooperation, fletcher2009simple}.  However, many other possibilities exist, including kin selection, reciprocity and group selection \\citep{Nowak1560, ubeda2011power, trivers1971evolution, nowak2005evolution, wilson1975theory, smith1964group}. \n\nLately the emergence of cooperation among self-interested agents has become an important topic in multi-agent deep reinforcement learning (MARL). \\cite{leibo17} and \\cite{hughes2018inequity} formalize the problem domain as an {\\em intertemporal social dilemma} (ISD), which generalizes matrix game social dilemmas to Markov settings. Social dilemmas are characterized by a trade-off between collective welfare and individual utility. As predicted by evolutionary theory, self-interested reinforcement-learning agents are typically unable to achieve the collectively optimal outcome, converging instead to defecting strategies \\citep{leibo17, perolat17}.\nThe goal is to find multi-agent training regimes in which individuals resolve social dilemmas, i.e., cooperation emerges.\nPrevious work has found several solutions, belonging to three broad categories: 1) opponent modelling \\citep{foerster17, KleimanWeiner2016}, 2) long-term planning using perfect knowledge of the game's rules  \\citep{lerer17, peysakhovich2018} and 3) a specific intrinsic motivation function drawn from behavioral economics \\citep{hughes2018inequity}. These hand-crafted approaches run at odds with more recent end-to-end model-free learning algorithms, which have been shown to have a greater ability to generalize (e.g. \\citep{espeholt2018impala}). We propose that evolution can be applied to remove the hand-crafting of intrinsic motivation, similar to other applications of evolution in deep learning.\n\nEvolution has been used to optimize single-agent hyperparameters \\citep{jaderberg2017population}, implement black-box optimization \\citep{wierstra2008natural}, and to evolve neuroarchitectures \\citep{miller1989designing, stanley2002evolving}, regularization \\citep{chan2002alleviating}, loss functions \\citep{jaderberg2018human, houthooft2018evolved}, behavioral diversity \\citep{conti2017improving}, and entire reward functions \\citep{singh2009rewards, singh2010intrinsically}. These principles tend to be driven by single-agent search and optimization or competitive multi-agent tasks. Therefore there is no guarantee of success when applying them in the ISD setting. More closely related to our domain are evolutionary simulations of predator-prey dynamics \\citep{yong2001cooperative}, which used enforced subpopulations to evolve populations of neurons which are sampled to form the hidden layer of a neural network.\\footnote{See also \\cite{potter2000cooperative} and \\cite{panait2005cooperative} for reviews of other evolutionary approaches to cooperative multi-agent problems.}\n\nTo address the specific challenges of ISDs, the system we propose distinguishes between optimization processes that unfold over two distinct time-scales: (1) the fast time-scale of learning and (2) the slow time-scale of evolution \\citep[similar to][]{hinton1987learning}. In the former, individual agents repeatedly participate in an intertemporal social dilemma using a fixed intrinsic motivation. In the latter, that motivation is itself subject to natural selection in a population. We model this intrinsic motivation as an additional additive term in the reward of each agent \\citep{chentanez2005}. We implement the intrinsic reward function as a two-layer fully-connected feed-forward neural network, whose weights define the genotype for evolution. We propose that evolution can help mitigate this intertemporal dilemma by bridging between these two timescales via an intrinsic reward function.\n\nEvolutionary theory predicts that evolving individual intrinsic reward weights across a population who interact uniformly at random does not lead to altruistic behavior \\citep{axelrod81}. Thus, to achieve our goal, we must structure the evolutionary dynamics \\citep{Nowak1560}. We first implement a ``Greenbeard'' strategy \\citep{dawkins1976selfish, jansen2006} in which agents choose interaction partners based on an honest, real-time signal of cooperativeness. We term this process {\\em assortative matchmaking}. Although there is ecological evidence of assortative matchmaking \\citep{keller98}, it cannot explain cooperation in all taxa \\citep{grafen1990animals, henrich2003, gardner2010greenbeards}. Moreover it isn't a general method for multi-agent reinforcement learning, since honest signals of cooperativeness are not normally observable in the ISD models typically studied in deep reinforcement learning.\n\n\n\n\n\\begin{figure*}[htb]\n  \\centering     \n  \\includegraphics[width=0.7\\textwidth]{figures\/ssd_tasks}\n  \\vspace{-.25cm}\n     \\caption{Screenshots from (a) the Cleanup game, (b) the Harvest game. The size of the agent-centered observation window is shown in (b). The same size observation was used in all experiments.}\n     \\vspace{-.25cm}\n     \\label{fig:gallery}\n\\end{figure*}\n\n\n\nTo address the limitations of the assortative matchmaking approach, we introduce an alternative modular training scheme loosely inspired by ideas from the theory of multi-level (group) selection \\citep{wilson1975theory, henrich2003}, which we term {\\em shared reward network} evolution. Here, agents are composed of two neural network modules: a policy network and a reward network. On the fast timescale of reinforcement learning, the policy network is trained using the modified rewards specified by the reward network. On the slow timescale of evolution, the policy network and reward network modules evolve separately from one another. In each episode every agent has a distinct policy network but the same reward network. As before, the fitness for the policy network is the individual's reward. In contrast, the fitness for the reward network is the collective return for the entire group of co-players. In terms of multi-level selection theory, the policy networks are the lower level units of evolution and the reward networks are the higher level units. Evolving the two modules separately in this manner prevents  evolved reward networks from overfitting to specific policies. This evolutionary paradigm not only resolves difficult ISDs without handcrafting but also points to a potential mechanism for the evolutionary origin of social inductive biases.\n\nThe paper is structured as follows. \nIn Section 2, we define our problem domain, and describe in detail our agent architecture and training methods. In Section 3, we present results from our experiments and further analyses of agent policies. Finally in Section 4, we discuss interpretations of our model as well as make suggestions for future work.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the mid-2000s, mobile telephones have become an inseparable part of our lives. The constant communication between a device and the mobile network leaves an involuntary trace of our activities. The spatio-temporal logs of our journeys and additional communication device information establish a promising model to evaluate human mobility and socioeconomic customs.\n\nFor the last two decades, uncovering underlying information from mobile phone records has been a developing research field. Data scientists, spatial data analysts, physicists and applied mathematicians pay more and more attention to discovering cellular data. Interpreting large amounts of Call Detail Records (CDRs) into useful information requires various tools and expertise. In the last ten years, dozens of research groups have published several major research journals discussing different applications of mobile network data analysis.\n\nSt. Stephen's Day is celebrated in Hungary every year on the 20$^{th}$ of August. Tens of thousands of people visit the capital for its all day long celebrations and the main event, a 30-minute-long firework show. The main area of the event includes three bridges and the embankments on the Danube for approximately three kilometres. With great views from the Buda and Pest embankments and the Castle District, these areas should show a significant spike in cellular activity and will be the primary subject of the analysis.\n\n\n\\section{Related works}\nUsing call detail records spanning over 52 weeks, accumulated over two-week-long periods, Gonzalez \\textit{et al.} analysed 100,000 randomly selected individuals' movements over half a year in Europe. They introduced basic human mobility patterns and discovered that most individuals travel only short distances, and just a few move over hundreds of kilometres. The study approximated the probability density function of travel distances with a truncated power-law \\cite{gonzalez2008understanding}.\n\nCandia \\textit{et al.} carried out a comprehensive study on the mean collective behaviour of individuals and examined how space and time deviations can be described using known percolation theory tools. They also proved that the inter-event time between consecutive calls is heavy-tailed, agreeing with previous studies on other human activities \\cite{candia2008uncovering}.\n\nA typical application of CDR processing is the sizeable social event detection and estimating the attendance during mass gatherings \\cite{wirz2013probing, mamei2016estimating, barnett2016social, hiir2019impact}.\n\nThis study builds upon previous works at John von Neumann Faculty of Informatics, \u00d3buda University. The 2014 State Foundation Day data set has already been analysed \\cite{pinter2018evaluation} regarding the large social event. Whereas in this work, the socioeconomic status of the attendees is studied.\n\nUsing mobile phone prices as Socioeconomic Status (SES) indicators has been proved to work well by Sultan \\textit{et al.} in \\cite{sultan2015mobile}. They identified areas in Pakistan where more expensive phones appear more often using indicators of accessibility to services, infrastructure, hygiene and communication. Their model performed with an absolute Pearson's correlation coefficient $> 0.35$ and p-value $< 0.01$.\n\nIn an earlier study, Pint\u00e9r \\textit{et al.} evaluated the connection between individuals' financial status and mobility customs. The authors used the radius of gyration, entropy, and Euclidean distance between home and work locations as mobility indicators and applied data fusion methods with average real estate prices to determine the influence of wealth on mobility customs \\cite{pinter2021evaluating}.\n\nRegarding socioeconomic status analysis, Pint\u00e9r \\textit{et al.} evaluated football fans' SES using their mobile phone details in Budapest during the 2016 UEFA European Football Championship. They eliminated CDRs from Subscriber Identity Modules (SIMs) during data preprocessing, which did not operate in mobile phones using Type Allocation Code (TAC) databases. \\cite{pinter2021analyzing} In another work, they analysed subscribers' wake-up times and explained how it correlates with their socioeconomic status. The analysis demonstrated a strong positive connection between the two indicators. They also showed that the mobile phone prices in the TAC database might have depreciated \\cite{pinter2022awakening}.\n\n\n\\section{Mobile network data}\n\nA CDR data set usually contains a caller ID, the cell tower it is connected to, its location, and a timestamp. Additional information on the purpose of communication, device type, and SIM holders' details help investigate more than just trajectories and cell densities.\n\nThe data set used for this research was obtained by Vodafone Hungary Ltd. The number of active SIMs was 11,540,058 in Hungary, of which Vodafone had an estimated 22.7 per cent of the market share in June 2014 (Hungarian National Media and Infocommunications Authority). These CDRs contain anonymous logs of customers' calls and text messages in Budapest, Hungary and its suburban areas, over approximately 525.14 km$^2$ (Hungarian Central Statistical Office).\n\nThe data set was collected between the 18$^{th}$ and 22$^{nd}$ of August 2014. A total amount of 191,528,883 records have been logged, between 8,890 cells. Three comma-separated value files have been acquired for the analysis in this research. The first one contained call data records, the second and third are supplementary information about cells and devices.\n\nThe CDRs in this data set consist of a timestamp, a hashed device identification (ID), a hashed cell ID and a type allocation code. TAC is the initial eight-digit segment of a device's International Mobile Equipment Identity (IMEI), uniquely identifying a particular device. In this data set, CDRs are of active call record type, meaning a record was made when the user was making a call or sending\/receiving a text message. Unfortunately, the data set does not contain information on cell switching, which would make more granulated data possible.\n\nThe supplementary cell lookup table contains cell IDs and positions as 2D coordinates in decimal degrees format. Cells at the same location were merged into base stations for further analysis, and the corresponding CDR's cell ID values were updated. Uniting particular cells is necessary for more straightforward data analysis. Nevertheless, cell tower antennas planted at an exact location might face different directions, but we do not have this information.\n\nThe device table contains a hashed device ID, the customer's age, gender, whether it is an individual or a business, and the subscription type (prepaid or postpaid). Some age and gender information is missing due to privacy restrictions.\n\n\n\\section{Methodology}\n\nDuring data cleansing, unnecessary spaces have been removed from the end of the lines. Furthermore, device and cell IDs were hashed information, a fundamental step for user privacy. On the contrary, it has a low information density and thus has been replaced with incrementing integer values. The reassigning does not affect the contained information, but is favourable to reducing data size. Due to the considerable amount of CDRs, the comma-separated value text files have been loaded into an SQLite database, with the scheme illustrated in Figure \\ref{fig:database}. To speed up data acquisitions, indices were created on timestamps, IDs and TACs.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.5\\linewidth]{images\/database.pdf}\n\t\\caption{The database structure showing CDR, device and cell tables with foreign key connections.}\n\t\\label{fig:database}\n\\end{figure}\n\nAlthough latitude and longitude values are in string format, there is no need for the original precision of 13 decimal points. This would mean $\\mu m$ resolution, while anything above the sixth decimal place is useless in this application. The unnecessary information can be discarded to save space and increase query speeds.\n\nAs a socioeconomic status indicator, the analysis used relative mobile phone ages in months to the event (August 2014) and phone release prices in EUR. Information on resolving the TACs is from the data provided by 51Degrees fused \\cite{pinter2021analyzing} with the GSMArena database \\cite{gsmarena}.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.65\\linewidth]{images\/cells_firework.pdf}\n\t\\caption{The selected cells for the large social event analysis.}\n\t\\label{fig:cellsfirework}\n\\end{figure}\n\nA data processing framework has been developed to acquire the relevant CDRs with the paired SES indicators. The first criteria are spatially being in a position that could indicate attendance at the fireworks. Cells along the two sides of the Danube are expected to be the primary cells servicing the attendees' mobile phones on the embankments. These cell IDs were selected manually due to the uneven separation line on the river. Any other cells that might support the selected areas are determined by a 250 m radius around the main event area, as visible in Figure \\ref{fig:cellsfirework}. Four cells were removed from the evaluation due to the insignificant activity count (less than 500) during the event. Extracting the selected cell IDs helps filter and transform the large CDR database table into a more manageable format. The temporal filtration rule for the fireworks data will be $\\pm$ 30 mins to the actual event. This will include possible additional users who did attend the venues but did not use their phones during the half-hour show.\n\nUser data from the device database table can be joined on the CDRs' device ID fields. This gives us the ability to analyse age and gender distributions in selected groups. The selected CDRs have been joined on the corresponding TAC values from the merged mobile phone property database for the SES indicator aggregation. For further analysis, relative ages of the appearing phones have been calculated from release dates and months to the event date (August 2014).\n\n\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{subfigure}[t]{0.49\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{images\/event_normal.pdf}\n\t\t\\caption{Daily cell activities in the studied area.}\n\t\t\\label{subfig:event_cells}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[t]{0.49\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{images\/event_cells.pdf}\n\t\t\\caption{Daily cell activities between the studied cells.}\n\t\t\\label{subfig:event_normal}\n\t\\end{subfigure}\n\t\\caption{Cell activities showing the extra mobile network load due to the State Foundation Day celebrations on the river Danube embankments and the Castle District in Budapest.}\n\t\\label{fig:event_day}\n\\end{figure}\n\n\n\\section{Results}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\begin{subfigure}[t]{0.49\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{images\/firework_price_in_eur.pdf}\n\t\t\\caption{Mobile phone prices, a higher value means higher SES.}\n\t\t\\label{subfig:firework_price}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[t]{0.49\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{images\/firework_relative_age.pdf}\n\t\t\\caption{Mobile phone relative ages to the event date (August 2014) in months, a higher value means lower SES.}\n\t\t\\label{subfig:firework_age}\n\t\\end{subfigure}\n\t\\caption{Average socioeconomic status indicator distributions by riverside cells among the large social event attendees.}\n\t\\label{fig:firework}\n\\end{figure}\n\nThe analysis focused on the socioeconomic status indicator distribution among the State Foundation Day celebratory firework viewers in Budapest, on the banks of the river Danube and in the Castle District in August 2014. The time frame is 20:00 -- 21:30, including half an hour before and after the 30-minute show, marked with vertical lines in Figure \\ref{fig:event_day}.\n\nA cell-by-cell average of mobile phone prices and relative ages was calculated for the SES indicator distribution. Figure \\ref{fig:firework} shows the spatial distribution of these indicators using Voronoi polygons generated around the cell tower locations. Cells are coloured by the average SES indicators; the higher the value, the darker the colour.\n\nFigure \\ref{fig:firework} demonstrates an opposite trend between mobile phone price and age. The scatter plot of the same data is shown in Figure \\ref{fig:correlation}, where the Pearson correlation coefficient $= -0.7329$. Data points are coloured based on their location in the city, but there are no visible groups based on SES.\n\nThe expectation was that there would be a significant contrast between Buda and Pest in socioeconomic indicator distribution. However, there are only minor differences in this spatial resolution, from which it can be concluded that those interested in the event do not divide drastically into socioeconomic groups.\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.4\\linewidth]{images\/correlation.pdf}\n\t\\caption{The correlation between average phone prices and ages in cells, where Pearson's $r = -0.7329$.}\n\t\\label{fig:correlation}\n\\end{figure}\n\n\\section{Conclusions and future work}\n\nThis paper presented a concept of socioeconomic data analysis on a large social event using call detail records and mobile phone details. This work fits into current research tendencies, fusing mobile network generated mobility data with socioeconomic descriptors that make it possible to deduce socioeconomic status from anonymous call detail records.\n\nWe expected that in Buda, where the housing prices are higher \\cite{pinter2021evaluating}, more expensive and newer phones would generate the majority of the activity. Nonetheless, we found that slightly more expensive phones were active in Pest, but the difference, on average, was not substantial. The base station level aggregation may have partly caused this result, or the attendees might not have watched the fireworks isolated from each other based on social status.\n\nFor future work, the firework attendees could be grouped into visitors and homeowners in the activity areas. Calculating home positions on CDRs has already been demonstrated to be helpful in \\cite{pinter2018evaluation} and could make a difference in the conclusion of this study.\n\n\n\n\\section*{Author contributions}\n\nMethodology, K.S. and G.P.;\nConceptualisation, G.P. and K.S.;\nSoftware -- data processing, K.S.;\nSoftware -- data preprocessing, G.P.;\nValidation, K.S. and G.P.;\nVisualisation, K.S.;\nWriting, K.S.;\nSupervision, G.P. and I.F.\n\n\n\\section*{Acknowledgement}\n\nThe authors would like to thank Vodafone Hungary and 51Degrees for providing the Call Detail Records and the Type Allocation Code database used in this study. Map tiles by CartoDB, under CC BY 3.0.\n\n\n\\printbibliography\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA simple model of itinerant antiferromagnets is provided by electrons on a\nlattice with short-range repulsion. In the low temperature phase, the system\nis in a Spin Density Wave (SDW) state. In three dimensions, above the\ntransition temperature, the electrons form a so-called {\\it nearly\nantiferromagnetic Fermi liquid}. Traditional mean-field techniques for\nstudying SDW instabilities of Fermi liquids fail completely in low\ndimension. In two dimensions for example, the Random Phase Approximation\n(RPA) predicts finite temperature antiferromagnetic transitions while this\nis forbidden by the Mermin-Wagner theorem. Nevertheless, one can study\nuniversal critical behavior using various forms of renormalization group\ntreatments appropriate either for the strong\\cite{Chakravarty}\\cite{Sachdev}%\n\\cite{Chubukov} or the weak-coupling limits\\cite{Hertz}\\cite{Millis}. The\nself-consistent-renormalized approach of Moryia\\cite{moriya} also satisfies\nthe Mermin-Wagner theorem in two dimensions. Since cutoff-dependent scales\nare left undetermined by all these approaches they must be found by other\nmethods. For example, in the strong-coupling limit, the spin-stiffness\nconstant of the non-linear $\\sigma -$model must be determined from Monte\nCarlo simulations. In the weak-coupling case however, Monte Carlo\nsimulations are limited to very small systems, of order $10\\times 10$ that\ndo not allow one to study much of the critical regime.\n\nRecently, the Two-Particle Self-Consistent approach\\cite{Vilk} was developed\nto obtain from a microscopic model a {\\it quantitative} description of\nitinerant electrons not only far from phase transitions, but also in the\ncritical regime. It was shown\\cite{Vilk} that in this approach the\nMermin-Wagner theorem is satisfied and that, away from the critical regime,\nthe approach gives quantitative agreement with Monte Carlo simulations of\nthe nearest-neighbor\\cite{Vilk} and next-nearest neighbor\\cite{Veilleux}\nHubbard model in two dimensions. Quantitative agreement is also obtained as\none enters the narrow critical regime accessible in Monte Carlo simulations.\nThe approach is restricted to the one-band Hubbard model with on-site\ninteraction, but is valid for arbitrary dispersion relation. The TPSC\napproach also allows one to study the case where the instability of the\nitinerant electrons is at an incommensurate wave-vector, but in this paper\nwe restrict ourselves to the case where the order is at the\nantiferromagnetic wave vector. The self-consistent-renormalized approach of\nMoryia\\cite{moriya} cannot deal with the incommensurate case without {\\it a\npriori }information. Even though it has the same critical behavior as the\nTPSC approach it does not allow one to obtain quantitative parameter-free\nresults from a microscopic Hamiltonian.\n\nWe first show in full generality that the TPSC approach gives the leading\nterm of the critical behavior in a $1\/n$ expansion. In other words, it gives\nthe $n\\rightarrow \\infty $ limit of the $O\\left( n\\right) $ model where $n=3$\nis the physically correct (Heisenberg) limit. It will be apparent that there\nis no arbitrariness in cutoff so that, given a microscopic Hubbard model, no\nparameter is left undetermined. One can go with the same theory from the\nnon-critical to the critical regime.\n\nWe then show that the previously studied two-dimensional critical regimes,\nnamely quantum-critical\\cite{Sachdev} and renormalized classical\\cite\n{Chakravarty} are reproduced here to leading order in $1\/n$. In the quantum\ncritical regime, one usually distinguishes two cases\\cite{Sachdev}: Model A,\nwhere the paramagnetic Fermi surface does not intersect the magnetic\nBrillouin zone, and Model B where it does. This distinction is important in\nthe quantum critical regime because it changes the dynamical critical\nexponent. In this paper, we also give results on Model C, the case of\nperfect nesting. In this case, the microscopic approach shows that\nmodifications to frequency-independent thermodynamic properties can arise.\nIn particular, in the two-dimensional perfect-nesting case the usual\nexponential dependence of correlation length on temperature $\\exp \\left(\ncst\/T\\right) $ can be modified to be roughly $\\exp \\left( cst\/T^3\\right) $\nin some temperature region of the renormalized classical regime.\n\nThen we study the renormalized-classical crossover from $d=2$ to $d=3$ in\nthe highly anisotropic case of weakly coupled planes.\\cite{Konno} The\ngeneral theory of such crossover is given in Appendix D, along with a\ndiscussion of universal crossover functions. In the main text it is shown\nthat in the highly anisotropic case the crossover can occur in a rather\nunusual regime, namely $t_{\\Vert }\\ll k_BT_N\\ll t_{\\bot }$ where $t_{\\Vert\n}\\left( t_{\\bot }\\right) $ is the inter (intra) plane hopping integral and $%\nT_N$ is the three-dimensional N\\'{e}el temperature. This regime is unusual\nbecause even though one is dealing with an itinerant fermion system, the\ninequality $t_{\\Vert }\\ll k_BT_N$ means that the smallest fermionic\nMatsubara frequency is larger than the dispersion in the parallel direction,\nmaking the three-dimensional band structure irrelevant for one-particle\nproperties. In the language of Refs.\\cite{BourbonnaisCaron}\\cite{Boies},\nthere is ``no coherent band motion'' in the parallel direction. Physically,\nthe extent of the thermal de Broglie wave packet in the direction\nperpendicular to the planes is smaller than the distance between planes, a\nsituation that does not occur in a usual Fermi liquid since in the isotropic\ncase the inequality $k_BT\\ll E_F$ implies that the thermal de Broglie\nwavelength is much larger than the lattice spacing. Another way of\ndescribing this $t_{\\Vert }\\ll k_BT_N\\ll t_{\\bot }$ situation is to say that\nthe itinerant electrons become unstable at the two-particle level while\ntheir motion in the third direction is still quasi-classical, or quantum\nincoherent, at the single-particle level because of thermal fluctuations. In\nthe more usual situation, coherence at the one-particle level is established\nbefore the phase transition, namely $k_BT_N\\ll t_{\\Vert }\\ll t_{\\bot }$.\nThese two regimes have been extensively discussed in the $d=1$ to $d=3$\ncrossover of Luttinger liquids by Bourbonnais and Caron\\cite\n{BourbonnaisCaron}\\cite{Boies}.\n\nThe above single-particle incoherent regime $t_{\\Vert }\\ll k_BT_N\\ll t_{\\bot\n}$ is likely to be the relevant one for high-temperature superconductors.\nWhile the parent insulating compound $La_2CuO_4$ has been extensively\nstudied in the strong coupling limit, this type of compound is expected to\nbe in an intermediate-coupling regime. Hence, it is legitimate to approach\nthe problem not only from the strong-coupling limit\\cite{Keimer} but also\nfrom the weak-coupling side, especially with the TPSC approach where all\ncutoffs are determined by the microscopic model. This problem is commented\non at the end of the paper. More detailed quantitative comparisons with\nexperiment will appear later.\n\n\\section{Two-Particle Self-Consistent approach}\n\nWe start from the Hubbard model,\n\n\\begin{equation}\nH=-\\sum_{<ij>\\sigma }t_{i,j}\\left( c_{i\\sigma }^{\\dagger }c_{j\\sigma\n}+c_{j\\sigma }^{\\dagger }c_{i\\sigma }\\right) +U\\sum_in_{i\\uparrow\n}n_{i\\downarrow \\,\\,\\,\\,}.  \\label{Hubbard}\n\\end{equation}\nIn this expression, the operator $c_{i\\sigma }$ destroys an electron of spin \n$\\sigma $ at site $i$. Its adjoint $c_{i\\sigma }^{\\dagger }$ creates an\nelectron. The symmetric hopping matrix $t_{i,j}$ determines the band\nstructure. Double occupation of a site costs an energy $U$ due to the\nscreened Coulomb interaction. In the present section, the hopping parameters\nneed not be specified. We work in units where $k_B=1$, and $\\hbar =1$. As an\nexample that occurs later, the dispersion relation in the $d$-dimensional\nnearest-neighbor model when the lattice spacing is $a$ is given by \n\\begin{equation}\n\\epsilon _{{\\bf k}}=-2t\\sum_{i=1}^d\\left( \\cos k_ia\\right) .\n\\end{equation}\nThe nearest-neighbor quasi-two dimensional case will be another case of\ninterest later, \n\\begin{equation}\n\\epsilon _{{\\bf k}}=-2t_{\\bot }\\left( \\cos k_xa_{\\bot }+\\cos k_ya_{\\bot\n}\\right) -2t_{\\Vert }\\cos k_za_{\\Vert }.\n\\end{equation}\n\nThe TPSC approach\\cite{Vilk},\\cite{Vilk2} can be summarized as follows. One\napproximates spin and charge susceptibilities $\\chi _{sp}$, $\\chi _{ch}$ by\nRPA-like forms but with two different effective interactions $U_{sp}$ and $%\nU_{ch}$ which are then determined self-consistently. Although the\nsusceptibilities have an RPA functional form, the physical properties of the\ntheory are very different from RPA\\ because of the self-consistency\nconditions on $U_{sp}$ and $U_{ch}$. The necessity to have two different\neffective interactions for spin and for charge is dictated by the Pauli\nexclusion principle $\\langle n_\\sigma ^2\\rangle =\\langle n_\\sigma \\rangle $\nwhich implies that both $\\chi _{sp}$ and $\\chi _{ch}$ are related to only\none local pair correlation function $\\langle n_{\\uparrow }n_{\\downarrow\n}\\rangle $. Indeed, using the fluctuation-dissipation theorem in Matsubara\nformalism we have the exact sum rules, \n\\begin{equation}\n\\langle n_{\\uparrow }^2\\rangle +\\langle n_{\\downarrow }^2\\rangle +2\\langle\nn_{\\uparrow }n_{\\downarrow }\\rangle -n^2=\\frac 1{\\beta N}\\sum_{\\widetilde{q}%\n}\\chi _{ch}(\\widetilde{q})\n\\end{equation}\nand \n\\begin{equation}\n\\langle n_{\\uparrow }^2\\rangle +\\langle n_{\\downarrow }^2\\rangle -2\\langle\nn_{\\uparrow }n_{\\downarrow }\\rangle =\\frac 1{\\beta N}\\sum_{\\widetilde{q}%\n}\\chi _{sp}(\\widetilde{q})\n\\end{equation}\nwhere $\\beta \\equiv 1\/T$, $n=\\langle n_{\\uparrow }\\rangle +\\langle\nn_{\\downarrow }\\rangle $, $\\widetilde{q}=({\\bf q},iq_n)$ with ${\\bf q}$ the\nwave vectors of an $N$ site lattice, and with $iq_n=2\\pi inT$ the bosonic\nMatsubara frequencies. The Pauli principle $\\langle n_\\sigma ^2\\rangle\n=\\langle n_\\sigma \\rangle $ applied to the left-hand side of both equations\nwith our RPA-like forms for $\\chi _{sp}$, $\\chi _{ch}$ on the right-hand\nside lead to \n\\begin{equation}\nn+2\\langle n_{\\uparrow }n_{\\downarrow }\\rangle -n^2=\\frac 1{\\beta N}\\sum_{%\n\\widetilde{q}}\\frac{\\chi _0(\\widetilde{q})}{1+\\frac 12U_{ch}\\chi _0(%\n\\widetilde{q})},  \\label{sumCharge}\n\\end{equation}\n\\begin{equation}\nn-2\\langle n_{\\uparrow }n_{\\downarrow }\\rangle =\\frac 1{\\beta N}\\sum_{%\n\\widetilde{q}}\\frac{\\chi _0(\\widetilde{q})}{1-\\frac 12U_{sp}\\chi _0(%\n\\widetilde{q})},  \\label{sumSpin}\n\\end{equation}\nwith $\\chi _0(\\widetilde{q})$ the susceptibility for non-interacting\nelectrons.\n\nIf $\\langle n_{\\uparrow }n_{\\downarrow }\\rangle $ is known, $U_{sp}$ and $%\nU_{ch}$ are determined from the above equations. This key quantity $\\langle\nn_{\\uparrow }n_{\\downarrow }\\rangle $ can be obtained from Monte Carlo\nsimulations or by other means. However, it may be also be obtained\nself-consistently\\cite{Vilk} by adding to the above set of equations the\nrelation \n\\begin{equation}\nU_{sp}=g_{\\uparrow \\downarrow }(0)\\,U\\quad ;\\quad g_{\\uparrow \\downarrow\n}(0)\\equiv \\frac{\\langle n_{\\uparrow }n_{\\downarrow }\\rangle }{\\langle\nn_{\\downarrow }\\rangle \\langle n_{\\uparrow }\\rangle }.  \\label{Usp}\n\\end{equation}\nEqs.(\\ref{sumSpin}) and (\\ref{Usp}) define a set of self-consistent\nequations for $U_{sp}$ that involve only two-particle quantities. We call\nthis approach Two-Particle Self-Consistent to contrast it with other\nconserving approximations like Hartree-Fock or FLEX\\cite{FLEX} that are\nself-consistent at the one-particle level, but not at the two-particle\nlevel. The above procedure\\cite{Vilk} reproduces both Kanamori-Brueckner\nscreening as well as the effect of Mermin-Wagner thermal fluctuations,\ngiving a phase transition only at zero-temperature in two dimensions, as\ndiscussed in the following section. Quantitative agreement with Monte Carlo\nsimulations on the nearest-neighbor\\cite{Vilk} and next-nearest-neighbor\nmodels \\cite{Veilleux} is obtained\\cite{Vilk} for all fillings and\ntemperatures in the weak to intermediate coupling regime $U<8t$.\n\nWe emphasize that deep in the critical regime, the {\\it ansatz} Eq.(\\ref{Usp}%\n) fails in the sense that $g_{\\uparrow \\downarrow }(0)$ eventually reaches\nzero at $T=0$ in the nearest-neighbor Hubbard model at half-filling while\nthere is no reason to believe that this really happens. The physically\nappropriate choice in the renormalized classical regime described below, is\nto keep the value of $g_{\\uparrow \\downarrow }(0)$ fixed at its\ncrossover-temperature value. In the numerical calculations also described\nbelow, we are never far enough from $T_X$ to have to worry about this. The\nvalue of $g_{\\uparrow \\downarrow }(0)$ is the one that is determined\nself-consistently.\n\n\\section{Critical behavior of the TPSC approach in arbitrary dimension}\n\nIn this section we discuss the critical behavior of the TPSC approach in\narbitrary dimension for hypercubic systems. It is convenient to set the\nlattice spacing to unity.\n\nAs one approaches a phase transition, one enters the {\\it renormalized\nclassical }regime,\\cite{Chakravarty} where classical thermal fluctuations\ndominate. In this case, the universality class for {\\it static} properties\nis fully determined by two exponents. Dynamics must also be considered so\nthat one introduces a dynamical critical exponent.\n\nWe consider the case where the transition is at the antiferromagnetic wave\nvector ${\\bf Q}_d$ in $d$ dimensions: ${\\bf Q}_2{\\bf =}\\left( \\pi ,\\pi\n\\right) ,$ ${\\bf Q}_3{\\bf =}\\left( \\pi ,\\pi ,\\pi \\right) $ etc. Since ${\\bf Q%\n}_d$ is at the corner of the Brillouin zone, the spin susceptibility $\\chi\n_0\\left( {\\bf Q}_d\\right) $ is always, by symmetry, an extremum. This\nextremum is the absolute maximum at half-filling not only in the\nnearest-neighbor hopping model, but also in more general models with\nnext-nearest-neighbor hopping\\cite{Veilleux}\\cite{Benard}. The\nnearest-neighbor model is discussed in more details at the end of this\nsection. It has some special features resulting from the additional nesting\nsymmetry. In the two-dimensional case, we also comment on peculiarities of\nnesting and on quantum-critical behavior\\cite{Chubukov}\\cite{Sachdev}.\n\n\\subsection{Renormalized classical regime.}\n\nAs one decreases the temperature sufficiently close to the phase transition,\nthere appears a small energy scale $\\delta U$ that measures the proximity to\nthe phase transition as determined by the Stoner criterion. This scale is\ndefined more precisely in Eq.(\\ref{DeltaU}). The key physical point is that\nthis energy scale is the smallest. In particular, it is smaller than the\ntemperature \n\\begin{equation}\n\\delta U\\ll T\n\\end{equation}\nso that the zero Matsubara frequency representing classical behavior\ndominates all others. The self-consistency conditions Eqs.(\\ref{sumSpin})(%\n\\ref{Usp}) then lead to a strong temperature dependence of $\\delta U$. This\nis the renormalized-classical regime\\cite{Chakravarty}. In this regime, the\nantiferromagnetic correlation length $\\xi $ becomes so large that\\cite{Vilk2}\n\\begin{equation}\n\\xi \\gg \\xi _{th}\n\\end{equation}\nwhere \n\\begin{equation}\n\\xi _{th}\\equiv \\frac{\\left\\langle v_F\\right\\rangle }{\\pi T}\n\\end{equation}\nis the single-particle thermal de Broglie wavelength and $\\left\\langle\nv_F\\right\\rangle $ is the Fermi velocity averaged over the Fermi surface.\nThis provides a partial justification for the usual procedure\\cite{Millis}%\n\\cite{Hertz} that eliminates completely the Fermionic variables and\ndescribes the system in terms of collective Bosonic variables, as is usually\ndone in Hubbard-Stratonovich types of approaches.\\cite{Millis}\\cite{Hertz}\n\nWe first show that when most of the temperature dependence of the\nsusceptibility comes from the temperature dependence of $\\delta U$, the\nRPA-like form that we have implies that {\\it in any dimension }the dynamical\nexponent is $z=2$ while the classical exponent $\\gamma \/\\nu =2-\\eta $ takes\nthe value $\\gamma \/\\nu =2.$ The other classical exponent $\\nu $ is\ndetermined from the self-consistency condition Eq.(\\ref{sumSpin}). We show\nthat the corresponding universality class is the same as the $n\\rightarrow\n\\infty $ limit of the $O\\left( n\\right) $ classical model. This universality\nclass is known in turn to be the same as that of the spherical model.\\cite\n{Stanley} We conclude this discussion with the lower critical dimension $d=2$%\n. There the exponent $\\nu $ cannot strictly be defined since, as was shown\nbefore\\cite{Vilk}, the correlation length diverges exponentially at zero\ntemperature instead of diverging as a power law at finite temperature. This\nbehavior is also the one expected from the $n\\rightarrow \\infty $ model,\nalthough nesting leads to different temperature dependences that are\nexplained further.\n\n\\subsubsection{Exponents $\\gamma \/\\nu $ and $z$ in arbitrary dimension}\n\nThe antiferromagnetic transition is characterized by the appearance of a\nsmall energy scale, or equivalently a large correlation length, in the\nretarded spin susceptibility \n\\begin{equation}\n\\chi _{sp}^R({\\bf q,}\\omega )=\\frac{\\chi _0^R({\\bf q,}\\omega )}{1-\\frac 12%\nU_{sp}\\chi _0^R({\\bf q,}\\omega )}.  \\label{GeneralRPA}\n\\end{equation}\nThe small energy scale is set by \n\\begin{equation}\n\\delta U=U_{mf,c}-U_{sp}  \\label{DeltaU}\n\\end{equation}\nwhere the temperature-dependent ''mean-field critical'' interaction \n\\begin{equation}\nU_{mf,c}\\equiv 2\/\\chi _0\\left( {\\bf Q}_d,0\\right)\n\\end{equation}\nis the temperature-dependent value of $U_{sp}$ at which a phase transition\nwould occur according to mean-field theory. In the vicinity of this point\nthe small energy scale $\\delta U$ allows us to approximate $\\chi _{sp}^R(%\n{\\bf q,}\\omega )$ by expanding the denominator near ${\\bf q\\approx Q}_d$ and \n$\\omega \\approx 0$ to obtain,\n\n\\begin{equation}\n\\chi _{sp}^R({\\bf q+Q}_d{\\bf ,}\\omega )\\approx \\xi ^2\\frac 2{U_{sp}\\xi _0^2}%\n\\left[ \\frac 1{1+{\\bf q}^2\\xi ^2-i\\omega \\xi ^2\/D}\\right]  \\label{chiRPA}\n\\end{equation}\nwhere the antiferromagnetic correlation length is defined by \n\\begin{equation}\n\\xi \\equiv \\xi _0\\left( \\frac{U_{sp}}{\\delta U}\\right) ^{1\/2}  \\label{ksi}\n\\end{equation}\nwith the microscopic length scale set by \n\\begin{equation}\n\\xi _0^2\\equiv \\frac{-1}{2\\chi _0\\left( {\\bf Q}_d\\right) }\\left. \\frac{%\n\\partial ^2\\chi _0\\left( {\\bf q,}0\\right) }{\\partial q_x^2}\\right| _{{\\bf q=Q%\n}_d}.  \\label{ksi02}\n\\end{equation}\nThe microscopic diffusion constant $D$ is defined on the other hand by \n\\begin{equation}\n\\frac 1D\\equiv \\frac{\\tau _0}{\\xi _0^2}\n\\end{equation}\nwhere the microscopic relaxation time is, \n\\begin{equation}\n\\tau _0=\\frac 1{\\chi _0\\left( {\\bf Q}_d\\right) }\\left. \\frac{\\partial \\chi\n_0^R\\left( {\\bf Q}_d{\\bf ,}\\omega \\right) }{\\partial i\\omega }\\right|\n_{\\omega =0}.  \\label{Gamma0}\n\\end{equation}\nThis relaxation-time is non-zero in both models $B$ and $C$ where the Fermi\nsurface intersects the magnetic Brillouin zone.\n\nIn the presence of a large correlation length $\\xi $ the scaling $q\\sim \\xi\n^{-1}$ and $\\omega \\sim \\xi ^{-2}$ justifies the neglect of higher-order\nterms in the expansion Eq.(\\ref{chiRPA}). Comparing the approximate form Eq.(%\n\\ref{chiRPA}) with the general scaling expression \n\\begin{equation}\n\\chi _{sp}^R({\\bf q+Q}_d{\\bf ,}\\omega )\\approx \\xi ^{\\gamma \/\\nu }X\\left( \n{\\bf q}\\xi ,\\omega \\xi ^z\\right)  \\label{GeneralScalingChi}\n\\end{equation}\nwhere $X\\left( {\\bf q}\\xi ,\\omega \\xi ^z\\right) $ is a scaling function, we\nimmediately have the announced results, \n\\begin{equation}\n\\frac \\gamma \\nu =2\\quad ;\\quad z=2.\n\\end{equation}\nThe Fisher scaling law $\\eta =2-\\frac \\gamma \\nu $ shows that the anomalous\nexponent $\\eta $ vanishes as in mean-field theory. In the following\nparagraphs, we compute the remaining exponent $\\nu $ to show that above four\ndimensions we do recover mean-field theory $\\nu =1\/2$ while for $2<d<4$, we\nhave the $n\\rightarrow \\infty $ result $\\nu =1\/\\left( d-2\\right) $.\n\n\\subsubsection{Exponent $\\nu $ in $2<d<4$ and equivalence to spherical $%\n\\left( n\\rightarrow \\infty \\right) $ model.}\n\nThe correlation length exponent is determined by solving self-consistently\nEqs.(\\ref{sumSpin}) and (\\ref{Usp}) for the quantity $\\langle n_{\\uparrow\n}n_{\\downarrow }\\rangle =U_{sp}\/U$. In general, we do this numerically using\nsome technical tricks that are discussed below. With this procedure, no\narbitrariness is left in the cutoffs, that are entirely determined from the\nmicroscopic Hubbard model. However, to study analytically the critical\nbehavior, we notice that there is a crossover temperature $T_X$ below which\nthe presence of the small energy scale $\\delta U$ $<<T$ makes the zero\nMatsubara frequency component in the sum rule Eq.(\\ref{sumSpin}) much larger\nthan all the others. This is the {\\it renormalized classical regime}\ndiscussed above. Its existence is a manifestation of critical-slowing down, $%\n\\omega \\sim \\xi ^{-2}$ $\\sim \\delta U$ near a phase transition. Using the\napproximate Lorentzian form Eq.(\\ref{chiRPA}) for the $i\\omega =iq_n=0$\ncomponent we rewrite Eq.(\\ref{sumSpin}) as follows, after a trivial shift of\nintegration variables, \n\\begin{equation}\n\\tilde{\\sigma}^2=\\frac{2T}{U_{sp}\\xi _0^2}\\int \\frac{d^dq}{(2\\pi )^d}\\frac{%\n\\xi ^2}{1+q^2\\xi ^2}.  \\label{Consistency}\n\\end{equation}\nIn this equation, \n\\begin{equation}\n\\tilde{\\sigma}^2=n-2\\langle n_{\\uparrow }n_{\\downarrow }\\rangle -C\\leq 1\n\\label{sigma2-2}\n\\end{equation}\nis the local moment $n-2\\langle n_{\\uparrow }n_{\\downarrow }\\rangle $ minus\ncorrections $C$ that come from the sum over non-zero Matsubara frequencies\n(quantum effects) and from the terms neglected in the Lorentzian\napproximation, namely those coming from short distances $({\\bf q-Q)}^2\\gg\n\\xi ^{-2}$.\n\nContrary to the strong-coupling case, and contrary to more usual approaches%\n\\cite{Sachdev}, $\\tilde{\\sigma}^2$ {\\it here is temperature dependent}\nbecause both $\\langle n_{\\uparrow }n_{\\downarrow }\\rangle $ and $C$ are.\nNevertheless, to find the critical behavior analytically, it suffices to\nnotice that this dependence is regular. In fact, we have that when $T\\simeq\nT_X$, the double occupancy can be approximated by $\\left\\langle n_{\\uparrow\n}n_{\\downarrow }\\right\\rangle =U_{sp}\/U\\approx U_{mf,c}\/U$ when $\\delta\nU\\rightarrow 0$.\n\nAt the N\\'{e}el temperature, $T_{N\\text{, }}$ the correlation length\ndiverges, $\\xi =\\infty ,$ so that Eq.(\\ref{Consistency}) determines the\nN\\'{e}el temperature through \n\\begin{equation}\n\\tilde{\\sigma}^2=\\frac{2T_N}{U_{sp}\\xi _0^2}\\int \\frac{d^dq}{(2\\pi )^d}\\frac \n1{q^2}.  \\label{Neel}\n\\end{equation}\nThe wave vector integration is cutoff at large $q$ by the Brillouin zone $%\n(-\\pi <q_i<\\pi $ for any component $q_i)$ so that the only divergence occurs\nfrom $q=0$ in $d\\leq 2$ ($q=0\\rightarrow q={\\bf Q}_d$ in the original\nintegration variables). Since the left-hand side of this Eq.(\\ref{Neel}) is\nfinite, this divergence prevents the existence of a finite-temperature\nantiferromagnetic phase transition in two dimensions or less.\n\nTo find the correlation length exponent in $2<d<4$, one rewrites Eq.(\\ref\n{Consistency}) in the form, \n\\begin{equation}\n\\tilde{\\sigma}^2=\\frac{2T}{U_{sp}\\xi _0^2}\\int \\frac{d^dq}{(2\\pi )^d}\\left[ \n\\frac{\\xi ^2}{1+q^2\\xi ^2}-\\frac 1{q^2}\\right] +\\frac{2T}{U_{sp}\\xi _0^2}%\n\\int \\frac{d^dq}{(2\\pi )^d}\\left[ \\frac 1{q^2}\\right] .\n\\end{equation}\nUsing the expression for the N\\'{e}el temperature Eq.(\\ref{Neel}), this last\nexpression becomes, \n\\begin{equation}\n\\tilde{\\sigma}^2\\left( 1-\\frac T{T_N}\\right) =\\frac{2T}{U_{sp}\\xi _0^2}\\xi\n^{2-d}\\int \\frac{d^d\\left( q\\xi \\right) }{(2\\pi )^d}\\left[ \\frac 1{1+q^2\\xi\n^2}-\\frac 1{q^2\\xi ^2}\\right] .  \\label{XsiT-TN}\n\\end{equation}\nSince the integral converges at $q\\xi \\rightarrow \\infty $ for $2<d<4$, it\ncan be replaced by a $\\xi $-independent negative number and one finds, \n\\begin{equation}\n\\xi \\sim \\left( \\frac T{T_N}-1\\right) ^{1\/\\left( 2-d\\right) }\\sim \\left( \n\\frac T{T_N}-1\\right) ^{-\\nu }\n\\end{equation}\nwhich gives, \n\\begin{equation}\n\\nu =\\frac 1{d-2}.\n\\end{equation}\nThis exponent and $\\gamma \/\\nu $ $=2$, found in the previous section, are\nthe same as the one for the Berlin-Kac spherical model\\cite{Stanley} or\nequivalently for the generalized Heisenberg model where spins are $n$\ncomponents vectors and $n\\rightarrow \\infty $. In three dimensions, this\nleads to \n\\begin{equation}\n\\nu =1,\\quad \\gamma =2,\\quad \\alpha =-1,\\quad \\beta ={\\frac 12},\\quad \\eta\n=0\\ \\text{{\\rm and}}\\ \\delta =5.  \\label{exponents}\n\\end{equation}\nFor comparison, numerical results\\cite{Pfeuty} for the $3D$ Heisenberg $%\n\\left( n=3\\right) $ model give $\\nu \\sim 0.7$ and $\\gamma \\sim 1.4$.\n\nAbove $d=4$, one recovers the mean-field results $\\gamma =2\\nu $ and $\\nu\n=1\/2$. This last result follows from the fact that in $d>4$, the integral in\nEq.(\\ref{XsiT-TN}) is dominated by the large momentum cutoff so that for $%\n\\xi >>1$, $\\left( 1-\\frac T{T_N}\\right) \\sim \\xi ^{-2}\\int d^dq\/q^4$.\n\n\\subsubsection{Two-dimensional case}\n\nWe have already proven in the last subsection that the transition\ntemperature vanishes in two dimensions. The correlation length may be found%\n\\cite{Vilk} in the renormalized classical regime directly by performing the\nintegral Eq.(\\ref{Consistency}) in $d=2$,\n\n\\begin{equation}\n\\xi =\\xi _0\\left( U_{sp}\/\\delta U\\right) ^{\\frac 12}\\sim \\Lambda ^{-1}\\exp\n(\\pi \\tilde{\\sigma}^2\\xi _0^2U_{sp}\/T)  \\label{expo}\n\\end{equation}\nwhere $\\Lambda \\sim \\pi $ is usually of the order of the size of the\nBrillouin zone, but not always as we discuss below.\n\nIn $d=2$, we call $T_X$ the temperature at which $\\delta U\\ $is much smaller\nthan temperature and the magnetic correlation length $\\xi $ grows\nexponentially. While in higher dimensions a phase transition occurs at\nfinite temperature, in $d=2$ the critical regime with an exponentially\nincreasing $\\xi $ extends all the way to zero temperature. For example, the\ntemperature $T_X$ is plotted as a function of filling in the two-dimensional\nnearest-neighbor Hubbard model for $U=2.5$ in Fig.1 of Ref.\\cite{Vilk}. In\nthis reference, $T_X$ is called a quasi-critical temperature. We stress that\nthere is a range of fillings near half-filling where at $T_X$ it is the\nantiferromagnetic wave vector that grows, despite the fact that at\nzero-temperature the phase transition would be at an incommensurate wave\nvector.\n\nThe exponential growth of the two-dimensional $\\xi $ clearly suggests that\nsmall $3D$ effects existing in real systems may stabilize long-range order\nat ${\\bf Q}_{d=3}$, before $T=0$. We later characterize the crossover driven\nby a small $3D$ hopping parameter $t_{\\Vert }\\ll t_{\\bot }$ from\ntwo-dimensional critical behavior to three-dimensional critical behavior.\nBut before, we comment on the two-dimensional quantum-critical regime and on\npeculiarities induced by nesting in the renormalized-classical regime.\n\n\\subsection{Quantum-critical regime}\n\nWhen there is a critical value of the interaction $U_c$ {\\it at zero\ntemperature} where one finds a paramagnet for $U<U_c$ and an antiferromagnet\nfor $U>U_c$, then the $T=0$, $U=U_c$ point of the phase diagram is a quantum\ncritical point.\\cite{Hertz} The vicinity of this point in two dimensions has\nbeen studied again recently\\cite{Sachdev}. In order to study such a regime\nwithin the Hubbard model at half-filling, one must introduce\nnext-nearest-neighbor hopping since $U_c\\left( T=0\\right) =0$ at this\nfilling in the nearest-neighbor model. One finds that the TPSC approach has\nprecisely the $n\\rightarrow \\infty $ model A or model B quantum critical\nbehavior\\cite{Sachdev}, depending on the specific microscopic model. In\nparticular, $\\xi \\ $scales as $1\/T$ as one approaches the two-dimensional\nquantum critical point from finite temperature. Again, in the TPSC approach\nthe cutoffs are specified without ambiguity. Model C, the perfect nesting\ncase, is relevant only to the renormalized-classical case, as we now discuss.\n\n\\subsection{Peculiarities induced by perfect nesting in the\nrenormalized-classical regime, especially in $d=2$.}\n\nThe dispersion relation of the nearest-neighbor Hubbard model on hypercubic\nlattices in arbitrary dimension satisfies $\\epsilon _{{\\bf k+Q}_d}=-\\epsilon\n_{{\\bf k}}$. Furthermore, at half-filling the particle-hole symmetry implies\nthat the Fermi surface is fully nested, namely $\\mu =0$ so that the equality \n$\\epsilon _{{\\bf k+Q}_d}-\\mu =-\\left( \\epsilon _{{\\bf k}}-\\mu \\right) $ is\nsatisfied for all wave vectors ${\\bf k}$. Slightly away from half-filling,\nnesting in the form $\\epsilon _{{\\bf k+Q}_d}-\\mu \\sim -\\left( \\epsilon _{%\n{\\bf k}}-\\mu \\right) $ is also a good {\\it approximation} at finite\ntemperature as long as $T>\\mu $, as discussed above. When there is perfect\nnesting, the zero-temperature critical interaction vanishes $\\left(\nU_c=0\\right) $. Hence the fully nested Fermi surface, referred to as Model C\nabove, does not have the simple quantum-critical point described in the\nprevious sub-section.\n\nWhen there is perfect nesting, the microscopic interaction-independent\nquantities $\\xi _0^2$ and $\\tau _0$ have a peculiar temperature dependence.\nThis occurs because they are derivatives of the susceptibility which itself\ncontains logarithmic singularities in the zero-temperature limit. These\nquantities are evaluated in two dimensions and in the quasi two-dimensional\ncase in Appendix A. Dimensional arguments that follow simply from this\nappendix show that in $d>2$ \n\\begin{equation}\n\\xi _0^2\\sim 1\/\\left( T^2\\ln T^{-1}\\right)  \\label{xsi0T}\n\\end{equation}\n\\begin{equation}\n\\tau _0\\sim 1\/\\left( T\\ln T^{-1}\\right) .\n\\end{equation}\nIn $d=2$, the $\\ln T^{-1}$ is replaced by $\\ln ^2T^{-1}$.\\cite{NoteT3}\n\nBy contrast, in the case of second-neighbor hopping, nesting is lost and the\nabove quantities are temperature independent for a wide range of values of\nthe second-neighbor hopping constant. The above temperature dependencies are\nthen a special property of nesting. In $d>2$ however, the above temperature\ndependencies are completely negligible in the critical regime since near the\nphase transition one can replace $T$ in the above expressions by $T_N$.\n\nThe only issue then is in two dimensions where the phase transition occurs\nat zero temperature. Even neglecting logarithms for the moment, one sees\nthat since $\\xi _0^2$ scales as $1\/T^2$ over a wide temperature range the\ncorrelation length in Eq.(\\ref{expo}) scales as $\\exp \\left( cst\/T^3\\right) $%\n. By contrast, in strong coupling, or in the non-nesting case of the\nweak-coupling limit, the correlation length scales as $\\exp (cst\/T)$.\n\nThe $\\exp \\left( cst\/\\left( T^3\\ln ^2T^{-1}\\right) \\right) $ behavior is\nhowever largely an unsolved problem. Indeed, in the critical regime in two\ndimensions, fluctuations remove the quasiparticle peak and replace it by\nprecursors of the antiferromagnetic bands, as shown in Ref.\\cite{Vilk2}. It\nis possible then that, in this regime, a more self-consistent treatment\nwould lead to $\\xi _0^2$ independent of temperature, as in the strong\ncoupling case or the non-nested weak-coupling case. It is also likely that\nthere will be an intermediate temperature range where the $\\exp \\left(\ncst\/\\left( T^3\\ln ^2T^{-1}\\right) \\right) $ regime prevails, even if deep in\nthe critical regime self-consistency leads to $\\exp \\left( cst\/T\\right) $\nbehavior.\n\nIt is important to recall that in practical calculations in the TPSC\napproach, one obtains a numerical value for the correlation length without\nadjustable parameter. For example in Fig.1 we present the temperature\ndependence of the correlation length for the two-dimensional\nnearest-neighbor Hubbard model. As discussed in Appendix A, in this case \n\\begin{equation}\n\\xi _0^2\\simeq {0.021U}_{mf,c}t_{\\bot }^2a_{\\bot }^2\/T^2  \\label{NumXsiZero}\n\\end{equation}\nand $U_{sp}\\simeq {U}_{mf,c}$ so that from the slope of the plot and from\nEq.(\\ref{expo}) one finds $\\tilde{\\sigma}^2\\simeq 0.21$. From the plot we\ncan also extract $\\Lambda ^{-1}\\simeq 0.022$ so that $\\xi $ is known without\nadjustable parameter. Appendix B explains physically the orders of magnitude\ntaken by $\\tilde{\\sigma}^2$ and $\\Lambda ^{-1}$ in this model. Similar\ncalculations can be done for arbitrary band structure. In strong-coupling\ncalculations,\\cite{Chakravarty}\\cite{Chubukov} one obtains $\\xi \\sim \\Lambda\n^{-1}\\exp (2\\pi \\rho _S\/T)$ with $\\rho _S$ a cutoff-dependent quantity that\ncan be evaluated only with Monte Carlo simulations.\n\nAnother consequence of the temperature behavior of $\\xi _0$ in Eq.(\\ref\n{xsi0T}) is that {\\it above} $T_X$ there is a range of temperatures for\nwhich the antiferromagnetic correlation length scales as $\\xi $ $\\sim \\xi\n_0\\sim 1\/T$. This behavior should not be confused with quantum-critical\nbehavior, even though the power-law scaling of the correlation length is the\nsame. Indeed, one finds that the argument of the exponential in Eq.(\\ref\n{expo}) is larger than unity in the corresponding regime while in the\nquantum-critical regime the argument of the exponential should be much less\nthan unity.\\cite{Sachdev} In fact the temperature dependence of the\nstaggered susceptibility for $T>T_X$ is also-different from the quantum\ncritical result.\n\n\\section{Quasi two-dimensional systems: Renormalized classical crossover\nfrom $d=2$ to $d=3$.}\n\nThe general discussion of universality in the renormalized-classical\ncrossover from $d=2$ to $d=3$ appears in Appendices C and D. In the present\nsection, we first clarify the various regimes of crossover, according to\nwhether or not single-particle coherence in the third dimension is\nestablished before the phase transition. Then, we go on to discuss the case $%\nt_{\\Vert }\\ll T_N<T_X$ where the SDW instability occurs before interplane\nsingle-particle coherence is established. More specifically, we find the\nscaling of the N\\'{e}el temperature with $t_{\\bot }\/t_{\\Vert }$ as well as\nthe size of the three-dimensional critical regime with the corresponding\nexponents, showing that the results are those of the $n\\rightarrow \\infty $\nlimit. We restore the lattice spacing units $a_{\\Vert }$ along the\nthree-dimensional axis and $a_{\\bot }$ in the planes. We assume however that\nthe ratio $a_{\\Vert }\/a_{\\bot }$ is usually of order unity and numerical\ncalculations are done for $a_{\\Vert }\/a_{\\bot }=1$\n\n\\subsection{One-particle and two-particle crossover from $d=2$ to $d=3.$}\n\nWe consider in this section the highly anisotropic situation where hopping\nbetween planes $t_{\\Vert }$ is much smaller than in-plane hopping, $t_{\\bot\n} $, \n\\begin{equation}\nt_{\\Vert }\\ll t_{\\bot }\n\\end{equation}\nas might occur in the high-temperature superconductor parent compound $%\nLa_2CuO_4$. In this case, we have that the three-dimensional transition\ntemperature to long-range order $T_N$ is always less than the crossover\ntemperature $T_X$ to the characteristic exponential behavior of the\ncorrelation length in two dimensions \n\\begin{equation}\nT_N<T_X.\n\\end{equation}\nThis is so because the microscopic in-plane $\\xi _0^{\\perp }$ and out of\nplane $\\xi _0^{\\Vert }$ lengths satisfy $\\xi _0^{\\perp }$ $\\gg \\xi _0^{\\Vert\n}$.\n\nThe crossover temperature to two-dimensional behavior for itinerant\nantiferromagnet always satisfies \n\\begin{equation}\nT_X<t_{\\bot }.\n\\end{equation}\nTwo limiting cases are then possible, depending on interaction and on\nhopping parallel to the three-dimensional axis $t_{\\parallel }$:\n\n{\\em a) Weak coupling or small anisotropy limit:} \n\\begin{equation}\nT_N<T_X\\ll t_{\\Vert }.\n\\end{equation}\nIn this case, when the transition to three-dimensional behavior occurs the\nthree-dimensional Fermi surface is relevant since the thermal de Broglie\nwavelength $v_F^{\\Vert }\/\\left( \\pi T\\right) \\sim $ $t_{\\Vert }a_{\\Vert }\/T$\nis larger than the distance between planes. In other words, the\nthree-dimensional band structure is relevant to the behavior of {\\it %\nsingle-particle} propagators in Matsubara frequencies before the phase\ntransition occurs. Fermions are quantum-mechanically coherent over more than\na single plane and nesting generally plays a role in the value of the\nordering wave vector. The crossover from two to three dimensional {\\it %\ncritical} behavior would occur in a manner analogous to the anisotropic\nHeisenberg model\\cite{Soukoulis}\\cite{Singh}\\cite{Tesanovic}\\cite{Kosterlitz}%\n.\n\n{\\em b) Intermediate coupling or very large anisotropy:} \n\\begin{equation}\nt_{\\Vert }\\ll T_N<T_X.  \\label{IncoherentRegime}\n\\end{equation}\nHere, long-range order is established before the single-particle coherence\noccurs between planes. A phase transition occurs only because of {\\it %\ntwo-particle} (or particle-hole) coherent hopping. When the phase transition\noccurs, thermal fluctuations are still large enough that coherent\nsingle-particle band motion in the parallel direction has not occured yet.\nThere are several ways of explaining physically what this last statement\nmeans. For example, it is clear that when $t_{\\Vert }\\ll T,$ features of the\nband structure in the parallel direction are irrelevant for single-particle\nproperties since the first Matsubara frequency is larger than the bandwidth\nin that direction. The motion {\\it between} planes is still in that sense\nquasi-classical when the phase transition occurs. Another way of saying what\nthis means is that the thermally induced uncertainty in the parallel wave\nvector is equal to the extent of the Brillouin zone in that direction,\ncorresponding, via the uncertainty principle, to a confinement within each\nplane.\n\nWe do not discuss the intermediate case $T_N<t_{\\Vert }<T_X$ but concentrate\ninstead on the very large anisotropy - intermediate coupling limit just\nintroduced.\n\n\\subsection{Numerical solutions}\n\nThe scaling behavior of the N\\'{e}el temperature $T_N$ and of the\nthree-dimensional crossover temperature are derived in the following two\nsections. We first present the numerical results obtained from the solution\nof the self-consistency relations Eqs.(\\ref{sumSpin}) and (\\ref{Usp}). The\nnumerical integration in Eq.(\\ref{sumSpin}) is made possible by rewriting\nthis equation in the form,\n\n\\begin{equation}\nn-2\\langle n_{\\uparrow }n_{\\downarrow }\\rangle =Ta_{\\Vert }a_{\\bot }^2\\int \n\\frac{d^3q}{\\left( 2\\pi \\right) ^3}\\sum_{iq_n}\\left[ \\chi _{sp}\\left( {\\bf q,%\n}iq_n\\right) -\\chi _{sp}^{as}\\left( {\\bf q,}0\\right) \\delta _{n,0}\\right]\n+Ta_{\\Vert }a_{\\bot }^2\\int \\frac{d^3q}{\\left( 2\\pi \\right) ^3}\\chi\n_{sp}^{as}\\left( {\\bf q,}0\\right) .  \\label{SelfQ2d}\n\\end{equation}\nThe sum over large Matsubara frequencies can be approximated by an integral\nin a controlled manner. The subtraction in the first integral removes\nsingularities of the integrand and makes the integral well behaved. Since\nthe transition occurs in the single-particle incoherent regime Eq.(\\ref\n{IncoherentRegime}), the integrand in square brackets is independent of $%\nq_{\\Vert }$ above the N\\'{e}el temperature. All quantities involving $%\nt_{\\parallel }$ come from the second integral over the asymptotic expression\nfor the susceptibility \n\\begin{equation}\n\\chi _{sp}^{as}\\left( {\\bf q+Q}_3{\\bf ,}0\\right) \\equiv \\frac 2{\\delta U}%\n\\frac 1{1+\\xi _{\\parallel }^2q_{\\parallel }^2+\\xi _{\\perp }^2q_{\\perp }^2}\n\\label{ChiAs}\n\\end{equation}\nwhere \n\\begin{equation}\n\\xi _{\\perp }\\equiv \\xi _0^{\\perp }(U_{sp}\/\\delta U)^{1\/2},  \\label{ksi2per}\n\\end{equation}\n\\begin{equation}\n\\xi _{\\parallel }\\equiv \\xi _0^{\\parallel }(U_{sp}\/\\delta U)^{1\/2}.\n\\label{ksi2par}\n\\end{equation}\nTo have sufficient precision for large two-dimensional correlation lengths,\nit is important to evaluate analytically $\\xi _0^{\\perp }$, $\\xi\n_0^{\\parallel }$, as well as the integral of Eq.(\\ref{ChiAs}) appearing in\nthe consistency equation Eq.(\\ref{SelfQ2d}). This is done respectively in\nAppendices A and C. To perform the second derivatives in the definition of $%\n\\xi _0^{\\perp }$, $\\xi _0^{\\parallel }$, we expand $\\chi _0({\\bf Q}_3)$ in\npowers of $t_{\\parallel }\/T$, keeping only the first non-zero term: thus $%\n\\xi _0^{\\perp }$ does not differ from the one already presented in Eq. (\\ref\n{NumXsiZero}). It is shown in Appendix A that over a wide range of\ntemperatures we have \n\\begin{equation}\n\\frac{\\partial ^2\\chi _0\\left( {\\bf Q}_3\\right) }{\\partial q_{\\bot }^2}\\sim\na_{\\bot }^2\\left[ \\frac{t_{\\bot }^2}{T^2}+O(t_{\\Vert }\/T)\\right] ,\n\\label{derivx}\n\\end{equation}\nand \n\\begin{equation}\n\\frac{\\partial ^2\\chi _0\\left( {\\bf Q}_3\\right) }{\\partial q_{\\Vert }^2}\\sim\na_{\\Vert }^2\\left[ \\frac{t_{\\Vert }^2}{T^2}+O(t_{\\Vert }^3\/T^3)\\right] .\n\\label{derivz}\n\\end{equation}\nThe inter-plane hopping $t_{\\parallel }$ in Eqs.(\\ref{SelfQ2d}) and (\\ref\n{ChiAs}) occurs explicitly only in $\\xi _0^{\\parallel }$ and the above\nresults imply that, \n\\begin{equation}\n\\frac{\\xi _0^{\\parallel }}{\\xi _0^{\\perp }}\\sim \\frac{\\xi _{\\parallel }}{\\xi\n_{\\perp }}\\sim \\frac{t_{\\parallel }a_{\\Vert }}{t_{\\bot }a_{\\bot }}.\n\\label{tzot}\n\\end{equation}\n\nWe present numerical results for the nearest-neighbor Hubbard model in units\nwhere $a_{\\Vert }=a_{\\bot }=1$ and $t_{\\bot }=1$. The value of $T_N\\left(\nt_{\\parallel }\\right) $ appears in Figs.2 and 3 for $U=4$. In Fig.2, we\nclearly see that $T_N$ becomes almost equal to $T_X\\approx 0.2$ for $%\nt_{\\Vert }$ still quite small. The scaling of $T_N\\left( t_{\\parallel\n}\\right) $ shown in Fig.3 is explained in the following subsection. Fig.4\nshows the variation of the in-plane correlation length $\\xi _{\\perp }$ as a\nfunction of temperature for various $t_{\\parallel }$, again for $U=4$. For\nthe purely two-dimensional case $t_{\\parallel }$ $=0$, one can observe for $%\nT\\leq T_X\\simeq 0.2$ the exponential behavior mentioned in the preceding\nsection. At a temperature about $0.16$, the in-plane correlation length $\\xi\n_{\\bot }$ is already as much as $10^3$ (in units where lattice space is\nunity). For $10^{-3}\\leq t_{\\Vert }\\leq 10^{-1}$, $\\xi _{\\bot }$ diverges at\nthe N\\'{e}el temperature located in the narrow range $0.16\\leq T_N\\leq 0.2$.\nFor $t_{\\parallel }$ $\\neq 0$, it is also clear that the crossover to\nthree-dimensional behavior occurs in an extremely narrow temperature range.\nThis is explained below. Note that the last few curves on the right-hand\nside are at the limit of validity of our approximations.\n\nWe note that for $U=4t$, we find that at $T=T_N$ the local moment is equal\nto three quarters of the full moment in the atomic limit, {\\it i.e.} $%\nn-2\\langle n_{\\uparrow }\\rangle \\langle n_{\\downarrow }\\rangle g_{\\uparrow\n\\downarrow }(0)=0.75$,{\\it \\ } $g_{\\uparrow \\downarrow }(0)=\\langle\nn_{\\uparrow }n_{\\downarrow }\\rangle \/\\langle n_{\\uparrow }\\rangle \\langle\nn_{\\downarrow }\\rangle .$ This number is only weakly dependent on\ntemperature in the range studied.\n\n\\subsection{Dependence of the N\\'{e}el temperature $T_N\\,$on $t_{\\Vert }$.\nCrossover exponent.}\n\nFrom the discussion of the previous section, Eqs.(\\ref{SelfQ2d})(\\ref{ChiAs}%\n)(\\ref{Consistency}) we know that the singular part of the self-consistency\ncondition may be written in the following form in the quasi-two dimensional\ncase,\n\n\\begin{equation}\n\\tilde{\\sigma}^2=\\frac{2Ta_{\\Vert }a_{\\bot }^2}{U_{sp}\\left( \\xi _0^{\\perp\n}\\right) ^2}\\int \\frac{d^3q}{(2\\pi )^3}\\frac 1{q_{\\perp }^2+\\xi _{\\perp\n}^{-2}+\\left( \\frac{\\xi ^{\\parallel }}{\\xi ^{\\perp }}\\right) ^2q_{\\parallel\n}^2}.  \\label{sigma2-3}\n\\end{equation}\nThe integral can be done exactly, as in Appendix C, and all the results\nobtained in this subsection and the following one can be obtained from\nlimiting cases of this general analytical result, as shown in Appendix D.\nHere we make approximations directly on the integrals since this makes the\nPhysics of the results more transparent. Although arbitrary cutoffs appear\nin the analytical expressions, we re-emphasize that in the numerical\ncalculations of the previous section the cutoffs are simply given by the\nBrillouin zone and there is no arbitrary scale in the results.\n\nAt $T_N$, we have $\\xi _{\\perp }^{-2}=0$. Furthermore, from Eqs.(\\ref\n{ksi2per}) and (\\ref{ksi2par}) we find $\\xi _{\\Vert }\/\\xi _{\\perp }=\\xi\n_0^{\\parallel }\/\\xi _0^{\\perp }$ so that the above integral Eq.(\\ref\n{sigma2-3}) takes the form \n\\begin{equation}\n\\tilde{\\sigma}^2=\\frac{2T_Na_{\\Vert }a_{\\bot }^2}{U_{sp}\\left( \\xi _0^{\\perp\n}\\right) ^2}\\int \\frac{d^2q_{\\bot }}{(2\\pi )^2}\\left[ \\int \\frac{dq_{\\Vert }%\n}{2\\pi }\\frac 1{q_{\\perp }^2+\\left( \\frac{\\xi _0^{\\parallel }}{\\xi _0^{\\perp\n}}\\right) ^2q_{\\parallel }^2}\\right] .\n\\end{equation}\nUsing the mean-value theorem for the integral over $q_{\\Vert }$ we have \n\\begin{equation}\n\\tilde{\\sigma}^2=\\frac{2T_Na_{\\bot }^2}{U_{sp}\\left( \\xi _0^{\\perp }\\right)\n^2}\\int \\frac{d^2q_{\\bot }}{(2\\pi )^2}\\left[ \\frac 1{q_{\\perp }^2+\\left( \n\\frac{\\xi _0^{\\parallel }}{\\xi _0^{\\perp }}\\right) ^2\\widetilde{\\Lambda }^2}%\n\\right]\n\\end{equation}\nwhere $\\widetilde{\\Lambda }$ is a constant that we need not specify. It is\ncontained in the range $0<$ $\\left| \\widetilde{\\Lambda }\\right| <\\pi\n\/a_{\\Vert }$. The above integral is the same as the one that determines the\ncorrelation length in two dimensions Eq.(\\ref{Consistency}), hence at $T_N$\nwe have that \n\\begin{equation}\n\\xi _{\\perp }^{-2}\\left( T_N\\right) =\\left( \\frac{\\xi _0^{\\parallel }}{\\xi\n_0^{\\perp }}\\right) ^2\\widetilde{\\Lambda }^2\\propto \\left( \\frac{t_{\\Vert\n}a_{\\Vert }}{t_{\\bot }a_{\\bot }}\\right) ^2\\widetilde{\\Lambda }^2.\n\\label{XsiTn}\n\\end{equation}\nComparing with the general theory of Appendix D where it is argued that $\\xi\n_{\\bot }^{-\\phi \/\\nu }\\left( T_N\\right) \\sim t_{\\Vert }^2$, we see that $%\n\\phi \/\\nu =2$. In other words, $\\phi \/\\nu =\\gamma \/\\nu =2$ and the crossover\nexponent\\cite{Pfeuty} $\\phi $ is here equal to$\\ \\gamma $ as is usually the\ncase in the $n\\rightarrow \\infty $ model.\\cite{Riedel} We obtain, using the\nexpression (\\ref{expo}) for the correlation length in two dimensions, \n\\begin{equation}\n\\frac 1{T_N}=\\frac{a_{\\bot }^2}{\\pi \\widetilde{\\sigma }^2U_{sp}\\left( \\xi\n_0^{\\perp }\\right) ^2}\\left[ \\ln \\frac{t_{\\bot }}{t_{\\Vert }}+c\\right]\n\\label{1otn}\n\\end{equation}\nwhere $c$ is a non-universal constant of order unity.\n\nIn the special case of perfect nesting (half-filled nearest-neighbor hopping\nmodel), the microscopic length $\\xi _0^{\\perp }$ is temperature dependent,\nas shown in Appendix A \n\\begin{equation}\n\\left( \\xi _0^{\\perp }\\right) ^2\\sim a_{\\bot }^2\\frac{0.085}{T^2}\\frac 1{%\n2\\chi \\left( {\\bf Q}_2\\right) }.\n\\end{equation}\nUsing this result as well as $U_{mf,c}\\equiv 2\/\\chi \\left( {\\bf Q}_2\\right)\n\\approx U_{sp}$ at $T_X\\approx T_N$ gives the scaling illustrated in Fig.3\nfor the case $a_{\\bot }\/a_{\\Vert }=1$ namely \n\\begin{equation}\n\\frac 1{T_N}\\sim \\frac{T_N^2}{U_{mf,c}^2}\\left| \\ln \\frac{t_{\\Vert }}{%\nt_{\\bot }}\\right|\n\\end{equation}\n\nThe logarithmic behavior in Eq.(\\ref{1otn}) is typical of systems that\nundergo a dimensional crossover from their lower critical dimension. For\nexample, the analog of Eq.(\\ref{XsiTn}) in the anisotropic Heisenberg case\nwould read,\\cite{Soukoulis} \n\\begin{equation}\n\\xi _{\\perp }^{-2}\\left( T_N\\right) \\sim \\left( \\frac{J_{\\Vert }}{J_{\\bot }}%\n\\right) \\widetilde{\\Lambda }^2  \\label{AH}\n\\end{equation}\nleading to\\cite{Soukoulis} $T_N^{-1}\\sim \\ln \\left( \\frac{J_{\\Vert }}{%\nJ_{\\bot }}\\right) .$ The above results Eqs.(\\ref{XsiTn}) and (\\ref{AH}) are\nsuggested by the simple RPA-like form $\\chi _{3d}\\sim \\chi _{2d}\/\\left(\n1-J_{\\Vert }\\chi _{2d}\\right) $ with $J_{\\Vert }\\chi _{2d}\\sim 1$ at the\ntransition and $\\chi _{2d}\\sim \\xi _{\\perp }^{\\gamma \/\\nu }\\sim \\xi _{\\perp\n}^2\\sim \\exp \\left( J_{\\bot }cst\/T\\right) $. As in the previous section, the\nquantity $U_{sp}\\left( \\xi _0^{\\perp ,\\Vert }\\right) ^2$ plays a role\nanalogous to the exchange constants $J_{\\bot \\parallel ,}$. In the perfect\nnesting case, these effective exchange constants would be temperature\ndependent since $U_{sp}\\left( \\xi _0^{\\perp ,\\Vert }\\right) ^2$ $\\sim\nU_{sp}\\left( t_{\\perp ,\\Vert }\\right) ^2\/T^2$. Note that in the crossover\nfrom one-dimensional Luttinger liquid behavior to three-dimensional\nlong-range order, the effective exchange constant $J_{\\bot }$ also scales%\n\\cite{BourbonnaisCaron}\\cite{Boies} as $ut_{\\bot }^2\/T^2$, with $u$ a\nrunning coupling constant. The one-dimensional Fermi surface is always\nnested.\n\n\\subsection{Size of the three-dimensional critical region.}\n\nThe singular temperature dependence of the correlation length is obtained\nfrom the equation\n\n\\begin{equation}\n\\tilde{\\sigma}^2=\\frac{2Ta_{\\Vert }a_{\\bot }^2}{\\delta U}\\int \\frac{d^3q}{%\n(2\\pi )^3}\\frac 1{1+\\xi _{\\perp }^2q_{\\perp }^2+\\xi _{\\parallel\n}^2q_{\\parallel }^2}.  \\label{sigma2-4}\n\\end{equation}\nSince the ratio $\\xi _{\\Vert }\/\\xi _{\\perp }=\\xi _0^{\\parallel }\/\\xi\n_0^{\\perp }$ is temperature independent, a simple change of integration\nvariables shows that near $T_N$ the scaling of both correlation lengths with\ntemperature is identical to the isotropic three-dimensional case. In other\nwords, the critical behavior near the phase transition is that of the\nthree-dimensional system. However, as one increases the temperature away\nfrom $T_N$, the correlation lengths can decrease until $\\xi _{\\parallel }\\ll\na_{\\Vert }$ while at the same time $\\xi _{\\perp }\\gg a_{\\bot }$. When $\\xi\n_{\\parallel }\\ll a_{\\Vert }$, the integral Eq.(\\ref{sigma2-4}) is\nessentially two-dimensional and for $\\xi _{\\perp }\\gg a_{\\bot }$ one should\nobserve the characteristic exponential temperature dependence of the\ntwo-dimensional correlation length.\n\nAs usual the definition of crossover contains some arbitrariness, so let us\nchoose \n\\begin{equation}\n\\xi _{\\parallel }\\left( T^{*}\\right) =a_{\\Vert }  \\label{Xsi=1}\n\\end{equation}\nas the definition of the crossover temperature $T^{*}$ between $d=2$ and $%\nd=3 $ critical behavior. In that regime, the correlation length Eq.(\\ref\n{Xsi=1}) scales with temperature as in the $d=2$ regime Eq.(\\ref{expo})\nexcept that, as argued before, $\\xi _{\\Vert }$ is smaller by a factor $%\n\\left( \\xi _0^{\\Vert }\/\\xi _0^{\\bot }\\right) =\\left( t_{\\Vert }a_{\\Vert\n}\\right) \/\\left( t_{\\bot }a_{\\bot }\\right) $, hence we obtain for $T^{*}$ \n\\begin{equation}\n\\frac 1{T^{*}}=\\frac{a_{\\bot }^2}{\\pi \\widetilde{\\sigma }^2U_{sp}\\left( \\xi\n_0^{\\bot }\\right) ^2}\\left[ \\ln \\left( \\frac{t_{\\bot }}{t_{\\Vert }}\\right)\n+c^{\\prime }\\right]  \\label{CrossExp}\n\\end{equation}\nwhere $c^{\\prime }$ is a non-universal constant of order unity. The size of\nthe crossover region is thus \n\\begin{equation}\n1-\\frac{T_N}{T^{*}}=T_N\\frac{a_{\\bot }^2}{\\pi \\widetilde{\\sigma }%\n^2U_{sp}\\left( \\xi _0^{\\bot }\\right) ^2}\\left( c-c^{\\prime }\\right) =\\frac{%\n\\left( c-c^{\\prime }\\right) }{\\ln \\left( \\frac{t_{\\bot }}{t_{\\Vert }}\\right)\n+c}\\sim \\frac 1{\\ln \\left( \\frac{t_{\\bot }}{t_{\\Vert }}\\right) }.\n\\label{SizeCrossover}\n\\end{equation}\nThe above results Eqs.(\\ref{Xsi=1}) to (\\ref{SizeCrossover}) are as expected\nfrom the usual theory of critical phenomena exposed in Appendix D. In\nparticular, the scaling of $T^{*}$ with $t_{\\Vert }\/t_{\\bot }$ is the same\nas that of $T_N.$ The smallness of the crossover region from $d=2$ to $d=3$\ncritical behavior in Fig.4 follows from the above considerations. The\nsmaller is $T_N$, the smaller is $T^{*}$. The above situation should be\ncontrasted with the problem of crossover from $d=3$ to $d=2$ in Helium\nfilms, studied by Fisher and Barber\\cite{Fisher} many years ago. In that\ncase, power law scaling occurred everywhere, giving quite different\nexpressions for the scaling of $T^{*}$ and $T_N$.\n\nGiven $\\xi _{\\parallel }\\left( T^{*}\\right) =\\xi _0^{\\parallel }\\left(\nT^{*}\\right) \\left( U_{sp}\/\\delta U\\left( T^{*}\\right) \\right) ^{1\/2}$, and $%\n\\xi _0^{\\parallel }\\sim t_{\\parallel }$, the above relation $\\xi _{\\parallel\n}\\left( T^{*}\\right) =a_{\\Vert }$ means that $\\delta U\\left( T^{*}\\right) $\nshould scale as $t_{\\parallel }^2$. Similarly we should have $\\delta U\\left(\nT_N\\right) \\sim t_{\\parallel }^2.$ We checked numerically\\cite{DareThese}\nthat the scaling with $t_{\\Vert }$ holds for $t_{\\Vert }<0.05$ in the\nhalf-filled nearest-neighbor model with $U=4t_{\\perp }.$\n\n\\section{Conclusion}\n\nWe have shown that the TPSC approach allows one to study all aspects of\nnearly antiferromagnetic itinerant electrons in one-band Hubbard models. The\nmethod is in quantitative agreement with Monte Carlo simulations in the\nnon-critical regime\\cite{Vilk}\\cite{Veilleux} while in the critical regime,\n(renormalized classical or quantum critical) the relatively weak temperature\ndependence of the local moment leads to the same critical behavior as\nstrong-coupling models to leading order in the $1\/n$ expansion, namely in\nthe $n\\rightarrow \\infty $ limit. There is no arbitrary cutoff so that all\nresults can be obtained as a function of lattice spacing, hopping integral\nand interaction parameter. Fermi surface effects are apparent, in particular\nin the case of perfect nesting where the two-dimensional\nrenormalized-classical correlation length diverges as $\\exp \\left(\ncst\/\\left( T^3\\ln ^2T\\right) \\right) $ instead of $\\exp \\left( cst\/T\\right) $%\n.\n\nWe have applied the method to a detailed study of the renormalized-classical\ncrossover from two to three dimensions where we have highlighted the\nexistence of a regime where the three-dimensional N\\'{e}el instability\noccurs before thermal fluctuations become small enough to allow coherent\nsingle-particle band motion between planes. An analogous phenomenon occurs\nin quasi-one dimensional systems\\cite{BourbonnaisCaron}\\cite{Boies}. The\nTPSC approach can be applied to study realistic cases. For $La_2CuO_4$ we\nwill show in a subsequent publication that with second-neighbor hopping one\ncan fit experiments on the magnetic structure factor.\n\nThe generalization of the TPSC approach beyond leading order in $1\/n$ is\nleft open. Also, the effect of self-energy feedback\\cite{Vilk2} on $\\exp\n\\left( cst\/\\left( T^3\\ln ^2T\\right) \\right) $ behavior of the correlation\nlength in the two-dimensional nesting case should be cleared in further\nstudies. Finally, the universal $d=2$ to $d=3$ crossover discussed in\nAppendix D should be investigated beyond leading order in $1\/n$.\n\nWe are indebted to C. Bourbonnais for numerous discussions and key ideas. We\nalso thank David S\\'{e}n\\'{e}chal for discussions and D.S. Fisher for\npointing out Ref.\\cite{Kosterlitz}. We acknowledge the support of the\nNatural Sciences and Engineering Research Council of Canada (NSERC), the\nFonds pour la formation de chercheurs et l'aide \\`{a} la recherche from the\nGovernment of Qu\\'{e}bec (FCAR) and (A.-M.S.T.) the Canadian Institute for\nAdvanced Research (CIAR) and the Killam foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLow surface brightness (LSB) disk galaxies represent a class of galactic\nsystems which have experienced very slow evolution since their formation\nepoch. Their low surface brightnesses ($\\; \\buildrel > \\over \\sim \\;$ 1 mag\/arcsec$^2$ below\nthe canonical Freeman (1970) value of $\\mu_0^B=21.65 \\pm 0.3$ mag\/arcsec$^2$)\nindicate that, over the age of the Universe, their mean stellar birthrate per \nunit area has been significantly lower than that of typical high surface \nbrightness (HSB) disks. Their current rate of star formation is similarly\nlow --- while some \\ion{H}{2} regions do exist in LSBs, the global\nstar formation rate in LSBs is lower by an order of magnitude than \ncomparably sized HSBs (McGaugh 1992; Knezek 1993; McGaugh \\& Bothun\n1994; R\\\"onnback \\& Bergvall 1994; de Blok, van der Hulst \\& Bothun 1995;\nde Blok 1997). The lack of\nsignificant star formation is reflected in the low metallicities of\nLSBs, which are typically $\\; \\buildrel < \\over \\sim \\;$ 1\/3 solar (McGaugh 1994;\nR\\\"onnback \\& Bergvall 1995; de Blok \\& van der Hulst 1998a).  Not \ncoincidentally, LSBs are also very gas-rich systems. McGaugh \\& de Blok \n(1997) found that the gas mass fraction of galaxy disks correlates strongly \nwith surface brightness. In LSBs, as much as 50\\% of the disk mass \nis in the form of gas, compared to $\\sim$ 10\\% at high surface brightnesses. \nTheir low surface brightnesses, low star formation rates, low metallicities,\nand large gas fractions all argue that LSBs are systems which are forming \nstars much more slowly than their HSB counterparts.\n\nThe suppressed rate of star formation in LSB disks must ultimately be connected\nto the differing physical conditions of the interstellar medium (ISM)\nbetween LSB and HSB disk \ngalaxies. As star formation is presumed to take place in molecular clouds,\nthe molecular content of LSBs is of particular interest.  In typical HSB\nspirals, the mass of molecular gas is comparable to that in neutral \n\\ion{H}{1} (e.g.,\\  Young \\& Knezek 1989). The situation in LSBs may be quite \ndifferent --- while several CO surveys of LSBs have been made (e.g.,\\  Schombert \net~al.\\  1990 (S90); Knezek 1993; de Blok \\& van der Hulst 1998b (dBvdH)), CO \nemission has not been detected in any LSB disk galaxy. If CO emission traces \nmolecular gas content in the same way as in normal HSB galaxies, then the upper \nlimits on molecular gas in LSBs are typically $M_{H_2}\/M_{HI}$ $\\; \\buildrel < \\over \\sim \\;$ 0.1,\nand are more severe in a few cases. These \nupper limits have led to the speculation that the low disk surface\ndensities in LSBs preclude molecular cloud formation and, in turn, inhibit\nstar formation (e.g.,\\  S90; van der Hulst et~al.\\  1993, Bothun et~al.\\  1997).\nAlternatively, the lack of CO detection may simply reflect the fact that\nthe CO\/H$_2$\\ conversion factor is not a universal constant, so that perhaps\nlarge quantities of molecular H$_2$\\ exist despite the lack of detected CO \nemission.\n\nUnfortunately, an observational answer to the question of the molecular\ncontent of LSBs is inexorably tied to the CO\/H$_2$\\ conversion factor\nand its dependency on environment. For example, Wilson (1995) and\nIsrael (1997) recently showed that the CO\/H$_2$\\ conversion factor was a strong \nfunction of metallicity; this dependency raises the upper limits on the \nderived molecular content of LSBs. Nonetheless, even accounting for \nmetallicity effects, previous CO surveys should have detected CO\nin LSBs if they had $M_{H_2}\/M_{HI}$ ratios similar to HSBs.\nOther dependencies should also play a role. For example, the local gas\ndensity and temperature can affect CO\/H$_2$\\\n(e.g.,\\  Maloney \\& Black 1988, Scoville \\& Sanders 1987).\nThese are in turn affected by the ionizing radiation field and the\ndensity structure (``clumpiness'') of the ISM. In LSBs all these \nfactors may well be significantly different than expected for HSBs,\nsuch that the true molecular-to-atomic gas mass ratio ($M_{H_2}\/M_{HI}$) \nis only weakly constrained by direct CO measurements.\n\nTo explore the ISM properties of LSB galaxies in a manner independent\nof the CO\/H$_2$\\ conversion factor, we take a complementary, theoretical \nroute towards understanding the molecular content of LSB galaxies. \nWe construct models of an inhomogeneous ISM under varying physical \nconditions, spanning a range of disk galaxy types. The models employ a \nMonte Carlo approach to radiative transfer (see Spaans 1996), and explicitly \nsolve for the CO emissivity and $M_{H_2}\/M_{HI}$ ratio in galactic disks. \nWe investigate models on a grid of metallicity, surface brightness, and \nISM density structure, tracking the changing physical conditions between\nLSB and HSB disk galaxies.  In particular, we\naddress the questions of how much molecular \nH$_2$\\ is expected in LSB disks, and whether the lack of observed CO in LSBs\nin fact indicates a lack of molecular gas.\n\n\\section{ISM Modeling}\n\\subsection{Modeling Technique}\n\nThe code developed by Spaans (1996) and its extensions as discussed in\nSpaans \\& van Dishoeck (1997), Spaans \\& Norman (1997), and Spaans \\& Carollo\n(1998) is used to\nderive the physical and chemical structure of the ambient ISM in LSBs.\nThe interested reader is referred to these papers for a detailed description\nof the code's structure. The main features can be summarized as follows.\n\n1) For a given metallicity, geometry, global pressure structure and\ndistribution of illuminating (ultraviolet) sources, the thermal and chemical\nbalance of the medium is computed in three dimensions. The continuum\n(dust attenuation) and line transfer is modeled through a Monte Carlo\nmethod. The self-shielding of H$_2$ and CO and the shielding of CO by H$_2$\nabsorption lines is explicitly included. The heating processes include\nphoto-electric emission by large molecules like Polycyclic Aromatic\nHydrocarbons (PAHs) and dust grains (Bakes \\& Tielens 1994), cosmic ray\nheating, collisional de-excitation of ultraviolet pumped H$_2$, and H$_2$\ndissociation heating.\nIt is assumed that 10\\% of the gas phase carbon is incorporated into PAHs.\nThis yields roughly equal photo-electric heating\ncontributions from carbonaceous particles larger\nand smaller than $10^{-6}$ cm. Generally photo-electric emission dominates\nthe heating rate unless the visual extinction exceeds 3 mag.\nThe cooling processes include fine-structure emission of C$^+$, C and O,\nrotational line emission of CO and vibrational (v=1-0) H$_2$ emission. All\nlevel populations are computed in statistical equilibrium and the line emission\nis again modeled through a Monte Carlo technique.\n\n2) The solutions to the thermal balance equations allow, for a given\nhydrodynamic pressure, multiple solutions. These constitute the possible\nmulti-phase structure of the ISM as first suggested by Field, Goldsmith, \\&\nHabing (1969). If multiple solutions exist, then one finds from a stability\nanalysis that there is a $\\sim 10^4$ K diffuse medium and a $\\sim 50$ K\ndense component. It is the density structure derived from these solutions\nwhich couples strongly with the chemical balance of interstellar gas, and\ntherefore with the amount of molecular gas which is supported by the stellar\nradiation field and the ambient pressure of the galaxy. This thermal\nstability approach does not incorporate the effects of hydrodynamic phenomena\nsuch as shocks or gravity. The cold component has a typical density of\n$\\sim 50-300$ cm$^{-3}$ and is representative of diffuse and translucent\nclouds in the Milky Way. To allow the inclusion of shocks and gravity in a\nphenomenological way, the dense phase is allowed to exhibit inhomogeneities.\nThat is, the ambient pressure determines the {\\it mean} density of this phase,\nwhile gravity as well as shocks drive perturbations in it.\n\n\\subsection{Model Parameters and Their Implementation}\n\nTo investigate the molecular content of the ISM the following model parameters\nare considered: average gas density, the average UV interstellar radiation \nfield (ISRF), metallicity, surface density, and ISM density structure.\nThese parameters are not all independent. To capture the essential\ndependencies of the ISM structure on ambient physical conditions the following\nscaling relations are adopted.\n\nThe HI volume density $n_{\\rm HI}$ correlates with HI surface density \n$\\Sigma_{HI}$ according to\n$$n_{\\rm HI} = \\Sigma_{HI}\/H,\\eqno(1)$$\nwhere $H=300$ pc is the scale height of the galaxy model. Using data from\nde Blok et~al.\\  (1996), one can derive a rough correlation between \nlocal surface brightness $\\mu^B$ and local HI density: \n$$\\log \\Sigma_{HI} \\approx -0.12*\\mu^B + 3.6.\\eqno(2)$$\nWith this relationship, the HI surface density and stellar surface brightness \ndo not drop off in lockstep; instead, the HI surface density falls off\nmore slowly.  While this is generally true, it should be emphasized that\nthis relation is admittedly crude with a lot of real scatter.  The aim\nis more to characterize the general behavior of disks to search for\nphysically meaningful trends rather than to attempt to model specific\nindividual galaxies.  In global terms, the gas mass fraction of the \ndisk increases as surface brightness decreases\nsuch that very low surface brightness disks ($\\mu^B_0$ $\\; \\buildrel > \\over \\sim \\;$ \n23) can have half their baryonic mass in the form of gas (McGaugh \\&\nde Blok 1997), even assuming a trivial amount of molecular gas mass.\n\nThe luminosity profiles of disk galaxies (especially LSBs) are generally\nexponential,\n$$\\mu^B(r) = \\mu_0^B + 1.086*(r\/h),\\eqno(3)$$\nwith scale length $h$ and central surface brightness in B\nmags per square arcsecond $\\mu_0^B$.\nCombining equations (2) and (3), one finds\n$${\\rm log}\\Sigma_{HI} \\approx -0.12*\\mu^B_0 -0.13*(r\/h) + 3.6.\\eqno(4)$$\nAgain, the relationship implies that, as a function of radius, the\nHI surface density drops off more slowly than the stellar surface\nbrightness, reproducing the extended gaseous disks observed in disk galaxies. \nIn this parameterization, the gas surface\ndensity is exponential, but with a scale length 3.3 times larger than\nthat for the stars. While real gas disks are not as well described by\nexponentials as the stellar component, we again stress this is merely\na convenient approximation for modeling purposes.  Deviations from\nthis approximation will alter only details and not the general trends\nof interest, and are probably small compared to the uncertainty\nin the modeling process.\nBecause equation (4) describes the HI surface density, while the model\ninputs are in terms of total (HI $+ H_2$) gas surface density,\nwe use an iterative scheme\nto arrive at the final model. First we calculate the model assuming\na total surface density given by equation (4). From this initial model, we \nderive the H$_2$ mass profile, then add this profile to the original\nHI profile to produce a total gas mass profile. This total profile is\nthen used as input to calculate a new, consistent ISM model.\n\nTo parameterize the strength of the ISRF in our models, we assume that\nthe ISRF is dominated by the contribution from the stellar populations\nin galaxies. Under this assumption, the ISRF scales with surface brightness:\n$$I_{UV} = I_{UV}(MW) * 10^{0.4*(\\mu_0^B({\\rm MW})-\\mu_0^B)}, \\eqno(2)$$\nwhere $I_{UV}(MW)$ is the strength of the ISRF in the Milky Way given\nby Draine (1978), and $\\mu_0^B({\\rm MW})$ is the central surface brightness\nof the Milky Way disk (assumed to be 21 mag arcsec$^{-2}$).\nThe wavelengths in the UV relevant to our results are between 912 and 1110\n\\AA\\ where lie all the H$_2$ and CO absorption lines which lead to\nphoto-dissociation of the molecules.  By scaling the UV ISRF with B-band\nsurface brightness, we are assuming that the spectral shape is\n{\\it independent\\\/} of surface brightness.  That is, we assume\nthat the stellar populations which give rise to the ISRF\ndo not drastically change as a function of surface brightness. This\nassumption is perhaps suspect.  Since there is generally less star formation\nin LSB than in HSB galaxies, one might suspect the UV ISRF to be relatively\nweaker in LSBs than implied by the difference in B-band surface brightness.\nOn the other hand, LSBs do tend to be blue, late type galaxies which have\nharder spectral shapes in the optical.  So one might equally well expect\nthis trend to continue into the UV, resulting in the opposite effect:\nthe difference in B-band surface brightness might overstate that in the UV.\nWithout strong constraints on the UV properties of\nLSBs we choose simply to hold the shape of\nthe ISRF fixed with optical surface brightness.\nIf the UV ISFR is relatively greater [less] than we assume, more [fewer]\nmolecules will be destroyed and so on balance there will\nbe less [more] gas mass in molecular form.\n\nWith the gas density and UV ISRF defined in terms of the disk surface\nbrightness, we can similarly define a parameter closely akin\nto the ionization parameter:\n$$\\log{U}=\\log(I_{UV}\/\\Sigma_{HI})=-0.28\\mu^B_0 + {\\rm constant},$$\nwhich essentially measures the number of ionizing photons per atom.\nBecause of our assumption that the ISRF scales linearly with\nsurface brightness, while the gas density drops more slowly, LSB\ngalaxies should have lower values of $U$ than HSBs. If,\nhowever, LSBs have a harder spectral shape than HSBs (due perhaps\nto a younger, hotter mean stellar population), this assumption\nmay underestimate $U$ in LSBs. While we \nuse surface brightness as a fundamental input parameter for the\nmodels, we note that with the pseudo-ionization parameter $U$ defined this\nway, models with central surface brightnesses 0, 1, 2, and 3 mag \narcsec$^{-2}$ below that of the Milky Way correspond to values\nof $U\/U_{MW}$ = 1.0, 0.5, 0.28, and 0.15, respectively.\n\nFinally, we need to characterize the inhomogeneity of the dense phase, if\nit is supported, in the models.\nThis inhomogeneity can be parameterized by {\\it choosing} a certain volume\nfraction $F$ of the gas in high density clumps with a fixed density contrast\n$C$. The size of the clumps is not varied and assumed equal to 2 pc, typical\nfor translucent clouds in the Milky Way. By investigating a range of density\ncontrasts, and therefore clump extinction, this somewhat arbitrary length\ndoes not strongly influence the results. We calculate one model (``H'', see\nTable 1) which\nis completely homogeneous and lacks any density structure, representing\na limiting extreme. Two more models\nare explored which have modest amounts of structure (``I1, I2'', with \nsmall $C$ and large $F$). Finally, the clumpy ISM models (``C1, C2'', large $C$\nand small $F$; see Table 1) are chosen to represent our own Galaxy at high ISM\npressure.\n\nWith these parameterizations, we are left with three variables\ndescribing the model galaxies: metallicity, ionization parameter,\nand ISM clumpiness. We create a grid of models spanning a range of plausible\nvalues: central surface brightness $\\mu_0^B = 21 \\to 24$, metallicity\n$Z\/Z_{\\sun} = 1 \\to 0.1$, and ISM types H (homogeneous, $P\\sim 10^3$ K cm$^{-3}$),\nI1 and I2 (intermediate, $P\\sim 2\\times 10^3$ K cm$^{-3}$), and C1 and C2 (clumpy,\n$P\\sim 10^4$ K cm$^{-3}$). These models thus capture\nthe properties of both high surface brightness spirals as well as low\nsurface brightness disks. For each model we calculate the H$_2$ gas mass\nfraction as a function of radius, as well as the CO emissivity\nand mass averaged gas temperature. From these models, we can\nanalyze ISM trends with surface brightness and address the question of\nmolecular gas content in low surface brightness disks.\n\n\\section{Results}\n\n\\subsection{Molecular Gas Fractions}\n\nFigure 1 shows $\\Sigma_{H_2}\/\\Sigma_{HI}$ as a function of radius for\nseveral characteristic models. Several trends are immediately obvious:\n\\begin{itemize}\n\\item At fixed metallicity and ISM structure, lower surface brightness\nmodels have {\\it higher} molecular fractions (Figure 1a). Because the number\nof ionizing photons per hydrogen atom decreases with decreasing surface \nbrightness, the molecules in the low surface brightness models are less apt \nto be dissociated by the background ISRF.\n\\item  At fixed surface brightness and ISM density structure, models with \nlower metallicity have lower molecular hydrogen gas content (Figure 1b). \nThis result is due to the fact that dust grains act as formation sites \nfor molecules; lower metallicities mean fewer dust grains to drive molecule \nformation.\n\\item At fixed surface brightness and metallicity, clumpier ISM models have\nhigher molecular gas fractions (Figure 1c). In clumpy models, a larger\nmass fraction of the gas is found in denser cores, and are shielded\nfrom the background ISRF. Molecules in diffuse ISM models lack this shielding,\nand are more easily dissociated by the UV background.\n\\end{itemize}\n\nHow well do these models describe actual disk galaxies? One point of\nconstraint is provided by the Milky Way ISM. The high surface brightness,\nsolar metallicity, and clumpy ISM model shows a mean H$_2$\/HI mass ratio\n$\\sim 1$ averaged across the inner scale length of the disk, similar to\nthat inferred for Milky Way-like Sb galaxies (Young \\& Knezek 1989). This \nresult is not surprising, since the ISM models were scaled to the ISRF and \nstructure of the Milky Way's ISM, but nonetheless it is reassuring that we \nrecover the correct physical description for the given model inputs.\n\nAssigning a model to LSB galaxies is not as straightforward. Certainly\nLSB disks are lower in metallicity (Webster et~al.\\  1983; McGaugh 1994;\nde Blok \\& van der Hulst 1998a) than HSB galaxies such as \nthe Milky Way. Their reduced surface brightnesses also probably\nresults in lower ionization parameters, although stellar population\ndifferences may modify this somewhat.\nThe density structure of the ISM in LSBs is not well determined,\nprecisely due to the fact that CO measurements have not yielded any\ndetections. Because of the lowered mass surface density of LSB disks\n(de Blok \\& McGaugh 1996, 1997), it is likely that the ISM pressures\nare too low to support the amount of multiphase structure found in the\nMilky Way. Such was the case in hydrodynamical models of LSB galaxies\nby Gerritsen \\& de Blok (1998), where a multiphase ISM was virtually \nabsent. Models H (homogeneous) and I1 and I2 (intermediate) are therefore\nlikely candidates to describe the density structure of LSB galaxies.\n\nFigure 2 shows the H$_2$\/HI mass ratio averaged over the inner disk\nscale length as a function of central surface brightness for the entire\ngrid of models. For metallicities typical of LSBs ($Z\/Z_{\\sun}$=0.1--0.3),\nthe models are lower in molecular content than the Milky Way, as \nexpected. Interestingly, though, the models are far from being void\nof molecular gas; mass fractions of 0.25 -- 0.5 are typical. Again,\nthe lowered ionization parameter as a function of surface brightness\nresults in\n{\\it higher} molecular fractions (at fixed metallicity and ISM structure) \nfor lower surface brightness galaxies. In fact, for very low surface \nbrightnesses, the molecular \ncontent can rival that of HSBs if they have any significant degree\nof clumpiness in their ISM. However, at such low surface brightnesses,\nthe ISM pressures are probably too low to support this level of\nstructure.\n\nNonetheless, our models suggest that typical LSB galaxies have\nmolecular contents which are only factors of 2--3 below that of\nnormal HSB spirals. The CO mass averaged gas temperatures in the molecular\nphase are presented in Figure 3 as a function of radius.\nIt is immediately obvious that the molecular gas in LSBs is by no means very\ncold, in contrast with their multi-phase counterparts. Typical temperatures\nare around 30--50 K, similar to Spitzer-type HI clouds in our own Milky Way. \nIn Figure 4, we show the cumulative H$_2$ gas mass fraction averaged over the\ninner scale length as a function\nof temperature for a Milky Way-like model ($\\mu^B_0 = 21, Z\/Z_{\\sun} = 1,$\nISM C2), a typical LSB model ($\\mu^B_0 = 23, Z\/Z_{\\sun} = 0.3,$ ISM I1),\nand a very low surface brightness model ($\\mu^B_0 = 24, Z\/Z_{\\sun} = 0.1,$ ISM H).\nFor the Milky Way model, nearly 50\\% of the molecular gas is at or below\n30 K, compared to 20\\% and only a few percent for the typical and\nextreme LSB models. Coupled with the decrease in total molecular\ncontent in the LSB models, our calculations suggest that LSBs should\nhave very small total amounts of {\\it cold} molecular gas.\n\nSuch high temperatures argue against efficient star formation in LSBs, but \nself-consistent rates of the order of $\\sim$ 0.05 M$_\\odot$ yr$^{-1}$ appear \nfeasible in these low metallicity environments (Norman \\& Spaans 1997;\nGerritsen \\& de Blok 1998).\nThis star formation rate is similar to observed star formation rates in LSBs\n(McGaugh \\& Bothun 1994; R\\\"onnback \\& Bergvall 1994; de Blok et~al.\\  1995).\nIn conclusion, the lack of detected CO emission in LSBs\ndoes not preclude the presence of modest amounts of\nmolecular H$_2$ gas. The CO detectability of an LSB depends on both the\nCO abundance and excitation in the galaxy; we turn now to predictions\nof the CO intensity of LSBs in order to directly compare to searches\nfor CO emission from LSBs.\n\n\\subsection{CO Intensity and the CO\/H$_2$\\ Conversion Factor}\n\nTo calculate the CO intensity of the models,\nthe root mean square velocity of the interstellar clouds, the vertical\nvelocity dispersion, is taken equal to 10 km s$^{-1}$, a typical value in the\nMilky Way and other galaxies. The turbulent velocity width of individual\nclouds is assumed equal to 3 km s$^{-1}$, consistent with the observed\ncorrelation between cloud size and line width for the\nMilky Way (Maloney \\& Black 1988). We calculate the face-on CO intensities \nfor our different ISM models, integrated over the inner scale length. \n\nFigure 5 shows the variation in I(CO), the CO intensity in K km s$^{-1}$, \nas a function of metallicity, surface brightness, and ISM structure.\nAs with the H$_2$\/HI\nmass ratio, several trends are immediately apparent: lower metallicity,\nhigher surface brightnesses, and a more diffuse ISM all act to lower the\nCO intensity in the models. All these trends are as expected. Lower \nmetallicities mean fewer carbon and oxygen atoms are available to form \nthe CO molecule; higher surface brightnesses result in a stronger\nISRF which destroys the CO molecule; and a diffuse ISM is less effective\nat shielding the CO molecules against radiative dissociation.\n\nAlso plotted on Figure 5 are the observational upper limits to the CO \nintensity of LSB galaxies determined by S90 and dBvdH. If LSBs have \nsolar metallicity, these\nobservations should have detected CO emission. But the subsolar\nmetallicities of LSBs (McGaugh 1994) result in lowered CO intensities,\nmaking detection difficult. At $Z\/Z_{\\sun} \\sim 0.3$, the CO emission\nis only a factor of $\\sim 2 - 5$ below the observational limits,\nsuggesting that deeper CO mapping may in fact reveal the molecular\nISM of moderately metal poor LSBs. However, reducing the metallicity\nby another factor of three reduces the CO emission to levels 30 times\nfainter than the current observational limits; detecting these LSBs\nin CO will be very hard indeed.  This drop in CO emission occurs in\nspite of the presence of a fair amount of H$_2$ in the models.\n\nPerhaps most germane to the observational status of molecular gas\nin LSB disk galaxies is the conversion factor $X = n({\\rm H}_2)\/I({\\rm CO})$\n(in units of $10^{21}$ cm$^{-2}$ (K km s$^{-1})^{-1})$. Figure 6 shows \nthis value calculated for the grid of ISM models. As expected,\n$X$ shows significant and systematic variation between \nthe different models.  At solar metallicities, $X\\sim 0.1-1$, spanning \nthe ``standard'' value of X derived from Milky Way observations ($\\sim$\n0.2 -- 0.5; see, e.g.,\\  Scoville \\& Sanders 1987). Because the CO intensity \nscales non-linearly with density, and in a different manner\nfrom the H$_2$ mass, X has a strong dependence on the \ndensity structure of the ISM. Our models\ncalculate the properties of the ISM over the inner disk scale length,\naveraging over both cloud and inter-cloud regions. As the\nISM becomes more clumpy, X decreases as the CO intensity rises faster\nthan the H$_2$ mass fraction. The value of X determined in the Milky\nWay may therefore be quite different from that applicable to \ngalaxies with a more homogeneous ISM.\n\nAside from the dependence on ISM density structure, there is also\na clear correlation between X and metallicity: as metallicity drops,\nthe value of X increases. Such a trend has also been seen in observational\ndata (e.g.,\\  Wilson 1995; Israel 1997), and in models of low metallicity\nclouds (Maloney \\& Black 1988). The strength of this trend is still\nquite uncertain. Israel (1997) finds a strong dependence on metallicity \n($\\partial \\log X\/ \\partial \\log Z = -2.7\\pm 0.3$), whereas Wilson (1995) derives a weaker \nrelationship, $\\partial \\log X\/ \\partial \\log Z = -0.67\\pm 0.1$.\nIn our models, the relationship is dependent on the ISM phase structure, but \nfalls in the range $\\partial \\log X\/ \\partial \\log Z = -1 {\\rm\\ to} -2$.\nAgain, however, it is difficult to directly compare our theoretical \nvalues with those determined observationally due to the different\nphysical scales involved. \n\nGiven the strong dependence on metallicity and ISM density structure,\nit is clear that use of the standard Milky Way value of X is suspect in \nLSB galaxies. We can instead turn the problem around and ask, given \nour theoretical calculation of X, what are the inferred constraints on the \nmolecular gas fraction of LSBs from the CO studies of S90 and dBvdH. If our \nmodels are correct, X in LSBs may be greater than the ``standard value''\nby as much as a factor of 10, significantly raising the upper limits on LSB\nmolecular gas content. A similar conclusion was reached by dBvdH, who \nexplored the consequences of a non-standard value of X. In that study,\na value of X of four times the galactic value was favored, resulting in\nupper limits for LSB molecular contents of $M_{H_2}\/M_{HI} < 0.25$. \nOur models favor the use of a high value of $X$ for LSBs, and indicate\nthat the correct value may be even a factor of two higher than that \nfavored by dBvdH. If so, the current non-detections of CO in LSBs still \nallow for significant molecular component of the ISM.  More stringent limits \non the molecular content of LSBs must await deeper CO observations.\n\n\\section{Discussion}\n\nOur models indicate that even very low surface brightness galaxies may not be \ncompletely void of molecular gas -- instead, the ISM may contain 10--20\\% \nof molecular gas (and perhaps more, depending on the detailed physical structure\nof the ISM). The physical conditions in this gas may be very \ndifferent from the conditions in the molecular ISM of the Milky Way. If\nthe ISM pressure is extremely low, as might be expected due to the low\nsurface mass density of LSB disks, the molecular phase of the ISM will\nbe diffuse and generally warmer than found in Galactic giant molecular\nclouds. The warm\ntemperature is due largely to the lack of shielding from the ISRF\nin a diffuse ISM; even a modest multi-phase ISM can self-shield the\nmolecular gas and lower the gas temperature. However the low\nsurface densities and star formation rates of LSB galaxies make it\nhard to generate and\/or sustain such a multiphase ISM (e.g.,\\  Gerritsen\n\\& de Blok 1998).\n\nAside from the explicit dependencies of the models on ISM structure,\nionization parameter, and metallicity, other more implicit model \ndependencies should also be reiterated. Our models assume that\nthe UV ISRF scales with optical surface brightness. Stellar population\ndifferences between LSB and HSB galaxies are not well-determined,\nbut the blue colors of LSBs argue that their stellar populations\nmay be hotter than those of HSBs. If so, we may underestimate the\nISRF in LSBs, thereby overestimating their molecular content. Similarly,\nwe have modeled an ISM where the neutral gas density increases\ncontinually into the center of the model, whereas many LSBs show\ncentral depressions of gas density. Again, this effect may push us\ntowards artificially high molecular contents (by underestimating\nthe ionization parameter). However, the HI mass profiles of LSBs\nare varied, so rather than acting as a systematic effect in our\nmodels, the dependency on gas profile is perhaps better viewed as \na caution against over-interpreting our results as they apply to \n{\\it individual} LSBs. A third model dependency worth noting is the\nassumption that the dust-to-gas ratio of the galaxies scales linearly \nwith metallicity.  One expects something very close to this from\nsimple considerations of chemical evolution (Edmunds \\& Eales 1998), and\nsuch a relationship is supported by\nobservational data (e.g.,\\  Issa, MacLaren, \\& Wolfendale 1990).\nThese dependencies are all\ntied to the systematic properties of LSBs which remain ill constrained.\nRather than attempting any further iteration on\nthe models, we leave these effects as a caveat to the ensuing\ndiscussion.\n\nThese uncertainties not withstanding, our models may also shed light \non the lowered efficiency of star formation\nin LSB disks. Compared to HSBs, LSB galaxies have a lower fraction\nof molecular material from which they can produce stars. In addition,\nwhatever molecular gas exists, it is in a more diffuse, warmer\nstate than is typical for molecular material in HSBs.  These warm \ntemperatures and low densities act to help stabilize any existing molecular \nclouds against gravitational collapse. Indeed, since the Jeans length \nscales as $\\sqrt{T\/\\rho}$, the size scale for the collapse of ISM \nsubstructure is quite large in LSBs. The larger size of any unstable \npatches makes them very susceptible to differential shear in the rotating \ndisks, so that gravitational collapse and subsequent star formation in the \nISM of LSBs will be quite difficult. Even in the solid body portion of\nthe rotation curve, where rotational shear is not a factor, the star\nformation rates remain low due to the increased collapse time of low\ndensity structure.\n\nThis stability has been parameterized (e.g.,\\  Quirk 1972; Kennicutt 1989)\nin a form very similar to the Toomre Q parameter for the growth of \naxisymmetric modes (Toomre 1964). Under such prescriptions, star formation\noccurs when the gas surface density exceeds some critical value:\n$\\Sigma_{gas} > \\alpha \\kappa \\sigma \/ 3.36 G$, where $\\kappa$ is the\nepicyclic frequency of the disk, $\\sigma$ the velocity dispersion of\nthe gas, and $\\alpha$ is a constant $\\sim 1$. Studies of LSB galaxies\nhave shown that the HI surface density is generally below this critical\nthreshold for star formation (van der Hulst et~al.\\  1993). In fact, the\ninnermost regions of LSB disks are often suppressed in HI; adding\ndiffuse, undetected H$_2$ increases the gas surface density and may make\nLSB galaxies somewhat more susceptible to induced star formation (e.g.,\\ \nMihos et~al.\\  1997; O'Neil, Bothun, \\& Schombert 1998).  However, the required\namount is not very reasonable.  There are some LSBs with star formation at\nsmall radii where the HI gas is sub-critical by a factor of 4 or more\n(de Blok, private communication), quite a bit more than can be made up by\nmolecular gas for reasonable model parameters.  Whether this is a failure of\nour models or of the Quirk-Kennicutt criterion (or both) is unclear.\n\nSimilar to the local stability criteria, parameters exist to describe\nthe stability of disks to growing global bar modes. One such\nparameterization is the Toomre $X_2$ parameter: \n$X_2 ={ {\\kappa^2 R}\\over{4\\pi G \\Sigma_d} }$, where $\\Sigma_d$ is the\ntotal disk mass surface density. If $X_2 >> 1$, disks are stable\nagainst $m=2$ perturbations (Toomre 1981). Mihos et~al.\\  1997 showed that \nbecause of their lowered disk surface density and increased dark matter \ncontent (relative to HSB disks), LSB galaxies are quite stable against such\ninduced bar modes. The inclusion of additional disk mass in the form\nof molecular ISM reduces this disk stability, but sufficient dark\nmatter exists in LSB galaxies to make them stable against all but\nthe strongest perturbations.\n\nWe note in passing that the quantity of mass in this (as yet undetected) \nmolecular \nISM is not nearly sufficient to account for all the dark matter in LSB\ndisks. Even under the dubious assumption of a maximum (stellar) disk,\nde Blok \\& McGaugh (1997) showed that the mass deficit in the inner\nregions of LSB galaxies is quite severe -- significant amounts of dark\nmatter must exist all the way into the centers of LSB disks. Under\nreasonable assumptions for the physical conditions in LSBs, our models\nsuggest that the molecular ISM can increase the disk surface \ndensity {\\it at most} by $\\; \\buildrel < \\over \\sim \\;$ 50\\%. To account for all the mass deficit implied\nby the rotation curve fitting of de Blok \\& McGaugh (1997), the molecular\nISM would need to be very cold and very clumpy, raising questions of\nwhy LSBs remain stable and how disk star formation is quenched. \n\nThe different evolutionary histories of HSB and LSB galaxies can be\ntraced to differences in their disk surface densities and in the \nconditions of their ISMs. A plausible evolutionary scenario for\nHSB galaxies has been outlined by Spaans \\& Norman (1997). In this\nscenario, once the proto-HSB gas disk forms, star formation begins\nat a retarded rate in the primordial molecular hydrogen ISM. This\nstar formation generates supernovae and enriches the ISM, leading to\na multiphase ISM that is able to cool and form stars efficiently --\nan HSB disk galaxy is born. In contrast, when a proto-LSB forms,\nit, too, forms a molecular ISM, but with a smaller molecular mass\nfraction and at lower surface density. At these low surface densities,\nit is difficult to trigger star formation or form\/maintain a multiphase\nISM. As a result, the LSB evolves little from its primordial conditions,\nmaintaining its low surface brightness and metallicity, and high gas\nfraction. \n\nUnder ``critical density'' conditions for star formation,\none might expect some bimodal surface brightness distribution for\ndisk galaxies, as galaxies will naturally follow one of two alternative\npaths depending on their surface density.  There is a claim of\na bimodal surface brightness distribution in one cluster (Tully\n\\& Verheijen 1997), but this does not appear to be a general property\nof field galaxies (de Jong 1996).\nInstead, it is more likely that there is a continuum of physical\nconditions in disk galaxies driven ultimately by surface density.\nLow density environments result in lowered star formation activity\n(as $t_{\\rm dyn} \\sim \\rho^{-{1\\over 2}}$ even the absence of any critical\ndensity models)\nand suppress the formation of a multiphase ISM; \nas surface density increases along a galactic sequence, star formation \nand surface brightness increase, accompanied by a rise in the\namount of complex phase structure (and higher molecular fractions) \nin the ISM. It is through this interplay that galaxy evolution\nand ISM processes shape the (cosmological) star formation rate.\n\n\\acknowledgements\n\nWe thank Erwin de Blok and Greg Bothun for valuable discussions.\nM.S. and J.C.M. have been supported by NASA through Hubble Fellowship grants\n\\#~HF-01101.01-97A and \\#~HF-01074.01-94A, respectively, awarded by the \nSpace Telescope Science Institute,\nwhich is operated by the Association of Universities for Research in\nAstronomy, Inc., for NASA under contract NAS 5-26555.\n \n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\\label{INTRODUCTION}\n\nRecent Hi-C imaging and sequencing technology have elucidated the importance of 3-D chromatin structure and epigenetics in gene regulation \\cite{nora2012spatial,rao20143d,lieberman2009comprehensive}. In addition to containing compartments of active, gene rich euchromatin and compartments of inactive, gene poor heterochromatin, chromatin is spatially partitioned into topologically associating domains (TADs) \\cite{pombo2015three}. TADs are insulated regions of chromatin, where sequences within each respective region have more frequent interactions than with sequences in other regions of the genome. Borders of TADs are often marked by the presence of CCCTC-binding factor (CTCF) \\cite{ji20163d}. CTCF is a highly conserved zinc finger protein that recognizes 50 base pair variant sequences throughout the genome \\cite{ohlsson2001ctcf}. CTCF is thought to facilitate TAD formation by binding to two distant locations of DNA and then binding to itself, creating a loop of chromatin \\cite{phillips2009ctcf,rao20143d}. Recent studies of CTCF have shown it is essential to loop formation, driving epigenetic forces of gene expression, but it is not essential to compartmentalization of chromatin into active and inactive regions \\cite{nora2017targeted}. \n\nAlthough CTCF's role in loop formation is well characterized, its role in gene regulation is less well understood. Previous studies have noted CTCF's importance in gene regulation during development, showing that disruption of CTCF affects gene transcription in mouse oocytes \\cite{wan2008maternal}. Other studies have shown that disruption of CTCF affects essential genetic pathways of cell proliferation, differentiation, and apoptosis \\cite{torrano2005ctcf}. Recent studies have linked CTCF to alternative splicing of nearby genes. In mammalian CD45 genes, CTCF is thought to promote inclusion of exon 5 by pausing RNA polymerase II \\cite{shukla2011ctcf}. Genome-wide, CTCF is thought to facilitate exon inclusion, or alternate exon usage, during RNA splicing by bringing exons in closer proximity to their promoters \\cite{ruiz2017ctcf}. However, these studies remain limited in the scope of genes investigated or largely correlative, demanding a functional investigation of the effect of CTCF on alternative splicing.\n\nHere, we used previously published ChIP-seq and mRNA-seq data from a CTCF knockdown mouse embryonic stem cell (mESC) model to examine the extent of CTCF dependent alternative splicing events  \\cite{nora2017targeted}. Specifically, we compared exon usage in genes that contain a CTCF binding site in mESC lines tagged with an auxin-inducible degron (AID) for CTCF and untagged wild type, before and after treatment with auxin. We provide evidence that the presence of intragenic CTCF alters exon usage in a transcription direction dependent manner. We show that degradation of CTCF in an AID system results in a higher proportion of upstream exon usage in alternative splicing. These results support the direct role of CTCF in regulating alternative splicing during embryogenesis, and nominate a heritable epigenetic system that can be probed to better understand the pathology of alternative splicing driven diseases that arise during development.\n\n\\section{METHOD}\n\\label{METHOD}\n\\subsection{Data Retrieval}\n\\label{Data}\nAll data analyzed in this study were from the previously published Nora et al., 2017 paper \\cite{nora2017targeted}. Expression levels for mRNA fragments were retrieved from the National Center for Biotechnology Gene Expression Omnibus (GSE98671). Experimental parameters and total reads were obtained from the supplements of Nora et al., 2017. CTCF ChIP-seq peak locations and magnitudes were provided by the Mirny Lab located at the Massachusetts Institute of Technology (mirnylab.mit.edu). Mouse genome mappings (NCBI37\/mm9) were available from the University of California Santa Cruz (UCSC) Genome Browser.\n\n\\subsection{Identification and Ranking of Gene Bound CTCF Sites}\n\\label{Identification}\nAll calculations were done on R (Version 3.4.1) using tools from the Bioconductor project. First, the most prominent CTCF sites that were successfully degraded by auxin were isolated. From the 43,607 CTCF ChIP peaks in the untreated sample, 13,131 peaks remained or were not fully degraded in the treated sample. These 13,131 peaks were identified by genomic location using the findOverlaps function and subsequently removed from analysis. The 30,554 remaining peaks were then cross referenced with the known gene locations of the mm9 assembly to find CTCF sites that were located within protein coding sequences. This final pool of 16,665 peaks were ranked by peak magnitude and the highest five thousand were examined in this study. \n\n\\subsection{Quantifying mRNA-Seq Reads}\n\\label{Quantifying}\nFor each of the 5,000 CTCF sites selected, the gene containing the CTCF site was found and the isoform with the most comprehensive selection of exons was selected. The locations of all exons upstream and downstream of the site were then identified. For each RNA-seq tag Density file, the signals in the exons upstream were summed. The resulting sum was divided by the total signal for the entire RNA-seq file and multiplied by the total number of reads in the experiment to estimate the number of reads in the upstream exons:\n$$R_{mRNA} = \\frac{\\sum_{exon} Signal }{\\sum_{total} Signal } R_{total} ,$$\nwhere $R_{mRNA}$ is the mRNA-seq Reads and $R_{total}$ the total number of reads in our experiment. The same calculation was done for the downstream exons. Estimated reads were rounded to the nearest whole number and pooled with data from experimental replicates under the same conditions. \n\n\\subsection{Statistical Analysis}\n\\label{Statistical}\nThe statistic used to compare distribution of isoforms around CTCF sites is the proportion ($P$) of reads upstream ($U$) compared to reads downstream ($D$), $P = \\frac{U}{D}$. This statistic will hereafter be referred to as the proportion of a CTCF site. The final set of sites with valid proportion values consisted of 2,636 sites. \n\n\n\\section{RESULTS}\n\\label{RESULTS}\n\\subsection{Orientation and Shift in Proportions}\n\\label{Orientation}\nBefore the influence of CTCF on alternative splicing can be examined, directionality effects due to CTCF and transcriptional direction have to be accounted for. Effects of various experimental conditions were quantified by dividing the proportion after treatment by the proportion before treatment. Kernel Density plots for the log change in proportions are plotted in Figure \\ref{fig1}. As Figure \\ref{fig1a} shows, changes in log expression ratio around the CTCF site does not depend on CTCF orientation. On the other hand, Figure \\ref{fig1b} shows that changes in log expression ratios are symmetric with respect to transcriptional direction. A two-sample t-test shows significant difference ($p.value = 2E-63$) between the distributions. Thus, comparisons must be made with respect to transcription orientation, with upstream of a CTCF site being defined as transcriptionally upstream and downstream as transcriptionally downstream. Once corrected for transcriptional direction, change in log expression ratio is positive (Figure \\ref{fig1c}).\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig1a}\n\t\t\\caption{}\n\t\t\\label{fig1a}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig1b}\n\t\t\\caption{}\n\t\t\\label{fig1b}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig1c}\n\t\t\\caption{}\n\t\t\\label{fig1c}\n\t\\end{subfigure}\n\t\\caption{Log change in proportions in CTCF-AID tagged cells from untreated to auxin 2 days ($N=2636$). (a) Distributions grouped by CTCF orientation overlap and show no significant difference. (b) Distributions grouped by transcription orientation are mirrored and show significant difference ($p.value=7E \u2013 63$). (c) When proportions are recalculated to account for transcriptional orientation, log change in proportions show significant deviation from zero ($p.value=2E \u2013 63$).}\n\t\\label{fig1}\n\\end{figure}\n\n\\subsection{Contingency Tables and Tests for Significance}\n\\label{Contingency}\nTo get an understanding of specific changes in isoform distribution and evaluate the significance of the change at specific CTCF sites, contingency tables were built for each site. Observations consist of the number of fragments detected upstream and downstream of the CTCF site with Fisher's exact test conducted to evaluate the difference. Fisher's exact test was preferred over chi-squared test because reads of mRNA fragment tend to be skewed and vary wildly between very large and very small counts. Multiple testing correction was performed using Bonferroni correction, resulting in a more conservative alpha for significance testing, $\\alpha=0.05\/2636=1.9E-5$.\nGiven the noisiness of the data, the reduced power owing to increased conservativeness of the test and FDR control were acceptable.\nThree contingency tables were constructed for each site to evaluate the influence of CTCF degradation on alternative splicing. The parameters for the tests are expressed in Table \\ref{tab1}. \n\n\\begin{table}[H]\n\t\\centering\n\t\\begin{adjustbox}{width=1\\textwidth}\n\t\t\\begin{tabular}{|l|l|l|l|}\n\t\t\t\\hline\n\t\t\t\\textbf{Test}   & \\textbf{Sample}  & \\textbf{Comparison} & \\textbf{Observed}\\\\ \n\t\t\t\\hline\n\t\t\t1  &  CTCF-AID tagged cells  & Untreated vs auxin 2 days  & \\thead{Distribution of fragments \\\\upstream and downstream\\\\ of the CTCF site}\\\\ \n\t\t\t\\hline\n\t\t\t2  &  Wildtype untagged cells & Untreated vs auxin 2 days& \\thead{Distribution of fragments \\\\upstream and downstream\\\\ of the CTCF site}\\\\\n\t\t\t\\hline\n\t\t\t3  &  Untreated tagged and untagged cells  & CTCF-AID tagged vs wildtype untagged& \\thead{Distribution of fragments \\\\upstream and downstream\\\\ of the CTCF site}\\\\\n\t\t\t\\hline \t\t\t\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{Description of tests conducted on contingency tables}\n\t\\label{tab1}\n\\end{table}\n\n\\subsection{Changes in Proportions in CTCF Bound Genes}\nThe distribution of p-values show anti-conservative trends for all three tests (Figure \\ref{fig2}), suggesting that the alternative hypothesis of equal exon usage in genes with CTCF binding sites may be true for some genes. In CTCF-AID tagged cells, treatment with auxin resulted in significant changes in proportions at 464 CTCF sites (Figure \\ref{fig2a}). Surprisingly, 356 sites in wildtype untagged cells also showed significant change after auxin treatment (Figure \\ref{fig1b}) even though treatment wouldn't result in CTCF depletion. Moreover, comparing untreated CTCF-AID to untreated untagged cells shows 483 significant sites (Figure \\ref{fig2c}). \n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig2a}\n\t\t\\caption{}\n\t\t\\label{fig2a}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig2b}\n\t\t\\caption{}\n\t\t\\label{fig2b}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig2c}\n\t\t\\caption{}\n\t\t\\label{fig2c}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{fig2d}\n\t\\caption{}\n\t\\label{fig2d}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{fig2e}\n\t\\caption{}\n\t\\label{fig2e}\n\t\\end{subfigure}\n\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{fig2f}\n\t\\caption{}\n\t\\label{fig2f}\n\\end{subfigure}\n\t\\caption{Factors that Affect Alternative Splicing. (a \u2013 c) Distribution of p-values from Fisher's exact test of factors influencing splicing. (a) Test 1 evaluates the effect of auxin on CTCF-AID tagged cells. (b) Test 2 evaluates the effect of auxin on wildtype untagged cells. (c) Test 3 evaluates the effect of CTCF-AID tagging. (d \u2013 f) scatter plots mapping proportions in control and experiment and color coded by significance of p-values. (d, e) Untreated vs auxin 2 days on tagged and untagged WT cells. (f) Untagged WT vs CTCF-AID tagged untreated cells. }\n\t\\label{fig2}\n\\end{figure}\n\nA large magnitude of change in proportion isn't always significant while a large number of site that showed moderate proportion changes were significant. Significant sites showed both positive and negative change in proportions (Figure \\ref{fig3a}). Nonsignificant points showing large changes had relatively small numbers of reads, making it possible for small variations to show a large magnitude change but still give high p-values; meanwhile, sites showing significance close to the center have a large number of reads (Supplementary Table Sites). A number of sites overlap in the tests for which they are significant (Figure \\ref{fig3b}). There are more overlaps than expected for such stringent selection, suggesting that there may be common mechanisms causing these sites to display greater variation. \n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig3a}\n\t\t\\caption{}\n\t\t\\label{fig3a}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig3b}\n\t\t\\caption{}\n\t\t\\label{fig3b}\n\t\\end{subfigure}\n\t\\caption{CTCF sites grouped by tests they are significant for. (a) Scatter plot mapping log change in proportion in wildtype untagged cells to log change in CTCF-AID tagged cells. (b) Venn diagram showing overlaps in test significance. There were 1718 sites which were not significant for any of the tests.}\n\t\\label{fig3}\n\\end{figure}\n\n\\subsection{Distribution of Proportion Change}\n\\label{Distribution}\nAssuming that all CTCF sites are in one population and impacted similarly, the distribution of proportion changes can be viewed as a whole to analyze how treatment affected splicing. Exposure to auxin for 2 days caused proportions to significantly increase in CTCF-AID tagged cells and to significantly decrease in wild-type untagged cells (Figure \\ref{fig4a}). As expected, treated tagged cells showed significant increase in proportions compared to treated untagged cells. Surprisingly, untreated tagged cells also showed significant increases over untreated untagged cells. However, it is of note that these changes were smaller in magnitude (Figure \\ref{fig4b}). \n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[b]{0.8\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig4a}\n\t\t\\caption{}\n\t\t\\label{fig4a}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[b]{0.8\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{fig4b}\n\t\t\\caption{}\n\t\t\\label{fig4b}\n\t\\end{subfigure}\n\t\\caption{Log Change in proportions across treatments. (a) Comparing untreated to auxin 2 days. Untagged cells show decrease with $p.value=3E-16$. Tagged cells show increase with $p.value=2E-63$. (b) Comparing wildtype untagged to CTCF-AID tagged. Untreated cells show increase with $p.value=2E-51$. Treated cells show increase with $p.value=5E-171$.}\n\t\\label{fig4}\n\\end{figure}\n\n\\section{DISCUSSION}\n\\label{DISCUSSION}\nWhile CTCF has been noted as a key player in shaping the 3D structure of chromatin, its direct effects on gene regulation, and in particular alternative splicing, is less well characterized. As noted, this study investigated the direct effects of CTCF on alternative exon usage in mouse embryonic stem cells. Using previously published ChIP-seq and RNA-seq data from Nora et al., 2017, we investigated alternative splicing in genes containing a CTCF binding site. We quantified and compared the changes in alternative exon usage after the CTCF was removed from a CTCF bound gene was degraded. \n\nWe found that degrading CTCF using auxin in CTCF-AID tagged cells resulted in an increase in the proportion of upstream fragments used in final mRNA transcripts. This finding supports Shukla et al, 2011's hypothesis that CTCF binding to DNA blocks transcription proteins and causes pauses in mRNA transcription. Splicing occurs concurrently with transcription, and thus when transcription is paused due to CTCF, splicing elements are able to act upon RNA upstream that were already transcribed with greater frequency. Depletion of CTCF likely prevents these pauses in transcription, giving splicing elements less opportunity to act on exons upstream of CTCF sites. Here, we provide evidence for this mechanism by showing that CTCF depletion results in greater upstream exon usage in mRNA formation. \n\nAlthough these findings seem promising, it should be noted that significant differences in the proportions of exon usage were also observed in control cases that should exhibit none. In particular, there were 384 genes that exhibited a significant change in alternative splicing in the wild type, untagged cells after treatment with auxin. When comparing wild type, untagged cells with CTCF-AID tagged cells without the presence of auxin, we found 483 CTCF bound genes that exhibited significantly different alternative splicing among these two conditions. Although the magnitudes of the changes were smaller than those between the experimental condition and control, these changes are alarming as they suggest that the AID tagging method itself may cause changes in gene expression. One explanation is that the tagging process affected the expression levels of genes related to splicing factors and transcription controls. Another is that although ChIP-seq shows tagged CTCF still bound to DNA, the binding efficiency may be impacted to an extent that differences in splicing may be observed. What may be even harder to explain is why untagged cells showed lower proportions after exposure to auxin. Untagged cells did not have their CTCF degraded, showed no difference in gene expression levels, and did not suffer from the same cytotoxic effects that tagged cells displayed. Nonetheless, untagged cells showed an anti-conservative distribution of p-values and a significantly negative change in proportion. These observations should be examined further, or they may undermine the conclusions made above.\n\nThe effect of CTCF binding on alternative splicing in mouse embryonic stem cells is apparent in this study. We showed a number of genes that exhibit a change in upstream exon usage after depletion of CTCF, suggesting a functional role of CTCF in determining alternatively spliced mRNA transcripts. As noted by Li et al., 2016, alternative splicing and the resulting isoforms have great impact not only on biodiversity and genetic variation but also on disease \\cite{li2016rna}. Understanding the mechanisms behind alternative splicing can provide insight into the pathology of diseases such as developmental disorders and cancers. This has already been hinted by Filipova et al., 1998 when the authors associated CTCF binding with deletions resulting in breast and prostate cancers \\cite{filippova1998widely}. Perhaps more exciting, understanding the decision-making machinery behind alternative splicing can expose potential vulnerabilities in alternative splicing driven mechanism and inform potential targets for therapy. Future work should focus on characterizing the types of genes and pathways affected by CTCF mediating alternative splicing. \n\n\\bibliographystyle{elsarticle-num}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nSince oxygen is the third most abundant element in the protosolar nebula \\citep{An89,Lo09}, this naturally makes water as the most abundant volatile compound in planetary bodies of our solar system, if one excepts the hydrogen and helium presents in the envelopes of the giant planets \\citep{En08,Boc17,Gr17}.\n\nWater-rich worlds (Europa, Titan, Enceladus, Pluto, Triton, etc) are ubiquitous in our solar system, and the building blocks of Uranus and Neptune are also supposed to be water rich \\citep{Mo18}. These properties led astronomers to consider the possible existence of massive water-rich planets around other stars, i.e. the so-called ocean planets \\citep{Le04}. Those planets would have grown from ice-rich building embryos formed beyond the snowline in protoplanetary disks, and would have subsequently migrated inward up to their current orbital location nearby their host star \\citep{Ra18a,Ra18b}. This motivated the implementation of an H$_2$O layer to existing internal structure models, in which the liquid water had a simple prescription for the temperature profile (often isothermal), which often led to the coexistence of liquid water with high pressure ices \\citep{So07,Va07,Fo07,Ze13,Ze19}. At that time, it was believed that the temperature structure had a minor impact on the radii as it is the case for telluric planets \\citep{Va06,Fo07}.\n\nHowever, exoplanets considered today as good candidates for being water-rich worlds are also subject to important irradiation from their host star due to their short orbital periods. For such conditions at the surface of the planet, assuming an adiabatic temperature gradient produces very shallow P(T) profiles \\citep{Th16}. As a consequence, water is not in condensed phase, but rather in supercritical state in most of their hydrospheres, making ocean planets way more inflated with an adiabatic prescription compared to an isothermal one \\citep{Tu20,Mo20,Ha20}.\n\nThe inflated hydrospheres of irradiated supercritical ocean planets have been recently shown to be good candidates to account for the large radii of sub-Neptunes planets \\citep{Mo20}. They could also provide a possible explanation for the bimodal distribution of super-Earth and sub-Neptune populations, also known as the Fulton gap \\citep{Fu17}. These physical properties, along with the availability of several sets of thermodynamic data for H$_2$O \\citep{Wa11,Du06,Ma19,Jo20}, has recently motivated the modeling of the equation of state (EoS) of water in conditions relevant to planetary interiors, from 0 to a few TPa, the latter value corresponding to a Jupiter-mass planet fully made of water \\citep{Ma19,Tu20,Ha20}.\n\nFor the sake of precision, mass-radius relationships of supercritical ocean planets must be calculated via the simultaneous use of atmosphere and interior structure models that are both connected at their boundaries. For example, \\cite{Tu20} focused on planets of masses $0.2$--$2$ $M_\\Earth$ and water contents of $0.01$--$5$ wt\\%, to investigate the presence of water in the planets of the TRAPPIST-1 system. They added an irradiated steam atmosphere on top of rocky cores, using tabulated mass-radius relationships of \\cite{Ze16}. These latters were computed at a 1 bar surface pressure, and might become invalid in the case of heavy H$_2$O layers (surface pressures considered up to 10 GPa). In the approach presented in \\cite{Mo20}, the atmosphere model from \\cite{Mar19} only considers the uppermost part of the hydrosphere up to a given pressure. The rest of the interior structure, including extreme phases of H$_2$O, is computed via an interior model \\citep{Br17}, allowing to compute planets with any water content. The aim of our work is to update this model by using state of the art equations of state, and to include a better connection between the atmosphere and the interior models.\n\nTo do so, we combine the three parts of an hypothetical supercritical planet (refractory interior, condensed-fluid H$_2$O layer, and steam atmosphere) in a self-consistent framework to provide analytical descriptions of mass-radius relationships, which depend on the planetary mass, water mass fraction (WMF) and the equilibrium temperature. Such a derivation will allow estimating the WMF of irradiated ocean exoplanets from ground- or space-based mass-radius observations.\n\nWe also discuss the possible existence of these supercritical planets in light of hydrodynamic and Jeans' atmospheric escapes, and provide the mass-radius domains where escape is efficient. We finally use our model to compute the WMF of exoplanets b, c, and d of the system GJ-9827, chosen as a test case, and find that planet d could be a planet in supercritical state made of $20\\pm 10\\%$ of H$_2$O by mass.\n\nSection \\ref{sec:previous} reviews the model from \\cite{Mo20}, presenting its main features, inputs and outputs. Section \\ref{sec:improvements} details the work that has been made to update the model's EoS and make a consistent connection between the interior and the atmosphere model. Results are shown in Section \\ref{sec:results} in the form of mass-radius relationships, and ternary diagrams, and a conclusion is made in Section \\ref{sec:ccls}.\n\n\\section{Underlying interior and atmospheric models} \\label{sec:previous}\n\nWe follow the approach of \\cite{Mo20} consisting in coupling a Super-Earth interior model derived from \\cite{Br17} and the atmospheric model described in \\cite{Ma17,Mar19}. Here we recall the basic assumptions of these models.\n\n\\subsection{Interior Model}\n\nOur model solves iteratively the equations describing the interior of a planet:\n\n\\begin{eqnarray}\n\\frac{\\mathrm{d} g}{\\mathrm{d} r}&=&4 \\pi G \\rho-\\frac{2 G m}{r^{3}}, \\label{eq:gauss}\\\\\n\\frac{\\mathrm{d} P}{\\mathrm{d} r}&=&-\\rho g, \\label{eq:hydrostatic}\\\\\n\\frac{\\mathrm{d} T}{\\mathrm{d} r}&=&-g\\gamma T\\frac{\\mathrm{d} \\rho}{\\mathrm{d} P}, \\label{eq:temp_grad}\\\\\nP &=& f(\\rho,T), \\label{eq:solve_eos}\n\\end{eqnarray}\n\n\\noindent which correspond to the Gauss's theorem, hydrostatic equilibrium, adiabatic profile with use of Adams-Williamson equation, and the EoS of the considered medium, respectively. $g$, $P$, $T$ and $\\rho$ are gravity, pressure, temperature and density profiles, respectively. $m$ is the mass encapsulated within the radius $r$, $G$ is the gravitational constant, and $\\gamma$ is the Gr\\\"uneisen parameter. The Grune\u00efsen parameter is key to compute the thermal profile of the planet, and the literature sometimes refers to the adiabatic gradient instead, expressed as follows \\citep{Ki12,Ma19,Ha20}:\n\n\\begin{eqnarray}\n\t\\nabla_\\mathrm{ad} = \\left(\\frac{\\partial \\ln T}{\\partial \\ln P}\\right)_S = \\gamma \\frac{P}{\\rho} \\frac{1}{c^2},\n\\end{eqnarray}\nwhere $S$ is the entropy, and $c$ the speed of sound.\\\\\n\nThe interior model can display up to five distinct layers, depending on the planet's characteristics:\n\n\\begin{itemize}\n\t\\item a core made of metallic Fe and FeS alloy;\n\t\\item a lower mantle made of bridgmanite and periclase;\n\t\\item an upper mantle made of olivine and enstatite;\n\t\\item an ice VII phase;\n\t\\item a hydrosphere covering the whole fluid region of H$_2$O.\n\\end{itemize}\n\nThe Vinet EoS \\citep{Vi89} with thermal Debye correction is used for all solid phases:\n\\begin{eqnarray}\n\\begin{aligned}\nP\\left(\\rho, T\\right)=& 3 K_{0}\\left[\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{\\frac{2}{3}}-\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{\\frac{1}{3}}\\right] \\times \\\\\n& \\exp\\left\\{\\frac{3}{2}\\left(K_{0}^{\\prime}-1\\right)\\left[1-\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{-\\frac{1}{3}}\\right]\\right\\}\\\\\n& + \\Delta P,\n\\end{aligned}\n\\end{eqnarray}\n\n\\noindent with\n\n\\begin{eqnarray}\n\\begin{aligned}\n\\Delta P = &9 \\frac{\\gamma \\rho R}{M_\\mathrm{mol} \\theta^{3}} \\times\\\\\n&\\left[T^{4} \\int_{0}^{\\frac{\\theta}{T}} \\frac{t^{3}}{e^{t}-1} d t-T_{0}^{4} \\int_{0}^{\\frac{\\theta}{T_{0}}} \\frac{t^{3}}{e^{t}-1} d t\\right],\n\\end{aligned}\n\\end{eqnarray}\n\n\\noindent where $\\theta=\\theta_{0}\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{\\gamma}$, $\\gamma=\\gamma_{0}\\left(\\frac{\\rho}{\\rho_{0}}\\right)^{-q}$, $R$ the ideal gas constant and $M_\\mathrm{mol}$ the molar mass of the considered material. All quantities with an index 0 are reference parameters obtained by fit on experimental data, given in table \\ref{tab:parameters}. The EoS used by \\cite{Mo20} to solve Eq. (\\ref{eq:solve_eos}) is the one formulated by \\cite{Du06}, valid up to 10 GPa and 2573.15 K.\n\nAll thermodynamic and compositional parameters of mineral layers are taken equal to those of Earth \\citep{St05,So07,So10}, and summarized in table \\ref{tab:parameters}. We refer the reader to \\cite{Br17} to get all the computational details. \n\n\\cite{Mo20} computed the Gr\\\"uneisen parameter for water via a bilinear interpolation in a grid generated from the python library of the IAPWS formulation\\footnote{https:\/\/pypi.org\/project\/iapws\/\\#description}, computing the Gr\\\"uneisen parameter in the form $\\gamma = f(\\rho,T)$ with $\\rho$ and $T$ varying in the 316--2500 kg.m$^{-3}$ and 650--10,000 K ranges, respectively. An important issue is that the density range is very limited, since this quantity can easily vary from $\\sim$10 kg.m$^{-3}$ at the planetary surface to $\\sim$5000  kg.m$^{-3}$ at the center of a 100\\% water planet of 1 $M_{\\Earth}$, implying that the computation of $\\gamma$ is erroneous at the top and at the bottom of the hydrosphere. A solution for overcoming this limitation is provided in Section \\ref{sec:gruneisen}.\n\nApart from compositional inputs, the main physical inputs of the model are the core mass fraction (CMF) $x_\\mathrm{core}$ and water mass fraction (WMF) $x_{\\mathrm{H}_2 \\mathrm{O}}$, the mantle mass fraction is then $x_\\mathrm{mantle} = 1 - x_\\mathrm{core} -x_{\\mathrm{H}_2 \\mathrm{O}}$. Pressure and temperature profiles are integrated from outside, and require the inputs of the boundary pressure $P_\\mathrm{b}$ and boundary temperature $T_\\mathrm{b}$. Finally, the model also requires the input of the planet's mass $M_\\mathrm{b}$ (subscript $b$ denotes the mass encapsulated within the boundary of the interior model, excluding the contribution of any potential atmosphere). Once defined, these input parameters allow for the computation of the planet's internal structure and associated boundary radius. In the case of the Earth ($x_\\mathrm{core}=0.325$, $x_{\\mathrm{H}_2 \\mathrm{O}}=0.0005$, $M_\\mathrm{b}=1$ M$_\\Earth$), the model computes a radius $R_\\mathrm{b}$ equal to 0.992 $R_\\Earth$, which is less than 1\\% of error, indicating that errors from the model are negligible compared to errors on measurements. In the following, subscript $b$ refers to quantities at the boundary between the interior model and the atmosphere model, such as bulk mass $M_\\mathrm{b}$, radius $R_\\mathrm{b}$, gravity $g_\\mathrm{b}$, pressure $P_\\mathrm{b}$ and temperature $T_\\mathrm{b}$.\n\n\\begin{table*}[!ht]\n\t\\movetabledown=5cm\n\t\\begin{rotatetable*}\n\t\\centering\n\t\\caption{List of thermodynamic and compositional parameters used in the interior model.} \n\t\\label{tab:parameters}\n\t\\begin{tabular}{llllllllllll}\n\t\t\\hline\n\t\t\\multicolumn{2}{l}{\\textit{Layer}}                     & \\multicolumn{2}{l}{\\textit{Core}}   & \\multicolumn{4}{l}{\\textit{Lower mantle}}         \n             & \\multicolumn{4}{l}{\\textit{Upper Mantle}}                  \n           \\\\ \\hline\n\t\tPhases                              &                  & \\multicolumn{2}{l}{Iron rich phase} & \\multicolumn{2}{l}{Perovskite} & \\multicolumn{2}{l}{Periclase} & \\multicolumn{2}{l}{Olivine}   & \\multicolumn{2}{l}{Enstatite}         \\\\\n\t\tComposition (\\%)                    &                  & \\multicolumn{2}{l}{100}             & \\multicolumn{2}{l}{79.5}       & \\multicolumn{2}{l}{20.5}      & \\multicolumn{2}{l}{41}        & \\multicolumn{2}{l}{59}    \n            \\\\ \\cline{3-12} \n\t\tComponents                          &                  & Fe             \n  & FeS              & FeSiO$_3$      & MgSiO$_3$     & FeO           & MgO           & Fe$_2$SiO$_4$ & Mg$_2$SiO$_4$ & Fe$_2$Si$_2$O$_6$ & Mg$_2$Si$_2$O$_6$ \\\\\n\t\tComposition (\\%)                    &                  & 87             \n  & 13               & 10             & 90            & 10            & 90            & 10            & 90            & 10                & 90     \n           \\\\ \\cline{3-12} \n\t\tMolar mass (g.mol$^{-1}$)           & $M_\\mathrm{mol}$ & 55.8457        \n  & 87.9117          & 131.9294       & 100.3887      & 71.8451       & 40.3044       & 203.7745      & 140.6931      & 263.8588          & 200.7774          \\\\\n\t\tReference density (kg.m$^{-3}$)     & $\\rho_0$         & 8340           \n  & 4900             & 5178           & 4108          & 5864          & 3584          & 4404          & 3222          & 4014              & 3215   \n           \\\\\n\t\tReference temperature (K)           & $T_0$            & 300            \n  &                  & 300            &               & 300           &   \n            & 300           &               & 300               &         \n          \\\\\n\t\tReference bulk modulus (GPa)        & $K_0$            & 135            \n  &                  & 254.7          &               & 157           &   \n            & 128           &               & 105.8             &         \n          \\\\\n\t\tPressure derivative of bulk modulus & $K_0^\\prime$     & 6              \n  &                  & 4.3            &               & 4             &   \n            & 4.3           &               & 8.5               &         \n          \\\\\n\t\tReference Debye temperature (K)     & $\\theta_0$       & 474            \n  &                  & 736            &               & 936           &   \n            & 757           &               & 710               &         \n          \\\\\n\t\tReference Gr\u00fcneisen parameter       & $\\gamma_0$       & 1.36      \n       &                  & 2.23           &               & 1.45         \n &               & 1.11          &               & 1.009             &    \n               \\\\\n\t\tAdiabatic power exponent            & $q$              & 0.91           \n  &                  & 1.83           &               & 3             &   \n            & 0.54          &               & 1                 &         \n          \\\\ \\hline\n\t\\end{tabular}\n\t\\tablecomments{Thermodynamic data are summarized in Table 1 of \\cite{So07}, and compositional data are the results of model calibration for Earth \nby \\cite{St05,So07}.}\n\t\\end{rotatetable*}\n\\end{table*}\n\n\\subsection{Atmospheric model} \\label{sec:atmo}\n\nThe atmospheric model generates the properties of a 1D spherical atmosphere of H$_2$O by integrating the thermodynamic profiles bottom to top. The model takes as inputs the planet's mass and radius, as well as the thermodynamic conditions at its bottom. We choose to connect the atmospheric model with the interior model at a pressure $P$ = $P_\\mathrm{b}=300$ bar (slightly above the critical pressure $P_\\mathrm{crit}=220.67$ bar) and at a temperature $T$ = $T_\\mathrm{b}$. $(P,T,\\rho)$ profiles are then integrated upward via the prescription from \\cite{Ka88} in the case of an adiabat at hydrostatic equilibrium. Once the temperature reaches the top temperature of the atmospheric layer, here set to $T_\\mathrm{top}=200$ K, an isothermal radiative mesosphere at $T=T_\\mathrm{top}$ is assumed. Figure \\ref{fig:profiles} shows several $(P,T)$ profiles representing the whole hydrospheres of planets (under Ma19+ parametrization, see section \\ref{sec:gruneisen}) for masses and irradiation temperatures in the range 1--20 $M_\\Earth$ and  $T_\\mathrm{irr}=$400-1300 K (see Eq. (\\ref{eq:teq})), respectively.\n\nThe atmosphere transition radius is controlled by the altitude of the top of the H$_2$O clouds, corresponding to the top of the moist convective layer, assumed to be at a pressure $P_\\mathrm{top}=0.1$ Pa. We choose this limit as the observable transiting radius, assuming that results are similar for cloudy and cloud-free atmospheres \\citep{Tu19,Tu20}. The EOS is taken from the NBS\/NRC steam tables \\citep{Ha84}, implying the atmosphere is not treated as an ideal gas. The discontinuities in $(P,T)$ profiles occuring for $T_\\mathrm{irr}=1300$ K are due to the limited range of these tables, but the height of this region ($P=$ 100-300 bar) is negligible compared to the thickness of the atmosphere. Increasing the $T_\\mathrm{top}$ temperature will impact the final structure of the atmosphere, decreasing both the thickness of the atmosphere and the interior. Numerical tests with $T_\\mathrm{top}$ varying from 200 K to $T_\\mathrm{skin}=T_\\mathrm{eff}\/ 2^{0.25}$ decrease the final radius of the planet of at most $\\sim 200$ km for the cases considered in this study. It corresponds to a difference of $2\\%$ in radius at most, but this difference is mainly below $1\\%$.\n\nShortwave and thermal fluxes are then computed using 4-stream approximation. Gaseous (line and continuum) absorptions are computed using the $k$-correlated method on 38 spectral bands in the thermal infrared, and 36 in the visible domain. Absorption coefficients are exactly the same as those in \\cite{Le13} and \\cite{Tu19} which includes several databases, specificaly designed for H$_2$O-dominated atmospheres. Rayleigh opacity is also included. This method computes the total outgoing longwave radiation (OLR, in W.m$^{-2}$) of the planet that gives the temperature that the planet would have if it was a blackbody:\n\n\\begin{eqnarray}\n\tT_\\mathrm{p} = \\left(\\frac{\\mathrm{OLR}}{\\sigma_\\mathrm{sb}}\\right)^{1\/4}, \\label{eq:tp}\n\\end{eqnarray}\n\n\\noindent with $\\sigma_\\mathrm{sb}$ the Stefan-Boltzmann constant. In order to quantify the irradiation of the planet by its host star, we define the irradiance temperature\n\n\\begin{eqnarray}\n\tT_\\mathrm{irr} = T_\\mathrm{eff} \\sqrt{\\frac{R_\\star}{2a}}, \\label{eq:tirr_obs}\n\\end{eqnarray}\n\n\\noindent where $T_\\mathrm{eff}$ and $R_\\star$ are the host star effective temperature and radius, respectively, and $a$ is the semi-major axis of the planet. The atmospheric model computes the Bond albedo from the atmosphere's reflectance \\citep[][using the method presented in]{Pl19}  assuming a G-type star linking both temperatures:\n\n\\begin{eqnarray}\n\tT_\\mathrm{irr} = \\left(\\frac{\\mathrm{OLR}}{(1-A)\\sigma_\\mathrm{sb}}\\right)^{1\/4} = \\frac{T_\\mathrm{p}}{(1-A)^{1\/4}}. \\label{eq:teq}\n\\end{eqnarray}\n\nThe literature often approximates $T_\\mathrm{irr}$ to the equilibrium temperature $T_\\mathrm{eq}$, which is the temperature the planet would have for an albedo $A=0$ (all the incoming heat is absorbed and re-emitted by the planet). Since it is the observable quantity, our results will be presented in term of $T_\\mathrm{irr}$. Equation \\ref{eq:teq} assumes that the planet is in radiative equilibrium with its host star. Any heating source in the planet interior would add an additional term in the radiative equilibrium of the planet with its host star, increasing the effective temperature of the planet for the same received irradiation \\citep{Ne11}. In this work, we model the structure of planets that have either no interior heating source, or that had time to cool off.\n\nFor a given planet mass, boundary radius and irradiation temperature, the atmosphere thickness and boundary temperature are retrieved from the atmospheric model. The latter is then used to compute the interior structure, and the former is taken into account to compute the total (transiting) radius.\n\n\\begin{figure*}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{profiles.pdf}}\n\t\\caption{$(P,T)$ profiles for 100\\% H$_2$O planets of masses $M_\\mathrm{p}=$1--20 $M_\\Earth$, and irradiation temperatures $T_\\mathrm{irr}=$400--1300 K with the Ma19+ parametrization (see section \\ref{sec:gruneisen}). Cases corresponding to smallest masses and highest temperatures are not shown, as their surface gravities are below the limit fixed in Sec. \\ref{sec:connect}. Phase transitions of H$_2$O are taken from \\cite{Wa11} at low temperatures (solid turquoise lines) and from \\cite{Ne11} at high temperatures (dashed turquoise lines), with labels IF (ionic fluid), SI (super ionic), P (plasma), and iN for the ice N.}\n\t\\label{fig:profiles}\n\\end{figure*}\n\n\\section{Model update} \\label{sec:improvements}\n\nThis section presents the improvements made on the existing model to push further its physical limitations. Since we are interested in planets with substantial amounts of water, we define a specfic CMF, which is only related to the mass budget of the rocky part:\n\n\\begin{eqnarray}\nx_\\mathrm{core}^\\prime = \\frac{x_\\mathrm{core}}{1-x_{\\mathrm{H}_2 \\mathrm{O}}}.\n\\end{eqnarray}\n\n\\noindent where $x_\\mathrm{core}$ is  the ``true'' CMF. $x_\\mathrm{core}^\\prime$ will be used to compare planets that have different WMF, but with similar refractory contents. For example, $x_\\mathrm{core}^\\prime=0.325$ corresponds to an Earth-like CMF, regardless the amount of water present in the planet.\n\n\\subsection{Used EoSs}\n\nThe choice of the EoS is critical, as it strongly impacts the estimate of the mass-radius relationships. Three EoS are then considered in this study:\n\n\\begin{itemize}\n\t\\item EoS from the latest revision of the IAPWS-95 formulation from \\cite{Wa02}\\footnote{http:\/\/iapws.org} (hereafter WP02). This reference EoS gives an analytical expression of the specific Helmholtz free energy $f(\\rho,T)$. Any thermodynamic quantity (pressure, heat capacity, internal energy, entropy etc.) can be computed by taking the right derivative of $f$, and those quantities have analytical expressions.\n\t\\item EoS from \\cite{Du06} (hereafter DZ06). This EoS is corrected around the critical point, and gives an analytical expression for pressure as function of density and temperature $P(\\rho,T)$.\n\t\\item EoS from \\cite{Ma19} (hereafter Ma19). This formulation was developed for planetary interiors by extending the IAPWS-95 EoS with ingredients from statistical physics allowing transition to plasma and superionic states. The authors created a fortran implementation\\footnote{http:\/\/cdsarc.u-strasbg.fr\/viz-bin\/qcat?J\/A+A\/621\/A128} that computes pressure, specific Helmholtz free energy, specific internal energy and specific heat capacity for a given couple $(\\rho,T)$.\n\\end{itemize}\n\n\\begin{figure*}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{aux_EOS_650.pdf}\\includegraphics[angle=0,width=5cm]{aux_EOS_2000.pdf} }\n\t\\caption{Pressure as function of density calculated with WP02 (blue), DZ06 (red) and Ma19 (green) in the cases of two different temperatures. The solid horizontal lines indicate the range of validity for WP02 and DZ06, and the dashed horizontal lines give the extended range. The black dotted line corresponds to the ideal gas law for water steam.}\n\t\\label{fig:comp_eos}\n\\end{figure*}\n\nThe validity ranges of the different EoSs, which rely on the availability of experimental data, are given in Table \\ref{tab:range}. Extended ranges proposed by \\cite{Wa02} and \\cite{Du06} are also indicated because the mathematical expressions of their EoSs allow for extrapolations beyond the corresponding validity ranges. However, they become invalid when phase transition occurs (e.g. dissociation of water). Other EoSs exist in the literature, covering various regions of the phase diagram of water, or being used for specific purposes. Our choice of EoSs among others is discussed in Sec. \\ref{sec:ccls}.\n\n\\begin{table}\n\t\\centering\n\t\\caption{Validity ranges of the different EoSs.} \n\t\\label{tab:range}\n\t\\begin{tabular}{lll}\n\t\t\\tablewidth{0pt}\n\t\t\\hline\n\t\t\\hline\n\t\tEoS \t& Valid \t\t& Extended \t\\\\\t\n\t\t\\hline\n\t\tWP02 & $P<1$ GPa & $P<100$ GPa   \\\\ \t\n\t\t&$T<1~273$ K & $T<5~000$ K \\\\\n\t\tDZ06  \t& $P<10$ GPa & $P<35$ GPa\\footnote{Limit given by \\cite{Du96}.\\label{limdz}}  \\\\\n\t\t&$T<2~573.15$ K& $T<2~800$ K\\footref{limdz} \t \\\\\n\t\tMa19   \t&$\\rho<100\\times 10^3$ kg.m$^{-3}$\t&  not specified \\\\\n\t\t&$T<100~000$ K\t&  not specified  \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nFigure \\ref{fig:comp_eos} shows the $P(\\rho)$ profiles derived from the considered EoSs at different temperatures. All EoSs present minor differences in their validity range, regardless the considered temperature. \\cite{Ma19} find that the WP02 overestimates the pressure beyond its extended range. For a given pressure in a planet's interior, this would underestimate the density, and then overestimate the total radius of the planet. A more pronounced deviation is visible for DZ06 above its validity range. Around the critical point ($\\rho\\sim350$ kg.m$^{-3}$, mostly visible at 650 K), WP02 is closer to DZ06, compared to Ma19, as expected. In the low density limit, all EoSs behave following the ideal gas law $P\\propto \\rho T$, which has a characteristic slope of 1 in log-log scale. \n\n\n\\subsection{Gr\\\"uneisen parameter for fluids} \\label{sec:gruneisen}\n\nThe Gr\\\"uneisen parameter $\\gamma$, already introduced in Eq. (\\ref{eq:temp_grad}), has many definitions. For solids, it gives the rate of change in phonon frequencies $\\omega_i$ relative to a change in volume $V$ \\citep{Gr12}:\n\n\\begin{eqnarray}\n\\gamma_i = -\\left(\\frac{\\partial \\ln \\omega_i}{\\partial \\ln V}\\right)_T.\n\\end{eqnarray}\n\nBy averaging over all lattice frequencies, it is possible to obtain a thermodynamic definition (using the internal energy $U$ and entropy $S$) of the Gr\\\"uneisen parameter \\citep{Ar84} via the following expression:\n\n\\begin{eqnarray}\n\\gamma = V \\left(\\frac{\\partial P}{\\partial U}\\right)_V = \\frac{V}{C_V} \\left(\\frac{\\partial P}{\\partial T}\\right)_V = \\frac{\\rho}{T} \\left(\\frac{\\partial T}{\\partial \\rho}\\right)_S. \\label{eq:gamma-therm-id}\n\\end{eqnarray}\n$\\gamma$ relates a pressure (or density) variation to a temperature change. Although initialy defined for solids, the meaning of $\\gamma$ holds for fluids. In planetary interiors, adiabatic heat exchange is mostly driven by convective heat transfer \\citep{St19}. At planetary scales, the Gr\\\"uneisen parameter can thus be used for both solids and fluids. From identities in Eq. (\\ref{eq:gamma-therm-id}), $\\gamma$ can be expressed using other thermodynamic constants such as the thermal expansion coefficient $\\alpha$, the isothermal bulk modulus $K_T$, and the specific isochoric heat capacity $c_V$\n\\begin{eqnarray}\n\\gamma = \\frac{\\alpha K_T }{\\rho c_V}. \\label{eq:gruneisen-expression}\n\\end{eqnarray}\n\n\\noindent  $\\gamma$ is assumed to be temperature-independent in solid phase, and its value is fitted from experimental data, taking into account small density variations. In this study, we use the Helmholtz free energy $F$ given in \\cite{Wa02} and \\cite{Ma19}. In the IAPWS95 release, the specific Helmholtz free energy $f$ in its dimensionless form $\\phi$ is divided into its ideal part (superscript $\\circ$) and a residual (superscript ``r\") via the following expression:\n\n\\begin{eqnarray}\n\\frac{f(\\rho,T)}{RT} = \\phi(\\delta,\\tau) = \\phi^\\circ(\\delta,\\tau)+\\phi^\\mathrm{r}(\\delta,\\tau),\n\\end{eqnarray}\n\n\\noindent with $\\delta=\\rho\/\\rho_c$ and $\\tau=T_c\/T$, $\\rho_c$ and $T_c$ being the supercritical density and temperature, respectively. After defining the derivatives of the ideal and residual part:\n\n\\begin{eqnarray}\n\\phi_{m n}^{\\circ}=\\frac{\\partial^{m+n} \\phi^{\\circ}(\\tau, \\delta)}{\\partial \\tau^{m} \\partial \\delta^{n}}, \\\\\n\\phi_{m n}^{\\mathrm{r}}=\\frac{\\partial^{m+n} \\phi^{\\mathrm{r}}(\\tau, \\delta)}{\\partial \\tau^{m} \\partial \\delta^{n}},\n\\end{eqnarray}\n\n\\noindent where integers $m$ and $n$ define the order of the derivative with respect to $\\tau$ and $\\delta$, respectively. From these expressions, \none can derive:\n\n\\begin{eqnarray}\n\\gamma_- = -\\frac{1+\\delta \\phi^\\mathrm{r}_{01}-\\delta \\tau \\phi^\\mathrm{r}_{11}}{\\tau^2 \\left(\\phi^\\circ_{20}+\\phi^\\mathrm{r}_{20}\\right)}, \\label{eq:grun-iapws}\n\\end{eqnarray}\n\n\\noindent where $\\gamma_-$ is the formulation of the Gr\\\"uneisen parameter computed following the approach of  \\cite{Wa02}.\n\nThe fortran implementation of \\cite{Ma19} computes $F(\\rho,T)$, along with other useful quantities such as $\\chi_T = \\left(\\frac{\\partial \\ln P}{\\partial \\ln T}\\right)_V$ and the specific isochoric heat capacity $c_V$. In this case, the Gr\\\"uneisen parameter is expressed as:\n\\begin{eqnarray}\n\\gamma_+ = \\frac{P(\\rho,T) \\chi_T (\\rho,T)}{\\rho c_V T}, \\label{eq:grun-mazevet}\n\\end{eqnarray}\n\n\\noindent where $\\gamma_+$ is the formulation of the Gr\\\"uneisen parameter derived from the quantities calculated via the approach of  \\cite{Ma19}.\n\nIn the case of an ideal gas, one can derive the theoretical value $\\gamma=\\frac{2}{l}$, where $l$ is the number of degrees of freedom for a given molecule. For H$_2$O,  $\\gamma \\simeq \\frac{1}{3}$, since $l=6$ (3 rotational and 3 vibrational degrees of freedom).\n\nThe Gr\\\"uneisen parameter is crucial to compute the adiabatic temperature gradient inside a planet's interior.  However, because temperature has low impact on EoSs used in solid phase, it is possible to assume isothermal layers in interior models when thermodynamic data are lacking, and generate internal structures close to reality \\citep{Ze19}. In the case of fluids (here H$_2$O), temperature rises sharply with depth. This strongly impacts the EoS and leads to different phase changes that are not visible in the case of isothermal profiles.\n\nEach computation for the interior model can be performed by using any of the three EoS (WP02, DZ06, Ma19) to solve Eq. (\\ref{eq:solve_eos}), and WP02 or Ma19 EoS to solve Eq. (\\ref{eq:temp_grad}) (i.e. computing $\\gamma$ with EoS WP02 or Ma19). In the following, we will use the name of the EoS used to solve Eq. (\\ref{eq:solve_eos}), and add + or - depending on the EoS used to compute the Gr\\\"uneisen parameter, $\\gamma_+$ (Ma19) or $\\gamma_-$ (WP02) respectively. For example, Ma19- indicates that the Ma19 EoS was used to solve Eq. (\\ref{eq:solve_eos}), and that the WP02 approach was used to solve Eq. (\\ref{eq:temp_grad}).\n\nFigure \\ref{fig:gruneisen_val} shows the values of $\\gamma_+$ and $\\gamma_-$ in the H$_2$O phase diagram. Since $\\gamma$ is integrated to obtain the temperature gradient, a small difference leads to different paths in the $(P,T)$ plane. The indiscernability between the WP02- and the Ma19- profiles shows that the internal structure (and thus mass-radius relationships) is more impacted by the temperature profile than the difference in the EoS in the case of a water layer. The difference in temperature between Ma19+ and Ma19-\/WP02- profiles is as high as $\\sim$2000 K, which also results in a difference of $\\sim 200$ kg.m$^{-3}$ in density at the center of the planet.\n\n\\begin{figure}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{grunmap+.png}}\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{grunmap-.png}}\n\t\\caption{Color maps showing $\\gamma_+$ (top panel) and $\\gamma_-$ (bottom panel) in the H$_2$O phase diagram. The phase diagram of water is  identical to the one shown in Figure \\ref{fig:profiles}. Ma19+, Ma19- and WP02- are the $(P,T)$ profiles defined in the case of a 1 M$_\\Earth$ planet fully made of H$_2$O, with the atmosphere part shown with short dashes. Ma19- and WP02- interiors are almost indistinguishable, hence represented by the same color, and all atmospheric profiles are identical, although the  gravity at the boundary is different for each case.}\n\t\\label{fig:gruneisen_val}\n\\end{figure}\n\n\\subsection{Connection between interior and atmospheric models} \\label{sec:connect}\n\nAtmospheric properties (OLR, albedo, mass and thickness) are all quantities that evolve smoothly. To enable a smooth connection between the two models, we implemented a trilinear interpolation module that can estimate atmospheric properties for a planet whose physical parameters $g_\\mathrm{b}$, $M_\\mathrm{b}$, and  $T_\\mathrm{b}$ are in the 3--30 m.s$^{-2}$,  0.2--20 $M_\\Earth$, and 750--4500 K ranges, respectively. This allows us to correct the slight deviations from nods of the grid, and trilinear interpolation ensures that properties computed at a nod are exactly those at the nod, which would not be the case if a polynomial fit was performed on data. Details of the connection between the two models are given in Appendix \\ref{sec:connection}.\n\nFigure \\ref{fig:teq_tp} shows $T_\\mathrm{irr}$ as a function of $T_\\mathrm{p}$ for a set of fixed $g_\\mathrm{b}$ and $M_\\mathrm{b}$. Due to a strong greenhouse (or blanketing) effect from the steam atmosphere, most cases lead to $T_\\mathrm{b} > 2000$ K. As previously stated, this consequence discards any EoS that does not hold for such high temperatures. A second observation is that at low temperatures, one input irradiation temperature $T_\\mathrm{irr}$ can correspond to two different planet temperatures $T_\\mathrm{p}$ (and atmospheric properties). Since our work focuses on highly irradiated exoplanets, we will only investigate cases with $T_\\mathrm{irr}~>~400$ K to bypass this degeneracy.\n\n\\begin{figure}[ht!]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{tp_teq.pdf}}\n\t\\caption{Irradiation temperature $T_\\mathrm{irr}$ as a function of the planet's temperature $T_\\mathrm{p}$. Several curves are obtained due to different values of $g_\\mathrm{b}$ and $M_\\mathrm{b}$ in the available parameter range. Coldest planets exhibit a degeneracy, as the same amount of irradiation is consistent with two different atmospheric structures. As shown by the color bar, when taking the ``hot\" solution for $T_\\mathrm{p}$, the temperature at the bottom of the atmosphere $T_\\mathrm{b}$ is $>2000$ K.}\n\t\\label{fig:teq_tp}\n\\end{figure}\n\n\\section{Atmospheric escape} \\label{sec:atmos-escape}\n\nPlanetary atmospheres are subject to two types of instabilities : hydrostatic and thermal escape. The former is encountered when the gravity at a given height is insufficient to retain the gas. In this case, the atmosphere cannot exist in hydrostatic equilibrium and atmospheric models fail to produce static $(P,\\rho)$ profiles. The choice of $g_\\mathrm{b}>3$ m.s$^{-2}$ is arbitrary, but allows to avoid these cases. The latter occurs when the thermal energy of gas molecules exceeds the gravitational potential, allowing their escape. Escape rates are then computed, indicating which molecules can remain in an atmosphere. Several mechanisms of non-thermal escape exist as well, involving collisions between atoms and ions producing kinetic energy that leads to knock-off \\citep{Hu82}, but they rely on processes that are beyond the scope of this study.\n\n\\subsection{Jeans' escape}\n\nOne widely known process of atmospheric escape is the Jeans escape. Gas molecules have a velocity distribution given by the Maxwell-Boltzmann distribution, which displays an infinite extension in the velocity space, meaning that \\textit{some} particles have velocities greater than the escape velocity. By integrating this distribution, one can derive the Jeans' particle flux (particles per time unit per surface unit) escaping the atmosphere at the exobase \\citep{Je25}:\n\n\\begin{eqnarray}\n\t\\Phi_J = \\frac{n_\\mathrm{e} v_\\mathrm{esc}}{2 \\sqrt{\\pi}} \\frac{1}{\\sqrt{\\lambda}} \\left(1+\\lambda\\right) \\mathrm{e}^{-\\lambda}, \\label{eq:jflux1}\n\\end{eqnarray}\n\n\\noindent where $n_\\mathrm{e}$ is the particle number density at the top of the atmosphere (exobase), $v_\\mathrm{esc} = \\sqrt{2 g_\\mathrm{b} R_\\mathrm{p}}$ is the escape velocity (we assume $R_\\mathrm{p}\\simeq R_\\mathrm{b}$ and $M_\\mathrm{p}\\simeq M_\\mathrm{b}$). $\\lambda = \\left(\\frac{v_\\mathrm{esc}}{v_\\mathrm{th}}\\right)^2$ is the escape parameter, with $v_\\mathrm{th}=\\sqrt{2R_g T_\\mathrm{e}\/\\mu}$ the average thermal velocity of molecules of mean molar mass $\\mu$ at the exobase temperature $T_\\mathrm{exo}$, and $R_g$ is the ideal gas constant. \n\nWe wish to provide an estimate of the physical characteristics of the planets that would lose more than a fraction $x_\\mathrm{lost}=0.1$ of water content over a typical timescale of $\\Delta t= 1$ Gyr. This condition is met when\n\n\\begin{eqnarray}\n4\\pi R_\\mathrm{p}^2 \\frac{\\mu}{\\mathcal{N}_\\mathrm{A}} \\Phi_J \\ge \\frac{x_\\mathrm{lost} M_\\mathrm{p}}{\\Delta t}, \\label{eq:Jescape0}\n\\end{eqnarray}\n\n\\noindent with $\\mathcal{N}_\\mathrm{A}$ the Avogadro number. Solving Eq. (\\ref{eq:Jescape0}) with Earth's properties ($n_\\mathrm{e} = \\frac{P_\\mathrm{top} \\mathcal{N}_\\mathrm{A}}{Rg T_\\mathrm{exo}} \\sim 10^{19}$, $R_\\mathrm{p}=R_\\Earth$,  $M_\\mathrm{p}=M_\\Earth$) yields $\\lambda \\le 100$. Due to the exponential term, the result is poorly sensitive to changes in parameters, including the exact location of the exobase. Assuming $T_\\mathrm{exo}=T_\\mathrm{irr}$, this condition can be rewritten as\n\n\\begin{eqnarray}\n\tR_\\mathrm{p} > \\frac{1}{\\lambda} \\frac{G \\mu}{R_g T_\\mathrm{irr}} M_\\mathrm{p}, \\label{eq:Jescape}\n\\end{eqnarray}\n\n\\noindent with $G$ the gravitational constant. This estimate is consistent with today's composition of planets of the solar system (see Fig. \\ref{fig:final}). Equation \\ref{eq:Jescape} gives an indication of the properties of the planets that are subject to H$_2$ or H$_2$O escape, implying that their atmospheres should be dominated by heavier molecules (H$_2$O, CO$_2$, O$_2$, CH$_4$, etc) or be rocky planets, respectively.\n\n\\subsection{Hydrodynamic escape}\n\nHydrodynamic escape, also referred to as hydrodynamic blowoff, occurs when upper layers of the atmosphere are heated by intercepting the high energy irradiation (Far UV, Extreme UV and X-ray fluxes, the sum of which is often called XUV flux) from the host star. This heating induces an upward flow of gas, leading to mass-loss at a rate \\citep{Er07,Ow13}\n\n\\begin{eqnarray}\n\\dot{M} = \\epsilon \\frac{L_\\mathrm{XUV} R_\\mathrm{p}^3}{G M_\\mathrm{p} (2a)^2}, \\label{eq:Hescape}\n\\end{eqnarray}\n\n\\noindent where $L_\\mathrm{XUV}$ is the host star XUV luminosity, $a$ is the planet's orbital distance and $\\epsilon$ is a conversion factor between incident irradiation energy and mechanical blowoff energy. Note that Eq. (\\ref{eq:Hescape}) is only true in the energy-limited case. Heating occurs by absorption of high-energy photons by molecules which are dissociated in the upper atmosphere, meaning that blowoff can be limited by i) the number of photons as 1 photon breaks 1 molecule, and ii) recombination time as a dissociated molecule may recombine before being able to absorb the XUV irradiation again. Boundaries between these regimes have been explored by \\cite{Ow16}, who showed that the sub-Neptune population undergoes mostly energy-limited mass-loss, validating the use of Eq. \\ref{eq:Hescape} in our case.\n\nFor our estimate, we use the X-ray and UV luminosities obtained by fits on observational data for M to F type stars by \\cite{Sa11}:\n\n\\begin{eqnarray}\n&L_\\mathrm{EUV} = 10^{3.8} L_\\mathrm{X}^{0.86},&\\\\\n&L_\\mathrm{X} = 6.3\\times 10^{-4} L_\\star,& \\qquad \\tau<\\tau_\\mathrm{sat} \\\\\n&\\phantom{L_\\mathrm{X}xxx} = 1.89\\times 10^{21} \\tau^{-1.55},& \\qquad \\tau>\\tau_\\mathrm{sat} \\nonumber\n\\end{eqnarray}\n\n\\noindent where $\\tau$ is the host star age in Gyr and $\\tau_\\mathrm{sat} \n= 5.72\\times 10^{15} L_\\star^{-0.65}$ \\citep{Sa11}. To estimate the XUV luminosity, the star's bolometric luminosity is assumed constant, a hypothesis supported by the stellar evolution tracks of \\cite{Ba15}. Integrating the XUV luminosity in the saturation regime ($0<\\tau < \\tau_\\mathrm{sat}$) and beyond, gives the finite quantity $E_\\mathrm{XUV}=\\int_{0}^{+\\infty} L_\\mathrm{XUV}~dt = 1.8 \\times 10^{39} $ W for a solar type star.\n\nAgain, we look for planets that could lose more than 10\\% of their mass over a 1 Gyr period, due to atmospheric blowoff:\n\n\\begin{eqnarray}\n\\epsilon \\frac{E_\\mathrm{XUV} R_\\mathrm{p}^3}{G M_\\mathrm{p} (2a)^2} \\ge x_\\mathrm{lost} M_\\mathrm{p}. \\label{eq:Hescape1}\n\\end{eqnarray}\n\n\\noindent Combining Eq. (\\ref{eq:tirr_obs}) and Stefan-Boltzmann's law $L_\\star~=~4\\pi R_\\star^2 \\sigma_\\mathrm{sb} T_\\mathrm{eff}^4$ gives\n\\begin{eqnarray}\n\t(2a)^2 = \\frac{1}{T_\\mathrm{irr}^4} \\frac{L_\\star}{4 \\pi \\sigma_\\mathrm{sb}}.\n\\end{eqnarray}\nSubstituting this expression in Eq. (\\ref{eq:Hescape1}) yields to the condition:\n\n\\begin{eqnarray}\nR_\\mathrm{p} \\ge M_\\mathrm{p}^{\\frac{2}{3}} \\left(\\frac{x_\\mathrm{lost} G}{\\epsilon 4\\pi \\sigma_\\mathrm{sb} T_\\mathrm{irr}^4 E_\\mathrm{XUV}}\\right)^{\\frac{1}{3}}. \\label{eq:Hescape2}\n\\end{eqnarray}\n\n\\noindent This condition only gives an indication of the planets that are subject to substantial hydrodynamic escape. All arbitrary quantities such as $\\epsilon\\simeq1$ \\citep{Ow12,Bo17} and $E_\\mathrm{XUV}$, are affected by a power of $1\/3$, resulting in a low dependency on the chosen values.\n\nThe nature of escaping particles is not considered in Eq. (\\ref{eq:Hescape2}), meaning the computed quantity is the total lost mass. \\cite{Bo17} developed a method to quantify the hydrodynamic outflow $r_\\mathrm{F}$ (how many atoms of oxygen leave for each hydrogen atom). Based on their work, we compute $r_\\mathrm{F}\\sim 0.2$, indicating substantial loss of both H and O, with an accumulation of O$_2$. Mass loss of water content and accumulation of O$_2$ have several implications for the habitability of exoplanets \\citep{Ri16,Sc16}. The power laws for mass is 1 and $\\frac{2}{3}$ in the cases of Eqs. (\\ref{eq:Jescape}) and (\\ref{eq:Hescape2}), respectively. This implies that hydrodynamic escape is more efficient for less dense planets. In contrast, Jeans escape is dominant in the case of denser planets. The power-law for $T_\\mathrm{irr}$ is $-1$ and $-\\frac{4}{3}$ in the cases of Eqs. \\ref{eq:Jescape} and \\ref{eq:Hescape2}, respectively, implying that hydrodynamic escape will take over Jeans escape at higher irradiation temperatures. As shown in Fig. \\ref{fig:mr+}, Eqs. \\ref{eq:Jescape} and \\ref{eq:Hescape2} leave a window for planets that lost their H$_2$ reservoir but kept heavier volatiles from which they formed \\citep{Ze19}. This result highlights the consistency between the possible existence of irradiated ocean planets and atmospheric escape.\n\n\\section{Results} \\label{sec:results}\n\n\\begin{figure*}[!ht]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{mr400+.pdf} \\includegraphics[angle=0,width=5cm]{mr600+.pdf}}\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm,]{mr800+.pdf} \\includegraphics[angle=0,width=5cm]{mr1000+.pdf}}\n\t\\caption{Mass-radius relationships for planets with Earth-like properties regarding their rocky part (see Table \\ref{tab:parameters}), and computed with $\\gamma_+$, for multiple temperatures and water contents. Colors correspond to the three used EoSs: DZ06 (red), WP02 (blue) and Ma19 (green). Dashed lines correspond to regions where the atmosphere model is extrapolated beyond the available grid (see Appendix \\ref{sec:trilinear}). Filled circles correspond to cases where both $P$ and $\\gamma$ remain in the range of validity of used EoS. Open circles correspond to cases where $P$ or $\\gamma$ are computed in the extended range. Crosses correspond to cases where $P$ or $\\gamma$ are in the extrapolated range. Shaded areas correspond to H$_2$ (gray), H$_2$O (pink) and hydrodynamic escape (shaded) (see Sec. \\ref{sec:atmos-escape}).}\n\t\\label{fig:mr+}\n\\end{figure*}\n\n\\begin{figure*}[!ht]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{mdr400.pdf} \\includegraphics[angle=0,width=5cm]{mdr1000.pdf}}\n\t\\caption{Relative difference on radius between Ma19+ and Ma19- parametrizations showing the large impact of the temperature profile on mass-radius relationships.}\n\t\\label{fig:mdr}\n\\end{figure*}\n\nThe aim of this paper is to quantify the impact of the choice of EoS and $\\gamma$ computation on mass-radius relationships. With three EoSs and two $\\gamma$ parametrizations, 6 cases are considered : WP02$\\pm$, DZ06$\\pm$ and Ma19$\\pm$, with the +\/- sign standing for $\\gamma_+$ and $\\gamma_-$, respectively. For each case, three validity domains are explored: true validity range, extended range, or extrapolated range. A case is valid when the $(P,T)$ profile remains strictly in the true validity range of the used EoS and $\\gamma$ computation. It is extended if the EoS and\/or $\\gamma$ computation reaches the extended range. If either the EoS or $\\gamma$ reaches the extrapolated region, the whole case is considered extrapolated. For example, the Ma19+ parametrization (see Fig. \\ref{fig:mr+}) is always valid due to the important validity range of Ma19 EoS and $\\gamma_+$ computation. On the other hand, the Ma19- parametrization is always extrapolated because the computed $\\gamma_-$ is out of its validity and extended range.\n\n\\subsection{Mass-radius relationships and choice of EoS}\n\nFigure \\ref{fig:mr+} presents computed mass-radius relationships for the $\\gamma_+$ parametrization, and assuming Earth-like properties for the rocky part (see Table \\ref{tab:parameters}). As predicted from the shape of EoSs curves, WP02 and DZ06 EoSs underestimate the density and thus produce larger planets. This effect is accentuated for more massive planets with a larger amount of water, corresponding to cases where water pressure reaches the highest values. The radius is also overestimated for low-mass planets, because the hydrosphere becomes extended due to the low gravity, implying that a slight underestimation of the density can still lead to a substantial difference in radius. These results show the incontestable asset of the EoS developed by \\cite{Ma19}, and rule out the possibility of using WP02 or DZ06 EoSs to produce reliable mass-radius relationships for planets with substantial amounts of water. To remain in the true validity ranges of WP02 or DZ06 EoS, one should consider a few \\% of water content at most in the planet.\n\nAs discussed in Sec. \\ref{sec:gruneisen}, $\\gamma_+$ is always lower than $\\gamma_-$ in Earth-sized planets fully made of water. As a result $(P,T)$ profiles for $\\gamma_+$ parametrizations are steeper than for $\\gamma_-$ parametrizations (see Fig. \\ref{fig:gruneisen_val}), meaning the interior is colder for $\\gamma_+$. In turn, colder planets will be denser and thus smaller. The impact of the choice between $\\gamma_+$ and $\\gamma_-$ is shown in Fig. \\ref{fig:mdr}, where the relative difference on the radius between Ma19+ and Ma19- parametrizations is presented. In all cases, the relative difference between the models is 10\\%  at most.\n\nAs the mass of a planet increases, its gravity becomes more important, and its hydrosphere (interior structure and atmosphere) consequently thinner. Thinner hydrospheres, especially in the case of massive planets, lead to smaller relative differences in radii. Moreover, values of $\\gamma_+ $ and $ \\gamma_-$ become closer (and even equal) in the 10$^1$--10$^2$ GPa pressure range (see Fig. \\ref{fig:gruneisen_val}), thus reducing even more significantly the radii differences between the Ma19+ and Ma19- parametrizations.\n\nThe value of $\\gamma$ increases when the ($P$,$T$) curves of a hydrosphere approaches the liquid--Ice VII transition, which leads to a more important temperature gradient that prevents the formation of high pressure ices. This observation is in major disagreement with models assuming isothermal hydrospheres \\citep{Va06,Va07,Se07,Ze13,Br17,Ze19}, an hypothesis often justified by assuming that temperature has a secondary impact on EoSs, \nwhich remains a valid statement  for solid phases but not in the case of the hydrosphere. A correct treatment of the temperature gradient \\citep{Mo20} leads to the presence of high-temperature phases for H$_2$O (ionic, super ionic, plasma), which are more dilated, impacting significantly the mass-radius relationships.\n\nIn the following, we use $\\gamma_+$ and Ma19 to compute the mass-radius relationships. Indeed, the pressure and temperature ranges in the hydrospheres of sub-Neptunes-like planets lie well in the region for which the Ma19 formulation was developed. Also, due to the blanketing effect of the atmosphere, even the coldest planets irradiated at $T_\\mathrm{irr}~=~400$ K have a temperature of more than 2000 K at the 300 bar interface (see Fig. \\ref{fig:teq_tp}), which corresponds to the pressure at which the atmospheric and the internal model are connected. This interface is already located well above the range of validity of $\\gamma_-$.\n\n\\subsection{Planetary composition}\n\n\\begin{figure}[!ht]\n\t\\resizebox{0.92\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{ternary-gjb.pdf}}\\\\\n\t\\resizebox{0.92\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{ternary-gjc.pdf}}\\\\\n\t\\resizebox{0.92\\hsize}{!}{\\includegraphics[angle=0,width=5cm]{ternary-gjd.pdf}}\n\t\\caption{From top to bottom: possible compositions of planets b, c and d of the GJ 9827 system \\citep{Ri19} in the forms of compositional ternary diagrams. Ternary diagrams were computed for the central masses of the planets, and contours are plotted for the measured radius and 1$\\sigma$ error bar.}\n\t\\label{fig:ternary}\n\\end{figure}\n\nMass-radius relationships only provide an order of estimate of the possible exoplanet composition. A more precise assessment is achieved via the use of compositional ternary diagrams. For a given planet mass and irradiation temperature, such a diagram shows the radius as a function of the planet's WMF and CMF. Possible compositions as thus retrieved from the contour at the level of the planet's measured radius. Computations presented here use only the central value of the mass of each planet, thus not taking into account the measurement error on the planet's mass.\n\nPossible compositions of the three planets of the GJ 9827 system are shown in Fig. \\ref{fig:ternary}, based on the planets parameters measurements made by \\cite{Ri19}. Planet b exhibits an Earth-like interior without the need of invoking a significant steam atmosphere. The presence of a thick steam atmosphere is rather consistent with the low-density measurements made for planet c, with a water content ranging from 1 to 8$\\%$. Physical properties (mass, radius and temperature) of planet c lead to important Jeans' escape (with our criterion in Eq. \\ref{eq:Jescape0}, see Fig. \\ref{fig:final}), suggesting the absence of H$_2$ and He in the atmosphere. Moreover, planet c is unlikely to accrete substantial amount of H$_2$ and He due to its low mass. Although planet d is consistent with a Jupiter-like interior due to their similar bulk densities, again, its high irradiation temperature suggests the presence of a H$_2$-He free atmosphere. Isochrones used by \\cite{Ri19} fix a lower limit on the age of 5 Gyr on the age of the system, which makes an H$_2$-He atmosphere less likely as Jeans' escape would remove them. Applying our model to the current measurements yields a WMF in the 5--30\\% range for planet d. These results are summarized in Table \\ref{tab:gj}.\n\n\\begin{table}\n\t\\centering\n\t\\caption{Planetary parameters of the GJ 9827 system used as input for the model, and estimated WMF using ternary diagrams (Fig. \\ref{fig:ternary}).} \n\t\\label{tab:gj}\n\t\\begin{tabular}{llll}\n\t\t\\tablewidth{0pt}\n\t\t\\hline\n\t\t\\hline\n\t\tPlanet \t& b \t\t& c & d \t\\\\\t\n\t\t\\hline\n\t\t$M_\\mathrm{p} ~(M_\\Earth)$ & $4.91\\pm0.49$ & $0.84\\pm0.66$ & $4.04\\pm0.83$ \\\\ \t\n\t\t$R_\\mathrm{p}~(R_\\Earth)$ & $1.58\\pm0.03$ & $1.24\\pm0.03$ & $2.02\\pm0.05$ \\\\\n\t\t$T_\\mathrm{irr}$ (K) & 1184 K & 820 K & 686 K \\\\ \\hline\n\t\tWMF (\\%) & 0--5 & 1--5 & 5--30 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nTernary diagrams presented here do not take into account the uncertainty on each planet's mass, and were computed for the central value only. If a planet's mass is slightly higher (resp. lower), its density increases (resp. decreases), while the estimated WMF diminishes (resp. grows). This implies that the mass and radius of a planet must be measured with extreme accuracy to constrain the WMF properly. Additional constraints can be applied from observational data such as the stellar elemental ratios (Fe\/Si, Mg\/Si) that could help constraining the core to mantle mass ratio \\citep{Br17}, and methods such as MCMC can be performed to simultaneously determine all parameters \\citep{Ac21}.\n\nFigure \\ref{fig:final} represents the computed mass-radius relationships for WMF of 0.2, 0.5 and 1. In this figure, the condition for substantial atmospheric loss due to Jeans' escape is derived by solving equation (\\ref{eq:Jescape0}) for each planet. One already known effect is that steam atmospheres are very extended \\citep{Mo20}, allowing to compute compositions without invoking small H$_2$-He enveloppes (1-5\\% by mass). The second effect is heating due to the adiabatic gradient, which decreases the density, and then increases the radius. In the 10--20 M$_\\Earth$ range, the radius of a planet with a WMF of 50\\% made of liquid H$_2$O is equal to that of a planet with a WMF of 20\\% constitued of supercritical H$_2$O. Also, the radius of a planet fully made of liquid H$_2$O is equivalent to that of a planet with half of its mass constituted of supercritical H$_2$O. This shows how important the error on the computation of WMF can be, depending on the physical assumptions made. In the figures presented in \\cite{Mo20}, where the DZ06 EoS was used, the model was able to match Neptune's mass (17 $M_\\Earth$) and radius (3.88 $R_\\Earth$) with a $95\\%$ H$_2$O interior at 300 K. With the Ma19 EoS, a 100\\% water planet presents a radius of 3.25 $R_\\Earth$ at $T_\\mathrm{irr}=400$ K and 3.6 $R_\\Earth$ at $T_\\mathrm{irr}=1300$ K.\n\n\\subsection{Analytical expression of mass-radius relationships}\n\n\\begin{figure*}[!ht]\n\t\\resizebox{\\hsize}{!}{\\includegraphics[angle=0,width=10cm]{mazevet.pdf}}\n\t\\caption{Comparison between mass-radius relationships computed with the Ma19+ model and those existing in literature. Our mass-radius relationships were computed for WMF of 20\\%, 50\\% and 100\\% with no metallic core, and temperatures of 400, 600, 800 and 1000 K. Thin solid lines and thin dashed lines are from  \\cite{Ze16} and \\cite{Br17}, respectively. Empty triangles, solid circles and stars correspond to planets subject to no atmospheric escape, to escape of H$_2$ only and to escape of both H$_2$ and H$_2$O (Jeans or blowoff), respectively. Planetary data are taken from the NASA exoplanet archive and updated to July 2020.}\n\t\\label{fig:final}\n\\end{figure*}\n\nAll produced mass-radius relationships are very well approximated by an equation of the form\n\\begin{eqnarray}\n\t\\log R_\\mathrm{p} = a \\log M_\\mathrm{p}+b\n\t+\\exp \\left(-d(\\log M_\\mathrm{p}+c)\\right), \\label{eq:fit1}\n\\end{eqnarray}\n\n\\noindent where log denotes the decimal logarithm, and $R_\\mathrm{p}$ and $M_\\mathrm{p}$ are normalized to Earth units. $a$, $b$, $c$ and $d$ are coefficients obtained by fits, and have one value for each composition $(x_\\mathrm{core},x_{\\mathrm{H}_2 \\mathrm{O}})$ and each temperature $T_\\mathrm{irr}$. For each fitted curve, we define the mean absolute error between data and fit as\n\\begin{eqnarray}\n\t\\mathrm{MAE} = \\frac{1}{N}\\sum_{i=1}^{N} \\left|\\frac{R_\\mathrm{p,model}-R_\\mathrm{p,fit}}{R_\\mathrm{p,model}}\\right|,\n\\end{eqnarray}\nValues of the MAE are 0.01--1\\% for all fits, indicating a good accuracy. The largest deviation between one point $(M_\\mathrm{p},R_\\mathrm{p})$ and the fitted curve is of $2.3\\%$, meaning the deviation between data and fit can be neglected. Fitted coefficients vary smoothly with respect to the three parameters $(x_\\mathrm{core},x_{\\mathrm{H}_2 \\mathrm{O}},T_\\mathrm{irr})$, allowing a good interpolation of the intermediate values. The produced grid uses the compositional parameters for the core and mantle calibrated for Earth (see Table \\ref{tab:parameters}), and data may be different if Fe\/Si or Mg\/si ratios are different.\n\n\\section{Discussion and conclusion} \n\\label{sec:ccls}\n\nThis work aimed at describing a model that computes a realistic structure for water-rich planets. This was achieved by combining an interior model with an updated EoS for water, and an atmospheric model that takes into account radiative transfer.\n\nVarious EoSs were investigated, and we find that results are identical when all of them are used within their validity range. However, the pressure profile rises sharply for planets with substantial amounts of water, invalidating the use of WP02 and DZ06 EoSs for WMF $>5\\%$. The blanketing effect due to the presence of the atmosphere leads to boundary temperatures greater than $2000$ K, leaving even less room for the DZ06 EoS to work properly. Both non-valid EoSs lead to the common result of overestimating the planetary radius by up to $\\sim$10\\%. Inexact computation of the Gr\\\"uneisen parameter yields another $\\sim$10\\% of error on the radius, at most. This requires to use an EoS that holds for pressures up to a few TPa and temperatures of $10^4$ K (conditions at the center of a pure water sphere of 1 Jupiter mass), such as \\cite{Ma19}.\n\nOther EoSs exist in the literature, such as those proposed by \\cite{Br18} and \\cite{Ha20}, which are functions either fitted or derived from the Gibbs or Helmholtz free energy. The range of validity for the EoS of \\cite{Br18} is less extended than that of \\cite{Ma19}, justifying our choice of EoS. \\cite{Ha20} presents a unified EoS for water from the connexion of already existing EoSs in their validity range, incuding \\cite{Ma19}. This EOS is then consistent with ours in the range of temperature and pressure explored here. The implementation of such an EoS is interesting for future works, especially when combining high pressure ices. \n\nIt should be noted that the most accurate EoS possible is not sufficient to produce precise mass-radius relationships for such planets. Assuming an adiabatic profile for the atmosphere (i.e. not taking into account radiative transfer) results in more extended atmospheres, as heat is transported solely by convection. Isothermal water layers seem closer to reality, but they produce the same mass-radius relationships as for liquid water \\citep{Ze16,Br17,Ha20}. Atmospheric models are essential for computing the atmosphere thickness and the energy that is transported to the interior.\n\nDerived MR relationships produce radii that match well those of the population of sub-Neptunes (1.75--3.5 $R_\\Earth$). This population corresponds to the second peak of the bimodal distribution of planet radii highlighted by \\cite{Fu17}, thus suggesting that irradiated ocean planets are good candidates to represent such planets \\citep{Mo20}. This bimodal distribution in planet radii has been predicted by \\cite{Ow13} and \\cite{Lo13} who investigated the atmospheric mass loss for Jupiter-like planets. However, the authors focused mainly on the loss of the enveloppe of a H\/He rich atmosphere. More recently, \\cite{Ow19} pointed out the need to extend this work to steam atmospheres. Our calculations aimed to do so in a very simplistic manner. Due to its greater density, we find that water is much less subject to atmospheric escape than H\/He. This suggests that highly irradiated planets could have lost their H\/He content through atmospheric loss processes, and the remaining matter led to either super-Earths ($R_\\mathrm{p} =$ 1--1.75 $R_\\Earth$) or a sub-Neptunes ($R_\\mathrm{p}=$ 1.75--3.5 $R_\\Earth$), depending on the final WMF.\n\nThe data grid can be used to assess a planet's composition once its mass and radius are known. Interpolating between the values can provide better precision. For a very precise computation, the full model is required since compositional parameters such as Fe\/Si and Mg\/Si ratios are required as well and depend on the star spectral analysis.\n\nTabulated mass-radius relationships and the coefficients obtained by fit for analytical curves can be found at \\url{https:\/\/doi.org\/10.5281\/zenodo.4552188} or \\url{https:\/\/archive.lam.fr\/GSP\/MSEI\/IOPmodel}. Explored parameter ranges are large enough to constrain planetary compositions for any WMF and CMF, and interpolate between given values without the need for the full model. We used the GJ 9827 system as a test case for our new relationships. Measured masses and radii of planets b and c of the GJ 9827 system indicate Earth-like or Venus-like interiors. We find that planet d could be an irradiated ocean planet with a WMF of $20\\pm10\\%$.\n\nIn the present model, only H$_2$O as a volatile is considered. Other volatiles such as CO$_2$, CH$_4$ or N$_2$ are expected to have similar densities as H$_2$O, thus producing similar mass-radius relationships. However, using a different gas will highly impact radiative transfer. Efficient radiative transfer for gases such as N$_2$ could keep the interior cold enough for maintaining a liquid water ocean, as it is the case for the Earth. An atmosphere dominated by gases such as H$_2$O or CO$_2$ lead to important blanketing, resulting in a Venus-like case.\n\nAtmospheric escape has motivated our focus on H\/He-free atmospheres. The addition of H$_2$ to the atmosphere is the scope of future work. The addition of O$_2$ as the product of water photodissociation will be considered as well.\n\n\\section*{Aknowledgements}\nOM and MD acknowledge support from CNES. We thank the anonymous referee for useful comments that helped improving the clarity of our paper and added important discussion.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\nAntihydrogen, the bound state of an antiproton and a positron, is the simplest pure antimatter atomic system.\nThe first cold (non-relativistic) antihydrogen atoms were synthesised by the ATHENA experiment in 2002 by combining antiprotons and positrons under cryogenic conditions in a Penning trap \\cite{ATHENA_Nature}.\nThe neutral antihydrogen atoms formed were not confined by the electric and magnetic fields used to hold the antiprotons and positrons as non-neutral plasmas, but escaped to strike the matter of the surrounding apparatus and annihilate.\nDetection of the coincident antiproton and positron annihilation signals was used to identify antihydrogen in these experiments.\nHowever, before performing high-precision spectroscopy, it is highly desirable, perhaps even necessary, to confine the antihydrogen in an atomic trap.\n\n\\section{Atom Trap}\n\nAtoms with a permanent magnetic dipole moment $\\vec{\\mu}$ can be trapped by exploiting the interaction of the dipole moment with an inhomogeneous magnetic field.\nA three-dimensional maximum of magnetic field is not compatible with Maxwell's equations, but a minimum is.\nThus, only atoms with $\\mu$ aligned antiparallel to the magnetic field (so-called `low-field seekers') can be trapped.\n\nALPHA creates a magnetic minimum using a variation of the Ioffe-Pritchard configuration \\cite{Ioffe_Pritchard}, replacing the transverse quadrupole magnet with an octupole \\cite{ALPHA_magnet}.\nThe octupole and the `mirror coils' that complete the trap are superconducting and are cooled to 4~K by immersing them in liquid helium.\nThe depth of the magnetic minimum produced is approximately 0.8~T, equivalent to a trap depth of $0.6~\\mathrm{K}\\times k_\\mathrm{B}$ for ground state antihydrogen.\n\nALPHA's scheme to detect trapped antihydrogen is to quickly release trapped atoms from the atomic trap and detect their annihilation as they strike the apparatus.\nHaving the antihydrogen atoms escape over a short time minimises the background from cosmic rays that can mimic antihydrogen annihilations (see section \\ref{sec:detector}), so the magnet systems have been designed to remove the stored energy in as short a time as possible.\nThe current has been measured to decay with a time constant of 9~ms for the octupole and 8.5~ms for the mirror coils.\n\nThe atom trap is superimposed on a Penning trap, which is used to confine the charged particles used in antihydrogen production.\nThe Penning trap electrodes are also cooled by a liquid helium reservoir and reach a temperature of approximately 7~K.\nIn the absence of external heating sources, the stored non-neutral plasmas should come into thermal equilibrium at this temperature.\n\nIntroduction of the multipolar transverse magnetic field modifies the confinement properties of the Penning trap.\nIn the most extreme case, this manifests as a `critical radius' \\cite{CriticalRadiusTheory}, outside which particles can be lost from the trap simply because the magnetic field lines along which the particles move intersect the electrode walls.\nEven if particles are not lost, the transverse field results in a higher rate of plasma diffusion \\cite{GilsonFajans_Diffusion}.\nAs the plasma diffuses and expands, electrostatic potential energy is converted to thermal energy, resulting in a temperature higher than would be otherwise expected.\n\nALPHA chose to use an octupole instead of the prototypical quadrupole in its Ioffe trap to reduce the transverse fields close to the axis of the Penning trap, where the non-neutral plasmas are stored.\nThough this choice can significantly ameliorate the undesirable effects, it does not eliminate them entirely.\nOther sources of heating, notably the coupling of the particles to electronic noise \\cite{NoiseTemperature}, will also increase the temperature.\nThis highlights the importance of direct, absolute measurements of the particle temperature to accurately determine the experimental conditions.\n\n\\section{Cooling and temperature measurements of antiprotons}\n\nThe temperature of a plasma can be determined by measuring the distribution of particles in the tail of a Boltzmann distribution - a technique common-place in non-neutral plasma physics \\cite{TemperatureMeasurement}.\nThis measurement has the advantage of yielding the absolute temperature of the particles without recourse to supporting measurements (for example, of the density distribution), unlike measurements of the frequencies of the normal plasma modes \\cite{modes}, which can only give a relative temperature change.\nThe plasmas typical in ALPHA have densities in the range $10^6$ to $10^8~\\mathrm{cm^{-3}}$, with collision rates high enough to ensure that the plasma comes to equilibrium in a few seconds.\nIn equilibrium, the energy of the particles conforms to a Boltzmann distribution.\n\nTo measure the temperature, the particles are released from a confining well by slowly (compared to the axial oscillation frequency) reducing the voltage on one side of the well.\nAs the well depth is reduced, particles escape according to their energy; the first (highest-energy) particles to be released will be drawn from the tail of a Boltzmann distribution.\nAs the dump progresses, the loss of particles causes redistribution of energy and, at later times, the measured distribution deviates from the expected Boltzmann distribution.\nThe escaping particles can be detected using a micro-channel plate as a charge amplifier, or for antimatter particles, by detecting their annihilation.\nThe temperature is determined by fitting an exponential curve to the number of particles released as a function of energy, such as in the example measurement shown in Fig. \\ref{fig:temperature}.\n\n\n\\begin{SCfigure}[1.0][h]\n\\centering\n\\input{prettyTemp}\n\\caption{An example temperature measurement of approximately 45,000 antiprotons, after separation from the cooling electrons and with the inhomogeneous trapping fields energised. The straight line shows an exponential fit to determine the temperature, which in this case, is $\\left(310~\\pm~20\\right)~\\mathrm{K}$}\n\\label{fig:temperature}\n\\end{SCfigure}\n\nThe actual process of manipulating the trap potentials can change the temperature of the particles as the measurement takes place.\nParticle-in-cell (PIC) simulations of the measurement process have predicted that the temperature obtained from the fit is around 15\\% higher than the initial temperature for a typical antiproton cloud.\nFor the denser electron and positron plasmas, the measured temperature can be as much as factor of two higher than the initial temperature.\nWe can apply the corrections determined from these simulations to the measured temperature to find the true temperature.\nThis temperature diagnostic has been applied to all three particle species used in ALPHA - antiprotons, positrons and electrons.\nThe lowest temperatures measured for electron or positron plasmas at experimentally relevant densities $\\left(10^6~\\mathrm{cm^{-3}} \\text{or more}\\right)$ is of the order of 40~K.\n\nElectrons are used to collisionally cool the antiprotons, which, due to their larger mass, do not effectively self-cool via synchrotron radiation.\nBefore mixing the antiprotons with positrons to produce antihydrogen, the electrons must be removed.\nIf the electrons were allowed to remain, they could potentially deplete the positron plasma by forming positronium, destroy antihydrogen atoms through charge exchange, or destabilise the positron plasma by partially neutralising it.\n\nElectron removal is accomplished through the application of electric field pulses.\nThese pulses remove the confining potential on one side of the well holding the antiproton\/electron two-component plasma, typically for 100-300~ns.\nThe electrons, moving faster than the antiprotons, escape the well.\nThe well is restored before the antiprotons can escape, so they remain trapped.\nHowever, the process does not avoid disturbing the antiprotons.\nThe electron removal process has been the focus of a significant portion of experimental effort at ALPHA, and the coldest antiproton temperatures obtained have been around 200-300~K.\n\n\\section{Evaporative Cooling}\n\nAntiprotons at a few hundred Kelvin will have a very small probability of forming low-energy, trappable, antihydrogen atoms.\nTo further cool the antiprotons, ALPHA has implemented a technique of forced evaporative cooling.\nEvaporative cooling is a common-place technique in neutral particle trapping, and has been instrumental in the production of Bose-Einstein condensates \\cite{EVC_in_atoms}.\nHowever, evaporative cooling has found limited application to charged particles.\n\nBefore evaporative cooling, a cloud of antiprotons, containing 45,000 particles, with a radius of 0.6~mm, density $7.6\\times10^6~\\mathrm{cm^{-3}}$, and initial temperature of $\\left(1040~\\pm~45\\right)~\\mathrm{K}$ was prepared in a 1.5~V deep potential well.\nThe collision rate between antiprotons was of order 200~$\\mathrm{s}^{-1}$, high enough to ensure that the temperatures in the parallel and perpendicular degrees of freedom had equilibrated before evaporative cooling commenced.\n\nTo perform evaporative cooling, the confining potential on one side of the well is slowly (with respect to the equilibration rate) lowered.\nParticles with kinetic energy higher than the instantaneous well depth escape the trap, carrying with them energy in excess of the mean thermal energy.\nThe distribution then evolves towards a Boltzmann distribution with lower temperature, and the process continues.\n\nStarting with $45,000$ antiprotons at 1040~K, we have obtained temperatures as low as (9~$\\pm$~4)~K with $\\left(6\\pm1\\right)\\%$ of the particles stored in a 10~mV deep well.\nMeasurements of the temperature, number of particles and transverse size of the clouds were made at a number of points between the most extreme well depths.\nThe temperatures and number of particles remaining at each measurement point are shown in Fig. \\ref{fig:EVC_data}.\n\n\\captionsetup[subfloat]{position=top,captionskip=-10pt, justification=raggedright, singlelinecheck=false, margin=20pt}\n\n\\vspace{-0.5cm}\n\\begin{figure}[h]\n\\centering\n\\subfloat[]{\\input{evcTempCor}}\n\\subfloat[]{\\input{evcEff}}\n\\caption{The temperature (a) and the fraction of the initial number of particles (b) after evaporative cooling to a series of well depths. The minimum temperature is (9 $\\pm$ 4)~K}\n\\label{fig:EVC_data}\n\\end{figure}\n\nThe evaporation process can be described using simple rate equations for the number of particles $N$ and the temperature $T$;\n\n\\vspace{-0.2cm}\n\\begin{subequations}\n\\begin{center}\n\\begin{tabular}{p{0.4\\textwidth} p{0.4\\textwidth}}\n\t\\begin{equation}\n\t\t\\frac{\\mathrm{d}N}{\\mathrm{d}t} = - \\frac{N}{\\tau_{ev}}, \n\t\\end{equation} \t&\n\t\\begin{equation}\n\t\t\\frac{\\mathrm{d}T}{\\mathrm{d}t} = - \\alpha \\frac{T}{\\tau_{ev}} .\n\t\\end{equation}\n\\end{tabular}\n\\end{center}\n\\end{subequations}\n\\vspace{-0.5cm}\n\n\\noindent Here, $\\tau_{ev}$ is the characteristic evaporation timescale and $\\alpha$ is the excess energy carried away by an evaporating particle, in multiples of $k_\\mathrm{B} T$.\nAt a given time, the distribution of energies can be thought of as a truncated Boltzmann distribution, characterised by a temperature $T$, and the well depth $U$.\n$\\tau_{ev}$ is linked to the mean time between collisions, $\\tau_{col}$ as \\cite{EVC_Theory}\n\\begin{equation}\n\t\\frac{\\tau_{ev}}{\\tau_{col}} = \\frac{\\sqrt{2}}{3} \\eta e^\\eta,\n\t\\label{eqn:tau}\n\\end{equation}\nwhere $\\eta = U\/{k_\\mathrm{B}T}$ is the rescaled well depth.\nWe note the strong dependence of $\\tau_{ev}$ on $\\eta$, indicating that this is the primary factor determining the temperature in a given well. \nWe find values of $\\eta$ between 10 and 20 over the range of our measurements.\nThe value of $\\alpha$ can be calculated using the treatment in reference \\cite{ketterleReview}.\nWe have numerically modelled evaporative cooling in our experiment using these equations and have found very good agreement between our measurements and the model \\cite{ALPHA_EVC}.\n\n\nMeasurements of the transverse density profile were made by ejecting the particles onto an MCP\/phosphor\/CCD imaging device \\cite{ALPHA_MCP}.\nIt was seen that, as evaporation progressed, the cloud radius increased dramatically - see Fig. \\ref{fig:radius}.\nWe interpret this effect to be due to escape of the evaporating particles principally from the radial centre of the cloud, and the conservation of the total canonical angular momentum during the subsequent redistribution process.\nInside the cloud, the space charge reduces the depth of the confining well.\nThis effect is accentuated closer to the trap axis, with the result that the well depth close to the axis can be significantly lower than further away.\nThe evaporation rate is exponentially suppressed at higher well depths (eqn. \\ref{eqn:tau}), so evaporation is confined to a small region close to the axis, causing the on-axis density to become depleted.\nThis is a non-equilibrium configuration, and the particles will redistribute to replace the lost density.\nIn doing so, some particles will move inwards, and to conserve the canonical angular momentum, some particles must also move to higher radii \\cite{confinementTheorem}.\nAssuming that all loss occurs at $r=0$, the mean squared radius of the particles, $\\left< r^2 \\right>$, will obey the relationship\n\\vspace{-0.5cm}\n\\begin{equation}\n\t\\label{eq:expansion}\n\tN_0\\left< r_0^2\\right> = N \\left< r^2 \\right>,\n\t\\vspace{-0.5cm}\n\\end{equation}\nwhere N is the number of particles, and the zero subscript indicates the initial conditions.\n\nAs seen in Fig. \\ref{fig:radius}, this model agrees very well with the measurements.\nThis radial expansion can be problematic when attempting to prepare low kinetic energy antiprotons to produce trappable antihydrogen atoms, as the energy associated with the magnetron motion grows with the distance from the axis, and the electrostatic potential energy released as the radius expands can reheat the particles.\nThe effect can be countered somewhat by taking a longer time to cool the particles, resulting in a higher efficiency and, thus, a smaller expansion, but we find that the efficiency depends very weakly on the cooling time.\n\n\\begin{SCfigure}[1.0][h]\n\n\t\\input{EVCSize}\n\t\\caption{The measured size of the antiproton cloud using a MCP\/phosphor\/CCD device as a function of the number of particles lost. This is compared to the size predicted from eqn \\ref{eq:expansion}}\n\t\\label{fig:radius}\n\\end{SCfigure}\n\nColder antiprotons are of great utility in the effort to produce cold antihydrogen atoms.\nAntihydrogen production techniques can be broadly categorised as `static' - in which a cloud of antiprotons is held stationary and positrons, perhaps in the form of positronium atoms are introduced \\cite{positronium}, or `dynamic' - where antiprotons are passed through a positron plasma \\cite{Nested}.\nIn the first case, the advantages of cold antiprotons are obvious, as the lower kinetic energy translates directly into lower-energy antihydrogen atoms.\nIn the second case, the colder temperature allows the manipulations used to `inject' the antiprotons into the positrons to produce much more precisely defined antiproton energies.\nIndirectly, this will also permit these schemes to produce more trappable antihydrogen.\n\n\\section{Annihilation vertex detector}\\label{sec:detector}\n\nAmong the most powerful diagnostic tools available to experiments working with antimatter are detectors capable of detecting matter-antimatter annihilations.\nAntiproton annihilations produce an average of three charged pions, which can be detected by scintillating material placed around the trap.\nThe passage of a pion through the scintillator produces photons, which trigger a cascade in a photo-multiplier tube to produce a voltage pulse.\nIndividual voltage pulses can be counted to determine the number of annihilations.\n\nA further technique uses a position-sensitive detector to reconstruct the trajectories of the pions and find the point where the antiproton annihilated (usually called the `vertex').\nThe ALPHA annihilation vertex detector comprises sixty double-sided silicon wafers, arranged in three layers in a cylindrical fashion around the antihydrogen production and trapping region.\nEach wafer is divided into 256 strips, oriented in orthogonal directions on the p- and n- sides.\nCharged particles passing through the silicon result in charge deposits, and the intersection of perpendicular strips with charge above a defined threshold marks the location a particle passed through the silicon.\n\nEach module is controlled by a circuit that produces a digital signal when a charge is detected on the silicon.\nIf a coincidence of modules is satisfied in a 400~ns time window, the charge profile is `read-out' and digitised for further analysis.\nEach readout and associated trigger and timing information comprises an `event'.\nThe pion trajectories are reconstructed by fitting helices to sets of three hits, one from each layer of the detector.\nThe point that minimises the distance to the helices is then identified as the annihilation vertex.\nAn example of an annihilation event is shown in Fig. \\ref{fig:vertex}(a).\n\n\\vspace{-0.5cm}\n\\begin{SCfigure}[1.0][h]\n\\includegraphics[width=0.6\\textwidth]{Verticesdrawing}\n\\caption{(a) an example reconstruction of an antihydrogen annihilation and (b) a cosmic ray event. The diamond indicates the position of the vertex identified by the reconstruction algorithm, the polygonal structure shows the locations of the silicon wafers, the dots are the positions of the detected hits, and the inner circle shows the radius of the Penning trap electrodes. Also shown are annihilation density distributions associated with antihydrogen production (c, e) and deliberately induced antiproton loss (d, f). (c) and (d) are projected along the cylindrical axis, with the inner radius of the electrodes marked with a white circle, while (e) and (f) show the azimuthal angle $\\phi$ against the axial position $z$}\n\\label{fig:vertex}\n\\end{SCfigure}\n\\vspace{-0.5cm}\n\nExamination of the spatial distributions of annihilations can yield much insight into the physical processes at work.\nATHENA established that antihydrogen production resulted in a characteristic `ring' structure - an azimuthally smooth distribution concentrated at the radius of the trap electrodes \\cite{ATHENA_imaging}, shown in \\ref{fig:vertex}(c) and (e).\nIn contrast, the loss of bare antiprotons occurred in spatially well-defined locations, called `hot-spots', examples of which are shown in \\ref{fig:vertex}(d) and (f).\nThis was interpreted to be due to microscopic imperfections in the trap elements.\nThese produce electric fields that break the symmetry of the trap and give rise to preferred locations for charged particle loss.\nWhen antihydrogen is produced in a multipole field, antiprotons generated by ionisation of weakly-bound antihydrogen also contribute small asymmetries \\cite{ALPHA_HbarOct}.\nThese features are present in Fig. \\ref{fig:vertex}(c) and (e).\n\nThe vertex detector is also sensitive to charged particles in cosmic rays.\nWhen passing through the detector, they are typically identified as a pair of almost co-linear tracks (Fig. \\ref{fig:vertex}(b)), and can be misidentified as an annihilation.\nCosmic-ray events when searching for the release of trapped antihydrogen thus present a background.\n\nTo develop a method to reject cosmic ray events, while retaining annihilations, we compared samples of the events using three parameters, shown in Fig. \\ref{fig:distributions}.\nCosmic rays have predominantly two tracks, while antiproton annihilations typically have more. 95\\% of cosmic events have two or fewer identified tracks, while 58\\% of antiproton annihilations have at least three.\nA significant number of antiproton annihilations can have only two tracks, so it is not desirable to reject all these events as background.\n\n\\vspace{-0.2cm}\n\\begin{SCfigure}[1.0][h]\n\\includegraphics[width=0.6\\textwidth]{vertexDistributions}\n\\caption{Comparison of the distributions of event parameters for antiproton annihilations (solid line) and cosmic rays (dashed line). Shown are (a) the number of identified charged particle tracks, (b) the radial coordinate of the vertex, and the squared residual from a linear fit to the identified positions for the events with (c) two tracks and (d) more than two tracks. The shaded regions indicate the range of parameters that are rejected to minimise the p-value as discussed in the text}\n\\label{fig:distributions}\n\\end{SCfigure}\n\\vspace{-0.4cm}\n\nWe determine if the tracks form a straight line by fitting a line to the hits from each pair of tracks, and calculating the squared residual value.\nAs seen in Fig. \\ref{fig:distributions}(c) and (d), cosmic events have much lower squared residual values than annihilations.\nThis is to be expected, since particles from cosmic rays have high momentum and pass through the apparatus and the magnetic field essentially undeflected, while the particles produced in an annihilation will, in general, move in all directions.\nIn addition, annihilations occur on the inner wall of the Penning trap, at a radius of $\\sim$2.2~cm, and as shown in Fig. \\ref{fig:distributions}(b), reconstructed annihilation vertices are concentrated here, whereas cosmic rays pass through at a random radius.\n\nBy varying the ranges of parameters for which events are accepted, we could optimise the annihilation detection strategy.\nThe point where the `p-value' -- the probability that an observed signal is due to statistical fluctuations in the background \\cite{PDG} --  was minimised requiring the vertex to lie within 4~cm of the trap axis, and the squared residual value to be at least 2~$\\mathrm{cm}^2$ or 0.05~$\\mathrm{cm}^2$ for events with two tracks and more than two tracks, respectively.\n\nThese thresholds reject more than 99\\% of the cosmic background, reducing the absolute rate of background events to 22~mHz, while still retaining the ability of identify $\\sim 40\\%$ of antiproton annihilations.\nWhile this method effectively removes cosmic rays as a source of concern, other background processes, including mirror-trapped antiprotons must also be considered when searching for trapped antihydrogen.\nOur cosmic-ray rejection method has been applied to data taken from the 2009 ALPHA antihydrogen trapping run, and a full discussion of the results obtained will be made in a forthcoming publication.\n\n\\section{Conclusions and outlook}\nIn this paper we have described two of the most recent techniques developed by the ALPHA collaboration in our search for trapped antihydrogen.\nEvaporative cooling of antiprotons has the potential to greatly increase the number of low-energy, trappable atoms produced in our experiment.\nThe use of our unique annihilation vertex imaging detector to discriminate with high power between annihilations and cosmic rays will be a vital tool to identify the first trapped antihydrogen atoms.\nWe have integrated both of these techniques into our experiment and are hopeful of soon being able to report detection of trapped antihydrogen.\n\n\\begin{acknowledgements}\nThis work was supported by CNPq, FINEP\/RENAFAE (Brazil), ISF (Israel), MEXT (Japan), FNU (Denmark), VR (Sweden), NSERC, NRC\/TRIUMF, AIF (Canada), DOE, NSF (USA), EPSRC and the Leverhulme Trust (UK).\nWe are also grateful to the AD team for the delivery of a high-quality antiproton beam, and to CERN for its technical support.\n\\end{acknowledgements}\n\n\\subsection*{Note added in proof}: Since the preparation of this article, trapping of antihydrogen atoms has been\nachieved by the ALPHA collaboration \\cite{ALPHA_Nature}\n\n\n\t\\bibliographystyle{aipnum4-1}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\n\\label{intro}\n\nThe recent developments in our understanding of the nuclear regions of\nnearby galaxies  provide us with a  new framework in  which to explore\nthe classical issue of the connection between host galaxies and AGN.\n\nAll evidence now points to the idea that most galaxies\nhost a supermassive black hole (SMBH) in their centers\n\\citep[e.g.][]{kormendy95} and that its mass is closely linked to the\nhost galaxies properties, such as the stellar velocity dispersion\n\\citep{ferrarese00,gebhardt00}.  This is clearly indicative of a\ncoevolution of the galaxy\/SMBH system and it also provides us with\nindirect, but robust, SMBH mass estimates for large sample of objects.\nFurthermore, the innermost structure of nearby galaxies have been\nrevealed by HST imaging, showing the ubiquitous presence of singular\nstarlight distributions with surface brightness diverging as\n$\\Sigma(r)\\sim r^{-\\gamma}$ with $\\gamma>0$ \\citep[e.g.][]{lauer95}.\nThe distribution of cusp slopes \\citep{faber97} is bimodal, with a\npaucity of objects with $0.3<\\gamma<0.5$.  Galaxies can then be\nseparated on the basis of their brightness profiles in the two classes\nof ``core'' ($\\gamma \\leq 0.3$) and ``power-law'' ($\\gamma \\geq 0.5$)\ngalaxies, in close correspondence to the revision of the Hubble\nsequence proposed by \\citet{kormendy96}.\n\nBut despite these fundamental breakthroughs we still lack a clear\npicture of the precise relationship between AGN and host galaxies.\nFor example, while spiral galaxies preferentially harbour radio-quiet\nAGN, early-type galaxies host both radio-loud and radio-quiet\nAGN. Similarly, radio-loud AGN are generally associated with the most\nmassive SMBH as there is a median shift between the radio-quiet and\nradio-loud distribution, but both distributions are broad and overlap\nconsiderably \\citep[e.g.][]{dunlop03}.\n\nIn this framework, in two senses\nearly-type galaxies appear to be the critical class of\nobjects, where the transition between the two profiles classes occurs\n(i.e. in which core and power-law galaxies coexist) and \nin which they can host  \neither radio-loud and radio-quiet AGN. We thus started a\ncomprehensive study of a sample of early-type galaxies (see below for\nthe sample definition) to \nexplore the connection between  the multiwavelength \nproperties of AGN and the characteristics of their hosts. \nSince the 'Nuker' classification can only be obtained when the \nnuclear region, potentially associated with a shallow cusp,\ncan be well resolved, such a study must be limited to nearby galaxies.\nThe most compact cores will be barely resolved at a\ndistance of 40 Mpc (where 10 pc subtend 0\\farcs05)\neven in the HST images. Furthermore, high quality radio-images \nare required for an initial selection of AGN candidates. \n\nWe then examined two samples of  nearby objects for\nwhich  radio observations combining  relatively high  resolution, high\nfrequency and sensitivity  are available,  in order  to  minimize the\ncontribution from radio emission not related to the galaxy's nucleus\nand confusion from background sources.\nMore specifically we focus on the samples of early-type galaxies \nstudied by \\citet{wrobel91b}  and \\citet{sadler89} both observed with\nthe VLA at  5 GHz with a  flux limit  of\n$\\sim$  1 mJy.   The two  samples were  selected with  a  very similar\nstrategy. \\citet{wrobel91a} extracted  a northern\nsample  of  galaxies from  the  CfA  redshift survey  \\citep{huchra83}\nsatisfying  the following  criteria: (1)  $\\delta_{1950} \\geq  0$, (2)\nphotometric magnitude  B $\\leq$ 14; (3) heliocentric  velocity $\\leq$ 3000\nkm s$^{-1}$, and (4) morphological  Hubble type T$\\leq$-1, for a total\nnumber of  216 galaxies.  \\citet{sadler89} selected  a similar southern sample\nof 116 E and S0 with  $-45 \\leq \\delta \\leq -32$. \nThe only difference between\nthe  two  samples  is  that  \\citeauthor{sadler89} did  not  impose  a\ndistance limit.   Nonetheless,  the  threshold in  optical  magnitude\neffectively limits the  sample to a recession velocity  of $\\sim$ 6000\nkm s$^{-1}$. \n\nIn \\citet[ hereafter Paper I]{capetti05}, we focused on the 116 \ngalaxies detected in these VLA surveys to boost the fraction\nof AGN with respect to a purely optically selected sample.\nWe used archival HST observations, available for 65 objects, to study their \nsurface brightness profiles and to separate these early-type \ngalaxies into core and power-law galaxies following the Nukers scheme, \nrather than on the traditional morphological classification (i.e. into E and\nS0 galaxies). \nHere we focus on the sub-sample formed by the 29 ``core'' galaxies. \n\nWe adopt a Hubble constant H$_{\\rm o}=75$ km  s$^{-1}$ Mpc$^{-1}$.\n\n\\section{A critical analysis of the classification as core galaxies.}\n\\label{sersic}\n\nIn \\citetalias{capetti05} we adopted the classification into power-law and\ncore galaxies following the scheme proposed by \\citet{lauer95}.\nWe then separated early-type galaxies on the  basis of the \nslope of their  nuclear  brightness\nprofiles obtained using the Nukers law (i.e. a double power-law \nwith innermost slope $\\gamma$)\ndefining as core-galaxies all objects with $\\gamma \\leq 0.3$.\nSince this strategy has been subsequently challenged by\n\\citet{graham03}, who introduced a different definition of\ncore-galaxies,\nit is clearly important to assess whether the identification\nof an object as a core galaxy is dependent on the fitting scheme adopted.\n\n\\citeauthor{graham03} argued that a S\\'ersic model\n\\citep{sersic68}\nprovides a better\ncharacterization of the brightness profiles of early-type galaxies.\nIn particular they pointed out that, among other issues, \ni) the values of the Nukers law\nparameters depend on the radial region used for the\nfit, ii) the Nukers fit is unable to reproduce the large scale\nbehaviour of early-type galaxies \nand, most importantly for our purposes, iii) the identification of\na core galaxy from a Nuker fit might not be recovered by a S\\'ersic\nfit. Conversely, they were able to fit power-law galaxies (in the\nNukers scheme), as well\nas dwarf ellipticals \\citep{graham03b}, with\na single S\\'ersic law over the whole range of radii. \nThey also suggested a new definition of \ncore-galaxy as the class of objects \nshowing a light deficit toward the center with respect to the S\\'ersic\nlaw \\citep{trujillo04}.\n\nIn this context,\nwe discuss in detail  here the behaviour of the most critical\nobjects, i.e. the two core galaxies for which the Nuker\nlaw returns the smallest values for the break radius,\nnamely UGC~7760 and UGC~7797 for which $r_b = 0\\farcs49$\nand $r_b = 0\\farcs21$ respectively. We fit both objects with\na S\\'ersic law. The final fits, shown in Fig. \\ref{sersicfig},\nwere obtained iteratively, fitting the external regions\nwhile flagging the innermost points\nout to a radius at which the residual from the S\\'ersic law exceeded\na threshold of 5 \\%. The S\\'ersic law  in general provides a remarkably good\nfit to the outer regions, with typical residuals of $\\sim$ 1\\%, but \na substantial central light deficit is clearly\npresent in both objects.\nThis indicates that both objects can be \nclassified as core-galaxies in the Graham et al. scheme.\n\nUsing the brightness profiles for the core-galaxies for which\nwe obtained Nuker fits in \\citetalias{capetti05} (14 additional objects) \nwe obtained similar results. Very\nsatisfactory fits can be obtained with a S\\'ersic law on the external\nregions of these galaxies, but \nthey all show an even clearer central light\ndeficit, as expected given the presence of well resolved shallow cores.\n\nWe conclude that, for the galaxies of our sample,\nthe objects classified as core-galaxies in the Nuker scheme \nare recovered as such with the Graham et al. \ndefinition.\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f1a.ps,width=0.50\\linewidth}\n\\psfig{figure=f1b.ps,width=0.50\\linewidth}\n}\n\\caption{S\\'ersic fits for the two core galaxies of our sample\nwith the smallest values for the core radius. A substantial central light \ndeficit is clearly present in both objects, conforming to\nthe ``core'' classification in the Graham et al. scheme.}\n\\label{sersicfig}\n\\end{figure*}\n\n\\section{Basic data and nuclear luminosities}\n\\label{nuc}\n\n\\input{tab1.tex}\n\nThe basic\ndata for  the selected galaxies, namely the recession velocity  \n(corrected for Local Group infall onto Virgo), the K\nband  magnitude from  the Two  Micron  All Sky  Survey (2MASS),\nthe galactic extinction and the total and\ncore radio fluxes were given in \\citetalias{capetti05}.\n \nIn the following three subsections, \nwe derive and discuss the measurements for the\nnuclear sources in the optical, X-ray and radio bands.\n\n\\subsection{Optical nuclei.}\n\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f2a.ps,width=0.33\\linewidth}\n\\psfig{figure=f2b.ps,width=0.33\\linewidth}\n\\psfig{figure=f2c.ps,width=0.33\\linewidth}\n}\n\\caption{\\label{hstnuc} Brightness profile and its derivative for \nthree objects of the sample, namely IC~4296, NGC~4373 and NGC~1316.\nThe first two galaxies show, at decreasing significance level, the\ncharacteristic up-turn in the profile associated with a nuclear source.\nThis is not seen in the third object which is then considered as a\nnon-detection.}\n\\end{figure*}\n\nThe detection and measurement of an optical nuclear source at the \ncenter of a galaxy is a challenging task particularly  \nwhen it represents only a small \ncontribution with respect to the host emission, \nas is likely often to be the case of the weakly active galaxies\nmaking up our sample.\n\nDifferent approaches have been employed in the literature.\nThe most widely used method is to fit the overall brightness \nprofile of a galaxy\nwith an empirical functional form and to define a galaxy as\n``nucleated'' when it shows a light excess in its central region\nwith respect to the model \\citep[e.g.][]{lauer04,ravi01}.\nThe drawback of this ``global'' approach is that it \nassumes that the model can be extrapolated inwards\nfrom the radial domain over which the fit was performed.\nFurthermore, the measurement and identification of the nuclear\ncomponent are coupled with the behaviour of the brightness profile at all radii\nand with the specific choice of an analytic form. \nAlthough this is not a significant issue for bright\npoint sources, it is particularly worrisome for the faint nuclei\nwe are dealing with.\nNonetheless, \\citet{rest01} pointed out that in general nuclear light excesses\nare associated with a steepening of the profile as the HST resolution\nlimit is approached. Indeed this is expected in the presence of\na nuclear point source, since the convolution with the Point Spread \nFunction produces a smooth decrease of the slope toward the center\nwhen only a diffuse galactic component is present.\nWe therefore preferred to adopt a ``local'' approach\nto identify nuclear sources, based on the characteristic up-turn\nthey cause in the nuclear brightness profile.\n\nMore specifically, we  evaluated the  derivative  of  the\nbrightness profile in a log-log representation \nfor the sources of our sample. In order to increase the \nstability of the slope measurement this has been estimated \nby combining the brightness \nmeasured over two adjacent points on each side of the radius of interest,\nyielding a second order accuracy. We then look for  \nthe presence of a nuclear up-turn in the derivative \nrequiring for a nuclear detection a difference larger than\n3 $\\sigma$ from the slopes at the local minimum and maximum.\nThis a rather conservative definition since \nthe region over which the up-turn is detected \nextends over several pixels while we only consider the\npeak-to-peak difference. \n\nTo illustrate this we focus on three\ncases. In the HST image as well as in the brightness profile of IC~4296 \na nucleus clearly stands out against the underlying background and\nthe central  steepening at about $r=0\\farcs15$ is highly significant. \nNGC~4373 is the detection with the least significance of our sample,\nin  which the presence of a  nucleus is uncertain\nfrom just the visual inspection of \nthe optical image,  but the\nderivative of the brightness profile reveals the effect  of the point\nsource with an increase of 0.013 $\\pm$ 0.004 from\n$r=0\\farcs1$ and $r=0\\farcs07$.  \nInstead  in NGC~1316 we do not  have evidence for any\ncompact point source, both in the image and in the brightness profile\nderivative, and it is considered as a non-detection.\n\nAdopting this strategy in 18 out of 29 objects we identify an optical nucleus,\nwith a  percentage of $\\sim$ 60\\% of the total sample. \nIn seven objects we did not find any upturn and these are \nconsidered upper limits.\nNote that this is again a conservative approach, since \na nuclear source can still be\npresent but its intensity might not be sufficient to compensate the\ndownward trend of the derivative sets by the host galaxy. \n\nIn the remaining 4 objects the central regions have a complex \nstructure and no estimate\nof the optical nucleus intensity can be obtained. \nIn two cases (UGC~8745 and UGC~9723) \nthe central regions are completely hidden by a kpc scale edge-on disk, \nwhile in NGC~3557 the study of its nuclear regions \nis hampered by the presence \nof a circumnuclear dusty disk. In UGC~9655, the innermost region\n($r<0\\farcs1$) has a lower brightness than its surrounding; since\nonly a single band image is available we cannot assess \nif this is due to dust absorption or\nto a genuine central brightness minimum \nas in the cases discussed by \\citet{lauer02}.\n\nWe measured the  nuclear luminosity with the task RADPROF  in IRAF,\nchoosing as the  extraction region a circle centered on the  nucleus with\nradius set at the location of the up-turn and as the background region \na circumnuclear annulus, 0.1\\arcsec\\ in width. \nFor the undetected nuclei we set as upper limits the light excess\nwith respect to the starlight background\nwithin a circular aperture 0.1\\arcsec\\ in diameter.\nThen  we use  the PHOTFLAM and  EXPTIME keyword in  the image\nheader to convert  the total counts to fluxes. \nErrors on the measurements of the optical nuclei are dominated\nby the uncertainty in the behaviour of the host's profile, \nwhile the statistical and absolute calibration errors amount\nto less than 10 \\%.h\nThe very presence of the nucleus prevents\nus from determining accurately the host contribution within\nthe central aperture. Our strategy is to remove the background measured\nas close as possible to the nucleus, i.e. effectively we adopted\na constant starlight distribution in the innermost regions. \nAn alternative approach would be to extrapolate the profile with\na constant slope instead. Our definition of\nnuclear sources (an increase in the profile's derivative) \nimplicitly requires that the observed profile lies\nabove this extrapolation, but the resulting flux is reduced\nby at most a factor of 2 (with respect to the case of constant\nbackground) for the nuclei with the smallest contrast\nagainst the galaxy. As will become clear in the next sections,\nerrors of this magnitude only have a marginal impact on our conclusions.\nThe resulting fluxes are reported\nin Table \\ref{tabsample2}.\nWe finally derived all the\nluminosities referred to 8140 \\AA\\ (see Table \\ref{lum}),\nafter correcting for the Galactic extinction as tabulated in \n\\citetalias{capetti05} and\nadopting an optical spectral \nindex\\footnote{We define the spectral index $\\alpha$ with the spectrum\nin the  form F$_\\nu  \\propto \\nu^{-\\alpha}$} $\\alpha_o = 1$.\n\n\\subsection{X-ray nuclei}\n\nFor the measurements of the X-ray nuclei we concentrate \nonly on the Chandra measurements, as this telescope provides \nthe best  combination of\nsensitivity and resolution necessary to detect the faint nuclei expected in\nthese weakly active galaxies.\nData for 21 core galaxies are available in the Chandra public archive.\n\nWhen available, we used the results of the analysis of the X-ray\ndata from the literature. \nWe find estimates of the luminosities of the nuclear sources (usually defined\nas the detection of a high energy power-law component)\nbased on  Chandra data  for 16\nobjects of our sample (12 of which are detections and 4 are upper limits) \nwhich we rescaled to our adopted distance \nand converted to the 2-10 keV  band, using  the\npublished power law index. In\nTab \\ref{tabsample2} we give a summary of the available\nChandra data, while references and details on the X-ray \nobservations and analysis are presented in Appendix \\ref{notes}. \n\nWe also considered the Chandra archival data for the 5 unpublished objects,\nnamely UGC~5902, UGC~6297, UGC~7203, NGC~3557 and NGC~5419. We analyzed these\nobservations  using the Chandra  data analysis  CIAO v3.0.2,  with the\nCALDB version 2.25, using the same strategy as in\n\\citet{balmaverde05}. \nWe reprocessed all the data from level 1 to level 2,\nsubtracting the bad pixels,  applying ACIS CTI correction, coordinates\nand pha randomization.  We searched for background flares and \nexcluded some period of bad aspect.\n\nWe then extracted the spectrum in a circle region centered on the nucleus \nwith a radius of 2\\arcsec\\ and we take the background in an annulus\nof 4\\arcsec. We grouped the spectrum to have at least 10 counts \nper bin and applied Poisson statistics. \n     \nFor two  objects  (NGC~3557 and NGC~5419) we obtain  a detection of a\nnuclear power-law source by\nfitting the spectrum using an absorbed power-law plus a thermal\nmodel,  with  the  hydrogen  column  density  fixed  at  the  Galactic\nvalue. Details of the results are given in Appendix \\ref{notes}.\nFor the remaining 3 galaxies we \nset an  upper  limit to  any  nuclear  emission, with  the\nconservative   hypothesis   that   all   flux  that   we   measure   is\nnon-thermal. We then fit the  spectrum with an  absorbed (to  the galactic\nvalue)  power law  model with  photon index  $\\Gamma=2$.  \n\nThe X-ray luminosities for all objects are given in Table \\ref{lum}.\n\n\\subsection{Radio nuclei.}\n\nThe radio data available for all objects of our sample \nare drawn from the surveys by \\citet{wrobel91b} \nand \\citet{sadler89}, performed with the VLA at 5 GHz with a resolution of\n$\\sim$ 5\\arcsec. Although these represent the most uniform\nand comprehensive studies of radio emission in early-type galaxies,\nthey do not always have a resolution sufficient to separate the core\nemission from any extended structure. \\citet{sadler89} \nargued that at decreasing radio luminosity there is a corresponding\nincrease of the fractional contribution of the radio core.\n\nTo verify whether the VLA data overestimate the core flux, we searched the literature  for radio  core measurements  obtained at\nhigher  resolution  (and\/or  higher  frequency) than  our  data.  This\nwould  improve  the  estimate  of the  core  flux  density,\navoiding the contribution of  extended emission or spurious sources to\nthe nuclear  flux as  well as revealing  any radio  structure.  Better\nmeasurements, from  VLBI data  or from  higher  frequency\/resolution VLA\ndata, are available for most CoreG  (23 out of 29) and compact cores\nwere detected in all but 2 objects. The radio core\nfluxes are taken from \\citet{nagar02} (15 GHz VLA data and 5 GHz VLBI\ndata), \\citet{filho02} and \\citet{krajnovic02} (8.4 GHz at the VLA),\n\\citet{jones97} (8.4 GHz VLBI data) and\n\\citet{slee94} (PTI 5 GHz interpolated data).\n\nIn Fig. \\ref{cfr} we\ncompare the radio  core flux density used in our  analysis \nagainst observations made at higher resolution.\nOverall there  is a  substantial agreement between the two datasets,\nwith a  median difference  of only\n$\\sim$0.25 dex (a factor  $\\sim$1.6), with only two objects\nsubstantially offset (by a factor of $\\sim$ 10).\nHowever, since  these data are  highly inhomogeneous and  given the\ngeneral  agreement with the  5 GHz VLA measurements, we\nprefer to retain the values of \\citeauthor{wrobel91b} and \\citeauthor{sadler89}.\nNonetheless, we always checked that using these higher resolution \ncore fluxes our main results are not significantly affected \n(see Appendix \\ref{radionuc} for a specific example).\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f3.ps,width=1.00\\linewidth}\n}\n\\caption{Radio core flux density for CoreG obtained at \n5 GHz with VLA (used in this work) compared to higher resolution data from\n\\citet{slee94,nagar02,filho02,\nkrajnovic02,jones97}.\nThe dotted line is the bisectrix of the plane.}\n\\label{cfr}\n\\end{figure}\n\n\n\\section{The multiwavelength properties of nuclei of core galaxies.}\n\\label{nuclei}\n\n\\input{tab2.tex}\n\nHaving collected the multiwavelength  information for the nuclei of our\ncore  galaxies we  can compare  the emission  in the  different bands.\nFirst of all, we can estimate the ratio between the radio, optical and\nX-ray  luminosities: the  median values  are Log$(\\nu L_r\/\\nu L_o)  \\sim -1.5$\n(equivalent to  a standard radio-loudness parameter Log  R $\\sim$ 3.6)\n\\footnote{R =  L$_{5\\rm {GHz}}$ \/ L$_{\\rm B}$. As in\n  Sect. \\ref{nuc} we transformed the optical fluxes to the B band\nadopting an optical spectral index $\\alpha_o = 1$.}\nand Log R$_{\\rm X}$  = Log$(\\nu L_{\\rm r}\/L_{\\rm X})  \\sim -1.3$,  \nboth with a  dispersion of  $\\sim$ 0.5\ndex.  These  ratios are clearly  indicative of a radio-loud  nature for\nthese  nuclei when compared  to both  the traditional  separation into\nradio-loud  and radio-quiet  AGN \n\\citep[Log R =  1, e.g.][]{kellermann94}, as well  as with  the\nradio-loudness threshold introduced by \\citet{terashima03}\nbased on  the X-ray to radio luminosity  ratio  (Log R$_{\\rm X}$ = -4.5).  \nFurthermore, the nuclear\nluminosities  in   all  three   bands  are clearly correlated (see\nFig. \\ref{corr1} and Table \\ref{tab0} for a summary of the results of\nthe statistical analysis): the generalized (including the presence\nof upper limits) Spearman rank correlation coefficient\n$\\rho$ is 0.63 and 0.89 for $L_{\\rm r}$ vs. $L_{\\rm o}$ \nand $L_{\\rm r}$ vs. $L_{\\rm X}$ respectively, with probabilities that\nthe correlations are not present of only 0.002 and 0.0001.\n\nBoth results are reminiscent of what is  observed for the\nradio-loud nuclei \nof low luminosity radio-galaxies (LLRG). \\citet{chiaberge:ccc} \nand \\citet{balmaverde05} reported on similar multiwavelength\nluminosity trends for the sample of\nLLRG formed  by the  3C sources with  FR~I morphology.\nThe connection between the CoreG and LLRG becomes \nmore evident if we add LLRG  in the diagnostic\nplanes (see  Fig. \\ref{corr} and Table \\ref{lumfr1}). \nThe early-type core  galaxies \nfollow the same behaviour of the stronger radio galaxies,\nextending it downward by 3 orders  of magnitude in radio-core\nluminosity as they reach levels\nas low as $L_{\\rm r} \\sim 10^{36}$ erg s$^{-1}$.\n\nWe estimated the best linear fit for the combined CoreG\/LLRG sample\nin both the \n$L_{\\rm r}$ vs. $L_{\\rm o}$ and $L_{\\rm r}$ vs. $L_{\\rm X}$ planes. \nThe best  fits were  derived  as the\nbisectrix of the  linear fits using the two  quantities as independent\nvariables following the suggestion by \\citet{isobe90} that this is\npreferable for problems needing symmetrical treatment of the variables. \nThe presence of upper limits in the independent variable\nsuggests that we could take advantage of the \nmethods of survival analysis proposed by e.g. \\citet{schmitt85}.\nHowever, the drawbacks discussed by \\citet{sadler89}\nand, in our specific case, \nthe non-random distribution of upper limits, argue against this approach. \nWe therefore preferred to exclude  upper limits from the  linear\nregression analysis. Nonetheless, a posteriori, \n1) the  objects  with an undetected nuclear\ncomponent in  the optical or X-ray are\nconsistent  with the correlation  defined by  the detections  only;\n2) the application of the Schmidt methods provides correlation\nslopes that agree, within the errors, with our estimates.\n \nWe obtained (indicating the Pearson \ncorrelation coefficient with $r$ and slope with $m$)\n$r_{ro}$=0.90 and $m_{ro}=0.89  \\pm 0.07 $, \n$r_{rx}$=0.89 $m_{rx} = 1.02\\pm 0.10 $ for the radio\/optical and\nradio\/X-ray correlations respectively.\nThe slopes and normalizations derived for CoreG, LLRG and the\ncombined CoreG+LLRG sample \n(see Table \\ref{tab0}) are consistent within the errors\nand this indicates that there is no significant change in the\nbehaviour between the two samples. \nOnly the dispersion is slightly larger for the CoreG nuclei \nbeing a factor of $\\sim 4$ rather than $\\sim 2$ for the LLRG sample alone. \n\n\\citet{chiaberge:ccc} first reported the presence of a correlation\nbetween radio and optical emission in the LLRG and they concluded that\nthis is most likely due to a common non-thermal jet origin for the\nradio and optical cores. \nRecently \\citet{balmaverde05} extended the analysis to the X-ray cores;\nthe nuclear X-ray luminosity also correlates with those of the radio\ncores and with a much smaller dispersion \n($\\sim$ 0.3 dex) when compared to similar\ntrends found for other classes of AGN \\citep[see e.g.][]{falcke95}, \nagain pointing to a common origin for the emission in the three bands.\nFurthermore, the broad band spectral indices of the 3C\/FR~I cores \nare very similar to those measured in BL Lacs objects (for which a jet\norigin is well established) in accord with the\nFR~I\/BL~Lacs unified model \n(we will return to this issue in Section \\ref{bllac}).\n\nThe core galaxies of our sample thus appear to smoothly extend the results\nobtained for LLRG to much lower radio luminosity, expanding the\nmultiwavelenght nuclear correlations to a total of 6 orders of\nmagnitude. This strongly argues in favour of a jet\norigin for the nuclear emission also in the core galaxies and that they \nsimply represent the scaled down versions of these already low luminosity AGN.\n\n\\begin{table}\n\\caption{Correlations summary}\n\\begin{tabular}{l l c c c l } \\hline \\hline\nSample  & Var. A    & Var. B & r$_{AB}$& Slope & rms     \\\\\n\\hline                                       \nCoreG     & L$_{O}$ & L$_{r}$   & 0.59 & 0.76$\\pm$0.21  & 0.62  \\\\\n          & L$_{X}$ & L$_{r}$   & 0.78 & 1.36$\\pm$0.20  & 0.59  \\\\\nLLRG      & L$_{O}$ & L$_{r}$   & 0.94 & 0.82$\\pm$0.11  & 0.32  \\\\\n          & L$_{X}$ & L$_{r}$   & 0.95 & 0.99$\\pm$0.11  & 0.33  \\\\\nLLRG+CoreG& L$_{O}$ & L$_{r}$   & 0.90 & 0.89$\\pm$0.07  & 0.56  \\\\\n          & L$_{X}$ & L$_{r}$   & 0.89 & 1.02$\\pm$0.10  & 0.58  \\\\\n\\hline\n\\end{tabular}\n\\label{tab0}\n\\end{table}\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f4a.ps,width=0.50\\linewidth}\n\\psfig{figure=f4b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{corr1} Radio core luminosity for the early-type galaxies\nwith a ``core'' profile versus the optical (left) \nand X-ray (right) nuclear luminosities.}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f5a.ps,width=0.50\\linewidth}\n\\psfig{figure=f5b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{corr} Comparison of radio and optical \n(left) and X-ray (right) nuclear luminosity\nfor the sample of core-galaxies (filled circles) and for the \nreference 3C\/FR~I sample of low luminosity radio-galaxies (empty\ncircles). \nThe three sources in common are marked\nwith a filled square. The solid lines reproduce the best linear fits.\n}\n\\end{figure*}\n\n\\subsection{Core galaxies vs. low luminosity radio-galaxies.}\n\\label{cg-fri}\n\n\\input{tab4.tex}\n\nThe results presented above indicate that the nuclei of the CoreG\nshow a very similar behaviour to those of LLRG.\nHere we explore in more detail how CoreG and LLRG\ncompare in their other properties, such\nas the structure of the host, black hole mass, radio-morphology\nand optical spectra.\n\nOur sample was selected to include only\nearly-type galaxies with a core profile, e.g. with an asymptotic slope \n(toward the nucleus) of their surface brightness profiles $\\gamma < 0.3$. \nRecently \\citet{deruiter05} showed, from the analysis of a combined\nsample of B2 and 3C sources, that they are all hosted\nby early-type galaxies and that the presence of a flat core\nis a characteristic of the host galaxies of all nearby radio-galaxies.\n\nA strong similarity between CoreG and LLRG emerges when \ncomparing the mass of their\nsupermassive black holes. \nWhen no direct measurement \n\\citep[taken from the compilation by][]{marconi03} \nwas available, we estimated $M_{BH}$ using\nthe relationship with the stellar velocity dispersion\n(taken from the LEDA database) \nin the form given by \\citet{tremaine02}.\nThe distributions of $M_{BH}$ (see Fig. \\ref{mbhhis}) of\nthe two samples are almost indistinguishable\\footnote{The \nprobability that the two samples are drawn \nfrom the same parent distribution is 0.32, \naccording to the Kolmogorov-Smirnoff test.}, as they \nhave median values of Log $M_{BH} = 8.54 $ and Log $M_{BH} = 8.70 $,\nfor CoreG and LLRG respectively,\nand they also cover  \nthe same range, with most objects with Log $M_{BH} = 8 - 9.5$.\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f6b.ps,width=0.5\\linewidth}\n\\psfig{figure=f6a.ps,width=0.5\\linewidth}\n}\n\\caption{\\label{mbhhis} Distributions for CoreG (shaded histograms) \nand for LLRG (empty histogram) of \n(left panel) black hole mass M$_{BH}$ and \n(right panel) absolute magnitude M$_K$.  \nThe LLRG histograms have been re-normalized multiplying \nby a factor 29\/19 for M$_{BH}$ and 29\/17 for M$_K$ respectively, \ni.e. the number of objects\nin the two samples for which estimates of these parameters are available.}\n\\end{figure*}\n\nFurther indications  of the  nature of CoreG  cores and  their connection\nwith LLRG come from  the emission lines in their optical spectra.\nLLRG are characterized as a class by their LINER spectra\n\\citep[e.g.][]{lewis03} and this\nis the case also for the CoreG of our sample.\nIn the NED  database, although about half of the CoreG do not have\na spectral  classification, 13 objects are classified \nas LINERs\\footnote{This result provides further support to the \nsuggestion by \\citet{chiaberge05}\nthat a dual population is associated with galaxies with a LINER\nspectrum, being formed by both radio-quiet and by radio-loud objects.\nThe CoreG are part of this latter sub-population of radio-loud LINER.}.  \nThe only exception is UGC~7203, with a Seyfert spectrum, but its\ndiagnostic line  ratios are borderline  with those of LINERs \\citep{ho97}.\nConcerning the emission line luminosity, \\citet{capetti:cccriga} found a\ntight relationship  between radio core  and line luminosity  studying a\ngroup of LLRG formed by the 3C\/FR~I complemented by\nthe sample of 21 radio-bright ($F_r > 150$ mJy) \nUGC galaxies defined by \\citet{noelstorr03}. Line luminosity for \nour CoreG clearly follow the same trend defined by LLRG, \nalthough with a substantially larger  dispersion, not unexpected given their\nlow line luminosity and the non uniformity of the data used for this\nanalysis. \n\nConsidering the radio structure,\nseveral objects of our CoreG sample have a radio-morphology with well developed \njets and lobes: UGC~7360, UGC~7494 and UGC~7654 are FR~I\nradio-galaxies part of the 3C sample (3C~270, 3C~272.1 and 3C~274), \nwhile in the Southern sample we\nhave the well studied radio-galaxies NGC~1316 (Fornax A), a FR~II\nsource, NGC~5128 (Cen A) and IC~4296.\nA literature search shows that at least another 11 sources \nhave extended radio-structure indicative of a collimated outflow,\nalthough in several cases this can only be seen in high\nresolution VLBI images, such as\nthe mas scale double-lobes in UGC~7760 or the one-sided jet of \nUGC~7386 \\citep{nagar02,falcke00}. \n\nConversely, hosts of 3C\/FR~I radio-sources are on average more \nluminous than core-galaxies\n(see Fig. \\ref{mbhhis}, left panel) although there is\na substantial overlap between the two groups: the median values are\n$M_K=-24.8$ and $M_K=-25.7$ for CoreG and 3C\/FR~I respectively,\nwith a KS probability of only 0.003 of being drawn from the same population. \nThis reflects the well known trend,\nalready noted by \\citet{auriemma77}, for which \na brighter galaxy has a higher probability of being a stronger\nradio emitter, and which is present also in our sample \\citepalias{capetti05}. \nThe selection of \nrelatively bright radio sources, such as the 3C\/FR~I, corresponds\nto a bias toward more luminous galaxies. Indeed, within our sample,\nimposing a threshold in total radio-luminosity of \n$L_{\\rm tot} > 10^{39}$ erg\/s,\\footnote{The 5 GHz luminosity was\nconverted to 178 MHz for consistency with the 3C\/FR~I values adopting\na spectral index of 0.7.} the low end for LLRG,  \ndecreases the median magnitude to -25.1,\nin closer agreement with the 3C\/FR~I value.\n\nWe conclude that the properties of our low radio luminosity CoreG show\na remarkable similarity to those of classical LLRG, in particular,\nthey share the presence of a flat core in their host's brightness profiles,\nthey have the same distribution in black hole\nmasses, as well as analogous properties concerning their optical\nemission lines and radio-morphology. \nThese results indicate that core galaxies and LLRG\ncan be considered, from these different point of view, as being drawn\nfrom the same population of early-type galaxies. They can only be\nseparated on the basis of their different level of nuclear activity,\nwith the LLRG forming the tip of the iceberg of (relatively) \nhigh luminosity objects. Furthermore, the emission processes\nassociated to\ntheir activity scale almost linearly over 6 orders of magnitude in\nall bands for which data are available.\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f7.ps,width=1.00\\linewidth}\n}\n\\caption{Emission line vs. radio core luminosity for CoreG galaxies (filled circles)\nand for the LLRG 3C\/FR~I sample (empty circles), from Capetti et al. (2005).}\n\\label{line}\n\\end{figure}\n\n\\section{Black hole mass and radio luminosity.}\n\nThe issue of the relationship between\nthe black hole mass and the radio-luminosity has been discussed by\nseveral authors, taking advantage of the recent possibility to measure\n(or at least estimate) $M_{BH}$.\n\\citet{franceschini98} pioneered this field showing, \nfrom a compilation of objects with\navailable black hole estimates, that the radio-luminosity tightly\ncorrelates with the black hole mass, with a logarithmic index of\n$\\sim$ 2.5. This result was subsequently challenged, by\ne.g. \\citet{ho02}.\nWe here re-explore this issue limiting ourselves to the sample\nof core early-type galaxies; while this substantially restricts\nthe accessible range in $M_{BH}$ and it applies only to radio-loud\nnuclei, it has the\nsubstantial advantage of performing the analysis on a complete sample\nwith well defined selection criteria and\ncovering a large range of radio-luminosity.\n\nThe comparison of  the radio-core luminosity with the  black hole mass\nis presented in  Fig.  \\ref{mbh}. Apparently,\na dependence of $L_r$ on  $M_{BH}$ is present,\nalthough   with  a  substantial   scatter. However, \nthe radio flux limit of the samples exclude objects with lower radio\nluminosity, potentially populating the lower part of the \n$L_R$ vs. $M_{BH}$ plane.\nFurthermore, the\ninclusion of LLRG (which, as discussed above, represent the high\nactivity end of the early-type population) radically changes the picture, as they populate the\nwhole upper portion of this plane. This indicates that  a very large\nrange (at least 4 orders  of magnitude) of radio-power can correspond to\na given $M_{BH}$ \n\\footnote{With respect to previous studies we report the\nnuclear   radio   emission   only,   instead  of   the   total   radio\nluminosity.  However, since  the fraction  of extended  emission grows\nwith radio luminosity, using the  total power would just move the LLRG\nupward, further increasing  the spread.}. \nThis is a clear\nindication that, not unexpectedly, \nparameters other than the black hole mass play a\nfundamental role in determining the radio luminosity of a galaxy.\n\nMore notable is the lack of sources with \n$M_{BH} < 10^8 M_{\\sun}$ \n(with only one exception). \nThe  effects  produced  by  our\nselection  criteria must be  considered before  any conclusion  can be\ndrawn.  In particular the correlation between the black hole mass and\nthe  spheroidal   galactic  component,  combined   with  the  limiting\nmagnitude,  translates into  a threshold  in the  accessible  range of\nblack  hole masses.  Using the limit in apparent  magnitude of our\nsample ($m_B <  14$), an\naverage color  of B-K =  4.25 \\citep{mannucci01}\nand the best fit to the relationship between\n$M_{BH}$ and $M_K$  from \\citet{marconi03} we  obtain that at distances\nlarger  than  20 Mpc  (corresponding  to 7\/8 of  the  volume covered\nin the Wrobel's sample) we  do include galaxies with expected\nblack hole masses $M_{BH} < 10^8 M_{\\sun}$. This represents\na severe bias against the inclusion of galaxies with low values\nof $M_{BH}$, regardless of their radio emission.\nThe lack of low black hole mass LLRG seems to favour  \nthe reality of this effect, as they\nare  not directly selected imposing an optical threshold; \nhowever, the already discussed statistical trend linking\nradio and optical luminosity might represent a more subtle\nbias leading to the same effect.\nThe existence of a minimum black hole mass\nto produce a radio-loud nucleus must be properly\ntested extending the analysis to a  sample  of less luminous\ngalaxies, likely to harbour less massive black holes.\n\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f8.ps,width=1.00\\linewidth}\n}\n\\caption{Nuclear radio-luminosity vs. black hole mass M$_{BH}$ for CoreG galaxies (filled circles)\nand for the LLRG 3C\/FR~I sample (empty circles).}\n\\label{mbh}\n\\end{figure}\n\n\\section{Constraints on the radiative manifestation of the accretion process}\n\\label{adaf}\n\nTaking advantage of the estimates of black hole mass we \ncan convert the measurements of\nthe nuclear luminosities to units of the Eddington luminosity. \nAll CoreG nuclei are associated with a low fraction of $L_{\\rm {Edd}}$, \nbeing confined to the range $L\/L_{\\rm {Edd}} \\sim 10^{-6} - 10^{-9}$\nin both the X-ray and optical bands (with only one X-ray exception), \nsee Fig. \\ref{eddihis}. \nFurthermore, as discussed in Sect. \\ref{nuclei},\nthe tight correlations between radio, optical and X-ray nuclear \nluminosities extending across LLRG and CoreG strongly argue in favour of a jet\norigin for the nuclear emission also in the core galaxies.\nIf this is indeed the case, the observed nuclear emission \ndoes not originate in the accretion process and the\nvalues reported above should be considered\nas upper limits.\n\nOur results add to the already vast literature reporting \nemission corresponding to a very low Eddington fraction associated with \naccretion onto supermassive black holes. \nThese results prompted the idea that in these objects accretion occurs\nnot only at a low rate but also at a low radiative efficiency,\nsuch as in the Advection Dominated Accretion Flows \n\\citep[ADAF,][]{narayan95} in which most of the\ngravitational energy of the accreting gas is advected into the\nblack hole before it can be dissipated radiatively, thus reducing the\nefficiency of the process with respect to the standard models of\ngeometrically thin, optically thick, accretion disks.\nThe ADAF models have been rather successful in modeling the observed\nnuclear spectrum in several galaxies, such as e.g. the\nGalactic Center and NGC 4258 \\citep{narayan95,lasota96}. \nConversely, ADAF models substantially \nover-predict the observed emission in the nuclei of nearby bright elliptical\ngalaxies \\citep{dimatteo00,loewenstein01}.\n\nThis suggested the possibility that a substantial fraction\nof the mass included within the Bondi's accretion radius\n\\citep{bondi52} might not\nactually reach the central object,\nthus further reducing the radiative emission from the accretion\nprocess with respect to the ADAF models. \nThis may be the case in the presence of an outflow\n\\citep[Advection Dominated Inflow\/Outflow Solutions, or ADIOS,][]{blandford99}\nor strong convection \\citep[Convection Dominated Accretion Flows,\n  or CDAF,][]{quataert00} in which most gas circulates in \nconvection eddies rather than accreting onto the black hole. \n\nUnfortunately, in the case of the galaxies under investigation,\nthe comparison of the theoretical predictions \nwith the observations so as to get constraints\non the properties of the accretion process is quite difficult. \nThis is due to the presence of different competing models, all\nof these with several free parameters, and to the observational data, in particular to\nthe scarce multiwavelength coverage of the nuclear emission\nmeasurements which\nprevents us from deriving a detailed Spectral Energy Distribution of these objects.   \nAs discussed above, this is more complicated for our radio-loud\nnuclei in which  the emission is most likely dominated by the\nnon-thermal radiation from their jets.\n\nNonetheless, \\citet{pellegrini05} recently studied in detail a sample\nof nearby galaxies for which the Chandra observations provide an\nestimate of the temperature and density of the gas in the nuclear\nregions, thus enabling one to derive the expected Bondi \naccretion rate, $\\dot{M}_B$. It is interesting to note that the estimates \nof $\\dot{M}_B$ for three sources common to both samples\nwith the lowest X-ray luminosity (namely NGC~1399, UGC~7629 (AKA\nNGC~4472) and UGC~7898 (AKA NGC~4649)) are relatively large, \n$\\dot{M}_B\/\\dot{M}_{\\rm Edd} = 10^{-2} - 10^{-4}$,\nwhile their X-ray luminosities are $L_X\/L_{Edd} = 10^{-8} - 10^{-10}$\n(see her Fig. 3). These luminosities are between\n3 and 5 orders of magnitude lower than expected from an\nADAF model, and they should be considered\nonly as upper limits. \nThese results argue in favour of an effective accretion rate\nsubstantially smaller than expected in the case\nof spherical accretion, suggesting that \nan important role is played by mass loss due to an outflow or \nby convection.\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f9a.ps,width=0.50\\linewidth}\n\\psfig{figure=f9b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{eddihis} Distributions of the nuclear luminosities \nmeasured as fraction\nof the Eddington luminosity in the X-ray (left) and optical (right)\nbands. }\n\\end{figure*}\n\n\\section{CoreG and the BL~Lacs\/LLRG unifying model}\n\\label{bllac}\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=f10.ps,width=1.00\\linewidth}\n}\n\\caption{Broad band spectral indices, calculated \nbetween 5 GHz, 5500 \\AA\\ and 1 keV, for core galaxies (filled\ncircles), low luminosity 3C\/FR~I radio-galaxies (empty circles), \nLow energy peaked BL Lacs (stars) and High energy peaked BL Lacs\n(squares).\nSolid lines mark the regions within 2\n$\\sigma$ from the mean $\\alpha_{ro}$ and $\\alpha_{ox}$ for BL Lacs \ndrawn from the DXRB and RGB surveys.\nThe dashed lines\nrepresent constant values for the third index, $\\alpha_{rx}$.}\n\\label{roox}\n\\end{figure}\n\n\\begin{figure*}\n\\centerline{\n\\psfig{figure=f11a.ps,width=0.50\\linewidth}\n\\psfig{figure=f11b.ps,width=0.50\\linewidth}\n}\n\\caption{\\label{spix} \nBroad band spectral indices vs. extended radio luminosity \nfor core galaxies (filled\ncircles), LLRG (empty circles), \nLBL (stars) and HBL (squares). The CoreG luminosity has been extrapolated\nto 178 MHz adopting a spectral index of 0.7.}\n\\end{figure*}\n\nIn Section \\ref{cg-fri} we presented evidence that \n``core'' galaxies and LLRG are drawn from the same\npopulation of early-type galaxies. They can only be\nseparated on the basis of their different level of nuclear activity,\nwith CoreG representing the low luminosity extension of LLRG. \nThe CoreG nuclei appear to be the scaled down versions of those of LLRG\nwhen their multiwavelength nuclear properties are considered. \nThus here we are sampling a new \nregime for radio-galaxies in terms of nuclear power and\nit is important to explore the implications of this result \nfor the model unifying BL Lac objects and radiogalaxies.\n\nUnification models ascribe the differences between the observed\nproperties of different classes of AGN to the anisotropy of the\nnuclear radiation \n\\citep[see e.g.][ for reviews]{antonucci93,urry95}.\nIn particular, for low\nluminosity radio-loud objects, it is believed \nthat BL Lac objects are the pole-on counterparts of radio-galaxies,\ni.e. their emission is dominated by the radiation \nfrom the inner regions of a relativistic jet seen \nat a small angle from its axis which is thus strongly\namplified by relativistic Doppler beaming. \nIn FR~I, whose jets are\nobserved at larger angles with respect to the line of sight,\nthe nuclear component is strongly de-amplified.  \nContrary to other classes of AGN there is growing evidence that\nobscuration does not play a significant role in\nthese objects \\citep{henkel98,chiaberge:ccc,donato04,balmaverde05}.\n\n\\citet{balmaverde05} found that there is a close similarity of the\nbroad band spectral indices between LLRG and the \nsub-class of the BL~Lacs, the Low energy peaked BL~Lac \n\\citep[LBL, ][]{padovani95}, in agreement with the\nunified model\\footnote{The small offsets between the two classes\ncan be quantitatively \naccounted for by the effects of beaming since\nDoppler beaming not only affects the angular pattern\nof the jet emission, but it also causes a shift in frequency of the\nspectral energy distribution \\citep[see ][]{marco3,trussoni03}}.\nWe performed the same comparison \n\\footnote{We used the standard definition of spectral indices,\nmeasured between 5 GHz, 5500 \\AA\\ and 1 keV. Optical fluxes have been\nconverted from 8140 \\AA\\ to 5500 \\AA\\ using a local slope of $\\alpha =\n1$; 1 keV fluxes are\ndirectly derived from the spectral fit.} including CoreG,\nsee Fig. \\ref{roox}. \nWe considered the radio selected BL Lacs sample\nderived from the 1Jy catalog \\citep{stickel91}\nand the BL Lac sample selected from the\n{\\it Einstein} Slew survey \\citep{elvis92,perlman96}. \nWe used the classification into \nHigh and Low energy peaked BL Lacs (HBL and LBL respectively), as well\nas their multiwavelength data \ngiven by \\citet{fossati98}. We also report the regions \n(solid lines) of the plane within 2\n$\\sigma$ from the mean $\\alpha_{ro}$ and $\\alpha_{ox}$ for the BL Lacs \ndrawn from the Deep X-Ray Radio Blazar Survey (DXRBS) \nand the ROSAT All-Sky Survey-Green Bank Survey (RGB) \\citep{padovani03}.\n\nCore galaxies are found to be located in  the same region covered by  LLRG. \nThis   is  not  surprising  since  they extend\nthe behaviour of LLRG in  the radio\/optical and radio\/X-ray planes,\nfollowing Log-Log linear correlations whose slope is close to\nunity, implying only a small dependence of spectral indices on\nluminosity. More importantly, they populate the same area\nin which LBL are found.\n\nWe also compared \nthe spectral indices of the different groups taking into\naccount the extended radio-luminosity L$_{ext}$\n(see Fig. \\ref{spix}) which does not depend on orientation. This enables us \nto properly relate objects from the same region of the luminosity\nfunction of the parent population. \nIndeed, the strongest evidence in favour of \nthe FR~I\/BL~Lac unifying model comes from the similarity in\nthe power and morphology of the\nextended radio emission of BL Lacs and FR~I \n\\citep[see e.g.][]{antonucci85,kollgaard92,murphy93}.\n\nThe CoreG reach radio-luminosities $\\sim$ 100 smaller \nthan in LLRG and the 1 Jy LBL.\nIn addition, in 13 CoreG the available radio-maps do not allow us to separate\ncore and extended radio-emission and $L_{ext}$ must be considered as\nan upper limit.\nThis suggests that the CoreG represent the counterparts of \nthe large low luminosity population of BL~Lac of LBL type which is\nnow emerging from the low radio flux limit surveys such as the DXRBS \n\\citep{landt01}. Clearly, this still requires measurements of the\nextended radio-luminosity of these low power BL~Lac.\nA ramification of this possible extension of the unified model\ntoward lower luminosities \nwould be the presence of relativistic jets also in our \nsample of quasi-quiescent early-type galaxies, as this is a prerequisite \nto produce a substantial dependence of the luminosity on the viewing angle.\n\nWe did not find any CoreG with spectral properties similar\nto those of the High energy peaked BL~Lac (HBL), even though HBL have\nextended radio-emission values of $L_{ext}$ similar to CoreG.\nThe spectral indices of CoreG imply \na difference  in both the radio-to-optical and\nradio-to-X-ray flux ratios of an average factor \nof  $\\sim$100 with respect to HBL.\nThe same result applies to LLRG, as all have a LBL-type SED, with the\nonly exception of 3C~264 \\citep{fr1sed}.\nOur optical selection criteria did not exclude the  parent population of  HBL \nsince their host galaxies are early-type  sufficiently luminous \n\\citep[$M_R < -22.5$,][]{scarpa00} to be included  in our\nsample. Most likely, the dearth of HBL-like CoreG is induced by the radio\nthreshold. Purely radio selected\nsamples of BL~Lacs are known to strongly favour the inclusion of LBL; \ne.g. in the 1 Jy sample\nthere are only 2 HBL out of 34 objects \\citep{giommi94}. \n\n\n\\section{Summary and conclusions}\n\nThe aim of this series of papers is to explore\nthe classical issue of the connection between host galaxies and AGN,\nin the new light shed by the recent developments \nin our understanding of the nuclear regions of\nnearby galaxies. \n\nWe thus selected a samples of nearby early-type \ngalaxies comprising 332 objects. We performed an initial\nselection of AGN candidates requiring a radio detection \nabove $\\sim$1 mJy leading to a sub-sample of 112 sources.\nArchival HST images enabled us to classify 51 of them\ninto core and power-law galaxies on the basis\nof their nuclear brightness profile.\nWe here focused on the 29 core galaxies.\n\nWe used HST and Chandra archival data to isolate their nuclear\nemission in the optical and X-ray bands, \nthus enabling us (once\ncombined with the radio data) to study the\nmultiwavelength behaviour of their nuclei.\nThe detection rate of nuclear sources is 18\/29 in\nthe optical (62 \\%, increasing to 72\\% if the sources affected by large\nscale dust are not considered) and 14 in the X-ray, out \nof the 21 objects with available Chandra data (67 \\%).\nOur selection criteria required a radio detection \nin order to select AGN candidates;\n26 CoreG are confirmed as genuine active galaxies \nbased on the presence of i) an optical (or X-ray) core, ii) a AGN-like\noptical spectrum, or iii) radio-jets, with only 3 exceptions,\nnamely UGC~968, UGC~7898 and NGC~3268.\n\nThe most important result of this analysis is that\n``core'' galaxies invariably host a radio-loud nucleus.\nThe radio-loudness parameter $R$ for the nuclei\nin these sources is on average Log R $\\sim$ 3.6, a factor \nof 400 above the classical threshold between radio-loud and\nradio-quiet nuclei. The X-ray data provide a completely independent\nview of their multiwavelength behaviour leading to the same result,\ni.e. a large X-ray deficit, at the same radio luminosity, \nwhen compared to radio-quiet nuclei.\n\nConsidering the multiwavelength nuclear diagnostic planes, \nwe found that optical and X-ray nuclear luminosities are \ncorrelated with the\nradio-core power, reminiscent of the behaviour\nof low luminosity radio-galaxies. The inclusion of CoreG\nindeed extends the correlations reported for LLRG toward much lower\npower, by a factor of $\\sim 1000$.\n\nThe available radio maps show that in 17 CoreG\nthe extended radio morphology is clearly indicative of a collimated outflow,\nin the form of either double-lobed structures or jets, \nalthough in several cases this can only be seen in high\nresolution VLBI images. This finding, combined with the \nanalogy of the nuclear properties, leads us to the conclusion that\nminiature radio-galaxies are associated with all core galaxies\nof our sample. \n\nThe similarity between CoreG and classical low luminosity\nradio-galaxies extends to other properties. Recent results show that\nLLRG are always hosted by early-type galaxies with a shallow cusp in their\nnuclear profile, and this is the case, by definition, for our CoreG.\nWhile the distributions of black hole masses, $M_{BH}$,\nof the two classes are indistinguishable,\nhosts of 3C\/FR~I radio-sources are on average slightly more \nluminous than CoreG but there is\na substantial overlap between the two groups.\nCoreG and LLRG also share similar properties from the point of view of\ntheir emission lines, as all sources with available data conform\nto the definition of a LINER on the basis of the optical line ratios\nand they follow a common dependence of line luminosity\nwith radio core power.\nCoreG and LLRG thus appear to be drawn\nfrom the same population of early-type ``core'' galaxies. They host\nactive nuclei with the same multiwavelength characteristics\ndespite covering a range of 6 orders of magnitude in\nluminosity. Thus LLRG represent the tip of the iceberg of (relatively) \nhigh luminosity objects.\n\nIt is unclear what mechanism is driving the level of nuclear\nactivity. As noted above, there is a marginal difference (less than 1\nmag) in the host\ngalaxies of CoreG and LLRG; this reflects the well known (but as yet\nunexplained) trend for which \na brighter galaxy has a higher probability of being a stronger\nradio emitter. As described in \\citetalias{capetti05},\nthis effect is present also within our sample of CoreG\nbut it cannot\nbe simply described as a correlation between  L$_r$ and  M$_K$.\n\nWe explored if there is a relationship between the black\nhole mass and the radio-luminosity.\nAgain, a  very large  range of\nradio-power corresponds  to a  given $M_{BH}$. We do not  find any\nrelationship   between  radio-power  and   black  hole   mass, clearly\nindicating  that parameters  other than  the  black hole  mass play  a\nfundamental role in determining the  radio luminosity of a galaxy.  No\nsources with $M_{BH} < 10^8  M_{\\sun}$ are found.  However, this might\nbe due to a  bias induced by the  sample's selection criteria. \nThe limit in optical  magnitude translates  into a\nthreshold of  accessible black  hole masses.  \nOnly by extending  this study to a sample of less\nluminous  galaxies (harbouring,  on average,  smaller black  holes) \nwill it be possible  to test the reality of a minimum  black hole mass to\nproduce a radio-loud nucleus.\n\nOur data can also be used to set constraints \non the radiative manifestation of the accretion process.\nThe nuclear luminosities of CoreG correspond, in units of the\nEddington luminosity, to the range $L\/L_{\\rm {Edd}} \\sim 10^{-6} -\n10^{-9}$ in both the optical and X-ray bands. \nIn analogy with the scenario\nproposed for LLRG, the available data support a common\njet origin for the nuclear emission in these observing bands also for CoreG.\nThus, the above values should be considered as upper limits to\nthe radiative manifestation of the accretion process, suggesting\nthat accretion occurs both at a low accretion level and at a low efficiency.\nIt is difficult to derive from these results clear \nconstraints on the properties of the accretion flow. In part this is due to\nthe limited information on the Spectral Energy Distribution of the CoreG\nnuclei and by the fact that in these\nradio-loud nuclei the observed emission is most likely dominated by \nradiation from their jets rather than from the accretion.\nThis is further complicated by \nthe presence of several competing accretion models\nwhose predictions of the emitted spectra depend on parameters that\nare not well constrained by the observations.\nNonetheless, in the galaxies with the least luminous nuclei, \nthe estimates of the accretion rate  from the literature\n(derived for the case of spherical accretion), combined with\nthe very low level of X-ray emission, suggest that \nan important role is played by outflows (or \nby convection) in order to substantially suppress\nthe amount of gas actually reaching the central object.\n\nAs reported above, the CoreG can be effectively considered\nas miniature radio-galaxies, in terms of nuclear luminosity,\nthus we are sampling a new \nregion in terms of luminosity for radio-loud AGN.\nIt is interesting to explore the implications of this result \nalso for the model unifying BL Lac objects and radio-galaxies.\nThe broad band spectral indices of CoreG present a very close similarity\nto those of Low Energy Peaked BL Lac, suggesting the extension\nof the unified models to these lower luminosities. The CoreG might \nrepresent the mis-aligned counterpart of the large population of low\nluminosity BL Lac emerging from the recent\nsurveys at low radio flux limits. Clearly, a more detailed comparison,\ntaking into account e.g. the (as yet not available) information on the\nextended radio power and morphology, is needed before this result\ncan be confirmed. An important ramification \nof this possible extension of the unifying model\ntoward lower luminosities would be the presence \nof relativistic jets, \nthe essential ingredient of this model, \nalso in our quasi-quiescent early-type galaxies.\n\nIn the third paper of the series we will explore the properties\nof the AGN hosted by galaxies with a power-law brightness profile.\n\n\\begin{acknowledgements}\nThis work was partly supported by the Italian MIUR under \ngrant Cofin 2003\/2003027534\\_002.\nThis research has made use of the NASA\/IPAC Extragalactic Database (NED)\n(which is operated by the Jet Propulsion Laboratory, California Institute of\nTechnology, under contract with the National Aeronautics and Space\nAdministration), of the NASA\/ IPAC Infrared Science Archive\n(which is operated by the Jet Propulsion Laboratory, California\nInstitute of Technology, under contract with the National Aeronautics\nand Space Administration) and of the LEDA database.\n\\end{acknowledgements}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn two dimensions, the configuration space\nof $n$ point-like particles ${\\cal C}_n^{2D}$\nis multiply-connected. Its first homotopy\ngroup, or {\\it fundamental group},\nis the $n$-particle braid group,\n${\\pi_1}({\\cal C}_n^{2D})={\\cal B}_n$.\nThe braid group ${\\cal B}_n$ is generated\nby counter-clockwise exchanges $\\sigma_i$\nof the $i^\\text{th}$ and $(i+1)^\\text{th}$ particles\nsatisfying the defining relations:\n\\begin{eqnarray}\n{\\sigma_i} {\\sigma_j} &=& {\\sigma_j} {\\sigma_i} \\hskip 0.5 cm\n\\mbox{for } |i-j|\\geq 2\\cr {\\sigma_i} \\sigma_{i+1} {\\sigma_i} &=&\n\\sigma_{i+1} {\\sigma_i}\\, \\sigma_{i+1} \\hskip 0.5 cm \\mbox{for }\n1\\leq i \\leq n-2\n\\label{eq:braidrelation1}\n\\end{eqnarray}\nThis is an infinite group, even for only two particles,\nsince $(\\sigma_i)^m$ is a non-trivial element of the\ngroup for any $m>0$. In fact, even if we consider\ndistinguishable particles, the resulting group,\ncalled the `pure Braid group' is non-trivial.\n(For two particles, the pure braid group\nconsists of all even powers\nof $\\sigma_1$.)\n\nIn quantum mechanics, the equation\n${\\pi_1}({\\cal C}_n^{2D})={\\cal B}_n$\nopens the door to the possibility of anyons\\cite{Leinaas77,Wilczek82a}.\nHigher-dimensional representations\nof the braid group give rise to\nnon-Abelian anyons \\cite{Bais80,Goldin85,Frohlich90}.\nThere has recently been intense effort directed towards\nobserving non-Abelian anyons due, in part,\nto their potential use for fault-tolerant quantum\ncomputation \\cite{Kitaev97,Nayak08}.\nOne of the simplest models of non-Abelian\nanyons is called {\\it Ising anyons}. They arise\nin theoretical models of the $\\nu=5\/2$ fractional quantum\nHall state \\cite{Moore91,Nayak96c,LeeSS07,Levin07}\n(see also Ref. \\onlinecite{Bonderson08}),\nchiral $p$-wave superconductors \\cite{Read00,Ivanov01},\na solvable model of spins on the honeycomb lattice\n\\cite{Kitaev06a}, and interfaces between\nsuperconductors and either 3D topological\ninsulators \\cite{Fu08} or spin-polarized\nsemiconductors with strong spin-obrit coupling \\cite{Sau09}.\nA special feature of Ising anyons, which makes\nthem relatively simple and connects them to BCS\nsuperconductivity, is that they can be understood\nin a free fermion picture.\n\nA collection of $2n$ Ising anyons has a $2^{n-1}$-dimensional\nHilbert space (assuming fixed boundary condition).\nThis can be understood in terms of $2n$ Majorana fermion\noperators ${\\gamma^{}_i}={\\gamma_i^\\dagger}$,\n$i=1,2,\\ldots,n$, one associated to\neach Ising anyon, satisfying the anticommutation rules\n\\begin{equation}\n\\label{eqn:clifford}\n\\{{\\gamma^{}_i},{\\gamma^{}_j}\\}=2\\delta_{ij}\\,.\n\\end{equation}\nThe Hilbert space of $2n$ Ising anyons with fixed boundary condition furnishes a representation\nof this Clifford algebra; by restricting to fixed boundary condition, we obtain\na representation\nof products of an even number of $\\gamma$ matrices, which has minimal dimension $2^{n-1}$.\nWhen the $i^\\text{th}$ and $(i+1)^\\text{th}$ anyons\nare exchanged in a counter-clockwise manner,\na state of the system is transformed according to the action of\n\\begin{equation}\n\\label{eqn:Ising-braid}\n\\rho({\\sigma_i})=e^{i\\pi\/8}\\,e^{-\\pi{\\gamma^{}_i}\\gamma^{}_{i+1}\/4}\\,.\n\\end{equation}\n(There is a variant of Ising anyons, associated with\nSU(2)$_2$ Chern-Simons theory,\nfor which the phase factor $e^{i\\pi\/8}$\nis replaced by $e^{-i\\pi\/8}$. In the fractional quantum\nHall effect, Ising anyons are tensored with\nAbelian anyons to form more complicated models\nwith more particle species; the phase factor depends\non the model.) A key property, essential for applications\nto quantum computing, is that {\\it a pair} of Ising anyons\nforms a two-state system. The two states\ncorrespond to the two eigenvalues $\\pm 1$\nof ${\\gamma^{}_i}\\gamma^{}_{j}$. No local degree of freedom\ncan be associated with each anyon; if we insisted on doing so,\nwe would have to say that each Ising anyon has $\\sqrt{2}$\ninternal states. In superconducting contexts,\nthe ${\\gamma^{}_i}$s are the Bogoliubov-de Gennes operators\nfor zero-energy modes (or, simply, `zero modes')\nin vortex cores; the vortices\nthemselves are Ising anyons if they possess a single\nsuch zero mode ${\\gamma^{}_i}$.\nAlthough the Hilbert\nspace is non-local in the sense that it cannot be decomposed\ninto the tensor product of local Hilbert spaces associated\nwith each anyon, the system is perfectly compatible\nwith locality and arises in local lattice models and\nquantum field theories.\n\nIn three or more dimensions,\nthe configuration space of $n$ point-like particles\nis simply-connected if the particles are distinguishable.\nIf the particles are indistinguishable, it\nis multiply-connected,\n${\\pi_1}({\\cal C}_n^{3D})=S_n$.\nThe generators of the permutation group\nsatisfy the relations (\\ref{eq:braidrelation1})\nand one more, ${\\sigma_i^2}=1$. As a result\nof this last relation, the permutation group\nis finite. The one-dimensional representations\nof $S_n$ correspond to bosons and fermions.\nOne might have hoped that higher-dimensional\nrepresentations of $S_n$ would give rise to\ninteresting 3D analogues of non-Abelian anyons.\nHowever, this is not the case, as shown in\nRef. \\onlinecite{Doplicher71a,Doplicher71b}: any higher-dimensional\nrepresentation of $S_n$ which is compatible with\nlocality can be decomposed into the tensor product\nof local Hilbert spaces associated\nwith each particle. For instance, suppose we\nhad $2n$ spin-$1\/2$ particles but ignored\ntheir spin values. Then we would have $2^{2n}$\nstates which would transform into each other\nunder permutations. Clearly, if we discovered such a system,\nwe would simply conclude that we were missing\nsome quantum number and set about trying to\nmeasure it. This would simply lead us back\nto bosons and fermions with additional\nquantum numbers. (The color quantum number of quarks\nwas conjectured by essentially this kind of reasoning.)\nThe quantum information contained in these\n$2^{2n}$ states would not have any special protection.\n\nThe preceding considerations point to\nthe following tension. The Clifford algebra\n(\\ref{eqn:clifford}) of Majorana fermion zero modes\nis not special to two dimensions. One could imagine\na three (or higher) dimensional system with topological defects supporting such zero modes. But the Hilbert space of these\ntopological defects would be $2^{n-1}$-dimensional, which\nmanifestly cannot be decomposed into the tensor product\nof local Hilbert spaces associated\nwith each particle, seemingly in contradiction with the results of \nRefs. \\onlinecite{Doplicher71a,Doplicher71b} on\nhigher-dimensional representations of the permutation group\ndescribed above. However, as long as\nno one had a three or higher dimensional system\nin hand with topological defects supporting\nMajorana fermion zero modes,\none could, perhaps, sweep this worry under the rug.\nRecently, however, Teo and Kane \\cite{Teo10}\nhave shown that a 3D system which is simultaneously\na superconductor and a topological insulator \\cite{Moore07,Fu07,Roy09,Qi08}\n(which, in many but not all examples, is\narranged by forming superconductor-topological insulator\nheterostructures) supports Majorana zero modes at\npoint-like topological defects.\n\nTo make matters worse, Teo and Kane \\cite{Teo10}\nfurther showed that exchanging these\ndefects enacts unitary operations on this\n$2^{n-1}$-dimensional Hilbert space which are\nessentially equal to (\\ref{eqn:Ising-braid}).\nBut we know that these unitary matrices\nform a representation of the braid group,\nwhich is not the relevant group in 3D.\nOne would naively expect that the relevant group is\nthe permutation group, but $S_n$ has no such\nrepresentation (and even if it did, its use in this\ncontext would contradict locality, according to Ref. \\onlinecite{Doplicher71a,Doplicher71b}\nand arguments in Ref. \\onlinecite{Read03}).\nSo this begs the question: what is the group ${\\cal T}_{2n}$\nfor which Teo and Kane's unitary\ntransformations form a representation?\n\nWith the answer to this question in hand,\nwe could address questions such as the following.\nWe know that a 3D incarnation of Ising anyons\nis one possible representation of ${\\cal T}_{2n}$;\nis a 3D version of other anyons another\nrepresentation of ${\\cal T}_{2n}$?\n\nAttempts to generalize the braiding of anyons\nto higher dimensions sometimes start with\nextended objects, whose configuration space\nmay have fundamental group which is\nricher than the permutation group.\nObviously, if one has line-like defects\nin 3D which are all oriented in the same direction,\nthen one is essentially back to the 2D situation\ngoverned by the braid group. This is too\ntrivial, but it is not clear what kind of extended\nobjects in higher dimensions would be the best starting point.\nWhat is clear, however, is that Teo and Kane's topological\ndefects must really be some sort of extended objects.\nThis is clear from the above-noted contradiction\nwith the permutation group. It also follows from\nthe `order parameter' fields which must deform\nas the defects are moved, as we will discuss.\n\nIn this paper, we show that Teo and Kane's\ndefects are properly viewed as point-like\ndefects connected pair-wise by ribbons.\nWe call the resulting $2n$-particle configuration\nspace $K_{2n}$. We compute\nits fundamental group ${\\pi_1}(K_{2n})$, which we denote by\n${\\cal T}_{2n}$ and find that\n${\\cal T}_{2n}={\\mathbb Z} \\times {\\cal T}^r_{2n}$.\nHere, ${\\cal T}^r_{2n}$ is the `ribbon permutation group',\ndefined by ${\\cal T}^r_{2n} \\equiv {\\mathbb Z}_2\n\\times E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$.\nThe group $E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$\nis a non-split extension of the permutation group $S_{2n}$ by ${\\mathbb Z}_2^{2n-1}$\nwhich is defined as follows: it is the\nsubgroup of $(\\mathbb{Z}_2)^{2n} \\rtimes S_{2n}$\ncomposed of those elements for which the total parity of\nthe element in $(\\mathbb{Z}_2)^{2n}$ added to\nthe parity of the permutation is even.\nThe `ribbon permutation group' for $2n$ particles,\n by ${\\cal T}^r_{2n}$, is the fundamental group of the\n reduced space of $2n$-particle configurations.\n\nOur analysis relies on the topological\nclassification of gapped free fermion Hamiltonians\n\\cite{Ryu08,Kitaev09} -- band insulators\nand superconductors -- which\nis the setting in which Teo and Kane's 3D defects\nand their motions are defined.\nThe starting point for this classification is reducing the problem\nfrom classifying gapped Hamiltonians defined on a lattice to\nclassifying Dirac equations with a spatially varying mass term.\nOne can motivate the reduction to a Dirac equation as\nTeo and Kane do: they\nstart from a lattice Hamiltonian and assume\nthat the parameters in the Hamiltonian vary smoothly in space, so that\nthe Hamiltonian can be described as a function of both the momentum $k$\nand the position $r$.  Near the minimum of the band gap, the\nHamiltonian can be expanded in a Dirac equation, with a position-dependent\nmass term.  In fact, Kitaev\\cite{Kitaev09} has shown that the reduction\nto the Dirac equation with a spatially varying mass term can be derived\nmuch more generally:\ngapped lattice Hamiltonians, even if the parameters\nin the Hamiltonian do not vary smoothly in space, are {\\it stably equivalent}\nto Dirac Hamiltonians with a spatially varying mass term.  Here, equivalence\nof two Hamiltonians means that one can be smoothly deformed into the other\nwhile preserving locality of interactions and the spectral gap, while\nstable equivalence means that one can add additional ``trivial\" degrees of freedom (additional\nsites which have vanishing hopping matrix elements) to the original lattice\nHamiltonian to obtain a Hamiltonian which is equivalent to\na lattice discretization of the Dirac Hamiltonian.\n\nSince this classification of Dirac Hamiltonians\nis essential for the definition of $K_{2n}$, we give\na self-contained review, following Kitaev's\nanalysis \\cite{Kitaev09}. Our exposition parallels\nthe discussion of Bott periodicity in Milnor's book\n\\cite{Milnor63}. The basic idea is that each additional\ndiscrete symmetry which squares to $-1$ which we\nimpose on the system is encapsulated by an\nanti-symmetric matrix which defines a complex structure\non $\\mathbb{R}^N$, where $N\/2$ is the number of\nbands (or, equivalently, $N$ is the number of bands\nof Majorana fermions). For any given system,\nthese are chosen and fixed. This leads to\na progression of symmetric spaces $\\text{O}(N)\\rightarrow\n\\text{O}(N)\/\\text{U}(N\/2) \\rightarrow \\text{U}(N\/2)\/\\text{Sp}(N\/4)\n\\rightarrow \\ldots$ as the number of such symmetries is increased.\nFollowing Kitaev \\cite{Kitaev09}, we view the Hamiltonian\nas a final anti-symmetric matrix which must be chosen (and, thus,\nput almost on the same footing as the symmetries); it is defined by a choice\nof an arbitrary point in the next symmetric space in the progression.\nThe space of such Hamiltonians is topologically-equivalent\nto that symmetric space.\nHowever, as the spatial dimension is increased, $\\gamma$-matrices\nsquaring to $+1$ must be chosen in order to expand\nthe Hamiltonian in the form of the Dirac equation\nin the vicinity of a minimum of the band gap. These halve the dimension\nof subspaces of $\\mathbb{R}^N$ by separating it\ninto their $+1$ and $-1$ eigenspaces and thereby\nlead to the opposite progression of symmetric spaces. Thus,\ntaking into account both the symmetries of the system and\nthe spatial dimension, we conclude that the space of gapped\nHamiltonians with no symmetries in $d=3$ is topologically\nequivalent to $\\text{U}(N)\/\\text{O}(N)$. (However, by the preceding\nconsiderations, the same symmetric space also, for instance, classifies\nsystems with time-reversal symmetry in $d=4$.)\nAll such Hamiltonians can be continuously deformed into each\nother without closing the gap, $\\pi_{0}(\\text{U}(N)\/\\text{O}(N))=0$.\nHowever, there are topologically-stable point-like defects\nclassified by $\\pi_{2}(\\text{U}(N)\/\\text{O}(N))=\\mathbb{Z}_2$.\nThese are the defects whose multi-defect configuration space\nwe study in order to see what happens when they are exchanged.\n\nThe second key ingredient in our analysis\nis 1950's-vintage homotopy theory, which we use to compute\n${\\pi_1}(K_{2n})$.  We apply the Pontryagin-Thom\nconstruction to show that $K_{2n}$, which\nincludes not only the particle locations but also\nthe full field configuration  around the particles\n(i.e. the way in which the gapped free fermion\nHamiltonian of the system explores $\\text{U}(N)\/\\text{O}(N)$),\nis topologically-equivalent to a much simpler\nspace, namely point-like defects connected\npair-wise by ribbons. In order to then\ncalculate ${\\pi_1}(K_{2n})$, we rely on the long\nexact sequence of homotopy groups\n\\begin{equation}\n\\label{eqn:long-exact-sequence}\n\\ldots \\rightarrow \\pi_{i}(E)\\rightarrow\n{\\pi_i}(B)\\rightarrow\\pi_{i-1}(F)\\rightarrow\\pi_{i-1}(E)\n\\rightarrow ...\n\\end{equation}\nassociated to a fibration defined by\n$F \\rightarrow E\\rightarrow B$.\n(In an exact sequence, the kernel of each map\nis equal to the image of the previous map.)\nThis exact sequence may be familiar to some readers\nfrom Mermin's review of the topological theory of\ndefects \\cite{Mermin79}, where a symmetry associated\nwith the group $G$ is spontaneously broken to $H$,\nthereby leading to topological\ndefects classified by homotopy groups ${\\pi_n}(G\/H)$.\nThese can be computed by (\\ref{eqn:long-exact-sequence})\nwith $E=G$, $F=H$, $B=G\/H$, e.g.\nif $\\pi_{1}(G)=\\pi_{0}(G)=0$,\nthen $\\pi_{1}(G\/H)=\\pi_{0}(H)$.\n\nThe ribbon permutation group is a rather weak enhancement\nof the permutation group and, indeed, we conclude\nthat Teo and Kane's unitary operations are\n{\\it not} a representation of the ribbon permutation\ngroup. However, they are a {\\it projective} representation\nof the ribbon permutation group. In a\n{\\it projective} representation, the group\nmultiplication rule is only respected up to\na phase, a possibility allowed in quantum mechanics.\nA representation $\\rho$ (sometimes called a linear\nrepresentation) of some group $G$ is\na map from the group to the group of linear transformations\nof some vector space such that\nthe group multiplication law is reproduced:\n\\begin{equation}\n\\rho(gh)=\\rho(g)\\cdot\\rho(h)\n\\end{equation}\nif $g,h\\in G$. Particle statistics arising as a projective\nrepresentation of some group\nrealizes a proposal of Wilczek's \\cite{Wilczek98},\nalbeit for the ribbon permutation group rather than\nthe permutation group itself. This difference\nallows us to sidestep a criticism of Read \\cite{Read03}\nbased on locality, which Teo and Kane's\nprojective representation respects.\nThe group $(\\mathbb{Z}_2)^{2n-1}$ is generated\nby $2n-1$ generators ${x_1}$, ${x_2}$,\n\\ldots, $x_{2n-1}$ satisfying\n\\begin{eqnarray}\n{x_i^2} &=& 1\\cr\n{x_i} {x_j} &=& {x_j} {x_i}\n\\label{eq:Z_2-def}\n\\end{eqnarray}\nHowever, the projective representation of\n$(\\mathbb{Z}_2)^{2n-1}$, which gives a subgroup of\nTeo and Kane's transformations, is an ordinary\nlinear representation of a ${\\mathbb Z}_2$-central extension,\ncalled the extra special group $E^1_{2n-1}$:\n\\begin{eqnarray}\n{x_i^2} &=& 1\\cr\n{x_i} {x_j} &=& {x_j} {x_i} \\hskip 0.5 cm\n\\mbox{for } |i-j|\\geq 2\\cr\n{x_i} x_{i+1} &=& z \\,x_{i+1} {x_i}\\cr\n{z^2}&=&1\n\\label{eq:extra-special}\n\\end{eqnarray}\nHere, $z$ generates the central extension, which we may\ntake to be $z=-1$. The operations generated\nby the ${x_i}$s were dubbed `braidless operations'\nby Teo and Kane \\cite{Teo10} because they could\nbe enacted without moving the defects. While these\noperations form an Abelian subgroup of ${\\cal T}_{2n}$,\ntheir representation on the Majorana zero mode\nHilbert space is {\\it not} Abelian -- two such operations\nwhich twist the same defect {\\it anti-commute} (e.g. $x_i$ and\n$x_{i+1}$).\n\nThe remaining sections of this paper will\nbe as follows. In Section \\ref{sec:strong-coupling},\nwe rederive Teo and Kane's zero modes and unitary transformations\nby simple pictorial and counting arguments in a `strong-coupling'\nlimit of their model. In Section \\ref{sec:free-fermion},\nwe review the topological classification of free-fermion\nHamiltonians, including topological insulators and\nsuperconductors. From this classification, we obtain\nthe classifying space relevant to Teo and Kane's model\nand, in turn, the topological classification of\ndefects and their configuration space.\nIn Section \\ref{sec:tethered}, we use a toy\nmodel to motivate a simple picture for the defects\nused by Teo and Kane and give a heuristic\nconstruction of the ribbon permutation group.\nIn Section \\ref{sec:Kane_space}, we give a full\nhomotopy theory calculation.\nIn Section \\ref{sec:projective}, we compare\nthe ribbon permutation group to\nTeo and Kane's unitary transformations and conclude that\nthe latter form a projective, rather than a linear,\nrepresentation of the former. Finally, in Section\n\\ref{sec:discussion}, we review and discuss our results.\nSeveral appendices contain technical details.\n\n\n\n\n\\section{Strong-coupling limit of the Teo-Kane Model}\n\\label{sec:strong-coupling}\n\nIn this section, we present a lattice model\nin $d$ dimensions which has,\nas its continuum limit in $d=3$, the model discussed by\nTeo and Kane \\cite{Teo10}. In the limit that the mass terms\nin this model are large (which can be viewed as\na `strong-coupling' limit), a simple picture of\ntopological defects (`hedgehogs') emerges.\nWe show by a counting argument that hedgehogs\npossess Majorana zero modes which evolve as the\nhedgehogs are adiabatically moved. This adiabatic evolution\nis the 3D non-Abelian statistics which it is the main purpose\nof this paper to explain.\n\nThe strong coupling limit which we describe is the\nsimplest way to derive the existence of Majorana zero\nmodes and the unitary transformations of their Hilbert space\nwhich results from exchanging them. This section does\nnot require the reader to be {\\it au courant} with the\ntopological classification of insulators and superconductors\n\\cite{Ryu08,Kitaev09}. (In the next section, we will\nreview that classification in order to make our exposition\nself-contained.)\n\nWe use a hypercubic lattice in $d$-dimensions, with a single Majorana degree of freedom at each site.\nThat is,\nfor $d=1$, we use a chain, in $d=2$ we use a square lattice,\nin $d=3$ we use\na cubic lattice, and so on.\nWe first construct a lattice model whose continuum limit is the Dirac equation with $2^d$-dimensional\n$\\gamma$-matrices to reproduce the Dirac equation considered by Teo\nand Kane; we then show how to perturb this model to open a mass gap.\nWe begin by considering only nearest neighbor couplings.  The Hamiltonian $H$ is an anti-symmetric\nHermitian matrix.  In $d=1$, we can take the linear chain to give a lattice model with the Dirac equation as its continuum\nlimit.  That is, $H_{j,j+1}=i$ and $H_{j+1,j}=-i$.  To describe this state in\npictures, we draw these bonds as oriented lines,\nas shown in Fig.~(\\ref{figDirac}a), with the orientation indicating the\nsign of the bond.  The continuum limit of this Hamiltonian is described by a Dirac equation with $2$-dimensional $\\gamma$ matrices.\nWhile this system can be described by a unit cell of a single site, we instead choose to describe it by a unit\ncell of two sites for convenience when adding mass terms later.\nIn $d=2$, we can take a $\\pi$-flux\nstate to obtain the Dirac equation in the continuum limit.\nA convenient gauge to take to describe the $\\pi$-flux state is shown in Fig.~(\\ref{figDirac}b), with all the vertical bonds\nhaving the same orientation, and the orientation of the horizontal bonds alternating from row to row.  The continuum limit\nhere has $4$-dimensional $\\gamma$ matrices and we use a $4$-site unit cell.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.5in]{dirac.pdf}\n\\caption{(a) A lattice model giving the Dirac equation in $d=1$.\n(b) A lattice model in $d=2$.}\n\\label{figDirac}\n\\end{figure}\n\n\nIn general, in $d$ dimensions, we can obtain a Dirac equation with $2^d$-dimensional $\\gamma$ matrices\nby the following iterative\nprocedure.  Let the ``vertical\" direction refer to the direction of the $d$-th basis vector.\nHaving constructed the lattice Hamiltonian in $d-1$ dimensions, we stack these Hamiltonians vertically on top of each other,\nwith alternating signs in each layer.  Then, we take all the vertical bonds to be oriented in the\nsame direction.\nThis Hamiltonian is invariant under translation in the vertical direction by distance $2$.  Thus,\nif $H_{d-1}$ is the Hamiltonian in $d-1$ dimensions, the Hamiltonian $H_d$ is given by\n\\begin{equation}\nH_d=\\begin{pmatrix} H_{d-1} & 2\\sin(k\/2) I  \\\\ 2\\sin(k\/2) I& -H_{d-1} \\end{pmatrix},\n\\end{equation}\nwhere $I$ is the identity matrix and $k$ is the momentum in the vertical direction.\nNear $k=0$, this is\n\\begin{equation}\n\\label{Hdcontinuum}\nH_d \\approx H_{d-1} \\otimes \\sigma_z + k \\otimes \\sigma_x.\n\\end{equation}\n\nThis iterative construction corresponds to an iterative construction of $\\gamma$-matrices.  Having constructed $d-1$\ndifferent $2^{d-1}$-dimensional $\\gamma$-matrices $\\gamma^{}_1,...,\\gamma^{}_{d-1}$, we construct $d$ different $2^d$-dimensional $\\gamma$-matrices,\n$\\tilde \\gamma^{}_1,...,\\tilde \\gamma^{}_{d}$, by $\\tilde \\gamma^{}_i=\\gamma^{}_i \\otimes \\sigma_z$ for $i=1,...,d-1$, and $\\tilde \\gamma^{}_d=I\\otimes \\sigma_x$.\n\nIn one dimension, dimerization of bonds corresponds to alternately strengthening and weakening the bonds as shown in\nFig.~(\\ref{figDimer}).  In two dimensions, we can dimerize in either the horizontal or vertical directions.\nIn $d$-dimensions, we have $d$ different directions to dimerize.  Dimerizing in the ``vertical\" direction gives, instead\nof (\\ref{Hdcontinuum}), the result\n\\begin{equation}\n\\label{eqn:dimerizations}\nH_d \\approx H_{d-1} \\otimes \\sigma_z + k\\otimes \\sigma_x + m_d \\otimes \\sigma_y,\n\\end{equation}\nwhere $m_d$ is the dimerization strength.  This corresponds to an iterative construction of mass matrices, $M_i$, as follows.\nIn one dimension, we have $M_1=i\\sigma_y$.  Given $d-1$ different mass matrices in $d-1$ dimensions, $M_i$, we construct $\\tilde M_i$ in\n$d$-dimensions by $\\tilde M_i=M_i \\otimes \\sigma_z$, for $i=1...d-1$, and $\\tilde M_d=iI\\otimes \\sigma_y$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{dimer.pdf}\n\\caption{Dimerization in $d=1$.}\n\\label{figDimer}\n\\end{figure}\n\nIf the dimerization is non-zero, and constant, we can increase the dimerization strength without closing the gap until a strong\ncoupling limit is reached.  In one dimension, by increasing the dimerization strength, we eventually\nreach a fully dimerized configuration, in which each site has one non-vanishing bond\nconnected to it.\nIn two or more dimensions, the dimerization\ncan be a combination of dimerization in different directions.  However, if the dimerization is completely in one direction, for example\nthe vertical direction, we increase the dimerization strength until the vertical bonds are fully dimerized.  Simultaneously, we reduce the\nstrength of the other bonds to zero without closing the gap.\nThis is again a fully dimerized state, the columnar state, with each site having one non-vanishing bond.\nAny configuration with uniform, small dimerization can be deformed into this pattern without closing the gap by rotating the\ndirection of dimerization, increasing the strength of dimerization, and then setting the bonds in the other directions to zero.\n\nIt is important to understand that the ability to reach such a strong coupling\nlimits depends on the perturbation of the Dirac equation that we consider;\nfor dimerization, it is possible to reach a strong coupling limit, while if\nwe had instead chosen to open a mass gap by adding, for example, diagonal\nbonds with imaginary coupling to the two-dimensional Dirac equations, we\nwould open a mass gap by perturbing the Hamiltonian with the term $i\\gamma_1\\gamma_2$, and such a perturbation cannot be continued to the strong coupling\nlimit due to topological obstruction.\n\nFurther, if the dimerization is non-uniform then it may not be possible to reach a fully dimerized state without\nhaving defect sites.  Consider the configurations in\nFig.~(\\ref{figHedgehog}a) in $d=1$ and in Fig.~(\\ref{figHedgehog}b) in $d=2$.  These are the strong coupling limits of the\nhedgehog configuration,\nand each contains a zero mode, a single unpaired site.\nThis is one of the central results of the strong-coupling\nlimit: {\\it topological defects have unpaired sites which,\nin turn, support Majorana zero modes}.\n\nSuch strong-coupling hedgehog configurations can be constructed by the following iterative process in any dimension $d$.\nLet $x_d$ correspond to the coordinate in the vertical direction.  For $x_d\\geq 0$, stack $d-1$-dimensional hedgehog configurations.\nAlong the half-line given by $x_d>0$ and $x_i=0$ for $1\\leq i \\leq d-1$, arrange vertical bonds, oriented to connect\nthe site with $x_d=2k-1$ to\nthat with $x_d=2k$, for $k\\geq 1$.  Along the lower half plane, given by $x_d<0$, arrange vertical bonds oriented to connect\na site with $x_d=-(2k-1)$ to that with $-2k$, for $k\\geq 1$.  This procedure gives the $d=2$ hedgehog in\nFig.~(\\ref{figHedgehog}b) from the $d=1$ hedgehog in\nFig.~(\\ref{figHedgehog}a), and gives a strong coupling limit of the\nTeo-Kane hedgehog in $d=3$.  That is, the Teo-Kane\nhedgehog can be deformed into this configuration,  without closing the gap.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{hedgehog.pdf}\n\\caption{(a) A one-dimensional hedgehog.\n(b) A two-dimensional hedgehog.}\n\\label{figHedgehog}\n\\end{figure}\n\n\nSo long as we consider only nearest-neighbor bonds,\nthere is an integer index $\\nu$ describing different dimerization\npatterns in the strong-coupling limit.  This index, which\nis present in any dimension, arises from the sublattice symmetry\nof the system, and is closely-related to the U(1) symmetry\nof dimer models of spin systems\\cite{Rokhsar88}.  Label the two\nsublattices by $A$ and $B$.  Consider any set of sites, such that every site in that\nset has exactly one bond connected to it.  (Recall that,\nin the strong coupling\nlimit, every bond has strength $0$ or $1$ and every site\nhas exactly one bond connected to it, except for defect sites.)\nThen, the number of bonds going from $A$ sites in this set to $B$ sites outside the set\nis exactly equal to the number of bonds going from $B$ sites in this set to $A$ sites outside the set.  On the other hand,\nif there are defect sites in the set, then this rule is broken.\nConsider the region defined by the dashed line\nin Fig.~(\\ref{figU1}a).  We define the ``flux\" crossing the dashed line to be the number of bonds crossing that boundary which leave starting on an $A$ site, minus the number\nwhich leave starting on a $B$ site. The flux is non-zero in this case, but is unchanging as we increase the size of the region.\nThis flux is the index $\\nu$. By the argument given above for\nthe existence of zero modes, $\\nu$ computed for any region\nis equal to the number of Majorana zero modes contained\nwithin the region.\n\nThe index $\\nu$ can be defined beyond the strong-coupling\nlimit. Consider, for the sake of concreteness, $d=3$.\nThere are 3 possible dimerizations, one for each\ndimension, as we concluded in Eq. \\ref{eqn:dimerizations}.\nIn weak-coupling, the square of the gap is equal to the sum of\nthe squares of the dimerizations. Thus, if we assume\na fixed gap, we can model these dimerizations by a unit vector.\nThe integer index discussed above is simply the total winding\nnumber of this unit vector on the boundary of any region.\n\nHowever, once diagonal bonds are allowed,\nthe integer index $\\nu$ no longer counts zero modes.\nInstead, there is a $\\mathbb{Z}_2$ index,\nequal to $\\nu(\\text{mod} 2)$ which\ncounts zero modes modulo 2. To see this in\nthe strong-coupling limit,\nconsider the configuration in Fig.~(\\ref{figU1}b).\nThis is a configuration with $\\nu=2$\nbut no Majorana zero modes. However, a $\\nu=1$\nconfiguration must still have a zero mode and, thus,\nany configuration with odd $\\nu$ must have at least one zero mode.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{U1.pdf}\n\\caption{(a) Defect acting as source of $U(1)$-flux.  Bonds are oriented from $A$ to $B$ sublattice.  There is a net\nflux of one leaving the region defined by the dashed line.\n(b) Configuration with diagonal bond added, indicated by the undirected line\nconnecting the two circles; either orientation of this line, corresponding to different choices of the sign of the\nterm in the Hamiltonian, would lead to the same result.  There is a net of flux of two leaving the region\ndefined by the dashed line.}\n\\label{figU1}\n\\end{figure}\n\n\nIn Fig.~(\\ref{figU1}), we have chosen to orient the\nbonds from A to B sublattice\nto make it easier to compute $\\nu$.  However,\nthe $\\nu$ and its residue modulo 2,\ndefined above are independent of the orientation of\nthe bonds (which indicate the sign of terms in the Hamiltonian) and depend only on which sites are connected by bonds (which indicate which terms in the Hamiltonian are non-vanishing).\n\nThe $\\nu(\\text{mod} 2)$ with diagonal bonds\nis the same as Kitaev's ``Majorana number\"\\cite{Kitaev06a}.\nWe can use this to show the existence of\nzero modes in the Teo-Kane hedgehog even outside the\nstrong-coupling limit.\nConsider a hedgehog configuration.\nOutside some large distance $R$ from\nthe center of the hedgehog, deform to the\nstrong coupling limit without closing the gap.  Then, outside a distance $R$, we can count $\\nu(\\text{mod} 2)$ by counting bonds leaving\nthe region and we find a nonvanishing result relative to a reference configuration: if there are an even number of sites in the\nregion then there are an odd number of bonds leaving in a hedgehog configuration, and if there are an odd number of sites then there\nare an even number of bonds leaving.\nHowever, since this implies a nonvanishing\nMajorana number, there must be a zero mode inside the region, regardless of what the Hamiltonian inside is.\nWe note that this is a highly non-trivial result\nin the weak-coupling limit, where\nthe addition of weak diagonal bonds, all oriented\nthe same direction, to the configuration of Fig.~(\\ref{figDirac}b)\ncorresponds to adding the term\n$i\\gamma^{}_1 \\gamma^{}_2$ to the Hamiltonian in $d=2$.\nBy the argument given above, even this Hamiltonian\nhas a zero mode in the presence of a defect\nwith non-zero $\\nu(\\text{mod} 2)$.\n\nGiven any two zero modes, corresponding to defect sites in the strong coupling limit, we can identify a string of sites\nconnecting them.  If we have a pair of defect sites on opposite sublattices, corresponding to opposite hedgehogs, then one particular string\ncorresponds to the north pole of the order parameter, as in Fig.~(\\ref{figstring}a).  However, we\ncan simply choose {\\it any} arbitrary string.\nLet $\\gamma^{}_i,\\gamma^{}_j$ be the Majorana operators at the two defect sites.  The operation $\\gamma^{}_i\\rightarrow -\\gamma^{}_i,\\gamma^{}_j\\rightarrow\n-\\gamma^{}_j$ can be implemented as follows.  We begin with an adiabatic operation on one of the defect sites and the nearest $2$\nsites on the line.  The Hamiltonian on those three sites is an anti-symmetric, Hermitian matrix.  That is, it corresponds to\nan oriented plane in three dimensions.  We can adiabatically perform orthogonal rotations of this plane.  Thus, by rotating by $\\pi$\nin the plane corresponding to the defect site and the first site on the string,\nwe can change the sign of the mode on the defect and the orientation of the\nbond, as shown in Fig.~(\\ref{figstring}b).  This rotation is an adiabatic transformation\nof the three site Hamiltonian\n\\begin{equation}\n\\begin{pmatrix}\n0 & 0 & i\\sin(\\theta) \\\\\n0 & 0 & i\\cos(\\theta) \\\\\n-i\\sin(\\theta) & -i\\cos(\\theta) & 0\n\\end{pmatrix}\n\\end{equation}\nalong the path $\\theta=0\\rightarrow \\pi$.\nWe then perform rotations on consecutive triples of sites along the defect line, which changes\nthe orientation of pairs of neighboring bonds, arriving\nat the configuration in Fig.~(\\ref{figstring}c).  Finally, we rotate by $\\pi$\nin the plane containing the other defect site and the last site.\nThis returns the system to the original configuration,\nhaving effected the desired operation.\n\nSince we only consider adiabatic transformation, we can only perform orthogonal rotations with unit determinant.  Thus, any transformation which swaps two defects and returns the\nbonds to their original configuration, must change the sign of one of\nthe zero modes: $\\gamma^{}_i \\rightarrow \\gamma^{}_j, \\gamma^{}_j \\rightarrow -\\gamma^{}_i$.  Indeed, any orthogonal\ntransformation with determinant equal to minus one would change the sign of the fermion parity in the system, as the\nfermion parity operator is equal to the product of the $\\gamma_i$ operators.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{string.pdf}\n\\caption{(a) Pairs of defects connected by a string.\n(b) First rotation applied to the configuration in (a)  Open circle replaces filled circle to indicate sign change of the Majorana mode on the site.  (c)After rotating along the string. (d) Rotating the last site and restoring the string to its original configuration}\n\\label{figstring}\n\\end{figure}\n\nWe used the ability to change the orientation of a pair of bonds in this\nconstruction.\nThe fact that one can only change the\norientation of bonds in pairs, and not the\norientation of a single bond,\nis related to a global $Z_2$\ninvariant: the Hamiltonian is an anti-symmetric matrix and the sign\nof its Pfaffian cannot be changed without closing the gap.  Changing the direction of a single bond changes the sign of this Pfaffian and so is not possible.\n\nThe above discussion left open the question\nof which zero changes its sign, i.e. is the\neffect of the exchange  $\\gamma^{}_i \\rightarrow \\gamma^{}_j, \\gamma^{}_j \\rightarrow -\\gamma^{}_i$ or  $\\gamma^{}_i \\rightarrow -\\gamma^{}_j, \\gamma^{}_j \\rightarrow \\gamma^{}_i$? The answer is that it depends on how the bonds are returned to their original configuration\nafter the exchange is completed (which is a clue that\nthe defects must be understood as extended objects,\nnot point-like ones). For the bonds to be\nrestored, one of the defects must be rotated by\n$2\\pi$; the corresponding zero mode acquires a minus sign.\nWe will discuss this in greater detail in a later section.\nThe salient point here\nis that the effect of an exchange is a unitary transformation\ngenerated by the operator $e^{\\pm\\pi \\gamma^{}_i \\gamma^{}_j \/4}$.\nThis is reminiscent of the representation of braid group\ngenerators for non-Abelian quasiparticles in the quantum\nHall effect \\cite{Nayak96c} and vortices in chiral\n$p$-wave superconductors \\cite{Ivanov01},\nnamely the braid group representation realized\nby Ising anyons \\cite{Nayak08}.\nBut, of course, in 3D the braid group is not\nrelevant, and the permutation group, which is associated with\npoint-like particles in $d>2$, does not have non-trivial\nhigher-dimensional representations consistent with\nlocality \\cite{Doplicher71a,Doplicher71b}. As noted in the introduction,\nthis begs the question:\nwhat group are the unitary matrices\n$e^{\\pm\\pi \\gamma^{}_i \\gamma^{}_j \/4}$ representing?\n\n\n\n\\section{Topological Classification of Gapped Free\nFermion Hamiltonians}\n\\label{sec:free-fermion}\n\n\\subsection{Setup of the Problem}\n\nIn this section, we will briefly review the topological\nclassification of translationally-invariant or slowly\nspatially-varying free-fermion Hamiltonians following Kitaev's analysis\nin Ref. \\onlinecite{Kitaev09}. (For a different perspective,\nsee Schnyder {\\it et al.}'s approach in Ref. \\onlinecite{Ryu08}.)\nThe 3D Hamiltonian of the previous section is a specific example which\nfits within the general scheme and, by implication,\nthe 3D non-Abelian statistics which we derived at\nthe end of the previous section also holds for an entire\nclass of models into which it can be deformed without\nclosing the gap. Our discussion will\nfollow the logic of Milnor's treatment of Bott periodicity\nin Ref. \\onlinecite{Milnor63}.\n\nConsider a system of $N$ flavors of electrons\n${c_j}({\\bf k})$ in $d$ dimensions. The flavor index\n$j$ accounts for spin as well as the possibility of multiple\nbands. Since we will not be assuming charge conservation,\nit is convenient to express the complex fermion operators\n${c_j}({\\bf k})$ in terms of real fermionic operators\n(Majorana fermions),\n${c_j}({\\bf k})=(a_{2j-1}({\\bf k})+ia_{2j}({\\bf k}))\/2$\n(the index $j$ now runs from $1$ to $2N$).\nThe momentum ${\\bf k}$ takes values in\nthe Brillouin zone, which has the topology of the\n$d$-dimensional torus $T^d$. The Hamiltonian\nmay be written in the form\n\\begin{equation}\n\\label{eqn:basic}\nH = \\sum_{i,j,{\\bf p}} iA_{ij}({\\bf p}){a_i}({\\bf p}){a_j}(-{\\bf p})\n\\end{equation}\nwhere, by Fermi statistics, $A_{ij}({\\bf p})=-A_{ji}(-{\\bf p})$.\nLet us suppose that the Hamiltonian\n(\\ref{eqn:basic}) has an energy gap $2\\Delta$,\nby which we mean that its eigenvalues $E_{\\alpha}(p)$\n($\\alpha$ is an index labeling the eigenvalues of\n$H$) satisfy $|E_{\\alpha}(p)|\\geq \\Delta$.\nThe basic question which we address in this section\nis the following. What topological obstructions\nprevent us from continuously deforming one\nsuch gapped Hamiltonian into another?\n\nSuch an analysis can apply, as we will see,\nnot only to free fermion Hamiltonians, but also\nto those interacting fermion Hamiltonians which,\ndeep within ordered phases, are well-approximated\nby free-fermion Hamiltonians. (This can include\nrather non-trivial phases such as Ising anyons,\nbut not Fibonacci anyons.) In such settings, the\nfermions may be emergent fermionic quasiparticles;\nif the interactions between these quasiparticles\nare irrelevant in the renormalization-group sense,\nthen an analysis of free-fermion Hamiltonians\ncan shed light on the phase diagrams of\nsuch systems. Thus, the analysis of free fermion\nHamiltonians is equivalent to the analysis of\n{\\it interacting fermion ground states} whose\nlow-energy quasiparticle excitations are free fermions.\n\nLet us begin by considering a few concrete examples,\nin order of increasing complexity.\n\n\\subsection{Zero-Dimensional Systems}\n\nFirst, we analyze\na zero-dimensional system which we will not assume\nto have any special symmetry. The Hamiltonian\n(\\ref{eqn:basic}) takes the simpler form:\n\\begin{equation}\n\\label{eqn:zero-dim}\nH = \\sum_{i,j} iA_{ij}{a_i}{a_j}\n\\end{equation}\nwhere $A_{ij}$ is a $2N\\times 2N$ antisymmetric matrix,\n$A_{ij}=-A_{ji}$. Any real antisymmetric matrix can be written\nin the form\n\\begin{equation}\nA =  O^T \\left( \\begin{array}{rrrrr}\n0 & -{\\lambda_1} &   &  & \\\\\n{\\lambda_1} & 0  &    &  &  \\\\\n   &    & 0  & -{\\lambda_2}&  \\\\\n   &    & {\\lambda_2}  &  0&  \\\\\n   &    &     &    &  \\ddots\\end{array} \\right) O\n\\end{equation}\nwhere $O$ is an orthogonal matrix and\nthe $\\lambda_i$s are positive. The eigenvalues\nof $A$ come in pairs $\\pm i\\lambda_i$;\nthus, the {\\it absence of charge conservation} can\nalso be viewed as the {\\it presence of a particle-hole symmetry}.\nBy assumption, ${\\lambda_i}\\geq\\Delta$ for all $i$.\nClearly, we can continuously deform $A_{ij}$\nwithout closing the gap so that ${\\lambda_i}=\\Delta$ for all $i$.\n(This is usually called `spectrum flattening'.)\nThen, we can write:\n\\begin{equation}\nA = \\Delta\\,\\cdot \\,{O^T} J O\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eqn:canonical-J}\nJ =  \\left( \\begin{array}{rrrrr}\n0 & -1 &   &  & \\\\\n1 & 0  &    &  &  \\\\\n   &    & 0  & -1&  \\\\\n   &    & 1  &  0&  \\\\\n   &    &     &    &  \\ddots\\end{array} \\right)\n\\end{equation}\nThe possible choices of $A_{ij}$\ncorrespond to the possible choices of $O\\in\\text{O}(2N)$,\nmodulo $O$ which commute with the matrix $J$.\nBut the set of $O\\in\\text{O}(2N)$ satisfying ${O^T} J O~=~J$\nis U($N$)$\\subset$O($2N$).\nThus, the space of all\npossible zero-dimensional free fermionic Hamiltonians\nwith $N$ single-particle energy levels\nis topologically-equivalent to the symmetric space O($2N$)\/U($N$).\n\nThis can be restated in more geometrical terms\nas follows. Let us here and henceforth\ntake units in which $\\Delta=1$.\nThen the eigenvalues of $A$ are $\\pm i$.\nIf we view the $2N\\times 2N$ matrix $A$\nas a linear transformation on $\\mathbb{R}^{2N}$, then\nit defines a complex structure.\nConsequently, we can view $\\mathbb{R}^{2N}$ as\n$\\mathbb{C}^{N}$ since multiplication of\n$\\vec{v}\\in \\mathbb{R}^{2N}$ by a\ncomplex scalar can be defined as\n$(a+ib) \\vec{v} \\equiv a\\vec{v} + b A\\vec{v}$.\nThe set of complex structures on $\\mathbb{R}^{2N}$\nis given by performing an arbitrary O($2N$) rotation\non a fixed complex structure, modulo the rotations\nof $\\mathbb{C}^{N}$ which respect the complex structure,\nnamely U($N$). Thus, once again, we conclude that\nthe desired space of Hamiltonians is topologically-equivalent\nto O($2N$)\/U($N$).\n\nWhat are the consequences of this equivalence?\nConsider the simplest case, $N=1$. Then, the\nspace of zero-dimensional Hamiltonians is topologically-equivalent to\nO($2$)\/U($1$)$=\\mathbb{Z}_2$: there are two classes\nof Hamiltonians, those in which the single fermionic level\nis unoccupied in the ground state,\n$c^\\dagger c = (1+i{a_1}{a_2})\/2=0$,\nand those in which it is occupied. For larger $N$,\nO($2N$)\/U($N$) is a more complicated space, but it\nstill has two connected components,\n${\\pi_0}(\\text{O}(2N)\/\\text{U}(N))=\\mathbb{Z}_2$,\nso that there are two classes of free fermion Hamiltonians,\ncorresponding to even or odd numbers of occupied\nfermionic levels in the ground state.\n\nSuppose, now, that we restrict ourselves to time-reversal\ninvariant systems and, furthermore, to those time-reversal\ninvariant systems which satisfy ${T^2}=-1$, where\n$T$ is the anti-unitary operator generating time-reversal.\nThen, following Ref. \\onlinecite{Kitaev09}, we\nwrite $T {a_i} T^{-1} = ({J_1})_{ij}{a_j}$. The matrix\n${J_1}$ is antisymmetric and satisfies ${J_1^2}=-1$.\n$T$-invariance of the Hamiltonian requires\n \\begin{equation}\n{J_1} A = -A {J_1}\n\\end{equation}\nSince ${J_1}$ is antisymmetric and satisfies ${J_1^2}=-1$,\nits eigenvalues are $\\pm i$.\nTherefore, ${J_1}$ defines a complex structure\non $\\mathbb{R}^{2N}$ which may, consequently,\nbe viewed as $\\mathbb{C}^{N}$.\nNow consider $A$, which is also\nantisymmetric and satisfies ${A^2}=-1$, in addition\nto anticommuting with $J_1$. It defines\na quaternionic structure on $\\mathbb{C}^{N}$\nwhich may, consequently, be viewed as\n$\\mathbb{H}^{N\/2}$.\nMultplication of $\\vec{v}\\in \\mathbb{R}^{2N}$\nby a quaternion can be defined as:\n$(a+bi+cj+dk)\\vec{v}\\equiv a\\vec{v}+b{J_1}\\vec{v}\n+cA\\vec{v}+d {J_1}A\\vec{v}$.\nThe possible choices of $A$\ncan be obtained from a fixed one\nby performing rotations of $\\mathbb{C}^{N}$,\nmodulo those rotations which respect the quaternionic structure,\nnamely Sp($N\/2$).\nThus, the set of time-reversal-invariant zero-dimensional free\nfermionic Hamiltonians with ${T^2}=-1$\nis topologically-equivalent to U($N$)\/Sp($N\/2$).\nSince ${\\pi_0}(\\text{U}(N)\/\\text{Sp}(N\/2))=0$,\nany such Hamiltonian can be continuously\ndeformed into any other. This can be understood\nas a result of Kramers\ndoubling: there must be an even number\nof fermions in the ground state so the division\ninto two classes of the previous case does not exist here.\n\n\n\\subsection{2D Systems: $T$-breaking\nsuperconductors}\n\nNow, let us consider systems in more than\nzero dimensions. Once again, we will assume\nthat charge is not conserved, and we will also\nassume that time-reversal symmetry is not\npreserved. For the sake of\nconcreteness, let us consider a single band of\nspin-polarized electrons on a two-dimensional lattice.\nLet us suppose that the electrons condense into\na (fully spin-polarized) $p_x$-wave superconductor.\nFor fixed superconducting order parameter,\nthe low-energy theory is a free fermion Hamiltonian\nfor gapless fermionic excitations at the nodal\npoints $\\pm\\vec{k}_{F}\\equiv (0,\\pm p_F)$. We now ask the question,\nwhat other order parameters could develop\nwhich would fully gap the fermions? For fixed values\nof these order parameters, we have a free fermion\nHamiltonian. Thus, these different possible order\nparameters correspond to different\npossible gapped free fermion Hamiltonians.\n\nThe low-energy Hamiltonian of a fully\nspin-polarized $p_x$-wave superconductor\ncan be written in the form:\n\\begin{equation}\nH = {\\psi^\\dagger}\\left( i{v_\\Delta}{\\partial_x}{\\tau_x}\n+ i{v_F}{\\partial_y}{\\tau_z}\\right)\\psi\n\\end{equation}\nwhere $v_F$, ${v_\\Delta}$ are, respectively, the\nFermi velocity and slope of the gap near the node.\nThe Pauli matrices $\\tau$ act in the particle-hole\nspace:\n\\begin{eqnarray}\n\\psi(k)\n\\equiv \\left(\n\\begin{array}{c}\nc_{\\vec{k}_{F}+\\vec{k}}\\\\\nc^{\\dagger}_{-\\vec{k}_{F}+\\vec{k}}\n\\end{array}\n\\right)\n\\end{eqnarray}\nThis Hamiltonian is invariant under the U(1):\n$\\psi\\rightarrow e^{i\\theta} \\psi$ which corresponds\nto conservation of momentum in the $p_y$ direction\n(not to charge conservation). Since we will be considering\nperturbations which do not respect this symmetry,\nit is convenient to introduce Majorana fermions\n${\\chi^{}_1}$, ${\\chi^{}_2}$ according to\n$\\psi={\\chi^{}_1}+i{\\chi^{}_2}$. Then\n\\begin{equation}\nH = i{\\chi^{}_a}\\left( {v_\\Delta}{\\partial_x}{\\tau_x}\n+ {v_F}{\\partial_y}{\\tau_z}\\right){\\chi^{}_a}\n\\end{equation}\nwith $a=1,2$. Note that we have suppressed the\nparticle-hole index on which the Pauli matrices\n$\\tau$ act. Since $\\chi^{}_1$, $\\chi^{}_2$ are each\na 2-component real spinor, this model has\n4 real Majorana fields.\n\nWe now consider the possible mass terms which\nwe could add to make this Hamiltonian fully gapped:\n\\begin{equation}\n\\label{eqn:Dirac+mass}\nH = i{\\chi^{}_a}\\left( {v_\\Delta}{\\partial_x}{\\tau_x}\n+ {v_F}{\\partial_y}{\\tau_z}\\right){\\chi^{}_a} + i{\\chi^{}_a}M_{ab}{\\chi^{}_b}\n\\end{equation}\nIf we consider the possible order parameters\nwhich could develop in this system, it is clear\nthat there are only two choices: an imaginary\nsuperconducting order parameter $ip_y$ (which\nbreaks time-reversal symmetry) and\ncharge density-wave order (CDW). These take the form:\n\\begin{equation}\nM^{ip_y}_{ab} = \\Delta_{ip_y}\\,i{\\tau^y} \\delta_{ab}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eqn:CDW-mass-eg}\nM^{CDW}_{ab} = \\rho^{}_{2k_F}\n{\\tau^y} \\left(\\cos\\theta\\,  \\mu^z_{ab}+\n\\sin\\theta\\, \\mu^x_{ab}\\right)\n\\end{equation}\nwhere $\\mu^{x,z}$ are Pauli matrices and $\\theta$\nis an arbitrary angle.\nFor an analysis of the possible mass terms\nin the more complex situation\nof graphene-like systems, see, for instance,\nRef. \\onlinecite{Ryu09}.\n\nLet us consider the space of mass terms\nwith a fixed energy gap $\\Delta$ which is\nthe same for all 4 of the Majorana fermions in the model\n(i.e. a flattened mass spectrum).\nAn arbitrary gapped Hamiltonian can be continuously\ndeformed to one which satisfies this condition.\nThen we can have\n$\\Delta_{ip_y}=\\pm\\Delta$,\n$\\rho^{}_{2k_F}=0$\nor $\\rho^{}_{2k_F}=\\Delta$, $\\Delta_{ip_y}=0$\n(in the latter case, arbitrary $\\theta$ is allowed).\nIf both order\nparameters are present, then the energy gap\nis not the same for all fermions.\nIt's not that there's anything wrong with\nsuch a Hamiltonian -- indeed, one can imagine a\nsystem developing both kinds of order.\nRather, it is that such a Hamiltonian\ncan be continuously deformed to one with\neither $\\Delta_{ip_y}=0$ or $\\rho^{}_{2k_F}=0$\nwithout closing the gap. For instance, if\n$\\Delta_{ip_y}>\\rho^{}_{2k_F}$,\nthen the Hamiltonian can be continuously deformed to\none with $\\rho^{}_{2k_F}=0$. (However if we try to deform\nit to a Hamiltonian with $\\Delta_{ip_y}=0$, the gap will\nclose at $\\Delta_{ip_y}=\\rho^{}_{2k_F}$.)\nHence, we conclude that the space of possible mass terms is\ntopologically-equivalent to the disjoint union\nU($1$)$\\cup\\mathbb{Z}_2$: a single one-parameter family\nand two disjoint points.\n\nSince ${\\pi_0}(\\text{U}(1)\\cup\\mathbb{Z}_2) = \\mathbb{Z}_3$,\nthere are three distinct classes of quadratic Hamiltonians\nfor $4$ flavors of Majorana fermions in $2D$.\nThe one-parameter family of CDW-ordered\nHamiltonians counts as a single class since\nthey can be continuously deformed into each other.\nThe parameter $\\theta$ is the phase of the CDW,\nwhich determines whether the density is maximum\nat the sites, the midpoints of the bonds, or somewhere\nin between. It is important to keep in mind, however,\nthat, although there is no topological obstruction to\ncontinuously deforming one $\\theta$ into another,\nthere may be an energetic penalty which makes it costly.\nFor instance, the coupling of the system to the lattice may prefer\nsome particular value of $\\theta$.\nThe classification discussed here accounts only for topological\nobstructions; the possibility of energetic barriers must\nbe analyzed by different methods.\n\nWe can restate the preceding analysis in the following,\nmore abstract language. This formulation will\nmake it clear that we haven't overlooked a\npossible mass term and will generalize to more\ncomplicated free fermion models.\nLet us write\n${\\gamma^{}_1}={\\tau_x}\\delta_{ab}$,\n${\\gamma^{}_2}={\\tau_z}\\delta_{ab}$.\nThen\n\\begin{equation}\n\\{{\\gamma^{}_i},{\\gamma^{}_j}\\}=2\\delta_{ij}\n\\end{equation}\nThe Dirac Hamiltonian for $N=4$\nMajorana fermion fields takes the form\n\\begin{equation}\n\\label{eqn:Dirac-eqn-generic}\nH = i\\chi({\\gamma_i}{\\partial_i} + M)\\chi\n\\end{equation}\nThe matrix $M$ plays the role that the\nmatrix $A$ did in the zero-dimensional case.\nAs in that case, we assume a flattened spectrum\nwhich here means that each Majorana fermion\nfield has the same gap and that this gap is\nequal to $1$. (It does {\\it not} mean that the energy\nis independent of the momentum ${\\bf k}$.)\nIn order to satisfy this, we must require that\n\\begin{equation}\n\\{{\\gamma^{}_i},M\\}=0 {\\hskip 0.3 cm} \\text{and}\n{\\hskip 0.3 cm} {M^2}=-1\n\\end{equation}\n\nNote that it is customary to write the Dirac Hamiltonian\nin a slightly different form,\n\\begin{equation}\n\\label{eqn:Dirac-to-conventional}\nH = \\overline{\\psi}(i{\\gamma_i}{\\partial_i} + m)\\psi\n\\end{equation}\nwhich can be massaged into the form of (\\ref{eqn:Dirac-eqn-generic})\nusing $ \\overline{\\psi}=\\psi^\\dagger \\gamma_0$:\n\\begin{eqnarray}\nH &=& {\\psi^\\dagger}(i{\\gamma_0}{\\gamma_i}{\\partial_i} +\nm{\\gamma_0})\\psi\\cr\n&=&  {\\psi^\\dagger}(i{\\alpha_i}{\\partial_i} +\nm\\beta)\\psi\\cr\n&=&i{\\psi^\\dagger}({\\alpha_i}{\\partial_i} -\nim\\beta)\\psi\n\\end{eqnarray}\nwhere ${\\alpha_i}={\\gamma_0}{\\gamma_i}$ and\n$\\beta={\\gamma_0}$. Thus, if we write\n${\\gamma_i}\\equiv {\\alpha_i}$ and $M\\equiv\n-im\\beta$ and consider Majorana fermions\n(or decompose Dirac fermions into Majoranas),\nwe recover (\\ref{eqn:Dirac-eqn-generic}).\nWe have used the form (\\ref{eqn:Dirac-eqn-generic})\nso that it is analogous to (\\ref{eqn:zero-dim}), with\n$({\\gamma_i}{\\partial_i} + M)$ replacing $A_{ij}$\nand the $i$ pulled out front. Then, the matrix $M$\ndetermines the gaps of the various modes\nin the same way as $A$ does in the zero-dimensional\ncase. Similarly, assuming a `flattened' spectrum\nleads to the condition ${M^2}=-1$.\n\nHow many ways can we\nchoose such an $M$? Since ${\\gamma^2_2}=1$,\nits eigenvalues are $\\pm 1$. Hence, viewed as a\nlinear map from $\\mathbb{R}^4$ to itself, this matrix\ndivides $\\mathbb{R}^4$ into two 2D subspaces\n$\\mathbb{R}^4={X_+}\\oplus{X_-}$\nwith eigenvalue $\\pm 1$ under\n${\\gamma^{}_2}$, respectively.\nFor ${\\gamma^{}_2}={\\tau_z}\\delta_{ab}$,\nthis is trivial:\n\\begin{equation}\n{X_+}= \\text{span}\\left\\{\n\\left(\\scriptstyle{\\begin{array}{c}\n1\\\\ 0 \\end{array}} \\right) \\otimes  \\left(\\scriptstyle{\\begin{array}{c}\n1\\\\ 0 \\end{array}}\\right),\n \\left(\\scriptstyle{\\begin{array}{c}\n1\\\\ 0 \\end{array}} \\right) \\otimes  \\left(\\scriptstyle{\\begin{array}{c}\n0\\\\ 1 \\end{array}}\\right)\\right\\}\n\\end{equation}\nwhere $\\tau_z$ acts on the\nfirst spinor and the second spinor is indexed by $a=1,2$,\ni.e. is acted on by the Pauli matrices $\\mu^{x,z}$\nin (\\ref{eqn:CDW-mass-eg}).\nThis construction generalizes straightforwardly\nto arbitrary numbers $N$ of Majorana fermions, which is\nwhy we use it now.\n\nNow ${\\gamma^{}_1} M$ commutes with\n${\\gamma^{}_2}$ and satisfies ${({\\gamma^{}_1} M)^2}=1$.\nThus, it maps ${X_+}$ to itself and defines\nsubspaces ${X_+^1}, {X_+^2}$\nwith eigenvalue $\\pm 1$ under ${\\gamma^{}_1} M$\n(and equivalently for ${X_-}$). ${X_+}$ can decomposed\ninto ${X_+^1}\\oplus{X_+^2}={X_+}$.\nChoosing $M$ is thus equivalent to choosing\na linear subspace ${X_+^1}$ of ${X_+}$.\n\nThis can be divided into three cases.\nIf ${\\gamma^{}_1} M$\nhas one positive eigenvalue and one negative one\nwhen acting on ${X_+}$ then the space of possible choices\nof ${\\gamma^{}_1} M$ is equal to the space of 1D linear subspaces\nof $\\mathbb{R}^2$, which is simply U(1). If, on the other hand,\n${\\gamma^{}_1} M$ has two positive eigenvalues, then\nthere is a unique choice, which is simply\n$M={\\gamma^{}_1}{\\gamma^{}_2}$. If ${\\gamma^{}_1} M$\nhas two negative eigenvalues, then\nthere is again a unique choice,\n$M=-{\\gamma^{}_1}{\\gamma^{}_2}$.\nTherefore, the space of possible $M$s is topologically\nequivalent to $\\text{U}(1)\\cup\\mathbb{Z}_2$.\n\nNow, suppose that we have $2N$ Majorana fermions.\nThen $\\gamma^{}_2$ defines $N$-dimensional\neigenspaces\n${X_+},{X_-}$ such that\n$\\mathbb{R}^{2N}={X_+}\\oplus{X_-}$\nand ${\\gamma^{}_1} M$ defines eigenspaces\nof ${X_+}$: ${X_+^1}\\oplus{X_+^2}={X_+}$.\nIf ${\\gamma^{}_1} M$ has $k$ positive eigenvalues\nand $N-k$ negative ones, then the space\nof possible choices of ${\\gamma^{}_1}M$\nis O(N)\/O(k)$\\times$O(N-k), i.e we can\ntake the restriction of ${\\gamma^{}_1}M$\nto ${X_+}$ to be of the form\n\\begin{equation}\n{\\gamma^{}_1}M = O^T  \\left( \\begin{array}{cccccc}\n1 &  &   &  & & \\\\\n & \\ddots  &    &  &  &\\\\\n   &    & 1  & &  &\\\\\n   &    &   &  -1&  &\\\\\n   &    &     &    &\\ddots &\\\\\n      &    &     &    &  & -1  \\end{array} \\right) O\n\\end{equation}\nwith $k$ diagonal entries equal to $+1$\nand $N-k$ entries equal to $-1$. Thus, the space of Hamiltonians for\n$N$ flavors of free Majorana fermions\nis topologically equivalent to\n\\begin{equation}\n\\label{eqn:BO-def}\n{\\cal M}_{2N} = \\bigcup_{k=0}^{N} \\text{O}(N)\/(\\text{O}(k)\\times\n\\text{O}(N-k))\n\\end{equation}\nHowever, since ${\\pi_0}(\\text{O}(N)\/(\\text{O}(k)\\times\\text{O}(N-k)))\n=0$, independent of $k$ (note that $0$ is the group with a single\nelement, not the empty set $\\emptyset$),\n${\\pi_0}({\\cal M}_{2N})=\\mathbb{Z}_{N+1}$.\n\nIn the model analyzed above, we had\nonly a single spin-polarized band of electrons.\nBy increasing the number of bands and allowing\nboth spins, we can increase the number of\nflavors of Majorana fermions. In principle, the number of\nbands in a solid is infinity. Usually, we can introduce\na cutoff and restrict attention to a few bands\nnear the Fermi energy. However, for a purely\ntopological classification, we can ignore energetics\nand consider all bands on equal footing.\nThen we can take $N\\rightarrow\\infty$,\nso that ${\\pi_0}({\\cal M}_\\infty)=\\mathbb{Z}$.\nThis classification permits us to\ndeform Hamiltonians into each\nother so long as there is no topological obstruction,\nwith no regard to how energetically costly\nthe deformation may be. Thus, the $2N=4$ classification\nwhich we discussed above can perhaps be viewed as\na `hybrid' classification which looks for topological\nobstructions in a class of models with a fixed set of\nbands close to the Fermi energy.\n\nBut even this point of view is not really\nnatural. The discussion above took as\nits starting point an expansion about\na $p_x$ superconductor; the $p_x$ superconducting\norder parameter was assumed to be large and\nfixed while the $ip_y$ and CDW order parameters\nwere assumed to be small. In other words, we\nassumed that the system was at a point in parameter\nspace at which the gap, though non-zero, was small\nat two points in the Brillouin zone (the intersection\npoints of the nodal line in the $p_x$ superconducting\norder parameter with the Fermi surface). This allowed\nus to expand the Hamiltonian about these points\nin the Brillouin zone and write it in Dirac form.\nAnd this may, indeed, be reasonable in a system\nin which $p_x$ superconducting order is strong\n(i.e. highly energetically-favored) and other orders\nare weak. However, a topological classification should\nallow us to take the system into regimes in which\n$p_x$ superconductivity is small and other orders\nare large. Suppose, for instance, that we took\nour model of spin-polarized electrons (which\nwe assume, for simplicity, to be at half-filling\non the square lattice) and went into a regime in\nwhich there was a large\n$d_{{x^2}-{y^2}}$-density-wave (or `staggered\nflux') order parameter \\cite{Nayak00b}\n$\\langle c^\\dagger_{{\\bf k}+{\\bf Q}} c_{\\bf k} \\rangle\n= i\\Phi(\\cos{k_x}a-\\cos{k_y}a)$, where $a$ is the lattice\nconstant and $\\Phi$ is the magnitude of the order parameter.\nWith nearest-neighbor hopping only,\nthe energy spectrum is $E^2_{\\bf k}=\n(2t)^2 (\\cos{k_x}a+\\cos{k_y}a)^2\n+ \\Phi^2 (\\cos{k_x}a-\\cos{k_y}a)^2$.\nThus, the gap vanishes at 4 points,\n$(\\pm\\pi\/2,\\pm\\pi\/2)$ and $(\\mp\\pi\/2,\\pm\\pi\/2)$.\nThe Hamiltonian can be linearized in the\nvicinity of these points:\n\\begin{multline}\nH = {\\psi^\\dagger_1}\\left( i{v_\\Delta}{\\partial_x}{\\tau_x}\n+ i{v_F}{\\partial_y}{\\tau_z}\\right){\\psi_1}\\\\\n+ {\\psi^\\dagger_2}\\left( i{v_\\Delta}{\\partial_y}{\\tau_x}\n+ i{v_F}{\\partial_x}{\\tau_z}\\right){\\psi_2}\n\\end{multline}\nwhere $v_F$, ${v_\\Delta}$ are, respectively, the\nFermi velocity and slope of the gap near the nodes;\nthe subscripts 1,2 refer to the two sets of nodes\n$(\\pm\\pi\/2,\\pm\\pi\/2)$ and $(\\mp\\pi\/2,\\pm\\pi\/2)$;\nand $\\psi_{A}$, $A=1,2$ are defined by:\n\\begin{eqnarray}\n\\psi_{1,2}(k)\n\\equiv \\left(\n\\begin{array}{c}\nc_{(\\pi\/2,\\pm\\pi\/2)+\\vec{k}}\\\\\nc_{(-\\pi\/2,\\mp\\pi\/2)+\\vec{k}}\n\\end{array}\n\\right)\n\\end{eqnarray}\nIf we introduce Majorana fermions\n$\\psi_{A}=\\chi^{}_{A1}+i\\chi^{}_{A2}$, then\nwe can write this Hamiltonian with possible\nmass terms as:\n\\begin{multline}\nH = i\\chi^{}_{1a}\\left( {v_\\Delta}{\\partial_x}{\\tau_x}\n+ {v_F}{\\partial_y}{\\tau_z}\\right)\\chi^{}_{1a}\\\\\n+ i\\chi^{}_{2a}\\left( {v_\\Delta}{\\partial_y}{\\tau_x}\n+ {v_F}{\\partial_x}{\\tau_z}\\right)\\chi^{}_{2a}\\\\\n+ i\\chi^{}_{Aa}\\, M_{Aa,Bb} \\,  \\chi^{}_{Bb}\n\\end{multline}\nWe have suppressed the spinor indices\n(e.g. $\\chi_{11}$ is a two-component spinor);\nwith these indices included, $M_{Aa,Bb}$\nis an $8\\times 8$ matrix. However, in order for\nthe gap to be the same for all flavors, the\nmass matrix must anticommute with $\\tau_{x,z}$.\nThus, $M_{Aa,Bb}={\\tau_y} {\\tilde M}_{Aa,Bb}$.\nThe matrix ${\\tilde M}_{Aa,Bb}$ can have\n$0,1,2,3$, or $4$ eigenvalues equal to $+1$\n(with the rest being $-1$). The spaces\nof such mass terms are, respectively,\n$0$, $\\text{O}(4)\/(\\text{O}(1)\\times\\text{O}(3))$,\n$\\text{O}(4)\/(\\text{O}(2)\\times\\text{O}(2))$,\n$\\text{O}(4)\/(\\text{O}(3)\\times\\text{O}(1))$,\nand $0$. Mass terms with $0$ or $4$ eigenvalues\nequal to $+1$ correspond physically to\n$\\pm id_{xy}$-density wave order,\n$\\langle c^\\dagger_{{\\bf k}+{\\bf Q}} c_{\\bf k} \\rangle\n= \\pm\\sin{k_x}a\\,\\sin{k_y}a$.\nMass terms with $2$ eigenvalues\nequal to $+1$ correspond physically to\nsuperconductivity, to $Q'=(\\pi,0)$ CDW order,\nand to linear combinations of the two.\nMass terms with $1$ or $3$ eigenvalues\nequal to $+1$ correspond to\n(physically unlikely) hybrid orders with, for instance,\nsuperconductivity at $(\\pm\\pi\/2,\\pm\\pi\/2)$ and\n$\\pm id_{xy}$-density wave order at $(\\pm\\pi\/2,\\mp\\pi\/2)$.\nClearly, this is the $2N=8$ case of the general\nclassification discussed above. Thus, the same\nunderlying physical degrees of freedom -- a single\nband of spin-polarized electrons on a\nsquare lattice -- can correspond to either\n$2N=4$ or $2N=8$, depending on where\nthe system is in parameter space. One\ncan imagine regions of parameter space where\nthe gap is small at an arbitrary number $N$\nof points. Thus, if we restrict ourselves to systems\nwith a single band, then different regions of the parameter\nspace (with different numbers of points at which the gap is\nsmall) will have very different topologies.\nAlthough such a classification may be a necessary evil\nin some contexts, it is far preferable, given the choice,\nto allow topology to work unfettered by energetics.\nThen, we can consider a large number $n$ of bands.\nSuppose that the gap becomes small at $r$ points\nin the Brillouin zone in each band. Then, the low-energy\nHamiltonian takes the Dirac form for $2N=2rn$\nMajorana fermion fields.\nAs we will see below, if $N$ is sufficiently large,\nthe topology of the space of possible mass terms\nwill be independent of $N$. Consequently,\nfor $n$ sufficiently large, the topology of the space\nof possible mass terms will be independent of $r$.\nIn other words, we are in the happy situation\nin which the topology of the space of Hamiltonians\nwill be the same in the vicinity of any gap closing.\nBut any gapped Hamiltonian can be continuously\ndeformed so that the gap becomes small at some\npoints in the Brillouin zone. Thus, the problem of classifying\ngapped free fermion Hamiltonians in $d$-dimensions\nis equivalent to the problem of classifying possible mass terms\nin a generic $d$-dimensional Dirac Hamiltonian\nso long as the number of bands is sufficiently large \\cite{Kitaev09}.\nThis statement can be made more precise and\nput on more solid mathematical footing using ideas\nwhich we discuss in Appendix \\ref{sec:dimension}.\n\n\n\\subsection{Classification of Topological Defects}\n\nThe topological classification described above\nholds not only for classes of translationally-invariant\nHamiltonians such as (\\ref{eqn:Dirac-eqn-generic}),\nbut also for topological defects within a class.\nSuppose, for instance, that we consider\n(\\ref{eqn:Dirac-eqn-generic}) with a mass\nwhich varies slowly as the origin is encircled\nat a great distance. We can ask whether such a\nHamiltonian can be continuously deformed into\na uniform one. In a system in which the mass term\nis understood as arising as a result of some\nkind of underlying ordering such as superconductivity or\nCDW order, we are simply talking about topological\ndefects in an ordered media, but with the caveat that\nthe order parameter is allowed to explore a very large\nspace which may include many physically\ndistinct or unnatural orders, subject only to the condition that\nthe gap not close.\n\nLet us, for the sake of concreteness,\nassume that we have a mass term with $N\/2$\npositive eigenvalues when restricted to the $+1$ eigenspace of\n$\\gamma^{}_2$. (For $N$\nlarge, the answer obviously cannot depend on the\nnumber of positive eigenvalues $k$ so long\nas $k$ scales with $N$. Thus, we will denote the\nspace ${\\cal M}_{2N}$ defined in Eq. \\ref{eqn:BO-def}\nby $\\mathbb{Z}\\times\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nwhere the integers in $\\mathbb{Z}$ correspond to the number\nof positive eigenvalues of the mass term when restricted\nto the $+1$ eigenspace of $\\gamma^{}_2$.)\nThen $M(r=\\infty,\\theta)$ defines a loop in\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nwhich cannot be continuously unwound if it\ncorresponds to a nontrivial element of\n${\\pi_1}(\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2)))$.\n\n\nTo compute ${\\pi_1}(\\text{O}(N)\/(\\text{O}(N\/2)\\times\n\\text{O}(N\/2)))$, we parametrize\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nby symmetric matrices $K$ which satisfy ${K^2}=1$\nand $\\text{tr}(K)=0$. (Such matrices decompose\n$\\mathbb{R}^{N}$ into their\n$+1$ and $-1$ eigenspaces:\n$\\mathbb{R}^{N}={V_+}\\oplus{V_-}$.\n$K$ can be written in the form:\n$K={O^T} {K_0} O$,\nwhere $K_0$ has $N\/2$ diagonal entries\nequal to $+1$ and $N\/2$ equal to $-1$, i.e\n${K_0}=\\text{diag}(1,\\ldots,1,-1,\\ldots,-1)$.)\nNote that any such $K$ is itself an orthogonal matrix,\ni.e. an element of O$(N)$;\nthus $\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\ncan be viewed as a submanifold of O$(N)$\nin a canonical way. Consider a curve $L(\\lambda)$\nin $\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\nwith $L(0)=K$ and $L(\\pi)=-K$.\nWe will parametrize it as $L(\\lambda)=K\\, e^{\\lambda A}$,\nwhere $A$ is in the Lie algebra of $\\text{O}(N)$.\nIn order for this loop to remain in\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$,\nwe need ${(K\\, e^{\\lambda A})^2}=1$.\nSince ${(K\\, e^{\\lambda A})^2}=K\\, e^{\\lambda A} K\\, e^{i\\lambda A}\n= e^{i\\lambda K A K} \\, e^{\\lambda A}$, this condition\nimplies that $KA=-AK$.\nIn order to have $L(\\pi)=-K$, we need ${A^2}=-1$.\nSuch a curve is, in fact, a minimal\ngeodesic from $K$ to $-K$.\nEach such geodesic can be represented by its\nmidpoint $L(\\pi\/2)=KA$, so the space of such geodesics is\nequivalent to the space of matrices $A$ satisfying\n${A^2}=-1$ and $KA=-AK$. As discussed in\nRef. \\onlinecite{Milnor63}, the space of minimal geodesics\nis a good enough approximation to the entire space of loops\n(essentially because an arbitrary loop can\nbe deformed to a geodesic) that we can compute\n$\\pi_1$ from the space of minimal geodesics\njust as well as from the space of loops. Thus,\nthe loop space of $\\text{O}(N\/2)\/(\\text{O}(k)\\times\\text{O}(N\/2-k))$\nis homotopically equivalent to the space of matrices $A$ satisfying\n${A^2}=-1$ and $KA=-AK$.\nSince it anticommutes with $K$, $KA$\nmaps the $+1$ eigenspace of $K$ to\nthe $-1$ eigenspace. It is clearly a length-preserving map\nsince ${(KA)^2}=1$\nand, since the $\\pm 1$ eigenspaces of $K$ are isomorphic to\n$\\mathbb{R}^{N\/2}$, $KA$ defines an element of\nO$(N\/2)$. Thus a loop in\n$\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2))$\ncorresponds to an element of O$(N\/2)$\nor, in other words:\n\\begin{equation}\n\\label{eqn:Bott-step-1}\n{\\pi_1}(\\text{O}(N)\/(\\text{O}(N\/2)\\times\\text{O}(N\/2)))=\n{\\pi_0}(\\text{O}(N\/2)).\n\\end{equation}\nThe latter group is simply $\\mathbb{Z}_2$\nsince $\\text{O}(N\/2)$ has two connected components:\n(1) pure rotations and (2) rotations combined with a reflection.\n\nIt might come as a surprise that we find a $\\mathbb{Z}_2$\nclassification for point-like defects in two dimensions.\nIndeed, if we require that the superconducting order parameter\nhas fixed amplitude at infinity, then vortices\nof arbitrary winding number are stable and we have a\n$\\mathbb{Z}$ classification. However,\nin the classification discussed here, we require a weaker\ncondition be satisfied: that the fermionic gap remain constant.\nConsequently, a vortex configuration\nof even winding number can be unwound without\nclosing the free fermion gap by, for instance, `rotating'\nsuperconductivity into CDW order.\n\n\n\\subsection{3D Systems with No Symmetry}\n\nWith these examples under our belts, we now turn to\nthe case which is of primary interest in this paper:\nfree fermion systems in three dimensions\nwithout time-reversal or charge-conservation symmetry.\nWe consider the Dirac Hamiltonian in $3D$\nfor an $2N$-component Majorana fermion field\n$\\chi$:\n\\begin{equation}\n\\label{eqn:3D-Dirac+mass}\nH = i\\chi\\left( \\partial_{1}{\\gamma^{}_1}\n+ \\partial_{2}{\\gamma^{}_2} +\n\\partial_{3}{\\gamma^{}_3}\\right)\\chi + i{\\chi}M{\\chi}\n\\end{equation}\nIn the previous section, we discussed\na lattice model which realizes (\\ref{eqn:3D-Dirac+mass})\nin its continuum limit with\n$2N=8$. Different mass terms\ncorrespond to different quadratic perturbations\nof this model which open a gap (which can be viewed\nas order parameters which we are turning on at the mean-field level).\nWe could classify such terms by considering,\nfrom a physical perspective, all such ways of opening a gap.\nHowever, we will instead determine the topology\nof the space of mass terms (and, thereby, the space\nof gapped free fermion Hamiltonians) by\nthe same mathematical methods by which we analyzed the\n$2D$ case.\n\nSince ${\\gamma_1^2}=1$ and has vanishing\ntrace, this matrix decomposes $\\mathbb{R}^{2N}$\ninto its $\\pm 1$ eigenspaces:\n$\\mathbb{R}^{2N}={X_+} \\oplus {X_-}$.\nNow ${({\\gamma^{}_2}{\\gamma^{}_3})^2}=-1$\nand $[{\\gamma^{}_1},{\\gamma^{}_2}{\\gamma^{}_3}]=0$.\nTherefore, ${\\gamma^{}_2}{\\gamma^{}_3}$ is a\ncomplex structure on ${X_+}$ (and also on ${X_-}$),\ni.e. we can define multiplication of vectors\n$\\vec{v}\\in{X_+}$ by complex scalars\naccording to $(a+bi)\\vec{v}\\equiv\na\\vec{v} + {\\gamma^{}_2}{\\gamma^{}_3}\\vec{v}$.\n(Consequently, we can view ${X_+}$ as\n$\\mathbb{C}^{N\/2}$.)\nNow, consider a possible mass term $M$,\nwith ${M^2}=-1$. ${({\\gamma^{}_2}M)^2}=1$\nand $[{\\gamma^{}_1},{\\gamma^{}_2}M]=0$.\nLet $Y$ be the subspace of ${X_+}$\nwith eigenvalue $+1$ under ${\\gamma^{}_2}M$.\nSince $\\{{\\gamma^{}_2}M,{\\gamma^{}_2}{\\gamma^{}_3}\\}=0$,\n${\\gamma^{}_2}{\\gamma^{}_3}Y$ is the subspace of ${X_+}$\nwith eigenvalue $-1$ under ${\\gamma^{}_2}M$.\nIn other words, ${X_+}=Y\\oplus{\\gamma^{}_2}{\\gamma^{}_3}Y$,\ni.e. $Y$ is a real subspace of ${X_+}$.\nHence, the space of choices of $M$ is the space\nof real subspaces $Y\\subset{X_+}$ (or,\nequivalently, of real subspaces\n$\\mathbb{R}^{N\/2}\\subset\\mathbb{C}^{N\/2}$).\nGiven any fixed real subspace $Y\\subset{X_+}$,\nwe can obtain all others by performing $\\text{U}(N\/2)$\nrotations of ${X_+}$, but two such rotations\ngive the same real subspace if they differ only\nby an $\\text{O}(N\/2)$ rotation of $Y$. Thus,\nthe space of gapped  Hamiltonians\nfor $2N$ free Majorana fermion fields\nin $3D$ with no symmetry is topologically-equivalent\nto $\\text{U}(N\/2)\/\\text{O}(N\/2)$. In the remaining\nsections of this paper, we will be discussing\ntopological defects in such systems and their motions.\n\n\\subsection{General Classification and Bott Periodicity}\n\\label{sec:bott}\n\nBefore doing so, we pause for a minute\nto consider the classification in other dimensions\nand in the presence of symmetries such as\ntime-reversal and charge conservation.\nWe have seen that systems with no symmetry\nin $d=0,2,3$ are classified by the spaces\n$\\text{O}(2N)\/\\text{U}(N)$,\n$\\mathbb{Z}\\times\n\\frac{\\text{O}(N)}{\\text{O}(N\/2)\\times\\text{O}(N\/2)}$,\nand $\\text{U}(N\/2)\/\\text{O}(N\/2)$. By similar methods,\nit can be shown that the $d=1$ case is\nclassified by $\\text{O}(N)$. As we have seen,\nincreasing the spatial dimension increases the\nnumber of $\\gamma$ matrices by one.\nThe problem of choosing $\\gamma_{1},\\ldots,\\gamma_{d}$\nsatisfying $\\{{\\gamma_i},{\\gamma_j} \\}=2\\delta_{ij}$\nand $M$ which anti-commutes with the $\\gamma_i$s\nand squares to $-1$ leads us to subspaces of\n$\\mathbb{R}^{2N}$ of smaller and smaller\ndimension, isometries between these spaces,\nor complex of quaternionic structures on these spaces.\nThis leads the progression of spaces\nin the top row of Table \\ref{tbl:classifying}.\n\nAt the same time, we have seen\nthat a time-reversal-invariant system in $d=0$\nis classified by $\\text{U}(N)\/\\text{Sp}(N\/2)$.\nSuppose that we add a discrete anti-unitary symmetry\n$S i S^{-1} =-i$ defined by\n\\begin{equation}\n\\label{eqn:symmetries}\nS {a_i} S^{-1} = ({J})_{ij}{a_j}\n\\end{equation}\nwhich squares to ${J^2}=-1$.\nIt must anti-commute with the mass term\n\\begin{equation}\n\\label{eqn:anti-comm}\n\\{J,M\\}=0\n\\end{equation}\nin order to ensure invariance under the symmetry,\nso choosing a $J$\namounts to adding a complex structure,\nwhich leads to the {\\it opposite} progression of\nclassifying spaces.\nConsider, as an example of the preceding statements,\na time-reversal invariant system in $d=3$.\nThen time-reversal symmetry $T$ is an\nexample of a symmetry generator $J$\ndiscussed in the previous paragraph.\nWe define a real subspace $Y\\subset{X_+}$,\nin a similar manner as above, but now\nas the subspace of ${X_+}$\nwith eigenvalue $+1$ under ${\\gamma^{}_2}T$,\nrather than under ${\\gamma^{}_2}M$. Once again,\n${X_+}=Y\\oplus{\\gamma^{}_2}{\\gamma^{}_3}Y$.\nNow, $\\{{\\gamma^{}_3}M,{\\gamma^{}_2}T\\}=0$,\nand ${({\\gamma^{}_3}M)^2}=1$, so the\n$+1$ eigenspace of ${\\gamma^{}_3}M$\nis a linear subspace of $Y$.\nThe set of all such linear subspaces is\n$\\mathbb{Z}\\times\n\\frac{\\text{O}(N\/2)}{\\text{O}(N\/4)\\times\\text{O}(N\/4)}$.\nBut this is the same classifying space as for a system\nwith no symmetry in $d=2$ (apart from a reduction\nof $N$ by a factor of $2$). Thus, we are led\nto the list of classifying spaces for gapped\nfree fermion Hamiltonians\nin Table \\ref{tbl:classifying}.\n\n\\begin{table*}\n\\begin{tabular}{c | c c c c c c c}\ndim.: &  0    &   1 & 2 & 3 & 4 & \\ldots\\\\\n\\hline\\hline\nSU($2$), $T$, $Q$& $\\mathbb{Z}\\times\n \\frac{\\text{O}(N)}{\\text{O}(N\/2)\\times\\text{O}(N\/2)}$\n & $\\text{U}(N\/2)\/\\text{O}(N\/2)$ & $\\text{Sp}(N\/4)\/\\text{U}(N\/4)$ &\n $\\text{Sp}(N\/8)$ & $\\mathbb{Z}\\times\\frac{\\text{Sp}(N\/8)}\n {\\text{Sp}(N\/16)\\times\\text{Sp}(N\/16)}$ \\ldots\\\\\nSU($2$), $T$, $Q$, $\\chi$&  $\\text{O}(N\/4)$& $\\mathbb{Z}\\times\n \\frac{\\text{O}(N\/4)}{\\text{O}(N\/8)\\times\\text{O}(N\/8)}$\n & $\\text{U}(N\/8)\/\\text{O}(N\/8)$ & $\\text{Sp}(N\/16)\/\\text{U}(N\/16)$ &\n $\\text{Sp}(N\/32)$ &\\ldots\\\\\nno symm.& $\\text{O}(2N)\/\\text{U}(N)$ &  $\\text{O}(N)$& $\\mathbb{Z}\\times\n \\frac{\\text{O}(N)}{\\text{O}(N\/2)\\times\\text{O}(N\/2)}$\n & $\\text{U}(N\/2)\/\\text{O}(N\/2)$ & $\\text{Sp}(N\/4)\/\\text{U}(N\/4)$ &\n \\ldots\\\\\n$T$ only & $\\text{U}(N)\/\\text{Sp}(N\/2)$  & $\\text{O}(N)\/\\text{U}(N\/2)$\n&$\\text{O}(N\/2)$ & $\\mathbb{Z}\\times\n \\frac{\\text{O}(N\/2)}{\\text{O}(N\/4)\\times\\text{O}(N\/4)}$ & \\ldots\\\\\n$T$ and $Q$ &\n$\\mathbb{Z}\\times\\frac{\\text{Sp}(N\/2)}{\\text{Sp}(N\/4)\\times\\text{Sp}(N\/4)}$\n& $\\text{U}(N\/2)\/\\text{Sp}(N\/4)$ & $\\text{O}(N\/2)\/\\text{U}(N\/4)$\n& $\\text{O}(N\/4)$\\\\\n$T$, $Q$, $\\chi$ & $\\text{Sp}(N\/4)$ &\n$\\mathbb{Z}\\times\\frac{\\text{Sp}(N\/4)}{\\text{Sp}(N\/8)\\times\n\\text{Sp}(N\/8)}$\n& $\\text{U}(N\/4)\/\\text{Sp}(N\/8)$ & $\\text{O}(N\/4)\/\\text{U}(N\/8)$\\\\\n&\\vdots & & & &$\\ddots$\n\\end{tabular}\n\\caption{The period-$8$ (in both dimension and number of\nsymmetries) table of classifying spaces for free fermion Hamiltonians\nfor $N$ complex $=2N$ real (Majorana) fermion fields in dimensions\n$d=0,1,2,3,\\ldots$ with no symmetries; time-reversal symmetry ($T$) only; time-reversal and charge conservation\nsymmetries ($T$ and $Q$); time-reversal, charge conservation,\nand sublattice symmetries ($T$, $Q$, and $\\chi$);\nand the latter two cases with SU($2$) symmetry.\nAs a result of the period-$8$ nature of the table,\nthe top two rows could equally well be the\nbottom two rows of the table.\nMoving $p$ steps to the right and $p$ steps down leads to\nthe same classifying space (but for $1\/2^p$ as many fermion\nfields), which is a reflection of Bott periodicity, as explained\nin the text. The number of disconnected components of any such\nclassifying space -- i.e. the number of different phases in that\nsymmetry class and dimension -- is given by the corresponding\n$\\pi_0$, which may be found in Eq. \\ref{eqn:stable-pi-0}. Higher homotopy groups, which classify defects, can be computed using Eq. \\ref{eqn:Bott periodicity}. Table \\ref{tbl:unitary-classifying}, given in Appendix\n\\ref{sec:QnotT}, is the analogous table\nfor charge-conserving Hamiltonians without time-reversal symmetry.}\n\\label{tbl:classifying}\n\\end{table*}\n\nIn this table, $Q$ refers to charge-conservation symmetry.\nCharge conservation is due to the invariance of\nthe Hamiltonian of a system under the U(1) symmetry\n${c_i}\\rightarrow e^{i\\theta}{c_i}$. In terms of\nMajorana fermions $a_i$ defined according to\n${c_j}=(a_{2j-1}+ia_{2j})\/2$, the symmetry takes the\nform $a_{2j-1}~\\rightarrow~\\cos\\theta a_{2j-1}\n+ \\sin\\theta a_{2j}$, $a_{2j}~\\rightarrow~-\\sin\\theta a_{2j-1}\n+ \\cos\\theta a_{2j}$. However, if a free fermion Hamiltonian\nis invariant under the discrete\nsymmetry ${c_i}\\rightarrow i{c_i}$ or, equivalently,\n$a_{2j-1} \\rightarrow a_{2j}$, $a_{2j} \\rightarrow - a_{2j-1}$,\nthen it is automatically invariant under the full U(1) as well,\nand conserves charge \\cite{Kitaev09}. Thus, we can treat charge\nconservation as a discrete symmetry $Q$\nwhich is unitary, squares to $-1$, and commutes\nwith the Hamiltonian (i.e. with the $\\gamma$ matrices and $M$).\nSince $Q$ transforms ${c_i}\\rightarrow i{c_i}$, it\nanti-commutes with $T$.\nNote further that if a system has time-reversal\nsymmetry, then the product of time-reversal $T$ and\ncharge conservation $Q$ is a discrete anti-unitary symmetry,\n$QT$ which anti-commutes with the Hamiltonian and with $T$\nand squares to $-1$. Then $QT$ is defined by a choice of matrix $J$,\nanalogous to $T$, as in Eq. \\ref{eqn:symmetries}.\nIf the system is not time-reversal-invariant,\nthen charge conservation is a unitary symmetry.\nIt is easier then to work with complex fermions,\nand the classification of such systems falls into an entirely\ndifferent sequence, as discussed in Appendix \\ref{sec:QnotT}.)\n\nIf a system is both time-reversal symmetric\nand charge-conserving, i.e. if it is a time-reversal\ninvariant insulator, then it may have an additional\nsymmetry which guarantees that the eigenvalues\nof the Hamiltonian come in $\\pm E$ pairs, just as in a superconductor.\nAn example of such a symmetry is the sublattice\nsymmetry of Hamiltonians on a bipartite lattice in\nwhich fermions can hop directly from the $A$ sublattice\nto the $B$ sublattice but cannot hop directly between sites on the\nsame sublattice. In such a case, the system is\ninvariant under a unitary symmetry\n$\\chi$ defined as follows. If we block diagonalize\n$\\chi$ so that one block acts on sites in the $A$ sublattice\nand the other on sites in the $B$ sublattice, then\nwe can write $\\chi=\\text{diag}(k,-k)$, i.e.\n${a_i}({\\bf x})\\rightarrow -k_{ij}{a_j}({\\bf x})$ for\n${\\bf x}\\in A$ and\n${a_i}({\\bf x})\\rightarrow k_{ij}{a_j}({\\bf x})$ for ${\\bf x}\\in B$.\nThis symmetry transforms the Hamiltonian to minus itself\nif ${k^2}=1$ or, in other words, if ${\\chi^2}=1$. Then\n$\\chi Q$ is a unitary symmetry which squares to $-1$\nand anti-commutes with the Hamiltonian, $T$,\nand $QT$. Hence $\\chi Q$, too, is defined by a choice of matrix $J$,\nas in Eq. \\ref{eqn:symmetries}. We will call such a\nsymmetry a sublattice symmetry $\\chi$ and a system satisfying\nthis symmetry a `bipartite' system, but the symmetry\nmay have a different microscopic origin.\n\nIn an electron system, time-reversal ordinarily squares\nto $-1$, because the transformation law is \n${c_\\uparrow}\\rightarrow {c_\\downarrow}$,\n${c_\\downarrow}\\rightarrow -{c_\\uparrow}$,\nas we have thus far assumed in taking ${J^2}=-1$.\nHowever, it is possible to have a system of\nfully spin-polarized electrons which has an\nanti-unitary symmetry $T$ which squares to $+1$.\n(One might object to calling this symmetry time-reversal\nbecause it doesn't reverse the electron spins,\nbut $T$ is a natural label because it is a symmetry\nwhich is just as good for the present purposes.)\nThen, since ${J^2}=1$, a choice of $J$ is similar to\na choice of a $\\gamma$ matrix. In general,\nsymmetries (\\ref{eqn:symmetries})\nwhich square to $+1$ have the same effect\non the topology of the space of free fermion Hamiltonians\nas adding dimensions since each such $J$\ndefines a subspace of half the dimension within the\neigenspaces of the $\\gamma$ matrices.\nThis is true for systems with ${T^2}=1$.\n\nSU(2) spin-rotation-invariant and time-reversal-invariant\ninsulators (systems with $T$ and $Q$) effectively\nfall in this category. The Hamiltonian for such\na system can be written in the form $H=h\\otimes {I_2}$\nwhere the second factor is the $2\\times 2$ identity\nmatrix acting on the spin index. Then time-reversal\ncan be written in the form $T= t \\otimes i{\\sigma_y}$,\nwhere ${t^2}=1$, and $Q$ can be written in\nthe form $Q= q \\otimes {I_2}$,\nso that $QT =qt \\otimes i{\\sigma_y}$, where ${(qt)^2}=1$.\nThus, since the matrix $i\\sigma_y$ squares to $-1$,\nthe symmetries $T$ and $QT$ have effectively become\nsymmetries which square to $+1$. They now move the\nsystem through the progression of classifying spaces\nin the same direction as increasing the dimension,\ni.e. in the opposite direction to symmetries which\nsquare to $-1$. Thus, SU(2) spin-rotation-invariant and\ntime-reversal-invariant insulators in $d$ dimensions\nare classified by the same space as\nsystems with no symmetry in $d+2$ dimensions.\nHowever, in a system which, in addition, has\nsublattice symmetry $\\chi=x\\otimes {I_2}$,\nwe have $(qx)^{-1}$. Thus, sublattice symmetry is\nstill a symmetry which squares to $-1$.\nSince the two symmetries which square to $+1$\n($T$ and $QT$) have the same effect as increasing\nthe dimension while the symmetry which squares\nto $-1$ has the same effect as decreasing the dimension,\nSU(2) spin-rotation-invariant and time-reversal-invariant insulators\nwith sublattice symmetry in $d$ dimensions\nare classified by the same space as\nsystems with no symmetry in $d+1$ dimensions\n(but with $N$ replaced by $N\/4$).\nSimilar considerations apply to superconductors with\nSU(2) spin-rotational symmetry.\n\n\\begin{table*}\n \\begin{tabular}[t]{|c|c|c|c|c|}\n \\hline\nSymmetry classes&Physical realizations&$d=1$&$d=2$&$d=3$\n \\\\\\hline\n\\hline D&SC&{\\color{blue}$p$-wave SC}&{\\color{blue}$(p+ip)$-SC}&0\n\\\\\\hline\nDIII&TRI SC&{\\color{red} ${\\rm Z_2}$}&{\\color{blue} $(p+ip)(p-ip)$-SC}&He$^3$-B\n\\\\\\hline AII&TRI\nins.&0&HgTe Quantum well&${\\rm Bi_{1-x}Sb_x}$, ${\\rm Bi_2Se_3}$, etc.\n\\\\\\hline\nCII&Bipartite TRI ins.&Carbon nanotube&0&{\\color{red} ${\\rm Z_2}$}\n\\\\\\hline\nC&Singlet SC&0&{\\color{blue} $(d+id)$-SC}&0\n\\\\\\hline\nCI&Singlet TRI\nSC&0&0&{\\color{red} Z}\n\\\\\\hline\nAI& TRI ins. w\/o SOC&0&0&0\n\\\\\\hline\nBDI&Bipartite TRI\nins. w\/o SOC&Carbon nanotube&0&0\n\\\\\\hline\n \\end{tabular}\n  \\caption{Topological periodic table in physical dimensions $1,2,3$. The first column contains 8 of the 10 symmetry classes in the Cartan notation\n adopted by Schnyder {\\it et al.}\\cite{Ryu08}, following Zirnbauer\n \\cite{Zirnbauer96,Altland97}.\n The second column contains the requirements for physical systems which can\n realize the corresponding symmetry classes. ``w\/o\" stands for ``without\". SC stands for superconductivity, TRI for time-reversal invariant, and SOC for spin-orbit coupling. The three columns $d=1,2,3$ list topological states in the spatial dimensions $1,2,3$ respectively. $0$ means the topological classification is trivial. The red labels {\\color{red} Z} and {\\color{red} ${\\rm Z_2}$} stand for topological states\nclassified by these groups but for which states corresponding\nto non-trivial elements of $\\mathbb{Z}$ or $\\mathbb{Z}_2$\nhave not been realized in realistic materials.\nThe blue text stands for topological states for which\na well-defined physical model has been proposed but convincing experimental candidate has not been found yet. (See text for more discussions on the realistic materials.) }\n  \\label{tbl:periodic}\n\\end{table*}\n\nIn order to discuss topological defects in\nthe systems discussed here,\nit is useful to return to the arguments which led to\n(\\ref{eqn:Bott-step-1}).\nBy showing that the space of loops in\n$\\text{O}(N\/2)\/(\\text{O}(N\/4)\\times\\text{O}(N\/4))$ is\nwell-approximated by $\\text{O}(N\/4)$,\nwe not only showed that\n${\\pi_1}(\\text{O}(N\/2)\/(\\text{O}(N\/4)\\times\\text{O}(N\/4)))=\n{\\pi_0}(\\text{O}(N\/4))$ but, in fact, that\n${\\pi_k}(\\text{O}(N\/2)\/(\\text{O}(N\/4)\\times\\text{O}(N\/4)))=\n\\pi_{k-1}(\\text{O}(N\/4))$ (see Ref. \\onlinecite{Milnor63}).\nContinuing in the same way, we can approximate\nthe loop space of $\\text{O}(N\/4)$ (i.e. the space of loops in\n$\\text{O}(N\/4)$) by minimal\ngeodesics from $\\mathbb{I}$ to $-\\mathbb{I}$:\n$L'(\\lambda)=e^{\\lambda A_1}$ where ${A_1^2}=-1$.\nThe mid-point of such a geodesic, $L'(\\pi\/2)=A_1$\nagain defines a complex structure ${A_1}={O^T}JO$,\nwhere $J$ is given by (\\ref{eqn:canonical-J}) so that\n$\\pi_{k}(\\text{O}(N\/4))=\\pi_{k-1}(\\text{O}(N\/4)\/\\text{U}(N\/8))$.\nIn a similar way, minimal geodesics in\n$\\text{O}(N\/4)\/\\text{U}(N\/8)$ from $A_1$ to $-A_1$\ncan be parametrized by their mid-points $A_2$,\nwhich square to $-1$ and anti-commute\nwith $A_1$, thereby defining a quaternionic\nstructure, so that the loop space of\n$\\text{O}(N\/4)\/\\text{U}(N\/8)$ is equivalent to\n$\\text{U}(N\/8)\/\\text{Sp}(N\/8)$ and, hence\n$\\pi_{k}(\\text{O}(N\/4)\/\\text{U}(N\/8))=\n\\pi_{k-1}(\\text{U}(N\/8)\/\\text{Sp}(N\/8))$.\nThus, we see that {\\it the passage from\none of the classifying spaces to its loop space\nis the same as the imposition of a symmetry\nsuch as time-reversal to a system classified by that space}:\nboth involve the choice of successive anticommuting\ncomplex structures.\nContinuing in this fashion (see Ref. \\onlinecite{Milnor63}),\nwe recover {\\it Bott periodicity}:\n\\begin{multline}\n\\label{eqn:Bott periodicity}\n{\\hskip -0.5 cm} {\\pi_k}(\\text{O}(16N))=\\\\\n\\pi_{k-1}(\\text{O}(16N)\/\\text{U}(8N))\n= \\pi_{k-2}(\\text{U}(8N)\/\\text{Sp}(4N)) \\\\\n= \\pi_{k-3}(\\mathbb{Z}\\times\n\\text{Sp}(4N)\/(\\text{Sp}(2N)\\times\\text{Sp}(2N)))\n= \\pi_{k-4}(\\text{Sp}(2N))\\\\\n= \\pi_{k-5}(\\text{Sp}(2N)\/\\text{U}(2N))\n= \\pi_{k-6}(\\text{U}(2N)\/\\text{O}(2N))\\\\\n = \\pi_{k-7}(\\mathbb{Z}\\times\n \\text{O}(2N)\/(\\text{O}(N)\\times\\text{O}(N)))\\\\\n = \\pi_{k-8}(\\text{O}(N))\n\\end{multline}\nThe approximations made at each step require\nthat $N$ be in the {\\it stable\nlimit}, in which the desired homotopy groups are\nindependent of $N$. For instance,\n${\\pi_k}(\\text{O}(N))$ is\nindependent of $N$ for $N>k\/2$.\n\nIt is straightforward to compute $\\pi_0$ for each of these\ngroups:\n\\begin{eqnarray}\n\\label{eqn:stable-pi-0}\n{\\pi_0}(\\text{O}(N))&=&\\mathbb{Z}_2\\cr\n\\pi_{0}(\\text{O}(2N)\/\\text{U}(N))&=&\\mathbb{Z}_2\\cr\n\\pi_{0}(\\text{U}(2N)\/\\text{Sp}(N))&=&0\\cr\n\\pi_{0}(\\mathbb{Z}\\times\n\\text{Sp}(2N)\/\\text{Sp}(N)\\times\\text{Sp}(N))&=&\\mathbb{Z}\\cr\n\\pi_{0}(\\text{Sp}(N))&=&0\\cr\n\\pi_{0}(\\text{Sp}(N)\/\\text{U}(N))&=&0\\cr\n\\pi_{0}(\\text{U}(N)\/\\text{O}(N))&=&0\\cr\n\\pi_{0}(\\mathbb{Z}\\times\n \\text{O}(2N)\/(\\text{O}(N)\\times\\text{O}(N)))\n &=&\\mathbb{Z}\n\\end{eqnarray}\nCombining (\\ref{eqn:stable-pi-0}) with (\\ref{eqn:Bott periodicity}),\nwe can compute any of the stable homotopy groups\nof the above $8$ classifying spaces. As discussed above,\nthe space of gapped free fermion Hamiltonians in\n$d$-dimensions in a given {\\it symmetry class}\n(determined by the number modulo $8$ of symmetries squaring to $-1$\nminus the number of those squaring to $+1$) is\nhomotopically-equivalent to one of these classifying spaces.\nThus, using (\\ref{eqn:stable-pi-0}) with (\\ref{eqn:Bott periodicity})\nto compute the stable homotopy groups of these classifying spaces\nleads to a complete classification of topological states and\ntopological defects in all dimensions and\nsymmetry classes, as we now discuss.\n\nGapped Hamiltonians with a given symmetry and dimension are classified by\n$\\pi_0$ of the corresponding classifying space in Table \\ref{tbl:classifying}. Due to Bott periodicity, the table is periodic along both directions of dimension and symmetry, so that there are 8 distinct symmetry classes.\nRyu {\\it et. al}\\cite{Ryu08} denoted these classes using the\nCartan classification of symmetric spaces, following\nthe corresponding classification of disordered systems and\nrandom matrix theory \\cite{Zirnbauer96,Altland97} which\nwas applied to the (potentially-gapless) surface states of\nthese systems. In this notation, systems with no symmetry\nare in class D, those with $T$ only are in DIII, and those\nwith $T$ and $Q$ are in AII. The other 5 symmetry classes,\nC, CI, CII, AI, and BDI arise, arise in systems which\nhave spin-rotational symmetry or a sublattice symmetry.\nThere are actually 2 more symmetry classes (denoted by A and AIII in the random matrix theory) which lie on a separate $2\\times 2$ periodic table, which is less relevant to the present work and will be discussed in the Appendix \\ref{sec:QnotT}. In Table \\ref{tbl:periodic} we have listed examples of\ntopologically-nontrivial states in physical dimensions 1,2,3\nin all 8 symmetry classes. To help with\nthe physical understanding of these symmetry classes, we have also listed the physical requirements for the realization of each symmetry class.\nIn each dimension, there are two symmetry classes\nin which the topological states are classified by integer invariants and\ntwo symmetry classes in which the different states are\ndistinguished by $\\mathbb{Z}_2$ invariants. In all the\ncases in which a real material or a well-defined physical model system is known \nwith non-trivial $\\mathbb{Z}$ or $\\mathbb{Z}_2$ invariant,\nwe have listed a typical example in the table.\nIn some of the symmetry classes, non-trivial examples\nhave not been realized yet, in which case we leave the topological classification\n$\\mathbb{Z}$ or $\\mathbb{Z}_2$ in the corresponding position in the table.\n\nIn one dimension, generic superconductors (class D) are classified by $Z_2$, of which the nontrivial example is a $p$-wave superconductor with a single Majorana zero mode on the edge. The time-reversal invariant superconductors (class DIII) are also classified by $Z_2$. The nontrivial example is a superconductor in which spin up electrons pair into a $p$-wave superconductor and spin down electron form another $p$-wave superconductor which is exactly the time-reversal of the spin-up one. Such a superconductor has two Majorana zero modes on the edge which form a Kramers pair and are topologically protected. The two integer classes are bipartite time-reversal invariant insulators with (CII) and without (BDI) spin-orbit coupling. An example of the\nBDI class is a graphene ribbon, or equivalently a carbon nanotube with a zigzag edge. \\cite{Fujita96,Nakada96}. The low-energy band structure of graphene and carbon nanotubes is well-described by a tight-binding model with nearest-neighbor hopping on a honeycomb lattice, which is bipartite. The integer-valued topological quantum number corresponds to the number of zero modes on the edge, which depends on the orientation of the nanotube. Because carbon has negligible spin-orbit coupling, to a good approximation it can be viewed as a system in the BDI class, but it can also be considered as a system in class CII when spin-orbit coupling is taken into account. In two dimensions, generic superconductors (class D) are classified by an integer, corresponding to the number of chiral Majorana edge states on the edge. The first nontrivial example was the $p+ip$ wave superconductor, shown by Read and Green\\cite{Read00} to have one chiral Majorana edge state.\nNon-trivial superconductors in symmetry class D\nare examples of {\\it topological superconductors}.\nSome topological superconductors can be consistent with spin rotation symmetry; singlet superconductors (class C) are also classified by integer, with the simplest physical example a $d+id$ wave superconductor. Similar to the 1D case, the time-reversal invariant superconductors (class DIII) are classified by $Z_2$, of which the nontrivial example is a superconductor with $p+ip$ pairing of spin-up electrons and $p-ip$ pairing of spin-down electrons.\\cite{Roy06,Qi09,Ryu08} The other symmetry class in\n2D with a $Z_2$ classification is composed of time-reversal invariant insulators (class AII), also known as quantum spin Hall insulators\\cite{Kane05A,Kane05B,Bernevig06a}. The quantum spin Hall insulator phase has been theoretically predicted\\cite{Bernevig06b} and experimentally realized\\cite{Koenig07} in HgTe quantum wells. In three dimensions, time-reversal invariant insulators (class AII) are also classified by $Z_2$. \\cite{Fu07,Moore07,Roy09} The $Z_2$ topological invariant corresponds to a topological magneto-electric response with quantized coefficient $\\theta=0,\\pi$\\cite{Qi08}. Several nontrivial topological insulators in this class have been theoretically predicted and experimentally realized, including ${\\rm Bi_{1-x}Sb_x}$ alloy\\cite{Fu07b,Hsieh08} and the family of ${\\rm Bi_2Se_3}$, ${\\rm Bi_2Te_3}$, ${\\rm Sb_2Te_3}$\\cite{Zhang09,Xia09,Chen09}. In 3D, time-reversal invariant superconductors (class DIII) are classified by an integer, corresponding to the number of massless Majorana cones on the surface.\\cite{Ryu08} A nontrivial example with topological quantum number $N=1$ turns out to be the B phase of He$^3$.\\cite{Qi09,Roy08b,Ryu08} The other classes with nontrivial topological classification in 3D are singlet time-reversal invariant superconductors (CI), classified by an integer; and bipartite time-reversal invariant insulators (CII), classified by $Z_2$. Some models have been proposed\\cite{Schnyder09} but no realistic material proposal or experimental realization has been found in these two classes. We would like to note that different physical systems can correspond to the same symmetry class. For example, bipartite superconductors are also classified by the BDI class.\n\nThe two remaining symmetry classes (unitary (A) and chiral unitary (AIII)) corresponds to systems with charge conservation symmetry but without\ntime-reversal symmetry, which forms a separate $2\\times 2$ periodic table. For the sake of completeness, we carry out\nthe preceding analysis for these two classes in Appendix \\ref{sec:QnotT}.\n\nTopological defects in these states are classified by higher homotopy groups of the classifying spaces. Following the convention of Ref. \\cite{Kitaev09}, we name the classifying spaces by $R_q,q=0,1,2,...,7$, with $R_1=O(N),~R_2=O(2N)\/U(N),~...~R_7=U(N)\/O(N),~R_0=\\mathbb{Z}\\times\n \\text{O}(2N)\/(\\text{O}(N)\\times\\text{O}(N))$ in the order of Eq. (\\ref{eqn:stable-pi-0}). The symmetries in Table \\ref{tbl:periodic} can be labeled by $p=0,1,2,...,7$, so that in $d$ dimensions and $p$-th symmetry class, the classifying space is $R_{2+p-d}$. A topological defect with dimension $D$ ($D<d$) is classified by\n\\begin{eqnarray}\n\\pi_{d-D-1}\\left(R_{2+p-d}\\right)=\\pi_0\\left(R_{p-D+1}\\right)\\label{eq:defectclassification}\n\\end{eqnarray}\nwhich is determined by the zero-th homotopy groups in Eq. (\\ref{eqn:stable-pi-0}). The important conclusion we obtain from this formula is that the classification of topological defect is determined by the dimension of the defect $D$ and the symmetry class $p$, and is {\\em independent} of the spatial dimension $d$. For $p$ and $D$, we obtain the same $8\\times 8$ periodic table for the classification of topological defects. \\cite{noteaboutKane}\n\nIn the following\nwe will focus on the 3D system with no symmetry, and discuss the generic cases in Sec. \\ref{sec:discussion}.\nSince a 3D system with no symmetry is classified\nby $R_7=\\text{U}(N)\/\\text{O}(N)$, point-like defects in\nsuch a system are classified by\n${\\pi_2}(\\text{U}(N)\/\\text{O}(N))$.\nIn the stable limit,\n${\\pi_2}(\\text{U}(N)\/\\text{O}(N))=\\mathbb{Z}_2$.\nHowever, we note that, for smaller values of\n$N$ below the stable limit, the classification is a little different,\ne.g. ${\\pi_2}(\\text{U}(2)\/\\text{O}(2))=\n{\\pi_2}(\\text{U}(1)\\times{S^2})=\\mathbb{Z}$.\nThe `8-band model' in Ref. \\onlinecite{Teo10},\nwhich we discussed in the strong-coupling\nlimit in the previous section,\nis an example of this particular `small $N$'\ncase, which is why the hedgehogs\nin that model are classified by a\nwinding number $\\in\\mathbb{Z}$.\n\n\n\\section{Exchanging Particles Connected\nby Ribbons in 3D}\n\\label{sec:tethered}\n\nIn this section, we will take an O(3) non-linear\n$\\sigma$ model, i.e. one with target space\n$S^2$, as a toy model for our problem.\nIt is essentially the `8-band' model\ndiscussed in Ref. \\onlinecite{Teo10}\nand in the strong-coupling limit in Section\n\\ref{sec:strong-coupling}. As noted above,\nit is the $2N=8$ limit of the classification\nreviewed above. It will be more familiar to\nmost readers and easier to visualize\nthan the full problem which we discuss in\nthe next section.\nWe will give a heuristic explanation\nof $\\pi_1$ of the configurations of\ndefects and will make a few comments about\nwhere our toy model goes wrong, compared\nto the full problem. In the next section,\nwe will undertake a full and careful calculation\nof $\\pi_1$ of the configurations of\ndefects of a model with classifying space\n$\\text{U}(N)\/\\text{O}(N)$.\n\nA free fermion Hamitonian with no\nsymmetry, but limited to 8 bands,\ncan be expanded about its minimum\nenergy point in the Brillouin zone as\n\\begin{equation}\nH = i\\chi ({\\partial_i}{\\gamma_i} +\n{n_i}{\\Gamma_i})\\chi\n\\end{equation}\nFrom the considerations in the previous\nsection, we learned that the space of mass terms\nis ${S^2}$, which is why we have written the mass\nterm in the above form with a unit vector $\\vec{n}\\in{S^2}$.\n(More precisely, the space of mass terms is\n$\\text{U}(2)\/\\text{O}(2)=\\text{U}(1)\\times{S^2}$,\nand the mass term can be written\n$e^{\\theta{\\gamma_1}{\\gamma_2}{\\gamma_3}}{n_i}{\\Gamma_i}\ne^{-\\theta{\\gamma_1}{\\gamma_2}{\\gamma_3}}$.\nHowever, the extra U(1) plays no role here and can\nbe ignored.)\nIn the model of Section \\ref{sec:strong-coupling},\nthe three components of $\\vec{n}$ correspond to dimerization\nin each of the three directions on the cubic\nlattice. In Teo and Kane's\nmodel they correspond to the real and imaginary\nparts of the superconducting order parameter\nand the sign of the Dirac mass at a band inversion.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=3in]{north-pole.pdf}\n\\caption{The pre-image of the North pole in the $S^2$\ntarget space of $\\vec{n}$ is a collection of arcs\nconnecting hedgehogs. A fixed tangent vector at the North\npole defines a vector field along the arcs, thereby making\nthem framed arcs, which may be viewed as ribbons.}\n\\label{fig:north-pole}\n\\end{figure}\n\nNow consider defects in the $\\vec{n}$ field,\nwhich are classified by ${\\pi_2}({S^2})=\\mathbb{Z}$.\nDefects with winding number $\\pm 1$ are positive and\nnegative hedgehogs. For simplicity, we will focus on\nthese; higher winding number hedgehods can be\nbuilt up from these. (In the real model, as opposed to the\ntoy model, the `hedgehogs' have a $\\mathbb{Z}_2$\nclassification so there are no higher winding number\nhedgehogs and, in fact, they do not even have a sign.)\nAs noted by Teo and Kane \\cite{Teo10}, the\n$\\vec{n}$ field around a hedgehog can be\nvisualized in a simplified way, following Wilczek\nand Zee's discussion of the Hopf term in\na $2+1$-D O(3) non-linear $\\sigma$ model\\cite{Wilczek83}.\nThe field $\\vec{n}$ can be viewed as a map from\nthe physical space where the electrons live,\n$\\mathbb{R}^3$. If we assume\nthat the total winding number is zero\n(equal numbers of + and - hedgehogs)\nand that $\\vec{n}$ approaches a constant\nat $\\infty$, we can compactify the physical space $\\mathbb{R}^3$\nso that it is $S^3$.\nThe target space of the map is $S^2$. The pre-image\nof the north pole $N\\in S^2$ is a set of arcs\nand loops. The choice of the north pole $N$\nis arbitrary, and any other point on the sphere\nwould be just as good for the following discussion.\nLet's ignore the loops for the moment\nand focus on the arcs.\nSince $\\vec{n}$ points in every direction at\na hedgehog, the arcs terminate at hedgehogs.\nIn fact, each arc connects a $+1$ hedgehog to\na $-1$ hedgehog.\nWe now pick an arbitrary unit vector in the tangent\nspace of the sphere at $N$. This vector can be\npulled back to $S^3$ to define a vector field\nalong the arcs which is clearly normal\nto the arcs. This is a {\\it framing}, which allows us to define,\nfor instance, a self-winding number for an arc. Intuitively,\nwe can think of a framing as a thickening of\nan arc into a ribbon.\nThus, the field $\\vec{n}$ allows\nus to to define a set of framed arcs connecting\nthe hedgehogs -- in other words, a set\nof ribbons connecting the hedgehogs. As the\nnormal vector twists around an arc, the ribbon twists,\nas shown in Fig. \\ref{fig:north-pole} (Although we will draw the ribbons\nas bands in the physical space, their width should\nnot be taken seriously; they should really be viewed as\narcs with a normal vector field.)\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2in]{ribbons-_arcs.pdf}\n\\caption{We will depict framed arcs or ribbons\nas arcs with twists accounted for by drawing\nkinks in the arcs, as shown above.}\n\\label{fig:ribbons->arcs}\n\\end{figure}\n\nAlthough these ribbons are strongly reminiscent\nof particle trajectories, it is important to keep in\nmind that they are not. A collection of ribbons\nconnecting hedgehogs defines a state of the system\nat an instant of time. Ribbons, unlike particle trajectories,\ncan cross. They can break and reconnect as the system\nevolves in time. As hedgehogs are moved, the\nribbons move with them.\n\nA configuration of particles connected pairwise\nby ribbons is a seemingly crude approximation\nto the full texture defined by $\\vec{n}$.\nHowever, according to the Pontryagin-Thom\nconstruction, as we describe in the next Section\n(and explain in Appendix \\ref{sec:appendix_pontryagin_thom_construction}),\nit is just as good as the full texture for topological\npurposes. Thus, we focus on the space of\nparticles connected pairwise by ribbons.\n\n\nWe now consider a collection of such particles\nand ribbons. For a topological discussion,\nall that we are interested in about\nthe ribbons is how many times they twist, so we will\nnot draw the framing vector but will, instead, be careful to\nput kinks into the arcs in order to keep track\nof twists in the ribbon, as depicted in Fig. \\ref{fig:ribbons->arcs}.\nThe fundamental group of their configuration space is\nthe set of transformation which return the particles\nand ribbons to their initial configurations,\nwith two such transformations identified if they\ncan be continuously deformed into each other.\nConsider an exchange of two $+1$ hedgehogs, as depicted\nin Fig. \\ref{fig:exchange1}.\nAlthough this brings the particles back to their\ninitial positions (up to a permutation, which\nis equivalent to their initial configuration\nsince the particles are identical), it does not\nbring the ribbons back to their initial configuration.\nTherefore, we need to do a further motion\nof the ribbons. By cutting and rejoining them\nas shown in Fig. \\ref{fig:exchange2}a, a procedure\nwhich we call `recoupling', we now have the ribbons\nconnecting the same particles as in the initial configuration.\nBut the ribbon on the left has a twist in it. So we rotate\nthat particle by $-2\\pi$ in order to undo the twist,\nas in Fig. \\ref{fig:exchange2}b.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.5in]{exchange1.pdf}\n\\caption{When two defects are exchanged, the\n$\\vec{n}$-field around them is modified. This is encapsulated\nby the dragging of the framed arcs as the defects are\nmoved.}\n\\label{fig:exchange1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.5in]{exchange2.pdf}\n\\includegraphics[width=1.75in]{exchange3.pdf}\n\\caption{(a) In order to restore the framed arcs so that\nthey are connecting the same defects, it is necessary\nto perform a recoupling by which they are reconnected.\nIn order to keep track of the induced twist, it is easiest\nto perform the recoupling away from the overcrossing.\n(b) The particle on the left must be rotated by\n$-2\\pi$ in order to undo a twist in the framed arc\nto which it is attached.}\n\\label{fig:exchange2}\n\\end{figure}\n\nLet us use $t_i$ to denote such a transformation, defined by\nthe sequence in Figs. \\ref{fig:exchange1},\n\\ref{fig:exchange2}a, and \\ref{fig:exchange2}b.\nThe $t_i$s do not satisfy the multiplication rules of the\npermutation group. In particular, ${t_i}\\neq t_i^{-1}$.\nThe two transformations ${t_i}$ and $t_i^{-1}$\nare not distinguished by whether the exchange\nis clockwise or counter-clockwise -- this is immaterial\nsince a clockwise exchange can be\ndeformed into counter-clockwise one -- but rather\nby which ribbon is left with a twist which must be\nundone by rotating one of the particles.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{exchange4.pdf}\n\\caption{The sequence of moves which defines\n$t_i^{-1}$. (Here, the $i^{\\rm th}$ particle is\nat the top left and the $(i+1)^{\\rm th}$ is at the\ntop right.) This may be contrasted with\nthe sequence in Figs. \\ref{fig:exchange1},\n\\ref{fig:exchange2}a, and \\ref{fig:exchange2}b,\nwhich defines ${\\sigma_i^1}$.}\n\\label{fig:exchange4}\n\\end{figure}\n\nTo see that the operations $t_i$, defined by\nthe sequence in Figs. \\ref{fig:exchange1},\n\\ref{fig:exchange2}a, and \\ref{fig:exchange2}b,\nand $t_i^{-1}$, defined by the sequence in \\ref{fig:exchange4},\nare, in fact, inverses, it is useful to note that when they are performed\nsequentially, they involve two $2\\pi$ twists of the same hedgehog.\nIn \\ref{fig:exchange2}b, it is the hedgehog on the left\nwhich is twisted; this hedgehog moves to the right in the first step of\n\\ref{fig:exchange4} and is twisted again in the fourth step.\nOne should then note that a double twist\nin a ribbon can be undone continuously by using\nthe ribbon to ``lasso'' the defect, a famous fact\nrelated to the existence of spin-$1\/2$ and the\nfact that ${\\pi_1}(SO(3))=\\mathbb{Z}_2$.\nThis is depicted in Fig. \\ref{fig:lassomove} in Appendix\n\\ref{sec:Postnikov}. It will be helpful for our late\ndiscussion to keep in mind that $t_i$ not only\npermutes a pair of particles but also rotates one\nof them; any transformation built up by multiplying\n$t_i$s will enact as many $2\\pi$ twists as pairwise\npermutations modulo two.\n\nThus far we have only discussed the $+1$ hedgehogs.\nWe can perform the similar transformations which\nexchange $-1$ hedgehogs. We will not repeat the above\ndiscussion for $-1$ hedgehogs since the discussion\nwould be so similar; furthermore, in the $N\\rightarrow\\infty$\nmodel which is our main interest, defects do not carry\na sign, so they can all be permuted with each other.\n\nWe have concluded that ${t_i}\\neq t_i^{-1}$\nand, therefore, the group of transformations which bring\nthe hedgehogs and ribbons back to their initial configuration\nis not the permutation group. This leaves open the question:\nwhat is ${t_i}^2$? The answer is that $t_i^2$ can be continuously\ndeformed into a transformation which doesn't involve\nmoving any of the particles -- Teo and Kane's\n`braidless operations'. Consider the transformation\n$x_i$ depicted in Fig. \\ref{fig:twist-transfer}. Defect $i$\nis rotated by $2\\pi$, the twist is transferred\nfrom one ribbon to the other,\nand defect $i+1$ is rotated by $-2\\pi$.\nSince a $4\\pi$ rotation can be unwound, as depicted\nin Fig. \\ref{fig:lassomove}, ${x_i^2=1}$.\n\nIntuitively, one expects that ${x_i}={t_i}^2$\nsince neither ${x_i}$ nor ${t_i}^2$\npermutes the particles and both of them involve\n$2\\pi$ rotations of both particles $i$ and $i+1$.\nTo show that this is, in fact, the correct, we need\nto show that the history in Fig. \\ref{fig:twist-transfer}\ncan be deformed into the sequence of Figs.\n\\ref{fig:exchange1}, \\ref{fig:exchange2}a,\n\\ref{fig:exchange2}b repeated twice.\nIf the history in Fig. \\ref{fig:twist-transfer} is\nviewed as a `movie' and the sequence of Figs.\n\\ref{fig:exchange1}, \\ref{fig:exchange2}a,\n\\ref{fig:exchange2}b repeated twice is\nviewed as another `movie', then we need a\none-parameter family of movies -- or a `movie\nof movies' -- which connects the two movies.\nWe will give an example of such a `movie\nof movies' shortly. With this example in hand, the\nreader can verify that  ${x_i}={t_i}^2$ by drawing\nthe corresponding pictures, but we will not do so here\nsince this discussion\nis superseded, in any case, by the\nthe next section, where a similar result is shown for\nthe $N\\rightarrow \\infty$ problem by more general\nmethods. We simply accept this identity for now.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.25in]{twist-transfer.pdf}\n\\caption{The sequence of moves which defines\n$x_i$: the defect on the left is rotated by $2\\pi$,\nthe twist is transfered to the ribbon on the right by two\nrecouplings, and then the defect on the right is\nrotated by $-2\\pi$. (Here, the $i^{\\rm th}$ defect is\nat the top left and the $(i+1)^{\\rm th}$ is at the\ntop right.) The defects themselves are not moved\nin such a process.}\n\\label{fig:twist-transfer}\n\\end{figure}\n\nWe now consider the commutation\nrelation for the $x_i$s. Clearly, for $|i-j|\\geq 2$,\n${x_i} {x_j} = {x_j} {x_i}$. It is also\nintuitive to conclude that\n\\begin{equation}\n{x_i} x_{i+1} = x_{i+1} {x_i}\n\\end{equation}\nsince the order in which twists are transferred is\nseemingly unimportant.\nHowever, since this is a crucial point, we verify it\nby showing in Figure \\ref{fig:movie-of-movies}\nthat the sequence of moves which\ndefines ${x_i} x_{i+1}$ (a `movie') can be continuously\ndeformed into the sequence of moves which\ndefines $x_{i+1}{x_i}$ (another `movie'). Such a deformation\nis a `movie of movies'; going from left-to-right in\nFig. \\ref{fig:movie-of-movies} corresponds to going\nforward in time while going from up to down corresponds\nto deforming from one movie to another.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=6in]{grid.pdf}\n\\caption{The sequence of moves which defines\n${x_i}x_{i+1}$ is shown in the top row.\nThe sequence of moves which defines\n$x_{i+1}{x_i}$ is shown in the bottom row.\nThe rows in between show how the\ntop row can be continuously deformed\ninto the bottom one. Such a deformation\nof two different sequences is a `movie of movies'\nor a two-parameter family of configurations.\nMoving to the right increases the time parameter while\nmoving down increases the deformation parameter\nwhich interpolates between ${x_i}x_{i+1}$ and\n$x_{i+1}{x_i}$.}\n\\label{fig:movie-of-movies}\n\\end{figure*}\n\nThus, we see that the equivalence\nclass of motions of the defects (i.e. $\\pi_1$\nof their configuration space) has an Abelian\nsubgroup generated by the $x_i$s. Since ${x_i^2}=1$\nand they all commute with each other, this is simply\n$n-1$ copies of $\\mathbb{Z}_2$,\nor, simply, $(\\mathbb{Z}_2)^{n-1}$.\n\nIn order to fully determine\nthe group of transformations which bring\nthe hedgehogs and ribbons back to their initial configuration,\nwe need to check that the $t_i$s generate the full\nset of such transformations -- i.e. that the transformations\ndescribed above and those obtained by combining them\nexhaust the full set. In order to do this, we need\nthe commutation relations of the $t_i$s with\neach other. Clearly, ${t_i}{t_j}={t_j}{t_i}$ for\n$|i-j|\\geq 2$ since distant operations which\ndo not involve the same hedgehogs nor the same\nribbons must commute. On the other hand\noperations involving the same hedgehogs or\nribbons might not commute. For instance,\n\\begin{equation}\n\\label{eqn:t-x-comm}\n{t_i} x_{i+1} = x_{i}x_{i+1}t_i\n\\end{equation}\nTo see why this is true, note that\nif we perform $x_{i+1}$ first, then defects $i+1$ and\n$i+2$ are twisted by $2\\pi$. However, $t_i$ then permutes\n$i$ and $i+1$ and twists $i$ by $2\\pi$. Thus, the left-hand-side\npermutes $i$ and $i+1$ and only twists $i+2$. The\n$(i+1)^{\\rm th}$ hedgehog was twisted by $x_{i+1}$\nand then permuted by $t_i$ so that it ended up in the\n$i^{\\rm th}$ position, where it was twisted again in\nthe last step in $t_i$; two twists can be continuously deformed\nto zero, so this hedgehog is not twisted at all.\nThe right-hand-side similarly permutes $i$ and $i+1$ and\nonly twists $i+2$ by $2\\pi$. The reader may find it instructive\nto flesh out the above reasoning by constructing\na movie of movies.\n\nThe multiplication rule which we have just described\n(but not fully justified) is that of a semi-direct product,\nwhich is completely natural in this context:\nwhen followed by a permutation, a transfer of twists\nends up acting on the permuted defects.\nThe twists $x_i$ form the group $(\\mathbb{Z}_2)^{n-1}$\nwhich we can represent by $n$-component vectors\nall of whose entries are $0$ or $1$ which satisfy the\nconstraint that the sum of the entries is even.\nThe entries tell us whether a given hedgehog is\ntwisted by $2\\pi$ or not. In any product of $x_i$s,\nan even number of hedgehogs is twisted by $2\\pi$.\nNow consider, for $n$ odd, the group elements given by\n\\begin{equation}\n\\label{sigma-i-odd}\n{\\sigma_i} = x_{n-1} x_{n-3} \\ldots x_{i+3}x_{i+1} x_{i-2}x_{i-4}\n\\ldots x_{1}\\, t_{i}\n\\end{equation}\nfor $i$ odd and\n\\begin{equation}\n\\label{sigma-i-even}\n{\\sigma_i} = x_{n-1} x_{n-3} \\ldots x_{i+2}x_{i} x_{i-1}x_{i-3}\n\\ldots x_{1}\\, t_{i}\n\\end{equation}\nfor $i$ even.\nFrom (\\ref{eqn:t-x-comm}), we see that ${{\\sigma_i}^2}=1$.\nThe group element $\\sigma_i$ permutes the\n$i^{\\rm th}$ and $(i+1)^{\\rm th}$\nhedgehogs and twists all of the hedgehogs.\nThus, the $\\sigma_i$s generate\na copy of the permutation group $S_n$. The $\\sigma_i$s do not\ncommute with the $x_i$s, however; instead they act\naccording to the semi-direct product structure noted above.\nOn the other hand, the situation is a bit different for\n$n$ even. This may be a surprise since one might expect\nthat $n$ even is the same as $n$ odd but with the last\nhedgehog held fixed far away. While this is true, exchanging the\nlast hedgehog with the others brings in an additional layer of\ncomplexity which is not present for $n$ odd. The construction\nabove, Eqs. \\ref{sigma-i-odd}, \\ref{sigma-i-even}, does not\nwork. One of the hedgehogs will be left untwisted by such\na construction; since subsequent $\\sigma_i$s will permute this\nuntwisted hedgehog with others, we must keep track of the\nuntwisted hedgehog and, therefore, the $\\sigma_i$s will\nnot generate the permutation group. \nIn the even hedgehog number case, the group of\ntranformations has a $(\\mathbb{Z}_2)^{n-1}$ subgroup,\nas in the odd case, but there isn't an $S_n$ subgroup,\nunlike in the odd case. To understand the even case,\nit is useful to note that in both cases, every transformation\neither (a) twists an even number of ribbons,\nwhich is the subgroup $(\\mathbb{Z}_2)^{n-1}$;\n(b) performs an even permutation, which is the\nsubgroup $A_n$ of $S_n$; or (c)\ntwists an odd number of ribbons and performs\nan odd permutation. Another way of saying this is\nthat the group of transformations is the `even part' of\n$(\\mathbb{Z}_2)^{n}\\rtimes S_n$: the subgroup of\n$(\\mathbb{Z}_2)^{n}\\rtimes S_n$ consisting of\nthose elements whose $(\\mathbb{Z}_2)^{n}$ parity\nadded to their $S_{n}$ parity is even. In the\nodd hedgehog number case, this is the\nsemidirect product $(\\mathbb{Z}_2)^{n-1}\\rtimes S_n$;\nin the even hedgehog number case, it is not.\nAs we will see in Section \\ref{sec:projective}, the\ndifference between the even and odd hedgehog number\ncases is related to the fact that, for an even number\nof hedgehogs, the Hilbert space decomposes into\neven and odd total fermion number parity sectors.\nBy contrast, the situation is simpler for an\nodd number of hedgehogs, where the parity\nof the total fermion number is not well-defined and\nthe representation is irreducible.\n\n\nTo summarize, we have given some plausible\nheuristic arguments that the `statistics' of $+1$ hedgehogs\nin a model of $2N=8$ Majorana fermions\nis governed by a group\n$E((\\mathbb{Z}_2)^{n}\\rtimes S_{n})$,\nthe `even part' of $(\\mathbb{Z}_2)^{n}\\rtimes S_{n}$:\nthose elements of $(\\mathbb{Z}_2)^{n}\\rtimes S_{n}$\nin which the parity of the sum of the entries of the element\nin $(\\mathbb{Z}_2)^{n}$ added to the parity of the\npermutation in $S_{n}$ is even.\n(The same group governs the $-1$\nhedgehogs). Rather than devoting more time\nhere to precisely determining the group for\nthe toy model, we will move on to the problem\nwhich is our main concern here,\na system of $2N\\rightarrow\\infty$ Majorana fermions.\nThis problem is similar, with some important differences.\n(1) The target space is no longer $S^2$ but is, instead,\n$U(N)\/O(N)$. (2) Consequently, the defects do not carry a sign. There\nis no preferred pairing into $\\pm$ pairs; the defects\nare all on equal footing. All $2n$ of them can be exchanged.\n(3) The group obtained by computing $\\pi_1$\nof the space of configurations of $2n$\ndefects then becomes the direct product\nof the `ribbon permutation group'\n${\\cal T}^r_{2n}$ with a trivial ${\\mathbb Z}$,\n${\\cal T}_{2n}={\\mathbb Z} \\times {\\cal T}^r_{2n}$.\nThe ribbon permutation group ${\\cal T}^r_{2n}$ is given by\n${\\cal T}^r_{2n} \\equiv {\\mathbb Z}_2 \\times E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$, where $E((\\mathbb{Z}_2)^{2n}\\rtimes S_{2n})$\nis the `even part' of $(\\mathbb{Z}_2)^{2n}\\rtimes S_{2n}$.\n\n\n\n\\section{Fundamental Group of the Multi-Defect Configuration Space}\n\\label{sec:Kane_space}\n\nIn Section \\ref{sec:free-fermion} we concluded that\nthe effective target space for the order parameter\nof a system of fermions in 3D with no symmetries is\n$U(N)\/O(N)$ -- which, as is conventional, we will simply call $U\/O$,\ndropping the $N$ in the large-$N$ limit. This enables\nus to rigorously define the space of topological configurations,\n$K_{2n}$, of $2n$ hedgehogs in a ball, and calculate\nits fundamental group $\\pi_1(K_{2n})$, thereby elucidating\nTeo and Kane's \\cite{Teo10} hedgehog motions and\nunitary transformations.\n\nWe now outline the steps involved in this calculation:\n\n\\begin{itemize}\n\n\\item We approximate the space $U\/O$ by a {\\it cell complex}\n(or CW complex), ${\\cal C}$, a topological space constructed by\ntaking the union of disks of different dimensions and\nspecifying how the boundary of each higher-dimensional\ndisk is identified with a subset of the lower-dimensional disks.\nThis is a rather crude approximation in some respects,\nbut it is sufficient for a homotopy computation.\n\n\\item We divide the problem into (a) the motion of the hedgehogs\nand (b) the resulting deformation of the field configuration\nbetween the hedgehogs. This is accomplished by expressing\nthe configuration space in the following way. Let us call the\nconfiguration space of $2n$ distinct points\nin three dimensions $X_{2n}$. (For the sake of mathematical convenience,\nwe will take our physical system\nto be a ball $B^3$ and stipulate that the points must lie\ninside a ball $B^3$. Let's denote the space of field configurations\nby ${\\cal M}_{2n}$. This space is the space of maps to $U\/O$\nfrom $B^3$ with $2n$ points (at some standard locations) excised.\nThe latter space is denoted by\n${B^3}\\,\\backslash\\, 2n\\mbox{ standard points}$.\nSince we will be approximating $U\/O$ by ${\\cal C}$,\nwe can take ${\\cal M}_{2n}$ to be the space of\nmaps from ${B^3}\\,\\backslash\\, 2n\\mbox{ standard points}$\nto ${\\cal C}$ with boundary conditions at the $2n$ points\nspecified below. Then, there is a {\\it fibration} of spaces:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{M}_{2n}$}\n    \\obj(2,2)[b]{${K}_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\item We introduce another two fibrations which further\ndivide the problem into more manageable pieces:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{${R}_{2n}$}\n    \\obj(2,2)[b]{${K}_{2n}$}\n    \\obj(2,1)[c]{$Y_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{N}_{2n}$}\n    \\obj(2,2)[b]{${Y}_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\nThe original fibration is kind of a \"fiber-product\" of the two new fibrations.\nHere, $R_{2n}$ is essentially the space of order parameter textures\ninterpolating between the hedgehogs, and $Y_{2n}$ is the space of\nconfigurations of $2n$ points with infinitesimal spheres\nsurrounding each point and maps from each of these\nspheres to ${\\cal C}$. ${\\cal N}_{2n}$ is the space\nof maps from $2n$ infinitesimal spheres to ${\\cal C}$,\nwith each one of the spheres surrounding a different\none of the $2n$ points (at some standard locations)\nexcised from $B^3$. We call these order parameter\nmaps from infinitesimal spheres to ${\\cal C}$\n``germs''.\n\n\\item Having broken the problem down into smaller\npieces by introducing these fibrations, we use the\nfact that a fibration\n$F \\rightarrow E\\rightarrow B$\ninduces a long exact sequence for homotopy groups\n\\begin{equation*}\n\\ldots \\rightarrow \\pi_{i}(E)\\rightarrow\n{\\pi_i}(B)\\rightarrow\\pi_{i-1}(F)\\rightarrow\\pi_{i-1}(E)\n\\rightarrow ...\n\\end{equation*}\nFor instance, applying this to the\nfibration ${\\cal M}_{2n}\n\\rightarrow K_{2n} \\rightarrow\nX_{2n}$ leads to the exact sequence\n$\\ldots \\rightarrow \\pi_1({\\cal M}_{2n})\n\\rightarrow \\pi_1(K_{2n}) \\rightarrow\n\\pi_1(X_{2n})\\rightarrow 1$.  It follows that $\\pi_1(K_{2n})$ is an extension of the permutation\ngroup $S_{2n}=\\pi_1(X_{2n})$.\nBy itself, the above long exact sequence is not very helpful\nfor computing any of the homotopy groups involved\nunless we can show by independent means that\ntwo of the homotopy groups are trivial. Then the homotopy\ngroups which lie between the trivial ones in the sequence\nare tightly constrained.\n\n\\item We directly compute that\n${\\pi_1}({\\cal N}_{2n})=(\\mathbb{Z}_2)^{2n}$\nand ${\\pi_1}(X_{2n})=S_{2n}$. We show that\nthe homotopy exact sequence then implies that\n${\\pi_1}(Y_{2n})=(\\mathbb{Z}_2)^{2n}\\rtimes S_{2n}$.\n\n\\item We compute the homotopy groups of $R_{2n}$,\ndefined by the fibration $R_{2n}\\rightarrow\nK_{2n} \\rightarrow Y_{2n}$.\nThis computation involves a different\nway from the cell structure of thinking about the topology\nof a space, called the ``Postnikov tower'', explained in\ndetail Appendix \\ref{sec:Postnikov}. The basic idea is to approximate\na space with spaces with only a few non-trivial homotopy groups.\n(This is analogous to the cell structure, which has only a few\nnon-trivial homology groups.) The simplest examples of such spaces\nare Eilenberg-Mac Lane spaces,\nwhich only have a single non-trivial homotopy group.\nThe Eilenberg-Mac Lane space $K(A,m)$ is defined for a\ngroup $A$ and integer $m$ as the space with homotopy group\n${\\pi_m}(K(A,m))=A$ and ${\\pi_k}(K(A,m))=0$ for all $k\\neq m$.\n(The group $A$ must be Abelian for $m>1$.)\nSuch a space exists and is unique up to homotopy.\nA space $T$ with only two non-trivial homotopy groups can\nbe constructed through the fibration\n$K(B,n) \\rightarrow T \\rightarrow K(A,m)$. The space $T$ has ${\\pi_m}(T)=A$\nand ${\\pi_n}(T)=B$, as may be seen from the\ncorresponding long exact sequence for homotopy groups.\nContinuing in this fashion, one\ncan construct a sequence of such approximations\n$M_n$ to a space $M$. They are defined by\n${\\pi_k}({M_n})={\\pi_k}(M)$ for $k\\leq n$ and\n${\\pi_k}({M_n})=0$ for $k>n$. They can be constructed\niteratively from the fibration\n$K(A,n) \\rightarrow {M_n} \\rightarrow M_{n-1}$, where ${\\pi_n}(M)=A$.\n\n\\item With ${\\pi_1}(Y_{2n})$, ${\\pi_2}(Y_{2n})$,\n${\\pi_0}(R_{2n})$ and  ${\\pi_1}(R_{2n})$ in hand,\nwe compute the desired group ${\\pi_1}(K_{2n})$\nfrom the homotopy exact sequence.\n\n\\end{itemize}\n\n\nWe now go through these steps in detail.\n\\vskip 0.25 cm\n\n{\\bf Approximating U\/O by a cell complex}.\nDepending on microscopic details,\ngradients in the overall phase of the\nfermions may be so costly that we wish\nto consider only configurations in which this\noverall phase is fixed. We will refer to this\nas the scenario in which `phase symmetry is broken'.\nIn this case, the effective target space is $SU\/SO$,\nthe non-phase factor of $U\/O \\cong U(1)\/O(1) \\times SU\/SO$.\nIn this case, we simplify matters by replacing $SU\/SO$\nby $\\widetilde{U\/O}$, the universal cover of $U\/O$.\n$\\widetilde{U\/O}$ is homotopy equivalent to $SU\/SO$,\nso this substitution is harmless.  This substitution\nresults in a reduced configuration space $\\widetilde{K}_{2n}$\nand we will concentrate first on calculating $\\pi_1(\\widetilde{K}_{2n})$.\nIn an appendix, we show that this\nreduction essentially makes no difference:\n$\\pi_1(K_{2n}) = \\pi_1(\\widetilde{K}_{2n})\\times {\\mathbb Z}$.\n\nWe now define a cell complex ${\\cal C}$\napproximating $\\widetilde{U\/O}$.\nIn constructing this cell structure, we are not interested\nin the beautiful homogeneous nature of $\\widetilde{U\/O}$\nbut rather only its homotopy type.  The homotopy type\nof a space tells you everything you will need to know\nto study {\\em deformation classes} of maps either\ninto or out of that space.  An important feature of any\nhomotopy type is the list of homotopy groups\n(but these are by no means a complete characterization\nin general). For $\\widetilde{U\/O}$, the\nhomotopy groups are $\\pi_i(\\widetilde{U\/O}) =\n0, {\\mathbb Z}_2, {\\mathbb Z}_2, 0, {\\mathbb Z}, 0, 0, 0, {\\mathbb Z}$ for $i = 1, \\dots, 9$ and\nthereafter $\\pi_i(\\widetilde{U\/O})$ cycles through the last eight groups.\n(For $U\/O$, the first group would be ${\\mathbb Z}$.)\n\nBecause $\\widetilde{U\/O}$ is simply-connected,\nbut has nontrivial $\\pi_2$, it natural in building a\ncellular model for its homotopy type to begin with $S^2$.\nSince $\\pi_2 (S^2) = {\\mathbb Z}$ and we only need\na ${\\mathbb Z}_2$ for $\\pi_2(\\til{U\/O})$, we should kill off the even\nelements by attaching a 3-cell $D^3$ using a\ndegree-2 map of its boundary 2-sphere to the original $S^2$.\nFor future reference, take this map to be\n$(\\theta, \\phi) \\to (2\\theta, \\phi)$ in a polar coordinate\nsystem where the north pole $N = (\\pi,0)$.\nSimilarly, a 4-cell is attached to achieve\n$\\pi_3(\\til{U\/O}) \\cong {\\mathbb Z}_2$.  The necessity of the\n4-cell is proved (Fact 1) below.\n\nThe preceding logic leads us to the cell structure:\n\\begin{equation}\n\\label{eq:U\/O_cell_structure}\n{\\cal C} = S^2 \\bigcup_{\\text{degree}=2} D^3 \\bigcup_{\\text{2Hopf}} D^4 \\bigcup \\text{cells of dimension} \\geq 5\n\\end{equation}\n\nSince we are only trying to compute the fundamental group $\\pi_1(\\widetilde{K}_{2n})$ from our various homotopy long exact sequences, we do not have to figure out the higher cells (dimension $\\geq 5$) of $\\til{U\/O}$.  We will, however, verify that $\\pi_3(\\til{U\/O})$ is generated by the Hopf map into the base $S^2 \\subset \\til{U\/O}$.\n\nTo summarize, we will henceforth assume that the order\nparameter takes values in the cell complex ${\\cal C}$.\nAlthough ${\\cal C}$ is a crude approximation for\nU\/O, it is good enough for the topological calculations\nwhich follow.\n\n{\\bf Dividing the problem into the motion of the hedgehog\ncenters and the deformation of the field configuration.}\nLet us assume that our physical system is a ball\nof material $B^3$. Let $n \\geq 0$ be the number of\nhedgehog pairs in the system.\nA configuration in $\\widetilde{K}_{2n}$ is a texture in the\norder parameter, $\\Phi(x): {B^3}\\rightarrow {\\cal C}$,\nwhich satisfies the following\nboundary conditions at the boundary of $B^3$\nand at the $2n$ hedgehog locations (which\nare singularities in the order parameter).\nThe order parameter has winding number $0$\nat the boundary of the ball, $\\partial B^3$\nand winding number $1$ around each of the\nhedgehog centers. (Recall that ${\\pi_2}({\\cal C})=\\mathbb{Z}_2$,\nso the winding number can only be $0$ or $1$).\n\nFrom its definition, $\\widetilde{K}_{2n}$ is the total space of a fibration:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{M}_{2n}$}\n    \\obj(2,2)[b]{$\\widetilde{K}_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\noindent The above diagram suggests that we\nshould think of the fibration\n$\\mathcal{M}_{2n}\n\\rightarrow \\widetilde{K}_{2n} \\rightarrow X_{2n}$\nin the following way: above each point in\n$X_{2n}$ there is a fiber $\\mathcal{M}_{2n}$;\nthe total space formed thereby is $\\widetilde{K}_{2n}$.\n(This is not quite a fiber bundle, since we\ndo not require that there be local coordinate\ncharts in which $\\widetilde{K}_{2n}$ is simply the\ndirect product.) Here, $X_{2n}$ is the simply the\nconfiguration space of $2n$ distinct points in $B^3$.\nWe write this formally as\n$X_{2n} = \\prod_{i=1}^{2n} B^3 \\setminus \\text{big diagonal}$.\n(The big diagonal consists of $2n$-tuples of points in $B^3$\nwhere at least two entries are identical.)\nThe space $\\mathcal{M}_{2n}$ consists of maps\nfrom $B^3 \\setminus 2n \\text{ points in a fixed standard position}$\nto ${\\cal C}$ with the prescribed winding numbers given\nin the preceding paragraph.\n\n\n\\begin{figure}[htpb]\n\\centering\n\\includegraphics[scale=0.5]{M2n.pdf}\n\\caption{$B^3 \\setminus 2n$ points in standard position.\nThe space $\\mathcal{M}_{2n}$ consists of maps from\nthis manifold to ${\\cal C}$.}\n\\label{fig:M2n}\n\\end{figure}\n\n{\\bf Germs of order parameter textures.}\nIt is helpful to introduce an intermediate step in the fibration.\nDefine a point in $Y_{2n}$ as a configuration in\n$X_{2n}$ together with a ``germ''\nof $\\Phi(x)$, which we call $\\widetilde{\\Phi}(x)$,\ndefined only near $\\partial B^3$\nand the $2n$ points. The idea behind the germ\n$\\widetilde{\\Phi}(x)$ is to forget about the order parameter\n$\\Phi(x)$ except for its behavior in an infinitesimal\nneighborhood around each hedgehog center\nand at the boundary of the system.\n$\\widetilde{\\Phi}(x)$ must satisfy the same boundary conditions as\n$\\Phi(x)$ itself. We take $\\widetilde{\\Phi}(x)$ to be\nconstant on $\\partial B^3$ and to have\nwinding number $1$ around each of the\nhedgehog centers. With this definition,\nwe now have the fibration:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$\\mathcal{N}_{2n}$}\n    \\obj(2,2)[b]{$Y_{2n}$}\n    \\obj(2,1)[c]{$X_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\noindent where $\\mathcal{N}_{2n}$ is the space of\n(germs of) order parameter textures $\\widetilde{\\Phi}$ from the\nneighborhoods of the $2n$ fixed standard points\nand $\\partial B^3$ to ${\\cal C}$.\nWe will henceforth replace discussion of germs\nwith the equivalent and simpler concept of maps\non $\\partial B^3 \\cup \\left( \\bigcup_{i=1}^{2n} S_i^2 \\right)$\nwhere $S_i^2$ is a small sphere surrounding\nthe $i$th standard point.  Thus,\n\\begin{equation}\n\\mathcal{N}_{2n} \\subset\n\\text{Maps}\\biggl(\\Bigl(\\partial B^3 \\cup \\bigcup_{i=1}^{2n} S_i^2\n\\Bigr) \\to {\\cal C} \\biggr).\n \\end{equation}\n\nWe now define $Q_{2n}$ as the ball $B^3$\nwith small balls (denoted below by interior$(S_i^2)$)\ncentered about the hedgehogs\ndeleted:\n\\begin{equation}\nQ_{2n}= \\Bigl(B^3 \\setminus \\bigcup_{i=1}^{2n} \\text{interior}(S_i^2)\n\\Bigr)\n\\end{equation}\nfor fixed standard positions $i = 1, \\dots , 2n$.\nThen $R_{2n}$ is the space of order parameter textures on\n$Q_{2n}$ which satisfy the boundary condition\nthat the winding number is $0$ on $\\partial B^3$\nand $1$ on each of the small spheres.\nWith this definition, we have the fibration:\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$R_{2n}$}\n    \\obj(2,2)[b]{$\\widetilde{K}_{2n}$}\n    \\obj(2,1)[c]{$Y_{2n}$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n    \\label{eqn:R_2n-def}\n\\enddc\\]\n\n\\noindent Given the cell structure ${\\cal C}$, we can\nspecify the boundary conditions for the order\nparameter precisely. On $\\partial B^3$,\nthe order parameter is equal to the North Pole\nin $S^2$. (Recall that ${S^2}\\subset{\\cal C}$ is\nthe bottom cell of the structure ${\\cal C}$ which\nwe are using to approximate U\/O.)  On each of\nthe spheres $S_i^2$, the order parameter\n$\\Phi(x)$ defines a map from ${S_i^2}\\rightarrow S^2$\nwhich is the identity map\n(where, again $S^2$ is understood as a subset\nof the order parameter space ${S^2}\\subset{\\cal C}$).\nThis ensures that the order parameter has the\ncorrect winding numbers at the boundaries of $Q_{2n}$.\nIn essence, what we have done in writing\nEq. \\ref{eqn:R_2n-def} is to break up an\norder parameter texture containing hedgehogs\ninto (a) the hedgehogs together with the order parameter\non infinitesimal neighborhoods around them (i.e. `germs')\nand (b) order parameter textures in the intervening regions\nbetween the hedgehogs. The space of configurations (a)\nis $Y_{2n}$; the space of configurations (b) is $R_{2n}$.\n\nThe name $R_{2n}$ is for ``ribbons.''\nAs we saw in Section \\ref{sec:tethered},\nif the order parameter manifold were $S^2$,\nwe could summarize an order parameter texture\nby looking at the inverse image of the North Pole\n$N \\subset S^2$ and a fixed tangent vector at the North Pole.\nThe inverse images form a collection of ribbons.\nNow, the order parameter manifold is actually\n(approximated by) ${\\cal C}$, but the bottom cell\nin ${\\cal C}$ is $S^2$. The effect of the 3-cell\nis that hedgehogs lose their sign, so  there is no\nwell-defined ``arrow'' running lengthwise along the ribbons.\nThe 4-cell allows the ``twist'' or framings of ribbons to be\naltered at will by $\\pm 2$.\n\n\n{\\bf Long exact sequence for homotopy groups.}\nIt is very convenient to use fibrations to calculate homotopy groups.\n(For those interested in $K$-theory, the last two chapters of\nMilnor's {\\em Morse Theory} \\cite{Milnor63}\nare a must read and exhibit these methods with clarity.) As noted above,\nfibrations have all the homotopy properties of fiber bundles\nbut are (often) found arising between function spaces\nwhere it would be a lot of work -- and probably a distraction from important business -- to attempt to verify the existence of\nlocally trivial coordinate charts. Operationally, fibrations share with fiber bundles the all-important ``homotopy long exact sequence'':\n\n\\[\\begindc{0}[30]\n    \\obj(1,2)[a]{$F$}\n    \\obj(2,2)[b]{$E$}\n    \\obj(2,1)[c]{$B$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n\\enddc\\]\n\n\\noindent we have:\n$$\\cdots \\to \\pi_{i+1}(B) \\to \\pi_i(F) \\to \\pi_i(E) \\to \\pi_i(B) \\to \\pi_{i-1}(F) \\to \\cdots$$\n\nWe now compute $\\pi_1(Y_{2n})$ from the exact sequence:\n  \n  \n  \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n\\begin{equation}\n\\label{eq:pi_1(Y_2n)_computation}\n\\begindc{0}[3]\n    \\obj(10,70)[a]{$\\pi_2(X_{2n})$}\n    \\obj(28,70)[b]{$\\pi_1(\\mathcal{N}_{2n})$}\n    \\obj(45,70)[c]{$\\pi_1(Y_{2n})$}\n    \\obj(62,70)[d]{$\\pi_1(X_{2n})$}\n    \\obj(80,70)[e]{$\\pi_0(\\mathcal{N}_{2n})$}\n    \\mor{a}{b}{$\\partial$}\n    \\mor{b}{c}{}\n    \\mor{c}{d}{}\n    \\mor{d}{e}{$\\partial$}\n\\enddc\n\\end{equation}\n\n\\noindent\nWe can compute two of the homotopy\ngroups in this equation by inspection.\n$\\pi_1(X_{2n})$ is clearly the symmetric group of point exchange:\n\\begin{equation}\n\\label{eqn:pi_1_X}\n\\pi_1(X_{2n}) = S_{2n} .\n\\end{equation}\nMeanwhile, $\\pi_1(\\mathcal{N}_{2n})$ amounts to (products of)\nloops of maps from the $S_i^2$ to ${\\cal C}$\nand reduces to $2n$ copies of the third homotopy group\nof ${\\cal C}$ (and, therefore, to ${\\pi_3}(\\til{U\/O})$).\nThus, $\\pi_1(\\mathcal{N}_{2n})=(\\mathbb{Z}_2)^{2n}$:\n\\begin{eqnarray}\n\\label{eqn:pi_1_N}\n\\pi_1(\\mathcal{N}_{2n}) = \\prod_{i=1}^{2n} \\pi_1(\\text{Maps}(S_i^2,\\til{U\/O})) &=& \\prod_{2n\\text{ copies}}\\pi_3(\\til{U\/O})\\cr &=&\n({\\mathbb Z}_2)^{2n} .\n\\end{eqnarray}\n\nTo proceed further, we need to evaluate {\\em boundary maps} in the homotopy exact sequence. In Appendix \\ref{sec:hopf-map},\nwe explain boundary maps through the example of the Hopf map.\nConsider Eq. \\ref{eq:pi_1(Y_2n)_computation}.\n$\\pi_2(X_{2n})$ is generated by the $2n\\choose{2}$\ndifferent 2-parameter motions in which a pair of hedgehogs\ncome close together and explore the 2-sphere of possible relative positions around their center of mass.  This 2-parameter family of motions involves no ``rotation'' of the maps $\\widetilde\\Phi$\nwhich describe $\\pi_1(\\mathcal{N}_{2n})$\n(i.e the order parameter configuration in the neighborhood\nof each hedgehog does not rotate as the hedgehogs are moved),\nso the left most $\\partial$ map in Eq. \\ref{eq:pi_1(Y_2n)_computation}\nis zero.  Similarly, a simple exchange of hedgehogs produces no twist\nof the order parameter configuration in the neighborhood\nof either hedgehog, so the second $\\partial$\nmap of Eq. \\ref{eq:pi_1(Y_2n)_computation} is also zero.\nThus, we have a short exact sequence:\n\\[\\begindc{0}[3]\n    \\obj(10,30)[a]{$1$}\n    \\obj(23,30)[b]{${\\mathbb Z}_2^{2n}$}\n    \\obj(41,30)[c]{$\\pi_1(Y_{2n})$}\n    \\obj(60,30)[d]{$S_{2n}$}\n    \\obj(75,30)[e]{$1$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{$\\alpha$}\n    \\mor{c}{d}{$\\beta$}\n    \\mor{d}{e}{}\n\\enddc\\]\nTo derive this short exact sequence, we used\nthe triviality of the boundary maps noted above\nand Eqs. \\ref{eqn:pi_1_X}, \\ref{eqn:pi_1_N}\nto simplify Eq. \\ref{eq:pi_1(Y_2n)_computation}.\n\nThere is a natural group homomorphism $s$:\n$$s:S_{2n}\\rightarrow\\pi_1(Y_{2n})$$\nwhich associates to each permutation a motion\nof hedgehogs which permutes the hedgehogs in $Y_{2n}$\nbut does not rotate the order parameter configurations\n$\\widetilde\\Phi$ near the hedgehogs. Then\n$\\beta \\circ s = id_{S_{2n}}$. In other words,\nthe sequence is {\\em split}:\n\\[\\begindc{0}[3]\n    \\obj(10,30)[a]{$1$}\n    \\obj(25,30)[b]{${\\mathbb Z}_2^{2n}$}\n    \\obj(40,30)[c]{$\\pi_1(Y_{2n})$}\n    \\obj(60,30)[d]{$S_{2n}$}\n    \\obj(75,30)[e]{$1$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{$\\alpha$}\n    \\mor{d}{e}{}\n    \\cmor((46,31)(51,33)(57,31)) \\pright(51,35){$\\beta$}\n    \\cmor((57,29)(51,27)(46,29)) \\pleft(51,25){$s$}[\\atleft,\\dasharrow]\n\\enddc\\]\nThus, $\\pi_1(Y_{2n})$ is a semi-direct product.\nTo determine $\\pi_1(Y_{2n})$ completely, it only remains to\nidentify how $s(S_{2n})$ acts on the twist factors ${\\mathbb Z}_2^{2n}$\nunder conjugation. It is quite clear that this action is the only\nnatural one available: $s(p)$ acts on ${\\mathbb Z}_2^{2n}$ by applying\nthe permutation $p$ to the $2n$ coordinates of $Z_2^{2n}$.\nSo, $\\pi_1(Y_{2n}) \\cong {\\mathbb Z}_2^{2n} \\rtimes S_{2n}$ with group law:\n\n\\begin{equation}\\label{eq:pi_1(Y2n)_group_multiplication}\n(v,p) \\circ (v',p') = (v + p(v'), p \\circ p')\n\\end{equation}\n\n\\noindent where $v \\in {\\mathbb Z}_2^{2n}$ is a ${\\mathbb Z}_2$-vector, $p \\in S_{2n}$ a permutation, and $p(v')$ the natural action of $S_{2n}$ on ${\\mathbb Z}_2^{2n}$ applied to $v'$. Note that this is precisely the multiplication\nrule which we obtained pictorially in Section \\ref{sec:tethered}.\n\n{\\bf Computing the homotopy groups of $R_{2n}$,\nthe space of order parameter textures interpolating\nbetween the hedgehogs}.\nOf course, computing $\\pi_1(Y_{2n})$ only gets us part\nof the way home. Our ultimate goal is to compute\n$\\pi_1({\\widetilde K}_{2n})$. Thus, we now turn to\nthe homotopy long exact sequence:\n$$\\pi_2(Y_{2n}) \\overset{\\partial_2}{\\longrightarrow} \\pi_1(R_{2n}) \\rightarrow \\pi_1(\\widetilde{K}_{2n}) \\rightarrow \\pi_1(Y_{2n}) \\overset{\\partial_1}{\\longrightarrow} \\pi_0(R_{2n})$$\nFirst consider $\\partial_1$.  The kernel of $\\partial_1$ is represented by loops in $Y_{2n}$ which extend to loops in $R_{2n}$.\nA loop $\\gamma$ in $Y_{2n}$ is a motion of the hedgehogs\ntogether with rotations (about the spheres $S_i^2$)\nof $\\widetilde\\Phi$ which brings the system back to\nits initial configuration. If a loop $\\gamma$ is in\nkernel of $\\partial_1$, then\nthere is a corresponding loop in the configuration space of\nribbons in $B^3$ (obtained by lifting $\\gamma$ to\n$\\widetilde{K}_{2n}$).\n\nThe next steps are to compute ${\\pi_0}(R_{2n})$\nand ${\\pi_1}(R_{2n})$. These computations are\ndetailed in Appendix \\ref{sec:Postnikov}, where we see\nthat using the cell structure\n${\\cal C}$ which we introduced for $\\til{U\/O}$ is\ntricky as a result of the higher cells.\nThus, we instead introduce the\n``Postnikov tower'' for $\\til{U\/O}$ which allows\nus to make all the calculations we need.\nWe find that $\\pi_0(R_{2n})=\\mathbb{Z}_2$\nand ${\\pi_1}(R_{2n})=(\\mathbb{Z}_2)^{2n}$.\n\nThus, $\\pi_1(\\widetilde{K}_{2n})$ sits in the following exact sequence:\n\\[\\begindc{0}[3]\n    \\obj(10,20)[a]{$\\pi_2(Y_{2n})$}\n    \\obj(28,20)[b]{$\\pi_1(R_{2n})$}\n    \\obj(45,20)[c]{$\\pi_1(\\widetilde{K}_{2n})$}\n    \\obj(62,20)[d]{$\\pi_1(Y_{2n})$}\n    \\obj(81,20)[e]{$\\pi_0(R_{2n})$}\n    \\obj(28,15)[b1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(62,15)[c1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(81,15)[c1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(28,10)[b2]{$(\\mathbb{Z}_2)^{2n}$}\n    \\obj(62,10)[d2]{$({\\mathbb Z}_2)^{2n} \\rtimes S_{2n}$}\n    \\obj(81,10)[e2]{${\\mathbb Z}_2$}\n    \\mor{a}{b}{$\\partial_2$}\n    \\mor{b}{c}{}\n    \\mor{c}{d}{}\n    \\mor{d}{e}{$\\partial_1$}\n\n\\enddc\\]\nRecall that when we studied $\\pi_2(Y_{2n})$, we found\n(exactly as in the case of $\\pi_2(X_{2n})$) that there are\n$2n\\choose{2}$ generators corresponding to relative\n2-parameter motions of any pair of hedgehogs\naround their center of mass.  This can be\nused to understand the map\n$\\partial_2: \\pi_2(Y_{2n}) \\to \\pi_2(R_{2n})$.\nThe image of any center of mass 2-motion is the ``bag''\ncontaining the corresponding pair of hedgehogs.\nThus, $\\text{coker}(\\partial_2) \\cong {\\mathbb Z}_2$; it is\n${\\mathbb Z}_2^{2n}$ modulo the even sublattice\n(vectors whose coordinate sum is zero in ${\\mathbb Z}_2$).\nThus, we have the short exact sequence:\n\\begin{equation}\\label{eq:extension_problem}\n\\begindc{0}[3]\n    \\obj(10,20)[a]{$1$}\n    \\obj(27,20)[b]{$\\text{coker}(\\partial_2)$}\n    \\obj(44,20)[c]{$\\pi_1(\\widetilde{K}_{2n})$}\n    \\obj(61,20)[d]{$\\text{ker}(\\partial_1)$}\n    \\obj(78,20)[e]{$1$}\n    \\obj(27,15)[c1]{\\begin{sideways}$\\cong$\\end{sideways}}\n    \\obj(27,10)[b2]{${\\mathbb Z}_2$}\n    \\mor{a}{b}{}\n    \\mor{b}{c}{}\n    \\mor{c}{d}{$\\pi$}\n    \\mor{d}{e}{}\n\\enddc\n\\end{equation}\n\nThe kernel $\\text{ker}(\\partial_1)$ consists of the part of $\\pi_1(Y)$ associated with even ($2\\pi$) twisting.  As shown in Figure \\ref{fig:exchange2}, a simple exchange is associated to a total twisting of ribbons by $\\pm 2\\pi$.\n\n\nThus, $\\partial_1(v,p) = \\sum_{i=1}^{2n} v_i + \\text{parity}(p) \\in {\\mathbb Z}_2$.  We use the notation $E(\\mathbb{Z}_2^m \\rtimes {S_m})$ for $\\text{ker}(\\partial_1)$.\n\n{\\em Note:} If $m=2$, ${\\mathbb Z}_2^{m} \\rtimes S_{m}$ is the dihedral group $D_4$ and its ``even'' subgroup $\\text{ker}(\\partial_1) \\cong {\\mathbb Z}_4$.  This shows that for $m$ even, the induced short exact sequence does not split, and the extension is more complicated:\n\n\\begin{equation*}\n1 \\longrightarrow {\\mathbb Z}_2^{2n-1} \\longrightarrow E(\\mathbb{Z}_2^{2n} \\rtimes S_{2n}) \\longrightarrow S_{2n}\\rightarrow 1\n\\end{equation*}\n\nThere is a final step required to solve the extension problem \\ref{eq:extension_problem} and finish the calculation of $\\pi_1(\\widetilde{K}_{2n})$.  We geometrically construct a homomorphism $s:\\text{ker}(\\partial_1) = E(\\mathbb{Z}_2^{2n} \\rtimes S_{2n}) \\to \\pi_1(\\widetilde{K}_{2n})$ which is a left inverse to the projection.\n\nThis will show that $\\pi_1(\\widetilde{K}_{2n})$ is a semidirect product ${\\mathbb Z}_2 \\rtimes \\text{ker}(\\partial_1)$, but since ${\\mathbb Z}_2$ has no nontrivial automorphism, the semidirect product is actually direct:\n\n\\begin{equation}\\label{eq:pi_1(K_2n)_direct_product}\n\\pi_1(\\widetilde{K}_{2n}) \\cong {\\mathbb Z}_2 \\times E(\\mathbb{Z}_2^{2n} \\rtimes S_{2n})\\end{equation}\n\nTo construct $s$, note that all elements of $\\text{ker}(\\partial_1)$ can be realized by a loop $\\gamma$ of maps into the bottom 2-cell of Eq. \\ref{eq:U\/O_cell_structure} $S^2$.  Still confining the order parameter (map) to lie in $S^2$, such a loop lifts to an arc $\\widetilde{\\gamma}$ of ribbons representing an arc in $\\widetilde{K}_{2n}$.  We may choose the lift so that as the ribbons move, they never ``pass behind'' the $2n$ hedgehogs.  (For example, we may place the hedgehogs on the sphere of radius $=\\frac{1}{2}$ inside the 3-ball $B^3$ (assumed to have radius $=1$) and then keep all ribbons inside $B_{\\frac{1}{2}}^3$.  These arcs may be surgered (still as preimages of $N \\subset S^2$) so that they return to their original position except for a possible accumulation of normal twisting $t2\\pi$.  Since $\\gamma \\in \\text{ker}(\\partial_1)$, $t$ must be even.  Now, allowing the order parameter (map) to leave $S^2$ and pass over the 4-cell (of Eq. \\ref{eq:U\/O_cell_structure}), attached by $2\\text{Hopf}:S^3 \\to S^2$, we may remove these even twists.  (The 4-cell can introduce small closed ribbons with self-linking $=2$ in a small ball.  These small ribbons can be surgered into other ribbons.)  This lifts a generating set of $\\text{ker}(\\partial_1)$ into $\\pi_1(\\widetilde{K}_{2n})$ as a set theoretic cross section (left inverse to $\\pi$). But what about relations?  Because the entire loop is constant outside $B_{\\frac{1}{2}}^3$, the corresponding homology class in $H_2(Q \\times I;{\\mathbb Z}_2) \\cong \\pi_1(R)$ is trivial, so $s$ is actually a group homomorphism.\n\n\n\n\\section{Representation Theory of the Ribbon Permutation Group}\n\\label{sec:projective}\n\nIn this section, we discuss the mathematics of the group\n${\\cal T}_{m}$ and its representation. The\npurpose of this section is to show that\na direct factor of ${\\cal T}_{m}$,\ncalled the even ribbon permutation group,\nis a ghostly recollection of the braid\ngroup and the Teo-Kane representation of the even ribbon permutation group is a projectivized version of\nthe Jones representation of the braid group\nat a $4^{th}$ root of unity, i.e. the representation\nrelevant to Ising anyons.\n\n\\subsection{Teo-Kane fundamental groups}\n\nIn Section V, we consider only even number of hedgehogs for\nphysical reasons.  In this section, we will include\nthe odd case for mathematical completeness.\n\\iffalse For each integer $m \\geq 0$, let $B^3_m$ be the closed complement\nof $m$ disjoint $3$-balls $D^3_i, i=1,2,\\cdots, m$ in the interior\nof $B^3$.  A Teo-Kane configuration is a continuous map $\\Phi:\nB^3_m \\longrightarrow U\/O$ such that $[\\Phi|_{\\partial B^3}]=0\\in\n\\pi_2(U\/O)$, and $[\\Phi|_{\\partial D^3_i}]=1\\in\n\\pi_2(U\/O), i=1,2,\\cdots, m$,  i.e. the map $\\Phi|_{\\partial B^3}: S^2\\rightarrow U\/O$ represents $0$ in\n$\\pi_2(U\/O)\\cong \\mathbb{Z}_2$, and $\\Phi|_{\\partial D_i^3}, i=1,2,\\cdots, m$ representing $1$.\n Then the Teo-Kane configuration space $K_m$ is\nthe topological space of all such maps with the compact-open topology, where the $3$-balls $D^3_i, i=1,2,\\cdots, m$ are not\nfixed.\\fi\n\nThe Teo-Kane fundamental\ngroup is the fundamental group of the Teo-Kane configuration\nspace $K_m$.  As computed in Section V, ${\\cal T}_{m}=\\pi_1(K_m)\\cong \\mathbb{Z}\\times \\mathbb{Z}_2 \\times\nE(\\mathbb{Z}_2^m \\rtimes {S_m})$, where\nthe subgroup ${\\cal T}^r_{m}=\\mathbb{Z}_2 \\times\nE({\\mathbb{Z}_2^m \\rtimes S_m})$ is called the ribbon permutation group. Here, $E({\\mathbb{Z}_2^m \\rtimes S_m})$ is\nthe subgroup of ${\\mathbb{Z}_2^m \\rtimes S_m}$ comprised of\nelements whose total parity in $\\mathbb{Z}_2^m$\nadded to their parity in $S_m$ is even.\nIn the following, we will call\nthe group $G_m=E(\\mathbb{Z}_2^m \\rtimes {S_m})$ the {\\it even} ribbon permutation group because\nit consists of the part of $\\pi_1(Y_m)$ associated with even ($2\\pi$) twisting.\nFor the representations of the Teo-Kane fundamental\ngroups, we will focus on the even ribbon permutation groups $G_m$.\nNo generality is lost if we consider only irreducible representations\nprojectively because irreducibles of $\\mathbb{Z}$ and $\\mathbb{Z}_2$ contribute only overall phases.\nBut for reducible representations, the relative phases from representations of $\\mathbb{Z}$\nand $\\mathbb{Z}_2$ might have physical consequences in interferometer\nexperiments.\n\nThe even ribbon permutation group $G_m$ is an index$=2$ subgroup of\n$\\mathbb{Z}_2^m \\rtimes S_m$.  To have a better understanding of $G_m$, we\nrecall some facts about the important group $\\mathbb{Z}_2^m \\rtimes S_m$.\nThe group $\\mathbb{Z}_2^m \\rtimes S_m$ is the symmetry group of the\nhypercube $\\mathbb{Z}_2^m$, therefore it is called the hyperoctahedral group, denoted as\n$C_m$.  $C_m$ is also a Coxeter group of type\n$B_m$ or $C_m$, so in the mathematical literature it is also\ndenoted as $B_m$ or $BC_m$.  To avoid confusion with the braid\ngroup ${\\cal B}_m$, we choose to use the hyperoctahedral group notation $C_m$.\nThe group $C_m$ has a faithful representation as signed\npermutation matrices in the orthogonal group $O(m)$: matrices with\nexactly one non-zero entry $\\pm 1$ in each row and column.\nTherefore, it can also be realized as a subgroup of the\npermutation group $S_{2m}$, called signed permutations: $\\sigma:\n\\{\\pm 1, \\pm 2, \\cdots, \\pm m\\} \\rightarrow \\{\\pm 1, \\pm 2, \\cdots, \\pm\nm\\}$ such that $\\sigma(-i)=-\\sigma(i)$.\n\nWe will denote elements in $C_m$ by a pair $(b,g)$, where\n$b=(b_i)\\in \\mathbb{Z}_2^m$ and $g\\in S_m$.  Recall the multiplication of\ntwo elements $(b,g)$ and $(c,h)$ is given by $(b,g)\\cdot (c,h)=(b+g.c, gh)$, where $g.c$\nis the action of $g$ on $c$ by permuting its coordinates.  Let $\\{e_i\\}$ be the standard\nbasis elements of $\\mathbb{R}^m$.  To save notation, we will also use it\nfor the basis element of $\\mathbb{Z}_2^m$.  As a signed permutation matrix $e_i$ introduces\na $-1$ into the $i^{th}$ coordinate $x_i$.  Let $s_i$ be the transposition\nof $S_m$ that interchanges $i$ and $i+1$.  As a signed permutation matrix, it\ninterchanges the coordinates $x_i, x_{i+1}$.\nThere is a total parity map $det: C_m\\rightarrow \\mathbb{Z}_2$ defined as\n$det(b,g)=\\sum_{i=1}^m b_i +parity(g)\\; mod \\; 2$.  We denote\nthe total parity map as $det$ because in the realization of $C_m$\nas signed permutation matrices in $O(m)$, the total parity is just the\ndeterminant.  Hence $G_m$, as the kernel of $det$,  can be identified as a subgroup of\n$SO(m)$.  The set of elements\n$t_i=(e_i,s_i), i=1,\\cdots, m-1$ generates $G_m$.  As a signed permutation matrix,\n$t_i(x_1,\\cdots,x_i,x_{i+1},\\cdots, x_m)=(x_1,\\cdots,-x_{i+1},x_i,\\cdots,\nx_m)$.\n\nGiven an element $(b,g)\\in C_m$, let $\\mathbb{Z}_2^{m-1}$ be\nidentified as the subgroup of $\\mathbb{Z}_2^m$ such that $\\sum_{i=1}^{m} b_i$ is\neven.  Then we have:\n\n\\begin{proposition}\\label{presentation}\n\n\\begin{enumerate}\n\n\\item For $m\\geq 2$, the even ribbon permutation group $G_m$ has a presentation as an abstract group\n\\begin{equation*}\n\\begin{split}\n<t_1, \\cdots, t_{m-1}| t_i^4=1, i=1,\\cdots, m-1,\\\\\n(t_it_{i+1}^{-1})^3=(t_i^{-1}t_{i+1})^3=1, i=1, \\cdots, m-2>.\n\\end{split}\n\\end{equation*}\n\n\\item The exact sequence $$1\\rightarrow \\mathbb{Z}_2^{m-1} \\rightarrow G_m \\rightarrow S_m\n\\rightarrow 1$$ splits if and only if $m$ is odd.\n\n\\item When $m$ is even, a normalized $2$-cocycle\n$f(g,h):  S_m\\times S_m \\rightarrow \\mathbb{Z}_2^{m-1}$ for the extension of $S_m$ by $\\mathbb{Z}_2^{m-1}$\nabove can be\nchosen as $f(g,h)=0$ if $g$ or $h$ is even and $f(g,h)=e_{g(1)}$\nif $g$ and $h$ are both odd.\n\n\\end{enumerate}\n\\end{proposition}\n\nWe briefly give the idea of the proof of Prop. \\ref{presentation}.  For $(1)$, first\nwe use a presentation of $C_m$ as a Coxeter group of type\n$B_m$: $<r_1,\\cdots, r_m|r_i^2=1, (r_1r_2)^4=1, (r_ir_{i+1})^3=1,\ni=2,\\cdots, m-1>$.  Then the Reidemeister-Schreier method \\cite{Magnus76} allows\nus to deduce the presentation for $G_m$ above.  For $(2)$, when\n$m$ is odd, a section $s$ for the splitting can be defined as\n$s(g)=(0,g)$ if $g$ is even and $s(g)=((11\\cdots 1),g)$ if $g$ is\nodd.  When $m$ is even, that the sequence does not split follows from\nthe argument in \\cite{Jones83}.  For $(3)$, we choose a set map\n$s(g)=(0,g)$ if $g$ is even and $s(g)=(e_1,g)$ if $g$ is odd.\nThen a direct computation of the associated factor set as on page $91$ of \\cite{Brown82} gives rise\nto our $2$-cocycle.\n\nAs a remark, we note that there are another two obvious maps from the\nhyperoctahedral group  $C_m$ to $\\mathbb{Z}_2$.  One of them is the\nsum of bits in $b$ of $(b,g)$.  The kernel of this map is the\nCoxeter group of type $D_m$, which is a semi-direct product of $\\mathbb{Z}_2^{m-1}$\nwith $S_m$.  The two groups $D_m$ and $G_m$ have\nthe same order, and are isomorphic when $m$ is odd (the two splittings induce the same\naction of $S_m$ on $\\mathbb{Z}_2^{m-1}$), but different\nwhen $m$ is even.  To see the difference, consider the order $2$ automorphism $\\phi: C_m\\rightarrow C_m$  given by $\\phi(x)=det(x) x$.\nIts restriction is the identity on $G_m$, but non-trivial on $D_m$.\n\n\\subsection{Teo-Kane unitary transformations}\n\nSuppose there are $m$ hedgehogs in $B^3$.  A unitary transformation\n$T_{ij}=e^{\\frac{\\pi}{4}\\gamma_i \\gamma_j}$ of the\nMajorana fermions is associated with the interchange of the ($i$,$j$)-pair\nof the\nhedgehogs. Interchanging the same pair twice results in the\n\\lq\\lq braidless\" operation $T_{ij}^2=\\gamma_i \\gamma_j$.  The\nMajorana fermions $\\gamma_i, i=1,2,\\cdots, m$ form the Clifford\nalgebra $Cl_m(\\mathbb{C})$.  Therefore, the Teo-Kane unitary\ntransformations act as automorphisms of the Clifford algebra\n$$T_{ij}: \\gamma \\rightarrow T_{ij} \\gamma T_{ij}^{\\dagger}.$$\nThe projective nature of the Teo-Kane representation rears its\nhead here already as an overall phase cannot be constrained by\nsuch actions.\nDo these unitary transformations afford a linear representation\nof the Teo-Kane fundamental group $K_m$? If so, what are their images?\nThe surprising fact is that the Teo-Kane\nunitary transformations cannot give rise to a linear representation of the\nTeo-Kane fundamental group.  The resulting representation is\nintrinsically projective.   We consider only the even ribbon permutation group\n$G_m$ from now on.\n\nTo define the Teo-Kane representation of $G_m$, we use the presentation of $G_m$\nby $t_i, i=1,\\cdots, m-1$ in Prop. \\ref{presentation}.  The associated unitary matrix for $t_i$ comes from the\nTeo-Kane unitary transformation $T_{i,i+1}$.  As was alluded above, there is a phase\nambiguity of the Teo-Kane unitary matrix.  We will discuss this ambiguity more in the\nnext subsection.  For the discussion below, we choose any matrix realization of the\nTeo-Kane unitary transformation with respect to a basis of the Clifford algebra $Cl_m(\\mathbb{C})$.\nA simple computation using the presentation of $G_m$ in Prop. \\ref{presentation}\nverifies that the assignment of $T_{i,i+1}$ to $t_i$ indeed\nleads to a representation of $G_m$.  Another verification follows from a relation to the Jones\nrepresentation in the next subsection.  We can also check directly that\nthis is indeed the right assignment for\n$T^2_{i,i+1}: \\gamma_i \\rightarrow -\\gamma_i,\\gamma_{i+1} \\rightarrow -\\gamma_{i+1}$.\nIn the Clifford algebra $Cl_m(\\mathbb{C})$, $\\gamma_i, \\gamma_{i+1}$\ncorrespond to the standard basis element $e_i, e_{i+1}$.  The\nelement $t_i^2$ of $G_m$ is $(e_i+e_{i+1},1)$.  As a signed\npermutation matrix, $t_i^2$ sends $(x_1,\\cdots, x_i,\nx_{i+1},\\cdots, x_m)$ to $(x_1,\\cdots, -x_{i},-x_{i+1},\\cdots, x_m)$,\nwhich agrees with the action of $T^2_{i,i+1}$ on $\\gamma_i,\n\\gamma_{i+1}$.\n\nTo see the projectivity, the interchange of the ($i$,$i+1$)-pair hedgehogs corresponds\nto the element $t_i=(e_i, s_i) \\in G_m$.  Performing the\ninterchange twice gives rise to the element $t_i^2=(e_i+e_{i+1},1)$, denoted as $x_i$.  Since\n$x_i$'s are elements of a subgroup of $G_m$ isomorphic $\\mathbb{Z}_2^{m-1}$,\nobviously we have\n$x_ix_{i+1}=x_{i+1}x_i$.  On the other hand,\n$T^2_{i,i+1}T^2_{i+1,i+2}=\\gamma_i \\gamma_{i+1} \\cdot \\gamma_{i+1}\n\\gamma_{i+2}=\\gamma_i \\gamma_{i+2}$, and $T^2_{i+1,i+2}T^2_{i,i+1}=\\gamma_{i+1} \\gamma_{i+2} \\cdot \\gamma_{i}\n\\gamma_{i+1}=-\\gamma_i \\gamma_{i+2}$.  Since it is impossible\nto encode the $-1$ in the $x_i$'s of $G_m$, the representation has to\nbe projective.  Note that an overall phase in\n$T_{ij}$ will not affect the conclusion.\nIn the next subsection, we will see this projective representation\ncomes from a linear representation of the braid group---the Jones\nrepresentation at a $4^{th}$ root of unity and the $-1$ is\nencoded in the Jones-Wenzl projector $p_3=0$.\n\nTo understand the images of the Teo-Kane representation of $G_m$,\nwe observe that the Teo-Kane unitary transformations $T_{ij}=e^{\\frac{\\pi}{4}\\gamma_i \\gamma_j}$ lie inside\nthe even part $Cl_m^0(\\mathbb{C})$ of $Cl_m(\\mathbb{C})$.\nTherefore, the Teo-Kane representation of $G_m$ is just the spinor\nrepresentation projectivized to $G_m \\subset SO(m)$.  It follows\nthat the projective image group of Teo-Kane representation of the even ribbon permutation group is $G_m$ as an\nabstract group.  Recall $Cl_m^0(\\mathbb{C})$ is reducible into two irreducible sectors if $m$\nis even, and irreducible if $m$ is odd.  When $m$ is even, it is important to\nknow how the relative action of the center $Z(G_m)\\cong \\mathbb{Z}_2$ of $G_m$ on the two\nirreducible sectors. The center of $G_m$ is generated by the element $z=((1\\cdots 1), 1)$, whose\ncorresponding Teo-Kane unitary transformation is $U=\\gamma_1 \\gamma_2 \\cdots\n\\gamma_m$ up to an overall phase.  As we show in\nAppendix \\ref{sec:braid-group},\nthe relative phase of $z$ on the two sectors is always $-1$.\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe now review and discuss the main results derived in this\npaper. Using the topological classification of\nfree fermion Hamiltonians \\cite{Ryu08,Kitaev09},\nwe considered a system of fermions in 3D which is allowed to\nhave arbitrary superconducting order parameter\nand arbitrary band structure; and is also allowed to\ndevelop any other possible symmetry-breaking\norder such as charge density-wave, etc. -- i.e. we do\nnot require that any symmetries are preserved.\nWe argued in Section \\ref{sec:free-fermion} that the space\nof possible gapped ground states\nof such a system is topologically equivalent to U($N$)\/O($N$)\nfor $N$ large (for large $N$, the topology of U($N$)\/O($N$)\nbecomes independent of $N$).\nBy extension, if we can spatially vary the superconducting\norder parameter and band structure at will with no\nregard to the energy cost, then there will be topologically\nstable point-like defects classified by\n${\\pi_2}(U($N$)\/O($N$))=\\mathbb{Z}_2$.\n\nThis statement begs the question of whether\none actually can vary the order parameter and\nband structure in order to create such defects.\nIn a given system, the energy cost may simply\nbe too high for the system to wind around\nU($N$)\/O($N$) in going around such a defect.\n(This energy cost, which would include the condensation\nenergy of various order parameters, is not taken into\naccount in the free fermion problem.) If we\ncreate such defects, they may be so costly\nthat it is energetically favorable for them\nto simply unwind by closing the gap over large\nregions. (The energy cost associated with such an unwinding\ndepends on the condensation energy of the order parameters\ninvolved, which is not included in the topological classification.)\nThus, U($N$)\/O($N$) is not the target space\nof an order parameter in the usual sense because\nthe different points in U($N$)\/O($N$) may not correspond to\ndifferent ground states with the same energy.\nHowever, in Section \\ref{sec:strong-coupling},\nwe have given at least one concrete model of free fermions with\nno symmetries in 3D in which the topological defects\npredicted by the general classification are present\nand stable. Furthermore, Teo and Kane \\cite{Teo10}\nhave proposed several devices in which\nthese defects are simply superconducting vortices\nat the boundary of a topological insulator.\n\nIn order to understand the quantum mechanics of\nthese defects, it is important to first understand their\nquantum statistics. To do this,\nwe analyzed the multi-defect configuration\nspace; its fundamental group governs defect\nstatistics. The configuration space of $2n$ point-like\ndefects of a system with `order parameter' taking values\nin U($N$)\/O($N$) is the space which we call $K_{2n}$.\nIt can be understood as a fibration.\nThe base space is $X_{2n}$, the configuration\nspace of $2n$ points (which we know has fundamental\ngroup $S_{2n}$ in dimension three and greater).\nAbove each point in this base space there is\na fiber ${\\cal M}_{2n}$ which is the space of maps from\nthe ball $B^3$ minu $2n$ fixed points to\nU($N$)\/O($N$) with winding number $1$ about each\nof the $2n$ points. $K_{2n}$ is the total space of\nthe fibration. In Section \\ref{sec:Kane_space},\nwe found that its fundamental group is\n${\\pi_1}(K_{2n})=\\mathbb{Z}\\times\\mathbb{Z}_2\n\\times E({\\mathbb{Z}_2^{2n} \\rtimes S_{2n}})$, where\n$E({\\mathbb{Z}_2^{2n} \\rtimes S_m})$ is\nthe subgroup of ${\\mathbb{Z}_2^m \\rtimes S_{2n}}$ comprised of\nelements whose total parity in $\\mathbb{Z}_2^{2n}$\nadded to their parity in $S_{2n}$ is even.\n\nThe fundamental group of the configuration\nspace is the same as the group of equivalence\nclasses of spacetime histories of a system\nwith $2n$ point-like defects. Since these different\nequivalence classes cannot be continuously\ndeformed into each other, quantum mechanics\nallows us to assign them different unitary\nmatrices. These different unitaries form\na representation of the fundamental group of the configuration\nspace of the system. However, we found in Section \\ref{sec:projective}\nthat Teo and Kane's unitary transformations are not\na linear representation of ${\\pi_1}(K_{2n})$,\nbut a {\\it projective representation}, which is to say that\nthey represent ${\\pi_1}(K_{2n})$ only up to a phase.\nEquivalently, Teo and Kane's unitary transformations are\nan ordinary linear representation of a {\\it central\nextension} of ${\\pi_1}(K_{2n})$, as discussed in\nSection \\ref{sec:projective}.\n\nThis surprise lurks in a seemingly innocuous set of\ndefect motions: those in which defects $i$ and $j$ are rotated\nby $2\\pi$ and the order parameter field surrounding\nthem relaxes back to its initial configuration. This has the\nfollowing effect on the Majorana fermion zero mode operators\nassociated with the two defects:\n\\begin{equation}\n{\\gamma_i}\\rightarrow -{\\gamma_i}\\, , \\hskip 0.5 cm\n{\\gamma_j}\\rightarrow -{\\gamma_j}\n\\label{eqn:gamma-trans}\n\\end{equation}\nOne might initially expect that two such motions, one affecting\ndefects $i$ and $j$ and the other affecting $i$ and $k$,\nwould commute since they simply multiply the operators\ninvolved by $-$ signs. However, the unitary operator which\ngenerates (\\ref{eqn:gamma-trans}) is\\cite{footnote1}:\n\\begin{equation}\nU^{ij} = e^{i\\theta} \\,{\\gamma_i} {\\gamma_j}\\, .\n\\end{equation}\nThus, the unitary operators $U^{ij}$ and $U^{ik}$ do not commute;\nthey anti-commute:\n\\begin{equation}\nU^{ij} \\, U^{ik}= -U^{ik}\\,U^{ij} \\,  .\n\\end{equation}\nHowever, as shown in Figure \\ref{fig:movie-of-movies}, the\ncorresponding classical motions can be continuously deformed\ninto each other. Thus, a linear representation of the fundamental group\nof the classical configuration space would have these two\noperators commuting. Instead, the quantum mechanics of this\nsystem involves a projective representation.\n\nProjective quasi-particle statistics were first proposed by\nWilzcek\\cite{Wilczek98}, who suggested a projective representation of the\npermutation group in which generators $\\sigma_j$ and $\\sigma_k$ anti-commute\nfor $|j-k|\\geq 2$, rather than commuting.  Read\\cite{Read03} criticized\nthis suggestion as being in conflict with locality.  We can sharpen Read's\ncriticism as follows.\nSuppose that one can perform the operation $\\sigma_1$ by acting\non a region of space, called $A$, containing particles $1$ and $2$,\nand one can perform $\\sigma_3$ by acting on a region called $B$, containing\nparticles $3$ and $4$, and suppose that regions $A$ and $B$ are disjoint.\nConsider the following thought experiment: let Bob perform operation\n$\\sigma_3$ at time $t=0$ and let Bob then repeat operation $\\sigma_3$ at\ntime $t=1$.  Let Alice prepare a spin in the state $(1\/\\sqrt{2})(|\\uparrow\\rangle+|\\downarrow\\rangle)$ at time $t=-1$, and then let Alice perform the following\nsequence of operations.\nAt time $t=-\\epsilon$, for some small $\\epsilon$, she\nperforms the unitary operation $|\\uparrow\\rangle\\langle \\uparrow| \\otimes \\sigma_1+|\\downarrow\\rangle\\langle\\downarrow | \\otimes I$, where $I$ is the\nidentity operation, leaving the particles alone.\nAt time $t=+\\epsilon$, she\nperforms the unitary operation $|\\uparrow\\rangle\\langle \\uparrow| \\otimes I\n+|\\downarrow\\rangle\\langle\\downarrow | \\otimes \\sigma_1$.   Thus, if\nthe spin is up, she performs $\\sigma_1$ at time $t=-\\epsilon$, while\nif the spin is down, she does it at time $t=+\\epsilon$.\nFinally, at\ntime $t=2$, Alice performs the operation $\\sigma_1$ again.  One may then\nshow that, due to the anti-commutation of $\\sigma_1$ and $\\sigma_3$, that\nthe spin ends in the state\n$(1\/\\sqrt{2})(|\\uparrow\\rangle-|\\downarrow\\rangle)$.  However, if Bob had\nnot performed any operations, the spin would have ended in the original state\n$(1\/\\sqrt{2})(|\\uparrow\\rangle+|\\downarrow\\rangle)$.  Thus, by performing these\ninterchange operations, Bob succeeds in transmitting information to a space-like\nseparated region (if $A$ and $B$ are disjoint, and the time scale in the\nabove thought experiment is sufficiently fast, then Alice and Bob are space-like\nseparated throughout).\n\nHaving seen this criticism, we can also see how Teo and Kane's construction\nevades it.  The fundamental objects for Teo and Kane are not particles, but\nparticles with ribbons attached.  One may verify that, in every case where\noperations in Teo and Kane's construction anti-commute, the two operations\ndo not act on spatially disjoint regions due to the attached ribbons.  That is,\nthe interchange of particles also requires a rearrangement of the order parameter field.\n\nWhile this argument explains why a projective representation\ndoes not violate causality, it does not really explain why\na projective representation actually occurs in this system.\nPerhaps one clue is the fact that the hedgehogs have long-ranged\ninteractions in any concrete model. Even in the `best-case scenario',\nin which the underlying Hamiltonian of the system is U($N$)\/O($N$)-invariant, there will be a linearly-diverging\ngradient energy for an isolated hedgehog configuration.\nThus, there will be a linear long-ranged force between hedgehogs.\nConsequently, one might adopt the point of view that, as a result\nof these long-ranged interactions, the overall phase associated with\na motion of the hedgehogs is not a purely topological quantity\n(but will, instead depend on details of the motion) and, therefore,\nneed not faithfully represent the underlying fundamental group.\nAs one motion is continuously deformed into another\nin Figure \\ref{fig:movie-of-movies}, the phase of the wavefunction\nvaries continuously from $+1$ to $-1$ as the order parameter\nevolution is deformed. It is helpful to compare this to another\nexample of a projective representation: a charged particle in\na magnetic field $B$. Although the system is invariant under the\nAbelian group of translations, the quantum mechanics of the\nsystem is governed by a non-Abelian projective representation\nof this group (which may be viewed as a linear representation of\nthe `magnetic translation group'). A translation by $a$ in the $x$-direction,\nfollowed by a translation by $b$ in the $y$-direction differs in its\naction on the wavefunction from the same translations in\nthe opposite order by a phase $abB\/\\Phi_0$ equal to the magnetic flux through\nthe area $ab$ in units of the flux quantum $\\Phi_0$.\nIf we continuously deform these two sequences of translations\ninto each other, the phase of the wavefunction varies continuously.\nFor any trajectory along this one-parameter family of trajectories\n(or `movie of movies'), the resulting phase of the wavefunction\nis given by the magnetic flux enclosed by the composition of\nthis trajectory and the inverse of the initial one. In our\nmodel of non-Abelian projective statistics, the phase changes\ncontinuously in the same way, but as a result of the evolution\nof the order parameter away from the defects, rather than as a result of a magnetic field.\n\nAn obvious question presents itself: is there a related theory in which the hedgehogs are no longer confined? Equivalently, could 3D objects with non-Abelian ribbon permutation statistics ever be the weakly-coupled low-energy quasiparticles of a system? The most straightforward route will not work: if we had a U($N$)-invariant system and tried to gauge it to eliminate the linear confining force between hedgehogs, we would find that the theory is sick due to the chiral anomaly. If we doubled the number of fermions in order to eliminate the anomaly, there would be two Majorana modes in the core of each\nhedgehog, and their energies could be split away from zero by a local interaction. This is not surprising since ribbon permutation statistics would violate locality if the hedgehogs were truly decoupled (or had exponentially-decaying interactions). On the other hand, if the hedgehogs were to interact through a Coulomb interaction (or, perhaps, some other power-law),\nthey would be neither decoupled nor confined, thereby satisfying the requirements that they satisfy locality and are low-energy particle-like excitations of the system. Elsewhere, we will describe a model which realizes this scenario \\cite{Freedman10}\n\nThe non-Abelian projective statistics studied in the 3D class with no\nsymmetry can be generalized to arbitrary dimension.\nAs shown in Eq. (\\ref{eq:defectclassification}), the classification of topological defects is independent of the spatial dimension.\nThus, in any dimension $d$ with no symmetry ($p=0$), point-like ($D=0$) topological defects are classified by\n$\\pi_0(R_{p-D+1})=\\pi_0(R_1)=\\mathbb{Z}_2$.\nMoreover, it can be proved that analogous topological defects in different dimensions not only carry the same topological quantum number, but also have the same statistics. In Sec.\n\\ref{sec:Kane_space}, we have defined the configuration space\n$\\mathcal{M}_{2n}$ which is the space of maps from\n$B^3\\setminus\\bigcup_{i=1}^{2n} B_i^3$ to $R_7=U\/O$,\nwith specific boundary conditions. Now if we consider point defect\nin the class with no symmetry in 4D, the configuration space\n$\\mathcal{M}_{2n}^{d=4}$ is defined by maps from\n$B^4\\setminus\\bigcup_{i=1}^{2n} B_i^4$ to the classifying space\n$R_6=Sp\/U$. Noticing that $B^4\\setminus\\bigcup_{i=1}^{2n} B_i^4$\nis homotopy equivalent to the suspension of\n$B^3\\setminus\\bigcup_{i=1}^{2n} B_i^3$, we obtain that\n$\\mathcal{M}_{2n}^{d=4}$ is equivalent to the space of maps from\n$B^3\\setminus\\bigcup_{i=1}^{2n} B_i^3\\rightarrow \\Omega\\left(Sp\/U\\right)$, where $\\Omega\\left(Sp\/U\\right)$ is the loop space of\n$Sp\/U$. Since  $\\Omega\\left(Sp\/U\\right)\\simeq U\/O$,\nwe obtain $\\mathcal{M}_{2n}^{d=4}\\simeq \\mathcal{M}_{2n}$. Thus we have proved that the configuration space $\\mathcal{M}_{2n}$ is independent of the spatial dimension $d$. On the other hand, the fundamental\ngroup of the configuration space $X_{2n}$ of $2n$ distinct points in $B^d$ is independent of $d$ as long as $d>2$. Consequently, the space $K_{2n}$ defined by the fibration $\\mathcal{M}_{2n}\\rightarrow K_{2n}\\rightarrow X_{2n}$ is also topologically independent of spatial dimension $d$ for $d>2$. Thus, the proof we did for $d=3$ applies to generic dimension, and non-Abelian projective statistics exist in any spatial dimension $d>3$ for point defects\nin the no-symmetry class. A similar analysis applies to extended\ndefects with dimension $D>0$. When the spatial dimension is increased by $1$ and the symmetry class remains the same, the classifying space is always changed from $R_{2+p-d}$ to $R_{2+p-d-1}\\simeq \\Omega^{-1}(R_{2+p-d})$. Consequently, at least for\nsimple defects with the topology of $S^D$, the statistics is independent of spatial dimension $d$ as long as $d$ is large enough. For point defects, the ``lower critical dimension\" is $d=3$, while for line defects, i.e. $D=1$, the ``lower critical dimension\" is at least $d=4$ since in $d=3$ we can have braiding between loops.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is well known that the canonical quantization procedure is consistent\nonly in Cartesian coordinates \\cite{1}. For most physically\nrelevant systems, it turns out to be possible to find a Cartesian system\nof axes and, hence, successfully\napply canonical quantization. Nevertheless, the\nHamiltonian dynamics of a classical system apparently exhibits,\nat first sight, a\nlarger symmetry than the associated canonically quantized system. Indeed,\nHamiltonian equations of motion are covariant under canonical\ntransformations, while the Heisenberg equations of motion are covariant\n under unitary transformations. Unitary transformations preserve the\nspectrum of the canonical quantum operators, while in the classical case\n canonical\ntransformations do not generally preserve the range of the canonical\nvariables.\n\nIt is worth mentioning in this regard\nthat the old Bohr-Som\\-mer\\-feld quantization postulate\n\\begin{equation}\n\\oint\\limits_{}^{}pdq=2\\pi \\hbar (n+1\/2),\\ \\ n=1,2,\\ldots\n\\label{1.1}\n\\end{equation}\nis invariant with respect to canonical transformations\n\\begin{equation}\np\\rightarrow {\\bar p}(p,q),\\ \\ q\\rightarrow {\\bar p}(q,p)\n\\label{1.2}\n\\end{equation}\nbecause\n\\begin{equation}\n\\oint\\limits_{}^{}pdq =\\oint\\limits_{}^{}{\\bar p}d{\\bar q}\\ .\n\\label{1.3}\n\\end{equation}\n As a consequence, since the result is identical in all canonical\ncoordinate systems, the Bohr-Som\\-mer\\-feld\nquantization is in fact ``coordinate-free\".\nThe characteristic properties of the quantum theory, like the energy\nspectrum, will be independent of the choice\nof canonical coordinates. In this respect, the old quantum dynamics enjoys\nthe same symmetry as classical dynamics.\n\nIn contrast to the Bohr-Sommerfeld procedure, canonical quantization leads\nto a result that is not covariant with respect to the initial\nchoice of canonical coordinates. For example, for a single degree of freedom,\nthe coherent-state phase space path integral representation of the\nevolution operator\n\\begin{eqnarray}\n&\\ &\\hskip-1cm\\<  p'',q'',t|p',q'\\> =\\< p'',q''|\ne^{-it{\\cal H}\/\\hbar}|p',q'\\>  \\nonumber \\\\\n&= & \\int\\limits_{}^{} \\prod\\limits_{\\tau =0}^{t}\\left(\n\\frac{dp(\\tau )dq(\\tau )}{2\\pi \\hbar}\\right)\\exp \\frac{i}{\\hbar}\n\\int\\limits_{0}^{t}d\\tau \\left[p\\dot{q}-h(p,q)\\right]\\ ,\n\\label{1.4} \\\\\n{\\cal H}&=&{\\textstyle\\int} h(p,q)\\,|p,q\\>\\<p,q|\\,dp\\,dq\/(2\\pi\\hbar)\\ ,\n\\label{4.2}\n\\end{eqnarray}\nis not covariant with respect to canonical transformations, although\nthe measure, being the product of local Liouville measures at each moment\nof time, is invariant under canonical transformations. The contradiction\nfollows from the observation that all classical dynamical systems with\npositive energy and one\ndegree of freedom are equivalent to, say, a free particle ($h={\\tilde p}^2\/2$ after\na suitable canonical transformation). Making such a canonical\ntransformation in (\\ref{1.4}), we seem to arrive at the same conclusion\nfor quantum systems because the integral (\\ref{1.4}) is formally\ninvariant. Such a conclusion is certainly incorrect.\n\n\\subsubsection*{Coordinate-free quantization}\n\nTo resolve this paradox, it has been proposed \\cite{2} to\ninterpret the ill-defined path integral (\\ref{1.4}) by means of\nthe regularized expression\n\\begin{equation}\n\\int\\limits_{}^{}{\\cal D}p{\\cal D}q\\,(\\,\\cdot\\,)\n\\rightarrow\\lim _{\\nu \\rightarrow\n\\infty}\n{\\cal M}_\\nu \\int\\limits_{}^{} {\\cal D}p{\\cal D}q\\,(\\,\\cdot\\,)\\, e^{-\n\\frac{1}{2\\nu}\\int _{0}^{t}d\\tau (\\dot{p}^2+ \\dot{q}^2)}\\ ,\n\\label{1.5}\n\\end{equation}\nwhere $\\ {\\cal M}_\\nu $ is a suitable normalization\n and the limit $\\nu\n\\rightarrow \\infty$ must be taken {\\em after} evaluation of the\npath integral. Various factors in (\\ref{1.5})\ncombine to give a Wiener measure\non the two-dimensional\n phase space. In contrast to the integral (\\ref{1.4}), the\ncoherent-state path integral with the regularized measure can be regarded as\na sum over trajectories of a  Brownian particle whose\nflat, two-dimensional configuration space\nis the original phase space of the system.\n\nThe spectrum $E$ of the system can be obtained from\nthe pole structure of\n the Fourier transform\nof the trace of the transition amplitude\n\\begin{equation} Z_t={\\rm\ntr}e^{-it{\\cal H}\/\\hbar}=\\sum\\limits_{E}^{}e^{-itE\/\\hbar}=\n\\int\\limits_{}^{}(dp'dq'\/2\\pi\\hbar)\\< p',q',t|p',q'\\>\\,,\n\\label{1.6}\n\\end{equation}\nwhere $\\< p',q',t|p',q'\\>$ is given by the corresponding path integral.\nUnder canonical transformations (\\ref{1.2}) the\nBrownian motion on a flat two-dimensional phase space\nremains such a Brownian motion, and if one interprets the stochastic\nintegral $\\int pdq$ in the Stratonovich sense,\n then the spectrum of the system is invariant under canonical\ncoordinate transformations.\n\nIn other words, the coherent-state path integral regularized with the\nhelp of the Wiener measure (\\ref{1.5}) provides a ``coordinate-free\"\ndescription of quantum theory \\cite{2}. Such a regularization\nprocedure applies to general theories without constraints.\n\n\\subsubsection*{Gauge theories}\n\nHamiltonian path integrals are often used to quantize\n gauge theories \\cite{3}. We now have in mind a system of $J$ degrees of\nfreedom $p=\\{p_j\\}$, $q=\\{q^j\\}$, $1\\leq j\\leq J$. A\nmain feature of gauge systems is the existence of nonphysical canonical\nvariables. In the standard formulation,\nthe formal path integral (\\ref{1.4}) is divergent because the\nHamiltonian action for gauge systems is invariant with respect to\ntransformations\n\\begin{equation}\nq\\rightarrow q^\\omega ,\\ \\ p\\rightarrow p^\\omega\n\\label{1.7}\n\\end{equation}\nwhose parameters $\\omega$ depend on the time, that is, there are\n orbits traversed by the gauge transformations (\\ref{1.7}) in the phase\nspace along which the action is constant and\ntraditionally have an infinite volume. The nonphysical variables can be\nassociated with these ``gauge\" directions in phase space.\n\nTo factor out such divergencies of the path integral, one should integrate out\nthe nonphysical variables and obtain a measure on the physical phase\nspace\n\\begin{equation}\n[PS]_{ph}=[PS]\/{\\cal G}\\ ;\n\\label{1.8}\n\\end{equation}\nhere ${\\cal G}$ consists of all transformations (\\ref{1.7}).\nTechnically, the\nprocedure amounts to a canonical transformation such that the generators of\n(\\ref{1.7}) become some elements of a new\nset of canonical momenta \\cite{3}. This canonical\ntransformation introduces explicit symplectic coordinates $p^*$ and $q^*$\non the physical phase space (\\ref{1.8}). However, it is important to realize\nthat the canonical coordinates on $[PS]_{ph}$\nare themselves defined only up to a\ncanonical transformation, i.e., the parametrization of the physical phase\nspace is not unique. As we have argued above, the formal integral in the\nHamiltonian path integral cannot provide a\ngenuine invariance with respect to\ncanonical transformations. In the framework of gauge theories, this\ninvariance implies\ngauge invariance because the spectrum of a gauge theory\ncannot depend on one or another particular parametrization of the physical\nphase space.\n\nThus, the regularization of the path integral measure with the help of a\nWiener measure and the invariance\nunder canonical coordinate transformations\nit offers should be extended to gauge theories. The aim of this\nletter is to address this problem. Hereafter, we use units where $\\hbar=1$.\n\n\\section{The projection method}\n\\setcounter{equation}0\n\n\\subsubsection*{Special constraint class}\n\nLet $\\varphi _a=\\varphi _a(p,q)$ be a set of  independent closed first-class\nconstraints, i.e.\n\\begin{equation}\n\\{\\varphi _a,\\varphi _b\\}=f_{abc}\\varphi _c\\ ,\n\\label{2.1}\n\\end{equation}\nand for convenience we also suppose that\nthe Poisson bracket of $\\varphi _a$ with the system Hamiltonian\nvanishes. The constraints\ngenerate gauge transformations on phase space which in their infinitesimal\nform are given by\n\\begin{eqnarray}\n&p &\\rightarrow p+\\delta p=p+\\delta \\omega ^a\\{p,\\varphi_a\\}\\equiv\np^{\\delta \\omega } \\label{2.2} \\\\\n&q & \\rightarrow q+\\delta q=q+\\delta \\omega ^a\\{q,\\varphi_a\\}\\equiv\nq^{\\delta \\omega} \\  ,\\label{2.3}\n\\end{eqnarray}\nfor general $\\{\\omega^a\\}$.\nFrom (\\ref{2.2}) and (\\ref{2.3}) it follows that the infinitesimal\ngauge transformations\ngenerated by the constraints are also infinitesimal canonical transformations\n\\begin{equation}\n\\{p^{\\delta \\omega},q^{\\delta\\omega}\\}=\\{p,q\\}+O(\\delta\\omega ^2)\\ .\n\\label{2.4}\n\\end{equation}\nA finite gauge transformation can be obtained\nby applying the operator $\\exp[ -\n(\\omega ^a ad\\, \\varphi _a)]$, $ ad\\,\\varphi _a\n=\\{\\varphi _a,\\ \\cdot \\}$, to phase\nspace variables.\n\nAs noted at the outset, canonical quantization singles out Cartesian\ncoordinates for special attention. We formulate a special class of\nclosed first-class constraint systems---which we shall refer to\nas constraints of ``Yang-Mills type\"---in such a favored set of\ncoordinates. Specifically, we choose\n\\begin{equation}\n\\varphi_a(p,q)=f^j_a(q)p_j\\equiv(f_a(q),p)\\,,\n\\label{c.1}\n\\end{equation}\nwhere $(\\,,\\,)$ denotes a scalar product in a Euclidean space, and\n$f_a(q)$ {\\it are linear functions of $q$ chosen so that the\nconstraints (\\ref{c.1}) are of the first class}, i.e. they\nsatisfy (\\ref{2.1}). With this choice, the\ngauge transformations (\\ref{1.7}) are linear canonical transformations.\nIt follows for such constraints that\n\\begin{equation}\np_j\\{\\varphi_a,q^j\\}=\\varphi_a(p,q)\n\\label{c.2}\n\\end{equation}\nholds as an identity, which we shall find useful.\nWe also assume that there is no operator ordering ambiguity in\nthe constraints after quantization. This situation is in fact\n entirely realized for a gauge theory based on a compact semi-simple\ngauge group $\\footnote{The formalism applies also to gauge groups\nbeing the direct product a semi-simple and some number of Abelian\ngroups.}$.\n\nSuch constraints enjoy an additional useful property. If\n\\begin{equation}\n|p,q\\>\\equiv e^{-iq^jP_j}e^{ip_jQ^j}|0\\>\\,,\n\\label{c.3}\n\\end{equation}\n where $|0\\>$ is the ground state\nof an harmonic oscillator, i.e, $(Q^j+iP_j)\\,|0\\>=0$ for all $j$,\ndenotes the coherent states in the same Cartesian coordinates,\nthen it follows that\n\\begin{equation}\ne^{-i\\Omega^a{\\hat\\varphi}_a(P,Q)}\\,|p,q\\>=|p^\\Omega,q^\\Omega\\>\\,,\n\\label{c.4}\n\\end{equation}\nnamely the action of any finite gauge transformation is to map one\ncoherent state into another.  Here $\\{{\\hat\\varphi}_a\\}$ denote\nthe constraint operators that generate the gauge transformations.\n\n\\subsubsection*{Coherent state propagator}\n\nThe total Hilbert space of a gauge system can always be split\ninto an orthogonal sum of a subspace formed by gauge invariant states\nand a subspace that consists of gauge variant states.\nTherefore an averaging over the gauge group automatically leads to\n  a projection\noperator onto the physical subspace of gauge invariant\nstates. The physical transition amplitude\nis obtained from the unconstrained propagator by averaging\nthe latter over the gauge group,\n\\begin{eqnarray} \\hskip-.5cm\n\\< p'',q'',t|p',q'\\>^{ph}&\\equiv &\n\\int\\limits_{G}^{} \\frac{d\\mu(\\omega)}{Vol\\ G} \\< p'',q'',t|e^{-i\\omega^a\n\\hat{\\varphi}_a}|p',q'\\>\n \\label{2.6a} \\\\\n&\\equiv &\n\\< p'',q'',t|\\hat{P}_G|p',q'\\>\\\\\n&=& \\int (d^J\\!pd^J\\!q\/(2\\pi)^J)\n\\< p'',q'',t|p,q\\>\\< p,q|\\hat{P}_G |p',q'\\>\\ ,\n\\label{2.6b}\n\\end{eqnarray}\nwhich is a quantum implementation of the classical initial value equation\nfor first-class constraints.\nHere $d\\mu(\\omega)$ is the invariant measure on the space of gauge group\nparameters, and $Vol\\ G = \\int_G d\\mu(\\omega)<\\infty$ is the gauge group\nvolume.\nIn what follows we also adopt a shorthand notation for the\nnormalized Haar measure\n\\begin{equation}\n\\delta \\omega \\equiv \\frac{d\\mu(\\omega)}{Vol\\ G} \\ ,\n\\ \\ \\ \\ \\ \\int\\limits_{G}^{}\\delta\\omega =1\\ .\n\\label{haar}\n\\end{equation}\nThe operator $\\hat{P}_G$\nis a projection operator onto the gauge invariant subspace.\n Its kernel is determined as the\ngauge group average of the unit operator kernel\n\\begin{equation}\n\\<p'',q''|p',q'\\>^{ph}\\equiv\\< p'',q''|\\hat{P}_G|p',q'\\> =\n\\int\\limits_{G}^{} \\delta\\omega\\, \\< p'',q''|e^{-i\\omega_a\n\\hat{\\varphi}_a}|p',q'\\>\\ .\n\\label{2.7}\n\\end{equation}\nFor some gauge systems, it can be calculated explicitly as well as\nthe kernel (\\ref{2.6a}) \\cite{pr}.\n\n\\subsubsection*{The path integral based on the projective method}\n\nApplying the projective formula (\\ref{2.6a}) to an infinitesimal\ntransition amplitude $t\\rightarrow \\epsilon =t\/N$ and making\na convolution of $N$ physical infinitesimal evolution operator\nkernels, we arrive at the following representation of the\namplitude (\\ref{2.6a})\n\\begin{eqnarray}\n\\< p'',q'',t|p',q'\\>^{ph} =&\\ &\n\\int \\prod\\limits_{l=1}^{N-1}(dp^J_ldq^J_l\/(2\\pi)^J)\n\\< p'',q'',\\epsilon |p_{N-1},q_{N-1}\\>^{ph}\n \\nonumber\\\\\n&\\times &\n\\< p_{N-1},q_{N-1},\\epsilon |p_{N-2},q_{N-2}\\>^{ph}\n\\cdots \\< p_1,q_1,\\epsilon |p',q'\\>^{ph}\\  .\n\\label{3.1}\n\\end{eqnarray}\nIn the continuum limit,\nwhere $N\\rightarrow \\infty,\\ \\epsilon\\rightarrow 0$, while\nthe product $t=N\\epsilon$ is kept fixed,\nthe convolution (\\ref{3.1}) of the kernels (\\ref{2.6a})\n$(t=\\epsilon)$ results\nin the coherent state path integral \\cite{kl2}\n\\begin{eqnarray}\n\\< p'',q'',t|p',q'\\>^{ph}&= &{\\cal M}\n\\int {\\cal D}C(\\omega) {\\cal D}p{\\cal D}q\\, e^{iS_H}\\ ,\n\\label{3.6} \\\\\nS_H\n&=&\\int\\limits_{0}^{t}dt'[\\left\n(p,\\dot{q}) - \\omega^a \\varphi_a(p,q) - h(p,q)\n\\right]\\ ,\n\\label{3.7}\n\\end{eqnarray}\nwhere ${\\cal D}C(\\omega)= \\prod_t \\delta\\omega(t)$\nis a formal  (normalized) measure for the gauge group\naverage parameters (cf (\\ref{2.6a})),\nand the symbol $h(p,q)$ is defined in (\\ref{4.2}).\nThus, the gauge group averaging\nparameters $\\omega^a$ become the Lagrange multipliers of the\nclassical theory in the continuum\nlimit.\n\nA relation between the path integral (\\ref{3.6}) and the projective\nformula (\\ref{2.6a}) is found in the boundary condition for the\npath integral. Recall that the integral (\\ref{3.6}) is taken over\nphase space trajectories that obey the boundary conditions\n\\begin{eqnarray}\np(0)&= &p'\\ ,\\ \\ \\ \\ q(0) = q'\\ ; \\label{3.9} \\\\\np(t)&= &p''\\ ,\\ \\ \\ \\ q(t)=q''\\ . \\label{3.10}\n\\end{eqnarray}\nIt is not hard to  find a gauge transformation such that\n\\begin{equation} (p^\\omega,{\\dot q^\\omega})-\n\\omega^a\\varphi_a(p^\\omega,q^\\omega)\n= (p,\\dot{q})\\ .\n\\label{3.11}\n\\end{equation}\nIt is equivalent to solving a linear equation\n\\begin{equation}\n\\dot{q}^\\omega + \\omega^a f_a(q^\\omega) = \\dot{q}\\ .\n\\label{3.12}\n\\end{equation}\nHaving found $q^\\omega$ one easily determines $p^\\omega$ as its\ncanonical momenta.\n\nThe path integral measure is formally\ninvariant under canonical transformations\nand, hence, the explicit dependence on the Lagrange multipliers of the\naction $S_H$ disappears after the canonical transformation constructed\nabove. The residual coherent state path integral represents a\ntransition amplitude in the unconstrained Hilbert space. However the integral\n$\\int {\\cal D}C(\\omega)$ cannot be factored out because a nontrivial\ndependence on the Lagrange multipliers survives at the boundaries.\nTo maintain the boundary conditions (\\ref{3.9}) and (\\ref{3.10}),\none can, say, require\n\\begin{equation}\np^\\omega(t) = p''\\ ,\\ \\ \\ \\ q^\\omega(t) = q''\\ .\n\\label{3.13}\n\\end{equation}\nThen it is impossible to satisfy the boundary condition (\\ref{3.9})\nbecause equation (\\ref{3.12}) admits only one boundary condition,\nsay, at the final time point. Thus, after the canonical transformation\nthe path integral\nmust be taken with boundary conditions that depends on $\\omega_a$\n\\begin{equation}\np^\\omega(0) = p'^\\Omega\\ ,\\ \\ \\ \\ q^\\omega(0) =q'^\\Omega\\ ,\\ \\\n\\ \\ \\Omega = \\Omega[\\omega] \\ ,\n\\label{3.14}\n\\end{equation}\nthat is, one gauge group average ``survives\" the canonical transformation\nthat removes the Lagrange multipliers from the action and provides\nthe equivalence of the path integral (\\ref{3.6})  to the projective\nrepresentation (\\ref{2.6a}).\n\n\\section{Gauge fixing and the path integral over physical phase space}\n\\setcounter{equation}0\n\nIn practice, it often turns out to be useful to integrate out the nonphysical\nphase-space variables associated with pure gauge degrees of freedom\nand work with the path integral over the physical phase space\n(\\ref{1.8}). For this purpose one usually fixes a gauge \\cite{3}\n\\begin{equation}\n\\chi_a(q) = 0\\ .\n\\label{g.1}\n\\end{equation}\nBy a necessary assumption, each gauge orbit $q^\\omega$ must\nintersect the gauge condition surface (\\ref{g.1}) (at least)\nonce. Under this assumption a generic configuration $q$ can be\nparametrized via lifting it onto the gauge condition surface\nalong a gauge orbit passing through $q$\n\\begin{equation}\nq = q_\\chi^\\theta(q^*)\\ ,\n\\label{g.2}\n\\end{equation}\nwhere $\\theta_a$ parametrizes the lift along a gauge orbit, and\npoints $q=q_\\chi(q^*)$ form the surface (\\ref{g.1}), i.e., $q^*$\nparametrizes the surface (\\ref{g.1}).\n\nIn the curvilinear coordinates (\\ref{g.2}) associated with\nthe chosen gauge condition, the constraints are linear combinations\nof canonical momenta for $\\theta_a$, and the Poisson bracket of\nthe canonical variables $p^*$ and $q^*$ with the constraints vanishes,\nthat is, $p^*$ and $q^*$ are gauge invariant according to (\\ref{2.2})\nand (\\ref{2.3}).\nThe $\\theta$-dependence\nof the action can be absorbed by a shift of the Lagrange\nmultipliers $\\omega^a$\non a suitable linear combination of the velocities\n$\\dot{\\theta}_a$ because the canonical one-form assumes the form\n\\begin{equation}\np\\dot{q} +\\omega^a\\varphi_a = p^*\\dot{q}^* + p_\\theta^a\\dot{\\theta}_a\n+\\omega^a\\varphi_a\n\\label{g.3}\n\\end{equation}\nand the Hamiltonian is gauge invariant (the $\\theta_a$'s are\ncyclic variables).\n\nThe integral over $\\theta_a$ yields the gauge group volume that\ncancels the one sitting in the measure ${\\cal D}C(\\omega)$. Finally,\nthe integrals over $\\omega^a$ and $p_\\theta^a$ can also be done, and\none ends up with the integral over physical phase space spanned\nby local symplectic coordinates $p^*, q^*$.\n\nThis result is usually achieved by a formal restriction of the\npath integral measure support in (\\ref{3.6}) to a subspace of\nthe constraint surface $\\varphi_a(p,q) =0$ selected by the gauge\n(supplementary) condition (\\ref{g.1}) \\cite{3}:\n\\begin{equation}\n{\\cal D} p{\\cal D} q{\\cal D} C(\\omega)e^{-i{\\textstyle\\int} dt \\omega^a\\varphi_a}\n\\rightarrow {\\cal D} p{\\cal D} q \\prod_t \\left(\\Delta_{FP}\\prod_a\n\\delta(\\chi_a)\\delta(\\varphi_a)\\right)\\;,\n\\label{f.1}\n\\end{equation}\nwhere $\\Delta_{FP} = \\det \\{\\varphi_a,\\chi_b\\}$ is the Faddeev-Popov\ndeterminant. After the canonical transformation associated with (\\ref{g.2})\nthe Faddeev-Popov measure assumes the form \\cite{3}\n\\begin{equation}\n{\\cal D} p^*{\\cal D} q^*{\\cal D} p^\\theta{\\cal D} \\theta \\prod_t \\delta(p^\\theta)\\delta(\\theta)\\;,\n\\label{f.3}\n\\end{equation}\nand the integration over the nonphysical variables $p^\\theta$ and\n$\\theta$ becomes trivial.\n\nTwo important observations are in order.  First, the procedure\n(\\ref{f.1}) corresponds to a canonical quantization {\\it after}\nthe elimination of all nonphysical degrees of freedom (the so called\nreduced phase-space quantization). As shown above, the physical\nvariables are associated with curvilinear coordinates, while\ncanonical quantization is consistent only in Cartesian coordinates.\nAs a result canonical quantization and the elimination of\nnonphysical degrees of freedom generally do {\\em not} commute \\cite{christ}.\nIn other words, the procedure (\\ref{f.1}) is not, in general, equivalent to\nthe Dirac quantization scheme \\cite{dir} where nonphysical degrees\nof freedom are removed after quantization.\n\nSecond, the geometry and topology of gauge orbits may happen to be\nsuch that there exists no unique gauge condition \\cite{grib}, meaning\nthat for any given $\\chi_a$ the system\n\\begin{equation}\n\\chi_a(q) = \\chi_a(q^{\\omega_s}) =0\n\\label{f.2}\n\\end{equation}\nalways admits nontrivial solutions with respect to $\\omega_s^a$. From\nthe geometrical point of view, the latter implies that the gauge orbit\n$q^\\omega$ intersects the gauge fixing surface more than once, namely, at\npoints $q^{\\omega_s}$. Discrete gauge transformations associated with\nthe gauge variables $\\omega_s^a$ do not reduce the number of physical\ndegrees of freedom, but they do reduce the ``volume\" of the physical\nconfiguration and phase spaces. Therefore the formal measure\n${\\cal D} p^*{\\cal D} q^*$ can no longer be Euclidean and the corresponding path\nintegral should be modified. If the residual discrete gauge transformations\nare explicitly known, then in such cases it appears to be possible to find\na modified path integral formalism that is equivalent to the\nDirac method \\cite{sha2}.\n\nFinally we remark that\nthe Liouville measure ${\\cal D}p^*{\\cal D}q^*= \\prod_tdp^*(t)dq^*(t)$\nis invariant with respect to canonical transformations. This freedom\nin the path integral over physical phase space can be interpreted\nas  gauge invariance. Indeed, another choice of a gauge condition\n(\\ref{g.1}) would induce another parametrization of the physical phase\nspace that is equivalent to the former via a canonical transformation.\nOn the other hand, we have argued in Section 1 that the formal invariance\nof the Liouville measure in the path integral is not sufficient to ensure\nthe invariance of the quantum theory\nwith respect to canonical transformations.\nIn the framework of gauge systems, it implies that, to achieve  gauge\ninvariance of the path integral over physical phase space, the measure\n should be regularized {\\em before} integrating out pure gauge degrees of\nfreedom with the help of a canonical transformation associated\nwith a chosen parametrization of the physical phase space by local\nsymplectic coordinates.\n\nIn the next section we propose a generalization of the path integral\nmeasure regularization with a Wiener measure to gauge theories.\n\n\\section{The Wiener measure for gauge theories}\n\\setcounter{equation}0\n\nThe Wiener measure regularized phase space path integral for a\ngeneral phase function $G(p,q)$ is  given by\n\\bn\n&&\\hskip-.3cm\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\n \\int\\exp\\{i{\\textstyle\\int}_0^T[p_j{\\dot q}^j+{\\dot G}(p,q)-h(p,q)]\\,dt\\}\\nonumber\\\\\n&&\\hskip1.5cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[{\\dot p}^2\n+{\\dot q}^2]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q\\nonumber\\\\\n &&\\hskip.3cm=\\lim_{\\nu\\rightarrow\\infty}(2\\pi)^J e^{J\\nu T\/2}\n \\int\\exp\\{i{\\textstyle\\int}_0^T[p_jdq^j+dG(p,q)-h(p,q)dt]\\}\\,d\\mu^\\nu_W(p,q)\\nonumber\\\\\n&&\\hskip.3cm=\\<p'',q''|e^{-i{\\cal H} T}|p',q'\\>\\;\\ ,\n\\label{4.1}\n\\en\nwhere the last relation involves a coherent state matrix element.\nIn this expression we note that ${\\textstyle\\int} p_j\\,dq^j$ is a\n{\\it stochastic integral}, and as such we need to give it a definition.\nAs it stands both the It\\^o (nonanticipating) rule and the Stratonovich\n(midpoint) rule of definition for stochastic integrals yield the same\nresult (since $dp_j(t)dq^k(t)=0$ is a valid It\\^o rule in these\ncoordinates). Under any change of canonical coordinates,\nwe consistently will interpret this stochastic integral\nin the Stratonovich sense because it will then obey the ordinary\nrules of calculus.\n\nWhy does the representation of the propagator as well as the Hamiltonian\noperator involve coherent states\n\\bn |p,q\\>\\equiv e^{-iG(p,q)}e^{-iq^jP_j}\ne^{ip_jQ^j}|0\\>\\;,\\hskip1cm(Q^j+iP_j)|0\\>=0\\;?\n\\label{4.3}\n\\en\nOne simple argument is as follows. The Wiener measure\nis on a flat {\\it phase space}, and is pinned at both ends thus resulting\nin the boundary conditions $p(T),q(T)=p'',q''$ and $p(0),q(0)=p',q'$.\nHolding this many end points fixed is incompatible with a Schr\\\"odinger\nrepresentation, which holds just $q(T)$ and $q(0)$ fixed, or with a\nmomentum space representation, which holds just $p(T)$ and $p(0)$ fixed.\nIt turns out, as a consequence of the Wiener measure regularization,\nthat the propagator is {\\it forced} to be in a coherent state\nrepresentation. We also emphasize the covariance of this expression\nunder canonical coordinate transformations. In particular,\nif ${\\overline p}d{\\overline q}=pdq+dF({\\overline q},q)$ characterizes a canonical\ntransformation from the variables $p,q$ to ${\\overline p},{\\overline q}$,\nthen with the Stratonovich rule the path integral becomes\n\\bn\n&&\\<{\\overline p}'',{\\overline q}''|e^{-i{\\cal H} T}|{\\overline p}',{\\overline q}'\\>\\nonumber\\\\\n&&\\hskip.3cm=\\lim_{\\nu\\rightarrow\\infty}(2\\pi)^J e^{J\\nu T\/2}\\int\n\\exp\\{i{\\textstyle\\int}_0^T[{\\overline p}_jd{\\overline q}^j+d{\\overline G}({\\overline p},{\\overline q})-{\\overline h}({\\overline p},\n{\\overline q})dt]\\}\\,d\\mu^\\nu_W({\\overline p},{\\overline q})\\nonumber\\\\\n&&\\hskip.3cm=\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}_0^T[{\\overline p}_j\n{\\dot{\\overline q}}^j+{\\dot{\\overline G}}({\\overline p},{\\overline q})-{\\overline h}({\\overline p},{\\overline q})dt]\\}\\nonumber\\\\\n&&\\hskip3.2cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[d\\sigma({\\overline p},{\\overline q})^2\/dt^2]\\,\ndt\\}\\,{\\cal D}{\\overline p}\\,{\\cal D}{\\overline q}\\,,\n\\label{4.4}\n\\en\nwhere $\\overline G$ incorporates both $F$ and $G$.\nIn this expression we have set $d\\sigma({\\overline p},{\\overline q})^2=dp^2+dq^2$,\nnamely, the new form of the flat metric in curvilinear phase space\ncoordinates. We emphasize that this path integral regularization\ninvolves Brownian motion on a flat space whatever\nchoice of coordinates is made. Our transformation has also made\nuse of the formal -- and in this case valid -- invariance of the\nLiouville measure.\n\nIf we have auxiliary terms in the classical action representing\nconstraints, then the expression of interest would seem to be\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}_0^T[p_j\n{\\dot q}^j-h(p,q)-\\omega^a\\varphi_a(p,q)]\\,dt\\}\\nonumber\\\\\n&&\\hskip1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[{\\dot p}^2+{\\dot q}^2]\\,\ndt\\}\\,{\\cal D} p\\,{\\cal D} q\\,{\\cal D} C(\\omega)\\;,\n\\label{4.5}\n\\en\nwhere the formal measure ${\\cal D} C(\\omega)=\\prod_t\\delta\\omega(t)$\nmay be proposed.\nWe expect some expression of this sort to hold; however,\nthe explicit proposal in (\\ref{4.5}) is incorrect\n as we now proceed to demonstrate.\n\nAccording to the discussion of the previous sections it is clear\nthat the physical propagator may also be given by\n\\begin{equation}\n\\lim_{\\nu\\rightarrow\\infty}\n{\\cal M_\\nu}\\int\\limits_{G}^{}\\delta\\Omega\n\\int\\exp\\{i{\\textstyle\\int}[p_j\n{\\dot q}^j-h(p,q)]\\,dt\\}\\exp\\{-(1\/2\\nu){\\textstyle\\int}{[\\dot p}^2+{\\dot q}^2]\n\\,dt\\}\\,{\\cal D} p{\\cal D} q\\ ;\n\\label{4.6}\n\\end{equation}\nhere all the paths satisfy $p(T),q(T)=p'',q''$ and $p(0),q(0)=p'^{\\,\n\\Omega},q'^{\\,\\Omega}$, where following the notation introduced in\nSection 2, we define\n\\bn\np^\\Omega=e^{-\\Omega^aad\\,\\varphi_a}p\\;,\\hskip1.5cm\nq^\\Omega=e^{-\\Omega^aad\\,\\varphi_a}q\\;.\n\\label{4.7}\n\\en\nIn short, we have used the fact that the unitary operators\nrepresenting the finite gauge group transformations satisfy\nthe condition (\\ref{c.4})\nmapping any coherent state into another coherent state.\n\nBased on the mapping property (\\ref{4.7}), we can give another formulation\nto the path integral (\\ref{4.6}). With the Wiener measure regularization\npresent, the path integral for any finite $\\nu$ is well defined,\nand as such we are free to change variables of integration.\nIn particular, let us make a canonical change of variables so that\n\\bn\n&&p(t)\\rightarrow e^{{\\textstyle\\int}_t^T ds\\omega^a(s)ad\\,\\varphi_a}p(t)\\;,\\nonumber\\\\\n&&q(t)\\rightarrow e^{{\\textstyle\\int}_t^T ds\\omega^a(s)ad\\,\\varphi_a}q(t)\\;,\n\\label{4.9}\n\\en\nwhere $\\omega^a$ are functions of time subject only to the\nrequirement that\n\\bn\n{\\textstyle\\int}_0^T\\omega^a(s)\\,ds\\equiv\\Omega^a\\;.\n\\label{4.10}\n\\en\nClearly, there are infinitely many functions $\\omega^a$ that will\nsatisfy such a criterion, and in a certain sense we will be led to\naverage over ``all'' of them. Note what this change of variables\naccomplishes. In the new variables, whatever the choice of\n$\\omega^a$ may be, the final values remain unchanged, $p(T),q(T)=p'',\nq''$, while the initial values have become $p(0),q(0)=p',q'$ since\n$(p'^{\\,\\Omega})^{-\\Omega}\\equiv p'$ and $(q'^{\\,\\Omega})^{-\\Omega}\n\\equiv q'$. Thus we have transformed all the gauge dependence\nfrom the initial points $p'^{\\,\\Omega},q'^{\\,\\Omega}$ and have\ndistributed it throughout the time interval $T$. This discussion is\nreminiscent of that in Sections 2 and 3.\n\nIt should be remarked that the condition (\\ref{4.10})\n may also be avoided if so desired.\nSuppose we drop the condition (\\ref{4.10}). Let $\\bar{\\Omega}^a$\nbe the value of the integral in the right-hand side of (\\ref{4.10}).\nSince the integral (\\ref{4.6}) involves the average over the gauge\norbit that goes through the initial point $p',q'$, the explicit\ndependence of the boundary condition on $\\bar{\\Omega}^a$ at the initial\ntime can be removed by an appropriate shift of the average parameters\n$\\Omega^a$. The initial boundary condition remains intact\n$p(0),q(0) = p'^\\Omega, q'^\\Omega$ in contrast to the case when\nthe condition (\\ref{4.10}) is imposed. Nevertheless, we proceed on the\nbasis of (\\ref{4.10}).\n\nLet us next see\nwhat are the consequences for the path integral of such a change\nof integration variables. We first observe that\n\\bn\n&&{\\dot p}(t)\\rightarrow{\\dot p}(t)-\\omega^aad\\,\\varphi_ap(t)={\\dot p}\n(t)-\\omega^a\\{\\varphi_a,p\\}(t)\\ ,\\nonumber\\\\\n&&{\\dot q}(t)\\rightarrow{\\dot q}(t)-\\omega^aad\\,\\varphi_aq(t)={\\dot q}(t)-\n\\omega^a\\{\\varphi_a,q\\}(t)\\;.\n\\label{4.11}\n\\en\nSuch a change leads to a new form for the path integral given by\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\n\\int\\limits_{G}^{}\\delta\\Omega\n\\int\\exp\\{i{\\textstyle\\int}[p_j({\\dot q}^j\n-\\omega^a\\{\\varphi_a,q^j\\})-h(p,q)]\\,dt\\}\\nonumber\\\\\n&&\\hskip-1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}[({\\dot p}-\\omega^a\\{\\varphi_a,\np\\})^2+({\\dot q}-\\omega^a\\{\\varphi_a,q\\})^2]\\,dt\\}\\,{\\cal D} p\\,{\\cal D} q\\;.\n\\label{4.12}\n\\en\nThis relation holds because the formal measure remains invariant\nunder this canonical transformation of coordinates. We recall\nthat in this form the fixed end points are $p(T),q(T)=p'',q''$\nand $p(0),q(0)=p',q'$.\nThis equation is true for any choice of $\\omega^a$ which fulfills\nthe required integral condition (\\ref{4.10}),\nand {\\it a fortiori} it is\nstill true if we average (\\ref{4.12}) over ``all'' functions which satisfy\nthe required integral condition. In so doing let us at the same\ntime incorporate the integral over $\\Omega$ and simply average\nover ``all'' functions $\\omega^a$ directly without any\ncondition on the overall integral value. For now let us continue\nto treat such an average in a formal manner; we will return to\nthe question of a proper average at a later stage. Thus we may\nreplace (\\ref{4.12}) by\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}[p_j({\\dot q}^j\n-\\omega^a\\{\\varphi_a,q^j\\})-h(p,q)]\\,dt\\}\n\\label{4.13}\\\\\n&&\\hskip-1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}[({\\dot p}\n-\\omega^a\\{\\varphi_a,p\\})^2+({\\dot q}-\\omega^a\\{\\varphi_a,q\\})^2]\\,dt\\}\\,\n{\\cal D} p\\,{\\cal D} q\\,{\\cal D} C(\\omega)\\;, \\nonumber\n\\en\nwhere $C(\\omega)$ denotes a measure which averages over all\nfunctions $\\omega^a$ as required.\nSince the object under discussion is manifestly gauge invariant,\nit is noteworthy that we can explicitly display such invariance\nunder the gauge transformations\n\\bn\n\\delta p=\\{\\varphi_a,p\\}\\delta\\lambda^a\\;,\\hskip1cm\n\\delta q=\\{\\varphi_a,q\\}\\delta\\lambda^a\\;,\\hskip1cm\n\\delta\\omega^a=\n\\delta\\dot{\\lambda}^a-f_{abc}\\omega^b\\delta\\lambda_c\\;,\n\\label{4.14}\n\\en\nfor general infinitesimal functions $\\delta\\lambda^a(t)$\nwhich vanish at the end points, and\nfor which the indicated path integral is invariant for\nall values of $\\nu$, hence in the limit. Although the\npath integral is invariant under the gauge transformations\nindicated, the reader should not jump to the conclusion that\nthe path integral diverges. In fact, the integral over the\ngauge functions $\\omega^a$ is an {\\it average},\nthat is, ${\\textstyle\\int} {\\cal D}C(\\omega)$ is finite, as we have\nstressed, and for a bounded integrand no divergences are possible.\n\nEquation (\\ref{4.13}) represents a manifestly gauge invariant\nexpression that is covariant under a general canonical\nchange of variables. For the class of constraints under\ndiscussion, we can also present another useful expression.\n Using the identity (\\ref{c.2}) leads to the equivalent relation\n\\bn\n&&\\lim_{\\nu\\rightarrow\\infty}{\\cal M_\\nu}\\int\\exp\\{i{\\textstyle\\int}[p_j\n {\\dot q}^j -\\omega^a\\varphi_a(p,q)-h(p,q)]\\,dt\\}\n \\label{4.15}\\\\\n&&\\hskip-1cm\\times\\exp\\{-(1\/2\\nu){\\textstyle\\int}_0^T[({\\dot p}-\\omega^a\n\\{\\varphi_a,p\\})^2+({\\dot q}-\\omega^a\\{\\varphi_a,q\\})^2]\\,dt\\}\\,{\\cal D}\np\\,{\\cal D} q\\,{\\cal D} C(\\omega)\\;, \\nonumber\n\\en\n and once again we recognize the parameters $\\{\\omega^a\\}$ as the\nLagrange mutipliers of the classical theory.\n\nAdditionally, we observe that the drift terms in the\nWiener measure cannot be neglected. For the Brownian motion\nwe have the It\\^o rule $dp(t)^2=\\nu dt$, and the connected expectation value\n$E(p(t)p(s))_{\\rm conn}=\\nu s(1-t\/T)$ for $s<t$. Thus the (Stratonovich)\nstochastic integral $(1\/\\nu){\\textstyle\\int} \\omega^a\\{\\varphi_a,p\\}\\,dp$\nand the term\n$(1\/2\\nu){\\textstyle\\int}[\\omega^a\\{\\varphi_a,p\\}]^2\\,dt$ are both of order\nunity for all values of $\\nu$ since in the general case\n$\\omega^a\\{\\varphi_a,p\\}\\simeq p$. A similar discussion holds\nfor $q$ as well. It is for this reason that our initial naive\nproposal (\\ref{4.5}) is not acceptable.\n\n\\subsubsection*{Choice of measure for the gauge variables}\n\nFinally we take up the question of the choice of the measure\n$C(\\omega)$ and its associated integral. Although we have\nloosely stated that ${\\cal D} C(\\omega)=\\prod_t\\delta\\omega(t)$ and that\nwe should integrate over all functions\n$\\omega^a$, this is still an imprecise concept. Despite appearances, there\nis actually a great deal of choice in this measure.\nThis freedom arises\nbecause the only real requirement on this measure is that it\nsimulates a {\\it single group invariant integral over the initial\nparameters} $p'^{\\,\\Omega},q'^{\\,\\Omega}$ as discussed in (\\ref{4.6}).\nWe shall consider two possible choices. The first one will provide\nus with a manifestly gauge invariant measure, while the second choice\ngives an example of a gauge noninvariant measure which nonetheless leads\nto the gauge invariant transition amplitude. The latter amounts to\nsome specific gauge fixing that is manifestly\n{\\it free} of any Gribov problem.\n\nTo define an appropriate measure\nthat is invariant under general gauge transformations,\nwe appeal to the classical theory of Kolmogorov \\cite{kol}\non stochastic processes, which will ensure that we obtain\na well-defined probability measure on the gauge path space.\nKolmogorov's theorem asserts that an\nunderlying probability measure on paths exists\nprovided the set of multi-time joint probability\ndensities satisfies certain basic consistency conditions.\nTo show the needed consistency let us again use $\\delta\\omega\n$ as the normalized Haar measure (\\ref{haar})\nfor the compact semi-simple gauge group under consideration.\nThen let us introduce a stochastic process defined by the following\nset of multi-time joint probability densities\n\\bn\n{\\cal P}_n(\\omega_n,t_n;\\ldots;\\omega_2,t_2;\\omega_1,t_1)\\equiv 1\n\\label{k.1}\n\\en\nfor all $n\\geq 1$. Here $T\\geq t_n>\\cdots t_2>t_1\\geq 0$.\nThe left-hand side of this equation is\nthe joint probability density for the gauge field\nto have value $\\omega_1$ at time $t_1$, value $\\omega_2$\nat time $t_2$, etc.\nIn this terminology $\\omega=\\{\\omega^a\\}$.\nThe right-hand side of this joint probability density\nrelation is simply unity, meaning that {\\it any} set of values\nat {\\it any} set of distinct times is equally likely.\nThis is the proper mathematical statement of a uniform\naverage over all gauge paths.\nConsistency of the given joint probability\ndensities is simply the trivial\nobservation that\n\\bn\n&&\\int {\\cal P}_n(\\omega_n,t_n;\\ldots;\\omega_r,t_r;\\dots;\\omega_1,t_1)\\,\n\\delta\\omega_r\\nonumber\\\\\n&&\\hskip1cm=1\\nonumber\\\\\n&&\\hskip1cm={\\cal P}_{n-1}(\\omega_n,t_n;\\ldots;\n\\omega_{r+1},t_{r+1};\\omega_{r-1},t_{r-1};\\dots;\\omega_1,t_1)\\;,\n\\label{k.2}\n\\en\nfor any choice or $r$, $n\\geq r\\geq 1$, and all $n$,\n$n\\geq2$; for $n=1$ the last line should be ignored.\nThe evident consistency of this set of joint\nprobability densities is then sufficient to guarantee for\nus a (countably additive) probability measure\non gauge fields, which we denote by $\\rho(\\omega)$,\nthat exhibits these joint probability distributions.\n\nAccepting this choice for the integration over gauge fields\nleads to the fact that the physical propagator\nmay be given the mathematically well-defined formulation\n\\bn\n\\hskip-1cm&&\\<p'',q''|e^{-i{\\cal H} T}|p',q'\\>^{ph}\\nonumber\\\\\n&&=\\lim_{\\nu\\rightarrow\\infty}(2\\pi)^J\ne^{J\\nu T\/2}\\int\\exp\\{i{\\textstyle\\int}[p_jdq^j+dG(p,q)\n-\\omega^a\\varphi_a(p,q)dt-h(p,q)dt]\\}\\nonumber\\\\\n&&\\hskip3.5cm\\times\\,d\\mu^\\nu_W(p,q,\\omega)\\,d\\rho\n(\\omega)\\;;\n\\label{k.3}\n\\en\nhere we have added $\\omega$ to the argument of $\\mu^\\nu_W$\nto acknowledge the presence of the drift terms.\nThe result only depends on the initial and final values of\n$p$ and $q$ since we have integrated over the set of gauge paths\nwithout any boundary conditions;\nthis result is still invariant under\ncontinuous and differential gauge transformations (\\ref{4.14}).\n\nFinally we note that the relation between\nthe physical Hamiltonian operator and\nthe classical expression $h(p,q)$ is given by\n\\bn\n{\\cal H}_{ph}\\equiv\n\\int h(p,q)\\,|p,q\\>^{ph}\\,^{ph}\n\\<p,q|\\,d^J\\!p\\,d^J\\!q\/(2\\pi)^J\\;,\n\\label{k.4}\n\\en\nwhere the physical coherent state $|p,q\\>^{ph}$ is obtained\nby the average of the coherent state (\\ref{c.4}) over the group\n$G$ with the normalized measure $\\delta\\Omega$.\n\nFormally, the measure $d\\rho(\\omega)$ constructed above comes naturally\nfrom the convolution formula (\\ref{3.1}) where each infinitesimal\ntransition amplitude is to be replaced by the corresponding infinitesimal\namplitude (\\ref{4.6}) with the Wiener measure. In this construction\nthe projection operator is inserted at each moment of time, that is, formally,\n$d\\rho(\\omega) = \\prod_t \\delta \\omega(t)$. Clearly, this formal measure\nsatisfies the conditions (\\ref{k.1}) and (\\ref{k.2}), and in addition it is\nmanifestly gauge invariant and normalized ${\\textstyle\\int} d\\rho(\\omega) =1$.\n\n  However, from the calculational point of view the measure $\\rho(\\omega)$\nis not always convenient. Sometimes it is also useful to have\na measure for the gauge variables that is not explicitly gauge\ninvariant (gauge fixing). A conventional gauge fixing discussed\nin Section 3 may suffer from Gribov ambiguities. Next we\nshow an example of a Gaussian probability measure free of such a disease.\n\nSince we want the measure to have at least one average over\nthe group manifold $G$, it is natural to assume\nthat for any time slice the measure must be the  group\ninvariant measure, but what is at our disposal is the\nrelationship of the functions at neighboring points of time.\nAs one set of examples, it would suffice to restrict our\nintegration to the set, or even a subset, of {\\it continuous\nfunctions}. A natural way to achieve it is to choose ${\\cal D} C(\\omega)$\nto be a Wiener measure on the manifold $G$\n\\bn\n{\\cal D} C(\\omega)= d\\rho_W(\\omega)=\n{\\cal N}\\exp[-\\textstyle{\\frac{1}{2}}{\\textstyle\\int} g_{ab}(\\omega)\n{\\dot\\omega}^a{\\dot\\omega}^b\\,dt]\\,\\Pi_t\\,\\delta\\omega(t)\\;.\n\\label{b.1}\n\\en\nHere the metric $g_{ab}(\\omega)$ is the positive-definite\nmetric associated with a homogeneous space determined by the\ncompact semi-simple gauge group.\nThe measure can also be regarded as the imaginary time quantum dynamics\n of a free particle propagating on the compact\nhomogeneous manifold $G$.\n\nLet us now establish a relation between the projection formula\n(\\ref{4.6}) and (\\ref{4.15}) with the choice (\\ref{b.1})\nfor the measure. Let $g_\\omega$ be an element of the gauge group in a\nmatrix representation. Then the action in the exponential\nin (\\ref{b.1}) can also be rewritten as\n\\begin{equation}\nS_W =-c\\,{\\rm tr}\\int\\limits_{0}^{T}(\\dot{g}_\\omega\ng_\\omega^{-1})^2\/2dt\\ ,\n\\label{b.2}\n\\end{equation}\nwhere $c=1\/{\\rm tr}(1)$ is a normalization factor.\nConsider a transition amplitude of\na free particle on the manifold $G$\n\\begin{equation}\nK_T(g_\\Omega, g_{\\Omega'}) ={\\cal N}\n\\int\\limits_{g_\\omega(0)=g_{\\Omega'}}^{g_\\omega(T)= g_\\Omega}\n\\prod\\limits_{t=0}^T \\delta\\omega(t) e^{-S_W}\\ ,\n\\label{b.3}\n\\end{equation}\nnormalized so as to satisfy\n\\begin{equation}\nK_T(g_{\\Omega''},g_{\\Omega'})=\\int K_{T-t}\n (g_{\\Omega''},g_\\Omega)K_t(g_\\Omega,g_{\\Omega'})\\,\\delta\\Omega\\,.\n\\label{b.3a}\n\\end{equation}\nDue to the global invariance of the action with respect to the left and\nright shifts, $g_\\omega\\rightarrow g_0g_\\omega$ and\n$g_\\omega\\rightarrow g_\\omega g_0$,  the amplitude (\\ref{b.3}) is\nalso invariant under these transformations\n\\begin{equation}\nK_T(g_\\Omega, g_{\\Omega'}) =\nK_T(g_0g_\\Omega,g_0 g_{\\Omega'}) =\nK_T(g_\\Omega g_0, g_{\\Omega'}g_0) \\ .\n\\label{b.4}\n\\end{equation}\nFrom (\\ref{b.4}) we deduce the identity\n\\begin{equation}\n\\int\\limits_{G}^{}\\delta\\Omega''\nK_T(g_{\\Omega''}, g_{\\Omega'}) =\n\\int\\limits_{G}^{}\\delta\\Omega'\nK_T(g_{\\Omega''}, g_{\\Omega'}) =  1\\ ,\n\\label{b.5}\n\\end{equation}\n which can be easily seen from the Feynman-Kac representation\nof the transition amplitude (\\ref{b.3}) as a spectral sum.\nThe integral (\\ref{b.5}) determines an action of the evolution\noperator on the ground state of the system. So, only the ground\nstate will contribute to the integral. We naturally assume that\nthe Casimir energy (the ground state energy) can always be\nsubtracted and included into the path integral normalization.\n\nNow we insert the identity (\\ref{b.5}) into the measure of the\npath integral (\\ref{4.6}) and then proceed with the change\nof variables (\\ref{4.9}). Since in the identity (\\ref{b.5})\neither $\\Omega''$ or $\\Omega'$ is a free parameter, we can\nalways choose it to coincide with the parameter $\\Omega$\nof the $G$-average in (\\ref{4.6}). Substituting\nthe path integral representation of $K_T$ (\\ref{b.3})\nin the appropriately transformed integral (\\ref{4.6}), we arrive\nat the expression (\\ref{4.15}) with the Wiener measure\n(\\ref{b.1}) for the gauge variables.\n\nTypically we\nencounter Wiener measures that are pinned at either the\ninitial time or at both end points; in the present case,\nthe measure for gauge variables\nis neither pinned at the initial nor the final\ntime as seen from the derivation of (\\ref{4.15}).\nSince the group is compact, the group volume is finite\nand we may therefore normalize such a Wiener measure that is\nnot pinned; our normalization is such that\n\\begin{equation}\n\\int{\\cal D} C(\\omega)=\n\\int\\limits_{G}^{}\\delta\\Omega''\\delta\\Omega'\nK_T(g_{\\Omega''}, g_{\\Omega'}) =  1\\ .\n\\label{b.6}\n\\end{equation}\nIn that case the formal measure ${\\cal D} C(\\omega)$ is actually a\nwell-defined (countably additive) probability measure which\nwe denote by $d\\rho_W(\\omega)$.\nWith this choice we note that\nthe physical propagator may also be given the well-defined\ndefinition (\\ref{k.3}) where $d\\rho(\\omega)\\rightarrow\nd\\rho_W(\\omega)$.\nThe result only depends on the\ninitial and final values of $p,q$ since we have integrated\nover the set of continuous $\\omega^a$ paths without any\nboundary conditions.\n\nThe measure is not invariant under the gauge transformations\n(\\ref{4.14}), nonetheless the transition amplitude is gauge\ninvariant because the measure provides the necessary projection\nonto gauge invariant states. In contrast to the conventional\nprocedure of section 3, there is no explicit gauge condition\nimposed on the system of phase space variables, and hence the\nGribov problem is avoided.\n\nOne should add that two such propagators, one from $t=0$ to\n$t=T$ and the second from $t=T$ to $t=2T$, for example,\nseems to\nnot compose to a propagator of the same form as (\\ref{k.3})\ndue to the discontinuity of paths at the interface.\nHowever, the resultant propagator  is nonetheless correct;\nit simply involves another acceptable form for the measure\n${\\cal D}C(\\omega)$.\n\n\n\\subsubsection*{Conclusion}\n\n\nWith (\\ref{4.15}) and two  choices\nof the measure for the gauge variables, we have arrived at\nour coordinate-free and mathematically\nwell-defined formulation for the path integral representation of\nthe special class of first-class constraints that was our goal.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nNATIVE AMERICAN FAMILY LIFE\n\n_**Colleen Madonna Flood Williams**_\n\n**NATIVE AMERICAN LIFE**\n\n**E UROPEANS AND NATIVE AMERICANS**\n\n**H OMES OF THE NATIVE AMERICANS**\n\n**H UNTING WITH THE NATIVE AMERICANS**\n\n**N ATIVE AMERICAN CONFEDERACIES**\n\n**N ATIVE AMERICAN COOKING**\n\n**N ATIVE AMERICAN FAMILY LIFE**\n\n**N ATIVE AMERICAN FESTIVALS AND CEREMONIES**\n\n**N ATIVE AMERICAN HORSEMANSHIP**\n\n**N ATIVE AMERICAN LANGUAGES**\n\n**N ATIVE AMERICAN MEDICINE**\n\n**N ATIVE AMERICAN RELIGIONS**\n\n**N ATIVE AMERICAN RIVALRIES**\n\n**N ATIVE AMERICAN SPORTS AND GAMES**\n\n**N ATIVE AMERICAN TOOLS AND WEAPONS**\n\n**W HAT THE NATIVE AMERICANS WORE**\nNATIVE AMERICAN FAMILY LIFE\n\n_**Colleen Madonna Flood Williams**_\n\n**Miniconjou men, women, and children pose between two teepees, circa 1890**.\n\n**SENIOR CONSULTING EDITOR DR. TROY JOHNSON PROFESSOR OF HISTORY AND AMERICAN INDIAN STUDIES CALIFORNIA STATE UNIVERSITY**\n\nAs always, to Paul R.Williams and Dillon J. Meehan with all my love. Especially for Kayleigh and Michael Baker and for Mathew and Maureen.\n\n  |\n\nMason Crest  \n450 Parkway Drive, Suite D  \nBroomall, PA 19008  \nwww.masoncrest.com\n\n---|---\n\n\u00a9 2014 by Mason Crest, an imprint of National Highlights, Inc.\n\nAll rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, taping, or any information storage and retrieval system, without permission from the publisher.\n\nPrinted and bound in the United States of America.\n\nCPSIA Compliance Information: Batch #NAR2013. For further information,  \ncontact Mason Crest at 1-866-MCP-Book\n\nFirst printing  \n1 3 5 7 9 8 6 4 2\n\nLibrary of Congress Cataloging-in-Publication Data\n\nWilliams, Colleen Madonna Flood.\n\nNative American family life \/ Colleen Madonna Flood Williams.\n\npages cm. \u2014 (Native American life)\n\nIncludes bibliographical references and index.  \nISBN 978-1-4222-2969-9 (hc)  \nISBN 978-1-4222-8856-6 (ebook)\n\n1. Indians of North America\u2014Kinship\u2014Juvenile literature.\n\n2. Families\u2014United States\u2014Juvenile literature. I. Title.\n\nE98.K48W55 2013  \n649'.108997\u2014dc23\n\n2013007470\n\nNative American Life series ISBN: 978-1-4222-2963-7\nTABLE OF CONTENTS\n\nIntroduction: Dr. Troy Johnson\n\n1 Native American Families\n\n2 The Iroquois Family of Nations\n\n3 Parrots for Pets and Postball as a Pastime\n\n4 Buffalo Hunters, Sheep Herders, and Desert Farmers\n\n5 Families of Abundance and Subsistence\n\n6 Family Ties in Central and South America\n\nChronology\n\nGlossary\n\nFurther Reading\/Internet Resources\n\nIndex\nINTRODUCTION\n\nFor hundreds of years the dominant image of the Native American has been that of a stoic warrior, often wearing a full-length eagle feather headdress, riding a horse in pursuit of the buffalo, or perhaps surrounding some unfortunate wagon train filled with innocent west-bound American settlers. Unfortunately there has been little written or made available to the general public to dispel this erroneous generalization. This misrepresentation has resulted in an image of native people that has been translated into books, movies, and television programs that have done little to look deeply into the native worldview, cosmology, and daily life. Not until the 1990 movie _Dances with Wolves_ were native people portrayed as having a human persona. For the first time, native people could express humor, sorrow, love, hate, peace, and warfare. For the first time native people could express themselves in words other than \"ugh\" or \"Yes, Kemo Sabe.\" This series has been written to provide a more accurate and encompassing journey into the world of the Native Americans.\n\nWhen studying the native world of the Americas, it is extremely important to understand that there are few \"universals\" that apply across tribal boundaries. With over 500 nations and 300 language groups the worlds of the Native Americans were diverse. The traditions of one group may or may not have been shared by neighboring groups. Sports, games, dance, subsistence patterns, clothing, and religion differed\u2014greatly in some instances. And although nearly all native groups observed festivals and ceremonies necessary to insure the renewal of their worlds, these too varied greatly.\n\nOf equal importance to the breaking down of old myopic and stereotypic images is that the authors in this series credit Native Americans with a sense of agency. Contrary to the views held by the Europeans who came to North and South America and established the United States, Canada, Mexico, and other nations, some Native American tribes had sophisticated political and governing structures\u2014that of the member nations of the Iroquois League, for example. Europeans at first denied that native people had religions but rather \"worshiped the devil,\" and demanded that Native Americans abandon their religions for the Christian worldview. The readers of this series will learn that native people had well-established religions, led by both men and women, long before the European invasion began in the 16th and 17th centuries.\n\nGender roles also come under scrutiny in this series. European settlers in the northeastern area of the present-day United States found it appalling that native women were \"treated as drudges\" and forced to do the men's work in the agricultural fields. They failed to understand, as the reader will see, that among this group the women owned the fields and scheduled the harvests. Europeans also failed to understand that Iroquois men were diplomats and controlled over one million square miles of fur-trapping area. While Iroquois men sat at the governing council, Iroquois clan matrons caucused with tribal members and told the men how to vote.\n\nThese are small examples of the material contained in this important series. The reader is encouraged to use the extended bibliographies provided with each book to expand his or her area of specific interest.\n\nDr. Troy Johnson  \nProfessor of History and American Indian Studies  \nCalifornia State University\n\n**This Native American family is dressed in traditional clothing, called regalia, which is worn for special occasions or tribal ceremonies. Family relationships are very important to Native Americans, and social life often revolves around the family.**\n**1** **Native American Families**\n\nFamilies are the building blocks of society. Thus, it is no wonder that Native American societies of the past and present revolve around the family. Traditionally, Native American family ties are traced through blood relationships, clans, bands, and tribes. Sometimes, adoptions establish family ties.\n\nThe varying family structures of Native Americans are quite complex. There are nuclear and extended Native American families. A nuclear family, or one generational family, is composed of a mother, father, and their children. An extended family is composed of two or more generations of a family living together.\n\nClans are another form of Native American families. A totem or a common ancestor generally links a clan. Clan traditions, customs, and organizational patterns vary greatly from region to region and people to people.\n\n**This photograph from the 19th century shows a Native American family of the Plains outside of their home.**\n\nObviously, Native American family life is a broad and complicated subject. The following chapters of this book will look at some of the different aspects of the family lifestyles of select groups of pre-Columbian and early-contact Native American cultures. These groups have been chosen to provide a representation of the differing lifestyles of pre-Columbian and early-contact Native American families throughout the Americas.\n\nIn this book, the past tense is used not to suggest that Native American families are a thing of the past. It is used solely to signify that these family descriptions are referring to Native American families from pre-European contact to early-European contact periods.\n\nTo learn more about Native American families, the author recommends that you study in greater depth the family structures, traditions, and customs of individual tribes. Keep in mind that no two families are exactly alike. This applies to Native American families, just as it does to all other families. It is important to note that while the families of the cultural regions depicted in this book shared many things in common, they were also vastly different in many ways. \u00a7\n\n**A shaman wearing a mask peers through the doorway of an Iroquois longhouse. When the first French and English settlers arrived in northeastern North America during the early 17th century the Iroquois Nation\u2014actually a group of tribes that included the Onondaga, Mohawk, Seneca, Oneida, Cayuga, and Tuscarora tribes\u2014controlled much of the region.**\n**2 The Iroquois Family of Nations**\n\nThe Haudenosaunee (Iroquois) people envisioned themselves as being one giant family. In fact, they formed a family of nations. The name, Haudenosaunee, which is used to refer to this Iroquoian nation, means, \"People of the Longhouse.\" The symbolic longhouse they shared was the northeast woodlands. The Mohawk were the Guardians of the Eastern Door of the Haudenosaunee's great figurative longhouse. The Seneca were the Guardians of the Western Door. Lineage was traced through the maternal, or mother's side, of the longhouse family.\n\nIn descriptions of their large, extended family, the Haudenosaunee referred to the tribes within their nation as brothers. The Elder Brothers of the Haudenosaunee are the Onundagaono (Onondaga), Kanienkahagen (Mohawk), and Onondowahgah (Seneca). The Younger Brothers are the Onayotekaono (Oneida), Guyohkohnyoh (Cayuga), and Ska-Ruh-Reh (Tuscarora).\n\nThe Haudenosaunee taught their children to respect and honor both their younger and elder brother tribes. They also taught their children to be thankful for their \"Three Sisters\"\u2014corn, squash, and beans. These three important food staples supported, sustained, and nurtured the growth of the Haudenosaunee nation.\n\nIroquois children inherited the clan symbol and ties of their mother. When a man and a woman married, the man moved from his mother's longhouse to the longhouse of his wife. He owned only personal items, clothing, and weapons.\n\nThe true center of longhouse family relationships revolved around the fireside family. A fireside family, like a nuclear family, was made up of a mother, a father, and their children. The Iroquois family then branched out to include extended family or clan members. Tribal nationality was comprised of clans. Finally, the intertribal family of the nation was made up of the members of all of its smaller tribes, bands, or \"brother\" nations.\n\nAnother group of Native Americans that lived on the Atlantic Coast of North America was the Algonquians. The members of the Algonquian-language family formed alliances similar to the Iroquois Confederacy. However, these alliances were never as structured as that of the Haudenosaunees. They tended to be looser groupings of small bands of Algonquian peoples that joined together only during battle or to trade with one another.\n\nGrand sachems, or chiefs, led Algonquian bands joined together as confederacies. The lesser chiefs of the individual tribal bands were known as sagamores. The grand sachems often served as negotiators for the sagamores. Algonquian tribes were often matrilineal **;** however, some, like the Mi'kmaq, were patrilineal.\n\nDaily work was divided between the two moieties of a clan. When a member of one moiety died, the members of the other moiety would take care of the chores associated with the death of a family member. This allowed the deceased person's immediate moiety members to attend solely to their mourning.\n\n**A group of Native American boys practice with bows and arrows. In Iroquois tribes, boys learned archery and other hunting skills at a young age. This training would serve them well when they became adults, and therefore responsible for providing food for their family.**\n\nThroughout the Northeast, women did the cooking, raised the children, and cared for the elderly members of their tribes. They did the weaving, tanning, pottery, and basket making. Women also made footwear and clothing for their families.\n\nWomen made cradleboards to make carrying their babies as they worked easier. The cradleboards were often decorated with ornate designs. Relatives and friends gave the woman special totems to hang from her child's cradleboard. This was done in order to help keep the baby healthy and safe.\n\n**These small figures depict a Native American storyteller entertaining a group of young listeners. Storytelling was a popular pastime among the Native Americans of the northeast. Both the adult figure and the young girl to the right are holding infant children in cradleboards.**\n\nNortheastern mothers often bathed their children in cold lakes and streams. This was believed to be necessary to help ensure a young child's survival through his or her first winter. The woodlands could be a cold and harsh place to live during the winter. Young children needed to be acclimated to the cold. Being bathed in cold water helped to acclimate the young.\n\n**The Mi'kmaq decorated their clothing with their clan symbol. They also painted it on their canoes, snowshoes, and other possessions.**\n\nThe Haudenosaunee women owned all property, and a respected elder matron led each family clan. Female clan leaders chose the sachems, or clan chiefs, for their tribes. If the woman who chose a sachem became displeased with the way he was representing their people, she could have him removed from his leadership position.\n\nHaudenosaunee women were the keepers of the \"Three Sisters.\" Mothers, daughters, and grandmothers sowed, tended, prayed over, and harvested the beans, squash, and corn that helped to feed their families. It was their responsibility to weed the fields and to store supplies of food for the family in underground caches.\n\nThe men of the Haudenosaunee nation hunted and trapped animals. They were brave warriors who protected their families and their nation. Haudenosaunee men were also responsible for making weapons and canoes. The clan sachems were men of the Haudenosaunee who represented their tribes at meetings of the Iroquois Confederacy. (If the men were from Algonquian tribes, then their sachems represented them at meetings of their own confederacies.)\n\n**The majority of the Haudenosaunee were divided into three main clans: the turtle, bear, and wolf clans. Animals played an important role in Haudenosaunee spiritual life. A clan's totem animal was considered to be its kindred spirit. Animals were also often the main characters of religious and teaching stories told by the elders to the young.**\n\nMaternal uncles played an especially big role in the life of young male Haudenosaunees. Clans were linked through the mother's bloodlines, so a young man learned his family history from his mother's brothers. Maternal uncles were responsible for preparing their young nephews for clan life, important rituals, marriage, and the responsibilities of adult life.\n\nAround the age of seven or eight, Haudenosaunee children were expected to begin performing chores. The chores that they were given helped ease the day-to-day workload of their fireside family. More importantly, the chores helped to prepare the children for their roles later on in life.\n\nYoung girls would help haul water and tend to the crops. They would also assist their mothers, aunts, and grandmothers with sewing and cooking. They learned to make clothing and footwear. They also learned to make cornhusk dolls and other toys for their younger siblings. Their aunts, grandmothers, and mothers shared stories and bits of wisdom with them as they worked. In this way, young girls gradually learned the skills necessary to be a responsible, productive wife and clan mother.\n\nHaudenosaunee men were often away from home for long periods of time. Because of this, young boys were encouraged to \"play\" at fighting, hunting, and camping. This type of \"play\" helped them to learn important life skills.\n\nTheir mothers often made toy war clubs for them out of corn silk. The boys would use these soft clubs to participate in mock battles against their friends. This allowed them to learn important hand-to-hand combat skills.\n\nTo be capable providers and effective warriors, the young boys needed to perfect their use of snares, bow and arrows, blowguns, and other weapons. When their grandfathers, fathers, and uncles were home from hunting or war parties, the young boys learned how to make canoes, weapons, and tools. The elder men gave the younger men advice on how to become respected hunters, warriors, husbands, and fathers.\n\nAfter a young boy killed his first deer, he was allowed to join the adult hunting parties. When he reached puberty, he was sent into the woods on a vision quest. A respected elder would often accompany the adolescent on this sacred journey.\n\nAt this time, the young man would demonstrate his physical strength, intelligence, and manliness. These were important assets, as life in the woodlands could be quite demanding. A young man would strive to demonstrate that he was capable of withstanding great challenges in order to protect and provide for his family and nation.\n\n**Around the year 1570, two influential men, Hiawatha and Deganawida, first encouraged the Cayuga, Mohawk, Oneida, Onondaga, and Seneca to unite. The Huron prophet Deganawida spoke of a vision of the tribes united beneath a \"Great Tree of Peace.\" Hiawatha, a Mohawk medicine man, traveled by canoe throughout the Haudenosaunee territories speaking about the importance of unity.**\n\n**Hiawatha and Deganawida's efforts were rewarded with the formation of the Iroquois Confederacy. This confederacy of tribes, or Iroquois nations, started out as a union of five nations. In the early 1700s, the Tuscarora joined the Haudenosaunee family of nations, bringing the number to six. Many historians now believe that the United States of America owes a debt of gratitude, not only to the Roman ideals of democracy and the republic, but also to the political models of the Haudenosaunee's Pine Tree Sachems, Grand Council, and Grand Council Fire.**\n\nThe young man might also reveal his dreams to his elder companion. The elder would consider the young man's dreams carefully and identify the guardian spirit that was trying to show itself to the young man. The young man would then find or carve a symbol of that spirit to keep as a personal totem.\n\nA young girl generally came of age around the same time that she became capable of bearing children. At this time, the girl's mother would build a small wigwam outside the main village. The girl would retreat into the wigwam for several days. She would pray for the blessings of healthy children, a good husband, and a long life. She would fast during this time, drinking only water.\n\nDelaware boys participated in a coming-of-age initiation rite called the Youth's Vigil. A boy would enter the woods and fast for a number of days. Hopefully, he would have a vision during this time that revealed the spirit guides and forces that would protect and direct him from young manhood to old age.\n\nFor the Delaware tribe of the Algonquian people, marriage and divorce were simple matters. A man and woman who moved in together were considered to be husband and wife. If either one wanted a divorce, the man would leave the family dwelling. It was the wife's right to retain custody of the home and children.\n\nFamily life in the woodlands culture area was not all work. Although the children owned few possessions, their parents did make toys for them. Young girls and boys used sticks to roll hoops made out of birch bark along the ground. Animal skin balls were stuffed with feathers or fur. Men and boys played a demanding game, much like modern-day lacrosse, called baggataway.\n\nStory telling was a favorite family activity that included young and old alike. The stories shared might be exciting, funny, scary, or sad. They were most certainly always entertaining, as the Haudenosaunee were known for their oratorical skills. Family elders used these stories to teach young people the secrets of successful living and the art of story telling, so that they might pass these things on to their own children. \u00a7\n\n**A child watches from his position tied into a cradleboard. Native American women used the device to carry their children while they worked at their daily chores: planting and cultivating crops, preparing meals, and making household items.**\n**3 Parrots for Pets and Postball as a Pastime**\n\nIn the region of the American South and the Caribbean Sea, clans were the most common family system. Among Native American tribes of the southeast, clans were mainly matrilineal. This was also the case with the Tainos, who lived in the Caribbean before arrival of Europeans in the late 15th century and early 16th century.\n\nIn the Caribbean, male rulers of Taino villages were known as caciques. If a female was the ruler of the village, she was called a cacias. The eldest son of the ruler's eldest sister traditionally succeeded a Taino leader. This practice made the Taino society primarily patriarchal, but also matrilineal.\n\nDuring labor, a Southeastern woman often retreated to the privacy of a birthing hut. After the child was born, a ritual cleansing took place. Animal grease was rubbed all over the child's body and then the infant was placed in a cradleboard. Much of a child's first year of life would be spent in this cradleboard.\n\nSoutheastern women nursed their babies until they were about two years old. They were gentle mothers who did not often resort to physical punishments. More often than not, these wise mothers disciplined their children verbally. Good behavior was promoted through the use of high expectations, wit, and wisdom. through the use of high expectations, wit, and wisdom.\n\nMothers tried to instill the traits of humility, industriousness, common sense, and cool-headedness in the young women of their families, clans, or tribes. They did this through story telling, by being good role models, and by having their young girls assist them with the family chores.\n\nWomen did the planting, tended the gardens, and harvested the crops. Women made clothing for themselves, their husbands, and their children. They also made pottery, tanned animal skins, and wove fine baskets and mats. These skills were all patiently taught to their granddaughters, nieces, and daughters.\n\nWomen prepared the meals. Corn chowders, cornbreads, or some other corn dish was a part of almost every family meal. Family meals often featured wild game, such as bear, deer, fish, or turkey. Pumpkins, beans, peas, potatoes, and squash might also accompany such meats.\n\nTaino women wove, made hammocks, cooked, and prepared the cassava. They also attended to their husbands' hair and painted their bodies according to ceremonial rules. If single, a women went without clothes; if married, she would wear a kind of apron that covered her from her waist to the middle of her legs.\n\n**A French explorer and artist named Jacques le Moyne was assigned the task of documenting the lifestyles and homes of the natives of Florida, as well as mapping the terrain. His journals and drawings have become an important historical resource on the Timucua people.**\n\n**The Timucua and other Indians native to the Florida region lived in houses such as this one made of grass and reeds over a wooden framework. When the first Spanish explorers arrived in the region during the 16th century, they found many Timucua settlements. Some were walled, as in this drawing, to keep out enemy tribes or wild animals.**\n\nMen were the family guardians and bravely protected their families and homes. They stood up for the rights of their family, both within their communities and within the world at large. They held the members of their family accountable for poor behavior, but also celebrated the achievements of their wives and children.\n\n**This 16th-century illustration by Jacques LeMoyne shows Taino natives paddling a dugout canoe. To make these canoes, the Taino cut down a large tree and hollowed it by using stone tools as well as careful applications of fire. Men used the canoes to hunt and fish.**\n\nThe men of the Southeast did the arduous work of clearing the fields so that the women could farm. Creek boys helped women in the gardens until they reached puberty. More often than not, however, men were usually off hunting, fighting, or fishing.\n\nMany Southeastern men were permitted to have more than one wife. A good provider or a wealthy man might have as many wives as he could support. However, his first wife always retained her social status as being the first. Frequently, it was the right of a first wife to approve of or disapprove of her husband's choice of subsequent wives.\n\nTimucua and other men of the Southeast often tattooed their bodies. A man's family status and skill as a warrior was often indicated through such tattoos. The Timucua would severely punish a man who tattooed himself with tattoos that reflected a higher status than he had truly earned. The offender was forced to remove the tattoos from his body.\n\nFurther south, in the Caribbean, the Taino men wore a breechcloth of cotton or palm fibers cut to a specific length to show rank. This made it easy for other members of his home and village to see his status among them.\n\n**The name the Cherokee use for their people is Ani-Yun'Wiya. This means the \"main people.\" The name \"Cherokee\" is thought to have come from a word used by the Choctaw to label the Ani-Yun'Wiya as \"the people of the land of many caves.\"**\n\nLarge groups of Taino men lived in houses called bohios. One bohio might house from 10 to 15 Taino men, as well as their entire families. It was not uncommon for a Taino bohio to serve as home to up to 100 people.\n\nTaino men cleared the land so that their women could farm it. They hunted, fished, and made wooden dugout canoes. They used their canoes to hunt ducks and sea turtles.\n\nTaino men trained dogs to assist them in their hunts. Taino men were peaceful and engaged in battle primarily only for self defense from the warrior Caribs that raided their villages.\n\nThe Carib men were known as fierce warriors. They traveled about the Caribbean, raiding the villages of neighboring communities. They often took prisoners, and it is said that they were cannibals. Male prisoners would be eaten. Female prisoners became the wives or slaves of their captors. Carib men were also expert sailors and fishermen whose dugout canoes were sometimes fitted with sails.\n\nCarib communities traced their relationships through matrilineal kin groups. They did not have chiefs. Carib wives were often women who had been taken from other island communities, like the Tainos, that the Carib had raided.\n\nChildren played together for the first few years of their lives. It was not unusual for children under four or five years of age to go without clothing during the warm months of spring and summer. Around five or six, children were expected to begin learning their respective life skills.\n\nBoys learned from their uncles, fathers, and grandfathers. They had close relationships with their maternal uncles, who oversaw their growth from boys to men. Boys were taught how to fish, hunt, and trap. They learned how to make bows and arrows. They carved spears and clubs to be used in battle. They were taught how to clear the land for farming and how to build palisades to protect their villages.\n\nIn their spare time, when they weren't helping their families or their villages, boys learned to do fine carvings and make tobacco pipes and ceremonial drums. A top-quality drum, ornately carved war club, or a first-rate tobacco pipe was a treasured item. Skillful craftsmen who could make these types of items were held in high regard.\n\nWomen of the village trained young girls in the arts of weaving, sewing, cooking, courting, marriage, and motherhood. Grandmothers, aunts, and mothers all spoke to the girls about the qualities of a good wife and mother. They talked to the girls about traditional family values, religious beliefs, and acceptable social behavior.\n\nThe little girls watched, learned, and helped as their mothers and other women of the tribe made clothing and blankets. They learned to make pottery, sleeping mats and floor mats out of cane, and finely woven grass baskets. The girls were also taught to cook, dry, and store corn and other food staples.\n\n**The Caribs were a fierce people that lived in the Caribbean region. They often attacked the villages of other tribes, as this Theodore DeBry illustration of 1570 shows.**\n\nFor many young women of the Southeast, coming of age meant starting their menstrual cycles. This great milestone in a young girl's life was marked by a special ceremony. At this time, young girls began to practice a ritual that they would observe monthly, until late in life when they reached menopause. The day their period began, they were separated from the rest of their family and village.\n\nA hut was set aside to house a menstruating woman. She used this time to rest, pray, and meditate. When her period ended, she took a ritual cleansing bath. Afterwards, she would put on clean clothing and return to her family home.\n\nThere were many things that a young boy needed to know and do to become a man. His uncles, father, and grandfathers tried to help him understand his place in the family, community, and world at large. Young men in the Southeast were expected, in some tribes, to participate in a battle and prove their skill as warriors before they would be declared men. Other tribes dictated that a young man who wished to marry must build a home and kill a bear or a deer, proving to all that he could be a good provider. The transition from boyhood to manhood was always cause for great family celebration.\n\nMarriage for the Seminoles was a fairly casual relationship. If a man wanted to divorce his wife, he merely left their home. Other Southeastern tribes planned arranged marriages for their children.\n\nA series of talks would take place between the women of the two clans. They would let the fathers of both children know what was going on, but the men had no real influence in the matter. If the women decided it was a good match, the young girl would be asked if she would consent to the marriage.\n\nNext, the girl would place a bowl of food outside her home. The young man who wished to marry her would ask if he might have a taste of the food. If the young girl said yes, then the marriage was arranged. If she said no, the young man knew that he had been turned down.\n\nSoutheastern families made time to have fun together. Men, women, and children alike would gather around a post in the village center to play a game called postball. The women were given the advantage of being able to use their hands. The men had to use sticks, similar to those used in lacrosse. The object of the game was to hit the post with a ball. After the game was over, everyone would share a festive meal and dance.\n\nTaino families enjoyed having fun, too. Taino parents allowed their children to have parrots as pets. Together, families watched and participated in recreational wrestling matches, foot races, and archery contests. Singing, swimming, and dancing were other ways that Taino families entertained themselves in their tropical homeland. \u00a7\n\n**The interior of a Mandan home is shown in this drawing from the 1840s. The Mandan were a tribe of Native Americans that lived on the prairies when the first European settlers arrived in the West.**\n**4** **Buffalo Hunters, Sheep Herders, and Desert Farmers**\n\nThe tribes of the American West are a diverse lot, ranging from the Native Americans of the Great Plains like the Sioux and Pawnee to the pueblo-dwelling Hopi and Zu\u00f1i of the southwest. There were some similarities in family relationships among these many tribal groups, however.\n\nThe Hopi were bound by the family ties of 12 clan groups. Called _phratries_ , these 12 main clans have many smaller clans within them. Men have always been Hopi religious leaders, but children inherit the clan of their mother. Hopi mothers, daughters, and granddaughters lived together with their children and husbands in one-room homes that were built side by side or atop the one-room dwellings of other Hopi families. The Spanish called these adobe apartment-style complexes \"pueblos.\"\n\nHopi women made clay pottery for their families, wove baskets from various plants, harvested edible wild plants, and did the cooking. They also maintained the family home, continually smoothing the adobe walls to keep them from weakening due to erosion. Hopi men farmed and hunted. They wove cotton for use in blankets and clothing. A Hopi man even made his wife's wedding outfit for her.\n\nThe men of Hopi communities belonged to special religious groups that met in kivas. They dressed as spirits called kachinas for religious ceremonies. Hopi men, dressed as kachinas, rewarded or disciplined Hopi children based on each child's behavior. Uncles were also the family disciplinarians.\n\nAlthough they did not do the farming, Hopi women owned the crops farmed by their men. Hopi women also owned their houses, but were only allowed in kivas when invited by the men. Hopi mothers raised both their sons and daughters until the age of six, when the boys were taken into the company of the village men.\n\nHopi girls had a special squash-blossom hairstyle that announced to the world that they were young women, eligible to marry. A young Hopi maiden had to earn the right to wear this special hairstyle. In order to do so, she participated in a coming-of-age ritual. During part of this ceremony, the young girl had to demonstrate her skill at grinding corn. If she successfully completed her coming-of-age ceremony, she was no longer regarded as a child.\n\nAnother tribe of the southwest was the Apache. They were more likely to wander than were the Hopi. Apache men were responsible for hunting and for protecting the family. In some tribal groups, Apache men also were responsible for providing farm labor for their in-laws. The men sometimes dressed up like gans, or mountain spirits, in order to ward off evil spirits, cure illness, or to ask these special spirits to watch over their crops and game.\n\nMost Apache men had only one wife. However, if a man decided to have a second wife, she would usually be one of his first wife's sisters or cousins. He could bring this second wife into the wickiup where his first wife lived, or build a second wickiup for his new wife.\n\nApache women gathered wood for their extended families, worked together to prepare feasts, gathered acorns, picked mulberries, and collected pine nuts. They made tools, clothing, and baskets for their families. After contact with Europeans, they started sheep herding and passed this skill onto their daughters. They taught their children respect for their elders, the land, and the sacred religious traditions of the Apache nation.\n\n**A Hidatsa woman grinds grain inside of a building made of wooden beams.**\n\nApache boys had to learn all of the many skills necessary for participation in their band's gathering, hunting, and raiding parties. They usually began war training around the age of 10. Over the course of their pre-teen and early teenage years, Apache boys were taught the qualities and skills of a wise and honorable Apache father, husband, and warrior.\n\n**This photograph shows the famous Sioux medicine man Sitting Bull with his family in front of their home. Sitting Bull was an influential Sioux leader during the 19th century. He opposed the invasion of white settlers into tribal lands, especially the Black Hills of South Dakota, which the Sioux considered a sacred place. Sitting Bull took part in many battles, including the battle of the Little Bighorn where the Sioux and Cheyenne won a great victory over the U.S. 7th Cavalry. After the Sioux were forced to stop fighting and move to a reservation, Sitting Bull traveled throughout the country, asking the government to respect the rights of his people. He was killed in December 1890.**\n\nLike the Apache, the tribes living on the Great Plains were nomadic. Until the late 19th century, most Plains tribes traveled through their territories following the buffalo herds. Men of the Plains spent much of their lives in pursuit of buffalo. The buffalo was the source of food, shelter, and clothing for their people, and these men had to be brave, skillful hunters to bring down such large game animals. When they were not hunting, Plains men were kept busy making tools and weapons to help them hunt and protect their families. Although they spent much time away from home, when they were home, Plains fathers were often playful and loving.\n\nWomen of the Plains tribes were responsible for child rearing, tepee construction, and transportation. They did the cooking and processed buffalo into dried and fresh meat, storage containers, blankets, tools, clothing, tepee covers, and other essential items. They were also responsible for repairing and maintaining both summer and winter camps. Tepees generally belonged to women.\n\nBeautifully decorated clothing given to Lakota Sioux family members was and still is a sign of affection and respect from wives, mothers, sisters, and grandmothers. Lakota women were quite accomplished in quillwork. The women converted porcupine quills into elaborate decorations for clothing and personal items.\n\n**Blackfoot fathers taught their daughters how to be proper ladies. They would spend a great deal of time telling their daughters jokes and silly stories over and over again. When their daughters could listen to their teasing and joking without giggling like silly little girls, their fathers praised them generously.**\n\n**In Hupa villages, women were the healers. They were paid for their services, but only retained their fees if their treatments were successful. Hupa healers were highly regarded within their village social structures. They were often wealthy, and Hupa social standing was based upon wealth.**\n\nIn many of the families throughout the Plains area, mothers and daughters were quite close. Because mothers had so much work to do processing buffalo meat and hides, grandmothers often oversaw the education of young girls. These honored elder women taught their young granddaughters all the skills necessary to become a respected wife, mother, and adult family member. Grandmothers also taught their granddaughters the tribe's moral values, history, and traditions.\n\nThe organization of families differed from tribe to tribe. For example, the Omahas were organized into 10 paternal clans, while the Blackfoot were bilateral. This meant a Blackfoot could trace his or her lineage through both sides of the family. A Blackfoot woman generally joined her husband's band upon marriage. Like other tribes, chiefs led the extended Skidi Pawnee families. The chief's sons usually followed in his footsteps and became family leaders after their fathers' deaths. Unlike the Blackfoot tribe, however, once a young Pawnee man married he lived in the earth lodge of his wife's family.\n\nThe tribes of the Great Plains also had different traditions of courting\u2014what we would call dating today. Courting with blankets was part of the Cheyenne lifestyle. A single girl who was of age might stand in front of her family tepee wrapped in a blanket. Young men who were attracted to her would stand in line for a chance to enter the blanket with her. The young girl would invite the man of her choice to stand with her inside the folds of the blanket. There, they would stand face to face, sharing a few innocent moments together.\n\n**On the Great Plains, the men of the tribe were responsible for hunting buffalo. This animal provided the tribe with meat for food. The buffalo's hide could be used to make clothing, blankets, or tepee coverings and its bones could be made into tools. Women of the tribe dried the meat and prepared the skins, as this 19th-century photograph shows. The women in the center and on the right have staked out the hides and are beating them with sticks as they dry in the sun.**\n\n**Hopi women check on a meal they are baking in an adobe oven.**\n\nA young Sioux man who was interested in a young woman might attempt to win her favor by playing love songs for her on a flute. He could also use the flute to send her special, secret messages that were in a code only he and his beloved understood. If she indicated that she was interested in him, the young man might begin to pursue her more openly.\n\nThe parents of a Cora groom would make a number of ritual requests to the bride's parents. The bride's parents could then accept or decline the marriage proposal. The Cora people also gave their young adults a form of premarital counseling. Elders from the family relayed marital advice of the ancestors to the couple in a ceremonial meeting.\n\nChildren living in the southwest played a variety of games. One shared by several tribes was a game of skill that involved a hoop and some darts. A player would roll a hoop across the ground as fast as he or she could. Other players would try to throw darts through the moving hoop. Each player had different-colored darts to make keeping score easier.\n\nSouthwest girls and boys often played running games. Kick-ball was played, and contests in relay racing were often held between villages, tribes, or bands. Sometimes, different villages staged races during which gambling and feasting took place. The racers often ran distances of up to 25 miles.\n\nPlains families sometimes played shinny. This was a game that used special sticks and a small ball. It was played much like modern-day field hockey. Young boys were also expected to practice with their bows, lances, and other weapons so that they would become good hunters and help provide food for the tribe one day. \u00a7\n\n**These Athabaskan parents stand proudly behind their children, who hold up the otters they have trapped. Family life was especially important to the Native Americans of the far north.**\n**5 Families of Abundance and Subsistence**\n\nThe Native Americans who lived in what today are the northwestern United States and Canada had very diverse lifestyles. The Northwest was a land of abundance. The people living here did not have to work as hard as the natives living in the far north. The Arctic and subarctic regions, on the other hand, were areas in which farming was impossible and survival was difficult.\n\nTraditionally, bilateral extended families formed the basis of most Arctic Inuit societies. Although Inuit society was primarily arranged around the nuclear family, the Inuits also maintained close relationships with their extended family members.\n\nNuclear and extended families often lived together. If they did not share the same winter sod home or summer caribou-skin tent, they often lived close to one another in small communities. Individual survival in the Arctic regions was dependent upon the cooperative contributions of all family members, especially during whaling season.\n\nAbove all else, Inuit men needed to be hunters and fishermen of the highest quality. Life in the Arctic depended upon stockpiling foods such as whale, seal, caribou, and arctic char. In coastal whaling villages, whaling captains were highly respected and admired. In villages that did not whale, the best hunters were held in high regard. Respected and admired above all Inuit men, however, was the family shaman.\n\n**Women of the northwestern tribes were well known for the watertight baskets they wove from grasses and plants, as well as their for beautifully decorated wooden boxes. The boxes were so finely fitted together that they could be used to cook food. Very hot rocks were put into the box around or under the food. The box was then closed and the food was left to cook in this small wooden oven. The tighter the box was fit together, the more heat it would hold in and the better it would cook food.**\n\nThe Inuit shaman possessed a gift for contacting spirits. He was also a wise man, able to instruct his people in times of trouble. The Inuit shaman was a healer and teacher who had great knowledge of humans, animals, and the spirits. A good shaman knew many herbal remedies and other types of treatments for illnesses.\n\n**Members of an Inuit family sit down together for a meal. Traditionally, the Inuit maintained close relationships with both immediate and extended family members.**\n\nInuit women of the Arctic were mothers, wives, homemakers, and gatherers. They kept track of the family food supplies. They sewed fine fur parkas to keep their husbands and children warm. They sewed seal-and walrus-gut raincoats to keep their husbands and sons dry during whale and seal hunting season. They collected salmonberries and other edible tundra plants in the late spring, summer, and early fall.\n\n**This painting of Inuit snow cottages, also called igloos, was made around 1834 by a man who had sailed with Captain James Ross's polar expedition. Ross was a 19th century English explorer who traveled to both the Arctic and Antarctic regions. After returning from the Arctic in 1833, he described how the Inuit people he met lived.**\n\nInuit children were often adopted out of large families and into smaller ones. This was done so that they would receive the amount of food, shelter, attention, and clothing necessary for survival in the harsh environment of their homeland. Children who were adopted into another family often grew up knowing both sets of parents. As this was an accepted and understood practice, they did not feel ashamed or unwanted because of their adopted status.\n\nAn Inuit girl who was fully grown exchanged the clothes of her childhood for those of an adult woman. In some Inuit groups, she was also tattooed to mark her status as an adult. Inuit boys also changed their style of dress when they became recognized as grown men.\n\nWhen an Inuit boy was deemed ready for marriage, two slits were cut in his cheeks or near the corners of his mouth. Ivory or bone cheek or lip plugs were placed in the openings. These cheek plugs visually announced to all that the boy was now a man, eligible for marriage.\n\nAnother people of the Arctic were the Aleut, who lived in the far northwest. The Aleut traced their family relatives through their mother's family bloodlines. Aleut society had hereditary classes of high nobles, commoners, and slaves. The leaders were recruited from the high nobles' class. They lived in communal homes called _barabaras_.\n\n**GAMES OF THE NORTH**\n\n**Native American children who lived in the Far North and Northwest played a variety of games. In some areas of the Arctic children played a game called \"polar bear.\" One child pretended to be a sleeping old lady. Another pretended to be her sleeping child. The \"polar bear\" snuck up to them and took away the sleeping child. The \"child\" had to hide wherever the \"polar bear\" put him or her. The \"old lady\" was then awakened and had to search for her child.**\n\n**Inuit kick-ball was played with a stuffed ball made from caribou or other animal skin. Two teams opposed each other and tried to maintain control of the ball. There were no goals like in soccer, but the game resembled modern-day soccer in many other ways.**\n\n**Story telling was another form of Inuit entertainment. At night, when the family was safe inside their home, a father would entertain his children and wife with tales from his life or the lives of his ancestors. During this special family time, he would share stories that taught his children the history of their people, the values of their culture, and the teachings of their religion.**\n\n**Northwest boys competed against each other in foot races, archery contests, and wrestling matches. The girls played with carved wooden dolls. Both girls and boys swam and played on the beaches.**\n\nIn comparison to the subsistence families of these other regions, the people of the Northwest lived in relative splendor.\n\nThe Northwest Haida were divided into two phratries: the Raven and the Eagle. Each of these was divided into a large number of clans. One or more clans formed a village. Haida clans were matrilineal.\n\n**Boys and girls alike played a game much like modern-day horseshoes. Stakes were driven into the ground at opposite ends of a playing field. The players tossed roots at these stakes and tried to see who could land their root closest to the stake.**\n\n**Families in the subarctic culture region played a game that was similar to the shell game. Players would each be given an equal number of tokens or markers of some sort. A player would take one token and then with both hands out of his or her opponent's view, place it in one hand. Now, he or she would display two clenched fists. The opponent would guess in which hand the token was being held. If he or she guessed correctly, the prize was the marker. When one player ran out of markers, the other player was declared the winner.**\n\n**Young Sanpoil girls of the plateau area played with dolls. They made these dolls by sticking clay balls atop sticks and wrapping them in bits of animal skin. The boys played with tops that were made from available local hardwoods. Sanpoil girls and boys turned somersaults, practiced walking on their hands, and did handstands. Their parents liked to play gambling games. Entire families enjoyed playing several games of skill that used hoops and poles.**\n\nHaida men's titles and wealth were inherited through the mother's side of the family. Men, however, owned the houses in Haida society. The fathers' clans built these wooden-plank longhouses. When a man died, his sister's eldest son inherited his house.\n\nHaida clan chiefs were men. All of the clan chiefs of a village would be members of a council. The wealthiest clan chief was recognized as the village chief.\n\nHaida marriages were often arranged. The two parties who were to be married had to be from different clans. They also were expected to be from the same social class. Sometimes, marriages were arranged when the husband- and wife-to-be were still small children. Unless one of the two children disgraced themselves or their families, they would usually be married to their arranged spouse when they came of age.\n\nIf a Haida man neglected or abused his wife, her parents had the right to take her and their grandchildren away from him. If he left his wife for another woman, then he was forced to pay reparations to his first wife. If he refused, he was punished severely, sometimes by death. However, if there was no other woman involved, a Haida man could leave his wife and family without fear of retribution.\n\nAnother tribe of the Northwest was the Cayuse. In this tribe men and boys hunted for food and to provide hides for clothing. They also provided their people with animal bones and antlers for making tools and utensils. Elk, deer, bear, antelope, and mountain sheep were hunted across their north-central North American homeland. The Cayuse men were accomplished hunters who learned to be comfortable hunting with bows, spears, knives, nets, and traps. Cayuse men also fished for salmon. This was often done from shore, using long-handled dip nets. The nets were used to reach out into the rushing streams and capture salmon that were swimming upstream to spawn.\n\nCayuse women cleaned salmon and hung them on long racks to dry in the sun. They dug for roots using wooden or bone digging tools. They mashed roots together and shaped them into little balls to make small sun-dried biscuits. When there was time, the Cayuse women took their children into the mountains to go berry picking. There they would pick huckleberries for drying. They also picked chokecherries to mix with dried meat or fish to make pemmican.\n\nChildren of the Salish tribe, which lived on the coast of North America, spent a lot of time with their grandparents while their parents were hunting, fishing, or gathering food. In fact, grandparents were the primary teachers. They taught family history, as well as the qualities and skills the young Salish children would need to be a good member of their society. \u00a7\n\n**The members of this family are Maya Indians of Central America. Around 1,600 years ago, their ancestors established a highly evolved society in Mexico and Central America. Today Mayan families still follow some of the same practices of the ancient Maya.**\n**6 Family Ties in Central and South America**\n\nBefore the arrival of Europeans during the 16th century in the region today called Latin America, two of the major native groups were the Maya of Mexico and Central America and the Inca of South America.\n\nThe Inca of South America had a hierarchical, patriarchal class structure. Men were the heads of their families and of the Incan society. Families were divided into classes. The ruling class was composed of the ruler and his family members. Temple priests, architects, and regional army commanders were lower in class only to the elite members of the ruling-class family. The two lowest classes were made of artisans, soldiers, and peasant farmers. These farmers grew all of the crops necessary to feed their own families, as well as the families of the upper classes.\n\nRuling-class women sometimes had power, but it was more usual for rulers to be men. The sapa (high priest or ruler) and the army commander were the most important men in any Incan village.\n\nInca women used llama wool and cotton to weave cloth. Dyes were made from indigo and other plants. The finest dyed and woven cloths were given to the ruling class. A particularly fine weaving might be given to the gods during an important religious ceremony. Cloth was also used as a material for bartering. Fine cloth might be traded for cocoa, turkey, or even gold.\n\n**Both the Inca and Maya enjoyed a special ball game. Most cities had ball courts close to their centers, and ball games were a big event. To play, two teams faced off on the ball court. The players could not touch the ball with their hands\u2014they could only use elbows, knees, or hips. The players scored by touching special markers or passing the ball through their team's ring.**\n\nInca children did not always live long lives. Sometimes, they were chosen to serve as sacrifices to the Sun God. These children were taken high into the mountains and buried alive with food, corn beer, and coca leaves during annual ceremonies that were held to ensure a good harvest and a happy Sun God. To be chosen to serve as a sacrifice was considered to be a great honor.\n\nInca men were expected to marry by the age of 20. Brides and grooms would exchange sandals at their wedding ceremonies. Inca leaders married their sisters to keep the blood of their families pure. Their sisters became their first, or principal, wives. The Inca tradition was for the son of the ruler and his principal wife, or sister, to be the heir to the throne.\n\nLike the Inca, the Maya of Central America also had a family-based caste system. Caste membership was hereditary. The elite noble class was made up of the ruler, his family, nobles, and priests. Upon the death of a ruler, his son or brother took his place. Mayans who were not born into the elite noble class were divided into a class of warriors, a middle class of tradesmen and craftsmen, or a lower-class peasantry.\n\n**Inca men loosen the soil with a taccla (foot-plow), while women behind them sow seeds in this drawing from an Inca codex. The book, which dates to 1565, is the only codex in existence that shows drawings of 16th-century Peruvian life.**\n\nMayan working-class men were skilled tradesmen. The peasant men were farmers. They spent most of their days in the fields. They grew maize (corn), cotton, beans, squash, and cacao. Mayan peasant women were often skilled weavers as well. They used cotton to weave cloth. The dyes used by Mayan women included indigo, brazil wood, logwood, annatto, and iron oxide. They also made baskets.\n\nSoon after birth, a Mayan infant's head was pressed between two boards. It was secured and left this way for several days. The pressure reshaped the child's skull. It is believed that this was done to make the shape of the head resemble that of an ear of corn.\n\nUpper-class Mayan children were taught to read and write using pictograms and hieroglyphics. They studied religion, math, and astronomy. They also studied the Mayan calendar and learned to count and write out the Mayan numeral system.\n\n**If an Inca child went to school, the child was taught to make and read the Inca _quipu_. A _quipu_ was made of many colored knots tied together. The way that the knots were spaced and the colors of the cotton rope used to make them all had special meanings. Religion, law, and math were also studied.**\n\nWhen Mayan boys and girls became teenagers, they participated in a celebration called the \"Descent of the Gods.\" After this, a matchmaker could negotiate their marriages. The bride's father expected to receive a good price for his young daughter's hand in marriage. The bridegroom's father would try to negotiate a fair amount of time for his son to work for his father-in-law. After the marriage, the husband would have to work for his wife's parents for five to six years. \u00a7\nCHRONOLOGY\n\nBefore 10,000 B.C. | **Paleo-Indians migrate from parts of Asia and begin settling throughout the Americas.**  \n---|---  \n10,000\u20135000 B.C. | **Medicine-wheel spiritual sites are built in the Great Basin region.**  \n6000\u20135000 B.C. | **The subarctic regions are settled as the climate begins to warm with the waning of the last Ice Age.**  \n5000\u20133000 B.C. | **Earliest-known organized Native American settlements are built in the Southeast.**  \n1400 B.C.\u2013A.D. 1500 | **Northeastern woodland cultures rise and prosper.**  \nA.D. 300\u2013900 | **Maya civilization reaches its highest point.**  \n300 | **Native Americans begin settling in the Plains region and migrating with the buffalo herds and the seasons of the year.**  \n1400\u20131521 | **The Aztecs dominate Mesoamerica.**  \n1492\u20131502 | **Columbus explores the West Indies and Central America.**  \n1740\u20131780 | **European wars in the Northeast severely affect lifestyles of Native Americans in this region.**  \n1760\u20131848 | **Growing Spanish influence around California begins to have impact on the lives of Native Americans in this region; missionaries begin attempts to \"civilize\" and \"Christianize\" Native Americans of this area.**  \n1830 | **Congress passes the Indian Removal Act, calling for Native Americans living east of the Mississippi River to be moved to a government-established Indian Territory located in what is present-day Oklahoma.**  \n1838 | **Cherokee are forced to move from the Southeast to Oklahoma on the \"Trail of Tears.\"**  \n1887 | **The Dawes\/General Allotment Act divides reservations into 80- and 160-acre tracts; these land parcels are to be owned by individual Indians.**  \n1952 | **Federal Relocation Policy is passed; this policy seeks to terminate all government services for Native Americans, negate treaty agreements, and relocate Native Americans from reservations to inner cities.**  \n1971 | **Congress passes Alaska Native Claims Settlement Act.**  \n1972 | **\"Trail of Broken Treaties\" organized by AIM results in a weeklong occupation of the Bureau of Indian Affairs headquarters in Washington, D.C.**  \n1992 | **This year marks the 500th anniversary of Columbus' entry to the West Indies, prompting many Native American artists to create artwork expressing their feelings about Columbus and subsequent Europeans and their effects upon the Native American culture.**  \n2013 | **Recent census figures indicate that there are more than 5.2 million Native Americans living in the United States and Canada.**\nGLOSSARY\n\n**acclimate** to adapt to a new temperature, altitude, climate, environment, or situation.\n\n**bilateral** having two sides.\n\n**breechcloth** a cloth worn about the loins or hip area.\n\n**cassava** a type of melon.\n\n**caste** a division of society based on differences of wealth, inherited rank or privilege, profession, or occupation.\n\n**char** a small-scaled trout with light-colored spots.\n\n**clan** a group of people tracing descent from a common ancestor.\n\n**erosion** the wearing away of soil, usually by wind or water.\n\n**fast** to go without food for a period of time.\n\n**kiva** a ceremonial structure that is usually round and partly underground.\n\n**lineage** descent in a line from a common progenitor.\n\n**longhouse** a long communal dwelling.\n\n**matrilineal** relating to, based on, or tracing descent through the maternal line.\n\n**moiety** one of two basic complementary tribal subdivisions.\n\n**oratorical** relating to speaking in public eloquently or effectively.\n\n**palisade** a fence of stakes designed for defense.\n\n**paternal** of or relating to the father.\n\n**patriarchal** characteristic of a social organization marked by the supremacy of the father in the clan or family.\n\n**patrilineal** relating to, based on, or tracing descent through the paternal line.\n\n**pemmican** a concentrated food consisting of lean meat dried, pounded fine, and mixed with melted fat.\n\n**sagamore** a subordinate chief of the Algonquian Indians of the North Atlantic coast.\n\n**totem** an object serving as the emblem of a family or clan and often as a reminder of its ancestry.\n\n**vision quest** a personal spiritual search undertaken by an adolescent Native American boy in order to learn the identity of his guardian spirit.\n\n**wickiup** a hut with a usually oval base and a rough frame covered with reed mats, grass, or brushwood.\n\n**wigwam** a hut having typically an arched framework of poles overlaid with bark, rush mats, or hides.\nFURTHER READING\n\nBrasser, Theodore. _Native American Clothing: An Illustrated History_. Buffalo, N.Y.: Firefly Books, 2009.\n\nCarew-Miller, Anna. _Native American Confederacies_. Philadelphia: Mason Crest Publishers, 2014.\n\nHoffman, Elizabeth DeLaney. _American Indians and Popular Culture_. 2 vols. Santa Barbara, Calif.: ABC-CLIO, 2012.\n\nOberg, Michael Leroy. _Native America: A History_. Malden, U.K.: Blackwell Publishing, 2010.\nINTERNET RESOURCES\n\n**http:\/\/www.csulb.edu\/colleges\/cla\/departments\/americanindianstudies\/faculty\/trj\/**\n\nWebsite of the American Indian Studies program at California State University, Long Beach, which is chaired by Professor Troy Johnson. The site presents unique artwork, photographs, video, and sound recordings that accurately reflect the rich history and culture of Native Americans.\n\n**http:\/\/www.nativeweb.org\/resources\/**\n\nThis Web site features a collection of resources and links to informative Native American Web sites.\n\n**http:\/\/nmai.si.edu\/home**\n\nThis site contains fascinating information collected by the Smithsonian Institution about Native American history and culture.\n\nPublisher's Note: The websites listed on this page were active at the time of publication. The publisher is not responsible for websites that have changed their address or discontinued operation since the date of publication. The publisher reviews and updates the websites each time the book is reprinted.\nINDEX\n\nAleut,\n\nAlgonquians, , ,\n\nApache, \u2013\n\nArctic, , ,\n\nAtlantic Ocean,\n\nCanada,\n\nCaribbean Sea, ,\n\nCaribs,\n\nCayuse,\n\nCentral America, ,\n\nCheyenne,\n\nCora,\n\nCreeks,\n\nDelawares,\n\n\"Descent of the Gods,\"\n\nEuropeans, ,\n\nGreat Plains, , \u2013\n\nGuyohkohnyoh (Cayuga),\n\nHaida, \u2013\n\nHaudenosaunee (Iroquois), \u2013\n\nHopi, \u2013\n\nIncas, \u2013\n\nInuit, \u2013\n\nKanienkahagen (Mohawk),\n\nLakota Sioux,\n\nLatin America,\n\nMaya, , \u2013\n\nMexico,\n\nMi'kmaq,\n\nNative American families adoption, \u2013\n\ncaste system,\n\nclan, , , , , , , , \u2013\n\nextended family, , , \u2013, \u2013, , ,\n\nintertribal family,\n\nkin groups,\n\nnuclear family, , , , \u2013, , \u2013, ,\n\nphratries,\n\nNorth America, , , , ,\n\nOnayotekaono (Oneida),\n\nOnondowahgah (Seneca),\n\nOnundagaono (Onondaga),\n\nPawnee,\n\nSalish,\n\nSeminoles,\n\nSioux, ,\n\nSka-Ruh-Reh (Tuscarora),\n\nSouth America,\n\nSpanish,\n\nSun God,\n\nTainos, , , \u2013,\n\nThree Sisters, ,\n\nTimucua,\n\nUnited States,\n\nYouth's Vigil,\n\nZu\u00f1i, \nPICTURE CREDITS\n\n 3: Library of Congress\n\n 8: David and Peter Turnley\/Corbis\n\n: Library of Congress\n\n: Nathan Benn\/Corbis\n\n: Jim Richardson\/Corbis\n\n: AINACO\/Corbis\n\n: Library of Congress\n\n: (top) \u00a9 OTTN Publishing; (bottom) Library of Congress\n\n: Hulton\/Archive\/Getty Images\n\n: Hulton\/Archive\/Getty Images\n\n: Historical Picture Archive\/Corbis\n\n: Richard A. Cooke\/Corbis\n\n: Library of Congress\n\n: (top) Hulton\/Archive\/Getty Images; (bottom) Bettmann\/Corbis\n\n: Bettmann\/Corbis\n\n: Joel Bennett\/Corbis\n\n: Richard A. Cooke\/Corbis\n\n: Hulton\/Archive\/Getty Images\n\n: Jeremy Horner\/Corbis\n\n: Werner Forman\/Art Resource, NY\nCONTRIBUTORS\n\n**Dr. Troy Johnson** is chairman of the American Indian Studies program at California State University, Long Beach, California. He is an internationally published author and is the author, co-author, or editor of twenty books, including _Wisdom Spirits: American Indian Prophets, Revitalization Movements, and Cultural Survival_ (University of Nebraska Press, 2012); _The Indians of Eastern Texas and The Fredonia Revolution of 1828_ (Edwin Mellen Press, 2011); and _The American Indian Red Power Movement: Alcatraz to Wounded Knee_ (University of Nebraska Press, 2008). He has published numerous scholarly articles, has spoken at conferences across the United States, and is a member of the editorial board of the journals _American Indian Culture and Research and The History Teacher_. Dr. Johnson has served as president of the Society of History Education since 2001. He has won awards for his permanent exhibit at Alcatraz Island; he also was named Most Valuable Professor of the Year by California State University, Long Beach, in 1997 and again in 2006. He served as associate director and historical consultant on the award-winning PBS documentary film _Alcatraz Is Not an Island_ (1999). Dr. Johnson lives in Long Beach, California.\n\n**Colleen Madonna Flood Williams** is the wife of Paul R.Williams, mother of Dillon Joseph Meehan, and daughter of Patrick and Kathleen Flood. She lives in Alaska with her husband, son, and their dog, Kosmos Kramer. She has a bachelor's degree in elementary education with a minor in art.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**EARLY BIRD BOOKS**\n\n**FRESH EBOOK DEALS, DELIVERED DAILY**\n\nLOVE TO READ?\n\nLOVE GREAT SALES?\n\nGET FANTASTIC DEALS ON BESTSELLING EBOOKS\n\nDELIVERED TO YOUR INBOX EVERY DAY!\n\n# Crazy in Berlin\n\n### Thomas Berger\n\n_TO JEANNE_\n\n# Contents\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\nAuthor's Note\n\nA Biography Of Thomas Berger\n> _Du bist verr\u00fcckt, mein Kind;_\n> \n> _Du musst nach Berlin..._\n> \n> You are crazy, my child;\n> \n> You must go to Berlin...\n> \n> \u2014Old song\n\n# _CHAPTER 1_\n\nIN THE TWILIGHT, THE bust appeared to be that of some cocked-hat Revolutionary War hero of not the very first rank, that is, not G. Washington but perhaps one of those excellent Europeans noted in fact and apocrypha for throwing their weight on our side, Lafayette, say, or von Steuben. _Fun Shtoyben_ was the right way to say it, which Reinhart knew and was certain that Marsala didn't, being his dumb but lovable buddy who was now gurgling at what was left of the bottle and would shortly hurl it away, maybe hurting someone, for a few Germans sat around in the park; he must warn him. But too late, there went the crash and narrowly missing a Kraut who merely smiled nervously and moved off, some difference from a movie Nazi who, monocled and enraged, would have spat in your face, and they were already taking a leak on Lafayette or whoever\u2014no, \"Friedrich der Grosse,\" the pedestal said, for Reinhart had a lighter that could be worked with one hand.\n\nAnd it was a gross thing to do, he decided in one of those drifts of remorse that blow across a drunk\u2014because he was just educated enough to recall vaguely old Frederick out at Sans Souci with Voltaire, writing in French, representing the best, or the worst, of one tradition or the other\u2014a part of the punkhood from which he had just this day legally departed, and which he was, in fact, at this very moment celebrating.\n\nFastening the fly, all one hundred buttons, no zippers in the Army because you might get caught in one as the enemy crept close, he said, just as sad as he had before been exuberant: \"What a way to pass your twenty-first birthday!\"\n\n\"Well,\" answered Marsala, twenty-four, looking forty, and always fit whatever his condition, spitting, not taking out his cigarette, and miffed, \"we could of made you a party from the messhall: them cooks are all my friends. What are you, griping?\"\n\nAs they turned to leave the park, a German nipped up and snatched the butt. There stood a woman by a tree. \"Honey,\" Marsala shouted, \" _schlafen mit_ me, ohhh won't you _schlafen mit_ me!\" A kind of music the making of which was his satisfaction, for having crooned it he moved on indifferently.\n\nOn the street they encountered a Russian soldier, far from home, needless to say unkempt and weary, destination unknown most of all to himself. In the friendly light of his hound's eye they accepted, and Reinhart returned, a salute; he went on in a hopeless, probably Slavic, manner. A two-car streetcar braked to a glide and they swung aboard, paying no fare because they were Occupation; and a good thing they hadn't to, for in a moment the son of a bitch stopped and everybody detrained and walked around a bomb crater to another car waiting on the other side, Marsala all the while looking truculently hither and yon: he was amiable only to his friends.\n\nThe ride on the new car Reinhart forgot even as it was in progress, for he had now reached that secondary state of inebriation in which the mind is one vast sweep of summer sky and there is no limit to the altitude a kite may go, the condition in which one can repair intricate mechanisms at other times mysterious, solve equations, craft epigrams, make otherwise invulnerable women, and bluff formidable men, when people say, \"Why, _Reinhart_!\" and rivals wax bitter. Here he was in Berlin\u2014the very name opened magic casements on the foam of seas perilous not two months since: Hitler was rumored to be still at large, the C.O. had been briefly interned by the Russians, and Art Flanders, the \"crack foreign correspondent whose headquarters were in the saddle\" and column in 529 dailies from Maine to the Alamo, had already called at the outfit for human-interest sketches.\n\nIndeed, the sheer grandeur of his geographical position had overwhelmed Reinhart until this very day, for he was an irrepressible dreamer. Marsala had been out screwing and playing the black market for a week, with already a dose of crabs and a wad of Occupation marks to show for it, and at the same time bitching incessantly that they might be stuck there forever\u2014and nonpartisan in his disinterest in any place but Home. No, he wouldn't have liked to be stuck in Calabria any better, besieged by his indigent relatives and wallowing in the dirt and backwardness for which no one could tell him of all people, his father having come from there, that Italy was not famous. \"I got your Roman ruins and your art right here,\" he would sometimes say, grabbing his clothes in the area of the scrotum\u2014the same place, in fact, that the Romans had had them\u2014\"You take that crap and give me the United States of America.\"\n\nNow, on the car, Marsala was once again the sound of unadorned naturalism, his hard voice, the one for enemies, piercing Reinhart's shoulder, for that was all the higher he came, like a rusty blade: \"You call it. I'll kick it out of you wherever you want.\"\n\nHis target was sealed with them in the crowd on the rear platform: an American soldier, between whom and Marsala stood, swaying with the general rhythm of the rocking car, a female citizen. Her visible part was a head of blonde hair, with a good washing probably as pale as Reinhart's own, but at present long estranged from soap and comb and as stringy as an Assyrian's beard. Notwithstanding that he had barracked with the man for two years, in whatever land, Reinhart supposed, first, a mistake, and, second, that Marsala was wronged, but these suppositions could not dwell long even in a flushed mind, for he saw the face of the other soldier charged with righteous outrage. A big man, maybe six even, with his weight from front to back, rather than in width, if one could tell from a limited view of his shoulders and fat, seedy head. He struck you right off as a lousy guy, a type who had been drafted from the driver's seat of a big-city bus, where he cursed _sotto voce_ at proffered dollar bills and depressed the door-lever on latecomers; a journeyman in the Shit-heels' Guild whose meanness was, after years, instinctive\u2014but all this was irrelevant beside the fact that in a quarrel involving a woman Marsala invariably stood on the bad side. He had surely with one of his sexual instruments, voice, hands, or groin, sought an unsubtle connection\u2014for him a crowded streetcar was as good as an alley and being caught out only a minor inconvenience soon adjusted in his favor: he had a friend, while the other man was alone.\n\nThus was Reinhart's euphoria wrenched away; what Marsala expected of him was by the known pattern of his friend's code so obvious as to go unstated. When the car stopped at the next bomb crater and the German passengers, all slumped and carrying bundles, duly filed around its margins to still another vehicle, the three soldiers and one girl drew apart and, out of a sudden sense of national delicacy, waited until the new car started away and the old reversed trolleys and started back. Then Marsala snarled, \"Let's get him,\" pitching in before the other man, now manifestly regretting where amour propre had led him as he saw Reinhart's large figure on the hostile side, had got ready: he was in the act of removing his blouse, newly pressed, perhaps by the girl, and bearing the triangle of the Second Armored Division which had fought all the way from Africa\u2014while Reinhart and Marsala were goofing off in Camp Grant, Illinois, Devonshire, and some tent city in long-liberated Normandy.\n\nAs a medic, and rear-area at that, Reinhart had no moral guts to oppose a combat man, even for cause, even when alcohol had anaesthetized his rational-young-man's disinclination to violence\u2014and as for two setting upon one, its morality threw him into a state of shock. He stood in his tracks, feeling undue exposure, lighting a smoke, and out of a complex shame not looking at the girl, and saw Marsala imprison the opponent's arms with the half-removed jacket and call: \"Okay, Carlo, in the nuts!\" Saw him, not able to resist his advantage until help arrived, give the man one with the knee.\n\nHis reaction to Reinhart's coming and pulling him loose was pure astonishment, hopefully as yet unalloyed with bitterness\u2014he must have supposed it the prelude to a more cunning mayhem\u2014and he had just time to begin \"What the fu\u2014\" before the freed adversary got a hammerlock on his throat and booted Reinhart from the field.\n\nThey fought on the site of a ruin. As Reinhart lay on the crushed-masonry ground, beneath a roof-to-basement cross section of fourteen flats, their cavities spilling tubs and bedsteads, he could not even have said where. On to two weeks in the city, and this was his first trip off base. His old buddy, for his birthday, had taken him to a black-market contact with Russian hootch to sell, his old buddy who in the grunting ranges overhead was at this moment being slaughtered. So he raised himself, hot and vital and clear, seized the traitorous and ugly bastard from the Armored by the back of the shirt-neck, turned him, and delivered two hundred three and a half pounds to the gut, to the eye and into the mouth. The man's meat broke wetly under his fists and yet retreated at one point to bulge at another, like some hateful sack of liquid, and it was for a time a joyful rage to work for a simultaneous and general recession. But where it took him was too terrible\u2014all at once he gave it up. The enemy, in a vast cobweb of blood, still stood. Odd, he appeared old, perhaps forty; his cap had gone, showing an area of baldness pitifully made conspicuous by a strand of hair deviated to hide it. He was standing\u2014but it was suddenly obvious that he was very dead.\n\nReinhart had broken both hands at the wrist. His lungs were gone, as well; his stomach was acid and his wit beclouded once more. It was so frightening that a corpse should remain upright. He watched Marsala come round and head-butt it in the midsection. It revived, and it fell, simply a beaten man, with an awful, beaten groan.\n\n\"Jesus,\" said Marsala. \"Not a car in sight. We might have to walk all the way back just because of this prick of misery. You did good, Carlo,\" he went on, rubbing his sore neck, which made a rasping noise, for he had an emery-paper beard. Kept rubbing, but he was in some awe.\n\nReinhart had not been in a fight since early grammar school and therefore had never known how it felt to kill a man and what, when done, was the peculiar scandal. He looked to the girl, who was some distance removed in the capacity of spectator, and who in return looked at him with stupid wonderment, and commanded her to approach. Which she did with a senseless caution, as if to ask: is my turn next?\n\n\"Why don't you attend to him?\"\u2014approximately; his German was at best uncertain and now surely further corrupted by the intermittent buzzer in his skull.\n\n\"Well, yes, if you wish,\" she replied, still showing wonder, and speaking from a face in which the ages were so mixed that one knew not whether oldness or youth was the essence. She knelt in worn clothing more suited to that attitude than the standing and examined the felled opponent, who even at her touch was coming painfully around. Who when he arrived came up slowly and resentfully from the supine, crying: \"Keep your whore off me!\" With more effort he was arisen and deliberately, crazily, gone across the ruin and onto the sidewalk, where it could be found and where not, the street, where, alone, he could be seen for a great time, despite the darkness now settled.\n\n\"Hadn't you better follow him?\" Reinhart asked incredulously. \"If the car does come he may be hit.\"\n\n\"Must I?\" She was nearer him now and, it struck him repugnantly, believed herself a transferred spoil of war.\n\n\"No, of course not, not if you haven't any _decency_.\" The last word in English; he didn't know it in German, and she didn't know it this way or, in truth, hadn't any, for she smiled.\n\nNothing smelled ranker than disloyalty. He had wanted so much to approve of the first German girl he met\u2014for this was she, not counting the women seen from train and truck on the journey to Berlin or the cleaning and secretarial help\u2014if for nothing else, as an act of anti-piety against the established faith. The very faith of which, curiously enough, he found himself at least a part-time worshiper, one of those half-agnostics who go to church without believing or stay home and believe; whatever, he had waited two weeks before going on pass, since this was the best manner in which to avoid Germans and still ache, with trepidation and even a kind of love, to see them.\n\n\"If I must go\u2014 _wiederschau'n_!\" She extended her hand in the genuine enthusiasm displayed by all Europeans, not just the French, upon arrivals and departures, as if for all their hatreds they love one another, or do for a moment at making and breaking contact, and at this first touch in ten minutes not motivated by hostility, Reinhart suddenly felt drunken again and feared that he might weep\u2014for the sore opponent vanished alone in the night, for his friend who did not understand fighting fair, for the girl now under his compulsion, and for the material things in waste all about them, all the poor, weak, assaulted and assaulting people and things, and of course for himself, isolated by a power he didn't want.\n\nBut certainly he did not cry. Instead he gave her his dizzy eyes and said:\n\n\"It's a terrible thing to desert a friend.\"\n\n_\"Bitte?\"_\n\nHe repeated it, as near as he could come, in German, and she replied:\n\n\"There was nothing else you could do. He would have killed the small man.\"\n\n_She_ was granting _him_ absolution! But his anger did at any rate conquer the sadness. He barked his ill will towards a woman who leaves a beaten man, and in a moment found his only ease in the thought that with luck his speech had been too bad for her full comprehension. For she had answered:\n\n\"I was not _with_ the soldier! Believe me, in all my life I have never before seen him!\"\n\nSo much for that. And Marsala, who had had his rest, was prepared to reassume the command so lately transferred; he might defer to Reinhart, from now on, in matters of personal combat, but surely never in affaires d'amour.\n\n\"C'mon, why mess with this one? A pig,\" he said without malice, perhaps kindly, if you wished to look at it that way, for there was no point in stirring up the girl's hopes, but anyhow with the candor of the unembellished man, which was just what Reinhart prized highest in Marsala and why he associated with him rather than with the refugees from college. The cruelty was an inseparable element of the greater value, a unique honesty and a kind of honor: Marsala never assumed an ethical superiority to anyone else. But he was generous in granting one, for now no sooner were his comments out than he showed with a bored jerk of the head that the girl's pigness was suddenly understood as Reinhart's precise interest.\n\n\"If the girl didn't belong to him,\" asked Reinhart, \"what in hell was the beef?\"\n\n\"How should I know?\" Marsala's swarthy head revolved in unworried failure to understand the provocations constantly offered; the world was full of enemies, that was all. You watched, you took care of them, they took care of you\u2014you did not look for a reason and you didn't actually feel any lasting grudge. \"There's all kinds of bastards around.\" He scratched his boot-toe in the rubble. \"If you really want to know, he called me a guinea.\"\n\nReinhart, missing the sly grin of mock piety and having learned Marsala's elaborate code regarding these nicknames\u2014Marsala himself habitually used them and especially those applying to his own kind, but denied the latter to non-guineas except people like Reinhart who held an honorary card\u2014merely said \"Oh\" and turned to the Kraut girl.\n\nAt close view he decided she was young and that her longitudinal lines of cheek and veteran eyes were from lifelong residence in a sanguinary country, but the darkness forbade one's being sure. She had merely come to watch a fight?\n\n\"One has to admit that it was interesting.\" She had moved very close to him, perhaps because of the dark, and there was enough light to see that just a chance remained to make her attractive, at least to get by. Reinhart would have liked to seize and scrub and comb and color and dress her\u2014to straighten her out; the world was filled with people who out of simple inertia wouldn't make a move to fulfill their own promise.\n\n\"But,\" she went on, \"you are a noncommissioned officer. Please, may I ask you: how does one get a job with the Americans?\"\n\n\"There are places for such things,\" he disappointedly replied. He hadn't known _what,_ yet had hoped for something other than the humdrum, perhaps an unexpected birthday gift. His parents' package had not arrived, very likely never would, the occasion being one on which their undependability was notable. Besides, the girl became more attractive as she talked; her voice was pitched low and had a melancholy music and her whole manner was submission to the male principle. \"You want me to get you a job, is that right?\" She was within a hair of contact with his belt buckle, and he had come under a compulsion at once to fuse into her body and not move his, which could be done by easing forward the belly usually, as a matter of vanity, held back.\n\n\"For Christ Almighty sake,\" said Marsala, in the testiness of one whose judgment has gone unheeded, \"the soldier has gone horny.\" He was right by being wrong; he assumed their conversation to be a bargaining.\n\nThey were now touching, the girl standing firm and, madly, as if unconscious of anything strange, pursuing her first interest: \"It's all so confusing. I am ready to do any kind of work\u2014as cleaning woman if need be. ... Do you have a _bottmann_?\"\n\nIt was much too rare for his simple vocabulary. He had not learned it in two years of college where he ostensibly majored in the language but in fact moped lonely around bars and crowded, smoky places with small string combos, with no real stomach for liquor and no real courage with women, drinking much, nevertheless, and fumbling at some tail. On the margin of a flat flunk he had enlisted in the Army. At any rate, he could deliver correctly not a single long sentence in German and could translate nothing beyond very short strings of words with exact English equivalents.\n\n_You are a bad man_ was maybe what she meant; if so it was a weak remonstrance, as when you are small and exchange exposures with the neighbor girl, who coyly says \"You are a bad boy,\" all the while pulling up her dress. He had slipped his arm into her worn coat, where a missing button made it easy, and around her narrow waist, and she came full into him, saying still, so madly!: \"Is a female _bottmann_ allowed?\"\n\nSuddenly and so nuttily did its sense at last arrive that he released her and retreated a step. _Lives of a Bengal Lancer, Four Feathers,_ \"but when it comes to slaughter you will do your work on water, an' you'll lick the bloomin' boots of 'im that's got it.\" When you were in the field against Mohammed Khan, Oxford-educated Pathan who returned to his mountain fastness to lead the tribes against the Crown, you had a batman; when, that is, you were a British officer serving Victoria, to whom you drank health and broke the glass, or for all he knew even at present, you had an orderly.\n\n\"Well, I'm going, _I'm on my way,_ \" Marsala groaned reproachfully, and hypocritically, for he scraped away only a short distance and sat down on a wasted wall, lighted a cigarette with a great flare and coughed.\n\nHer look had no defenses: \"I know just a few English words.\"\n\n\"That one is very rare,\" replied Reinhart with all his gentle forces. He added: \"I am just an _Unteroffizier,_ a corporal, a nobody, a silly fellow\u2014\"\n\nShe returned his smile in the exact degree of wryness with which it had been given her, aiding him, in tune with him, so that when the exchange was completed he had been purged of self-pity and satisfaction with the vision of himself as uncorrupted by efficacy; and furthermore was not made sore by its loss. He was forgiven all the way down the line, and most of all for thinking it was forgiveness\u2014she was far beyond that, standing there before him on the mound of trash, without vanity, making no judgments, facing facts.\n\n_\"Also,\"_ he said. \"You will have your job. You have my word.\"\n\nHis oath was no doubt meaningless in German; one certainty of alien languages was that each had its own way, untranslatable, for the moral expressions. But its effect was not needed by her, who it could be seen in the clearness of her eye admitted no doubt towards him.\n\nMarsala was back, seizing Reinhart's elbow and, this time with unbelievable modesty, whispering in his ear: \"You're not thinking of slipping her the tool? I mean, it's all right with me, I just wanted to get things straight, no sense for me to stand here, just gimme the sign\u2014\"\n\n\"Old buddy,\" said Reinhart, \"friends may come and friends may go and some may peter out, I know, but I'll be yours through thick or thin.\"\n\n\"Yeah, I know,\" came Marsala's hoarse whisper, which was louder than his normal voice, \" 'peter out or peter in.' Well, what will it be? It's boring to stay here. I mean, for me.\" \"Boring\" was a word he had learned from Reinhart, using it with weak authority and only as a favor to his teacher. It took quite enough effort, however, to have its power.\n\n\"You must come to see me at my organization,\" Reinhart told the girl. \"Now can you remember this? It is the 1209th\u2014but the number doesn't matter. We are a military hospital and are in a school building in Zehlendorf, at the corner of Wilskistrasse and Hartmannsweiler Weg. Across the street in a wooden building is headquarters. You come in the door and turn left. You go all the way to the end of the hall, to the last room, and there I am.\"\n\nThe language became easier to use as he spoke, and he found himself on better terms with talking than he had been in years; in German even directions were a kind of success, precise and scientific. Still in temper with him, the girl, now three feet away and in deep shadow, said: \"You have a good accent!\"\n\n\"Now you _will_ remember?\"\n\n\"Oh yes! But please, what is your name?\"\n\nHe had turned to leave with Marsala, like a monarch\u2014in all the world there are no good departures\u2014and now, kingly, gave: \"Carlo.\"\n\nShe was stepping towards him in an eager courtesy. \"As in Monaco\u2014I shall remember.\"\n\nMarsala grinned like a possum at the traditional repast; honor was being done his old rule: give them only your first name, which cannot be traced. His lack of civilization had suddenly become repulsive.\n\n\"No\u2014Reinhart, Carlo Reinhart. _Es ist ein deutscher Name_.\"\n\n\"Certainly.\"\n\nHe shook hands with her and in American fashion held it too long, so that hers wilted and sought to escape.\n\nThey were still a party of three at the streetcar stop. But before one came, if ever it did, a jeep throttled up out of the blackness, bearing MPs on their eternal quest for miscreants. Like all American police, they stayed at their remove of faint hostility even after Reinhart and Marsala identified themselves and proved blameless; indeed, even after the constables took them aboard for a ride to their billeting area, which since it lay off the beat was a considerable favor, it seemed needful for the sake of an institutional pride that all pretend it was a kind of arrest and sit silent on the way. The four men, that is, for the girl had not been considered, had not, properly speaking, been seen, the non-fraternization policy being neither quite repealed nor, beyond the flagrant, enforced.\n\nShe might stay there for hours\u2014this was a thought of Reinhart's, which he answered with the familiar indictment: so many millions of non-Germans would lie dead forever. Yet it had been so appropriate to pity her; he aggressively presented that claim to himself. \"Certainly\" was her answer to the characterization of his name. From birth he had been a good, sturdy _German_ type, lived in a solid _German_ house, on a diet of _G._ potato salad (with vinegar), _G._ cole slaw (with bacon grease), _G._ coffeecake (with butter-lakes), run on a regimen of _G._ virtue (bill-paying, bedding and rising early, melancholia), and whenever he left it was met by approving people who said: \"Ah! He's going to be (or is) a big _G._ like his grandfather.\"\n\nThe term, however, had a double meaning, which was honored even by near-illiterates; its other use was as synonym for a kind of foulness. And now there was a third, for here he was in the ancient homeland, and he was something different.\n\n\"Home\" was a compound in Zehlendorf from which the 1209th General Hospital had evicted the German residents, a block of three long apartment buildings arranged in an open rectangle around a private park. The latter, in some former time almost a university campus with green and wanton walks for rambling, had been converted at the order of the 1209th's commanding officer into a junkyard for the disposal of property the Germans left behind. The colonel was neither opposed to comfort for his men nor a partisan of pain and deprivation for the owners who after all would one day return; he was nothing, no Savonarola, no crypto-fascist symbol of the military mind, not even, because he was a medic, quite a soldier, nor, because he was commanding officer, quite a doctor\u2014but owing to this he wished grievously to be something, if only a converter of matter from one form to another. Thus he periodically had put to the torch, had resolved into carbon and the immaterial gases, the giant cairn of objects which Reinhart and Marsala now skirted on their way to the south building: couches and loveseats, dining tables, bedsteads, chaises longues, sideboards, three pianos, fourteen wind-up victrolas and two thousand records; eight thousand books; rugs, pictures, tablecloths, postcard collections, skis, jewelry boxes, letters, diaries, journals, manuscripts, apologias, Nazi party cards, memoranda, paper, paper, paper; and one little souvenir plaque from the Western Hemisphere: an electric-pencil sketch of a pickaninny sitting in a Chic Sale, inscribed \"Best wishes from Savannah, Ga.\" Another pile held the noncombustibles, mainly cooking utensils and fifty more or less complete china services from the royal house of an imaginary principality.\n\nFive men had been busted in rank when caught salvaging items from the auto-da-f\u00e9. On the other hand, the colonel did not lack in a rude sense of justice: if you could make away with an overstuffed chair or an alarm clock without being seen, it was yours and beyond all future confiscation.\n\nThe tenants of Building A, first floor right, had furnished a very decent little flat, the cynosure, as Reinhart might say, of all eyes in their section. Particularly those of Buck Sergeant Tom Riley, their next-door neighbor and late technician third grade, who had made so free in their absence as not only to enter their home but also to sink his big ass into the mohair couch and fall asleep. When awakened by the crudest means they could summon on such short notice, he arose complaining, \"You must of got your nose up the colonel to get to keep this furniture. Our living room looks like a Mexican cat house.\" He lumped fatly to the hall door. \"And why only two guys here instead of three?\" His swollen nose tensed with authentic peevishness, and small wonder, for he had been reduced one grade for unsuccessful pilferage from the trashpile; but more than that, he was by instinct a petulant man, with the face of an old baby.\n\n\"Don't be bitter,\" said Reinhart, who had followed him out. \"It's my birthday.\"\n\n\"Oh.\" As might have been expected, he missed the ironic import of the non sequitur; he would be resentful, thank you, on his own time and ground; instead, he grasped Reinhart's hand in embarrassed but genuine feeling, as you could tell by his nose, which went soft, saying: \"You don't mean to bird-turd me? Many happy returns of the day.\" And already at his own door, he turned: \"Ain't it sad? Here we are, getting older by the minute.\" Without change of expression or girding himself for the effort, he suddenly screamed, in a voice like a jazz cornet, an obscene epithet which, though it went up the concrete stairwell like a skyrocket, made no public stir, being heard all over the 1209th so frequently that it had lost its force as description: if you ever found that fellow to whom it was originally applied, you should have to think up a new one.\n\nBecause nothing succeeded like the envy of others, Reinhart returned to the living room in a, now sober, swagger. Off the top floor of the building lay an attic stuffed with furniture\u2014the colonel lighted the bonfire only after his space ran out\u2014and Marsala and Reinhart had picked the lock on its door and secretly helped themselves while everybody else was away at work. As to tenants, the orders demanded three per flat; given free choice, the buddies shrewdly compacted with Doyle, who three days later left on detached service.\n\nSo they had a proper home now, Marsala, a slum boy, confessing it was the nicest he ever had and the most spacious. Parlor, kitchen, and bath were luxuries militarily undreamed of. You could, see, get some rations from a cook, have a little lunch in the kitchen, then take your broad to the couch, knock off a piece, and then wash up in the bath. Marsala had indeed gone through the series three times in recent days, failing only in the last because he could not make a swift transition from pleasure to hygiene. Hence the jar of blue ointment in the bathroom medicine chest, yet another evidence the place was truly home.\n\nThe green tiles of the corner stove were still warm from a small fire they had built just after chow; July nights were cool this far north. A fall of maroon drapery concealed the big window above the couch, which in the daytime showed their private balcony and, beyond, a green promenade between their block and the next, an _all\u00e9e_ in the old, European sense, banned to cars and wastecans\u2014and to the colonel, for natives retained ownership of the adjacent buildings. The enormous sideboard on the east wall had almost ruptured them to carry, and had little utility when in place, but great authority. Central was a round table of oak and six attendant chairs. No tablecloth. Other deficiencies were: nothing matched; no pictures on the wall; no knickknacks placed around; no doilies to protect the arms of the couch; no magazine rack with _The Woman's Home Companion_ and last week's _This Week_ and the publication General Motors sends gratis to Chevrolet-drivers. But it was Reinhart's own home and he believed it was nice.\n\nHe seated himself at the table to brood on the folly of early-evening drinking, his close-cropped blond head propped on a red fist still tremulous from the fight, pale-blue eyes charged with red, one trouserleg loose from the knotted condom round the boot-top. Marsala, who didn't know it was not nearly time to go to bed, had ignorantly gone and tomorrow would awake extra soon and, it went without saying, loud.\n\nReinhart all his life had detested birthdays; they were like Sundays in the middle of the week, outlawing the ordinary by a promise of the special, never fulfilled. Until this moment, for he could never think while in motion, the twenty-first had been another of the same. But, ruminating, he saw now that it had, indeed, a touch of the exotic. He had drawn blood and spilled some of his own. He rose and went again to the hall mirror in which he had inspected himself on the route back from Riley's leaving. No, no hallucination: a nice scratch-cum-bruise on the left cheekbone, made easier to see if you tweezered your fingers about it. Riley, still sleepy, no doubt had laid it to shaving.\n\nIn addition to Marsala's salve, the bathroom cabinet held Reinhart's collection of medicines. Once every two months he had a slight complaint, each time in a different organ, never serious\u2014whichever doctor was on duty in the 1209th dispensary would smile, prescribe, and likely as not give in to the urge to punch the tight belly and caution him jokingly not to worry, the undertaker would be a stranger to him for years.\n\nThe large merthiolate badge had dried and was almost ready to flake before he finished his final self-examination in the bathroom mirror, in the course of which his spirits curved downwards again. The pompous, pink-and-blond-faced creep who stared back at him had been endured enough for one day. He got out his rubbing alcohol, cotton, and applicator sticks, and wiped away the crimson fraud, threw the evidence out the window, went to the bedroom, kicked his clothes in the corner, fell on the bed next to Marsala's, and was immediately in sleep.\n\n# _CHAPTER 2_\n\nFIRST LIEUTENANT NATHAN SCHILD, a traitor, handed a sheaf of papers to the German known as \"Schatzi,\" a courier to\u2014well, above all, to an impossibility, since the measure of truth is what a man will give for it and Schild would have walked to the noose to deny that this Thing had any existence outside the mind of a malignant halfwit. Schatzi himself was just barely possible, being a returned traveler from that until the twentieth century undiscovered bourne; the most efficient of men, who could answer all questions with: \"I was four years in Auschwitz.\" Behind that, darkness, and not, according to the code of the underground, to be searched by Schild. Although Schatzi wore the garb of a petty bureaucrat\u2014felt hat, stiff collar, briefcase\u2014he suggested a ruin; although the night was warm, he trembled and winced, as if the whole of his skin had been sandpapered and recorded in pain the blows of the air's molecules.\n\n\"What is this stuffing?\" asked Schatzi, roaming Schild's person with his free hand like a restless lover's, probing the fly-front of the blouse.\n\nSchild extracted a folder. \"I forgot it because it's more or less negligible.\"\n\n\"Needless remark,\" said Schatzi, pleasantly nasty. He struck Schild on the elbow with his finger-tips like a row of icepicks, of course hitting the nerve.\n\nIf a man could be said to have earned a broader latitude of eccentricity than most, it was Schatzi. He had once shown Schild the scars on his back, and if it had been day one could have seen the terrible commentary of his face, a kind of scorecard of the times. And that he had undergone torture and was not a Jew made it all the more criminal for Schild to detest him.\n\n\"Please,\" said Schild, with genuine, if exaggerated feeling, for, although it was not that important, neither was it of no matter, and Schatzi discounted precision, \"I must have this back by tomorrow noon at the outside. Captain St. George just asked for it.\"\n\n_\"Er kann mich im Arsche lecken!\"_ Sooner or later Schatzi related everything to himself, and scatologically, which was not perverse given his late experience of life reduced to essentials. Nevertheless, while understood, the common-denominatorship of Schatzi was hardly winning. Also in character, Schatzi walked in a cloud of food odors; tonight it was herring. But who was so degraded as to gainsay his right to courtly fare, let alone such simple meals as he was no doubt issued at some Soviet mess?\n\nThey stood on a strip of Wannsee shore near a wrecked pleasure pavilion, the salient feature of which was a tin Coca-Cola sign hanging crazily in the light of Schatzi's torch, the patented slogan of its own International in German here; downtown Schild had seen the red and gold standard of Woolworth's in a similar death-agony of capitalism. Beyond the symbolism it was a remote and even foolish place to meet. Two men upon a dark bench, one a German civilian... it was not Schild's job to pass upon the point of rendezvous\u2014while not forgetting for a moment that Schatzi was his superior, his mind was also wired in another circuit with the alternating current of forgetting and remembering\u2014but he did, anyhow, make his apprehensions known, and Schatzi suggested next time bringing a girl for protective coloration. Such was Schatzi's circuit: belly-rectum-pubis. The simplest interpretation was that he lived somewhere near the lake; perhaps, like a rat, in a hole just above the water line. Sometimes, in fact, as if caught by a sudden high tide, he appeared damp at their appointments. Schild bit his mind's tongue; at least Schatzi said what Schatzi thought, suffered no internal wrestling matches with an indestructible malice like an extra organ.\n\nIn a sudden surge of self-remonstrance, Schild said: \"We get a ration of candy, you know. In the American Army there are constant pressures upon the soldiers to be the same conspicuous consumers they were in civilian life. It's not enough just to fight a war.\"\n\nSchatzi took the chocolate bar from his hand, undid its wrapper, scrutinized it in the light's ray with an invisible jeweler's lens, tested its friability with a thumbnail.\n\n\"I really found one last week that had all of it been shaped from brown clay. Think on that! The astonishing industriousness required, it having been a first-order job. All that work for fifty mark and you Amis are burdened with actual ones that you cannot eat till the day of doom.\"\n\nHe pushed five ten-mark notes, already rubber-banded into that amount\u2014yes, the Japs held the Malay Peninsula, but his bands were pure gum\u2014into the hand of Schild which had come forward on its own volition, on its own idiotic hand's sense that it would be shaken, just as a foot will all at once assume sovereignty and stub itself as punishment for some foot-crime or kick the girl's shoe across the way out of some foot-lust. Schild's face meanwhile was performing, in a vacuum, a squalid drama.\n\n\"But, my dear fellow, you shall not get more than this at the Tiergarten and think on the length of miles from here to there. You are an actual Greek for business, comrade!\"\n\nSchatzi returned the candy and took back the money, signifying the end of his joke.\n\n\"I know people who are pleased at your work,\" he resumed, \"and if you are not so careful they shall give you one of these posterboards to take home to Tennessee: 'The Hitlers Come and Go Away But the German People Stay Always!' \"\n\n\"If you get some pleasure from insisting that I'm from Tennessee, go ahead\u2014\" began Schild, but the flashlight quivered and sank, and Schatzi groaned from the ground: \"I like the name.\"\n\n\"An attack of dizzy,\" he went on as Schild bent to aid, \"brought on by four heavy suppers every day and eight hours of sleep the night, still in a warm bed. _Gem\u00fctlichkeit_ is killing me. Give me some pills in the right pocket, they will make me miserable.\" There was nothing in either pocket but Occupation marks. \"Make all the money you canst,\" said Schatzi, on his feet again but breathing with a whistle and coughing bubbly.\n\nAt such moments the call of Schild's guilt echoed through the great tombs of the martyrs. Schatzi was indeed dying, yet he continued to serve. This, Schild knew in the final, serious level of the self, was why he hated him so: out of his own incapacity for a like magnitude of effort.\n\n\"You know,\" said Schatzi, shaking his body like a dog, blowing air from his nostrils, combing his hair, long as a woman's, with his fingers, \"when Kurt brought you first, to my opinion you were a double agent. Arrived at by irrational methods, it is possible, but there was something with your eyes.\" He took a flask from his briefcase and tasted of it. \"Ah!\" He spat. \"Good drink will make a cat speak! They are, isn't it true, simply myopiac? A fat small man, a little soccer ball of a man with eyes like that\u2014and thicker spectacles\u2014came into the K.Z. along with myself. I watched, in vainly, to see the existence there bring him down.\" He spat again, making a nasty sound on the sand. \"Ah! Cuts the flame.\" Schild preferred to assume his version of \"phlegm\" was a portmanteau word: he must be burning inside. \"One gets obsessions when you are a captive. But at the end of three months still he bounced. I never did see him on a work gang. He disappeared days somewhere but in the nights returned to the barracks. I had been convinced that he was a police spy and I am in fact yet. He lay in the bed each night end-to-end with my own bed and stared at me over his round belly and through his feet. It was terrorizing, I tell you, a man could never once find him asleep. When I awokened in the morning he looked, still; perhaps he did not close his eyes the night long. Because I do not know, you see, because I tell you I slept, I functioned as usually I do, under the watching of sixty-six devils I could do as always, because I tell you that beyond a club to my genital members there is nothing which a man can do which will touch me at all.\"\n\nSchatzi's voice had taken on the authority usual to his concentration-camp reminiscences. On Schild's refusal he pocketed the flask, but not before illustrating its quality, heavy silver; its feature, a spring cap worked with the thumb. As always, he withheld the d\u00e9nouement until Schild in the double dread\u2014the tedious responsibility of the auditor to help dramatize, the terrible certitude that the small fat man, whether bona fide police spy or hero, would like all the other creatures of Schatzi's memory meet an unspeakable end\u2014until, cold in July, he must urge him to go on.\n\nBut Schatzi had got a sudden subtlety. \"Is Captain St. George the ass you take him as?\" He pronounced \"St.\" as _Sankt:_ no cue for worry, a man fills out abbreviations in his native tongue.\n\n\"He's a Republican.\"\n\n\"Are not we all? What does this mean? I don't understand, I don't understand.\"\n\n\"I'm sorry, I forgot. He's nonpolitical, an aboriginal American type. I thought all the world knew. Let me explain: If I express so much as simple approval of a labor union, he will say, 'Well, I'm not against unions but you've got to admit sometimes they go too far. I understand if a light bulb burns out in a factory the place stays in darkness until an authorized member of the electrical workers' union comes to replace it.' But if he saw me leading a mob on the White House under a red flag he would lay it to money or some private passion. Do you understand now? We have billeted together for two years, he knows how I look in my underwear and that I use a soap stick instead of tube lather for shaving. He knows whatever my eyebrows do when I'm puzzled, the contents of my musette bag\u2014\"\n\n\"I understand now that _you_ are the ass. How does a police agent operate if not this way? Fritz, Fritz, it is a little wonder that after four years of duty you are yet a first lieutenant!\" It was not clear that Schatzi meant more than chaff. He had himself taken irresponsible risks near St. George, more than once lingering before Schild's billet on a bicycle. What was obvious now, though, was his unease at Schild's developing a point, hence the underground name, a remonstrating symbol of the overwhelming awareness and power which they both served and before which elaboration was ludicrously futile.\n\n\"With all this knowledge,\" Schild finished defiantly, \"what could not be forgiven? He would trust a man forever whom he had watched cutting his toenails.\"\n\n_\"Also,\"_ said Schatzi. \"I used to swim at this place but I do not mourn it\u2014any more than I need to play the piano again. Have I told you I once have played the piano in a splendid club where the tables were connected each to each in a system of tubes from which the air is exhausted\u2014what do you say for them?\u2014vacuum, _so,_ vacuum tubes, through which the people in this place could communicate on little pieces of paper\u2014this was the same place where Emil Jannings was controller of the W.C. Haha! Did you hear of this film _The Last Laugh_?\" He allowed the insatiable black space over the water to swallow his light's beam for a moment, then reclaimed it to thrust into Schild's eyes. \"You will never drown in the water to mock the hangman, not you. I will just as soon choke myself as to have you know my real name. Without respect for this famous na\u00efvet\u00e9, there is something sinister about an American.\"\n\n\"You seemed to have no worry about Kurt.\"\n\n\"Kurt lived until aged ten in Budapest, Paris to the age of eleven and a half, Budapest again for three years, then Rome to the age of twenty, and finally Washington. His father is in the diplomatic service, his mother is an Hungarian and the influence. Do you know Kurt's actual identity? In yesterday's _Stars and Stripes_ \u2014a queer journal, by the bye! What are these letters at the lower-left hand of page two, this so-called 'B-Bag'?\"\n\nThe damp had begun an osmotic affection for Schild's feet. \"Oh,\" he answered in a momentary quicksand of sorrow which sucked the life from his voice but was all to the good for the present purpose: \"That's supposed to be the uninhibited feelings of the enlisted men with complaints, the vox populi of ersatz democracy. The name comes from an expression, 'Blow it out your barracks bag,' let off steam, air your gripes. The enlisted men used to carry their gear in two bags, one labeled 'A,' the other, 'B.' \"\n\n\"I tell you that tells nothing. I have read a letter yesterday which said\"\u2014he broke off and produced the very clipping, holding the light for Schild to read:\n\n> You can search the whole Enclave until your goddam corns are thumping and you won't find one place where EM can get anything better to drink than flat beer that the Krauts made when Hitler was a PFC. Yet every ninety-day wonder in my outfit wallows in Haig & Haig. The chickens are getting bigger and I don't have to say what is getting deeper. Yours for World War III,\n> \n> T\/5 P.....-OFF\n> \n> _Bremen_\n\n\"Yes, that's the sort of thing.\"\n\n\"Do you ever use it?\"\n\n\"No,\" said Schild. \"As I say, it's essentially for enlisted men. Besides,\" smiling in irony, his profession, place, and time's surrogate for good humor, which Schatzi could not see because he was again being nervous with the flashlight, \"my complaints are not so simple.\"\n\nSchatzi laughed, for a change in a pleasant tone, perhaps owing to the fact that he had nothing to gain or lose from the passage: \"As to this B-Bag, obscure name still, I do not believe from a swift look that the code would be too hard to break. It is not a device without imagination, but surely American Espionage has better means for important messages. I think these are no more than general intelligences for each sector. However, it would be that one can do worse than to attempt to decode the letters signed Berlin, a damp finger to the wind, one could say.\"\n\nWhen Schatzi spoke like a neurasthenic spinster he was not fooling, even though it was only at such times that he amused Schild, an extraordinary achievement. In good Middle European style Schatzi was most suspicious of what was most innocuous, and perhaps the reverse, although in that he had not been tested. Almost to Schild's disappointment, there was nothing dangerous, complex, or oblique in the Berlin situation. As American Intelligence analyst, he inspected confiscated Nazi correspondence files; as something else entirely, he chose interesting items for transmission to Schatzi, the jobs meshing beautifully.\n\nBut Schild was a great over-preparer, despite the persistence with which, while he stood smeared with grease, Hellesponts shriveled to birdbaths. Having been alerted for his present function since he first reported to his draft board, having been, by unseen hands, guided to and through infantry OCS and later transferred to Intelligence by like means\u2014Schatzi could suspect American na\u00efvet\u00e9, Schild could not afford to: the Party, with all its resources, could perform the miraculous only with the aid of history's buffoons\u2014sent to France and then to Berlin with the first Occupation troops, having on a word from X, a nod from Y, and a furtive motion of the elbow joint from Z, been put in touch with \"Kurt,\" who conducted him twice to the presence of \"Schatzi\" and vanished forever; symbolizing in his very position at this juncture, this square foot of wet sand, the energy and infinite pains of the agency whose creature he was\u2014but that was just it, what small service he rendered! Two or three sheafs of trash a week, available to anybody who would walk into a bombed building and pick up a handful of scattered papers. Not to mention that the Red Army, which had got to Berlin a month before the Western powers, had surely missed little of consequence. Still, this seemed somehow his own deficiency notwithstanding the clear directions that limited him to the role, and he was conscious, in all the weak jealousy of the impotent, that herein lay another motive for aversion to his courier of the wide horizons.\n\nSchatzi left off his nonsense about the counterspies' use of the B-Bag, or what, had it not arisen from his total dedication, would have been nonsense and resumed his original aim. \"In any rate, in yesterday's _Stars and Stripes,_ on page number three, you will find a little item to announce the appointment of Nicholas G. Pope, civilian military-government official, as licenser of German newspapers in Bavaria. Kurt, Pope. The very man.\"\n\nHow loudly he spoke, how careless with the light. The very fact that the beach was abandoned and dark made it more conspicuous than the stage at the Titania Palast. Schild instinctively resisted the exposure of Kurt's identity, learned it, that is, and didn't learn it, a technique by which information could give comfort but not be divulged even under torture.\n\nFortunate in all his cautions and fears, for they served, after all, to give him a constant business that his larger function did not, Schild arranged with Schatzi for their next assignation and sought to move off towards the broken timbers of the pavilion and the jeep on the forest road beyond. Schatzi's hail was very like a shiv into the small of his back. As he turned to hear the not-forgotten-for-a-moment fate of the soccer-ball man, between the sound and the sense he saw in his memory Schatzi's earlier flashlight motions. Across the water lay Kladow. Who there received his signals?\n\n\"... so this guard made off with his cap and threw it over the top of the wire into this area that was not permitted for the prisoners and ordered him on the pain of death instantly to go and bring it back. The man climbed with the strange nimbleness of the fat, quite indifferent with the barbs going into the palms of the hands, got this cap, made it free of the snow with his underneath side of the arm, and then brought it down over his skull, which was of course shaven clean, down to the ears. On the climb again back, he was slow and breathed hard; on the top strand of wire, he let out some steam, for it was very cold, and at the time when just more than half of his weight was over\u2014the guard had planned it well, you see, to ensure that the fall might be on the near side\u2014the machine-pistol bullets released the air in him and the man did not fall as planned but shriveled and stayed on the wire like a soccer-ball bladder without the air. The wind even moved him. I think still he was a concealed policeman, shot in mistake. The guard vanished some time later.\"\n\nSchild's fingers crept to the button of his holster in a parody of a poorly remembered Hoot Gibson at bay. He would not really draw his pistol on Schatzi; for one thing, in the world outside the concentration camp this was not done, or at least not by him. If Schatzi were a double agent, the worst course was a show of violence; yet his hand would not cease its histrionism, did not indeed for some moments after Schatzi wound up the anecdote and went to the heart of the matter, for his private intelligence system extended even to the reports of Schild's nerves.\n\n\"Look,\" he said and boldly morsed his light at Kladow, from which, as if ignited by his, another was briefly born, died, lived, died. \"I go in a minute to Potsdam with a boat full of food from an American Army kitchen. For everything I know, this can be served to Herr Truman at Stalin's villa on the next mealtime. ... But please do not think\"\u2014he held the torch at the point of his chin, splashing the beam up across the peaks and declivities of his face, as a child might make a satan's mask in the mirror of a darkened room.\n\n\"Of course I didn't think\u2014\"\n\n\"\u2014that I could do this for anything but money.\" Upon this summit of innocence Schatzi snapped off and withdrew. In the willows by the water he made noises of effort; and well before Schild had got to his own vehicle he heard the outboard churn off in such a splash that his courier might have been on the way to quench Hell. And above and beyond the lovely organization of the jeep in first gear, he heard the boat engine climb hysterically towards its extreme velocity and, reaching it, miss and backfire and belch and puke, and his heart worked with it in the shore man's empathy until it eventually leveled into a continuum of asymmetrical impulses, like a laughter hopelessly mad, hopelessly free.\n\n# _CHAPTER 3_\n\nWHEN ONE WAS TEN, nobody, least of all the boys of German stem, served willingly on the Kaiser's side in war games. The little kids and younger brothers gunned Fokkers through the back yards and crashed flaming against the garage as Rickenbacker and his hat-in-ring squadron of Spads roared overhead piloted by the big boys. Then landing at the Allied aerodrome, which was quite a different thing from an airfield, and into the flight office (again the garage) swaggered they, tightbooted, helmet straps swinging free, demanding coggnack from the Frog wench behind the bar which without transition gave onto the office; while the de-Germanized younger brothers greased the planes, for this was also a hangar.\n\nShortly Richthofen might pay another call in his craft painted all checkers like a taxi, pitching to the tarmac a black gauntlet, showing a brief glimpse of grim but noble face, black-goggled, over the fuselage; mocking ailerons just clearing the high-tension wires at the field's end but not the undercarriage, which was severed. Jerry now could not land; with nothing to lose, this dogfight was for keeps. Inside the glove, which a mechanic fetched in, a note in Teutonic script: \"My compliments to your gallant command. I issue an open challenge.\" Cross of Malta. Signed, Baron R. Aloft, fabric tearing in Immelmann turns, oil-line burst spewing goggle lenses with black slime, Browning jammed, you dropped the foe with your sidearm and looked down in long salute at his incendiary spiral into the chrysanthemum bed. _Ave, atque, vale,_ brave adversary! I slew you as you would have slain me, your cause was hopeless but not contemptible, we share in that community which the whey-faced civilian regardless of nation cannot enter.\n\nTo be sure, not every enemy was a Richthofen. G-8 the master spy, commuting behind German lines in the limitless disguises from his armpit cosmetic chest, was certain to meet the pigface Hun, bayoneter of Belgian babies, violator of maidens; cabbage for a head, sausage-limbed, cheeks of ass like dirigibles kissing, he waddled in cruel insolence before the helpless or groveled in fright before his master. G-8, whose trunk formed a triangle standing on its apex, was right to destroy this creature if only for aesthetic reasons.\n\nReinhart's paternal grandfather, who looked like one of G-8's victims, with a somewhat better distribution of weight, was more charitably described as a double for Hindenburg, a sound man. He cut meat on weekdays and on Sundays read the _Volks-Zeit-ung_ and decanted in the cellar a thin, tasteless homebrew of which he was inordinately proud. He could support a leg of beef at full arm's length with one hand; he was a source of blutwurst slices on the sly; his place of work was floored with wood shavings and blood lay in pools on the butcher's block and dripped from the joints on the hook. He had left the Old Country to beat the draft; he had reportedly thought the Nazis were fools and was fortunate to have departed the world before they could disabuse him. These, and a small store of mispronunciations, were all Carlo could remember beyond an enormous, kindly, mustached face that smelled of beer and pipe, that was less articulate in the general tongue than himself at nine, that seemed, for all its implicit power and the massive hands that could not touch his shoulders so gently as not to bruise, a distant presence.\n\nWhen in adolescence Reinhart was suddenly overwhelmed with the purposelessness of the bleak journey from pablum to embalming fluid\u2014not for himself, but everybody else; he would somehow, alone, escape and was now investigating the various modes of exit\u2014he once asked his father, and querulously, because the reason he alone would escape was that he alone had the guts or intelligence to ask questions rather than weakly submit, just as the power of his will would protect him, alone, from eyeglasses, baldness, false teeth, poverty, a wife: \"Why didn't Grandpa ever go back to Germany?\"\n\nCould the Old Country, so remote, so rare, fail to exceed the here and now?\n\nHis father was shaving, or repairing the outside cellar door that hadn't been true to its frame for a decade, or washing the automobile, and perhaps before the answer came had done each of these tenfold, because for him a question must be repeated many times, by reason of his apathy, which was a superb thing in its way and could have been heroic if behind him Rome were burning or half of London keeling over with the plague.\n\nFinally, as he rose from whatever job, scant of breath, with stated or implied senility of leg, he replied: \"Use your head. What was there for him _there_?\"\n\nThere had, of course, been so much for him _here._ He hadn't even owned his own butchershop, but was wage-slave in a native American's, a man twenty years younger. He built a brick house in an end of town that shortly thereafter suffered an encroachment of lower-class cotton-pickin' refugees from Kentucky, know locally as Briarhoppers, with their rusty cars and back-yard shacks and incestuous five-to-a-room, brought in by an absentee landlord named Horace Remington, who everyone was sure had changed his name from Levy, although they had never seen him\u2014any more than they had seen for themselves that the earth was round. Grandma, native-born, whom he had carried off from a ten-dollar-a-week job at the pencil factory, survived him one year, and left material holdings of the house and two hundred fifty dollars. His one son, the man Carlo watched return with steel wool to the corroded hubcaps, was far from a raving success. This car, for example, was bought used with the two-fifty bequest\u2014which showed how far back it went. And as to the house, it had gone to join its neighbors in the pocket of \"Remington,\" who, despicable though he was, paid a third again as much as any other offerer and therefore was able to seduce even the good people into aiding his effort to destroy the respectability of such areas. Now if one passed the old homestead he heard guitar music and saw degenerate faces at the windows.\n\nCarlo left his father at the task\u2014he would have been glad to help on request, but would not volunteer; he received fifty cents a week in allowance, for which his obligation was to cut grass in summer and shovel snow in winter\u2014and went indoors, to the dark cavity at the end of the basement, which was really not so much cellar-end as dead space, a kind of tomb, under the front porch. Here in a cardboard box soft with moisture were his grandfather's few effects, having for half a decade been in chemical mixture with the insatiable air: a twenty-five-cent pipe, \"real bruyere\" stamped on the shank; a straight razor, broken; a dollar watch, scarred chromium, its intestines locked in rust; cufflinks with the Oddfellows' paralytic eye. A letter postmarked Berlin-Something, Berlin-Smear, April 12, 1927. My God, three years after Carlo was born; he was still astonished at evidence that the world had been up and around when he was so young. Inside, handwriting that not only was in another tongue but also in an alien alphabet, even the figures were queer. Finally, a single book, the leather of which some dry past time infected with an eczema that the basement damp had treated with a salve of blue mold. _N\u00fcrnberger_ something, the golden letters just visible on the granulated cover. Within, it was all pictures of that splendid medieval town of towers, castles, moats, rivers flowing through buildings, dolls' dwelling places hung over sallyports, and ironbound doors, four feet high, for dwarfs' abodes in the bottom of the city wall; labyrinthine ways among steep houses with a little extra roof sprouting over each attic window, the general roofs themselves nowhere true, everywhere splaying, overshooting, cutting back, growing dormers and loggias and lookouts and hexagonal capsules, restrained from soaring off their timbered plaster only by the weight of these execrescences and a million tiles fine as the scales of a trout.\n\nMagic and fabulous\u2014no, it was not so much these as the Ohio street outside his own window, with its covering of smooth tar down which if you rolled for ten minutes you would pass a flat, dun high school, a raw Presbyterian church without a steeple, and fifty lawns so level and unobstructed that you could some Sunday push a roller from one limit to the next and squash nothing but a row of homeowners trimming the edges. In N\u00fcrnberg female angels ringed the city fountain gushing water from their breasts, but what was extraordinary was that America could be so ugly-dull; that was the fairy tale, along with radio programs for the housewife, movie cowboys who never kissed a girl, public drivel about shut-ins, mothers, flowers, the favorite prayers of celebrities, ministers being tough guys and businessmen wise ones, the stupid arrogance of newspaper reporters who wrote \"grass roots\" and \"with the arrival of spring, usually blas\u00e9 New Yorkers set aside their sophistication and frolicked gaily as children.\" And sports. At the end of his sophomore year in high school he was the largest boy in the building and was invited to come out for football by the coach, a witless man adored by his teams apparently because he cursed them at half-time. He took Reinhart's refusal in good grace, and was clever to do so; Carlo had some time before sent away to York, Pa., for a set of dumbbells and was already more muscular than the coach, except in the head. In Europe they did such things as weight lifting and gymnastics, although he was a touch too unwieldly for the latter.\n\nHe thought about Harvard and Amherst, places of old stone and vines and fireplaces in each room and tutors, as at Heidelberg and Leipzig, but it was revealed to him his last year in high school that he must go to the nearest state university, on account of the free tuition. At this institution he was permitted neither a lodging in town nor a single in the dormitory; with the unction peculiar to the tribe, the dean of men said he must mix and placed him with a roommate who hung pennants on one side of their metal-and-concrete cell, worked a water wave in the front of his hair, and crooned popular songs in idle moments. When shortly he applied again for a single, he discovered it was already on the way: the roomie had reported that Reinhart didn't mix.\n\nAlthough the architects had designed the few single rooms to be a constant punishment for the social deviates assigned them, Reinhart lived happy in his. He had loathed the college before he saw it and after a month's residence knew his prior feeling as too mild: it was in sum a flat green mall overrun with round pink faces saying \"Hi!\"\n\nHe had read much as a boy, but only in the literature of the imagination. Expository writing was rough, almost impossible going; he had never been easy with the language of documents and directions on packages, and was not now with that of the natural and social sciences. Philosophy was somewhat better because it didn't, and didn't really pretend to, get anywhere. English was a book of contemporary readings about, on the one hand, the underprivileged and, on the other, initially irresponsible people coming to a sense of social obligation: there was a story, in the form of a letter to her parents, about a rich girl who married a labor organizer, the compassion going all one way\u2014inward, to the letter-writer and spouse\u2014and no passion at all. And German, that hard and very real tongue, proved difficult and dreary, with twelve cases for each noun, insanely irregular verbs, and perverse genders that made a door a female and a maiden a neuter, defying even that principle of nature by which, according to a neighbor in psychology, projecting objects seen in dreams are male sex symbols and receptacles female, for _the pen_ was in Deutsch as feminine as _the box_ was not.\n\nFor these reasons he grew fond of his little room, last floor back, next to the toilets, with an air conduit passing first over the ranges in the kitchen four stories lower, and came not to hear the staccato flushings and smell the lemon sauce for the semolina pudding. He sometimes hid out there all day, cutting classes on the motive of the little ills\u2014sinus, swollen Achilles tendon, foreign matter in the eye, etc.\u2014for which the dispensary would write a note, eschewing meals with the aid of Oreo cookies and those stale, soap-flavored cheese-cracker sandwiches one buys at the drugstore, and reading books of his own choice.\n\nHe had now grown to six-two, still an inch below his grandfather\u2014which he might yet attain\u2014and as much above his father as that mark exceeds five-nine. The set of dumbbells had given way to a barbell with changeable weights; in the \"clean-and-jerk\" lift he could handle two hundred pounds, five more than he weighed, yet he was inclined to solid beef rather than the sharp definition of muscle permitted more wiry types; and he was clumsy, tripping over roots on tardy runs to eight-o'clock classes, tending to enter a doorway with poor aim and collide with its frame, sometimes splintering the wood. A recluse, but when he emerged, a recipient of good will and that friendly fealty paid to large men in jabs-in-the-ribs and blows-on-the-upper-body, which along with the strain of trying to better his mark in the clean and jerk every afternoon kept him always sore of skin.\n\nThe books of his choice were _The Invisible Man,_ which he was at any given time rereading; a volume printed at the author's expense called A _Life in the Field,_ by an Englishman who had towards the end of the nineteenth century scouted both in Matabeleland and along the Big Horn River in Montana; and Middle European short stories in English translation, in which the characters tended to live in the mountains or the valleys between them, walking to school on rutted cowpaths, sometimes getting lost in the forest\u2014or had departed all this for the garish, quick life of the cities, which had gone to ashes in their mouths, and now yearned for the pastoral long ago forsaken; had a quiet but desperate passion for a girl who did not know they were alive or held them in sororal affection; attended day school, oppressed by a severe master and a fat bully; kept a faithful dog. Always a single sensibility, sometimes misunderstood, usually not even taken account of, by the insensate many; and in an atmosphere of mist, distant sounds, and if in the mountains, of course the silent, imperturbable snow, deceptively serene and treacherous, and on the glacier, a frozen rainbow. The stories were to be found in collections under one rubric or another but could take place in any of a variety of Central European areas at any given time under diverse political registrations: Bohemia, the Burgenland, Silesia, even Switzerland, anywhere that had a Germanic color and preferably a castle on some steep over placid water and in the foreground a cottage with a roof of straw.\n\nMeanwhile he was almost flunking out in his course of study in German. For one thing, it was at eight o'clock, and he was most nights up till three, reading; for another, the language as taught had no relations to the tales, being at first Herr Schmidt exchanging the time of day with his neighbor and then simple scientific excerpts for the premedical students, which the instructor decided it wouldn't hurt the few general people to read, either. At the end of the year he just, but made it in German and the other courses, low C's with the exception of zoology lab, where in the interests of a moody, fitful romance a girl friend had made his dissection drawings, upping him a grade.\n\nSimultaneous with Reinhart's entrance into junior year in high school, the Wehrmacht had invaded Poland; at the end of the term they took France; upon his graduation, entered Russia; just before his first-year college Christmas holiday, were at war with America. By one means or another, he was aware of these events although he never read the papers. He was furthermore aware that wars were wrong and foolish and the official ways of nations, always stupid and often wicked; that propaganda, regardless of side, was an absolute lie: for example, as everyone knew, the German \"atrocities\" of World War I were fabrications of the British and French, who moreover did not let up in the ensuing twenty years, thereby giving Hitler some excuse for his silly ravings. Hitler held no appeal for him, having an unmistakable aroma of the tramp and no dignity, and, discounting their portrayal in Hollywood movies, the Nazis _were_ preposterously vulgar; but opposed to the little, venomous, weak French and the British, thin and effeminate, they could hardly be assigned the exclusive evil in an intestine European quarrel over markets and territories.\n\nYet when America came into the war, it was a man's place to go soldiering, and the ideals concerned were not public ones dreamed up by journalists and pompous bores in high office but private matters. He felt himself a kind of German, yet he would cheerfully have slain the whole German army in fair combat and exposed himself to the same fate. On this principle he almost presented his person to the enlistment office early Friday morning, December 12, 1941; doubtless would have, had there been such a bureau in his little college town; but there wasn't, and the closest city, the place he crept alone every weekend to oppose his harsh weekday regimen with whiskey and coke, was eight miles off\u2014it was impossible to hitchhike there, enlist, and get back without missing classes.\n\nHis second thoughts were confirmed three days later when the mincing dean told the assembled men that being educated people they could better serve by pursuing their studies with renewed vigor. It was not only his idea, he averred, but that of the Armed Forces, who as reward would commission every man to graduate. However, six weeks later, when the first fine fire had cooled and it was too late to volunteer from a position of enthusiasm, the male students were reconvoked. Now anyone wishing to stay _out_ of the Army must _enlist_ \u2014in the reserve.\n\nReinhart called at the dean's office posthaste, already having been the target of remarks in bars, inarticulate grumbles by gray-sideburned potguts on the theme of why so much meat was not yet sacked in olive drab. The dean's secretary, one of those tight-rectumed persons whose every little motion is spite against some subject so long vanished that every other human being has taken on his-her appearance, after consulting the records told him with much satisfaction that the Enlisted Reserve Corps had a certain academic standard to which he failed to measure up. He cut classes and went to town and got stinking, which was not easy to do in an otherwise deserted tavern on Wednesday afternoon with no music. A fortyish waitress named Wanda some time in the next six hours told him _I knock off work at eleven_ and at eleven-thirty, in a one-room apartment where a leaky faucet dripped a quick rhythm to which no one could have kept stride, displayed unusually kittenish ways and a pair of thick thighs marbled with blue veins. The First Time he had ever really Got In; as usual the popular consensus, which in his dormitory held that the experience was persistently overrated, was a lie; indeed, it had been in all his years the lone achievement; a pity that our society offered no male career in that direction.\n\nIn the late spring, just before the end of the year, another alteration in his university's theory of the reserves. If they limited membership to the bright students, the campus would soon be depopulated by the draft; so now a simple passing grade became a ticket of entry. Reinhart was permitted to sign up and given a little wallet-card signed by the Secretary of War as an assurance that he would do his service in the classroom. Actually, he was still ahead of time, was still not old enough to register for the draft. He had been a clever fellow in grammar school, doing eight years in seven, before the rot set in, and was yet only seventeen.\n\nSophomore German was _Wilhelm Tell,_ tough to read, maudlin of sentiment. Reinhart now had a lodging in town and in consideration of the low rent went without maid service; a _gem\u00fctlich_ sty except on those monthly occasions when his nihilism grew strong enough to annihilate itself temporarily and he borrowed the landlady's carpet sweeper. He read _The Sorrows of Werther_ on his own, in English of course, and went so far as to get lent the German text by his professor, who after the fashion of the kind supposed that only good students had such ambitions and was at once wary, impressed, and all the more condescending for the pretense that he was not. But it was far too tedious to go line for line with the original; he pooped out on page two.\n\nAs to the other courses, American history was worst, debunking all the colorful legends and filling the vacuum so made with a thick Cream-of-Wheat of\u2014as usual\u2014economics; tariffs and taxes and indentured servants and land grants, and a general agreement that every one of the wars could have been avoided had these items not been mishandled by well-meaning but inept statesmen.\n\nAt the end of the fall term Reinhart made _I_ -for-incomplete in history, as an alternative to the F he would have received had he not one morning in February absentmindedly cut his toenail too deep, inadvertently generating a wound which kept him from the exam. Presently the _I_ stood for infinity: along about the beginning of March his gorge rose for the last time and would not come down; he went to the campus headquarters of the reserve and signed on for active duty.\n\nHis parents protested in their pallid way, finding everything a rejection of them and at a loss to see that their weak representations made self-counsel necessary; as if an impalpable father were not enough, he had a mother with whom nothing succeeded like failure. She would have preferred his staying in school, especially now that he was flunking. He tried to convince her that the Army life held promise of far more squalid drudgery than did college, that it was likely a person of his delicate constitution would collapse in training, and she was to a degree mollified.\n\nOf course he didn't really say this; he seldom talked to his parents at all, simply, on his holiday visits home, communicating silently through the shoulder blades, a language he had learned from his father. When he was a small boy Reinhart had often wished for a temporary catastrophe from which he could rescue his folks\u2014an unarmed burglar or minor fire\u2014not only to show he cared but also to see if they did, if they could honor triumph as well as defeat, but the occasion never came, and just as well, for it might have come during one of his frequent illnesses\u2014at which time, anyway, he _had_ their interest.\n\nThey were German too, one generation closer than he, and celebrated the fact in their tastes\u2014must have, because they could hardly have invented them on their own: heavy, flavorless food, limited ambitions, disapproval of the maverick, funeral-going, trust in people with broad faces, and belief in the special virtue of a dreary breed known as the German mother. \"German\" as a lifelong malady that was without hope but never serious; as the thin edge above want and far below plenty; as crepe-hanging; as self-pity\u2014yet from these compounding a strange morality that regarded itself as superior to all variant modes. He had been encouraged since infancy to think of himself as an average man, but in a harshly restricted community where some were less average than others; if wealthy, had immorally taken too much from the world; if very poor, were immorally lazy; if taking pleasure in the material, ostentatious; if ascetic, holier-than-thou. But never \"German\" as the lofty vision, the old and exquisite manners of prince and peasant, battlements and armor, clear water splashing down from high, blue rocks, wine named for the milk of the Virgin, maleness, the noble marriage of feeling and thought.\n\nBut they sent him to college, on an insurance policy which his father, being an agent, had sold to himself, and the premiums for which, lean year in and out, had claimed all their unencumbered money, and Reinhart had first opted for Liberal Arts instead of Business Administration and now left even that. As he departed for camp he carried, along with his toilet articles and change of socks in a miniature suitcase, an acute suspicion that he would come to nothing, and... a marvelous sense of relief.\n\nAt the induction center an interviewer saw the _B_ in zoology on the record and put him down for a medic, asking him first, though, for as a volunteer you had some faint choice. And he agreed, suddenly finding his bloodthirsty fancy had paled; a superior and sensitive person deplored violence; it didn't, as every retroactive commentator on past wars insisted, \"settle anything.\" He personally had made himself so strong with the weights that no one bothered him, and if they did, he generally gave way in the conviction that not only were they probably right but that also anger and hostility were degrading. Under the Geneva Rules medical troops were all but neutral, and in recognition of this were not intentionally shot at and if captured were obliged to go on treating wounded, theirs or the enemy's, it made no difference; they were above the taking of sides.\n\nThe Germans honored this convention\u2014that was admitted by the most rabid. For after he had been in the service a few months, Reinhart began to seek reasons why the Germans, while wrong\u2014they warred against the U.S., for one thing, and it was probably true that Czechoslovakia and Norway and Holland, little harmless Holland!, had inoffensively not deserved invasion; true as well that, even discounting for cheap newsmen and their \"copy,\" there had been regrettable brutalities by the extremist, Nazi units, although in view of the Belgian babies of World War I you should go cautiously here; they were surely wrong to torture Jews, who he had discovered in college were, at least in their American branch, a pretty good bunch of fellows given certain peculiarities, and who apparently had not during the German inflation of the twenties enriched themselves while gentiles starved, as alleged by Hitler & Co., although one must be careful here, too, in simple justice, for anyone who had ever traded in a Hebrew haberdashery knew the Jew as far from a na\u00efve man\u2014he had come under an obligation to find reasons why the Germans, though mistaken, though bullies, though bad, if you will, were yet not _bad,_ were not to be allowed that case which the greatest writers assure us even Satan has.\n\nThe Army, oddly enough, was filled with superior people, the universities being then in the process of emptying to that purpose. Every barracks had its circle of cultivation, and while its membership was still outnumbered by the gross herd playing cards, shooting dice, and shouting incessantly fuck this, fuck that, it in the strength of unity read newspaper editorials, went on pass to hear the nearest city's philharmonic, and discussed international political events. At every post where Reinhart served, this circle in fact had been semi-officialized, meeting at least once a week with the authority and encouragement of an intellectual officer. Since he was channeled in that direction by cultural imperatives and nobody else seemed interested in him, Reinhart willy-nilly frequented this society, attending a few concerts, where he felt unpleasantly conspicuous as the middle-aged civilian audience beamed benevolently on the display of high-minded soldiery, and sitting in on some discussions, quaking with terror that he might be called upon to add his half-cent. If that sum were indeed low enough to symbolize the content of his head as he sat surrounded by his frighteningly articulate comrades.\n\nThe prevailing sentiment was, as one intense, red-haired, hollow-cheeked PFC (they were all privates and PFCs) put it, \"just left of center, like FDR.\" Reinhart literally did not know what this meant, except that while in grammar and high schools, when he took his father's cue in politics, he had detested Roosevelt, had at campaign times worn little buttons against him, one for Landon pinned to a sunflower head of yellow felt, another reading simply: \"We don't want Eleanor either.\" And still, even after he lost all interest in that sort of thing, carried a vague distaste for the man which was renewed at every picture of the teeth, the cape, the cigarette holder, the dog, the wife melting in good will, the sons drooping in false modesty, the desk ornaments, and Sarah Delano R., the grim progenitor of all these. Yet it was not subsequently hard to swallow that he had been an improvement on old Hoover, starched-collar, pickle-faced, the personified _No._ And whatever left-of-center now meant\u2014he had always supposed it a kind of radical creed presided over by kindly-looking cranks like Norman Thomas who were understood to be not serious and a more extreme variety represented by Earl Browder with his mustache and dark shirt and faintly alien air, which might be sinister if it ever got its most improbable chance\u2014what it meant now could only be something respectable, if somewhat strangely motivated, for these young men professed a constant concern for victims of one social outrage or another, in which company they themselves could not be counted, so that it was not a demonstration of self-interest.\n\nReinhart was impressed, even cowed, by their easy yet earnest assurance and disturbed by the shrinking of his hitherto supposed wide horizon. How he had wasted his faculties to date! Even if his sympathies had been all along on the right side: these people too were opposed\u2014and from a far more intelligent point of vantage\u2014to the double-breasted, cigar-smoking deities of business, the devotional poems in Sunday supplements, Mother's Day, Congressmen, and the suburban imagination. In college he had been too apathetic to find this out, confined in the circle of self as he was then. Beneath the surface pall there was meat in the political and economic disciplines; as approached by these acute young men, they were adventurous and splendid and, he soon saw, were far fitter areas for the mature moral effort than the gross physical projects he had earlier honored.\n\nFor example, one's build. These men, by his earlier standards, were usually physical wrecks, if small, skinny, if large, flabby, shoulders slumped, belly, if they had one, bulging, the whole man hung with garments as a point of merit shabby as the Army would allow. And no pride of carriage even in the shower, where if he met one of them Reinhart was thrown into confusion: embarrassed by his undulating biceps as he soaped the scalp, yet unwilling to loose the arm's tension if there was also present one of the common sort of soldier who didn't applaud intellect.\n\nIt was stupid, perhaps mean, to be a good soldier in any manner, although he had been right to get appointed to the medics on motives of nonviolence. All these people had been drafted, so that they had no choice, but they _would_ have chosen the medical department. Some even had friends immensely admired who would not serve in anything but conscientious-objector enclosures; some others confessed that while that was going too far for them, it was a thing most noble for a man to hold fast at any sacrifice to what he believed right and true, against _the mob,_ by which they certainly did not mean _the people,_ who were always r. and t, but rather the crowd who ran things. Reinhart would earlier have supposed the latter meant Roosevelt and his entourage, with everything but Maine and Vermont, four terms without hindrance, no end in sight, but he soon found this a misapprehension, the situation being precisely the reverse, with all such good folk victims. Indeed, the persons to be admired were invariably victims, and the degree of their victimization was the degree of one's approval. The unfortunates even included some staggeringly rich men, who however were \"liberal\" and therefore smeared, earning the herohood into which poor men were enlisted at birth.\n\nReinhart had never used his head for much but dreams, he knew, and this new employment of the brain was exciting as well as good, for neither did it ignore the heart as it surveyed the vast panorama of the evil that men had made in the world and recommended sensible alleviations. The underfed coolies of Asia alongside the oversated warlords; the black and twisted miner deep in the earth's entrails, considered with the flabby oyster of a mineowner in his house on the hill; the poor little have-not, next to the arrogant, pudgy have. These contrasts were inexcusable in a world where education should be within everyone's reach, where it was now technically feasible for every man to be served by the machine rather than vice versa; they were wicked and what was worse, silly, most of the wrong people not wishing to be bad so much as not understanding what was good.\n\nYou take the Germans, for example, or really to test these intelligent new ethics, take Hitler. You at least had to grant that, terrible as they were, he had stuck to his ideals. If that awful energy could have been diverted into virtuous channels, if he could have stopped after solving the problem of unemployment and building the wonderful net of highways!\u2014No, you most assuredly did not take either the Germans or Hitler; and if you did, there were strong grounds for popping you in the booby hatch. At least, so said without words the faces of the others to whom Reinhart, breaking his long silence, introduced this application of the theory they had so generously trained him to use. The trouble was that they had forgetfully omitted one clause from the grand code: no Germans need apply.\n\nReinhart was quick to know the justice in this, too, for, awakening from his long sleep, he had begun to see the terrible landscape of actuality. It was false to think that the Nazis were an accidental, noxious but temporary weed upon a permanently rich German ground of the essence, which might one day be cleared. No, go as far back as you would, the wars of 1914-18 and '70 against freedom-loving, culture-cradle France, the rise of brutal Prussia, way back to the war lasting thirty years and further to the razing of the magnificent Roman civilization by the tribes which Tacitus had earlier observed as being without mercy. Martin Luther overthrew the wickedness of popery; Frederick the Great sponsored the culture of the Age of Reason; Goethe was spokesman for the liberties of the heart and mind; Schopenhauer and Nietzsche, revolted by their time's cruel and shallow materialism, drafted prescriptions for the free personality\u2014even these were but masks for more Germanic creeds, or the same old one, of tyranny, militarism, suicide, irresponsibility, and madness.\n\nWhen in the last months of the war American troops went choking through the fell streets of Buchenwald and other camps, passed the vast trenches of slack human skins, the bones inside all loose from their connections, and oven-grates of gray human ash, took in their nostrils that bouquet of burned man which for recognition it is unnecessary ever to have smelled before, and for sleep impossible to forget after\u2014when the pictures and accounts were published, for Reinhart as usual was not there\u2014the most malevolent indictment by the anti-Germans had not been enough, the righteous people who wished to reduce the land to a pastoral community were too mild, perhaps not even another Flood would suffice. For the outrage had been done to him, Reinhart, who had trusted in his origins.\n\nThere is no native American but the redskin; we others are something else at a slight remove, which cannot be changed; our names and looks and surely some complexion of the corpuscles themselves are to some old line peculiar, else we should blow away without identity. So he believed\u2014his only belief, along with an idea of the possibility of simple decency\u2014and thus, with his deep relation to what the superior, bright young men in the discussion groups were pledged to destroy, he disqualified himself from their company and took up instead\u2014well, what was fun, booze and snatch and other pursuits generally pointless and amoral, and was forever delighting such people as Marsala with his adaptability.\n\nHis one secret was that he liked the Army, where the petty decisions were provided and the major ones ignored, and where you could live as if you had been born the day you put on the uniform.\n\n# _CHAPTER 4_\n\nJUST AS IT HAD ARRIVED in England after the great mass of troops assembled there for the Continental assault was gone, so did the 1209th cross the Channel and proceed eastward against the stream of real soldiers returning. At the outset, the assignment to Germany was seen as punishment cruel and perverse. For a year they had run an enormous Nissen-hut hospital in Devonshire, tending casualties flown straight there from the fields of battle, wounds yet hot and reeking. They were veterans of the European Theater and should have been let to cross the water and swagger before the slobs on Stateside duty, to mix undelineated with the repatriated combat regiments, back in the frame where the greater category enveloped the smaller, overseas versus home.\n\nInstead, the score was to stay grievously unjust: for more than a year the 1209th had had to stand holding its portable urinals while patients lay smug with honorable wounds, relating the grand experiences denied to people of the rear areas. Charging the Siegfried Line; streetcars filled with explosives rolled down the hill into Aachen; the bridge at Remagen, with its sign: \"You are crossing the Rhine by courtesy of the \u2014th Infantry Division\"; the bombs falling on the ball-bearing works at Peenem\u00fcnde, courtesy of the Eighth Air Force; the Ardennes, where even company clerks and cooks took up their virgin rifles and joined the defense and even a general proved a hero, courtesy of the 101st Airborne Division; and at the very end, \"Germany\" itself made commonplace by courtesy of the Third Army, who got to Pilsen in Czechoslovakia and burst into the famous brewery to fill their helmets with beer. By courtesy of the 1209th General Hospital, Colonel Roy Fester commanding, one passed his water, told his stories, took a pain pill, and went to sleep.\n\nJust at the point, though, where the responsible latrine intelligence had disqualified the hysterics who insisted the 1209th would any day be shipped to the Pacific, and established beyond a peradventure that it would settle in the Helmstedt field where the unit was then resting as an alleged transient, and stay there forever\u2014just at this point where the wailing was loudest, there being nothing else to do except peer through the single set of field glasses at the nurses' tents across the meadow, came a courier of unquestioned authority with the word.\n\n_Berlin,_ it was to be Berlin, so long as something had to be accepted, a horse of a different hue from mere Germany; considerably better, in fact, since the combat forces had never got there. It would be at the courtesy of only the Russians, and the Russians themselves, with the Germans downed, were now a kind of enemy and face to face with their allies kept weapons at port arms. Already they had sealed the Helmstedt checkpoint, and when, after a week of negotiations, the colonel was permitted to pass with jeep, driver, and one aide, he made only fifteen miles before another Soviet unit arrested and held him twenty-four hours incommunicado.\n\nAll this, not to mention Berlin of the Nazi mythos: old Hitler screaming crazy garbage; creepy little Goebbels, dark and seamed, scraping along on his twisted foot; fat, beribboned Goering, more swollen joke than menace; swastikaed bruisers maltreating gentle little Jews; the Brandenburg Gate and Unter der Linden Trees; and acres of the famous blonde pussy, whom twelve years of Nazism had made subservient to the man in uniform: one heard that an SS trooper could bend down any girl on the street and let fly. And, once in the city, little work conjoined with a peculiar honor: the crap-house spokesmen who in England had been privy to a document from higher headquarters listing the 1209th as the biggest and best hospital in the Communications Zone, saw another now which said, approximately: the 1209th, selected because it was the biggest and best in the Communications Zone, would be the only general hospital in Berlin District.\n\nBerlin was not the worst place to end a war; better, surely, than the gooks in the islands or France where pigs lived in the same houses as people.\n\nAs Reinhart had promised the girl, he could have been found in the frame building across the street from the hospital almost any morning if the visitor came late, and any afternoon, provided the visitor came early; he put in a good four hours of daily attendance, give or take an hour either way, and had much impressed his superior, Lieutenant Harry Pound, by his drive. Pound was not properly a medical-administrative officer at all, but an infantryman, had in fact waded in on Omaha Beach on D-Day, H-Hour plus two, and shortly thereafter led a patrol into a hedge row filled with Krauts and their armament, collecting Mauser slugs and souvenirs of grenades in all four limbs and, later, the Silver Star. Under treatment by the 1209th he had healed into limited service and was transferred from bed to staff. Their job, Pound and Reinhart, was \"Special Services,\" recreation, diversion, amusement both for patients and medics, things that had meaning in the long, pastoral days in England but which now were needless, except insofar as they satisfied the rules of organization. However, there was in the works a plan for Sunday guided tours of the Nazi ruins; Pound ostensibly was always out somewhere arranging for permission to enter with a force of sightseers into the Soviet Sector but had not yet got even an admission that he existed\u2014if indeed he _was_ really trying, for he had a girl friend in the nurses' contingent and was often seen with her when officially he was understood to be elsewhere\u2014and besides, individuals could go across the border on their own hook without hindrance by the Russians, without the shepherding of Pound, which was to say he and Reinhart had no motive for an undue haste in consummating their project, especially since their desks were littered with schedules and itineraries and manifests and notes to show the colonel if he snooped.\n\nReinhart's obligation was to write up a guidepaper listing the principal Nazi monuments, their late tenants, and a fact or two, to be mimeographed and distributed to the tourists. He was not, at the outset of each period of composition, a facile writer, thinking first that here was his chance to show off, second that here was where he would be shown up, and third that it didn't matter either way because the jerks who went on the tours would immediately spiral the papers into little piccolos and toot obscenities through them at passing broads, if the experience at the Cheddar Caves and Exeter Cathedral had been representative.\n\nHowever, with stage three he reached the firm ground of the professional artist and could compose with enthusiasm. The only difficulty here was that when he got fluent, he inclined towards the poetic, and when that, put aside his proper work and began a letter to a female in the States who was at once a sort of girl of his and a kind of estranged wife of another soldier on European duty, as near as he could tell no precise love existing in either relation but friendship and interest all around: he always knew where Ernie was stationed and what he was doing, and vice versa, according to Dianne, and there was even some talk, now that the war was over, for a get-together between Ernie and him, arranged through their intermediary three thousand miles off.\n\nA week after his birthday, no more fights but a couple of drunks since\u2014now, he thought as he looked into the bathroom mirror that morning at the pouting aftermath of dissipation, you must take it easy, greasy, and you'll slide twice as far\u2014Reinhart sat alone in his office, with pen to foolscap, well into a new letter:\n\n> DI MY DEAR,\n> \n> I certainly understand why the Princess was late with my birthday present, and will look forward with lots of pleasure when it arrives in Berlin after a long transatlantic voyage, which will make it only sweeter to the undersigned. ... Well, I've gotten where I always wanted to be, Di, to the heart of Europe and just wish I could be holding your hand while we look down from the battlements of some old palace with the peasants going along with their oxcarts down below\u2014Ha Ha, the real peasants I mean, not the kind you always call me!! And I'd just as soon we left old E. playing baseball or whatever somewhere, because frankly Di, while I really like him, as you know, from what you tell me I don't think he shares our tastes and maybe that was the trouble between you. ... \n> \n> To go from the ridiculous to the sublime\u2014all pardons asked\u2014there are lots of exciting things transpiring here. The Intelligence Officer in our outfit, who is a friend of mine, is certain Hitler is still hiding somewhere around the city. I met a Tyrolean Count the other day, the kind of fellow you would love\u2014I hope not literally! With an ascot tie, and all. He invited me to hunt on his estate in Bavaria which perhaps I'll get around to doing when I'm not needed here\u2014but that will be quite awhile. You see, no one else in the outfit can translate the Nazi documents we captured. I'm just attached to this medical outfit now for eating and sleeping arrangements. I wish I could tell you just what my job is, but even though the war is over in this Theater, there are still plenty of secrets. ... \n> \n> Oh Di, when I look at your picture I think perhaps when I get home we won't be so platonic! Like to have your reactions to this. ...\n\nHe was moving along as magisterially as the Ohio River off Cincinnati, and as impurely\u2014but Ernie was in the paratroops and had shot nine Germans and taken as prisoner twenty more, and wore the Purple Heart\u2014when a spot of color not olive-drab came into the corner of his eye, stuck there, not moving but vital, and since composition was the product only of solitude, his drain was corked.\n\nThe color was yellow of hair and rose of skin on a girl, just plump and no more, like a peach, who stood diffidently in the doorway. She was small, wearing spectacles with lenses large and exactly round and an abundance of drab clothing, including high woolen stockings and thick, awkward shoes that made her walk as if deformed, for under his even look she had moved gimpily into the room. Rather, was moved: the thin arm of another party could be seen as far as the elbow, at which point it disappeared round the doorframe. An inch off the arm's furthermost extension she stopped and smiling as gloriously as one can and still show no teeth, said in a high-pitched and cowardly voice:\n\n\"Razher nice vezher ve are hoffing today!\"\n\nFrom behind the door, a whisper, and again the disembodied arm, this time making much of its hand, after a moment of which the girl moved by the use of her own muscles. Her walk was now pleasantly normal, if prim with perhaps an aim to restrict the swinging of her long blonde braids. The latter she caught one in each hand as she halted still far enough from Reinhart's desk so that he could see her down to the round knees which the skirt did not quite reach, where although at rest she yet maintained some slight side-to-side movement as if she were still walking in the mind. The effect was curiously provocative and perverse, for she appeared to be a kind of large child rather than a small adult, and he regarded her severely.\n\n\"Tischmacher Gertrud,\" was her next sound. Her little fist had come loose from the right braid and was available for the shaking if he so required.\n\nSomebody was pulling something weird. Reinhart rose and went around the desk, first going towards her to throw them off guard, and at the last minute executing a left-oblique turn of a smartness he had never been up to when in formation. Popped through the doorway, his head met that of the other girl, the one of the ruin, whose name he had not originally got and who now, though still nameless and taken in a suspicious act for which there was no apparent motive, greeted him like a friend and he had a handshake after all.\n\nHe asked her in and invited both of them, she and Fra\u00fclein Tischmacher, to chairs, of which they cornered the market, since there was only enough furniture in the shabby, rickety place to service his and Pound's narrow purposes. He even opened one of the French windows on the sand-and-crabgrass side lawn, to clear the air, for his series of cigarettes, the sine qua non for writing, had tinted the inside atmosphere gray-blue and it surely stunk to someone just entering.\n\nThe niceties owed to his guilt about not having turned a finger for her job. He had even \"given his word,\" he remembered, whatever that was; he said such things when under the influence he became formal and constricted. In real life, as now, he was, he knew, deft, volatile, witty. Sitting on an old wooden box, his legs up on the desktop, rough-skin boots, size 11-C, murdering the papers there\u2014oops! the letter to Dianne was ruined, but no matter\u2014grinning easily, he lighted another cigarette and blew a process of smoke rings, each smaller than the last and spurting through it, each round as Gertrud's eyes, as she watched them with honest awe.\n\n\"I am sorry I took so long to come,\" said the other girl, very slow and clear so that he could understand the German. Her hair sent no message of having had a wash since the night he first saw it could stand one; similarly, her dark-green beret and gray coat with breast ornamentation of Cossack's cartridge loops. But miraculously, the fresh sunlight which marched through the open window in a brutality that made Reinhart wince, was kinder to her used face than the night had been. Something could be made of her, if you took the trouble.\n\nReinhart had the courage to admit that he had not yet found the right thing for her, that he had of course been working on the problem for two weeks and would no doubt soon reach a satisfactory end. Not a day passed that he didn't arise painfully, come slowly through shaving, two portions of powdered eggs, a pint of coffee, and a lungful of Zehlendorf's pine air to health and good prospects and then feel drop over it all the shadow of his given \"word.\" The trouble was he never knew how to get things done, how to make deals, how to \"see\" people who could arrange. At the same time he had no hope that anything could ever be done in a straightforward way.\n\nThe girl spoke fast, and incomprehensibly to him, to Gertrud, and Gertrud then said: \"She vants\u2014wwants you to believe she is grrateful for zis. She wwants to say sank you.\"\n\n\"You speak English!\" Reinhart was not so astonished as he made out, but she was charming, although too young for one to admit to himself that he might find some use for the charm.\n\nHer eyes, bluer than the high, immaculate sky revealed when he opened the window, bluer than a broken bird's egg you might find if you went behind the building and searched the pine grove, than, if you walked far enough westward you would see, the Havel; blue, the quintessence of blue, so that if the color in all its other uses had faded, Zeus might take from Gertrud's store enough to renew the blue everywhere in the world and not leave her one whit of blue the less. These remarkable eyes, surely kept behind spectacles not because they were poor of vision but rather as protection against some thief who might pluck and sell them as sapphires in Amsterdam, showed their stars to Reinhart as, below, the small pink mouth said:\n\n\"Yes, yes, I know English zo wwell, having studied it zix yearss. I sink I do not too badly, do you?\"\n\nOh, marvelous, marvelous, he agreed, and would have preferred her over Churchill addressing Commons.\n\n\"You have acted so kindly to my cousin,\" she went on. \"Perhaps I do not seem especially rude when I ask, do you sink there is also available for me a chob\u2014do you sink for me\u2014do you sink there is also a job for me? There.\" Not covering her knees when she stood, her skirt did not pretend to when she sat but made a soft frame for the round thighs that it was no doubt a grave evil to look at.\n\nSo he looked away quickly, looked at the other girl's sad, sweet, and honest way, and suddenly heard his own voice saying: _\"Warten Sie eine Weile,\"_ Lovett, he would see a lieutenant named Lovett, who was chairman of everything out of the usual course, or if not he, then another officer named Nader, whose duties were similar. To the girls, however, he said only \"Wait,\" and in a tone which they considered too masterful to nod to, instead following his departure with heads neatly turning.\n\nThe building had no rhyme or reason. Nobody could tell what function it had served before the Fall; it may have been the only place in Germany where one could hide from the Gestapo, or perhaps on the other hand was a Gestapo-designed labyrinth through which their captives were permitted to wander free and moaning, madly seeking a nonexistent egress. Three weeks in Germany now, and Reinhart had yet to see his first right angle, true line, and square space. Outside, he regularly got lost en route to the Onkel Tom movie theater, ten minutes' walk away, and strolling of an evening over to the Grunewald park, to the body of water called by the Krauts \"Krumme Lanke\" and by the GIs \"Crummy Lake,\" he could not be less certain of his position in space were he in Patagonia.\n\nSomehow he reached the foyer and assuming the fresh soul of one who had just entered from the street, struck out to the right, passing the orderly room and detachment commander's office, a treacherous area in which, although he had a certain immunity from the worst of its menaces, it was not wise to linger. From there on, he looked in sundry doors, sniffed up divers halls, consulted with acquaintances encountered in passing, most as bewildered as he, and at length spied Lovett himself, the sissy, in a large room on the northeastern corner.\n\nThe lieutenant stood willowy beside an ancient desk to which a gnarled Kraut, in a peaked Wehrmacht cap, applied cloth and a white fluid from a long brown bottle.\n\n\"I want you to bring out the highlights of the _carving,_ \" Lovett was saying with his arbitrary, Bible-like stresses. And then, \"Highlow, Reinhart,\" although he had not yet looked up to see him.\n\n\"I am being willfully misunderstood,\" he continued, in a very lowww voice indeed, which quickly rose to a kind of screech to say: \"But who knows German?\"\n\nNader, dark and thuglike, sat at another desk and relieved himself of what, asking public pardon, he called \"The Return of the Swallow,\" by Belch. You seldom saw one without the other, and never saw either without wondering at their compact, which was surely queer and yet, on the same assurance\u2014namely, that you simply knew it\u2014was not _queer._\n\n\"Well, I do, somewhat,\" Reinhart admitted. He was not equipped to tell the man what Lovett wished, but ordered merely: _\"Polieren, polieren!\"_ which the fellow was doing anyway, and added \"please\" and \"yet\" and \"still\" and \"to be sure,\" the little words Germans hang on everything.\n\nSatisfied, even pleased, Lovett lowered himself into a chair in the way one might drop a length of garden hose and listened to Reinhart's requests with a crooked eyebrow, replying when they were done: \"Wanna come to a party tonight?\"\n\n\"Really?\"\n\n\"Certainly, really. Do you _think_ we are snobs? Of course we _are,_ but you look civilized. A little house-warming at our billet. American _girls_ \u2014if that's what you can call our nurses. Wine\u2014if that's what you can call this German cat-peepee. And songs. You know our place. Any time after eight.\"\n\nSince there was almost no finish left, it was impossible to shine the desk. The German knew that as well as anybody, but he kept working humorlessly as a sociologist, now moving right between Reinhart and Lovett so that they could see only unimportant parts of each other, and Lovett, usually so quick to be waspish, suffered the obstruction\u2014perhaps in the idea that any sound from him would be received by the dolt as a countermand of his previous order. Reinhart had begun to wonder about the man and what impressions he must receive, there with his bottle and rag between two aliens speaking of nothing\u2014for two words in a foreign tongue are double too many if you don't get their drift\u2014when Nader came over swinging his simian arms and said:\n\n\"Take off like a big-ass bird, Jack.\"\n\nThe German looked vulnerably from him to Reinhart, his nose long and tapered like a carrot, cheeks marred, black-brown eyes screwed in deep sockets. He was an old man, maybe fifty, commanding sympathy.\n\n_\"Er sagt, dass Sie herausgehen m\u00fcssen.\"_ said Reinhart, shrugging the blame off himself.\n\n\"You see, Dewey, that can't be polished,\" said Nader to Lovett, who flipped his hand negligently and then bit at a finger. All his nails were chewed down to the nub; the fingers were long and white and maneuverable as rubber.\n\nThe German put his cleaning materials into a wooden box, the contents of which on his route to the door he stopped to rearrange. The wine bottle, being too tall, gave him trouble; he transferred it from box to armpit, then had to lower the box to the floor for resituation. As he bent, tilting the bottle, the liquid poured off on Nader's desktop, and by the time he got his cloth to work, his head shaking stupidly, the papers thereon clung to one another in gluey fraternity.\n\nSo far as Nader went, what was done, was done. He calmly watched the man wad the papers and drop them in his box, mop up the excess fluid and then with a clean cloth rub up a high gloss, the kind Lovett had wanted on _his_ desk.\n\n\"Just _look_ ,\" cried Lovett, biting his tongue.\n\nThe German crept out, and upon his heels went Nader, returning shortly with the clotted papers, which he arranged for drying across an extra chair and table near the open window.\n\n\"The only thing that burns,\" he said to Lovett, \"is that that bastard could pull such a cheap trick and think he fooled me.\" He had not as yet recognized Reinhart's presence. Reinhart had no feeling towards him but distaste.\n\nLovett blithely ignored him and said to Reinhart: \"Send this woman to me. The nurses' quarters can probably use another maid.\"\n\n\"Course I coulda let him take this stuff for ass-wipe,\" Nader went on to himself, aloud, describing one document as a report to the colonel on how many butt cans had been placed around the area.\n\n\"And what about the girl, sir? She knows English. We could use an interpreter over in our office. For one thing, we've got this tour of Berlin coming up, and neither Lieutenant Pound nor I know anything about the city. She could\u2014\"\n\n\"Oh _God_!\" screamed Lovett. \"Is Pound going to start that awful Cook's Tour doodoo again? Tell him to forget about it, _please_! He'll lose twenty men, just as he did at Stonehenge and I'll have to go hunt them behind every Druid altar, or here, I suppose, falling down drunk with some filthy Russian. But then I suppose nothing _I_ say ever matters to that sloppy creature. All right, I'll see about the girl tomorrow, but if I ever come to your office and see you with your hands or anything else where it shouldn't be, I will know you for a deceitful person. Be at the party. Bye!\" His eyes closed like the lid of a rolltop desk, and when shut were as seamed.\n\nOn the return to his own office, Reinhart crept noiselessly the last fifty steps to the door and, unobserved, studied for a few moments the backs of the girls within, who had apparently not so much as twitched in muscle-ease since he left, and not spoken, certainly, which you could tell from the long-established set of their heads. Looking at the clean pink of Gertrud's scalp-parting which ran from crown to nape without a disordered hair, he regretted the evil will that had asked Lovett for her assignment to himself; she was probably about fourteen. Under civil law it would no doubt have been a crime to employ her, but then he was under the power of no such ordinance, and besides had no criminal aims.\n\nHe entered and stopped between their chairs, saying to the older, \" _Alles ist in Ordnung,_ everything is arranged. You must see Lieutenant Lovett tomorrow. Ask for him at the front of this building.\" Her eyes were soft brown over violet shadows as she showed a gratitude that made him as uneasy as if he were wearing sweaty underwear; it was too much, in the light of his mixed motives. And still it did not make her happy, but rather more sad. Indeed, everything about her broke one's heart to see.\n\n\"I hope you don't mind being a cleaning woman for our nurses. There was nothing else available, since you have no English.\"\n\nAnd now a kind of pride appeared for the first time, as she answered: \"No, I can do that.\"\n\nWhile he spoke, he felt near the back of his left hand the proximity of something warm and alive, not quite touching, but there: a piece of Gertrud, but whether hair, cheek, or hand he could not tell and did not wish to look. Finally, having finished his one report, he had no choice. It was her cheek, with its beautiful petal flush, extended in curiosity.\n\nHe teasingly seized the thick braid nearby and said in English: \"As for you, Miss Tischmacher, how would you like to help us here, in this office, with translations?\" And the soft young face moved sideways, towards a nest in his palm, but he had dropped the hand before it got there and went to sit on his box behind the desk.\n\nThere had suddenly come to him an explanation for the whole works, from start to finish, the perverse ways of this child, the other girl's melancholy poetry and strange demand on his conscience. They had successfully taken him along both roads without resistance. And the only sense it made was that they sought what could not have been gained in a direct and open appeal to the fixed authorities. Why else apply to a corporal, one lone and powerless jerk in an army of thousands? Because he looked first like a fool and second like a German, and because she, the older, was a Nazi who would have been put at picking street-rubble if she had made her appeal elsewhere. Perhaps it was even a kind of treason to get her a job under the cover of which she could hide from her proper deserts. It was what they had been warned against: the Germans did it before and would do it again if you were not vigilant, sack the world and then when beaten ask the pity for themselves, but this time we will not be duped.\n\nNow it was done. Lovett, particularly, was not the man before whom one could change a tune and retain face. Reinhart rubbed his head, sensing he had gone white from the discovery, and two short blond hairs drifted separately down the air past his eyes. He could see them until they reached the dark floor. For an altogether different reason he was going bald: you had constantly to wear a cap in the service, the hair couldn't breathe. And then he remembered another reason why he liked the Army: no cause was ever wholly lost.\n\nSimulating casualness, he asked: \"You know, of course, that you must fill out the political questionnaire, the _Fragebogen_.\"\n\nShe had arisen when he sat down, perhaps in a counterfeit respect\u2014which, if so, failed; it made him feel like a fool to sit before a standing woman, especially one so small and shabby, and the fact that Gertrud held to her chair meant that the other was the only adult present.\n\nBut he had got her, there. In a reflex of sudden worry, she turned to Gertrud and said something in a rapid German he could not make out.\n\n_\"Das schadet nichts\"_ the pretty girl replied, smiling bluely, guilelessly, at Reinhart.\n\nNo, it couldn't have mattered to her; she was luckily young enough to be disqualified from the rolls of mischief; but the other had been stirred. He knew, as she said \" _Komm,_ Trudchen,\" and walked slowly to the door, that he would not see her again. And well, in a way, it _was_ sad, and because at bottom he hated to win out over any person, he finally asked her name.\n\n\"Bach Lenore.\" It was Trudchen who answered. \"As wiss the great composer. Lori is a direct descendant.\"\n\nLori studied her in amiable suspicion. _\"Was hat sie gesagt?\"_ she asked Reinhart, and when he told her, said: \" _Es ist kein wahres Wort daran,_ there's not a word of truth in it. Trudchen is a good girl, but she exaggerates too much.\"\n\n\"Goodbye,\" said Reinhart, and despite himself, \"Good luck.\" As a pair they were at once pathetic and amusing. Now it was Trudchen who was all for going, and Lori who lingered.\n\n_\"Reinhart.\"_ In her throat was the rich and authentic quality with which the name had been spoken two generations ago. _\"Rrreinhaht,\"_ an old possession become new and attractive, suggesting ancient connections between them. \"It is surely a German name. Have you found any relatives here?\"\n\nIt was of course a new idea, to go with the name, exotic and adventurous; to find an identity in a far place, among the enemy. Yet here she was again giving him another obligation, goddamn her.\n\n\"Oh, I wouldn't know how,\" he said in weary helplessness. \"I'm not German any nearer than my grandfather, and it must be forty years or more since he came to the States.\"\n\nHe got up and reclaimed his rightful chair, booting the box into a corner filled high with similar junk and obstructing a closet door which if opened would reveal more. The buildings they had inherited hardly supported the Krauts' reputation for tidiness and order. The closet boxes held ream on ream of papers carrying the letterhead of something called the National Socialist Volkswohlfahrt. Bureaucratic crap, very like the material in the 1209th's own files. He had nevertheless informed Pound, who lazily said to see the Intelligence officer. This passed on to Lovett became: \"An _intelligent_ officer? Don't hold your breath till I find one.\"\n\n\"You don't care, then,\" said Lori, only to establish the fact, without a hint of reproach.\n\nTrudchen poked her arm. \" _Ach,_ they should be the ones to look for _him_!\"\n\nExactly so, but his answer was: \"Of course I care, but I don't even have their last address.\"\n\n\"Oh, then your family used to correspond with them? If you can just remember the city, you can apply to the burgomaster's office, who has records that date back, I suppose, to the Middle Ages.\"\n\nTrudchen poked her again. \"But if it is in the East Zone, Leipzig or Dresden, somewhere like that, he might as well forget about it.\" When she was serious, her little mouth puckered and extended like the jaws of a pink snapdragon.\n\n\"Yes,\" said Reinhart quickly. \"I'm sure now it was Dresden.\"\n\nOh, goddamn her sadness! When she heard that, Lori's eyes disappeared again into the heavy violet shadows. \"Then it is very likely they are dead.\"\n\nYes, he knew of the purposeless total bombing of Dresden, a nonmilitary target, and by not the barbarous Russians but the impeccable Western Allies. It was a very good place to have had relatives, and to have had them so disposed of, whatever their sins, by a crime of the righteous. Yes, if that was what she wanted, we are all depraved.\n\n\"I must get back to work,\" he said, and routinely added: \"Shall I see you again?\"\n\nTrudchen answered, in English: \"Shooly, I shall come to commence my job!\" She warded off his interruption with a small hand. \"After I shall have seen Lieutenant Lofatt first.\"\n\nHe had not really meant her, poor little instrument that she was. Besides, he realized now that she was too young to be hired, anyway. The morning which had begun so favorably with his letter to Di had ended in a thorough waste.\n\nLori said only _\"Wiederschau'n,\"_ and was almost gone before he called her back.\n\n\"Why do you always say that?\" he asked, irritably. \"I thought it was _wiedersehen._ \"\n\n\" _It's_ the same thing!\" she said in a voice bright with melody. And he wondered that such a small thing could lift her spirits.\n\nWhen they had gone Reinhart sought to recover the letter he had ruined with his boots, but no luck. He should have to re-copy two whole pages, as thankless a job as sorting used laundry. Better to start all over again. He never lacked in invention, with the right audience.\n\nHe had just taken out a clean sheet of onionskin from the office supply when the old German who had made the mess over at Lovett's stopped in front of the window to light a cigarette, a miserable stub of a cigarette that he had taken from a small tin box. He fired the match in trembling fingers which brought it so slowly to the butt-end that the contact was charcoal to charred tobacco, dead to dead. He stared witlessly at both for a time, then returned the stub to the tin.\n\nReinhart's sudden arrival through the window took him unawares. Scared, he hastened to leave, but his weak old legs were poor servants of his intent. His right shoe was busted out at the juncture of upper and sole, issuing a string of gray stocking.\n\n\"Don't run away,\" said Reinhart, jovially. \"I just wanted to give you these.\" A five-cigarette pack of Fleetwoods, the abominable brand included in K rations, which had lain in his field-jacket pocket since the first day in Berlin, when the cooks had not yet been set up for hot chow.\n\nWhat a shabby gift for such a wealth of gratitude! The man lost his speech, the corners of his mouth twisted in emotion, in awed delight he even forgot he was old and infirm, and disappeared round the corner with the vigor of a stripling.\n\nWatching him go, Reinhart thought, in satisfaction with his own courageous realism: certainly, he too could have been a Nazi, but now he was old and sick and defeated, become by the processes of cruel time himself a victim. Humanity is not the rights and wrongs of politics but a more general lottery of success and failure, and even more fundamental than those were youth and age and how one is constantly becoming the other.\n\nIf he had relatives they too would be old\u2014for he thought of them in terms of his grandfather\u2014and so far separated from him that the blood connections must be taken on theory... yet he was not a Laplander or Lestrygonian; if he had any structure beneath the meretricious American veneer, it was one he shared with them. If Nazism was a German disease of the bone, his own marrow, even at two generations' remove, could hardly be spotless. How many times had he felt within himself a black rage at existence-as-it-was and the eunuchs who prospering in it made its acceptance a standard of virtue?\n\nJust the other Sunday he had gone with Marsala out to Wannsee to prowl the deserted mansions on the lakeside. These had already been looted by the Russians, but there remained sufficient evidences of the genteel life: sunken bathtubs in washrooms as big as stables; roofed terraces of tile, for dancing; genuine oil paintings; one home had an iron portcullis which at the instance of an electric switch ascended from the basement to guard the door. The houses were in that intermediate state of ruin asking for more. If they had been untouched, he would have looked and left. As it was, the job needed completion. The Russkies had stolen rugs and furniture, roweled the floor, spattered the walls, had multiple diarrhea in the bathtubs and washstands. But still whole were most windows, the pictures, some glassware and vases and other fragile objects prime for the breaking. He smashed everything that came to hand, assisted only feebly by Marsala, who had been a juvenile delinquent when young and in America but here and now turned delicate, as if God were watching, and occasionally said, as he witnessed a crystal goblet pulverize against the fireplace: \"We maybe shoulda mailed that home instead.\"\n\nYes, that was surely Nazism, that passion to destroy simply because it could be got away with, because one had been trained all his life to respect and abide by the constraints and then found in a crisis that they held no water. Who wouldn't be a criminal if it weren't for the police?\n\nHe would find his relatives. If they were Nazis\u2014but why suppose that? Because, although otherwise so stupid, he knew one truth, knew it so well he habitually tried to evade it\u2014perhaps that is the definition of a dreamer, he thought, a man with an unusual sense of reality. Facts must be faced. There was such a thing as Nazism. It was a product of human beings, not some exotic heresy of the anthropoid apes that, owing to simian muteness, you must judge without ape-defense. The Nazis had first been clowns, and then almost without transition, devils. His parents, like their neighbors, had burned on sight the literature mailed by the Bund to German-sounding names, as they did the product of California box numbers which peddled data on the life-force; yet with match in one hand they might loosely say on the other that Hitler had a point when it came to the Jews. At college Klaus Greiner, a gentile refugee from Frankfurt\u2014his father had been some kind of political writer\u2014described his first two encounters with American strangers: a girl at a dance gave him an invidious lecture on democracy; a man in a cafeteria admired his being a national of the country that had at last settled the kikes' hash.\n\nBut whatever his relatives were, they were his. In almost every way but the accepted idea of common decency, he felt himself at odds with the world, a kind of Nazi without swastika, without revolver and gas ovens, without the specific enemies\u2014indeed, it was a crazy feeling, an apparently motiveless identification, for although it did not include the trappings, it did comprehend the evil, as when you awoke from a nightmare of murder and for hours afterward despite the evidence of daylight and routine believed yourself an assassin; and worst of all, coexistent with the guilt, the memory of a terribly depraved yet almost romantic pride: _once, anyway, you were not a victim._\n\nLori, with her quiet European authority, had no doubt known from the first that he would come to this. He must look for her again and say: I am neither pious nor indifferent\u2014he could even, as if from outside, see himself in the attitude of gentle yet strong and manly conviction and hear his firm voice purged of boyish tenor\u2014I will find my relatives, because no man is an island.\n\nThere remained a minor problem. His maternal grandparents, who had died when he was too small to know them and were therefore of no interest, came from what indeed was an eastern province, he was not sure just where but now in Soviet hands. His father's father had been native Berliner\u2014but where? He vaguely recalled the old letter in the box beneath the front porch, postmarked Berlin-hyphen-blot. Ah, it was hopeless. He went, anyway, back through the window to his desk and wrote a V-mail home, although it were more sense to poke in tea leaves or consult a necromancer than ask his folks.\n\nOn the way through the labyrinth to the mailroom he thought of Lovett's party and hoped he was not the only enlisted man invited; he might be taken for a fruit. And finally, another tinge of Trudchen: how old must a girl be before you may desire her?\n\n# _CHAPTER 5_\n\nA THIN DUST OF TALCUM lay on Lieutenant Schild's sallow cheeks, notwithstanding which the beard's threat was darkly manifest. He had shaved a half-hour earlier, taking his usual care at the sharp angles of the chin and the mastoid region, peering nearsightedly, vulnerably, into the magnifying mirror, studying the giant's face which regarded him with similar scrutiny. The steel-rim, GI-issue eyeglasses were back in place now, and with them the correlative look: tranquil, remote, mathematical, self-sufficient. He buttoned himself into the blouse, which, though it corresponded at all points to the measurements of his upper body, seemed uncongenial to it, and placed upon his curly poll the cap whose forward point was subtly out of line with the rear one. Finally, he folded the tie in a knot that was lumpily gauche.\n\nHis toilet was preparation for Lovett's party, which he assumed to be well under way, the time standing at 9:30. Only slightly acquainted with his host, whom he had met the day before in official work, he had delayed his arrival until it could be unobtrusive. After checking his watch again, he strapped on the pistol that Occupation rules required\u2014in this the medics, legally unarmed, had it better\u2014and left the house, conscious of the housekeeper's green eyes on him. She was a comely war widow in her late thirties and slept in. He would not acknowledge her as a human being,\n\nAs he descended the outside stair, his superior, Captain Roderick St. George, came bovinely up the walk.\n\n\"Hi, Nate. Goin' out cattin'?\" He had, it seemed, determined on the long ritual, not having seen Schild since just before evening chow. Schild muttered some half-audible nonsense, and it was all the same to St. George, whose low estimate of himself was necessarily transferred to his associates.\n\nThe captain withdrew a cigarette, fired his lighter and then held it aflame interminably without using it.\n\nSquinting, Schild made known a mild complaint, and St. George forthwith snapped shut the pigskin-jacketed Ronson. He disliked giving hurt, but took a modest pleasure in being an agent of mercy.\n\n\"Oh, _I'm_ sorry,\" he said, victoriously. \"Something exciting on the string?\"\n\nSchild received a malicious impulse, gave it its head.\n\n\"Look, she has a friend, a small blonde with fine skin and breasts like lemon-halves...\"\n\n\"Haha...\"\n\n\"You know these Germans are unleashing the bottled-up passions of years.\"\n\n\"Hahahaha...\" St. George thought this a genuine joke. He could not place himself in any kind of relation with illicit sex but the comic. Besides, he regarded all Jews as humorists.\n\nHe chuckled a stanza and then said sanctimoniously: \"No, Nate, you go and have fun. I've got work to do. By the way, you haven't seen, have you, a missing folder of Kraftfahrkorps correspondence?\"\n\n\"I'll look in my files tomorrow. Offhand, though, I can't remember it.\"\n\n\"No hurry. It was among that load I brought in today. I may have it myself.\"\n\nThe odd thing was that although St. George would have been a success as a civilian, he had never been one; not ever, if one didn't count his eighteen years or so as a legal infant, after which he entered the Point. He was now forty-five, with twenty-odd years of garrison and administrative duty in the States and the peacetime colonies behind him. In the war he led a small Intelligence team that during the hostile phase jeeped company to company on the Third Army front and interrogated German prisoners. At present its business was more sedentary. In many of the buildings occupied by the American forces there were great stores of abandoned Nazi correspondence. Somebody had to assemble and classify them: in the reams of paper that had fallen behind the police state like dung from a plodding horse, St. George and his little crew were put to picking straws.\n\nThe captain now reiterated his counsel about having fun and squished into the house on his crepe soles.\n\nIt had taken Schild months to accept the reality of the captain's stupidity, for despite Marx's and Lenin's examples to the contrary, Schild generally tended to overvalue people. But St. George's was far too crude a role for a double agent. And this was, in the smallest way, that is to say, personally, regrettable, providing nothing against which to sharpen the teeth.\n\nDarkness again, lightly flavored with the smell of growing things. He resented this street which showed no mark of war. Lovett's house across the way was marked by a globed light above the door, but the window blinds were drawn\u2014indicating, no doubt, the conservative character of a party made by medical men. At the gate, it struck him that he had no motive whatever for accepting the invitation. Lovett had rudely flung it at him as he was about to leave the office after speaking with Lieutenant Nader, a preposterous, almost illiterate officer who assured him that yes, the place was loaded with papers but he better claim them fast because the colonel had fifteen men on permanent assignment to burn all the useless trash in the building and they were already halfway through.\n\nNext he had gone to see the colonel, to whose inner office he was conducted by an insolent sergeant-major with a border-Southern accent who announced him as \"Child,\" and stayed to listen to his business. On his entrance the colonel, who had been sitting in deep study of the ejection device on a mechanical pencil, snatched up a huge bolo knife from beneath his desk and sprang to the open casement, screaming: \"Look sharp there, private!\" Handing the weapon to an unseen soldier outside: \"Here's the only thing for that crabgrass\u2014wait a minute, what are you doing with that butt? _What?_ Field strip it, balls! Carry it around to the can! Wait a minute, where's Lovett? _Where?_ An hour ago I saw, God damn him that nance, a lid missing from one of the garbage cans in back of the hospital. You tell Lovett to mince over there and find it. No, not you, _him_ \u2014a gold bar doesn't make him too good for that.\"\n\nThe colonel, Nader had told him, was scared shitless of anybody, even a corporal, from another headquarters, invariably assuming it to be a higher one that had him under surveillance for suspicion of untidiness. Schild's request to impound what remained of the enemy documents scarcely salved his nerves.\n\n\"Don't tell me Lovett hasn't been sending them to you all the while! That silly pimp!\"\n\nSchild sternly put down in himself the dirty little pleasure that it was probably not abnormal to feel at Lovett's being abused\u2014but why does the girl-man stimulate sadism rather than pity?\u2014and made a defense.\n\n\"That is true, colonel, today's the first time I've seen him,\" said the sergeant-major, neutral and hateful at the same time; he was that kind of man, just as he was the sort to turn accusingly a confession upon its maker. This worthy, it was clear, held the reins of authority; typical suburban, neat-haired, office-manager type, probably from some middle place like St. Louis or Lexington.\n\n\"I have it,\" said the colonel, nervously popping the eraser from his pencil, scattering across the green blotter the contents of the reserve-lead reservoir. \"I'll assign Sergeant Shelby here to complete responsibility for the allocation of whatever it is you require. Shelby's your man, Lieutenant Shields, want something around here, ask the enlisted men. My officers just weren't there when the brains were passed out.\" He retrieved the leads one by one, a neat trick with hands sheathed in white gloves. He answered Schild's stare with a smile that vanished as quickly as oil into leather.\n\n\"Eczema,\" he said ruthlessly. \"On all ten fingers.\" He tore off a glove and showed his right hand, which looked as if it were made of rusty metal. \"Neurodermatitis\u2014terrible for a man of action.\"\n\nShelby grunted \"Yeah\" and grandly proceeded Schild into the outer office where he imperturbably took a seat behind his desk and began to read Sad Sack in _Yank,_ from time to time calling one of the clerks to witness an especially funny turn.\n\n\"Sergeant,\" called Schild, after a few moments had defined the insolence, \"I want you to show me where the papers are stored.\"\n\n\"Well yes, I will.\" Not looking up from the page. \"If you'll tell me when.\" But already he was weakening, that shadow of the coward was stealing across his eyes.\n\n\"Now.\" Schild spoke it in his smallest voice, to demonstrate to the man and his lackeys what a small, two cents' worth of force was needed to bring him to heel.\n\nShelby sullenly arose and led him out, smelling of after-shave lotion. In the hall Schild told him he had changed his mind, would come another day, smiled, and left almost lazily.\n\nBut the outfit was a nest of madmen and clowns, a traveling medicine show rather than a hospital. And he realized, at Lovett's gate, that this condition of comedy was what lured him to the party, that he could handle it or let it go at his pleasure, without, as it were, a tab to pay. He had already freed the latch, was stepping into the yard, when a low, evil whisper, as if from the conscience, said: \"Enjoy yourself. Why not?\"\n\nHe drew away in the illusion that he had collided with a kind of animate bush which, weightless and retreating, yet aggressed with whipping branches in a hundred quarters, and although he stepped to the side, off the path onto the lawn, Schatzi continued to press him. Thus, without a word, he was forced to return to the public walk, where a hand jerked his sleeve in the direction of the street corner and left off, and he followed.\n\nAt the corner, where in sound underground practice they could survey all paths of approach\u2014or in the darkness, hear them\u2014Schatzi spoke in a queer tone that was loud while pretending to be low, an undertone which must have been audible behind Lovett's closed door a hundred meters off.\n\n\"Yes, my good sir,\" he said. \"I am authorized to buy from you five cartons of cigarettes. Payment on delivery.\"\n\nIf they were overheard, it was a black-market deal\u2014more than that, if an enemy operative lurked behind the tree, he was forced to hear what was after all the description of a crime towards which the Allied authorities were turning severe, and might ignore it in favor of the larger, for which he had insufficient evidence, only at the cost of his clear duty. The beauty of the method was Schatzi's acting in worse and more furtive conscience than when he met Schild unmasqueraded, as at the Wannsee contacts.\n\nHowever, having gone so far to establish urgency, stealth, and a suggestion of controlled hysteria, Schatzi began to talk quite banally of Lovett's party.\n\n\"I have sold them some glassware, very lovely crystal glasses which I am relying upon you to guard over. Some persons may get drunken, you see, and it will be a scandal to break these glasses which cannot be replaced all over Germany. I speak not of my own convenience, since they have paid me, but namely of the uselessness to destroy pretty objects which also have their place in the world, or don't you agree?\"\n\nThis preface out of the way, he thrust himself under Schild's nose and in a passion of distrust asked: \"What are your relations with Lieutenant Nader? I know yesterday you have seen him!\"\n\nIt was degrading that Schatzi, with his own active assistance, managed always to take him by surprise.\n\n\"He's Intelligence officer for the 1209th General Hospital and therefore the logical man to see about the German documents in their area.\"\n\n\"Of course, _Intelligence officer_ \u2014does not that mean to you something odd?\"\n\nSchild regretted saying \"German\"; he was commonly careful to use \"Nazi\" or \"Hitler,\" rather than the adjective that comprehended an entire people, not only because the distinction figured importantly in Soviet policy, but also because Schatzi was a non-Hitlerite German. And finally because he could not truly believe in the separation and clung to it all the more, in an effort towards self-mastery.\n\n\"As a matter of fact, it does.\" He made a joke: \"He has no intelligence.\"\n\nSchatzi hooked into his elbow with murderous fingers. \" _Was, was?_ I don't understand!\" And still claimed not to on repetition. \"Don't smile!\" he whispered angrily. \"If you do not think this is serious, something can perhaps be done about you.\"\n\nHe had never spoken this way before. True, he was Schild's superior, but for purposes of organization rather than discipline. And he was a German. ... How easily vileness slips in when one is momentarily weak with indignation! Yes, Schatzi was a German, a good one, which in his time meant a hero it was a privilege to know, an honor to be rebuked by, and thus Schild accepted the onus: What error had he made with Nader?\n\n\"The responsibility of an Intelligence officer is that of an open police spy, no?\" asked Schatzi. \"Therefore you present yourself to him conveniently. He can simply sit in his desk and you walk into his hands. This leads a person to say there are two possibilities: you might be a fool or you might be a counter-agent.\" He floated an inch away, and returned to his earlier, crafty voice: \"But I cannot pay more to you, since Captain Josephson of the Engineers Department has promised already to sell me all I would need for a thousand mark the carton.\"\n\nNot until he finished did Schild hear the footfalls, deliberate, soft, and yet massive as a lion's on the route of his bars. As they approached, the courier grew ever more spurious, and when at last the organism that made them, in his own agonizingly good time, arrived in closeup, Schatzi sprang dramatically to the curb and found on his forehead a sweat so heavy it required both hands to dry. Now the melodrama was inflated beyond all sane proportion, and it was Schild who felt wet all over in genuine perspiration, certain, in a dread moment as the newcomer stopped before him and he saw a face as puffed and insensitive as a medicine ball, that it was an arrest.\n\n\"Lovely evening, men. May I trouble you for a light?\"\n\nA great curved pipe like Sherlock Holmes's, like Stalin's, and by the flare of the match, a golden lapel-cross. He continued to intake and expel till the flame seared Schild's fingers, and then, with one last cumulus of smoke straight into Schild's eyes, he padded on with a clabbering \"good night.\"\n\n\"A holy man,\" said Schild derisively, regulating his breath as Schatzi returned. And then, as Schatzi said nothing, stood rather in silent, corrosive accusation as the minutes vibrated through the watch on Schild's wrist, up his forearm, biceps, shoulder\u2014 _\"Yes. That would be the perfect disguise!\"_\n\n\"Don't be ridiculouse,\" Schatzi answered in a very low voice. \"That was the Protestant chaplain for the 1209th Hospital. He is quite likely looking for girls, the younger the better, the dirty old man. ... You have then no explanation.\" It was not a question. \"Among the papers of Nader was concealed a memorandum which read 'Documents\u2014Schild.' They all go to him before you deliver them to me, yes?\"\n\nTo be frightened by a fat, buttery, strolling chaplain! Schild recovered so rapidly that he all but made another small joke. \"Ridiculouse,\" how ridiculouse it was. Schatzi was after all accusing him of treachery; of all imaginable moments it should have been the most terrible, yet he could barely withhold laughter. Nader, Lovett, the colonel, Shelby, the chaplain, and, in his own house, St. George, with their uniforms and pipes and insignia and parties and cleanup details and evening walks\u2014who but Schatzi could envision that fat, genial toad with the gold cross pinching some German teen-ager's behind, or Nader's playing the deep game?\n\n\"I had to go to Nader, you see,\" he whispered. \"Anything else would have been suspicious. I assure you he's a buffoon.\"\n\n\"Now I must not again hear you say that of anyone,\" said Schatzi, \"or I will know you for a traitor. I have told you those are the most dangerous persons. But even so, you do not have a connection with Nader, _you say,_ however, you go from him to the office of the commanding colonel and insult Sergeant Shelby.\"\n\nSurely he did not presume to direct Schild's official relations with enlisted men; he was getting now clearly beyond his limits, and Schild forbore from righteous protest only because his intuition told him Schatzi had not yet reached the serious argument of which this was preface.\n\n\"Shelby?\" Caution made him pretend briefly not to recall the name.\n\n\" _Shelby,_ yes!\" Schatzi's breath into his ear was like a long needle piercing the drum. \"He is a sympatizer, but he will not forever be one with rudeness.\"\n\n\"How was I supposed to know that?\"\n\n\"You might have smelled it\u2014but that is not the point. A source in the very headquarters of the major American medical hospital in Berlin. If Major General Floyd Parks becomes ill, where does he go? To _1209th_! If the deputy commander, Colonel Frank Howley? _To 1209th_! Eisenhower comes to Berlin, twists his ankle\u2014even you should see the actualities. There is no brains in making enemies of someone who has the slightest power. That is the first rule. The second is, give to a man a chance.\"\n\nGive to a man a chance! It was a touching slogan of wonderful, innocent charity, like the creed of some early social reformer, some Robert Owen, now outworn but fond in memory. He had not looked for such a sentiment in Schatzi and finding it was not quite sure he got its sense, unless beneath that scarred and charred carapace there was an old idealism that had remained impervious to the arrests and tortures and the subtler ravages of the illegal life.\n\n\"I'm afraid I still don't understand. Do you wish me to identify myself to him?\"\n\n\"He was told someone would come to ask his assistance.\"\n\nSchild did not think it wisdom to remind Schatzi his directions had been simply to go to the 1209th, with no mention of a source; the injustice being done him stirred more caution than hurt.\n\n\"Now they must give him another carton-box of vodka.\"\n\n\"He is bribed?\"\n\n\"To be sure, he is only a sympatizer and not under discipline. But tell me what time is it? Ah, so late! One more detail: in a room on the south front of Shelby's building is a closet filled to top with papers of the old _Winterhilfe_ \u2014this was a Nazi agency to deal with the poor, clothing and food for charitable distribution. In the last years of the war, one hears, they gave out clothing of the Jews exterminated in the camps, sometimes even forgetting to cut off the yellow badges. Haha, cynicism could not be carried farther on...\"\n\nHow innocuous Schild's own little joke had been in contrast to this, the authentic, vintaged gallows-humor.\n\nSchatzi continued: \"Now there is a boy in that office, with some kind of entertainment service for the Ami troops. A great lout, with him you would be correct when you said clown. Just go there and get the files from him, no need to let Shelby know.\"\n\n\"No need to let Shelby know?\" Schild could do no more than parrot the sentence.\n\n\"Certainly not! Anything you can get without him, all the better. One shouldn't wish that _Scheisskerl_ to become too self-important. As I told you, he is not even a member of the Party, he cannot, in the end, be controlled\u2014 _Achtung_!\" He reared back and harked with hand to ear. But it was only someone entering Lovett's door down the block.\n\n\"Isn't that a Wehrmacht cap?\" asked Schild, seeking a moment's respite, for prolonged exposure to Schatzi's undiluted presence was very like being worked over with a blowtorch; one had left only short breath, and that was filled with the smell of scorching. However, Schatzi's own habit of disregarding nothing was so influential that he found himself eager for the answer, for some clue as to his frequent changes of costume, which were more likely to achieve publicity than disguise.\n\nBut Schatzi gave every notice he could, in silence and darkness, that the question was a faux pas, social and not conspiratorial, nonetheless offending.\n\n\"Go now to your party,\" he whispered coldly. \"And for heaven's sake don't be rude to anybody. Have pleasure, dancing and drinking, show yourself to be a normal person. What there is to lose but your chains?\" With the latter he moved into better humor, saying in what no doubt was a friendly way: \"Here is a little gift from me to you.\"\n\nHe pressed a small, flat package in Schild's hand. What was it, rubbers? Schatzi didn't understand and Schild, laughing, didn't know the term of the German-in-the-street.\n\n_\"Empf\u00e4ngnisverh\u00fctende Mittel?\"_\n\nThe dictionary formality got a laugh even from Schatzi. \"Cigarettes of the Fleetwood brand. _Also,_ unless emergency, the usual time and place.\"\n\nHe left, or rather he was no longer there. Nor was there a sound that could not have been made by a leaf crashing onto a pillow of moss. The \"gift\" lying uneasily in his pocket\u2014as if it were soaked in phosphorous-water which when dry would explode\u2014Schild went again towards Lovett's gate. Just before it he met the stout chaplain, whose pipe had once more gone out, this time, however, without appeal. A soft girl of about fifteen and in long braids stood swaying at his side.\n\n\"Good evening, men,\" he voiced richly. \"I don't suppose\u2014no, I see you're busy.\" Peering. \"Oh, just one of you! Well, to the party, eh? I may look in later, but just now I must act the Samaritan to this child, who is out all alone after curfew.\" He reached for a braid. \"You don't suppose you\u2014no, go on in and have fun. This is what chaplains are for. But, I say, have you any idea where Jugenheimer Weg lies?\"\n\nTelling him, Schild thought he heard a distant, hideous snicker, a passage of air through corrupted channels; and so it was, and no more: the chaplain sucked on his dead pipe.\n\n\"Sank you from zuh bottom of my hot,\" said the girl, with a pliant little moue very visible in the glow of Lovett's porchlight.\n\nWere they all dead drunk or had he got the wrong night? He knocked interminably without result, and no doubt would have given up had he not received so many counsels, nay, commands, to go there and enjoy himself. At length Lovett unbarred the portal, showing a face painted by Dante Gabriel Rossetti, slightly under the weather, and donating three white fingers towards hospitality.\n\nIn the living room, which was the smaller and more airless for it, sat a number of guests in contemplation of their belt buckles. As Schild came in from the hall they looked up as one and stared ominously, hatefully, man and woman, and as quickly looked away in instant boredom. Lovett gave him a green tumbler, and Nader, who it appeared was also _en menage,_ presented a hard look from beneath the one long eyebrow like a caterpillar across his forehead, and went into a corner from which presently a phonograph began to bray. The company arose to dance claustrally, the rug having been cleared away but not the furniture. A nurse or two, lacking partners, watched Schild in a hopeless expectancy which soon settled into resentment, for he had discovered a sofa in the opposite quarter where one might settle to view the passing parade.\n\nAt one end of the couch sprawled a young man, who had managed by a disorderly arrangement of large limbs to command nine-tenths of the surface that should have been free; as if this were not enough, Schild was astonished to see he wore corporal's stripes.\n\nWith a failure of consideration he immediately regretted, since the fellow on his approach made ample room, he asked: \"Is this affair for enlisted men, too?\"\n\nTaking no seen offense, the young man grinned and waved his hand abroad. There were, in truth, several noncoms scattered through the crowd. A sergeant waltzed by at that moment with a bony nurse, in some danger of being impaled on her chin.\n\n\"Oh, I didn't mean anything was wrong. I've heard these medic outfits practice a good deal of democracy.\"\n\n\"More in the breach than in the observance.\" But the corporal's very impudence belied this. Schild anticipated trouble. He warily took stock of this soldier, whose olive-drab hulk slouched into the couch as if he owned it. Assuming comparable _sang-froid,_ Schild without looking lifted the drink in his hand and took a modest draught. It was not a green glass at all, but a clear tumbler containing a viscuous green fluid, an oleaginous, minty, sweetish ooze. His tongue curled in revulsion. While he fancied desperate measures, the liquid crawled across the palate and drained into his throat.\n\n\"God, that fool gave me a full glass of cr\u00e8me de menthe!\"\n\n\"That's probably all he had left. They pooled the officers' liquor rations for the party, but most of it was wines and liqueurs. I got Chablis.\" The young man, lips parted in good humor, lifted the glass that stood between his feet. \"It's awful, too, if that makes you feel better.\"\n\nStrangely, it did. There was a generosity in the corporal's ease which minimized his impudence. With a wry nod at his orders, and despite a sense of imminent nausea, Schild organized himself for fun. But when the opportunity appeared, it was in the corporal's name.\n\n\"Reinhart!\"\n\nA large nurse, constructed on the plan of Rubens' second wife, stood before them offering her heroic body with a slight upthrust of the hips. High above, gigantic breasts made bold, made brutal, and threatened the poor weak seams of her olive dress. One listened for the _ping_ of parting threads deep in her armpits. Very likely, said fun-loving Schild to Schild, she cocks her hips to balance the bulk of those incredible glands.\n\n\"Come on, Reinhart,\" she cried and rowdily assaulted the corporal's arm. \"Don't be sticky.\" She got him on his feet and into the amplitude of her fa\u00e7ade. From the phonograph wailed a niggardly statement of love denied. And Schild sat in his standard condition: alone.\n\n# _CHAPTER 6_\n\nNUESE LIEUTENANT VERONICA LEARY presided over the nut ward of the 1209th. Reinhart knew her by sight and name; was not, however, in the least acquainted. Were the standards of rank, which he approved, now to be swept aside?\n\nThis was his first mixed-grade party, and he had so far found it difficult to put off his snobbery, even though most of the officers were from the medical staff, which meant an amiable, unsoldierly, democratic lot whose professional view of man as viscera saved them from megalomania. The administrative officers, having got wind of the conglomerate guest-list, stayed away, victims of poor judgment. For of the enlisted personnel Lovett had invited only notorious brown-nosers whose obsequiousness no intimacy could corrupt.\n\nWith qualms about his own status, Reinhart had soon retreated to the isolation of the sofa. Now here was Leary, legitimizing him by her substantial presence, like Europa and the ox with functions reversed. But since he was, despite nature's perverse generosity, larger still than she, the issue was a push-pull in a progressively diminishing tempo, and a nettled comment, charged with liquorous and sexy odors, blown into his ear.\n\n\"Do you know, you're really a punk dancer.\"\n\n\"Never claimed to be a good one.\"\n\nWhich she went for in a large way, with a splendor of teeth and a marvel of air-blue eyes, in a demonstration of the frequently altering but at any given moment perfect dominion of her withal fragile, sentient face over the dumb classicism of that body.\n\n\"D'yuh know what?\" she asked, giddy again but kindly, \"Really, when it comes right down to it, you don't have much fun, do you?\"\n\n\"I'm having fun right now,\" he said, so pitifully that his heart cracked right through at the vibration and hung like a sundered glacier about to plunge into the sea.\n\n\"Aw, kiddy, come on and cheer up! When I used to see you I would think there's the very nicest boy in the 1209th. And also the saddest, because who knows what secrets lu-r-r-r-k in the hearts of corporals.\"\n\nThis was actually a horror to Reinhart: as he walked in dignity and rectitude, strange eyes had marked him, had abstracted a piece of him, as it were, that, insensitive fool that he was, he had never missed.\n\n\"All right, it was just an idea,\" she said then, surprisingly enough, eyes bright with the fool's-gold of ennui, mouth parodying good humor. This came from hither-and-yon; she was revolving her head, apparently surveying the room for another candidate to storm, who, not capitulating instanter, would get the same short shrift as Reinhart.\n\nFor he had her sized up, and stood enjoying bitterness confirmed. When a nurse smiles at a corporal, _caveat_ would-be lover! A nurse is an ill integration of woman and officer, with one of the roles appearing wherever the ordinary lines of human deportment would ordain the other; so that you are always puckering to kiss a golden bar or saluting a breast, a stranded sycophant between sex and power.\n\nThe music having suddenly pooped out, the rest of the crowd clogged the rear of the room, where Nader gave first aid to the record player with loud frustration at the complexities of wire. No man being opposed\u2014even the dark, nervous officer had vanished\u2014they returned to the couch, where Lieutenant Leary announced her name to Reinhart as \"Very,\" and plumped down proprietor-close. He had then, by default, been chosen.\n\nSince high school Reinhart had made it a principle to avoid really pretty girls, with their detestable and arrogant ignorance of the principle: they're all the same upside down. He played courtier to no one, and was gratified in college to see that the lackeys of the prom queens were to a man spectacled and pimply, usually students of science. However, although she was beautiful, Very was reclaimed by her size; it was a near-deformity, being almost divine, and made her human.\n\n\"Do you have the time?\" he asked, for in spite of all, he was horribly bored.\n\n\"Sure, but who'll hold the horse?\" Very answered brightly. \"An old joke that if my father's said once, he's said a thousand times. But I can't tell you because yesterday I sold my watch to the Russians for two hundred and fifty dollars.\"\n\nShe showed a fine, empty wrist\u2014at such narrowing places she was as slim as she was generous in the areas for expanse\u2014and went on to add that really the watch was sold through an agent, who no doubt had kept a sizable commission since timepieces went for about five hundred; but to her it was well worth the missing half: she feared the Russians, who were reputed to prefer large women.\n\n\"And I'm not what you'd call petite.\" Robustly she snorted.\n\n\"The Russians like 'em fat,\" Reinhart said gallantly. \"And that's not you.\"\n\nShe looked away with a hint of pain, as if the remark were out of order, and then returned to, anyway, do best by it: \"One thing I know, it's sure hard to lose it when you put on blubber. Cripes, you get so hungry, sometimes!\" Her extraordinary grin over nothing, open, unafraid, witless, was more splendid than anyone else could make for cause. Trying unsuccessfully to match it, he cursed the fate that had led him early in life and from a false psychology to cultivate the impassivity of an Oriental. Now, in a time to be bravura, he found himself instead sneakily edging his knee over against hers, laying his hand on the cushion where hers, he had observed, habitually flew at punctuations in her speech, studying the rich mouth as it carved words from the adamant of the northern Midwest. If he wished to touch her, he should do it; there was a bond between large people, as among Negroes, Greek-Americans, soldiers, etc., by means of which their secrets were kept only from the outside world; thus, if he had such a wish, she already knew it and sitting there unprotestant was not offended.\n\nBut there went the music again, and since girls genuinely like to dance\u2014so much so that they will partner another of their own sex rather than sit aside\u2014Reinhart patiently rose and returned to the grappling, this time, however, since he was prepared, getting the initiative before she did, encircling her waist with a tensed forearm the muscle of which, though she did not complain, surely put a rope burn in the small of her back, manhandling her, on the turns raising her whole weight off the floor on just that single arm.\n\nCoincident with her total surrender he went into tumescence and regretted that for the sake of slim hips he had worn the tight OD trousers which would show the most meager change of contour. He must hold his lower body away and cast the mind on some serene subject matter. The phonograph played \"Long Ago and Far Away,\" from some movie faintly recalled, abominably corny yet sad and sweet. No doubt most of them here were led to thoughts of home, and he was in this mood charitable, retaining for himself an achingly beautiful sense that it was he who was far away and long ago, like someone who lingers in the theater after the performance has ended, amid the discarded programs and slowly vanishing odors and the houselights extinguishing bulb by bulb.\n\nThus as the hour fled, when the record player broke down and they returned to the couch, Reinhart felt nostalgia for the dance, and when it began again to revolve and they danced, he thought of the distant perfection of the time on the sofa, and was always ready to pull Very one way or the other like a great anthropomorphic balloon, for she had become incredibly light on her feet. In these activities, he got his hands on her in various quasi-legal ways: against the side of a knocker, as they went off the floor; slipping down from the waist and swooping like a swallow across the buttock-swell as they waited, swaying, between records; up her grooved back to the hard metal juncture of the strained brassiere, over which slip, shirt, and jacket provided no more cover than wallpaper over the last tenant's picture-hooks.\n\nThe society of girls is a very delightful thing, as he recalled someone had told David Copperfield, not professional, but very delightful. Reinhart had not for years, excepting caf\u00e9 encounters and alleyway contacts with foreigners, which was something else again, had it to enjoy. Those American scents and sounds, one's own language speaking of nothing, but understood; the thousand familiar references in matter and spirit; the absence of ambiguity\u2014Europe was suddenly squalid, skinny, crooked, and dark, he would not have taken it for _eine Mark_ or _cinq francs._ With this Yankee smooth-warm cheek against his he thought of Lori and her little cousin towards whom as late as this afternoon he had had inclinations which, because he saw them now as a product of the time and place rather than himself, could be definitely labeled strange and discordant, the whole business devious and gnarled.\n\nHe decided he was in love, or that he would assume he was tonight and decide on its permanence the next morning. This, and the fact that no officer in the company looked disapproval\u2014indeed, in the crush on the floor none could have if he would\u2014the fragrance of cosmetics, the shadows when some excellent person turned down the lights for the dreamy songs, the warm wall of humanity around the tight little cell of their mutual interest, the yielding of his artificial will to the natural magnetism of her mass\u2014in the strength of these he closed with her all the way to the shins, lost false modesty and with his lower-middle, that had become sensitive as the tips of both hands, could feel the very mount of Venus, while his mouth in the movement of the music slowly followed the round of her cheek to the lip-crevice and made entry.\n\nThis bliss was disrupted when some bastard inaugurated a series of hot records on the turntable. Reinhart knew as he led Very back to the sofa that it was an opportunity, even an obligation, to take her outside, perhaps in supreme audacity to make a headlong rush through the back yards to his flat just around the corner, taking the tide at its flood. Yet his feeling was more delight than desire; he wished rather to prolong this time which had come fortuitously than replace it with his own initiative, which held no surprises at all.\n\nHow marvelous it is to be singled out, and spared the tight-wire balance of establishing favor! But it also makes a man a good deal more cautious than a shy girl would believe, forwardness in small things being a fortress against large. Girls who are bold can better withhold. It is as if the tiger dug the pit, fixed the net, arranged the camouflage, and crouched laughing by while Frank Buck stumbled through and captured himself. The other side of the coin was the pleasant fantasy that you could sit very still, coining banalities when necessary, but nothing coarse or even really interested, it went without saying, sneaking through the requisite time, playing cherry, so to speak, until the girl was so wild with unrequited passion that she would positively drag you to her bed.\n\nOn still another hand, Very bore all the moldmarks of a nice girl, the sort whose intimacies were flagrant because her intentions were innocent, like some Samaritan who courts denunciation as a pickpocket by reseating your slipping watch. She could very well get you all the way to the Beautyrest only to repair a loose spring, and nothing upset him more than basing an effort on principles not understood until its miscarriage.\n\nHe again could have stood a drink, and Very had just as soon, but a difficult negotiation through the dance-floor athletes who despite their average age of thirty to forty were astonishingly spry at the fast music, discovered only a wet table of empty bottles. He returned to see Very sharing the couch with a gloomy captain whose collar caduceus bore a _D_ for Dentist. He was known to Reinhart as a relatively good egg, as well as a painless practitioner, but he now wore the pious look of that partygoer who makes a fetish of his loneliness and searches grimly all evening for fellow worshipers, thus partaking in much more community than the busiest extrovert. Skinny and fuzzy, as if he had been twisted together from pipe cleaners, he sat grumbling in an undertone. However, he had dropped there only in quest of a seat, not trouble, and his scowl of greeting as Reinhart sat on the other end carried no hint of malice.\n\n\"You know,\" Reinhart said as she moved comfortably against his shoulder, \"I used to be air-raid warden in the nurses' quarters in England. Did you ever see me there?\"\n\n\"Gosh, I hope you didn't see me! I'm always a mess around the barracks, especially in England. Wasn't it awful there! The continual fog and rain, and that horrible tea all blue with milk, and fish and chips, and sausages filled with oatmeal. You know how many times I went to town? Once. Once, and I had enough.\"\n\n\"Weren't you ever to London?\"\n\n\"Oh cripes no. That was dumb of me, wasn't it?\u2014I should have gone to London, anyway, because as everyone says you'll never get the chance again. I bet you did, though\u2014were the Piccadilly commandoes really pretty? Go on,\" she dug an unbelievably hard elbow into his side, \"you can tell _me_.\"\n\nAt this the captain, whom he could see beyond her, ostentatiously repressed a grin and cast his eyes on the ceiling, and Reinhart was suddenly embarrassed at the public disclosure of her stupidity. A statistical friend once told him that one out of every ten girls is pretty, and one out of every ten girls is intelligent; ergo, one out of every one hundred girls is pretty and intelligent. Pooling the women he knew with those of the statistical friend made a grand total of eighty-five; they had had every expectation that somewhere in the remaining seven and a half to each man would appear the rare combination, but Reinhart had since lost track of the friend, in whose consignment the marvel would have to be, for his own quota was exhausted. As to Very, to balance the proposition she should have had to be a Mme Curie, a George Eliot, for only genius could be commensurate with her beauty, which he realized sitting there finding fault where none was appropriate, had become ever more glorious with use. She was fantastically beautiful, there was no other possible description, and comparable to nothing, lake, sky, gems, or flowers, but an until-now masturbation dream of the female essence.\n\nSo he began to lie, not grossly like a politician but subtly like a statesman, referring to his parents' street as a _road,_ and to his college as _school;_ spoke familiarly of dinner clothes, of riding boots, fencing, martini cocktails and the sediment of sherry wine, and of the possibility of buying oil paintings from ruined Germans for a song. He talked of love, not particularized but general, yet with a hint that behind him lay the wreckage of a hundred hearts, each keeping with it a piece of his own, for he was more passionate-impulsive than cruel. And finally, of the manly arts: boxing, judo, water polo\u2014and creeping through the poison-gas chamber in basic training.\n\nHe was about to bear down on the last\u2014and with justice, for it was quite true that his gas mask had sprung a leak, letting in the smell of deadly chlorine\u2014when he reflected that since nurses had had the same training, Very was not likely to see it as exotic.\n\n\"Well, _go on,_ \" she screamed as the story bogged, flashing the long lashes which had some time earlier\u2014it being, after all, a long evening after a full day's work\u2014begun to lower.\n\n\"It wasn't really anything\u2014\" On the contrary, he sensed, awfully, that it was the only thing for a whole half-hour that _did_ interest her, and, to a degree humiliated, he determined almost vengefully for once to give the banal truth.\n\n\"Why, the instructor said if we so much as imagined we smelled chlorine to get the hell from the shed. Which I did. There was some sand in the valve of my mask, it turned out.\"\n\n\"But were there any toxic effects afterwards?\" She pressed harder against him and seemed to study his right nostril.\n\n\"Just a little dizziness.\" Off on another lie.\n\n\"But you had a blood count, surely?\"\n\n\"Not exactly, but\u2014\"\n\n\"How long ago was this?\"\n\nChrist, he could smell it still, that odor of laundries and swimming pools, fiercely clean, implacably antiseptic, inhuman, the same stink outer space must have beyond the farthest planet. Three years ago, when he was a punk recruit, his suntans yet shiny, his fatigues still dark green, his gas mask clotted; it gave him no pleasure to dwell on that era.\n\n\"Oh, well then, I guess it wasn't serious.\"\n\n\"Of course not, that's what I said.\"\n\n\"No mental effects?\"\n\n\"Mental! You mean _crazy_? I hope not, nuts as I already am\u2014but you can't be serious. Chlorine attacks the lungs, if it gets you, and you are so busy dying you haven't got time to go mad.\"\n\n\"When you've been around as many weirdies as I have, kiddy,\" she said in a bluff, coarse way that made him recoil, \"you would know that many people don't have time for anything else.\"\n\nThis was a grisly turn indeed, and his feelings rumbled in his stomach as he pursued it. \"Seriously, can chlorine gas\u2014?\"\n\n\"Oh, you're not worrying now, after this long? There now, I've upset you. Maybe I was joking a little. To tell the truth, I don't know anything about poison gas except what they told us.\" She laughed a little too violently and a touch too long, and if at its peak you had taken a still picture with a very fast camera, you might have seen that she herself for a moment looked deranged. \"But are you aware that many kinds of internal medication taken to excess can produce a psychosis? Sulfa drugs, for example.\"\n\n\"They can!\" He said it in so terrible a voice that the dentist, still there on the other side of Very, jerked in professional, hypocritical dismay, as if his drill had slipped and lacerated a tongue.\n\n\"Well, only _temporary_.\" Into his face she had pushed hers wide with the most glorious grin of the evening, at once splendid and grotesque, and so near that with almost no effort he could have sunk his incisors into her velvet nose. Then she drew back and laughed, laughed, laughed. \"Oh, I've got you so scared! Now the next time you contract nasopharyngitis you won't take sulfa, and then you'll catch pneumonia, blaming it all on me.\"\n\nHe asked, somewhere between joke and real, \"Does that mean I'd be put on your ward?\"\n\n\"For pneumonia? No.\"\n\n\"I mean\u2014the other.\" He deplored euphemism, but he fancied that her mirth had become briefly acid with malevolence.\n\n\"Why not get to know me off duty, instead?\" She patted his hand, but it was not in the least provocative. \"I'm nicer then. Besides, we don't need you, we're all filled up, got more patients now than when the war was on. Bet you thought it would be the other way around. That's because it's never the real things that crack people, but the imaginary.\"\n\nThe music had at last become soft. Because someone tripped over it, lightly cursing, Reinhart crossed his restless, foot-tapping leg over the quiet one; his trunk inclined in a long plastic crescent: a smooth-leather couch would have ejected him to the floor. Very when solemn was not very Very, he said to himself, and to her: \"I was always in a funny position.\"\n\nHe was in enough of one now for her to hesitate and then produce a question he had not heard for three years, an idiotic cuteness nevertheless poignant, fragrant of Tom Collinses floating Maraschino cherries and cheeseburgers dripping catsup but with no onion, because of the necking to come, in some congested, clamorous pleasure palace on the great Midwestern plain.\n\nAnd when she asked \"Funny haha or funny peculiar?\" he was caught, with all the force of his past, in the iron fist of love, and would inevitably have been drawn sideways in a most funny uncomfortable position to crush the charming folly against her lips with his own\u2014had not the dentist at that moment peered ugly around her exquisite right breast and called:\n\n\"Hey, Reinhart, why don't you scout around for some hooch! That damned Lovett has some in the kitchen, I know. Do me a favor and go look.\"\n\n\"Why don't you go yourself?\" Specialist officers would accept almost anything that was simultaneously assured and good-humored.\n\n\"That lousy skinflint faggot!\" mumbled the dentist, and mixed himself again into the cushions.\n\nVery rolled her eyes. \"You were saying, when we were so r. i.?\"\n\n\"That I always wanted to be in combat, but frankly, I was too cowardly to volunteer for the infantry. What I wished would happen was that I would simply be assigned there through no voluntary act of my own. Then my conscience would have been clear, as it were.\"\n\n\"Conscience? Who lets himself in for danger unless he has to?\"\n\n\"That's it,\" he groaned. \"I'm very sensible. I _didn't_ volunteer, and I'm not sorry that I did not. My regret is that somebody else didn't make me. When I say conscience, I don't mean it bothers me now, but that it would have if I volunteered, so much so that I would probably have been killed.\"\n\n\"Obscure.\"\n\n\"Don't you get it? I would have felt I was committing suicide.\"\n\n\"Don't talk like that!\" She manipulated his hand, as if this perversion had settled in that member, and could be worked out, like a cramp. It was only too clear that he wasn't getting through, and he understood that he very likely never would. Anyway, her failure was in itself a kind of success. Having essayed this theory with others\u2014if you haven't heroism to bring to a woman, you have to lay your intentions at her feet\u2014he had tasted many times the ultimate indifference most people have to the imagination's projections, especially in the hypotheses of somebody else's morality.\n\n\"I'm sometimes embarrassed at fighting the war as a kind of Broadway press agent.\"\n\n\"Special Services are certainly necessary, or the Army wouldn't have it. Besides, think how human it is to entertain people. Think how fine it would be if each side fought with entertainers, with the victory going to whoever made most people laugh.\" It was obvious from the jolly bell in her own throat, which she now tolled, who would win. And Reinhart, with this revelation of the open secret of her force\u2014that she would always be victor, from an inability to imagine loss\u2014knew that he must have her.\n\nSo, with mock impatience, he said: \"You're not serious.\" And slid his arm around her splendid waist, as the captain's face hove into view once more, saying:\n\n\"You wanna dance?\"\n\nThree or four times, to Very's blind shoulder. When he eventually registered, she declined, and considering the situation, perhaps too rudely, Reinhart thought. To make up for which he grinned amenity at the man.\n\n\"I didn't ask _you_ ,\" the dentist groused, and ambled off in the half-bitter, half-stoical slump of a panhandler.\n\n\" _You_ wanna dance?\" she mimicked, and meant it, moving to draw Reinhart to his feet.\n\n\"How can we now, if you just refused him?\"\n\nBut she didn't, genuinely, see why not.\n\n# _CHAPTER 7_\n\nIDLY, BUT WITH GREAT care, Schild marked the room and its furnishings: solid pieces, dark; dual escutcheon lamps on the wall at various points in an academic rhythm: fireplace, for example, bracketed by a pair whose vertical members swelled like pregnant bellies to the points of the switches. A corner stove, ceramic, beige, built up of molded doughnuts of ever-diminishing circumferences, baroque welts and carvings, small black door amidships for introduction of fuel. Which was those bricks stacked neatly by.\n\n\"Compressed coal dust, very tidy,\" said Lovett, who had come up silently and followed the direction of Schild's eyes. \"These Germans are the most technically advanced people in Europe, damn them, if that's a recommendation. The throne room upstairs has a pushbutton flusher\u2014what I mean is, no chain!\" He threw himself gingerly on the very edge of a sofa cushion, giving the impression of artificial vivacity, and staring, said: \"No, I _don't_ know you\u2014you're surely new in the outfit.\"\n\n\"Look, Lieutenant, you invited me yesterday. Frankly, I wish you'd remember it.\" Schild spoke in the sharp tone of eminent reason. Yet he seldom used it for so slight a cause as this, and he wondered now at himself: whether his motive had been on Schatzi's example, or that from some hint of unconscious fear he had suddenly needed exterior proof of his identity.\n\n_\"Certainement,\"_ said Lovett quickly. \"You're the Nazi-hunter. I'm sorry. Have you met any of these lovely people?\" Upon the negative he rose, saying \"How lucky you are,\" and snatching Schild's arm, led him through the crowd to the kitchen, which was 1920's-modern, with a gas stove up on four legs, like the one everybody's mother once had, including Schild's, and a bright, yet enclosed breakfast nook sprinkled with painted rosebuds, fir trees, and goody-goody gnomes in Lederhosen and dirndls, a sovereign little house within the house. Peering inside, Schild made out the witch, a small worn person of feminine gender, smiling bereavement, expropriation, and sycophancy.\n\n\"Atrocious old bitch of a Nazi housekeeper,\" said Lovett. \"That's who.\" His attenuated index finger signaled dismissal, and the old woman trotted her carpet slippers up the back stairs to the second floor.\n\nWhen they were inside the booth Lovett produced, by an elaborate act of spontaneous creation, a full bottle of Scotch and two paper cups.\n\n\"And what are _you_?\" he asked without warning, lolling his head and transforming his eyes into little knife-cuts intended to symbolize high interest. \"I mean, what are you _really_?\"\n\nTo have reproduced exactly what Schild's mother had once asked, Lovett should have gone on: \"You are still a good boy?\" And should have been lying on a hospital bed, the white, segmented, cranked-and-rodded dais of pain, flanked by electrical nurse-alarms, half-filled vessels of water, folded cardboard sputum cups; should have worn magnifying glasses which projected eyes in terrible, bloated particularity, showing the iris as not a smooth round but rather an uneven burst of pigment threads darning into the void of the pupil, repeating silently the accusation so often voiced in earlier times of health: that their vision was lost in the pregnancy that engendered him. A queer, cruel, lifelong lie, that not until she was under ground did he, consulting the old schoolgirl snapshots, expose. Indeed, it was only by the spectacles that you could know her amid the anonymity of fifty middy blouses.\n\nCome tell me the truth, they ask, of which you are manifestly a walking denial: what crimes lie concealed behind your fa\u00e7ade, who are you to be closed when we, the rest of us, are open? And how determined they are to wonder forever, how implacable is their will to ignorance! \"I am nothing that I wished to be: chronologically, not a fireman, not a cowboy, not a gentile, philosopher, lover, nor revolutionary. But what are _your_ failures?\"\n\nNo purpose in asking that of Lovett, who was really a kind of success, who besides had wanted only a simple statement of civilian occupation, doctor, lawyer, Indian chief, against which to set his own probably rare, surely very dear calling. He saw in no pride that Lovett had chosen him, of all the crowd, for cahoots, just as the lone Negro in a company would draw to him, or the person with a lisp or one arm, the girl with the hair-lip, and it happened twice at parties for Russian war relief that he attracted the pariah of that context, the lost Republican who cornered him to trust, conspiratorially, that the aid would not strengthen communism in that forlorn country. Whatever the pariahhood, it unerringly found and clove to him: he must stink of separateness. From this final, subtlest of variations on anti-Semitism, which built its Dachau in the heart, there was no refuge, and he foresaw the day he would be assigned to infiltrate the B'nai Brith and at the first meeting be pulled aside to receive the confidence of some disguised Nazi.\n\n\"I was a teacher at a private school.\"\n\nLovett smirked triumphantly. \"In New York?\"\n\nWhere else? Schild felt himself capable of the accent of East Broadway and the Houston Street shrug, but was proved right in his restraint by Lovett's next question.\n\n\"Fashionable?\"\n\nPerhaps it was the Scotch which had sheared the falls and rises from the usually schizophrenic voice, leveled it into an even plain of clay, for what Schild heard was \"fashionable,\" and he was disinclined to believe it, even of Lovett.\n\nBut a timid knock on the back door freed him from the issue, a weak knock, but followed rapidly by an entry in the opposite character, bold, brutal, hinge-torturing.\n\n\"Oh, _why_ do they use the _yard_?\" Lovett wailed. \"We have a pretty john!\"\n\nIn a moment his despair sharpened into fright. A freckled Soviet face, mounted on tunic shoulder-boards and wearing a cap awry, poked jovially into the entrance of the dinette.\n\nAnd roared: _\"Herr Leutnant, ich bin hier. Was f\u00fcr ein Haus! Sch\u00f6n, Sch\u00f6n!\"_\n\nThe Russian was a little lieutenant of artillery, dressed in high-neck tunic, flared breeches like displaced wings, and boots. His good brown eyes searched for an object that did not elicit admiration, and failed. A line of dirt across his prominent Adam's apple showed how far he had washed. His hair had been shaved up to the temples, and obviously with a dull scissors, by himself and that very afternoon. He saluted Lovett, Schild, and the house. Saying _\"Verzeihung,\"_ he stepped to the sink and took a drink of water through his hand.\n\nLovett had met him on a black-market mission to the Kurf\u00fcrstendamm, picked him up for a souvenir, for who had ever known a _Russian_?; had written out the address\u2014who ever thought he would _find_ it? The only trouble was he only spoke _German,_ who could talk to him my _Gawd_! Finding that Schild could, Lovett sniffed in pique and vanished.\n\nIn the living room the lieutenant shook off Schild's patronage and charged the cautious company with outthrust arm, announcing \" _Leutnant_ Lichenko!\" And prevailed, pumping hands and snowing compliments, and when he had taken care of even the humblest, he turned the approbation on himself. He explained the three medals in a Venetian-blind overlap on his thin chest, the deeds of valor which they marked, and expressed curiosity that the Americans were not equipped with boots. His own, he averred, were of a superb workmanship and quality beside which the German-army issue could not dare to show its pressed-paper grain. He applied the same judgment to his tunic, breeches, belt, and cap. The latter he removed extravagantly for the ladies but replaced directly. He wore it as he and Schild studied the phonograph, which by means of a small device on either side of the turntable had knowledge of each record's duration and released new ones at the proper intervals from the stack it bore.\n\n\"Goes round, _herum,_ push button to play again, to puh-lay a-gain, pu-ush button. _Sechs_ records only, will it take. Compree? _Sechs_ records,\" said Nader, winking stupidly at Schild.\n\n\"What's that, Dwight Fiske?\" asked a thin dentist who had joined the elbow crowd. \" 'The Colonel's Tropical Bird.' There's a sex record for you, Leek!\"\n\nWho was a plump kibitzer of a nurse that threw him a dreadful smile and said, dreadfully: \"You're so-o-o-o gay.\" Pointing at Lichenko, she asked Schild, \"Suppose he'd like to dance?\" But Schild had already given her his back.\n\nLichenko took off his cap, scratched his head, inspected the fingernail, replaced the cap, and begged Leek's pardon. Throughout, his left eye, which seemingly he could work independently of the right one, was fixed towards a picture on the next wall, and his feet were casually screwing him that way. At last he was ready to go directly before it, to abrade its surface\u2014in an inconspicuous corner, so that if damage were caused none would show\u2014and to proclaim: \"A genuine oil painting. Oh, very _sch\u00f6n,_ indeed. Private property, yes? But roses in a bowl and nothing else! Where can the philosophy be in something like this?\" Did Schild know Repin? Oh, _ausgezeichnet,_ excellent, excellent!\" \"Ivan the Terrible Kills His Son.\" Bloody picture. _Angst, Angst!_ \"This we call the _Russkaya dusha,_ the Russian soul. Or did you know that already?\"\n\nWith the question, for which since he was not just making noise he wished an answer, he gave specific notice to his benevolent patron: \"You _do_ know, _das ist sehr gut:_ all these things can be useful for friendship. My German is fluent, yes? And my accent is unusually accurate. That is because I worked at it both in theory and in application. _Ach,_ it is not easy to do things the right way, but it is always possible, _ja_?, always possible, my friend.\"\n\nHe had said \"my friend\" and taken Schild's hand. Russian male friends kissed on meeting and walked hand-in-hand, yet since 1917 homosexuality had all but vanished in the Soviet Union.\n\nHis calluses torturing Schild's smooth palm, Lichenko approved: \"Your German, you know, is excellent. Was this learned in the wonderful American schools?\"\n\nOh, partly, and in part from a grandfather. Schild was pleased and apprehensive at once, the latter from questioning, any questioning.\n\n\"You are of German descent, then? Does it give you a queer feeling to return to your old motherland as an enemy? The Russian word is _rodina. Rodina_ \u2014motherland. I will teach you the Russian language in this manner, term by term, although I am Ukrainian. But Russian nowadays is more useful, yes? But you are German?\"\n\nSchild smiled lazily to let it pass, but Lichenko ripped at his fingers: \"Tell me, tell me!\"\n\nSneezes, orgasms, interrogations, their irrevocable end is ordained in their beginning.\n\n_\"Ich bin j\u00fcdisch.\"_\n\n\"I see, I see! Then it is not queer but pleasant!\" Lichenko grinned\u2014indeed, he had not stopped grinning: the ravines in his face were grin-grooves, his irregular nose was lumpy with grin, these along with his winged thighs making him a Mercury of mirth. It was, frankly, a private thing, which he could respect while not losing any skin from his own ass.\n\nAnd he was soon away to other pictures and _objets d'art,_ furniture, rugs, and the dark-blue wallpaper with its silver suggestions of flower petals dissolving in ink. He bounced into an obese chair, which bounced him halfway out. He sneaked carefully back into place, and the chair submitted to good manners. As for Schild, he sat crosslegged at its side on the floor.\n\nLichenko's German was very good, too, for he was an educated man, an engineer, in fact, although the war had caught him before he finished school. His intended specialty concerned dams and sluiceways, the diversion of streams, paradises from deserts, the transformation of the face of the earth, or anyway one-sixth of it. Nor was Soviet engineering a cultural Siberia. He slid easily from an apostrophe to steamshovels into American writing, where he was better than oriented.\n\n\"We read American books in the Soviet Union!\" he shouted happily, digging his hard heels into the floor. \"More than we do Russian. Also, more than you read in the United States. You know, of course, that American authors would starve but for the money they get from Soviet sales.\"\n\n\"I know,\" said Schild, who _did,_ at that moment; did, because fact can be countermanded by wish and hope and generosity and brotherhood, else we are lost.\n\n\"Have you read Dreiser?\" he asked.\n\n\"The greatest American writer,\" replied Lichenko. \"But Upton Sinclair Lewis is _sch\u00f6n,_ too, and Jack London's _Babbitt_ and _The Iron Heel._ \"\n\n\"I met Dreiser once.\"\n\n_\"Also\"_ Lichenko mumbled, removing his cap and testing his forehead with a sweaty palm.\n\n\"He spoke at the school where I taught. A great, majestic man, a champion of humanity, and a friend of the Soviet Union, as I suppose you know.\"\n\n_\"Es interessiert mich das zu wissen,\"_ said Lichenko vaguely, but hopped to his feet positively, and bowed. For there stood Leek.\n\n\"Now, you can't keep our ally out of circulation!\" she chided, and led Lichenko to the dancing area\u2014though not before Schild was constrained to translate some small-talk, including a Soviet tribute to womanhood in an ornate German that englished as something Albert might have said to Victoria; nor before he volunteered to hold the doffed but troublesome cap, Lichenko having been at a loss for a cache where that article would be neither crushed nor stolen.\n\nA universal sense of fun could not be withstood. The troublesome shortage of fuel reached Lichenko's apparatus, without benefit of Schild, and he laughed long and loud, tore himself from the Siamese coupling with Leek, and shot into the kitchen and out the back door. He reappeared with a shoulder sack of bottles: vodka, schnapps, whisky, and other fluids.\n\nGlasses flowed, music tinned, everybody danced. Reinhart, Schild saw, glided about as if on figure skates with the large nurse, in the perfect attitude of the adolescent sexual captive: closed eyes, back arched _affetuoso._ A more direct confirmation of the reputed egalitarianism of medical units could not be imagined.\n\nSchild thought about Lichenko, who had so little and so much. Throw the switch of the time machine and there he was, with his cartload of firewood, ankle-deep in mire, on his weary animal way to the sod hut, the black bread, and the cabbage; the trashy icons; the spent wife; the ravished, if pretty, the prematurely aged, if plain, daughter; swearing so vilely that if his master had the ill fortune to canter past, he might have got another crop's end for his lifelong collection of abuse. Born old, senile at twenty, dead at forty, without ever having passed through the human. A lump of dung, hating love and beauty and intelligence because he was defined by their absence. So, gorge the rotten potatoes, let the grease befoul the lips and drip from the chin, fill the gut with the stolen bottle, and when it explodes in the head, give the wife a fist to her decaying teeth and the daughter a hand between her thighs, because you are beyond judgment, beyond hope. But not beyond history, which moves not for revenge or profit or virtue, but for the negation of negation, the arrangement of disorder, above all, for an end to waste.\n\nThese things mature without ever having been formally born. Lichenko was, of a sudden, disorderly. Only a moment earlier he had displayed high, good, and legitimate spirits; now he had enrolled in that brotherhood of savage peoples from whom firewater should be, and often is, legally withheld. His maw sagged; his eye carmined; he drank as if, oblivious to the torrent that washed his face and gummed his tunic, he hungered for glass, and proved it by incising a piece of the tumbler's rim. A pencil line of blood traced out the groove of his chin, like the after-punch make-up of movie martyr-priests who may invite their adversary to put on the gloves but never return the favor _ad hoc._\n\nAmidst all those medics his wound went unattended, on the theory that if he lost enough blood he might collapse peaceably. But he had just begun to play. After checking Leek in an armchair, he detonated into the frenzy of a solo jackknife dance. The floor quaked, for although he was small, the boards were simpatico with his rhythm; the rolled carpet against the far wall, which was not, had its long, heavy belly rent by a boot-heel.\n\nWith such sport, with Lovett impotently aflutter, with Nader enveloping himself in the phonograph wires, with the awed company's disengagement, the room was progressively demolished. Lichenko's success with the carpet sent him to Leek's chair, which, after removing her, he ruined with a single jump, hard heels forward; he smashed the mantlepiece mirror, pelted coal cakes in black bursts against a carved lowboy, got a painting once out of six throws with as many tumblers, and shouted _\"Bezbol!\"_ Then, to Schild, he said: \"Fascists.\"\n\n\"Yes,\" answered Schild.\n\n\"Not the Americans.\"\n\n\"No.\"\n\n\"The Germans. Why should this house be spared? If you saw what they did in my country...\" He sank wearily into the chair and passed a hand across his face. He signaled to Leek, pointing at his lap. She sat there submissively, bovinely, as he read her idly in Braille.\n\nStill on the floor, Schild shifted his weight from left to right ham and adjusted his pistol belt.\n\n\"Will you permit me to see your weapon?\" asked Lichenko. He ejected Leek. He looked feverish as he palmed the Colt. A vein in his forehead pumped into prominence. He worked the action, stroked the barrel, warmed it with his cheek, and peeped down the muzzle. He found the clip, extracted it, loosened a bullet, felt its slug, replaced the bullet, replaced the clip, pointed the piece at the still proximate Leek, went \"Boom, boom!\" or thereabouts in Russian vowels, finger on trigger guard.\n\nIgnoring her squeak, squeaking himself in delight, he took out his own pistol and thrust it butt first at Schild.\n\n\"We'll exchange. Then we'll each have a souvenir of a time we will never forget.\"\n\nIt was what the Russians called a _Nagan,_ a cheaply made cap-gun affair, but Schild accepted it reverently, and, though he detested firearms, gave an earnest imitation of Lichenko's ecstasy over the Colt.\n\n\"You _agree_? You will _exchange_?\" shouted Lichenko. \"My dear friend, I salute you!\"\n\nOn his feet, he highballed with the right hand, a dwarf against the bas relief of the large Americans who had started to surge genially upon him through the dismembered furniture. Schild was rising in honor of the moment, at the very least to return the salute, when Lichenko began to discharge the .45. He perhaps intended to squeeze off only one cartridge as an additional salute to fortune, but the kick from the one convulsed his hand into another, and he was drawn _nolens volens_ into a full tribute.\n\nBelow the scored ceiling and within the vermiculated walls, Lichenko, now-spent gun drooping, sniffed the atmosphere of powder, appeared about to sneeze, did not, and pushed a reproachful lower lip like a coal chute at Schild.\n\n\"Ah, my friend, this American pistol!\"\n\nHe tore at his choker collar. \"Excuse me, Vasya is ill...\" And fell prone into the plaster, which, of course, was now the common ground.\n\n\"You win the medal,\" said Lieutenant Leek, an unjolly snowman, to Schild.\n\nHe blew clean his glasses and inquired silently with blurred vision.\n\n\"For Number One Horse's Ass, Berlin District.\"\n\nNader touched Schild's elbow and whispered: \"Look, Jack, Lovett went for the MPs. Get your ass-hole buddy out of here. We'll con them. He was just drunk.\" He knelt beside Lichenko and fingered back an eyelid, peered at the red orb. \"Drunk as a skunk.\" He ordered Corporal Reinhart to lend a hand.\n\nReinhart lifted Lichenko by the belt and pulled him over a shoulder.\n\n\"Veronica, why don't you go along?\" asked Nader. \"Jesus, maybe he's dead.\"\n\n\"O.K. Thanks for the party,\" said she. \"I had a lovely time anyhow.\"\n\nReinhart navigated through the rear door and into the garden. Already military-police sirens sounded in the distance. Through the back yards, having trouble at every sonbitching fence, and around the block to Schild's house was their silent way. Reinhart's calves were tired on the stairs, but he gripped the banister and made it in good shape to the room, where as his burden was lowered to the bed it came to life briefly, displaying a revolving eye, and returned to dreamland with a mouthful of bedspread.\n\nVeronica examined the body from a distance, found it hale. Reinhart said: \"You're a good fellow, sir, to do this for that Russian. If his army found out about it he'd be headed for Siberia.\" And Very seconded that.\n\n\"I think,\" said Schild, \"the American Army is what we have to worry about at the moment.\" He took a tiny package from his pocket. \"Can you use these?\"\n\nAnybody who ever opened a K ration had Fleetwood cigarettes to dispose of, yet Reinhart was sure these were his own come home. He was too weary, the evening had been too extravagant, to inquire by what route.\n\nNew relations consisting so fiercely in the precise time of day and the specific mise en sc\u00e8ne, the sudden dislocation of these threw both Very and Reinhart into a diffidence, especially now as, their task ended, they went out of step down the sidewalk to Very's house. From the side of his eye Reinhart could see her shoulder bag swinging off the divisions of silence. An occasional officer or nurse, not breaking the peace, breathed past them in the frenetic diaspora from the party. Down the street, its siren dying like a throttled pussycat, another MP jeep arrived.\n\nAt her door, Reinhart chickened out of trying for a good-night kiss, perhaps with a view towards establishing his independence, which, in the pale simulacrum of _post coitum tristis_ that was his after-party letdown, he felt had been compromised. Or perhaps it was a defense against the progressive frigidity she gave off as they approached the front step.\n\n\"Oh this is where you live?\" he asked numbly.\n\n\"Haha! Were you going to charge me with breaking and entering?\"\n\nThis cruel parody of his own earlier fantasies on the mansion of her person, despite the false laughter\u2014whose spuriousness was advertised by its miraculous lack of resonance; it was as if a great bell rang so shallowly that nothing trembled\u2014suddenly elicited his overdue response to Lichenko's rampage. To assault an entity of order, to register a spontaneous nay against the sanctioned and authorized; mean, but it were meaner than never to be so moved.\n\nBut Very skipped insouciantly inside the doorway. Shortly, her other end appeared, saying: \"See you in the funny papers.\"\n\n# _CHAPTER 8_\n\nCAPTAIN ST. GEORGE'S SURPRISE WAS limited. He had rapped at and opened Schild's door on his regular schedule of unnecessary morning information\u2014\"The bath is free\"\u2014and looked upon a scrawny, alien fundament. Lichenko, rump to door, was bent in a study of his big toe.\n\nSchild's guest had awakened with a refusal to recognize his benefactor. Something Schild however took in good grace and did not sully with a word as the Russian scratched an elusive cap-a-pie itch, lip-farted at his own image in the mirror, and spat a long drizzle of saliva out the window in droll reproof of the sunlight that made him wince. Then he turned, said with ill humor: _\"Da\"_ and undressed for an examination of his pelt.\n\nSt. George recovered with expedition. \"I thought you had a woman in here,\" he chortled to Schild, who was elbow-propped on the floorbound blanket that had been his bed.\n\nArising with the aid of the dresser corner, khaki undershirt and shorts clinging to him like old crepe paper, sallow, hairy, shivering\u2014Berlin's air in the shadows stayed cold till noon\u2014Schild said as St. George averted his eyes: \"So what else is new?\"\n\nHe watched the dull pain fill the captain's eyelids, distend his cheeks, sag the loose mouth, and lower the chin, going down like mercury in a chill. It was a kind of crying just beneath the epidermis; years of it had made his face one big bag.\n\nSt. George addressed himself to the oval mirror on the dresser. \"Do you know?\" His ebullience had left. \"Is it necessary to have an electric shaver honed every so often? Mine is beginning to pull.\" He held the device towards Schild, retaining the pillbox affair which made any current American.\n\n\"Duncroft,\" he went on, \"says it can't be done. He says the razor companies aren't going to sharpen old ones and thereby put themselves out of business. They're going to want to sell you a new one, he says. But he's always cynical.\"\n\nLichenko left off his big toe and went over the others, as if in count; midway through the left foot, he lost his sum and began again.\n\n\"Did you see a strange man as you entered this room?\" asked Schild.\n\n\"A refugee from the party, I take it.\" St. George twisted the top of a battered tin of GI foot powder, sifted a quantity into his palm, and traced rhomboids with his fingertip.\n\nLater, as he and Schild crossed the street towards breakfast at the 1209th officers' mess, to which St. George had got their small unit attached for rations, he said: \"Strange how when you meet a man who's naked you don't get to know much about him. Maybe we all rely too much on externals, but that's the way it goes.\"\n\nAt the mess Schild was daily juxtaposed with many of the to him anonymous faces of the party, and occasionally received a curious but cordial nod. He was also aware, from time to time, of the regard of multiple eyes and the drone of comment at adjoining tables as he forked in his Harvard beets and masticated the grainy roast, but was conscious of no ill will, nor, in his sense of the word, suspicion. Once he saw Lovett across the tent, fragmenting bread with neurasthenic fingers and spotting the pieces individually over his tray, which could also be interpreted as evidence of placidity.\n\nBut after one lunch, Nader met him outside.\n\n\"Did you pack off the Russky? Dewey's going to make trouble, mark my words, because the Old Man's about to hang it on him.\"\n\n\"I'll make it good,\" said Schild. \"It was my fault. I gave him the gun.\"\n\n\"Well by Jesus we'll put in a complaint through the Kommandatura,\" Nader said. \"Just write down his name and outfit. Did you get the outfit?\" He rummaged in his pockets for writing materials.\n\nBoth extremities of Nader's pencil bore chewmarks. His paper was the reverse of a snapshot carrying the pale-violet emblem of an Oklahoma developer. Schild wrote, slovenly: \"Lt. Krylenko, Engrs Corps, Red Army.\"\n\n\"I don't have much hope this will find him.\" He smiled commiseration. \"But you won't have to. Just let me know the amount of the damage, and I'll see you get reimbursed in full. I'll see your C.O. myself this afternoon. You'll be fully cleared.\"\n\n\"Now,\" said Nader, \"I know you're oke, but you'll do best by us if you crap out early. What I mean is\u2014\" He broke off until the pencil was reseated in his breast pocket and the flap buttoned. \"Let me as somebody who was soldiering when you, Dewey Lovett, and anybody else in this road show was still sucking titty. Know the Old Man since Jesus was a PFC: hates anything involved. We'll never find the Russian, but the thing will be wrapped up and Dewey will stop pissing and moaning.\"\n\n\"I thought it was simple decency to get him out of your place ahead of the MPs,\" said Schild. \"That officer fought all the way from Stalingrad to Berlin. It would be outrageous if his first defeat came from blowing out some German plaster.\"\n\n\"As long as we're off the hook, old buddy, that's all, as long as we've got something to hand them. The phony name you wrote down here will do as well as another.\" He showed his chipped teeth in amiable pride. \"What time I have left from being personnel officer, unit censor, permanent O.D., fire officer, postal officer, and post-exchange officer, I put to provost marshaling. You'd never make a successful crook.\" Nader's hairy wrist disposed of all possible demurrers, embarrassments, confusions. \"Hell, I remember his name, I just don't know to spell it.\"\n\nExposure of his minor deception freed Schild to submit the whole affair to a lens, under which he saw: being caught out by such a man and in such a way was not a demerit.\n\nNader pushed the snapshot at him. \"You still didn't look at the picture on the other side.\"\n\nA ripe piece of girl in her late teens sat on the first step of a rotting porch. In the background depended the rusty chains of an old slat swing; through one link, the shaft of a raffia fan. As she cuddled a stuffed panda, her deep lower lip sat in transitory melancholia, or what may have been counterfeit desire. Pleats of voile skirt fell gradually away at the thigh on the viewer's side, as if at the border of some proscenium arch through which shortly a young man with haircut, black suit, and cascaded necktie would heel-and-toe as commencement valedictorian. She was relaxed or prepared to spring, according to which evidence had more weight: the tension of the femoral muscles or the flaccid pubis in the valley that knew no drawers.\n\nThis was an exercise to put Schild on his mettle. How to communicate in the proper measure somewhere between coarseness and patronage. For Nader, in the classic manner of such picture-passers, watched him as a bank guard observes an unshaven man; that self of his which dwelt in Nader's mind was about to acquire a habitation and a name. Schild suddenly ached with regret that he could not miraculously reduce his size and quality to the mode of the image and plunge into that seedy, sweaty, alien world of desiccated lumber, rusty metal, the treasures of shooting galleries, the failures of fabric, unshaven armpits, sagging wash-lines, off-stage radios, that universe of the enervated Sunday, and make animal love to the girl. Not having that option, he grinned wryly and said: \"Choice\"; watching it join the community in the wallet. \"I believe the idea now is to ask who is she?\"\n\n\"You don't know any less than I do,\" Nader replied. \"This was found behind a chair after an enlisted men's dance in the service club at Camp Grant, Ill. I carry it for laughs.\"\n\nSchild returned to the tent for another cup of coffee, the dregs of which he hung over until the last eater had departed and it was seemly to draw and transport Lichenko's rations. He did not know why his guest remained, but it could hardly have been for \"his big brown eyes.\" A phrase from his father's code, applied to those business associates whose sudden appearance of friendship logic and experience exposed as conspiracy. A \"friend\" dropping into the office for a smoke was of course a spy who mentally photographed a new button and, within half an hour after his departure, set up machinery to reproduce the plagiarism in quantity, to steal the orders of the first party and libel _him_ to the stolen customers as a thief to confound.\n\nHe carried the laden tray towards the billet. It could hardly have been for his big brown eyes that Lichenko lingered. This old-Jew's suspicion would not be put down, despite his violent attempts at negation that, as he crossed the street, became exterior, the ultimate ineffectuality. He shook his whole body as if in a chill, and the spinach in the end compartment, having by nature no integrity, easily lost its coherence and slipped over the rim like a string of mucus.\n\nLichenko could be on furlough, on extended pass, on some perfectly uncomplicated special duty from which he had legally or illegally gleaned four days of liberty; he may have been lost, have searched in vain the wilderness of Berlin for his unit, temporarily have given up. He was perhaps a liaison man between Red and American Intelligences, who\u2014the Soviets being no fools in these matters\u2014could better prosecute his purpose by four or five days' discretion, especially around an idiot like St. George, who, it could quickly be seen by a shrewd fellow, would fly all to pieces at the first suspicion that his outfit was to have a serious role. There were, too, the possibilities of amnesia, outright absence without leave... and even, of course, desertion. Notwithstanding Schild's automatic rejection of anti-Soviet messhall gossip that \"hundreds\" of Russian soldiers had decamped to the Allied sectors, which if true would have been a lie, he was certain that it _was_ true in cases. It was something that could be faced without equivocation. Renegades ye will always have with you.\n\nIn the middle of the street, a two-and-a-half ton truck nearly ran him down, the driver leaning out to carp, spotting the silver bar, recovering. A fine midsummer sun crafted suburban shadows which lay only slightly to the northeast of their objects, the time standing not far beyond one o'clock; that extreme portion of the sky that to the grounded seems at last palpable but to the winged is merely the middle distance towards another intangibility, hung unusually high even for Berlin. A soldier in fatigues lingered on the stair of one of the officers' homes, in the grip of an internal monologue that he broke off to inspect Schild with academic superiority, which indicated he was on an errand for a captain or above. Shortly such a person, wearing a khaki undershirt, appeared at the door and bellowed: \"Bugger off, Wilbur!\" As Wilbur without acknowledgment merged with the shrubbery, this undershirt shouted: \"Bugger you, too, Rosenthal!\"\n\nOne thing was certain: Schild's eyes did not improve over the years.\n\n\"Can't you see? It's Young! ...Oh, I'm sorry, I thought you were Rosenthal. Rosenthal's a DOCTOR IN OUR OUTFIT.\"\n\nCollege Joe type with a grin. Back to bed, my boy, the world will run very well without you, or very badly; in any case, without you. An angular girl, with hairy legs, pumped decorously past on her bicycle. It was Schatzi in transvestite disguise\u2014of course it was not, but Schild, too, could be permitted an error of identification. In the past three days he had seen Schatzi in every bony face; he had recognized him falsely with greater assurance than he had yet seen him in actuality. It was only because the genuine article never appeared in daylight that the apparitions could be ignored.\n\nThe real Schatzi\u2014he had left him an hour before Lichenko had come to the party. Certainly if he, Schild, were Schatzi, he would not fail to trace a connection between these events, to draw up the disjunctive proposition so favored by his courier: either... or; either Schatzi knew of Lichenko or he did not. If he knew, there could be no doubt that what had happened was with his connivance. If he did not, all the rules commanded that he be told. But this clause was the emergency measure, necessary now because he had ignored the first principle of the code: never to get into such a situation. Thus to obey the second was to admit a transgression of the first, and Schatzi already suspected him\u2014or pretended to; in practical effects there was no important difference between reality and appearance\u2014of mishandling his contact with the 1209th, either foolishly or from a motive of treason.\n\nIf, on the other hand, Lichenko had been planted on him\u2014had _they_ nothing better to do than keep him under surveillance? It was a preposterous idea. Still, since that first morning Lichenko had been sullen and unresponsive, lurking in the bathroom when Schild was home and going through his belongings when he was away: a pair of OD socks left separate in the footlocker tray had been united in a neat ball some time between breakfast and lunch. Fortunately, Schild had long been in the habit of destroying his letters\u2014not that he received many; he had luckily cut off from Waslow when he went underground, Waslow who was not long afterwards expelled as an infantile leftist when he resisted the change of line from hard to soft vis-\u00e0-vis the bourgeois democracies; but he occasionally got communications from his sister, who typically had not only again changed husbands but again swapped gods, with the end of the war conceiving a perverse attraction towards the doctrine of Jung, whom she suggested Nathan visit, as long as he was stationed in Germany. Jung, the anti-Semite.\n\nAnd then Lichenko's queer behavior over the chessboard. His visible emotions while playing could only be called ferocious; he groaned cavernously at momentary setbacks, howled at each little triumph, and upon the general victory\u2014which he was never long in gaining, for Schild was not only an inferior player at best but would have been almost afraid in these circumstances to be a good one\u2014Lichenko became most invidious, arrogantly shoving the board across the table like a dirty cafeteria plate and rising to swagger about the room on hard heels.\n\nLichenko's larger game was surely something more than chess, and unpleasant as it was to think that in this, too, they were adversaries, to that degree the mind would not accept another possibility. As to the heart: it could not endure a second enemy among the two men with whom he held a common purpose. Whatever Lichenko's menace, Schild forgave him for it.\n\nWhy should a citizen of the United States of America be a Communist?, thought Lichenko, all itchy again, a quarter-hour after his fourth bath in as many days. He felt large lice loping on his back. Off came the tunic. Spine presented to mirror. Not a beast in view. Imagination. They would leave that final place when he next wore a civilian shirt, even a dirty civilian shirt, even a lice-infested civilian shirt. Did the old holy men really wear hairshirts? What then was their lie? Surely a truth was what you gave for it. Yet everyone, and from what he could see, particularly the big-spenders of belief, had their lie. Believe in very little, said his mother, and your disappointments will be as small. This had seemed funny to him when he was ten but had grown more grave with age. Old people know more than they can tell directly. His mother then had not been old in years, but some people are born old. He had seen many a baby of whom, if you squinted your eyes, you could get a picture as an old man with cap and pipe, taking the sun in the park.\n\nAs he returned from the bathroom, the German woman moved correctly down the hall, as if on little wheels. Sluts walk so, being so large between the legs that their organs would fall out if they took long steps. You see! he grinned silently at her back, there's no need to be so grand! Next time you pass, Vasya's fingers will pinch your bottom!\n\nHe had never, in his belly, believed in the existence of foreign Communists\u2014Communists where the Bolsheviks were not in power? No sense to that. Besides, foreign comrades were not taken seriously even by Soviet Party members, as he knew from his brother. The largest Party outside the USSR, the German C.P., had been puffed out like a match when Hitler arrived. And as to the Americans, hahaha! Who already owning an automobile, a ten-room apartment, a motion-picture projector, short-wave radio, and probably an airplane, became a Communist? Lichenko knew so much about America, had had so many fantasies about it, he oftentimes forgot that he had never been there and rather owed his data to the Soviet news agency's New York correspondent, whose dispatches he of course translated in reverse. Thus: the American worker lived like an emperor, and there was no U.S. Communist Party.\n\nSince moving into Schild's billet Lichenko suspected he had been wrong about the latter. After all, the Bolsheviks had not always held power in his own country; everything started somewhere; if necessary, before one's own birth. The old czars, he believed despite his mother's testimony to the contrary, had not been first-rate people. The last one, he understood despite the Bolsheviks' like opinion, had no culture and was ruled by a woman herself the instrument of a corrupt monk. Therefore the Communists: who had begun as a small, weak band of, he supposed, idealists and martyrs\u2014except that Stalin, even that early, committed armed robbery for the furtherance of _his_ ideals; and no sooner had they kicked out the czar and won the Revolution than Lenin and Trotsky slaughtered the Kronstadt sailors who had helped them.\n\nPerhaps there could be American Communists, for Nathan Schild seemed to be one: who else would consistently praise the Soviet Union while finding fault with his own country? A normal man bragged of his motherland even if he detested its superstructure, as did Lichenko, because there was a personal pride that took no account of politics. And some of the things Schild claimed to believe: that the Moscow treason trials were genuine trials and concerned with real treason\u2014he was either a lunatic or a Communist.\n\nMore likely, both. For what Lichenko would never believe was that a gentle, generous, sweet man like Nathan could, in his right mind, give allegiance to a pack of murderers. On the counsel of his affection for him, then\u2014the heart does not lie\u2014he did not abandon his plan to defect to the West, but added to it a finer purpose: he would also save Nathan. It would be a finer game now, with rewards or disasters of greater magnitude, but the very irony of his situation\u2014leave it to Vasya to choose as cover the one Communist in a division of Americans!\u2014contributed to his courage.\n\nBack in the room, he thought he might permit himself another tubbing. Immersed, he could cogitate better than in the liberty of the bedroom. He still had no concrete plan. Time grew no longer. The NKVD would have had his name for three days; perhaps they had already traced him as far as the house party. And as yet he had not found the propitious moment to begin his labor of truth and love with Nathan. The trouble was, these considerations made for anxiety, which was assuaged only in the bathtub's warm wet trough.\n\nBut he could not go now. There, he saw from behind the curtains, came Nathan with lunch, and an excellent lunch it was, although Nathan never gave it any importance. Indifference to the material conditions of life must be unique with American Communists. Certainly it was unknown to the Russian Party! This handsome house, for example, which Nathan treated as if it were a pigsty.\n\nLichenko knelt and worked out a cigarette butt embedded in a bedside circle of rug mangy with other burns. The Red Army destroyed many things but nothing that could be put to use. However, reason was a crime for which no American would ever be shot. Was it a matter of distance? Three thousand miles away. You could talk all you wanted about the universal force of gravity, the iron ball and the feather dropped from the same height hitting the ground together. Just try it: by the time the feather comes to rest the iron will be a ball of rust. So with an elephant and an ant. Density, not volume and weight. So with an American; try as you may to drop him, it will be a launching. Lichenko had been a mediocre student of physics in the Kharkov technical school and insensible at the time to its multifold uses.\n\n# _CHAPTER 9_\n\nON HIS WAY TO WORK the morning after the party, Reinhart strolled down Very's way. An irregular blob of olive-drab descending her porch was soon fashioned by his eyes into her form, but as it came towards him on the sidewalk he saw it was not Very but her antithesis: the lieutenant who took in drunken Russians.\n\nHe was rather shorter than the evening before and indefinably seedy, with dust on his glasses; yet he had a more assured address, hard and bright. He was the kind of Jew before whom Reinhart felt very vulnerable, as if somewhere back he had done him a dirtiness which he, himself, did not remember but the Jew never forgot. He felt this while knowing it was not true, for not only had he not done them wrong: he had never done them anything one way or the other. None of his best friends were Jews. The species was unknown in his home town, which had no foreigners\u2014just another reason for its unspeakable dreariness. At college there were some, who had their own fraternity and seemed to go around en bloc, occasionally sitting next to one in classes, where they were usually witty and always clever; and some girls as well, who were either remarkably beautiful or characteristically ugly, never plain, and it was a pity the lovely ones were off-limits\u2014there had been a girl, forever enrolled on his list of classics, with sable hair, alabaster nose, cheeks of white iris, and an exquisite name, Esther Rosewater, which he used to say underbreath when she passed oblivious, _Esther Rosewater, how I love you, Esther Rosewater;_ she made him weak in the knees, and never knew it. For that was the other thing about Jews; when they weren't eying you with suspicion, they never saw you at all.\n\nAs to this lieutenant, Reinhart thought: I could break him in two. At the same time, he was vaguely afraid of him.\n\nBadly returning Reinhart's salute\u2014his fingertips not quite making it to the inferior rim of his spectacles\u2014the lieutenant referred briefly to their mission of the night before. He had found upon awakening that Miss Leary had dropped a comb in his rooms, and he had just returned it. Palpably of small value but it was her property and women care about such things, don't they?, smiling in the condescending conspiracy of the males. He could have been lying. Reinhart, who was unusually observant, remembered no loss. Yet losses remembered are hardly losses; moreover, an officer, unlike a noncom, had little reason to dissemble in courting a nurse.\n\n\"Was Miss Leary in?\"\n\n\"No, I left it with her roommate.\"\n\n\"What's her name, by the way?\"\n\nBut Schild didn't know and cared enough only to ask: \"Don't you know? Isn't Lieutenant Leary your girl?\"\n\nReinhart had a tendency to toss the ball to his superiors, to tell an excess of truth that would confront them with the damning fact of their authority. When he said sorrowfully \"How can she be?\" the lieutenant's response confirmed him. He, the officer, showed not only understanding but sympathy.\n\n\" _I_ have no objections, certainly.\"\n\nNow it was his apparent approbation that made Reinhart uneasy. He would have preferred to leave while he was ahead, but the lieutenant hung on, walking with him towards the administration building.\n\n\"The Russian\u2014did he recover all right? He was a crazy little fellow. Sometimes I think all Russians are mad, or is that Communism in action? Have you seen what they did in Wannsee?\"\n\nHe fancied that with his first word the lieutenant had shot an angry look: of course, one's big mouth had not considered that he might be a Russian Jew. Then, too, he had earlier observed that any mention of Russians not obvious praise never sat well with \"liberals,\" and he would have bet his duffel bag, with all its souvenirs, that his companion belonged to that breed. He had, therefore, found his weakness; he no longer felt gauche; he could not help falling before the temptation.\n\n\"No one who hasn't seen them would believe what a bunch of dirty tramps the Russians are. When we came in on the autobahn and met that crew, we thought first they were slave laborers for the Germans, and then service forces, maybe. But no, they were the cream of the combat troops.\"\n\nHe saw pure hate through the lieutenant's glasses\u2014or was it agony?\u2014the eyes were all watery.\n\nThe hell with him. He was not an officer in the 1209th, and you couldn't be court-martialed for an honest description of what you, and no doubt he as well, had seen. Everyone had his own chauvinism, the sacred affiliation that he would not suffer to be questioned, let alone criticized. And how disgustingly stupid, for, in this case, was it not their very uncouthness that made the Russians' victory all the more remarkable?\n\nSo he said something to that effect, but even then the lieutenant's manner did not improve, and since by that time they had arrived in the front hall of headquarters, they parted coolly, no salute being necessary under a roof.\n\n\"Goot morning, a very nice day ve are hoffing!\"\n\nTrudchen sat blooming behind Pound's big, messy desk against the forward wall, except that it was not messy but rather a place of truly stacked papers, dustless, and with a little bouquet of yellow pansies in a jam jar. On his own desk, similarly impeccable, was a pink rose. She was already flying her own colors.\n\n\"You are surprised, yes?\"\n\nRight, but his habit was never to show it. He thought, for the first time, that she might be uncomfortable to have around.\n\nShe arose and came towards him, the thick sweater, unbuttoned, swaying in its two parts equivalent to the braids.\n\n\"You see, I work for no payment until the opplication is officially opproved. But I also cannot eat at the mess until that time. Perhaps you can bring me somesing at lunchtime.\"\n\nReinhart tucked his cap under the belt and drifted into his chair.\n\n\"What age did you put down?\"\n\n\"Eighteen.\"\n\n\"And they believed it?\"\n\n\"Oh, vy not. It is only two years a lie!\"\n\nSixteen\u2014even those tender years seemed too many, but they did put her under the wire. Through her sweater halves he saw soft little breasts, very round, under the crocheted shirt. She was the kind of girl who in a movie would be asked by the hero, do you really need those glasses? No, she would say and fling them away forever. But Reinhart rather liked spectacles on a pretty girl; they were vulnerable-making, sexy.\n\n\"Let's see now, what can we find for you to do?\" He fished through the desk drawers, coming first upon the last letter from Di, which when he had put it away yesterday, having finished the answer, was open, with its envelope paperclipped to the back. Now the former was inserted in the latter, as if it had just arrived; for a moment, until he saw the slit in the envelope top, he thought it had: the outside of all her letters looked the same, with \"Mrs. Ernest Cooley\" in bright-purple ink in the return-address space. Ah well, Trudchen had made it neat, which reminded him to write the customary \"Ans.\" and the date on the face of the envelope. He reached for the fountain pen habitually kept in the righthand corner of the central drawer, and felt nothing. Nor was it elsewhere in the desk or in his pocket, and Trudchen had not seen it when she policed up.\n\nThe loss was serious. What with the black market, the PX stock of pens was exhausted, and it was not seemly to sign correspondence with a pencil. Reinhart felt an ill mood come down over him like a sack. The worst thing was that he could not, with depressed senses, find any work for Trudchen. The map of Berlin, on which she could have been employed to trace a route for the tour of the Nazi ruins, had also vanished. And he dreaded the coming of Pound, whom he had not told of their new employee, for the excellent reason that he himself had not believed she would be hired.\n\nAs if his nerves had created him prematurely, for it was only eleven o'clock, he heard Pound's footsteps in the hall.\n\n\"Quick!\" he said to Trudchen, \"start straightening out those boxes.\" He pointed, without looking, to the chaos in front of the closet, and grabbing a fistful of papers from his now-tidy \"out\" basket, fell on them with knitted brow and deliberative forefinger.\n\nPound sounded two feet from the door when Reinhart realized that Trudchen had moved not to the ordered task but rather closer to him. She had removed her sweater and was flexing her arms in a most provocative, catlike manner, her pink shirt everywhere in undulation.\n\n\"What are you doing!\" he said furiously.\n\n\"But you see, already I have arranged those boxes this morning before you arrived.\"\n\nHow irrefutably true, now that the eyes were turned in that direction: rank on rank, they pyramided almost to the ceiling, with not a loose paper showing, not a cartonflap awry. Impossible that one small girl could have done all that in a week, but there they were.\n\nAnd here also was\u2014not Pound. The liberal lieutenant, with an ingratiating smile, stood in the doorway.\n\n\"Too bad I didn't know when we came in that you were the fellow,\" he said.\n\n\"For what?\" Reinhart stood up.\n\n\"Yes, and here they are.\" The lieutenant pulled a box from the left slope of the pyramid, weakening the whole organization so that if Trudchen had not sprung to the gap the work of her morning would have been at naught.\n\n\"You must replace that at once!\" she shouted, and the lieutenant, walking from the pile with his box, showed her a look he might have given some vermin too ripe to crush.\n\nChrist, didn't he even know the simple principles of stress and strain? thought Reinhart, whose height permitted him to get the topmost carton and fill the hole.\n\n\"Okay, this one is small enough to carry with me now. I'll send a detail over for the rest.\"\n\nHe was halfway to the door when Reinhart, standing high and wide, blocked the route.\n\n\"I'm afraid, _sir,_ that you'll have to tell me what this is all about.\" He weighted the title with deliberate provocation\u2014for one thing, because he was wholly in the right; for another, to break the officer's damnable insolence.\n\nFor a moment, and for all his natural, seedy weakness and his fake amenity, the lieutenant's eyes were hostile.\n\n\"Get out of my way\u2014\" This at once calm, masterful, and most persuasive, and Reinhart would have complied had not Trudchen rushed up desperately to add her small person to the barrier. Not even the lieutenant could resist this preposterous event. He smiled, albeit in somewhat ill grace, and set his box on the floor.\n\n\"Schild is my name, Army Intelligence. Would you like a receipt?\"\n\nNo wonder now at his sang-froid. Army Intelligence! The very title had a splendid, piercing authority, far grander, because including brains, than even the paratroops, Rangers, or fighter squadrons: keen, intrepid operators in the very camp of the enemy, many-faced, anonymous; if caught, standing before the wall with a contemptuous smirk towards the rifles; if successful, only the gratification of knowing oneself supreme; no vulgar show, whatever medals were due must wait perhaps ten years hence, and perhaps not even then, for the secrets of the bureau can never be revealed.\n\n\"I'm sorry,\" said Reinhart with a mouth of contrition. \"You see, I didn't understand. I couldn't just let anyone take these things in the absence of Lieutenant Pound. He's in charge here. Actually I don't care about the stuff at all, and neither does he. What is it, or are we allowed to ask?\"\n\nIntelligence. No sooner had he got in the medics on his own request than Reinhart sought to escape. It was humiliating to be the one kind of soldier denied a gun. Intelligence. He even knew German, or enough for a start anyway, the rest he could pick up quickly in a training program. Psychologically he had probably all his life been a kind of undercover agent. In high school he used to follow certain girls in their Friday-evening walks, trail them from nine to midnight, at a distance, in and out of candy-store doorways, and, with the aid of evergreen bushes, right to their front steps, all unbeknownst to them, sometimes forever, sometimes only until the next morning's study-hall revelation. Intelligence. Its operations turned out to be very secret indeed; in three years of service he had never so much as learned where to apply.\n\n\"Just routine correspondence of some German agency, I should think,\" said Lieutenant Schild, staring grimly at Trudchen, who kept leaning into Reinhart and kept getting pushed away. \"It's pretty tedious to have to go through it, but we must.\"\n\nTrudchen leaned against Reinhart again and said, with great, solemn lashes, \"Anyvay, ve should vait for Lieutenant Pound!\"\n\n\"Truchen, I want you to do something for me. Go over and sit in that chair behind my desk and _schweigen Sie._ \"\n\n\"Vy so _formal_?\" she asked pouting, but did it.\n\nJust as he had hoped, Schild's interest was caught.\n\n\"You are fluent?\"\n\n\"Not really, but I have enough for a good base. I'd like to have an opportunity to brush up my German.\"\n\nSchild leaned close and said in an undertone, jerking his thumb towards Trudchen: \"Where did you get that little tot?\"\n\nFor a moment Reinhart thought: oh, but she's not that young; then he realized that what Schild had said, in his Eastern accent, was \"tart,\" a term out of old plays, meaning \"whore\" or thereabouts, perhaps not so strong; what he always saw when he read it was a circular piece of pastry with strawberry jam in the center, and hence, a girl whose person might symbolize such a sweet. That was Trudchen all right. Yet he was responsible for her, in a way, and although it was funny it was also nasty.\n\nHis remonstrance was lost before he found it, for Schild, very certain, proceeded.\n\n\"Take my word and get rid of her before she gets you in trouble.\"\n\n\"But she's just a kid!\" He said it too loud and dared not look towards her. \"I swear I haven't touched her, Lieutenant, I haven't even thought of her in that way\u2014\"\n\n\"Don't be foolish,\" said Schild, sharply, \"I'm not concerned about her welfare but yours.\"\n\nSo he had made an impression on him! Reinhart was almost ready to say: I'll trade Trudchen for a transfer to Intelligence; perhaps now the war was over, some of Schild's men would go home on points and there would be openings.\n\n\"She was peddling her little ass up and down the officers' street last night. Finally she got her prey, that fat\u2014, well, it's better not to say. ... One of these days she'll turn up pregnant and I don't think you'll want to be made the goat.\"\n\nHurt, Reinhart grunted thanks; the lieutenant clearly thought him na\u00efvet\u00e9 incarnate; the trouble was, his complexion was too fair, there were no shadows on his face, no lines of character, and his eyes being pale blue looked stupid; he had labored his life long under the prejudice of his appearance. He was like a big, bland baseball bat; Schild on the other hand resembled a pair of scissors, ugly, black, incisive. How he envied him, even to the tarnish on his silver bar, the dried fog at the edge of his glasses, and the bulge in the flyfront of his ETO jacket that betrayed an undone button. And, as he watched him leave, perhaps even his dirty mind, which was a symbol of freedom.\n\nA sob from Trudchen drew him to her, more in curiosity than pity.\n\n\"What are you doing!\"\n\nShe was crying, had her spectacles off for that purpose, was flushed and dripping, and presented so much misery that, despite himself, he gave her his olive-drab handkerchief, which luckily was clean.\n\n\"He has told you somesing evil about me, that\u2014\" Whatever followed went into the handkerchief. One braid had got twisted about her neck like a noose, and Reinhart, leaning across the desk, returned it to order. The flushed face came briefly out of the cloth to say \"Sank you.\"\n\n\"Trudchen, where were you last night?\"\n\n\"Wiss my cousin Lori in her cellar. This is all the place we have to live, in a cellar which is all cold and wet and without light. The dampness comes into one's bones and most nights one cannot sleep because of the pain...\"\n\nSuddenly arthritic and conscious that the sun had some moments before left the big window, he sat down on the desktop.\n\n\"No,\" he said gently, \"I mean earlier, before you went to bed\u2014were you in this area last evening?\"\n\n\"Most absolutely n\u2014\" She started to speak into the handkerchief but emerged to study him narrowly. \"You won't think that I must be a foolish or superstitious person? ...I consulted with your priest.\"\n\nPriest. By chance he knew that the Catholic priest of the 1209th was away on leave to Rome; his assistant, Joe Para, who was one of Tom Riley's roommates across the hall, had taken a shower that morning in Reinhart's bathroom. Yet there was surely an unintentional error here.\n\n\"Well, then,\" said Trudchen when he explained, \"this was the Protestant priest, a very large man, do you know him?\"\n\nOf course, Schild's \"fat\u2014\"; things were linking up in sweet reason.\n\n\"I wished to see him for guidance,\" she went on. \"I am alone in the world, without father and mother, sometimes it is all so confusing. Do you believe in God?\"\n\n\"I haven't made my mind up,\" said Reinhart. \"But I don't hold anything against someone who does.\" It were cruel, if Chaplain Peggott gave her comfort, to abuse that great, grinning, flabby sententious ass, and it certainly had nothing to do with God, who if He existed at all, Reinhart was sure, was an It rather than a He and altogether neutral. As for himself, who had been as infant a Lutheran and then, when a schism developed within that congregation over whether or not the Ladies Aid should amortize the church mortgage by serving public suppers and his parents left with the progressive element, a Presbyterian. He believed that Protestantism was deadly mediocrity, Catholicism weak-minded, and Judaism alien\u2014and all harmless. He was incapable of bigotry, on the ground that it was a massive bore\u2014like the convictions from which it sprang. On the other hand, if it were carried to fanaticism, to that ultimate degree in which to advance his cause the believer was willing to destroy himself rather than other people: gone this far, it was, as with a Joan of Arc, a heroism to which the original motive was irrelevant.\n\nGertrud of Berlin\u2014it was scandalous to be with the force which compelled a small girl towards martyrdom.\n\n\"Look, I'll see if I can get you a better place to live. I think they have rooms around this area for civilian employees, maybe right in this building, God knows there's room enough.\" He revolved his head in disgust at a vision of a thousand rooms unoccupied while girls slept in wet cellars. \"And for Lori, too.\"\n\n\"Oh, but there is a reason that she cannot,\" said Trudchen, and immediately began again to weep softly. \"But your priest is not a good man. He tried to have his vay viss me.\" She reached to Reinhart's hand. \"He touched me\u2014here.\" She cupped his hand very neatly around one of her breasts and, even though the illustration was clear, kept it there infinitely. \" 'I just must see if you are wearing your medal,' he said. 'But perhaps it slipped down.' And then, so quick as one could think of it, he\u2014\" In the quickness she described, Trudchen had stretched open the neck of her shirt and inserted Reinhart's hand on the bare skin underneath. \" 'Vair is it?' he asked, with a very horrible smile. This must be it\u2014' \"\n\nWithdrawing so swiftly that he unraveled a strand of pink crochet, he shouted: \"It's a lie, Trudchen, it's a contemptible lie!\"\n\nPiggy Peggott\u2014he had many sins, but they were of another kind of gluttony: he was famous in the officers' mess for seconds, thirds, and fourths; but all one had to do was look at him to see that somewhere back home he had the inevitable preacher's wife in dowdy, unkempt clothing and disorderly hair, to whom he was flagrantly faithful; it was simply a matter of definition.\n\nNot to mention that: \"Protestants don't wear medals!\"\n\nIf she had earlier cried in soft self-pity, she howled now in the most violent hatred, her face red and ugly, swinish.\n\n\"It vas this Jew who turned your feelings against me!\"\n\nHe felt himself tremble fearfully, thought for a second that he had hit her; indeed, his big hand hung tremulously in the air between them as if it had bounced there off her small face. But it had not\u2014at the instant it would have struck, the fist had been seized by the mind, for Trudchen, in her temper, was not silent.\n\n\"They have no respect. Of course none of this did happen, but that was what he told you, was it not?\u2014only he made me the bad person, that dirty, filthy creature, that foul\u2014\"\n\n\"Don't say it, Trudchen, it simply is not said. He must have made a mistake, anybody could do that. You have to admit that there are German girls who might\u2014well, anyway, it had nothing to do with his being of the Hebrew faith.\"\n\n\"But it does have something to be connected to that I am a German. At least for him it does. Because the Nazis do not like the Jews, I am made to suffer. In 1933 I am four yearss old; in 1938, nine yearss. They did not permit children to operate the concentration camps.\"\n\nReinhart had a weakness in the small of his back, which standing up did not relieve. He wished he had a grievance; being without one in the modern world was disabling. How gratifying to be the lowliest Negro in Alabama, with no person alive who was not in your debt. How satisfying to be a Jew, with a two-thousand-year claim or, now, a German who had got his medicine unjustly. He should have been in combat and had his foot shot off, so that when he was brought a complaint he could point to the stump and say: obviously, I can do nothing about it, I can't even walk.\n\nHe produced a roll of peppermint Lifesavers and, thumbnailing back the tinfoil, offered the first segment to Trudchen. Shortly it could be heard clicking against her little rabbit-teeth.\n\n\"This is very sweet and not at all\u2014what do you say?\"\n\n\" 'Hot,' I guess.\"\n\n\"It is 'not so hot'? But that means 'no good,' yes? That is not what I mean. I like it better than ours, which are more\u2014\"\n\n\" 'Hot.' That's another usage\u2014the word is good for almost anything.\"\n\nSuch as her face, which now, with glossy lines of tear, was cooling. He should have liked to stroke it. She was so helpless, yet at the same time, if that were possible, indomitable. It was the same combination of contraries he had seen in Lieutenant Schild.\n\nIn the afternoon a PFC sent by Schild began to remove the cartons, antlike\u2014with small burdens and many trips. Pound slopped in at two o'clock, looking for his sunglasses, which after a moment's search he remembered he had sold, listened to Reinhart's explanation of Trudchen and account of Schild's mission, saying to the first, \"Why not?\" and to the second, \"Good deal,\" punched Reinhart in the belly, and left at two-ten.\n\nNeither Lovett nor Nader was in their office, owing, Reinhart assumed, to the catastrophic finale to the party; the colonel no doubt was grinding them into powder. About which even if he had liked them Reinhart would have felt rather more pleasure than pain, he being an enlisted man to the core.\n\nSince, then, official authority could not be consulted, he prowled through the empty rooms in the furthermost reaches of his own wing and found a little closetlike chamber that would meet Trudchen's want. It was already outfitted with a tiny stove and a naked steel bedstead and spring; from the 1209th supply room he fought a mattress and sleeping bag out of the sergeant in charge.\n\nDelighted, Trudchen threw her arms about him when she saw the new quarters.\n\n\"Do you need some help to get your things from Lori's?\"\n\n\"Oh no, you must not bother!\" A brief crease flew through her clear brow. \"I have almost nothing. You will not go there?\"\n\n\"Not if you don't want me to.\"\n\n\"Ah, not I. But Lori would be _aim_ borrossed.\"\n\n\"By the way, did she get her job?\"\n\nTrudchen showed a sly look. \"Do you know, she did not tell me! She is a very odd human being. One must accustom oneself to her strangeness, but she is very nice.\"\n\nLeaving her there, he returned to the office. Four o'clock. The PFC had disappeared, after having taken away all of three cartons. On the point of calling it a day, himself, he saw the heavy sweater that Trudchen had left behind. He carried it to her room, but already she had gone, either by the window or some secret back door off the hall, whose existence he knew not of. Folding the sweater, with a view to placing it on the bed, he felt a hard, cylindrical object somewhere in the weave. It was his missing pen, along the bottom seam of a pocket. So funny: she could hardly have stolen it and then permitted so simple a discovery. Must, rather, first have borrowed and then retained in a slip of the hand. Yet if he reclaimed it now, she might remember, look, find it again with him, think he had caught her in a theft but for reasons of his own would not protest, be discomfited. He placed the sweater on her pillow and left.\n\nHe was inclined to visit Veronica, but rather than search the hospital building for her ward, which in his imagination had acquired a sinister aura, he strolled again down the street of her billet on the chance that her duty, too, was done.\n\nThe salmon-colored gauze had been removed from the glass of the front door; on the inside surface an unseen agency, swift and sure, manipulated a cleaning rag. Its movements were mesmerizing; he had an impulse to throw himself on the grass and watch it as the warm-cool late afternoon relaxed into calm evening. Beside the door grew a bush bearing round, white berries like small versions of those pure-sugar jawbreakers with a nut in the center. There was a bush like that in his parents' front yard, and next to it a weeping willow high in which he had once established an outpost for General Custer. Alone among the men of the 1209th he had been in no hurry to get back to the States, had in fact long planned to ask, in rakish defiance, for permanent assignment to the Occupation forces, was waiting only until it could be more than an empty, sour-grape gesture\u2014for, without combat points, he was more or less permanent as it stood. Now, just now, watching the rag fly across the pane, seeing the bush, recalling General Custer, and with the sudden, almost unbearably dear smell of grass\u2014he had not at first marked that the lawn was newly cut\u2014he ached for home.\n\nThe door opened just as the general bliss had given into the deadly specificities. He had come far since his first year in the Army when he frequently had such seizures; yes, he had enlisted to escape, but there was forever another present to flee from; in the summertime, especially, one craved elsewhere. But he had nothing to get back to. In the most literal sense: already in September 1943 his parents had let his room to a man who worked an electric drill in the local defense plant, a man who had remained, had settled down, who surely had dispensed with the arrowhead collection and the stuffed bass's head on the bookcase. And college: he simply could not face that again after three years of the expansive life.\n\nThe door had opened and a figure in head-handkerchief and apron came onto the step, saying: _\"Sind Sie nicht wohl?\"_\n\nIt was not unreasonable, since he had, after all, fallen on the lawn\u2014an event thus called to his own attention. The person was Lori.\n\n\"No, I'm quite well,\" he said in German. \"It is pleasant to sit here. I see you have your job, come tell me about it. Sit down here with me.\"\n\n\"I cannot sit on the grass!\" she said incredulously. \"I am the maid.\"\n\nAt any rate, if she looked no happier now, she was no sadder. Since he was on their level, he noticed that her legs, though dressed in coarse cotton, were finely turned and rather long for her height. But there was also something terribly competent in her appearance now that she wore working clothes, a hint of hard strength that reminded him of his suspicions.\n\n\"There were no difficulties about the _Fragebogen?\"_\n\n\"What you wish to ask is whether I was a Nazi, isn't it? You are more shy than your fellow Americans. ... There was no such thing as a Nazi\u2014you should know that if you have asked any other Germans. In all this great country there were no Nazis; not even Hitler, as you would hear if you could find him.\"\n\nHe sat up, aghast at her change from suppliance to this arrogant self-possession. It was the famous German alternation from serf to lord, no doubt, and he felt it cruelly there on the fresh grass.\n\nGetting to his feet, he said braggingly in English: \"What the hell do I care?\"\n\n_\"Bitte?\"_\n\n_\"Mit mir macht es nichts.\"_\n\n\"I know,\" she said. \"That is why\u2014\" She took a deep breath and suddenly finished it in English: \"\u2014I love you.\"\n\nYou couldn't stay angry when that was said to you, but you could look insane.\n\n\"Have I said something wrong?\" She took off her dusting cap, and a wealth of hair came forth, and golden it was and clean.\n\n\"You do not know English,\" he said in a voice full of augury\u2014as if he were to go on with: you can say awful things in it. \"What you mean is that you like me.\"\n\n\"It is not the same as in French?\"\n\n\"I think not.\"\n\n_\"Also.\"_\n\nIn the distance he heard a laugh like a great bronze chime. Unmistakable. He felt criminally that he didn't wish Very to see him with this girl. But Lori, too, had heard and was even more anxious to flee from him.\n\n\"There comes one of my mistresses,\" she whispered, already in backward motion. \"I must go.\"\n\n\"Veronica Leary? Lieutenant Leary?\"\n\n\"I do not know the name, but the laugh cannot be mistaken.\"\n\nIt was heard again, turning the corner only a few yards away, and Reinhart audaciously pushed into the house after Lori.\n\n\"Ah, what are you doing here, you mad fellow?\" she asked in confusion.\n\n\"I'm going into the kitchen.\" And so he did, and sat silently until Very and what sounded like a roommate entered the door and went upstairs.\n\n\"Where is Miss Leary's room\u2014the laughing one?\" he whispered.\n\nIn frightful wonder Lori answered: \"In the rear. You will go there?\"\n\n\"Certainly not. Then I can leave by the front and she won't see me out the window.\"\n\nBut instead of moving on that plan, he looked at Lori and said: \"You've washed your hair. It is very _sch\u00f6n._ \"\n\n\"Thank you. I didn't have any soap until I started this work. ... But please go now.\"\n\nDidn't have any soap\u2014he was terribly touched by that fact. One thought of the bombings and fires and loss of loved ones, the _G\u00f6tterd\u00e4mmerung,_ but not to be able to wash your socks or bathe, that was degrading and mean.\n\n\"And where you live, I understand, is in some wet cellar. Let me get you a better place\u2014\"\n\n\"All right, but now you must go.\" She took his hand in her small but very strong one and pulled him from the chair.\n\n\"Tonight, as soon as you leave work, we'll get you a new room. I won't leave now until you promise.\"\n\n_\"Ach, was kann ich tun!\"_ she breathed in despair. \"I cannot.\"\n\n\"Why not?\" He tightened his hand on hers.\n\n\"For reasons too long to explain now\u2014\"\n\n\"Promise! After work. What time do you finish?\"\n\n\"Ah, what can I do?\" she repeated. \"I'll lose my job if you do not go.\"\n\n\"Come with me!\" Now it was he who impelled her, through the hall to the foot of the stair.\n\n\"Hey Very, are you decent?\" he shouted in a tremendous voice which agitated a small vase on the foyer table. And in no time his large friend appeared at the top, blooming lavishly in a powder-blue dressing gown, a dea ex machina about to catch the next elevator down from Olympus.\n\n\"Kiddy!\" she screamed jovially. \"Did you break in here to violate me? You-all ain't supposed to be in nurses' quarters!\"\n\n\"That's what your maid insisted.\"\n\n\"Well, get out then, you fiend. I'll see you after chow\u2014outside.\"\n\n\"I can't, I've gotta work tonight.\" He turned to Lori, who looked very grave, and said as quickly as he could in German: \"Unless you meet me this evening I shall cause you trouble. What time?\"\n\n_\"Um sechs Uhr._ I eat at your mess after the soldiers are finished.\" She turned away in shame.\n\n\"Okay then,\" shouted Very. \"Don't go away mad. Hey, where did you learn German? Wait a minute.\" She disappeared, and returned with a piece of olive-colored apparel, pitched it downstairs, it taking the air like a parachute and falling to rest at Lori's feet.\n\n\"Would you tell her to press it and be careful not to use too hot an iron?\"\n\nWhich he did, adding: _\"Um sechs Uhr,_ outside the mess tent.\"\n\nChow was SOS, shit-on-a-shingle, ground beef and gravy slopped across a slice of bread, diced carrots and canned peas, rice pudding filled with raisins resembling dead flies. Reinhart ate a grimacing spoonful of each and then smoked two consecutive cigarettes, his only pleasure the dropping of their butts into the swill.\n\n\"Anyone ever tell you you eat like a goat?\" he asked Marsala, who was stuffing down seconds.\n\n\"I've got a right to, I worked all day,\" his roommate answered on a rising, plaintive note, missing the point.\n\nAt the garbage cans were two small boys who had temporarily ducked the guard. As Reinhart prepared to empty his full messkit, one of them, saying \"Pleasse,\" took it from him, with a spoon flipped out the cigarette-ends, poured the contents in a tin with jagged rim, and began ravenously to feed.\n\nMarsala pushed his boy roughly aside. \"Go on, you goddam Krauthead.\" But there was nothing in his kit but three drops of gravy, and when the guard appeared, sweating and worried, with his switch at the ready, Marsala stared into his bland face and threatened: \"Go on, you fuck, or I'll take ya apart. How do you like that,\" he went on to Reinhart, \"those kids belong to his own country.\"\n\n\"Well, we hired him to keep them away.\"\n\n\"Yeah, but who would really do it except a German?\"\n\nTheir natural anarchism saved Italians. They were, after all, the original fascists, but even Mussolini had inspired more laughter than hatred. Someone should take the guard aside and say: Sit down, Hans, have a smoke. Now I'll give you the rundown on life. People are worth more than things, and abstractions have almost no worth at all. When you get an order your sole responsibility is to _act_ as if you are carrying it out. Hypocrisy is the better part of competence. It is foolish, I know, and defies everything you and I were taught; but in the degree to which you serve others and not yourself, the others will forsake you. However, comprehending neither Marsala's threat nor Reinhart's interior monologue, \"Hans\" had driven the children out of range, lashing their meager shins in the most dispassionate manner.\n\nReinhart had delayed taking his meal, and Marsala with him, until the tent was almost empty of soldiers and the queue of civilian workers had begun to form at the front flap, and en route to and at the apartment he dawdled for twenty minutes, part of which was aimed to bore Marsala with his company. It worked: the buddy at last drifted across the hall to needle Riley, and Reinhart returned to the mess area. Almost too late: the Army trucks used for workers' transport idled at the curb. He spotted Lori, carrying a small, lidded pail, about to mount a tailgate.\n\n_\"Also, Sie sind falsch!\"_ he accused.\n\n\"I looked for you,\" she stoically replied. \"I have either to ride this truck or walk many kilometers.\"\n\nWithin, the side benches were loaded with women who gave off chattering to stare at Reinhart.\n\n\"Go on.\" He lifted her up in one strong action, getting on his jacket a bit of splash as the cover jarred from her can, and vaulted himself in with a terrible noise on the metal floor.\n\nWhich prompted the driver to peer through his spy-window and call: \"Haul ass, kid. No riders.\"\n\n\"The Lover sent me, Eberhard. I have to get new quarters for this woman.\"\n\n\"Lovett never told me about it.\"\n\n\"All right, all you have to do is tell him when you get back.\"\n\n\" _You_ tell him, for Jesus' sake,\" grunted Eberhard, dropping the isinglass trapdoor.\n\nThey had squeezed onto the bench between a very fat girl and a very skinny woman, so that Lori was compressed and Reinhart slashed by sharp elbows.\n\n\"Tell me now,\" he asked. \"Why all the strange reactions? I think you should want to have a better place to live. Trudchen told me this afternoon about your cellar\u2014how she couldn't sleep there for the wet\u2014\"\n\n\"Trudchen? She doesn't live with me! ...I warned you about her untruths, but I suppose not enough. She lives with her parents in a pleasant flat, not bombed, near the hospital.\"\n\n\"And I got her a room in headquarters building! What game is she playing?\"\n\n\"That's just it, you see, a game. She is very young and willful. It is not easy to be an adolescent girl in the present time.\"\n\nNo, he supposed not; for that matter, it had not been easy for him to be an adolescent boy, five years and three thousand miles back, in a smooth place where the only craters were excavations for new bungalows. At least Trudchen had no pimples.\n\n\"And then, too, perhaps her family are not all that could be desired\u2014but that's another story. As to me, well, frankly, I have a husband.\"\n\n\"Oh, that's all right. You see, I'm not\u2014\" He had intended to say: interested in you in that way. But it would have been insulting.\n\n\"He is very strange\u2014as now it seems I am helpless to prevent your seeing for yourself.\"\n\nThe truck was under way, clanking, creaking, and in clouds of blue exhaust, which defying the principles was drawn stinking into their compartment. Under cover of his conversation in the other direction, Bony Elbows waxed friendly, cutting her sharp patella into the outer surface of his thigh. She was, he had seen on entering, at least forty-five years of age.\n\n\"Was he in the war?\"\n\n\"He had an odd role.\" That was her last word until a half hour later when, after various stops, one of which freed them of Fat and Thin at the same time, the vehicle came to rest at what seemed to him a purely arbitrary point in nowhere and she and he detrucked.\n\nThey stood before a hill of waste whose farthest margin must have, spilling over Asia's width, been forever eroding into the Pacific. The sun, elsewhere on this day so rich, voided this dark field, and the sweet air had long ago sold out to its competitor gases. On this range figures thin and slumped roamed crumbling through its Brenners, sack-bearing, searching, genitors of no sound. But on the summit a small girl, a ragged head above a cotton bag, called shrill and disconsolate to nobody below: _\"Wo is der Heinrich?\"_\n\n\"Behold,\" said Lori. \"N\u00fcrnberger Strasse.\"\n\nFive minutes' impossible trek and they teetered on the powdered brick at the entrance to a subterranean passage. Reinhart fired his lighter, but Lori hastily lowered its cap. \"There may be escaping gas.\" She drew him, now blind, down the prairie-dog way.\n\n# _CHAPTER 10_\n\nON THE SOFA LAY AN amorphous lump to which was appended a great pale ham. Lori slammed the door. A hollow groan issued from the ham, and two apertures appeared in its wan surface. After a time a mouth revealed itself, as if in one of those motion-picture cartoons where inanimate objects come to life through lines from nowhere, with the breezy implication that humanity is some sleight-of-hand. However, the present process was not flippant, but ponderous and awesome.\n\nLori put down her pail and fired more oil lamps, and in the richer light the great object rose gradually and with tremendous deliberation, like a sinking ship preparing for the final and irrevocable plunge, to an attitude of sitting.\n\n_\"Herr Reinhart, mein Mann,\"_ Lori waved loosely at the hulk.\n\n_\"Sehr angenehm.\"_ The voice was full, sonorous, making a grand thing of the words, and the eyes which the light showed to be as large and ripe as purple-black plums honored Reinhart directly and briefly, then shifted within the largesse of lid to Lori, who stood before the table, one hand at the base of a lamp, her left side from flank to hair bright in its refulgence.\n\n\"Here is your dinner.\"\n\nHe ignored her to revolve his head to Reinhart, saying in English without accent: \"Ah, this is your corporal!\"\n\nAs Reinhart closed on the cold sponginess of the extended hand, he felt with surprise that his own was not being shaken in acquaintance but rather used as a purchase whereby this large figure was lifting itself from the couch, and the weight was such as to compel him to throw his rear feet wide, lest he be toppled forward.\n\n_\"S'il vous plait,\"_ his burden wheezed with difficulty on the way up, and then, all at once, was upright before him, or rather looming over him, for the man was a good seven feet tall and bulky as the great Kodiak bear. Reinhart was cast into the, for him, rare feeling of slightness. The pull left the hand, but it stayed clammily and, oddly, weightless, in his own, until he opened his fingers and gravity, not its parent body, moved it to fall slowly away.\n\nUsing that language, Lori's man noted that he could speak English, and would, as a courtesy. Swaying a bit, he said that it was all but impossible for him to stay erect, but that he insisted on doing so until his guest was seated. Lori having furnished a chair, he sank again to the sofa, and drew the dressing gown that was his lone garment more snugly about him.\n\nLying still, pale, and full, like a sack of mozzarella, he tasted of the air with porcine nostrils, and began:\n\n\"Now we can converse at our ease. My name is Bach, which as you perhaps know, signifies 'brook' in German, and, naturally, to every German, and very likely to others as well, simply to utter the name is to conjure up the image of the master of the Thomasschule and the three most eminent of his twenty offspring\u2014for his loins were apparently as prolific as his brain\u2014who were also composers of a high rank, but not quite so well known outside their own land. So far as I know, I am not a descendant of that noble line. And you are called...?\"\n\n\"Reinhart.\"\n\n\"The name, of course, means 'pure of heart,' _Hart_ being the Low German variant of _Herz._ But I have a feeling that you, like so many Americans, have no great interest in etymology. Unfortunately, it is one of my many weaknesses. And I do have more than my share.\" He indicated his body with a sweep of the hand. \"The main among them being a physical impuissance, if you'll permit the word, in spite of a monstrous size. This misfortune has caused my energy to be diverted directly to my brain, which as a result is extraordinarily active and frequently denies me sleep, occupied as it forever is with a thousand and one theories, ideas, and bits of information which it should like to synthesize. I speak of this brain as if it acts of its own volition, has a life, as it were, of its own. For indeed it seems to have such an independent existence\u2014awe-inspiring, to say the least. I\u2014it is ridiculous, is it not, to speak of an 'I' separate from one's brain? but it really seems that way to me\u2014I conceive of my own identity as relating more closely to the emotions, for I am their creature and toil under the dominion of the harsh ambassadors they send to the external world, the senses.\" Here he snorted: \"Smell!\" Poked a pair of spread fingers into his eyes: \"Sight!\" Extended a fat, pink tongue, swollen as a bladder: \"Taste! And so on. Do I make myself clear?\" He stared for a while at Reinhart, as if he had forgotten him, then asked, shyly: \"I say, do you smoke?\"\n\nReinhart offered the cigarettes, saying, \"Please keep them all. I have many.\"\n\n\"Oh, kind, kind. I cannot thank you enough.\" He seemed about to rise, but decided against it. He dropped a tear. Wiping his nose on his dressing gown of dirty-orange cotton, he reverently chose a cigarette from the pack, called for a light from Lori, and getting it puffed luxuriously, his huge bald skull reflecting light like a mirrored ball upon a lawn.\n\n\"Now where were we? Oh yes, I believe some biography may be in order. Perhaps you would like to hear of my term of years in the Orient, where I served as cultural attach\u00e9 in the Embassy in Tokyo. A strange people, the Japanese, rather stolid, in spite of their reputation for wit. Their art is curiously constipated. Nevertheless, it has a kind of mordant humor all its own, in its juxtaposition of human limitation and the infinity of nature. But perhaps I'm doing them an injustice. They have, like all peoples, much to recommend them. Good clear skins, for example; one never finds them shriveling up in later years, and scrupulously clean. Absolutely no odor! This may owe to their arriving at puberty earlier than we. Our Western pubescence, which, although we think it consonant with some divine ordinance, is the slave of social, rather than natural, imperatives, has certain unpleasant concomitants: the foul stink of perspiration, for example. Children, you will note, never stink, even in the heat of strenuous play. The Japanese, arriving at adulthood still in the vigor of extreme youth, consume the life-stuff _in toto,_ while in us a certain excess accrues which maturates. Your excellent English verb, by the way, expresses beautifully both aspects of this process: the fructification and the rot. Orientals maintain that white men smell like corpses.\"\n\nThe slight movement of Bach's trunk, as he pitched the cigarette butt to the cracked concrete floor, where Lori stamped it dead, communicated a tremor to his lower extremities; the robe slipped away, exposing a view of verdigrised leg braces, complex in rods, wires, and articulations.\n\n\"The Japanese have an unusual poetry, which resists qualitative judgment. So long as a _haiku_ is written in accordance with the traditional seventeen-syllable form, it is the peer of every other constructed in the past, or to be constructed in the future. If it violates the form, it is not a _haiku._ A Westerner at his first exposure is nearly driven mad by the question as to whether this is the beginning or the end of art, not to mention morality and history. Of course, this question is of no concern to the Japanese: it simply _is,_ without qualification. They are wise and courageous enough to accept the given. Westerners can approach this knowledge only by burlesque, as when the Englishman says the great thing about the Order of the Garter is that no damned merit's involved.\"\n\nBach repeated the phrase, fondling it word by word, with the lust of a gourmand measuring off the links of a sausage, and developing an amusement which terminated in a high-pressure giggle, half-audible; the remainder being in the upper, silent-dog-whistle ranges, where it worked a secret violance on Reinhart's nerves, so that forboding ballooned the membranes of his heart as might a seizure of gas.\n\nBach gasped and grunted a tongue which Reinhart took for Japanese. \"Let me translate:\n\n> The snow crowns pale Fuji\n> \n> Here below, it is spring.\n\nThat is of my own authorship, but it will do.\"\n\nHe spoke Orientally again, in an altered voice.\n\n\"Chinese. Their verse is considerably different, but I am too exhausted to explore the subtleties of the difference at this time.\"\n\nDespite his growing nausea, Reinhart asked for a translation. He was determined not to permit this strange man to elude fact, believing that the secret of power lay in its mystery.\n\n\"Oh, yes, that is Meng Hsien-Wong.\n\n> Like a shimmer of bird calls\n> \n> The petals of the pear-flower drift\n> \n> Through the late clear air;\n> \n> Already since the morning rain\n> \n> The blossoms have grown older.\n> \n> So does the pear-branch, snow-perfumed,\n> \n> Hold a bright mirror up to man.\n\n\"You can see right off that this is not so pure as the _haiku,_ being corrupted with morality. You perceived the moral, of course?\"\n\nReinhart did not attend to this bit of malice. He had, at the mention of the \"pear-flower,\" discovered a primary cause of his illness: the room stank of rotten fruit.\n\n\"The latter was a tributary verse to an incomparable thirteenth-century painting by Chien Hsuan which I once owned but was confiscated, supposedly for some use in the advancement of the war, but how such an item could be used for such a purpose, I have no idea.\"\n\n\"No, Bach,\" said Lori, still standing by the table. \"You sold it, don't you remember?\"\n\nHe narrowed his eyes at Reinhart, and his massive face became mean, piggish, as he spoke to Lori in German: \"Manners, manners! We here speak the language of our guest.\"\n\n\"But I have no English, as you know.\"\n\nReinhart, working at a piece of gum, felt slightly relieved. He explained that he knew German and suggested that it be spoken for Lori's benefit.\n\n\"To be sure,\" said Bach, \"I am at your command. Yet, I am about to tell you something in confidence. However, I wonder if I dare? She understood enough of my comments on the painting to correct me.\"\n\n\"You probably have told the story before,\" observed Reinhart, in a schoolmasterish voice.\n\n\"Of course! That's exactly it.\" He peered sagely at Reinhart. \"You look like quite a decent fellow. Tell me, how many Germans have you shot?\"\n\nReinhart enjoyed a brief daydream of cutting down rank upon rank of blond men with a Thompson submachine gun. But he lacked in nerve to carry it off. He sheepishly admitted:\n\n\"I've never fired a gun since I put on the uniform. I'm a medic, a sort of half a soldier. Geneva Convention...\"\n\nBach made the best of it. \"An appropriate office for an American, really; an exemplary role.\" With a beatific smile, \"A marvelous people: one-hundred thirty millions of decent chaps spread out across that strange Siberia. I have been there, of course, so I will not amuse you by asking if you know my friend Smith in New York.\"\n\n\"My home is in Ohio,\" said Reinhart, dolefully.\n\n\"Quite so. Very near Chicago. You see, I do know. I once, with a friend, took a motor trip from that city to Michigan. We passed a number of persons who hailed us with leafy stalks, and felt like Christ entering Jerusalem through the palm branches. However, when we were eventually brought to halt by an exceptionally violent signal, we were asked to purchase celery. But excuse me for a moment, won't you?\"\n\nHe called Lori, and with the same kind of help Reinhart had rendered earlier, performed the impressive ritual of rising. By a tottering, brink-of-disaster, Humpty-Dumpty locomotion he arrived at the door, where he leaned briefly against the jamb, while that frail member moaned at the weight, and then went out. The door stood open. His voice boomed in the hall in a complaint about the lack of light, and another door could be heard to open, but not afterwards to close. The rich rush of his water was audible.\n\nLori sat on the edge of the couch, extending both hands in supplication. \"I fear he's been drinking. It's horribly embarrassing, you must forgive me.\"\n\nReinhart was also embarrassed\u2014for his own membership in the sex that made noise at the toilet. To cover up, he said, \"It's true, then, that he knows Chinese, and so on.\"\n\n\"Yes,\" said Lori, smiling wearily. \"For some years he was assistant curator of Oriental art at the museum. I am sorry we have no paintings or china left for him to show you. He can be very illuminating. But most of our own collection had to be sold and what few things remained went to the incendiary bombs.\"\n\n\"You sold them to pay for his medical treatment?\"\n\n\"Oh no\u2014it is another long story.\"\n\n\"You have so many.\"\n\n\"Yes, life is merely several long stories laid end to end.\" She reached across and patted his knee maternally. \"They must not trouble you.\"\n\nAlthough they should have, at that moment they did not. His distress owed rather to the dreadful odor, which was on the point of laying him low. Life takes precedence over courtesy.\n\nLori shook her head at his apologetic question. \"That is one of Bach's conceits. He read in Eckermann that Schiller was stimulated by the aroma of rotting apples.\" She opened a drawer in the table. _\"Voila!\"_ Exposing, sure enough, three blackened, scabrous fruits.\n\nBach was missing for a long time after he could no longer be heard. When he reappeared he stated that, having taken the air, he was much refreshed, which claim was supported by his looking a shade stronger on his pins, though still not by any means competent.\n\nReinhart was not sure as to what proportion of Bach's weakness could be attributed to gigantism and what to drink. Indeed, the drunkenness referred to by Lori had taken Reinhart by surprise, for Bach, given his odd body, had not spoken in a way that would have seemed, to Reinhart, out of consonance with it had Bach been sober.\n\nAt any rate, Bach swayed in, regained the sofa, where now with his new-found strength he sat rather than reclined, and confessed to copious draughts of potato schnapps; had, in truth, drained the bottle, otherwise he would have offered some to his guest. A pity, grievously hard to get; for the past few years in Germany, there were few potatoes to eat, let alone drink. He gauged the present state of his inebriation to be at the half-saturation point, but rapidly clearing.\n\n\"If you stay with me throughout the period of sobering, you will no doubt see the engine run diminuendo and eventually cough dead, so I had better make the most of what articulate time's left.\n\n\"Now I am not unconscious of my failure to ask you of yourself, but your status is not in question. I have reason to believe that the American Intelligence, when it finds time, will be unusually interested in mine. You see in me one whose only engagement for the future is with Judgment Day, for, frankly, I was a National Socialist.\"\n\nReinhart straightened in his chair, crossed his legs the other way, tugged at the margin of his jacket, and checked his buttons. How seldom we meet the real thing!\n\n\"It would be silly for me to do anything else, my record being readily available. For I was no more tepid in my political convictions than in anything else. In short, if I was a Nazi, I was an absolute one. My only wish is to tell my story without rancor, without extenuation, and submit myself to your mercy. Will you, as a matter of simple humanity, grant me this favor? _Hier stehe ich_ \u2014\"\n\n\"Only you are reclining, Bach,\" interrupted Lori, with a foolish giggle which made Bach frown and even Reinhart to turn his head in impatience. She had brought up another chair when dispossessed of the sofa, and slumped there like a discarded rag doll.\n\n\"Please, please,\" Bach replied in German, \"none of your _Quatsch._ This is a sober affair.\"\n\n\"How can I hide it?\" asked Bach. \"What is done is done. Nazism might be defined as an extreme attempt to alter the relations of Jews and gentiles, in the latter's favor. All the other involvements start from this, and this is one of those sublime simplicities that achieve the miracle of fecundity in reduction, like the Cartesian _cogito._ It requires the utmost in intellectual courage to accept the proposition that all human beings are either Jews or non-Jews; with anything less, the whole thing collapses into absurdity.\n\n\"Yes, exactly, you smile. So should have I genuinely, not too many years ago, and so do I now, with the hypocrisy of courtesy, and also with real sympathy, for I can deny myself no indulgence in my present state. But I did not begin life as an anti-Semite. There were no Jews in my birthplace, a small village in Bavaria, and it was not until I entered the Gymnasium at ten years old that I ever saw a Jew, and not until I entered the university that I ever, to my knowledge, heard Jews remarked on in a special way. In short, for many years I thought of Jews as simply Germans of a religious persuasion different from my own. Such hostile attitudes towards the Israelites as I came across from time to time, I believed to be the by-products of doctrinal differences of the sort that obtain between Catholics and Protestants\u2014of which I was thoroughly aware, as a Protestant Bavarian.\n\n\"I continued in my na\u00efve tolerance throughout the university years. A force to support it was my personal status as an aesthete. I avoided the drinking and the fraternal societies and the other nonsense, and consequently did not escape being marked as an odd one. Finding myself in the same category as the Jews, I went so far as to make some friends among their ranks. They were, naturally, excellent scholars, and their scholarship was conditioned with the sort of finesse that is so sadly lacking among the Germans. In my reaction against _Spiessb\u00fcrgertum,_ I shortly became infatuated with the Jews, and with their culture. And surely no culture is more attractive to the young man than the Jewish, just as there is no more repugnant than the German. Besides, Orientalism was my pursuit, and the Jew was the earliest flowering of the East. I gradually became aware of the indictment brought against my pets by the Germans, in all of its ramifications, but I still was not be moved. Indeed, I became more pro-Jewish than before. And I did this in an unusual mode. I accepted the accusations as truth, taking issue only with the interpretation. That is to say if the charge was that Jews owed allegiance only to their international Jewish state, I agreed and approved. For, thought I, what else could they do, when throughout history they have been rejected from the Christian society they sought innocently and sincerely to join? In the Twenties, as you may know, the Germans were in narrow straits, while the Jews allegedly flourished. Now it took some nerve to hold, as I did, that it was natural and just that they should tighten the screws against the gentile, for would not the latter have done so had the situation been reversed? I could never see anything peculiarly evil in the Jew's economic behavior. Should the executioner be blamed for the practice of hanging?\n\n\"As for the arguments on racial grounds, they were sheer foolishness, only to be believed in by the kind of people who take up vegetarianism, Rosicrucianism, and other crackbrained schemes to evade paying the piper. I paid little attention to them, and I think this was also true of most convinced anti-Semites, whatever certain loudmouths said. This phase of Nazism was sheer spectacle; this was the Nazis' analogy to Christianity's graven images, saints' relics, etc., and a central vulgarity on which Protestants and Catholics could agree. Never since Luther, whose nationalistic fury vis-\u00e0-vis Rome withered his sense of psychology, had the national appetite for histrionics been so appeased.\n\n\"Well, then, in the light of all this, why did I eventually reverse myself and become ineluctably anti-Semitic, even to the extent of joining the National Socialist party, which I had from the first abominated as an unholy alliance of gangsters and buffoons? I became an anti-Semite, not for the usual reason\u2014because of the anti-Semites\u2014but because of the Jews, and I joined the united front against the Jews because there was nowhere else to go.\n\n\"When I emerged from the university into the great world, moved to Berlin and entered its intellectual life\u2014which in that time was almost uniformly Jewish\u2014I did not change my stand. I still baited the anti-Semites, and, as one will when in the grip of a self-righteous obsession, in the absence of suitable adversaries engaged in dialogues with myself, acting first as _advocatus diaboli_ in the presentation of the strongest possible case against the Jew, then demolishing it with my better arguments. I would probably never have had cause to change had I kept company only with gentiles, and certainly never, had my associates been anti-Semites. But I found myself in ever-closer relationships with Jews, whom I attracted as my philo-Judaic position became known, and whom I of course sought out. And thus the foundation came to be built, stone by stone, for the mansion of knowledge. For I found that no matter how well disposed a gentile is towards a Jew it can never be sufficient, for the Jew will not stop short of the total debasement of his friends. The Jew does not want, and does not ask for simple understanding. He craves only total victory, and rewards anything less with corrosive hatred.\n\n\"I was not permitted by the Jews to deplore the persecutions they had suffered at Christian hands. In their arrogance they asserted that this very act of deploring was a form of anti-Semitism because it credited their enemies with efficacy, and no matter how superficially well-intentioned the gentile who entertained such sentiments, he could not avoid unconscious _Schadenfreude,_ no more than can the athlete who sympathizes with the cripple.\n\n\"This is an excellent example of the Jew's ability to pursue his end by contradictory means. Sometimes he will object to the very designation of 'the Jews,' maintaining that no such racial, cultural, religious, ethnic, or whatnot entity exists, that it is the sheerest invention, the most fantastic lie. If you point out that if this were so, anti-Semitism would also be nonexistent, he will say, 'Exactly, that supports my claim that the whole affair is simply the eternal struggle between the mob and the elite, with no relation to Jewishness.' At other times, and under other guises, he will present the argument that only the Jews exist, and no other people, because of all the peoples of the earth, only the Jews have been able to preserve their identity in every milieu. He can disclaim Jewish influence on any culture, or assert that the Jew is the _Ern\u00e4hrer_ of our heritage, and cite Abraham, Moses, and Jesus. Yes, he will cite Jesus with the composure which is his forte! The modern world is, for him, a theater in which the Jews are anonymous members of the cast\u2014unless the situation requires the reverse strategy, in which case he produces a list of leading performers beside whom the gentiles are relegated to the beer-hall stage: Marx, Freud, Einstein.\n\n\"He can assume any position at will, for he believes in none. And he hates the sympathizer because sympathy implies melioration, and melioration is change. The Jew's real aim is to bring Time to a stop. Like all Asiatics, he has a horror of motion, process, becoming\u2014whatever name you like\u2014for us Occidentals, the superior Deity. When _is_ replaces _to be,_ he will have won. Humanitarianism, liberalism, evolution, tolerance, understanding, these he rightly sees as temporal devices to frustrate him, whereas he delights in the antagonism of fanatics. The anti-Semite is his darling, just as the atheist is the theist's sweetheart, the murderer the victim's beloved. The Jew would be a god. How near to success is he then when called a devil! And how he writhes in hatred when a slobbering, mealy-mouthed humanitarian addresses him as Man!\n\n\"Totalitarianism provides his most congenial society, with its stupid calls and alarms, its aping of the Jew's own tricks, such as the obliteration of time and the fierce attack on moderates, and\u2014 _its persecution of the Jew!_ When he becomes an obsession, he is on the threshold of victory.\"\n\nBach retracted his big head into layers of neck-flesh, recovered, then began again, right forefinger extended:\n\n\"I do not mean to claim that I quickly saw the light. Young and innocent as I was, I determined after each rejection to redouble my efforts at understanding, feeling still that it was _our_ responsibility that this strange people faced the world in a crabbed, distorted way. How very close was I to the truth! A human being if thrown into water at birth can swim. A few years of growth and this talent is gone, to be regained only by artifice. Yet, this is as it should be. Artifice is what makes us human. It is morally necessary to withhold this function from a child until he has lived long enough to learn the properties of water and the human body, and to experience a sense of achievement in placing them in a new relationship. So with me. By the heart, I had arrived at the proper relationship with the Jews, the masterly one, but I was condemned to tread the earth for some years in ignorance before returning to it by ratiocination.\n\n\"But, to proceed. I told myself again that the Jews had no reason to think kindly on their oppressors, and that it was only natural they would out of pride decline any aid that tended to imply a lack of self-competency on their part. I summoned up my resources of love, decency, intelligence. They might deny me, but I would not deny them. I suddenly took on, through the force of my commitment, the identity of a Jew; and the soma reflected the psyche: the cartilaginous tissue of my nose thinned, my eyebrows thickened, and my shoulders developed a nervous twinge.\n\n\"At first, my gentile friends derived much fun from this state of affairs, and would jokingly call upon me for the Jewish point of view on every question (this \"Jewish opinion\" is a favorite delusion of gentiles, and one which while ostensibly deploring, Jews enjoy enormously). But it did not take long for them to discover that what was an idle jest to them, was deadly serious to me. As my philo-Semitism became firmer and firmer, I felt a wall rising between us. The last brick fell into place when a story began to go around that I was really half-Jewish, and had thrown my lot with the alien part of my heritage. This fiction, I realized, was only their defense against accepting the terrible fact that I had, in free will, abdicated from the gentile's estate.\n\n\"But, of course, neither was I received as a fellow by the Israelites. Here there existed no solid wall\u2014this people could not have survived all those agonizing centuries by material means. (The Jew, by the way, has always deluded his enemies into thinking he is materialistic. Nothing could be farther from the truth, which you can appreciate when you observe that he has flourished in the West under capitalism, a philosophy which above all others is abstract and visionary, and based on the intangibles of faith and spirit. He is, however, naturally opposed to the recent developments of capitalism. If it becomes humane, that is to say, evolves into true socialism\u2014which is absolute materialism\u2014we have a chance of conquering him. Vain thought!)\n\n\"So I was with and around the Jews without being of them. Oh, they don't hold secret meetings, like the mythical Elders of Zion (that wonderful legend, which is far too gross to be of Jewish origin\u2014you know they, themselves, 'plant' most of the anti-Semitic fairy tales\u2014is an example of the gratuitous aid they are often rendered by moronic gentiles), they have no arcane signs or handgrips, no insigne. How they communicate their identity to each other is so mysterious that it exceeds mystery, as does the manner in which a single spermatozoon out of ten thousand penetrates the egg. The important thing is that it happens. And, if _we_ cannot grasp it, no Jew can fail to. Which is why no Jew can truly forsake his people, and why the Jews display that odd combination of mockery and pity towards those of their fellows who vainly toy with religious 'conversions' and facial surgery.\n\n\"The great reversal (from philo- to anti-Semitism) came, as those things do, all at once. I was in the habit at that time of spending the evenings with my Jews in a cellar-cafe where over a single glass of beer or cheap wine we would exhaust hours talking art, literature, philosophy, and those other diversions of the young, including politics, of which ours was, in that day\u20141927\u2014communism. All in what I cherished, despite numerous disillusionments, as the intimate atmosphere of brotherhood. One evening a newcomer appeared at our table, a fierce, hideous, wooly-headed young Israelite, looking like the pictures one sees of Trotsky as a youth. He was discoursing passionately on some topic, political I should imagine, but as I took a seat, he terminated abruptly. 'It's all right,' one of the others told him, 'Bach is all right.' He nodded amiably at me, and rather transparently began to comment on inconsequential matters. Later, when I had left the table briefly to speak with a friend across the room, I saw on returning that he and Schwartz, whom I regarded as my closest comrade, had their heads together, snickering. The object of their amusement was obvious. Now, lest you think me hypersensitive, I must explain to you that the Jew's humor is concerned solely with satire; he does not laugh at things, but always at people. That is to say, he finds funny not what occurs by chance, such as a stout man's tumble on the ice, but what has taken place by human will, and the involvements therefrom, such as, say, a gentile posing as a Jew. This temper stems first from the Jews themselves having suffered too much from chance to find comedy in the fortuitous, and, second, from their great reverence for the given, the inanimate, the timeless. One might almost say the Jew would see the ice mocked by the stout man's hindquarters.\n\n\"I felt a rush of loathing at that moment, as one about to vomit feels the bile-bitter fluid rise in his throat, and not at the Jews, but at myself. For a moment I had seen in those mirrors of degradation that dreadful, abominable specter that no one can face with composure: my naked self. But I choked it down and took my seat, for the deepest self-knowledge bears with it the deepest cowardice. The impulse to action was to come almost an hour later. The conversation had continued in the same silly direction the newcomer had indicated: tastes in wine, the beasts at the Zoo, a job a friend had lately got on one of the Ullstein papers\u2014 _B.Z. am Mittag,_ I believe\u2014and so on. Finally, the group began, only half-seriously, to plan an outing in the next week. Half-seriously, I say, because we were all unemployed, and could not have raised the money for the elaborate refreshments listed as the minimum fare. 'Where shall we go?' cried Schwartz. Someone named a favorite section of the Grunewald. The eternal dupe, I had been swept up again into the warmth of the fraternity, and was adding my bit. I noted with good humor that we should avoid the spot named, because on a recent Sunday stroll I had marked that it was uncomfortably crowded.\n\n\" 'Yes,' said the young Trotsky, 'too many Jews.' I think now that he was merely passing a harmless, if masochistic jest, as Jews often do, but, then, it struck the spot that had been worked raw by the earlier incident. I broke down and wept. God, there is nothing more terrible than a young man's sorrow! But not even that will move a Jew! I sprang to my feet. 'Yes,' I sobbed, 'just as here,' and fled from the caf\u00e9. From that moment on, the battle was joined.\n\n\"I had been a fool, but my greater folly was yet to come. I fell prey to the subtlest device of this devil, and joined the ranks of his greatest ally, the National Socialist Party.\n\n\"To war on decency, love, truth, freedom, is to permit the Jew to mask himself with the Good, and thus to embrace him. Through our aid, the Jew was able to achieve what in all the anguished millenia before he was not. Si _monumentum requiris, circumspice!_ We weeded out his weaklings, while increasing his moral capital with every one we destroyed. We hardened him with our tortures. _We_ tempered him, refined him in our fires, _we_ polished him down to the indestructible core. Today you can see the results of our craftsmanship: he is pure hard diamond, and his radiant leer sparkles in triumph over his fallen forge-slave.\"\n\nIn conclusion Bach reached over and dropped his hand on Reinhart's knee with a startling weight. Startling because when he had held it earlier in greeting, it was light, and since it was clammy as well, reminiscent of a damp sponge. Now it hit with a _plop_ like a waterlogged sponge, and, sure enough, when Reinhart looked down he saw a faint wet stain melting the crease from his trouser-knee. This oddity was as full of liquid, it occurred to Reinhart, who remembered both the tears of gratitude at the cigarettes and the weeping in the story, as a cheese is full of \"whey\" in all the best fairy tales. As the first occasion on which he had come across anyone whose hands genuinely dripped with perspiration, it was worthy of cataloguing. Then too, in this damp cellar nothing dried. His own sweat, while not as plentiful as Bach's, sheathed him like a trout's mucous envelope. A strip of stagnant water lined the base of the wall; the concrete blocks above had sieved out a patina of mineral salts.\n\nBach's rhetoric had made poor Reinhart's head reel, from amusement through indignation to logical vertigo. He repeated the process, this time at greater cost, that he had undergone in Philosophy 100, where the splendid promise of the fall catalogue\u2014\"The major traditions of European thought\"\u2014was blighted by the inevitable petty-Machiavelli of a lecturer with his _cul-de-sacs:_ \"Epimenides, a Cretan, said all Cretans are liars. Was he telling the truth?\" And even if he understood, he was lost, and guilty, guilty.\n\nAt last, in desperation, he said: \"Just let me get hold of this. You want to kill the Jews with kindness?\"\n\nBach made his giggle, and the hairs rose on Reinhart's neck.\n\n\"Leave it to the American to put things without equivocation!\"\n\nReinhart took his advantage to steer into the congenial area of behaviorism. \"But all this is in terms of _feelings_ and _ideas._ What exactly did you _do,_ as a Nazi?\"\n\nBach withdrew the sponge to his own knee, his eyes bagging in disappointment.\n\n\"I should have thought the intellectual history to be the more valuable. Well, then, if you insist, I can produce a few crumbs of physical activity. Humiliating, but perhaps useful as an index to the nightmare from which it took me so long to awaken.\n\n\"I joined the Party in November, 1938. I shan't dwell on the scruples the conquering of which took me an entire decade from the aforementioned events. I placed the button under my lapel a few days before the celebrated _Kristallnacht_ when, in retaliation for the murder by the Polish Jew Grynszpan of an attach\u00e9 at the German Embassy in Paris, the Nazis instituted an action against Jews and Jewish property throughout the Reich. It may sound queer to you that I participated in some of the raids in Berlin. Yes, I the aesthete! My request for a role was most suspiciously received, the storm troops being constituted of the most ungodly scum you can imagine, whose motivation was not a holy passion against Jews but a simple nihilistic lust for destruction. However, a fanatical eye is an effective persuader. I managed to win a position on one of the flying squads that swooped on the Jewish shops in the Kurf\u00fcrstendamm. You cannot understand, nor can I describe the exultation with which I plied my axe, even astonishing the thugs whom I accompanied, so that by the end of the night there was a tacit agreement among these canaille that I was their leader.\n\n\"In a china shop, where we had done a job worthy of your proverbial wild bull and were ready to depart, one of my companions came upon a hidden safe, buried in the rear wall. We had to send out for explosives to open it, it being impervious to the pick, and I was all for abandoning the project for better work elsewhere. But the cupidity of these swine was aroused; they were convinced the Jew had cached his treasure there. The door was eventually burst, revealing an empty chamber save for a single object, a small vase, which on examination I determined to be a piece of thirty-pfennig trash from Woolworth's.\n\n\"Now why the Jew would have placed such a thing in his vault I could not at first explain, and, indeed, was about to pitch it aside, when the thought struck me that the scoundrel had got intelligence of the raid, and, lacking anything better, had employed this means of retaliation, with a sense that Nazis of the common stripe would be certain to think it valuable and demonstrate their idiocy by confiscating it unbroken. A very deep joke, typically Jewish. But I knew, with the penultimate hatred which is, as I now know, stupidity, but which then seemed wisdom, that at last I had in my hands an instrument to enable me to top the Jew at his own game. I led my men in another diligent round of razing. When we had done, the showcases were flinders, the walls demolished to the lath, the woodwork a pile of faggots for the stove, the wiring ripped out\u2014in short, a reduction that could have qualified us journeymen house-wreckers. A tiny table was spared, and placed in the center of the room. On it, I centered the vase, filled to the rim with my ordure.\"\n\nFor all the foliage, thought Reinhart, he is a clown at the core. \"That was sort of childish, wasn't it?\"\n\nBach could not be conned. \"No, with gratitude, if you mean 'and therefore not responsible.' \" And, not seeing Reinhart's grin grow dim, he struggled to his feet without assistance; swaying over him, face contorted, arms rising and falling like a crazy windmill, he screamed in the voice of his giggle, piercing, forceful, but not loud: _\"Oh no, no. Can't you understand? In Auschwitz we of the SS could kill two thousand head in half an hour, but it was burning the bodies that took time.\"_\n\nHe produced a cavernous belch that shook him to the fundament, and toppled backwards, ever so slowly, onto the couch, which recoiled to the floor and recovered. Massively he slept.\n\nLori, too, slept in the chair, but the absence of sound as Reinhart rose and prepared to creep away, awakened her.\n\n\"No, are you leaving?\"\n\nReinhart pointed to the sofa.\n\n\"He is spent, poor man!\"\n\nAt the purity of her look, Reinhart seized her bony shoulders and shook them violently as he might have washed his overcoat with air. When he had exhausted the brutality of his violated virtue and summarized Bach's dissertation, she tossed back her head and laughed extravagantly.\n\n\"Bach in the SS! Pardon my rudeness. Perhaps one must be German to see the joke. The SS had most severe physical requirements.\"\n\n\"Why would he tell such a story?\" asked Reinhart, aloud but to himself, as the chair again received his mass. \"If he concocts this out of the thin air the man is surely mad.\"\n\n\"No, he is not insane. The minds of the insane run in straight lines, not always Euclidean, but always straight. The job there is to find the geometrical system by which to measure them. Here, if you insist, we have something eccentric, twisted but normal. In fact,\" she added, \"normal is twisted.\"\n\n\"But why evil?\" he wailed. \"When people lie they make themselves better, not worse.\"\n\n\"No, you foolish boy!\" She thrust her face up at him. \"No, they first make themselves something, whether good or bad, but something. A man cannot live without a function. Can you understand that, you _American_?\"\n\nHe had never in all his life heard the national adjective pronounced with contempt. _Amerikaner:_ he loathed it for a moment himself, but there was yet one more hateful.\n\n\"You _German_!\" he ranted. \"Can you understand this: I am ashamed to be of German descent! It makes me sick to my stomach. I might lie to make myself worse, as you say, but not to claim I hurt defenseless people. You once asked about my relatives\u2014I hope they were killed in the bombing! And if they weren't, they are dead anyway in their souls. Do you know what you did when you murdered the Jews? You committed suicide, all of you!\" Of course, no sooner was it out than he realized he echoed Bach to the letter, and was ashamed.\n\n\"Don't talk of things you cannot understand.\" She turned her back.\n\nHe reclaimed from the table the pack of cigarettes he had given Bach and made for the door. Lori pursued him. In the dank passageway, in the pale light that reached there from the lamps within, they grappled, she shrilling: \"I must make you understand about Bach. It is simply an overactive anterior pituitary. Not only does this outlaw of a gland produce great size, but it also eliminates the sexual urge!\"\n\n\"I don't care, I don't care.\" Saying which again and again, he nevertheless permitted her to pull him back inside. He knew now of his own impotence: his great moral address had been delivered, every word, in English.\n\nLori drew him to the chair and notwithstanding their differences in size, literally knocked him into it, all that was necessary being one good push in the midsection.\n\n\"Now,\" she cried, standing militantly before him. \"It was you who insisted on coming here. You forced me to bring you against my will. Therefore you will stay until I finish. Bach has done as much for me as one human being can do for another. He has saved my life, my very life!, every single day for three years.\"\n\n\"You were anti-Nazi?\" asked Reinhart in rapturous awe, but she paid him no mind.\n\n\"And it involved more than simply not turning me in to the Gestapo\u2014you perhaps think in your na\u00efve way that that much could be expected of a husband; you have not lived in Germany\u2014and more than concealing me, too, although that at the daily risk of his own life.\"\n\nBeneath the vast, important feelings Reinhart had a little tickle of pride, no less important, at her ceasing to speak so as to favor his imperfect knowledge of the tongue. She spoke swiftly and with the full resources of idiom and construction, and he did not miss a word.\n\n\"The long story of his art collection is pitifully short. He sold it, piece by piece, to pay for day by day of my life.\"\n\n\"The Gestapo then could be bribed.\" The idea made that dread agency less terrible.\n\n\"The money went elsewhere. Who got it does not matter.\"\n\n\"Excuse me, I am so stupid, foolish as you say\u2014if you don't wish to answer you don't have to. Why, when he has this to tell, does Bach pretend to be the reverse?\"\n\n\"Because the meaningful things are never said. Because he is infected with the Berliner's disease, irony and gallows-humor. Because\u2014\" She moved intensely near, and he was afraid she might call him _American_ again, with all that scorn. \"No, _I_ shall not lie to you. ... Because the time when he could do something for me has now passed.\"\n\nHe found that, idiotically, he had replaced the wretched cigarettes on the table.\n\n\"I came here tonight to take pity on you,\" he said. \"I have to ask it instead for myself. Believe me, it is not easy to be a fool. You have to work hard at it.\" He went again to the door, this time unaccompanied, from which distance he looked long at her minor, crumpled figure, and said: \"You are a Jew.\"\n\nBach groaned lightly in his sleep while Lori with careful hands arranged a quilt on his recumbent hulk. Then, extinguishing all the lamps but the one on the central table, she came once more to Reinhart.\n\n\"That saves me, _nicht wahr_? That one-half Jewishness, that mongrel portion which so short a time ago condemned me, is now my salvation. And enlightened people no longer believe in miracles! Yet within oneself, one is always just a person. Even Hitler. Do you know, his favorite meal was corn-on-the-cob and jelly omelet. Think of that: there were moments when his sole concern was to retain a bit of slippery jam on a fork.\"\n\nHe opened the door and stared forlornly into the gloomy passage.\n\n\"Shall I light you out?\"\n\n\"No thanks. I'll try to manage that much on my own. May I come again?\"\n\nBriefly she was against him, her small head in the hollow of his rib cage.\n\n\"You are a fool, a good fool, a kind fool.\"\n\nShe gave his hand the short, one-shot European shake and said no more as he began the tortuous ascent to the mid-world.\n\n# _CHAPTER 11_\n\nONE HAD HIS CHOICE in the officers' liquor ration, but one could not command what was not available. The fifth of gin represented an impressive trial of even Captain St. George's noted patience, not to mention the vermouth.\n\n\"Eleven months, Nate, it took me to assemble the ingredients of a martini, with the olives still to come. And the funny part is that I never eat the olives. Still, a drink looks naked without it, and I think, don't you, that an olive adds a certain essential something. And the ice! There's something, where can a fellow get ice in this stricken city?\"\n\nNo, pickled onions would never do, and although the fashion was passing to lemon peel, St. George had read, he held with the olive.\n\n\"Besides, the question is academic: I haven't seen pearl onions or lemons, either.\"\n\nThe captain was therapeutic, a plump, well-padded bandage. Why the medics did not use him as resident healer in some recuperation camp was beyond Schild's reason.\n\nSunset in the back yard with St. George, an awful thought as recent as two weeks ago, now was Schild's hope. He had fallen into an attendance on the captain's problems as one tormented by the rash might lower himself into a pool of warm oil and, comforted, in debt to the oil, so to speak, take up its study. A new approach for Schild, who hitherto might instead have gone into dermatology. But he had at once lost his strength, not by a slow erosion but at once, as if someone had opened a valve.\n\nLast night, undoing his tie before the mirror\u2014an atypical incident; Schild was so little concerned with his appearance that he rarely stood before the glass to put on his necktie, let alone remove it\u2014standing, then, at the dresser, the room's interior precisely reversed from the real, he was overcome by a quick delight, like some small-flat resident with his persistent discovery in dreams of a door behind the bookshelf that opens into another room no one knew about. Admiring the new figure in the wallpaper, all the fresh textures, the dimensions not yet contemptible by use, his eye swept to Lichenko and arrest. He had not, in the farthest reach of the new landscape, forgotten his guest; indeed, deliberately he had sought him out, as if, since the mirror worked a comprehensive reversal, it would also reverse Lichenko; as if from a novel aspect he would be seen again as he had entered Schild's affairs, the subsequent devious patterns now revealed as a foible of the stale vision.\n\nInstead, on Lichenko's face reigned a supreme and splendid peace, a glutton's on arising from the board, a sadist's on hanging up his whip, a drunkard's on counting his corks. He looked not at all at Schild but into the obesity of the featherbed that collapsed and reasserted itself under his experimental hand as if in breathing. His own serene breaths followed suit. He had got heavier in fourteen days, a fact that his reversed image stated with a kind of hostile assurance not apparent right side front. The German woman did up his clothes: his blouse a thing of smooth planes, his boots another mirror, the trouser crease a lethal edge. His eyes hung heavy, as if he could not bear the weight of the lids. Give him six months and he would be a little fat man, a fat little tradesman, retaining his cheek-furrows but in the discontent of prosperity. The fact was that an agent provocateur satisfied, however that surfeit sat upon his features, was an agent whose mission went well. To watch him was like looking into the mouth of a clogged toilet.\n\nThe violence remained in the glass, as, turning away, he began to measure his position, he, Schild, the man of limited ambitions, commonplace talents, one who served, a rational man, mistaken now by who knew what compound of humanity and history and place for the reverse of these qualities; for although he may have been guilty it was not of the transgression for which he was now under surveillance, could not be, for it was precisely his crime that he was incapable of a crime, and unfortunately no secret police or central committee was yet in search of that kind of deviation.\n\nAs a boy he had developed a feat of controlled consciousness, a triumph over what was initially a disability, almost, he had feared at the time, a madness. Under duress of his parents, who were tyrants for the communal life\u2014at least in theory, at least for the children; when it came to themselves it was another thing: \"when you grow up you got to work don't forget it\"\u2014 expelled from home, he had gone with a children's group, Jewish, always and forever Jewish, to Coney Island for \"fun\"; always and again \"fun,\" a separate and distinct endeavor from one's other pursuits, to be sought only by congress with, in this case, a million other organisms who had not forgotten to leave behind their flesh and sweat glands. Despite the sun's inability to plumb the floor of this forest of skin where he lay, he had, impossibly, got a bad burn. He did not tan and therefore unless forced avoided the sun; once in it by necessity, however, he tested that familiar theory that stern resolution overcomes all, arriving at the invariable and, for one with dark hair and eyes, peculiarly shameful rebuttal of scorched hide. The end of this day found the flames mounting to his head; he was lost for an hour and when found was sick enough to warrant being taken all the way to Manhattan in a taxi\u2014for which when he was well he heard \"somebody had to pay guess who?\"\u2014and, home, was in a half-delirium for three days while his pelt grew leprous.\n\nOnly half a delirium, because he knew what he was about and didn't, simultaneously; or did but suspected he shouldn't; feeling so queer in the head one hadn't the right, so to speak, to be on the same terms with reality as other people. Lying on the sofa whose brown plush marked with patches of psoriasis the points of humanity too often stated\u2014head, hips, and heels\u2014watching his mother drop the wooden egg into the stocking toe, beneath the bridge lamp whose transverse member had long ago been separated from its upright by nervous readjustments and now bore its oiled-paper shade on the frayed cord alone, he felt a dread that she might not look up on schedule and carp. People don't read on the flat of their backs holding the book over them like a sunshade, it blinds them at twenty. He yearned to receive the admonition that in normal times would have driven him out to the library and its public privacy; now to assure himself of actuality, it was a value from the past that he would have given even his assent to establish. His sister, in bed fifteen minutes, failed once to call for water; for once his father, who rarely left the house after dark, was out for cigars, which he rarely smoked. Across the airshaft from the kitchen, the Kaminskys were anaesthetized, for the first time in history they forbore to exchange insults in heavy Polish accents.\n\nIn his three days of partial coma Schild learned what his twelve years of comparative clarity had not laid clear: in the degree of one's need, one's companions and surroundings become negative, fail to comply. One seizes the bathroom in a quiet hour and instantly sets a fashion for the others, who appear at the door in force; one goes to the movies on Monday night and cannot win a seat from the horde of people who will miss work next morning to perform this natural service of denial; one is hungriest the evening the rest of the family, ill, dine on rye and cream cheese and admire each other's lack of appetite. These schemes are seen with the semiconsciousness as colors are more vivid and distances overcome with squinted eyes. In health, we are tormented by reality's presence; sick, by its loss.\n\nWhen well, Schild commanded this partial coma as it had in illness mastered him, for when the prisoner holds the key, a cell changes from a proscription to a defense of liberty. It later became his obsession that experience in the large could be controlled\u2014nay, must be, which had no regard for could\u2014the alternative was fascism. But experience in the small, in the disparate grooves where the larger powers, so to speak, lacked elbow room, what could not be ordered could at least be converted.\n\nIn the present circumstance, Schild looked at Schild from a distance, finding him not small but different, as if in dramatic irony he knew something that his double did not. One Schild was doomed; the other could stand here invisible and invulnerable, if necessary watch from asylum the first go to the noose.\n\nSchatzi had not appeared in ages. Schild had made two fruitless trips to Wannsee on consecutive nights; on the third he had posted a note on a tree in the Fasanenstrasse, in Wilmersdorf, used as a bulletin board by bombed-out persons seeking their families. This schedule, to be put into effect in case of emergency, had been ready for use since his first engagement with Schatzi. The note read: \"Seek whereabouts of Oskar Reichel, formerly of Ludwigkirchstrasse 32. His wife is alive and well. Apply Bauer, Weddingweg 8, Lichterfelde.\"\n\nIn theory, contact having been broken, Schatzi was thus directed to come to Schild, at the former's discretion. In practice, he did not: clearly, because Schild had all at once acquired a bad odor. That Schatzi had himself fallen afoul of one power or another was unlikely, if not impossible; further, it could not be imagined. If for no other reason, fate would keep him free to taunt Schild, whatever the extremity.\n\nSchild felt comfortably like a corpse, with Captain St. George droning on as sole mourner at the wake. The evening lurked in the population of leaves overhead, pandering for nightfall, which in Berlin appeared later than elsewhere and then only after extensive preliminaries, leaving early: at half-past three you could read your watch without a flashlight. St. George's cigar-end periodically bloomed with fire, drooled a thin bouquet. His voice, deficient in resonance, permitted the transmission of noises from small live things rustling in the bush.\n\n\"Even the lizards, or whatever that is, are going, as our British friends say, to Bedfordshire. That's my idea, too. I am so [yawn] sleepy. Must be the [yawn] air. Goo [yawn] night.\"\n\nThe captain rose and snapped-to his folding camp stool, tonight with satisfaction. He liked smooth-snapping things, as being both easy and smart; but his taste was often impeded by the stool, for which he frequently cursed Aberfitch & Crombie: the extra ten dollars they added to every item had no justification when things stuck.\n\nSchild sorely wished him to stay, should have liked to make the brief sound which would have accomplished that, for it was no great or difficult feat to convince St. George his presence was wanted. Yet he could not, as it were, solemnize his conversion to the invertebrate. Assuming Schild was asleep or, more likely, forgetting him simultaneously with his decision to retire, the captain stumbled into the rear door of the house.\n\nThe night chill found Schild's marrow. A bulky insect dropped into the hair of his forearm and was entangled to frenzy. Fool that it was, it interpreted his humane efforts as malicious and died shortly of its false assumption. He sat alone within the limitations of the fence; a small yard, smaller still without St. George; a corral with like squares on either side, extending on to the ends of the block: multiple petty-bourgeois cells. A creeper rose embraced the pickets. He once knew an Eagle Scout who was born and raised five stories above the sidewalk on 191st Street but could identify every plant within a ten-mile radius of the camp at Alpine, New Jersey.\n\nTheir landlady slept on two biscuit mattresses on the floor of the kitchen. The easiest thing in the world, if she had not been German, would be to walk right in and have her. If she had not been German, however, it would not be the easiest thing. He fancied his contempt had some sexual attraction for her. The angles of her face sharpened as he passed, and he sensed furtive movements behind doors when St. George was out of the house. But he had already had his gentile, and of the classic kind, complete with rabbit nose, soap-and-water cheeks, and anti-Semitic Daddy and Mother, and he had no strong sadism even towards Germans.\n\nHe turned his slat chair, property of the house, squealing rustfully in the joints, so that he could not see the kitchen window. A dog whined in the distance; a Southerner or a Negro, passing on the sidewalk out front, described to a mute companion a succession of events that were invariably _mothafuhn;_ the faint odor of pine, which was everywhere at night in the western quarter of Berlin, was superseded by the sudden smell of candy, moreover, a precise candy: Mary Janes, nut-flavored, tallowlike caramel on the outside, peanut butter in the center. Mrs. Grossman gave you one less for a penny than her competitors; Milton, when she took ten minutes off for raspberry soda and plain cake, gave one more. Amazing so shrewd a woman never knew this for fifteen years. It was after Milton had been killed in Spain that she received this knowledge, along with the other things that had to be said, fat and old and bitter, knotting a cord end from the drawer of saved string. \"Better they were poison!\" Although she should have known that her son was his master, and not vice versa. A boy, buying some candy, slipped a nickel underneath a newspaper on the counter, for it was Sabbath and Mrs. Grossman would not touch money until the sun had set. \"Go read your books!\" she shouted to Schild in Yiddish, the language, it struck him suddenly, for many things but, above all, humor; he could feel nothing beyond a terrible impulse to laugh and a sense of how terrible was his impulse. In the doorway, he bumped into the boy, who had been delayed a moment, stripping the wrapper from his bit of sweet; it was a Mary Jane. Schild's teeth clogged in empathy, he could almost taste the peanut butter, it had always been very like putty.\n\nAll this in the second before he knew, like a bat, that a human being stood in the darkness behind his chair. He sat easy, who else could it be but Schatzi?\n\n\"And here we have Herr Schild, _zu Hause_ like any merchant.\"\n\nWhere had he been? was an obvious question, but could not be asked. Schatzi made succulent sounds over his candy, offering some to Schild which when taken was a disappointment not living up to its odor: so much for the memory. He was called back to the present; the atmosphere was less charged at once. There sat Schatzi, renewing the link. He had been a fool to concoct an elaborate, sinister design from meaningless coincidence, for it was through Schatzi that his line of authority led. How could a simple lieutenant in the Red Army be concerned? One must face it that Lichenko was even rather pitiful. It was no disrespect, it took nothing from his honor, was no adverse implication on the triumph of the society that had given him his chance for manliness and heroism, that Lichenko was\u2014he almost thought, was no Milton Grossman; but Lichenko, too, had charged the fascist guns.\n\n\"I suppose you wonder where have I been,\" said Schatzi, careful to keep it low. It was so dark now that he was seen as black on black, improbable but not impossible, which could not have been more appropriate to him. His voice, his rustlings, the thump of his butt-bones to the ground\u2014all expressed an unusual geniality. In his own way, he apparently felt it good to see Schild again. He refused an offer of the chair, although it must have been uncomfortable for an ill man to sit on the cold earth.\n\n\"I saw your note some days ago; although, I was otherwise preoccupied... no, really, I am just pleased, stay where you are sitting. The ground is not unpleasant. Beneath the grass you know is sand, it does not hold the water. Berlin is one single great island of sand on the Brandenburg plain, yet it permits a lush growth of plant life, no? In spite of its architecture, Berlin is a beautiful city, but so few foreigners know it and that is sorrowful.\"\n\nSchild assumed the cynicism of his answer would be _de rigueur:_ \"There is little left to see now.\"\n\n\"So much the worse for all,\" said Schatzi, not shouting but giving the illusion he was and with the kind of conviction that Schild recognized as having originated in higher chancelleries than either of them would ever subordinate. \"It was senseless of the Americans to destroy the city. The most unfortunate way to win the German people away from fascism.\"\n\nLichenko had assured Schild that Russian artillery, particularly that multibarreled weapon known by the Germans as Stalin's Organ, had leveled more of Berlin than the Eighth and Royal Air Forces together, and he had agreed then as he did now, yet neither time in hypocrisy.\n\nSchatzi, making liquid, furry, catlike noises, swabbed his gums with his tongue, and then went with the keen tip of his smallest finger for a molar in the recalcitrant love of caramel. Actually he could not be seen with such precision: his ring flashed in transit, his cuff rustled, and Schild supplied the other details from memory.\n\n\"Not to speak of reconstruction,\" continued Schatzi. \"The Soviet Union has been given the most horrible section of the city.\" To go any farther would be to imply that Stalin had been hoodwinked by Churchill and Roosevelt. Indeed, he had already gone too far. \"You must indulge me in my English. Certainly I did not intend to say 'horrible' except in regard to the bombing there. It is, on one's other hand, the sector most worth for rehabilitation.\"\n\nThey had from the first always spoken English together, although Schild had often sought to turn to German, partly from a masochistic pride in his fluency\u2014and partly from the vicarious nostalgia in which he looked back on the time before his majority: the tongue of the old International had been German. But Schatzi had resisted, not so much from a pride of his own, Schild thought, as from his sixth sense for conspiracy, which told him that obscurity had as great a role as precision in underground technique. In one's second language, facts are never finally established; when blame must be cast, it can thus fall on the vocabulary and not the man, or if the man, then first on him who by birth qualifies for absolute comprehension. There is at any rate a possibility for such miscarriage, and the professional asks for no more, from his own side or the foe.\n\nExploiting, to himself, his ambivalent pleasure with the present confirmation of the hypothesis, he received another notice\u2014as he often did and as often was unarmed against, for it was his constant failure that though he had the imagination of disaster, he had not the mind. His suspicions rose the faster for his inability to believe in them. Not only had Schatzi never before transmitted the \"line\"; he had never been so generally obliging. His manner asked for forbearance, as if, getting that, he would go on to request ten dollars, repayable on demand. He was not, for once, in haste: he had never before sat at an interview; he had never before come to Schild's billet; _he had never before been pleasant._\n\nHe continued to be, despite his theme, which was the occasion for neither grace nor evil but the neutrality of fact. And the first fact laid in Schild's head by Schatzi multiplied within the minute; in the same minute that his heart multiplied its reasons for foreboding, his reason produced offspring, like some woman in Asia, or what-have-you _mise-en-sc\u00e8ne_ for the current classic instance of futile misery, who continues to reproduce like a mink notwithstanding the famine.\n\n\"In the Western zones,\" Schild said, \"all the ex-Nazis are getting jobs with the Military Government.\"\n\nThis cut off Schatzi for a moment of aggrieved silence. Now Schild had perhaps gone too far. His question was put in a voice that suggested this was the first time he had ever been brought to this turn of the road, one nearer the hairpin than was comfortable, and unless Schild could produce Automobile Association sanction he would drag his feet.\n\nThe sudden caution, standard operating procedure for anyone else to be met in Schild's professional circles, was unusual for Schatzi, a piece with the rest of his tonight unique demeanor. But that this was the norm and Schatzi's usual manner the oddity, had no force, for the ordinary Schatzi, who was extraordinary, was precisely what Schild had been prepared to meet in Europe. He had hated him, true, and he had just now begun to like him, but these nervous reactions were beside the point that Schatzi had been absolutely authentic.\n\nOr perhaps so directly to the point that they were invisible, integrated in the drama of hatred and fear and fascination of which Schatzi was a walking pr\u00e9cis. Whatever his temporary odors, to Schild he stank of the concentration camp; he had acquired there a beastliness which but for the final morality could not be separated from that of his captors. Hideous to think so; but only moral realism to know that the difference between saint and devil was frequently never revealed until the last judgment.\n\n\"By the way,\" Schild said at last, when Schatzi's moment of silence had lengthened into an evident volition not to speak at all, \"I saw the 'big lout' and have begun to go through the papers from his office. Only the top two or three cartons have Winterhilfe files. The rest is material from the Bund Deutscher M\u00e4del.\"\n\n\"A female division of the Hitler Youth.\"\n\n\"Yes, I know that.\" Schild was as usual irritated at being told what he already knew. \"At any rate, I have filtered out some things for you.\"\n\nNo answer. The sounds of Schatzi's breathing became quickly like the aspiration of a rubber pillow crushed by a thigh, and died. The crickets sang madly below the fence\u2014or wherever; if you went there to find them, they would instead be at the place you had left, and back there again only to hear their song in the bush. Behind Schild, a casement had its fastenings undone, its halves slithering open in slow provocation, followed swiftly by a broad drift of light that created a visible Schatzi but did not animate him. He wore bicycle cuff-guards resembling money clips. His shoes were swarthy, pebble-grained, and had long Italian points. The sole of one, showing a medallion of chewing gum in the arch, hung directly before Schild; danced to a rhythm that owed more to emaciation and senility than tacit music; the leg within its frayed sheath of woolen underwear was surely bare tibia and fibula and a snarl of ancient sinews. Long underwear in the middle of summer: for his pants cuff had ridden high, one bicycle clip, being sprung, failing; and he lay on his back in the grass, with one leg arched high, the other looped over it. Had he suffered a seizure? Schild rose to see beyond the bridge of legs, saw Schatzi's eyes wide open, bland and insensitive as two bottle caps, paralytic. Dread had just put down his immediate, instinctive disbelief, he had just received the full import of the underwear shroud, when Schatzi belched like a cannon and with a sudden effort of overbearing vitality raised to the sitting position.\n\n\"Queer person who lives in your quarters,\" he said.\n\nSchild turned expecting that the German woman, _d\u00e9shabill\u00e9,_ could be seen framed in the window\u2014not in concupiscence of his own but in amused anticipation of Schatzi's; he was captivated by the sudden transition from imagined death to carnality. Instead he saw Lichenko, in undress rightly enough, but Lichenko! Who, bent at the waist, lips funneled and eyes squinted in bestial ill humor, swung one arm apelike. He was naked. The other arm crooked in menace. In his paw was, again, Schild's .45.\n\nIn the haste to the door Schild yet had attention for the nimble Schatzi, who had sprung up beside him and maintained the pace at his elbow. He saw in his courier's action that which relieved his greater worry: would Schatzi, knowing of Lichenko, show the innocent curiosity of a boy chasing a ladder wagon?\n\nThey symbolically broke into the kitchen, for its door was open and only the oppressive light of the interior barred entry. The German woman lay stiff and still on her mattress in the corner, frozen in contempt, not fear, her handsome face fierce, free, and remote as an eagle's. She had, it was clear, cowed Lichenko with no more than her moral advantage.\n\nLichenko had jumped behind a high cabinet at the first sound of intrusion, where he thrust the pistol, or sought to thrust it, into the space between cabinet and wall. He was apprehended before this was managed. But, as if in that brief moment with himself he had taken a realistic account of his project's miscarriage, seen it, that is, as a mere limited venture gone awry with no permanent blot on the amour propre, he met Schild straight on, handing him the pistol butt-first\u2014to show, by its empty slot, that it was not loaded\u2014and offering his guileless face, open and unafraid.\n\nWas he drunk again? Schild had taken care to keep whiskey from his own room and Lichenko without direction had set a personal off-limits on St. George's quarters; he had in fact developed an unusual delicacy towards the house in general, which Schild found more difficult to excuse than the expected barbarism. Yet here was the return of the barbaric, and he, Schild, had run to brook it, in his reflexes one with the scared calves at Lovett's party.\n\nSchatzi, temporarily forgotten, spoke to the woman\u2014had been speaking to her and was now heard reacting to her consistent silence: _\"Keine Antwort is auch eine Antwort.\"_ No answer is also an answer: for what reason was she working with Schild?\n\nBut Lichenko was not drunk. He began to shiver from the cold and adjusted the cinch in the towel about his waist, for neither was he wholly nude. He was, indeed, suddenly nothing he had been, neither victim nor captor nor na\u00efve nor sinister, and as he prepared to speak from this new person, Schild struck him in the mouth. He had meant to knock him unconscious, so that Schatzi could not hear the Russian accent, but he had never before struck a person with this intent; he had never, since boyhood, struck any person for any reason, even comedy. He now punched too high and tore his third knuckle on Lichenko's teeth.\n\nIt had been as hard a blow as he could summon in cold blood, but with only the free-swinging arm and no body behind it, did no physical damage. Lichenko, however, was whipped, all the more for his initial show of dignity. He grasped again at his towel, grinned in coy brutishness, rolled his head like a fawning dog. And then he whined, in German, and all was lost: \"My friend, this whore tempted me!\"\n\nFrom the other corner Schatzi burst into his aspirant snigger, and an oxlike plodding at the door announced St. George, who, in maroon robe with white piping, slippers with elastic inserts, and pajamas a continuum of pale-blue hounds-teeth, after some deliberation had formulated his amiable comment.\n\n\"This looks like Grand Central Station!\"\n\nThe pistol in his right fist, Schild furiously cut its barrel into Lichenko's cheek and, as he went to the floor, followed him down, hacking him down, not ceasing his awful work until St. George, whose cries had gone unheeded, fell on him and stilled him with his bulk.\n\n# _CHAPTER 12_\n\n\"CAN'T FIND ANY LETTERS of Grandpa's you asked for\u2014stuff all cleaned out from under the porch to provide place for screens years ago,\" wrote Reinhart's father. \"Maybe you even did it yourself\u2014if you were paid for it. If my advice means anything, tho, I'd drop the idea\u2014your just asking for trouble\u2014as soon as you find any German relations they will want to borrow money from you... ,\" etc., typed on a V-mail blank, small as the Lord's Prayer engraved on the head of a pin.\n\nAsk a stupid man, get a stupid answer. When he told Trudchen about it\u2014for, despite her peculiarities, she was still around, still without pay, reporting to the office every morning long before he arrived\u2014she said: \"Ah, vy bozzuh! I will be your relative.\"\n\nThat was all he had told her. He did not seek to expose her pitiful lies; he let stand the assumption that she lived, orphaned, in the little back room in the office building, where indeed she did report at the end of each workday. Above all, he remained silent on the visit to Lori's. What he had learned there was for adults only, and he was not at all certain he could stand to think of it himself.\n\nHe at last understood that the complement to his long self-identification with Germanness had been a resolve never to know the German actuality. Knowledge had exhausted his options; he now had no choice but to seek out, if still they existed, his links to what, a brief half-century after Gottfried Reinhardt took ship for the New World, had disintegrated in murder and betrayal.\n\nHe had not really believed the witness of the Buchenwald photographs; mass exterminations were incredible. Real deaths were your friend Bill, one moment live, the next run down by a drunken driver; Al killed by pneumonia; Roy, his heart full of Jap metal, taking the Iwo Jima bastion and expiring a hero; or someone's brother, well known, electrocuted by the state for the crime of homicide, and his victim; these corpses were believable in sight and mind; despite the mortician's garish art, beyond the mystery of any death, were the concrete memories of impediments of speech, casts of eye, a rolling gait, a red Ford with a two-tone horn, and the only four-button suit in southern Ohio.\n\nSimilarly with the violet shadows under Lori's eyes. Whoever had sold her safety from incineration had seen them upon every payment, must have had the queer guts to imagine their transformation into white ash and his own agency in the burning. And the man who would have fired the oven, dressed in his black SS-suit, with his blond crown and his blue eyes, the model to which every boy aspires, the handsome soldier fearless before the enemy, gentle with women. ... These types were not explained by the simple, pious indignation of: two kinds of man, one good and one bad; we of course are the first; they, the second.\n\nNor by the lack of a democratic tradition: was this what men did when denied the vote? Nor militarism: you mean that the great Frederick mounted his stallion and rode down women and children and unarmed men, and that the old knights of N\u00fcrnberg swung their blades against little ghetto-tailors?\n\nReinhart had been reared in what he assumed to be (since everything else was) the German code; there are two kinds of cowards: one who will not fight a man his own size or larger, and one who will fight only someone weaker; sometimes, but by no means always, the same person. But the validity of this, too, was here outmoded, for the SS man, fresh from his ravages on the helpless, stood fast against the superior enemy; was, to be sure, the fiercest soldier met by the Allied troops.\n\nAs to the anonymous blackmailer, Reinhart insisted that his, too, was a strange, mad kind of courage, for beyond gentleness and humanitarianism and a deficiency of passion, what stays the normal man from murder or even its threat is fear, not of the godly or human law or vengeance or nightmares, but of the suggestion of his own mortality.\n\nHere all the known qualities of humanity had been united with their contradictions. This was what Bach dramatized in his monstrous monologue of truth in falsehood, that guilt could be confessed to only in a lie of the guiltless, that the first loss of the criminals had been in their human imaginations. Where Reinhart had looked in Germany for life, first in dreams of ancient glory and then, after the Nazis, for a vitality at least of evil, he saw only a horror of deadness, of which the literal corpses, the loose skins of Dachau, were but the minor part.\n\nYet more important than this moribund nation were the good people, those \"good Germans\" on whom the sanctimonious propaganda of Our Side did its work of slaughter, the mature ones like Bach who by conscious volition stayed decent and sought no fanfare for it now, and children like Trudchen who willy-nilly were clean. Were his relatives to be counted with them?\n\nEven in his duty of conscience, however, he was balked by the same ineptness which characterized his dealings in the humdrum; when fountain pens were hard to get, people like Marsala had pocketsful; similarly with liquor, broads, and passes; he, Reinhart, so damned special, one of the ought-ought per cent of the American population to go to college, a member of the owning and stable class, could manage nothing.\n\nIt was very well to say loosely, as Lori had, to go to the burgomaster's office. He tried just that, visited the town hall in Sch\u00f6neberg, which he was astonished to see employed as many bureaucratic flunkies as it were an American city untouched by war, who notwithstanding that he was Occupation showed much the same bored insolence and then when pressed claimed a search of the birth records back to 1850 turned up no Gottfried Reinhart. Of course, there was always the Russian Sector, which the eyeglassed clerk recommended snottily-reproachful, as if to say: that's what you get for dividing our city. There was what Reinhart would earlier have identified as certainly a Nazi; now he thought it more likely the man might turn out to be an unsung hero of the anti-Hitler opposition and this job his reward.\n\nHe got aid from an unconsidered quarter. Although when he had first revealed it to Trudchen his project left her cold, she greeted him one morning with sudden interest and suggestion.\n\n\"You must have a _dett_ -ek-tive! And I have just your person. The man who makes some work about here\u2014he with the scarred face. He is called\u2014so silly!\u2014Schatzi, that means 'sweetheart,' did you know? Do you know which I intend?\" She had her own table now, a jittery-jointed piece which swayed like a drunken spider when she assaulted the old Underwood. \"He is very active in the black market. This takes him everywhere and in consequence to that he knows everyone.\"\n\n\"Not the old man in the Wehrmacht cap?\"\n\nA regrettable concomitant of Trudchen's employment was false tint laid on thickly over her natural color, and Reinhart also bore the guilt of that. He had bought her lipstick and rouge from the PX, on her request for the \"raddest of the rad.\"\n\n\"Oh, he has worn it, yes, but also many other costumes. When he sells one thing, he attires himself in another.\"\n\n\"You don't mean that old man who works in Lovett's office?\"\n\n\"Not regularly. He makes much money on the black market\u2014why should he vorry?\"\n\nAnd he had thought the old fellow pitiful; it was a true instance of what one, disinclined to contribute, says of street cripples with their tin cups: they could probably buy and sell any of us poor working stiffs.\n\n\"I don't suppose he was a National Socialist?\" Reinhart could no longer use \"Nazi\"; with the passing of each German day the term became more like the name of a soap powder, some slick and vulgar \"Rinso\" invented by Americans, who eventually reduce everything to that level: \"Nazi,\" the cute name for a pack of buffoons, played always by the same actors, regularly thwarted by some clean-shaven Beverly Hills Boy Scout whom a ruptured eardrum disqualified from the real war.\n\nHowever, he was not wholly serious even in putting it the long way, since in this area Trudchen's unreliability was massive. Perhaps understandably, to her the history of modern times was a catalogue of her own losses and the responsible instrument, fate in general.\n\nShe lifted her little painted clown's-face, the freshness obscured by the rouge but the innocence still there, and said: \"Not he! He was a prisoner in a concentration camp.\"\n\nWhich was a flat lie\u2014although perhaps not hers but the old man's; the surviving martyrs of the camps were hardly thrown into menial jobs and black-marketeering.\n\n_\"Ausgezeichnet! Prima!_ Then he should be just the man to find _die Familie_ Reinhart,\" he said in an irony that she did not receive. \"Of course it isn't likely he'll find anybody. There's a separation of fifty years. Think of that, Trudchen, the last time I was German my father hadn't yet been born.\"\n\n\"Please?\"\n\nInstead of clarifying it, he fell to work with his pencil\u2014which was blunt and unpleasant to use; if she didn't soon return his pen he must come right out and ask her to\u2014on the long-delayed Guide to the Ruins for the sightseeing tour.\n\n> The Olympic Stadium, built for the Olympic Games in 1936, has a seating capacity\u2014\n\nOr was it more graceful to say \"seats\"? Or \"seats\" as a noun: \"stadium, etc., has 124,000 seats.\" \"Capacity\" of course had a more serious tone. This was one of those days when nothing sounded right, which unluckily had begun to outnumber those on which nothing sounded wrong.\n\n\"You do not wish to hire this man?\" asked Trudchen, starting to type the stencil for Page One, which, for Pound had decided on a grandiose project that would impress the colonel, was to stand as title sheet.\n\nHe had to grin. All European girls spoke with an animation at once funny and delightful, an excess of feminine vitality that juiced each word. If this held even in a sadness like Lori's, with Trudchen, who was never less than gay, who was young and unmarred and in a perpetual celebration of ripeness, it was the very model of unalloyed girlship; you never, as sometimes at Home, suspected that you confronted a transvestite boy.\n\n\"Ah,\" she went on, \"how hoppy you will make zem! In these timess to have an American cowsin!\"\n\nIn mock grimness he answered: _\"Our American Cowsin._ I hope for better luck. That's the name of the play Lincoln was watching when he was shot.\"\n\n\"By Chon Vil-kes Boat, yes?\" This in an eagerness which threw a tremble into her physical establishment. \"And the year, 1864, yes? The day I do not know.\"\n\n\"Don't ask me!\" He ambled to the French window to look on as perfect weather as the earth offered, the life-enhancing air of the Brandenburg plain, full of golden light and green smells. \"My family wasn't in the country at the time. They were here.\"\n\nCould Jews have been killed on such a day, or had they waited for rain?\n\nThe great pines stood high in the adjacent grove, and seeing down among their feet he recognized the steel-gray, crosshatched shadows from old German engravings, which were not artist's strategy but the true lay of the land. He could have watched without doubt a delegation of trolls emerge from some root-home and bear away the Nibelungs' lode, but impossible to the mind's eye were the long sallow lines of victims.\n\n\"This man, this good German, how can I get in touch?\"\n\nTrudchen giggled like a spring: \"Tahch\u2014this is very vivid and so clear that no explanation is needed\u2014baht he vill come here some time. I have taken the freedom to ask him that you might... vould... could\u2014oh well, that you want to see him.\"\n\nAlong with the cosmetics she wore a peek-a-boo white blouse disclosing an eyelet-margin slip and, beyond, the rim of a brassiere which carried larger burdens than formerly had hung upon her chest, and the pigtails no longer swung free but were entwined about her head in a yellow cocoon. In the aggregate, this was also a lie: that she was a mature girl.\n\n\"If he was in a camp, then he must be a Jew?\"\n\nAsking which he returned to his desk and fell into the chair with the noise of a beef haunch flung onto a butcher's block.\n\n\"Oh no!\" cried Trudchen with candid enthusiasm. \"You are incorrect when you think only Jews were mistreated. You do not know of the Resistance?\"\n\nSure, the plot to kill Hitler of 20 July 1944. This had already been exposed in his discussion groups as a conspiracy of reactionary generals, scarcely better than der F\u00fchrer himself, whose motives were suspect and results, a failure; and who were eleven years late.\n\nOf course there was that\u2014she took no notice of the negating conditions, perhaps because he lost his nerve while talking to her, who was blameless, and presented them weakly\u2014but what she meant was something of a greater scope and duration, embracing all of the non-Nazi population: a total rejection of Hitler and all of his works, dating back to 1933 and earlier. She as a German could tell him that, even though she took no interest in politics, being young and silly.\n\n\"And what did they do about it?\"\n\n\"Ah, what can anybody do against beasts who are ruthless? The SS and the Gestapo, their first job was to control Germans, not Jews.\"\n\nHe sat upright and brought down his fist upon the desk, not in anger but rather a kind of pleading.\n\n\"That is understood. But it is over now. National Socialism turned out to be nothing. You couldn't find one German today who would say a good word about it. Yet it was a _German_ thing, wasn't it? I don't mean the war, or the Axis, but what went on here: a horrible, dreadful thing that was completely new. Old Genghis Khan and Attila the Hun were saints alongside of this. The whole history of man is disgusting, I grant you, but why would the Germans try to set a new record? But no, I don't even want to ask that. God knows if I had been a German what I would have been. _But why can't someone at least say he is sorry_?\" He looked into space, for he had no wish or reason to make it personal.\n\nHe was an idiot to speak of this to Trudchen, and she was quite right to look calfly insensate and say: \"One cannot be sorry for what one has not done.\"\n\n\"You must pay me no attention,\" she went on, \"because I am not clever, but what I can see is that God makes people suffer.\" Her mouth and eyes went into round wonder, which made her, there behind the crazy lines of lopsided table and old typing machine, a complex of circles: head, eyes, glasses, mouth, breasts, hips. \"At eleven o'clock in the morning of 3 February, this year, I had the fortune to be in the Bayerischer Platz Underground station when your planes came over making a direct strike with an aerial mine that blew a thirty-feet hole out of the bottom of the tube. So suddenly I did not feel anysing, no wownd, and knew only what occurred when this baby in the arms of the vo-man in front of me, now, with the blast, on top of me, this baby stared down and tried to cry at me but instead of the cry this string of blood dripped quietly from its mouth. It was alive, but dead, also; both at the same time\u2014how can I explain this terrible sing that I mean! Your planes had come to kill Nazis, but the bombs cannot tell good from bad. A little chilt of eight months old, it had to suffer. Is it not the same way with God's vengeance for the murder of Jesus Christ?\"\n\nIt was wackily, harmlessly funny, as when the village crank says of the cyclone-torn bungalow: this is what they get for all that drinking. But she was growing into a big girl, and it was time to be set straight\u2014which no one had bothered to do for him when he was on that level.\n\n\"You don't\u2014\" he began, when Lieutenant Pound appeared in the doorway and Trudchen hurriedly flung back into her story.\n\n\"So when this blood began to descend upon me I reached towards my sleeve for the handkerchief but my hand could not go far, being halted by a soft, varm, cling-ging mass such as one's hair after washing it, and I thought: so I have lost an arm, how easier in the fact than in the worry. Limbs, limbs, I have always feared losing them most.\"\n\n\"Don't bullshit, Trudchen,\" said Pound, patiently genial, closing the door which was in his absence never closed, demonstrating his talent for violently hurling it to without its latching: he \"pulled\" it, as one does a punch in a false fight. \"You've got two bigfat white arms today.\"\n\nAlthough his monastery was now neat, this abbot had stayed slovenly; as he went briskly to his desk below the little window, his loose shoelaces clicked, his tie end flapped over his shoulder, his bowlegs like two lips endlessly yawned away from each other and gulped shut.\n\nPerhaps it was Pound's own experience in violence: he never believed anything she said. And by his example, Reinhart, too, invariably lost belief. Although, given her time and place, the tale had been credible enough at the outset, with the introduction of self it became fiction like all the others. She was, he had to face it, the most incredible liar he had ever met.\n\nWith a significant look at Pound, who was too bored to register it, Reinhart said: \"Go on, Trudchen. What happened then?\"\n\n\"Well, it was really an arm, but blown off from someone other and lodged between mine and my ripps, as if it were robbing my pocket.\" She placed a rolled-up stencil in the position described; buff backing to the outside, it was a painfully authentic replica.\n\nHer attention was now directed exclusively towards Pound, and Reinhart, in half-conscious jealousy, went to block her line of vision.\n\n\"You know what? You are a prevaricator!\"\n\nSilently, Trudchen unrolled the third arm in the enormous self-confidence the mythomaniac shares with the artist, while at the same time her round nose sharpened as if in death, as if for a moment she really tested that condition the truthful call life, and rounded again as quickly; she had been there before and did not like it.\n\n\"Stop pissing around with the kid,\" Pound ordered irritably. He was in a rare short mood, probably connected with the miscarriage of certain affairs of money, towards which these days he had developed an obsession. The black market had denuded him of watch, pen, pocket knife, cigarette case, lighter, ring, identification bracelet, all bedding but one blanket, all ties, shirts, drawers, undershirts, socks, and caps beyond one each, towels, writing paper, the leather frame of his wife's picture, and his musette bag. Three days earlier he had received by mail a new pipe and pouch: the latter had already metamorphosed into a paper envelope. Which he rustled in now, spilling much, but onto a page of _Yank,_ which when done he coned to funnel back the overflow, his narrow eyebrows shimmering ever upwards like heat waves fleeing a summer pavement.\n\n\"Haven't you finished that guidebook yet?\" he went on, with querulous twitchings. \"The colonel has a wild hair in his asshole ever since Lovett's Folly. He might put us on cleaning butt cans any minute.\"\n\nBecause he was properly a cigarette man, he smoked a pipe the wrong way, inhaling great mortifying draughts which after a time in his innards came back through every superior aperture, mouth, nose, ears, eyes, suggesting that his head was afire.\n\n\"I'll finish it today,\" Reinhart answered sullenly, not unmindful of Trudchen's spectacular show of industry; she socked so loudly at the typewriter you couldn't hear the clearing of your own throat. No sooner did a third person come than he felt odd man out; his maximum for rapport was one being at a time. Thus it was fine with Pound alone, or alone with Trudchen, but with three people he invariably sensed a conspiracy against him.\n\n\"Oh good,\" said Pound. \"If you are that close to the end, you can put the fucker aside for fifteen minutes and write me a letter to the wife. You know, this and that, etc., and I'm short on dough because we had to buy new winter uniforms this month.\"\n\n\" _You_ short on dough?\"\n\nPound made a sighing descent into his swivel chair. \"Come over here,\" he said confidentially. \"I don't know why I can't tell you, since you know all my other chicken-shit business. The thought of going back to that woman\u2014the one you write for me\u2014is more than I can stomach. You know, when I was wounded I made kind of an agreement with Fate that if I didn't die I would be somebody new. I never told this before to anybody in the service, but I used to be, before I was drafted, a bank teller for thirty-seven fifty a week, a creepy little rectum-kissing rabbit with two snot-nosed kids and a dog with some kinda skin rash that made his hair fall out in pink spots\u2014he also used to sit around on the rug in the evening and fart all the time\u2014and this woman, see. Well, she isn't the worst person in the world, but she is set on making a man a coward. She even wanted to scare me out of using a blowtorch to take off the old paint on the outside of the house\u2014which I was only doing cause who can afford those prick union painters and if you hire scabs the others will come by and bomb your house\u2014you'll start a fire, she said. And by Christ I went ahead and did it anyway, and you guessed it, it did start a fire that burned off one wall. I never missed Bob Hope's radio show on Tuesday nights for five years\u2014Professor Colonna: 'that's what I keep telling them down at the office'; Brenda and Cobina, and the rest of them. Think of that: 259 straight; once they were off because of a special news feature, something about that fucking shitbum Hitler. I tell you I was yellow as they come, but after basic they sent me to OCS where they thought that was just being cautious, I guess, a good quality for a officer. Well, we were pinned down along this hedge row in Normandy and I was dumping in my pants for fear, but still I noticed my top fly button was loose and I fastened it. And then I thought what a dirty little turd I was: with your ass about to be blown off and you button the barn door\u2014do you get the picture? I was more afraid of my dong showing than of the German 88s. So I thought all of a sudden: World, you got twenty-eight years from me, you can keep all the rest and stick them up your giggy, and I jumped up and went across there and took that Kraut platoon, and I don't mean to say I wasn't scared, but anyway for once there was a reason. Shit.\"\n\nHe had puffed so hard on his pipe that already its tobacco was exhausted and the air made noxious.\n\n\"You know what I made so far on the black market? Thirteen thousand, two hundred and twenty-two dollars, and it's all gone back to the States to a bank in L.A., California. That's where my nurse Anne Lightner is from, L.A., where they go in for the beach living. I'm going to get sprung from this woman as soon as I get home, and then I'm going out there and buy a used-car lot. That's the kind of thing they go big for out there, with all that beach living. Everybody drives a car, that's what Anne says.\"\n\nSo was another idea exploded. It was sad, in a way, that nobody, simply nobody was what he seemed. To Reinhart, Pound had been the classic type of swashbuckler. Now he saw the late bank-clerk lines of worry and doubt, faded but still visible, at the corners of mouth and eyes, and he even liked him better for them\u2014for daring has no unusual moral worth if you have lived with it from the cradle\u2014yet there was no discounting the loss of something rare.\n\n\"But I have to play it cool with Alice till I get back and can defend myself,\" said Pound, refilling his pipe. \"So write her nice. I don't have to tell you what to say, you have enough crap to snow anybody.\" This was admiringly put, with the quick wink he must have learned in his new life, but looking sharply Reinhart saw the hint of a quaver in it, as if, in at least the most minor part, there was still a tinge of bluff.\n\nAll the while Trudchen had been typing with fanatic energy\u2014faking madly, for the guidebook manuscript lay on Reinhart's desk.\n\nAs he passed her on the return route, a doorknock sounded, and notwithstanding his shouted \"Enter!\" she leaped up and teetered to the knob\u2014high heels, yet!\n\nIt was a soldier, for Pound. She made him wait while she proceeded to the lieutenant with a formal announcement, working her body in a queer movement which Reinhart first believed was an effort to balance on the high spikes and then recognized as an amateur version of a whore's undulations. Her breasts were hard metallic cones, yet she still wore the thin, little-girl's skirt ending an inch above the knees, and still the owlish, juvenile spectacles. Involuntarily he burst into a loud, barking laugh, which hideous though it was nobody but himself seemed to hear.\n\nLieutenant Schild's judgment had been correct, only a bit premature (as an Intelligence man, of course, he was expected to be one jump ahead of events); if she was not on her way to tartdom, then Reinhart was an orangutan.\n\n\"Dearest Alice,\" he scrawled on the yellow pad, taking in return a warm thrill of fancy that this unseen proxy wife was really his own, that he had entered her in the connubial bed and that she had borne him two small resemblances of himself, albeit snot-nosed.\n\nOn Pound's indifferent grunt Trudchen wobbled back to her table. Reinhart had also purchased the mascara which gave her an appearance of sore, fire-tinged eyes, but the high heels were from another protector, he now had no doubt.\n\nThe soldier had gone. In his stead, in the hall shadows beyond the half-open doorway stood a shrouded representation of a human figure, crepuscular, mysterious. Upon Reinhart's look it slid noiselessly out of range. Sauntering, Pound took Trudchen's typewriter from beneath her very pounding fingers, ripped out and discarded the paper, and saying \"At last I found the Kraut who can fix this old machine,\" left.\n\n\"Darling Alice: Sweetheart, I\u2014\" Reinhart began again.\n\n\"You try alvays to hoomiliate me...\" Trudchen's lips were fashioned into a little red crossbow, through which slid the pink bolt of her tongue, in and out, tasting the lipstick.\n\nHe threw down the pencil in disgust, said malevolently: \"How about returning my pen?\"\n\n\"Vy do you always do this? Because I am only this little German girl?\"\n\nHe strode massively across and bruised his fist on her table: \"Right now, I want that pen!\"\n\n\"Oh, Gee whiz!\"\n\nFind who taught her that and you had the whoremaster: Reinhart had never said \"Gee whiz\" in all his life long. But the tears were her own. He had last seen them when she cursed that poor Jew for telling the truth.\n\n\"Well, Gee, take it beck again, and don't say I vas shtealing it.\" Engulfed by the mixture of water and words, dissolving mascara, smeared rouge, falling hairpins\u2014for in the grief she tore her hair down into the old pigtails\u2014she opened the middle drawer and drew away.\n\nReinhart came round behind her. There it lay, the old black Parker, that gallant, veteran instrument of romance and adventure on two continents, vicarious cannon, sceptre, phallus. He seized it, already feeling the brute, and when her blue eyes peeped sideways at him over their scorched rims and she said \"I _opp_ ologize\"\u2014by this time he had long forgotten what the beef was and took the pen merely so as to return it to her formally, as a permanent gift.\n\n\"So kind,\" she cried, smiling-through-tears. \"Do you care for my shoes? I have yesterday traded them with the chocolate you gave me.\"\n\n_\"Fabelhaft!\"_ He stood behind her, hands lightly riding her narrow shoulders, eyes descending into the sweet crevasse of the pectoral range, very clear through the thin cloud of blouse.\n\n\"And I have somesing for you,\" she said, \"so you will not think so bad of this little Germany.\"\n\nFrom the drawer she withdrew a handbill of cheap European paper, weightless, the color and grain of whole-wheat bread, infamously inked. All he could read from where he stood was a headline: ES LEBE MENSCHLICHKEIT!\n\n\"Proclamation of the Resistance,\" she crooned victoriously. \"I have found it in this very room, in the carton-boxes. Perhaps this selfsame room in which we sit was nothing but head-quarters!\"\n\n_Long Live Humanity!,_ no doubt to be understood in the sense of Hitler's Peace, a peculiar German cruelty. He received her greatest whopper with an enervation so profound as to be almost pity.\n\n\"Trudchen, I can read German...\" he groaned, his hands rising heavily from her shoulders and more heavily returning.\n\n\"Then read!\" she screamed, turning in frenzy, and his left hand traveled into her blouse at the open neck and down the breasts' warm canyon. Her mouth, open throughout the quick transformations of fury, fear, awe, and finally, madness, rose to his neck like the sucker of a great vampire fish surfacing from the depths of the sea, fastening to the elbow of his windpipe, so that, prohibited from breathing he fisted a tail of blonde hair and pulled as if to sever her head from the shoulders. In a moment his large right arm proved stronger than small-girl lips; he had her loose and held her gaping, an interval for bullying mastery, and then turned her, brought her forward and up, the nether hand taking a purchase within her fat furrow, hot beneath cool cloth, and carried her to cover the light snow of tobacco grains on Pound's clean desk.\n\nHe had come so far in what had seemed desperate comedy, as in school when the kids steal your cap and you tolerate their passing it just out of reach until the smallest boy is the bearer and you engulf and batter him to the point at which his incipient grief takes the laugh off you. But Trudchen now had fear least of all, and laughed, herself, as one does whose will is consonant with the world's; the little witch's face in a garish disorder of evil, yet her odor was childlike, of soap.\n\nIn endless pursuit of pride, then, he became fastidious, working his way through the jungle of queer fasteners and ribbons, and the three buttons which at the crucial junction of her parts secured the last guardian triangle of doveskin fabric, beaching finally upon a little round belly incapable of further discovery.\n\nThe key in the devil's lock, _entrez monsieur, enchant\u00e9 de faire votre connaissance,_ excruciating, pain, pain, pleasure\u2014well into that groove of unification where the senses are harnessed towards a single fanatical end, his suddenly lost purpose. Ah, it was all so crazy. A small window broke the wall above Pound's desk, high above\u2014standing at full height Reinhart could just frame his face in it\u2014and absurd, fit only for some lazy postman on stilts to pass a parcel in from the outside, to save a trip through the labyrinth. It was from this glass that he got an immaterial signal into the corner of the eye, and as if to breathe and moisten the throat, he straightened and turned his head, saw close up to the pane the feathered neck of a man who wanted a barber, Pound's; beyond and lower, a face like a contour map of an asteroid, ripped and pitted by hot chips flying off Jupiter; two had by accident embedded collaterally and, still smoking, were eyes: ostensibly directed at Pound, but seeing him, knowing him and what he was at, not caring, not even amused, but knowing. The old German, now named: Sweetheart. In exchange for the typewriter he presented a thick wad of notes. Pound buttoned them in an upper pocket and, one-breasted like an Amazon, vanished.\n\n_\"Mein Tiger!\"_ whispered Trudchen. Looking down, melting, Reinhart felt rather than saw he had unwittingly been a success. He had also forgotten all precautions, and swift through his mind like an Army documentary ran the series of awful upshots.\n\n_\"Ach,\"_ said Trudchen, yet hypersensitive, opposing partition, \"I have taken care...\" Not knowing to what she referred, he accepted her assurance.\n\nCrumpled in her fist, the old handbill, taken in surprise like everything else, was still their partner. He tore it from her and read the first line below the bold title, read it twice as with his unoccupied hand he returned himself to order. It did not change: \"The appeal of Hitlerism is to the eternal _Schweinhund_ in man.\" Of course it was anti-Nazi; no matter by whom or where, it had been produced in honor and conscience and at cost, and its anonymous author, if he had eluded his compatriot enemies, had lived perhaps only to drown in the same foreign flood that swamped them.\n\nHe kissed her, long and exploratory, for the first time, and saying \"Ah, I must be crazy, anyone could have walked in,\" he burst away, she moaning in the sudden isolation. He ran through the French window and around the corner, and saw that Schatzi had not, because of his heavy burden, got farther than the public sidewalk.\n\nSchatzi accepted the inevitable cigarette and slipped it between his ear and the drooping rim of the workman's cap that with neckerchief and soiled jacket and weary trousers formed his present costume, which he would surely have had trouble in selling to a naked man.\n\n\"Do you need some conversation?\" he asked, with a tremble of his nose, \"or is it simply generosity? Excuse my lack of strength.\"\n\nHe placed the typewriter upon the octagonal stones of the sidewalk. No sooner was it done than a woman rode by on the adjacent bicycle path and they felt the slipstream of her passing.\n\n\"Into the mechanism no doubt this blew some sand,\" said Schatzi, his voice like a dumping of gravel. \"So much longer to clean!\" He elevated his hands in a Jewish shrug, and while the right one was up, put out a finger and ran it across his upper lip, making a gargoyle mouth.\n\nSeeing him now in reality and close-up, Reinhart could not doubt his girl friends' tales were true: if Schatzi were not from the concentration camp, then that establishment was illusion. True, he was more than mere skin, but give an unfilled pelt a few months' meals and you would have Schatzi. He lived, but just lived and no more, with not one breath beyond the essential. His face was dreadful, romantically hideous, in the ugliness only supreme virtue permits, perhaps creates, as with the old saints; and though his angles were sharp, his constant tremble blurred and made them remote.\n\nConfronted with this overwhelming authenticity, Reinhart on the instant forgot his purpose and, instead of speaking, sent a grin. He watched Schatzi catch it, warp it with the secret they shared, and send it back.\n\n\"Your breathing is labored,\" he said. \"Exercise shortens the span of life. He lives most long who lies in one place without movement, like a piece of warm bacon, all his life long, _ja_?\"\n\n\"I never thought of that,\" answered Reinhart. It seemed so marvelously reasonable; he put from his mind the obvious reference to the tumble with Trudchen and worried about the years gone in nailing down his coffin with a barbell. He had never before talked with an authority on mortality\u2014who yet, he saw with a happy loss of trepidation, was also a human being, whose smile was only superficially diabolic.\n\nFor a great sweetness was exuded by Schatzi's hard person as he suddenly stared into Reinhart's face and said: \"You wish to send me on a qvest, _ja_? She told me, this little piece of sausage, this Gretchen\u2014\"\n\n\"Trudchen.\"\n\n\"So. You search for your kinfolk\u2014this is correct, 'kinfolk' or simply 'kin'?\"\n\nSo close was he, perhaps by reason of defective hearing, he almost climbed Reinhart's frame. It was disconcerting, especially since Reinhart judged from his clothing that he must stink and drew always away, until on the fifth circle of their patch of walk he envisioned how from a distance their two figures must look in revolution and permitted himself to be captured. He had been quite wrong: Schatzi put forth the distinct odor of eau-de-cologne.\n\n\"Wwwwell,\" said Schatzi, \"you have come to the right potty. Ve vill\"\u2014successful pronunciation of the first _w_ satisfied him in perpetuity\u2014\"simply look for all the Reinharts who are not yet dead and there you are!\" He actually winked, which is to say one eye was swallowed whole by the lids, like a ravenous bird ingesting a black cherry.\n\nImpossible to think the concentration camps had not been serious; therefore what Reinhart saw before him now was the human triumph, a wit which had faced the dreadful and survived, no cloistered humor like his own. He himself was suffering depression, feeling wet and dirty and unusually exposed, and indeed, since Schatzi had taken the initiative he was no longer interested in his own mission.\n\n\"I don't want to remind you of your troubles,\" he said, though of course he did, \"but would you say the concentration camp was the worst thing that could be imagined?\"\n\nIf Schatzi had earlier been ebullient, he now went into a positive delight that Reinhart, because he had no experience of the world, found very grisly.\n\n\"Ah, no, no, not the worst! The worst, my young friend, is to die. Just that simple. Two added to two makes four, always. The living and the dying, and nothing else, makes ray-oll-ity.\"\n\nSo Reinhart, conscious it was asinine but getting no other suggestions, gave him another cigarette. Which went behind the other ear.\n\n\"Now you must tell me an answer,\" Schatzi said. \"Why must you find these relatives? Of course,\" he went on before Reinhart could speak, \"to help them. You Amis are a decent lot. You do not become happy to see anyone starve, let by themselves relations of blood, _ja_? This gives one faith for the future of the world in your hands.\"\n\nHard as Reinhart looked among the rocks which clicked together in Schatzi's voice, he could find no insincerity, therefore he stifled the impulse to say \"Horseshit!\" He had at last, there could be no mistake this time, found the man with a right to say anything and it be valid. Not even Bach and not even Lori, not even when he had learned their truth, had so impressed him.\n\n\"I'd think you would hate the Germans.\"\n\n\"I hate them? My friend, _I am myself a German._ \" Saying which Schatzi bent to the typewriter, on the way down adjusting his cap, the crown of which was dark with oil. Someone had borrowed his tie to hang a felon and returned it with a frozen knot that would never undo; no doubt he had it wired to his collar or to that frail armature on which his pennyworth of skin was hung.\n\nA marvel that he could pick up such a weight. Reinhart moved to aid him but was waved off.\n\n\"But one detail\u2014\"\n\n\"Of course.\" In this regard Reinhart never admitted another as master. He produced his wallet and counted off five hundred-mark notes, fifty dollars, from the wad of five thousand which Marsala had got from a Russian soldier for Reinhart's graduation watch.\n\n\"I didn't mean you to do this for nothing.\"\n\n\"Now,\" said Schatzi, \"you have shamed me with your generosity. Ray-olly, I cannot\u2014\" He drew from his pocket a brilliant blue handkerchief and snorted into it, thin and airy like a fife badly played. He took the money. \"This is not what I purposed to say\u2014which at any pace, I have now forgotten.\"\n\nReinhart watched him go down the walk with his burden. Twenty feet away, he turned and shouted, \"You shall hear of me!\" And then he moved off the pavement into the trees, where he spat fiercely and vanished.\n\nReinhart had neglected to give him his grandfather's name! Hot on the trail he ran, through the patch of forest to the wide prospect of Argentinische Allee, and surveyed the feasible directions. But Schatzi was gone.\n\n# _CHAPTER 13_\n\nSCHILD'S FATHER'S BUSINESS WAS concerned with buttons\u2014well, you know how capitalism works on the petty levels, he neither made them nor used them, but stood in the middle between maker and user, collecting a profit.\n\nLichenko, however, did not know these things, which was why he asked. He was especially interested in the money: were the earnings large from such a trade?\n\n\"He never thought so,\" said Schild, \"But they were considerably better than working-class wages.\" His smile was both bitter and genial\u2014the first towards the distasteful topic; the second for Lichenko, to whose will he was now committed.\n\n\"Oh, but the workers, we will not speak of them,\" Lichenko said contemptuously. \"You surely are of a superior class.\" This was the kind of thing he had been saying, in one way or another, for three days, and Schild could not yet gauge the degree of its subtlety.\n\nLichenko closed his eyes now and breathed profoundly, as if he were falling off. Sometimes he did; sometimes, after the same indications, not. The game hinged on whether or not Schild rose to go: if he did, Lichenko awakened; if he did not, Lichenko slept.\n\nThe bed was a chaos of stale sheets decorated with brown blood and streaks of St. George's iodine salve. Lichenko had not left it since they laid him there on the night of the beating. Not that he had been seriously hurt: his actual wounds\u2014a slash of the cheek, an abrasion of the lower lip\u2014had, after the excitement was done, proved superficial. The rest were bruises, ugly, indigo-and-lavender, but bruises, and had already begun to pale under the application of St. George's paste. And he had been struck only in the face, so that his body was as sound as ever and could have no special need for this perpetual pillowing.\n\nYet there he lay, sometimes straight and stiff as a corpse, suppressing breath; sometimes curled like a foetus, in which position he made bubbly noises; sometimes with limbs wanton and torn mouth wearing a wan, roguish smile, as if he had dropped there exhausted from a saturnalia.\n\nSchild felt towards him a strange, new emotion: not, as in the case of Schatzi, loathing compounded of fear and envy, and certainly not the fierce hatred which was the sudden motive for the beating\u2014indeed, the latter had been transformed in his memory to a distant episode involving two strangers who bore no resemblance to the Lichenko and the self he knew. Rather, this strange new feeling was the sad, sour regret of a father towards an offspring he can neither endure nor discard. He would have liked, in a moment when his own back was turned, to have had him obliterated in some bloodless, painless fashion, with no noise.\n\nHis blows had pierced the mask. He at last faced that issue he had hitherto obscured with romantic moralizing. Lichenko had originally stayed on at the billet to grovel in comfort like a pig in a slough, although admittedly deserved. But the fact of his second breach of peace indicated not all of him had yet gone soft. The fine, progressive elements in his conscience had rebelled against the ease, not with sufficient force to carry him back to duty, but at least enough to generate a protest, which appropriately had been directed towards the German woman. At that point a deft understanding might have restored him to manhood. Instead, Schild had pushed him back again, perhaps forever beyond redemption.\n\nBut in destroying him, he had also cemented Lichenko to himself. If his earlier hosthood, which he recognized as having been too permissive, owed to simple courtesy, it had since the beating become a nurseship, bonded by the obligations of guilt and limited by nothing. He found it ethically impossible even to object when Lichenko, who certainly could walk as well as ever, preferred the bedpan to the bathroom, and that only when transported by Schild\u2014he would not suffer the _Hausfrau_ in the room. Although at other times he showed great facility in bed-positions\u2014the ass mountain, the pretzel, the scissors, the beached fish, the dismembered Osiris, the solipsist ostrich\u2014at mealtime Lichenko would not elevate from absolute supine, so that there was nothing to do but spoon-feed him like an infant. His back itched fiercely every quarter-hour and would admit no cure but the application of Schild's hairbrush, wielded by Schild, to the trough of his spine.\n\nThe problem of washing, which offended Schild most, even more than the bedpan, had been rather more simply resolved: Lichenko left it behind when he took to invalidism. A person, he believed, did not get dirty in bed. With the passing of the days, his decision seemed less fortunate. After three, in a room from which Lichenko also had decided to bar fresh air on the ground that in his weakened condition he might contract a disease of the lungs, Schild had ceased to dread, might even in two more days have come to yearn, the call for soap and water.\n\nNaturally, a man in sickbed needed recreation. Lichenko required an oral reading of each day's _Stars and Stripes,_ first in the original\u2014so that he could \"study English\"\u2014and then in German translation. The comics were to be read with full gesture and if possible in voices simulating the spirit and sex of each character, especially the female ones, like Miss Lace and Daisy Mae, to whom it was impossible to give credence if they spoke in baritone. Furthermore, it was cruelly difficult to understand the narrative without a sense of what had gone before\u2014before, that is, Lichenko had come West\u2014synopses must be furnished, and definitions. For example, who really was Skeezix? A typical American? A character to identify with, or one to hold in _secret_ contempt? He insisted grimly on _secret:_ one was not so stupid as to think you could sneer openly at a feature of an official Army publication.\n\nAfter the reading came the cards\u2014he claimed to be too weak nowadays for chess\u2014which Lichenko scattered across the foul sheets in Russian arrangements, for games that three hours hence Schild would savvy no better than at the outset except to know he was loser and must pay, the fee being invariably fifty marks, arrived at by a computation as exotic as the game.\n\nNursing his patient of course demanded more time than Schild's Army duties would allow, and no one was quicker to see this than St. George, as soon as the morning after the beating.\n\n\"Oh Nate,\" he said, looking away, for he could not have met Schild's eye with anything but reproach, and he was the soul of tolerance, \"Nate, take a few days off to look after the little fellow.\"\n\nConjure with this: a captain of Intelligence, the commanding officer of a unit of the United States Army, a career officer\u2014he still had never inquired why Lichenko was a guest in the first place. One kind of charge placed against the revolutionary by the voices of petrifaction, was arrogance: 'He asks us to believe that he, and he alone, knows the Way, and if we do not admit this, he will not admit that we are fellow human beings.' Schild had read that somewhere long ago, had banned its source from his memory\u2014very likely some renegade, they were always eloquent; of course if he wished he read them, too, he was no Catholic with an Index\u2014but afterwards carried its indictment with him, like a pocket rule, speaking to it on occasion: You talk of arrogance, you, in your arrogant assumption that we suppress all doubt; we at least have the humility to abandon our selves.\n\nHe asked it now: And what of St. George, _l'homme moyen sensual,_ could there be a more ruthless overbearing than that on which his bovine assurance was fixed? In his mood Schild held it outrageous that St. George had not that first morning after Lovett's party turned in Lichenko to the MPs as a deserter from the Red Army. Which was his clear duty, the Yalta Agreement standing as witness. Indeed, St. George could be court-martialed for malfeasance of office, were it known, and reduced to his permanent rank of PFC or whatever was the breath-taking altitude to which he had mounted in the fifteen years before Pearl Harbor.\n\nThus as always, Schild in his deliberations surrendered to irony, the only weapon whose victories were won exclusively from its wielder, the sword with which the Jews, like Samurai, disembowel themselves to spite their enemies. He knew now, in retroactive projection, that he had always known Lichenko was a deserter, even as early as that first rap on Lovett's door, and in full cognizance encouraged him in the defection. He, Schild, was a traitor; he denounced himself in the dock, took himself to the cellar, shot a revolver into the base of his own skull, and did not weep over the loss of one more counterrevolutionary.\n\nWho wept for a Jew? He derived from the question a brutal, hurting pleasure, of the kind one feels as a child, scratching an itch till it bleeds. And whether it was the pain, the pleasure, or the warmth of blood that gave him courage to press on, on he went with sharp nails through the soft flesh and webbed sinews to the nerve core. In twenty-eight years, among the regiments of shadows which had come and gone, wearing whatever badge of unit\u2014no matter whether Star-of-David or even hammer and sickle; no matter whether in love or hatred, sympathy or suspicion\u2014he had met one man alone who did not treat him as a Jew.\n\nWho would weep for a Jew? _Lichenko would not._ Deserter, drunkard, schnorrer, leech, to the undeluded eye he was a compound of the baser failings\u2014indeed he was what Schild's father had always predicted Schild himself would grow up to be\u2014and very likely a liar as well, for when a man is one thing, it is natural to suppose he completes the series, and it seemed appropriate to Schild, perhaps desirable, that Lichenko had not been a valiant warrior, either, but was rather a coward wearing counterfeit or stolen medals. If he would grant him all, he must begin by giving him nothing.\n\nThe final solution will have arrived that day on which one man admits to another that he is a Jew and the second neither laughs nor draws his revolver nor melts in feigned, or more dreadful yet, authentic sympathy, but rather collapses in boredom\u2014as Lichenko at the party indicated he might if Schild said another word on the subject. In Lichenko's egocentric vision he knew now that he had never been more, or less, than a host fat for the parasiting, a mere object, a thing to be used, not comrade nor ally, not even a man\u2014 _and therefore not a Jew._\n\nLichenko was the new man who had sprung, unarmed, from the forehead of the Idea, with no chains, no history, and a concern only for himself, the product of a proposition that worked. Never say that new kinds of creation are impossible; if you can build a bridge, you can make a man with the sensibility of a bridge, without debts, incapable of guilt, and lacking all purpose beyond his immediate function\u2014and therefore neither a Jew nor interested in one. It had been worth the effort, was Schild's thought, and the thought was also new: for not one moment of his service had he sought any manner of payment, any proximate hope.\n\nOne day in August 1939, Ribbentrop's plane descended on Moscow, where the airport building flew the swastika and the band played the Horst Wessel Song, Molotov called fascism a matter of taste, and Stalin signed the pact with Hitler. In New York, Schild straightaway joined the Party. Truth is never literal: he was already a member for some months, and his first response to the Pact was a suicide of all that was not his body.\n\n'If a universal proposition is true, the particular which stands under it is also true; but if the universal is false, the particular may or may not be true.' The merciless clarity of the Greek logic; before it, the Hebraic superstitions were quaintly impotent. If you say A, you must also say B. Those who are not with us are against us. What does it matter, said Lenin, how the chicken is carved, so long as it is finally in pieces?\n\nAlternatives to these were the Munich Agreement; Roosevelt in his wheel chair; the furniture of the Seder\u2014roasted egg, bitter herbs, piece of bone, eye of newt and toe of frog, wool of bat and tongue of dog, presided over by Schild's father, an unbeliever; and millions of weak little Jews chanting the _Kaddish_ for the dead. Now they could pray for the latest corpses, those \"anti-fascists\" who fled this Party and its compact with the devil.\n\nSchild would stay. And he did not simply stay but joined, took that second breath to which all earlier belonging was mere apprenticeship. For a cause, a real cause, a man first forsakes all others to become one; and then, if he has the true vocation, denies the one to become many. First gives up women, if he is a monk, and then gives up the desire for women; if a Nazi, first tolerates the murder of the Jews, and then after that second breath, himself shoots the revolver.\n\nIf a Communist... the only virtue Schild would grant himself was that in his internal dialogues he never lied: that was for the liberals. Certainly the NKVD, like the Gestapo, pays its call without warning, in the small hours; surely \"we\" have our concentration camps, our dictator, our elite, our peculiar truth which denies the witness of the uninstructed eye, and if your m\u00e9tier is opposition to the regime, you did no better to migrate to the Worker's Homeland than had you tried it with Hitler. We wish to hear no exotic points of view; we will not suffer variety; our conscience, too, is corporate. Now we have entered into a pact with what we so much resemble in our means that the cowards and opportunists can cry: all enemies of \"decency\" are together in one basket.\n\nWe do not cavil: it is precisely your \"decency,\" a world in chains, that we would destroy, and if Hitler can hasten its end, he will be used until history is ready to fling him aside. The difference between us and you is that we will do anything to prevail; between us and Hitler, that _we are right._\n\nThus had Schild accepted reality. In the destructive element, immerse! To create that future life in which there will be no separations of one man from another, which is to say that time when no one is a Jew real or symbolic, when all the old rises and falls are planed away and men are simply man and he a stranger to passion, one must first, in the now, act upon the reverse of that vision, be separate\u2014be a Jew, that is, in extremis; if necessary, as it was, ally with an anti-Semite.\n\nSchild's progress had not been easy, or of short duration, and whether the end was serene he did not know, to date not having reached it. In particular, he was corrupted by a special feeling towards the Germans, throughout and in spite of the ideological transformations. With Hitler's invasion of the USSR the pact of course fell from memory. From then until the final victory was apparent, the eye was shifted from sharp focus on fascism-versus-the-Socialist-ideal to the less demanding _Gestalt_ of Germans-against-humanity, the latter represented most crucially by the Russian people, who incidentally had a government which tried new things but was essentially a Slavic branch of that general democracy now menaced by barbarism.\n\nTrue, to the professionals Nazism was still finance capital in the last terrible flush before death, and the Western Powers, temporarily useful, were the same thing not yet so far advanced that they themselves knew it. Nazism was fascism and fascism, capitalism; nowhere was the specific quality of Germanness material. And no sooner did the Red Army take Berlin than it erected its billboards: The Hitlers Come and Go, But the German People Remain.\n\nInsofar as the populace had connived with the Nazis they had seriously erred and must not now resent their rightful punishment by the Soviet troops. But more important, what was past was past and the future stretched out bright and grand, offering that great opportunity which so seldom comes to a people: to start out new, from nothing. Crushed and smoking lay everywhere at foot the best evidence of the failure of all hitherto existing societies. The Germans were wrong, and guilty\u2014guilty of following an extreme reactionary in his mad-dog assault on the Socialist homeland\u2014but were neither fundamentally mad-dog themselves (for peoples can be misguided but are never bad) nor in any way hopeless of reclamation; indeed, by so simple a measure as prompt adherence to the correct ideology they could enter immediate partnership with the Soviet Union itself, as magnanimous in victory as it was invincible in war.\n\nA historical crisis, admittedly, capitalism being done in by its inherent contradictions\u2014yet why Germany? No, excuse me, that is of course understood: the most advanced capitalist country of Europe; inevitably the agony would there have its nucleus. But why the one peculiar feature. Why the Jews?\n\nTo answer the first question is not to need the second: in its desperation, crumbling capitalism will seek a scapegoat on whom to hang its failure. As simple as that, comrade, nothing Dostoyevskian\u2014unless you will admit that Dostoyevsky, too, was a byproduct of the social decay preceding the Revolution\u2014and above all do not quote me Heine: \"It is indeed striking, the deep affinity between these two ethical nations, Jews and old Germans. This affinity has no historical origin... basically the two people are so similar that one might regard the Palestine of the past as an Oriental Germany...\"\n\nWith full respect to all cultures and races, comrade\u2014after all, it was Lenin who with the brilliant collaboration of Stalin, always the foremost of his colleagues, drew up that system by which for the first time in history Russia's many and diverse subnational cultures live today in peace and harmony, each with its own autonomous state, including even the Volga Germans (unfortunately the presence of certain fascist agents provocateurs and counterrevolutionaries concealed among the predominantly loyal mass of the latter made necessary certain rearrangements when the area was threatened by the Hitlerite invasion, and the patriotic Volga Germans themselves requested to be transported elsewhere in the Soviet Union, which plea was granted; a far cry from the concentration camps to which the Nisei were sent in America). With full respect to all cultures, comrade, and to their interesting and colorful traditions each of which symbolizes some old socio-economic thesis or antithesis, it is fruitless and perhaps heretical to stagnate with the past. Not what peoples have been but what they will be, is our sole concern.\n\nHatred of the Germans, therefore, is not valid, and if persisted in might become a dangerous malady. Similarly with the obsession that one is a Jew, which incorrectly puts too much stress on two delusions: (1) that Jews are that important, and (2) that oneself is.\n\nBy the Central Committee in his own skull, then, the first, the last, and the most ruthless of the Party's disciplinary boards, Schild had long before the arrival of Lichenko been granted only one more chance to rectify his errors. Had there been a thousand, he now realized, he would have spoiled them all, because he was not, and could never be, pure, adamant, resolute, unilateral; that is, could not be a Lenin. Lenin was not a Jew.\n\nBut Trotsky\u2014yes, regard that classic example, that bright needle of a Jewish mind and its corrosion from pride, which is a Christian sin. And Milton Grossman, who at twenty-five had collected no excess in his passage through the world, who had seemed only a disembodied conscience and a pair of black eyes fixed on a morning horizon. He was to leave for Spain on a tramp merchantman of which he would say no more than that it sailed soon from Halifax. In his room behind the shop he had packed the knapsack which yet bore the symbol of the Boy Scouts of America. The irony of this had been funny, and Schild laughed, but then seeing that Milton did not, he knew it was no irony, which is the tension between the way things are and the way they are imagined, but rather another marker on Milton's undeviating and dedicated road.\n\nSchild, too, had been a scout, in the same troop. It had of course degenerated by his time\u2014bullying by the patrol leaders, petty thefts in the tents at the Alpine camp, obscene language and practices\u2014Milton, with his thirty-six merit badges, was by then only a distant legend, and it meant nothing to the others that Schild was his friend. Not until years later did Schild come to know that at the arrival of the Miltons, too, the _grosse M\u00e4nner,_ the troop is always in decay\u2014and falls again upon their passing from the scene, because without a constant image of strength before their eyes men, or boys, see nothing.\n\nAt nineteen, Schild was big enough to go to Spain himself, that is, old enough and large enough in size to be in his first year at City College, to sit through purposeless lectures, to sign petitions and stand with a claque at anti-fascist rallies and peace movements and enlist in involved conspiracies to stop the Socialist candidates for student council, to study the terrain of the essential American ground: folk songs, baseball, comic strips\u2014and to report at four o'clock each afternoon to the squalid office which his father kept on Broadway just above the northern boundary of Union Square, there to involve himself for two and a half hours in the commerce of buttons.\n\nBut he was not big enough to go to Spain. It was characteristic of his friend that Milton did not suggest it. What we admire in those who stand above us is their assurance that they do, truly, see over our heads. He had similarly never suggested that Schild join the Young Communist League, never indeed that he so much as become intellectually a Marxist. Milton went towards the truth, the true was the necessary, follow if you will. Of the pre-Marxian thinkers Milton's favorites were the Stoics, whom he had read as a college freshman and shared with Schild, then on the bottom rung of high school and still a simple idolator of athletes and a noisy drinker of cokes at Mrs. Grossman's counter. \"Fate leads the willing and drags the unwilling by the neck. _But they both go_.\"\n\nIt was the first genuine idea that Schild had ever heard, and its function in his _Bildung_ was that upon its movement he had twice passed from adolescence into maturity. The second time\u2014and not a piece with his second breath of commitment to the Party, because it both pre- and post-dated that event, had really no fixed duration, continued still\u2014his second transformation began when he understood its heresy.\n\nMilton Grossman died in Spain, in July 1937. As to the means of his death there could be no question; this was one of the rare times the fact followed from the simple conditions of time, place, character. He had achieved the herohood for which his progress through twenty-five years, from Washington Heights to a Catalan field, had been apprenticeship. And so it was assumed, without the spelling out, for reports were necessarily fragmentary and cryptic, the Lincoln Brigade was outside the law of the country of its origin, and Spain under its cumulus of gunsmoke lay three thousand miles across the sea.\n\nAnother year, and someone was returned, or someone knew someone who had come back, who knew someone in Valencia who had seen Milton in the hospital, felled by tetanus, and since the shortage of serum was notorious. ... Yet the achievement was not diminished; Byron, who had gone to fight for Greek liberty, died of meningitis, and the shorter literary dictionaries, with no space for elaboration, read: \"died, for the freedom of Greece, Missolonghi, 1824.\"\n\n... but what did those rarer reference books on the shelf of some terrible agency tell of Grossman, Milton? \"Found guilty of Trotskyist wrecking. Liquidated.\" \"Executed after investigation uncovered his role in the conspiracy of the Fascist gang known as P.O.U.M.\" \"Agent provocateur in the pay of Franco. Sentence carried out, July 1937.\" Or perhaps only a sparrow-track of cypher. The world had not become more cruel since Byron, but its truths were more devious, less capable of proof, yet, for all that, truer. The real story of Byron, the _concrete_ one\u2014a term of Milton's for a quality he always sought beneath the capitalist veil of lies\u2014might be of another order, the mission to Greece a shabby quest of ego, a Trotskyism of that time, and who knew but what the meningitis were some Aesopian code-name for the 'control of disorderly elements'?\n\nBut surely it was unprecedented that at home a friend dare not speak his name. For Schild naturally had gone with his questions to those who returned. The cause had been lost and they were weary and older than their years, but they were also proud and illuminated with what could only be called the sad joy of men who have wet their comradeship in blood. They sang fierce, exuberant songs, were curt, succinct, yet eloquent in a language which was properly half-alien to the beneficiaries of their sacrifice.\n\nBut for Milton Grossman not even Spanish idiom would serve. There had not, to their memory, ever been such a person, or if there had, no doubt he was overlooked in the terrible struggle against the open fascists on the other side of no-man's-land and the _fifth columnists behind our own lines._\n\nOf course Schild knew of the wreckers, the anarchists, the hirelings of Trotsky, those worst of all enemies because they are one's own kind, who extend a hand as comrades and with the other clasp their dagger. The greatness of a cause can be measured by the decadence of its adversaries; we can be proud of the very rottenness of those we have cast out. For all their mumbo-jumbo, and all matters of clerical fascism aside, the Catholics have a valid principle: he who embraces the incorrect faith in ignorance may be saved; only he who knows the true faith and rejects it is certain to be damned. It could never be said that Milton Grossman was ignorant; like Trotsky he was all mind, his mind all blade, and that all edge, the Jewish edge... and behind it, the abysmal weakness.\n\nTo continue the inquiry was to make oneself suspect. And needlessly\u2014for Schild asked the questions only to test the answers already in his possession. No doubt the flaw had always been there, waiting for the day when the force of concrete, historical events would burst it wide. But it had been the earlier Milton in whom Schild had seen the Way, who had armed him with the weapons. There was ironical justice, but justice, in turning them now against the too-competent teacher. And his oddest feeling was that in so doing he did Milton an honor greater than he deserved; that in the measure of its being undeserved, Milton would be pleased; that, finally, he deserved to be pleased.\n\nIt was then, when he thought of Milton, though dead, though discredited, though renegade, as someone still to be taken into account, that Schild realized his sole defense against insanity was the Party. The acceptance of one's own complicity in the Party's crimes was the only escape from knowing oneself a criminal. Fate leads the willing and drags the unwilling by the neck: ostensibly Greek, but how much closer to the long, moaning servitude of the Jews, with whom in the end Milton chose to identify.\n\nFor he had written Schild one letter from Spain, a strange letter, in the early spring of '37. Strange even for Milton, who was more talker than writer\u2014\"like Sophocles, Jesus of Nazareth, and Hitler,\" as he used to say in his Bren-gun voice and then stop to catch breath before throwing it away again, eyes rising through an atmosphere of mixed slyness and purity, \"all seekers of oral gratification; you will notice none of us smoked. _O vanitas_!\"\u2014and hence never wrote proper letters but rather short scrawls discontinuous in thought and calligraphy, on whatever surface lay at hand and could be mailed, cigarette packages, cereal boxtops, the reverse of one's own note to him; and in Spain, until now, no letters at all.\n\nThis one was pencil, on an unbleached, glazed strip, serrated across the midsection, of\u2014Spanish toilet paper. \"If I should not be at large by next Yom Kippur, read this.\" On the religious holidays in New York, Milton's observance was, dragging Schild along, to go to some lunch counter and stuff himself with pork; his ambition at twenty had been to lay a girl between afternoon and evening prayers on the Day of Atonement; he had never yet done this, he said, because he could not determine which was the greater sin, to screw a Jewish girl or to commit racial shame with a shiksah, for which he used the Nazi term _Rassenschande._ Once on that day, sitting on a bench in the middle island of Broadway, watching the promenaders in their best clothes, he said: \"When at last the Messiah comes, he will be an anti-Semite.\"\n\nThe letter therefore fell within the known context, had besides the familiar mordant-shading-into-mortuary wit, the _Galgenhumorische_ pun like Mercutio's: if I should not be _at large,_ that is, if I should not be a gross _Mann;_ he anticipated his death. Then followed a translated quotation from a Hebrew religious poem of the eleventh century. He had returned to God. Small wonder he could not have made that candid.\n\nBut an old mutual admiration of theirs had been Poe's \"Purloined Letter\" and Dupin's theory of deception, which he explains by a game of puzzles played upon a map. One player requires his opponent to find the name of a certain town. A novice will invariably choose the \"most minutely lettered names; but the adept selects such words as stretch, in large characters, from one end of the chart to the other. These, like the over-largely lettered signs and placards of the street, escape observation by being excessively obvious.\"\n\nIt was a faith that Milton spoke of, but rather one lost than another gained:\n\n> ... thou didst vouchsafe to give me a perfect creed, to believe that thou art the God of Truth and thy prophets are true, and when thou didst not place my portion among those who rise up and rebel against thee; among the foolish people who blaspheme thy name; who deride thy law; chide thy servants, and deny the truth of thy prophets. They assume innocence, but underneath is deceit; they make a show of a pure and clean soul, whilst the bright spots of the leper are concealed underneath... SOLOMON BEN GABIROL, _died Valencia, c. 1057_\n\nLichenko stayed. To keep him was to abet a desertion from the Soviet Union. To turn him in was an admission that the hideous sacrifices which had gone into his making were not finally criminal, but useless. Milton had never been able to forgive a confusion of the two.\n\n# _CHAPTER 14_\n\nLICHENKO STAYED. AND IN that staying Schild ironically discovered a focus for that energy he had ever kept on call against the grand mission. For he was, or had been, a romantic, a man to whom time now and past were ancillary to time's end, and while he saw history as a continuous process and within that process himself as nothing, with the other eye he looked on the personal life as a series of choices culminating in an absolute, a supreme of either victory or martyrdom, a storming of some Winter Palace or a fell day like that in 1933 when the Gestapo was unleashed on the German Communist Party.\n\nInstead, his future had arrived in the form of\u2014Lichenko; and time had stopped. Schatzi, who for all his shrewdness had not known of Lichenko until the beating, for all his eccentricity was a good Communist and had made his report; and what he, for all his hatefulness yet a hero of the camps, thought of the newest traitor did not figure in Schild's reveries so markedly as that Schatzi's long-held, unjust, fantastic suspicions of him had been confirmed and, finally, that, for the first time personally liable for an actual crime, he felt less guilt than serenity and lacked absolutely the sense of being hunted.\n\nAnd most corrupt, his sense of humor had despite his efforts to brook it begun to prevail over the conscience. Lichenko's invalidism had required only his attendance to be supremely ludicrous\u2014to be, in fact, lunatic. Objectively the situation was simply a Russian slob nursed by a nervous Jew; the first was not ill, but the second was; and since each in his present arrangement was necessary to the other in just that condition, both were mad.\n\nOr perhaps only Schild was, for he noticed that Lichenko these days never laughed; indeed, since entering into permanent bed, and despite his abominable appearance, he had developed a dignified gravity which one who knew only the earlier Lichenko would have believed impossible. One lunchtime when, carrying the loaded tray and an under-arm burden of newspapers and magazines, Schild had difficulty at the door, Lichenko sprang from the sheets to his relief, showing not only vigor but incredible strength for so small a man: he took the heavy tray in one hand and the papers in the other and, studying _Life's_ cover tit-girl as he went, walked silently, stately, to the bed as if it were the high altar in St. Basil's and he Patriarch; and immediately upon reclining was again the man so infirm that Schild must needs not only spoon the mashed potatoes into his mouth but also support his head simultaneously. Lunacy, to be sure, but Lichenko's were not so much the doings of a lunatic\u2014Schild realized, as he heard himself laughing without accompaniment from his patient\u2014as those of a sane man who is humoring a lunatic.\n\nSimilarly, Lichenko of an evening had invented a new amusement. He had fallen off his taste for public reading and even for cards. As to the former, he had been disillusioned by the knowledge that Skeezix lived in time, more or less relative to the limited days of actual people, had years ago at the beginning of the story been a baby, was now in his twenties, would grow old. Since fictional persons are a lie to begin with, he said, they are only interesting if they stick to it and do not pretend to have the dull troubles of real people; otherwise you did better to have true stories, which of course are always boring but then don't pretend not to be. Like\u2014he broke off to peer at Schild in a kind of suspicion and remark that it was possible he, Schild, would not agree, and immediately launched an attack on L'il Abner from the opposing ground: nobody could tell him an American peasant acted in that fashion.\n\nAs to the cards, it was immoral to win from a man ignorant of the game; had he known that at the outset he would not have played; he might even return the winnings, as he was not a _gengster,_ unless\u2014again he stopped abruptly as if to give prominence to his expression, which was this time a sneer; one so broad, however, that surely its purpose was rather mock than serious. And again Schild laughed, and again Lichenko's face returned to wood.\n\nConversation appeared as the new entertainment. It was hardly more, being Lichenko's questions and Schild's answers; but it was not less, and since Schild had never known speech could be employed for amusement, at least not by him\u2014he had listened to Milton; in both the Party and the Army the human sound was used only to assent to orders from above and command what lay below; to St. George it was the minimum of small-talk to get rid of him; his parents and sister had been great talkers in disregard of the defenseless tympanum, which was why he was not\u2014since his voice had no resonance in this small room with the peaked ceiling which in the corners joined the wall a scant five feet above the floor, crowded with furniture and now with the warm congestion of dependent humanity; although Lichenko was not ill, it _did_ make a difference to him that Schild was there to serve\u2014these were reasons enough, if still morally inadmissible, why he should enjoy their mutual discourse.\n\nBut more important was the fact that after the initial ten minutes at Lovett's they had never really talked. An ordinary citizen of the Soviet Union, that person who to an American existed only in theory, he had had one under his roof for three weeks and never yet found the propitious moment to ask: what is it like, the experience of that citizenship? Indeed, to place the query was not only an opportunity but, in the present context, an obligation, just as in Party circles in America one was under the reverse imperative _not_ to question the mysterious figures who were manifestly Russian but carried passports bearing names like T. Smith.\n\nBefore the beating, Lichenko had obviously never been in the mood for talk; he had been eating, or sleeping, or washing, or scratching, or hanging over magazines or the chessboard; and that was the answer to the question never asked: there was no question about life in the USSR, it was life with incessant activity and without doubt, and even a deserter from it, one who could not meet its demands, yet carried with him its energy. In his very exploitation of Schild, Lichenko honored his society: a bourgeois gone bad would not have had the guts to go so far.\n\nAs to the other question\u2014why had he deserted?\u2014the science of dialectics admitted no such concern; Schild was not permitted to receive it into his mind; a person was either this or that; if that, he should either be ignored or destroyed; the alternative, if one did neither of these, was to relate it to the fact that oneself was lost. And this Schild had already done.\n\nNow that conversation had finally come, it was appropriately on the theme of, not Lichenko, but Schild; not on the simple deserter but on the more complex; and Lichenko's half of it was so shrewd that Schild briefly considered whether after all he had not been wrong about him.\n\nAccording to Schild's wristwatch\u2014which was strapped to Lichenko's bony arm, having been the stake in one last game of cards before the no-gaming resolution went into effect\u2014the time measured seven in the evening. The tray had been washed and lay gleaming on the dressertop, against next morning's breakfast time when it would vanish briefly to reappear heavy with eggs, melting yellow in the centers, and oatmeal porridge slushed with milk and sugar; in the end compartment, two pieces of white bread, thick as bricks, coated so lavishly with golden butter you could not lift them without smearing your fingers, which was as pleasant a sensation as running your hands over a woman, and though he knew that with this abundance the Americans had developed a culture of eating\u2014it was some old law or another that when there was overproduction on the one hand and a shortage of markets on the other, a society tried to fill the gap with elaborate manners\u2014and while he approved of this whole lovely ensemble of errors, he could not forbear from licking his hands. And Nathan made no objection.\n\nThat, indeed, was Schild's reaction to all of life, so far as Lichenko could see, and he wondered again why a man with such tolerance would join a band of evil thugs whose only difference from the other group just defeated lay in the latter's being German. Although there were German Communists, too, and surely many among them who were but lately Nazis, and wait and see if it was not exactly those who were raised to power in the East Sector. Ah, Nathan, you fool!, you who were rightly so quick to act when Vasya disgraced your house, in the big things you are truly like the silly comic strips you so dearly love to read. Look into the mirror and you will see the living Small Abner.\n\nThough not hurt (and it was an awful strain to continue to pretend he had been; actors justly earned higher wages than a fellow who operated a lathe); although in fantastically better physical condition than he had ever before enjoyed (for the first time he had hopes of one day becoming handsomely stout), the kind it was shameful to have to hide under a mock illness rather than announce with much noise and movement; in spite of the great rewards at hand and the greater ones in promise (if the United States proved but a vain dream, then perhaps merely some sleepy hamlet in the Black Forest and a German woman with a nice round ass and a little craft like decorating Christmas-tree ornaments, and a garden of cabbage and beets\u2014if one lived too high he got only boils and the gout) ...despite every reason for being up and about, for seizing life and making it groan, he had instead chosen to play the sick hog. And it had begun to work.\n\nNathan was a queer fish. For some reason he had buried his humanness so deep that one could bring it to the surface only by outraging him. Yet Lichenko had always known it was there, else he would not have taken the trouble to find it\u2014and it _was_ trouble, and Nathan was very lucky to have him. For now he had, at who could tell what final cost, at last established the conditions for that intimacy in which the truth could be aired.\n\nSeven o'clock, the good air outside the closed window, which he had not had in his nose for weeks, still bright and full of August. But Nathan had turned on the dresser lamp like the indoor man he was, and come to sit by the bed to await his, Vasya's, pleasure. The room which had on first sight looked so grand that he assumed Nathan must share it with a regiment had truly become a home. With use, the very bedsheets, so white and hard when first entered, had softened and lost their harsh odor of bleach. Even Nathan's sloppiness, which until the \"illness\" intervened he had constantly opposed, had worked to the homely purpose, the rug dark with scorchings, the rent in the curtains, the deep scratches of footboard and dresser-front catching the shadows like old scars on the faces of your loved ones. ...\n\nWith a scissor-kick, as if in the water, he shot himself backwards, conking his skull on the headboard, which was not intentional but certainly claimed Nathan's attention. Instantly his friend was up and arranging the pillow.\n\n\"Are you hurt?\"\n\nCould Nathan truly be as pained as he looked, at the possible hurt of another?\n\n\"Ah, no!\" Lichenko tried to joke. \"The bedstead is undamaged!\"\n\nHe had got him there: Nathan fell back laughing. He himself of course did not, it being a kind of vulgarity to laugh at one's own jokes, and instead, with serious mien, fixed the pillow from which Schild had puffed out all the good head-hollows.\n\n_\"Was f\u00fcr Kn\u00f6pfe macht Ihr Vater?\"_ he asked.\n\nNathan could sit in a straight chair for hours without so much as crossing his legs, and he was thin, too, so that this ability owed nothing to the padding on his rump.\n\n\"Oh, he doesn't make the buttons. He buys them from a buttonmaker and sells them to a manufacturer of women's dresses. If that's confusing, don't bother with it. Your country is mercifully free of the middleman.\"\n\nAgain Lichenko shot himself backward, but this time the pillow dulled the thud of his head hitting oak, and this time Schild did not rise, for simultaneously with the action away, Lichenko had shot his hand forward and asked: \"What is that you say, free...?\"\n\n\"From the middleman.\"\n\n_\"So.\"_ The olive-drab undershirt, which with drawers of the same cloth, on loan from Schild, was his costume of illness, had with the movements ridden up and constricted about his narrow chest like a dog harness. \"I should have told you earlier, my dear friend, I hear what is said, but between the words sometimes comes the _swoosh_ of the rockets: 'free'\u2014 _swoosh_ \u2014'from the middleman'\u2014 _swoosh,_ so that what goes into my mind is often different from what has been spoken. For example, I thought just now I heard you say someone was free in the Middle Ages.\"\n\n\"Haha,\" laughed Nathan, but wryly. \"Not being a Jesuit, I could hardly say that.\"\n\n\"Ah, again, another example: what I heard then was something about Jesus Christ! ...You see what I mean.\" He threw his feet about under the bedclothes, which commotion looked as if a small animal were trapped there, and smiled helplessly.\n\nJust the thing to replace Nathan's nervousness with the responsibility of a job; he could never endure being misunderstood. It was a relief to see him break the stiff column of his spine as he leaned forward and said very slowly and with the enunciation of him who speaks into an ear trumpet: \"Je-su-it\u2014a religious order which invented a kind of fascism four hundred years before Mussolini.\"\n\n\"Of course I knew it was something old,\" Lichenko answered. \"But you see a hydraulic engineer does not have time to learn much beyond the principles of his science. I..., my dear friend, should you be angry if I confessed to a dishonesty?\"\n\nBehind Schild's genial facade he saw an emotion begin at the throat and descend\u2014a giraffe would look like that if it swallowed a melon\u2014either hatred or fear, since these were the only feelings a person might find politic not always to reveal, but which of these was here operative Lichenko could not say, there being no apparent reason for either. Wishing no lies to stand between them, he had prepared merely to admit he had not read the American books they discussed at Lovett's party, so long as the truth was out that an engineering student had no spare time.\n\nInstead, he said quickly: \"I borrowed another of your handkerchiefs while you were gone to the dining hall. I shall send you a dozen when I go\u2014\" For a moment he imagined he had heard himself continue with: \"to America,\" for suddenly that was where his fancy had fled, in just that wink of the eye he had seen himself at the handkerchief stall in a store big as a sports arena, had gone back further to park his yellow Ford at the curb outside. He wore a tight blue suit of narrow gray stripes and a black felt hat low over his brow; the woman at the counter believed him a suave but dangerous racketeer, a pearl-handled revolver encased in a silk glove, as he smiled with sharp white teeth and said \"Enchanted,\" or whatever was proper at such a moment, which he would know.\n\n\"\u2014a dozen. Tell me which color do you prefer? Always this olive, or should you like some of blue with narrow gray stripes?\"\n\nThis was what he really said while Nathan loosened, sat back, and finally crossed his legs, one trouser riding up to uncover a pale shin whipped with dark hair.\n\n\"You know you may take anything of mine,\" said Schild, \"and I'll be disturbed only if you try to pay me back.\"\n\nHis incredible generosity! It had, more than any other single thing, been the cause of Lichenko's delay. He understood that far back around the time of Jesus Christ the first Communists worked on that motive and no other, when, that is to say, they were weak and victims rather than victimizers, and it must have been splendid to live then, when good and bad were easy to isolate. Some time since, they had become so mixed that one could no longer take the sayings of one's mother as a serious guide to life. For example, of Schild his mother would first make some old-peasant observation such as that a man with a high bridge to his nose was untrustworthy, or that ears set at that angle caught only evil wisdom. But if he showed his manners she would think him fine as a \"nobleman,\" which in her lexicon took on ever more precious connotations as she grew older and had further to look to see the lovely time of her youth when her father had one hundred per cent more land than her husband had now, since the latter owned none at all, and when the fields were the property of a handsome count who never cursed rather than a gang of rude bullies who stole nine-tenths of every harvest in the name of some swindler they called \"the people.\"\n\nLichenko's mother had been illiterate. She had gone under orders to night school and learned to read and write, but she had still been illiterate\u2014according to his brother, who belonged to the Party and, being very literate, wrote articles on agricultural matters for a newspaper in Kiev, which Lichenko, perhaps because he himself was only moderately literate, could never read beyond the first paragraph: \"The representative liaison committee from the Stalin Collective Farm at Rusovo yesterday presented to the Central Organization of Rural Co-operative Societies a voluntary petition from the Third Link of field workers on the Stalin Collective Farm that it be permitted to raise its quota in regard to the harvest of wheat. Now, what does this mean relative to the development of large-scale socialist production in the sphere of agriculture? This means...\"\n\nOr take his brother\u2014now you would assume he and Schild, being political comrades, would hit it off. But, ah no, his brother had no respect for foreigners, Communists or not, as he had once admitted to Vasya; indeed, he placed little value on any people but the Great Russian and had got so that just before the war he would speak Ukrainian only with the greatest distaste.\n\nNo, to understand Nathan one must regard him with one's own eyes: it was the generosity, not the Communism, that was native to him, and if you said well, the Americans have so much they can afford to give some away, you had only to compare him with another like Captain St. George to see the difference. Nathan lived like a holy man of yore.\n\n\"I suppose your dearest wish is to return to your family now the fighting is over,\" he said, straightening the undershirt. \"Tell me of them. Your sister\u2014is she beautiful? Is she so slender? You have a photograph, of course.\"\n\n\"No\u2014well, I did have,\" Schild spoke in concern, \"but in the area of Metz my belongings were stolen.\"\n\n\"And your mother\u2014can she read and write? No, don't answer. How silly of me to ask! A fine, cultivated noble\u2014gentleman like you! Besides, certainly everybody in the United States is literate.\"\n\nThis seemed to soothe Schild, and his black eyes glowed behind the lenses as he protested happily: \"Not at all. There are about ten or twelve million Americans who cannot read and write. We are not speaking now of the Soviet Union, Vasili Nikolaievitch.\"\n\n\"But then it is not necessary for everyone to read and write,\" said Lichenko, shrugging with his voice. \"All one really needs is something to eat and wear\u2014protection from the _golod_ and _kholod,_ as one says in Russian\u2014girls to love, maybe a drink of spirits now and again, and the policeman not on your tail. I mean, if one belongs to the common people.\"\n\nSchild assented by his silence.\n\n\"The uncommon ones,\" Lichenko went on, \"take care of themselves. Then there are the ones between, who don't know what they want, _nicht wahr?_ Something different, anyway; this is not right and that is not right. Nothing is right for them!\" he exclaimed in a kind of joyful hopelessness, pedaling his legs rapidly as if riding a bicycle. \"But look at a big oak tree: it loves no girls, drinks only water, does not eat at all, lasts longer than the oldest man, and is satisfied throughout.\"\n\n\"And is chopped down by the first fellow who needs wood,\" said Schild, nodding pleasantly. His shirt pocket might be unbuttoned, but his tie and collar were fast and most uncomfortable to see through the heavy, still air. Keeping the windows shut had been a phase of Lichenko's scheme of absolute pressure to the body as well as the spirit, and while no effect could be discerned in Schild, he himself was sweating like a plowhorse.\n\n\"Yet,\" Nathan continued, not so much as a gloss on his steep forehead, \"isn't even that oak better than a worker under capitalism?, who is chopped down when he is _not_ needed.\"\n\n\"Stupid!\"\n\n\"Yes, stupid is a better word for it than evil.\"\n\nStupid Nathan! He saw even a tree politically, and no doubt would be the first to cut down an oak, to make paper for pamphlets to celebrate someone else's sowing of the reclaimed ground, or to denounce them for seeding the wrong thing, whichever would be most bleak and deadly and contradictory of his generous heart. There was a difference of thousands of meters, in more than land and sea, between him and Lichenko's brother, in spite of their similar faiths. His brother had, all to himself, a four-room apartment with a refrigerator and a private bathroom, but what had Schild to gain? He even disapproved of his father's wealth.\n\n\"It would be better, I think, if the window were open.\" Lichenko scrubbed his face with the undershirt tail, which when he pulled it down again was wet as a swimming suit, and since by that time Schild had opened one half of the casement and the evening air made chill entry, his belly was shortly cramped with cold.\n\n\"Good, that is just enough. Now please close it.\"\n\n\"You haven't a fever?\" asked Schild as he came back to his chair softly as a cat.\n\n\"Frankly, I don't know. I feel very strange. Perhaps I should take a bath. ... Of course you have a bathroom in your home in the U.S.A. And with hot water, no? _Sch\u00f6n_!\"\n\n\"But there are many people who have not. My grandparents lived in the working-class quarter of New York City, in unbelievable slums. They had nothing but a cold-water flat, one room for living and sleeping, and the other a combination kitchen-bath. The tub had a wooden cover that served as dining table.\"\n\n_\"Wundervoll!\"_ Lichenko chortled. \"I knew it! They were workers and yet had a private bath, and their son grew up to be a great industrialist of buttons and _his_ son became a fine intellectual.\" He saw a cruel angle develop in the corner of Schild's mouth, at odds with a sad cast of the eye. He, Vasya, had been carried away as usual: fact, fact was wanted and not his opinions, which only irked his friend in the proportion they were genuine.\n\nHe writhed about until his feet hung over one side of the bed and his head, the other. In upside-down vision Schild looked like a baldheaded man with a beard\u2014indeed, somewhat like a Lenin with glasses. He had played this game as a boy: if you frowned, the lines of the forehead resembled a mouth; the real mouth you must ignore, and also that the nose opens in the wrong direction; with the remainder you had a fairly credible face which gave to the expressions what the Moscow radio gave to the truth\u2014an odd twist, both human and not. It was years since he had played it, however, and he had lost his old proficiency in interpretation.\n\n\"What are you doing now?\"\n\nThe mouth in the center of Schild's head answered: \"I'm smiling.\"\n\n\"Forgive me, one gets restless in bed. To entertain myself while you are gone I have remembered certain boyhood amusements.\" He righted himself, all hot above the neck, and sighed. \"When I was sick as a child my mother sang little songs to me. They were always about food. For the life of me I cannot now recall a note, or I should sing one. They only come back when I am hungry.\"\n\nSchild bathed in a pond of jocularity as he said: \"Then we shall have to starve you.\"\n\n\"No,\" Lichenko answered, \"that has already been done, and believe me, my friend, just for the singing it is not worth it.\"\n\nIt was aired, his first open attack on the regime of his country; he felt excellent well for having made it, and he stared fearlessly at Schild, who appropriately cast his eyes aside in deep embarrassment. Which meant he knew, then, of the Kremlin-made famine of 1933, and it meant as well that he was not so corrupt as to try to defend it. Yet if Nathan did know and, regardless of a disapproval however sincere, continued to work for those devils who had not only created the famine but standing on two million corpses denied they were there... Lichenko lost the path as all at once he found he wanted Schild to be both innocent and guilty, for only in that combination could he forgive him.\n\nBut Nathan was neither. So solemnly eloquent he almost cracked one's heart, yet with a peculiar elation that seemed to swell his own, he spoke of Hitler's assault on the USSR and the scorched-earth tactics and withdrawals which, because of the treacherous surprise, had been at first the Soviets' only defense. He spoke well; indeed, so well that Lichenko almost believed the hunger here at issue was rather that of 1941 than 1933. No question that the invasion by the Germans had been worse than living under Stalin: they were foreigners. Yet, although the data was of course suppressed, hundreds of thousands of his compatriots had had another opinion, hung garlands on the invaders and enlisted in General Vlasov's anti-Kremlin army or even in the Wehrmacht. They were wrong. If you must have a tyrant, why not keep your own?\n\nHe could not help it, he still had scruples about disabusing Schild. The Red Army, as Nathan was saying, _had_ done a magnificent job; they _were_ heroes; he, Vasya, was a hero and it was just and proper to hear someone say so. The Soviet Union was the greatest country in the world: there lay no contradiction between believing that and fleeing to America, or the Black Forest, or some southern land where dark-complexioned people drank wine and slept all day in the shade. And it was very probable that the Party elite represented a new and superior kind of man. He even believed Bolshevism would triumph in the long run, everywhere, because he could see in it no weaknesses and knew by experience it would stop at nothing. Even Hitler had a limit: the Germanic \"race,\" by which he measured everything, including his Ukrainian allies, and in the end this folly brought down his house. He was wrong.\n\nThe Communists, however, were right\u2014oh yes, no doubt even the famine was correct from the high point of vantage, the Kremlin had its eye always on the main chance, for there in the grave lay Lichenko's father and mother, who starved, yet there was he, son and heir, fewer than ten years later at the breech of the rocket gun, fighting loyally to save Moscow, and Stalin, from the enemy.\n\nCommunism, Nathan, is never wrong\u2014as you would immediately agree but not understand\u2014because its only principle is success. Just as yours is failure; what you really love is not the Red Army's victories but the sacrifices and agony required to achieve them. How you would have approved of the famine! ...But the point I wish to make is that Stalin and his gang neither liked nor disliked starving two million people. They saw it as necessary to their plan that they requisition more foodstuffs than the peasants produced. If as a result the peasants died, they simply did not care. Communism is never wrong, Nathan, because it has no feelings at all, certainly no good ones, but no bad ones either\u2014none at all. It is difficult to tell you that, because I have and you have, and furthermore I am a man without ambition and thus discredited.\n\nThe unspoken rang so loudly against his frontal bone that Lichenko could hardly believe Schild had not heard it, too; crystalline, cold, and true it was, like the sound of a gong made of glass. And he had never been a great one for thinking, which was his brother's talent.\n\nOnce before the war his brother in a literary phase had read a book called _The Idiot_ by a writer towards whom his brother had mixed feelings\u2014saying on the one hand he did show a consciousness of something, although on the other he was of course hopelessly something and you could not look to him for something else\u2014at any rate, in an unusually amiable mood he quoted to Vasya the very kernel of what in this writer he thoroughly disapproved: this Idiot, who if that were not enough was also a prince, appropriately found everything strange; but one evening in Switzerland, where typical of the decadent Russian nobility having nothing else to do he went to drink sulfur-water or whatnot, he heard the bray of an ass in the marketplace: \"I was immensely struck with the ass, and for some reason extraordinarily pleased with it, and suddenly everything seemed to clear up in my head.\"\n\nFollowing the quotation his brother observed that heavy silence which means such nonsense speaks for itself. To Vasya it had said nothing until this moment more than five years later when, without the ass's aid, he found himself in the princely condition. Everything seemed to clear up. ... He had stayed on not to save Schild but to understand him, not because Schild was good but rather because he was interesting. It was the game of the Communists, who were never wrong, to save people. For an ordinary man, an idiot, it was enough to know how the next fellow used the privilege and obligation of life, which was not the best thing imaginable, but we none of us\u2014his brother, Stalin, Hitler, the Americans, the prince\u2014had anything else.\n\nNaturally, Nathan had not heard. That inner ear through which the rest of humanity hears the most important sounds is confiscated when one joins the Communists. He had often confirmed this by speaking silently to his brother: \"You bastard, the only reason I wouldn't shoot you if I had the chance is that we have the same blood.\" Results always negative, despite his brother's noted gift for smelling out heresy.\n\nHowever, Schild had picked up a subtler noise which Lichenko missed. His voice became furtive as he left the siege of Stalingrad to warn: \"St. George is coming upstairs.\"\n\nAt last Lichenko heard the footsteps, which being both heavy and soft like those of any large animal but the horse, were unmistakable: those shoes which he so coveted, with their fat soles of yellow gum rubber; shod so, a man could run right up a smooth wall. Why Schild should think St. George a menace, however, was far from clear\u2014if at the same time, as Nathan insisted, and Lichenko had to agree, the captain was also a fool. But a good fool, a jovial one, at least wise enough not to try to be clever. He did not even suspect he had a political as second-in-command, and was the happier for it. In a Russian company the most harmless-looking boob was invariably the secret-police informer. The wonderful American invention was a man who looked his role.\n\nHe lay badly in need now of just the neutrality that St. George dispensed. He readied his mouth to call \"Kom een!\" his pronunciation of which the captain never failed to approve; he was already enmired in St. George's warm sludge, that secure, absolute, fool's medium in which all was forever orderly\u2014when, just as the footfalls reached the door, darkness smothered him in its close sheet.\n\nOutside the window night had come unnoticed, but the room was blacker still, for even a night swollen and dim with cloud has its suggestions of distant fire. Damn you, Nathan, for extinguishing the lamp on a friend! Now what had been merely necessary became imperative. He called to St. George and could not hear his own voice; he strove to rise but lost the first fall to inertia, the second to his knotted bedclothes, and won the third only to hear his quarry pad beyond the bend of the hall. Nevertheless he got to the lamp, eerily not meeting Schild on the way, choked the button in its narrow throat, making light\u2014of which he had the conviction it would reveal nothing but a chamber enclosing only himself.\n\nYet there sat Nathan on his hard chair, on his cast-iron behind, and looking not at all guilty, when for once he should have, but rather self-righteous.\n\n\"Yes, it's all right now,\" he said. \"He's gone to his room.\"\n\nIn the interval of darkness the lamp had prepared for a success, developing its weak yellow into a splendid flare\u2014only to lose the contest to Schild's face, which like unpolished bone claimed all the light and gave none back. He had never looked more saintly.\n\n\"But come,\" he said, rising to Lichenko's aid and fading quickly into his old contrition. \"You shouldn't be up\u2014you'll take a chill.\" He offered to support him and, when that was spurned, walked before, as if he were clearing a channel through some invisible marsh between the dresser and bed; alone and unwitting he went, and no one followed.\n\nFor Lichenko had turned to the big clothes cabinet in the corner next the window, turned the key, and peered into its cavern which gave the illusion of a vaster space than the surrounding room. At one end of the rod Schild's uniforms hung unruly, as if rifled by a thief. At the other, his own, which seemed unusually small upon its hanger; and his boots, bow-legged, slumped, wanting straight heels.\n\n\"My cap, I do not see my cap, and I cannot go without it,\" he said, into the depths but to Schild.\n\n\"Oh yes,\" Schild answered, in a strangely strong voice. \"You will want your cap. Isn't it there on the shelf?\"\n\nSurely it was; he had forgotten the single shelf across the top of the cabinet, perhaps because he was too short to use it, but the edge of the cap's shiny visor poked an inch beyond the board, like the nose of a midget peeking down from hiding, and he seized it. Upon his head the cap was tight, since he had not had a real haircut for three weeks, only Nathan's trim-job around the ears with a little sewing scissors. He also got into his boots, balancing badly on one leg at a time\u2014you cannot live abed for more than a day, even faking, and not feel giddy on your feet\u2014and then seeing in the mirror a soldier on tropics-duty, for he wore cap, olive-drab shorts and undershirt, and boots, he groaned at his stupidity and sat upon the floor.\n\nSchild came to him and, bending over, grasped his left heel and toe.\n\n\"I'll pull and you pull, and off it comes. Ready?\" Before he could answer, Nathan did his part unaccompanied; off it came and then the other.\n\n\"Now,\" said Schild, \"we'll just put these back into the cabinet where they can't be scuffed. And the cap, too. You won't want to get it full of lint.\" He plucked it from Lichenko's head and ran his elbow across it twice.\n\n\"Don't crush my cap,\" Lichenko shouted.\n\n\"Ah no, this is how they brush hats in the fine American stores.\"\n\n\"How am I to know that?\"\n\nSeizing his hand, Schild brought him upright.\n\n\"What you do know is that I have no reason to ruin it, _nicht wahr_? Therefore what I do must be to its advantage.\" He looked very scholarly as he replaced the cap on the shelf. At the angle Lichenko saw that his glasses were covered with a film of dust and at least one fingerprint, distinct in oil.\n\n\"Why don't you clean your spectacles?\" he shouted angrily. \"You can't see out of your own head!\"\n\nCarefully, Schild unhooked the temple pieces from behind his ears, and painstakingly shined the lenses with the small end of his straw-colored necktie, which tonight as usual was twisted ahead of the larger.\n\nLichenko turned aside, embarrassed by the naked face, saying: \"You should not have done that to the captain.\"\n\n\"Then come,\" Schild offered, the glasses yet in his fingers, \"we shall go and apologize to him; I mean, we'll go and _I_ will apologize, and you can see his feelings haven't been hurt.\"\n\n\"Oh, I'm sure of that.\" He reached up under the tunic and drew his breeches from the crossbar of the hanger. No matter where he wandered hence he would never find another man so alert to his moods and purposes, but was that not the trouble?\n\n\"Yes,\" Schild reassured, \"he is just a person. ... But whatever are you doing? You are ill, my friend, and must not worry about your uniform. As you can see I have taken good care of it. Look at the blouse\u2014as clean and pressed as new, eh? And the medals\u2014only yesterday I sponged the ribbons with gasoline. How bright their colors are! See the Order of the Red Banner\u2014\"\n\nLichenko sidestepped him and struggled into the breeches. After the fly was fastened he could hardly get a hand into his pocket, so American had been three weeks of meals\u2014and that, too, was the trouble. He withdrew a wad of marks and thrust them at Schild.\n\n\"Here is payment for the underwear and handkerchiefs and whatever else I have taken, and also the winnings from the cards. You see, I cheated in those games\u2014silly, no?, since I could have beaten you anyway, still I could not resist when it was so easy. But there you have it all back again.\" He threw the bills upon the dresser.\n\n\"Yes, the cards!\" Schild said, desperately exuberant. \"We'll have a three-handed game of something and get old St. George\u2014you'll see he isn't hurt in any way\u2014and take his money. He'll like that, he'll do anything for company.\"\n\n\"As to your personal kindness,\" Lichenko continued, reaching for his blouse, \"there is no repaying that, not when one understands what kindness is, a thing which should make the giver feel good or he should not do it.\" He said more as he crumpled the blouse over his head, but could not hear it, himself. He was so sick of himself he feared he might vomit on the very uniform whose smartness he also owed to Schild. He had learned in fifty seconds that cowardice may be a slow disease but is felt as an instant affliction, and comes more violently in rooms than on the fields of battle; at Kursk, when a Tiger tank broke rumbling and malignant through to their artillery position, he had leaped upon the deck and dropped a grenade down its throat; in _gem\u00fctlich_ Zehlendorf he could not even stave off the insulting of a fool, much less tell the cold truth to a friend.\n\n\"Come,\" said Schild, who looked now as if he were drunk or, rather, pretending to be drunk and wild, in the manner of some honor student ostentatiously letting down his hair at the end of term. \"St. George has a bottle...\" He rolled his eyes in what he surely meant as license, but to Lichenko they suggested those of a horse gone mad with fright.\n\nFright? Why should _he_ be afraid, the one who wasn't taking a risk? Or did his odd sympathy even extend to Lichenko's future troubles in the great world outside?, where, after all, most people had had to struggle all their lives without his help. For the first time he was struck by Nathan's incredible arrogance.\n\nHe buckled on the wide dress belt and strung the breast strap through the epaulette on his right shoulder, and reached again for the boots, which Nathan still held.\n\nBut Schild swung them behind his back, like a child, saying: \"First let's have that drink.\"\n\n\"No, Nathan, I am not fooling any more.\" He took the boots from him and this time sat down upon the bed to pull them on. \"I shall say goodbye to the captain but I want no drink.\" He needed only three drops of spirits to fall unconscious; his head already felt like an electric-light bulb, hot, light, empty, fragile, and loose where it screwed onto his neck; a moment somewhere back he had discovered he was ironically and genuinely ill.\n\n\"Goodbye?\" asked Schild, his voice very ugly, so nasty it caught him up a bit, himself, and he pressed it out sweeter for the rest: \"Where can _you_ go?\" He did not wait for a reply\u2014being already in possession of all answers to all questions; indeed, it was mere courtesy that he had put the statement in the interrogative.\n\n\"Almost anywhere but home,\" said Lichenko, grinning weakly, trying to, at any rate, as his head slowly unscrewed and Schild's image kaleidoscoped with the vivid colors of the hair-lotion bottles on the dressertop. Nevertheless his mind stayed clear.\n\n_\"You son of a bitch.\"_\n\nNathan had spoken in English, that flat, nasal language in which nothing sounded either interesting or important; and so far as he could see him through the spinning, his expression followed suit. Lichenko grinned again, hard and acid, but this time within his own heart and on the terms of his own failure. In the end, how he had conducted himself did not matter, that was the funniness of it and also the horror; in the end, the great truths could not pass through the neck of the smallest one: you cannot stir the curiosity of a corpse.\n\nHe would leave in a moment. As soon as he recovered his balance he would get his cap from the cabinet and walk through the door, down the stairs\u2014the German woman, he reflected, handsome if too thin, would continue to go to seed\u2014and stand upon the threshold, facing outward. One could hope the night was not windy; the world seemed larger when the wind blew, especially if the sky was dark and you could see so little that was permanent. Other persons feared lighting bolts, sunstroke, drowning, snakebites\u2014he had always had fantasies of being blown away in a gale.\n\nIn a moment... already he could feel the strength rising from somewhere down about his ankles, which were firm in the good old boots. You couldn't beat boots, which would hold you erect when you were limp with exhaustion. He could not believe that the Americans, in their low shoes, had much endurance.\n\nAfter looking at him a long time in the same blank way, Nathan had suddenly turned towards the dresser lamp, seized the wad of Occupation marks, and begun to count. It would be an impressive sum, for what Lichenko had won in the cards from Schild were just a few negligible leaves around the fat core of the bonus he had been paid on the day of Lovett's party. The regular pay, in rubles, was allegedly deposited at home against one's return; these marks, intended to be spent in Germany, had on some guarantee of the Americans been printed wholesale and cost the Red Army nothing. They also, if he knew his bureaucrats and their ingenious scheme of allotments, were very likely all one would ever get in his hand. For him, of course, the matter was now academic.\n\nHe would face the world with empty pockets and without a plan. This, he realized, in a chill about the kneecaps which was closer to a falling nerve than a rising strength, was absolute freedom.\n\n\"Yes, Nathan, all of it is yours,\" he said faintly, for part of him was in that state of freedom while the rest held tenaciously to the here-and-now, and his voice was not strong enough to sound both places with the same volume. \"Count it, keep it, spend it. Money is a good thing, _especially for a person of your type._ \" He meant: it may not be grand or powerful, but it is human to know the price of beans.\n\nAs if he had arrived at the total, Schild nodded to himself and rerolled the bills.\n\n\"Thank you,\" he said quietly. \"We are quits. And now if you can spare a minute I must get St. George to come and say his _Lebewohl_.\"\n\n_\"Lassen Sie sich Zeit,\"_ Lichenko answered, \"take your own good time.\" He lay back across the bed and closed his eyes; he felt a small object drop upon his chest and separate like a broken egg; he heard Nathan leave the room. He would sleep a minute.\n\n\"Well,\" St. George had said to Schild, \"I did wonder if he had permission to stay this long away from his company. I did think it was funny.\" In his pajamas\u2014his alternate set, of vertical green and white stripes\u2014lipping an unlighted pipe, smelling of mouthwash, he stood sagging near his window just opened over the black-quiet yard. \"But desertion! I hope you're certain about that. Or rather, I hope you are wrong, because he is a nice fellow.\" He anyway had to sleep the night on it.\n\nSchild neither slept nor tried to, nor could have said how he passed the hours of darkness, for they were too grievous small: a turn of the corridor and already the bathroom window was mother-of-pearl; another, and five o'clock had surely come. Silently he crept into St. George's room and took up the wrist-watch from the bedside table, held the cold snake of its expansion bracelet: only four o'clock in Berlin's delusive and too-early light. Nevertheless he woke the captain, who took his warnings with a face like a stale onion roll and at last rose, puffing and aged, to stuff himself into the uniform.\n\n\"Boy oh boy,\" said St. George when he was dressed. \"Here's a time I would give these bars to anyone who would take them. This is a lousy business I have to do, Nate. You should be glad you're out of it.\" He made a pot of his overseas cap and drew it on. \"God knows what they'll do to him. I don't think Russia's much of a place.\"\n\n\"But then you didn't make the regulations, did you?\" asked Schild, as he pressured him, without touching, to the door.\n\n\"I guess that's how to look at it.\" With a foot into the hall, though, he recoiled and, whispering, brushed Schild's ear with his earnest, bulbous nose: \"But does he know yet?\"\n\nSchild answered harsh: \"Now I would hardly tell him.\"\n\nHe ate this thought like a caramel and, swallowing it, grimaced, and then going into a profound melancholy moved with heavy hump of shoulders towards the staircase.\n\nWithin the hour two military policemen\u2014Americans: Schild had somehow believed they would be Russian\u2014came in tall, thin, and bored from the street, mounted the stair with drawn pistols on white lanyards... and soon descended supporting Lichenko between them, for, still in half-sleep, he could not walk erect and would not try to see with his eyes. Yet at the threshold he straightened, jerked his arms from captivity to fix his cap, said _\"Ladno!\"_ the Russian okay, and walked unassisted in the new, barren day.\n\nSt. George had not returned. His mouth metallic with want of rest, Schild mounted to the room which he had not seen since the evening before and in which he had not been alone for three weeks. Scattered across the bed he saw the roll of marks in the pattern in which it had burst when he threw it. He believed that he should burn them straightaway, but as he stooped to the gathering the door downstairs made its sound and he was hailed by a raucous American voice.\n\nThe taller MP stood wide-legged and screamed up the stairwell: \"Lootenant, did that fuckin' Communist steal your wrist-watch? He's wearing a gold Bulova.\"\n\n\"No,\" said Schild, after a moment. \"I sold it to him.\"\n\nHe thought: I will never know how long it might have gone on if he had not made that crack about Jews and money.\n\n# _CHAPTER 15_\n\nCONSIDERED AS A UNIT, REINHART and Very were some twelve feet, three hundred and forty pounds of person, and, as the beast with two backs, would have ranked in the hierarchy of animal size just after the whale, the Indian elephant, and the hippopotamus. Their coupling, however, was apparently not to come\u2014unless it was she who overwhelmed Reinhart\u2014for all day now he ached with the surfeit obtained in another quarter. Discretion ruled out any further sport at the office, but immediately after work each afternoon he had been calling at Trudchen's little room down the hall, to vault between her soft legs in a ferocity which, though it had long left reason behind, never stayed her call for more and worse. Indeed, it had become S.O.P. for her, just before the climax, to scream into his ear: \"You don't hurt me enough!\" and drive her small fangs into the lobe, which, while it is that portion of the human surface with the fewest nerve endings and correspondingly insensitive, still feels pressure and can swell fat and red with mistreatment and make you look odd as you go about your other business.\n\nBut all in all Reinhart felt very natural and right about the arrangement, as one can only when he so adjusts his life as to be dirty on the one hand and clean on the other\u2014a sort of Renaissance ideal\u2014and therefore hypocritical on neither. With Trudchen there was no pretense of love; with Very, very little of sex; although, not being a brute or a pervert, with the former he did not withhold \"love\"\u2014he was very kind to Trudchen\u2014and with Very his imagination was not so barren as to exclude \"sex\"\u2014he after all kissed her rather more than he did Trudchen, if not in so French a style, and who knew what random transport might seize her in some propitious time and place? Meanwhile, it was satisfaction of a kind of lust merely to be with her, to have her seen at his side by resentful others. Though they were not flagrant: in public they never held hands.\n\nAnd usually they were in public: for one, because even in Berlin, with its acres of forests and ruins, even if you could drag a respectable girl through stocking-snagging jungles, people abounded\u2014Germans of course did not count, but Americans were behind each tree and in the hollow of every bomb crater\u2014for another, having no strong need to tumble her, a man had to find public amusements with his woman.\n\nFor example, the Nazi monuments. Pound's and his tour had at last moved from paper to actuality. One Sunday shortly past noon two of the small vehicles termed \"weapons carriers,\" the parallel benches in their roofless beds creaking with packed behinds in olive drab, tooled from Zehlendorf to the now deranged nerve center of Hitler Germany.\n\nVery's turn was like the stately movement of a world-ball on its axis\u2014not a petty soccer-sized globe, mind you, but the grand sphere that dominates some centennial exposition\u2014as she descended from the truck on the same helpful hand that Reinhart, as official guide, had granted the other nurses in the party. Her other difference was that she gave his fingers a pronounced squeeze, which not only brought pain to his knuckles but also impatience to his heart: there they were, in the great chaotic plaza before the ruined Chancellery and she was obviously unmoved. Not to mention that she had given, he had seen\u2014for on general grounds it was a pleasure to watch her\u2014only perfunctory notice to the legend incarnate of the series: Brandenburg Gate, Unter den Linden, entrance to the Wilhelmstrasse, Hotel Adlon, Foreign Office, Propaganda Ministry; had instead touched her cap, flicked her lapel, straightened her skirt, and coughed ladylike behind satiny nails.\n\nNow she nicely picked, with the others in the party of fourteen, across the center island nasty with torn Volkswagens and an Opel, on its side, showing naked steel supports for a roof long gone, and a lamppost twisted and wilting like a licorice whip on end; in her turn presented the long red pass to the inevitable tommygun Russians at the Chancellery door and was, with stupid, mammary ogling, admitted.\n\nReinhart clove to her side, and the others, officers, nurses, and enlisted men, clung to his; shortly they were all lost together in a choppy surf of crushed marble through which black wires squirmed like sea-snakes. And as quickly were again found, in a vast chamber of pale-gray mosaic, where a skylight of ten thousand broken panes still dribbled glass fragments down the golden incline of sun that met the shrapnel-pitted wall. They stood there, the fourteen, in a noisy, echoing silence of rubber heels abrading marble, inhaling the sour white dust which floated on the air like steam in winter, in their awe daring nothing but to take this polluted breath and give it back at the proper intervals. Over the doorway, a mile down a runway of litter fifty feet wide and to the depth of a horse, the Nazi eagle of stone-and-gilded-bronze. Besides themselves, no man.\n\nNaturally, thoughts of a mighty morality spilled into Reinhart's mind, through, as it were, the skylight: if you seek his monument, look around you; Ozymandias, king of kings, etc.; living and dying with and by the sword. And PFC Farnsworth T. Cronin, who had majored in political science, in Massachusetts, and who at this moment subtly wedged himself between Reinhart and Veronica, intoned softly: \"Power corrupts, and absolute power corrupts absolutely.\"\n\nSidestepping, Reinhart eased over to Very. \"Can you imagine him walking down the middle of this vault, his bootheels echoing for ten minutes before you could see him? He must have looked pretty insignificant in his own house.\"\n\n\"Who?\" asked Very, throwing highlights off the undercushion of her scarlet lip. \"Oh you mean Hitler. But did he live right here in the Reichstag? Must have been drafty, haha.\"\n\nIf you were careless you might identify as imbecility that which was rather inattentiveness; before penetrating the Brandenburg Gate, they had swung left up the squalid lane to the old Reichstag ruin, a long, columned cinder surmounted by a dome burned to chicken wire and facing a park of weeds. In his cicerone remarks Reinhart attributed its burns to a fire set by the Nazis in 1933. Cronin corrected in a voice flat with certainty: \"No, it was restored after that. What you see here came from the bombings and the Russian assault this spring\"\u2014he had apparently snooped in all these places before the tour got under way, while the real men in the outfit were out getting tail. Anyway, perhaps it was just as well this all had eluded Veronica, who also probably failed to notice that thereupon Reinhart suppressed the remainder of his own commentary, not only for the Reichstag but the succeeding buildings as well.\n\n\"My fault,\" he said now, manfully. \"This is the Chancellery.\"\n\n\"The What-cellery?\" But he saw in her blue eyes a candid fooling.\n\n\"Of course,\" said Cronin, studying the mosaics with his bland face, \"we are in the New Chancellery which Hitler built circa 1938-39. The Old one, dating from the time of the Hohenzollerns, is next door.\" Cronin never put his eyes on a person; meeting one on the arctic tundra, with nothing else to look at, he no doubt would try to inspect the wind. A tedious creep, yet you could tell by the measure of his tediousness that he did know whereof he spoke; it were destructive vanity not to use him for what he could provide.\n\n\"If you know the place, Farnie, tell us what else is worth seeing.\"\n\n\"Well, the terrace and garden are certainly _there_ ,\" Cronin answered, almost, in his pleasure, giving one a fair shot at his face, but not quite.\n\n\"Then lead on, McSnerd, and make a trail through the swamp!\" said Very, sending her chime like a bowling ball down the marble gallery. And this time Cronin looked full face, demonstrating above it a dun-colored scalp parted dead center, like a statesman of the Harding era, and wondering, Amherst eyes: wondering not only who she was but why.\n\nEventually they crossed a hall of massive pillars, where Russian names, in their queer letters sometimes just eluding comprehension by a hair, were scratched into the bomb-sprayed walls and from a ceiling of bare girders loose power cables swung like thin pythons anxious to drop upon a meal. And, as a thick, sifting carpet, the usual litter of broken stone, plaster powder, splintered wood, and piecemeal metal, in a quantity which if reassembled, by divine act or motion-picture film run backwards, into its original forms would twice exceed them, for no fecundity can match disintegration's.\n\nReinhart thought about this, but it was Very, with her fine intuition, who said: \"Why when things are broken do they seem like more than when they're together?\"\n\n\"Dunno,\" answered Cronin, who had apparently determined her quality and was peculiarly intrigued by it\u2014he was breaking a trail through the trash, as she had asked, and just for her, while the others mushed ankle-deep\u2014\"no doubt the air between the pieces.\"\n\n\"I don't read you.\" She stepped to the French window, of which Cronin opened and held the shutter and then caught her arm: beyond its threshold was a two-foot drop to the terrace floor.\n\nMeanwhile, Reinhart bulled on through and nearly broke both ankles but recovered with the gay veldt-bound of a springbok. Coming back, he raised his hands under Very's elbows and lowered her like a light barbell, effortlessly, then in malice offered the same to less-than-average-size Cronin, who took it!, being indecently beyond that kind of vanity.\n\nAs the others tumbled through each in his own fashion, a nurse named Lieutenant Leek despite support turning her foot, the trio of leaders waded across the terrace and into the junkyard garden of sand, dismembered trees, disjunctive wheels and pipes and tin air ducts, disjected planks; blooming out of these, in the dirty fungus-white of sunless growths, two concrete structures, pocked by shot, seared by flame, sprouting excrescences of scaffold and webbed iron, yet squatted conditionally whole.\n\nOn the left\u2014they had come round to the far side\u2014was Hitler's bunker, according to Cronin, who named the other, a cylinder with conical roof, as a sentry blockhouse manned by the SS until the eleventh hour. In the deep embrasure of the bunker entrance a detached steel door stood angled; next to it at the same degree slouched a Mongol guard, who at their appearance sullenly presented his shoulder blades and a view of trousers-rear seemingly heavy with a load.\n\n\"Slav slob,\" wittily noted Reinhart.\n\n\"Yes,\" said Cronin to a length of corroded pipe lying at his feet, \"he should be wearing a J. Press jacket and white bucks.\" Although his statement was cryptic, his emotion was not: when he looked towards the guard his eyes were filmy with approval. Then, in the self-congratulatory manner of a white man extending common courtesy to a Negro, he plunged across the debris to the doorway, open pack of cigarettes at the ready position, loudly saying: \"Z-DRAHST-voo-ee-tee, ta-VA-reesch, KAHK pa-jee-VA-yee-tee?\" You could hear all the stresses of the little Russian phrasebook distributed a month earlier by Reinhart's department.\n\nThe Mongol revolved instantly and gave him the submachine-gun muzzle big as a megaphone and all perforated with dime-sized air vents, more death-ray than gun, and if a man ever meant to squeeze the trigger, it was he. But Cronin was a stranger to cowardice; with inexorable good will he advanced, and the Mongol, though snarling imprecations in a tongue that sounded nothing like Russian and never lowering his equalizer, gave ground. Reaching the entrance, Cronin pressed the smokes at him as one might a cross upon a devil, engaged him in a going-and-coming, frustrating inquiry, and was at last driven by him into the waste of loose planks before the SS turret, where Reinhart and Very waited.\n\n\"I'm afraid it's forbidden to enter the bunker,\" he said pridefully, stepping up, as if myopic, so near that Reinhart, always uncomfortable in close approach, backed off, caught his heel in the fork of a grounded tree-branch, and freeing it too violently threw away his balance and fell backwards into a shallow trench which till then no one had marked.\n\n\"But that's the next best thing,\" said Cronin, pretending not to see, or perhaps really, in his odd way, not noticing, as Very howled vulgarly and the rest of the party, clattering through the ventilating ducts, joined her in sadistic mirth, \"that's the ditch where they burned the bodies of Hitler and Eva Braun.\"\n\nReinhart's back-skin bubbled in gooseflesh, more historical than personal, as he scrambled slowly upwards: he had felt a distinct and depraved wish to continue lying there for a while.\n\n\"Ostensibly,\" Cronin went on.\n\nOn the bank now, Reinhart saw in the trench's sandy gutter only an ambiguous rubbish of dead leaves, board-ends, and fragments of paper coarsened and grayed by dried rain. Already he had become as reluctant to kneel and rummage in it as he had been, a moment earlier, to leave its placid bed.\n\n\"Here, in this ditch?\" No, it was too much, along with the imperial chaos inside the Chancellery, to believe; was rather lock, stock, and barrel a vast hoax of propaganda and journalism; normal people like himself not only did not make history but did not see its leavings firsthand.\n\n\"I said ostensibly,\" Cronin answered. \"In my opinion it was a not too ingenious device to cover his escape to South America.\"\n\n\"Oh you don't think so?\" asked Veronica, the corners of her mouth yet remembering the laugh on Reinhart, as did her wet eyes, life-blue in this landscape of neutral tones canescing into time past.\n\n\"If so, it worked.\" Said by a newcomer to the area of the three, a stout captain in green trenchcoat, his shirt collar wearing the doctor's bare caduceus, and Lieutenant Leek hobbled up, and in another moment the others, too, lining round the pseudo grave.\n\n\"Unfortunately yes,\" said Cronin, although the captain had not properly addressed the comment to him, \"anything German always succeeds famously with us. Give Hitler a year and we'll welcome him back to defend us against the 'Reds.' \"\n\nReinhart had got interested in watching the captain, whom he did not know, whose face was manifestly German-American, wide-cheeked, beer-florid, piggishly nostriled, stupid and good\u2014what had it to say in defense of that old seed sprung from this ground and carried across the ocean to form it?\n\n\"Do you think the Reds are a real danger now?\" the captain asked in utter innocence, coloring more, for to see Cronin he had to send his eyes across Very's Himalayan front.\n\n\"Only the American Legion and the vigilantes can save us from them. First, the unions must be stamped out...\" Cronin's face became a mask of crafty evil, apparently mimicking a memory of Goebbels'. \"FDR has already been got rid of, thank God.\"\n\nHow much of the sarcasm, which Cronin injected with real ferocity, astonishing Reinhart who had not believed he could show much feeling towards anything, how much of it reached the captain it was difficult to say. Too little, Reinhart feared, and he hastened to spread it abroad that Cronin spoke in jest.\n\n\"Well,\" the captain answered, humorlessly shaking his thick jowls, \"I don't know it's anything to kid about, if these Reds are going to make trouble just as soon as we get rid of this fellow.\" He pointed into the depression. \"If he means the Communists, I don't think in the end they'd be much better than Hitler. Didn't they make a pact with him which gave him a green light to start the war? And then proceeded to divvy up poor old Poland, even though they later became our allies. Killing people is all any of those fellows know, robbing people and killing them, year in and year out, for no reason at all. I've been a physician for seventeen years but I've never been able to figure out what makes fellows like that\u2014because that's what they are, aren't they, just fellows, people like anybody else in the beginning.\"\n\nExcellent fat captain, with your wide, honest, Nordic face: you have come through! Reinhart watched him kneel like a barrage balloon folding, ever threatening to burst upwards again, and poke into the trench with graceful, doctor-sensitive hands that bore not an ounce of the excess flesh he carried elsewhere. Soon he discovered nothing and rose, despite his weight, easily, saying: \"The American Legion stamping out the unions? I have five or six patients at home who belong to both and I also think they were good Roosevelt-Truman men, and so you've got me all confused.\"\n\n\"Truman, ha!\" snorted Cronin and suddenly gave Reinhart a knowing eye; in this matter he was willing to grant that they shared a community, and what was shameful was that Reinhart had no courage to indicate him nay, from a combination of guilt and vanity was yellow to reveal he stood with the doctor, two dense and heavy light-complexioned oafs who saw the mellow where the bright boys detected the sinister. On the other hand, he, Reinhart, would as soon cut his throat as join the American Legion or a union or the Communists or the Republicans or the New Deal, or any other outfit the joining of which prohibited one the next day from being malignantly anti-Legion, anti-union, etc., which alternation, irresponsible as it might be, to him signified, as nothing else, the precious quality of humanness.\n\nThe others by now had lost interest\u2014in the trench; towards Cronin and the doctor they had shown none to start with or end\u2014and broke their ring, meandering into the rubbish towards the Chancellery terrace and the wall through which they had earlier issued, a series of high windows, shutters in all degrees of angle and caries, above each its own _oeil-de-boeuf_ like the dot to an exclamation.\n\n\"We can split up, if you want,\" Reinhart cried before the dispersal had gone absolute. \"Everybody meet in one hour at the truck outside!\" Two persons made a noise of despair. \"All right, forty-five minutes then. There's lots more to see inside: you won't be bored for a moment.\" Nevertheless he again heard the groans.\n\n\"If little Harry Truman's all that will stand in his way, expect Hitler back next month from Argentina,\" said Cronin, \"and back in the saddle.\"\n\n\"No,\" the captain answered, not looking into the ditch now and not with corny, self-conscious moral majesty, but with majesty nonetheless, the placebo-prescription majesty of the American general practitioner, famed source of the basic wisdom. \"No,\" he said, looking directly and honestly at Cronin, the punk kid yet with downy lip, probably still with Onan as his model, \"no, _he must never happen again_.\"\n\nFrom the exodus one figure lingered back, a first lieutenant who held his doffed cap and scratched a graying sideburn with the same hand. He studied something in the litter on which he stood and called, without raising his head: \"Hey Bernstein!\"\n\nCronin's captain was named Bernstein. He joined his friend, who had found a Wehrmacht belt buckle in the sand, translated for him its inscription, _Gott mit uns,_ and with him, two men whose race was half run, walked out of sight beyond the SS tower.\n\n_Bernstein:_ his forebears, like so many Jews, forced by the census takers to assume cognomens, had gone to gems, precious metals, and flowers for their names, had chosen that crystallized juice of ancient trees on the Baltic coast of Prussia, the sherry-golden amber, for theirs, and, fortune's dupes, brought it to America where its sound was more rasping than lovely on the air, if not downright comic, signifying bagels and upthrust hands. But it was another crime to be laid at the German door and not against the Jews, whose old desert tongue contained no word like \"Bernstein\"\u2014or \"Reinhart,\" which in Reinhart's sudden view was scarcely better and only different from the doctor's in that no one owed it apology\u2014or \"Schicklgruber-Hitler,\" the funniest and ugliest of the lot.\n\nWhat could a man called Cronin say to Bernstein, Reinhart & Schicklgruber, attorneys at law, delicatessen owners, or what have you, who fell out over the fratricide practiced by the last-named? Germans, Hitler's first victims were Germans!, for that's what German Jews were, no mistake; else, observing no loyalty but to their own tribe everywhere alien, they would better have defended themselves. Nay, might have taken the offensive, their noted acumen more than compensating for deficiency in numbers, and launched their own Hitler. But no they had been too trusting, too na\u00efve, too German and not Jewish enough. Jews shrewd? They were rather the rubes and boobs of history; after two thousand years they were still fresh from the sticks, assuming booblike that even in the city men were men and life was what you made it.\n\nSuch innocence was almost wicked. Watching Bernstein's shoulders too heavy to be moved by the effort of walking, Bernstein's too-solid flesh which, if some ambitious or perhaps merely desperate forebear had not shipped the Atlantic, would have by his fellow Germans been resolved into a dew, still hearing oxlike Bernstein's simple, hateless statement that Hitler must not happen again\u2014Reinhart himself, pure Teuton, on the margin of this ditch would have condemned German, no, the world's gentiles to eternal fire and thought it too cool\u2014considering Bernstein the good, the innocent, the Jew, obsessed by Bernstein, Reinhart hated him.\n\nJews really were the chosen, the superior people. This had been Bach's final meaning, only put in the queer, inside-out logic with which the truth was approached by Middle-Europeans, who really were sapient and deep and lived on an old ground ever fertilized by fresh gore. Poor cloistered Cronin, poor dear Veronica, they could not understand irony, that means to confront the ideal with the actual and not go mad, that whip which produced the pain that hurts-so-good, so that in the measure to which it hurt it was also funny. Finally, having flogged and laughed yourself to the rim of death's trench, you looked within and saw irony's own irony: the last truth was the first.\n\nPoor Cronin's hitherto mobile mouth fell open, static and silent at the incantatory syllables of \"Bernstein.\" He could be read like a highway poster: 'Can a Jew, vis-\u00e0-vis Hitler's ghost, be wrong?' Far easier to accept that oneself is an ass. When he closed his lips again he wore a smile bespeaking relief; when he returned to the ivy he would switch his major to natural science.\n\n\"Politics,\" said Very, pressing her bosom like an armload of soccer balls against Reinhart's arm\u2014accidentally?: to study that was Reinhart's own relief\u2014\"thank God that's an Irish trait I don't have! Find a politician, find a crook, as the man says.\" Poutingly she flung away from his side, as if he were sure to hold the opposing view, swinging capelike her soft fall of hair which, seining the sun, caught a sudden amber shaming old Prussia with its clarity and fire.\n\nIn the combination of Very and Trudchen, Reinhart's needs were met. Such a thing was thereby proved possible, contrary to the popular wisdom which crepe-hangingly warned that man, the questing beast, was never satisfied, that worse than not achieving your aim was getting it. Indeed, he was living high off the hog in Berlin. He was rich: Marsala had sold all his gadgets in the black market and, each week, the candy ration. He did even less work than before: now that the tour was set it ran itself and Pound was gone off on leave to Switzerland, of course in the company of Nurse Lightner, where he intended to buy and transport to Berlin as personal luggage a footlocker full of wrist-watches.\n\nOrganize your sex life and all else followed, the phallus being the key to the general metropolis of manhood, which most of the grand old civilizations knew but we in America had forgotten. For example, in Ohio carnal knowledge of a sixteen-year-old girl was a prelude to the penitentiary; they could stick their pointed tits like crayon-ends in your face, wag their sloping little behinds, in summer wear shorts to the junction of belly and thigh, but if you rolled an eyeball towards them you were a pervert. He never entered Trudchen without tremors of retroactive revenge.\n\nWith Very, on the other hand, he was getting back at Germany and all its exoticism gone nasty. That was the great thing about women: with one, you had a place in a context. He had begun to think of himself as the kind of fellow who might one day get married; at least he detected the future inclination. Writing to Pound's wife he had felt vicariously that peculiar pleasure of having an attachment one owed to and was owed by. Love as a mutual debt\u2014certainly it was new to him as he grew old.\n\nNo longer did he spring from bed at Marsala's eight-o'clock clarion, but lingered for a second and a third and then the thrust of a hard hand against his head, at which, still unconscious\u2014which was his excuse\u2014he punched out wildly at his disturber, and even though he usually missed, Marsala stayed sullen all day at these thanks. A mature man should not live with another, but with a woman from whose soft lump beside him under the steaming, odorous blankets he can take a motive to rise, the sooner to be off to honest work, the sooner to be home again as evening falls to meet this sweet dependent, now the smiling presence of the succulent table, prepared for two and not five hundred.\n\nYet what honest work? Had the war not come he would now have been for a year and a quarter a Bachelor of\u2014what? A process beginning in Central Europe in 1933 (Carlo had a popgun, wanted an air rifle), or 1924, then: Hitler, having failed to capture Bavaria with his private army of cranks and loafers, sits in the prison of Landsberg am Lech dictating to Hess a lunatic statement of aims which two decades later when they have been realized to the letter are still unbelievable (umbilical cord severed and tied, Doctor slaps Carlo's bare bottom, Carlo wails, he is human and alive), a process whose origins are in the mists of the past, whose products are millions of dead and a continent made garbage\u2014this same process, the blowing of an ill wind, solves for one young man a dilemma, what to do with himself?, but only for the nonce.\n\nInsurance? His father could get him in at Ecumenical Indemnity (Laughter). The campus again?; this time indebted to nobody: \"they\" were going to make it free for veterans, no selling apples this postwar. Which meant either of two: either everybody would go to college, and being mass it would be mean; or none of the ex-servicemen would go, leaving the same old collection of pubescent punks he had got his fill of long before coming to occupy Berlin. Germany itself. Take out papers, if you could find a government to become a national of. Pose as a mustered-out SS man, for which you had the proper appearance, make a living in chocolate bars and Lucky Strikes, pimping for Trudchen. Or merely sit in some congenial ruin and weep away to a skeleton, for what as a German you did, as an American you did not do, and as a man you saw no fit atonement for.\n\nSince his needs were met\u2014women, riches, life of leisure, _gem\u00fctlich_ flat, loyal friend (who else but a true-life Horatio would dodge punches to do one a favor?), his connection with history (American news correspondents staged a spontaneous demonstration of Berlin GIs celebrating the Japanese surrender; photographed it; Reinhart stood upper left dutifully tossing his cap towards the sky)\u2014since all these holdings were verifiable to the senses, euphoria must, by definition, ensue.\n\nYet, within the very seed of comfort he detected an inimical, corrosive juice which like the acid in a hand grenade waited tirelessly on the pulling of a pin to begin incendiary mixture. Satisfaction was his, but so also was a growing conviction it should not be: why should he alone be rewarded when the rest of the world was taxed? Even the other Americans had their troubles, wanted grievously to go home, suffered in what he so grossly enjoyed.\n\nHe began to fear his own compulsions; if he did not hurt Trudchen enough, neither did she him, and it was not because each did not try. Violent as it was, that plunging to explosion only suggested a damage he could imagine but never yet achieve, that catastrophic end the reaching of which he came, in a kind of pride of horror, to believe was his true vocation. Truly, Trudchen was too depraved to defile and too small a mount to ride to victory.\n\nVirtually unused went his murderous-muscled body, the welted hands with one of which he could have lifted Hitler and cracked off that weak neck like a sparrow's, penetrated Goering's breadbasket as a thumb would sink into a rotten pear. Where was the game worth the candle, where now, standing in the empty stadium, too late, alone, a lackey groundskeeper amid discarded programs and ticket stubs, where now to find another contest?\n\nTime had fled. _Berlin bleibt doch Berlin,_ as the natives said, but for the original occupiers\u2014the 82nd Airborne having replaced the 2nd Armored, Reinhart's medics were seniors in service and disenchantment\u2014as September approached, it was a different city from that Newfoundland into which their trucks had rolled on a sun-swept afternoon in July. The aftermath of war had shaded into the onset of peacetime. Regiments of women in kerchiefs and dark stockings labored to clear the bomb-sites and reclaim sound bricks. The Russians freed and dumped into the Allied sectors some thousands of Wehrmacht prisoners, who staggered along the main thoroughfares tattered, hollow-eyed, embarrassing civilians, panhandling American passersby. The black market shrunk from too-flagrant spectacle. The newest currency regulations were difficult to evade: Pound converted his Swiss watches into Occupation marks\u2014and because the going price had fallen with the replacement of Soviet combat troops by a more conservative element, nonrapists, small spenders, of a dour respectability, got only half as much as he would have in July\u2014but could not get them into dollars and home. \"Here I sit,\" he said to Reinhart, once for every hour they spent together, \"with my finger in my ass and one hundred thousand marks.\"\n\nWith the new Russians came fewer explosions from their sector, although incidents, frequently mortal, continued. Earlier they had shot Allies and one another in jest; now the motive had changed to a solemn dislike. Americans were counseled to avoid the eastern quarters of the city, were seduced to remain on home ground by a grandiose Red Cross Club on the Kronprinzenallee, where in the stately dining room a string ensemble in threadbare tuxedoes ingratiatingly whined and the fare was sinkers and coffee in individual silver pots; by the Uncle Tom movie theater on Onkel-Tom-Strasse which led to a structure called Uncle Tom's Hut in the Grunewald Forest, the name German-given, long before VE-Day, for a reason no GI could grasp; by the Berlin Philharmonic, at concerts in the Titania Palast in Steglitz, though soon its conductor, out legally one night after curfew, was misunderstood by an American sentry and shot dead.\n\nPersonnel who numbered their years in the late thirties or more were shipped back to the States as senile. So went Reinhart's friend Ben Pluck, in civil life a lawyer; in the Army, having declined to serve, eternal PFC. Others left on longevity points; thus transported was Tom Riley, from across the hall, saddening everyone whose flat lay adjacent to the stairwell; no more would the iron treads echo his jovial filth.\n\nIn the latrines they predicted the 1209th would go to Osaka, Japan, where the bearded clam ran crosswise, or the Azores, as in the limerick about sores, or as a kind of liaison force to the Turks in Istanbul. On the wards were one hundred twenty complainants of nasopharyngitis, all on the light diet. The colonel ate out the assembled officers and nurses on the subject of fourteen spent contraceptives spiked off the hospital grounds on the lances of his sanitation crew, directing his remarks principally to Chaplain Peggott and Major Clementine Monroe, the superannuated chief nurse.\n\nEverybody in Reinhart's apartment building had a local mistress save two ethereal privates from Supply, who had each other. Don Mestrovicz, technician fourth grade of the EENT clinic, had two in the same family: a mother still young enough, a daughter just old enough, to whom he was the filling in their sandwich. Corporal Toole from the motor pool owned a big round woman with a behind like the belly of a lute. Bruce Freeman, of X-Ray, had an ash-blonde named Mimi Hammerschlag who played bit parts in Ufa pictures; Jack Eberhard, company clerk, a dishwater blonde who like him made strange noises when drunk. Sergeant Deventer's girl could do a take-off on Hitler with a comb for a mustache; Bill Castel's woman, an artist, cut out his silhouette in orange paper. Ernie Wilson's piece was three weeks pregnant; Roy Savery's, one month; five others professed falsely to the condition, three of whom named the same sire, T-3 \"Plumber\" Cobb\u2014he laid a lot of pipe\u2014but were duly unmasked. And Farnsworth Cronin was sometimes seen with a boyish girl whose name was spelled _Irene_ and pronounced _Ee-ray-nuh;_ he, however, called her _Boo._\n\nSupply outfitted everyone with short jackets, like Eisenhower's, calling in the old skirted blouses. All noncommissioned officers in the ETO were granted a liquor and beer ration; in the 1209th these were consumed on the rear balconies, feet on ledge, cigars in jaw, and in the company of the girls, who giggled much and sometimes sang in English. No Werewolves having turned up, the district order that US personnel carry arms when off compound\u2014the medics, their red-cross sleeve bands\u2014was rescinded. Under the authority of the Information and Education Program, Gerald Gest was sent to Paris for a month to study French civilization at the Sore-bone, and a class in basic psychology, meeting once a week in an empty storeroom in headquarters, was offered to qualified enlisted men, which meant everybody; in its chair, PFC Harvey Rappaport, MA from NYU.\n\nA sandy-haired corporal named Gladstone, who worked in the post exchange, blew out his brains there one night after closing, leaving no note. Veronica's neuropsychiatric ward, already so crowded that three patients bunked in a supply room, somehow stuffed in five more beds. Walking past its door you never heard a sound, although by her account half a dozen patients wept all day and another man made squealing noises with a finger against his teeth. A paratrooper, under observation for persistent bed-wetting, was discovered to be a poseur\u2014in the wee hours he did not really wee-wee but soaked his mattress with H20 from the bedside glass\u2014and sent back to his outfit on charges of malingering. A tentatively diagnosed schiz struck Lieutenant Llewellyn, assistant psychiatrist, in the nape, knocking off his glasses, then sought to crush them but couldn't with bare feet. Another patient, a brawny man with the hair of a goat, incessantly planned to become a novice in the Carmelite nuns.\n\nNo doubt it owed to such spectacular persons and events that Veronica by the fourth week of their acquaintance had lost her bloom, or rather that part of it which was rosy towards Reinhart, who suspected that being normal he bored her. And he could not very well divulge the doings of that other self who lodged with Trudchen, the mad one, the one with passions which, being there resolved, freed this one, the front man, to be so smooth and bland. Back there, Himmler did his dreadful work; up here was elegant Ribbentrop, kissing hands.\n\nFor years he had cultivated the art of surrender to women to offset his bulk, which sometimes on its approach caused, particularly small, girls to look for cover. The brute tamed by gentility, the handsome and moral equilibrium of opposites. No, its validity consisted only in the abstract, never in practice. For instance with Very: he cared little about the destination of a date, so long as it was not an official Army entertainment where they must be separated by rank. But Very had for the movies the insatiable hunger with which it was said expectant mothers went to dill pickles\u2014a touch of madness for them, really, Western, gangster, comical, historical, pastoral, pastoral-comical, historical-pastoral, any passage of arc light through celluloid into minuscule glass beads generating counterfeit life, but especially Dramas in which any one of those actresses with big eyes and hard white jaws, dressed in jodhpurs, carrying whips, riding stallions, of course gelded, consummated a union with the scion of a swell family of old Virginia and ate for breakfast grapefruit in a bed of ice, her nostrils flaring.\n\nIn the absence of an enlisted boy friend this taste would have carried Very every evening to the Onkel Tom Kino, to that roped-off centrally situated block of seats exclusive to officers, there for two hours to shuffle off the coil of banal mortality.\n\nNow if this was her pleasure, and as a gentleman his was in seeing she received hers, why now admit obstruction? \"Look, honey, don't worry about me. I'll sit back in the enlisted section and meet you outside somewhere when the show's over.\" \"Now you're sore.\" Of course he was not angry, just piqued at her resistance to civility. With the best intent in the world she went on: \"No, tonight we'll do what you want.\" \"I want to see the movie.\" and usually they did, segregated for two hours and when afterwards they met, Veronica, and not he, looked miffed.\n\nAt other times when there had been no conflict of wishes, when they had taken long night strolls sometimes as far as the walled villas of Dahlem's tree-murmuring walks (almost the only alternative he could offer to the movie-show, which was another reason for his reluctance to prevail), necessarily avoiding society, when they should have formed not two but one, in a sealed capsule of mutual affection, Veronica had lately seemed, not exactly withdrawn, but at least preoccupied. Working with these lunatics all day\u2014apparently her thesis that they got worse in peacetime was daily confirmed\u2014what could you expect? At first he tried to jolly her out of it, but in itself it is a morbid thing to have to cheer a woman, a transposition of the proper roles, she being by nature equipped to bring joy, while man is the rightful brooder.\n\nAnd considering the precise Very, unfortunately her physical design was not for melancholy. When not in the mobile oval of laughter, her mouth formed a horizontal too broad; her chin appeared square and somewhat virile; when not quivering, her nose was a mere cartilaginous organ, not altogether true, for the induction of breath, and one could understand that it might turn crimson with the grippe. Her eyes when solemn were too pale a blue, the little skeins of iris-color patchily breaking unity, and was not the right one a lash-breadth off the zero aim? Not stimulated, her blood declined to flood her cheeks, and once, at the corner of Max-Eyth-Strasse, in the side apron of his flashlight beam he saw her face was ashen.\n\nVaguely desperate\u2014for he _was_ extremely fond of Very; not in love, actually: that was just something he had thought\u2014Reinhart conceived a plan to get her into the fresh daylight air with a view of water and woods, away from minds, anyway, for one afternoon. He organized some hardboiled eggs, canned meat, and other junk from the mess sergeant, even borrowed the still half-full jars of mustard pickle and mayonnaise Bruce Freeman's mother had mailed that gourmet, and one Wednesday, which that week was Very's day off, with her set out for a picnic on the shore of the Havel.\n\nFrom the beginning, from the moment Corporal Toole let them out of his jeep at the woodland corner of Pfaueninselchaussee and Koenigstrasse, everything went right. The better part of an hour went before they gained the shore, but Very's color improved with each brisk step. At intervals Reinhart hopped off the road into the forest, to bring back talismans: a spray of lace fern, pine cones, a root like the trunk of an elf-woman, a stone resembling an eye, and of course, even out there, a clip of rifle cartridges. Excepting the latter, he gave them one by one to Very, who by the fourth presentation complained of loaded hands, twitted him for his idiocy, and, at last, laughed\u2014perhaps only a snicker, but her first in a week. He was rapidly bringing her back.\n\nOn the beach, of which, wandering to the right from the spit pointing towards Peacock Island, they found a length unoccupied by military wreckage, Reinhart brought the goodies from his musette bag. In a messkit bottom Very mashed the eggs with mayo. When Reinhart bit into the first sandwich a fragment of shell cracked between his teeth, just as if he were home. He ate two, and then one of Spam, and then three pairs of saltines enclosing a hard cheese the color and taste of GI soap, and then an orange\u2014for he had brought nothing else for thirst\u2014and Veronica joked about his capacity. The scorings he had lately noticed in her cheeks were but night shadows, already dispersed by the sun.\n\nHe lowered his head against a massive log half-buried in the sand and extended his legs luxuriously, out, out, out, toes towards the lake, taking the pleasure of a prolonged stretch, rather like that of a mild orgasm, grunting, eyes narrowed, arms going back over the log. Five yards away the water munched quietly on the sand. Across against its far margin, the dark horizontal of the Kladow shore, a white sail quivered. On the left, and so near that in his view it seemed not an island but rather the other side of an unbroken bay, lay the Pfaueninsel. A suspicion of autumn, a certain chill filament woven into the otherwise still very warm fabric of sunlight, rather imagined than felt, and as yet too thin to penetrate vegetable nature, was felt by Reinhart, in whom it engendered a sad, sweet deliberation on the coming death of the year; and since the end of anything is peace, his heart, too, like Very's, fell placid.\n\n\"Ah,\" he cried suddenly, sitting up, \"we forgot the mustard pickles!\" He unscrewed the jar and offered it.\n\nVery, while he had unfeelingly stuffed himself, had not eaten a bite, he now noticed retroactively; and the flush in her cheek was nearer the introduction of illness than health returned, as she stared with terrible white eyes into the jar and said, feebly: \"They look like alligators in the mud.\"\n\nShe raised her stare to him, and he saw in it a catastrophe from which he would fain have run, had it not been intermixed with a beautiful weakness towards which his manhood inexorably flowed as all streams to the sea. She had essayed a joke, but tears caught her hard upon the last word. Against his chest he brought her weeping, fragrant head, and told close into her ear the platitudes of comfort.\n\nShe shortly pushed him off in a kind of anger and, with eyes still melting, assigned all guilt to him.\n\n\"If this isn't anything, nothing ever was: I am pregnant.\"\n\nAt the edge of the beach, a fish, or a frog, or some other animate and lonely thing, loudly slapped the water and sank through a necklace of air bubbles.\n\n# _CHAPTER 16_\n\nNEXT CAME THE INSECTAL hum of a far-off engine, in perfect rhythm with the prickling of Reinhart's hide. Unless nocturnal fancies could inseminate, his tremors belonged to another man, for he, Reinhart, had been no closer to Very's reproductive area than the line of her belt. To put down the guilt, he developed a fury: And I, he raged in secret, I have always acted as if she didn't have a \u2014\u2014 (the good old bare word from the honest Anglo-Saxon culture of artisans and farmers, dating from a time before the mincing French crossed the Channel, before the eunuch scholars began to drone in tedious Latin, and eons before small Reinhart belatedly learned from a schoolmate that females are not smooth between their legs and do not produce young by unwinding at the navel).\n\nAnd by extension, the term applied not only to the orifice but also to that woman who made free with hers. In love with a\u2014 but he would not think it again, this short, blunt syllable which in barracks was aired as frequently as exhausted breath: he would not because \u2014\u2014 ness was not here at issue. Suddenly he envied her her achievement, lusted not for her body but for her trouble; wished he could weep for having committed a grand foolishness and be comforted by a big disinterested horse's ass who never took a chance; began himself to grieve for all the errors he never made, all the disasters that all at once he strickenly knew would never ruin him\u2014except that, so far as was apparent to the outside eye, he stayed slick and bland. Control. How detestable it was, control; how uncontrollable. How selfish!\n\nWildly he seized her again, this woman attractively defiled with adventure, for the first time his hand went where the eye's fingers had so often dawdled, to the great hemisphere of her right breast and then to the left, circumnavigating like a Renaissance explorer. Licensed, it was a disappointment; and indeed he knew not why he toyed there, since his purpose was, with lump in throat, self-sacrifice.\n\nHis one hand still mobile and encountering more brass button than pendulous woman\u2014at least, that must be what was cutting him\u2014with his other Reinhart lifted her rinsing face, now of a more poignant beauty, pale, implying the sanctity of a plaster, Hibernian-featured Virgin, so much more moving than must have been the real one, dark and muttering in guttural Hebrew. Deep into her eyes he was careful not to look, as he said: \"Very, I will marry you!\"\n\nWhat did he expect?: at minimum that the rivulets would cease to flow? Rather were they renewed, as like the heat of summer reaching the highest snow, a brilliant flush mounted to her forehead and a greater rush of water came down.\n\n\"How can you!\" she wailed. \"You are not Catholic!\" Repeatedly she struck his chest with her balled fist, no doubt leaving bruises.\n\nJesus Christ. Like a mongoloid he stared expressionlessly at the lake. The hum of the engine had grown to a still-distant roar.\n\n\"Isn't there anything I can do?\" He heard himself say it and was astonished by the mousiness of his tenor.\n\n\"Hold me.\"\n\nHe did, with static hands, and squeezed her, and pushed his nose into her soft hair and breathed relief that he had not really loved her but only thought so for a while.\n\n\"But Him, what about Him?\"\n\nBecause he had clearly pronounced the capital, she germanely asked: \"God?\"\n\n\"Christ sakes, I mean the guy, who certainly wasn't me! The Invisible Man, because I was under the impression I saw you in all your off-time\u2014unless of course it was one of your psychopaths during duty hours. Is that now part of the therapy?\"\n\nIn more abandonment than she ever showed while necking, Veronica snuggled into him. \"Go on,\" she whispered, \"say anything, I deserve it.\"\n\nNo, with just that quantity of censure he was done. Reinhart on the judicial bench would have freed all malefactors who pleaded guilty, for what could subsequent punishment do but incriminate the judge? Besides, he recognized in his coarseness the tedious old suburban lie that the sexual life was to be regulated by a middle-aged housewife's sense of right and wrong. Screw, screw, screw, if you wanted to, he was proud to think was his credo; and that his own girl friend sported on that plan was the sincerest form of tribute.\n\nStill, if that were her taste, why had she to look elsewhere from him? He was not repulsive to women; time past, he had actually spurned unsolicited advances.\n\n\"Who am I to say anything?\" he asked, now looking into her eyes. \"I'm nobody\u2014as you have proved.\" He attempted to loosen their connection, but she had clasped her arms about his waist and locked her fingers, as in that test of strength in which you try to crack the other fellow's spine.\n\n\"You're the best friend I've ever had. Do you think I could tell anyone else?\" Her mouth with its liberal lipstick was crushed against his blousefront; on the journey home he would look as if he had been shotgunned in the chest. In her rich hair, which was no longer his property, was caught a fragment of twig, which, nevertheless, he plucked out. That dear fragrance which in the old days clung to his cheek for hours after leaving her, which during a night of sleep transferred to the hood of his sleeping bag, where he could smell it next bleak morning, now penetrated his nostrils as he supposed a sister's might, stirring mild affection but also thoughts of silly stench.\n\n\"I know, I'm like a brother to you\u2014but God damn it, Very, I never knew that till now. You've made a fool of me.\"\n\nThe chopped-egg sandwich Very had made for herself and not eaten, already slightly wilted, lay upon the green canvas of the musette bag. Still holding her, he took it and began to bite off the valances of squeezed-out filling. He did this theatrically, playing the conscious role of a person who vulgarly stuffs himself at high moments, learned from the motion pictures.\n\n\"Well, now what are your plans?\" he asked. A bit of egg fell, narrowly missing the gold bar on Very's right epaulet, the tracking of which brought his glance to a side view of her cheek and his attention, since her eyes were closed, to the matter of whether or not she had gone to sleep. \"Hey,\" he said, striking her roughly with the blunt of his palm, \"recover! What are you going to do now? Look, first, are you dead sure? You know swimming will delay it, and an illness too, I think. Didn't you have a cold last week?\" Without physical intimacy he yet knew very well the schedule of her menses: the laugh was not so broad in that quartet of days, and she sometimes complained of headaches. It was just, or should have been, over.\n\n\"I've had the Curse enough years to know all its tricks,\" she answered wryly, cocking up a brow that suggested the old, witty Very's\u2014and he would have liked to catch her there, saying That's it, hold it right at that point and nothing is lost, but she was seized sooner by her own voice, which wavered and ended brokenly: \"This time it's for real.\"\n\nAh, Reinhart thought, means business, does it?; isn't kidding around; no joke; on the level; for real. Perhaps he believed that nothing ever happened to him because when it did, its effects were stated in barbarous language. Once in Piccadilly talking to a streetwalker he heard overhead the Model T chatter of a buzz-bomb and thought he might die, there in the thronging black street, while a whore said \"Coo, ain't it a loud one? Four pounds for awnight, I'm no bawgen basement.\" His apprehension proved baseless; the bomb sailed on to detonate in some working-class quarter, where the survivors climbed from the smoking ruins to say \"Gor, that wasn't 'alf close.\" Poets are never bombed and, if women, never knocked up.\n\n\"Well, what am I to do about it, since you so nicely included me in? I don't want in\u2014you might say, I never _got_ in.\"\n\nObviously with Very it had been love; and hardly the kind he talked of to himself in his childish way\u2014never again! Very in love, a victim of the conquistador passion; what would ten minutes earlier have been impossible to accept was now only difficult. At least she displayed one requisite of the authentic state: sorrow.\n\n\"You?\" she answered. \"It isn't your trouble, Carlo. You could just get up and walk away from me\u2014you ought to. I can't ask anything of you.\" She finished crying and sat stolidly in that neutral condition which precedes the return of vanity; unlike Trudchen she looked older without make-up. She _was_ older than Reinhart by three years; his girls always, saving Trudchen, were; one's elders are kinder than his juniors and peers. Maureen Veronica Leary, from a suburb of Milwaukee fittingly named St. Francis, graduate of a grade school, high school, and hospital (on the three-year wartime crash program) each under the rubric of another saint. She had a brother and three sisters, all older, and her father, a retired street-railway motorman, told stories of Galway, where the supernatural was commonplace, although he hailed from a town near Dublin named Blackrock. By the time toys and clothing had come down to Very, stuffing leaked from the dollbabies and the stocking-toes were lumpy with darning; but once when she was twelve her father, drunk, brought home a pair of rollerskates her very own. Her mother stood five-ten and her number two sister, just under six and was unmarried at thirty.\n\nSo much had Reinhart, in the normal course of events, learned. But who was the real, the essential Very?, to the exterior of which he had been attracted by its air of simpleminded jolly Catholic health. Perhaps a pagan. Not only did he usually go for older girls, he also had a weakness for Roman Catholics, who even when Irish remembered the Latin basis of their persuasion and were very feminine, seldom prudes even when they would not submit: to them a man's appetites, being natural, being God's splendid trick to ensure the race's continuance, were never, even when illicit, loathsome. Now he realized he probably could have, with Very; too late he remembered her constant surrender; indeed, she at the outset had chosen him and then waited in vain.\n\n\"You're in love with this guy, is that it? I suppose you don't want to tell me who he is.\" She still sat within the enclosure of his right arm, while his left, propped against the ground, suffered a slow paralysis. The sand worked up under his fingernails and shortly into a small, smarting cut on the first joint of his thumb, on which he had forgotten to put a new Band-Aid after removing the old one that morning. He had damaged himself on the clip of Trudchen's brassiere, which lousy German thing was not a simple hook-and-eye but a pronged buckle with criminal tines. Since in his subsequent lust he had not taken care, if she had VD he with his open wound was a goner. Ah, accept it, we are all submerged in filth up to our heads. Accepting which, he saw the one man who was exempt emerge from the high thicket of marsh grass thirty yards down the beach.\n\nPreceding this, the engine noise he had unregisteringly heard earlier, had grown loud, identifying itself as an outboard motor; had come so thundrous close to their position that Reinhart expected momentarily to be swamped but didn't care; had, just as it must either become visible or explode, shut itself off with two loud farts in the marsh, yet unseen. Within the minute, this person in black beret and bulky coat appeared, stamping down the last few rushes which denied him clearance to the beach and, that done, seeing them\u2014not necessarily looking towards them, but seeing. By the man's use of this unusual faculty Reinhart recognized, despite the altered outline, Schatzi; who surveyed the four points of the world and approached.\n\nFearfully Reinhart promised Very they would return to her business later and ruthlessly withdrew his arm. Not adjusting to the new arrangement, Very stayed numbly huddled, leaning against\u2014nothing, for Reinhart was already on his feet, obstreperous in greeting.\n\n_\"Guten Tag! Wie befinden Sie sich? Wir machen ein_ \u2014 _wie sagt man_ 'picnic'?\"\n\n\"Just so,\" replied Schatzi, ten yards away and apprehensively halting there as if he foresaw an attack. \" _Picknick,_ one and same.\" He winced obsequiously. \"Do I receive your permission to come there?\"\n\n_\"Wo?\"_ Reinhart was still yelling as though his auditor were in Kladow.\n\n\"In the vicinity of you and your lady.\"\n\n_\"Warum nicht?\"_\n\nSchatzi came, still cautious, shoulders thrown high and hands buried in the pockets of the great overcoat, which was a dirty teddy-bear plush and fell to his ankles; scarred face contorted below the beret like a withered acorn in its cap; feet, of which only the neat little brown toes were visible, scuttling forward, one-two, pause, one-two\u2014his overcoat fell past his ankles to brush between his footprints the spoor of a tired fox that drags its tail.\n\n\"Fancy occurring here with you,\" he said when he arrived, gauging on the balances of his eyes a specimen of flesh from both the large lumps before him: the one on the ground and Reinhart.\n\n\"Whatever are you doing here\u2014way out here?\" asked Reinhart, altering to superiority. At the same time his unease grew more severe: he was certain Schatzi thought he had again caught him in a screw, or just after; for a crazy moment he suspected Schatzi trailed him for just that purpose.\n\n\" _Ach,_ business, always business\u2014your gentle lady, she is ill?\" At last Veronica acknowledged his arrival, looking up with forlorn-beagle visage, saying naught. He removed his beret.\n\n\"Just tired,\" said Reinhart. \"We had a long\u2014\" he lost his voice as he watched Schatzi prepare and deliver a massive, obscene, hideous wink. However offensive, it was mesmeric: the lid flattened and then went concave, seeming to close upon a hole rather than a ball. More horrible yet, Reinhart helplessly felt his own eye return the favor.\n\n\"We had a long walk,\" he said quickly.\n\n\"Exercise, ceaseless exercise,\" said Schatzi benignly. \"Well, why not?, as you say. You are yet top young for a _Herzschlag, ja_? _Auf Englisch hei\u00dft das_ 'failure of the heart,' am I correct?\" With fingers like wire-clippers he pinched a bit of jacket, shirt, and skin on Reinhart's forearm. \"Your lady as well, though. Whatever will be her difficulty? Paleness! Ah, right in the pocket I have this brandy which will make the trick.\"\n\nIgnoring Reinhart's weak verbal opposition, he withdrew a silver flask, unscrewed the cap and let it dangle upon its little chain, hitting the body of the vessel _tok-tok,_ and stared down onto Veronica's crown.\n\n\"She doesn't want any,\" said Reinhart, but so as not to offend Schatzi he offered to take a draught himself.\n\n\"Who doesn't?\" Very, who had been playing no heed, now with violent interest seized the same forearm that had been pinched and pulled herself up. Daintily accepting the flask, she arched her neck like an old grad under the stadium and drained off quite a large slug, then paused to take air and would have returned to kill what was left had not Schatzi, deft as a mongoose, leaped into the breach between her movements and reclaimed his property, saying: \"Already better, _ja_?\"\n\n\"Whew!\" whistled Very towards Reinhart. \"That went down like a whole loaf of bread.\" She gulped five times and smoothed the sitting-wrinkles from her skirt; lingeringly, with some evident pleasure in the touch of her own belly and thighs. Schatzi averted his face, as if offended.\n\nIf so, how right he was to be. \"At least thank the man,\" Reinhart muttered low.\n\n\"I don't know Kraut.\"\n\n\"Haven't you just heard him speaking English?\"\n\n\"I'll take your word for it.\" Turning to Schatzi, she asked, naturally very loud: \"Hey, didn't I spot you hanging around in front of Lieutenant Schild's place the other night?\"\n\nSchatzi twisted his neck to favor the left ear. \"Spot? Hanging?\"\n\n\"On the level now, weren't you? Oh, you weren't in that getup. You had on a cap with a beak and you sat on a bike. Well what I mean is, it was you, I know. Come clean, I won't blow the whistle on you for being out after curfew.\"\n\nReinhart made the sound of a bellows. \"Damn it, Veronica, what do you think our friend will be able to understand of that?\"\n\nBy way of answer she simply smirked: drunk, apparently, or pretending to be from the moment her lips had kissed the flask. It was a way out of her difficulties.\n\n\"She thanks you very much,\" said Reinhart to Schatzi, who was confusedly repeating under his breath, 'cap with a beak, sat on a bike.' \"And she thinks she saw you last night in front of the house where\u2014where lives an officer she knows.\"\n\nWhere lived, indeed, to admit to himself the complete data, Lieutenant Schild. _Lieutenant Schild._ Which, admitted to the mind, was instantly transformed to: _that Jew._ Who hath usurped my office twixt the sheets? The Jew, the Jew...\n\n\"Lieutenant Schild,\" repeated Schatzi, wonderingly, pointing his ridged carrot-nose towards the lake, high; meanwhile his eyes went everywhere else. \"Lieutena\u2014wait a moment, I think\u2014no. _Also! A_ great fat beast of a man, with a mustache like a broom, and an implement to his speech, so that when he says something he makes this sound between the words: _shicksh, shicksh._ Now tell me am I right?\"\n\n\"He is thin and dark,\" Reinhart said evenly.\n\n\"Yes, an Italian,\" Schatzi smiled his recognition. \"Yes, I was able to obtain for him\u2014you will pardon me, Madame\u2014some items of which we shall only say they are worn by the ladies and cannot be seen unless one is\u2014please pardon me, Madame\u2014in a relationship of intimacy.\" He laughed quaintly: \"Hahahaheeheehee,\" colored, and said: \"Now I have gone too far.\"\n\n\"That isn't possible in the present company,\" Reinhart answered hatefully.\n\n\"Oh, he speaks English all right, but I don't get a word,\" said Veronica to herself, and then to Reinhart: \"Ask him if I can have another drink of that radiator fluid.\" She threw up her hand. \"I'll pay him for it, don't look so ghoulish.\"\n\n\"Nonsense,\" said Schatzi, already presenting the flask. \"My compliments. You are a friend of Lieutenant Schilda? Please, I do not mean to offend.\" Again he laughed, this time in a very horsy manner with open, serrated mouth. \"Herr Unteroffizier Reinhart, please tell the lady what is the joke.\"\n\n\"I wish I knew it myself,\" Reinhart said sullenly. \"I wish I knew what was so goddamned funny.\"\n\n\" _Also,_ Schilda is the town where the fools live. What is it in the States?\"\n\n\"Reinhartville,\" said its exclusive inhabitant, watching Very swallow the rest of the brandy. With gelid courtesy he accepted the weightless flask and gave it to Schatzi. \"Well,\" he said, turning to her, \"your troubles are solved. Since Schild is Italian, he is also Catholic. He can marry you, and may I say no one would be more appropriate.\"\n\nShe failed to answer. Already her eyes were distorted, as if one saw them under water.\n\n\"You will mah-ree Lieutenant Schild? How lovely,\" crooned Schatzi, moving in upon her, thin jowls tremulous, as an ambitious chihuahua might approach a mastiff bitch. \"I can furnish food and drink for the feast. But you must both soon go back to the U.S. When?\"\n\n\"See what I mean?\" murmured Very. \"If that's English, I'll eat it.\" Now her eyes looked as though a hair were drawn across each retina. By age thirty her figure would be throughout, like a Balkan peasant woman's, the diameter of her chest; her abdomen in permanent pregnancy; thighs, like jodhpurs. The catalogue of Reinhart's malice continued through her parts, which in the here and now were flawless... and the receptacle for a Jew. Evil, evil, evil\u2014with evil he flagellated himself while there was still time. For of course he had this deep feeling about Jews, deeper than any he had had for Very; indeed, he recognized now, in the core of his hatred, that it was love. He loved the dead of the camps, and Bernstein, and half of Lori, and... Schild; and the dearer the possession, the dearer it was to lose it to them; nay, the dearest were not enough. Thus had Schild been in his presence then, he might have killed him as his wedding gift: Jews were too good to live.\n\n\"With all my resources am I trying to be understandable,\" said Schatzi to Reinhart, pathetically. \" _So,_ you tell me please, Herr Reinhart, when is this mar-ee-ahzh?\" He replaced the beret which he whipped off whenever he spoke to Very and drew Reinhart aside. In an undertone he asked: \"And is not this queer?, this little _f\u00eate champ\u00eatre_ without the fianc\u00e9? You rogue! The little Trudl is not sufficient for your capacity. And then the Bach woman, too, I believe, as well. Extraordinary. Soon you will have exceeded the Swiss Ambassador, Herr V\u00f6gli von M\u00f6gli T\u00e4gli.\" As a period to his joke he again whinnied. \"Did you grab it? _V\u00f6gle von moglich t\u00e4glich._ Ah, no matter.\"\n\n\"I had nothing to do with Lori,\" Reinhart stated gravely, \"at least not in that way.\" Nevertheless he was grateful for the accusation. He might have resented another man's combining the disparate ideas of sensuality and Frau Bach and projecting them upon him; but he saw at this juncture that rather than the deed it is the nature of the doer that rules moral judgment. Schatzi, the good German, the gentile, the witness that martyrdom was not exclusively Jewish; was it not a glorious truth of humanity that one virtuous man reclaimed a multitude of sinners? Looking at Schatzi\u2014this twisted, blackened wire, never again to charge chandeliers, to make possible the splendors of filament or the shrewdness of connection; but _wire it still was; honor cannot be annihilated_ \u2014looking at him in homage, Reinhart said: \"Why were you sent to Auschwitz?\"\n\n\"Because I was a criminal,\" Schatzi said mercilessly. \"But now as concerning this present matter: who actually\u2014he switched to German\u2014\"Who is this female lieutenant? Is she really going to marry this Schild? And, if so, when? Pardon my unusual curiosity, but the man owes me a considerable amount of money\u2014enough, let us say, to give me an interest in any major activity of his. I suspect he's a slippery customer. You know these Italians.\"\n\nHow unfeeling of Reinhart to have stimulated these unpleasant memories! With an agitation painful to see, Schatzi babbled on in rapid and incomprehensible German, blinking, panting, wiping his nose.\n\n\"My friend,\" said Reinhart, placing his big hand on Schatzi's shoulder cap, encountering nothing there but bunched teddy-bear plush, withdrawing it lest the weight fell the poor ill person, \"my friend, I did not mean to disturb you. I just want to say: is it not tragic that in our time it came to pass that a man had to be a criminal to remain decent?\"\n\n\"No, please, I'm not\u2014\"\n\n\"No,\" said Reinhart, \"I won't say anything more about it, I promise.\" He sat down on the log he had earlier used as head-stop. \"Here, have a cigarette with me.\" He took one himself and pressed the remainder of the pack upon Schatzi, who, still upset, struck it away. \"Go on, you can keep them, I mean it,\" Reinhart said and with sweet exasperation looked to Very for support, and saw her ambling drunkenly up the trail off the beach.\n\nSchatzi marked her too and in a kind of fear choked: \"She leaves!\"\n\n\"Yes,\" Reinhart answered dully. \"And I don't think she knows the way back.\" Aware of his responsibility, he nevertheless took his own good time in mounting a pursuit, so that when at last he arose she had for some moments been out of vision, beyond a bristling turn of bush. And then the essential sadness struck him like an instant fever: a woman abandoned, unloved, stumbling off alone. In this matter he could be of some use, all the more because of the late harm done his vanity: for once put aside your goddamned self!\n\nPelting round the bush, squashing pine cones, whipped by green streamers, he spied her moving particularly, whoopsily up a bank of firs ten yards from the path where stood motionless a substantial animal showing the outline of a wolf, as well as its immediate difference: a tiny twig between its monster jaws. Seeing Reinhart, the dog spat out and as soon recaptured the twig, danced, made as if at him then away, and suddenly losing guts and idea, dropped the stick and with lifted leg discharged a high stream of urine against a sapling.\n\n\"Come back, Veronica,\" Reinhart called. \"The dog is harmless.\"\n\nBut first to respond was the animal. Kicking back a spray of sand and leaves, it advanced on him sportively, threw great paws upon his blouse, and sought to lave his face with a tongue big as a towel.\n\n\"Down, boy!\" By a mistake of tactics he was drawn into the game of shove-return; the more forcefully he flung the heavy body back, the more joyfully did it thrust in again, with salivary grin and mock-ferocious tusks. From her place among the firs Very peeped through the hairy branches and screamed.\n\n\"Cut it out!\" Reinhart yelled. \"I told you he was harmless. Down, you fool! Get down, damn you. Oh, damn you. _Heraus_!\"\n\nBut nothing served till Schatzi, coming up silently behind, barked: _\"Pfui!\"_ Midway in its spring, the dog at once closed in upon itself like a jackknife and folded to the ground.\n\n\"What we call a German shepherd,\" said Reinhart, brushing himself clean. \"Does he belong to you?\" The dog looked from Schatzi to him in the quick, simple changes of canine emotion, from a loyal shame to a disloyal expectation, and slunk its great head forward in a neutral direction.\n\n\"Oh my goodness gracious!\" Schatzi said in exasperation. \"It follows me about\u2014but swiftly now before she returns back...\"\n\nIt appeared he had taken as an expression of fact Reinhart's wry remark that this Schild should marry Veronica; was concerned about the money Schild owed him: \"married men have spare marks\" were his words. \"But if they marry they must leave soon for the States, _ja_? Married couples are not permitted by the military laws to exist while giving service into the American Army\u2014do I make this clear?\"\n\nReinhart backed away a step: in the high emotion of his interest Schatzi had begun to spray a mist of spit. Curious fellow; but then if Schild, whom it seemed everybody had a case against, was in his debt, no wonder. Owing money to an alumnus of Auschwitz was a good deal rottener than any sexual transgression. He decided it was impeccable to detest Schild, and since that detestation had no intercourse with anti-Semitism, it was generative of power.\n\n\"Look,\" he said in a strong, new voice. \"I will get your money back. _I_ will. Just don't you worry. You shall have it.\" He seized the man's birdlike right hand and crushed it in pledge. Then seeing Schatzi's emotion rise rather than fall and not wishing the embarrassment of maudlin thank-you's, he slipped away to fetch Very, who while they talked had gingerly emerged from her green shelter and reached the trail.\n\n_\"Pfui teufel!\"_ said Schatzi, behind him, and the dog, who had presumably offered to be out of order, whined like the slow splitting of a board.\n\nPicking the briars from Very's uniform, brushing her with disinterested, whisked hands, he counseled her not to brood upon and surrender to misfortune; for his promise to retrieve Schatzi's money had been but a prefatory resolution to the main, to the one with which he assumed the obligation of her rescue.\n\n\"Don't tell me what to do,\" said Very, as a drunk does, _de haut en bas._ \"And watch your hands.\" Since he was at that moment in the region of her shoulder blades and had not gone lower, this could hardly be the complaint of modesty outraged.\n\nSo he laughed and smacked her full on the bum and repeated: \"Don't worry!\"\n\n\"You're vulgar,\" she said with dignity and marched on to the log, and sat, and coolly went a-fishing in her bag.\n\nReinhart threw up his hands in light despair, for the benefit of Schatzi, who was looking the very devil. The dog leaped to its feet; Schatzi cried: _\"Pfui!\"_ It subsided.\n\n\"Why does he keep making that hideous sound?\" asked Very, pinching her face into a tiny looking glass while her other hand screwed a scarlet bullet of lipstick from its golden shell.\n\n\"Madame,\" he answered, instantly restored, definitely bowing. \"This ahneemal is an undisciplined rahscal without a code of ethical manners. One feels that one must give apologies.\"\n\n\"I think,\" said Reinhart as he saw her face sour, \"that she wants to know why you say 'Fooey.' Is that the dog's name?\"\n\n\"Ah! Ohhohoho, _jetzt verstehe ich._ But no, this is how in German we speak to docks. What must you say? We say _Pfui!_ This means 'stop what you do!' 'dezist!' and so forth.\"\n\n\"It seems to work very well,\" said Reinhart. \"I've never seen a dog trained so well.\"\n\n\"Why not?\" Very said to her mirror in weary disgust. \"That's the way the people are here, except that instead of 'Fooey' it was Der Fooey. Heil, Der Fooey!\" She fascist-armed her lipstick.\n\n\"Knock it off, Veronica. You don't know what you're saying.\"\n\n\"Heil Reinhart,\" she cried, playing on him the sun's little spotlight off the mirror.\n\n\"Ignore her,\" he told Schatzi, only to see the man vastly amused and himself raise a flat palm and say: \"Heil Reinhart!\" and laugh with stained teeth.\n\n\"Excuse me,\" said Schatzi, repeating the salute but this time only mouthing the address. \"However, it is very funny to see Americans do this.\" He clapped himself upon the skull. \"Of course! I am forgetting! Soon I may find your relatives, dear boy.\"\n\nReinhart counterfeited an excitement he did not feel: \"You don't mean it.\"\n\n\"Most surely I do.\" Schatzi shrugged in his coat, cast an ominous glance upon the dog, and grimaced at the lapping margin of the water unclean with minor driftwood. \"I am on the tracks, it is as much as to say.\" He inspected Reinhart to see what he had aroused, put a finger in the aperture of his own ear, and said: \"Ah well, perhaps I am interfering with your afternoon.\"\n\n\"Not at all! Sit down on this log and tell me about it. What could be more important!\" Or more inconvenient? He sat down, himself, a weariness having caught him in the reins. He envisioned his kin as tattered, hungry, and cellar-dwelling, that much responsibility added to his present chores. Arrange an abortion for a Catholic, retrieve money from a Jew, accept as family a tribe of Germans, go and catch a falling star, get with child a mandrake root\u2014here at last were things to do, God wot. The dog, he noticed, was inching towards him on its belly, great gray lout of a thing, beseeching.\n\n\"Well,\" said Schatzi, continuing to stand. \"I have my look upon a certain family right there in Zehlendorf, who I know had had some great-uncle go to America many years ago.\"\n\n\"But is their name Reinhart?\"\n\n\"That is simply the whole point. No. But could not have your grandfather a sister?\u2014who would quite naturally change her name when she married a husband?\"\n\nIn relief, Reinhart said: \"Now you're joking.\" Behind Schatzi's back the dog had crept forward two feet; now it paused and slavered amiably. \"No doubt thousands of Germans have relatives in America, and none of them named Reinhart. But it's my fault. I never gave you my grandfather's first name and date of birth.\"\n\n_\"Also,\"_ Schatzi reacted. \"You had better do that, so that we can put to shame the false persons who will try to claim your blood.\" Jamming his fists deep into his pockets, he shuddered.\n\n\"Ill write it down,\" said Reinhart, \"\u2014are you cold in the sunlight?\"\n\n\"Mere _Angst,_ \" Schatzi smiled. \"Freedom is difficult to endure. But you must use my pen.\" He brought forth one of those American fountain pens that profess to last a lifetime\u2014Reinhart wondered if he had owned it in Auschwitz: \"Mr. Schatzi of Berlin, Germany, used this Superba Everlasting Masterwriter for three years in the living death of a concentration camp. Yet when he was liberated _it still wrote good as new_!\" He also produced a writing surface: a matchbook cover, also Yankee, on the outside a riot of yellow and red exhortation; within, a cooler plea terminating in a tiny coupon one could, if his name were no longer than Li Po's, mail in with ten cents for a sample of accessories to shaving.\n\nUnder the salutation provided by the advertiser\u2014 _Dear Allah Shavecream Folks: Yes, I want to take advantage of your generous offer. Please rush sample kit to:_ \u2014Reinhart had no alternative but to write:\n\n> Gottfr. Reinhart, b. Aug 14, '61.\n\n\"Thus!\" said Schatzi, reading the script close to his face. \"An old man.\"\n\n\"He died more than ten years ago.\"\n\n\"Therefore one can die in America just as anywhere else, _ja_? This is sometimes doubted in Europe, and then it is too suggested you stuff your dead as hunting trophies and mount them round the parlor, but I am sure this is peculiar to Kah-lee-for-nia, if there.\"\n\nQuick to catch his mood of levity, Reinhart jokingly commiserated: \"Too bad, I spoiled your coupon.\"\n\nSchatzi reclaimed his pen so quickly that Reinhart's fingers felt as if struck by the beak of a carnivorous bird. \"Whatever do you mean?\" he cried, reading the matchbook, then slyly cocked his head: \"This is a swindle, _ja_? The persons at this postal box will still keep your ten cents and send you nothing, _ja_? Hahaha...\"\n\nNo doubt it lay very deep, but Reinhart was never hindered by such a concern when it would be mean and ill-mannered to withhold a reply in kind. He made laughter, too, and as Schatzi's increased in volume his own increased in racket, sobbing for breath, and was joined in the second chorus by Very's golden instrument\u2014how healthy it was to hear her!\n\nScreaming with laughter as one does when he finds the joke is that there is none, Reinhart watched the dog worm in under the clamorous cover and, taking from the general amusement a fool's license, roll upon its spine and wave great ludicrous paws.\n\nExpecting Schatzi to begin laughing all over again, he saw him instead hide the matchbook in a fastness of his coat and, stooping, attempt to take the dog unawares with a hook of fingers to its upside-down head. To scratch was apparently his intent; but the dog held to the first appearance, rolled upright, and fell back snarling into the fence of Reinhart's legs.\n\n\"I offer to this thing love,\" Schatzi said, \"and receive back only ill humor. _Was kann man tun?_ Worse than a human woman.\" He stared at Very. \"I must be about to my business, now. What did you say Madame's name is?\"\n\nSince in her present state Very would no doubt take unkindly an oral answer; since her head was at the moment turned away, Reinhart picked up a stick and scratched VERONICA LEARY in the sand, feeling somehow, against his better judgment, as if he were selling her to a white-slaver; and in atonement to Schatzi\u2014for how wickedly misguided was a heart which was queasy towards _him_ \u2014offered another five hundred marks to finance his assignment.\n\n\"You must not at all times be so ready with your purse,\"\n\nSchatzi adjured. \"What have I done for you as yet? Besides, do you know, there could be only a single payment if I give you satisfaction. Namely, that when you return back to the Oo Ess Ah you find my kinsfolk there.\"\n\n\" _You_ have American relatives?\" Reinhart wished instantly he had not sounded incredulous.\n\n\"That is my only trouble. But I should accept some, nonetheless, and they could be gangsters or anything, I would not care.\"\n\nLittle, wistful man, he shook goodbye with Reinhart and then with Very\u2014yes, she too put out her hand\u2014and ambled up the trail. With no more clowning, with frequent backward faces of reproach, the dog followed.\n\n# _CHAPTER 17_\n\nIT WAS A FALLACY to confuse animals with human beings. Schatzi thought for a moment, as if he were counting seconds on his fingers, then gave the dog another taste of the stick. He had a technique of whipping refined to maximum sting, minimum bruise. The dog was his property and to disable it were no sense. Simultaneously with the blow, he said soberly and with no great volume:\n\n\"Guard the boat.\"\n\nThe beast cried out, as was to be expected, and, as Schatzi knew, far in excess of what the pain would require, for it was not without a limited intelligence. Its chief want was constancy: a singular defect in one of a breed noted for just that virtue. But then, in justice to the dog, it had come into his possession no longer than a fortnight ago, and there was reason to believe its primary loyalty was still fastened to the former owner, a fellow countryman who had reciprocated to the extent that he recoiled from an offer of two packs of cigarettes but sold him for three.\n\nHe, Schatzi, had fed it well, had made obvious a capacity for return affection relative to what the beast showed him, had shown tolerance to the first few miscarriages of the dog's assignments, was now nearing the margin of estrangement. When the disobedience could be interpreted as willful, he understood, even approved: the finest organisms are those with a recalcitrant substance which when tamed by its master does not dissolve but compounds with his own. Thus Russian cavalrymen he had seen who were one with their horses, not so much riders as centaurs. This in fact was the timeless sense of the ancient myth. But he had begun to think otherwise of the dog, to see in it a fundamental baseness which said not: \"I refuse to guard your cursed boat until you associate it with my being.\" But rather, \"What boat? I chase hares and sport in the sand, and you beat me for your pleasure.\"\n\nThe latter it was saying now, with the voice of its large craven eyes, its great back hunched against the forward seat; and with the repugnant knowledge that it had duped him, that he had given it what it most wished and so confirmed its appraisal of him, he in fury threw the stick far across the water. The dog went over the side and into the surf with a sudden displacement of weight that put the deck awash, and before Schatzi could work with the bilge can, was back and rearing its wet snout to the gunwale, the dripping stick between its jaws. Yes, it was not without intelligence, he admitted, reluctantly amused, but see what it could make of this! He spun the flywheel and the motor caught, and gunned away full throttle. Looking behind, he saw the dog strike out valiantly in pursuit, in a violent battle with the water, which as the distance grew between them it slowly lost but would not admit. Halfway across the Havel and far enough, he assumed, to make his point\u2014he could no longer see its commotion\u2014Schatzi bent back towards the Tiefehorn, the apex of the Wannsee peninsula.\n\nTwo hundred yards from shore he stilled the motor and rowed in, an eye on the woods. Which after a time satisfied him that they were deserted, and he beached and concealed the craft. Ten minutes' walk through the forest brought him to a compound of four brick buildings around a garden: a peacetime tuberculosis hospital and during the war a school for air-raid wardens. In the garden were half a dozen Russian graves with their red stars of wood.\n\nHis goal was the great radio-transmitter building a hundred meters beyond the compound proper. Built of pallid concrete diseased with green-and-black camouflage splashes, the structure bulked four floors tall, was as long as high, as deep as long, and had no windows. To a median groove in its rear face rose a pile of rusting Wehrmacht helmets, taken off German heads by Russian hands. Also in the environs were: cartridges and shells, both unexploded and used, all calibers; hand grenades, both as loose eggs and, with the wooden handles attached, potato mashers; bayonets; _Panzerfaust_ bazookas; elements of the imitation Luger called P-38; corrugated gas-mask canisters; gray-green tunics with a thread of red decoration through a middle buttonhole\u2014the rusted and patinaed and mildewed and rotten, already forgotten, material particularities of an obliterated army. Which meant nothing to Schatzi\u2014he remained.\n\nAs he entered, Schatzi took a noseful of the unique odor of the interior, a blend of urine, feces, damp, fire, and electrical effluvium from a transmitter that through it all\u2014last stand, Russian plunderers, American snoopers\u2014retained a deep, visceral stream of life. Its inexorable hum, issuing from the second floor but audible throughout, with the odor and, where the bulbs remained, the dim lights still burning in the halls from which humanity had fled and yet remained in the characteristic carpet of litter and excrement, had spent its force on Schatzi. Once inside he passed into a calm, and picking his way down a concrete stairway clogged with junk, which two steps before its bottom connection surrendered and itself melted into waste, he descended to the basement. Where, since he had earlier extinguished the ceiling lamps and smashed their sockets, he worked his passage with the hand torch that had once disturbed Schild.\n\nAlready he thought of Schild in the past tense, no feat for him who had but lately served in that enclosure where the present was so difficult to establish and all Jews looked alike. There had in fact been a uniform diminution of indentities as one went down through the categories of prisoners, from the green breast-triangle of the professional criminals to the yellow and black superimposed to form the Star of David. On his morning work gang the faces were the same for three years, yet a good five thousand individuals, by the record, had come into that lineup and shortly gone, without distinguishing themselves in transit. ... It was most unlikely that Schild was a fairy; to be a Jew was enough, and a Communist to boot. (In Auschwitz his breast patch would have been superimposed yellow and red, to show his double affiliation.) Therefore all the less was his keeping the Russian understandable. After Schild and the preposterous Sankt George had carried the Russian, all bloody, upstairs, a piece of chocolate oiled the woman's tongue: the fellow was a deserter, of course.\n\nThe Red Army had dismantled the dynamos and shipped them off to the Soviet Union\u2014this made all the more mysterious the live transmitter on the floor above; its power supply was gone\u2014but in their great haste to complete the job before the Americans took over the sector, they built huge crates that couldn't clear the basement door, then had to take them down and start again. The detritus from their work\u2014boards, tarpaper, peels of metal housing\u2014Schatzi had assembled wantonly to fill a shallow corridor off the end of the main cellar-hall. Which barred the nosey without teasing them to burrow through it, and also caused Schatzi himself some trouble in his arrivals and departures. For this reason he placed a rigid limitation on the latter, and returned now, breaking his own rules, only to fetch some Meissen china for the last deal with Lieutenant Lovett. That worthy, who had been as the Americans said \"framed,\" was flying out of Tempelhof tomorrow in the direction of the U.S.A. and wanted a souvenir for his mother.\n\nIt took both time and care to pass through the barrier, since one had to close in the tunnel behind, and before he could draw after him the last length of coat tail, above the noise of his entrance he heard quick paw-sounds on the stairway and in the hall and, unseen but heard, his dog announced itself without the blind. Nothing to do but grasp rearward to its collar and pull it in, and forestall in oneself the impulse towards congratulation, which was what, with idiot tongue and rolling eyes, it sought and getting would store up as merit against future failures.\n\nBack of the debris, a door unlocked directly into Schatzi's quarters. For earlier tenants it had served as storeroom; its walls were continuous metal shelves from floor to ceiling. They now were heavy with stores of another nature, the materials of Schatzi's major trade: cigarettes, confections, cosmetics, and the mechanical instruments of utility-pleasure: fountain pens, watches, lighters. And also: china from Meissen; Black Forest cuckoo clocks; unique beer steins, hand-crafted and -colored, each with a history; Hitleriana: signatures of the Man, photos of same with notable associates, counterfeit currycomb scrapings of his dog Blondi's coat, and two books from his personal library: a sob-sister romance by Hedwig Courts-Mahler and one volume of Ranke's _History of the Popes,_ the latter with marginal annotations in the F\u00fchrer's hand, often simply _Scheisse_!: souvenirs from a broader range of Nazidom; and finally a sheaf of small paintings on cardboard by an old man who lived under a heap of rubble in the Soviet Sector, whose wife had been killed in the bombings, whose daughter was raped and VDed by the Russians, and whose pictures\u2014calendar landscapes painted in saccharine and molasses\u2014were moving slowly even with Ami soldiers.\n\nSchatzi took some teacups and saucers and wrapped them in pages of the Red Army paper _T\u00e4gliche Rundschau._ Although his collection held twenty-four, he had chosen only five sets and, moreover, had in a sharp, glancing blow against the edge of a shelf chipped the rim of one saucer. It was just those persons who claimed sophistication in _objets d'art,_ like Lovett, who were the easiest marks, who could be relied on to call it \"Dresden\" and be suspicious only of the price.\n\nWarehouse, yes, but the place was also home; there was a cot for Schatzi and a length of chain for the dog (where, having enough foolishness for the day, he now secured it), and a little iron stove whose pipe issued through a chink in the wall, emerging outside beneath the cairn of helmets. Even so, he did not dare to keep a fire in the daytime\u2014giving precedence of mind over body, for he was always cold notwithstanding summer. Cold always, a feeling to which one never adjusts, the history of which is the history of the person and in his case almost a history of the times.\n\nThe first, the only, personal comment the man had ever made him, and he could not recall it without its full complement: 1919, seven idealists in the private room of a cheap caf\u00e9 in M\u00fcnchen. The sour and insidious stench of beer, not only in the air, one's own mouth, and the breath of the others, but in clothing dropped on the foot of the flophouse bed and donned again the next day (one could not go naked while they were washed), and doubtless also in Harrer's briefcase, which along with the cigar box for funds made up the Party office.\n\nHarrer was president; had he remained so, it must be admitted that events would have been less interesting. For one thing, Schatzi's clothes would likely have stunk of beer to this very day! He enjoyed such reflections, trifling with times past and irrefrangible; they were the only feasible control\u2014which surely even that other early member, he who drank no beer, would admit now, granting for the moment that he could be assembled from the ashpit in the _Kanzleigarten._\n\nBut the cold. ... This fellow, about thirty, voice roughened by poison gas in the war, clothes neat but knot of tie off center, capable of incredible fury in abstract argument, but when the _Ober_ splashed beer (which he would not suffer in his mouth) on the green fedora upon light raincoat on the adjacent chair, unruffled and gracious. In the discussion he moved that invitations to meetings be printed on the gelatine-duplicating machine and, further, that the cigar box be opened to buy three rubber stamps. On this matter Schatzi cautiously stood with the majority, thinking it over till next week; he had understood the group's aim was to put a little money in his pocket, rather than take it out\u2014the latter, however, being only academic at the time, for he was a month in arrears in dues and on the point of ejection from the \"Home for Men\" for nonpayment of rent. No, it wasn't true that he had joined to make his fortune in the narrow sense\u2014unless one is a German bourgeois or an American of any class, money is an obsession only when one is poor\u2014but a country is putrid and needs airing when it gives no justice to him who still carries enemy shrapnel in the meat of his thigh.\n\nHe, Schatzi, seldom spoke at meetings. He was never strong on ideology, and for that reason his associates treated him with a certain condescension. But earlier in the year when the premier of Bavaria, the Red Jew Eisner, was shot in the street and the ensuing proletarian revolution caught Schatzi's friends of the Thule Society momentarily planless, he had got a chance to show his talent. The Reds hung a picture of Eisner on the wall against which he fell and mounted a guard nearby to force passersby to salute. Schatzi bought a sack of flour, soaked it in the urine of a bitch in rut, accidentally dropped it while passing the portrait. The bag burst, and the stuff powdered the base of the wall: soon all the male dogs in Munich were congregated there to whine and sprinkle their eulogy.\n\nIn crushing the revolution there was some loss of blood. The Thule Society\u2014a different order from that of the seven caf\u00e9-gatherers, though with like sympathies; the fellow of the rubber stamps, for one, had been elsewhere\u2014fought as underground shock troops within the city, while the Whites besieged it from without, and in a matter of weeks the revolutionary forces had dwindled to a rabble of left-wing soldiers in the 19th Infantry barracks, of whom the Whites executed every tenth man, and a swinish lot of prisoners in the courtyard of the Munich slaughterhouse, some hundreds of whom were formally shot and the rest battered, pierced, crushed, mutilated, and otherwise coaxed to enter the land of the shadows. Having won some merit in this action, Schatzi came to the attention of Captain Ernst R\u00f6hm, who was always on the lookout for young men\u2014for more purposes than one, as it turned out\u2014and was recruited for one after another of R\u00f6hm's private armies, some disbanded the day they were formed, by the Jew-Socialist traitors in the government, the same that had betrayed Germany in the last days of the war. Schatzi had been very young in that time, twenty-two and three, and a prisoner of the feverish passions of the callow: for example, when he thought of the government he saw a single face, wan, spectacled, hooknosed, showing sly sanctimony that broke quickly into womanish fear as a good fist smashed into it.\n\nBut the cold. ... After this meeting, when the cigar box had been replaced in the briefcase and the briefcase snapped and the reckoning paid, R\u00f6hm paying Schatzi's, the seven rose to leave. Even as he buttoned his poor outer clothing\u2014he had no overcoat\u2014Schatzi trembled before the thought of the late-fall wind in the street; and the man beside him, getting into the raincoat, stared from the deepset eyes since famous.\n\n\"I don't know why, I am always cold,\" Schatzi apologized, and didn't know why he did that, either, for it was honest enough, but the man without opening his mouth seemed to demand it.\n\nIn answer Adolf Hitler said: \"You no doubt eat meat, which oxidizes too fast in the stomach and the warmth is dissipated. The German nation as a whole consumes meat in the manner of a pack of hogs at their swill, and can never be strong until all that is at an end\u2014not to mention that, as Schopenhauer observed, it smokes instead of thinks. I oppose all that.\" He pulled the hat very low over his brow and left the caf\u00e9 with quick steps as R\u00f6hm, smiling with his mutilated nose, took Schatzi's arm.\n\nIt was a tablecloth of many colors, handwoven, fringed, and according to Lovett, who folded it briskly and placed it in the wooden crate, an article of Holland. Although the old lady of the house swore that she had bought it once on holiday in that country, he had pronounced it contraband of war, first for her and now for himself. And now, in the incredibly solipsist way that only Americans can do well, he related the details to Schatzi, as if expecting congratulations.\n\nAfter a cursory inspection of the china, he drew five hundred Occupation marks from a fat wallet and thrust them at Schatzi, for all the world as if he, Lovett, had got the better. For who feels he has got it, has it\u2014added to which, by the look of the billfold, if when Lovett arrived in the States his purchase was exposed, the expenditure had been small and the swindle might even give him an aura of adventure. Standing there before him, Schatzi could conjure up a little narrative two months hence in which his own image would appear as a quaint, Old World rascal. And its force was sufficient to alter, for a moment, the long, straight direction of his life.\n\n\"No, no,\" he said, returning to Lovett half the sheaf of bills. \"The price that we agreed upon was two hundred and fifty marks. You are so careless of your money!\"\n\nIn Lovett delight and dismay contested, with the latter ultimately victorious. For, while he took the money, he now for the first time studied Schatzi and then applied the same inspection to the china.\n\n\"This chip,\" he said. \"Oh! It isn't old at all\u2014\"\n\n\"But,\" Schatzi broke in happily, \"it is not the age of the chip that must trouble one, but instead the age of the china. As it does happen, I know the late history of these pieces. They were on the estate of the Graf von Halsbach zu Willmark in East Prussia for decades of many years. Unfortunately for him, the count remained until the last hour in the face of the Russian advance, and is it necessary to relate further of his outcome? His daughter alone escaped, with means of certain compromises\u2014\" He slowed down, watching Lovett's doubt metamorphose into a sexless, vicarious lust\u2014whether fastened to the count, the Russians, the daughter, the china, or the unspecified violence, he could not say\u2014and continued: \"But one can never be for long uncertain in these cases.\" He turned over the saucer in question, and pointed to the moldmark of a factory in southeast Berlin: \"You see, unmistakable. Every piece of genuine Dresden ware carries that age-old stamp.\"\n\n\"Yes,\" said Lovett, \"unmistakable. I hope I didn't offend you, but the price, well frankly it's so modest. You see\u2014\" laughing girlishly\u2014\"I'm not one to usually complain about something costing too little, but some things you just _know_ you have to put out a good price for, or they're no good and there's no use in puttin' out your money...\"\n\nLooking through the living-room curtains, Schatzi saw Nader, whom he feared, on the other side of the street and about to cross.\n\n\"The price,\" he said hurriedly, \"is set at that level because I cannot in all decency take a commission on this selling. The count's daughter is in a bad state of life, ill and needs the money for drugs and food.\"\n\n\"You must take the rest of this.\" Lovett, who had also seen his roommate, thrust the bills into his hand, and they had disappeared into Schatzi's shaggy pocket before Nader entered.\n\nNader scraped about dispiritedly in the hall for some moments\u2014time enough for Lovett to conceal his purchase beneath the tablecloth in the box\u2014before he came into the room with a sad look for his friend and a hard one for Schatzi, who prepared to leave.\n\n\"It's no dice, Dewey. The Old Man's had it in for you for a long time. He told me frankly that he has been looking for an excuse to ride you out ever since you joined the outfit. And he also said he always thought\u2014he said\u2014I don't want to hurt your feelings, Dewey, but he said, 'Nader, I soldiered with you for ten years. I'd hate to think you turned queer when you got your commission.' \"\n\n\"Well, Wally, you tried, and I am very grateful,\" said Lovett. He flipped away like a doffed glove and began to stuff the crate with his enormous stock of extra underwear. \"I told you before that I'm quite actually happy to be going. By the time I get back to the States I'll be up for discharge, and I'm so anxious to get out of this horrid uniform and back to the shop. Mother's been going it alone there for three years and just hasn't been able to cope. The _qui vive_ is what one must always be on in antiques.\"\n\n\"That prick!\" said Nader. \"His trouble is he just hates culture. You know his idea of fun? Throwing down glass after glass of booze and telling stories about toilets. Hour after hour. I used to have to listen to all that trash without opening my mouth in the old days when I was top for the station-hospital company at Bliss. One time I signed up for a correspondence course on how to improve my English. When he saw it he said: 'Now, Nader, you can't make a silk purse out of a piece of sowbelly.' How I used to ache to get that dirty muff diver in an alley and slam the poison outen him\u2014why can't a man improve himself?\"\n\nNader's body took on the temper of the grievance; the trapezius muscle at the base of his neck threatened to burst from the shirt.\n\nLovett fussed the rough top onto the crate. \"It's all right, Wally,\" he said. \"One can't right all the world's wrongs\u2014Ouch!\" He had got a splinter in his pinkie.\n\n\"The point is,\" answered Nader, taking over the job, nailing down the top with sixteen nails, precisely one hammerblow for each, and without a break in rhythm getting after a nose-itch with his left hand, \"the point is, a guy does all he can for a friend.\" He manhandled the great carton to his shoulder and fought it to the porch, being at the door the recipient of Schatzi's courtesy.\n\nRid of his burden, and having no gratitude, he blocked Schatzi's exit. \"What the hell are you doing here?\"\n\nSchatzi's arms flew up to guard his face.\n\n\"You know why I don't like you?\" Nader continued, glaring. \"You always look like you want somebody to kick your ass.\"\n\n\"But I have authority to come now to this house,\" said Schatzi \"The Lieutenant\u2014\"\n\n\"Well, if you're finished, screw.\"\n\nSchatzi glanced up and down the street and moved closer. \"My dear Captain\"\u2014his upgrading Nader had a gross purpose for what he assumed to be a gross person\u2014\"I have been told that the Lieutenant Lofatt's difficulty can be traced to a Russian officer, is it true, and perhaps if this Russian can be discovered, your friend will not suffer for it.\"\n\nNader was not attracted. \"I said blow.\" He offered Schatzi assistance in negotiating the stair.\n\nAt the same moment, however, and before Schatzi had begun to move\u2014looking at Nader with an odd smile that the lieutenant did not understand was admiration\u2014he saw the old housekeeper issue from the door and hail him with her fingers.\n\n\"The blond officer has stolen my tablecloth and I don't know what to do,\" she cried. \"You are their friend? Then you can help me.\"\n\nShe was breathless, fat, and wheezing, and what was left of her reason and passion obviously had its locus in the thick and tasteless furnishings of her home. It was precisely this kind of person that the movement in its early days had been pledged to get rid of; Hitler had instead purged R\u00f6hm and dispensed with the Strassers, and Goebbels, degenerate, maimed opportunist that he was, had submitted to the policies under which the bourgeoisie flourished.\n\n\"You are a widow, no?\" he asked, noticing that on her second sentence Nader went within.\n\n\"My husband was office manager of a fine company, small but fine. It was a direct hit. Afterwards we couldn't even find his body. Please, I have no one to help!\"\n\n\"My good lady\u2014\" he drew back as she clutched at his sleeve. Anybody with a brain in his head would have anticipated Lovett by offering to sell the cloth before he had ever touched it. \"\u2014the Americans are honest enough. There has no doubt been a misunderstanding. You must speak with him again.\"\n\nHe was already on the bottom step, but the woman followed him down and, unless he broke away immediately, would surely weep, and that he must be spared.\n\n\"With their parties they already have destroyed everything else,\" she wailed. \"Did you see, the living room is empty, and I have not yet been paid. The ceiling\u2014the ceiling was shot away for no reason.\"\n\nNow that it was called to his attention, he remembered a certain damage in the room\u2014yet still not enough for his tastes; they would have had to burn off the roof and knock out at least one wall for it to seem anything like a home to him.\n\n\"But of course, this is why the officer is being sent away. You do not go unrevenged.\"\n\nThe silly bitch listened to nothing he said. \"Please speak to him,\" she cried. \"I have no English.\"\n\n\"Ah, my good lady, neither have I, you see.\"\n\nHe had left his bicycle in a clump of bushes around the corner\u2014not from fear of theft, for in spite of all it was still an honest land, but out of caution; avoiding the neighbor Schild, who might have seen an unconcealed vehicle.\n\nOn the long trip to the Soviet Sector, Schatzi had to show a different combination of papers at each checkpoint, American, British, and Russian, and there was always the possibility that some illiterate of the last type might shoot him. Sergeyev would of course spring him from an arrest, but, in his own words, could \"grant no immunity from a bullet.\" However, he once again without incident went through the waste of Potsdamer Platz and the barricade on its east side, and although the crowds delayed him\u2014where did they all come from, and why?\u2014it was far better than to chance a remote and less-peopled entrance, with guards accordingly more primitive.\n\nFrom here on he found it more politic to walk much of the way, wheeling the bike beside him: it was unwisdom in this area to distinguish oneself on a wheeled possession. Even so, he was stopped once by a Russian private, not a guard but one of the many roaming at large, who would have confiscated the vehicle had not Schatzi thrust in his face the pass from Sergeyev that read: \"The bearer, L. K. Burmeister, German national, registration number 2XL-1897340-C, is on special business for the Army of the Union of Soviet Socialist Republics. \u2014V. Sobko, 10th Section, Hdqrs., Red Army.\" At which the fool saluted as smartly as a Russian could, and went on.\n\nArriving before a relatively sound four-story building off Alexander Plata, Schatzi parked his bike in the iron rack, and this time snapped a small lock around two spokes and an upright of the frame, and with still another pass issued him by Sergeyev, with data as false as the other, persuaded the soldier on door-duty to lower his machine pistol, and entered under the long red banner that said: \"The unity of all Antifascists is the guarantee for the construction of a democratic Germany.\"\n\n\"Yes, do that,\" Sergeyev was saying into the telephone as Schatzi entered the smallest room, at the farthest end, on the last floor. \"Show him no more mercy than you would a fly that had fallen into your soup, and at the same time deal with care, just as you would with the fly, for if you crushed him then and there you might have to empty the entire bowl.\"\n\nTo one who on first meeting Hitler had seen an eccentric small-town sissy, there had come in ensuing years a disbelief in anything but the unlikely. However, it must be faced that Sergeyev differed from the commissar of legend only in his wearing mufti rather than the high, tight collar that was wanted to set off his bullet head. But then, even the civilian clothes were regulation for the type: the paradoxical jacket, both too tight and too baggy; dark-green tie and blue-striped shirt, both clean but looking dirty; and on a folding chair in the corner, a black felt hat with a little pond of light dust in its crown-dent and the brim lowered all the way round.\n\nReplacing the instrument in its cradle and without giving Schatzi any sign that he was received, Sergeyev revolved his squat weight, the ancient swivel chair croaking like a frog, to a low wooden shelf on the wall behind the desk, where, as usual, the articles from his pocket were scattered: crushed packet of Russian cigarettes, cheap brass lighter that took forever to catch, mean collection of zinc small money, and a nail file in a mock-leather case frayed here and there to its subcutaneous paper. On his own initiative\u2014for he wished to give Sergeyev no more than he already had, and the best strategy to that end was no unrequested sound\u2014Schatzi had assumed this foible owed to a stout man's difficulty in getting to his pockets while seated. As to the drawers of the desk: Sergeyev had no more considerable paunch than a keg, which is to say from shoulders to thighs he was one thick swell with no protuberance, but he sat so tight against the furniture before him as to join it to his person. Thus the shelf. Facing which he now stayed and, to it, said in German: \"This is not your day.\"\n\nThe diameter of his cropped skull widened as head gave way invisibly to neck, and both head and neck, and face, as he now made his return revolution, and the thick, hairless hands that grasped the cigarette and smashed at the lighter's wheel, were crimson as an angry baby. Sergeyev, however, was never angry. If anyone threatened to make him so, he had him destroyed. As, anyway, he had once told Schatzi, who was disinclined to demand proof.\n\n\"You have no answer to that? Ah well, in the east country you'll have years to talk all you wish to the ice and snow, unless a guard puts you out of your madman's misery with a rifle butt.\"\n\nSchatzi waited through the inevitable joke about Siberia, as one does for the amenities. \"Yes, quite true, it's not my regular time, but I have something you should want to know.\"\n\n\"And you've come here last of all, not having been able to sell it elsewhere. You think you can fool me, you piece of filth? You still do not believe that I can fling you down the stairs at my pleasure?\"\n\nSergeyev arose and pounded to the door, opened it and thrust half of himself into the hall. \"Yes, that stair down there, the one you came up,\" this part of his speech itself being flung in that direction and thus scarcely audible to Schatzi. Who nevertheless had heard the threat clearly enough in the past to know it was habitually delivered in a voice devoid of all emotion; but not that he believed it empty\u2014somehow one knew without evidence that Sergeyev was the type to say it a hundred times and do it on the hundred and first, or the thousand and twelfth, or not at all, for he had no rhythm and no limit and, indeed, beyond the pocket articles, no discernible self. These on the return trip he cleared from the shelf with one hand as sweep and one as scoop and buried in his clothing, as if in anachronistic worry that Schatzi might swipe them while his head was out of the room.\n\n_\"Also!\"_ He threw himself into the chair and grasped either end of the desk. \"Proceed!\"\n\n\"Lieutenant Schild\u2014\"\n\n_\"Are you insane?\"_ shrieked Sergeyev. He sprang up again, went again to the door, looked out, came back on a circuitous route of examination\u2014his office was small as a private washroom, with no window, the streaked beige walls marred by no ornament, and no furniture beyond the desk and the lone extra chair that Schatzi occupied\u2014and disengaged the telephone's handpiece from its berth.\n\n\"Do you think me so na\u00efve that you can inform to American Intelligence before my face?\" he screamed, still with no emotion, stamping out his cigarette in a little glass bowl evil with tobacco tar.\n\n\"Fritz, then,\" said Schatzi. \"A _Russian deserter is living at Fritz's billet._ \" He stirred in his chair, smiling ill in fright and pride. \"Also: Fritz is going to marry an American nurse.\"\n\nHe was met with an open mouth of short teeth, which appeared to be a smile. \"Tell me,\" said Sergeyev, his voice liquid with unction, \"confess to me\u2014you were Ernst R\u00f6hm's very favorite fairy-boy of all, _nicht wahr_? This is the sole reason why you were present on that famous 30 June 1933 when your lover and his faction were purged. We know all this already, so I can tell you it is useless to continue your mad resistance!\"\n\nA half year before, in the middle of January, when the Russian forces were rolling through Poland, the SS closed the Auschwitz camp and herded the dangerous prisoners\u2014mainly such Jews as were left, and politicals\u2014on a long death march to the enclosure at Mauthausen, in Germany. Schatzi was permitted to escape. Right off, he was almost murdered by Polish vigilantes who came in to fill the vacuum. For the uniform he had continued to wear as protection against just such a hazard, snipping the green triangle off the breast, was precisely what identified him to a Polish tradesman who had made deliveries to the camp and seen Schatzi in his privileged role as \"professional criminal\" leading work gangs to dig their own mass graves. This Pole, until two weeks before a collaborator, was now applying the same industry to preserve Number One in the new arrangements; in which he was as unsuccessful as a man can be: the guerillas shortly knocked _him_ off, but saved Schatzi for the oncoming Soviet authorities. Who in turn not long after arrival dispensed with the vigilantes themselves, so neatly that no trace of the bodies was found by three Swiss Red Cross delegates searching for eighteen days, _but saved Schatzi._\n\nIn neither case were the Russians wanton: the Polish guerillas, having shown enterprise once, would likely have proved troublesome in the Soviet occupation, and Schatzi, being officially an unperson and by personal history an advocate of no live cause, a friend of no man, totally dependent upon his captors and nicely shaped by years of captivity\u2014it would be almost indecent to get rid of a man who could be used, and for no payment, beyond not taking that which had no absolute value: his life.\n\nHe was taken to Berlin and assigned to Sergeyev, who notwithstanding the mufti, was apparently an officer in an Intelligence section of the Red Army: _apparently,_ because this was never mentioned, Sergeyev's office being this shabby, airless cube in a building tenanted otherwise by the German Communist Party.\n\n\"And we know, believe me, that you make daily reports to American Espionage,\" Sergeyev continued in his genial way. \"Must I remind you once more that you are no safer in that sector than this one? How much sufferance would the Americans show if we informed them of your past, _Misterrrr_ 'Burmeister,' _sirrr_.\" On the English words he did a humorous imitation of the American _r,_ which at the same time was very accurate.\n\n\"The only American agent I deal with is Fritz.\"\n\n\"Never mind about that!\" Nevertheless, Schatzi saw him write FRITZ on the back of a used envelope\u2014which he pulled from a wastebasket; there was nothing on the desktop but the glass bowl for ashes and a pile of paperclips artfully arranged to appear loose but really joined into a two-foot chain, as he had discovered on an earlier visit when Sergeyev suddenly hurled it at him\u2014FRITZ, he wrote it a second time and began to elaborate its lines with the pencil as he started again on R\u00f6hm.\n\nR\u00f6hm, R\u00f6hm! From Sergeyev's badgering at every visit, each time with a different angle of attack\u2014the last time, of all things, he had been accused of being a spy in R\u00f6hm's camp _for Hitler_ \u2014one could see that beneath the surface foolishness they knew everything already. And if they knew everything, they must surely know he had not been with R\u00f6hm's personal party on the terrible night of June 30, 1933, when Hitler and company burst into the Bavarian hideaway and carted them off to the slaughter. And, to boot, Sergeyev had once asked him for an account of the executions at the Lichterfelde Cadet School in Berlin. But surely they knew that if he had been with R\u00f6hm he would have been taken to Munich, if not killed on the spot, as some were, in the sanatorium at Wiessee.\n\nAs to his erotic associations with R\u00f6hm\u2014it was impossible to explain to anyone who had never known him the dynamism of the man, the virility which made denying him his pleasure almost shameful. Schatzi had not been given to the practice before he met him, and did not continue it extensively after the purge\u2014indeed, although he had tried most of them, he had yet to find a kind of sex that was not tedious.\n\nHis not having been with R\u00f6hm's party on that historical night was a piece of the strange kind of luck that blessed him his life long\u2014or plagued him, for with his leader's death perished a purity that he had found neither before nor since in the walks of men, a hard, clean, uncompromising resolution, honor, and bravery that the foul little Austrian upstart had betrayed to a moral leper like Goering, a weak-minded fanatic like Himmler, the antediluvian cowards of the Reichswehr, and the reactionaries of the Ruhr who had given niggardly money to the Party with the sole aim of getting more in return.\n\nWhat was there to tell? Schatzi stayed in Berlin at headquarters, keeping a finger on developments, while R\u00f6hm and the other SA leaders conferred in the Bavarian retreat. Aware that they were incessantly calumniated by the evil voices at Hitler's ear, sensing that they, the private army of the National Socialist revolution, the oldest fighters, the idealists, the conscience of the movement, had already been made superfluous in the general corruption, they were yet unprepared for the ferocity of their blood-brothers. R\u00f6hm was expecting a visit from Hitler on July 1, at which he intended to plead again to his old comrade-in-arms the case for the SA. He had a touching little gift for the F\u00fchrer, a handsome bookplate. He waited in trust, with no guards; he was after all the only man in the Party who called Hitler by his first name, not to mention that he had been a Nazi even before Adolf. But when Hitler arrived, it was with a band of thugs and in the dead of night.\n\nSimultaneous with the raid in Bavaria, Himmler and Goering took the headquarters in Berlin, capturing a hundred and fifty officers, whom they imprisoned in the Cadet School coal cellar in Lichterfelde and shot in quartets throughout the next twenty-four hours. The condemned men kept precise count of the executions; guessing whose turn came next was insurance against despair. They sang the song named for Horst Wessel. And, in innocent trust, heiled Hitler and went to their deaths faithful to his memory, for they supposed him also to have been a victim of the reactionary plot to crush the revolution.\n\nIn the twenty-seventh group-of-four Schatzi's name was called\u2014not, of course, \"Schatzi,\" but \"Ernst, Friedrich Paul, _Ober-sturmbannf\u00fchrer_ ,\" and even at that moment he thrilled to the crisp drumroll of his title: he had been a poor lance-corporal in the army for three years of the war, owing to the petty jealousy of a sergeant who consistently blocked his promotion. As he was marched with three others out into the mild morning and across the yard to the execution-wall, he saw some of the faces of his remaining comrades pressed against the cellar window, those old veterans of the Putsch, of a thousand caf\u00e9 and street fights, of the Freikorps, and, before that, spotted here and there in the army of the Western Front. They had been fighting somewhere for almost twenty years, against impossible odds, for much of it ill fed and ill clothed, and always betrayed. Not one had broken down in the cellar. That was pretty good for the \"pack of fairies\" that so revolted Goering.\n\nThe wall was a dripping stucco of human flesh; fired from six yards away, the bullets blew the heart through a man's back. An SS guard opened their clothing at the breast. Having difficulty with Schatzi's woolen undershirt, he parted it with the ceremonial dagger from his belt and, inadvertently nicking the skin, excused himself. On Schatzi's right hand was Appel, whom he had never liked. He caught his eye now as the guard went down the line drawing charcoal target-circles around their left nipples, and said softly, \"Ahoy!\" the old Freikorps greeting. Appel had been one of R\u00f6hm's especial favorites; he smiled now over the gravity of his girlish face.\n\n\"By order of the F\u00fchrer, FIRE!\" The four prisoners stiff-armed the salute to Hitler and cried his name so loud they did not hear the order, and their chests were blasted through their backs. Or rather, three died not hearing\u2014or if they did, were in a second beyond knowledge. Schatzi, falling with the others, heard, and knew that R\u00f6hm was dead, that Hitler had betrayed them, and that from then on he would give credence and fealty to no movement but that of his own pulse\u2014which he heard now in the wrist crumpled beneath his ear, for he was not dead, had not indeed been hit, but rather was pulled down by the unity with his fellows. Lying with slit-eye at the level of the concrete, he saw the approach of the sole of a boot, was turned over and tested by it. A pistol slug fractured the pavement near his nose, the sharp chips whipping his face, already bloody from the liquid of Appel's heart.\n\nSchatzi preferred later, with his last ration of sentimentality, to believe that the officer had missed on purpose\u2014it was said the executioners' squad had to be changed frequently because of nerve failure\u2014but he dared not see who it was. Shortly the disposal wagon, borrowed from a local butcher, returned from its last trip, and he along with the lifeless others was sacked into its tin bed. The rear doors were secured. The deliberate horse wheeled it creaking to the gate, which, opening, had its own sound. Fortunately, he had been thrown on top of the pile and was not crushed by the other bodies. Giving fate five minutes, as near as he could estimate, he tried the doors and found his hands too weak to manipulate the catch. Treading back, the wagon swaying, Appel and two more soft underfoot, he hurled himself forward. The wagon stopped\u2014he had been conscious of the awful silence only as something to flee, but of course his movement broke it for the guards up front. The blond face of a horrified SS private was a circle in the bursting doors. Gory and wild\u2014he had come so far since that he could smile at the remembered terror of that young calf\u2014Schatzi flung out, felling the boy. They were on a deserted side street near the Stich Canal. He knew the area well, and escaped unpursued.\n\n\"I can only repeat what I have told you before,\" he said to Sergeyev's smile, which was turning more grisly. \"My associations with the Nazis ceased on June 30, 1933, except that for the next twelve years I was their victim like so many others. What we in the early SA wanted was much the same as the Communists; we were even called 'brown on the outside, red on the inside.' \"\n\n\"Don't insult me with your filthy comparisons between the international workers' movement and a reactionary-mystico-homosexual cult,\" Sergeyev shouted. \"That was the only intelligent thing that Hitler ever did, to crush that foulness without mercy. What I want from you is the truth about those intervening years. In reality, you all the time were working with the Hitlerites as underground agent, _nicht wahr_? Or were you even that early taking American money?\"\n\nSchatzi patiently went through it all once more: after his escape he had lived for two years under a variety of aliases, outside both civil and Nazi law, until discovered by the Gestapo; after which he was kept in places of confinement for ten years.\n\n\"Excellent, excellent,\" said Sergeyev. \"Go right ahead with your resistance. But when you collapse into a quivering, boneless mound, remember it was your own doing....\" He put down the pencil and, with the difficulty Schatzi had foreseen, dug into his pocket and found the nail file, put it to work with minute attention on the fingers of the right hand. One by one; it seemed hours before he finished and started on the left. Finally, though, it was done, and he brushed off the fall of nail dust\u2014only to go into his breast pocket for a toothpick and clean around the little pegs which served him for teeth.\n\nSchatzi ever so slightly changed his position in the chair, which made a loud, splitting, flatulent sound. He was genuinely embarrassed. Sergeyev bit through the toothpick, chewed it up, in fact, and blanched.\n\n_\"Did you\"_ he said, for once not acting, and thus showing that everything heretofore had been dramatics, and in a voice so mad with anguish that it seemed afraid, and Sergeyev afraid was so fearsome that in another moment Schatzi might have flung himself from the window, had there been one, _\"did you have the audacity to fart?\"_\n\nPerhaps because this time he had really been moved, he accepted the explanation, took up his pencil again, and twitched it in dismissal.\n\n\"Report on your regular day.\"\n\n# _CHAPTER 18_\n\nNOW TO THE SAVING of Veronica. Of all women for fertilization of the egg, a nurse; of all for illicit impregnation, a Catholic. Finally, a professional worker in the branch of healing to which problems of love were fundamental, herself love's dupe.\n\nSince on her own terms her infraction was inexcusable, his job would not be simple. It was even possible she would resist being saved\u2014as she had refused that first, hysterical offer of marriage\u2014and absolutely certain she would not admit the mode he had determined on. To a Catholic the mere use of a Trojan, he understood, was the denial to a new soul of its right to incarnate, reach puberty, and disapprove of contraception. Either you suppressed lust at its first tingling or, embracing it, you were obliged to stay for the d\u00e9nouement. Abortion, of course, was downright murder.\n\nTherefore would his guile be summoned to sally forth from the imaginary fields where it had so often bested Machiavelli. And his ingenuity: not even in a military hospital with a hundred doctors could one hope easily to recruit an abortionist, another nurse was unthinkable, and although he could name as many unscrupulous enlisted technicians as there were wards he had little faith in their Army-learned craft.\n\nIn the last he was terribly confirmed by a story of Marsala's. Roy Savery, an enlisted assistant in the operating room, had just yesterday performed an abortion on his German girl friend and she bled to death.\n\n\"His trouble was,\" said Marsala in high disgust, \"oh my aching back!, he loved her. If he didn't of, she would still be alive and he wouldn't be up for court-martial. Any girls I knock up I do them a favor and don't see them again.\" He chuckled and screwed his gangster's face around the stump of cigar on which his large incisors were clamped, smoke and speech intermingling from the side alleys of his mouth: \"Why not? I never raped anybody long as I lived. Am I right?\"\n\nThey sat _en famille_ in their living room, at the round table beneath a chandelier of five dead bulbs and one live. Marsala took off his undershirt, revealing a natural vest of hair, from deep in the tangles of which glinted a silver religious medal as might a fragment of broken airplane within the jungles of the Mato Grosso. He had too extravagantly stoked the corner stove an hour before, and the air was at that temperature in which the skin weeps and philosophy proliferates.\n\n\"Poor girl.\" Reinhart groaned, in part because he was miserable with perspiration.\n\n\"Yeah, he held a gun on her to make her haul his ashes,\" Marsala growled disingenuously, suspending from his index fingers the dancing, ghostlike undershirt, which he inspected for cleanliness and finding insufficient balled and cast under the sideboard. \"He should of done it, see? Then he woona owed her nothing at all, if you get my meaning.\"\n\n\"It's burning in here. I'd better open the window.\"\n\n\"No, whadduh yuh crazy? I take pneumonia with no shirt on, you dumb dong.... So get yaself a American girl like you got, huh you big dummy Carlo? Knows how to take care of her humping self, huh? Now don't tell your old buddy you don\u2014\"\n\n\"You got any extra money, buddy?\"\n\nMarsala snatched from a back pocket and propelled across the table his old brown billfold fat as a squab and said, while drawing on the stogie: \"Take whadduhyuh want.\"\n\nReinhart chose a sheaf of one thousand marks from a store of twice that much. \"Can I have this?\" He fanned the bills so Marsala could count them.\n\n\" _What_ is this?\" his friend answered, outraged, and bending over, seized the wallet's remaining notes and threw them in his face. \" _What_ is this, fuck-your-buddy week? _La putana Maria!_ You won't take my money, I give you a shot inna head.\"\n\n\"I can probably pay you back next month\u2014\"\n\n\"Okay, say one more word and I go rub shit in your sack,\" shouted Marsala, dilating his hirsute nostrils. \"Don't hump me with them college insults, Carlo Kraut. My cash's not good enough for you, okay, okay, OKAY!\" He paced furiously around the room, having his great noisy pleasure.\n\nIn a moment he marched into the hall, flung wide the outer portal, and bellowed Riley's old call up the stairwell, hearing which Jack Eberhard came out upon the top-floor landing and cried in riposte: \"You like cake? Take this, it's raisin.\" Then more doors opened and some of the other good old boys popped out shouting all the grand old irreverences on the genito-urinary tract, the oestrous cycle, the gastro-intestinal system, and their heresies, and when someone mock-flatulated with a hand in bare armpit, someone else whooped: \"Kiss me again, sweet lips!\"\n\n\"Where's Reinhart?\" called the guys from the third floor, and the cry was taken up by throats on all levels to the roof: \"Rein-hart! Reinhart!\" Inside, Reinhart listened, a kind of warm cramp in his stomach, and then rose, went into the hall, and looked up through the spiral of shining comrade-faces whom one day it would be a death to leave.\n\n\"Short arm!\" he shouted. \"Marsala, get the flashlight!\" And everywhere sounded the cheers and catcalls and boisterous generosity, and the third-floor guys fetched a pitcher of water and poured it down in a great quivering sheet, really funny because they really aimed to hit them and only narrowly failed. Marsala got angry and had to be held back by real force from climbing up and kicking the bowels out of the whole bunch. To soothe him Reinhart recited everybody's favorite poem:\n\n> When the nights are hot and sultry\n> \n> Is no time to commit adultery.\n> \n> But when the frost is on the punkin,\n> \n> That's the time for Peter Duncan.\n\nA society grounded on common inconvenience, where friendship was innocent of opportunism and tolerance flourished without manifesto: no crime could outlaw you from this company; no merit beyond the grossest went recognized; where sensitivity was soon reduced to coarseness and ambition stifled; where lethargy was rewarded and disenchantment celebrated; this cul-de-sac off the superhighway to the glorious Houyhnhnm of the future where a chicken would stew in every pot and each man be his own poet, unarmed, owing allegiance to one world\u2014this splendid, dear, degrading society, here as nowhere else Reinhart felt at home and loved.\n\nAn invitation from an _ad hoc_ party headed for the noncoms' club, there to swallow strong German beer, cuss, spit, smoke, and perhaps, about closing time, to plunge into a sharp dispute on a subject of no permanent importance (such as Marrying a Virgin) and nearly come to fracas, poignantly tempted him\u2014as in college when a gang formed in the recreation room, he had never been. At the moment, having a role, he saw his mission to save Veronica as only arbitrary, but manhood's job could be defined as that which replaced the known and comfortable with the difficult and unpleasant.\n\nBeing a man, he went inside to the bathroom and spread his available money in series along the washstand lip, which being European did not seek to stint on marble and extended flat and wide for ten inches on either side of the basin. Last week, unsuspecting next week's extremity, he had mailed home a money order for his maximum allowance, corporal's pay plus ten per cent, roughly eighty bucks. Remaining were three thousand marks, to which were now added Marsala's twenty-two hundred, totaling the equivalent of five hundred twenty American dollars. Vis-\u00e0-vis such a sum, a German physician of the present day could ill afford to stand upon his ethics. How Reinhart would lure Veronica to a foreign operating table, unsuspecting, he had not as yet studied. But the means by which the doctor would be gained were as close as belowstairs in Very's very billet.\n\nLori Bach\u2014Lori and Bach, who in their combination, in their-cellar, in his conscience, localized a grief which, unable to admit, he for a month had pretended was not there. Also manly was his resolve to go, on the strength of a concrete purpose, and look it in the eye.\n\n\"Ah!\" said Bach from the sofa. \"So kind of you to bring a friend, Mr. Corporal Reinhart, and if I am not deceived by my failing vision\u2014although the cooked carrots brought almost nightly by my good wife from the American mess, if indeed stewing does not destroy their sight-giving properties, are restoring it\u2014he is an officer; and where but among you excellent Yanks could be possible such a friendship: corporal and lieutenant, splendid, splendid.\"\n\nFalling from Reinhart's hand, his cow-teat fingers in a feat of levitation floated to the lieutenant. \"Bach. So good to know you.\"\n\n\"Schild,\" answered the officer who bore that name. _\"Es freut mich.\"_ And then his eyes, pained, confused, bugged at Reinhart and seemed to ask approval for himself.\n\nInstead, Reinhart recommended Bach. \"He is a good man, Lieutenant, he is a better man than I can say. I am very proud to know him.\"\n\nSchild stared dully, said plaintively: \"Yes.\" Without waiting for the invitation Bach already prepared upon soundless, moving lips, he fell into the nearest chair and put a grim surveillance on his own feet.\n\n\"My wife,\" said Bach, beaming on Schild but speaking to Reinhart, \"has not yet returned with him you require. Let us then, over three of the cigarettes you so kindly sent along to me, commingle our thoughts. The packet is just there upon the table. Please serve Mr. Lieutenant Schild\u2014which of course means 'shield'; and one is happy to see, ha!, that he has come _with_ rather than _upon_ it; every American, how singular!, seems to be of German descent\u2014and yourself, and then I shall be so bold as to ask one.\"\n\nAnd there was all of it again, like the landscape of a recurrent nightmare: the concrete tomb, the sweet smell of garbage, the white monster; all awful and yet familiar, like Xmas with the relatives, or for that matter, life, simply life in general, from whose calculated ills we do not fly to seek others known not of but surely worse, because unchosen.\n\nChoice: make this one and you must also make the next, and once begun you have the habit. A mere hour ago he had sought out Lori as she came from the mess tent, given her, right there on the plain thronged with her colleagues making for the trucks, his problem, bald and coarse; and so forthright was his temper that he left uncorrected the implication Veronica was his mistress.\n\nCertainly she knew a German doctor, and her wise-weary eyes took no stock of him at all, seeing him as end, not means, yet were fond in recognition, attended on him specially and without demand. Yes, that very evening, if he liked, she could bring the physician to her cellar for a meeting, Bach's and her cellar on N\u00fcrnberger Strasse, which since he could never have found it again she placed in relation to the Kurf\u00fcrstendamm and the ruin of the Kaiser Wilhelm Church. She would see him there at eight o'clock, as simple as that. He shook her fine-boned hand, her small, dynamic hand, and saw her hair again could use a soaping, that her beret was frayed, that her stockings of rough brown cotton sagged at the ankle and the gray coat wanted its central button; and each deficiency was another focus for his sudden love.\n\nHaving chosen action, then, having chosen love, clothed in the warmth of his volitions he had wandered through the slowly chilling late-afternoon light, in the time of day for gentle melancholy, the hour when perhaps even devil and saint are briefly, postprandially imperfect; when colors, which had been subdued by its noontime flare, spring defiantly at the sun in its decline, radiantly false as Kodachromes; when Reinhart in his earlier self had been wont to dream of being ruthless Tamerlaine, or Don Giovanni severing a maidenhead, or a poet with flashing eyes and floating hair.\n\nNow, however, in the realization that he had, in the only sense harsh actuality permits, done these, been these, or didn't wish to be, he forgave himself and plunged into the palpable present. Schild. He would go upon this moment to Lieutenant Schild and squeeze from him Schatzi's money. Moreover, since morally speaking it was beside the point and tactically an obstacle, he dispensed with the identification of Schild as Jew, thin, dark, sharp, arrogant, and deceitful as the man incontestably was. To dispense with it he had first to make it; and then must congratulate himself on its not making a difference, and then say a thanks to fate for at last coughing up a Jew who had trespassed against gentiles.\n\nBut the first-lieutenant's bar was quite another thing. To beard an officer, a corporal armed only with right's might was ill weaponed, and the technique of obsequious insolence which in three years' service Reinhart had made his own was a device rather more for survival than dominance.\n\nHe moved along the street of officers' billets, a short block of the little toy houses of Zehlendorf with terracotta-tiled tentroofs, tight fences, and playing-card lawns. How queer it must be for Schild to live in such a house and look out upon a provincial street through white curtains; whereas Reinhart himself had done it for years; how contemptible to Schild's keen senses. How could Schild forgive the neat-meshing casements and the correct dun stones in the walk? A spreading evergreen bush flanked his stoop, from the lintel above his door sprouted a night light like a globed mushroom. Had a Jew ever lived in such a house, and had he been ripped screaming from it by pink-and-blond young men?\n\nAs, asking, he lingered at the gate, the answer opened the door and stood uncertainly upon the threshold, _ecco homo,_ Lieutenant Schild, and the response of Reinhart's heart, in the same vocabulary lately used for loving Lori, said: even had he raped Veronica and murdered Schatzi, I could never raise my hand against him.\n\nMan, man, one cannot live without pity. What Reinhart proposed to feel was the general emotion, but as he watched Schild come pitiably down the walk in his forlorn movements and crummy uniform, wiry hair bushing his cap, opaque spectacles, blousefront a home for lint, splay-shoed, wrinkle-pocketed, choking on a necktie with a dirty knot, insignia corroded and awry, haddock-faced\u2014as he made these sorrowful entries in the ledger, Reinhart's sympathies became particular. Whatever pity Schild deserved for simply being a Jew, he required more for merely being Schild. The decent thing to do was leave.\n\nBut before sluggish Reinhart could get under way, Schild had reached the gate and, with its faded pales between them, said stoically, for all the world as if he knew of the mission which Reinhart had just abandoned: \"Yes, Corporal, you came for me?\"\n\n\"I was taking a walk,\" Reinhart answered shamefully. \"And I saw this house and remembered the crazy Russian we took upstairs last month\u2014\" he broke off and in concern came back: \"Did you get all the files from the office?\"\n\n\"You did me a kind service that night,\" said Schild, cloudily, fingering the gate's catch; but though it was a simple rod and slot he could not work it, stopped trying and capped his hands on the picket-points. \"I wish I could do something about repayment, but you see I am not in your company.\"\n\nTo Reinhart, too, it seemed a tragedy; he felt his cheeks lose their blood and fall in, to match Schild's; like Schild's his voice sounded as if it crossed a body of water: \"I'm sorry, very sorry. ... That German kid hasn't bothered you any more, has she, when I'm not there? Dirty little whore, she makes me sick.\"\n\nWithout trying the exterior handle, without hands he applied his hip against the gate and pressed inexorably in: the hardware ground, bent, was sprung free, shooting its several parts and screws tinkling to the walk; and Schild came through to the pavement, unheeding what had been necessary for his egress\u2014which, done, struck Reinhart as regrettable and clarified his mind. He gathered the fragments of the lock and after a quick determination that they could never be reassembled, at least left them available on the cap of the gatepost.\n\n\"Since I can't repay your favor,\" said Schild, perhaps, in the public air, a breath or two less mad\u2014for mad is what he was, or had just been; as clear a, as well as the only, case of depressive mania Reinhart had ever seen and which, now thank God it had begun to pass away, he was able to identify and reflect that he had answered it correctly\u2014\"since I can't repay your favor, may I ask another one?\"\n\nA formidable non sequitur, yet suggesting an idea not at all lunatic; irony, rather, and one had only to look at Schild, in whatever condition, to understand the authority with which he manipulated that instrument. Half-dead from some despondent cancer, he yet in one short question, in a failing voice, exposed the skeleton of charity: he who takes a favor returns it by asking another; he who gives one is repaid by the commission to do a second; and the score is even throughout, unless, indeed, the giver has the better.\n\n\"Of course,\" granted Reinhart, foreseeing the little drama without passion in which he would deliver to Very a billet-doux from Schild, foretasting his own humiliation and perversely enjoying its savor.\n\n\"May I come along with you on your walk?\"\n\nSo. Again he had persuaded another to play him for a fool, for, make no mistake, people use us as we ask them to: this is life's fundamental, and often the only, justice. If he understood that, on the other hand he saw that to Schild it was not a mocking, ass-making request. The lieutenant actually waited on his approval, head down, his cap points echoing the general wilt of his body.\n\nDid his grief owe to Veronica's, of which he had been agent? Surely no man, whatever his responsibility and whatever the upshot, would lose his nerve by this. It was rather Schatzi's money; but again, would a bad debt to a _German,_ even a good one, so resound in the sane conscience?\n\n\"Would you like to borrow some money?\" asked Reinhart, without a warning to himself. \"I have an awful lot.\" He drew his wallet as if it were a gat and with one finger triggered it open. \"See, over five hundred.\"\n\nInterminably Schild stared into the note-clogged leather breech, and so near Reinhart could have snapped it to and clipped off the end of his nose. When at last his eyes lifted, their fright was giving way to the old, cold certainty that they, and no one else's, owned all truth and virtue.\n\nSuperiorly turning his head towards the house, he said to the yard: \"Now I know how a whore feels. Everybody offers me money.\"\n\nOh, he was a fellow who could be rubbed the wrong way, and certainly he had his reasons; society had slipped him the shaft, he had doubtless been diddled by the dangling digit of destiny; there was some extenuation for his own failings, but none for those who trespassed against him\u2014and for once to all this Reinhart, rebuffed, said _balls,_ and with as much offensive familiarity as he could summon from dead start, clapped him smartly on the shoulder and announced:\n\n\"Why sure, come along. I'm going to see a man about an abortion. A broad I know, as they say, has bread in the oven. How do you like the size of this wad? If that won't buy the job, nothing will.\"\n\nNot waiting for Schild's reaction, he businesslike stored the wallet and marched down the sidewalk. At the corner he was pleased to hear the hurrying footsteps behind, but still he gave no quarter, and who knew how far his calves might have propelled him unaccompanied\u2014for in contrast to Schild's they were muscled as an oak root, tireless as pistons, and at the moment the body they supported was, inflated with purpose, a lighter-than-air craft\u2014in reality, he was detained at the curb by the passing trucks of Lori's caravan.\n\nReaching him, Schild spoke breathlessly above the roar: \"You can't be serious! That's against the law; besides, it's dangerous. You could be tried for murder.\"\n\nHis concern, if innocent, was madly out of proportion to their acquaintance; if disingenuous, nonsensical\u2014if he knew Very was the object of the plan, why should he, who already had fled from his own responsibility, complain? Anyway, God damn the man for his officiousness.\n\n\"But you see,\" said Reinhart. \"I'm not doing the job myself. I'm going to hire a German doctor.\"\n\n\"So.\" Schild gave him a face of regretful sadism\u2014a smile of malice taking pleasure in itself Reinhart had seen, but never a _frown._ \"So, the little blonde got to you even though you were warned.\"\n\nNow here, where Reinhart should have felt anger, he did not. His reply was simple sullenness: \"You're not even an officer in my company.\"\n\nHard upon the tailgate of the last truck he stepped into the street, into a cloud of blue exhaust, choking. Thus, with his eyes closed, he did not see the jeep which turned the corner and, also blinded by oil smoke and carbon monoxide, might have injured him, or he it, had not his persistent saviour this time succeeded. Schild's thin fingers, he felt with smarting arm, were strong.\n\n\"Nate!\" shouted the man behind the wheel, a fattening captain who wore a knitted OD tie, \"I thought I might _run_ into you, a-ha-ho, mpf, mpf! Give you a ride?\"\n\nHand still hooked into Reinhart's swelling forearm, and applying a force whose aim was the other side of the street, Schild answered curtly: \"I have some private business with this corporal.\"\n\n\"So be it,\" spoke the captain, reaching for the gearshift, gathering in at the mouth the drawstrings of his barracks-bag face. \"Nowadays you're always arresting someone.\"\n\n\"If you can't use that ride, I can,\" Reinhart told Schild, shaking him off. \"Going towards the Ku-damm, Captain?\"\n\n\"Could be, if you'll tell me where it is, unless that's in the Russky sector. Brr, I wouldn't chance it there, and do you know, Nate, I still can't get that poor devil out of my\u2014\"\n\nSchild interrupted: \"We've changed our minds.\" He produced the kind of smile, with much evidence of teeth, that one shows when his underwear is torturing his privates. \"I'll confess to you, St. George, if you won't tell my CO., the corporal and I have something cooking on the black market.\" He lowered the back of the front seat so Reinhart could hop in back.\n\n\"I know your commanding officer\u2014a real son of a bitch,\" replied St. George, going into uproarious mirth. \"A dirty son of a bitch! Corporal,\" he said, gagging on the _r's,_ \"in case you didn't know, I'm cussing out myself. You got that kind of C.O.?\"\n\nReinhart grunted icily at the silly slob. Tyrannical officers, who were candid about their power, were preferable to jovial ones in love with their own decency. As to Lieutenant Schild, whose head snapped back on his fragile neck as St. George jerked the car into forward movement, he defied classification: who was doing what for whom and how was one to feel about it?\n\nSo started they towards the enormous cairn of rubble underneath which lay N\u00fcrnberger Strasse, in whose name Reinhart for the first time recognized a memorial to his old city of legend and determined to lay the symbolism before Bach, the specialist in things undreamt of by other philosophers.\n\nHowever, now that he faced Bach, with Schild in the adjacent chair, Reinhart could worry over nothing but that Bach would begin where they had last left off, on the Jews; or, before an audience half of which was virgin to his dramaturgy, repeat the farce so successful at its opening, while Reinhart sat paralyzed by the ethics of entertainment: _please do not tell your friends the surprise ending._ Hastily he began to collect the differences between this visit and his last, as the man lying down to rest adds up and tries to cherish the details which differentiate this night from last, the nightmare-ridden: tonight I am lying on my other side, the pillow slip is fresh, the moonlight does not shine upon the window\u2014oh, but God, I have the same head and I am scared.\n\nFirst he noted an increase in illumination. The oil lamps were in their old positions but unfired. From the center of the stained ceiling, the nucleus of a web of hairline fissures, hung a hot electric bulb. Augmenting whose glare Bach's reflecting, porcelain head irradiated his immediate area. His sofa, at the principal surfaces worn to the bare hemp of warp and woof, in the hitherto obscure corners shone now in a pattern of emerald, turquoise, white, and scarlet: a scene, a world, the edge of some equatorial swamp profuse with hot flowers and curving flamingos and reed-green water, and, on the lip of the depression behind Bach's shoulder, the great throat-cup, here in ruby, of the bird whose beak can hold more than his bellican.\n\nBach himself wore a green suit, a moss-colored huntsman's suit with oval bone buttons and odd straps and trimmed in gray beading; in the lapel slot, a spray of edelweiss, fake, showing its wire stem. His trousers were cuffless, bell-bottomed, seam-striped like a uniform, and between them and the floor lay ankle-high shoes of reversed leather, pine-cone brown, fastened with tasselled cords threading through a series of bright chromium clips.\n\n\"You are feeling better now,\" said Reinhart.\n\n\"Oh thank you, thank you,\" Bach replied, the inside of his mouth red as the flesh of a blood-orange, here and there the yellow seeds of teeth. \"But the clothing produces an illusion at odds with reality. In truth, I believe that I am dying\"\u2014he threw his hands at Reinhart to dispose of a reaction which had not come\u2014\"please do not grieve: 'by my troth, I care not; a man can die but once; we owe God a death and let it go which way it will he that dies this year is quit for the next.' \" He looked slyly towards Schild and returned, fingering the blossoms in his lapel. \"This flower, by the way, is quite false.\"\n\n\"It is very nice, anyway,\" Reinhart replied.\n\n\"You are always kind,\" said Bach, coyly raising his right trouserleg to reveal an inch of brace, as a Victorian coquette might have exposed her ankle, \"but the chiefest consideration will always be: where could a man with my infirmity get the real thing? Mountains, my good Corporal, one must climb the highest peaks to reach this noble plant, which is as difficult to come by as an honest man.\" Having exhausted his air on the speech, Bach took more through the tube of his cigarette, in a long and intense suction which burned back an inch of ash. \"The same characterization that applies to me does as well for the so-called building under which we sit at this moment\"\u2014swinging his head over the sofa arm, he spat the smoke at the floor, as if it were a mouthful of milk\u2014\"despite its apparent improvement\u2014the laying on of electricity\u2014it is quite likely, I am led by all my senses to believe, to collapse without warning.\"\n\nReinhart, who would never again be taken in, said in swaggering irony: \"Not while I am here, at any rate; because I have a charmed life.\"\n\n\" _Is_ that true!\" exclaimed Bach, fanatically interested, seeking to rise unaided, in the violence of his attempt giving the illusion that he had almost made it and lost by a hair; whereas in fact he had not moved a centimeter.\n\n\"Indeed it is. May I sit down?\"\n\n\"Ohhhh\u2014\" Bach began a long gasp at his own poor manners, not waiting for the completion of which Reinhart fell beside him at the end of the couch and ground out a space with his hips, at Bach's yielding expense.\n\nHe plucked at the threads of upholstery on the sofa arm. \"Handsome, very tasteful.\"\n\n\"Gobelin,\" said Bach, with difficulty twisting his neck, upon a static body, to face him.\n\n\"I certainly know his name,\" Reinhart replied, crossing his legs and inadvertently fetching Schild, whose chair was a good four feet away, a kick in the shins, smiling absolution for himself for that, smiling then at Bach in self-admiration which quickly shaded into joke as he saw upon the great grapefruit a polite confusion that told him he had guessed wrong. \"Don't mind me,\" he said rakishly. \"I told a Catholic friend last week that I had never read Father Douai's translation of the Bible.\"\n\n\"Yes,\" Bach answered, still perplexed. \"May I ask, however, of the charm upon your life?\"\n\n\"Oh of course.\" He glanced covertly at Schild, who had, as late as a Stanley Laurel, just begun to rub his injured shin. \"Well, I think I felt it first when we were pinned down along a hedgerow in Normandy. There was the enemy a bare hundred yards away in the next hedge, laying down a withering machine-gun fire. Well, they were bottling up the whole American advance; somebody had to do something. And I must confess, our leaders had failed us completely. The company commander, the platoon lieutenants, the NCOs, they all proved to be perfect cowards. You see, this was our baptism of fire\u2014\"\n\n\"You were infantry?\" asked Bach, a hand against his left cheek, as if he restrained his head from swinging back to the frontal position. Despite the evidence of a similar, internal attempt to control his eyeballs, they were their own masters and veered continually towards Schild, until at last they fixed in that direction as a lecher's will upon a maiden.\n\n\"Glider infantry,\" Reinhart corrected, \"a unit in the 101st Airborne Division, later to become the so-called 'Battling Bastards of Bastogne.' \"\n\n\"How terrible!\" interjected Bach.\n\n\"And I don't mean to say that at first I wasn't scared myself. But then, crouching there, staring across that new field of rye through the hedge, towards that line of green blooming like roses with gun flashes, I suddenly looked down and saw my trousers were open. I put down my submachine gun to button them\u2014and then I thought: 'What a wretched little swine I am to care about this when I might be killed in two minutes!' And then, just as quickly, I _knew_ I would not be killed, got this absolute certainty that I could stand up and walk slowly across the field and never be hit. So I did just that, climbed up over the top and began to walk slowly towards the German line. After I had got about ten meters out, slugs whizzing all around me but never hitting, the Krauts stopped firing! Stopped cold. I think now they thought I was coming to surrender; and it is true that the end of a white handkerchief was showing from the breast pocket of my field jacket. Anyhow, when they stopped I gave a big holler and discharging my Thompson advanced on them as fast as I could run. Behind me the rest of the company came whooping forward, not shooting, though, because I was in their line of fire. And do you know\u2014\" he slapped his hand upon Bach's green knee and felt, rather than the expected quivering of aspic, a hard and sharp junction of almost naked bone and metal brace\u2014\"do you know, those Germans sat paralyzed behind their guns and did not shoot once more, and when I looked over the hedge, down into their trench, all fifty-three of them threw up their hands and yelled _'Kamerad'!_ And of all things they turned out to be a crack unit of the SS, you know, the SS, fiercest fighters of all, who never surrender.\"\n\n\"Oh yes,\" Bach answered, lowering his hand; his head, as promised, instantly swung away like some half-door between a kitchen and dinette. \"I surely know of the SS and can only say that the fact must have been as you suggest, that they anticipated the surrender would be vice versa. For to them fear meant as little as does memory to an ingrate. In the Warsaw Ghetto the SS fought on until the last schoolboy put down his penknife and the last little housewife dropped her paving brick.\"\n\nSo of course there it was, Schild raised his eyes, the curve of Bach's fat cheek glistened with triumphant sweat, and Reinhart's big feet began to punish each other for the humiliating failure. \"Ghetto,\" that beastly ugly word the pronunciation of which began in the deepest throat and worked forward like a piece of phlegm\u2014he had heard nothing else. The loathsome Germans and the damnable Jews: the plague that had befallen both their houses was kind beside the one he now wished upon them. He also wished for nerve to direct Schild to the booby hatch and for courage to tell Bach he intended to carry off his wife, with whom he was in earnest love.\n\nYes, spiting all his wishes, he forced himself to say: \"Were you in the SS at that time?\"\n\nBach again pushed round his head, but before he made a word Schild rose and spoke ferociously to Reinhart: \"I'm not going to let you do it, you understand? If anything goes wrong they'll put you in Leavenworth for twenty years. According to your stupid middle-class morals, I suppose, better to take a chance on ruin rather than beget a child out of wedlock. You are an idiot!\"\n\nThere was no longer a question that he had gone nutty as a fruitcake: with hard steps he strode to the end of the cellar and leaned against the wall and gravely examined its waterstains.\n\nBach began to speak in a low, grating, regular tone, like an electric drill needing oil: \"The SS? My friend, I\u2014\"\n\n\"What business is it of yours what I do?\" Reinhart screamed at Schild, notwithstanding the poor fellow was mad. \"I can get through life without your help!\" And notwithstanding that Bach, poor chap, was an invalid, he turned on him viciously: \"For Christ's sake can't you talk of something other than the Jews?\"\n\n\"Curious,\" said Bach, smiling mildly, \"the manner in which a member of the other ranks may speak to an officer, in the American Army.\"\n\n\"He's not in my unit,\" Reinhart answered, lowering his voice. In the corner of his eye he saw Schild return.\n\nBut the words were kind; the face, gentle: \"Because I am your friend. Isn't that reason enough?\"\n\n\"Sure it is, sure it is.\" Reinhart swallowed. \"I suppose it is the only good one for doing anything in the world.\" He dared not admit to himself how deeply he was touched, how much sense lay in madness, how heroic was decency's response to brutality's negation. For this he could repay Schild only with candor.\n\n\"Things have got all complex,\" he said, \"simply because I let them slide.\" Schild, standing, hovered before him blankly, nervously. Beside him Bach breathed with a slight moan. Perhaps, after all, now that truth was having its day, he _was_ about to die. The winding stain which Schild had traced on the wall was not, he could see now, a decoration of seeping water but rather a weakening division in the concrete which seemed to widen as he looked and perhaps _would_ bring down the house\u2014there had been sense in that, to him, lunatic action as well.\n\n\"I should have told you before. But maybe it wasn't eccentric to think you might know. Veronica Leary is pregnant.\"\n\n\"That big nurse?\" Schild shrugged, splayed his hands in impatient despair.\n\n\"Why put it that way?\" Reinhart was angry all over again and himself despaired that relations with his newly found friend could ever be on the unswerving line of constant respect. \"Can't you even call her by her name?\"\n\nBach snorted as if he, too, had never witnessed an outrage of that magnitude, but turning to him in alliance one saw him chatteringly blow his snout into an aquamarine handkerchief.\n\nSchild's feet, too, were splayed, and his head forward and depressed below the level of his shoulders; sitting before him Reinhart could look down the back of his head to where the collar, too large, yawned out from the hairy neck.\n\n\"So,\" said Schild, \"that's worse yet than the German girl, isn't it? How could you get in such a predicament?\" Fierce yet charged with loving concern, like that sonorous old actor who always played the father of an East Side boy torn between the life of the spirit and the life of matter, when that theme was \u00e0 la mode.\n\n\"Ah God!\" sighed Reinhart. \"Finally, I see. I didn't knock her up, if that's what you mean. I'm just trying to help Veronica out of it.\"\n\n\"Why?\" Schild again chose his chair.\n\n\"On the basis of friendship. She was never my girl. She didn't 'betray' me. But I might do the same if she had. Common humanity is more important than sex. What matters is, she's in trouble\u2014just as, a moment ago, you thought I was. I'd rather aid than be aided any day, just like you.\"\n\n\"Then I am out of order,\" Schild replied, \"and there's no help for that.\" After the briefest illness, his face lay down and died.\n\nBut why take so hard a simple error that in the end had done no harm? In his statement was implied a personal doom, unpeopled, glacial, bone-white, so much more terrible than Bach's presages of a technicolor disaster. Was he serious? Reinhart looked at Bach, the absurd man, the absolutely useless man, who even if he were restored to health, if he had ever been there, would only stand and gawk at Oriental art, crap like that, and rant foolishly. But was he frivolous? And finally, did it matter?\n\nHe asked the last question only to make sense of his ready answer\u2014for that was truth: first the answer and then the question, so that while we wonder we can continue to live\u2014did it matter?, oh hell yes, for all we have in this great ruined Berlin of existence, this damp cellar of life, this constant damage in need of repair, is single, lonely, absurd-and-serious selves; and the only villainy is to let them pass beyond earshot.\n\n\"Do you know what we could use right now?\" he said unwaveringly to Bach.\n\nThat huge fellow swung round his enormous head, his pale eyebrows climbing in inquiry, his second chin reluctantly altering its seat in his collar, which was white and overlaid with peasant embroidery in red thread.\n\n\"A good laugh. Say something funny.\"\n\n\"Very well.\" And Bach was as good as his word in that at least he tried. He told a story of two friends, Palmstr\u00f6m and Korf. Who, finding a mouse in their house, built a cage of latticework, into which Palmstr\u00f6m climbed at twilight and began to play the violin. As night fell, the mouse was lured in by the music. Palmstr\u00f6m went to sleep, then so did the mouse. In the morning Korf put the cage into a furniture wagon and hauled it out to the country. The mouse was there released. He loved his new home. Korf and Palmstr\u00f6m, delighted, returned to town.\n\n\"That's the end?\" asked Reinhart.\n\n\"Of course,\" said Bach, wincing in amusement, suffused with rose color. \"What more could we wish? Consummate art, which I can assert only because it is certainly not of my own creation\u2014needless to say, for what is?\"\n\nFor Christ's sake what a story. Then he heard Schild snicker, and saw him laugh with a na\u00efve mouth of which the upper lip flattened and glistened tightly midway across his upper teeth, and his ears protruded like a schoolboy's.\n\nThrough water-brimming eyes\u2014his spectacles were tiny fish-bowls\u2014Schild finally looked at Bach.\n\n\"That is Christian Morgenstern.\"\n\n\"Exactly,\" answered Bach. \"How exciting that you also know his work!\"\n\n\"Do you remember this one?\" Eagerly he moved to the edge of his chair and rapidly, in an accent which to Reinhart sounded perfect, quoted in a German of which Reinhart understood nothing.\n\n\"All right,\" said Reinhart testily, watching them howl at each other, \"what does that mean?\"\n\nBach began: \"This gentleman named Korf\u2014\"\n\nBut Schild, impatient in his high levity, broke in: \"He has invented a kind of joke, you see, that works by delayed action. The people he tells it to are horribly bored. But later that night, when they are in bed, they suddenly wake up and laugh like babies.\"\n\nAll very German and although remote from Reinhart's old medieval visions, somehow not alien to them. At least he was gratified by the alliance of Schild and Bach, Jew and German, in a common cause.\n\nAt this point he heard a distant, cavernous sound, as one in the bottom of a sewer would hear a person scratching at the manhole. Lori approached through the passageway.\n\n# _CHAPTER 19_\n\nBY THE SLEEVE, LORI ushered in a man wearing dark glasses and carrying a cane, a meager man concealed within an enormous overcoat. This doctor, if such he was, would be splendid for the job. He was blind.\n\nReinhart heard and felt the slow removal of Bach's weight from the couch and, staring up the rising underbelly of the green Zeppelin, towards the gondola, discerned that respect drew him up. Schild, too, had risen, was already, being nearest the door, in an introduction.\n\n\"Sir,\" said Lori in her to Reinhart always lucid German, \"I am Frau Bach. I should like to have you meet my brother, Dr. Otto Knebel.\"\n\n_\"Herr Doktor, es freut mich,\"_ said Schild, shaking the hand which groped for his, showing an unsuspected command of the gracious forms, even slightly bowing. \"Oberleutnant Schild. _Ich bin ein Amerikaner_.\"\n\n_\"Ich bin dessen gewiss.\"_ The reply was in a high, aspirant voice, not ugly or unpleasant but strangely lingering within the innermost channel of the ear, as if a bug had crawled in there to die and, caught, changed its mind. \"I am certain of that.\"\n\nFor the second time in his life Reinhart had heard \"certainly\" as answer to an American's self-definition. It no longer seemed strange, but because he had already got his he did not wish another. Therefore when the doctor was moved to meet him, he, mimicking Schild's handshake and suspicion of a bow, rumbled low and uvular, authentically, _\"Sehr angenehm!\"_\n\n\"This surely,\" said the doctor to Lori, \"is your Ami corporal.\"\n\nCompared to his, Schatzi's hand had been full-fleshed, hamlike; one thought not of bare bones: one held tendons and a complex of thin vessels through which slow and miserly came corpuscles one by one. Behind the glasses, in front of the tall back collar of the coat, was a real head: small, stark, but real, and so marked with life, so marking life, that the memory of other faces was rank on rank of dummies. The lenses were too black to see through, in compensation for which they themselves were animate. Finely amused now, they dramatized the implications of the breathy voice which rendered stout German as if it were the tongue of dragonflies.\n\n\"Yes,\" said the doctor. \"But you should have seen me six months ago!\" Bach made a giddy noise. \"What, Bach, my good fellow! You have held captive these Americans? Then there is still a chance that we may win the war! Very well, my dear Lenore. Now that I have located Bach I can manage alone.\"\n\n\"Ah Doctor, you come right to me and take this seat, _my_ seat, _bitte, bitte._ May I help you? Please, please.\"\n\nIn the strength of his schoolgirl agitation Bach took two quavering steps and grasped for the doctor's arms, which that gentleman, moving efficiently behind a probing cane, ignored with a blind man's insouciance. On his left sleeve he wore a yellow armband carrying the rubric, three black balls, of a vicious, violent, antidemocratic cause. (Reinhart remembered his colleague Cronin's description of such an insigne, such a movement, the week before on Cronin's boarding an airplane which would fly him back to France as a case of chronic athlete's foot. \"Open your eyes,\" Cronin said, upon no provocation whatever, \"the same old thing's starting up all over again.\" The boring ass; why had one come along on the truck to say goodbye? But then Cronin hit him lightly on the shoulder and said, \"So long, Reinhart. You're the only one I could ever talk to.\" Very simple: one had come along because one, all in all, had liked and would miss him.)\n\nSitting down in Bach's corner of the sofa, the doctor said:\n\n\"Lenore, have you some of those excellent American pastilles? The anticipation of talking dries my throat.\"\n\n\"Oh, here,\" said Reinhart, who happened to be carrying a cylinder of Lifesavers. \"Take these... and keep them.\"\n\nThe eyeglasses widened their circles. _\"Vielen dank_ \u2014 _sprechen Sie Deutsch?_\n\n\"A little. _Ich verstehe besser ah ich spreche_.\" He grinned in self-deprecation, though his auditor could not see it.\n\n\"Your accent is very good.\" The doctor's mouth was a pale pink cave, toothless; moreover, showing no evidence that teeth had ever been.\n\n\"Not good enough to fool you.\"\n\n\"Why you should wish that, especially nowadays...\" A Life-saver tumbled over the doctor's tongue, glinting orangely. \"But I knew you from your hand, not your speech. ... Now Bach, are you still standing there with your misplaced courtesy? Kindly be seated. And Lenore and the Lieutenant, and you, Corporal, please. As to the Lieutenant, now, I should think that though he has been in the U.S.A. some years, he was born in Germany, no?\"\n\nSince seeing the armband Reinhart had been occupied with nothing but worry for Schild. From the data of his first visit to the cellar he could hardly suppose Bach and Lori were the doctor's fellows in a neo-Nazi faction. Who then was the doctor but the blackmailer that had preyed on them during the Hitler years and still today somehow retained his evil power? And how compelling he was: Reinhart had brought forth the Lifesavers like an automaton. Schild, the eternal do-gooder, was already captured by the man's infirmity; Schild, the Intelligence officer, did not see the armband; Schild, the Jew, already was impaled on the doctor's fascist needle.\n\nSchild, the innocent fool, looked sadly pleased. \"Is my accent so good?\" He sat down, as he had been ordered to. \"I am a native American, doctor. I am one of the lucky Jews.\"\n\n\"I too am lucky, but I have not been able to decide whether my luck owes to my Aryan mother or my Jewish father,\" said the doctor. \"This is the kind of thing which confuses everybody but the Nazis.\" He closed on his candy, swallowed it, and took another from the pack. \"Why do your countrymen waste so much paper, Mr. Corporal? Really, these fruit drops would not grow stale exposed to the air. Really, what are they but crystallized sugar-water? But won't you, all of you, join me? Bach, you must! I prescribe sweets as a substitute for that abominable ersatz-schnapps with which you are destroying your liver.\"\n\n\"Doctor, I have sincerely tried to stop drinking,\" Bach said, his face a quivering sack of shame as he lowered himself onto a folding camp chair which he overhung in every dimension. \"I will conquer it, I will, you shall see.\"\n\n\"May your reformation not wait upon my seeing,\" replied the doctor, lightly. And Bach's despair was as if a truck passed overhead.\n\nOf course, if the doctor was Lori's brother he was but half a gentile, had but half the aptitude for corruption. Of course, Reinhart had not forgotten that so much as ignored it in his quest for a villain to save someone from. Yet why the brazen badge?\n\n\"Now Corporal, I think you and I have a private matter to discuss,\" said the doctor, placing his cane on the floor and in so doing offering a view of his full profile in the various perspectives of slow movement. His right eye, seen in the harsh knife of light which, as he bent, thrust in from the side, behind the dark glasses\u2014my God, an eye? A navel, rather, a belly-button of the head, baby-new and pink within the old foxed leather which bound the skull.\n\nOne's own eyes indrew behind the barrier of cheekbones, hid in scarlet darkness, as nevertheless one's more courageous mouth asked: \"Doctor, what is the meaning of your armband?\"\n\n\"It means\"\u2014the black circles swung round and established order\u2014\"that if you drive an automobile French fashion, use it, that is to say, as a projectile with which to aim at pedestrians, I am your perfect target. I cannot see you come.\"\n\n_\"Wie bitte?\"_\n\n\" _Es tut mir leid._ I was having a bit of a joke, most unfairly. The sleeveband is of course the sign of the disabled person. Unfortunately I do not have Bach's gift for foreign languages. Bach, could you perhaps\u2014\"\n\n\"No, it is not necessary,\" said Reinhart. \"I understand. It is an excellent thing\u2014\"\n\n\"They do not have it in America,\" Bach cried eagerly. \"Never, nowhere have I seen it.\"\n\nLori, still standing, chided: \"Now Bach, if you do not permit Otto to have some privacy, he will not talk with you later.\"\n\n\"Quite so, quite so,\" Bach mumbled, turned laboriously, and to Schild instituted a speech which began: \"However\u2014\"\n\n\"Lenore,\" the doctor said, \"There is no reason why you should not sit here and assist us with your good sense. Also, working for the Americans you should have learned some English by now, unless Father's old claim was true, that we were the champion dunces of Dahlem.\"\n\nAlthough, because of the difference between the doctor's and Bach's girths, there was now a good seat and a half to the left of Reinhart's port hip, Lori sat down so close against him that, for the comfort of both, he had to lay his arm along the back ridge of the sofa. His love for her was just in the degree to which it remained intactile. Introduce desire and you would soon have the same old two-backed animal scuffling in the dirt, into which he and Trudchen transformed themselves daily, destructive, nightmarish, impermanent, having nothing to recommend it but necessity. With Lori he mixed spirits, was embarrassed by the flesh... but she rested, almost lay, within his arm-hollow, her hard, thin bones piquing him, the shoulder of her thick old prickly-wool sweater, carrying a scent of spice, touching his cheek. And he, who involuntarily rose at a woman's smell\u2014as a sleeping cat erects its ears at every sound\u2014almost any woman, any smell, sometimes, in the street, at pure cloud of odor, the woman having long gone by, was shortly, or longly, risen.\n\n\"Now,\" said the doctor, to see whom Reinhart had to clasp Lori more closely in looking round her blondeswept head, \"this young woman you have got in trouble\u2014\"\n\n\"Ah no, Otto, it was not he,\" Lori broke in far too eagerly for the pride of the fellow she had made her cave.\n\n\"How do you know that?\" asked Reinhart, arrested in his drawing away by the sofa arm in the small of his back; because of this his irritation became briefly paranoid: how dare he be boxed in?\n\n\"Because maids, like concierges, know everything,\" said Lori, mock-mysteriously, without trying to turn. \"The Gestapo of belowstairs...\"\n\nThe doctor disintegrated another Lifesaver and swallowed its rubble. He chose a third, perhaps a fourth, since the pack appeared to stand currently at three-quarter size. His thin lips, opaque glasses, and traces of eyebrow expressed satisfaction. His hair was a thick bush, one finally noticed as one continued to creep so tightly against Lori that when she spoke he heard the vibrations in his own chest. Bach, remote in a spirited monologue to Schild, Reinhart worried over not, nor did he despise him.\n\n\"There, there,\" said Lori, patting his nearer knee with a twinkle in her hand, \"everybody knows you could have.\"\n\nThe doctor stared exhaustively, sightlessly, at Reinhart. Finally he spoke in his loud whisper: \"Let me for a change be honest. Obviously I cannot perform the operation. I could find a colleague to do it, of course. But I intend not to. I have come here and taken your time, and your pastilles, under false colors. My motive was simply to 'see' an American. Are you angry with me?\"\n\n\"No,\" Reinhart answered. \"Surely not.\"\n\n\"But you should be.\" The doctor was impatient. \"I can solve your problem, yet I will not. And as far as you know, for a capricious reason.\"\n\nReinhart smiled tolerance and dropped his cupped left hand on Lori's shoulder. \"I can't force you, can I?\"\n\n\"Then you are not serious?\" asked the doctor in dramatic astonishment. \"Disgrace for the lady, shame for you\u2014for although you may not be the other principal in the catastrophe, your honor is somehow involved, yes?, or you wouldn't be here. Come now, at least try to bribe me.\"\n\nSmiling again, Reinhart answered, conscious that when he had to speak without preparation his damnable German was certainly ungrammatical and, despite his \"good accent\" Americanized in pronunciation\u2014you cannot take care of everything simultaneously\u2014so that to these Germans he was ludicrous for another reason. _In their reality_ he sounded:\n\n\"I donnt tink dot so easy to corrupt you are.\"\n\n\"On the contrary, I am supremely corruptible. I have no honor whatsoever. For example, I would do anything to save my life.\"\n\nReinhart felt Lori stir against him, and he released her sweater-shaggy shoulder. \"Oh well, wouldn't anybody do that?\"\n\nFor the first time, but briefly, the doctor lost what had all this while been more nearly ebullience than anything else. And then, taking another, a purple, Lifesaver, he said, with the old aplomb and in the voiceless voice Reinhart had come to hear as oddly beautiful, \"On the other hand, if by necessity you have learned this fact about yourself, it is nice to know. Some American writer\u2014have you read him?\u2014wrote a verse about seeing a man eat of his own heart. 'Is it good?' he asked. 'Well,' said the man, 'it is bitter\u2014but I like it. First, because it is bitter, and second, because it is mine.' \"\n\nReinhart did not understand. And Lori had _not_ learned much English, therefore could only repeat the words more slowly, in her low-pitched music.\n\nHe shook his head. \"The funny thing is that I know all the words; it must be the combination.\"\n\n\"Bach!\" cried the doctor. \"Excuse me for a moment. Please give us the English for this.\"\n\nBach did, with an attitude of excessive expectation; sought to explicate, was halted.\n\n\" _Danke sehr._ Now just return to your lieutenant. We did not wish to disturb you.\"\n\nReinhart determined to read, when he went home, this author whom a non-English-speaking German knew better than he. However, the doctor had turned out to be the usual lunatic, in love with his own rhetoric. He returned to the subject which had become a great bore to Reinhart, who had decided at the first resistance to seek another physician through Schatzi.\n\n\"I have no scruples against abortion in itself\u2014\"\n\nHis speech came within an interval of breath-taking on the part of Bach, who heard it and answered: _\"Die meisten meiner Mitmenschen sind traurige Folgen einer unterlassenen Fruchtabtreibung.\"_\n\n\"Bach, don't you realize you are interrupting?\" chided Lori, seizing the hand with which Reinhart, bending forward, traced his trouser crease in the area of the shin. \"I assure you that if you persist Otto will avoid you. ... Please do not do that,\" she said to Reinhart. \"A hard object in your breast pocket jabs into my back.\"\n\nA pencil, which he removed to the other side. Nevertheless, he disliked a carping woman.\n\nBach desisted, and when Schild spoke, cautioned him with wrinkled forehead.\n\n\"Did you get that?\" Schild sadly asked Reinhart. \"Most men are the sad results of abortions never undertaken.\"\n\nBut by now, having adjusted to German, Reinhart heard English as somewhat dull upon the ear and difficult to follow. He believed that Schild was repeating his old objections to the plan for Very's salvation, and assured him resignedly that it was all off. \"You can stop worrying.\" He should, in the first place, have hired Schatzi and thus given no one an opportunity for humanitarianism, friendship, theory, oratory, and so forth: that was the way with intellectuals; from his old uneasiness towards them, for which he had blamed himself, he was at last liberated; worse than boring, they were of absolutely no utility; if you want a barrel built, hire a cooper.\n\n\"You have changed my mind,\" said Reinhart to the doctor. \"Forget it. I was foolish. I don't want to get into trouble.\"\n\nBach, still actively desisting from interruption, wrestling with himself, gave up suddenly on an interval of losing and said, with hysterical bravery: \"Tell him, Doctor, tell him about the Russian concentration camps! They were worse than the German ones!\"\n\nLori wrenched angrily within Reinhart's surround, Schild recoiled sickly upon himself, as if someone had hurled towards him a bucket of filth, and the doctor sighed.\n\nHis weary answer: \"Ah Bach, you take what you choose. But so be it, we shall leave it at that.\"\n\nReinhart somewhat rudely thrust Lori from his line of vision. She pushed back with unusual strength for so small a body, crumpling his outstretched fingers, and if in that second of pain he had been asked, do you still love her?, he would have said, sorely, because she is as tough as a root. Gently this time he raised himself from the slump and looked over her head.\n\n\"Are the Communists as bad as the Nazis? Were you in a Russian camp? I didn't even know the Russians had concentration camps.\" Saying which he looked haughtily at Schild, whom he had gauged as a pro-Russian liberal, and saw thereupon what he should have known from experience was more to be pitied than defied. He would never be able to match his moods, to meet aggression with the same, and humility in kind.\n\n\"Bach provides a much more effective torture than either,\" said the doctor genially. \"Whatever theories of coercion are developed in the future, they must take account of his method: admiration of the nonadmirable. He believes that because I was a prisoner I have a special and heroic wisdom. He is wrong, but my vanity insists otherwise; therefore, in my sense, which is nobody else's business, he is right. Why, however, should you permit me, or him, to inflict this nonsense on you? ...Now tell me, is it true that one can enter an American cinema while the motion picture is in progress? Isn't it queer to see middle, end, and then the beginning?\"\n\n\"Yes,\" Reinhart answered, \"yes, one can enter at any time. But American movies are made for an audience whose average mental age is twelve years old. You should have seen the pictures they made on Nazism. Such trash is almost criminal.\"\n\n\"The Nazis were presented as good men?\"\n\n\"Oh no, but either they were monsters who did not resemble human beings or they were ridiculous buffoons.\" He was making out all right with his primitive, do-it-yourself German, for the doctor seemed to understand.\n\n\" _Also,_ this was an error: too realistic. I agree with you, this theme should be dealt with as fantasy. Lenore, do the privileges of your job include Ami films?\"\n\n\"Not exactly, Otto,\" Lori answered brightly. \"But do you recall the old joke of Father's about the man who was asked if he had ever eaten hare? 'Not exactly,' he said. 'But yesterday I shook hands with a fellow whose cousin's brother-in-law lives next door to a widow whose late husband once saw someone eating hare.' It's not as bad as that with me. I make the beds of persons who see the pictures every night.\"\n\nThe reference to Veronica could not have been more obvious. Reinhart intended his response to be equally obvious in disregard.\n\n\"Your father has a good sense of humor?\" he asked. How strange for a German! But then he remembered that her father was a Jew.\n\n\"Well, yes,\" answered Lori, looking at him from the corner of her eye, he thrusting himself to the side so that she could do it, \"I have never thought about it so seriously, but I suppose he had.\"\n\n_Had?_ Yes, dumb Reinhart, not everyone is always young and American enough to have two living parents. Besides, he was a Jew. Yet he had to speak, he, Reinhart, one in five in this subterranean, brightly lighted urinal\u2014monstrosity, Jew, half-Jew, half-Jew, Siegfried.\n\n\"He was killed\u2014\"\n\n\"He is dead,\" said Lori.\n\n\"\u2014by the Nazis.\"\n\n\"He is dead.\"\n\n\"And who else, who else?\" If in all his life he had reached no goals, he would take this one.\n\nFrom his implacable face she turned away in embarrassment and towards the doctor gave her dirge: _\"Voter, zwei Br\u00fcder, Schw\u00e4gerin, Neffe, Nichte.\"_\n\nIn English, thus excluding his wife and brother-in-law, Bach cried: \"There is no wit like that of Berlin, of which since I am not a native I can assure you without immodesty. Hitler and his damned barbarians hated this city because they could never break its spirit, because they could not transform it into a N\u00fcrnberg. I confess to you that I am a separatist. I fervently hope we remain forever isolated from the Fatherland.\" He slapped his knee\u2014too hard, and winced.\n\n\"I wish I could do something,\" Reinhart said. \"I wish I could say something\u2014\"\n\n\"You can indeed,\" the doctor answered, impatiently stripping the paper tube from the remainder of the fruit drops, catching five of the six in the wire whisk of his left hand: one fell to the concrete and broke into three golden arcs and a modicum of sugar dust. \"A lemon, _ja_? I can smell it now it is crushed.\"\n\nEither Schild or Bach made a sound like the winding of a watch.\n\n\"You can,\" the doctor repeated, and whatever else he said wound through the holes of the five candies in his mouth and expired before finding the orifice of speech.\n\n_\"Ich habe ihn nicht verstanden,\"_ Reinhart whispered into Lori's hair.\n\n\" 'You can say something,' \" Lori answered loudly. \" 'You can tell us what you will make of yourself now the war is over.' \"\n\nHe raised his meditation to point at the ceiling, to macerate his vision on the fierce lightbulb: father, two brothers, sister-in-law, nephew, niece, like the roster of a holiday reunion.\n\n\"Well, I cannot bring them back, whatever I do,\" he shouted quietly. \"But in my own small way I can fight all hatreds based on race, color, or creed. In my own small way I can say: we must love one another or die!\" When he was moved, words came from nowhere, inspired; yet he was conscious of the falsity of those which had just arrived. It was fairly certain that of the six victims in Lori's roll he could not have loved at least one, so goes the world. And how did a fellow go about loving any of those who killed them? For a principle means either what it says or nothing; if we love one another, we love the murderers, every one. And finally, was love really the sole alternative to massacre?\n\n\"One must love himself,\" said the doctor. \"The men who killed my family did not. What are totalitarians but people who have no self-love and self-respect, who believe that the humanity into which they are born is contemptible?, who believe a thing is preferable to a person, because a thing is absolute.\"\n\n\"But a thing,\" said Bach, \"has a sense of its thingness. The Will works in inanimate as well as animate objects. That sofa may know very little, but it knows that it is a sofa.\"\n\n\"Of course I agree, Bach, that this sofa has a self: I have heard it most painfully groan when you sat upon it and chuckle when you arose, but we shall wait forever if we expect it to will itself into a chair. This poor couch is so predictable.\" He actually looked sad and patted its arm. \"If you prick it, will it not bleed? But that is not necessarily true of a man, who may spit in your eye, or, having a taste for pain, beg you to prick him again, only harder. And what might he not make of it as a moral act? That by taking his life you have confirmed his conviction that you are inferior to him, and for some men life is a small price to pay for such reward. Or that by causing him to die well you have relieved him of the need to live well, for any victim is willy-nilly a success. Or that by divesting him of everything but the naked self you have made it possible for him to accept that self. In the end he may have used you as you believed you were using him, and who can say who was the victor?\"\n\n\"Oh no,\" cried Reinhart, even though he thought it likely he had misunderstood, \"you cannot build some elaborate theory that in the end Nazism did good. That sounds like the idea of those old fellows in Neuengland\u2014the northern U.S.A.\u2014Rolf Valdo Emerson, _und so weiter,_ who wore frock coats and walked in the woods and never cared about women, and therefore had this dry belief that evil was only the servant of a greater good.\"\n\nDuring this\u2014how fluent?\u2014speech Lori twisted round and studied him, trying, he supposed, to be unnerving: a person without experience should sit silent as a vegetable. Well that, last time with Bach, he had done. He felt now as if _he_ were drunk, and finishing his representations to the doctor, he stared defiantly at her strong, straight nose.\n\n\"Otto can say anything he likes. You see, he has paid for the right.\"\n\n\"There you have the corruptive results of working for the Amis,\" laughed the doctor. \"If I paid for the privilege to be theoretical, then I was cheated, my dear Lenore. All other German males are born with that right and obligation. But how true if you imply that this chap from over the sea should not be permitted to speak further without paying tribute! Come, Herr Unteroffizier, surely you have some more candy about you.\" The doctor retrieved his stick from the floor and brandished it. \"Here comes some English\u2014you did not know I had some? _Komm on you dirty rat hand ovuh zuh goods._ This is what the racketeers order, no? Bach has a detective novel which he reads aloud to me\u2014\"\n\n\" _Ja, ja,_ I have it just here,\" Bach said eagerly, struggling to rise. \"I read with simultaneous translation\u2014\"\n\nReinhart grandly waved him down. \"That won't be necessary at present.\" He did, of course, have in his clothing another piece of sweet: a chocolate bar foolishly stored in his shirt pocket, over the heart. It was now limp. He gave it to the doctor and apologized.\n\n_\"Sehr gut,\"_ the doctor responded. He smelled it. \" _Schokolade!_ I will not eat this. I shall present it to the widow who lives across the hall from me.\" He placed his cane on the concrete, giving Reinhart another sight of the umbilicus of his right eye. \"I am trying to seduce her.\"\n\nReinhart grinned anxiously and withdrew an inch from Lori, as if it were a mistaken but justified statement of his own aims, but when the doctor's glasses were turned on him again he saw their terrible wistfulness.\n\n\"Oh God, Doctor, eat it, eat it,\" he said, his voice ragged in pity. \"Next time I can bring you a carton for your widow.\"\n\n\"If your motive is kindness, please do not. Such largesse, if I gave it to her, would earn me only contempt. And if I kept it for myself I would eat it all immediately and fall ill. In either case I should curse you. But why do you now wish to bribe me without profit for yourself when earlier you refused to do it for gain?\"\n\n\"Because he is a good man.\" It was Schild who spoke, and pleadingly, and Reinhart suffered for him in anticipation, for the doctor _was_ a kind of demon, after all; in revenge for his having been tortured by evil and falsity he would torture goodness and truth.\n\n\"And I suppose you are, too,\" sharply replied the doctor. \"I don't trust a man who would rather give than receive. I can't stand his damned pretense that he is too good for the world. He is mad. I disapprove of lunacy, illness, disability, and failure.\"\n\nReinhart could no longer contain himself. The mad doctor's ranting left him personally untouched, but poor Schild gulped it all down, sounding again and again that watch-winding noise in his throat, and poor Lori was limp against his shoulder, used, no doubt, to the habitual insane rhetoric of the cellar; she had, as before, gone to sleep, but the constant strain!; he would rescue her from it before the hour was out; if need be, kick Trudchen's cheap little ass into the street and give Lori her room. Meanwhile he must catch Schild before he disappeared round the bend.\n\nHe shouted: _\"Das ist National-sozialismus!_ I don't know what you are trying to do, Doctor\u2014I sympathize with you, I would give my own eyes to get yours back, believe me, I would give my life if your family could come back again, I have never done anything\u2014I couldn't even hold a gun because of the Geneva Convention\u2014but don't say the Nazis were right. If that is true, then it was all useless; your loved ones died for nothing. All those corpses\u2014I saw them in the photographs. Those beloved people, they were too good for the world. The rest of us are too bad for it...\" His voice had broken, broken, as he knew ever more poignantly that with whatever motive he had begun his defense of general reason, he continued it for the sake of his own.\n\nTherefore was the doctor right, even as he sought to repudiate him; therefore was he cramped with guilt for a crime he had not perpetrated and agonized by a suffering he had not had to endure. To be vicarious always is always to be base.\n\n\"Why do healthy people believe there is wisdom in a wound? _Mumble, mumble_...\" The doctor slipped the envelopes from the Hershey bar, which in his stark handbones had lost its borrowed warmth-of-Reinhart and returned to brittleness, and segment on segment inserted it into slit-mouth quick-lips, munched, munched, munched. Soon was the lower third of his face childishly smeared with brown. His hair, dark-blond, high, luxuriant-grown as a Zulu's, had burst forth from the cropped skull of the camps. Against Schild his whisper had gone hard, cruel; towards Reinhart, Reinhart now decided, it had always been a snicker.\n\nOn he went in the idiom of masticating chocolate, with a necessarily greater show of gesture than when he spoke audibly, which nevertheless stayed Greek. Schild, who had been slumped, wired up his spine and sat straight, neurasthenic. Lori slept, heavy for so light a girl.\n\nBach, however, listened eagerly and when the doctor, the last bit of candy down the hatch, gave off, the giant bobbed his peeled egg at Reinhart and said in English: \"There you have the doctor's world-outlook in a nutshell!\"\n\nThen it was that Reinhart realized the doctor was fake from the word go; that he was no more an alumnus of a concentration camp than Schild was a hangman.\n\nThe latter suddenly glared at him and snapped: \"Very well! Russian 'concentration camps.' _Sehr gut,_ ask the doctor about them! Simply the Buchenwalds of another fascism...\"\n\nThe doctor wiped his mouth on a handkerchief as holey as a net dishrag, to get which he had opened his coat and revealed the necklines of, at quick count, four gray sweaters and a shirt collar of brown.\n\n\"Certainly they are not,\" he said good-naturedly. \"If you won't let me avoid the subject\u2014it is not offensive to me, since it is mine, but it should be, I insist, to you. If I must talk on this theme, I'll take my stand on precision. Young Corporal, you talk of love. But perhaps love is for boys and girls and old ladies who love their dogs. For us professionals, consider precision. Love one another or die? But we die anyway, _ja_?\"\n\n\" 'The subject is not offensive to me,' \" Reinhart suspiciously repeated.\n\n\"I did not say pleasant or without pain. I said not offensive,\" said the doctor, impatient. \"Now you interpret it as you wish.\" He resumed: \"The Soviet camps: as you must know, Lieutenant, they have quite another purpose than the Nazis', which latter were in their most extreme form mere extermination-places. The aim of the Soviet camps is to change people. Sometimes, inadvertently, live men are there changed into corpses; well, at least they are no longer counterrevolutionaries.\n\n\"Each kind of camp has a favorite kind of prisoner. The Nazis preferred the man who by existence was a criminal, that is, the Jew. Good Jews, bad Jews, Jews who as individuals _were_ criminals by the usual definition, even those Jews who would have agreed with everything in the code of fascism but that all Jews should be exterminated\u2014no, this is not yet enough: even those Jews who might have helped the Nazi cause\u2014were murdered indiscriminately. There was some early plan for 'useful Jews,' but it was soon abandoned. An Einstein perhaps could have been forced or tricked into giving Hitler the atomic bomb. Nevertheless he would not have been saved from the gas chamber.\n\n\"Where in all history can we find another idealism comparable to this? Hitler did what we have always been told is the supreme glory of man, and apparently impossible to a god\u2014for what chance did Jesus of Nazareth take if he was immortal?\u2014Hitler sacrificed himself for that which is greater than the self, for he stuck to his guns and he is dead now. I think a name for that is Love.\n\n\"With the Soviets, however, no man is judged by what he is but rather by what he can become. Their favorite prisoner is the man capable of learning the error of his ways. He must do this through hard labor on projects useful to the state, and hence to mankind, and thus there is no waste. The penal system of a faith as inclusive as the Nazis' was exclusive, and for that reason psychologically superior to the latter. Here at last the Jew, for example, is not a second-class citizen: he can be as great a swine as a gentile. Did you know, until that desert tribe of Hebrews found the one authentic God and that they were His chosen, an exclusive religion had never been invented? Ever since, the gentiles, who never could take a joke, have been punishing the Jews for being so damned clever. To the Communists, however, this old strife is a great bore. A man's a man, and is capable of anything. And of course when you believe that, you are loving one another.\"\n\n\"Excuse me, Doctor,\" said Reinhart, adjusting a prickling shoulder under Lori's weight, \"when you opened your coat I saw your shirt. It looks like part of a uniform, but not quite the color of a U.S. Army shirt\u2014\"\n\n\"I should not suppose it does.\" The doctor's whisper lost strength in extended speech; Reinhart really helped him by interrupting. He cleared his throat with the soft yet dynamic sound one might make shaking out a floormop. \"You are wrong if you think the average German feels no guilt; he simply will not dance it to the tune of you people who were not involved. My widow gave me this shirt. I suspect it is a storm-trooper's garment, but naturally I cannot see it. ...\n\n\" _Also,_ Lieutenant, we have looked precisely at the differences. My brother-in-law insists, however\u2014since he cannot forgive himself for being a gentile German [Bach flushed and looked at his legs]\u2014on their similarities. His interest lies in proving Communism worse. Because I was once a Communist I am inclined to agree. The conscience is a Himmler as demented as the real one. Remorse, whose seat is in the memory, has a purpose. Guilt, the product of the conscience, is always useless, the wrong kind of self-concern, cheating, cowardly, immoral.\"\n\nSince the doctor's comments on his shirt, which had proved him as false as anything could, Reinhart had been rather nursing his shock than listening. He came back now to strike another blow for virtue.\n\n\"It isn't hard to be a murderer. The tough thing is to be a victim.\" He smiled so bitterly that Lori woke up on his shoulder, saying _\"Wie bitte?\"_ to which he answered, _\"Nichts, schlafen Sie noch.\"_\n\nFor the first time, Bach, who had been frozen in wonder and delight, noticed her.\n\n\"Rude!\" he cried in outrage. \"Your brother is speaking!\"\n\n_\"Ach,\"_ she said, _\"was kann man tun?_ He hasn't stopped since I was a little girl.\" Her head sank again.\n\nThe doctor laughed and laughed at the awful thing\u2014if he _was_ an authentic ex-prisoner\u2014she had said to him. \"When we were small she used to punch me if I talked too much. In the solar plexus. Very effective when struck just right: I couldn't speak for half an hour. Therefore would I take revenge by playing the Leonore Overture on the gramaphone, which, because I insisted she was named for it, she detested. Then in would come brother Leo, who couldn't study mathematics for the din, and he would shout in his shrill voice: 'Twins have only half a brain each.' But if the altercation continued until Father had to come upstairs, we were all for it. Father had a face like a weapon. He was a very severe man. I can recall nothing loving about him but much that was precise.\n\n\"Once when at table I spoke without permission he afterwards beat me so strenuously he sprained his arm. Feeling guilty, as I usually did upon such an event, but not remorseful, I offered to fit him out with a sling\u2014already, you see, the future physician. 'Do you want another whipping?' he asked. 'This time for being a fool? From your point of view my sprain is richly deserved.' That is to say, he was a self-respecting man. I hated him for years. But now I think he must have died well.\"\n\n\"That old Prussian authoritarianism,\" said Reinhart, remembering an argument of Cronin's. \"There you have the origin of Nazism.\"\n\n\"Except that my father was a Jew,\" said the doctor.\n\n\"Jews can be tyrants, too.\" Reinhart was earnest, no longer baited or tested the doctor. \"Isn't that what we mean when we say racism is a lie? Everybody gets his chance to be a bastard.\"\n\n\"Yes, and we should not deny it even to, especially to, a victim. For there are victims and there are victims. If you read _Mein Kampf_ you will find Hitler believed himself a victim, and because when he became a master he failed to do his job well, I am still able to agree with him in that early appraisal of himself.\"\n\n\"Haha,\" jeered Reinhart. \"Victim of what?\"\n\n\"Of indifference. The German people never understood what he wanted of them. Being normal people, they were always interested principally in themselves.\"\n\n\"While the innocent were being murdered all around them... to you this is right?\"\n\n\"If you think I shall tell you what is right or wrong, my friend, you are mistaken. That is your own affair. I care only for practical matters.\"\n\nReinhart rubbed his head. Fresh from yesterday's close haircut, it felt to his hand small, hard, monkeylike, and shiny as a convict's.\n\n\"I give up,\" he said, without knowing whether the idiom was feasible in German. \"The trouble is, Doctor, I just don't know what you want. If everything we have always thought is decent, is wrong, false, misguided, or useless, what alternative is there? The only thing I can see is the contradiction of decency; Nazism is as good a name as any, so long as we understand that Nazism in this sense is not just a German but a human thing. The Russians, then, if they have concentration camps, are Nazis. Perhaps there was some Nazism in dropping the atomic bomb on Japan, which must have killed a lot of women and children and at least some Japanese who never wanted to go to war in the first place.\n\n\"The British, someone once told me, invented the concentration camp during the Boer War. The French, so I heard, put German refugees in concentration camps at the beginning of this war. In democracies there are white people who lynch Negroes; there is anti-Semitism. _I_ have been guilty of Nazism when I used force or threatened to on someone weaker than I or outnumbered, or when I had bad thoughts about Jews and other defenseless people\u2014because I have done these things.\" He looked proudly guilty.\n\n\"I should hope so,\" said the doctor. \"What's good enough for everybody else should be good enough for you.\"\n\n\"But isn't selfishness the terrible crime of the modern era, selfishly being concerned with oneself and therefore thinking the other fellow is garbage?\" He took his arm off Lori's shoulder so that he could rub his head with both hands. \" _I_ want power, _I_ want money, _I_ want to be superior to a man with a colored skin or with a hooked nose\"\u2014from his tumult he was able to call time, to say \"Excuse me, it was just an example\" to Schild, who, in the reverse of Schatzi's habit, was looking at him but not seeing\u2014\"therefore I tell myself he does not matter, is not even human. Then I can go on to do what I wish with him, slavery, torture, murder.\"\n\n\"Imagine yourself a citizen of the American South,\" said the doctor, \"a person who is in daily contact with Negroes and thus must come to terms with the fact of their existence. Would you mistreat them?\"\n\n\"God, I should try not to.\"\n\n\"You might occasionally fail, _ja_?\"\n\n\"I am just a human being.\"\n\n\"No question of that, and so was Julius Streicher, as Hitler, who was no man's fool, said so well: 'He may have his faults, but well, probably none of us is entirely normal, and no great man would pass.' Yes. But why would you try not to mistreat Negroes? Is there profit in it?\"\n\n\"It would mean something to me,\" said Reinhart.\n\n\"So there is a profit after all.\" The doctor spoke as if he, himself, were making the discovery. To be sure, his manner throughout had been rather seeker than owner of fact; did he lack the courage of his confusions? \"You cannot get respect for yourself by robbing it from another man. As to the Negroes, they might not know or understand what you were doing and therefore show no gratitude, _ja_? But to a healthy man this would make no difference. The self is not a gallery with a claque. And it would not be necessary for him to love the Negroes or hate the brutal whites, or worship a god or history, or be a radical or conservative. Just to be a man were sufficient, _ja_?\"\n\n\"Your example is too easy. Excuse me for trying to tell _you_ about life, but is it not more complicated than that? I am not likely to live where Negroes are mistreated. I did not live in Germany in Hitler's time or in Russia. I am not a Jew, my father is not an oppressed worker or sharecropper. On the other hand, neither am I a fascist or a boss\u2014well, let's face it, _I_ am nothing in particular, but you know what I mean. What would I do in a situation where an Auschwitz is possible? ...I have not told you\u2014somewhere in Berlin, if they are still alive, I have some relatives. I hired a man to find them, but just now I realized I have always hoped he never could. What if they were Nazis?\"\n\nAs a further twist of the knife, the doctor removed his glasses and began to clean them with breath and handkerchief. Reinhart averted his eyes.\n\n\"Since I can't see through these things,\" said the doctor, \"I clean them from a motive of pure vanity. I do not wish to be thought a sloven.\" He replaced his spectacles and took up his cane. \"I should like to meet your relatives if you find them. By various accidents and choices, I have a foot in everybody's camp. I am a halfbreed of every persuasion. You claim to have done nothing. I have done everything. Every individual life is a questioning of the validity of all others.\"\n\n\"And also a confirmation of it?\" asked Schild.\n\n\"Ah now,\" answered the doctor, \"that is irrelevant, for why should I need you, or you need me, or either of us need, say, Hitler or Stalin to tell us what we are? _Ich bin kein Weltverbesserer und lasse Sie liegen._ \"\n\n\"Then you should be satisfied with your lot,\" said Reinhart, \"neither were the Germans who were not Nazis world-reformers, and they let you lie.\"\n\n\"True,\" said the doctor cruelly, \"and they were not the ones who killed my family and took away my freedom, were they? They heard the cries and turned away, but at least they did not come and help fire the ovens.\"\n\nReinhart had chewed his gum too long. It disintegrated. He tried to reassemble it with his tongue. He failed. Ashtrays here were unknown; the smokers had crushed out their butts on the floor. He swallowed his fragmented Spearmint and said\u2014\n\nBut the doctor had not waited for him: \"There is but one demand we can make on others: that they let us alone. Anything beyond that is a corruption or will be one within the hour.\" He rose easily and hunched over his cane, which his fingers grasped as an owl a branch. \"Do you think I say this because of what the others did to me? The others, I tell you, are irrelevant.\n\n\"I was a Communist. The day after Hitler came to power I fled to the Soviet Union with my family. Thanks to the tactics imposed on it by Stalin, the German Party was shortly wiped out by the Nazis. But we all knew that history was using the Nazis for our ends, so we\u2014those of us who got out in time, that is\u2014did not despair. The Jews? A kind of vermiform appendix on the body of history. An illusion. Science knows no definition of Jew or gentile. ... In Moscow I had a good job in the Medical Institute, doing research on skin cancer. I won two decorations for my work and soon rose to head my section. My family and I, four of us, lived in a modern apartment of four rooms\u2014had four times the space, that is, of the average Russian family. After the required time, we became citizens.\n\n\"My chief assistant, at whose cost I had been promoted, for he had worked there since its founding, was an old Russian Jew with, like so many of them, a German name: Kupstein. He was the sort who would always be an underling. He did nothing well, but what was worse he knew and admitted it. He broke slides, he misread calibrations, once he managed to fracture the lens in a microscope\u2014rather a difficult thing to do under ordinary conditions.\n\n\"But we human beings were not so ready to exploit our power over him. Obviously he could not help it, and his constant contrition! He could, naturally, not only have been discharged but also imprisoned for his failures. Indeed, in the Soviet view he should have been; insofar as I made allowances for his good intent I was a bad Communist and perhaps an outright traitor\u2014and when I say this I do not refer to the disguised GPU informer on our staff. I speak of my Communist conscience. The secret police are given too much credit; for the important things we never need them.\"\n\nCrazed old man, leaning on a cane, rasping in _Deutsch._ Why had Reinhart almost flunked German 2? He understood every word, every nuance. The doctor condemned guilt in others but loved his own. He suffered retroactively for being sloppy years ago in Russia. ... If he had been in Russia how could the Nazis have got at him? Lori stirred. Without prior planning he whispered in her ear: \"I love you.\" She smiled sleepily and closed her lashes again, muttering _\"Knorke mit Ei.\"_ Something with an egg. Total misunderstanding.\n\n\"... after that episode I had no choice but to relieve him of his duties. We could all have been killed. Yet I still could not report him, sentimentalist that I was. And quite rightly was I punished for that weakness. With nothing to do he hung about my elbows all day and interfered with my own work. Titration tube in mouth, I would hear his squeaky voice and almost swallow some septic liquid. Bending over the microscope I would suddenly smell his breath, vile from some horrible cheese, as he bent alongside.\n\n\"And what did he speak of? Palestine, which he called Israel. He had been there for two weeks in the 1920's with a Soviet scientific team and was terribly impressed by everything from communal farms to climate. 'Believe me, my dear Doctor,' he would squeak, 'working on the _kibutzim_ seems a pleasure for these strange Jews. Imagine Jews as farmers! The sun turns their skin black as Africans' and has bleached the hair of some as blond as a Pole's\u2014or, as your own. Sabotage is unknown, yet one never sees a policeman. Is this possible? I doubt it. But it is the witness of my own eyes. And oranges! As many as you can eat. And the young people. Imagine happy Jews!'\n\n\" 'Hirsch Davidovitch,' said I, 'your satires are very clever but they may be misunderstood. Besides, you are interfering with the experiment. Really, this sort of time-wasting is more appropriate to a bourgeois-capitalist laboratory'\u2014I spoke that way in those days, and not simply for the GPU informer\u2014'we work here for the health of the international working classes and have not a moment to spare, please.' But next day he would start in again: 'My dear Doctor, the olives! I have seen them large as this.' Pointing to the bulb of a Florence flask, he would knock over a rack of test tubes and then, sponging up the mess, strike the flask from table to floor.\n\n\"Kupstein, Kupstein, of course you were winning,\" said the doctor, sinking an inch into the orifice of his coat collar; he had once been a tall man, but that too was now a memory. \"From the first time I had tolerated his statements without an effective rebuttal, I was a fellow conspirator. It was 1938. In Germany the N\u00fcrnberg Laws had sealed the fate of the Jews\u2014foolish Jews, one beats another and shouts 'help,' as the saying goes. My father, the lifelong reactionary who ordered me from his house when he found my copy of Marx, loses his department store to the Nazis, brings suit in the bourgeois courts he trusts so much, leaving the Nazis no choice but to send him to Buchenwald. Almost did I ask: well, what does he expect? With my brothers Leo and Viktor, who had given neurotic importance to their Jewish halves and turned active Zionists, they had been doomed by their stupidity and cowardice. Marxism, they agreed, was 'no answer.'\n\n\"In the Soviet Union, meanwhile, the great purges which had begun in 1936 were now in full fury; among the high Government and Party officials only Stalin seemed secure. Could our entire leadership, except Stalin, be corrupt? Yes, no question that it _could be._ Communism, as I said before, admits unlimited possibility. A man can be anything history needs him to be. No chosen people here, either for good or evil. For example, among the condemned officials were many Jews, and of course the commander-in-chief of the whole plot was Trotsky, born Lev Bronstein. He had conspired with Nazi Germany to destroy the Soviet state. Impossible? _But nothing is!_ By definition a state built and maintained by the proletariat is just, and whom it charges with a crime is guilty.\n\n\"When the rosters of the eminent were depleted, the purge began to claim the malefactors among the technicians and managerial workers. I at last discovered who had been the police informer in my department\u2014Rostov, a biochemist\u2014for he disappeared soon after Yezhov, the head of the GPU, was purged. The director of the institute had not survived through 1937; three successors, with only a month or two between turns, followed him to the wall or to Siberia. Dr. Narovkin, in effect my chief assistant, though Kupstein still held the title, was called to a corridor telephone one afternoon and never came back. His replacement, a simian type by the name of Gorky, sent by the personnel section without consulting me, did not bother even to imitate a scientist. All day long he sat in a corner of the laboratory, behind two carboys of acid, watching the rest of us.\n\n\"Dr. Narovkin's work had been essential to the experiment. He had done months of research on malignancies in lymphoidal tissues. If I could at least have had his notes! But they too had vanished, the day after his own disappearance. The project was hopeless? You must remember that this was a Soviet laboratory. We had been _ordered_ to discover, first, a preventive against sarcoma and, second, a cure for it. I reinstated Kupstein in his old post. What difference could it make now? None but for the better. Kupstein had worked in Soviet laboratories since 1919, and one thing he could do well was write reports. On his own initiative and with a perfectly straight face he now composed a manuscript of fifty thousand words reporting the successful achievement of our goal: we had found both a preventive and a cure for fleshy malignancies, and in one year less than our allotted time. I solemnly read and appended my signature to this handsomely written nonsense and forwarded it to the newest director of the institute. Not long afterward I received another decoration.\"\n\n\"Doctor,\" said Reinhart. \"Aren't you uncomfortable standing up?\"\n\n_\"Schweigen Sie!\"_ Schild ordered in an offensive, Prussian manner, so startling Reinhart that he answered, as Prussianly, _\"Jawohl!\"_ and did shut up most smartly. Bach smothered a giggle behind a trembling hand.\n\n\"Now there was no restraining Kupstein,\" the doctor went on. \"Defying Gorky's unwavering surveillance, he no longer whispered. Now he spoke his heresies in the tone of normal conversation. 'Do not despair over your loved ones back in Germany, Doctor, every death there is a life for Israel. The Jews one day will leave the cities and return to the land. Olives, lemons, palm trees!' I could not admit that he was mad, you see, because then I should have had to accept that I also was a lunatic\u2014for from the first his rantings had taken malignant growth in my imagination, like that very sarcoma which he and I so successfully defeated in our report. Damn the Jews\u2014my relation to them had always been an embarrassment; now it became a poison. 'The Law,' Kupstein would sometimes say, 'the immutable Law. The Jews have little else, but they have the Law and it does not change.'\n\n\"One day, speaking so, he followed me into a storage room at the other end of the laboratory from Gorky, who as usual sat at his table, but would come after us if we did not soon reappear. I took quick advantage of the situation. I seized Kupstein and said: What would you have had me do? Stay in Germany and die like a fool? You know how Nazis deal with Communists!'\n\n\"Brushing my hands away, he answered in a loud voice: 'Just yesterday there was an unopened crate of new test tubes right there. Now where could they be?' His eyes were innocent behind their pince-nez.\n\n\"I seized him again. 'Kupstein, have mercy, I beg of you. We here in this country, in this very laboratory, are working for not only the Jews but the entire human race.'\n\n\" 'I don't understand,' he said, again very loud, 'since the end of the sarcoma project one can't find a thing here. How can we proceed to defeat carcinoma without test tubes?'\n\n\"He referred to our new assignment: a preventive for bone cancer. At any moment Gorky would come snooping. 'I warn you, Kupstein,' I whispered. 'I have heard enough to have you sent away for twenty years, if not executed, as a foreign agent. Have you forgotten that Palestine is a British colony?'\n\n\"In astonishment he answered: ' _I_? Have you forgotten'\u2014there was the slightest pause, perhaps not really in Kupstein's speech but rather in my hearing\u2014'that central supply holds you responsible for every piece of equipment?'\n\n\"Gorky stood in the doorway, his thick eyebrows gathered in upon the root of his nose. 'Doctors,' he said, 'I must confess I have those test tubes in back of my table. I have been taking them out one by one from the straw and shining them with a bit of cloth, being ever so careful. They are now ready to go into the sterilizer\u2014may I operate the sterilizer, Doctor? You will see I can do a good job.' His face, menacing until a moment ago, was a cretin's.\n\n\"Two days later, at three o'clock in the morning, I was arrested by the NKVD and taken to 22 Lubianka Street. I never saw my family again. For what I estimate to be seventy-two hours\u2014there was no window in the room\u2014I was interrogated without pause. I received no food, and water was administered\u2014a glassful was dashed against my face\u2014only when I attempted to collapse. For at least forty-eight of those hours I was given no idea of the charges against me. The NKVD officer\u2014he was replaced by another from time to time, but they all looked the same\u2014insisted again and again that I confess, that my crimes were known to him but, consonant with the just laws of the Soviet Union, he must hear the details from me. By turns he addressed me as villain, child, poor idiot, honorable but misguided patriot, personal friend. At the idiot level I had an opportunity to think... that devil Kupstein! Obviously he, and not Gorky, had been the police spy. Being a loyal and convinced Communist, I knew too well I had no hope this side of a full admission, but of what? Kupstein had surely turned me in on a charge of Jewish chauvinism. I could not confess to that, of all things. In an access of shame and hatred I asked for pen and ink. I wrote a statement which in style, if not quite in length, rivaled the sarcoma report. I revealed myself as an espionage agent for the National Socialist government of Germany.\n\n\"My interrogator read it with satisfaction. 'Excellent,' he said. 'I'm sure you feel better for having got this off your chest.' He flipped through the pages. 'You see, you cannot fool us, although you are a most clever man. As a half-Jew you did not think we would suspect you of working for the Nazis, eh? And marrying a Jewish wife was also shrewd, eh? But we are shrewder yet, eh? Now name your accomplices and we will be finished with this unpleasant business.'\n\n\"My accomplices. To be sure, I had neglected this all-important matter. I wrote fifteen pages more, implicating Kupstein. This was a grim joke for which I was prepared to pay: since he worked for the NKVD, I had no doubt they would reject it. But here I intended to take my stand if it killed me, as well it might.\n\n\" 'Splendid.' The interrogator smiled. 'Now your conscience is clean. You understand that we already knew everything about the entire conspiracy. Your fellow agents in Leningrad and Kiev were arrested last week. Kupstein is also certainly no news to us; for years we have known of his fascist, Zionist intrigues as an agent of Trotsky.'\n\n\"Which in the language of the NKVD meant precisely the opposite. Somehow Kupstein the _Verderber,_ the spoiler, had blundered through two decades in his own kind of peace\u2014until I betrayed him.\n\n\"I was sentenced to fifteen years of hard labor\u2014a mild sentence considering my grave crimes\u2014and sent to the Kotlas camp in the region of Arkhangelsk. The details of that servitude are not as relevant as Bach maintains. The Nazi camps were worse. To make a comparison of the two, Lieutenant, is pointless. A single principle applies to both: in both the prisoners properly are innocent. I represented a grievous error on the part of the Soviet authorities. As you have heard, I was guilty.\"\n\n\"And Kupstein?\" asked Schild.\n\n\"And,\" said Reinhart, \"will you kindly explain how you got to Germany from Siberia?\"\n\nThe doctor pulled a blue muffler from inside his coat and draped it around his head as a woman would; but when he brought down his hands Reinhart saw he looked rather like Mahatma Ghandi.\n\n\"Through the camp intelligence, I heard that Kupstein was executed. My wife and children were not arrested, but they had to leave the apartment and it was made difficult for my wife to find work. I don't know how they survived. Before long the question was academic. I was arrested in July, 1938. A year later, when Stalin and Hitler signed their pact and divided Poland, I was brought back from the camp and deported to Germany.\"\n\n\"Oh _no_ ,\" Reinhart gasped, a sound applicable to whichever judgment he would finally make on the doctor's tale.\n\n\"My Soviet citizenship was revoked upon my conviction for the crime of espionage. In the pact each side agreed to return the other's nationals it held prisoner. The Gestapo met us at the border between German and Russian Poland. My wife and children were included in the transport, I understand. I was not allowed to see them. ... They died, I believe, at Buchenwald, where my father and brothers had earlier.\n\n\"The Nazi methods of interrogation were second-rate\u2014exclusively physical brutality; there has really been nothing new in that line since the ancient Chinese.\" He shook his head almost regretfully. \"The Nazis were a mediocre lot with only one idea: audacity succeeds; the _id\u00e9e fixe_ of the suicide. Where, other than poor stupid Germany, could they have got twelve years to discredit it? ...To the Nazis I was the same kind of embarrassment that the Jews had been to me. I repudiated my Communist affiliation, citing as evidence my treason to the Soviet Union, and demanded to be held as a Jew. If I had been interested in preserving my life, I chose the correct strategy. For here was another point of difference between the two systems.\n\n\"In the USSR one is given just what he asks for: at the end of my confession I asked for punishment. My request was honored. Not so with the Nazis. In their Neanderthal psychology a man asks for one thing to conceal his aim in another direction. Besides, they thought, who being something better would ask to be a Jew? I went into their dossier as Communist first, Jew second; and that took my eyes\u2014convinced I could give them information on the Communist underground, they tortured me\u2014but saved my life.\"\n\nPlucking at the floor with his cane, the doctor walked to the door. They all rose. Reinhart reached him first and took his arm.\n\nThe doctor shook him off irritably, then repented, saying with a smile: _\"Es geht allein schon schwer genug!,_ it is hard enough alone.\"\n\nTough old cuss, said Reinhart sotto voce, and then he saw him pass a skeleton hand across the dark glasses, as if to verify he was indeed sightless, but the very movement was evidence of an unextinguished hope that he was not.\n\n\"Twins have but half a brain each.\" The doctor grinned and pointed in Lori's direction. \"She still sleeps. _Knorke,_ I go.\"\n\nTwins, he and Lori. Which meant the doctor looked twenty years older than he was and Lori was twenty years older than she looked. Unless it was another lie.\n\nThe doctor shook Reinhart's hand, and then Schild's, and finally that of Bach, who had just reached them.\n\n\"Gentlemen, I say good night. You no doubt agree with me that an inconvenient means to self-respect is to undergo punishment for a crime you have not committed\u2014as you tonight have been punished. What a lunatic way for two young men to spend an evening! Have we nothing here in our Germany with which to entertain you? Especially you, old chap.\" He punched at Reinhart with his cane. \"Why so solemn? Doesn't it bore you?\"\n\nReinhart had imperfectly understood the doctor's story (his mark in German 2 had, after all, approached justice), but on the basis of the experience with Bach, he smelled the self-hatred in it and understood, anyway, that people in their most serious monologues depreciate rather than celebrate themselves, and are given to exaggeration besides.\n\n\"Well,\" he said. \"Do you expect me to laugh at life in our time?\"\n\nInstead of answering\u2014he should have known better than to expect him to\u2014the doctor said: \"Perhaps it will be as well if your relatives turn out to be Nazis; they have nothing further to lose.\"\n\nReinhart said: \"I personally don't think Schatzi will find them.\"\n\n\"What was that name?\" asked the doctor.\n\n\"Clever fellow,\" said Bach. \"He gives a job to his sweetheart. But she won't try very hard if it means the food you give her must be shared with them.\"\n\n\"No, 'Sweetheart' is this man's name. Don't ask me why.\"\n\nReinhart raised his nose nobly. \"And who cares? He was three years in Auschwitz.\"\n\n\"There could be only one,\" the doctor murmured, as if to himself, and then he gave a succinct reminiscence of Schatzi. Which, Reinhart observed as he fell through space, yet clubbed Schild harder.\n\n\"I will kill him,\" he said quietly. The great cables in his biceps expanded and split both sleeves at the seam.\n\n\"Good,\" said the doctor. \"But I hope not in ignorance. Kill him because he, as much as any of us, is a victim.\"\n\nHe insisted that Bach not disturb Lori: he knew the contour of every broken brick between this cellar and his own, which was close by. Again he said _es geht allein schon schwer genug,_ and went out.\n\n# _CHAPTER 20_\n\nSCHILD THOUGHT: HOW AWFUL for Reinhart, now he knows how it feels to be a Jew. He himself was weary of trust and mistrust, weary of hatred, of victims, especially weary of Jews, as, he thought, only a Jew can be. His predominant emotion towards Schatzi still was envy, now unconditional: the freedom he had seen in him was no illusion.\n\nWhen the door banged behind the departing doctor, Bach's wife woke up. A plain girl, but Reinhart, as unrepresentative an American GI as you could wish, seemed taken with her. He was not as innocent as he had seemed. Perhaps he was even sinister, now that Schatzi no longer was. What did he want of Schild? asked Schild unfairly, for it was he who had pressed himself on Reinhart, but unfairness is also a freedom. Schild liked Bach, therefore he must keep Reinhart from seducing his wife. But illicit love is also love, which must not be opposed. _Ah, but we die anyway, ja?_ said the doctor, forgetting to add: _alone._\n\nWho weeps for a Jew? he had asked with respect to Lichenko, one of the little men, symbolic Jews, for the love of whom we\u2014they\u2014control experience. Lichenko did not, but Reinhart and Bach did. Perhaps even Schatzi did. _Give to a man a chance,_ he had said so plaintively. He also was a victim, a kind of Trotskyite of Nazism, and though privileged\u2014for the Nazis were more tolerant of their heretics than were these others\u2014though a labor supervisor, also a prisoner. Schatzi's present allegiance signalized his reformation. Communism excludes no one, denies nobody his opportunity to alter, recognizes no people intrinsically chosen or condemned.\n\nStanding large and slumped before the sofa, Reinhart spoke low to the girl. So as not to jinx him, Schild made his cong\u00e9 to Bach, whose great kindly face looked down like a benevolent Buddha's, and opened the door\u2014or tried to. Five minutes ago a blind skeleton had flung it back as easily as if it were a curtain; for him, Schild, the door was frozen. The knob, a European type, a curved lever, broke off in his hand. And no putative seduction stayed Reinhart's, Mr. Fixit's, prompt assistance.\n\nUsing his elbows like Schild's father commandeering the telephone, Reinhart forced the _Brecher_ to give ground, examined the damage, described it as negligible, made temporary repairs.\n\n\"It will come off again if it is pulled too hard,\" he said to Bach, in German. \"Now if you had a bit of wire\u2014\"\n\nBach answered in English: \"My dear fellow, do not concern yourself about that. We live beneath a heap of ironmongery. Tomorrow, in the full sun, I shall grub in it for wires. What gauge is to be recommended for this purpose?\"\n\n_\"Bach hat kein Draht,\"_ said Mrs. Bach, who had a certain animation, but Schild decided Reinhart's interest in her owed to the incapacity of her husband; thus it was a sinister thing, the sexual excitement of betrayal, in which she herself at least connived: \"Bach has no wire.\"\n\nLichenko's way had been wholesome, to take the German woman by force. Last night when in sleepless midnight clarity he labored on the pillow, adding sums, he believed he had denied her to Lichenko because he wanted her for himself. Holy as a monk dreaming of the Virgin, he crept down to the kitchen and sacrificed himself upon the altar between her hard legs, she soundless except for piston hips upon the mattress. At seven o'clock, tame, she knocked upon his bedroom door and entered bearing breakfast on the last tray with which he served Lichenko and had no stomach to return for the last time to the messhall. In another land it would have been touching: bread, jam, coffee, from her own meager rations\u2014her pantry was no Army larder\u2014but the old hatred, now compounded, moved him rather to strike it to the floor.\n\n\"Bach has no wire,\" Frau Bach repeated, and now Schild heard the contempt fall on Reinhart, not the giant. \"If you wish something in this place, you must ask _me_.\"\n\nA flush of embarrassed lust suffused Reinhart's skin, although she proceeded to define the precise limits of her statement. She drew a pin from her hair and threaded it through the lever's empty screwhole. _\"Also.\"_ Tense with pride, she opened the door.\n\nReinhart shook Bach's hand. \"We must go. Did I tell you that I like your suit?\"\n\nBach perspired with gratitude. As high above Reinhart as the latter towered over Schild, as Schild himself loomed over Schatzi\u2014but there ended the stairway of heads, whose lower steps would bear most weight, carrying as they did the others. But he had excluded Lichenko, smaller than he, larger than Schatzi, a truly free man who would fit in no progression.\n\n\"A gift,\" said Bach, \"of my kind wife. She adorns me rather than herself, probably because I am good for nothing else. But that, too, one learns to accept. The mystery remains, for whom was such a garment made? For it is my perfect size, and no tailor came to call with his tape measure. Singular!\"\n\nReinhart, lifting away up, felt a lapel, and Schild remembered an old anti-Semitic routine: 'Sam, the customer wants a green suit. Toin on the green light!' Bach's horn buttons were his proper interest: how much the gross, less the usual two percent for cash?\n\nHe supposed he saw in Frau Bach's smile, which was entitled to it by half, the Hebraic celebration of a shrewd purchase as she spoke to Reinhart: \" _M\u00f6gen Sie den Anzug,_ do you like the suit? I bought it from this 'Schatzi,' little Trudchen's friend.\"\n\nReinhart gave her his large, gentile blandness: \"It is beautiful.\" Schild shook hands with everyone.\n\n\"You are always welcome here,\" said Bach from the heart. \"Next time perhaps things will be better and we can serve coffee!\"\n\nSchild permitted himself briefly to see that vision of Schild to which Bach had given, and offered to give again, hospitality; it was not unloving and it was not unloved, it was not institutional. Perhaps it also was free\u2014but it passed too quickly into the dark cloaca of the cellar hall, and he had time only to call, in simulated enthusiasm: _\"Knorke mit Ei!\"_\n\n\"Berlin slang, meaning 'Splendid.' \" He answered Reinhart's question as they clung, Alpinists, to the summit of Monte Klamotte, Mount Rubble, and searched in darkness for the comb so marked in daylight.\n\n\" 'Splendid,' \" Reinhart repeated, \" 'splendid with an egg.' There's something about Berlin that gets you, isn't there?\"\n\n\"Me?\" asked Schild.\n\n\"That gets a person, I mean.\" Reinhart turned his ankle on a broken brick, starting a minor avalanche. \"It always used to have an evil ring\u2014also awesome and faraway, like 'Mars,' or 'Jupiter.' But here it is, and it is real. Strange to say, I just realized I love it.\"\n\n\"Because it is broken,\" said Schild.\n\n\"I guess so. All the crap has been blasted away, leaving something honest, and I think what the doctor meant was that honesty really does win out in the end. That is horrible and at the same time funny. ... Funnier yet because I believe the doctor himself is a fake.\" By the poor, cloud-filtered light of an introvert moon he checked Schild's face. \"You see, I have been to that cellar before. The other time Bach told me a long story which turned out to be a lie.\"\n\n\"A lie?\"\n\n\"The whole cloth. Imagine him in the SS!\"\n\n\"I can't imagine anyone in the SS,\" Schild lied. \"Maybe that was a fake, too.\" He did not understand why he could not speak straight to Reinhart; the good intent was there.\n\n\"Would to God it had been,\" Reinhart answered fervently, and tripped himself up on a naked concrete-reinforcing rod, fell, kept talking: \"Like the murder of the Belgian babies in World War I\u2014give me a hand please?, I feel a hollow under here that I'll break into if I make a commotion myself. ... Thanks\u2014which was a propaganda lie. Dirty Nazis! They made it impossible to lie about the Germans. Thus Martin Luther and Frederick the Great and Johann Wolfgang von Goethe are swine, too, because they helped to make all this. N\u00fcrnberg, were you ever in N\u00fcrnberg? I used to think there was something fine there\u2014\" He crashed through the intervals of a grounded metal bedstead.\n\nSchild took a lower way, through a shallow trough which yielded underfoot as if he walked across a human body. \"When?\" he asked.\n\n\"Never,\" said Reinhart. \"I was never there, _naturally._ I saw a book once. Albrecht D\u00fcrer's house stands to this day, Albrecht D\u00fcrer, the old artist of the Middle Ages. He made one etching called 'Ritter, Tod, und Teufel.' When I first saw it I couldn't read German, I didn't know what that name meant, I knew only _Teufel,_ and he was easy to spot: a face like a wolf, with mad eyes and one crescent-shaped horn in the back of his head and two like a ram's curving out from under his ears. His ears were donkey's. On the other side of the picture is Death, on a crummy, melancholy old horse. He has a long white beard, a hole for a nose, and wears a crown full of snakes, holds an hourglass. The scene takes place in a gully full of junk, lizards, skulls, tree-roots, etc.; it looks something like Berlin today. A sneaky-looking hound runs along the bottom, and there is D\u00fcrer's trademark and the date on a little sort of tombstone.\"\n\nThey had reached the bottom of their own declivity, which egressed to nowhere, and attacked the next smoking slope, Reinhart continuing to walk point.\n\n\"But in the distance you can see the towers of a great castle. Death and the Devil may have entree everywhere, but they are not in that castle, which I believe must represent a heaven. And neither is the Knight, who I'm coming to in a minute. Because he would not be a knight unless he served his time in the gully of death and the devil. Well, the Knight\u2014there he is in the foreground, on his splendid charger walking stately through the crap, the Devil leering over the horse's rump, old Death wheezing at him in front, the dog _sehleichend_ along below, the castle far away\u2014they could do him in and nobody in those towers would know it until too late, but even if they did, what good would it be? What help can anybody else give you against Death and the Devil?\n\n\"The castle is not relevant, as the doctor would say.\" Reinhart passed through a doorway and was immediately again in the free air, for the wall stood alone in the world with no building as relative; Schild followed.\n\n\"Welcome to mine house,\" said Reinhart. \"I wish I knew where the hell we are, I think we're coming back to Bach's cellar.\"\n\nHe stopped abruptly and Schild bumped into him and excused himself and said: \"If you'd wait a minute I could show you.\" He knew the way and wondered why he did not seize the leadership from Reinhart.\n\nBut Reinhart pistoled a hand and shot at a great concrete box on the dim horizon, an entire basement blown intact from the earth. \"No, I see what I've been looking for. ... Neither are the Death and Devil relevant. The Knight rides through the gully as if he doesn't see them. Of course he does\u2014the style D\u00fcrer draws in, there's not room for the enormous horse let alone anything else; they are packed in that lousy gulch like a frankfurter in its skin. Therefore the Knight sees them\u2014but he walks on. And I tell you, they look pretty squalid. If you glance quickly at the picture you won't see anything but the Knight, with his long straight spear, a bit of fur towards the tip, the splendid armor with which he is, as they say, caparisoned, but most of all that wonderful tough face, sure of itself, looking not at the airy castle or horseshit Death or the mangy Devil, because they'll all three get him soon enough, but he doesn't care. He is complete in himself\u2014isn't that what integrity means?\u2014and he is proud of it, because he is smiling a little.\"\n\nReinhart reached the caisson, where he waited till Schild climbed the rise and stood puffing beside him.\n\n\"And he is not en route to do combat with an unarmed enemy. He is a man and needs no helpless victim to give him respect. When I think of him there, walking forever across the pages of a moldy old book\u2014and I guess not even there now, since my father burned it\u2014I could... I could smile, I suppose, because I do not feel sorry for him.\"\n\nSchild smiled wryly and said: \"You never saw the serf who had to help him into that heavy armor and take care of the splendid horse, or the bonded peasants who tilled his field, so that the knight could strut about as he pleased while the underlings did the work.\" Perversely he clung to his loyalties while still older ones besieged him: stifling summer on the ramparts above Manhattan, windows sealed and blinds lowered as antithermal charm, faint sounds of street serfs playing stickball, Sir Nathan riding the rug, charging through a bowdlerized Malory in which Launcelot and King Arthur's wife exchanged ethereal admiration. _For the French book saith that Sir Servause had never courage nor lust to do battle against no man, but if it were against giants, and against dragons, and wild beasts._\n\n\"No,\" said Reinhart. He tore off a chunk of loose mortar from the wholesale cellar\u2014astonishing that such strength was accompanied by any mind at all\u2014and pitched it like a baseball, although it must have weighed fifteen pounds, far across the rubble range and down night's black throat without a murmur.\n\n\"No,\" Reinhart repeated, \"you don't get the idea. There were no serfs or vassals in this picture. This Knight was real, but not real. How can I say it? I just thought, he was not necessarily even a German. He is just a drawing\u2014just art, is all\u2014a lie, if you like. He belongs as much to a serf as he would to a real knight. A picture belongs to anybody who looks at it. It can even be burned, and somebody will still have it in his mind. Besides, you admit anyway that Death and the Devil are free to all\u2014why not then the Knight?\"\n\nBecause Jews were never knights, even though they had lived in Germany since long before the Middle Ages; was it in Heine that one read of the ancient Jewish communities along the Rhine?, who said: Don't blame us for the killing of Christ, we were living here at that time! But riding the rug, working at the exalted old language to which he then had not yet realized he was historically a newcomer (but so, in his day, was Reinhart), neither did Sir Nathan admit his native disqualification for the quest of the Holy Grail. _Sir Launcelot let them say what they would, and straight he went into the castle, and tied his horse to a ring in the wall; and there he saw a fair green court, and thither he dressed himself, for there him thought was a fair place to fight in. So he looked about, and saw much people in doors and windows, that said, Fair knight thou art unhappy._\n\n\"But,\" Reinhart said unhappily, \"if you want to say they don't make them like that nowadays, I agree with you. That's progress for you: get rid of the whole works, serfs, peasants, castles\u2014and knights, not to mention _Tod_ and _Teufel._ Where do these kinds of Death and Devil fit in the doctor's story\u2014even if he is a fake?\"\n\n\"He was real, all right,\" Schild snapped. \"I'm not sure about you and me, but he was real.\"\n\n\"Nobody in that cellar ever shows you any evidence.\"\n\nSchild laughed in sharp anger and answered in his birthright idiom: \"So whadduh you, district attorney?\" It sounded authentic; he had not come so far; his temper softened. \"You just said it is impossible to lie about the Nazis\u2014\"\n\nReinhart had found a chairleg and now slowly, inexorably bruised it against the concrete wall, until its end was fibrous as a brush.\n\n\"The Germans, I said, but I am glad to hear _you_ think there is a difference.\"\n\nQuite right, the error was his, but why the special punishment? And why should Reinhart bring it, whom he trusted, to whom he was in a unique relationship of owing nothing and vice versa, his friend. ... The moon had eluded its cloud but was still niggardly, showing Reinhart as a large pale blob belonging to the powdered landscape. The gentile is everywhere at home. Reinhart leaned against the basement as if he owned it, waiting for the Jewish opinion.\n\n\"Why _me_?\" Schild shouted in fear and loathing.\n\nReinhart was hurt, but calm. \"Because you're the only other German-American I've got to talk to. We have a common interest in those potato pancakes we were fed as boys.\" His irony surprised him; he grinned and wrinkled his brow low, like an ape.\n\n\"For Christ's sake,\" said Schild, \"don't tell me you don't know I'm a Jew.\"\n\nHe had been wrong about Reinhart's face; its contempt was as acute as its good feeling had been blunt.\n\n\"All right,\" said Reinhart. \"You're better than I am, you know everything without having to try, and you can stick it up your ass.\"\n\nHe shuffled along the basement wall, kicking up brick dust, which filtered through the hairs of the inner nose smelling like cordite. He now looked rather more resigned than angry, and at the corner of the concrete he threw up his head, pointed, and called: \"The path is here!\"\n\nHe had known where it was all the while. Why had he led them to wander? He was sinister, but he was also good. He descended an excavation, his round head falling evenly from sight.\n\n\"Wait!\" Schild shouted, pelting after, through the crying, broken turf. When he reached the bombhole Reinhart's broad back was laboring across the other rim. \"I gave you an order!\" He suffered fear that the man would deny him again, this time in insubordination\u2014the first irregularity had been merely personal\u2014and he would be required to turn him in for arrest. 'You are always arresting someone,' St. George, whose Army it was and not Schild's, had complained.\n\nHe scrambled across the chasm as Reinhart, obeying, waited. He had trouble, too, at the rim, and not being as tall as Reinhart, could not have made it without help. Which he received, unrequested. Reinhart's hand was cold and dirty.\n\nReaching the upper level, he began to speak his amends, which, as always, altered during their travel from source to mouth. Hysteria was, finally, the only cause he had ever served, but at least he was loyal to that. He accepted his uniqueness, and remembered an old story told him by a fellow traveler undergoing the transition to simple liberal and eventually no doubt to worse\u2014the typical American politics of _pis aller_ \u2014and that was his respects to Reinhart.\n\n\"When Trotsky and Stalin first fell out, the Politburo met to resolve their differences. Since Stalin controlled a majority of its members, it soon decided in his favor and demanded that Trotsky recant. 'You are ordered,' the decision read, 'to stand up and say: \"Comrade Stalin, you are right. I am wrong. I apologize.\" 'Very well,' Trotsky answered, 'I accept the decision.' He stood up and said in a heavy Yiddish rhythm: 'You are _right_? I am _wrong?_ I _apologize_?\"\n\nReinhart grinned. \"Neither do I, sir. ... Since we are speaking freely, I can say I knew that whatever else might be said of you, you weren't chicken-shit. Jewish officers never are. They have too much pride to be. They are free.\"\n\n\"No,\" Schild answered quietly. \"If you believe that you believe in a lie and you make it too hard on the Jews.\"\n\n\"But I have seen it. I have three years' service\u2014I enlisted,\" said Reinhart in pride. \"If you don't mind my saying it, Jews are sometimes know-it-alls and their manners could stand improvement, but that doesn't have anything to do with decency and is anyway a proof of their freedom\u2014\" He checked on Schild's reaction with the defiant self-righteousness, nose slightly flared, of the man who by his general benevolence is sanctioned to be specifically offensive; he wished to hurt Schild, Schild could see, in the interests of some comprehensive good that would finally bankrupt him, Reinhart, but first he would take a small profit.\n\n\"\u2014and don't tell me that is anti-Semitism,\" said Reinhart, cowering, for all his size. \"I'm sick of being made to feel a swine because I'm of German descent. I'm sick of being in the privileged class that nothing ever happens to. I'm tired of being big and healthy, but I can't help it, I was born that way. If you would be a prisoner in any concentration camp ever made, I would be a guard. Now, you know everything\u2014but do you know that? How that makes a person feel? Do you know what it is to be in debt to everybody? Not you, you are always right.\"\n\n\"I?\" said Schild. He sat down on a ridge of waste. The sudden armistice within had relaxed his muscles. He repeated the grammatical fiction almost genially: _I,_ the pronoun of rectitude: \"I am a murderer.\"\n\nReinhart took seat beside him, and with the added weight the ridge of brick-halves squashed out about their ankles.\n\n\"Ah,\" said Reinhart, \"you should have a pair of these boots. Now your shoes will be filled with that junk.\"\n\n\"That's all right,\" Schild said, although he too, with a sense of expansiveness, granted its tragedy; he, the rude _Besserwisser,_ accepted this Middle Western, gentile horror of discomfort and unrespectability, opened his shoes and dumped them clean. His right sock had a large hole revealing his largest toe.\n\n\"Why don't you turn that in to salvage?\" paternally asked Reinhart, pointing rudely.\n\n\"No salvage for officers,\" he answered, self-consciously pitiful. \"We have to buy our own.\"\n\n\"I keep forgetting.\" Reinhart searched his pockets. \"You got a butt?\u2014wait, by God, here's that little pack of Fleetwoods you yourself gave me last month. Well, they're as good now as ever. They are made stale.\"\n\nSchild took one and found he was quite right; Reinhart knew everything.\n\n\"Now don't you worry,\" Reinhart said, \"all that was just talk. Berlin does something to everybody; makes one want to accuse himself.\" He blew a smoke-mustache from his nostrils. \"In a war there's no such thing as murder. It's kill or be killed. I don't blame the regular German army, for example, for fighting against the Allies\u2014even if their cause was wrong; that's a very different deal from the particular Nazi outrages. To be precise\u2014when I said the doctor might be a fake, I meant in the unimportant things, such as whether or not he was in those camps, whether or not he was a Communist or a twin of Lori, and so on. I never for a minute doubted he was honest in the fundamental human things\u2014you see he could be an ex-Nazi and still be straight on those. Did you ever think of Hitler as just a man eating jelly omelets, needing a haircut, clearing his throat, getting out of bed in the morning and yawning? Did you ever think of someone saying to him at such a time: 'Come on, Adolf, I see a bit of dandruff on your collar and I heard you belch, and I know you have your troubles. Come on now, you can't crap me, you're a man like any other.'\n\n\"But I started by wanting to be precise. Precisely, I can conceive of an honorable German hating Hitler yet fighting for his country in the Wehrmacht. I can also imagine a German Jew who in spite of what was done to him thinks of Germany as his own country, for he is a _German,_ isn't he? And if he has permitted the Nazis to convince him he isn't, he has let them win\u2014in a way they never did with all their bullies and gas ovens. _They_ are the non-Aryans, _they_ are the degenerate race who rotted and betrayed a great people, not the Jews. I can conceive of such a man, I don't mean I expect any particular individual to be one, you can't blame a man for _not_ being a hero.\"\n\nDespite his fervor Reinhart spoke slowly, and Schild for once was not impatient. Having confessed, he had awaited the question of a pure-hearted fool, which, the old legends promised, would heal his wound. Instead he found himself cured of Germanic whimsy. He, and not Reinhart, was the romantic; fools there are in abundance, but not one is innocent.\n\n\"Reinhart,\" he said evenly, \"now listen to me. I forgive you. Do you understand? I forgive you.\"\n\n\"That is not what I want\u2014\"\n\n\"But that is what you get from me, nevertheless. And if you won't take it\"\u2014he grinned and shot his cigarette-end in a high rocket which no sooner exploded on the wasteland than two shadowy children filtered from behind a rubble hillock and claimed it as prize, quarreling on who should pinch its ember, whose ragged smock-pocket should tote it to their used-tobacco Shylock\u2014\"you can stick it up your ass.\"\n\n\"It doesn't do me any good,\" said Reinhart. \"Now them\u2014forgive those kids. They really had nothing to do with it, unless you believe with Hitler that a whole people can be degenerate.\"\n\nBut he would not let a gentile be sanctimonious with him. On the other hand, he again cleaned his shoes for Reinhart's sake and rose, saying: \"Do you know we have to walk back to Zehlendorf?\"\n\n\"Unless we can hitch a ride.\"\n\n\"This late?\" asked Schild, looking at his bare wrist. In what bleaker field was his watch ticking now? To Reinhart, he knew all the answers, yet why was his every emotion another question? \"Do those children stay awake all night on the chance an American will come by and throw away a butt?\"\n\n\"What else have they got to do?\" Reinhart asked toughly. He field-stripped his own cigarette and hooved it into the ground.\n\n\"What do _you_ have against them?\"\n\n\"A private grudge,\" said Reinhart, \"that's my own damned business. Well, if we have to walk, that makes it easy, no choice.\" He rebloused his pants, tightened his belt, adjusted the jacket, made his cap smart, and, ready for any D-Day, motioned Schild to take the lead.\n\nOnce they were out of the rubblefield and in the open gray streets gulching the ruins, Schild fell back beside him in an aesthetic revulsion against captaining one man all the way to the Grunewald Forest. With no one before him to control the pace, Reinhart increased his stride, measuring off a yard per step. Schild fell behind. On the bicycle path of the Hohenzollerndamm, in Wilmersdorf\u2014they were beyond the congestion and, hence, the worst damage, on a wide thoroughfare becoming suburban, with streetcar tracks, bounded by greenery and particular rather than mass ruins\u2014Schild leaned against a poster-pillar and took air.\n\nMarching with loud slaps of his rubber soles, head fixed as if he were in ranks, Reinhart went on unheeding. Schild watched him for a hundred yards in the light of Berlin's dawn, which came early in the small hours\u2014therefore it was later than he had supposed. Reinhart would soon look back. On a childish impulse Schild stepped behind the pillar and waited. The footsteps rapidly tramped beyond earshot.\n\nHe found that inadvertently he had kept Reinhart's veteran Fleetwoods. Going in through the cellophane and limp pasteboard, his fingers made inordinate noise, and had he still been a nervous man he might have mistaken the sounds for those of someone creeping out of ambush behind him. He fired his cigarette and took a lungful of corrosive smoke, toying with a paradox: the one man he knew who was the ambush type had least need of concealment. In proof of this he saw Schatzi standing nearby on the sidewalk, hiding in the open air and light, a concrete apparition.\n\n\"I have followed you like a sickly conscience,\" said his courier, who wore a motley of olive-drab clothing.\n\n\"You are out of uniform,\" Schild answered, laughing softly. \"If the MPs come along I can have you arrested.\"\n\n\"I'm doing you no harm,\" Schatzi said in some worry. Then he smirked weakly. \"Come Fritz, you make the joke with your old comrade with whom you have already deceived, so that I am in trouble across the boundary.\" He pointed over his shoulder. \"The Russian is gone, _ja_?\"\n\nSchild asked: \"Do I throw off an odor, that you can follow me with your nose?\"\n\n\"Perhaps you will not believe, what can one do?\" He shrugged. \"Having some business on the Tauentzien Strasse\u2014very well, being exact, in the basement of the KaDeWe\u2014ah Fritz, what a pity that excellent compartment store must be bombed!\" Tears coursed the runoffs on either side of his crag-nose. \"Ah, Fritz, I must confess I have had a drop\u2014I am in my glasses, as it were.\" He wove across the bicycle path and rested against the pillar. \" _Verzeihung, Herr Litfass! ..._ Why should I care about this ugly place? What have they done to my N\u00fcrnberg? Because I am not _Saupreuss,_ a Prussian pig, _beileibe nicht._ Pure Bavarian, _verfluchte Scheiss_!\"\n\nHis American overseas cap was pulled low and round as a sailor's. His nose began to run; he cleared it onto the dark green of his new field jacket in two short swipes marking the chevrons of snot-corporal. \"From the Ranke Platz I saw you creeping over the ruins with that oaf and I thought, this Fritz has lost his Russian fairy-boy and got him a nice young American in its stead. You have been foolish, Fritz, and they know about it\u2014they know everything\u2014you don't deal now wiss stupid Nazis.\"\n\nHe reached for Schild's sleeve and, missing, fell to the ground on his hands, yet caught himself arched, and backed spiderlike till his rump was against the pillar. From the point of contact he rose inch by inch along his spine, cleaving to the shaft.\n\nErect, coughing vacantly, he whined: \"They killed my dear dog, Fritz, with a machine pistol shot him through the head. That is their kind of people! I loved that creature, to which they should not have done this harm. I gave to him food from out my own mouth.\"\n\nSchild began to walk away, in peace.\n\n\"Come again here and listen, you bit of turd!\" Schatzi screamed. \"In Auschwitz I liquidated better men as you by the thousands. _Du kannst mich im Arsche Lecken._ \"\n\n\"No,\" said Schild, calmly smoking, \"no, you did not. You only buried them. You were forced to, you yourself were a prisoner.\"\n\n\" 'Forced to,' \" Schatzi repeated drunkenly. \"I carried a club, Fritz, but one must be careful how hard one beats them, or the SS will rage with jealousy and take the post away. Then too, these thin bones were easily cracked, which meant the job would be nonsense because they must remain strong enough to dig\u2014I always knew you were a double agent, didn't I always say so? You yourself are Intelligence! Pity me, Fritz, they have murdered my dog. Dirty Russians! It unsecured itself from the chain and came by my heels already, unknown to me until I was stopped by this sentry at Sergeyev's building. The dear dog has been thinking, 'Ah my master is attacked!' He sprang at the soldier and the Russian shot him.\"\n\n\"I am sorry. Really I am,\" said Schild.\n\n\"Brown on the outside, red on the inside like a beefsteak, we were in the early _Sturmabteilung._ We had many similarities to the Communists, Fritz. Idealism, we were idealists, and we died for it\u2014like the Jews. We were the first Jews. Thus I can understand you. I too hate this filthy Germany.\" He wiped his nose again, promoting himself to snot-sergeant. \"You are a soldier, but you were shrewd enough to get for yourself a safe position behind the lines of battles. Why should you not, if you are clever enough? I do not criticize. In the Great War I was not a shrewd fellow like you, but a simple foot-soldier. Just see this.\" He raised his trouser cuff and lowered the stocking. \"Verdun, February 1916\"\u2014a blind, purple hole in his calf. He opened the jacket sleeve and that of the wool shirt beneath it, and drew back the arm of heavy, dirty underwear: \"Verdun, September 1916\" \u2014a masticated chicken leg was his left forearm. \"August 1918, mustard gas in the lungs, the forest of the Argonne. As you known, I still today cough. While I collected these thanks, I must not tell you what was occurring behind our back in Germany, you would believe me insulting to you and your peo\u2014no, one does not say that, but there existed fat swine who profited by our blood. And when after the war we went to settle wiss them, these Nazis killed us instead. _Berufsverbrecher,_ professional criminal, I am called in the camp\u2014\" The liquid discharges of the eyes and nose left prison-bar traces on his dusty face.\n\nSchatzi continued the catalogue of wrongs done him, and Schild thought, they are as real as anyone's, as Reinhart's, as his own: _who among us is not a Jew?_\n\nHe said: \"All right. Now you'd better get yourself together. Perhaps you can find another dog\u2014\"\n\n\"The only thing in years I wished to love!\" Schatzi threw his head back against the pillar and unabashedly wailed.\n\nTwice tonight Schild had been chosen to hear a candid heart; it was the old choice and fitted his old gift. For the first time he knew it as a tribute to what he was, or what they supposed him to be, and perhaps after all these were one and the same. If Hitler had not died in the Chancellery garden he would one day seek Schild out and tell him, weeping, of being twice denied a career in art by the academic examiners; rejected by his sweetheart niece who then blew out her life; a bum in a Vienna flophouse, befriended by a Jewish old-clothes dealer; gassed in the war; and, finally, of his last defeat as the Red Army swarmed over the Spree. And Schild would say, All right, Hitler, we shall weep together.\n\nTo Schatzi he repeated: \"Collect yourself, man. Sergeyev will have your head if you are reported drunk.\"\n\nImmediately Schatzi dried from within and became one hard instrument of suspicion. \"Whom did you say?\"\n\n\"Sergeyev. You just mentioned going to Sergeyev's building\u2014\"\n\n\"Ah, but did I mention why?\" He fell into his usual semi-crouch, which put his head four feet off ground, and Schild, who had always believed this the stance of attack, realized at last its purpose was rather to make Schatzi a small target.\n\nWith supreme distaste, but nothing else would serve, he grasped Schatzi's jacket and pulled him upright and vulnerable. \"The next time you see him, report that Fritz is finished.\" He watched the red respect flood Schatzi's eyes. \"Do you understand?\" Schatzi trembled in admiring assent. \"If you do not tell him, I shall let Corporal Reinhart beat you to death.\"\n\n\"This great beast?\" whispered Schatzi.\n\n\"All I have to do is nod to him.\" Schild released his grip, requiring all his muscles to hold his face stern. \"He also knows your story and, because he is a gentile, holds a grudge. Do you remember a Dr. Otto Knebel in Auschwitz?\"\n\n\"They all looked alike to me,\" Schatzi answered without thought, then taking a sober one: \"Perhaps he was in Monowitz, another section of the camp from mine.\"\n\n\"He remembers you.\"\n\n\"Maybe he lies. The SS marched them out to Germany when the Russians approached, and who would survive such an ordeal? Lucky, I escaped. Luckier yet, you were never there in any case.\" He was returning to his old self, with both relief and disappointment at Schild's apparent decision not to molest him further. \"But accept from me this warning, Fritz. Serg\u2014 _they_ do not recognize luck. And they have also their camps. You did better to transfer from Berlin, where they can easily get at you, before playing the renegahd.\"\n\nWithout feeling, Schild said: \"I'm no renegade. You can also tell him I won't talk.\"\n\n\"Fritz,\" said Schatzi. He came close in a reek of liquor, eyes drifting: \"I have some regret for mistakes in my life. Wiss my family was not the love you Jews have for each another. You can not understand how my father was beating me always. I have had another dog at ten years of age. My father struck that dog to death when it slipped its chain and entered the house and fed upon his slippers.\"\n\nBehind him Schild heard the noise of a vehicle in Hohenzollern Platz.\n\n\"Was that in N\u00fcrnberg?\" he asked.\n\nSchatzi was caught up short, made his eyes keen, and answered: \"Precisely. Do you know the city? In the Altstadt, below the cahstle wall.\"\n\n\"Near the _D\u00fcrerhaus_?\"\n\n\"In fact, overlooking,\" Schatzi answered. \"I have heard D\u00fcrer's house is _kaputt_ from the bombing. Is it so?\"\n\n\"I don't know. I have never been there.\"\n\n_\"D\u00fcrerhaus, Scheisshaus,_ what should we care, eh Fritz?\"\n\nSchild backstepped from his camaraderie, turned and saw the jeep bumping over the streetcar tracks. It was now as light as an overcast afternoon. The tall MP beside the driver saluted, and Schild knew a moment's qualm. But it was not he who arrested Lichenko.\n\n\"You want a ride, Lieutenant? Is that crumb bothering you? Hey Hitler, spricken see English?\" He smacked his billyclub into his palm. \"C'mere. I'll give you some democracy right in the nuts.\"\n\n\"He's with me,\" said Schild, officerly factual, showing his ID card. \"If you drive on down this street you will see a corporal. I want you to give him a lift if you are going that way. He too was with me on official business, so don't bother him about a pass.\"\n\nThe MP obsequiously lowered his club. Likely had he seen Schatzi alone, he would not even have made the threat; he wished merely to be appreciated.\n\n\"I hope,\" Schild continued nevertheless, \"you don't speak in that loose fashion to every German you meet. You might run into an anti-Nazi.\"\n\nAgain the MP assented, careful in his policeman way to give excuse without a show of confusion. \"I didn't know there were any.\"\n\n\"Neither did I,\" said Schild. \"But we can't let that make a difference.\"\n\nThe jeep snorted down the vacant Hohenzollerndamm.\n\n\"You Amis are strangest of the strange,\" said Schatzi. By means of the MP's menace he had regained full dignity. \"This I first believed was a weakness of the mind, and next for me it was a sinister thing. Finally, I see, and it is harmless: you really believe that you are the master race.\" This time he wiped his nose on a handkerchief. \"The Germans, you know, never did, and least of all when this crazy sissy Adolf, and this cripple Goebbels, and that fat Zeppelin with the large mouth Goering, told them they were. A German knows he is not anysing. Instead for a time he thinks that _they_ are, Adolf _und Gesellschaft._ Never himself. A French waiter makes a German feel like an ox. An Englishman makes him feel ill dressed. His great philosophers either talk so he cannot understand them, like Hegel, or tell him what a disgusting lout he is, like Nietzsche. And then there are the Jews, always so clever and so successful. See the magnificent land where they run things, America! ...Fritz, I am speaking earlier of my mistakes. I work for the Communists because they force me. When I am a young man I spilled much of their blood, but now I am old and weak. Unless I serve they will denounce me to the American police\u2014this thing with R\u00f6hm and the early SA. I will be treated like a Nazi, _ja_? At least this way I am free. But can you get me to America, I shall not inform _them_ you will go. Is it an honest arrangement?\"\n\n\"Perfectly,\" said Schild, \"except that I don't get a profit. Now I'll make you a deal. First, you report to Sergeyev that Fritz is done. Second, you make certain I never see you again. On my part I'll keep Corporal Reinhart from killing you. Now\"\u2014he seized Schatzi and turned him around\u2014\"that way is east. Go already, in peace and freedom.\"\n\nSchatzi went, looking back from time to time with the reproach and puzzlement of an exiled pet, but going. Schild watched him as far as the Platz and, reminded by his animal progress, pitied him again for the loss of his dog.\n\nReinhart was at Roseneck, Rose Corner, when he heard the jeep engine and, because his permanent pass was good not later than 0100 hours, he crept into an empty beer garden and hid behind a tree. The car made the turn and vanished into Rheinbabenallee. Emerging, he saw the darkness of the Grunewald woods a couple more football-field lengths down the street. He had lead in his ass and his feet were aflame. Distances elsewhere standard, in Berlin were triple; and he had taken no real exercise in years. As well he was a chair-medic, the Rangers and paratroopers were lucky not to have him. He regretted having pulled on ahead of Schild, for not only was he tired, he was lonely. But the Jews and their mad pride, he would never learn to cope with it.\n\nCrossing Kronprinzenallee at last, he saw where someone had chopped down a tree in the Grunewald. He walked in and sat on the fallen trunk. He searched in vain for a cigarette, but the Pall Malls were at Bach's, the Fleetwoods with Schild. No matter, his lungs were weak enough. He struck himself in the chest and coughed histrionically, feeling a certain softness in the pectorals. Weight lifters out of training develop breasts like women, look worse than the ninety-pound weaklings they originally were, he remembered. Undoubtedly the same thing happens to the muscles of the trained mind: in time intellectuals' heads grow flabby. The morality of Puritans becomes mushy. Life mocks those who try.\n\nNow that he had found Kronprinzenallee he knew the way home: straight down it about fifty miles to Argentinische Allee, around that crescent about halfway, another twenty, until you reached a patch of trees and sandbags and excavations, traversing which you came finally upon the farthermost limbs of the detachment headquarters building. Trudchen would be long gone to the bosom of her family, obeying her parents' ukase against staying in bed with a man after ten o'clock at night. He had his joke; actually, she told them she had to work overtime at the office. They were, he supposed, a typical German familial unit, of which he should make a sociological investigation\u2014except that he knew all about normal people, who are everywhere the same.\n\nRest in his condition only made it more difficult to return to movement. He checked the blouse of his trousers, that precise indicator of a soldier's smartness\u2014Schild, for example, had he worn boots would have stuffed the cuffs crudely into their mouths and buckled the straps. The contraceptive around his right boot proved to be frayed. He took a new one from his watch pocket, peeled it, tied it in place. The Kraut who found the discarded rubber would never figure it out.\n\nReady to move, he saw in Hohenzollern's distance the insignificant form of Lieutenant Schild, walking steadily, nothing ambitious but with a certainty in his carriage that he would get there, wherever it was. A tough little guy, in his own way. If I could be like Schild, Reinhart believed, I would not complain. So he waited for him.\n\nArriving, Schild said: \"The MPs didn't find you?\"\n\nReinhart boasted: \"I was too quick for them.\"\n\n\"They were going to give you a ride.\"\n\n\"I can make it all right. Why didn't you take one?\"\n\n\"I have an aversion to the police.\"\n\n\"You and me both.\"\n\n\"I wanted to ask you,\" said Schild, \"what are you going to do about your friend Nurse Leary?\"\n\n\"That's a difficulty.\" He was getting nervous again at having to walk so slowly. \"Let her go to hell, I guess. Except that I gave my word.\"\n\n\"What do you owe huh?\" Schild asked, New Yorkly.\n\n\"Nothing whatsoever. That is exactly why I cannot go back on my word.\"\n\n\"I'd think it would be the other way around.\"\n\nReinhart felt the newly arranged cuff working loose with his stride. Screw it. He smiled down at Schild and said: \"What you mean is you thought _I_ would think so. But I don't. It would be letting the other fellow decide what you yourself should feel. I never have been able to stand that. That's what I like about the Army, where you are told what to do and eat and wear, but never what to think and feel. Everybody but me seems to hate it, on the grounds that it takes away their 'freedom.' When did you last see a free man in civilian life?\" He pushed back his sweat-heavy cap and snorted. \"Look at me. I alone am right. Ha! Join the Reinhart Party!\"\n\nPaired, the travel had improved; already they had crossed the sandy P\u00fccklerstrasse and the apartment houses of Argentinische Allee lay in the field of vision.\n\nSchild said: \"The knights of Cornwall are no men of worship, as other knights are. And because of that, they hate all men of worship.\"\n\n\"What is that?\"\n\n\"From King Arthur. You put me onto him with your 'Ritter, Tod, und Teufel.' \" Schild slowed. \"Here, let's have the last two of your Fleetwoods. ... You think a lot, don't you?\"\n\n\"Too much,\" Reinhart complained with a mouthful of smoke. \"Indeed I am a drone. However, I don't usually do the talking. By nature I am a listener.\"\n\n\"Do you ever get any snatch?\"\n\nAstounded, Reinhart asked him to come again. The same, and this intellectual was not even grinning.\n\n\"Since you ask, Lieutenant, only that little Trudchen\u2014just as you predicted. She was no virgin. She is depraved, in fact, was already, which makes me feel less badly about it.\"\n\n\"Well,\" said Schild, \"if she's old enough to have it, she's old enough to use it.\"\n\n\"War is terrible when it corrupts a young kid like that,\" Reinhart said piously.\n\nCigarette in the side of his mouth, head cocked so the smoke could not catch in his glasses, Schild asked: \"Are you serious?\"\n\n\"No.\"\n\nThey joined in pragmatic, male laughter, the kind that would have stimulated the heroines of those movies Very loved to heat up their castrating irons.\n\nThen it was only fair that Reinhart interrogate Schild. Who confessed to having been a teacher in civil life, specializing in English, in a private school that Reinhart suspected was \"progressive,\" where the students did whatever they wished and called the instructors by their first names. Was it? In part, Schild admitted, laughing more than ever.\n\n\"Frankly, that's just the kind of thing I would have loathed as a kid. I always got a lot of satisfaction from believing I was more progressive than the school. Your place would take that away. Don't you think it is better for superiors not to understand the people under them too well? I like a rather stern authority that I can hate and feel morally better than.\"\n\n\"And overthrow?\" Because he had done less talking, Schild's cigarette was down to a nub carrying a long, quivering ash. A sudden burst of fresh morning wind snowed it across his blouse.\n\n\"Sure, sure, but there will always be something else.\" Throat parched, Reinhart threw away his half-smoked smoke and said: \"I accept life. Some things in it are by nature hateful.\"\n\n\"You are the most extreme reactionary I have ever met,\" said Schild, but pleasantly; almost, one could imagine, with approval.\n\nAs they turned into Argentinische they met a light, drifting film of rain, which was refreshing to the warm cheeks but also a douche on Reinhart's spirit. He had run off at the mouth, revealed the location of his defenses and their strength, while wise Schild had really said nothing. Worse, he could not have faith in his own honesty: in action he had always proved to be the least independent of men, not reactionary at all. This evening, for example, he had needed Schild's open ear.\n\nThey cut through the little woods and came out by quiet detachment headquarters, passed the sleeping hospital, and in the now generous rain took each other's leave at the corner of officers' row. Schild dripped water from the end of his obvious nose and smelled of wet wool, even as Reinhart did himself, and had trouble with his spectacles.\n\n\"I'm supposed to wear glasses myself,\" Reinhart said. \"For reading. I am farsighted. But I broke them in England. ... Do you feel all right now, Lieutenant? Frankly, I thought you acted a bit odd early this evening.\"\n\nSchild removed his cap and wrung it like a sponge, put it back all wrinkly. Then he saw Reinhart's frown, took it off again and smoothed it across his knee. \"Okay now? Reinhart, you are a fop. Why yes, I feel good. But I thought you were interested only in yourself.\"\n\n\"Never did I say that,\" Reinhart answered dolefully. \"You have any idea of how late it is? We must have walked ten or fifteen miles.\"\n\n\"Three-thirty perhaps, four, who knows? If you get in trouble at your outfit I'll fix it.\" He returned Reinhart's salute and said: \"I never have anybody to talk to, either. Thank you for letting me come along.\"\n\nReinhart watched him all the way to the gate of his billet, thinking, I have done you a favor? His own waterladen cap weighed on his head like a sandbag; he removed and wrung it \u00e0 la Schild. When he looked again through the rain his friend was stepping safely over the threshold.\n\nReinhart approached the rear of his apartment building on the alley that was but an unpaved continuation of the officers' street. He stifled an impulse to climb up and enter his flat through the balcony, which might have incited the half-awakened Marsala to witless mayhem. Similarly, before going around front he took a leak in the bushes lest toilet-flushing would wake his buddy. While in midstream he heard a door down the street and there, inexplicably, came out Schild again, walking to the corner at which they had parted, and then out of sight up Possweg. Cursing the capacity of his bladder, he at length finished. At the corner he saw Schild enter the dark grove where the mess tents were pitched. It's too early to get coffee! The guard there will shoot him! were among his angry self-expostulations. Reaching there himself, he saw the tents were farther to the left than he had remembered; besides, Schild was fifty yards into the trees and still going.\n\nLike an Iroquois, Reinhart crept silently from pine to pine until there were few left and a plain like a soccer field showed light before him. At the edge of the grove Schild stood with two civilian persons, one of thickset middle height and the other a great lump of fellow larger than Reinhart by three inches in every dimension and, by the set of his massive shoulders, no Bach. Drawn up on the soccer field beyond, a black European sedan.\n\nPerhaps because the wind was wrong\u2014at that moment he realized the rain had stopped and the wind risen, cold against damp clothing\u2014he heard nothing of their speech, knew only that they looked at Schild and Schild at the ground. But the evil voice sounded within him: _The black market, how like a Jew._ As if upon that signal, the two men seized his friend, each on an arm, and dragged him towards the car.\n\nFear's fat serpent squirmed down Reinhart's throat, circled the belly, and undulated through his intestines. Even the smaller guy looked as if he were built of bricks\u2014the larger one was a monster; he could not have felled him with a baseball bat. They wore cloth caps and neckerchiefs and dark clothing, were some European kind of thugs, ranged against a little Jew and an over-sized boy. Oh, unfair! he whimpered and had every intent to hide, but was too limp to stand still, too weak to walk away, so no choice was left for a coward but to run towards them.\n\nHe squished over the wet sand and was nowhere near when they heard him and turned. The large one released Schild and stalked forward.\n\n\"Go away, boy,\" he growled in German. \"This is no affair of yours.\"\n\nReinhart slowed but kept coming, still too scared to stop.\n\n\"He means no harm,\" said Schild. \"Let me talk to him.\"\n\nBrick-built, maintaining his hold, answered: \"All right, but not in English.\" And told Monster to stand aside.\n\n\"This is private business,\" Schild said harshly. \"Intelligence work. I'll thank you to keep your nose out of it.\"\n\nReinhart stopped three feet from the giant. It was so strange to be addressed in German by his friend. Groping for vocabulary, he said: \"What kind of Intelligence needs the capture of an American?\"\n\n\"Damn you,\" Schild yelled furiously, shaking off Bricks's arm. \"Don't come here with your na\u00efve bungling. I give you a direct order to leave this minute if you value your stripes.\"\n\nPlanning something, an old trick of movie combat, Reinhart trembled in anticipation\u2014for it never could be worked in real life, on this damp plain, in wet clothing, by a coward who was sure he had misjudged appearances. With relief he heard the threat to his rank. A man in authentic danger would hardly be so precise.\n\n\"All right.\" Shamefaced, he added: \"I forgot to tell you earlier: I have never understood the Jews, but I'm not proud of it.\"\n\nSchild answered: \"Neither have I, neither am I.\" Yet his head rose in pride.\n\nMonster mockingly repeated: _\"Die Juden habe ich nie verstanden\u2014\"_\n\nKnowing his fist would shatter, and caring desperately about it\u2014he hated to be hurt\u2014certain that as usual he was wrong and lost and impotent, he released what little reason he had accumulated in twenty-one years, wheeled badly, plunged slippingly, and weakly struck into the giant's armored belly. Wondrously he felt his hand prevail as if it had punched a cushion. The man's deep guts wormed against his knuckles. Monster buckled, retching. Reinhart kicked his face.\n\nHe turned towards Bricks, just in time to save Schild's blood, for the man exposed a paratrooper's gravity knife, dropped the blade, and waded in low.\n\nHe screamed: \"Get away you fool!\" Which brave little Schild ignored, waiting defenseless, calmly, Jewlike, for his fate.\n\nReinhart sprang, Bricks did quick footwork, Reinhart fell flat in muck and looking up saw Schild professionally elude the blade, simultaneously knee Brick's groin and chop his neck. Wilting, Bricks cried in pain, to which Schild's cruel answer was a hacking at his forearm, precision hands above and below, that snapped the bone.\n\nBut Monster, face of gore, had meanwhile lumbered over. With his hobnailed sole he opened Reinhart's cheek as you would boot a melon. On all fours, Reinhart took another, ill aimed; still, his cropped skull was grooved from fore to aft and red fluid flowed out and blinded him. A third and he would be dead. It was all very real.\n\nUnseeing, he crabbed as Monster swung again; he tumbled over, cleared his eyes. His life's dear blood left gout on gout; his bare cheekbone sorely caught the wind; but he got up. He stalked Monster, who great as he was now retreated; who could not have been an ex-SS man, for they were not craven. He caught him, took the battering of the leaden fists, bore in inexorably, embraced him in murderous love. Monster's hands belabored his kidneys, tried for his neck, but he was now in too close; therefore they tightened around his own small-of-back and sought to break it.\n\nHis head within the hollow of Monster's neck, he bit for the jugular. The skin was tough as chain mail, and besides he could not close his jaws, having an obsession this would push his senses through that cranial wound which Monster was opening further with each chin-stab. Then did he turn his face sidewise, upon the good cheek, and join his hands\u2014he could just barely, around that iron barrel\u2014and compressed, and it was hopeless.\n\nFrom his good eye he saw Schild leave the fallen, whining Bricks and come to punish Monster's back, futilely, without a weapon. The rabbit punches which felled Bricks bounced harmlessly from Monster's invulnerable nape. Slowly Reinhart's spine began to crack. But then within the clasp he believed so weak, so did Monster's ribs, fibrously, like celery, yet not in sufficient time. For there was Bricks rising from the ground, crippled, twisted, knife in a left hand sufficient for the job on Schild's unguarded back.\n\n\"Lieutenant... behind you,\" he panted, his voice loose air and too little of that. At the cost of his life he loosed the hold on Monster that curtailed his own breath, and Monster exploited it. Off his axis now he shouted: \"Schild for the love of Christ\u2014\" Still was Schild deaf, the bastard; he would give his life for Reinhart's which was already gone. A last hope was to call the worst he could; to stir his friend, if only by hate, to preserve his own hide. And surely he hated him enough then to frame the cry, as one can only hate him who makes you beneficiary of his total sacrifice. _Jew, I want them to kill you._\n\nHe never said it aloud; too authentically had it sounded through the chambers of his heart. Rather have him dead than hear that knell.\n\nSo, looking into his eyes, he saw Schild get it once, twice, thrice, in terrible thumps up to the knife's haft, and squint in agony and sink. His thick glasses slid down his thin nose. He fell behind Monster, sparing Reinhart further sight, but his feet were in view and writhing.\n\nNow came Reinhart's turn. Monster was killing him as it was, but Brick's dagger were quicker. Yearning for peace he awaited its first nick; getting it, heard a queer noise which he supposed was his heart ticking out. Monster loosed him slightly, turned head to look at Bricks in the sedan, stripping gears, driving off.\n\nMonster roared, and was cut short by the vise of Reinhart's arms. He pleaded, shedding tears and blood from his raw face, and Reinhart was moved but could not oppose his own awful will. Not even when he heard the tearing of the vertebral column beneath his wrists could he free poor Monster, who thus died in his embrace. Life all gone, he let him fall. He knelt by Schild and searched for a pulse. He found none. He retrieved the spectacles, which were unbroken, and mounted them on his friend: he recognized him again.\n\nHe stood up, victor, and surveyed the field. Then jealous Death, who wins all battles, wound him in its dark sheet.\n\n# _CHAPTER 21_\n\nEACH NIGHT AFTER HIS error with the agent Fritz, Major Sergeyev went to bed in his street clothes. Nail file, lighter, toothpicks, etc., remained in the pockets. Checking their location was his first concern on arising, unarrested, in the morning. On the nearby camp table, a foxed and worn edition of _The Foundations of Leninism,_ by J. V. Stalin, lay ready to be seized as one went to answer the knock upon the door. Beside it, a toothbrush, a bit of salt twisted in a paper, a safety razor, a sliver of soap, and a hand towel, in a small cloth bag.\n\nSeven mornings he arose in a Soviet officers' billet in Lichtenberg, ate breakfast at a mess in the basement of a commandeered factory, was driven to his office in a confiscated German Opel sedan.\n\nEmerging from the messhall on the eighth morning, he saw, ten yards away, that his driver not only had a different profile from his usual but wore a blue cap. The man got out and politely opened the rear door of the automobile. His trousers were piped in red and blue. Within were two more bluecaps. Silently they cleared a space for Sergeyev to sit between them. He said: \" _Spasibo,_ thanks.\"\n\nThey drove for a time on the wide thoroughfare which when he had first come to Berlin as an agent in the Thirties was called Frankfurter Allee, but now he understood was to be renamed for Stalin. It had been badly bombed. He tried to go to sleep, but whenever the bluecaps saw his eyelids droop they jolted him with their elbows, in silence and without malice. Thereafter he slept with his eyes open, a technique he had developed while interrogating Social Fascists, Trotskyists, members of the POUM, and other mad-dog wreckers and counterrevolutionary jackals in Spain.\n\nHe awoke as the car stopped before a prison in Pankow. The bluecaps accompanied him to the gate house, where they signed him over and he was searched. Two prison guards, also in blue caps, were his companions on a walk of moderate length inside the building, down a damp corridor, into a bare room with a boarded window. Before they gave him the order he had begun to disrobe. Naked, he pressed himself against the concrete wall. The guards examined the seams of his garments, looked between his toes, searched the hair of armpits and pubis, peered into his mouth, probed his anus. They confiscated the pocket articles.\n\nHe dressed. They conducted him to a small cell four floors above the ground. Its window was boarded; in its iron door, a spyhole and a letter-sized slot. After some time a pan of gruel was passed through the latter. He wished he had his packet of salt. To keep fit he strolled occasionally from the door to the slop bucket at the back wall of the cell, three steps, then from his bunk to the other wall, one step. He wished he had his _Foundations of Leninism._ The ceiling bulb burned all night.\n\nNext day he got a boiled potato, and at another meal found a bubble of fat in his hot-water soup. He wished he had his manual for espionage agents, but it was at the office. The toilet articles he did not yearn for, not having been permitted to wash.\n\nOn what he estimated, by the number of times he had been fed, seven, to be the fourth day\u2014not having seen the sky since he entered, he did not try to fix the hour\u2014the guards took him to a room containing a desk, one empty chair and one filled one. In the latter, behind the desk, sat a uniformed man, lean and elegant, drinking from a china cup. He wore long hair, graying at the temples. He motioned Sergeyev to sit.\n\nHaving swallowed, he said pleasantly: \"My name is Chepurnik. Of course you can see my rank.\"\n\n\"Yes, Comrade Major.\"\n\n\"You yourself were once a major, were you not? Of course you were!\" He poured himself another cup of water from the glass carafe, drank it off, and said: \"Now Sergeyev, tell me like a good fellow, have you been fed decently here?\"\n\n\"Excellently, Comrade Major.\"\n\n\"And have you been given something to read, to put in the hours profitably?\"\n\n\"I made no request, sir. My book was left at the billet.\"\n\nChepurnik opened a drawer. \"I believe I have it just here. ... Ah yes, _The Foundations of Leninism._ Splendid.\" He leafed through the pages. \"I see you follow the practice of placing little brackets about certain passages that you should like to turn to again. Here is something: 'the proletariat cannot and ought not to seize power if it does not itself constitute a majority in the country.' \"\n\n\"Yes, sir. But you have not read what comes just before: 'The opportunists assert that the proletariat, etc.' \"\n\nWith no change in his expression of encouragement Chepurnik half-rose and threw the contents of the carafe in Sergeyev's face. \"You see,\" he said, with fox-bark laughter, \"it is not vodka, but pure water!\"\n\nReaching for his handkerchief, Sergeyev remembered it had been taken from him in the search; no doubt, so that he could not knot it about his neck and hang himself. He wished he had his towel. He was surprised by Chepurnik's sudden offer.\n\n\"Go on, use it!\" The major tossed a handkerchief at him, which floated just out of reach. As he bent to fetch it, Chepurnik rose again and, leaning across the desk, with an inexorable meter-stick pushed the chair-edge slowly backward: Sergeyev fell to his rump. Chepurnik came around and helped him up.\n\n\"There,\" he said, returning, \"is a lesson. Observe how lean and muscular fact upsets dull brutishness. And you a major in Red Army Intelligence! Poor fellow, you could not have been an officer's orderly in the NKVD. You, with your low grade of cunning. Use Stalin's _Foundations_ as your codebook, leave it out where anybody can see, this will fool everyone! Poor chap, I would pity you were it not that I am nauseated by treason. ... Your only hope is to convince me you were misled in your criminal ventures.\"\n\nIt was still too soon for Sergeyev to know how and what he must confess to, as experienced in the area, from his years on the other side of the desk, and willing as he was. He lowered his head and said nothing.\n\nChepurnick raged: \"You upset my stomach. You look like a syphilitic chancre. Get out!\"\n\nThe guards came in and took Sergeyev back to his cell. He asked for writing materials. They were provided. He composed a _mea culpa_ referring to sins as far back as his residence in a seminary just before the Revolution. And admitted having at that time entered into a long-term conspiracy of priests and fellow novices to overthrow the rule of the proletariat in favor of a clerical dictatorship. As an agent in Hitler Germany he betrayed sub rosa Communists to the Gestapo. In Spain he took Franco money. Currently, at the time of his arrest, he had been employed by the Joint Distribution Committee, a Zionist branch of the American FBI. Yet his capture had revealed to him the overwhelming might and right of the International Workers' Movement and he expressed contrition. He asked that he be punished mercilessly.\n\nFinally he again faced the examiner. Chepurnik now wore golden pince-nez. For some reason he seemed to cultivate the appearance of an Old Bolshevik. Sergeyev could not understand why, since they had all been put to death as traitors.\n\nThe major drank from his cup, but the carafe was missing. Wiping his mouth on a beige silk handkerchief, filmy as a cobweb, he said: \"Ah yes. I have read your lies.\" This time it was vodka, and not water, that dashed against Sergeyev's cheeks and fierily into his eyes. \"Your face reminds me of a pig's rectum.\"\n\nBack in the cell, as soon as his sight returned, Sergeyev began to write a new confession. He admitted having been lured by the American Intelligence agent Schild (\"Fritz\") and the Fascist homosexual Ernst (\"Schatzi\") into a plot to restore the Nazi regime in Germany. The leader of the movement was Hitler's lieutenant Martin Bormann, whose whereabouts the Allies had sought unsuccessfully since the fall of Berlin. Sergeyev could reveal that Bormann had been flown out to Lisbon and there awaited recall.\n\nAt the next interview Chepurnik gave him a cigarette.\n\n\"My pitiable fellow,\" he said, \"you have convinced me of one thing: that you are honestly trying to tell the truth. But it is most difficult when everything one has to work with is as corrupt as a boil. For example, your so-called information about Bormann. I can assure you that he was captured by the NKVD on the second day of the fall of Berlin and has long ago been executed. And while we are on the subject, you probably also do not know that Hitler did not die in the Kanzlei as is popularly supposed. The NKVD got him and he has since been held secretly in the Kremlin, pending Stalin's decision on how best to make use of him, which is to say, as prisoner or as corpse. Also the Braun woman.\" Removing his pince-nez, he winked. \"She was imprisoned here for a week before being sent on to Moscow. French underwear. Too skinny. But... _o\u00f9 sont les neiges d'antan?_ This is how the French say, all that is past.\" Chepurnik was indeed a good-looking man, with a high forehead and long jaw. Sergeyev could smell his cologne.\n\n\"But to your problem,\" said the major, \"from what principal did you get orders to kill the agent Fritz?\"\n\nSergeyev collected his forehead sweat on a bladed hand and rubbed it into his frayed, stained trousers. \"A terrible error,\" he answered. \"I sent my men, these Germans, to bring Fritz for an interview with me. He resisted. In the ensuing fight he and my man Hans were killed.\"\n\n\"If you insist on telling me what I already have, and lying to boot, the Soviet people will be merciless. Fritz was your agent, and yet you did not know he went everywhere with a bodyguard, a professional athlete named\u2014\" he brought a dossier from a drawer and leafed through it\u2014\"yes, named Reingart. This man is in the hire of the Counterintelligence Department of the United States Army. Before the war he lifted weights in the Radio Music Hall, which is a sports arena in New York, U.S.A.\"\n\n\"No, I did not suspect that, Comrade Major,\" Sergeyev answered in profound shame. \"I did not know that the bringing in of Fritz would entail difficulties. Those two worthless Germans should have been able to take care of anything. Just as well for them they did not come back.\"\n\n\"And why did they not, Sergeyev?\"\n\nHe could not meet the major's eye. \"Well, as you know, the one was murdered by this weight lifter. The other, I am afraid, had fled to the British Zone.\"\n\nChepurnik shook his head in tragedy. \"The Soviet people gave you great responsibility, high rank, absolute trust. And you cannot live up to the most primitive concept of honor: a superior never hangs the blame on his underlings. Aren't you ashamed of yourself? What do you suppose these unfortunate Germans thought of the Soviet Union when they looked at you, its representative, a heap of garbage. ... No, I refuse to listen to any more.\" He put his hands over his ears and summoned the guards.\n\nAgain and again Sergeyev rewrote the confession. He admitted his complicity in every crime against the Soviet state since its founding. He begged to be sentenced to lifelong imprisonment at hard labor. He implicated his wife and children, who lived in Leningrad, and asked that they be arrested. In his eighth revision he requested the death penalty for himself.\n\nAll to no avail. Chepurnik took every admission as evidence of a greater concealment. On the other hand, Sergeyev's infrequent and weak protestations of a particular innocence\u2014in the context of the established general crime\u2014outraged the major.\n\n\"How dare you assert you did not trade in American cigarettes on the black market!\" he would scream, hurling the contents of his cup and carafe, or when those were not at hand, his meter-stick, which was rather more dangerous. \"We discovered a hundred packages in the stuffing of your mattress!\" And Sergeyev would return to his cell and incorporate this new failing into his latest confession. Which meant: in detail, for Chepurnik cast an especially cold eye on his prisoner's use of anything, he, Chepurnik, had given him. He was, that is, a man who could not be defrauded. Sergeyev, who had never seen a packet of American cigarettes, had to invent a brand: \"Tom Smith Variety\"; and to be precise in his count: \"At the period of my greatest activity in this illegal and treacherous commerce\u2014the purpose of which was to sow the seeds of dissension among Red Army troops to whom I distributed the cigarettes\u2014I kept on hand approximately 523 packets, which, at 24 the packet, would in sum amount to 12,552 cigarettes.\"\n\nChepurnik hurled it into his face. \"What kind of trick is this? Everybody in the entire world knows Tom Smith Brand is sold only in packages of ten cigarettes each! ...By the way, don't you ever wash? You give off the odor of horse urine.\"\n\nChepurnik was still reading the nineteenth confession when the guards brought Sergeyev into the room. He scowled, and Sergeyev's heart mounted like steam from a samovar. No matter how enraged, the major had never scowled. Therefore it could mean only that he approved.\n\nHe did. He turned the manuscript to face Sergeyev, who saw with delight that it had been typed in the official form, and presenting a fountain pen asked him to sign it.\n\n\"Now,\" said the major, with the pursed lips of a French dandy blowing the ink dry, \"now that should take a great weight off you.\"\n\n\"Sir, may I ask one question?\"\n\nChepurnik began an ominous half-smile. \"Careful now... just make certain it is not some punishable folly, and yes, you may.\"\n\n\"Has the Fascist homosexual Ernst, alias 'Schatzi,' been arrested?\"\n\nChepurnik laughed: \"Alas, you force me to do it. Your rations will be reduced by half for one week. I warned you! Herr Ernst is a loyal Communist of impeccable character and I advise you to discontinue your persistent attempts to involve him in your criminal activities. Without Germans of his kind we should have a most grievous struggle to create a Socialist society here. I shall not tolerate your Great Russian chauvinism!\"\n\nThe major grasped his cup as if he might throw it, but instead he took a drink of water or vodka and wiped his lips. He went on: \"Had it not been for Ernst we might have taken longer to uncover your crimes.\" He brought the old dossier from a drawer and leafed through its early pages. \"Do you recognize this? 'Hitler cleverly crushed without mercy that foulness.' \"\n\n\"No, sir.\"\n\n\"NO?\" Chepurnik screamed. \"How dare you say no? I did not speak these words, Ernst did not say them. Who did?\"\n\nSergeyev remembered. \"I am sorry, sir. I did, to the agent Ernst, in reference to the R\u00f6hm movement.\"\n\n\"Now do you doubt our tolerance? Can you imagine what would happen to an American or an English officer who praised Adolf Hitler? ...Secondly, Herr Ernst learns of the defection of the agent Schild. Immediately, in the prescribed manner, he comes to give his report\u2014only he cannot find you. The great Sergeyev is not at his desk, as the Soviet people expect their officers to be every minute of the day. No, the megalomaniac Sergeyev has, totally uninformed as to the condition, the whereabouts, the state of mind, and the companions of the agent Schild, already sent his men to fetch him for a 'talk,' and then himself gone with a German whore and a liter of vodka to the billet. Where did you plan to interview the agent Schild, between the whore's legs?\n\n\"No, even that would have been more sensible. You were going to have him held all night for your 'talk' in the morning, so that he would be reported missing from his unit and the whole American nation, from President Truman down, would protest to Comrade Stalin about another so-called kidnapping.\"\n\n\"My crime grows ever larger,\" Sergeyev muttered. He held himself erect only with supreme effort; so skinny was he the chair wished to eject him.\n\n\"I see,\" Chepurnik said. He closed his green eyes for a moment. Before opening them again, he slowly tore the confession in half. \"Then this, too, is false. _Il faudra toujours recommencer._ That is French for 'we must begin all over again.' When will you learn that you do not have the intelligence for a large crime?\"\n\n# _CHAPTER 22_\n\nREINHART TURNED ON HIS side in bed and played it cool, studying the little blue light at the end of the ward. He saw Very's large, maternal shadow in the darker cavity of the office doorway. She had been his first concern on awakening from that positive death on the bloody field\u2014somewhere back, weeks, he supposed; part of his trouble was a derangement of the sense of time. ... No, he would not lie. Be fair to yourself, the doctors told him, but surely that did not mean believing falsehoods. His first occupation had been resentment and self-pity, for Schild stayed dead.\n\nHe passed out again when his wounds were treated in the emergency clinic, not from pain but in another attempt to die. In vain. You could not beat the Jews at sacrifice; it was their profession. Next he had come conscious as a patient in the ward for Superficial Wounds and Contusions, of which he did not remember Marsala was master until he heard the buddy's raucous complaints issuing from the treatment room: an underling wardboy had loused up the bandage-count. But the evidence of normality in the world outside his head terrified him, and when Marsala came to the bedside Reinhart simulated coma.\n\nLater, en route to the night shift on the psycho ward, Veronica stopped in. Marsala was goofing off someplace, so Reinhart showed life. Then it was that he thought of her troubles and forgot for a moment his own.\n\n\"I'm sorry, Very, I tried as I promised but ran into a snag.\"\n\nHe felt her pat his brush and wondered where the wound up there had gone. The gash in his cheek he still had; its bandage refracted the vision of his left eye.\n\n\"Honey...\" Poor Very suffered tears. \"... was this terrible fight because of me?\" Yet she once again looked full-blown, her splendid flesh bursting, wherever it could, from the commonplace seersucker uniform.\n\n\"No, I never fight for any purpose. If I had wanted something from those Germans I would have licked their boots.\" He now licked his lips and found the cuts towards the left corner and the bitter medication upon them.\n\n\"Don't be so fresh with yourself,\" she said. Later, in his need to tabulate things, he recalled this as the first of a series of such admonitions from practically everybody.\n\n\"Anyway, I'm stuck here and I don't know for how long.\" He reached for her hand and then, remembering the other patients, gave it up.\n\nBending, she displayed the partition of her breasts and whispered: \"Don't worry about me any more\u2014I'm O.K. now\u2014false alarm.\"\n\nHe laughed in a hysterical, private place, for one reason, because he could not in public: his face was cracked. There you had a capsule history of human affairs, amidst the larger one in which they participated with everyone else, including the nearby patients who gawked innocently at Veronica's body: what began in birth ended in death. Except that this one was all a mistake.\n\n\"Schild\u2014you know I thought for a while that Schild was your lover\"\u2014she caught at her mouth\u2014\"no, that had nothing to do with this trouble, believe me... and he thought I was, and now none of it matters. ... Very, look at the top of my head and tell me what you see.\"\n\nShe did more, exploring with a finger. \"I think it is a slight bruise and nothing worse. You'll be better in no time.\"\n\n\"Are you sure? He kicked me bad. I was certain my scalp was laid open.\"\n\n\"You know it's all bone, with only a thin layer of skin and hair\u2014\" Her big teeth glistened in fun. \"Excuse me, well, everybody has a bonehead, me too.\"\n\n\"Only mine is solid\u2014but are you sure?\"\n\n\"Clear the way for added cargo!\" She bounced-sat on the edge of the bed and waylaid her rising hem at the bulge of rolled nylon. \"It's pretty hard to 'lay open' a scalp; there's not much to cut and it would take something awful to break the bone wide open. Though of course there's fractures, but you don't usually know they're there until you look at the x-ray. ... Know who's ward nurse here? Eleanor Leek, the cute little plump girl from the party where we met. She danced with that Russian. Sure you do! I'll put in a good word for you. Now I have to go, the schizzes all start crying if I'm not on time. Oh, did you want to talk about the fight? But I don't think it's good yet. Rest awhile. Now kiddy, I'll say goodbye. I would kiss you but everybody's lookin'.\"\n\nAs the _Santa Maria_ must have swept from the Bahaman harbor, past the awed little brown men in their crummy dugouts, proceeding westward still, so Veronica sailed in splendor up the ward aisle and out the door, flying the high standard of her winged cap, and the natives returned to their fishing and coconuts.\n\nSoon afterwards Lieutenant Leek, whom when he saw her he remembered, the fat, homely, merry person, hurried in from the corridor.\n\n\"Good, you're conscious. Holy cow, you kept the brass waiting all day. Can you talk to them now?\"\n\n\"Who?\" With Very's departure he stepped on a merry-go-round which turned slowly to no music. \"I don't feel so good.\"\n\n\"I'll give you a pain pill soon as they leave.\" She left his sight and in a moment or two he was ringed by many male uniforms.\n\nFor one, the commander of the 1209th General Hospital, Lieutenant Colonel Fester, whom in three years' service with the outfit he was seeing for the fourth time, despite the legend that Fester was ubiquitous.\n\n\"Now Steinhart, look sharp,\" the colonel said in a clarion tenor. He wore white gloves, like a doorman. \"This is a terrible thing you are a party to, but I know any man of mine can defend his actions. Handpicked, every man Jack of you. Remember how we kept those niggers out of our latrine on the troopship coming over? Medics, maybe. They don't let us carry guns, but by Jesus we've got fists. I'd put my hand in the fire\u2014if I didn't have this eczema\u2014for any one of you and know you'd return the favor\u2014say, are you awake? Here you, Teats, or whatever your name is. Nurse! Give this man a hypo of something. What do you use nowadays? You've got all the drugs in the world nowadays. It wasn't like that in the old days. By God, APC capsules, gentian violet, and you had it; after that, the pine box. Remember the old days, Major, or was that before your time?\"\n\nReinhart opened his eyes. Next to the colonel stood a major with a young face, yet old gray sideburns.\n\n\"We'll talk about that later, sir,\" the major answered with short patience. \"Here, the fellow is awake\u2014\"\n\n\"Coming around, Steinhart?\" the colonel cried. \"Good, now stay awake for a moment. This is Major Koenig from G-2, Berlin District. I don't have to tell you he is a Very Important Person so far as we are concerned here in our humble way. He wants to ask you about last night.\"\n\nReinhart also saw Captain St. George, lachrymose; an enlisted man, PFC Walter Walsh, swelled with gravity; Lieutenant Nader, resentfully watching the colonel; and finally, a brutish-faced captain who wore the crossed-pistols badge of the military police.\n\n\"Your name is Reinhart, correct?\" asked Major Koenig. \"How are you feeling, Corporal?\"\n\n\"Carlo T., 15302320, and a little dizzy, sir.\"\n\n\"Well, who isn't? And I haven't just broken anybody's back, either,\" Koenig said. \"By the way, how did you do that?\"\n\n\"Only because he would have broken mine if I didn't.\"\n\n\"I asked _how_.\"\n\n\"Just caught him around the waist and bent him backwards.\"\n\n\"Good fellow!\" broke in the colonel. \"I used to watch you wrestle at Camp Pickett every Friday night, Reinkoenig. You made me lots of money. That's why I chose you for the 1209th. You still a corporal? Months ago I told that goddamned Lovett to put you on orders for a third stripe.\"\n\n\"Thank you, sir, but I had basic at Camp Barkeley\u2014\"\n\n\"Quite all right, my boy. Just answer these questions\u2014\"\n\nThe major asked: \"Do you feel well enough to get up for a few minutes? It's difficult to talk here. The nurse said we may use the ward office.\"\n\nReinhart lay quiet for a moment, his right as a casualty, then indicated he would try. Blackness flooded his brain as he sat upright, and he heard their voices as if through the closed window of a ballroom. His wrist was seized.\n\n\"Come on, fellow,\" the colonel shouted.\n\nIn transit Reinhart slept awhile and when he came to, watching the red-black ocean recede, thought they all had gone.\n\n\"You need a pick-me-up!\" the colonel roared. \"Teats! Mix an ounce of medical alcohol with grapefruit juice and bring it to this soldier.\" But Lieutenant Leek had crept off to hide.\n\n\"It's all right, Major,\" Reinhart said to Koenig, who offered no aid nor sympathy. \"I can make it.\"\n\nThe major walked smartly towards the office. Reinhart shook himself, feeling a twinge in his cheek, slipped into the dirty canvas slippers waiting below the bed, rose and followed. With all manner of noise the colonel dogged him but was denied at the door.\n\n\"This,\" said Koenig, standing just within, \"is an Intelligence matter, sir. Captain St. George and I will have to speak privately with Corporal Reinhart. If you gentlemen can give us a few moments?\"\n\n\"Me neither?\" The MP officer's growl betrayed a fright at his exclusion. He put a broken cigarette into his mouth and tried to light it, getting only air. From the patients out in the ward arose an anonymous murmur of ridicule, which was tonic to the colonel.\n\nHe said: \"As to me and Nader and PFC Walter, we'll mix here with the men till you need us.\" Followed by his reluctant entourage he went back down the aisle between the beds and shortly, among his bursts of loud, merry scatology, came the obstreperous derision which he was famous for misunderstanding as popularity.\n\nMaking an effort, Reinhart indeed felt better and stronger. The major, however, who sat upon a white-enameled chair, directed him to lie upon the operating table.\n\nThere was no allowance for discussion, so he did. St. George\u2014he saw the genuine sadness on the fat face, and pitied him, and liked him\u2014slumped against the wall.\n\n\"Now, PFC Walsh,\" said Koenig, \"who was mess guard last night, heard noises on the sports field and went to investigate. By the time he got there you were unconscious and the two other men were dead. He states he heard an automobile engine, and there were tire tracks in the sand. May we hear your account?\"\n\nOn his back, unable to see his interrogator, Reinhart spouted perspiration and anticipated a nameless catastrophe. Unknown enemies held him supine and prepared to work upon him an obscene damage; he felt womanly, about to be raped. Yet all through this, his fluent voice romped on, as if it were rather the child of another man. The voice told of the chance, friendly encounter early in the evening between Reinhart and Schild, of a purposeless wander to the Tiergarten, since the black-market contact never appeared and they thus could not buy their Meissen china. Then they took a drink in a jerry-built cafe on the Ku-damm, looked for a ride home and finding none began to walk. They lost their way temporarily. Finally they reached Zehlendorf. Then the straight story of seeing Schild enter the grove, and the fight.\n\n\"There must be more,\" said Koenig in his ominous, factual way.\n\nSt. George spoke for the first time: \"Lieutenant Schild had some of his own contacts whom I knew nothing about. We worked in that manner. Perhaps it wasn't SOP\u2014\"\n\n\"You said all that.\"\n\nBut once begun, St. George was briefly invincible. He owed it to his late colleague. And Reinhart thought, Schild was closer to him than to me; he was never really my friend, yet I did what I could to save him; why do I tell myself I was his killer?\n\n\"Perhaps it wasn't SOP, but I respected his intelligence. Then he was a Jew, you know. I never thought about it before, but it could hardly have been pleasant service here where they did such horrible things to his people. ... One of his German contacts was a little old fellow dressed like a workingman, who rode a bike. He came once to the billet when the Russian stayed with us. But perhaps he wasn't there for Nate. When I saw him he was talking with the landlady.\"\n\n\"Russian!\" blurted Koenig. He reassumed self-control. \"You can tell me all that later, Captain. At the moment we are interested in the corporal.\"\n\nKoenig did not trust Reinhart, so much was clear. But Reinhart had changed since he killed a man. Earlier he would have hated Koenig. Now he was beholden to him; wished he would let him rise, but knew he deserved no favors.\n\nThe major asked: \"This little old fellow that Captain St. George mentions\u2014was he one of the two Germans on the sports field?\"\n\nReinhart answered no. Schatzi\u2014Schild knew him too. He remembered the doctor's revelation and his own bombastic threat to kill this complex person both victimizer and victim. Doubtless no one took it seriously. Could they have seen him a few hours later! But Schild had, and thought him inadequate, and come to help, and died.\n\n\"I take it then you know the man St. George means, if you are sure he was _not_ there.\"\n\n\"I believe so,\" said Reinhart, \"he hangs out around here. He is a big wheel in the black market.\"\n\n\"Could this have been a dispute over a black-market deal?\"\n\nSt. George answered, scandalized: \"Certainly not! I never knew a man less interested in money than Nate.\"\n\nKoenig sighed. \"The corporal has just testi\u2014stated he and Lieutenant Schild went to the Tiergarten, were driven to that area by you, yourself, in which you concur, to meet someone who offered a set of Meissen china for sale.\"\n\n\"Sure,\" St. George laughed indignantly, \"but that was for Reinhart!\"\n\n\"That's right,\" said Reinhart to the major he could not see, endeavoring to meet St. George's eye with a message of loyalty. But the captain seemed to avoid him.\n\n\"A do-gooder, the late Lieutenant Schild.\"\n\nSt. George answered: \"Always,\" and hung his head.\n\nKoenig continued his keen probing, to which his immaculate and subtle contempt was an additional tool. Reinhart dissembled in the only way he could, by blondly, wholesomely baring all but his suspicions, which anyway was legally impeccable. Later, the MP officer was let in, and Nader and Walsh, and, perhaps with an idea to stop his noise, briefly, the colonel.\n\nKoenig suddenly finished; whether for good and all, naturally he did not indicate. Nader and the MP, although they showed a personal distaste for Reinhart, seemed in the absence of contrary evidence to believe his story and agree he must stand a court-martial which would formally find him guilty of homicide in the line of duty, sentence one dollar. This to forestall an attempt by Monster's heirs, if they could be found\u2014he remained unidentified\u2014to bring charges.\n\nEverybody having left, Reinhart assumed he could get up, and did so, and was frightened by the appearance of St. George, who lingered behind the operating table.\n\n\"Listen,\" said the captain, with suppressed dislike, \"maybe you know. Does Lieutenant Schild have any family? We don't know who to inform. For some reason he gave as next-of-kin the name of a prostitute in Paris, Texas. I never had occasion to look at the record until this morning, and then I recalled her name from when we served together at Camp Maxey.\"\n\n\"I never knew him well,\" said Reinhart, whoozily standing.\n\n\"Who did? ...He had a strange sense of humor, and this shows he would go all the way with it.\" Still with obvious unfriendliness, he came to Reinhart's support. \"You see, we were in combat zones since D-Day. He could have been killed at any time. But for the joke he gave the name of this streetwalker.\"\n\nReinhart took the offered arm. They moved together into the ward. At his bed, first one on the right-hand line, he thanked the captain and shook hands, and saw astonished gratitude, and understood merely another error: St. George did not dislike, but rather fearing being disliked.\n\n\"Oh that's all right,\" said the captain, pumping his fingers. \"If you don't mind, maybe I can drop in from time to time to see you. But you'll be better soon.\" He left anxiously. He returned and placed a just-opened package of Parliament cigarettes on the night table. \"Have a luxury smoke.\" At the foot of the bed he turned and said: \"He was a good fellow,\" and waited.\n\n\"Yes, he was,\" Reinhart answered and blacked out. He dreamed he was twelve years old; selling newspapers from door to door he accumulated money for a bicycle; someone stole the money but when he went outside there was the bike on the porch; a Negro applied Simonize to the fenders; _so you're a Negro,_ he said, _isn't that fine!;_ the black man rose in terror, great white eyeballs gleaming, and ran down the street. When Reinhart awakened, the same Negro, whom he had never before seen in life, carrying a tray of food walked past the bed and to his own, number five, where he sat near its head and ate rather insolently, winding spaghetti into a spoon.\n\nLeek appeared with a tray for Reinhart containing various forms and colors of mush. He was suddenly horny; she was not so bad. He invited her to come sit upon the palm of his hand. She thought he waved her off, and went. He called her back and asked the time. Six-ten. He slept for an hour. Awakening, he asked the time of the fellow in the next bunk, who had a wart in the center of a bushy red eyebrow. Six-eleven. He denigrated the fellow's watch, cursed its owner. The fellow slyly turned his back, and Reinhart cried into the pillow because he could not hurt him.\n\nWhen everybody had gone to sleep he wandered into the toilets and counting them again and again at last made his choice of one to sit upon and read a comic book. He had not finished a page when a wardboy whom he did not recognize entered and ordered him back to bed.\n\n\"You been in here two hours.\"\n\nReinhart snarled contempt. \"I've been in the Army since Christ was a PFC, and don't you forget it, you prick of misery.\"\n\nBut he went. Someone kept putting a flashlight in his eyes and forced him to eat sulfa pills and drink four glasses of water.\n\nOne day Marsala, whom he was forced to admit knowing, sat on the edge of the bed and whispered, so that the bastard with the red eyebrows couldn't hear: \"Carlo, whadduh yuh doin'? You been here three weeks, your cheek is almost healed. But you keep acting crazy, they send you to Psycho. I don't shit you, pal. I heard Captain Cage talk about it this morning.\"\n\nReinhart had all he could do to keep from spitting in his ugly face. \"Here's what I say to you: go to hell.\"\n\n\"See what I mean. Whadduh yuh want to be sore for? Jesus, I'm distracted.\" He rubbed his thick and whorled temples. \"So you took too much sulfa, but that don't stay on permanent. Your white-corpuscle count is okay\u2014\"\n\n\"You flunky, what do you know about medicine? You traitor, you make me puke.\" He stared fiercely at the quondam buddy, stared through him as if he were cellophane\u2014a gift he had nowadays and would have done anything to get rid of.\n\n\"Okay,\" said Marsala. \"Okay then. You and I are through.\n\nGet it? As soon as I get back to the apartment your crap goes out inna street.\" He symbolically spat. \"And the same goes for your lousy _putana,_ that little kid who's young enough to be ya granddaughter, dirty guy.\"\n\nThe return on his aggression soothed Reinhart, convinced him that even wounded and mad he was potent. In delight, he said: \"I hurt your feelings, didn't I?\"\n\nMarsala looked at him a long time, his ferocity melting into a kind of grief. \"Nah, I consider the source. ... Look, what for did you tell that kid Trudy she could have your extra OD shirt and pants? Don't you think you will ever get out of here?\"\n\n\"Did you give them to her?\" He saw the whole thing and was serene.\n\nMarsala clicked his teeth in lascivious regret. \"You didn't say she could have them, you ain't even seen her since you're here, am I right? She talked me into it, kid. I'm sorry. When we get through and are in there washing up she says she is thirteen. Jesus I'm a dirty guy.\" He slapped his wrist with two fingers.\n\nHappily, Reinhart lied: \"Of course I did. I sent her a note. I'm going home soon and don't need that extra stuff. Give it all to her and take her for yourself. She is really sixteen and if she's old enough to have it, that's old enough to use it. I apologize for being nasty. You're the best buddy I ever had, and when I get home I'll write you a letter once a month. ...\" He closed his eyes and kept on talking. When he opened them again Marsala was gone and Red Eyebrows, who he also noticed had red hair, was peering at him most curiously.\n\n\"Hey Red,\" he called. \"I didn't mean what I said yesterday about your watch. I was still a little dizzy from this trouble I had the other night. Have a luxury cigarette.\" He picked up the Parliaments and handed them over. They were strangely dry and friable for a fresh pack.\n\n\"Yesterday?\" asked Red. \"That was almost a month ago.\" He laughed as if he were insane.\n\nBut when later in the day they transferred a patient to the Neuropsychiatry Ward, it was not Red.\n\nVeronica's shadow flowed back into the black cavity of the office, meaning all was quiet on the ward. Reinhart himself certainly never made a sound. In the latrine he had even perfected a technique for micturating without noise. He wished to call no attention to himself, because he was altogether mad.\n\nIt owed to that kick in the head which couldn't be proved. The staff in Superficial Wounds assured him he suffered from a mild toxic psychosis, the effects of an overdose of sulfa, for which they took full responsibility. Having admitted their guilt, they insisted it make him free, especially since the technical manual, _Guides to Therapy for Medical Officers,_ called the reaction rare. And as to Reinhart's confession that he habitually swung the pills into the deep socket of his jaw, drank the water, and spit them out when the nurse turned her back: in their view this illusion was rather an index to how many he had swallowed and been deranged by. Apologizing, they force-fed him gallons of water. Drowning, he was still mad.\n\nThe Psycho people, on the other hand, kept their convictions secret. Lieutenant Llewellyn walked on eggs from bed to bed on his morning tour of inspection, wearing the silky, untrimmed mustache to make himself look older and the plastic-rimmed eyeglasses for wisdom, carrying his mouth slack and moist in an advertisement of patience. He was rather leery of Reinhart; few indeed of his patients had killed a man with their bare hands.\n\nCaptain Millet, the chief, stayed always in the office and one went to see him at intervals. From crown to temple he shone bald as Bach; around the ears, a ballet tutu of salt-and-pepper hair. As Llewellyn was listener, Millet questioned, and had a talent for the irrelevant: Do you like girls, did you ever play with yourself, do you have headaches, did you dress up in your mother's clothes when a boy, what do you think other people think of you, what do you want to be?\n\nTo the last Reinhart invariably answered: \"Able to tell time again.\" For this was the heart of the matter, but Millet, bored, toyed with his pen and never took a note.\n\nHis head he had stopped bothering to mention; if Superficial Wounds, in whose area of interest it lay, could not find that seepage of brain fluid, Psycho, devoted to the impalpable, would hardly. On the basis of many motion pictures about amnesiacs he drew up his own strategy of treatment: he could be cured by another raking blow on the skull. But owing to the queer angle, he could not slug himself sufficiently hard, and he was afraid to ask one of the nuts to do it, who might kill him. Which brought to mind an essential feature of his condition: a lack of interest in death as therapy.\n\nOnce he had tried strenuously to die, and again the next morning, when it had seemed necessary to the _Gestalt_ of himself-Schild-and-Germany. Now it would be a simple missing of the point, for the self within him was already unearthly and losing the rest were impertinent. If someone sought to kill him he might not resist, but he would not raise his hand against himself.\n\nHis inner cautions to the contrary, in a burst of bravado he delivered this information to Captain Millet. Who blankly answered: \"That is comforting.\" And Reinhart was ashamed of his vanity and of his suddenly revealed wish for Millet's affection, whom he didn't even like and to whom he had bragged only because, he thought, Millet didn't care.\n\nThe captain went on: \"You mean, you will not commit suicide by violence. That is too easy, whereas what always attracts you is the difficult.\"\n\n\"No,\" Reinhart confessed. \"The impossible.\"\n\nAn ear fringe grew as fast as a full head of hair; Millet needed a haircut, which deficiency, however, and now a vulnerable smile\u2014his teeth were crooked\u2014canceled the disapproval from his next remark: \"Why do you think you are so important?\"\n\n\"Because\u2014\" Reinhart groped for something smashing; in his bare cupboard was one bone; he seized it\u2014\"because I am insane.\"\n\nMillet said seriously: \"The Army may make errors in assignment, but they were right about me. I can show you my diplomas. I assure you, you are not insane and will not be.\"\n\n\"Then I am a fake.\"\n\nMillet's pen scratched upon his notepad, but Reinhart saw only doodles, and not imaginative ones at that. \"As late as the nineteenth century they used to chain patients to the wall and whip the disorders out of them. The treatment was oftentimes successful. It might be used today except that its good effects were, I believe, only temporary and it required enormous physical exertion on the therapists' part. Now we have the lazy man's method. When you decide whether or not you are a fake, come in and tell me.\"\n\nWell, he guessed he _had_ made a mark on Millet, if Millet talked to him in that ironical way. The captain was softer with the other patients, according to them\u2014for Reinhart sometimes conversed with those who were articulate. The enuretic poseur of a paratrooper, for example, whom Very talked about back in August, had returned. Perhaps he falsified the one symptom, but he had plenty more, couldn't use a tableknife, thought people were after him, etc.\n\nBut having got his special notice from Millet, Reinhart went back to where he started. Because his case was irrelevant to the fundamental proposition by which lunatics and psychiatrists are one: Life goes on. And not only proceeds to the measure of the ticking second hand, but also abides in wondrous detail which perhaps one can only know when on furlough from the process.\n\nWho ever before had opportunity to study the fabric of a pillowcase? The close and naked eye saw no two threads the same. And the canvas slippers: their weave black with dirt, rich with memories of various feet walking diverse floors in many lands; their old human smell, sour, interesting. Chipped was the white paint on the bedstead, revealing multiform corrosions, patina, wounds in ancient iron; whom had it supported and in the aftermath of what: Caesarean sections, irrigations of the maxillary sinus, removal of the vermiform appendix, mere hang-overs, some deaths. Blankets of wealthy white wool, shrunken gray pajamas gaping at the fly and over the heart USAMD embroidered in red; maroon robe of weary corduroy, too short; night table all one's own offering water in pitcher and glass, and beneath, on the low-slung shelf, colored books random and in shadow.\n\nAnd through the window\u2014lingering on the glass itself, marvelous substance almost invisible, metamorphosis of gritty sand, just as the butterfly comes from a worm and remembers it not\u2014a view of this side of the grove on the other side of which, on the sand, some fellows murdered his only friend while he watched through another window, that glass bell under which he was wax fruit.\n\nIn this celebration of matter he got through his mornings, which were worst. From within came spontaneous improvement as the sun traveled towards America; afternoons were fair, he could sleep, and wake up in early evening, glad of life's refuge from dreams. As natural light failed and Edison's took over, he was better. Finally, each dark midnight, the absolute cure: bored with the soul's business he lay unsleeping, yet not sleepless, believing in these hours of exuberant health that everybody should have a goal. During such a period of clarity and courage he decided to take his discharge, when it came, in Berlin and marry Lori.\n\nOr join with her in whatever other relationship external conditions would allow, the mode was irrelevant; internal coherence was all. Even _m\u00e9nage \u00e0 trois,_ in thinking of which he signified his affection, respect, and pity for Bach, nay his downright love for this man who needed and had a nurse, not a wife. She would continue in that role, plus which he would gain a friend to listen interminably. Reinhart spoke frankly to himself of its romanticism: this boy from distant and simple Ohio oh beautiful for spacious skies and amber waves of grain, in a rathole under the rubble of a dark and evil idea, living in adultery and cuckoldry sanctioned by mutual love, talking art and philosophy: there at last was the old German idealism he had searched for so long.\n\nOn the third night of his planning, in the second week of his residence in Psycho, he believed it politic to leave the bed and steal to the office. The paratrooper, next to him, sobbed peacefully; from elsewhere sounded placid onanistic rustles; the coast was clear.\n\nVeronica, back to doorway, sorted files. Before Reinhart could announce himself\u2014because he meant no harm\u2014he was seized from behind by two husky wardboys leaping out of ambush, who were lucky to remind him poignantly of Monster's dread embrace, and he offered no resistance.\n\n\"Oh that's okay, fellows,\" said Very when she saw, holding up parallel fingers\u2014for some reason, all five instead of the standard two\u2014\"this kid and I are just like that.\" She directed them to the treatment room, to prepare wet cocoons against an expected need towards dawn. They left, grumbling fealty.\n\n\"I was wondering when you'd come, you ingrate. I got put on nights so you could.\"\n\nShe almost leered at him, and he thought: working here gets them all eventually. They sat on either side of the desk, which had kneeholes on both sides, so that his and her legs touched. Soon she squeezed one of his between both of hers and crooned: \"I've been feeling rational guilt lately. I thought of all I owe you, Carlo, and I could just cry. You were my friend when I had nobody else to turn to, and the fact that it all came to nothing doesn't make any difference.\" She slapped on top of his the right hand of the high-scorer on St. Something's girls' basketball squad of 1940.\n\n\"Glad to do it, Very,\" he answered quickly, for she was in the mood for something and he had little time; at that cursed early first light of Berlin he would lose his wits again. \"I\u2014\"\n\n\"And I thought regardless of what went on you've always been my best boy friend, right here.\" She indicated her left melon, underneath which lay the heart. \"Because you stuck with me, and whether you ever said it or not, that is love.\" She winced in ecstasy and pile-drove her linked fists into the back of that same old hand of his, flat upon the table. \"Oh darling, you won't regret it!\"\n\nHe withdrew; another such love and he was crippled for life. \"Look, Very, what I wanted to talk about was\u2014\"\n\n\"I've been a fool,\" she said. \"All the other men I ever knew wanted only one thing, like, excuse my French, this son of a bitch\u2014\"\n\n\"I don't want to know who he was,\" Reinhart interrupted. \"And I'll tell you something about me.\" He had to do this, it was the only means by which he would ever get to his business. \"So do I. Whatta you think I am, a fairy? I just use a different strategy. Now this thing about the abortion. If I could have arranged that, then I would always have something over you, get it? You'd be forced to let me have what I want.\" He feared it was too strong: what business was so pressing that one must for its sake kick another human being in love's groin?\n\nHer quick answer spared his recantation. Through tears of happiness she sang, \"Darling, I knew you loved me. You will never get a job in Hollywood, you are a terrible actor! You're afraid your feelings will be hurt, poor dear. But I tell you it's all right, I dote on you. I don't care if you are an enlisted man!\"\n\nSo. One should not always walk against the wind. He rose and called her to him. Checking the door, she came. She had found him out, he admitted. He must have her, for love, and _now._\n\nOh, oh, oh.\n\n_Now._\n\nNo, no, no.\n\n_Now._\n\nImpossible. Duty. No. Where?\n\nHer bailiwick. He was only a visitor.\n\nMaybe\u2014but no.\n\n_Now._\n\nOhhh. Terrible. Yet love was good.\n\nTell Tweedledum and Tweedledee to watch the shop.\n\nThey'll know.\n\nSo? A couple of privates.\n\nSo while he waited in the corridor, Very stopped by the treatment room and gave her excuse for ten minutes' absence. Then, down the hall, they found an empty, moon-illuminated ward whose patients had been transferred to another on the first floor, but the beds remained.\n\nNot in love, he made gentle approaches. In love, sitting on the bed-edge, she grew fiercely reluctant. He had to take every button of her uniform as it were a separate fort. In her white slip she cried for shame and changed her mind, and having no effect on his, slapped his face but left her hand there and coyly squashed his nose. Dead weight he must lift to draw slip skirt over bottom. Opposing her own divestiture, she nevertheless stripped off his pajama shirt. His cheek wound throbbed in each little red hole left by the stitches; the invisible scalp-ache ached as if it would have a separate orgasm. In the sexiest presentation she did not match: brassiere of pink, pants of blue. A coldness on his back foretold that the swelling former was a hoax of sponge rubber or wadded powder puffs, and he knew he could not perform. But as he retreated, so did she charge. She liberated the strained hammock and held it conquered and limp as an enemy flag. Real? By Magellan, if they were not real, there were no Eastern and Western Hemispheres, which had sat as their models. _En voyage,_ his trousers were suddenly gone, and also the canvas slippers. He dove towards her sea-blue pants. Then did she bound, like the well-known main, and give him a struggle beside which poor Monster's had been as a schoolgirl's in a pillow fight. Trudchen's old lacerations, which had healed, were reopened in his earlobes, and Very's teeth were twice as large. His solar was plexed and his clavicle cleaved. She probed for that rib which Adam had given to make her. When they fell to the floor, he hit lowmost, heel-ass-head. His only purpose was tenderness. Under incessant punishment he got her again to the upper level, dropped her on the mattress, _goyng, goyng_ said the springs as she bounced with flying members. She gathered herself and shuddered, blurted: \"My God, the night supervisor is due any minute!\" She crawled towards the footboard and her clothes. \"Shh,\" said Reinhart, \"I think I hear her now!\" She closed her legs, ivory in the half-light, pillars of the world; stood up and listened with flowing hair and sensate, rose-marked, goddess breasts, with belly-swell of satin and velvet groin. On Olympus sounded ox-eyed Hera's jealous thunder and Aphrodite practiced _Selbstmord._ \"Where?\" She heard the silence of the corridor, and was caught _en face_ to Reinhart and borne down through the wine-dark sea.\n\nLater, beached and dripping, she kissed his ten fingers and remarked: \"You didn't hear the supervisor at all.\" He grinned splendidly, Zeus-like, and with his godlike ear heard her all over again.\n\nNext morning he arose without his symptoms and with a grand conviction they were no more. In hubris he dared them to appear, dared also to draw up their roster: the quivering wire in his bowels; the cold sweat and hot chill; the vacancy of a head yet heavy; the apprehension of attack; the sense he shared the skin of another man; a feeling of insupport towards the sacroiliac, as if two segments there were rather gristle than bone; the famous derangement of time; a horror over possible events: Would Trooper, writing a letter in the next bunk, ever finish that line? Would Lieutenant Llewellyn start his tour on right or left side? Would, next time he looked, a straight pin still gleam from the crack in the floor?\n\nHard as he tried, he stayed healthy. Llewellyn, when he came and heard, was happy in his innocent, brotherly way and advised him to counsel with Millet. Trooper, waiting with his own troubles, in his turn told the lieutenant of the old dream of jumping without a chute.\n\nReinhart dawdled for more self-tests. When finally he was ready to see the captain, he was last in line. Waiting, he felt his euphoria fall into a dull headache of resentment towards the other patients, leavened by a sharp distrust of Millet.\n\nMillet turned his head and blew a large nose into an olive-drab handkerchief, very insulting. \"Of course your problems are important,\" he said. \"More important than any other patient's. And so are those of each one of them\u2014when they sit before this desk. That is a fact. Do you know what a fact is?\"\n\nReinhart looked defiant. \"Yes, something that cannot be changed.\"\n\nThe captain had finally got a haircut, the ballet skirt was trimmed to a gray fuzz. He sloppily put away his handkerchief.\n\n\"Isn't it rather, something that should not be ignored?\"\n\nAll right, he knew everything. But Reinhart had seen him blow his nose; he was omnipotent and all-knowing, but he also caught colds.\n\n\"Well, I'll tell you a fact,\" Reinhart said. \"In this fight, which I got into to help my friend, I killed the big German. It is a terrible thing to kill a man with your bare hands. I regret it, I think from now on I'll turn the other cheek, but I didn't have a choice then. But I don't feel _guilty,_ if you know what I mean. It was a good fight, a fair one, I mean, for him. He was bigger and stronger than me, and we all take our chances. This is a fact. Why then instead do I feel a guilt for Schild's death? Ill tell you. Because I could have saved him.\"\n\nMillet drew some rectangles on his pad. Nothing touched him. He said indifferently: \"Why didn't you?\"\n\nReinhart's symptoms had returned, full flood. While his chest shivered, bloodless, his suffused head burned; he could not recall whether he had sat in this interview one minute or all day; his whole spine was superimposed rings of lard; he looked for Millet any moment now to draw a long, keen blade and leap upon him.\n\nStill looking away, Millet suddenly ordered him, as captain, not doctor, to answer.\n\nReinhart shouted, in glorious hatred. \"Because I wanted you to die, you bastard.\"\n\n\"Not me,\" said Millet. \"I was not there. That is a fact.\"\n\nTerrible, deep remorse for the error. Not Millet, certainly: Reinhart wished to kill the only man he could ever talk to, and that was Millet, who wasn't even a Jew.\n\n\"Well, I don't feel so good. I'm not even sure I'm sitting here right now. Maybe I died back there on the field. What living man always feels guilty of what he hasn't done?\"\n\n\"Every single one of us,\" Millet answered, although it had not been a question. \"You are not special in that regard. In the heat of combat, soldiers always wish their comrades would get hit, as a kind of charm against their being hit themselves, and then experience guilt if their wish comes true\u2014but not till the battle is over and they have time and security enough to brood about, rather than preserve, themselves.\"\n\n\"But,\" cried Reinhart, \"that is all just personal. What I am involved in is the murder of a whole continent of Jews by my people.\"\n\n\"Excuse me.\" Millet reddened his nose with the handkerchief.\n\n\"Do you know what is good for that?\" Reinhart asked. \"One of those benzedrine inhalers.\"\n\n\"Thank you,\" said the doctor. \"I will try one.\"\n\n\"Only not just before you go to bed. It will keep you awake. The benzedrine, you know.\"\n\nMillet again expressed gratitude, and said: \"The personal thing is very interesting. In the three weeks of our talks\u2014\"\n\n\"Is it so long? I can't remember, you see.\"\n\n\"Oh you will, you will. ... Not once have you spoken of Lieutenant Schild as a person. Was he really your friend?\"\n\nHe understood Millet's technique well enough. It was always to oppose the freely given and dig for, what was on Millet's terms and not his own, the withheld. Simulating anger and hurt\u2014for his true feeling was of challenge to a battle of wits\u2014he concluded the interview.\n\nThen, back in bed, which had new, icy sheets, he found as usual he could not organize his thinking. Instead he tasted, in fancy, Veronica's aphrodisiac body and felt relief that tonight was her night off. Too rich for the blood. There was another crime for which he knew no remorse: taking advantage of her love. And now he compounded it. He wrote a note to Lori, asking her to come see him after work, and had it delivered by a wardboy, PFC Remington, whom he had done a favor for in England. \"And,\" he cautioned, while Remington studied him with the same uncomfortable eyes he had used since Reinhart became a patient, \"the German girl, and for Christ's sake not Miss Leary.\"\n\nLori came to the ward door an hour after the patients' suppertime. Nurse Bronson, Very's substitute, would have turned her away, but fortunately Lieutenant Llewellyn, permissive, dreamy type, had lingered late over his reports.\n\n\"A girl,\" he said encouragingly to Reinhart. \"A visitor. Well, isn't that kind.\"\n\n\"It is most important that I see her for at least five minutes. I believe, sir, it would be a kind of therapy for my disorder. Of course I know it is against the rules, and if you say so, I must get rid of her.\"\n\nPain, like a wave of heat, warped Llewellyn's plastic glasses. \"Oh no, Reinhart, we are absolutely opposed to duress! I hope you think better of us than that. I tend to agree with your feeling that you should see this kind lady.\"\n\n\"You won't think I'm a malingerer?\"\n\n\"Please free yourself of any such apprehension. Perhaps you will let me prescribe a mild sedative?\" He went, lankily nervous, to a glass-front cabinet.\n\n\"No thank you, Lieutenant, she is not upset! Hahaha,\" laughed Reinhart.\n\nAfter a moment Llewellyn bewilderedly smiled. Then he worried: \"Normal interpersonal relationships are too rare in this somewhat false environment. No, I agree with you in your feeling that you should see her. But may I ask, if you don't mind, it is better in my judgment, whatever that is worth, that you and she do not confer out in the ward. It would disturb the other poor fellows. I should be glad to have your opinion on that, however.\"\n\n\"That's quite right, sir, quite right.\"\n\n\"Oh good. Well then, I am sure Miss Bronson, who has a fine sense of these things, will be happy to let you borrow the treatment room.\"\n\n\"We could be in love for a while,\" said Lori, her strong, realistic face undisturbed by his proposal, the harsh light, or Miss Bronson's periodic doorchecks one of which still echoed in the metal furniture. \"But nothing else is practical.\"\n\nSuffering a partial aphasia for the German language, he asked her to repeat.\n\nIngenuously, she shook her hair and said in his own tongue: \"Uh little surp-rise: I study English already. ... Wiss one another we may to love for some time. But alvays is _untunlich._ \"\n\nIn his anticipatory fancies he had alternated between smugly hearing an absolute acceptance and listening with shame to an unqualified turndown. Instead, he had the usual compromise. He must learn, damn him, that people out there in the universe beyond his head were real and unimaginable. Had Schild really been his friend? What more could he, Reinhart, have done to prove it? Did he really love Lori? If not, why was he willing to live in that abominable cellar?\n\n_\"Liebling\"_ he answered in German, \"I don't understand what you mean.\"\n\n\"Then you never will if I try to explain, Unteroffizier Carlo Reinhart, because my talents are not in analysis. You have met my brother and my husband. They never have allowed me to do anything but listen.\"\n\nReinhart left his table-seat, went to her, and took her hand. \"Nobody has ever appreciated you.\"\n\nLaughing astonishment, she squeezed him back. \"What a strange remark! But I assure you that I have nothing to say, I have no ideas, why should I want to speak of nothing?\"\n\n\"That's the old Prussian nonsense, children, kitchen, church is all a woman's good for.\" He was made reckless by his indignation: \"Then come to America! I will send for you as soon as I arrive.\" Till that moment he had never believed he would go there again himself, but the sensible man, the Rotarian, the Philistine, who resides in the liver, somehow survives all blows to head and heart.\n\n_\"Kinder, K\u00fcche, Kirche...,\"_ she repeated. \"I have none of them. However, you would think my idea of American women just as funny, no doubt, but I shall spare you. America! This will interest Bach. You surely did not suspect he has long dabbled in technology. _Also,_ he now announces to me that he has invented some means to make a glass which withstands heat. He has done this without a laboratory, simply mathematical equations in a notebook. Is this possible?\u2014no. But in America someone will give money for it, perhaps. Is that likely?\u2014no. But\u2014\"\n\n\"I didn't mean you could bring Bach!\" Reinhart thought: besides, we already have Pyrex.\n\nShe shaped her thin lips as if to pronounce _o_ umlaut.\n\n\"You understand,\" he said. \"I even like him, but be fair once to yourself. That is no kind of life for a young woman. It isn't right to sacrifice oneself for somebody else, no matter who.\"\n\nPlacing upon him her famous direct look\u2014that for which he loved her\u2014she answered: \"Certainly. So do not do it for me. I don't know what 'Teutonic efficiency' is, since I have lived in Berlin all my life, but here you have an imitation: one, I love Bach very much; two, think of your self-respect! I am old enough to be your mother.\"\n\nIn confusion's rage, he shouted: \"Then what did you mean by all your hints? If you love Bach why do you say you and I can love each other for a little while? Either way it's a betrayal of something or other. ... I hate things that are dishonest and secret.\"\n\nHearing the nurse at the door, he withdrew his hand. Miss Bronson's beet-face, pickled, cautioned against further noise and gave him five minutes to conclude.\n\n\"Then you must hate love,\" said Lori, \"not to mention life\u2014no, I don't mean that. It has been a long day for me. Fr\u00e4ulein Leary stayed home from duty and had numerous requests\u2014to press her clothes and so on. I think she is in love again. Why not get yourself an American girl? People from different countries really don't understand each other, as Bach says.\"\n\nWith one or two other more important items to check off his list, Reinhart put them by, to insert here: \"That is why we have these tragic wars.\"\n\nLori rose and gathered her old coat about her waist. \"According to Otto, no. War is the one time when they really do understand one another. Therefore he champions obscurity in human affairs\u2014no,\" she raised a hand, \"I will not discuss it; they both say I never can get anything right.\"\n\nHe had not only recaptured his sense of time: he had got back a better one than he lost, with a precise second hand. Exactly four minutes of Bronson's ration remained.\n\n\"Lieutenant Leary is in love _again._ Who was it the first time?\"\n\nThe candid eyes were now impure. \"Yourself, of course.\"\n\n\"Don't do me any favors. I'm not really sick. I've been faking all the while. And by the way, does nothing affect you? If you remember, Schild and I were at your home only a few hours before he was killed.\"\n\n\"If you remember, I have lived in Berlin for twelve years of Hitler and five years of war.\"\n\n\"I'm sorry,\" he said truly. \"I just want you to tell me who was Miss Leary's lover in August.\"\n\n\"Facts, always facts, what will you do with them? _Oh-kay, it's ah deal!_ This captain who lives, lived, with Oberleutnant Schild.\"\n\nHe began loudly to laugh, then choked it off for fear of Bronson. In mirth he used his own language: \"He is shorter than her!\"\n\n\"Please? As yet I don't understand so much English.\"\n\n_\"Knorke,_ it doesn't matter. ... So you are old enough to be my mother. You really are a twin of the doctor? Which would make you sixty.\" He shook his head. Two minutes left.\n\n\"Sixteen February, 1905, for both of us. Otto was the younger-looking before he went to Russia. I did not see him again until after he had gone through the camps.\" She shook Reinhart's hand, once up, once down. \"If at eighteen I had had a son like you, and he survived, I would not be disappointed now.\"\n\nBronson stuck her head in and called time, and Reinhart growled: \"Go away or I'll tell Lieutenant Llewellyn you applied duress.\" She winced and left.\n\n\"However,\" Lori went on, \"I didn't and I'm not disappointed either.\"\n\n\"You would say then, life goes on.\"\n\nShe pushed a sportive lock behind her ear; she stood in need of a washing and combing; in the center of a general relief which he could not explain, Reinhart felt a twinge which he could: he would never provide the brush and soap.\n\n\"I do not!\" she answered fiercely. \"Life can do as it pleases.\" Still feral, she leaping captured his neck, drew down his head, and kissed his mouth.\n\nHe concealed his momentary anguish of regret: \"Thanks, anyway, for never caring about me.\"\n\n\"Have it your own way.\" Tough, small, unkempt, Lori marched modestly into the corridor, uneasy Bronson voiding the route. Once through the outer door, she returned to hovering, vacant, liberated Reinhart for her formal peroration.\n\n\"I have forgotten! Here comes some English: 'Ve mawrn zuh death of a man of honor, First Lieutenant Schild, zalute zuh gallant Corporal Reinhart, shall effer keep green zuh memory of the former, and await with affection and respect the... re-choining by the latter of our fellowship, Knebel, M.D., your dear Lenore, and yours truly Bach, Ph. D.' _Also,_ I have remembered every word!\"\n\nOld Sad Sack St. George had topped Veronica. Now Reinhart understood why the captain had not tried to visit him on the psycho ward. He should notify St. George that Very worked nights, up to suppertime the coast was clear; except that this action was as much as to admit he was privy to their quondam goings-on. And he detested having the goods on anybody, which were always squalid. Besides, the captain, who physically favored his father, carried to Reinhart a suggestion of what he, himself, would be in twenty years: middle-aged, ingratiating, secretly prurient. He didn't try to get in touch.\n\nWith Marsala, too, contact had been broken, which he laid to a primitive, Italianate superstition towards bats in the belfry: extravagance was permitted only in the service of lust, drink, and anarchy. Well, it had been an accident anyway that he was quartered with the damn guy; he was never so close to Marsala as the fellow ginzos with their home packages of sausage and cherry peppers.\n\nVeronica returned to duty the next evening and squirted him loving, guarded looks as she went about ward business, but not having taken his afternoon nap, he dropped off to sleep at lights-out and so did not get to the office. Next morning he suspected having been touched on the face sometime during the night, but it could have been just a dream. However, he did find under his pillow a note on unlined paper, which read: \"I'm going to knock myself off\u2014Jesus destroy this mesage.\"\n\nHe actually believed it was Veronica's until Trooper reached over a long, thin arm and tore it away, saying, \"I changed my mind.\"\n\nReinhart looked at his bird-dog face and said: \"That wouldn't have settled anything.\"\n\n\"You're wrong,\" Trooper answered. \"It would so have, but I just realized I don't want anything settled. That's my trouble.\"\n\n\"Well, what do I care about you?\" Reinhart said irritably.\n\n\"That's all right. Nobody does. That's why I used to piss the bed, to get somebody to. But they didn't.\" He pulled the sheets over his head and said, underneath: \"I don't care any more.\"\n\nFurious, Reinhart jumped across the aisle and re-exposed the forsaken face. \"Knock off that crap, Trooper. Tell me, is it true you got the Silver Star in the Holland jump?\"\n\n\"I didn't deserve it. The ones who did were all killed.\"\n\n\"Don't hand me that. The fact is I understand you were screwed. Anybody else who singlehanded bumped off ten Germans and captured fourteen more would have got the DSC. An officer would have got it.\"\n\n\"He would?\" He crept up on the pillow but still disbelieved, and they argued, Reinhart temporarily winning. However, it would be a long fight to get Trooper to understand that the world, and not himself, was wrong.\n\nHe concealed his new mission from Millet when they talked, and observed the captain's techniques. Afterwards, with his own variations, since Trooper was not so sophisticated a case as he, he used them on Trooper. Trooper ate a good lunch for the first time since he had come on the ward. Another two days, Reinhart had him traducing the doctors.\n\nHe said, almost smiling: \"Reinhart, you ought to take up this psycho stuff when you get out of the Army.\"\n\nReinhart winked. \"Have you been reading my mind?\"\n\nTrue, like knighthood, this profession gave you a permanent upper hand; like the priesthood, it made everybody else feel guilty and also grateful; like the Jews it was much reviled yet indispensable and always right. For example, Millet as a person was probably not much\u2014he looked as if in civil life his sport was golf; his tips to caddies, meager\u2014here he sat as universal daddy.\n\nHe sensed a certain competition with Millet in succeeding interviews, but was forced to simulate his old quest for approbation.\n\nTowards the end of the week he reported: \"I am all right again, Captain. I am sure now. I sleep well and in the regular hours, I realize that wound in my head was just imaginary, time is again just as it used to be, and I am not suspicious of anybody. Your treatment has been successful.\"\n\n\"That's good,\" admitted Millet. \"Why do you refuse the recreational therapy?\"\n\n\"Because I don't like to weave baskets and I already have a billfold.\"\n\nMillet said permissively: \"Uh-huh. Nurse Reynolds tells me\"\u2014 he found the place in a document\u2014\"you stated a wish to make a shoulder holster. Which she opposed. Were you\u2014\"\n\n\"Oh that was a kind of joke. I wanted to give it to my roommate Marsala. His brother's a hood in Murder, Incorporated.\"\n\n\"No, I don't question that. I merely wished to know if Reynolds' refusal made you angry.\"\n\n\"Yes\u2014well goddammit, she said no in that sweet, tolerant manner used towards psychos, yet I know what she was thinking\u2014don't let Reinhart do anything that suggests violence.\"\n\n\"What did you do then?\"\n\n\"I walked away and didn't say anything.\" He stared at Millet's pale, bored eyes and shouted: \"What in hell do you think I did, beat her up?\"\n\n\"Did you want to?\"\n\n\"If you don't mind my saying it, sir, you know about as much of a man's mind as a golfball.\" He glanced at Millet's desk, insultingly clear of letter openers, etc., even pencils, for obvious reasons. \"If you mean deep down, I probably wanted to screw her. All men want to make love to every woman and kill every man. Man is a savage only partially tamed.\"\n\nMillet smiled. \"Is that your own theory?\"\n\n\"I read it somewhere, and then consulted my own soul. I know you people don't think a man can help himself, but I have.\"\n\n\"Was Lieutenant Schild your friend?\"\n\nReinhart sighed. \"You want me to say no. But one thing I will not do to get out of here: lie. It seems very clever to look only for the deep secrets. What we see of a person is supposedly only the false exterior; what he _really_ is, is underneath and hidden. Thus a man who appears generous is really selfish, great lovers are secretly queers, and heroes are really cowards covering up. A fellow who feels guilty about the Jews is actually the worst anti-Semite of all, and so on. No doubt this is true. But out in the world we have no time to check these things. If a bully comes towards you with a club, you have no chance to reflect that he is actually not frightening but pitiful, that if you gave him love and understanding he would be your friend.\n\n\"Because if the front is a lie, so are the depths when taken alone. For himself, a hero may be a coward. For you, if he is on your side in a battle you do not want to know what he is in some reality outside yours. A Don Juan may be a fairy, but in practice he will make love to your girl and not to you. Do you see what I mean? The fa\u00e7ade, too, has a reality and truth. You sit here in front of me like a god, asking me questions which I cannot ask you. Why? For reasons of your own. Somewhere back no doubt you grew a guilt towards people with mental troubles because you really have contempt for them. But I don't want to hear about it and obviously you don't want to tell it.\n\n\"Now the Jews and me. My feelings about them are irrational. Actually the Jews bore me stiff. And so do the Germans. All I ever cared about was old medieval N\u00fcrnberg, and that is long gone. Italy, I think, is what I like, with sunshine and that melodic language. I also hate politics and sociology and all that crap that deals with people as groups. I hated those mobs of idiots screaming _Sieg Heil!_ and who didn't?, but I also dislike those hordes of Russians in Red Square, who in spite of Communism are supposed to be generally good, and also the 'starving multitudes' of Asia and the 'laboring masses' everywhere. I name these examples in an effort to be honest. I don't like conventions of generals and bosses any better, but there everybody agrees with me.\n\n\"So with the Jews, who seem to be a persistent mob throughout history, only acting in the reverse of the usual mob; they storm nothing but are stormed. They are always around with their dull troubles and their rituals and their foods, feeling special everywhere and superior. I confess I used to think it was a trick for the Jews to always complain about mistreatment. They seemed a race of gripers.\n\n\"Then the Nazis came. Or rather, I finally noticed the Nazis. And they were something new. When you speak badly about the Nazis you cannot tell a lie. Maybe, secretly, every gentile wants to kill every Jew, but the Nazis did it in practice and the other Germans, or many of them, didn't care. But you see, _someone must care._ \"\n\nMillet raised his head, which had been lowered as if in sleep, and asked: \"Why?\"\n\n\"So that Germany will not perish.\"\n\n\"But you were concerned about the Jews.\"\n\n\"If you want to understand anything, you must listen,\" Reinhart chided. \"I am concerned about myself.\"\n\nMillet's head sank again.\n\n\"So I met Schild that night. He forces himself on me. The motives get all mixed up, who is doing what for whom. We listen to a man who is himself confused. It is a grotesque evening, like everything in Berlin turns out to be: giants, twins who are apparently twenty years apart, blind men, would-be abortionists, experts on art, turncoats, Communists, ex-prisoners of the concentration camp, good Germans who turn out to be bad, and vice versa, and Schild and I.\"\n\n\"How many people were there?\"\n\n\"In addition to Schild and me, only three. I assure you it was fantastic and ridiculous. And all in this damp cellar, but we sat on a Goblin couch worth a thousand dollars. And afterwards we get in this mortal combat. But I'll tell you this: it all happened and is still easier to believe than the concentration camps\u2014which, by the way, the Russians have, too.\n\n\"Now I'm ready to answer. Was Schild my friend? On one hand, yours, no. I used him. If he hadn't been a Jew I wouldn't have given him a minute, for he was a kind of creep. I felt this definite satisfaction when he got it in the back, and it wasn't the one you spoke of. More complex than that. I felt it because, fighting for him as I was, _nobody could blame me for his death._ Well, here comes a joke: no one does but myself.\n\n\"But was he my friend? In my sense, yes. He was someone I could talk to, and not the way I am talking to you, which is a sort of fraud since you are invulnerable and never talk back. And then for another reason. When you hear it you will never let me out of Psycho, because I guess it means I really am nuts. When Schild was a boy he read the King Arthur stories. And he still believed them up to the time he died.\"\n\nMillet asked lazily: \"What's 'nuts' about that?\"\n\nReinhart groaned: \"Because so do I. Really.\"\n\n# _CHAPTER 23_\n\nTEMPELHOF AIRPORT WAS STILL A mess of cracked-eggshell buildings, but the Air Force had policed up the field and laid its steel-mesh runways. Grounded craft sat dirty and rather larger than they looked in the sky, on what Reinhart believed was in the jargon of the trade called the \"apron.\" Identification of airplanes was a prideful talent with some people, not him. The nearest fellow in the party, a tall thin T\/5 with heavy eyebrows arched in perpetual curiosity, pointed out a Liberator, joked: \"It looks like a pregnant dachshund.\"\n\nUpon application the T\/5 confessed to being a case of chronic dermatitis, showed a bandaged right hand, said with a smirk of self-hatred: \"I guess you were wounded.\"\n\n\"No,\" said Reinhart, pointing to Trooper, who stood sickly at his left, \"he and I are psychos.\" Succeeding that, Trooper dug him with an elbow of embarrassment; he turned to the gutless fellow and bawled him out; therefore he did not see the T\/5's reaction.\n\nBut he heard him say: \"Well, this skin trouble is supposed to be psychosomatic, so we're all in the same boat.\"\n\n\"It's nothing to brag about,\" Reinhart answered. Crushed, the T\/5 monkeyed with his duffle bag.\n\nStaff Sergeant Owens, in charge of the patient group from door of hospital to hatchway of plane, again called the roll and lost four names in the roar of revving engines and the braggadocio bellowing of mechanics.\n\n\"See that rusty heap over there behind the Liberator?\" Trooper asked Reinhart. \"I bet you beaucoup marks that's what we got to ride. And we won't get any chutes. Oh my busted back, I feel bad.\"\n\n\"Now Trooper, I'll tell you a thing. Know that Air Corps gunner down at the end of the ward? When I told him we were shipping out to Paris by plane, he said, 'Then I'll never get out of Berlin. I love to fly but one time two months ago I dreamed the engines conked and I had to jump.' That's why he ended up in Psycho\u2014the dream has haunted him ever since. He says he would rather go down in flames than hit the silk. Everybody has his own horror. You don't mind the jump but are leery of the plane.\"\n\n\"That's on account of my training, Carlo. Those instructors never had any elasticity.\"\n\nInstead of wondering what that meant, Reinhart hung himself up on his own term: he was very leery of Very Leary. Although as late as the evening before she had betrayed no knowledge of his leaving, he, the old victim of guile, now practitioner, would not feel safe until he soared the air. He looked towards the buildings and would not have been astonished had Veronica come sailing from them and taxied up the mesh, dwarfing the Liberator. What a piece to run away from!; he supposed again that he _was_ nuts. The T\/5 there, who stared hornily at the female member of a professional party heading towards them, would sell his country for a Veronica.\n\nHe was torn by a distant shout and in involuntary panic tried to hide behind emaciated Trooper, who, a blotter for anybody else's emotion, almost wept: \"They after you, too?\"\n\nReinhart regained his self-respect. \"Don't get your balls in an uproar. I'm looking out for you.\"\n\n\"It's not,\" said Trooper, \"that I'm afraid to get hurt. I just don't like the humiliation.\"\n\nThe shout was closer, and clearer, and in the two syllables of Reinhart's cognomen. He turned to face the firing squad and saw\u2014Marsala. He ran to meet him, ignoring Sergeant Owens' howl.\n\nTen yards away, Marsala slowed to a measured, inexorable pace, waded in, hosing his target with an abuse of great variety and color: scatophagous Reinhart, the traitor, the Oedipus Rex, the fornicator of infants, the defiler of graves, the double-barreled international bastard and revolving son of a bitch.\n\n\"Fuck you and fuck your friends,\" he at length concluded. He turned away. His eyes were wet.\n\n\"I was going to write you from Paris, buddy. They only told me yesterday I was on shipment, and wouldn't let me get out to say goodbye.\"\n\nMarsala wiped his nose and fired up a big black cigar, throwing the match over a shoulder.\n\n\"Well whaddo I care where you go in Paris, huh? What's it to me you die like a turd? You goin' on a plane, huh? I don't mean what I said then, I take off the curse, you die and ya blood will be on my heart. Here, you wanna cigar? I ain't mad no more. Besides you're a poor dumb cuckoo. ... You're shit, too. How'd you work it? I'll say this for you, you got a lot of guts and talent to fool that old Millet. I been wrong about you for years. I thought you was this college type. But for Christ in heaven, after looking at you I think we run my brother from Murder Inc. for chief of police. You make a whore out of this young kid, you knock off a guy with your bare hands, then you play nuts and get home before everybody else. What college you come from, I think it was reform school.\"\n\nReinhart modestly smiled at this somewhat inflated pr\u00e9cis of his career, which however was sound in the essentials\u2014including the last-named. Millet had been on the verge of sending him back to duty\u2014believing in the actuality of King Arthur turned out to be quite O.K. mentally\u2014when in the interests of a new scheme Reinhart relapsed into the old symptoms. Just as he had foreseen, the ward needing his bed for new arrivals, Millet got rid of him.\n\n\"I don't know if this is the best way. I might get hung up in Paris for months, but I have to take care of that paratrooper over there. Where do you think they'll send us?\"\n\nMarsala spit out a fragment of wet tobacco. \"The 179th General. That's on the north side in a place called Clitchy.\"\n\n\"Stinks, I guess.\"\n\n\"Oh no.\" Marsala's eyebrows climbed. \"I saw it. I ran down there when we was in Normandy. It ain't a sow pen like the 1209th, I tell you that much. And then you get a pass, you go to Pig Alley\u2014and don't try to crap me any more you would go to this Loo museyum.\"\n\nThe party of flight nurse, co-, and full pilot had passed them, and sure enough, skirted the Liberator for the rusty cargo plane behind it. Reinhart felt imminent-departure gas in his stomach.\n\n\"Well, this is it, buddy. I'll write you a letter, tell you how it goes. You'll get home yourself any day now, you got more than enough points.\" He shook his hand.\n\nMarsala diffidently picked his nose. \"I never answer.\"\n\n\"Why not? You going to hold a grudge?\"\n\n\"Whaaat grudge, you rummy?\" He punched Reinhart's hand aside. \"Give the Princess a smooch for me. ... Now I guess with all your twat here you forgot her: you know, that married chicken with a husband in the paratroops.\"\n\nOh by God, Dianne Cooley. He owed her a letter for three months. And her husband Ernie, in the 82nd Airborne, the bright shoulder patch of which he saw, across the field, on the narrow shoulder of Trooper. It was an off chance.\n\n\"To tell you the truth,\" he said. \"I'm not convinced that I ought to leave. I like the Army, as I always told you, and I like Berlin, but it seems to be a good time to get out of both. But I'll have to play it straight in Paris. I don't want a Section Eight discharge. That might affect my career.\"\n\nMarsala shivered in his overcoat. \"It's gonna be a cold winter here for Jesus sake. A week yet to Halloween and it's already like a witch's tit. They're cutting down trees in the Tier Garden. No coal. ... Career? I got your career.\" He squeezed himself in the crotch. \"You're gonna be a hood, that's what.\"\n\n\"A psychiatrist. How about that? Except for a couple of people, including you, everybody I see is sick, boy, and bad.\"\n\n\"Especially you,\" Marsala complained. \"I take back what I said about you fakin'. I knew old Millet couldn't be fooled.\"\n\nReinhart shook the buddy's hand again and saw the black eyes swim. \"So long, Jimmy, you were the best of them all. Have fun with Trudchen.\"\n\nHolding his back stiff, he rejoined the group of patients. Time, where did old Time go, what were its mysteries? Not the narrow time towards which some weeks ago he was disordered, but that great gray fog behind us and before us, into which our lovers and friends vanish, events pass, and which claims even our old selves as we stand here in the limited clearing, nude in our newest one. Having learned from the Italians that crying is no reflection on a fellow's manliness, Reinhart dropped a tear or two for the summer of 1945, already gone; the war, long gone; the Army, soon to go; his twenty-first year, going drop by drop; and inventoried in water the dear people known and lost in this adventure, Schild, Lori, Bach, even including Veronica, whom he now believed he really did love but even so intended to remain adamant towards, and that was the sadness of it.\n\nBy the time he rejoined Trooper his face had dried. He anyway reached for his handkerchief. A tractor was towing the cargo crate to a clear vista on the landing strip.\n\n\"Yep, that's it,\" said Trooper. \"Like I said, the old C-47. I'm gonna faint.\"\n\n\"Like hell you are,\" Reinhart muttered. \"Wait a minute, anyway, I'm looking for something.\" No handkerchief in his pocket. He opened his duffle bag. On top lay a little Red Cross sack of dirty socks and pinned to it was a note which he could, and did, read without undoing.\n\n> Kiddy\u2014thanks for not making a scene about leaving. You have always been decent about a person's feelings. I will always think of you close to my heart but I can't get in touch with you anymore like this or any other way, _because_ \u2014well, I never did tell you the name of my real boy friend and I better not now\u2014ha! what I called him in front of you was s.o.b.\u2014nasty to laugh tho, because I do love him very much and now we are back together. He has to get a divorce from his wife but is not a Roman Catholic so its O.K. I have sinned but true love conquers all. We plan a double wedding with Ann Lightner & Lt. Pound who is in the same fix as X. Don't know where we'll live\u2014all around, I expect, since X is Regular Army. Oops, maybe I told you too much. Anyway, I send you all the love I can without being disloyal to my Husband To Be. Your intimate friend,\n> \n> VERY\n\nHe got a handkerchief and blew his nose just as Owens called the roll again and, conveniently, just at his own name, and the sergeant took it as answer. They filed towards the open doors in the plane's belly. Trooper didn't collapse, because Reinhart threatened to forsake him in Paris if he did.\n\nReinhart was last in line. He took a final noseful of Berlin air, which was cold and fresh and yet carried a faint dust of ruin. German rubble-workers around the administration building conversed in their native argot, and by some acoustical principle he could hear them.\n\n_\"Kommste imma erst so sp\u00e4t nach Hause?\"_\n\n_\"Nee, nur wochentags. Der Sonntag jeh\u00f6rt meiner Familie_ \u2014 _da schlafe ick'n janzen Tag.\"_\n\nHe did not understand a word. From the air the city would look like the crater-pocked, man-void moon. Finally one foot, and then the other, stepped from Brandenburg sand-plain to echoing metal floorboard.\n\nThe flight nurse, who wore a long green coverall, took the roster from departing, sycophantic Owens and began her own count. This time Reinhart watched for Trooper's response. It came on \"Poteet, Hastings F., Jr.,\" a name Reinhart had heard as many times as the roll was called and never connected with his patient for the simple reason that Trooper did not answer properly but rather raised one finger and coughed.\n\nReinhart shouted \"ho\" at his own name, which came right after, and asked Trooper: \"Did you know a guy in the 82nd named Ernie Cooley?\"\n\nAnd Hastings F. Poteet said instantly, with no sense whatsoever of the coincidence: \"Oh you know him, too. They'll never get Cooley.\"\n\n\"I'll be damned. You really know him, Ernie Cooley, from Norwood, Ohio? I used to go out with his wife, if it's the same one.\" He expatiated on the theme of one small world.\n\nWhich made no impression on Trooper, whose delusion was that while the world is infinite, all things are simultaneous. He waited with his polite, beagle eyes until he had his chance to say that Cooley had deserted in Normandy in June, 1944, almost as soon as the chutes hit the ground in their first jump of the war, and that, again, they would never get him\u2014because if they did it was curtains.\n\n\"Why?\"\n\n\"Desertion in the face of the enemy. They shoot you in the heart for that\u2014and for other things too,\" he added darkly.\n\nThe T\/5, who sat on Reinhart's left and had been listening, struck his curious brows into their business: \"That's almost as bad as what sometimes happens in the hospital: desertion in the face of an enema. When they catch you for that, you get shot in the ass.\"\n\nReinhart thought it very funny\u2014he was near hysteria, anyway, at leaving Berlin to rejoin the earth-people; the doors were still open, the engine quiet, the pilot outside on the ground, lazily joking with a mechanic; he could still burst away and regain the great, ruined, dear city\u2014but a single slight smile and the T\/5 would own him for the rest of the flight.\n\nHe looked about for another victim on which to stick this adhesive fellow. Against the other wall, on the line of metal seats which paralleled his own, sat eight or nine types; on his side, eight or so more. In addition to Trooper and himself there were five other psychos, all quiet cases, whom he knew only by sight; one obvious traveler from the skin ward\u2014a sergeant of limited dimensions, whose acned cheeks were relief maps of Berlin; beyond himself, nobody from the staff of the 1209th.\n\nThe pilot climbed in and sauntered forward. Ground crew without, and nurse inside, sealed the doors, and probability surrendered to necessity. His chance was forever gone. Only Trooper's whine saved Reinhart from claustrophobic frenzy.\n\n\"We won't need chutes, old buddy,\" he told him with a pat on the shoulder patch. \"These machines never fail.\" He looked across to Sergeant Acne for confirmation and saw, in spite of the eruptions, the crewcut showing at one side of a cocked overseas cap, the OD overcoat\u2014he saw\u2014well, he saw, but sick in the gut from his hallucination he begged forgiveness of Jehovah Millet and would have given blood sacrifices to have Him there to say he did not see\u2014Schatzi. \"Do they, Sarge?\" he nevertheless asked.\n\nSarge silently turned his head towards the pilot's cabin. Poor chap, if he wasn't Schatzi\u2014in his profile, corrupted by the malady, there was little likeness\u2014he was to be pitied, for a hideous boil lived in the very orifice of his left ear. He had not heard.\n\nBut why should he be Schatzi? How could he be? From the second button of his coat hung the medical tag with which they each had been labeled, like laundry sacks, at the hospital. Then there were dogtags, medical records, shipping orders, and duplicate copies of the roster for everybody from Eisenhower down. And what of the ludicrous, revealing accent? Of course, some GIs were refugees, and many native Americans went through the Army with never a public word but \"sir.\"\n\n\"Y'all want some chewin' gum?\" The nurse stood before them, offering Juicy Fruit. \"Y'ears won' have diffi-culty with th'air preshah, you chew gum.\" Tall, serpentine, rather slack-titted in the coverall, wearing tawny hair a bit long for a servicewoman, she handed a slice to each\u2014her fingers touched Reinhart's and did they not linger?\u2014and turned to the other row.\n\nSarge accepted his piece, unwrapped it with enormous care, folded the paper and placed it in his breast pocket!, put the gum in his mouth, chewed\u2014and the largest boil beneath his left eye loosened and fell to the floor, where it stuck like the actor's putty it was.\n\nNow was Reinhart astonished at himself: despite this proof he could not believe sergeant and Schatzi were synonymous. In Berlin he had learned to doubt all appearances, which must also include a false one: that is, its falsity might consist in its being real. The world was strange\u2014and interesting.\n\nAnd difficult. For of course he recognized Schatzi and his problem was what to do about it, which he would rather avoid. Crucial times were not at an end with the simple killing of a Monster, the dying of a Schild, with an unrequited proposal, or with leave-taking of lovers and friends; nor even with personal _non compos mentis._\n\nBut the gods, to whom he was dear, no matter how far they had permitted him to wander alone, finally furnished aid. Apollo resolved himself into a sunbeam, came down through the livid overcast and penetrated the Plexiglas window, striking the T\/5 in the medulla oblongata, inspiring him to jar Reinhart with his elbow and add: \"That's what the sentries say in the hospital: 'Halt. Who goes there, friend or enema?' \"\n\n\"What did we do to deserve you!\" Reinhart cried in burlesque despair. \"Talk to Trooper. I see a guy I know.\" He rose and crossed the cabin.\n\nThe seat at Sarge's left had been used briefly by the nurse to sort her gear; now it was empty. Sarge turned the other way when Reinhart took it. Seen at six inches, the make-up was an outrageously poor job, the acne an obvious work of mucilage, eyebrow pencil, and lipstick (Trudchen's 'raddest of the rad'). Whom could it have taken in for a moment? Answer: Every typical person, who would no sooner see the disability than avert his eyes, so as not to embarrass the sufferer; so as for health's sake to suppress an interest in the corrupt; so as\u2014but Reinhart, enough! The typical person simply would not imagine such a fake and therefore would not see one.\n\nUnder the cover of the other passengers' conversation, which was amplified in the metal tube, he asked into the sergeant's pseudo-foul ear: \"What did you hope to gain?\"\n\nSchatzi faced front, seemingly watched the T\/5 across the aisle, and answered, quietly venomous: \"I vill get to America and you vill not try to stop me.\"\n\nReinhart checked: the nurse had gone into the cargo compartment in the rear; Trooper, the T\/5, and a redheaded fellow were pooling complaints about the Army.\n\n\"You are mad,\" he whispered, \"hopelessly, utterly mad, and I pity you.\"\n\nSchatzi choked on his gum, which he had been chewing all wrong anyway\u2014too consciously, like all Europeans, as if it were candy rather than a substitute for twitching\u2014choked and responded in desolation and fear: \"Let me alone or you'll be sorry.\"\n\nReinhart covertly withdrew a roll of Occupation marks from his pants pocket\u2014they had been too many for the wallet\u2014and without a glance at the denominations pulled off and retained two, and placed the rest under Schatzi's tight arm.\n\n\"I promised I would get your money from Schild. There it is.\"\n\nSchatzi was truly overcome; among the patches of false acne grew areas of real emotion's rash, mottled, hot. He grappled with his American uniform, then with himself. He wiped his chin and drew away a palm of smeared cosmetic. His eyes sprinkled. Yet he managed to stay inconspicuous. They still had no one's notice.\n\n\"I do not understand your tricks,\" he whisperingly wailed. \"Are you the new agent? But being an American still comes first\u2014I cast myself upon your merciness. Oh God do not give me to Chepurnik. You cannot know what they are like, they are not people as we are. They are objects without blood. See what happens to Schild. With this I had nothing to do, believe me.\" But towards the end he had forgotten and raised his voice. The T\/5 heard and in a mock soprano began to sing \"They Wouldn't Believe Me.\" Schatzi turned on him the old death-ray eyes and he shriveled in midnote.\n\n\"I don't care about your squalid black-market deals,\" said Reinhart.\n\nInstantly Schatzi dried and hardened. \"Oh yes, your lovely friend Schild, for whom you would, and did, kill. You saw none of his profits, _ja_? He used you as a sexual rubber.\" At last he gave him the whole hideous face. But it was more ludicrous than repulsive. He stank of Juicy Fruit. \"Black market! Black market was my trade. This swine Schild sold his country. This fine land America that we poor victims of to-tah-li-tahrianism would die for, he died to betray. With good fortune I happen to learn of these facts in the course of my business. I report them to the Ami FBI, who are ready to seize him just when comes this well-known fight.\"\n\nCalmly, Reinhart enjoyed the lies, a souvenir of Europe. They would be all too rare in America. But he must get to work before the engine started, the propellers revolved, and his initiative was gone. If he knew the pedantic ways of people who do such things as fly planes, the C-47 would not kill its motor, once started, for the end of the world. And just this time he did not want Schatzi to succeed in an imposture. Those of the past he forgave him\u2014yes, truly forgave, not like a god but like a man; he expected no reward\u2014but this one was too vulgar.\n\n\"What shall we do about you?\" he asked, preparing to rise.\n\n\"You harm me at your own cost!\"\n\n\"My dear fellow,\" said Reinhart. \"With all good will, I cannot understand you.\"\n\nSchatzi began a sneer of victory\u2014or, at any rate, what _he_ thought was one\u2014but the Juicy Fruit clogged in his teeth. Humiliated, he plucked it out and held it in his right hand. Obviously it felt nasty there, and no better in the left. He brought the paper wrapper from his breast pocket and was on the point of rolling the little gum ball therein, when the nurse came out of the cargo chamber and, undulating bow-wards, saw his heresy and warned: \"You don' chew, you gon' be dog-sick.\" He guiltily returned it to his mouth, where, according to his expression, it grew to baseball size.\n\n\"Why you grinnin'?\" she pseudo-sternly asked Reinhart. She bent and read his tag. \"What's this, an Eyetalian name? Carlo. Kind of cute, though. You a psycho? Well then what do the normal ones look like?\"\n\nWhen she left Schatzi threw in the towel. He gagged on the gum, finally swallowed it, and again begged for mercy in the name of the United States of America: \"You people believe a man is what he will become rather than what he has been, _ja_? I tell you I have reformed. Just this minute, sitting here amidst your fine comrades who love one another. I do not ever belong to anysing. Loneliness! Lack of love! These can make a man to a criminal. Have I been a rascal?\u2014no. Yet had impulses come to me which have been dangerously near. After I arrange with great difficulty and money to be included in this shipping-to-Paris I discover your name on the list thereof. Wal, I thought, this Reinhart causes trouble and I strike back. His German relatives! Yes, I have found them, Heinz Tischmacher, son of his grandfather's sister and second cousin to him. Tischmacher, Heinz: office worker, member of the Nazi Party. Tischmacher, Frau Emmi: likewise. Tischmacher, Reinhold: twenty-one years of age, graduate of Hitler Youth to the Waffen SS, killed on the Eastern Front in 1944. Tischmacher, Gertrud, Trudl, Trudchen: sixteen, member of the Jungm\u00e4del, girls' branch of Hitler Jugend.\n\n\"On the other side of his descent Tischmacher has a distant relation to this monstrosity Bach, whom he takes money from, for four years, not to reveal to his Nazi comrades that Bach, Lenore, _Mischling_ Jew, has not gone to Switzerland so much as does she hide in a closet of her man's flat so as to avoid the Gestapo and subsequent killing.\"\n\n\"These then are my people,\" said Reinhart. But he worried only over whether he had committed incest with Trudchen. \"You have done a good job. I'm afraid this is all I have left to pay you.\" He brought forth his last two notes.\n\nChuckling madly, Schatzi refused them. \"No, these are not your kinsfolk, dear boy! Is not this evidence of my reform? This is my untruth with which I prepared to threaten you. Which now I reveal and confess. This evil impulse to destroy you which I have conquered. So Christ said, _'Die Wahrheit wird Euch frei machen.' \"_\n\n\"Destroy _me_?\" asked Reinhart. He banged his head back against the fuselage, denting it (the fuselage), and guffawed.\n\nSchatzi smiled, frightened to death but also hopeful. \"Ah then,\" he whispered, \"mirth and good feeling. You will not expose me, _ja_? In the States we must make a partnership: you belong and have the handsomeness and the muscles\u2014however did you break that great swine's back?\u2014I provide the mental.\"\n\nThe nurse appeared at the door of the pilot's cabin, her hip reared to catch her sexy, sinous wrist. She said: \"Y'all settle down and connect your seat belts. We go in two seconds. ... Ah got my eye on you, laughin' Carlo!\" She would be a different kind of piece.\n\nReinhart shook Schatzi's hand and winked elaborately. He whispered: \"My friend, you have my word on it.\" Then he went to the nurse, and for another reason feeling her supple arm, betrayed him.\n\n## AUTHOR'S NOTE\n\nAs many readers will have recognized, I am indebted to Konrad Heiden's classic work _Der Fuehrer_ (tr. by Ralph Manheim, New York, 1944) for some of the events in the career of one of my imaginary people.\n\n# A Biography of Thomas Berger\n\nThomas Louis Berger (1924\u20132014) was an American novelist best known for his picaresque classic, _Little Big Man_ (1964). His other works include _Arthur Rex_ (1978), _Neighbors_ (1980), and _The Feud_ (1983), which was nominated for a Pulitzer Prize.\n\nBerger was born in Cincinnati, Ohio, the son of Thomas Charles, a public school business manager, and Mildred (n\u00e9e Bubbe) Berger. Berger grew up in the town of Lockland, Ohio, and one of his first jobs was working at a branch of the public library while in high school. After a brief period in college, Berger enlisted in the army in 1943 and served in Europe during World War II. His experiences with a medical unit in the American occupation zone of postwar Berlin inspired his first novel, _Crazy in Berlin_ (1958). This novel introduced protagonist Carlo Reinhart, who would appear in several more novels.\n\nIn 1946, Berger reentered college at the University of Cincinnati, earning a bachelor's degree two years later. In 1948, he moved to New York City and was hired as librarian of the Rand School of Social Science. While enrolled in a writer's workshop at the nearby New School for Social Research, Berger met artist Jeanne Redpath; they married in 1950. He subsequently entered Columbia University as a graduate student in English literature, but left the program after a year and a half without taking a degree. He next worked at the _New York Times Index_ ; at _Popular Science Monthly_ as an associate editor; and, for a decade, as a freelance copy editor for book publishers.\n\nFollowing the success of Rinehart in Love (1962), Berger was named a Dial Fellow. In 1965, he received the Western Heritage Award and the Richard and Hinda Rosenthal Award of the National Institute of Arts and Letters for _Little Big Man_ (1964), the success of which allowed him to write full time. In 1970, _Little Big Man_ was made into an acclaimed film, directed by Arthur Penn and starring Dustin Hoffman and Faye Dunaway.\n\nFollowing his job as _Esquire_ 's film critic from 1972 to 1973, Berger became a writer in residence at the University of Kansas in 1974. One year later, he became a Distinguished Visiting Professor at Southampton College, and went on to lecture at Yale University and the University of California, Davis.\n\nBerger's work continued to appear on the big screen. His novel _Neighbors_ (1980) was adapted for a 1981 film starring John Belushi and Dan Aykroyd. In 1984, his novel _The Feud_ (1983) was nominated for a Pulitzer Prize; in 1988, it too was made into a movie. His thriller _Meeting Evil_ (1992) was adapted as a 2012 film starring Samuel L. Jackson and Luke Wilson.\n\nIn 1999, Berger published _The Return of Little Big Man_ , a sequel to his literary classic. His most recent novel, _Adventures of the Artificial Woman_ , was published in 2004.\n\nBerger lived in New York's Hudson Valley.\n\nIn 1966, two years after he wrote _Little Big Man_ , Berger stands at Little Bighorn Battlefield National Monument, the site of Custer's last stand in 1876. This was Berger's first visit to the famous battlefield.\n\nThis black-and-white image became the readers' vision of Berger: dark and esoteric. (Photo courtesy of Gerry Bauer.)\n\nA snapshot of Berger with his friend Zulkifar Ghose, taken in midtown Manhattan in the summer of 1974. (Photo courtesy of Betty Sue Flowers.)\n\nThis marked-up manuscript page comes from a story called \"Gibberish,\" from Berger's original short story collection _Abnormal Occurrences._\n\nIn this 1984 letter to his agent, Don Congdon, Berger tells Congdon that he was mentioned on _The David Susskind Show_ , a television talk show.\n\nIn this 1997 letter, Berger writes to Roger Donald, his editor at Little, Brown, about characters, props, and plot points in _The Return of Little Big Man._\n\nIn 1997, Berger wrote to Congdon about communications from Michael Korda, editor in chief of the publisher Simon & Schuster, and Donald.\nAll rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this ebook onscreen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of the publisher.\n\nThis 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, businesses, companies, events, or locales is entirely coincidental.\n\nAn earlier version of Chapter Ten appeared in New World Writing Number 8 under the title \"Confession of a Giant.\" Copyright \u00a9 1955 Thomas Berger.\n\ncopyright \u00a9 1958 by Thomas Berger\n\ncover design by Michael Vrana\n\n978-1-4804-0090-0\n\nThis edition published in 2013 by Open Road Integrated Media\n\n345 Hudson Street\n\nNew York, NY 10014\n\nwww.openroadmedia.com\n\n**THOMAS BERGER**\n\nFROM OPEN ROAD MEDIA\n\nFind a full list of our authors and\n\ntitles at www.openroadmedia.com\n\nFOLLOW US\n\n@OpenRoadMedia\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nPatricia Cleveland-Peck\n\nIllustrated by David Tazzyman\n\nYou\n\ncan\n\n'\n\nt\n\ntake an elephant on the bus . . .\n\nIt would simply cause a terrible fuss!\n\nElephants\n\n'\n\nbottoms are heavy and fat,\n\nand would certainly squash the seats quite flat.\n\nAnd\n\ndon\n\n'\n\nt\n\nsit a monkey in a\n\nshopping trolley . . .\n\nFor monkeys are naughty and find it jolly\n\nto snatch your shopping and chuck it about.\n\nNo, leave monkey at home when you go out.\n\nNor should a tiger travel by train . . .\n\nThink of the panic. Think of the pain.\n\nTigers are built to spring and to leap.\n\nThink of the passengers half-asleep.\n\nAnd\n\ndon\n\n'\n\nt\n\nhail a taxi if the\n\ndriver\n\n'\n\ns\n\na seal . . .\n\nWith such slippery flippers, he\n\ncan\n\n'\n\nt\n\ngrasp the wheel.\n\nThe taxi will slither and probably swerve,\n\nthen throw everyone out at the very next curve.\n\nA centipede on roller skates is rather bizarre . . .\n\nWith one hundred feet,\n\nhe\n\n'\n\nd\n\ngo fast and go far.\n\nBut to put on his boots would take him an\n\nage\n\n\u2013\n\nhe\n\n'\n\nd\n\nget in a temper,\n\nhe\n\n'\n\nd\n\nget in a rage.\n\nAnd don't put a camel in a sailing boat . . .\n\nIt\n\n'\n\ns\n\nfar too tricky to keep afloat.\n\nHis hump and his feet would, I think,\n\ncapsize the vessel\n\nand\n\nmake it\n\nsink.\n\nA giraffe in an aeroplane\n\nwouldn\n\n'\n\nt\n\nbe right . . .\n\nThe roof of a plane just\n\nhasn\n\n'\n\nt\n\nthe height.\n\nWith legs and a neck so bony and long,\n\na giraffe on a plane would simply be wrong.\n\nAnd\n\ndon\n\n'\n\nt\n\nask a whale to ride a bike . . .\n\nJust imagine what it would be like -\n\nwithout a bottom to sit on the seat.\n\nAnd how would he pedal without any feet?\n\nA\n\npig\n\non a skateboard?\n\nAnother mistake . . .\n\nAnd I\n\nwouldn\n\n'\n\nt\n\nput a hippo in a hot air balloon . . .\n\nThe\n\nbasket\n\n'\n\ns\n\ntoo small, there\n\nwouldn\n\n'\n\nt\n\nbe room.\n\nAnd if it did fly, with\n\nhippo\n\n'\n\ns\n\ngreat weight,\n\nit would come crashing down in a terrible state.\n\nAnd never let a bear near an ice cream van . . .\n\nBears gobble up ice cream as fast as they can.\n\nAnd if\n\nthey\n\n'\n\nre\n\nstopped they get annoyed,\n\nand an angry bear is one best to avoid.\n\n\" Then how can we\n\ntravel?\"\n\nthe animals shout.\n\n\"How can we animals get carried about?\"\n\n\"What\n\n'\n\ns\n\nthe best vehicle?\n\nWe\n\nhaven\n\n'\n\nt\n\na\n\nclue.\n\n\"\n\nWell,\n\nI\n\n'\n\nve\n\ngot an idea . . .\n\nHow about you?\n\nYes, animals on rollercoasters are good for a laugh . . .\n\nThere\n\n'\n\ns\n\nroom here for EVERYONE \u2013 even giraffe!\n\nSo\n\nit\n\n'\n\ns\n\ngoodbye to skateboards, balloons and THAT bus,\n\nfor we now have a conveyance that suits ALL OF US!\n\nwhee\n\nTo Isabel, with love ~ PC-P\n\nFor Mum and Dad x ~ DT\n\nBloomsbury Publishing, London, New Delhi, New York and Sydney\n\nFirst published in Great Britain in 2015 by Bloomsbury Publishing Plc\n\n50 Bedford Square, London, WC1B 3DP\n\nThis eBook edition first published in Great Britain in 2015\n\nText copyright \u00a9 Patricia Cleveland-Peck 2015\n\nText copyright \u00a9 David Tazzyman 2015\n\nThe moral rights of the author and illustrator have been asserted\n\nAll rights reserved\n\nNo part of this publication may be reproduced or transmitted by any means,\n\nelectronic, mechanical, photocopying or otherwise, without the prior permission of\n\nthe publisher\n\nA CIP catalogue record for this book is available from the British Library\n\nISBN 978 1 4088 4980 4 (HB)\n\nISBN 978 1 4088 4982 8 (PB)\n\nISBN 978 1 4088 4981 1 (eBook)\n\nVisit www.bloomsbury.com to find out more about our authors and their books\n\nYou will find extracts, author interviews, author events and you can sign up for\n\nnewsletters to be the first to hear about our latest releases and special offers\n\nwww.bloomsbury.com\n\nBLOOMSBURY is a registered trademark of Bloomsbury Publishing Plc\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nGerbail T. Krishnamurthy and Shakuntala KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7(C) Springer-Verlag Berlin Heidelberg 2009\n\nGerbail T. Krishnamurthy and Shakuntala Krishnamurthy\n\nNuclear HepatologyA Textbook of Hepatobiliary Diseases\n\nGerbail T. Krishnamurthy\n\nTuality Community Hospital, Hillsboro, 97123, OR, USA\n\nShakuntala Krishnamurthy\n\nTuality Community Hospital, Hillsboro, 97123, OR, USA\n\nISBN 978-3-642-00647-0e-ISBN 978-3-642-00648-7\n\nSpringer Dordrecht Heidelberg London New York\n\nLibrary of Congress Control Number: 2009926189\n\n(C) Springer-Verlag Berlin Heidelberg 2009\n\nThis work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law.\n\nThe use of general descriptive names, 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 protective laws and regulations and therefore free for general use.\n\nProduct liability: The publishers cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature.\n\nCover design: eStudioCalamar, Figueres Berlin\n\nPrinted on acid-free paper\n\nSpringer is part of Springer Science+Business Media (www.springer.com)\n\nDedicated with love and affection to our grandson Sachin Thomas Wheatley\n\nPreface to the Second Edition\n\nSince the publication of the first edition of Nuclear Hepatology almost 10 years ago, new advances have taken place, both in our understanding of liver pathophysiology and various imaging modalities. A unique feature of imaging of physiology is that it enables quantification. Since liver physiology is complex, quantification has been a challenge. Many publications in the past have often included quantification based on home-made software not available for others. Thus, the comparison of results from one center to another becomes difficult, if not impossible. The second edition of Nuclear Hepatology addresses these issues. Sophisticated software for liver and gallbladder function has been tested and validated through many of our previous publications. Now that software is FDA approved and available for others through a commercial company, it is hoped that future publications of imaging of liver and gallbladder physiology would routinely include quantification.\n\nAlthough the function of the hepatocyte is complex, it can be broadly divided into two main categories, substrate uptake at the basolateral border and intracellular transit prior to excretion into the canaliculi. The earliest manifestation of hepatocellular injury occurs in the form of prolongation of the intracellular transit time, which occurs long before the disruption of the basolateral border with subsequent rupture and death. The liver enzymes are released into circulation after the death of the hepatocyte. Hepatocyte function can be restored completely by treating the patient as soon as the intracellular transit is altered, but not after the rupture of the basolateral border (cell death). Measurement of intracellular transit time from a Tc-99m HIDA study enables early detection of the hepatocyte injury and early therapeutic intervention.\n\nMost of the chapters include updated information. The second chapter describes the latest information related to liver physiology and Chap. 5 provides sophisticated software quantification of pathophysiology. Cholescintigraphic images are correlated with morphologic images obtained with ultrasound, CT or MRI. Since PET\/CT imaging has currently become a standard in the care of the cancer patient, a detailed description is provided in Chap. 12. Latest information on biliary dyskinesia in adults and neonatal hepatitis in infants is updated. It is our firm belief that physicians, surgeons, and pediatricians caring for patients with liver and gallbladder disease would become more familiar with the latest advances in imaging technology and provide the best care for their patients, based on evidence from objective parameters.\n\nContents\n\n1 Morphology and Microstructure of the Hepatobiliary System1\n\n1.1 Morphology1\n\nReferences17\n\n1.2 Microstructure19\n\nReferences25\n\n2 Liver and Spleen Function27\n\n2.1 Liver Function27\n\nReferences44\n\n2.2 Spleen Function45\n\nReferences47\n\n3 Imaging Agents49\n\n3.1 Morphology and Physiology Imaging Agents49\n\nReferences67\n\n3.2 Radiolabeling of Red Blood Cells and Leucocytes70\n\nReferences73\n\n3.3 Gallium-67 Citrate73\n\nReferences74\n\n3.4 Somatostatin Receptor Imaging Agent75\n\nReferences76\n\n3.5 Fluorine18, 2-Flouro-2-deoxy-D-glucose (18F-FDG)77\n\nReferences82\n\n4 Imaging of Liver and Spleen Morphology85\n\n4.1 Imaging with Radiocolloid85\n\nReferences98\n\n4.2 Adenoma and Focal Nodular Hyperplasia99\n\nReferences105\n\n4.3 Hemangioma106\n\nReferences113\n\n4.4 Somatostatin Receptor Scintigraphy114\n\nReferences122\n\n5 Imaging and Quantification of Hepatobiliary Function125\n\n5.1 Hepatobiliary Imaging125\n\nReferences152\n\n5.2 Measurement of Hepatic Arterial vs. Portal Venous Blood Flow153\n\nReferences160\n\n5.3 Hepatopulmonary Syndrome160\n\nReferences165\n\n5.4 Duodenogastric Bile Reflux166\n\nReferences170\n\n5.5 Imaging and Quantification of Hepatocyte Asialoglycoprotein Receptors with Tc-99m Galactosyl Human Serum Albumin171\n\nReferences172\n\n6 Gallbladder, Sphincter of Oddi, Cholecystokinin, and Opioid Relation175\n\n6.1 Effect of Cholecystokinin on the Gallbladder and Sphincter of Oddi175\n\nReferences187\n\n6.2 Opioids190\n\nReferences194\n\n7 Intrahepatic Cholestasis197\n\n7.1 Imaging with Tc-99m HIDA197\n\nReferences205\n\n7.2 Imaging with Tc-99m Galactosyl Human Serum Albumin207\n\nReferences209\n\n8 Extrahepatic Cholestasis211\n\n8.1 Intraluminal Causes211\n\nReferences223\n\n8.2 Wall Thickening224\n\nReferences233\n\n8.3 Combined Intrahepatic and Extrahepatic Cholestasis (Sclerosing Cholangitis)235\n\nReferences239\n\n8.4 Extrinsic Compression239\n\nReferences241\n\n9 Diseases of the Gallbladder243\n\n9.1 Chronic Calculous Cholecystitis246\n\nReferences254\n\n9.2 Chronic Acalculous Cholecystitis257\n\nReferences270\n\n9.3 Acute Cholecystitis272\n\nReferences287\n\n9.4 Management of Gallbladder Disease290\n\nReferences296\n\n10 Biliary Dyskinesia299\n\nReferences315\n\n11 Pediatric Nuclear Hepatology319\n\n11.1 Congenital Biliary Atresia vs. Neonatal Hepatitis319\n\nReferences331\n\n11.2 Cystic Diseases of the Hepatobiliary System332\n\nReferences339\n\n12 Malignant Liver Lesions341\n\n12.1 Management346\n\nReferences346\n\n13 Liver Transplantation347\n\n13.1 Types of Liver Transplantation347\n\n13.2 Normal Functioning Liver Transplant351\n\nReferences360\n\nIndex363\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_1(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 1. Morphology and Microstructure of the Hepatobiliary System\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nLiver and gallbladder disease is a common entity around the world and according to the World Health Organization estimation accounts for 46.1% of global disease [1]. Liver transplantation for end-stage liver disease, segmental resection for tumor, and therapeutic interventional maneuvers has made it necessary to have thorough knowledge of the morphology of the liver and biliary system in much greater detail than ever before [2]. With the widespread application of segmental liver resection for living donor liver transplantation or radioablation of liver tumors, thorough knowledge of internal structures is critical for radiologists, nuclear medicine physicians, and surgeons. Since the publication of the first edition of our book in 2000, many advances in liver disease therapy have taken place, making it necessary to provide more detailed anatomic and histopathological information. Anatomical details of the vascular and ductal structures are well depicted on a multi-detector computer tomography (MDCT), magnetic resonance cholangiopancreatography (MRCP), and endoscopic retrograde cholangiopancreatography (ERCP), enabling identification of the third or fourth order branches on the images. Such detailed structural information is necessary for surgeons for segmental resection of the liver.\n\n## 1.1 Morphology\n\nLiver and gallbladder disease is a common entity around the world and according to the World Health Organization estimation accounts for 46.1% of global disease [1]. Liver transplantation for end-stage liver disease, segmental resection for tumor, and therapeutic interventional maneuvers has made it necessary to have thorough knowledge of the morphology of the liver and biliary system in much greater detail than ever before [2]. With the widespread application of segmental liver resection for living donor liver transplantation or radioablation of liver tumors, thorough knowledge of internal structures is critical for radiologists, nuclear medicine physicians, and surgeons. Since the publication of the first edition of our book in 2000, many advances in liver disease therapy have taken place, making it necessary to provide more detailed anatomic and histopathological information. Anatomical details of the vascular and ductal structures are well depicted on a multi-detector computer tomography (MDCT), magnetic resonance cholangiopancreatography (MRCP), and endoscopic retrograde cholangiopancreatography (ERCP), enabling identification of the third or fourth order branches on the images. Such detailed structural information is necessary for surgeons for segmental resection of the liver.\n\n### 1.1.1 Embryology\n\nThe liver and biliary systems develop from an endodermal bud that arises during the 5th week of intrauterine life when the embryo is about 3 mm in length [3]. The bud originates from the ventral surface at the junction of the foregut with the midgut, and soon divides into cranial (pars hepatica) and caudal (pars cystica) branches. The ventral pancreas sprouts near the caudal bud, and the dorsal pancreas arises on the opposite side at the foregut-midgut junction (Fig. 1.1.1). When the embryo is about 5 mm in length, the cranial and caudal branches become further connected by a common stalk, which later becomes the common bile duct. By the time the embryo is about 7 mm in length, the cranial branch (pars hepatica) divides into two cellular columns, which later become the physiologic right and left hepatic lobes [4]. The gallbladder develops from the caudal branch (pars cystica) and continues its connection with the common stalk through a channel, which later becomes the cystic duct. Canalization takes place to form a patent biliary tree (gallbladder, cystic duct, right and left hepatic duct, common hepatic duct, and the common bile duct). When the embryo is about 10 mm in length, the gut begins to make a 90\u00b0 clockwise rotation and completes it by the time it is 12 mm in length, which brings the ventral pancreas in close contact with the dorsal pancreas, facilitating their fusion into a single pancreas before birth [5]. This rotation brings the common bile duct posterior to the duodenum. Many congenital abnormalities around this region are secondary to mal-rotation at this junction. The caudate lobe arises separately close to the inferior vena cava, independent of the right and left hepatic lobes. The liver begins bile secretion by the 12th intrauterine week, thus completing the formation of the hepatobiliary system, the most complex metabolic factory in the human body. In adults, the liver weighs about 1,500-1,800 g and forms about 1\/50 of the body weight. In children, however, it forms a relatively much larger fraction (1\/20) of the total body weight.\n\nFig. 1.1.1\n\nEmbryology of the hepatobiliary system. The hepatic bud arises from the endoderm of the primitive foregut at its junction with the midgut when the embryo is about 3 mm in length. The hepatic bud divides into cranial (pars hepatica) and caudal (pars cystica) branches when the embryo reaches 5 mm size. The ventral pancreas arises from the pars cystica, which later gives rise to the biliary system. After a 90\u00b0 clockwise rotation (10 mm embryo), the ventral pancreas fuses with the dorsal pancreas. This rotation brings the common bile duct posterior to the duodenum (12 mm). The common bile duct opens into the duodenum at the postero-medial wall at an elevation called the papilla\n\n### 1.1.2 Liver Lobes and Surfaces\n\nMost of the liver is situated in the right upper quadrant of the abdomen underneath the right hemi-diaphragm, with the superior border situated at the level of the right fifth intercostal space. Liver consists of five surfaces: anterior, posterior, right lateral, superior, and inferior. The anterior, right lateral, and posterior surfaces are smooth, and superior and inferior surfaces are rough with grooves and fissures for entry and exit of the vascular and biliary structures (Fig. 1.1.2). The liver is divided into right and left lobes on the basis of either anatomic or physiologic markers. Anatomically, the liver is divided into right and left lobes on the basis of the line of attachment of the falciform ligament. The anatomic division serves no useful purpose in the management of liver disease and hence has been given less importance in recent years. The physiologic division, on the other hand, has gained much more popularity as it follows embryologic development and delineates functional lobes and segments, whose line of demarcation is used for liver resection during transplant or tumor surgery. The physiologic division into right and left lobes is indicated on the inferior surface by a deep fissure called the plane of Serege-Cantlie that passes from the gallbladder fossa along the inferior border to the inferior vena caval groove along the superior border [5].\n\nFig. 1.1.2\n\nSurfaces, segments, and lobes of the liver. The anatomic left lobe (divided by the falciform ligament) is much smaller than the anatomic right lobe, but the physiologic left lobe is much larger than the anatomic left lobe. The quadrate lobe, which forms a part of the anatomic right lobe, forms a part of the physiologic left lobe. Liver consists of three lobes (including the caudate lobe), four segments, and eight areas. HA hepatic artery, BD bile duct, PV portal vein, GB gallbladder\n\nThe falciform ligament, the marker of the anatomic right and left lobes, lies to the left of the deep fissure (Serege-Cantlie line) that divides the liver into physiologic right and left lobes. The caudate lobe situated posteriorly and quadrate lobe situated anteriorly form part of the anatomic right lobe, but both of these structures belong to the physiologic left lobe (Fig. 1.1.2). The anatomic right lobe is approximately six times larger (85%) than the anatomic left lobe (15%), whereas the physiologic right lobe is 70% and physiologic left lobe 30% in size. Thus, the physiologic left lobe is much larger than its anatomic counterpart. It is important, therefore, to be very specific while describing lobes of the liver, whether one is referring to the anatomic or to the physiologic division. Throughout this book, we will refer to the physiologic division unless otherwise mentioned.\n\n### 1.1.3 Segments and Areas\n\nThe liver is divided into lobes, segments, and areas. The division is made on the basis of either the vascular [6, 7] or the bile duct branches [8]. As both the vessels (hepatic artery and portal vein) and bile ducts travel together to every lobe, segment, and area of the liver, it usually makes no difference which structure is chosen as the reference point for the division. Segments are named by either numbers or given a directional nomenclature, similar to the division of the lungs into lobes and segments. Couinaud [6] and Bismuth [9] used numbers, whereas Healey and Schroy [8] preferred directional nomenclature. The imaging technology (nuclear medicine, CT, MRI, and ultrasound) and surgeons prefer to use numerical segments for the division [10-16]. Table 1.1.1 shows names and numbers adopted by different authors. Starting from the caudate lobe (I), the numbers go clockwise covering the lateral (II, III) and medial (IVA, IVB) segments of the left lobe, and inferior (V, VI) and superior (VII, VIII) half of the right lobe. The directional nomenclature used by Healey and Schroy [8] is much easier to remember, whereas the numbers require memorization. Recent authors seem to prefer the numbers proposed by Couinaud and Bismuth [6, 9, 17].\n\nTable 1.1.1\n\nNomenclature for hepatic segments and lobes as adopted by different authors\n\nLiver lobes | Hjortsjo (5) | Healey and Schroy (8) | Couinaud (6) | Bismuth (9)\n\n---|---|---|---|---\n\nCaudate lobe | Dorsal segment | Lobus caudatus | I | I\n\nLeft lobe | Dorso-lateral segment Ventro-lateral segment Central segment Dorso-ventral segment | Lateral superior area Lateral inferior area Medial superior area Medial inferior area | II III IV IV | II III IVA IVB\n\nRight lobe | Ventro-caudal segment Dorso-caudal + intermedio-caudal Dorso-cranial + intermedio-cranial Ventro-cranial | Anterior-inferior area Posterior-inferior area Posterior-superior area Anterior-superior area | V VI VII VIII | V VI VII VIII\n\nThree major hepatic veins drain the liver blood into the inferior vena cava, left hepatic vein, middle hepatic vein, and right hepatic vein. The middle hepatic vein, which follows the direction of the Serege-Cantlie plane (from the gallbladder fossa below to the inferior vena caval groove above), divides the liver into physiologic right and left lobes. The right hepatic vein divides the right lobe into anterior (V and VIII) and posterior (VI and VII) segments. The left hepatic vein divides the left lobe into medial (IVA and IVB) and lateral (II and III) segments. The right and left branches of the portal vein, hepatic artery, or common hepatic duct divide the liver segments into superior and inferior areas. The liver, therefore, consists of two large lobes, four segments, and eight areas (Fig. 1.1.3). Recent studies, however, suggest that in the case of extended liver resection for malignancy, Hjortsjo's segmental division may provide better anatomic detail than Couinaud's classification [18]. The caudate lobe (I), despite being small in size, is considered as a separate lobe mainly because of its unique embryology and vasculature [19, 20]. It lies between the hilar structures and inferior vena cava and consists of three parts: Spiegel's lobe (left), the paracaval (middle) and caudate process (right).\n\nFig. 1.1.3\n\nHepatic and portal venous system. The left hepatic vein, carrying blood from the lateral segment of the left lobe, joins the middle hepatic vein, carrying blood from the medial segment of the left lobe and anterior segment of the right lobe, to form a single venous trunk before joining the inferior vena cava. The right hepatic vein carries blood mostly from the posterior segment of the right lobe and joins the inferior vena cava separately. The portal vein is formed by the union of the superior mesenteric vein and splenic veins. It divides into right and left portal branches that travel in opposite directions and enter the parenchyma in the middle, dividing the liver into superior and inferior areas. (CBD common bile duct, IVC inferior vena cava)\n\n### 1.1.4 Hepatic Artery\n\nAfter arising from the celiac axis, the hepatic artery runs between the two layers of the hepatogastric ligament (lesser omentum) and enters the liver at the hilum. It is situated anterior to the portal vein and on the left side of the common bile duct (Fig. 1.1.3). In 90% of patients, the hepatic artery divides into right and left branches before entering the hilum of the liver. In the remaining 10%, the hepatic artery divides into three terminal branches, with the third branch entering the quadrate lobe (IV) directly. Wide variability is seen in the branching of the hepatic artery, portal vein, and bile ducts, and their clear delineation by imaging procedures (MDCT, ERCP, MRCP) is essential prior to segmental liver resection [21]. The hepatic artery supplies about 400 ml of arterial blood per minute at 100 mmHg systolic pressure and accounts for 25% of the total liver blood flow [22].\n\nThe cystic artery to the gallbladder usually arises from the right hepatic artery in the triangle of Calot, which is bordered by the inferior liver surface above, common hepatic duct to the left, and the cystic duct to the right. Often the cystic artery may arise from the main hepatic artery, left hepatic artery, or the gastroduodenal artery. Soon after its origin, the cystic artery enters the gallbladder at its neck and divides immediately into a superficial and a deep branch, both of which were first identified by Vesalius in 1564 [5]. The superficial branch supplies blood mostly to the free inferior wall covered by the peritoneum, and the deep branch supplies blood to the superior wall, which lies in direct contact with the inferior liver surface.\n\n### 1.1.5 Portal Vein\n\nThe portal vein is formed by the union of the splenic and superior mesenteric veins and measures about 5.5-8 cm in length; it enters the liver at the porta hepatis (Fig. 1.1.4). It usually divides at a 180\u00b0 angle into the right and left portal vein branches, which enter the liver parenchyma in the middle, dividing the liver into superior and inferior areas. The right portal vein gives off anterior and posterior segmental branches, and the left portal vein gives off medial and lateral segmental branches (Fig. 1.1.5). At least one branch enters each area of the liver segment. On average, three small portal vein branches (varying from one to six) enter the caudate lobe. They may arise from the left, right, or portal vein bifurcation. The portal vein supplies about 1,200 ml blood per minute to the liver (75% of the total liver blood supply) at 7-10 mmHg systolic pressure [22].\n\nFig. 1.1.4\n\nPortal vein. Coronal section CT (with contrast agent) shows the formation of the portal vein (PV) by the union of the superior mesenteric vein (SMV) with splenic veins\n\nFig. 1.1.5\n\nPortal vein branches. Coronal section CT (with contrast agent) at the porta hepatis shows the branching of the portal vein (PV) into right portal vein (RPV) and left portal vein (LPV) at almost 90 \u00b0. Middle hepatic vein (MHV) is seen entering the inferior vena cava at the superior liver margin\n\n### 1.1.6 Hepatic Veins\n\nThere are three distinctly separate hepatic veins: right, middle, and left (Fig. 1.1.6). They interdigitate and overlap with the portal venous system. The left hepatic vein has two tributaries, the medial and lateral branches. The medial branch drains blood from segments IVA and IVB, and the lateral branch drains blood from segments II and III (Fig. 1.1.6). The middle hepatic vein runs along the gallbladder fossa-inferior vena cava plane (Serege-Cantlie) and divides the liver into physiologic right (V-VIII) and left lobes (I-IV). It receives blood from the superior (IVA) and inferior (IVB) areas of the medial segment of the left lobe, and the superior (VIII) and inferior (V) areas of the anterior segment of the right lobe. The middle and left hepatic veins unite to form a single trunk before joining the inferior vena cava (Fig. 1.1.6). The right hepatic vein is the largest of the three veins and drains blood mostly from the superior and inferior areas of the posterior segment (VI and VII) of the right lobe (Fig. 1.1.7). Small tributaries are also received from the anterior segment. The right hepatic vein joins the inferior vena cava directly as a separate branch.\n\nFig. 1.1.6\n\nHepatic veins. Axial CT (with the contrast agent) at the upper part of the liver shows three hepatic veins. Left hepatic vein (LHV) has a medial (MB) and lateral (LB) branch and unites with the middle hepatic vein (MHV) before joining the inferior vena cava (IVC) as single trunk. Right hepatic vein (RHV) joins the IVC directly\n\nFig. 1.1.7\n\nRight hepatic vein. A coronal section CT with contrast agent at the middle of the right lobe shows the right hepatic vein (RHV) draining blood from the posterior (VI and VII) segment\n\nThe major portion of the venous blood from the caudate lobe drains directly into the inferior vena cava through three short veins [19, 20]. Smaller veins also drain into the middle and left hepatic veins. Because of this separate venous pathway directly into the inferior vena cava, the caudate lobe often maintains normal function during thrombosis of the hepatic veins (Budd-Chiary syndrome). The patients with Budd-Chiary syndrome demonstrate a normal pattern of Tc-99m-sulfur colloid uptake by the caudate lobe, whereas the rest of the liver parenchyma may show marked reduction in radiocolloid uptake. The venous blood from the gallbladder drains directly into the inferior vena cava. As the major portion of the caudate lobe falls on the left side of the Serege-Cantlie plane, it is usually included with the physiologic left lobe.\n\n### 1.1.7 Bile Ducts\n\nThe biliary tree consists of a network of ductal systems progressively increasing in size and originating at the hepatocytes as the bile canaliculus (Fig. 1.1.8). Bile canaliculi from the adjoining hepatocytes unite to form the cholangioles (<20 \u00b5m), which in turn combine to form the interlobular ducts (20-100 \u00b5m) and later the area ducts (100-400 \u00b5m). Ducts from each area unite to form the segmental ducts (400-800 \u00b5m). The segments are positioned anterior and posterior in the right lobe, and medial and lateral in the left lobe [23]. The anterior segmental duct unites with the posterior segmental duct (72%) to form the right hepatic duct (Fig. 1.1.9). The right posterior segmental duct unites with the left hepatic duct directly in 22%, and in the remaining 6% of the cases, the right anterior segmental duct joins the left hepatic duct directly [8].\n\nFig. 1.1.8\n\nIntra- and extra-hepatic ducts and gallbladder. 1 Gallbladder, 2 common bile duct, 3 common hepatic duct, 4 right hepatic duct, 5 anterior segmental duct, 6 anterior superior area duct, 7 anterior inferior area duct, 8 posterior segmental duct, 9 posterior superior area duct, 10 posterior inferior area duct, 11 left hepatic duct, 12 medial segmental duct, 13 medial inferior area duct, 14 medial superior area duct, 15 lateral segmental duct, 16 lateral superior area duct, 17 lateral inferior area duct, 18 falciform ligament\n\nFig. 1.1.9\n\nVariations in bile drainage from the right lobe. The anterior segmental duct (ASD) usually joins with the posterior segmental duct (PSD) forming the right hepatic duct (RHD) in 72%. Sometimes either PSD (22%) or ASD (6%) may join the left hepatic duct directly. Both RHD and LHD unite to form the common hepatic duct\n\nIn the left lobe, the medial segmental duct usually unites with the lateral segmental duct (62%) to form the left hepatic duct. The left medial segmental duct joins the left inferior area duct in 25% of the cases. Other variations are less frequent (Fig. 1.1.10). It is rare for either the left medial or the left lateral segmental duct to join the right hepatic duct directly. The medial segmental duct drains bile from the quadrate lobe (IVA, IVB), and the lateral segmental duct drains bile from the entire anatomic left lobe (II, III). In a Tc-99m-HIDA study, the medial segmental duct and the lateral segmental duct appear as separate trunks in an anterior view image, but the anterior and posterior segmental ducts of the right hepatic lobe are superimposed on one another in the anterior or posterior view of the liver. A right lateral view, therefore, is necessary to the separate bile drainage pattern of the anterior segmental duct from the posterior segmental duct of the right lobe (Fig. 1.1.2).\n\nFig. 1.1.10\n\nVariations in drainage of bile from the left lobe. The medial segmental duct (MSD) usually unites with the lateral segmental duct (LSD) in 62%, forming the left hepatic duct (LHD), which later unites with the right hepatic duct (RHD), forming the common hepatic duct. In the remaining 38% of patients, the main difference in bile drainage pertains to variations in area ducts joining with each other. It is rare for either the left medial or the left lateral segmental duct from the left lobe to join directly with the right hepatic duct (RHD). (LSAD lateral superior area duct, LIAD lateral inferior area duct, MIAD medial inferior area duct, MSAD medial superior area duct, PSD posterior segmental duct, ASD anterior segmental duct)\n\nThe caudate lobe (I) is divided anatomically into three parts: (1) the caudate process (right), (2) the paracaval (middle), and (3) Spiegel's (left) lobe (Fig. 1.1.11). Ducts from the caudate process are small, difficult to identify, and drain bile mostly into the right posterior segmental duct or right hepatic duct, and less often into the left hepatic duct. The duct from Spiegel's lobe is much larger in size, easy to identify at surgery, and drains bile mostly into the left hepatic duct. Ducts from the paracaval portion are small and variable in course [19, 20]. Because bile drainage from the caudate lobe is very variable and can occur into the left hepatic duct, right hepatic duct, or bifurcation, cancer of the hilar region can spread directly into the main lobes through the caudate lobe. Because of this unique feature, the caudate lobe is usually resected with a major lobectomy for Klatskin's tumor [19].\n\nFig. 1.1.11\n\nCaudate lobe. Axial section CT with contrast shows the caudate lobe situated posteriorly, in between the inferior vena cava and vessels at the porta hepatis. The caudate lobe consists of three parts: (1) right (caudate process), (2) middle (paracaval), and (3) left (Spiegel's) lobe\n\n### 1.1.8 Lymphatics\n\nThe lymph vessels from the liver parenchyma generally follow the course of the blood vessels and bile ducts and join with the gallbladder lymph vessels at the porta hepatis, and later divide into two main lymph channels; one channel follows the course of the common bile duct, and the other follows the course of the hepatic artery. Both channels pass through several lymph nodes. The channel following the course of the hepatic artery drains lymph primarily into nodes around the celiac axis, and the other channel, which follows the course of the common bile duct, drains lymph into lymph nodes around the pancreas [5]. The lymph from the liver parenchyma and bile ducts ultimately reaches the cisterna chyli (Fig. 1.1.12). The location of the liver lymph nodes is highly variable with the exception of one node situated at the junction of the gallbladder neck with the cystic duct (node of Mascagni). Enlargement of the node of Mascagni may cause cystic duct obstruction and block bile entry into the gallbladder, mimicking acute cholecystitis. Small lymph vessels around the central vein accompany the hepatic veins and inferior vena cava and drain lymph directly into the thoracic duct [24].\n\nFig. 1.1.12\n\nNerve supply and lymphatic drainage of the hepatobiliary system. The vagus (parasympathetic) nerve from the medulla descends along each side of the neck and mediastinum and enters the abdomen. The left vagus gives off branches to the anterior gastric plexus (AGP), from whence the vagal branches reach the liver and gallbladder and intrahepatic ducts. The right vagus gives off branches to the posterior gastric plexus (PGP), from whence the vagal branches are given off mainly to the common bile duct and the sphincter of Oddi. The phrenic nerve from cervical 3-4 supplies the liver capsule and the peritoneum covering the liver and gallbladder. Sympathetic nerve fibers reach the biliary system via the splanchnic nerves (T7-11) after passing through the celiac ganglion. The lymph from the gallbladder is drained to a node near the neck (node of Mascagni) and then to the nodes along the common bile duct, which also receive lymph from the lower half of the liver. The liver and gallbladder lymph vessels enter the peripancreatic lymph nodes and ultimately drain lymph into the cisterna chili\n\n### 1.1.9 Nerves\n\nThe hepatobiliary system is supplied by both the somatic and autonomic nervous system. The somatic nerve supply comes from the lower thoracic intercostal nerves (T7-11) and the right phrenic nerve (C3, 4). The lower thoracic intercostal nerves supply the parietal peritoneum. The right phrenic nerve supplies the diaphragm and the peritoneum covering the liver and the gallbladder. The pain sensation due to distension of the liver capsule, gallbladder wall, and bile ducts is transmitted through these nerves (Fig. 1.1.12).\n\nThe autonomic nerve supply consists of both the sympathetic and parasympathetic nervous systems. The parasympathetic nerve fibers travel via the vagus, which arises from the medulla and traverses down on each side of the neck and mediastinum to reach the abdomen. Because of the clockwise rotation of the gut during early intrauterine life (Fig. 1.1.1), the left vagus nerve becomes the anterior and the right vagus the posterior trunk. Both of these nerve trunks were first identified and correctly described by Vesalius in 1543 [25]. The anterior trunk (left vagus) enters the anterior gastric plexus at the gastro-esophageal junction. A branch from the anterior gastric plexus, the anterior hepatic nerve, enters the porta hepatis and bifurcates; one branch supplies the intrahepatic and proximal extrahepatic bile ducts and blood vessels, and the other branch supplies the gallbladder. The posterior vagal trunk (right vagus) passes behind the stomach and enters the posterior gastric plexus. A few of its branches enter the celiac ganglion. From the posterior gastric plexus, the nerve fibers enter the distal common bile duct and the sphincter of Oddi (Fig. 1.1.12).\n\nThe parasympathetic nerve supply to the gallbladder comes primarily from the anterior trunk (left vagus). The distal common bile duct and the sphincter of Oddi receive their parasympathetic nerve supply primarily from the posterior trunk (right vagus). The parasympathetic motor function of the gallbladder, therefore, is controlled mainly through the anterior trunk (left vagus) and that of the sphincter of Oddi mainly through the posterior trunk (right vagus). An injury or section of the anterior trunk (left vagus) alone during vagotomy, therefore, has the same effect as complete vagotomy (cutting of both vagii) as far as the motor function of the gallbladder is concerned. Parasympathetic ablation enhances muscular relaxation and increases bile stasis within the gallbladder, promoting cholelithiasis [26-28]. Stimulation of the parasympathetic system, on the other hand, increases the tonicity of the gallbladder and promotes complete bile emptying.\n\nThe hepatobiliary system receives the sympathetic nerve supply via the greater splanchnic (T7-9) and lesser splanchnic nerves (T10-11). The axons from preganglionic sympathetic nerve cells in the lateral horn of the thoracic T7-11 segments travel via the greater and lesser splanchnic nerves and enter the celiac ganglion. The postganglionic sympathetic fibers from the celiac ganglion enter the liver, gallbladder, common bile duct, and the sphincter of Oddi. Sympathetic nerve stimulation relaxes the gallbladder wall. The hepatic artery and its branches are supplied by sympathetic nerves that control the vascular tone. The pain from the hepatobiliary system is carried to the central nervous system via both the splanchnic nerves and the branches of the right phrenic nerve (Fig. 1.1.12).\n\n### 1.1.10 Biliary Apparatus\n\n#### 1.1.10.1 Intrahepatic Ducts\n\nThe hepatobiliary system is analogous to a tree. The hepatocytes can be considered to represent the leaves and canaliculi the trunk; area and segmental ducts represent the branches of the tree. The common hepatic duct and the common bile duct represent the trunk and the sphincter of Oddi the roots of a tree. The gallbladder is akin to a fruit [29]. Bile canaliculi from the hepatocytes join to form the cholangioles, which unite to form the interlobular bile ducts and then into larger area ducts. Ducts from each area unite to form the segmental ducts. The anterior and posterior segmental ducts from the right lobe unite to form the right hepatic duct, and the medial and lateral segmental ducts from the left lobe unite to form the left hepatic duct (Fig. 1.1.7).\n\n#### 1.1.10.2 Extrahepatic Ducts\n\nThe right and left hepatic ducts leave the liver parenchyma and proceed inferiorly for a distance of 0.5-1.5 cm before joining to form the common hepatic duct [8]. The common hepatic duct is 2-7 cm in length and joins with the cystic duct to form the common bile duct.\n\n#### 1.1.10.3 Gallbladder\n\nThe gallbladder lies along the inferior liver border in a groove between the right lobe and the quadrate lobe (medial segment of the left lobe; IVA, IVB). It consists of a fundus, body, and neck (Fig. 1.1.13). The gallbladder measures about 7-10 cm in length, 3-5 cm in width, and 40-50 ml in volume [30]. The superior third of the wall is in direct contact with the liver and hence lacks the peritoneal covering. The rest of the gallbladder wall is covered with the visceral peritoneum. The gallbladder is a pear-shaped, single-chamber organ with the extension of the fundus below the inferior liver margin. The position of the fundus, on the body surface, corresponds to the point of intersection of the lateral margin of the right rectus muscle with the right costal margin. A line drawn between the left anterior superior iliac spine and umbilicus points to the fundus of the gallbladder when it is extended upwards to meet with the right costal margin. Gallstones often lodge in a pouch-like structure (Hartmann pouch) at the neck, initiating cystic duct obstruction and subsequent acute cholecystitis. The mucous membrane thrown into folds in the neck and cystic duct often acts as a valve (Heister valve) for bile entry into and exit out of the gallbladder. The cystic duct measures about 3-8 cm in length and less than 3 mm in diameter and joins the common hepatic duct at a 45\u00b0 angle (80%) to form the common bile duct. Sometimes, the cystic duct may directly join the right hepatic or the left hepatic duct. There are many variations in the way the cystic duct joins with the common hepatic duct, right hepatic, or left hepatic duct [31]. The artery to the gallbladder (cystic artery) usually arises from the right hepatic artery (72%) or from its accessory branches (13%), and less often directly from the hepatic artery (6%) or the gastroduodenal artery.\n\nFig. 1.1.13\n\nExtrahepatic biliary tract. The distal one-third of both right and left hepatic ducts, and the entire common hepatic and common bile ducts, are extrahepatic in location. The gallbladder consists of a fundus, body, neck, and cystic duct. The mucous membrane in the neck and cystic duct is thrown into folds, acting as a valve (Heister's valve) for bile entry and exit. Gallstones often lodge in a pouch near the neck (Hartmann's pouch). The sphincter of Oddi (enlarged) at the distal end of the common bile duct consists of three separate sphincters. The distal common bile duct is surrounded by the choledochal sphincter (sphincter of Boyden). The distal pancreatic duct (duct of Wirsung) is surrounded by a separate pancreatic sphincter. The common channel formed by the union of two ducts is surrounded by the ampullary sphincter. The common channel (ampulla of Vater) opens into the duodenum at an elevation called the papilla. The name sphincter of Oddi refers to all three sphincters\n\nThe gallbladder is supplied by both the sympathetic and parasympathetic nervous systems, which control its tone, contraction, and relaxation. Parasympathetic stimulation causes contraction and bile emptying, and sympathetic stimulation results in relaxation and bile stasis. The gallbladder lymph, after passing through a node at the neck (node of Mascagni), enters the hepatic plexus and ultimately reaches the cisterna chyli (Fig. 1.1.7). The gallbladder is absent in 1:1,600 live births, and in about 8% of cases, it is entirely intrahepatic in location [32]. The C loop of the duodenum and the hepatic flexure of the colon are in close proximity to the gallbladder.\n\n#### 1.1.10.4 Common Bile Duct\n\nThis duct is formed by the union of the common hepatic duct with the cystic duct. It varies in length from 7 to 17 cm, and the lumen is less than 8 mm in diameter (Fig 1.1.13). The common bile duct is divided into the supraduodenal, retroduodenal, intrapancreatic, and intraduodenal segments. The supraduodenal segment is the longest and measures 2-4 cm in length. The retroduodenal and intrapancreatic segments each measure about 2.5-3 cm in length. The intradoudenal segment is the narrowest part of the common bile duct. The retroduodenal segment lies behind the duodenal bulb and may be non-visualized in a Tc-99m HIDA study if there is fluid collection in the duodenal lumen. The common bile duct enters the postero-medial wall of the second part of the duodenum and unites with the pancreatic duct (duct of Wirsung), forming a common channel, the ampulla of Vater, which finally enters the duodenal lumen at an elevation called the \"papilla.\" The papilla lies about 8-10 cm away from the pylorus of the stomach. The common bile duct joins with the pancreatic duct, forming a common channel of 10-12 mm length in 86% of cases. The two ducts join together at the ampulla just before opening into the duodenum in 6% of cases, and in the remaining 8%, both ducts open separately into the duodenum [32].\n\n## 1.2 Microstructure\n\nThe liver is the largest organ in the body and consists of an intricate structure to carry out the complex exocrine (bile secretion) and endocrine (protein synthesis) functions efficiently. The organ shape and structure are well maintained despite the paucity of connective tissue.\n\nThe concept of the lobule as the basic micro-architectural unit of the liver prevailed for well over a century. A lobule is hexagonal in shape and consists of a tributary of the hepatic vein at the center surrounded by six portal triads (consisting of the terminal branch of the hepatic artery, portal vein, and bile duct) at the periphery, and the space in between is occupied by a single cell plate of hepatocytes and cells of the sinusoidal space. A new concept of an acinus as the basic liver unit was first proposed in 1958 and is now well accepted because it explains the function and regeneration of the liver in a much better way than the lobule model [1]. An acinus is diamond shaped with a portal triad at one corner and central vein (hepatic vein) at the other three corners. Hepatocytes are arranged in three zones. The cells in zone 1 towards the portal triad receive a much higher concentration of oxygen and other nutrients than cells in zone 3 towards the central vein. Zone 1 cells, therefore, can withstand hypoxia better and regenerate much faster than cells in zone 3. The liver consists of the polygonal cells (hepatocytes), sinosoidal cells, canalicular cells, and other supporting cells (Fig. 1.2.1). An adult liver consists of well over 250 billion cells, of which hepatocytes constitute 78% of the volume, non-hepatocytes 6.3% of the volume, 2.8% by endothelial cells, 2.1% by Kupffer cells, and 1.4% by stellate cells. The remaining 16% of liver volume is made up of extracellular space [2].\n\nFig. 1.2.1\n\nMicrostructure of the liver. The vascular space is divided into two compartments by the endothelial cells: (1) endothelial or sinusoidal space and (2) the perisinusoidal space of Disse. Kupffer cells are located in the sinusoidal space and stellate cells (Ito cells) in the perisinusoidal space of Disse. The basolateral border of the hepatocyte faces the space of Disse. Canaliculi are invaginations of the lateral wall of two adjacent hepatocytes. Canaliculi join to form the canal of Hering, which in turn unites with others to form the interlobular ducts. The portal triad consists of a branch of the bile duct, hepatic artery, and portal vein. Substrates move from the basolateral border to the canaliculi through the hepatocyte.SER smooth endoplasmic reticulum, RER rough endoplasmic reticulum\n\n### 1.2.1 Vascular Compartment\n\nThis compartment is situated between two single layer plates of hepatocytes and consists of two spaces: (1) the sinusoidal space and (2) the perisinusoidal space of Disse (Fig. 1.2.1). The sinusoidal space is the much larger of the two and accounts for 2\/3 of the vascular compartment. Four types of cells are found in the vascular space: (1) endothelial cells (sinusoidal lining cells), (2) Kupffer cells, (3) stellate cells (lipocyte or Ito cells), and (4) pit cells [3]. These four cells together constitute the functional unit of the hepatic sinusoid.\n\n### 1.2.2 Endothelial Cell\n\nThese are flat cells lining the vascular space and consist of numerous holes or fenestrae in between [4]. The fenestrae vary in size from 0.1 to 3 \u00b5m in diameter and selectively allow certain plasma constituents to pass through from the sinusoidal space to the perisinusoidal space of Disse. The number of fenestrae decreases and the diameter of the hole increases in patients with alcoholic liver disease and cirrhosis [5-7]. The main function of the fenestrae is to act as a selective filter, allowing only those constituents that need to enter the perisinusoidal space of Disse and excluding others from the entrance. Red blood cells and leucocytes are excluded, whereas the electrolytes, plasma proteins (albumin), vitamins, and other essential nutrients are allowed free entry into the perisinusoidal space of Disse.\n\n### 1.2.3 Kupffer Cell\n\nKupffer described two types of cells, one in 1876 [8] and the other in 1899 [9]. It was first thought that the cell described in 1876 was the reticuloendothelial cell bearing his name (Kupffer cell). However, in 1951, Ito showed that this cell is the fat-storing (lipocyte, Ito cell) cell situated in the perisinusoidal space of Disse, recognized now as the stellate cell [10]. The cell described by Kupffer in 1899 is now considered the Kupffer cell. The Kupffer cells are highly specialized macrophages distributed irregularly within the sinusoidal space. There is no direct connection between two adjacent Kupffer cells. Their surface is irregular with folds and ruffles. They are highly mobile scavenger cells often found within the space of Disse or may lie free within the sinusoidal space, unattached to the endothelial cell. The cytoplasm is rich in lysosomes, Golgi apparatus, and rough endoplasmic reticulum (Fig. 1.2.1). The Kupffer cells proliferate locally to maintain their population, but at times of greater need they increase their number by calling reinforcement from the bone marrow.\n\n### 1.2.4 Stellate Cell\n\nThese cells, \"the sternzellen of Von Kupffer,\" first described in 1876, are the resting fibroblasts in the perisinusoidal space of Disse that are now organized as the stellate cells [8]. They have been called by many names, such as lipocyte, hepatic pericyte, or Ito cells [10, 11]. For many years, it was thought that these star-shaped cells did not have any function in humans. Stellate cells are rich in vitamin A and are the storage site for retinoids. They become activated during liver injury and play a dominant role in angiogenesis, vascular remodeling, repair, and fibrosis [12]. The activation of stellate cells increases production of membrane metalloproteinase 1 and 2 (MMP-1 and MMP-2) and tissue inhibitors of metalloproteinases (TIMP). Metalloproteinases promote matrix degradation and subsequent replacement by interstitial collagen in the subendothelial space of Disse. Other accompanying effects include a reduction in the number of fenestrae and a loss of microvilli along the perisinusoidal surface of the hepatocytes, both of which result in decreasing the delivery of organic anions (including HIDA) into the perisinusoidal space of Disse.\n\n### 1.2.5 Pit Cell\n\nThese cells are large granular T lymphocytes or natural killer cells found within the sinusoidal space. They are highly mobile killer lymphocytes and contain organelles necessary for removal of tumor cells and virus-infected hepatocytes. Activated natural killer cells promote hepatocyte proliferation and regeneration of the liver [14].\n\n### 1.2.6 Hepatocyte\n\nThe hepatocyte is the largest cell in the liver and varies in size from 13 to 30 \u00b5m with an average diameter of 25 \u00b5m (Fig. 1.2.2). Each milligram of liver tissue consists of 202,000 cells. There are about 250 billion hepatocytes in an adult liver [15]. A hepatocyte is polyhedral, multifaceted (as many as eight surfaces), and measures about 11,000 cubic \u00b5m in volume. Because of their relatively larger size, hepatocytes account for 78% of the liver by volume. Sinusoidal cells account for 6%, and the extracellular space occupies the remaining 16% of the liver volume. On the basis of function, the hepatocyte plasma membrane is divided into three domains: (1) the basolateral domain, (2) contact or contiguous domain, and (3) canalicular domain.\n\nThe basolateral domain faces the perisinusoidal space of Disse and accounts for 70% of the cell wall. It consists of 30-50 microvilli, which increase the absorptive surface by six times [16]. The microvilli extend along the paracellular space until the two adjacent hepatocytes come in close contact at the point of the desmosome. This surface is bathed in plasma filtered through the fenestrae of the sieve plate of the endothelial cells and allows metabolic exchange between the plasma and the hepatocyte. Organic anions, including Tc-99m-HIDA, enter the hepatocyte along the basolateral domain.\n\nThe contact domain accounts for 15% of the plasma membrane and is placed between two adjacent hepatocytes. The tight junction situated near the canaliculus prevents plasma constituents from entering the canalicular space directly from the space of Disse. The gap junction can be located anywhere along the lateral border and provides direct communication between adjacent hepatocytes on a selective basis.\n\nThe canalicular domain accounts for the remaining 15% of the cell surface and is situated at the center of two adjacent hepatocytes. It represents a specialized part of the hepatocyte. The canalicular wall consists of microvilli that increase the functional surface.\n\nHepatocyte contains a nucleus and a nucleolus that are rich in deoxyribonucleic acid. The smooth endoplasmic reticulum (SER) is made up of tubular structures containing microsomes and carries out bilirubin conjugation and detoxification of drugs and other organic anions. SER is steroid sensitive and participates in enzyme induction when phenobarbital is administered. The rough endoplasmic reticulum (RER) contains ribosomes and is responsible for protein synthesis, including albumin [13]. Lysosomes are cytoplasmic particles close to bile canaliculi and contain hydrolytic enzymes, including acid phosphatase [16]. Lysosomes perform a scavenger function and remove from blood excess material, including ferritin, bile pigments, and metals such as copper. The Golgi apparatus consists of particulates and vesicles and lies close to the bile canaliculus. The lysosomes and Golgi apparatus together perform the tasks of storage, entrapment, and final excretion into bile of various non-essential body constituents. The mitochondria, which participate in oxidative phospharilation, heme synthesis, and citric acid cycle, are scattered throughout the cell. The hepatocytes and cells in the sinusoidal and canalicular space are supported by a cytoskeleton consisting of microtubules and micro filaments. The cells of the sinusoidal and perisinusoidal space of Disse are supported by collagen, laminin, protoglycon, fibrinonectin, and heparan sulphate [15, 16].\n\n### 1.2.7 Bile Canaliculus\n\nBile canaliculi are a simple convoluted border of the hepatocyte and account for 13% of the hepatocyte wall. The convolutions are microvilli that increase the surface area for bile secretion (Fig. 1.2.1). A canaliculus varies from 0.1 to 1 \u00b5m in diameter. At the periphery of a hepatic lobule several bile canaliculi join to form the canal of Hering, which acts as a transitional zone between intralobular and interlobular ductal systems. The interlobular ducts unite to form area ducts that drain bile from an area of the liver. Area ducts unite to form the segmental ducts, which in turn unite to form the lobar ducts. The cells lining the ducts (cholangioles) are cuboidal in shape and contain apical microvilli, which project into the duct lumen [17].\n\n### 1.2.8 Gallbladder and Cystic Duct\n\nThe gallbladder wall consists of three layers: (1) the serosal layer, (2) fibromuscular layer, and (3) mucosal layer. The serosal layer is the peritoneum, which covers about 2\/3 of the gallbladder wall with the exception of the superior 1\/3, which is in direct contact with the liver. The fibromuscular layer consists mostly of elastic fibrous tissue. The muscular layer is irregular and consists of longitudinal and circular muscle fibers, which are well developed around the fundus and infundibulum, and scanty over the body and neck of the gallbladder. The mucosa consists of a single layer of cells with three different cell types: columnar, pencil, and basal. The mucosal layer is thrown into folds that increase the surface area of the gallbladder. The columnar cells have microvilli, 0.7-0.8 \u00b5m in length, along the luminal surface. The intercellular space between two adjacent columnar cells is narrow at the luminal end, but is widely open towards the fibromuscular layer [18]. Water and electrolytes are continuously absorbed from the lumen, through these intercellular spaces, making room for entry of fresh hepatic bile into the gallbladder during fasting [19]. The wall is impermeable to bile acids, bilirubin, radiocontrast agents, and other organic anions, including Tc-99m-HIDA. The neck contains mucous-secreting cells. Pencil cells, which are long and narrow, are found mainly in the body and extend from the basement membrane to the lumen. Basal cells are small and are concentrated more towards the fibromuscular layer.\n\nThe water absorbed from the gallbladder lumen through the lateral intercellular spaces enters venous blood through several small veins that drain into the hepatic or portal veins. There is no one single large cystic vein. The venous blood from the free wall of the gallbladder drains into venous radicals, which eventually enter the portal venous system.\n\nThe cystic duct, common hepatic duct, and common bile duct possess a mucosa, submucosa, and muscularis. The mucosal cells are a continuation of the gallbladder mucosal cells and contain the columnar cells with a ciliated border. The walls of the common hepatic duct and common bile duct are composed of a thick layer of connective tissue interspersed with few muscle cells. This structural configuration is well suited for the bile ducts to participate in bile concentration and discharge.\n\n### 1.2.9 Sphincter of Oddi\n\nFor well over a century, the study of this small segment of the biliary tract has fascinated anatomists, physiologists, and microscopists [20]. More recently, it has gotten the attention of gastroenterologists, hepatologists, and electron microscopists. For many years it was believed that the sphincter was a mere extension of the smooth muscle from the duodenal wall. Boyden's detailed work in 1937 clearly established it as a true sphincter, separate from the duodenal wall musculature [21]. The common bile duct enters the postero-medial wall of the second part of the duodenum tangentially and traverses for about 2 cm before joining the pancreatic duct (duct of Wirsung) to form a common channel, the ampulla of Vater. The common channel travels for 10-15 mm before entering the duodenal lumen at an elevation, the papilla, which is about 7-10 cm from the pylorus of the stomach (Fig. 1.1.13). The circular and longitudinal muscle fibers form a sphincter called the sphincter choledochus (sphincter of Boyden) at the distal end of the common bile duct before it joins the pancreatic duct. The sphincter is about 10-15 mm in length and mostly intramuscular. The contractions of this sphincter prevent reflux of pancreatic secretions into the common bile duct. The distal pancreatic duct (duct of Wirsung) is also surrounded by a distinct sphincter called the pancreatic sphincter whose contractions prevent bile reflux into the pancreatic duct. The pancreatic sphincter is absent in about 20% of the subjects [22]. The common channel formed by the union of the common bile duct and the pancreatic duct is called the ampulla of Vater and is surrounded by a weak sphincter called the sphincter of ampulla (pylorus of Westpal). The name, \"sphincter of Oddi,\" commonly refers to the function of all three sphincters situated at the distal end of the common bile duct and the pancreatic duct. Variations in the length, its union with the pancreatic duct, and opening into the duodenum of the common bile duct are frequent and are related to embryologic development [23]. The common bile duct opens through the ampulla of Vater into the lumen of the second part of the duodenum in 82% of the cases. The opening is lower than usual in 5%, or opens at an angle in 7%, or opens into the transverse duodenum (third part) in the remaining 6% of the patients [24].\n\nTable 1.2.1\n\nFunction of various cells of the liver [15]\n\nHepatocyte | Endothelial cell | Kupffer cell | Stellate cell | Canalicular cell\n\n---|---|---|---|---\n\nBile secretion | Plasma filtration | Phagocytosis of colloids, bacteria, endotoxin, tumor cells | Storage of fats, vitamin A, and retinoid | Bile transit\n\nPhospharilation | Endocytosis | Receptor for Fc fraction, C3b complement | Collagen secretion | Water secretion\n\n|\n\nRemoval of collagen Fc fragment of IgG, C3b complement\n\n|  |  |\n\nHem synthesis\n\n|  |  |  |\n\nElectrolyte transfer\n\nProtein synthesis\n\n|  |  |  |\n\nBeta oxidation\n\n|  |  |  |   \n|  |\n\nSecretion of TNF-a\n\n|  |\n\nGlycogenesis\n\n|  |\n\nCollagenase\n\n|  |\n\nBilirubin conjugation\n\n|  |\n\nInterleukins\n\n|  |\n\nDrug detoxification\n\n|  |\n\nArachidonic acid\n\n|  |\n\nLipoprotein synthesis\n\n|  |\n\nErythroblastosis\n\n|  |\n\nStorage of ferritin, vit B-12\n\n|  |\n\nStorage of iron, ferritin, hemosiderin, immune complex\n\n|  |\n\nReferences\n\n1.\n\nWilliams R. Global challenges in liver disease. Hepatology 2006;44:521-526PubMedCrossRef\n\n2.\n\nStarzl TE. The saga of liver transplantation, with particular reference to the reciprocal influence of liver and kidney transplantation (1955-1967). J Am Coll Surg 2002;195:587-610PubMedCrossRef01498-9)\n\n3.\n\nLinder HH, Green RB. Embryology and surgical anatomy of the extra hepatic biliary tract. Surg Clin N Amer 1964;44:1273-1285\n\n4.\n\nNetter FH. CIBA collection of medical illustration, vol. lll, Digestive system, part lll. Liver, biliary tract, and pancreas. CIBA, Summit NJ, 1964\n\n5.\n\nHess W. Surgery of the biliary passages and the pancreas. D Von Nostrand Company Inc, Princeton NJ, 1965\n\n6.\n\nCouinaud C. Le foie etudes anatomiques et chirurgicales, Masson, Paris, 1957\n\n7.\n\nReifferscheid M. Chirurgie der leber. Georg Thieme Verlag, Stuttgart, 1957\n\n8.\n\nHaley JE, Schroy PC. Anatomy of the biliary ducts within the human liver. Analysis of the prevailing pattern of branchings and the major variations of the biliary ducts. Arch Surg 1953;66:599-616CrossRef\n\n9.\n\nBismuth H. Surgical anatomy and anatomical surgery of the liver. World J Surg 1982;6:3-9PubMedCrossRef\n\n10.\n\nGoldsmith MA, Woodburn RT. Surgical anatomy pertaining to liver resection. Surg Gynecol Obstet 1957;141:429-437\n\n11.\n\nSugarbaker PH. En bloc resection of hepatic segments 4B, 5 and 6 by transverse hepatectomy. Surg Gynecol Obstet 1990;170:250-252PubMed\n\n12.\n\nSugarbaker PH, Nelson RC, Murray DR, Chezmar JL, Barnardino MF. A segmental approach to computerized tomographic portography for hepatic resection. Surg Gynecol Obstet 1990;171:189-195PubMed\n\n13.\n\nNelson RC, Chezmar JL, Sugarbaker PH, Murray DR, Bernardino MF. Pre-operative localization of focal liver lesions to specific liver segments: utility of CT during arterial portography. Radiology 1991;181:443-448\n\n14.\n\nLaforture M, Madore F, Patriquin H, Breton G. Segmental anatomy of the liver: a sonographic approach to the Couinaud nomenclature. Radiology 1991;181:443-448\n\n15.\n\nSoyer P. Segmental anatomy of the liver: utility of a nomenclature accepted worldwide. AJR. Amer J Roentgenol 1993;161:572-573\n\n16.\n\nDodd GD III. An American's guide to Couinaud's numbering system. AJR. Amer J Roentgenol 1993;161:574-575\n\n17.\n\nSutherland S, Harris J. Claude Couinaud-A passion for the liver. Arch Surg 2002;137:1305-1310CrossRef\n\n18.\n\nKogure K, Kuwano H, Fujimaki N, Ishikawa H, Takada K. Reproposal for Hjortsjoo's segmental anatomy on the anterior segment in human liver. Arch Surg 2002;137:1118-1124PubMedCrossRef\n\n19.\n\nAbdalla EK, Vauthey J, Couinaud C. The caudate lobe of the liver. Implications of embryology and anatomy for surgery. Surg Oncol Clin N Am 2002;11:835-848PubMedCrossRef00035-2)\n\n20.\n\nKitagawa S, Murakami G, Hata F, Hirata K. Configuration of the right portion of the caudate lobe with special reference to identification of its right margin. Clinical Anat 2000;13:321-340CrossRef13%3A5<321%3A%3AAID-CA2>3.0.CO%3B2-R)\n\n21.\n\nVarotti G, Gondoles G, Goldman J, Wayne M, Florman SS, Schwartz ME, Miller MM, Emre S. Anatomic variations in right liver living donor. J Am Coll Surg 2004;198:577-582PubMedCrossRef\n\n22.\n\nSherlock S, Dooley J. Diseases of the liver and biliary system. 10th edition, Blackwell Science, Malden, MA, 1997\n\n23.\n\nRoberts SK, Ludwig J, LaRusso NF. The pathology of biliary epithelia. Gastroenterology 1997;112:269-279PubMedCrossRef70244-0)\n\n24.\n\nJones AL. The architecture of bile secretion. A morphological perspective of physiology. Dig Dis Sci 1980;25:609-629PubMedCrossRef\n\n25.\n\nHolle F. Historical outline. In: Holle F, Anderson S. eds. Vagotomy: latest advances. Springer, New York, 1974\n\n26.\n\nLaRusso D, Misciagno G, Noviello MR, Torantino S. Cholelithiasis after billroth ll gastric resection. Surgery 1988;103:579-583\n\n27.\n\nGriffiths JMT, Holmes G. Cholecystitis following gastric surgery. Lancet 1994;2:780-782\n\n28.\n\nTurunen M, Antila L. Gallbladder disease following gastrectomy. ACTA Chir Scand 1964;127:134-137PubMed\n\n29.\n\nKrishnamurthy S, Krishnamurthy GT. Evolution of nuclear hepatology as a clinical subspeciality. J Nucl Med Technol 1995;23(Suppl):35S-45S\n\n30.\n\nGunnarson E, Investigation of the distension capacity of the human gallbladder. ACTA Radiol (Stockh) 1961;56:161-165CrossRef\n\n31.\n\nGoor DA, Ebert PA. Anomalies of the biliary tree. Report of a repair of an accessory bile duct and review of the literature. Arch Surg 1972;104:302-309PubMedCrossRef\n\n32.\n\nBoyden EA. The accessory gallbladder-An embryological and comparative study of aberrant biliary vesicles in man and domestic mammals. Am J Anat 1926;38:177-231CrossRef\n\nReferences\n\n1.\n\nRapapport AM. The structural and functional unit in the human liver (liver acinus). Anat Rec 1958;130:673-689CrossRef\n\n2.\n\nBloulin A, Bolender RP, Weibel ER. Distribution of organelles and membranes between hepatocytes and non-hepatocytes in the rat liver parenchyma. A stereological study. J Cell Biol 1977;72:441-455CrossRef\n\n3.\n\nGendrault J, Montecino-Rodriguez, Cinqualbre J. Structure of the normal human liver sinusoid after perfusion fixation. In: Knook DL, Wisse E (eds). Liver sinusoidal cells. Elsevier Biomedical Press, Amsterdam, 1982;pp 93-100\n\n4.\n\nWisse E. An electron microscopic study of the fenestral endothelial lining of rat liver sinusoid. J Ultrastructure Research 1970;31:125-150CrossRef90150-4)\n\n5.\n\nToner PG, Carr KE. Ultrastructure of the liver and biliary apparatus. In: Marlow S, Sherlock S. Eds, Surgery of the gallbladder and bile ducts. Butterworth & Co Ltd, Boston, 1981, pp 19-65\n\n6.\n\nWisse E, DeZanger RB, Charles K, Van Der Smissen P, McCuskey RS. The liver sieve: considerations concerning the structure and function of endothelial fenestrae, the sinusoidal wall and the space of Disse. Hepatology 1985;5:683-692PubMedCrossRef\n\n7.\n\nHorn T, Christoffersen P, Henrikseen JH. Alcoholic liver injury: defenestration in non-cirrhotic livers. A scanning microscopic study. Hepatology 1987;7:77-82PubMedCrossRef\n\n8.\n\nKupffer C Von. Uber die Sternzallen der Leber. Arch Mikr Anat 1876;12:353-357CrossRef\n\n9.\n\nKupffer C Von. Uber die sogenanten Sternzellen der sangetheier Leber. Arch Mikr Anat 1899;54:254-260CrossRef\n\n10.\n\nIto T. Cytological studies on stellate cells of Kupffer and fat storing cells in the capillaries of the human liver. ACTA Anat Nippon 1951;26:42 (abstract)\n\n11.\n\nLee JS, Semela D, Iredale J, Shah VH. Sinusoidal remodeling and angiogenesis: a new function for the liver-specific pericyte? Hepatology 2007;45:817-825PubMedCrossRef\n\n12.\n\nFriedman SL, Rockey DC, Bissell M. Hepatic fibrosis 2006: report of the third AASLD single topic conference. Hepatology 2007;45:242-249PubMedCrossRef\n\n13.\n\nFriedman SL, Basal MB. Reversal of hepatic fibrosis-fact or fantasy? Hepatology 2006;43:S82-S88PubMedCrossRef\n\n14.\n\nNakashima SL, Inui T, Habu Y, Kinishita M, et al. Activation of mouse natural killer cells accelerates liver regeneration after partial hepatectomy. Gastroenterology 2006;131:1573-1583PubMedCrossRef\n\n15.\n\nSherlock S, Dooley J. Diseases of the liver and biliary system, 10th edn. Blackwell Science, Malden, MA, 1997\n\n16.\n\nFeldman G. The cytoskeleton of the hepatocyte. J Hepatology. 1989;8:380-384CrossRef90038-X)\n\n17.\n\nJones AL, Schmucker DL, Reston RH, Murakami T. The architecture of bile secretion. A morphological perspective of physiology. Dig Dis Sci 1980;25:609-629PubMedCrossRef\n\n18.\n\nEveff RD, Higgins JA, Brown AC. The fine structure of normal mucosa in human gallbladder. Gastroenterology 1964;47:49-60\n\n19.\n\nTormey JM, Diamond JM. Studies on the structural basis of water transport across epithelial membranes. Federation Proceedings 1966;25:692-707\n\n20.\n\nOddi R. D'une disposition a sphincter del'ouverture du canal cholique. Arch. Ital Biol 1887:8:317-322\n\n21.\n\nBoyden EA. The sphincter of Oddi in man and certain representative mammals. Surgery 1937;1:25-37\n\n22.\n\nBoyden EA. The anatomy of the choledochoduodenal junction in man. Surg Gynecol Obstet 1957;104:641-652PubMed\n\n23.\n\nLinder HH. Embryology and anatomy of the biliary tree. In: Way LW, Pelligrini CA. (Eds) Surgery of the gallbladder and bile ducts. WB Saunders Co, Philadelphia, 1987;pp 3-22\n\n24.\n\nLinder HH, Pena VA, Ruggieri RA. A clinical and anatomical study of anomalous termination of the common bile duct into the duodenum. Ann Surg 1976;184:626-632CrossRef\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_2(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 2. Liver and Spleen Function\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nThe liver is the largest organ and carries out the most complex biological functions in the body. It secretes bile, synthesizes proteins, metabolizes nutrients, hormones, and drugs, and detoxifies noxious endogenous and exogenous substrates. To accomplish all of these functions, the liver is located centrally in the body and endowed with well-designed architecture with a generous amount of blood supply. Secretion of bile is one of many important liver functions, and bile promotes digestion and absorption of essential nutrients; it also serves as a vehicle to get rid of biological waste products from the body. The biliary tree is designed not only for continuous bile secretion and flow, but also for periodic bile storage and discharge (gallbladder) at the time when food enters the small intestine. The spleen carries out many functions whose importance has been recognized only recently. This chapter will discuss the various functions of these two organs.\n\nThe liver is the largest organ and carries out the most complex biological functions in the body. It secretes bile, synthesizes proteins, metabolizes nutrients, hormones, and drugs, and detoxifies noxious endogenous and exogenous substrates. To accomplish all of these functions, the liver is located centrally in the body and endowed with well-designed architecture with a generous amount of blood supply. Secretion of bile is one of many important liver functions, and bile promotes digestion and absorption of essential nutrients; it also serves as a vehicle to get rid of biological waste products from the body. The biliary tree is designed not only for continuous bile secretion and flow, but also for periodic bile storage and discharge (gallbladder) at the time when food enters the small intestine. The spleen carries out many functions whose importance has been recognized only recently. This chapter will discuss the various functions of these two organs.\n\n## 2.1 Liver Function\n\nThe liver secretes about 600-800 ml of bile per day, at 25-33 ml h-1 or 0.42-55 ml min-1 [1]. Under normal hydration, osmolality of the hepatic bile is similar to that of plasma and ranges between 290-320 mOsm -l. Bile secretion is independent of hepatic perfusion pressure; thus, it differs from that of urine formation, which is very much dependent upon the glomerular filtration pressure in the kidneys [2]. The hepatic bile is composed of 98% water and 2% solutes, which include bile acids, phospholipids, cholesterol, conjugated bilirubin, electrolytes, and proteins (Table 2.1.1). Almost all of these functions are carried out mainly by the hepatocyte, which possesses all of the ingredients necessary for performing complex functions [3]. After extraction from blood in the space of Disse, some substrates are metabolized and transported through the hepatocyte to be secreted into canaliculi, while others may be secreted without any changes. All three domains of the hepatocyte plasma membrane participate in the uptake and excretion [4]. The uptake from the blood in the space of Disse takes place along the basolateral domain, which accounts for approximately 40% of the hepatocyte border. Bile secretion occurs at the canalicular domain, which constitutes 10% of the cell border. The lateral domain, the wall facing two adjacent hepatocytes, forms the remaining 50% of the cell border and plays a major role in water and solute transport and regulation of the total volume of bile secreted per day (Fig. 2.1.1).\n\nFig. 2.1.1\n\nMechanism of bile secretion. Uptake of solutes from the space of Disse occurs along the basolateral domain of the hepatocyte through many different pathways controlled by different transporter proteins. Four organic anion transporter proteins (OATPs) and organic anion transporter-2 (OAT2) and organic cation transporter proteins (OCT1) control uptake of anions and cations along the basolateral border (red). Sodium-dependent bile salt uptake occurs via sodium taurocholate protein (NTCP). Five multidrug resistance-associated proteins (MRPs) control secretion (reflux) from the hepatocyte back into space of Disse (blue). After intracellular transit, solute secretion into bile takes place along the canalicular domain through MRP2, multidrug- resistant-1 p-glycoprotein (MDR1 and MDR3), bile salt export pump (BSEP), breast cancer-resistance protein (BCRP), and flippases (ABCG5\/ABCG8). Chloride channel, glutathione (GSH) transporter, and chloride (Cl-)\/ bicarbonate (HCO3 -) exchange transporters also control excretion. Cholangioles absorb bile salts from the lumen through apical sodium-dependent bile salt transporter (ASBT) protein, and other anions are absorbed through OATP1A2. Exchange of Cl- for HCO3 - takes place. These substrates are secreted into the peribiliary plexus at the basolateral domain of the cholangiocytes mediated by MDR3, OST-1, and OST-2. Water enters the canalicular bile by three different routes: (1) transcellular, (2) paracellular, and (3) a combination of paracellular and transcellular\n\nTable 2.1.1\n\nComposition of human hepatic and gallbladder bile\n\nWater (g\/dl-1) | 98 | 92\n\n---|---|---\n\nNa+ (mEq\/l-1) | 150 | 130\n\nK+ (mEq\/l-1) | 5 | 10\n\nCa2 + (mEq\/l-1) | 5 | 23\n\nC1\u2212 (mEq\/l-1) | 100 | 25\n\nHCO3\u2212 (mEq\/l-1) | 28 | 10\n\nBile salts (g\/dl-1) | 1 | 6\n\nBilirubin (g\/dl-1) | 0.05 | 0.3\n\nCholesterol (g\/dl-1) | 0.1 | 0.6\n\nFatty acids (g\/dl-1) | 0.12 | 0.8\n\nLecithin (g\/dl-1) | 0.04 | 0.3\n\nBile formation in the hepatocyte can be divided into four major phases: phase I, uptake of substrates from blood (space of Disse) along the basolateral border; phase II, metabolism (hydroxylation); phase III, detoxification (conjugation); phaseIV, excretion into bile canaliculi along the canalicular border. Both phases II and III take place intracellularly within the hepatocyte. A disease process can affect one, two, three, or all four phases at a time. The entire metabolic process is controlled by genes mainly through their nuclear receptors.\n\n### 2.1.1 Basolateral (Sinusoidal) Domain\n\nIn the past decade, new knowledge has contributed to better understanding of the mechanisms involved in bile formation and flow. Basolateral and canalicular domains of the hepatocyte possess many polypeptide transporters that regulate the entrance and exit of substances across the plasma membrane (Fig. 2.1.1). Both sodium-dependent and sodium-independent pathways control the uptake of substrates along the basolateral border. The sodium-dependent pathway is regulated by sodium taurocholate cotransporting polypeptide (NTCP), which controls the majority of conjugated bile salts, a few sulfated steroids, and to a minor extent uptake of unconjugated bile salts. The sodium-dependent pathway also uses the Na+\/K+ ATPase pump, which enables net movement of three Na+ ions out of the hepatocyte for every two K+ ions moving into the hepatocyte from the space of Disse [5]. The net effect of these electrolyte exchanges results in a higher concentration of K+ inside than outside, and a higher concentration of Na+ outside than inside of the hepatocyte. Due to this ion imbalance, the inside of the hepatocyte carries a much higher negative charge (-35 mV) than its outside environment [6].\n\nSodium-independent pathways are represented by several members of the superfamily of organic anion-transporting polypeptides (OATPs). Four of the OATPs located along the basolateral border include OATP1B1 (formerly OATP-C), OATP1B3, OATP1A2, and OATP2B1 [7]. In humans, the highest concentration of OATP1B1 and OATP1B3 is found in the liver. Bilirubin uptake is controlled mainly by OATP2B1. Three of the OATPs (OATP1B1, OATP1B3, and OATP1A2) have overlapping functions for conjugated and unconjugated bile salts, bromosulphophtalein, sulfates, glucoronides, and selected organic anions and organic cations. OATPs control uptake of many drugs, including radiotracers (Tc-99m-HIDA). OATP2B1 also controls uptake of bromosulfaphtalein (possible Tc-99m HIDA) and steroid sulfates. Other sodium-independent uptake systems, separate from OATPs, are the organic anion transporter2\/organic cation transporter1 (OAT2\/OCT1) gene family members, and they control uptake of organic anions and organic cations, respectively.\n\nBesides the above group of uptake transporters, the basolateral domain has transporters that control excretion of substrates from the hepatocyte into blood (reflux) in the space of Disse (Fig. 2.1.1). These transporters belong to the adenosine triphosphate (ATP) binding cassette (ABC) family transporters called multidrug resistance-associated proteins (MRPs). Of the six known MRPs, five (MRP1, MRP3, MRP4, MRP5, and MRP6) are located along the basolateral border and play a role in the efflux of drugs and their metabolites (MRP4 and MRP5), drug conjugates (MRP1), and bile salts (MRP3) into the space of Disse. The sixth MRP (MRP2) is located along the canalicular border and controls excretion of bilirubin, bile salts, and other organic anions into the canaliculi.\n\n### 2.1.2 Nuclear Receptors\n\nOf the more than 100 nuclear receptor super-family members known in the mammalian cells, 49 have been identified in the human cell and are divided into four major classes [8]. Class I nuclear receptors are homodimers and include most steroid receptors (glucocorticoids, estrogens, androgens, mineralocorticoids, and progesterone). Class II receptors consist of eight heterodimer partners of RXR (retinoid X receptor) and include constitutive androstane receptor (CAR), farnesoid X receptor (FXR), liver X receptor (LXR), peroxisomal proliferator receptor (PPAR), pregnane X receptor\/steroid, and xenobiotic receptor (PXR\/SXR), retinoic acid receptor (RAR), thyroid hormone receptor (TR), and vitamin D receptor (VDR). Class II nuclear receptors play a major role in controlling the function of the hepatocyte. Class III nuclear receptors are orphan homodimers and include retinoid X receptor (RXR), chicken ovalbumin upstream promoter (COUP-TF), and hepatocyte nuclear factor 4 (HNF4), and class IV is monomers and includes liver receptor homologue1 (LRH1). The short heterodimer partner (SHP) is a nuclear receptor separate from the other class four types described (Table 2.1.2).\n\nTable 2.1.2\n\nNuclear receptors, ligands, and their target genes that influence uptake and excretion of organic anions by the hepatocyte\n\nNuclear receptor | Ligand (s) | Major target gene\n\n---|---|---\n\nRXR partners (class II receptors)\n\nFXR (farsenoid X receptor) | Bile acids, bilirubin | BSEP, SHP, UGTs, SULTS, MRP2, MDR3\n\nPXR (pregnane X receptor) | Xenobiotics, UDC | CYP3A, OATP1B1, MRP2, MRP4, GST\n\nCAR (constitutive  X receptor) | Xenobiotics, phenobarb | CYP3A, OATP1B1, MRP2, MRP4, UGT, SULTs, GSTs\n\nLXR (liver X receptor) | Oxysterols (metabolites of cholesterol) | CYP7A, CYP8B, ABCG5\/8\n\nRAR (retinoic acid receptor) | All-trans retinoic acid | NTCP, MRP2\n\nPPAR (peroxisomal  proliferators receptor)\n\nOthers\n\nSHP-1 (short heterodimer partner) | None | Inhibits NTCP, CYP7A, CYP8B.\n\nHNF-\u03b1 (hepatocyte nuclear factor \u03b1) | None | NTCP, CYP7A.\n\nModified from Boyer [9]\n\nA typical nuclear receptor consists of five functional domains (Fig. 2.1.2). At the aminoterminal (N) end is the activation function 1 (AF1) region, which is responsible for ligand-independent transcriptional activation and coordination. DNA-binding domain (DBD) controls high affinity recognition with specific response. The hinge region in the middle facilitates coordination of multiple domains. Ligand-binding domain (LBD) determines specificity and affinity and is responsible for manifesting species variability. The C-terminal end contains activation function 2 (AF2), acts as a control switch, and maintains ligand-dependent transcriptional function [8].\n\nFig. 2.1.2\n\nStructure of class II nuclear receptors. A nuclear receptor consists of five functional domains. The aminoterminal (N) end is the activation function 1 (AF1) region, which controls the ligand-independent transcriptional activation and coordination. The DNA-binding domain (DBD) controls high-affinity recognition with a specific response. The middle region (HINGE) facilitates coordination of multiple domains. The ligand-binding domain (LBD) determines specificity, affinity, and species variability. The C-terminal end contains activation function 2 (AF2) and acts as a control switch to maintain ligand-dependent transcriptional function\n\n### 2.1.3 Regulation of Basolateral Transporters\n\nClass II nuclear receptors control many of the functions of the hepatocyte. By definition, class II nuclear receptors cannot act alone or as homodimers. They act by forming a heterodimer partner with retinoid X receptor (RXR). After complexing with RXR, class II receptors acquire DNA-binding capacity and regulation of transcriptional activity. Five (FXR, CAR, PXR, LXR, and RAR) of eight class II nuclear receptors play major roles in influencing hepatic uptake and excretion of organic anions (Table 2.1.2).\n\nCholestasis suppresses sodium taurocholate-cotransporting polypeptide (NTCP) through FXR (farnesoid X receptor)-mediated induction of short heterodimeric partner 1 (SHP1), which prevents further uptake of bile salts that may otherwise reach toxic levels. Cholestasis also downregulates OATP1B1 through bile acid-mediated activation of SHP1, which decreases hepatocyte nuclear factor \u03b1 (HNF\u03b1). HNF\u03b1 is one of the major activators of OATP1B1. Although cholestasis in general downregulates nuclear receptors, it stimulates OATP1B3 via activation of FXR, and thus provides an escape mechanism for clearance of xenobiotics from the body (Fig. 2.1.2). After the uptake, intracellular transit of both sodium-dependent and sodium-independent substrates is regulated by the cyclic adenosine monophosphate-mediated dephosphorylation process, which is controlled by protein kinase B. Uptake of toxins like phalloidin and microcystin is mediated by OATP1B1 and OATP1B3, whereas the uptake of the most toxic natural substance known, amanitin, is controlled solely by OATP1B3 [7].\n\n### 2.1.4 Transport Through the Hepatocyte\n\nAfter uptake along the basolateral border facilitated by NTCP, bile acids are hydroxylated and transported through the hepatocyte via a 33-kDa cytosolic protein, 3-\u03b1-hydroxysteroid dehydrogenase in phase II (Fig. 2.1.3.). After its uptake, bilirubin binds to glutathione-S-transferase and undergoes conjugation (phase III) by hepatic microsomal enzyme uridine-diphosphonate glucoronyl transferase (UGT) to form monoglucoronide and then diglucoronide [9]. The conjugation process converts hydrophobic salts into hydrophilic salts, which facilitates rapid excretion into bile canaliculi. Many organic anions (including Tc-99m-HIDA) are transported through the hepatocyte by yet undefined mechanisms.\n\nFig. 2.1.3\n\nMicroenvironment and transporter proteins of the hepatocyte. Substrates are carried in blood loosely bound to albumin and delivered into the space of Disse. Basolateral domain of the hepatocyte has many transporter proteins that control uptake (red) and excretion (blue) of substrates from and into blood in the space of Disse. Uptake proteins (red) include four organic anion transporter proteins (OATPs), sodium taurocholate cotransporting peptide (NTCP), and sodium\/potassium ATPase. It has five multidrug resistance-associated proteins (MRPs) that control reflux of substrates from the cytoplasm into blood. After the uptake, substrates are metabolized (phase 2), conjugated (phase 3), and excreted (phase 4) unchanged into bile canaliculi. Excretion is controlled by canalicular domain proteins, such as multidrug resistant p-glycoproteins (MDR1, MDR2), MRP2, bile salt export protein (BSEP), breast cancer-resistant protein (BCRP), organic cation (OC), and flippases (ABCG5\/ABCG8), FIC1 protein\n\n### 2.1.5 Secretion into Bile Canaliculi\n\nThe quantity of solute transported across the canalicular membrane is the rate-limiting step in the volume of bile secreted per day. The canalicular membrane contains several ATP-dependent and ATP-independent transport proteins (pumps) to enable secretion of solutes from the hepatocyte into canaliculi [7, 10]. These pumps include ATP-dependent multidrug-resistance-1 p-glycoprotein (MDR1) and phospholipid transporter multidrug-resistance p-glycoprotein 3 (MDR3). The canalicular membrane also localizes multidrug resistance-associated protein 2 (MRP2), canalicular-bile-salt-export pump (BSEP or SGPG) and ABC half transporters, and breast cancer-resistance protein (BCRP), all of which facilitate excretion of bile salts and xenobiotics into bile canaliculi. Flippases (ABCG5 and ABCG8) control cholesterol metabolism and excretion both by hepatocytes and intestinal cells. The ATP-independent transport systems include a chloride channel, a chloride-bicarbonate-anion exchanger, and a glutathione (GSH) transporter (Fig. 2.1.1). Secretion of other organic anions, such as bilirubin, BSP, indocyanin green, and glutathione (Tc-99m-HIDA), is controlled mainly by MRP2. The secretion of organic cations (e.g., cancer chemotherapy agents, cyclosporin A, calcium channel blockers, and other drugs) is mediated by MDR1-a 170 gene product [11]. After passing through the tight junction, sodium and other cations enter the canaliculi through the space between the lateral domain of two adjacent hepatocytes, and the water then simply follows the electrolytes by passive diffusion.\n\n### 2.1.6 Regulation of Canalicular Membrane\n\nProduction of bile salt export protein (BSEP) and multidrug resistance glycoprotein 3 (MDR3) is controlled by a FXR-mediated process resulting in increased bile salt excretion into canaliculi, thereby facilitating formation of mixed micelles (Fig. 2.1.3). This action protects hepatocytes and cholangiocytes from toxic levels of bile salts. FXR also upregulates MRP2 and increases bile salt excretion into canaliculi. PXR (pregnane X receptor) upregulates MDR1, which is a key transporter protein in cellular excretion of many drugs and xenobiotics. LXR (liver X receptors \u03b1 and \u03b2) controls flippases, ABCG5 and ABCG8, and transport proteins [9].\n\nCholangiocytes along the bile ducts further modify bile composition by selective absorption or secretion of bile solutes and water. Cholangiocytes absorb bile salts through apical sodium-dependent bile salt transporter (ASBT) and OATP1A2. After their uptake, bile salts are excreted through the basolateral membrane (via MRP3) into the peribiliary plexus where they reach the portal circulation. Bile acids and sterols also use organic solute transporters (OST\u03b1\/OST\u03b2) to be secreted into the peribiliary plexus through the basolateral domain. Unlike the basolateral border of the hepatocyte, which lacks MRP2, the basolateral border of the cholangiocyte contains MRP2 that controls excretion of organic anions into peribiliary plexus (Fig. 2.1.1).\n\n### 2.1.7 Aquaporins\n\nOf the daily total bile volume of 600 ml, 450 ml (75%) is secreted by hepatocytes, and 150 ml (25%) is added by the cells lining the canaliculi (cholangioles or cholangiocytes). Of a total of 450 ml hepatocellular bile, 225 ml (50%) is bile salt dependent, and the remaining 225 ml (50%) is bile salt independent (Fig. 2.1.1). Although the cholangiocytes account for only 3 to 5% of liver volume, they play a major role controlling daily bile volume mainly by secretion or absorption of water by either direct passive membrane passage or through special water-channel proteins called aquaporins (AQPs) located within the hepatocytes and cholangiocytes. AQPs are small (25-34 kDa) hydrophobic protein molecules consisting of 263-323 amino acids. Both amino and carboxy terminals are located intracellularly within the cytoplasm (Fig. 2.1.4). Each AQP monomer has six \u03b1 helical transmembrane domains connected by five loops, A-E (Fig. 2.1.4A). Three of the connecting loops (A, C, E) are extracellular, and two (B, E) are intracellular [12]. When one intracellular and one extracellular loop, each containing Asp-Pro-Ala (NPA) amino acids, are juxtaposed to within the plasma membrane, AQP forms a tetramer containing four single water channels and allows molecules of 3 \u00c5 size or less to pass through (Fig. 2.1.4B). This enables water molecules (2\u00c5) to pass through the tight junction readily. Figure 2.1.5 summarizes the mechanisms and carrier proteins that play a major part in the transport of water from blood into the hepatocyte and finally into bile canaliculi.\n\nFig. 2.1.4\n\nStructure of aquaporins. Aquaporins are membrane proteins that regulate water movement. Each aquaporin has six alpha helical transmembrane domains connected by five loops, A-E E (top). Three of the connecting loops (A, C, E) are extracellular and two (B, E) are intracellular. Each aquaporin forms a tetramer (four single water channels) when one intracellular and one extracellular loop, each containing Asp-Pro-Ala (NPA) amino acids, are juxtaposed within the plasma membrane and allows molecules of 3 \u00c5 size or less to pass through (bottom). This enables water molecules (2\u00c5) to pass through the tight junction readily\n\nFig. 2.1.5\n\nWater movement within the hepatocytes and cholangiocytes. In the basal state, aquaporin 8 and aquaporin 1 are distributed free within the cytoplasm of the hepatocytes and cholangiocytes, respectively. Water passes passively through the lipid plasma membrane, facilitated by AQP9 and AQP4 water channels located on the membrane, respectively. When stimulated by cholecystokinin or secretin, hepatocyte AQP8 translocates to the canalicular border, and cholangiocyte AQP1 translocates to the apical border, promoting rapid passage of water from the cell into the bile duct lumen. In the stimulated state, each AQP channel can increase its water passage by more than ten times at its basal state\n\nWater can pass either through the cell (transcellular) or between two adjacent (paracellular) cells. When electrolytes, bile salts, and other organic anions are transported from the basolateral to the canalicular border through the hepatocyte, and secreted into canaliculi, water follows passively through the cell via both channel-mediated and non-channel-mediated pathways. Of the 13 AQPs that have been identified in the mammalian cells, seven (AQP0, AQP1, AQP4, AQP5, AQP8, AQP9, AQP11) are found within the hepatobiliary system [13]. Three of the AQPs (AQP0, AQP8, AQP9) are found in the hepatocytes, and two (AQP1, AQP4) are localized in the cholangiocytes. Four of them (AQP1, AQP4, AQP8, AQP9) play a major role in controlling water transport in the hepatobiliary tree. In the basal state, AQP8 and AQP1 are distributed free within the cytoplasm of the hepatocytes and cholangiocytes, respectively, and water diffuses passively through the lipid plasma membrane, facilitated by AQP9 and AQP4 water channels on the membrane (Fig. 2.1.5). In the stimulated state (CCK or secretin), hepatocyte AQP8 translocates to the canalicular border, and cholangiocytes AQP1 translocates to the apical border, allowing rapid passage of water from the cell into the bile duct lumen. Each AQP maintains its own rate of water transport; AQP0 is the slowest, while AQP1 and AQP4 allow 50-80 times more rapid flow of water. In the stimulated state, each AQP channel can increase its water transport ten times more rapidly than at the basal state.\n\n### 2.1.8 Protein Secretion\n\nLiver is the main source of plasma proteins, albumin, and globulins. Serum albumin synthesized entirely by the hepatocytes accounts for 60% of plasma proteins, is composed of 585 amino acids, and does not contain any carbohydrate moiety. In addition to controlling osmotic pressure, albumin functions as a carrier protein for drugs, metals, vitamins, amino acids, steroid, fatty acids, and Tc-99m-HIDA. Other proteins secreted by hepatocytes include \u03b1-1 anti trypsin, \u03b1-fetoprotein, \u03b1-2 macroglobulin, antithrombin lll, ceruloplasmin, C-reactive protein, fibrinogen, haptoglobin, hemopexin, and transferrin (Table 2.1.2). Most of these proteins are composed of a carbohydrate moiety and hence are called glycoproteins [14]. Hepatic bile is relatively more dilute (contains fewer bile salts, bilirubin, chloride, and bicarbonate per liter) than the gallbladder bile (Table 2.1.3).\n\nTable 2.1.3\n\nPlasma proteins secreted by hepatocytes [11]\n\nProtein | Molecular weight (kDa) | Function | Ligand binding | Plasma conc.  (mg dl-1)\n\n---|---|---|---|---\n\nAlbumin | 66 | Osmotic pressure | Hormones amino acids fatty acids, vitamins | 4,500-5,000\n\n|\n\nCarrier protein\n\n|\n\nAlpha-1 anti-trypsin | 54 | Trypsin and general | Present in serum & tissue secretions | 1.3-1.4\n\n|\n\nProtease inhibitor\n\n|\n\nAlpha Feto adults protein | 72 | Osmotic regulation | Hormones | Undetectable in adults\n\n|\n\nCarrier protein | Amino acids | Present in fetal blood\n\nAntithrombin lll | 65 | Protease inhibitor | Binding to Protease | 15-60\n\nCeruloplasmin | 134 | Copper transport | 6 copper atoms per mol | 15-60\n\nC-reactive protein | 105 | Tissue inflammation | Complement C1q | Increased in inflammation\n\nFibrinogen | 340 | Fibrin precursor during hemostasis.\n\n|  |\n\n200-450\n\nTransferrin | 80 | Iron transport | Two iron atoms per mol | 3-6.5\n\nHaptoglobin | 100 | Transport of hemoglobin | Hemoglobin | 40-180\n\nProthrombin | 72 | Hemostasis | Calcium chelation | 10.0\n\n### 2.1.9 Cell Death: Apopstosis or Necrosis\n\nLiver, being metabolically very active, has mechanisms to increase or decrease the number of cells needed to carry out complex biological functions. It can get rid of unwanted cells through cell death or recruit new cells through regeneration. Liver can regenerate to its original size within a few weeks after resection of 60-70% of its volume. Cell death is a common phenomena associated with varieties of liver diseases, including viral hepatitis, cholestasis, ischemia\/reperfusion injury, and hepatotoxins. Cell death occurs in one of two forms: necrosis or apopstosis. Both forms utilize a common pathway, but with different end results (Fig. 2.1.6). They involve participation of the plasma membrane, mitochondria, nucleus, endoplasmic reticulum, and lisosomes [15]. Necrosis is an acute process involving contiguous liver cells and takes place rapidly within a few minutes. Cells swell, forming blebs along the plasma membrane, disrupting its permeability, which ultimately results in its rupture with the release of cytosolic proteins, such as aspartate aminotransfarase (AST), alanine aminotransfarase (ALT), alkaline phosphatase (Alk.Phos), and lactic dehydrogenase (LDH), into blood. Necrosis is associated with acute inflammatory cell infiltration. Apoptosis, on the other hand, is a programmed cell death that occurs at a relatively a slow pace, requiring ATP. It affects cells at discontinuous locations, and death is characterized by chromatin condensation, DNA degradation, and shrinkage of cells with very little acute inflammatory cell infiltration [16]. Cytochrome c plays a central role in both processes (Fig. 2.1.6).\n\nFig. 2.1.6\n\nCell death through apoptosis or necrosis. Both forms utilize a common pathway, but with different end results. The process is initiated by interaction of death ligands with its corresponding receptor, which converts procaspase 8 to caspase 8. Caspase 8 can release caspase 3 directly in the type 1 mechanism. In the type 2 mechanism, caspase 8 converts Bid to tBid, which acts on the mitochondrial membrane, which releases cytochrome c. In the absence of ATP, necrosis sets in, involving contiguous liver cells, and takes place rapidly within a few minutes. Cells swell, forming blebs along the plasma membrane, disrupting its permeability, resulting in a rupture with the release of cytosolic proteins into the blood stream. Necrosis is followed by acute inflammatory cell infiltration. Apoptosis, on the other hand, is a programmed cell death that requires the presence of ATP and occurs at a relatively a slow pace. It affects cells at discontinuous locations, and death is characterized by chromatin condensation, DNA degradation, and shrinkage of cells with very little acute inflammatory cell infiltration (adopted from [16])\n\nHepatocytes, cholangiocytes, sinusoidal endothelial cells, stellate cells, and Kupffer cells express death receptors, such as tumor necrosis factor-alpha receptor 1 (TNF\u03b1-R1). The plasma membrane receives death signals through injury by toxins, ischemia\/reperfusion injury, or toxic biological waste products. Death ligand and receptor interactions release adaptor proteins, TRADD and FADD, which in turn lead to activation of caspase 8 ( cysteine-asp artate proteases ). Caspase 8 can directly release the end product in the form of caspase 3 (type 1 signaling), which induces apoptosis (Fig. 2.1.7). Activated caspase 8 converts cytosolic inactive protein Bid to its active form, tBid, which translocates to the mitochondria and releases cytochrome c through activation of Bak and Bax, two members of the Bcl2 family (type 2 signaling). Type 2 signaling leads to opening of membrane permeability transition (MPT) pores in the inner membrane of the mitochondria. Opening of MPT starts the process of cell death, either through necrosis or apoptosis, depending upon the next sequence of events that follows [17]. If the MPT opening is sudden due hepatotoxins, ischemia\/reperfusion injury, or calcium overload, all of which lead to ATP depletion, the process leads to mitochondrial depolarization, disruption of oxidative phosphorylation, and mitochondrial swelling, rupture of the outer membrane, and necrosis [18]. In the presence of an adequate amount of ATP, cytochrome c releases caspase 9, which in turn releases caspase 3, which promotes apoptosis (Fig. 2.1.6).\n\nFig. 2.1.7\n\nDeath ligand and receptor interaction. Plasma membrane receives death signals through injury by toxins, ischemia\/reperfusion injury, or toxic biological waste products. Death ligand and receptor interactions release adaptor proteins, TRADD and FADD, which in turn lead to activation of procaspase 8 (cysteine-aspartate proteases) to caspase 8. Caspase 8 can directly release the end product caspase 3 (type 1 signaling) that induces apoptosis. Activated caspase 8 converts cytosolic inactive protein, Bid, to its active form, tBid, which translocates to the mitochondria and releases cytochrome c through activation of Bak and Bax (type 2 signaling). Type 2 signaling leads to opening of membrane permeability transition (MPT) pores in the inner membrane of the mitochondria. Opening of MPT starts the process of cell death, either through necrosis or apoptosis, depending upon the next sequence of events that follows. If the MPT opening is sudden due to death receptor-ligand interaction, which leads to ATP depletion and rupture of the outer membrane, necrosis is the outcome. Nuclear factor k beta in the genes controls cell death. In the presence of adequate amounts of ATP, cytochrome c releases caspase 9, which in turn releases caspase 3, which promotes apoptosis\n\n### 2.1.10 Bile Entry, Storage, and Concentration by the Gallbladder\n\nHepatic bile that enters the gallbladder is of much lower osmolality than the bile that leaves the gallbladder after a meal (Table 2.1.1). A major portion (60-70%) of the hepatic bile secreted during fasting enters the gallbladder, and a minor fraction enters the duodenum directly, depending upon the tonus of the sphincter of Oddi (Fig. 2.1.8). During a period of 10 h in the night, the liver produces about 250 ml of bile (25 ml h-1 or 0.42 ml h-1), of which 175 ml (70%) enters the gallbladder (17.5 ml h-1 or 0.3 ml min-1). An empty gallbladder can fill to its maximum capacity of 50 ml during a period of 6 h. A fully filled gallbladder continues to receive hepatic bile secreted during fasting by absorbing 0.3 ml of water per minute. This is accomplished primarily by absorption of water and electrolytes through the gallbladder wall. As much as 90% of water can be absorbed through the gallbladder wall during a 6-h period. Dietschy demonstrated that 100 ml of hepatic bile placed inside the gallbladder reduces to less than 10 ml in 6 h mainly through absorption of water (Fig. 2.1.9). As a consequence of this selective water absorption, the concentration of solutes increases in the gallbladder bile [19]. This process of selective absorption of water resulting in higher concentration of bile salts, bilirubin, cholesterol, fatty acids, and lecithin in the gallbladder bile than hepatic bile is called the concentration function of the gallbladder [20, 21]. A normal gallbladder can sequester all of 3-6 g total body bile salts within it after an overnight fast. Fresh hepatic bile enters the gallbladder along its central long axis and moves laterally to reach the wall as the space is made available through removal of water. It takes nearly 30 min for the fresh hepatic bile to reach the wall from the central long axis [21].\n\nFig. 2.1.8\n\nBile entry mechanism into the gallbladder. A higher mean (range) pressure at the sphincter of Oddi (15 cm of H2O) than the pressure inside the gallbladder lumen and absorption of water through the wall during fasting facilitate constant hepatic bile inflow into the gallbladder\n\nFig. 2.1.9\n\nBile concentration. Mainly through different rates of absorption, gallbladder (rabbit) increases the concentration of sodium and bile salts and decreases the concentration of chloride and bicarbonate when compared to hepatic bile. Gallbladder bile volume decreases by 90% of its basal volume to achieve these concentrations [20]\n\nThe volume of hepatic bile that enters the gallbladder during fasting is controlled by two mechanisms, the tonus of the sphincter of Oddi and absorption of water through the gallbladder wall (Fig. 2.1.10). Under basal conditions, the mean pressure in the sphincter of Oddi is 15 cm of water, in the common bile duct, 12 cm of water, and in the gallbladder, 10 cm of water. By choosing the path of least resistance, the hepatic bile enters the gallbladder. The tonus of the sphincter of Oddi is dependent upon the frequency and amplitude of phasic wave contractions. The phasic wave frequency ranges from 1 to 13 per minute with an average of six contractions per minute [22]. High frequency contractions (> 8 min-1) are relatively rare (14%). Sphincter of Oddi phasic wave contractions occur in conjunction with phase III of the migrating motor complex, which begins in the stomach and traverses through the duodenum and jejunum [23].\n\nFig. 2.1.10\n\nAbsorption of water through the gallbladder epithelium. Lateral intercellular channels between the columnar epithelial cells are widely open during fasting, allowing free passage of water from the lumen into the connective space. Water absorption creates space for constant entry of hepatic bile into the gallbladder during fasting [20]\n\nThe second mechanism that promotes hepatic bile entry into the gallbladder is the rate of absorption of water through the gallbladder wall. During fasting, the lateral intercellular channels between the columnar epithelial cells are widely opened, allowing the passage of sodium and water from the gallbladder lumen into the interstitial spaces and then the hepatic venous blood (Fig. 2.1.4). Constant removal of water allows steady entry of hepatic bile into the gallbladder (Fig. 2.1.11). These lateral intercellular spaces collapse when the gallbladder wall contracts following ingestion of a meal or injection of cholecystokinin. Quabain inhibits water transport across most epithelial membranes, including the gallbladder [20]. Spontaneous gallbladder emptying and refilling play a minor role in making room for the fresh hepatic bile. In large number of normal subjects monitored continuously for 2 h with Tc-99m-HIDA by the authors, spontaneous gallbladder emptying to the extent of 5-10% was noticed in less than 2% of the subjects. The bile that enters the duodenum during fasting normally moves antegrade towards the jejunum. Normally, there is no bile reflux from the duodenum into the stomach in the basal state.\n\n### 2.1.11 Gallbladder Emptying\n\nEmptying of the gallbladder is under both hormonal and nervous control, with hormonal control playing the major role. Nervous control is exerted through both the sympathetic and parasympathetic system as shown in Fig. 1.1.12 of Chap. 1. The cholinergic parasympathetic nerve fibers to the gallbladder come primarily from the anterior gastric plexus (left vagus) and control its contraction and emptying. The gallbladder response to sham feeding is mediated through these cholinergic nerve fibers and can be blocked with atropine or after vagotomy [24]. The parasympathetic cholinergic nerve fibers to the sphincter of Oddi come mainly from the posterior gastric plexus (right vagus). Sympathetic post-ganglionic nerve fibers from the celiac ganglion reach the gallbladder and the sphincter of Oddi. Sympathetic nerve stimulation relaxes the gallbladder wall and promotes hepatic bile entry.\n\nThe hormone-induced contraction and emptying of the gallbladder occur mainly through endogenous release of cholecystokinin (CCK), demonstrated first in 1928 by Ivy and Oldberg [25]. Cholecystokinin, a linear 33 or 39 amino acid polypeptide, is produced from a large precursor protein with 114 amino acids. Most of the biological functions of the hormone are confined to the last four carboxy terminal amino acids. A much shorter cholecystokinin with only eight carboxy terminal amino acids (CCK-8) carries out most of the biological functions of the parent molecule with 33 amino acids [26]. The contraction and emptying of the gallbladder begin within 2-3 min after an intravenous administration of CCK-8, but it may take as long as 10-20 min to begin emptying following a meal [27]. This time delay in gallbladder emptying after a meal is due to a combination of time taken for the meal to pass from the stomach into the duodenum and the time taken for the CCK-secreting cells in the duodenal mucosa to release enough hormones into the blood stream. Cerulein, motilin, and other gastrointestinal hormones with an identical carboxy terminal tetrapeptide also induce gallbladder contraction and emptying [28].\n\nFig. 2.1.11\n\nPattern of bile transit within the gallbladder. Fresh hepatic bile enters the gallbladder along its central long axis and moves laterally to reach the wall as water is removed by the lateral intercellular channels. It takes about 30 min for the fresh hepatic bile to reach the wall from the central long axis. Fully filled gallbladder outline is superimposed onto early frames to show how radiolabeled fresh bile moves inside. Radiolabed bile first enters at 12 min and reaches the wall at 48 min [21]\n\n#### 2.1.11.1 Spleen Function\n\nThe spleen is composed mostly of reticuloendothelial cells and lymphoid tissue. The spleen is usually seen during radionuclide imaging with radiocolloids and radiolabeled red blood cells, leucocytes, monoclonal antibodies, or peptides (somatostatin). As the spleen does not concentrate Tc-99m-HIDA, it is not seen during functional hepatobiliary imaging.\n\nA normal spleen weighs between 80-200 gm with an average of 150 gm [1]. In the posterior view of a radiocolloid scan, the spleen normally measures 10.5 cm along the oblique axis. The spleen has to enlarge more than 2.5 times its normal size before it is palpable in the left upper quadrant during a routine physical examination. Most of the palpable spleens are enlarged, but not all non-palpable spleens are normal in size.\n\nThe spleen carries out many important functions (Table 2.2.1) that include: (1) hemopoiesis, (2) destruction of senescent red blood cells, leucocytes, and platelets, (3) culling and pitting, (4) phagocytosis, (5) reservoir function, and (6) immunologic function [2].\n\nTable 2.2.1\n\nFunctions of the spleen\n\n(1) Hemopoiesis\n\n---\n\n(2) Destruction of senescent red blood cells, leucocytes and platelets\n\n(3) Culling and pitting\n\n(4) Phagocytosis\n\n(5) Reservoir function\n\n(6) Immunologic function\n\nDuring the first 6 months of intrauterine life, the spleen functions as a major hematopoietic organ, and this function normally disappears by birth. In thalassemia and myeloid metaplasia, the spleen is capable of resuming its intrauterine function in adult life to produce red blood cells. The normal function in adults is one of destruction of senescent red blood cells, leucocytes, and platelets. Excessive destruction of blood cells by the spleen results in anemia, leucopenia, or thrombocytopenia. The volume of red blood cells in the body and their destruction by the spleen are measured by labeling the cells with chromium-51 [3]. During labeling, hexavalent chromium-51 (Na2CrO4) crosses the red cell membrane and attaches to hemoglobin after reduction into a trivalent form [4]. The physical half life of Cr-51 is 27 days. Random labeling, elution of Cr-51 and death of senescent red cells all account for the mean RBC survival half-time of 35 days (normal life span of RBC is 120 days). Chromium-51 released after RBC death is taken up by the reticuloendothelial cells, and it does not label other red blood cells in blood; hence, it serves as an ideal marker for RBC survival studies.\n\nThe selective removal of abnormal red cells by the spleen is called culling, and removal of intra-erythrocytic inclusions without destroying the RBC is called pitting. Howell-Jolly bodies, remnant of RBC nucleus, are removed from the red blood cells by the spleen. The appearance of Howell-Jolly bodies in the peripheral blood, therefore, is an indication of either splenectomy or of a non-functioning spleen [5]. Hypersplenism is documented by showing increased spleen\/liver and increased spleen\/precordial count ratio in association with decreased RBC survival time. Return of splenic function after splenectomy is attributed to auto-transplantation (due to spillage) of the splenic tissue on the peritoneal surface or an accessory spleen [6]. Imaging of the spleen with radiocolloids, a popular modality in the 1970 and 1980s, is largely replaced today by computerized tomography and ultrasound. Radiocolloid spleen imaging is now obtained occasionally to clarify an abnormality that has already been detected with CT or ultrasound or to confirm splenomegaly in the diagnosis of polycythemia rubra vera [7].\n\nThe spleen capsule has a thin muscle layer whose contraction squeezes out the sequestered RBC, WBC, and platelets into peripheral circulation. Injection of epinephrine induces smooth muscle contraction with a subsequent rise in peripheral cell count [3]. The spleen plays a protective role on the lung tissue, and pulmonary hypertension is shown to develop after splenectomy in some patients [8]. The immunologic function of the spleen is carried out by the lymphocytes and reticuloendothelial cells (phagocytosis). The spleen produces antibodies against many microorganisms, especially polysaccharide-encapsulated bacteria, such as pneumococci. Overwhelming infection sometimes follows after splenectomy in children. Human splenic autotransplantation produces significant antipneumococcal antibody in response to administration of pneumococcal vaccine [9].\n\nReferences\n\n1.\n\nFitz JG. Cellular mechanism of bile secretion. In: Zakim D, Boyer TD, eds. Hepatology.  A textbook of liver disease. Philadelphia, WB Saunders, 1996, pp 362-376\n\n2.\n\nBrauer RW, Leong SF, Holloway RJ. Mechanism of secretion. Effect of perfusion pressure and temperature on bile flow and bile secretion pressure. Am. J Physiology 1954;177:103-106\n\n3.\n\nBoyer JL, Klatskin G. Canalicular bile flow and bile secretary pressure. Evidence for a non-bile salt dependent fraction in the isolated perfused rat liver. Gastroenterology 1970;59:853-859PubMed\n\n4.\n\nJones AL, Schmucker DL, Renston RH, Murakami T. The architecture of bile secretion. A morphological perspective of physiology. Dig Dis Sci 1980;25:609-629PubMedCrossRef\n\n5.\n\nSellinger M, Barrett C, Malle P, Gordon ER, Boyer JL. Cryptic Na+, K+-ATPase activity in the rat liver canalicular plasma membranes: evidence for its basolateral origin. Hepatology 1990;11:223-229PubMedCrossRef\n\n6.\n\nMeier PJ. Molecular mechanisms of hepatic bile salt transport from sinusoidal blood into bile. Am J Physiol 1995;269:G801-G812PubMed\n\n7.\n\nPauli-Magnus C, Meier PJ. Hepatobiliary transporters and drud-induced cholestasis. Hepatology 2006;44:778-787PubMedCrossRef\n\n8.\n\nKarpen SJ. Nuclear receptor regulation of hepatic function. J Hepatology 2002;36:832-850CrossRef00129-0)\n\n9.\n\nBoyer JL. Nuclear receptor ligands: rational and effective therapy for chronic cholestatic liver disease. Gastroenterology 2005;129:735-740PubMed\n\n10.\n\nTrauner M, Meier PJ, Boyer JL. Molecular pathogenesis of cholestasis. N Eng J Med 1998:339:1217-1227CrossRef\n\n11.\n\nKamimato Y, Gatmaitan Z, Hsu J, Arias IM. The function of Gp 170, the multidrug resistance gene product, in rat liver canalicular membrane vesicles. J Biol Chem 1989;264:11693-11698\n\n12.\n\nMasyuk AI, Marinelli RA, LaRusso NF. Water transport by epithelia of the digestive tract. Gastroenterology 2002;122:545-562PubMedCrossRef\n\n13.\n\nMasyuk AI, LaRusso NF. Aquaporins in the hepatobiliary system. Hepatology 2006;43:S75-S81PubMedCrossRef\n\n14.\n\nDonohue TM Jr., Tuma DJ, Sorrell MF. Plasma proteins metabolism. In: Zakim D, Boyer TD. Hepatology. A text book of liver disease, 3rd edn. Philadelphia, WB Saunders, 1996, pp 130-148\n\n15.\n\nLemasters JJ. Necrapopstosis and the mitochondrial permeability transition: shared pathways to necrosis and apoptosis. Am J Physiol 1999;276:G1-G6PubMed\n\n16.\n\nLemasters JJ. Dying of a thousand deaths: redundant pathways from different organells to apoptosis and necrosis. Gastroenterology 2005;129:351-360PubMedCrossRef\n\n17.\n\nMalhi H, Gres GJ, Lamaster JJ. Apoptosis and necrosis in the liver: a tale of two deaths? Hepatology 2006;43:S31-S44PubMedCrossRef\n\n18.\n\nKrishnamurthy G, Krishnamurthy S. Cholescintigraphic measurement of liver function: how is it different from other methods? Eur J Nucl Med Mol Imaging 2006;33:1103-1106PubMedCrossRef\n\n19.\n\nDietschy JM. Water and solute movement across the wall of the everted rabbit gallbladder. Gastroenterology 1964;47:395-408PubMed\n\n20.\n\nWheeler HO. Concentrating function of the gallbladder. Am J Med 1971;51:588-595PubMedCrossRef90283-X)\n\n21.\n\nKrishnamurthy S, Krishnamurthy GT. Hepatic bile entry into and transit pattern within the gallbladder lumen: a new technique quantitative cholescintigraphic technique for measurement of its concentration function. J Nucl Med 2002;43:901-908PubMed\n\n22.\n\nThune A, Scicchitano J, Roberts-Thompson I, Toouli J. Reproducibility of endoscopic sphincter of Oddi manometry. Dig Dis Sci 1991;36:1401-1405PubMedCrossRef\n\n23.\n\nLee SK, Kim MH, Seo DW, Yoo BM, Lee MH, Myung SJ, Min YI. Frequency of phasic wave contraction is variable during long-term sphincter of Oddi manometry. Am J Gastroenterol 1996;91:2395-2398PubMed\n\n24.\n\nFisher RS, Rock E, Malmud LS. Gallbladder emptying response to sham feeding in humans. Gastroenterology 1986;90:1854-1857PubMed\n\n25.\n\nIvy AC, Oldberg E. A hormone mechanism of gallbladder contraction and evacuation. Am  J Physiology 1928;86:599-613\n\n26.\n\nMutt V. Cholecystokinin: isolation, structure, and function. In: Jerzy Glass GB. ed. Gastrointestinal hormones. New York, Raven, 1980, pp 169-221\n\n27.\n\nBobba VR, Krishnamurthy GT, Kingston E, Turner FE, Brown PH, Langrell K. Gallbladder dynamics induced by a fatty meal in normal subjects and patients with gallstones: Concise communication. J Nucl Med 1984;25:21-24PubMed\n\n28.\n\nBloom SR, Adrian TE, Mitchenere P, et al. Motilin induced gallbladder contraction. A new mechanism. Gastroenterology 1981;80:1113\n\nReferences\n\n1.\n\n1. Mattsson O. Scintigraphic spleen volume calculation. Acta Radiol [Diagn] (Stockh) 1982;23:471-477\n\n2.\n\nSpencer RP, Pearson HA. The spleen as a hematological organ. Semin Nucl Med 1975;5:95-102PubMedCrossRef80007-9)\n\n3.\n\nGray SJ, Sterling K. The tagging of red blood cells and plasma proteins with radioactive chromium. J Clin Invest 1950;29:1604-1613PubMedCrossRef\n\n4.\n\nPollycove M, Tono M. Blood volume. In: Sandler MP, Patton JH, Coleman RE, Gottschalk A, Wackers FJTh, Hopper PB. Diagnostic nuclear medicine, 3rd edn. Williams and Wilkins, Baltimore, 1996, pp 827-834\n\n5.\n\nCrosby WH. Normal functions of the spleen relative to red blood cells. A review. Blood 1959;14:399-408\n\n6.\n\nPearson HA, Johnston D, Smith KA, Touloukian RJ. The born-again spleen. Return of splenic function after splenectomy for trauma. New Eng J Med 1978;298:1389-1392PubMedCrossRef\n\n7.\n\nSpencer RP. Spleen imaging. In: Sandler MP, Patton JH, Coleman RE, Gottschalk A, Wackers FJT, Hopper PB. Diagnostic nuclear medicine, 3rd edn. Williams and Wilkins, Baltimore, 1996, pp 865-874\n\n8.\n\nHoeper MM, Niedermeyer J, Hoffmeyer F, Flemming P, Fabel H. Pulmonary hypertension after splenectomy? Ann Intern Med 1999;130:506-509PubMed\n\n9.\n\nLeemans R, Manson W, Snijder JAM, Smit JW, Klasen HJ, The TH, Timens W. Immune response capacity after human splenic autotransplantation. Restoration of response to individual pneumococcal vaccine subtypes. Ann Surg 1999;229:279-285PubMedCrossRef\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_3(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 3. Imaging Agents\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nHistorical evolution The introduction of radiocolloids in the 1940s, whose rate of clearance from the circulation was used as an indicator of liver function, gave birth to nuclear hepatology [1]. Imaging of the liver morphology began in 1954 with gold-198 colloid using an automated rectilinear scanner developed by Cassen [2, 3]. Imaging of morphology was supplanted by imaging of liver physiology with the introduction of I-131 rose bengal in 1955 by Taplin et al. [4]. Morphology imaging gained wide clinical popularity after the introduction of technetium-99m colloid in 1965 [5], and a rapid leap in imaging of physiology occurred with the introduction of technetium-99m-HIDA agents in 1976 [6].\n\nHistorical evolution The introduction of radiocolloids in the 1940s, whose rate of clearance from the circulation was used as an indicator of liver function, gave birth to nuclear hepatology [1]. Imaging of the liver morphology began in 1954 with gold-198 colloid using an automated rectilinear scanner developed by Cassen [2, 3]. Imaging of morphology was supplanted by imaging of liver physiology with the introduction of I-131 rose bengal in 1955 by Taplin et al. [4]. Morphology imaging gained wide clinical popularity after the introduction of technetium-99m colloid in 1965 [5], and a rapid leap in imaging of physiology occurred with the introduction of technetium-99m-HIDA agents in 1976 [6].\n\n## 3.1 Morphology and Physiology Imaging Agents\n\nThe liver carries out the most complex biological functions through the hepatocytes, Kupffer cells, endothelial cells, fat cells (Ito cells), blood vessels and biliary epithelial cells. The liver imaging agents are classified broadly into two groups: (1) those that identify pathology mainly through detection of changes in liver morphology and (2) those that identify pathology by identifying changes in liver physiology (Table 3.1.1). A wide variety of imaging agents now available enable the study of specific cell functions (Table 3.1.2). Radiocolloids are the most common agents for imaging morphology, and Tc-99m-HIDA compounds are the most preferred agents for physiology. Since liver morphology imaging is now carried out mostly with CT or ultrasound, imaging of physiology is currently the most common diagnostic procedure in nuclear hepatology. Since the publication of the first edition of our book, fluorine-18-labeled 2-fluoro-2-deoxyglucose (18-FDG) has become a major factor in the assessment of primary and metastatic lesions of the liver and is included in the current edition.\n\nTable 3.1.1\n\nClassification of liver imaging agents\n\n(1) Morphology imaging agents | Radiocolloids (gold-198, Tc-99m-S colloid) gallium-67 citrate\n\n---|---\n\n(2) Physiology imaging agents | These agents can be further divided into four subgroups based on specific function\n\n(a) Hepatobiliary agents | Tc-99m-HIDA, Tc-99m-PG, I-131 rose bengal, etc.\n\n(b) Blood pool agents | Tc-99m-RBC, Tc-99m-albumin\n\n(c) Receptor-specific agents | Tc-99m-GSA, In-lll octreotide, In-lll monoclonal antibody, Tc-99m-monoclonal antibodies\n\n(d) Miscellaneous agents | Indium -lll WBC, Tc-99m-WBC(HMPAO), fluorine-18 2-fluoro-2-deoxyglucose (18FDG) and other agents that are used primarily for imaging of other organs, but pass through the liver during their elimination, enabling imaging of the hepatobiliary system, e.g., Tc-99m-sestamibi, Tc-99m-tetrofosmin, Tc-99m-HMPAO, etc.\n\nTable 3.1.2\n\nThe cell, mechanism of uptake, disposition and dose of radiolabeled agents for imaging of the liver and spleen\n\nAgent | Uptake by | Mechanism of uptake | Excretion into bile | Dose\n\n---|---|---|---|---\n\nTc-99-S colloid | Kupffer cell | Phagocytosis | No | 2-5 mCi\n\nTc-99m-GSA | Hepatocyte | ASGP receptor | No | 2-8 mCi\n\nTc-99m-HIDA | Hepatocyte | RME | Yes | 2-8 mCi\n\nTc-99m-RBC | Hemangioma | Blood pool | No | 10-20 mCi\n\nTc-99m HMPAO | Infection | Chemotaxis | No | 10-20 mCi\n\nIn-111 WBC(oxine) | Infection | Chemotaxis | No | 0.5-1.0 mCi\n\nIn-111 Octreotide | Somatostatin- | Receptor-ligand receptor | No interaction | 3-8 mCi\n\nIn-111 MOAB | TAG-72 | Antigen-antibody Interaction | No | 2-5 mCi\n\nGa-67 citrate | Hepatocyte | Unknown | No | 2-10 mCi\n\nF-18 FDG | Hepatocyte | Glucose receptor | No | 10-15 mCi\n\nASGP Asialoglycoprotein, GSA, galactosyl human serum albumin, HIDA hepatic iminodiacetic acid, RME receptor-mediated endocytosis.\n\n### 3.1.1 Radiocolloids\n\nRadiocolloids are small particles, 5-1 \u00b5m size, that are removed from circulation by the reticuloendothelial cells of the liver, spleen and bone marrow. They carry a negative charge of -30 mV and do not pass through the dialysis membrane, which is permeable to ions. However, they readily pass through an ion-exchange column and show no movement on paper chromatography [7]. Although Au-198 is no longer used for imaging, thorough understanding of its pharmacokinetics provides a sound basis for understanding the biokinetics of Tc-99m-S colloid and is described briefly below.\n\n### 3.1.2 Gold-198 Colloid\n\nGold (Au)-198 colloid is one of the smallest radiocolloid particles known to nuclear medicine. The particles vary in size from 5 to 50 m\u00b5, with an average size of 30 m\u00b5, and clear from blood with a T \u00bd of 3 min. Gold-198 has a physical half life of 2.7 days, decays by a beta minus emission and emits a monoenergetic gamma photon of 411 keV for imaging. The adult dose ranges from 100 to 150 \u03bcCi in 25 \u03bcg to 2.5 mg gold. The three organs concentrating Au-198 colloid are the liver, spleen and bone marrow, with the smallest particles preferentially taken up by the bone marrow and the largest particles by the spleen [9]. The percent dose localized in each organ depends upon its function and size, and the size and electrochemical character of the radiocolloid particles. Normally, the liver weighs about 1,500-1,800 g, the spleen 150-200 g and the bone marrow \u223c1,500 g (Table 3.1.3). Normally, about 90% of the Au-198 colloid dose is taken up by the liver, 7% by the reticuloendothelial (RE) cells of the bone marrow and 3% by the spleen [10]. In moderate hepatocellular disease, the liver mass increases (2,400 g) because of fatty infiltration, and the mass of the spleen increases (250 g) because of portal hypertension. In advanced cirrhosis, the liver actually shrinks in size (1,400 g) because of fibrosis, whereas the spleen continues to increase in size (400 g) because of the increase in portal hypertension. The mass of reticuloendothelial cells in the bone marrow does not change. In moderate liver disease, the uptake by the liver decreases to 70%. In advanced cirrhosis when the liver shrinks in size, the radiocolloid uptake may fall below 35%. As the uptake of radiocolloid by the liver decreases, the uptake by the spleen and bone marrow increases in direct proportion to the degree of portal hypertension.\n\nTable 3.1.3\n\nLiver, spleen and bone marrow mass in normal subjects and patients with liver disease [10]\n\nClinical status | Mass of the organ in grams\n\n---|---\n\nLiver | Spleen | Bone marrow\n\nNormal | 1,807 | 174 | 1,500\n\nEarly to intermediate liver disease | 2,400 | 250 | 1,500\n\nIntermediate to advanced liver disease | 1,400 | 400 | 1,500\n\n### 3.1.3 Technetium-99m-Sulfur Colloid\n\nTechnetium-99m-sulfur colloid particles are much larger, with a wider variation in size than Au-198 colloid particles, with an average size of about 300 m\u00b5 (Table 3.1.4). The usual dose for planar imaging in adults is 2-5 mCi. Liver perfusion and SPECT studies require a much larger dose, in the range of 5-10 mCi. Technetium-99m has a physical half life of 6 h and emits a gamma photon of 140 keV energy, which makes it ideal for imaging. Intravenously injected particles clear from blood with a T \u00bd of 2.5 min.\n\nTable 3.1.4\n\nCharacteristics of Tc-99m-S colloid\n\nParameter\n\n|   \n---|---\n\nHalf life | 6.03 h\n\nBlood clearance T\u00bd | 2.5 min\n\nParticle size range (\u03bcm) | 100-1,000\n\nParticle mean size (\u03bcm) | 300\n\nDecay constant | 0.1149 h-1\n\nMean disintegration | 87%\n\nG-rad \u03bcci-1 h-1 | 0.0369\n\nAdult dose | 2-8 mCi\n\nDose to liver from a study | 0.68-2.72 rads\n\nNormally, about 85% of the injected dose of Tc-99m-S colloid is taken up by the liver, 7% by the spleen, 5% by the bone marrow and the remaining 3% by other organs, such as the lungs, stomach, etc. [11]. In moderate parenchymal liver disease, the radiocolloid uptake by the liver decreases, and the uptake by the spleen and bone marrow increases. In severe parenchymal liver disease (cirrhosis), when the liver shrinks in size, \u223c30% of the dose is taken up by the liver, and as much as 30% by the spleen, 25% by the red marrow and the remaining 13% of the dose by other organs, including the lungs, kidney and stomach (Table 3.1.5). As most of liver morphology imaging is currently obtained with CT, MRI or ultrasound, radiocolloid imaging is chosen primarily for delineating the functional characteristics of the lesion that have already been detected with one of the other imaging modalities. Radiocolloid spleen imaging is often obtained for accurate measurement of spleen size in the diagnosis of polycythemia rubra vera and other myeloproliferative disorders. Technetium-99m-S colloid preparations are also used in the measurement of gastric emptying time, evaluation of shunt (LeVeen) patency and for the detection of acute GI bleeding, gastro-esophageal reflux or pulmonary aspiration of gastric contents, etc.\n\nTable 3.1.5\n\nRelative distribution of Tc-99m-sulfur colloid (% injected dose) in normal subjects and patients with liver disease [11] | Liver | Spleen | Marrow | Other\n\n---|---|---|---|---  \n|  |  |\n\nClinical status\n\nNormal | 85 | 7 | 5 | 3\n\n|\n\nMild to moderate liver disease | 67 | 13 | 12 | 8\n\n|\n\nModerate to severe liver disease | 32 | 30 | 25 | 13\n\n|\n\nThe radiocolloid dose is used within 6 h of its preparation, and the particles are shaken vigorously just prior to injection to prevent settlement at the bottom of the vial. As free Tc-99m pertechnetate appears in breast milk, mothers are instructed not to feed their infants with breast milk for the next 24 h and to use formula milk instead. The breast milk expressed prior to radiocolloid injection and stored is preferable to formula milk. The usual pediatric dose is 15-75 \u03bcCi kg-1 (0.56-2.78 MBq kg-1). For bone marrow imaging in pediatrics, the dose is 30-150 \u03bcCi kg-1 (1.1-5.6 MBq kg-1). The minimum dose per study is 600 \u03bcCi (22.2 MBq). For gastroesophageal reflux and pulmonary aspiration studies, the Tc-99m-S colloid dose is given orally, mixed with a liquid or a semisolid meal.\n\n### 3.1.4 Mechanism of Radiocolloid Uptake\n\nMetchnikoff in 1884 first observed a process where highly mobile polymorphonuclear leucocytes ingested (phagocytosis) foreign particles, microorganisms and cellular debris, removing them from the circulation. Similar phenomena were observed in the cells lining the liver sinusoids [12]. Kupffer in 1899 described the nature of these cells in the hepatic sinusoids which today bear his name, the Kupffer cells [13]. Aschoff in 1924 introduced the concept of the reticuloendothelial system consisting of the mesenchymal cells distributed throughout the body [14]. Radiocolloid uptake is seen primarily in three organs in the body: the liver, spleen and bone marrow. Lung and other organ uptake normally is not high enough to be seen on the images.\n\nThe uptake of radiocolloids by the RE system cells is dependent upon various factors, including particle size, charge, dose, chemical composition and other factors. The maximum phagocytic capacity in humans is 1.07 mg min-1 kg-1 body weight [15]. The RE system acts as a biological filter by removing effete cells and foreign material from blood and thus restricting the general toxic effect on the body. The filtering mechanism becomes increasingly efficient in the spleen as the particle size increases. The smallest particles are removed preferentially by the bone marrow, medium-size particles by the liver and the largest particles by the spleen. As the Au-198 colloid particles are the smallest (30 \u03bcm), most of them are taken up by the liver and bone marrow, and very few particles by the spleen, explaining the reason why the spleen is usually not visualized well in the scans obtained with gold-198.\n\nUpon intravenous injection, radiocolloid particles are coated by plasma opsonins, making them susceptible to phagocytosis. Radiocolloid-opsonin complex attaches to the RE cell membrane and initiates phagocytosis (Fig. 3.1.1). The cytoplasm of the RE cell flows around the opsonized radiocolloid particles as pseudopods, encircles the particle and ultimately incorporates into the cytoplasm, forming a phagosome [16, 17]. The ultimate fate of the radiocolloid in the phagosome depends upon its nature. The phagosome is either destroyed and the contents stored in the cytosol or the phagosome is destroyed, digested and its contents released back into circulation. Gold-198 radiocolloids are stored intracellularly, whereas the Tc-99m-S colloids are digested and released into the circulation [18].\n\nFig. 3.1.1\n\nMechanism of radiocolloid uptake. After intravenous injection, radiocolloid comes in contact with plasma proteins forming a radiocolloid-opsonin complex. The complex is encircled by pseudopodes of the Kupffer cells in the liver and reticuloendothelial (RE) cells in the spleen and bone marrow. After complete engulfment, the radiocolloid forms a phagosome within the cytosol\n\n### 3.1.5 Dosimetry of Radiocolloids\n\nThe liver is the critical organ for radiocolloids and receives 39 rads mCi-1 of Au 198 and 0.34 rad mCi-1 of technetium-99m-S colloid (Table 3.1.6). From one imaging study, the liver receives 5.8 rads from Au-198 (200 \u03bcCi dose) and 0.68 rads from Tc-99m-S colloid (2-mCi dose). Radiation to the liver decreases as the severity of liver disease increases. The total body receives 1.4 rads mCi-1 with Au-198 and 0.019 rads mCi-1 with Tc-99m-S colloid. Introduction of Tc-99m-sulfur colloid in 1965 was a major breakthrough that revolutionized liver morphology imaging in the late 1970s and through the early 1980s.\n\nTable 3.1.6\n\nAbsorbed dose (rads mCi-1) from Tc-99m-sulfur colloid in normal subjects and patients with liver disease [11] | Normal | Moderate disease | Severe disease\n\n---|---|---|---\n\nOrganLiver | 0.34 | 0.21 | 0.16\n\nSpleen | 0.21 | 0.28 | 0.42\n\nRed marrow | 0.02 | 0.04 | 0.07\n\nOvaries | 0.005 | 0.008 | 0.012\n\nTestes | 0.001 | 0.002 | 0.003\n\nTotal body | 0.019 | 0.019 | 0.019\n\n### 3.1.6 Technetium-99m HIDA Agents\n\nFunctional imaging nuclear hepatology took a quantum leap with the introduction of Tc-99m-HIDA agents by Loberg et al. in 1976 [6]. Hepatobiliary diseases are quite common, and they often present clinically with dramatic suddenness requiring immediate diagnosis and therapy (acute cholecystitis). Technetium-99m HIDA agents fulfill the requirement for a rapid diagnosis.\n\n### 3.1.7 Labeling\n\nTechnetium-99m-HIDA agents are lidocaine analogues. Lidocaine has been used in clinical practice for many years, and its pharmacokinetics are well understood. Liver is the primary site of uptake and metabolism of lidocaine and indocyanin green [19-21]. Lidocaine (C14H22N2O) has a molecular weight of 270.80 and clears from blood in a biexponential fashion; the fast component has a T\u00bd of 7 min, and the slow component a T \u00bd of 108 min. Lidocaine shows a high first-pass extraction by the liver. About 90% of extracted lidocaine is metabolized into monoethylglycylxylidine, and the remaining 10% is excreted in urine unchanged. Age does not affect the biokinetic behavior of lidocaine.\n\nBased upon its wide clinical application in cardiology and with the knowledge that lidocaine is metabolized primarily in the liver, it was hypothesized that a technetium-99m-labeled lidocaine or its analogues could find potential imaging application in either cardiology or hepatology. Since technetium-99m could not be attached to lidocaine directly, iminodiacetic acid (IDA) was chosen as a bifunctional chelate to carry the ligand (lidocaine) at one end and the radiotracer (Tc-99m) at the other end [6]. Each molecule of labeled Tc-99m-HIDA complex, therefore, consists of two molecules each of the ligand and the chelate and an atom of technetium-99m in the middle (Fig. 3.1.2). When Tc-99m-HIDA was injected into mice and gamma camera imaging began, the investigators were startled not to find any myocardial uptake, but were immensely pleased to see radioactivity first in the hepatocytes and later throughout the biliary system [22]. For lack of any other better terminology, the investigators settled on a new name, Hepatic IminoDiacetic Acid, or HIDA. Thus, the major portion of the name (HIDA) reflects the chelate (IDA) more than it does the ligand (lidocaine), which provides the most critical functional information.\n\nFig. 3.1.2\n\nMolecular structure and hepatobiliary transit of Tc-99m-HIDA. Lidocaine (green) is the ligand with the biologic function, technetium-99m (black) is the radiotracer, and iminodiacetic acid (IDA) is the chelate (red) that binds them together. A labeled whole complex consists of an atom of Tc-99m, two molecules of lidocaine and two molecules of IDA. Albumin delivers the radiotracer to the space of Disse where the dissociation takes place. Tc-99m HIDA is taken up by the hepatocyte and secreted into bile canaliculi in free form where it mixes with the hepatic bile and serves as an ideal in vivo tracer for imaging of the entire hepatobiliary tree [27]\n\nRadiolabeling with technetium-99m does not affect the blood clearance or the hepatic uptake of lidocaine, but it alters the intrahepatic transit. Unlike lidocaine, Tc-99m-HIDA is not metabolized during its transit through the hepatocyte. It is secreted as native Tc-99m-HIDA into the bile canaliculi. This feature is readily demonstrated by re-injection of radiolabeled gallbladder bile intravenously into the same animal where the exact kinetics of the original injection are reproduced. Lidocaine forms a dimer with technetium-99m, which increases the molecular weight from 270.8 for lidocaine to 833 for Tc-99m-HIDA complex (Fig. 3.1.2). Increase in molecular weight enhances hepatocyte uptake and excretion. Labeling with technetium-99m imparts hepatic specificity. Lidocaine labeled with C14 or Tin (Sn)-113m, however, does not show hepatic uptake and excretion; instead, it is excreted mostly through urine [6].\n\n### 3.1.8 Structure-Function Relationship\n\nThe basic configuration of all Tc-99m-HIDA agents is very similar (Fig. 3.1.3). The bifunctional chelate, IDA, attaches to a molecule of lidocaine at one end and to an atom of technetium-99m at the other. Technetium-99m is the radiotracer, and the biological function resides with lidocaine. The biokinetic behavior of the labeled complex can be altered by making substitutions in the benzene ring at positions 2,4,6 (ortho), 4 (para) or 5 (meta) with a methyl, ethyl, isopropyl or isobutyl group. A halogen is attached in the meta position (mebrofenin). More than 30 new compounds were created by making various substitutions at different positions [23, 24, 25]. Six of the compounds have undergone critical clinical trials, and three have been approved for routine use by the United State's Food and Drug Administration.\n\nFig. 3.1.3\n\nMolecular configuration of six Tc-99m HIDA agents. Biological function is altered (varying liver T 1\/2 ) by making different chemical substitutions at positions 2,4,6 of the benzene ring. Mebrofenin (TMB) has a bromine at position 5, which makes it highly resistant to displacement by bilirubin [26]\n\nHepatic uptake of Tc-99m-HIDA agents varies from a low of 82.5-98.1% [26]. Hepatic uptake of Tc-99m-disofenin is 89% and mebrofenin 98% (Table 3.1.7). The uptake and excretion of Tc-99m-HIDA agents are dependent upon various factors, of which molecular structure, weight, lipid solubility and protein binding are important parameters. Radiolabeled complexes with a molecular weight between 300 and 1,000 are preferentially taken up by the hepatocytes and rapidly secreted into bile. An isopropyl substitution at 2 and 6 positions (disofenin) makes it a better agent than a dimethyl or diethyl substitution at the same locations, or an isopropyl substitution at the para position (PIPIDA). A methyl substitution at positions 2, 4, and 6 and a bromine atom at 5 (mebrofenin) result in creating the best agent of all. Tc-99m-mebrofenin shows the highest liver uptake (98%) and strongly resists displacement by a high bilirubin level. Both agents are secreted from the liver rapidly into bile with a mean excretion half time of 17 min with mebrofenin and 19 min with disofenin. The dose not taken up by the liver is excreted through the kidneys. Urinary excretion of mebrofenin is 2% and disofenin 11% of the dose injected.\n\nTable 3.1.7\n\nBiokinetic features of Tc-99m mebrofenin and Tc-99m disofenin\n\nAgent | Liver uptake (% dose) | Urine excretion (% dose) | Liver excretion (T\u00bd, min) | Radiation (mrad) to\n\n---|---|---|---|---\n\nLiver | Gallbladder\n\nTc-99m-Mebrofenin (Choletec) | 98 | 2. | 17 | 70 | 410\n\nTc-99m-Disofenin (Hepatolite) | 89 | 11 | 19 | 75 | 370\n\n### 3.1.9 Biokinetics of Tc-99m-HIDA Agents\n\nBiokinetic behavior of Tc-99m-HIDA agents can be divided into six functional phases: (1) blood transport, (2) uptake by the hepatocyte, (3) transit through the hepatocyte and secretion into bile canaliculi, (4) flow through the intrahepatic and extrahepatic ducts, (5) entry into the gallbladder and (6) final discharge into the small intestine [26, 27]. The dose not taken up by the liver is excreted in urine.\n\n### 3.1.10 Blood Transport\n\nTc-99m-HIDA agents are transported in blood bound to serum albumin, forming an albumin-Tc-99-HIDA complex [26]. Protein binding enhances hepatic delivery and hepatocyte uptake and decreases renal excretion. The agents clear from the blood at variable rates (Fig. 3.1.4). Hypoalbuminemia decreases hepatic delivery and increases renal excretion. The affinity of Tc-99m-HIDA to bind with albumin is much lower than that of bromsulfalein and bilirubin [28]. A substitution at the para position improves both albumin binding and lipid solubility. Methyl substitution at positions 2, 4 and 6 and a bromine at position 5 (mebrofenin) increase both hepatic delivery and hepatocyte uptake, and markedly reduce renal excretion. A butyl substitution at the para position has a similar effect on the uptake [25]. Albumin-bound Tc-99m-HIDA leaves the sinusoidal space through the fenestrae of the endothelial cells and enters the perisinusoidal space of Disse, a space unique for liver capillaries. Disassociation between albumin-Tc-99m-HIDA takes place in the space of Disse very close to the basolateral border of the hepatocyte (Fig. 3.1.2). Only Tc-99m-HIDA enters the hepatocyte, leaving albumin behind in the blood. This mechanism is common for most organic anions [28].\n\nFig. 3.1.4\n\nBlood clearance of six Tc-99m HIDA agents. Note the fastest blood clearance with mebrofenin, TMB [26]\n\n### 3.1.11 Hepatocyte Uptake\n\nThe mechanism of uptake of Tc-99m-HIDA by the hepatocyte is similar to those of other organic anions [30]. Liver concentrates three types of organic anions, including bile acids, free fatty acids and non-bile acid cholephils (bromsulfalein, bilirubin, rose bengal, indocyanin green and Tc-99m-HIDA). Non-bile acid cholephils are taken up by the hepatocyte by a mechanism called receptor-mediated endocytosis (RME). RME has been well documented for low density lipoprotein, IgA, insulin, transferrin, asiologlycoprotein and cholesterol [31]. It is believed that Tc-99m-HIDA compounds follow the RME pathway, followed by bilirubin and other organic anions. By subjecting cultured rat hepatocytes to different in vitro experimental conditions, Okuda et al. and Lan et al. have demonstrated three possible pathways for uptake of Tc-99m-HIDA: (1) organic anion pathway probably through RME, (2) bile acid (bile salt) pathway and (3) free fatty acid pathway [28, 32, 33].\n\n### 3.1.12 Receptor-Mediated Endocytosis\n\nAfter dissociation from albumin in the space of Disse, Tc-99m-HIDA attaches to its specific receptors through receptor protein (ligandin). These receptors are located along the basolateral border and along the walls of the coated pits, which are mere invaginations of the basolateral membrane into hepatocyte cytosol [31, 34]. The ligand (Tc-99-HIDA) and the receptor protein (ligandin) cluster in the coated pit (Fig. 3.1.5). This collection is one of the primary requirements for RME to progress [34]. After separation from the surface membrane the coated pit forms a coated vesicle, and Tc-99m-HIDA gets internalized within the hepatocyte. After losing the protein covering, the vesicle becomes an endosome. In the case of LDL, IgA, insulin and transferrin, the protein coating the vesicle is clathrin. Co-transport through bile acids and free fatty acids occurs simultaneously with RME.\n\nFig. 3.1.5\n\nSchematic representation of receptor-mediated endocytosis for uptake and excretion of Tc-99m HIDA by the hepatocyte. Primary uptake occurs via receptor-mediated endocytosis (RME). After detaching from albumin in the space of Disse, the radiotracer attaches to the ligand in receptors within the coated pits [1], which are invaginations of the basolateral border of the hepatocyte. A coated vesicle [2] is formed when it separates from the surface membrane. The coated vesicle rapidly loses its clathrin coat, forming an endosome [3]. Two endosomes combine together to form a fused endosome [4]. Hydrogen is pumped into the fused intra-vesicular space, initiating uncoupling of the receptor and ligand (CURL). Ligand enters the bile canaliculi, and the receptor moves to the surface for recycling. Tc-99m HIDA in addition uses free fatty acid (FFA) and bile acid (BA) pathways for uptake and excretion in free form into bile canaliculi (modified from Steer [31])\n\nDespite sharing a common mechanism of uptake, there are a few differences in hepatocyte uptake between Tc-99m-mebrofenin and Tc-99m disofenin. In cultured rat hepatocytes, bilirubin reduces the uptake of disofenin much more profoundly than that of mebrofenin. At 20 \u03bcM bilirubin in the culture medium, the hepatocyte uptake of Tc-99m disofenin reduces to 34% from a basal value of 100% without bilirubin in the culture medium. In contrast, the uptake Tc-99m mebrofenin remains at 70% of the basal value under identical experimental conditions. Both disofenin and mebrofenin show a reduction in hepatocyte uptake when bile acids or free fatty acids are added to the culture medium, suggesting the existence of other uptake pathways. These results indicate that Tc-99m-HIDA agents share a common pathway with organic anions (bilirubin), free fatty acid and bile acids. Bromsulfalein (BSP) inhibits uptake by the hepatocyte of Tc-99m disofenin much more profoundly than Tc-99m mebrofenin (Table 3.1.8).\n\nTable 3.1.8\n\nEffect of 20\u03bcM of various organic anions on the uptake (%) of Tc-99m disofenin and Tc-99 mebrofenin by cultured rat hepatocytes [32]\n\nAgent | Basal uptake | BSP | Bili | Tauro cholate | Glyco cholate | Cholate | Deoxy cholate | Chenodeoxy cholate\n\n---|---|---|---|---|---|---|---|---\n\nTc-99m-Disofenin | 100 | 59 | 34 | 61 | 59 | 66 | 42 | 42\n\nTc-99m-Mebrofenin | 100 | 107 | 70 | 69 | 85 | 80 | 62 | 71\n\n### 3.1.13 Transit Through the Hepatocyte and Secretion into Bile Canaliculi\n\nUnlike bilirubin and other organic anions, Tc-99m-HIDA agents are secreted into bile canaliculi in their native state, without undergoing any conjugation during their transit through the hepatocyte [6]. The mechanisms by which Tc-99m-HIDA is transported through the hepatocyte and then secreted into the bile canalicular lumen are not clear. The general belief is that the mechanism is similar to those of non-cholephil organic anions, free fatty acids and bile acids. Vesicular transport and receptor-ligandin transport are thought to be involved. After losing the clathrin coat, the vesicle forms an endosome. Two endosomes together form a fused endosome. At this point the ligand (Tc-99m-HIDA) and the receptor (ligandin) separate and start moving in two different directions. The ligand (Tc-99m-HIDA) enters the bile canaliculi, and the ligandin moves to the hepatocyte surface for recycling [35]. Rough endoplasmic reticulum and the Golgi complex are known to synthesize the receptor. After entering the canaliculi, Tc-99m-HIDA mixes thoroughly with the hepatic bile, and from then on it serves as an ideal in vivo tracer for delineation of the entire hepatobiliary tree (Fig. 3.1.2). The curves generated over the liver provide a measure of the rapidity of uptake and excretion of Tc-99m HIDA (Fig. 3.1.6). The dose not taken up by the liver is excreted through urine (Fig. 3.1.7).\n\nFig. 3.1.6\n\nHepatic uptake and excretion of six Tc-99m HIDA agents. Fastest uptake and excretion are noted with disofenin (DISIDA) and mebrofenin (TMB)\n\nFig. 3.1.7\n\nUrinary excretion of six Tc-99m HIDA agents. Less than 2% of the injected dose of mebrofenin (TMB) is excreted in 24-h urine. Other agents show an increasing amount of urinary excretion [26]\n\nCurrently, Tc-99m-disofenin and Tc-99m-mebrofenin are the most popular agents (Fig. 3.1.8). Tc-99m mebrofenin, which has bromine at position 5 (meta), clears from the liver much more rapidly than Tc-99m disofenin (Fig. 3.1.9).\n\nFig. 3.1.8\n\nMolecular structure of Tc-99m-disofenin and Tc-99m-mebrofenin. An atom of bromine at 5 and three methyl groups at 2, 4, 6 positions in mebrofenin change its biological behavior from that of disofenin with an isopropyl group at 2 and 6 positions\n\nFig. 3.1.9\n\nHepatic uptake and excretion of Tc-99m disofenin and Tc-99m mebrofenin. The uptake and excretion by the liver of Tc-99m mebrofenin are significantly faster when compared to Tc-99m disofenin\n\n### 3.1.14 Flow Through Intrahepatic and Extrahepatic Ducts\n\nThe hepatic bile is radiolabeled instantly, as soon as Tc-99m-HIDA secreted by the hepatocyte enters the bile canaliculi. In vivo bile radiolabeling under total basal conditions allows delineation of the entire intrahepatic and extrahepatic bile ducts. Of the 600 ml total bile produced per day, 450 ml is secreted by the hepatocytes and 150 ml by canalicular cells [36]. The bile within the ducts is radiolabeled as the hepatic bile passes through the ducts.\n\n### 3.1.15 Gallbladder Storage\n\nAbout 70% of the hepatic bile secreted during fasting hours enters the gallbladder (0.3 ml min-1), and the rest enters the duodenum directly [37]. A fully filled normal gallbladder can accommodate up to 50 ml of bile. It would, therefore, take approximately 180 min (6 h) for a completely emptied gallbladder to refill to its full capacity. Having the ability to accommodate a constant inflow of 0.3 ml bile min-1, the hepatic bile during fasting is made possible by absorption of an equal volume of water through the gallbladder wall (about 0.3 ml min-1). The gallbladder wall absorbs water, chloride and bicarbonates at a much faster rate than sodium, bile salts and cholesterol from hepatic bile. By this selective absorption, the gallbladder can sequester all of the total body bile salts within it during 10-12 h of fasting. This selective process of solute concentration is called the concentration function of the gallbladder.\n\nHighly concentrated gallbladder bile is discharged into the duodenum upon the arrival of food into the small intestine, where bile salts facilitate digestion and absorption of nutrients into the blood stream. There is a rapid rise in Tc-99m-HIDA counts when radiolabeled hepatic bile enters the gallbladder. Accumulation of a very high specific activity bile in a relatively small volume (50 ml) accounts for the gallbladder being the critical organ in a Tc-99m-HIDA study [38], receiving about 908 mrad mCi-1. The upper large intestine, lower large intestine and small intestine receive decreasing doses (Table 3.1.9). Radiation to the gastrointestinal tract decreases in liver failure when kidneys become the preferential route of excretion for Tc-99m-HIDA.\n\nTable 3.1.9\n\nRadiation-absorbed dose (mrad mCi-1) to various organs from Tc-99m-HIDA in normal subjects and patients with increasing severity of liver disease [38]\n\nBilirubin level | Normal | Patients with increasing serum bilirubin level\n\n---|---|---  \n|\n\n<1 mg dl-1 | < 1 mg dl-1 | 1-5 mg dl-1 | 5-10 mg dl-1 | >10 mg dl-1\n\nOrgan\n\n|\n\nGallbladder | 908 | 728 | 617 | 309 | 101\n\nUpper colon | 302 | 235 | 198 | 100 | 36\n\nLower colon | 199 | 154 | 131 | 68 | 27\n\nSmall intestine | 189 | 147 | 125 | 65 | 25\n\nLiver | 76 | 91 | 90 | 47 | 18\n\nOvaries | 62 | 50 | 43 | 25 | 13\n\nKidneys | 43 | 58 | 67 | 105 | 132\n\nU. Bladder | 35 | 46 | 53 | 87 | 111\n\nBone marrow | 24 | 21 | 19 | 13 | 9\n\nSpleen | 9 | 8 | 7 | 5 | 3\n\nTestes | 4 | 4 | 4 | 4 | 5\n\nTotal body | 16 | 15 | 14 | 9 | 6\n\n### 3.1.16 Final Discharge into the Duodenum\n\nWhen the food leaves the stomach and enters the duodenum, it stimulates CCK-secreting cells in the mucosa of the duodenum and jejunum to release endogenous cholecystokinin into the circulation. It usually takes about 6-26 min (mean 16 min) after a meal for serum CCK levels to rise above the threshold to induce contraction and emptying of the gallbladder [39]. Once the gallbladder contraction is initiated, bile emptying is maintained for 1-2 h post-meal. In addition to initiating gallbladder contraction and emptying, cholecystokinin stimulates water secretion by cholangiocytes lining the bile ducts and hastens bile flow by directly stimulating smooth muscle of the bile ducts [40]. Cholecystokinin also increases intestinal peristalsis and facilitates movement of bile emptied from the gallbladder antegrade towards the jejunum and ileum. By inducing contraction of the pylorus of the stomach, it prevents duodeno-gastric bile reflux.\n\n### 3.1.17 Other Tc-99m-Labeled Hepatobiliary Agents\n\nMany Tc-99m-labeled potential hepatobiliary agents have not been approved by the United States Food and Drug Administration for routine clinical use. Some are approved in other countries. These agents include Tc-99m-labeled penicillamine [41, 42], dihydrothiooctic acid [43], tetracycline [44] and pyridoxylidineglutamate [45]. Tc-99m-Pyridoxylidineglutamate has been used extensively in Australia, Japan and other Asian countries [46]. Some of the agents used for imaging of other organs pass through the liver and are secreted into bile, and often they provide information about the hepatobiliary function (Fig. 3.1.10). The myocardial perfusion imaging agent, Tc-99m-sestamibi, is taken up by hepatocytes and secreted into bile, and it provides an opportunity to measure the gallbladder ejection fraction with cholecystokinin in those rare patients where acalculous chronic cholecystitis (cystic duct syndrome) mimics an anginal type of pain.\n\nFig. 3.1.10\n\nSecretion of technetium-99m sestamibi into bile. Myocardial (H) perfusion imaging agent shows accumulation in the liver (L), bile ducts and gallbladder (GB). CBD common bile duct, INT intestine\n\n### 3.1.18 Technetium-99m-DTPA Galactosyl-Human Serum Albumin\n\nThe basolateral and lateral (not canalicular) border of the plasma membrane of the hepatocyte is rich in asialoglycoprotein (ASGP) receptors, which serve as a binding site for Tc-99m DTPA-galactosyl human serum albumin (Tc-99m GSA). These receptors specific for the hepatocytes are not found in any other cells in the body [31]. After an intravenous injection, Tc-99m GSA circulates in the blood, is extracted by the hepatocyte plasma membrane and is transferred to the lysosomes through receptor-mediated endocytosis. Blood disappearance parallels the clearance of indocyanin green, considered a gold standard agent for hepatocyte function [47]. Studies in Japan have shown a great potential for the measurement of the hepatic reserve prior to resection of the liver in patients with hepatocellular carcinoma, cholangiocarcinoma, cirrhosis and metastatic disease. Rapid sequence SPECT imaging allows quantitative measurement of functional reserve and enables prediction of prognosis [48-50].\n\n### 3.1.19 New SI Units for Measurement of Radiation\n\nThe International Commission on Radiation Units and Measurements recommended in 1974 that new SI (system international) units replace old CGS units for all scientific work after 1984. The old and new units of measurement and their relationship are shown in Tables 3.1.10-3.1.12 [51].\n\nTable 3.1.10\n\nOld CGS and new SI units for measurement of radioactivity [51]\n\nOld units curie | New SI units. Becquerel | Disintegrations per second (dps)\n\n---|---|---\n\nMegacurie (MCi) | 37 PBq | 3.7 \u00d7 1016\n\nKilocurie (KCi) | 37 TBq | 3.7 \u00d7 1013\n\nCurie (Ci) | 37 GBq | 3.7 \u00d7 1010\n\nMillicurie (mCi) | 37 MBq | 3.7 \u00d7 107\n\nMicrocurie (\u03bcCi) | 37 kBq | 3.7 \u00d7 104\n\nNanocurie (nCi) | 37 Bq | 3.7 \u00d7 10\n\nPicocurie (pCi) | 37 mBq | 3.7 \u00d7 10-2\n\nTable 3.1.11\n\nRelationship between new Becquerel and old Curie units [51]\n\n1 Becquerel (Bq) = 1 dis sec-1 = 27.03 \u00d7 10-12 Ci = 27.03 pCi\n\n---\n\n1 Kilobecquerel (kbq) = 103 Bq = 27.03 nCi\n\n1 Megabecquerel (MBq) = 106 Bq = 27.03 \u03bcCi\n\n1 Gigabecquerel (GBq) = 109 Bq = 27.03 mCi\n\nTable 3.1.12\n\nRelationship between old and new SI units for measurement of radiation [51]\n\nParameter | Old unit | New SI unit | Conversion factor\n\n---|---|---|---\n\nRadioactivity | Curie (Ci) = 3.7 \u00d7 1010 dps | Becquerel (Bq) = 1 dps | 1 Ci = 3.7 \u00d7 1010 Bq\n\n1 Bq = 2.7 \u00d7 10-11 Ci\n\nRadiation exposure | Roentgen (R) = 2.58 \u00d7 10-4 C kg-1 | Coulomb kg-1 C kg-1 | 1 R = 2.58 \u00d7 10-4 C kg-1\n\n1C kg-1 = 3.88 \u00d7 103 R\n\nRadiation absorbed dose | rad = 100 erg g-1 | Gray(Gy) = 1 J kg-1 | 1 rad = 0.01 Gy\n\n1 Gy = 100 rad\n\nRadiation dose equivalent | rem = QF \u00d7 rad | Sievert (Sv) = QF \u00d7 Gy | 1 rem = 0.01 Sv\n\n1 Sv = 100 rem\n\n## 3.2 Radiolabeling of Red Blood Cells and Leucocytes\n\nRadiolabeled red blood cells are used in nuclear medicine mainly for the assessment of vascular spaces, measurement of left ventricular volume and ejection fraction, and for localization of bleeding sites. Radiolabeled leucocytes are used primarily for the detection and localization of abscesses and delineation of sites of diffuse or focal infection. Major applications in nuclear hepatology include differentiation of hemangiomas from other focal liver lesions with Tc-99m-labeled red blood cells and differentiation of abscess from other cystic liver lesions with indium-111 oxine or Tc-99m-HMPAO labeled leucocytes.\n\n### 3.2.1 Red Blood Cell Labeling with Tc-99m\n\nTechnetium-99m red blood cells (RBC) are used for blood pool imaging in the separation of vascular malformation (hemangiomas) from other non-vascular liver lesions. Radiolabeling of RBCs with chromium-51 was first accomplished by Gray and Sterling in 1950 for the main purpose of measuring their survival in the diagnosis of hemolytic anemias [1]. Technetium-99m-labeled RBCs are used primarily for blood pool imaging, and survival studies cannot be carried out due to the short physical half life of Tc-99m of only 6 h [2]. Three methods of RBC labeling are available: the in vivo method, in vitro method and combined in vivo and in vitro method. The in vivo method is technically simple, but results in a slightly higher background radiation. The in vitro method, on the other hand, is technically more involved, but gives excellent lesion to non-lesion contrast because of low background radiation.\n\n### 3.2.2 In Vitro Method (Ultratag Kit made by Mallincrodt, St. Louis, MO)\n\nAbout 1 ml of patient blood is drawn into a heparinized syringe and added to the mixing vial, which contains 50 \u03bcg stannous chloride, 3.7 mg sodium citrate, 5.5 mg dextrose and 0.11 mg sodium chloride. Stannous (2+) chloride readily crosses the red cell membrane and attaches to the heme component of the hemoglobin molecule, ready to do its job of reducing valence of Tc-99m-pertechnetate from 7+ to 4+. About 0.6 ml of 0.1% sodium hypochlorite is added later as an oxidizing agent that converts all excess stannous chloride (2+) into stannic (4+) state in plasma, but not inside the red blood cell because of its inability to cross the intact red cell membrane. About 20-30 mCi of 99m-TcO4 (pertechnetate) is added to the reaction vial. Tc-99m-pertechnetate readily crosses the red cell membrane and attaches to the globin molecule after it gets reduced from 7+ to 4\\+ valence by the waiting stannous (2+) chloride on the heme fraction of the hemoglobin molecule. Once Tc-99m is reduced from the 7+ to 4\\+ valence state, it does not come out of the hemoglobin molecule. The entry of Tc-99m through the red cell membrane thus becomes one-way traffic. After 15-20 min of gentle incubation at room temperature, most of the Tc-99m is trapped inside the red blood cell [2]. The in vitro method is a very efficient labeling technique and does not necessitate blood centrifugation to remove plasma, because 97% of technetium-99m is associated with the red blood cell and less than 3% remains in plasma. Hemoglobin contains 95% of the total red cell radioactivity, of which 77% is associated with globin and the rest with the heme fraction of the hemoglobin molecule [3].\n\n### 3.2.3 In Vivo Method\n\nStannous chloride (10-20 \u03bcg kg-1) from a commercial pyrophosphate kit is injected intravenously. Tin (2+) enters the red cell membrane and attaches to hemoglobin to be ready to act as a reducing agent when Tc-99m-pertechnetate enters the cell. About 30 min later, 20-30 mCi Tc-99m-pertechnetate in saline is injected intravenously. Tc-99m-pertechnetate crosses the red cell membrane, gets reduced by stannous chloride and later binds firmly to hemoglobin.\n\n### 3.2.4 Combined In Vivo and In Vitro Method\n\nPre-tinning is accomplished first by intravenous injection of cold stannous pyrophosphate made for bone scanning. Fifteen to 20 min later, about 3-4 ml patient blood is drawn into a 50-ml-volume syringe, and 20-30 mCi Tc-99m-pertechnetate is added to the syringe, shaken gently for 10 min and re-injected through the same catheter. Because of the higher efficiency of labeling and lower background, the in vitro method may be preferred over the in vivo method in the detection of hepatic hemangiomas (blood pool) and gastrointestinal bleeding [4, 5].\n\n### 3.2.5 Leukocyte Labeling\n\nLeukocytes labeled with either In-111 oxine or Tc-99m-HMPAO are used in nuclear hepatology for the diagnosis of liver abscess, empyema or acute cholecystitis [6, 7].\n\n### 3.2.6 Indium-111 Oxine\n\nAbout 50-60 ml of patient blood is drawn into a heparinized syringe, and 10 ml of 0.9% sodium chloride containing 6% hetastarch is added to blood and mixed thoroughly [8]. The mixture is allowed to form sediment for 40 min. After the sedimentation, leukocyte-rich plasma is collected, diluted with saline in a ratio of 2:1 and centrifuged at 150 g for 10 min. The leukocyte-rich pellet is washed twice with 10 ml saline and centrifuged each time for 10 min at 150 g after washing. Washed cells are then resuspended in 4.5 ml saline. About 0.3-0.5 ml of indium-111 oxine containing 0.4-0.5 mCi (14.8-18.5 MBq) In-111 is added to the reaction vial and incubated gently for 15 min at room temperature. Indium-111 oxine penetrates the leukocyte membrane, and once inside the cell, indium-111 and oxine dissociate. Oxine diffuses out of the cell into the labeling medium, leaving indium-111 behind trapped inside the cell [6]. The suspension is then centrifuged for 8 min at 90 g, and the supernatant is discarded and the pellet containing labeled leukocytes is resuspended in platelet-poor plasma. About 6 ml of the suspension containing 0.3-0.4 mCi of In-111 leukocytes is injected intravenously. Indium-111-labeled leucocytes clear from blood with a T 1\/2 of 6 h, which is very close to the value of 7 h for cells labeled with the gold standard P-32, di-isopropylfluorophosphate. After labeling, each leukocyte receives approximately 1,480 rad (14.8 Gy) radiation from Auger electrons of 0.6-25.4 keV energy from In-111. This amount of radiation from its own source does not affect its function [8].\n\nThe major advantage of In-111 oxine-labeled leukocytes over Ga-67 citrate or Tc-99m-HMPAO WBCs for imaging abscesses is that the entire abdomen, with the exception of the liver and spleen, remains free of any secreted radioactivity into the bowel [9]. Abdominal infection is diagnosed by taking early images at 2-10 min and repeat images within 30-60 min after injection. Delayed images at 24 h may be needed in some patients. In the case of chest infection, the earliest images that will provide any useful information are taken at 3-4 h after injection. Due to usual margination of leucocytes in the lungs, early chest images are usually not diagnostic [10].\n\n### 3.2.7 Tc-99m-HMPAO\n\nThe blood separation step for leucocyte labeling with Tc-99m HMPAO remains identical to that of labeling with In-111 oxine described above. After separation, leukocytes are resuspended in 20% plasma\/ACD solution. About 20-25 mCi of Tc-99m-HMPAO is added to the reaction vial and incubated gently for 10 min at room temperature, washed with plasma and resuspended in plasma and reinjected. The labeling yield is between 50 and 60% with 80% activity bound to granulocytes [11]. The labeled leukocytes clear from blood with a T \u00bd of 4 h, which is slightly shorter than for In-111-labeled granulocytes. The main advantage of Tc-99m HMPAO is that a much larger dose can be given, yet the total body and organ radiation dose is much less. The agent is easily available at all times of need at a very reasonable cost. The main disadvantage is that free Tc-99m-HMPAO, not bound to leucocytes, is taken up by the hepatocytes and secreted into bile. Intestinal luminal activity interferes with the diagnosis in images taken beyond 30 min after injection. A negative scan rules out abdominal infection. Increasing numbers of studies in recent years have been being done with Tc-99m-HMPHO [11]. Interference from secreted bile radioactivity is avoided by taking abdominal images early, within 10-20 min after injection. The absorbed radiation dose from labeled red blood cells and leucocytes is shown in Table 3.2.1.\n\nTable 3.2.1\n\nRadiation absorbed dose from Tc-99m-labeled RBC and WBC and indium-111-labeled WBC\n\nOrgan | Tc-99-m-RBC | Tc-99m-WBC (HMPAO) | In-111 WBC\n\n---|---|---|---\n\nrad mCi-1 (Gy\/37 MBq) | rad mCi-1 (Gy\/37 MBq) | rads mCi-1 (Gy\/37 MBq)\n\nLiver | 0.07 | 0.15 | 5.0\n\nSpleen | 0.05 | 0.22 | 40.0\n\nLungs | 0.06 | - | -\n\nKidneys | 0.05 | - | -\n\nOvaries | - | 0.03 | 0.4\n\nTestes | - | 0.19 | 0.02\n\nBone marrow | 0.03 | 0.16 | 4.0\n\nWhole body | 0.02 | 0.03 | -\n\n## 3.3 Gallium-67 Citrate\n\nThe tumor detection capability of gallium-67 citrate was described first by Edwards and Hays in 1969 [1]. Very soon it was recognized that in addition to the detection of varieties of tumors (lymphomas, hepatomas, malignant melanoma and squamous cell carcinomas), Ga-67 was also taken up in high concentration by infectious lesions and abscesses. Benign conditions, such as pseudonodules of cirrhosis and other non-specific inflammations, do not concentrate the agent [2]. This differential uptake by benign versus malignant lesions and the ability to detect infection led to widespread clinical application of Ga-67 citrate imaging in the 1970s and 1980s, before CT and ultrasound became popular imaging techniques for detection of focal hepatic lesions. Gallium-67 imaging is reemerging primarily for determining therapeutic response to various chemotherapeutic agents in various malignancies.\n\n### 3.3.1 Pharmacokinetics\n\nGallium-67 is produced in an accelerator by bombarding zinc target with protons as represented by the equation [67Zn (p, n) 67Ga]. It has a physical half life of 78.3 h and decays by electron capture to stable zinc. It has three principle gamma photons with the mean energy of 93.7 keV (36%), 185 keV (20%) and 300 keV (16.0%). Upon intravenous injection, Ga-67 citrate is transported in blood bound to plasma proteins, mainly transferrin, and to a minor extent to lactoferrin and ferritin. Two different mechanisms are involved with reference to cellular uptake: normal soft tissues concentrate Ga-67 bound to transferrin, whereas tumors concentrate the fraction that is either free or bound loosely to other proteins [3]. In the hepatocytes, Ga-67 uptake is seen mostly in association with lysosomes and mitochondria and to a lesser extent in the cytosol. The nucleus concentrates less than 10% of the total hepatocyte uptake. The exchange of Ga-67 takes place between the lysosomes and the cytosol. After intravenous injection, blood pool activity remains high for up to 24 h [4]. Beyond 48 h, most of Ga-67 is cell bound. About 10% of injected activity is excreted in stool during the first week.\n\nIncreased capillary permeability facilitates Ga-67 entry into the site of acute inflammation, abscess or tumor. In the case of an abscess, Ga-67 is found in association with the leukocytes, siderophores and bacteria. Staphyllococcus aurius, E. coli and other microorganisms are shown to ingest Ga-67, accounting for high uptake by the abscess [5, 6]\n\nBecause of normal uptake by bone, and slow excretion by mucosa into the colon, the bone and lower large bowel receive the largest radiation dose from Ga-67: 18 mGy\/37 MBq to bone marrow and 8.4 mGy\/37 MBQ to the lower large bowel (Table 3.3.1). Currently, Ga-67 citrate imaging has re-emerged in the evaluation of chemotherapy for lymphomas.\n\nTable 3.3.1\n\nRadiation absorbed dose from gallium-67 citrate\n\nOrgan | mGy\/37 MBQ | rads mCi-1\n\n---|---|---\n\nMarrow | 18 | 1.8\n\nLower colon | 8.4 | 0.84\n\nSpleen | 7.0 | 0.70\n\nLiver | 6.3 | 0.63\n\nUpper colon | 5.5 | 0.55\n\nKidneys | 5.4 | 0.54\n\nOvaries | 3.0 | 0.30\n\nTestes | 2.5 | 0.25\n\nWhole body | 2.6 | 0.26\n\nModified from MIRD pamphlet no. 11, 1975\n\n## 3.4 Somatostatin Receptor Imaging Agent\n\nSomatostatin is a small peptide found in many normal tissues and in benign and malignant tumors. It was originally thought to regulate primarily the release of growth hormone [1]. Now it is evident that in addition to influencing growth hormone release, somatostatin regulates the release of cyclic neuropeptides and inhibits release of thyrotropin, insulin, glucagon, gastrin, secretin and cholecystokinin [2]. The somatostatin receptors are present in the brain, thyroid, lung, gastrointestinal tract, liver, gallbladder, pancreas, adrenals and activated leukocytes [3]. Two molecular forms of somatostatin have been identified: one consists of 14 (Fig. 3.4.1) and the other 28 amino acids in the molecule. The longer one with 28 amino acids is a dimer formed by the union of two shorter molecules. The serum half life of somatostatin is less than 3 min. Five subtypes of somatostatin receptors (SSTR 1-5) are recognized in the body. Both somatostatin 14 and 28 attach to all five subtypes with equal affinity. Octreotide is an analog of somatostatin with eight amino acids (octapeptide). It binds mainly to somatostatin receptor subtypes 2, 3 and 5, but not to subtypes 1 and 4 [3, 4].\n\nFig. 3.4.1\n\nMolecular structure of somatostatin and its radiolabeled analogues. Somatostatin consists of 14 amino acids. The receptor-binding sites are located on four sequentially arranged amino acids. In labeled octreotide, radioiodine, I-123, attaches directly to tyrosine (Tyr), whereas indium-111 requires DTPA as chelate for its attachment\n\n### 3.4.1 Indium-111 Pentetreotide (OctreoScan)\n\nRadiolabeled octreotide binds to somastostatin receptors on the surface of tumors, enabling their detection [5]. Radiolabeling is achieved with either iodine I-123 or indium-111. When I-123 is chosen, the amino acid tyrosine, a normal component of the molecule, is radiolabeled [6]. In the case of indium-111, a bifunctional chelate, diethylenetriaminepentaacetic acid (DTPA), is required for radiolabeling (Fig. 3.4.1). As a bifunctional chelate, DTPA attaches to indium-111 (radiotracer) with one arm and to the amino acid, d-phenylalanine of somatostatin analogue (ligand), with the other arm. Four amino acids (phenylalanine, tryptophan, lysine and threonine) arranged in a sequence provide the binding site for the radiolabeled analogue [7]. It binds with high affinity to SSTR2 and with lower affinity to SSTR3\/5.\n\nFollowing intravenous administration, In-111 pentetreotide clears from blood rapidly. About 33% of the administered dose remains in circulation at the end of 10 min. Kidneys are the major route of excretion. About 50% of the injected dose is excreted in urine in 6 h, 85% in 24 h and 90% in 48 h. Less than 2% of the dose is excreted in feces in 3 days. Stool radioactivity mostly represents biliary excretion. Kidneys are the target organs for radiation. About 7% of the dose accumulates in the kidneys by 4 h [8]. The recommended dose for scanning is 3 mCi for planar and 6 mCi for SPECT study. Clinical studies usually require a SPECT study necessitating a 6-mCi (222 MBq) dose [9]. Radiation to various organs from a diagnostic dose depends both upon the age and dose. The kidneys, spleen and urinary bladder receive the most radiation [10].\n\n### 3.4.2 Technetium-99m Depreotide\n\nTc-99m depreotide is a somatostatin analogue with a molecular formula of C65H95N16O12S2 and structure of R-Tyr-(D-Trp)-Lys-Val-R-(\u03b2-Dap)Lys-amide [11]. This agent is not superior to In-111 octreoscan, but due to its higher affinity for SSTR3 may be useful in those patients who show less or no uptake of In-111 octreoscan. In one study, 8 of 25 (32%) patients with negative In-111 octreoscan showed positive uptake with Tc-99m depreotide [12].\n\n## 3.5 Fluorine18, 2-Flouro-2-deoxy-d-glucose (18F-FDG)\n\nGlucose is the primary source of energy for most living cells in the body, especially for the neurons of the central nervous system. Carbohydrates in the diet are the main source of glucose. Each glucose molecule has 6 carbon (hexose), 6 oxygen and 12 hydrogen atoms and can be shown in the form of either a stick or ring diagram (Fig. 3.5.1). After intravenous administration, glucose diffuses readily out of the intravascular space through the capillary membrane and enters the interstitial space from where it enters cells either through active or passive diffusion. Its uptake is controlled by many transporter proteins on the plasma membrane and uses both sodium-dependent glucose transporter (SGLT) and facilitative sodium-independent glucose transporter pathways [1]. Once inside the cell, glucose is subjected to a ten-step metabolic process (glycolysis) that yields the end product (Fig. 3.5.2). Glucose is converted first into glucose-6-phosphate (G-6-P) by adding a phosphate group to the sixth carbon atom mediated by the enzyme glucokinase (hexokinase). Isomarase enzyme converts it into fructose-6-phosphate, which in turn is converted into fructose 1-6-diphosphate by the addition of another phosphate group at the carbon 1 position. Through the interaction with other enzymes, pyruvate is released, which may enter into the citric acid cycle or be converted into lactic acid. The final products are water and carbon dioxide. Most of these reactions take place rapidly within the cell. By reversing the metabolic process, glucose is re-synthesized (gluconeogenesis). In the reverse process, G-6-P is converted into glucose by the enzyme glucose-6-phosphatase (G-6-Ptase).\n\nFig. 3.5.1\n\nStructure. Molecular structure of glucose is shown in stick and ring forms\n\nFig. 3.5.2\n\nGlycolysis. Glucose breaks down into carbon dioxide and water through various intermediary steps (modified from [3])\n\n### 3.5.1 Structure\n\nRemoval of one oxygen atom of the hydroxyl group at the second carbon position converts glucose into 2-deoxyglucose (2DG), a glucose analogue (Fig. 3.5.3), which goes through an initial metabolic process similar to that of glucose. By replacing hydrogen with the fluorine (F-18) atom at the second carbon position of 2DG, a new compound, 2-fluoro-2-deoxyglucose (F-18 FDG), is created that behaves quite differently from glucose while going through the rest of the metabolic process [2]. The first metabolic step, however, remains intact, with 2-fluoro-2-deoxyglucose behaving like glucose with the conversion into 2DG-6-P (Fig. 3.5.4). Labeling with fluorine-18 enables the new compound (F18-FDG) for use in positron emission tomography (PET).\n\nFig. 3.5.3\n\nGlucose analogues. Removal of oxygen atom from C-2 position of glucose creates 2-deoxyglucose (2DG), which enables labeling with fluorine-18 relatively simple at the C-2 position, resulting in 2-fluoro-2-deoxyglucose (F-18 FDG)\n\nFig. 3.5.4\n\nMechanism of glucose and F-18 FDG transfer across the cell. Both leave the intravascular space readily and enter the cell. Through the enzyme hexokinase, both get converted into their respective 6-phosphate form. Unlike glucose, F-18 FDG does not undergo any further metabolic process enabling imaging. Glucose is converted into either glycogen or CO2 \\+ H2O, depending upon the body's needs\n\n### 3.5.2 Biodistribution\n\nImmediately after intravenous injection, F-18 FDG distributes throughout the body and readily crosses the blood-brain barrier. It diffuses out of the intravascular space into the interstitial space rapidly and enters the cell. Initial entry is directly proportional to organ blood flow. Many glucose transporter proteins on the plasma membrane control its entrance. Sodium-dependent glucose transporter (SGLT) proteins use the Na + K + ATPase pump. The sodium-independent pathway uses facilitative transporter (GLUT) proteins. Of the 13 known GLUT proteins, liver cells have four: GLUT2, GLUT7, GLUT9 and GLUT10 [3, 4]. Although initial uptake reflects organ blood flow, retention depends upon a subsequent metabolic pathway used by each organ. Organs with high hexokinase content show much higher F-18 FDG uptake than those with low hexokinase activity. The hexokinase content of organs is as follows: brain > heart > kidney > lung > liver [1]. The brain concentrates 6.9%, liver 4.4% and heart 3.3% of the injected dose (Fig. 3.5.5). Organs with high glucose-6-phosphatase enzyme activity show more rapid clearance than those with low enzyme activity. Brain and heart with high hexokinase content show increasing uptake during the first hour, whereas other organs with high G-6-Ptase show declining activity [5]. Increasing uptake by brain and heart is due to rapid conversion of F-18 FDG into F-18 DG-6-P mediated by the enzyme glucokinase (hexokinase). Compared to brain and heart, liver has much less hexokinase activity and more G-6-Ptase activity; both acting together contribute to rapid clearance of F-18 FDG from the liver. In the liver cell, G-6-Ptase converts F-18 FDG-6-P back into F-18 FDG, which diffuses back into blood [1]. Neither F18-FDG nor F-18 FDG-6-P enters the bile. This is evident from the fact that bile ducts and gallbladder are not seen in whole body PET study.\n\nFig. 3.5.5\n\nUptake and retention of F-18 FDG by various organs. Brain and heart show continuous uptake for 1 h, whereas other organs begin to show rapid clearance\n\nIn humans, the brain shows the highest uptake of F-18 FDG, followed by the liver, heart, red bone marrow, etc., as shown in Table 3.5.1. Kidneys excrete 20% of the injected dose in urine in 1 h and 21% in 2 h [5]. The initial uptake continues to increase in the brain and heart during the first hour, whereas it begins to decrease in the liver, kidney, spleen, pancreas and lung (Fig. 3.5.5). Continued uptake by the brain and heart reflects their high hexokinase activity relative to other organs, which begin to show a decrease. Once F-18 FDG is converted into 2DG-6-P, it is trapped inside the cell without much clearance by the heart and brain. The trapping rate is much less in the liver, kidney, spleen, pancreas and lung. Most of the radiotracer cleared from these organs is excreted in urine in its original form as F-18-FDG. Normally, glucose filtered by glomeruli is completely reabsorbed by the tubules. This shows that replacement of one hydroxyl group at the C-2 position in the glucose molecule by one atom of F-18 in FDG changes the biological behavior of the radiotracer to a great extent.\n\nTable 3.5.1\n\nOrgan uptake and radiation dose from F-18 FDG in human organs\n\nOrgan | Uptake [5] | Radiation dose [4]\n\n---|---|---\n\n(% Injected dose) | (mGy\/185 MBq) | (rad 5mCi-1)\n\nBrain | 6.9 | 4.81 | 0.48\n\nLiver | 4.4 | 2.22 | 0.22\n\nHeart | 3.3 | 12.03 | 1.20\n\nRed marrow | 1.7 | 2.04 | 0.20\n\nKidneys | 1.3 | 3.88 | 0.39\n\nLungs | 0.9 | 2.04 | 0.20\n\nSpleen | 0.4 | 2.22 | 0.22\n\nPancreas | 0.3 | 2.22 | 0.22\n\nTestes | 0.04 | 2.78 | 0.28\n\nOvary | 0.01 | 2.78 | 0.28\n\nBladder wall | 6.3 | 31.45 | 3.15\n\nRest of the body | 74.4 | -\n\n|\n\nTotal body | 99.95 | -\n\n|\n\nFrom Mejia [5]\n\n### 3.5.3 Radiation Dose\n\nOn a per organ basis, the urinary bladder wall receives the largest dose (0.091mGy\/MBq), followed by the heart, brain, kidneys, liver and other organs, as shown in Table 3.5.1. 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J Nucl Med 1995;36:542-549PubMed\n\n10.\n\nStabin MG, Kooji PPM, Bakker WH, Inoue T, Endo K, Coveney J, deJong R, Minegishi A. Radiation dosimetry for indium-111-pentetreotide. J Nucl Med 1997;38:1919-1922PubMed\n\n11.\n\nLebtahi R, LeCloirec J, Houzard C, Daou D, et al. Detection of neuroendocrine tumors: Tc-99m-P829 scintigraphy compared with In-111 pentetreotide scintigraphy. J Nucl Med 2002;43:889-895PubMed\n\n12.\n\nShah T, Kulakiene I, Quigley A, Warbey VS, Srirajaskanthan R, et al. The role of Tc-99m-depreotide in the management of neoroendocrine tumours. Nucl Med Commun 2008;29:436-440PubMedCrossRef\n\nReferences\n\n1.\n\nLong C. Studies involving enzymatic phosphorylation 1. The hexokinase activity of rat tissue. Biochem J 1952;50:407-415PubMed\n\n2.\n\nGallagher BM, Fowler JS, Gutterson NI, MacGregor RR, Wan C, Wolf AP. Metabolic trapping as a principle of radiopharmaceutical design: some factors responsible for the biodistribution of 18F] 2-deoxy-2-fluoro-D-glucose. J Nucl Med 1978;19:1154-1161[PubMed\n\n3.\n\nOehr P. Transport and metabolism of glucose and FDG. In: Oehr P, Biersack HJ, Coleman RE (eds) PET and PET-CT in onclogy. Springer, Berlin, 2004CrossRef\n\n4.\n\nOehr P. Radiopharmaceutical production and safety of 18F] FDG. In: Oehr P, Biersack HJ, Coleman RE (eds) PET and PET-CT in onclogy. Springer, Berlin, 2004[CrossRef\n\n5.\n\nMejia AA, Nakamura T, Masatoshi I, Hatazawa J, Masaki M, Watanuki S. Estimation of absorbed doses in humans due to intravenous administration of fluorine-18-fluorodeoxyglucose in PET studies. J Nucl Med 1991;32:699-706PubMed\n\n6.\n\nDowd MT, Chen C, Wendel WJ, Faulhaber PJ, Cooper MD. Radiation dose to the bladder wall from 218F] fluoro-2-deoxy-D-glucose in adult humans. J Nucl Med 1991;32:707-712[PubMed\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_4(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 4. Imaging of Liver and Spleen Morphology\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nIn most nuclear medicine departments throughout the world, gamma cameras have completely replaced rectilinear scanners. A large field of view gamma camera with a single or dual head, fitted with a low-energy, parallel hole, all-purpose collimator, is generally chosen for most planar studies using technetium-99m-labeled agents. The main advantage of the dual-head gamma camera is patient convenience, where the data collection time is reduced by one-half as it allows simultaneous anterior and posterior image acquisition. The reduction in imaging time often improves the spatial resolution because of less patient movement during data collection. A triple-head gamma camera, when available, is preferred for SPECT images. The gamma camera image acquisition parameters are identical for both the liver and spleen.\n\nIn most nuclear medicine departments throughout the world, gamma cameras have completely replaced rectilinear scanners. A large field of view gamma camera with a single or dual head, fitted with a low-energy, parallel hole, all-purpose collimator, is generally chosen for most planar studies using technetium-99m-labeled agents. The main advantage of the dual-head gamma camera is patient convenience, where the data collection time is reduced by one-half as it allows simultaneous anterior and posterior image acquisition. The reduction in imaging time often improves the spatial resolution because of less patient movement during data collection. A triple-head gamma camera, when available, is preferred for SPECT images. The gamma camera image acquisition parameters are identical for both the liver and spleen.\n\n## 4.1 Imaging with Radiocolloid\n\n### 4.1.1 Patient Preparation\n\nIn general, there is no special patient preparation required for morphology imaging with radiocolloids. This is a distinct advantage over functional imaging with Tc-99m HIDA where the patient preparation requirements are quite stringent. The morphology imaging studies obtained immediately after a full meal may cause image artifacts and obscure the spleen in the anterior view because of photon attenuation by the gastric contents. The posterior and left lateral view images usually clarify this situation.\n\nRadiocolloid imaging introduced in the early 1950s with the use of rectilinear scanners marked the birth of nuclear hepatology. Morphology imaging with radiocolloids remained very popular until the early 1980s, when it was overtaken by CT and ultrasound. Diagnosis of various diseases is made from assessing the changes in image pattern of the organ morphology. A thorough knowledge of normal morphology of the liver and spleen and normal variant patterns is critical to be able to detect disease. This section deals mainly with liver and spleen morphology as seen on a radiocolloid image (Table 4.1.1).\n\nTable 4.1.1\n\nData acquisition with Tc-99m-sulfur colloid\n\nInstrument | Single, dual, or a triple head, large field of view gamma camera\n\n---|---\n\nCollimator | Low-energy, parallel hole, all purpose collimator\n\nAgent | Technetium-99m-sulfur colloid or Tc-99m-albumin colloid\n\nDose | Adults, 2-4 mCi for planar images, 4-10 mCi for perfusion or SPECT images. Children, 30-50 \u03bcCi Kg -1 (minimum 300 \u03bcCi)\n\nSpectrometer setting | 140 keV photo peak at 15-20% window\n\nPerfusion images | At 1 or 2 s\/frame seconds for 60 s\n\nPlanar image data collection | 500,000-100,000 counts for each of anterior, posterior, and two lateral views. Additional views are obtained as needed. A 5- or 10-cm-long lead marker is placed along the right costal margin to facilitate measurement of liver size and position. The same marker is used for measurement of spleen size using the posterior image\n\nSPECT data collection\n\nMatrix | 64 \u00d7 64 word or 128 \u00d7 128 word mode. Rotation, clockwise 360\u00b0, angle 3-6\u00b0 for each stop. (360\/64 = 6\u00b0)\n\nFilters | Depends upon the computer and gamma\n\nViews | Transaxial, coronal, and sagittal\n\nPerfusion images are obtained immediately after the injection of the radiotracer at 1 or 2 s per frame for 60 s using the setup for the planer mode described above and reformatted at 4-8 s\/image for clinical interpretation. The planar and SPECT image data are collected 15-30 min after intravenous injection when all of the radiocolloid particles from blood are extracted by the RE cells\n\n## 4.2 Normal Liver\n\n### 4.2.1 Perfusion\n\nLiver and spleen perfusion images are obtained either in the anterior or posterior view using Tc-99m in any form. The aorta is the first abdominal organ to be seen, followed by the spleen, kidneys, and liver (Fig. 4.1.1). Aorta to spleen transit time is 2-4 s, and aorta to kidney transit time is 3-6 s. Despite the fact that both the hepatic artery and the splenic artery arise from a common celiac artery, there is an apparent delay in the appearance of liver perfusion due to dilution of hepatic artery radioactivity by the cold portal venous blood, as the portal vein receives its radioactivity much later than the splenic artery. The delay in arrival of the portal vein radioactivity is due to a delay in intestinal capillary phase. The liver receives 25% of its blood supply through the hepatic artery and 75% through the portal vein. The liver portal perfusion is clearly seen 10-14 s after the appearance of the abdominal aorta [1].\n\nFig. 4.1.1\n\nLiver perfusion. Anterior perfusion image shows abdominal aorta (frame no. 6), followed by spleen and kidneys (no. 7). Liver appears faintly in the beginning (no. 8), representing hepatic artery perfusion, and becomes clear in late (no. 9) images due to arrival of portal venous blood flow\n\n### 4.2.2 Shape, Surface, and Borders\n\nThere are wide variations in the shape of a normal liver, and most of the variations occur along the superior and inferior margins [2, 3]. The most common shape of a normal liver is that of an approximate triangle consisting of a superior, inferior, and a right lateral border (Fig. 4.1.2). Superior and right lateral borders form a smooth rounded contour at the right upper quadrant. The superior and inferior borders meet at an acute angle at the left tip of the left hepatic lobe. The inferior and right lateral borders meet at various angles. Often the inferior tip of the right hepatic lobe is elongated, forming a Riedel's lobe. The superior border of the liver is \"S\" shaped, and the contour is influenced by the size of the heart, exit of the hepatic vein, and the elevation of the dome of the right hemi-diaphragm. The shape, which looks like a gendarme's hat, is primarily due to the elevation of the right hemi-diaphragm.\n\nFig. 4.1.2\n\nNormal variation in liver shape and frequency (%). Triangular shape is the most common type [2-4]\n\nThe contour of the inferior border is affected by the fossa for the gallbladder and the entrance of the structures at the porta hepatis. The gallbladder fossa is usually located at the junction of the lateral 1\/3 with the medial 2\/3 of the inferior border (Fig. 4.1.3). The position of the gallbladder can be located during physical examination by drawing a straight line between the left anterior superior iliac spine and the umbilicus and extending it superiorly to meet the right costal margin. The fundus of the gallbladder corresponds to the point where this line meets the right costal margin. The gallbladder can be located anywhere along the inferior border and may even be intrahepatic. The intrahepatic gallbladder causes a filling defect in a radiocolloid scan and looks like a focal hot spot in a Tc-99m-HIDA study. Alteration in liver shape is usually due to the effect of extrinsic compression. The anterior and right lateral surfaces are smooth in outline, whereas the inferior surface has grooves and fissures for the entrance and exit of vessels and ducts, respectively.\n\nFig. 4.1.3\n\nPlanar image. The liver (L) is seen clearly, and the spleen (s) faintly in an anterior (ANT) view. The spleen may project anterior to the liver in the right lateral (RL) view. In the left lateral (LL) view, the spleen appears clearly with the left lobe of the liver (L) projecting faintly anterior to it\n\n### 4.2.3 Lobes\n\nThe liver is divided into the right and left lobe on the basis of either morphology or physiology. The line of attachment of the falsiform ligament marks the boundary between the morphologic right and left lobe. The division of the liver into physiologic right and left lobes is based on the embryologic development of the bile ducts, hepatic artery, and portal vein. The line of physiological division runs between the fossa for the gallbladder in the front and the deep fissure for the inferior vena cava in the back. The point of entrance of the porta hepatis also indicates the boundary between the physiologic right and left lobes. The line of physiological division (porta hepatis) is situated on the right side of the anatomic line of division (falsiform ligament). Therefore, the physiologic left lobe is much larger than the morphologic left lobe. In the anterior view, the line of attachment of the falciform ligament is shown by a slight decrease in counts because of photon attenuation.\n\nThe anterior view usually shows the entire liver and inferior part of the spleen. In the posterior view, the liver appears triangular in shape with a rounded supero-lateral corner. The posterior border in the right lateral view often shows a concave impression because of the right kidney. A wide vertical photopenic area between the right and left lobes is due to absorption of photons by the dense thoracic vertebrae (Fig. 4.1.3). Normally, the uptake of the radiocolloid by the reticulo-endothelial (RE) cells of the vertebral body is seen very faintly. The caudate lobe is situated posteriorly along the superior border, but is not usually seen in the planar posterior image, mainly because of absorption of photons by the vertebrae.\n\n### 4.2.4 Dimensions of the Normal Liver and Spleen\n\nThe size of the liver is usually measured with an anterior view image and the size of the spleen with a posterior view (Fig. 4.1.4). A calibrated radioactive marker placed along the costal margin during data acquisition enables measurement of the organ size. The position of the liver is assessed with reference to the right costal margin. A normal liver is situated above the right costal margin and the normal spleen above the left costal margin. The right lobe of the liver measures 10.0-17.5 cm in the right mid-clavicular line. The maximum vertical height from the superior to inferior border of the right lobe is 10-20 cm The horizontal length from the right lateral border to the tip of the left lobe is 14-24 cm [4, 5]. Because of wide variations in shape at the tips, dimensions that include the tips tend to be less reliable as a measure of overall liver size. The measurement along the right mid-clavicular line serves as a better reference point for estimating true liver size than measurements along other lines. For accurate measurement of spleen size, an oblique line is drawn from the superior to inferior tip using a posterior view image. The upper limit of the normal spleen is about 10.5 cm along this oblique axis [6].\n\nFig. 4.1.4\n\nDimensions of the normal liver and spleen. In the anterior view, the right lobe measures 10.0-15.5 cm in the midclavicular line (a) and 10.0-20.0 cm from the dome to the inferior tip (c). The horizontal length from right lateral border to the tip of the left lobe ranges from 14.0 to 24.0 cm (b). In the posterior view, the upper limit of the normal spleen is 10.5 cm along the posterior oblique axis\n\n### 4.2.5 Pattern of Radiocolloid Uptake\n\nA normal liver weighs 1,500-1,800 g and the spleen 150-200 g. The bone marrow is estimated to weigh about 1,500 g [7]. The liver takes up about 90% of radiogold colloid (Au-198) and 85% of Tc-99m-sulfur colloid. The spleen concentrates 3% of Au-198 colloid and 7% of Tc-99m-sulfur colloid. Bone marrow takes up about 7% of Au-198 colloid and 5% of Tc-99m-sulfur colloid (Table 3.1.4). The difference in radiocolloid uptake among these three organs is a function of their overall size and the physico-chemical nature of the radiocolloid particles [8]. The smallest of the particles are preferentially taken up by the bone marrow, medium-size particles by the liver, and the largest particles by the spleen. As the liver disease progresses, there is a shift in radiocolloid uptake from the liver to the spleen and bone marrow. In moderate severity cirrhosis, radiocolloid uptake by the liver decreases (65-70%), and the uptake by the spleen and bone marrow increases. There is also uptake by lungs and other organs. In advanced cirrhosis, liver uptake may be as low as 30-35%. The spleen and bone marrow continue to show an increase in uptake of radiocolloid up to as high as 25-30% of the dose by each organ. The remaining 10-12% of the injected radiocolloid dose is taken up by activated RE cells in lungs and other organs.\n\n### 4.2.6 Spleen and Bone Marrow Uptake\n\nIn the posterior view, the intensity of radiocolloid uptake by a normal spleen is usually equal to that of the normal liver [9]. In cirrhosis, the intensity of splenic uptake increases in direct proportion to the degree of portal hypertension (Fig. 4.1.5). In advanced cirrhosis, the intensity of spleen and bone marrow uptake often exceeds that of the liver. Often, the intensity of thoracic and lumbar vertebral body radiocolloid uptake by the RE cells may match that of Tc-99m-MDP uptake by the bone mineral matrix. Decreased radiocolloid uptake by the spleen is also abnormal and often is noticed in patients with Hodgkin's disease, reticulum cell sarcoma, poorly differentiated lymphocytic lymphoma, mucosis fungoidis, or other types of lymphomas. In polycythemia rubra vera, the normal spleen\/liver ratio is maintained despite an enormous increase in the size of the spleen.\n\nFig. 4.1.5\n\nSitus inversus. In the anterior (ANT) view, the spleen is located on the right and liver on the left side of the body. In the right lateral (RL) view, the spleen is superimposed at the postero-inferior part of the liver. Posterior (POST) view shows liver on the left and spleen on the right, separated by a photon-deficient column because of absorption by the vertebral bodies\n\nThe liver moves 1-3 cm up and down with each respiration. The size of the liver, therefore, may falsely appear large if the patient takes deep breaths during scanning. When the liver enlarges and extends below the right costal margin, the radiocolloid uptake in the liver tissue extending below the right costal margin may appear greater than the tissue uptake behind the costal margin. This apparent effect is due to attenuation of photons by the ribs, and the right breast exaggerates the difference in women. The effect of extrinsic compression on the liver is ascertained by taking one image during deep inspiration and another during deep expiration. An intrinsic liver defect will not change its position, whereas a defect due to extrinsic compression moves with respiration [10]. Decreased radiocolloid uptake by the liver and increased uptake by the spleen, lungs, and other soft tissues are indicators of poor prognosis [11, 12]. Fatty infiltration of the liver causes irregular uptake of radiocolloid and accumulation of xenon-133 during a V\/Q study obtained for the diagnosis of pulmonary embolism.\n\n### 4.2.7 Appearance of the Normal Liver and Spleen on SPECT Images\n\nIncreased spatial resolution and high lesion to non-lesion contrast on single photon emission computed tomography (SPECT) show some of the normal structures much more prominently than in a planar image. Often such normal structures are misinterpreted as cold lesions in a radiocolloid scan or as hot lesions on a blood pool or Tc-99m-HIDA study. The right and left branches of the portal vein, hepatic artery, and bile ducts pass through the middle of the liver in the opposite direction, 180\u00b0 apart (Fig. 4.1.6). The coronal, sagittal, and transaxial slices passing through these structures may depict these normal structures as cold or hot lesions depending on the radiotracer used for the study. A cold lesion on a radiocolloid scan that fills in with Tc-99m RBC suggests a vessel or a hemangioma. A cold lesion that fills in with Tc-99m-HIDA suggests a biliary origin (choledochal cyst). In the sagittal slice passing through the middle of the right lobe, the right portal vein may appear as a curvilinear defect in the radiocolloid scan. Sagittal slices over the lateral aspect of the right lobe show a linear defect that is usually due to the posterior branch of the right portal vein. The anterior branch of the right portal vein is slightly smaller and is not always evident on sagittal slices. These anatomical variants may not be evident in the planar images due to overlying or underlying liver tissue. The right and left portal veins and the right posterior segmental vein are 1.5-2 cm in diameter, large enough to be seen on the SPECT images as defects (Fig. 4.1.6). The medial and lateral segmental veins of the left lobe are too small to cause a defect [13]. Two or three of the most anterior coronal slices may show only the left lobe, confirming its more anterior location. Lower transaxial slices show the gallbladder fossa as a pear-shaped defect along the anterior margin.\n\nFig. 4.1.6\n\nPortal vein. Transaxial CT with contrast shows division of the portal vein in the middle of the liver into right portal vein (RPV) and left portal vein (LPV). LPV divides into medial (M) and lateral (L), and RPV into anterior (A) and posterior (P) segmental branches\n\n## 4.3 Abnormal Liver\n\n### 4.3.1 Hepatomegaly\n\nA general response of the liver to any injury is one of diffuse enlargement. Hepatomegaly usually disappears when the offending agent is removed. Metabolic diseases in children are generally associated with hepatomegaly (Table 4.1.2). Most normal-size livers do not extend below the right costal margin. In patients with chronic obstructive lung disease, the liver may be pushed downwards to be palpable below the right costal margin. It is important to establish if a clinically palpable liver is due to displacement or enlargement.\n\nTable 4.1.2\n\nCauses of hepatomegaly\n\nCommon causes | Uncommon causes | Rare causes\n\n---|---|---\n\n(1) Hepatitis | (1) Hepatoma | (1) Glycogen storage disease\n\n(2) Fatty infiltration | (2) Hemochromatosis | (2) Gaucher's disease\n\n(3) Cirrhosis | (3) Granuloma | (3) Cystic fibrosis\n\n(4) Metastatic tumors | (4) Drug-induced hepatitis | (4) Galactosemia\n\n(5) Congestive heart failure | (5) Wilson's disease | (5) Kwashiorkor\n\n(6) Leukemia | (6) Infections | (6) Budd-Chiari syndrome\n\n(7) Lymphoma | (7) Hemangioendothelioma | (7) Amyloidosis\n\n(8) Abscess | (8) Polycystic disease | (8) Gangliosidosis\n\n(9) Biliary obstruction | (9) Cholangiocarcinoma\n\n|\n\nThe space-occupying lesions of the liver can be divided into four categories: (1) marginal lesions, (2) intrahepatic solitary focal lesions, (3) intrahepatic multiple focal lesions, and (4) intrahepatic diffuse lesions (Table 4.1.3).\n\nTable 4.1.3\n\nSpace-occupying lesions of the liver\n\nMarginal lesions | Intraparenchymal lesions\n\n---|---\n\nCompression by adjacent organs | Single focal | Multiple focal | Diffuse\n\nRight kidney (hypernephroma) | Hepatoma | Metastasis | Hepatitis\n\nEnlarged gallbladder | Abscess | Polycystic disease | Cirrhosis\n\nPancreatic Ca, Pseudocyst | Adenoma | Fatty infiltration | Fatty infiltration\n\nIntracapsular hematoma | Hemangioma | Hemangioma | Wilson's disease\n\nCholedochal cyst | FNH | \\-------- | Lymphoma\n\nColon mass in hepatic flexure | Metastasis | Multiple mets | Chemotoxins\n\nRight costal margin | Simple cyst | Multicystic disease | Hemochromatosis\n\nProminent portahepatis | Hydatid cyst | Hydatid cyst | Sclerosing cholangitis\n\nAscites | Infarction | Amoebic abscess | Biliary cirrhosis\n\nRight breast | Intrahepatic GB | - | -\n\nEnlarged heart | Pseudo tumor | - | -\n\nBile leak into GB fossa | - | -\n\n|\n\nFNH = Focal nodular hyperplasia\n\n### 4.3.2 Marginal Lesions\n\nMost marginal lesions are caused by compression of the liver margin by adjoining normal organs or lesions arising from them. A normal right kidney placed high in the posterior abdomen or a mass arising from it (hypernephroma) may cause a defect along the posterior border of the right lobe of the liver. Such lesions are seen as a focal defect in the posterior planar view or as a concave posterior border in the planar right lateral view. In a SPECT image, it may appear as a round defect in the posterior slices of the coronal view and in the mid and lower slices of the transaxial views. A fully filled gallbladder may cause a cold defect in a radiocolloid scan along the inferior border in the anterior planar image. The gallbladder fossa appears as a pear-shaped defect along the anterior border in the right lateral planar image. A large choledochal cyst arising from the common bile duct may compress the middle of the inferior liver margin, causing a cold defect near the porta hepatis in a radiocolloid scan.\n\nA normal stomach after a full meal or a mass arising from the stomach or pancreas (pseudocyst, cancer) may cause a defect along the inferior border of the left lobe. Depressed right lateral ribs or a hematoma following an auto accident may cause a defect along the lateral surface of the right lobe. A lesion arising from the hepatic flexure of the colon may cause compression along the inferior liver margin of the right lobe. Post-traumatic subcapsular hematomas cause defects along any surface depending upon the point of impact, but most tend to occur along the right lateral or the anterior surface [14]. A large pendulous right breast or breast prosthesis causes a concave defect along the superior border that may be misinterpreted as a filling defect or as hypoconcentration of radiocolloid. Breast artifact is corrected by taking a repeat image after lifting the pendulous breast or removing the prosthesis (Fig. 4.1.7). An enlarged heart because of congestive heart failure or cardiomyopathy causes an impression along the superior margin at the junction of the right and left lobes.\n\nFig. 4.1.7\n\nLiver artifact. A large defect over the superior part of the right lobe caused by a breast prosthesis (a) disappears in the repeat image after its removal (b)\n\nMarginal lesions of the spleen are seen mostly along the medial (hilum) or the lateral surface. Lateral surface lesions are mostly hematomas secondary to trauma due to auto accidents. Medial surface lesions are due to a large pancreatic pseudocyst, choledochal cyst, or a postprandial fluid-filled stomach. Most post-traumatic hematomas along the lateral surface resolve in 3-4 weeks [14].\n\n### 4.3.3 Intrahepatic Solitary Focal Lesions\n\nThese are the most common type of liver lesions seen when scintigraphy is obtained early in the course of liver disease. They tend to be associated with normal or mildly abnormal liver function tests. Most focal liver lesions appear to be round. Infarction appears wedge shaped, and radiation necrosis causes a characteristic appearance (quadrangular) depending upon radiation port. A thorough knowledge of the mode of clinical presentation, clinical findings, and changes in liver function tests are necessary to arrive at an etiological diagnosis. This is true for all types of liver imaging procedures, including radiocolloid scan, computerized tomography, ultrasound, and magnetic resonance imaging. No single imaging test or even a combination of imaging tests is capable of making an etiological diagnosis all of the time [15-18].\n\n### 4.3.4 Intrahepatic Multiple Focal Lesions\n\nMultiple metastases, polycystic or multicystic liver disease, and fatty infiltration are examples of multiple intrahepatic focal lesions. Multiple metastatic lesions are the most common (Fig. 4.1.8). In a patient with a known primary cancer elsewhere, multiple liver lesions are considered secondary until proven otherwise. Polycystic liver disease is associated with cysts in the kidney, spleen, and other organs and has an autosomal dominance pattern of inheritance. Fatty infiltration may cause focal or diffuse defects. On ultrasound, fatty infiltration appears as a hypoechoic mass with angulation and interdigitating margins. [19]. Liver with fatty infiltration shows xenon-133 retention after a lung ventilation study obtained to rule out pulmonary embolism. Xenon-133 retention by the liver is specific for fatty infiltration [20].\n\nFig. 4.1.8\n\nMultiple metastatic liver disease. Transaxial, coronal, and sagittal SPECT images with radiocolloid show multiple intrahepatic metastasis from colon cancer (courtesy of Dr. Kumaresan, Hyderabad)\n\n### 4.3.5 Intrahepatic Multiple Diffuse Lesions\n\nAlcoholic hepatitis, viral hepatitis, and cirrhosis are the most common causes followed by other rare diseases, such as hemochromatosis, Wilson's disease, primary biliary cirrhosis, primary sclerosing cholangitis, and drugs (chemotherapy) or toxin-induced hepatitis. Many of these diseases are associated with a changing image pattern, depending upon the phase of parenchymal liver disease. Viral and alcoholic hepatitis often heal completely when the offending agent is removed completely. Defects due to chemotherapy usually disappear after the completion of the course. The role of radiocolloid imaging in the detection of liver lesions has diminished recently and has been replaced by CT and ultrasound, which have better spatial resolution and provide additional information about structural changes in organs around the liver [21]. Radiocolloid imaging is now requested mostly to clarify an abnormality already seen with the CT or ultrasound study to confirm whether the lesion is hepatic in origin. The posterior view image shows all three organs that take up radiocolloid, the liver, spleen, and bone marrow, in the thoracic and lumbar vertebral bodies and provides information related to their relative function. Bone marrow replacement by a tumor causes a \"cold\" defect in the vertebra in a radiocolloid image and a \"hot\" lesion in a Tc-99m MDP at the corresponding site (Fig. 4.1.9).\n\nFig. 4.1.9\n\nDiscordant Tc-99m colloid and Tc-99m MDP uptake by a vertebra. A metastatic tumor in T-12 vertebra causes a \"cold\" defect (D) on the radiocolloid image (a) and a \"hot\" lesion on Tc-99m MDP bone scan (b)\n\n### 4.3.6 Relative Merits of the Diagnostic Tests\n\nThe accuracy of positive predictive and negative predictive values is similar for CT, ultrasound, and radiocolloid scan. The sensitivity of CT and radiocolloid scan is similar, but slightly low for ultrasound (Table 4.1.4). Any one of the three imaging modalities is capable of detecting a solitary liver lesion, but the choice of one over another depends upon local expertise, availability, reproducibility, and cost [22]. Because CT and ultrasound studies provide additional information about nearby organs, they have become clinically popular and have almost completely replaced radiocolloid scans for liver imaging today. Once a lesion is detected by one imaging modality, incremental information from the second or third imaging test is not substantial enough to justify obtaining more than one morphologic imaging procedure. This factor is observed more critically by insurance companies and other third party payers than by physicians and patients. The information obtained from a combination of one morphological and one physiological imaging may be better for patient management than the information gathered from a combination of two or more morphological imaging procedures.\n\nTable 4.1.4\n\nSensitivity, specificity, accuracy, positive predictive and negative predictive value of radiocolloid scan (RN), CT, and ultrasound (US) studies in the detection of space-occupying lesions of the liver\n\nNo. of patients | Sensitivity | Specificity | \\+ Predictive value | \u2212 Predictive value | Ref.\n\n---|---|---|---|---|---  \n|\n\nRN | CT | US | RN | CT | US | RN | CT | US | RN | CT | US\n\n|\n\n1,438 | 86 | 75 | 75 | 79 | 91 | 82 | 82 | 90 | 81 | 83 | 77 | 76 | [15]\n\n122 | 86 | 93 | 82 | 83 | 88 | 85 | 83 | 83 | 76 | 86 | 86 | 78 | [16]\n\n80 | 79 | 76 | 61 | 81 | 89 | 94 | 74 | 83 | 87 | 84 | 84 | 77 | [17]\n\n1,640 | 84 | 81 | 73 | 81 | 89 | 87 | 80 | 85 | 81 | 84 | 82 | 77\n\n|\n\nBiochemical tests specific for a disease may offer clues in arriving at an etiological diagnosis from an imaging procedure. Most hepatomas are associated with a rise in serum alpha-fetoprotein, and the tumor size correlates with serum levels [23]. However, a slight rise in serum alpha-fetoproteins levels is often seen in viral hepatitis, post-necrotic cirrhosis, chronic active hepatitis, drugs, and a variety of other liver diseases. Hepatomas may cause diffuse irregular radiocolloid uptake or present as a solitary or multiple focal liver lesion [24].\n\n## 4.4 Adenoma and Focal Nodular Hyperplasia\n\nAdenoma and focal nodular hyperplasia (FNH) are relatively rare solitary tumors of women in their reproductive years and show a close association with the use of oral contraceptives [1]. The tumors remain mostly asymptomatic (Fig. 4.2.1). They are discovered incidentally during abdominal CT and ultrasound examinations obtained for some other medical indication. These tumors may suddenly become symptomatic and present with acute onset of abdominal pain, hemorrhage, and hypotension [2]. Adenoma bleeds more frequently than focal nodular hyperplasia. One study reported bleeding incidence at presentation of 4 in 27 patients with adenoma and none of 23 patients with focal nodular hyperplasia [3]. They may extend beyond the liver border in some patients and become clinically palpable on routine examination. Liver function tests are usually normal in both tumors.\n\nFig. 4.2.1\n\nA schematic diagram (top) of focal nodular hyperplasia and liver cell adenoma. Focal nodular hyperplasia consists of radiating dense fibrous tissue with all types of liver cells, but without any capsule. Histology (bottom, left) shows the hepatocytes, Kupffer cells, and fibrous tissue with formation of small nodules. Adenoma (bottom, right) consists of a capsule with monotonous sheets of hepatocytes without any Kupffer cells or fibrous tissue (courtesy of Dr. Ted Pinkert)\n\nBoth tumors appear as non-specific space-occupying lesions in a radiocolloid scan, ultrasound, MRI, or CT examination. Arteriography, contrast CT, or blood pool imaging may show hypervascularity in FNH. Adenoma, on the contrary, usually shows no hypervascularity. But these morphologic and arteriographic characteristics are not distinct enough to separate FNH reliably from adenoma (Table 4.2.1) or from other space-occupying lesions of the liver listed in Table 4.1.2.\n\nTable 4.2.1\n\nCharacteristic features of hepatic adenoma and focal nodular hyperplasia\n\nParameter | Adenoma | Focal nodular hyperplasia\n\n---|---|---\n\nAge | 30-40 years | 30-40 years\n\nFemale:male ratio | 30:0 | 10:1\n\nOral contraceptive association | +++ | ++\n\nWell-defined capsule | Present | Absent\n\nBile ducts | Absent | Decreased and narrowed\n\nFibrosis | Absent | Present\n\nHypervascularity | ++ | ++++\n\nHepatocyte number (in lesion) | Normal or increase | Normal or decrease\n\nKupffer cells | Absent or decrease | Normal or decrease\n\nRadiocolloid uptake | Absent | Normal or decrease\n\nTc-99m-HIDA uptake | Increase or normal | Normal\n\nTc-99m-HIDA excretion | Absent | Slow\n\nTc-99m-NGA receptors | not known | Increase or normal\n\nHIDA = Hepatic iminodiacetic acid, NGA = galactosyl neoglyco-albumin\n\n### 4.4.1 Adenoma\n\nAdenoma is a well-encapsulated, round, solitary lesion, varying in size from 8 to 15 cm. It consists of sheets of hepatocytes without any bile ducts or fibrous septa (Fig. 4.2.1). Kupffer cells are either completely absent or markedly decreased in number. Few of those Kupffer cells that are present lack phagocytic capacity and hence the nodule appears as \"cold\" in a radiocolloid scan [4]. The hepatocytes are able to concentrate Tc-99m-HIDA, but cannot secrete into bile because of the lack of bile ducts [5]. About 80% of adenomas show mixed echogenecity on the ultrasound examination, and about 86% appear as a hypodense mass on non-contrast CT examination [4]. The most consistent pattern of hepatic adenoma on a multimodality imaging is one of a well-circumscribed solitary mass lesion on CT or US with no uptake on a radiocolloid scintigraphy. Atypical forms with functioning Kupffer cells are rare [6]. Some adenomas may show a nodule-in-nodule pattern on CT and MRI and are indistinguishable from hepatocellular carcinoma. Adenomas may be separated from carcinomas by depicting its functional characteristics. Lack of Ga-67 citrate uptake (in contrast to hepatocellular carcinoma) and no radiocolloid uptake, as well as late uptake with no excretion of Tc-99m HIDA, may distinguish adenoma from hepatoma and FNH and other types of focal liver tumors [7].\n\n### 4.4.2 Focal Nodular Hyperplasia\n\nIn contrast to hepatic adenoma, which consists mostly of sheets of hepatocytes, focal nodular hyperplasia (FNH) usually contains almost all of the cellular components of a normal liver, including the hepatocytes, Kupffer cells, bile ducts, and other supporting cells. There is a central area of fibrosis with radiating fibrous connective tissue that divides the nodule into several smaller sections of varying size, mimicking the pattern seen with pseudonodules of cirrhosis [8]. A capsule is absent, a clear distinction from adenoma (Fig. 4.2.1). Bile ducts show proliferation with narrow lumens impeding bile flow through them. The hepatocytes show a clear to dark cytoplasm with sparse bile canaliculi [2]. Most FNHs are found just beneath the liver surface, and some are pedunculated. The size of the lesions varies from 1 to 15 cm. Malignant transformation is uncommon. About 90% of FNH are hypervascular, and the rest are hypovascular on the angiogram. On the ultrasound examination, 50% are hyperechoic, 40% hypoechoic, and 10% show a mixed pattern [9, 10].\n\n### 4.4.3 Scintigraphic Features\n\nBoth focal nodular hyperplasia and a regenerating nodule of cirrhosis contain Kupffer cells. Most of the other focal liver lesions (adenoma, hemangioma, primary and metastatic liver lesions) do not contain any Kupffer cells, or those few Kupffer cells that are present lack phagocytic function. A normal radiocolloid uptake pattern is seen in 64% of FNH, and the remaining 34% show decreased uptake [11]. A characteristic scintigraphic feature of FNH includes a hypervascularity seen on contrast CT (Fig. 4.2.2a). About 76% of them showed hypervascularity in a perfusion study with Tc-99m-HIDA (Fig. 4.2.2b). Early phase functional images (within 10 min) show Tc-99m-HIDA uptake by the nodule equal to that of the adjoining normal liver (Fig. 4.2.2c). Late phase images (after 45 min) show slow clearance by the nodule and a normal clearance from adjoining normal liver. At 60 min, the radioactivity retained by FNH is about twice as much as the normal liver and hence appears as \"a hot nodule\" on the late image (Fig. 4.2.2 d and e). Mean excretion half time of 18 min for the normal liver increases to 40 min for FNH, explaining the reason for the appearance of a hot lesion [11]. The slow excretion is due to narrowing of the cholangioles draining the FNH. The time of appearance of the hot nodule is a function of both how rapidly the radiotracer is cleared from the normal liver and how slowly it is cleared from the nodule. The nodule is usually seen clearly between 18 and 60 min after injection of Tc-99m-HIDA. Routine inclusion of the perfusion phase as an integral part of hepatobiliary imaging enhances the nodule detectability by showing its hypervascularity. The radiocolloid image depicts the lesion as \"cold\" (Fig. 4.2.2f). The overall detectability of FNH is 92% with Tc-99m-mebrofenin cholescintigraphy, 84% with CT, and 84% with MRI [10] (Fig. 4.2.3).\n\nFig. 4.2.2\n\nFunctional characteristics of focal nodular hyperplasia. It is hypervascular as seen on a contrast CT (a) and perfusion study with Tc-99m-HIDA (b). Early images show normal uptake (c), and late images at 45 min (d) and 60 min (e) show retention of Tc-99m HIDA (e). The lesion appears as \"cold\" on a radiocolloid image (f). (Reproduced with permission from the publisher, [11])\n\nFig. 4.2.3\n\n(a- c) Blood pool features. A large focal nodular hyperplasia in the posterior right lobe appears as hypodense on a CT scan without contrast (a) and as non-uniform hyperdense lesion with the contrast agent (b). Blood pool imaging with autologous Tc-99m RBCs shows a non-filling center (c)\n\n### 4.4.4 Surface Receptors\n\nThe hepatocytes of FNH show an increase in the number of surface receptors for Tc-99m-galactosyl-human serum albumin much more intensely than the adjoining normal liver parenchyma. In a study comprising 12 patients with focal nodular hyperplasia, 9 with hepatocellular carcinoma, and 3 patients with metastatic liver cancer, the authors were able to distinguish FNH from primary and metastatic liver lesions by calculating a nodule\/normal liver ratio obtained with Tc-99m- neoglycoalbumin. Eight of 12 patients with FNH showed an increase in ratio, and the remaining 4 showed an uptake equal to that of the adjoining normal liver tissue. The nodule\/normal liver mean ratio was 1.7 \u00b1 0.3 in patients with FNH and 0.4 \u00b1 0.2 in patients with primary and metastatic cancer, enabling clear separation of benign from malignant liver lesions [12]. Multimodality imaging with tumor-seeking agents like Ga-67 citrate and Tl-201 chloride, combined with Tc-99m phytate (colloid) and a hepatobiliary agent (Tc-99m-pyridoxyl-5-methyl tryptophan), often aids in separating benign from malignant hepatic lesions [13, 14].\n\n### 4.4.5 Treatment\n\nA flexible treatment approach is recommended, and asymptomatic patients are managed conservatively [15]. Some nodules regress spontaneously on discontinuation of oral contraceptives [16]. Surgical resection is preferred when the nodule grows rapidly in size or becomes acutely symptomatic. Because both hepatic adenoma and FNH occur more frequently in young women during their child-bearing years and are often detected serendipitously on US or CT, done for unrelated reasons, proper management requires clear differentiation of adenoma from FNH before deciding on a specific form of therapy [17, 18]. Biopsy is not mandatory.\n\nAdenomatosis (multiple small adenomas) is a distinct entity and behaves much differently from adenoma and focal nodular hyperplasia in terms of its clinical presentation. Adenomatosis occurs in both sexes, shows no association with oral contraceptives, and is associated with elevation of serum alkaline phosphatase and gamma glutamyl transpeptidase. Hemorrhage is uncommon, and it requires no specific form of therapy [19].\n\n## 4.5 Hemangioma\n\nHemangiomas are the most common liver lesions, accounting for nearly 5-7% of all benign tumors [1]. They are congenital vascular malformations at birth that increase in size with the growth of the liver. Hemangiomas affect both sexes and occur at all ages, but manifest clinical symptoms usually in the 3rd-5th decades of life. They are relatively more common in women than in men, with a ratio of 4:1-6:1 [2]. They are more frequent in multiparous women and increase in size during pregnancy and after administration of estrogens [3].\n\n### 4.5.1 Histopathology\n\nCavernous hemangioma is usually a solitary lesion consisting of multiple vascular channels, lined by a single layer of flat endothelial cells, and supported by an intervening fibrous tissue. They vary in size from few millimeters to several centimeters, some as large as 20-30 cm in size. Hemangiomas larger than 4 cm are referred to as giant cavernous hemangiomas. Most are sessile without a capsule and are located deep inside the superior part of the liver. Occasionally, a pedunculated lesion may be seen well separated from the main liver mass. Hemangiomas consist of a large blood pool, but show a decreased blood flow (ml gm-1 min-1) through the lesion when compared to the adjoining normal liver tissue [4].\n\n### 4.5.2 Clinical Presentation\n\nMost hemangiomas are asymptomatic. In the past they were usually discovered at autopsy [2]. Today hemangiomas are discovered serendipitously while performing an ultrasound or CT scan of the abdomen, often when these imaging studies are obtained for assessment of other abdominal organs [5]. Large hemangiomas may cause abdominal pain because of compression of the adjoining organs or thrombosis within the lesion. Acute onset of severe abdominal pain may indicate hemorrhage.\n\n### 4.5.3 Diagnosis\n\nMost hemangiomas are discovered today during their asymptomatic phase with the ultrasound or CT as space-occupying lesions of the liver. The real challenge, therefore, is not one of discovery, but that of differentiation from other types of liver lesions, such as adenoma, focal nodular hyperplasia, primary or metastatic tumors. Blood pool imaging with Tc-99m RBC, contrast CT, MRI, and contrast angiography have been used for diagnosis [6-9]. Imaging with Tc-99m-labeled autologous red blood cells carries the highest specificity and is considered the diagnostic modality of choice for confirmation of hemangiomas [10].\n\n### 4.5.4 Blood Pool Scintigraphy\n\nPatient red blood cells are labeled with Tc-99m using an in-vitro method as described in detail in Chap 3.2. A rapid sequence perfusion study is obtained after injecting 15-30 mCi of autologous Tc-99m RBC (or Tc-99m pertechnetate for in vivo method) by collecting data at one frame\/2 s for 60 s. Planar blood pool images in the anterior, posterior, and two lateral views are obtained after the perfusion study. Delayed images are obtained at 30 or 60 min by collecting at least 1 million counts per image. When not seen clearly in planar images by 60 min, SPECT imaging study is obtained between 1 and 2 h. The SPECT data are collected over a 360\u00b0 rotation by obtaining 64 frames at 30 s\/frame using a 64 \u00d7 64 \u00d7 8 matrix. After appropriate correction for camera nonuniformity and center of rotation deviation, the projections are reconstructed with the vendor-specific filtered back-projection algorithm. Transaxial, coronal, and sagittal slices are obtained with known pixel size to be able to measure size accurately. Images are recorded on 8 \u00d7 10-inch X-ray film.\n\n### 4.5.5 Scintigraphic Features\n\nA normal study (Fig. 4.3.1) clearly shows a blood pool in the abdominal aorta, inferior vena cava, two kidneys, liver, and spleen. Generally, the inferior vena cava is seen much more prominently than the aorta. The heart is the hottest organ in the chest, and the spleen is the brightest organ in the abdomen in the posterior view.\n\nFig. 4.3.1\n\nNormal abdominal blood pool. Anterior (ANT) and posterior (POST) planar images show inferior vena cava (I), aorta (A), liver (L), spleen (S), and two kidneys (K). Liver appears faint in the right lateral (RL) view and spleen appears bright in the left lateral (LL) view\n\nHemangiomas typically show a decreased perfusion followed by delayed filling, reflecting their basic pathophysiology. Despite an enormous increase in the blood pool, blood flow (ml gm-1 min-1) is generally reduced relative to surrounding normal liver tissue. Due to reduction in blood flow, mixing of radiolabeled RBC with the unlabeled RBC within the hemangioma occurs slowly over several minutes or hours, depending upon the size of the hemangioma. Very large hemangiomas are often missed when sufficient time is not allowed for complete mixing or equilibration, prior to acquisition of SPECT data. Giant hemangiomas are delineated better in delayed SPECT images obtained 2-4 h after injection of the labeled red blood cells.\n\nSmall hemangiomas at the peripheral part of the liver equilibrate early and are seen faintly on planar blood pool images and become very clear on the SPECT images (Fig. 4.3.2). Complete equilibration with unlabeled RBC takes a much longer time as the size of the hemangioma increases. They manifest varied characteristics on the ultrasound and CT image. Many appear as hypodense on CT without contrast and become isodense (Fig. 4.3.3) or hyper-dense after the contrast. Some fill in only partially and others not at all. These varied features make it difficult to distinguish hemangioma from other benign and malignant liver lesions from a CT scan.\n\nFig. 4.3.2\n\nSolitary hemangioma. Planar images at 10, 20, and 40 min show blood pool at the tip of the right lobe equal to that of the rest of the liver. Coronal SPECT image at 60 min clearly shows increased blood pool at the tip, establishing hemangioma\n\nFig. 4.3.3\n\nVariable pattern hemangioma. A CT without contrast (upper left) shows a hypodense lesion in the right lobe that becomes isodense (upper right) after injection of the radiocontrast agent. Immediate planar Tc-99m RBC blood pool scintigraphy shows hemangioma faintly, which becomes much clearer at 5 and 15 min and in the right anterior oblique (RAO) view\n\nGiant hemangiomas often involve one complete segment or lobe of the liver (Fig. 4.3.4). Rapid sequence perfusion study shows markedly decreased blood flow, followed by delayed equilibration. Some may equilibrate only partially even at the end of 3-4 h and may require images at 18-24 h. Biopsy of such large lesions is a high risk procedure and should be avoided.\n\nFig. 4.3.4\n\nGiant cavernous hemangioma. A perfusion scintigraphy (upper left) shows no blood flow through a giant hemangioma involving the superior part of the right lobe and the medial left lobe in a patient with breast cancer. Coronal view SPECT at 60 min shows hemangioma with a small supero-lateral region not yet equilibrated (upper right). An anterior (ANT) and posterior (POST) planar image at 2.5 h shows complete filling in of the hemangioma (middle left). A CT with contrast (lower left) shows a variable blood pool within the hemangioma and an ultrasound study (lower right) shows mixed hyper- and hypoechoic regions (courtesy of Dr. Ronald Hagelman, Tucson, AZ)\n\nThe sensitivity and accuracy of an imaging test depend very much on the size of the hemangioma. For lesions larger than 2 cm, the sensitivity and accuracy of Tc-99m RBC perfusion and SPECT blood pool imaging varies from 89 to 92% and 89 to 94%, respectively. For similarly sized lesions, MRI sensitivity varies from 85 to 100% and accuracy from 81 to 100% [10]. For lesions less than 2 cm, scintigraphic sensitivity is 58% and accuracy 60%, and MRI carries a sensitivity of 83% and accuracy of 84% (Table 4.3.1). As MRI often fails to differentiate hemangioma from hypervascular neoplasm or focal nodular hyperplasia, blood pool imaging with SPECT is considered as the method of choice for confirmation of hemangioma [6, 10]. Angiography is necessary only in those patients who do not show a blood pool pattern typical for hemangioma. Needle biopsy is avoided in most patients unless the lesion is small and the diagnosis is not confirmed from blood pool imaging. MRI with contrast may be preferred over angiography and needle biopsy for small lesions not seen on the blood pool study [10].\n\nTable 4.3.1\n\nComparison of Tc-99m red blood cell SPECT and MR imaging in the diagnosis of hepatic hemangioma [10] | Hemangioma size in cm (n = 64)\n\n---|---  \n|\n\n1.0-1.9\n\n(n = 24) | 2.0-2.9\n\n(n = 13) | 3.0-13.0\n\n(n = 27)\n\nSPECT\n\nNo. positive (n = 50) | 14 | 12 | 24\n\nNo. negative (n = 14) | 10 | 1 | 3\n\nSensitivity (%) | 58 | 92 | 89\n\nAccuracy (%) | 60 | 94 | 89\n\nMR imaging\n\nNo. positive (n = 58) | 20 | 11 | 27\n\nNo. negative (n = 6) | 4 | 2 | 0\n\nSensitivity (%) | 83 | 85 | 100\n\nAccuracy (%) | 84 | 81 | 100\n\nAbout 90% of hemangiomas are single, found in both hepatic lobes, and in the right lobe nine times more frequently than the left (Table 4.3.2). Most are situated along the superior margin of both lobes [11]. Multiple hemangiomas may involve both lobes at different locations (Fig. 4.3.5). The typical pattern of a \"cold\" lesion on blood flow and a delayed filling (hot) on the blood pool image is found in 66% of the hemangiomas. Others show atypical patterns of early incomplete and delayed complete filling. For lesions between 1 and 2 cm, SPECT increases sensitivity by 11% over planar images. On ultrasound examination, 59% appear as hyperechoic, 11% hypoechoic, and 30% show a mixed pattern.\n\nFig. 4.3.5\n\nMultiple hemangiomas. Anterior (ANT) and posterior (POST) view planar images (top) show three hemangiomas of variable sizes. Superior (S) and inferior (I) cut transverse SPECT images (left bottom) show one hemangioma in the left lobe (no. 1) and another in the posterior part of the right lobe (no. 2). Anterior (A) and posterior (P) coronal slices confirm no. 1 lesion, and the posterior slice shows an additional (no. 3) hemangioma. A CT with contrast agent shows three hypodense lesions. Only lesion no. 1 shows slight enhancement at the periphery\n\nTable 4.3.2\n\nFeatures of 130 hemangiomas [11]\n\nFeature | Number (%)\n\n---|---\n\nDistribution\n\nSingle | 116 (89)\n\nMultiple | 14 (11)\n\nRight lobe | 118 (91)\n\nSubdiaphragmatic | 103 (87)\n\nLeft lobe | 12 (9)\n\nSuperficial | 109 (84)\n\nDeep | 21 (16)\n\nPosterior | 83 (64)\n\nAnterior | 47(36)\n\nScintigraphic features\n\nTypical filling pattern | 86 (66)\n\nAtypical filling pattern | 44 (34)\n\nUltrasound features\n\nHyperechoic | 76 (59)\n\nHypoechoic | 14 (11)\n\nMixed pattern | 39 (30)\n\n### 4.5.6 Treatment\n\nNo treatment is required for most patients without symptoms. Lobectomy or segmentectomy is recommended for patients with recent onset of pain or with a rapidly expanding lesion [12, 13].\n\n## 4.6 Somatostatin Receptor Scintigraphy\n\n### 4.6.1 Somatostatin Source\n\nSomatostatin belongs to a multigene family peptide and is synthesized, stored, and released by many cells in the body [1]. Upon release from cells, somatostatin acts as an autocrine, paracrine, and endocrine hormone. Two forms of somatostatin are identified: one with 14 (Chap. 3, Fig. 3.4.1) and the other with 28 amino acids. The larger form is a dimer formed by the union of two shorter molecules, attached at the N-terminal end [2, 3]. The cells secreting somatostatin are distributed throughout the body. High concentration of somatostatin is found in the anterior pituitary, thyroid, lungs, liver, spleen, gastrointestinal tract, pancreas, kidneys, adrenal medulla, and paraganglions of the nervous system. Somatostatin-secreting cells were once called APUD (amine precursor uptake decarboxylation) cells and now have been renamed neuroendocrine cells [4].\n\n### 4.6.2 Action of Somatostatin\n\nIn contrast to most hormones, which generally have a stimulatory effect on their target cells in the body, somatostatin has predominantly an inhibitory effect on its target cells. It is a short peptide with a serum half life of less than 3 min. It inhibits secretion of: (1) growth hormone and thyrotropin from the anterior pituitary gland; (2) insulin, glucagon, and exocrine secretion from the pancreas; (3) gastrin, vasoactive intestinal polypeptide (VIP), secretin, and cholecystokinin from the gastrointestinal tract, and (4) hormones secreted by tumors arising from various organs (Table 4.4.1). By its ability to inhibit the production and release of the hormone, somatostatin either reduces or totally abolishes the hormonal effect on the target cells and often reduces the size of the primary and metastatic tumors. Other physiological actions of somatostatin include reduction of hepatic blood flow, inhibition of gallbladder contraction and bile emptying, and inhibition of gastrointestinal motility. It increases absorption of water and electrolytes from the intestine [5].\n\nTable 4.4.1\n\nSomatostatin receptor-positive tumors\n\nGastrinoma | Carcinoid\n\n---|---\n\nInsulinoma | Medullary thyroid cancer\n\nGlucagonoma | Pituitary adenoma\n\nSmall cell lung cancer | Neuroblastoma\n\nVIPoma | Paraganglioma\n\nCholangiocarcinoma | Meningioma\n\nPheochromacytoma | Motilinoma\n\n### 4.6.3 Somatostatin Receptors\n\nFive subtypes of somatostatin receptors have been recognized [5]. Each subtype has its own chromosome location and manifests a different level of affinity for somatostatin uptake (Table 4.4.2). Normal organs that are seen faintly on a indium-111 pentetreotide (OctreoScan) scan include the lungs, anterior pituitary, and G-I tract, and the organs that are seen intensely include the thyroid, liver, and spleen (Fig. 4.4.1). Neuroendocrine tumors with the heaviest concentration of somatostatin receptors and visible on In-lll pentetreotide (OctreoScan) scans are shown in Table 4.4.3.\n\nFig. 4.4.1\n\nTwo foci of carcinoid tumor seen with In-111 OctreoScan. Both lesions seen in the 4-h image become much crisper in the 24-h image. Colon seen at 24 h represents bile and often makes it difficult to separate from the tumor\n\nTable 4.4.2\n\nCharacteristics of human somatostatin receptor subtypes [5]\n\nCharacter | Subtype 1 | Subtype 2 | Subtype 3 | Subtype 4 | Subtype 5\n\n---|---|---|---|---|---\n\nChromosome no. | 14 | 17 | 22 | 20 | 16\n\nG protein binding | + | + | + | + | +\n\nReceptor affinity to:\n\nSomatostatin-14 | +++ | +++ | ++ | ++++ | +++\n\nOctretide | \u00b1 | +++ | ++ | \u00b1 | ++\n\nVapreotide | \u00b1 | ++ | + | + | +++\n\nLantreotide | \u00b1 | ++ | + | + | ++++\n\nDistribution in normal organs | Brain, lungs, stomach, jejunum, kidneys, liver, pancreas | Brain, kidneys, bile ducts | Brain, pancreas | Brain, lungs | Brain, heart, adrenals, pituitary, small- intestine, skeletal muscle\n\nTable 4.4.3\n\nSomatostatin receptor positivity by in-vivo scintigraphy and in-vitro autoradiography [12]\n\nTumor type | In-vivo scintigraphy (%) | In-vitro autoradiography (%)\n\n---|---|---\n\nMeningioma | 100 | 98\n\nParaganglioma | 100 | 92\n\nSmall cell lung cancer | 100 | 57\n\nHodgkin's disease | 98 | 100\n\nCarcinoid | 95 | 88\n\nGastrinoma | 93 | 100\n\nUnclassified APUDoma | 89 | 100\n\nNeuroblastoma | 89 | 65\n\nPheochromacytoma | 87 | 73\n\nNon-Hodgkin's lymphoma | 83 | 87\n\nNon-functioning pituitary adenoma | 75 | 55\n\nPituitary GH producing tumor | 70 | 98\n\nMedullary thyroid cancer | 69 | 38\n\nBreast cancer | 68 | 46\n\nInsulinoma | 46 | 67\n\nSarcoidosis | 100 | 100\n\nTuberculosis | 100 | 100\n\nRheumatoid arthritis | 100 | 86\n\n### 4.6.4 Octreotide\n\nSomatostatin with a serum half life of less than 3 min is unsuitable for diagnostic or therapeutic application. This problem is solved by making molecular substitution of its basic structure and creating somatostatin analogues of longer serum half life. The first successful synthetic compound was octreotide with eight amino acids. It behaves much like the parent hormone with 14 or 28 amino acids in inhibiting somatostatin secretion by the tumor. The serum half life of octreotide is about 2 h after a subcutaneous administration. In order to accomplish both the task of binding to the receptors and at the same time carry its physiological action, all somatostatin analogues must retain the basic loop structure and four receptor-binding amino acids, phenylalanine (Phe), tryptophan (Trp), lysine (Lys), and threonine (Thr). Octreotide differs from somatostatin in not only having only eight amino acids, but also by carrying D-Phe in place of -Phe. Vapreotide (RC-160) and lanreotide (BIM-23014) are newer synthetic analogues of somatostatin used in the treatment of somatostatin receptor-positive tumors [5].\n\n### 4.6.5 Radiolabeling of Octreotide\n\nOctreotide was radiolabeled first with radioiodine I-123. The loop structure and three of the four receptor-binding amino acids were retained; the fourth receptor binding amino acid, phenylalanine, was replaced by tyrosine to enable easy radioiodination [6, 7]. The radiolabeling technique was cumbersome and the cost of production and delivery very high due to the short physical half life of I-123 (13 h). Because of these drawbacks, imaging did not become very popular clinically. Replacement of radioiodine I-123 with indium-lll appeared logical and was readily accomplished (Chap. 3, Fig. 3.4.1).\n\n### 4.6.6 Indium-111 Pentetreotide (OctreoScan)\n\nA bifunctional chelate, diethylenetriaminepentaacetic acid (DTPA), attaches to the ligand (octreotide) at one end and to the radiotracer indium-111 at the other end [8]. The loop structure and the four receptor-binding amino acids, Phe, D-Trp (in place of Trp), Lys, and Thr, are retained. To facilitate firm chelation with DTPA, Phe is replaced by D-Phe [9]. Following intravenous injection, In-111 pentetreotide (OctreoScan) distributes rapidly in the extravascular space with only 33% of the dose remaining in the intravascular pool at the end of 10 min and less than 1% at the end of 20 h. About 50% of the dose is excreted in urine in 6 h, 85% in 24 h, and more than 90% in 72 h. A very small amount is secreted into bile, which is excreted in the stool, with about 2% of the dose in 72-h stool (Fig. 4.4.1). The biokinetics of In-lll pentetreotide differ slightly from those of C-14 octreotide. In rats, most of C-14 octreotide enters the liver to be excreted in bile [9], whereas most of In-lll pentetreotide enters the kidney to be excreted in urine [9, 10]. The usual scan dose of In-lll pentetreotide is 3-6 mCi bound to 10 \u03bcgm of the peptide. The biological half time of In-111 pentetreotide is about 6 h [9]. Soon after injection, most of In-111 is attached to the receptors on the surface of the cell. After about 6 h, 50% of the cell radioactivity gets internalized. Auto radiographic studies show that In-111 inside the cell remains with the cytoplasm and the nucleus of cultured cells from carcinoid and glucagonoma [11]. This feature indicates a therapeutic potential for In-lll pentetreotide.\n\n### 4.6.7 OctreoScan Imaging Protocol\n\nFor patient preparation, discontinue therapy with somatostatin analogues for 24-48 h before injection and throughout imaging. Hydrate the patient with 16-24 oz of liquids just before and after the dose. For image acquisition, see Table 4.4.4.\n\nTable 4.4.4\n\nOctreoScan imaging protocol\n\nAgent | In-lll pentetreotide\n\n---|---\n\nDose | 3 mCi for planar and 6 mCi for SPECT images\n\nRoute | Intravenous\n\nImaging time | 4 and 24 h. Repeat at 48 or 72 h if necessary. SPECT images are taken at 4 h\n\nBowel preparation | Laxatives (Bisacodyl, fleet enema) before scan\n\nImage acquisition parameters\n\nGamma camera | Large field of view camera\n\nCollimator | Medium energy parallel-hole collimator\n\nSpectrometer settings | One set over 172 keV and the other over 245 keV energy with 20% window\n\nMatrix | 64 \u00d7 64 or 128 \u00d7 128 word matrix\n\nCounts per image | 300,000 for head and neck or 10-min image\n\n|\n\nChest and abdomen, each 500,000 counts or 10 min per view\n\nSPECT protocol\n\nMatrix | 64 \u00d7 64 or 128 \u00d7 128 matrix, 360\u00b0 rotation, with 4 or 6\u00b0, 60-90 stops\n\nFilter | Hemming or Wiener\n\n### 4.6.8 Somatostatin Receptor-Positive Tumors\n\nSuccessful detection of hepatic and extrahepatic somatostatin receptor positive tumors with In-lll OctreoScan depends upon several factors, including tumor size, histology, location, and receptor subtype within the tumor (Fig. 4.4.2). A clear distinction must be made between tumor somatostatin receptor content versus tumor positivity with In-lll OctreoScan. Tumors may have differing concentrations of receptor subtypes with varying affinity levels for somatostatin-14 or somatostatin-28 vs. In-lll pentetreotide (Table 4.4.2). Octreotide has a relatively higher affinity for receptor subtypes 2, 3, and 5 than for subtypes 1 and 4. Since In-lll pentetreotide differs in structure slightly from that of octreotide, it could have a slightly different affinity for tumor uptake than octreotide and may explain lower or higher rate of scintigraphic tumor detection rate in clinical studies.\n\nFig. 4.4.2\n\nSomatostatin receptor-positive neuroendocrine tumor with liver metastasis. Metastatic lesion in the liver is seen better in the posterior than in the anterior view. Note normal bowel activity with focal collection in the cecum mimicking a carcinoid lesion. Normal uptake is seen in the kidneys, liver, spleen, and genitalia\n\nOverall detection and localization of somatostatin receptor-positive tumors with In-lll pentetreotide scintigraphy alone vary from 98 to 100% for Hodgkin's disease, meningioma, lung cancer, and paraganglioma, 90-95% for gastrinoma and carcinoid, and 80-89% for pheochromacytoma and neuroblastoma [12]. The detection and localization of metastatic gastrinoma in the liver are 92% by In-lll pentetreotide scintigraphy alone compared to a sensitivity of 83% by multimodality imaging with ultrasound, CT, MRI, and contrast angiogram [13]. Overall, in-vivo scintigraphy shows an excellent correlation with in-vitro autoradiography, but wide variations are seen in a few specific types of tumors (Table 4.4.3). In vivo scintigraphy with In-lll pentetreotide often may show results much better than those predicted from in vitro autoradiographic studies, suggesting a role for factors other than somatostatin receptor binding alone. Such findings are commonly seen in the case of carcinoid, small cell lung cancer, paraganglioma, neuroblastoma, pheochromacytoma, and medullary thyroid cancer. In the case of insulinoma, growth hormone-producing pituitary adenoma, and unclassified APUDomas, scintigraphy may detect lesions less often than predicted from autoradiographic results. Both planar and SPECT images are obtained early at 3-4 h when the likelihood of bowel activity due to bile is less (Fig. 4.2.1). Segmented bowel activity can mimic focal tumor, and SPECT images taken at a later time (beyond 4 h) may not be able to separate the two.\n\nSeveral factors, both technical and physiological, contribute towards false-positive results, which occur in one in ten patients with Zollinger-Ellison syndrome [14]. High specificity of In-111 somatostatin receptor scintigraphy can be achieved by reducing false-positive results through clear understanding of the disease and circumstances that cause false-positive scans. An intrahepatic gallbladder may mimic hepatic metastasis by accumulating the radiotracer secreted into bile. An improvement in specificity aids in the proper management of patients. Planar images may identify only a few of the intrahepatic lesions, whereas SPECT may identify many more intrahepatic lesions (Fig. 4.2.3).\n\nFig. 4.4.3\n\nMetastatic gastrinoma. Planar images (top) in the anterior and posterior view show lesions in parapancreatic and paragastric lymph nodes. Normal activity is seen in the kidneys and spleen. Coronal view SPECT images show (bottom) multiple intrahepatic lesions in the liver, not shown by planar images (Post = posterior; Ant anterior slices)\n\n### 4.6.9 Radiation Dosimetry\n\nFollowing intravenous injection, In-111 OctreoScan is distributed diffusely throughout the body. Good hydration and frequent urination reduce the radiation dose to the organs [15]. A dose of 3 mCi In-lll pentetreotide is sufficient for planar imaging, and a dose of 6 mCi is required for a SPECT study. SPECT imaging has now become a routine standard for achieving the best sensitivity and specificity of the test. SPECT of the chest, abdomen, and pelvis are obtained routinely. Kidneys are the critical organs receiving the largest dose, followed by the spleen, urinary bladder, and liver [16]. The radiation to the kidneys from a 6-mCi (222-MBq) dose is 11.4 rad (115.44 mGy). The effective dose equivalent is 2.28 rem 6 mCi (22.2 mSv 222-1 MBq). Internalization of In-lll into the cytoplasm and nucleus of the cells from carcinoid and glucagonoma from In-lll pentetreotide raises the therapeutic potential for In-lll [11].\n\nSomatostatin receptor status of the tumor may have great impact on clinical diagnosis and management, especially in the selection of the type of therapy. Documentation of somatostatin receptor positivity can reduce the number and necessity of other diagnostic imaging tests. It is suggested that somatostatin receptor scintigraphy be used as the imaging modality of first choice in the case of gastrinoma because of its cost-effectiveness and also for its impact on patient management [17, 18]. Somatostatin receptor-positive tumors have been successfully treated medically with a combination of octreotide and prednisone [19]. Although somatostatin has been labeled with technetium-99m, it has not gained widespread clinical acceptance [20].\n\nReferences\n\n1.\n\nDeNardo GL, Stadalnik RC, DeNardo SJ, Raventos A. Hepatic scintiangiographic patterns. Radiology 1974;111:135-141PubMed\n\n2.\n\nQuinn JL lll. Nuclear medicine in gastroenterology. Hospital practice 1972, pp 115-122\n\n3.\n\nMcAfee JG, Ause RG, Wagner HN Jr. Diagnostic value of scintillation scanning of the liver. Follow-up of 1,000 studies. 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Somatostatin receptor scintigraphy: its sensitivity compared with that of other imaging methods in detecting primary and metastatic gastrinoma. A prospective study. Ann Intern Med 1996;125:26-34PubMed\n\n14.\n\nGibril F, Reynolds JC, Chen CC, Yu F, Goebel SU, Serrano J, Doppman JL, Jensen RT. Specificity of somatostatin receptor scintigraphy: a prospective study and effects of false-positive localization on management in patients with gastrinomas. J Nucl Med 1999;40539:553\n\n15.\n\nYamada T, Niinuma K, Lemaire M, Terasaki T, Sugiyama Y. Mechanism of the tissue distribution and biliary excretion of the cyclic peptide octreotide. J Pharmacol Expt Ther 1996;279:1357-1364\n\n16.\n\nStabin MG, Kooji PPM, Bakker WH, Inoue T, Endo K, Coveney J, deJong R, Minegishi A. Radiation dosimetry for indium-111-pentetreotide. J Nucl Med 1997;38:1919-1922PubMed\n\n17.\n\nTermanini B, Gibril F, Reynolds JC, Doppman JL, Chen CC, Stewart CA, Sutliff VE, Jensen RT. Value of somatostatin receptor scintigraphy: A prospective study in gastrinoma of its effects on clinical management. Gastroenterology 1997;112:335-347PubMedCrossRef\n\n18.\n\nJamar F, Fiasse R, Laners N, Pauwels S. Somatostatin receptor imaging with indium-lll-pentetreotide in gastroenteropancreatic neuroendocrine tumors: Safety, efficacy and impact on patient management. J Nucl Med 1995;36:542-549PubMed\n\n19.\n\nPalmieri G, Lastoria S, Colao A, Vergara E, Varrella P, Biondi E, Selleri C, Catalano L, Lombardi G, Bianco AR, Salvatore M. Successful treatment of a patient with a thymoma and pure red-cell aplasia with octreotide and prednisone. N Engl J Med 1997;336:263-265PubMedCrossRef\n\n20.\n\nVallabhajosula S, Moyer BR, Lister-James J, McBride BJ, Lipszyc H, Lee H, Bastidas D, Dean RT. Preclinical evaluation of technetium-99m-labeled somatostatin receptor-binding peptides. J Nucl Med 1996;37:1016-1022PubMed\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_5(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 5. Imaging and Quantification of Hepatobiliary Function\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nThe liver is one of the most frequently imaged organs in the body, using ultrasound, CT, MRI, or scintigraphy. The first three imaging techniques depend upon morphological changes to detect disease, whereas scintigraphy uses both morphological and physiological changes to discover liver pathology. Since physiological changes usually precede morphological alterations by several weeks or months, there is great potential for early diagnosis by scintigraphy, well before irreversible functional changes take place. Once very popular, imaging with radiocolloids now has been almost completely replaced by quantitative hepatobiliary functional imaging with Tc-99m HIDA [1].\n\n## 5.1 Hepatobiliary Imaging\n\nThe liver is one of the most frequently imaged organs in the body, using ultrasound, CT, MRI, or scintigraphy. The first three imaging techniques depend upon morphological changes to detect disease, whereas scintigraphy uses both morphological and physiological changes to discover liver pathology. Since physiological changes usually precede morphological alterations by several weeks or months, there is great potential for early diagnosis by scintigraphy, well before irreversible functional changes take place. Once very popular, imaging with radiocolloids now has been almost completely replaced by quantitative hepatobiliary functional imaging with Tc-99m HIDA [1].\n\nFunctional studies are obtained with the protocol outlined in Table 5.1.1, by using any one of several Tc-99m-HIDA agents that delineates the entire hepatobiliary tree as it travels from the hepatocyte to enter the duodenum (Chap. 3). Time of appearance of various normal biliary structures depends upon many factors, of which the tonus of the sphincter of Oddi plays the major role[2]. Following intravenous injection of Tc-99m-disofenin or Tc-99m-mebrofenin, the liver peak uptake is reached within 10 min, and the common bile duct is seen by 20 min. Both the gallbladder and small intestine are seen within 30 min in the great majority of patients (Table 5.1.2). Some patients show a reciprocal relationship between the time of appearance of the gallbladder and small intestine, with the intestine appearing late when the gallbladder appears early and vice-versa. Normally about 50% of gallbladders are seen within 15 min, 90% within 30 min, and 95% within 50 min. In the remaining 5%, gallbladders appear between 50 and 60 min after injection of the radiotracer [3].\n\nTable 5.1.1\n\nHepatobiliary imaging protocol\n\nPatient preparation | Patient should fast overnight (6-10 h optimal), minimum 4 h and maximum 24 h. Routine pre-emptying of the gallbladder with cholecystokinin is unnecessary and should be avoided. Pre-emptying may be needed only when the patient is fasting for longer than 24 h or on parenteral nutrition. When a pre-emptying protocol is used, one should wait for 60 min after CCK-8 to begin hepatic phase imaging with Tc-99m-HIDA\n\n---|---\n\nAgent | Tc-99m-mebrofenin or Tc-99m-disofenin\n\nDose | 2-4 mCi for serum bilirubin less than 2 mg%\n\n|\n\n5-8 mCi for serum bilirubin 3-10 mg%\n\n|\n\n10 mCi for serum bilirubin greater than 11 mg%\n\nChildren | 200 \u03bcCi kg -1 (minimum 1 mCi)\n\nPosition | Supine\n\nCamera | Large field of view dual-head gamma camera\n\nCollimator | Low energy, all purpose, parallel hole\n\nWindow | 20% at 140 keV photopeak\n\nMatrix | 128 \u00d7 128 \u00d7 16\n\nComputer data acquisition | Data acquisition is separated into two parts: (1) Hepatic phase and (2) gallbladder phase\n\nHepatic phase | Obtain 1 frame 2 s-1 for 30 frames (first minute) and 1 frame per min for 59 min (total 60 min). Obtain a right lateral view when gallbladder appearance is not certain at 60 min\n\nGallbladder phase | 1 frame per min for 30 min (61-90 min)\n\nBeginning at 5 min, CCK-8 is infused at a dose rate of 3 ng kg-1 min-1 for 10 min through an infusion pump\n\nTable 5.1.2\n\nTime of appearance of normal common bile duct, gallbladder, and small intestine\n\nAppearance of the common bile duct within 20 min | 100%\n\n---|---\n\nAppearance of the gallbladder: | within 15 min | 50%\n\n|\n\nwithin 30 min | 90%\n\n|\n\nwithin 50 min | 95%\n\n|\n\nwithin 60 min | 100%\n\nAppearance of the small intestine within 60 min | 80%\n\n### 5.1.1 Identification of Lobes, Segments, and Areas from Tc-99m-HIDA Images\n\nPhysiologically, the liver is divided into three lobes (including the caudate lobe), four segments, and eight areas (Fig. 5.1.1). The right lobe is divided into anterior and posterior segments, and the left lobe into medial and lateral segments. The right and left branches of the portal vein travel through the middle of the liver in opposite directions and divide the liver into superior and inferior areas. The right and left lobe each consists of four areas. The caudate lobe is situated along the superior margin of the posterior surface and is considered an independent lobe because of its unique blood supply. Its veins drain blood directly into the inferior vena cava.\n\nFig. 5.1.1\n\nBiliary anatomy. The bile ducts from the medial (nos. 4a, 4b) and lateral (nos. 2, 3) segments of the left lobe unite to form the left hepatic duct (LHD) and are seen separately in an anterior image. The ducts from the anterior (nos. 5, 8) and posterior (nos. 6, 7) segments of the right lobe join to form the right hepatic duct (RHD), and a right lateral view is required to separate them and their branches. RHD and LHD unite to form the common hepatic duct, which in turn forms the common bile duct after joining with the cystic duct from the gallbladder. The caudate lobe in the posterior view is usually obscured by photon attenuation by vertebral bodies\n\nDivision of the liver into physiologic right and left lobes is currently preferred over anatomic division because it serves as a useful boundary line during surgical resection of the liver. The lobes, segments, and areas are identified on an anterior, right lateral, and posterior view image, much like identifying the lobes and segments of a lung from a perfusion scan. Obstruction of a segment or an area bile duct is seen as bile stasis over the corresponding region, similar to identifying a lack of perfusion of a lobe or segment in a lung scan due to pulmonary embolism. The numbers shown in Fig. 5.1.1 are adopted from Table 1.1.1 of Chap. 1, as developed by different authors over the years. Healy and Schroy suggested a directional nomenclature after careful dissection of nearly 100 cadaver livers [4].\n\nThe medial segment of the physiologic left lobe (4a, 4b) forms a part of the right lobe when the liver is divided morphologically into right and left lobes on the basis of the attachment of the falciform ligament. The caudate lobe (no. 1) is posterior in location, but usually not seen on a posterior view image due to attenuation of photons by the vertebrae. The inferior area of the medial segment of the physiologic left lobe (4b) is often referred to as the quadrate lobe.\n\nThere are many variations in the manner in which the segmental ducts unite to form lobar ducts. The right hepatic duct is formed by the union of its anterior and posterior segmental ducts and the left hepatic duct by the union of its medial and lateral segmental ducts in 72% of the patients. The posterior segmental duct from the right lobe joins the left hepatic duct directly in 22% of patients. In the remaining 6%, the anterior segmental duct from the right lobe joins with the left hepatic duct directly [4]. It is very rare for the bile from either the medial or lateral segmental duct of the left lobe to drain directly into one of the segmental ducts of the right lobe (Fig. 1.1.4, Chap. 1).\n\nThe distal 0.5-1.5 cm of the right and left hepatic ducts, and the entire common hepatic and common bile ducts are outside of the liver parenchyma (extrahepatic), although they may appear as intrahepatic in an anterior view Tc-99m-HIDA study. This apparent depiction is due to extension of the inferior liver border in front of these ducts. The common hepatic duct usually measures about 2-7 cm, and the common bile duct is about 7-17 cm in length. The total length of the common duct (common hepatic plus common bile ducts) from its origin at the union of the right and left hepatic ducts to its termination into the duodenum varies from 10 to 20 cm. On a Tc-99m-HIDA image, however, total length of the common duct measures about 6.6 \u00b1 1.3 cm [3]. This foreshortening is caused by the position of the gamma camera (placed anteriorly) and its relationship with the direction of the common hepatic and common bile duct.\n\n### 5.1.2 Duct Asymmetry\n\nAsymmetry in appearance of the right and left hepatic duct is common on cholescintigraphy. Normally, the left hepatic duct appears much more prominent than the right in 55% of the subjects; the right hepatic duct is more prominent than the left in 13%, and both ducts appear as equal in 10% of the subjects (Table 5.1.3). In the remaining 22% of the subjects, neither duct is seen because of rapid bile flow through a lax sphincter of Oddi [3]. Several reasons are offered for duct asymmetry, including: (1) a more anterior location of the left liver lobe than the right, (2) a more anterior location of the left hepatic duct in comparison to the right hepatic duct, (3) a variant bile flow pattern where either the posterior or the anterior segmental duct from the right lobe drains bile directly into the left hepatic duct, (4) an undivided portion of the left hepatic duct is twice as long as the right duct and hence prone to tortuosity, impeding bile flow through it, and (5) a short right hepatic duct that lies in direct line with the common hepatic duct offers less impedance to bile flow through it. A slower bile flow allows relatively more counts to accumulate in the image to manifest as an apparent prominence.\n\nTable 5.1.3\n\nVariations in the appearance the normal right and left hepatic ducts\n\nLeft hepatic duct more prominent than right hepatic duct | 55%\n\n---|---\n\nRight hepatic duct more prominent than the left hepatic duct | 13%\n\nRight hepatic duct = left hepatic duct | 10%\n\nNeither duct seen clearly | 22%\n\nHepatobiliary studies are usually carried out 4-10 h after the fast, which establishes the basal state. Fasting less than 4 h or more than 24 h is avoided. Cholecystokinin reaches its lowest serum level during fasting, promoting maximum increase in the tonus of the sphincter of Oddi and maximum relaxation of the smooth muscle of the gallbladder wall. Acting together, both of these factors promote rapid filling of the gallbladder to its normal volume of 50 ml. The liver normally secretes about 600 ml bile per day (0.4 ml min-1) continuously, of which about 70% (0.3 ml min-1) enters the gallbladder, and the remaining 30% (0.1 ml min-1) enters the duodenum directly during fasting [5, 6]. A fully filled gallbladder is still able to accommodate this constant inflow of hepatic bile (0.3 ml min-1) by absorbing an equal volume of water through the wall. The lateral intercellular spaces between the columnar epithelial calls are kept widely open during fasting, allowing free passage of water from the gallbladder lumen into the interstitium. Fasting for less than 4 h results in either non-visualization or low Tc-99m-HIDA counts within the gallbladder because of the preferential flow of bile directly into the duodenum through a lax sphincter of Oddi.\n\n### 5.1.3 Effect of Food Intake on Uptake and Excretion of Tc-99m-HIDA\n\nPhysiological changes that take place soon after a meal affect some of the functional parameters obtained with Tc-99m-HIDA study and should be taken into account during data interpretation. In the immediate postprandial state, time to peak hepatic uptake of Tc-99m-HIDA decreases compared to studies performed after 6-10 h of fasting (Fig. 5.1.2). These changes are attributed to an increase in postprandial liver blood flow and faster extraction of Tc-99m-HIDA by the liver [7]. The radiotracer also clears from the liver parenchyma much more rapidly in the immediate post-prandial state, which is attributed to an increase in ductal bile flow induced by endogenous release of secretin, cholecystokinin, and other gastrointestinal hormones. Faster uptake combined with rapid liver clearance shifts the peak of the hepatic curve to an earlier time. The gallbladder does not fill in because of its contraction induced by post prandial release of endogenous cholecystokinin. When the gallbladder is the organ of interest for study, it is essential to maximize hepatic bile entry into it by fasting for at least 8-10 h, when both the absorption of water through the wall and the tonus of the sphincter of Oddi are at their peak [8]. Longer duration of fasting provides much better counting statistics because of preferential entry of Tc-99m HIDA mixed hepatic bile into the gallbladder [9]. Fasting longer than 24 h has an adverse effect on gallbladder filling because of the formation of bile sludge, which decreases water absorption through the wall.\n\nFig. 5.1.2\n\nEffect of feeding on the kinetics of Tc-99m-HIDA: In the fasting state (top), increased tonus of the sphincter of Oddi diverts more of the hepatic bile into the gallbladder than intestine. There is residual radioactivity in the liver at 60 min. In the post-prandial state (bottom), there is more rapid hepatic uptake and excretion, shifting the peak of the curve to an earlier time, and very little residual radioactivity remains in the liver beyond 15 min. The gallbladder does not fill in because of a lax sphincter\n\n### 5.1.4 Quantification of Liver Function\n\nBesides providing an excellent morphology of the entire hepatobiliary system, imaging with Tc-99m-HIDA enables simultaneous quantification of liver and gallbladder function as an integral part of imaging. Traditionally, the liver function is monitored by obtaining the serum level of various substances produced by the liver, including albumin, bilirubin, alkaline phosphatase, and transaminase. Serum levels are indirect indicators of liver function and are influenced by the status of the cardiac and renal function, and hence may not reflect accurately the hepatocyte function in the presence of heart or kidney failure. Measurement of radioactivity emitted from the liver serves as a direct indicator of the physiological status of the hepatocyte, which is unaffected by renal function [10]. The biokinetic pathway of Tc-99m-HIDA involves several steps before its elimination from the liver into the small intestine (Chap. 3.1). Quantification of function allows differentiation of hepatocyte from biliary disease. New FDA-approved software called Krishnamurthy Hepato-Biliary Software (KHBS) is now available for quantification of liver and gallbladder function and can be loaded onto a PC. It has seven main components for analysis: Hep-function, HEP-segments, HEP-static, GB-function, GB-segments, GB-static, and HPS (hepatopulmonary syndrome). Setup block allows entering local parameters and the option to edit images. GB-phase parameters are entered for CCK-8 and fatty meal stimulation (Fig. 5.1.3). Hepatic extraction fraction and excretion half time are two of the functional parameters measured routinely as an integral part of hepatobiliary imaging [1]. Hepatic extraction fraction measures the integrity of the basolateral border, and excretion halftime reflects intracellular transit from the basolateral border to the canalicular border, secretion into canaliculi, and flow through the small and large bile ducts.\n\nFig. 5.1.3\n\nSetup and GB-input data for KHBS. Activation of the setup block provides the option to enter desired parameters pertaining to local need. User login, data location (or study location), image size, color option, hospital details, font size, agents, and normals, etc., are entered into the SetUp menu once for permanent storage. Images can be resized with magnification or minimizing buttons. Annotation and move option are available for image editing for report or for power point presentation. Gallbladder parameters are changed depending upon CCK-8 or fatty meal stimulus\n\n### 5.1.5 Hepatic Extraction Fraction (HEF)\n\nStudies are acquired as the patient lies in the supine position underneath a large field of view dual-head gamma camera, and the data are collected as described in Table 5.1.1. Both Hep-phase and gallbladder-phase data are transferred from the gamma camera terminal to the PC where KHBS is located. By activating the browse button, data location is identified, and both phase image data of a patient are uploaded onto KHBS. Report block at the top creates desired images for interpretation. The perfusion, HEP-phase, GB-phase, and multiple image presentation formats are changed to meet local preference. The perfusion part of the study is analyzed subjectively using 30 frames of the first minute data. HEP-function block at the top of the screen is then activated. The software sums up the first 30-frame perfusion data, labels it as frame one, and uses the remaining 59 min frames (total 89 frames) for assessment of the liver function. When interrupted data are acquired, instead of continuous data, the HEP-static format is activated for image presentation.\n\nAfter activating the HEF-function block at the top, seven regions of interest (ROI) are drawn, one each over the heart, background (spleen), right lobe, left lobe, gallbladder, stomach, and intestine, as shown at the bottom of the screen (Fig. 5.1.4). Each ROI consists of at least 50-75 pixels. Heart and background regions are usually drawn on the first minute frame. The heart ROI may include one or both ventricles. Background is drawn over the spleen. Raw data curve from each ROI is displayed instantly after completion of the ROI. Often intestinal activity may overlay the spleen region in later images and give a false background. Normally, the spleen curve parallels the heart curve when there is no intestinal overlap. A different background region is chosen if there is intestinal overlap of the spleen. The liver ROIs include only the parenchyma of the superior right lobe and left lobe, and are usually drawn on the 10- to 20-min frames when ducts become visible. Care is taken to avoid inclusion of heart or bile ducts with the liver ROI. Superior and right lateral liver margins, and gallbladder, which move in and out of the ROI during deep inspiration or upon coughing, are excluded from the liver ROIs. The spleen, which appears clearly in the first minute summed frame, disappears in the later frames. Stomach, gallbladder, and intestinal ROIs are usually drawn on the last (60-min) frame. After selecting all ROIs, the entire 60-min study is reviewed in cine mode to ensure separation of the regions of interest. Regions of interest are altered if there is superimposition by other structures. After checking all seven ROIs for accuracy, the HEP-function result block is activated to obtain the result (Fig. 5.1.5).\n\nFig. 5.1.4\n\nHepatic phase ROIs. Heart and background (Bkg) regions of interest are chosen on the first-minute frame. Right and left lobe ROIs are drawn over the liver parenchyma in 10-20-min frame by excluding bile ducts. Gallbladder, stomach, and intestinal ROIs are drawn on the last 60-min frame. Curves on the right represent the same color ROIs on the left\n\nFig. 5.1.5\n\nResults of Hep-phase: Hepatic extraction is calculated by deconvolutional analysis using the heart for the input and liver for the output function. The excretion half time is measured by non-linear least-square fit using spleen as the background. Excretion T 1\/2 is slightly longer for the left than the right liver lobe (statistically not significant). Basal differential hepatic bile flow and DGBR values are shown on the right. Middle panel shows the ROIs for each structure to check for accuracy\n\nThe software first smoothes the liver curves and then subjects them to deconvolution. The liver true response curve is deconvoluted from the input (heart) and output (liver) curve using Fourier transformation. Since the hepatic curve represents the sum of counts from the hepatocyte and hepatic blood pool, the deconvolution corrects for hepatic blood pool [11]. A smoothed decreasing long tail in the shape of one-half of a cosine wave is appended to the end of the input and output curves. This long tail avoids errors due to abrupt data termination at 30 min. The append begins at the 30th frame and falls to zero at the 128th added frame, giving approximately six times the duration of acquisition [12]. Duration of such a length is a minimum required to avoid artifacts in the deconvolution process. The formula for the appended tail in frame i of the heart or liver curve is given by:\n\n![$$\n{\\\\rm Tail} = 0.5 \\\\times {\\\\rm AMP} \\\\times \\\\left\\( {{\\\\rm Cos\\[}\\\\pi  \\\\times \\\\left\\( {{\\\\rm i - 30}} \\\\right\\)\\\\left\\( {128 - 30} \\\\right\\)\\] + 1} \\\\right\\)$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ1_5.gif)\n\nwhere i = 30-128, and AMP is the heart or liver curve value (counts per pixel) at frame 30.\n\nThe hepatic extraction fraction (HEF) is the ratio of the Y intercept of the exponential fit to the maximum data point in the liver response curve. The exponential fit is by linear least square, working in reverse from 30 min to the first frame, which visually departs significantly from the exponential fit. HEF is calculated by the following equation.\n\n![$$\n{\\\\rm Hepatic}\\\\,{\\\\rm Extraction}\\\\,{\\\\rm Fraction}\\\\left\\( {{\\\\rm HEF}} \\\\right\\) = \\\\frac{{{\\\\rm{Y\\\\ intercept\\\\ exponential\\\\ fit\\\\ liver\\\\ response\\\\ curve}}}}{{{\\\\rm Y - MAX}\\\\,{\\\\rm data}\\\\,{\\\\rm value\\\\ liver\\\\ response\\\\ curve}}}$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ2_5.gif)\n\nThe deconvolutional technique separates hepatocyte counts from the hepatic blood pool counts. A deconvoluted liver curve, therefore, represents a hypothetical true response as though Tc-99m HIDA was injected directly into the hepatic artery or portal vein [11, 12]. The normal hepatic extraction fraction with Tc-99m-mebrofenin and Tc-99m-disofenin varies from 92% to 100%. The extraction fraction decreases as the hepatocyte function decreases. Functional changes show good correlation with Child's classification (Table 5.1.4) of the severity of liver disease [13].\n\nTable 5.1.4\n\nChild's classification of liver disease\n\nClinical and laboratory findings | Child's class\n\n---|---\n\nA | B | C\n\nAscites | None | Controlled | Uncontrolled\n\nNeurological Findings | None | Minimal | Advanced\n\nNutrition | Excellent | Good | Poor\n\nBilirubin (mg%) | <2.0 | 2.0-3.0 | >3.0\n\nAlbumin (gm%) | >3.5 | 3.0-3.5 | <3.0\n\n### 5.1.6 Hepatic Excretion Half Time\n\nAfter the uptake, Tc-99m-HIDA is secreted by the hepatocytes in its native state into bile canaliculi, where it mixes thoroughly with the canalicular bile. Secretion of radioactive bile into canaliculi serves as the direct in vivo, non-invasive method of radiolabeling hepatic bile, without altering basal liver physiology. After mixing with the canalicular bile, Tc-99m- HIDA follows the path taken by the hepatic bile. Excretion half time is a measure of how fast the radiotracer is secreted by the hepatocyte into bile canaliculi and how rapidly it flows through the intrahepatic bile ducts to enter the common hepatic duct and common bile duct before entering the gallbladder and duodenum. Measurement of excretion half time uses the liver and spleen ROI (Fig. 5.1.4). Since both the liver and spleen receive their arterial blood supply from a common source, the celiac artery, the spleen serves as the most appropriate organ for subtraction of the blood background. The background subtracted liver curve is modeled by an uptake and excretion compartment and is given by the following equation [12].\n\n![$$\n{\\\\rm L}\\\\left\\( {\\\\rm t} \\\\right\\){\\\\rm  = k}\\\\left\\( {{\\\\rm e}^{{\\\\rm  - 0}{\\\\rm .693t\/TE}} {\\\\rm  - e}^{{\\\\rm 0}{\\\\rm .693t\/TU}} } \\\\right\\)$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ3_5.gif)\n\nwhere L(t) = background corrected liver counts\/pixel at time t,\n\nk = a constant of the model, TE = excretion effective T \u00bd, and TU = uptake effective T\u00bd.\n\nAlthough the hepatic phase data are collected for 60 min, calculation of HEF uses only the first 30-min data, whereas excretion T 1\/2 uses all 60-min data points. Patient values are shown along with normal range (Fig. 5.1.5). Although in most patients with normal liver function, both HEF and T 1\/2 excretion can be obtained by collecting total data only for 45 min, in patients with moderate to severe liver disease (serum bilirubin levels above 5 mg%), it is necessary to collect data for a minimum of 60 min to obtain a reliable excretion half time [13]. Normal T \u00bd excretion values obtained with KHBS using Tc-99m mebrofenin range from 6 to 38 min for the right lobe and from 6 to 58 min for the left lobe (Fig. 5.1.5). A longer excretion T 1\/2 value of the left lobe is due to influence of bile within the left hepatic duct, which is more superficial than the right hepatic duct. Both HEF and T 1\/2 excretion values are highly reproducible, both within and between institutions, and among different technologists [14]. HEF values remain within a normal range in early biliary disease, but decrease in hepatocellular disease. When the liver disease is severe and progressive, the HEF value begins to decrease, and excretion half time increases in both hepatocyte and biliary disease [15]. Both parameters provide a reliable measure of the severity of hepatobiliary diseases, irrespective of the etiology (Fig. 5.1.6).\n\nFig. 5.1.6\n\nRelationship between hepatic extraction fraction (HEF) and excretion half time in health and disease. HEF remains near normal level in early disease and begins to decrease as bilirubin increases (a). Excretion half time increases from the very beginning and raises further as serum bilirubin value increases (b). HEF and excretion half times show an inverse relationship (c)\n\n### 5.1.7 Effect of Bile Duct Obstruction on Liver Function\n\nIn the case of hyperacute or acute total obstruction of the common bile duct, HEF values remain within a normal range for 4-5 days, as long as the serum bilirubin level remains below 8 mg% (Fig. 5.1.6A). HEF begins to decrease when obstruction persists and the serum bilirubin level rises above 8 mg% [16]. Excretion half time, on the other hand, increases from the very beginning (Fig. 5.1.6B). Prolongation of the excretion half time in biliary obstruction is due to bile stasis within the canaliculi and intrahepatic bile ducts. In congenital biliary atresia, a condition resulting from extrahepatic bile duct obstruction, HEF remains high for about 2 months after birth and then begins to decrease if the obstruction is not relieved [17, 18]. Excretion half time and HEF show an inverse relationship; as the HEF decreases, excretion half time increases, and vice-versa (Fig. 5.1.6C). High serum bilirubin levels decrease HEF by two mechanisms: first, it displaces Tc-99m-HIDA from its carrier protein, albumin, and second, bilirubin competes with Tc-99m-HIDA for the hepatocyte uptake by receptor mediated endocytosis.\n\n### 5.1.8 Basal Differential Hepatic Bile Flow\n\nThis parameter is obtained as a part of HEP-phase data analysis as shown in Figs. 5.1.3 and 5.1.4. After leaving the liver, the hepatic bile in the basal state enters the gallbladder and duodenum, with volume depending upon the tonus of the sphincter of Oddi. Normal gallbladders are usually visualized within 60 min, and 10-100% of the hepatic bile enters the gallbladder. In about 20% of normal subjects, all of the hepatic bile enters the gallbladder and none the intestine, so that intestinal bile entry at 60 min remains at 0%. Intestinal bile entry, therefore, normally ranges from 0 to 100%. Longer duration of fasting increases the sphincter tone and diverts more of the hepatic bile to enter the gallbladder than intestine. Total bile produced during 60 min of hepatic phase imaging is obtained by adding gallbladder and intestinal counts at 60 min. Percent of the hepatic bile entering the gallbladder is obtained by the following formula.\n\n![$$\\\\begin{array}{rl}\n{\\\\rm Hepatic\\\\ bile\\\\ flow\\\\ into\\\\ gallbaldder}\\\\,\\\\left\\( {\\\\rm \\\\% } \\\\right\\) & = \\\\frac{{{\\\\rm Gallbladder}\\\\,{\\\\rm total}\\\\,{\\\\rm counts}\\\\,{\\\\rm at 60 min } \\\\times {\\\\rm 100\\\\_}}}{{{\\\\rm Gallbladder\\\\ total\\\\ counts\\\\ at\\\\ 60\\\\ min  +  Intestinal\\\\ total\\\\ counts\\\\ at\\\\ 60\\\\ min}}}\\\\\\\\\n{} &  = \\\\frac{{49,700 \\\\times 100}}{{49,700 + 50,300}} = 49.7\\\\%\n\\\\end{array}$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ4_5.gif)\n\n![$$\\\\begin{array}{rl}\n{\\\\rm Hepatic\\\\ bile\\\\ flow\\\\ into\\\\ Intestine}\\\\,\\\\left\\( {\\\\rm \\\\% } \\\\right\\) & = \\\\frac{{{\\\\rm Intestinal}\\\\,{\\\\rm total}\\\\,{\\\\rm counts}\\\\,{\\\\rm at 60 min } \\\\times {\\\\rm 100\\\\_}}}{{{\\\\rm Gallbladder\\\\ total\\\\ counts\\\\ at\\\\ 60\\\\ min  +  Intestinal\\\\ total\\\\ counts\\\\ at\\\\ 60\\\\ min}}}\\\\\\\\\n{} & = \\\\frac{{50,300 \\\\times 100}}{{49,700 + 50,300}} = 50.3\\\\%\n\\\\end{array}$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ5_5.gif)\n\n### 5.1.9 Basal Duodeno-Gastric Bile Reflux (Basal-DGBR)\n\nThis parameter is obtained as a part of HEP-phase data analysis as shown in Figs. 5.1.4 and 5.1.5. Bile entering the duodenum during fasting flows forward to enter the jejunum. Normally, there is no bile reflux into the stomach. The stomach ROI is selected carefully by avoiding inclusion of the left lobe of the liver or duodenal loop in the stomach ROI, both of which will result in a falsely high DGBR. Normal DGBR values up to 8% represent mostly background activity. The DGBR value should be ignored when there is no visible bile entry into the intestine. A high DGBR value in such cases represents only the background activity in the stomach and intestine. DGBR values are checked with the images to confirm bile reflux. DGBR values higher than 8% are usually associated with visible bile reflux into the stomach (Fig. 5.1.5)\n\nThe DGBR value is obtained by dividing total stomach counts at 60 min by the sum of stomach and intestinal counts at 60 min.\n\n![$$\n{\\\\rm DGBR}\\\\left\\( {\\\\rm \\\\% } \\\\right\\) = \\\\frac{{{\\\\rm Stomach\\\\ counts\\\\ at\\\\ 60\\\\ min} \\\\times {\\\\rm 100}}}{{{\\\\rm Stomach\\\\ counts +  Intestinal\\\\ counts\\\\ at\\\\ 60\\\\ min}}}$$](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ6_5.gif)\n\nBoth differential hepatic bile flow and basal DGBR values are shown on the right side of the Hep-function result along with normal values (Fig. 5.1.5). Three figures in the middle show the ROIs for each region to enable the physician to check for accuracy. Since radiolabeled bile has to enter the duodenum to have any reflux into the stomach, all DGBR values are ignored when there is no bile entry into the intestine. Occasionally, DGBR may empty in the middle of data acquisition; in such an instance, the peak reflux value is obtained by changing the frame number as shown in the option block at the bottom (Fig. 5.1.5).\n\n### 5.1.10 Quantification of Gallbladder Function\n\nA normal gallbladder measures about 50 ml in volume and requires 3-4 h to fill to its full capacity after complete emptying. Since the gallbladder is already filled to its full capacity after 4-6 h of fasting, two mechanisms are responsible for accommodating a constant inflow of 0.3 ml min-1 of the hepatic bile: (1) absorption of water through the gallbladder wall and (2) an increase in tonus of the sphincter of Oddi. During fasting, the epithelium of the gallbladder wall absorbs water from the lumen through widely opened lateral intercellular spaces between the columnar cells of the mucosa. Bile salts, bile pigments, cholesterol, and other bile consti-tuents are not absorbed. This process of selective absorption of water resulting in a higher concentration of solutes is called the concentration function of the gallbladder [8]. After an overnight fast, the mean pressure within the sphincter of Oddi is 15 mmHg, 12 mmHg in the common bile duct, and 10 mmHg inside the gallbladder. Because of the pressure difference at various levels, hepatic bile follows the path of the least resistance and enters the gallbladder.\n\nBile from the common hepatic duct enters the gallbladder in a step-wise fashion and corresponds to the phasic waves passing through the sphincter of Oddi [19]. Radioactive hepatic bile first enters the gallbladder along its central long axis and moves slowly laterally towards the wall [20]. After entering the central long axis, radiolabeled bile usually takes at least 30 more minutes to reach the wall (Fig. 5.1.7). One should wait till the entire gallbladder bile is radiolabeled before embarking on measurement of its emptying. For example, when a gallbladder appears first at 45 min of the hepatic phase imaging, the measurement of the ejection fraction should start at 75 min (45 + 30 = 75 min) to allow sufficient time for radiolabeling of the entire gallbladder bile. After completion of 60 min of HEP-phase data collection, in such an instance, the patient is made to wait for an additional 15 min before starting GB-phase imaging with cCK-8. The ejection fraction value would be falsely high if partially radiolabeled gallbladder bile were measured.\n\nFig. 5.1.7\n\nFilling of the gallbladder: Gallbladder fills from inside out. Radiolabeled bile first enters the gallbladder at 12 min along its central long axis and moves laterally to reach the wall at 48 min, taking a total of 36 min. One must wait for all of the gallbladder bile to be radiolabeled before embarking on study emptying. Partially radiolabeled gallbladder bile may result in overestimation of its ejection fraction\n\nFig. 5.1.7\n\nGallbladder bile volume and Tc-99m HIDA count relationship. A rubber balloon filled with 50 ml water and Tc-99m representing the gallbladder is placed inside a container placed underneath a gamma camera. One milliliter of water is removed at a time through the syringe, and counts are taken with the gamma camera after each withdrawal (a). Counts show a linear relationship with the bile volume (b). Ejection fraction measured by volume and Tc-99m-HIDA count methods show a perfect linear (c) relationship [23]\n\n### 5.1.11 Gallbladder Ejection Fraction\n\nMeasurement of gallbladder emptying is clinically popular primarily due to the availability of a simple, non-invasive, and reliable quantitative diagnostic test. In the past, gallbladder emptying was measured with oral cholecystogram by applying a technique called the sum of the cylinder method, which was introduced in 1949 by De Silva [21]. On the oral cholecystogram, the gallbladder image is divided into series of small cylinders, and the volume of each cylinder is computed by applying a mathematical formula for the volume of a cylinder. By summing the volume of all cylinders, gallbladder total volume is obtained. A slightly modified version of sum of the cylinder method has been adopted for measuring the gallbladder volume with the ultrasound [22]. Both oral cholecystogram and ultrasound are geometric techniques and are based on the assumption that the gallbladder is a cylinder before, during, and after contraction. It is frequently seen that the gallbladder is not a cylinder before contraction, and it often changes its shape during and after contraction. A count-based, non-geometric technique overcomes these shortcomings [23]. Cholescintigraphy with Tc-99m-HIDA is a non-geometric method and enables precise measurement of both the ejection fraction and ejection rate. The scintigraphic method uses Tc-99m HIDA counts to represent bile volume as there is a direct linear relationship between bile volume and counts within the gallbladder (Fig. 5.1.8).\n\nFig. 5.1.8\n\nGallbladder phase ROIs: Gallbladder, bkg, CHD, and CBD ROIs are drawn usually on the first-minute frame, stomach, and intestinal ROIs on the last-minute frame. Curves on the right represent the counts from regions of the same color\n\n### 5.1.12 Data Collection\n\nThe gallbladder phase data are acquired using the protocol outlined in Table 5.1.1 at 1 frame per minute for 30 min when CCK-8 is used as the stimulus for contraction. The data are collected for a minimum of 60 min, preferably 60-120 min, when a fatty meal is used as the stimulus to induce gallbladder contraction. A magnification factor of 1.2-2.4 may be used during data acquisition. The gamma camera angle is changed to a degree that maximally separates the CBD from the gallbladder and the duodenum [9]. Infusion of CCK-8 is begun at 3 min at a dose rate of 3 ng kg-1 min-1 and infused for 10 min through an infusion pump. The first 5 min counts prior to CCK-8 infusion represent the basal volume of the gallbladder.\n\n### 5.1.13 Dose Rate and Duration of Cholecystokinin Infusion\n\nIn the United States, a fragment of the hormone cholecystokinin is available for clinical use as CCK-8 (Kinevac). In Europe and other countries, the entire molecule with 33 or 39 amino acids is made available. The hormone is prepared according to the manufacturer's instructions in the package insert (Kinevac, Sincalide, Bracco Diagnostics, Princeton NJ). Volume is adjusted with saline to achieve the desired dose rate and duration of infusion. Selection of a proper dose and dose rate within the physiologic range is critical for accurate results. Duration of the infusion is a matter of personal preference. We routinely use a dose rate of 3 ng kg-1 min-1 and infuse for a total duration of 10 min through an infusion pump. Before the test begins, the patient is given instructions to raise the hand as soon as the symptoms begin and to raise the hand again when symptoms abate. The patient is asked to grade pain intensity as mild, moderate, or severe. The technologist notes down the time of onset, total duration, and intensity of pain on a work sheet for interpretation.\n\n### 5.1.14 Effect of Non-Physiologic Dose of Cholecystokinin\n\nThe package insert (Sincalide, Kinevac by Bracco Diagnostics, Princeton, NJ) suggests a CCK-8 dose of 0.02 \u03bcg kg-1 (20 ng kg-1) infused over 30 or 60 s. This dose rate was originally developed in 1970s with the use of oral cholecystogram, and studies since have demonstrated that this dose rate is too large for cholescintigraphy and often produces a falsely low ejection fraction in 20-26% of normal subjects [24-28]. A change in dose rate and\/or duration of infusion produces different values for the gallbladder ejection fraction (Table 5.1.5). Ideally, one should establish local values when the technique chosen is different from the published literature [29].\n\nTable 5.1.5\n\nEffect of dose and duration of CCK-8 infusion on gallbladder ejection fraction (mean \u00b1 S.E)\n\nCCK-8 dose rate (ref) | Duration of CCK-8 infusion\n\n---|---  \n|\n\n3 min | 30 min | 60 min\n\n0.5 ng kg-1 min-1 [27] | - | 79.3% \u00b1 6.9% | 91.3% \u00b1 5.3%\n\n3.3 ng kg-1 min-1 [23] (10 ng kg-1 3 min-1) | 59.4% \u00b1 4.0% | - | -\n\n5.0 ng kg-1 min-1[27] | - | 82.4% \u00b1 6.7% | 94.0% \u00b1 5.1%\n\n### 5.1.15 Calculation of Gallbladder Ejection Fraction\n\nAfter uploading the gallbladder phase data onto KHBS, the gallbladder function button at the top is activated. Six ROIs are drawn: gallbladder, background, common hepatic duct (CHD), common bile duct (CBD), stomach, and intestine (Fig. 5.1.8). The background ROI is drawn over the liver, superior and lateral to the gallbladder. The CHD region includes both the right hepatic duct and left hepatic ducts in the form of a letter T or Y. The gallbladder, background, CHD, and CBD regions are drawn using the first frame. Stomach and intestinal ROIs are drawn on the last frame. Net gallbladder counts are obtained after subtraction of the background counts and corrected for decay [9]. Region of interest over the CHD enables identification of possible bile reflux in the presence of critical obstruction of the common bile duct. The software also calculates post-CCK-8 duodeno-gasrtic bile reflux (Fig. 5.1.9).\n\n![$$\n{\\\\rm Gallbladder\\\\ ejection\\\\ fraction}\\\\left\\( {{\\\\rm GBEF}} \\\\right\\)\\\\left\\( \\\\%  \\\\right\\) = \\\\frac{{{\\\\rm GB\\\\ Peak\\\\ counts-First\\\\ minimum\\\\ GB\\\\ counts }\\\\left\\( {\\\\rm B} \\\\right\\) \\\\times 100}}{{{\\\\rm GB\\\\ Peak\\\\ counts}}}\n$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ7_5.gif)\n\n![$$\n{\\\\rm GBEF = }\\\\frac{{{\\\\rm A--B} \\\\times {\\\\rm 100}}}{{\\\\rm A}}$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ8_5.gif)\n\n  * Latent period (LP) = Time from beginning of CCK - 8 infusion to beginning of gallbladder emptying\n\n  * Ejection period (EP) = Time from beginning of gallbladder emptying to the first minimum counts\n\n  * Normal values for 10 min infusion of 3ng \/ kg \/ min of CCK - 8\n\n  * Ejection rate (ER) = % Ejection fraction \/ Ejection period = % \/ minute\n\n  * Normal values for 10 min infusion of 3ng \/ kg \/ min of\\ CCK - 8\n\n  * Latent period (LP)\n\n  * Ejection period (EP) = 7-19 min\n\n  * Ejection period (EF) \u2265 50%\n\n  * Ejection rate (ER) \u2265 3.5% \/ min\n\nFig. 5.1.9\n\nGB-phase results. Occasionally gallbladder counts may increase just before it begins to empty (first vertical bar). First minimum counts represent the end of gallbladder emptying (second vertical bar). Ejection fraction (EF) is calculated by dividing counts emptied between two vertical bars by counts at the first vertical bar. Time between two vertical bars is the ejection period (EP). Ejection rate (ER) is calculated by dividing % EF by EP. Post-CCK DGBR is obtained by dividing stomach counts by the sum of stomach and intestinal counts\n\nPost-CCK-8 duodeno-gastric bile reflux is calculated by dividing stomach counts by the sum of stomach and intestinal counts. Normally cholecystokinin acts on the gallbladder, promoting its contraction and emptying. It acts on the pylorus sphincter of the stomach and prevents bile reflux. Occasionally, one may find a large amount of DGBR with CCK-8 that may explain a patient's symptoms.\n\n### 5.1.16 Ejection Fraction with Fatty Meal Stimulation\n\nFatty meal stimulates the release of endogenous cholecystokinin from the endocrine cells lining the mucosa of the duodenum, jejunum, and upper ileum. It may take as long as 6-26 min after a meal for the endogenous CCK level to rise above the serum threshold to initiate gallbladder contraction and emptying. Once the gallbladder begins to empty, it continues ejection for 1-3 h post-meal. The duration of data collection with the fatty meal, therefore, should be for a minimum of 60 min, preferably for 120 min. The data are collected at 1 frame per min for 60 or 120 min (Fig. 5.1.10). A standardized test meal (8 oz 70 kg-1 body weight) of known nutritional contents and caloric value is ingested at 5 min, such that a baseline count prior to the meal represents basal gallbladder volume [30, 31]. Mean (\u00b1 SD) ejection fraction for 1 h fatty meal stimulation study is 50% \u00b1 20%. Individual values can be as low as 32%. In the same group of normal subjects, 10 min infusion of 3 ng kg-1 min-1 of CCK-8 produces a mean EF of 70% \u00b1 17% [32]. In the USA, the nutritional value of Half and Half (H & H) milk varies from city to city (Table 5.1.6). One must standardize the technique and establish local normal values.\n\nFig. 5.1.10\n\nGallbladder emptying with fatty meal stimulation: Meal is ingested at 10 min. There is a latent period of 16 min before the gallbladder begins to empty. Ejection fraction is 73%, ejection period of 34 min, and an ejection rate 2.1% per min [31]\n\nTable 5.1.6\n\nVariations in nutritional contents and total calories of Half-and-Half milk in six US cities [33]\n\nNutrient content per 30 ml | Portland, OR (Alpenrose) | New York, NY (America choice) | Detroit, MI (CFBerger) | Jackson, MS (Dairy fresh) | Los Angeles, CA (Altadena) | Miami, FL (McArthur dairy)\n\n---|---|---|---|---|---|---\n\nTotal fat (g) | 3.0 | 3.0 | 3.0 | 3.5 | 3.0 | 3.0\n\nCholesterol (mg) | 10.0 | 15.0 | 10.0 | 15.0 | 15.0 | 15.0\n\nCarbohydrate (g) | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0\n\nProtein (g) | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0\n\nSodium (mg) | 10.0 | 20.0 | 10.0 | 20.0 | 15.0 | 15.0\n\nTotal calories (30 ml) | 40.0 | 40.0 | 35.0 | 40.0 | 40.0 | 40.0\n\n### 5.1.17 Data Analysis\n\nSoftware is adapted for measurement of the ejection fraction with fatty meal. Two regions of interest, one over the gallbladder and the other over the liver as background, are drawn. Due to the slow rate of bile emptying, CBD and CHD are not well delineated with the fatty meal stimulation. The value of the gallbladder ejection fraction with a fatty meal is reported with a reference to its total duration of measurement, i.e., at 60 min or 120 min, post-meal. Both a 30-min continuous infusion of CCK-8 or a 60-min post fatty meal appear to produce a mean ejection fraction of about 80% (Table 5.1.7). The latent period (the time from meal ingestion to beginning of gallbladder emptying) is variable depending upon the time for release of endogenous CCK [28]. A delay in gastric emptying may also cause late release of cholecystokinin. Since there are no CCK-secreting cells in the esophagus and stomach [33], hormone-induced gallbladder emptying does not begin until food reaches the duodenum. The gallbladder ejection fraction is low in patients with gallstones and diabetes [34]. Obese diabetics have a much more pronounced reduction in EF compared to non-obese diabetic patients [35]. A cephalic phase emptying because of cholinergic nerve stimulation, independent of hormonal action may induce gallbladder emptying [36]. A comparison of fatty meal stimulation with cholecystokinin in the same group of normal subjects showed wide variation in the ejection fraction [32]. The fatty meal ejection fraction at the end of 1 h ranges from 23 to 91% (mean 54%), whereas a 10-min infusion of CCK-8 in the same group of normal subjects produces values ranging from 37 to 91% (mean 76%). ROI over the stomach region enables calculation of duodeno-gastric bile reflux.\n\nTable 5.1.7\n\nEffect of type of meal and duration of emptying on gallbladder ejection fraction (%)\n\nMeal type | Meal volume | (Mean \u00b1 SE)\n\n---|---|---\n\nDuration from meal ingestion\n\n30 min | 60 min | 120 min\n\nHalf & Half [28] | 8 oz 70 kg-1 | - | 64.4 \u00b1 6.7% | -\n\nLipomul [27] | 15 ml | 49.8 \u00b1 5.2% | 71.7 \u00b1 3.6% | 77.8 \u00b1 2.0%\n\n### 5.1.18 Gallbladder Segmentation\n\nOften a septa or fold can divide the gallbladder chamber into two separate compartments, proximal and distal (Fig. 5.1.11). Ultrasound studies usually show the length, thickness, and position of the septa inside the gallbladder. Carefully done histopathological examination (Fig. 5.1.12) confirms the folds. Recognition of such compartments is essential during hepatic phase imaging as the two compartments may show a different degree of filling and emptying. Radiolabeled hepatic bile enters the proximal compartment first, followed 5-10 min later by entry into the distal compartment. Two compartments are easy to recognize with cholescintigraphy in the early images as the radiolabeled bile enters the gallbladder (Fig. 5.1.13). It becomes much more difficult to recognize it in late images as bile radioactivity equilibrates in both compartments. For calculation of the gallbladder segmental function, GB-Seg block at the top is activated, and the patient data are loaded. One ROI is drawn around the entire gallbladder and another between the two compartments (Fig. 5.1.14). The program calculates the total EF and EF for each compartment separately (Fig. 5.1.15). The distal compartment usually empties poorly because of septa acting as a one-way valve. Because these findings are new and not well appreciated, ultrasonographers should mention the presence of the septa or fold in their ultrasound report of the gallbladder to draw the attention of clinicians who may then request a Tc-99m HIDA study for further evaluation of abdominal pain [37].\n\nFig. 5.1.11\n\nGallbladder segmentation. CT scan shows a constriction near the neck dividing the gallbladder into proximal (GB-neck) and distal (GB-body) compartments. Gallstones are layered in the distal compartment\n\nFig. 5.1.12\n\nGallbladder fold or septa. A transverse fold divides the gallbladder into proximal and distal compartments. This fold acts in vivo as a barrier between two segments and reduces bile emptying from the distal segment\n\nFig. 5.1.13\n\nFilling of a segmental gallbladder (GB). The proximal segment appears first at 20 min (top row right), and distal segment begins to appear at 25 min (middle row left). Left hepatic duct (LHD) is seen much more prominently than the right hepatic duct (RHD). The distal segment is much larger in size than the proximal segment (bottom row right)\n\nFig. 5.1.14\n\nRegions of interest for gallbladder segmentation. One region is drawn around the entire gallbladder and the other over the septum that divides it into proximal and distal compartments. Counts and curves from the proximal and distal compartment and total gallbladder are displayed\n\nFig. 5.1.15\n\nResults of gallbladder segmentation. By drawing one region of interest around the entire gallbladder and another over the fold or septum (bottom), the software calculates the ejection fraction for each compartment separately along with the gallbladder total ejection fraction (top)\n\n## 5.2 Measurement of Hepatic Arterial vs. Portal Venous Blood Flow\n\nThe liver has a dual blood supply through the hepatic artery and portal vein. The hepatic artery supplies about 400 ml of arterial blood per min at 100-120 mmHg systolic blood pressure. Thus, approximately 25% of the total liver blood supply comes via the hepatic artery and the remaining 75% through the portal vein. After injection into a peripheral vein, the radiotracer arriving via the portal vein takes 7-10 s more to reach the liver than the radiotracer arriving via the hepatic artery. This delay via the portal vein is primarily due to the time taken for the radiotracer to traverse the intestinal veins and portal vein.\n\nThe hepatic arterial versus portal venous blood supply to the liver is measured using a first-pass curve obtained with any radiotracer that passes through the liver [1]. Technetium-99m pertechnetate, Tc-99m-DTPA, and Tc-99m-MDP have been used. These tracers that merely pass through, but are not retained by the liver do not allow any additional liver imaging. The agents that both pass through and are retained by the liver (Tc-99m S-colloid or Tc-99m HIDA) allow follow-up imaging of the liver. Immediately upon an antecubital vein injection, the radiotracer mixes thoroughly with blood in the ventricles before its arrival at the liver. The quantity of the radiotracer reaching the liver is directly proportional to the volume of blood supplied to the liver. The differential blood flow to the liver, via the hepatic artery versus portal vein, is calculated by two methods. One method is based on analysis of the slope of the uptake and washout [2], and the other method employs the area under the curve by deconvolutional analysis [3].\n\n### 5.2.1 Data Collection\n\nA fasting patient is positioned supine underneath a large field of view gamma camera fitted with a low-energy, all-purpose, parallel-hole collimator. The position of the gamma camera in either the anterior or posterior view is adjusted to make sure that it covers the lower part of the lungs, entire liver, spleen, and kidneys. A simultaneous anterior and posterior perfusion can be obtained with a dual-head gamma camera (Fig. 5.2.1A,B). About 10-15 mCi (370-555 MBq) of Tc-99m-labeled agent is injected intravenously as a rapid bolus followed by a 30-ml saline flush. Data are collected on 64-by-64 word mode matrix at 1 frame per 0.5 s for 100 s [2]. Data acquisition is begun just before the injection of the radiotracer. The first 30 frames are summed to form a composite image on which four regions of interest are drawn: (1) liver (mid part of the right lobe excluding the right kidney and aorta), (2) lower right lung, (3) spleen, and (4) cross talk region between the right lung and the liver (Fig. 5.2.2). The left kidney is substituted for the spleen in patients with splenectomy.\n\nFig. 5.2.1\n\nLiver perfusion study. Anterior (top) and posterior (bottom) view liver perfusion study obtained with 5 mCi Tc-99m-HIDA shows the passage of radiotracer serially through the right ventricle (RV), left ventricle (LV), abdominal aorta (A), spleen (S), kidneys (K), and liver. Left lobe perfusion is seen better in the anterior view, and the perfusion of the spleen and kidneys is seen better in the posterior view\n\nFig. 5.2.2\n\nRegions of interest. Regions of interest (ROI) are drawn over the middle of the liver, right lower lung, spleen, left kidney, and cross-talk region between liver and right lung. ROIs should exclude liver margins and aorta\n\n### 5.2.2 Slope Method\n\nTime-activity curves are generated over all four regions as outlined above (Fig. 5.2.3A). The cross-talk curve from the region between the right lung and liver (Fig. 5.2.3B) is scaled to the same height as the early part of the liver curve (before the arrival of the liver arterial phase, usually between 0 and 20 s) and subtracted from the liver and spleen curve to generate the corrected liver and spleen curves (Fig. 5.2.3C). Corrected liver and spleen curves are displayed separately (Fig. 5.2.3D,E). Three time points are identified on the corrected time-activity curve. The initial arrival of radioactivity at the liver is denoted by Ta, which corresponds to the beginning of the hepatic arterial phase. Because the hepatic and splenic arteries have a common origin from the celiac artery, the time of peak activity on the spleen curve (Sp) is considered the end of the arterial phase and beginning of the portal venous phase, and corresponds to time, Tp, on the liver curve (Fig. 5.2.3D). Normally, there is 7 \u00b1 2-s delay between the arrival of the hepatic arterial and portal venous blood supply to the liver. The slope of the arterial phase, La, is measured from Ta to Ta + 7 s, and the slope of the portal venous phase, Lp, is measured from Tp to Tp + 7 s. Total counts are integrated, and percent arterial blood flow to the liver is measured using the following formula from a linear fit to the curve [2].\n\nLa = slope of the hepatic arterial phase; Lp = slope of portal venous phase.\n\nFig. 5.2.3\n\nHepatic arterial vs. portal venous blood flow by the slope method. Time-activity curves are generated over the liver (a) and cross-talk region between the lung and liver (b). Both curves are scaled to the same height, and the corrected liver curve (c) is obtained after subtracting the cross-talk counts. The onset of the arterial (Ta) and portal venous (Tp) phases are noted on the corrected liver curve (d). La and Lp represent the slope of the arterial and portal venous phases measured over 7 s from Ta and Tp, respectively. Spleen curve (e) shows the onset of arterial (Ta) and peak arterial flow (Tp); Sa and Sp represent the slopes of the curve in a manner similar to La and Lp [2, 4]\n\nTo correct for the fraction of hepatic arterial component still present during the portal venous phase in the liver curve, the slope of the splenic curve is used with the following modification. The ratio of Sa\/Sp of the splenic curve is assumed to represent the fraction of the arterial flow present during the portal venous phase of the liver curve [2].\n\n### 5.2.3 Area Method\n\nThis method uses the deconvolutional analysis [2, 4]. Like the slope method, it is dependent upon the temporal separation between the arrival of the hepatic arterial and portal venous blood to the liver. It takes into account the role of recirculation of the injected radiotracer. Data acquisition is identical to the slope method described above (Fig. 5.2.1). Using a large of field of view gamma camera, computer data are collected on a 64 \u00d7 64 word matrix at one frame per 0.5 s for 100 s. The data collection is started just before injection of 10-15 mCi (555-740 MBq) of the radiotracer. On the first 30-frame composite image, regions of interest are drawn, and time\/activity curves are generated over: (1) liver, mid right lobe excluding the right kidney and aorta, (2) lower right lung, (3) spleen, and (4) cross-talk region between the right lung and liver (Fig. 5.2.2). The cross-talk curve is scaled to the same height as the initial part of the liver curve (before the arrival of the radiotracer to the liver) and subtracted from the liver and spleen curve to generate the corrected liver and spleen curve (Fig. 5.2.4A). Liver and spleen curves are deconvoluted with the lung curve using the modified Fourier transformation technique (Fig. 5.2.4B). The curve is expanded from 200 to 1,024 data points by addition of an exponentially decreasing tail that eliminates artifacts because of a sharp cutoff at the end of data termination. An assumption is made that the spleen blood flow pattern is similar to that of the hepatic artery. The spleen curve is used to approximate hepatic arterial blood flow. The spleen curve is multiplied by a constant so that the up-slope superimposes over the early part of the liver curve (Fig. 5.2.4C). The liver and modified spleen curves are integrated to give the areas under the curve, A L and A s, respectively\n\nFig. 5.2.4\n\nHepatic arterial vs. portal venous blood flow by the area method. Corrected spleen and liver curves are obtained first (a) after subtracting the background counts from the lung to liver cross-talk region and then subjected to deconvolutional analysis (b). Magnitude of the spleen curve is modified so that its upslope matches with that of the liver (c). A s and A L represent the area under the modified spleen and liver curves, respectively\n\nBoth the slope and area methods are found clinically useful and have been validated in experimental animals [5]. The slope method technically is much simpler, but carries wider variability between studies. The area method using the deconvolutional analysis provides the best separation between normal patients and those with increasing severity of liver disease. The arterial-to-venous ratio increases as the severity of liver disease increases from Child's class A to C (Table 5.1.3). In cirrhosis, low pressure (7-10 mmHg) portal venous blood flow is affected much earlier than high pressure (100-120 mmHg) hepatic arterial blood flow. Portal venous blood flow decreases as the portal venous pressure raises (Fig. 5.2.5). Measurement of the hepatic arterial vs. portal venous blood flow is found useful in the diagnosis of hepatic vein thrombosis and also in following patients treated for portal hypertension [4].\n\nFig. 5.2.5\n\nRelationship between portal venous blood flow vs. portal venous pressure. There is an inverse relationship between the two; as the portal venous pressure increases, the portal venous blood flow decreases [7]\n\nThe final shape of the liver and spleen curve depends upon whether or not the chosen agent is retained by these organs. Technetium-99m sulfur colloid, which is retained by both organs, and Tc-99m-HIDA, which is retained only by the liver but not by the spleen, produce curves whose shape is different from each other and also from those agents (Tc-99m labeled MDP, albumin, or pertechnetate) not retained by either organ. The choice of the agent does not usually affect the values of arterial vs. venous blood flow, because the first-pass study is independent of the biokinetic behavior of the radiotracer [5].\n\nNormal median portal venous blood flow is 78% and the median hepatic artery flow 22%. The median portal venous blood flow decreased to 68% in mild liver disease and remains below 49% in severe liver disease. The median portal venous blood flow falls below 4% in patients with portal vein thrombosis. Scintigraphic technique is shown to be 90% sensitive and 100% specific in patients with portal vein thrombosis when portal blood flow falls below 20% [4]. Perfusion changes and indices of hepatic arterial blood flow in patients with cirrhosis can be measured by comparing them to the renal or splenic arterial peak, thus avoiding the influence of portal venous reduction on the hepatic arterial peak [6]. Calculation of the hepatic arterial-to-portal venous ratio provides a non-invasive method to evaluate objectively the benefits of transjugular intrahepatic portosystemic shunt (TIPS) therapy. In a study involving 28 patients, the mean (SD) portal venous pressure of 25.5 \u00b1 4.6 mmHg before TIPS decreased to 18.5 \u00b1 3.9 mmHg after TIPS. The portal venous blood flow of 29.2 \u00b1 11.1% before then increased to 38.2 \u00b1 13.4% after TIPS, indicating its therapeutic benefits. Portal venous flow shows an inverse relationship with the portal venous pressure; portal venous blood flow decreases as the portal venous pressure increases. The relationship between portal venous flow and pressure tends to normalize after a successful TIPS procedure [7, 8]\n\n## 5.3 Hepatopulmonary Syndrome\n\nThe functional relationship between the liver and lung originally recognized in 1935 is now called hepatopulmonary syndrome (HPS). The syndrome consists of a triad of: (1) liver disease, (2) increased alveolar-arterial oxygen gradient, and (3) intrapulmonary vasodilatation [1, 2].\n\n### 5.3.1 Clinical Presentation\n\nPatients usually present with a combination of symptoms indicative of both liver and lung disease: esophageal varices, gastrointestinal bleeding, spider nevi, ascites, and splenomegaly, indicative of liver disease, and dyspnea, clubbing, platypnea, and orthodeoxia, indicative of lung disease [3, 4]. Platypnea is dyspnea in an upright position, which is relieved by assuming a supine position [5]. Orthodeoxia is arterial deoxygenation, exaggerated in the upright position and relieved by recumbency. Platypnea and orthodeoxia, which were found only in a small percentage of patients with cirrhosis (5%), are much more frequent and severe in intensity in patients with HPS, often reaching as high as 88-100% [6]. Spider nevi of the palms and around the umbilicus are considered the cutaneous markers of HPS [7]. As the liver reaches its end stage, the patients develop ascites, generalized edema, pleural effusion, and interstitial fluid accumulation in the lungs. Chest X-ray changes consist of either finely diffuse spidery infiltrates or focal arteriovenous malformations [8].\n\n### 5.3.2 Pathophysiology\n\nArterial deoxygenation due to an intrapulmonary shunt is the hallmark of HPS. Pulmonary artery pressure remains normal or slightly low [6]. Severe hypoxemia (PaO2 <60 mmHg), in the absence of primary lung disease, in combination with liver disease clinically should raise the suspicion of HPS. The pulmonary capillaries, which normally measure 8-15 \u03bcm, dilate up to 100 \u03bcm in diameter, often forming spider nevi on the pleural surface [9]. Radiolabeled macroaggregates of albumin (MAA) of 15-150 \u03bcm in diameter normally get trapped almost completely within the pulmonary capillary bed after intravenous injection. These radiolabeled Tc-99m MAA particles readily pass through the dilated pulmonary capillaries in patients with HPS and enter the systemic circulation to be trapped in normal size capillaries of the brain, liver, kidney, and other organs, in proportion to their blood supply [10, 11, 12]. Normally less than 6% of Tc-99m-MAA particles bypass the lung to lodge in other organs [11, 13]. It is theorized that the liver in HPS either produces vasodilators or is incapable of inactivating vasodilators produced elsewhere. Incriminated vasodilators include prostaglandins, vasoactive intestinal polypeptide, calcitonin, glucagon, nitric oxide, and atrial natriuretic factor, etc. [6].\n\n### 5.3.3 Diagnosis\n\nArterial blood gases are obtained to document hypoxemia (Table 5.3.1). Contrast 2D echo-cardiography (2D echo) with indocyanin green or agitated saline is the most preferred initial diagnostic imaging procedure [14, 15]. Saline agitation creates microbubbles of 60-90-\u03bcm size, which opacify the right heart chambers within one to three cardiac cycles after injection into an antecubital vein. The left heart chambers are not opacified due to filtration of all bubbles by the normal lung capillaries [16]. The bubbles pass through dilated intrapulmonary capillaries in HPS to enter the pulmonary veins and left heart chambers. A confirmatory diagnosis of HPS requires documentation of arterial hypoxemia (PaO2 less than 70 mmHg), normal pulmonary function tests, typical chest X-ray findings, and opacification of left heart chambers in the absence of a right-to-left cardiac shunt. Right-to-left cardiac shunt is suggested on a 2D echo when both right and left ventricular chambers are opacified simultaneously, within one to three cardiac cycles after intravenous injection of agitated saline [17].\n\nTable 5.3.1\n\nDiagnostic workup for hepatopulmonary syndrome (modified from [18])\n\nCirrhosis with hypoxemia\n\n|   \n---|---  \n|\n\nChest X-ray\n\n|\n\nNormal | Abnormal (treat, if hypoxemia persists)\n\nContrast echo (EC) and pulmonary function tests (PFTs)\n\n(\u2212) CE and normal PET | (+) CE and normal PFT's | (+) CE and abnormal PFT's\n\n\u2193 | \u2193 | \u2193\n\nNo HPS | HPS | Tc-99m MAA Scan\n\n|  |\n\nShunt >6%Shunt <6%\n\n|  |\n\n\u2193\u2193\n\n|  |\n\nHPSNo HPS\n\n### 5.3.4 Scintigraphic Quantification\n\nA mild form of HPS is found in as many as 4-17% of patients with varieties of chronic liver diseases [18]. Despite being very sensitive, 2D echo lacks the specificity and ability to quantify the degree of shunt. A perfusion scintigraphy supplements 2D echo by providing both quantification and specificity for a definitive diagnosis of HPS.\n\n### 5.3.5 Procedure\n\nThe patient is made to sit upright for 5 min to maximize the degree of intrapulmonary shunt. Macroaggregates of Tc-99m-albumin particles are prepared carefully as per instructions provided in the package insert. A drop of the prepared material is fed into a hemocytometer chamber and examined under the light microscope to ascertain that at least 90% of the particles are in the 15-90-\u03bcm size range. A radiochromatogram is obtained to confirm that there is better than 90% radiolabeling. About 2-3 mCi Tc-99m-MAA is injected into an antecubital vein while the patient is seated. After injection, the patient is made to lay supine. A large field of view dual-head gamma camera, fitted with a low-energy, all purpose, parallel-hole collimator, is positioned laterally on each side, or in front and behind the head. The spectrometer is set for 140-keV photon peak energy with a 20% symmetrical window. The camera heads are positioned above the shoulders to avoid counts below the neck. The counts are taken for a preset time of 5 min with each head and recorded on a 64 \u00d7 64 computer matrix. After taking 5-min head counts, the detectors are moved over to the chest to the anterior and posterior position. Preset 5-min counts are taken again in the anterior and posterior view simultaneously and recorded on a 64 \u00d7 64 computer matrix.\n\n### 5.3.6 Data Analysis\n\nOn the lateral or anterior and posterior views of the head, regions of interest are drawn to cover the entire brain, excluding the scalp and superior sagittal sinus when they are visible. On the anterior and posterior view chest images, regions of interest are drawn to cover both lungs. Care is taken to avoid the liver or kidneys in the lung ROI (Fig. 5.3.1). Geometric mean counts are calculated using the following formula:\n\n![$$\\\\begin{array}{l}\n{\\\\rm Geometric\\\\ mean\\\\ brain\\\\ counts }\\\\left\\( {{\\\\rm GMBC}} \\\\right\\)\\\\\\\\\n\\\\quad = \\\\sqrt {{\\\\rm right\\\\ lateral\\\\ or\\\\ anterior\\\\ view\\\\ brain\\\\ counts } \\\\times \\\\,{\\\\rm left\\\\ lateral\\\\ or\\\\ posterior\\\\ view\\\\ brain\\\\ counts}}\\\\\\\\\n\\\\quad = \\\\sqrt {1022 \\\\times 1262}  = 1,135\\\\\\\\\n\\\\end{array}$$](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ23_5.gif)\n\n![$$\\\\begin{array}{l}\n{\\\\rm Geometric\\\\ mean\\\\ lung\\\\ counts }\\\\left\\( {{\\\\rm GMBC}} \\\\right\\)\n\\\\qquad = \\\\sqrt {{\\\\rm anterior\\\\ view\\\\ lung\\\\ counts } \\\\times \\\\,{\\\\rm posterior\\\\ view\\\\ lung\\\\ counts}}\\\\\\\\\n\\\\qquad = \\\\sqrt {637,417 \\\\times 970,115}  = 786,363\\\\\\\\\n\\\\end{array}$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ24_5.gif)\n\nFig. 5.3.1\n\nQuantification of hepatopulmonary shunt: Regions of interest are drawn over the brain in two lateral views of the head and over the lungs in the anterior and posterior views of the chest. Total counts in 5 min are noted in each view. The geometric mean counts are obtained by using the equation given in the text. Hepatopulmonary shunt ratio is obtained by dividing brain counts by the sum of brain and lung counts\n\nAn assumption is made that about 13% of the cardiac output is delivered to the brain [19]. By applying this correction factor, the hepatopulmonary shunt is calculated by the following equation:\n\n![$$\\\\begin{array}{rl}\n{\\\\rm Hepatopulmonary\\\\ shunt }\\\\left\\( {{\\\\rm HPS}} \\\\right\\) & = \\\\frac{{\\\\frac{{{\\\\rm GMBC}\\\\left\\( {{\\\\rm Brain}} \\\\right\\)}}{{0.13}}}}{{\\\\frac{{{\\\\rm GMBC}\\\\left\\( {{\\\\rm Brain}} \\\\right\\)}}{{0.13}} + {\\\\rm GMLC}}}\\\\left\\( {{\\\\rm Lung}} \\\\right\\)\\\\\\\\\n{} & = \\\\frac{{{\\\\rm 1,135 \/ 0}{\\\\rm .13}}}{{{\\\\rm 1,135 \/ 0}{\\\\rm .13 786,363}}}\\\\\\\\\n{} & = {\\\\rm 8730 \/ 795,093 0}{\\\\rm .02 2\\\\% }\\\\\\\\\n\\\\end{array}$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ25_5.gif)\n\nIn normal subjects and in patients with intrinsic lung disease or with cirrhosis but without shunt, the hepatopulmonary shunt ratio varies from 3 to 6% [11, 17]. A value higher than 6% is considered indicative of HPS [17, 18]. Patients with Child's class A, B, or C liver disease without HPS generally show values below 5%. The mean shunt value is 30 \u00b1 4% in patients with hepatopulmonary syndrome. A shunt value of 37% was found in an 8-year-old child (Fig. 5.3.2) waiting for liver transplantation [20].\n\nFig. 5.3.2\n\nHepatopulmonary shunt. An 8-year-old boy waiting for liver transplantation has 37% right-to-left shunt due to intrapulmonary vasodilatation. In addition to brain, kidneys and liver are seen in a Tc-99m MAA scan [20]\n\n### 5.3.7 Standardization\n\nThe technique of measuring hepatopulmonary shunt is very simple, and often there may be room for complacency. It is necessary to check rigidly for Tc-99m-MAA particle size and percent labeling. Greater than 90% of the particles should be between 15 and 90 \u03bcm in size, and labeling efficiency better than 90%. A preparation not meeting these two requirements is discarded. Injection of too many small particles (below 15 \u03bcm) overestimates the degree of the shunt, and the shunt is underestimated when too many particles are much larger than 90 \u03bcm in size.\n\n### 5.3.8 Treatment\n\nTreatment is medical in early liver failure and surgical for end-stage liver disease. Medically, the patients are treated with indomethacin [21], almitrine bismesylate [22], octreotide, or other drugs [6]. Surgical treatment involves liver transplantation for end-stage liver disease. Liver transplantation, once listed as a clear contraindication, is now considered an optimal therapy for HPS and is shown to improve both liver and pulmonary functions [23]. In the Cleveland Clinic study, the mean ratio of 18.7% in HPS patients decreased to 4.5% after liver transplantation with marked improvement of pulmonary blood gases [24].\n\n## 5.4 Duodenogastric Bile Reflux\n\nGastrointestinal peristalsis begins at the gastric pacemaker located near the gastro-esophageal junction and travels antegrade towards the gastric fundus, body, and pylorus, and progresses further along the small and large intestine. Peristaltic waves move the gastrointestinal intraluminal contents in an antegrade fashion. Duodenal contents are thus prevented from entering the stomach by the dual action of the antegrade peristalsis and contraction of pyloric sphincter [1]. Patients with atrophic gastritis, gastric ulcer, and esophagitis are often found to have bile in the stomach (duodeno-gastric bile reflux) raising an etiological relationship between dyspepsia and bile reflux [2].\n\nDetection of bile by chemical analysis in the gastric juice aspirated through a nasogastric tube has been used over the years as a test for duodeno-gastric bile reflux. Chemical analysis is not only cumbersome, but also the insertion of the naso-gastric tube itself may cause duodeno-gastric bile reflux (DGBR). Counting or imaging of radiolabeled bile in the stomach makes the test technically much simpler and avoids the necessity of chemical analysis and intubation [3]. Gamma camera imaging makes the test readily acceptable, enables detection, provides quantification of the degree of duodeno-gastric (D-G) bile reflux, and avoids the necessity of gastric juice aspiration through the N-G tube in both children and adults [4].\n\n### 5.4.1 Rationale\n\nAfter its secretion by the liver, the hepatic bile enters the gallbladder or duodenum, or both. During fasting, about 70% of the hepatic bile enters the gallbladder, and the remaining 30% enters the duodenum directly [5]. Facilitated by the peristaltic waves, bile entering the duodenum moves forward with the rest of the duodenal contents received from the stomach. Since only the amount of bile entering the duodenum is available for reflux into the stomach, a technique that enables quantitative measurement of bile entry into the duodenum will be ideal for measuring bile flow forward into the jejunum or backward into the stomach (D-G reflux). Bile entering the gastrointestinal tract is quantified by selecting two regions of interest: one over the stomach and another over intestinal tract (excluding liver, gallbladder, and bile ducts). Duodeno-gastric bile reflux is calculated by dividing total stomach counts by total counts in the stomach and intestine. The technique allows measurement of D-G bile reflux both during fasting and after administration of cholecystokinin or after feeding.\n\n### 5.4.2 Data Collection and Analysis\n\nHepatic phase imaging data are used for calculation of D-G bile reflux during fasting, and gallbladder phase imaging data for calculation of post-cholecystokinin or post-prandial D-G bile reflux. Data are collected as outlined in Table 5.1.1. By 60-min post-injection, most of Tc-99m HIDA clears from the liver, leaving the nearby gastric region free of interference by cross-talk counts from the left lobe of the liver (Fig. 5.4.1). All 60-min data are carefully reviewed in a cine display for any overlap of intestinal radioactivity onto gastric ROI [6]. A frame between 50 and 60 min (usually the 60-min frame) that does not contain any superimposition of intestinal loops onto the gastric bed is chosen for selection of gastric and intestinal ROIs. Gastric ROI includes the traditional gastric bed, below the left lobe of liver and to the left side of the distal common bile duct, and extending laterally up to the splenic bed. The intestinal ROI includes the rest of the upper abdomen, excluding the liver, gallbladder, and common bile duct. The urinary bladder is excluded from intestinal ROI (Fig. 5.4.2). Time\/activity curves are generated from both ROIs. By 60-min post-injection, most of radioactivity clears from the cardiac blood pool and kidneys. Sometimes the D-G reflux may occur early and empty in the late hepatic phase imaging. In such circumstances the frame with the peak D-G reflux is chosen for ROI selection. The counts are corrected for physical decay.\n\n![$$\\\\begin{array}{l}\n{\\\\rm Duodeno - gastric\\\\ bile\\\\ reflux }\\\\left\\( {\\\\rm \\\\% } \\\\right\\)\\\\\\\\\n\\\\qquad \\\\quad = \\\\frac{{{\\\\rm Total\\\\ counts\\\\ in\\\\ gastric\\\\ ROI} \\\\times {\\\\rm 100}}}{{{\\\\rm Total\\\\ counts\\\\ in\\\\ gastric\\\\ ROI  +  Total\\\\ counts\\\\ in\\\\ intestinal\\\\ ROI}}}\\\\\\\\\n\\\\qquad \\\\quad = \\\\frac{{54,173 \\\\times 100}}{{54,173 + 74,066}}\\\\\\\\\n\\\\qquad \\\\quad = 42\\\\%\\\\\\\\\n\\\\end{array}$$\n](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ26_5.gif)\n\nFig. 5.4.1\n\nBasal duodeno-gastric bile reflux. Duodeno-gastric bile reflux is rare in normal subjects. Reflux in this patient occurs early (frame no. 7) and continues throughout. It is calculated as a part of Hep-phase data analysis\n\nFig. 5.4.2\n\nPost-CCK-8 duodeno-gastric bile reflux. It is calculated as an integral part of GB-phase data analysis. The gastric ROI (S) is drawn below the left lobe of the liver, on the left side of the distal common bile duct extending laterally to the splenic region. The intestinal ROI (I) encompasses the rest of the abdomen. Basal D-G bile reflux of 42% (Fig. 5.4.1) increased to 58% with CCK-8\n\nBile reflux after CCK-8 is similarly calculated using data obtained during the gallbladder phase. Cholecystokinin normally induces the contraction and emptying of the gallbladder, stimulates contraction of the pyloric sphincter, and simultaneously increases small intestinal peristalsis, facilitating rapid antegrade bile movement through the small bowel. Contraction of the pyloric sphincter normally prevents D-G bile reflux. Post-CCK images are checked in the cine display to ascertain that there is no movement of bile radioactivity beyond the field of view of the gamma camera. If there is, then the gamma camera position is adjusted to include all bile regions, and an additional view over the lower abdomen is taken. Cine display of all frames is essential to ascertain that no intestinal loop is encroaching upon the gastric region, and when found, such frames are excluded for ROI selection [7]. Often there is no D-G bile reflux during fasting; rather, it occurs only after CCK-8 (Fig. 5.4.3).\n\nFig. 5.4.3\n\nDuodeno-gastric bile reflux only after CCK-8. There is no basal D-G bile reflux (0-2 min). Bile reflux of 10% occurs after CCK-8 administration (G stomach)\n\n### 5.4.3 Bile Reflux in Health and Disease\n\nThe data shown in Table 5.4.1 were collected in 22 patients without gallstones who were referred for measurement of the gallbladder ejection fraction. They did not have any symptoms of gastric dyspepsia. Mean D-G reflux in these subjects was 2.4% (95% CI = 0.4-4.4%). D-G reflux during fasting or after CCK-8 administration is rare and usually does not exceed 5%. When significant reflux is found, it suggests bile gastritis may be responsible for patient symptoms. Prokinetic agents are often prescribed for patients with large volume D-G reflux.\n\nTable 5.4.1\n\nBile flow into stomach vs. small intestine in 20 normal subjects\n\nNo. | Subject | Total stomach counts | Total intestinal counts | % Stomach | % Intestine\n\n---|---|---|---|---|---\n\n1 | HC | 93,729 | 3,830,883 | 2.3 | 97.7\n\n2 | H | 71,380 | 3,012,831 | 2.3 | 97.7\n\n3 | CC | 36,521 | 2,684,925 | 1.3 | 98.7\n\n4 | E | 51,226 | 1,483,318 | 3.3 | 97.3\n\n5 | EJ | 103,127 | 3,651,893 | 2.7 | 97,3\n\n6 | GE | 43,555 | 1,091,999 | 3.8. | 96.2\n\n7 | AP | 67,967 | 2.661,020 | 2.5 | 97.5\n\n8 | B | 42,169 | 1,309127 | 3.1 | 96.9\n\n9 | BB | 85,603 | 2,063,219 | 3.9 | 96.1\n\n10 | WD | 20,095 | 616,171 | 3.2 | 96.8\n\n11 | HR | 40,526 | 1,169,047 | 3.3 | 96.4\n\n12 | LC | 23,421 | 3,343,890 | 0.7 | 99.3\n\n13 | KD | 19,204 | 731,084 | 2.5 | 97.5\n\n14 | KJ | 15,746 | 2,107,383 | 0.7 | 99.3\n\n15 | KC | 26,342 | 2,259,335 | 1.1 | 99.9\n\n16 | MV | 16,093 | 459,163 | 3.4 | 96.6\n\n17 | MK | 20,084 | 1,299,986 | 1.5 | 98.5\n\n18 | NL | 27,480 | 680,653 | 3.8 | 96.2\n\n19 | O | 28,626 | 1,847,370 | 1.5 | 98.5\n\n20 | PJ | 24,585 | 1,545,024 | 1.6 | 98.4\n\n|  |  |\n\nMean \u00b1 SD | 2.4 \u00b1 1.05 | 97.6 \u00b1 1.05\n\n|  |  |\n\nS.E | 0.23 | 0.23\n\nNote: No background subtraction was made from either the stomach or intestine\n\n## 5.5 Imaging and Quantification of Hepatocyte Asialoglycoprotein Receptors with Tc-99m Galactosyl Human Serum Albumin\n\nHepatocyte plasma membrane is rich in asialoglycoprotein (ASGP) receptors, which are not found in any other cell in the body. The receptor is located along the basolateral and lateral domain, but not along the canalicular domain [1]. Technetium-99m-DTPA-galactosyl-human-serum albumin (Tc-99m GSA) binds to these receptors, and the amount bound varies inversely with the severity of liver disease [2-4]. It is not taken up by the spleen and like radiocolloid is not secreted into bile. ASGP receptor concentration on the membrane reflects the functional integrity of the hepatocyte accurately, much like indocyanin green, cholinesterase, serum albumin, and hepaplastin [3]. The quantity of Tc-99m GSA uptake correlates well with blood clearance of indocyanin green (Fig. 5.5.1). Technetium-99 GSA uptake decreases in patients with varieties of liver diseases including chronic hepatitis, cirrhosis, cholangiocarcinoma, hepatocellular cancer, metastasis, fulminant hepatic failure, and space-occupying benign lesions of the liver [2-4].\n\nFig. 5.5.1\n\nCorrelation between Tc-99m-GSA uptake (LU15, %) and indocyanin green blood retention. There is a good inverse relationship between Tc-99m GSA uptake by the liver with plasma clearance of indocyanin green at 15 min [2]\n\nAlmost all of the injected radiotracer clears from the blood and is taken up exclusively by the liver normally within 15 min. Several parameters have been developed to express the hepatocyte function, the most common one being the extraction index at 15 min. It represents the percentage of the integral of the cumulative counts in the liver for 1 min between 15 and 16 min to the total dose [2].\n\n### 5.5.1 Data Collection and Analysis\n\nAfter 4-6 h of fasting, with the patient in supine position, a gamma camera (single, double, or a triple head) fitted with a low-energy, high-resolution, parallel-hole collimator is positioned anterior to the liver. Sequential anterior planar images (128 \u00d7 128) at 1 frame per 30 s for 20 min are obtained immediately after a bolus injection of 5 mCi Tc-99m GSA (185 MBq) into the antecubital vein. Immediately after the planar images, SPECT data are acquired on a 128 \u00d7 128 \u00d7 16-matrix computer for 64 stops at 10 s per stop at a 5.6\u00b0 interval [2]. The spectrometer is set for 140 keV at a 20% window. Liver uptake at 15 min is calculated by using the following formula.\n\n![$$\n{\\\\rm Liver}\\\\,{\\\\rm uptake at 15min }\\\\left\\( {{\\\\rm LU15}} \\\\right\\) = \\\\frac{{\\\\int\\\\limits_{15}^{{\\\\rm 16}} {{\\\\rm C}\\\\left\\( {\\\\rm t} \\\\right\\){\\\\rm dt} \\\\times {\\\\rm 100\\\\% }} }}{{{\\\\rm Total injected dose}}}$$](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ27_5.gif)\n\nWhole liver counts and residual liver counts are obtained by measuring organ volume from SPECT images. Whole organ volume is determined by detecting the edge for each slice and then adding all of the slices. The volume of each lobe is obtained by selecting the gallbladder fossa -inferior vena cava plane (plane of Serege-Cantele), which divides the liver into physiologic right and left lobes, or by referring to the CT or MRI references [2]. After obtaining the volume, each lobe is divided into its physiologic segments: the right lobe into the anterior and posterior segments, and the left lobe into the medial and lateral segments [2-4]. The ratio of the counts in each lobe to the whole liver counts provides LU15 for the lobe.\n\nResidual count ratio (RCR) is determined by the following formula.\n\n![$$\n{\\\\rm RCR  =  RC \/ WC}$$](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ28_5.gif)\n\nwhere RC = residual liver count from SPECT images (counts from region of the liver left not to be resected), and WC = whole liver count calculated from SPECT images.\n\n![$$\n{\\\\rm Index\\\\ of\\\\ residual\\\\ liver\\\\ function\\\\ \\(RLU15\\)  =  LU15 \\\\times  RCR}$$](A978-3-642-00648-7_5_Chapter_TeX2GIF_Equ29_5.gif)\n\nSome authors express uptake as hepatic GSA clearance using the Patlak plot method and generate functional images of clearance (Ku). This method is powerful, but some may raise an issue with the chosen terminology \"clearance\" for an agent that does not clear from the liver [3, 4]. The uptake shows an excellent correlation with indocyanin green plasma clearance. Estimation of predicted postoperative residual function after resection of the liver for hepatocellular carcinoma, cholangiocarcinoma, and metastatic liver tumors shows good correlation with postoperative liver function (Fig. 5.5.2). This functional parameter may be ideal for predicting end-stage liver disease and timing of liver transplantation.\n\nReferences\n\n1.\n\nKrishnamuthy S, Krishnamurthy GT. Technetium-99m iminodiacetic acid organic anions: review of biokinetics and clinical application in hepatology. Hepatology 1989;9:139-153CrossRef\n\n2.\n\nBobba VR, Krishnamurthy GT, Kingston E, Brown PH, Eklem M, Turner FE. Comparison of biokinetics and biliary imaging parameters of four Tc-99m iminodiacetic acid derivatives in normal subjects. 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Detecting intrapulmonary right-to-left shunt with contrast echocardiography. Observation in patients with diffuse pulmonary arteriovenous fistulas. Mayo Clin Proc 1976;51:81-84PubMed\n\n16.\n\nKrowka MJ, Tajik AJ, Dickson ER, Wiesner RH, Cortese Da. Intrapulmonary vascular disorders (IPVD) in liver transplant candidates. Screening by two-dimensional contrast-enhanced echocardiography. Chest 1990;97:1165-1170PubMedCrossRef\n\n17.\n\nAbrams GA, Jaffe CC, Hoffer PB, Binder HJ, Fallon MB. Diagnostic utility of contrast echocardiography and lung perfusion scan in patients with hepatopulmonary syndrome. Gastroenterology 1995;109:1283-1288PubMedCrossRef90589-8)\n\n18.\n\nAbrams GA, Nanda NC, Dubovsky EV, Krowka MJ, Fallon MB. Use of macroaggregated albumin lung perfusion scan to diagnose hepatopulmonary syndrome: a new approach. Gastroenterology 1998;114:305-310PubMedCrossRef70481-0)\n\n19.\n\nWade OL, Bishop JM. Cardiac output and regional blood flow. Blackwell Scientific, Oxford, England, 1962\n\n20.\n\nMurakami JW, Rosenbaum DM. Right-to-left pulmonary shunting in pediatric hepatopulmonary syndrome. Clin Nucl Med 1999;24:897PubMedCrossRef\n\n21.\n\nAndrivet P, Cadranel J, Housset B, Harigault R, Harf A, Anot S. Mechanism of impaired arterial oxygenation in patient liver cirrhosis and severe respiratory insufficiency. Effect of indomethacin. Chest 1993;103:500-507\n\n22.\n\nKrowka MJ, Cortese DA. Severe hypoxemia associated with liver disease. Mayo Clin experience and the experimental use of almitrine bismesylate. Mayo Clin Proc 1987;62:164-173PubMed\n\n23.\n\nScott VL, Dodson F, Kang Y. The hepatopulmonary syndrome. Surg Clin North Am 1999;79:23-41PubMedCrossRef70005-0)\n\n24.\n\nLange PA, Vogt DA, Carey WB, Stroller JK. Prevalence and reversibility of hepatopulmonary syndrome following liver transplantation. Am Rev Respir Dis 1993;147:540A (abstract)\n\nReferences\n\n1.\n\nFisher RS, Cohen S. Physiological characteristics of the human pyloric sphincter. Gastro-enterology 1973;64:67-75\n\n2.\n\nCapper WM, Airth GR, Kilby JO. A test for pyloric regurgitation. Lancet 1966;2:621-623PubMedCrossRef91930-1)\n\n3.\n\nRokkjaer M, Marqversen J, Kraglund K, Peterson J. Quantitative determination of pyloric regurgitation in response to intraduodenal bolus injection. Scand J Gastroenterol 1977;12:827-832PubMedCrossRef\n\n4.\n\nNicolai JJ, Silberbusch J, vanRoon F, Schopman W, Berg JWO. A simple method for quantification of biliary reflux. Scand J Gastroenterol 1980;15:775-780PubMedCrossRef\n\n5.\n\nKrishnamurthy GT, Bobba VR, McConnell D, Turner FE, Mesgarzadeh M, Kingston E. Quantitative biliary dynamics: introduction of a new non-invasive scintigraphic technique. J Nucl Med 1983;24:217-223PubMed\n\n6.\n\nSorgi M, Causer D, Wolverson RL, Mosimann F, Tulley N, Ghosh SK, Donovan IA, Alexander-Williams J, Harding LK. Quantification and the elimination of errors in bile reflux tests using a gamma camera. Scand J Gastroenterol 1984;19:33-35\n\n7.\n\nThomas WEG, Jackson PC, Cooper MJ, Davies ER. The problems associated with scintigraphic assessment of duodenogastric reflux. Scand J Gastroenterol 1984;19 (suppl 92):36-40\n\nReferences\n\n1.\n\nSteer CJ. Receptor-mediated endocytosis: mechanisms, biologic function, and molecular properties. In: Zakim D, Boyer TD (eds) Hepatology. A textbook of liver disease. WB Saunders, Philadelphia, 1996, pp 149-214\n\n2.\n\nUetake M, Koizumi K, Yagawa A, Nogata H, Tezuka T, Kono H, Ozawa T, Kusano T, Miyaburuko M, Hosaka M. Use of Tc-99m DTPA galactosyl human serum albumin to predict postoperative residual liver function. Clin Nucl Med 1999;24:428-434PubMedCrossRef\n\n3.\n\nHwang E, Taki J, Shuke N, Nakajima K, Kinuya S, Konishi S, Michigishi T, Aburano T, Tonami N. Preoperative assessment of residual hepatic functional reserve using Tc-99m-DTPA-galactosyl-humen-serum albumin dynamic SPECT. J Nucl Med 1999;40:1644-1651PubMed\n\n4.\n\nSasaki N, Shiomi S, Iwata Y, Nishiguchi S, Kuroki T, Kawabe J, Oci H. Clinical usefulness of scintigraphy with Tc-99m-galactosyl-human serum albumin for prognosis of cirrhosis of the liver. J Nucl Med 1999;1652-1656\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_6(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 6. Gallbladder, Sphincter of Oddi, Cholecystokinin, and Opioid Relation\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nThe bile, essential for digestion and absorption of nutrients, is secreted by the liver continuously, stored in the gallbladder during fasting, and discharged into the duodenum intermittently after each meal. The bile salts, which are an important component of bile, promote efficient digestion and absorption of essential nutrients from the intestine. The gallbladder has a unique mechanism to sequester almost all of the bile salts during fasting and release them into the duodenum after the arrival of food. Food stimulates the release of endogenous cholecystokinin into circulation from the endocrine cells lining the mucosa of the duodenum, jejunum, and upper ileum. Cholecystokinin (CCK) acts on its receptors located in the smooth muscle and initiates the contraction and emptying of concentrated bile from the gallbladder; it simultaneously relaxes the sphincter of Oddi to allow smooth passage of bile into the duodenum.\n\nThe bile, essential for digestion and absorption of nutrients, is secreted by the liver continuously, stored in the gallbladder during fasting, and discharged into the duodenum intermittently after each meal. The bile salts, which are an important component of bile, promote efficient digestion and absorption of essential nutrients from the intestine. The gallbladder has a unique mechanism to sequester almost all of the bile salts during fasting and release them into the duodenum after the arrival of food. Food stimulates the release of endogenous cholecystokinin into circulation from the endocrine cells lining the mucosa of the duodenum, jejunum, and upper ileum. Cholecystokinin (CCK) acts on its receptors located in the smooth muscle and initiates the contraction and emptying of concentrated bile from the gallbladder; it simultaneously relaxes the sphincter of Oddi to allow smooth passage of bile into the duodenum. During the evolutionary process it appears as though nature has placed these three organs very close to each other anatomically to obtain the maximum efficiency of the biological system (Fig. 6.1.1).\n\nFig. 6.1.1\n\nThe relationship among the sphincter of Oddi, gallbladder, and cholecystokinin. The sphincter of Oddi consists of three components: choledochal sphincter, pancreatic sphincter, and ampullary sphincter. Gallbladder wall contains mainly stimulatory (contraction) and the sphincter of Oddi inhibitory (relaxation) receptors for cholecystokinin. CCK-secreting endocrine cells are distributed densely in the mucosa of the duodenum [44]\n\n## 6.1 Effect of Cholecystokinin on the Gallbladder and Sphincter of Oddi\n\n### 6.1.1 Cholecystokinin\n\nA hormone being responsible for contraction and emptying of the gallbladder was proposed first by Ivy and Oldberg in 1928 [1]. A year later, Ivy and associates identified the hormone and named it \"cholecystokinin\" (from the Greek: chole = bile, cysto = sac, kinin = move; move the bile sac) because of its primary motor effect on gallbladder contraction [2].\n\n#### 6.1.1.1 Source\n\nCholecystokinin is found in various organs of the body, including the gastrointestinal, endocrine, genitourinary, and nervous systems [3]. The highest concentration is found in the upper intestinal tract, cerebral cortex, and anterior pituitary gland [4, 5]. In the gastrointestinal tract, the cells secreting cholecystokinin are distributed primarily in the mucosa of the duodenum, jejunum, and upper ileum (Fig. 6.1.2). There are about 40 million (1.3 million cm-1) CCK-secreting cells in the duodenum, 80 million (0.55 million cm-1) in the jejunum, and about 5 million cells in the entire ileum [4]. There are no CCK-secreting cells in the esophagus, stomach, distal ileum, colon, and rectum. The cerebellum and posterior pituitary gland do not contain any CCK-secreting cells. Two types of hormone-secreting cells are found in the intestinal mucosa: open endocrine cells and closed endocrine cells. Open endocrine cells are tall and flask shaped with microvilli along the free border, which enable direct contact with the digested nutrients passing through the intestinal lumen. Closed endocrine cells are basket shaped without any microvilli and do not reach the mucosal surface and hence have no direct contact with the nutrients passing through the intestinal lumen [6].\n\nFig. 6.1.2\n\nDistribution of cholecystokinin and fibroblast growth factor-secreting cells in the alimentary canal. The cholecystokinin-secreting cells are concentrated mainly in the duodenum, jejunum, and proximal ileum. There are no CCK-secreting cells in the esophagus, stomach, and intestinal tract beyond the proximal ileum [4]. Fibroblast growth factor stimulating cells are distributed mainly in the terminal ileum\n\n#### 6.1.1.2 Structure\n\nAt least five molecular forms of cholecystokinin have been identified. All five are linear- chain polypeptides, each with a varying number of amino acids in its molecule [3, 7]. The longer hormones with 33-58 amino acids can be cleaved at different locations to yield hormones of shorter molecular length. The component that retains the carboxyl terminal tetrapeptide exhibits most of the biological function of the parent molecule (Fig. 6.1.3). Component I with more than 40 and component II with 33-39 amino acids in the molecule are the most abundant forms. ComponentdIII contains 12, component IV has eight (octa-peptide or CCK-8), and component V consists of four amino acids. Component V with four amino acids is the shortest biologically active form.\n\nFig. 6.1.3\n\nMolecular structure of cholecystokinin. It consists of 33-39 amino acids that can be cleaved (scissors) at several locations to produce shorter fragments. Amino terminal tetrapeptide (CCK-4) is necessary for biological actions of the fragments [44]\n\nHormone CCK-8 has been synthesized and is available for clinical use as Sincalide (Kinevac). The terminal amino acid, phenylalanine, is amidated, and the seventh amino acid, tyrosine, is sulfated. Sulfation of the amino acid tyrosine is essential for retention of the biological potency of the hormone. To be biologically active, each component must possess the carboxy terminal tetrapeptide. When the amino acids are numbered serially beginning at the carboxyl terminal (CO-NH2) end, the hormonal fragments are called CCK-4, CCK-8, CCK-12, CCK-33, CCK-39, etc. The molecular weight of CCK-8 is 1,143. Because of its shorter length and lesser mass, CCK-8, on a molar basis, is four to five times biologically more potent than CCK-33. All five components of CCK are found in the small intestinal mucosa and the central nervous system. Currently, it is believed that each component is secreted, in-vivo, by a specific group of CCK-secreting cells. The concentration of larger components (CCK-33, CCK-39) is higher in the central nervous system and the gastrointestinal tract, whereas the concentration of smaller components (CCK-8) is higher in the anterior pituitary gland [5].\n\n#### 6.1.1.3 Cerulein\n\nOther peptides with a carboxyl terminal tetrapeptide identical to that of CCK also possess a cholecystokinetic effect. Gastrin and cerulein are two such peptides with an identical carboxy terminal tetrapeptide, and both exhibit a cholecystokinetic effect. Cerulein, isolated first from the skin of an Australian frog, is not found in humans. It closely resembles the CCK-8 structure and consists of ten amino acids (decapeptide), two more than CCK-8. The seventh amino acid, tyrosine, is sulfated. The sixth amino acid is threonine in cerulein, and methionine in CCK-8 (Fig. 6.1.4). Biologically, cerulein is ten times more potent than CCK-33 on a molar basis, and 47 times more potent on the basis of its net weight. The molecular weight is 1,352 [8].\n\nFig. 6.1.4\n\nStructure of CCK-8 and cerulein. Both contain an identical N-terminal tetrapeptide. The sixth amino acid is methionine in CCK-8 and threonin in cerulein. Both are sulfated on the seventh amino acid, tyrosine. Cerulein has two amino acids (decapeptide) more than CCK-8. Molecular weight of CCK-8 = 1,142 and cerulein = 1,352 [32]\n\n#### 6.1.1.4 Release of CCK\n\nCholecystokinin is released into circulation soon after the arrival of food into the duodenum. Fat is the most potent stimulant of all nutrients. Serum CCK level begins to rise 8-10 min after a meal and reaches the peak level by 30-60 min [9, 10]. The serum level remains above the basal level for 2-4 h post- meal, depending upon the nature of food ingested. The serum half life of CCK is 2.5 min [11].\n\n#### 6.1.1.5 Receptors\n\nCholecystokinin acts through two types of receptors, CCK-A (CCK-1) and CCK-B (CCK-2). CCK-A receptors are distributed predominantly in the smooth muscle of the gut and a few areas of the brain, and CCK-B receptors are distributed mainly in the brain. The gallbladder contains mostly CCK-A and the pancreas mostly CCK-B type receptors [12, 13]. These receptors are located on the surface of the cell and provide easy access to cholecystokinin. Cholecystokinin binds to CCK-A receptors in the gallbladder smooth muscle and initiates its contraction and bile emptying immediately. By binding to CCK-B type receptors, cholecystokinin stimulates pancreatic enzyme and bicarbonate secretion. CCK-B receptors manifest an inhibitory effect on the motility of the distal colon and pylorus of the stomach, acting via nitric oxide pathway [14]. Both cholecystokinin and leptin influence eating by acting on the brain as inhibitory hormones [15]. In humans, plasma cholecystokinin levels are associated with satiety [16]. Cholecystokinin antagonists (loxiglumide, devazepide, and TP-680) either block or reverse its actions by competitively binding to CCK receptors. These antagonists increase appetite, food intake, and weight gain in animals by blocking CCK brain receptors [17, 18].\n\n#### 6.1.1.6 Actions of CCK\n\nThe stimulation of gallbladder contraction and subsequent bile emptying is one of the most important and the best known actions of cholecystokinin. Cholecystokinin simultaneously relaxes the sphincter of Oddi and facilitates smooth passage of bile through it into the duo-denum [19]. Cholecystokinin also exhibits other biological actions listed in Table 6.1.1. It increases both the volume and flow of hepatic bile by increasing water secretion by bile canaliculi and bile ducts. For a short time, it was thought that a separate hormone (pancreozymin), different from CCK, acted on the pancreas to increase its enzyme and bicarbonate secretion [20]. It is now well recognized that a single hormone, cholecystokinin, is responsible for both actions: induction of gallbladder contraction and stimulation of pancreatic enzyme secretion [21]. In the pancreas, both cholecystokinin and secretin bind to the same receptor and increase enzyme and bicarbonate secretion [22]. Cholecystokinin increases intestinal peristalsis and promotes antegrade flow of bile and nutrients through the lumen. By inducing contraction of the pyloric sphincter, CCK prevents duodeno-gastric bile reflux.\n\nTable 6.1.1\n\nActions of cholecystokinin\n\n(1) Contracts and empties the gallbladder\n\n---\n\n(2) Increases pancreatic enzyme and bicarbonate secretion\n\n(3) Relaxes the sphincter of Oddi and lower esophageal junction\n\n(4) Increases secretion of insulin, glucagon, and somatostatin by Islet cells\n\n(5) Contracts the pyloric sphincter, preventing duodeno-gastric bile reflux\n\n(6) Increases hepatic bile secretion\n\n(7) Increases intestinal peristalsis\n\n(8) Increases intestinal blood flow\n\n(9) Suppresses appetite\n\n(10) Relaxation of lower esophageal sphincter.\n\n(11) Protection of gastric mucosa through release of somatostatin\n\n(12) Decreases systolic blood pressure\n\nCholecystokinin binds to stimulatory CCK-A receptors in the gallbladder smooth muscle and initiates its contraction [13]. Simultaneously, it binds to inhibitory receptors in the sphincter of Oddi smooth muscle and promotes its dilatation. These combined, but paradoxical, actions promote smooth passage of bile from the gallbladder into the duodenum [23]. CCK-A receptor-rich smooth muscle is distributed mostly in the fundus and body of the gallbladder. Very few receptors are found in the smooth muscle of the neck and the cystic duct. When serum CCK level rises above the threshold, the fundus contracts first, followed by the body. The relaxation of the sphincter of Oddi occurs at the same time, allowing smooth passage of bile through it [24, 25].\n\nCholecystokinin and cerulein both act on the same receptors [26-29]. The cystic duct smooth muscle, which has only a few CCK receptors, does not usually contract, because its threshold for contraction is set at a much higher level than the threshold for contraction of the smooth muscle in the body and fundus. When a large bolus dose of CCK-8 is injected, however, the cystic duct often contracts and prevents gallbladder emptying [30]. In chronic acalculous chronic cholecystitis (cystic duct syndrome), not only is there a decrease in the total number of CCK receptors in the body and fundus, but also the cystic duct smooth muscle threshold is lowered, allowing contraction of the body, fundus, and cystic duct, all at the same time, resulting in non-emptying of the gallbladder [31].\n\nDue to the low serum concentration of cholecystokinin during fasting, most CCK receptors in the smooth muscle are free, facilitating maximum relaxation of the gallbladder wall and a maximum increase in the tonus of the sphincter of Oddi. Both of these factors acting together promote preferential hepatic bile entry into the gallbladder during fasting. Upon CCK release post-meal, these receptors get saturated, initiating contraction of the gallbladder (stimulatory receptors) with simultaneous relaxation (inhibitory receptors) of the sphincter of Oddi. The degree of gallbladder bile emptying (ejection fraction) correlates directly with the total number of CCK receptors in the gallbladder smooth muscle [32].\n\n#### 6.1.1.7 Dose Response\n\nThe degree of gallbladder emptying is dependent both upon the dose rate and duration of infusion of cholecystokinin or cerulein [33]. The higher the dose is, the greater the degree of emptying, as long as the administered dose is within the physiological range (Fig. 6.1.5). The threshold for the beginning of gallbladder contraction and emptying lies between 0.5 and 1.0 ng kg min-1 of CCK-8 or cerulein. Peak emptying is noted between 3 and 5 ng kg min-1 dose [34, 35]. A further increase in CCK-8 dose actually decreases gallbladder emptying. An infusion of 0.02 \u03bcg kg-1 (20 ng kg-1) or 0.04 \u03bcg kg-1 (40 ng kg-1) over a 3-min period results in an ejection fraction that is much lower than that obtained with 0.01\u03bcg kg-1 over 3 min (Fig. 6.1.6). The CCK-8 doses listed in Table 6.1.2 are shown to be within the physiological range and promote smooth contraction and emptying of the gallbladder. For an identical dose, dose rate, and duration, it appears that ceruletide may be more potent than CCK-8. For a 3-min infusion of 5 ng kg-1, the gallbladder mean ejection fraction is 61% with ceruletide and 38% with CCK-8 (Fig. 6.1.6).\n\nFig. 6.1.5\n\nDose response curve. The gallbladder ejection fraction increases as the dose of ceruletide (or CCK-8) increases [32]\n\nFig. 6.1.6\n\nEffect of non-physiologic CCK-8 or ceruletide dose on gallbladder emptying. After the peak response at 5 ng kg-1 3 min for ceruletide and 10 ng kg-1 3 min for CCK-8, the gallbladder ejection fraction begins to decrease for any further increase in dose [31]\n\nTable 6.1.2\n\nCholecystokinin-33 (CCK-33), octa-peptide of cholecystokinin (CCK-8), and ceruletide dose for measurement of gallbladder ejection fraction\n\nHormone (Ref) | Route | Dose rate | Trade name | Manufacturer | GBEF (%)\n\n---|---|---|---|---|---\n\nCCK-33 (PI) | IV | 1 IDU\/kg\/min | CCK-33\/Kabi | Pharmacia Laboratory, Piscataway, NJ | -\n\nCCK-8 [31] | IV | 10 ng kg-1 3 min | Kinevac | Bracco Diagnostics, Princeton, NJ | >35%\n\nCeruletide [32] | IV | 5 ng kg-1 3 min | Tymtran | Adria Laboratory, Columbus, OH | >40%\n\nCeruletide (PI) | I M | 300 ng kg-1 | Tymtran | Adria Laboratory Columbus, OH | -\n\nIDU = Ivy dog unit, PI = package insert\n\n#### 6.1.1.8 Effect of a Large Dose of CCK-8\n\nThe dose required during quantitative cholescintigraphy, in general, is found to be much lower than the hormonal dose recommended in the package insert (Kinevac) by the vendor. The package insert dose originally was meant for oral cholecystogram or for stimulating pancreatic enzyme secretion [33]. The recommended dose in the package insert (0.02 \u03bcg kg-1 or 20 ng kg-1) often produces abdominal pain and low ejection fraction (Fig. 6.1.7) when used in control subjects [34-35]. A low ejection fraction response obtained with a larger dose of CCK-8 is attributed to reaching the cystic duct smooth muscle threshold for contraction, with the resultant effect of non-emptying of the gallbladder [36-37]. Cholecystokinin dual action of antegrade intestinal peristalsis and simultaneous contraction of the pyloric sphincter normally produces forward bile flow through the intestinal lumen and prevents duodenal-gastric bile reflux [38-40].\n\nFig. 6.1.7\n\nEffect of sequential CCK-8 doses on gallbladder emptying. A 5 ng kg-1 3 min CCK-8 dose given sequentially four times on a single occasion, with 30 min between doses, produces similar ejection fractions. Note that a 20 ng kg-1 3-min CCK-8 dose given on a different day produces a lower ejection fraction [38]\n\n#### 6.1.1.9 Sequential CCK Doses\n\nThe gallbladder ejection fraction remains constant for a fixed dose of cholecystokinin or ceruletide, given on two separate occasions (Fig. 6.1.7). A second identical dose of CCK-8 or ceruletide given 20-30 min after the first dose produces an ejection fraction similar to that of the first dose. There is neither a potentiation nor inhibition effect from the first dose when a period of 20-30 min is allowed between doses. A short serum half life of 2.5 min does not seem to leave any significant residual CCK activity from the first dose to influence the effect of the second dose [41, 42]. This unique feature enables cholescintigraphy to study the effect of various drugs on the sphincter of Oddi and the gallbladder after a single dose of Tc-99 m-HIDA. The dose of CCK-8 (Kinevac, Bracco Laboratory, Princeton, NJ) and ceruletide (Tymtran, Adria Lab, Columbus, OH) is measured in microgram units, and cholecystokinin-33 (CCK\u2122, Pharmacia Laboratory, Piscataway, NJ) in Ivy dog units. One Ivy dog unit is defined as the amount of CCK-33, when injected intravenously over 10-15 s, that raises the pressure within the gallbladder by 1 cm of H2O [43].\n\n### 6.1.2 Sphincter of Oddi\n\nRugero Oddi in 1887 first proposed the sphincter mechanism at the distal end of the common bile duct that today bears his name [44]. The existence of the sphincter remained controversial for many years, when Boyden in 1937 put an end to the controversy by showing both macroscopic and microscopic details of the sphincter in both animals and humans [45, 46].\n\n#### 6.1.2.1 Structure and Function\n\nThe human sphincter of Oddi is about 10-15 mm in length, situated within the muscular layer of the media of the duodenum (Fig. 6.1.1). It consists of three distinct segments: (1) sphincter choledochus, (2) sphincter pancreaticus, and (3) ampullary sphincter. The choledochal sphincter covers the distal end of the intraduodenal part of the common bile duct before it joins with the pancreatic duct (duct of Wirsung). The pancreatic sphincter is located at the distal end of the pancreatic duct. The ampullary sphincter covers the distal end of both ducts after they unite to form a single common channel that opens into the duodenal lumen at an elevation called the ampulla of Vater [46]. The term \"sphincter of Oddi\" refers to all three sphincters.\n\nThe main function of the sphincter of Oddi is regulation of bile flow through it and prevention of the reflux of duodenal contents into the common bile duct and pancreatic duct. Internally, it prevents bile reflux into the pancreatic duct and reflux of pancreatic enzymes into the common bile duct. In the majority of humans (86%), the distal common bile duct and the distal pancreatic duct join together, forming a common channel of 10-12 mm length that opens into the duodenum at the papilla of Vater. In 6% of patients, the two ducts join together just before opening into the duodenum with a common channel. In the remaining 8% of patients, the two ducts open separately into the duodenal lumen at the ampulla of Vater [46]. The sphincter consists of a circular and a longitudinal layer of muscle. In humans, the entire sphincter is located within the duodenal wall, and the wall has to be cut opened to expose the sphincter of Oddi for clinical and experimental studies. In American and the Australian opossums, however, the entire sphincter is situated outside of the duodenal wall, making it easy to study its function, without any need to cut open the duodenal wall.\n\n#### 6.1.2.2 Sphincter Pressure\n\nThe basal pressure within the sphincter Oddi is 15-18 mmHg and rises with the arrival of periodic phasic waves. Phasic waves occur at an average of 4 waves\/min, and each wave lasts for 10-15 s (Fig. 6.1.8). During the passage of a phasic wave, the pressure within the sphincter of Oddi rises sharply, reaching a peak amplitude as high as 90-140 mmHg [47, 48]. Normally, 80% of phasic waves progress antegrade (towards the duodenum), 9% retrograde (towards the liver), and 13% occur simultaneously (Table 6.1.3) at the proximal, middle, and distal segments of the sphincter. The wave normally begins proximally and travels distally within the sphincter. The sphincter of Oddi basal pressure, phasic wave frequency, and direction of propagation remain unchanged after cholecystectomy. Wide variations in normal sphincter of Oddi pressures reported in the literature are often due to technical differences among the studies and should be taken into account when comparing results of one study with the other. Pressure changes within the sphincter of Oddi remain constant between repeat studies when a standardized technique is applied [49].\n\nFig. 6.1.8\n\nManometric pressure changes in the common bile duct (CBD) and sphincter of Oddi. Sphincter of Oddi basal pressure of 15 mmHg raises to as high as 100-150 mmHg at the peak of a phasic wave. Pressure within common bile duct remains at 15-20 mmHg [46]\n\nTable 6.1.3\n\nPressure and wave frequency, sequence, and amplitude changes in a normal sphincter of Oddi [48]\n\nParameter | Median | Range\n\n---|---|---\n\nBasal pressure (mmHg) | 15 | 5-35\n\nWave amplitude (mmHg) | 135 | 95-195\n\nWave frequency (no\/min) | 4 | 2-6\n\nWave sequence (%):\n\n|  |\n\nAntegrade | 80 | 12-100\n\nSimultaneous | 13 | 0-50\n\nRetrograde | 9 | 0-50\n\n#### 6.1.2.3 Action of Cholecystokinin on the Sphincter of Oddi\n\nCholecystokinin acts on the sphincter of Oddi smooth muscle and immediately abolishes the phasic wave activity (Fig. 6.1.9). The hormone reduces the sphincter of Oddi wave amplitude and pressure from a peak 130-140 mmHg to less than 10 mmHg pressure and reduces the wave frequency from four to less than one. Abolition of waves and reduction in wave amplitude decrease the pressure inside and simultaneously promote dilatation of the sphincter orifice [50, 51]. Reduction in wave amplitude and frequency reaches the nadir within 2-4 min after a single bolus injection of CCK-8, and the basal state is reestablished after 8-10 min, reflecting the effect of a short serum half life (2.5 min) of the hormone [11]. The effect on the sphincter is maintained throughout the duration of infusion of the hormone. The sphincter remains open for 2-3 h post-meal because of the longer duration of endogenous CCK release.\n\nFig. 6.1.9\n\nAction of CCK-8 on normal sphincter of Oddi. CCK-8 abolishes the phasic waves in the sphincter of Oddi and increases the number of waves and pressure in the duodenum [46]\n\n### 6.1.3 Gallbladder\n\nThe main function of the gallbladder is to store and concentrate (bile salts) bile during fasting and discharge into the duodenum soon after the arrival of food from the stomach. A normal gallbladder holds up to 50 ml and empties almost completely following a fatty meal. Since bile salts are very essential for efficient digestion and absorption of nutrients, the gallbladder sequesters almost all of them by selectively absorbing water and electrolytes through the wall during fasting. Absorption of water takes place through widely open lateral intercellular spaces between columnar epithelial cells lining the mucosa. Although several hormones, including cholecystokinin, are known to act on gallbladder contraction and emptying, until recently not much was known about its filling. A new hormone called fibroblast growth factor (FGF19) in the portal blood that facilitates gallbladder relaxation and filling was identified recently [52]. After discharge into the duodenum, bile salts travel through the jejunum and ileum, helping the digestion and absorption of nutrients. After absorption in the terminal ileum, bile salts activate nuclear farnesoid X receptor (FXR), which stimulates the production and release of FGF19 (Fig. 6.2.2). Acting through cyclic adenosine monophosphate (cAMP), FGF19 promotes gallbladder relaxation, increases its volume, and facilitates filling [53].\n\nReduction in gallbladder ejection fraction in patients with biliary dyskinesia is attri-buted to either a decrease in the total number of receptors in the body and fundus or a decrease in the threshold for contraction of the CCK receptors in the neck and cystic duct or both [54, 55]. Because of this phenomenon, it is essential to keep the CCK-8 dose within the physiological range during quantitative cholescintigraphy in the diagnosis of biliary dyskinesia. Cholecystokinin antagonists like loxiglumide, devazepide, and TP-680 decrease gallbladder emptying by competitively occupying the CCK-A receptor in the gallbladder smooth muscle. Animals given CCK antagonists show a reduction in gallbladder emptying and an increase in appetite and weight gain [14, 15]. Cholecystokinin protects gastric mucosal integrity through the release of somatostatin, and it relaxes the lower esophageal sphincter through activation of CCK-A receptors at the distal end of the esophagus [56, 57]. Controversy about its action on the pancreas, whether acting on the pancreatic acinar cells directly or acting indirectly through the vagus nerve (cholinergic), seems to have been settled in recent studies by using freshly prepared normal human pancreatic cells. At physiologic concentration, cholecystokinin in humans stimulates enzyme secretion by pancreatic acinar cells directly through calcium signaling and mitochondrial activation. Blockade by atropine and tetrodotoxin does not inhibit the direct action [58]. After a century of trials, some feel that a judgment can now be made about the direct action of cholecystokinin on the pancreatic acinar cells [59].\n\n## 6.2 Opioids\n\nOpioids are the mainstay in the management of moderate to severe intensity pain of diverse etiology. They are the drugs of first choice for the treatment of postoperative pain. Opioids are often mixed with other pain medications, including many non-steroidal anti-inflammatory drugs, such as aspirin acetaminophen, and neproxin. Opioids raise the pressure in the sphincter of Oddi by acting on its smooth muscle and thus interfere with cholecystokinin- or fatty meal-stimulated gallbladder ejection fractions. The action of many opioids on the sphincter lasts much longer than their serum half-life would indicate, suggesting that the metabolites may also possess a constrictive action on the sphincter of Oddi. The interaction between the opioids and the sphincter of Oddi calls for their careful scrutiny during quantitative cholescintigraphy.\n\n### 6.2.1 History of Opioids\n\n\"Opioid\" is a generic name that refers to both natural and synthetic compounds with a morphine-like action. They are derived from the opium plant, which in Greek means juice. The juice comes from the capsule of the unripe seed of Papaver Somniferum. The milky juice from the seed is dried and made into opium powder, which contains more than 20 alkaloids. Serturner first isolated a pure substance from this powder in 1806, which produced somnolence first, followed by a dream state. He called it morphine after the Greek god of dreams, Morpheus. Codeine was isolated from opium powder in 1832 and papaverine in 1848 [1]. The analgesic action of all opioids is graded with reference to morphine, which serves as the gold standard for pain control. Table 6.2.1 shows different opioids available for pain control and their effect on the sphincter of Oddi.\n\nTable 6.2.1\n\nEffect of opioids on the sphincter of Oddi. The drugs are grouped under four major categories into marked, moderate, mild, or no effect based on the extent of sphincter of Oddi pressure rise | Generic name | Proprietary name\n\n---|---|---\n\nMarked rise in sphincter pressure\n\n|\n\nMorphine | Morphine\n\n|\n\nLevorphanol | Dromoran\n\n|\n\nMeperidine (methadone) | Demerol\n\nModerate rise in sphincter pressure\n\n|\n\nDextromoramide | Palfium\n\n|\n\nDiacetyl morphine | Heroin\n\n|\n\nCodeine | Codeine\n\n|\n\nFentanyl | Sublimaze\n\n|\n\nButorphanol | Stadol\n\n|\n\nNalbuphine | Nubain\n\n|\n\nHydrocodone | Hycodon\n\n|\n\nOxycodone | Roxicodone\n\n|\n\nPropoxyphene | Darvon\n\n|\n\nBupremorphine | Buprenex\n\nMild rise in sphincter pressure\n\n|\n\nDextropropoxyphene HCl | Doloxene\n\n|\n\nPentazocine | Talwin\n\n|\n\nPhenazocine | Norphen\n\n|\n\nPhenoperidine | Peridine\n\nNo effect on the sphincter\n\n|\n\nHydroxyzine | Hydroxyzine\n\n### 6.2.2 Biokinetics of Morphine\n\nMorphine consists of an OH group at positions 3 and 6 and a CH3 group at 17 (Fig. 6.2.1). Other opioids differ structurally from morphine by having different chemical substitutions at positions 3, 6, and 17. Morphine is administered by oral, subcutaneous, intramuscular, or intravenous routes. Many dermal patches are now available for long-term continuous application. Intravenous injections are preferred for hospital patients, especially during the immediate postoperative period. Upon intravenous injection, morphine distributes in an initial volume of 71.8 l kg-1 and clears from plasma with biexponential components: component I with a half time of 0.16 h and component II with a half time of 2.5 h (Table 6.2.2). Morphine is metabolized in the liver and converted into morphine glucuronide, which clears from plasma much more slowly than the parent molecule. About 60% of injected morphine is excreted in urine in 24 h, and 73% in 3 days. The excretion rate does not change in patients with cirrhosis of the liver [2].\n\nFig. 6.2.1\n\nStructure of morphine. Note OH group at positions 3 and 6 and CH3 at position 17. The structure of other opioids differs mainly in having different chemical substitution at these positions\n\n### 6.2.3 Dose\n\nA minimum of 0.04 mg kg-1 of intravenous morphine is recommended during Tc-99 m-HIDA cholescintigraphy [3]. Intravenous doses as large as 15 mg are well tolerated [4]. In adults, a total intravenous dose of 3-4 mg of morphine sulfate is usually adequate for confirmation of acute cholecystitis with cholescintigraphy. Liver extracts 60-80% of morphine on first pass and converts rapidly into a glucuronide form. Because of high extraction and rapid metabolism in the liver, oral morphine is not as effective as an intravenous dose.\n\n### 6.2.4 Action of Morphine on the Sphincter of Oddi\n\nMorphine acts on the sphincter of Oddi within 2-3 min after an intravenous injection, and increases the sphincter basal wave frequency from 4 to 10-12 min-1 and the wave amplitude from 70 mmHg to 136 mmHg (Fig. 6.2.2). The common bile duct basal pressure increases from 10 mmHg to 29 mmHg [4]. Low doses increase only the rate and amplitude, whereas higher doses increase basal pressure as well [5]. During a Tc-99 m-HIDA study, most gallbladders with a patent cystic duct are visualized within 5-15 min after an intravenous dose of morphine. The effect of morphine on the sphincter of Oddi is mediated by all types of opioid receptors. Naloxone and atropine slightly modify, but do not completely abolish the effect of morphine on the sphincter.\n\nFig. 6.2.2\n\nEffect of morphine on the sphincter of Oddi. Morphine increases both the number and the amplitude of wave pressure throughout the sphincter of Oddi and forces hepatic bile entry into the gallbladder when the cystic duct is patent [4]\n\nMorphine is the most potent of all opioids. The effect of other opioids on the sphincter of Oddi is a function of their structural configuration (Table 6.2.2). The effect of levorphonol, dextromoramide, and meperidine on the sphincter of Oddi is equivalent to that of morphine [6, 7]. Fentanyl, nolbuphine, hydrocodon, propoxyphene, and bupremorphine show a moderate spasmodic effect on the sphincter. Pentazocine, phenozocine, and phenoperidine show a mild effect, and hydroxyzine has no effect on the sphincter of Oddi [8].\n\nTable 6.2.2\n\nPharmacokinetics of morphine [2]\n\nInitial volume of distribution (l kg-1) | 71.8 \u00b1 65.2\n\n---|---\n\nClearance half time of the fast component (h) | 0.16 \u00b1 0.1\n\nClearance half time of the slow component (h) | 2.5 \u00b1 1.5\n\nPlasma protein binding (%) | 19.8 \u00b1 6.1\n\nUrinary excretion in 72 h (% dose) | 72.9 \u00b1 10.0\n\n### 6.2.5 Screening for Opioid Intake\n\nQuantitative cholescintigraphy is frequently used in the diagnosis of biliary dyskinesia, which consists of two disease entities: cystic duct syndrome (CDS) or chronic acalculous cholecystitis (CAC) and sphincter of Oddi spasm (SOS). The most consistent quantitative functional abnormality in patients with biliary dyskinesia is a reduction in gallbladder ejection fraction in the case of cystic duct syndrome and bile reflux into the hepatic duct followed by a rapid refilling of the gallbladder in the case of sphincter of Oddi spasm [9]. Because of the spasmodic effect on the sphincter of Oddi, it is critical to ensure that the patient is not on any opioids prior to performing a quantitative cholescintigraphy [10]. This issue becomes even more complex in outpatients where many patients are unaware that a non-steroidal anti-inflammatory drug they take for various indications (arthritis, headache, peptic ulcer, or primary or metastatic cancer) may also contain an opioid. This problem is relatively easy to recognize in hospitalized patients by careful review of charts for a list of current medications.\n\nAs only about 60-75% of the administered dose of morphine is excreted in 24-72 h urine, a safe rule is to allow a minimum of 24 h for inpatients and 48 h for outpatients to be off of opioids before performing a quantitative cholescintigraphy for the diagnosis of biliary dyskinesia [2]. If the ejection fraction is within normal range in a patient who has received an opioid, then there is no need to repeat the test. The test should be repeated if the ejection fraction is low and the patient has received an opioid within the last 48 h. The ejection fraction measured after discontinuation of an opioid for longer than 48 h often shows a dramatic increase in value when compared to the value before discontinuation (Fig. 6.2.3).\n\nFig. 6.2.3\n\nEffect of CCK-8 on gallbladder ejection fraction with and without opioids. Note an ejection fraction of 21.16% while on hydrocodone and a value of 41.88% while off of hydrocodone\n\n### 6.2.6 Morphine or Cholecystokinin: One, Neither, or Both?\n\nVarieties of protocols are used in the performance of a Tc-99 m HIDA study. The clinical indication for a Tc-99 m-HIDA study is established first in order to determine when to give or not to give morphine or cholecystokinin [11, 12]. In patients with suspected acute cholecystitis, giving morphine before or during cholescintigraphy is appropriate. When the gallbladder is not seen in the clinical setting of acute cholecystitis, giving morphine at 60 min reduces the total time required for final diagnosis. In a patient with suspected biliary dyskinesia, however, it is inappropriate to give morphine or cholecystokinin prior to cholescintigraphy. In suspected biliary dyskinesia, it is better to wait and take delayed images at 2-4 h than to give morphine at 60 min after injection of Tc-99 m-HIDA. If the gallbladder fills late without morphine, then cholecystokinin is given to measure the gallbladder ejection fraction to rule out biliary dyskinesia. Poor emptying of the gallbladder is a characteristic feature of biliary dyskinesia. The gallbladder also empties poorly in patients given an opioid (Fig. 6.2.3). Questions often arise about whether one should measure the ejection fraction with CCK when the gallbladder appears after morphine administration. In some patients given morphine, the gallbladder empties normally with CCK and thus excludes biliary dyskinesia [13]. Once the gallbladder ejection fraction reduces below the normal value, it never regains its normal function unless the reduction was due to prior opioid administration. Dual studies have confirmed that the ejection fraction is highly reproducible in both normal subjects and patients with biliary dyskinesia [14].\n\nReferences\n\n1.\n\nIvy AC, Oldberg E. 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Digestion 1976;14:415-423PubMedCrossRef\n\n28.\n\nWetzner SM, Vincent ME, Robbins AH. Ceruletide-assisted cholecystography: a clinical assessment. Radiology 1979;131:23-26PubMed\n\n29.\n\nHedner P, Rorsman G. On the mechanism of action for the effect of cholecystokinin on the choledochoduodenal junction in the cat. Acta Physiol Scand 1969;76:248-254PubMedCrossRef\n\n30.\n\nCourtney DF, Clanachan AS, Scott GW. Cholecystokinin constricts the canine cystic duct. Gastroenterology 1983;85:1154-1159PubMed\n\n31.\n\nTorsoli A, Romarino ML, Alessandrini A. Motility of the biliary tract. Rendic R Gastroenterol 1970;2:67\n\n32.\n\nUpp JR Jr, Nealon WH, Singh P, Fagan CJ, Jonas AS, Greeley GH Jr, Thompson JC. Correlation of cholecystokinin receptors with gallbladder contractility in patients with gallstones. Ann Surg 1987;205:641-648PubMedCrossRef\n\n33.\n\nSturdevant RA, Stern DH, Resin H, Isenberg JI. Effect of graded doses of octapeptide of cholecystokinin on gallbladder size in man. Gastroenterology 1973;64:452-456PubMed\n\n34.\n\nKrishnamurthy GT, Bobba VR, Kingston E. Optimization of octapeptide of cholecystokinin (OP-CCK) dose for gallbladder emptying. In: Raynaud C (ed) Proceedings of the III World Congress of Nuclear Medicine and Biology. Paris, France. Pergamon Press, 1982, pp 2244-2247\n\n35.\n\nKrishnamurthy GT, Turner FE, Mangham D, Bobba VR, White SA, Langrell K. Ceruletide intravenous dose-response study by a simplified scintigraphic technique. AJR Am J Roentgenol 1985;144:733-737PubMed\n\n36.\n\nKrishnamurthy S, Krishnamurthy GT. Gallbladder ejection fraction: a decade of progress and future promise (editorial). J Nucl Med 1991;32:542-544\n\n37.\n\nZeissman HA, Fahey FH, Hixson DJ. Calculation of a gallbladder ejection fraction: advantage of continuous sincalide infusion over the three-minute infusion method. J Nucl Med 1992;33:537-541\n\n38.\n\nToftdahl DB, Hojgaard L, Winkler K. Dynamic cholescintigraphy: induction and description of gallbladder emptying. J Nucl Med 1996;37:261-266PubMed\n\n39.\n\nValenzuela JE, Defilippi C. Inhibition of gastric emptying in humans by secretin, the octapeptide of cholecystokinin, and intraduodenal fat. Gastroenterology 1981;81:898-902PubMed\n\n40.\n\nFraser R, Fone D, Horowitz M, Dent J. Cholecystokinin octapeptide stimulates phasic and tonic pyloric motility in healthy humans. Gut 1993;34:33-37PubMedCrossRef\n\n41.\n\nKrishnamurthy GT, Bobba VR, Kingston E, Turner FE. Measurement of gallbladder emptying sequentially using a single dose of 99mTc-labeled hepatobiliary agent. Gastroenterology 1982;83:773-776PubMed\n\n42.\n\nSostre S, Canto MI, Kalloo AN. Gallbladder response to a second dose of cholecystokinin during the same imaging study. Eur J Nucl Med 1992;19:964-965PubMedCrossRef\n\n43.\n\nTorsoli A, Romarino ML, Colagrande C, Demaio G. Experiments with cholecystokinin. Acta Radiologica 1961;55:193-206PubMedCrossRef\n\n44.\n\nOddi R. D'une disposition a sphincter speciale del'ouverture du canal cholique. Arch Ital Biol 1887;8:317-322\n\n45.\n\nBoyden EA. The sphincter of Oddi in man and certain representative mammals. Surgery 1937;1:25-37\n\n46.\n\nBoyden EA. The anatomy of the choledochoduodenal junction in man. Surg Gynecol Obstet 1957;104:641-652PubMed\n\n47.\n\nGeenen JE, Hogan WJ, Dodds WJ, Stewart ET, Arndorfer RC. Intraluminal pressure recording from the human sphincter of Oddi. Gastroenterology 1980;78:317-324PubMed\n\n48.\n\nCarr-Locke DL, Gregg JA. Endoscopic manometry of pancreatic and biliary sphincter zones in man. Basal results in healthy volunteers. Dig Dis Sci 1981;26:7-15PubMedCrossRef\n\n49.\n\nThune A, Scicchitano J, Roberts-Thomson IC, Toouli J. Reproducibility of endoscopic sphincter of Oddi manometry. Dig Dis Sci 1991;36:1401-1405PubMedCrossRef\n\n50.\n\nToouli J, Roberts-Thompson C, Dent J, Lee J. Manometric disorders in patients with suspected sphincter of Oddi dysfunction. Gastroenterology 1985;88:1243-1250PubMed\n\n51.\n\nToouli J, Hogan WJ, Geenen JE, Dodds WJ, Arndorfer RC. Action of cholecystokinin-octapeptide on sphincter of Oddi basal pressure and phasic wave activity in humans. Surgery 1982;92:497-503PubMed\n\n52.\n\nChoi M, Moschetta A, Bookout AL, Peng L, Umetani M, Holstrom SR, et al. Identification of a hormonal basis for gallbladder filling. Nat Med 2006;12:1253-1255PubMedCrossRef\n\n53.\n\nPortincasa P, DiCaula A, Wang HH, Palasciano G, van Erpecum KJ, Moschetta A, Wang DQH. Coordinate regulation of gallbladder motor function in the gut-liver axis. Hepatology 2008;47:2112-2126PubMedCrossRef\n\n54.\n\nHogan WJ, Geenen JE. Biliary dyskinesia. Endoscopy 1988;20:179-183PubMedCrossRef\n\n55.\n\nKrishnamurthy S, Krishnamurthy GT. Biliary dyskinesia: role of the sphincter of Oddi, gallbladder, and cholecystokinin. J Nucl Med 1997;38:1824-1830PubMed\n\n56.\n\nKonturek JW, Hengst K, Konturek SJ, Sito E, Stachura J, Domschke W. Physiological role of cholecystokinin in gastroprotection in humans. Am J Gastroenterol 1998;93:2385-2390PubMedCrossRef\n\n57.\n\nBoeckxstaens GE, Hirsch DP, Fakhry N, Holloway RH, D'Amato M, Tytgat GNJ. Involvement of cholecystokinin-A receptors in transient lower esophageal sphincter relaxations triggered by gastric distension. Am J Gastroenterol 1998, 93:1823-1828PubMedCrossRef\n\n58.\n\nMurphy JA, Criddle DN, Sherwood M, Chvanov M, Mukherjee R, McLaughlin E, Booth D, et al. Direct activation of cytosolic Ca2 + signaling and enzyme secretion by cholecystokinin in human pancreatic acinar cells. Gastroenterology 2008;135:632-641PubMedCrossRef\n\n59.\n\nSaluja A, Longsdon C, Garg P. Direct versus indirect action of cholecystokinin on human pancreatic acinar cells: is it time for a judgment after a century of trial? Gastroenterology 2008;135:357-360PubMedCrossRef\n\nReferences\n\n1.\n\nJaffe JH, Martin WR. Opioid analgesics and antagonists. In: Gilman AG, Rall TW, Niles AJ, Taylor P (eds) Goodman and Gilman's: The pharmacological basis of therapeutics, 8th edn. Pergamon, New York, 1990, pp 485-521\n\n2.\n\nPatwardhan RV, Johnson RF, Hoyumpa A Jr, Sheehan JJ, Desmond PV, Wilkinson GR, Branch RA, Schenker S. Normal metabolism of morphine in cirrhosis. Gastroenterology 1981;81:1006-1011PubMed\n\n3.\n\nChoy D, Shi EC, McLean RG, Hoschl R, Murray IPC, Ham JM. Cholescintigraphy in acute cholecystitis: use of intravenous morphine. Radiology 1984;151:203-207PubMed\n\n4.\n\nHelm JF, Venu RP, Geenen JE, Hogan WJ, Dodds WJ, Toouli J, Arndorfer RC. Effects of morphine on the human sphincter of Oddi. Gut 1988;29:1402-1407PubMedCrossRef\n\n5.\n\nVenu R, Toouli J, Geenen JE, Hogan WJ, Helm J, Dodds WJ, Arndorfer RC. Effect of morphine on motor activity of the human sphincter of Oddi. Gastroenterology 1982;84:1342 (Abstract)\n\n6.\n\nJones RM, Fiddian-Green R, Knight PR. Narcotic induced choledochoduodenal sphincter spasm reversed by glucagon. Anesth Analg 1980;59:946-947PubMed\n\n7.\n\n7. Hopton DS, Torrance HB. Action of various new analgesic drugs on the human common bile duct. Gut 1967;8:296-300PubMedCrossRef\n\n8.\n\nChessick KC, Black S, Hoye SJ. Spasm and operative cholangiography. Arch Surg 1975;110:53-57PubMedCrossRef\n\n9.\n\nKrishnamurthy S, Krishnamurthy GT. Biliary dyskinesia: role of the sphincter of Oddi, gallbladder, and cholecystokinin. J Nucl Med 1997;38:1824-1830PubMed\n\n10.\n\nJoehl RJ, Koch KL, Nahrwold DL. Opioid drugs cause bile duct obstruction during hepatobiliary scans. Am J Surg 1984;147:134-138PubMedCrossRef90047-3)\n\n11.\n\nWestlake PJ, Hershfield NB, Kelly JK, Kloiber R, Lui R, Sutherland LR, Shaffer EA. Chronic right upper quadrant pain without gallstones: does HIDA scan predict outcome after cholecystectomy? Am J Gastroenterol 1990;85:986-990PubMed\n\n12.\n\nKrishnamurthy S, Krishnamurthy GT. Cholecystokinin and morphine pharmacological intervention during 99mTc-HIDA cholescintigraphy: a rational approach. Semin Nucl Med 1996;26:16-24PubMedCrossRef80013-4)\n\n13.\n\nAchong DM, Oates E. Normal gallbladder ejection fraction after morphine augmentation. Clin Nucl Med 1999;24:837-841PubMedCrossRef\n\n14.\n\nKrishnamurthy GT, Krishnamurthy S, Brown PH. Constancy and variability of gallbladder ejection fraction: impact on diagnosis and therapy. J Nucl Med 2004;45:1872-1877PubMed\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_7(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 7. Intrahepatic Cholestasis\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nCholestasis is retention of bile within the hepatobiliary system resulting in an accumulation of bile products in the body fluids. It is divided into intrahepatic and extrahepatic types, depending upon the location of the pathology. Intrahepatic cholestasis is defined traditionally as bile retention within the hepatocytes, bile canaliculi, cholangioles, small or large bile ducts down to and including the proximal two thirds of the right hepatic and left hepatic ducts. Extrahepatic cholestasis is bile stasis due to a pathology located at the distal third of the right hepatic and left hepatic ducts and beyond, down to and including the sphincter of Oddi. The previous classification into obstructive and non-obstructive jaundice is no longer popular clinically [1].\n\nCholestasis is retention of bile within the hepatobiliary system resulting in an accumulation of bile products in the body fluids. It is divided into intrahepatic and extrahepatic types, depending upon the location of the pathology. Intrahepatic cholestasis is defined traditionally as bile retention within the hepatocytes, bile canaliculi, cholangioles, small or large bile ducts down to and including the proximal two thirds of the right hepatic and left hepatic ducts. Extrahepatic cholestasis is bile stasis due to a pathology located at the distal third of the right hepatic and left hepatic ducts and beyond, down to and including the sphincter of Oddi. The previous classification into obstructive and non-obstructive jaundice is no longer popular clinically [1].\n\nBile secretion: Formation of bile is an osmotic pressure-dependent uptake and secretory process that occurs along the basolateral and canalicular domains of the hepatocyte, respectively. Of a total of approximately 600 ml of bile produced by the liver per day, 450 ml is secreted by the hepatocytes and the remaining 150 ml by cells lining the bile canaliculi and bile ducts (Chap. 2, Fig. 1). About 50% of bile secreted by the hepatocytes (225 ml) is bile acid dependent, and the remaining 50% (225 ml) is independent of bile acid secretion. The uptake of bile acids, non-bile acid organic anions, cations, and other solutes by the hepa-tocytes from blood in the space of Disse occurs along the basolateral border. Of the several mechanisms utilized for the solute uptake, Na+-K+ ATPase pump, Na+-BS\u2212 (bile salt) co-transport, sodium-hydrogen exchange, Na-HCO3 \u2212 symporter, and organic anion endocytosis play the major roles [2]. Bile salts (taurocholate) are the most important and abundant solutes in bile. Transport of bile salts from blood into hepatocytes is mediated by the sodium-taurocholate co-transporter (NTCP) system [3]. ATPase of the basolateral border regulates the extracellular and intracellular ion concentration gradient [4].\n\n## 7.1 Imaging with Tc-99m HIDA\n\nActive transport of solutes across the canalicular membrane is the rate-limiting step in the amount of bile produced per day. The tight junction between the two adjacent hepatocytes is functionally leaky (a leaky tight junction) and allows passage of water and electrolytes from blood into the canalicular lumen. The canalicular lumen situated between two adjacent hepatocytes is about 0.75 \u03bcm in diameter, and the wall contains microvilli of 500-1,000 nm in length and 100 nm in diameter. The canaliculi and their microvilli contract and propel bile towards larger bile ducts. Secretion of large lipophilic cations (anticancer drugs, calcium-channel blockers, cyclosporin A, and other drugs) is mediated by a transporter protein called multidrug-resistance-1 P-glycoprotein (MDR1). Secretion of phosphatidylcholine is facilitated by another protein called multidrug-resistance-3 P-glycoprotein, called MDR3. Transport of organic anions, including bromosulphalein, glucuronides, and possibly Tc-99m HIDA, across the canalicular membrane is attributed to multidrug resistance protein 2 (MRP2). Endotoxin lipopolysaccharide induces cholestasis by an early retrieval followed by down-regulation of MRP2, which moves from the canalicular membrane to the interior of the cell [5]. Inhibition of contraction of the canalicular wall and microvilli results in atony and dilatation of the canaliculi, leading to intrahepatic cholestasis (Fig. 7.1.1). Tc-99m-HIDA is secreted into canalicular bile in free form, without undergoing any biotransformation during its intracellular transit. This free status can be confirmed by reinjecting Tc-99m-HIDA-labeled gallbladder bile intravenously into the same animal, reproducing the original biokinetic parameters [6]. Reduction in uptake and slow transit through the hepatocytes, and collection in the dilated canaliculi produced a low hepatic extraction fraction and prolongation of excretion half-time in a Tc-99m HIDA study.\n\nFig. 7.1.1\n\nPathogenesis of intrahepatic cholestasis. Normal canaliculi and microvilli contract and propel bile forwards. Infections, drugs, and other toxins cause bile stasis by inactivating contractile function of the bile canaliculi and microvilli\n\n### 7.1.1 Etiology of Intrahepatic Cholestasis\n\nThree types of injury are recognized as being responsible for cholestasis: (1) direct injury, (2) immunologic, and (3) cholestatic. These injuries are caused by microorganisms, poisons, drugs, and metabolic, granulomatous, veno-occlusive, ischemic, and other diseases (Fig. 7.1.2). The injury may involve the endoplasmic reticulum, lysosomes, mitochondria of the hepatocytes or the canalicular cells. In the case of drugs, the injury may be caused directly by the offending agent or by one of its metabolic products [7].\n\nFig. 7.1.2\n\nLocation of intrahepatic vs. extrahepatic cholestasis. Intrahepatic cholestasis occurs between the hepatocyte and the proximal right hepatic and left hepatic ducts. Pathology of the extrahepatic cholestasis lies between the proximal right hepatic and left hepatic ducts and the sphincter of Oddi\n\n### 7.1.2 Viral Hepatitis\n\nAll types of viral hepatitis (A, B, C, D, E, and G) cause intrahepatic cholestasis, and all produce an identical image pattern. The diagnosis is usually made by getting a detailed clinical history and by obtaining a serum hepatitis profile. Bacterial and fungal infections and toxins (mushroom poisoning) also cause intrahepatic cholestasis primarily by affecting the hepatocytes and cholangioles. The bile secretion rate is reduced, hence bile ducts and the gallbladder appear late in a Tc-99m HIDA study (Fig. 7.1.3).\n\nFig. 7.1.3\n\nHepatitis A. Diffuse radiotracer retention with no bile secretion for 1 h in a patient with hepatitis A. Gallbladder is seen at 18 h, and the intestinal activity has moved away from the field of view of the gamma camera\n\n### 7.1.3 Drugs\n\nDrugs are the most common cause of intrahepatic cholestasis, which is classified into two major types [8]: (1) hypersensitive (exudative) and (2) canalicular. The hypersensitive type is characterized by fever, anorexia, marked eosinophilia, and increased levels of serum cholesterol and alkaline phosphatase. The canalicular type is characterized by intense pruritus, associated with a normal eosinophil count and serum alkaline phosphatase level [9]. Sulfonamides, quinidine, and allopurinol are incriminated in the hypersensitivity type of intrahepatic cholestasis. Sex hormones, phenothiazines, antibiotics, (erythromycin), and nitrofurantoin are common causes of the canalicular type of intrahepatic cholestasis. The drug or one of its metabolite secreted into bile canaliculi interferes with canalicular contraction, causing intrahepatic cholestasis [10].\n\nIt is estimated that about 2% of hospitalized patients in the USA suffer from some form of intrahepatic cholestasis related to a prescription drug. Up to 1% of patients receiving chlorpromazine develop intrahepatic cholestasis [9]. Various drugs that cause intrahepatic cholestasis are listed in Table 7.1.1. Most of the drugs react with cytochrome p450 and undergo conjugation with glucuronic acid, sulfate, amino acids, or glutathione prior to elimination from the body. The cytochrome p450 system is under genetic control and produces cytotoxic effects mainly through formation of electrophiles and free radicals [11, 12]. A detailed drug history aids in identifying the offending agent responsible for intrahepatic cholestasis. A cholescintigraphic image pattern remains non-specific in intrahepatic cholestasis [13].\n\nTable 7.1.1\n\nLocalization of drug-induced intrahepatic cholestasis [9]\n\nHepatocyte (A) | Canaliculus (B) | Hepatocyte and canaliculus (A + B) | Ducts (C)\n\n---|---|---|---\n\nCarbon tetrachloride | Estrogens | Phenytoin | Benoxyprofen\n\nHalothane | Testosterone | Quinidine | Ascending cholangitis\n\nChloroform | Phenothiazine | Allopurinol | Floxuridine\n\nAmiodarone | Erythromycin | Butazolidine | -\n\nMethyldopa | Nitrofurantoin | Thiobendozole | -\n\nIsoniazide | Azathioprine | Furasemide | -\n\nKetoconazol | Cyclosporin A | Sulfonamide | -\n\nAcetaminophen | Tranquillizers | - | -\n\nRifampin | Antipsychotic agents | - | -\n\nIndomethacin | - | - | -\n\nTetracycline | - | - | -\n\nValproic acid\n\n|  |  |\n\nFlexin\n\n|  |  |\n\nNovobiocin\n\n|  |  |\n\nCocaine\n\n|  |  |\n\nFlutamide\n\n|  |  |\n\n### 7.1.4 Metabolic Causes\n\nHemochromatosis due to iron overload, Wilson's disease due to copper overload, and fatty infiltration and deficiency of several enzymes are known metabolic causes of intrahepatic cholestasis [14, 15].\n\n### 7.1.5 Alcoholic Hepatitis\n\nAlcohol is metabolized primarily in the liver and converted into acetaldehyde through the enzyme alcohol dehydrogenase. Acetaldehyde in turn is converted into acetate by the enzyme aldehyde dehydrogenase. Each gram of alcohol produces seven calories of energy. Most of the liver injury from alcohol occurs primarily due to its metabolite, acetaldehyde, whose accumulation exerts a toxic effect on the plasma membrane, tubulin, and other cytoskeletons of the hepatocyte [16]. When alcohol intake continues for many years, other toxic effects are manifested in the form of fatty infiltration, fibrosis, and ultimately cirrhosis of the liver and portal hypertension.\n\n### 7.1.6 Primary Biliary Cirrhosis (PBC)\n\nIt is a disease of unknown etiology and affects primarily the canaliculi, cholangioles, and small bile ducts. Clinically, PBC presents with pruritus and malaise accompanied by elevation of serum liver enzymes and lipoproteins. It is more common in middle-aged women than men. Serum anti-mitochondrial antibody is found in almost 100% of the patients [17]. Both planar and SPECT cholescintigraphy with Tc-99m-HIDA shows diffuse parenchymal retention with uniform prolongation of excretion half time from all regions of the liver. The extrahepatic biliary tree remains normal, a common feature for all intrahepatic cholestasis. Cholescintigraphic features enable differentiation of PBC from primary sclerosing cholangitis, where the parenchymal retention is non-uniform [18]. The gallbladder remains normal in size, but shows a reduction of CCK-8 stimulated ejection fraction. In a middle-aged woman with a positive serum antimitochondrial antibody who clinically presents with pruritus and malaise, a Tc-99m-HIDA study is confirmatory for PBC if it shows a diffuse prolongation of excretion half time and no obstruction of the extrahepatic biliary tree. A normal extrahepatic biliary tree may even preclude the need for a contrast cholangiogram to exclude primary sclerosing cholangitis, which PBC often clinically mimics [18]. Characteristic cholescintigraphic features of primary sclerosing cholangitis and primary biliary cirrhosis enable differentiation between two relatively rare conditions [18, 19].\n\n### 7.1.7 Benign Recurrent Intrahepatic Cholestasis (BRIC)\n\nBRIC is a rare disorder affecting patients in their 30s and is often found in several members of the same family. Intense pruritus with elevation of liver enzymes, accompanied by spontaneous recovery, is its characteristic feature. The extrahepatic biliary tree is normal [20]. There are no identifiable serum markers for BRIC. The hepatocytes are unable to get rid of bile acids and other organic anions from the body [21]. Uptake of Tc-99m-HIDA by the hepatocyte is normal, but secretion into bile is delayed or absent, resulting in nondelineation of the entire biliary system, often mimicking total acute common bile duct obstruction. The image pattern appears much like a radiocolloid scan without the spleen.\n\n### 7.1.8 Total Parenteral Nutrition (TPN)\n\nThe major liver enzyme abnormalities in patients with TPN are elevation of gamma glutamyl transferase, ALT, and alkaline phosphatase. Peak enzyme elevation occurs 1-4 weeks after initiation of TPN. Early pathological changes include fatty infiltration in adults and intrahepatic cholestasis in children. When TPN is continued for many years, fibrosis and cirrhosis set in [22]. Uptake of Tc-99m-HIDA by the hepatocyte is maintained, but secretion is markedly delayed or even absent, mimicking total CBD obstruction. The gallbladder is often non-visualized [23].\n\n### 7.1.9 Ischemic Hepatitis\n\nThis condition is often due to an acute reversible hypotension or cardiac failure. Following the ischemic episode there is a rapid rise in aspartate amino transferase and lactic dehydrogenase, and mild elevation of bilirubin and glucose. Liver enzyme abnormality returns to normal, usually within 8-10 days [24]. Functional abnormalities tend to be much more severe in intrahepatic cholestasis than in patients with early partial obstruction of the common bile duct [25].\n\n### 7.1.10 Acute Cholangitis\n\nAcute cholangitis clinically presents as a Charcot triad, which consists of fever with chills, jaundice, and abdominal pain, as described originally by Charcot in 1877 [26]. Obstruction of the common bile duct first initiates the process and then sustains acute cholangitis. Obstruction leads to bacterial overgrowth and reflux of infected bile from the liver into the blood stream. Up to 90% of patients may remain asymptomatic when the obstruction of the common bile duct is not critical or complete [27, 28]. Charcot triad sets in mostly when the obstruction becomes complete. Both gram-positive and gram-negative organisms are found with equal frequency. Serum bilirubin rises above 5 mg% in more than 50% of the patients. Leucocytosis is frequent (63%), and elevation of serum amylase occurs in 30% of the patients [29]. Elevation of serum bilirubin above 5 mg% is a common feature of acute cholangitis, but not of acute cholecystitis (Table 7.1.2).\n\nTable 7.1.2\n\nClinical and laboratory findings in acute cholangitis vs. acute cholecystitis [29]\n\nParameter | Acute cholangitis | Acute cholecystitis\n\n---|---|---\n\nClinical features:\n\nMean age (years) | 60 | 50\n\nMale:female | 1:1 | 3:2\n\nFever (%) | 90 | 90\n\nChills (%) | 75 | 50\n\nJaundice (%) | 95 | 30\n\nRUQ pain (%) | 80 | 95\n\nRUQ tenderness (%) | 80 | 95\n\nLaboratory findings:\n\nWBC >10,000 (%) | 70 | 70\n\nWBC >20,000 (%) | 10 | 5\n\nBilirubin >1.5 mg (%) | 95 | 30\n\nBilirubin >5.0 mg (%) | 55 | 5\n\nElevation of alkaline phosphatase (%) | 90 | 40\n\nIncrease in SGOT and SGPT (%) | 95 | 50\n\nIncrease in serum amylase (%) | 35 | 15\n\nIn the past, choledocholithiasis and postoperative stricture were the most common causes of acute cholangitis. In series reported prior to 1980, stone or postoperative stricture accounted for almost 90% of acute cholangitis; now they account for only 26%. Instru-mentation (ERCP) of the biliary tract is now the most common cause of acute cholangitis (35%), indicating a shift in the etiology of acute cholangitis in more recent years [29].\n\n### 7.1.11 Scintigraphic Features of Intrahepatic Cholestasis\n\nInjury to the hepatocyte, bile canaliculi, and small bile ducts produces a nonspecific image pattern in a Tc-99m-HIDA study. Location of the exact site of injury among these three cannot be made from the images. An etiologic diagnosis, therefore, requires a thorough knowledge of clinical presentation, liver function tests, and a serum viral profile for hepatitis. The diseases that combine the features of both intrahepatic and extrahepatic cholestasis (e.g., sclerosing cholangitis) have a very characteristic image presentation and often enable an etiologic diagnosis [18, 19].\n\nThe dose of Tc-99m-HIDA for imaging is increased to 5 and 10 mCi to compensate for a decrease in hepatocyte uptake secondary to a high serum bilirubin, which competes with Tc-99m-HIDA. The hepatic extraction fraction decreases and excretion half time increases in direct proportion to an elevation of serum bilirubin (Fig. 7.1.4). Delayed images at 4 or 24 h are necessary to show intestinal activity, which establishes patency of the bile duct. In a patient who presents clinically with a Charcot triad, nonvisualization of the gallbladder with features of intrahepatic cholestasis in a TC-99m-HIDA study suggests acute cholangitis (Fig. 7.1.5).\n\nFig. 7.1.4\n\nCholescintigraphic features of intrahepatic cholestasis. Hepatocytes show decreased extraction and delayed excretion of Tc-99m-HIDA. Images are shown between 20 and 36 min (a). Hepatic extraction is only 28% (normal, 92-100%), and T 1\/2 excretion is 278 min (normal, 11-33 min) (b)\n\nFig. 7.1.5\n\nAcute cholangitis secondary to acute cholecystitis. The gallbladder is non-visualized due to obstruction of the cystic duct, and there is diffuse retention at 4 and 16 h. Common bile duct is patent and allows bile entry into small intestine\n\n## 7.2 Imaging with Tc-99m Galactosyl Human Serum Albumin\n\nIdentification of unique asialoglycoprotein (ASGP) receptors on the plasma membrane of the hepatocyte [1] and introduction of technetium-99m-DTPA-galactosyl-human-serum albumin (Tc-99m GSA), which binds to it, allow non-invasive functional imaging and quantification. Asialoglycoprotein receptor concentration on the plasma membrane is an indicator of the functional integrity of the hepatocyte. The quantity of Tc-99m GSA uptake correlates inversely with blood retention of indocyanin green [2]. Liver uptake of Tc-99 GSA decreases in patients with chronic hepatitis, cirrhosis, cholangiocarcinoma, hepatocellular cancer, metastasis, fulminant hepatic failure, and space-occupying benign lesions of the liver [2-5].\n\nData collection and analysis: Most of the clinical studies reported in the literature are from Japan, and the agent is not available in most other countries, including the United States. Functional imaging and quantification require standardization to keep the variables to a minimum. Patient fasting, which is not necessary during imaging with Tc-99m-S-colloid, is very essential for imaging with Tc-99m GSA, because the uptake may be variable due to postprandial hyperperfusion of the GI tract. After 4-6 h of fasting, the patient is positioned supine underneath a gamma camera (single, double, or a triple head) fitted with a low-energy, high-resolution, parallel-hole collimator that is positioned anterior to the liver. Sequential anterior planar images (128 \u00d7 128) at one frame per 30 s for 20 min are obtained immediately after a bolus injection of 5 mCi Tc-99m GSA (185 MBq) into the antecubital vein. Immediately after the planar images, SPECT data are acquired on a 128 \u00d7 128 \u00d7 16 computer matrix for 64 stops at 10 s per stop at 5.6\u00b0 intervals [2]. The spectrometer is set for 140 keV at a 20% window. The liver uptake index at 15 min is calculated as described in Chap. 5.\n\nSome authors express uptake as hepatic GSA clearance using the Patlak plot method and generate functional images of clearance [3, 4]. The receptor index decreases in patients with chronic hepatitis and cirrhosis (Fig. 7.2.1). Since the cells other than the hepatocytes do not take up any Tc-99m GSA, preoperative imaging allows accurate measurement and prediction of residual liver function postoperatively after resection of hepatocellular carcinoma, cholangiocarcinoma, and metastatic liver tumors (Fig. 7.2.2). It is not known at this time if this functional parameter can serve as an ideal marker for end-stage liver disease and timing of liver transplantation.\n\nFig. 7.2.1\n\nRelationship between Tc-99m GSA receptor index and liver disease. Normal hepatocytes rich in ASGP receptors show high index. Reduction of Tc-99m GSA receptor index occurs in patients with chronic hepatitis and cirrhosis of liver [4]. GSA galactosyl human serum albumin\n\nFig. 7.2.2\n\nAbsence of Tc-99m GSA uptake by tumors. CT with contrast shows dilatation of bile ducts in the left lobe (a, b, arrow) of a patient with cholangiocarcinoma. T2-weighted MR shows hyper-intensity of left lobe bile ducts (c, d, arrow). Liver SPECT shows no uptake of Tc-99m GSA (e) by the left lobe of. GSA galactosyl human serum albumin [3]\n\nReferences\n\n1.\n\nSherlock S, Dooley J (1997) Cholestasis. In: Diseases of the liver and biliary system, 10th edn. Blackwell Science Inc, Malden, MA, pp 217-237\n\n2.\n\nTrauner M, Meier PJ, Boyer JL. Molecular pathogenesis of cholestasis. N Eng J Med 1998;339:1217-1227CrossRef\n\n3.\n\nHagenbuch B, Meier PJ. Sinusoidal (basolateral) bile salt uptake system of hepatocytes. Semin Liver Dis 1996;16:129-136PubMedCrossRef\n\n4.\n\nSellinger M, Barrett C, Malle P, Gordon ER, Boyer JL. Cryptic Na+, K(+ ) -ATPase activity in rat liver canalicular plasma membrane: evidence for its basolateral origin. Hepatology 1990;11:223-229PubMedCrossRef\n\n5.\n\nKubitz R, Wettstein M, Warskulat U, Haussinger D. Regulation of the multidrug resistance protein 2 in the rat liver by lipopolysaccharide and dexamethasone. Gastroenterology 1999;116:401-410PubMedCrossRef70138-1)\n\n6.\n\nLoberg MD, Cooper M, Harvey E, Callery P, Faith W. Development of new radiopharmaceuticals based on N-substitution of iminodiacetic acid. J Nucl Med 1976;17:633-638PubMed\n\n7.\n\nSherlock S. Patterns of hepatic injury in man. Lancet 1982;1(8275):782-786PubMedCrossRef91822-0)\n\n8.\n\nLee WM. Drug-induced hepatotoxicity. N Engl J Med 1995;333:1118-1128PubMedCrossRef\n\n9.\n\nSherlock S, Dooley J (1997) Drugs and the liver. In: Diseases of the liver and biliary system, 10th edn. Blackwell Science Inc, Malden, MA, pp 337-369\n\n10.\n\nCorcoran GB, Racz WJ, Smith CV, Mitchell JR. Effects of N-acetylcysteine on acetaminophen covalent binding and hepatic necrosis in mice. J Pharmacology Exp Ther 1985;232:864-872\n\n11.\n\nSmith CV, Hughes H, Mitchell JR. Free radicals in-vivo. Covalent binding to lipids. Mol Pharmacol 1984;26:112-116PubMed\n\n12.\n\nSeeff LB, Cuccherini BA, Zimmerman HJ, Adler E, Benjamin SB. Acetaminophen hepatotoxi-city in alcoholics. Ann Intern Med 1986;104:399-404PubMed\n\n13.\n\nLieberman DA, Krishnamurthy GT. Intrahepatic versus extrahepatic cholestasis. Discrimination with biliary scintigraphy combined with ultrasound. Gastroenterology 1986;90:734-743PubMed\n\n14.\n\nPowell LW, Halliday JW, Cowlishaw JL. Relationship between serum ferritin and total body iron stores in idiopathic hemochromatosis. Gut 1978;19:538-542PubMedCrossRef\n\n15.\n\nBearn AG. Wilson's disease. An inborn error of metabolism with multiple manifestations. Am J Med 1957;22:747-753PubMedCrossRef90125-0)\n\n16.\n\nLauterburg BH, Bilzer H. Mechanisms of acetaldehyde hepatotoxicity. J Hepatol 1988;7:384-390PubMedCrossRef80012-6)\n\n17.\n\nWalker JG, Doniach D, Roitt IM, Sherlock S. Serological tests in diagnosis of primary biliary cirrhosis. Lancet 1965;1:827-831PubMedCrossRef91372-3)\n\n18.\n\nKeeffe EB, Lieberman DA, Krishnamurthy S, Krishnamurthy GT, Gilbert SA. Primary biliary cirrhosis: Tc-99m-HIDA planar and SPECT scanning. Radiology 1988;166:143-148PubMed\n\n19.\n\nRodman CA, Keeffe EB, Lieberman DA, Krishnamurthy S, Krishnamurthy GT, Gilbert SA, Eklem MJ. Diagnosis of sclerosing cholangitis with Tc-99m-labeled iminodiacetic acid planar and single photon emission computed tomographic scintigraphy. Gastroenterology 1987;92:777-785PubMed\n\n20.\n\nLesser PB. Benign familial recurrent intrahepatic cholestasis. Am J Dig Dis 1973;18:259-264PubMedCrossRef\n\n21.\n\nMinuk GY, Shaffer EA. Benign recurrent intrahepatic cholestasis. Evidence for an intrinsic abnormality in hepatocyte secretion. Gastroenterology 1987;93:1187-1193PubMed\n\n22.\n\nQuigley EM, Marsh MN, Shaffer JL, Markin RS. Hepatobiliary complications of total parenteral nutrition. Gastroenterology 1993;104:286-301PubMed\n\n23.\n\nShuman WP, Gibbs P, Rudd TG, Mack LA. PIPIDA scintigraphy for cholecystitis: False-positives in alcoholism and total parenteral nutrition. Rm J Roentgenol 1982;138:1-5\n\n24.\n\nGitlin N, Serio KM. Ischemic hepatitis: Widening horizons. Am J Gastroenterol 1992;87:831-836PubMed\n\n25.\n\nLima J, Brown R, Krishnamurthy S, Krishnamurthy GT. Reliability of quantitative Tc-99m-mebrofenin scintigraphy (QMS) in the diagnosis of intrahepatic cholestasis(IHC). J Nucl Med 1991;32:976-977\n\n26.\n\nCharcot JM: Lecons sur les maladies du foie des voies biliaires at des veins. Paris Faculte' de medecine. Recueilleies et publices. Bournelle et Senestre, 1877\n\n27.\n\nFlemma RJ, Flint LM, Osterhout S. Bacteriologic studies of biliary tract infection. Ann Surg 1967;166:563-572PubMedCrossRef\n\n28.\n\nPitt HA, Postier RG, Cameron JL. Biliary bacteria: significance and alterations after antibiotic therapy. Arch Surg 1982;117:445-450PubMedCrossRef\n\n29.\n\nPitt HA, Cameron JL. Acute cholangitis. In: Way LW, Pelligrini CA, eds. Surgery of the gallbladder and bile ducts. WB Saunders, Philadelphia, 1987, pp 295-313\n\nReferences\n\n1.\n\nSteer CJ. Receptor-mediated endocytosis: mechanisms, biologic function, and molecular properties. In: Zakim D, Boyer TD (eds) Hepatology. A textbook of liver disease. WB Saunders, Philadelphia, 1996, pp 149-214\n\n2.\n\nUetake M, Koizumi K, Yagawa A, Nogata H, Tezuka T, Kono H, Ozawa T, Kusano T, Miyaburuko M, Hosaka M. Use of Tc-99m DTPA galactosyl human serum albumin to predict postoperative residual liver function. Clin Nucl Med 1999;24:428-434PubMedCrossRef\n\n3.\n\nHwang E, Taki J, Shuke N, Nakajima K, Kinuya S, Konishi S, Michigishi T, Aburano T, Tonami N. Preoperative assessment of residual hepatic functional reserve using Tc-99m-DTPA-galactosyl-human-serum albumin dynamic SPECT. J Nucl Med 1999;40:1644-1651PubMed\n\n4.\n\nSasaki N, Shiomi S, Iwata Y, Nishiguchi S, Kuroki T, Kawabe J, Ochi H. Clinical usefulness of scintigraphy with Tc-99m-galactosyl-human serum albumin for prognosis of cirrhosis of the liver. J Nucl Med 1999;10:1652-1656\n\n5.\n\nKoizumi K, Monzawa S, Shindo C, Hosaka M. Primary hepatic amyloidosis well delineated by Tc-99m DTPA galactosyl HSA liver SPECT. Clin Nucl Med 1999;24:271-273PubMedCrossRef\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_8(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 8. Extrahepatic Cholestasis\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nExtrahepatic cholestasis is the retention of bile products within the body secondary to a pathology located outside of the liver parenchyma, usually beyond the middle third of the right and left hepatic ducts, including the common hepatic and common bile ducts. Causes of extrahepatic cholestasis can be divided into four major categories depending upon the exact site of location of the pathologic process: (1) intraluminal, (2) wall thickening, (3) extrinsic compression, and (4) combined intrahepatic and extrahepatic causes (Table 8.1.1). The mode of clinical presentation depends very much upon the duration, degree of obstruction, and exact site of pathology\n\nExtrahepatic cholestasis is the retention of bile products within the body secondary to a pathology located outside of the liver parenchyma, usually beyond the middle third of the right and left hepatic ducts, including the common hepatic and common bile ducts. Causes of extrahepatic cholestasis can be divided into four major categories depending upon the exact site of location of the pathologic process: (1) intraluminal, (2) wall thickening, (3) extrinsic compression, and (4) combined intrahepatic and extrahepatic causes (Table 8.1.1). The mode of clinical presentation depends very much upon the duration, degree of obstruction, and exact site of pathology.\n\nTable 8.1.1\n\nCauses of extrahepatic cholestasis\n\n(1) Intraluminal causes\n\n---\n\n(a) Gallstones (choledocholithiasis)\n\n(b) Hemobilia (blood clots)\n\n(c) Parasites (round worm)\n\n(d) Inspissated bile or concretions\n\n(e) Miscellaneous causes\n\nTumor emboli\n\nPapilloma\n\nAdenomyoma\n\nFibroma\n\nCystedenoma\n\n(2) Wall thickening\n\n(a) Benign (stricture)\n\n(b) Malignant (cholangiocarcinoma)\n\n(3) Combined intrahepatic and extrahepatic cholestasis\n\nSclerosing cholangitis\n\n(4) Extrinsic compression\n\nNormally, the biliary structures appear in a sequential pattern on a Tc-99m HIDA study; the right and left hepatic ducts appear first, followed by the common hepatic duct, gallbladder, common bile duct, and duodenum. Quantitative parameters show the normal hepatic extraction fraction and excretion half-time (Fig. 8.1.1). Ducts proximal to the right and left hepatic ducts are not usually seen clearly. They become prominent only when there is obstruction to bile flow distally.\n\nFig. 8.1.1\n\nNormal cholescintigraphy. Liver shows excellent uptake and excretion of Tc-99 HIDA. The right hepatic duct and left hepatic ducts, common hepatic duct, and gallbladder are seen early (top left), followed by the common bile duct and duodenum (top right). Hepatic extraction fraction (HEF) and excretion values are normal (bottom)\n\n## 8.1 Intraluminal Causes\n\n### 8.1.1 Clinical Presentation\n\nIntraluminal obstruction is usually caused by gallstones, inspissated bile, parasites, blood clots, or pedunculated tumors arising from the wall (Table 8.1.1). The onset of signs and symptoms depends upon the degree and duration of obstruction. A patient with acute complete obstruction of the common bile duct usually presents with sudden onset of epigastric or right upper quadrant pain, which often radiates to the chest, shoulder, or interscapular region in the back. When complete obstruction persists for longer than 3 days, acute cholangitis may set in with onset of clinical Charcot triad, which consists of fever with chills, jaundice, and biliary colic. The time for the serum liver function tests to become abnormal after the onset of obstruction depends upon the presence or absence of the gallbladder; in the absence of the gallbladder, the liver function tests become abnormal within 4-6 h, whereas they may take as long as 48-72 h with an intact gallbladder. In presence of the gallbladder, the liver function tests become abnormal much earlier when the level of obstruction lies proximal to the junction of the cystic duct with the common hepatic duct than when the obstruction lies distal to their junction. Patients with partial bile duct obstruction usually make a benign clinical presentation.\n\n### 8.1.2 Choledocholithiasis\n\nFormation of gallstones is initiated with secretion of lithogenic bile, which is the first indication of a breakdown of normal mechanisms responsible for keeping bile solutes in solution [1]. The majority of gallstones are composed of cholesterol, and few consist of bile pigments. The stone may form de novo within the common bile duct (primary stone), or it may originate in the gallbladder or liver and then move down into the common bile duct (secondary stone). About 85-90% of all stones are secondary, and the rest are primary. Secondary stones are more common in western China, Hong Kong, and other Asian countries. In certain ethnic groups, as much as 50% of the CBD stones are primary [2].\n\n### 8.1.3 Pathophysiology of Bile Duct Obstruction\n\nThe pathophysiologic changes that follow bile duct obstruction depend upon several factors, of which the mode of onset (sudden vs. gradual) and severity (partial vs. complete) are the most important. Much of our knowledge about post-obstructive pathophysiologic changes is derived from ligation of the common bile duct in experimental animals. Following complete ligation of the common bile duct in the rat, the serum alkaline phosphatase and alanine aminotransferase (ALT) levels begin to rise rapidly within 5 h, reach peak levels by 20 h, and later begin to decline to reach a steady state by 4 days. Serum bilirubin begins to rise by 5 h, reaches peak levels by 4 days, and then begins to decline to reach a new steady state level, which is often set at a much lower level than the peak. The serum alkaline phosphatase level begins to rise and reach peak levels much earlier than the serum bilirubin level. The number of bile canaliculi doubles in 5 h and then begins to decrease after 4 days. The hepatocytes shrink in size and reach a steady state by 2 days. Post-obstructive shrinkage of the hepatocytes and swelling of bile canaliculi together lead to a bile leak into the blood stream [3]. Bile flow decreases to 65% of the basal value when the pressure within the ducts exceeds 16-17 cm of water, and the bile secretion ceases completely when the pressure rises above 20-30 cm of water [4, 5].\n\nIn piglets, the serum levels of aspartate aminotransferase (AST), ALT, alkaline phosphatase, total bilirubin, and direct bilirubin more than double by 24 h after ligation of the common bile duct. Elevation of bile duct pressure occurs much faster than the dilatation of the common bile duct. Common bile duct pressure rises from a basal 7.2 cm H2O to 19.0 cm H2O, whereas the duct diameter increases from a basal 5.6 mm to 9.8 mm, and the bile concentration of ciprofloxacin reduces to one-seventh of the original concentration, emphasizing the importance of pressure changes on the rate of secretion of bile and bile components [6].\n\nIn dogs, liver function tests may remain normal for up to 3 h, but become abnormal by 24 h following the complete ligation of the common bile duct. The hepatocyte uptake and secretion into bile of Tc-99m-HIDA continue for up to 3 h post-ligation, enabling cholescintigraphic delineation of the gallbladder and hepatobiliary tree, proximal to the site of obstruction. The hepatobiliary tree often simulates features seen in a physiologically tight sphincter of Oddi, where all of the hepatic bile may be diverted into the gallbladder, and none enters the duodenum. By 24 h, the hepatocytes continue to concentrate Tc-99m-HIDA, but are unable to secrete it into bile, which results in total non-visualization of the entire biliary tree [7]. Scintigraphy then appears much like a radiocolloid liver scan without the spleen.\n\n### 8.1.4 Cholescintigraphic Features of Total Obstruction of the Common Bile Duct\n\nThe cholescintigraphic features merely reflect the pathophysiological changes that follow obstruction of the common bile duct as described above. The normal mean secretory pressure of the hepatocytes in humans is about 35 cm of water, the mean pressure within the gallbladder is 10 cm, in the common bile duct (CBD) is 12 cm, and in the sphincter of Oddi is 15 cm of water [8]. Bile secretion continues at a normal rate as long as all these normal pressure differentials are maintained. Bile secretion begins to decrease when CBD pressure begins to rise, and the secretion ceases completely when CBD pressure equals or exceeds the secretory pressure of the hepatocytes [9]. Normally about 30% of the hepatic bile that enters the duodenum during fasting enables cholescintigraphic visualization of the small intestine within 60 min in about 80% of the subjects. In the remaining 20% of normal subjects, the duodenum is not seen due to entry of all of the hepatic bile into the gallbladder [10-12]. The flow of the entire hepatic bile into the gallbladder and none into the duodenum during fasting in normal subjects is simply a reflection of an increase in the tonus of the sphincter of Oddi.\n\nAfter 6-8 h of fasting, a normal gallbladder is usually filled to its full capacity of 50 ml, and its ability to accommodate a constant inflow of hepatic bile (about 0.3 ml min-1) is due to absorption of an equal volume of water through the wall. When only the gallbladder is seen by 60 min, but not the duodenum, the reasons for non-visualization may include a physiologically tight sphincter of Oddi or a complete obstruction of the common bile duct. After CCK-8 infusion, the gallbladder contracts and empties bile normally into the duodenum in the case of a physiologically tight sphincter of Oddi [10], but no bile enters the duodenum in the case of complete obstruction of the CBD [13, 14].\n\n### 8.1.5 Bile Duct Obstruction for Less than 48 Hours (Hyper Acute)\n\nA gallstone dislodged from either the gallbladder or intrahepatic ducts or an acute edematous acute pancreatitis is the most common cause of acute complete obstruction of the CBD. A dislodged gallstone usually gets trapped just above or within the sphincter of Oddi, the narrowest part of the CBD. The time interval between the onset of acute CBD obstruction and cessation of hepatic bile secretion is variable, depending upon the level (below or above the entrance of the cystic duct into the common hepatic duct), degree (complete or partial), and presence or absence of the gallbladder. Following acute complete obstruction of the CBD, the liver continues to secrete bile slowly for 24-48 h. The uptake of Tc-99m-HIDA remains relatively high (Fig. 8.1.2a), but the secretion slows, resulting in delayed visualization of biliary structures (Fig. 8.1.2b, c). The gallbladder and bile ducts (proximal to the level of obstruction) are seen, but not the duodenum. Quantitative parameters show a normal hepatic extraction fraction and prolongation of the excretion half-time (Fig. 8.1.3). Administration of CCK-8 increases bile secretion and flow by the intrahepatic bile ducts and forces the hepatic bile to enter the gallbladder; also, there is no bile entry into the duodenum. Due to high pressure in the common bile duct, the gallbladder fails to empty in response to CCK-8 infusion. Some patients show an immediate bile flow into the gallbladder, while others may take a few more minutes after CCK-8 (Fig. 8.1.4). Filling of the gallbladder in response to CCK-8 is a physiological paradox and confirmatory feature for acute total obstruction of the common bile duct. The only other condition that shows a similar feature is spasm of the sphincter of Oddi (Chap. 10).\n\nFig. 8.1.2\n\nHyperacute CBD obstruction (less than 48 h). Obstruction of less than 48 h shows excellent uptake of Tc-99m-HIDA by the hepatocytes. Bile secretion is delayed (a). The gallbladder is seen at 4.5 h (b). Cholangiogram confirms obstruction due to stone at distal end of non-dilated CBD (c)\n\nFig. 8.1.3\n\nEffect of acute complete duct obstruction on functional parameters. In acute obstruction, hepatic extraction fraction (HEF) remains normal, but excretion half-time increases. Gallbladder appears late and fills more after CCK-8. There no bile entry into the duodenum\n\nFig. 8.1.4\n\nParadoxical filling of the gallbladder. Cholecystokinin administration in patients with hyperacute complete CBD obstruction increases bile secretion by intrahepatic ducts and forces the hepatic bile to enter the gallbladder, instead of eliciting its emptying. In patient no. 1, gallbladder fills in immediately, and it takes few more minutes to fill in patient no. 2. Gallbladder size and counts increase after CCK-8\n\n### 8.1.6 Obstruction for More than 48 Hours\n\nExtraction of Tc-99m-HIDA by the hepatocytes is maintained at a relatively high level, but the secretion into bile ceases completely. Cholescintigraphy obtained after 48 h of obstruction, therefore, shows good liver uptake without delineation of any of the biliary structures (Fig. 8.1.5a). Hepatic extraction fraction values in the range of 50-85% are common between 2 and 5 days, and the values fall below 50% when total obstruction persists for more than 5 days (Fig. 8.1.5b). Cholescintigraphic images and quantitative parameters accurately reflect histopathological changes of bile duct obstruction as described above in experimental animals. In the rat, for example, a mean HEF value of 97% before bile duct ligation falls to 70% by 2 h and 16% by 48 h after complete ligation of the common bile duct. Long-standing CBD obstruction ultimately compromises hepatocyte functioning and results in a low HEF value [15-17].\n\nFig. 8.1.5\n\nComplete obstruction of the duct for more than 48 h. There is uptake by the hepatocytes, but no secretion of the radiotracer into bile. The entire biliary tree and the small intestine are nonvisualized for up to18 h (top). Hepatic extraction fraction (HEF) is reduced to 28% and excretion half-time shows infinity (bottom)\n\nBoth bilirubin and Tc-99m-HIDA share a common organic anion receptor-mediated endocytosis for uptake by the hepatocyte. When serum levels are high, bilirubin occupies most of the available receptor sites and blocks the uptake of Tc-99m-HIDA by the hepatocytes and produces a low HEF value. Technetium-99m mebrofenin competes with bilirubin for hepatocyte uptake much better than Tc-99m-labeled diisopropyl IDA (DISIDA), diethyl IDA, para-isopropyl-IDA (PIPIDA), or para-butyl IDA. The reciprocal relationship between Tc-99m-HIDA uptake and serum bilirubin level is well documented in both animal and human studies [17, 18]. In a patient with acute onset of biliary pain, a HEF value above 65% combined with cessation of bile secretion (non-visualization of the bile ducts and gallbladder) is a reliable indicator of a pure biliary disease (complete CBD obstruction), whereas an HEF value below 50% under similar circumstances indicates accompanying severe hepatocellular damage [19, 20].\n\n### 8.1.7 Partial Obstruction\n\nPatients with a partial bile duct obstruction usually present with mid-abdominal or right upper quadrant pain. Elevation of serum alkaline phosphatase or gamma glutamyl transferase is the earliest biochemical abnormality. Cholescintigraphic features of partial CBD obstruction include normal hepatic extraction fraction, bile stasis in ducts proximal to obstruction, and prolongation of excretion half-time values (Fig. 8.1.6a). Intestinal acti-vity may be seen within 60 min or delayed slightly in a few of the patients. Diagnosis of obstruction of the common bile duct can neither be excluded nor confirmed solely on the basis of the appearance or non-appearance of the duodenum, respectively.\n\nFig. 8.1.6\n\nPartial obstruction of the common bile duct. The uptake of the radiotracer remains normal, but the excretion is slowed. Common bile duct is poorly visualized. There is bile pooling in the intrahepatic ducts. The gallbladder and small intestine are seen (top). Hepatic extraction fraction (HEF) remains normal, but the excretion half-time increases to 237 min (bottom)\n\nQuantitative parameters indicate the severity of the disease, and an etiological diagnosis (biliary vs. hepatocellular) is usually made in combination with the image pattern. A mean HEF value of 94% was obtained in 14 patients with isolated partial CBD obstruction and a value of 96% in 13 patients with sclerosing cholangitis. In contrast, 14 patients with cirrhosis had a mean HEF of 60%. Hepatic extraction values below 30% are more common in patients with Child class C than class A and B cirrhosis [20]. Complete CBD obstruction of longer than 5-day duration, however, compromises hepatocyte function and lowers HEF values to levels seen in patients with cirrhosis [15]. Excretion half-time values begin to increase immediately after the onset of CBD obstruction and reach values as high as 200-300 min (Fig. 8.1.6b). Many patients with critical obstruction may show only an up-slope to the curve with T1\/2 value of infinity [15, 20]. Cholescintigraphic features of partial CBD obstruction are non-specific, and an etiological diagnosis (stone vs. stricture) can be made only in a few of the patients based solely on an image pattern. A gallstone lodged in the CBD may cause a filling defect with a convex superior border enabling an etiological diagnosis in a few of the patients [21].\n\n### 8.1.8 Functional Recovery Pattern\n\nThe liver recovers its function rapidly after relief of CBD obstruction. Generally, serum bilirubin values decline much more rapidly than serum alkaline phosphatase levels, a feature that is the reverse of what takes place during the onset of obstruction [3]. Serum bilirubin levels decline by 50% of the preoperative value within 2 days, reach 25% by 1 week, and 10% by 3 weeks (Fig. 8.1.7). By 10 days after the relief of obstruction, the serum bilirubin level declines to 25%, whereas the alkaline phosphatase level may decline to only 60% of the preoperative value [9].\n\nFig. 8.1.7\n\nRecovery of liver function. After relief of bile duct obstruction, bilirubin level declines much more rapidly than serum alkaline phosphatase. Serum bilirubin declines to 50% of preoperative value in 2 days, to 25% value by 1 week, and to less than 10% by 3 weeks. Alkaline phosphatase level, however, remains above 50% of the preoperation value at the end of 1 week [8]\n\n### 8.1.9 Hemobilia\n\nThis is a unique name that combines both Greek (haima, meaning blood) and Latin (bilis, meaning bile) to convey the exact meaning of the pathologic process: a mixture of blood and bile. Blunt and penetrating liver trauma is the most common cause, followed by iatrogenic factors secondary to instrumentation or surgery on or around the biliary tract. Liver biopsy, endoscopic retrograde cholangiopancreatography (ERCP), or transhepatic cholangiography is the most common iatrogenic cause [22]. Gunshot injury is becoming a more common cause in recent years because of the rapid rise in gun-related crimes. Penetrating injury may rupture an artery or a vein and the accompanying bile duct, establishing a direct communication between blood and bile.\n\n### 8.1.10 Parasites\n\nObstruction of the common bile duct due to parasites is usually caused by migration of the round worm (Ascaris Lumbricoids) through the sphincter of Oddi. A female round worm is about 25-35 cm in length and 3-6 mm in diameter. A male worm is slightly smaller in size, measuring 15-30 in length and 2-4 mm in diameter. The upper small intestine is the usual habitat, and often live worms are passed in the stool. When the worm migrates into the common bile duct, the usual symptoms of bile duct obstruction (right upper quadrant pain, vomiting, fever, and jaundice) occur. A history of passing the parasites in the stool is the most helpful diagnostic clue. Ova are found on the microscopic examination of the stool specimen [23]. The appearance of a \"bull's eye\" sign in an ultrasound study of the common bile duct suggests a round worm, especially in countries with a high prevalence of ascariasis [24].\n\n### 8.1.11 Miscellaneous Causes\n\nTumor emboli from cholangiocarcinoma (near the bifurcation), foreign bodies, inspissated bile, papilloma, adenomyoma, fibroma, cystedenoma, and giant cell tumors are some of the rare causes of intraluminal obstruction [25].\n\n### 8.1.12 Segmental Biliary Obstruction\n\nAn intraluminal segmental or lobar bile duct obstruction produces a characteristic scintigraphic pattern. Obstruction due to a gallstone is more common in China, Hong Kong, and other Far Eastern countries than in the western hemisphere [26]. The scintigraphic pattern depends upon whether the obstruction is partial or complete. In the case of a partial segmental bile duct obstruction, the uptake and excretion of Tc-99m-HIDA by the hepatocytes into hepatic bile continues manifesting a bile pool within the duct, proximal to the level of obstruction [27]. A focal retention is seen over the involved segment, while the rest of the normal liver clears most of its radioactivity in the images obtained between 45 and 60 min after injection [28]. In the right lobe, where the segmental ducts are positioned anterior and posterior, a right lateral view is necessary to identify the obstructed duct. In the case of the left lobe, where the segmental ducts are positioned medial and lateral, an anterior view image alone will suffice to identify the obstructed duct. When the segmental or area duct obstruction is complete and persists for a longer duration, uptake of Tc-99m-HIDA decrease or ceases completely, resulting in a filling defect.\n\n### 8.1.13 Measurement of Duct Size on Cholescintigraphy\n\nMeasurement of actual duct size from the cholescintigraphic images is prone to inaccuracy due the effect of specific activity, scatter, and pooling of radiolabeled bile within the biliary system. An in-vitro experiment (Fig. 8.1.8) clearly illustrates the effect of changing of the dose or intensity of image settings on duct size. A 4-mm internal diameter tube is measured in different sizes when the dose and\/or intensity settings are changed. High specific activity hepatic bile in a normal subject or bile pooling within the duct in a patient with distal bile obstruction increases the photon scatter, depicting the duct as though it is dilated when in actuality it is not. Ultrasound is far superior for measurement of ductal dilatation. Cholescintigraphy supplements contrast cholangiopancreatography (ERCP) when it fails to identify the intrahepatic segmental duct obstruction primarily because of lack of retrograde flow of the contrast [26, 29]. Magnetic resonance cholangiopancreatography (MRCP) is an emerging new technology that may eventually replace ERCP in some patients with obstruction of the extrahepatic biliary tree. Magnetic resonance does not require any contrast agent to visualize the bile ducts, but depends upon ductal dilatation to be able to detect the level of obstruction [30, 31].\n\nFig. 8.1.8\n\nEffect of bile pooling on duct size. Measurement of duct size from cholescintigraphy is subject to error due to the effect of the scatter and image intensity (I). Numbers 1-4 denote increase in radioactivity. A 4-mm internal diameter tube appears at various sizes depending upon activity and intensity (I) settings\n\n## 8.2 Wall Thickening\n\nWall thickening occurs gradually over weeks or months before causing impediments to bile flow because of narrowing of the lumen. Wall thickening may be caused by a benign or malignant cause.\n\n### 8.2.1 Benign Causes\n\n#### 8.2.1.1 Stricture\n\nMany conditions that cause benign stricture are listed in Table 8.2.1. Wall injury that leads to benign stricture occurs more commonly following a diagnostic or therapeutic instrumentation of the biliary tree. Most strictures occur at the middle of the common hepatic duct, which lies directly across the point of ligation of the cystic duct during an open or laparoscopic cholecystectomy [1]. Laparoscopic cholecystectomy, which has almost replaced open surgical cholecystectomy, is the most common cause of benign stricture [2]. Small gallstones migrating from either the gallbladder or liver may temporarily lodge in the common bile duct and cause constant irritation and subsequent stricture formation. Ischemia [3], infusion of chemotherapy agents [4], infection, radiation [5], chronic pancreatitis [6], and liver transplantation [7] are other well-established causes of benign stricture.\n\nTable 8.2.1\n\nCauses of benign stricture of the bile duct\n\n(1) Open cholecystectomy\n\n---\n\n(2) Laparoscopic cholecystectomy\n\n(3) Migrating stones\n\n(4) Ischemia\n\n(5) Hepatic intra-arterial chemotherapy\n\n(6) Radiation\n\n(7) Infection\n\n(8) Chronic pancreatitis\n\n(9) Liver transplantation\n\n(10) Blunt or penetrating trauma\n\n#### 8.2.1.2 Pathophysiology\n\nThe pathophysiological changes that follow a benign stricture are quite different from those that occur after an intraluminal obstruction. In contrast to an intraluminal obstruction, which occurs suddenly, stricture formation takes months or years after the initiating injury. Post-transplant CBD stricture occurs at or just above the anastomotic site in 10-20% of patients with liver transplantation. Duct ischemia prior to transplantation is responsible for the stricture [7]. The cause of benign stricture in non-transplant patients often remains unknown.\n\n#### 8.2.1.3 Clinical Features\n\nThe clinical presentation of benign stricture often remains vague for many weeks or months because of its slow progression. Intermittent upper abdominal pain is the most common presenting symptom. Elevation of serum alkaline phosphatase or transaminase is the earliest liver function abnormality, followed by elevation of serum bilirubin, especially when the stricture becomes critical with dilatation of the bile duct.\n\n#### 8.2.1.4 Cholescintigraphic Diagnosis\n\nEarly in the course of benign stricture, the uptake (HEF) and excretion of Tc-99m-HIDA by the hepatocytes are maintained within normal range, but the bile flow through the stricture is reduced (Fig. 8.2.1a). The reduction in bile flow leads to bile stasis within the intra-hepatic ducts with delineation of the segmental and area ducts (Fig. 8.2.1b). There is an increase in excretion half-time, which serves as a quantitative measure of the degree of obstruction (Fig. 8.2.1c). The delay in gallbladder filling may indicate an increase in its intra-luminal pressure. The distal common bile duct initially shows a smooth tapering towards the stricture, but this appearance changes when the duct above the stricture dilates (Fig 8.2.1d). The stricture of the common bile duct diverts most of the hepatic bile to enter the gallbladder, and very little bile enters the duodenum [8]. The hepatic extraction fraction remains high in early stricture, but begins to fall when the obstruction becomes severe and persists for a longer duration [9].\n\nFig. 8.2.1\n\nEarly benign stricture. Liver shows excellent uptake, but slightly delayed secretion of Tc-99m HIDA into bile (a). Later, there is intense bile pooling within the intrahepatic ducts (b, c). Common bile duct and the gallbladder appear late, indicating high pressure within them. There is smooth tapering of the CBD towards the stricture (d)\n\nLow extraction fraction with complete cessation of bile secretion occurs in critical stenosis with serum bilirubin levels above 10 mg% (Fig. 8.2.2.).\n\nFig. 8.2.2\n\nCBD critical stricture. Bile secretion ceases and the bile ducts, gallbladder, and intestine are not seen even at 24 h (top). The hepatic extraction decreases and excretion T1\/2 increases (bottom left). Cholangiogram confirms critical stenosis of the common bile duct (bottom right)\n\nUltrasound is used frequently for the diagnosis, and its accuracy depends very much upon the presence or absence of ductal dilatation [10]. A normal common bile duct usually measures up to 10 mm in diameter, and the ultrasound is unable to detect the stricture when the dust size remains within the normal range (Fig. 8.2.3). Ultrasound accuracy, however, increases when the diameter of the common bile duct exceeds 10 mm, serum bilirubin level raise above 10 mg%, and jaundice persists for longer than a week [11]. Dilatation of the extrahepatic ducts occurs much earlier than dilatation of the intrahepatic ducts, as dictated by the law of Laplace, which states that the stress is directly proportional to the internal diameter of the tubing [12]. Because of their larger internal diameter, the extrahepatic ducts sustain more stress and dilate much earlier than the intrahepatic ducts, which are much smaller in size.\n\nFig. 8.2.3\n\nRelationship of common bile duct size with serum bilirubin and duration of jaundice. There is usually no dilatation of the common bile duct (CBD) with bilirubin levels below 5 mg%; dilatation occurs when serum bilirubin rises above 10 mg%; dilatation is variable between 5 and 10 mg% (a). Diameter of the CBD increases (b) as the duration of jaundice increases [11]\n\nStricture impedes bile flow and reduces gallbladder emptying in response to CCK-8 infusion. There may be prolongation of the latent period with a reduction in gallbladder ejection fraction [8]. The bile emptied from the gallbladder may reflux into the common hepatic duct and right and left hepatic ducts, instead of flowing forward into the duodenum. Bile reflux into the common hepatic and right and left hepatic duct is one of a reliable signs of obstruction and is referred to as Krishnamurthy-Bobba sign [13, 14]. Refluxed bile reenters the gallbladder immediately after cessation of CCK-8 infusion, and the gallbladder curve often shows a rapid refilling. In patients with right upper quadrant pain, mild elevation of serum alkaline phosphatase, and normal bilirubin, cholescintigraphy maintains a much higher sensitivity for the detection of obstruction than ultrasound. Ultrasound, however, becomes a preferred modality for diagnosis in cases with high serum bilirubin and dilatation of ducts [11, 15, 16].\n\nIn normal subjects, the mean time of appearance of the common bile duct is 16 min, of the gallbladder 20 min, and of the duodenum 24 min. The gallbladder ejection fraction of 35% or higher is obtained in response to a 3-min infusion of 10 ngm kg-1 (3.3 ngm min-1 for 3 min) of CCK-8. The latent period is less than 3 min, and the ejection period is between 8 and 12 min. The curve over the common bile duct may show a peak corresponding to the second half of the gallbladder ejection period. The common hepatic duct curve usually does not show any peak as the bile emptied from the gallbladder flows antegrade through the sphincter of Oddi and enters the duodenum [17-19]. Because of forward bile entry into the duodenum, there is no rapid refilling of the gallbladder.\n\nDuring quantitative cholescintigraphy, patients with bile duct obstruction often experience a mild to moderate degree of abdominal pain because of either contraction of the gallbladder or distension of the common bile duct. Pain is experienced mostly during the gallbladder ejection period. Pain experienced after completion of the gallbladder ejection is considered non-biliary in origin. Since cholecystokinin increases intestinal peristalsis, late onset abdominal pain (after the gallbladder ejection period) is considered to indicate pain of intestinal origin due to increased peristalsis. A precise documentation of the temporal relationship between the onset of pain and the phase of gallbladder emptying is essential for defining the exact origin of abdominal pain [20, 21].\n\n#### 8.2.1.5 Cholescintigraphic Accuracy of Biliary Obstruction\n\nOf a total of 214 patients with well-documented biliary obstruction (Table 8.2.2), cholescintigraphy accurately detected obstruction in 200 patients, for a sensitivity of 93% [12, 19, 22-28]. Differentiation between functional vs. anatomic obstruction is made by using either amyl nitrite or a calcium channel blocker, nifedapine. These drugs, which relieve functional but not anatomic obstruction, promote smooth passage of bile only in case of a functional obstruction [29, 30]. The quantitative cholescintigraphic parameters are technically simple to measure, reliable, and easily reproducible within and between indi-viduals and among institutions [31, 32]. Simultaneous quantification provides an objective parameter that may alter patient management strategy as to the timing of therapy and also allows later testing for whether or not the chosen therapy has achieved its intended goals. In most patients, nuclear hepatology studies supplement morphological imaging studies; however, in patients where functional alterations precede morphological changes, one may have to depend mostly on the findings of cholescintigraphy [33]. An etiological diagnosis of biliary obstruction, whether it is intraluminal, wall thickening, or extrinsic compression, often cannot be made from cholescintigraphy alone. An ultrasound, CT ERCP, or MRCP is required for an etiological diagnosis. An ERCP may be ideal for confirmation of an intraluminal obstruction, ultrasound for wall thickening, and CT for an extrinsic compression.\n\nTable 8.2.2\n\nSensitivity of Tc-99m-HIDA cholescintigraphy in the detection of biliary obstruction\n\nAuthor (ref.) | Number of patients with biliary obstruction | Obstruction identified with cholescintigraphy\n\n---|---|---\n\nZeman [22] | 60 | 59\n\nKrishnamurthy [19] | 14 | 13\n\nLecklitner [24] | 25 | 23\n\nLieberman [26] | 19 | 14\n\nDarweesh [23] | 15 | 10\n\nBrown [12] | 14 | 14\n\nJuni [25] | 10 | 10\n\nLieberman [27] | 13 | 13\n\nLee [28] | 44 | 44\n\nTotal | 214 | 200 (93%)\n\n### 8.2.2 Malignant Causes\n\n#### 8.2.2.1 Cholangiocarcinoma\n\nCholangiocarcinoma is the most common malignancy of the biliary system, causing wall thickening and obstruction to bile flow. Incidence of cholangiocarcinoma appears to have been rising in recent years, which is partly related to the higher frequency of its detection by using more sophisticated diagnostic methods [34]. Cholangiocarcinoma occurs more commonly in the 6th decade of life and affects men more frequently than women in the ratio of 1.2:1. The proximal common hepatic duct or one or both of its bifurcating branches are the most common locations [35]. About 58% of cholangiocarcinomas occur in the proximal third, 17% in the middle third, 18% in the distal third, and the remaining 7% are distributed diffusely throughout the common hepatic and common bile duct (Fig. 8.2.4). The site and frequency of occurrence of cholangiocarcinoma has changed in the more recent reports, probably reflecting early diagnosis by non-invasive imaging methods and early confirmation by cholangiopancreatography [36, 37].\n\nFig. 8.2.4\n\nLocation of cholangiocarcinoma. Most occur in the common hepatic duct at its bifurcation [37]\n\n#### 8.2.2.2 Clinical Presentation\n\nPresenting symptoms and signs include weight loss, jaundice, and pruritus. Serum bilirubin levels may fluctuate, but the general trend is one of sustained increase. Patients usually seek medical attention late in the course of the disease, many after the onset of jaundice. By this time there is dilatation of the bile duct proximal to the site of cancer, and it is readily detected on ultrasound or CT examination [38]. Bismuth classifies cholangiocarcinoma into four types based on its location and extent (Fig. 8.2.5). Type I involves the common bile duct or the distal common hepatic duct. The proximal common hepatic duct and the right hepatic and left hepatic ducts are free of cancer. In type II, the carcinoma extends up to the level of the junction of the right and left hepatic ducts, but the proximal part of both the right and left hepatic ducts are free of cancer. In type III, the cancer involves either the right hepatic (IIIa) or the left hepatic duct (IIIb). One of the lobar ducts is free of cancer. Type IV extends to both the right and left hepatic ducts and their segmental branches, and the cancer is unresectable [39].\n\nFig. 8.2.5\n\nStaging of cholangiocarcinoma. In stage I, only the common duct is involved; both the right hepatic duct (RHD) and left hepatic duct (LHD) are free of cancer. In stage II, both the proximal hepatic duct and distal RHD and LHD are involved. Stage III involves the common hepatic duct along with either the right hepatic duct (IIIA) or the left hepatic duct (IIIB). In stage IV, entire extrahepatic biliary tree is involved, including both the right and left hepatic ducts along with their segmental branches\n\n#### 8.2.2.3 Carcinoma of the Ampulla of Vater\n\nOf the 18% of the cholangiocarcinoma arising from the distal third of the common bile duct (Fig. 8.2.5), 75% are less than 4 mm from the ampulla of Vater [36]. The clinical presentation of the peri-ampullary cholangiocarcinoma is distinctly different from those arising from the common hepatic duct and its bifurcation. Periampullary carcinoma presents more often as painless jaundice. Courvoisier first made the observation in 1890 that a palpable, painless gallbladder in a patient with jaundice has either a periampullary or a pancreatic cancer [40]. Periampullary cholangiocarcinoma accounts for less than 1-3% of obstructive jaundice. The common age group is between 55 and 65 years, with equal prevalence between men and women. There is frequent association with primary sclerosing cholangitis [41].\n\nAbout 50% of patients with cholangiocarcinoma already have intra-abdominal and intrahepatic metastases at the time of their first presentation to a clinician [42]. Obstruction manifests as two types of image pattern on Tc-99m-HIDA study, depending upon the degree (partial vs. complete obstruction) and duration of obstruction. Early segmental duct partial obstruction may show a normal or slightly reduced Tc-99m-HIDA uptake with bile stasis proximal to obstruction [43]. Complete obstruction, however, shows marked reduction or no uptake at all in the region affected by the tumor [44]. Involvement of the left hepatic duct, for example, may show no uptake at all over the entire left lobe.\n\nMost patients with cholangiocarcinoma present late, after the onset of jaundice, when hyperbilirubinemia and ductal dilatation are evident. A typical pattern consists of bile pooling in the duct proximal to the site of obstruction and increased liver to duodenal transit time, often as long as 24 h. Multimodality imaging, including ultrasonography, cholescintigraphy, CT, ERCP, and MRCP, are complementary imaging modalities [45]. Biopsy is required prior to deciding on a definitive therapeutic strategy.\n\n#### 8.2.2.4 Stent Patency\n\nPlacement of a palliative endo-prosthesis to relieve bile duct obstruction due to malignancy is now a standard treatment [46]. Endo-prosthesis relieves the obstruction promptly and establishes bile flow immediately. The plastic stents tend to occlude prematurely or move ferred, and they function much better and for a longer period of time than the plastic variety [47, 48]. The stent patency is easily documented with Tc-99m-HIDA cholescintigraphy.\n\nSurgery is recommended for early stages. Some prefer surgery even for tumors located proximally at the bifurcation (Klatskin tumor). Pylorus-preserving partial pancreatoduodenectomy or the Kausch-Whipple resection is the preferred surgical procedure [49, 50]. Insertion of an endo-prosthesis is now a popular therapeutic option, especially in the terminal stage. Migration, infection, and occlusion are frequent complications of the endo-prosthesis. Recent progress in stent design and improvements in stent insertion techniques have contributed towards longer duration of stent function, for up to 3-4 months. Occluded stents are easily replaced, and it is currently not a one-time procedure [51].\n\n## 8.3 Combined Intrahepatic and Extrahepatic Cholestasis (Sclerosing Cholangitis)\n\nBy involving both intrahepatic and extrahepatic ducts sequentially, or simultaneously, sclerosing cholangitis is a typical example of a disease that combines features of both intrahepatic and extrahepatic cholestasis. Sclerosing cholangitis (SC) is a slowly progressive disease of unknown etiology characterized by chronic inflammation, fibrosis, and narrowing of both intrahepatic and extrahepatic ducts at multiple levels [1]. Several contributing factors have been identified (Table 8.3.1). Multiple strictures with normal duct in between give the characteristic beaded appearance on the contrast cholangiogram. The term \"primary\" is used when all other possible causes of cholangitis have been excluded. Cholangitis associated with choledocholithiasis, biliary tract surgery, carcinoma, chemotherapy, or acquired immunodeficiency syndrome (AIDS) is called secondary sclerosing cholangitis. Only after exclusion of these secondary causes is the term primary applied.\n\nTable  8.3.1\n\nCholescitigraphic features of primary sclerosing cholangitis, isolated common bile duct obstruction, and primary biliary cirrhosis\n\nParameter | Primary sclerosing cholangitis | Isolated common bile duct obstruction | Primary biliary cirrhosis\n\n---|---|---|---\n\nLevel of CBD obstruction | Proximal | Distal | None\n\nNumber of obstructions | Multiple | Single | None\n\nBeading of the ducts | Common | Rare | None\n\nBile pooling proximal to obstruction | Uncommon | Very common | Uncommon\n\nCystic duct obstruction | Common | Rare | Rare\n\nLiver clearance half time | Marked increase | Moderate increase | marked increase\n\nRegional variations in clearance half time | Wide variation from region to region | Uniform increase from all regions | Uniform increase from all regions.\n\n### 8.3.1 Primary Sclerosing Cholangitis\n\nPrimary sclerosing cholangitis usually affects patients in their 40s, and men more often (75%) than women [2]. An association with inflammatory bowel disease is found in 75% of the patients [3, 4]. The clinical onset is insidious, with fatigue, pruritus, and malaise. Biochemical changes are characterized by moderate elevation of serum alkaline phosphatase and mild elevation of alanine amino transferase. Elevation of serum bilirubin is a late event. Urinary copper excretion is increased [3]. Unlike primary biliary cirrhosis, there is no specific serological marker for primary sclerosing cholangitis.\n\n### 8.3.2 Pathophysiology\n\nHistologically, four stages are identified. Stage I represents the initial degenerative changes in the duct-lining epithelial cells and infiltration by lymphocytes and neutrophils. Stage II is characterized by peri-portal necrosis, stage III consists of portal to portal fibrous septa and disappearance of bile ducts, and stage IV represents the end stage with cirrhosis and onion skin appearance on biopsy, a characteristic histological finding. The cholangiogram shows a typical beaded appearance of the extrahepatic ducts. Both intrahepatic and extrahepatic ducts are involved, but the contrast cholangiogram is often incapable of detecting the involvement of the intrahepatic ducts [5]. In 100 patients with primary sclerosing cholangitis, 87% had involvement of both intrahepatic and extrahepatic ducts, 11% had involvement of only the intrahepatic ducts, and 2% had involvement of only the extrahepatic ducts [4].\n\nThe disease progresses slowly over 10-15 years before reaching the stage of cirrhosis, portal hypertension, liver failure, and death. The median survival is 12 years [5]. The course of the disease is reversible during stage I, may be reversible in stage II, but is irreversible at stages III and IV. The following criteria are required for the diagnosis: (1) alkaline phosphatase increased by more than 1.5 times the normal amount for at least 6 months and (2) multiple strictures or beading of the ducts on the cholangiogram. Characteristic planar and SPECT images on Tc-99m-HIDA study may be included as one of the required criteria [6].\n\n### 8.3.3 Secondary Sclerosing Cholangitis\n\nSecondary sclerosing cholangitis follows an identifiable cause, such as surgery on the biliary tract, choledocholithiasis, intra-arterial infusions of chemotherapeutic agents [8], cholangiocarcinoma [9], acquired immunodeficiency syndrome [10], and congenital biliary abnormalities. The diagnosis of primary sclerosing cholangitis is made only after excluding all of the above causes. Inflammatory bowel disease (ulcerative colitis or Crohn's disease) is now considered an accompaniment of primary sclerosing cholangitis, and when found either before or in association with cholangitis, the condition is still labeled primary sclerosing cholangitis. The scintigraphic findings of secondary sclerosing cholangitis have not been studied as thoroughly as primary sclerosing cholangitis. The ERCP findings of secondary sclerosing cholangitis are indistinguishable from those of the primary sclerosing cholangitis.\n\n### 8.3.4 Cholescintigraphic Diagnosis\n\nCholescintigraphic planar images are obtained for 60 min with 3-5 mCi of Tc-99m-HIDA. SPECT images are obtained between 60-90 min [6, 11]. Early planar images (within 10 min) show patchy uptake by the liver (Fig. 8.3.1). Late images show a typical pattern of beading of the right hepatic, left hepatic, common hepatic, or common bile ducts, depending upon the extent of the disease (Fig. 8.3.1). The hepatic extraction fraction remains within the normal range early in the course of the disease and begins to decrease in advanced stages [6, 11]. The excretion half time increases from the very beginning. Due to multiple duct involvement at various levels, the excretion half time varies widely from region to region, often reaching values as high as six to seven times the normal value. The gallbladder is often not visualized due to obstruction of the cystic duct (Table 8.3.1).\n\nFig. 8.3.1\n\nSclerosing cholangitis. Planar image shows beading of right hepatic, left hepatic duct, common hepatic and common bile ducts. Gallbladder is not seen due to obstruction of the cystic duct\n\nSPECT images show characteristic features of varying parenchymal retention among the different regions. Liver parenchyma with normal bile ducts shows rapid clearance, while the parenchyma drained by obstructed bile ducts shows retention, giving the characteristic scintigraphic images (Fig. 8.3.2). The involvement and extent of cholangitis as indicated by cholescintigraphy may be more critical and much larger than the disease shown on the contrast cholangiogram. This underestimation of severity and extent by the cholangiogram is due to the inability of the contrast to enter most of the intrahepatic ducts. Several features characteristic of primary sclerosing cholangitis that help to differentiate it from primary biliary cirrhosis and isolated common bile duct obstruction are shown in Table 8.3.1. Magnetic resonance cholangiopancreatography is an emerging new technique that is capable of demonstrating involvement of both intrahepatic and extra-hepatic ducts [12, 13].\n\nFig. 8.3.2\n\nSPECT in sclerosing cholangitis. Transaxial (a) and coronal (b) slices, obtained between 60-90 min, show multiple areas of focal parenchymal retention over the involved ducts along with regions of complete clearance from ducts not affected\n\n### 8.3.5 Therapy\n\nThe medical therapy for primary sclerosing cholangitis includes such drugs as D-penicillamine, cyclosporine, methotrexate, corticosteroids, azathioprine, colchicine, or cholesteramine. Antibiotics are prescribed when there is superimposed infection [5]. Endoscopic stenting is shown to be safe and effective, and it works for several years until end-stage liver disease sets in [14]. Liver transplantation is recommended for end-stage liver disease when medical therapy fails.\n\n## 8.4 Extrinsic Compression\n\nExtrinsic compression of the bile duct can occur at any point along their course from the small intrahepatic duct to the termination of the common bile duct (Table 8.4.1). Compression due to an enlarged lymph node occurs more frequently at the porta hepatis where there are many lymph nodes. Lymphosarcoma, reticulum cell sarcoma, and Hodgkin's disease account for the majority of such obstructions [1]. Acute edematous pancreatitis is a frequent cause. Carcinoma of the stomach, pancreas, and gallbladder may invade the bile duct directly, whereas cancer of other distant organs spreads via a hematogenous route [2]. Chronic pancreatitis, which causes obstruction due to thickening of the intraduodenal portion of the duct wall, may also cause obstruction by an extrinsic compression due to fibrosis around the distal common bile duct [3]. Duodenal diverticulae are found along the medial duodenal wall, very close to the ampulla of Vater, in about 2% of patients undergoing barium meal examination [4]. On rare occasions, the ampulla of Vater may be found within the diverticulum. Acute cholecystitis and acute pancreatitis often co-exist and cause non-visualization of the gallbladder during a Tc-99m-HIDA study [5, 6]. Non-visualization of the gallbladder in a patient with a marked elevation of serum lipase or amylase may indicate acute biliary pancreatitis. Elevation of serum lipase and amylase aids in differentiating pancreatitis from acute cholecystitis. Cancer of the head of the pancreas is a well-recognized cause of extrinsic compression (Fig 8.4.1).\n\nFig. 8.4.1\n\nExtrinsic compression. Common bile duct obstruction because of extrinsic compression by cancer of the head of the pancreas. Uptake is excellent (left). Distal common bile duct ends abruptly (middle), and the gallbladder is seen late (middle) and does not empty even at 24 h (right)\n\nTable 8.4.1\n\nCauses of bile duct obstruction due to extrinsic compression\n\n(1) Peri-ductal lymph node enlargement\n\n---\n\n(2) Carcinoma of head of the pancreas\n\n(3) Chronic pancreatitis\n\n(4) Acute edematous pancreatitis\n\n(5) Annular pancreas\n\n(6) Duodenal diverticula\n\n(7) Pseudocyst of the pancreas.\n\n(8) Mirizzi syndrome\n\n### 8.4.1 Mirizzi Syndrome\n\nThis is a rare type of common bile duct obstruction resulting from extrinsic compression from a large gallstone impaction in the Hartmann's pouch or at the neck of the gallbladder. Patients with Mirizzi syndrome usually have a long cystic duct that runs parallel to the common hepatic duct before joining it to form the common bile duct [7, 8]. Two types of Mirizzi syndrome are described. Type I Mirizzi syndrome consists of extrinsic compression of the common hepatic duct by the stone in the gallbladder neck or in Hartmann's pouch. In type II Mirizzi syndrome, the stone erodes into the common hepatic duct forming a cholecystocholedochal fistula. Recent studies have shown a role for non-invasive diagnosis with MR imaging [9]. Laparoscopic cholecystectomy is the preferred initial approach in uncomplicated cases, and open cholecystectomy is recommended when anomalies exist around the Callot triangle [10].\n\nReferences\n\n1.\n\nAdmirand WH, Small DM. The physiological basis of cholesterol gallstone formation in man. J Clin Invest 1968;47:1043-1052PubMedCrossRef\n\n2.\n\nMadden J. Common duct stones. Their origin and surgical management. Surg Clin North Am 1973;53:1095-1113PubMed\n\n3.\n\nVital A, Bioulac-Sage P, Iron A, Balabaud C. Morphologic structure of bile canaliculi after bile duct ligation in the rat. A time-course study. Arch Pathol Lab Medicine 1982;106:464-467\n\n4.\n\nAccatino L, Contreras A, Fernandez S, Quintana C. The effect of complete biliary obstruction on bile flow and bile acid secretion: post-cholestatic choleresis in the rat. J Lab Clin Med 1979;93:706-717PubMed\n\n5.\n\nJohnson G, Sundman L. Bile and dry matter output at elevated liver secretion pressure. Acta Clin Scand 1964;128:153\n\n6.\n\nChen C, Shiesh S, Wu M, Lin X. The effect of bile duct obstruction on the biliary secretion of ciprofloxacin in piglets. Am J Gastroenterol 1999;94:2408-2411PubMedCrossRef\n\n7.\n\nKlingensmith WC III, Whitney WP, Spitzer VM, Klintmalm GB, Koep LM, Kuni CC. Effect of complete biliary-tract obstruction on serial hepatobiliary imaging in an experimental model: concise communication. J Nucl Med 1981;22:866-868PubMed\n\n8.\n\nOster MJ, Csendes A, Funch-Jensen P, Skjoldborg H. Intra-operative pressure measurements of the choledocho-duodenal junction, common bile duct, cystico-choledochal junction, and gallbladder in humans. Surg Gynecol Obstet 1980;150:385-389PubMed\n\n9.\n\nPelligrini CA. Pathophysiology of biliary obstruction. In: Way LW, Pelligrini CA (eds). The surgery of the gallbladder and bile ducts. Philadelphia, WB Saunders, 1987, pp 103-117\n\n10.\n\nWilliams W, Krishnamurthy GT, Brar HS, Bobba VR. Scintigraphic variations of normal biliary physiology. J Nucl Med 1984;25:160-165PubMed\n\n11.\n\nKrishnamurthy GT, Bobba VR, McConnell D, Turner F, Mesgarzadeh M, Kingston E. Quantitative biliary dynamics: introduction of a new non-invasive scintigraphic technique. J Nucl Med 1983;24:217-223PubMed\n\n12.\n\nDoo E, Krishnamurthy GT, Eklem MJ, Gilbert S, Brown PH. Quantification of hepatobiliary function as an integral part of imaging with technetium-99m-mebrofenin in health and disease. J Nucl Med 1991;32:48-57PubMed\n\n13.\n\nBlue PW. Hyperacute complete common bile duct obstruction demonstrated with Tc-99m-IDA cholescintigraphy. Nucl Med Commun 1985;6:275-279PubMedCrossRef\n\n14.\n\nLee AW, Ram MD, Shin WJ, Murphy K. Technetium-99m BIDA biliary scintigraphy in the evaluation of the jaundiced patient. J Nucl Med 1986;27:1407-1412PubMed\n\n15.\n\nJuni JE, Reichle R. Measurement of hepatocellular function with deconvolutional analysis: application in the differential diagnosis of acute jaundice. Radiology 1990;177:171-175PubMed\n\n16.\n\nLima J, Shankaran H, Krishnamurthy GT, et al Evaluation of hepatocyte function following acute common bile duct obstruction (ACBDO) by Tc-99m-mebrofenin scintigraphy in rats. Eur J Nucl Med 1992;19:687 (abstract)\n\n17.\n\nLima J. Quantifica\u00e7ao em hepatologia nuclear. Thesis submitted to the University of Coimbra as partial fulfillment for a doctoral degree. University of Coimbra, Portugal, 1997\n\n18.\n\nKrishnamurthy S, Krishnamurthy GT. 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Arch Surg 1983;118:1221-1223PubMedCrossRef\n\n9.\n\nKim PN, Outwater EK, Mitchel DG. Mirizzi syndrome: evaluation by MR imaging. Am J Gastroenterol 1999;94:2546-2550PubMedCrossRef\n\n10.\n\nSare M, Gurer S, Taskin V, Aladag M, Hilmioglu F, Gurel M. Mirizzi syndrome: choice of surgical procedure in the laparoscopic era. Surg Laparosc Endosc 1998;8:63-67PubMedCrossRef\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_9(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 9. Diseases of the Gallbladder\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nLiver and gallbladder diseases are two of the most common digestive system problems around the world [1]. In the United States, there are about 20.5 million people with gallbladder disease, with an estimated annual cost for medical care of more than 6.4 billion dollars [2]. Gallstones account for the majority of gallbladder problems. Women are affected two to three times as frequently as men [3]. Race, heredity, gender, age, and obesity are some of the important known risk factors for gallstones (Table 9.1.1). Between the ages 60 and 74, the prevalence of gallbladder disease is as high as 25.3% in men and 33.1%% in women (Table 9.1.2), and it is relatively more common among the Mexican Americans (Table 9.1.3). The highest rate among the Americans is found in the Pima Indians of Arizona [4]. By the teenage years, as many as 10-13% of Pima Indian girls develop lithogenic bile, and by 35-44 years, about 71% develop gallstones. Almost 90% of Pima Indian women over the age of 65 develop gallbladder disease, and the prevalence is much higher than in Pima Indian men.\n\nLiver and gallbladder diseases are two of the most common digestive system problems around the world [1]. In the United States, there are about 20.5 million people with gallbladder disease, with an estimated annual cost for medical care of more than 6.4 billion dollars [2]. Gallstones account for the majority of gallbladder problems. Women are affected two to three times as frequently as men [3]. Race, heredity, gender, age, and obesity are some of the important known risk factors for gallstones (Table 9.1.1). Between the ages 60 and 74, the prevalence of gallbladder disease is as high as 25.3% in men and 33.1%% in women (Table 9.1.2), and it is relatively more common among the Mexican Americans (Table 9.1.3). The highest rate among the Americans is found in the Pima Indians of Arizona [4]. By the teenage years, as many as 10-13% of Pima Indian girls develop lithogenic bile, and by 35-44 years, about 71% develop gallstones. Almost 90% of Pima Indian women over the age of 65 develop gallbladder disease, and the prevalence is much higher than in Pima Indian men. Such a high prevalence is also found in Mestizo Hispanics of Chile and the high-altitude rural population of Peru [5, 6]. The incidence of gallbladder disease varies from country to country; the Italians [7], British [8], Scottish [9], and Swedish [10] people have a much higher rate than people from other parts of Europe.\n\nTable 9.1.1\n\nRisk factors for gallstones\n\n(1) Heredity\n\n---\n\n(2) Obesity\n\n(3) Gender\n\n(4) Diabetes\n\n(5) Age\n\n(6) Low socioeconomic status\n\n(7) Parity\n\n(8) Pregnancy\n\n(9) Drugs, e.g., somatostatin analogues (octreotide)\n\nTable 9.1.2\n\nPrevalence of gallstones and gallbladder disease in men and women in the United States [2]\n\nGallstones | Gallbladder disease\n\n---|---\n\nAge (years) | Men (%) | Women (%) | Men (%) | Women (%)\n\n20-29 | 1.3 | 4.4 | 1.3 | 6.5\n\n30-39 | 1.1 | 5.2 | 1.9 | 10.2\n\n40-49 | 5.9 | 8.2 | 7.3 | 15.7\n\n50-59 | 7.3 | 11.9 | 11.7 | 25.0\n\n60-74 | 17.2 | 16.4 | 25.8 | 33.1\n\nMean | 5.5 | 8.6 | 7.9 | 16.6\n\nTable 9.1.3\n\nEffect of gender and ethnicity on the prevalence of gallstones and gallbladder disease in the United States [2]\n\nGender, ethnicity | Gallstones | Gallbladder disease\n\n---|---|---\n\nMen\n\nWhites (non-Hispanics) | 5.8 | 8.6\n\nBlacks (non-Hispanics) | 3.9 | 5.3\n\nMexican Americans | 6.1 | 8.9\n\nWomen\n\nWhites (non-Hispanics) | 8.6 | 16.6\n\nBlacks (non-Hispanics) | 7.9 | 13.9\n\nMexican Americans | 12.8 | 26.7\n\nNatural history of gallstones : Most gallstones initially remain asymptomatic. The frequency of pain developing in an asymptomatic gallstone patient is 10 \u00b1 3% at end of 5 years, 15 \u00b1 4% at 10 years, and 18.4 \u00b1 4% at 15 years (Fig. 9.1.1), which remains unchanged at 20 years [11]. Asymptomatic gallstone patients develop biliary pain (become symptomatic) at an average rate of 2% per year [12, 13]. The frequency of gallstone formation and onset of biliary pain both increase during pregnancy because of elevation of serum estrogens, which induce bile stasis, crystallization, and stone growth within the gallbladder. Gallstones form seven times more frequently in obese women than in non-obese women of comparable age [10]. Stones form more frequently in patients with increased body mass index and less physical activity than in lean people who maintain vigorous physical activity [14]. Stones smaller than 10 mm in size tend to pass through the cystic duct and common bile duct spontaneously during or after delivery [15].\n\nFig. 9.1.1\n\nNatural history of asymptomatic gallstones. Cumulative probability of onset of biliary pain in patients with asymptomatic gallstones increases with time: 10 \u00b1 3%, 15 \u00b1 4%, 18 \u00b1 4%, and 18 \u00b1 4%, at the end of 5,10, 15, and 20 years, respectively [11]\n\nComposition of gallstones: Gallstones are generally classified into two main types: cholesterol and pigment stones [16-18]. The color of the mucosal surface of the gallbladder depends on the nature of the gallstone composition (Fig. 9.1.2). Cholesterol stones are the most common, are soft in consistency, and consist of layers of cholesterol alternating with mucin and glycoproteins (Fig. 9.1.3). Although cholesterol is insoluble in water, it is kept in solution in bile by the formation of micelles, which are composed of cholesterol, bile salts, and phospholipids (lecithin). As the bile cholesterol concentration increases, a large number of multilayered vesicles and crystals are formed, which aggregate and grow into large stones [19, 20]. Pigment stones are more common in patients with cirrhosis or chronic hemolytic anemia. They are black or brown in color, hard in consistency, and contain bilirubin or calcium salts [21]. Stones formed within the common bile duct usually belong to the pigment type. The great majority of gallstones (65-70%) seen on a plain abdominal X-ray are pigment stones [22].\n\nFig. 9.1.2\n\nGallbladder color. Combination of light-colored cholesterol and dark-colored pigment crystals and stones may cast different hues on the surface (left) and inside of a gallbladder (right)\n\nFig. 9.1.3\n\nGallstone color. Cholesterol stones (left) are yellowish and soft, and pigment stones (right) are dark in color and hard in consistency. Pigment stones have multiple facets\n\n## 9.1 Chronic Calculous Cholecystitis\n\n### 9.1.1 Clinical Presentation\n\nPatients with chronic calculous cholecystitis usually present with mild- to moderate- intensity upper abdominal pain without fever or leucocytosis. They ignore pain in the beginning and seek medical aid only when pain becomes intolerable. Pain usually begins 15-30 min after a meal, and this time interval roughly corresponds to the time taken for the serum endogenous cholecystokinin level to rise above the threshold to induce gallbladder contraction and emptying. The site of origin and direction of propagation of biliary pain are variable. The most common location is the epigastrium or the right hypochondrium (Fig. 9.1.4). The pain beginning in the right lower quadrant may mimic an acute appendicitis, or that beginning on the left side of the abdomen may simulate gastritis or diverticulitis. The pain may radiate to the chest and shoulder, mimicking an anginal attack. The pain deep in the abdomen and radiating to the interscapular region may simulate an attack of acute pancreatitis [23]. The pain following dinner usually reaches its peak intensity at midnight [24]. Biliary pain is attributed to distension of the gallbladder or common bile duct wall when they contract against obstruction to bile flow. Pain intensity correlates directly with the degree of wall distension [25]. Stones lying free within the gallbladder lumen do not cause pain. Pain is intermittent, with weeks or months of pain-free intervals between attacks. In a given patient, however, the pain frequency, duration, and intensity remain constant despite wide variations among patients [26].\n\nFig. 9.1.4\n\nLocation and propagation of biliary pain. Epigastrium is the most common location, followed by the right hypochondrium. The pain may radiate to the chest and shoulder, mimicking an anginal attack, or to the back, mimicking pancreatitis. Radiation to the lower quadrants of the abdomen may mimic appendicitis or diverticulitis [22]\n\n### 9.1.2 Biliary Pain Pathway\n\nThe liver and biliary tract receive both somatic and autonomic (sympathetic and parasympathetic) nerve supply. The somatic supply is provided through the thoracic intercostal nerves from spinal segments T8, 9, and 10, which supply the parietal peritoneum. The phrenic nerve (cervical third and fourth segments) supplies the diaphragm and the underlying parietal peritoneum and the gallbladder. Pain originating from the gallbladder, bile ducts, and peritoneum is also carried centrally (afferent) through the branches of the right phrenic nerve. The axons from the sympathetic preganglionic neurons located in thoracic spinal segments T7 to T10 travel along the ventral root and the corresponding sympathetic ganglion to reach the celiac ganglion via the greater and lesser splanchnic nerves. Postganglionic sympathetic fibers from the celiac ganglion supply the liver and the biliary tree (Fig.1.1.7, Chap. 1). Biliary tract pain is carried centrally through these afferent sympathetic fibers. Afferent nerve fibers originating from the corresponding somatotomes and dermatomes join the liver and biliary tract afferent fibers in the spinal cord and serve as a common channel for referred pain. Afferent fibers cross the midline to enter the opposite spino-thalamic tract before reaching the thalamus where pain is processed and sent to higher centers in the cortex.\n\nThe parasympathetic nerve supply is via the right and left vagal nerves, which originate in the medulla. After entering the abdomen, vagal fibers pass through the posterior and anterior hepatic plexus before entering the liver, bile ducts, and gallbladder. The parasympathetic branches to the gallbladder and intrahepatic ducts come mainly from the left (anterior) vagus, and branches to the extrahepatic bile ducts and the sphincter of Oddi arrive mainly from the right (posterior) vagus. The parasympathetic system supplies the efferent nerve fibers and controls motor function of the gallbladder, bile ducts, and sphincter of Oddi. Pain from the gallbladder and common bile duct is referred to the epigastrium, right hypochondrium, right shoulder, or anterior chest (Fig. 9.1.4).\n\n### 9.1.3 Histopathologic Features\n\nThe gallbladder of patients with chronic cholecystitis is often small in size and consists of thickening, fibrosis, and microcalcification of the wall. A single large stone or multiple small stones are common (Fig. 9.1.2). Bile is usually clear, but when it is viscous and contains debris, it settles at the bottom of the fundus of the gallbladder as sludge. The wall shows chronic inflammation and infiltration with lymphocytes, without any hemorrhage or necrosis, which are features of acute cholecystitis. The mucous membrane shows ulceration and scarring. Fibrosis and wall thickening reduce absorption of water through the wall, which accounted for delayed visualization or non-visualization of the gallbladder in the days of oral cholecystogram. Yet most of such non-visualized gallbladders in oral cholecystograms were seen in a Tc-99m-HIDA study, because the entry of even a single drop of high specific activity hepatic bile enables visualization of the gallbladder in a cholescintigram.\n\n### 9.1.4 Diagnosis of Cholelithiasis\n\nUltrasonography is the diagnostic procedure of choice for detection of gallstones. The gallstones are hyperechoic and produce an acoustic shadow beyond the wall (Fig. 9.1.5). A confirmatory diagnosis of cholelithiasis by ultrasound requires demonstration of stone movement with gravity by taking gallbladder images at several different angles [27]. Often a calcified polyp may mimic a gallstone by casting an acoustic shadow, but it does not move with change of patient position. The stones are readily demonstrated in a fully filled gallbladder, but are often missed when the gallbladder is contracted and small in size, incompletely filled with bile, or completely filled with stones. Small stones mixed with sludge settle at the fundus and may fail to produce an acoustic shadow. Bile sludge is a thick, echo-dense fluid containing mucin, protein, calcium bilirubinate granules, and cholesterol monohydrate crystals [28, 29]. Bile sludge produces an irregular dense echo pattern within the gallbladder and does not produce an acoustic shadow (Fig. 9.1.6). Sludge formation increases after prolonged fasting or parenteral nutrition, and it usually clears after a fatty meal. Sludge formed during pregnancy disappears after delivery [15]. The sludge may collect at the dependent part of the gallbladder in a layered fashion and simulate settlement of the contrast agent (Fig. 9.1.7). About 15% of gallstones contain enough calcium to be seen on a plain abdominal X-ray film or on a CT scan [30].\n\nFig. 9.1.5\n\nDetection of gallstone with the ultrasound. Typically, the gallstone appears as hyperechoic within an acoustic shadow beyond the wall. Note one stone in the neck and another in the body of the gallbladder\n\nFig. 9.1.6\n\nBile sludge. On the ultrasound study the sludge appears as an irregular echo-dense mass without an acoustic shadow. Also note thickening of the wall\n\nFig. 9.1.7\n\nLayering of radiocontrast. High-density radiocontrast agent settles at the dependent part of the gallbladder and appears as a separate layer from bile\n\n### 9.1.5 Bile Aspiration\n\nWhen standard imaging procedures fail to show gallstones, duodenal bile aspiration is performed to demonstrate cholesterol crystals, the forerunner of gallstones. After positioning the tip of an orally passed tube in the second part of the duodenum, CCK-8 is infused intravenously to induce contraction and emptying of the gallbladder. The bile emptied from the gallbladder into the duodenum is aspirated through the naso-duodenal tube and examined under the microscope for stones or subjected to biochemical analysis [31]. The test is relatively invasive and expensive and used mostly for research purposes when all other simple methods fail to demonstrate gallstones.\n\n### 9.1.6 Quantitative Cholescintigraphy\n\nCholescintigraphy is used for measuring gallbladder motor function (ejection fraction) and for differentiating symptomatic from asymptomatic gallstones by closely monitoring the temporal relationship between the onset of pain and the phase of gallbladder emptying. A geometric technique for measurement of gallbladder volume by using an oral cholecystogram was first introduced in 1948 by deSilva [32]. A slightly modified method was adopted for volume measurement with the ultrasound [33]. Both oral cholecystogram and ultrasound are geometric methods and require that the gallbladder is of a particular shape (pear) before contraction and that it maintains that shape during and after contraction. It is widely recognized that the gallbladder has many different shapes when it is full and that it changes its shape to different forms during and after its contraction.\n\nTo overcome the theoretical disadvantages of a geometric technique, a counts-based, non-geometric cholescintigraphic method was introduced in 1981 [34]. The cholescintigraphic method does not require that the gallbladder be of a particular shape. Counts originating from the gallbladder accurately represent the bile volume (see Fig. 5.1.4, Chap. 5). Emptying is expressed as an ejection fraction by dividing counts emptied by the total counts before emptying. The technique is highly reproducible either with ingesting a fatty meal or after intravenous administration of cholecystokinin [35, 36]. The short serum half life (2.5 min) of cholecystokinin enables measurement of serial ejection fractions by administering two to four sequential doses of CCK-8 on a single occasion [37]. Such techniques enable the study of the effect of various drugs on the gallbladder or on the sphincter of Oddi function (Fig. 6.1.5, Chap. 6).\n\nCholescintigraphy is usually obtained 6-10 h after fasting (minimum 4 h) to ensure a maximum tightness of the sphincter of Oddi and maximum relaxation of the gallbladder. In patients with serum bilirubin less than 2 mg%, a dose of 2-3 mCi (74-111 MBq) of HIDA will suffice. Larger doses are required in patients with hyperbilirubinemia. During the first 60 min of data collection (hepatic phase imaging), a relatively larger fraction of the hepatic bile enters the gallbladder in both normal subjects and patients with gallstones. A gallbladder count rate of more than 35,000 min-1 by 30 min and more than 70,000 min- 1 by 45 min-1 is usual with a 5-mCi dose of Tc-99m-HIDA (Table 9.1.4). A gallbladder that appears early (within 10-15 min) shows a much higher count rate at 60 min (Fig. 9.1.8) than a gallbladder that visualizes late (between 30 and 60 min). In those 5-10% of the patients in whom the gallbladder appears late (after 50 min), it is necessary to wait 90 min to ensure adequate counts within the gallbladder. Variations in the time of appearance and rate of filling of the gallbladder are common and should be taken into account before proceeding with quantitative cholescintigraphy [38].\n\nFig. 9.1.8\n\nNormal cholescintigram. Gallbladder, intrahepatic and extrahepatic bile ducts, and duodenum all appear within 30 min in one patient (left), whereas the gallbladder appears by 60 min in another patient (right). Distal common bile duct has a convex left margin, and the first part of the duodenum stays closer to the gallbladder\n\nTable 9.1.4\n\nGallbladder counts following intravenous injection of 5 mCi Tc-99-HIDA in normal subjects and patients with gallstones [34]\n\nTime | Normal subject (n = 15) | Gallstone patients (n = 15)\n\n---|---|---\n\n15 min | 5,100 \u00b1 70 | 4,200 \u00b1 60\n\n30 min | 40,600 \u00b1 520 | 38,700 \u00b1 490\n\n45 min | 75,200 \u00b1 680 | 70,800 \u00b1 630\n\n60 min | 72,000 \u00b1620 | 78,100 \u00b1 710\n\n### 9.1.7 Fatty Meal Stimulation\n\nGallbladder emptying is under both nervous and hormonal control [39]. Nervous control is exerted via the cholinergic (vagus) and adrenergic nervous system. Gallbladder emptying in response to a sham feeding occurs via the cholinergic nerve stimulation and can be blocked completely with atropine or after vagotomy [40]. Gallbladder emptying after a meal, however, occurs predominantly through the release of endogenous cholecystokinin (hormonal control), while the nervous control plays a minor role.\n\nIt can take 10-20 min after a meal for the serum CCK level to rise above a threshold to induce contraction and emptying of the gallbladder. The mean latent period (time from fatty meal ingestion to the beginning of gallbladder emptying) is 15.5 min. Post-prandial serum CCK levels stay above the threshold for 2-3 h post-meal [41, 42], and the gallbladder emptying is maintained as long as the serum CCK levels remain above the threshold for contraction (Fig. 9.1.9).\n\nFig. 9.1.9\n\nRelationship between serum cholecystokinin (CCK) level and gallbladder emptying. Serum CCK level begins to rise soon after a meal, peaks by 10-30 min, and the level is maintained above the threshold for contraction for more than 90 min. Gallbladder maintains its emptying as long as the post-prandial serum CCK level remains above the threshold for contraction (modified from [41])\n\n### 9.1.8 Effect of Nutrients on Gallbladder Emptying\n\nThe degree of gallbladder emptying post-meal is dependent upon various factors, including the quality, quantity, total calorie intake, and the proportion of proteins, fats, and carbohydrates in the meal. A mean ejection fraction of 82% has been reported with a liquid meal consisting of 300 ml meritene, 15 ml lipomul with 36 g carbohydrates, 29 g of fats, and 10 g of proteins, and an ejection fraction of 64% with half-and-half milk [40, 43]. These results clearly show that one must strictly follow the investigator's protocol, while adopting values reported in the literature, or must establish local values if a different type of meal is used.\n\n### 9.1.9 Causes of Gallbladder Low Ejection Fraction\n\nMany conditions associated with a gallbladder low ejection fraction are listed in Table 9.1.5. A decrease in emptying is attributed to various factors, including wall fibrosis, a decrease in the total number of CCK receptors in the gallbladder wall, or inactivation of cholecystokinin by anti-cholecystokinin peptides in the serum [44]. The total number of CCK receptors in the gallbladder smooth muscle shows a direct correlation with the degree of emptying; the higher the receptor number, the greater is the ejection fraction, and vice versa [41]. Although a decrease in gallbladder emptying is a common feature of patients with cholelithiasis, a normal or even an exaggerated emptying has been reported in a few selected patients with cholelithiasis [45].\n\nTable 9.1.5\n\nCauses of gallbladder low ejection fraction\n\n(1) Cystic duct syndrome (chronic acalculous cholecystitis)\n\n---\n\n(2) Diabetes\n\n(3) Chronic calculous cholecystitis\n\n(4) Obesity\n\n(5) Common bile duct obstruction\n\n(6) Sphincter of Oddi spasm\n\n(7) Opiates\n\n(8) Somatostatin analogues (octreotide)\n\n(9) Anticholinergic drugs (atropine)\n\n(10) Congenital anomalies (septum, bilobed, etc.)\n\nA delay in the appearance of the gallbladder or low ejection fraction in response to CCK stimulation indicates a functional abnormality [46, 47]. The efficiency with which the wall absorbs water, sodium, and other electrolytes from the lumen is a reflection of the concentration function of the gallbladder and determines the time of appearance of the gallbladder in a Tc-99m-HIDA study. It has been argued in the past whether the abnormal concentration and contraction functions of the gallbladder precede or follow the development of the gallstones. Most of the conditions associated with a decrease in gallbladder emptying are also associated with a higher incidence of gallstones (Table 9.1.5). Patients with ulcerative colitis, for example, show a lower gallbladder ejection fraction and a higher incidence of gallstones when compared to a control group [48]. Treatment with octreotide, a somatostatin analogue, which decreases gallbladder emptying, also increases the incidence of cholelithiasis and cholecystitis [49]. Octreotide decreases gallbladder emptying by interfering with the release of both acetylcholine (cholinergic stimulation) and cholecystokinin (hormonal stimulation), and also by directly blocking the effect of cholecystokinin on the gallbladder wall [50]. The recent literature, therefore, appears to support the hypothesis that the functional abnormalities precede the development of gallstones by months or years.\n\nSome of the investigators suggest using ultrasound for measurement of both the gallbladder volume and emptying, and claim that it can substitute cholescintigraphy for measurement of emptying [51]. Ultrasound emptying measurements are subject to error due to change of shape, shift of axis, and other technical artifacts introduced during contraction and emptying of the gallbladder. Combining the geometric ultrasound technique for measurement of the resting volume (after making necessary volume correction) and non-geometric cholescintigraphy for measurement of gallbladder emptying overcomes many of the theoretical limitations of a geometric method. Combination offers the best features of both techniques for evaluation of gallbladder morphology and function [52].\n\n## 9.2 Chronic Acalculous Cholecystitis\n\nChronic acalculous cholecystitis (CAC), also known as cystic duct spasm or cystic duct syndrome, is one of two diseases that come under the broad category of 'biliary dyskinesia,' the other one being spasm of the sphincter of Oddi (Chap. 10). Biliary dyskinesia is a purely functional abnormality without an easily identifiable morphologic abnormality. The term cystic duct syndrome merely indicates the major site of functional abnormality. The syndrome was first described by Cozzolino et al. in 1961, but did not receive much clinical recognition until a reliable and easily reproducible diagnostic procedure became available for measurement of the gallbladder ejection fraction [1].\n\n### 9.2.1 Clinical Presentation\n\nPatients with chronic acalculous cholecystitis usually present with symptoms very similar to those with chronic calculous cholecystitis. Intermittent mild to moderate intensity right upper quadrant or epigastric pain is the most common presenting symptom. Pain begins 30-60 min after a meal and is often associated with nausea and vomiting. Post-prandial pain after dinner reaches its peak intensity at midnight. Women in their 30s and 40s are affected three to four times more frequently than men of similar age. The physical examination, liver function tests, blood counts, and serum amylase and lipase levels are normal. Gallbladder morphology remains normal, and no gallstones are found within the gallbladder.\n\n### 9.2.2 Histopathologic Features\n\nHistopathologic abnormalities are confined mostly to the cystic duct. There is kinking and narrowing of the cystic duct lumen due to thickening and fibrosis of the wall [2]. Adhesion and angulation of the infundibulum are common. The connection of the cystic duct with the common hepatic duct normally shows wide variations (Fig. 9.2.1). About 50% of cystic ducts join with the common hepatic duct along its right margin at a 45 degree angle, 18% pass behind the common hepatic duct before joining it along its left margin, and the remaining 32% show a spiral form before joining the common hepatic duct either in front or behind [3]. Microscopic examination of the cystic duct shows infiltration with chronic inflammatory cells, concentric thickening, and fibrosis of the media and adventitia with narrowing of the duct lumen (Fig. 9.2.2).\n\nFig. 9.2.1\n\nVariations in the union of the cystic duct with the common hepatic duct. The cystic duct usually joins the common hepatic duct at 45\u00b0 angle along its right margin (50%). It may pass behind the common hepatic duct before joining it on the left side (18%). A spiral cystic duct joins in front in 33% of the patients [3]\n\nFig. 9.2.2\n\nCystic duct histopathology in normal subjects and patients with cystic duct syndrome (CDS). Normal cystic duct wall shows thin mucosa, muscular layer, and adventitia. All three layers show inflammation and concentric thickening with narrowing of the lumen in CDS\n\n### 9.2.3 Functional Abnormality\n\nThe major abnormality pertains to the concentration and contraction functions of the gallbladder. Bile aspirated from the duodenum in patients with CAC shows a lower concentration of bile salts and phospholipids than in normal subjects [4]. Hepatic bile enters the gallbladder slowly through the narrowed cystic duct. Although the gallbladder may appear within 30 min in a Tc-99m-HIDA study (Fig. 9.2.3), a delayed appearance (after 40 min) is more common (Fig. 9.2.4). Some of the gallbladders may take as long as 2-3 h for their appearance. Late appearance is attributed to delayed and decreased entry of hepatic bile into the gallbladder primarily due to slow absorption of water and electrolytes through the wall. Gallbladder low ejection fraction in response to intravenous cholecystokinin or after a fatty meal ingestion is the characteristic feature. Low ejection fraction is due to a combination of several factors, of which wall fibrosis, a decrease in the total number of CCK receptors in the gallbladder wall, and a decrease in the cystic duct threshold for CCK are considered very important contributing factors (Fig. 9.2.5). A decrease in the cystic duct threshold with a paradoxical contraction is attributed to an activation of inhibitory CCK receptors, which induce contraction of the cystic duct instead of normal relaxation [5, 6]. A decrease in the smooth muscle threshold for CCK induces cystic duct contraction before the contraction of the fundus and body, with subsequent non-emptying of the gallbladder [7]. Unable to empty bile against a spasmodic cystic duct, the gallbladder contracts and assumes many different shapes, with globular being the most common form. Usually there is no bile reflux into the common hepatic duct or to the right and left hepatic ducts [8].\n\nFig. 9.2.3\n\nNormal bile formation and flow. Bile flow into the gallbladder may remain normal in some of the patients with cystic duct syndrome. The gallbladder, intrahepatic ducts, extrahepatic ducts, and small intestine are seen within 40 min. Radioactivity clears almost completely from the liver and enters the gallbladder at 60 min\n\nFig. 9.2.4\n\nDelayed gallbladder appearance. Gallbladder that does not appear by 30 min fills in by 60 min with fewer counts than normal. Intrahepatic and extrahepatic ducts are normal\n\nFig. 9.2.5\n\nGallbladder cholecystokinin (CCK) receptors in health and disease. A normal gallbladder rich with CCK receptors produces almost complete emptying, while the gallbladder with chronic cholecystitis elicits poor emptying because of paucity of CCK receptors\n\nDuring fasting, the serum endogenous cholecystokinin reaches its lowest level, which promotes the maximum increase in the tone of the sphincter of Oddi and maximum relaxation of the gallbladder wall. An increase in sphincter tone raises the sphincter of Oddi mean basal pressure to 15 cmH2O. The common bile duct and the gallbladder pressure are maintained at 12 cmH2O and 10 cmH2O, respectively. The pressure differences at these three levels promote preferential entry of the hepatic bile into the gallbladder as the bile seeks the path of the least resistance. During cholescintigraphy, Tc-99m-HIDA simply follows the path taken by the hepatic bile [8].\n\nAfter 4-6 h of fasting, a normal gallbladder is completely filled to its maximum capacity of 50 ml, but it continues to receive hepatic bile at the rate of 0.3 min. A completely filled gallbladder is able to accommodate an additional 0.3 ml of hepatic bile per minute simply by absorbing an equal volume of water through the wall. Absorption of water takes place along the lateral intercellular channels situated between the columnar cells of the mucosa (see Fig. 2.5, Chap. 2). These lateral intercellular channels are open widely during fasting when the gallbladder is completely relaxed, but they close tightly when the gallbladder contracts after a meal. These two factors, an increase in sphincter tone and absorption of water through the wall, are the primary mechanisms by which the gallbladder sequesters most bile salts during fasting. The process by which the gallbladder concentrates bile salts during fasting by selective absorption of water and electrolytes through the wall is called the concentration function of the gallbladder [9]. The concentration function can be measured quantitatively and non-invasively with cholescintigraphy using Tc-99m HIDA as described in Chap 2.\n\n### 9.2.4 Measurement of Gallbladder Ejection Fraction\n\n#### 9.2.4.1 Imaging Procedure\n\nThe patient preparation and data acquisition and analysis are monitored carefully. The patient fasts for a minimum of 4 h, preferably for 8-10 h. Detailed drug history is taken to ensure that the patient is not currently taking any medications that act either on the gallbladder or sphincter of Oddi. Opioids, calcium channel blockers, nitrates, sympathetic and parasympathetic agonists or antagonists, and other drugs that are known to act on the gallbladder or the sphincter of Oddi are withdrawn for a day or two before scheduling the patient for measurement of the gallbladder ejection fraction.\n\nThe data are collected with the large field of view gamma camera interfaced to an online computer in two separate phases: the hepatic phase between 0 and 60 min and gallbladder phase between 61 and 90 min. The hepatic phase data are collected on a 64 \u00d7 64 matrix as one frame per minute for 60 min, following intravenous injection of 2-3 mCi of Tc-99m-HIDA. After ascertaining the adequacy of counts within the gallbladder (Table 9.1.4), the gallbladder phase data are acquired. When the gallbladder appears late and does not contain adequate counts at the end of 60 min, a second dose of Tc-99m-HIDA is given at 60 min, and the gallbladder phase data collection is delayed for an additional 30-60 min. This delay ensures adequate counts within the gallbladder for measurement of the ejection fraction. Images are carefully scrutinized in cine display to check for superimposition of structures over one another, especially the superimposition of the gallbladder and duodenal radioactivity. In the supine position, the gallbladder fundus lies anteriorly and the neck posteriorly. Small gallstones often gravitate to the dependent posterior part and settle in the neck, blocking bile entry into the gallbladder. Changing the patient position to a right lateral decubitus position or asking the patient to walk for few minutes usually dislodges the stones from the neck and allows bile entry into the gallbladder. A septum at the neck may allow filling of the proximal segment and delay entry of radiolabeled bile into the distal segment (Fig. 9.2.6).\n\nFig. 9.2.6\n\nSepta at the neck (S). Bile enters the small proximal segment at the neck first (left) followed by delayed filling of the body and fundus (right) forming the distal segment (A). Ultrasound shows septa at the neck\n\n#### 9.2.4.2 Gallbladder Phase Data Acquisition and Analysis\n\nThis data collection usually occurs between 61 and 90 min after Tc-99m-HIDA injection, unless there is a delay in filling of the gallbladder [10-12]. If there is a superimposition of structures during the hepatic phase data collection, the gamma camera angle is changed to the position that maximally separates the gallbladder from the common bile duct and the duodenum. When the gallbladder and duodenum are superimposed, drinking a glass of water moves the duodenal radioactivity away from the gallbladder region of interest [13]. The data are collected at one frame per minute for 30 min. Beginning at 3 min, 3 ng kg-1min-1 of cholecystokinin-8 (sincalide, Bracco Diagnostics Inc., Princeton, NJ) is infused over 10 min through an infusion pump [8, 14].\n\nBefore the test begins, the patient is instructed to raise a hand when pain is experienced and raise the hand again when pain is relieved. The time of onset and relief of pain are noted on the gallbladder time\/activity curve to critically establish the temporal relationship between pain and the phase of gallbladder emptying. Biliary pain typically occurs during the gallbladder ejection period. Pain experienced after the ejection period is considered non-biliary in origin [8]. Saline may be infused as a placebo prior to cholecystokinin infusion. The gallbladder ejection fraction depends upon various factors, including the dose, dose rate, and the duration of infusion of sincalide. These variables are controlled strictly to obtain a consistent result. The gallbladder ejection fraction is calculated using the counts before and after emptying [14, 15]. The normal gallbladder ejection fraction is 35% and higher when 3 ng kg-1 min-1 CCK-8 is infused over 3 min, and 50% and higher when the identical dose rate is infused over 10 min. Large bolus doses of sincalide produce a non-physiological response and should be avoided.\n\nContraction and emptying of the gallbladder are initiated when sincalide binds to its receptors located in the smooth muscle, which are distributed irregularly in the wall (Fig. 9.2.7). The smooth muscle is much thicker in the fundus and neck than in the body and cystic duct. Cholecystokinin receptors located in the smooth muscle of the body, fundus, and cystic duct show varied thresholds for contraction. In dogs, for example, the smooth muscle in the cystic duct shows a much higher threshold (is less sensitive) than the smooth muscle in the fundus [5]. The cystic duct, therefore, does not contract when the fundus and body contract in response to a physiologic dose of sincalide. A dose rate within a physiologic range causes smooth, sustained, and coordinated contraction and emptying of the gallbladder.\n\nFig. 9.2.7\n\nValentine gallbladder. A bi-lobed gallbladder high in position fills with bile (top left). Each segment empties at different rate in response to CCK-8 infusion, with a total ejection fraction of 47.5% (top right). Cholangiogram (bottom) shows the septa in between two segments [18]\n\n#### 9.2.4.3 Sincalide Dose\n\nA low ejection fraction is often obtained in normal subjects when a large, non-physiologic dose rate of sincalide is given rapidly. In the package insert, the manufacturer of sincalide (Bracco Laboratory, Princeton, NJ) recommends a dose of 0.02 \u03bcg kg-1 (20 ng kg-1) given in 30-60 s. The recommended dose in the package insert for sincalide originally was developed for an oral cholecystogram or for stimulation of pancreatic enzyme secretion. This dose rate is too large for cholescintigraphy and often causes a low ejection fraction in normal subjects and should be avoided [7, 16, 17]. An optimal dose rate for cholescintigraphy is 3.0 ng kg-1 min-1 infused over 3 or 10 min. This dose rate (3 ng kg-1 min-1) is much lower than the dose recommended in the package insert. Currently, we have standardized the sincalide dose as 3 ng kg-1 min-1 for 10 min and consider GBEF of 50% or greater as normal. We hope that others will follow this procedure in total such that it becomes a universal standard. Local normal values should be established when a different sincalide dose rate or duration of infusion is chosen.\n\n#### 9.2.4.4 Congenital Abnormalities of the Gallbladder\n\nCongenital folds or septa often cause abnormalities in filling and emptying of the gallbladder [18]. An intrahepatic gallbladder may cause a filling defect in the liver in the early images (within 10 min) that fills later with radiolabeled bile and may not empty normally in response to CCK because of adhesion of its wall to the liver parenchyma. Often a gallbladder is divided into two segments of equal or unequal size (Valentine gallbladder) by a fold, and each segment may fill differently and empty at different rates in response to sincalide (Fig. 9.2.7). An epithelial fold or septa near the neck creates a pouch-like compartment (Hartmann pouch) where gallstones may lodge and delay or prevent bile entry into the distal compartment (Fig. 9.2.8). Septa in the middle of the gallbladder may allow normal filling of the proximal segment but delay the filling of the distal segment for 10-15 min. In such instances, only the proximal segment may empty bile normally, while the distal segment is prevented from emptying by the septa, which acts as a one-way valve (Fig. 9.2.9). Imaging at a shorter frame rate (one frame\/minute) enables easy recognition of such morphologic variations [19, 20]. Failure to recognize such morphologic abnormalities may provide false functional information. The distal segment, which fills late, may empty bile poorly, while the proximal segment shows a normal ejection fraction. Poor emptying of the distal segment may promote formation of gallstones. When the shape of the gallbladder in the anterior view mimics the duodenal loop, a right lateral view helps to separate them (Fig. 9.2.10). The gallbladder projects at the middle of the anterior liver border, while the common bile duct remains posterior in location. Cholescintigraphy usually does not show gallstones as a filling defect within the gallbladder due to mixing of Tc-99m-HIDA with gallbladder bile. A large gallstone, however, may occasionally become visible within the gallbladder when most of the bile empties in response to CCK-8 (Fig. 9.2.11).\n\nFig. 9.2.8\n\nHartmann pouch. Anterior (Ant) and posterior (Post) view cholescintigrams (top) show a pouch (H) at the neck (Hartmann pouch) of the gallbladder (GB). An ultrasound of the gallbladder (bottom) confirms the pouch at the neck\n\nFig. 9.2.9\n\nSepta in the body of the gallbladder. An ultrasound study shows a prominent septa in the body (top left).The proximal segment fills in by 58 min (top right), whereas the distal segment takes 3 h to fill in the cholescintigram (bottom). After CCK-8, mostly the proximal segment empties, with a total ejection fraction of 31%\n\nFig. 9.2.10\n\nCurved gallbladder. A comma-shaped gallbladder (GB) may mimic first part of the duodenum (DU). A right lateral view (RL) helps to visualize the long axis of the gallbladder and relationship with the common bile duct (CBD)\n\nFig. 9.2.11\n\nGallstone within the gallbladder. Radiolabeled hepatic bile enters the gallbladder (35 min), surrounds the gallstones, and obscures it (pre-CCK). The stones become obvious as a filling defect only after CCK-8-induced gallbladder emptying\n\n#### 9.2.4.5 Irritable Bowel\n\nAt the end of the hepatic phase imaging (60 min), most of the bile entering the duodenum moves and collects at the ligament of Trietz, jejunum, and proximal ileum. In a patient with irritable bowel syndrome, the bile moves much more rapidly through the small and large bowel, resulting in the visualization of the distal ileum and proximal colon. Administration of CCK-8 during the gallbladder phase in such patients increases the intestinal peristalsis and moves the bile much further away, resulting in the delineation of the entire ascending, transverse, and descending colon. This recognition of colon appearance is essential in separating the biliary from bowel pain (Fig. 9.2.12).\n\nFig. 9.2.12\n\nIrritable bowel. Bile transit is hastened by CCK-8 infusion. The bile emptied from the gallbladder after CCK-8 mixes with rest of the intestinal bile and moves rapidly through the ascending (AC), transverse (TC), and descending colon\n\n#### 9.2.4.6 Therapeutic Response to Cholecystectomy\n\nLaparoscopic or open cholecystectomy is the most appropriate therapy for cystic duct syndrome (Table 9.2.1). Of a total of 320 patients with cystic duct syndrome who underwent cholecystectomy solely on the basis of a low ejection fraction, 287 patients (90%) had relief of pain [21-27]. Histopathological examination showed evidence of gallbladder or cystic duct disease in 285 patients (89%). Such a high therapeutic success rate, however, is not universal. One report showed pain relief in only 69% of patients with the cystic duct syndrome [28]. Recent studies have identified Helicobacter DNA in both bile and the gallbladder wall of patients with chronic cholecystitis, raising the possibilities of an infection as a forerunner of chronic cholecystitis and of antibiotics having a role in the treatment [29, 30]. These therapeutic options may add new dimensions to the role of quantitative functional imaging in the diagnosis and management of patients with various types of gallbladder disease [31].\n\nTable 9.2.1\n\nCholescintigraphic, histopathologic, and post-cholecystectomy results in patients with cystic duct syndrome (chronic acalculous cholecystitis)\n\nAuthors [Ref] | No. of patients with pain and low EF (cholecystectomy) | No. of abnormal histopathology of the gallbladder | No. of patients with post-cholecystectomy relief of pain\n\n---|---|---|---\n\nPickelman et al. [20] | 19 | 11 | 18\n\nFink-Bennet et al. [21] | 124 | 115 | 105\n\nZack et al. [22] | 59 | 54 | 56\n\nMisra et al. [23] | 67 | 60 | 58\n\nHalverson et al. [24] | 12 | 10 | 10\n\nSorenson et al. [25] | 11 | 11 | 11\n\nKleiger et al. [26] | 28 | 26 | 27\n\nTotal | 320 | 287 (90%) | 285 (89%)\n\n#### 9.2.4.7 Standardization of Technique\n\nIn the literature, many different values have been reported for the post CCK-8 gallbladder ejection fraction. This wide variation is primarily due to differences in methodology. Variations in the dose, dose rate, and duration of infusion of CCK-8 influence the gallbladder ejection fraction. Unlike the left ventricular ejection fraction, which is controlled solely by the ventricle itself through its pace maker, the gallbladder ejection fraction can be controlled to any desired level simply by controlling the dose, dose rate, and duration of infusion of CCK-8.\n\nThe gallbladder contracts and empties bile as long as the serum cholecystokinin level is maintained above the threshold level. After cessation of CCK-8 infusion, the gallbladder emptying continues for an additional 8-12 min and then stops. The emptying resumes upon CCK-8 reinfusion. One can obtain as many as two to four sequential gallbladder ejection fractions following a single dose of Tc-99m-HIDA [14]. The CCK-8 contractile receptors in the cystic duct usually do not respond when the hormonal dose is within the physiologic limit. Large, non-physiologic doses, however, induce cystic duct contraction with subsequent non-emptying of a normal gallbladder [17]. The physiologic dose rate of cck-8 ranges from 1 to 4 ng kg-1 min-1. A steady-state serum cholecystokinin level can be achieved for 1-2 h by continuously infusing doses as low as 0.3 ng kg-1 min-1. A basal serum cholecystokinin level of less than 1 pmol l-1 raises to above 4 pmol l-1 by 10 min, reaches a peak level of 6.5 pmol l-1 by 30 min, and a steady-state level between 5 and 6 pmol l-1 can be maintained for as long as 70 min-1 by a constant infusion [16]. Infusion of a smaller dose over a longer period of time simulates postprandial CCK release and is much more effective than infusion of a larger dose as a bolus [7, 17].\n\n## 9.3 Acute Cholecystitis\n\nAcute cholecystitis is one of the common abdominal emergencies that require immediate diagnosis and therapy. It usually presents with right upper quadrant pain, fever, and leucocytosis. Since gallstones are found in nearly 10% of all Americans, a mere association between gallstone and abdominal pain with fever cannot be equated with a diagnosis of acute cholecystitis [1, 2]. The great majority of gallstones remains silent, and only about 15-20% ever become symptomatic [3]. About 85-90% of acute cholecystitis associated with gallstones is called acute calculous cholecystitis, and the remaining 10-15% without gallstones is called acute acalculous cholecystitis [4].\n\n### 9.3.1 Clinical Presentation\n\nAcute cholecystitis clinically presents as right upper quadrant or epigastric pain and fever, mimicking other abdominal and lower thoracic acute emergencies (Table 9.3.1). Pain may radiate to the right shoulder, mediastinum, or lower anterior chest, mimicking an acute myocardial infarction [5]. The pain radiation to the right lower quadrant may mimic an acute appendicitis or that radiating to the left lower quadrant may mimic an acute diverticulitis. Pain is due to distension of the inflamed gallbladder wall. On deep inspiration when the liver moves downwards, the fundus of the gallbladder extends below the right costal margin and is felt as a soft mass during deep palpation. When the palpating finger touches the inflamed gallbladder wall, the patient experiences pain and abruptly stops breathing. This is called Murphy's sign.\n\nTable 9.3.1\n\nDifferential diagnosis of acute cholecystitis\n\n(1) Acute pancreatitis\n\n---\n\n(2) Acute hepatitis\n\n(3) Gastritis\n\n(4) Acute appendicitis\n\n(5) Acute cholangitis\n\n(6) Acute nephritis\n\n(7) Angina or acute myocardial infarction\n\n(8) Pleurisy\n\n(9) Right lower lobe pneumonia\n\n(10) Acute diverticulitis\n\nAcute cholecystitis is three to four times more common in women than men, and the incidence increases with age. The clinical presentation of a calculous or an acalculous acute cholecystitis is very similar. A combination of fever with chills, right upper quadrant abdominal tenderness, and clinical jaundice is called Charcot triad. The triad is found more frequently when acute cholecystitis occurs in association with obstruction of the common bile duct. The place of maximum tenderness over the gallbladder fundus corresponds to the point where a line drawn between the left superior iliac spine and umbilicus meets the right costal margin when the line is extended upwards (Fig 9.3.1). This point also corresponds to the intersection of the lateral border of the right rectus abdominus muscle with the right ninth costal cartilage. Murphy's sign is readily elicited by deep palpation at this point. Clinical presentation of acute cholecystitis is variable and mimics varieties of other abdominal diseases, and hence, clinical diagnosis alone is not reliable to plan for a definitive therapy. The accuracy of clinical diagnosis alone ranges from 45 to 77% [6].\n\nFig. 9.3.1\n\nClinical location of the gallbladder fundus and Murphy's sign. A straight line drawn between the left anterior superior iliac spine and the umbilicus points to the fundus of the gallbladder when extended upwards to meet the right costal margin. This is also the point of Murphy's sign\n\nAcute cholecystitis is usually accompanied by leucocytosis; about 75% of patients have more than 10,000 white cells dl-1. Liver function tests usually remain normal, especially when the patients present early in the course of the disease. A few of the patients who present late may show a mild elevation of serum alkaline phosphatase and transaminase. An elevation of serum bilirubin indicates a complicated acute cholecystitis, such as acute cholangitis or obstruction of the common bile duct.\n\n### 9.3.2 Pathophysiology\n\nMacroscopic appearance of the gallbladder wall as seen directly by the surgeon just prior to cholecystectomy may reflect the true pathology much more accurately than the histopathological changes seen under the microscope after laparoscopic cholecystectomy. The microscopic findings after laparoscopic cholecystectomy may show changes secondary to tissue damage sustained during surgical manipulation and not necessarily the changes because of acute cholecystitis.\n\nThe histopathologic changes are divided into six evolving stages: (1) edema, (2) congestion, (3) focal necrosis, (4) suppuration, (5) gangrene, and (6) perforation. Edema is the earliest change initiated by infection or a gallstone impaction in the cystic duct or in Hartman's pouch. Edema leads to obstruction of the cystic duct and distension of the gallbladder [7]. When the gallbladder distends, the stone may get dislodged and fall back into the lumen, and the pathologic changes initiated by the stone may then subside completely with spontaneous recovery. When inflammation and edema of the cystic duct continue, a full blown picture of acute cholecystitis sets in, and edema extends to involve the rest of the gallbladder wall. Either wall edema or stone or a combination of both is the most common cause of obstruction of the cystic duct, which prevents entry of hepatic bile into the gallbladder, resulting in its non-visualization in a Tc-99m-HIDA study [8]. Acute inflammation increases the vascularity of the gallbladder wall and infiltration with inflammatory cells: neutrophils and monocytes during the acute phase and lymphocytes as it enters the subacute phase. Edema and infiltration with inflammatory cells produce thickening of the gallbladder wall, which is seen with ultrasonography [9].\n\nAs the focal necrosis progresses and becomes more diffuse, it may lead to the development of suppuration and abscess formation. Inflammation from the superior gallbladder wall, which lies in direct contact with the liver, often spreads to the adjoining liver tissue and causes edema and pericholecystic fluid collection, manifesting a \"rim sign\" of acute cholecystitis on cholescintigraphy [10]. When the necrosis becomes diffuse, the pain receptors often lose their sensation and produce a negative Murphy's sign. A negative Murphy's sign, therefore, is an ominous feature and calls for an immediate therapeutic intervention. Necrosis leads to ulceration and occasional perforation and bile leak. Due to its faraway location from the entrance of the cystic artery, the fundus is more vulnerable to ischemic necrosis, rupture, and bile leak than the body and neck. When the diagnosis and treatment of acute cholecystitis are delayed, the incidence of perforation and bile peritonitis increases to as high as 12% [11].\n\n### 9.3.3 Cholescintigraphic Approach\n\nThe patient should fast for a minimum of 4 h, preferably for 8-10 h, but not more than 24 h. Serum endogenous cholecystokinin reaches its lowest level during fasting, which promotes a maximum increase in the tone of the sphincter of Oddi and maximum relaxation of the gallbladder wall. A wide basal pressure difference between these two structures during fasting promotes preferential bile entry into the gallbladder when the cystic duct is patent, and Tc-99m-HIDA simply follows the path taken by the hepatic bile [12, 13].\n\n### 9.3.4 Hepatic Phase Imaging\n\nWith the patient in the supine position, a large field of view dual-head gamma camera fitted with a low-energy all-purpose collimator is positioned anteriorly and posteriorly over the upper abdomen to cover the entire liver. In patients with a normal bilirubin level, a dose of 2-3 mCi (74-111 MBq) Tc-99m-HIDA is injected intravenously, and the data are acquired at 1 frame\/minute for 60 min. Hypervascularity of the acutely inflamed gallbladder wall can be demonstrated by obtaining a radionuclide perfusion study by collecting the first minute data at 1 frame per 2 s [14]. A higher dose (3-6 mCi or 111-222 MBq) is needed when a perfusion study is desired. After completion of 60 min data collection, the images are viewed in cine mode display and reformatted at 2 frames\/image and recorded on X-ray film for interpretation.\n\n### 9.3.5 Delayed Imaging vs. Morphine Administration\n\nTwo options are available when the gallbladder is not seen by 60 min (during hepatic phase imaging) in a patient with clinically suspected acute cholecystitis. One option is to choose a delayed imaging protocol by obtaining images at 3-4 h after injection of Tc-99m HIDA, some times even at 24 h. The other option is to administer morphine (0.04 mg kg-1) intravenously at 60 min and take images immediately for an additional 30 min (total imaging time 90 min). In both cases, a second dose of Tc-99m-HIDA (1-3 mCi) is administered if the radiotracer from the first dose clears almost completely from the liver by 60 min.\n\nThe diagnostic sensitivity, specificity, and accuracy of cholescintigraphy using the delayed imaging protocol vary from 92 to 100% [15-20]. A mean sensitivity of 97%, specificity of 96%, and accuracy of 97% have been reported from a total of 1,426 patients from six reports (Table 9.3.2). The major disadvantage of the delayed imaging protocol is that it simply delays the diagnosis and hence the therapy.\n\nTable 9.3.2\n\nSensitivity, specificity, and accuracy of Tc-99m -HIDA cholescintigraphy in the diagnosis of acute cholecystitis by using a delayed imaging protocol (without using morphine)\n\nAuthors (Ref.) | No. of patients | Sensitivity (%) | Specificity (%) | Accuracy (%)\n\n---|---|---|---|---\n\nFonseca et al. [15] | 113 | 100 | 100 | 100\n\nFreitas et al. [16] | 186 | 97 | 87 | 94\n\nMatolo et al. [17] | 619 | 92 | 97 | 95\n\nMauro et al. [18] | 95 | 100 | 94 | 96\n\nSzlabick et al. [19] | 117 | 100 | 98 | 99\n\nWeissman et al. [20] | 296 | 95 | 99 | 98\n\nTotal | 1,426 | 97% | 96% | 97%\n\n### 9.3.6 Morphine Dose\n\nThe most popular imaging protocol for suspected acute cholecystitis is to administer morphine intravenously at 60 min if the gallbladder is not seen by then. Morphine acts immediately on the sphincter of Oddi and raises the sphincter pressure by increasing the frequency and amplitude of phasic waves [21]. The usual dose of morphine is 0.04 mg kg-1, infused intravenously over 1 min. A standard 70-kg weight patient would require a minimum total dose of 2.8 mg. Since morphine sulfate for intravenous use is packaged in a 2-mg unit dose, it is convenient to dispense a 4-, 6-, or 8-mg dose for adults. When the cystic duct is patent, morphine increases the sphincter of Oddi pressure and forces the hepatic bile entry the gallbladder [22-30]. Hepatic bile does not enter the gallbladder, despite morphine, if the cystic duct is obstructed, as is the case in patients with acute cholecystitis. A sensitivity of 96% and a specificity of 94% have been shown with the morphine protocol (Table 9.3.3). The major advantage of intravenous morphine is that it enables an early diagnosis, usually within 90 min, allowing the referring physician to plan for an appropriate therapy strategy immediately. The disadvantage of intravenous morphine is its central sedation. Caution is exercised while giving morphine to outpatients, and the patient is instructed not to drive for at least 8-10 h after receiving morphine.\n\nTable 9.3.3\n\nSensitivity and specificity of Tc-99m HIDA cholescintigraphy in the diagnosis of acute cholecystitis with morphine augmentation\n\nAuthors (Ref.) | Sensitivity | Specificity\n\n---|---|---\n\nChoy et al. [22] | 23\/24 (96%) | 35\/35 (100%)\n\nKim et al. [23] | 11\/11 (100%) | 18\/18 (100%)\n\nKeslar et al. [24] | 19\/19 (100%) | 12\/12 (100%)\n\nMehta et al. [25] | 18\/18 (100%) | 13\/13 (100%)\n\nVasquez et al. [26] | 10\/10 (100%) | 22\/26 (85%)\n\nFig et al. [27] | 12\/12 (100%) | 14\/17 (96%)\n\nKistler et al. [28] | 13\/14 (93%) | 14\/18 (78%)\n\nFink-Bennett et al. [29] | 35\/36 (97%) | 115\/117 (98%)\n\nKim et al. [30] | 24\/28 (86%) | 15\/17 (88%)\n\nTotal | 165\/172 (96%) | 258\/273 (94%)\n\n### 9.3.7 Scintigraphic Features of Acute Cholecystitis\n\nObstruction of the cystic duct is the salient pathophysiologic feature of acute cholecystitis (Fig. 9.3.2A). The most common cause of cystic duct obstruction is edema of the wall [7, 8]. Gallstones, which may initiate inflammation, often fall back into the lumen when the gallbladder distends, but the edema continues. A diagnostic test that establishes the status of the cystic duct (patency or obstruction), therefore, carries a much higher sensitivity and specificity than other morphology imaging modalities that only show gallstones, wall thickening, or pericholecystic fluid collection as indicators of acute cholecystitis (Fig. 9.3.2B). Documentation of the cystic obstruction in an appropriate clinical setting confirms the diagnosis of acute cholecystitis with a high degree of certainty [31]. Hypervascularity of an acutely inflamed gallbladder wall is seen in 72% of patients with acute cholecystitis and can be shown by including a Tc-99m-HIDA perfusion study as a part of the imaging protocol. Hypervascularity of the wall correlates with the severity of acute cholecystitis [14, 32].\n\nFig. 9.3.2\n\nAcute cholecystitis. Cholescintigraphic non-visualization of the gallbladder due to obstruction of the cystic duct is the most characteristic feature. Liver shows normal uptake excretion and the bile enters the duodenum (a). Ultrasound shows gallstones in the neck and thickening of the gallbladder wall (b)\n\n### 9.3.8 Pericholecystic Hepatic Retention of Tc-99m-HIDA or \"Rim Sign\"\n\nAn acute inflammation along the superior gallbladder wall often spreads to the adjoining liver parenchyma, causing focal hepatitis where pericholecystitic hepatocytes show a normal pattern of uptake, but a delayed excretion of Tc-99m-HIDA relative to the hepatocytes far away from the gallbladder, manifesting a thin strip of increased radioactivity along the gallbladder fossa in the late images (Fig. 9.3.3A). This is called a \"rim sign\" [33]. The spread of infection from the gallbladder wall into the adjacent liver tissue takes place via a special set of bile ducts called the \"aberrant ducts of Luschka\" [34]. These aberrant ducts connect the pericholecystic hepatocytes with the adventia of the gallbladder wall where they end blindly (Fig. 9.3.3B). An increase in intraluminal pressure because of inflammation produces pseudo-diverticulum in the gallbladder wall. Microorganisms from the superior gallbladder wall travel along these aberrant channels and pseudo-diverticulum and infect the hepatocytes and Kupffer cells of liver parenchyma.\n\nFig. 9.3.3\n\nRim sign of acute cholecystitis. Hepatocytes adjacent to the superior gallbladder wall show normal uptake, but a delayed excretion of Tc-99m-HIDA, resulting in a thin rim of increased radioactivity (arrow) along the gallbladder fossa (a). The gallbladder infection spreads to the liver via the aberrant bile ducts (Luschka) and inflammatory pseudo diverticulum (b), which end blindly in the adventia of the gallbladder wall [34]\n\nOn histopathological examination, the liver tissue from the rim sign region shows edema, sinusoidal congestion, hyperplasia of Kupffer cells, and obliteration of the canalicular lumen, impeding bile flow from the rim sign region [35]. Rim sign is found in 34-60% of patients with acute cholecystitis. Despite a strong positive predictive value, the rim sign alone does not carry a high enough specificity for acute cholecystitis to terminate cholescintigraphy at 60 min when the gallbladder is not seen. Morphine augmentation is necessary to increase the sensitivity, specificity, and overall diagnostic certainty (Table 9.3.3) before planning for a definitive therapeutic strategy [36]. Gangrene, abscess, and perforation of the gallbladder wall with bile leak are some of the complications of acute cholecystitis [37, 38]. Among patients with a definite rim sign, the frequency of complication is often as high as 45%. A gangrenous acute cholecystitis on the ultrasound study shows an edematous wall with thick sludge or pus within the gallbladder (Fig. 9.3.4).\n\nFig. 9.3.4\n\nGangrenous acute cholecystitis. Ultrasound shows edematous gallbladder wall, and the lumen contains hyper-echoic pus or bile sludge\n\n### 9.3.9 Gallbladder Pre-Emptying with CCK\n\nSome advocate intravenous cholecystokinin with the notion that pre-emptying, prior to cholescintigraphy, facilitates rapid gallbladder filling when the cystic duct is patent. Administration of cholecystokinin prior to cholescintigraphy is counterproductive physiologically as it creates conditions that are just the opposite of what is required for diversion of hepatic bile into the gallbladder. By stimulating contraction of the gallbladder and relaxation of the sphincter of Oddi, cholecystokinin promotes free bile flow directly into the duodenum (Chap. 6). Comparison of studies with and without CCK-8 pre-emptying shows a much lower specificity for studies obtained with a CCK-8 pre-emptying protocol. The specificity of cholescintigraphy without CCK-8 pre-emptying is 94%, in contrast to 81% with a CCK-8 pre-emptying protocol. Pre-emptying with CCK-8, however, does not alter the sensitivity (94%) of cholescintigraphy [39]. In a study of 86 patients using a CCK-8 pre-emptying protocol, the gallbladder did not visualize by 60 min in 43 patients. In 18 of these 43 patients, the gallbladder filled in only after the use of intravenous morphine, indicating the necessity of maximizing the tonus of the sphincter of Oddi prior to cholescintigraphy [40]. Pre-emptying is appropriate in patients on hyperalimentation or those who have waited for longer than 24 h (Fig. 9.3.5).\n\nFig. 9.3.5\n\nProtocol for Tc-99m-HIDA study. Four-hour fasting is the minimum, 8-10 h is ideal, and more than 24 h fasting should be avoided. Morphine or CCK-8 use depends upon the clinical situation at hand\n\nThe decision as to when to use morphine vs. cholecystokinin is dependent upon the clinical challenge at hand. Administration of morphine is appropriate in a clinical setting of acute cholecystitis, and its use is inappropriate in a clinical setting of biliary dyskinesia. Administration of cholecystokinin, on the other hand, is appropriate in a clinical setting of biliary dyskinesia, but not in patients with acute cholecystitis [39]. Rarely, there is a need to use both agents sequentially in a given patient [41, 42].\n\n### 9.3.10 Cholangitis\n\nThe superior gallbladder wall not covered by the peritoneum lies directly against the inferior surface of the liver (the bare area). The aberrant ducts of Luschka and inflammatory pseudo-diverticulum serve as the shortest and most direct route for spreading infection from the gallbladder wall to the biliary canaliculi and hepatocytes (Fig. 9.3.3B). The microorganisms responsible for acute cholecystitis travel along these aberrant ducts and cause acute cholangitis. Clinically acute cholangitis is suggested by the onset of the Charcot triad of fever and chills accompanied by gallbladder tenderness and jaundice in the absence of obstruction of the common bile duct. The characteristic features of a combined acute cholecystitis and acute cholangitis on cholescintigraphy include non-visualization of the gallbladder (despite morphine), delayed clearance of Tc-99m-HIDA from the entire liver (intrahepatic cholestasis), leucocytosis, right upper quadrant pain, fever, and jaundice (Fig 9.3.6). Rim sign is typically absent due to diffuse retention of Tc-99m-HIDA by the liver, not just by the pericholecystic hepatocytes.\n\nFig. 9.3.6\n\nAcute cholecystitis and acute cholangitis. The gallbladder is non-visualized, and there is diffuse retention of Tc-99m HIDA by the liver parenchyma. Rim sign is absent, and bile enters the duodenum\n\n### 9.3.11 Differential Diagnosis of Acute Cholecystitis\n\nMany diseases are to be considered in the differential diagnosis in any patient who presents clinically with abdominal pain, fever, and leucocytosis (Table 9.3.1). Pain due to a right renal stone is colicky in nature and very severe in intensity when compared to pain of acute cholecystitis, which tends to be of moderate intensity. A renal stone often causes hematuria as it passes through the ureter. The pain due to viral hepatitis is mild in intensity and is accompanied by abnormal liver function tests. The viral hepatitis profile confirms the diagnosis. Acute pancreatitis and acute cholecystitis often coexist. Non-visualization of the gallbladder in association with a rise in serum amylase or lipase indicates the co-existence of acute cholecystitis and acute pancreatitis [43, 44]. A peptic ulcer may cause epigastric pain mimicking cholecystitis. Infection with Helicobacter pylori is now the most common cause of peptic ulcer. These bacteria produce the enzyme urease that splits C-14 labeled urea in the stomach and liberates C-14-labeled carbon dioxide, which is eliminated in the expired breath. A positive C-14 urea breath test confirms active H. pylori infection. Gallbladder pain radiation upwards into the chest and shoulder may mimic an anginal pain. An abnormal electrocardiogram, elevation of cardiac enzymes, or abnormal Tc-99m pyrophosphate myocardial image helps confirm a cardiac origin of acute pain. A thorough clinical evaluation along with the measurement of an appropriate biochemical profile aids the primary care physician in requesting the most appropriate imaging test to either confirm or rule out each diagnostic possibility.\n\n### 9.3.12 Comparison of Cholescintigraphy and Ultrasound\n\nImmediately after the introduction of Tc-99m-HIDA agents in 1976 and up until 1984, most cholescintigraphic studies were carried out with the first-generation agents using a delayed imaging protocol, which often delayed diagnosis, sometimes for up to 24 h. The diagnostic time interval using the current generation Tc-99m-HIDA agents is now reduced to less than 90 min by administering morphine intravenously at 60 min when the gallbladder is not seen in a patient with suspected acute cholecystitis. A meta-analysis of 2,466 patients from 30 reports revealed cholescintigraphy as the test of choice for acute cholecystitis and ultrasound for cholelithiasis [45]. For acute cholecystitis, the cholescintigraphic sensitivity is 97% (95% confidence interval, 0.96-0.98) and specificity 90% (95% confidence interval, 0.86-0.95). Morphine administration shortens the diagnostic time interval, but does not alter the sensitivity or specificity when compared to delayed imaging protocols (Tables 9.3.2 and 9.3.3). Adjusted sensitivity of ultrasound for acute cholecystitis is 88% (95% confidence interval, 0.74-1.0) and specificity 80% (95% confidence interval, 0.62-0.98). Adjusted sensitivity for detection of cholelithiasis with the ultrasound is 84% (95% confidence limit, 0.76-0.92) and adjusted specificity is 99% (95% confidence interval, 0.97-1.0). Since the salient pathophysiology of acute cholecystitis is cystic duct obstruction, cholescintigraphy, which establishes the status of the cystic duct (patency or obstruction), is much more reliable than ultrasound or CT, both of which depend upon wall thickening as an indicator of acute cholecystitis.\n\n### 9.3.13 Application of Baye's Analysis\n\nClinical diagnosis is often a probability estimate based on the frequency of the disease in the study population. Because it is impossible to study the entire population, the estimates are made from studies comprising a limited number of samples from the population at risk. After the clinical examination, a clinician makes a rough estimate of the disease probability (pre-test probably) based on the frequency of that disease found in that particular population. Imaging tests are requested either to confirm the clinical diagnosis or rule out the disease from consideration with a high degree of certainty. The clinicians feel more comfortable to proceed with a definitive therapeutic strategy after confirmation of acute cholecystitis by an imaging procedure. When the clinical diagnosis is not confirmed by the imaging test, then the clinicians have the option of accepting the results of the imaging test as final or ignore it completely and obtain additional diagnostic tests.\n\nMost imaging procedures are interpreted subjectively, and they provide a dichotomous result in the form of either the disease being present or absent. The probability of a disease being present or absent after an imaging test (post-test probability) depends upon various factors, including the sensitivity, specificity, false-positive, and false-negative ratio of the test, and the prevalence of the disease in the study population. Probabilities are assessed by applying Bayesian analysis [46, 47]. After conducting a thorough clinical evaluation and obtaining a basic biochemical profile, the clinician arrives at a tentative clinical diagnosis and then chooses one of the diagnostic imaging tests for confirmation. In the case of acute cholecystitis, Tc-99m-HIDA cholescintigraphy and ultrasound are two of the most common imaging options, and the clinician may decide to choose either one or both for confirmation.\n\n### 9.3.14 Target Disease | Target disease\n\n---|---\n\nTest Result | Present | Absent\n\nPositive | a | b\n\nNegative | c | d\n\n<<Probability of acute cholecystitis when the imaging test is positive or negative is given by the following formulas.\n\n![\n$$\\\\begin{array}{l}\n{\\\\rm Probability\\\\ of\\\\ acute\\\\ cholecystitis\\\\ when\\\\ the\\\\ testispositive}\\\\left\\( {{\\\\rm abnormal}} \\\\right\\) \\\\\\\\\n= \\\\frac{\\\\left\\[ \\\\begin{array}{c}\n{\\\\rm Sensitivity} \\\\times {\\\\rm Prevalence\\\\ of\\\\ acute}\\\\\\\\\n{\\\\rm cholecystitis\\\\ in\\\\ the\\\\ study\\\\ population}\\\\\\\\\n\\\\end{array}\\\\right\\]}{\\\\left\\[ \\\\begin{array}{l}\n{\\\\rm Sensitivity} \\\\times {\\\\rm Prevalance}\\\\\\\\\n{\\\\rm of\\\\ acute\\\\ cholecystitis}\\\\\\\\\n{\\\\rm in\\\\ the\\\\ study\\\\ population}\\\\\\\\\n\\\\end{array} \\\\right\\] + \\\\left\\[ \\\\begin{array}{l}\n{\\\\rm Falsepositive}\\\\\\\\\n{\\\\rm fraction} \\\\times {\\\\rm Prevalence\\\\ of\\\\ no\\\\ acute}\\\\\\\\\n{\\\\rm cholecystitis\\\\ in\\\\ the\\\\ study}\\\\\\\\\n{\\\\rm population}\\\\\\\\\n\\\\end{array} \\\\right\\]} \\\\\\\\\n\\\\end{array}$$\n](A978-3-642-00648-7_9_Chapter_TeX2GIF_Equ1_9.gif)\n\n![\n$$\\\\begin{array}{l}\n{\\\\rm Probability\\\\ of\\\\ acute\\\\ cholecystitis\\\\ when\\\\ the\\\\ testispositive }\\\\left\\( {{\\\\rm abnormal}} \\\\right\\) \\\\\\\\\n= \\\\frac{{\\\\left\\[ \\\\begin{array}{c}\n{\\\\rm False - negative\\\\ fraction} \\\\times {\\\\rm Prevalence\\\\ of\\\\ actue\\\\ cholecystitis\\\\ in\\\\ the}\\\\\\\\\n{\\\\rm study\\\\ population}\\\\end{array} \\\\right\\]}}{{\\\\left\\[ \\\\begin{array}{l}{\\\\rm False - negativefraction} \\\\times {\\\\rm prevalence}\\\\\\\\\n{\\\\rm of\\\\ acute\\\\ cholecystitis\\\\ in}\\\\\\\\\n{\\\\rm the\\\\ study\\\\ population}\\\\end{array} \\\\right\\] + \\\\left\\[ \\\\begin{array}{l}\n{\\\\rm Specificity} \\\\times\\\\\\\\\n{\\\\rm Prevalence\\\\ of\\\\ no}\\\\\\\\\n{\\\\rm acute\\\\ cholecystitis\\\\ in}\\\\\\\\\n{\\\\rm the\\\\ study\\\\ population}\\\\end{array} \\\\right\\]}} \\\\\\\\ \\\\end{array}$$\n](A978-3-642-00648-7_9_Chapter_TeX2GIF_Equ2_9.gif)\n\nPre-test probability (prevalence): The proportion of patients who have the target disease, as determined before the test is carried out = a + c\/a + b + c + d.\n\nPost-test probability: The proportion of patients with the target disease with a positive test.\n\nLikelihood ratio (LR): It is the ratio of the probability of a test result among the patients with the target disease to the probability of the same test result among patients who do not have the target disease.\n\nThe likelihood ratio for a positive test = sensitivity\/1-specificity.\n\nThe likelihood ratio for a negative test = 1-sensitivity\/specificity\n\nAfter knowing the pre-test (clinical evaluation) and post-test (after an imaging procedure) probability, the clinicians can calculate the Bayes factor or likelihood ratio of the target disease prior to subjecting the patient to a definitive therapy. The Bayes factor is considered more objective and can substitute for a p value of a null hypothesis [48, 49]. By referring to the nomogram (Fig 9.3.7), the post-test likelihood ratio is obtained directly without need for an actual calculation [50, 51]. The likelihood ratio corresponds to the point where a straight line drawn between the pre-test probability and post-test probability intersects the likelihood ratio line in the middle. Let us assume that in a patient with right upper quadrant pain and fever, the pre-test probability of acute cholecystitis (after the clinical examination) is 20%. If the gallbladder is not visualized in a Tc-99m-HIDA study (97% sensitivity), the likelihood ratio of acute cholecystitis is approximately 200. The cost effectiveness of a positive (GB not seen) Tc-99m HIDA test is greater when the pre-test probably is lower, and the cost effectiveness decreases as the pre-test probably increases to a very high level.\n\nFig. 9.3.7\n\nNomogram for likelihood ratio. A line drawn between the pre-test (clinical diagnosis) and post-test (after imaging test) probability intersects the line in the middle and indicates the likelihood ratio [51]\n\nThe discriminating power of a test is the difference in post-test probability between a positive (gallbladder not seen) and a negative test (gallbladder seen) for acute cholecystitis [31]. The wider the difference between positive and negative tests, the greater is the discriminating power (Fig. 9.3.8). Cholescintigraphy, which establishes the patency or obstruction of the cystic duct (the basic pathology of acute cholecystitis), provides a much better discriminating power than ultrasound. Ultrasound, which detects gallstones and thickness of the gallbladder wall but does not establish the status of the cystic duct, therefore, provides a much lower discriminating power for acute cholecystitis (Fig. 9.3.9). The discriminating power of a test works at its best when the prevalence of disease in the study population is around 50%. The discriminating power decreases precipitously when the prevalence is below 20% or above 80% [31].\n\nFig. 9.3.8\n\nRelative merits of imaging tests for acute cholecystitis. Radionuclide imaging (cholescintigraphy) and oral cholecy-stogram, which establish the status of the cystic duct, show a wider difference between a positive (cystic duct not patent) and negative test (cystic duct patent) when compared to ultrasound, which shows only gallstones and wall thickening\n\nFig. 9.3.9\n\nDiscriminating power. The difference between a positive (gallbladder not seen) and a negative test (gallbladder seen) probability is the discriminating power of the diagnostic test. A positive cholescintigraphy and oral cholecystography in an appropriate clinical setting confirm acute cholecystitis. A negative test (gallbladder seen) rules out acute cholecystitis. Ultrasound shows the least discriminating power for acute cholecystitis [31]\n\nApplication of Baysian analysis requires the establishment of true sensitivity and specificity of an imaging test. The ultimate diagnosis of acute cholecystitis, however, is based upon the histopathological criteria chosen by the pathologist. Some pathologists are satisfied with the mere presence of transmural inflammation of the gallbladder wall, while others insist upon the presence of hemorrhage and necrosis before assigning the definitive diagnosis of acute cholecystitis. The best combination of sensitivity and specificity for cholescintigraphy is obtained when the presence of any one of the above two histological criteria is chosen as the basis for the diagnosis of acute cholecystitis [52]. Under an appropriate clinical setting, most clinicians are satisfied with the cholescintigraphic documentation of cystic duct obstruction as sufficient evidence for the diagnosis of acute cholecystitis in planning for a definitive therapy.\n\n### 9.3.15 Acute Acalculous Cholecystitis\n\nClinically, it presents very similarly to acute calculous cholecystitis, but has few unique clinical features of its own. The patients are usually much older and have additional risk factors, such as a recent surgery, crush injury, burns, sepsis, or hyperalimentation. About 10-15% of all acute cholecystitis belongs to the acalculous variety. The incidence of acute acalculous cholecystitis appears to be increasing recently, which may be attributed to increased clinical awareness and sensitivity of the diagnostic tests [53]. The morbidity and mortality of acute acalculous cholecystitis are much higher than in acute calculous cholecystitis. As obstruction of the cystic duct is common for both conditions, the diagnostic sensitivity of cholescintigraphy is similar for both calculous and acalculous acute cholecystitis [54, 55]. Edema (and not the stone) as the primary cause of cystic duct obstruction is easy to appreciate in the case of acute acalculous cholecystitis, as there are no gallstones associated with it.\n\n### 9.3.16 Gallbladder Cancer\n\nGallbladder cancer is a relatively rare, but highly lethal disease. In the United States, about 7,100 new cases per year are found, and 3,500 deaths are attributed to it annually [56]. There are no specific symptoms that draw special attention, and the majority of cancers are advanced at the time of diagnosis. Many are discovered incidentally during microscopic examination of the gallbladder specimen submitted after a routine cholecystectomy for a benign cause (gallstone). Adenocarcinomas account for 80-95% cases, and the rest include squamous cell and undifferentiated or anaplastic type. Cholecystectomy is the initial therapy, but prognosis is poor [57]. The role of chemotherapy and\/or radiation therapy is not well established.\n\n## 9.4 Management of Gallbladder Disease\n\nCholecystectomy is the treatment of choice for most diseases of the gallbladder. The very first cholecystectomy was performed in 1882 by a German surgeon, Langenbach [1]. Today as many as 600,000 cholecystectomies are performed annually in the United States with either laparoscopic or open cholecystectomy. Open cholecystectomy carries a mortality rate ranging from 0 to 2.4% and morbidity ranging from 4.1 to 14.7%. The incidence of bile duct injury varies from 0.06 to 0.45% [2]. Laparoscopic cholecystectomy, which has now almost replaced open cholecystectomy, carries a similar rate of mortality, but a slightly lower morbidity (1.6-12%) than open cholecystectomy [3]. Laparoscopic cholecystectomy in the beginning was restricted mostly to symptomatic cholelithiasis, but now it is performed for varieties of gallbladder diseases, including acute cholecystitis, gangrenous cholecystitis, hydrops, and empyema, etc. [4]. The young as well as the old patients tolerate the laparoscopic cholecystectomy procedure well, and the hospital stay is much shorter than with an open cholecystectomy [5]. Administration of prophylactic antibiotics does not seem to affect the rate of wound infection in case of elective laparoscopic cholecystectomy [6]. Laparoscopic cholecystectomy for symptomatic gallstone disease during pregnancy is considered safe both for the mother and fetus, and is preferred over conservative management for all trimesters [7].\n\n### 9.4.1 Complications\n\nThe complications of acute cholecystitis can occur either before or after treatment. The complications of delayed or untreated acute cholecystitis include acute cholangitis, gallbladder perforation, and bile leak. Post-operative complications include bile leak, cholangitis, infection, and stricture formation. Bile fistula occurs as a chronic complication.\n\n### 9.4.2 Perforation of the Gallbladder\n\nPerforation of the gallbladder is rare and is found in 10% of patients undergoing open cholecystectomy for acute gangrenous cholecystitis [8]. The bile may leak into a small closed space or leak freely into the peritoneum, causing bile peritonitis. Localized leak is the most common type and accounts for 50% of all perforations (Fig. 9.4.1). Most of the localized leaks and those that form fistula remain undiagnosed as they usually do not produce symptoms. Free perforation with bile peritonitis presents with the most disabling symptoms and is responsible for many deaths if not treated immediately. Most perforations into the subhepatic space are walled off and manifest very few symptoms. Perforation near the neck or Hartmann pouch may form a biloma and compress the common bile duct externally, manifesting a Mirizzi syndrome.\n\nFig. 9.4.1\n\nTypes of gallbladder perforation. Subhepatic bed is the most common site, followed by leak into free peritoneum and colon. Perforation near the neck and Hartmann pouch may form a biloma and compress the common bile duct extrinsically, manifesting a Mirizzi syndrome [7]\n\n### 9.4.3 Bile Leak\n\nThe most common complication of laparoscopic cholecystectomy is bile leak through an unligated cystic duct or from an injury to the common hepatic or common bile duct (Fig. 9.4.2). Leak is best defined as bile collection in an unanticipated location or in an anticipated location at an unanticipated time. The common hepatic duct, which lies directly in front of the tip of the laparoscopic instrument, sustains injury more frequently than other bile ducts. The incidence of bile duct injury from laparoscopic cholecystectomy is twice as frequent as from open cholecystectomy. The consequence of bile leak depends upon its volume and location. A small volume subhepatic bile leak (Fig. 9.4.2A) or large volume bile leak into the stomach (Fig. 9.4.2B) causes few of the symptoms. Large volume hepatic subcapsular leak (Fig. 9.4.2C) or leak into the peritoneal cavity (Fig. 9.4.2D) requires treatment. Since the leak prefers the path of least resistance, major bile ducts often are not visualized beyond the point of the leak (Fig. 9.4.2).\n\nFig. 9.4.2\n\nLocations of bile leak. Bile may leak into subhepatic space (top left), stomach (top right), right subdiaphragmatic (bottom left), or peritoneal space (bottom right)\n\n### 9.4.4 Treatment of Bile Leak\n\nSmall volume bile leaks close spontaneously if the tonus of the sphincter of Oddi is maintained at its minimum by treating patients with antispasmodics (Fig. 9.4.3). Use of opioids is avoided. Large volume bile leaks may require sphincterotomy. Sphincterotomy creates the path of least resistance for hepatic bile and promotes closure of the leak (Fig. 9.4.4A, B). Often the leak from the common hepatic or common bile duct may require stents or surgical repair. Malignant strictures usually require stents [9].\n\nFig. 9.4.3\n\nSpontaneous closure of bile leak. Bile leak into the lateral aspect of the right lobe 2 days after cholecystectomy (left) stops spontaneously after 12 days (right). Narcotics that constrict the sphincter of Oddi were discontinued during this time period\n\nFig. 9.4.4\n\nClosure of bile leak after sphincterotomy. A large volume bile leak through an unligated cystic duct (top) required sphincterotomy for its stoppage (bottom)\n\n### 9.4.5 Bile Duct Stricture\n\nThe frequency of postoperative bile duct stricture is much higher after laparoscopic than after open cholecystectomy (Table 9.4.1). Most postoperative strictures occur in the proximal one half of the common hepatic duct, often involving its bifurcation into right and left hepatic ducts. Strictures near the bifurcation are treated with Hepp-Couinaud hepaticojejunostomy using the left hepatic duct [10]. Lately, the design of laparoscopic instruments has improved, and it is now possible to perform cholecystectomy safely with needles as small as 3 mm in diameter [11]. Postoperative strictures commonly occur at the site of ductal injury and are classified into four grades depending upon its location from the bifurcation into left hepatic with right hepatic duct [12]. Grade I strictures are more than 2 cm, grade II less than 2 cm from the union, grade III are at or just below the union, and grade IV at the site of union. Well-designed prospective studies have shown that immediate cholecystectomy is much more cost-effective than delayed elective cholecystectomy [13-16].\n\nTable 9.4.1\n\nComplications of open vs. laparoscopic cholecystectomy [3]\n\nComplication | Open cholecystectomy (%) | Laparoscopic cholecystectomy (%)\n\n---|---|---\n\nCBD injury | 0.1 | 0.2\n\nBile leak | 0.8 | 5.6\n\nPancreatitis | 1.0 | 1.1\n\nJaundice | 1.2 | 1.0\n\nAtelectasis | 10.0 | 5.4\n\nIleus | 12.0 | 5.3\n\nUrinary retention | 17.4 | 6.5\n\nPostoperative subhepatic fluid collection is common and is noted in as many as 25% of the patients following an open cholecystectomy [17], but only about 30-45% of this subhepatic fluid collection is due to bile leak [18]. Bile fistulae as a consequence of acute and chronic complications occur in the subhepatic space, duodenum, stomach, or hepatic flexure of the colon (Fig. 9.4.5).\n\nFig. 9.4.5\n\nCommon sites of biliary fistula. Fistulas occur as a complication of either acute or chronic cholecystitis. Most of them remain asymptomatic\n\n### 9.4.6 Prophylactic Cholecystectomy\n\nPatients with asymptomatic gallstones or diabetes mellitus usually show a decrease in gallbladder emptying, but generally do not require cholecystectomy when either disease exists alone [19, 20]. However, patients with a combination of asymptomatic gallstones and diabetes mellitus develop acute cholecystitis and its complications 2.2 times more frequently than with either disease alone. Based upon such findings, prophylactic cholecystectomy is recommended for asymptomatic gallstone patients with diabetes mellitus [21]. Prophylactic cholecystectomy is also recommended for patients undergoing cardiac transplantation, but not for those undergoing renal transplantation [22, 23]. Newer therapy in the form of oral dissolution, contact dissolution, and extracorporeal biliary lithotripsy has been used less and less frequently in recent years [24]. Some of the preventive measures are known to reduce the incidence of gallstones and accompanying complications. A 30-min exercise five times a week reduces symptomatic gallstones by 34% in men [25], and recreational physical activity (jogging, running, bicycling) reduces the incidence of symptomatic gallstones and cholecystectomy significantly in women [26].\n\nReferences\n\n1.\n\nWilliams R. Global challenges in liver disease. Hepatology 2006;44:521-526PubMed\n\n2.\n\nSandler RS, Everhart JE, Donowitz M, Adams E, Cronin K, Goodman C, Gemmen E, Shah S, Avdic A, Rubin R. The burden of selected digestive diseases in the United States. Gastroenterology 2002;122:1500-1511PubMed\n\n3.\n\nEverhart JE, Khare M, Hill M, Maurer KR. Prevalence and ethnic differences in gallbladder disease in the United States. 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Pathogenesis and treatment of gallstones. N Engl J Med 1993;328:412-421PubMed\n\n25.\n\nLeitzmann MF, Giovannucci EL, Rimm EB, Stampfer MJ, Spiegelman D, Wing AL, Willett WC. The relation of physical activity to risk for symptomatic gallstone disease in men. Ann Intern Med 1998;128:417-425PubMed\n\n26.\n\nLeitzmann MF, Rimm EB, Willet WC, Spiegelman D, Grodstein F, Stampfer MJ, Colditz GA, Giovannucci E. Recreational physical activity and the risk of cholecystectomy in women.N Engl J Med 1999;341:777-784PubMed\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_10(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 10. Biliary Dyskinesia\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nBiliary dyskinesia consists primarily of two disease entities, the sphincter of Oddi spasm (SOS) and cystic duct spasm (CDS). In the past when its pathophysiology was not clearly understood, cystic duct spasm was called cystic duct syndrome. These two entities are anatomically separate, but physiologically connected by an abnormal functional response to cholecystokinin. The sphincter of Oddi spasm is located within the sphincter at the distal end of the common bile duct, whereas the cystic duct spasm is confined mostly to the cystic duct of the gallbladder [1, 2]. In the literature, sphincter of Oddi spasm is also known by such names as papillary stenosis or bile duct dyskinesia, and cystic duct spasm as chronic acalculous cholecystitis. Both the sphincter of Oddi and gallbladder contain receptors for cholecystokinin, which is released endogenously into circulation from the intestinal mucosa upon entry of food into the duodenum [3, 4]. By binding to CCK-A type receptors, cholecystokinin normally stimulates the contraction of the gallbladder with simultaneous relaxation of the sphincter of Oddi. The cystic duct, which also contains CCK receptors, normally does not respond, because its serum threshold for contraction is set at a much higher level than the threshold for contraction of the fundus and body of the gallbladder [5].\n\nBiliary dyskinesia consists primarily of two disease entities, the sphincter of Oddi spasm (SOS) and cystic duct spasm (CDS). In the past when its pathophysiology was not clearly understood, cystic duct spasm was called cystic duct syndrome. These two entities are anatomically separate, but physiologically connected by an abnormal functional response to cholecystokinin. The sphincter of Oddi spasm is located within the sphincter at the distal end of the common bile duct, whereas the cystic duct spasm is confined mostly to the cystic duct of the gallbladder [1, 2]. In the literature, sphincter of Oddi spasm is also known by such names as papillary stenosis or bile duct dyskinesia, and cystic duct spasm as chronic acalculous cholecystitis. Both the sphincter of Oddi and gallbladder contain receptors for cholecystokinin, which is released endogenously into circulation from the intestinal mucosa upon entry of food into the duodenum [3, 4]. By binding to CCK-A type receptors, cholecystokinin normally stimulates the contraction of the gallbladder with simultaneous relaxation of the sphincter of Oddi. The cystic duct, which also contains CCK receptors, normally does not respond, because its serum threshold for contraction is set at a much higher level than the threshold for contraction of the fundus and body of the gallbladder [5]. Controversy over the real existence of the entity of biliary dyskinesia prevailed for many years [6]. Manometric studies, however, have helped to understand the basic sphincter mechanism and the action of cholecystokinin on it in both health (Chap. 6) and disease, especially in patients with SOS (Table 10.1.1). Both quantitative cholescintigraphy and biliary manometry have contributed to clear understanding of the bile secretion and flow mechanisms and the pathophysiological changes associated with biliary dyskinesia [7-9].\n\nTable 10.1.1\n\nBiliary pressure profile in asymptomatic control subjects and patients with sphincter of Oddi spasm [9]\n\nStructure | Control subjects (n = 9) | SOS patients (n = 10) | P value\n\n---|---|---|---\n\nCBD pressure (mmHg) | 8.6 + 1.02 (6-13) | 17.0 + 1.25 (12-22) | 0.001\n\nSp. Oddi pressure (mmHg) | 14.9 + 0.99 (11-21) | 36.9 + 5.44 (20-70) | 0.005\n\nPhasic contractions:\n\nAmplitude (mmHg) | 113.0 + 8.6 (95-125) | 109.5 + 9.3 (85-160) | 0.37\n\nFrequency (per min) | 6.89 + 0.20 | 7.56 + 0.41 | 0.46\n\nDirection of propagation:\n\nAntegrade | 54% + 6% | 29% + 7% | 0.04\n\nSimultaneous | 28% + 4% | 0% + 3% | 0.34\n\nRetrograde | 18% + 6% | 41% + 9% | 0.01\n\nValues are mean + SE. Ranges within parentheses. Adopted from Meshkinpour et al. [9]\n\n## 10.1 Pathophysiology\n\nA normal sphincter of Oddi exhibits an average of six to seven contractions per minute with a wave amplitude of 113 mmHg. The mean pressure in the sphincter of Oddi is about 15.0 mmHg, and 9.0 mmHg in the common bile duct. About 54% of the contraction waves progress antegrade, 18% retrograde, and 28% occur simultaneously in the proximal, middle, and caudal segment of the sphincter (Table 10.1.1). Administration of cholecystokinin abolishes the contraction waves and relaxes the sphincter of Oddi immediately, and promotes smooth passage of bile emptied from the gallbladder into the duodenum (Fig. 10.1.1). Sphincter of Oddi spasm is characterized manometrically by an increase in basal pressure of the sphincter to 37 mmHg, and the common bile duct to 17 mmHg. The total number of basal contractions increases to 17 min-1, and the number of retrograde contraction increases to 41% (Fig. 10.1.2A). In the case of SOS, usually dormant contractile receptors in the sphincter become activated. These activated receptors paradoxically induce contraction of the sphincter in response to cholecystokinin [10, 11]. Cholecystokinin administration increases both the number of waves and the number of retrograde waves per minute (Fig. 10.1.2B). Because of this paradoxical response (contraction) of the sphincter, the bile either fails to empty from the gallbladder or the emptied bile refluxes into the common hepatic duct and right and left hepatic ducts, and reenters the gallbladder soon after cessation of CCK-8 infusion [12].\n\nFig. 10.1.1\n\nAction of CCK-8 on normal sphincter of Oddi. Normal contraction waves are abolished immediately after CCK-8 administration, which facilitates dilatation of the sphincter for smooth passage of bile through it [11]\n\nFig. 10.1.2 a, b\n\nPropagation of contraction waves and effect of CCK-8 on the sphincter in SOS. Contraction begins first at the caudal end and travels retrograde towards the middle and cephalic end of the sphincter (top). Administration of CCK-8 paradoxically increases the number of retrograde contractions (bottom), resulting in closure of the sphincter [11]\n\nCystic duct spasm is often associated with kinking, fibrosis, and thickening of the wall with narrowing of the of the cystic duct lumen [13]. Hepatic bile enters the gallbladder slowly, but cannot get out readily in response to cholecystokinin (Chap. 9.2). Due to a decrease in the total number of CCK receptors, there is poor contraction and emptying of the gallbladder, post-meal [14, 15]. The threshold of the cystic duct contraction receptors is presumably lowered, which results in its contraction when cholecystokinin is released.\n\n## 10.2 Clinical Presentation\n\nBiliary dyskinesia patients are mostly women in their 3rd or 4th decades of life, and present with post-prandial right upper quadrant or epigastric pain. Pain usually begins 30 min after a meal and reaches its peak intensity 2-3 h later. Liver function tests and blood counts remain within normal range. The gallbladder and intrahepatic and extrahepatic bile ducts are morphologically normal, and gallstones are usually absent on the ultrasound examination. Because of these normal biochemical and morphologic findings, the diagnosis of biliary dyskinesia is frequently delayed for months or years, and patients often go from physician to physician seeking relief from post-prandial abdominal pain.\n\n## 10.3 Cholescintigraphic Diagnosis\n\nAlthough biliary manometry is considered the gold standard, it is invasive, expensive, and not readily availability in many community hospitals. Cholescintigraphy provides a non-invasive and less expensive alternative. Imaging and quantitative analysis are carried out by acquiring data in two separate phases: the hepatic phase and gallbladder phase. Patients are studied after 4-6 h of fasting. Detailed clinical and drug history is taken to exclude the intake of opioids, smooth muscle relaxants, calcium channel blockers, or other drugs that act either on the sphincter of Oddi or gallbladder. In hypertensive patients, calcium channel blockers are discontinued for 2-3 days prior to the test. With the patient in the supine position, a large field of view dual-head gamma camera fitted with a low-energy, all-purpose, parallel-hole collimator is positioned anteriorly and posteriorly, and 3-5 mCi Tc-99m HIDA is injected intravenously. The data are acquired on the computer at one frame\/minute for 60 min [16-19]. Hepatic phase images usually appear normal (Fig. 10.1.3, top) unless the disease is far advanced, in which case there may be bile stasis in the extrahepatic ducts and delay in entry of bile into the duodenum. If the gallbladder is not seen by 60 min, delayed images are obtained at 3-4 h after injection of a second dose of Tc-99m HIDA. Injection of opioids to induce contraction of the sphincter of Oddi to fill the gallbladder is avoided.\n\nFig. 10.1.3\n\nCholescintigraphy in SOS. The uptake and excretion of 99mTc-HIDA are normal with delineation of the gallbladder and entire biliary tree. Bile enters the small intestine normally (top). After CCK-8 infusion, the gallbladder contracts and empties bile, which refluxes into the common hepatic duct (KB). Refluxed bile reenters the gallbladder immediately after cessation of CCK-8 infusion with an increase in size of the gallbladder (bottom)\n\n## 10.4 Gallbladder Phase Imaging\n\nThe data are acquired with the gamma camera. Hepatic phase images are carefully scrutinized for superimposition of duodenal or common bile duct radioactivity over the gallbladder. If there is any superimposition, the gamma camera angle is altered to separate the gallbladder from other structures. Often this may require changing the gamma camera angle by 10-15\u00b0 to the left or right anterior oblique position (Fig. 10.1.3, bottom). Most of the time, no such angulation is required. The data are collected at one frame\/minute for 30 min, (between 61-90 min) after injection of Tc-99m HIDA. Cholecystokinin at a dose of 3 ng kg-1 min-1 is infused intravenously for 10 min beginning at 65 min after HIDA injection The patient's symptoms (nausea, abdominal cramps, and\/or pain) are monitored carefully, and their time of onset, severity, and time of relief are noted on the data sheet or gallbladder time\/activity curve. ROIs are drawn over the gallbladder, liver (background), common hepatic ducts including the right hepatic and left hepatic ducts, common bile duct, stomach, and intestine (Fig. 10.1.4), and gallbladder ejection fraction, ejection period, ejection rate, and duodenogastric bile reflux are obtained (Fig. 10.1.5). The gallbladder, common hepatic duct (including the right and left hepatic ducts), common bile duct, and stomach and their corresponding curves are scrutinized carefully. Sphincter of Oddi spasm is differentiated from cystic duct spasm (Fig. 10.1.6) by careful analysis of both the image pattern and quantitative functional parameters [12, 13]. In most cases, the gallbladder ejection fraction remains normal. In many patients, the post-CCK-8 paradoxical refilling volume exceeds the basal volume by showing much higher counts after refilling than basal counts before hormone injection (Fig. 10.1.7). A true paradoxical filling recurs with the second dose of CCK-8 (Fig. 10.1.8) and must be differentiated from pseudo-paradoxical filling that may occur due to superimposition of intestinal activity on the gallbladder region of interest (Fig. 10.1.9).\n\nFig. 10.1.4\n\nRegions of interest for gallbladder phase study. One ROI is drawn over each of the gallbladder, liver (background), common hepatic duct, including right hepatic and left hepatic ducts, common bile duct, stomach, and intestine. Raw curves on the right represent the region of the same color\n\nFig. 10.1.5\n\nGallbladder phase results. The length of horizontal bar at the top represents the duration of CCK-8 infusion. The software calculates the results automatically after the ROIs are drawn. Two vertical bars denote the beginning and end of gallbladder emptying. The operator makes adjustments if automatically calculated results do not agree with the expected results\n\nFig. 10.1.6\n\nBiliary dynamics in biliary dyskinesia. In the case of sphincter of Oddi spasm, the gallbladder contracts and empties bile, which refluxes into the common hepatic and intrahepatic ducts. Most of the refluxed bile reenters the gallbladder (top). In patients with cystic duct syndrome, the gallbladder empties poorly, and no refilling occurs (bottom). Pain or discomfort occurs mostly during the gallbladder ejection period [12]\n\nFig. 10.1.7\n\nTypes of post-CCK-8 paradoxical filling curve patterns seen in patients with sphincter of Oddi spasm. Most gallbladders show normal ejection fraction followed by different patterns of refilling. Many gallbladders show much larger bile volume after refilling than the basal volume as shown by overshooting of the refilling curve [18]\n\nFig. 10.1.8\n\nTrue paradoxical filling with CCK-8 given twice. In both cases, the gallbladder empties normally (EF 92% and 86%), followed by paradoxical filling\n\nFig. 10.1.9\n\nAn apparent paradoxical filling after normal emptying (EF75%) of the gallbladder (GB) due to superimposition of intestinal (INT) activity\n\n## 10.5 Sphincter of Oddi Spasm\n\nCholescintigraphic findings vary depending upon the presence or absence of the native gallbladder. There is usually no bile pooling within the intrahepatic ducts in SOS, a feature distinctly different from those with stricture of the common bile duct [17]. Several types of functional abnormalities have been shown by different authors, depending upon the method of data collection and analysis [20, 21]. Shaffer et al. measured the hepatobiliary peak uptake time and calculated the percent excretion at 45, 60, and 90 min. Cholecystokinin at a dose of 0.02 units kg-1 min-1 was infused slowly between 60 and 90 min. The region of interest included both the liver and bile ducts. In 22 subjects with previous cholecystectomy with no current symptoms (control), the hepatobiliary mean peak time was 9.8 \u00b1 0.7 min. Percent excretion at 45, 60, and 90 min was 50.6 + 3.6, 66.3 + 2.8, and 84.7 + 1.3, respectively. The emptying rate between 0 and 60 min, and 60-90 min (after CCK) was 1.46 + 0.07 min-1 and 0.73 + 0.6 min-1, respectively. In nine patients with post-cholecystectomy SOS, the mean peak time was increased to 23 min, and percent excretion at 45 and 60 min was decreased to 20 and 39, respectively. The emptying rate between 0 and 60 min was 1.05 + 0.13 min-1 and increased to 1.12 + 0.19 min-1 after CCK infusion between 60 and 90 min [21]. Others have shown prolongation of the hepatic hilum to duodenal transit time [22, 23]. Most of the functional parameters improved after papillotomy (Fig. 10.1.10).\n\nFig. 10.1.10\n\nHepatic bile clearance in SOS. Normally, peak uptake of 99mTc-HIDA occurs within 10 min and 66% is excreted within 60 min. CCK enhances excretion. In patients with SOS, the peak is delayed and 60-min excretion is decreased. CCK administration has only a partial effect before papillotomy. Both uptake and excretion improve after papillotomy [21]\n\nThe hepatic phase images appear normal in most patients with early SOS (Fig. 10.1.3), and the gallbladder shows a normal ejection fraction. Cholescintigraphy in SOS is characterized by reflux of gallbladder bile into the common hepatic and right and left hepatic ducts with re-entry of refluxed bile into the gallbladder immediately after cessation of CCK-8 infusion. The curve shows rapid refilling of the gallbladder almost to its original volume, often even exceeding the original volume because of stimulation and expulsion of hepatic bile by cholecystokinin. Intrahepatic and extrahepatic bile ducts can accommodate almost all of 40-50 ml of bile emptied from the gallbladder without being dilated. Common hepatic duct bile reflux also occurs in patients with stenosis of the common bile duct and is referred to as the Krishnamurthy-Bobba sign [24]. When patients complain of pain during the gallbladder dynamic phase, the pain duration usually corresponds to the gallbladder ejection period. Not every patient with biliary dyskinesia, however, complains of pain at the time of the study. The extrahepatic ducts may appear early and persist for a longer duration of time, resulting in a marked delay in peak hepatic uptake in advanced cases [22, 25].\n\nSphincter of Oddi spasm is differentiated from anatomic stricture of the common bile duct by assessing both the image pattern and quantitative curve analysis after a pharmacologic intervention. Amyl nitrite, nitric oxide, and nifedipine relieve the spasm and dilate the sphincter, promoting rapid bile flow through it in patients with SOS, but not in those with an anatomic stricture (Fig. 10.1.11) of the common bile duct [26-28]. Sensitivity of cholescintigraphy in SOS varies from 70 to 90% [29].\n\nFig. 10.1.11\n\nCholescintigraphic differentiation of SOS from CBD stricture. The normal common bile duct time\/activity curve shows an early peak with rapid decline before and after amyl nitrite (top). The peak is delayed in both stricture and SOS. Following amyl nitrite, there is a rapid decline in patients with SOS (bottom), but not with stricture (middle) [26]\n\n## 10.6 Cystic Duct Spasm\n\nThis spasm is often characterized by a slow filling of the gallbladder during the hepatic phase with poor emptying response to cholecystokinin in the gallbladder phase. Although most of the gallbladders appear within 60 min after injection of Tc-99m HIDA, some are delayed for 3-4 h. Reduction in the gallbladder ejection fraction with CCK-8 is the salient feature. Many patients, but not all, experience mild- to moderate-intensity abdominal pain during the gallbladder ejection period. The gallbladder curve appears flat in patients with ejection fractions below 15%, and it is often difficult to note the exact time of the onset and end of gallbladder emptying (Fig. 10.1.4B). There is usually no bile reflux into the intrahepatic or extrahepatic ducts, and the gallbladder curve does not show refilling after cessation of CCK-8 infusion [12, 18]. The direction of the latter half of the curve is assessed carefully to make sure that there is no refilling of the gallbladder, which would suggest SOS. A large, non-physiological CCK-8 dose when given as a rapid intravenous bolus often produces a low ejection fraction and abdominal pain in many of the normal subjects, mimicking cystic duct syndrome. This effect is attributed to exceeding the cystic duct threshold for contraction and should be avoided.\n\n## 10.7 Differentiation of SOS from CDS\n\nThe differentiation is critical for appropriate patient management because cholecystectomy is the preferred treatment for CDS and antispasmodics or sphincterotomy for SOS. Cholecystectomy for SOS is inappropriate and may make the patient's condition get worse by taking away its low pressure reservoir function [30]. Both conditions are relatively more common in women than men. Proper attention is given during patient preparation, data collection, and analysis. The hepatic phase imaging data appear similar in both conditions, and there is bile entry into the duodenum in early cases. Only during the gallbladder phase is a difference seen between SOS and CDS. Cholecystokinin-induced bile reflux into the common hepatic duct and paradoxical refilling of the gallbladder are typical for SOS. In the authors' experience, another condition that may cause such a paradoxical filling is hyperacute (within 48 h) complete obstruction of the common bile duct (Chap. 8). Cholecystokinin stimulates bile secretion by cholangiocytes lining the intrahepatic bile ducts and expels bile from the liver by inducing contraction of the bile ducts. The hepatic bile is forced to enter the gallbladder in the presence of complete obstruction of the common bile duct. Other features of hyperacute CBD obstruction include its acute clinical presentation, abnormal liver function tests, and no bile entry into the duodenum during the hepatic phase. On rare occasions, superimposition of jejunal or ileal loop activity over the gallbladder region may cause false elevation of the gallbladder curve, which is readily recognized with the cine display of the gallbladder phase data. Patients with CDS do not exhibit paradoxical filling of the gallbladder, and there is no bile reflux into the intrahepatic ducts (Fig. 10.1.12).\n\nFig. 10.1.12\n\nDifferentiation of CDS from SOS. Cholecystokin (CCK) receptors on the gallbladder smooth mucle remain free in the fasting state (left). In cystic duct syndrome (middle), the gallbladder empties poorly in response to CCK-8 infusion because of a decrease in the total number of CCK receptors in the gallbladder body and fundus and activation of contractile receptors in the cystic duct. In the case of the sphincter of Oddi spasm (right), the sphincter contracts because of activation of the contractile receptors, and the bile emptied from the gallbladder refluxes into the common hepatic and intrahepatic ducts, later reentering the gallbladder (CCK-induced paradoxical filling of the gallbladder) after cessation of CCK-8 infusion [12]\n\n## 10.8 Treatment\n\nTreatment for SOS is different from that of CDS. A normal sphincter of Oddi function is modulated by local generation of nitric oxide [27]. Medical treatment with nitrites, nitrates, calcium channel blockers (nifedipine), and other smooth muscle relaxants has been shown to relieve the spasm and dilate the sphincter in patients with SOS [28]. Endoscopic sphincterotomy is reserved for those patients who fail to respond to medical therapy. Sphinctero-tomy is invasive, expensive, and carries a high rate of procedural complications. The overall endoscopic sphincterotomy complication rate in 2,347 patients in a multicenter study from 17 institutions in the United States was 9.8%. The complication rate is much higher (21.7%) in patients with sphincter of Oddi dysfunction [31]. Acute pancreatitis (5.4%) and hemorrhage (2.0%) account for the great majority of complications from endoscopic sphincterotomy. Special training and frequent experience in doing large numbers of patient procedures reduce complication rates [32]. Sphincteroplasty through an endoscopic balloon dilatation is effective in patients with bile duct stones [33]. However, its role in sphincter of Oddi spasm has not been studied critically in a large number of patients. Cholecystectomy is a relative contraindication in SOS. An intact gallbladder acts as a reservoir and accommodates bile that fails to pass through the sphincter in patients with SOS. Cholecystectomy takes away this reservoir function, and the symptoms persist or may even get worse, manifesting a post-cholecystectomy syndrome [30, 34].\n\n## 10.9 Cholecystectomy\n\nThis is the treatment of choice for CDS. Due to its low morbidity and low mortality, laparoscopic cholecystectomy has become more popular and has replaced open cholecystectomy as the preferred therapy [35]. Of a total of 320 patients with CDS who underwent cholecystectomy solely on the basis of a gallbladder low ejection fraction, 90% had relief of abdominal pain [36-42]. Histopathological examinations showed microscopic changes of disease in the gallbladder wall or cystic duct in 90% of patients (Chap. 9.2). Abdominal pain is known to persist in about 20% of patients undergoing cholecystectomy for various reasons [43], and the majority of these patients may have SOS. Clinically, SOS is divided into three types. Only type I responds to endoscopic sphincterotomy, and types II and III have a varied response. Type III is considered as duodenal visceral hyperalgesia, and the pain may not originate from the biliary tree at all [44]. Sphincterotomy is more effective than placebo in patients with sphincter pressure greater than 40 mmHg than in those without pressure elevation [45].\n\n## 10.10 Post-cholecystectomy Syndrome\n\nThis can now be considered an iatrogenic disease exaggerated by an inappropriate therapy (cholecystectomy) for patients with SOS. Removal of the gallbladder, which serves as a low pressure reservoir by accommodating hepatic bile, exaggerates the symptoms in patients with SOS. This syndrome can now be studied non-invasively by combining magnetic resonance cholangiopancreatography (MRCP) with quantitative cholescintigraphy. Rubini and Dimonte reported five patients with post-cholecystectomy syndrome and showed dilatation of the common hepatic and common bile ducts in four and cystic duct remnant in three (two remnants with gallstones) on MRCP. Cholescintigraphy showed bile pooling in both intrahepatic and extrahepatic ducts, with no bile entry into the duodenum at 60 min in all five patients. Bile did not enter the duodenum for another 60 min after a high-fat meal. Both peak hepatic uptake time and excretion half time were increased. The hepatic extraction fraction stayed above 80%, suggesting primarily a biliary disease. All four patients who underwent endoscopic sphincterotomy responded well with relief of symptoms [46].\n\n## 10.11 Pitfalls\n\nCholescintigraphy plays an important role in the evaluation of patients with biliary dyskinesia. Clear understanding of the mechanism of bile secretion and flow and factors that alter bile flow is important in arriving at a proper diagnosis. Functional imaging and measurement of parameters quantitatively require appropriate patient preparation and attention to technical details during data acquisition and analysis. Unlike the left ventricle, whose ejection fraction remains constant, the gallbladder ejection fraction can be controlled to any desired level simply by controlling the dose, dose rate, and duration of infusion of cholecystokinin. Cholecystokinin is available in two different molecular forms: an intact molecule with 33 amino acids (CCK-33) and a fragment with 8 amino acids (CCK-8 or sincalide). The dose recommended in the package insert of Kinevac (Bracco Diagnostic Inc., Princeton, N.J.) is found to be too large and often elicits a non-physiologic response during cholescintigraphy [47-49]. The dose should be reduced for cholescintigraphy (Chap. 6). When cholecystokinin is not available, the test is standardized using a fatty-meal stimulation, where the total calorie, fat, carbohydrate, and protein contents are controlled strictly for the chosen meal. Use of opioids for pain control is quite common in both inpatients and outpatients, and their use reduces the gallbladder ejection fraction. The gallbladder ejection fraction returns to normal after the discontinuation of opioids (Fig. 10.1.13). A commitment to high technical quality control, test standardization, and direct interaction with the patient before, during, and after the procedure produces the best clinical outcome [50].\n\nFig. 10.1.13\n\nEffect of opioids on gallbladder ejection fraction. A patient on vicodin for pain during viral hepatitis A shows slow clearance of the radiotracer from the liver (T 1\/2 = 92 min) and an ejection fraction of only 3% (top). 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Cholecystokinin cholescintigraphy: Detection of abnormal gallbladder motor function in patients with chronic acalculous gallbladder disease. J Nucl Med 1991;32:1695-1699\n\n38.\n\nZech ER, Simmons LB, Kendrick RR, et al Cholecystokinin enhanced hepatobiliary scanning with ejection fraction calculation as an indicator of disease of the gallbladder. Surg Gynecol Obstet 1991;172:21-24PubMed\n\n39.\n\nMisra DC, Blossom GB, Fink-Bennet D, et al Results of surgical therapy for biliary dyskinesia. Arch Surg 1991;126:957-960PubMedCrossRef\n\n40.\n\nHalverson JD, Garner BA, Siegel BA, et al The use of hepatobiliary scintigraphy in patients with acalculous biliary disease. Arch Intern Med 1992;152:1305-1309PubMedCrossRef\n\n41.\n\nSorenson MK, Francher S, Lang NP, Eidt JF, Broadwater JR. Abnormal gallbladder nuclear ejection fraction predicts success of cholecystectomy in patients with biliary dyskinesia. Am J Surg 1993;166:672-675PubMedCrossRef80677-4)\n\n42.\n\nKleiger PS, O'Mara RE. The clinical utility of quantitative cholescintigraphy: the significance of gallbladder dysfunction. Clin Nucl Med 1998;23:278-282CrossRef\n\n43.\n\nBlack NA, Thompson E, Sanderson CFB, ECHSS group. Symptoms and health status before and 6 weeks after open cholecystectomy: a European cohort study. Gut 1994;35:1301-1305PubMedCrossRef\n\n44.\n\nDesautels SG, Slivka A, hutson WR, chun A, Mitrani C, DiLorenzo C, Wald A. Postcholecy-stectomy pain syndrome: pathophysiology of abdominal pain in sphincter of Oddi type III. Gastroenterology 1999;116:900-905PubMedCrossRef70073-9)\n\n45.\n\nCraig AG, Toouli J. Sphincterotomy for biliary sphincter of Oddi dysfunction. Cochrane databse of systematic reviews 2001, issue 1, art. no.:CD001509. DOI:10.1002\/14651858.CD001509\n\n46.\n\nRubini G, Dmonte M. Postcholecystectomy syndrome. Evaluation by biliary cholescintigraphy and MR cholangiopancreatography. Clin Nucl Med 1999;24:784-789PubMedCrossRef\n\n47.\n\nKrishnamurthy GT, Bobba VR, Kingston E. Optimization of octapeptide of cholecystokinin (OP-CCK) dose for gallbladder emptying. In: Raynaud C, (ed) Proceedings of the third world congress of nuclear medicine and biology. Paris, France: Pergamon Press; 1982, pp 2224-2247\n\n48.\n\nToftdahl DB, Hojgaard L, Winkler K. Dynamic cholescintigraphy: Induction and description of gallbladder emptying. J Nucl Med 1996;37:261-266PubMed\n\n49.\n\nZiessman HA, Fahey FH, Hixon DJ. Calculation of a gallbladder ejection fraction: advantage of continuous Sincalide infusion over the 3-minute infusion method. J Nucl Med 1992;33:537-541PubMed\n\n50.\n\nKrishnamurthy S, Krishnamurthy GT. Gallbladder ejection fraction: a decade of progress and future promise (editorial). J Nucl Med 1992;32;542-544\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_11(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 11. Pediatric Nuclear Hepatology\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nThe liver and biliary system develops from a bi-lobed (cephalic and caudal) endodermal bud that sprouts along the ventral surface of the distal end of the foregut at its junction with the midgut. The cephalic bud divides into two branches, which later form the right and left lobes of the liver. The caudal bud gives rise to biliary tract and the gallbladder (Fig. 11.1.1, Chap. 1). The biliary tract starts first as a patent tubular structure, but turns into a solid core during early intrauterine life through the proliferation of epithelial cells. After recanalization of the solid core by 12-14 weeks, bile secretion begins and flows into the small intestine. Most of the congenital abnormalities of the biliary tract are due to either failure of recanalization of the ducts (biliary atresia) or faulty recanalization with cystic dilatation of intrahepatic (Caroli's disease) and extrahepatic ducts (choledochal cyst).\n\nThe liver and biliary system develops from a bi-lobed (cephalic and caudal) endodermal bud that sprouts along the ventral surface of the distal end of the foregut at its junction with the midgut. The cephalic bud divides into two branches, which later form the right and left lobes of the liver. The caudal bud gives rise to biliary tract and the gallbladder (Fig. 11.1.1, Chap. 1). The biliary tract starts first as a patent tubular structure, but turns into a solid core during early intrauterine life through the proliferation of epithelial cells. After recanalization of the solid core by 12-14 weeks, bile secretion begins and flows into the small intestine. Most of the congenital abnormalities of the biliary tract are due to either failure of recanalization of the ducts (biliary atresia) or faulty recanalization with cystic dilatation of intrahepatic (Caroli's disease) and extrahepatic ducts (choledochal cyst).\n\nPhysiologic jaundice occurs in 65% of near- or full-term infants, and usually clears within few weeks after birth. Persistent uncomplicated long-term hyperbilirubinemia is a concern, and past studies suggested some link with neurological sequelae, such as mental retardation, autism, and cerebral palsy. The effects of uncomplicated high serum bilirubin levels have been addressed well in more recent studies. In a Canadian study involving 56,019 near- or full-term infants with serum bilirubin levels greater than 20 mg%, no evidence of bilirubin toxicity (kernicterus, gaze palsy) was found, lessening the anxiety over uncomplicated hyperbilirubinemia [1]. There is also no increased risk of autism due to hyperbilirubinemia [2]. However, persistent hyperbilirubinemia beyond 2 weeks should raise concern for several other possibilities (Table 11.1.1), some requiring immediate intervention, while others may be managed by careful follow-up without a specific therapy. Neonatal hepatitis and congenital biliary atresia are two of the diseases that account for the great majority of persistent jaundice beyond 2 weeks in a neonate.\n\nTable 11.1.1\n\nCauses of persistent jaundice beyond 2 weeks after birth\n\n1. Neonatal hepatitis\n\n---\n\nViral\n\nCytomegalo virus\n\nHepatitis B\n\nRubella\n\nHerpes\n\nCoxsackie\n\nBacterial\n\nE. coli\n\nKlebsiella\n\nPseudomonas\n\nProteus\n\nBacteroids\n\nStreptococcus\n\nTreponema pallidum\n\nParasitic\n\nToxoplasma gondii\n\n2. Biliary atresia\n\nExtra-hepatic ducts (congenital biliary atresia)\n\nIntra-hepatic ducts.\n\nCongenital rubella\n\nTrisomy 17\n\nTrisomy 18\n\nAlagille's syndrome (arterio-hepatic dysplasia)\n\n3. Genetic or metabolic\n\nAlpha 1 anti-trypsin\n\nGalactosemia\n\nHereditary fructosemia\n\nHypothyroidism\n\n## 11.1 Congenital Biliary Atresia vs. Neonatal Hepatitis\n\nCongenital biliary atresia (CBA) is a progressive, sclerosing, inflammatory disease, primarily affecting the extrahepatic bile duct from the region of the portahepatis to its termination in the duodenum. It is estimated to affect 1 in 12,000 live births [3]. Untreated, the disease progresses relentlessly, leading to biliary cirrhosis and death. Hepatoportoenterostomy (Kasai procedure) is the definitive therapy in early stages. The rate of success of establishing an adequate bile flow by the Kasai procedure depends very much on the duct size at birth and the time interval between birth and hepatoportoenterostomy (HPE). Effective bile flow is established in 77-82% of the infants when HPE is performed within 60 days after birth. The success rate decreases to 45-59% when HPE is delayed for 60-90 days and decreases further to 10-28% when HPE is delayed for 3-4 months. The success rate increases to 92% when the duct lumen is greater than 150 \u03bcm in size [4]. Early diagnosis, therefore, plays a critical role in determining the outcome of therapy (Table 11.1.2).\n\nTable 11.1.2\n\nEffect of timing and duct size on rate of success of hepatoportoenterostomy (HPE) for congenital biliary atresia [4]\n\nAge at HPE | Success rate\n\n---|---\n\n<60 days | 77-82%\n\n61-90 days | 45-59%\n\n90-120 days | 10-28%\n\nDuct size\n\n>151 \u03bcm | 92%\n\n50-150 \u03bcm | 81%\n\nNo patent ducts | 18%\n\n### 11.1.1 Etiology\n\nThe exact etiology of CBA is unknown. Two theories have been proposed; one theory is that of metaplasia of the hepatocytes, and the other is that of faulty metaplasia combined with ingrowth from extrahepatic ducts [5, 6]. Ischemia, toxins, and duct injury are well recognized risk factors. Recent studies suggest that atresia may be due to a failure of remodeling at the hepatic hilum with a continuation of the fetal-type bile leak due to poor mesenchymal support. Bile leak through the ducts initiates an intense inflammatory reaction with a subsequent obliteration of the bile duct lumen. The characteristic findings on liver biopsy include ductal proliferation, canalicular and cellular bile stasis, periportal edema, and fibrosis. The title \"atresia\" is reserved for patients with complete obliteration of the duct lumen. Hypoplasia is a transition phase before complete obliteration of the duct lumen [7].\n\nCongenital biliary atresia is divided into two clinical forms, fetal and perinatal. The perinatal form is the most common and is characterized by late-onset neonatal jaundice, with a jaundice-free time interval after birth. There are no other accompanying congenital abnormalities. Remnants of the bile duct are found within the hepatoduodenal ligament [4]. The fetal form is less frequent and accounts for less than 25% of the cases. It is characterized by an early onset of neonatal jaundice, with no clearance of physiologic jaundice (no jaundice-free time interval). It is frequently associated with other congenital anomalies, including polysplenia, asplenia, cardiovascular defects, or abdominal situs inversus, etc. No bile duct remnants are found in the hepato-duodenal ligament.\n\n### 11.1.2 Clinical Presentation\n\nInfants with CBA are usually born at full term with a normal birth weight and show normal growth pattern in the immediate neonatal period. It is more common in girls than boys. Jaundice usually starts 2 weeks after birth with acholic stool. Serum shows nonspecific elevation of direct bilirubin, alanine aminotransferase (ALT), and gamma- glutamyl transferase (GGT). About 90% of infants show a serum conjugated bilirubin (direct bilirubin) level greater than 4 mg dl-1 [8, 9]. Aspiration of bile through a naso-duodenal tube (after instillation of 25% magnesium sulfate into the duodenum to stimulate gallbladder contraction and bile emptying) establishes patency of the bile ducts and thus excludes the diagnosis of congenital biliary atresia and confirms the diagnosis of neonatal hepatitis. Bile is aspirated from the duodenum in more than 80% of the infants with neonatal hepatitis, but not from infants with biliary atresia [10]. Intubation and duodenal bile aspiration are relatively invasive in a newborn, and a negative test (no bile aspiration) is not always specific for congenital biliary atresia.\n\nHepatomegaly is unusual at birth and begins to develop 6-8 weeks later. Cholestasis, fibrosis, and cirrhosis develop as the infant grows. The incidence of biliary atresia is higher among Chinese and Filipinos (2-3 per 10,000) than in Japanese and Caucasians [11]. Serial liver function tests provide an indication of the severity of the disease. Serum values of total bilirubin, direct bilirubin, GGT, and alkaline phosphatase, and the alkaline phosphatase\/GGT ratio do not reliably distinguish congenital biliary atresia from neonatal hepatitis [12]. Laparoscopy has no diagnostic role as it cannot assess the patency of the bile ducts. Cholangiography is invasive and technically difficult to do because of the small size of the ducts.\n\n### 11.1.3 Cholescintigraphy\n\nThe indication for cholescintigraphy in a neonate is persistent jaundice beyond the 2nd week of life or new jaundice that develops 3 weeks after birth. An ideal patient preparation includes treatment with 5 mg kg-1 per day of phenobarbital, in two equally divided doses, for 5-7 days prior to cholescintigraphy. Phenobarbital stimulates bile production [13] and increases the secretion of the radiotracer into bile, enabling better delineation of bile ducts and duodenum in infants with neonatal hepatitis, but not in those with CBA [14-16]. Cholestyramine or ursodeoxycholic acid also promotes bile secretion. The infant should not be fed for an hour before and an hour after injection of the radiotracer. The duration of fasting prior to cholescintigraphy should be increased in older children in proportion to their age.\n\n### 11.1.4 Data Collection\n\nThe agent of choice in neonatal cholestasis is Tc-99-m mebrofenin, because it competes with serum bilirubin for hepatocyte uptake much more effectively than Tc-99m disofenin or any other Tc-99m-HIDA agent [17]. A dose of 100 \u03bcCi kg-1 (1 mCi minimum) of Tc-99-m-mebrofenin is injected intravenously. A gamma camera, preferably with a small field of view, fitted with a low-energy, parallel-hole, all-purpose collimator, is positioned anteriorly over the upper abdomen. The computer data are collected on a 64 \u00d7 64 word mode matrix at one frame per minute for 60 min (Fig. 11.1.1). Delayed images are obtained between 2-4 h and 20-24 h when intestinal activity is not seen in early images.\n\nFig. 11.1.1\n\nCholescintigraphy in the neonate. There is normal extraction and rapid excretion of 99mTc-HIDA in a normal neonate (top left). Extraction is decreased (persistent heart activity), and there is delayed excretion in neonatal hepatitis (top right). In biliary atresia, extraction is good, but no excretion into small bowel is seen (bottom left). Kidneys form the alternate route of excretion [19]\n\n### 11.1.5 Data Analysis\n\nA normal neonate shows rapid liver uptake and excretion compared to the infant with CBA or hepatitis (Fig. 11.1.1). For quantification, three regions of interest are drawn, and time-activity curves are generated separately over the (1) heart, (2) right upper lobe of the liver, and (3) spleen. The spleen ROI is used for blood background. Because both the hepatic and splenic arteries arise from the same celiac artery, the spleen serves as an ideal organ to represent the liver blood background. The hepatic extraction fraction and excretion half times are obtained by subjecting data to deconvolutional analysis, very similar to the procedure for adults (Fig. 11.1.2) [18, 19]. Measurement of the hepatic extraction fraction by deconvolutional analysis is equivalent to measuring the first pass extraction by injecting the tracer directly into the hepatic artery or portal vein. The hepatic extraction fraction provides an objective criterion for separating biliary atresia from neonatal hepatitis in early cases of CBA. The liver extraction fraction and excretion half time provide a measure of the severity of hepatobiliary disease. For a comparable level of serum bilirubin, infants with congenital biliary atresia show a relatively higher hepatic extraction fraction than infants with neonatal hepatitis [19, 20]. Obstruction over a longer period of time ultimately compromises hepatocyte function, and the hepatic extraction fraction begins to decrease and loses its power to differentiate biliary from hepatocyte decrease. The diagnosis of congenital biliary atresia or neonatal hepatitis is also made by assessing both the images and the shape and direction of the curves (Fig. 11.1.3).\n\nFig. 11.1.2\n\nQuantitative biliary dynamic studies in pediatrics. Hepatic extraction fraction (HEF) and excretion half-time (HCT) in a normal neonate (left), congenital biliary atresia (middle), and neonatal hepatitis (right). HEF remains relatively normal in biliary atresia, whereas both HEF and HCT are abnormal in neonatal hepatitis (modified from [19])\n\nFig. 11.1.3\n\nHeart and liver simple time-activity curves in three children. In a normal child (a), both heart and liver curves show a rapid decline. In biliary atresia (b), the liver curve shows slow uptake and no excretion. In neonatal hepatitis (c), the liver curve shows slow excretion without an uptake peak, indicating mainly background reduction [12]\n\n### 11.1.6 Normal Neonate\n\nCholescintigraphy in a normal neonate is characterized by rapid hepatocyte uptake and secretion of Tc-99 m-HIDA. Peak hepatic uptake occurs within 5 min, the gallbladder appears within 10 min, and intestinal activity is seen between 20-30 min. Unlike in adults and older children, the common hepatic duct, common bile duct, and the cystic ducts are not seen in a neonate [15]. After reaching the peak within 5 min, the liver shows a rapid decline. Peak cardiac counts occur in the first frame (1 min) followed by a rapid decline, reaching a background activity level within 5 min. Both liver and heart curves converge towards the end [12]. A subjective analysis of the shape of the curves helps to separate normal infant from congenital biliary atresia or neonatal hepatitis (Fig. 11.1.3). Calculation of the hepatic extraction fraction and excretion half time provide better and more powerful objective functional parameters than a simple subjective analysis of the curves (Fig. 11.1.2). Two of the commercial vendors now are planning to provide hepatobiliary software for routine clinical use on their gamma camera and computer systems, but for others, the users have to develop their own software and database for clinical application.\n\nIn a normal neonate, the hepatic extraction fraction value ranges between 87 and 100% with an average of 99.0 \u00b1 3.6% (Table 11.1.3). A value below 92% (mean, 2 SD) is considered abnormal. Mean excretion half time with Tc-99 m disofenin is 23.6 \u00b1 7.7 min; a value above 40 min is considered abnormal. An artifact in the calculation of the hepatic extraction fraction due to an abrupt truncation of data collection is avoided by adding an appendage at the end of the curve [21].\n\nTable 11.1.3\n\nHepatic extraction fraction and excretion half time (mean \u00b1 SD) in normal subjects and children with congenital biliary atresia, hepatitis, and miscellaneous hepatobiliary diseases [18]\n\nParameter | Normal | CBA | Hepatitis | Miscellaneous\n\n---|---|---|---|---\n\nNumber | 12 | 9 | 16 | 8\n\nHepatic extraction fraction (%) | 99.0 \u00b1 3.6 | 79.3 \u00b1 25.5 | 51.5 \u00b1 20.6 | 87.8 \u00b1 13.3\n\nHepatic excretion halftime (min) | 23.6 \u00b1 7.7 | 183.1 \u00b1 201.1 | 101.9 \u00b1 77.0 | 44.8 + 35.7\n\n### 11.1.7 Congenital Biliary Atresia\n\nNeonates with congenital biliary atresia maintain a relatively high extraction Tc-99m-HIDA with clear delineation of the hepatic morphology with well-defined borders and contour. Major abnormalities are confined to bile secretion. Despite a good extraction by the hepatocytes, there is total lack of secretion of Tc-99m-HIDA into the bile, resulting in non-visualization of the entire biliary tree. The liver image appears much like a radiocolloid scan without the spleen. Delayed images at 24 h do not show any evidence of bile entry into the small intestine [15, 16]. The liver time-activity curve shows good uptake, but no excretion. Excretion half time values often indicate infinity (Fig. 11.1.2). The hepatic extraction fraction value is maintained at a high level despite high serum bilirubin, especially when cholescintigraphy is obtained within 45 days after birth. When the extraction fraction value begins to decline in patients with CBA, however, it indicates functional compromise because of persistent bile duct obstruction over a longer period of time [20]. A diagnosis of congenital biliary atresia is highly probable in a neonate with conjugated (direct) hyper-bilirubinemia, a relatively high hepatic extraction fraction in association with non-visualization of the entire biliary tree and the small intestine. Cholescintigraphy is unable to differentiate intrahepatic (Alagille's syndrome) from extrahepatic (congenital biliary atresia) ductal obstruction.\n\nThe incidence of persistent jaundice beyond 2 weeks after birth, in general, is estimated to be 1 in 5,000 live births; neonatal hepatitis occurs 1 in 8,000, and biliary atresia, 1 in 10,000-15,000 live births. Pure intrahepatic biliary atresia or hypoplasia is rare and occurs in 1 in 70,000 births [22]. The extrahepatic bile ducts are patent in most patients with intrahepatic biliary atresia. Intrahepatic biliary atresia is commonly associated with intrauterine infection and genetic abnormalities as listed in Table 10.1.1. Alagille's syndrome is a form of intrahepatic biliary hypoplasia or atresia in association with pulmonary stenosis, vertebral body abnormalities, sexual and mental underdevelopment, and a typical facial feature [23].\n\n### 11.1.8 Neonatal Hepatitis\n\nNeonatal hepatitis is usually caused by viral, fungal, or bacterial infection (Table 10.1.1). Jaundice persists beyond 2 weeks after birth, and the liver function tests show non-specific elevation of enzymes with conjugated hyper-bilirubinemia [11]. Cholescintigraphic features depend upon the severity of hepatocellular disease at the time of imaging. Uptake of Tc-99m-HIDA by the hepatocyte is delayed and decreased. Both the time-to-peak hepatic uptake and excretion half time are increased. Intestinal and gallbladder visualization time may be either normal or markedly delayed. Due to the marked reduction in hepatocyte extraction of Tc-99m HIDA, the liver morphology is poorly defined with indistinct borders and contour. Heart and liver time-activity curves run in parallel, indicating mostly background reduction (Fig. 11.1.3). The hepatic extraction fraction value shows an inverse relationship with the serum bilirubin level. Extraction fraction values as low as 30% are common [19]. Intestinal activity is seen by 24 h. An apparent appearance of good uptake by the liver is merely a reflection of hepatic blood pool radioactivity and is not due to true uptake by the hepatocytes, as reflected by low hepatic extraction fraction value.\n\nThe success of cholescintigraphy is variable, depending upon the time interval between birth and imaging (Table 11.1.4). A sensitivity of 100%, specificity of 80%, and accuracy of 93% have been reported from some centers with technetium-99m mebrofenin [24]. High accuracy is attributed partly to the agent's ability to compete effectively with serum bilirubin for uptake by the hepatocytes. One cannot always separate congenital biliary atresia from neonatal hepatitis solely on the basis of the HEF value, emphasizing the importance of combining quantification with the image pattern to increase diagnostic accuracy [20].\n\nTable 11.1.4\n\nEfficacy of cholescintigraphy, ultrasonography, and liver biopsy in the differential diagnosis of congenital biliary atresia from neonatal hepatitis [24]\n\nParameter | Cholescintigraphy | Ultrasonography | Liver biopsy\n\n---|---|---|---\n\nSensitivity (%) | 100 | 70 | 95\n\nSpecificity (%) | 80 | 70 | 100\n\nAccuracy (%) | 93 | 70 | 97\n\nPositive predictive value | 91 | 82 | 100\n\nNegative predictive value | 100 | 54 | 91\n\n### 11.1.9 Management of Neonatal Cholestasis\n\nNeonatal hepatitis is treated medically with good nutrition, anti-viral, anti-bacterial, or anti-parasitic agents, depending upon its etiology. Most patients recover uneventfully (Fig. 11.1.4A). The American Academy of Pediatrics recommends phototherapy for healthy term infants [24]. The Kasai procedure is the treatment of choice for congenital biliary atresia [26, 27]. The neonate is prepared for surgery with preoperative antibiotic prophylaxis, optimal hydration, and good nutrition. The abdomen is explored through a high transverse incision, and a fine catheter is inserted into the gallbladder by piercing the fundus. Fibrosis of the wall with no bile within the gallbladder or obliteration of the common bile duct lumen confirms biliary atresia. The presence of bile inside the gallbladder with patency of the common bile duct, on the other hand, indicates neonatal hepatitis. A cholangiogram is obtained by injecting the radiocontrast through the catheter.\n\nFig. 11.1.4\n\nResponse to therapy in neonatal cholestasis. An infant with giant cell neonatal hepatitis responds to medical therapy and shows bile secretion into intestine (left). An infant with congenital biliary atresia (right) shows bile secretion into the intestine after a Kasai procedure. (Courtesy of Dr. Vasundara Tolia, Detroit, MI [20])\n\nIn the original Kasai procedure, the jejunum is cut beyond the ligament of Trietz, and a portoenterostomy is established. The proximal part of the jejunum is anastomosed to the side wall of the Roux-en-Y limb, below the portoenterostomy. The most common complication of portoenterostomy is acute ascending cholangitis due to reflux of intestinal contents into bile ducts. Several modifications to the original portoenterostomy were made mainly with the hope of preventing acute ascending cholangitis (Fig. 11.1.5). The temporary jejunostomy stoma created for decompression is usually closed at 3 months after confirming adequacy of bile flow. Most pediatric surgeons now seem to prefer the original Kasai procedure (Kasai 1) and avoid creating an external fistula. Administration of appropriate antibiotics reduces the incidence of acute or recurrent cholangitis. A successful Kasai procedure establishes free bile flow into the small intestine (Fig. 11.1.4B)\n\nFig. 11.1.5\n\nPortoenterostomy for congenital biliary atresia. In Kasai I, an open end of a loop of the jejunum is attached directly to the undersurface of the liver to serve as a bile conduit. In Kasai II and Kasai III, a temporary external fistula is created. Other types of anastomoses include Segura I, Ueda I, and Lilly-Altman (modified from [11])\n\nThe rate of success of portoenterostomy for congenital biliary atresia depends primarily upon the time interval between birth and surgery and also upon the size of the duct lumen at birth (Table 11.1.2). Poor results are generally attributed to delayed portoenterostomy. A success rate of 89% and a 10-year survival rate of 33-74% are obtained when the neonate with congenital biliary atresia receives portoenterostomy within 60 days after birth (Table 11.1.5).\n\nTable 11.1.5\n\nClinical outcome after Kasai procedure for congenital biliary atresia [3]\n\nParameter | Success rate (%)\n\n---|---\n\nEffective bile excretion | 89\n\nClearance of jaundice | 62\n\nJaundice-free survival | 53-62\n\n10-year survival | 33-74\n\nOnset of esophageal varices | 29-73\n\nLiver transplantation is the ultimate therapy when HPE fails. The usual cause for HPE failure is continued fibrosis secondary to recurrent cholangitis. Approximately 65% of infants who receive HPE for biliary atresia ultimately require liver transplantation [28, 29]. Biliary atresia is the most common indication and accounts for 54% of all liver transplantation in children. Other indications include metabolic diseases (14%), acute hepatic necrosis (11%), autoimmune and other cirrhosis (7%), malignancy, and other miscellaneous conditions [30]. Split liver and living donor liver transplantations have helped to overcome, to some extent, the shortages of donor livers in the pediatric population [31, 32]. The high cost and limited availability of donors still remain two of major factors limiting liver transplantation in children (see Chap. 12). The Kasai operation remains the first choice of treatment, and its application early at experienced centers reduces the need for liver transplantation [33]. Non-invasive functional imaging with quantification is ideal for children, for evaluation of both native and transplant livers. Congenital biliary atresia, metabolic diseases, and acute hepatic necrosis all together account for nearly 80% of all transplants in children (Fig. 11.1.6). In the case of congenital biliary atresia, cholescintigraphy is suitable not only for diagnosis, but also for assessing function after the Kasai procedure or transplant liver function when the Kasai procedure fails [34].\n\nFig. 11.1.6\n\nIndications for liver transplantation in children. Congenital biliary atresia is the most common indication, followed by metabolic diseases, acute hepatic necrosis, and others [33]\n\n## 11.2 Cystic Diseases of the Hepatobiliary System\n\nCysts are abnormal spaces within the liver parenchyma or along the biliary tract (choledochal cysts). They vary in size and location and contain mostly clear liquid. Choledochal cysts are rare, occurring in 1 in 100,000-150,000 live births. About 80% of choledochal cysts become symptomatic early and are diagnosed before the age of 10. Cysts are classified in many ways, and the classification provided in Table 11.2.1 is primarily for diagnostic purposes using various imaging techniques, including ultrasound, CT, MRCP, and cholescintigraphy. Cystic diseases due to parasites (hydatid cyst) and bacteria (abscess) are covered separately under space-occupying lesions of the liver in Chap. 4. This section deals primarily with those congenital cysts that have direct communication with the bile ducts and hence fill in with radiolabeled bile. The diagnosis of a non-communicating cyst is inferred when radioactive bile does not enter the cyst.\n\nTable 11.2.1\n\nClassification of cystic diseases of the hepatobiliary system\n\n1. Parasitic (hydatid cyst)\n\n---\n\n2. Non-parasitic cystic disease\n\na) Intrahepatic\n\n1) Solitary or multiple intrahepatic cysts with no communication with the bile ducts (adult polycystic liver disease)\n\n2) Mixed variety where few cysts communicate with the bile ducts\n\n3) Diffuse or segmental cystic dilatation of the bile ducts (Caroli's disease)\n\nb) Extrahepatic.\n\n1) Choledochal cyst\n\na) Cystic\n\nb) Fusiform\n\nc) Cylindrical\n\nd) Rosary (multiple cysts)\n\ne) Diverticular\n\nf) Choledochocele\n\nCysts are either solitary or multiple. Solitary cysts are rare, found at all ages, and most remain asymptomatic [1]. The cysts occur mostly along the hepatobiliary tract, starting at the smallest intralobular duct to the termination of the common bile duct into the duodenum. Most of the cysts are congenital, but a few of the acquired cysts follow an infection, trauma, or obstruction of a duct. The cyst may be entirely extrahepatic or intrahepatic in location.\n\nThe occurrence of both intrahepatic and extrahepatic cysts in a patient is rare, but is well recognized [2].\n\n### 11.2.1 Etiology\n\nThe exact cause of cystic dilatation of the bile ducts is unknown. Various hypotheses have been proposed, including weakness of the duct wall, angulation of the duct, presence of a valve-like structure within the lumen, inflammation, or sphincter dysfunction, all of which increase in intraductal pressure with impedance to bile flow [3]. The dilatation is initiated by an irregular and unequal proliferation of duct cells during recanalization of the solid core during early intrauterine life. Cysts are relatively more common in the Japanese and other Asians than in people elsewhere in the western hemisphere. An incidence of 1 in 13,000 hospital admissions has been reported in Japanese children [4]. The cyst wall ranges in thickness from 1 to 10 mm and consists mostly of fibrous tissue lined with cuboidal cells, or often with no lining cells at all. The cyst wall lacks mucous glands and a muscle layer. The cyst volume varies from a few milliliters to several liters [3].\n\n### 11.2.2 Classification of Choledochal Cysts\n\nCysts are classified mainly into the intrahepatic and extrahepatic variety. Kami et al. and Todani et al. recommend classification based upon the shape, size, extent, and location of the cyst [5, 6]. The cystic type is the most common variety and accounts for 54% of all choledochal cysts (Fig. 11.2.1). Fusiform is the next most common (24%), followed by the cylindrical (15%) and rosary type (4%). Together, these four account for nearly 97% of all cystic lesions. The diverticular type arises from one side of the duct wall, consists of a narrow opening, and may compress the adjoining structures, manifesting clinical symptoms of obstruction. A cyst at the distal end of the common bile duct, situated within the duodenal wall close to the ampulla of Vater, is called a choledochocele and is the rarest type (0.5%). The gallbladder remains normal in size, and gallstones are relatively rare in patients with choledochal cysts [3].\n\nFig. 11.2.1\n\nClassification of extrahepatic bile duct cysts. Cystic type is the most common (54%), followed by fusiform (24%), cylindrical type (15%), rosary (4%), and diverticular form (2.5%). Choledochocele is cystic dilatation of the intraduodenal part of the common bile duct before its entrance into the duodenum\n\n### 11.2.3 Intrahepatic Cystic Dilatation (Caroli's Disease)\n\nCaroli et al. in 1958 first described an entity of pure intrahepatic cystic dilatation [2]. Numerous authors have since described frequent association of intrahepatic cysts with cysts of the extrahepatic ducts. Studies from Japan report as many as 28-42% of patients with extrahepatic cysts having simultaneous intrahepatic cysts [3, 5, 7]. The fusiform and cylindrical types are more common in patients with intrahepatic cysts (Fig. 11.2.2).\n\nFig. 11.2.2\n\nClassification of intrahepatic bile duct cysts. Fusiform and cylindrical types are typical examples of Caroli's disease. Cystic type may affect both the intrahepatic and extrahepatic ducts. Simultaneous involvement of both intrahepatic and extrahepatic ducts is found in 28-42% of patients\n\n### 11.2.4 Diagnosis\n\nCholedochal cysts are three times more common in women than men, and more common in Asia and the Orient than in the western hemisphere [3]. Cysts remain asymptomatic for many years. When symptoms occur, more than one half of the patients present with a triad of abdominal pain, abdominal mass, and jaundice. Pain is usually localized to the right upper quadrant. The onset of symptoms usually coincides with the enlargement of the cyst, and the symptoms subside when the cyst decreases in size or drains spontaneously into the duodenum. Liver function tests show mild non-specific elevation. Ultrasonography or CT is the initial diagnostic procedure of choice. Once a cyst is identified with one of the imaging modalities, cholescintigraphy confirms the diagnosis. A cyst by ultrasound or CT in the vicinity of the biliary tract that fills with radiolabeled bile is specific for a choledochal cyst (Fig. 11.2.3). Choledochal cysts must be differentiated from cysts in other nearby organs, including cystic lesions of the kidney (renal cyst, hydronephrosis, and Wilm's tumor), pancreas (pseudocyst), or duodenum (diverticulum), etc. The malignant potential of a choledochal cyst is about 2.4% [6].\n\nFig. 11.2.3\n\nCholescintigraphic features of a choledochal cyst. A choledochal cyst fills in with radiolabeled bile and appears as a round structure in the middle of the common duct. The gall\u00adbladder is not seen\n\n### 11.2.5 Cholescintigraphy\n\nThe imaging study is obtained 4-6 h after fasting. Usually there is no need for pretreatment with phenobarbital, unless the child is severely jaundiced. Any one of the Tc-99m-HIDA agents may be chosen when serum bilirubin is within normal range. Routinely 1 min per frame images are obtained for 60 min (Fig. 11.2.3). Delayed images are obtained at 4 and 24 h when necessary.\n\nThe most common cholescintigraphic feature of a choledochal cyst is that of an early defect (90%) in the vicinity of the common hepatic or common bile duct that fills with radioactive bile at 4 or 24 h (Table 11.2.2). Intrahepatic ductal prominence or bile pooling is seen in 22% [8]. The gallbladder usually appears late. Only about 25% of the gallbladders are seen within 60 min, and up to 25% remain nonvisualized, even at 24 h [8, 9, 10].\n\nTable 11.2.2\n\nCholescintigraphic features of choledochal cyst\n\nParameter | Frequency (%)\n\n---|---\n\n1. Prolonged intrahepatic ductal prominence | 22\n\n2. Visualization of the gallbladder within 1 h | 25\n\n3. Non-visualization of the gallbladder by 2-4 h | 71\n\n4. Non-visualization of the gallbladder by 24 h | 25\n\n5. Filling defect in the region of the cyst in early images | 90\n\n6. Filling of the defect with the radiolabel in late images | 89\n\nDelayed filling or non-filling of the native gallbladder in a patient with choledochal cyst suggests that the cyst may act as a low pressure reservoir for bile. Gallbladder histopathological changes indicative of acute cholecystitis are found in 7% and those of chronic cholecystitis in 59% of patients with choledochal cysts [8]. This raises the possibility that the presenting symptoms may be due to onset of cholecystitis and not the choledochal cyst itself. Large cysts (one measured 400 ml) may compress adjoining structures and present clinically as an acute emergency [11]. Combined application of ultrasound and Tc-99m-HIDA cholescintigraphy is the most cost-effective approach in the diagnosis of choledochal cysts [12].\n\n### 11.2.6 Management of Choledochal Cysts\n\nSurgical excision of the cyst is the treatment of choice [13]. The cyst is excised with the hepatic duct Roux-en-Y jejunal anastomosis. This approach results in the most favorable postsurgical outcome, with a low incidence of ascending cholangitis. Total excision of the cyst is recommended because of its high malignant potential, which is 20 times higher than in the general population [3]. Surgical management of pure intrahepatic cysts (Caroli's disease) remains controversial. Lobectomy is advocated when cysts involve only one lobe of the liver. In the severe form of Caroli's disease, recurrent ascending cholangitis often leads to fibrosis and portal hypertension. About 20% of the choledochal cysts diagnosed in adults are treated much like those in children with total surgical excision [14].\n\n### 11.2.7 Cystic Disease of the Liver Parenchyma\n\nLiver cysts can be either solitary or multiple. Solitary cysts are uncommon, non-hereditary, and not associated with cysts in other organs. The Mayo Clinic reported only 38 solitary asymptomatic liver cysts during a follow-up period of 47 years [15]. Frequent use of abdominal ultrasound examination now can detect many more asymptomatic liver cysts than before. Symptomatic solitary liver cysts are four times more common in women than men and are found more frequently in the anterior-inferior area of the right lobe [16]. The cyst surface is usually smooth with wall thickness less than a centimeter. The cyst size varies from a few milliliters to several liters. One reported cyst contained 17 l of fluid [17]. Most of the cysts contain clear fluid. Some cysts may contain blood, mucin, proteins, and cholesterol. Bile is rarely found in a solitary cyst, indicating its non-communication with the bile ducts. A solitary cyst on ultrasound or CT that does not fill with radiolabeled bile indicates a non-communicating simple liver cyst (Fig. 11.2.4A).\n\nFig. 11.2.4\n\nLiver cyst. A simple liver cyst appears as a round defect early and does not fill in with radiolabeled bile later (top). A polycystic liver shows multiple small and large liver cysts on CT (middle). Cholescintigraphy in polycystic liver disease shows functioning liver tissue along the margins of right and left lobes (bottom). No filling of the cysts occurs. The cyst fills in only when there is wall rupture\n\n### 11.2.8 Polycystic Liver Disease\n\nPolycystic liver disease is classified into two main categories, non-communicating, and communicating. Non-communicating cysts are usually multiple, common in adults (adult polycystic liver disease), and show an autosomal dominant pattern of inheritance. It is associated with cysts in other organs, especially of the kidneys [18]. Cysts become symptomatic when they rupture or get infected. Tc-99m-HIDA cholescintigraphy can establish reliably whether or not the abdominal pain is due to rupture of the liver cyst [19]. In patients with polycystic liver disease on CT or ultrasound, non-filling with radiolabeled bile indicates a non-rupture (Fig. 11.2.4B, C).\n\nReferences\n\n1.\n\nJangaard KA, Fell DB, Dodds L, Allen AC. Outcome in a population of healthy term and near-term infants with serum bilirubin levels of \u2265325 \u00b5mol\/l (>19 mg\/dl) who were born in Nova Scotia, Canada, between 1994 and 2000. Pediatrics 2008;122:119-124PubMedCrossRef\n\n2.\n\nCroen LA, Yoshida CK, Odouli R, Newman TB. Neonatal hyperbilirubinemia and risk of autism spectrum disorders. Pediatrics 2005;115(2):e135-e138PubMedCrossRef\n\n3.\n\nBalistreri WF. Neonatal cholestasis. J Pediatr 1985;106:171-184PubMedCrossRef80282-1)\n\n4.\n\nBalistreri WF, Grand R, Hoofnagle JH, Suchy FJ, Ryckman FC, Perlmutter DH, Sokol RJ. Biliary atresia: current concepts and research directions. Summary of a symposium. Hepato\u00adlogy 1996;23:1682-1692CrossRef\n\n5.\n\nCocjin J, Rosenthal P, Buslon V, Luk L Jr, Barajas L, Geller SA, Ruebner B, French S. Bile ductule formation in fetal, neonatal, and infant livers compared with extrahepatic biliary atresia. Hepatology 1996;24:568-574PubMedCrossRef\n\n6.\n\nDesmet VJ. Congenital diseases of intrahepatic bile ducts: variations on the theme \"ductal plate malformation\". Hepatology 1992;16:1069-1083PubMedCrossRef\n\n7.\n\nTan CEL, Davenport M, Driver M, Howard ER. Does the morphology of the extrahepatic biliary remnant in biliary atresia influence survival? A review of 205 cases. J Pediatr Surg 1994;29:1459-1464PubMedCrossRef90144-9)\n\n8.\n\nMieli-Vergani G, Howard ER, Portman B, Mowat AP. Late referral for biliary atresia-missed opportunities for effective surgery. Lancet 1989;1:421-423PubMedCrossRef90012-3)\n\n9.\n\nMowat AP, Davidson LL, Dick MC. Earlier identification of biliary atresia and hepatobiliary disease: selective screening in the third week. Arch Dis Child 1995;72:90-92PubMedCrossRef\n\n10.\n\nGreene HL, Helinek GL, Moran R, O'Neill J. A diagnostic approach to prolonged obstructive jaundice by 24-hour collection of duodenal fluid. J Pediatr 1979;95:412-414PubMedCrossRef80519-3)\n\n11.\n\nDe Lorimier AA, Harrison MR. Congenital biliary atresia. In: Way LW, Pelligrini CA (eds). Surgery of the gallbladder and bile ducts. WB Saunders, Philadelphia, 1987, pp 581-607\n\n12.\n\nLeonard JC, Hitch DC, Manion CV. The use of diethyl-IDA Tc-99m clearance curves in the differentiation of biliary atresia from other forms of neonatal jaundice. Radiology 1982;142:773-776PubMed\n\n13.\n\nJaffe SJ, Jachau MR. Perinatal pharmacology. Ann Rev Pharmacology 1974;14:219-238CrossRef\n\n14.\n\nThaler MM. Effect of phenobarbital on hepatic transport and excretion of 131 I-rose bengal in children with cholestasis. Pediatr Res 1972;6:100-110PubMedCrossRef\n\n15.\n\nMajd M, Reba RC, Altman RP. Effect of phenobarbital on 99mTc-IDA scintigraphy in the evaluation of neonatal jaundice. Semin Nucl Med 1981;11:194-204PubMedCrossRef80004-9)\n\n16.\n\nMajd M, Reba RC, Altman RP. Hepatobiliary scintigraphy with 99mTc-PIPIDA in the evaluation of neonatal jaundice. Pediatrics 1981;67:140-145PubMed\n\n17.\n\nKrishnamurthy S, Krishnamurthy GT. Technetium-99m-iminodiacetic acid organic anions: review of biokinetics and clinical application in hepatology. Hepatology 1989;9:139-153PubMedCrossRef\n\n18.\n\nBrown PH, Juni JE, Lieberman DA, Krishnamurthy GT. Hepatocyte versus biliary disease: a distinction by deconvolutional analysis of technetium-99m IDA time-activity curves. J Nucl Med 1988;29:623-630PubMed\n\n19.\n\nHowman-Giles R, Moase A, Gaskin K, Uren R. Hepatobiliary scintigraphy in a pediatric population: determination of hepatic extraction fraction by deconvolution analysis. J Nucl Med 1993;34:214-221PubMed\n\n20.\n\nTolia V, Kottamasu SR, Tabassum D, Simpson P. The use of hepatocyte extraction fraction to evaluate neonatal cholestasis. Clin Nucl Med 1999;24:655-659PubMedCrossRef\n\n21.\n\nJuni JE, Thrall JH, Froelich JW, Wiggins RC, Campbell DA Jr, Tuscan M. The appended curve for deconvolutional analysis-method and validation. Eur J Nucl Med 1988;14:403-407PubMedCrossRef\n\n22.\n\nShim WKT, Kasai M, Spence MA. Racial influence on the incidence of biliary atresia. In: Bill MA, Kasai M (eds) Progress in pediatric surgery, Urban, Scharzenberg, 1979\n\n23.\n\nAlagille D, Odievre M, Gautier M, Dommergues JP. Hepatic ductular hypoplasia associated with characteristic facies, vertebral malformation, retarded physical, mental, and sexual development, and cardiac murmur. J Pediatr 1975;56:63-71\n\n24.\n\nMaini A, Khanduri A, Gambhir S, Yacha SK, Das BK. Role of Tc-99 m-mebrofenin in evaluation of neonatal cholestasis syndrome. Indian J Nucl Med 1997;12:84-87\n\n25.\n\nAmerican Academy of Pediatrics, provisional committee for quality improvement and subcommittee on hyperbilirunemia. Practice parameter: management of hyperbilirunemia in the healthy term newborn. Pediatrics 1994;94(4):558-562\n\n26.\n\nKasai M, Kimura S, Asakura Y. Surgical treatment of biliary atresia. J Pediatr Surg 1968;3:665-668CrossRef90897-X)\n\n27.\n\nKasai M, Suzuki H, Ohashi F, Ohi R, Chiba T, Okamoto A. Technique and results of operative management of biliary atresia. World J Surg 1978;2:571-579PubMedCrossRef\n\n28.\n\nOtte JB, deVille de Goyet J, Reding R, Hausleithner V, Sokal E, Chardot C, Debande B. Sequential treatment of biliary atresia with Kasai portoenterostomy and liver transplantation: a review. Hepatology 1994;20:41S-48S (suppl)PubMed\n\n29.\n\nRyckman F, Fisher R, Pedersen S, Dittrich V, Heubi J, Farrell M, Balistreri W, Ziegler M. Improved survival in biliary atresia patients in the present era of liver transplantation. J Pediatr Surg 1993;28:382-385PubMedCrossRef90236-E)\n\n30.\n\nReyes J, Mazariegos GV. Pediatric transplantation. Surg Clin North Am 1999;79:163-189PubMedCrossRef70013-X)\n\n31.\n\nEgawa H, Uemoto S, Inomata Y, Shapiro J, Asonuma K, Kiuchi T, Okajima H, Itou K, Tanaka K. Biliary complications in pediatric living related liver transplantation. Surgery 1998;124:901-910PubMedCrossRef70015-7)\n\n32.\n\nReding R, Goyet J, Delbeke I, Sokal E, Jamart J, Janssen M, Otte J. Pediatric liver transplantation with cadaveric or living related donors: comparative results in 90 elective recipients of primary grafts. J Pediatr 1999;134:280-286PubMedCrossRef70450-6)\n\n33.\n\nChardot C, Carton M, Spire-Bendelac, Pommelet CL, Golmard J, Auvert B. Prognosis of biliary atresia in the era of liver transplantation: French national study from 1986 to 1996. Hepatology 1999;30:606-611PubMedCrossRef\n\n34.\n\nReyes J, Mazariegos GV. Pediatric transplantation. Surg Clin North Am 1999;79:163-189PubMedCrossRef70013-X)\n\nReferences\n\n1.\n\nWitzleben C. Cystic disease of the liver. In: Zakim D, Boyer TD (eds) Hepatology. A textbook of liver disease, vol 2. WB Saunders, Philadelphia, 1996, pp 1630-1649\n\n2.\n\nCaroli J, Soupault R, Kossakowski J et al. La dilatation polycystic congenitale des voies biliares intra-hepatiques. Sem Hop Paris 1958;34:488-491PubMed\n\n3.\n\nFonkalsrud E. Biliary cystic disease in children and adults. In: Way LW, Pelligrini CA (eds) Surgery of the gallbladder and bile ducts. WB Saunders, Philadelphia, 1987, pp 609-629\n\n4.\n\nKimura K, Tsugawa C, Ogawa K, Matsumoto Y, Yamamoto T, Kubo M, Asada S, Nishiyama S, Ito H. Choledochal cysts. Etiological considerations and surgical management in 22 cases. Arch Surg 1978;113:159PubMedCrossRef\n\n5.\n\nKomi N, Udaka H, Ikeda N, Kashiwagi Y. Congenital dilatation of the biliary tract; new classification and study with particular reference to anomalous arrangement of the pancreaticobiliary ducts. Gastroenterol JPN 1977;12:293-304PubMed\n\n6.\n\nTodani T, Watanabe Y, Narusue M, Tabuchi K, Okajima K. Congenital bile duct cysts: classification, operative procedure, and review of 37 cases including cancer arising from choledochal cyst. Am J Surg 1977;134:263-269PubMedCrossRef90359-2)\n\n7.\n\nSaito S. Surgical treatment and long-term follow-up results of congenital dilatation of the biliary duct. Presented at the International Symposium on Cholestasis in Infancy-Its pathogenesis, diagnosis, and treatment. Sendai, Japan, 1978, June 7-9\n\n8.\n\nKao PF, Huang MJ, Tzen KY, You DL, Liaw YF. The clinical significance of gall-bladder non-visualization in cholescintigraphy of patients with choledochal cysts. Eur J Nucl Med 1996;23:1468-1472PubMedCrossRef\n\n9.\n\nCamponovo E, Buck JL, Drane WE. Scintigraphic features of choledochal cyst. J Nucl Med 1989;30:622-628PubMed\n\n10.\n\nPadhy AK, Gopinath PG, Basu AK, Upadhyay P. Hepatobiliary scintigraphy in congenital cystic dilatation of biliary tract. Clin Nucl Med 1985;10:703-707PubMedCrossRef\n\n11.\n\nWilliams LE, Fisher JH, Courtney RA, Darling DB. Preoperative diagnosis of choledochal cyst by hepatoscintigraphy. N Engl J Med 1970;283:85-86PubMedCrossRef\n\n12.\n\nGates GF, Miller JH. Combined radionuclide and ultrasonic assessment of upper abdominal masses in children. AJR Am J Roentgenol 1977;128:773-780PubMed\n\n13.\n\nTsardakas E, Rdonett AH. Congenital cystic dilatation of the common bile duct. Arch Surg 1956;72:311-315CrossRef\n\n14.\n\nWeyant MJ, Maluccio MA, Bertagnolli MM, Daly JM. Choledochal cysts in adults. A report of two cases and review of the literature. Am J Gastroenterol 1998;93:2580-2583PubMed\n\n15.\n\nHenson SW Jr, Gray HK, Dockery MB. Benign tumors of the liver III. Solitary cysts. Surg Gynecol Obstet 1956;103:607-609PubMed\n\n16.\n\nCaplan LH, Simon M. Nonparasitic cysts of the liver. Am J Roentgenol Radium Ther Nucl Med 1966;96:421-428PubMed\n\n17.\n\nFlagg RS, Robinson DW. Solitary nonparasitic hepatic cysts. Report of oldest known case and review of the literature. Arch Surg 1967;95:964-973PubMedCrossRef\n\n18.\n\nGabow PA, Ikle DW, Holmes JH. Polycystic kidney disease: prospective analysis of nonazotemic patients and family members. Ann Intern Med 1984;101:238-247PubMed\n\n19.\n\nSalam M, Keeffe EB. Liver cysts associated with polycystic kidney disease: role of Tc-99m hepatobiliary imaging. Clin Nucl Med 1989;14:803-807PubMedCrossRef\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_12(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 12. Malignant Liver Lesions\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nThe liver is a common site for both primary and metastatic malignant lesions. Although the metastatic lesions are the most common, primary malignancies like hepatocellular carcinoma (HCC) and cholangiocarcinoma (CC) have become increasingly more common in recent years in the United States [1]. HCC arises from the hepatic parenchymal cells, the hepatocytes, and CC from the cells lining the major bile ducts and gallbladder, the cholangiocytes. In Asian countries like China, Taiwan, and Japan, HCC is one of the three most common causes of death due to malignancy. HCC has a serum marker in the form of \u03b1-fetoprotein, and no such marker exists for CC. Gallium-67 citrate, which has been an imaging agent for HCC over the years, still remains popular in places where F-18 fluorodeoxyglucose (F-18 FDG) is not readily available. A filling defect on a radiocolloid liver scan (Fig. 12.1.1) associated with intense Ga-67 uptake (Fig. 12.1.2) and increased serum \u03b1-fetoprotein in a patient is more likely to be HCC than any other type of malignancy. F-18 FDG shows avidity for CC, HCC, metastatic lesions, and abscesses. Being a common imaging agent for many different types of liver lesions, F-18 FDG imaging provides no specificity for any one particular type of malignancy.\n\nThe liver is a common site for both primary and metastatic malignant lesions. Although the metastatic lesions are the most common, primary malignancies like hepatocellular carcinoma (HCC) and cholangiocarcinoma (CC) have become increasingly more common in recent years in the United States [1]. HCC arises from the hepatic parenchymal cells, the hepatocytes, and CC from the cells lining the major bile ducts and gallbladder, the cholangiocytes. In Asian countries like China, Taiwan, and Japan, HCC is one of the three most common causes of death due to malignancy. HCC has a serum marker in the form of \u03b1-fetoprotein, and no such marker exists for CC. Gallium-67 citrate, which has been an imaging agent for HCC over the years, still remains popular in places where F-18 fluorodeoxyglucose (F-18 FDG) is not readily available. A filling defect on a radiocolloid liver scan (Fig. 12.1.1) associated with intense Ga-67 uptake (Fig. 12.1.2) and increased serum \u03b1-fetoprotein in a patient is more likely to be HCC than any other type of malignancy. F-18 FDG shows avidity for CC, HCC, metastatic lesions, and abscesses. Being a common imaging agent for many different types of liver lesions, F-18 FDG imaging provides no specificity for any one particular type of malignancy.\n\nFig. 12.1.1\n\nTc-99m radiocolloid scan (top) shows HCC as a filling defect in segment 4A and 4B of the left lobe. A CT (bottom left) shows it as a low density lesion and MRI with contrast (bottom right) shows the lesion as hypervascular\n\nFig. 12.1.2\n\nGallium-67 citrate scan of patient shown in Fig. 12.1.1 shows high intensity uptake in segment 4A and 4B of the left lobe. As the lesion is anterior in location, it is seen better in the anterior (left) than in the posterior view (right). Bone marrow and transverse colon activity below the liver is normal. Patient had increased serum alpha-fetoprotein, and surgical resection confirmed hepatocellular carcinoma\n\nDetection of primary and metastatic malignancy depends upon uptake and retention of F-18 FDG by the normal liver and tumor (Fig. 12.1.3). Uptake is dependent upon the enzyme glucokinase, which facilitates F-18 FDG uptake and immediate conversion into F-18 FDG-6 phosphate inside the cell. High levels of FDG-6 phosphatase enzyme in the hepatocyte promote conversion of F-18 FDG-6 phosphate back into F-18 FDG, which may exit the hepatocyte rapidly, especially in delayed images taken beyond an hour after injection. The balance between uptake and exit determines the sensitivity for tumor imaging. The serum level of \u03b1-fetoprotein and tumor stage greatly influence the sensitivity of F-18 FDG PET\/CT imaging (Fig. 12.1.4). Patients with a serum \u03b1-fetoprotein level of less than 20 ng ml-1 show a sensitivity of 44%, whereas those with levels greater than 400 ng gl-1 show a sensitivity of 86%. Stage II and III tumors show F-18 FDG PET\/CT sensitivity of 31-62%, whereas stage IVa and IVb show much higher sensitivity at 68-85%. Overall sensitivity of F-18 FDG PET\/CT for HCC is 61% and for metastatic lesions 86%. Higher sensitivity for metastatic lesions is related to the slow rate of conversion of F-18 FDG-6-phosphate back into F-18 FDG. Although colon cancer frequently metastasizes to the liver, other types of cancer, including breast and lung, also involve the liver (Fig. 12.1.5). Primary liver tumors less than 2 cm in size are rarely detected, and 78% of tumors \u22655 cm are readily identified with PET\/CT [2]. Risk factors for HCC include primary sclerosing cholangitis, hepatolithiasis, chronic infections, including typhoid and parasites, drug exposure, and genetic influence.\n\nFig. 12.1.3\n\nNormal PET\/CT. Brain and heart show the highest F-18 FDG uptake (middle panel). Liver, spleen, kidneys, and bone marrow show low uptake, and lungs show no uptake. Top (CT) and middle (PET) panel images are fused in the bottom panel\n\nFig. 12.1.4\n\nColon CA with liver metastases. Multiple metastatic lesions are found in both liver lobes\n\nFig. 12.1.5\n\nBreast CA with metastases to both lobes of the liver. Right lobe involves segments 5, 7, and 8. Segment 7 lesion shows central necrosis with no uptake in the center. Left lobe has lesions in segments 2 and 3\n\nCC is a malignant epithelial tumor of the biliary tree and is less common than HCC [3]. About 60% of CC tumors arise from the extrahepatic part of the biliary tree and spread upwards to invade the liver parenchyma (Klatskin tumor). Hilar lesions are classified into five different types, depending upon the extent of the tumor (see Chap. 8.2). Hilar tumors progress slowly, and the diagnosis is often delayed for weeks or months. Although the sensitivity for detection of the primary cholangiocarcinoma with F-18 FDG is slightly higher than for HCC, metastatic lesions show no difference between the two types (Table 12.1.1). CT remains the primary diagnostic procedure of choice, and PET\/CT adds valuable information with regard to the extent of the primary tumor and its resectability, especially in separating intrahepatic from common bile duct lesions [4, 5].\n\nTable 12.1.1\n\nSensitivity and accuracy (%) of F-18 FDG PET\/CT imaging in the detection of hepatocellular carcinoma and cholangicarcinoma | Hepatocellular carcinoma [1] | Cholangicarcinoma [3]\n\n---|---|---  \n|\n\nPrimary | Metastatic | Primary | Metastatic\n\nSensitivity | 61 | 86 | 84 | 83\n\nAccuracy\n\n|  |  |\n\n61 | 79\n\nEarly detection of metastatic liver lesions plays a crucial role in the proper management of the patient. A single lesion in either lobe is locally resected, whereas multiple lesions in one or both lobes call for more aggressive surgery and require clear delineation of involvement of each of eight segments of the liver. Couinaud's classification divides the liver into eight segments [6], and Bismuth divides the liver into nine segments by separating Couinaud's segment 4 into 4A and 4B [7]. Resection of four or more lesions followed by chemotherapy frequently results in 3-year survival [8].\n\n## 12.1 Management\n\nThe therapy for early uncomplicated primary tumor is surgical resection. Five-year survival for margin-negative resections ranges from 30 to 45 years. Margin-positive lesions show poor 5-year survival, ranging from 0 to 13 years [3].Only segments 6 and 7 (posterior half of the right lobe) are left behind in cases where other segments show metastatic involvement. Other forms of therapy include photodynamic therapy, intraductal high-intensity ultrasonography, radiofrequency ablation, and chemo-embolization. Liver transplantation, which was considered in the past as a contraindication for HCC and CC, is now accepted as a life-saving measure for both.\n\nReferences\n\n1.\n\nWood CB, Gillis CR, Blumgart LH. Retrospective study of the natural history of patients with liver metastases from locorectal cancer. Clin Oncol 1976;2:285-288PubMed\n\n2.\n\nPark JW, Kim JH, Kim SK, Kang KW, Park KW, Choi JI, Lee WJ, Kim CM, Nam BH. A prospective evaluation of 18F-FDG and 11C-Acetate PET\/CT for detection of primary and metastatic hepatocellular carcinoma. J Nucl Med 2008;49:1912-1921PubMedCrossRef\n\n3.\n\nUstundag Y, Bayraktar. Cholangicarcinoma: a compact review of the literature. World J Gastroenterol 2008;14(42):6458-6466PubMedCrossRef\n\n4.\n\nKim JY, Kim MH, Lee TY, Hwang CY, Kim JS, Yun SC, Lee SS, Seo DW, Lee SK. Clinical role of 18F-FDG PET\/CT in suspected and potentially operable cholangiocarcinoma: a prospective study compared with conventional imaging. Am J Gastroenterol 2008;103:1145-1151PubMedCrossRef\n\n5.\n\nMoon CM, Bang S, Chung JB, Park SW, Song SY, Yun M, Doo J. Usefulness of F18-fluodeoxyglucose positron emission tomography in the differential diagnosis and staging of cholangiocarcinomas. J Gastroenterol Hepatol 2007;23(5):759-765PubMedCrossRef\n\n6.\n\nCouinaud C. Le foie etudes anatomiques et chirurgicales. Masson, Paris, 1957\n\n7.\n\nBismuth H. Surgical anatomy and anatomical surgery of the liver. World J Surg 1982;6:3-9PubMedCrossRef\n\n8.\n\nTanaka K, Shimada H, Ueda M, Matsuo K, Endo I, Togo S. Role of hepatectomy in treating bilobar colorectal cancer metastases. Surgery 2008;143:259-270PubMedCrossRef\nGerbail T. Krishnamurthy and S. KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7_13(C) Springer-Verlag Berlin Heidelberg 2009\n\n# 13. Liver Transplantation\n\nGerbail T. Krishnamurthy1 and Shakuntala Krishnamurthy1\n\n(1)\n\nTuality Community Hospital, 97123 Hillsboro, OR, USA\n\nAbstract\n\nLiver transplantation is an optimal therapy for various types of end-stage liver disease [1]. The first liver transplantation was performed in a dog by Welch in 1955, and the first human liver transplantation was performed by Starzl et al. in 1963 [2, 3]. Although the early clinical results were disappointing, modern immunosuppressive agents have improved survival rates in both children and adults. The survival rates in children at 1, 3, 5, and 10 years are 82%, 80%, 78%, and 76%, respectively [4]. In the Model for End-Stage Liver Disease (MELD) survey, the 5-year survival rate after liver transplantation has now increased to 88% [5].\n\nLiver transplantation is an optimal therapy for various types of end-stage liver disease [1]. The first liver transplantation was performed in a dog by Welch in 1955, and the first human liver transplantation was performed by Starzl et al. in 1963 [2, 3]. Although the early clinical results were disappointing, modern immunosuppressive agents have improved survival rates in both children and adults. The survival rates in children at 1, 3, 5, and 10 years are 82%, 80%, 78%, and 76%, respectively [4]. In the Model for End-Stage Liver Disease (MELD) survey, the 5-year survival rate after liver transplantation has now increased to 88% [5]. The indications for liver transplantation for various end-stage liver diseases are shown in Table 13.1.1.\n\nTable 13.1.1\n\nIndications for liver transplantation\n\nParenchymal diseases\n\n---\n\nPost-necrotic cirrhosis\n\nPrimary biliary cirrhosis\n\nPrimary sclerosing cholangitis\n\nAlcoholic liver failure\n\nViral hepatitis\n\nInborn errors of metabolism\n\nFulminant hepatic failure\n\nAutoimmune hepatitis\n\nCystic fibrosis\n\nNeonatal hepatitis\n\nCholestatic diseases\n\nBiliary atresia\n\nCystic fibrosis\n\nBiliary cirrhosis\n\nSclerosing cholangitis\n\nFamilial cholestasis\n\nGraft vs. host disease\n\nChronic hepatic rejection\n\nTumors\n\nHepatoma\n\nHepatoblastoma\n\nApudomas\n\nMiscellaneous\n\nBudd-Chiari syndrome\n\nTrauma\n\n## 13.1 Types of Liver Transplantation\n\nThere are essentially three types of liver transplantation: (1) cadaver liver, (2) living-donor liver, and (3) auxiliary liver. In cadaver liver transplantation, the recipient's entire liver is removed and replaced (orthotopic) by a cadaver liver (Fig. 13.1.1). In the case of a living-donor liver transplantation, either the lateral segment (segments 2 and 3) or the entire left lobe (segments 4A, 4B, 2, and 3), anterior section (segments 5 and 8), or posterior section (segments 6 and 7) of the right lobe or the entire right lobe (segments 5, 6, 7, and 8) from the donor liver replaces the entire native liver of the recipient. In the case of an auxiliary liver transplantation for patients with fulminant hepatic failure, the donor liver acts as a transient bridge until the native liver recovers its function. The donor liver is placed either adjacent to the native liver or replaces the left lobe of the recipient liver [6-8]. The transplantation procedure usually involves four stages: (1) hepatectomy of the diseased native liver, (2) the anhepatic stage, which is the time interval between liver removal and interruption of blood flow through the vena cava, portal vein, and hepatic artery, (3) the reperfusion stage when the donor liver is being revascularized in the recipient, and (4) the biliary reconstruction stage in which a choledochocholedochostomy (adults) or a choledochojejunostomy (in children less than 35 lbs) is performed.\n\nFig.13.1.1\n\nLiver transplantation. Orthotopic liver transplant with end-to-end anastomosis is the most common type. In auxiliary transplantation, the donor liver is placed in either an orthotopic or heterotopic position\n\nNuclear medicine procedures are used for the evaluation of liver function both in the pre- and post-transplant period. In the case of a living-donor liver transplantation, both the donor and the recipient may undergo pre-transplant evaluation, which generally includes assessment of the functional reserve of the donor liver and determination of the ideal time for transplantation in the recipient [9]. It is essential to establish end-stage liver disease in the recipient prior to considering transplantation. Since no single test is adequate for either determining end-stage liver disease or indicating the ideal time for transplantation, serial imaging and non-imaging diagnostic procedures are performed over many months or years. Persistent poor function over many months or years despite appropriate therapy is an indication of end-stage liver disease. Just prior to liver transplantation, a multiple-gated acquisition (MUGA) study is obtained for evaluating the left ventricular ejection fraction and wall motion and a myocardial perfusion study for testing the adequacy of the coronary circulation in the recipient. After the transplantation, nuclear medicine studies are used to detect immediate and late postoperative complications (Table 13.1.2) and also for the assessment of functional recovery of the donor liver (Fig. 13.1.1). In the immediate postoperative period, cholescintigraphy is used to exclude bile leak or to assess transplant function. Blood pool studies are obtained to identify vascular complications, such as bleeding and thrombosis [10, 11]. Post-transplant chronic ischemia of the liver causes diffuse bile duct stenoses, manifesting as a vanishing bile duct syndrome in some patients [12].\n\nFig.13.1.2\n\nMagnetic resonance cholangiopancreatogram (left) shows the gallbladder, common bile duct, right hepatic duct (RHD), and left hepatic duct (LHD) and their segmental branches. RHD receives bile from anterior (A) and posterior (P) and LHD from medial (M) and lateral (L) segmental branches\n\nTable 13.1.2\n\nComplications of liver transplantation [10]\n\nComplication | Frequency (%)\n\n---|---\n\nInfections | 30.0-60.0\n\nBiliary obstruction | 15.0-25.0\n\nRejection | 10.0-15.0\n\nBile leak | 7.5-10.0\n\nBleeding | 5.5-6.5\n\nRenal failure (requiring dialysis) | 5.0\n\nArterial thrombosis | 3.0-5.0\n\nLymphoproliferative disorder | 2.0-2.5\n\nPortal vein thrombosis | 1.0-2.0\n\nIn the case of living-donor liver transplantation, the volume of the donor liver and its segmental morphology are measured with a single or multi-detector spiral CT. For a multi-detector spiral CT study, 750 ml of water is ingested as a negative contrast, and 120-150 ml of a non-ionic contrast is injected intravenously at 4-5 ml s-1 using scanning collimation of 1 mm, and 1-mm-thick images are reconstructed for interpretation [13]. The intraoperative volume in milliliters and weight in grams of the right lobe are estimated by using the preoperative volume [14]:\n\nIntraoperative volume = 0.656 \u00d7 preoperative volume + 87.629 ml\n\nIntraoperative weight = 0.678 g\/ml \u00d7 preoperative volume + 143.704\n\nSince one or two segments from an adult are adequate for a child, living-donor liver transplantation for post-Kasai biliary atresia has gained widespread acceptance with 5-year survival of nearly 90% [15]. Currently, hepatitis C infection is the most common cause of chronic liver disease in adults and accounts for 42% of cases, followed by hepatitis C and alcohol in 22%, alcohol alone in 8%, non-alcoholic fatty liver disease in 9%, hepatitis B in 3%, and other causes in 16% [16]. Hepatitis C in adults and congenital biliary atresia in children are the two most common indications for liver transplantation.\n\n## 13.2 Normal Functioning Liver Transplant\n\nTransplant liver maintains its normal shape and size in the right upper quadrant and functions much like a native normal liver. The gallbladder is usually absent as cholecystectomy is routinely performed as an integral part of transplantation to the prevent formation of as well as future complications from gallstones. The hepatic extraction fraction and excretion half time values of the transplant liver are maintained within the normal range established for the normal native liver. Common hepatic and common bile ducts show normal features. Duodeno-gastric bile reflux is rare (Figs. 13.1.3 and 13.1.4).\n\nFig. 13.1.3\n\nLiver segments derived from a planar Tc-99m-HIDA study. A line passing through the common bile duct (CBD) and common hepatic duct (CHD) when extended upwards to meet the superior margin divides the liver into right and left lobes. A line passing through the right hepatic duct and left hepatic duct divides each lobe into superior and inferior segments. In a planar image, segment 5 is superimposed on segment 6 and segment 7 on segment 8 of the right lobe. The segments are clearly separated in the left lobe. The caudate lobe (segment 1) is added to the right or left lobe depending upon user preference. Normal values for segments are shows within parentheses. Planar image depicts the biliary anatomy similar to MRCP shown in Fig. 13.1.2\n\nFig. 13.1.4\n\nNormal liver transplant. Early images beginning at 2 min show normal morphology and bile formation (top). Late images beginning at 38 min show free bile flow into intestine. Gallbladder is absent due to cholecystectomy (bottom). Hepatic extraction fraction and excretion half time are normal\n\n### 13.2.1 Complications\n\nLiver transplant complications occur both in the immediate and delayed postoperative period (Table. 13.1.2). Immediate complications include bile leak and arterial or venous thrombosis, and delayed complications (rejection) are immunological in nature.\n\n### 13.2.2 Bile Leak\n\nLeak is inferred when bile enters an unanticipated region or an anticipated region at an unanticipated time [17]. Bile leak at the anastomotic site is relatively common and accounts for 10% of all complications. The leak may occur into the liver parenchymam or into the subhepatic space forming a biloma, or the leak may enter the peritoneal space, causing bile peritonitis. Biloma produces a filling defect within the liver in early images and fills in with Tc-99m-HIDA-radiolabeled bile in late images. In contrast, a benign or malignant space-occupying lesion remains a \"cold\" defect throughout. Bile enters an abscess cavity when the abscess connects with a bile duct. Healing of the abscess cavity is demonstrated with serial imaging (Fig. 13.1.5). Bile leak at the anastomotic site (choledocho-choledochal) is usually small in size and closes spontaneously. Bile leak at the choledocho-duodenal junction, on the other hand, tends to be large in volume and carries a much higher morbidity and mortality due to simultaneous leakage of intestinal contents into the peritoneal space. In the supine position, leaked bile gravitates to the right or left paracolic gutter and later enters the pelvic cavity. Opioids prescribed for pain control often increase the volume of bile leak because of their constrictive effect on the sphincter of Oddi.\n\nFig. 13.1.5\n\nAbscess in a transplant liver. An abscess causes a large filling defect within liver parenchyma (a) and fills with bile at 4.5 h (b). A repeat study 3 months later shows shrinkage of the abscess, but fistula persists\n\n### 13.2.3 Bile Duct Stricture\n\nStricture is a late complication and accounts for 15 to 25% of the total complications of liver transplantation. Stricture is attributed to bile duct ischemia caused by arterial stenosis or thrombosis. Cholescintigraphy shows bile pooling in ducts proximal to the stricture. Segmental and area ducts are seen with unusual prominence [18]. Denervation during transplant surgery alters sympathetic-parasympathetic-hormonal control over the ducts and sphincter of Oddi. Nervous and hormonal imbalance may cause biliary dyskinesia. Distinction between anatomic stricture and biliary dyskinesia is made with the use of amyl nitrite or a calcium channel blocker (nefidapine), very similar to studies obtained in non-transplant patients [19, 20].\n\n### 13.2.4 Primary Non-Function Versus Rejection\n\nPrimary non-function is non-recovery of donor liver function in the recipient despite good surgical technique establishing adequate portal vein and hepatic artery blood flow. Clinically, it presents as liver failure in association with encephalopathy, persistent acidosis, and severe coagulopathy, and requires retransplantation. Acute rejection occurs within 3 months of transplantation, and chronic rejection follows later. Unlike the kidneys, the liver does not manifest a hyperacute rejection. Rejection is a microvascular phenomenon where fibrin, immune and inflammatory cells deposit within and around the capillaries, manifesting an obliterative angiopathy and chronic ischemia. Chronic ischemia leads to obliteration of the bile ducts (vanishing bile duct syndrome) and intrahepatic cholestasis. Both rejection and intrahepatic cholestasis resulting from other causes manifest similar cholescintigraphic findings, and hence one entity cannot be separated from the other [21]. Rejection is managed with immunosuppressive agents like cyclosporine and tacrolimus [22]. Serial studies enable evaluation of liver function and aid in the adjustment of the immunosuppressant dosage [23].\n\n### 13.2.5 Cholescintigraphy\n\nThe data are collected with a large field of view gamma camera fitted with a low-energy, all-purpose, parallel-hole collimator using 3-8 mCi of Tc-99m-HIDA. Technetium-99m mebrofenin is the best agent both in normal as well as jaundiced patients. Liver perfusion images are obtained at 2 s per frame during the first 1 min, followed by functional images at 1 min per frame for 59 min, for a total of 1 h, and recorded on a 128 \u00d7 128 word mode matrix. A short imaging interval is essential for location of the exact site of the bile leak. The first place of unexpected bile radioactivity usually indicates the site of origin of the leak. Once there is collection of a large volume of bile, it becomes difficult to identify the exact site of the bile leak.\n\n### 13.2.6 Auxiliary Liver Transplantation\n\nIn an auxiliary transplant, the donor liver is placed in a heterotopic position adjacent to the native liver or in the space created by partial hepatectomy of the native liver (Fig. 13.1.6). Auxiliary liver transplantation is restricted mostly to patients with fulminant hepatic failure where the native liver has the potential for full functional recovery if the patient survives the acute episode. Fulminant hepatic failure consists of liver failure and encephalopathy occurring within a 2-8 week period in a previously healthy person. Fulminant hepatic failure is relatively rare, and accounts for 6% of all adult liver transplants and 11% of all pediatric liver transplants each year in the United States [24]. Mortality is 100% in untreated patients. Full functional recovery is due to the generous regeneration capacity of the native liver in those patients who can withstand an acute insult [25].\n\nFig. 13.1.6a,b\n\nPattern of recovery of native liver function following auxiliary liver transplantation. Cholescintigram (a) and a corresponding schematic diagram (b) are shown in a 2-year-old child with auxiliary liver transplant for fulminant hepatic failure. At 10 days, the native liver shows poor uptake, and the auxiliary liver has good uptake of 99mTc-HIDA. Function appears equal by 7 months. Function of the auxiliary liver begins to decrease by 8.5 months and disappears completely at 11.5 months. The native liver recovers its function completely with 100% extraction fraction. (Courtesy of Dr. Muriel Buxton-Thomas, Kings College Hospital, London, UK)\n\nFulminant hepatic failure heals slowly over several months to attain full functional recovery (Table 13.1.3). When serum liver function tests show either an improvement or deterioration in a patient with auxiliary liver transplantation, it is not clear from the serum levels whether the changes reflect the function of the native or the donor liver. The immunosuppressive dose needs to be increased if there is rejection of the donor liver. An improvement of the native liver function, on the other hand, calls for either a reduction in dosage or total discontinuation of immunosuppressive agents. The donor liver undergoes spontaneous regression and atrophy when the native liver recovers its full function, and steroids are withdrawn safely at this time. The donor liver is surgically removed if it does not undergo spontaneous atrophy and interferes with the function of the native liver [26].\n\nTable 13.1.3\n\nDifferentiation of the native from donor liver function following auxiliary liver transplantation for fulminant hepatic failure\n\nStudy no. | Time post-transplant | Native liver function | Auxiliary liver function\n\n---|---|---|---\n\nHEF(%) | Ex. T\u00bd(min) | function(%) | HEF(%) | Ex. T\u00bd(min) | function(%)\n\n1 | 10 days | 37 | 27 | 15 | 73 | 21 | 85\n\n2 | 7 months | 51 | 18 | 44 | 54 | 17 | 56\n\n3 | 8.5 months | 78 | 16 | 64 | 50 | 53 | 36\n\n4 | 11.5 months | 100 | 8 | >99 | - | - | <1\n\n(Courtesy Dr. Muriel Buxton-Thomas, Kings College Hospital, London)\n\n### 13.2.7 Differentiation of Native vs. Donor Liver Function\n\nCholescintigraphy is able to measure the total as well as the function of the donor and native liver separately. The differential liver function information helps the clinician in adjusting the dose of immunosuppressive agents.\n\nCholescintigraphy is obtained in adults with a large field of view and in children with a standard or even small field of view gamma camera, fitted with a low-energy, general-purpose, parallel-hole collimator. With the patient in a supine position, the camera is positioned anteriorly to cover the entire native and the donor liver. A dynamic 60-frame image at minute intervals are obtained with 4 to 6 mCi of Tc-99m HIDA, and recorded on a 128 \u00d7 128 computer matrix. The preferred imaging agent in jaundiced patients is Tc-99m mebrofenin, which has superior biokinetic behavior in patients with hyperbilirubinemia [27]. One can obtain a liver scan with Tc-99m-S colloid to outline the liver borders, but it is not essential in most cases, as early Tc-99m-HIDA images, obtained within the first 5 to 6 min, provide similar morphologic information (Fig. 13.1.2).\n\n### 13.2.8 Quantification\n\nMany different methods have been used for quantification of transplant function. Deconvolutional analysis (Chap. 5) is applied to measure the extraction fraction of the native and auxiliary liver separately, by choosing heart for the input and liver for the output function. Excretion half time is measured with a monoexponential fit. Some express the function as uptake of the injected dose by the native versus donor liver. Four regions of interest are drawn; two ROIs over the liver (one over the entire native liver and the other over the entire donor liver) and two ROIs for background just lateral to each liver. Background counts (from Tc-99m-S colloid when a preliminary scan is obtained) are subtracted from the liver counts, and the net liver counts and curve are displayed. From the net liver curve, three functional parameters are generated [28].\n\n(1) T-max = time to maximum liver counts.\n\n(2) T 1\/2 = clearance half time using a mono-exponential fit to the data points from peak counts.\n\n(3) Relative uptake (RU) by the liver. RU is obtained by noting the area under the time-activity curves of the native and graft livers between 2 and 10 min.\n\n![$$\\\\begin{array}{l}\n{\\\\rm RU\\\\ by\\\\ native\\\\ liver = }\\\\frac{{{\\\\rm Native\\\\ liver\\\\ area\\\\ between\\\\ 2 - 10\\\\ min} \\\\times {\\\\rm 100}}}{{{\\\\rm Native\\\\ liver\\\\ area\\\\ between\\\\ 2 - 10\\\\ min  +  donor\\\\ liver\\\\ are\\\\ between\\\\ 2 - 10\\\\ min}{\\\\rm .}}}\\\\\\\\\n{\\\\rm RU\\\\ by\\\\ native\\\\ liver = }\\\\frac{{{\\\\rm Graft\\\\ liver\\\\ area\\\\ between\\\\ 2 - 10\\\\ min -- 100}}}{{{\\\\rm Graft\\\\ liver\\\\ area\\\\ between\\\\ 2}{\\\\rm .-- 10\\\\ min  +  native\\\\ liver\\\\ area\\\\ between\\\\ 2 -- 10\\\\ min}{\\\\rm .}}}\\\\\\\\\n\\\\end{array}$$](A978-3-642-00648-7_13_Chapter_TeX2GIF_Equ1_13.gif)\n\nIn one study, authors followed patient recovery for up to 30 months and calculated the percent uptake by the native liver by using the area under the curve between the 2nd and 10th min, and validated the scintigraphic method in the management of the patient with auxiliary liver transplantation [28].\n\nDeconvolutional analysis is applied when the software is available. In children, where the liver physiology is much faster than in adults, the data are collected at a much shorter interval, 30 s per frame for 30 min. Hepatic extraction fraction calculation uses only the first 30 frames (15 min), and the excretion T1\/2 is calculated using all 60 frames. Native and donor liver HEF and excretion half times are measured separately for both. The contribution of each liver towards total function is calculated and expressed as percent (Table 13.1.3).\n\n### 13.2.9 Recovery Pattern in Auxiliary Liver Transplantation\n\nImmediately after the transplantation, the function of the native liver remains depressed with a median uptake value of 27% (range 4%-36%) during the first 4 weeks. The median uptake value increases gradually to twice the baseline value 6-12 months after transplantation. When serum liver function tests show stable normal values, the uptake by the native liver raises to above 90%. The native liver T-max value may normalize within a month, but the excretion T1\/2 value usually remains elevated for a much longer period of time. The immunosuppressive drug dose may be decreased when the relative uptake value raises above 30%, and the drug may be withdrawn completely when the uptake value reaches above 90% [28]. The pattern of recovery of native liver function following an auxiliary liver transplantation in a 2-year-old child is shown in Table 13.1.3. The hepatic extraction fraction and excretion half time of the native liver and auxiliary liver are measured separately for each liver (Fig. 13.1.6). As the function of the native liver improves gradually and recovers fully, the auxiliary liver function deteriorates with regression of functional liver volume.\n\n### 13.2.10 Living-donor Liver Transplantation\n\nAlthough the liver is one solid organ, its distinct segmental and lobar anatomy and physiology allow resection into smaller portions for transplantation [29]. Resection of the donor liver is performed along the physiological planes (Fig. 1.1.3, Chap. 1). The lateral segment of the left lobe (areas 2 and 3) or the entire left lobe (areas 2, 3, 4A, and 4B) is resected from the donor for transplantation in the recipient (Fig. 13.1.1). The first living-donor liver transplantation was performed by Raia et al. in Brazil in 1989 [30]. The left lobe of the liver from a mother was transplanted successfully into her son in Australia in 1989 [31]. There are numerous centers around the world performing living-donor liver transplantations [32-34]. Major advantages of living-donor liver transplantation are: (1) reduction in waiting time, (2) high quality of the donor liver, and (3) immunological similarity in haplo-identical donors.\n\n### 13.2.11 Complications\n\nPostoperative complications are a major concern both in the donor and the recipient. Among the first 100 living-donor liver donor transplantations performed at the University of Chicago, 91 donors had resection of the lateral segment of the left lobe (areas 2 and 3), and the remaining 9 had resection of the entire left lobe (areas 2, 3, 4A, and 4B). Thirteen of the 100 donors had complications, including bile leaks, infection, injury to the bile ducts and spleen, etc., as listed in Table 13.1.4 [35]. Bile leak, bleeding, thrombosis, or infection may occur in either the recipient or the donor, or both. Rejection is an additional complication in the recipient. The living-donor liver transplantation procedure has been found to be safe both for the donor and the recipient. Donor survival is 100%, and the recipient survival ranges from 80 to 94% in centers that perform more than 15 living-donor transplantations in a year [22]. Common complications within 30 days after transplantation include bleeding, extrinsic obstruction, hemobilia, and bile leak around the T-tube or at the anastomotic site. Bile leaks around the T-tube usually close spontaneously, and those that continue to leak may require sphincterotomy or insertion of a biliary stent. Complications occurring after 30 days include stricture at or proximal to the site of anastomosis [36]. Viral, bacterial, and fungal infections are always a concern in an immunocompromised liver transplant patient [37].\n\nTable 13.1.4\n\nComplications in living-liver donors [31]\n\nBile leak from cut end\n\n---\n\nInjury to bile ducts\n\nInjury to spleen\n\nStricture of the common bile duct\n\nIleus\n\nWound infection\n\nUrinary tract infection\n\nHepatic artery thrombosis\n\nAbscess\n\n### 13.2.12 Future Directions\n\nThe major advantage of hepatobiliary imaging is the opportunity to detect functional abnormality early before irreversible morphological changes take place. Simultaneous quantification of function provides a measure of the severity of disease and enables the clinician to determine the appropriate time to intervene with therapy, and also to test later whether or not the chosen therapy has achieved the intended goals. Following a thorough clinical evaluation and analysis of liver function tests, Tc-99m-HIDA cholescintigraphy is an appropriate initial imaging procedure for detection of most pathophysiologic changes associated with bile formation and flow [38]. Cholescintigraphic results can guide the clinician in developing a management strategy and also aid in determining the need for other imaging (CT or MR) or non-imaging procedures that will lead to the ultimate diagnosis. Liver biopsy is the most appropriate next step for patients with intrahepatic cholestasis. For extrahepatic cholestasis, the clinician may choose percutaneous transhepatic cholagiography for obstruction at or proximal to the union of the right hepatic and left hepatic ducts. ERCP is preferred for obstructions in the common hepatic or common bile duct. Magnetic resonance cholangiopancreatography (MRCP) is being used more often as a non-invasive alternative to invasive ERCP [39]. Receptor-based imaging may enable the selection of a receptor-specific therapy. Quantitative asialoglycoprotein receptor imaging with Tc-99m DTPA galactosyl human serum albumin enables to predict residual liver function after resection of hepatocellular cancer and cholangiocarcinoma and to forecast prognosis of liver cirrhosis [40-42]. A somatostatin receptor-positive metastatic gastrinoma may respond to therapy with octreotide.\n\nReferences\n\n1.\n\nNational Institute of Health Consensus Development Conference statement: Liver transplantation. June 20-23, 1983, Hepatology 1983;3:(Suppl) 107S-110S\n\n2.\n\nWelch CS. A note on transplantation of whole liver in dogs. Transplant Bull 1955;2:54\n\n3.\n\nStarzl TE, Marchioro TL, Von Kaulla KN, Hermann G, Brittain RS, Waddell UR. Homo\u00adtransplantation of the liver in humans. Surg Gynecol Obstet 1963;117:659-676PubMed\n\n4.\n\nGoss JA, Shackleton CR, McDiarmid SV, Maggard M, Swenson K, Seu P, Vargas J, Martin M, Ament M, Brill J, Harrison R, Busuttil RW. Long-term results of pediatric liver transplantation. An analysis of 569 transplants. Ann Surg 1998;228:411-420PubMedCrossRef\n\n5.\n\nPfitzmann R, Schwenger J, Rayes N, et al. Long-term survival and predictors of relapse after orthoptic liver transplantation for alcoholic liver disease. Liver Transpl 2007;13:197-205PubMedCrossRef\n\n6.\n\nStarzl TE, Demetris AJ, Van Thiel D. Liver transplantation (1). N Engl J Med 1989;321:1014-1022PubMedCrossRef\n\n7.\n\nKawasaki S, Makuuchi M, Matsunami H, Hashihura Y, Ikegami T, Nakazawa Y, Chisuwa H, Terada M, Miyagawa S. Living related liver transplantation in adults. Ann Surg 1998;227:269-274PubMedCrossRef\n\n8.\n\nSudan DL, Shaw BW Jr, Fox IJ, Langnas AN. Long-term follow-up of auxiliary orthotopic liver transplantation for the treatment of fulminant hepatic failure. Surgery 1997;122:771-778PubMedCrossRef90086-6)\n\n9.\n\nShah AN, Dodson F, Fung J. Role of nuclear medicine in liver transplantation. Semin Nucl Med 1995;25:36-48PubMedCrossRef80005-4)\n\n10.\n\nStieber AC, Gordon RD, Galloway JR. Orthotopic liver transplantation. In: Zakim D, Boyer TD (eds). Hepatology. A text book of liver disease. WB Saunders, Philadelphia, 1996, pp 1759-1780\n\n11.\n\nBrown RK, Memsic LD, Busuttil RW, Pusey E, Ray RA, Kangarloo H, Hawkins RA. Accurate demonstration of hepatic infarction in liver transplant recipients. J Nucl Med 1986;27:1428-1431PubMed\n\n12.\n\nSanchez-Urdazpal L, Gores GJ, Ward EM, Maus TP, Wahlstrom HE, Moore SB, Wiesner RH, Krom RA. Ischemic-type biliary complication after orthotopic liver transplantation. Hepatology 1992;16:49-53PubMedCrossRef\n\n13.\n\nPannu HK, Warren WR, Fishman EK. Liver transplantation: preoperative CT evaluation. RadioGraphics 2001;21:S133-S146PubMed\n\n14.\n\nLamke AJ, Brinkman MJ, Schott T, Niehaus P, Felix R. Living donor right liver lobes: Preoperative CT volumetric measurement for calculation of intraoperative weight and volume. Radiology 2006;240:736-742CrossRef\n\n15.\n\nKyoden Y, Tamura S, Sugawara Y, Yamashiki N, Matsui Y, Togashi J, Kaneko J, Kokudo N, Makuuchi M. Outcome of living donor liver transplantation for post-Kasai biliary atresia in adults. Liver Transpl 2008;14:186-192PubMedCrossRef\n\n16.\n\nBell BP, Manos M, Zaman A, Terrault N, Thomas A, Navarro VJ, Dhotre KB, et al. The epidemiology of newly diagnosed chronic liver disease in gastroenterology practices in the United States: Results from population-based surveillance. Am J Gastroenterol 2008;103:2727-2736PubMedCrossRef\n\n17.\n\nVon Moll LK, Juni JE, Merion RM. Scintigraphic demonstration of accessory hepatic duct leak following liver transplantation. J Nucl Med 1988;29:259-262PubMed\n\n18.\n\nLantsberg S, Lanchbury EE, Drolc ZA. Evaluation of bile duct complications after orthotopic liver transplantation by hepatobiliary scanning. Nucl Med Commun 1990;11:761-769PubMedCrossRef\n\n19.\n\nMadacsy L, Velosy B, Lonovics J, Csernay L. Differentiation between organic stenosis and functional dyskinesia of the sphincter of Oddi with amyl nitrite-augmented quantitative hepatobiliary scintigraphy. Eur J Nucl Med 1994;21:203-208PubMedCrossRef\n\n20.\n\nBhatnagar A. Nifedipine interventional cholescintigraphy. A new method for assessing sphincter of Oddi? Indian J Nucl Med 1997;12:93-96\n\n21.\n\nJain A, Khanna A, Molmenti EP, Rishi N, Fung JJ. Immunosuppressive therapy. Surg Clin North Am 1999;79:59-76PubMedCrossRef70007-4)\n\n22.\n\nParker BM, Cywinski JB, Alster JM, Irefin SA, Popovich M, Beven M, Fung JJ. Predicting immunosuppressant dosing in the early postoperative period with noninvasive indicyanine green elimination following orthoptopic liver transplantation. Liver Transpl 2008;14:46-52PubMedCrossRef\n\n23.\n\nHawkins RA, Hall T, Gambhir SS, Busuttil RW, Huang SC, Glickman S, Marciano D, Brown RK, Phelps ME. Radionuclide evaluation of liver transplant. Semin Nucl Med 1988;18:199-212PubMedCrossRef80028-X)\n\n24.\n\nShakil AO, Mazariegos GV, Kramer DJ. Fulminant hepatic failure. Surg Clin North Am 1999;79:77-108PubMedCrossRef70008-6)\n\n25.\n\nBoudjema K, Cherqui D, Jaeck D, Chenard-Neu MP, Steib A, Freis G, Becmeur F, Brunot B, Simeoni U, Bellocq JP, et al. Auxiliary liver transplantation for fulminant and subfulminant hepatic failure. Transplantation 1995;59:218-223PubMed\n\n26.\n\nMetselaar HJ, Hesselink EJ, de Rave S, ten Kate FJ, Lameris JS, Groenland TH, Reuvers CB, Weimar W, Terpstra OT, Schalm SW. Recovery of failing liver after auxiliary heterotopic transplantation. Lancet 1990;335:1156-1157PubMedCrossRef91158-7)\n\n27.\n\nKrishnamurthy S, Krishnamurthy GT. Technetium-99m-iminodiacetic acid organic anions: review of biokinetics and clinical application in hepatology. Hepatology 1989;9:139-153PubMedCrossRef\n\n28.\n\nBuyck D, Bonnin F, Bernuau J, Belghiti J, Bok B. Auxiliary liver transplantation in patients with fulminant hepatic failure: hepatobiliary scintigraphy follow-up. Eur J Nucl Med 1997;24:138-142PubMedCrossRef\n\n29.\n\nBismuth H, Houssin D. Reduced-size orthotopic liver graft in hepatic transplantation in children. Surgery 1984;95:367-372PubMed\n\n30.\n\nRaia S, Nery JR, Mies S. Liver transplantation from live donors. Lancet 1989;2:497PubMedCrossRef92101-6)\n\n31.\n\nStrong RW, Lynch SV, Ong TH, et al. Successful liver transplantation from a living donor to her son. N Eng J Med 1990;322:1505-1507CrossRef\n\n32.\n\nEmond JC. Clinical application of living-related liver transplantation. Gastroenterol Clin North Am 1993;22:301-315PubMed\n\n33.\n\nDonovan J. Nonsurgical management of biliary tract disease after liver transplantation. Gastroenterol Clin North Am 1993;22:317-336PubMed\n\n34.\n\nReding R, Goyet JV, Delbeke I, Sokal E, Jamart J, Janssen M, Otte J. Pediatric liver transplantation with cadaveric or living related donors: comparative results in 90 elective recipients of primary grafts. J Pediatr 1999;134:280-286PubMedCrossRef70450-6)\n\n35.\n\nGrewal HP, Thistlethhwaite JR Jr, Loss GE, Fisher JS, Cronin DC, Siegel CT, Newell KA, Bruce DS, Woodle ES, Brady L, Kelly S, Boone P, Oswald K, Millis JM. Complications in 100 living-liver donors. Ann Surg 1998;228:214-219PubMedCrossRef\n\n36.\n\nStratta RJ, Wood RP, Langnas AN, Hollins RR, Bruder KJ, Donovan JP, Burnett DA, Lieberman RP, Lund GB, Pillen TJ, et al. Diagnosis and treatment of biliary tract complications after orthotopic liver transplantation. Surgery 1989;106:675-684PubMed\n\n37.\n\nMazariegos GV, Molmenti EP, Kramer DJ. Early complications after orthotopic liver transplantation. Surg Clin North Am 1999;79:109-129PubMedCrossRef70009-8)\n\n38.\n\nKrishnamurthy S, Krishnamurthy GT. Nuclear hepatology: where is it heading now? J Nucl Med 1988;29:1144-1149PubMed\n\n39.\n\nFulcher AS, Turner MA, Capps GW, Zfass AM, Baker KM. Half-Fourier RARE MR cholangiopancreatography: experience in 300 subjects. Radiology 1998;207:21-32PubMed\n\n40.\n\nSasaki N, Shiomi S, Iwata Y, Nishiguchi S, Kuroki T, Kawabw J, Ochi H. Clinical usefulness of scintigraphy with Tc-99m Galactosyl-human serum albumin for prognosis of cirrhosis of the liver. J Nucl Med 1999;40:1652-1656PubMed\n\n41.\n\nHwang E, Taki J, Shuke N, Nakajima K, Kinuya S, Konoshi S, Michigishi T, Aburano T, Tonami N. Preoperative assessment of residual hepatic functional reserve using Tc-99m-DTPA-galactosyl-human serum albumin dynamic SPECT. J Nucl Med 1999;40:1644-1651PubMed\n\n42.\n\nUetake M, Koizumi K, Yagawa A, Nogata H, Tezuka T, Kono H, Ozawa T, Kusano T, Miyabukuro M, Hosaka M. Use of Tc-99m DTPA galactosyl human serum albumin to predict postoperative residual liver function. Clin Nucl Med 1999;24:428-434PubMedCrossRef\nGerbail T. Krishnamurthy and Shakuntala KrishnamurthyNuclear HepatologyA Textbook of Hepatobiliary Diseases10.1007\/978-3-642-00648-7(C) Springer-Verlag Berlin Heidelberg 2009\n\nIndex\n\nA\n\nAbscess\n\nAquaporins\n\nApopstosis\n\nAbsorption of water\n\nAbsorbed doses\n\nAdenoma\n\nAction of somatostatin\n\nArea method\n\nAlcoholic hepatitis\n\nAcute cholecystitis\n\nAcute cholangitis\n\nAcute acalculous cholecystitis\n\nAuxiliary liver transplantation\n\nAmpula of Vater\n\nB\n\nBismuth\n\nBile canaliculous\n\nBile concentration\n\nBile ducts\n\nBudd-Chiar syndrome\n\nBiliary apparatus\n\nBasolateral border (domain)\n\nBile secretion\n\nBile entry\n\nBasolateral transport\n\nBone marrow uptake\n\nBlood pool\n\nBile duct obstruction\n\nBenign recurrent intrahepatic cholestasis (BRIC)\n\nBenign stricture\n\nBiliary pain pathway\n\nBile aspiration\n\nBaye's analysis\n\nBile leak\n\nBiliary dyskinesia\n\nBiliary atresia\n\nC\n\nCanalicular membrane\n\nCystic duct\n\nCholecystectomy\n\nCaudate lobe\n\nCouinaud\n\nCaudate process\n\nCommon bile duct\n\nCholangiocytes\n\nChild's classification\n\nCholecystokinin\n\nCerulein\n\nCisterna chili\n\nCholangiocarcinoma\n\nComposition of gallstones\n\nChronic acalculous cholecystitis\n\nCholelithiasis\n\nCongenital abnormalities\n\nCystic duct spasm\n\nCongenital biliary atresia\n\nCholedochal cyst\n\nCystic diseases of the gallbladder\n\nCaroli's disease\n\nD\n\nDucts\n\nDifferential bile flow\n\nDuodeno-Gastric bile reflux\n\nDuodenum\n\nE\n\nEmbryology of liver and gallbladder\n\nExtrahepatic obstruction\n\nEndothelial cell\n\nEndocytosis\n\nExcretion\n\nExtraction fraction\n\nExcretion halftime\n\nEjection fraction\n\nExtrensic compression\n\nF\n\nFalciform ligament\n\nFocal lesions\n\nFocal nodular hyperplasis\n\nFatty meal\n\nFluorine-18 FDG\n\nFilling of the gallbladder\n\nG\n\nGallbladder\n\nGallbladder emptying\n\nGallium-67 citrate\n\nGallbladder segmentation\n\nGalactosyl human serum albumin (GSA)\n\nGallbladder cancer\n\nGallstones\n\nGold-198\n\nGastrinoma\n\nH\n\nHepatobiliary imaging\n\nHepatopulmonary syndrome\n\nHemobilia\n\nHepatocellular carcinoma\n\nHepatic artery\n\nHepatic veins\n\nHepatocyte\n\nHemangioma\n\nHepatic extraction fraction\n\nHartmann pouch\n\nI\n\nImaging agents\n\nIntrahepatic cholestasis\n\nIrritable bowel\n\nIndium-111 pentetreotide (OctreoScan)\n\nJ\n\nJaundice\n\nK\n\nKlatskin's tumor\n\nKupffer cell\n\nKHBS\n\nKasai procedure\n\nL\n\nLiver function\n\nLiver lobes\n\nLymphatics\n\nLiver function\n\nLeukocytes\n\nLilli-Altnman\n\nLiver transplantation\n\nLiving-donor liver transplant\n\nLigand\n\nLiver cysts\n\nLiver abscess\n\nM\n\nMicrostructure\n\nMultidrug resistance glycoprotein (MDR)\n\nMorphine\n\nMalignant liver diseases\n\nMorphology of liver\n\nMagnetic resonance cholangiopancreatography (MRCP)\n\nN\n\nNerve supply\n\nNuclear receptors\n\nNecrosis\n\nNeonatal hepatitis\n\nNeonatal cholestasis\n\nO\n\nOrganic anions\n\nOrganic cations\n\nOctreotide\n\nOctreoScan\n\nOpioids\n\nP\n\nParasites\n\nPitt cell\n\nPortal vein\n\nProtein secretion\n\nPlasma proteins\n\nPseudopod\n\nPhagosome\n\nPrimary biliary cirrhosis\n\nPrimary sclerosing cholangitis\n\nPerforation of the gallbladder\n\nPost-cholecystectomy syndrome\n\nQ\n\nQuadrate lobe\n\nQuantification of function\n\nR\n\nRadiocolloid\n\nReticuloendothelial cell\n\nRelative merits of tests\n\nRed blood cell\n\nRadiation dose\n\nRim sign\n\nS\n\nSerege-Cantlie line\n\nSclerosing cholangitis\n\nSegments of the liver\n\nSincalide dose\n\nSlope method\n\nSegmental biliary obstruction\n\nSpiegel's lobe\n\nStellate cell\n\nSphincter of Oddi\n\nSpleen function\n\nStructure-function relation\n\nS-I units\n\nSomatostatin receptors\n\nSolitary focal lesion\n\nStricture\n\nStent patency\n\nSurface receptors\n\nSomatostatin scintigraphy\n\nSugura procedure\n\nT\n\nTransport proteins\n\nTechnetium-99m-sulfurcolloid\n\nTechnetium-99m HIDA agents\n\nTechnetium-99m mebrofenin\n\nTechnetium-99m-disofenin\n\nTechnetium-99m-HMPAO\n\nTechnetium-99m --depreotide\n\nTechnetium-99m-GSA\n\nTotal parenteral nutrition\n\nTransplantation\n\nU\n\nUptake\n\nUeda I\n\nUltrasound\n\nV\n\nViral hepatitis\n\nVascular compartment\n\nW\n\nWall thickening\n\nWirsung duct\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nFirst published by Zero Books, 2013   \nZero Books is an imprint of John Hunt Publishing Ltd., Laurel House, Station Approach,   \nAlresford, Hants, SO24 9JH, UK  \noffice1@jhpbooks.net  \nwww.johnhuntpublishing.com  \nwww.zero-books.net\n\nFor distributor details and how to order please visit the 'Ordering' section on our website.\n\nText copyright: Paul Gordon 2012\n\nISBN: 978 1 78099 327 0\n\nAll rights reserved. Except for brief quotations in critical articles or reviews, no part of this book may be reproduced in any manner without prior written permission from the publishers.\n\nThe rights of Paul Gordon as author have been asserted in accordance with the Copyright,   \nDesigns and Patents Act 1988.\n\nA CIP catalogue record for this book is available from the British Library.\n\nDesign: Stuart Davies\n\nPrinted and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY\n\nWe operate a distinctive and ethical publishing philosophy in all areas of our business, from our global network of authors to production and worldwide distribution.\n\n# CONTENTS\n\nAcknowledgements\n\nVictor Serge - a life line\n\nIntroduction: the art of not dying away\n\n1. In the shadow of madness: the experience of prison\n\n2. Yesterday we were nothing: the revolutionary experience\n\n3. Building on corpses: the repressive state\n\n4. A stranger to no land: the experience of exile\n\n5. Magicians of word and thought: the Russian cultural revolution\n\n6. A certain sort of courage: the writer as witness\n\n7. Shared veins: hope and the soul\n\nWritings by Victor Serge\n\nFurther reading\n\nEndnotes\nFor John Berger  \nwith love and gratitude\n'What does not change \/ is the will to change'  \nCharles Olson, 'The Kingfishers'\n\n# Acknowledgements\n\nThis book simply would not have come into being without Gareth Evans. Love and thanks to him and to Melissa Benn and Graham Music for their encouragement and critical engagement; also to Hannah and Sarah, for being who they are. If I may borrow some words from Serge, I am grateful to them all for existing.\n\nThe music of Dino Saluzzi and Mal Waldron, brooding, tender, searching, endlessly evocative, provided the perfect soundtrack.\n\nPaul Gordon\n\nJuly 2012\n\n_Permissions_\n\nThe quotations from Serge's novels are used by kind permission of the translator, Richard Greeman. The lines from Serge's poems are reprinted by kind permission of City Lights Books, translation (c) 1989 by James Brook. The lines from 'The Kingfishers' are from Charles Olson, _The Collected Poems of Charles Olson_ , _Excluding the Maximus Poems_. (c) 1987 by the Regents of the University of California, reprinted by kind permission of the University of California Press.\n\n# Victor Serge - a life line\n\n1890 - Born 30 December, Victor Lvovich Kibalchich, into Russian revolutionary \u00e9migr\u00e9 family in Brussels\n\n1912 - Sentenced to five years in French prison for association with armed anarchist gang\n\n1915 - Marries Rirette Maitrejean\n\n1917 - Released from prison, takes part in syndicalist uprising in Barcelona  \nRussian Revolution\n\n1918 - Interned in concentration camp in France trying to get to Russia\n\n1919 - Arrives in Petrograd to join the revolution, marries Liuba Russakova\n\n1920 - Son, Vlady, born\n\n1923 - Comintern agent in Berlin and Vienna\n\n1924 - Death of Lenin, succession of Stalin\n\n1928 - Expelled from Communist Party, begins writing fiction, _Men in Prison_ (1930), _Birth of our Power_ (1930) and _Conquered City_ (1932), all published in France\n\n1933 - Exiled to Orenburg, near Kazakhstan, writes _Midnight in the Century_ (1939) and much of his poetry\n\n1935 - Daughter, Jeannine, born\n\n1936 - International solidarity campaign secures his exit from Soviet Union, lives in Brussels and Paris\n\n1937 - Publishes _Destiny of a Revolution_ and _From Lenin to Stalin_\n\n1940 - Fall of France; assassination of Trotsky in Coyocoan, Mexico\n\n1941 - Begins exile in Mexico where he writes three more novels, _The Long Dusk, The Case of Comrade Tulayev_ and _Unforgiving_ _Years_ , and _Memoirs of a Revolutionary_ , marries Laurette Sejourne\n\n1947 - Dies of heart attack, 17 November, stateless, he is buried as a Spaniard in the French cemetery\n\n# Introduction:  \nthe art of not dying away\n\n_Dazzling_. That was the last word Victor Serge ever wrote. In his poem, 'Hands', a meditation on a sixteenth century terra cotta, which he had finished and typed up in the early hours of the day that would be his last. That evening, Tuesday 17 November 1947, he had gone out to see his artist son, Vlady, but he wasn't at home. He met Julian Gorkin, his old comrade, who had traveled from revolutionary Barcelona to greet him in Brussels when he was finally allowed to leave Russia. They talked for a bit before shaking hands and parting. Probably feeling unwell, he decided to take a cab home. His heart gave way before he even had time to tell the driver where he wanted to go.\n\nGorkin went to the police station to identify him just after midnight:\n\nIn a bare shabby room with grey walls, he was laid out on an old operating table, wearing a threadbare suit and a worker's shirt, with holes in his shoes. A cloth bandage covered the mouth that all the tyrannies of the century had not been able to shut. One might have thought him a vagabond who had been taken in out of charity. In fact, had he not been an eternal vagabond of life in search of the ideal? His face still bore the stamp of bitter irony, an expression of protest...\n\nHe had arrived here six years before with Vlady - his companion, Laurette, and his daughter, Jeannine, would come later - fleeing from Nazi-occupied Europe, leaving behind a lifetime of political activity - an anarchist in France, a syndicalist in Spain, a critical Bolshevik in Russia, an agent of the Comintern in Germany and Austria, years of internal exile in Russia, a supporter of the revolutionary POUM in Spain, of which Gorkin had been one of the leaders. The journey had taken six months on a cargo ship he described as 'an ersatz concentration camp of the sea'. There were more than 300 of them on it, including the surrealist Andre Breton, with whom Serge had campaigned in Paris against the Stalinist show-trials, and Breton's wife, Jacqueline. Oddly, the passengers included the anthropologist Claude Levi-Strauss, on his way to a new post in New York. Levi-Strauss was clearly in awe of Serge: he was a man, after all, who had known Lenin, but he found his physical presence at odds with his preconceptions. 'more like an elderly and spinster aunt...with an asexual quality...very far removed from the virile and superabundant vitality commonly associated...with subversive activities.' This, surely, says more about Levi-Strauss than it does about Serge. (Although he complained, Levi-Strauss traveled in comparative comfort compared to most of the passengers, as he was one of the few to have the use of the ship's two cabins.)\n\nBy stark contrast, the young man who would become Mexico's greatest writer and who would eventually win the Nobel Prize, Octavio Paz, had met Serge in Paris not long before and was 'immediately and powerfully' drawn to him. 'I spent hours talking with him, 'he recalled. 'Serge's human warmth, his directness and generosity, could not have been further from the pedantry of the dialecticians. A moist intelligence. In spite of his sufferings, setbacks. and long years of arid political arguments, he had managed to preserve his humanity....I was not moved by his ideas, but by his person...an example of the fusion of two opposing qualities: moral and intellectual intransigence with tolerance and compassion.' (It was Serge who introduced Paz to the work of the French painter\/writer, Henri Michaux, 'a discovery of capital importance for me'.)\n\nTheir journey involved yet another period in prison (this time in French-run Martinique) and, then bizarrely, his first ever flight in a plane. Mexico was not his choice but there was nowhere else, and it had become a home of sorts to thousands of political refugees, thanks to Lazaro Cardenas, the former president whose government was one of the few to have supported the Republican side in Spain, and opened the country to many Spanish and other exiles.\n\nBut the exiles brought their disputes with them, and these were often violent. Only a few months before Serge got there, Trotsky had finally fallen to the assassins who had pursued him for so long. (A previous attempt had involved an armed raid on his compound led by the Communist artist David Siqueiros, who would eventually find refuge in Chile, thanks to the intervention of the Chilean consul-general in Mexico, a poet who would come in time to speak for thousands in his hatred of oppression, one Pablo Neruda. Strange days indeed.)\n\nSerge's and Gorkin's own lives were threatened and on more than one occasion they had to go into hiding. The country's president received appeals from US and British politicians and intellectuals calling on him to protect them. At the same time, the editor of a magazine to which they had contributed was being told by the Interior Minister, Miguel Aleman, later to be President, that the governments of Britain and the USSR were demanding that all platform be withdrawn from these 'agents of Hitler'. Meetings were broken up and, on one occasion, Gorkin and another comrade were stabbed.\n\nAlways poor, and often indescribably lonely, (as he wrote to a friend), he explored the country. And he wrote and wrote and wrote - for _Partisan Review, Politics, Horizon, New Leader_ (as their Mexican correspondent), _The New International, Mundo, Libertad y Socialismo_. And three more novels: his best, his most ambitious yet, against all the odds to add to the four he'd already published, _The Long Dusk, The Case of Comrade Tulayev_ and _Unforgiving Years_. And a life of Trotsky with his widow, Natalia Zedova, a real act of generosity given Trotsky's abuse of Serge when he had dared to disagree. Only one of these would be published while he was still alive. Writing for the desk drawer, he called it.\n\nWith characteristic lack of self-pity, Serge wrote in his notebook: 'It is terribly difficult to create in a void without the slightest support, without the least ambience...at the age of 50...facing an unknown future which does not exclude the possibility that the dictatorships will last longer than the rest of my life.' And he was living at an altitude, over 7,000 ft above sea-level, that could do him no good whatsoever because of his heart, and learning to live with endless _temblores_ (earthquakes), 2000 recorded each year.\n\nAnd he read and read, making sense of the world as the ever worse news reached him from Europe. And he developed his interest in psychology. He'd always believed in the soul, what Vlady called his 'materialist spirituality'. This interest was encouraged by his friendship with the revolutionary German psychiatrists, Fritz Fraenckel and Hubert Lennhof. He was grief struck by Fraenckel's early death - he was only 52 - in June 1943, and it was a year before he was able to write about him, forcing himself to do so, knowing 'only too well the frailty of memory and the iniquitous and impoverishing omission which entombs the dead'. 'I owe a great deal,' Serge wrote, to his 'intelligent equilibrium in a time of instability and to his intellectual richness, which the malicious and the foolish weren't able to appreciate on account of his ways as an amused, sad and irresolute Bohemian...'How light he was on the earth!' At his funeral Serge said, 'Nobody who came close to him escaped his influence, everyone has been made at least a little better.' (Lennhof whispered, 'You don't know how much hostility there was towards him.')\n\nAnd it was here in exile that Serge wrote his single, greatest work, the incomparable _Memoirs of a Revolutionary_ , a work of political witness and engagement and solidarity, unparalleled since. The very first lines speak of a deep feeling he had known from childhood, of 'living in a world without any possible escape, in which there was nothing for it but to fight for an impossible escape'. (The very same existentialist dilemma was being articulated thousands of miles away by Sartre, Camus and de Beauvoir...)\n\nLooking back on his life, Serge said:\n\nI give myself credit for having seen clearly in a number of important situations. In itself, this is not so difficult to achieve, and yet it is rather unusual. To my mind, it is less a question of an exalted or shrewd intelligence, than of good sense, goodwill and a certain sort of courage to enable one to rise above the pressures of one's environment and the natural inclination to close one's eyes to facts, a temptation that arises from our immediate interests and from the fear which problems inspire in us.\n\nAmong the countless things he had seen clearly: that the creation of the Cheka (the secret police) in 1917 was one of the Bolsheviks' 'gravest and most impermissible errors'; that the Bolshevik leadership had lied about the nature of the Kronstadt rebellion in 1921; that the Communists would turn on the independent revolutionary left in Spain; that Stalin would eliminate the entire 1917 revolutionary generation in order to safeguard his power; that Russia had become a vast prison camp. (It was Serge who first described the state as totalitarian.)\n\nAnd yet he never lost hope, that ordinary people would act for themselves and take control of their own lives. On the ship taking him away from Europe he recalled, 'The Russians and Spaniards among us know what it is to take the world into their hands, to set the railways running and the factories working...no kind of predestination impels us to become the offal of the concentration camps.' He wasn't one of those former communists, whose 'god had failed', for he had never had one. For him there was no inevitability in how the revolution had turned out, any more than there was inevitability in anything. History was made by men and women who made choices.\n\nHoping to return to Europe, Serge envisaged a democratic renewal, 'of traditional democratic freedoms made revolutionary once again', in order 'simply to practice the art of not dying away'.\n\n* * *\n\nIt would be wrong to suggest that Serge was completely unrecognized during his life. Far from it; it was precisely his standing internationally that protected him from Stalin and brought about his and his immediate family's release from the Soviet Union, although it could not save his remaining relatives, most of whom would die in camps. And, because he wrote in French, several of his novels and his poetry were published in France, although obviously not in Russia. (One, _Midnight in the Century_ , was even nominated for the Prix Goncourt, the prestigious French literary prize, in 1939, although it didn't win.)\n\nAnd he had his influential admirers and supporters who included the radical US critic and editor, Dwight Macdonald, and George Orwell, who tried to find a publisher for his _Memoirs_. (Serge was so poor he had only one copy of the manuscript which he was, understandably, reluctant to trust to the transatlantic mail.) And he is even a character in a poem 'Aesthetique du mal', by Wallace Stevens. The first of his books to be translated into English, _The Long Dusk_ , was translated by Ralph Manheim, who would go on to become the foremost translator of contemporary German writers, including Brecht and Gunter Grass. And, in Britain, some of his books were published in mass-market editions by Penguin in the 1960s.\n\nBut since his death his work has had a precarious existence, going in and out of print, and waiting a long time to be translated into English. While _The Long Dusk_ was translated by Ralph Manheim the year before Serge died, all his other novels waited a long time. It was 20 years before Roger Trask's translation of _The Case of Comrade Tulayev_ appeared. Richard Greeman's translations of _Men in Prison, Birth of Our Power_ and _Conquered City_ all appeared in the early 1970s. But it was a further ten years before _Midnight in the Century_ appeared in translation, with _Unforgiving Years_ doing so only in 2008, nearly sixty years after it was written. And while the _Memoirs_ were published in 1963 by Oxford University Press in Peter Sedgwick's pellucid translation, they insisted he cut the manuscript by one-eighth before they would do so. It is only now, nearly 50 years later in 2012, that we have a complete English version. (Richard Greeman tells the story of Sedgwick's labors, an act of love and of solidarity, in the new edition.)\n\nAnd since his death he has, of course, had his prominent partisans. His 'unrepentant humanism' was celebrated by Christopher Hampton in his quietly thoughtful book, _Socialism in a Crippled World_ (1981), while Christopher Hitchens acknowledged him as 'the first person to recognize and comprehend the roots of the emerging Stalinist regime'. For John Berger, there is 'no other writer with whom Serge can very usefully be compared'. This singularity, for Berger, the 'essence of the man and his books', is to be found in his attitude to the truth; 'for Serge the value of the truth extended far beyond the simple (or complex) telling of it...The truth for Serge was something to be undergone.' In her introduction to a new edition of _The Case of Comrade Tulayev_ , Susan Sontag was moved to a generous celebration of the man:\n\nThere was nothing, ever, triumphant about his life...except[s] the triumph of being immensely gifted and industrious as a writer...of being principled and also astute and...incapable of keeping company with the faithful and cravenly gullible... of being incorruptible as well as brave and therefore on a lonely, different path from the liars and toadies and careerists, the triumph of being, after the 1920s, right.\n\nBut Serge has also been the subject of almost casual misrepresentation: the maverick British Trotskyite, David Widgery, dedicated his impressive account of the left in Britain to his memory, describing him as a 'Syndicalist, Bolshevik, Trotskyist', conveniently forgetting the anarchist he had been for many years, which got him several years in prison and which was _always_ part of who he was, while the epithet 'Trotskyist' is just misleading. A biography of the 'American Pimpernel', Varian Fry, who helped Serge and very many other intellectuals and artists to escape from France in 1940, describes him as a man 'who had travelled the revolutionary road from violence to tolerance', a gross oversimplification, that is also a misrepresentation. And it's a surprise to see the radical, and usually inspired, critic, John Leonard, say of him, that he 'had helped the Russian Revolution but then run away from it'. Really? Serge didn't run from anything, other than the Nazis in 1940 when his life and the lives of his family were in danger and he had done everything he could to put the revolution right, or at least stop it going wrong. And even Sontag, in what is otherwise a marvelous appreciation of the man, somehow manages to depoliticize him, which is quite a feat, given that he was born, lived and died a political being. It is this, more than anything else, that has stopped him getting the recognition that is his due. And while we must be grateful to the individuals and groups on the far left who have done so much to keep Serge's words and spirit alive, he is much too important to be a figure of the margins.\n\n* * *\n\nFor myself, I first read Victor Serge as a disaffected teenager in the late 1960s\/early 1970s, when I was trying to make sense of this thing called communism which seemed to be doing such terrible things in the world, despite its claim to be acting for the benefit of humankind, an idea to which I was strongly drawn. One of the books on our shelves at home was a volume called _The God That Failed_ , a collection of essays by former Communists, about why they had become disillusioned. The book, edited by the Labour MP Richard Crossman, with contributions by Richard Wright, Ignazio Silone and Arthur Koestler and others, was a popular volume with a catchy title, that seemed to damn the communist cause pretty conclusively. Equally damning were books like Alexander Solzhenitsyn's _One Day in the Life of Ivan Denisovich_ , Koestler's _Darkness at Noon_ , and George Orwell's _Animal Farm_ and _Nineteen Eighty Four_ , which I read around the same time. It was at the same time that I came across Serge's, _The Case of Comrade Tulayev_ , then published in its iconic Modern Classics series by Penguin. But this was telling a much more subtle and complex story that I was only able to appreciate with time.\n\nSerge's _Memoirs_ is the book I've read more often, and given to more people, than any other. Many people will recognize themselves in Serge biographer Susan Weissman's account of her total absorption. She first read them on a long train journey but had not finished them when she reached her destination. She sat on a bench in the station and read right to the end. Even after many readings, I still have to tear myself away once I pick them up, they are so engrossing. It was also one of the books I took traveling with me after I left university, including to Portugal, then in the throes of its own revolution, along with the wonderful Penguin volume of John Berger's _Selected Essays_ , and John Dos Passos' massive _USA_ , an astonishing counter-story of that country's development. (I learned much later that Serge admired Dos Passos and was clearly influenced by him.)\n\nI have never had a need for heroes, but Serge is a man I have admired greatly for most of my life and my admiration has deepened with my reading and re-reading of him over the years. That he wrote at all given the circumstances of his life, the persecution and the poverty, is remarkable; that he wrote so creatively, tellingly, movingly, is little short of astonishing, as is the fact that he never lost hope that people could take their destiny in their own hands and make a better world. And I've always loved the fact that this man - novelist, poet, historian, witness, revolutionary activist, humanist - who spoke several languages, never set foot inside an institution of formal learning; his father thought it 'stupid bourgeois instruction for the poor'. He was truly educated in museums, libraries and churches and in 'ransacking encyclopedias' and in, what he calls somewhere, 'the university of the streets'.\n\n* * *\n\nSo what follows is an introduction to a man who really shouldn't need it and who, in many ways, does not; all we have to do, ultimately, is to read him. But he does need to be contextualized and that is what I've done, as well as provide a critical appreciation of his main writings, following, roughly, the chronology of his novels. If I've quoted him at length in places that's because he says things so much better than any paraphrase or summary and also because he often achieves what he does by an accumulation of images. (That's why I quote rarely from the _Memoirs;_ as someone said, the problem in writing about Serge's life is that he says it all so much better than we can.) And, yes, this is also a celebration of a unique and inspiring man and repayment of a debt of gratitude that is long overdue.\n\n# 1  \nIn the shadow of madness: the experience of prison\n\n'I was no longer a man', Serge wrote of the moment of sentence, but 'a man in prison. An inmate'. The process of dehumanization started then, 'I had crossed the invisible boundary'. Admission to the prison itself began with the inmates losing their few personal effects, 'We start out by being robbed.' And whatever individuality they have is also taken away when their heads are shaved and they are 'sufficiently depersonalized' to appear before the administrative authorities. Nothing distinguishes one man from another, 'the same stubbly chins, the same shaved skulls - and doubtless the same look of the hunted man'. The mass photographing, 'a gallery of lost souls', further reduces every one to two or three varieties of expression: 'animal passivity, confusion, humiliation - each modified by anger, despair, defiance, or taciturn sullenness...'\n\nWhen he was in his early twenties, Victor Serge was implicated by the French police in the activities of the notorious 'Bonnot Gang', a group of illegalist anarchists who had been causing havoc in parts of the country. Believers in the 'propa-ganda of the deed', they carried out armed robberies and made their getaway by car, which was then something of a novelty. Serge had expressed his support for them and when police raided the offices of the newspaper he edited, they found a couple of revolvers. He refused to give evidence against the robbers, was convicted and given what could only be seen as a vindictive sentence of five years imprisonment. Writing nearly 30 years later, Serge wrote, with an honesty and self-evident pain, of the emotional legacy this had left him:\n\nIt burdened me with an experience so heavy, so intolerable to endure, that long afterwards, when I resumed writing, my first book (a novel) amounted to an effort to free myself from this inward nightmare, as well as performing a duty towards all those who will never so free themselves...\n\nAlthough written in the first person, narrated by a man who is never named, _Men in Prison_ is, Serge makes clear at the outset, a work of fiction, but it is also true. 'Everything in this book is fictional and everything is true.' Serge said that what he was trying to do, was through literary creation, to bring out the general meaning and human contents of a personal experience. In other words, Serge is saying, don't make the mistake of thinking that this book, is about me. The book is called, after all, _men_ in prison, not _a man_ in prison. It's about all men 'crushed in that dark corner of society'. And the only way open to Serge to do it at that time, with the creative resources available to him, was as a first person narrative. He was being true to his own experience, but also that of the many people whose lives he had shared or observed.\n\nThe rules of the prison can be summarized, Serge wrote, in three peremptory words: _'Living is forbidden!'_ Everything in prison came down to this. Above all, life was extinguished through time. Prisoners often speak of 'doing time'. They are punished through the theft of time; it is not theirs and what time there is, is either structured, as in meaningless work, or dead, empty time that destroys the soul. Serge was able to articulate this with an emotional depth which is truly frightening. Faced with 'the mystery of time's passage', the experience of time becomes like a torture. There are minutes and hours which have no end, 'the eternity of the instant'. But there are also empty hours, endless days, and weeks which pass without 'leaving the least memory behind them, as if they had never been. I cannot distinguish the years that are behind me'.\n\nWhen they are repeated indefinitely, the same feelings grow dull. One loses count of the hours and the days. Even something that moved or terrified during the first days no longer has any effect: 'Suffocation? Drowning? A torpor sneaks into your veins, between your temples: All of life takes on the faded ocher-hue of the cell. You can no more escape this torpor than you can escape these four walls. The rhythm of your inner life slows down.'\n\nNight brings no respite:\n\nEven the city's rumble seems to have stopped. Nothing. Sleep is impossible...I am already in a sort of tomb. I can do nothing. I see, hear, and feel nothing. I only know the next hour will be exactly like this one. The contrast between this vacant, empty prison time and the intense rhythms of normal life is so violent that it will take a long and painful adaptation to slow down the pulse of life, to deaden the will, to stifle, blot out, obliterate every unsettling image from my mind.\n\nThis experience comes to feel like being buried, a terrifying sensation that occurs again and again in the book: 'Burial. Each hour is like a shovelful of earth falling noiselessly, softly, on this grave...I am alone in a numbered sepulchre...Stretched out on my cot, like a dead man in his shroud (I even like to cross my hands over my chest like a dead man)...'\n\nLife is also extinguished through hard work, forced labor, 'poured onto the limit of your strength', twelve hours a day, from seven in the morning to seven in the evening. Not only that, but this work was supposed to be done in absolute, perpetual silence. The absurdity of that rule was, Serge commented, equaled only by its cruelty. 'If it were actually enforced and respected, it would be the simplest way to drive the prisoners quietly mad'. In practice, it was not rigidly applied, but its very existence gave the guards huge power as they could resort to it as and when they wanted, at any moment.\n\nThe prison also did all it could to kill any living relationship between prisoners. Communication between the inmates, as in the use of the 'telephone', using the wastepipes of the building, was especially severely punished. But even something as banal as whistling, humming, talking to yourself out loud were prohibited, as these too were signs of life, however pathetic. Of course, prisoners disobeyed, but they did so in a situation of considerable fear that they would be caught.\n\nSerge was an acute observer of how the modern prison _as a building_ served its function beautifully. The only 'perfect and irreproachable work of architecture in the modern city', he said, its perfection lay in the total subordination of its design to its function: 'Its perfection is revealed at first glance: it is impossible to mistake it for any other kind of edifice. It is proudly, insularly, itself.' It was able to do the seemingly impossible task of housing a vast crowd, while achieving the total isolation of each individual in that crowd: 'Busier than a beehive, it is able to accomplish, silently and systematically, as many different tasks as there are lives tossed into its grinding cogs'.\n\nThe hold of the modern, urban prison was total. No one ever escaped from them. While people even used to escape from the Bastille, that symbol of pre-revolutionary despotism, and even from the French overseas territories of New Caledonia and Guiana, where people were transported, 'no one escapes from the model jail'.\n\nSerge resisted completely any tendency to romanticize his fellow prisoners. One encountered, he remarked, just the same proportion of 'weak spirits, human scum, average types, and exceptional men, gifted with some spark of divinity', as anywhere else. As for the warders, they too, he said, were criminal, but respectably so, and with a guaranteed immunity from punishment for anything they might do. These people, who included 'sadists, inflexible hypocrites, morons, racketeers', could look forward to a pension at the end of their unspeakable lives. Serge drew a devastating picture of this person:\n\nThe hybrid of bureaucratic scribbler and turnkey...a singularly monotonous breed...their bodies grown fat from sitting...faces...stamped by the murky grayness of the prison bureau...eyes...grown dim with the horrid dullness that emanates from the forms, receipts, registers and filing cabinets, where the same inanely bureaucratic descriptions of hopeless victims and miserable wretches pile up _ad infinitum_...\n\nAnd yet, 'incredible as it may seem', there were also warders who were good and 'almost intelligent', men who were capable of acts of kindness, when, for instance, they would respond to the pleadings of a man in isolation who is 'too weary of solitude', and allow him a yard companion during exercise. On one such occasion, the narrator himself met a man who, when hearing he was an anarchist, shook his hand effusively. This man had known Peter Kropotkin and Pierre Martin, prominent Russian and French anarchists respectively, who had signed the 1883 Manifesto of the Anarchists and been imprisoned for their politics.\n\nThe warders and inmates ended up in a bizarre, shared existence, 'the same life on both sides of the same bolted door'. (The use of the familiar _tu_ , as a form of address, was evidence of an instinctive recognition of a common existence and a common mentality.) And yet, of course, warders had power and prisoners were powerless; few men in modern society, Serge observed, wielded 'such absolute power' over their fellow men as did prison wardens. They had, in effect the power of life and death over the prisoner:\n\nAll it takes from him is a suggestion to the chief Guard...and the prisoner...is constantly harassed with petty discipline and loaded down with penalties. The Warden can inflict penalties up to ninety days in the hole; more than enough to send the man...to the infirmary, eyes ruined, lungs ravaged by tuberculosis, throat swollen, and ears dripping pus...in effect, several sentences of slow death...\n\nThose disciplined were subjected to an extremely harsh regime, with their rations reduced to dry bread and morning soup; they have to march for twenty minutes with their arms folded across their chests, then for twenty minutes with their arms folded behind their backs. Even when they are allowed to rest, they have to sit with their legs together, their elbows pressed to their sides. They marched 30 miles a day.\n\nEvery man who is thrown into a cell, Serge said, immediately begins to live 'in the shadow of madness'. Serge was an acute observer of the many ways in which people fended off this ever-present threat. Something as unremarkable as walking was something everyone discovered within a few hours of being inside, 'He begins to walk', he says simply. Some, like him, even write poems as they do so. Others, like the narrator too, turned to books, but the bulk of the prison library consisted of bad adventure novels, probably bought because not even the public would buy them, but then there was also a collection of Balzac, 'heaven-sent'. At breaks in work everyone would read, but some books had to be disguised so as to appear as if they came from the library. A strange type of contraband came into existence, for which the prisoners even invented a word, _camelote_ , such as a volume of Casanova's Memoirs, while the narrator reads the philosopher, Taine, on intelligence. Each, in its own way, provides 'rich spiritual nourishment'.\n\nSerge would also say later that it was the possibility of writing about his experience that helped keep him sane. As he wrote to his friend, the Romanian writer Panait Istrati, 'I already saw a kind of justification of that infernal voyage in the possibility of describing it'.\n\nVisits from families, too, helped men survive, but these, also, could be a source of torment, loaded with a weight of expectation they could not possibly bear, accompanied by shame and embarrassment. People were unable to touch each other, but there were times too when, because of the darkness, people could not even see each other clearly. _'The other person is there:_ corporeal, yet ghostly; present, yet inaccessible'. Letters were 'tiny wings in the hands of the men'. Each one had its soul, its character, its voice, 'a woman's love on the page' can lift someone out of his misery.\n\nSo too can a glimpse of life outside: 'The silent _quai_ under the tall, green poplars, the shimmering water where trees trembled in reflection among patches of sky, the path along the bank where a child was running: This peaceful vision of life, glimpsed suddenly through the window calmed him.'\n\nEven the coming of spring, 'bittersweet season' that it is, is impossible not to feel:\n\nIn April, with the first buds on our stunted trees, the first clear skies, the first warm days, such a powerful call seemed to come from the very heart of life that we all felt we were emerging, our nerves raw, from some great lethargy. April quickened our failing energies. Three hundred wooden shoes beat more smartly against the pavement in our round; broken marionettes began to straighten up again; grey faces were uplifted...\n\nIn the same way, a thought or memory coming unbidden might bring some comfort, but it could never last:\n\nSome times, especially in the evening, a noise from the street may reach the prisoner in his cell. An automobile sounds its horn. The bell of a trolley car rings out in the distance. Instantaneously, the image of the illuminated streets and of that trolley car appears in your mind's eye. You see the conductor taking the steering bar into his wool-gloved hands. You see everything. You breathe in the smell of asphalt and gasoline. And then everything vanishes...the calm grayness of time sooner or later resumes its usual hue.\n\nSooner or later, everyone went to see the doctor. If he was lucky, a man might get some extra bread or even medicine, but what made a difference to so many people, Serge said with some insight, was that this involved moments of contact with _other men_. It was _this_ that made their ailments subside. Serge was not being judgmental here; he's not saying people are pretending to be ill, but showing a subtle appreciation of the true nature of the psychosomatic. If the only way we can get care is by being ill, we become ill.\n\nThe prison hospital, too, provided considerable respite; one was allowed to walk in the infirmary 'garden'. The term garden is used here, Serge says, in its 'most perfunctory sense'; the barren pond, the stunted trees, the cropped bushes - this could be the square, he said evocatively, in some mining village in the north, the men survivors of a gas explosion, 'After all, one form of suffocation is much like another.' But even there, one could walk up and down these paths at will, 'Divine freedom of the body! We meet, we talk; divine freedom of the soul!' Here, too, even the barest of the natural world provides comfort; here are the white quartz pebbles that the men look for in the gravel, 'so velvet, so soft to the touch, so white that they bring to mind many forgotten things'. And the trees are not abstract, but 'real, accessible trees, whose bark we love to touch and on which we love to watch the ants climbing'.\n\nSerge was clear about what was involved in the struggle of those who chose to resist, as opposed to going under. In a chapter called, 'Drunken boat', taking its title from the poem by the young Rimbaud, Serge said, in almost unbearable prose, that it was:\n\n...a voyage toward the unknown. The march will be long, so long that there is nothing by which to measure its duration, through a relentless night strewn with pitfalls. Falling along the way would be like sinking into a dark lake on a moonless night under a leaden sky in solitude: No cry would ever be heard. So be silent, then, whatever happens. The fleeting circles would barely break the surface of the still water, which would soon close without a ripple over the drowned man.\n\nThis silence is not the silence of submission, but neither is it a silence of pointless resistance, however understandable. Something else is called for, an acceptance that is, at the same time, a refusal, of the status of prisoner - 'I will be Number 6731...a robot programmed to obey', because one has no choice, but not to be confined by this - 'I will be myself - a free man'. The stakes in this struggle were high:\n\nA victory over jail is a great victory. At certain moments you feel astonishingly _free_. You sense that if this torture has not broken you, nothing will ever be able to break you...And when a broad ray of sunlight inundates the barred window, when good news comes in from the outside, when you have succeeded in filling the dismal day with useful work, an inexpressible joy may ascend within you, like a hymn.\n\nBut he is stunned, when he had served his sentence in full, not to feel the great joy he had anticipated, 'at this dark threshold to life', a brilliant formulation, capturing all the contradictory feelings of the moment. He felt, instead, a sort of oppressive anxiety and, at the same time, a certain joy in a negative sense, 'like a fleeting light at the bottom of a well', that if he were not, after all, going to be freed, he would kill himself. At the same time, Serge had the insight that nothing could be more disappointing than the long-awaited fulfillment of a wish, because the reality itself was too concrete. One came to the realization that one had been living on exaltation which, once gone, left in its place 'a great void in which things appear only as they are, nothing more'.\n\nFor a time, a brief time, he experiences what we know now as institutionalization, that state of mind in which individuals have become so used to their confinement, whatever its nature, that they find it very difficult, or even impossible, to leave. And if they do succeed, it is what takes them, sooner or later, back inside, to what is familiar:\n\nI will be free in a few hours. Free. The enormous word is written in flaming letters before me...I am about to enter the unreal...I can no longer imagine what life is...I begin to love this darkness...where every stone is familiar to me...a world that has become a deep unforgettable part of me...My heart aches as a corner of the workshop and certain faces appear before me. I am leaving, they are staying. The Mill is eternal. That must be it; I feel the mark of jail too deeply within. They no longer brand your shoulder with a hot iron; it is an inner wound that will start to ache tomorrow.\n\nBut he struggles, resolving to overcome this, not to carry away with him any defeat. The Mill has not worn him down. He will leave it with his mind intact, stronger for having survived, tempered by thought. He has not lost the years taken from him.\n\nThe narrator has, like Serge, been in prison while a long, destructive and bitter war has been fought outside; the prison has been 'an unbelievable island, cut off from the movement of history'. Once outside, the first human figure he comes across is like a specter, shockingly evoked, a ghost in uniform who transforms into reality the unreal landscape, 'this landscape of the world between', which he will never see again: 'Our ringing footsteps fall in together. The first man I meet on the threshold of the world is a man of the trenches.'\n\n* * *\n\nWithin a year or so, Serge was in prison once again. This time, after the failure of the syndicalist uprising in Barcelona when he was in France trying to get to Russia in the early throes of revolution, he found himself interned in a concentration camp, in a disused convent, in the west of the country. He would make use of this experience in his second novel, _Birth of our Power_ , which was published in France in 1931. This prison was nothing like the one in _Men in Prison;_ it was, relatively speaking, lightly guarded, barbed wire, a row of sentries, a low wall covered by fragments of broken bottles. Even so, no one had yet managed to get out of it. The prisoners were free to associate with one another, to read, to discuss things, to walk about within the camp confines, and no one had to work. And yet they were still prisoners.\n\nThis little world within the world is powerfully depicted by Serge. The camp, he said, might have been the main square of 'a bizarre village where there aren't any women'.\n\nIn this 'little piece of Europe', there were Greeks, Macedonians, Bulgarians, Chetniks, Serbs, Russians, Alsatians, Spaniards, Belgians and Romanians - 'thieves, marauders, phony foreign noblemen, probable spies, certain victims, unlucky people, vagabonds, second offenders, undesirables, Germanophiles, simple-minded people, rebels, revolutionaries'. And, as the powerful in Europe started to fear more and more the threat to their power posed by the communists, there are 'Jewish tailors and restaurant owners' who were 'guilty of having, elbows on the counter, maintained the integrity of the Bolsheviks'. (Serge himself was interned as a Bolshevik sympathizer.)\n\nThe camp was a reflection of the society in which it sat. It was even possible to get rich there, as one of the men, Maerts, 'the buccaneer', does. There were the decent and the corrupt, the highly intellectual and those who have lost their minds; some eat well while others, like Antoine, a traveler, driven from his roads by the war, lived on potato peelings, carrot leaves and even-gnawed bones found in the garbage every night. At night, one of the barracks becomes like 'an inn of olden times, in an old port haunted by pirates'.\n\nWhat makes life tolerable is the existence of a community of like-minded people and the continued existence of the life of the mind: 'We formed a world apart within this city. It sufficed for one of us to call the others together with that magic word 'Comrades', and we would feel united, brothers without even needing to say it, sure of understanding each other even in our misunderstanding.'\n\nIn their dormitory, there were always people pouring over their 'endlessly annotated, commented, summarized texts'. There, revolutionary figures from the past like Saint-Just, Robespierre, Babeuf, Blanqui, Bakunin, were spoken of 'as if they had just come down to take a stroll under the trees.' Even those with 'no great sense of history', still had some great stories to tell, like Dmitri, the Russian sailor, who almost got his whole crew of an English steamer to mutiny, after throwing inedible soup in the face of the first mate, but ended up in chains instead. And in the late afternoon, a few people from each room would assemble to hear someone read from the newspapers. This is how they hear what is happening in Russia, in fragments, which are often inaccurate, as when they hear that Lenin has been assassinated. In fact, he survived. But this also serves to emphasize their isolation from everything that is going on in the world, and their impotence, and to instill a sense of futility, of time running out, while great things were being achieved elsewhere. And these feelings turn into rage. They think about an uprising, but know it would be futile. With some insight Serge remarked on how one of their fears was precisely that of being cowardly and they were therefore afraid of 'throwing ourselves into an adventure...out of our own impotence'.\n\nAnd there was the ever-present threat of death, which came 'without fanfare, simply, faceless, without terror'. The camp is decimated by an epidemic of typhus, but the political Russians do better than others. Even in this regard, their solidarity makes them stronger. The emergency fund they had set up, though meager, provides just about enough for the least fortunate 'to keep the flame of life glowing, if only as an ember'. And, they do not allow anyone to be moved to the infirmary, which has been nicknamed the Morgue, because that is where people go to die.\n\nOnce again, it was time which was taken from the inmates and time which they had to contend with. And unlike criminal prisoners who know when they will be released however far in the future, they had no idea at all when they might be freed, and, if so, what will then happen to them:\n\nLong walks in the yard, to kill the time. Rare were those, in this forced leisure, who still knew the value of time, who read, who sketched, who studied, Equally rare the obstinate ones who refused to let themselves go...We stayed alive. The days passed by. The weeks, the months, the seasons, the battles, the revolution, the war passed by. Life passed by.\n\nWhen the armistice comes, 'it exploded above us, like a dazzling rocket, tracing a meteoric curve through the sky of our gray life'. Suddenly there is joy at the prospect of freedom, 'the end of the nightmare'. But this is only the beginning of the end of their captivity as they must wait for negotiations about an exchange of prisoners over which they have no control whatsoever. But, in the end, they leave.\n\n* * *\n\nIt is a century since Victor Serge began his years in prison. It may seem incredible now, but there was a time, not so long ago, when many of us on the left believed in the abolition of prison, that prison might be done away with, for all but the most dangerous people. _(The Politics of Abolition_ (1974) was the title of radical criminologist, Thomas Mathiesen's fine book about the prisoners' movement in Norway and his involvement in it.) We argued that sending people to prison was unjust and unfair, that it was counterproductive, and that it was a mark of a disciplinary state. Serge's writings were among those that inspired us.\n\nThings could not have turned out more differently. Our societies have become more, not less, punitive; almost everywhere, prison populations are higher than they ever have been. The provision of prison and associated services has become very big business indeed, one of the few growth industries. Prisoners are no longer transported around the country in the prominent and threatening 'black marias' of old, but in anonymous vehicles, which might easily be delivering groceries. The concentration camp, which Serge must have been one of the first to describe, has become a central feature of life in many authoritarian regimes.\n\nThe physical conditions of prison may well have improved in some countries, the specific details may be different from Serge's time, but _Men in Prison_ today is completely relevant to our modern understanding of the central nature of imprisonment. It is a genuine classic of prison writing that transcends its time. Rarely has the brutality, the pointlessness, the tedium, the dehumanization of prison, been so memorably depicted, but without bitterness or self-indulgence.\n\nModern jails, Serge concluded, 'are imperfectible, since they are perfect. There is nothing left but to destroy them.'\n\n# 2  \nYesterday we were nothing:  \nthe experience of revolution\n\nOn his release from prison in 1917, Victor Serge went to Barcelona to take part in the planned syndicalist uprising there. This failed, bloodily, and Serge lost many friends and comrades. He then traveled to Russia to join the revolution there, eventually reaching Petrograd in 1919. Despite his anarchist past and libertarian convictions, he had decided that he would join the Bolsheviks, but do so 'independently, without renouncing thought or critical sense'. Serge believed that the Bolsheviks were profoundly mistaken in many ways, in their centralizing leanings and bureaucratic tendencies, their intolerance of difference and their unquestioning belief in the state. But, for Serge, they still represented the best hope for the revolution. One had to be with them and among them if one were to counter them 'with freedom of the spirit and the spirit of freedom'. Serge was widely respected outside the Party and enjoyed good, if sometimes strained relations with the non-Bolshevik revolutionaries, such as the anarchists and the Mensheviks, and with the cultural intelligentsia. From the outset he was actively involved in the revolution, both as an organizer and as a soldier. (Serge took part in the battle of Pulkovo Heights in October 1919, which would prove decisive in the defense of the city and of the revolution itself.) It was these experiences of revolution that would be the subject of his next two novels, _Birth of our Power_ , published in France in 1931, and _Conquered City_ , published the following year.\n\nWith no strong narrative to engage the reader, _Birth of our Power_ is, to my mind, the least successful of Serge's novels. Sometimes one feels his struggle to find a new form for what he is saying. Serge was trying to take the novel away from a central subject, to capture a process, and he isn't always successful. There are times, too, when it feels as though he has been swept up in his creative processes, but forgets about his readers. And that is a pity because there are always passages of great lyricism, and Serge's descriptions - of people, of environments - are often arresting. And no one described the values of solidarity and courage, or changes in political consciousness, as he did.\n\nHere, for example, is Serge's description of the group his narrator found himself in:\n\nThere were at least forty or fifty of us, coming from every corner of the world - comrades, that is to say, more than brothers by blood or law, brothers by a common bond of thought, habit, language, and mutual help. No profession was foreign to us. We came from every conceivable background. Among us, we knew practically every country in the world, beginning with the capitals of hard work and hunger, and with the prisons. There were among us those who no longer believed in anything but themselves. The majority were moved by ardent faith; some were rotten - but intelligent enough not to break the law of solidarity too openly. We could recognize each other by the way we pronounced certain words, and by the way we had of tossing the ringing coin of ideas into any conversation. Without any written law, we comrades owed each other (even the most recent newcomer) a meal, a place to sleep, a hide-out, the peseta that will save you in a dark hour, the _douro_ (a hundred sous) when you're broke (but after that it's your own look out!). No organization held us together, but none has ever had as much real and authentic solidarity as our fraternity of fighters without leaders, without rules, and without ties.\n\nAnd there is the portrait of Dario, the agitator; up at 6am, he has time for a coffee only while standing in the street, and then a day of leafleting, meetings, speaking, standing on a chair, at factories. But the most tiring hour is when he has to meet Portez the cement worker and argue his positions once again. When the narrator thinks of expressing his doubts, he knows Dario would laugh, 'You distrust intellectuals, especially those who have tasted the poison of Paris'.\n\nSerge was also trying to capture the change in political consciousness of those who wish to make their own future. The shoe-shine stand operator, Sanche _el Tuerto_ ('One-eye'), who 'usually sees men only from the knees down', is barely literate, struggles to read, but when he picks up the revolutionary paper _Soli_. 'A sort of smile twists his mouth. He wouldn't be able to repeat or to explain what he is reading, but a great contentment flows into the marrow of his bones.' And when he says, simply, 'No time' to 'a rich French shoe', he is doing something momentous and the customer goes away with the understanding that 'something is happening in the world'. Sanche's 'No time', both 'worries and enlightens him immeasurably', much more even than the day's news of a German sinking of a Brazilian boat and the bombing of London by Zeppelins. He knows, or at the very least senses, that something is afoot, something that he does not yet understand.\n\nBut the changing consciousness is not confined to individuals. It is something that affects the collective which is transformed by an increasing awareness of its own power, which surges through the city like 'new blood injected into the arteries of an old organism', unseen but vividly experienced by those in the know:\n\nWorkers stream out through the dazzling city toward their houses in the poor quarters, their steps lightened, shoulders thrown back with a new feeling of power. Their hands never tire of caressing the weapons' black steel. And waves of pride flow from that steel into their muscular arms...to those precincts of the brain where, by a mysterious chemistry, that essential life force we call the Will is distilled...\n\nAs elsewhere, Serge is particularly attuned here to people's subjective experience, what it _feels_ like to be poor, to _feel_ degraded, as the workers do, by 'the contrast between their sloppy old suits or overalls and bourgeois dress', when they pass expensive restaurants and luxurious cafes they never enter, or shop windows with astonishing displays of objects, 'so beyond their means as to be not even tempting'. It is here they 'encounter the women of that other race, sheathed in precious fabrics, their complexions colored by good health and luxury as if by a soft inner light...the well-fed men with relaxed faces, haughty, superior looks, under broad felt hats'.\n\nAnd he evokes powerfully the sheer physicality of being in a mass of people: 'Heads, bodies, hands are growing all around us like tropical vegetation; a powerful odor of warm and vibrant flesh - the smell of masses of men and of sunlight - makes our nostrils throb. I also breathe in the avid smell of the oranges being eaten greedily by a young girl...'\n\nSerge was also a keen observer of the places he found himself in and powerfully evoked these topographies, these physical contexts, as he does in this book, which opens with the words, 'This city and us'. And later on he remarks, 'Every city contains many cities. This was ours. We did not penetrate into the others.' Serge beautifully describes the landscape, the 'craggy mass of sheer rock - shattering the most beautiful of horizons', that towers over this city. It is 'crowned by an eccentric star' of jagged masonry cut centuries ago into the brown stone which now conceals secret constructions under the innocent grassy knolls. The secret citadel underneath lends 'an evil aspect' to the rock, which, between' the limpid blue of the sky, the deeper blue of the sea, the green meadows of Llobregat and the city, resembles a strange primordial gem'. We would have loved this rock, Serge says, had it not been for the way in which it had become an instrument of oppression, for the mountain was 'a prison - subjugating the city, blocking off its horizon with its dark mass under the most beautiful of suns'. The landscape, in other words, which might seem natural to most people, is itself political and saturated with history. It is a continual reminder of oppression, and even death:\n\nOur voices would suddenly drop off, when, at a bend in the path, the stark, grass-covered corner of the citadel's ramparts loomed up before us. The name of a man who had been shot was on all our lips. [Francisco Ferrer, libertarian educator, executed in 1909] We used to stop at certain places from which we could see the narrow confines of the dungeons. Somewhere within these fortifications, men like us, with whom each of us at one time or another identified ourselves, men whose names we no longer remembered, had undergone torture not long ago...\n\nPeople's names and faces may have been forgotten with the passing of time, but what remains is 'a searing, confused feeling for the indignities suffered in the cause of justice'. The pain these men suffered comes to be remembered 'as one remembers something one has suffered oneself...And, from that, I had an even greater sense of the communion between their lives and ours'.\n\nAnd there is the constant awareness of the danger faced by those at the forefront of the struggle:\n\nAnd the idea I am trying to get rid of pierces me, like an electric needle, from one temple to the other: Dario will be killed, for that city, for us, for me, for the future. Every morning when he leaves the house where he has slept, every evening when he enters the back rooms of little cafes where fifteen men - including one traitor - are waiting for him, at every moment of his patient agitator's labor, he moves toward that end marked out for him.\n\nAnd even in the maelstrom of political upheaval there is room for love:\n\nHe pressed herself close to him without speaking a word. He sought her face and found only her ardent lips. \"Let me look at you,\" he said. He struck a phosphorous match against the wall. A sputtering blue star, hissing and spidery, burst into flame at his fingertips, her delicate, soft-toned face - with its huge dark eyes, each now lighted by a spark, shining from out of their deep-set, dusky orbits - was nestled in the hollow of his arm, with a poor, gentle, worried smile. Dario gazed at it until the ephemeral light singeing his fingers went out. They made love in total darkness - in silence, for he was hurried and tired, and she always felt on the verge of losing him.\n\n* * *\n\nThe uprising having failed, the narrator is on the run in Paris whose beauty 'smiles on implacably like summer'. It is a bad day, he says, and he runs all over Paris in search of refuge but his 'addresses are running out, and time is moving on'. He worries that the papers he has will not stop him being arrested if he is forced to register at a hotel. 'Where to find a roof tonight? A few hours under shelter, time to recharge my nervous equipment, and the future is saved.' He eventually ends up at a sixth-floor door in 'this enemy city' where he is told he is in luck, that the man's girlfriend walked out on him recently, leaving him more room. He reassures him wonderfully saying, 'You'll be all right here. I have an excellent reputation; you can sleep with both eyes shut.'\n\nThe man with the 'excellent reputation' is Broux, who despite 'his worn-out lungs, his obstinate self-effacement, his bookish timidity', is a strong man, strong by means of his 'awareness of how impossible it is to live, he raises himself precisely to a higher possibility of living, to an endurance which is more sure of itself because it believes it has nothing more to lose. From his weakness he was able to create a strength, from his despair, an acquiescence, from his acquiescence, a hope...'\n\nOutside, and almost as a backdrop, there is a war going on. Somewhere, up there above these clouds, 'men dressed in leather are trying to fix this city in their bombsights'. The anti-aircraft guns produce in response, 'explosive blossoms'. Serge remarks, without any irony, that civilization reaches its high point in this 'senseless combat' above the Louvre, as 'Masterpieces of ingenuity, summing up the work of all races in all times - millions of men suffering, striving, daring - seek each other out, with the greatest human lucidity, in order to destroy each other...'\n\nWhen they eventually reach the Russian border, there is no great welcome, only the 'indifferent expressions, undernourished looks' of the guards:\n\nNever could the idea come to anyone to rush toward them with outstretched hand saying _Brothers!_ for they belonged entirely to a world where words, feelings, fine sentiments shed their prestige immediately on contact with primordial realities. One could only have talked to them about a fire in front of which you could warm up, about shoes to be mended, about flannels to keep your empty stomach warm, about hot soup with which to fill it. I stared intently at these silent men, standing there in such great distress. I thanked them for teaching me about true fraternity, which is neither in sentiments or words, but in shared pain and shared bread. If I had no bread to share with them I must keep silent.\n\nThe novel ends with the narrator remembering a letter he had received from Spain just before the final journey, that he has not yet read. As old books burn in the fire, he recalls 'the city we had not been able to take, our hope, our will, our power, our real power since I was about to go to sleep in a conquered city where...this moment, this shelter, this warmth which allow me to think of you [his former comrades].' He imagines Dario walking in at that moment, to 'shrug his invisible burden off his shoulders', when his eyes alight on one line, 'no different from all the others in the forest of symbols...\"they killed our Dario\".'\n\n* * *\n\nIf _Birth of Our Power_ doesn't always work well as a novel, _Conquered City_ which followed, must be _the_ novel of the revolutionary experience. Again, there is no central story but a series of narratives, different perspectives. Serge's use of montage or jump-cuts can be disorienting, but he makes us _feel_ we are in the unnamed city, Petrograd, at a crucial point for the city, and the revolution for which it stands. The novel takes place in 1919\/20 at a time when revolutionary Russia was threatened by its enemies within, who were backed by the major powers from outside, Britain, the United States, France who were appalled, and profoundly threatened, by this radical challenge to the established order, their order. 1919 saw the creation of socialist republics in Hungary and Bavaria, a workers' uprising in Germany, a general strike in Barcelona, and a general strike in Glasgow which brought tanks on to the streets of the city: 'How they long for our death back there, for the death of this Republic...which is still the greatest hope...the work of those who have always been vanquished, always duped first and then massacred...'. Only yesterday, someone says, 'we were nothing...we counted only as statistics: labor force, emigration, death rate, crime rate, suicide rate.' This is what the revolution challenges.\n\nThe title, of course, is ironic, but sadly so. The city, as the novel shows clearly, is far from conquered: it reflects the growing awareness of one of the narrators who has arrived in the country with great expectations only to see a very different reality at almost every turn:\n\nWe have conquered everything and everything has slipped out of our grasp. We have conquered bread, and there is famine. We have declared peace in a war-weary world, and war has moved into every house. We have proclaimed the liberation of men, and we need prisons...and we are the bringers of dictatorship. We have proclaimed fraternity, but it is the 'fraternity and death' in reality.\n\nEven the snow is not bright, but grey. Everywhere there is a prehistoric gloom because there is no power. Everywhere people are hungry, everywhere the people are cold. They keep their old fur coats on and they rip up floorboards in old houses to keep their fires going. The factory chimneys no longer smoke unless there is some emergency work to be done. People work in semidarkness because there is no power. There is typhus and the threat of cholera. And the spring thaw will, within a few days, make cesspools of the frozen piles of excrement in the courtyards. We feel the cold along with Professor Lytaev, who still teaches a class at the university in the evening as if 'in a city of another time, in the middle of an abandoned monastery'. He has to keep his fur cloak and hat on while his audience listen 'frozen, in their coats'. 'Hard rectangles of night pressed in through the white frost-ferns on the windows.' We experience the constant plotting and threats to the revolution from within: 'Everywhere the enemies of the revolution are active: The Counterrevolutionary Centre Right organization could count on 146 confederates in the city, organized in groups of five, and a thousand sure sympathizers. These forces could be mobilized in a single night.' Chapter 11 starts, 'The 1st Estonian Regiment went over to the enemy on May 24. The 3rd Infantry of the Second Brigade turned traitor on May 28...'\n\nThe revolution is already over-reacting massively, 'spider webs knocked down with ax blows'. An anarchist bomb attack is used as a pretext to execute no fewer than sixty-seven people, 'spies, counterrevolutionaries, foreign agents' but also, 'ex-financiers, ex-high officers, monarchistic professors, vice-den operators, and unlucky adventurers'. The report of their deaths fills two tiny columns in the barely legible newspapers plastered on the walls. At a time when the southern front in the civil war was going badly, 'Sixty-seven? The price in blood of a skirmish...'\n\nAnd all the time, people's hearts and minds are being hardened. When a vote is taken at a Special Commission (Cheka) hearing, only one person, Kirk, dares to vote against the death sentence. The chair, Osipov, dismisses him saying they must be pitiless and share the responsibility. 'You're a Don Quixote', he tells Kirk, 'with your lone horsemen's ways...this affair no longer has any importance. No more than your death or mine would have this week.' So too, Danil, who has nothing but contempt for intellectuals, who haven't 'the least idea of the stench of a sacked town or the look of an open belly full of fat green flies over which poppies droop their heads', dismisses historian Platon Nikolaevich's resort to Dostoyevsky. The Karamazovs, he says, 'split hairs with their beautiful souls; we are carving flesh itself and the beautiful soul doesn't mean a damn thing to us. What is serious is to eat, to sleep, to avoid being killed, and to kill well. There's the truth.'\n\nA 'monstrous state' is arising, devouring its enemies, real and imagined, as happens to many of the people we encounter here. And at the same time, the new administration must also address the ordinary problems of everyday life in a city and Serge vividly conveys the enormity of the tasks facing those who want to build a new order:\n\nOther oases of electricity burning from dusk to dawn: the Committees, Committees of Three, of Five, of Seven, of Nine, the Permanent, Temporary, Special Subaltern, Superior, Supreme Committees deliberating on the problem of nails, on the manufacture of coffins, on the education of preschool children, on the slaughter of starving horses, on the struggle against scurvy, on the intrigues of the anarchists...on road transport...So much thought straining and working every-where in these messy rooms under the same portraits, in that same atmosphere of neglect characteristic of conquered places where people are always rushing in and out.\n\nSerge is also alert to the new bureaucracy that is taking shape and which is wonderfully captured in this passage, with its echoes of the depictions of city life by other modernist writers, notably TS Eliot and Virginia Woolf:\n\nAround ten o'clock the street took on a feeble animation. People suddenly rushed by on urgent, necessary, imperious, deadly tasks. They moved quickly, similar in their diversity - uniforms and black leather - men and women alike, young or ageless, carrying overstuffed briefcases under their arms: dossiers, decrees, transcripts, theses, orders, mandates, absurd plans, grandiose plans, senseless paperwork and the quintessence of will, intelligence, and passion, the precious first drafts of the future, all this traced in little Remington or Underwood characters, all this for the task, for the universe; plus two potato pancakes and a square of black bread for the man carrying these burdens. This was also the hour when those who had accomplished the tasks of the previous night returned homeward, chilly and agitated with oddly wrinkled yellowed faces, yet feeling a final rush of energy mixed with their fatigue.\n\nAnd all the time there is the stunning contrast between the grandeur of the city, built by the modernizing Peter the Great in the early years of the eighteenth century as his new capital, and the misery which now prevails. Grimy red flags hang from the eighteenth century Italianate palaces, 'Nobility and grandeur still showed through the rags and tatters'.\n\nLaundry hangs from dirty windows on the main boulevard, while the chimney pipes of little iron stoves, spitting out their puffs of dirty black smoke, poke through broken windows:\n\nMud-spattered shop fronts, crumbling facades, shopwindows full of bullet-holes and held together with tape, splintered shutters, watchmakers' shopwindows displaying three watches, an old alarm clock, and one fancy pendulum; unspeakable grocery stores; herb teas packaged to look like real tea, as if there were still fools so stupid as to be taken in by these labels, tubes of saccharine, dubious vinegar, tooth powder - brush your teeth carefully, citizens, since you have nothing to use them on...\n\nIn the countryside, source of the grain which the city needs if it is not to starve, murderous divisions are emerging all the time and families are divided against one another:\n\nThe harvest had been brought in the countryside. It was being hidden. Tillers who fought under the red flags with their old scythes buried the wheat and sounded the tocsin at the approach of the Anti-Christ. Others, their sons, with red stars sewn into their old Imperial Army caps, arrived to search their barns. Workers, fearful of being stoned, harangued village elders. They were men caught between hunger, hatred, discipline, faith, war, fraternity, typhus, and ignorance.\n\nAnd in the chaos that prevails, the most ordinary, unexceptional individuals acquire an appalling power, 'a non-com transformed into an ataman had railroad workers thrown into locomotive boilers alive. But, a son of the people, he gave the daughters of his old generals to his exasperated soldiers...'\n\nAt the same time, Serge is alert, as he always is, to the beauty and joy that are present despite the cold, the hunger, the fear: 'Scattered bursts of laughter hung in the woods among the slim white trunks of birches. Specks of dull silver seemed to hang in the air.' (This has echoes of those amazingly condensed poems of the imagists, such as Pound's poem about the Paris Metro.) And after Xenia, the young Party member, has carried out the searches assigned to her one night, she decides to go for a walk. She stops to look at a single white cloud, reflected in the water of the canal, floating 'in the sky of that water' as if above the city:\n\nWhen we are dead, thought Xenia, when everything is finished, perhaps a similar cloud will pass through a similar sky at this very spot. What eyes will see it reflected in this water, eyes that will have known neither war, nor famine, nor fear, nor anguish, nor night patrols, that will not have seen man strike down man? I can't even imagine it.\n\nAnd people still are in love, as are Olga and Arkadi:\n\nShe was happily aware of the red glow of his cigarette in the semidarkness. She loved to move about in the invisible light of eyes following her from the far end of a dimly-lighted room. Nowhere in the world could anyone give this man a greater feeling of calm, a more secure rest, a surer joy. She knew this. And the warmth of his eyes resting on her, soft at first, then imperious like a magnet, enveloped her wholly, imparted new suppleness to her movements. Somewhere deep inside, her whole being cried out that this was an immense happiness...she laughed silently.\n\nAlthough even this most private realm is not totally removed from the political context, as when Xenia worries if she should wear French perfume when she goes to meet Rhyzik, 'Was the use of these luxuries invented by the depravity of the rich not unworthy?...Wouldn't he be angry at this bit of refinement in her?'\n\n* * *\n\nIf _Birth of our Power_ is a novel about the unsuccessful bid for power, and _Conquered City_ the struggle to hold onto power, part of Serge's last novel, _Unforgiving Years_ , is about the revolution gone wrong. D, a long-time Communist, has decided while in Paris that he must, finally, break with the Party. Using the skills he learned as a Comintern agent, he is on the run from his former masters but he is taking so many precautions that he starts to wonder if by the very act of doing so, for example by changing taxis, he is drawing attention to himself. And, in any case, how easily he might have been seen, just by chance, by one of his numerous faceless pursuers. When he is hiding out in a small hotel, everything he hears or sees may be significant and dangerous - a time given for a taxi, or the time of arrival of a telegram may be codes. 'I'm going crazy', he thinks.\n\nHere, Serge was using his own experience as an agent of the Communist International (Comintern), set up by the Bolsheviks in 1919 to promote revolution abroad. He worked with revolutionary organizations in Germany and Austria and reported to the International on the developing situation. (His published reports are collected in the volume, _Witness to the German Revolution.)_. This was, at times, incredibly dangerous, and some pages of the _Memoirs_ dealing with this period read like a thriller.\n\nFoolishly, D has announced his resignation before escaping. This is not formally a capital offense but only because it is simply unthinkable. An unwritten law, Serge writes, dictated the elimination of agents who disobeyed, and disapproval of the regime was the worst form of disobedience since it implied the use of the individual conscience whose existence could not even be brooked, for what it might do to the ruling edifice of iron discipline. The idea that anyone might bow out without betraying, or withdraw into a state of simple insignificance, any superior believing this would be thought a lunatic or an accomplice to be himself eliminated.\n\nEven as he looks forward to his rendezvous with Nadine, he worries that she too is being caught up in an invisible net, precisely because she is to be trusted and because she is bound to him by 'a friendship more definitive than love'. The bare fact of her connection to him puts her in extreme danger.\n\nAnd yet the ordinary life of the city goes on and is beautifully evoked: 'Morning purity of cobbles and asphalt...a dappled light under which one would wish to live for a long time, meditating'. At the same time, there is the appalling complacency of the city in the face of its future, with its 'windows of clockmakers, cobblers, and booksellers, the elaborate food stuffs, the color postcards full of gross jokes and sexual innuendo...in which human beings have attained the maximum possible degree of self-indulgence, and thus the height of freedom, of relaxation'. This city in which people are being hunted cannot see its own future which is so evident, 'A dangerous thing, relaxation...'\n\n# 3  \nBuilding on corpses: the repressive state\n\nNo one grasped the awful reality of repression in the Soviet Union more than Victor Serge did, and no one was better able to show the truth of what was happening than he did. Twenty years before Soviet leader Nikita Khrushchev would admit _some_ of the truth of the Stalinist terror on 25 February 1956, in his speech to the 20th Party Congress, Serge was describing _in detail_ what was going on in this country, a country he was the first to describe as totalitarian. (Although obviously of huge significance, equally important was the fact that Khrushchev's speech was itself given _in secret_. It was only published in full in Russia in 1989.) Serge's writings, such as _Destiny of a Revolution_ and _From Lenin to Stalin_ , also predate by more than 30 years Alexander Solzhenitsyn's monumental _The Gulag Archipelago_ , a book that shocked the west with its horrendous account of the camps. (Solzhenitsyn says in his preface that, when he started to write the book in 1958, 'I knew of no memoirs or works of literature dealing with the camps'.)\n\nSerge was the first person to describe in any detail the vast prison camp system in Russia. It was in 1937 that he wrote that the country had the largest concentration camp system in the world. There were forced-labor camps, dungeons, lumber camps, mines, even secret camps, camps notorious for their brutality or for their hunger, 'filthy corners from which there is no return'. There were camps, such as the SLON, the Solovietsky Special Camp, so vast they occupied 'an entire, vast northern country'. There were even model camps, 'attractive reform colonies for the edification of foreign investigators and movie-goers'. (Serge is here referring to a propaganda film made for foreign consumption.) The location of these, in the most remote and inhospitable areas, was not just to keep them away from the mass of the population. More important, the environment itself was a crucial feature of the punishment, killing many, but demoralizing and dividing those who managed to survive. Serge summed it up well:\n\nWhat cannot happen in a detachment of condemned men lost in the Siberian brush, including bandits, desperate or exasperated peasants, stool-pigeons ready for anything, intellectuals and technicians, harshly treated politicals, all of them bound to a hard task, badly fed, and submitted to the absolute power of a policeman who is himself a condemned man?\n\nSerge also quoted from a brief newspaper report, the message of thanks sent to Stalin by miners in Karaganda to Stalin. Nowhere is it said that this mine is at the centre of a vast concentration camp, that the work is done by prisoners, that perhaps some of the signatories themselves are condemned men, that the Karaganda is 'one of the most dreaded camps for the hunger that reigns there, for its remoteness from all civilization, for the harshness of its inner rule'.\n\nSerge also saw the crucial importance to the Soviet state of internal passports, probably unique in modern times, which served to control, not just the movement of ordinary people as the name suggests, but their everyday lives. Passports were always refused to families of people who were executed or sent to prison for long periods. So anyone without a passport was automatically suspect when looking for housing or work or, simply trying to buy anything, let alone in the eyes of the police. And the passport could always be marked at work to the serious detriment of the holder. Serge cites examples of workers who didn't turn up for days of 'voluntary labor', that is unpaid work to help meet targets, and who were sacked for 'sabotage of the production plan'. There was never any secret about the internal passport, yet it still came as a shock to many in the West to know these existed and what their effect was.\n\nAnd, of course, Serge understood the complete power wielded by Stalin, a power that extended far beyond the borders of the country he ruled. Trotsky was, of course, killed by his agents thousands of miles away in Mexico, but he was only the most famous to be murdered on Stalin's orders. Andres Nin, Serge's friend and comrade from the POUM, was abducted on the streets of Barcelona and never seen again. In September 1937, Serge himself had been due to meet the NKVD agent, Ignace Reiss, who wanted to defect. He failed to keep the rendezvous arranged in Rheims and his bullet-ridden body was later found, with a ticket to Rheims in his pocket.\n\n* * *\n\nAfter the 1917 revolution, the new government found itself in possession of the files of the Tsarist secret police, the Okhrana, and one of the many tasks Serge was entrusted with was to make a study of these. What he and his colleagues found provided an astonishing insight into the workings of a repressive regime. In a secret room in the Okhrana building in Petrograd, accessible only to the chief of police and the officer in charge of the files, they found dossiers on more than 35,000 provocateurs. These Tsarist agents included a man who had been the exiled Lenin's spokesman in the Russian parliament, Malinowsky, and also the head of the Social Revolutionary party's terrorist organization, at a time when it was engaged in carrying out assassinations. (When a Commission was established after the revolution to question former police officials, it turned out that two members were themselves former Tsarist agents.)\n\nSerge also discovered a series of files on himself, even though he had returned to Russia only in 1919, after the fall of the Tsar. The Okhrana clearly had paid great attention to the activities of revolutionaries abroad; when the Russian embassy in Paris was handed over to the Provisional Government, files there showed that 15 Tsarist agents had been working in that city alone and, what's more, that a prominent French journalist, head of the foreign desk at the leading paper, _Le Figaro_ , was also on the Okhrana payroll.\n\nThe files made Serge think about the psychology of the provocateur or informer, what it was that made people betray their supposed comrades and friends. Serge used as an example a woman, Serova, who had informed on revolutionary groups, betraying the location of literature and weapons, as well as individuals, who were then arrested. She was, Serge said, 'a weak character, living in poverty, who works courageously as a party member'. She is arrested. 'Abruptly torn out of her normal existence, she feels lost.' Fearing forced labor or even death, she realizes that she can avoid this fate if she says 'a word, just one word', about someone who actually had done her some wrong. She hesitates. But 'an instant of cowardice is enough; and there is plenty of cowardice in the depths of a human being. The most terrible thing is that from now on, she will no longer be able to turn back...They have her now.'\n\nFar more dangerous for Serge, however, were those who were not cowards, but who believed in nothing and who cared nothing for the cause they served, taken by the idea of danger, intrigue, conspiracy, 'a complicated game in which they can make fools of everyone'.\n\nIn the files, Serge also found what he described as among 'the saddest of human documents'. - receipts for payments involved in the execution of political opponents; the cost of setting up a gallows, the travel, return, of a priest. and, of course, the hangman's traveling expenses.\n\n* * *\n\n_Midnight in the Century_ , written by Serge while himself in exile, is the story of a group of oppositionists, in internal exile in the fictional place, Chernoe. A central part of the story is how they are ensnared in the murderous machinations of the leader. The novel begins with the lecturer, Mikhail Kostrov, having an uneasy feeling that something is about to happen to him. He is arrested in the street and thrown for several weeks into, what Serge calls, 'Chaos', in a hideously over-crowded prison, cut off from the world with no information about the charges against him. ('If the houses of detention are overcrowded', a judge tells him, 'it is not the fault of the proletarian dictatorship but that of the counter-revolution which assails us on every side'.) One day, he gets a parcel from his wife of boiled eggs and, even though it has been broken open by the guards with a dirty knife, he is relieved because he understands the messages implicit in the package; he knows this would not be allowed in serious cases. It also means his wife is still working.\n\nAfter several months detention he is finally questioned and it quickly becomes clear that he has been watched closely. He is confronted with statements he has made in lectures and articles he has written which are taken to be critical of the regime. When he refers to disorders in Uzbekistan he is warned of 'domestic espionage', and even the 'scornful way' he pronounced certain names in private is evidence of his oppositional tendencies. Kostrov is also confronted with a joke against the leader that he has told. About the actress to whom he shared this joke, he is told, 'You sent her far, to a very cold climate'. (Elsewhere, Serge relates the true story of how a seemingly private joke can have the most appalling consequences. Two couples who are out for a drive and are slightly drunk. They get a puncture and one of them makes a joke about an explosion under Stalin's behind. Several months later, the two women fall out and the remark is reported; the man who made the joke gets ten years in prison for terroristic agitation, and the two women five years for failing to report it.)\n\nThe next day, Kostrov decides to give in, 'and wrote out one more surrender' with 'all the right words...the edification of socialism, the great wisdom of the CC [Central committee], the correctness of its tactics, the repudiation of errors due to lack of understanding, to the petty bourgeois spirit, to the counter-revolutionary influence of comrades now denounced and capitulated'. When he finishes he hears himself say, 'Go on, you rotten fraud.'\n\nThe fourth part of the book, 'Directive', is a chilling account of a meeting of the Politburo. Stalin is thinking about how to use the forthcoming party conference to his best advantage. He needs to install 30 new regional secretaries, involving 30 dismissals in disgrace, which threaten 300 influential local secretaries, 3000 less influential local secretaries, 30,000 even less influential local secretaries - none of which will be discussed. The dismissals are necessary to get rid of people who implemented a plan that resulted in two failed harvests. That the plan was his is fatally beside the point.\n\nSerge shows how the directive will have to be carefully worded for maximum political effect, to take account of all the convolutions of the leader's mind; it must be both very obscure and very precise at the same time; it must foresee eventualities, even if these are contradictory; it must command definite actions, while suggesting various others so as to permit effective repudiation of anyone who might carry them out. Those around Stalin understand that the greatest danger was not the visible one, it is the one 'which cannot be unmasked because it does not yet exist in the facts: analysis reveals it lying latent among the masses'. One man's paranoia had become a whole system of thought.\n\nIn charge of the Chernoe exiles is GPU officer Fedossenko. When he receives the directive, it is immediately clear to him what he must do, prepare a case against several of them and, even as he reads the order, 'faces appeared clearly before his eyes, faded, re-appeared...Ryzhik, Elkin, Varvara...Tabidze, Avelii...Kostrov'. Fedossenko 'was satisfied with himself'. In fact, his unfailing intuition had already in effect anticipated the directive. He already has a hold over Varvara, who was sacked over a stolen loaf (it was stolen by the delivery man; Kostrov is implicated in the alleged sabotage of a consignment of school notebooks, (they have a simple misprint in a table), and is suspected of 'duplicity toward the central committee' (Fedossenko has a copy of his statement of repentance). Moreover, a woman with whom Kostrov has become friendly, Maria Ismailova, is an informer.\n\nAt first, Fedossenko tries a friendly approach to Kostrov, 'I'm talking to you as a comrade', telling him he is sure the Party will re-admit him soon. But Fedossenko is up against time, he needs to get his report done in time for it to be useful in the preparations for the party conference. He is also ambitious and wants very much to see his report published in the monthly bulletin of the Security Department. But even Kostrov is resisting, despite the lack of news from his wife and child, the filthy cell he is kept in, and the fact that every day he looks older. Meanwhile, Rodion says he will confess, 'For everything. I'm the one who did everything. Alone, I confess!' This, of course, is completely useless as there has to be a conspiracy.\n\nIn the end, Fedossenko himself is accused of incompetence, of chasing after minor crimes, while a 'secret Committee of Five' was carrying out its activities. A loose association of comrades has become the 'counter-revolutionary Trotskyite centre of Chernoe'. He is also accused of allowing the most dangerous Trotskyist to escape; in truth, Rodion, who managed to get away, is the least political of the group. He has also failed to inform Moscow of the 'active and organized counter-revolutionary Right opportunists in the distribution service of the Public Education Department right in Moscow'. He is relieved of his command, his physical being shrinks before his colleagues and, like a 'puppet or ghost', the 'demolished Fedossenko' is locked in a cell. (Serge was good at showing the rivalries among state officials, in this case how visibly relieved everyone is at the meeting when the blame is being pinned on Fedossenko, 'Six pairs of lungs exhaled the same _ouf'_ and the same thoughts of 'pig-face', big-shot'.)\n\nSerge was also acute about the character types that inhabit the state machine, like the head of security, 'a middling-minded man', who sits slightly back from the table as a sort of self-effacement and whose cropped moustache reminded one that 'he shaved every morning, like an ordinary man, that he probably desired a woman, that he too lived an ordinary life'. And, at the same time, he is all too aware of his great importance, 'the eye and hand of the Party - The hand that searches. The hand that holds the handcuffs. The hand that holds the poison. The hand that holds the revolver in the service of the Revolution'. And if he didn't say this, his whole manner expressed it, 'shadow of the great men over whom he watches day and night, formidable shadow over the subordinates he commands in the name of danger and of safety, deadly shadow over the captives he sends to their fate in the name of a magnificent future'.\n\nOf the accused exiles, only Rodion manages to escape although he nearly drowns in a river and is saved by a wild man, living completely outside society. He eventually gets to a town where he gets a job on a building site, in the 'Socialist Emulation' brigade. He is befriended by a woman who shows him how to carry the most number of bricks, in the most secure way, 'There was no time to breathe, to exchange a few words, or to smoke'. He discovers that they are building a new district HQ for state security.\n\n* * *\n\nSeveral years after _Midnight in the Century_ , in _The Case of ComradeTulayev_, Serge took the political novel to a new level; compelling, deeply disturbing, above all, sophisticated in its understanding of the political maneuvering of a ruthless dictator who rules, not just by fear, but with the acquiescence or collusion of those around him and his subjects. Tulayev, a leading official, is being dropped off one night at his mistress's flat, when an office worker, Kostia, walking home, sees the official car of a man who is obviously important. He hears his driver say, 'Good night Comrade Tulayev' and immediately realizes who he is, the man responsible for the purges in the universities and for mass deportations. He takes out the gun he has just been given (slightly improbably, it might be said) and shoots him, 'The explosion was deafening and brief'. Kostia carries on home through the snow. We do not hear about him again until the end of the book.\n\nSuspicion immediately falls on Tulayev's driver, who is brutally questioned and tortured. The account of this is particularly harrowing. The fact that he has been a good and trusted employee, is not only no help to him, but itself becomes a cause of suspicion when it is discovered that there are testimonials in his file from Nikolai Bukharin, whose driver he once was. Once a leading Bolshevik, Bukharin now stands convicted as a traitor and shot.\n\nThe case is being investigated by Erchov who, before his appointment, had thought himself 'happily forgotten' by the Personnel Service. In a particularly chilling scene, he is appointed by The Chief himself, amid a sea of smiling faces in the Kremlin. 'A heavy responsibility, Comrade Erchov.' he says, 'Bear it well.' It is as if Erchov knows, just as we imagine, his likely fate. Erchov had reached 'the pinnacle of his life', but is now afraid, now he faces three thousand dossiers all calling for the death penalty, 'three thousand nests of hissing vipers'. Erchov comes under increasing pressure to investigate Rublev, a former leading Bolshevik, associate of Bukharin and others now dead, but he knows there is nothing to link him to the case, 'not a shadow of a connection'. What there is rather is a trap for him, Erchov.\n\nEventually, Erchov is ordered to take a rest. Chillingly, his superior quotes Goethe to him, 'Kennst du das Land wo die Zitronen bluhn?' ('Do you know the land where the lemon trees blossom?') And he and his wife, Valia, travel on a special train to a resort reserved for party officials. When the train stops at a deserted station, he is summoned to take an important phone call, but is instead shown the order for his own arrest and stripped of his uniform and his authority. He has become such a professional that he even finds himself thinking that whoever organized this has to be complimented. The train, meantime, goes on with his wife. Assurances to him about her future serve only to underline her certain fate.\n\nThe murder investigation also implicates Makeyev, a peasant who has risen in the ranks of the Party and who was known as an enemy of Tulayev. A true mediocrity, brilliantly described, he had 'learned the official phrases which bring peace to the soul' and was 'exceptionally gifted in the art of forgetting in order to grow greater'. Of the little peasant he once he was, he 'preserved only a rudimentary memory, just enough to make him proud of his transformation'. Violent to his wife, he has nevertheless managed to secure a good post for his mistress. He boasts at one point, 'Men like myself have to have hearts of stone. We build on corpses, but we build'.\n\nSerge brilliantly describes how the investigation almost takes on a life of its own:\n\nThe case ramified in every direction, linked itself to hundreds of others, mingled with them disappeared in them, reemerged like a dangerous little blue flame from under fireblackened ruins. The examiners herded along a motley crowd of prisoners, all exhausted, all desperate, all despairing, all innocent in the old legal meaning of the word, all suspect and guilty in many ways; but it was in vain that the examiners herded them along, the examiners always ended up in some fantastic impasse.\n\nFormer investigator Erchov, meanwhile, has himself become one of the main suspects, and is being held in a timeless hell. He doesn't know whether it has been four weeks or five or six since his arrest and, anyway, what did normal time have to do with 'the fermentation of a brain between the concrete walls of a secret prison in the age of the rebuilding of the world'. (Serge himself once spent 85 days, nearly 3 months, in an inner GPU prison without reading or occupation of any sort, 70 of those were spent in total solitude, 'without even taking the air in the grey courtyard reserved for the more tractable prisoners'.) When he is eventually interrogated, he claims the allegations are 'absolutely insane...sheer madness'. And yet, when The Chief himself questions him, he says 'exactly what Erchov would have said in his place, what Erchov, in his despair, ought to be thinking'. His voice is so like Erchov's own inner voice that 'it restored Erchov to complete lucidity, and even to a sort of assurance'. The Chief asks him for 'The objective truth...'\n\nWith considerable insight, another prisoner tells Erchov that the Party cannot possibly admit that it is 'impotent before a revolver shot fired from no one knows where, perhaps from the depths of the people's soul...' and that The Chief is fully aware that a shot fired at Tulayev is aimed also at him. He tells Erchov he should simply confess. For his part, Makeyev does confess: 'He was loyal, body and soul. Adaptable too and he knew the Central Committee was always right, the Political Bureau always right, The Chief always right - the errors of power compel recognition, become Truth'. But when he signs his statement saying he wishes to confess and cease all resistance to the Party, only the M of his name 'was still strong, the other letters looked crushed'.\n\nThe case becomes the case of Makeyev, Rublev and Erchov. An attempt is made to implicate a fourth man, Ryzhik, an old Trotskyist who is living completely alone but still under intense surveillance in a hamlet of five houses called Dirty Hole, 'at the junction of two icy rivers lost in solitude'. (When writing letters he uses the address, 'the Brink of Nothing'.) Ryzhik would normally have been an immediate suspect because of his 'moral solidarity with the guilty', but he is added to the case only for tactical purposes, those of public presentation, 'to make the case more convincing to foreigners', while the prosecution even discuss allowing him to protest his innocence. Ryzhik, however, refuses even to take part in the interrogation and decides to die by hunger strike. News of his dying causes great panic, 'I order him saved!' The Chief says, as though his will must prevail whatever the reality.\n\nIt is because of his political understanding and experience that Ryzhik is able to decipher 'the hieroglyphics...branded...into the very flesh of the country', in other words, to understand the real meaning of what is happening. But this ability to make sense of all the trials, and the endlessly elaborate conspiracies and machinations they claimed to expose, is also a terrible burden, leaving him with an agonizing feeling of vertigo. Moreover, what he sees is not just the machinations of power but the people behind it all, '...each hieroglyphic was human: a name, a human face with changing expressions, a voice, a portion of living history'.\n\nWhen it becomes clear that The Chief wants a trial that will link Tulayev's killers to the Trotskyists in Spain, attention turns to Kondratiev, a Communist who has returned from Spain and who embodies the last of the genuine revolutionary spirit. Kondratiev has spent years doing the unglamorous work of organizing river transport; 'at a time when abandoned barges rotted along the banks, he harangued crafty and discouraged fisherman in forgotten settlements, got together teams of young men, appointed captains seventeen years old...created a School of River Navigation...'\n\nSerge chillingly describes how Kondratiev's colleagues at work at the Combustible Trust start behaving just slightly oddly to him once word gets around that he has fallen under suspicion in some way. He himself has no inkling. His secretary comes into his office 'too silently', her mute lips outlined in 'too harsh a red', her eyes looked frightened and she does not use the word 'comrade'. Others avoid him; a man, once a prot\u00e9g\u00e9, leaves by a back entrance after a lecture in order not to have to speak with him, a colleague shakes his hand in such a strange fashion that Kondratiev rubs his hands to get rid of the feeling, and he does not offer him a lift as he usually does. Only the young female students who genuinely know nothing sit with him. (In the end, Kondratiev is temporarily reprieved with a post in gold production in Siberia.)\n\nXenia Popov, who is in Paris on Party business, hears about the case and tries desperately to save Rublev's life. 'Grace, grace for Kiril Rublev, grace', she wants to telegraph, but to whom? Only The Chief can possibly save him and he will not get the letter in time, even if he reads it. She sends a message to her father, asking him to intervene, and she calls on Professor Passereau, 'famous in two hemispheres', President of the Congress for the Defence of Culture, and a member of the Moscow Academy of Sciences, asking that he send a message of support for Rublev. While claiming to be more moved than he can express, Passereau proceeds to give every reason he can for doing nothing - that he respects Russian justice, that he met Rublev only once, that his committee meets only once a month, that he has little influence, that they have other cases to pursue, and so on.\n\nXenia is also warned off her activities by a senior embassy official, who accosts her in a cafe, reminding her that what she is doing will have consequences for her father who is still in Russia. She is tricked into returning home, where she is of course arrested, while her parents are put under house arrest and their phone disconnected, their contact with the outside world cut off. A friend, Gordeyev, warns him of the seriousness of possible charges, attempted desertion during a mission, 'activities contrary to the interests of the Union' 'Shivering into himself, Popov became so old that he lost all substance'.\n\nMeanwhile Kostia, Tulayev's actual assassin, is living in the 'Road to the Future' kolkhoz, in love with his girlfriend, Maria, and struggling to meet the production targets set by the Plan. It is only when he comes to see his friend, Romachkin, from whom he obtained the gun, and happens to see the newspaper with its brief item about the confession of three men for espionage, treason and murder and their execution, that he learns the consequences of his action. Romachkin tells him it is too early for justice, 'What we have to do is work, believe in the Party, feel pity. Since we cannot be just, we must feel pity for men...' Romachkin had learned of the case at a party meeting and, after an initial moment of hesitation, had raised his hand in support of the verdict and sentence, along with everyone else.\n\nKostia feels utterly alone. There is no one he can talk to about this. Maria does not want to know and, when she senses the true meaning of what he is saying, tells him, 'I know how much harm is done when the struggle is desperate...there is a great and pure force in you...'. In utter desperation, he writes an anonymous letter of confession. Comrade Fleischman, who is sorting out the files, 'thousands of pages, gathered into several volumes', reads it and, sensing the truth of what it says, finds himself close to tears. He lights the candle used for sealing letters, which is stained with red, and 'In the flame of the bloodstained candle' he burns it and crushes the ash in his hands. He drinks some tea and, 'Half aloud, with as much relief as gloomy sarcasm' announces, 'The Tulayev case is closed.'\n\nSerge based his novel on the actual assassination of Sergei Kirov, a party functionary in Leningrad in December 1934. The assassin was a party member, Leonid Nikolayev, who would be tried in secret and shot before the end of the year. But Stalin used the murder for a brutal suppression of the opposition, described by Serge in a particularly harrowing chapter in _Destiny of a Revolution_ , his detailed account of the state of the country 20 years on. He tells us that 114 prisoners, imprisoned before the shooting on terrorism charges, mostly people who had simply entered the country illegally, were summarily shot after the retroactive application of a decree speeding up executions after sentence. Later that month, 15 former leaders of the Leningrad Opposition were arrested, including Zinoviev and Kamenev, both old Bolsheviks and even associates of Stalin. As they could not be clearly linked to the murder, they were sentenced to long periods in prison for having formed a tendency in the Party. There were also mass deportations from all the major cities. Serge quotes a French technician living in Leningrad who estimated the number at 'close to one hundred thousand...The railway stations were bottled up for two weeks...The unfortunates sold their personal property on the railway platforms and eight days later the State stores could be seen chock-full of second-hand furniture'. As to those deported, this technician who knew several described them as 'very honest collaborators in Soviet technique and science'.\n\nThe novel is a terrifying account of how a state can exercise power over individuals leaving them helpless, using the threat of reprisals against their loved ones: Xenia acts according to her conscience, but in so doing she puts her father and mother at grave risk. And how, seemingly effortlessly, it can weave a web of accusations to suit whatever purpose it has in mind. Also how personally terrifying the leader had become; Kondratiev's secretary takes a call from him in 'terrified reverence', and Erchov, as we saw, has internalized him so much he has lost his own identity. And also how the state implicates thousands of ordinary citizens in its behavior in seeking public demonstrations of support for its actions and, in the process, eliminates the individual conscience.\n\n# 4  \nA stranger to no land: the experience of exile\n\nSerge spent many years of his life in some form of exile, first in Orenburg, near Kazakhstan, 1,000 miles from the Soviet capital, then in Europe and, finally, in Mexico. Not only was he able to understand what this actually meant as a human experience; he was able to capture it in his writing in ways that no one else had done. In _Midnight in the Century_ (1939) and in _The Long Dusk_ (1946), both written in exile, and his poetry, most of which was written in Orenburg, he brought to life the endless fear, uncertainty, loneliness and poverty of exile, but also the incredible solidarity, and love, that was found there.\n\nAlthough those in internal exile had a freedom those in prison did not, the conditions of their lives were generally appalling. They were forced to live in remote, inhospitable places, and were usually ostracized by the local communities. They were expected to survive on even less than the already minimal rations. And they were subject to endless surveillance by the various authorities, and severely restricted in their movements. Their relatives were also subject to constant harassment.\n\nAs mentioned in the last section, _Midnight in the Century_ is the story of five oppositionists in internal exile, their struggles to survive, to keep alive a belief in the possibility of the better kind of world they hoped they were building, and whose exile offers them no protection against forces larger than themselves. The miserable livelihoods which are permitted them are always at the mercy of a minor official's whim. Rodion, who has been working collecting rubbish, is sacked suddenly for no reason. His boss tells him, 'It's not my fault, you understand. I have an order.' This prompts Ryzhik's sardonic observation, 'The Salvageable Rubbish Co-op no longer needs to draw up plans', and he asks if the rubbish is now escaping or if they are simply drowning in it. He is forced to rent just a corner of a room, with a mattress on it. Varvara is blamed for the disappearance of a loaf of bread and is also sacked. (In fact, it was taken by a delivery man.) And when a long-awaited consignment of school notebooks arrive and are found to have a misprint in a multiplication table, Kostrov, who is working in the local education department, is accused of sabotage. Even to take wood out of the river was a serious crime, as it belonged to the State Forest Trust and taking it, as Kurochkin did for firewood, meant 'risking jail or perhaps worse'.\n\nEven doing something as ordinary as buying cigarettes, when a long-awaited consignment arrives at the town, becomes an occasion for them to be harassed and discriminated against. 'They're workers' cigarettes,' they're told, 'The counter-revolution has no right to them.' Others in the shop are glad, as it means there are fewer people to be served and more cigarettes to go round. Elkin is not bothered. 'We'll buy them in the private sector', he quips, 'there are some right here'. (Elkin is mocking Lenin's New Economic Policy, NEP, which allowed a limited form of private enterprise in the economy. and which many believed was a retrograde step.) The 'private sector' in this case is, in fact, 'the hands of a ragged, sunbrowned curly-headed urchin of under twelve'. 'The future of our country', Elkin remarks sardonically. (Elkin works in the State Fishery Trust working on plans for catches, storages and distribution. 'I know how many fish are supposed to be caught in five years,' he says at one point, 'Alas, nobody knows how many will be caught.')\n\nThe exiles decide to stop for a while in the sunlight which streams down on them, on the town, on the food queues and the fuel queues, and on 'the grey newspapers pasted on the wall to proclaim the triumphs of industrialization'. Elkin says, 'One day you'll lie down on a cot in a disheartening darkness. Then remember the sunshine of this moment. The greatest joy on earth, love apart, is sunshine in your veins.'\n\nAnd, there is an ever-present fear of the larger forces against which exile is no defense and which can devour them at any moment. Ryzhik says at one point, 'Listen, brother, I'm uneasy. There are five of us - and not one informer! Do you think that's possible? And if it's like that, what do you think they're preparing for us, those bastards, with their thirty-six thousand dossiers?' What is being prepared for them becomes appallingly clear as the novel progresses.\n\nAs well as being about the terrible conditions of internal exile, this is very much a novel about solidarity. It is this that makes it a book of hope, as well as of tragedy. This solidarity is a bond among the exiles, but also a bond with other exiles and prisoners elsewhere. At one point in the story, Varvara receives a parcel. In it are food, cigarettes and a picture of her daughter, Katia, and also some books, including a copy of _The Arabian Nights_. The package has obviously been unwrapped, then redone by the secret police. Varvara and her friends are sure there is a message contained in it, but they cannot find it. Just as they are about to give up looking, something makes them cut open the back cover of the book: 'From the torn linen emerged, folded lengthwise, the thin slips which Ivanov had covered with microscopic calligraphy in Projects Office No.4 of the SPCC. Special Purpose Concentration Camp, on Kola Peninsula. Avelii could not have felt more joy watching his trained falcon swoop down on a hare in the sweet-smelling grass.' The color drains from Varvara's face, as she looks at these letters, 'words, thoughts, truths for the Revolution. The meaning of our lives, since nothing else is left'.\n\nAnd even in these awful conditions it is still possible for people to love one another; the novel contains one of Serge's most touching depictions of a relationship between a man and a woman in these times:\n\n'Don't bruise me', is all Varvara can think to say to Avelii, 'seeking a tender word for him but not finding one...They didn't go to sleep until they had talked for hours about so many things that it later seemed to them that they had tried to empty their lives in order to mingle them. They would never remember everything. These words exchanged, breath to breath, bodies entwined, hands seeking hands, would always yield new aspects, poignant and revealing. Like clouds scattered by a strong wind into momentary and never-quite graspable shapes.\n\nThe next day, Kostrov takes Varvara's arm in the street. He does so, saying 'How nice it is, Varvara Platonova,' when what he really feels is, 'a kind of gratefulness, as if he had said to her: I thank you for having those bright eyes, this slender neck, for carrying I don't know what joy within you.'\n\nIt is this sense of not being alone, of being part of something bigger than them, which gives them hope and makes it possible for them to live in this darkest hour, at midnight. In the same way, Elkin is elated when he was being transferred, to be allowed by his guard, 'a rather decent bugger', to go and look at an old stall selling all sorts of bits and pieces. There he finds an old copy of a book by Trotsky, the author's name scraped off the cover, 'They turned the first pages together smiling.' And when Rodion escapes and reaches the house of Galia, although she is disappointed that he is _not_ Elkin, as she first imagines when she sees a figure in the bushes, she gives him everything she has, bread, onions, dried fish, a green apple, matches, some money, 'She filled his pockets, happy to touch him'. For his part, he felt 'overwhelmed by a happiness he did not yet deserve...'\n\nAnd once again we see that active acceptance of a given situation that is so often a feature of Serge's characters. When they are sitting in the sun with their cigarettes bought in the private sector, Rodion asks Elkin about the place of thought in these terrible times. Elkin replies that, 'it's something of a midnight sun piercing the skull' and wonders aloud, 'What's to be done if it's midnight in the century?' Rodion replies, 'Midnight's where we have to live then.' and, as he says it, he feels 'an odd elation.' To embrace reality, however awful, can bring a feeling of freedom.\n\nSolidarity, for Serge, was not some romanticized, idealistic notion. It was very much a reality and something that he experienced, again and again, throughout his life. Recalling some of the people he met while in exile in Orenburg, most of them probably dead, he says, 'I am grateful to them for having existed, and because they incarnated an epoch'. He and his family had been able to leave the Soviet Union because of the campaign waged on their behalf by friends and supporters. A 'miracle of solidarity', Serge himself described it, but he was only too aware that others had not been so fortunate and would not be: 'It is humiliating to think that a certain sort of literary solidarity worked in my favor which won't work for others, simple and great revolutionaries without inkwells...No writers' congresses are likely to want to know about them'. And, of course, Serge used his freedom to work in solidarity, on behalf of those left behind, trying to tell people the truth about what was really happening there. He campaigned against the trials, setting up an international committee of enquiry whose members included the surrealist Andre Breton, the writer and militant Magdeleine Paz, and many others. (Serge insisted on the long title of 'Committee for Inquiry into the Moscow Trials and the Defense of Free Opinion in the Revolution' on the grounds that the revolution was not confined to the Soviet Union nor was Stalin's repression limited to there.)\n\nAnd, later, in Marseilles and sliding, despite himself, into a state of despair as he waited for a place of refuge which he thought might never materialize, letters from the writer and editor, Dwight Macdonald, in New York, and from the poet, J-P Sansom, in Switzerland, 'seem to clasp my hands in the dark' and give hope. These were two men that Serge never met. And Serge and his family were able to escape from Europe because of these and many other people who sent him money, gave him work, and worked to get him papers.\n\n* * *\n\nIt was while he was in internal exile in Orenburg, near Kazakhstan, that Serge wrote most of his poetry. The manuscript was one of the things stolen from him as he left the Soviet Union and he had to recompose the poems from memory. They were published as a volume, _Resistance_ in 1938, with an English translation by James Brook in 1972. In his introduction to the English edition, Richard Greeman finds echoes of many of the most interesting French modernist poets including Mallarme, Rimbaud, Appollinaire and Verlaine, along with many Russian writers with whose work he was well-acquainted. Serge was, he says, 'receptive to every mode of poetic intervention'. He was also, as we shall see in the next chapter, an insightful reader of his Russian contemporaries. Serge had loved poetry since he was a young man. It struck him, he said in his _Memoirs_ , as 'a substitute for prayer...so greatly did it uplift us and answer our constant need for exaltation'. The young Serge particularly loved the work of Paul Verhaeren, a precursor of Walt Whitman, for his 'anguished thoughts' on modern life. In his own poetry we see the same forces and values at work as in his other writings.\n\nLike his fictional exiles, Serge in exile was still able to appreciate the natural wonder of the place he found himself in, despite the poverty and harshness of daily existence:\n\nVast, vast horizons, pure, distant, and light\n\nsoft grass, under the hot shimmering air,\n\nvast, vast sky, forgotten blinding sky, impossible to look at.\n\n('On the Ural river')\n\nAnd watching a group of girls crossing the river, laughing and shrieking, he is moved by their delight, although he has to accept that it is, in the end, a simple one:\n\nWhat party, what love, what desire, what pleasure are they talking\n\nabout to have that tinkling bell-like laughter?\n\nProbably none, they're laughing just\n\nbecause it's a nice day...\n\nAnd he is too much of a realist to be caught up in any fantasy:\n\nI know that they will not have their promised joy,\n\nhappiness is not on the other side of the river,\n\nthe other face of the world will stay closed to them...\n\nNevertheless he carries this vision with him:\n\nThey are on the other shore, four real girls,\n\nfrom my village of exile\n\nand their image has not failed in me. ('Four girls')\n\nExile was also a place for remembering, in particular, comrades from the past who are no longer alive, like Andre in Riga and Dario in Spain:\n\nO rain of stars in the darkness\n\nconstellation of dead brothers!\n\nI owe you my blackest silence...('Constellation of dead brothers')\n\nAnd he recalls, in two particularly telling lines, the personal sacrifices revolutionaries like him have had to make, unable even to live their own lives:\n\nWe have never been what we are,\n\nthe faces of our lives are not our own...('Confessions' section of 'History of Russia')\n\nIt was while he was in exile that he learned of the death of his friend, the Romanian writer, Panait Istrati. It was Istrati who had written the original preface to _Men in Prison_ and who had published under his own name a book by Serge. Istrati had returned to Romania where he died from TB. Serge wrote a deeply-felt and moving tribute to him, which is partly an incantation around the word 'Finished':\n\nFinished: the romances, dark lips and golden eyes...\n\nFinished: the paprika dishes and the slightly rough red wine shared with beggars while swapping tall tales...\n\nFinished: the books written...\n\nFinished: the insults.\n\nThey did not stint there.\n\nThe poem ends:\n\nI am listening in your stead,\n\nwhat radiant silence falls on the clamor.\n\n* * *\n\nSerge's long, panoramic novel, _The Long Dusk_ is the story of a different kind of exile, this time about a disparate group of refugees who think they have found refuge in Paris, but are forced to flee once again once when German occupation is imminent. Simon Ardatov, a Russian physician, banned from working in France as a doctor, is forced to make a living of sorts working for a scientific cuttings agency. A man who had 'stood under the gallows without trembling', he is now in fear of his landlady demanding the rent, and his brown suit is presentable only in certain, inelegant areas of the city. Pepe Ortiga, a young refugee from the Spanish civil war, had witnessed the 'senseless massacre' at Teruel, and seen 'in a week more bodies than a man should see in a lifetime'; his smoldering black eyes looked out with insolence at the world. And Moritz Silber(stein), a minor wheeler dealer, his affability was intolerable, took his rent payments lightly.\n\nThe novel brilliantly conveys the moods of fear and panic, the unbearable tension, as people await the inevitable French capitulation, and yet hope it somehow doesn't happen. Shops start to close up and overloaded cars start to leave the city. Some are driven to suicide. Others wrap up important or valuable books and bury them. There are some who carry on as though nothing has happened. And there are always those who welcome the new rulers. 'Well, the medicine's been swallowed. It feels better now, doesn't it?' says one woman. The Jewish area of the city has become an 'asylum for the wretched of all nations'.\n\nThe swift, and violent, Germanization of the city is well-evoked:\n\nSeveral cities, several different and mutually hostile lives were super-imposed, one upon the other. There were the military staffs, the pleasures of the military staffs, the whisperings of government intrigues, the surveillances...There was espionage, counter-espionage, commissions, sub-commissions, inspections, secret police in factories, banks, offices, railroad stations...reports, memoranda, dossiers, classifications, planning, Order. There were perilous clandestine traffickings, messages to prisoners, liberations at a price, mail to the other Zone at ten francs a letter...Trains rolled eastwards, loaded with machinery and raw materials, luxury articles, Normandy apples, potatoes, furloughed soldiers invigorated by the Paris nights - and in the somber blue light of camouflaged stations, trains full of severe casualties disgorged their loads of the burned and torn, the maimed and the blinded.\n\nAnd when fuel merchant, Augustin Charras, hero of the First World War, goes for a stroll around the city he loves, which is 'all clean, in mournful Sunday dress', he sees all the signs of the occupation - the swastikas flying from official buildings, the Nazi salutes, soldiers having their photos taken beside the tourist attractions. Chaurras, who 'always seemed to come from far away when spoken to', fears more and more for the safety of his teenage daughter, Angele, whom he sends south. And when he puts himself in danger for hiding a deserter, decides that he too must leave the city.\n\nOrtiga and Ardatov, 'experts in defeat', know the best ways out of the city, the safer routes to the south, but they can do nothing about the new threat to populations on the move, attacks from the air. The highway to Moulins was filled with 'a human river' as all across the country, from 'wounded Champagne and the orchards of Normandy', the roads of France are given over to this movement of human ants heading for escape:\n\nToo many autos riddled with bullets, too many brains still filled with the nightmare of columns of refugees machine-gunned by enormous low-flying planes grazing the treetops, too many families carrying away a cold little body under a blanket...In the silent night, everyone seemed to hear the droning of distant planes that surely were flying toward this road, capable of spotting these trucks, prepared to let loose meteors of horror.\n\nWhen they eventually reach the south, there is no safety but only a temporary haven; the exiles have exchanged one form of fear for another and they face the seemingly impossible task of securing papers to leave the country altogether. Serge brilliantly conveys the sheer desperation of people who do not have the right documents, who get over one hurdle just to be faced with another. Leaving the country was never just a question of getting one visa; a whole set of documents had to be put in place, like a suit of cards. One country might well grant a visa, but only to nationals of certain countries; others might grant visas but only for transit and only on proof of onward travel, as well as a visa in place for the final destination. And, of course, papers, authentic or forged, cost money and were usually valid for a short time only: 'The word _visa_ could make asthmatics breathe again, relieve sufferers from heart-trouble, cure neuroses, dispel the temptation of suicide; it reigned over condemned horizons as _mirage_ reigns over a desert strewn with bones - but this word also wrought devastation, giving rise to diseases of the personality hitherto unknown to psychiatry.'\n\nIn this situation, people will go virtually anywhere that will take them, even to countries they haven't heard of. Gaetani, one of the refugees, says, 'I remember the days when we carried a map of the [Paris] Metro around with us. Today, it's the planisphere. Our vision is broadening.'\n\nThere are people of all nationalities, from all possible backgrounds, 'Jews of a hundred nations, the last republicans of strangled republics, the last socialists of banned parties, the last revolutionaries of defeated revolutions, the last liberals of conservative democracies...the last parliamentarians of discredited parliaments...'\n\nAfter 30 days waiting, a woman, clutching her handbag, rushes across the square with enough of her wits intact to demand a lethal dose of veronal. Others have not managed to keep their wits intact, like the 'bearded young American, born in Hungary', who kept 'stamping frantically and shouting in cadence: _Kon-sul ame-ri-cain! Kon-sul ame-ri-cain!'_ He is dragged off somewhere, 'No way to shut him up, you couldn't hit an American'.\n\nArdatov hears of the German attack on Russia and knows that he is now 'doubly hunted', as a Russian and as a communist. He eventually gets French papers, but, knowing he is almost certainly being watched, avoids seeing friends so as not to put anyone remotely at risk, 'Isolation thickened around him.' He had prepared his departure 'as meticulously as a Chinese artisan chisels an ivory charm'. He says goodbye only to Chaurras, who will not leave his country even if he could. Meanwhile, Moritz Silber, has used all his skills to get papers in the name of Silver, a Lithuanian Catholic, and hopes to get to Ecuador. But the possession of papers is no protection against the individuals who have taken it on themselves to inform on people. In this case, he falls victim to the deeply anti-semitic Vibert, the one-time bar-owner. 'Crafty in the performance of duty', Vibert has a fearsome intuition for people who are hiding something and relishes the fact that people called him Viper. Sensing Silber's change of mood to one of hope, once he has secured his false papers, he strikes to deadly effect.\n\nOrtiga finds himself arrested yet again. 'Modern man goes to prison as he takes a train or bus,' he thinks. It's not a problem as long as there is the chance of escape. In prison, he befriends the wonderfully named, Nihil Cervantes, an anarchist who long ago forgot his real name, who offers to share his straw mattress with him. Cervantes tells him they will be sent to build a railway across the Sahara and that, already, he has a plan of escape and invites him to join: 'I know how to say \"I am a man friendly to men\" in all the languages of the desert and since it's the truth it will get us through wherever we go'. When Ortiga gets a small parcel from Angele, including a tin of sardines, Cervantes takes nothing but some of the oil on a piece of bread, leaving half a sardine for each of the weakest prisoners. (Even though he is faint with hunger, Ortiga wishes he could keep the tin intact because the name, 'Angele', has been scratched on it with a knife.) Ortiga's heroic efforts to keep the malnourished Cervantes alive come to nothing.\n\nSerge brilliantly conveys the endless anxiety of those who have to live in this time. An act as simple or ordinary as opening a door is laden with fear, 'Every time I open the door' Angele says, 'I know that bad news is coming. When someone goes out you can't tell if he's coming back. Or if you open the door, you never know who or what. 'Apart from anything else, being in possession of false papers was a very serious offense. In this time and in this place nothing is certain, 'The planks of safety over the abyss of perdition' seemed perpetually on the point of collapsing. At one point, Moritz Silber sees the trucks loaded with a 'human potpourri', the result of raiding parties:\n\nThe - Jewish - director of an art theatre in Berlin; the - Jewish correspondent of Amsterdam's (defunct) leading daily;... the pretty little Catalan girl, who had also escaped - but for love; the aging, crockety German intellectual woman with an expired residence permit; the Tunisian with no papers at all...; the Viennese - Jewish - psychoanalyst whom this absurd arrest might kill, because his papers would never be in order, because his American visa would expire in a week; the lame, painted widow of a pre-Hitlerian German playwright, looking like a figure in a _danse macabre;_ the Italian Freemason who had come to Marseille without a safe-conduct and was wanted by the Armistice Commission...\n\nAnd, then at the same time, something wonderful could happen, 'At the very moment of despair a letter arrived bearing stamps from the other world...'\n\nThe novel ends with the start of armed resistance in the area, a resistance which Augustin Charras has decided to join. Serge closes with, 'The end - (but nothing is ended)'.\n\n# 5  \nMagicians of word and thought: the  \nRussian cultural revolution\n\nStupefied, that was the word Serge used to describe his response to the literary revolution that followed the political revolution, 'this glittering debut of Soviet literature...the audacity and the candor of the writers under a regime barely emerged from the terror'. Small circles of 'young men with hollow cheeks', he wrote, 'wearing the grey greatcoat of the Red Army came together in the evening in very cold (but gold paneled) rooms around poets and prose writers such as Gumilev and Zamyatin, who stoically taught them the art of writing'. This was the time, Serge reminded his readers, 'of the barbed wire fence around Russia, the death of the weak, the conspiracies, the allied intervention, and the war where no prisoners were taken...This time left us no books. And yet, 'entire galaxies of young writers' sprang up in Russia nevertheless, 'new men shaped by the storm'.\n\nThis 'literary renaissance' was the time of novelists like Andrei Bely, Mikhail Bulgakov, Ilya Ehrenburg, Vassily Grossman, Boris Pilnyak, Yevgeny Zamyatin, and Serge himself, as well as the short story writers, Isaac Babel, Andrei Platonov, Varlam Shalamov and Mikhail Zoschenko, and of poets like Anna Akhmatova, Alexander Blok, Osip Mandelstam, Vladimir Mayakovsky, Boris Pasternak and Sergei Yesenin. Also breathtaking was the the revolution in other art forms - painting, architecture, theatre, music and, the still relatively new form of cinema, the men and women involved in these producing the finest work of their lives, people like El Lissitsky, Kasimir Malevich, Lyubov Popova, Alexander Rodchenko and Alexander Tatlin; in theatre, Vsevolod Meyerhold and Constantin Stanislavsky; and in cinema, Alexander Dovzhenko, Sergei Eisenstein and Dziga Vertov.\n\nSerge wrote a great deal about the writers of the time; his articles, mainly for French leftist journals, constitute a sort of literary history of the period, and he was, often, perceptive and generous. Of Mayakovsky, who is probably today the best known poet of the time, Serge was particularly insightful. He acknowledged his energy and commitment: Mayakovsky did what he said, 'helping the Republic out of the mud, and working as a living poet on the street'. (At demonstrations he would emerge from the crowd to declaim slogans - one of them was 'Lion of Britain, whine!' - some of these poems were 'little master-pieces'.) And in the 'heroic years', he agitated, designed posters, wrote a great work, 'One Hundred and Fifty Million', a unique work, Serge commented, in its 'originality, breadth and power of expression'. And yet even this revealed, in every line, what Serge called 'the internal tragedy of Mayakovsky the Futurist', among other things an overwheening ego; maybe 100 lines start with 'me'. What one was seeing in Mayakovsky's life and work was 'the drama of a poet who could not, in spite of his keen desire to do so, escape from the past'. The fatal flaw of futurism, the artistic doctrine to which Mayakovsky subscribed, was that, despite its name, it was 'tragically linked to the past, in its naive pretense of avoiding it'.\n\nThe symbolist poet, Alexander Blok, was Serge said, one of the three or four great lyric poets whom 'nature gives to a privileged people in a hundred years'. He was 'the first, the most admired and the most loved of the master musicians of the Russian language'. Serge praised, in particular, two poems, 'The Twelve' and 'The Scythians', for their understanding of, and support for the revolution. And when he died at only 41, Serge's grief was palpable at the loss of 'one of the magicians of word and thought'.\n\nSo too when Andrei Bely was 'devoured' at the age of 53 (in 1934), Serge spoke of him as 'a musician, seeking to give words a sound which were pure, rich and precise'; he possessed all 'the knowledge of the day without it having become a dead letter for him'. Bely's masterpiece, _Petersburg_ , published just before the revolution, is now widely regarded as one of the founding works of literary modernism, comparable to Joyce's _Ulysses_. It was very influential on Serge's own writing, especially _Conquered City_. Serge said of his legacy: 'When they have conferred, by the reorganization of everything, a new value on life and on living beings, men will feel quite different and much less powerless in the face of death. But even at this boundary, there is no other hope. Bely knew it better than most of us.'\n\nOf the popular novelist Boris Pilnyak, Serge wrote a long appreciation, which acknowledged his gifts - his originality of talent, his dynamic view of things, the breadth of his outlook and so on - but concluded that the reader was saddened by not finding in him 'anything more than intuitive insight and primitive admiration'.\n\nIn the theatre, Serge acknowledged the genius of Vsevolod Meyerhold. Even the propaganda plays he put on, like Mayakovsky's _The Bedbug_ , were 'the only great ones', because he still managed to imbue them with 'so frenzied a spirit', that they became something else.\n\nBut, of course, this explosion of creativity could not last. There had always been strong currents of philistinism opposed to what these men and women were trying to do, as well as narrow-minded views of what art should be about, especially in a period of revolution These merged in the figure of Stalin who was philistine to his core, denounced anything innovative as 'formalism', and regarded creative artists as tools of the revolution; writers he once famously remarked were 'engineers of the human soul'. For Serge, the key point of the official suffocation of creativity was the year 1929. 'There is always an hour,' he wrote, 'when the redeeming choice between cowardice and choice is possible. It was in 1929 that the Soviet writers abdicated their dignity.' Creativity was replaced by 'sterility, of spiritless official propaganda, of stereotypes approved by the bureaus'.\n\nTypically, Serge himself only _started_ writing fiction at this point, when he was in his late 30s, after he came close to death in 1928, from an intestinal occlusion. It was also, significantly, the year he was expelled from the Party, which was a different kind of death sentence. Until then, writing of this kind had seemed secondary, but he became increasingly aware of the limitations of historical work which did not permit showing men as they really were, their inner workings and 'penetrating deep into their souls'.\n\nActual death was the fate of many writers. Isaac Babel, best known for his stories from the civil war, Boris Pilnyak and Meyerhold were all shot. Yesenin and Mayakovsky were driven to suicide, as was the poet, Andrei Sobol. Osip Mandelstam found himself in a camp because of a poem he had written about Stalin which he had shared with a few people only, one of whom betrayed him. He died as a result of his time in prison. The student, Varlam Shalamov, who was involved in distributing Lenin's last testament which had become an illegal document, survived a total of 17 years in prison camps, where he composed the truly shocking stories which would be collected as _Kolyma Tales_ , but they were never published in the Soviet Union during his life time.\n\nThe price of survival, albeit temporary, for others was public humiliation, as it was for Pilnyak who was forced to rewrite his novel, _Mahogany_. Officially criticized for its 'pessimistic and counterrevolutionary' portrayal of provincial life he had to make it more optimistic, although even this did not save his life, but only deferred his execution. And when he learned in 1937 of Pilnyak's disappearance, Serge remarked that it was not possible to understand from the outside the terrible pressures of a totalitarian regime. One could not, therefore, judge someone because of their 'small retreats, the little acts of pusillanimity' they engaged in to preserve themselves. (As well as being a writer, Pilnyak had been an important publisher, bringing out works by Bely, Babel, Ehrenburg, Yesenin, Mayakovsky, Mandelstam, Pasternak, Zamyatin, and many others.)\n\nAnd one hears the sadness when Serge writes in 1937 about the fate of the poet Bezymensky, 'in the flood of terrifying news which comes incessantly to us from Moscow'. One of the most remarkable brains of our younger generation, Serge recalled how ten years before, he had always 'joyfully made his way' to his 'miserable dwelling'. He adapted and maneuvered and became a sort of laureate, even going so far as to praise the execution of Marshal Tukachevsky, the leading Soviet general. But not even this could save him.\n\nYevgeny Zamyatin was the only writer besides Serge who was allowed to leave the Soviet Union. Ostensibly permitted to go to Paris for medical treatment, he never returned and died there in 1937. Much influenced by HG Wells, of whom he was a great admirer, his novel, _We_ , is an account of a totally controlled, totally administered world, 'the One State', where people have no names, but numbers, and their lives are completely ruled by a principle of rationality. It's a parody of Taylorism, the principle of 'scientific management' (much admired by Lenin): 'such precise beauty: not one superfluous gesture, deviation, turn. Yes, this Taylor was, beyond doubt, the greatest genius the ancients had'. The book ends with the words 'For rationality must conquer.' _We_ , which was translated into English in 1924, was itself a huge influence on Aldous Huxley's _Brave New World_ , and especially on George Orwell.\n\nOthers were condemned to artistic silence. Mikhail Bulgakov even wrote to Stalin personally in 1929 asking to be allowed to leave the country if he wasn't allowed to work; he was given some dull work in the theatre as a result. At the same time, he was writing his greatest work, _The Master and Margarita_. Now universally acclaimed as a masterpiece, in advance of its time, and hugely influential, it was published only in 1966, six years after Bulgakov's death, and even then could only come out abroad. (One of its most quoted lines is 'Manuscripts don't burn'. Bulgakov himself had burned a version of the book in fear of the consequences; there was no Professor Woland to return it to him with those words, as he does for the Master.)\n\nSo too, around this time, Sigizmund Krzhizanovsky was writing _The Letter Killers Club_ , but this too was never published in the author's lifetime; few of his writings were. The letter killers are a secret group of 'conceivers' who meet on a Saturday and who, to preserve their conceptions, commit nothing to paper, 'if writers prevent each other from writing, they don't allow readers even to form an idea...libraries have crushed the reader's imagination'. The group meet in a room with empty bookshelves. Krzhizanovsky was not writing a political allegory but was aiming at something other than that, something metaphysical, and yet he cannot escape the brutal context of his creation, any more than his club can. The book ends, 'The police may pay a visit. Let them: no one searching emptiness has ever managed to find anything'.\n\nAndrei Platonov chose not to write fiction in the early years of the revolution, believing he could serve it better through his practical skill as an electrical engineer. (His first publication was a pamphlet extolling the values of electrification, which would change the nature of work and the nature of humanity.) He started writing fiction in the late 1920s but few of his writings were published. It is only relatively recently that we are able to read him, this man who, according to John Berger, understood 'living modern poverty more deeply than any other storyteller I have come across', a poverty different from others because its 'desolation contained shattered hopes'.\n\nVassily Grossman, now rightfully acclaimed as a great chronicler of his times, was able to publish some stories in the 1930s, but his monumental masterwork, _Life and Fate_ , was seized in 1959 and only published in Switzerland in 1980. Grossman, who had died in 1964, was told his novel could not be published for two or three hundred years. Ilya Ehrenburg abandoned his innovative writing, such as the truly-groundbreaking, _The Life of theAutomobile_ (1929), and wrote work which was officially approved of, and even got him a Stalin Prize.\n\nSerge himself was told in 1928 by the director of literary publications, Ilya Yenov, a one-time friend, 'Even if you produce a masterpiece every year, not a line of yours will appear!'. The Russian translation of _Men in Prison_ , officially approved, set up to be printed in 10,000 copies, was destroyed. Serge was saved by the fact that he wrote in French and could seek publication abroad, but even this was problematic as he could never be sure that his manuscripts would teach their destinations. He developed the practice of sending detached chapters to France that could be published together, even if some did not arrive!\n\nThe great poet Anna Akhmatova was also forced into a kind of silence. Her former husband, Nikolai Gumilev, had been implicated in an anti-Bolshevik plot in 1921 and shot along with 61 others and, even though they had been divorced for three years, this seemed to taint her in the eyes of the authorities. Her later husband, Nikolai Punin, died after many years in the camps; her son was also harassed and imprisoned. She herself would be publicly insulted by Stalin's cultural commissar, Andrei Zhdanov. Although she was unable to publish, she never stopped writing, and her poem sequence, 'Requiem', spoke for many, 'one hundred million voices cry' through her 'tormented mouth'. Akhmatova recalled how, once in a Leningrad prison queue, she was approached by a woman with blue lips who, in a whisper, asked if she would be able to describe 'this', and when she said yes, 'something like the shadow of a smile crossed what had once been her face'.\n\nIf the greatest minds were prevented from publishing, the Soviet state, Serge caustically remarked, could make 'almost worldwide reputations' in a few days, for people prepared to do as they were asked, using publicity methods 'borrowed from the American trusts'. The order just had be given to all the sections of the Communist International to have their publishing houses translate a seventh rate work; the entire Communist and Communist-inspired press will proclaim its merits.\n\nIn the field of the visual arts, the revolution was, if anything, even more astonishing in its creativity than in literature. The artists Alexander Rodchenko and El Lissitsky were Serge's contemporaries, as was Vladimir Tatlin. Kazimir Malevich and Lyubov Popova were of an older generation, but threw their energies behind the revolution and were far in advance of their time artistically. It was in 1918 that Malevich produced 'Suprematist Composition: White on White', as revolutionary in its own way as much that was happening around him, and which anticipated so-called color field painting by almost four decades.\n\nThis artistic freedom was not limited to individual artists but was extended to the whole field of art education. Vkhutemas, the Higher Art and Technical Studios, was the state sponsored school of art set up in 1920, 'to prepare master artists of the highest qualifications for industry, and builders and managers for professional-technical education'. It was similar in conception to the much better known Bauhaus, set up by Walter Gropius in Weimar in 1919 and forced to close in 1933, many of its staff having gone into exile and the place itself being attacked as degenerate by the Nazis. Like the Bauhaus, Vkhutemas sought to merge the craft tradition with the most modern technology. The constructivist, Alexander Rodchenko, was dean of the metal work faculty. The cubo-futurist painter, Lyubov Popova, taught the use of color. Malevich, advocate of what he called suprematism, taught there, joining the staff in 1925, as did Lissitsky. The textile department was run by the constructivist designer, Varvara Stepanova. At one point it had a teaching staff of 100 and more than 2,500 students. Svomas, free state art studios, were set up in several Russian cities to spread awareness of and competence in the arts to workers and peasants. Entrance examinations were abolished, art history courses were optional, the faculty was replaced by avant-garde artists, and students were free to choose their professors.\n\nTatlin's monument to the Third International, for which Victor Serge would work in a few years time, was designed at Vkhutemas; never built, perhaps unbuildable, wonderfully grandiose, the dream of a time that refused to be constrained by reality. 'Made of glass, iron and Revolution', Viktor Shklovsky the literary theorist said, while for Mayakovsky it was 'the first monument without a beard'. Taller than the Eiffel Tower, and towering over Petrograd, it would have housed a conference centre, a radio station, a cinema, its different sections revolving at different speeds, the proposal has become an icon and highly influential. (Hints of it can even be seen in Val Myer's Broadcasting House, 1932, in central London.)\n\nConstructivists, like Rodchenko, wanted to make objects that were useful, as well as aesthetically pleasing, from cups and plates and cigarette packets to furniture and social clubs for workers, things that would contribute to the new way of life being constructed. Others, like Lissitsky and Malevich, were more interested in pushing the boundaries of what was aesthetically possible, although early on in his career Lissitsky designed the first post-revolutionary Russian flag and he is also, of course, the creator of the well-known propaganda poster, 'Beat the whites with the red wedge', hugely influential and much replicated, even today.\n\nThis was also a period of cross-fertilization in the arts, of breaking down boundaries between different forms. Lyubov Popova, the painter, worked in the theatre with Meyerhold, designing sets for his productions, before her premature death in 1924. Ivan Kudriashev, sent by Malevich to Orenburg in 1919 to set up a branch of Svomas, got involved in doing decorative work for the First Soviet Theater there. Rodchenko and Stapanova worked in theatre too, and Rodchenko also did work for the cinema with Dziga Vertov. He and Lissitsky also designed Mayakovsky's books, and their work influenced generations of book and magazine designers.\n\nThere is an awe-inspiring energy about so much of the work in this period, a seemingly endless creativity, astonishing fertility, that, even today, many of the works from this time still seem ready to burst out of their frames, off the paper. One of Lissitsky's greatest compositions, 'Untitled (Rosa Luxemburg)', dedicated to the German revolutionary, murdered in 1919 by paramilitaries, and her body thrown into a canal. is small, and executed with just pencil, brown paper and some gouache. Karel Ioganson's 'Construction', the polar opposite of Tatlin's tower in its quietness and scale, too is just a sheet of paper marked with some ink and colored pencils, while Varvara Bubnova's 'Untitled' of 1920-21, is just ink and brown paper. How much more creative is it possible to be, one wonders, with these materials, at this time?\n\nOf course, it's impossible for us to look at this work now and not be aware that's its time was very short, that by the end of little more than a decade it would all be over. Maybe this is part of what is moving, this brief flowering, this sense of people seizing an opportunity of real freedom, that they know won't come again, to do something useful to the revolution, however they saw it, but without compromising. And of course, it is deeply dismaying to learn that Lissitsky and Rodchenko both ended up, in effect, as propagandists for the regime. Both worked on the official magazine _USSR in Construction_ , published monthly throughout the years of the terror and whose main aim was to promote a favorable image of the USSR abroad. Rodchenko even took pictures of slave workers on the White Sea Canal project, a huge propaganda effort to show both the success of the first Five Year Plan, and the use of prison labor as a reforming activity.\n\nAs for music, Igor Stravinsky, one of the most important composers of the twentieth century, had left Russia some years before the revolution and was hostile to it when it took place, but he retained a strong artistic connection to the country. He composed some of his most important music of the period for Sergei Diaghilev's Ballets Russes, whose dancers included the legendary Nijinsky and Pavlova, notably _The Firebird_ (1910), _Petroushka_ (1911) and _The Rite of Spring_ (1913), which famously provoked a riot when first performed in Paris. He once quipped that music was 'new just _before_ the Soviets', meaning he was the true revolutionary, but this ignored the work of Nikolai Roslavets, a modernist composer who was persecuted for his artistic beliefs, not to mention Dmitri Shostakovich, who in the most challenging personal circumstances, produced a body of work unequalled in its creativity, poetry and humanity. (Fifty years after the revolution, Shostakovich recalled the revolutionary poet, Alexander Blok, in his song cycle, 'Seven Romances'.) It was during this period, too, that Lev Theremin produced the electronic instrument that bears his name today.\n\nIn cinema, Sergei Eisenstein's _The Battleship Potemkin_ (1925) was hugely innovative and much admired and, indeed, referenced, especially its 'Odessa steps' sequence. Dziga Vertov's _Man with a Movie Camera_ (1929), is even more remarkable in its creativity. A documentary without sound about the lives of ordinary people in the Ukraine and elsewhere, the film is celebrated for its use of many technical innovations and it was much influenced by the photomontage techniques being developed by Rodchenko and Lissitsky. So too was Vsevolod Pudovkin's groundbreaking film, _Mother_ (1928), based on Maxim Gorky's novel of the same name, about a woman's struggle for justice in pre-revolutionary Russia. Rodchenko and Vertov worked on the journal _Kino-fot_ together, and Rodchenko designed posters and tickets for Vertov's films, as well as doing the advertising for Potemkin.\n\nEven in the field of architecture for which, obviously, there was very little scope in a time of extreme shortages, the new order was looking to be as modern as possible. When a trade union body wanted a new headquarters in Moscow it was the great master of modernism himself, Le Corbusier, who they asked to design it. Two years later, however, a competition to build the new Palace of the Soviets in Moscow rejected Corbusier's designs, as it did those of other leading modernists, the preferred design being more bombastic and combined with a memorial to Lenin. (Construction was stopped in 1941 on the outbreak of war.)\n\nArchitects found themselves designing buildings for the rapidly expanding secret police. The impressive constructivist Iset Hotel in Ekaterinburg - still in use - was built for the Cheka, and the city also had specially-built accommodation for Cheka staff and their families. Serge notes without a hint of irony that the traveller visiting a commune in the Middle Ages would have stopped to contemplate the belfry tower or the town hall, rising above the poor dwellings of the artisan and the bourgeois. The traveller visiting the cities of the USSR today, 'stops involuntarily, in Moscow, at the top of the Kuznetsky Most, the liveliest artery of the capital, to take in at a glance the latest architectural ensemble, the most imposing edifices of the GPU...A building of fifteen storeys, huge co-operative stores, dwellings and offices; in the basement and at the rear of vast courts are perfectly silenced prisons; and somewhere behind those facades of fine, polished stone, those shop fronts, those screened windows where the lamplight flares up in the evening, somewhere at the rear of a cellar lighted by neon lights, are the cleverly conceived execution rooms...\n\nNot even the old palaces of St Petersburg, Serge said, could bear comparison with the new GPU building which dominated the Neva and the Volodarsky Prospect; 'The same is true in all of the centres of the USSR. The most imposing building is that of the GPU.'\n\nAnd we remember Rodion at the end of _Midnight in the Century_ who has found work as a laborer on a secret police building...\n\n* * *\n\n# 6  \nA certain sort of courage: the writer as witness\n\n'This age must be witnessed,' Victor Serge said in his _Memoirs_ , 'The witness passes, but his testimony manages to endure - and life still goes on.' Serge was the first of the great historical witnesses of the twentieth century. Witnessing was, for him, 'a means of expressing for people what most of them live without being able to express, as a means of communion, as a testimony about the vast life that flows through us and whose essential aspect we must try to fix for the benefit of those who come after us'. It was something that informed everything he wrote, including his fiction, and was not confined to his reportage, reviews or analysis. Writing was an _act_ of solidarity and nowhere is this more true than in his _Memoirs of a Revolutionary_ , written in exile in Mexico and, of course, never published in his lifetime.\n\nFrom the very first lines the reader is captive. Serge tells how, at an early age, he realized that there was no possible escape from the world and yet there was no alternative but to fight for such an escape. He also tells us early on that he learned from the political milieu into which he was born that the only purpose in life was 'conscious participation in the making of history'. On the walls of their apartment were pictures of men and woman who had been executed for their beliefs. (It was a distant relative who had made the bomb used to kill the Tsar, Alexander II in 1881.)\n\nSerge never attended school. Instead his father worked with him, 'not often and not well' he said, but his 'passion for knowledge and the radiance of a constantly armed intelligence', which had never allowed itself to stagnate, or 'to recoil from an inquiry or conclusion', affected him so powerfully, that he was hypnotized by it. Learning was not separate from life, but life itself, and he learned the concrete meaning of words like, 'bread' and 'rent' and 'hunger'. But he also learned at an early age the value of literature as something which could lift people out of the mundane and give real sustenance.\n\nLooking back on his early years, Serge said he had no personal regrets at all; indeed, he said he pitied those who did not know the cruel side of life and the necessity of fighting for mankind. His only regrets were for the energies wasted in struggles which were bound to be fruitless.\n\nSerge had a knack for catching important things in a few words - 'a time of pot-bellied peace' (France and Belgium just before the war), or 'the sudden conversion to fratricidal patriotism' of the German socialists, French syndicalists and anarchists on the outbreak of war in 1914. And when he reached Russia in 1919 to be part of the revolution, it was 'a world frozen to death'; Petrograd was 'the metropolis of Cold, of Hunger, of Hatred, and of Endurance'. Winter itself was 'a torture (there is no other word for it) for the townspeople': no heating, no lighting, and the ravages of famine'. The young and the old were killed in their thousands.\n\nBecause of his anarchist and syndicalist past, and independent mind, he was respected by people across the non-Bolshevik left, and because of his literary connections and interests he was known in those circles as a man of culture. As a result, he was often asked to intercede for people who had been arrested; 'The telephone became my personal enemy, at every hour it brought me voices of panic-stricken women'. He also came into contact with many of the key figures in the political and cultural life of the times and he wrote arresting little portraits of them; he was repelled by the affectation and calculated elegance of the poet Yesenin, but he could not resist the 'positive sorcery of that ruined voice'; Georg Lukacs, the Marxist philosopher, 'lived bravely in the general fear', yet would not shake Serge's hand in public (Serge says this with not one note of bitterness); 'a single glance was enough to tell the caliber' of men like Joaquin Maurin and Andres Nin, both teachers who would give their lives for the cause in Spain; the Italian communist, Antonio Gramsci, 'fitted awkwardly into the humdrum of everyday existence, losing his way at night in familiar streets, taking the wrong train' but was intellectually 'absolutely alive'; the US anarchist, Emma Goldman, with 'her organizing flair and practical disposition, her narrow but generous prejudices, and her self-importance'.\n\nSerge's recollections convey, as no one else does, the endlessly appalling conditions, physical and emotional, in Russia from the civil war and war communism onwards, the New Economic Policy, ('The sordid taint of money is visible on everything again'), the slow slide of the country into psychosis (Serge's own word), the arrival of Thermidor, the onset of reaction _from within the bureaucracy_ , in November 1927, ('the exhausted Revolution had turned full circle against itself'), the slide also of individuals into despair, ('Nowadays even my friend...has stopped thinking'). Serge took it upon himself to investigate the desperate underbelly of the society and discovered 'the social inferno', where nothing had changed since Dostoyevsky's time. But when Comintern secretary, Angelica Balabanova, sends through the diplomatic bag, an orange and a bar of soap to Serge and his wife, on the birth of their son, these are 'delicacies from another world'.\n\nSerge watched in hope, as did others, the Chinese revolution begin in 1926. A victory there would have meant the salvation of the Soviet Union, but the official line was that the Communists should accept the rule of the Kuomintang under Chiang Kaishek. As a result, Communists were massacred in their thousands. With great courage, Serge spoke out against Stalin himself, who had abandoned support for revolutionary movements abroad, because they would destabilize him, by encouraging renewed revolutionary activity in the Soviet Union, and because they would pose external challenges to his dominance in the Communist world.\n\n'Our crime as Oppositionists', Serge wrote, 'lay simply in existing, in not disowning ourselves, in keeping our friendships and talking freely in each other's company'. As the noose around any kind of opposition tightened, he prepared a document which he sent to friends in Paris asking them to publish it if he were to disappear. It was in this that he was the first to describe the Soviet Union as a totalitarian state. And because of his connections with sympathetic people in the bureaucracy, Serge was made aware of the extent to which the Russian secret police and military were collaborating with the Gestapo. A formal pact between the two countries could not be far off. And, of course, he was right.\n\nToday reading _Destiny of a Revolution_ , Serge's account of Russia after 20 years, one is struck by the appalling accumulation of evidence about the everyday lives of ordinary Russian people - the food shortages, the alcoholism, the fear, the ruthless oppression of any opposition, however minor, the impoverished public culture - the constant lies, the massive gap between the official version and the lived reality. Serge had an eye not just for the terrible conditions but what they led to; it was terrible that people slept 'in the corridors, in garrets, in lofts, in cellars', but also what these led to, 'the ignorance, the alcoholism and the informing...the bitter struggles that can occur, for example, over a room whose occupant, an old woman, seems to be on the point of dying', how people were endlessly divided against each other.\n\nSerge brilliantly describes (in _Conquered City)_ how letters of denunciation were put together; 'unknown hands, laboring in profound secrecy. Obstinately cut alphabets of all dimensions from the papers, collected them, aligned them on notebook sheets: it would take at least five hundred characters for the contemplated letter.' The patient labor of creating the 'demonic alphabet' is carried out in solitude and in silence. The cut papers then have to be sent with a stone to the bottom of a well, for burning would create smoke and 'where there's smoke there's a fire, don't they say?'\n\nBut, for Serge, the greatest evil was not the poverty of means and of men. It was the bureaucratic spirit that prevailed in the schools and which was translated into suspicion, informing, the repetition of formulae devoid of all content, the lessons of pure Stalinism crammed into children eight years old, the stifling of any critical spirit, the repression of all thought, and the hypocritical dissimulation to which the child accustoms himself out of necessity'.\n\nFor Serge witnessing wasn't just about reporting something; it was about understanding. So he didn't just describe what was happening, but offered explanations, and these explanations were always drawn from his first-hand experience, his lived knowledge, not just of the material facts which was always considerable, but of the people and the forces involved. Serge would have been the first to admit that the democratic forces within the revolution had, indeed, been roundly defeated very early on, but he was clear about what had brought this about: 'the revolution hemmed in by its foes, undermined at home by Vendees, by conspiracies, by sabotage, by epidemics, by schisms...the conflict between the battling vanguard of the working-class and its backwards elements, the least conscious and most selfish, those least inclined to sacrifices demanded by the general interest.' In addition there were the activities of dissident groups which served the counter-revolution, for instance the general strike attempted in 1919 by the Mensheviks and social revolutionaries, something which would have been suicidal for the revolution. But in the end it was the failure of the revolution to happen anywhere else, especially in Germany, that was to be the decisive factor. Instead of breaking the 'iron circle in which the soviets were suffocating', the revolution faced 'lasting isolation, increased economic difficulties, a moral depression, the weakening of the internationalist revolutionary tendencies, the strengthening of bureaucratic nationalistic, moderating tendencies...'\n\nSerge was in no way romantic about people; he had had too much experience of personal ambition, betrayal, venality and so on for that. But he was also reluctant to judge, as we saw when he refused to condemn other writers who had maneuvered this way and that to try to save themselves. But he understood, as few people did, the forces at work at particular moments. This is why Serge's account of the Russian revolution is so important today because it stands as a corrective to the popular view that the revolution was somehow fated from the start and, moreover, that this is how all revolutions will end, must end. Not at all, says Serge; there were always choices to be made, courses of action to be decided on. It could have been different. Serge is always also a reminder that one of the reasons the revolution degenerated was the massive external forces ranged against it.\n\n* * *\n\nDaria, in Serge's last novel, _Unforgiving Years_ , working as a teacher in a village in Kazakhstan, has been a witness of sorts. But, because of the times in which she lives, her writings have had to be of a strange kind indeed:\n\nA curious document, this journal, whose carefully chosen words sketched out only the outer shapes of people, events, and ideas: a poem constructed of gaps cut from the lived material, because - since it could be seized - it could not contain a single name, a single recognizable face, a single unmistakable strand of the past, a single allusion to assignments accomplished...No expression of torment or sorrow (this for the sake of pride), no expression of doubt or calculation (for the sake of prudence), and nothing ideological, naturally, for ideology is the sludge at the bottom of the pitfall...\n\nThe construction of the 'featureless record' was like a 'thought puzzle in three dimensions turned entirely toward some undefinable fourth dimension', had become an exhilarating occupation. So there is nothing about the passionate relationship she is having with Klimentii, nothing of the 'surrenders of the flesh, phosphorences of the spirit transmuted into inner riches, she'd had no inkling of before'. Also, she sees that 'no contoured shoulder, no quiver of eyelashes can ever be wholly expressed...'But even this minimal record is potentially too dangerous for the time and must be burned when she is transferred. Her last act there is to distribute 'her riches', some bread and sugar and a bar of rose-scented soap which she carves into slices so it can be distributed more fairly.\n\nOnce in Leningrad, Daria is shocked at how even the buildings seem to have aged by 'a couple of centuries in a few short seasons', just as the men and women 'looked decades older in only a few months', while the children had 'aged a lifetime before knowing what life was'. But it is what she sees in people's faces when they glance at her that disturb her even more. She has never seen this before; it is not the gaze of the hungry:\n\n...this look was inexpressibly different from the looks of the past. She hadn't known that eyes could change so, and cry out so loudly in silence something intolerable. It was neither pain, nor hallucination...What were all those eyes saying? That they had weathered day and night, indefinitely, the storms of snow and terror, of filth, exhaustion, cold, hunger, fright, sickness, with no hope of escape, no hope of healing...That they were watching life die away within themselves.\n\nShe sees the dead being transported, each corpse tied to a sled pulled on a string by its next of kin; a 'new breed of resourceful specialists' earned their food by sewing discarded sheets or squares of sackcloth around the remains. Daria passed several 'such mummies on the street. Rigid pods floating just above the trodden snow. A living man or woman to pull the string, and sometimes a child behind...'\n\nThis part of the novel ends with Daria saving the life of a neighbor by giving her last vodka, vitamins, 'a tin of fish in brine, the half-eaten bar of stale chocolate', and then going home to face her own feelings that she, too, is beginning to die, 'Hunger and loneliness, two tentacles of death'.\n\nThe third section of the novel, 'Brigitte, Lightning, Lilacs', is a stunning achievement of witnessing. It is a shocking picture of Germany during the final days of the war, when the country has become a waste land, devastated by aerial bombardment, as well as by the war on the land; a country laid waste, where not only topography has been destroyed, cities flattened, landmarks destroyed, but time too has been altered, the future itself has been wiped out, 'the old was obliterated forever'.\n\nRichard Greeman points out in his introduction that Serge was in advance of his time in showing the Germans as themselves victims, something that was unthinkable until relatively recently, when it has become possible to acknowledge the horrors afflicted on ordinary Germans, as in the fire-bombing of cities like Dresden which had no strategic purpose but was designed purely to terrify, and also the use of mass rape of women by Russians, again as a form of terror. But when Serge was writing, in the years just after the war, this was a courageous act, another gesture of solidarity with the victims of history.\n\nSerge evokes the fear that is always present: 'There were daylight raids, nighttime raids, twilight raids, dawn raids, and errors in the warning system, which announced a bombing raid when it had already begun and sounded the all-clear as it was starting over again'.\n\nSerge himself, of course, had spent time in Germany in the 1920s, working for the Comintern and had lived among these people. To him, they were not some faceless enemy, but real men and women, many of whom had thoughts and ideas similar to his own, and who had come very close to making their own socialist revolution. Brigitte of the section title is not a Nazi, but from a social democratic family, and her fianc\u00e9, a soldier, we learn, has himself been executed along with the rest of his tank crew, because of their 'bad attitude', in not supporting the war wholeheartedly. 'A desert is what we have made', he says to her in one of his letters. He describes what it is to like to ride over a group of men who have been hiding in the snow - 'they screamed like mice being crushed' and the tank treads are clogged with bleeding flesh. He also tells her how his unit attacked an enemy tank, even though they wished to surrender. He watches a blonde 20 year old burning, 'I watched his face twist like a paper mask tossed onto a bonfire'.\n\nHe, too, has had to become a witness. He tells Brigitte, 'I had to see it all, since I'm the observer...I told myself: Look at what you're doing, you must look without blinking, you're not allowed to close your eyes'. He vividly describes the gradual brutalization that he sees taking place in his fellow soldiers. Fear, he says with some insight, comes from a surprise inflicted on the imagination, 'Once the surprise has worn off, a hanged man seems perfectly natural'. There were so many, they no longer frightened anyone. 'Was it necessary to unleash hell on earth', he asks.\n\nAnd for Gunther, who has brought Brigitte the letters, 'the only natural coupling is a rape in the barn of some smoldering farm', as he recalls the 'skinny black-haired Slovene' who had tried to hide under some sacks and then opened her mouth to scream, but didn't because others were already doing so. Serge paints an appallingly vivid picture of the inferno that Germany has become:\n\nThunderclaps sent huge waves through the earth, crackling outbursts transmuted into great surges of heat, as though invisible ripples of fire were pulsing outward from a fiery oven, somewhere nearby, to one side, deep underground. \"We're going to be baked like potatoes in ashes,\" an old man calmly remarked...The earth shuddered, smoke crept across it, people dwelt in a volcanic realm of sudden explosions, smoldering dormant fires, smoky eddies of soot, dust clouds, the stench of reeking corpses, charred and splintered trees that persisted in budding and even put out, here and there, tender pale-green leaves as though nothing were amiss.\n\nLife continues, just as the lilac bushes tended by Herr Schiff, the schoolteacher, have survived the heat and the dust, 'The force of simple vegetal vitality. 'Schiff goes back into his house and finds a pillow case which he ties to a ruler and puts it above the door: 'Already white rags were flocking across the ruins, some floating with the gay flutter of doves. As far as the eye could see, the whole city was covering itself with white birds, captives who would never take wing.'\n\n* * *\n\nSerge would be followed as a witness by people such as Nadezhda Mandelstam, widow of the poet, Yevgenia Ginzburg and Alexander Solzhenitsyn who reported, in their own brave ways, from that same place of desolation. And of course Primo Levi, _the_ witness of the Shoah, who, in _If This is a Man, The Truce_ , _The Drowned and the Saved_ and other books, dared to speak of the unspeakable, all the time reminding us that the real witnesses, 'those who saw the Gorgon', were dead; he and other survivors were the exception, an anomaly. It was this which haunted him till he too joined them. And let's remember too that Levi was not always the hugely popular and widely-read figure he became. When he first wrote about his experiences, no major publisher was interested in his book and, when eventually published by a small imprint, it quickly went out of print. People did not want to know.\n\nI think too of Eduardo Galeano, the self-described 'Magical Marxist - one half reason, one half passion, a third half mystery' - who, in his massive counter-history of the Americas, _Memory of Fire_ , as well as in other books like _Mirror_ and _The Book of Embraces_ , has redefined the writing of history, mixing historical fact, popular myth, fiction and poetry, to achieve a deeper truth. I think also of Simon Leys, the Sinologist who became a political commentator in the 1960s and 1970s, only because no one else seemed willing to do it. Indeed, Leys compares the refusal of the Western media to hear the truth about what was happening in Mao's China with Serge's experience regarding Stalin's Russia. Leys remarks that his sources were invariably public, and that his only expertise was his knowledge of Chinese, as well as knowing _how_ to read what is said and not said, crack the code of the official jargon, the 'secret language full of symbols, riddles, cryptograms, hints, traps, dark allusions, and red herrings.'\n\nBut of all the writers who have followed, it was the Spanish Communist, Jorge Semprun, who most embodied the legacy of Serge. Like Serge he came from a political family; his father was the Spanish republic's ambassador to the Netherlands, although his maternal grandfather had been the Conservative prime minister, Antonio Maura. Born in Madrid in 1923, Semprun lived in France after the defeat of the Republic, was a member of the Spanish Communist Party in exile. With the German occupation, he joined the resistance, was arrested in 1943 and sent to the concentration camp at Buchenwald. The five day train journey there would provide the framework for his first novel, _Le Grand Voyage_ , translated as _The Cattle Truck_ , which won the 1963 Formentor prize. Semprun was presented with copies in 12 languages, but the Spanish edition, being specially set up in Mexico, hasn't yet arrived, is blank, 'Finally, I feel moved.'\n\nSemprun's allocation to the camp work administration, effectively run by the Communists there, along with his fluency in German, almost certainly ensured his survival. Bizarrely, horribly, it was this work that would lead many former camp inmates later to fall under suspicion, sometimes fatal, after the war was over. Josef Frank, a leading Czech Communist, found himself in the dock in Prague accused of being a war criminal and a Nazi collaborator. He was hanged a few days later and his ashes scattered 'on the snow somewhere around Prague, so that no trace of his passage on earth would remain'.\n\nIt was his experience at Buchenwald that Semprun used as the centre for his fictionalized memoir, _What a Beautiful Sunday_ (1983). Semprun was, of course, familiar with Goethe's visits to the area the previous century with his friend, the poet Eckermann: 'Not without some degree of intellectual perversity, I was pleased to imagine Goethe's conversation with Eckermann on the subject of the Buchenwald camp. What would Goethe have said if he had noticed, as he walked along the Avenue of Eagles, one December Sunday, for instance, the wrought-iron inscription on the monumental camp gates, Jeden das Seine, TO EACH HIS DUE?. The book begins with the narrator, who has strayed from the area permitted to inmates, gazing at a beech tree from the camp when, behind him, he hears a German officer cocking his pistol, getting ready to kill him. In response to the question what is he doing there, he thinks for a second and says, 'Das Baum, so ein wunderschones Baum.' He then jumps to attention and shouts his number in German, 'Haftling vier-und-vierzig-tausend-neun-hundert-vier', which probably saves his life. 'Between SS Warrant Officer Kurt Krauss and No. 44904 there is all the distance created by the right to kill.'\n\nAfter the war, Semprun became a translator for Unesco, but started working clandestinely in Spain for the Communist Party in 1953, an activity he would carry on courageously for many years. How dangerous the work was shown by the fact that Semprun's replacement was arrested, tortured and executed. He was also member of the party central committee and of the Politburo. His account of this period of his life, _The Autobiography of Frederico Sanchez_ , his _nom de guerre_ , won the prestigious Planeta prize in 1977. But Semprun became increasingly critical of the Party's position, as it more and more lost touch with what was actually happening in Spain. He, and his friend Fernando Claudin, would eventually be denounced at an executive committee meeting in Moscow by the legendary 'La Pasionaria' (Dolores Ibarrurri) as 'feather-brained intellectuals', and expelled from the Party. (Semprun also used this experience for the screenplay of Alain Resnais' subtle and compelling 1966 film, _La Guerre est Finie_ , although he is probably better known for his work with Costa-Gavras.)\n\nIn many ways Semprun and Serge could not have been more different. Semprun was from a well-off family, had an elite education, and was an uncritical, orthodox Communist for many years; he said himself he did _not want_ to know the truth of Soviet Russia and elsewhere. When he did confront the truth, as he did when he wrote the screenplay for Costa-Gavras' 1970 film, _L'Aveu (The Confession)_ about the Czech show trials, he did so with an unrivaled power. With Serge, he shared an emotional honesty - 'There is no such thing as an innocent memory. Not for me any more.' he wrote, as well as humanity and courage. And like Serge, Semprun had postponed his writing, preferring political activity to the word. Hence the title he chose for his memoir, _Litterature ou la vie_ , although as someone remarked, this really ought to have been translated as _Writing or Life_. Above all, they shared a commitment to truth and to witnessing for others, and it was always in the service of these that they wrote.\n\n# 7  \nShared veins: hope and the soul\n\nSerge was in no way a personal writer; there is very little ego in his writing and this is true even of the _Memoirs_. He has a very clear sense of what is private and what is public. When he writes that the worst intimacy of prison was not that of bodies, although that wass awful, but 'not being able to be with yourself...to remove your face from the prying glance of others', we see what a torment this was to a private man. The word 'I' was, he once said, repellent to him, 'a vain affirmation of the self which contains a large measure of illusion and another of vanity or unjustified pride'. He preferred to think instead of a 'we', as his experience illuminated that of the people, 'to whom I feel tied'. Unlike the 'we' of Zamyatin's dystopia in which the individual does not exist, Serge's 'we' is the ground, the context of individuality. It is this sense of a real connection to, and authentic communion with, others, a deep humanism, always on the side of the powerless but which never slides into sentimentality, that gives his work much of its power. For his prose is of a piece with that view, always congruent He had a rare ability to convey things powerfully in ordinary language, without any obvious artifice, literary conceit or stylishness. There is not one word that sends the reader to the dictionary, or a formulation whose meaning is not clear to the attentive reader.\n\nAnd he was always writing about real people, never about abstractions. This was much rarer on the left at the time he wrote than it ought to have been. It is this that makes books like _From Lenin to Stalin_ and _Destiny of a Revolution_ engaging still, 75 years after he wrote them. This short passage from _Conquered City_ about conditions in the countryside, merges both the awful facts of what is happening, with the terrible personal consequences, 'So began the black years. First expropriated, then deported, some seven per cent of the farmers left the region in cattle cars amid the cries, tears, and curses of urchins and disheveled women and old men mad with rage. Fields lay fallow, cattle disappeared, people ate the oil cake intended for the stock...'\n\n* * *\n\nOne of the things that gives depth to Serge's understanding of people was his belief in the soul. This was unusual for someone who was in no way religious and probably had a lot to do with the anarchism, with which he had grown up, and which formed him politically. Whatever its roots, Serge had always believed in, what he called, a 'materialist spirituality'. 'The immaterial,' he wrote in his notebook, 'is not in the least unreal' even if it was 'unexplainable by yesterday's scientific rules'. And the soul, for him, was an embodied soul, 'The soul would be nothing if it were not flesh.'\n\nSerge's interest in these matters was deepened by his friendship in exile in Mexico with the German psychiatrists, Fritz Fraenckel and Hubert Lennhof, both veterans of the German socialist movement and the Spanish civil war. Fraenckel, who was once mistaken by a waitress for Einstein, believed that Freud's thinking was equal to that of Marx with its 'new revelations' of man. Their conversations touched on the role of character and personal psychology in the beginning of the conflict between Trotsky and Stalin, as well as the psychological roots of Nazism and the emotional foundations of totalitarianism. The two men would discuss their dreams on the bus. Serge wrote in his notebook:\n\n_Men are psychological beings;_ impossible to act with them, on them, without taking this fact into account, in the most serious sense of the definition. Socialist schematism didn't...take their souls into account...'No psychology!' I heard this little sentence thousands of times in Russia. It meant, 'We're fighting, we're working, efficiency first, material objectivity!' and it came out of the most narrow-minded industrial pragmatism...The striking thing is that the Russian Revolution came to an end through a psychological drama. The whole of contemporary history revolves round that drama and around the Nazi phenomenon which is both economic and psychological at one and the same time...Psychology will perhaps be the revolutionary science of totalitarian times; socialism will no longer be able to do without it without lowering itself and reducing itself to a kind of sterility.\n\nThese are private notes only, written for himself, but we can see Serge groping towards something important, a whole dimension of being human, one which had been fatally missing from the socialist project. Even if he never theorized it, it was always present in what he wrote, that socialism was not about abstractions, but about real people, and it is real people who inhabit his writing.\n\nSerge knew of psychologist Bruno Bettelheim's courageous article on how different individuals responded to the terrible shock of finding themselves thrown into concentration camps, based on his observations while himself an inmate in Dachau and Buchenwald. He also reviewed the Marxist psychoanalyst, Erich Fromm's, pioneering 1941 book, _Escape from Freedom_ , welcoming it as a 'valuable contribution to what might be called our \"intellectual rearmament\"'. Agreeing with Fromm that people accepted authoritarian regimes partly because they were afraid of their own freedom, Serge argued that totalitarian regimes created, in turn, a new kind of insecurity, 'even worse than the one it remedied'; Germans and Russians now lived in a state of 'permanent catastrophe'. And Serge had seen in his own life what persecution could do to individual people; his wife Liuba Russakova had been driven mad by it; on one occasion, offered a cup of tea by the writer, Boris Pilnyak, she thought she was being poisoned. She spent time in psychiatric clinics in Russia (where the secret police were active) and in France, where, tragically, she had to be left behind when Serge and their children went to Mexico. She died in 1985.\n\n* * *\n\nSerge was not an optimist; he never saw silver linings in the dark clouds, never believed that everything would somehow turn out for the best. Not for him the 'optimism of the will', advised by Gramsci, even if combined with the 'pessimism of the intellect' (whatever that might mean). What he did have was _hope_ , which is something different, and which always carries within it the possibility of disappointment. And unlike optimism, which is just a kind of wishful thinking, hope is based on experience, and, for Serge, this was not just his own but the experience of history, that people would always strive for a better life: 'The course is set on hope', as he ended one of his poems in exile.\n\nAnother contemporary, the free-thinking Marxist philosopher, Ernst Bloch, devoted much of his life to documenting the many manifestations of what he called, 'the principle of hope'. This was evident, Bloch argued, in a great deal of culture, from fairy tales to popular songs, to drama and opera. Unlike the Freudian unconscious which was directed to the past, what Bloch called the 'Not-Yet-Conscious', was disposed 'towards the side of something new that was dawning up'. It was the utopian element in human culture, utopian in the sense of what Bloch called the 'forward dream', the hope for, and anticipation of, something better. Human beings, Bloch argued, were not just beings who craved or had needs, but were beings capable of wishing, and it is this wishing, whether for something personal or something social, that gives rise to desire and hope. (It was this that Marx himself had in mind when he spoke of the 'dream of the matter' that the world had long ago possessed but which it did not know how to bring into being.)\n\nStrangely, it is precisely Serge's hopefulness, Susan Sontag remarked with some insight, that has contributed to his marginality. His work, she said, 'refused to take on the expected cargo of melancholy'. This, she implies, is much preferred, particularly on the left. Sontag brings to mind another of Serge's contemporaries, Walter Benjamin, and about whom she had herself written a perceptive and moving article, where she spoke of his 'unquenchable gloom'. The contrast between the two men is striking. A small industry has grown up around the figure of Benjamin. There have been biographies, a film, endless commentaries, even novels and an authorized collected works, now in several volumes. There is also a major memorial at the place of his death.\n\nThe difference between the reception of the two men is not hard to understand. Benjamin is the man of almost pure theory and, as such, the kind of man feted in our time with its love of 'theoretical practitioners'. (The fact that he was also more than a little mystical in his thinking adds a certain exoticism.) The manner of his death in September 1940 has lent weight to the myth: the hard, and painful, trek over the Pyrenees, the loss of a briefcase with an unpublished manuscript, 'more precious than my life', suicide by poison in a hotel room in Port-Bou when he feared he would be sent back to certain death.\n\nI have no wish here to deny Benjamin's importance, or the value of his thought. His essay on the work of art in the age of mechanical reproduction was genuinely groundbreaking, although even he could not have foreseen that the photograph, which he thought was displacing the original work of painting, would itself become a commodity, despite being, as he remarked, 'infinitely reproducible'. It's impossible also to forget, once encountered, the figure of the angel of history, even if it doesn't really stand up to too much scrutiny. And impossible, too not to be charmed by the idea of this man, the most exacting of intellectuals, doing a series of radio broadcasts for children! Nor do I wish to deny the terrible personal tragedy of his death, but simply to note the acute disparity between the after-lives of the two men.\n\n(Serge did, of course, also write theoretically but, to my mind, this was not his strength. Serge writing about individual writers, like Blok or Mayakovsky, is far more interesting than Serge developing a theory of the proletarian novel.)\n\n* * *\n\nIt is in his hopefulness, that Serge makes me think of John Berger, who has himself sustained hope for very many years, in everything he has done. Himself a long-time and insightful admirer of Serge, as we saw earlier, Berger has always been on the side of the oppressed, not just in his active solidarity with the powerless from Mexico to Gaza, but in his writing. Whatever the form he has inhabited - he has been novelist, poet, artist, critic, film-maker - he has been a storyteller, often in powerful collaboration with others, 'against the great defeat of the world'. His has been a persistent voice, intransigent and angry but also compassionate, that a more just ordering of the world is, not just desirable, but possible. His belief in the power of art has sustained many, whether in his understanding of poetry, with its echoes of Serge, as a secular form of prayer against the cruelty and indifference of the world, or the other voices he has, through his generous celebration, helped make known, like Anne Michaels, Juan Gelman, Andrei Platonov, or the work of the artists he has championed and helped us to appreciate, like Giacometti, Zadkine, Leger, Juan Munoz. This is why so many people feel they have this direct relationship with him, as though he is writing to and for each one of us. This is what makes a book, like _Bento's Sketches_ (2011), which takes, as its starting point, the lost sketchbook of a seventeenth-century philosopher (!), with its drawings of a cat sleeping, a dead badger, some flowers, people in Berger's life, its stories of exile, its meditations on art and the state of the world, so inspiring, like a message of hope. 'Some fight because they hate what confronts them;' Berger wrote many years ago, 'others because they have taken the measure of their lives and wish to give meaning to their existence. The latter are likely to struggle more persistently.' These words about another contemporary of Serge, Max Raphael, the German Marxist art critic, he might equally have said of Serge himself.\n\n* * *\n\nSerge also never forgot that human beings lived in a physical world. Just as there was never an 'I' who was outside of a 'we', so we humans, by definition, inhabited a natural world that is not of our making, that has been given to us. This was a constant source of wonder to him, as well as sustenance, and he never stopped celebrating it. Remember the tree bark the prisoners liked to touch, or the pebbles to hold, or the arrival of spring, or the sunlight on the exiles' faces. Or the lilac that survives the endless bombing. Here is a short passage from _Birth of Our Power_ , which is like a little prose poem:\n\nThe blue waters mirror a pure sky in their shimmering silk folds. Invisible strings tremble on the burning air like the flight of bee swarms. The light hums. In the distance are white sails. Flights of seagulls describe curves of whiteness which fade like a light caress in the crystal blue air. The rocks of Montjuich are tinted with amber.\n\nAnd it is the 'splendidly simple world' of nature that Daria is delighted to find herself connected to, once again, in the final section of his last novel, _Unforgiving Years_. Like Serge, Daria has found a refuge of sorts in Mexico and, standing in her yard in early morning, she sees:\n\nPurple sprays of bougainvillea poured over the broken walls. A thicket of menacing nopals - fleshy green - bristled vehemently, and they bore bulbous flowers of a delicate red. A yellow campanile rose above its surround of tall trees, hairy with creepers trailing from every branch. The brightness of the morning was expanding into a vivid symphony of color that promised to intensify almost beyond endurance after this hour of exquisite softness. A monumental joy...conjoined earth and sky in the embrace of the light.\n\n* * *\n\nIt is true that Serge's hope was, at times, misplaced. He greatly underestimated the power of Stalinism and its successors. Stalin was removed from power, not by a rebellion of those he subjugated, as Serge had imagined he would be, but only by death, in March 1953, while The authoritarian state itself remained in place foralmost a half century after Serge, and countless others, had hoped for its demise; the statue of Felix Dzerzhinsky, first head of the Cheka, remained in the feared Lubyanka Square until 1989. But the regimes were not unchallenged. Many people recall the Prague Spring and Poland's Solidarity movement, but we need to remember, too, the uprisings in East Germany in 1953, in Hungary in 1956, in Poland in 1970, all unbelievably courageous, all ruthlessly crushed.\n\nParticularly in the wake of the collapse of the Soviet empire in 1989, so much has been talked about a supposed end of history. But history is never over; it is always in the making, and people still hope for a better world, for a better future, in which they can be the subjects, not the objects of their lives, and they are prepared to give life itself for it. This much, at least, must be clear from recent events in the Arab world, and not only there. Wherever they are, people are constantly trying to change their lives, to challenge oppression. They may not do so under the banner of socialism, far less communism - has an idea ever been so twisted out of meaning as this one? - but they still want something other than what they have. I'm reminded of the words of William Morris about people fighting for something and when it comes about is not what they meant, 'and other men have to fight for what they meant under another name'.\n\nSo Serge's words about conscious participation in the making of history, of being on the side of all that enlarges human beings and against what diminishes them, and of just requiring a certain sort of courage to speak the truth, are as vital today as they ever have been.\n\n* * *\n\nThe last word he wrote, _Dazzling_.\n\n# Writings by Victor Serge\n\n## Novels\n\n_Men in Prison_ , 1930, translated by Richard Greeman, 1969.\n\n_Birth of Our Power_ , 1931, translated by Richard Greeman, 1967.\n\n_Conquered City_ , 1932, translated by Richard Greeman, 1975.\n\n_Midnight in the Century_ , 1939, translated by Richard Greeman, 1982.\n\n_The Long Dusk_ , 1946, translated by Ralph Manheim, 1946.\n\n_The Case of Comrade Tulayev_ , 1948, translated by Roger Trask, 1950.\n\n_Unforgiving Years_ , 1971, translated by Richard Greeman, 2008.\n\nShort stories\n\n_Le Tropique et le Nord_ , 1972, translated by John Manson at <http:\/\/www.pennilesspress.co.uk>\n\n## Poetry\n\n_Resistance_ , 1938, translated by James Brook, 1986.\n\n## Political writings\n\n_Carnets_ , 1985, translated by John Manson at <http:\/\/www.victorserge.net>.\n\n_Collected Writings on Literature and Revolution_ , translated and edited by Al Richardson, 2004.\n\n_Destiny of a Revolution_ , translated by Max Shachtman, 1937. (Sometimes published as _Russia: twenty years after.)_\n\n_From Lenin to Stalin_ , translated by Ralph Manheim, 1937.\n\n_Memoirs of a Revolutionary_ , translated by Peter Sedgwick, 1963. (New edition with restored material, 2012.)\n\n_Revolution in Danger: writings from Russia 1919 -1921_ , translated by Ian Birchall, 1997.\n\n_What Every Radical Should Know About State Repression_ , anonymous translation, nd.\n\n_Witness to the German Revolution: writings from Germany 1923_ , translated by Ian Birchall, 1997.\n\n_Year One of the Russian Revolution_ , translated by Peter Sedgwick, 1972.\n\n# Further reading\n\n(The literature on Serge is now substantial; these are the things I've found most interesting.)\n\nMurray Armstrong, 'The searchers', _Guardian_ , 22 September 1990.\n\nJohn Berger, 'Victor Serge', _Selected Essays and Articles: the look of things_ , Penguin, 1972.\n\nJulian Gorkin, 'The last years of Victor Serge, 1941-47', _Revolutionary History_ , vol. 5, no. 3, autumn 1994.\n\nRichard Greeman, Introductions\/afterwords to the individual novels. -: 'The Victor Serge affair and the French literary left', <http:\/\/www.marxists.org\/history\/etol\/revhist\/backiss\/vol5\/no3\/greeman.html> -: 'Victor Serge and the novel of revolution', <http:\/\/www.marxists.de\/culture\/greeman\/sergenovel.htm>\n\nJames Hoberman, 'Orphan of history', _New York Review of Books_ , 22 October 2009.\n\nBill Marshall, _Victor Serge: the uses of dissent_ , Berg, 1992.\n\nSusan Sontag, 'Unextinguished: the case for Victor Serge', Introduction to 2004 NYRB edition of _The Case of Comrade_ _Tulayev_ , also in _At the Same TIme: essays and speeches_ , Penguin Books, 2008.\n\nSusan Weissman, _Victor Serge: the course is set on hope_ , Verso, 2001.\n\n# Debt to translators\n\nEnglish language readers of Victor Serge are inevitably indebted to his many translators; without them his work simply would not exist. We are particularly indebted to Richard Greeman and to Peter Sedgwick, not just translators but partisans, key members of what Serge's French publisher, Francois Maspero called, the 'secret international of Serge admirers', trying to make sure he is given his rightful place in the world.\n\nThanks also to Ian Birchall, James Brook, Max Eastman, Ralph Manheim, John Manson, Al Richardson, Roger Trask and the, sadly anonymous, translator of _What Every Radical Should Know about State Repression_. Thanks, too, to the many small publishers who have done so much in the face of commercial adversity to keep Serge in print, in particular, Francis Boutle, Bookmarks, Haymarket, Journeyman, Pluto Press, Redwords and Writers and Readers.\n\n# Endnotes\n\n For Gorkin's account see, 'The last years of Victor Serge, 1941-47', _Revolutionary History_ , vol. 5, no. 3, autumn 1994.\n\n Paz speaks of Serge in his memoir, _Itinerary: an intellectual journey_ (trans, Jason Wilson), Harcourt, New York, 1999.\n\n I still find it difficult to use the word 'soviet' in anything other than its true meaning - the workers' councils and factory committees thrown up in the revolution as expres- sions of genuine popular power. They were opposed by the Bolsheviks as early as late 1917, and eventually dominated by them. (See Maurice Brinton, 'The Bolsheviks and Workers' Control', in _For Workers' Power: the selected writings of Maurice Brinton_ , edited by David Goodway, AK Press, 2005). The word also came, especially in the mouths of US politicians and military people, to be a term of cold war abuse: 'the soviets' were the enemy. As for the Union of Soviet Socialist Republics, 'four words, four lies', the great, and greatly-missed, political and social theorist, Cornelius Castoriadis, once quipped, according to Milan Kundera.\n\n John Berger, 'Victor Serge', in _Selected Essays and Articles: the look of things_ , Penguin, 1972, pp. 75-77; Christopher Hitchens, 'Victor Serge: pictures from an inquisition' in _Arguably_ , Atlantic Books, 2011; Susan Sontag, 'Unextinguished: the case for Victor Serge', Introduction to 2004 NYRB edition of _The Case of Comrade Tulayev_ , also in _At the Same TIme: essays and speeches_ , Penguin Books, 2008.\n\n Davd Widgery, _The Left in Britain 1956-68_ Penguin, 1976; Andy Marino, _American Pimpernel_ , Century Hutchinson, 1999, p. 224; John Leonard, _The Last Innocent White Man in_ _America_ , New Press, 1993, p 124.\n\n In the middle of the previous century the great Russian writer Dostoyevsky had spoken of his own imprisonment in similar terms. Having served four years for his part in a political conspiracy, Dostoyevsky wrote to his brother, that he felt he had 'been buried alive and shut up in a coffin...it was an indescribable and unending agony, because each hour, each minute weighed upon my soul like a stone'. Dostoyevsky too would write a novel, _The House of the Dead_ , to come to terms with his experience.\n\n The embassies and military missions of the Allies were, in reality, 'centers of permanent conspiracy', as Serge describes them in his report on state repression, where counter-revolu- tionaries of all kinds 'found subsidies, weapons, political direction'. One of those active against the revolution was the British diplomat and spy, Robert Bruce Lockhart, who was eventually arrested. He was one of those whose freedom was being negotiated for in exchange with the camp inmates in France, who included Serge.\n\n The story is recounted in Susan Weissman, pp.210-213.\n\n Serge's researches were first published in France in 1926 as _Les Coulissses d'une Surete Generale: ce que tout revolutionnaire doit savoir de la repression_. Various editions have been published since, usually dropping the title and using only the sub-title, _What every radical should know about state repression_ , which gives a rather misleading idea of the book's contents. (The first edition I came across was published by one of the more paranoid groups on the far left.)\n\n The Cheka, or Extraordinary Commission, created in 1917 by Lenin, was reorganized in 1922 as the State Political Directorate or GPU, of the NKVD, the People's Commisariat for Internal Affairs.\n\n In his short story, 'The Leningard Hospital', Serge showed how the network of repression extended to the psychiatric system. The patient, Iouriev, has been admitted for a strange misfortune indeed, because he has overcome fear. His psychiatrist explains:  \n\"Iouriev, humble citizen of our times, was ravaged by it for a long time, like you and me. The workers, he explains, are afraid of dying of hunger it they don't steal, afraid of stealing, afraid of the Party, afraid of the Plan, afraid of themselves. The guilty are afraid to own up, the innocent are afraid of their innocence and of having nothing to confess. The intellectuals are afraid of understanding and afraid of not understanding, afraid of seeming to understand or not seeming to understand...The people are afraid of the authorities and the authorities are afraid of the people...the men of the Politburo are afraid of each other, afraid to act, afraid not to act...The Leader is afraid of his subordinates, his subordinates are afraid of him...The revelation came to Iouriev one morning. He woke up, delivered. No fear of anything - anything. The very illumination of feeling clean. He felt he had no right to keep the secret to himself so he spent several days writing forty Appeals to the People...He spent a night sticking them up in the centre of the city, under the eyes of the militia and belated passers-by. They took him for an ordinary bill-poster who was working a little late to have more peace and quiet. And he went home to bed. The next morning he is arrested but his suitcase is already packed.\"  \nSerge was here anticipating the practice whereby the Soviet state would declare 'insane', dissidents and opponents, an abuse of psychiatry that was condemned worldwide. Prominent figures treated in this way included the poet Joseph Brodsky, the scientists Zhores Medvedev and Andrei Sakharov, and people like Natalya Gorbanevskaya who, with unbelievable courage, demonstrated, with a handful of others, in Red Square against the Soviet invasion of Czechoslovakia in 1968.  \nThe story is probably the only work of his to have been adapted for the screen; an adaptation, directed by Sarah Maldoror, was broadcast on French TV in 1983. The cast included Rudiger Vogler, best known for his work in several Wim Wenders films.\n\n _Midnight in the Century_ is dedicated to several comrades, some dead, some in prison. Serge himself says, 'I dedicate these messages' to, Kurt Landau, Andres Nin, Erwin Wolf; to Joaquin Maurin in prison, and to Juan Andrade, Julian Gorkin, Katia Landau and Olga Nin. Andres Nin, one of the leaders of the POUM, kidnapped and murdered in Spain in June 1937 by Stalinist agents, was a hugely important figure also to George Orwell, who had served with a POUM brigade in Spain. In his classic biography, Bernard Crick states, 'The memory of the martyred Nin stayed with Orwell', and the 'hate' figure of Goldstein in _Nineteen Eighty-_ _Four_ 'is Nin quite as much as Trotsky'; much of his testimony in the novel derives from pamphlets in Orwell's possession by, or about, Nin, rather than from Trotsky, as is usually assumed. _(George Orwell: a life_ , Penguin Books, 1992, pp. 227, 365).\n\n A vivid account of Serge's expulsion from Russia, the theft of his poetry and other papers, and the hopeful search for them after the opening up of the Soviet Union, is given by Murray Armstrong in 'The searchers', _Guardian_ , 22 September 1990.\n\n These were helpfully brought together by Al Richardson in _Collected Writings on Literature and Revolution_ , 2004.\n\n Kolyma, an area of north-eastern Siberia, was a source of gold and a vast prison camp. It was administered by a trust, Far Northern Construction, set up by the Soviet state in 1931 to run forced labor in the area. The FNC expanded and may well have been running a prison camp the size of the whole of Western Europe. As many as three million people may have died there - from the cold, hunger, exhaustion, brutality. (Shalamov was asked by Solzhenitsyn to collab- orate on _The Gulag Archipelago_ but declined because of his age.) He died in 1982. (See John Glad's 'Foreword', to his translation of the stories, Penguin, 1994.)\n\n The best introduction to the art of the period remains Camilla Gray's pioneering, _The Russian Experiment in Art_ , _1863-1922_ , revised after her untimely death by Marian Burleigh-Motley, Thames and Hudson, 1962\/1986. See also Norbert Lynton, _Tatlin's Tower: monument to revolution_ , Yale University Press, 2009, and Victor Margolin, _The Struggle for_ _Utopia: Rodchenko, Lissitzy, Moholy=Nagy, 1917-1946_ , University of Chicago Press, 1997. For a different level of analysis from a radical perspective, see TJ Clark, _Farewell to an Idea; episodes in the history of modernism_ (Yale University Press, 1999), especially Chapter 5, 'God is not cast down'.\n\n The only instance I have come across of Serge writing about visual art was when he was in Berlin as a Comintern agent and went to the autumn exhibition of the Academy of Fine Art, where he found nothing but torment, suffering and darkness, and artists, including Barlach, Kollwitz and Kokoschka, who know 'nothing of joy'; it was, he said, the 'decadent art of a dying civilization' _(Witness to the German_ _Revolution_ , pp. 257-8).\n\n Eisenstein himself admired the writing of Charles Olson, whose words I've used as the epigraph, although he did not live to see this poem.\n\n Strangely, the avant-garde film-maker, Maya Deren, who was born in Russia in 1917, did her own translation of Serge's _Conquered City_. It was found in her papers after she died tragically in 1961 at the age of 41.\n\n An architect of a different kind, Berthold Lubetkin, eventually found himself in London where his work continues quietly to astonish. Born in 1901, the son of a railroad engineer who would be murdered in Auschwitz, Lubetkin had been a student of the Vkhutemas, where his teachers included Rodchenko and Tatlin, and later at Svomas in Petrograd. A reservist in the Red Army during the civil war, he had designed the USSR Trades Pavilion used at various locations in Europe in the late 20s. In 1932, he set up his practice, Tecton, which would last until 1948. Lubetkin may be best known for his 1934 penguin pool at London Zoo, but his Highpoint flats in north London continue to stun, especially on a bright day; they were praised by Le Corbusier himself, as 'an achievement of the first rank'. But Lubetkin was aware of the contradiction he found himself in, designing housing for middle class intellectuals. (The original brief by Zigmund Gestetner had been for a building to house his employees at his north London factory. Apart from its successor, Highpoint Two, Tecton would only build public buildings.) In 1935, Tecton was commissioned to build a health centre in the poor borough of Finsbury, in the heart of London; bringing together different facilities previously dispersed, the new centre would be, Lubetkin thought, a 'megaphone for health'. The building was poorly looked after, but continues in use, and is impressive still, as is Tecton's public housing, for instance the Hallfield Estate, near London's Paddington Station, living testimony to a belief in good quality housing, that is also architecturally innovative. They are, in architectural historian John Allan's beautiful summation, 'buildings that cry out for a world that has never come into being'.  \nAfter the war, Lubetkin agreed to be the architect\/planner for Peterlee, near Durham, one of the new towns being built as part of the reconstruction after the devastations of the war. But two years of opposition to his innovative ideas, and incessant bureaucratic wrangling, left him defeated. and eventually he resigned. The town was then built on the more usual lines of anonymous urban sprawl. He went into semiretirement, but still was involved in new work and in teaching, before his death in 1990. (See Malcolm Reading and Peter Coe, _Lubetkin and Tecton: an architectural study_ , Triangle Architectural Publishing, 1992).\n\n My copy of _Destiny of a Revolution_ was published by Hutchinson, along with something called the National Book Association and looks, at first sight, like one of those classic Left Book Club editions of the 1930s. It carries a printed leaflet by Arthur Bryant, a popular historian, who became an admirer of Hitler and the Nazis. (Two years after they published Serge, they published a new version of Hitler's _Mein Kampf!)_ Bryant says Serge is an 'unrepentant Communist', who fails to see that what he describes is the 'inevitable result of applying the Marxian doctrine to the real world'. (Even he praises Serge for 'lucidly and dispas- sionately' portraying the details of life in Russia: 'Not Dante himself painted such a picture of horror and inhumanity as this lifelong Communist'.) Bryant's corrective is necessary because there is not one word in the book that justifies its use as right-wing propaganda. Bryant was also involved in the Right Book Club, set up in 1937 by WA Foyle, owner of the prestigious London bookshop, and actively led by his daughter, the autocratic Christina who, infamously, never allowed trade unions in the shop.\n\n 'The art of interpreting nonexistent inscriptions written in invisible ink on a blank page' in Simon Leys, _The Angel and the Octopus: collected essays and articles 1983-1999_ , Duffy and Snellgrove, 1999, and _Chinese Shadows_ , Penguin, 1978.\n\n Dan Gretton, and others from Platform, the political arts organization, inspired by Semprun's book, walked from Goethe's house in Weimar to Buchenwald: 'the distance from humanism to barbarism is 10,166 steps'. Semprun's memoir, _Literature or Life_ (1997), also begins at Buchenwald on the day of its liberation. Semprun, who has not had a mirror for years, sees himself reflected in the faces of his British liber- ators, 'in that terror-stricken gaze, I see myself - in their horror'.\n\n The English translation was published with the title, _Communism in Spain in the Franco Era_ , which makes it sound like a politics text-book, which it most certainly is not.\n\n See the obituary by Michael Eaude, _Guardian_ 20 June 2011. Semprun also wrote the compelling and sophisticated novel of political intrigue, _The Second Death of Ramon Mercader_ (1973), as well as others never translated into English.\n\n An article by Fraenckel and Lennhof, 'On the psychology of National Socialism', (translated by Eileen Holly and Ursula Ott), was published in _Free Associations_ , 2002, pp. 216 - 226.\n\n Bettelheim's article was first published in Dwight Macdonald's journal, _Politics_ , to which Serge was also a contributor. He incorporated it into his classic book, _The_ _Informed Heart_ (1960).\n\n Fromm's book was published in Britain in 1942 as _The Fear of_ _Freedom_. A later book, _Marx's Concept of Man_ (1961), making use of Marx's early writings, in particular the 'Economic and philosophical manuscripts', helped bring the humanist Marx to a wide audience.\n\nContemporary culture has eliminated both the concept of the public and the figure of the intellectual. Former public spaces \u2013 both physical and cultural \u2013 are now either derelict or colonized by advertising. A cretinous anti-intellectualism presides, cheerled by expensively educated hacks in the pay of multinational corporations who reassure their bored readers that there is no need to rouse themselves from their interpassive stupor. The informal censorship internalized and propagated by the cultural workers of late capitalism generates a banal conformity that the propaganda chiefs of Stalinism could only ever have dreamt of imposing. Zer0 Books knows that another kind of discourse \u2013 intellectual without being academic, popular without being populist \u2013 is not only possible: it is already flourishing, in the regions beyond the striplit malls of so-called mass media and the neurotically bureaucratic halls of the academy. Zer0 is committed to the idea of publishing as a making public of the intellectual. It is convinced that in the unthinking, blandly consensual culture in which we live, critical and engaged theoretical reflection is more important than ever before.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztavo b/data_all_eng_slimpj/shuffled/split2/finalzztavo
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@@ -0,0 +1,5 @@
+{"text":"We LOVE making dogs' quality of life better and promoting anything that is fun for people and dogs to do together. If you love that stuff too, check out these links!\nAgile Canines - Next Door to Downtown Dogs!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With generous support from the Dallas Women's Foundation, the Crow Collection will be hosting workshops with local women artists throughout the year. Fill your day with everything female!\nLearn about the process of creating batiks from Evie Thompson, a traditional batik Indonesian designer, and make your own henna. Finish off your day hearing from first-generation minority women artists discussing their work and how it has been influenced by their experiences, culture, and heritage.\nDon't forget to purchase a ticket for a Persian miniature family workshop from 10:00 am to 12:00 pm with local artist Nida Bangash!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Microsoft have recently announced that the exam requirements for the Microsoft Dynamics CRM Partner Programs will be changing on September 1st 2016.\nThese changes are being implemented to take into account the new Microsoft Dynamics products which are being released. The following competencies will be affected: SPA, CSA, MPN, CRM and CRM Cloud.\nThis article will introduce the requirements for Partners to obtain or maintain a Dynamics CRM Silver or Gold competency, and list the changes that will come in September.\nDynamics CRM Silver and Gold competencies \u2013 how does it work?\nMicrosoft Dynamics Silver and Gold competencies allow Partners to gain access to a set of core benefits provided by Microsoft. For more information about these benefits, visit the Partner Network Core benefits and Requirements page.\nIn order to gain Dynamics CRM Silver and Gold competencies, Partners must meet a number of requirements in various categories (exams, the organisation's profile, customer evidence and company revenue).\nThis requirement makes it especially important for Partner companies to employ consultants who have passed the latest certification exams. This can be accomplished more easily by hiring contractors who have the right qualifications.\nWhat will be changing on September 1st 2016?\nExam requirements for Microsoft Dynamics competencies are regularly updated to include the newest products. Starting September 1st 2016, Microsoft Dynamics CRM 2015 or 2016 exams and assessments will be required \u2013 whereas CRM 2013 exams will no longer be accepted.\nDetailed requirements for Dynamics CRM MPN competency, Dynamics CRM MPN Cloud Competency, Dynamics CRM CSA (On-Premise) competency, Dynamics CRM CSA (Online) competency and Dynamics CRM SPA competency can be found on the PartnerSource article: Changes to Microsoft Dynamics Exam Requirements.\nHaving passed the latest Dynamics certifications is an interesting asset on a Dynamics consultants' resume. If you need help preparing for the exam, the best place to go is the Microsoft Dynamics Learning Portal (DLP). Dynamics CRM 2016 Offical courses and videos are currently available on the DLP.\nIf you are an independent Dynamics consultant who doesn't have access to DLP, we can help you get access to free specialised Microsoft Dynamics training.\nYou can keep track of the available certification exams, register to take them and actually take them online thanks to Microsoft's recent Online Proctored Certification Exams tool.\nThe information that you provide about your organisation to the Microsoft Partner Network must be pertinent.\nPartners will need to provide three (for Silver) or five (for Gold) unique customer references for solutions they have provided in the previous 12 months. The solutions will need to be based on the products associated with the sought competency.\nTo become an MPN member with a Silver or Gold competency, your organisation will need to meet a minimum revenue goal based on your market and track.\nDetails and additional information regarding each of these criteria can be found on the MPN Customer Relationship Management competency page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enjoy sitting on the covered composite deck with a view of the open space and large private back yard. Often deer will meander by and nestle in. The front porch has room to sit and take in an unobstructed view for miles. The upper level has 3 bedrooms a Jack and Jill bath, large master bedroom with an amazing view of the open space, and master bath and walk-in closet. There is a great room with lots of windows and a study with french doors. The spacious kitchen has a sit at island plus a roomy eating space. The see-thru fireplace can be enjoyed from the kitchen as well as the great room. The powder bath is conveniently located just inside the door leading to the 2 car garage with workbench. The finished basement can be used as a 4th bedroom or perfect man cave with bar area, small refrigerator and 3\/4 bath. Stay cool this summer with the newer central AC unit. This home sits on a cul-de-sac and is nicely located for quick access to downtown Parkers restaurants and entertainment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are we influenced by Society or are we CONTROLLED?\nThis post is based off of a discussion I had earlier that had me scratching my head. You see, we all try to be 'original,' but being 'original' nowadays is cliche, therefore, there is no originality. Society has created types and no matter how different you may 'think' you are, you ALWAYS fall in to some category.\nI understand humans are intellectual, have psychological depth, and are not two dimensional.\nAs we grow older we start to figure out, or attempt to figure out, the 'type' of person we are. So we choose the clothes we like, the music, our friends, etc. It's not as if 'WE' truly decided who we wanted to be, society sort of molded us in that way. We are influenced by artists, celebrities, writers, our parents, friends, yadayada- and these people are influenced by other people or things. We shape ourselves to be the person we need to be, or even want to be at that moment. But is this a way of Society controlling us? It may not be intentional or even something you think about because it comes so naturally. We have been born in to this Society and abide by the rules subconciously. Even criminals and killers follow the rules of their 'type' in Society. And this goes way back to the beginning of time, I suppose.\nSo my question still stands: Are we controlled by the Society around us or are we just influenced?\nIts like the old idea of how anarchists try to get together and form rallies.... The minute they do, they are no longer anarchists.\nThe best way to answer the question is just by saying, \"We aren't controlled per say, but to exist in society we must conform here and there.\" We can resist influences, we can resist 'control,' but at the end of the day we can't go too far out of the borders and constraints.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztedw b/data_all_eng_slimpj/shuffled/split2/finalzztedw
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+{"text":"John Jay Beauty College (159328) is a higher education institution located in Jefferson Parish, LA. In 2016, the most popular Bachelor's Degree concentrations at John Jay Beauty College (159328) were .\nIn 2016, 65 degrees were awarded across all undergraduate and graduate programs at John Jay Beauty College (159328). 98.5% of these degrees were awarded to women, and 1.54% awarded men. The majority of degree recipients were white (31 degrees), 2.07 times more than then the next closest race\/ethnicity group, black or african american (15 degrees).\nThe median undergraduate tuition at John Jay Beauty College (159328) is $NaN, which is $0 approximately the same the national average for ($11,400).\nAfter taking grants and loans into account, the average net price for students is $8,337.\nIn 2016, 51% of undergraduate students received federal grants, while N\/A% of undergraduate students received federal loans.\nIn N\/A, the cost of tuition at John Jay Beauty College (159328) was $NaN. The cost of tuition at John Jay Beauty College (159328) is $0 approximately the same than the overall (public and private) national average for ($11,400).\nThis chart compares the tuition costs of John Jay Beauty College (159328) (in red) with those of other similar universities.\nIn 2016 John Jay Beauty College (159328) had an average net price \u2014 the price paid after factoring in grants and loans \u2014 of $8,337. Between 2015 and 2016, the average net price of John Jay Beauty College (159328) grew by 2.85%.\nThis chart compares the average net price of John Jay Beauty College (159328) (in red) with that of other similar universities.\nThe average yearly cost of room and board at John Jay Beauty College (159328) was of $N\/A in N\/A. During the same period, the average yearly cost of books and supplies was $NaN. The cost of room and board N\/A between N\/A and N\/A. The cost of books and supplies remained the same by NaNN% during the same period.\nThis chart compares the average student costs at John Jay Beauty College (159328) (in red) with that of similar universities.\n51% of undergraduate students at John Jay Beauty College (159328) received grants or loans in 2016. This represents a decline of 16.4% with respect to 2015, when 61% of undergraduate students received financial aid.\nThis chart compares the average award discount at John Jay Beauty College (159328) (in red) with that of other similar universities.\nIn N\/A the default rate for borrower's at John Jay Beauty College (159328) was N\/A%, which represents N\/A out of the N\/A total borrowers.\nJohn Jay Beauty College (159328) received N\/A undergraduate applications in N\/A, which represents a 0% annual growth. Out of those N\/A applicants, N\/A students were accepted for enrollment, representing a N\/A% acceptance rate.\nThere were 54 students enrolled at John Jay Beauty College (159328) in 2016, and N\/A% of first-time enrollees submitted SAT scores with their applications.\nJohn Jay Beauty College (159328) has an overall enrollment yield of N\/A%, which represents the number of admitted students who ended up enrolling.\nIn N\/A, the undergraduate acceptance rate of John Jay Beauty College (159328) was N\/A% (N\/A admissions from N\/A applications). This is approximately the same as the acceptance rate of N\/A, which was N\/A%. Between N\/A and N\/A, the number of applicants stayed by N\/A%, while admissions stayed by N\/A%.\nThis chart compares the acceptance rate of John Jay Beauty College (159328) (in red) with that of other similar universities.\nN\/A% of enrolled first-time students at John Jay Beauty College (159328) in N\/A submitted SAT scores with their applications.\nJohn Jay Beauty College (159328) has a total enrollment of 54 students. The full-time enrollment at John Jay Beauty College (159328) is 54 students and the part-time enrollment is 0. This means that 100% of students enrolled at John Jay Beauty College (159328) are enrolled full-time.\nThe enrolled student population at John Jay Beauty College (159328), both undergraduate and graduate, is 46.3% White, 33.3% Black or African American, 13% Hispanic or Latino, 5.56% Asian, 0% Native Hawaiian or Other Pacific Islanders, 0% Two or More Races, and 0% American Indian or Alaska Native.\nStudents enrolled at John Jay Beauty College (159328) in full-time Undergraduate programs are majority White Female (44.4%), followed by Black or African American Female (31.5%) and Hispanic or Latino Female (13%). Students enrolled in full-time Graduate programs are majority N\/A, followed by N\/A and N\/A.\nThe total enrollment at John Jay Beauty College (159328), both undergraduate and graduate, is 54 students. The full-time enrollment at John Jay Beauty College (159328) is 54 and the part-time enrollment is 0. This means that 100% of students enrolled at John Jay Beauty College (159328) are enrolled full-time compared with 73.2% at similar .\nThis chart shows the full-time vs part-time enrollment status at John Jay Beauty College (159328) (in red) compares to similar universities.\nRetention rate measures the number of first-time students who began their studies the previous fall and returned to school the following fall. The retention rate for full-time undergraduates at John Jay Beauty College (159328) was 88%. Compared with the full-time retention rate at similar (78%), John Jay Beauty College (159328) had a retention rate higher than its peers.\nThis chart shows the retention rate over time at John Jay Beauty College (159328) (highlighted in red) compares to similar universities.\nThe enrolled student population at John Jay Beauty College (159328) is 46.3% White, 33.3% Black or African American, 13% Hispanic or Latino, 5.56% Asian, 0% Native Hawaiian or Other Pacific Islanders, 0% Two or More Races, and 0% American Indian or Alaska Native. This includes both full-time and part-time students as well as graduate and undergraduates. By comparison, enrollment for all is 43.3% White, 25.7% Hispanic or Latino, and 18.7% Black or African American.\nAny student who is studying in the United States on a temporary basis is categorized as a \"Non-Resident Alien\", and the share of those students are shown in the chart below. Additionally, 1 students (1.85%) did not report their race.\nIn 2016, 63 more women than men received degrees from John Jay Beauty College (159328). The majority of degree recipients at John Jay Beauty College (159328) are white (31 degrees awarded). There were 2.07 times more white graduates than the next closest race\/ethnicity group, black or african american (15 degrees).\nThe most common Bachelor's Degree concentration at John Jay Beauty College (159328) is N\/A, followed by N\/A and N\/A.\nThe most specialized majors across all degree types at John Jay Beauty College (159328), meaning they have significantly more degrees awarded in that concentration than the national average across all institutions, are Personal & Culinary Services (65 degrees awarded).\nThe most common jobs for people who hold a degree in one of the 5 most specialized majors at John Jay Beauty College (159328) are Chefs & head cooks (8,997 people), Morticians, undertakers, & funeral directors (3,936 people), Cooks (3,881 people), Food service managers (2,632 people), and Firstline supervisors of food preparation & serving workers (1,592 people).\nThe most specialized majors at John Jay Beauty College (159328) are Personal & Culinary Services (65 degrees awarded).\nThe most common industries for people who hold a degree in one of the 5 most specialized majors at John Jay Beauty College (159328) are Restaurants & Food Services (14,505 people), Funeral homes, & cemeteries & crematories (4,358 people), Traveler accommodation (2,622 people), Elementary & secondary schools (1,210 people), and Grocery Stores (1,160 people).\nIPEDS uses the Classification of Instructional Programs (CIP) standard, so the categories may not match the exact concentrations offered by John Jay Beauty College (159328).\nIn N\/A, the most common bachelors degree concentration at John Jay Beauty College (159328) was N\/A with N\/A degrees awarded.\nThis visualization illustrates the percentage of degree recipients from bachelors degree programs at John Jay Beauty College (159328) according to their major.\nIn 2016, 1 degrees were awarded to men at John Jay Beauty College (159328), which is 0.0156 times less than the number of degrees awarded to females (64).\nThis chart displays the gender disparity between the top 5 majors at John Jay Beauty College (159328) by degrees awarded.\nIn N\/A, 1 degrees were awarded to men at John Jay Beauty College (159328) in General Cosmetology, which is 0.0385 times less than the 26 female recipients with that same degree.\nIn N\/A, 38 degrees were awarded to women at John Jay Beauty College (159328) in Aesthetician & Skin Care Specialist, which is N\/A times more than the 0 male recipients with that same degree.\nIn 2016, 8% of students graduating from John Jay Beauty College (159328) completed their program within 100% \"normal time\" (i.e. 4 years for a 4-year degree). Comparatively, 77% completed their degrees within 150% of the normal time, and 77% within 200%.\nThe student demographic with the highest graduation rate at John Jay Beauty College (159328) is N\/A and N\/A (N\/A% graduation rate). Across all , Asian Female students have the highest graduation rate (67.2%).\nThe most common race\/ethnicity at John Jay Beauty College (159328) is white (31 degrees awarded). There were 2.07 times more white recipients than the next closest race\/ethnicity group, black or african american (15 degrees).\n4.62% of degree recipients (3 students) did not report their race.\nThe most common race\/ethnicity and gender grouping at John Jay Beauty College (159328) is white female (31 degrees awarded). There were 2.07 times more white female recipients than the next closest race\/ethnicity group, black or african american female (15 degrees).\nJohn Jay Beauty College (159328) has an endowment valued at nearly $N\/A, as of the end of the 2015 fiscal year. The return on its endowment was of $0 (N\/A%), compared to the 0.0261% average return ($3.98k on $15.2M) across all .\nIn 2015, John Jay Beauty College (159328) had a total expenditure of $N\/A. Of that $N\/A, they spent $316k on salaries and $N\/A on benefits.\nJohn Jay Beauty College (159328) employs N\/A N\/A, N\/A N\/A, and N\/A N\/A. Most academics at John Jay Beauty College (159328) are N\/A (N\/A), N\/A(N\/A), and N\/A (N\/A).\nThe most common positions for non-instructional staff at John Jay Beauty College (159328) are: N\/A, with N\/A employees, N\/A, with N\/A employees, and N\/A with N\/A employees.\nJohn Jay Beauty College (159328) has an endowment valued at about $N\/A, as of the end of the 2015 fiscal year. The endowment of John Jay Beauty College (159328) stayed 0% from the previous year. The value of their endowment was $N\/A approximately the same as than the median endowment of according to the Carnegie Classification grouping.\nThis line chart shows how the endowment at John Jay Beauty College (159328) (in red) compares to that of some similar universities.\nAs of 2015, John Jay Beauty College (159328) received $0 in grants and contracts from the federal government, $0 from state grants and contracts, and $0 from local grants and contracts.\nThe bar chart shows the share of the primary expenses at John Jay Beauty College (159328) over time, and the line chart shows the expenditure for solely salaries and benefits over time compared to the median for the Carnegie Classification grouping.\nThis tree map shows all of the primary expenses of Research at John Jay Beauty College (159328) as a share of total expenditure.\nIn 2015, John Jay Beauty College (159328) paid a median of $316k in salaries, which represents N\/A% of their overall expenditure ($N\/A) and a 16% growth from the previous year. This is compared to a 0% growth from N\/A and a 0% growth from N\/A.\nIn N\/A, John Jay Beauty College (159328) paid a total of $N\/A to N\/A employees working as instructors, which represents N\/A% of all salaries paid.\nIn N\/A, the most common positions for instructional staff at John Jay Beauty College (159328) were N\/A with N\/A employees; N\/A with N\/A employees; and N\/A with N\/A employees.\nIn N\/A, the most common positions for non-instructional staff at John Jay Beauty College (159328) were N\/A with N\/A employees; N\/A with N\/A employees; and N\/A with N\/A employees.\nIn N\/A, the most common demographic for instructional staff at John Jay Beauty College (159328) was N\/A with N\/A employees, N\/A with N\/A employees, and N\/A with N\/A employees.\nThis chart shows the gender split between each academic rank present at John Jay Beauty College (159328).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Watching Virat Kohli bat yesterday was a delight like never before.\nHe has scored so many hundreds in such short time that it almost feels like an every day occurrence. No doubt each of his century is a spectacle but yesterday's innings was in my view one of his best, if not THE best.\nHe was out there for 49 of the 50 overs, he faced more than half of those, and he remained unbeaten.\nBut that is not what made it such a delight to watch. Why I found this innings to be magnificent was the fact that Kohli scored only 60 runs in boundaries, meaning that 100 of his runs were scored by running between the wickets.\nYet he had a strike rate in excess of 100.\nI have never seen a batsman as consistent as Kohli. His fitness, his timing, his running between the wickets, the way he paces his innings, and his ability to hit big - it is all top class.\nSoon after his innings ended, I was amazed that he had made 100 runs in singles and doubles. I felt that it was quite a rare achievement and didn't think it happened many times in ODIs, if at all.\nSo this morning I tried to find out which other batsmen had achieved the feat of scoring 100 runs in an ODI by running between the wickets.\nTurns out, it has happened only 4 times before, and Kohli's was the 5th such instance. Only 5 times in 3,971 ODIs!\nWith every innings he inches closer to Sachin's record of hundreds in ODIs and overall. With every innings he shows why he is the best in the world.\nRCB will go upto the playoffs this season, I can assure you that!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What will reader get after reading the online book Inspired by Art: Fighting Goliath (The David Chronicles Book 4) By Uvi Poznansky? Actually, as a reader, you can get many lessons of life. It will be better if you read the bookalone. So, you can really feel content of the book deeply. From the lesson, you will know about the meaning of life and human around you. You will be smart in choosing... As noted earlier, Rana is the most passionate and outspoken one in the house, but timid Fatima and Zayna, the little facilitator, become more fully developed as we get to know their intimate thoughts and personal feelings.\nblack notes on the keyboard into action, and, along with the new forms in which music was expressing itself, insistently demanded a more adequate keyboard technique.\nA Simple Favor had an interesting concept.. but it was quite boring to listen to. I honestly zo Reminded me of Gone Girl, and I hated that book. Okay before I go into this review, I just wanted to say that when I saw the movie trailer for this I honestly wanted to read the book.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How long can I keep my spirulina ?\nSpirulina is a dehydrated product that keeps well. The packages in resealable bags that we offer allow optimum conservation, protected from light and moisture. You can keep at least 1 year after your order.\nIt is therefore quite interesting to order bigger packages (500gr or 1kg) to benefit from the best rates !\nHow long will last me a 100 gr bag ?\nThe recommended doses for spirulina is 3 to 5 grams per day. A bag of 100 gr so will last you about 1 month.\nWhy buy spirulina Spirulina Algah\u00e9 home ?\nBy buying from us, you ensure quality spirulina.\nAt all levels of production (harvesting, drying, packaging, sanitary analysis), particular attention is paid to obtain a product of the highest quality.\nNo binder nor caking agent is added to ensure a 100% Spirulina for all available forms (twigs, powder, tablets and petals).\nEach batch is subjected to sanitary analysis by an accredited independent laboratory.\nThe advantage of direct sales \"from farm to fork\" is also able to offer a high quality spirulina, at competitive rates by removing the margins taken by intermediaries.\nWhy she did not spirulina organic label ?\nJust because the organic label in France does not yet exist for spirulina ! One can find spirulina with organic labels in shops as international agreements allow spirulina produced abroad to be marketed with organic labels legally. But the quality is not necessarily to go with these labels.\nAlthough we do not have a label, we bring you the guarantee of Spirulina cultivated without pesticides, dried at low temperature to preserve the nutritional qualities. Furthermore, no coloring or preservatives are added, and spirulina is analyzed by a laboratory for bacteriological accept checks and to control the absence of heavy metals.\nSpirulina is considered as the natural ally of good health for all: children in times of growth, men, women, elderly. In addition the daily use of spirulina is advantageously integrated into the lifestyle of the athletes and vegetarians.\nIs Spirulina is suitable for children ?\nSpirulina provides children and youth with vital substances. This algae strengthens the immune system and brings their body the elements they need to grow well. They are now in a permanent state of stress: school, homework, tutoring, sports.\nAll this is challenging for them. Spirulina helps young people start the day with more energy, desire and mood. The little alga is a healthy snack that parents can give to their children without hesitation. They must consume about half of the amount of spirulina recommended for adults.\nCan you also give this algae animals ?\nCan we meet any side effects ?\nNo, Spirulina is not a medicine, but a nutrient. Just as one does not usually encounter side effects when consuming fruits and vegetables. At first, some people may experience mild nausea, stomach pain, headaches or rashes.\nThis is common in people whose metabolism is not working properly and those that are or have been, many drugs during their lives. This is a positive sign, it shows that spirulina is and detoxifies the body. If this is the case, it is advisable to reduce the dose or temporarily stop treatment.\nHow to consume twigs ?\nCan I order Spirulina if I am not in France ?\nYes, orders for deliveries outside of France are possible. They are not made through the online store.\nNote that shipping costs will be based on rates colissimo (www.colissimo.fr).\nThe shipping fees are offered from 49 euros of purchases for delivery in France.\nBelow, delivery charges apply. Note that shipping costs will be based on rates colissimo (www.colissimo.fr).\nThe secure Paybox system that uses SSL (Secure Socket Layer) so that the transmitted information is encrypted by software and no third party can read it in transit on the network.\nThe page where you enter your number is SSL secured and hosted by Paybox, ensuring optimal security and confidentiality of your bank details.\nFor those who do not want to use this online payment, an alternative solution is offered when ordering: payment by check. To do this, follow the traditional path of an order on the website, and select \"check\" when choosing the payment. In this case, your package will be mailed upon receipt of your payment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Provincial Administrator Eugenio Payongayong (leftmost) and Officer-in-charge Ma. Victoria Marcelo of the Provincial Youth, Sports and Public Employment Service Office (rightmost) present during the Monday Flag Raising Ceremony at the Bulacan Capitol Gymnasium, City of Malolos, Bulacan this morning the winners and National Qualifiers for Athletics category (L-R) Karen Mae Pelegria, 2nd place in both 200 and 400 meters, girls division and Marvilyn Canion, 1st place in both Long Jump and Triple Jump for girls in the 2019 Batang Pinoy Luzon Qualifying Leg Sports Competition held last March 16 \u2013 23, 2019 in Isabela Province. Also in the photo is their coach Mr. Lamberto Nicolas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"HERE COMES THE BRIDE \u2013 Let our UP-DO and MAKE-UP Specialists work their magic to make that walk down the isle a lasting impression. Beautiful and Glowing, a vision that will last forever! Don't forget your bridal party, bring them with you and enjoy our private upstairs area, bring food and drinks and relax while you prepare for your special day! Come a few days before to be sure your hands and feet look their best too!\nBRIDAL PAMPERING \u2013 Enjoy a one hour Hot Stone massage, our Signature Spa Facial, a Spa Manicure, and our Deluxe Pedicure. This is the PERFECT way to pamper the Bride while we take care of your lunch!\nBEAUTIFUL NAILS \u2013 Let's show off that AMAZING ring on your special day with a full set of Acrylic or Gel nails topped off with a coat of our brilliant shine Shellac.\nCOUPLES DAY \u2013 Share a day of \"TOGETHERNESS\", the Bride and her Husband-to-be, The Bride and her Mother, The Bride and her Sister\u2026enjoy a Couples Massage and Deluxe Pedicures (just the two of you). These services will be in the relaxing, quiet, and private atmosphere of our upstairs spa.\n* We, at Kali Co Court are not permitted to provide alcoholic beverages however, You ARE permitted to bring them with you.\n* All cancellations and changes must be made no later than 48 Hour before the day of services. When scheduling a Spa or Bridal party, we require that a credit card number be on file to secure your appointments. Should any changes or cancellations occur after the allotted time frame - your credit card will be charged in full for the services affected. Please also note that on the day of services, we are not able to rearrange or switch appointments in any manner due to the fact that everything is booked in a specific order to honor your requested timeline.\n* Call our salon for Spa pricing 740-455-2541 or contact Madisen Stiverson about your special event today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you an experience host looking for important information that you may need to file your tax return this year? Here's a short step-by-step guide to help you find the Gross Earnings section of your Airbnb account.\n1. From the Airbnb home page, click on your profile picture at the top of the page to reveal a drop-down menu. Select \"Account Settings\" or \"Settings\".\n2. On the Account Settings page, select \"Transaction History\" from the left sidebar.\n3. From the Transaction History page, select the \"Gross Earnings\" tab.\n4. On the Gross Earnings tab, choose the time period you want to view by selecting the month and year in the \"From\" and \"To\" fields. For instance, if your tax year is January to December 2018, select these in the From and To fields.\n5. To download and save a copy of this history for your tax filing purposes, select the \"Download CSV\".\n6. Open the CSV file in your preferred spreadsheet application. To calculate your earnings from hosting, add up all the numbers in the \"Amount\" column. The numbers in the Amount column represent your Gross Earnings (the listed price of your Experience) minus the Host Fee (the Airbnb service fee).\nIf you have any questions about how to report your Airbnb earnings, we recommend contacting a tax advisor in your area. You can also find helpful information on our Responsible Hosting pages, by selecting your country, language and the relevant tax article.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mens st michael bracelet evil eye black onyx catholic saint, st michael rosary bracelet, turquoise mosaic stone gold, religious jewelry archangel saint michael medal vintage. Archangel michael bracelet st michael patron saint of police. St michael the archangel saint medal charm bracelet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pediatrics is the generally the branch of medicine that involves the medical care of infants, children, and adolescents. Child care is the most important aspect of the infant's life and should not be neglected. As infants are more susceptible to infection, they should be taken care with utmost affection and be treated immediately in case they fall ill. So, here are the best Child Specialists in Chandigarh who can trust with your children or infants. If neglected now, they can be affected in the future. So, please get them treated as soon as you feel something is wrong with their health. Also, if you are looking at these, just because you are expecting a baby anytime soon, you will soon need one of the best gynecologists in Chandigarh.\nFinding the perfect paediatric for your little one is not a herculean task anymore. Here we give you a list of the best child specialists in Chandiagrh. Keep it handy for regular check-up or in-case of any unforeseen emergency condition. We wish your new-born baby good health and do not forget to share any feedback or additional information with us. The Paediatrics mentioned in the list are NOT ranked in any order. Please feel free to choose as each one them is highly qualified in his\/ his expertise and provides top-notch medical assistance to all patients.\nDr. R. P. Bansal is one of the famous Pediatrician in Chandigarh and has an experience of 31 years in this field. He completed MBBS from Government Medical College & Rajendra Hospital, Punjabi University, Patiala in 1982 and MD \u2013 Pediatrics from Government Medical College & Rajendra Hospital, Punjabi University, Patiala in 1986. He is a member of Medical Council of India (MCI), Indian Medical Association (IMA) and Indian Academy of Paediatrics (IAP). Some of the services provided by the doctor are Diabetes in Children, Growth & Development Evaluation \/ Management, Thyroid Disease in Children, Childhood Infections and Infant & Child nutrition etc. He is currently working in Children Health Clinic & Vaccination Center, Sector 34, from MON \u2013 SAT. Consultation fee of Rs.500 is charged by this doctor.\nDr. Ramneek Singh Bedi is one of the best Pediatrician in Chandigarh and has an experience of 32 years in this field. He completed MD \u2013 Paediatrics from Dayanand Medical College and Hospital (DMC and H) in 1983 and Diploma in Child Health (DCH) from Government Medical College, Amritsar in 1981.He is a member of Indian Academy of Paediatrics (IAP), Indian Medical Association (IMA) and National Neonatology Forum (NNF). Dr. Ramneek Singh Bedi practices at Bedi Hospital- A Mother and Child Hospital in Sector-33, from MON-SAT. He charges Rs.500 as consultation.\nDr. Mahesh Hiranandani is among the best chid specialists in Chandigarh and has an experience of 22 years in this field. He completed MBBS from Dr. Sampurnanand Medical College, Jodhpur in 1987 and MD \u2013 Pediatrics from POSTGRADUATE INSTITUTE OF MEDICAL EDUCATION AND RESEARCH, CHANDIGARH in 1992. He is a member of Indian Academy of Paediatrics (IAP) and National Neonatology Forum (NNF). Some of the services provided by the doctor are Newborn Jaundice, Chickenpox Treatment, Viral Fever Treatment, Childhood Infections and Vaccination\/ Immunization etc. Dr. Mahesh Hiranandani practices at Mamta Child Health Care Centre in Sector-18, from MON-SAT. He is charging Rs.500 per visit.\nDr. Rajiv Sehgal is one of the best Child specialists in Chandigarh and has an experience of 25 years in this field. He completed MBBS from Government Medical College & Rajendra Hospital, Punjabi University, Patiala in 1985 and MD \u2013 Pediatrics from Maharshi Dayanand University, Rohtak in 1992. He is a member of Medical Council of India (MCI), Indian Medical Association (IMA) and Indian Association of Pediatrics. Some of the services provided by the doctor are Infant & Child Nutrition, Diabetes in Children, Growth & Development Evaluation \/ Management, Childhood Infections and Thyroid Disease in Children etc. He currently works in Children Clinic & Vaccination Center, Sector 30, from MON-SAT. He charges consultation fee of Rs.350.\nDone MD (Pediatrics) from a prestigious institute of PGI, Chandigarh & also an experience of working as a senior resident at Fortis Hospital & as attending Consultant at Healthcare Multispeciality hospital. She works in Omni Clinics And Diagnostics, Sector 34, from MON-SAT. The consultation fees for this doctor is Rs.350.\nDr. Pankaj Raina is one of the top Pediatrician in Chandigarh and has an experience of 16 years in this field. He completed MD \u2013 Pediatrics from Government Medical College & Rajendra Hospital, Punjabi University, Patiala in 2004 and MBBS from Government Medical College & Rajendra Hospital, Punjabi University, Patiala in 1999. Some of the services provided by the doctor are Nebulization, Growth & Development Evaluation \/ Management, Vaccination\/ Immunization, and Management of common childhood illnesses etc. Dr. Pankaj Raina practices at Niramaya Pediatric Centre in Zirakpur, Chandigarh, and Chakravarty Nursing Home in Sector-10, Panchkula. Youcan consult him at Niramaya Pediatric Centre in Zirakpur, all days of the week. Consultation fees for this doctor is INR 300.\nDr. Diljot Singh Bedi is a popular Pediatrician in Chandigarh and has an experience of 14 years in this field. He completed MBBS from GMCH, Chandigarh in 2000 and MD \u2013 Pediatrics from Christian Medical College, Ludhiana in 2003. Some of the services provided by the doctor are Adolescent Medicine, Newborn Jaundice, Measles Treatment, Diabetes in Children and Limping child etc. He currently works in Dr. Bedi's Pediatric & Psychiatric Clinic, Sector 37 C, from MON \u2013 SAT. He charges consultation fee of Rs.800.\nDr. Gunjan Baweja is a Consultant Pediatrician in Sector-19, Chandigarh and has an experience of 8 years in this field of caring for children. She completed MBBS from Jammu in 2008, MD \u2013 Paediatrics from Manipal Hospital in 2012 and MRCPCH (UK) at RCPCH UK in 2015. She also did her training in Pediatrics from Emma Children's Hospital, AMC Amsterdam in 2015 and Neonatology training from PGIMER, Chandigarh in 2015. She did her training in Pediatric Echocardiography from Sir Ganga Ram Hospital, New Delhi. She is a Member of Royal College of Pediatrics and Child Health, London, European Academy of Pediatrics, Indian Academy of Paediatrics (IAP), Executive member IAP Chandigarh and Emergency Medicine Association. She practices at JP Medical Centre in Sector-19, every day in the week and charges Rs.400 per consultation.\nDr. Vikram Bedi is a popular Pediatrician in Chandigarh and has an experience of 4 years in this field. He completed MBBS from Sri Guru Ram Das Institute of Dental Sciences Research in 2012 and MD \u2013 Pediatrics from MGM Institute of Health Science, Bombay in 2016. He is a member of Indian Academy of Paediatrics (IAP) and National Neonatology Forum (NNF). Some of the services provided by the doctor are Growth & Development Evaluation \/ Management, Diet For Kids, New Born Care, Vaccination\/ Immunization and Infant & Child nutrition etc. He currently works in Bedi Hospital & Infertility Center which is owned by her, Sector-33, Monday to Saturday. The consultation fee for his doctor is Rs.500.\nDr. Manu Sharma words \u2013 Healthy child Nations pride. With this motto, we would strive to work for child care. He is a degree in MD \u2013 Pediatrics, MBBS, and Fellowship in Neonatology along with being a Pediatrician, Neonatologist. He has 6 Years of experience in this field. He consults at Child Health Care Clinic And Vaccination Centre, Sector- 37, from MON-SAT. The consultation fees for this doctor is INR 300.\nThe above are the top 10 best Child Specialists in Chandigarh who can treat your children and make sure they remain healthy. So don't delay and visit your nearest child specialist as soon as possible and make sure your child is healthy and hale by referring to the above addresses and phone numbers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Russian billionaire Alisher Usmanov was last night warned that his ongoing attempts to take over Arsenal could threaten the club's on-field progress. The message was delivered by the managing director, Keith Edelman, who, along with the rest of the Arsenal board, remains committed to stopping Usmanov gaining control of the Premier League leaders, a task made more difficult after he became their largest single shareholder.\n\"There are many examples of clubs who have been taken over and are not having as smooth a ride as first hoped for,\" said Edelman. \"Ars\u00e8ne [Wenger] has always said a board where there is no conflict or tensions helps the team because they are not distracted. It is important we continue that, we want there to be calm waters.\"\nUsmanov increased his stake to 24.2% last week and is now only 5.8% shy of what is required to launch a formal takeover bid. Such a move would prove difficult while the board, which includes the now former largest shareholder Danny Fiszman, remain tied to a lockdown agreement which prevents the sale of their shares until April 2009.\nUsmanov, an Uzbek-born steel magnate who is worth an estimated \u00a33bn, could prevail if he was able to acquire some of the 18% of shares which are owned by Arsenal's small stakeholders. He would need 51% to gain sole control of the club.\nUsmanov's increasing shareholding at Arsenal has raised the possibility of him being offered a seat on the board. \"That is something we will have to consider if it is requested,\" admitted Edelman. Should his takeover attempt prove successful, the club's former vice-chairman David Dein - who sold his shareholding to Usmanov last year - could make a controversial return to Arsenal as part of the new regime.\nArsenal's attractiveness is unsurprising while success on the pitch - they are five points clear at the top of the Premier League - is combined with financial growth off it. Figures released yesterday show Arsenal made a pre-tax profit of \u00a320m in the six months to November 2007, up from \u00a312.9m in the previous year. Key to that surge has been the \u00a341.4m of match-day income, \u00a33.4m more than they made in the six months to November 2006, only four of which had been spent at the 60,000-seater Emirates Stadium.\nThe extra revenue means Wenger has a budget of close to \u00a369m to spend on players, and the Frenchman will be backed should he decide to go against tradition and compete at the more expensive end of the transfer market in the summer. \"The board has always backed Ars\u00e8ne's judgment on players and we will continue to do that,\" said Edelman. \"Our job is to get the funds in line to match Ars\u00e8ne's ambitions and he will have a sizeable pot to do that.\"\nWenger insists his priority is keeping together the current squad which, should they beat Birmingham in the early kick-off today, will temporarily go eight points clear of Manchester United, who play Newcastle United this evening.\n\"This is a decisive period because it is the time when all the teams are going for something - either they do not want to go down or they are playing for Europe,\" the Arsenal manager said. \"But we are not anxious, we were under real pressure at the start of the season when people were saying we were absolutely nobody. The players stood up and now we have nothing to lose.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Terra Firma Academy is offering the only internationally accredited Carbon Footprint Course in Africa, in Johannesburg on 12 \u2013 14 February 2019.\nThis is a 3 day course, facilitated by Terra Firma Academy's in-house Carbon, Environmental and Sustainability Professional.\nHow to conduct a carbon footprint assessment in accordance with the Greenhouse Gas Protocol and ISO 14064.\nAn understanding of the economic effects of climate change on industry sectors.\nUnderstand carbon reduction opportunities and options to mitigate your tax risk.\nAccess to data gathering tools.\nThe course offers 3 SACNASP CPD points to qualifying members, along with a Terra Firma Solutions Certificate of Competence, available as an online exam after course completion, included in the course fee.\nThe below voluntary accreditation options are available. Attendees can book for one or both.\nIEMA assessment: The exam takes place one day after completion of the course. It is a two hour multiple choice exam at an additional R2 185 incl. VAT. If attendees successfully pass the exam, they will receive a certificate from IEMA \u2013 an internationally renowned sustainability institute.\nThe total course fee, excluding above accreditation, comes to R10 350 incl. VAT. Terra Firma Academy offers discounted rates for 5 or more team members attending.\nRegistrations close on 31 January.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Walton County Sheriff's Office reported the following Community Awareness Crime Report for the period Thursday, Feb. 7 \u2013 Sunday, Feb. 10, 2019. Please note an arrest or charge does not constitute a conviction.\nArrest- Sandra Conner was arrested in the area of Mountain Creek Church Road for Possession of Methamphetamine and Failure to Maintain Lane following a traffic stop.\nSuspicious Person\/Damage to Property- Deputies took a report at a residence on Watson Fain Road in reference to complainant reporting an unknown individual in a white in color Chevy Blazer opening a package left for the complainant, but leaving the package at the residence around 4:50 pm.\nArrest- Sheila Burnette was arrested at a residence on Mountain Creek Road for an outstanding warrant out of Auburn Police Department, an outstanding warrant out of Barrow County Sheriff's Office, and an outstanding warrant out of Monroe Police Department following a dispute.\nArrest- James Wigginton was arrested at a residence on Saddlebrook Drive for an outstanding warrant out of Dawson County following a Suspicious Person call.\nArrest- Cecil Morris was arrested in the area of Highway 81 for Obstruction after a brief foot chase following a dispute.\nArrest- Donald Mount was arrested in the area of Shoal Creek Road and Highway 11 for Driving While Out a License, No Proof of Insurance, and Failure to Obey Traffic Control Device following a traffic stop.\nArrest- Kaila Sewell was arrested in the area of James Powers Road for Driving While License Suspended following a traffic stop.\nArrest- Jason Wold was arrested in the area of Highway 81 near Corinth Christian Church for Driving While License Suspended following a traffic stop.\nArrest- Patricia Frady was arrested in the area of Snows Mill Road and Jim Daws Road for Loitering, Public Drunkenness, Possession of Contraband Across Guardlines, Possession of Drug Related Objects, and Possession of Methamphetamine following a Suspicious Person call.\nArrest- Christopher Seagraves was arrested in the area of Highway 81 and Forrest Falls Drive for Reckless Driving, Fleeing to Elude, and Theft by Receiving Stolen Property following a vehicle pursuit.\nEntering Auto- Deputies took reports at four residences on Magnolia Lane in reference to numerous unlocked vehicles being entered and several items being taken sometime between 10:00 pm February 9th and 9:00 am February 10th.\nEntering Auto- Deputies took reports at four residences on Brentwood Boulevard in reference to numerous unlocked vehicles being entered and several items being taken sometime between 10:00 pm February 9th and 9:00 am February 10th.\nArrest- Brianna Guerra was arrested in the area of Kings Court for Possession of Marijuana Less Than One Ounce, Possession of Methamphetamine, Possession of THC Oil, and Possession of Drug Related Objects following a Suspicious Person call.\nArrest- Lester Day was arrested in the area of Highway 81 and Bold Springs Road for Driving While License Suspended following a traffic stop.\nArrest- Catherine Lew was arrested at a residence on Stewart Road for an outstanding Battery warrant out of Walton County following a dispute.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This was my fourth year participating in the Blogging from A to Z Challenge, and possibly my most successful, in terms of getting new followers and visits to my blog. In 2012, while waiting for my first novel to be edited, I decided to take part (thanks to James Duckett), and used the theme of writers, poets, essayists, and lyricists for my posts. Ah, but I had done no planning, and found the task of keeping up with writing, proofing, and posting to be daunting (I did a lot of work on those no-post Sundays!). Lesson learned.\nBy keeping my posts very short this year (under 100 words, with a video under five minutes), I think I was able to get more readers (everybody's busy). And by creating and scheduling the posts in March, it freed me up to do more visiting. I found some great blogs that I will continue to follow.\nI prefer WordPress to Blogger, as a reader. It was easier to follow blogs that didn't require a code, a magic word, or triple confirms to follow. It was easier to \"like\" a post when I didn't feel a need to comment.\nThe A to Z team was fabulous! Thanks to co-hosts for following, liking, and commenting on my posts. I loved the inspirational posts throughout the month.\nI was very impressed by your April postings and am not altogether surprised to hear that preparation was key. I'm sure visiting all the other participants kept you more than busy.\nI'm working my way through the A to Z reflection post list. I wish I had been as organized as you. This was my fourth year, but my first year that I didn't prepare my posts in advance and it makes a world of difference. It seriously cut down on the number of blogs I could visit and my favorite part is interacting with new people. Next year I will drastically shorten my posts, as well.\nSince I use blogger , I found it more difficult to comment on WP. Actually the worst commenting system in my opinion is Disqus and ironically that's the system I use. I switched to it a while back and I hate it. I want to switch back, but have no idea how to do it without losing all my comments . Congratulations on making it to the end.\nThanks, Melissa. I appreciate your taking the time to write. Isn't it interesting about Blogger and WP? And why can't it be simpler?!\nGoing to stop over to yours now\u2026.\nCongratulations on finishing the challenge. This was my second year participating. This year I did most of the work up front, scheduled posts and edited or changed posts along the way. It was a lot easier. I wish I had been that prepared last year. Thank you for stopping by my site.\nSo glad to have connected again, Martha. Short posts always work for me during a busy month such as April, and yours were fantastic. Thank you. Congrats on completion and on a job well done.\nCongrats on finishing. The different blogging platforms are getting a lot of comments on reflection. As a Blogger user, I tire of adding all my ID to wordpress blogs. Each type has its drawbacks.\nOh, that's a good point, Susan. I didn't realize.\nI really enjoyed your musical posts (some more than others!) Thanks for those. I think short is good: people can check online if they want more. A few blogs I started following had such long posts that I gave up. I was interested, but just didn't have the time if I wanted to \"share the love\". Heartily concur with your views on comments \/ likes too.\nI not only enjoyed your short posts, I discovered some instruments I had never \"heard of\" even though I had \"heard\" their sound.\nThanks, Stepheny. Same here \u2013 your posts were easy to read, and I liked what you said in your reflection about the negativity. Right?\nYour reflection rings true for my experience too. I forgot to mention that having a LIKE button is a great way to connect to the site without always leaving a comment. I felt badly for bloggers who didn't make it easy to do either thing. I would feel like I was writing in a closet ALONE. Great job with your blog. I follow you, so I know.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Scammers stole the identity and banking data of Richard Overton, a 112-year-old World War II veteran living in Austin, Texas.\nHis family has set up a GoFundMe page to help cover his $15,000-a-month care expenses.\nSeniors lose an estimated $36 billion a year to financial scams and abuse.\nA photograph of Richard Overton in his military uniform is propped up against a folded American flag in his home.\nNo one is immune from identity theft and fraud. Not even, it seems, America's oldest World War II veteran.\nLast week, relatives of Richard Arvin Overton, a 112-year-old Austin, Texas man, reportedly discovered a number of withdrawals from his bank account made over the last several months.\nThe amount of money stolen was not disclosed.\nOverton requires round-the-clock medical care and resides at home. His care costs about $15,000 a month, and his family has set up a GoFundMe to help foot the bill. A phone call to Volma Overton, the veteran's cousin, was not immediately returned.\nIt's a harsh reminder that, while anyone can be a victim of fraud, older adults are especially vulnerable to scams.\nA 2015 report from True Link Financial showed that senior citizens lose more than $36 billion a year to financial scams and abuse, including identity theft.\nThieves have set their sights on scamming older Americans: They impersonate Medicare representatives and either ask you to submit payment for a new ID or they call to confirm your bank account and Social Security number.\nJust to be sure, Medicare sends all of its information via mail and does not contact beneficiaries on the phone. Avoid handing out your personal data to anyone who calls you.\nIn February, the Justice Department announced a broad enforcement sweep, charging more 250 defendants of defrauding seniors in a variety of schemes including mass mailing and telemarketing.\nLosses from these plots are estimated at more than $500 million.\nSee below for a list of common elder fraud cases.\nA portrait gifted to Richard Overton on his 110th birthday by a fellow veteran.\nEarlier this year, the Financial Industry Regulatory Authority put into effect two regulatory changes to help protect seniors from financial exploitation.\nOne requires broker-dealers to ask their clients for a trusted person whom the advisor can contact in the event of suspected fraud or cognitive decline.\nThe other rule change will require broker-dealers to place a temporary hold on a requested account withdrawal if they believe financial exploitation is at work. The firm would have to notify the contact of the hold.\nSee below for additional steps you can take to shore up your identity and banking information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Britain's Financial Conduct Authority (FCA) has begun an investigation into whether Deutsche Bank breached anti money laundering laws for its Moscow clients, it has emerged.\nFinancial watch dogs in Germany and UK will be analyzing how quickly Deutsche Bank reported the suspicious trades to the authorities.\nThis latest investigation by the FCA comes after the bank was fined a record $2.5bn by US and UK regulators in April for manipulating the Libor rate \u2013 the benchmark for interest rates on trillions of dollars of financial contracts.\nDeputy Prime Minister Igor Shuvalov explained at the time that the main idea of the Bill to allow people who have made any mistakes in their business activities to declare the property and then become completely law-abiding citizens.\n#AceMarketsNews \u2013 July.16: LONDON (Reuters) \u2013 The FTSE 100 edged higher on Thursday, though without matching the gains seen across the eurozone after Greece's parliament approved austerity measures, with U.S.-exposed stocks getting a lift.\nThe FTSE 100 index closed up 0.6 percent, at 6,796.45 points, lagging a 1.5 percent rise for the Euro STOXX 50, with peripheral euro zone markets benefitting from a drop in yields after the dramatic vote in the Greek parliament.\nAlso in focus were comments from Federal Reserve Chair Janet Yellen, who said on Wednesday the U.S. central bank remains on track to raise interest rates this year.\nHer comments sent the dollar higher, improving the prospects for U.S.-facing companies such as Pearson, Intercontinental and Wolseley, which rose between 1 and 3 percent.\n\"There seems to be an American theme on the leaderboard today,\" said Richard Hunter, head of equities at Hargreaves Lansdown.\nAmong other gainers, Rio Tinto added 1 percent after posting a sharp rise in second quarter iron ore output from a year ago, even as selling prices deteriorate and bad weather disrupted operations.\nDixons Carphone climbed 1.5 percent after beating forecasts with a 21 percent rise in yearly profit.\nThe electrical goods and mobile phone retailer, which was formed in a merger last year, also said its integration was progressing well.\n\"In its first full year as a merged company, Dixons Carphone is emerging as a worthy challenger to Amazon,\" Simon Johnstone, analyst at Kantar Retail, said in a note.\nOn the downside, Britain's biggest sporting goods retailer, Sports Direct, underperformed the market with a 0.3 percent rise. Although it posted a 21 percent rise in profit, the company said it would cut its bonus scheme earnings target for 2016.\n\"The company has revised down targets for 2015\/16 after failing to make acquisitions and the stock has slumped to the foot of the board as a result,\" said Tony Cross, market analyst at Trustnet Direct.\nBT closed up 0.6 percent after the British telecoms regulator said it might be made to spin off its networks unit, which wholesales capacity to rivals like Sky and TalkTalk, to boost competition in the broadband market.\n#AceFinanceNews \u2013 July.16: LONDON (Reuters) \u2013 Britain has cut its stake in Lloyds Banking Group by a further percentage point to under 15 percent, accelerating its drive to return the bailed-out lender to full private ownership.\nThe latest sale means the government has so far raised more than 13 billion pounds ($20 billion) from selling its shares in the bank, having pumped 20.5 billion pounds into Lloyds during the 2007\/09 financial crisis leaving it with a 43 percent stake.\nFinance minister George Osborne has said he wants to return Britain's banking assets to the private sector at a faster pace and is also looking to sell at least three quarters of the government's stake in Royal Bank of Scotland over the next five years.\nSince last December, Britain's shares in Lloyds have been sold on the stock market by Morgan Stanley through a trading plan that allows for regular disposals provided the price is above the government's target of 73.6 pence per share.\nThat has enabled the government to accelerate the rate of selling its shares and its stake has fallen to 14.9 percent from 24.9 percent at the start of the plan.\nThe plan is scheduled to run until the end of the year but its success raises the prospect that UK Financial Investments (UKFI), which manages the government's stake, may close it early in order to hold back shares to offer to private retail investors.\nThe final sale could see the government offer around 4 to 5 percent of Lloyds shares to retail investors at a discount to the market price and could take place next March, according to industry sources.\nThe value of the government's remaining stake stands at 9 billion pounds, based on the current share price. At 1445 GMT, Lloyds shares were up 1 percent at 86.49 pence.\nOsborne is determined to sell some of the shares to retail investors, mirroring the 1980s privatisation drive of then Prime Minister Margaret Thatcher aimed at encouraging ordinary Britons to invest in companies.\nLloyds is attractive to private investors because of the dividends it is expected to pay out in the coming years. Prior to its bailout, it was one of the highest dividend paying stocks in Britain, handing over half its profit to shareholders in 2005 and 2006.\nThe plans to return at least 50 percent of its sustainable earnings to shareholders and that could include extra one-off dividends, further increasing its appeal to retail investors.\nLloyds still faces outstanding issues that could yet cause a rethink to the sale plans, including a review of the industry by Britain's competition watchdog and the mounting bill of compensating customers mis-sold loan insurance, for which Lloyds has already set aside 12 billion pounds.\n#AceFinanceNews \u2013 RUSSIA\/TEHRAN:July.16: Moscow and Teheran are in the process of negotiations, which could see Russia supply Sukhoi Superjet passenger aircraft to Iran, Russia's Transport Minister Maksim Sokolov said on Wednesday.\n\"Such talks are being held,\" Sokolov said, as cited by Reuters. \"And not only Superjets, but also other technology is being met with certain interest from our Iranian comrades.\" The Sukhoi passenger aircraft commenced operations in 2011.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Durham University has been invited to join the Coimbra Group, an association of European universities committed to creating academic and cultural links to promote internationalisation, academic collaboration, excellence in learning and research, and service to society.\nSeveral of the Group's 38 existing members already have research or teaching links with Durham including Aarhus, Bologna, Heidelberg, Jena, Siena, Uppsala, the eponymous Coimbra and existing UK members Edinburgh and Bristol.\nDurham's membership will bring mutual benefits in terms of research collaboration, influence over European policy-making, staff and student exchanges between members and intelligence on funding opportunities.\nDurham also aims to increase its participation in impactful research that drives economic growth and Europe's global competitiveness.\nEstablished in 1985, the Coimbra Group has three policy pillars, research, education and outreach, and has working groups on topics that include student mobility, education innovation, heritage and STEM (sciences, technology, engineering and mathematics). It enjoys strong links with a sister network in Latin America.\nThe Group ratified Durham's membership on 10 June 2016 following negotiations led by Durham University's Dean for Internationalisation, Professor Danny Donoghue.\nDurham University is also a member of the Matariki and University of the Arctic international networks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"and I'm interested in this 2018 Honda Odyssey.\nNothing more better then to make the customer feel comfortable.\nI was born in New York City, Raised in Valrico, Florida. I am Fluent in both English and Spanish.My favorite hobby is Mountain Biking at our local trails such as Alafia State Park and Balm-Boyette scrub if you haven't been, you should definitely go! My favorite car of all time is the Acura NSX. My goal as an Auto Consultant is to give you the best and easiest experience in your journey to purchasing a new Vehicle. Whether it be new or old my goal is to make sure we get you in the right seat for the right price. Thanks for taking your time to look at my profile looking forward to seeing you soon!\nI was born and raised in Florida and played basketball and football most of my life. I am the proud father of one son, who is in the United States Navy, and two daughters, both of whom are students at the University of Florida. My wife and I enjoy boating, hanging out with our pets (four dogs and two cats), riding our Harley, traveling, and spending time at home. Customer service before, during, and after the sale is my number one priority. It is very important to me to build a relationship of trust with my customers and establish a no-hassle, no-pressure, honest, and pleasant car-buying experience.\nI was born in Colombia and was brought here by my parents at a young age, the car business has allowed me to help customers have a fun, easy and honest experience out of a process that is not everyone's favorite pastime. When I am not working I like visit any and every food establishment possible since I have been know to be a food connoisseur or at the gym burning off the delicacies I have indulged in. Please reach out to me with any questions or concerns via email, call or text or just visit me at Brandon Honda. Email: Jrios1@brandonhonda.com Cell:863-393-6255.\nOriginally from Upstate New York. I have been selling cars for over 5 years started out with Toyota. I have been with Honda, specifically Brandon Honda since March 2017. I try to make the process as easy and stress free as possible for my clients and create lifelong relationships with them.\nI will treat you the same way, i take care of my parents, all Love, Commitment, and Joy. Making sure you have an awesome experience. Sebastien Brandon Honda.\nWhen I am not helping put dreams in driveways, you can find me chasing my own. Whether that be bouncing my stand-up routines off friends, studying to further my career, or hiding my super hero identity from those close to me; I like to keep busy. I am a mom to one, and am excited to build my career here with Brandon Honda where the company emphasizes their employees well-being. Call, text, or e-mail me and I will be happy to help you out!\nOriginally from Egypt ,love travelling, boating and family time .\nI am originally from Connecticut, but been in Florida most of my life. I enjoy good food, beaches, music, and laughter. I believe in what you put in the universe, comes back!\nBrandon Honda Certified Real Nationwide Lifetime Warranty. Unlimited time, unlimited miles, Carfax 1 Owner Certified, Sunroof \/ Moonroof, Leather Seats, Super Clean, Clean Carfax Certified, Low Miles, Like New, Must See, This Is A BHO Car Brandon Honda Original !!, Purchased New and Serviced Right Here at Brandon Honda, Well Maintained, Dealer Serviced, Carfax Service Records Available, Push Button Start, Bluetooth, Remote Start, Backup Camera, Born and Raised at Brandon Honda !!., Priced Below Retail, Quite Possibly the Best Vehicle for the Money, Brandon Honda NO Gimmicks Just Great Service and Prices !!, Won't last long At This Price. Red 2018 Honda Odyssey EX-L FWD 9-Speed Automatic 3.5L V6 SOHC i-VTEC 24VReal Nationwide Lifetime Warranty. Unlimited time, unlimited miles honored anywhere in the USA and Canada. Recent Arrival! Odometer is 4748 miles below market average! 19\/28 City\/Highway MPGAt Brandon Honda we simplified the way you buy your next new or pre-owned vehicle. First we give you a REAL Nationwide Lifetime Warranty to protect your investment. Unlimited time, unlimited miles, honored anywhere in the country. At BrandonHonda.com, within seconds you can unlock your best price and configure your complete deal online. How about a cash offer for your vehicle? You will get your trade value and cash offer within minutes, and we will buy your car even if you do not buy from us. Free credit score, it's true; you will get your free credit score and pre-approval. Your credit will not show an inquiry and you do NOT need to provide a Social Security Number. And with our exclusive Showroom2Go program, you can arrange a test drive and complete your purchase at your home or office. With one visit to BrandonHonda.com, you can get ALL the information you need to make an informed decision. Give us a call 813-518-6048.\nYour inquiry has been submitted to Brandon Honda.\nYour inquiry has been submitted to at Brandon Honda.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Re: Cloud web hosting packages?\nTheir support is professional, communicative, and quite Fast. Hostforweb.com pricing, bandwidth, and storage are excellent.\nThey help you with any small issue you may have and will install stuff for you because Host-ed is professional web host.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This deeply moisturizing cr\u00e8me safely and effectively relieves scaly, flaky and itchy dry skin associated with Psoriasis and Eczema. It absorbs easily and gently calms and soothes hot, irritated skin. This product is made with a safe and effective blend of herbal extracts and skin vitamins such as Neem, Burdock, Bearberry and Vitamins A and E to help soften and soothe dry, irritated, scaling skin.\nIt penetrates deeply to help reduce redness and encourage healthy skin. This pleasant moisturizing formula does not contain steroids nor any coal or pine tars. For use on the scalp, in the evening, apply to affected areas using a cotton swab. Put a shower cap on to protect your sheets and clothing, and leave the cr\u00e8me on overnight. In the morning, wash and condition hair as normal. This may be used as often as needed.\nScoop a small amount of Derma E Psorzema Creme onto the fingertips and apply to any affected area of the skin. May be used as often as desired to relieve symptoms.\nPurified Water (Aqua), Caprylic\/Capric Triglyceride, Glycerin, Cetyl Alcohol, Stearic Acid, Peg-100 Stearate, Glyceryl Stearate, Melia Azadirachta (Neem) Leaf Extract, Arctium Lappa (Burdock) Root Extract, Zanthoxylum Zanthoxyloides (Fagara) Bark Extract, Coptis Chinensis (Chinese Golden Thread) Root Extract, Berberis Aquifolium (Barberry) Extract, Arctostaphylos Uva Ursi (Bearberry) Leaf Extract, Allantoin, Retinyl Palmitate (Vitamin A), Tocopheryl Acetate (Vitamin E), Organic Chamomilla Recutita (Matricaria) Flower Extract* (.5%), Organic Simmondsia Chinensis (Jojoba) Seed Oil* (.25%), Polysorbate 60, Dimethicone, Potassium Sorbate, Phenoxyethanol, Ethylhexylglycerin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You need a system that is fully functional to serve your customers. You want reliable access for administration. Reliability is what you get with Transform POS. As your one-stop-shop, we keep you stocked with the supplies you need-- paper rolls, ribbons, loyalty cards and more. Transform brings you exclusive pricing and scheduled shipping.\nTransformPOS brings over a decade of professional experience installing and servicing up-to-date systems designed for the hospitality industry. We partner with you to select the platforms best suited to your business. We offer a solid reputation for on-site service and support-- 24\/7\/365 -- to keep your systems running smoothly and securely. With TransformPOS as your partner in hospitality, you can keep customers happy, staff productive, and your business thriving.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyiug b/data_all_eng_slimpj/shuffled/split2/finalzzyiug
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index 0000000000000000000000000000000000000000..b07da7790141d6efd96276e52020a6840e6a3959
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"With the excellent performance of BIG-IP platform, enterprises are able to save all costs for management, power consumption, space and cooling and to lay the foundation for business growth on the basis of integrated system. F5 is the best enterprise switch with the top performance level for replacing an old L4 or L7 switch or for accelerating SSL traffic and securing security system visibility.\nBIG-IP hardware platforms are specially designed for application delivery. One system makes it possible to implement such functions as sever load balancing, global data center load balancing, web application firewall, HTTP acceleration, spam filtering, and WAN optimization.\nBIG-IP systems not only have high-performance SSL and compression function, but provide improved connection management in the way of removing intensively running processes of applications in order for these resources to be used more efficiently.\nBy adding an actual security x-x-layer to BIG-IP system, it is possible to provide the security and full-packet filter engine to block and limit access as default in a detailed way.\nBy setting BIG-IP hardware functions for easily managing out-of-band, front panel, worm upgrade, remote booting, and USB support, it is possible to save the cost for upgrade and maintenance.\nBy implementing hot-swappable factor, spare power supply & fan, compact flash, multi-boot support and permanent management, it is possible to guarantee the establishment of the essential enterprise infrastructure in reliable hardware.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Articulated by FDA Commissioner Scott Gottlieb, the new approach recognizes it's the deadly smoke caused by setting tobacco on fire sucking into the lungs that are responsible for the deaths of 480,000 Americans each year \u2014 not nicotine.\nPart of this new strategy, according to Gottlieb, is ensuring new and innovative products that are significantly less dangerous than smoking and can draw people away from cigarettes can come to market.\nThe FDA is now facing its first true test of this approach and has the opportunity to show its warm words for the role of innovation and consumer choice in reducing smoking-related deaths can be translated into action.\nA new generation of products that heat but don't burn tobacco is helping millions of smokers across the world kick the habit. Approximately three million smokers have switched to Philip Morris' heat-not-burn battery-powered product IQOS.\nCurrently sold in cities in 26 countries but not the U.S., IQOS and the tobacco sticks that are inserted into it are flying off the shelves, with new factories being set up in Europe to cope with demand.\nIn Japan, the only country where IQOS is sold nationwide, it already accounts for 10 percent of the tobacco market after little more than a year on the market. Critically, 72 percent of those using IQOS in Japan are doing so exclusively, dropping regular cigarettes altogether.\nWhy are these new tobacco products so appealing? For one thing, they use real leaf tobacco which gives users a taste and experience that mimics smoking more closely than other reduced-risk nicotine products.\nBut by far their biggest appeal is the opportunity they present to smokers who have so far been unwilling or unable to quit smoking.\nBecause there is no combustion in IQOS, there is no smoke, which means levels of lethal chemicals found in cigarette smoke are dramatically reduced.\nAccording to PMI Science, 18 chemicals whose concentrations the FDA deems as essential to evaluating the relative safety of tobacco products are reduced by 90-95 percent in IQOS.\nWhile all of this research is published, the public will be forgiven for treating health claims from Big Tobacco with extreme skepticism.\nThankfully, PMI Science is not the only source of the relative risks of IQOS. Konstantinos Farsalinos, a research fellow at the Onassis Cardiac Surgery Center in Athens and world-renowned researcher on e-cigarettes, conducted investigations into the toxicity of IQOS in a non-PMI lab and found broadly the same results.\nFurthermore, clinical trials carried out in Japan and the U.S. show smokers who switched exclusively to IQOS saw reductions in levels of the harmful and potentially-harmful chemicals similar to levels seen in smokers who ceased cigarette use for the duration of the study.\nU.S. smokers don't have access to IQOS at present but an application is working its way through the bowels of the FDA. But just as important as allowing IQOS onto the market in the first place is allowing consumers access to the full knowledge about its safety relative to cigarettes.\nIf IQOS and products like it are to going to be successful in transitioning people away from smoking it is imperative that smokers know switching exclusively to IQOS will massively reduce their exposure to lethal chemicals and could save their lives.\nTo that end, my colleague at the Reason Foundation, Brian Fojtik, and I filed a comment in support of PMI's application for a Modified Risk Tobacco Product application.\nShould the MRTP be granted, the full benefits of switching from cigarettes to IQOS could be communicated to consumers and therefore attract the maximum number of smokers who are interested in quitting smoking but have found other methods unsatisfactory.\nIf the FDA is serious about providing smokers the best possible options to quit and avoid an early grave, granting the MRTP for IQOS will be a major win for tobacco harm reduction and will provide the tools and information smokers desperately need.\nThis column first appeared in the Washington Examiner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":".: That's an Excellent Question - Klister Covered by Hard Wax :: Ski XCOttawa.ca :: Skiing in Ottawa and Gatineau Park :.\nQuestions #3 - What is the best base binder and how do you decide on klister or regular binder? And what is the point of covering with hard wax?\nAlso, how do you prepare or know the right wax to use all the time . . . are you going by logs and experience? I suppose you need to understand the snow structure, humidity, temp of snow, etc. Not all of that is available on the Gats web site, which is why I like your teams report. Hopefully I can learn to be a more effective skier.\nWhen to use regular wax, binder or klister and how to decide?\nklister - in some situations.\nBinders are always heated into the base so that they adhere better. Then you should allow them to cool before applying more wax for the day's conditions.\nUsually when new dry snow gets cold quickly the snow crystals stay sharp and abbrassive like sand paper. In these conditions, regular kick wax will work, but may wear off your skis. Here the solution is to iron in 1-2 layers of the \"wax du jour\" which will be green or blue. This will usualy suffice for 15-20km. If you are skiing 50km then you should consider special binder wax. Every company makes one, and they are rather inexpensive for the improved skiing experience you'll have when you can still stide uphill long after your friends have stopped to re-wax.\nWhen binder really becomes important is after a thaw-freeze and the tracks get icy. We're blessed with excellent grooming in the Gatineau park, and after all the rain we've had it's amazing that they can till the ice into a softer surface. But often those little chunks of ice that have been pressed into a track are quite abbrassive, and quickly tear the wax off your skis. This is the time when either regular binder or klister needs to be applied. The line between which to use isn't clear cut. Just a week ago a lot of XC Ottawa was skiing on Teho green (a binder), while others were on klister. You have to decide based on just how icy the track is. The icier it is the more you should think about klister. If it's really and fast icy, then klister for sure.\nNow the worst of scenarios: ice covered with a thin layer of new dry snow or big chunks of ice mixed\/covered with a little new snow. (These are the days for which skating was likely invented!) The trick here is to wax for both the ice and the new snow.\nApply a thin layer of klister as a binder, heat it in with either an iron, a torch, a heat gun or a blow drier. Use the palm of your hand or thumb to smooth the klister.\nCrayon hard wax on top of the klister (this helps prevent the klister from icing up due to the new snow). Use thin layers, smoothing them out gently with a cork. Gently means you are not corking hard enough to heat any of the klister so that it mixes with the covering hard wax. You want to apply just enough hard wax to completely cover the klister...otherwise it will grab that fresh snow.\nIn some cases when it's really cold the grip will come from the covering wax, in which case you will need to apply several thin layers.\nKlister covered with hard wax is a tricky art to learn. The reason for this is that ice and fresh snow will often be mixed together a different temperature! You might need blue klister as a base at -10\u00b0C and colder, violet from -5\u00b0C to -10\u00b0C, and red klister near 0\u00b0C. But the principle is the same; apply klister, freeze the klister, and then cover the klister without mixing. Follow those simple rules and hopefully you get glide, grip that lasts and doesn't drag or ice up.\nHow to know the wax of the day?\nThat comes with experience, trial and error, reading, and recording your successes and failures really helps. Waxing is an art you must practice.\nQ & A: That's an Excellent Question: What Makes a Good Wax Iron?\nQ & A: What's with the webcam?\nHumour: Tom, Ed and the Great Horn.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's where the now-official Anthony Bourdain Food Trail will stop in N.J.\nIt's Valentine's Day, Jersey. What's not to love?\nBest in Jersey: Can you guess this historic spot?\nBest in Jersey: Can you guess the name of this north Jersey landmark?\nBest in Jersey: Can you guess this scenic spot in south Jersey?\nThis is your spot for the latest health news, including insights from top industry experts. Learn about the latest in technology, wellness and nutrition, as well as Castle Connolly's annual powerhouse list of Top Doctors.\nFrom clever decor and home repair ideas to irresistible recipes and hosting tips, this is your home inspiration hub. And let's not forget our jaw-dropping home makeover features and favorite local real estate listings.\nFamily always comes first, so we put the focus on what matters most. Whether you're looking to find fun options for the kids, select a summer camp or locate the top neighborhoods and schools, we've got you covered.\nNew Jersey is home to fascinating places and personalities. Get the scoop on the hottest fashion and lifestyle trends, explore must-see events and get to know the people who are transforming the Garden State.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meet me at the Marion Public Library Author Fair on March 24th, 1-3 PM. We have some great local authors from around town.\nThis entry was posted in poetry, Uncategorized and tagged books, fiction, poetry, promotion, sale. Bookmark the permalink.\nI've always liked fairy tales and this humorous adaptation of Cinderella is no exception. The protagonist falling apart (from leprosy) was somehow both intriguing and at times emotional. I laughed a great deal at the subtle surprises thrown into the traditional narrative; especially the fate of the wicked stepmother!\nThis entry was posted in book review and tagged book review, books, cinderelleper, fiction, ford forkum. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..631c119c814899638d0fdec11ba84d57ef4c7c39
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@@ -0,0 +1,5 @@
+{"text":"Today's AM fix was USD 1,292.00, EUR 942.65 and GBP 764.81 per ounce.\nYesterday's AM fix was USD 1,291.50, EUR 943.46 and GBP 767.56 per ounce.\nGold climbed $1.10 or 0.08% yesterday to $1,294.70\/oz.\nSilver rose $0.03 or 0.15% to $19.42\/oz.\nGold is marginally lower today at $1,293.50\/oz and remains in lock down in an unusually tight range between $1,287\/oz and $1,306\/oz this week. Gold in Singapore, which often sets the price trend in Asia, traded at $1,292.23\/oz prior to a bounce to just over $1,295\/oz.\nGold has been in a very narrow range between $1,283\/oz and $1,310\/oz for a month now. There are a lot of things going on underneath the surface of the calm gold market this month. That superficial calm is likely to give way in the coming days as we appear on the verge of a sharp move to the upside or downside once gold breaks out of the recent range.\nA break below $1,283\/oz is possible and this could see gold quickly fall to test longer term support at $1,200\/oz. This is likely if the technical traders and computer manipulations continue to dominate. However, should physical demand pick up on rising geopolitical tensions and the return of Indian demand with the easing of import duties, gold should quickly challenge resistance at $1,385\/oz and $1,418\/oz.\nThe Russian central bank has again increased its gold reserves by another 900,000 ounces worth $1.17 billion in April.\nRussia's gold reserves rose to 34.4 million troy ounces in April, from 33.5 million troy ounces in March, the Russian central bank announced on its website yesterday. The value of its gold holdings rose to $44.30 billion as of May 1, compared with $43.36 billion a month earlier, it added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Co-founder of Domaine de la C\u00f4te, Rajat Parr is a two-time James Beard Award winner and internationally regarded as one of the world's foremost experts on wine. Rajat Parr grew up in Calcutta, India, before journeying to the United States where he chose to pursue the cellar instead of the kitchen. He honed his wine knowledge and tasting skills, while beginning a career that would see him become one the world's most celebrated sommeliers.\nParr not only frequently traveled abroad to European wine regions; he also became close with several of California's top winemakers. This ultimately translated into bottling his own wines, where Parr made the transition from sommelier to producer. During this period, he became closer to Santa Barbara County winemaker Sashi Moorman, as they found mutual interest in pushing the boundaries of California wine, every vintage seeing how much earlier they could harvest to achieve more balanced alcohols, racier acidity, and purer flavors. Most recently, in early 2014, Parr took over the historic and acclaimed Seven Springs Vineyard in Oregon. Beyond his wineries, today Parr remains partners with Michael Mina in his two RN74 restaurants. He also co-authored the James-Beard-Award winning book Secrets of the Sommeliers, and co-founded the trend-setting California non-profit wine group, In Pursuit of Balance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Wayne Main Street Office (located at 35000 Sims Street) will be closed from Friday, March 27th until Monday, April 6th. Executive Director Lindsey Wooten will be attending the National Main Street Conference in Atlanta, GA. She will have access to email and can also be reached by phone if you need anything (734) 629-6822. Otherwise, she will be back on Monday April 6th!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. Eligibility. The Summer Styles Sweepstakes (the \"Sweepstakes\") is open only to individuals who are legal residents and physically located in the fifty (50) states in the United States or the District of Columbia and who are at least 21 years of age at the time of entry. Employees, officers and directors of America's Collectibles Network, Inc., d\/b\/a Jewelry Television\u00ae (\"Sponsor\"), Sponsor's parent companies, and each of their respective affiliates, subsidiaries, advertising and promotion agencies, distributors and other prize suppliers (collectively with Sponsor, the \"Sweepstakes Entities\"), and each of such employees', officers' and directors' immediate family members and\/or those living in the same household (whether legally related or not) of each are not eligible to enter the Sweepstakes or win a prize. For purposes of this Sweepstakes, immediate family members are defined as spouse, partner, parents, legal guardians, in-laws, grandparents, siblings, children and grandchildren and those living in the same household shall mean people who share the same residence at least three (3) months a year, whether legally related or not. Void where prohibited by law. By participating in the Sweepstakes, each entrant unconditionally accepts and agrees to comply with and abide by these \"Official Rules\" and the decisions of Sponsor, including the interpretation of these Official Rules and its exercise of discretion, which will be final and binding in all respects.\n2. Entry Period. The Sweepstakes begins on or about 12:01 AM Eastern Time on May 21, 2018 and all entries must be received by the Sponsor on or before 11:59 PM Eastern Time on August 31, 2018 (the \"Entry Period\").\nGrand Prize Drawing: To enter the Grand Prize drawing, each person must fill out and submit one completed registration form during the Entry Period online at www.jtv.com\/summer (the \"Website\"). This registration form will enter each person into the Grand Prize drawing. Limit one (1) entry per person per each day of the Entry Period. If an entrant submits more than one (1) entry for any day during the Entry Period, only the first entry submitted for that day will be accepted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rabbits are far from cheap and easy pets. They have specific and often complex needs in regard to not only their diet, but housing, health and general well-being.\nWhilst we reinforce the message that Rabbits do NOT make good pets for children, they do in-fact make good FAMILY pets when cared for properly; your child should never have sole responsibility for any living being, it is our role as adults to not only teach our children the correct way to interact and provide for our companion animals, but have sole responsibility for their health and welfare too.\nThe following pages will provide a basic guide to those needs which we hope you will find helpful and allow you to give your bunnies a long, hoppy contented life as part of the family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadaly b/data_all_eng_slimpj/shuffled/split2/finalzzzadaly
new file mode 100644
index 0000000000000000000000000000000000000000..10022d0b107659ba499308d9812080f979bd0087
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The federal capital gains tax has been around in some form since 1916 and has occasionally been a hotbed of debate in some national elections. There have been very few big changes to the tax, but a few minor tweaks have occurred. These changes deal with reporting gains and losses, and they affect how much tax you'll pay to the federal government on gains.\nThe IRS rolled out a new tax form for reporting capital gains and losses from stocks, bonds, mutual funds, and similar investments during the 2011 tax year. Investment transactions are now reported on Form 8949, Sales and Other Dispositions of Capital Assets. The IRS also revised Schedule D and Form 1099-B to accommodate the new Form 8949.\nCongress passed the Emergency Economic Stabilization Act in 2008. The EESA requires that brokers report the cost basis of investment products to investors and to the IRS on Form 1099-B.\nIn theory, having brokers report cost basis along with sales proceeds was intended to reduce the burden on individual taxpayers to maintain extensive records on their investments. It was thought that it would simplify the tax-reporting process.\nBefore the EESA, the 1099-B only reported information about the sale of investments, such as the date of sale and sale proceeds. Taxpayers then had to provide the purchase date and the purchase price when reporting the transactions on their tax returns.\nMany brokers were already providing gain\/loss reports as supplemental information with their annual reports and 1099-Bs, but cost basis information was included directly on the 1099-B beginning in 2011 if the broker was required to supply that information.\nBrokers have been required to provide cost basis for stocks acquired beginning in 2011, and for mutual funds and stocks in a dividend reinvestment plan beginning in 2012. The reporting requirement began in 2013 for all other investment products acquired beginning in that year.\nThe IRS substantially revised Form 1099-B to facilitate this cost basis reporting, and Schedule D now functions as a summary of all capital gains transactions. Both of these forms are needed when filing a Form 8949.\nForm 8949 is the reporting form used by individuals, businesses, and estates and trusts to report capital gains. The IRS provides the form and instructions for completing it.\nTaxes on capital gains are based on the length of time the investment is held. Investments with realized capital gains occurring less than one year after purchase will have a higher tax rate than investments held for longer than one year.\nBeginning in 2018, the TCJA has broad-ranging changes for taxpayers including some provisions for capital gains. Investors will continue to be taxed at ordinary income tax rates on short-term capital gains. Long-term capital gains are taxed in three buckets: 0 percent, 15 percent, and 20 percent.\nForm 8949 provides for the reporting of both short term and long term capital gains. All types of taxpayers should receive the necessary tax reporting details for Form 8949 on a 1099-B statement.\nTo report certain transactions you don't have to report on Form 8949.\nTo report a gain or loss from a partnership, S corporation, estate, or trust.\nTo report capital gain distributions not reported directly on Schedule 1 (Form 1040), line 13 (or effectively connected capital gain distributions not reported directly on Form 1040NR, line 14).\nTo report a capital loss carryover from 2017 to 2018.\nBeginning in 2018, long-term capital gains have their very own tax brackets that will determine how much tax you will pay.\nLong-term gains were taxed at either 0 percent, 15 percent, and 20 percent before the passage of the Tax Cuts and Jobs Act (TCJA), and that's still the case. But these rates used to be tied to your ordinary income tax bracket.\nIn other words, if your ordinary income put you in a 33 percent tax bracket on your overall income, you'd fall into the 15 percent long-term capital gains bracket. The 20 percent rate was reserved for those who fell into the top ordinary income tax bracket of 39.6 percent.\nThe TCJA gives long-term gains and qualified dividends their very own tax brackets effective January 1, 2018, but these still correlate with your overall income. Currently, the 0 percent capital gains rate applies to income up to $38,600 if you're single, $51,700 if you qualify as head of household, or $77,200 if you're married and file a joint return.\nBeyond this, you'll jump into the 20 percent capital gains rate. This bracket covers incomes of up to $425,800 if you're single, $452,400 if you're head of household, or $479,000 if you're married and filing jointly. In other words, this bracket now covers a large swath of investors. You'll pay the 20 percent rate on your gains only if your overall income exceeds these levels.\nThe IRS is additionally issuing a whole new tax form for the 2018 tax year, a Form 1040 that will replace the old 1040, as well as Forms 1040A and 1040Ez. The new tax form is supposed to be shorter and simpler, but it comes with multiple schedules\u2014along with all the old forms and schedules like Schedule D and Form 8949 that still exist.\nRest assured that you'll still be entering the same information, but in different places\u2014as though preparing your taxes wasn't already complicated enough.\nCost-basis reporting by brokers will never fully and completely eliminate the need for taxpayers to maintain their own records because basis reporting applies only to newly acquired shares that have occurred since these changes were made. If you purchased stocks before 2011, mutual fund shares before 2012, or bonds before 2013, basis reporting on these assets won't be reported on Form 1099-B.\nThat information will likely be found in other reports or data, however, such as brokerage statements, year-end reports, or trade confirmations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00ab \u00bfQue tienen en com\u00fan Unicef, Gucci y Rihanna?\nThis entry was posted on Wednesday, November 5th, 2008 at 1:34 am by Patricia and is filed under Celebrities, Editoriales, fot\u00f3grafos, Portadas, Revistas, Vanity. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tiefenthal may have to cancel the party.\nBarbara took Sjaak in his arms.\nDonnie wants his mother's approval.\nTheodore aspires to be a teacher.\nMonty is no longer covered by his parents' health insurance.\nEarle and Wilmer both look worried and exhausted.\nStanly finished his coffee and went off to work.\nI wonder why Turkeer wanted us to be here.\nWhy do you think Jim prefers living in the country?\nThe chance that Heather will get arrested is very slim.\nKyu would rather swim than play tennis.\nAmy loves Rayan's sense of humor.\nMagnus won't be killing anybody else.\nIt's easy to understand why Eric doesn't like Dennis.\nRamesh took off his jacket.\nI heard you paid a visit to Suu.\nWhen are you going to see Galen again?\nI'm sure Alastair will know what's best.\nI thought Giovanni couldn't ride a bicycle.\nI plan to go to Boston next Monday to see Ti.\nCharlie still feels a little guilty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enjoy snacks, treats, and beverages inspired from the wizard world.\nAnswer trivia to win prizes!\nThis is an all ages event, and you must RSVP to attend. Please call the studio @ 403-786-8002 or facebook message. Seats are limited.\nThe only other cost is whichever pottery you choose to paint.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eleven SWOSU biology students and Associate Professor Dr. Lisa Boggs will present information on Wednesday, November 1, generated from an August trip to Yellowstone National Park.\nEleven biology students at Southwestern Oklahoma State University will present information on Wednesday, November 1, generated from an August trip to Yellowstone National Park.\nThe students, under the guidance of SWOSU Associate Professor Dr. Lisa Boggs, participated in an ecological study of the Greater Yellowstone Ecosystem and will present poster and oral presentations from 4-5 p.m. in Room 212 of the Old Science Building on the SWOSU campus in Weatherford. The public is invited to attend and interact with the students at the free event. Refreshments will be served.\nDuring the five days spent learning in the park, the students encountered: bison, wolves, grizzly bears, mountain goats, big horn sheep, elk, deer, moose, coyote, tundra swans, badgers and other wild animals. They took part in three citizen science days, collecting data for the National Park Service and assisting in invasive weed control, bison studies and stream quality sampling.\nStudents who participated in the course are: Chris Arganbright, Sayre; Sarah Dengler, Yukon; Rachel Fast, Ryan Johnson, Jennifer Prophet and Julie Thiessen, all of Weatherford; Bailee Fehring, Sterling; Cassie Hill, Comanche; Lyndsey Hendricks, Bessie; Whitnie Holten, Anadarko; and Allison Statton, Oklahoma City.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadghe b/data_all_eng_slimpj/shuffled/split2/finalzzzadghe
new file mode 100644
index 0000000000000000000000000000000000000000..6efde313610ad4475474cec4af3bc7306f769b4c
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@@ -0,0 +1,5 @@
+{"text":"Yet another motherboard from Shuttle that sounds like it should be a car... I mean with a name like AV45GTR who wouldn't think of some wicked little roadster?\nToday, we're going to be look at Shuttle's AV45GTR, a P4 motherboard based on the controversial P4X266A. Intel had claimed they did not licence VIA Technologies to build a P4 chipset, however VIA bought up S3 who did have a license for the P4 so they felt they didn't need get anther one from Intel. Intel took VIA to court over the matter, and now the two companies aren't exactly friendly.\nWhen the Pentium 4's were first released, they only worked with RDRAM which was very expensive. Intel later released the i845 chipset which allowed P4's to be used with SDRAM, but SDRAM just didn't have enough memory bandwidth to keep the P4 happy.\nEventually DDR was introduced as an alternative. DDR has twice the bandwidth of SDRAM, but not as much as RDRAM. The trade off comes in price, as DDR is quite a bit cheaper than RDRAM. SDRAM\/DDR RAM has shot up over 100% in price over the last few months and now the price of DDR and RDRAM is about the same. Now a days there are quite a few DDR chipsets available for the P4 - from Intel's own i845-D (DDR 266) to SiS's 645 (DDR 333) and ALi's ALLADiN-P4 (DDR 333).\nThe layout of the AV45GTR is pretty clean, the ATX power connector is up at the top of the board so it doesn't get in the way of anything, however the P4 connector and the Aux. power connector are on the other side of the board, which isn't that great especially with those cables hanging over the CPU socket.\nOne 4x AGP, five PCI's and three DIMM slots are pretty much standard equipment these days, it also has on board Hardware 5.1 audio care of the C-Media 8738 and on board Ultra\/133 RAID controller using the Highpoint HPT372. We were a little disappointed that the board had no free fan headers after installing the CPU heatsink.\nAs mentioned above, the P4X266A is basically the P4X266 with the same improved memory controller found in the KT266A Athlon chipset. With the competition having faster chip sets, VIA felt the pressure to release a new revision of the P4X266 in order to keep up to speed. It supports up to 3GB of PACE\/2100 DDR RAM.\nThe VT8233 southfield is basically a tweaked version of their notorious 686B. The VT8233 supports Ultra\/100 hard drives, 4 USB 1.1 ports and has both the VIA AC'97 codec integrated as well as the modem MAC codec.\nIt was nice to see that even though the VIA VT8233 southbridge has integrated audio, that Shuttle decided to use the C-Media 8738 5.1 as a sound card which has about the same audio quality as the SB Live! 5.1. We didn't experience any problems when watching DVD's or playing our favorite games in the office, RTCW and MOHAA.\nThe AK45GTR also has an IDE RAID controller that uses the HighPoint HPT372 chipset which allows for up to two IDE channels supporting RAID's 0, 1 and 0+1.\nRAID 0 is not considered a true RAID since, there's no data redundancy. What it does is take two drives of the same size\/configuration and stripes them. Meaning it makes one big drive out of two smaller ones.\nThis improves performance by cutting hard drive latency in 1\/2 since the data is divided equally and written on the two hard drives. This also has the added effect of increasing data bandwidth by two. The reason it's not considered a RAID is because if one drive fails, all data is lost.\nRAID 1 mirrors two drives of the same size. In theory, if one drive fails, the other will take over as the primary hard drive and the system will operate as normal. This is what is supposed to happen with a SCSI hard drive setup and it actually works pretty well here. However IDE doesn't like devices that are unstable and the system will usually crash if there is a fatal error with one drive. Since the data has been written on the other drive, all is not lost however.\nWith RAID 0+1, you need four hard drives of the same configuration\/size. What RAID 0+1 does is stripes two sets of two hard drives, one set for a RAID 0 configuration and the other for RAID 1. What this does is offer the best of both worlds, the high performance of RAID 0, with 100% data redundancy of RAID 1. Hence the name RAID 0+1. The only downside would be the need for four identical hard drives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Put on your play clothes, how that logo feel?\nYou don't wanna miss a moment, where's your NoDoz pills?\nYe and a gang of stars take a trip Uptown for the holidays on this new hip-hop Christmas carol. Add this one to the playlist for the holiday season.\nRun\u2013D.M.C. set the standard for hip-hop Christmas jams with \"Christmas in Hollis\" (referring to Hollis, Queens). The title \"Christmas in Harlem\" is presumably homage to the Run\u2013D.M.C. classic.\nFun fact: The iTunes version of this song only has Kanye's and Cyhi the Prynce's verses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On July 3rd, iPinYou declared that its core product: DSP Optimus\u2122 system will be free of use for advertisers and agencies. Any service charge will also be completely transparent, including the cost of media purchase. Meanwhile, the company also announced that it has completed 20 millionUSD of Round C financing, with the investment primarily used to to expand on mobile and to expand into foreign markets.\nThe strategy of providing free access to iPinYou's DSP is aimed at spurring the development of programmatic buying in the China market, lowering the cost for more advertisers to embrace and try programmatic. And at the same time, advertisers can experience transparency and real time optimization for their marketing campaigns. It is estimated that the advertisers who use the iPinYou's DSP system will see an improvement of at least 30% in media efficiency as well as enjoy the benefits of additional user insight reporting.\nIn the past two years, as programmatic buying becomes increasingly more popular, a number of companies with various backgrounds, such as SEM and Affiliate Networks, have started to label themselves as DSPs. This led most advertisers to believe that a DSP's core function is to aggregate and buy media. However, the real value of a DSP is to present a powerful matching system, helping advertisers increase efficiency of transactional media ad costs, and not on gaining profits off of an arbitrage media model. iPinYou aims to solve this issue by supporting media purchase transparency.\nGrace Huang, the founder and CEO of iPinYou, said that \"through the experience of serving over 1000 customers, iPinYou's DSP system is China's leading programmatic buying system with the most complete functionalities, largest reach in inventory, the most advanced optimization. This year witnessed iPinYou as the first DSP in China to launch PDB (Programmatic Direct Buy) functionalities. iPinYou decided to open up their DSP product system for free in order to help brands cut down on the cost of experiencing this new programmatic trend.\" Only by sufficient utilization of the DSP system can advertisers understand its benefits. Like taxi hailing apps, only through new user experiences will users learn its benefits and become long-term users.\nIn the era of media fragmentation, the concept of big data and Internet are increasingly embraced and accepted. Due to the high cost of development of DSP system, iPinYou has in the past only offered its DSP system at no profit to several 4A companies and large e-commerce customers. However, iPinYou is now offering its free DSP system for all advertisers and agencies, which will, to a large extent, make more brands really experience big data marketing and make programmatic buying a standard marketing strategy.\nTo coordinate this program, iPinYou will also provide certification training in optimization to help Advertisers operate programmatic systems. Since 2013, iPinYou jointly with other experts in the field: world-renowned 4A agencies- OMG, IPG, Aegis, ad exchange platforms- Sina's SAX third-party monitoring institutions - AdMaster, Miaozhen Systems, IResearch and DCCI Internet data center together hosted such professional training. It is estimated that during the year iPinYou will train nearly 1000 professional certified optimization analysts for the industry. Over the past few years, with the strategic cooperation of 4A companies, iPinYou has become their primary partner in programmatic technology service, and in effect, iPinYou helped train nearly 60 certified optimization analysts throughout China. For those customers who haven't used the iPinYou DSP system, programmatic buying can also be accessed through these important partners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This was a regular maintenance visit for our car. The service was great and I was done before the scheduled time.\nI brought in my Odyssey for an inspection and a recall that had to be done and was very pleased with the service and friendliness of everyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Smith & Ouzman has decades of experience in providing services and support for local authorities and government departments.\nFor democracy to work, strict procedures need to be adhered to. At Smith & Ouzman we have the products, processes and experience that are essential for the successful delivery of electoral and government print projects.\nWe work to your timetable, not ours, to put you and your team back in control of your schedule and we guarantee that our dedicated expert account manager will oversee your project from start to finish.\nachieve standardised output from many variable forms of data from all leading elections software providers.\nadapt to very high or low volumes always with the same result \u2013 on time delivery without exception.\nWe maintain numerous certifications in the areas of quality assurance, security, environmental and corporate social responsibility so you can be safe in the knowledge that you are working with a responsible, reputable and experienced supplier.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadtnc b/data_all_eng_slimpj/shuffled/split2/finalzzzadtnc
new file mode 100644
index 0000000000000000000000000000000000000000..c1eee200c0292089c667644e645c055a9f1e04cf
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Common Name: Milletia, Spatholobus stem, Millettia vine.\nCaulis Spatholobi is produced chiefly in the provinces of Guangdong, Guangxi, and Yunnan. The crude herb is cut into slices or decocted into gelatin for medication.\nFunctions: 1. To invigorate the blood; 2. To nourish the blood; 3. To relax and activate the tendons.\n1. Blood deficiency and blood stagnation manifested as irregular menstruation, dysmenorrhea or menorrhea. Spatholobus stem (Jixueteng) is used with Chinese angelica root (Danggui), White peony root (Baishao) and Chuanxiong rhizome (Chuanxiong).\n2. Soreness and painful joints caused by invasion of wind-cold-damp manifested as numbness of the limbs, or paralysis caused by poor nourishment of tendons and muscles due to deficient blood. Spatholobus stem (Jixueteng) is used with Chinese angelica root (Danggui), Chaenomeles fruit (Mugua), Chuanxiong rhizome (Chuanxiong) and Mulberry mistletoe (Sangjisheng).\nDosage: 10-15 g , or in a large dose up to 60 g, decocted in water for an oral dose.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"British charities are gearing up to assist in the disaster relief activities. Please give whatever you can.\nThe British Red Cross are accepting donations via their website or over the telephone (08450 53 53 53).\nCornwall charity ShelterBox is collecting donations to send out emergency shelters and supplies to people made homeless by the earthquake and tsunami.\nYou can also donate at globalgiving.co.uk.\nPlease let us know if you are aware of any other sites that should be listed here.\nPosted on Monday, March 14th, 2011 at 2:15 am, and filed under News. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. They can be charged wirelessly - It's a challenge to find ways to differentiate phones -- especially as once top-tier features become common even in lower-end phones -- but Samsung has built one into the S6 line. Both phones will be able to be charged wirelessly. \"With fully embedded WPC and PMA certified wireless charging technology, the Galaxy S6 and Galaxy S6 edge are setting a new industry standard for universal wireless charging,\" the company said in its press release. \"The devices work with any wireless pad available in the market that supports WPC and PMA standards.\" While that could be seen as a gimmick, it's actually a practical innovation that solves a problem for customers. Samsung also said that the S6 phone will offer fast wired charging, \"1.5 times faster than the Galaxy S5, providing about 4 hours of usage after only 10 minutes of charging.\" \"It's built right in,\" Shin said. \"No more fumbling for power cords in the dark. That's design with a purpose. That's the future.\"\nNext articleKumbe ashaoa! Khaligraph Jones' wife is pregnant!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is a soldering iron, and why do you need one?\nRight here in this article we will go through all of the facts you really must know about using soldering irons for your daily electronics tasks, like soldering circuits, soldering wire, etc. There is a lot to know about these hand tools that obviously cannot be covered completely in one article. So we'll only go over the most important details. For a start let's consider the question: what actually is a soldering iron?\nIn it's most basic form a soldering iron\/pencil is just a heated metal rod that's used to melt solder. You hold onto a plastic handle to prevent burning yourself while using it. The tip of the heated rod melts a special wire known as \"solder\".\nSolder is sometimes made of lead, but due to health concerns often isn't these days. The idea when you're soldering is to use the molten solder to join two pieces (often wires) together. Once they have been joined with solder that allows electrical current to flow between them.\nSo as you should have worked out by now - they are very useful tools. They are mostly used in the field of electronics repair, manufacture and installation. However they have some less well known uses also, such as making jewelry or creating artwork.\nDepending on the function you would intend to use your soldering iron for there are some things you must consider. If you're just a hobbyist having some fun, a cheap basic iron will do. However if your requirements are those of a professional you'll want something high-end. There's a large variety of different types to suit being used for various different vocations. The most basic form the much like a pencil, and is also known as a pencil iron.\nThese type of soldering iron are typically low wattage, often around 15 watts. This mean that they produce a lower heat, and have a long heat-up time. They also don't come with built in temperature control. Don't let all this put you off though, because these are good budget options. These are perfect for someone just learning how to soldering who isn't yet sure if they won't to become a pro.\nAt the other end of the price range is the temperature controlled \"Soldering Stations\". Soldering stations are perfect for you, if you're a professional who needs all the specialized features available incorporated into one high performance product. Thee come with adjustable temperature. They also sometimes have a sensor in the soldering tip, which can allow for precise digital adjustment of the heat. If you're a pro, and working with circuits you have to get one of these in your tool box.\nAnother option you might want to consider is soldering tweeters. These are particularly useful if you're working on very small circuits, and surface mount components. Hopefully you find plenty of value in this article. Please feel free to get in touch if you have more to add about soldering, or electronics.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is it possible to load a roaming profile from the WL500G? I have SAMBA enabled and it's working. I tried it but the computer tells me this: Coul'd not log you in. The is probably caused because there is already a copy of the profile on the server or the security settings won't allow it. I checked it, but there's only one profile on the WL500G. Does somebody know how to fix this?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaeoho b/data_all_eng_slimpj/shuffled/split2/finalzzzaeoho
new file mode 100644
index 0000000000000000000000000000000000000000..06b34c91634ed8f08b246ddad1be783b42101c7a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaeoho
@@ -0,0 +1,5 @@
+{"text":"Joe Krush. New York: Harcourt, Brace and Company, (1954). First edition. SIGNED by Welty on title page. Illustrated with drawings by Joe Krush. Very good with four light tape shadows on paper boards and rear endpaper a little rumpled with a shallow stain along outside edge and evidence of something removed near top. (Perhaps ex-library but there are no other signs.) In a near fine dust jacket with two inconspicuous tears and minor soiling to rear panel which is just slightly darkened along the lower edge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UnCiv is a turn-based strategy game based on the incredibly popular Civilization franchise. This open source game brings all the magic of the saga created by Sid Meier in 1991 to smartphones, and all for free!\nYour mission in UnCiv is to create the most successful civilization in history. To do this, you'll need to settle new cities, manage your resources, defeat your enemies, and act diplomatically. And if you want to win, you'll have to micromanage each aspect of the game, all with your ultimate strategy in mind.\nAlthough the graphics in UnCiv are relatively simple compared to the actual game, this game still has plenty to offer as you think ahead to make your civilization the best in the world. Not only will you have to explore the map and expand your civilization, but you'll also have to research new technologies to keep up with the changing times.\nUnCiv is a great open source version of this classic video games. You don't have to pay anything or watch any ads in order to play one of the best strategy games in the genre. What's more, it only takes up 6 megabytes, so it's the perfect alternative for smartphones with less storage space.\nNo opinions about UnCiv. Be the first!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Source tarballs from the various stable releases are also available in the downloads section.\nPatches sent upstream for review must be based on the current git tree and not on stable releases, unless the bug only affects a stable release.\nEvery submitted patch must be signed off by its author.\nYou may contribute to LXC either by sending a patch or patchset directly on the lxc-devel mailing-list.\nYou can use git format-patch to generate mailable patch.\nBeware of \"copy\/paste\" on mail clients as they can break tabs and lines (see git send-email or git imap-send).\nFork the repository, create a branch, commit you work (with -s !), and push it.\nThen follow the github's doc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When I was in middle school, our version of the \"Beatles vs. Stones\" debate was \"Nirvana vs. Pearl Jam\". Arguments were waged, lines were drawn, and there was no safe passage for fence riders. I was firmly in the Nirvana camp and decried Pearl Jam as a \"jam band\" with pompous 70's guitar solos and a poser front man. In truth, I really liked Pearl Jam too and just had a harder time learning their stuff on guitar. But as I said, you had to choose only one and Nirvana was it for me. I remember the first time I saw \"Smells Like Teen Spirit\" on MTV and I was blown away. It not only looked really cool, it looked like something I could be a part of; a concert in a gymnasium, jeans and a t-shirt, and a guitar style that didn't seem all that difficult or forced. I had to be tricky when buying Nevermind because my parents weren't in the habit of getting me albums with naked babies on the front. But I succeeded in scoring my very own copy and without being hyperbolic, I can say that it literally changed my life.\nNevermind was the first album that I ever remember really immersing myself in. Immersed as in constant, repeated listenings to get every lyric, vocal line and musical nuance. I didn't understand what the lyrics were supposed to actually mean most of the time, but you didn't have to. \"How\" Kurt sang and played was as important as \"what\" he was singing. His delivery conveyed the emotion he was going for much more than mere words could. Like tons of other kids my age, Nevermind, made me want to pick up a guitar and get it all out. Kurt made us believe that all of the confusion, isolation and general weirdness of growing up could somehow be waded through with a chuck of wood and some strings. The journey could even be advanced along if you were lucky enough to get your hands on a distortion pedal too!\nMy parents bought me my first guitar for $60 off of a friend from church. It was a cheap Les Paul knock-off, had a neck like a telephone pole and was the heaviest thing I had ever strapped on. I was in love. It was the most beautiful inanimate object I had ever seen. The blisters on my fingers were glorious war wounds and the cramping in my neck was a labor of love I gladly endured for the chance to be on this journey. While I was learning to play guitar, Nevermind, was one of my most trusted road maps. The first riff I ever learned to play was \"Come As You Are\". The first full song I learned to play from top to bottom was \"Smells Like Teen Spirit\". Sure I learned other people's songs, but I was determined to get the whole Nevermind album down. Even if I could never get my voice to get that razor-sharp wail in it, I could almost get my guitar to sound like it was speaking the same language as his. But that's like saying that a toddler and Shakespeare both speak English.\nThe first concert I ever played as a \"band\" was in the summer of 1995. I had started playing around after school with my friends Rusty, Ryan and Geoff and we were all learning together. Geoff's sister Rebecca was having a birthday party and wanted us to play for it. That's the kind of motivation you need when you are starting out so you can transform from aimless to motivated. We had to actually learn some songs! Somehow I got roped into singing as well and we did our best renditions of \"Smells Like Teen Spirit\", \"When I Come Around\" by Green Day, \"Come Out And Play\" by The Offspring, \"Zombie\" by The Cranberries, \"Plush\" by Stone Temple Pilots and a few others. I remember singing horribly off-key, playing the guitar line to \"Zombie\" on the wrong string, trying to stage dive onto people that where standing right in front of me and having a great time doing it! Everyone, including the awe-struck crowd, had a blast! Somewhere there is a video floating around of the show and hopefully it never sees the light of day again!\nEven now, whenever I hear those first few clean guitar chords being chunked out or watch the \"Smells Like Teen Spirit\" video, I am instantly transported back to a place of beginning and discovery. It's truly a magical thing. It speaks not only to the power of music but to the power of specific artists who's music can find a way into your life at crucial moments, guide you into new areas, and never leave you once they are done showing you things. Maybe that's the thing, maybe the illumination is not just a one time thing. I don't know what \"with the lights out, it's less dangerous\" means, but I know how I felt when I sang it then and how I feel when I sing it now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks for signing up to the tempertemper newsletter!\nYou can now sit back and look forward to the next mail-out. Don't forget, you can unsubscribe at any time via the footer on the email you receive or by heading to the unsubscribe page.\nGoogle have made a few changes lately. The most recent in the pipeline is the way URLs (web addresses) are displayed in their search results.\nI've made the informed decision to stop supporting browsers of a certain age. Is it time you did likewise?\nUploading an image to your website isn't without its pitfalls. Formatting images is often a combination of preparation and your CMS's wizardry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaetcu b/data_all_eng_slimpj/shuffled/split2/finalzzzaetcu
new file mode 100644
index 0000000000000000000000000000000000000000..68f94c467c088cfcee500d5f35ac32db0e1f2b63
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaetcu
@@ -0,0 +1,5 @@
+{"text":"The clinic is located in a pale blue mid terrance building with a small garden wall around it.\nThe building is on the right hand side of the road as the road curves round to the right (coming up the road from the castle roundabout) and opposite TJ Warehouse. Other landmarks are The Congregational Church, the Credit Union on Unity Street and Home and Away properties all on the opposite side of the road. Coming down the North road you pass Carrickfergus train station, pass under the railway bridge, past St. Brides car park (old Tesco's car Park) past a small parade of shops on your left then some terraced housing before you'll see the clinic on the left hand side of the street.\nFROM LARNE come along the Shore Road (A2) to Belfast, continue along the shore until you pass Carrickfergus Castle. At the roundabout take the 3rd exit up 'Albert Road' Bottom of the North road.\nTrains - Carrickfergus Station is a 4-6 Minute walk. Come out the station down to the main road cross at the level crossing turn right and walk down Albert road to the clinic on your left.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Calculate the present value of an annuity of 600.00 per quarter for 2 years, if interest is compounded every 6 months at the nominal rate of 9%.\nIn this problem, the payment (effective) period (quarterly) is not the same as the compounding period (6 monthly).\nCalculate the effective interest rate for the payment period.\nThe effective interest rate for the payment period is calculated using the effective interest rate formula.\nThe compounding period is 6 monthly, so there are 2 compounding periods in a year (m). The payment period is quarterly, so there is 1\/2 a compounding period in a payment period (n).\nThe present value is calculated using the present value of an annuity formula. The annuity is 600.00 per quarter for 2 years (8 quarters) at an effective interest rate of 2.225% per quarter.\nThe payments on this annuity are made quarterly (effective period) and do not coincide with the compounding periods which are six monthly.\nTo solve the problem it is necessary to first calculate the effective rate of interest for each payment period, and then use this effective rate in the present value of an annuity formula.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Outdoor Fireplace Kitchen For Your Place Of Residence Outdoor Fireplace Kitchen presented by Kenny on Thursday, August 25th, 2016 in category Modern.\nSee also Outdoor Fireplace Kitchen For Your Condo Outdoor Fireplace Kitchen from Modern Topic.\nHere we have another image Outdoor Fireplace Kitchen Presented To Your Place Of Residence Outdoor Fireplace Kitchen featured under Outdoor Fireplace Kitchen For Your Place Of Residence Outdoor Fireplace Kitchen. We hope you enjoyed it and if you want to download the pictures in high quality, simply right click the image and choose \"Save As\". Thanks for reading Outdoor Fireplace Kitchen For Your Place Of Residence Outdoor Fireplace Kitchen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I made this \"live action\" music video on Sunday! It's my first attempt at something like this. The cosplayer's name is Mindy Mica and she was kind enough to spare some time for this shoot. There was an event at Klang Parade and I noticed Christmas decorations were up so I thought of making a festive themed one. The song is Jingle Bells ga Tomaranai \u2013 the new Aqours collaboration song with the School Idol Festival mobile game. There was a snowglobe thing which had air pumped inside to inflate it into a spherical shape at the mall, this was where most of the \"snow\" scenes were done. You can actually go inside, it's RM 5 per pax and you can probably fit 3-4 people in there. The rest was filmed around the mall.\nThis is one of those by-fans-for-fans things, it's content I like to see as a Love Liver so I thought of producing some to put out there too.\nYeah, I was pretty surprised to hear about it too. It was posted by a fellow Love Liver earlier today so I immediately drove down to Avenue K to check it out. I later found out that the gashapon machines were brought in by Shojikiya. It's one of those dangly things which you can hang on stuff \u2013 the available characters are Chika, You, Riko, Dia and Kanan. Each roll is RM 15 (RM 5 tokens x 3) \u2013 if you're not familiar with gashapon machines, they are novelties or collectibles stored in a plastic \"egg\" which is then cranked out of a machine. What comes out is random, you can't choose the one you want, so it adds an element of chance (and gets collectors to spend more).\nI wanted to get 1 of each girl but my limit was RM 150, which gives me 10 attempts. I did manage to get one of each but it took me till the 9th try (RM 135) to get Riko. I also had a brain fart and kept on referring to her as Ruby in the video. To be fair, she has a similar hair color but as a Love Live fan, I shouldn't have made the mistake in the first place. Oh well. It did give me a chance to try out moving text though. I want to challenge myself to add a new effect that I learned into each video or make it better somehow. I've also been trying to add a lot of energy but I think this is a bit too much. Haha.\nI was actually quite hyped up and on a lot of caffeine since I was about to start my daily torture exercise regime before I rushed down for this.\nI know, it's early. Elichika's birthday is actually on the 21st of October but the Malaysian Love Livers celebrated it early last Sunday at Alpacasso Cafe @ AEON, Midvalley. I've always seen people on reddit celebrating various waifu birthdays and I thought it sounded like a lot of fun. I've just never seen any locally until I met Anson via the COSFAIR event where I filmed the MilkyWay girls. It might sound strange to celebrate a 2D anime character's birthday but it's actually a lot of fun to hang out with like-minded people and share something we're all interested in. I think the fellowship comes first, the character's birthday is just the pretext of meeting up. That's the best way I can explain it to someone who doesn't understand why people do this.\nI've never quite gotten into Eli Ayase (the character) but I do like the seiyuu (voice actress). Nanjou is one of my two \"best girls\" (an anime term denoting your favorite) \u2013 the other being Pile (voicing Maki). I even started listening to fripside lately, the songs for A Certain Scientific Railgun are pretty good.\nI just finished editing and putting together subtitles for the video of Eli's birthday celebrations with the Malaysian Love Livers. I know my Chinese isn't very good, and some of you might be surprised to learn that Mandarin is in fact my first language \u2013 I just can't read or write. I also can't speak it very well especially when people start using harder words so the subtitles might not be 100% accurate. I'm a little puzzled why I don't understand the harder words too since I spoke Mandarin until I went to New Zealand to do high school at 15. 15 years is a long time to pick up a language! I think it's mostly coz I never immersed myself in Chinese culture \u2013 I don't listen to Chinese songs etc so I never had any reason to go beyond simple conversational use.\nI still wanted to do the narration in Chinese though coz I like the idea of bilingual videos. However, I encourage you to turn on CC for English subtitles since they're probably more accurate and conveys what I mean better. I wrote the text in English first before attempting to translate it verbally into spoken Mandarin and needless to say, it took a lot of takes.\nSomeone even brought a Kit Kat + M&M custom made cake for Eli's birthday! We were the ones to eat it though. Haha.\nYes, I'm calling it a documentary coz I'm being pretentious and I spent 12 hours editing it. I was actually quite lucky coz I never read my emails (sorry guys but it's true). It just so happened that I couldn't sleep (see previous post) so I was going through all my stuff and I found a flyer about a cosplay event by Evolve Mall. I went to the event page and came across a promo video featuring the MilkyWay girls.\nI thought it was really good. I've been watching a lot of cosplay dance covers by Love Live groups coz I want to make my own and also to improve my video editing skills. I creeped on the MilkyWay posts and found a girl called Vicky, who turned out to be one of the leaders. I messaged her the night before (yesterday) and asked if I could tag along and film them, not just their performance, but also shadow them and eat with them and interview them.\nI gave her my number and she said she'll discuss it with the other group members and got back to me 20 minutes later saying it was okay with everyone. Thus, I went there this morning to film them. They actually had two performances but I only managed to get the first one since I had to leave at 3 pm. It was a really fun experience since I'm a huge fan of Love Live and it's always nice to see a local group do so well.\nI have been trying to make more YouTube videos lately \u2013 this is my second one specifically for this purpose. My better half and a LLSIF friend told me the first one was ultra shaky and barely watchable and I took that feedback to heart. This time, I spent 12 times the amount of post-production work on the video. Well, it probably won't take so long for a pro but I just installed Adobe Premiere CC a few days ago!\nI did learn a lot of things about video editing too, and I found out that I really like doing it. I like talking to people and getting good video and I like arranging the sequences to make a cohesive film. The music sequences in this one isn't as good as I want coz I didn't have a tripod with those swing\/tilt\/stabilize things. I'll need to get one so next time I do something like this, I'll have one static camera pointing at the stage (like how the music video is) on tripod while I actually hold the second camera.\nI did have a second one today but I couldn't operate both. The idea is the static one is for wide screen whole stage shots while the one you're holding (with stabilizer) goes for close ups. Thus, I can focus on the trio who's currently singing and this will allow me to cut from the full stage shot to a close up. Also, in the interview segment I had a lot of trouble deciphering the words.\nThat's completely my fault coz I didn't wire everyone up. I also didn't have 9 microphone units but they're actually quite cheap so I'll definitely get some if I want to explore more on doing videos like this. When you record something with external sound, it gives you a lot of flexibility especially with B footage and the voice overs can even go on unrelated stuff, which I wish I did for editing ease.\nAlso, my Chinese isn't very good. I can understand conversational Mandarin but once you get to difficult words, I'll be totally lost. I went to New Zealand at quite a young age to study and I didn't learn Chinese in Malaysia anyway. I had to puzzle out the meaning via context (very useful) and a bit of guesswork so I hope the translations are somewhat accurate.\nI wanted translations and closed captions for YouTube coz the video is almost entirely in Chinese. YouTube is a very English-centric medium, unlike Nico Nico Douga or Youku so English subs are necessary. I want it when I watch the Love Live Sunshine radio dramas and skits anyway, wouldn't be able to understand otherwise. Turn on CC if you want to read the English subs, I also spent a few hours writing them.\nI'm getting a lot better in many ways and even though I'm very frustrated at not being able to sleep (the insomnia is killing me) at least I'm learning new skills and doing stuff that I love. Thanks to Vicky and the rest of the MilkyWay cosplay group for letting me hang out with them today! It was a really interesting experience and I'll see you all at the next event.\nI'll be concentrating more on YouTube videos now so let me know what you think \u2013 feedback and criticism is always welcome. Cheers!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SKF W6201-2Z bearing original in Ireland are widely used in industrial drive, agriculture, compressors, motors and generators, construction, industrial fan, industrial transmission and other fields. SKF W6201-2Z bearing original in Ireland can ensure high quality for more than 80% raw material of SKF W6201-2Z bearing original in Ireland provided by the steel plant.\nExplorer series bearing of SKF W6201-2Z bearing original in Ireland are better than any brand bearings currently in bearing performance , lifetime, design, and the use of performance. Explorer series bearings of SKF W6201-2Z bearing original in Ireland have more advantages ,such as higher meet rated , lower noise, reducing warranty costs greatly, increasing running time of the machine.\nskf w6202 2z bearing is one of the best products we sell, our company is also one of the best skf w6202-2z bearing company. Expect us to cooperate.\nProduct Description. This 6203 2Z SKF bearing is a Shielded Deep Groove Radial Ball Bearing with a standard radial internal clearance The bearing's dimensions are 17x40x12.\n6201-2Z SKF bearings produced by the Japan's IKO company has become the representative of the famous brand product. 6201-2Z SKF have get high reputation from the users of 6201-2Z SKF.\nMore details about SKF W6201-2Z. Wholesale brand SKF W6201-2Z Spherical Roller Bearings, cheap SKF W6201-2Z price agent, sizes, supplier; Supplies SKF W6201-2Z ball bearings trader; We are the professional exporter and distributor of SKF W6201-2Z dealer, factory, cost.\nFind great deals on eBay for 6201 bearing skf and 6203 bearing skf. Shop with confidence.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaeuxn b/data_all_eng_slimpj/shuffled/split2/finalzzzaeuxn
new file mode 100644
index 0000000000000000000000000000000000000000..a75b89eac0b410a295149f0512128c1823b2de45
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaeuxn
@@ -0,0 +1,5 @@
+{"text":"When the engine is custom, and frequently replace the cylinder and piston.\nBasically it is a new use.\nCompression adjustment due to the thickness of the base gasket is easy.\nBase gasket has been commercially available, However, I like human beings Let's make cut out from the base gasket sheet.\nAlso, make a base gasket squeeze-out by increasing cut the part on purpose, The custom of secret\u2026 You can also use in applications such as.\nBase gasket sheet is cut out with a cutter.\nFor example, it is possible to vary the thickness about this.\nLeft is commercially available base gasket, Base gasket that right is made of me.\nMy base gasket is a little rough, but there is no problem in use. Is what important selection of thickness than that.\nRather , compression than the piston to be used will vary, and Cylinder head metal cutting of the piston alignment is a lot of work.\nIt is also important piston selection in the race.\nYou might be able to save money and find a hit while changing the thickness of the base gasket First.\nIt is easy to understand and compare.\nIt is OK you stack a thin base gasket, and to use, or It is also OK to use the thick ones. I like better to be adjusted in one piece.\nYou can use the clearance adjustment in such a case.\nI think the person who adjusts the thickness of the base gasket is most by such reason.\nIf and increasing the stroke of a long crank,Clearance adjustment will be required.\nWhen will teamed up to If irresponsible, It might be the connecting rod and the piston to hit somewhere.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Murder on the Orient Express (2017) Movie Torrent Download free BluRay 720p HD, . 2017 by twentieth Century Fox.. Best iptv channels from around the world. Full ultra hd iptv channels and Most updated iptv channel m3u list are waiting for you.\nShaatir The Fox 3 3gp Movie Download . Welcome to VideosZ.com! Unlimited HD downloads . 58cc09ba06 . free download bengali .. Fantastic Mr Fox (2009) Movies 720p Download , Fantastic Mr Fox (2009) Movies Download , Fantastic Mr Fox (2009) HD Bluray Movies 720p Download. Insaaf Ki Jung full movie hd 1080p kickass download . . .\nCompare 1080p Lg & Get Best Prices at LowPriceShopper!. download free the Har Raat Naya Saudagar full movie in hindi . . High-definition . online Shaatir The Fox part 2 full . full-hd-1080p-hindi .. Hindi Movie Shaatir The Fox Hai Full Movie Download .\nShaatir The Fox Hd Mp4 Download Shaatir The Fox Hd Mp4 Download. .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Magnesium Hanging Anode. Kit includes 4 meters of cable with an M10 terminal, 1x mounting bracket and 2x re-useable rubber tie wraps. For use in Fresh water.\nFixing Details: M10 Terminal for connection to wire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> If the third is a serious problem (which I doubt) then it is one that we are going to have to live with in the short term and which will go away by itself in time. The first two can be handled by deft editorial wording and quick reassurances, along the lines of \"No knowledge of the RDF spec documents is necessary in order to use JSON-LD.\"\n> But my main concern is not about how early the RDF connection is spelled out, but that it does get stated clearly and unambiguously and normatively in the specification document*somewhere*. This is after all a*standards specification*, not a propaganda or advertising effort. (Or a \"for dummies\" tutorial.) It needs to state the facts clearly and unambiguously, and to clearly state the relationships to other standards. To re-define the RDF data model, calling it by another name, and not stating that it is a re-statement of the RDF abstract graph syntax, is just wrong.\nresponse, the statement above is what I agree with.\nwell-intentioned the motives are for doing it. \"\nassociated with the letters R-D-F. That's it.\nNext message: Kingsley Idehen: \"RDF's challenge\"\nNext in thread: Markus Lanthaler: \"RE: Official response to RDF-ISSUE-132: JSON-LD\/RDF Alignment -- Sub-issue on the re-definition of Linked Data\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MEGHAN Markle has won the affection of Prince Harry's niece and nephews with a hidden talent.\nMeghan Markle married husband Prince Harry in May 2018, and it looks as though the Duchess of Sussex has made quite the impression on her husband's niece Princess Charlotte and nephews Prince George and Prince Louis.\nThe former Suits actress is reportedly in Kate Middleton and Prince William's children's good books, after impressing with her cookery skills.\nMeghan \u2013 who has been plagued by pregnancy rumours \u2013 made it known she was a talented cook on her website The Tig, which has since closed down.\nAnd it has been reported her culinary skills are what have wooed over the youngsters.\nA source told Entertainment Tonight: \"Meghan is an avid cook.\n\"Meghan often pops around to visit with Kate and the kids dropping off treats.\"\nGeorge, Charlotte and Louis are over the moon when they receive Meghan's delicious dishes, but they also enjoy getting involved in the cooking process too.\nDuring an appearance at a Commonwealth Big Lunch event in March this year, Kate reportedly said: \"I've done that with George and Charlotte\u2014making pizza dough.\n\"They love it because they can get their hands messy.\"\nAnd Kate's little ones love nothing more than a sweet treat, as Charlotte was spotted eating an ice cream on a family day out.\nAlthough Kate and William have access to a team of professional chefs, the couple do enjoy indulging in a takeaway.\nIn a previous interview with BBC Radio One the Duke and Duchess of Cambridge admitted their favourite fast food dish was a curry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaezmu b/data_all_eng_slimpj/shuffled/split2/finalzzzaezmu
new file mode 100644
index 0000000000000000000000000000000000000000..b88e561a5258b2591100c4c506956cca8012ce0e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaezmu
@@ -0,0 +1,5 @@
+{"text":"I would like take some time from my day to wish a swell Good Friday to any of the Christian or Catholic members of the forum, and I hope you all have fun today under His good graces.\nTo the rest of the community: two days until Easter. Prepare thyselves.\nGood Friday to you, too, sir!\nLocation : Queensland, Australia (Country), Australia (Continent), Terra, Sol, Milky Way, My Stomach.\nHmmdeed. I didn't get anything, so I just decided to snag myself an Easter Bilby (Australian Bunny) and feast on it's living corpse which is made of a delicious Cadbury dairy milk chocolate which is good for the entire family, your self image, and adding excess kilos to your ripped frame.\nI'm gonna be that guy and wish everybody a belated Happy Passover. I continued the tradition of slaughtering my neighbor's lambs and smearing the blood on the front door, because damnit I'm the firstborn and I'm not taking any chances.\nShade, don't make fun of him, it's not his fault. We both know your swag is so great even straight men want a piece of you.\ntoday was a good cheap candy day. I got cheap candy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Aquila, we believe that our network of now about 400 or so brokers and ISOs are more successful if we give them more data and information about the performance of our platform. It's the Aquila position that operational transparency is a competitive advantage to help our partners capture more revenue. We hypothesize that if we increase transparency of how we fund small businesses on our platform, we will make it easier, and cheaper, for all our ISO and broker partners to win more revenues and commissions.\nTherefore, today we publish a new post that we hope catches your attention: How to make $9,000 a day on Aquila in three easy steps. In the next paragraphs, we'll tell you the formulas that we used to put this model together and how we think you can make it work for your business \u2013 where you're just printing money, on Aquila.\nWhat is the Aquila Funding Funnel?\nOver the past two years, we at Aquila have completed over 5,000 small business financing underwriting via our automated underwriting and data clustering platform. Every day, our system uses bank transaction data from thousands of business bank accounts to predict credit and default risk. In this time, our funnel from registration toward funding a small business has become more and more predictable. We call this pattern our Aquila Funding Funnel. We've broken this funnel up into 3 steps, Registration, Cash Flow Analysis, and Wire deposit.\nThe Aquila Funding Funnel Registration step is a little different as it begins with an immediate bank login at Merchant registration. Immediate bank account login allows Aquila to analyze and underwrite each merchant in minutes. As such, Aquila can make decisions on merchants in as little as five minutes of a merchant registration or even faster.\nHow to use the Aquila Funding Funnel and make $9,000 a day?\nGiven that the Aquila conversion rate is a straightforward 10% to funding, the next question is: How do we get to $9,000.00 in revenues per day? That number is very large. Most ISOs should know that Aquila's median funding is now $15,000 for merchants with paper quality in the C \u2013 D range. Merchants with bank accounts that can sustain a daily debit of around $200 per day are a great fit for a minimum of $15,000 of funding. At 10 points in commissions, this means an ISO can win $3,000 in commissions from winning two wires out of 20 merchants. Thus, the problem of winning $9,000 a day in commissions is a marketing problem to find enough qualified merchants to fill the funnel down to the wire.\nAquila focuses on cash flow underwriting only. As such, we never pull FICO scores of merchants. We don't need to. Each new merchant that lands our platform makes our clustering algorithm (which uses a combined OLS-and-semi-supervised-classification model of gamma-rayed-eigen values blah blah blah) smarter and more accurate and faster to respond with each customer. As such, we have no need to pull the credit score of any merchant we evaluate for small business funding. This means, that sourcing merchants that will end up at the bottom of your funnel just got much easier because our sales partners don't have to wait for a credit check.\nSo what's the catch for the money?\nIf you've read this far, you're guessing there must be a catch. Making money is NEVER that easy! We agree. There is a catch! The catch is sourcing 80 qualified businesses, each day, that need a maximum of $15,000 in funding with weak or problematic credit histories. It's also important to source these 80 business, per day, at a cost that is less than $9,000 a day or $45,000 a week. As former brokers ourselves, we know that it's possible to spend at least $125 per lead and have a much lower conversion rate to a qualified lead to fund. However, when we were brokers we didn't know where to concentrate our marketing dollars. We guessed and spent it anywhere and everywhere. We did not target our marketing. Thus, we think that ISOs who have more accurate channels for their Aquila landing page will win links to 80 qualified businesses.\nWe recommend ISOs read our other blog posts on the most profitable small business NAICS codes for the first half of 2018 or the most profitable small business NAICS codes for 2017. These posts will give you intelligence to identity their most profitable marketing targets. There's no need to waste money on weakly performing NAICS codes! ISOs should market their Aquila landing pages dynamically to the best targets that exhibit either, seasonal or structural, changes to their business funding demand. You can't beat free marketing intelligence. So may we pay you your money?\nMay we pay you your money?\nISOs that can craft marketing and attention of their Aquila Landing page to 20 to 80 small business merchants seeking no more than $15,000 in small business financing, will be able to generate anywhere from $3,000 to $9,000 (or even more) in commissions from Aquila, each day. Math doesn't lie.\nAlso, many ISOs have dead paper or dead leads that aren't fundable due to a number of reasons. It's our view that many of those companies and merchants are not dead. They simply require ongoing monitoring that identifies when the business' cash flow performance recovers or when a merchant may start a new business. We launched Aquila with the mission Never Throw Paper Away. Aquila was built for small business cash flow performance monitoring at the lowest cost. We analyze and monitor thousands of business bank accounts to identify the next short-term funding opportunity. As such, ISOs should use their Aquila URL to continuously market to those merchants they could not fund. When the business becomes fundable, Aquila will send an email alert to the ISO to sell that next financing.\nIf you're not yet an Aquila ISO or broker, register to our platform here: https:\/\/aquilacashflow.com\/isos\/new. If you are already an ISO or Broker on our platform what are you waiting on? Login to Aquila and start using your Aquila landing page by giving it to any merchant that fits the box! Need help? Give us a call and let us help you get your Aquila landing page customized so it's inviting and ready for merchants to sign-up and link their bank accounts for funding. Making money should be easy and we'd love to help get you there.\nAquila is a small business, cash performance analytics, and receivables financing marketplace. In November 2016, we at Aquila, launched our first small business, bank analysis and receivables financing transactions. More than a year later, over 4,000 Small Business merchants and their Independent Sales Organization (ISO) partners come to us to seek cash flow support and receivables financing. In 2017, Aquila monitored over 20,000 merchant bank accounts, extracting bank data, and seeking cash flow patterns that would help small businesses survive. We studied small businesses ability to complete payments on short-term debt including, receivables financing, leases, and credit card debts. Today, we are excited to share the top five (5) SIC codes where small businesses have the best cash flow performance. Merchants in these codes are able to meet their short-term, debt financing payments, and without too much stress on business operations.\nProfessional, technical and consulting services usually include highly-skilled, professional services. Examples of small businesses in this segment include accounting services, security services, Information Technology (IT) consulting, and Human Resources (HR) consulting. These firms are usually led by one owner who has a \"key client\" contract, or relationship, driving a major share of their deposit cash flow to the business. When a consultant\/owner loses a key client, cash flows are disrupted and it may take time to identify a new client or to secure a new long-term contract. When a consultant gains a new client, owners often will finance their contract with advances against their future receivable payments. These businesses are also sometimes seasonal. Accountants and bookkeepers, for example, usually don't see their businesses pickup until after the Christmas and New Year Holidays. As such, slow cash days are very common for these businesses at the end of the summer. During these slow periods, this industry group is at its highest risk for slow pay or possibly defaults on their obligations. This industry segment enjoys its highest cash flow days are during the first 2 quarters, or six months of the year. We encourage brokers and ISOs to increase their marketing to this industry just before the end of the year and into January and to limit marketing before the slow summer months.\nThis industry segment consists of mostly insurance companies run by franchise business owners and may include insurance franchises as well as financial advisors. Insurance does not appear to suffer the strong seasonality as some other industries, such as bookkeeping and accounting services. Insurance products are sold all year round. The merchant bank data that we analyze, in this space, reflect consistent revenue deposits, tied to a major franchise partner's brand. These franchise enterprises appear to have sufficient marketing resources to help their franchisee drive continuous business sales all during the year. This group performs well on continuous average daily balances that are well over $2,000 per day and with monthly deposits that may easily surpass $20,000 per month, for single-franchise owners. The number of business owners who enter these businesses are consistent and reliable.\nOn Aquila, the majority of companies, we see, in the accommodation and food services group are retail food businesses. Size varies from large city restaurants, all the way to small coffee shops and even street food vendors. These small businesses are usually single owner and their merchants have consistent, localized, demand that they are able to support with their businesses unique location. This group also appears to be well targeted by alternative finance companies. Our analysis of their bank transaction records show a number of alternative financing companies that support them with various financing products. The low barrier to entry into this industry segment also makes this a very large and active industry segment. It's our view that this group may see the most competition from small business financing brokers, making acquiring clients in this segment the most expensive of all five. We are not confident that this segment will experience high growth, given competition intensity.\nMerchants in this group included several elderly care facilities and nursing homes. Rehabilitation centers and wellness centers were included as well. We continue to see more merchants from this segment come to Aquila for cash flow analysis. We are confident that as Baby Boomers continue to enter into retirement, this group will continue to grow as a small business opportunity. The barriers to launch facilities appear to be low and we have seen a number of single-owners of companies in this group. This group exhibited some of the highest, consistent deposit cash flows of all businesses on Aquila. However, these businesses appeared to have the lowest operating margins of our top five grouping, as most of their payables were to labor and real estate expenses. Nonetheless, we continue to see many new companies and growing established companies in this space. We expect it will continue to perform well in 2018 and beyond. We recommend that sales executives and brokers continue to target this industry group.\nIn 2017, real estate rental and leasing was Aquila's best performing small business NAICS code group. These small businesses generated substantial business revenues. Merchants in this group usually had the best cash flow performance and gross margins that allowed them to seek out and repay their business financing comfortably. This group also experience the lowest number of short-term defaults or cash flow problems of all industry groups. Out of the Aquila sample, this group exhibited high average daily balances over a sustained 3 month to 6 month period well over $10,000 in balances, per day. These merchants also had the least number of judgements and liens against their business, or owners, for prior non-repayment of financial obligations.\nGiven the threat of rising interest rates, in 2018, we are unable to predict whether this group will continue to perform as strongly in 2018 as it did in 2017. However, we encourage brokers, sales executives, and ISO teams to direct their marketing dollars to this industry and ancillary or related businesses that will depend on it success.\nFor the rest of February, Aquila offers a flat 10 points for all closed deals in this group to FIRST TIME FUNDING ISO partners. To get started, login to your Aquila Cash Flow ISO dashboard and share your ISO referral link with merchants that are from the best performing industry segments and let's fund.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"9 out of 10 based on 795 ratings. 1,571 user reviews.\nPamela Allen activities. What others are saying \"Activity Pack Bertie and the Bear by Pamela Allen $\" See more. More ideas. Author Studies Literacy Activities Literature Study Early Childhood Activities Education School English Writing Kids Story Books. A Pamela Allen Maths and Literacy Pack made by the M&M's. Donna Lawler.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Killer app, or just killer looks?\nYou will show off your Xbox One with Forza 5. Its wonderful vistas and gorgeous cars make it perfect for public display. But as great as it looks, it's Drivatar that's Forza's real evolution.\nDrivatar learns the way you and every player drives, taking that data and applying it to AI opponents, making every race feel unique. No longer will you smash your way into first place and coast along; drivers aggressively look to regain position, bump you off the road and refuse to concede simply because you are on the racing line.\nOpponents are equally as likely to make mistakes, too. Many times drivers crashed spectacularly, leaving me to avoid the incident and try and stay in the race. There is a thrilling sense of speed that keeps you engaged in every part of a race, alert to whichever challenge is rapidly approaching your bumper.\nDrivatar also allows for deeper difficulty customisation to suit your skill. The spread of medal rewards makes it all the more enticing to increase the challenge. It's no longer a chase for the podium, but a series of battles across the entire track. Battling to edge your way into the top 10 and earn a bronze is just as exciting as going for gold.\nBut as much as Drivatar learns how you drive, it doesn't learn how you like to race. It would be nice if Forza paired you with drivers who have a similar driving style to you, be it with finesse or smash-and-grab aggression. Spending half a lap looking for the perfect overtake only to be clattered a la Destruction Derby is infuriating.\nSome flaws can frustrate in prolonged sessions. The classical music adds unnecessary pretension and a disjointed sense of calm. Equally, every crash sounds identical, cheapening their intensity, and the lack of night races and dynamic weather is inexcusable, especially when racers on the previous generation offer both.\nDespite these problems, nothing takes away from what is a great game. Every moment is exhilarating, and Drivatar isn't just a buzzword but an example of what the next-gen can do.\nCheck out Sportra.com's beginner's guide on Forza 5 right now!\nThe bodywork is built for showing off your new machine, but there's something more substantial under the hood.\nDeep customisation to suit your skill level.\nLack of dynamic weather and night races.\nForza races into Xbox One's launch line-up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Download residential and commercial leases that allow a landlord and tenant to come to a binding agreement over the use of property upon terms such as start & end date, monthly rent, utilities, parking, common areas, and any other conditions negotiated upon. Once the document has been signed by all parties and acceptance has been given the form becomes legally valid.\nThe landlord is not required but recommended to screen all applicants via the rental application and to check their background. Once the review is complete the lessee should be given a written lease.\nStep 1 \u2013 Tenant shows interest in renting either commercial or residential property and inquires about the monthly rent along with other terms. Usually a verbal agreement is made.\nStep 2 \u2013 The landlord will take the tenant's personal information with the rental application and will verify their monthly income, employment (current and previous employers), references, and any other documents they may need such as previous year tax returns.\nStep 3 \u2013 After the verification process is complete the landlord will decide whether to approve or reject the applicant. If approved, the landlord will decide how much of a security deposit to charge that is to be held until the end of the term in case of non-payment of rent or any damage left on the premises after the vacancy is over.\nStep 4 \u2013 The tenant should be requested to meet and sign the lease. Once both parties have signed the agreement becomes legally binding and the lessee is given access to the premises at the start date unless a deal was made to move-in early for a pro-rated amount.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagedc b/data_all_eng_slimpj/shuffled/split2/finalzzzagedc
new file mode 100644
index 0000000000000000000000000000000000000000..a6578ff1cf08e32ee1149f9bcaf3a388ad1eff25
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@@ -0,0 +1,5 @@
+{"text":"Tender closing for residential site at Alexandra View (Parcel B). URA closed the tender for the residential site at Alexandra View (Parcel B) today. Singland Homes Pte. Ltd. submitted the highest tendered sale price of $332,680,000.00 or $10,442.92 psm\/ GFA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3 solid Teak duck figure with swivel head and glass eyes.\nDark teak body with lighter wood beak (beak maybe from birch or light walnut).\nHead can be positioned with hidden swivel band.\nMarked Skjode Denmark. Eames & Bojesen Era, Mid Century Modern.\nCondition: In excellent vintage condition. Please see photos for condition and details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bright contemporary home w\/ warm finishes on large lot up a private driveway and perfectly situated to capture city views. Open floorplan w\/ spacious chef's kitchen w\/ indoor counter seating & outdoor dining area. Top of the line stainless appliances, solid quartzite stone countertops and custom cabinetry opening to family room & fpl. Wide plank oak flooring create a warm and inviting feel to the surrounding tree filled setting. Dining area next to open large living space w\/ pool table & plenty of seating for entertaining. All rooms opening w\/ beautiful sliding walls of glass to enjoy tree top views by day and stunning skyline by night. Large backyard w\/ swimmers pool, fire pit, spacious entertaining decks & built-in barbecue. Master suite w\/ stunning city views, spa bath w\/ free standing soaking tub and separate steam shower w\/ large walk-in closet. 2nd suite with large outdoor deck w\/ treetop views and 2 additional suites. Smart home & garage equipped for electric cars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Despite their rather strange appearance and very long, twitchy nose, the tapir belongs to the same family as the horse, zebra and rhinoceros. The snout is long, but also very mobile, enabling the tapir to seek out the tastiest leaves, fruit and berries on trees and shrubs. In the Zoo, they are fed horse and pony pellets, fruit and vegetables.\nTapirs love to swim and their long snout is useful again as it helps them to breathe. They live in the forests of South America, but are often hunted for their meat and skins for leather which has significantly reduced their numbers in the wild. However, they can live until they are 30 years old.\nTapirs give birth to a single baby which is covered in a pattern of white spots and stripes. This provides great camouflage when they are hiding in the undergrowth if threatened by predators. As they eat vast amounts of plants, they spread seeds in their poo. This means that new plants grow in other parts of the forest providing food and shelter to different species.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"will BCaa's help with cutting?\nProtein Whats best for the time of day?\nBeneficial to take the same amount of whey on off days?\nShould I take Creatine with meal or just take it alone?\nSucked in by the GNC sales pitch?\nJust took first dose of creatine in loading phase...anyone else experience this...?\nUndecided! Thunder, Clout, Octane, CEE, BCAA? What a choice!\nWhere to get MAN Scorch, for a good price?\nAnyone heard of thise Gamma O?\nComments on my superdrol cycle?\nWhat should i get next?\nCould Rebound XT cause hair loss?\nIs taking just Ephedra+Caffiene as effective as EC+Aspirin?\n1.5 grams of protein per lb?\nTrying to find the old speed stack?\nany1 get headaches from lipo 6?\nserious mass = heartburn ??\nI want to know which is the newest and best NO product today?\nSupplement to strengthen the mind?\nTyrosine, ALCAR, OR Choline to stack with caffeine????\nwill taking a tylenol diminish the effects of lipo 6?\nis NO-Xplode waste of money?\nWhey protein and good ole asulflame k.\nM1T tabs in 180 bottle ?\nHelp Bodybuilding.com Decide: Best Canadian Shipping Service?\nGive me a Protein Power as complete as ON 100% Whey, but no flavor..\nHydrolyzed Protein that taste fantastic; Should I be concerned?\nPlease help review my workout\/supplement plan!\nTopical Caffeine and CNS effects?\nTaking gatorade with whey protien.\nWhos C.e.e. Is Better ???\nNeed help on starting out!!!\nGI index: rating for ON weight gainer?\nWhats so good about tea??\nCreatine article by Author Rea : why is Triacana in it?\nSuperdrol vs. real oral steroids (healthwise) ?\nGrowth Hormones- Methyl OHN-Need Help!!\nRebound XT, work for anyone?\nO\/T - Does studying make you feel the need to eat??\nStarting CEE Stack, what works?\nAny word on chosen CEE pro testers?\nThe best homemade endurance stack period.\nI m going on holidays in july what should i bring?\nCan you stack omega-thunder with no-xplode?\nI'm always sore : Supplments to help?\nEverybody better take their milk thissle!!! Everything to know.\nAnyone know the reason you cant take ephedrine and Cee together?\nWhat do you take with CEE?\nAnyone taken UL m1t capsules ?\nFirst time taking creatine....why go fancy?\nWhat's going on with these questions?\nWhen to switch to a different creatine??\nVitamin World Arginine and Ornithine, any good?\nScivation Xtend Watermelon Is HERE!!\nWhere are the M1T2 testers?\nEnough protein, not enough calories?\nSo much for CEE tasting so bad. I just tried it and it's no big deal at all..\nWhat do you think of GNC?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaihco b/data_all_eng_slimpj/shuffled/split2/finalzzzaihco
new file mode 100644
index 0000000000000000000000000000000000000000..b13d4a8cbf0a366bffbb9e8e1226cab5a86b3c12
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Hampton Inn & Suites Olympia Lacey, Wa is set in Olympia, 5 km from Hands on Children's Museum, and provides free WiFi. 6 km from Washington Center for the Performing Arts, the property is also 6 km away from Olympia Community Center. The accommodation offers a 24-hour front desk. Guests at the hotel can enjoy a continental breakfast. The nearest airport is Sea-Tac Airport, 72 km from Hampton Inn & Suites Olympia Lacey, Wa.\nthe hotel Hampton Inn & Suites Olympia Lacey, Wa so we can give you detailed rates for the rooms in the desired period.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all need to save money and natural resources.\nUsing Solar Energy instead of electricity is one very important way to save. It is also a great way to improve you home.\nThe money you will save using solar energy will help you to be able to spend your money on other improvements. The money you save by building it yourself will be staggering!\nCheck out the featured Solar Energy information that is located on this page. The choices here, are the best on the market.\nClick Here! for Access to the Best Way to save on your Electric Bill! Make sure that you watch the informative video after you get there!\nSo, you are interested in knowing how to generate power and reduce your electricity bills, than you have come to the right place. With the ever increasing costs of living and global warming, there should be better time when we stop throwing money out the window and save some by starting to generate our own electricity.\nWhether you want to simply cut your power bills to half or completely eliminate them \u2013 the one stop solution is Earth4Energy.\nWhy pay a huge amount like $1000's for utilization of solar or wind power when you can have the opportunity to build your own home made solar system for less than $200. That's right, there are guides that teach you everything you need to know about generating your own electricity by using wind and solar power. With the complete step-by-step setup fully illustrated manuals and easy to follow video instructions present in the guide you will be able to create renewable energy at home.\nSolar and wind power systems brought from the market costs higher but the guide and the program in it gives you the information that is required to get the same results at a tiny price.\nOnce you have learned and ready to build and install your alternate power supply, you will save hundreds of dollars on your electricity bill. You need not have to build a 500 hundred foot high wind turbine to save lots of money on energy, a much smaller can be built by following the instructions present in the Earth4Energy kit. The Earth4Energy kit is the best system for people who are looking to save money on their home energy bill and want to build a energy generator at home .There is no other kit out there in the market like Earth4Energy that explains clearly with manuals or even includes videos. The Earth4Energy kit is available online for purchase. Many people can now have access to alternative solar and wind energy. The Earth4Energy kit suggests one of those popular alternative energy devices that prove to drop home energy costs by 50-80%. The kit also shares information on where to purchase free batteries required for your project. These batteries store the alternative energy making it a portable energy resource. You can take them anywhere, for any use! Eartth4Energy is simply the best friendly system for preparing an alternative home energy. So if you are planning to save money on those high energy bills, then you should bring this guide and start working on it.\nMake power at home with solar and wind energy to eliminate your power bill. Get our complete guide at Why pay a huge amount like $1000's for utilization of solar or wind power when you can have the opportunity to build your own home made solar system for less than $200. That's right, there are guides that teach you everything you need to know about generating your own electricity by using wind and solar power. With the complete step-by-step setup fully illustrated manuals and easy to follow video instructions present in the guide you will be able to create renewable energy at home.\nMake power at home with solar and wind energy to eliminate your power bill. Get our complete guide at Click Here!\n...You can build your own Solar Panels, saving $1,000's off of retail price.\n...Almost anyone can do this, even if you have no solar experience.\nLots of extra bonus materials included!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NYC Starts Offering Free WiFi Via Payphone Kiosks\u2026But At What Cost to Your Online Privacy?\n24\/7 free WiFi access in New York City telephone kiosks? No, you're not dreaming!\nThe pilot program to offer free broadband access via public payphone kiosks is part of the Bloomberg administration's \"efforts to promote greater digital inclusion for New Yorkers,\" which also includes wireless at schools, libraries, and senior centers.\nThe first location is at West 58th Street and Broadway, with at least 10 other locations throughout Manhattan, Brooklyn (545 Albee Square and 2 Smith Street), and Queens (30-94 Steinway Street, Astoria).\nPlans are in place to open locations in the Bronx and Staten Island, and NYC officials hope to spread public WiFi throughout the city, which includes more than 12,000 payphones in about 9,000 locations.\nIt's costing about $2,000 per location, but what is this going to really cost the average citizen in terms of online privacy protections?\nTurns out, it could be a lot more than $2,000.\nToo bad the NYC technology officials behind this otherwise awesome idea are missing the point of potential crime. It's not the crime you can see \u2014 for example, graffiti \u2014 but the crime you can't see. Take identity theft, credit fraud, or any other type of cyber crime \u2014 all of those terrible things happen without seeing the actual dirty fingerprints of the criminal at work.\nThe signal would extend out a couple of hundred feet, but according to a report from McAfee and Guardian Analytics, cloud cyber crime is enjoying a boom. Criminal hackers are employing cloud resources to provide the data-crunching power that they previously relied on their victims' own computers to supply.\nWithout proper encryption and the use of a personal VPN, anyone else can see your personal details, passwords, user names, and other sensitive information. But a personal VPN works to communicate securely between two points. The security is provided by encryption, and the two sides must both use the same encryption algorithm and key for it to work. This means that no one else can understand, or more importantly, modify the information being communicated.\nThat is important because there is no control over radio waves that are bouncing around the city streets.\nIn 2011, the FBI reported that 300,000 identity theft victims lost a combined $1.1 billion to Internet criminals. That's an average of about $3,666 per victim.\nThe typical Internet criminal commits literally thousands of these crimes and almost never gets caught. According to the FBI, nearly 304,000 Internet crime complaints in 2010 resulted in 1,420 cases and only six convictions. So for every 50,000 victims, one cybercriminal was convicted.\nIn response to the decreased use of payphones, other options for out-of-date kiosks include touch-screen wayfinding panels, information kiosks, charging stations for mobile communications devices, or virtual bulletin boards.\nBut are touch-screen technologies practical in the all-weather environment of city sidewalks? Could touchscreen technology allow for a digital advertising panel to convert to a neighborhood map or a subway map at a pedestrian's touch? Would it be practical to offer powered mobile device plug-in or wireless power facilities where a pedestrian could connect a mobile device that is low on power in order to make a phone call or otherwise use the device?\nMore importantly, why are city officials not taking this time to remind its citizens to use a virtual private network to encrypt all of their personal communications while using the phone kiosks?\nThe city says it's free, but nothing in life ever is, so protect your privacy before a cyber-thief finds you first.\nWe have warned about these kinds of hotspot attacks for years, but it is significant that Cisco agrees with our analysis. After all, Cisco is the largest manufacturer of WiFi equipment, having shipped 10 million WiFi access points to customers. If they are saying that wireless hotspots are inherently vulnerable to attacks, you can be sure that they know what they are talking about.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Wednesday, March 6, Illinois legislators including State Senator Bill Brady (R-Bloomington) listened to the Governor lay out his proposed 2014 budget in his annual Budget Address.\nThe Governor's Budget Address is just the start of the process of crafting a budget. During the remainder of the spring session, budget bills will be introduced in both the Senate and the House as legislators work to create the budget that will ultimately dictate spending for the 2014 fiscal year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi you are looking at a BRAND NEW lot of pocket knives PRICE IS FOR ONE dozen This is an EXACT photo of what the item will look like. Perfect for resale or gift giving Price is for 12 Pocket Knives.\nQuality Metal Assisted Opening Folder. Blade, Titanium Rainbow Finished 440 Stainless Steel Tanto. Closed Length:4-1\/2\", Blade Length 3-1\/2 \". Handles: Metal with G-10 Scales, Metal clip on Back. Please make sure this is legal in your area.\nIt is your responsibility to make sure this is legal in your area. We only sell to members 18 years of age and older. All emails will be answered within 24 hours. Thanks For Looking And Have Fun Shopping! The item \"Titanium Spring Assisted Opening Pocket Knife G-10 Dozen 12 Knives Rainbow Cheet\" is in sale since Thursday, February 25, 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaituz b/data_all_eng_slimpj/shuffled/split2/finalzzzaituz
new file mode 100644
index 0000000000000000000000000000000000000000..1fe38cbfd67d4066eb6882671bffe08a96523b5c
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+{"text":"Vancouver's head coach conducted an end-of-year press conference with the city's media on Monday, after missing the playoffs in 2014.\nApr. 14, 2014 12:00 p.m.\nIf you have any doubts about John Tortorella, just give the guy five minutes.\nFans and pundits have gone back and forth about the Vancouver Canucks and their two administrators \u2013 Tortorella, the head coach, and Mike Gillis, the former general manager \u2013 as the team's 2014 season wound down. The coach's critics have said his system can't work in Vancouver, that he grossly misread the team in front of him, and that the zebra couldn't change his stripes and got eaten alive because of it.\nBut on Monday morning, one day after Vancouver's year-ending 5-1 win over the Calgary Flames, there he was\u2026 answering questions and seeming everything but complacent.\nHonest. Responsible. Both reflective and cautionary.\n\"I felt from Day 1 that it's stale,\" Tortorella said, answering a question from Vancouver Sun columnist Iain McIntyre. \"And that's not their (the players') fault. This is a group that's been together for a long time. It's stale. It needs youth. It needs a change.\n\"We're not in 2011. We have to stop talking about 2011.\nTortorella accepted the blame for his players' lost season, insisting that \u2013 while his critics have said he pushes his players too far, going back to his days in Tampa and New York \u2013 he wasn't on his players enough, that he wasn't active enough in the dressing room, and wasn't drilling his system into Vancouver's four lines like he should have.\nAnd while the coach has always come off as an old-fashioned curmudgeon \u2013 because of the yelling or the angry faces, and even his march down to Calgary's dressing room that netted him a pre-Olympic suspension \u2013 Tortorella is professing youth, even though he still sees blooming power forward Zack Kassian as a project.\nThe coach also insisted that he wants to play four lines \u2013 a criticism levelled against him was his over-playing of Canucks stars like the Sedin twins and Ryan Kesler \u2013 but that the team didn't have enough depth this season to do so.\n\"The team is older, it's trending,\" he said. \"I'm not gonna sit up here and pretend. You have to understand\u2026 it's getting older. It does need to be revitalized. You have to stop thinking about 2011.\n\"That's not an excuse for me. I could get gassed out of here today. It's the truth of what's going on.\nIn saying that, Tortorella acknowledged the rebuilds or retools that have happened with \"some of the teams in our division\" and how successful they've been. Perhaps he means the ascension to perennial contender status of the Los Angeles Kings or the Chicago Blackhawks \u2013 both once floundering franchises \u2013 or the prolonged excellence of teams in Anaheim and San Jose, and now St. Louis.\nTortorella is believed to be on his way out as head coach of the Vancouver Canucks after just one season behind their bench, with the firing of Gillis and the hiring of Trevor Linden as the club's President of Hockey Operations.\nThe coach is yet to have his exit interview with Linden while the latter makes his decision to retain or sack personnel before the start of next season.\n\"You can say, 'Okay, we're old, we've gotta get younger, but we're gonna win a Stanley Cup' when we get younger,\" Tortorella said, when asked what the Canucks need to do this offseason to correct the past year's wrongs.\n\"It doesn't happen that way. You need to go through the process\u2026 things change and teams change. Personnel changes.\n\"That's a decision that management has to make, as to how they go about it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sv\u00f8mmebassenget er tilgjengelig mellom kl. 08.00 og kl. 22.00.\nWhat a bad maintenance this hotel has!\nI had to wake up around 3:30 am in the mid of sleeping. Because water pipe of my bathroom ceiling was broken and all water came down to floor and flooded all over my room. What a terrible maintenance this hotel has! and they didn't apologize for this inconvenience stay..\nroom heating was not working, went to bed cold and waited to next morning to seek help Was told had to wait until 8:00am until engineer came in, he came up did minor adj in heating element it work for a moment and said he would have to replace unit if it didn't work. I did not call front desk I was checking out at 1200 noon saw him at check out talk him the heat wasn't working he said thanks for letting him know. I think the hotel was aware of this problem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Show the result of inserting the keys F,S,Q,K,C,L,H,T,V,W,M,R,N,P,A,B,X,Y,D,Z,E in order into an empty B-tree with minimum degree 2. Only draw 2 configurations of the tree just before some node must split, also draw the final configuration.\nWe have t = 2, therefore the most a node can hold 2t \u2013 1 = 3 (node is full when number of key = 3) and the least t \u20131 = 1. And the insertion has to be in order of F,S,Q,K,C,L,H,T,V,W,M,R,N,P,A,B,X,Y,D,Z,E.\nDuring a call to B-TREE-INSERT we can have a DISK-WRITE operation whenever the root node is changed. For example the insertion into a full node and the median key has to move up to the root node causing a change in root node.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We do! You may contact us here and we will set up a custom coupon code of your choosing that will give your customers a 5% recurring discount, and you will receive a 5% recurring commission if they use that coupon. This is a great way to make money on the side! Our largest affiliate makes over $1000\/mo by just referring customers to us!\nDo your proxies work with Nike or Ticketmaster?\nHow long until I receive my proxies?\nInstantly! Our system is fully automated and your proxies will be available within minutes.\nWhat limits do you have for bandwidth and threads?\nNONE! That's right, we allow unlimited threads and unmetered\/unthrottled bandwidth. Most proxy providers cut costs by openly limiting your threads or bandwidth, but a lot of them do it secretly without you knowing. For example, if they cut your bandwidth speed by half just for an hour each day they would cut a significant amount of their costs\u2026 and you wouldn't even know!\nHow fast are your proxies?\nAll our proxies are on 1 Gbps DEDICATED network lines. Unlike most other providers, we do not oversell our proxies and ensure all speeds test to the full 1 Gbps.\nDo you allow proxy replacements?\nWhat are proxies? What is it used for?\nA proxy is basically an IP that forwards your internet request\/data, without the websites that you are connected to, finding out your actual IP address.\nHow do I USE the proxies?\nHow do I TEST the proxies?\nWhat are the available proxy locations?\nCan my ISP see that I am using a proxy or block it?\nWe enforce high level proxy anonymity protocol to ensure that your IP does not leak out. Your ISP will not recognize our proxy IP usage.\nWhat can I use the proxy for? Is it legal?\nProxies are normally used to hide your computer's IP address or if you require multiple IPs to access a site, proxies can be used for such purposes. Proxies are indeed legal.\nKindly login your CLIENT AREA and browse to your available list of services, and select the service that you would like to cancel. Click \"Request Cancellation\", change the cancellation from immediate to at the end of the billing period, then write the reason why you are cancelling (as we appreciate any feedback as to why you are cancelling your service). You can also follow this video guide here. If you have any questions please create a support ticket.\nHow do I cancel my Paypal subscription?\nYou can cancel your recurring subscription payment profile in your PayPal account by following this guide HERE . Please make sure your subscription is properly cancelled in Paypal, in order to prevent Paypal from automatically sending payment on your behalf for renewal.\nHow do I setup a Paypal subscription?\nYou may setup a Paypal subscription by using the Paypal Subscribe option when paying your upcoming invoice.\nHow can I pay only once for my invoice?\nYou may pay your invoice once through the Paypal Check Out option.\nOur proxies are indeed at the right location as shown on your proxy dashboard. Web-based IP location checking tools do not regularly update their IP location database, and they may take anywhere from 1-5 weeks to update their records fully. You can read this ARTICLE for more details about IP locations.\nAre the proxies compatible with my software?\nOur proxies can work on most software. If your software supports IP, Port, most likely you will be able to use our proxies on the respective software.\nWhere is my data going?\nOur proxies simply forward your internet requests\/data to your targeted site and no data is stored on our server.\nWhen I check the location of my new proxy IP the result is a different location than what the proxy dashboard says.\nIf you put ANY of your IP addresses you have from us at the top right of this PAGE you will find that your IP address is in fact in the correct location. Since ARIN owns IP addresses, they govern this location and it is up to third parties, such as MaxMind, to honor this governance \u2013 which unfortunately is not the case.\nWhy and when should I replace my proxy's IP?\nIn rare occasions, the proxy may go offline due to unforeseen circumstances. If you urgently need to use all the available proxies in your account, you may replace the proxy that isn't working through your proxy DASHBOARD and you can also follow this video guide for reference here Also, if you find a particular IP address to be unsuitable to your usage, you may also replace the respective IP via your proxy dashboard.\nCan I use my proxy on multiple computers?\nAre your proxies anonymous, and will my IP leak out?\nOur proxies are indeed anonymous, and we enforce high level proxy anonymity to ensure that your IP does not leak out.\nCan I test your proxies?\nSure! Kindly submit a support ticket HERE and we will help you get started with the trial.\nUnfortunately, we do not accept payments from Bitcoin.\nCan I manually send payment?\nPlease send $(amount for package) to admin@blazingseollc.com through your PayPal account and confirm in this ticket after payment with your Paypal Email, Paypal Transaction ID and screenshot showing proof of payment (that contains the Transaction ID), and kindly create a support ticket and include the information above. Please let us know the package you are going for, and we will add access accordingly.\nCan I pay through my credit card?\nWe do accept credit card payments; however, credit card payments are handled by Paypal through their guest checkout option.\nPayPal is not letting the payment go through. What do I do now?\nIf you are having issues paying your invoice please contact support right away. You may send a support ticket HERE Please keep in mind we have no control over cancelling eChecks. eChecks may take up to 5 days to clear.\nHow do I pay my monthly invoice?\nYou can pay your monthly invoices via your CLIENT AREA . Please browse to the \"My Invoices\" section to view the latest invoice for your subscription.\nHow do I downgrade my proxy subscription?\nKindly login to your proxy DASHBOARD and browse to the \"Upgrade\" section, found on the left navigation panel. Enter the \"New Amount\" for your proxies and click on the \"Upgrade\" button to downgrade. If you presently have 20 proxies, and would like to downgrade to 10 proxies, the \"New Amount\" should be 10. After you select \"Upgrade\" then a window should pop up requesting the respective IPs you want to keep. Then after you provide the list you may proceed with the downgrade and credit will be applied to your account.\nHow do I upgrade my proxy subscription?\nKindly login to your proxy DASHBOARD and browse to the \"Upgrade\" section, found on the left navigation panel. Enter the \"New Amount\" for your proxies and click on the \"Upgrade\" button to upgrade. If you presently have 10 proxies, and would like to upgrade to 20 proxies, the \"New Amount\" should be 20.\nMy proxy expired, how do I renew?\nYou may renew your subscription through your control panel HERE . Once logged in, please browse to \"My Invoices\" section to pay the latest invoice for your subscription.\nWhy did my proxy IP change?\nIs there a free monthly refresh\/rotation for my proxy's IP?\nYou may manually opt for a free monthly refresh or IP rotation for your proxies by activating this feature on your proxy DASHBOARD Please browse to the \"Settings\" page on your proxy dashboard, and check the option that reads, \"Check this box if you want your static proxies rotated every 30 days. This option is not suited for people that use our proxies for social media and wish to keep their IPs for as long as possible.\" You can also follow this video guide HERE that will show you your respective rotation options in your dashboard.\nMy proxy subscription expired. I renewed, but why did my proxy's IP changed?\nWhen the proxy subscription expires, our system automatically removes the proxy IPs from your account due to the expiry. The IPs are then removed from our system entirely. You will be given a new set of IPs after the renewal.\nCan I recover the previously used proxy IPs?\nUnfortunately, when the proxy IPs are automatically removed from your account due to expiry, the IPs would be removed from our system entirely, and it is not recoverable.\nHow do I login to the control panel?\nHow do I login to the proxy dashboard?\nYou may login to the proxy dashboard HERE with your email and password you have set. For accounts created from April 2017 onward, kindly login using the W\/WHMCS option.\nWhat type of authorization methods your proxies support?\nOur proxies support IP AUTHORIZATION and USERNAME\/PASSWORD authorization, in which you may set your preferred authorization method via your proxy dashboard accordingly.\nHow do I authorize my IP?\nI authorized my IP, but the proxy is not working. Why?\nI am using IP authorization, but the proxy keeps asking for username\/password. Why?\nIP authorization uses port 3128. If you have incorrectly set the proxy's port to 4444 (which is for username\/password authorization), the proxies will continuously prompt you for the username\/password. Please make sure to set the correct authorization method on your proxy DASHBOARD. If you chose IP authorization, the proxy dashboard should be set to IP authorization accordingly. Be sure you have properly authorized your IP in your dashboard. You can simply follow this video guide HERE. Otherwise, if the proxy dashboard is set for username\/password authorization, and if you configure your proxies to use port 3128 for IP authorization, the proxies will continuously prompt you for the username\/password, as the proxies will follow your proxy dashboard's settings.\nI am using username\/password authorization, but the proxy keeps asking for the username\/password, even after I have entered it repeatedly. Why?\nYou may have entered the wrong username\/password \u2013 please login to your proxy DASHBOARD to obtain the correct username\/password for your proxies. If you have incorrectly set your proxy dashboard to use IP authorization, instead of username\/password authorization, the proxies will continuously prompt for the username\/password, even though you have entered the right username and password. Please make sure to set your proxy's port to 4444 when using username\/password authorization. You may also follow this video guide on how to use USERNAME\/PASSWORD authorization in your dashboard.\nWhat port does your proxy service use?\nFor IP authorization, our proxies utilize port 3128. For username\/password authorization, our proxies utilize port 4444. And for SOCKS connectivity, our proxies utilize port 1080.\nDoes the dedicated \/ semi-dedicated \/ rotating proxies work for foot sites?\nOur dedicated\/semi-dedicated\/rotating proxies DO NOT support shoe sites \/ foot sites, as it is not supported. If you do use dedicated\/semi-dedicated\/rotating proxies to access foot sites, most likely you will see a blank screen or a looping loading\/connecting screen which is blank, which indicates that it is most likely blocked by the foot site. On the other hand, our special dedicated SHOE PROXIES are special proxies that are specifically purposed and optimized for shoe sites instead. Only shoe proxies will provide you access to foot sites.\nDo your proxies support SOCKS protocol?\nYes, our dedicated and semi-dedicated proxies support SOCKS. You will need to change the proxy's port to 1080 for SOCKS connectivity.\nHow do I use SOCKS on the proxies?\nIs the username\/password authorization available for SOCKS?\nUnfortunately, username\/password authorization method is not available for SOCKS for our proxies.\nHow to change my proxy's location?\nKindly login to your proxy DASHBOARD and browse to the \"Location Preferences\" tab found on the left navigation panel, and you may assign your proxy's preferred location preferences accordingly. Next, browse to the \"Proxy Replacements\" tab and replace the proxy IP that you would like to change its location. Once replaced, our system will assign an IP based on your preferred location settings that you have set earlier. You may also follow this video guide HERE DISCLAIMER: Each replacement will use a single proxy replacement credit. If you purchased 10 proxies, you are entitled to 10 proxy replacements per billing cycle. The replacement credit will be replenished on the next billing cycle. Kindly use the replacements cautiously, as we are unable to replenish the credit manually.\nI want my proxy IPs to stay the way it is without changing every month.\nI want my proxy IPs to rotate\/refresh to a new list of IPs every month.\nAre your proxies suitable for social media?\nOur dedicated proxies are suitable for managing social media accounts. However, since site access security may change any time, we are unable to guarantee that the proxies will grant you access to a specific website. Our only guarantee is that the proxies can be connected to, and it should load Bing.com without issue.\nMy proxy is banned! What should I do?\nWe provide free proxy replacements in which you may use to replace the banned proxy IP. Kindly login to your proxy DASHBOARD and browse to the \"Proxy Replacements\" tab, and replace the proxy IP that you find unsuitable for your usage. Once replaced, our system will assign a new proxy IP as a replacement. You can also follow this video guide HERE on how to replace your proxies within your dashboard. DISCLAIMER: Each replacement will use a single proxy replacement credit. If you purchased 10 proxies, you are entitled to 10 proxy replacements per billing cycle. The replacement credit will be replenished on the next billing cycle. Kindly use the replacements cautiously, as we are unable to replenish the credit manually.\nWhy does Speedtest shows a slower speed after I use the proxy on the browser?\nDo you offer residential proxy IPs?\nWe only offer data center IP presently.\nIs it possible for me to get a diverse subnet for the proxy's IPs?\nOur system is automatically set to provide you with a diverse subnet for your proxy's IP. If you still find the proxy IPs to be lacking in diversity, you may proceed to replace the proxy's IP via your proxy DASHBOARD accordingly. Kindly login to your proxy dashboard and browse to the \"Proxy Replacements\" tab and replace the IP that you find unsuitable. You can also follow this video guide HERE on how to replace your proxies. DISCLAIMER: Each replacement will use a single proxy replacement credit. If you purchased 10 proxies, you are entitled to 10 proxy replacements per billing cycle. The replacement credit will be replenished on the next billing cycle. Kindly use the replacements cautiously, as we are unable to replenish the credit manually.\nHow long does it take to deliver the proxies after purchase?\nThe proxies are normally delivered almost instantly after purchase. However, you will need to set the proxy's location accordingly on your proxy DASHBOARD before the system assigns the IP into your account.\nDo your proxies support Google scraping?\nDue to the fact that Google modifies their website security regularly, we are unable to provide a guarantee that your proxies will provide you access to a specific website, such as Google, or can be used for Google Scraping purposes. Kindly read our knowledgebase articles here for more info on Google Scraping and sites that we do not support: GOOGLE SCRAPING UNSUPPORTED SITES Our proxies work extremely well on Bing, thus kindly switch your service to Bing, and you will find our proxies performing extremely well there.\nWe have a strict no refund policy for any purchase. That being said, we cannot guarantee that all proxies will grant you access to a specific site, and that if you experience circumstances where the proxies are unable to connect to \"X-website\", there will be no refund for the purchase \u2013 our only guarantee is that the proxies are able to be connected to.\nCan your proxies work on a Mac?\nYes, our proxies do work on a Mac. You may use the proxies on various operating systems and platforms. If your system supports using proxies in IP, Port format, you should be able to use the proxies.\nHow do I use the proxy tester in the dashboard?\nDid not receive verification code, how can I access my account?\nIf you have accidentally requested cancellation for your respective subscription you can remove it through your client area. Please follow the steps below on how to remove your cancellation request.\nIf you require any further assistance please do not hesitate, we would be more than happy to help!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bay's webpage hosts upwards of 100 million objects for public sale or sale at any given time. So that you spend good time. If you are considering of shopping for Tiffany jewelry on eBay you might need to know there's an excellent probability the diamond ring or silver necklace you are taking a look at is pretend. The piece that by no means loses its worth is all the time what folks most want. The approach is about the value of the piece on the secondary market versus viewing the merchandise intrinsically in opposition to non-branded related objects. Nevertheless this yr they notified me that I would have to pay twice the worth of our ring to be able to. Cost might be made by credit score cards via Pay Pal. One also can buy the Tiffany jewelry online. So what Tiffany valentine gift are you getting your love ones? So don't hesitate, get a box of chocolate to your beloved ones directly. Writer: Susan Silver It's an incredible reality that people love presenting gifts to their liked ones in occasions of particular occasions and anniversaries.\nRight here i will suggest the top 6 heart-formed designs Valentine Gifts from Tiffany & co Ideas, and some which means behind them. New fashion of gold ornaments, as three-dimensional circular arc modeling designed in bottle, pot, sun and other factor, has the symbolic and lucky meaning to the female Aquarius. White gold Cuban bracelet by maker George L'Enfant, c. This Bracelet Beaded Jewelry Patterns was related with this subjects. This DIY Jewelry Case was related with this subjects. Whereas it's possible you'll find good deals on-line, there are several shops that will guarantee the authenticity of their jewelry watches and give you an optimum service plan. Whereas, the Indian on-line jewelry business has seen main VC investments in on-line jewellery shops in recent times. His experimental nature paid off handsomely and he soon turned a recognised identify in the jewellery business. This No Jewelry was related with this I am Stronger Than My Excuses Charm matters.\nThis Antique Platinum Jewelry Markings was related with this subjects. This Jewellery Design College was related with this subjects. Los Angeles Jewellery Purchaser is where to promote a Tiffany diamond ring for money. As probably the most recognized and desired luxury manufacturers, Tiffany & Co. is very vulnerable to counterfeiters. With over 150 years of expertise in the jewellery trade, our experts know the brand equity of Tiffany jewelry and different luxurious manufacturers. Hyperlinks on Tiffany Jewelry needs to be soldered steady links. And a very powerful thing is that all the Links of London Jewelries there are very low-priced. I imply what's there not to like? Happily, there are various strategies to rapidly restore your silver to its unique magnificence. In 1868, Tiffany purchased the silver works of John & Edward Moore. How to buy tiffany lamp? Charles' son, Louis Comfort Tiffany, (February 18, 1848 \u2013 January 17, 1933)was a famous decorative glass and lamp designer most famous for his Artwork Nouveau pieces in stained glass. Clearly, tougher to seek out limited edition items will value extra. The jewelry kind of these amulets werent fully realized until Egyptian Pharaohs began creating charm bracelets from items of steel and treasured stones.\nSometimes individuals will positively give them feedback and say issues like, \"Wasn't authentic, however is pretty and quick transport.\" So verify and see if anyone says the jewelry wasn't authentic. As of the stone material, check out if it is natural, treated or stabilized. However should you store at Marshalls or Ross you will have to buy your personal present bag or wrapping paper and do it yourself. All the pieces about Tiffanys is classy from their regal doorman to the attractive blue bag your buy is enclosed in. Tiffany & Co. has opened its newest pop-up store in Los Angeles, within the Grove shopping center. It is exceedingly essential for every one beauty-loving female to methods to make use of tiffany store. This webpage has been established to inform and educate the general public about replica Tiffany jewelry and equipment. Tiffany Necklaces jewellery: Tiffany Film overview: the streets of latest York aboriginal within the morning, Holly seems to be admirable girl stood in foreground of the Tiffany and Co Agreeableness Armlet Adornment Outlet adornment abundance window. Escape will shut the window.\nYou're going to get my articles as quickly as they're printed. A few of them are crafted with distinct sorts of materials equivalent to steel, beads, zinc, copper, brass, wood, clay and plastic. Themed collections which can be nicely assembled, effectively completed, and well deliberate, can be highly valued for their entirety reasonably than for his or her individual parts. He had such a powerful work ethic that he walked to work through the blizzard of 1888; his 600 staff also walked as nicely. Be sure to factor within the condition as properly which can affect the resale price. Jade represents the essence of the earth and sky and the nimbus of the solar and the moon. Procuring on eBay will be a great way to seek out antique or distinctive jewellery that you simply won't discover anyplace else. Publisher: coolpurple These folks commonly see one thing concerning cubic zirconia as faux, gaudy, or bogus. And even in case you do not like it's a great sport about it. First, key pendants are very trendy and well-liked this 12 months. In America, stained glass grew to become very talked-about in the 1960's and 1970's. And, as we move forward we are seeing new types and designs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..1829f482b2dc4d9ad5600d43f8b243df69b386c8
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@@ -0,0 +1,5 @@
+{"text":"Android rooting is the most pretty technic to reach your device performances and functionalities without any doubt on your Androids. I'm sure you are also a big fan of Android rooting as well as being addicted to its benefits like me. Actually, Android rooting is helping all the system modification with respect to the device model and operating system version. There is Dozen of Android rooting tools and I have tried different rooting tools on my Android devices. But, couldn't satisfy with the functionality. However, until I found about Towelroot. If you feel interested in the Towelroot, do not miss the post here. Read this post to know get root access on your gadget with Towelroot.\nTowelroot is a kind of Android rooting tool that allows you to root your Android device in just a few seconds. Absolutely, it works amazingly on my device and gave me a rooted device in the blink of an eye. Whatever you need to root your Android devices, Towelroot lets to root access on your Android device without any issue. Now I am enjoying the benefits of rooting on my smartphone.\nBasically, Rooting is the #1 method we can take the admin rights to all length and weight of the Android operating system. On the other hand, the promises providing the best Superuser privileges in a one-click attempt. As the result of that Towelroot is a rooting application which takes the number of options with respect to the device in hand and various other perspectives. Therefore, it lets you take the Admin level permissions on your handsets and then you can enjoy root-only apps choosing from thousands of options which would result in the benefits of themes, wallpapers, powerful applications, games and so much interesting.\nThis is a rooting app that is compatible with all android mobile phone and tablets such as Motorola, Samsung, LG, HTC, Lenovo, Samsung Galaxy S3, S4 Active, Samsung Galaxy S5-AT and version, Nexus 5, LG G3, etc with any other Android OS. Remind that this app comes completely free of cost. The main fact of this app is Towelroot app can be rooted Android devices in no time also can be un-rooted easily. Also, you can use this app for a Custom ROMs, management of the stock apps, exclusive options to battery saving, block ads, system modification and so on from which you were barred in the non-root devices.\nIt is not available in the Google Play store so that you have downloaded the Towelroot APK from the official website.\nyour device's warranty will void once you go through this process.\nTake the complete back of your personal data on your Android before going through this process.\nTowelroot compatible with Android KitKat (4.4) Android versions and above versions.\nSome virus guards detect Andr.Exploit.Rat as a virus. So that you can bypass the warning because It cannot be fixed.\nThis is kind of one-click root software for your Android device. Especially, it allows you to take complete superuser privileges on Android levels even without the involvement of the PC. Hence, this app developed by Geohot for Root Android smartphones or tablets devices and all the developer credits go to him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many causes of autism have been proposed, but understanding of the theory of causation of autism and the other autism spectrum disorders is incomplete. Heritability contributes about 90% of the risk of a child developing autism, but the genetics of autism are complex and typically it is unclear which genes are responsible.In rare cases, autism is strongly associated with agents that cause birth defects. Many other causes have been proposed, such as exposure of children to vaccines; these proposals are controversial and the vaccine hypotheses lacks compelling scientific evidence.\nAutism has a strong genetic basis, although the genetics of autism are complex and it is unclear whether ASD is explained more by rare mutations with major effects, or by rare multigene interactions of common genetic variants. Complexity arises due to interactions among multiple genes, the environment, and epigenetic factors which do not change DNA but are heritable and influence gene expression. Studies of twins suggest that heritability is 0.7 for autism and as high as 0.9 for ASD, and siblings of those with autism are about 25 times more likely to be autistic than the general population. However, most of the mutations that increase autism risk have not been identified. Typically, autism cannot be traced to a Mendelian (single-gene) mutation or to a single chromosome abnormality like fragile X syndrome, and none of the genetic syndromes associated with ASDs have been shown to selectively cause ASD. Numerous candidate genes have been located, with only small effects attributable to any particular gene. The large number of autistic individuals with unaffected family members may result from copy number variations\u2014spontaneous deletions or duplications in genetic material during meiosis. Hence, a substantial fraction of autism cases may be traceable to genetic causes that are highly heritable but not inherited: that is, the mutation that causes the autism is not present in the parental genome.\nIs autism caused by synaptic disfunction?\nSeveral lines of evidence point to synaptic dysfunction as a cause of autism. Some rare mutations may lead to autism by disrupting some synaptic pathways, such as those involved with cell adhesion.Gene replacement studies in mice suggest that autistic symptoms are closely related to later developmental steps that depend on activity in synapses and on activity-dependent changes. All known teratogens (agents that cause birth defects) related to the risk of autism appear to act during the first eight weeks from conception, and though this does not exclude the possibility that autism can be initiated or affected later, it is strong evidence that autism arises very early in development.\nIs autism caused by environmental factors?\nAlthough evidence for other environmental causes is anecdotal and has not been confirmed by reliable studies, extensive searches are underway. Environmental factors that have been claimed to contribute to or exacerbate autism, or may be important in future research, include certain foods, infectious disease, heavy metals, solvents, diesel exhaust, PCBs, phthalates and phenols used in plastic products, pesticides, brominated flame retardants, alcohol, smoking, illicit drugs, vaccines,[and prenatal stress,although no links have been found, and some have been completely dis-proven.\nIs autism caused by vaccines?\nRecent brain studies show that autistic brains grow at an unusual rate between age 1 and 2, and then slow again to a normal rate of growth. Some imaging studies suggest that certain areas of the brain are larger than is typical. Research is ongoing to determine whether these differences in brain structure cause autism, are caused by autism, or are co morbid with autism and caused by something else.\nRecent brain imaging studies show that autistic people and typically developing people do not use their brains in the same way. Autistic people do not use their brains to \"daydream\" in the same way as most people, nor do they process information about faces in the same way. So far, while we know that this information is true, we don't know what causes these differences -- or whether these differences somehow cause autistic symptoms.\nChemicals in the brain transmit signals which allow the brain to function normally. Sophia Colamarino explains: \"Nerve cells communicate using electrochemical signals; there is evidence from many different domains that the ability of the brain to transfer information may be defective.\" An understanding of which transmitters are problemmatic may lead to effective treatments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In order to receive a more accurate quotation for Holden Caprice, please fill out all the missing fields below. Any individual details you can provide in the next step. The more details you provide the more definite quote you receive.\nThe booking will be sent to 2 of limo companies in Palm Cove QLD 4879, Australia that attend to your request by providing quotation through their inbox. You will be notified by email message about each price.\nThis particular booking is specifically designed for Holden Caprice, which is provided by Cairns Limousines in Palm Cove QLD 4879, Australia. Limousine company will provide you a free quote, that you can accept or reject. Limoscanner imposes on any obligations to purchase any of the services. And the quotation provided is totally complimentary.\nThis is a complimentary quotation request for Holden Caprice. You have no obligation to buy any of the services at this time. No credit card required.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tired of the same old salad? Try this flavorful main dish salad that features tender shrimp flavored with lime and garlic...it's light, refreshing and a great change of pace.\nGrate 2 teaspoons zest and squeeze 1 tablespoon juice from the lime.\nStir the lime juice, lime zest and garlic in a 2-quart shallow, nonmetallic baking dish or a gallon-size resealable plastic bag. Add the shrimp and toss to coat. Cover the dish or seal the bag and refrigerate for 30 minutes, turning the shrimp over several times during the marinating time.\nHeat the broth in a 2-quart saucepan over medium-high heat to a boil. Add the pepper and onion and cook until the vegetables are tender-crisp, stirring occasionally.\nReduce the heat to medium. Add the shrimp and marinade to the saucepan and heat to a boil. Cook until the shrimp are cooked through. Stir in the cilantro.\nDivide the lettuce, tomatoes and shrimp mixture among 4 serving plates. Season with the black pepper.\nOval shaped dinner rolls are filled with eggs, crumbled bacon, shredded cheese, and green onion for a unique addition to the breakfast table.\nThe Zesty Citrus-Thyme Marinade really makes this recipe. The kabobs go nicely over a bed of rice -brown basmati and wild rice would be a good blend. Heat the remaining marinade and drizzle on top of the kabobs before serving to flavor the rice as well.\nIcy treats for the dog days of summer!\nThe kids will have hours of fun making ornaments and decorating the tree!\nEat more fruits and veggies for good health!\nWorried you're consuming too much caffeine? Know the facts.\nKids love to mold this into fascinating shapes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'd like to organize it by local companies and services they offer, of course, you don't need to utilize these, but then everyone in the group can enjoy a taste.\n3 Different Companies, similar offerings.\nMonticello Wine Tour & Taste \u2013 This company has 5-stars on Google, with 64 reviews and nothing less than a 5-star rating! There are public tours available, starting at $79. You can also do a tour that includes lodging, and the price goes up from there to $550. If instead of a local winery tour you'd rather a brewery tour, they include that option in their list of public tours. If you'd rather build your own private tour they have three different sized vehicles you can choose from, and from there you build your own tour, and there is an hourly fee.\nThere are many ways to taste the local flavor in and around Charlottesville, including public & private winery tours.\n2. Wish Wish Wine Tours \u2013 This company also maintains excellent ratings across multiple platforms. There are 4 options for public tours all $79, and then the private tours range from $255 to $485. That includes vehicles that can hold anywhere from 3 people up to 14 people for 5 hours. There are 13 popular local winery and brewery tours as well as the option to build your own tour.\nWine tasting while in Charlottesville has easily become one of the most popular things to do while in the area. The beautiful views, excellent local wine, and community make for the perfect combination for a lovely afternoon.\n3. Blue Ridge Wine Excursions \u2013 With this company, you have the option of either paying hourly with choice of vehicle or paying per person, also with the choice of vehicle. They have a detailed list of the local wineries on their site. This includes specific notes about each local winery and noting the winemaker and owners.\nWhen you've successfully toured some of the local wineries for the day, be sure to grab dinner in town! Mas is always a delicious experience, and Lampo will not disappoint!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Read the article My House Shall Be Called A House of...Announcements by Constance Cherry. Write a response (by commenting on this post). Responses should be no less than 250 words and must be posted by midnight, Thursday, March 20th.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you all, for seeing me and addressing my needs.\nIt was of great warmth, welcoming and satisfactory.\nKristin is always a attentive. She provides excellent patient care. She listens intently, and she always follows up the next day to check on the patient. She is a winner.\nGreat service and knowledge from the whole staff. Thanks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At first glance, Sim Jia Wei looks no different from other boys his age. On closer scrutiny however, he suffers from global developmental delay and autism.\nJia Wei was first taken care of by his mother. After his parents' divorce in 2005, his mother returned to Malaysia. There had been no contact with his mother after that.\nIn 2005, Jia Wei was enrolled in the Towner Gardens School, one of the schools of the Movement for the Intellectually Disabled of Singapore. His teacher described him as having a good level of understanding. He responds through gesturing though he could be aggressive at times.\nHis father passed away in 2009 due to lung cancer. Since then, Jia Wei's paternal grandparents, Mdm Tan Yah Ho, 75, and her husband, Mr Sim Tiang Huat, 80, took on the parenting role in caring for Jia Wei.\nUnfortunately, after an accidental medication overdose prescribed for behavioural management in 2012, Jia Wei suffered significant functional decline and could not continue with schooling.\nIn 2012, KK Hospital recommended that Jia Wei can be placed at the Red Cross Home for the Disabled (RCHD) for better care and support. Mdm Tan and her husband subsequently enrolled Jia Wei at RCHD on 17 May 2012 when he was 12 years old.\nOver the past three years, Jia Wei has been cared for by the team of dedicated nursing professionals, therapy and healthcare aides, physiotherapists as well as occupational and speech therapists. At RCHD, Jia Wei, engages in activities designed to enhance his physical, creative and social well-being, just like the other residents of our Home.\nToday, 15 year old Jia Wei is a cheerful, responsive and occasionally mischievous boy who receives regular visits from his grandparents. Mr Sim takes 1.5 hours to get to RCHD from his home at Bedok. Yet despite his age, he visits Jia Wei at the Red Cross Home for the Disabled almost daily. Mdm Tan visits Jia Wei once a week. Such is their unconditional love for their grandson, Jia Wei.\nMdm Tan is very contented with the quality care provided for Jia Wei at RCHD. \"The staff at RCHD takes very good care of him. It is also very clean and hygienic. Though some people asked if we wanted to transfer him to a home, we decided against it because we are very happy with the way Jia Wei is cared for at RCHD,\" said Mdm Tan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jet lag is when your built in internal clock desynchronizes with the external time. Big words there, so let's break it down \ud83d\ude42 We all have internal clocks. Us and most of the mammal race, if not all. It's known as the Circadian Rhythm. It regulates when you wake up and sleep. It also affects how hungry you feel at certain times or how alert you are at other times. You know that annoying way you wake up minutes before your alarm clocks goes off? That's your internal clock in action. Its usually pretty accurate, that is, until you travel and cross timezones. Then everything rocks and destabilizes.\nJet lag is usually worse when you travel west to east and lose time than when you travel east to west and gain some hours.\nSo how do you deal with the symptoms? How do you protect yourself from excessive jet lag each time you travel? Following are some tips and tricks you can implement to shield yourself and reduce the impact of jet lag while improving your travel experiences.\nMove the time you sleep time forward if you are traveling West to East. Sleep earlier. Take your meals earlier and try to live out your days as you would if you were already at your destination. This will help your body adapt and move the internal clock to the timezone you are visiting.\nIf you are traveling East to West, delay your sleep schedule. This us usually easier to adapt to since we gain time moving East to West. By moving your sleep time forward or delaying it, you slowly adjust your internal clock to your destination's. This reduces the sleep disruption and jet lag symptoms when you travel.\nHowever, if your stay at your destination won't take more than two days, you would be better served sticking to your natural time. Most people require a day to adjust to each timezone crossed. A day wouldn't be worth disrupting your sleep cycle for.\nIt would seem trivial but you would be surprised by how dehydration messes sleep up. Stay well hydrated before, during and after your flight to beat jet lag. Avoid caffeine-laced drinks and alcohol which reduce water in the body. This causes you to feel patched up and fatigued. Cabin pressure and anxiety also affects your sleep cycle. Staying hydrated keeps you relaxed and helps better repair the disruption of your internal clock.\nFlying long distances means sitting for long periods of time in restricted spaces. This can reduce blood circulation. Combined with cabin pressure in the airplane, reduced oxygen levels and anxiety, it can lead to disruption of your sleep cycle. Have light excises before flight. Walk around during the flight or stretch from time to time. Wiggle your toes to help circulation from the lower limbs.\nThis should improve blood circulation and help you relax. Relaxation is conducive to sleep and will help you ease better to a natural sleep cycle at your destination and ease jet lag effects. However, avoid intensive workouts after landing and before sleep.\nConsider seeking medical advice if you are a frequent flier who experiences nasty jet lag episodes. Doctors can prescribe drugs that can help you sleep better or stay awake and alert when you need to. While this should be a last resort, sometimes it may be necessary.\nMelatonin can also be used under prescription to reduce jet lag related symptoms. Melatonin is secreted naturally by the body to regulate our circadian systems. 3-5 milligrams a few hours before sleep should do the trick. However, ensure you have exhausted the natural remedies and consult your doctor before going the medication route.\nA bath soothes your muscles and quietens your mind after a long flight. (Part of the reason why some of our best ideas occur in the shower \u2013 no distractions: total relaxation \ud83d\ude42 ) Turns out its also great for coping with jet lag. A bath also lowers your body temperature which is a prerequisite for sleep. It does help you sleep better.\nJet lag can disrupt your life whenever you travel. The symptoms are never pleasant. However, as pointed out you can take steps to mitigate it. Take steps today to prepare yourself for your travels. Experiment until you have your prefect formula for dealing with jet lag.\nHow else do you deal with jet lag and it's effects?\nAre you looking for a way to earn free flights, then Frontier Airlines World MasterCard is the best option. enjoy its Discounted companion tickets, availability and great miles award.\nWith the help of my fellow bloggers, i am delighted to review The Frontier Airlines World MasterCard.\nMiles Momma @ Miles Momma Review FRONTIER WORLD MASTERCARD \u2013 40,000 BONUS MILES. \"This card is offered through Barclay Bank and they are really stepping up in the mile and point world with their rewards credit cards.\nFirst, Frontier has the lowest domestic award mileage requirement of most all airlines at 20,000 miles roundtrip. Most all major airlines charge 25,000 miles.\nYou can book one-way tickets which is not allowed by airlines like US Airways, Delta, and others.\nFrontier doesn't charge fuel surcharges only government taxes.\nTorsten Jacobi @ Mighty Travels Review Earn up to 50,000 Bonus Miles with the Frontier Airlines World MasterCard \"Frontier Airlines offers to earn up to 50,000 Bonus Miles with its World MasterCard \u2013 good for 2 domestic Round Trips!\nGreat award chart \u2013 Frontier operates from it's hub in Denver to many US and non-US destinations. Flights within the United States are just 10,000 miles one-way. So you can get 4 one-ways anywhere in the US just for signing up.\nSome sweet spots involve the seasonal flights to Fairbanks, Alaska that shows a good amount of availability.\nGood availability \u2013 Most domestic routes have lots of availability \u2013 often every flight, every day. However the routes to Alaska, Mexico, Dominican Republic and Jamaica are much harder to get at a standard level.\nNo change fees \u2013 Award bookings with Frontier incur no penalty for changes up to 8 days before departure and $75 per ticket after that. That is teh most customer freindly policy of all domestic airlines.\nJust $500 spend required \u2013 No need for 'manufactured spending' for this card \u2013 $500 are each to achieve.\nZero percent intro APR is extremely rare among miles earning cards.\nIn general, you shouldn't use credit cards if you don't pay off the balance in full each month to avoid interest. In particular, you shouldn't use rewards cards because they tend to have higher interest rates than other cards.\nIf am guessing right, travel was part of your new year's resolutions. (Just guessing here \ud83d\ude42 ) Or perhaps somewhere on your bucket list. Most of us wish we can travel more, experience different places, food, cultures and people. And for good reason. Travel changes your perspectives on life. It opens your eyes to ideas and ways of life you never thought possible. Spend a week in Rio de Janeiro or Caracas and the warm spirit and zest for life of the South Americas will probably catch up on you too!\nSo how do you manage to fulfill your travel dreams without breaking the bank?\nNothing just happens on its own. Events are set in motion. Plan your travel well ahead of time and chances are that you can manage it without spending a fortune. Failure to plan is planning to fail. In your plans, research cheaper options for the basic necessities such as food, accommodation and transport. If you want to rock travel at an expensive destination, consider going off the normal beaten route. Instead, opt for other experiences. Cook your own food. Consider couch-surfing or staying with friends. All this requires planning well in advance and making your acquaintances earlier.\nThe high travel seasons are well known. They are in the summer and around major holidays in the US and Europe. These are also the times when traveling can be expensive. From airfare to getting decent and affordable accommodation. The high season makes it a nightmare. How do you get around this? Make off season your high season! You have October through April to enjoy travel and amazing experiences on a shoestring budget.\nDuring this time you can snag budget travel deals, accommodation and other offers designed to keep tourist businesses afloat during the off season.\nUnless you can teleport from JFK Airport to Bangkok's Suvarnabhumi Airport, you need an air ticket. This is an opportunity to cheapen out! That air ticket is a means to an end. The end being all the fun you are going to have on the other side. How do you get a cheap air ticket? Time to get creative.\nDo you require that taxi to take you around your destination? No you don't! In-fact, you are probably getting ripped off. The more \"touristy\" you look and act the higher the chance that prices are being inflated on you. And let's face it. It's not a good way to experience the local culture. What's the use of traveling if you will be holed up at your hotel's pool sipping mojitos?\nGet off your comfort zone. Go off the beaten path. Experience the local culture, cuisine and people firsthand. It's cheaper and fun this way and you get much more for your buck! Other expenses you can do away with include eating out. Try cooking for yourself. Consider couch-surfing or staying with a local family instead of hotels. Use the train or bike your way around.\nA smile changes everything. It creates positive experiences. It opens hitherto closed doors. It changes situations and warms up people to you. It's contagious and it even makes you look more intelligent and approachable. Smiling is a recipe to rock travel. A smile will buy you what no money can't. Local hospitality and friendship for example. Keep smiling\u2026its your currency to amazing experiences!\nWhatever travel is for you; a lifelong goal, a resolution, item on the bucket list. You can do it. You don't need that much money. What you need most is the boldness to make it happen and courage to go off the beaten path. Where there is a will there is a way, or you make one!\nStill wondering if it is possible to earn miles without setting foot in a plane? Yes! you can earn as many miles as possible by taking advantage of the following miles offered to you by the airlines from various programs.\nSome airlines will let you purchase miles and award you bonuses of up to 100 percent on purchased miles. Though it is always cheaper to earn miles with flying, sometimes it is much cheaper to buy miles especially when promotions abound than a ticket, so check the bonuses before booking.\nAlmost all airlines have an online shopping portal that allows you to earn miles on every purchase. Airlines have deals with stores like Crate and Barrel, Dell Computer, target, Best Buy, Walmart, Drugstore.com, and Barnes and Noble among others. These stores offer at least one mile per dollar spent, though sometimes they award more.\nThe fastest and most lucrative way to accumulate frequent miles is with credit cards. Various travel credit cards offer huge sign up bonuses worth as much as $500 in airfare in order to entice people. All you have to do is to spend a certain amount of money in a certain amount of time to qualify for the bonus.\nThough miles and points earning debit card are scarce, you can use them to earn points especially if you don't want to spend money you don't have. Delta and SunTrust banks still offer a debit card. You earn 1 mile per dollar spent on Delta purchases, 2 miles per dollar spent on direct delta purchases and 5,000 miles sign up bonus on your first purchase.\nYou might have others ways to Earn Frequent Flyer Miles. Please share with us in the comments.\nFirst of all give us a quick background of both of you.\nWe're Dave and Vicky from acoupletravelers.com, a travel blog that chronicles our backpacking journey through Europe and Asia. Both of us were living and working in DC and felt somewhat unfufilled with our jobs and decided that we need to push the reset button on life before it got too late. We quit our jobs in September 2012 and set off to Japan (we had been planning the trip for months before this). Since then we've been to almost 20 countries and are blogging and running businesses on the side! In addition to acoupletravelers.com I write about business at selfmadebusinessman.com and Vicky writes about food at avocadopesto.com.\nWhat gave you the inspiration to start a travel blog?\nIt was really just two things.\n1. We thought it would be nice to have a record of journey.\n2. We thought we could make some money or at least get discounted activities.\nLuckily we were successful in both!\nDo you find there are any challenges to blogging as a couple, such as who's turn it is to write?\nAt first we just took turns depending on who felt like writing about what. Later on we were a bit more systematic about it. I think it's important to fill roles. Luckily both Vicky and I have different strengths that compliment each other. I handle more of the technical side of the blog. Vicky handles more of the writing.\nSame deal with travel \u2013 how do you decide where to travel to next?\nWe had a route planned though we didn't stick to it 100%. Mostly because it just takes forever to get anywhere in Asia lol. It's fairly logical though as we just travel to neighboring countries, so when we were in Korea then we did China, Vietnam, etc. Nowadays we're more inclined to wing it. We just set off for Spain to visit family and then are going to Italy and the Balkans, again for family.\nWhat is it about travel that has you so obsessed about it?\nExperiencing new cultures, meeting new people, and finding out things about yourself you didn't know before.\nWhat have been your favorite places that you've visited so far?\nJapan and France are probably our top two. Japan is so interesting and unlike a lot of Asia. At the same time it's very safe to travel there and modern as well. France is just delightful. I highly recommend doing a driving tour.\nIt's probably too early right now for us to realize what our regrets are. I guess I wish we spent more time in Japan. Really though it's been a fantastic trip and for the most part I wouldn't change a thing!\nAny funny or embarrassing stories that you're willing to share from your travels?\nAt this point so much of this just seems incorporated into our daily life that we don't even take notice. On one of my first nights in Japan I accidentally walked in on my couch surfing host naked \u2013 that was a shock for us both and made for quite an awkward breakfast!\nFor the budget travelers out there, what would be your top tip for saving money when traveling?\nDon't buy things you don't need.\nSure we got a lot of credit cards before we started traveling. We use the Chase Premier Rewards and the Capital One Venture Card. Each of us have saved thousands on these cards alone. It's not that hard really.\nAside from budgeting, what's the best general travel advice you've ever received?\nI think just always be aware of what is going on around you and be vigilant.\nDo you plan to keep traveling indefinitely or do you see yourself settling down somewhere before long?\nNot sure when we'll settle down completely but we are already slowing things up by renting apartments for a month or two at a time. Soon that will probably turn into 2-3 months, then 3-6, and then you're a permanent resident!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The mother of a disabled boy has described her anger at finding a note on her car accusing her of \"laziness\".\nEmma Gearing, 26, a holder of a valid blue badge, said the person who left the note was \"spineless\" and \"too quick to judge\".\nHer disabled son is fed through his stomach.\nThe note read: \"Laziness is not a disability. Using a disabled badge when you don't need it could cost you \u00a32,000 and permanent removal of your badge. Don't take your good health for granted.\"\nSpeaking to the BBC, Mrs Gearing said she had parked in Maidstone on Tuesday afternoon while she visited the doctor and waited to collect her young son from school.\nShe found the note when they returned to the car.\nThe mum-of-two said: \"When I saw it I got really upset. They've just assumed it was me but they could have asked me.\n\"It upset my seven-year-old as well, people shouldn't be allowed to get away with it.\"\nMrs Gearing said she had faced abuse on previous occasions from people who thought she was pretending to be disabled; she had been \"blocked in\" by other motorists and had been the victim of verbal abuse.\nShe said: \"I have to show them the peg in my son's stomach to prove it.\"\nMrs Gearing's four-year-old son suffers from a range of conditions, including a heart defect and seizures, for which several pieces of medical equipment are needed.\nShe posted a photo of the note to Facebook, saying: \"To the spineless heartless living thing who left this note on my car I hope you can sleep peacefully tonight.\"\nDisability Rights UK deputy chief executive Sue Bott CBE said it \"wasn't always obvious\" why a person had a blue badge, adding that they \"may have a hidden disability or be the parent of a disabled child.\"\nHowever, Bott also said the blue badge scheme had been abused in the past.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Gnarly DUI chick and confusingly famous sister to a worthless pile of meat and bone, Stephanie Pratt, has said she'd like Ashley Tisdale to play her in the film version of the book version of the TV version of Lauren Conrad's life. It's fine. My head's melting trying to comprehend this too.\nAh, the merging of High School Musical and \"The Hills.\" This combination literally could not be more inane. So naturally I have to spend dozens of minutes of my day picking it apart.\nWhat intrigues me most is how distorted this telephone game of Conrad's life must eventually be. I can only imagine, in its various iterations, that her life will go from the story of a listless, passive SoCal girl trying to justify her existence to that of a fierce Dark Ages valkyrie slaying all those that stand in the way of avenging her family's death. Which, BTW, I'd totally pay $12 to see. And I think Vanessa Hudgens would do wonders for the role.\nMore power to Conrad for milking this phenomenon bone dry, I suppose. She's been paid lots of money to be on TV and has a novel under her name. I write blogs while temping, and I've just now made enough to see that valkyrie movie. Which is why I feel so awesome about myself. All the time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You ever feel like you are in the engine room of a submarine?\nAnd you are shoveling coal into the engine.\nAnd building the ship at the same time.\nYou are not sure where you are going.\nAnd you haven't seen the sun for some time.\nBecause you haven't hit anything yet?\nPage generated by WordPress 4.2-alpha-31516 in 0.326 seconds using 39 queries (postoffice.gristmillmedia.com).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The perfect complement to a smokey eye look, this kohl-pigmented mascara delivers high drama by creating extremely black, stretched-out looking lashes. Plus, the long-wearing formula holds its own against flaking, smudging, sweat and humidity.\nBuilds the appearance of volume lash by lash as opposed to traditional volumising formulas that create the appearance of volume by clumping lashes together. Plus, a thicker brush base evenly combs through eyelashes from roots to ends, while a tapered tip grabs those small, hard-to-reach lashes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Everybody knows that the main purpose of security cameras is to keep you aware of what is happening within a specific area and to alert you in case there is an impending danger to your life, to your loved ones' safety, and to your properties. Therefore, you need to make sure that you get and install efficient and reliable security cameras just like the Blink XT Camera System.\nWhat makes Blink XT Camera System an Efficient Security System?\nThe Blink XT Camera System stands out among the numerous security cameras available in the market today because of its features and specifications. The fact that it is wireless makes the camera system even more effective because other people might miss the existence of the security cameras. It also does not look messy so you can have security and keep your home or office pleasant to the eyes at the same time. It is lightweight so carrying it or moving it from one place to another if there is a need to do so is easy and convenient.\nWhat can Blink XT Camera System do for you?\nIt gives you peace of mind despite extreme weather conditions. While other security cameras are quite vulnerable especially when it gets wet. This is why most security camera system owners are worried when it's raining. However, if you own the Blink XT, then it is safe even from extreme weather conditions and can still perform its function well despite such scenarios.\nIt lets you use cloud storage. You do not have to worry about full memory cards because you can easily save footages through cloud storage for easy retrieval in the future.\nIt allows you to use a security camera system without much technical expertise. It is very easy to use so with just the basic technical skills, you can operate the system and keep your whole family as well as your properties safe. It only takes about ten minutes to install and set up this camera system so you can use it in no time.\nIt allows you to customize the system. With many different options, you can tailor fit your Blink XT Camera System according to your needs and preferences. Some of the settings that you can change are the name of your security camera, the kinds and frequency of the alerts that you will receive, the videos and footages that you prefer to watch, movement detection sensitivity levels, and duration of video footages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"American and Canadian fighters jets intercepted two Russian military aircraft that flew north of Alaska and Canada on Thursday night, U.S. Air Force officials said today.\nNorth American Aerospace Defense Command (NORAD) also confirmed there was another incident on Wednesday night with another pair of Russian aircraft that did not require an intercept, bringing the total number of sightings to four in as many days.\n\"Alaskan-based NORAD F-22 Raptors and Royal Canadian CF-18 Hornets intercepted and visually identified two Russian TU-95 bomber aircraft\" flying around the north coast of Alaska and Canada, said Mary Ann Clemons a NORAD spokesperson.\nThe Russian bombers did not enter American sovereign or Canadian airspace, Clemons said.\nThe U.S. military's Air Defense Identification Zone stretches 200 nautical miles from the Alaska coastline into international airspace. Aircraft entering that zone are asked to identify themselves as they transit through. American territorial airspace begins 12 nautical miles from American shores.\nOn Wednesday, two Russian IL-38 maritime patrol aircraft flew halfway up the Aleutian Islands chain, according to a U.S. official. Clemons said the Russian aircraft were identified during a maritime patrol close to Alaska. These aircraft were identified by NORAD, but no aircraft were scrambled to do so by visual means.\nOn Monday, two F-22 Raptor fighters and an E-3 AWAC reconnaissance aircraft intercepted two TU-95 Russian bombers that had flown into the ADIZ 100 miles south of Kodiak Island.\nOn Tuesday, two TU-95 bombers flying up the Aleutian Island chain were tracked by an E-3 AWAC aircraft as they flew 35 miles from the Alaska coast before turning around. A third aircraft, an IL-38 flying a different route briefly entered the ADIZ before turning back.\nEach encounter has received a different response from NORAD.\n\"The intercepts are professional ones in accordance with international norms,\" said Captain Scott Miller, the chief spokesman for NORAD.\nThis week's intercepts mark the first times since July 4, 2015 that NORAD aircraft have intercepted Russian military aircraft flying near the American ADIZ. Russian military aircraft have never strayed into American territorial airspace.\nMiller said the activity this week is not unprecedented given that the peak of long range Russian bomber flights into the AZID occurred in 2014.\nA year later that activity dropped off significantly, probably due to a 2015 safety stand down implemented by the Russian military following a slew of deadly crashes involving TU-95 Bear bombers.\nMiller noted that Thursday night's encounters highlighted \"the strength of the bi-national relationship of NORAD\", a joint American and Canadian command based in Colorado Springs, Colorado.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"BioWare GM Casey Hudson will be on hand to unveil a new trailer.\nPrepare to see a new side of the world of Anthem\u2122 as BioWare General Manager Casey Hudson takes the stage at the 2018 Game Awards to unveil an all-new trailer.\nTune in to the Game Awards on December 6 at 5:30 p.m. PST, 8:30 p.m. EST, and 2:30 a.m. CET, airing live from Los Angeles and streaming free at https:\/\/www.twitch.tv\/thegameawards.\nThe Game Awards will celebrate the most amazing moments in gaming and esports of 2018, and we are so excited to be a part of this event and debut this latest trailer.\n** CONDITIONS, LIMITATIONS AND EXCLUSIONS APPLY. SEE ea.com\/ea-access\/terms AND origin.com\/store\/origin-access\/terms FOR DETAILS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Ajanta mural depicts an ancient Indian ship with high stem and stern and three oblong sails attached to three masts. Steering-oars can also be seen.\nBased on Wikipedia content that has been reviewed, edited, and republished. Original image by Sweekar Bhushan Pamulaparthy. Uploaded by Dr Avantika Lal, published on 15 August 2018 under the following license: Public Domain. This item is in the public domain, and can be used, copied, and modified without any restrictions. Please note that content linked from this page may have different licensing terms.\nPamulaparthy, Sweekar B. \"Ancient Indian Ship.\" Ancient History Encyclopedia. Last modified August 15, 2018. https:\/\/www.ancient.eu\/image\/9103\/.\nPamulaparthy, Sweekar B. \"Ancient Indian Ship.\" Ancient History Encyclopedia. Ancient History Encyclopedia, 15 Aug 2018. Web. 24 Apr 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cucumber Drawing. Here presented 51+ Cucumber Drawing images for free to download, print or share. Learn how to draw Cucumber pictures using these outlines or print just for coloring.... Spread the slices on paper towels and salt them lightly, to draw out as much moisture as possible. After 30 minutes, rinse the slices with cold water and pat them dry on paper towels. After 30 minutes, rinse the slices with cold water and pat them dry on paper towels.\nWhat is your approach to salting cucumbers before making a cucumber salad? How much salt, for how much cucumber, for how long? I was surprised recently when I turned a huge 18\" cucumber into less than 3\/4 a cup of cucumbers after 2 days in the fridge salted and inside of paper towels. how to draw a rolling pin Drawings of Cucumber submitted by users. Learn how to draw Cucumber from these Draw Something Drawings. The Best of Draw Something exists to showcase the very best drawings in the OMGPOP game Draw Something and Draw Something 2 for iPhone, iPad, iPod and Android. Upload your drawings and vote for the best!\nBoth slices of cucumber are placed in the eye and let stand for 20 minutes Another way you can do is to use two cotton balls that have been soaked with cucumber juice, and place it in your eyes. The health benefits of cucumber is to eliminate dark circles in the eyes.\n8\/07\/2018 \u00b7 Watch video \u00b7 How to draw My Little Pony Equestria Girls - Twilight Sparkle step by step 5:06 Video Drawing How To Draw Cartoon Castle HD Step by Step drawing lesson for kids!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For fans who remember the days when a too-young-to-shave Joey McIntyre was the littlest New Kid on the Block, it might hardly seem possible that he's a father \u2014 but McIntyre and his wife, Barrett, added a new member to their growing brood on May 31, and he used Twitter to share a snapshot over the weekend.\nMcIntyre's moment of domestic bliss comes in the midst of a joint tour and recording project between the New Kids and the Backstreet Boys, who have christened themselves NKOTBSB. The collaboration has produced a joint compilation CD which includes two new songs, 'All in My Head' and the first single, 'Don't Turn Out the Lights.' The bands will be on the road across the US and Canada throughout the summer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"cultivated plant seized by the DEA costs taxpayers an average of $3.02.\nin all 50 states in 1996.\nwhen low-THC strains of marijuana were harvested for their fiber content.\nThis strain of marijuana will not intoxicate users when smoked.\nHouse Agricultural Committee, called the findings \"damning.\"\ncombat marijuana cultivation in Kentucky, Tennessee, and West Virginia.\nof the total plants eradicated there were hemp.\nArmentano of The NORML Foundation @ (202) 483-8751.\nRobert Evans in a letter to Congress. \"We urge the House to defeat H.\nseek to legalize marijuana [as a medicine.]\"\nKeith Stroup or Paul Armentano of NORML @ (202) 483-5500.\nsimple possession of marijuana to those that maintain criminal penalties.\nclinical policy director Robert Ali said.\nthat found \"no significant changes [in] ... patterns of cannabis use\"\nmarijuana law should be left up to individual jurisdictions.\nacquits Lance Corporal Kevin Boyd of charges that he smoked marijuana.\nfederal law to prohibit the possession and sale of hemp products.\ncould pose serious trouble for the military's random drug testing program.\nwho 'pops' on a drug test will argue this [defense,]\" he said.\nconcentrations of amino and fatty acids.\nFor more information, please contact either Paul Armentano or Allen St.\nPierre of The NORML Foundation @ (202) 483-8751.\nfigured he had a pretty good reason for it.\ntreat muscle spasms caused by multiple sclerosis.\nhis Hillsboro home was an appropriate \"choice of evils.\"\nlaw to be broken to avoid \"imminent public or private injury.\"\ncannabis club he was associated with.\nafter his arrest, although he knew it was illegal everywhere in the country.\ndisabled and uses a wheelchair.\nrubbery legs. The drugs also took a while to work.\n\"When I had a spasm,\" he said, \"I couldn't take a pill and make it go away.\nBut I could smoke marijuana and it would immediately subside.\"\nOne spasm was so severe that he kicked a step and broke a toe.\nmarijuana plants at his Hillsboro home.\nimplanting a pump that would feed drugs directly into his spinal column.\nOnly rarely, he said, did the marijuana make him high.\nAt first he bought marijuana off the street and from other persons with MS.\ntreating the muscle spasms associated with MS.\ntreatment\" of spasms, he testified.\n\"junk science.\" One of them, he noted, studied only one person.\ndrug that can be grown in the back yard.\n1970s before his MS diagnosis.\n\"but something needs to change.\"\nanother day for cross and re-direct this morning.\nEUGENE, Ore. (AP) -- Not quite old enough to buy beer or liquor? No problem.\nOne in three Oregon businesses that sell alcohol will sell it to minors.\nbooze is sold or served.\nbeer or a bottle of liquor while an adult OLCC inspector watched.\nsaid OLCC administrator Pamela Erickson.\nfound it was much more of an industry-wide problem.\"\nthe customer was under 21.\nmyself and found they aren't real attentive.\"\nestablishments, according to computer documents provided to The Register-Guard.\nobtain alcohol, although few actually try to buy it themselves.\nmany stores that will sell it, though.\"\n\"There's a sense of frustration for many of the owners who are licensees,\"\nseries of meetings about this issue beginning next month.\nSpringfield and two police officers for $2 million.\n19-year-old convenience store clerk Raymond John Oliver.\n\"This settlement represents our vindication, finally after all these years,\"\namount of money can give us back what we lost.\"\n34, has worked in a sawmill in Eugene.\ngo to trial in federal court in Eugene.\nand our families to be over forever.\"\nthe same with the investigation that eventually led to their release,\"\nShaver said. \"I don't think you need to apologize for good police work.\"\na country where mistakes can be eventually undone.\"\nwitnesses into false testimony and gave incorrect information on lab tests.\nshot three times in the head with a .22-caliber handgun.\nthe inside of the tape used to bind Oliver's hands.\nInvestigators said they had no evidence Proctor and Boots knew Kuppens.\nnotified police when he found the body.\nwrong, either through negligence or on purpose.\nwhat her son and Proctor went through.\nlived long enough to see Chris out.\"\nbuilding to evict his tenants.\nCenter] and I ordered him to vacate the property.\"\nreplying curtly that \"that's my business.\"\nboss had sent any agents to Zachariah's home.\ncops threatened him [Zachariah] with forfeiture.\"\nlandlord) get along very well. \"He really likes me,\" Peron said.\nactivities within the building constitute only \"inadmissible hearsay.\"\npreliminary injunction hearing,\" Cahill wrote in his decision.\nkeeping, or giving away controlled substances.\"\nsidewalks are kept free from litter, loiterers, and pot smokers.\nrenew its request for an immediate Temporary Restraining Order.\"\nmedical marijuana initiative [\"Pot grower runs afoul of ill defined law,\"\nOfficer Ferguson is a member of the Orange County Narcotic Officers Association.\nHughes spending scads of her own money.\nsurprise some people before the race is over.\nefforts would only make the failure more intense.\nincentives to achieve excellence in the school system.\nprivatizing the system after a transition period.\nfor such intrusions into private lives are politically popular.\nendorsed him and contributed $1,000 to his campaign.\nSubject: HT: Monthly Monday March To End Prohibition!\nMonthly Monday March To End Prohibition!\neducational protests aimed at hastening an end to prohibition.\ninsists that federal law must prevail.\nbode well for the people of California, should he become governor.\nalcohol was repealed, Westerville just shook its head and stayed dry.\nonce known as the \"dry capital of the world.\"\nin a different county, will remain alcohol-free.\nwho want to build hotels and restaurants in the area.\nTradition Over Profits, a citizens group.\nThe four measures all passed easily.\n-- Sales of liquor by the glass for on-premises consumption passed, 284-160.\nIt's quite a change for a city that's had dry laws on the books since 1858.\ntried to open four years later were both destroyed by dynamite.\nMay 7 - \"They had no idea. No clue at all,\" the undercover officer said.\nkid\" was who showed up in class last February.\nnot want us to use her real name or show her picture.\nwait until school's out so we can go get stoned.\"\n\"Jayne\" told us she would \"act up\" to get placed in \"in-school suspension.\"\nwe'd go back and forth until we decided how we were going to do it.\"\nask me 'where do you live?'.. That obviously was a lie.\"\nWilmot High. \"I'm sure they'll be a lot more careful with what they're doing.\"\nthree others will face juvenile charges.\nflowerpot that had been tossed from a sixth-story window, the police said.\nthough no one was immediately charged, said Chief of Department Louis R.\nofficer who was nearby ran over to help subdue him, Sergeant Cuneo said.\nflowerpot. It was a well-placed shot.\"\nCenter, where they were treated for head, neck and back pain.\nthey would be dead right now.\"\noverdoses of drugs,\" a number of them teenagers.\nthey knew their children were using drugs on their property; etc.\nthe drug prevention program administered by the police in our schools.\nwe resort to the unsuccessful policies that led to such drug use.\nwould-be cops formed friendships sharing marijuana during recess.\nwould find similar circumstances in many other Mexican cities.\ndifficult,\" said Adrian Lopez Rivera, who spent two years as a policeman.\nofficer candidates knew they were the subject of a study.\nsurprised by Nexos's characterization of the neighboring police force.\nwas quoted as saying he had a new girlfriend he \"beats for the heck of it.\nWhat's more, she has no brothers.\"\nwith three years of salary and bribes.\ndrugs, robbery or excessive violence.\nthe payoff was about $60 a day. Officers then earned about $360 US a month.\nawakening from this nightmare -- albeit in the teeth of US resistance.\ndeath-grip, out of irrationality and into sanity and humane public policy.\nPhase 1: Win a general election, of course, preferably with a sizable majority.\ncountry or reduce demand within their population.\non social and drug policy around the world have been advocating for years.\nflat-footed. Then move quickly to implement.\npolitical divides. This means coming clean with the Canadian people.\nthat did not and could not work.\nhave been non-violent offenders when they were first incarcerated.\npragmatic considerations displace black and white moral declarations.\ndeliver under any circumstances short of totalitarian rule.\nproblem were from the outset preferred over hysteria.\nI see very little likelihood of your having to reverse course.\nreverse 80-odd years of bad, racist and counter-productive public policy.\nferocious. But it is the right thing to do.\nConsequences of Prohibition in Canada.\"\nCraig McLeod Frank Jones, Ph.D.\nfarm. They said they also seized growing equipment.\ncontrolled substance is George Blancke, 51.\nan upstairs room, said Staff Sgt. Matt Torigian.\nbeen charged with possession of a narcotic.\nalleged to be No. 3 in the cartel.\ninternational awards. It has been shown in 15 countries.\nand that his air ticket was paid for by the program's producer.\nGuardian said he was stopped at London's Heathrow Airport and deported.\noperate Britain's main independent television channel.\nfaked the programme. They are considering legal action.\nCarlton were absolute rubbish,\" he added.\nhe stood by the integrity of his work.\nregulatory body - was scathing about the company's performance.\nCentral, the second largest ITV company, which is owned by Carlton.\nmoney. They decided to take no further action.\nprogramme was sold to 14 countries.\nin the face of their constantly changing allegations.\"\ninvestigated and the result made public.\nflow of drugs to the United States and the spread of lawlessness here.\nBut the helicopters can't fly. They are part of an aging fleet of 36 \"Hueys\"\nColombia's anti-drug police. \"There are 140 troops here with nothing to do.\nanything about it. This situation is unprecedented for us.\"\nground is virtually impossible and access by river is slow and dangerous.\ncocaine and a growing portion of its heroin.\nrebel forces and drug suppliers.\nurging the purchase was included in the fiscal 1997 budget.\ntraining or budget to maintain the Black Hawks.\nthe Colombians don't need that much capability.\"\nColombia and undermine regional success against the cocaine trade.\"\nnot purchase the Black Hawks. Gilman, in a letter sent Monday to Thomas R.\nadministration for the past two years but never carried out.\nour fight, at its source.\"\nit cannot be spent at all.\nfighting the war on drugs in Colombia.\"\nare more supportive of our position.\"\nenough funds for new ones.\npast few years . . .\nSubject: Heavy fine for man who sent drugs to French MPs.\nprison term of 125 days if he refuses to pay it, the court said.\nand presenting drugs in a favourable light.\nusers, according to official estimates.\nyears, a study by the United Nations Drug Control Programme has indicated.\nscheduled for New York between June 8 and 10.\nproduction, trafficking and consumption with their other ramifications.\ncontrols in other regions of the world.\nAmerica and Asia, the UNDCP stated.\n321,292 kilogrammes or 12 percent of worldwide cannabis seizures.\nand North American countries, the study pointed out.\nproduced in such cities as Lagos and Johannesburg.\nthe problem, the agency stated.\nlike the ports of entry.\nurban employment as key elements in drug control on the continent.\neducation programmes for rehabilitated addicts.\nCopyright (c) 1998 Panafrican News Agency. All Rights Reserved.\nwill become weekly and will be in French, Italian, English and Spanish.\nplace in Paris, 5\/7 June 1998.\nParliament elections of June 1999.\nthe members of CORA of 1998 will have the right to speak and vote.\nresources we have at our disposal.\nfinancial contribution for the production of CORAFAX.\nANTIPROHIBITIONIST OF THE ENTIRE WORLD ..\nwill take place in a hall of the National Assembly.\nwhere it would come in conflict with French and Swedish extremism.\nbeing now heard by the Magistrate.\nagainst cannabis consumers and to fight the \"War On Drugs\".\nanswers that in politics conditions have to be created.\nCommission about the antiprohibitionist views of European commissar Emma Bonino?\nhopes that parliament will confront such an absence.\nkeep this fact present if it intends to effectively fight crime.\nall in telecommunications and energy resources.\nambiguity. Others, instead, just think it's ugly.\nfunding will go to informing drug- attics and taking care of their health.\nmarijuana, although sanctions like suspension of driver's licence remain.\nthose who receive and distribute the drugs.\nwith the Talebanis, greatly because of Afghanigas and oil resources.\ntheme of organized crime, in order to find common strategies of action.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"When it comes to basic recipes, this is as basic as they come. It can be prepared quickly, to be used with that nights lunch or dinner. Being only one part water and one part mustard powder, it makes a quite hot mustard. If trying for a normal mustard flavour, all that needs to be done is let it stand for 15 minutes and the heat will start to decrease. This mustard recipe can be make prior to making you meal giving you that 15 minutes of standing time while the lunch or dinner meal it is going to be along side is being prepared and cooked.\nIn medium bowl combine 1\/2 cup of the mustard powder and 1\/4 cup water and mix well. Taste before adding second 1\/4 cup water and season with sea salt and fresh ground pepper to taste. Now when adding the second 1\/4 cup add slowly while mixing well to make sure the desire consistency is achieved and it does not become to watery.\nNow add if any or all of the optional ingredients. Mix well.\nLet the mustard stand for about 15 minutes before enjoying.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The water damage Chelsea agents at 911 Restoration of Westchester goes above and beyond to prove their dedication to their customer's by getting their home back on track after it has been damaged by a flood.\nOur mold removal Chelsea teams are here to safeguard your property when water floods your basement or your roof begins to leak during a storm. Every home and business owner should take precautions to keep their property protected against the effects of water damage.\nWith our exceptional home restoration skills, the main goal of our crew is to keep you calm during the reconstruction and assured that your home will be beautiful once we are finished with it.\nOur pros value efficiency and professionalism is when it comes to home restoration, so our mold removal Chelsea techs make sure to offer services in all types of water damage scenarios such as pipe bursts, flooding, roof leaks, sewage backups, and mold infestations.\nWhen you hire us, our crew guarantees to arrive within 45 minutes of receiving your emergency, and work with IICRC certified technicians in order to give you a thorough, free home inspection. So, in order to keep your home dry and free of mold, call the licensed, bonded and insured water damage Chelsea operation to come and take care of the water damage in your home today.\nBacked by many years of experience, the water damage Chelsea pros know how to restore any type of flooring after it has been saturated by moisture of any kind.\nOur specialists always act quickly when it comes to situations involving water because through all of our experience in the field, our pros know that prompt service it is crucial to a successful renovation and customer satisfaction.\nAlthough can make your property look dirty and dingy, it can also seriously affect the health of anyone breathing in the mold spores, which is why the problem needs to be addressed quickly.\nBreathing in mold can cause anyone living under the roof with it to begin feeling unpleasant symptoms. Our mold removal Chelsea experts are the most thoroughly trained professionals in the industry and we consistently aim to provide you with comprehensive mold and water damage restoration.\nOur specialists want you to be aware that your wood flooring is exceptionally at risk of obtaining water damage and mold growth with every spill and leak that it comes into contact with. If you spill some juice or notice a leak under the sink, there are ways to begin the cleanup process that will minimize the damage.\nDo not let liquid seep into the floor. If you notice water on the floor or have a spill, make sure to immediately use an absorbent towel or sponge to clean up the liquid.\nMake a mixture of ash and mineral oil to remove stains. Apply the mixture with a soft cloth and gently buff out the area. Toothpaste will also work.\nIf you can't get stains out, replace the finish. The Chelsea water damage removal crew can replace the finish of your floors to make the look like new again. They will look great without the hassle of getting completely new flooring.\nIn the occurrence that flooding affects your home, the pros will be at your property quickly after receiving your emergency call, ready to begin the water removal process with all the most advanced drying technology, so contact us today!\nIt is important to know that getting your water damage remediated will not break the bank. Our water damage Chelsea staff does whatever they can to make their services affordable to everyone.\nOur mold removal Chelsea professionals are proud to give our customers free water damage estimates, as well as keep competitive prices and work with all major insurance companies.\nOur offices are always open, because the water damage Chelsea specialists know how time is of the essence when you are dealing with a pipe burst or a water heater explosion.\nOur mold removal Chelsea professionals know that if too much time goes by it is very likely that a mold infestation will begin to surface, and our techs want to put a stop to that before it has a chance to happen.\nAlong with getting your home back into pristine condition, our water damage Chelsea pros also want to keep your emotional wellbeing in tact. By providing you with the utmost customer service, our mold removal Chelsea agents will make sure that you have peace of mind about the renovations that need to be done on your family home.\nOur staff members understand how important it is to get your house, filled with all your memories and treasured possessions, back to into the condition that you know and love. Our water damage Chelsea crew will work diligently until everything is back in its proper place.\nWhen we are renovating your living space you can be sure that we will treat it as if it were our own, so you never have to worry about having workers on the premises. If you find yourself knee deep in water, call branch owners Eli, Daniel, and the water damage Chelsea team with 911 Restoration Westchester so they can begin giving you the fresh start you deserve today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Although many people say the PlayStation 4 and PlayStation 4 Pro are best paired with a TV, computer monitors are also great displays for Sony's current-gen consoles. They have a smaller footprint, which is great if you have limited space and don't have room for a 50-inch TV. And they have lower input lag, which is especially beneficial in fast-paced competitive games.\nShopping for the best monitors for PS4 is a difficult task, though, mainly because most monitors on the market are directed toward PC gaming. Fortunately for you, we already got you covered with this guide. In this guide, we have compiled the top products currently out on the market, from budget options to 4K monitors.\nWhile there are also budget-friendly TVs, monitors are usually cheaper, which makes them better options for those who have a small budget. If you can only spend around $150 for a monitor, check out the HP N270h, a 27-inch display with a 1080p resolution and a 60 Hz refresh rate.\nFeaturing an IPS panel, the N270h is a gaming monitor with thin bezels at three sides and an anti-glare screen coating for reduced glare when using the monitor in a bright room. It has one HDMI 1.4 port, which allows for connectivity with the PS4. The default stand is basic and only offers tilt adjustment, so if you want a more flexible setup, you'll need to mount it to a third-party monitor stand or arm.\nThe N270h is a decent low-end gaming monitor, with good response time and input lag. It has good color accuracy and wide viewing angles, but it has subpar contrast. You can use it for different kinds of games, including first-person shooters and racing games. If you want a simple monitor for PS4 gaming, get this HP product.\nLike the HP N270h, the Dell P2417H is an affordable IPS monitor with a 1080p resolution and a 60 Hz native refresh rate, but it has a smaller 24-inch screen. While it's clearly designed for professional use, this Dell monitor also works well for gaming on PS4.\nSelling for about the same price as the HP monitor above, the P2417H has a minimalist design similar to most other Dell office monitors. It has narrow bezels and a clean back panel with most of the connection ports down-firing at the bottom area. For input, it has one DisplayPort, one VGA port, and one HDMI port, with the latter being the most important for PS4 gaming. In addition, it has a USB hub composed of four USB ports, which can be used for charging your controllers.\nThe P2417H has good picture quality for an IPS monitor selling for less than $200, with great color accuracy out of the box, decent brightness, and wide viewing angles. While not on par with VA monitors, the contrast ratio is also decent. The response time is good while the input lag is low, which is great if you love to play fighting games. Overall, the P2417H is a good monitor for gaming on PS4, especially if you want something that comes with a USB hub and a fully adjustable stand.\nIf you are into competitive gaming, the BenQ Zowie RL2455S will make for an excellent gaming monitor. It's a 24-inch TN monitor designed specifically for competitive gaming on consoles, boasting an extremely fast response time and a very low input lag. It's great for games such as Overwatch, Street Fighter V, and Rocket League.\nRegarding design, the RL2455S \u2013 which sells for about the same price as the Dell P2417H \u2013 is not going to win any awards, especially with those thick bezels. It has a plain design from top to bottom and comes with a basic stand. It has dual HDMI 1.4 ports, which is great if you also want to connect it to a second console. While the monitor has speakers, the sound quality is mediocre, as expected from a budget monitor.\nCompared to the Dell and HP budget monitors, this BenQ product has inferior picture quality, with poor contrast and narrow viewing angles. It's not a suitable display for games with pretty visuals such as God of War, Shadow of the Tomb Raider, and Persona 5. But for fast-paced competitive games where response times and input lag are more important than picture quality, it's an ideal monitor.\nThe Samsung C27F398 is a good alternative to the HP N270h if you want a monitor more suitable for survival horror games like the recently released Resident Evil 2 remake. It's a 27-inch VA monitor with a 1080p resolution and a 60 Hz refresh rate, both of which are perfectly fine specifications for gaming on PS4 consoles.\nLike many Samsung monitors, the C27F398 has a curved screen, which adds a bit of style to the otherwise basic-looking chassis. It has fairly narrow bezels all around and features an intuitive joystick control for navigating the on-screen display (OSD). For PS4 compatibility, it has one HDMI 1.4 port, located at the back, along with the DisplayPort. While the monitor has no USB hub, it has an audio out port for headphones, which is nice.\nThe C27F398 has better overall picture quality than the BenQ gaming monitor above. It has a high contrast ratio, which leads to more vibrant images, and boasts excellent color accuracy once fully calibrated. Regarding gaming performance, this Samsung display is good, featuring a fast response time for a VA monitor and a low input lag. The curved screen doesn't really make for a more immersive gaming experience, though, considering the screen size and aspect ratio.\nOften considered one of the best TN monitors currently out, the ViewSonic XG2402 is an excellent monitor for gaming, whether you game on PC or consoles. It's a much more versatile TN display than the BenQ Zowie RL2455S, featuring a fully adjustable stand and a USB hub. It sells for around $200-250, with the price sometimes dropping below $200 during a big sale.\nThe XG2402 is a 24-inch display with a 1080p resolution and a 144 Hz refresh rate, the latter of which is beneficial if you are also going to use the monitor for PC gaming. It has a fully adjustable stand with good adjustment ranges, which is important considering the viewing angle issues of TN monitors. It has two HDMI ports for PS4 connectivity and two USB 3.0 ports for charging your PS4 controllers.\nFeaturing an extremely fast response time and an exceptional input lag, this ViewSonic display is best used for fast-paced games. You can use it for first-person shooters, fighting games, and sports games, among others. The monitor's picture quality is also decent, which is the best-case scenario for a TN monitor. It has good color accuracy out of the box and good peak brightness. All in all, the XG2402 is a great monitor for PS4 gaming. You can't take advantage of the very high native refresh rate, though, since PS4 games don't go over 60 frames per second.\nIf you want a relatively affordable 4K monitor for your PS4 Pro, check out the LG 27UD58-B, an IPS display with a 27-inch screen and a 60 Hz refresh rate. You can get it for just around $300, which is a surprising price point for a 27-inch 4K monitor. LG also offers a 24-inch variant, but we don't really recommend that one since larger screens are better for a 4K resolution.\nConsidering the price, it's not surprising the 27UD58-B has minimal features. It has a basic design that won't turn heads, with a plain back panel and a stand with very limited adjustability. The menu control, which comes in the form of a small joystick, is located at the bottom of the monitor, while the input ports are outward-facing at the back. The monitor features two HDMI 2.0 ports, which can support a 4K resolution at 60 Hz.\nThis LG monitor is a great display for role-playing games, survival horror games, and adventure games, among other genres. It has good color accuracy out of the box, wide viewing angles, and decent brightness. The response time is good for an IPS monitor while the input lag is low whether the monitor is running in 4K or 1080p, which is great if you want to connect it to a regular PS4. It's a great product overall, offering excellent value for the money.\nThe Asus VP28UQG is a 28-inch 4K monitor selling for about the same price as the LG 27UD58-B, which makes it another good option if you want a budget-friendly 4K monitor. But unlike the LG monitor, it has a TN panel instead of IPS, which automatically translates to narrower viewing angles.\nCompared to high-end Asus gaming monitors, the VP28UQG has a subtler chassis design, with none of those futuristic aesthetics and customizable lighting systems. The screen, which has an anti-glare coating, is surrounded by fairly thick bezels. For menu navigation, the monitor has an intuitive joystick control at the back.\nFeaturing dual HDMI 2.0 ports for connectivity with both the regular PS4 and the more powerful PS4 Pro, this Asus 4K monitor is a suitable display for fast-paced games. It has a fast response time, which is a given considering the panel type, and a low input lag. If you are into first-person shooters such as Call of Duty: Black Ops 4 and Battlefield 1, you'll love this monitor. But if you value picture quality more than response times, we recommend the IPS and VA monitors on this list instead.\nConsidered by many as the best 4K monitor for Sony's current-gen consoles, the LG 27UK650-W is an outstanding monitor that works well for all-around use, including gaming and office work. It's a 27-inch IPS monitor with HDR support, dual HDMI ports, and an adjustable stand. You can get it for around $400-450.\nLike most other LG monitors, this particular model has a minimalist design, featuring thin bezels at the top and sides and an all-white back panel. It can be easily mounted to a third-party wall mount or monitor arm. The OSD control, in the form of a small joystick, is located at the bottom of the monitor and is easy to use.\nFor PS4 connectivity, there is a pair of HDMI 2.0 ports at the back, along with the DisplayPort for computers and the audio out port for headphones. There are no USB ports on this monitor, which might disappoint some of you considering the price. Moreover, the monitor does not come with internal speakers, but that's not really a big deal.\nThe 27UK650-W is an excellent 4K monitor for games such as God of War, Uncharted 4: A Thief's End, Persona 5, and Red Dead Redemption 2. The overall picture quality is great, with superb color accuracy, wide viewing angles, and decent HDR performance. The contrast is not as good as the contrast of VA monitors, though, so don't expect too much on that front.\nWhile not as good as the ViewSonic XG2402 and other TN monitors, this LG 4K monitor also offers a fast response time, which is great for fast-paced games. In addition, the input lag is excellent whether the monitor is running in 4K or 1080p. Overall, the 27UK650-W is a highly recommended 4K display for PS4 gaming, especially if you want something that also works well for productivity and general use.\nA 32-inch display, the ViewSonic XG3220 is a great option if you want a 4K monitor that uses a VA panel for significantly better contrast compared to IPS and TN monitors. It supports HDR content and features a USB hub for charging your DualShock 4 controllers, which is convenient.\nLike the other ViewSonic monitor on this list, the XG3220 has a stylish design, easily separating it from regular monitors. The default stand offers full adjustment options and features a headphones hook at the top and basic cable management. The monitor has dual HDMI 2.0 ports, which is great if you also own an Xbox One X and want a single monitor for console gaming. In addition, it has an audio out port for your gaming headphones.\nThe XG3220, which has a 60 Hz refresh rate, is best used for single-player games, especially those with lovely visuals. It has good overall picture quality, with excellent color accuracy, decent brightness, and a superb contrast ratio. The response time is also good for a VA monitor while the input lag will satisfy most gamers. You can also use this monitor for watching movies, which makes it an all-around machine for entertainment use.\nIf you like the LG 27UK650-W but prefer a bigger monitor with more features, get the LG 32UD99-W instead. It's a 32-inch IPS monitor with a 4K resolution and a 60 Hz refresh rate. It's a great monitor for both gaming and productivity, with the large screen and high resolution allowing for easy multitasking for the latter. It sells for around $700-800, which makes it the most expensive product in this guide to the best monitors for PS4.\nRegarding design, the 32UD99-W is not that different from the LG 27UK650-W, featuring thin bezels and a white back panel with outward-facing connection ports. It comes with an adjustable stand that's easy to remove if you want a wall-mounted setup. For PS4 connectivity, the monitor features a pair of HDMI 2.0 ports. In addition, it has two USB 3.0 ports for charging your PS4 controllers, along with a USB-C port, which is useful if you also want to connect the monitor to a laptop.\nThe 32UD99-W has great picture quality and good gaming performance. It has excellent color accuracy once calibrated, wide viewing angles, decent contrast, and good brightness. It also supports HDR content, but like the LG 27UK650-W, it's decent at best regarding HDR performance. Overall, this LG monitor is a recommended product, not only for PS4 gaming but also for productivity, watching movies, and general use.\nMonitors come in different sizes, from 24-inch monitors to massive 49-inch monitors. Since one of the advantages of monitors over TVs is their smaller footprint, we don't recommend monitors larger than 32 inches for PS4 gaming, with 24-inch and 27-inch monitors being the most ideal sizes.\nIf you have very limited space or mostly play competitive games, 24-inch monitors are ideal since they have a small footprint and allow you to easily see the entire screen, which can be advantageous in competitive gaming. If you mostly play single-player games and want a more immersive gaming experience, go for 27-inch monitors. They don't take up a lot of desk space and look good whether you sit less than four feet away from the screen or from around four to five feet out.\nIf you typically sit more than five feet away from the screen and want a display larger than 30 inches, consider 32-inch monitors. While they have a larger footprint, they are great for games with pretty graphics and are the most suitable if you want a large monitor that works great for both PS4 gaming and productivity.\nThere are also 34-inch, 35-inch, and 38-inch monitors, but those size categories are specific to ultrawide monitors, which have a 21:9 aspect ratio. Neither the regular PS4 nor the more powerful PS4 Pro support that kind of aspect ratio. If you connect an ultrawide monitor to a PS4 console, the output will be either horribly stretched out to fit the screen or displayed with large black bars on both sides.\nWhile they are uncommon, there are also computer monitors larger than 40 inches. Some are flat monitors with a standard 16:9 aspect ratio while others feature an ultrawide format. We don't really recommend monitors larger than 40 inches for PS4 gaming, though. If you are considering them, you might as well go for a TV instead.\nFor PS4 gaming, we recommend sticking to 1920\u00d71080 (FHD) and 3840\u00d72160 (4K UHD) monitors, with the latter being the more suitable if you game on a PS4 Pro. Unlike the Xbox One S and Xbox One X, Sony's current-gen consoles don't natively support a 2560\u00d71440 (QHD) resolution, although some games actually run in that resolution.\nFor 24-inch monitors, a 1080p resolution is perfectly fine. Many people consider a 4K resolution to be a waste on such a small screen. A 1080p resolution is also fine on a 27-inch monitor, but only if you sit at least three or four feet away from the screen when gaming. If you sit close to the screen, you may notice the low pixel density of the monitor. Meanwhile, a 4K resolution is best experienced on larger screens such as 27-inch and 32-inch monitors.\nOne of the most important aspects to consider is the monitor's panel technology. There are three panel technologies used in monitors: In-Plane Switching (IPS), Twisted Nematic (TN), and Vertical Alignment (VA). All of them have pros and cons.\nIPS monitors have the best color reproduction and viewing angles. And while they are not as responsive as TN monitors, they have good response times and are usually better than VA monitors on that front. IPS monitors are ideal if you mostly play single-player games with pretty graphics and care more about picture quality than response times. They are more expensive than TN and VA monitors, though, especially those featuring a 4K resolution.\nTN monitors are the cheapest, simply because TN panels are cheap to produce. They have mediocre color reproduction and narrow viewing angles. But on the other hand, they boast the fastest response times, which is beneficial in fast-paced competitive games. If you mostly play fast-paced competitive games and don't care about picture quality, go for TN monitors.\nVA monitors sit between IPS and TN monitors. They have better overall picture quality than TN monitors and have decent response times. They offer the best contrast ratio, which translates to deeper blacks and more vibrant images. Like IPS monitors, VA monitors are best used for story-driven or slow-paced games. They are more suitable for survival horror games than IPS monitors, though, since they have much better contrast, which is beneficial in games with a lot of dark scenes.\nListed in Hertz (Hz), the refresh rate is the number of times a monitor redraws or updates the image on the screen per second. A higher refresh rate translates to increased smoothness in games, which is great when playing fast-paced games. However, for PS4 gaming, a 60 Hz refresh rate is perfectly fine since PS4 games don't go past 60 frames per second. The monitor refresh rate is one of the least important aspects to consider when shopping for the best monitors for PS4.\nThere are different kinds of input ports found in monitors. Some monitors simply feature a DisplayPort and an HDMI port. Others feature a Mini DisplayPort, a USB-C port, or a VGA port, the latter of which is for connectivity with older computers. For PS4 gaming, the only input port that you need to worry about is HDMI, because that's what the console uses to interface with displays.\nNot all monitors feature the exact same HDMI port, though. Some monitors include an HDMI 1.4 port while others feature the newer HDMI 2.0 port. If you game on a regular PS4, an HDMI 1.4 port is fine since it can support a 1080p resolution at 60 Hz, which is the maximum setting for that console. If you game on the PS4 Pro and want to get a 4K monitor, the only products you should consider are those that feature at least one HDMI 2.0 port. Unlike the older HDMI 1.4, HDMI 2.0 can fully support a 4K resolution at 60 Hz.\nLastly, we'll talk about additional features. When it comes to additional features, not all monitors are the same. Some are loaded with a lot of features while others have a shorter list of features. However, not all monitor features are useful for PS4 gaming, so don't buy a monitor just because it has an impressive list of features. Below is a quick rundown of some monitor features and what they mean for PS4 gaming.\nHDR: High dynamic range (HDR) allows for more lifelike images, with higher luminosity, better contrast, and more colors. Both the PS4 and PS4 Pro support HDR, which is great. However, monitors are not that great regarding HDR implementation.\nThere are many monitors that support HDR content, but most of them are not fully equipped to display HDR content properly. Some don't support a wider color range while others can't get bright enough or don't feature a high contrast. While some monitors are fairly decent regarding HDR performance, many don't really add much when HDR mode is enabled.\nIf you want a monitor that actually offers good HDR performance, you'll need to cough up around $1,000-2,000 for one (see: Acer Predator X27, Philips Momentum 436M6VBPAB). If you have that kind of money to spend on a display, we strongly suggest going for an HDR TV instead.\nG-Sync\/FreeSync: G-Sync and FreeSync are adaptive synchronization technologies often featured in monitors, most especially the latter. They allow for variable refresh rates by dynamically adapting the monitor refresh rate to the graphics card output for smoother gaming. However, both the PS4 and PS4 Pro don't support G-Sync and FreeSync. Only consider whether a monitor has G-Sync\/FreeSync if you are also going to use your monitor for gaming on PC or Xbox One, which actually supports FreeSync.\nUSB Hub: If the monitor has a USB hub, you can use it to charge your DualShock 4 controllers. Some monitors feature up to four USB 3.0 downstream ports. While it's certainly convenient if the monitor has a USB hub, it's not that big of a feature since there are still other methods to charge your PS4 controllers.\nSpeakers: When shopping for the best monitors for PS4, one of the last things you need to worry about is the internal speakers. Built-in monitor speakers are not that great, with only a few monitors featuring decent-sounding speakers. For gaming, you'll want to use external speakers or headphones for a better audio experience, most especially if you love to play games with an excellent soundtrack \u2013 like Nier: Automata and Bastion. If you have limited desk space or want a more minimalist setup, go for the best computer soundbars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Being a doctor is everything to me, but I'm not going to let bureaucrats tell me how to practice medicine. It's about saving lives no matter what, screw them and their paperwork.\nWhen the one woman I could never forget, Ginger Crawford, shows up as my boss all bets are off. The heat between us is tough to ignore but I have to if I want a future at this hospital. It isn't easy. Every time we argue, I want her even more.\nBut I've got secrets she can never know, and she's always going to be way too good for me. So it's best if she hates me\u2026at least that's what I keep telling myself.\nDetective Jackson Cantrell never imagined that one night with an irresistible stranger would turn his life upside down. He's spent years living in the shadows, but Dr. Michaela Roarke awakened a passion inside him he'd buried years ago.\nBut when the stalker's attacks quickly escalate beyond mere photographs to bodily harm, Jackson must race to save Michaela's life. And he'll have to figure out how to keep her once she discovers his lie.\nAs a Black Ops assassin, \"Lucky\" Landon has had more than his fair share of close calls. Now he's turned in his sniper rifle for the simple life of his small hometown. So the last thing he ever expected was to end up at gunpoint. Or that the woman holding the gun would be his best friend's little sister and Lucky's on-again\/off-again lover.\nTaylor Elliott is Trouble, and she likes it that way. And seeing Lucky again? Well, he's been her dirty little secret for the past few years and everyone knows that secrets in a small town are almost impossible to keep. But Taylor has bigger problems on her plate. Like the local mob boss who wants her dead.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Family Fun Things to Make and Do Need some inspiration for family fun things to make and do? Then our list of 40 things to make and do should be a good starting point! Peakles and I share some of the things we like to do together at home and when we are out and about too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaogry b/data_all_eng_slimpj/shuffled/split2/finalzzzaogry
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+{"text":"RapidScale has positioned Mr. Salisbury to be responsible for all facets of IT and operations at RapidScale.\nRapidScale, a leader in managed cloud solutions, welcomes Andrew Salisbury to their team of IT and cloud experts as Director of Service Delivery. Andrew will be leading RapidScale's support and engineering teams and is responsible for all facets of IT and operations within the organization. He brings 17 years of experience in technology and management, and specializes in design, delivery and management of IT infrastructure. His skills contribute to his excellence in meeting complex business needs, and workforce and operational development.\nPrior to joining RapidScale (http:\/\/www.rapidscale.net\/), Andrew was the Senior Practice Director of Infrastructure Services for Prosum Technology Services, an IT consulting firm in Los Angeles. There he led a team of 35 IT professionals focusing on the overall business strategy and development, sales enablement, and growth of the practice. He is a proven leader in the IT industry, with demonstrated abilities to operate successfully in rapidly changing technology environments, increase revenue, maintain profit and drive high customer satisfaction. During his time with Prosum Technology Services, Andrew assisted customers through maturing processes, migrated key applications and workloads into the cloud, implemented mobility initiatives, modernized its information security, and developed a focus on data management.\nRapidScale, a managed cloud services provider, delivers world-class, secure, and reliable cloud computing solutions to companies of all sizes across the globe. Its state-of-the-art managed CloudDesktop platform and market-leading cloud solutions are the reasons why RapidScale is the provider of choice for leading telecommunications providers, VARs, MSPs, and agents throughout the United States. RapidScale is not only delivering a service but also innovating advanced solutions and applications for the cloud computing space. RapidScale's innovative solutions include CloudServer, CloudDesktop, CloudOffice, CloudMail, CloudRecovery, CloudApps, and more. For more information on RapidScale, visit http:\/\/www.rapidscale.net.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We announce the existence of a postdoctoral position to develop activities within the Thematic Project 2017 \/ 12004-1 (Edaphic mites: diversity, relation with the microbiota and ecology in natural and cultivated areas in Brazil, with elaboration of databases and keys to identify selected mite groups), financed by FAPESP and coordinated by Prof. Dr. Gilberto Jos\u00e9 de Moraes (ESALQ-USP).\nWe are looking for candidates with experience in the area of taxonomy of edaphic mesostigmatid mites, preferably with skill in the elaboration of taxonomic keys using Lucid\u00ae program. The fellow will also participate in field collections and processing of collected mites, collaborate in the preparation of taxonomic catalogs about families of edaphic Mesostigmata, in biosystematic and biological studies. Activities will be conducted at Departamento de Fitossanidade, FCAV-UNESP, Jaboticabal, S\u00e3o Paulo state, under supervision of Prof. Dr. Raphael de Campos Castilho, researcher of this thematic project.\nThe postdoctoral fellow will receive a fellowship from FAPESP for 24 months, with possibility for renewal for another 12 months, with the possibility of having installation assistance.\nExpected starting date: January 1, 2019.\nInterested parties should send a letter of interest, CV and contact of 3 references until November 4, 2018 to Raphael de Campos Castilho (raphael.castilho unesp.br).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Members of the Liaison Committee of the European Forum of Medical Associations (EFMA) met with WHO\/Europe staff for a full day to discuss future priorities for collaboration. These included antimicrobial resistance, immunization awareness and external communications. As a result, EFMA and WHO\/Europe identified several joint projects on these areas that will be launched in the coming months.\nMs Zsuzsanna Jakab, WHO Regional Director for Europe, met with EFMA members and highlighted the importance of advocacy and active partnerships to raise awareness of health issues. Ms Jakab praised the cooperation between EFMA and WHO\/Europe and reiterated the importance of bringing forward new projects.\nEFMA was originally established in 1984 to enable WHO to have an open and continuous dialogue with European national medical associations. EFMA is an independent organization that aims to improve the quality of health care across Europe and formulate consensus on health policies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wire Storage Trucks, Folding Security Trucks, Plastic Transfer Truck, Rolling Locking Computer Cart Cages, Mobile Lockers at Global Industrial.\nThe Mobility Of A Standard Truck And The Security Of A Lockable Storage Unit Are Combined For Maximum Versatility In These Security Carts.\nHeavy-gauge chrome plated 2-1\/2\" square wire mesh allows for visibility and air circulation, makes inventory control simple and minimizes dust accumulation. Dual hinged doors with padlock hasp for security. Choice of polyurethane wheel swivel stem casters with protective bumpers or extra capacity dolly base with corner bumpers and rubber wheel casters. Shelves not included. Assembles easily.\nAdditional Quick Adjust Shelves are available for greater interior shelf space. Shelves can be added, removed or adjusted without disassembling the truck.\nSecurity Carts Protect Against Theft and Damage!\nFolding trucks conserve valuable floor space, as they fold when not in use. Completely encloses all types of merchandise and lockable for maximum security. Constructed of 1\" square tubing and heavy gauge 1\" x 2\" wire mesh. Top, bottom deck and center shelf are 1\" square tubing covered with 20 gauge sheet metal for added strength and rigidity. Center shelf folds to allow storage of oversized cargo. Security carts maneuver easily on 6\" x 2\" hard rubber casters, 2 with brakes. Gray epoxy finish. Shipped assembled.\nHeavy Duty Lockable Security Trucks Secure Contents While Maintaining Visual Contact.\nSecurity cage trucks are made of flattened 13 gauge expanded metal mesh to provide maximum security with clear viewing of contents. Mesh doors include a padlockable hasp for security (lock sold separately). All welded steel security trucks have 12 gauge steel shelves, and are reinforced with 3\/16\" thick angle corners for strength. Tubular steel handle has a smooth radius bend for comfort. 12 gauge caster mounts accept bolt-on 6\" phenolic casters (2 rigid, 2 swivel) for easy mobility. Available with 2, 3 and 4 fixed shelf styles, or adjustable middle shelf to add storage versatility. Gray powder coat finish.\nLittle Giant\u00ae Mobile Storage Lockers Securely Transport Materials For Schools, Workshops, Storage Rooms And More.\nLittle Giant\u00ae Mobile Storage Lockers are solid steel carts with gray powder coat finish for long lasting durability and dependability. Flattened expanded metal panels with 3\/4 x 1-1\/2 inch diamond-shaped openings allow air circulation and easy viewing of contents while maintaining high-strength security. Double doors swing open a full 270\u00b0 and are made with a padlockable hasp. 6 or 10 inch diameter Phenolic, Polyurethane or Solid Rubber wheels provide easy mobility. Various configured Fixed or Adjustable type 1, 2 or 3 shelf units are available in different dimensions and load capacities. 1 Year Limited Warranty.\nSecurity Box Trucks Are Ideal For Secure Storage In Shipping Departments, Warehouses, Schools And More.\nSecurity Box Trucks are solidly made for long lasting durability and provide secure storage and transport of cargo. Expanded metal sides and top provide air circulation and easy visibility of contents. Hinged top cover has padlock hasp for added security. Side drop-gate folds down for quick loading and unloading. Solid steel bottom deck provides durability and strength for heavy loads. Casters provide smooth transport.\nStore, Protect and Transport Valuable Cargo.\nSecurity Trucks feature a unique locking latch system that can be padlocked to safeguard the cargo. Trucks are manufactured of 12 gauge steel on the top and base, sides and ends are constructed of \u00be\" flattened expanded metal surrounded by a double-welded 1-\u00bd\" x 1-\u00bd\" angle steel frame. Trucks roll effortlessly on 5\" x 2\" phenolic casters (2 swivel, 2 rigid). Shipped assembled. Made in USA.\nOptional Middle Shelf is constructed of 12 gauge steel for additional storage (31-\u00bd\" shelf clearance). Note: Must be ordered when Truck is ordered.\nOptional Fork Pockets allow the Truck to be lifted and moved by a forklift. Note: Must be ordered when Truck is ordered.\nSecurity Trucks Are Used For Secure And Efficient Commercial Or Industrial Material Handling And Transport Applications.\nSecurity Trucks are made of solid steel construction for long lasting durability and dependability. All-welded 2 x 2 inch steel wire panels provide total security and see-through visibility on cage type mobile material handling and transport carts. Feature removable push handles and 8 inch diameter mold-on casters, 2 rigid, 2 swivel. 3-point locking handle with steel rod locking mechanism. Digital Security Lock option available. 2000 lb. load capacity, 66 inch height trucks are available in 48 x 24 inch or 60 x 30 inch lengths and widths and choice of Red, Green, Blue or Gray color powder coat finishes. Optional Metal Shelves adjust easily along grid panel using included plastic shelf clips. 250 lb. capacity shelves are available in 14-gauge steel.\nDurable Plastic Truck Is Easy-To-Clean And Provides Heavy Duty Security.\nSecurity truck has a strong, low maintenance polyethylene body that is easy-to-clean and impact resistant. Durable safe and secure, these transfer trucks offer multiple options for use! Two \u00bd\" thick plywood doors with a durable textured ABS laminate finish provide ample security utilizing two lockable slide bolts. Heavy-duty plated 8\" door hinges and integral door holdbacks are included. Plastic security truck includes twin zinc-plated steel push handles and rolls easily on 6\" polyurethane casters (2 swivel with brakes and 2 rigid) mounted to a strong galvanized steel base for long life. Shipped assembled.\nOptional \u00be\" Painted Plywood Shelves are available for additional storage space. 34 Cu. Ft. Security Trucks can accommodate two shelves; 46 Cu. Ft. Trucks accommodate three shelves.\nThese Mesh Security Trucks Provide Content Ventilation While Maintaining Security!\nThese security trucks feature durable 14 gauge steel construction. Angle iron and expanded metal pattern on cabinet doors and sides allow visibility. Doors lock using recessed 3-point handle that is padlockable. (Lock sold separately.) Interior storage area measures 48\"H. Moves easily on 6\" phenolic casters. Ships fully assembled and ready for use. Durable gray powder coat finish.\nAluminum Security Storage Trucks are Lightweight and Corrosion Resistant.\nThese cage trucks are designed to provide safe, lockable security of valuables or dangerous materials. Hi-tensile, primary extruded aluminum is rust-proof for long life. Framework and base are constructed of .070\" thick extruded aluminum. Solid Panel Truck features 1\/8\" thick aluminum sheet that are louvered for ventilation; Mesh Panel Truck is constructed of 1\/8\" thick aluminum amplimesh for maximum airflow and easy viewing of contents. Doors have lockable latch to increase security. Aluminum security truck rolls smoothly with polyurethane or pneumatic casters. Limited Lifetime Warranty against rust and corrosion.\nMetro qwikSLOT Wire Security Storage Trucks Offer Easy Visual Inspection Of Contents For Efficient Inventory Management.\nStrong and durable cage trucks offer 2x2\" steel grid panels on three sides for easy access. Panels allow the user to move shelves as needed on the grid paneled sides. Shelves come with 4-shelf clips. Choose one shelf type from the (2) shelf options, either (1)250-pound capacity shelf in 14-gauge steel, or the 3\/4\" layered wood. Truck has 2-swivel, 2 rigid casters. Polyurethane and black mold-on rubber swivel casters include \"total-lock\" brakes. Extra shelves available. Made in the USA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today we past well over 400 views on this new site! Thanks to everyone visiting!\nThe biography section is almost completely transferred from the old format. The chords are under way. And next is discography.\nThe first followers of the blog has also turned up! We hope to keep you satisfied with updates on Johnny Cash! And also posts on related issues.\nHave a fantastic evening, and keep checking in! Things change here at http:\/\/www.unofficial-johnnycash.com all the time nowadays!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaomsz b/data_all_eng_slimpj/shuffled/split2/finalzzzaomsz
new file mode 100644
index 0000000000000000000000000000000000000000..89ca1e8397e52ca566b96101ce9b925a1aa46db2
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+{"text":"The EWC Mediwellness Centre was founded in 2004. We have since become well established and respected for our range of 'miracle' treatments. With EWC, clients can choose between a range of technologically innovative treatments for body, health and skin. With everyone's busy schedule these days, it is easy to neglect your overall well-being. Yet, you still want to look young, healthy, and fresh. We understand this dilemma, which is why we offer revolutionary health and beauty treatments at our wellness clinic. Through our services and aesthetic treatments you will be looking and feeling younger in no time.\nWe specialise in body contouring and reshaping treatments. Endermologie is our specialist treatment. This treatment will reshape your body through skin tightening, slimming and cellulite reduction, by manipulating the connective tissues underneath your skin. Therefore, without receiving surgery, endermologie can eliminate those unwanted fat cells, leaving you with a firmer and more toned figure.\nOur clinic is situated in the beautiful Vineyards Office Estate in the Northern Suburbs. We welcome everyone to come and relax and receive world-class treatments from our qualified team of beauty therapists and doctors.\nThe range of expert treatments available at EWC guarantee results, even after the first session. Visit our website for more information about each treatment.\nAbsolute Ablutions is a respected manufacturer of mobile ablutions in South Africa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tire World & Auto Service LLC provides Heating System Diagnostic services to Chattanooga, TN, Red Bank, TN, Hixson, TN, and other surrounding areas.\nWhy Should You Have Heating System Diagnostic Services Performed at Tire World & Auto Service LLC?\nWe proudly service the Heating System Diagnostic needs of customers in Chattanooga, TN, Red Bank, TN, Hixson, TN, and surrounding areas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Boil potatoes or cut plantain into two and boil. Remove the skin and crumble the vegetable nicely.\nIn a pan, heat oil and splutter mustard seeds. Add urad dal, green chilli pieces and curry leaves. Once the dal turns pinkish add the crumbled vegetable, turmeric powder and salt. Mix well and switch off.\nOptionally, Lemon juice can be added before serving.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To report abuse or to get help, contact your provincial health authority. Each province has resources to help.\nAt work, tell your supervisor and the human resources manager about your situation. Discuss scheduling options and other safety precautions to provide for your well-being. Give a recent photo of the abuser to your human resources manager, and if possible, ask to prohibit the abuser's access to your workplace. Tell human resources if there is a current protective order in place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WineO-Bingo Bingo Cards to Download, Print and Customize!\nAre you hosting a WineO-Bingo party? Your guests will love playing WineO-Bingo Bingo! Download 2 free pages of WineO-Bingo Bingo including instructions and a randomized call sheet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Chrysler Pt Cruiser Parts \u00bb thanks for visiting our site, this is images about chrysler pt cruiser parts posted by Maria Nieto in Chrysler category on Apr 22, 2019. You can also find other images like chrysler wiring diagram, chrysler parts diagram, chrysler replacement parts, chrysler electrical diagram, chrysler repair manuals, chrysler engine diagram, chrysler engine scheme diagram, chrysler wiring harness diagram, chrysler fuse box diagram, chrysler vacuum diagram, chrysler timing belt diagram, chrysler timing chain diagram, chrysler brakes diagram, chrysler transmission diagram, and chrysler engine problems. Please click next button to view more images.\nHere you are at our website. Nowadays we are delighted to announce that we have discovered an awfully interesting topic to be pointed out, namely chrysler pt cruiser parts. Lots of people looking for info about chrysler pt cruiser parts and definitely one of them is you, is not it?\nThere are lots of explanation why you are researching for specifics about chrysler pt cruiser parts, and surely, you are searching for new suggestions for your purposes. We determined this on the web sources and we feel this is one of many wonderful material for reference. And you know, when I first found it, we liked it, hopefully youre too. We know, we may have different opinions, but, what we do just plan to assist you in finding more recommendations concerning chrysler pt cruiser parts.\nAs Well As d dodge shadow together with d dodge daytona turbo z as well as d dodge shelby charger n further maxresdefault furthermore further leftfront moreover also t ec r zee s yzobrze swstw in addition s l together with as well as d dodge shelby charger shelby sale in addition fan relay module ab for chrysler voyager jeep wrangler tj radiator for town country rs dodge moreover d pt cruiser intercooler intercooler further part diagnosing p p p and p faulty pcm chrysler v lyos il rae. T Ec R Zee S Yzobrze Swstw besides Fan Relay Module Ab For Chrysler Voyager Jeep Wrangler Tj Radiator For Town Country Rs Dodge likewise S L likewise likewise D Dodge Daytona Turbo Z.\nchrysler pt cruiser parts diagram, chrysler pt cruiser parts tension pulley, chrysler pt cruiser parts and accessories, chrysler pt cruiser parts catalog, chrysler pt cruiser parts online, chrysler pt cruiser engine parts. 2005 chrysler pt cruiser parts. pt cruiser dash replacement parts. 2001 pt cruiser parts. 2002 pt cruiser parts manual. chrysler pt cruiser hood support. 2001 chrysler pt cruiser white. 2001 chrysler pt cruiser parts. pt cruiser dash kit. 2002 chrysler pt cruiser limited. pt cruiser a frame. pt cruiser hood support rod. pt cruiser o2 sensor replacement. chrysler pt cruiser transmission parts. chrysler pt cruiser convertible parts. 2001 pt cruiser dash lights. 2004 pt cruiser dash lights. chrysler pt cruiser back window trim parts. 2005 chrysler pt cruiser turbo. 2002 chrysler pt cruiser parts. 2005 pt cruiser rear suspension. pt cruiser turbo parts. chrysler pt cruiser air conditioner parts. 2006 chrysler pt cruiser parts. 2003 pt cruiser parts. 2003 chrysler pt cruiser parts. 2014 pt cruiser inside.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Travel Hub mission is to inspire people to explore the world.\nOur expert travel planners can help you in enjoying the perfect holiday experience.\nPartnering with all the major travel operators around the globe and with access to all global tourist attractions we can be your prime travel companion.\nCall us today and make your dream trip come true.\nWe can be your trusted partner in arranging all the travel documents, accommodation facilities, transportation and relaxed travel itineraries for making every journey with us a memorable one.\nSame Day Visa change Airport to Airport Package in best price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Loews Hotels & Co., The Cordish Companies and the St. Louis Cardinals announced the groundbreaking of the $65 million Live! By Loews \u2013 St. Louis Hotel. Set to debut in early 2020, the four-star hotel will be located in the Ballpark Village, the new dining and entertainment district located next to Busch Stadium in the city.\nDesigned by HKS, Inc., the hotel's exterior architecture and interior design will be influenced by local St. Louis character and the St. Louis Cardinals. Its 216 rooms will include 19 suites.\nA six-story mural on the building's exterior will display the footprints of Busch Memorial Stadium (Busch II) and the current Busch Stadium. Inside the hotel, design elements will be carried through by locally-inspired artwork, fabrics and textures, complimented by technology.\nOn the first floor of the hotel, The Great Room and Outdoor Terrace incorporates an indoor lounge and an outdoor terrace with food and beverage options, including private dining, as well as a digital AV wall. The open-air Gashouse Gang bar will be a nod to the baseball team of the St. Louis Cardinals in 1934. The exterior's wall will show the names of each player from this team in a custom art piece. On the second-level amenities deck, the Terrace bar's media wall and fire pits serve as focal points for the partially-covered venue.\nAlso located on the hotel's second level, 17,000 sq. ft. of meeting and special event space will offer views of the St. Louis skyline. The space's east side will display all glass doors that open to an event lawn. An executive board room and several break-out meeting rooms will complete the event space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The father of a man shot by a French soldier as he carried out a machete attack at the Louvre museum in Paris says his son is not a terrorist, according to BBC.\nRetired Egyptian police general Reda al-Hamamy accused the soldier who shot him of \"using brute force with a poor young man\".\nThe French authorities say his son Abdullah, 29, was shot when he attacked the soldiers, injuring one of them.\nThey say he shouted the Islamic phrase \"Allahu Akbar\" (\"God is greatest\").\nA French police source said Abdullah Hamamy was no longer in a critical condition but was still unable to communicate and could not yet be questioned.\nLe Pen kicks off campaign, promises French \"freedom\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"United States ambassador to the United Nations (UN) Nikki Haley maintained this week that America's recent attack on Syria was necessary because it will help prevent chemical weapons from being used in our country.\nThat's right \u2026 attacks on American soil.\nIn an interview ripped straight from an episode of Homeland, Haley channeled her inner Dar Adal when she claimed that America's latest unconstitutional intervention in Syria was warranted because it was defensive in nature.\nWhat was this attack? An ill-advised escalation of a proxy war \u2013 one that occurred less than two weeks after Trump announced his intention to remove U.S. troops from this war-torn nation.\nThat sort of de-escalation \u2013 which is precisely what Trump promised he would do during the 2016 campaign \u2013 was obviously unacceptable to the war gods.\nAnd so on cue, an alleged chemical weapons attack was launched on April 7 in a suburb of Damascus \u2013 ostensibly by the regime of Syrian president Bashar al-Assad.\nDo we buy this narrative? Not for one second.\nAside from the suspicious timing, as we've noted in the past prior chemical weapons attacks in Syria have been exposed as false flags.\nAgain, this is precisely the sort of thing ripped straight from an episode of Homeland \u2013 with shadowy deep state operatives puppeteering the mainstream media on behalf of the military-industrial complex. All while U.S. taxpayers shell out billions of dollars they don't have on a massive new military spending spree \u2013 including military spending for other nations.\nIt's sadly predictable. And embarrassingly transparent.\nUnfortunately, Trump rolled over for it. Just liked he did on taxes. Just like he did on his proposed border wall. And just like he did on the budget (which was the last straw as far as we were concerned).\nHe rolled over for it.\nPretty surprising, isn't it? Yes. This is the guy who beat both political parties \u2013 and the mainstream media \u2013 to become president. A guy who vowed in his inaugural address to fundamentally reorient the foreign policy of the United States away from costly, reckless and ineffective global interventionism.\nAmerica was no longer going to be the world's policeman, Trump said \u2026 yet our nation is once again picking up the billy club.\nIn fact, in addition to the recent airstrikes Haley told Fox News that the ground troops Trump was planning on removing from this sovereign Middle Eastern nation would be staying there indefinitely \u2013 at least until America had destroyed the Islamic State (ISIS), removed all chemical weapons and gained a better idea of what the Islamic Republic of Iran was up to in the region.\n\"We are not going to leave until we know we have accomplished those things,\" she said.\nIn other words, the coup is complete.\nWatching this administration cave to the neoconservative establishment isn't just disappointing \u2026 it's terrifying.\nSeriously: If they can get a guy like Trump to flip, there really is no hope.\nAlso, it's not even remotely lost on us that in between his announcement about pulling ground troops out of Syria and committing us to perpetual war there, the office (and temporary residence) of Trump's most prominent personal attorney was raided by agency of the Federal Bureau of Investigation (FBI).\nBottom line? Our nation has crossed another significant milepost on the road to perpetual enslavement at the hands of the military-industrial complex \u2013 which is exactly what U.S. president Dwight Eisenhower warned us against in his farewell address in 1961.\nIf they can get a guy like Trump to flip, there really is no hope.\nIn the 1991 movie JFK, there was a memorable scene filmed on the National Mall between a fictitious 1960s-era deep state operator (portrayed by Donald Sutherland) and Jim Garrison (portrayed by Kevin Costner). Garrison was the district attorney of Orleans Parish, Louisiana \u2013 a guy who doggedly pursued an alleged conspiracy into the assassination of U.S. president John F. Kennedy.\nIn that famous scene, the deep stater \u2013 \"Mr. X\" \u2013 shared some pearls of wisdom with Garrison about the underlying motivation of the deep state.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I wonder if you will remember?\nHow you brought frenetic joy and energy into every room you entered. How the room vibrates with anticipation when you enter it. How you mirror light and shimmer it around like a crystal hanging in the window.\nHow you watched a cartoon based on your favourite song to sing with Bede, 'We're Going on a Bear Hunt'. The characters sung 'my heart is still beating and I'm still thinking of you' as they longed for one another unable to overcome separation. Finally feeling safe enough to stop being brave, you begin to cry a guttural, uncontrollable, grief riddled cry. I wonder if you will remember how real your grief is. That it is not a manufactured memory it is a grief borne from a profound and humble love.\nI hope you will you remember that you gave him a new brand of normal.\nUnquestioning, unrelenting, unerring. His fierce, uncompromising, sister. Never allowing for his disability, but always protecting any vulnerability. There were never any excuses with you. You raised him up.\nIf he had the book. You wanted it. He was generous but you taught him to share.\nIf he wanted a quiet moment you wanted to invade it and be right there with him. He was kind and determined but you taught him tolerance.\nIf he was learning you wanted to conquer. He was determined but you made him keep pace.\nHe was our cherished baby but in your love he became the best kind of big brother.\nI hope you will remember the 'small' stuff. Like the walks to the park, the shared stories, the drives to kindy, the playing on the mat. The evenings spent at the beach watching the sunset, the spas floating together, the long drives with the two of you side by side \u2013 you asking to hold his hand, indignant at his refusal. Laughing, exploring, loving, living. In every way each other's equal.\nI know you will remember the stories. Like how at 2 you couldn't sort your colours but you could administer a subcutaneous infusion or a dose of heavy sedation no problems. How a hospital bed wasn't a border for you. It was a seat, to sit with your beloved.\nBefore you were born you would kick him while he was fighting for his life, he would snuggle into the kick. When he died you would still crawl into his cot and stroke his face as you said goodbye, leaning into him. From life to death uncompromising love, his sister in his space. Right by his side. Unmoving. A mirror to his light reflecting it right back to him.\nYou won't remember that when someone cried you would take their hand and walk them into see him so they could stroke his cool face and be reminded that although Bede was \"diff-ent\" he was still Bede and so he was OK.\nThose details that conjure the trauma, the grief, are irrelevant in a way. Eclipsed by the love that drove them.\nMay you always remember the love that allowed you to care for his body when his soul was gone. The love that allowed you to accept your brother just as he was even in the most challenging of circumstances. The same love that drove your friendship.\nYou will hear the stories of what a fierce, funny little thing you were forcing him to keep up with his therapy to keep up with you. But will you remember your tenderness, your gentleness if he was unwell, your care?\nWill you remember the intangibles?\nThat you were a core part of his everything.\nThat Bede was never as normal as when he was with you.\nHow much you adored him and admired him?\nWill you remember that along with G you were his world?\nWill you remember the chats, the reaches for a cuddle, the quizzical looks. The fed up sighs, the delighted squeals, the Christmas mornings.\nI know you won't remember the first time you met. Him supposedly not understanding the magnitude of the moment, you a day old supposedly knowing not a lot. Both of you reaching out across the bed to hold one another's hands.\nHave you ever seen a boy happier to have a sister to roll with?\nI hope you will remember how true it all was. That in 2 years the two of you packed in enough bonding, enough love, enough hope, enough tears, enough joy, enough light to last you both a lifetime.\nThis is not a memory we have created or reinforced for you.\nThe two of you were brother and sister.\nWith Gus by your side the three of you were everything. Gus enabled him, he shone and you my joyous little fairy taught him how to dance in the light.\nIn the end it doesn't matter if you remember.\nIn your formative years you were exposed to magic. That won't leave. That is imprinted on each fibre of your being. Your unique brand of defiant, nurturing, funny, joyful magic intermingles with his persistent, fortifying, hopeful, funny, light filled magic and it meets Gussy's pragmatic, funny, clever, nurturing, magic.\nAnd death can't separate that.\nAnd time can't diminish that.\nAnd forgotten moments don't erode that.\nI have no doubt you will forever be your own spectacular person. You will also forever be his sister.\nThis feeling that wakes you in the middle of the night crying for him. This feeling that he is being torn from us cell by cell. This feeling is the hard edge of love.\nMay all the softness of all the easy love he has shrouded you in comfort you and carry you and hold you. May his light shine the truth upon you. You were loved and you loved and you belonged to each other and you belonged together.\nI'm sorry that you're not.\nI'll meet you where we are whole and easy and glow.\nA Bede update will come \u2013 a response to your love.\nTonight is Cressida's. Just as Gus has his own reminder for all the he is and has been.\nAnd so he died as he lived with strength, dignity, grace, shining his light and in the arms of those that loved him the most.\nHe was signing for 'more' and smiling almost until the very end.\nGood naturedly defiant, triumphant. He did it his way.\nHe stayed true\u2026 A peaceful protest wrapped in warmth.\nBede was called into hospital last week when, as he smiled and laughed and attended rehab, we got the news his blood sodium was so high it would send a grown man into a coma. He continued to live on his terms. A few days into the admission, despite brain scans showing no acute change, we got the sense it was time to take our boy home. The doctors supported us but were also backing him to turn it around as he had done spectacularly so many times before.\nWe made it back to his happy place just in time on Tuesday. He died in the early hours of the next morning.\n12 days ago Bede was making pizza, smiling, attending kindy, hanging with his brother and sister, living the best quality of life we could have ever hoped for. Roy says it feels a bit like he was on a train and simply thought \"Yep. This is my stop, time to hop off.\" Simple as that. Whatever it was Bede once again outwitted brain cancer to do it his way.\nA gentle soul with a cheeky sense of humour. Determined to have the peace he so richly deserved. There was none of the pain or agitation we feared for him. Love won.\nThere was no raging against the dying of the light.\nJust a miraculous boy who shines on into the night.\nBede you remain the wonder that is keeping the stars apart.\nI carry your heart with me, We carry you in our hearts.\nThis news being delivered by blog and social media reflects the incredible number of incredible people that love Bede. It is an overwhelming task to inform you all in a way that does justice to the friendships Bede has cultivated. Please forgive us as we focus on caring for our three children. Details of his service on Wednesday will come in time.\nIt's tempting to say your effect has resonated far beyond what I could have ever imagined but that is simply untrue.\nWe have always had high expectations of you son, from the moment we met you.\nYou have risen to meet them.\nSo it comes as no surprise that when a world class neuro oncologist walks into the room his first words are not medical.\nHis first words are how incredibly proud he is of you. We are too kiddo.\n\"Well it's definitely bigger\" is one of his classic good natured opening line understatements.\nHe says he has never seen a tumour this size in an alive or functioning child before. It's impossible.\nHe says he is now even more blown away by the video of Bede on the treadmill learning to walk.\nHe says for you to be so well let alone learning and developing new skills is impossible.\nHe tells us that he's proud of us too and that the incredible level of love and care you are receiving are the only possible things that could be helping keep you so well, he says some other things I didn't know I needed to hear.\nYou are doing the impossible.\nSome describe the war, the fight. We have always characterised you as mounting a peaceful protest. I would never understate the brutality of brain cancer and all it has delivered upon you but there was no grand invasion. This is who you are and always have been \u2013 the most incredible, generous, accommodating human being. It does not surprise me you are the same with your cancer.\nI look over at your Daddy holding you as you are rousing from the haze of anaesthetic, resisting wakefulness. He is crying. Later he told me he thinks he's better than me at not getting carried away with the thought that we could have you in our arms forever. But he had. Who wouldn't be tempted by that the sweetest of thoughts? We love you so much and you have been doing so well.\nYour doctor says he won't give us time frames anymore.\nInstead he says \"It's impossible for Bede to be here right now, who's to say he won't keep defying the odds.\" He's finally got you pegged.\nI should be crying too but as we review the brutal images I can't help but smile. I'm buoyed by a deep pride in you very predictably achieving the impossible and I know where you are concerned there is ALWAYS faith to be had. The radiologist is confused and starts explaining to me that 'eventually Bede will reach his tipping point.' 'There's not much more the brain can take.' This is very obviously the first time he has met you. There is joy with you even in the hardest moments. He doesn't appreciate that.\nThe tumour is so big now small parts die leaving necrotic matter in your brain as it outstrips it's own blood supply.\nAs the next few days tick away you get sleepier. Reminding the deepest parts of my heart that you will die soon. I want to ask you to stay but that wouldn't be fair. You can't. I can't think of that right now. You have a neck that needs nuzzling, a pizza to make and a smile to be lured from the corner of your lips. There's a green sheep that needs to be found. So I push it all to the furthest corners of my brain.\nNow we sit on the beach together and I sing you the lyrics to mine and your Dad's wedding dance song.\nYou have achieved the impossible \u2013 I expected nothing less from you my beautiful boy.\nGo on now in whatever direction you choose. I will be here loving you and wishing you close.\nWe always have and we always will.\nPlease make sure you've also had a read of Part 1 \u2013 Bede is living! Bede deserves for all his joy to be seen.\nWe have done it, we have won, life is good. The sun is shining, his sly, cheeky grin fills our hearts and his steadfast love permeates every corner of our family. Our beautiful Bede has once again risen to the challenge and defined his own course.\nNo longer a fragile or struggling little boy that we need to stay hushed for \u2013 Bede has thrived and is embracing life. Bede loves being surrounded by different people, different sounds, smells, textures. He is making new friends and trying new things. He is gaining brand new skills, the skills they called impossible. Thanks to his persistence and our dedicated team's hard work he is now able to sit up completely unassisted! (yes! you read that right! He is so casual about it.) He went to a performance of The Very Hungry Caterpillar recently and sat up the front with all the other kids and got his face painted for the first time.\nBede's babbling voice with gentle, happy, inquisitive tone and intonation fill the air as he practises words and clumsily manages a few. His contentedness soothes any doubt I ever had.\nBede loves going to the dog park in the afternoon and seeing all his old and new canine friends as they bound over to his wheel chair and nuzzle him or the eager ones that slobber him in kisses to their owner's dismay. He loves attending kindy and being cared for by the staff who engage with him with such skill and genuine care it has blown us away. He loves swimming lessons and his music therapy and protests his devoted and beloved physio louder than ever!\nA workout this hard is serious business.\nHe loves pressing his switches with recorded messages or calling out to us with his laughter to remind us all he is at the centre of our world.\nBede sits on the water's edge at sunset, a total beach baby like the rest of his family, and lets the waves wash over him. The cool sensory tug of the waves comes and goes threatening to topple him over but Bede remains strong, determined and lapps up every bit of happiness that is washed his way. Bede's always been good at going with the flow. That hasn't changed.\nIt's been a year since I dared write the words that so many of my friends who have lived alongside brain cancer have thought but dared not say. A year that left me wrapping Bede and his siblings in a loving cloak of protection. Unable to share as I focussed on him and making it all better. A year that has left me timid and with no time to write but Bede was born to shine and I am privileged enough to be his voice. It's been a year of hard, constant, vigilant work. Both exhausting and energising it's been an absolute labour of love that we could not have done without our incredible support crew of carers, therapists, educators, family and friends. The good bits outweigh the bad. In that time as I dealt with whatever was in front of me I have also felt so overwhelmed by all the support we received from our local and the wider Australian communities. We have been able to give Bede support we never imagined. What thanks could ever be big enough? You joined us in changing Bede's experience of his life. My mum said the thanks would come from Bede. Perhaps the best thanks is a life well lived and he is here enjoying trying salted caramel for the first time and pulling his little sister in for a hug.\nSalted caramel and sunset celebrations.\nLife slowly returns to the hum of normal Roy returns to work, I return to study, Gus takes surf lessons and Cress is in daycare. Each day I am thankful for the ordinary and extraordinary and I love Bede even more.\nWhen you try and tolerate your sister for your mum's sake but she won't stop therapy-ing you!\nBede, it's been a year since I felt life was too hard for you. I was wrong. I should have known.\nYou are remarkable. Thank you for holding the faith when my own faltered.\nThank you for reminding me as you so often do that magic exists.\nYou are bigger than the ugliness of brain cancer and and as small as a tender kiss and whispered sigh.\nThe most complex life defined by the simplest love and purest happiness.\nYou are sitting on the beach with me. Right now. Perfect just as you are.\nOur triumphant king of the point.\nFor this moment and everyone that has gone before and lies ahead.\nThen the doctor walked in\u2026 that friendly face we know too well. \"I've looked at the scan\".\nTo be honest I have been buckling under that privilege and like for many people 2016 has at times been a hard one. Not least was within days of Bede's story airing on The Project we were told we would lose Bede any day. That Bede was in what is known as the honey moon period, even his wellness was bittersweet. That there was no medical explanation of why he was doing so well given his tumour had now grown so large. We were to prepare ourselves.\nSo we did. I withdrew from uni a week before my final exams. Roy once again put work on hold. We stopped pushing him as hard in therapy. We spoke to Gus. We worked with palliative care. We braced. True to form Bede lived, he shone, he grew.\nBut we were now in a new place having received so much attention and support and everyone having a different idea or expectation of how we should be feeling or responding to that attention, how we should be living, what we should be doing and all the while the doctors were certain that we would lose Bede imminently, it was time to plan his funeral.\nSo for the first time my anxiety took over. In the absence of adrenalin, dealing with the blunt force trauma of the news we were losing Bede, that his tumour was now the biggest it had ever been, combined with the chronicity of cares and doing my best to be a fun Mum for our other two kids I have been paralysed. With every ounce of strength I have had I have worked to maximise Bede's quality of life, hold down the fort at home and hold on to our marriage. I haven't been returning your calls, responding to your messages, answering your emails because it's taken everything we have had to just survive. It's been a brutal 6 months.\nBut shutting down doesn't allow for much light to creep in.\nI've had this concept percolating in my mind of the success story.\nMother's post images and stories of their children's lives once they have defeated cancer and they are out fishing, dancing, attending balls, going to school, achieving awards, getting drivers licences. Meanwhile renewed debates rage over quality of life versus quantity and what side effects of cancer render a life not worth living.\nSomehow I don't feel the permission to participate in any of it. Because my child will die or because I am perceived as bias. Either way we won't have that success story.\nBut the truth is we are all going to die. Our success is not in some predetermined outcome or measured in months or achievements it is in the way we live our lives. Bede lives brilliantly and boldly and he is seeing in 2017 surrounded by friends and family and with a smile on his face. I keep getting caught out by these moments of OK-ness where I think I wish someone had told me this would be possible back then.\nSo I want to talk to the mothers who have shared my heart break \u2013 who's children have no cure, there is just time. To the families who the 'success stories' don't resonate with.\nHere I am audaciously holding my terminally ill child up as hope.\nBecause the one promise they made us is that it was impossible that Bede would ever live until two. Now he is starting to drive his motorised car, we have a brand new wheel chair on the way, he is communicating and making choices with his switches, he has completed his kindy orientation. He has made new friends and his old friends love him even more. He is acquiring new skills and sharing the same old soothing light and when I lay on the mat with him the other day he laughed until he cried.\n2016 has been hard for many of us but you can have moments of despair, moments where you lose trust in the system, or faith in yourself, where the challenges seem insurmountable, where you lose your positivity or even moments where you wish for the mercy of death and you can move beyond them.\nBede will turn 4 in a matter of days. He will start kindy in a matter of weeks. His brand new wheel chair will arrive in a matter of months. He is once again slowly being surrounded by people who are backing him for the slightly longer haul. This year I will once again try to go back to uni, roy will try and return to his career, our other kids will each reach their own milestones.\n3 years after we ceased chemo for Bede's cancer, a cancer that was meant to retaliate by taking him almost immediately we are at home, watching the sunset and bathed in the glow of gratitude.\nWe want to wish everyone a happy, healthy, joyful new year. There is so much more to say but for tonight this will have to do.\nThis blog is dedicated to our family's greatest success story. My teacher, my friend, my uncle. A man who demonstrated to us all that a boy swathed in his parents love and his own strength and zest for life could defy the odds and never be defined by disability. A joker, a family man, a special olympian. The very best of us.\nRest in Peace now Dominic, you and your memory are so very cherished.\nThere has been so many beautiful and busy things happening in our lives lately. All the while there is Bede. Steady, unwavering. Our touchstone of love and hope and light that keeps us all grounded. That slows us all down. That makes sure we never get caught up in the business in life.\nIt's funny how elastic time is. Cress has just turned 1 and it feels like she was only born yesterday. That same year in Bede's life feels like an age. He has accomplished much but he has also endured much.\nLast post I wrote about how Bede's distress prompted me to ask the doctors to bring his MRI forward.\nSo there I sit. In that same room. With that same doctor. I am in the same clothes. Roy wears that same weathered look. But getting your heart broken is never the same.\nThe doctor asks if we need a support person. We laugh. We have made our home here in this moment. He laughs. I know what that question means though.\nIt's growing again. We've never had growth in this many scans in a row. Not even in the beginning.\nBut if having your heart broken is never the same what's the difference this time?\nThis time the aftermath is brutal.\nMy heart is raw. Because the truth is I want my son to die. I'm tired. He is tired. I want peace for him, rest. But I instantly hate myself because I know then that is all there will be. He will have no more growth, no more moments tenderly reaching out to his brother, laying next to his sister. He wont squawk away at his dad anymore, he won't let me kiss him in the way only I can.\nHe will just be gone. That is not enough for my special light filled boy. Life is hard but death seems worse.\nRoy likens it to having something you love more than anything else in the world, that you want more than anything else, that you treasure above all else but knowing you can't have it. Knowing it's better for everyone if you don't. Knowing you need to let it go. It's heart breaking.\nAll I want for Bede is happiness. I grieve that we don't all get that.\nI am angry that we feel so alone.\nRoy is confident about life after Bede, that he will be able to go forward living a life in testament to him. That his legacy will be brilliance.But I am scared. Scared that without my steady ship, my touchstone, my beautiful gentle soulful boy I will crumble because my world will never be the same.\nI am scared that the woman that will mother cress will be a stranger to who I am now. So I try to smile as much as I can and love as hard as I can in the hope that I can build her up enough that she will weather having me as her mum.\nWe go on trying to be present, trying to soak in every single moment, utterly mindful of how precious each one is while we spend sunny staurday afternoons choosing toddler sized coffins that would never, could never hold all that our son is. My heart bleeds and I grieve. I am tired. My bones and muscles and every tendon and ligament ache.\nBut as the bitterness of life rages the storms are rolling in across that familiar ocean. I know this too shall pass.\nAs I type tears stream down my face and I pound at the keyboard because life is hard and even writing isn't easy anymore because there is so many conflicting things to say. Because of course amongst all this grief there is light. Bede's interminable light.\nBede has grown more in the last few months than he has in the past few years and I am reminded that often despite the agitation, Bede is the fullest version of himself when his tumour is growing. Gus says now there is just more of Bede to love.\nHe is immersed in his therapy at least three days week. He is building his upper body strength and undeterred by previous failings he is attempting to crawl. This kid is the personification of determination He is relishing laying on the grass in the sunshine with his brother and sister. He is laughing fuller and deeper than ever before unable to catch his breath as the giggles pour out of him.\nOur little Bede who can sign and screech but can not speak mimicked the words 'I love you' back to me the other day and it was everything to me.\nThrough all this sadness and fear and joy and light and grief and hope and uncertainty and love we have found so much comfort in our friends and community. A fundraiser is being held for Bede on June 17 in North Beach. With an aim to making his life as meaningful and as happy as possible and his death as easy and as comfortable as possible.\nI will post more details later but for now you can check out some of the details for the event on the Facebook Event page (CLICK HERE) or on the booking website where you can also buy tickets if you'd like to come down and spend an eveing with us and say Hi. We'd love to see you!\nHe is small but he is mighty. Bede shines.\nI've gone back to uni. In the hope of completing my degree and Bede hanging around long enough for me to be able to afford all the services I wish for him.\nBede's been helping me study.\nOne of the units I'm studying has a strong emphasis on human biology.\nEverything featured in the first lecture on that aspect of the unit related to Bede.\nThe tentacle like villi that line the gut, the sympathetic and para sympathetic nervous systems, the thyroid, the structure of bones, the grey matter of the brain, the list went on.\nThen we moved to the lab and I started looking at some of these things under a microscope.\nCellular structures, as pretty as art.\nIt really hit me that Bede's body is failing him on the most basic, fundamental levels. That if we are the sum of our parts Bede is crumbling from within.\nNo wonder he has been crying so much, no wonder at times the fight has felt so insurmountable.\nSo what is it that has been sewing him together? His light, our love, your hope?\nIf our bodies are poetry in motion, cells signing to one another, are life's intangibles threading Bede back together into the now.\nIf it is all those beautiful intangibles then how strong is that thread?\nI'm not sure. But nonetheless here he is.\nResourceful as ever. Rebuilding himself from within and reinventing himself yet again. Shining his light and his resolve and his unconditional love.\nSo that even as we fall, as we fail, as we falter, Bede pulls us up and reminds us we are more than the mere sum of our parts. We are whatever we each choose to be.\nBede chooses to play with his brother and sister, to work hard in therapy, he chooses to love. He chooses to laugh in between his tears. He chooses to let it shine. He chooses to pull us in as hard and as close as he can.\nBede's last MRI showed the cancer was growing again, significantly.\nBede's symptoms have been intensifying but he chooses not to bow to them. Just as we feel as though it is time to surrender to the thought that he may be deteriorating he regroups, he dominates at life.\nSo tomorrow we go into the next MRI. Brought forward in response to his ever increasing upset.\nI am too tired to hope for a miracle, I am not na\u00efve enough to try but the thought of spending a moment of life without the glorious sum of his cells, without the villi that refuses to absorb nutrients, without the nervous symptoms that refuse to get organized, without the thyroid that refuses to make enough hormone, without the little body that refuses grow, without the bones that refuse to bear their load, without the eyes that refuse to sleep, without the brain that refuses to follow my plan for it, the thought of a moment without those stubborn clusters of cells just renders me useless.\nMaybe that's it after all. Maybe Bede's presence isn't in spite of his body's failings but due to the stubborn, loving, determination that body has generated.\nBecause at each moment he has chosen us. As I once again hold him before he goes under and repeat those well worn words \"you are strong, you are loved, you are important\" we will choose him. With every last ounce of strength. For now and forever.\nBut we are preparing ourselves for the reality that the choice might not be ours for the making.\nSo as he heads into the scanner yet again we hope you'll join us in hoping, praying, wishing for whatever is best for Bede.\nHe may be small, but he is mighty.\nI want to be your earth.\nI want to be your earth so no matter what happens gravity will always bring you back to me.\nI want to be your foundation, the ground upon which you build.\nThe dirt from which all the colours of your flowers bloom.\nI want to be your earth so my valleys bring you far back from the edge and my mountains let you reach your potential.\nLet you relax in my shade.\nSo my sweetness can balance out life's bitterness and the coolness of my seas will take the sting out of your pain and wash life clean.\nI want to be your earth, tie your feet to the ground and never set you free.\nBut deep down I'm not sure you exist for earth. If I'm your earth I'm scared that means you won't always be with me.\nSo let's be each other's song and together we can dance on the breeze. Let's be each other's breeze to carry us all the way home to you and me. Lets be each others light so in the darkness we can both see.\nBut really I just want to be your earth so gravity will always hold you close to me.\nOnce again there is so much to say that it is overwhelming.\nFirst off the long overdue news\u2026..\nYou stormed the heavens with us.\nYou loved, you hoped, you prayed.\nBede's tumour looks stable. It hasn't gone back down, it hasn't grown.\nThe 6 months we were given has been and gone without us even noticing it's passing because there was so much living to be done.\nI want to jump up and down and scream 'He's done it!' but I think the real proof will be in our next scan in December. Thank you for getting behind our bumble.\nIn reality we haven't had much time to think about it there has been too much work to be done!\nBede is a bright determined adventurer. He's an explorer.\nHe no longer exists in a cold, sterile hospital room. His presence dances through our days. His belongings spill from every surface in our home and his cries and babbles and laughter fill every crevice and surface of our home, our car, our lives and our hearts.\nHe is determined and he is strong. I suppose his perserverence is nothing surprising but in this new setting, in rehab and intensive therapies, it is awe inspiring. I watch him, our little boy, who fought for his life and who is now determined to live.\nBede loves story time, trying to climb things and music. One of his favourite things is standing in his walking frame and strumming the guitar.\nOur family has spent the last 5 months or so in our darkest time. Drifting from positivity. We repeatedly reached out needing a life rope and unfortunately for one reason or another none was there. Bede was well but despite this, we were not. We have been emotionally, practically, financially, physically at our breaking point. One thing has come after the other, a minor example of this was our car and belongings being stolen from our home.\nWhile we have been able to delight in our children and take solace in some of our friendships it is hard to explain just how hard it has been and it has been lonely. Because no one can do it for you.\nBut we have put one foot in front of the other, and clung to each other and now finally the clouds are parting. We have a carer for the next 6 weeks which is some long awaited help. We feel like we can breathe again and tackle our challenges together and there is space to blog now. So after some unexpected time away, time that we needed to take care of our family, we're back.\nThank you for asking after us.\nThank you for letting us know you miss the blog.\nI suppose the moment for this blog has long past.\nBut it's been too difficult to write.\nFollowing so much joy and hope and just spectacular everyday-ness Bede's last MRI wasn't great.\nWe had been talking about Bede walking, about him going to school one day.\nOur doctor walked into our rehab session.\nI clutch Bede to my chest.\nHow dare cancer do this to us now.\nNot when he finally visits the park, participates in family dinner, explores.\nNot when he is learning to live.\nI desperately want to put brain cancer on hold.\nThis is what the moment you know you're about to be told your child is dying looks like. You'd think I'd be an expert by now but I'm not.\nThis is what disconnected soul trauma looks like. A sunny friendly room, with two happy babies, a loving husband, and a man who is far too kind tasked with breaking my heart.\nI slump onto Bede's therapy mat frozen. Looking between my tiny newborn daughter and Bede. Talking to B, reassuring him even though he is oblivious.\nI can't bring myself to get up. I don't want to. I know they're waiting for me.\nBut I cannot cross that threshold. I don't want to go back into a world where even the golden moments are all about losing Bede. I know when I stand up I lose him all over again. I know what that is now and today I can't bear the grief. I don't want to cross that threshold. But here we are and mothering Bede as he deserves demands that I do. So I rise and I brace and I refuse to cry because there are questions be asked.\nThe answers are brutal. The tumour shows significant growth although we can't know for sure how quickly it will take Bede its estimated he only has 6-8 months.\nMy worst fear that Bede has hung around just long enough to meet his sister seems to have arrived.\nAt night Cress wakes me up to breast feed but I cant get back to sleep. Instead I lay there wondering how I will mother Bede in death. I know we have always wanted to keep him at home and I wonder if and how that will be possible. How I will honour him.\nI remember all we have is today.\nRoy and I focus. We regroup. We know we've been here before. Almost a year a go exactly. We once again book tickets to Sydney and we hold our breath\u2026.\nOnce again Sydney delivers a completely different perspective.\nIt lashes out exhausted hope.\nProfessor Sydney doesn't think Bede is necessarily terminal. He doesn't think we should rush the next MRI. He does think the tumour change that's clearly visible on the MRI is within the realms of normal fluctuations. He disagrees about its significance.\nHe reminds us to focus on therapy and rehabilitation and nutrition. He says we may just have a very long haul ahead of us and we need to focus on minimizing Bede's long-term disability.\nWe try and reconcile this with the prognosis Dr Perth gave us. Two men who are leaders in their fields. We decide to live each day is it comes, order a bottle of champagne and wake up the next morning and take the kids to Taronga.\nSo here we are again living in the in between. Waiting with bated breath.\nBut there is no waiting for Bede\u2026 he barrels ahead embracing life and learning and his brother and sister.\nSo now the next MRI is tomorrow. We're leaning towards no more treatment if the tumour has grown. That's a scary thing to put in writing in the public domain. It feels like a decision that is almost too deeply personal to share but that's what this blog has always been.\nIf we choose no more treatment for Bede it will be because our backs are to the wall. Because they are all too exceptionally horrendous due to the unique complications Bede would face. I spoke a long time a go about the fallacy of choice.\nSo let's call this the elusive tumour update. Not my best piece of writing, not my best piece of life. It is what it is and it is the best I've got right now. I have a Bede update ready to go but I wanted to separate him from this. I think because who he is now seems so separate from this news.\nI was at mothers group today and some beautiful catholic nuns help out looking after the babies to give the mums a chance to eat fruit toast and chocolate cake. I was talking to one of the sister's about Bede's scan tomorrow. She is a small gentle woman but with all the conviction in the world she said \"Let's pray and storm the heavens!\" So if you have any prayers, love, light, hope, good vibrations, sunshine on a rainy day, wishes to send our way please do.\nI just want our Bumble Bede, our brave little explorer, to be ok.\nHe may be small but he is mighty and more than ever he seems bigger than this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LEXINGTON, Mass., Jan. 22, 2019 (GLOBE NEWSWIRE) -- Translate Bio (Nasdaq: TBIO), a clinical-stage messenger RNA (mRNA) therapeutics company developing a new class of potentially transformative medicines to treat diseases caused by protein or gene dysfunction, today announced that the Company has received verbal notification from the U.S. Food and Drug Administration (FDA) that the Agency has completed its review of the Company's Investigational New Drug Application (IND) submission for MRT5201 for the treatment of ornithine transcarbamylase (OTC) deficiency and has additional clinical and nonclinical questions. The Company submitted the IND in December 2018 to support the initiation of a Phase 1\/2 clinical trial in patients with OTC deficiency. The FDA is placing the IND on clinical hold until these questions are resolved. The Company expects to receive formal written communication with additional information from the FDA in the near future and plans to work with the FDA in an effort to resolve its questions as promptly as possible.\nThis press release contains forward-looking statements within the meaning of The Private Securities Litigation Reform Act of 1995. Such forward-looking statements include, but are not limited to, those regarding: Translate Bio's expectations regarding the timing in which it will receive additional communications from the FDA regarding the clinical hold on MRT5201 and the scope of any questions and concerns regarding its IND, product candidate, platform and delivery system; Translate Bio's plans to work with FDA in an effort to resolve its questions and concerns as promptly as possible; Translate Bio's beliefs regarding the broad applicability of its MRT platform and its MRT delivery systems.; and Translate Bio's plans, strategies and prospects for its business, including its lead development programs. The words \"anticipate,\" \"believe,\" \"continue,\" \"could,\" \"estimate,\" \"expect,\" \"intend,\" \"may,\" \"plan,\" \"potential,\" \"predict,\" \"project,\" \"should,\" \"target,\" \"would\" and similar expressions are intended to identify forward-looking statements, although not all forward-looking statements contain these identifying words. Such statements are subject to numerous important factors, risks and uncertainties that may cause actual events or results to differ materially from current expectations and beliefs, including but not limited to: the risk that Translate Bio may not resolve this or any other clinical hold in the near term or at all, or that the FDA could make decisions that adversely impact the ability of Translate Bio to advance its programs in development; Translate Bio's ability to advance the development of its platform and programs under the timelines it projects, demonstrate the requisite safety and efficacy of its product candidates and replicate in clinical trials any positive findings from preclinical studies; the content and timing of decisions made by the U.S. Food and Drug Administration, other regulatory authorities and investigational review boards at clinical trial sites, including decisions that may arise as a result of the clinical hold on MRT5201; Translate Bio's ability to obtain, maintain and enforce necessary patent and other intellectual property protection; the availability of significant cash required to fund operations; competitive factors; general economic and market conditions and other important risk factors set forth under the caption \"Risk Factors\" in Translate Bio's Quarterly Report on Form 10-Q for the quarterly period ended September 30, 2018 filed with the Securities and Exchange Commission on November 8, 2018 and in any other subsequent filings made by Translate Bio. Any forward-looking statements contained in this press release speak only as of the date hereof, and Translate Bio specifically disclaims any obligation to update any forward-looking statement, whether as a result of new information, future events or otherwise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you or your company requires a private meeting room, larger conference space, or grounds for a fun filled training day let Groomes be your host. We have accommodation at Groomes to provide a boardroom environment, product launch or assessment centre.\nWe can help you plan a corporate function from a simple meeting with business lunch or a full day with team building activities, afternoon tea with home made cakes, dinner and accommodation.\nIf my company ever finds out how much I enjoyed myself they will fire me. Great stay, great food, great company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We value online security and want our customers to feel comfortable when purchasing from us. That's why we have ensured that all payment transactions on our site occur in a highly secure environment. So whether you pay using a card or through Paypal, all of these transactions will be protected by high security encryption technologies that will keep your personal details safe.\nWe accept all credit and debit cards that have the Visa, Mastercard or American Express logos on them. To ensure your safety, we use a highly secure payment system shielded by 128bit SSL technology. 'Secure Socket Layer' (SSL) essentially ensures that there is a secure connection between computers and encrypts the data that is being transferred. This security system ensures that when you order with us, your personal information stays safe. In addition to this, our system is designed so that we will never see your card details meaning that you can rest assured when paying with us.\nWe love Paypal and that's why we've made paying with your Paypal account really easy. At the checkout page, simply choose the Paypal payment method, enter your user email address and password and your payment will occur through your Paypal account. Simple! Paypal uses industry leading security technology to ensure your information is kept private as well.\nOnce we have received your payment, we will email you to confirm your order. This email will contain your payment receipt which will appear as a PDF document. After this, we will go ahead and process your order for shipping. Once complete, we will send you another email to inform you that your order has been processed and shipped.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home Archive by category \"Dominican Brides\"\nDominican brides make great wives. Dominican girls are among the most beautiful girls in the world. Meet the most beautiful Dominican women for dating online. The Dominican Republic is a great place to meet exotic women for dating. Meeting Dominican women is easy and fun. Find 1000's of Dominican women and girls seeking marriage, love and dating. Join a marriage tour to the Dominican Republic which allows you to meet beautiful Caribbean women. Date young and beautiful Latin brides online NOW.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"On March 7, 2019, Top 10 Cases Promoting Progress of Rule of Law in 2018 was released. It was coorganized by the Information Bureau of Supreme People's Court and China Central Television (CCTV ) Press Center. Dior's 3D Trademark Case represented by NTD was selected as the only intellectual property case.\nThe other listed cases are: Zhang Wenzhong's Acquittal Case; Sudden Death Caused by Discouraging Smoking in Elevator in Zhengzhou; Sun Zhengcai's Bribe-taking Case; General Ye Ting Offspring's accusation against Xi'an MoMo in a reputation infringement; Xu Shuiyun's accusation against Wu Cheng District Government for Illegal Demolition; A Hit-and-Run Death Caused by Zhu Zhenbiao's Chasing; Jiangsu Provincial Government accusation against Haide Company for Damaging Ecological Environment; Liu Yongtian's Gang-related criminal case; Ping'an Lutong ordered by Beijing's Shunyi District Court to pay compensation for Pipi's cripple.\nThe selection has been held for the fourth time since 2015. This year many netizens voted for 20 nominated cases, mainly through the media platforms covering the official Weibo, Wechat of the Supreme People's Court, the website of China Court, the official Wechat of China Trial Magazine, etc., and more than 40, 000, 000 valid votes were received. An expert jury was formed to evaluate the cases. They are law experts from Peking University, Tsinghua University, senior news experts from the People's Daily, the Xinhua News Agency, China Media Group, and the department heads from the Supreme People's Court. Ultimately, top 10 cases were selected based on the votes and experts opinions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What if we moms could see all that we're doing -- all the creating and training and coaching and supporting and loving -- rather than dwelling on all that we're not? What if we realized that we only need to fill our daily twenty-four hours with what we're called to do, not what we impose upon ourselves?\nWhat if we gave ourselves grace and redefined accomplishment?\nA productive day might look like a pile of books on the floor next to the couch where you read to your children for an hour, your cadence rising and falling like it has done hundreds of times before as you've turned those very same pages. It might resemble a bunch of sweaty-headed kids who never got a bath because you caught fireflies late into a summer night. It might be seen in a sink full of dishes after a family dinner. It might sound like siblings apologizing to one another for the fourth time that afternoon.\nAccomplishments in motherhood come in many forms, and rarely are they tidy and obvious. Redefine accomplishment. You'll discover that you're accomplishing an impressive amount.\nExcerpt from Then I Became a Mother. Available in Kindle and paperback editions for your summer reading pleasure!\nThank you for posting this perfect, perfect post. This is a message that we all need to internalize and remember.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's not easy to push for something in any organization.\nEither you do it the Old Way i.e. forceful 'mandatory compliance' and we all know where that will get you OR you do it the New Way i.e. Motivated discretionary effort starting from within the company.\nAnd the answer is: You initiate a movement! (Watch TED 3min video here).\nHave you ever started a movement in your company? Did people embrace your movement? Why or why not?\nAt internalDesk, we know how to start a movement. We know what it takes to reach the tipping point.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Down the rabbit hole for a night on the town, our heroine known only as 'The Black-Haired Girl' (Kana Hanazawa) meets new friends who show her an unexpected side of Kyoto's nightlife. As their night moves from tipsy to boozy, they slide into a trippy world of second-hand books, guerilla theatre, and chanceencounters. Meanwhile, fellowstudent 'Senpai' (Gen Hoshino)is desperately searching for The Black-Haired Girl. Tonight is the night he'll confess his romantic feelings for her!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tim Duff, Plutarch's Lives: Exploring Virtue and Vice. Oxford: Oxford University Press, 2000. Pp. 444. ISBN 0-19-815058-X. \u00a355.00.\nWhen Frankenstein's monster turns up in the hut of the poor old man -- and I am talking of Mary Shelley's version, not Mel Brooks' -- cultural education is the novel's evident agenda, both for the monster and for us. The old man has only three books, but how many do you need to civilize a monster? For religious feeling -- a necessary starting point -- there is Milton (a very Romantic and radical gesture this, to turn to the high peaks of poetry rather than the more obvious choice of the Bible). For sentimental education there is Goethe's The Sorrows of Young Werther (it would have to be Goethe for anyone in Shelley's circle). And for everything else? Plutarch. History, morality, science -- Plutarch has it all. He sums up a whole world of Western knowing, a stupendous body of work which, like Milton or Goethe, has the power to make a monster transcend himself.\nIt is snapshots like this that remind us just how much we have lost Plutarch. Over the centuries, for traditionalists and radicals alike, Plutarch could provide an essential educational and ethical resource. He was the paradigm of what made the classics classical. And nowadays? How many departments feel the need to hire a Plutarchan expert? How many courses or curricula take it for granted that knowing Plutarch is a bedrock of the value of classics? Plutarch, above all other authors, shows the fallacy of assuming that classics maintains a stable canon or that scholars across the generations can expect to share an intellectual horizon.\nThis is especially true of Plutarch's Lives. Once the staple read for a Shakespeare, a Lord Chesterfield, a Macaulay, they have so fallen out of fashion that they largely feature only in historians' footnotes, a pit of fragmentary source hunting from which Tim Duff's monograph -- the first monograph in English on the Lives for a quarter of a century -- hopes to lift them. He does not aim to put Plutarch back on his pedestal as the paradigm of classical learning, but he does want to read Plutarch's Lives seriously, as coherent texts in themselves and as key documents of the early Greek negotiation of the Roman Empire. He is building here on the work of Christopher Pelling and Simon Swain in particular, and his book length study offers an intelligent, controlled, and scholarly development of his central claims.\nThese claims can be simply expressed in nuce. First, the Lives are written to a moralizing programme: they set out to construct not full biographies but ethical discussions through the narrative of life stories. Second, this moralizing programme, although it has some normative principles -- ambition and anger are consistently 'the most deadly passions,' good education is a necessary tempering of both -- also aims at a more open-ended and exploratory didacticism. Hence Duff's subtitle, 'Exploring Virtue and Vice' -- the exploration is both Plutarch's and ours. Third, and most insistently, central to this process of moralizing and of exploration is the structuring device of synkrisis, the formal pairing of two Lives, and the act of comparing and contrasting which the synkrisis itself embodies. Duff is consistently -- almost pathologically -- polite in the book, but he does allow himself one moment of scornful anger when he considers modern scholars who have offered editions or translations of half of a pair of lives (Caesar without Alexander, as it were); and there is one moment of surprised regret expressed for those benighted souls who have thought that the synkriseis are of a lower level of prose from the biographical narrative. This reading of the Lives as an integral process of contrast and comparison feeds directly back into the moralizing agenda: it is by exploring such narratives that your own life finds both its paradigms and its evaluations. But also it leads to a fourth and final point that the process of synkrisis between Greek and Roman heroes constructs a political project of locating (contemporary) Greek life in and against the power of the Roman Empire.\nAs with so much of Plutarch, the proof of the pudding is not only in the gnomic conclusions but also in the narrative travelling which ends in such wisdom. Duff's book is structured to develop at length the readings necessary to justify and explore his claims. The first section of the book, entitled 'the moralizing programme', looks at the programmatic statements of Plutarch -- what he claims he is doing. The study of the past is 'a morally improving activity,' argues Duff, and 'through reading [Plutarch's] own literary work' you should learn to 'imitate virtuous men.' This leads to a treatment of the normative principles evident throughout Plutarch and his carefully constructed image of a 'hero's soul.' The second section of the book is four case studies (Pyrrhus\/Marius; Phokion\/Cato Minor; Lysander\/Sulla; Coriolanus\/Alciabiades), cases which are seen as increasingly complex in their exemplary function, until with Alcibiades we are offered a figure who is difficult to fit into any normal model, 'hard to define, hard to judge,' and thus: 'How could Alcibiades be a paradigm for anything -- except individuality itself?' Duff sees this open-endedness not as a paradox for his whole project but as the great strength of Plutarch's portraiture. The final section of the book, 'writing in parallel,' first looks at synkrisis as a device in the Lives and secondly considers the politics of conjoining Greek and Roman Lives in a world where Greek cultural capital was in stark contrast with Roman imperial authority.\nThe qualities of Duff's work are varied and sterling. The book is largely reliable and scholarly (with over 100 pages of bibliography and indices, and no page without a ringfence of footnotes). The readings in general are careful and precise, and the structure of the argument is intelligent and its conclusions convincing. It is particularly important that the last chapter, the briefest, on the politics of comparison, is there. It resituates the apparently controlled and judgmental intellectual world of 'compare and contrast' within the far messier business of cultural engagement and the vagaries of power-brokering. It opens a new vista for the literary work of the previous sections and reminds us of what can be at stake in Plutarch. The central claims of the book are admirably demonstrated, although there is too much rather plodding exposition of Plutarch's texts for my taste. If his general conclusions seem unadventurous, it remains a telling indication of the state of Plutarch studies that they still need to be forcibly made and defended at such length. It is a book that will be used, as well as read, by scholars, and if it succeeds in resuscitating the idea of actually reading Plutarch's corpus it will have proved itself a major achievement.\nCan one, then, ask for more? I think one can, and in at least two ways. First, on politics. Duff is, in my view, certainly right to see the Lives as a document in the politics of Greek enagagement with the Empire. But there is much more to be said. Although Duff sensibly turns to the Political Precepts to give a privileged guide, that text is only one of a host of relevant works in Plutarch and elsewhere. Like Simon Swain in Hellenism and Empire, Duff exclusively privileges texts that talk explicitly and overtly about Roman power, as if the business of negotiating with a dominant system is only conducted face to face. He does not discuss the reception of Plutarch's (political) writing, nor explore the multiple possibilities of reading Plutarch's self-representations and his representations of other figures. Greek culture\/Roman power is a necessary matrix for understanding the work of this period, but it needs more nuance, more sense of the differing strategies of self-definition and public display if it is to be adequate to the complexities of the cultural politics of this period. How to Tell a Flatterer from a Friend, say, or in a different way, Sympotic Questions, or even On Isis and Osiris are also about Greekness, civic identity under new circumstances, finding one's place in the order of things. They bear on how Plutarch represents himself as a Greek scholar in a Roman context. It would have been good to see Duff extend this final chapter and to try to work its insights back through and into the readings of the Lives themselves in a more systematic and developed way.\nThe second area is the ethical politics of imitation. Duff does not push very hard against the central concepts of 'imitation' or 'exemplarity,' and has little interest in following what Plato and others in the ancient world have to say, let alone Kant (among others among the moderns) with his distinction between Nachahmung and Nachmachung. He does not, for example, analyze in any depth what the implications of his view of the Life of Alcibiades are. If Alcibiades is an example of individuality, what are you doing when you read -- follow -- it? What lessons can or ought you draw? If 'hard to judge' is your judgment on Alcibiades what moralizing lesson follows? How do such questions relate to a 'moralizing programme?' Duff does not wish to look very hard at how models -- exemplarity -- work and\/or fail nor in Plutarch's strategies for controlling (mis)reading. There is a dangerous instability integral to the process of 'compare and contrast', especially when linked to an ideal of absorbing a lesson through imitation. The difficulties of Plutarch's project are, however, rather too domesticated in Duff's account.\nPlutarch was for many generations an author who provided moral inspiration, ethical education and narratives to live by. Duff, for all his talk of moralizing programmes, does not seem comfortable with this legacy. Unlike recent scholars of Plato and Aristotle (especially in political theory), Duff does not seek to find an agenda for the modern reader in Plutarch, nor does he extensively engage with that heritage. Plutarch, for Duff, is a good read, but not good for you to read. Perhaps it is our collective embarrassment at the dodgy moralizing use of classics that conspires to keep Plutarch obscure. But I suspect that not to engage more directly with that aspect of Plutarch's work sells his power short. The force of his prose to move, to teach, to inspire generations of readers should be re-explored and not forgotten nor embarrassedly ignored (however nobly motivated such embarrassment is). Duff has done a good job in giving a solid basis for the study of Plutarch's Lives, but Plutarch also need to be treated with a bit more passion if his star is going to rise again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Last week, I gave a talk at the Groupon weekly GeekFest event titled \"Smalltalk MVC Applied in Rails\". Though the talk briefly touched upon Smalltalk's original MVC pattern and how it is applied in desktop development, the main focus of the talk was on decoupling views from controllers in Rails the way they are decoupled in Smalltalk MVC, so I would like to elaborate more on that in this blog post.\nThe diagram above shows the relationship between Model, View, and Controller in Smalltalk MVC with desktop development.\nThe View refreshes itself automatically from the appropriate models it is providing a view of.\nNotice that nowhere in that flow does the Controller directly interact with the View to update it with Model data. In fact, well written desktop applications avoid that sort of coupling to ensure clean separation between control flow and presentation logic.\nNow, if we were to transfer all of these ideas transparently to the web, View state is simply the parameters (request or session) that are populated by user actions. Controllers get access to them temporarily on user actions to cause updates in Models, but then once they render a View, the View itself can be responsible for translating its state (parameters) into Model objects via a Presenter layer. The Presenter layer in Rails is nothing but the good old (badly named) Rails Helpers. They automatically get access to the View context (request and session parameters), allowing them to act as Adapters that neatly hide the details of converting request and session parameters into Model objects the View can rely on to render its contents. This frees Controllers to focus only on Model updates and routing control logic, avoiding the typical clutter with Model loading logic that we often see in Rails applications. This then decouples the Views from Controllers the way they are in original Smalltalk MVC.\nUnclutter Controllers from data loading logic for multiple objects that the View needs, allowing each part of the View to load its data directly.\nMake View partials easily reusable as they rely on Presenters\/Adapters (Helpers) to load their data by pull instead of having to include code in every reusing Controller to push the data into the Views.\nEasily test-drive and maintain the logic of Presenting\/Adapting View data in small cleanly separated methods instead of having that logic all mixed in Controllers.\nWhen Models needed for the View have dependencies in their load order, there is no need to explicitly order their loading in the Controller. Helper methods can be composed of other helper methods, resolving the dependencies automatically.\nAvoid the dissonance in View code caused by a mix of \"@object\" references and Helper \"object\" references. All objects in the View get populated from Helpers with \"object\" references or directly from model classes making the code more readable.\nTrivial extraction of partials from Views given that they do not contain any \"@object\" variables and all references are \"object\" references. Developers thus do not need to put any effort into error-prone switching of \"@object\" variables into \"object\" locals. The helper \"object\" references can already serve as locals.\nControllers already have access to the context of Presenters (Helpers) thus are able to reuse the View data loading logic without need to duplicate.\nThese are some of the benefits experienced in my last three Rails projects, giving the team great flexibility in maintaining Views, Models, Controllers, and Presenters without the mix of concerns typically experienced in Controllers, allowing for much easier test-driven development and flexibility in composing\/modifying features for customers.\np.s.2 Though \"before_filter\"s in Rails can be used to easily load data in controllers. They still put the onus on the controller to do the data loading by push, requiring developers to add such logic to every controller that will reuse a particular View partial, and adding complexity to reasoning about the code.\nSince erectors views are full ruby objects, this would support this new mvc approach even more.\nPretty cool. If it proves itself practical enough, it could become an alternative to Haml.\nI like the idea presented here but I've seen so much 'view-pushing' code in controllers that it seems very unnatural to see views essentially taking care of themselves.\nPerhaps the mindset of most traditional Rails developers is that views are not really there to be re-used -- models and controllers are -- therefore views should b just very 'thin' throwaways? Better not to put too much logic in there.\nBut I can definitely see great applications of this idea. During your talk I was thinking about a story I was working on where we needed to enable\/disable certain options in a pulldown based on a model's state -- all of that logic was in the controller which felt wrong. It would have been cool to put all that stuff in the views or within helpers used by the view.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We spoke with an expert about navigating at-home genetic testing kits, which are growing in popularity.\nWhile at-home genetic testing becomes more popular, given its convenience and low cost, people should still access proper resources and talk to genetic counselors who can offer clear advice on the results and what you need to know about these tests.\nIn this week's episode of CURE Talks Cancer, we spoke with Whitney Ducaine, who is the director of cancer genetics services at InformedDNA, to help explain exactly what you need to know about the FDA-approved 23AndMe test and why you should consult a genetic counselor about concerns and how to digest genetic testing results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Right click on the link and choose \"Save link as\" to download Show Das Poderosas by baixarmp3musica mp3 for free.\n3. Show Das Poderosas by baixarmp3musica - Menina M??\nShow Das Poderosas by baixarmp3musica free mp3 download.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oklahoma has had its share and then some of big names in the sports industry \u2013 Troy Aikman, Blake Griffin and Sam Bradford, to name a few. And then there's Adam Edwards. The former OSU wide receiver went on to a career in the NFL playing defense for the Dallas Cowboys. After sustaining a serious injury, his promising career was cut short. That's when he decided to return to his Oklahoma roots and start an exciting new chapter. We sat down with Adam recently for a look back at life in the big leagues and beyond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I believe the original Damon\/Helios Chrome only played live about 2 times. This is the ONLY time Chrome ever played live in the United States! LUCKILY someone there, who I would like to thank personally, recorded the show themselves. The sound quality is surprisingly good for an audience recording (or possible soundboard) way back in 1981. All kinds of samples and things from TV and things, it must have been trippy as hell to be at this show. I have heard that this could be a soundboard recording. Whatever way this recording was obtained, as Chrome fans we're INCREDIBLY lucky that this rare Chrome show was recorded as one of the only 2 times Chrome ever played live!\nNeed help printing CD Artwork? Click here for an easy how-to set of instructions!\nClick here to check out a super rare poster advertising this one and only time the original Helios\/Damon Chrome played in the U.S.!\nClick here to check out a flyer\/bulletin advertising this one and only time original Helios\/Damon Chrome played in the U.S.!\nClick here to see a cutout of the Chrome section of the same flyer!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaptky b/data_all_eng_slimpj/shuffled/split2/finalzzzaptky
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+{"text":"You rely on energy for your business. Rely on us for energy efficiency.\nLet our team of energy experts help you turn your existing building into a High Performing Facility.\nImproving energy efficiency can help lower utility bills allowing you put more money into your business.\nBetter lighting, temperature controls and indoor air quality make your building a better place to work or live.\nPACE, PPAs, incentives and tax deductions are just some of the options for financing energy upgrades.\nWe helped them. How can we help you?\nHow did Newman Consulting Group help Wayne State University save more than $20,000\/year in just one building? Three simple energy efficiency strategies that cost just $13,200. Click to learn more.\nNew construction, planned upgrades and previously implemented energy efficiency projects all may qualify for PACE financing. Ask Newman Consulting Group, Michigan PACE consultants, if your building project qualifies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was posted on Sunday, May 4th, 2014 at 15:47 and is filed under Uncategorized.\tYou can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.\ndo it \u2013 this will reduce your learning time down drastically and save you much annoyance.3. learn from a star who really earns a big salary as a skilled blogger \u2013 it's time to stop looking for the next greatest thing.\nOfficial Regenerative Institute Vimeo Channel.\n'thank you' from them in one form or another.the following two methods are more direct but more rewarding if successful. if you want to truly get noticed by other bloggers, just say hello to them. put yourself in front of them.\nplug-in) is to write using simple language.\ncan't beat it.grab your shaker cup, toss in some water first to stop the gunky buildup in the bottom. add forty grams or protein, forty grams of dextrose and 7 grams of creatine.you now have the best post workout supplement shake.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was about to leave the apartment to take stuff to my car and make a Target run, when I thought about leaving a light on. I went to turn the light over the stove on which apparently doesn't work now. So I just made a mental note to turn the light on by the door to make sure I don't trip over any of our stuff when I walk back in. As I walked to the door in the dark with my hands full of bags and a to-go cup of coffee, while thinking about not tripping when I get back home, I hit the Christmas tree box, AGAIN as I mentioned the first time in my last post Packing is Pitiful. I tried to recover which resulted in me falling to my knees on top of the box and completely skinning my shins, and now I have two matching bumps and bruises and I have to wear a dress tomorrow\u2026 great. Just another day at Tragic Girl Headquarters. Ouch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Is the Formula of a Compound Determined?\nThe empirical formula for a chemical compound is an expression of the relative abundances of the elements that form it. It isn't the same as the molecular formula, which tells you the actual number of atoms of each element present in a molecule of the compound. Different compounds with very different properties may have the same empirical formula. You can derive the molecular formula of a compound from its empirical formula only if you know the molar mass of the compound.\nIf you know the empirical formula of a compound, you know the elements present in the compound and their relative proportions. Calculate the molar mass based on the formula and divide this into the mass of the actual compound. The division gives you a whole number. Multiply the subscript of each element in the empirical formula by this number to get the molecular formula for the compound.\nChemists can determine the elements in a compound and their relative percentages by a chemical reaction with a known compound that produces products that they can collect and weigh. After doing so, they divide the mass of each element by its molar mass to determine the number of moles present in a particular amount \u2013 usually 100 grams. The number of moles of each element produces the empirical formula, which is the simplest expression of the elements present in a single molecule of the compound and their relative proportions.\nThe first step in determining the molecular formula of a compound is to calculate the empirical mass from its empirical formula. To do this, look up the mass of each element present in the compound, and then multiply that number by the subscript that appears after its symbol in the formula. Sum the masses to determine the molar mass represented by the formula.\nThe next step is to weigh a sample, then divide the empirical mass into the actual mass of the compound. This division produces a whole number. Multiply the subscripts in the empirical formula by this number to determine the molecular formula.\n1. Analysis of a compound reveals it contains 72 g carbon (C), 12 g hydrogen (H) and 96 g oxygen (O). What is its empirical formula?\nStart by dividing the mass of each element present in the compound by the molar mass of that element to find the number of moles. The periodic table tells you the molar mass of carbon is 12 grams (ignoring fractions), that of hydrogen is 1 gram and that of oxygen is 16 grams. The compound therefore contains 72\/12 = 6 moles carbon, 12\/1 = 12 moles hydrogen and 96\/16 = 6 moles oxygen.\nThere are 12 moles of hydrogen but only 6 moles of carbon and oxygen, so divide by 6.\nThe ratios of carbon to hydrogen to oxygen are 1 : 2 : 1, so the empirical formula is CH2O, which happens to be the chemical formula for formaldehyde.\n2. Calculate the molecular formula for this compound, given that the sample weighs 180g.\nCompare the recorded mass to that of the molar mass expressed by the empirical formula. CH2O has one carbon atom (12g), two hydrogen atoms (2g) and one oxygen atom (16g). Its total mass is thus 30 grams. However, the sample weighs 180 grams, which is 180\/30 = 6 times as much. You therefore have to multiply the subscript of each element in the formula by 6 to get C6H12O6, which is the molecular formula for the compound.\nThis is the molecular formula for glucose, which has very different properties than formaldehyde, even though they have the same empirical formula. Don't mistake one for the other. Glucose tastes good in your coffee, but putting formaldehyde in your coffee is likely to give you a very unpleasant experience.\nDeziel, Chris. \"How to Find Molecular Formula From Empirical Formula.\" Sciencing, https:\/\/sciencing.com\/how-to-find-molecular-formula-from-empirical-formula-13712170.html. 22 May 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our WordPress developer Los Angeles team believes WordPress should be the first choice for businesses wanting to establish an online presence to promote their products and services.\nWordPress is a highly versatile and customizable platform which when optimized correctly can deliver outstanding results. With many years of experience as a WordPress Developer Los Angeles company, we understand the importance that a website plays in the overall branding and reputation of a business.\nYour website is often the first interaction that customers have with your business and so being able to leave them with a positive impression and desire to return is paramount. It is, therefore, crucial to have a website that goes beyond normal standards and is able to transmit your brand and values to a wide audience, building trust while simultaneously making a lasting impression.\nAt Branding Los Angeles, our team of designers and marketers work with clients from day one to turn their vision into a reality. Through constant consultation and refinement during the design process, the perfect business logo can be created through one-on-one support and direct collaboration with the design team. Having the best logo design for your company will prepare you for your next marketing push knowing that you have a brand image that is optimized for both online and print marketing use.\nGiven the accessibility of the WordPress platform, many businesses owners initially set up their website themselves or with the help of a colleague. One of the strengths of WordPress web design is that it empowers businesses owners and reduces reliance on programmers. This is great for developing an initial web presence, however as a business grows it can quickly become overwhelming. Beyond maintaining a basic static site, a huge time investment is needed to learn how to effectively manage and optimize growth. The best solution is to hire a team of knowledgeable professionals whose expertise in the platform allows them to adapt and modify your WordPress site to your individual design and technology needs quickly and efficiently. Having a custom website design based on the needs of your particular business and customers optimizes conversion rates of site visitors, a key aspect of growing your business and brand presence online. The expert web design team at Branding Los Angeles has been working with the WordPress platform since its debut, giving us an unparalleled understanding of how to optimize its' performance for any number of different businesses. Contact our WordPress Developer Los Angeles team for more information on WordPress Web Design Services.\nThe adaptability of WordPress means that it can be customized for a huge range of businesses, something which we specialize in achieving through intensive one-on-one consultation with our clients, providing high-quality custom web design solutions.\nAll of the WordPress web design done by Branding Los Angeles is carried out with a bottom-up approach, through constant consultation and feedback with our clients, our team of experts create solutions that integrate your web presence and branding with your unique product or service. With Branding Los Angeles working closely with your business to develop and refine your web marketing and online brand presence, you will have more time to focus your attention on your customers.\nSeamless social media integration with your site has also become a must in today's world, with search engines placing greater emphasis on the social profile of a business. This is another crucial aspect of online branding which needs serious consideration and attention to anyone who is serious about brand development. WordPress web design offers considerable advantages regarding the integration of social media aspects and with a custom approach, the results can be outstanding. Search engine optimization (SEO) is another crucial factor in a business' online marketing and branding. WordPress is an extremely SEO friendly platform, meaning that with the right tweaking it can vastly improve how well your company website ranks on search engines such as Google, Bing, and Yahoo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarscw b/data_all_eng_slimpj/shuffled/split2/finalzzzarscw
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+{"text":"On June 14th and 15th, 2018, WHITRAP Shanghai organised a workshop on The Future of Small Settlements in China and the World Heritage Convention. The workshop aimed to present WHITRAP's on-going research on small settlements in China and the Asia Region, discuss village management issues in the context of the World Heritage Convention, and comment on the priorities of poverty alleviation and urbanisation.\nFifteen representatives from different professions and research fields (heritage studies, architecture, anthropology, law, urban planning, eco-tourism, agronomy, sociology, etc.) gathered at WHITRAP to discuss and share their experiences concerning issues and threats related to villages and small settlements.\nThe workshop addressed the dichotomous situation Chinese villages are facing in this historical moment - trying to achieve long-term sustainable development and a better quality of life, while simultaneously preserving their cultural diversities and values. Differences in terms of economic development, local agricultural systems, geographical areas, ethnic characteristics, and issues in defining authenticity and values were underlined, in particular within the on-going social transformations due to opposite trends of migration and gentrification.\nThe first day was devoted to the presentation of papers, while the second day was conceived as an open 'round-table' event to encourage dialogue and exchange among the participants.\nThis event was the first step for the Institute in developing a network of Chinese and international experts and scholars in related fields, and feedback from the workshop will permit the fine-tuning of WHITRAP's work and research on villages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The appreciation of the US dollar exchange rate against the Egyptian pound over the past few months has led to a lack of spare car parts, exceeding 70% of the market size due to the decline in the amount imported and the restrictions on spare parts at customs for violating the new prices.\nDesouky Saied, head of the spare car parts division in the Cairo Chamber of Commerce, said that the continuous price increase has prevented importers from buying goods due to the decline in the consumers' purchasing power. In addition, most of the goods in the market are currently found in companies' storage, and are not new goods.\nSaied explained that the deficit includes all types of spare parts and those for European, American, Chinese, Korean and Japanese models. The deficit is especially clear among Chinese, Korean, and European spare parts, particularly as these models are those most popular with car owners.\nMember of the spare car parts division in the Cairo Chamber of Commerce, Mohamed El-Battah, said that there is a clear deficit in the availability of spare car parts due to the high price of the US dollar, and the new prices imposed by customs on importing companies. Thus, those parts violating the new customs policies have not been released.\nHe added that the new prices are considered uncertified because they violate the real price of the spare car parts, where these prices are higher than the original ones by 20-30%.\nEl-Battah said that customs must communicate with producer companies in all countries in order to determine the global prices used and applied, and avoid the randomness of customs rules and the irregular price increases.\nHe explained that customs are still using old prices that were originally used when prices were higher and exceeded the current prices.\nHe noted that the most difficult-to-find parts in the market are shock absorbers, mirrors, brake linings, gasoline pumps, fuses, head and rear lights.\nDaily News Egypt has obtained a copy of the list of the trial prices in US dollars. The list revealed that diesel engines of Mercedes, Volvo, and BMW models have jumped in price to $650 for four-cylinder engines, $700 for five-cylinder, and $750 for six-cylinder.\nThe price of four-cylinder engines of Japanese cars has reached $550 and $500 for Peugeot, whereas it reached $450 for Fiat and its equivalents.\nFour cylinder engines of model 102 for Mercedes cars and their equivalent have reached the price of $950, whereas the 104 model engines of six cylinders reached $2,000.\nPeugeot model 505 of four-cylinder engines reached $450, 504 reached $425, 304 and 305 reached $350, 104, 2014 and 205 reached $300.\nAs for Fiat cars and their equivalent, four-cylinder engines of model 123 Argentina has reached $350; model 131 Regata, Bolognese, and Alfa Romeo reached $325; whereas the models of 124, 125, and 1300 reached $300.\nThe most prevalent car in the Egyptian market, model 128, 127, and Ritmo, of four-cylinder engines, reached the price of $275, exceeding model 133 by $25, which is $250.\nThe price of Volkswagen Jetta and Golf four-cylinder engines amount to $400, whereas the Fiat Tipo and Tempra engines reached $375, and Lada's engines reached $325. Volkswagen water-cooled four-cylinder engines reached $300, while air-cooled engines reached $275.\nEngine part prices for Volvo, Mercedes, and BMW reached for $10 for the crank, $50 for the carburetor, and $10 for the exhaust furnace.\nThe prices of rear parts for the same car brands reached $8, whereas Jetta and Golf reached $6, and $5 for Peugeot.\nThe front parts of Mercedes, Volvo, and BMW for the recent models reached $1,600, and other models reached $100. Peugeot Golf and Jetta reached $700, while passenger cars reached $600, and $500 for Fiat 132 and Argentina.\nThe prices of the rear parts for Mercedes, Volvo, BMW recent models reached $1400; $600 for Peugeot, Golf, and Jetta; $500 for Japanese cars; $400 for Fiat 132 and Argentina; $350 for Fiat 131 and Regata; and $300 for Fiat 128, Lada, Bolognese, and their equivalents.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Riosta Barcelona carries out guided visits to the buildings of the Museu de les Cultures del M\u00f3n and the Museu Etnol\u00f2gic de Barcelona that have the objective to highlight the important heritage value of these two buildings and explain the latest researches and interventions done.\nThe building where Museu Etnol\u00f2gic is located allows us to understand the architectural features of a modern building, built in the 70's. During this tour, you will be able to understand why unique modern buildings like this one are part of the architectural heritage of the city.\nSunday from 12.00 h to 13.15 h. Price:8\u20ac. Catalan.\nWe are finishing the online reservation system, for the moment, implement the following form and we will get in touch with you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Click the button below to add the Calcium Orotate to your wish list.\nWhen I use this, my , persistantly low calcium always moves to \"in range\"\nMost vitamin stores in Hawaii don\u02bbt carry calcium \u02bborotate\u02bb in fact the salespeople ususally don\u02bbt even know what it is.\nI trust Lloyd Wright\u02bbs products. That\u02bbs what\u02bbs most important for me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For over 40 years Robertson Geo has pioneered the development of wireline logging instrumentation technologies and techniques.\nThe comprehensive range of geophysical logging solutions and software is developed and built at its North Wales manufacturing and calibration centre. On-site test facilities include a 100m test borehole, calibration blocks, ovens and autoclaves capable of testing complete probe assemblies in \"real world\" conditions, providing a full suite of traceable calibration and quality assurance data.\nFrom the Arctic to Australia, from coal mines to windfarms, energy developments to sky-reaching superstructures, Robertson Geo technologies have been used to acquire quality geophysical data from over 160 countries around the globe. As an end user of its own products, the Robertson Geo engineering feedback loop from direct field experience fosters a continuous improvement culture driven by own use, leading to mature systems that are truly \"industry hardened\" innovative products.\nNo project is too small, no project too large or challenging.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasowc b/data_all_eng_slimpj/shuffled/split2/finalzzzasowc
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+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasowc
@@ -0,0 +1,5 @@
+{"text":"This Mystery of Trinity is rightly called the mystery of all the mysteries because the Holy Trinity is at the core of our Christian faith. Yet, the fundamental truth we believe is not only extremely difficult to understand, but in fact, it goes beyond our natural reasoning. How is it possible that we believe in three distinct Divine Persons, the Father, the Son and the Holy Spirit, and yet they remain One God? Some of the greatest minds like St. Augustine, St. Thomas Aquinas and Pope Emeritus Benedict XVI have attempted to shed a little light on the mystery. However, in the face of such immense truth, the best explanations would seem like a drop of water in the vast ocean.\nI have no illusion that I could explain the mystery better than the brightest minds of the Church, but we may reflect on its meaning in our ordinary lives. The joyful Easter season ended with the celebration of the Pentecost Sunday last week, and we resume the liturgical season of the year or simply known as the ordinary season. As we begin once again the ordinary season, the Church invites us to celebrate the Solemnity of the Most Holy Trinity or the Trinity Sunday. The Church seems to tell us that the unfathomable mystery of Trinity is in fact intimately closed to our day-to-day living, to our daily struggles and triumphs, to our everyday pains and joys. How is our faith in the greatest mystery of all connected to our ordinary and mundane lives?\nWe often have false images of God. We used to think that God or Trinity is the greatest person (or three persons) among things that exist He is like a universal CEO that manages things from an undisclosed location or a super big and powerful being that controls practically everything. Yet, this is not quite right. He is not just one among countless beings. God is the ground of our existence. He is the very reason why anything exists rather than nothing. Thus, the act of creation is not what happened at the Big Bang 13.7 billion years ago. It is fundamentally God's gift of existence to us. To be created means that we do not necessarily exist. Every single moment of our life is God's gratuitous gift.\nThe Scriptures reveals the mystery of our God. He is not solitary and self-absorbed God, but our God is one God in three divine persons. Our God is a community founded on creative mutual love and constant self-giving. Therefore, our creation is not a mere accident, but God's creative act and His gift of love. We exist in the world because God cannot but love us and wants us to share in the perfect life of the Trinity. St. Thomas Aquinas rightly says that we only believe two fundamental teachings, two credibilia : first, God exists, and second, we are loved in Jesus Christ.\nWe often take for granted our lives and immerse in daily concern of life; we rarely ask what the purpose of this life is. Yet, it does not diminish the truth that God lovingly sustains our existence and cares for us, even to the tiniest fraction of our atom. Whether we are busy doing our works, focus on our family affairs, or simply enjoying our hobbies, God is intimately involved. Thus, apart from God, our lives, our daily toils, and concerns, our sorrows and joys are meaningless and even revert to nothingness. Celebrating the Trinity Sunday means to rejoice in our existence as a gift, and to glorify God who is immensely loving and caring for us.\nReflection by: Bro. Valentinus Bayuhadi Ruseno, O.P.\nBy: Fr. Ramon T. Salibay, O.P.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The past week has been a blur. I'm behind on a few things. Sleep deprivation is through the roof. Even though I declared Sunday as an official Catch Up Day, I'm still struggling to keep up with things. This craziness is temporary. I've been through it with each addition to our family. It lasts for a few weeks. This time around may be a bit longer because Willa is in heat. Exhale. Despite all, I've managed to create a couple more Bodacious Patchwork Doggy Quilts.\nIt's taken me twice as long, but they're done.\nPrior to creating the first quilt-of-the-week, I wanted to make a dog bone template. I have other ideas for templates, but for now, my focus was on a dog bone. After about an hour, mission accomplished. Next week, I'll make a smaller dog bone template.\nYes, I make my own templates.\nBoth of the Bodacious Patchwork Doggy Quilts featured a dog bone applique. I loved how it turned out. It definitely motivated me to think of more. After all, if I'm going to sew up 2-3 quilts per week, I need a variety of appliques to add.\nWhen the first quilt was posted, our record was broken. During the Bodacious 7 Days Of Love, one of the quilts sold in 10 seconds. I didn't think that time could, or ever would, be beaten. However, it was. The blue hued quilt sold in 6 seconds.\nThe woman who purchased the quilt said she kept refreshing her screen.\n\"It was like purchasing Red Sox tickets. Refresh. Refresh. Refresh.\"\nI think it's safe to say that 6 seconds will be our record. That's the time it takes from point A (mouse click) and point B (screen refresh).\nFor the second Bodacious Patchwork Doggy Quilt, I decided to go down the route of Girly-Princess-Diva. I used prints with muted shades of mauve, peach, green, and purple. It sold within 2 minutes.\nAs with most of the Patchwork Doggy Quilts I've made, it's always hard to part with them. But, I do. Eventually, I'll make one for each of our fur-kids.\nYes, this includes Olivia too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was very simple and straight forward to place a free advert, and didn't take long to do.\nKTM 250, MOT till May, serviced December, tidy bike for age, likes to be on the back wheel, great bike, only selling due to getting a sports tourer. Reduced price no offers.\nExcellent condition, loads of fun to ride, car purchase forces sale.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Should you remember back ahead of the One-Room Concern I had my very own small task which was to obtain my home prepared to get a photoshoot. They've a nearby shop as well as an Antiques floor currently in Atlanta - Harrington had some very factors. I provide up this as it is Atlanta moment again and last September I had been hobbling along sporting athletic shoes hoping to get through all the walking! Writer lets you safely shop tens of thousands of pictures posts, and more with Google for free. Whether spreading your knowledge, breaking news, or whatever's in your thoughts, you're in excellent company on Blogger. In line with the comments, there seemingly have been a problem with Writer last night.\nThey've an Antiques floor now along with a nearby store within Atlanta - Huff Harrington had some things that are pretty. I bring this up since it is Atlanta occasion again and I hobbled along carrying athletic shoes looking to get through every one of the walking! Writer enables you to correctly store tens of thousands of posts, pictures, and more with Google for free. Whether revealing breaking news, your knowledge, or whatever's in your thoughts, you are in great firm on Blogger. On the basis of the comments, there seems to have been a problem with Writer yesterday evening.\nThey have a nearby store and an Antiques flooring currently here in Atlanta - Huff Harrington had some factors that are very. Since it is Atlanta moment, I deliver this up and I used to be hobbling along carrying tennis shoes hoping to get through all of the walking! Blogger lets wedding officiant services you safely store thousands of posts, photos, and much more with Google for-free. Whether revealing breaking news, your experience, or whateveris in your thoughts, you are in great business on Blogger. In line with the responses, there appears to have been an issue with Blogger yesterday evening.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cases that deal and lawsuits pertaining to transaction breaches, damages, compensations, malpractice or encroachments, etc are perhaps known as civil carrying cases. It is also known like a civil suit coupled with when taken to court, civil proceedings take add. They are meant for handle conflicts between people or possibly organizations. When another particular action or an company activity causes someone damage, the other party will most best civil lawyers in Chandigarh likely file a civil instance for relief.\nThese kinds of matters can be complicated and simply legal assistance is very important. The best civil lawyers in Chandigarh can a person to to get out involved with a hassle-filled legal circumstance and save you including tonnes of trouble. In that note, let everyone take a look for what kind of carrying case the best civil lawyers in Chandigarh can manipulate efficiently.\nMany a time, disputes arise involving tenants and lease to. A tenant trying when you need to sue a property manager for refunding this particular caution money. To do with the other hand, a tenant effectively not move information about even after your current landlord has shipped a notice.\nIf one single party hopes to slow down an technique coming coming from another special occasion that is regarded as damaging regarding the several other in a way, fair claims are already the method to use. For example, the wreckage of a house is an old but relevant case where to seek an end to the tennis court. Temporary restraining orders also belong for this category.\nIrrespective with the type with contract, person can lug the breacher to legal court. In most cases, there will have to be a compiled civil lawyer contract offering both typically the parties putting their signature to it. If you want to doesn't accept the terms, they could be sued.\nSo, had been the leading cases in which best remedied in commentator legal administration of virtually any civil professional.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasxgd b/data_all_eng_slimpj/shuffled/split2/finalzzzasxgd
new file mode 100644
index 0000000000000000000000000000000000000000..d167dfdacd5b0b3850a719c2a7167caa94dd7885
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasxgd
@@ -0,0 +1,5 @@
+{"text":"Raz's Midnight Macabre: You're So Cool, Brewster! The Story of Fright Night Documentary Needs Your Help!\nYou're So Cool, Brewster! The Story of Fright Night Documentary Needs Your Help!\nI'm a huge fan of \"Fright Night\" and seriously can't wait to see this documentary on the classic film.\n\"The Story of Fright Night\" will feature detailed interviews with those who helped shape and create this much loved movie. Guided by Tom Holland the makers of this new documentary will raise the stakes and unearth the behind the scenes stories including footage and images from Tom's personal archives.\nFully supported and produced by original \"Fright Night\" writer and director Tom Holland, this documentary is the ultimate insight into the making of this cult classic movie as well as a tribute to the late, great Roddy McDowall.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Axial Flow Fan private customized - DONGGUAN woo Electronics Co., Ltd.\n\"Axial Flow Fan\" must have before they can be called a printing printing equipment, and most major printer, printing shop operations Brush monochrome color, there are four color eight colors, according to the specifications have eight power points, four power half power and full power, currently on the market are imported in large localized \"Axial Flow Fan\", than Fang Lo-lan, Heidelberg, etc. Some simple printing can also use screen printing, just a screen to complete the printing, What, then results than large printing machine appearance. Some printers also have a large number of other printing equipment such as: bookbinder, Carbonless list of printers, active cutting machine, high speed folder gluer, etc., additional hair piece chamber, laser cutter room, \"Axial Flow Fan\"and binding paste table workshop. There are also small print, such as screen printing, just a screen to complete the printing, What, then results than large printing machine appearance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The item It was the night before Christmas, and all through the house : Christmas Eve traditions represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in Brigham Young University.\nCiccotti (1982- ) was a student at Brigham Young University. Due to her father's employment she was able to travel extensively through the United States, as well as through France and Italy.\nCreated by Lia Filomena Ciccotti while enrolled in English 356 during fall semester 2002.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is 1 public school and 1 private school listed on our site for North Ft Myers.\nThere are 39 full-time teachers educating a total of 620 students. 396 students receive a free lunch while 51 receive a reduced-price lunch.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Top Democrat slams brakes on confirming Phil Murphy's picks to lead education in N.J.\nNew Jersey's top lawmaker has slammed the brakes on the confirmation of Gov. Phil Murphy's nominees to lead the two state agencies in charge of education and colleges in New Jersey.\nState Senate President Stephen Sweeney told NJ Advance Media on Wednesday he's not satisfied with answers that Murphy's picks for state education commissioner and the secretary of higher education gave him as they wait to be confirmed.\nIt's the latest flap between Murphy, the state's new governor, and Sweeney, the fellow Democrat whom Murphy needs to install many of his initiatives in New Jersey -- including a higher minimum wage, a new tax on millionaires, and legal marijuana.\nThe Senate, the upper house the New Jersey Legislature, has to sign off on both nominees: Lamont Repollet as state education commissioner and Zakiya Smith Ellis as the state's higher education secretary.\nReppolet is the former superintendent of Asbury Park schools, while Smith Ellis was a White House education policy advisor to then-President Barack Obama.\nPhil Murphy faced anger and tears over school funding at a town hall. Here's how he reacted.\n\"I'm mad as heck, and I want to change this, too,\" Gov. Phil Murphy said during a town hall Wednesday night.\nEach have been serving in an acting capacity for months and can continue to do so.\nBut both have been approved by the state Senate Judiciary Committee and were expected to finally be confirmed when the full Senate holds a voting session Thursday at the Statehouse in Trenton.\nSweeney, though, said he never put them on the schedule.\nHe said he was bothered by how Repollet is approaching calls to rework the oft-criticized way New Jersey doles out state aid public school.\n\"When I heard that the (acting) commissioner of education said he needed six-to-eight months before we could come up with a fix for education funding, that was absolutely unacceptable,\" Sweeney, D-Gloucester, told NJ Advance Media.\n\"As a former superintendent, he's well aware what's wrong and the administration is slow walking us in a solution to funding and it's not acceptable,\" Sweeney added.\nMeanwhile, Sweeney said he gave Smith Ellis \"a list of questions that I had concerns with, and the answers back were not sufficient.\"\n\"They weren't answers,\" he said.\nRichard McGrath, a spokesman for Sweeney, clarified that \"we haven't seen or heard the level of willingness needed to make progress\" on school funding and making college more affordable in New Jersey.\nMahen Gunaratna, the head of Murphy's communication team, shot back in a statement.\n\"The Senate president has unilaterally decided to hold up the confirmations of two highly qualified cabinet nominees, despite overwhelming support from the Senate Judiciary Committee,\" Gunaratna said. \"There's no good reason why he's standing in the way of their confirmations.\"\nWhile both Murphy and Sweeney belong to the same party, the two have had a largely frosty relationship since Murphy took office in January. Sweeney has also pushed back on Murphy's plans to institute a millionaire's tax and raise the sales tax to bring in new revenue to the state.\nAs for school funding, Sweeney wants to fully fund the state's existing funding formula and remove all caps and limits on how much new aid a district can receive from one year to the next.\nTo achieve that, he's willing to reallocate state aid from districts considered overfunded to districts considered underfunded.\nMurphy pledged during his campaign last year to fully fund the formula but more recently has talked of working with lawmakers to revamp it -- though he hasn't detailed how he would do that.\nMurphy's first state budget proposal -- which must be signed by June 30 -- does not continue the reallocation of aid Sweeney started last year, and his idea of \"full funding\" appears to include the caps and limits Sweeney wants to abolish.\nSweeney's plan would eventually call for about $1 billion more in school aid than Murphy's.\n\"We need a fix now,\" Sweeney said. \"We're not waiting to six-to-eight months and getting into another school year.\"\nSweeney added that he expects the Senate to sign off on Repollet and Smith Ellis in the end.\n\"I don't expect that they won't get confirmed,\" he said. \"But I'm entitled to get my questions answered.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaulag b/data_all_eng_slimpj/shuffled/split2/finalzzzaulag
new file mode 100644
index 0000000000000000000000000000000000000000..7eaee3eb69a233c1c2a8a4ff7d626212d21aefc0
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaulag
@@ -0,0 +1,5 @@
+{"text":"Yes, you should clean it. Or better, have a professional clean it. Watch how they do it, and take notes. Then next time, you can clean it yourself. The technician may take the cover off the unit, and spray from the inside out.\nIf you're going to do it yourself, check the manufacturer's documentation for the approved cleaning method. It usually involves gently rinsing it with a garden hose, though may recommend a cleaning spray. There are loads of videos on YouTube, that describe how to properly clean the unit.\nWhatever you do, be gentle, and DO NOT scrub, brush, or in any way touch the fins. They are very delicate, and can be bent and damaged with the lightest touch. Even spraying them with a high pressure nozzle on your hose can cause damage, so be careful when cleaning the unit.\nWarning: Don't forget to turn the power off using the disconnect, before performing this, or any other maintenance on the unit.\nIn case you worry about you car's safety, you can ask the auto transport company to accommodate your care on the top level of the carrier truck. This will allow to avoid possible flui leaks from other vehicles or damage from small stones, but keep in mind that the driver will have to rearrange other cars on the carrier; thus, an auto transport company will usually require and additional fee for this service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"i ordered a cedar chest in june 2009, it arrived in july 2009 damaged, i returned it to mercantilia asking that they replace it. i have both e mailed them and phone called them at least 8 times and have not heard a word.\ni am ready to write the ceo and ask him\/her to intervene on my behalf. i am very unhappy with my experience with this company in more ways than one.\nReview #146303 is a subjective opinion of PissedConsumer146303.\nI spent enough money in the store to earn $75.00 in rewards. now I realize that my credit was good for a short time only.\nI own 4 vehicles and can spend my money anywhere but I choose to go to your store. I don't need things everyday. I keep my vehicles in fairly good shape and shouldn't be penalized. as I said I can take my money anywhere.\nThe people have always taken good care of my vehicles and would truly see this come to an end. I concider myself a valued customer and would like to remain one.\nReview #146243 is a subjective opinion of PissedConsumer146243.\nLast year I received a debt letter from Financial Consumer Services stating I owed money on a Chemical Bank Account. Your site listed them as a fraud. Not to mention I never had a Chemical Bank account.\nLast month I noticed that I received another debt collection letter showing the same acount number only slightly higher in $ amount from some entity calling itself ARP, also Il based.\nDo you have any information on this new scam? I would like to know how I can get off this listing, and how we can put this entity out of business. Pleas help.\nReview #146229 is a subjective opinion of poster.\nReview #146094 is a subjective opinion of PissedConsumer146094.\nOn 8\/20\/09 I ordered a turntable confirmation #179250096058.\nIt arrived and did not work, so I shipped it back.\nYou received it on 9\/11\/09 at 9:37 a.m. at the dock signed by F.Fence.\nI adked for a replacement and have heard nothing...please let me know what is happening.\nIf the item is not available, then I request a refund of $98.80.\nI'd also like a refund for the cost of the return postage of $16.50.\nThank you for all your help with my problem.\nReview #145997 is a subjective opinion of PissedConsumer145997.\nHello,i am martin,i live in birmingham,i bought a ticket to travel to china next month but from the look of my schedule i can make it then,is is possible i return the ticket and ask for refund of the money and if not what else can i do about it?i bought the ticket from agent but then this is my first time of using the airline so i dont really know how it works please tell me,you could write me on my id at,martindioxin@yahoo.co.uk,hope to hear from you soon.\nReview #145993 is a subjective opinion of poster.\naurora brooks I have not been able to get my money back after they were informed of my health problems. I was in a accident and was injured and became very sick with diferent health issues.\nI explain to them that I would not be able to afford the payments anymore. They said they could not refund me but they would give me catalog credit from all of their other producyd that were less expense and I could use that credit towards another thing.\nI said I need my money back. T hey have ignored my situation and ignored me.\nReview #145986 is a subjective opinion of PissedConsumer145986.\nI think YOU are dumb and wasted MY time for reading your idiotic post. Die.\nReview #145978 is a subjective opinion of PissedConsumer145978.\nThey are clever cheating lying money grabbers. Good at what they do.\n\"Out of sight out of mind\". I will never trust a \"overseas\" sneaky, greedy, lower than life oriental again. Not even in the \"good old U.S.\". I never did but I took a chance.\nThey know what they are doing and I became a foolish consumer trying to save a dime. I hope in my case the saying is true...........>\"what goes around comes around\".\nToo bad I won't know when that happens. If I could look the person in the eye.............dishonesty,immoral,selfish,indecent.\nReview #145976 is a subjective opinion of PissedConsumer145976.\nOther Company in Livingston, New Jersey - Businesses don't honor gift cards- don't get them!!!\nI won a \"car detail\" in a silent auction. No expiration date. When I took my certificate in a year and a half later, the business ownership changed (car dealership). They did not honor it.\nThis past Christmas, my husband gave me a $100 gift card to use at a gift\/decor store. Well, they changed ownership and did not honor it either, but offered a 10 percent discount.\nThat stinks! Is it a nobody's fault situation? I am NOT buying gift cards or silent auction items as a result and think you may want to consider the same.\nReview #145939 is a subjective opinion of PissedConsumer145939.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Washington, DC \u2013 The House Foreign Affairs Subcommittee on Asia and the Pacific marked up and reported the North Korea Travel Control Act (H.R. 2372), introduced by Rep. Schiff and Rep. Joe Wilson (R-SC), to the full committee. This bill would instruct the Secretary of State to restrict the use of U.S.\n\"My thoughts and prayers are with Senator John McCain and his family during this time. Senator McCain is an American hero, as evidenced by his service in Vietnam and in Congress.\nCongressman Joe Wilson (SC-02), Chairman of the Subcommittee on Readiness, issued the following statement after the House Armed Services Committee completed the markup of the National Defense Authorization Act (NDAA) for Fiscal Year 2018. This legislation would fund our military and vital national security projects.\nToday, Congressman Jim Clyburn (SC-06), dean of the South Carolina delegation, and Congressman Joe Wilson (SC-02) introduced Congressman Ralph Norman (SC-05) to Congress following his swearing in ceremony by Speaker Paul Ryan. Watch remarks by Congressman Clyburn, Congressman Wilson, and Congressman Norman here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One thing I find really cool about drifting besides the cars and the smoke and the tandems, are the places they're built. Now that I have a camera, I find absolutely no problem with barging into a shop to take some photos of the vehicles being prepped for carnage, provided I know the people of course. That took me out to North Hollywood to see my buddy Jef Groff. I met this cool dude in Top Drift this year and absolutely fell in love with his car. Working around a BMW all year sort of gave me a soft spot for them, and for good reason too!\nDrifting is about making your mark. You have to find a way to set yourself apart from all the other drivers and cars. 2 guys that manage to do just that, are Rapper Dan Savage and James Evans from the Sikky Manufacturing camp, who both came into Formula Drift this year with 2 cars that you definitely cannot miss. These guys saw success in the Pro 2 series with James taking a podium in Texas, and Dan taking home the Pro 2 overall championship. Besides the setup, which would obviously be tailored to either Dan or James, the cars are extremely similar.\nAs we learned in the last article, a beast of a machine under your foot is required to run Formula Drift. Formula D isn't a place where your 500whp car can make it, unless you have some serious skill or a beefy setup with enough grip that allows you to change the Earth's rotation at any given time. But, it's not just the car that makes you successful in Formula D, you have to know everything behind the scenes. It's more than just hopping in the car and driving. As a follow up to the first \"What It Takes\" article, I decided to go deeper with the drivers and show a little more behind the scenes, and how dedicated you have to be in order to run a season of Formula Drift.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today's photo is an abstract composition coming to you from -- my wall! Red, white, and a curvy, pointy dividing line. What do you think it is?\nToday's 365 thought: this is, if nothing else, getting me to try new things. The last several days have been photos that I might never even have thought of taking last year!\nI know what it is! I know what it is!\nI would guess a maple leaf flag, most likely the Canadian national flag. I almost guessed the Toronto hockey team flag, but that's blue and white.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavfxk b/data_all_eng_slimpj/shuffled/split2/finalzzzavfxk
new file mode 100644
index 0000000000000000000000000000000000000000..000f884d3f72ee96dfba815a87b9a3941a32098d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavfxk
@@ -0,0 +1,5 @@
+{"text":"Snowshoe is home to 251 acres of skiable terrain, 1500 vertical feet, 57 trails and the best snow around. Our three ski areas include the Snowshoe Basin, Western Territory and Silver Creek.\nOpening Week at Snowshoe Mountain!\nIt's #WhaleWeek at Snowshoe Mountain!\nSnowshoe Mountain's Ice Bucket Challenge To Support ALS!\nShavers Lake @ Snowshoe Mountain!\nSnowshoe Foundation Presents: Treasure on the Mountain!\nBallhooter Spring Break @ Snowshoe Mountain!\nMountaineer Parks @ Snowshoe Mountain - First Update!\nSnowshoe Mountain January 21st Powder!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We can help you identify residential property risk, benchmark performance and save money.\nTechnology solutions to residential property questions. Save money without increasing risk.\nFast access to accurate local property market analysis and trends, to inform your business decisions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If the port requires user input to build, configure, or install, set IS_INTERACTIVE in the Makefile. This will allow \"overnight builds\" to skip it. If the user sets the variable BATCH in their environment (and if the user sets the variable INTERACTIVE, then only those ports requiring interaction are built). This will save a lot of wasted time on the set of machines that continually build ports (see below).\nIt is also recommended that if there are reasonable default answers to the questions, PACKAGE_BUILDING be used to turn off the interactive script when it is set. This will allow us to build the packages for CDROMs and FTP.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"360 lace frontal is very popular now as it allows you to make versatile styles like high ponytail, bun, half-up half-down,etc. Our 360 lace frontal comes with pre-plucked wide hairline which help to achieve the super natural look.\nOur Brazilian virgin hair is highly recommended by the majority of hair designers and women. It comes with high quality grade and it can be bleached and dyed to any color. It is not only the good choice for hair extensions, but for perfect hair wigs.\nAll of our virgin hair weaves like brazilian hair extensions are top quality and at affordable price. Our other brazilian hair weave and brazilian virgin hair are also good hair weaves to let you hair become longer and thicker. If you are the fan of virgin Brazilian hair weave, you can look our brazilian hair extension, they are our very popular hair weave. We also have hair wigs and hair extensions what can make you have a new hairstyle. If you are interested in them, you can have a look on our homepage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The European Union should end Turkey's membership talks on only one condition: if Ankara brings back the death penalty, European Parliament President Antonio Tajani said Wednesday.\n\"It's very difficult today for Turkey to get into the European Union, but we cannot close the chapter,\" Tajani told POLITICO during an interview in his office. \"But we need to send clear messages: the death penalty is not acceptable for us\" and if Ankara decided to reintroduce it, then there's \"every chance\" dialogue would be closed.\nThe former European commissioner said there were a number of problems with Turkey, including its stance on \"personal freedom and freedom of opinion. It's unacceptable that there are journalists in jail,\" Tajani said, adding that he has raised these issues with Turkish President Recep Tayyip Erdo\u011fan.\nIn March 2015, the EU signed a deal with Turkey to take back refugees crossing into Europe in exchange for breathing fresh life into long-stalled EU membership talks. But tensions with Ankara have increased since an attempted coup last summer and Erdo\u011fan's subsequent crackdown on opponents. The sweeping new powers that Erdo\u011fan gained after a close-run referendum in April have made matter worse for some EU countries.\nOn Sunday, German Chancellor Angela Merkel said \"Turkey should not become an EU member,\" a line shared by her rival for the German chancellorship Martin Schulz and by the Austrian government.\nMerkel said during an election TV debate that she doesn't \"see a mandate to expand the customs union [deal with Turkey] under the current circumstances.\" An upgrade of the customs union agreement was one of the main points in the 2015 deal.\nTajani, who when elected president of the European Parliament said he wanted to be \"a good spokesperson\" for the assembly, took a different line from fellow MEPs. In June, the Parliament called for Turkey's accession talks to be suspended if Ankara implemented plans to expand Erdo\u011fan's powers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavlpk b/data_all_eng_slimpj/shuffled/split2/finalzzzavlpk
new file mode 100644
index 0000000000000000000000000000000000000000..521da496f7add85ab36d2f63437616a7869c206a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavlpk
@@ -0,0 +1,5 @@
+{"text":"A stay at Pattaya Sea View Hotel places you in the heart of Pattaya, steps from Pattaya Beach Road and Pattaya Beach. This 4-star hotel is 0.4 mi (0.7 km) from Art in Paradise and 0.5 mi (0.7 km) from Hard Rock Cafe.\nMake yourself at home in one of the 55 air-conditioned rooms featuring refrigerators and minibars (stocked with some free items). 32-inch LED televisions with cable programming provide entertainment, while complimentary wireless Internet access keeps you connected. Private bathrooms with showers feature complimentary toiletries and hair dryers. Conveniences include phones, as well as safes and coffee\/tea makers.\nBe sure to enjoy recreational amenities including an outdoor pool and a fitness center. Additional amenities at this hotel include complimentary wireless Internet access, concierge services, and a fireplace in the lobby.\nEnjoy a meal at the restaurant or snacks in the coffee shop\/caf\u00e9. The hotel also offers 24-hour room service. Wrap up your day with a drink at the bar\/lounge. Buffet breakfasts are available daily from 6 AM to 10 AM for a fee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I always thought you were Yellow!\nQuit stealing my 6 year old's drawings, she's gonna sue you for copyright infringement!!!!\nWow! Your daughter must be one awesome artist if she can draw like me!\nmartigan...you are my hero...i wish i could draw like you!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This was taken from http:\/\/www.unix.com\/unix-dummies-questions-answers\/59467-sunos-5-10-vi-arrow-keys-not-working-post302181425.html and I didn't want to lose it.\nEnter the following into $HOME\/.exrc file. First entry is to make DELETE work as Backspace and Remaining\/Subsequent 4 entries are traversal into file using Arrow-Key. All 5 entries are applicable for \"Editing\" mode of vi editor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A step-by-step introduction to how to build your own Local COM server and client.\nMany CPians have written about COM, why again?\nMost people learned COM by building a InprocServer, i.e., a DLL file first, and then continued to write a LOCAL COM server together with a client. At this point, a simple example showing a LOCAL COM server and client will be very helpful. However, believe it or not, I spent hours searching the net, trying to find a small example of a LOCAL COM server and client so I can grow my own system based on it, but I never found one! Most of the examples I found are Inproc servers. In some cases, I found LOCAL servers, but the document only lists the basic steps\/ideas without giving the code. As a matter of fact, from knowing the steps to building a real one is still quite a long way to go: you have to write the idl file, the reg file, you have to include different files at the right places (even in the right order! you will see), ..., and a single mistake will bring you very confusing error messages or results. I then found Andrew Troelsen's book and tried to follow his example. However, I moved his server code from the attached CD to my local PC only to find that his LOCAL server did not even compile - I finally figured out the reason (you will see if you continue to read) and I believe that he should have mentioned this problem in his book - this would have saved his readers lots of frustration and searching around. Meanwhile, he did not present the client code, so that is still not a whole story.\nSo I decided to write down all the necessary steps you should follow, all the real code you need to write, all the files you have to create, and all the things you want to remember if you decide to build a LOCAL COM server and client - this small example can then serve as a starting point for your much better and bigger (also ambitious) project. Still, let me clearly state that half of the credit should go to Andrew Troelsen: for the server side, I am using his example and his code (with the problem corrected), if you want to know more about this part, you can check his book out.\nTherefore, if you are interested in building a LOCAL COM server and client, this example might be helpful to you. Also, I assume you have the basic knowledge of COM: the architecture, the interfaces, COM run-time service functions, and the variety of tools that you can use to get the GUID and to inspect your components, so on and so forth. In the next section, a LOCAL COM server is constructed, and after that, a client is built. Enjoy!\nIn this section, the steps that are needed to build a LOCAL server are described. This simple server contains a single coclass called MyCar and the following three interfaces: ICreateMyCar, IEngine and IStats. The whole server is very much self-explained, so let the show begin!\nStep 1: Start VC6.0 and create a new WIN32 Application workspace, select \"a simple application\", and name it CarLocalServer (or whatever you want to).\nInsert this file into your project. Also, notice that all the GUIDs should be generated by using guidgen.exe, so your IDs will not look like the ones in the above file. I listed above every line of this file so you can have a clear picture about this simple server. For the rest of the files in the server, you can download them and read them through.\npay attention to the included .h files and remember, whenever you need to include the stdafx.h file, you want to include it first, before any other included files. If you don't do this, your compiler will give you some very confusing error messages (at least mine did).\nIf it does, make sure to get rid of it (comment it out). This is the other reason why I could not get the server successfully compiled (and it took me a while to figure this out).\nI really think Troelsen should have mentioned these two things. Well, if you happen to read his book, I hope you notice this article too, it might save you some time.\nyou can use different names (keep the coclass name MyCar unchanged).\nAfter you create this file, double click it, your Windows system should tell you that your components are successfully registered.\nStep 5: Build the whole project. You should have no compiling errors, and remember that you should at least run it once.\nNow you have it: a LOCAL COM server. Let me again say that this part of the code is from Troelsen's book and I just made little corrections as seen in step 3 (to get rid of the compiling errors). Also, I added some error protection in the server too; in case something goes wrong, you can get some error messages that make sense.\nNow, let us follow the following steps to build a LOCAL COM client. I will list the source code here and explain the important parts since Troelsen did not present the client in his book.\nStep 1: Start your VC6.0 again and create a new WIN32 Console Application workspace. Select \"an empty project\", and name it CarLocalClient (or whatever you want to).\ncout << header << \": Error(\" << hex << hr << \"): \"\nNotice the include statement: you don't really need to include anything like windows.h, you only need to include the CarLocalServerTypeInfo.h and CarLocalServerTypeInfo_i.c files, and these two files include all the necessary header files for you already. Notice that these two files are generated by your system after VC6.0 compiles CarLocalServerTypeInfo.idl file, and they are located in the workspace that you have created in Section 2. So you need to either copy them to your client workspace directory, or you can use a relative path like I did here.\nTo access your COM components (your coclass and interfaces), you first make the call to the COM service function CoGetClassObject(), and what you get from this call is the pointer to your IClassFactory interface. Once you have this interface, the rest seems to be obvious: you then make a call to CreateInstance() on this interface to start your journey. You may have noticed the CLSTX_LOCAL_SERVER parameter in CoGetClassObject() call, this is where we show that we are building a LOCAL server!\nAnother choice is to call CoCreateInstance() instead of CoGetClassObject(). This function in fact makes a call to CoGetClassObject() and then another call to CreateInstance() just like what we did in our client code. Perhaps you should do what we did here in this client code: besides the performance considerations, it seems to be safer to do so: by making the two calls yourself, you can put error protections after each call, so if anything goes wrong, you have a fairly clear idea of where the problem is. On the other hand, if you use CoCreateInstance(), if this call fails, you lose control of where the problem is. Notice in this client code, we just show the error, we did not put any exception handling code - I just assume that you will be the one who will add the appropriate exception handlings.\nStep 3: build the project and try it out. You should be able to \"talk\" to your COM server without making it a in-process DLL file.\nThese are the basic steps of how to build a LOCAL COM server and client. If you really try it out, you will see that it does involve a fairly large amount of work, and again, a single mistake will give you very confusing error messages. Perhaps, after trying this example out, you will start to appreciate ATL more.\nThis article presents a simple example of how to build your own LOCAL COM server and client. The real benefit of it, at least I hope, is that you can use this as a starting point to build much more practical and bigger projects. You might want to use ATL in your real work, but understanding what is going on under the hood is always good. Thank you for reading, and I certainly look forward to your comments and suggestions.\nIt always returns \"E_NOINTERFACE No such interface supported\"\nI know this post is pretty old, but I had the same issue, so I thought I'd update this thread a bit.\nThis interface error happens because the server example is registering the interfaces, but not the coclass itself.\nSo when windows is looking for the implementation of the CarLocalServer class, there is no entry in the registry telling it to load our EXE file for that.\nI have read elsewhere the solution is to make a RGS (registry) file with the keys to create.\nThe server program will then be modified to create these register keys when called with the \"-RegServer\" command line argument.\nAlternatively, you can have an installer create those keys or craft a REG file for regedit, it is up to you.\nSo let's begin by creating the registry file.\n\/\/ Let's see if we were started by SCM.\nfails in newer Visual Studio versions.\n- You will have to run the server EXE as Administrator, otherwise the RGS registry actions will fail silently.\nI mean , your first son should be lawyer and other two can be doctor.\nIn you case, your third son may not get license while other two sons start treating you. High probability to make mistake when they are fresh with less experience. You may die due to wrong treatment before your third son get license to sue other two. So, First son should be lawyer and should be ready to sue when your other son get license to taking care of you (free).\nthere has a error and the hr is 0x80040155:Interface not registered.\nWhy is this error occuring and what can be done to fix this problem?\nI had the same problem and it took me hours to find it.\nThe registration of the components said it had registred, but in fact it had not!!!!\n4) Run the samples: and all is well.\nPfoei. Buy a round of beer. Kick the cat. etc.\nI am tried this sample code.\nGet the class factory interface for the CFDownloadControler class...success.\nI want to display a dialog on LOCAL COM Sever and the clients to add a item to the dialog's list.\nI added following code in QueryInterface function.\n\/\/ Which aspect of me do they want?\nthis demo is fancy enough.\ndo not count references ?\nI am trying to run an out-of-process COM server as an NT service. I created the application as a Win32 console application in VC++ 6.0. Within the same workspace I added a DLL project which creates and registers the proxy dll for the COM server by compiling and linking the files output by MIDL.\nThe service starts up normally from within the service manager console. But when I try to get the factory object of my COM class from the client using CoGetClassObject(), I get E_ACCESSDENIED (General access failure returned from LoadLib\/CreateProcess).\nCould anyone please help me out ? Incidentally, if I run the COM server not as an NT service but simply as a console application from within Visual Studio, CoGetClassObject() succeeds and I am able to access the services of my COM class normally.\nAttached below is the ServiceMain() of my COM server. The interface of my COM class has just one function : fx(wchar_t ** lplpszOut), which copies the string \"Hello World\" into the string passed in.\nThanks in advance for any help. If this is not the correct forum to post this question, can someone please redirect me to the appropriate one ?\nThank you for an excellent article. I also noticed Mr. Troelsen's lack of sample code for a C++ CoCar client. I suspect that he expected you to know how to write it yourself, since he had previously done a similar client for his \"Shapes\" server.\nAnyway, I found that your sample code would compile, but not run, on my machine. I tried compiling the client to access the library as an in-process server (with CLSCTX_INPROC_SERVER), and then it ran just fine. When I try to run it as a local, out-of-process server (with CLSCTX_LOCAL_SERVER), I get a COM error 80040154 \"Class Not Registered\". I scrutinized my registry settings to make sure the CoCar COM server was registered, and everything was fine. If it weren't, I don't think I'd be able to run it as an in-process server either. I was even able to access the library from a VB client.\nI'm about to read the chapter on location transparency, and so I suppose I may figure out why I couldn't run your code without an adjustment. But in the meanwhile, you might update your article to add something about registration for use as an in-proc versus local server.\nThanks again for such a helpful article.\nHave you a code sample in VB that works for this case ?\nI get that error, Could you help me pls ?\nThis is because Microsoft loves to change its SDKs in ways that break backwards compatibility with code that is even just one compiler version old--often for no identifiable reason.\nYou're not done yet. You also need to add the statement using namespace std somewhere in your code prior to any statements that reference the iostream library (I recommend putting it right under your includes).\nConfused yet? That's just the very tip of the iceberg of chaos and confusion that is COM+ development. I come from a long Java background, so I've gotten spoiled. But now my job requires me to interface a bunch of legacy COM stuff to .Net through COM Interop. Heaven help me. After a month of banging my head on a wall I'm finally getting used to the idea that, instead of doing productive work, I'm now going to spend the bulk of my time tracking down moving-target library dependencies, evanescent compiler extensions, and mercurial macros, while also making sense of misleading error messages from the compiler and COM runtime, and diddling endlessly with ever-changing MIDL options just to get a few lines of code that's only a few years old to compile and run.\nHere is .header file snippet (header created by MIDLing the IDL file I have). I can call doit() function with no errors but I get errors when I try to call SayHorray() which returns a string.\nHere is my C application code snippet: NOTE: i am using c, not cpp! c app calling com object.\nhoorayObj->lpVtbl->SayHooray(hoorayObj); \/\/got compile error \"too many parameters\" when pass &str in last param, left it out.\nHow can the client get a LPWSTR list from the server?\nI found this example very useful to build a local server without using ATL.\nBut I'm realling having troubles for passing LPWSTR list from the server to the client.\nI tried to use a double pointer (LPWSTR**) but it doesn't work.\nI'm new to COM development and this article helps me a lot!!\nI have a question, when the client calls the server, a command line \"\/Embedding\" is passed to the winmain of the server. Could please tell me how??\nHow will I going to call the server on VB?\nExcellent program for to learn more about COM programming.\nI want do that the server sent to client any value like m_maxSpeed or curSpeed when those change automatically. i want do another interface that sent automatically several values of the variables to the client.\nnow, for running the client properly I need the server to be ran first.\nhow do I change the code so that the client will activate the server (so I need to run only the client itself) ?\n1. for the server to compile, I needed to uncheck the checkbox \"MkTypLib compatible\" in project->settings->MIDL tab (it was checked by default when I followed your instructions in creating the server project).\n2. for the server to run properly, I needed to write \"\/Embedding\" in project->settings->debug tab, in the Program Arguments line.\nhow this server will look?\nWill this local server run as background process or it will showup in dos screen?\nI understand what SysAllocString is doing.\nBSTR is a MFC data type. But I've watch the content of the variable (here: carName) and I don't have a glue what it is.\nMy second Question is about OLESTR. It is often used in MSDN. But there is no explenation about it.\nI can't say for sure not having read the mentioned book, but it appears from your two points that the book is not using MFC and precompiled headers and you are. For example, the first part about placing the stdafx.h first is only an issue if you're using precompiled headers and the second issue is probably due to him not using MFC and you using MFC.\nYou are totally right. Good point!!\nhere is also the missing register file: sorry for that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"wedding koozies ideas custom cheap awesome sayings for contemporary styles decorating bedroom.\nwedding koozies ideas beer decorating styles quiz.\nwedding koozies ideas best beer images on favors sayings decorating for bedroom.\nwedding koozies ideas can favors images decoration especially greenery stickers decorating games y8.\nwedding koozies ideas in addition casual sneakers a decorating outside for fall on budget.\nwedding koozies ideas personalized favor for summer weddings decorating a small bedroom to look bigger.\nwedding koozies ideas as decorating a studio apartment on budget.\nwedding koozies ideas funny sayings lovely best can images on decorating a small bedroom pinterest.\nwedding koozies ideas turquoise neoprene pocket can personalized with design and custom print decorating games didi.\nwedding koozies ideas funny best most popular designs s decorating living room with fireplace.\nwedding koozies ideas best on personalized inside favors decorating for fall meme.\nwedding koozies ideas best on decorating a small bedroom with queen bed.\nwedding koozies ideas cool decorating games unblocked.\nwedding koozies ideas hey i found this really awesome listing at wwwcom country count decorating games mafa.\nwedding koozies ideas something old new luxury favors decorating for fall 2018.\nwedding koozies ideas something old saying coolers decorating a studio apartment for christmas.\nwedding koozies ideas cups website inspiration s decorating styles for 2019.\nwedding koozies ideas gallery of rustic picture top projects for decorating living room with plants.\nwedding koozies ideas custom can coolers decorating styles that are out.\nwedding koozies ideas favors long distance by new decorating styles defined.\nwedding koozies ideas decorating games mafa.\nwedding koozies ideas archives interesting favors decorating a studio apartment videos.\nwedding koozies ideas funny design your own can cooler decorating living room.\nwedding koozies ideas funny finally married can cooler decorating living room.\nwedding koozies ideas surprising totally magnificent destination can coolers decorating styles quiz.\nwedding koozies ideas inspiration website with photo gallery decorating a small bedroom on budget.\nwedding koozies ideas quotes unique cute sayings info decorating small spaces on a budget.\nwedding koozies ideas via totally here to party decorating for small spaces.\nwedding koozies ideas and also greenery shawl for fall on a budget.\nwedding koozies ideas funny free booze can cooler decorating a studio apartment ikea.\nwedding koozies ideas best on personalized decorating a studio apartment for man.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxwxs b/data_all_eng_slimpj/shuffled/split2/finalzzzaxwxs
new file mode 100644
index 0000000000000000000000000000000000000000..b2784ba78d74d2d114706507d04a6393a4d60a76
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxwxs
@@ -0,0 +1,5 @@
+{"text":"Around half of all parents with a child on the spectrum describe their child's oral health as fair or poor, according to a study published by Pediatric Dentistry. As noted by the American Speech-Language Hearing Association, the findings are no surprise, considering that a dentist's office can overload the senses with a plethora of noises, sharp tools and new smells that can make the experience challenging for children with autism. But factors that contribute to dental care extend beyond a visit to the dentist, as well. Good oral care begins with diet, though some children tend to have marked preferences on a narrower list of foods. It's up to parents to make sure that dental care, especially for children with autism, is a priority that's not a challenge but easy to remember to do.\nStudies show that out of all the healthcare needs in America, oral health care is the one that's most often unmet among children with or without special needs. Prevention is the key to keeping health issues later in life at bay (like heart disease), so it is important for parents to teach good dental care early. Doing so will also avoid dental pain caused by decay and reduce the need for lengthy procedures, such as complicated fillings or root canal treatments. Try to establish a routine after meals and right before bedtime, using a variety of methods to show kids how to brush. Older children can help to 'model' a proper tooth cleaning routine; visual prompts can also show the angles in which a toothbrush should be held to ensure all parts of teeth are reached. Of course, diet is also key. Avoid sugary, refined foods, which increase the likelihood of cavity formation.\nChildren who are not overwhelmed by the sound of an electric toothbrush may benefit from dedicated sonic-style brushes that come with an app. The app, which can be downloaded onto a phone or tablet, features cartoon characters 'leading' your child through the brushing routine. Basically, every tooth should be brushed on the inside, outside and biting surface, and attention should be paid to the area touching the gum, since this is an area is where most plaque builds up. Apps will help kids brush patiently, keeping the brush on their teeth long enough for a thorough clean.\nYou many need to stand close to your child and prompt him\/her when it comes to brushing. Be patient if they don't get it right or grow impatient initially. Reinforce good attempts and reduce prompting to foster independence. Try to be flexible with respect to where your child likes to brush his\/her teeth. If they do so fantastically on the sofa, it might be a great idea to get creative considering that the consequences of poor brushing can be drastic. Some parents like using the 'first-then' approach, using visual or verbal cues to show kids that after brushing comes an activity they love.\nRegular teeth cleanings and check-ups are key from toddlerhood, so choosing a sensitive, supportive, well-informed dentist is key. Consider Planned Activity Training (PAT) with a behavioral psychologist if your child is very sensitive about opening their mouth, letting others inspect their teeth, etc. PAT is centered on establishing a step-by-step plan to build a positive routine. You might want to try practicing these steps at home or with the help of a dentist. You might start out just entering the dentist's office and saying hello to the staff, receiving a sticker or star from them. Visual supports can also be used to show the different stages of a dental visit. Some dentists offer headphones to patients, who can select music, TV shows and other types of entertainment to 'step out' mentally from a stressful dental visit. Some parents opt to have a dental check-up themselves and let their children watch them.\nA dentist that supports patients with autism will have specific protocols for visits. For instance, they will try to minimize waiting time and communicate very well with patients. They will explain what they are about to do and give the child all the time he\/she needs to process the information. They will respect any sounds or repetitive movements that result from nerves or anxiety and will always allow the parent or caregiver to take the lead.\nOptimal dental care from childhood is important, especially so children can avoid serious conditions associated with poor oral health in their adulthood. Starting early with a good routine is important for children with autism. Patiently use aids, such as visual prompts, role plays and the like, to help children understand how to clean teeth. Finally, select a recommended dentist who adapts as much as necessary to your child's needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Time for some selfies of me and Mum. As you can see, at first I wasn't in the mood. I just wasn't too pleased that Mum had the audacity to interrupt my nap time just to take a selfie.\nBut then I felt the love and gave in just for a few snaps. When Mum said it's for the blog, I couldn't resist. After all, how could I disappoint my fans?\nYou both look so sweet. The perfect Sunday Selfie!\nI think these are really sweet selfies - worth waking up for!\nLOVE the second one, Athena. I like the effect in black and white ... the love between you two is so obvious!\nAwww, these are beautiful selfies.\nGreat selfies, Athena! Hope you're having a wonderful Labor Day weekend!\nAdorable selfies! Glad you shared the spotlight with your mom for the day.\nYou really can't fault your mom, Athena. When you're as cute as you are, how could anyone resist taking some snuggly selfies with you? These really are such adorable selfies!\nYou always take such lovely selfies, Athena.\nThanks for giving an extra effort today, Athena!\nThose selfies were just beautiful Athena.\nWe can feel the love in those selfies!!\nWe're glad you made the effort but we totally understand! Moms always want us to wake up and smile for the flashy beast.\nYou both look beautiful. And together, even more so.\nYou both look very beautiful.\nAaaaaaaaaaaaaw Athena your mommy and you are so purretty. Have a pawsum week.\nI'm glad you did these, Athena! They're lovely selfies of you and your mum.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"has there been any examples of getting the CM19A transceiver to work? i had experience with a Firecracker working on mac using Heyu and a USB-Serial adapter, but id like to use the CM19a since it has built in usb?\nRe: CM19a in Mac OSX?\nSince Mac users usually are not DIY types, I did not think mochad would be of interest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"terms and conditions and privacy, oh my!\nWe understand that your data is personal, and important to you!\nEverything we do at Nudge is designed to create a better experience for you. Part of improving that experience is understanding who you are and what you're interested in. To learn this we collect data throughout the process.\nPlease reference the links below to fully understand what data we're collecting, why we think it's important and how it can help improve your experience. We'll update the information within the links when we see fit so be sure to keep an eye on them. If you have any questions, don't hesitate to reach out to us by emailing hello@nudgetv.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vodafone is offering a fixed wireless broadband service to customers waiting for a fibre connection. Customers signing for the Ultimate Home Fibre plan get an Ultra Hub Plus modem. Vodafone says this will give them a \"mobile broadband connection over Vodafone's 4G\/3G mobile network while they wait for their fibre broadband to be installed\".\nIn a press release Vodafone's outgoing consumer director Matt Williams says: \"Our customers tell us they are frustrated by installation wait times, while others say they are putting off a move to fibre because they simply don't want to be disconnected while they wait\".\nWilliams says there are significant delays as people wait for fibre. Yet, at the time of writing the average wait for a broadband connection on the Chorus network is 13 days. Enable says it generally connects customers inside two weeks.\nCustomers can use their Vodafone fixed wireless during the install and avoid any disruptions. After fibre is installed the Ultra Hub Plus modem acts as a backup. If there's a disruption on the fibre network traffic is automatically routed via the fixed wireless network.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazusw b/data_all_eng_slimpj/shuffled/split2/finalzzzazusw
new file mode 100644
index 0000000000000000000000000000000000000000..99930a3d3aa5b8a02247b778701770a628981f80
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazusw
@@ -0,0 +1,5 @@
+{"text":"Nat reflection is ALWAYS the worse option to choose.. I don't understand why anyone would ever want to nat reflect.. if host.domain.tld is on the same network next to you - then why would you not just resolve host.domain.tld to that IP.. Why would you ever want to go to the public IP to be reflected back in?? As to forwarding port X to port Y.. That is always a work around in itself to all to go to the same service with the limitation of napt and only 1 public IP, etc. If you want to go to host.domain.tld:port then go there where host.domain.tld resolves to the local IP and not the public ip..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have a staff of expert ac repair specialists in Lake Elsinore who are ready to help you with your next air conditioning installation or repair!\nHave your heating systems within Lake Elsinore, CA repaired, installed, and maintained by the experts at All About Air before the temperatures drop.\nAll About Air offers affordable and professional ac repair services in Lake Elsinore for all types of air conditioning and heating systems.\nMy spouse and I wishedto thank you for supporting us out last month. We value it without doubt you placing us up in a hotel for 2 nights while you needed to acquire the part required to switch our air conditioner. Within Lake Elsinore people certain can't remain or stay in your home with the temperature of 89-90 degrees. Thanks a lot again for your generosity. I have already supplied your phone number to few of my family and colleagues that tend to be in need for help with their ac repair system.\nDon't delay until right after your air conditioning unit stops running to give us a call. Our ac repair and installation service will help keep your unit operating smoothly when you need it most. All of us are about air, literally, and that we have been around in the business of keeping people cool and comfortable over twenty five years. When it comes to ac repair in Lake Elsinore we have some of the highest quality HVAC qualified professionals in the business.\nLake Elsinore provides a warmer climate year round, even though for many this is actually suitable living condition, it could actually ruin air conditioning units. All About Air incorporates a group of licensed, insured, and professional technicians that are skilled in every components of air conditioning technology, and we are available Twenty-four hours a day, 7 days a week for emergency ac repair in Lake Elsinore. Our goal is always to keep our consumers happy, and we have attained many esteemed awards over the years to prove it.\nThey are some of the labels that we offer installation and ac repair. Additionally, we service many other big brands. If you want a free estimate for a new unit while we are on a service call for your ac repair, we will thoroughly calculate the square footage of your property, select which room' will need approximately more air flow based on a technological approach, and determine the best size unit your house will require. If you realise you are regularly making calls for ac repair in Lake Elsinore, it might be your unit is either too small, outdated, or it could be something as simple as switching the thermostat to the appropriate position.\nMost units are meant to last up to 15 years; if your unit is older, then it might be time to replace it. Our ac repair technicians are able to determine the best approach to keeping your home comfortable. They attend ongoing HVAC workshops and classes as a way to stay on top of the latest technology accessible in ac repair, maintenance, and installation. There's a science associated with the appropriate installation of your unit, which can significantly lower the amount of ac repair in the future.\nInstalling a brand new unit could help you save money in over time on energy costs and can cut out ac repair on an older unit. Routine routine maintenance will also reduce service calls for your ac repair in Lake Elsinore. Ask our ac repair tech about an Energy-efficient unit. There might be tax credits available for new installations up to December 2013 for energy related improvements in your home. The next time you require us for ac repair in Lake Elsinore, make sure and ask our knowledgeable technician tips on how to be more energy efficient so you can cut back your utility costs in the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"German Chainsaw, Wholesale Various High Quality German Chainsaw ... Big Power Fast Cutting German Chain Saw Petrol ... Chinese 58CC Professional Chainsaw manufacturer.\nTabletop Milling Machines and ... High-quality parts from real materials; A big ... the World's Leading Robot Store For Personal and Professional Robot Technology.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From this page you will be able to access a range of reference materials and resources in areas relevant to the work of Significance International.\nThe resources themselves will often be links to webpages, or listings of references to access in more traditional ways. These Resources pages will build over time.\n1. SI Submissions, Commentaries and Grant Results: This page provides 'one-stop-shop' access to our submissions to national Inquiries, analysis of resulting policy documents, and appraisal of two grant programs of relevance to movable cultural heritage.\n2. Official Standards: a page with a selection of Australian and international Standards relevant to the management and care of cultural and scientific collections. From here you can easily read about a Standard and proceed to purchase it from official retailer SAI Global.\nClick here to read about Standards and to search in your more general fields of interest . Please note that if you click on this link SAI Global InfoStore will place a cookie on your computer which tracks your travel on their site, but does not record your identity.\n3. Risk resources: this is essentially the reference list for the Risk Management for Collecting Organisations Workshop, complete with as many hyperlinks as possible, to make your searching easier.\n4. Significance resources: this is the reference list for any Significance Assessment workshops we conduct. The critical resource for our teaching is Russell and Winkworth 2010 Significance 2.0: a guide to assessing the significance of collections.\nAlso included on this page is a .pdf file of the 'Significance 2.0 Summary Card' - a ready reference for collection fieldwork. You can download it and print it off to whatever standard you choose.\n5. The 'Collections Management in Focus Survey' webpage presents information about a survey conducted by Veronica Bullock under an ICCROM Fellowship in 2010. Published articles are listed and data are provided for anyone who wants to use it in future research projects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Evandro Gussi is the Comptroller General of the Brazilian House of Representatives. Gussi was elected as a congressman in 2014 (2015-2019 term) and served as the Green Party Chief Whip from 2016 to 2017. He is trained as a lawyer and is registered with the Brazilian Bar Association. He is currently a member of the House of Representatives and serves on the Constitution and Justice Comittee, as well as the Education and Culture Comittee. Under his leadership, the Brazilian Congress approved the \"RENOVABIO\" Act, an important landmark piece of legislation for the biofuels sector.\nGussi has a S.J.D. in Law from the University of S\u00e3o Paulo Law School (2009) and received a Master's degree in Law from the Law School of Federal University of Rio Grande do Sul (2005).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I must have walked by hundreds of mall kiosks this past holiday season. You know, those carts set up in the aisle-ways, designed to maximize rental revenues. Although I generally walk right by, this year my wife stopped in her tracks and exclaimed, \"Look at the cute TVs!\"\nTVs? I thought to myself. On a kiosk? Sure enough, right in front of us was a colorful kiosk full of flat-panel LCD TVs from Hannspree. Hannspree is a new division of an established Taiwanese company seeking to extend its product and brand presence with a lineup of highly stylized TVs. What s really neat about these products isn't necessarily the design, however, but rather the fact that Hannspree has decided to eschew traditional retail channels.\nAs we have increasingly become an experience-driven culture, smart brands are responding to this shift with branded retail locations, pop-up (i.e. temporary) stores, and kiosks. These outlets can be thought of as \"brand boutiques,\" places that sell a brand by creating an experience designed to engage, inform, and entertain the consumer. Far more than a traditional end-cap or store-within-a-store, the brand boutique allows for unprecedented control of the brand at the most critical customer touch point: the point of sale.\nMany of today s consumer electronics and other products roll off the same assembly lines used by their competitors. Huge companies like Flextronics crank out gadgets and gizmos for nearly every top consumer electronics brand. In Taipei, a single factory might make Dells in the morning, and then Macintoshes during the night shift. Similarly, when you boil it down, there is very little service-level differentiation between providers of services like cable and wireless. Companies like Apple know very well that the experience, the suite of products and services offered and how they work with each other, is just as if not more important than technical innovation in order to differentiate from the mass of similar products and services in the market.\nThe realization that consumers primarily seek experiences, as opposed to specifications of features and benefits, is now extending beyond the creation of products and into the sales channel, impacting how products and services are sold. Packaging, websites, and support are becoming more important; and so is the retail presence.\nIt wasn t so long ago that boutiques were strictly the realm of luxury retailers like Chanel, Gucci, and Coach. The retail trend for most other products was in the direction of big-box retailers ranging from Wal-Mart to Best Buy. Eventually, brand-centric organizations craving differentiation, like Gap and Nike, started to establish branded stores.\nAs companies have become more concerned about brand, they see a need to control how their products are sold and how the brand is perceived. Third shelf down in aisle four at Circuit City is no longer sufficient. Through direct one-on-one contact with buyers, manufacturers can also learn about consumer preferences very quickly, and finely tune their products and experience in response.\nThere has been a recent surge in brand boutiques, both temporary and permanent, for an incredible array of products and services, including Song Airlines, MTV, Sharp, Suave, Purina, and Palm. While Apple was not the first technology-centric brand to open brand boutiques (Sony and Gateway were faster out of the gate), it was arguably the first successful one due to the complete experience its stores offer. More than a sales channel, brand boutiques provide a brand identity for consumers to associate with.\nSo, do they work? Yes, but only if handled the right way. My friend Nish Nadaraja, a San Francisco-based marketing executive, helped Method Home with their brand boutique strategy in 2004. Method Home sells a line of home cleaning products mainly through Target Stores. While sales were good, Method felt it didn t have enough exposure. Being a new brand and a startup on a tight budget presented quite a challenge. In an effort to gain maximum exposure while still controlling the brand experience, Nish suggested that Method dip into the marketing budget and open a Union Square pop-up store; that is, a branded, interactive retail store that pops-up for a limited time and then goes away. The results were better than anyone could have imagined. Aside from creating buzz and sales, Method was also able to gain valuable customer feedback and even rotate employees through the store to get them more in touch with their customer base. For the budget of a decent print-marketing campaign, Method generated buzz, gained new customers, and even earned a profit. Nish says one of the secrets to a successful brand boutique is having the right evangelists out there supporting it, and planning exclusive fun events like wine tastings targeting specific demographics and engaging seminars in the space.\nThese efforts can clearly result in a better consumer experience. Boutiques are obviously more pleasant, approachable, and visually diverse than big box retailers. This adds to the fabric of neighborhoods that have been increasingly trying to keep big-boxes out. Aside from being nice places to shop, brand boutiques generally have salespeople that are incredibly knowledgeable about the products they are selling, and how they compare to those offered by competitors. As consumer electronics and services grow increasingly complex, boutiques may very well be the only way these products can be sold. Obvious drawbacks are that it s not as easy to compare competitive products when they aren t located in the same store, and of course it s not in the brand boutiques salespersons interest to have you buy anything elsewhere even if it is a better match for you.\nCompanies that establish successful brand boutiques will be able to use them to introduce new and complementary products and services. Just watch over the coming year as Apple takes advantage of its brand boutiques by launching a cellular phone and service.\nBrand boutiques are catering to the notion that people want to be entertained, engaged, and informed. As consumers, we can just sit back and enjoy the show!\nMahin Samadani is a Business Development Director for frog design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's been close to 6 months now since my kids and I have moved out of my parents' house and started living on our own. And it had been quite a struggle for us when it comes to living comfortably \u2013 I mean our new apartment is still so bare since we moved in. So I'm always looking to buy new home furniture and appliances but have been delaying it because of our tight budget.\nFor months, we settled with only a couple of appliances \u2013 a small desk fan, an electric stove, a single tub washing machine, and the rice cooker I got from my parent's house (kinda stole it! HAHA). That's it. We haven't managed to purchase anything else because I didn't want to just grab whatever cheap stuff we can afford \u2013 only to regret seeing it broken a few months later. Though I also don't wish for high-end premium products that may take us years to save up or pay for. Instead, I go for affordable stuff without compromising \"quality\" \u2013 either local or international brands.\nLast week, I chanced upon a big home appliance sale in one of the malls in the city, the Metro Store Appliance Blast 2017, and thought it would be the perfect time to take advantage of the huge discounts on our much-needed home appliances in the apartment.\nWhat I like about Metro's #ApplianceBlast2017 is their wide selection of featured products from different brands. From small kitchen appliances to large screen LED televisions, the store is packed!\nNow let me share a little secret about our new Asahi electric stand fan. Don't be fooled by its ordinary physical look. Do you know that its advanced technology comes with a blast sound system? It is equipped with a super bass, amplified megaphone that is best for your kids' mini concert!\nLet's talk a bit about the technical aspect of our brand new Asahi industrial stand fan.\nAnyhow, the Metro Appliance Blast 2017 is happening until August 31, 2017 at all Metro branches nationwide. So if you've been wishing and planning AND DELAYING to purchase some home appliance, it's your chance to save up to 50% off on selected items!\nOver to you, what home appliances have you been planning to take home? Share them in the comments below!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are looking for an Email Producer to join a global brand in London.\nYou will be supporting the Global Marketing & Digital team with the building of emails for both retail and eCommerce.\nThe Email Producer is responsible for the email coding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cleary Gottlieb represented the underwriters, led by Merrill Lynch, Pierce, Fenner & Smith Incorporated and Jefferies & Company, in a recent convertible senior debt offering by Salix Pharmaceuticals. Salix issued $345 million of 2.75% convertible senior notes due 2015. Cleary Gottlieb also represented Bank of America as the counterparty to Salix in capped call transactions hedging Salix's obligations relating to the notes. The offering (including full exercise of the related over-allotment option) closed on June 3.\nSalix, based in Morrisville, North Carolina, is a specialty pharmaceutical company that develops and markets prescription pharmaceutical products for the treatment of gastrointestinal diseases. Salix's strategy is to in\u2013license late\u2013stage or marketed proprietary therapeutic drugs, complete any required development and regulatory submission of these products, and market them through the Company's gastroenterology specialty sales and marketing team. Cleary Gottlieb previously represented the underwriters in a 2008 convertible bond offering by Salix.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the Canadian hockey goaltender, see Dave Kerr.\nDave Kehr (born 1953) is an American film critic. For many years a critic at the Chicago Reader and the Chicago Tribune, he later wrote a weekly column for The New York Times on DVD releases. He is now a curator in the Department of Film of the Museum of Modern Art.\nDave Kehr did his undergraduate work at the University of Chicago, where he studied English. He learned French in part to read the Cahiers pieces on film. At the time the university did not have a film studies curriculum. He started writing on film for The Maroon, the student newspaper, when he was president of the film society, Doc Films.\nFrom 1974\u20131985, Kehr wrote for the Chicago Reader, where he established a reputation for independent thinking and an understanding of visual style. Until 1978 he edited the Reader's question and answer column, \"The Straight Dope\".\nHe next wrote as a film critic for the Chicago Tribune (1986\u20131992). He moved to the East Coast to become film critic for the New York Daily News (1993\u20131998). In 1995, he was a member of the jury at the 45th Berlin International Film Festival. Kehr is a past chairman of the National Society of Film Critics, and a member of the National Film Preservation Board of the Library of Congress.\nBeginning in 1999, Kehr wrote a weekly column for The New York Times on DVD releases, also contributing occasional pieces on individual filmmakers or films. While at the Times, Kehr also produced a film blog, with reviews of films and occasional comment pieces.\nKehr left the New York Times in late 2013 to work as Adjunct Curator in the film department of the Museum of Modern Art. In 2017 he was promoted to Curator.\nKehr frequently refuses to conform to prevailing popular opinion and has expressed criticism of generally well-reviewed movies, examples including Das Boot, Raging Bull, Apocalypse Now, The Empire Strikes Back, Dog Day Afternoon and Alien, all of which, according to Rotten Tomatoes, currently hold a 96% or larger approval rating among movie critics.\nKehr, Dave (2011). When Movies Mattered: Reviews from a Transformative Decade. Chicago: University of Chicago Press.\n^ a b c Steve Erickson, \"Interview with Dave Kehr\" Archived 25 December 2010 at the Wayback Machine, Senses of Cinema, June 2001, accessed 4 May 2010.\n^ \"Berlinale: 1995 Juries\". berlinale.de. Retrieved 30 August 2018.\n^ Foundas, Scott (21 October 2013). \"Dave Kehr Named MoMA Adjunct Curator for Film\". Variety.\n^ \"Das Boot\". Retrieved 12 September 2013.\n^ \"Raging Bull\". Retrieved 22 February 2013.\n^ \"Apocalypse Now \u2013 Review Comments\". Rotten Tomatoes. Archived from the original on 7 November 2012. Retrieved 18 October 2011.\n^ \"Star Wars: Episode V \u2013 The Empire Strikes Back \u2013 Review Comments\". Rotten Tomatoes. Archived from the original on 28 March 2010. Retrieved 18 October 2011.\n^ \"Dog Day Afternoon \u2013 Review Comments\". Rotten Tomatoes. Archived from the original on 24 December 2012. Retrieved 18 October 2011.\n^ \"Alien \u2013 Review Comments\". Rotten Tomatoes. Archived from the original on 18 November 2011. Retrieved 18 October 2011.\n^ \"Das Boot\". Rotten Tomatoes. Retrieved 12 September 2013.\n^ \"Apocalypse Now\". Rotten Tomatoes. Retrieved 18 October 2011.\n^ \"Star Wars: Episode V \u2013 The Empire Strikes Back\". Rotten Tomatoes. Retrieved 18 October 2011.\n^ \"Dog Day Afternoon\". Rotten Tomatoes. Retrieved 18 October 2011.\n^ \"Alien\". Rotten Tomatoes. Retrieved 18 October 2011.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you have limited space and time but still want to try your hand at growing vegetables, think containers. No yard is necessary; a patio, deck, windowbox, or even a fire escape will do. All you'll need is a couple of containers, some soil, seeds or seedlings, and a little fertilizer. Your cash outlay will be minimal, and you'll have the satisfaction of growing something tasty to eat.\nYou won't be alone. According to a 2009 National Gardening Association Survey, about half of all food gardening households (48 percent) grow food in containers. I've always been blessed with enough space for an in-ground garden, so I never had to \"resort\" to growing vegetables in containers, but this year I decided to explore the topic.\nHollandia Nurseries in Bethel, Connecticut, offers gardening seminars every spring.\nNot the best or cheapest choice for growing vegetables, but you could grow them here.\nSomewhere out there in the marketplace, there's the perfect container for you in the material, shape, size, and price range you seek. Ceramic, plastic, terra-cotta, wood...the list goes on. You can purchase a container that's diminutive, or you can get one that requires a forklift to move. You can even get a railing planter with a bottom that's configured to sit snugly on a 2x4 or 2x6 deck railing. Or you can simply reuse something you already have, such as a whiskey planter, garbage can, joint-compound bucket, baskets, or windowbox.\nWhatever container you use, just make sure that it has drainage holes at or near the bottom and that it's deep enough for what you want to grow. Lettuces and herbs require the least depth; tomatoes, peppers, eggplant, broccoli, and cukes the most.\nA high-quality potting soil is your best bet, and Reelick recommends Hamptons Estate Professional Potting Soil. Another good choice is a commercial soil-less mix. Topsoil alone is too heavy and should be avoided. And it never hurts to add some compost. To retain moisture, Reelick suggests adding some Soil Moist to the bottom of the container to help retain moisture.\nLeaf lettuce is a good choice for a container garden. The plants continue to grow as you harvest individual leaves, and if you shelter the container from the heat of the sun, you'll extend the season into early summer.\nEither will work. Nearly any vegetable can be grown in containers (Reelick wouldn't suggest trying corn, pumpkins or watermelon), but it's best avoid \"mammoth\" varieties. Stick with \"patio\" tomatoes, Asian-style eggplants (long and skinny), and bush squash. Anything tall will need a trellis or stake and a deep container.\nIf your containers dry out completely, your plants are toast. Soak the containers before they reach that condition. Self-watering containers are sold by Gardener's Supply and many other garden sites. For fertilizing containers, Reelick swears by Osmocote and Jack's Classic Plant Food.\nWhen the food is ready, don't let it go to waste. Pick leafy lettuce by the leaf, and the plant will grow more. When the radishes are pulled, you can plant something else.\nEach year you'll want to put a fresh soil mix into your containers, so you need to empty them before spring rolls around. With ceramic containers, it's important to do this in the fall if you live in a cold climate, or the containers may crack.\nThis in-depth treatment of the topic includes a nifty design for a self-watering container you can make from a 5-gal. bucket.\nThere's a section on veggie container gardening indoors and a chart that lists minimum container size and plant spacing for several common vegetables.\nThis page suggests specific varieties suitable for container growing and also contains a troubleshooting chart for correcting common problems, such as spindly or stunted plants.\nA chart lists commonly available containers, along with suggestions on what to grow in each. There's also a recipe for a homemade soil mix and a list of commercially available mixes. And an account of how a central Florida gardener grew tomatoes in 5-gal. square cooking-oil cans.\nmicrowaveoven writes: most popular design.\nKayClayton writes: Really like this.\nKarenBudnick writes: Great post, Ruth - I have my seeds (container lettuce - Garden Babies Butterhead) from Renee's Garden and organic potting soil. Rain or not - there's a container in my garage with their name on it this week-end!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Making the image visible. The chemicals to do this are known as developer.\nMaking the visible image fixed, so it changes no longer.\nThere are different kinds of films. The most common of these is negative film. With negative film, all colors are reversed. This is corrected in an additional step of making a print of each image. So called reversal film, which is used for diapositives directly produces positive images (which are framed to become a diapositive).\nThis whole process is known as film processing.\nBlack and white negative processing is the chemical means by which photographic film and paper is treated after photographic exposure to produce a negative or positive image. Photographic processing transforms the latent image into a visible image, makes this permanent and renders it insensitive to light..\nFilm processing Facts for Kids. Kiddle Encyclopedia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First floor office in central Maroochydore, located on busy Wises Road, with easy access to the CBD, Sunshine Motorway, and beaches and schools. Surrounded by other successful businesses.\nFitout in place. Two offices, reception area, kitchen and common area.\nLease price $280 per square metre plus outgoings plus GST.\nFor further information or to arrange an inspection please contact Chris Sales.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An easy to use and safe way to pay for your child's school fees and expenses.\nWith the cold weather, we have a number of students who normally walk home being picked up from school. If your child is being picked up, kindly notify your child's teacher in the agenda planner, by e-mail or by phoning the school to ensure that the student and teacher are aware of the arrangements, especially when they are different than normal.\nConstable Petkau gave a presentation to our grade 5-8 students on drug awareness. Students learned about various types of drugs, their affects and dangers associated with use and misuse of a variety of drugs. We learned a lot and hope that our students make good decisions and remember what is at stake when using drugs. Just say no!\nWith the recent cold weather, we are again experiencing problems with parents parking in the staff lots or blocking staff in by pulling into the alley or the handicapped stall at the front of the school. For student safety, it is critical that parents and caregivers park on the west side of Horace across the street from the school or farther down on 3rd avenue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Sunday, I was elected to be the Executive Board Member for our church. Confession: I HAVE NO IDEA WHAT THIS MEANS.\nI attended my first church \"business meeting\" about two years ago. I had no idea it was happening; I just showed up for a Sunday night service, and all of a sudden people were passing out ballots. My stomach knotted up with the feeling you get when you totally forget about a test until the teacher starts passing out scantron forms for the multiple choice portion. They were voting on who the deacons would be for the upcoming year...as best I understood. I had never been to a church where there were \"deacons\" either. I grew up in North Carolina, and I thought Deacons were people who went to Wake Forest University.\nWhen people started reading \"minutes,\" and giving financial reports I almost passed out. People were all, \"according to the bylaws of the church constitution, we need to wait two weeks before we vote on this matter, so I make a motion that we call a special-order business meeting at that time.\"\nAnd then from the back, \"Second.\"\nI felt like someone was going to call on me and ask my opinion about something and I was going to have to get up and recite the first two paragraphs of the Declaration of Independence, or the Gettysburg Address or something.\nWell I've come a long way since my first business meeting, and I'm not so intimidated anymore. But I do cringe every time I see a visitor sitting in the congregation - I want to go sit next to them and whisper the Pledge of Allegiance, or list off Supreme Court justices to lighten the mood a little bit.\nMy game plan for business meetings is to sit there and keep my mouth shut. I don't vote on anything unless I feel strongly about it. I've never voted for any particular color of paint or carpeting, and I've never voted for or against anyone being on a safety committee or a transportation committee. I read the financial report and look for anything fishy; I never find anything - just good math and good choices, so I keep my mouth shut.\nWe had our quarterly business meeting during the Sunday night service because we needed to nominate\/elect\/vote on the people who would be in leadership this coming year. You know, who's in charge of VBS, who's on the financial oversight committee, who's the chairman of the deacons, who's the high priest who'll be making sacrifices on our behalf...wait, scratch that last one.\nAll wise, honest people of character were nominated. They agreed to serve, we voted them in. No sweat. But there were still two leadership roles that nobody had volunteered for. (I think you see where this is going.) One was Executive Board Member.\nDan asked if he could do it, since it will require meetings during the daytime that other working folks would have a hard time attending.\nPastor said, \"The church bylaws require that it be a lay person.\"\nWorship Pastor leaned across the aisle and said, \"But your wife can do it!\"\nEveryone laughed; I shrunk down 6 inches in my seat.\nWe moved on; I thought I was in the clear.\nThen someone piped in from the back, \"I think we would be well-represented by Kate. She would certainly make things more interesting.\"\nMore laughter. I'm not sure if that was supposed to be a compliment, a joke, or an insult.\nPastor says, \"Kate, would you be willing to serve?\"\nMy exact words were, sheepishly, \"I mean, I would do it.\"\nMore laughter. I'm almost laying down in my seat.\n\"I make a motion that Kate Conner be our new Executive Board Member.\"\nSo I suppose I have to figure out what the heck an Executive Board Member does. Apparently it's not a big deal. I'm pretty sure I just go to three meetings this year and represent our church. I'll probably be the only woman, and the only person under retirement age in attendance. I plan on wearing my skinny jeans and my biggest earrings just to make a splash.\nI'm just saying, if I'm going to do it, I'm going to OWN IT. Watch out, Kate Conner is about to vote on some stuff.\nDan said he would make me a T-shirt that says Executive Board Member on it. I hope he does.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"HT MEDIA share price is up 0.5% to Rs 40.0 on the NSE. The total volume of shares traded is 0.0 m.\nOverall, the broader S&P BSE TECK Index is trading down by 0.0%. And the benchmark NIFTY 50 is at 11,633.0 (up 0.3%).\nOver the last 30 days, the HT MEDIA share price is down 15.1%.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dies R Us: A Winter Scene . . .\nToday I have a quick & easy winter scene created for your inspiration!\nI'm always amazed at how quickly I put together a card with the use of these fabulous die cuts purchased at the Dies R Us Store.\nThe possibilities are ENDLESS when using die cuts!!\nUsing a white card stock panel, sponge your favorite shade of blue ink (I've used Tim Holtz 'broken china & faded jeans' distressed ink). ** I have circle punched a piece of masking medium (post it note will work too) and put it on my white card stock before sponging to create a moon.\nTear a couple scraps of white card stock and adhere one in front of the other, leaving the upper portion unattached so you can insert your trees.\nDie cut 2 sets of the Memory Box 'Evergreen Trio' #99240 die set in your choice of green card stock shade ... I've used SU Old Olive. Position one set in between your snowy panels and the second set in the background.\nDie cut your favorite woodland critter and position in the foreground. I have used the Marianne LR0377 Mama Moose and Baby (sorry Marianne dies are not routinely carried at our store.) But there are several beautiful DEER dies available that would work perfectly in this scene! Just type DEER in the 'search' tab of our store to see what's available.\nLastly add some 'snow'. I've used Liquid Pearls for my snow effect.\nI hope you've found inspiration in today's card creation and remember you can stop by this inspiration blog any week day (Mon-Fri) for a new inspiration sample from one of the design team members!\nThere's a new challenge up at the Dies R Us Challenge Blog!\nThe theme is a 'Photo Inspiration' using HOT COLORS ... be sure to come check it out and then join in the fun! YOU could be the next winner of a $10 gift voucher to the Dies R Us Store!!\nand I (Darlene) will see you once again on Tuesday!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract\u2014The aim of this study is to focus on the importance of students' motivation in learning English. Teachers of English who teach students of other subjects other than English should face the challenge of motivating their students to learn English. The teacher who tries to find out the causes of obstacles in front of students motivation and tries to remove them will succeed in his mission of teaching because the more we get motivated learners the more teaching is efficient. In Algeria, enhancing students' level of motivation is an ongoing challenge especially with the students of other subjects because during their previous school years they do not give importance to English as they do for example with Physics or Mathematics if these students belong to the scientific stream. In our study, we have used a questionnaire as a research tool. We have distributed 100 copies to our population; 1st year students of the Preparatory School to Economics, Commerce and Management Sciences of Annaba Algeria. We have collected 90 answered copies that constitute the basis of our study and analysis.\nIndex Terms\u2014English for specific purposes, teaching\/learning English, motivation.\nF. Guerid is with Setif II University Algeria, Algeria (e-mail: gueridfethi@yahoo.fr).\nCite:F. Guerid, \"Enhancing Students' Level of Motivation in Learning English: The Case of 1st Year's Economics Students,\" International Journal of Languages, Literature and Linguistics vol. 1, no. 3, pp. 188-192, 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two men, both hiding a deadly secret, are on a murderous rampage through the desert.\nNature of the Beast when it opens | Nature of the Beast full movie in Latin Spanish | Nature of the Beast movie online | Nature of the Beast online | Nature of the Beast online sub eng | Nature of the Beast full movie in Spanish Latin online | Nature of the Beast premiere | Nature of the Beast premiere argentina | Nature of the Beast premiere date | Nature of the Beast premiere in us | Nature of the Beast download | Nature of the Beast premiere in Europe | Direct download Nature of the Beast | Where can I watch Nature of the Beast online? | Where to watch full movie streaming?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pallet storage and pallet rack installation requires an experienced crew with the proper tools and equipment. The installation needs to adhere to the design and take into account load levels, environmental considerations, building codes and other important factors when designing and installing your system. Let us take the lead on handling these needs for you while you take care of what you know best, your business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you want a great study session, you need to get ready for it. You wouldn't rock up to a marathon wearing the wrong clothes, on two hours sleep, an empty stomach and with no plan. So, make sure you don't sit down to study when you're not ready to do great work.\nIn this blog post I'm going to share with you what French cooking has to do with productivity AND my five-minute, seven-step process to prepare yourself for a great study session.\nDon't forget to sign up for my free studying resource library.\n'Mise en place' is a French culinary term which means 'everything in its place'. This refers to the setup process a chef uses before they start cooking. If they want their cooking to go smoothly and their output to be incredible delicious, and still hot when they plate it, they have to get ready first.\nThey gather and prep their ingredients, collect the equipment they'll need, make sure their space is clean and get clear on the plan or recipe they're going to use. By getting ready they save time, improve their focus and reduce interruptions.\nWe can apply mise en place to our studying by introducing this simple but powerful five-minute process into our studying.\nNow I'm not a tidy person. My bedroom doesn't have a floor it has a floordrobe\u2026BUT my desk is super tidy because I value my productivity.\nTo have a great study session, spend a few minutes tidying your desk of anything that you don't need such as study materials, cups, plates\u2026etc. If you have lots of studying 'stuff' like spare notebooks, folders, stationery then see if you can move it off your desk for now and then back later because it's important to balance having a lovely study space and a study space you can do great work in.\nYou don't want to waste time rushing to grab a folder looking for a textbook so collect all the stationery and study materials you're going to need for a great study session before you start.\nThe most common mistake I see students make is keeping their phones next to them or non-study Internet tabs open while they study. Our devices are glued to us almost 100% of the time so we have to be intentional about unsticking them for awhile.\nNow I'm not sat over here like a martyr because I'm guilty of this too. I know I can only have a great study session if I let my phone go but it's difficult sometimes.\nIf you can learn to study intentionally, with total focus on your study tasks and nothing else, you will get your studying done quicker. Would you rather study for two hours while checking your phone and social media or one hour so you can relax guilt-free for the spare hour?\nIf you want to achieve more in less time, do what you need to do to remove distractions before you start studying. Put your phone on silent, airplane mode, do not disturb or move it to another room. Turn off the radio or TV, move somewhere quiet and close down any non-study Internet tabs or email that you do not need.\nThis is a hard habit to break but ask yourself what's more important \u2013 checking Instagram\/replying to that WhatsApp message, or doing well in your studies?\nThis is simple stuff but make sure you have food and drink before you study to keep your energy and hydration up. Make a hot drink and grab a glass of water as well as your favourite study snack. I'd like to tell you to pick a healthy snack but right now chocolate is my best friend so I pick this!\nGetting ready for a great study session requires you to remove any chance of your brain making excuses to stop studying. You don't want to get five minutes in and realise you need the bathroom, your bare feet are cold or your seat's uncomfortable.\nSo do what's necessary beforehand \u2013 wear comfortable clothes, take a bathroom break, grab a lamp or get a cushion for your chair.\nListening to music while I study is a non-negotiable for me. I have to have earbuds in, the music on low, and the music cannot have lyrics because lyrical music makes my brain feel squiggly and I can't think straight.\nA few studies appear to demonstrate that non-lyrical music can improve focus. For example, a 2012 study published in Work magazine found that performance in a work environment decreased when background music had lyrics. The participants' productivity and attention were worse with lyrical music compared to non-lyrical music.\nTo have a great study session, try adding non-lyrical music to your routine. There are lots of different types of instrumental music so don't think that you've tried to listen to some classical music before and it didn't work because you have a lot of options.\nMy go-to music is film or video game soundtracks because they're designed to engage you and help you focus on the content but not be so engaging that you switch your focus to the music instead.\nThe Theory of Everything by J\u00f3hann J\u00f3hannsson.\nTo have a great study session you need to make sure you're working on the right things. You may be busy the entire time and ticking off tasks here, there and everywhere, but you don't want to finish and realise you didn't move the needle on anything important. Your essay is still not started, you didn't email your tutor even though you need to, you didn't start your revision.\nSo your last step before you jump into studying is to get clear on your ONE priority for the session. What ONE task will have the most impact on your progress and grades?\n\u200bPick that, even if it's hard\/boring\/long and do that. Break it down if you need to but working on your priorities will help you reach your study goals quicker.\nI hope this process help you have a great study session. Try it out and let me know how it goes over on Instagram @chloe.burroughs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Maids a new adaptation of Jean Genet 'The Maids'. Written exclusively for One-Eighth Theater by Oscar nominee Jos\u00e9 Rivera. World premiere September, 2016.\nCast: Laura Butler Rivera, Charlie Munn, Casey Robinson, Folami Williams and Daniel Irizarry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's a typical shift in your community shop when you see a patient on the board that makes you nervous. 2 year old male with syncope. It's been awhile since you have treated someone born in the 21st century and you know this child's workup will likely involve an EKG. The closest pediatric hospital is 2 hours away. How comfortable do you feel interpreting the squiggle lines generated by this little heart?\n16yoF with 4 days of bilateral lower quadrant abdominal pain and diarrhea that was tachy, dry, and with a diffusely tender abdomen and some right-sided discomfort on pelvic exam with a mild leukocytosis.\nDr. Steuerwald's approach to listening to patient presentations: Pick out the main symptoms, get a time course, and listen for any other true \"weirdness\" then build your own timeline of events.\nDr. Steuerwald correctly identified the need to get a RLQ US to assess for appendicitis!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Succeed in Online Trading With Good Money Management - Hallo Home Based Business, n the article you read this time with the title Succeed in Online Trading With Good Money Management, we have prepared well for this article you read and download the information therein. hopefully fill posts Article stock market trading, you write you can understand. Happy reading.\nJust like any other type of business, the online trading is in need of capital. Moreover, a lot of money is already involved with trading. Online trading is only one aspect of the large and complex industry. If you will search the internet about day trading, you will be surprised that it is connected or somehow similar to handling a business. Trading revolves around money. Thus, effective money management is of vital importance in stock market.\nTrading is slowly becoming a popular business for people who are looking for extra income or an alternative source of income. Some people conduct trading online as a job on the side and earn extra cash while others do to earn large profits. People venturing in the stock market is nothing new. In fact they even engage in online day trading training before they completely embark in it.\nThere are a lot of people who recognize the potential of earning big in this type of business. Profits are fast and easily obtained in the trading industry as there are a lot of opportunities in the different aspects of the trading industry. However, adequate knowledge and essential skills are very important to people who would like to enter the realm of stock market trading.\nIf you are online trading against the odds, it is best to keep losses small, run profits and have confidence in what you are doing and believe that you can make a lot of money. This can be done only if you have good money management. Good and effective money management is all about setting rules and keeping the risks and losses at a level that you are comfortable with. These rules and guidelines must be polished and thought over even before you compete with the best online trading experts.\nIn good trading money management, you want to maximize your profits and minimize your losses. The two cardinal rules of superior trading money management are to run your profits and cut your losses short. This is also referred to as money management and trade management. Risk management is very important in the field of trading. It is the very core of trading.\nA trader's account will grow very fast when the trader does well in managing the trades and diminishing the bad trades that he will encounter. There are different free day trading tips on how a trader can successfully do this. By mastering the key elements in online trading, he will become a top player in the stock market industry over time.\nthat is all articles Succeed in Online Trading With Good Money Management This time, hopefully can provide benefits to all of you. Okay, see you in another article posting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Before moving on with a review of todays racing I hope you all have a wonderful Christmas and thank you for all your support throughout the year.\nI am currently rated UK Number 1 Tipster via tipsters review and I want to remain at the top. With your help I can do that.\nThe more reviews we get the quicker we can get to the waiting list where you will all be part of a closed group, which will help protect the prices for everyone involved.\nBowson Fred blew the start last night, he finished nicely for 3rd and will be winning soon when it all goes right.\nUbaltique won very nicely indeed. He hit the 3rd last which hampered him so the run can be upgraded.\nPetite Jack showed a great turn of foot to win nicely at Lingfield. Luke Morris was superb on board, kept him covered up for just long enough.\nRoundabout Magic gave us our 3rd winner on the day. He was ridden really well and finished strongly to win going away. Nothing went right for Sandfranskipsgo, he will be winning soon.\nWidnes was given an awful ride by Gordon. Sat nicely in 2nd she somehow seemed to lose concentration and negated herself to 6th turning for home. The pair flew home for 2nd but a poor ride has cost the horse not only a win but also his handicap mark.\nAs you all know there is no racing for the next two days, Boxing Day is a big day of racing and the important information for you all is that Boxing Day tips will be sent over between 3pm \u2013 4pm on Christmas Eve. We did this last year and it worked very well so I see no reason why it won't work well this year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfonk b/data_all_eng_slimpj/shuffled/split2/finalzzzbfonk
new file mode 100644
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--- /dev/null
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+{"text":"How Much Does it Cost To Fly To Lutherville Timonium?\nThe cheapest flights to Lutherville Timonium found within the past 7 days were $101. Prices and availability subject to change. Additional terms may apply.\nAlthough you may try and resist the urge, you know that when the travel bug comes, there is only one way to get rid of it. Thankfully, Travelocity is here to help nip that travel bug in the bud with great deals on flights to Lutherville Timonium. Browse through our inventory to explore a wide range of cheap flights to Lutherville Timonium to solidify your next big trip out of town. Whether you are heading out on a business trip, exploring a place you have never been for pleasure or are bringing the family on the annual vacation, we have the perfect getaway waiting for you. In no time, you will be day dreaming of your plane landing in Lutherville Timonium as you anxiously await to exit the plane and explore what Lutherville Timonium has to offer. Considering all the amazing local food and culture in Lutherville Timonium, we won't hold this against you.\nWhen you are ready to embark on your next big adventure, Travelocity will be right here waiting to open the doors to the garden so that you can go and smell the roses. When you decide to book and travel is entirely up to you, but don't wait too long because we have some of the best deals on the web and they won't last long. Take time to go through our huge selection of cheap flights to Lutherville Timonium to find the best deals that fit into your budget. If browsing on your phone while your boss isn't looking is more your thing (we promise we won't tell), consider checking out Travelocity's mobile app which allows you to take advantage of all our flights to Lutherville Timonium without having to compromise your work computer's search history. If you are looking to save even more, you can easily save up to $537 when you bundle one of our cheap flights to Lutherville Timonium with one of our hotels. So why are you waiting around? There is no better time than now to start booking that next trip. Start your search now to book one of our flights to Lutherville Timonium today!\nQuestion : Which hotels are located near airports or major attractions in Lutherville Timonium?\nAnswer : We've put together a list of hotels in Lutherville Timonium close to airports and popular sights to see. Check out the hotels below. All of them are within proximity to must-see attractions and major airports in and around the city of Lutherville Timonium.\nRadisson Hotel North Baltimore: Situated in the business district, this hotel is 2.4 mi (3.9 km) from Fire Museum of Maryland and 3 mi (4.9 km) from Towson Town Center. Towson University and Pine Ridge Golf Course are also within 6 mi (10 km). This hotel has a 3.00 star rating.\nRed Roof PLUS+ Baltimore North - Timonium: Situated in Lutherville Timonium, this hotel is 2.6 mi (4.2 km) from Fire Museum of Maryland and within 6 mi (10 km) of Towson University and Pine Ridge Golf Course. Greenspring Station and Johns Hopkins Health Care & Surgery Center are also within 6 mi ( This hotel has a 2.00 star rating.\nHoliday Inn Timonium: Situated in Lutherville Timonium, this hotel is within 6 mi (10 km) of Fire Museum of Maryland, Towson University, and Pine Ridge Golf Course. Oregon Ridge Nature Center and Greenspring Station are also within 6 mi (10 km). This hotel has a 3.00 star rating.\nExtended Stay America Baltimore - Timonium: Situated in Lutherville Timonium, this hotel is within 6 mi (10 km) of Fire Museum of Maryland, Pine Ridge Golf Course, and Oregon Ridge Nature Center. Greenspring Station and Johns Hopkins Health Care & Surgery Center are also within 6 mi (10 km). This hotel has a 2.50 star rating.\nQuestion : What are the top attractions in Lutherville Timonium?\nAnswer : Lutherville Timonium has over 14 points of interest for you to choose from. We suggest checking out Pimlico Race Course and Towson University first because they are the most popular attractions in Lutherville Timonium. Don't forget about other highly recommended spots like Baltimore Museum of Art, Washington Monument, and Towson Town Center. Get out there and see what the city has to offer on your trip to Lutherville Timonium!\nHave stayed here many times, although this time rooms were noticably smokey & musty smelling. The first room was totally unacceptable and the service person changed us to a better smelling room although it was not as clean smelling as rooms in years past.\nFriendly and helpful staff: front desk, housekeeping, culinary were all very nice. The room was clean.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wind turbines or solar panels, nuclear power plants or 'clean coal' solutions, what does it mean to find renewable, sustainable and green answers to the energy challenge, not only in Australia, but for the world? How can this transformation occur and what are the implications? Panel discussion on alternative energy with Ben McNeil, Prasad Menon, Michael Bielinski and Grant King.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free, public events for the EdTech community in the Greater Boston Area.\nsupporting network of friends, advisers and mentors is key for success.\nWe welcome all new members and encourage those who are interested to volunteer with the group or present at one of our upcoming events.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Look through the listings of Member singles that have joined Austin Flirt that are tagged with Soul. Dating others that have like minded interests is a pefect way to find things to do once you are dating. Create a Totally Free Account to Find your perfect match!\nI'm not much of a flirt. I was just looking for new friends my age, and looking for a local chat room. I am NOT looking for a hook-up, but just a mature man my age that wants a relationship.\n:) hi there how are you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Langar Near The Hall, shortlisted for the Nottinghamshire Tourism Awards 2017, was absolutely thrilled to be announced this year's winner in the Self Catering Property of the Year category!\nThe Awards evening, held at the Crowne Plaza Nottingham saw many thriving Nottinghamshire Tourism businesses competing in a number of categories.\nWe were shortlisted along with The Sherwood Hideaway and Premier Suites and couldn't believe it when our gorgeous cottage was announced as the winner!\nA huge thank you to all our guests, friends and family for their support and praise since we opened our doors in July 2014.\nThis entry was posted in Awards, Langar, little extra touches, News and tagged cottage, Holiday accommodation, Langar, little extra touches, Nottingham, Nottinghamshire, self catered, self catered accommodation, Tourism Awards. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfthw b/data_all_eng_slimpj/shuffled/split2/finalzzzbfthw
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@@ -0,0 +1,5 @@
+{"text":"Liede-Schumann, S. & Meve, U. 2006. Calciphila, a new genus in African Asclepiadeae (Apocynaceae, Asclepiadoideae), and taxonomic rectifications in Cynanchum. Novon: A Journal for Botanical Nomenclature 16(3): 368-373. doi: 10.3417\/1055-3177(2006)16[368:CANGIA]2.0.CO;2 Full text BHL Reference page.\nGovaerts, R. et al. 2017. Calciphila in World Checklist of Selected Plant Families. The Board of Trustees of the Royal Botanic Gardens, Kew. Published on the internet. Accessed: 2017 Sept. 2.\nInternational Plant Names Index. 2017. Calciphila. Published online. Accessed Sept. 2 2017.\nTropicos.org 2017. Calciphila . Missouri Botanical Garden. Published on the internet. Accessed: 2017 Sept. 2.\nThis page was last edited on 17 December 2018, at 05:24.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We believe that when people feel connected to the company's purpose, they give more of themselves and engage more with their role.\nOur approach is to work closely with the company to develop and facilitate events, training workshops andcoaching which connect people to the shared identity of the organisation and which focus their efforts on shared goals.\nBy creating this connection, organisations get more from and give more to their people: they keep them longer, use their talents better and direct their collective energy towards realising company ambitions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TAKE ONE HUNDRED PEOPLE ,\" said Isabel.\nIt was typical of the sort of trying question Isabel asked herself, in the way in which we sometimes ask ourselves questions that admit of no definitive answer. She was an optimist when it came to humankind, unfashionably so, and so she thought the answer was ninety-eight, possibly even ninety-nine. Jamie, the realist, after a few moments' thought, said eighty.\nBut this was not a question which could be disposed of so easily; it raised in its wake other, more troubling questions. Were those one or two people the way they were because of the throw of the genetic dice\u2014a matter of patterns and repeats deep in the chemistry of their DNA\u2014or was it something that went wrong for them a long time ago, in some dark room of childhood, and stayed wrong? Of course there was quite another possibility: they chose.\nShe was sitting in a delicatessen when she remembered this conversation with Jamie. Now, from that convenient vantage point, she looked out of the window\u2014that man who was crossing the road right then, for example; the one with the thin mouth, the impatient manner, and the buttoned collar was perhaps one of that tiny minority of the malevolent. There was something about him, she felt, that made one uneasy; something in his eyes which suggested ruthlessness, a man who would not wait for others, who did not care, who would suffer from road rage even while walking\u2026She smiled at the thought. But there was certainly something unsettling in his demeanour, a hint of poisoned sexuality about him, she felt; a whiff of cruelty, something not quite right.\nShe looked away; one did not want such a person to see one staring; nor, she reminded herself, did she want to catch herself engaging in such idle speculation. Imagining things about perfect strangers might seem a harmless enough pursuit, but it could lead to all sorts of ridiculous fantasies and fears. And Isabel was aware that of all her manifold failings, thinking too much about things was one of the most egregious.\nOf course a delicatessen in Edinburgh was not the most obvious place to entertain such thoughts on the nature of good and evil, but Isabel was a philosopher and knew full well that philosophical speculation came upon one in the strangest places and at the strangest times. The delicatessen was owned by her niece, Cat, and in addition to selling the usual things that such shops sold\u2014the sun-dried tomatoes and mozzarella cheese, the fresh anchovy fillets and the small bars of Austrian marzipan\u2014this delicatessen served coffee at the three or four small marble-topped tables that Cat had found on a trip to the Upper Loire valley and that she had carted back to Scotland in a hired self-drive van.\nIsabel was sitting at one of these tables, a freshly made cappuccino before her, a copy of that morning's Scotsman newspaper open at the crossword page. Her coffee had been made by Cat's assistant, Eddie, a shy young man to whom something terrible and unexplained had happened some time ago and who was still awkward in his dealings with Isabel and with others. Eddie had gained in confidence recently, especially since he had taken up with a young Australian woman who had taken a job for a few months in the delicatessen, but he still blushed unexpectedly and would end a conversation with a murmur and a turning away of the head.\n\"You're by yourself,\" said Eddie, as he brought Isabel's coffee to her table. \"Where's the\u2026\" He trailed off.\n\"I'd like to see him sometime,\" said Eddie vaguely. \"But I think that\u2026\" He left the sentence unfinished, yet Isabel knew what he meant.\nEddie moved away. A customer had entered the shop and was peering at the counter display of antipasti; he needed to return to his duties.\nIsabel sighed. She could have brought Charlie with her, but she had decided against it, leaving him instead at the house with her housekeeper, Grace. She often brought him to Bruntsfield, wheeling him, a wrapped-up cocoon, in his baby buggy, negotiating the edge of the pavement with care, proud in the way of a new mother, almost surprised that here she was, Isabel Dalhousie, with her own child, her son. But on these occasions she did not go into Cat's delicatessen, because she knew that Cat was still uncomfortable about Charlie.\nThere had been acceptance, later, and reconciliation, but by that stage Isabel had announced her pregnancy and Cat had retreated in a mixture of resentment and embarrassment.\nCat had looked at her with an expression that Isabel found impossible to interpret.\nCat said nothing, and Isabel realised that what she was witnessing was pure envy; unspoken, inexpressible. Envy makes us hate what we ourselves want, she reminded herself. We hate it because we can't have it.\nBy the time that Charlie arrived, tumbling\u2014or so it felt to Isabel\u2014into the world under the bright lights of the Royal Infirmary, Cat was talking to Isabel again. But she did not show much warmth towards Charlie; she did not offer to hold him or to kiss him, although he was her cousin. Isabel was hurt by this, but decided that the best thing to do was not to flaunt Charlie before her niece, but allow her to come round in her own time.\nch had made her feel bad, but the discomfort had been too great and she had found herself dreading the experience. That was not a way to bond with one's child, she thought; babies can pick up the physical tension in the mother, the drawing back from contact. So she had switched to a baby formula.\nIsabel would not leave the delicatessen without exchanging a few words with Cat, no matter how strained relations might be. Now she rose from her table and made her way to the half-open door to the office. Eddie, standing at the counter, glanced briefly in her direction and then looked away again.\nCat had a brochure in front of her, her pen poised above what looked like a picture of a jar of honey.\n\"Do people buy lots of honey?\" Isabel asked. It was a banal question\u2014of course people bought honey\u2014but she needed something to break the ice.\nThere was a silence. Cat stared at the photograph of the jar of honey. Isabel drew in her breath; this could not be allowed to go on. Cat might come round in the end\u2014and Isabel knew that she would\u2014but it could take months; months of tension and silences.\nCat continued to stare fixedly at the honey. \"I don't know what you mean,\" she said.\nShe was not sure why she should be asking her niece's forgiveness, but she was. When it came to forgiveness, of course, it did not matter whether somebody was wronged or not\u2014what counted was whether they felt wronged. That was quite different.\n\"I don't have to forgive you,\" said Cat. \"You haven't done anything wrong, have you? All you've done is have a baby. By my\u2026\" She trailed off.\nUs. She was getting used to the first person plural, but here, of course, in this atmosphere, it was heavy with significance, a land mine of a word.\nShe left Cat's office. Outside, from behind the counter, Eddie looked up and exchanged glances with Isabel. For a young man who everybody imagined understood nothing, he understood everything, thought Isabel.\nGrace had been cradling Charlie in her arms and now handed him over to Isabel in the hall.\nShe carried him through to her study and sat down in the chair near the window. It still felt strange to her to have Charlie in her study, her place of work. Babies belonged to a world of blankets, colour, softness, not to this place of paper, files, telephones. And philosophy, which is what Isabel did as editor of the Review of Applied Ethics, seemed so far removed from the world of infancy. Would Immanuel Kant have known how to hold a baby? she asked herself. It was highly unlikely; babies were too irrational, too messy for him, although he would have acknowledged, of course, that each baby should be treated as an end in its own right, and not as a means to an end. So one should have a baby because one wanted that baby to be born and to have a life, not because one wanted the pleasure oneself of having a baby; that was implicit in any Kantian view of the matter. But even if he had acknowledged the essential value of every baby, would Kant have had the faintest idea how to deal with a baby? Would he even have known which side was up? That was the cold and informal face of philosophy, Isabel thought; something far removed from the world that ordinary people knew; their world of struggle, and messy passions, and unresolved pointless differences, such as her difference with Cat. That was Kant's child-unfriendliness established; Hume would have known about children, she decided. He would have found babies good company because they were full of emotions, unexpressed perhaps, or made known only in the crudest of manners, but emotions nonetheless. And Hume, the Good Davey as he was called, was easy company, of course, and would have been liked by babies.\nContent in his mother's arms, Charlie now seemed to be dropping back to sleep. Isabel watched as his eyelids fluttered and then closed. She could watch him for hours, she thought, whether he was awake or asleep; it was difficult to believe that she\u2014and Jamie, of course\u2014had created this little boy, this person, had started a whole future on its track. That struck her as little short of a miracle; that a few small cells could multiply and differentiate and create an instrument of thought and language, a whole centre of consciousness.\nIsabel handed the sleeping child over to Grace and went to her desk. Charlie's birth had had little impact on the Review of Applied Ethics; in her determination to be prepared well in advance, Isabel had put together two special issues during her pregnancy\u2014an issue on moral particularism in the work of Iris Murdoch (\"the moral dilemmas of Oxford types,\" Jamie had called it) and one on the morality of boundary controls. The Murdoch issue had gone to press shortly after Charlie's birth, and the second would be published within a month or two. It had caused her some anxiety, this second one, because the topic was such an uncomfortable one. States are entitled to have some control over their borders\u2014there is general agreement on that\u2014but when they try to keep people out, then passions are raised and accusations of heartlessness made. Auden had a poem about this, which she quoted in her editorial. He had written from the perspective of a displaced person who hears the rhetoric of hate of his persecutors; and there is that arresting line that brings it home so strongly: \"He was talking of you and me, my dear,\" says the man to his wife. You and me: at the end of every bit of exclusion, every act of ethnic cleansing, every flourish of heartlessness, there is a you and me.\nIsabel found this intensely difficult, and had noticed that most people simply avoided the issue or did not discuss it in the open. The heart went one way\u2014those who want a new life should be helped to get it, if possible\u2014but the head might look in another direction, at the pragmatic impossibility of allowing the unfettered movement of peoples. So there were passports and quotas and restrictions, all of which amounted to a discouragement. Please don't come, these regulations said; please don't ask.\nShe looked at the room around her, at her desk, at her books. None of this would belong to her forever; it would change hands and somebody new would be here, somebody who would not even know who she had been, somebody who would look at her with astonishment if she came back, in some thought experiment, and said, That's my desk\u2014I want it. Our possessing of our world is a temporary matter: we stamp our ownership upon our surroundings, give familiar names to the land about us, erect statues of ourselves, but all of this is swept away, so quickly, so easily. We think the world is ours forever, but we are little more than squatters.\nStill deep in thought, Isabel stared at the pile of unopened mail. The harvest of just two days, it was neatly stacked in a red metal in-tray. Much of this was manuscripts; Isabel did not accept electronic submissions to the journal\u2014she disliked reading on-screen\u2014and required everything that was submitted to be printed out for her perusal. This meant that a river of paper entered the house each month, swirled in eddies about her study for a week or so, and then was guided out in a stream of recycling bags. The rejected manuscripts, those that she judged unworthy of the editorial board's scrutiny, were often the work of doctoral students, anxious for their first publication. Isabel was gentle in her rejection of these, expressing the hope that the authors would find somebody else willing to publish their work. She knew that this was unlikely, that the Review may have been their fifth or sixth port of call. But she could not be brutal; she had been a doctoral student herself once and remembered what it was like.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"5e Suitte : Ouverture, Allemande, Courante, Sarabande, Gavotte, Menuet en Rondeau, Gigue.\nMancini Francesco : XII Solos for a flute or a violin with a thorough bass.\nSonate VI : Largo, Allegro,Largo, Allegro.\nVirgiliano Aurelio : Il Dolcimelo.\nRicercar per Flauto : Cornetto : Violino : Traversa.\n- German National Library of Berlin (Germany).\n- Herzog August Library of W\u00f6lfenb\u00fcttel (Germany).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Curating a fresh music stream on Tumblr. Check it out and follow me http:\/\/freshreplay.tumblr.com. Tired of writing and just want to post about music. Plus this blog is fugly. May turn this site into a music web app but starting to enjoy MOG and have no time.\nRdio just released their streaming music service and I signed up here. It has a nice interface and integration with social apps but does not have the library of music that I am looking for. Out of about 12724 songs in my iTunes library Rdio matched 4341 not bad for a start but I think I will just host my entire library myself in the cloud. Waiting for Apple to put the Lala service they bought and shut down to use and host iTunes in the cloud. I believe there is going to be a lot of restrictions and limitations though to appease the recording studios. I want just a lightweight iTunes that I can run on my iPhone or any computer that has my entire music library with ratings and playlists. I do it right now with Dropbox but I only have 50 GB and want more and there is no good mobile app to sync with.\nI like MOG but am unlikely to pay for such a service. Also I am looking for fresh songs to listen to and preview that are not available through a music streaming service that requires licensing from the studios and generally is slow to update the music library. My best bet is listening to live sets, radio shows, blogs, Twitter and Youtube for fresh tracks. Hypem is probably the closest solution to what I am looking for but it suffers from the wisdom of crowds and songs that I have had on replay for weeks are suddenly number 1. The best tracks usually come from a curated list of blogs and friends one reason I started a blog. I tried the blog but I hate writing, this is probably the longest post I will make, and there are so many good music blogs out there already I would rather curate, consume, and reblog. All about more music and less writing.\nIn the end, I am leaning towards developing my own music streaming service but do not have the time and do not want to deal with copyright and recording studios. I most definitely support the artists by buying their albums, going to their shows, or sharing the tracks I enjoy. For now just enjoy this Tumblr and if you want a fire hose of my music interests and tastes follow me on Twitter @freshreplay.\nJack Beats new release set to release on the 28th of June on Herve's very own Cheap Thrills Record label. Jack Beats will be at EDC on Friday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgrrl b/data_all_eng_slimpj/shuffled/split2/finalzzzbgrrl
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+{"text":"Welcome to Balance Chiropractic & Wellness. From the minute you enter the office, you will experience patient centered health and wellness. Dr. Jennifer Rahe will listen to your concerns and will give recommendations that will benefit you, the patient, as an individual. Balance Chiropractic & Wellness welcomes patients of all ages, from 0 to 100. Our motto is \"It's your life\u2026Live it healthy!\" and we want to give every patient the resources and ability to live their healthiest life.\nWe look forward to seeing you in the office and hope that you take the time to look through the extensive information that can be found on our website. You will find a patient library, newsletter, and easy access patient forms for you to prepare prior to your arrival to expedite your first appointment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a port of Red Hat's Bluecurve theme to recent versions of GNOME and current versions of MATE.\nBluecurve Icon Theme from the Red Hat Art project.\nBlueCurve Metacity Theme from Fedora Project.\nI only changed one line in the Bluecurve-Looks theme to make it work with MATE. It does not effect how it works with GNOME.\nTested with GNOME 2.28 and up.\nAfter unpacking the main tarball, drag and drop included tarballs to the GNOME\/MATE Appearance preference panel.\nGTK 3.6 to 3.10 supported. May not work on 3.12.\nThank you very much for making Bluecurve usable on modern systems, it's a great addition to the FOSS world. Keep on your great work!\nNice work bringing this great theme to MATE. I do hope you will add a GTK-3 theme.\nThanks for submitting! Retro-themes aren't only fun, but they feel more usable. Newer themes often follow the extreme minimal route, and have no contrast.\nThe GTK3 theme is in private testing. Still some bugs to work out.\nFound it in xfce-look.org. What about Xfce?\nThese sites are all connected. This is probably on kde-look.org as well. That said, the icon and the gtk2 theme should work with xfce though I have no idea how. I don't use xfce.\nI wanted to create and account and say thank you for this.\nThe rpm is ready installed and uploaded.\nFor future release can you pls send me apackage with that structur.\nEnail adresse in in the src.rpm changelog.\nI'm the initiator and maintainer of the fedora-mate-desktop repo.\nAre you interest that i build and upload a rpm from your package?\nDone. I love how it just randomly stopped working.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Sitting Rooms image below is a part of Sitting Room Designs Ideas Collection Gallery. This digital photography of Sitting Rooms has dimension 1000 \u00d7 667 pixels. You can see a gallery of Sitting Room Designs Ideas below. And if you think that this image are interested to share with your friends, don't hesitate to share it on your Social Media account.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So whether you are male or female, athlete or gamer, our Posture Corrector is a must. BodyWellness\u2122 Best Posture Corrector fits comfortably and is adjustable to all body types. It firmly pulls the shoulders back, eliminating the \"rounding\" of shoulders which cause unhealthy poor posture. Additionally, our Posture Corrector is amazing for helping athletes maintain proper back form while working out to also help prevent injuries.\nBodyWellness\u2122 Best Posture Corrector is not about a temporary fix. When your body adjusts to wearing our Posture Corrector, muscle memory will form so that even when taking it off, your body will naturally adjust to its newer and healthier posture. The lightweight but firm material is adjustable and made of breathable material which means you can wear it all day for maximum effect. Even under clothing!\nAnswer: No, unless you are wearing the posture corrector incorrectly. If you experience pinching either you are wearing it too tight or you are slouching your shoulders into the posture corrector too much. Loosen the straps to your comfort level and pull those shoulders back! That's the beauty of BodyWellness\u2122 Best Posture Corrector, if you are slouching, it will become uncomfortable, forcing you to keep your shoulders back to regain comfort. It is the perfect way to strengthen your muscles to develop long-term healthy posture.\nAnswer: Yes! BodyWellness\u2122 Best Posture Corrector is designed to be sleek so that it fits under shirts or clothing, so you can be wearing it in public without people noticing. But for maximum comfort, consider wearing it over clothing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mermaid Adventures: the Frozen Time Free Game - GamesGoFree.com - Download and play for free!\nAlice and her friends need your help! Make everything that depends on you to save the water kingdom!\nStrange and incomprehensible things happen in the subaqueous realm. Now there is no any calm, because a spiteful octopus and his malicious servants have stolen it! They sow the bad seed everywhere, bother the harmless inhabitants of the ocean, disturb peace and destroy everything around. Their villainies are difficult to be stopped. Take part in the fight against evildoers! You are to help the brave mermaid Alice and her friends to interrupt them and restore peace! First wake Alice`s team with the help of the wise Turtle and then act! Bring happiness back to the underwater kingdom in this wonderful adventure game! Surmount all the obstacles and let the subaqueous dwellers be tranquil!\nFind lots of genuinely splendid games at GamesGoFree.com and play 100% for free! Enjoy Mermaid Adventures: the Frozen Time together with your friends without limitations and get an unprecedented gaming experience!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhemx b/data_all_eng_slimpj/shuffled/split2/finalzzzbhemx
new file mode 100644
index 0000000000000000000000000000000000000000..1a2b05baaa586718d06404cbad1c7120c8b71b05
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"7. Please rate your overall experience with Parker Home Services.\n8. What is the chance that you would recommend Parker Home Services to others?\nDo you have any comments, concerns or suggestions about your latest experience with Parker Home Services that you would like to share with us?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This listing is for a handmade leather pouch that can be used to secure your knife, multi-tool, or whatever else you want on your person, but not in your pocket.\nEach pouch is handsewn out of 8-10 ounce latigo cowhide, and features a snap closure and riveted belt loop.\nWe offer pouches in three different sizes, based on whatever you need to carry.\nHere are their dimensions, as well as a few examples of what will fit into them.\nSmall: 1.25\" X .75\" Wide 4\" Tall. Will fit small pocket knives, simple Swiss army knives, etc.\nThe leather pouch I received was well made better than the original pouch my leatherman came with. The pouch belt loop sits nicely on belt and the slight gap allows for easier access to contents. I highly recommend this pouch in any size.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Created for daily training and competition, GU Energy Gel packs energy-dense calories in a portable packet to help sustain energy demands so you can smash those PBs.\nThe 100-calorie packet contains carbohydrates (maltodextrin and fructose) that use non-competing pathways to help maximize carbohydrate absorption and utilization while diminishing stomach distress. Sodium, the primary electrolyte lost in sweat, aids in hydration by maintaining water balance. The branched-chain amino acids (leucine, valine, isoleucine) may reduce mental fatigue and decrease muscle damage.\nThis product is made with all vegan ingredients and are gluten free and kosher.\nIngredients: maltodextrin, water, fructose, l-leucine, natural flavors, potassium and sodium citrate, citric acid, calcium carbonate, l-valine, sea salt, gellan gum, l-isoleucine, sunflower oil, sodium benzoate (preservative), potassium sorbate (preservative).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mistresses Anna and Mia had both fallen victim to this loser and they had to do something about it. They did not want to let him go around conning people so they plotted to do something about it. They laid a trap for him and when he fell for it, they trampled him and punished him by spitting on him and humiliating him. He begged for mercy and promised never to do it again.\nLady Bertie has always taught her slave to be honest and it was a big surprise for her when she found out he had been lying to her. She was so mad at him that she had to punish him. She made him lie down and she spat on him as she smoked. To make the punishment worthwhile, she made him lick the spit and swallow. Then she dared him to lie to her again.\nMadame Marissa did not like the kind of spices the slave used to make food. Thus she did not enjoy the food as much as she thought she would. She had to punish her slave for it and she did. She did not have to look far for a way to punish him. She used her heels and her spit. She spat on them and his work was to lick them clean.\nThis mistress is a no nonsense mistress. When she realized that her slave was slacking, she knew she had to do something. She decided to use her spit fetish to humiliate and punish her slave. She spat on her high heels and she forced him to eat it. She did not give him an option and much as he was disgusted and humiliated, he had to do what she wanted him to do.\nMadame Marissa wanted to humiliate this loser with her spit but she knew her spit alone would not be enough for the kind of humiliation she wanted this guy to endure. She called her friends mistresses Virginia and Madison and together, they had fun taking turns to spit on him. The poor guy had to lick their spit and swallow it. By the time they were done with him, he felt like dying.\nMadame Madison did not like her slave's bad breath. She had always told him to observe proper hygiene but he seemed to be slacking off. So to make sure he never slacked off again, she took his tooth brush and she scrapped her dirty shoes, her sweaty and smelly feet and even spat on it before she gave it back to him. She wanted him to use her spit as toothpaste to brush her teeth.\nLady Kara does not like someone lying to her. When she realized her slave had lied to her, she understood why he had done it, but still she did not condone it. She punished him to make sure he never repeated the mistake. She made him lick her dirty soles after she had spat on them. She also used her boots to trample on him and humiliate him as punishment for lying to her.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An Oregon trade cooperative has released a report with recommendations that it says can help Oregon businesses facing logistics challenges due to the loss of weekly Port of Portland Terminal 6 container service and changes in the maritime and transportation industries.\n\u2022 Resolution of Terminal 6 labor-management issues and restoration of weekly container service.\n\u2022 Operational enhancements and actions to improve the existing logistics system.\n\u2022 Strategic investments in freight logistics to sustain services.\n\u2022 Policy actions to enhance Oregon trade and improve the transport of goods.\nOregon Gov. Kate Brown launched the International Trade and Logistics Initiative -- a partnership of the Port of Portland, the Oregon Department of Transportation, Business Oregon and the Oregon Department of Agriculture \u2013 in April 2015. Its mission: to identify trade-related, freight logistics solutions to help small- and medium-sized businesses across the state stay competitive.\nMore than 88 percent of Oregon exporters are small- and medium- sized businesses, according to state data, and many of these businesses are believed to be experiencing increased transportation costs, longer transit times, reduced reliability, and loss of markets and market share due to various factors, including the loss of weekly container service at Terminal 6, something that took place in April 2015 due in part, to labor strife and productivity issues.\nExisting support to shippers currently underway includes the creation of a drop yard for the monthly Westwood container service at Terminal 6, the restart of an upriver barge-rail shuttle to connect with farmers in eastern Oregon, southeastern Washington and Idaho; and exploration of both a container reuse pilot and intermodal rail facility for mid-Willamette Valley shippers.\nThe International Trade and Logistics Initiative's full report is available at www.oregontradesolutions.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblwqs b/data_all_eng_slimpj/shuffled/split2/finalzzzblwqs
new file mode 100644
index 0000000000000000000000000000000000000000..1e73fe1de97af4399cfc162b0f955b1a64ee9772
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblwqs
@@ -0,0 +1,5 @@
+{"text":"The molecular biology revolution is well underway. Dennis Paustenbach brought it up in 2008 at a DRI meeting, and I blogged about it then. But there is blissful ignorance among most boards, lawyers, futures representatives, trustees of asbestos trusts, and judges. Therefore, there can be and are more or less accidental discoveries of additional possible causes of mesothelioma, such as the new paper on mesotheliomas and vinylidene chloride (think Saran wrap), as described in the first post of this morning here on GlobalTort. Meanwhile, researchers are pushing forward on the role of the BAP1 mutation in mesothelioma, with a paper two weeks ago providing plenty of grist for future arguments by plaintiff's experts and lawyers, but not so much for defense experts and lawyers. Moreover, sober researchers and doctors are optimistic about soon significantly slowing mesothelioma deaths, and maybe even providing cures when the disease is found very early. Thus, last Friday, Dr. Farris Farassati shocked some lawyers with his molecular biology presentation at ACI's asbestos conference \u2013 he spoke about precision medicine treatments for mesothelioma. And, today, groups such as the Mesothelioma Applied Research Foundation are non-partisan and run by smart researchers and physicians, instead of lawyers. There will be more or less cures \u2013 the real question is when, and how many will die before that day.\nThe molecular biology revolution matters far beyond asbestos. Think benzene cases tried or settled based on genetic markers. Think microRNA signatures for cellular injuries caused by tobacco smoke, cadmium and other substances. Also think about the Human Exposome project. Think \"the next asbestos,\" but then think that 3,500 annual mesotheliomas are a drop in the ocean when compared the vast numbers of cancers every year. As a result, the \"next asbestos\" could actually be 2x- 10x the size of asbestos.\nHeadlines today are prominent and frequent regarding cybersecurity breaches, fired CEOS, and related liability concerns. Heck, even the SEC went out of its way to sound a wake up call by holding a conference on cybersecurity. Or, think about GM and recalls, with the CEO claiming: \"I did not know \u2013 no one told me.\" Will your manufacturing company, insurance company, asbestos trust, or law firm become the subject of a future headline because little or no attention was paid to the ongoing revolution in molecular biology, and the world's scientific knowledge?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am interested to know what you all think of the Seth Material or some of the core teachings. Has anyone read The Nature of Personal Reality here?\nThere is a lot to take in from the Seth materials, and for each person to internally query to figure out what works for them. It's a stream entry manual, but one where at least to my discernment you can tell where shifts occur in what was being relayed.\nI have and then revisited some of it more recently. It id good but somewhat heavy reading yo me. The Bashar stuff seems to put similar concepts in much simpler terms for people to understand on youtube I have found.\nI think that Seth (Roberts) is the best channeled material. I prefer the collections of quotes from Seth to the books; you get almost the same amount of information in a more condensed way. There are several good collections. The one I like the most is \"Jane Roberts - Seth - Quotes and Exercises\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Toronto (Canada) - The Association of Registered Graphic Designers of Ontario (RGD) and the Gladstone Hotel invite international submissions for the first-ever Social Good Design Awards, honouring graphic design projects done under the theme of communication design for social good; work with the power to incite action and make meaningful change in the way we live our lives.\nToronto (Canada) - The Association of Registered Graphic Designers of Ontario (RGD) and the Gladstone Hotel invite international submissions for the first-ever Social Good Design Awards. The \"So Good\" Awards honour graphic design projects done under the theme of communication design for social good; work with the power to incite action and make meaningful change in the way we live our lives.\nToday's graphic designers have an ever-greater responsibility to visualise the significant issues facing our society and find new and powerful ways to incite change in the way we behave. When we approach these issues creatively, we can make a real difference in the world. Ultimately, it is time for us to apply our creativity to shift thinking toward alternative solutions to ethical issues that impact our lives and the world we live in.\nThe Social Good Design Awards is an exhibit that gives voice to the important work designers are doing and can do to change the way we think and act.\nWinning projects will be on display in an exhibit at Toronto's Gladstone Hotel from 9-18 November 2012 and in a printed catalogue. An awards reception will take place on Saturday, 10 November.\nRGD and the Gladstone Hotel invite submissions of communication design projects from all designers worldwide. Projects must have been created, conceived or produced between 1 September 2007 and 1 September 2012. Conceptual concepts are welcome.\nEntries, forms and fees must be received by 19:00 on Monday, 24 September 2012.\nFor details on submission requirements, categories and selection criteria, visit www.rgdontario.com\/sogood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The reign of King Jamshid ushered in the first golden age in Iran's history. \"The imperial farr was his. The world submitted to him, quarrels were laid to rest \u2026.\" Jamshid introduced the sciences, arts and crafts, medicine, shipbuilding, mining, and the use of perfumes. He organized society in accordance with the professional classes, and he established Now-ruz (New Day) on the first of Farvardin (21 March), when he finally rested from his endeavors. For three hundred years Jamshid ruled over a country where \"men knew nothing of sorrow or evil \u2026 and the land was filled with music,\" but he became increasingly arrogant with his achievements.\nUngrateful, proud, forgetful of God's name.\nSuch hubris cost him divine glory (farr), the most important attribute of kingship. Iran was soon filled with dissension and revolt, and the once-glorious Jamshid went into hiding for a hundred years. When he reappeared on the shores of the Sea of China, he was killed by the demon-king Zahhak.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm working on the trip list for 2016. I have had a few requests including Norway Fiordland, Marseilles and Greece as well as beginner trips. Is there anything in particular you would like the club to do? If so, please let me know and I'll see what we can do. I'm aiming to have a range of trips with a variety of length, location and ability. If you would like help organising a trip, please let me know.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmmcj b/data_all_eng_slimpj/shuffled/split2/finalzzzbmmcj
new file mode 100644
index 0000000000000000000000000000000000000000..0792187896ec2f3052928b95b52bcdb8ebc67a0e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmmcj
@@ -0,0 +1,5 @@
+{"text":"Need a vacation? Chances are your skin does, too. Book a getaway and stash this convenient little kit in your carry-on. Whether you're a masque devotee, or you can't decide which one to try first, this kit has everything covered. Hydrate your skin, illuminate your complexion, wake up tired eyes, and smooth your lips for the true royal treatment, no matter when pampering calls.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the single most important ideas in organisation is the sales funnel. Sales funnels can increase your revenue online by automating your service. They offer a path of least resistance to expanding as well as scaling your company on auto-pilot. But after hanging around with some of the world's best online marketers, I can tell you with near certainty, not just just how sales funnels will certainly enable you to make more money online, but which specific funnels you ought to execute.\nNow, if you're anything like me, you may be averse to particular funnels. As an example, I'm never an ecommerce man, yet I do know that free-plus-shipping funnels are absolutely crushing it. Actually, just this previous week, while going to the Closing Table Mastermind with Perry Belcher and also Roland Frasier, I obtained a lot more insight into the alluring \"tripwire\" funnel, allegedly created by Perry Belcher.\nNevertheless, Belcher didn't develop the concept. He drew that from an old Columbia House ad that offered 13 CDs or tapes for the large cost of $1.00. Remember it? But all of us recognize that this alluring deal (aka tripwire) had way extra to it than just the $1.00 cost. Behind-the-scenes, there was a continuity program at work. In Columbia House's' design, this was called unfavorable choice payment. That means, you automatically obtain normal monthly deliveries (in addition to an expense) unless you expressly mention otherwise.\nSo why is this so powerful as well as what does it have to do with sales funnels? In fact, what you'll see is that Columbia House's sales funnel was quite basic. You simply selected the CDs or tapes you wanted for a dollar, then got a normal month-to-month shipment every single month at a pre-defined price. They would continually bill you for those CDs or tapes every month. There were variants of this offer as well where you can get hold of 11 CDs or tapes for one dime then get the 12th totally free.\nIn either case you check out it, these were all tempting deals. This was too good to pass up. However what Columbia House and also various other firms understood was that, although they would certainly lose loan out of the gate, they would, generally, in time, earn money turn over initially. And that is the power of the sales funnel. Prior to automation, this is exactly how sales funnels worked. And, if you hired, you would certainly get an online person efficiently strolling you with the funnel.\nThe Columbia House example provides deep understanding into the psychology of an offer that's irresistible. By first determining the lifetime worth of a customer as well as their retention rate, Columbia House figured out that it can invest a particular amount of money acquiring customers as well as still be very rewarding. That's essentially what's finished with sales funnels today. And also what really makes those sales funnels so powerful are tripwires.\nToday, it's the tripwire that produces the irresistible offer. Although Columbia House didn't quite have major up-sells or down-sells in their funnels, they developed a straightforward 2 step funnel that functioned. It worked due to the fact that the tripwire was so appealing. It appealed deeply to the consumer. After all, just how can you withstand an offer like that for 12 or 13 CDs or tapes for $1 or 1 cent when they were retailing at around $10 each at the time.\nAnother powerful instance of this was via Sports Illustrated. As a matter of fact, Sports Illustrated was successfully on the decrease at the time it presented its revolutionary tripwire. There's something concerning tangible items that individuals simply connect with. Also when you're supplying an electronic details item on the back end, by sweetening the deal with a substantial good, you can fairly actually send your conversions via the roof.\nIn the example of Sports Illustrated, take yourself back to 1990 when magazine subscription rates were beginning to leave. In that period, Sports Illustrated endured a 7% decline in their circulation in the very first half of the year. Currently, no matter what they did, they were unable to obtain the membership rates up. Also by providing deep discount rates on subscriptions, consumers weren't getting it. And also, if they discounted as well considerably, it would impact their capacity to get marketers paying costs rates without revealing that consumers were paying near the complete cover rate of the publication.\nSports Illustrated, possessed by Time, Inc., had not been the only publication that was suffering under the firm's umbrella. People magazine had likewise experienced an extreme decrease of 5%. Time, Inc., recognized that it needed to do something fast. So it started to supply a tripwire that included sports bloopers on VHS. Nonetheless, the cost was exorbitant as well as there were just a lot of bungles it might provide. So, they relied on the football phone as the new tripwire.\nWhy the football phone? First, it appealed to the target market. And, it was special and amazing. Even more amazing than a boring publication registration. The football-shaped phone concept was developed by Sports Illustrated's director of flow at the time, Michael Loeb. At that time, advertising on wire was also extremely cost-effective. And also, the phone, which was sourced from China, just cost a few dollars \"landed\". So all of it made sense.\nLoeb worked with the logistics of this deal for many months. Nonetheless, that had not been the only deal. They additionally devised a deal that contained a sneaker phone. If you'll reflect to 1990-1991, it was both uses that were running virtually concurrently. Talk about an early-on split-test, right?\nSo, what were the results of all this? Over 1 million brand-new customers. That's right. As well as those offers only ran for a minimal time. But it effectively saved the business. That's the power of a tripwire deal. It's a tangible item that gets people delighted about whatever else the core deal is. In fact, some people will just get what you're offering entirely for the uniqueness of the concrete tripwire offer.\nSales funnels are among the solitary most powerful principles in business today for a very certain reason. It's not just the example of the tripwire that's effective. It's the entire kitten play dirty, so to speak. There are so many components and also parts to what make a sales funnel work so well that I wished to break it down and also provide some even more examples to provide a much deeper understanding to exactly how you can utilize them to enhance your income online.\nCurrently, if you're not utilizing a sales funnel now, or if you're just uncovering what a sales funnel is, after that there are a few means you can quickly utilize this powerful concept in your service to create explosive outcomes. Primarily, if you do not have a ClickFunnels account, go below as well as sign up for the 14-day complimentary test. Once you've setup your account, grab a personalized domain name as well as begin developing your funnel. If you need assistance, see a few of the on-boarding video clips consisted of with ClickFunnels.\nIt does not matter if you have an existing company or you're working to configuration a new organisation, funnels are the trick to expanding and also scaling despite where you're beginning with. Nonetheless, there are 5 specific ways you can make use of sales funnels to raise your income today. It matters not what service you remain in or what you're selling, you can take advantage of these today, now, to produce eruptive outcomes.\nAt the Inner Circle mastermind this week, Russell Brunson dissected the hook-story-offer strategy that comes before developing the funnel. Can you build a sales funnel prior to you have a hook and also a story? Yes. However you may have trouble getting grip, particularly if you're completing in a red sea. Red seas are entrepreneur-infested waters contending for the very same pool of consumers, whereas blue oceans are a location where couple of business owners are swimming in.\nWhat does that inform you? Invest the time on the hook and also the story. Yes, the deal is likewise vital. However it's the hook as well as the tale that's mosting likely to reel prospects in. For instance, Brunson used the narrative of fit-to-fat-to-fat creator, Drew Manning. Manning, that remained in terrific shape, went out there and also acquired a number of weight over a 6 month period to much better understand what it feels like to be fat. But after that he headed out there as well as shed all the weight.\nThat right there is an incredible hook as well as tale. It's unique. And also, as a result of that, Manning has experienced a meteoric increase, although his deal isn't that excellent. Yet when the hook and the tale is just so good, the deal is practically unimportant. Properly, you can sell nearly anything when the hook and the tale simply crush it. An additional instance is the story of Robert Allen that would certainly constantly say he could be decreased in any city in the country and get a home no loan down within 60 days. And that's precisely what he did. His brand has actually corrected a billion bucks in sales as a result of it.\nThose are all fantastic hooks as well as tales. Without that, your offer will certainly have a hard time. If you can think of a special means to create your hook and also story, the rest will form. Since there is so much red ocean available, this is possibly one of the most important indicate make before you release any kind of sales funnel.\nTripwires are great low-barrier-of-entry products that make potential customers take out their bank card. Typically, these are too-good-to-pass-up products that are supplied at an unbelievable cost. Simply think of the tripwire supplied by Columbia House or Sports Illustrated, for example. Tripwires aid make it less complicated to get new customers, and then to rise them up your value ladder.\nAnything that makes client purchase more difficult, such as a costly product or service out of eviction, is mosting likely to make growing as well as scaling your company more difficult. The truth is that it costs 5 times as much to draw in a new client as it does to retain an existing one. That's why tripwires are so powerful. They lower the barrier-of-entry right into your business. As well as anything that can lower the barrier of entrance is always suitable.\nAnother terrific means to launch a sales funnel as well as easily raise your earnings no matter what service you remain in is with a lead magnet. The lead magnet needs to include genuine worth. However it can be in different layouts such as a PDF book, checklist, rip off sheet and so forth. After the download, you can provide an one-time offer. This suggests a three-step funnel. You might additionally create an up-sell if they take the one-time offer and even a payment-plan down-sell if they reject it.\nWhy do a lead magnet with lots of value? First, you're setting yourself up as a specialist. If that content is really workable, and it includes an enormous amount of worth to the prospect's life, then they're mosting likely to view you as the authority. However it does something far more than that. It refers micro-commitments. The theory? Obtain a person to agree to do three tiny things for you and also the 4th one is instantly an of course.\nAmong the most effective sales funnels you can construct is based around the webinar. As a matter of fact, it was the webinar that saved Brunson's organisation from bankruptcy a number of times. It was a combination as well as conclusion of expertise that bring about the perfect webinar manuscript, which creates the basis for the best sales funnel. A few of it originated from Jason Fladlien, while other items came from somewhere else in Brunson's trip.\nNevertheless, all that information integrated right into the layout of a best webinar and also funnel, has actually actually transformed the businesses of numerous entrepreneurs who have actually taken their knowledge as well as monetized it via the rapid pressure of webinars. Constructing a webinar is just one of the best courses to success with a sales funnel. As a matter of fact, the webinar funnel is fairly potentially one of the most transformative funnel you could introduce for any type of company, new or old.\nOne more means to raise your earnings with a sales funnel is to construct a high-ticket coaching funnel that goes from application to call to close.\nThe application permits you to remove those that aren't severe. And, on the phone call, you can close the deal. Now, this type of sales funnel jobs if you've already had results with clients. If not, benefit cost-free till you get outcomes for individuals. Then, collect endorsements as well as utilize those as component of the pitch for the training.\nProbably among the biggest as well as most prominent sales funnels these days is the free-plus-shipping offer funnel. It makes a great entry factor into any kind of funnel by creating a single offer or up-sell after the deal, comparable to a tripwire funnel. However, the distinction below is the power of the word complimentary. Yes, tripwires, which are extremely affordable product and services, work well. But there is actual power in the word complimentary.\nAs a matter of fact, people will certainly jump up and also down for a cost-free deal, also if they need to cover the delivery and also handling, in contrast to taking an offer that they even need to pay a tiny amount of loan for.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The UP CIDS Program on Social and Political Change, together with World Scientific Publishing (Singapore) and the UP Department of Political Science, is co-organizing the launch of and a panel discussion on the book \"Ensuring a Square Meal: Women and Food Security in Southeast Asia\" on October 19, 2018 (Friday), 2:00 to 4:00 PM, at the UP CIDS Conference Hall.\nThe UP Center for Integrative Studies (UPCIDS) Programs on Alternative Development, and Social and Political Change, together with the UP Department of Political Science will be holding a roundtable discussion on the book \"Islamic Populism in Indonesia and the Middle East,\" on 21 October 2017, 10:00 a.m. onwards at the UP Center for Integrative and Development Studies, Lower Ground Floor, Bahay ng Alumni, UP Diliman. The discussion is free and open to the public, but seating will be on a first-come, first-served basis.\nThe UP Department of Political Science, International IDEA, Constitution Transformation Network, and the UP-Center for Integrative and Development Studies Social and Political Change Program invite you to a forum entitled \"The Challenges of Constitution Building: The Fiji and Taiwan Experience\" on Thursday, October 5, 2017, from 8:30 am to 10:00 am at the UP-CIDS Conference Room located at the Lower Ground Floor of Ang Bahay ng Alumni at UP Diliman. Dr. Naidu Vijay from the University of South Pacific, Fiji and Dr. Wen Chen Chang from the National Taiwan University will be giving lectures on the subject.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today I ran across a modern misunderstanding that carries through to almost every science fiction and fantasy novel I have read. In A Clash of Kings Jon Snow prepares to go into the frozen north. He is comforted by the fact that the Seven (the gods of the southern people) have no power north of the wall since there are no churches (septs in Martin's world) or people who believe in them up there, but his people's gods, the nameless Old Gods of nature will be in the north.\nNo! I have never encountered that idea in any historical mythology that I have ever read. Oh, sure, people like the Romans had no problem making token sacrifices to other gods as well as their own, but never out of belief that their own gods were not present. If anything the belief of pagans seemed to be that other gods had turf to be respected or challenged, but their gods were true gods, meaning they were the gods of the world. The Greeks did not fear to sail beyond the Mediterranean because Poseidon could not see them there, but because of terrible things out at the world's edge. (And honestly, a lot of the times it seems like the Greeks would have been glad if Poseidon had passed them by.) There is no record of a Viking afraid to sail beyond Odin's sight, because in every mythology I can think of the gods extended throughout and beyond the whole world, Midgard (our world) being just part of Yggdrasil. Other gods were regarded with local caution, like don't mess with this Jehovah dude in Palestine but as long as you're a good Roman he won't bother you in Rome, but never blaspheme Jupiter anywhere or you're a dead man.\nThe difference between the actual worldviews of the middle ages and the medieval-esque ideas presented in fantasy comes from a simple question of of primacy. The supernatural tended to be the supreme reality in ancient belief systems, like the Chinese dynastic cycle where the fall of a dynasty marked that it had lost the favor of the heavens and therefore lost power while the new dynasty worked hard to show through prosperity that it had the divine mandate. Pluralistic modernism views religion as a soothing bit of selfish irrelevancy or a source of personal power and benefit. Gods are invented by and depend upon man, their creators, for their existence, power, and benefit. Moderns tend to treat beliefs as a sort of spiritual masturbatory act. Believe if you need to need to comfort yourself, but it isn't something you should talk about far less perform in public.\nWhy does this matter for fiction? This distinction is critical in terms of verisimilitude and characterization. Whether secular or not, the modern understanding of history, that we believed in gods when we were more at the mercy of the world around us and 'grew' past those beliefs as we developed a better understanding of nature has some truth to it. The argument about whether the change in pervasive belief was a virtue or a folly is beyond the scope of this essay. I want, in stead, to focus on how to write a non-humanist character as opposed to a subtly humanist character. Here are some examples of Early Modern Characters and Modern Characters.\nEMC: The gods existed before humanity. New religions are a discovery of what already existed. They can be forgotten or rediscovered but they have always been there.\nMC: The gods are made by people. When the people who believe in a god are gone, the god ceases to exist or have influence.\nEMC: The gods are in charge of nature, and their favor will result in nature working out for my benefit.\nMC: Nature is independent of the gods, so it is my actions about the physical world and not my interactions with the spiritual that determine my well-being.\nMC: Spirituality only exists for the benefit of those who believe. It is better to change religious beliefs to get an immediate benefit or abandon them than endure hardship.\nMC: What I can see and experience are the ultimate determinants of truth.\nActual conflicts between religions, which the Song of Ice and Fire books include, have tended in history to be conflicts of divine power instead of conflicts of reality. People met, and either the battle between armies or supernatural showdowns occurred. People then tended to believe in the victor as well as or more than the loser in the divine conflict. This melding and merging of religious beliefs is more consistent throughout the historical record than the belief that gods lost power or stopped existing. The sole exception to this are the three monotheistic faiths of Islam, Judaism, and Christianity. Even then the tendency has been for melding such as the mixture of African animism with Christian imagery in Vodoo and with Daoism in Aikido philosophy. This trend actually shows up in scripture as one of the big reasons why the people of Israel were to drive the nations out of the holy land or they and their children would want to get along with other faiths, eventually losing their own to their destruction.\nSo, when writing fiction it is critical that we determine what our characters believe. If I were to write an animist, like the Old Gods of Martin's fiction, then as long as there were natural events like streams, forests, and the wind, there would be the presence of the Old Gods. Martin writes that the Old Gods could not exist without their sacred groves, but his own back-story fiction states that they did not come from the sacred groves, so it does not follow that they would go away without them. Similarly, as the Red Woman brings her religion to Stannis Baratheon, the Sept should have opposed her, not the academics. Religious belief does carry power and influence, and if the legendary conflicts between saints and druids in my own people's history are any indication, those in power do not relinquish it easily.\nThis is especially important when writing about populations that undergo religious conversions. When the favor of the divine is essential to your family's life and welfare, the reaction to a convert is not, \"How distasteful but that is your choice.\" The reaction should be similar to someone bringing nuclear waste to store next door. Throughout history people can, have, and will kill over such things not out of a sense of evil, but out of the sincere belief that the other religious belief threatens themselves, their nation, and their loved ones. This is why missionaries get killed a lot, regardless of the faith in question. The only person who is going to be all right with a change in their beliefs is someone who did not really believe in the first place.\nThis is a glaring weakness in The Song of Ice and Fire and too much of modern fantasy. If we choose to write characters who believe, we must write characters who act on their fundamental truths just as surely as we do ours. OR we must explain why their belief system can accommodate other faiths. For the Romans, it was an hierarchical thing. Appease the local gods but don't forget that the gods of Rome are still on top. For a religion like Hinduism, other religions may be tolerated because they are reflections of elements of a deeper truth (which we have so don't talk to us about yours, please). For a humanist, if you feel that believing something makes you a more well-rounded person, that is fine. Just don't get any of your religion on the public furniture.\nBecause a character who sees their gods as limited by boundaries he can surpass (walk south for a month and Zeus can't hear you and you can do whatever you want) are not going to put much trust in their gods for very long. After all, they have proof in their own experience of the humanistic tenet: mankind is greater than gods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Du you want to hand out gospel tracts?\nMany of us travel in different countries for one or another reason and it is always good to have a tract on hands.\nWe have many Polish and Estonian (Estonian tract is coming soon) workers in our countries and they also have the right to hear about Jesus.\nWe don't sell tracts, but offer them to you for free to print\/copy and spread. You can also engage a Print shop, for example www.print24.com If you use a Print shop you will get much more professional result. We can help you with printing process and even to write your contact information in tracts, this we will do for free.\nOur tracts are in A5 size (before folding). If you want to print\/copy them yourself we recommend our A4-layout, it gives you 2 tracts per A4 paper. A4-layouts are not availble on our website, but if you want to have them you, just let us know and you will get them on email (pdf-file).\nHelp to write your contact information on tracts is free.\nIf you want us to help you with the printing process, we can help you and will do it without any profit, so you pay only for the actual cost for printing.\nPrice example from www.print.24.com in Sweden (including taxes, shipping, PayPal fee, Jan 2017). There are some small differences in prices between countries. Prices are just a small hint, there are even other mounts of tracts to order.\nTracts will be printed on 4-color on 135g glossy paper.\nContact us for more information if you are intrested to spread tracts.\nKlick a tract and it will open in a new tab.\nand thousands in Finland and in Spain !\nWe can help you with writing your own contact information on tracts: Who is spreading them (name of the congregation or organazation). Web address, Phone number, email, times for church services etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbolpq b/data_all_eng_slimpj/shuffled/split2/finalzzzbolpq
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index 0000000000000000000000000000000000000000..962f36d03127b42a3046c7f53fa29460f9a00bb1
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+{"text":"I am sorry to be posting at this sad time, when this amazing community will be closing.\nI just have a quick question regarding one of my decks, namely the bohemian cats tarot - gold edition.\nI have this set, still new and sealed, and have been trying to find more information on it, and specifically if anyone might know how to value this deck, should I wish to sell it.\nMany thanks for any information on the deck or on the value of this, as I know it was a limited edition print run of 500 decks.\nHave you the \"KIT\" with the journal, or the deck ? The kit is up on ebay Canada for $1500 CDN \"or best offer\". There's no knowing what it will sell for. What I do when I really want to know is watch a listing like that and see what happens.\nSo I followed that one up and I think the seller is a bit mad. Had it listed a month ago for $1000 CDN; and it didn't sell. So it is now listed again at 50% more. The deck (gold edged) alone has also failed to sell at $250 CDN.\nHOWEVER - a UK seller just sold the deck alone (one of the limited edition run of 500 which AFAIK was the total number of DECKS, but I may be wrong... and it's numbered - I don't know if they ever had the journal) for a mere 71 pounds - $117 CDN.\nI wouldn't get too excited; be realistic - and watch the $1500 one... . I think you can rest assured that $1000 is off the wall !\nSomeone here bought a sealed Baba deck just two weeks ago and it was missing a card... Just sayin' Sealed isn't ALWAYS a guarantee - and when it's LE, you can't be sure the publisher can help out with a missing card. That said - with a sealed one, it isn't the seller's fault if a card is missing. I do agree.\nYes you are right Gregory and I think you are talking about me buying a Baroque Bohemian Cats sealed deck and it missed the Page of Cups !\nSo that is probably why AJ says they would pay more for an UNsealed deck !\nUnsealed : the seller had a chance to make sure all the cards are there. But, then, the deck is not guaranteed to be new, and I have see unsealed decks that were described as \"in new condition\" but turned out to be very filthy..... so, unless you know the seller and trust them, I would be wary of buying a very expensive unsealed deck now.\nSealed : it's new and nobody put their filthy paws on it ! So I would probably be more comfortable paying a costly sum for a sealed deck, despite my earlier misadventure two weeks ago.\nOf course, buying a sealed OOP deck from an artist or a publisher that is not longer in activity, and discovering that one card is missing... there is not much you can do.\nBut the thing is : it is a Baba deck we are talking about, and Baba Studio is still there, with a TOP of the top customer service.\nI always feel better bidding on an opened OOP deck that has a lot of pictures.\nThank you to all of you that responded to my question, and sorry not to have replied sooner.\nYes, I suppose you are right, in that any deck is only worth what someone is willing to pay for it at the end of the day.\nMy one is the second edition, gold limited to 500 copies one, which has the blue cover on the box, and I believe it comes with the lovely journal inside, and that the box\/packaging itself was specially designed for this limited print run and edition.\nI agree that buying a sealed OOP deck is not as straightforward as one would hope, as I sold a sealed tarot box set on ebay a few months back, and the quality of the card printing inside was awful, with two cards having rips in them. I was of course mortified, and though I had 100% feedback, the buyer was quick to distrust me. However, we worked it out and I gave her a full refund etc... But I was the one who got stung in the end.\nAnyway, I have seen my bohemian cats version on Amazon for like two thousand pounds, crazy!\nI have hummed and ha about opening it, but thought this would de-value it somehow. I always take really good care of my opened decks anyway, but understand given the aforementioned experience, why people would be nervous to spend so much money on a sealed deck, in case of manufacturing errors. I am however, encouraged by baba studios professionalism and good customer care.\nWell, it still remains sealed for now, but I am soooo tempted to open it and look through the deck. It seems gorgeous what online images I have viewed. I think mine is the border less version too, so bigger images.\nThanks again for everyone's input, I appreciate all views.\nI see your point. Thought I think that there is no reason for the buyer to distrust you in such a matter, especially if you were helpful and worked with them to find a solution. In the past I have given super positive feedback to a seller (not for a deck) eventhough the item was not what was described, but from communication with the seller it was clear it was not their fault and they were very keen on finding a solution.\nBut I understand that in the end, it's a no-win\/no-win situation, one that you do not want to risk being in again !\nSo, if I were the one selling an expensive OOP deck, in the light of all that has been said now, I think I would open it to check on it, then I would list it as \"in new condition, has been recently opened and gently perused once to make sure all cards are there, but otherwise unused\" or something of the same kind.\n2000 \u00a3 sounds extreme ! I have also seen this set listed for 300 $ and it did not sell right away (maybe it did not sell at all, cannot remember), so I would be extremely surprised if it actually sold for 2000 \u00a3, but then again, you never know...... for my part it feels like the person listing it for 2000 \u00a3 does not really want to part from their deck !\nI wish you best luck if you set your mind on selling the deck It is a beautiful deck !\nAnyway, I have seen my bohemian cats version on Amazon for like two thousand pounds, crazy!.\nNever believe those Amazon listings. I've seen in print Jane Austen books listed for hundreds of dollars. Find a listing on ebay where it actually sold.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great Look and Quality ! We Love It!\nStimulated by classic Persian motifs, this piece is inspired by decades of timeless tradition. Add an elegant flair to your space with this artful homage to traditional rugs. Alexandria is beautifully Hand-Tufted of 100% New Zealand Wool in India and has a 1\/2' pile height .Cotton Backing. Made in India. Vacuum regularly & spot clean stains. Professional cleaning recommended periodically.\nDimensions 90.0 In. X 27.0 In. X 0.5 In.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: This apartment features air conditioning, a seating area, washing machine and flat-screen TV with cable channels.\nOffering free WiFi and air conditioning, Apartment on Nalbandyan Street is situated in Yerevan, 300 metres from the Rupublic Square.\nThis apartment features a seating area, washing machine and flat-screen TV with cable channels. Free housecleaning services including fresh bedding and towels are carried out every 3 days.\nA 24 hours customer support is available.\nGuests can cook in the kitchen equipped with fridge, stove, microwave, dishwasher and electric kettle, and various caf\u00e9s and restaurants are located within a 3-minute walk of the apartment.\nYerevan Train Station is 8 km away, and Zvartnots International Airport is 12 km from Apartment on Nalbandyan Street.\nPlease inform Apartment on Nalbandyan Street (New Building) in advance of your expected arrival time. You can use the Special Requests box when booking, or contact the property directly with the contact details provided in your confirmation.\nThis section presents all information about the hotel Apartment on Nalbandyan Street (New Building) located in Yerevan: photo of rooms, rates, amenities, etc. On the website you can choose and book suitable room of Apartment on Nalbandyan Street (New Building) without a fee.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bright, clever postcard features a banana slug in a gas mask asking for fresh air. Delivers the serious message that secondhand smoke is unhealthy for all animals. The postcard offers facts about how secondhand smoke is toxic for banana slugs, creatures that have a single lung and also breathe through their skin, and encourages animal lovers to stop smoking. The bottom of the postcard includes the California Smokers' Helpline contact information (easily trimmed for out-of-state use).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"October 12th Jin Yongnan, member of the Standing Committee of the Political Bureau of the Korean Workers' Party and the Standing Committee of the DPRK Supreme People's Assembly, met with the Chinese sports delegation headed by Qi Zhongwen on the 12th.\nKim Yong Nam said that this year, Chairman Kim Jong-un and General Secretary Xi Jinping had three historic meetings to jointly lead the DPRK-China relations into a new and higher stage of development. The visit of the Chinese sports delegation to the DPRK will further strengthen the friendly feelings of the people of the DPRK and China and promote the long-term, healthy and stable development of the DPRK-China relations.\nDuring the visit, the Chinese and Korean men's and women's basketball teams were mixed and friendly, and Cui Longhai, member of the Standing Committee of the Political Bureau of the Korean Workers' Party, and Vice Chairman Cui Longhai watched the match. The delegation also participated in the China-DPRK Friendship Tower, the Jinxiu Mountain Sun Palace, and the former residence of Wanjingtai, and went to the Wanjingtai Children's Palace and Jinricheng Comprehensive University to carry out friendly exchanges.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbourq b/data_all_eng_slimpj/shuffled/split2/finalzzzbourq
new file mode 100644
index 0000000000000000000000000000000000000000..ba1564dce11b5f3bb90c9caa4c80804b99fc16eb
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbourq
@@ -0,0 +1,5 @@
+{"text":"where the winds of the four seasons blow through refreshingly.\nthe finest sunrise gives vitality.\nOwner chef cooks fish everydays.\nHe cooks with his whole heart.\nBefore you use Hashimoto-ya, there is something I would like you to know.\nbut it can also lead to a small misunderstanding.\nI would like to tell you some things I would like you to know before booking to Hashimoto-ya.\nto touch, to eat, to enjoy with the five senses.\nwhere Abuko rises over the island.\nWelcome to our excellent accommodation.\nFrom any room, you can hear the sound of ripples in the sea, the healing effect is outstanding.Especially in the ocean side rooms, the sea is in front of you, you can see a wonderful view.\nHimen's nature, taste, city events, etc. are posted for you to browse.\nCopyright \u00a9 Hashimoto-ya. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A USB card\/device, the gateway functions much like a wireless router. But it can support up to 32 simultaneous connections, meaning that pending your base computer finding a proper bandwidth connection, it could supply a 32-person hotspot in a quasi-mobile operation. Or better yet, using a WiFi hotspot, you could extend said hotspot to your friends sitting at a cheaper cafe across the street. That's it, Gizmodo is forming a biker gang and we are officially buying leather pants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cailang is with full experience in making Luxury gift paper box, This divider paper box is a kind of design for products packaging, also as storage, is useful.\nCailang Paper Products has business line for every industry. Thus, Cosmetic paper box has Skin care kit box, Perfume\/Fragrance packaging box, Eyeshadow cardboard box, eyelash printing box, Essential Oil paper box; Luxury gift paper box has Top & Bottom box\/2pcs paper box with lid, Sliding drawer paper box, Magnetic Book paper gift box, Hinged lid rigid shoulder box, Custom wine paper box, Custom candle box, High-end clothing paper box; Collapsible paper box has Collapsible box with corner folding, Collapsible box with side way folding, Card coated box\/Tuck top box; Cylinder\/Round paper box has Round paper box, Paper Cylinders with rolled edge, Flat edge cylinder round tube, and so on.\nIf you are gift retailer or wholesaler, we are welcome your own ideas about Luxury Paper Packaging Boxes, and make you unique Luxury gift paper box!\nIf you are agency company, welcome be a business partner, trust you will surprised by price and quality!\nCustom Sample Time: Within 5-7 Days.\nYour Trusted Supplier of Branded Gift Boxes!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We're glad you asked! At CSC, we continuously strive to make our vision of \"transforming lives by the chiefessays.net power of God for the expansion of His kingdom\" a reality.\nTRANSFORMING LIVES BY THE POWER OF GOD FOR THE EXPANSION OF HIS KINGDOM.\nChristian Stronghold Church, a non-traditional church located in northeast Dallas, was founded in 1995 by Pastor Hal Johnson, Sr. Our name is based on Nahum 1:7, which reads: \"The Lord is good, a stronghold in the day of trouble, and He knoweth them that trust Him.\" Christian Stronghold has grown from a home Bible study group of seven people to over 600 members.\nJesus died as a voluntary substitute for sinful man.\nJesus was bodily raised http:\/\/www.essaywriterusa.com from the dead.\nGod imputes righteousness to the believer upon their acceptance by faith of His Son as Savior, and gives the believer eternal life.\nJesus will bodily return to the earth.\nThe Bible is the inerrant Word of God.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enjoy our selection of delicious fruit and ginger wines. Ideal for mixing or drinking chilled.\nFruit Wines are fermented fruits (not grapes) that form the basis of these alcoholic beverages. They may also contain herbs, spices and flower extracts and the fermentation process may include elements of honey and sugar.\nFact: As long as the base ingredient can be fermented, fruit wine can be made.\nFact: More commonly called 'Country Wine' in Great Britain.\nGinger Wine is a fortified wine using the fermentation of ground ginger and raisins. Can be drunk neat or with lemonade\/ginger ale.\nThe most common consumption of Ginger wine is the 'Whisky Mac' \u2013 why not try it today?","meta":{"redpajama_set_name":"RedPajamaC4"}}